diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzakjh b/data_all_eng_slimpj/shuffled/split2/finalzzakjh
new file mode 100644
index 0000000000000000000000000000000000000000..d965d27f04e45575fcf92e99b1df2b21e1aab607
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzakjh
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\nThe \\emph{Kepler} space telescope collected continuous photometric data for nearly 3.5 years from a small 0.25\\% patch of the celestial sphere \\citep{koc10,bor16}. This uniform monitoring of 150,000 stars enabled the discovery of over 4700 transiting exoplanet candidates through an automated detection pipeline \\citep{jen10}.\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/docs\/counts_detail.html}} Removing the human component from the signal vetting process ({\\tt Robovetter}; \\citealt{tho18}), enabled the first homogeneous transiting small planet sample suitable for exoplanet population studies. Through full automation, the sample completeness can be measured via transit injection\/recovery procedures \\citep{pet13b,chr15,dres15,chr17,chr20}. Here, artificial transit signals are injected into the raw light curves, enabling quantification of the pipeline's detection efficiency as a function of, for instance, the injection signal strength. In addition, the final \\emph{Kepler} catalog (DR25) also provided a measure of the sample reliability (the rate of false alarm signals) by testing the software's ability to remove systematics from the final catalog \\citep{cou17b}.   \n\nWith these catalog measurements available, the underlying planet population can be extracted by correcting for the detection and orbital selection effects. Many studies have used \\emph{Kepler} automated planet catalogs to identify a remarkable surplus of sub-Neptunes and super-Earths at short periods (FGK Dwarfs: \\citealt{you11,how12,pet13b,muld18,hsu19,zin19,he19}; M Dwarfs: \\citealt{dres13,mui15,dres15,har19}), despite their absence in the solar system. Several other population features have been discovered using the \\emph{Kepler} catalog. For example, the apparent deficit of planets near the 1.5--$2 R_\\Earth$ range, colloquially known as the ``radius valley'' \\citep{ful17}. This flux-dependent population feature indicates some underlying formation or evolution mechanism must be at play, separating the super-Earth and sub-Neptune populations \\citep{owe17,gup18}. Nearly 500 multi-planet systems were identified in the \\emph{Kepler} data, enabling examination of intra-system trends. Remarkably, very minor dispersion in planet radii is observed within each system \\citep{cia13}. By examining the intra-system mass values from planetary systems that have measurable transit-timing variations (TTVs), a similar uniformity has been identified for planet mass \\citep{mil17}. Moreover, the orbital period spacing of planets appears far more compact than the planets within the solar system. These transiting multi-planet systems also appear to have relatively homogeneous spacing, indicative of an unvaried dynamic history. This combination of intra-system uniformity in orbital spacing and radii is known as the \"peas-in-a-pod\" finding \\citep{wei18}. \n\n\nOne of the main objectives of the \\emph{Kepler} mission was to provide a baseline measurement for the occurrence of Earth analogs ($\\eta_\\Earth$). Despite the scarce completeness in this area of parameter space, several studies have extrapolated from more populated regions, providing measurements ranging from 0.1--0.4 Earth-like planets per main-sequence star \\citep{cat11,tra12,pet13b,sil15,bur15,zin19b,kun20,bry20b}. Currently, it remains unclear what planet features are important for habitability, making precise measurements unattainable. Moreover, these extrapolations are model dependent and require a more empirical sampling of the Earth-like region of parameter space to reduce our uncertainty. Unfortunately, the continuous photometric data collection of the \\emph{Kepler} field was terminated due to mechanical issues on-board the spacecraft after 3.5 years, limiting the completeness of these long-period small planets.\n\nUpon the failure of two (out of four) reaction wheels on the spacecraft, the telescope was no longer able to collect data from the \\emph{Kepler} field, due to drift from solar radiation pressure. By focusing on fields (Campaigns) along the ecliptic plane, this drift was minimized, giving rise to the \\emph{K2} mission \\citep{how14, cle16}. Each of the 18 Campaigns were observed for roughly 80 days, enabling the analysis of transiting exoplanet populations from different regions of the local Galaxy. However, the remaining solar pressure experienced by the spacecraft still required a telescope pointing adjustment every $\\sim$6 hours. This drift and correction led to unique systematic features in the light curves, which the \\emph{Kepler} automated detection pipeline was not designed to overcome. As a result, all transiting \\emph{K2} planet candidates to date have required some amount of visual assessment. Despite this barrier, nearly 1000 candidates have been identified in the \\emph{K2} light curves (e.g., \\citealt{van16b,bar16,ada16,cro16,pop16,dres17,pet18,liv18,may18,ye18,kru19,zin19c}). However, this assortment of planet detections lacks the homogeneity necessary for demographics research. Attempts have been made to replicate the automation of the \\emph{Kepler} pipeline for \\emph{K2} planet detection \\citep[e.g.,][]{kos19,dat19}, but these procedures still required some amount of visual inspection. \\citet{zin20a} developed the first fully automated \\emph{K2} planet detection pipeline, using the {\\tt EDI-Vetter} suite of vetting metrics to combat the unique \\emph{K2} systematics. This pilot study was carried out on a single \\emph{K2} field (Campaign 5; C5 henceforth) and detected 75 planet candidates. The automation enabled injection\/recovery analysis of the stellar sample, providing an assessment of the planet catalog completeness. Additionally, the reliability (rate of false alarms) was quantified by passing inverted light curves---nullifying existing transit signals---through the automated procedure, analogous to what was done for the \\emph{Kepler} DR25 catalog. Any false candidates identified through this procedure are indicative of the underlying false alarm contamination rate. \n\n\\begin{figure*}\n\\centering \\includegraphics[width=\\textwidth]{kepStell.pdf}\n\\caption{The distribution of \\emph{Kepler} \\citep{ber20} and \\emph{K2} \\citep{hub16,har20} targets as a function of surface gravity ($\\log(g)$), effective stellar temperature ($T_\\textrm{eff}$) and stellar metallicity ([Fe\/H]). \n\\label{fig:HR}}\n\\end{figure*}\n\nWith corresponding measures of sample completeness and reliability for C5, \\cite{zin20b} carried out the first assessment of small transiting planet occurrence outside of the \\emph{Kepler} field, finding a minor reduction in planet occurrence in this metal-poor FGK stellar sample. This provided evidence that stellar metallicity may be linked to the formation of small planets; however, the weak trend detected requires additional data for verification. \n\n\n\nThis paper is a continuation of the Scaling \\emph{K2} series \\citep{har20,zin20a,zin20b}, which aims to leverage \\emph{K2} photometry to expand our exoplanet occurrence rate capabilities and disentangle underlying formation mechanisms. The intent of this study is to derive a uniform catalog of \\emph{K2} planets for all 18 Campaigns (with exception to C9, which observed the crowded galactic plane, and C19, which suffered significantly from low fuel levels), and provide corresponding measurements of the sample completeness and reliability. The underlying target sample and the corresponding stellar properties are discussed in Section \\ref{sec:stellarSample}. \nIn Section \\ref{sec:pipe} we outline the automated pipeline implemented in our uniform planet detection routine. We then procure a homogeneous planet sample and consider the interesting candidates and systems in Section \\ref{sec:catalog}. In Sections \\ref{sec:complete} and \\ref{sec:reli} we provide the corresponding measures of sample completeness and reliability for this catalog of transiting \\emph{K2} planet candidates. Finally, we offer suggestions for occurrence analysis and summarize our findings in Sections \\ref{sec:suggest} and \\ref{sec:summary}. \n\n\n\\section{Target Sample}\n\\label{sec:stellarSample}\n\n\n\nWe first downloaded the most up-to-date raw target pixel files (TPFs) from MAST\\footnote{\\href{https:\/\/archive.stsci.edu\/k2\/}{https:\/\/archive.stsci.edu\/k2\/}}, which recently underwent a full reprocessing using a uniform procedure \\citep{cad20}. To ensure consistency among our data set, we used the final data release (V9.3) for all available campaigns considered in this paper.\\footnote{C8, C12, and C14 were released after our analysis was performed. Thus, a previous version (V9.2) was used for this catalog. Overall, we expect the differences to be minor and leave inclusion of these updated TPFs for future iterations of the catalog.} We began our search using the entire EPIC target list (381,923 targets; \\citealt{hub16}). Of this list, we found 212 targets have FITS file issues our software could not rectify. For all remaining targets we used the {\\tt K2SFF} aperture \\#15 \\citep{van16b}, which is derived from the \\emph{Kepler} pixel response function and varies in size in accordance with the target's brightness. Circular apertures exceeding $10\\arcsec$ radii are prone to significant contamination from nearby sources. Our vetting software, {\\tt EDI-Vetter}, is able to account for this additional flux, but the software's accuracy begins to decay beyond a radius of $20\\arcsec$. Additionally, large apertures have a tendency to contain multiple bright sources, providing further complications. To address this issue, we put an upper limit on the target aperture size (79 pixels; $\\sim20\\arcsec$ radius). This cut removed 4548 targets that have apertures we deemed too large. Of these rejected targets, 492 are in Campaign 11, which contains the crowded galactic bulge field. After applying these cuts, 377,163 targets remained and were then used as our baseline stellar sample.\n\n\n\\begin{figure*}\n\\centering \\includegraphics[height=7cm]{K2_Galaxy.png}\n\\caption{The galactic distribution of \\emph{K2} target stars. The disk structure follows the interpretation provided by \\cite{hay15}. The thick disk consists of stars with a heightened abundance of $\\alpha$-chain elements (O, Ne, Mg, Si, S, Ar, Ca, and Ti) compared to the stars in the thin disk \\citep{wal62}. Additionally, the $\\alpha$-poor disk begins to flare up beyond the solar neighborhood. The galactic coordinates presented here have been calculated using \\emph{Gaia} DR2 \\citep{gai18}.\n\\label{fig:galaxy}}\n\\end{figure*}\n\nTo parameterize this list of targets we relied on the values derived by \\cite{har20}, which used a random forest classifier, trained on LAMOST spectra \\citep{su04}, to derive stellar parameters from photometric data (222,088 unique targets). Additionally, the available \\emph{Gaia} DR2 parallax information was incorporated to significantly improve our understanding of the stellar radii. From this catalog, additional measurements of stellar mass, metallicity, effective temperature ($T_{\\text{eff}}$), and surface gravity ($\\log(g)$) were provided and used in this study. In cases where stellar parameters were not available in \\cite{har20} (typically due to their absence in the \\emph{Gaia} DR2 catalog or not meeting strict photometric selection criteria), we used the stellar parameters derived for the EPIC catalog \\citep{hub16} (94,769 targets). Finally, we used solar values for targets that lack parameterization in both catalogs (41,061 targets).\\footnote{\\cite{zin20b} showed that systematic differences between stellar catalogs are minor. Additionally, we emphasize our pipeline's agnosticism to stellar parameterization.}\n\n\nExcluding the 41,061 targets without stellar parameters, we isolated 223,075 stars that appear to be main-sequence dwarfs based on their surface gravity ($\\log(g)>4$). Within this subset we identified 48,702 targets as M dwarfs ($T_{\\text{eff}}<4000K$), 164,569 as FGK dwarfs ($4000<T_{\\text{eff}}<6500K$), and 9,731 as A stars ($T_{\\text{eff}}>6500K$). This abundance of M dwarfs is nearly 17 times that of the \\emph{Kepler} sample (2808 M dwarfs; \\citealt{ber20}) and can be clearly seen in Figure \\ref{fig:HR}. Another noteworthy feature of the \\emph{K2} stellar sample is the wide range of galactic latitudes covered by the 18 campaigns. This enabled \\emph{K2} to probe different regions of the galactic sub-structure. In contrast, a majority of the \\emph{Kepler} stellar sample was bounded within the thin disk (see Figure 1 of \\citealt{zin20b} for a comparison). In Figure \\ref{fig:galaxy} we show this galactic sub-structure span for \\emph{K2}. Making broad cuts in galactic radius ($R_g$) and height ($b$) we can distinguish thick disk ($R_g<8$kpc and $|b|>0.5$kpc) from thin disk stars ($|b|<0.5$kpc). Overall, we found 191,002 dwarfs located in the thin disk, while 17,123 dwarfs reside in the thick disk. This sample distinction is interesting and may warrant further consideration in future occurrence studies.\n\n\nThe underlying stellar sample is important because it is the population from which the planet candidates are drawn. However, the stellar parameters are subject to change as more comprehensive data and more precise measurements become available (i.e., the upcoming \\emph{Gaia} DR3). To ensure our catalog remains relevant upon improved stellar parameterization, we take an agnostic approach to the available stellar features throughout our pipeline. In other words, each light curve is treated consistently regardless of the underlying target parameters. The only caveat to this claim is our treatment of the transit limb-darkening. Our transit model fitting routine requires quadratic limb-darkening parameters. We derived the appropriate values using the ATLAS model coefficients for the \\emph{Kepler} bandpasses \\citep{cla12}, in concert with the available stellar parameters. This minor reliance on our measured stellar values will have negligible effects on the presented catalog. In the case where a signal is transiting with a high impact parameter, this limb darkening choice can be important, modifying the inferred radius ratio by an order of 1\\%. However, this boundary case is well within the uncertainty of our measured radius ratio values ($\\sim4\\%$) and therefore not significantly impacting our inferred radius measurements. All other detection and vetting metrics in our pipeline are independent of this parameterization. \n\n\n\n\\section{The Automated Pipeline}\n\\label{sec:pipe}\nOur automated light curve analysis pipeline consists of four major components: pre-processing, detrending, signal detection, and signal vetting. In Figure \\ref{fig:diagram} we provide a visual overview of this procedure, and now briefly summarize the execution of each step.\n\n\\begin{figure}\n\\centering \\includegraphics[width=\\columnwidth{}]{figureDiagramsmall.pdf}\n\\caption{ An overview schematic of the automated detection pipeline used in the current study to identify \\emph{K2} planet candidates. This follows the same procedure described in \\cite{zin20a}. The effects of pre-processing and detrending have also been depicted for EPIC 211422469, illustrating the importance of the corresponding steps. \n\\label{fig:diagram}}\n\\end{figure}\n\n\\subsection{Pre-Processing}\nThe raw flux measurements from \\emph{K2} are riddled with systematic noise components due to the thruster firing approximately every six hours (required to re-align the telescope pointing) and the momentum dumps every two days. This spacecraft movement smears the target across several different pixels, all with unique noise and sensitivity properties, leading to a significant increase in the overall light curve noise. We passed all raw light curves through the {\\tt EVEREST} software \\citep{lug16,lug18}, which minimizes this flux dispersion issue using pixel-level decorrelation (PLD) to fit and remove noise attributed to the spacecraft roll. In Figure \\ref{fig:diagram} we show how {\\tt EVEREST} reduces the noise, as measured by the root mean-square (RMS), of the EPIC 211422469 light curve by a factor of four. However, this pre-processing also has the ability to reduce transit signals or remove them completely. Thus, it is essential that any injection\/recovery tests address this concern by injecting signals into the data before this pre-processing, as we later discuss in Section \\ref{sec:complete}.\n\n\n\n\\subsection{Detrending}\nIn the example light curve for EPIC 211422469 (Figure \\ref{fig:diagram}) there is a clear long-term trend remaining in the data after pre-processing. The goal of detrending is to remove this red-noise component and any stellar variability, producing a clean, white-noise dominated, time series. We used two Gaussian process (GP) models, with ``rotation'' kernels, to remove these long- and medium-term trends. These kernels use a series of harmonics to match and remove periodic and red-noise trends in the photometry.  The first pass GP looked for general flux drifting (periods $>10$ days), subtracting the appropriate model. The second pass GP identified and removed medium-term trends (5 days $<$ period $<10$ days) often associated with stellar variability. In testing, we found these two GP passes were effective in removing red-noise from the data without significantly impacting transit signals. The Ljung-Box test (a portmanteau test for all autocorrelation lags; \\citealt{lju78}) found that $67\\%$ of our processed light curve residuals produced p-values greater than 0.001, indicating a lack of statistically significant short-term correlated structure. However, stellar harmonics with short periods (usually $<0.5$ days) continued to contaminate the remaining 33\\% of our light curves. To address these trends we fit and removed sine waves with periods less than $0.5$ days. These signals are all below the 0.5 period requirement threshold for planet candidacy, but still have the ability to contaminate the light curve. In an effort to minimize over-fitting, we required the signal amplitude to be at least $10\\sigma$ in strength. If this requirement was not met, the harmonic removal was not applied.  \n\nThe period limits used for both the GP models and the harmonic fitter attempted to preserve transit signals, avoiding periods which are prone to transit removal (0.5 days $<$ period $<5$ days). Unfortunately, some stellar variability exists in this forbidden period range, making it difficult to address every unique situation. Using an auto-regressive integrated moving average algorithm (as suggested by \\citet{cac19a,cac19b}) may reduce some of these remaining correlated residuals, but this methodology also has limitations in dealing with stellar variability and is beyond the scope of this paper.\n\nIn addition, these detrending mechanisms are capable of reducing or (in the case of very deep transits) removing the signal altogether (i.e., the \\emph{Kepler} harmonic removal highlighted by \\citealt{chr13}).We acknowledge these costs in performing our automated detrending procedure, knowing the effects will be accounted for in our catalog completeness measurements (see Section \\ref{sec:complete}).  \n\n\\subsection{Signal Detection}\nOnce the light curve has been scrubbed with our pre-processing and detrending routine, the flux measurements can be examined for transit signals with {\\tt TERRA} \\citep{pet13b}. This algorithm uses a box shape to look for dips in the light curve, enabling a quick examination of each target. To measure the signal strength we rely on the same metric used for the \\emph{Kepler} transiting planet search (TPS), the multiple event statistic \\citep[MES;][]{je02}, which assumes a linear ephemeris to indicate the strength of the whitened signal. For our detection threshold we require a signal MES value greater than $8.68\\sigma$. This threshold, which is higher than that of the \\emph{Kepler} TPS ($7.1\\sigma$), was arbitrarily selected to reduce the total signal count to 20\\% of our total target sample. We also bound the period search of {\\tt TERRA}, ranging from 0.5 days to a period where three transits could be detected (nominally 40 days, varying slightly from campaign to campaign). Any signal above our MES threshold and between these period ranges is given the label of threshold-crossing event (TCE).  \n\n\n\\subsubsection{Multi-planet systems}\n\nMulti-planet systems are rich in information, but require careful consideration when attempting to identify these asynchronous signals. Ideally, a model of the first detected signal would be fit and subtracted from the light curve before re-searching the data for additional signals. However, real data are noisy, making such a task difficult to automate. Should the signal be incorrectly fit, the model subtraction will leave significant residuals, which will continuously trigger a detection upon reexamination. Moreover, false positives that do not fit the transit model will also leave significant residuals. Finally, astrophysical transit timing variations (TTVs) are not easily accounted for in an automated routine \\citep{wu13,had14,had17}. Thus, deviations from simple periodicity will leave significant residuals. Dealing with these complex signals, without loss of data, remains a point of continuous discussion (e.g. \\citealt{sch17}).\n\nAs was done in many previous pipelines, we relied on masking of the signal after each iterative TCE detection \\citep{jen02,dres15,sin16,kru19,zin20a}. This method has its own faults, as it requires some photometry to be discarded upon each signal detection. \\citet{zin19} showed that this $3\\times$ the transit duration removal has the ability to make multi-planet systems more difficult to detect. To mitigate this loss of data, we used a mask of $2.5\\times$ the transit duration ($1.25\\times$ the transit duration on either side of the transit midpoint) after each signal was detected. At this point the light curve was reexamined for an additional signal. This process was repeated six times or until the light curve lacked a TCE, enabling this pipeline to detect up to six planet systems.\n\n\\subsubsection{Skye Excess TCE Identification}\n\n\\begin{figure}\n\\centering \\includegraphics[width=\\columnwidth{}]{Skyplot_C04.pdf}\n\\caption{ An illustration of the Skye excess TCE identification process for Campaign 4. The expected white noise range for this number of cadences is plotted in green ($3.63\\sigma$). The median number of TCEs is shown by a central blue dotted line. We highlight the rejected cadences with a red x. These 16 cadences exceed the Skye excess limit and are masked in our final signal search.  \n\\label{fig:sky}}\n\\end{figure}\n\nCertain cadences are prone to triggering TCEs within the population of light curves. These can usually be attributed to spacecraft issues and are likely not astrophysical. Inspired by the ``Skye'' metric used in \\citet{tho18}, we minimized this source of contamination by considering the total number of TCEs with transits that fall on each cadence. The median value and expected white noise range was then calculated using:\n\n\\begin{equation}\n\\sqrt{2}\\; \\text{erfcinv}(1\/N_{\\text{cad}}) \\; \\sigma,\n\\label{eq:sky}\n\\end{equation}\nwhere erfcinv is the inverse complementary error function, $N_{\\text{cad}}$ is the total number of cadences in the campaign, and $\\sigma$ is the median absolute deviation in the number of TCEs detected at each cadence. This calculated value represents the largest deviation expected from this number of cadences under the assumption of perfect Gaussian noise. Any cadences that exceed this limit are likely faulty and warrant masking. In Figure \\ref{fig:sky} we provide an example of this procedure for Campaign 4. To further assist future studies we made the Skye mask for each campaign publicly available.\\footnote{\\href{http:\/\/www.jonzink.com\/scalingk2.html}{http:\/\/www.jonzink.com\/scalingk2.html}}\n\nOnce established, the Skye mask was used to reanalyze all light curves, removing problematic cadences. The resulting list of significant signals were then given the official TCE label and allowed to continue through our pipeline. Overall, we found 140,046 TCEs from 52,192 targets. However, a vast majority of these signals are false positives and required thorough vetting.\n\n\n\\subsection{Signal Vetting}\nTo parse through the 140,046 TCEs and identify the real planet candidates, we employed our vetting software {\\tt EDI-Vetter} \\citep{zin20a}. This routine builds upon the metrics developed for the \\emph{Kepler} TPS ({\\tt RoboVetter}; \\citealt{tho18}), with additional diagnostics created to address \\emph{K2} specific issues (i.e., the systematics caused by the spacecraft). These metrics attempt to replicate human vetting, by looking for specific transit features used to discern false positive signals. We now briefly discuss our planet candidacy requirements, but encourage readers to refer to \\citet{zin20a} for a more thorough discussion. \n\n\n\n\\subsubsection{Previous Planet Check}\nThis test looks to identify duplicate signals in the light curve (as originally discussed in Section 3.2.2 of \\citealt{cou16}). This test was only applied if the light curve produced more than one TCE. In such cases, the period and ephemeris were tested to ensure the second signal was truly unique. If not, this repeat identification was labeled a false positive. The goal of this test was to remove detections of the previous signal's secondary eclipse and signals that were not properly masked, leading to their re-detection.\n\n\\subsubsection{Binary Blending}\n\\label{sec:bblend}\nThe goal of this metric is to remove eclipsing binary (EB) contaminants from our catalog. There are two attributes that make EB events unique from planet transits: a deep transit depth and a high impact parameter ($b$). Leveraging these two features, we used the formula first derived by \\citet{bat13} and then modified by \\cite{tho18} to identify these contaminants, while remaining agnostic to the underlying stellar parameters:\n\n\\begin{equation}\n\\frac{R_{pl}}{R_\\star} + b\\le 1.04,\n\\label{eq:EB}\n\\end{equation}\nwhere $R_{pl}\/R_{\\star}$ represents the ratio of the transiting planet and the stellar host radii. Should this metric be exceeded, the TCE would be flagged as a false positive, removing most EBs from our sample.\n\nHowever, this diagnostic assumes the transit depth provides an accurate measure of $R_{pl}\/R_{\\star}$. If an additional source is within the target photometric aperture, the depth of the transit can be diluted by this additional flux, leading to an underestimation of $R_{pl}\/R_{\\star}$. To address this potential flux blending issue, we cross-referenced our target list against the \\emph{Gaia} DR2 catalog \\citep{gai18}. If an additional source is in or near the aperture, we calculated the expected flux contamination using available photometry from \\emph{Gaia} and \\emph{2MASS} \\citep{skr06}, correcting the $R_{pl}\/R_{\\star}$ value accordingly. If Equation \\ref{eq:EB} was not satisfied, the TCE was labeled as a false positive.\n\nOur inferred flux dilution correction assumed the signal in question originated from the brightest source encased by the target aperture. \\footnote{This assumption was enforced by our pipeline, which selected the brightest \\emph{Gaia} target within the aperture.} Therefore, transits emanating from a dimmer background star would experience a greater flux dilution and could avert this metric. We provide further discussion of this potential contamination rate in Section \\ref{sec:astrFP}.\n\n\n\\subsubsection{Transit Outliers}\nThis diagnostic was developed to deal with systematics specific to \\emph{K2}. We expect real candidates to produce dips in the stellar flux, but retain comparable light curve noise properties during the eclipse. In contrast, systematic events can produce a dip with noise properties independent of the light curve, producing a heightened RMS during the event. By measuring the RMS in and out of transit, we can identify significant changes, and flag signals that have false positive-like RMS properties.\n\n\\subsubsection{Individual Transit Check}\n\\label{sec:ITC}\nUsing the formalism developed by \\citet{mul16} (the ``Marshall'' test), we look to identify individual transit events that appear problematic. If one of the transits is either dominating the signal strength or does not fit a transit profile appropriately, it is indicative of a systematic false alarm. By fitting each individual transit signals with any of the four common systematic models (Flat, Logistic, Logistic-exponential, or Double-logistic), we looked for events that did not have an astrophysical origin. If an individual transit fit a systematic model better, that transit was masked and the light curve was re-analyzed to ensure the signal MES still remained above the $8.68\\sigma$ threshold before proceeding. In addition, we made sure the total signal strength is spread evenly among each observed transit. In cases where a single event dominated the signal strength, the TCE was labeled as a false positive.  \n\n\\subsubsection{Even\/Odd Transit Test}\nThis vetting test looks for EBs that produce a strong secondary eclipse (SE). In some cases these SEs can be deep enough to trigger a signal detection at half the true period, folding the SE on top of the transit. To identify such contaminants, we separated every other individual transit into odd and even groups. We then re-fit the transit depth within each group and looked for significant discrepancies. In cases where the disparity is greater than $5\\sigma$, the signal was labeled as a false positive.\n\n\\subsubsection{Uniqueness Test}\nThis diagnostic is based on the ``model-shift uniqueness test'' \\citep{row15, mul15, cou16, tho18} and compares the noise profile of the folded light curve with that of the TCE in question. If the folded light curve contains several transit-like features, it is indicative of a systematic false alarm. We compared the strength of the next largest dip, beyond the initial signal and any potential secondary eclipse, in the phase-folded data to the transit in question, assessing the uniqueness and significance of the periodic event. If another similar magnitude dip existed in the folded time series, it likely originated from light curve noise and the TCE was deemed a false positive.  \n\n\n\n\\subsubsection{Check for Secondary Eclipse}\nFollowing the methodology presented in Section A.4.1.2 and A.4.1.3 of \\citet{tho18}, we examined the transit signal for a secondary eclipse (SE). As previously noted, deep SE signals are a notable signature of an EB. However, some hot Jupiters are also capable of producing a SE, therefore detection of a SE is not in itself justification for exclusion. TCEs with a meaningful SE must exhibit a transit impact parameter of less than 0.8 or the SE must be less than 10\\% of the transit depth. If neither of these criteria were satisfied, the TCE was labeled as a false positive.\n\n\\subsubsection{Ephemeris Wandering}\nHarmonic signals have the ability to falsely trigger a TCE detection, but their inability to match the transit model leads to significant movement of the measured ephemeris. If the signal's transit mid-point, as detected by {\\tt TERRA}, changed by more than half the transit duration when optimized with our MCMC routine, the TCE earned a false positive label. \n\n\\subsubsection{Harmonic Test}\nIn addition to the ephemeris wandering metric, we also attempted to fit the light curve with a sine wave at the period of the detected TCE. If the amplitude of this harmonic signal was comparable to the TCE depth, or the strength of the harmonic signal was greater than $50\\sigma$, the TCE received a false positive label. To avoid misclassification due to period aliasing, we examined periods of $2\\times$ and $\\frac{1}{2}\\times$ the TCE. Additionally, harmonic signals tend to trigger TCEs with long transit durations, which often correspond to the sine wave period. Therefore, we also tested harmonics with periods equaling $1\\times$, $2\\times$, and $3\\times$ the transit duration. If any of the examined phases exceeded our harmonic metric threshold, the TCE achieved a false positive label.\n\n \n\n\n\n\\subsubsection{Phase Coverage Test}\nIt is important that the transit signal has good phase coverage with the available \\emph{K2} data. In certain cases, a few outlier points can be folded, near an integer multiple of the \\emph{K2} cadence, and trigger a TCE. These limited data signals are not meaningful and warrant a false positive label. In addition, the masking applied by our automated pipeline may remove large fractions of the TCE. To ensure we have good signal phase coverage, we examined the gap sizes in the phase folded light curve. If large portions (more than roughly 30 mins.) of photometry are missing during ingress or egress, we labeled these signals as false positives. \n\n\n\n\\subsubsection{Period and Transit Duration Limits}\nWe limit the {\\tt TERRA} software to period ranges beyond 0.5 days. However, this search algorithm is still able to identify periodic signals just below this user defined threshold.\\footnote{This phenomenon is due to the method in which {\\tt TERRA} steps through period space, moving in cadence integers until it exceeds the period limits.} These boundary case signals are usually astrophysical false positives and are difficult to distinguish from real planet candidates. Thus, we remove any TCEs with periods less than 0.5 days. Furthermore, we imposed a strict limit on the transit duration. If the TCE transit duration was greater than 10\\% of the period of the signal, we deemed it a false positive, as these signals are often misidentified harmonics. Moreover, a transiting signal with a duration that exceeds this limit would represent a planet orbiting within three times  the radius of the host star for a short period eclipsing binary. Thus, removing these signals improves the purity of our sample.\n\n\n\\subsubsection{Period Alias Check}\nMis-folding a real transiting signal on an integer multiple of the true signal period will trigger many of the previously described false positive flags. To avoid this misclassification we tested period factors of $2\\times$, $\\frac{1}{2}\\times$, $3\\times$, and $\\frac{1}{3}\\times$ against the original signal. If any of the alternative models produced a likelihood value greater than 1.05 times the original period fit, we reran the entire vetting analysis, using the new corrected period.\n\n\\subsubsection{Ephemeris Match}\nDeep transit signals from bright sources (the parent) can pollute neighboring target apertures (the child) and produce transit artifacts. These signals can be identified by their nearly identical period and ephemeris measurements. We followed the procedures of \\citet{cou14b} and Section A.6 of \\citet{tho18} to identify these false positives, testing our candidate list against itself and our full TCE sample, ensuring previously rejected deep eclipsing binary signals were considered.\n\nWith an ephemeris match it is important to deduce the signal's true origin, since both the parent and child targets will be identified by the matching algorithm. It is expected that the child signal will be significantly polluted by stray starlight, reducing the expected transit depth. Thus, we assign parenthood to the target with the largest transit depth signal and label all additional matches as false positives.\n\nOverall, the implementation of the ephemeris match vetting metric only reduced our catalog by $\\sim2\\%$, considerably less than the $\\sim6\\%$ reduction found for the \\emph{Kepler} DR25. This discrepancy can be attributed to the \\emph{Kepler} field's higher target density, which was dominated by faint stars that are more susceptible to this type of false positive. Therefore, the increased average brightness of \\emph{K2} targets and the reduced target density, explain this reduction in ephemeris matches. \n\n\\begin{figure*}\n\\centering \\includegraphics[height=6.5cm]{hostStarBright.pdf}\n\\caption{The brightness and planet radius distributions of the \\emph{Kepler} and \\emph{K2} host stars, colored by host star effective temperature. The left panel shows the \\emph{Kepler} candidates as described in \\cite{ber20} and the right panel shows the catalog presented in this paper. \\emph{K2} targets tend to be slightly brighter---with larger planets, due to the reduced completeness---than \\emph{Kepler} targets, making them better candidates for follow-up surveys. \n\\label{fig:host}}\n\\end{figure*}\n\n\\subsubsection{Consistency Score}\nFinally, if the TCE passed all of the described vetting diagnostics without achieving a false positive flag, the light curve was reanalyzed---including detrending and signal detection---50 times. This final test measures the stochastic nature of the detrending and the MCMC parameter estimation. Any signal near one of the vetting thresholds would likely be pushed over during these reexaminations. The consistency score was then calculated as the number of times a given TCE was able to pass all the vetting metrics over the number of times tested. Any TCE able to achieve a consistency score greater than 50\\% was granted planet candidacy. Of the 1046 TCEs that initially passed all the {\\tt EDI-Vetter} thresholds, 806 met this final consistency requirement. \n\n\n\n\\section{The Planet Catalog}\n\\label{sec:catalog}\nUsing our fully automated pipeline, we achieved a sample of \\emph{K2} transiting planets suitable for demographics. Within this catalog we found:\n\\begin{itemize}\n    \\item 806 transit signals\n    \\item 747 unique planet candidates\n    \\item 57 multi-planet candidate systems (113 candidates)\n    \\item 366 newly detected planet candidates\n    \\item 18 newly identified multi-planet candidate systems (38 candidates).\n\\end{itemize}\nThe majority of new candidates were found in campaigns not exhaustively searched (C10-18), with a few new candidates identified with low MES in early campaigns. Of the 806 transiting signals identified, 51 signals were detected in multiple, overlapping, campaigns. In the subsequent sections we discuss how the parameters of the planets were derived and the unique systems and candidates found in this catalog. \n\n\n\\subsection{Planet Parameterization}\nTo maintain the homogeneity of our sample, we used an automated routine to fit and estimate the planet and orbital features. As discussed in Section \\ref{sec:stellarSample} we assume the stellar parameters derived by \\citet{har20}, but found 130 of the 747 detected planets did not have stellar characterization available. Fortunately, our pipeline is agnostic to changes in stellar features, enabling subsequent parameter revision. To avoid the assumption of solar values and provide the most current stellar parameters, we used the \\citet{har20} methodology, equipped with APASS (DR9; \\citealt{hen16}) and SkyMapper \\citep{onk19} photometry, to characterize 59 of the remaining host stars. The final 71 targets, which did not have the necessary photometric coverage and\/or parallax measurements for the aforementioned classification, were parameterized using the {\\tt iscochrones} stellar modeling package \\citep{mor15},\\footnote{In order to maintain stellar parameter uniformity we ran the other 628 planet hosts through {\\tt isochrones}. We fit a linear offset on these targets between our parameters and the {\\tt isochrones} parameters, and applied these offsets to the 71 {\\tt isochrones}-only targets.} ensuring all planets in our catalog have corresponding stellar measurements. Using the {\\tt emcee} software package \\citep{goo10, for13}, we measured the posterior distribution of the transiting planet parameters: The ephemeris, the radius ratio, the transit impact parameter, the period, the semi-major axis to stellar radius ratio (apl), the transit duration (tdur), and the vetting consistency score (score). These values were all derived under the assumption of circular orbits. Furthermore, we provide diagnostic plots for each candidate. In Figure \\ref{fig:newExample} we show an example plot for a new sub-Neptune (EPIC 211679060.01), enabling swift visual inspection. However, it is important to clarify that all planets in our sample have been detected through our fully automated pipeline. These plots are only meant to help prioritize follow-up efforts. Figure \\ref{fig:host} shows how the distribution of \\emph{K2} host targets is skewed toward brighter stars, compared to the \\emph{Kepler} candidate hosts, enabling follow-up efforts for a majority of our catalog. \n\n\nWith careful consideration of the transit radius ratio ($R_{fit}$), the planet radius can be extracted. $R_{fit}$ is directly measured by the MCMC routine, but estimates of the true planet radii ($R_{pl}$) required we take into account the stellar radii measurements ($R_{\\star}$) and potential contamination from nearby sources ($\\frac{F_{total}}{F_{\\star}}$). To attain our best estimate we used the following procedure:\n\\begin{equation}\nR_{pl} =R_{\\textrm{fit}}\\; R_{\\star}\\; \\sqrt{\\frac{F_{\\textrm{total}}}{F_{\\star}}},\\\\\n\\end{equation}\n where $\\frac{F_{total}}{F_{\\star}}$ was calculated using the \\emph{Gaia} DR2 stellar catalog to identify contaminants in and near the photometric aperture and a Gaussian point-source function was implemented to estimate the corresponding magnitude of contamination. \\footnote{We do not impose an upper $R_{pl}$ limit in our catalog, retaining our stellar parameter agnosticism. Therefore, 28 candidates exceed $30R_{pl}$. We provide suggestions for dealing with these candidates in Section \\ref{sec:summary}.} In addition to the aforementioned flux complications, \\cite{zin20a} also showed that the required detrending of \\emph{K2} photometry underestimates the radius ratio by a median value of 2.3\\%. We did not adjust our estimates of the planet radius to reflect this tendency, as the mode of this distribution indicated a majority of the measured radii ratios are accurate (see Figure 14 of \\citealt{zin20a}). However, we increased the uncertainty in our planet radius measurements to account for this additional detrending complication. The overall planet radius uncertainty ($\\sigma_R$) was calculated by assuming parameter independence and adding all of these relevant factors in quadrature,\n\\begin{equation}\n\\begin{aligned}[t]\n\\sigma_R &=  \\sqrt{\\sigma_{\\textrm{fit}}^2+\\sigma_{\\star}^2+\\sigma_F^2+\\sigma_{\\textrm{Off}}^2} \\;,&  \\textrm{where}\\\\\n\\sigma_{\\textrm{Off}} & =  0.023\\; R_{pl} & \\textrm{and}\\\\\n\\sigma_F & =  R_{pl}\\;\\frac{F_{\\textrm{total}}-F_{\\star}}{3.76\\; F_{\\textrm{total}}}\\;. &\n\\end{aligned}\n\\end{equation}\nHere, $\\sigma_{\\textrm{Off}}$ and $\\sigma_F$ represent the uncertainty due to detrending and flux contamination respectively (see Section 8.2 of \\citealt{zin20a} for a thorough explanation and derivation of these parameters). Despite these additional contributions, the majority of uncertainty stems from the radius ratio fit ($\\sigma_{\\textrm{fit}}$; $\\sim4\\%$) and the stellar radius measurement ($\\sigma_{\\star}$; $\\sim6\\%$). For most targets $\\sigma_F$ contributes of order $10^{-3}\\%$ uncertainty, however, the most extreme candidate (EPIC 247384685.01) had a measured $F_{\\star}$ that is only 70\\% of $F_{\\textrm{total}}$, contributing an additional $7\\%$ uncertainty to the radius measurements.\n\n\\begin{figure}\n\\centering \\includegraphics[width=\\columnwidth{}]{211679060.pdf}\n\\caption{A sample diagnostic plot for EPIC 211679060.01; the remaining candidate plots are available \\href{http:\/\/www.jonzink.com\/scalingk2.html}{online}. This planet is a new sub-Neptune found in C18. The grey points represent the light curve data used to extract the signal. The purple points show the binned average (with a bin width equal to 1\/6 the transit duration).    \\label{fig:newExample}}\n\\end{figure}\n\n\n\n\\begin{deluxetable*}{lcc}\n\\tablecaption{The homogeneous catalog of \\emph{K2} planet candidates and their associated planet and stellar parameters. The visual inspection flags were manually assigned to help prioritize follow-up efforts. These indicators had no impact on the analysis performed by this pipeline. The known planet (KP) flag indicates a confirmed or validated planet. The planet candidate (PC) flag identifies an unconfirmed candidate and the low priority planet candidate (LPPC) flag designates a weak or more difficult to validate candidate. Finally, the false positive (FP) flag specifies candidates that are likely not planets.      \\label{tab:catalog}}\n\\tablehead{\\colhead{Column} & \\colhead{Units} & \\colhead{Explanation}} \n\\startdata\n1 & --- &      EPIC Identifier  \\\\\n2 & --- &      Campaign   \\\\\n3 & --- &      Candidate ID\\\\\n4 & --- &      Found in Multiple Campaigns Flag   \\\\\n5 & --- &     Consistency Score \\\\\n6 & d   &     Orbital Period  \\\\\n7 & d   &     Lower Uncertainty in Period  \\\\\n8 & d   &     Upper Uncertainty in Period  \\\\\n9 & --- &     Planetary to Stellar Radii Ratio  \\\\\n10 & --- &     Lower Uncertainty in Ratio  \\\\\n11 & --- &     Upper Uncertainty in Ratio  \\\\\n12 & $R_\\earth$ &     Planet radius ($R_{pl}$)  \\\\\n13 & $R_\\earth$ &    Lower Uncertainty in $R_{pl}$  \\\\\n14 & $R_\\earth$ &    Upper Uncertainty in $R_{pl}$  \\\\\n15 & d    &    Transit ephemeris ($t_0$)  \\\\\n16 & d    &    Lower Uncertainty in $t_0$  \\\\\n17 & d    &    Upper Uncertainty in $t_0$  \\\\\n18 & ---  &    Impact parameter (b)  \\\\\n19 & ---  &    Lower Uncertainty in b  \\\\\n20 & ---  &    Upper Uncertainty in b  \\\\\n21 & ---  &    Semi-major Axis to Stellar Radii Ratio ($a\/R_\\star$)   \\\\\n22 & ---  &    Lower Uncertainty in $a\/R_\\star$  \\\\\n23 & ---  &    Upper Uncertainty in $a\/R_\\star$  \\\\\n24 & d    &    Transit Duration  \\\\\n25 & $R_\\sun$  &   Stellar Radius ($R_\\star$)   \\\\\n26 & $R_\\sun$  &   Lower Uncertainty in $R_\\star$  \\\\\n27 & $R_\\sun$  &   Upper Uncertainty in $R_\\star$  \\\\\n28 & $M_\\sun$ &   Stellar Mass ($M_\\star$) \\\\\n29 & $M_\\sun$ &   Lower Uncertainty in $M_\\star$  \\\\\n30 & $M_\\sun$ &   Upper Uncertainty in $M_\\star$  \\\\\n31 & K       &   Stellar Effective Temperature ($T_\\textrm{eff}$)  \\\\\n32 & K       &   Uncertainty $T_\\textrm{eff}$  \\\\\n33 & dex     &   Stellar Surface Gravity (log(g))  \\\\\n34 & dex     &   Uncertainty log(g)  \\\\\n35 & dex     &   Stellar Metallicity [Fe\/H]  \\\\\n36 & dex     &   Uncertainty [Fe\/H]  \\\\\n37 & ---     &   Stellar Spectral Classification \\\\\n38 & ---     &   Visual Inspection Classification \\\\\n\\enddata\n\\tablecomments{This table is available in its entirety in machine-readable form.}\n\\end{deluxetable*}\n\nWe provide our list of planetary parameters for this catalog with corresponding measurements of uncertainty in Table \\ref{tab:catalog}. A plot of the resulting planet period and radius distribution is provided in Figure \\ref{fig:catPl}. The deficit of planets near $2R_\\earth$ is in alignment with the radius gap identified with \\emph{Kepler} data \\citep{ful17} and with previously discovered \\emph{K2} candidates \\citep{har20}. This gap appears to be indicative of some planetary formation or evolution mechanism, the exact origin of which remains unclear. One theory suggests stellar photoevaporation removes the envelope of weakly bound atmospheres in this region of parameter space \\citep{owe17}, separating the super-Earths from the sub-Neptunes. Alternatively, the hot cores of young planets may retain enough energy to expel the atmosphere for planets within this gap (core-powered mass-loss; \\citealt{gup18}). However, these two mechanisms require additional demographic data to parse out the main source of this valley. The catalog derived here provides additional planets and the necessary sample homogeneity needed to examine this feature with greater detail, but such a task is beyond the scope of the current work.\n\n\n\n\\begin{figure}\n\\centering \\includegraphics[width=\\columnwidth{}]{CatalogPlanet.pdf}\n\\caption{The planet sample detected through our fully automated pipeline. The round markers show the new planet candidates (PCs) and + markers show the previously known PCs. The new candidates are uniformly distributed throughout the plot, because many of them come from campaigns not previously examined. The markers have been colored by consistency score to show that most candidates consistently passed the vetting metrics. \n\\label{fig:catPl}}\n\\end{figure}\n\n\n\n\n\n\n\n\\subsection{Multi-Planet Yield}\n\\label{sec:mult}\n\n\n\n\nMulti-planet systems provide a unique opportunity to understand the underlying system architecture and test intra-system formation mechanisms (e.g., \\citealt{owe19}). Furthermore, these planets are more reliable, due to the unlikely probability of identifying two false positives in a light curve \\citep{lis14, sin16}. In our sample we detected 57 unique multi-planet systems, with three systems independently identified in more than one campaign (EPIC 211428897, 212012119, 212072539). In Figure \\ref{fig:multHist} we show the total observed multiplicity distribution for our catalog. Within this sample we did not find any multi-planet systems around A stars, but we identify 17 M dwarf systems and 40 FGK dwarf systems. In consideration of galactic latitude, we found 21 systems that lie greater than $40\\degr$ above\/below the galactic plane. EPIC 206135682, 206209135, and 248545986 all host three planet systems, while the remaining 18 systems only host two. This multi-planet sample provides unique coverage of galactic substructure.\n\n\n\n\n\\subsubsection{Unique Systems}\n\nOur highest multiplicity system is EPIC 211428897, with four Earth-sized planets orbiting an M dwarf (system first identified by \\citealt{dre17}). Our pipeline found this system independently in two of the overlapping fields (C5 and C18), further strengthening its validity. In addition, \\citet{kru19} identified a fifth candidate with a period of 3.28 days. Despite the clear abundance of planet candidates, the \\emph{K2} pixels span $3.98\\arcsec$ and \\emph{Gaia} DR2, which {\\tt EDI-Vetter} uses to identify flux contamination, can only resolve binaries down to $1\\arcsec$ for $\\Delta \\mathrm{mag}\\lesssim3$ \\citep{fur17}. Thus, high-resolution imaging is necessary for system validation. \\citet{dre17} observed this system using Keck NIRC2 and Gemini DSSI, and found a companion star within $0.5\\arcsec$. Since these planets are small, the likely $\\sim \\sqrt{2}$ radii increase, due to flux contamination, will not invalidate their candidacy \\citep{cia17,fur17}. However, it remains unclear whether all of the planets are orbiting one of the stars or some mixture of the two. This will require further follow-up, and remains the subject of future work.\n\n\n\nThrough our analysis we identified 18 new multi-planet systems. EPIC 211502222 had been previously identified as hosting a single sub-Neptune with a period of 22.99 days by \\citet{ye18}. Our pipeline discovered an additional super-Earth at 9.40 days, promoting this G dwarf to a multi-planet host. The remaining 17 systems are entirely new planet candidate discoveries, existing in campaigns not exhaustively searched (C12-C18). Notably, the K dwarf EPIC 249559552 hosts two sub-Neptunes which appear in a 5:2 mean-motion resonance. These resonant systems are important because of the potential detection of additional planets through transit-timing variations (e.g. \\citealt{hol16}). EPIC 249731291 is also interesting since it is an early-type F dwarf (or sub-giant) system, with two short period gas giants.\n\n\\begin{figure}\n\\centering \\includegraphics[width=\\columnwidth{}]{multEdit.pdf}\n\\caption{ A histogram showing the observed system multiplicity distributions of our catalog (Scaling \\emph{K2}) and the \\emph{Kepler} DR25 \\citep{tho18}. The longer data span and reduced noise properties enabled \\emph{Kepler} to identify higher multiplicity systems. In addition, we provide the distributions as a function of stellar spectral type (M: $T_{\\text{eff}}<4000K$; FGK: $4000\\le T_{\\text{eff}}\\le 6500K$; A: $T_{\\text{eff}}>6500K$).    \n\\label{fig:multHist}}\n\\end{figure}\n\n\\subsection{Low-Metallicity Planet Host Stars}\nThe core-accretion model indicates a link between stellar metallicity and planet formation \\citep{pol96}. Observational evidence of this connection was first identified with gas giants \\citep{san04, fis05}. However, the recent findings of \\citet{tes19} complicates this narrative, by showing a lack of correlation between planetary residual metallicity and stellar metallicity. Additionally, direct comparison of metal-rich and metal-poor planet hosts in the \\emph{Kepler} super-Earth and sub-Neptune populations, found no clear difference in planet occurrence \\citep{pet18}. This is further indicative of a complex formation process. Thus, identification of low-metallicity planet hosts enables us to test our understanding of formation theories and to identify subtle features. Within our sample of planets, we found four candidates orbiting host stars with abnormally low stellar metallicity ([Fe\/H]$<-0.75$). Upon further examination, two of these candidates (EPIC 212844216.01 and 220299658.01) have radii greater than $30R_\\Earth$, indicative of an astrophysical false positive. The remaining two super-Earth sized planets orbit a low stellar metallicity M dwarf (EPIC 210897587; [Fe\/H]$=-0.831\\pm0.051$). This system has previously been validated using WIYN\/NESSI high resolution speckle imaging \\citep{hir18} and appears in tension with the expectations of core-accretion in-situ formation, providing constraints for planet formation models.\n\n\n\n\n\\section{Measuring Completeness}\n\\label{sec:complete}\n\nAny catalog of planets will inherit some selection effects due to the methodology of detection, limitations of the instrument, and stellar noise. These biases will affect the sample completeness and must be accounted for when conducting a demographic analysis. The selection of transiting planets can be addressed using analytic arguments, but the instrument and stellar noise contributions to the sample completeness are dependent on the stellar sample and the specifics of the instrument. With an automated detection pipeline this detection efficiency mapping can be achieved through the implementation of an injection\/recovery test. Here, artificial signals are injected into the raw photometry and run through the automated software to test the pipeline's recovery capabilities, directly measuring the impact of instrument and stellar noise on the catalog. Many previous studies have used this technique on \\emph{Kepler} data, yielding meaningful completeness measurements \\citep{pet13b,chr13,chr15,chr20,bur17}.  \n\n\n\\subsection{Measuring CDPP}\n\\label{sec:cdpp}\nThe first step in computing the detection efficiency is to establish a noise profile for each target light curve. With these values available, the signal strength can be estimated given the transit parameters. The ability to make these calculations enables one to understand the likelihood of detection for a given signal. \\emph{Kepler} used the combined differential photometric precision (CDPP) metric to quantify the expected target stellar variability and systematic noise, as described in \\citet{chr12}. For all targets we provide CDPP measurements for following transit durations: 1, 1.5, 2, 2.5, 3, 4, 5, 6, 7, 8, 9, and 10 hours, spanning the range of occultation timescales expected for \\emph{K2} planet candidates.\n\nTo compute these values we randomly injected a weak ($3\\sigma$) transit signal, with the appropriate transit duration, into each detrended light curve. The strength of the recovered signal, normalized by the injected signal depth, provided a measure of the target CDPP. This process was carried out 450 times to ensure a thorough and robust examination of the light curve, sampling the full light curve for four hour transits ($\\sim$20 day period) and 25\\% of the light curve for one hour transits ($\\sim$0.5 day period). To measure the impact of differing transit durations, this process was executed for each respective CDPP timescale. For a detailed account of this procedure see Section 4 of \\citet{zin20a}. \n\nIn Figure \\ref{fig:cdpp} we show the measured 8h CDPP for the targets within this sample.  Overall, there is a clear correlation with photometric magnitude, demonstrating our ability to quantify the light curve noise properties. Moreover, despite the introduction of additional systematic noise from the spacecraft pointing, the detrended \\emph{K2} photometry is still near the theoretical noise limit at the brighter magnitudes. The complete set of measurements are available in Table \\ref{tab:cdpp}.\n\n\\begin{figure}\n\\centering \\includegraphics[width=\\columnwidth{}]{cdppMag.pdf}\n\\caption{The light curve noise (the 8-hour CDPP measurements) for our target sample as a function of the \\emph{Kepler} broadband magnitude measurement. The median CDPP markers represent the median within a one magnitude bin and the corresponding bin standard deviation. The \\emph{Kepler} noise floor represents the shot and read noise expected from the detector alone \\citep{jen10b}.\n\\label{fig:cdpp}}\n\\end{figure}\n\n\\begin{deluxetable*}{lcc}\n\\tablecaption{Description of the CDPP measurements of each stellar target. \\label{tab:cdpp}}\n\\tablehead{\\colhead{Column} & \\colhead{Units} & \\colhead{Explanation}} \n\\startdata\n1 & ... & EPIC identifier\\\\\n2 & ... & Campaign\\\\\n3 & ppm & CDPP RMS Value for Transit of 1.0 hr \\\\\n4 & ppm & CDPP RMS Value for Transit of 1.5 hr \\\\\n5 & ppm & CDPP RMS Value for Transit of 2.0 hr \\\\\n6 & ppm & CDPP RMS Value for Transit of 2.5 hr \\\\\n7 & ppm & CDPP RMS Value for Transit of 3.0 hr \\\\\n8 & ppm & CDPP RMS Value for Transit of 4.0 hr \\\\\n9 & ppm & CDPP RMS Value for Transit of 5.0 hr \\\\\n10 & ppm & CDPP RMS Value for Transit of 6.0 hr \\\\\n11 & ppm & CDPP RMS Value for Transit of 7.0 hr \\\\\n12 & ppm & CDPP RMS Value for Transit of 8.0 hr \\\\\n13 & ppm & CDPP RMS Value for Transit of 9.0 hr \\\\\n14 & ppm & CDPP RMS Value for Transit of 10.0 hr \\\\\n\\enddata\n\\tablecomments{This table is available in its entirety in machine-readable form.}\n\\end{deluxetable*}\n\n\n\\subsection{Injection\/Recovery}\n\nThere are several points along the pipeline at which the signal can be injected. Ideally, injections would be made on the rawest form of photometry (at the pixel-level), but doing so is computationally expensive and provides a marginal gain in completeness accuracy (see \\citealt{chr17} for the effects on the \\emph{Kepler} data set). Moving just one step downstream, the injections can be more easily made at the light curve-level. Here, the artificial signal is introduced into the aperture-integrated flux measurements, followed by pre-processing, detrending and signal detection. Finally, the most accessible, but least accurate, method is by injecting signals after pre-processing (e.g., \\citealt{clo20}). Since the pre-processed {\\tt EVEREST} light curves are readily available, this method requires minimal computational overhead. However, it fails to capture the impact of pre-processing on the sample completeness. These effects are especially important for \\emph{K2} photometry, which undergoes significant modification before being searched. Following the procedures of our previous study \\citep{zin20a}, we injected our artificial signals into the aperture-integrated light curves (before pre-processing; see Figure \\ref{fig:diagram}). In the next few paragraphs we briefly outline our methodology, but suggest interested readers reference Section 5 of our previous work for a more detailed account.\n\nUsing the {\\tt batman} Python package \\citep{kre15}, we created and injected artificial transits in the raw flux data. For each target, we uniformly drew a period from [0.5, 40] days and an $R_{pl}\/R_{\\star}$ from a log-uniform distribution with a range [0.01,0.1]. The ephemeris was uniformly selected, with the requirement that at least three transits reside within the span of the light curve, and the impact parameter was uniformly drawn from [0,1].\\footnote{In the current iteration of this pipeline we have removed the eclipsing binary impact parameter limit mentioned in Section 5 of \\citet{zin20a}. In doing so, we provide a more accurate accounting of the impact of grazing transits.} All injections were assumed to have zero eccentricity. This assumption is motivated by the short period range of detectable \\emph{K2} planets, which many have likely undergone tidal circularization. In addition, eccentricity only affects the transit duration, making its impact on completeness minor. The limb-darkening parameters for the artificial transits were dictated by the stellar parameters discussed in Section \\ref{sec:stellarSample}. Using the ATLAS model coefficients for the \\emph{Kepler} bandpasses \\citep{cla12}, we derived the corresponding quadratic limb-darkening parameters based on their stellar attributes. In cases where stellar parameters did not exist, we assumed solar values.\n\nWe expect the pipeline's recovery capabilities scale as a function of the signal strength. In order to quantify this effect, one must have a measure of the expected injection signal strength (MES). Equipped with our CDPP measurements, this value is directly related to the transit depth (depth) and can be analytically found using:\n\\begin{equation}\n\\text{MES}=C\\;\\frac{\\text{depth}}{\\text{CDPP}_{\\text{t}_{\\text{dur}}}}\\;\\sqrt{N_{\\text{tr}}},\n\\label{eq:MES}\n\\end{equation}\nwhere $\\text{CDPP}_{\\text{t}_{\\text{dur}}}$ represents the targets CDPP measures for a given transit duration (achieved through interpolation of the measured CDPP values discussed in Section \\ref{sec:cdpp}) and $N_{\\text{tr}}$ is the number of available transits within the data span. The $C$ value is a global correction factor that renormalizes the analytic equation to match the detected signal values. For this data set we found $C=0.9488$. Using this equation, we calculated the expected MES values for all injections and measured the sample completeness.\n\n\nOnce our injections were performed, we passed this altered photometry through the software pipeline, testing our recovery capabilities. We considered a planet successfully recovered if it met the following criteria: the detected signal period and ephemeris were within $3\\sigma$ of the injected values and the signal passes all of the vetting metrics. The results of this test can be seen in the completeness map in Figure \\ref{fig:TCEcomplete}. To quantify our detection efficiency as a function of injected MES, we used a uniform kernel density estimator (KDE; width of 0.25 MES) to measure our software's recovery fraction. These values were then fit with a logistic function of the form:\n\\begin{equation}\nf(x)=\\frac{a}{1+e^{-k(x-l)}}.\n\\end{equation}\nThe best fit values of $a$, $k$, and $l$ are listed in Table \\ref{tab:complete}.\n\nWhile MES is closely related to completeness, additional signal parameters can play a role. \\citet{chr20} tested the effects of stellar effective temperature ($T_{\\text{eff}}$), period, $N_{\\text{tr}}$, and photometric magnitude on the completeness of the analogous \\emph{Kepler} injections, finding the strongest effect linked to $N_{\\text{tr}}$. In Figure \\ref{fig:TCEcomplete} we show two of these completeness features for the \\emph{K2} injections: spectral class and $N_{\\text{tr}}$. \\citet{chr15} noted a significant drop ($\\sim4\\%$) in detection efficiency for cooler M dwarfs, which exhibit higher stellar variability. Remarkably, we did not find a significant difference between the AFGK dwarfs ($T_{\\text{eff}}>4000K$) and the M dwarfs ($T_{\\text{eff}}<4000K$). It is likely that the increased systematic noise of \\emph{K2} blurs this completeness feature, making the two populations indistinguishable. We also considered the additional noise contributions expected for young stars. In looking at 8033 targets associated with young star clusters, as indicated by {\\tt BANYAN $\\Sigma$} \\citep{gag18}, we could not identify a significant completeness difference. Like the \\emph{Kepler} TPS, we found $N_{\\text{tr}}$ has the strongest effect on our \\emph{K2} planet sample completeness. Here, a significant drop in detection efficiency was expected for signals near the minimum transit threshold. In light of the pipeline's three transit requirement, these marginal signals were more susceptible to systematics and vetting misclassification. In other words, if any of the three transits were discarded by the vetting, it would immediately receive a false positive label. We also parsed the data into larger $N_{\\text{tr}}$ value bins, but found little difference in completeness between $N_{\\text{tr}}$ equal to four, five, and six; the vetting was unlikely to discard more than one meaningful transit. Furthermore, we considered the effects of the signal period. While this parameter is strongly correlated with $N_{\\text{tr}}$, it has the potential to describe period-dependent detrending and data processing issues. Separating the injected signals at a period of 26 days (see Table \\ref{tab:complete}), we found a loss of completeness ($\\Delta a\\sim0.20$) that was comparable to the $N_{\\text{tr}}$ partition ($\\Delta a\\sim0.21$). This similarity indicates a strong correlation between parameters, suggesting either of the two features would appropriately account for the reduced completeness in this region of parameter space. Since $N_{\\text{tr}}$ provides a slightly larger deficit, we suggest using this function for future demographic analysis of our catalog.  \n\nCompleteness measurements provide a natural test of {\\tt EDI-Vetter}'s classification capabilities. The introduction of systematics from the telescope makes signal vetting more difficult, when compared to the \\emph{Kepler} TPS, requiring more drastic vetting metrics. This could lead to significant misclassification, discarding an abundance of meaningful planet candidates. Fortunately, in Figure \\ref{fig:TCEcomplete} we identify a $\\sim13\\%$ loss of completeness due to the vetting metrics, which is comparable to the \\emph{Kepler} {\\tt Robovetter} ($\\sim10\\%$ loss; \\citealt{cou17b}). Ideally, this difference would be zero, but such a minimal loss is acceptable and, more importantly, quantifiable. \n\nWe have provided the completeness parameters necessary for a demographic analysis of this catalog. However, the injections carried out here are computationally expensive and hold significant value for research beyond the scope of this catalog. Therefore, we provide, in addition, the injected {\\tt EVEREST}-processed light curves and a summary table of the injection\/recovery test.\\footnote{\\href{http:\/\/www.jonzink.com\/scalingk2.html}{http:\/\/www.jonzink.com\/scalingk2.html}} These data products enable users to set custom completeness limits and to test their own vetting software with significantly reduced overhead. \n\n\n\\begin{figure*}\n\\centering \\includegraphics[width=\\textwidth{}]{completenessTable.pdf}\n\\caption{Plots of the measured completeness, resulting from our injection\/recovery test, for the presented planet sample. To show the effects of characteristic stellar noise, the number of transits, and the loss of planets due to our vetting software, we provide several slices of the data. The corresponding logistic function parameters are available in Table \\ref{tab:complete}. The heat map shows the overall vetted completeness as a function of planet radius ratio and the transit period. For our stellar sample, the radii ratios of 0.01, 0.03, and 0.1 correspond to median planet radii of 1.1, 3.4, and 11.3 $R_\\Earth$ respectively.  \n\\label{fig:TCEcomplete}}\n\\end{figure*}\n\n\n\n\n\\begin{deluxetable}{lccc}\n\\tablecaption{The logistic parameters for the corresponding completeness functions shown in Figures \\ref{fig:TCEcomplete} \\& \\ref{fig:complete_lat}. Additionally, we include the completeness parameters for a separation of 26 day period signals. It is important to highlight that $a$ represents the maximum completeness for high MES signals. \\label{tab:complete}}\n\\tablehead{\\colhead{Model} & \\colhead{a} & \\colhead{k} & \\colhead{l}} \n\\startdata\n\\textbf{Unvetted} & 0.7407 & 0.6859 & 9.7407 \\\\\n\\textbf{Vetted} & 0.6093 & 0.6369 & 10.8531 \\\\\n\\\\\n\\textbf{>3 Transits} & 0.6868 & 0.6347 & 10.9473\\\\\n\\textbf{=3 Transits} & 0.4788 & 0.6497 & 10.5598\\\\\n\\\\\n\\textbf{<26d Periods} & 0.6619 & 0.6231 & 10.9072\\\\\n\\textbf{>26d Periods} & 0.4635 & 0.6607 & 10.5441\\\\\n\\\\\n\\textbf{AFGK Dwarf} & 0.6095 & 0.6088 & 10.8986\\\\\n\\textbf{M Dwarf} & 0.6039 & 0.8455 & 10.5636 \\\\\n\\\\\n\\textbf{C1} & 0.3923 & 0.7654 & 11.3914\\\\\n\\textbf{C2} & 0.6430 & 0.7173 & 10.8544 \\\\\n\\textbf{C3} & 0.7462 & 0.6689 & 10.5701 \\\\\n\\textbf{C4} & 0.6734 & 0.6344 & 11.1443 \\\\\n\\textbf{C5} & 0.4425 & 0.5923 & 11.3923 \\\\\n\\textbf{C6} & 0.7654 & 0.5759 & 10.8772 \\\\\n\\textbf{C7} & 0.3941 & 0.6052 & 11.7002 \\\\\n\\textbf{C8} & 0.6669 & 0.5726 & 10.0560 \\\\\n\\textbf{C10} & 0.5572 & 0.6469 & 10.0056 \\\\\n\\textbf{C11} & 0.2171 & 0.4759 & 12.3882 \\\\\n\\textbf{C12} & 0.6192 & 0.7341 & 10.6272 \\\\\n\\textbf{C13} & 0.6853 & 0.5698 & 11.3878 \\\\\n\\textbf{C14} & 0.7505 & 0.6596 & 10.9776 \\\\\n\\textbf{C15} & 0.6067 & 0.6480 & 10.4673 \\\\\n\\textbf{C16} & 0.6809 & 0.7256 & 10.5453 \\\\\n\\textbf{C17} & 0.5848 & 0.6633 & 10.3635 \\\\\n\\textbf{C18} & 0.6116 & 0.4676 & 11.5783 \\\\\n\\enddata\n\n\\end{deluxetable}\n\n\\subsection{Completeness and Galactic Latitude}\n\\label{sec:comLat}\nEach \\emph{K2} campaign probed a different region along the ecliptic. These fields correspond to unique galactic latitudes, where distinct noise features may spawn differences in inter-campaign completeness. To address these potential variations, we consider the completeness as a function of campaign in Figure \\ref{fig:complete_lat}.\n\nOverall, there is significant scatter among the lower absolute galactic latitude campaigns ($\\mid b \\mid<40\\degr$). This trend is indicative of target crowding near the galactic plane \\citep{gil15}. In these low latitude fields the photometric apertures are more contaminated by background sources, contributing additional noise, variability, and flux dilution, making transit detection more onerous. This is highlighted by Campaign 11, which is the closest field to the galactic plane ($b\\sim9\\degr$) that we analyzed, thus providing the lowest completeness of all campaigns ($a=0.22$). Conversely, Campaign 6 with $b\\sim48\\degr$ has the highest completeness ($a=0.77$).\n\n\\begin{figure*}\n\\centering \\includegraphics[width=\\textwidth{}]{latCom.pdf}\n\\caption{The calculated vetted completeness for the low (left; $\\mid b \\mid<40\\degr$) and high (right; $\\mid b \\mid>40\\degr$) galactic latitude Campaigns. The corresponding logistic function parameters are available in Table \\ref{tab:complete}. \n\\label{fig:complete_lat}}\n\\end{figure*}\n\nIn the high galactic latitude campaigns ($\\mid b \\mid>40\\degr$), this crowding effect is less salient, reducing the inter-campaign completeness scatter. Furthermore, these well isolated targets, provide high quality photometry and yield the highest completeness of all \\emph{K2} campaigns ($a\\sim 0.70$).\n\nThese inter-campaign differences are meaningful, however the limited number of targets (and synthetic transit injections) within each campaign, subjects these completeness measurements to further uncertainty. Therefore, the values provided for the global completeness assessment (for Figure \\ref{fig:TCEcomplete}) are more robust and should be used for full catalog occurrence analysis.\n\n\n\n\n\n\\section{Measuring Reliability}\n\\label{sec:reli}\nDespite efforts to remove problematic cadences, instrument systematics pollute the light curves, creating artificial dips that can be erroneously characterized as a transit signal.\nTo measure the reliability in a homogeneous catalog of planets, the rate of these false alarms (FAs) must be quantified. \\citet{bry19} showed that proper accounting of the sample FA rate is essential in extracting meaningful and consistent planet occurrence measurements. \n\nThe main goal of {\\tt EDI-Vetter} (and its predecessor {\\tt Robovetter}) is to parse through all TCEs and remove FAs without eliminating true planet candidates. However, this process is difficult to automate and requires a method of testing the software's capability to achieve this goal. Accomplishing such a task necessitates an equivalent data set that captures all the unique noise properties, which contributes to FAs, without the existence of any true astrophysical signals. With such data available, the light curves can be processed through the detection pipeline. If the vetting algorithm worked perfectly, nothing would be identified as a planet candidate. Therefore, any signals capable of achieving planet candidacy would be an authentic FA and provide insight into the software's capabilities. \n\n\\citet{cou17b} explored two methods for simulating this necessary data using the existing light curves: data scrambling and light curve inversion. The first method takes large portions of the real light curve data and randomizes the order, retaining all of the noise properties while scrambling out true periodic planet signatures. Using this method \\citep{tho18} was able to replicate the real long period TCE distribution of the \\emph{Kepler} DR25 catalog. The second method inverts the light curve flux measurements. Upon this manipulation, the existing real transit signals will now be seen as flux brightening events, rendering them unidentifiable by the transit search algorithm. Under the assumption that many systematic issues are symmetric upon a flux inversion, the photometry will retain its noise properties without containing any real transit signals. This method was used by \\citet{tho18} to replicate the real short period TCE distribution of the \\emph{Kepler} DR25 catalog, while maintaining the quasi-sinusoidal features, like the rolling bands that repeat due to the spacecraft's temperature fluctuations (see Section 6.7.1 of \\citet{van16} for further detail on this effect).\n\n\\begin{figure}\n\\centering \\includegraphics[width=\\columnwidth{}]{FullCat.pdf}\n\\caption{The distribution of TCEs from the real light curves and the distribution of inverted TCEs from the FA simulation. This shows consistency among the two distributions with a minor surplus of inverted TCEs at longer periods and a deficiency at the 3rd harmonic of the thruster firing (18 hours).\n\\label{fig:TCEreliability}}\n\\end{figure}\n\n\nSince the spacecraft also underwent quasi-periodic roll motion during the \\emph{K2} mission, leading to cyclical instrument systematics, we chose the light curve inversion method. In Figure \\ref{fig:TCEreliability} we assess our ability to capture the noise features using this method. By comparing the distribution of TCEs from the real light curves with those of the inverted light curves, we can identify regions of parameter space where the inversions over- or under-represent systematic noise properties. Overall, the two distributions (108,379 TCEs and 110,548 inverted TCEs)\\footnote{The individual transit check described in Section \\ref{sec:ITC} was the dominant vetting metric for both the TCEs and the inverted TCEs, discarding non-transit shaped signals} are well aligned, providing an adequate simulation of the data set's noise characteristics. However, the inverted TCEs are slightly over-represented at long periods and under-represented at the 3rd harmonic of the 6 hour thruster firing. While we acknowledge these minor discrepancies, their impact will be small on the overall measure of the catalog reliability. \n\nAfter running the inverted light curves through our pipeline we identify 77 FA signals as planet candidates ( 806 candidates were identified in the real light curves). We used this information, alongside the formalism described in Section 4.1 of \\citet{tho18}, to quantify our sample reliability. The first step in achieving this measurement requires an understanding of the vetting routines FA removal efficiency ($E$). From the inverted light curve test we can estimate $E$ as\n\\begin{equation}\nE\\approx\\frac{N_{FP_{\\mathrm{inv}}}}{T_{TCE_{\\mathrm{inv}}}},\n\\end{equation}\nwhere $N_{FP_{\\mathrm{inv}}}$ is the number of TCEs that were accurately flagged as false positives and $T_{TCE_{\\mathrm{inv}}}$ is the total number of TCEs found in the inversion test. For the total data set, we found {\\tt EDI-Vetter} has a 99.9\\% efficiency in removing FA signals. However, this extreme competence must be balanced by the abundance of TCEs found by our pipeline. \n\n\\begin{figure}\n\\centering \\includegraphics[width=\\columnwidth{}]{Reliability_94.1.pdf}\n\\caption{The calculated reliability of our planet sample as a function of orbital period and planet radius. The reliability percent and the number of candidates have been listed in each corresponding box. The white regions represent areas of parameter space where the number of candidates and FAs are sparse, making accurate measurements of reliability unachievable. \n\\label{fig:reliability}}\n\\end{figure}\n\nWe can determine the reliability fraction ($R$) of our catalog using the number of TCEs flagged as false positive ($N_{FP}$) in the real light curves and the number of planet candidates ($N_{PC}$):\n\\begin{equation}\nR=1-\\frac{N_{FP}}{N_{PC}}\\Bigg(\\frac{1-E}{E}\\Bigg).\n\\label{eq:reli}\n\\end{equation}\nOverall, we found the planet catalog provided here is 91\\% reliable. This is slightly lower than the 97\\% reliability of the \\emph{Kepler} DR25 catalog \\citep{tho18}. However, this \\emph{Kepler} value is greatly improved by the extensive data baseline. A more appropriate comparison would consider a period range with a comparable number of transits (\\emph{Kepler} candidates with periods greater than 10 days have a similar number of transits). In this region of parameter space the {\\tt Robovetter} has a reliability of 95\\%, which is closer to the value reported for {\\tt EDI-Vetter}. Moreover, these broad summary statistics fail to capture the complexity of this metric.\n\n\\begin{figure*}\n\\centering \\includegraphics[height=7cm]{latReli.pdf}\n\\caption{The calculated reliability for the low and high galactic latitude Campaigns. The reliability percent and the number of candidates have been listed in each corresponding box. The white regions represent areas of parameter space where the number of candidates and FAs are sparse, making accurate measurements of reliability unachievable. \n\\label{fig:reliability_lat}}\n\\end{figure*}\n\nWe found 8 of the FAs detected in our inversion simulation are hosted by sub-giant and giant stars, which have notably more active stellar surfaces. By excluding all giants with $\\log(g)$ less than 4, we focus on the dwarf star population (in Figure \\ref{fig:reliability}) and consider how reliability changes as a function of period and planet radius. At longer periods the reliability drops. The reduced number of transits available for a given candidate more easily enables systematic features to line up and create a FA signal. Smaller planet radius regions are also more susceptible to FAs due to their weak signal strength, which can be replicated by noise within the data set. When accounting for this period and radius dependence we expect 94\\% of our dwarf host candidates to be real astrophysical signals.\n\n\\subsection{Reliability and Galactic Latitude}\n\\label{sec:galRelib}\n\n\nLike completeness, reliability may also exhibit inter-campaign differences. However, the number of FAs detected by inversion are not significant enough to enable thorough campaign by campaign analysis. Since Section \\ref{sec:comLat} showed that a majority of field differences could be attributed to galactic latitude, we considered differences in reliability for high ($\\mid b \\mid>40\\degr$) and low ($\\mid b \\mid<40\\degr$) absolute galactic latitudes in Figure \\ref{fig:reliability_lat}. Overall, we found that both low and high absolute galactic latitude campaigns produce a very similar reliability, 95\\% and 94\\% respectively. Upon examination of these two stellar populations, no significant differences were identified, thus these changes in reliability are likely due to statistical fluctuations from the reduced number of FAs (34 in low latitudes and 35 in high latitudes). Therefore, we encourage future population analysis to use the full reliability calculation provided in Figure \\ref{fig:reliability}.\n\n\n\\subsection{Astrophysical False Positives}\n\\label{sec:astrFP}\nIt is important to highlight that reliability is a measure of the systematic FA contamination rate. There exist non-planetary astrophysical sources capable of producing a transit signal. For example, dim background eclipsing binaries (EBs) may experience significant flux dilution from the primary target, manifesting a shallow depth transit. This planetary signal mimicry may lead to candidate misclassification \\citep{fre13,san16,mor16,mat18}.  The \\emph{Kepler} DR25 relied on the centroid offset test \\citep{mul17} to identify these contaminants. This test considered the TCE's flux difference in and out of transit for each aperture pixel. Larger differences near the edge (or away from the center) of the aperture indicated a non-target star origin, enabling the identification of contaminants to within $1\\arcsec$. Although it is possible that such signals were of planetary nature \\citep{bry13}, the expected flux dilution makes classification difficult. Thus, candidates with centroid offsets were removed from the DR25 candidate catalog. Employing a similar test for \\emph{K2} would be difficult given the spacecraft's roll motion. The {\\tt DAVE} algorithm \\cite{kos19} considered each individual cadence (instead of the entire transit) to carry out a similar procedure for \\emph{K2} and was able to identify 96 centroid offsets from the list of known candidates. While this method is helpful, it lacks the statistical strength of the original centroid test. Instead, we choose to leverage the \\emph{Gaia} DR2 catalog to identify these sub-pixel background sources. Doing so, we are able to identify potential contaminates to within $1\\arcsec$ of the target star \\citep{zie18}, the equivalent limit of the \\emph{Kepler} DR25 centroid offset test.\\footnote{\\emph{Gaia's} spatial resolution reduces to $2\\arcsec$ for magnitude difference greater than 5.}  \n\n\\begin{figure}\n\\centering \\includegraphics[width=\\columnwidth{}]{seperation.pdf}\n\\caption{The number of \\emph{Gaia} DR2 sources within a one arcminute radius of each planet candidate. This plot shows that \\emph{K2} candidate hosts (green) generally occupy more isolated fields than \\emph{Kepler} candidate hosts (blue). A characteristic aligned with the fact that $92\\%$ of our \\emph{K2} candidates are observed at an absolute galactic latitudes greater than the edge of the \\emph{Kepler} field ($\\mid b \\mid>22\\degr$), where stellar density subsides.\n\\label{fig:seperation}}\n\\end{figure}\n\nTo estimate the rate of contamination in our sample we consider the \\emph{Kepler} certified false positive (CFP) table \\citep{bry17} alongside the \\emph{Kepler} DR25 candidate catalog. The \\emph{Kepler} CFPs represents a sample of 3,590 \\emph{Kepler} signals not granted candidacy and thoroughly investigated to ensure a non-planetary origin. Carrying out all of our vetting procedures (including the contaminant identification from \\emph{Gaia} DR2) to both the CFP table and the DR25 candidate list, we would expect to find 169 CFPs and 3,470 DR25 candidates (a $\\sim4.6\\%$ contamination rate). However, this rate is likely an upper limit. The astrophysical false positive rate for background EBs should recede with increased distance from the galactic plane, given the expected change in aperture crowding as a function of galactic latitude. Overall, the \\emph{Kepler} target stars are closer to the galactic plane ($b \\lesssim 20\\degr$) compared to the \\emph{K2} targets, which largely occupy a higher absolute galactic latitude ($\\mid b \\mid\\gtrsim 20\\degr$). This apparent density distinction is shown in Figure \\ref{fig:seperation}, where we present the number of \\emph{Gaia} sources within a one arcminute radius of each candidate host. The \\emph{Kepler} candidates clearly occupy a more crowded field than a majority of our \\emph{K2} candidates, subjecting \\emph{Kepler} targets to a heightened occurrence of background contaminating sources. Overall, the CFP table is unique to the \\emph{Kepler} prime mission and differences in instrument performance, catalog construction, and stellar fields limit our ability to make precise contamination rate estimates for our \\emph{K2} catalog.\n\n\n\\begin{figure}\n\\centering \\includegraphics[width=\\columnwidth{}]{fluxDilution.pdf}\n\\caption{The expected Equation \\ref{eq:EB} values for the candidates within our catalog, conservatively assuming that the signals originated from the dimmest aperture-encased sources (Diluted). These quantities are plotted against the catalog values, which assume the signal host was the brightest aperture-encased source (Catalog). The yellow regions, which includes 18 candidates, indicate signals that could be astrophysical false positives ($R_{\\mathrm{pl}}\/R_{\\star}+b>1.04$) with their transit signal diluted. We also found an additional 11 candidates that would have produced a $R_{\\mathrm{pl}}\/R_{\\star}$ greater than 0.3 under this worse-case scenario assumption, presenting EB like properties and warranting rejection from our catalog. Signals that exceed either of these thresholds have been colored in green to indicate a potential background EB signal.\n\\label{fig:fluxDilution}}\n\\end{figure}\n\n\nTo further assess our ability to remove background EBs, we can perform a worst-case scenario test. The \\emph{Gaia} DR2 source catalog is essentially complete \\footnote{Visual companions, who are not spatially resolved, are an exception to this completeness claim.} for photometric $G$-band (comparable to the $Kepler$-band) sources brighter than 17 magnitudes, reducing to $\\sim80\\%$ completeness for $G=20$ targets \\citep{bou20}. Correspondingly, in Figure \\ref{fig:cdpp} it is shown that our CDPP values increase as a function of magnitude, reducing our pipeline's ability to detect signals around these dim targets. For example, a Kep.$=17$ star requires an occultation that exceeds $1\\%$ to qualify as a TCE ($8.68\\sigma$), increasing to $40\\%$ for targets with Kep.$=20$. When these dim stars are within the aperture of a brighter source, the photometric noise floor is further elevated by the noise contribution from both stars. By combining the low probability of greater than $40\\%$ eclipses and the $80\\%$ stellar completeness at $G=20$, \\emph{Gaia} provides sufficient coverage of the parameter space where background EBs could be hidden, yet still detectable by our pipeline. Thus, we can bound our contamination rate by making a conservative assumption that all of our candidate signals originate from the dimmest aperture-encased \\emph{Gaia} DR2 identified source. Currently, our candidate catalog assumes each transit signal corresponds to the brightest star within the aperture, however, such an assumption may misidentify diluted background EBs. Our worse-case scenario test helps quantify the magnitude of this contamination and the corresponding results are displayed in Figure \\ref{fig:fluxDilution}. We found 74 of our candidate targets contain additional aperture-encased \\emph{Gaia} source. Nine of these targets exhibit transits that can only be physically explained by a signal from the brightest star. In other words, the transit brightness reduction exceeds that of the entire flux contribution from the background stars. We found 17 candidates that would exhibit $R_{\\mathrm{pl}}\/R_{\\star}+b$ values greater than our catalog threshold (1.04) and an additional 13 candidates that would not meet our $R_{\\mathrm{pl}}\/R_{\\star}\/le0.3$ requirement. If we assume all 30 of these signals originated from a background source, we establish a background EB contamination rate of $4.0\\%$. However, the true parent source of these candidates remains unclear, thus this estimation is again an upper limit. In addition, we acknowledge that our detection metrics could remain averted by small radius-ratio non-grazing EBs, but we expect such cases to be rare. Overall, we expect contamination from background EBs to make up less than 5\\% of our candidate list.\n\n\n\n\\section{Occurrence Rate Recommendations}\n\\label{sec:suggest}\nAll exoplanet demographic analyses require defining an underlying stellar sample of interest. Our catalog used nearly the entire \\emph{K2} target sample, which may be too broad for future analysis. The completeness measurements given here may not accurately reflect that of a reduced target list. Therefore, we recommend users select their own stellar sample and consult the  injection\/recovery summary table to assess the completeness of their selected targets for the highest degree of accuracy.\\footnote{\\label{noteWeb}\\url{www.jonzink.com\/scalingk2.html}} However, the completeness parameters provided in Table \\ref{tab:complete} are robust to minor sample selection modifications. Evidence of this claim is provided by the fact that the M dwarf and AFGK dwarf samples provide consistent results, see Figure \\ref{fig:TCEcomplete}, despite known differences. \n\nIn addition, 41,061 targets in our sample lacked stellar parameters, requiring the assumption of solar values. This may modify the results once these parameters become available. The remaining stellar parameters were provided by \\cite{hub16} for 94,769 targets and \\cite{har20} for 222,088 targets. While \\cite{zin20b} showed the offset between these two methods is minimal, making this mixture of catalogs reasonable, a uniform stellar parameterization would provide more homogeneous results. Fortunately, the pipeline's stellar agnosticism makes parameter updating uncomplicated. As additional data from the forthcoming \\emph{Gaia} DR3 becomes available, it will likely modify many of these stellar values and alter the underlying sample of dwarf stars. We suggest users implement the most up-to-date stellar parameters along with the injection\/recovery summary table to evaluate sample completeness.\n\nUpon close inspection, the \\emph{K2} population appears more stochastic along the main sequence (see Figure \\ref{fig:HR}). This is largely attributed to the guest observing selection process for the \\emph{K2} fields, where individual proposals each applied their own target selection criteria to construct target lists that addressed their specific science goals. This could lead, for instance, to situations where the G dwarfs in a given campaign represent a distinct population from the G dwarfs in another campaign (e.g. probing different ranges of stellar metallicity, which is known to impact planet occurrence rates), or from a given population of field G dwarfs. \\citet{zin20b} looked at the stellar population around Campaign 5 and found this latter selection effect did not provide a biased sample of FGK dwarfs for C5. However, for occurrence rate calculations, similar inspection of other campaigns should be carried out to ensure each campaign provides a uniform representation of their respective region of the sky. Where that does not appear to be true, users of this catalog are encouraged to independently select a set of targets from the full set of available targets (using, for instance, \\emph{Gaia} properties) that more clearly represents an unbiased sample of the desired population.\n\n\nIncorporating \\emph{Gaia} DR2 into our pipeline enabled us to provide more accurate planet radii measurements. \\citet{cia17} showed that non-transiting stellar multiplicity can artificially reduce transiting planet depths, leading to an overestimation in the occurrence of Earth-sized planets by 15-20\\%. Our pipeline used the \\emph{Gaia} DR2 to account for neighboring flux contamination, improving the precision of our radii measurements and the accuracy of future occurrence estimates. However, planet radius is markedly dependent on the underlying stellar radius measurements and our catalog is derived independent of such parameterization. Therefore, we did not impose any strict upper limits on planet radius and found 28 of our candidates have planet radii exceeding $30R_\\Earth$. These candidates are likely astrophysical false positives. We suggest users consider an upper radius bound when carrying out occurrence analyses. Users may also consider using the \\emph{Gaia} renormalized unit weight error (RUWE) values to further purify their sample of interest, as suggested by \\citep{bel20}.  \n\nIdeally, the reliability would also be updated as additional information on stellar and planetary parameters are made available. This is possible using the reliability summary table, but given the small number FAs found we expect very minor changes to occur.\n\nIt is important to note that our completeness measurements do not address the window function, which requires three transits occur within the available photometry. All injections were required to have at least three transits occurring within this window, removing this detection probability from the calculated completeness. In testing, we found most light curves follow the expected analytic probability formula ($prob$):\n\n\\begin{equation}\n\\label{eq:win}\n\\begin{aligned}[t]\nprob & =1; & P<t_{\\text{span}}\/3\\\\\nprob & =\\frac{t_{\\text{span}}}{P}-2;  & \\frac{t_{\\text{span}}}{3}\\le P \\le \\frac{t_{\\text{span}}}{2}\\\\\nprob & =0;  & P> t_{\\text{span}}\/2,\n\\end{aligned}\n\\end{equation}\nwhere $P$ is the signal period and $t_{\\text{span}}$ is the total span of the data (see Figure 11 of \\citealt{zin20a}). However, intra-campaign data gaps exist and should be carefully considered in any occurrence rate calculations.\n\n\\section{Summary}\n\\label{sec:summary}\nWe provide a catalog of transiting exoplanet candidates using \\emph{K2} photometry from Campaign 1-8 and 10-18, derived using a fully automated detection pipeline. This demographic sample includes 747 unique planets, 366 of which were previously unidentified. Additionally, we found 57 multi-planet candidate systems, of which 18 are newly identified. These discovered systems include a K dwarf (EPIC 249559552) hosting two sub-Neptune candidates in a 5:2 mean-motion resonance, and an early-type F dwarf (EPIC 249731291) with two short period gas giant candidates, providing an interesting constraint on formation and migration mechanisms. Follow-up observations and validation of these and a number of the other new candidates presented in this catalog is currently underway (Christiansen et al. in prep).\n\nWe employed an automated detection routine to achieve this catalog, enabling measurements of sample completeness and reliability. By injecting artificial transit signals before {\\tt EVEREST} pre-processing, we provide the most accurate measurements of \\emph{K2} sample completeness. Additionally, we used the inverted light curves to measure our vetting software's ability to remove systematic false alarms from our catalog of planets, providing a quantitative assessment of sample reliability. Using this planet sample, and the corresponding completeness and reliability measurements, exoplanet occurrence rate calculations can now be performed using \\emph{K2} planet candidates, which will be the subject of the next papers in the Scaling K2 series. With careful consideration of each data set's unique window functions, the \\emph{Kepler} and \\emph{K2} planet samples can now be combined, maximizing our ability to measure transiting planet occurrence rates throughout the local galaxy.\n\n\\section{Acknowledgements}\n\nWe thank the anonymous referee for their thoughtful feedback. This work made use of the gaia-kepler.fun crossmatch database created by Megan Bedell. The simulations described here were performed on the UCLA Hoffman2 shared computing cluster and using the resources provided by the Bhaumik Institute. This research has made use of the NASA Exoplanet Archive and the Exoplanet Follow-up Observation Program website, which are operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This paper includes data collected by the \\emph{Kepler} mission and obtained from the MAST data archive at the Space Telescope Science Institute (STScI). Funding for the \\emph{Kepler} mission is provided by the NASA Science Mission Directorate. STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5\u201326555. J. Z. acknowledges funding from NASA ADAP grant 443820HN21811. K. H-U and J. C. acknowledge funding from NASA ADAP grant 80NSSC18K0431.\n\n\n\\software{{\\tt EVEREST} \\citep{lug16,lug18}, {\\tt TERRA} \\citep{pet13b}, {\\tt EDI-Vetter} \\citep{zin20a}, {\\tt PyMC3} \\citep{sal15}, {\\tt Exoplanet} \\citep{for19}, {\\tt RoboVetter} \\citep{tho18}, {\\tt batman} \\cite{kre15}, {\\tt emcee} \\citep{for13}}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe game of Cops and Robbers (defined, along with all the standard notation, later in this section) is usually studied in the context of the {\\em cop number}, the minimum number of cops needed to ensure a winning strategy. The cop number is often challenging to analyze; establishing upper bounds for this parameter is the focus of Meyniel's conjecture that the cop number of a connected $n$-vertex graph is $O(\\sqrt{n}).$ For additional background on Cops and Robbers and Meyniel's conjecture, see the book~\\cite{bonato}.\n\nA number of variants of Cops and Robbers have been studied. For example, we may allow a cop to capture the robber from a distance $k$, where $k$ is a non-negative integer~\\cite{bonato4}, play on edges~\\cite{pawel}, allow one or both players to move with different speeds~\\cite{NogaAbbas, fkl} or to teleport, allow the robber to capture the cops~\\cite{bonato0}, make the robber invisible or drunk~\\cite{drunk1,drunk2}, or allow at most one cop to move in any given round~\\cite{oo, hypercube, lazy_gnp}. See Chapter~8 of~\\cite{bonato} for a non-comprehensive survey of variants of Cops and Robbers.\n\n\\bigskip\n\nIn this paper, we consider a variant of the game of Cops and Robbers, called \\emph{Containment}, introduced recently by Komarov and Mackey~\\cite{komarov}. In this version, cops move from edge to adjacent edge, the robber moves as in the classic game, from vertex to adjacent vertex (but cannot move along an edge occupied by a cop). Formally, the game is played on a finite, simple, and undetected graph. There are two players, a set of \\emph{cops} and a single \\emph{robber}. The game is played over a sequence of discrete time-steps or \\emph{turns}, with the cops going first on turn $0$ and then playing on alternate time-steps.  A \\emph{round} of the game is a cop move together with the subsequent robber move.  The cops occupy edges and the robber occupies vertices; for simplicity, we often identify the player with the vertex\/edge they occupy. When the robber is ready to move in a round, she can move to a neighbouring vertex but cannot move along an edge occupied by a cop, cops can move to an edge that is incident to their current location. Players can always \\emph{pass}, that is, remain on their own vertices\/edges. Observe that any subset of cops may move in a given round. The cops win if after some finite number of rounds, all edges incident with the robber are occupied by cops. This is called a \\emph{capture}. The robber wins if she can evade capture indefinitely. A \\emph{winning strategy for the cops} is a set of rules that if followed, result in a win for the cops. A \\emph{winning strategy for the robber} is defined analogously.  As stated earlier, the original game of \\emph{Cops and Robbers} is defined almost exactly as this one, with the exception that all players occupy vertices.\n\nIf we place a cop at each edge, then the cops are guaranteed to win. Therefore, the minimum number of cops required to win in a graph $G$ is a well-defined positive integer, named the \\emph{containability number} of the graph $G.$ Following the notation introduced in~\\cite{komarov}, we write $\\xi(G)$ for the containability number of a graph $G$ and $c(G)$ for the original \\emph{cop-number} of $G$.\n\n\\bigskip\n\nIn~\\cite{komarov}, Komarov and Mackey proved that for every graph $G$, \n$$\nc(G) \\le \\xi(G) \\le \\gamma(G) \\Delta(G), \n$$\nwhere $\\gamma(G)$ and $\\Delta(G)$ are the domination number and the maximum degree of $G$, respectively. It was conjectured that the upper bound can be strengthened and, in fact, the following holds.\n\n\\begin{conjecture}[\\cite{komarov}]\\label{con:komarov}\nFor every graph $G$, $\\xi(G) \\le c(G) \\Delta(G)$. \n\\end{conjecture}\n\n\\noindent Observe that, trivially, $c(G) \\le \\gamma(G)$ so this would imply the previous result. This seems to be the main question for this variant of the game at the moment. By investigating expansion properties, we provide asymptotically almost sure bounds on the containability number of binomial random graphs ${\\mathcal{G}}(n,p)$ for a wide range of $p=p(n)$, proving that the conjecture holds for some ranges of $p$ (or holds up to a constant or an $O(\\log n)$ multiplicative factors for some other ranges of $p$). However, before we state the result, let us introduce the probability space we deal with and mention a few results for the classic cop-number that will be needed to examine the conjecture (since the corresponding upper bound is a function of the cop number).\n\n\\bigskip\n\nThe \\emph{random graph} ${\\mathcal{G}}(n,p)$ consists of the probability space $(\\Omega, \\mathcal{F}, \\mathbb{P})$, where $\\Omega$ is the set of all graphs with vertex set $\\{1,2,\\dots,n\\}$, $\\mathcal{F}$ is the family of all subsets of $\\Omega$, and for every $G \\in \\Omega$,\n$$\n\\mathbb{P}(G) = p^{|E(G)|} (1-p)^{{n \\choose 2} - |E(G)|} \\,.\n$$\nThis space may be viewed as the set of outcomes of ${n \\choose 2}$ independent coin flips, one for each pair $(u,v)$ of vertices, where the probability of success (that is, adding edge $uv$) is $p.$ Note that $p=p(n)$ may (and usually does) tend to zero as $n$ tends to infinity. All asymptotics throughout are as $n \\rightarrow \\infty $ (we emphasize that the notations $o(\\cdot)$ and $O(\\cdot)$ refer to functions of $n$, not necessarily positive, whose growth is bounded). We say that an event in a probability space holds \\emph{asymptotically almost surely} (or \\emph{a.a.s.}) if the probability that it holds tends to $1$ as $n$ goes to infinity.\n\n\\bigskip\n\nLet us now briefly describe some known results on the (classic) cop-number of ${\\mathcal{G}}(n,p)$. Bonato, Wang, and the author of this paper investigated such games in ${\\mathcal{G}}(n,p)$ random graphs and in generalizations used to model complex networks with power-law degree distributions (see~\\cite{bpw}). From their results it follows that if $2 \\log n \/ \\sqrt{n} \\le p < 1-\\eps$ for some $\\eps>0$, then a.a.s. we have that\n\\begin{equation*\nc({\\mathcal{G}}(n,p))= \\Theta(\\log n\/p),\n\\end{equation*}\nso Meyniel's conjecture holds a.a.s.\\ for such $p$. In fact, for $p=n^{-o(1)}$ we have that a.a.s.\\  $c({\\mathcal{G}}(n,p))=(1+o(1)) \\log_{1\/(1-p)} n$. A simple argument using dominating sets shows that Meyniel's conjecture also holds a.a.s.\\  if $p$ tends to 1 as $n$ goes to infinity (see~\\cite{p} for this and stronger results). Bollob\\'as, Kun and Leader~\\cite{bkl} showed that if $p(n) \\ge 2.1 \\log n \/n$, then a.a.s.\n$$\n\\frac{1}{(pn)^2}n^{ 1\/2 - 9\/(2\\log\\log (pn))  }  \\le c({\\mathcal{G}}(n,p))\\le 160000\\sqrt n \\log n\\,.\n$$\nFrom these results, if $np \\ge 2.1 \\log n$ and either $np=n^{o(1)}$ or $np=n^{1\/2+o(1)}$, then a.a.s.\\ $c({\\mathcal{G}}(n,p))= n^{1\/2+o(1)}$. Somewhat surprisingly, between these values it was shown by \\L{}uczak and the author of this paper~\\cite{lp2} that the cop number has more complicated behaviour. It follows that a.a.s.\\ $\\log_n  c({\\mathcal{G}}(n,n^{x-1}))$ is asymptotic to the function $f(x)$ shown in Figure~\\ref{fig1} (denoted in blue).\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=3.3in]{zig-zag}\n\\end{center}\n\\caption{The ``zigzag'' functions representing the ordinary cop number (blue) and the containability number (red).}\\label{fig1}\n\\end{figure}\n\nFormally, the following result holds for the classic game.\n\n\\begin{theorem}[\\cite{lp2, bpw}]\\label{thm:zz}\nLet $0<\\alpha<1$ and $d=d(n)=np=n^{\\alpha+o(1)}$.\n\\begin{enumerate}\n\\item If $\\frac{1}{2j+1}<\\alpha<\\frac{1}{2j}$ for some integer $j\\ge 1$, then a.a.s.\\\n$$\nc({\\mathcal{G}}(n,p))= \\Theta(d^j)\\,.\n$$\n\\item If $\\frac{1}{2j}<\\alpha<\\frac{1}{2j-1}$ for some integer $j\\ge 2$, then a.a.s.\\\n\\begin{eqnarray*}\nc({\\mathcal{G}}(n,p)) &=& \\Omega \\left( \\frac{n}{d^j} \\right), \\text{ and } \\\\\nc({\\mathcal{G}}(n,p)) &=& O \\left( \\frac{n \\log n}{d^j}  \\right)\\,.\n\\end{eqnarray*}\n\\item If $1\/2 < \\alpha < 1$, then a.a.s.\\\n$$\nc({\\mathcal{G}}(n,p)) = \\Theta \\left( \\frac {n \\log n}{d} \\right).\n$$\n\\end{enumerate}\n\\end{theorem}\n\nThe above result shows that Meyniel's conjecture holds a.a.s.\\ for random graphs except perhaps when $np=n^{1\/(2k)+o(1)}$ for some $k \\in {\\mathbb N}$, or when $np=n^{o(1)}$. The author of this paper and Wormald showed recently that the conjecture holds a.a.s.\\ in ${\\mathcal{G}}(n,p)$~\\cite{PW_gnp} as well as in random $d$-regular graphs~\\cite{PW_gnd}.\n\n\\bigskip\n\nFinally, we are able to state the result of this paper.\n\n\\begin{theorem}\\label{thm:main}\nLet $0<\\alpha<1$ and $d=d(n)=np=n^{\\alpha+o(1)}$.\n\\begin{enumerate}\n\\item If $\\frac{1}{2j+1}<\\alpha<\\frac{1}{2j}$ for some integer $j\\ge 1$, then a.a.s.\\\n$$\n\\xi({\\mathcal{G}}(n,p))= \\Theta(d^{j+1}) = \\Theta(c({\\mathcal{G}}(n,p)) \\cdot \\Delta( {\\mathcal{G}}(n,p) ) )\\,.\n$$\nHence, a.a.s.\\ Conjecture~\\ref{con:komarov} holds (up to a multiplicative constant factor).\n\\item If $\\frac{1}{2j}<\\alpha<\\frac{1}{2j-1}$ for some integer $j\\ge 2$, then a.a.s.\\\n\\begin{eqnarray*}\n\\xi({\\mathcal{G}}(n,p)) &=& \\Omega \\left( \\frac{n}{d^{j-1}} \\right), \\text{ and } \\\\\n\\xi({\\mathcal{G}}(n,p)) &=& O \\left( \\frac{n \\log n}{d^{j-1}} \\right) = O(c({\\mathcal{G}}(n,p)) \\cdot \\Delta( {\\mathcal{G}}(n,p) ) \\cdot \\log n )\\,.\n\\end{eqnarray*}\nHence, a.a.s.\\ Conjecture~\\ref{con:komarov} holds (up to a multiplicative $O(\\log n)$ factor).\n\\item If $1\/2 < \\alpha < 1$, then a.a.s.\\\n$$\n\\xi({\\mathcal{G}}(n,p)) = \\Theta(n) = \\Theta (c({\\mathcal{G}}(n,p)) \\cdot \\Delta( {\\mathcal{G}}(n,p) ) \/ \\log n ) \\le c({\\mathcal{G}}(n,p)) \\cdot \\Delta( {\\mathcal{G}}(n,p).\n$$\nHence, a.a.s.\\ Conjecture~\\ref{con:komarov} holds.\n\\end{enumerate}\n\\end{theorem}\n\nIt follows that a.a.s.\\ $\\log_n  \\xi({\\mathcal{G}}(n,n^{x-1}))$ is asymptotic to the function $g(x)$ shown in Figure~\\ref{fig1} (denoted in red). The fact the conjecture holds is associated with the observation that $g(x) - f(x) = x$, which is equivalent to saying that a.a.s.\\ the ratio $\\xi({\\mathcal{G}}(n,p)) \/ c({\\mathcal{G}}(n,p)) = d n^{o(1)} = \\Delta({\\mathcal{G}}(n,p)) \\cdot n^{o(1)}$. Moreover, let us mention that Theorem~\\ref{thm:main} implies that the conjecture is best possible (again, up to a constant or an $O(\\log n)$ multiplicative factors for corresponding ranges of $p$).\n\n\\bigskip\n\nNote that in the above result we skip the case when $np=n^{1\/k+o(1)}$ for some positive integer $k$ or $np=n^{o(1)}$. It is done for a technical reason: an argument for the lower bound for $\\xi({\\mathcal{G}}(n,p))$ uses a technical lemma from~\\cite{lp2} that, in turn, uses Corollary 2.6 from~\\cite{Vu} which is stated only for $np=n^{\\alpha+o(1)}$, where $\\alpha \\neq 1\/k$ for any positive integer $k$. Clearly, one can repeat the argument given in~\\cite{Vu}, which is a very nice but slightly technical application of the polynomial concentration method inequality by Kim and Vu. However, in order to make the paper easier and more compact, a ready-to-use lemma from~\\cite{lp2} is used and we concentrate on the ``linear'' parts of the graph of the zigzag function.  Nonetheless, similarly to the corresponding result for $c({\\mathcal{G}}(n,p))$, one can expect that, up to a  factor of $\\log^{O(1)}n$,  the result extends naturally also to the case  $np=n^{1\/k+o(1)}$ as well.  \n\nOn the other hand, there is no problem with the upper bound so the case when $np=n^{1\/k+o(1)}$ for some positive integer $k$ is also investigated (see below for a precise statement). Moreover, some expansion properties that were used to prove that Meyniel's conjecture holds for ${\\mathcal{G}}(n,p)$~\\cite{PW_gnp} are incorporated here to investigate sparser graphs.  \n\nThe rest of the paper is devoted to prove Theorem~\\ref{thm:main}.\n\n\\section{Proof of Theorem~\\ref{thm:main}}\n\n\\subsection{Typical properties of ${\\mathcal{G}}(n,p)$ and useful inequalities}\n\nLet us start by listing some typical properties of ${\\mathcal{G}}(n,p)$. These observations are part of folklore and can be found in many places, so we will usually skip proofs, pointing to corresponding results in existing literature. Let $N_i(v)$ denote the set of vertices at distance $i$ from $v$, and let $N_i[v]$ denote the set of vertices within distance $i$ of $v$, that is, $N_i[v] = \\bigcup_{0 \\le j \\le i} N_j(v)$. For simplicity, we use $N[v]$ to denote $N_1[v]$, and $N(v)$ to denote $N_1(v)$. Since cops occupy edges but the robber occupies vertices, we will need to investigate the set of edges at ``distance'' $i$ from a given vertex $v$ that we denote by $E_i(v)$. Formally, $E_i(v)$ consists of edges between $N_{i-1}(v)$ and $N_i(v)$, and within $N_{i-1}(v)$. In particular, $E_1(v)$ is the set of edges incident to $v$. Finally, let $P_i(v,w)$ denote the number of paths of length $i$ joining $v$ and $w$.\n\n\\bigskip \n\nLet us start with the following lemma. \n\n\\begin{lemma}\\label{lem:elem1}\nLet $d=d(n) = p(n-1) \\ge \\log^3 n$. Then, there exists a positive constant $c$ such that a.a.s.\\ the following properties hold in ${\\mathcal{G}}(n,p) = (V,E)$.\n\\begin{enumerate}\n\\item Let $S \\subseteq V$ be any set of $s=|S|$ vertices, and let $r \\in {\\mathbb N}$. Then\n$$\n\\left| \\bigcup_{v \\in S} N_r[v] \\right| \\ge c \\min\\{s d^r, n \\}.\n$$\nMoreover, if $s$ and $r$ are such that $s d^r < n \/ \\log n$, then\n$$\n\\left| \\bigcup_{v \\in S} N_r[v] \\right| = (1+o(1)) s d^r.\n$$\n\\item ${\\mathcal{G}}(n,p)$ is connected.\n\\item Let $r = r(n)$ be the largest integer such that $d^r \\le \\sqrt{n \\log n}$. Then, for every vertex $v \\in V$ and $w \\in N_{r+1}(v)$, the number of edges from $w$ to $N_r(v)$ is at most $b$, where \n$$\nb = \n\\begin{cases}\n250 & \\text{ if } d \\le n^{0.49} \\\\\n\\frac {3 \\log n}{\\log \\log n} & \\text{ if } n^{0.49} < d \\le \\sqrt{n}.\n\\end{cases}\n$$\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nThe proof of part (i) can be found in~\\cite{PW_gnp}. The fact that ${\\mathcal{G}}(n,p)$ is connected is well known (see, for example,~\\cite{JLR}). In fact, the (sharp) threshold for connectivity is $p = \\log n \/ n$ so this property holds for even sparser graphs. \n\nFor part (iii), let us first expose the $r$th neighbourhood of $v$.  By part (i), we may assume that $|N_r[v]|=(1+o(1)) d^{r} < 2 d^r$. For any $w \\in V \\setminus N_r[v]$, the probability that there are at least $b$ edges joining $w$ to $N_r(v)$ is at most\n$$\nq := {2 d^r \\choose b} p^b \\le \\left( \\frac{2ed^r}{b} \\right)^b \\left( \\frac dn \\right)^b = \\left( \\frac{2ed^{r+1}}{bn} \\right)^b.\n$$\nIf $d \\le n^{0.49}$, then\n$$\nq \\le \\left( \\frac{2e d \\sqrt{n \\log n}}{bn} \\right)^b \\le n^{-0.005 b} = o(n^{-2}),\n$$\nprovided that $b$ is large enough (say, $b = 250$). For $n^{0.49} < d \\le \\sqrt{n}$ (and so $r=1$), we observe that\n$$\nq \\le \\left( \\frac{2e}{b} \\right)^b = \\exp \\left( - (1+o(1)) b \\log b \\right) = o(n^{-2}),\n$$\nprovided $b = 3 \\log n \/ \\log \\log n$. The claim follows by the union bound over all pairs $v, w$. The proof of the lemma is finished.\n\\end{proof}\n\n\\bigskip \n\nThe next lemma can be found in~\\cite{lp2}. (See also~\\cite{lazy_gnp} for its extension.)\n\n\\begin{lemma}\\label{lem:elem2}\nLet $\\eps$ and $\\alpha$ be constants such that $0<\\eps<0.1$, $\\eps<\\alpha<1-\\eps$, and let $d=d(n)=p(n-1)=n^{\\alpha+o(1)}$. Let $\\ell \\in {\\mathbb N}$ be the largest integer such that $\\ell < 1\/\\alpha$. Then, a.a.s.\\ for every vertex $v$ of ${\\mathcal{G}}(n,p)$ the following properties hold.\n\\begin{enumerate}\n\\item [(i)] If $w\\in N_i[v]$ for some $i$ with $2\\le i \\le \\ell$, then $P_i(v,w) \\le \\frac {3}{1-i\\alpha}$.\n\\item [(ii)] If $w\\in N_{\\ell+1}[v]$ and $d^{\\ell+1} \\ge 7 n \\log n$, then $P_{\\ell+1}(v,w) \\le \\frac{6}{1-\\ell \\alpha}\\frac{d^{\\ell+1}}{n}$.\n\\item [(iii)] If $w\\in N_{\\ell+1}[v]$ and $d^{\\ell+1} < 7 n \\log n$, then $P_{\\ell+1}(v,w) \\le \\frac{42}{1-\\ell \\alpha} \\log n$.\n\\end{enumerate}\nMoreover, a.a.s.\\\n\\begin{enumerate}\n\\item [(iv)] Every edge of ${\\mathcal{G}}(n,p)$ is contained in at most $\\eps d$ cycles of length at most $\\ell+2$.\n\\end{enumerate}\n\\end{lemma}\n\n\\bigskip\n\nWe will also use the following variant of Chernoff's bound (see, for example,~\\cite{JLR}):\n\\begin{lemma}[\\textbf{Chernoff Bound}]\nIf $X$ is a binomial random variable with expectation $\\mu$, and $0<\\delta<1$, then \n$$\n\\Pr[X < (1-\\delta)\\mu] \\le \\exp \\left( -\\frac{\\delta^2 \\mu}{2} \\right)\n$$ \nand if $\\delta > 0$,\n$$\n\\Pr [X > (1+\\delta)\\mu] \\le \\exp \\left(-\\frac{\\delta^2 \\mu}{2+\\delta} \\right).\n$$\n\\end{lemma}\n\n\\subsection{Upper bound}\n\nFirst, let us deal with dense graphs that correspond to part (iii) of Theorem~\\ref{thm:main}. In fact, we are going to make a simple observation that the containability number is linear if $G$ has a perfect or a near-perfect matching. The result will follow since it is well-known that for $p=p(n)$ such that $pn - \\log n \\to \\infty$, ${\\mathcal{G}}(n,p)$ has a perfect (or a near-perfect) matching a.a.s. (As usual, see~\\cite{JLR}, for more details.)\n\n\\begin{lemma}\\label{lem:perfect}\nSuppose that $G$ on $n$ vertices has a perfect matching ($n$ is even) or a near-perfect matching ($n$ is odd). Then, $\\xi(G) \\le n$.\n\\end{lemma}\n\\begin{proof}\nSuppose first that $n$ is even. The cops start on the edges of a perfect matching; two cops occupy any edge of the matching for a total of $n$ cops. All vertices of $G$ can be associated with unique cops. The robber starts on some vertex $v$. One edge incident to $v$ (the edge $vv'$ that belongs to the perfect matching used) is already occupied by a cop (in fact, by two cops, associated with $v$ and $v'$). Moreover, the remaining cops can move so that all edges incident to $v$ are protected and the game ends. Indeed, for each edge $vu$, the cop associated with $u$ moves to $vu$. \n\nThe case when $n$ is odd is also very easy. Two cops start on each edge of a near-perfect matching which matches all vertices but $u$. If $u$ is isolated, we may simply remove it from $G$ and arrive back to the case when $n$ is even. (Recall that the cops win if all edges incident with the robber are occupied by cops. As this property is vacuously true when the robber starts on an isolated vertex, we may assume that she does not start on $u$.) Hence, we may assume that $u$ is not isolated. We introduce one more cop on some edge incident to $u$. The total number of cops is at most $2 \\cdot \\frac {n-1}{2} + 1 = n$; again, each vertex of $G$ can be associated with a unique cop and the proof goes as before.\n\\end{proof}\n\n\\bigskip\n\nNow, let us move to the following lemma that yields part (i) of Theorem~\\ref{thm:main}. We combine and adjust ideas from both~\\cite{lp2} and~\\cite{PW_gnp} in order to include much sparser graphs.  Cases when $\\alpha = 1\/k$ for some positive integer $k$ are also covered. \n\n\\begin{lemma}\nLet $d=d(n) = p(n-1) \\ge \\log^3 n$. Suppose that there exists a positive integer $r=r(n)$ such that \n$$\n(n \\log n)^{\\frac {1}{2r+1}} \\le d \\le (n \\log n)^{\\frac {1}{2r}}.\n$$\nThen, a.a.s.\\ \n$$\n\\xi({\\mathcal{G}}(n,p)) = O(d^{r+1}).\n$$\n\\end{lemma}\n\n\\begin{proof}\nSince our aim is to prove that the desired bound holds a.a.s.\\ for ${\\mathcal{G}}(n,p)$, we may assume, without loss of generality, that a graph $G$ the players play on satisfies the properties stated in Lemma~\\ref{lem:elem1}. A team of cops is determined by independently choosing each edge of $e \\in E(G)$ to be occupied by a cop with probability $Cd^r\/n$, where $C$ is a (large) constant to be determined soon. It follows from Lemma~\\ref{lem:elem1}(i) that $G$ has $(1+o(1)) dn\/2$ edges. Hence, the expected number of cops is equal to\n$$\n(1+o(1)) \\frac {dn}{2} \\cdot \\frac {Cd^r}{n} = (1+o(1)) \\frac {Cd^{r+1}}{2} .\n$$\nIt follows from Chernoff's bound that the total number of cops is $\\Theta(d^{r+1})$ a.a.s.\n\nThe robber appears at some vertex $v \\in V(G)$. Let $X \\subseteq E(G)$ be the set of edges between $N_r(v)$ and $N_{r+1}(v)$. It follows from Lemma~\\ref{lem:elem1}(i) that \n$$\n|X| \\le (1+o(1)) d |N_r(v)| \\le 2 d^{r+1}.\n$$\nOur goal is to show that with probability $1-o(n^{-1})$ it is possible to assign distinct cops to all edges $e$ in $X$ such that a cop assigned to $e$ is within distance $(r+1)$ of $e$. (Note that here, the probability refers to the randomness in distributing the cops; the graph $G$ is fixed.) If this can be done, then after the robber appears these cops can begin moving straight to their assigned destinations in $X$. Since the first move belongs to the cops, they have $(r+1)$ steps, after which the robber must still be inside $N_r[v]$, which is fully occupied by cops. She is ``trapped'' inside $N_r[v]$, so we can send an auxiliary team of, say, $2 d^{r+1}$ cops to go to every edge in the graph induced by $N_r[v]$, and the game ends. Hence, the cops will win with probability $1-o(n^{-1})$, for each possible starting vertex $v \\in V(G)$. It will follow that the strategy gives a win for the cops a.a.s.  \n\nLet $Y$ be the (random) set of edges occupied by cops. Instead of showing that the desired assignment between $X$ and $Y$ exists, we will show that it is possible to assign $b(u)$ distinct cops to all vertices $u$ of $N_{r+1}(v)$, where $b(u)$ is the number of neighbours of $u$ that are in $N_r(v)$ (that is, the number of edges of $X$ incident to $u$) and such that each cop assigned to $u$ is within distance $(r+1)$ from $u$. \n(Note that this time ``distance'' is measured between vertex $u$ and edges which is non-standard. In this paper, we define it as follows: edge $e$ is at distance at most $(r+1)$ from $u$ if $e$ is at distance at most $r$ from some edge adjacent to $u$.)\nIndeed, if this can be done, assigned cops run to $u$, after $r$ rounds they are incident to $u$, and then spread to edges between $u$ and $N_r(v)$; the entire $X$ is occupied by cops. In order to show that the required assignment between $N_{r+1}(v)$ and $Y$ exists with probability $1-o(n^{-1})$, we show that with this probability, $N_{r+1}(v)$ satisfies Hall's condition for matchings in bipartite graphs. \n\nSuppose first that $d \\le n^{0.49}$ and fix $b=250$. It follows from Lemma~\\ref{lem:elem1}(iii) that $b(u) \\le b$ for every $u \\in N_{r+1}(v)$. Set \n$$\nk_0 = \\max \\{ k : k d^{r} < n \\}.\n$$ \nLet $K \\subseteq N_{r+1}(v)$ with $|K|=k \\le k_0$. We may apply Lemma~\\ref{lem:elem1}(i) to bound the size of $\\bigcup_{u \\in K} N_r[u]$ and the number of edges incident to each vertex. It follows that the number of edges of $Y$ that are incident to some vertex in $\\bigcup_{u \\in K} N_r[u]$ can be stochastically bounded from below by the binomial random variable ${\\rm Bin}(\\lfloor c k d^r \\cdot (d\/3) \\rfloor, Cd^r\/n)$, whose expected value is asymptotic to $(Cc\/3) k d^{2r+1} \/ n \\ge (Cc\/3) k \\log n$. Using Chernoff's bound we get that the probability that there are fewer than $bk$ edges of $Y$ incident to this set of vertices is less than $\\exp(-4k \\log n)$ when $C$ is a sufficiently large constant. Hence, the probability that the sufficient condition in the statement of Hall's theorem fails for at least one set $K$ with $|K|\\le k_0$ is at most\n$$\n\\sum_{k=1}^{k_0} {|N_{r+1}(v)| \\choose k} \\exp( - 4 k \\log n)  \\le \\sum_{k=1}^{k_0} n^k \\exp( - 4 k \\log n) = o(n^{-1}).\n$$\n\nNow consider any set  $K \\subseteq N_{r+1}(v)$ with $k_0 < |K| = k \\le |N_{r+1}(v)| \\le 2 d^{r+1}$ (if such a set exists). Lemma~\\ref{lem:elem1}(i) implies that the size of $\\bigcup_{u \\in K} N_r[u]$ is at least  $cn$, so we expect at least $cn \\cdot (d\/3) \\cdot Cd^r\/n = (Cc\/3) d^{r+1}$ edges of $Y$ incident to this set.  Again using Chernoff's bound, we deduce that the number of edges of $Y$ incident to this set is at least $2 b d^{r+1} \\ge b |N_{r+1}(v)| \\ge bk$ with probability at least $1-\\exp(- 4 d^{r+1})$, by taking the constant $C$ to be large enough. Since\n$$\n\\sum_{k=k_0+1}^{|N_{r+1}(v)|} {|N_{r+1}(v)| \\choose k} \\exp( - 4 d^{r+1} ) \\le 2^{2 d^{r+1}} \\exp( - 4 d^{r+1} ) = o(n^{-1}),\n$$\nthe necessary condition in Hall's theorem holds with probability $1 - o(n^{-1})$. \n\nFinally, suppose that $d > n^{0.49}$. Since Lemma~\\ref{lem:perfect} implies that the result holds for $d > \\sqrt{n}$, we may assume that $d \\le \\sqrt{n}$. (In fact, for $d > \\sqrt{n}$ we get a better bound of $n$ rather than $O(d^2)$ that we aim for.) This time, set $b = 3 \\log \\log n \/ \\log n$ to make sure $b(u) \\le b$ for all $u \\in N_{r+1}(v)$. The proof is almost the same as before. For small sets of size at most $k_0 = \\Theta(n\/d)$, we expect $(Cc\/3) k d^{3} \/ n \\ge (Cc\/3) k n^{0.47}$ edges, much more than we actually need, namely, $b k$. For large sets of size more than $k_0$, we modify the argument slightly and instead of assigning $b$ cops to each vertex of $N_{r+1}(v)$, we notice that the number of cops needed to assign is equal to $\\sum_{u \\in K} b(u) \\le |X| \\le 2 d^{r+1}$. (There might be some vertices of $N_{r+1}(v)$ that are incident to $b$ edges of $X$ but the total number of incident edges to $K$ is clearly at most $|X|$.) The rest is not affected and the proof is finished.\n\\end{proof}\n\n\\bigskip\n\nThe next lemma takes care of part (ii) of Theorem~\\ref{thm:main}.\n\n\\begin{lemma}\nLet $d=d(n) = p(n-1) \\ge \\log^3 n$. Suppose that there exists an integer $r=r(n) \\ge 2$ such that \n$$\n(n \\log n)^{\\frac {1}{2r}} \\le d \\le (n \\log n)^{\\frac {1}{2r-1}}.\n$$\nThen, a.a.s.\\ \n$$\n\\xi({\\mathcal{G}}(n,p)) = O \\left( \\frac {n \\log n}{d^{r-1}} \\right).\n$$\n\\end{lemma}\n\\begin{proof}\nWe mimic the proof of the previous lemma so we skip details focusing only on differences. A team of cops is determined by independently choosing each edge of $e \\in E(G)$ to be occupied by a cop with probability $C \\log n \/ d^r$, for the total number of cops $\\Theta(n \\log n \/ d^{r-1})$ a.a.s.\n\nThe robber appears at some vertex $v \\in V(G)$. This time, $X \\subseteq E(G)$ is the set of edges between $N_{r-1}(v)$ and $N_{r}(v)$ and $|X| \\le 2 d^r$. We show that it is possible to assign $b=250$ distinct cops to all vertices $u$ of $N_r(v)$ such that a cop assigned to $u$ is within ``distance'' $r$ from $u$. The definition of $k_0$ has to be adjusted. Set \n$$\nk_0 = \\max \\{ k : k d^{r-1} < n \\}.\n$$ \nLet $K \\subseteq N_{r}(v)$ with $|K|=k \\le k_0$. The expected number of edges of $Y$ that are incident to some vertex in $\\bigcup_{u \\in K} N_{r-1}[u]$ is at least $(ckd^{r-1}) (d\/3) (C \\log n \/ d^r) = (Cc\/3) k \\log n$, and the rest of the argument is not affected. Now consider any set  $K \\subseteq N_{r}(v)$ with $k_0 < |K| = k \\le |N_{r}(v)| \\le 2 d^{r}$ (if such a set exists). The size of $\\bigcup_{u \\in K} N_{r-1}[u]$ is at least  $cn$, so we expect at least \n$$\n(cn) \\left(\\frac {d}{3} \\right) \\left( \\frac {C \\log n}{d^r} \\right) = \\frac {Cc d^r n \\log n}{3 d^{2r-1}} \\ge \\frac {Cc d^r }{3} \n$$ \nedges of $Y$ incident to this set. Hence, the number of edges of $Y$ incident to this set is at least $2 b d^{r} \\ge b |N_{r}(v)| \\ge bk$ with probability at least $1-\\exp(- 4 d^{r})$, by taking the constant $C$ to be large enough. The argument we had before works again, and the proof is finished.\n\\end{proof}\n\n\\subsection{Lower bound}\n\nThe proof of the lower bound is an adaptation of the proof used for the classic cop number in~\\cite{lp2}. The two bounds, corresponding to parts (i) and (ii) in Theorem~\\ref{thm:main}, are proved independently in the following two lemmas. \n\n\\begin{lemma}\\label{lemma:lowerbound1}\nLet $\\frac{1}{2j+1}<\\alpha<\\frac{1}{2j}$ for some integer $j\\ge 1$, $c=c(j,\\alpha)=\\frac{3}{1-2j \\alpha}$, and $d=d(n)=np=n^{\\alpha+o(1)}$. Then, a.a.s.\\\n$$\n\\xi({\\mathcal{G}}(n,p)) > K := \\left( \\frac{d}{30c(2j+1)} \\right)^{j+1}\\,.\n$$\n\\end{lemma}\n\n\\begin{proof}\nSince our aim is to prove that the desired bound holds a.a.s.\\ for ${\\mathcal{G}}(n,p)$, we may assume, without loss of generality, that a graph $G$ the players play on satisfies the properties stated in Lemmas~\\ref{lem:elem1} and~\\ref{lem:elem2}. Suppose that the robber is chased by $K$ cops. Our goal is to provide a winning strategy for the robber on $G$.  For vertices $x_1,x_2,\\dots,x_s$, let $\\textrm{C}^{x_1,x_2,\\dots,x_s}_i(v)$ denote the number of cops in $E_i(v)$ (that is, at distance $i$ from $v$) in the graph $G \\setminus \\{x_1,x_2, \\dots,x_s\\}$.\n\nRight before the robber makes her move, we say that the vertex $v$ occupied by the robber is \\emph{safe}, if for some neighbour $x$ of $v$ we have $\\textrm{C}^x_1(v) \\le\\frac{d}{30c(2j+1)}$, and\n$$\n\\textrm{C}_{2i}^x(v), \\textrm{C}_{2i+1}^x(v)\\le \\left( \\frac{d}{30c(2j+1)} \\right)^{i+1}\n$$\nfor $i=1,2,\\dots,j-1$ (such a vertex $x$ will be called a \\emph{deadly neighbour} of $v$). The reason for introducing deadly neighbours is to deal with a situation that many cops apply a greedy strategy and always decrease the distance between them and the robber. As a result, there might be many cops ``right behind'' the robber but they are not so dangerous unless she makes a step ``backwards'' by moving to a vertex she came from in the previous round, a deadly neighbour! Moreover, note that a vertex is called safe for a reason: if the robber occupies a safe vertex, then the game is definitely not over since the condition for $C_1^x(v)$ guarantees that at most a small fraction of incident edges are occupied by cops.\n\nSince a.a.s.\\ $G$ is connected (see Lemma~\\ref{lem:elem1}(ii)), without loss of generality we may assume that at the beginning of the game all cops begin at the same edge, $e$. Subsequently, the robber may choose a vertex $v$ so that $e$ is at distance $2j+2$ from $v$ (see Lemma~\\ref{lem:elem1}(i) applied with $r=2j+1$ to see that almost all vertices are at distance $2j+1$ from both endpoints of $e$). Hence, even if all cops will move from $e$ to $E_{2j+1}(v)$ after this move, $v$ will remain safe as no bound is required for $\\textrm{C}_{2j+1}^x(v)$. (Of course, again, without loss of generality we may assume that all cops pass for the next round and stay at $e$ before starting applying their best strategy against the robber.) Hence, in order to prove the lemma, it is enough to show that if the robber's current vertex $v$ is safe, then she can move along an unoccupied edge to a neighbour $y$ so that no matter how the cops move in the next round, $y$ remains safe.\n\nFor $0 \\le r \\le 2j$, we say that a neighbour $y$ of $v$ is {\\em $r$-dangerous} if\n\\begin{itemize}\n\\item [(i)] an edge $vy$ is occupied by a cop (for $r=0$)\\,, or\n\\item [(ii)] $\\textrm{C}^{v,x}_r(y) \\ge \\frac 13 \\left( \\frac{d}{30c(2j+1)} \\right)^{i}$ (for $r=2i$ or $r=2i-1$, where $i = 1,2, \\ldots, j$)\\,,\n\\end{itemize}\nwhere $x$ is a deadly neighbour of $v$. We will check that for every $r \\in \\{0, 1, \\ldots, 2j\\}$, the number of $r$-dangerous neighbours of $v$, which we denote by $\\textrm{dang}(r)$, is smaller than $\\frac {d}{2(2j+1)}$. Clearly, since $v$ is safe, \n$$\n\\textrm{dang}(0) \\le \\textrm{C}^x_1(v) \\le \\frac{d}{30c(2j+1)} \\le \\frac {d}{2(2j+1)}.\n$$\nSuppose then that $r=2i$ or $r=2i-1$ for some $i \\in \\{1,2,\\ldots, j\\}$. Every $r$-dangerous neighbour of $v$ has at least $\\frac 13 \\left( \\frac{d}{30c(2j+1)} \\right)^i$ cops occupying $E_{\\le (r+1)}(v)$. On the other hand, every edge from $E_{\\le (r+1)}(v)$ is incident to at most $2$ vertices at distance at most $r$ from $v$. Moreover, Lemma~\\ref{lem:elem2}(i) implies that there are at most $c$ paths between $v$ and any $w \\in N_{\\le r}(v)$. Finally, by the assumption that $v$ is safe, we have $\\textrm{C}^{x}_{2i}(v), \\textrm{C}^{x}_{2i+1}(v) \\le  \\left( \\frac{d}{30c(2j+1)} \\right)^{i+1}$, provided that $i \\le j-1$; the corresponding conditions for $\\textrm{C}_{2j}^x(v)$ and $\\textrm{C}_{2j+1}^x(v)$ are trivially true, since both can be bounded from above by $K$, the total number of cops. Combining all of these yields\n\\begin{eqnarray*}\n\\frac 13 \\left( \\frac{d}{30c(2j+1)} \\right)^i \\cdot \\textrm{dang}(r) &\\le& 2c \\cdot \\textrm{C}_{\\le (r+1)}^x(v) \\le 2c \\cdot (2+o(1)) \\textrm{C}_{r+1}^x(v) \\\\\n&\\le& 5 c \\cdot \\left( \\frac{d}{30c(2j+1)} \\right)^{i+1},\n\\end{eqnarray*}\nand consequently $\\textrm{dang}(r) \\le \\frac {d}{2(2j+1)}$, as required. Thus, there at most $d\/2$ of neighbours of $v$ are $r$-dangerous for some $r \\in \\{0,1,\\dots,2j\\}$. \n\nSince we have $(1+o(1)) d$ neighbours to choose from (see Lemma~\\ref{lem:elem1}(i)), there are plenty of neighbours of $v$ which are not $r$-dangerous for any $r=0,1,\\dots,2j$ and the robber might want to move to one of them. However, there is one small issue we have to deal with. In the definition of being dangerous, we consider the graph $G \\setminus \\{v,x\\}$ whereas in the definition of being safe we want to use $G \\setminus \\{v\\}$ instead. Fortunately, Lemma~\\ref{lem:elem2}(iv) implies that we can find a neighbour $y$ of $v$ that is not only not dangerous but also $x$ does not belong to the $2j$-neighbourhood of $y$ in $G\\setminus \\{v\\}$. It follows that $vy$ is not occupied by a cop and $\\textrm{C}^{v}_r(y) < \\frac 13 \\left( \\frac{d}{30c(2j+1)} \\right)^{i}$ for $r=2i$ or $r=2i-1$, where $i = 1,2, \\ldots, j$. We move the robber to $y$.\n\nNow, it is time for the cops to make their move. Because of our choice of the vertex $y$, we can assure that the desired upper bound for $\\textrm{C}_r^v(y)$ required for $y$ to be safe will hold for $r \\in \\{1,2,\\dots,2j-1\\}$. Indeed, the best that the cops can do to try to fail the condition for $\\textrm{C}_r^v(y)$ is to move all cops at distance $r-1$ and $r+1$ from $y$ to $r$-neighbourhood of $y$, and to make cops at distance $r$ stay put, but this would not be enough. Thus, regardless of the strategy used by the cops, $y$ is safe and the proof is finished.\n\\end{proof}\n\n\\begin{lemma}\\label{lemma:lowerbound2}\nLet $\\frac{1}{2j}<\\alpha<\\frac{1}{2j-1}$ for some integer $j\\ge 1$,  $\\bar c=\\bar c(\\alpha)=\\frac{6}{1-(2j-1) \\alpha}$ and $d=d(n)=np=n^{\\alpha+o(1)}$. Then, a.a.s.\\\n$$\n\\xi({\\mathcal{G}}(n,p))\\ge \\bar{K} := \\left( \\frac{d}{30 \\bar c (2j+1)} \\right)^{j+1} \\frac{n}{d^{2j}}\\,.\n$$\n\\end{lemma}\n\n\\begin{proof}\nThe proof is very similar to that of Lemma~\\ref{lemma:lowerbound1}. The only difference is that checking the desired bounds for $\\textrm{dang}(2j-1)$ and $\\textrm{dang}(2j)$ is slightly more complicated. As before, we do not control the number of cops in $E_{2j}(v)$ and $E_{2j+1}(v)$, clearly $\\textrm{C}^x_{2j}(v)$ and $\\textrm{C}^x_{2j+1}(v)$ are bounded from above by $\\bar{K}$, the total number of cops. We get\n\\begin{eqnarray*}\n\\frac 13 \\left( \\frac{d}{30\\bar c(2j+1)} \\right)^j \\cdot \\textrm{dang}(2j-1) &\\le& 2\\bar c \\cdot \\textrm{C}_{\\le (2j)}^x(v) \\le 2\\bar c \\cdot (2+o(1)) \\bar{K} \\\\\n&\\le& 5\\bar c \\cdot \\left( \\frac{d}{30\\bar c(2j+1)} \\right)^{j+1},\n\\end{eqnarray*}\nand consequently $\\textrm{dang}(2j-1) \\le \\frac {d}{2(2j+1)}$, as required. (Note that we have room to spare here but we cannot take advantage of it so we do not modify the definition of being $(2j-1)$-dangerous.)\n\nLet us now notice that a cop at distance $2j+1$ from $v$ can contribute to the ``dangerousness'' of more than $\\bar c$ neighbours of $v$. However, the number of paths of length $2j$ joining $v$ and $w$ is bounded from above by $\\bar c d^{2j}\/n$ (see Lemma~\\ref{lem:elem2}(ii) and note that $d^{2j} = n^{2j \\alpha + o(1)} \\ge 7 n \\log n$, since $2j \\alpha > 1$). Hence,\n\\begin{eqnarray*}\n\\frac 13 \\left( \\frac{d}{30\\bar c(2j+1)} \\right)^j \\cdot \\textrm{dang}(2j) &\\le& \\frac {2 \\bar c d^{2j}}{n} \\cdot \\textrm{C}_{\\le (2j+1)}^x(v) \\le \\frac {2 \\bar c d^{2j}}{n} \\cdot (2+o(1)) \\bar{K} \\\\\n&\\le& 5 \\bar c \\cdot \\left( \\frac{d}{30\\bar c(2j+1)} \\right)^{j+1},\n\\end{eqnarray*}\nand, as desired, $\\textrm{dang}(2j)\\le \\frac{d}{2(2j+1)}$. Besides this modification the argument remains basically the same.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\\subsection{The spherical sup-norm problem}\nThe sup-norm problem on arithmetic Riemannian manifolds is a question at the interface of harmonic analysis and number theory that intrinsically combines techniques from both areas. Let $X = \\Gamma \\backslash G\/K$ be a locally symmetric space of finite volume, where $\\Gamma$ is an arithmetic subgroup. Arithmetically and analytically, the most interesting functions in $L^2(X)$ are joint eigenfunctions $\\phi$ of all invariant differential operators and the Hecke operators: these are precisely the functions that arise from (spherical) automorphic forms.   The sup-norm problem asks for a quantitative comparison of the $L^2$-norm ${\\|\\phi\\|}_2$ and the sup-norm ${\\|\\phi\\|}_{\\infty}$, most classically in terms of the Laplace eigenvalue $\\lambda_{\\phi}$, but depending on the application  also in terms of the volume of $X$ or other relevant quantities. Upper bounds for the sup-norm in terms of the Laplace eigenvalue are a measure for the equidistribution of the mass of  high energy eigenfunctions which  sheds light on the question to what extent these eigenstates can localize (``scarring''). Besides the quantum mechanical interpretation, the sup-norm problem in its various incarnations has connections to the multiplicity problem, zero sets and nodal lines of automorphic functions, and bounds for Faltings' delta function, to name just a few. See \\cite{Sarnak2004Morawetz, Rudnick2005, GhoshReznikovSarnak2013, JorgensonKramer2004}.\n\nIf $X$ is  compact, the most general upper bound is due to Sarnak \\cite{Sarnak2004Morawetz}:\n\\begin{equation}\\label{sa}\n{\\|\\phi\\|}_{\\infty} \\ll_X  \\lambda_{\\phi}^{(\\dim X - \\rk X)\/4}{\\|\\phi\\|}_2,\n\\end{equation}\na bound which does not use the Hecke property and is in fact sharp (for general $X$) under these weaker assumptions. Sarnak derives this bound from asymptotics of spherical functions. A slightly different but ultimately related argument proceeds via a pre-trace inequality that bounds ${\\|\\phi\\|}_{\\infty}^2$ by a sum of an automorphic kernel over $\\gamma \\in \\Gamma$. If the test function is an appropriate Paley--Wiener function, only the identity contributes to this sum, and one obtains as a (``trivial'') upper bound for ${\\|\\phi\\|}_{\\infty}$ the square-root of the spectral density as given in terms of the Harish-Chandra $\\textbf{c}$-function. If the Langlands parameters of $\\phi$ are in generic position, this coincides with \\eqref{sa}.\n\nTo go beyond \\eqref{sa}, one uses a test function that localizes   not only the archimedean Langlands parameters, but in addition the parameters at a large number of finite places (where ``large'' means a function tending to infinity as a small and carefully chosen power of  $\\lambda_{\\phi}$). This is called the amplification technique and leads, after estimating the automorphic kernel, to a problem in the geometry of numbers: count the elements of $G$ which appear in Hecke correspondences and lie in regions of $G$ according to the size of the kernel (such as counting rescaled integer matrices lying close to $K$). It has been implemented successfully in a variety of cases, see e.g.\\ \\cite{IwaniecSarnak1995, HarcosTemplier2013, BlomerPohl2016, BlomerMaga2016, Marshall2014a, Templier2015, Saha2017a, BlomerHarcosMagaMilicevic2020} and the references therein.\n\n\\subsection{Automorphic forms with $K$-types}\nIn this paper we want  to open a new perspective on the sup-norm problem and propose a version of higher complexity. The sup-norm problem makes perfect sense not only on the level of symmetric spaces, but also on the level of groups, and a priori there is no reason why one should restrict to spherical, i.e.\\ right $K$-invariant automorphic forms. Let $\\tau$ be an irreducible unitary representation of $K$ on some finite-dimensional complex vector space $V^{\\tau}$, and consider the homogeneous vector bundle over $G\/K$ defined by $\\tau$. A cross-section may then be identified with a vector-valued function $f:G\\to V^{\\tau}$ which transforms on the right by $K$ with respect to $\\tau$:\n\\[ f(gk) = \\tau(k^{-1})f(g),\\qquad g\\in G,\\quad k \\in K. \\]\nIt is now an interesting question to bound the sup-norm of $f$ or, more delicately, its components as the \\emph{dimension}   of $V^{\\tau}$ gets large. Such a situation cannot be realized in the classical case $G = \\SL_2(\\RR)$, since $K = \\SO_2(\\RR)$ is abelian, hence each $V^{\\tau}$ is one-dimensional. In this paper, we offer a detailed investigation of the first nontrivial case $G = \\SL_2(\\CC)$. For concreteness, we choose the congruence lattice $\\Gamma = \\SL_2(\\ZZ[i])$, although our results extend to more general arithmetic quotients of $G$ using the techniques in \\cite{BlomerHarcosMagaMilicevic2020}.\n\nNontrivial irreducible unitary representations of $G$ are principal series representations parametrized by certain pairs $(\\nu, p) \\in \\mfa^{\\ast}_{\\CC} \\times \\frac{1}{2}\\ZZ$, where as usual $\\mfa$ is the Lie algebra of the subgroup of positive diagonal matrices; see \\S\\ref{SL2C-subsec}. (By a small abuse of notation we will later interpret $\\nu$ simply as a complex number.) Each representation space $V$ of $G$ decomposes as a Hilbert space direct sum\n\\begin{equation}\\label{decomp}\nV = \\bigoplus_{\\substack{\\ell\\geq|p|\\\\\\ell\\equiv p\\one}}  V^{\\ell}\n= \\bigoplus_{\\substack{\\ell\\geq|p|\\\\\\ell\\equiv p\\one}} \\ \\bigoplus_{\\substack{|q| \\leq \\ell\\\\q\\equiv\\ell\\one}}V^{\\ell,q},\n\\end{equation}\nwhere $V^{\\ell,q}$ is one-dimensional. Here and later, $\\ell\\in\\frac{1}{2}\\ZZ_{\\geq 0}$ parametrizes the $K$-type, i.e.\\ the $(2\\ell+1)$-dimensional representation $\\tau_{\\ell}$ of $K$, and the diagonal matrix $\\diag(e^{i\\varrho}, e^{-i\\varrho})\\in K$ acts on $V^{\\ell,q}$ by $e^{2qi\\varrho}$. (The upper index $\\ell$ in $V^\\ell$ should not be mistaken for $\\ell$-th power.) \n\nRepresentations occurring in $L^2(\\Gamma\\backslash G)$ consist of even functions on $G$ and have $p\\in\\ZZ$. A representation  contains a spherical vector only if $p = 0$. In particular, the forms with $p\\neq 0$ are untouched by any of the spherical sup-norm literature. For $p \\neq 0$, no complementary series exists, so $\\nu \\in i\\mfa^{\\ast}$.\n\n\\subsection{Main results I: vector-valued forms}\\label{main-results-intro-sec}\nAs explained above, we are interested in ``big'' $K$-types which occur for all representation parameters $|p| \\leq \\ell$, but arguably the most interesting case is when the $K$-type is ``new'' and no lower $K$-types appear in the same automorphic representation space. Hence from now on we restrict to $p = \\ell$. The sup-norm problem for large $\\nu$ was studied in detail in \\cite{BlomerHarcosMagaMilicevic2020}, so here we keep $\\nu$ in a fixed compact subset $I\\subset i\\RR$ and let $\\ell$ vary.  The spectral density is a constant multiple of $p^2 - \\nu^2$. In particular, for a  given $K$-type $\\tau_{\\ell}$, there are $\\OO_I(\\ell^2)$ cuspidal automorphic representations $V\\subset L^2(\\Gamma\\backslash G)$ with spectral parameter $\\nu\\in I$ and $p = \\ell$  (see \\cite{dever2020ambient}), and in the light of the trace formula this bound is expected to be sharp. In each of these we consider the $(2\\ell+1)$-dimensional subspace $V^{\\ell}$. Let us choose an orthonormal basis $\\{ \\phi_q : |q| \\leq \\ell\\}$ of $V^{\\ell}$, with $\\phi_q\\in V^{\\ell,q}$ as in \\eqref{decomp}. The function $G\\to\\CC^{2\\ell+1}$ given by\n\\begin{equation}\\label{eq:vectorvalued}\ng\\mapsto\\left(\\phi_{-\\ell}(g), \\dotsc, \\phi_\\ell(g)\\right)^{\\top}\n\\end{equation}\nis a vector-valued automorphic form for the group $\\Gamma$ with spectral parameter $\\nu$ and $K$-type $\\tau_{\\ell}$.\nThe Hermitian norm of this function,\n\\[\\Phi(g):=\\Bigl(\\sum_{|q|\\leq \\ell} |\\phi_q(g)|^2\\Bigr)^{1\/2},\\qquad g\\in G,\\]\nis independent of the choice of the orthonormal basis, and it satisfies ${\\|\\Phi\\|}_2=(2\\ell+1)^{1\/2}$. Let us fix a compact subset $\\Omega\\subset G$. Our remarks on spectral density and dimension suggest that\n\\begin{equation}\\label{3\/2-exp}\n {\\| \\Phi|_{\\Omega} \\|}_{\\infty} :=\\Bigl\\| \\sum_{|q| \\leq \\ell} |\\phi_q|_{\\Omega}|^2\\Bigr\\|^{1\/2}_{\\infty}\\ll_{\\Omega, I} \\ell^{3\/2}\n \\end{equation}\nshould be regarded as the ``trivial'' bound; cf.~Remark~\\ref{non-arith}. Our first result is a power-saving improvement.\n\n\\begin{theorem}\\label{thm1} Let $\\ell\\geq 1$ be an integer, $I\\subset i\\RR$ and $\\Omega\\subset G$ be compact sets.\nLet $V\\subset L^2(\\Gamma\\backslash G)$ be a cuspidal automorphic representation with minimal $K$-type $\\tau_{\\ell}$ and spectral parameter $\\nu_V\\in I$. Then for any $\\eps>0$ we have\n\\[ {\\| \\Phi|_{\\Omega} \\|}_{\\infty} \\ll_{\\eps,I,\\Omega} \\ell^{4\/3+\\eps}. \\]\n\\end{theorem}\n\nWe will explain some ideas of the proof in a moment, but we remark already at this point that the exponent is  the best possible, given that we sacrifice cancellation of the terms on the geometric side of the pre-trace formula and given our\ncurrent knowledge on the construction of the most efficient amplifier. In other words, under these conditions we solve the arising matrix counting problem in a best possible way.\nSince we trivially have ${\\|\\Phi\\|}_{\\infty}\\gg\\ell^{1\/2}$, the above bound is one-sixth of the way   from the trivial down to the best possible exponent (absent the possibility of some escape of mass into a cusp). This matches (after a renormalization) the original and still the best available subconvexity exponent $5\/24$ of Iwaniec--Sarnak~\\cite{IwaniecSarnak1995} for the sup-norms of spherical Maa{\\ss} forms of large Laplace eigenvalue on arithmetic hyperbolic surfaces.\n\n\\subsection{Main results II: individual vectors}\nIt is a much more subtle endeavour to investigate the sup-norm of the individual basis elements $\\phi_q$. Here one must contend with the inherent high multiplicity, a known serious barrier in the sup-norm problem. Indeed, a straightforward construction \\cite{Sarnak2004Morawetz} shows that\nsome scalar-valued $L^2$-normalized form $\\phi\\in V^{\\ell}$ (essentially the projection of the vector-valued form \\eqref{eq:vectorvalued} in the modulus-maximizing direction) has sup-norm on $\\Omega$ as large as ${\\|\\Phi|_{\\Omega}\\|}_{\\infty}$ in Theorem~\\ref{thm1}. However, our natural basis $\\{\\phi_q : |q| \\leq \\ell\\}$ of $V^{\\ell}$ is distinguished by consisting of eigenfunctions under the action of the group $\\{\\diag(e^{i\\theta}, e^{-i\\theta}) : \\theta \\in \\RR\\}$ of diagonal matrices in $K$. This is the classical basis with respect to which the representation $\\tau_\\ell$ is given by the Wigner $D$-matrix. By a similar heuristic reasoning as for \\eqref{3\/2-exp}, one might expect that the baseline bound should be ${\\| \\phi_q |_{\\Omega} \\|}_{\\infty} \\ll \\ell$. Indeed, we prove this bound up to a factor of $\\ell^{\\eps}$ (cf.\\ Remark~\\ref{non-arith} below), noting that it is not ``trivial'' in any sense other than that it does not require arithmeticity. Moreover, as shown by the next theorem, we are in fact able to break this barrier.\n\n\\begin{theorem}\\label{thm2}\nUnder the assumptions of Theorem~\\ref{thm1}, we have\n\\[\\max_{|q|\\leq\\ell}\\, {\\| \\phi_q |_{\\Omega} \\|}_{\\infty}  \\ll_{\\eps,I,\\Omega} \\ell^{26\/27+\\eps}.\\]\n\\end{theorem}\n\nFor special values of $q$ we can improve on the exponent considerably. The central vector $\\phi_{0}$ is distinguished as can  the ``archimedean newvector'' \\cite{Popa2008} in the sense that its Whittaker function determines the archimedean $L$-factor of the underlying representation. Another interesting situation is the  extreme case of the vector $\\phi_{\\pm\\ell}$.\n\n\\begin{theorem}\\label{thm3} Keep the assumptions of Theorem~\\ref{thm1}. \n\\begin{enumerate}[(a)]\n\\item\\label{thm3-a}\nFor $q = 0$ we have\n\\[ {\\| \\phi_{0} |_{\\Omega} \\|}_{\\infty} \\ll_{\\eps,I,\\Omega} \\ell^{7\/8+\\eps}.\\]\n\\item\\label{thm3-b}\nSuppose that $V$ lifts to an automorphic representation for $\\PGL_2(\\ZZ[i])\\backslash\\PGL_2(\\CC)$. For $q = \\pm \\ell$ we have\n\\[{\\| \\phi_{\\pm \\ell} |_{\\Omega} \\|}_{\\infty} \\ll_{\\eps,I,\\Omega} \\ell^{1\/2+\\eps}. \\]\n\\end{enumerate}\n\\end{theorem}\n\nThe strong numerical saving in the case $q = \\pm \\ell$, going far beyond the Weyl exponent, is quite remarkable, in particular in view of the seemingly weaker saving in Theorem~\\ref{thm1} which might be regarded as an easier case. We will discuss this in \\S \\ref{KS-intro}. The assumption that $V$ is associated to a representation of $\\PGL_2$ rather than $\\SL_2$ is only for technical simplicity and not essential to the  method, cf.\\ \\S \\ref{RSSection}. This assumption holds if and only if the elements of $V$ are fixed by the Hecke operator $T_i$ (which is an involution on $L^2(\\Gamma\\backslash G)$).\n\n\\begin{remark} In the case of the spherical sup-norm problem, Sarnak~\\cite{Sarnak2004Morawetz} put forward the purity conjecture that the accumulation points of the set\n\\[ \\left\\{\\frac{\\log {\\|\\psi\\|}_{\\infty}}{\\log \\lambda_{\\psi}}:\\text{$\\psi$ is a joint eigenfunction}\\right\\} \\]\nlie in $\\frac{1}{4}\\ZZ$. It would be very interesting to see if an analogous conjecture may be expected in the $K$-aspect, and even if there may be examples exhibiting different layers of power growth as in \\cite{Milicevic2011,Blomer2020,BrumleyMarshall2020}. In particular, the savings in Theorem~\\ref{thm3} produce already a considerable ``exponent gap''.\n\\end{remark}\n\n\\begin{remark}\\label{non-arith}\nWe record that our essentially best possible estimates on the spherical trace function in \\S\\ref{gen-sph-fun-intro-sec}, which are of purely analytic nature, coupled with the formalism of the pre-trace inequality, yield what might be considered ``trivial'' geometric estimates: for any co-finite Kleinian subgroup $\\Gamma\\leq G$, without any arithmeticity assumption, we have\n\\[ {\\|\\Phi|_{\\Omega}\\|}_{\\infty}\\ll_{I,\\Omega,\\Gamma}\\ell^{3\/2}\\qquad\\text{and}\\qquad\n{\\|\\phi_q|_{\\Omega}\\|}_{\\infty}\\ll_{\\eps,I,\\Omega,\\Gamma}\\ell^{1+\\eps} \\]\nfor any $L^2$-normalized vector-valued Maa{\\ss} eigenform $(\\phi_{-\\ell},\\dots,\\phi_{\\ell})^{\\top}$ with spectral parameter $\\nu\\in I$ and $K$-type $\\tau_{\\ell}$ (with $\\phi_q\\in V^{\\ell,q}$ as before).\n\\end{remark}\n\nOur Theorems~\\ref{thm1}--\\ref{thm3} above, and the non-spherical sup-norm problem in general, come with several novelties of representation theoretic, analytic and arithmetic nature that we discuss briefly in the following subsections.\n\n\\subsection{Generalized spherical functions}\\label{gen-sph-fun-intro-sec}\nThe classical pre-trace formula features on the geometric side the Harish-Chandra transform $\\widecheck{h}$ of the test function $h$ on the spectral side. This transform is a bi-$K$-invariant function obtained by integrating $h$ against the elementary spherical functions (which themselves are bi-$K$-invariant, and hence in the case of $G = \\SL_2(\\CC)$ simply a function of one real variable). In typical applications there is no cancellation in this integral, so an asymptotic analysis of spherical functions is the first key step (see \\cite{BlomerPohl2016} for a general result in this direction). Our set-up requires a generalized version for homogeneous vector bundles over $G\/K$. For $G = \\SL_2(\\CC)$, the corresponding \\emph{spherical trace function} equals (see \\S \\ref{section:sphericaltransform} for details)\n\\begin{equation}\\label{spherical-def}\n\\varphi_{\\nu,\\ell}^{\\ell}(g) =\n(2\\ell+1)\\int_K \\psi_{\\ell}(\\kappa(k^{-1} g k))\\,e^{(\\nu-1)\\rho(H(gk))}\\,\\dd k,\n\\end{equation}\nwhere $\\dd k$ is the probability Haar measure on $K$, $\\rho$ is the unique positive root, $\\kappa$ (resp.\\ $H$) is the $KAN$ Iwasawa projection onto $K$ (resp.\\ $\\mfa$), and\n\\begin{equation}\\label{chi-ell}\n\\psi_{\\ell}\\left( \\begin{pmatrix} \\alpha  &\\beta \\\\ - \\bar{\\beta} & \\bar{\\alpha} \\end{pmatrix} \\right) := \\bar{\\alpha}^{2\\ell}, \\qquad \\left( \\begin{matrix} \\alpha  &\\beta \\\\ - \\bar{\\beta} & \\bar{\\alpha} \\end{matrix} \\right) \\in K.\n\\end{equation}\nThe trivial bound is $|\\varphi_{\\nu,\\ell}^{\\ell}(g)|\\leq 2\\ell+1$, which is sharp for $g = \\pm\\id$, and the key question is how quickly $\\varphi_{\\nu,\\ell}^{\\ell}(g)$ decays, uniformly in $\\ell$, as $g\\in G$ moves away from $\\pm\\id$. We observe that $\\varphi_{\\nu,\\ell}^{\\ell}(g)$ is  invariant under conjugation by $K$, hence it suffices to investigate it for upper triangular matrices $g\\in G$. We shall use the Frobenius norm\n$\\|g\\|:= \\sqrt{\\tr(gg^*)}$, and we note that for $g\\in G$ this is always at least $\\sqrt{2}$. The following bound is new and most likely sharp for fixed $\\nu\\in i\\RR$ (up to factors $\\ell^{\\eps}$ and powers of $\\| g \\|$).\n\n\\begin{theorem}\\label{thm4} Let $\\ell\\geq 1$ be an integer, and let $g = \\left(\\begin{smallmatrix} z & u\\\\ & z^{-1}\\end{smallmatrix}\\right) \\in G$ be upper triangular. Then for any $\\nu\\in i\\RR$, $k \\in K$, $\\eps>0$, we have\n\\[ \\varphi_{\\nu,\\ell}^{\\ell}(k^{-1}gk) \\ll_{\\eps} \n\\min\\left(\\ell, \\frac{\\ell^\\eps\\|g\\|^6}{|z^2 - 1|^2}, \\frac{\\ell^{1\/2+\\eps}\\|g\\|^3}{|u|}\\right). \\]\n\\end{theorem}\n\nThe proof shows that the factor $\\ell^{\\varepsilon}$ can be replaced with a suitable power of $\\log 2\\ell$. The same remark applies to Theorems~\\ref{thm6} and \\ref{thm5} below. \n \nThe spherical trace function $\\varphi_{\\nu,\\ell}^{\\ell}$ can be used to analyze the vector-valued function\n\\eqref{eq:vectorvalued}. It is, unfortunately, unable to identify the individual components $\\phi_q$, and there does not seem to exist a general theory of spherical functions covering such cases. As the components are eigenfunctions of the action of the diagonal elements, we can single out $\\phi_q$ by considering\n\\begin{equation}\\label{spherical-averaged}\n\\varphi_{\\nu,\\ell}^{\\ell,q}(g):=\\frac1{2\\pi}\\int_0^{2\\pi}\\varphi_{\\nu,\\ell}^{\\ell}\\left(g\\diag(e^{i\\varrho},e^{-i\\varrho})\\right)\\,e^{-2qi\\varrho}\\,\\dd \\varrho.\n\\end{equation}\nThe function $\\varphi_{\\nu,\\ell}^{\\ell,q}$ is an interesting object that does not seem to have been considered before. It is not conjugation invariant anymore, so it needs to be analyzed on the entire $6$-dimensional group $G = \\SL_2(\\CC)$, and little preliminary reduction is possible. When restricted to $K$, it is not hard to see that $\\varphi_{\\nu,\\ell}^{\\ell,q}(k)$, for $k = k[u,v,w]\\in K$ written in terms of Euler angles (cf.\\ \\eqref{decomp-K}), is essentially a Jacobi polynomial in $\\cos 2v$. We refer to \\S \\ref{thm5a-proof-sec} for a more detailed discussion. In particular,\n$\\varphi_{\\nu,\\ell}^{\\ell,q}(\\pm\\id)=1$. Therefore, at least heuristically, a safe baseline bound should be\n\\begin{equation}\\label{trivial-q}\n\\varphi_{\\nu,\\ell}^{\\ell,q}(g) \\ll_\\eps \\ell^\\eps.\n\\end{equation}\nUnlike in the bi-$K$-invariant case, where the trivial bound is just an application of the triangle inequality and hence is indeed trivial, the expected baseline bound \\eqref{trivial-q} turns out to be hard to prove. It requires very strong cancellation in the $\\varrho$-integral, along with the decay properties of $\\varphi_{\\nu,\\ell}^{\\ell}$.  Taking \\eqref{trivial-q} for granted, we wish to investigate in what directions and with what speed we can identify decay as we move away from $\\pm\\id\\in G$. Interestingly, this is extremely sensitive to the value of $q$.\n\nLet $\\mcD\\subset G$ be the set of diagonal matrices, $\\mcS$ the normalizer of $A$ in $K$ (which consists of the diagonal and the skew-diagonal matrices lying in $K$), and\n\\begin{equation}\\label{adbc}\n\\mcN:=\\left\\{\\begin{pmatrix}a & b\\\\ c & d \\end{pmatrix}\\in G: |a| = |d|, \\ |b| = |c|\\right\\}.\n\\end{equation}\nIt is clear that  $\\mcS\\subset K\\subset\\mcN\\subset G$. For $g \\in G$ and non-empty $\\mcH \\subset G$, we shall write $\\dist(g,\\mcH)$ for their distance $\\inf_{h\\in\\mcH}\\|g-h\\|$. Note that here $\\|g-h\\|=\\|g^{-1}-h^{-1}\\|$, hence also\n\\begin{equation}\\label{distinvariance}\n\\dist(g,\\mcH)=\\dist(g^{-1},\\mcH^{-1}).\n\\end{equation}\nAs an alternative to $\\dist(g,\\mcN)$, we shall also use\n\\begin{equation}\\label{Dgdef}\nD(g):=\\left||a|^2-|d|^2\\right|+\\left||b|^2-|c|^2\\right|.\n\\end{equation}\nFor orientation, we remark the elementary inequality\n\\[\\dist(g,\\mcN)^2\\leq D(g)\\leq 2\\|g\\|\\dist(g,\\mcN).\\]\nIn the spirit of \\cite[Th.~2]{BlomerPohl2016}, we use a soft argument that provides some decay of $\\varphi_{\\nu,\\ell}^{\\ell,q}(g)$ in generic ranges and considerable uniformity.\n\\begin{theorem}\\label{thm6} Let $\\ell,q\\in\\ZZ$ be such that $\\ell\\geq\\max(1,|q|)$. Let $\\nu\\in i\\RR$ and $g \\in G$.\nThen for any $\\eps>0$ and $\\Lambda>0$, we have\n\\begin{equation}\\label{thm6bound}\n\\varphi_{\\nu,\\ell}^{\\ell,q}(g) \\ll_{\\eps,\\Lambda}\\ell^{\\eps}\\min\\left(1,\\frac{\\| g \\|}{\\sqrt{\\ell}\\dist(g,K)^2\\dist(g,\\mcD)}\\right) + \\ell^{-\\Lambda}.\n\\end{equation}\n\\end{theorem}\nIn the special case $q\\in\\{-\\ell,0,\\ell\\}$, we use more elaborate arguments for stronger bounds.\n\n\\begin{theorem}\\label{thm5} Let $\\ell\\geq 1$ be an integer, $\\nu\\in i\\RR$ and $g\\in G$. Let $\\eps>0$ and $\\Lambda>0$ be two parameters.\n\\begin{enumerate}[(a)]\n\\item\\label{thm5-a}\nWe have\n\\begin{equation}\\label{thm5boundq=0}\n\\varphi_{\\nu,\\ell}^{\\ell,0}(g) \\ll_{\\eps,\\Lambda}\n\\ell^{\\eps}\\min\\left(1, \\frac{1}{\\sqrt{\\ell} \\dist(g, \\mcS)}\\right)+\\ell^{-\\Lambda}.\n\\end{equation}\nMoreover, $\\varphi_{\\nu,\\ell}^{\\ell,0}(g) \\ll_{\\Lambda} \\ell^{-\\Lambda}$ holds unless $D(g)\\ll_\\Lambda\\|g\\|^2(\\log\\ell)\/\\sqrt{\\ell}$.\n\\item\\label{thm5-b}\nWe have\n\\begin{equation}\\label{thm5boundq=ell}\n\\varphi_{\\nu,\\ell}^{\\ell,\\pm \\ell}(g) \\ll_{\\eps} \\| g \\|^{-2+\\eps} \\ell^{\\eps}.\n\\end{equation}\nMoreover, $\\varphi_{\\nu,\\ell}^{\\ell,\\pm \\ell}(g) \\ll_{\\Lambda} \\ell^{-\\Lambda}$ holds unless\n$\\dist(g, \\mcD) \\ll_\\Lambda\\| g \\|\\sqrt{\\log\\ell}\/\\sqrt{\\ell}$.\n\\end{enumerate}\n\\end{theorem}\n\nWe expect that the bounds in Theorem~\\ref{thm5} are essentially best possible, possibly up to powers of $\\ell^{\\varepsilon}$ and $\\| g \\|$. The proof requires detailed analysis that could in principle be applied to all values of $q$ and would detect, for instance, further Airy-type bumps in certain regions and for certain choices of parameters.\n\n\\begin{remark}\\label{remark3}\nLess precise results but in a more general setting were obtained by Ramacher~\\cite{Ramacher2018} using operator theoretical methods. Combined with an argument of Marshall~\\cite{Marshall2014a}, these were applied by Ramacher--Wakatsuki~\\cite{RamacherWakatsuki2017a} to the sup-norm problem with $K$-types. For compact arithmetic quotients of $\\SL_2(\\CC)$, and for $\\phi\\in V^{\\ell}$ as before,\n\\cite[Th.~7.12]{RamacherWakatsuki2017a} yields ${\\| \\phi \\|}_{\\infty} \\ll \\ell^{5\/2 - \\delta}$ with an unspecified\nconstant $\\delta>0$; this does not even recover the baseline bound.\n\\end{remark}\n\n\\subsection{Paley--Wiener theory}\nFor a reductive Lie group $G$, Paley--Wiener theory characterizes the image of $C_c^{\\infty}(G)$ under the Harish-Chandra transform. For bi-$K$-invariant functions, this is a famous result of Gangolli~\\cite{MR289724}: the image consists of entire, Weyl group invariant functions satisfying certain growth conditions. For general $K$-finite functions, the picture is much more complicated: any linear relation that holds for the matrix coefficients of generalized principal series also needs to hold for the matrix coefficients of the operator-valued Fourier transform (and hence for the $\\tau$-spherical transforms for $\\tau\\in\\widehat{K}$). A complete list of these ``Arthur--Campoli relations'' requires a full knowledge of all the irreducible subquotients of the non-unitary principal series, which in general is not available. Arthur~\\cite{MR697608} describes them as a sequence of successive residues of certain meromorphic functions; see also \\cite{Campoli1979}. Needless to say, a good knowledge of available functions on the spectral side is crucial for the quantitative analysis of the pre-trace formula in the sup-norm problem.\n\nFor the case of $G = \\SL_2(\\CC)$, in a somewhat neglected paper, Wang~\\cite{Wang1974} devised an elegant argument to establish a completely explicit Paley--Wiener theorem for the $\\tau_{\\ell}$-spherical transform acting on $C_c^{\\infty}(G)$: in addition to the Weyl group symmetry, we have the additional symmetry $(\\nu, p) \\leftrightarrow (p, \\nu)$ whenever $\\nu \\equiv p\\pmod{1}$ and $|\\nu|, |p| \\leq \\ell$; see Theorem~\\ref{thm10} in \\S \\ref{section:sphericaltransform}. The additional symmetry is counter-intuitive at first (the pairs $(\\nu,p)\\neq(0,0)$ satisfying $\\nu\\equiv p\\pmod{1}$ correspond to a discrete set of non-unitary representations), but it enters the picture as it fixes the eigenvalues $\\nu^2+p^2$ and $\\nu p$ of two generators of $Z(\\mathcal{U}(\\mfg))$, and hence the infinitesimal character. See \\cite[Cor.~2]{Wang1974} and its proof. A more conceptual explanation, along the lines of irreducible subquotients, can be found after \\eqref{eq:tau-ell-isotypical-decomposition-algebraic}. Wang's remarkable result is that these are \\emph{all} relations.\n\nThe extra symmetry makes the application of the pre-trace formula more delicate. For instance, it appears impossible to single out an individual value of $p$ by a manageable test function on the spectral side. We circumvent this problem by employing a carefully chosen Gaussian \\eqref{eq:def-gaussian-spectral-weight} that at least asymptotically singles out our preferred value $p=\\ell$. The price to pay for this maneuver is that we lose compact support.  As a result of independent interest, we prove a new Paley--Wiener theorem for $K$-finite Schwartz class functions on $G = \\SL_2(\\CC)$. For the notation, see  \\S \\ref{section:sphericaltransform}.\n\n\\begin{theorem}\\label{thm:pws} For $f\\in\\mcH(\\tau_{\\ell})$, the following two conditions are equivalent (with implied constants depending on $f$).\n\\begin{enumerate}[(a)]\n\\item\\label{pws-a} The function $f(g)$ is smooth, and for any $m\\in\\ZZ_{\\geq 0}$ and $A>0$ we have\n\\begin{equation}\\label{eq:mAbound}\n\\frac{\\partial^m}{\\partial h^m}f(k_1 a_h k_2)\\ll_{m,A} e^{-A|h|},\\qquad h\\in\\RR,\\quad k_1,k_2\\in K.\n\\end{equation}\n\\item\\label{pws-b} The function $\\widehat{f}(\\nu,p)$ extends holomorphically to $\\CC\\times\\tfrac12\\ZZ$ such that\n\\begin{equation}\\label{eq:symmetry}\\widehat{f}(\\nu,p)=\\widehat{f}(p,\\nu),\\qquad\\nu\\equiv p\\one,\\quad|\\nu|,|p|\\leq\\ell,\n\\end{equation}\nand for any $B,C>0$ we have\n\\begin{equation}\\label{eq:BCbound}\n\\widehat{f}(\\nu,p)\\ll_{B,C} (1+|\\nu|)^{-C},\\qquad|\\Re\\nu|\\leq B,\\quad p\\in\\tfrac12\\ZZ.\n\\end{equation}\n\\end{enumerate}\n\\end{theorem}\n\nThe Schwartz space offers a lot more flexibility in applications. A less precise result for more general groups is given in \\cite[Th.~3]{MR1111747}, and we refer the reader to the introduction of that paper for additional discussion and motivation of Paley--Wiener type theorems for rapidly decaying functions.\n\n\\subsection{Beyond the pre-trace formula: a fourth moment}\\label{KS-intro}\nWe still owe an explanation for the sub-Weyl exponent in Theorem~\\ref{thm3}\\ref{thm3-b}, where $q = \\pm \\ell$. The proof of this bound is different from the other results: it is inspired by a brilliant recent idea of Steiner and Khayutin--Steiner~\\cite{St, KhayutinSteiner2020} in the \\emph{weight} aspect for the groups $\\SO_3(\\RR)$ and $\\SL_2(\\RR)$. The starting point is the desire to choose the amplifier so long that it works as self-amplification. In this way, the amplifier can be made independent of the well-known but inefficient trick of using the Hecke relation $\\lambda_p^2 - \\lambda_{p^2} = 1$. A self-amplified second moment is in effect a fourth moment, and the key observation is that it can be realized as the diagonal term in a \\emph{double} pre-trace formula. This only has a chance to work if the corresponding geometric side can be analyzed sufficiently accurately, and to this end, two extra features are necessary: a special behaviour of spherical functions with rapid decay conditions (such as, for instance, the Bergman kernel for $\\SL_2(\\RR)$) and the possibility for a \\emph{second moment} count on the geometric side, i.e.\\ pairs of matrices, in a best possible way.\n\nFor the proof of Theorem~\\ref{thm3}\\ref{thm3-b}, we implement this idea for first time in the context of principal series representations. Our proof proceeds differently than both of \\cite{St} and  \\cite{KhayutinSteiner2020}. We avoid the theta correspondence and instead detect the diagonal term in the double pre-trace formula by an argument that is reminiscent of the Voronoi formula for  Rankin--Selberg $L$-functions over $\\QQ[i]$, cf.\\ \\S \\ref{sec28}. As we lose positivity, we have to use the full power of the pre-trace formula, unlike our other results where the softer pre-trace inequality suffices. The argument is analytically subtle, since we also lose the possibility to choose the test function in the pre-trace formula freely:  part of it is now  given to us by the gamma kernel in the Voronoi summation formula (a new feature compared to \\cite{St} and \\cite{KhayutinSteiner2020}). At this point we need a very precise understanding of the Harish-Chandra transform in Theorem~\\ref{thm:pws} with complete uniformity in the auxiliary complex parameters, and the reader may observe that in the end only the strong $g$-dependence in \\eqref{thm5boundq=ell} saves the final bound.\n\n\\subsection{Matrix counting}\nHaving discussed some of the analytic and representation theoretic novelties, we finally comment briefly on the arithmetic part. In all previous instances of the sup-norm problem, the analysis of the geometric side of the pre-trace formula  amounts to counting matrices close to $K$, because the elementary spherical function is bi-$K$-invariant and decays away from $K$. Given the results on spherical trace functions in \\S \\ref{gen-sph-fun-intro-sec}, it is clear that from an arithmetic point of view the sup-norm problem with big $K$-types is conceptually very different from the spherical sup-norm problem.\n\nThe localization behaviour of generalized spherical functions has distinct features as reflected by Theorems~\\ref{thm4} and \\ref{thm5}. The spherical trace function $\\varphi_{\\nu,\\ell}^{\\ell}$ concentrates close to the identity. The functions $\\varphi_{\\nu,\\ell}^{\\ell,\\pm\\ell}$ localize sharply around diagonal matrices (but not necessarily within $K$). For $\\varphi_{\\nu,\\ell}^{\\ell,0}$, there is localization on diagonal and skew-diagonal matrices within $K$, then there is a gradual transition to a second layer in a neighbourhood of the 4-dimensional manifold $\\mcN$ defined by \\eqref{adbc}, and outside this neighbourhood we see sharp decay. Theorem~\\ref{thm6} is in some sense a combination of these two extreme cases. Correspondingly, the counting techniques in \\S\\S \\ref{thm1-proof-sec}--\\ref{sec-proof2} are still based on the geometry of numbers, but they differ conceptually and technically from the earlier treatment of the spherical sup-norm problem. In particular, as mentioned in \\S \\ref{KS-intro}, for the proof of Theorem~\\ref{thm3}\\ref{thm3-b}  we have to achieve a best possible double matrix count, cf.\\ Lemma~\\ref{lemma-ell-count}.\n\n\\subsection{Acknowledgements}\nThis work began during D.M.'s term as Director's Mathematician in Residence at the Budapest Semesters of Mathematics program in the summer of 2018; D.M. would like to thank BSM, the Alfr\\'ed R\\'enyi Institute of Mathematics, as well as the Max Planck Institute for Mathematics for their hospitality and excellent working conditions.\n\n\\section{Preliminaries}\n\n\\subsection{Representations of $\\SU_2(\\CC)$}\\label{SU2-subsec}\nIn this subsection, we review the representation theory of the maximal compact subgroup\n\\[ K=\\SU_2(\\CC)=\\left\\{ k[\\alpha,\\beta]:=\\begin{pmatrix}\\alpha&\\beta\\\\-\\bar{\\beta}&\\bar{\\alpha}\\end{pmatrix}:|\\alpha|^2+|\\beta|^2=1\\right\\} \\]\nof $G=\\SL_2(\\CC)$.  We use \\cite[\\S2.1.1,2.2]{Lokvenec-Guleska2004} as a convenient reference.\n\nFor $u,v,w\\in\\RR$, we parametrize $K$ using essentially Euler angles $(2u,2v,2w)$ as follows:\n\\begin{equation}\\label{decomp-K}\nk[u,v,w]:=\\begin{pmatrix}e^{iu}&\\\\&e^{-iu}\\end{pmatrix}\\begin{pmatrix}\\cos v&i\\sin v\\\\i\\sin v&\\cos v\\end{pmatrix}\\begin{pmatrix}e^{iw}&\\\\&e^{-iw}\\end{pmatrix}.\n\\end{equation}\nGenerating an equivalence relation $\\sim$ on $\\RR^3$ by\n\\begin{equation}\\label{angleequiv}\n(u,v,w)\\ \\sim\\ (u+2\\pi,v,w),\\ (u,v,w+2\\pi),\\ (u+\\pi,v+\\pi,w),\\ (u+\\pi\/2,-v,w-\\pi\/2)\n\\end{equation}\nwe may parametrize $\\SU_2(\\CC)$ by $\\RR^3\/\\!\\sim$, or by a specific fundamental domain such as $[0,\\pi)\\times[0,\\pi\/2]\\times[-\\pi,\\pi)$, in which each point in $\\SU_2(\\CC)$ has exactly one pre-image other than those with $v\\in\\frac{\\pi}2\\ZZ$. The probability Haar measure on $\\SU_2(\\CC)$ is given by\n\\begin{equation}\\label{dk}\n\\dd k=(2\\pi^2)^{-1}\\sin 2v\\,\\dd u\\,\\dd v\\,\\dd w.\n\\end{equation}\n\nThe irreducible representations of $K=\\SU_2(\\CC)$ are classified as $(2\\ell+1)$-dimensional representations $\\tau_{\\ell}$, for $\\ell\\in\\frac12\\ZZ_{\\geq 0}$, described explicitly as the space $V_{2\\ell}$ of polynomials of degree at most $2\\ell$, with a basis given by $\\{z^{\\ell-q}:|q|\\leq\\ell,\\,q\\equiv\\ell\\pmod{1}\\}$ and $\\SU_2(\\CC)$ action given by\n\\begin{equation}\\label{matrix-coeff}\n\\tau_{\\ell}(k[\\alpha,\\beta])z^{\\ell-q}=(\\alpha z-\\bar{\\beta})^{\\ell-q}(\\beta z+\\bar{\\alpha})^{\\ell+q}=\\sum_{\\substack{|p|\\leq\\ell\\\\p\\equiv\\ell\\one}}\\Phi_{p,q}^{\\ell}(k[\\alpha,\\beta])z^{\\ell-p}.\\end{equation}\nA $K$-invariant scalar product on $V_{2\\ell}$ is given by $(z^{\\ell-q},z^{\\ell-p})=(\\ell-q)!(\\ell+q)!\\delta_{q=p}$, so that $\\Phi_{p,q}^{\\ell}$ are (unnormalized) matrix coefficients of $\\tau_{\\ell}$. Moreover,\n\\[\\left\\{\\Phi_{p,q}^{\\ell}\\,:\\,\\text{$p,q,\\ell\\in\\tfrac12\\ZZ$ and $|p|,|q|\\leq\\ell$ and $p,q\\equiv\\ell\\one$}\\right\\}\\]\nis an orthogonal basis of $L^2(K)$. In harmony with \\cite[\\S4.4.2]{Warner1972a}, we denote by $\\xi_{\\ell}$ the character of $\\tau_{\\ell}$, by $d_{\\ell}=2\\ell+1$ the dimension of $\\tau_\\ell$, and by $\\chi_{\\ell}=d_{\\ell}\\xi_{\\ell}$ the normalized character of $\\tau_\\ell$. Finally, we denote by $\\widehat{K}=\\{\\tau_{\\ell}:\\ell\\in\\frac12\\ZZ_{\\geq 0}\\}$ the unitary dual of $K$.\n\n\\subsection{Representations of $\\SL_2(\\CC)$}\\label{SL2C-subsec}\nFor compatibility with the existing literature, we shall use the Iwasawa decomposition of $G=\\SL_2(\\CC)$ in two forms, $G=NAK$ and $G=KAN$, where $N$ (resp.\\ $A$) is the subgroup of unipotent upper-triangular (resp.\\ positive diagonal) matrices, and $K=\\SU_2(\\CC)$ is the standard maximal compact subgroup.\n\nWe fix a Haar measure on $G$ by setting\n\\[\n\\dd g=|\\dd z|\\frac{\\dd r}{r^5}\\dd k\\quad\\text{for } g=\\begin{pmatrix}1&z\\\\&1\\end{pmatrix}\\begin{pmatrix}r&\\\\&r^{-1}\\end{pmatrix}k, \\quad z\\in\\CC,\\,\\,r>0,\\,\\,k\\in K,\n\\]\nwhere $|\\dd z|=\\dd x\\,\\dd y$ for $z=x+iy$, $x,y\\in\\RR$, and $\\dd k$ is as in \\eqref{dk}.\n\nWe write $\\mfa\\simeq\\RR$ for the Lie algebra of $A$,  $\\rho$ for the root on $\\mfa$ mapping\n$\\big(\\begin{smallmatrix}x&\\\\&-x\\end{smallmatrix}\\big)$ to $2x$,\n$\\exp:\\mfa\\to A$ for the exponential map, and $\\kappa:G\\to K$ and $H:G\\to\\mfa$ for\nthe projection and height maps defined by $g\\in\\kappa(g)\\exp(H(g)) N$ for every $g\\in G$.\nThus explicitly, for $g=\\left(\\begin{smallmatrix}a&b\\\\c&d\\end{smallmatrix}\\right)\\in G$ we have\n\\begin{equation}\\label{eq:kappaH}\n\\kappa(g)=\\begin{pmatrix}a\/\\sqrt{|a|^2+|c|^2}&\\ast\\\\c\/\\sqrt{|a|^2+|c|^2}&\\ast\\end{pmatrix},\\quad \\exp(H(g))=\\begin{pmatrix}\\sqrt{|a|^2+|c|^2}&\\\\&1\/\\sqrt{|a|^2+|c|^2}\\end{pmatrix}.\n\\end{equation}\nFinally, let $M\\simeq S^1$ be the centralizer of $A$ in $K$, which consists of diagonal matrices in $K$.\n\nFollowing \\cite[Ch.~III]{GGV}, we introduce for every pair $(\\nu,p)\\in\\CC\\times\\frac12\\ZZ$\nthe (generalized) principal series representation $\\pi_{\\nu,p}$. Let us denote by $C^\\infty(\\CC)$ the set of functions $\\CC\\to\\CC$ that are smooth when regarded as functions $\\RR^2\\to\\CC$. The representation space $V_{\\nu,p}$ consists of those functions $v\\in C^\\infty(\\CC)$ for which the transformed functions\n\\begin{equation}\\label{transformedfunctions}\n\\pi_{\\nu,p}\\left(\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}\\right)v(z)\n=|bz+d|^{2p+2\\nu-2}(bz+d)^{-2p}v\\left(\\frac{az+c}{bz+d}\\right),\\qquad\n\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}\\in G,\n\\end{equation}\nextend to elements of $C^\\infty(\\CC)$. The above display then actually defines the representation $\\pi_{\\nu,p}:G\\to\\GL(V_{\\nu,p})$. The space $V_{\\nu,p}$ is complete with respect to the countable family of seminorms\n\\[\\sup\\bigl\\{\\bigl|v^{(a,b)}(x+yi)\\bigr|+\\bigl|\\widehat{v}^{(a,b)}(x+yi)\\bigr|:x^2+y^2\\leq c\\bigr\\},\\qquad (a,b,c)\\in\\NN^3,\\]\nwhere we abbreviate $\\widehat{v}:=\\pi_{\\nu,p}\\left(\\big(\\begin{smallmatrix}&-1\\\\1&\\end{smallmatrix}\\big)\\right)v$ for $v\\in V_{\\nu,p}$. The action of $G$ is continuous in the topology induced by these seminorms; thus, $\\pi_{\\nu,p}$ is a Fr\\'echet space representation.\n\nUsing the action of $K=\\SU_2(\\CC)$ and its diagonal subgroup $\\left\\{\\diag(e^{i\\varrho},e^{-i\\varrho}):\\varrho\\in\\RR\\right\\}$, we can decompose the $K$-finite part of $V_{\\nu,p}$ into an \\emph{algebraic direct sum} of finite-dimensional subspaces and further into one-dimensional subspaces:\n\\begin{equation}\\label{eq:tau-ell-isotypical-decomposition-algebraic}\nV_{\\nu,p}^{\\text{$K$-finite}} = \\bigoplus_{\\substack{\\ell\\geq|p|\\\\\\ell\\equiv p\\one}}V_{\\nu,p}^{\\ell}\n= \\bigoplus_{\\substack{\\ell\\geq|p|\\\\\\ell\\equiv p\\one}}\n\\ \\bigoplus_{\\substack{|q|\\leq\\ell\\\\q\\equiv\\ell\\one}}V_{\\nu,p}^{\\ell,q}.\n\\end{equation}\nPrecisely, $V_{\\nu,p}^\\ell$ is a $(2\\ell+1)$-dimensional subspace on which $\\pi_{\\nu,p}|_K$ acts by $\\tau_\\ell\\in\\widehat{K}$.\n\nIf $\\nu\\not\\equiv p\\pmod{1}$ or $|\\nu|\\leq|p|$, then $\\pi_{\\nu,p}\\cong\\pi_{-\\nu,-p}$ is irreducible, and these are all the equivalences among the representations $\\pi_{\\nu,p}$. If $\\nu\\equiv p\\pmod{1}$ and $|\\nu|>|p|$, then $\\pi_{\\nu,p}$ and $\\pi_{-\\nu,-p}$ are reducible. Assume $\\nu>0$, say. Then the sum of $V_{\\nu,p}^{\\ell}$ with $|p|\\leq\\ell<\\nu$ is a closed invariant subspace of $V_{\\nu,p}$, and the representation induced on the quotient is irreducible. The closure of the sum of $V_{-\\nu,-p}^{\\ell}$ with $\\ell\\geq\\nu$ is an invariant subspace of $V_{-\\nu,-p}$, and the representation induced on it is irreducible. Both of these representations of $G$ are isomorphic to $\\pi_{p,\\nu}\\cong\\pi_{-p,-\\nu}$. This observation will become relevant in \\eqref{eq:spherical-function-symmetry-2} below.\n\nThe space $V_{\\nu,p}$ has a $G$-invariant Hermitian inner product if and only if $\\nu\\in i\\RR$, or $p=0$ and $\\nu\\in(-1,0)\\cup(0,1)$. In the first case, we say that $\\pi_{\\nu,p}$ belongs to the (tempered) unitary principal series. In the second case, we say that $\\pi_{\\nu,p}$ belongs to the (non-tempered) complementary series. In either case, the Fr\\'echet space representation $\\pi_{\\nu,p}$ induces an irreducible unitary representation on the Hilbert space completion\n$\\widehat{V_{\\nu,p}}$ that we shall still denote by $\\pi_{\\nu,p}$. The only equivalences among these unitary representations are $\\pi_{\\nu,p}\\simeq\\pi_{-\\nu,-p}$. The equivalence classes, along with the trivial representation, form the unitary dual $\\widehat{G}$ of $G$.\n\nFor $\\pi\\simeq\\pi_{\\nu,p}\\in\\widehat{G}$ we write\n\\[V_\\pi:=\\widehat{V_{\\nu,p}},\\qquad V_\\pi^\\ell:=V_{\\nu,p}^\\ell,\\qquad V_\\pi^{\\ell,q}:=V_{\\nu,p}^{\\ell,q},\\]\nand then \\eqref{eq:tau-ell-isotypical-decomposition-algebraic} is equivalent to the orthogonal Hilbert space decomposition\n(cf.~\\eqref{decomp}):\n\\[V_\\pi = \\bigoplus_{\\substack{\\ell\\geq|p|\\\\\\ell\\equiv p\\one}}V_\\pi^{\\ell}\n= \\bigoplus_{\\substack{\\ell\\geq|p|\\\\\\ell\\equiv p\\one}}\n\\ \\bigoplus_{\\substack{|q|\\leq\\ell\\\\q\\equiv\\ell\\one}}V_\\pi^{\\ell,q}.\\]\nThe projection $V_\\pi\\to V_{\\pi}^{\\ell}$ is realized by the operator\n\\begin{equation}\\label{eq:projectionbychi}\n\\pi(\\ov{\\chi_\\ell}):=\\int_K \\ov{\\chi_\\ell}(k)\\pi(k)\\,\\dd k\\in \\End(V_\\pi),\n\\end{equation}\nwhere $\\End(V_\\pi)$ denotes the Hilbert space of Hilbert--Schmidt operators on $V_\\pi$ endowed with the Hilbert--Schmidt norm. This leads to the ``block matrix decomposition''\n\\begin{equation}\\label{eq:block-decomposition}\n\\End(V_\\pi)=\\bigoplus_{\\substack{m,n\\geq|p|\\\\ m,n\\equiv p\\one}}\\Hom(V_{\\pi}^{m},V_{\\pi}^{n}),\n\\end{equation}\nwhere the direct sum is meant in the Hilbert space sense. Hence, for $f\\in C_c(G)$, the $(m,n)$-component of the Hilbert--Schmidt operator (cf.\\ \\cite[Th.~2]{GelfandNaimark})\n\\begin{equation}\\label{eq:pif}\n\\pi(f):=\\int_G f(g)\\pi(g)\\,\\dd g\\in\\End(V_\\pi)\n\\end{equation}\nequals\n\\begin{equation}\\label{eq:tau-projection}\n\\pi(\\ov{\\chi_n})\\pi(f)\\pi(\\ov{\\chi_m})=\\pi(\\ov{\\chi_n}\\star f\\star\\ov{\\chi_m})\\in\\Hom(V_{\\pi}^{m},V_{\\pi}^{n}),\n\\end{equation}\nwhere the convolutions are meant over $K$.\n\n\\subsection{Plancherel theorem}\\label{subsec:plancherel}\nIn this subsection, we review the Plancherel theorem for $G=\\SL_2(\\CC)$ pioneered by Gelfand and Naimark, following the original sources \\cite{GelfandNaimark,GelfandNaimark2} and their translations \\cite{GelfandNaimarkTranslated,GelfandNaimark2Translated}. We note that the list of unitary representations given in \\cite{GelfandNaimark2} is incomplete for higher rank groups (cf.\\ \\cite{Stein,Vogan,Tadic}), but this does not affect the results we are quoting. In addition, we warn the reader that the translations contain some misprints not present in the originals, e.g.\\ in the crucial formulae \\cite[(137)--(138)]{GelfandNaimarkTranslated}.\n\nWe identify once and for all (non-canonically) the \\emph{tempered unitary dual} $\\Gtemp$ with the set\n\\[\\left\\{\\pi_{it,p}:(t,p)\\in\\left(\\RR_{>0}\\times\\tfrac12\\ZZ\\right)\\cup\\left(\\{0\\}\\times\\tfrac12\\ZZ_{\\geq 0}\\right)\\right\\},\\]\nwith topology inherited from the standard topology on $\\RR^2$. The \\emph{Plancherel measure} on $\\widehat{G}$ is supported on $\\Gtemp$, and it is given explicitly as\n\\begin{equation}\\label{eq:concrete-plancherel-measure}\n\\dd\\mupl(\\pi_{it,p}):=\\frac{1}{\\pi^2}(t^2+p^2)\\,\\dd t.\n\\end{equation}\nFor $\\pi_{it,p}\\in\\Gtemp$, the underlying Hilbert space $\\widehat{V_{it,p}}$ is independent of the parameters: it equals $\\Vcan:=L^2(\\CC)$. On this common representation space, \\eqref{transformedfunctions} defines the unitary action $\\pi_{it,p}:G\\to\\U(\\Vcan)$ that agrees with \\cite[(65)]{GelfandNaimark} for $(n,\\rho)=(2p,2t)$. The \\emph{operator-valued spherical transform} of $f\\in C_c(G)$ is the map $\\Gtemp\\to\\End(\\Vcan)$ given by $\\pi\\mapsto\\pi(f)$ as in \\eqref{eq:pif}. The Plancherel theorem for $G$ concerns the extension of this transform to $L^2(G)$, and characterizes its image.\n\n\\begin{theorem}[Gelfand--Naimark]\\label{thm:abstract-isomorphism}\nThe map given by \\eqref{eq:pif} extends (uniquely) to an $L^2$-isometry\n\\[L^2(G)\\longrightarrow L^2(\\Gtemp\\to\\End(\\Vcan)),\\]\nwhere the operator-valued $L^2$-space on the right-hand side is meant with respect to the Hilbert--Schmidt norm ${\\|\\cdot\\|}_{\\mathrm{HS}}$ on\n$\\End(\\Vcan)$ and the Plancherel measure $\\mupl$ on $\\Gtemp$. In particular, for every $f\\in L^2(G)$, the following Plancherel formula holds:\n\\begin{equation}\\label{eq:abstract-plancherel}\n\\int_G |f(g)|^2\\,\\dd g = \\int_{\\Gtemp} {\\|\\pi(f)\\|}_{\\mathrm{HS}}^2\\,\\dd \\mupl(\\pi).\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nThe theorem follows from \\cite[Th.~5]{GelfandNaimark}; we only need to check that our Plancherel measure corresponds to the one in \\cite[(137)]{GelfandNaimark}. We do this in four steps.\n\\newline\\emph{Step 1.}\nWe observe that the constant $(8\\pi^4)^{-1}$ in \\cite[(137)]{GelfandNaimark} should be $(16\\pi^4)^{-1}$ due to a small oversight in the derivation of \\cite[(130)]{GelfandNaimark} from \\cite[(129)]{GelfandNaimark}. The oversight is that the change of variables\n\\[(w_1,w_2,\\lambda)\\mapsto(\\zeta_1,\\zeta_2,\\zeta_3):=(w_2,w_1\\bar\\lambda+w_2\/\\bar\\lambda,w_1)\\]\ncoming from \\cite[(123)]{GelfandNaimark} is not 1-to-1 but 2-to-1.\n\\newline\\emph{Step 2.}\nWe rewrite the corrected right-hand side of \\cite[(137)]{GelfandNaimark} as a sum over $p\\in\\frac{1}{2}\\ZZ$ and an integral over $t>0$, keeping in mind that $(n,\\rho)$ in \\cite{GelfandNaimark} is $(2p,2t)$ in our notation.\n\\newline\\noindent\\emph{Step 3.}\nWe observe that the Haar measure $\\dd\\mu(g)$ used by Gelfand--Naimark is $2\\pi^2\\dd g$. Indeed, applying\n\\cite[(40)]{GelfandNaimark} to a right $K$-invariant test function $f\\in C_c(G)$, we obtain by several changes of variables that\n\\begin{align*}\\int_G f(g)\\,\\dd\\mu(g)\n&=\\int_{\\CC\\times\\CC^\\times\\times\\CC}f\\left(\\begin{pmatrix}w^{-1}&z\\\\&w\\end{pmatrix}\n\\begin{pmatrix}1&\\\\v&1\\end{pmatrix}\\right)|\\dd v|\\,|\\dd w|\\,|\\dd z|\\\\[4pt]\n&=\\int_{\\CC\\times\\CC^\\times\\times\\CC}f\\left(\\begin{pmatrix}w^{-1}&z\\\\&w\\end{pmatrix}\n\\begin{pmatrix}1\/\\sqrt{1+|v|^2}&\\bar v\/\\sqrt{1+|v|^2}\\\\&\\sqrt{1+|v|^2}\\end{pmatrix}\\right)|\\dd v|\\,|\\dd w|\\,|\\dd z|\\\\[4pt]\n&=\\int_{\\CC\\times\\CC^\\times\\times\\CC}f\\left(\\begin{pmatrix}w^{-1}&z\\\\&w\\end{pmatrix}\\right)\n\\frac{|\\dd v|\\,|\\dd w|\\,|\\dd z|}{(1+|v|^2)^2}\\\\[4pt]\n&= \\pi \\int_{\\CC^\\times\\times\\CC}f\\left(\\begin{pmatrix}1&z\\\\&1\\end{pmatrix}\n\\begin{pmatrix}w&\\\\&w^{-1}\\end{pmatrix}\\right)\\frac{|\\dd w|\\,|\\dd z|}{|w|^6} = 2\\pi^2\\int_G f(g)\\,\\dd g.\n\\end{align*}\n\\emph{Step 4.}\nPutting everything together, the corrected version of \\cite[(137)]{GelfandNaimark} yields\n\\[\\int_G |f(g)|^2\\,2\\pi^2\\dd g = \\frac{1}{16\\pi^4}\\sum_p\\int_0^\\infty{\\|2\\pi^2\\pi_{it,p}(f)\\|}_{\\mathrm{HS}}^2\\,(4t^2+4p^2)\\,2\\dd t.\\]\nThis formula is equivalent to \\eqref{eq:abstract-plancherel}, hence we are done.\n\\end{proof}\n\n\\begin{remark}\\label{remark:Plancherel} In the proof above, we claimed that the Plancherel measure in \\cite[Th.~5]{GelfandNaimark} is off by a factor of $2$. For double checking this claim, we looked at \\cite[Th.~11.2]{Knapp}, and we found (to our dismay) that the Plancherel measure there is off by a factor of $\\pi$. For example, for the test function $f(g):=1\/\\tr(gg^*)^2$, the Fourier transform given by \\cite[(11.14)]{Knapp} equals $F_f^T(t)=\\pi\/\\tr(tt^*)$, hence in \\cite[(11.17)]{Knapp} the left-hand side is $\\pi^2$, while the right-hand side is $\\pi$. For triple checking our claim, we verified that our Plancherel measure yields the correct inversion formula for the classical spherical transform (for bi-$K$-invariant functions), as in \\cite[\\S 3.3]{FHMM}.\n\\end{remark}\n\n\\begin{theorem}[Gelfand--Naimark]\\label{thm:general-inversion-formula}\nLet $f\\in C_c^\\infty(G)$. For every $\\pi\\in\\Gtemp$, the operator $\\pi(f)\\in\\End(\\Vcan)$ is of trace class, and the following inversion formula holds:\n\\begin{equation}\\label{eq:general-inversion-formula}\nf(g)=\\int_{\\Gtemp} \\tr(\\pi(f)\\pi(g^{-1}))\\,\\dd\\mupl(\\pi).\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nThe theorem follows from \\cite[Th.~19]{GelfandNaimark2} applied to $n=2$ and $x=R(g)f$, or from \\cite[Th.~11.2]{Knapp}, with appropriate correction of the Plancherel measure (cf.\\ Remark~\\ref{remark:Plancherel}).\n\\end{proof}\n\n\\subsection{The \\texorpdfstring{$\\tau_{\\ell}$}{tau-ell}-spherical transform}\\label{section:sphericaltransform}\nFor a given $\\ell\\in\\frac{1}{2}\\ZZ_{\\geq 0}$, it is interesting to see what Theorems~\\ref{thm:abstract-isomorphism} and \\ref{thm:general-inversion-formula} yield for test functions $f\\in L^2(G)$ with the following property: for almost every $\\pi\\in\\Gtemp$, the operator $\\pi(f)$ acts by a scalar on $V_{\\pi}^{\\ell}$ and by zero on its orthocomplement $V_{\\pi}^{\\ell,\\perp}$. In the light of \\eqref{eq:block-decomposition}, \\eqref{eq:tau-projection}, \\eqref{eq:abstract-plancherel}, and Schur's lemma, these test functions form the Hilbert subspace $\\mcH(\\tau_{\\ell})\\subset L^2(G)$ defined by the conditions\n\\begin{itemize}\n\\item $f(g)=f(kgk^{-1})$ for almost every $g\\in G$ and $k\\in K$;\n\\item $f=\\ov{\\chi_\\ell} \\star f \\star\\ov{\\chi_\\ell}$.\n\\end{itemize}\n\nLet $\\Gtemp(\\tau_{\\ell})$ be the set of $\\pi\\in\\Gtemp$ whose restriction to $K$ contains $\\tau_{\\ell}$. For $f\\in\\mcH(\\tau_{\\ell})$, the operator-valued function $\\pi\\mapsto\\pi(f)$ is supported on $\\Gtemp(\\tau_{\\ell})$, and there it is simply determined by the scalar-valued function $\\pi\\mapsto\\tr(\\pi(f))$ via\n\\begin{equation}\\label{eq:opfromtr}\n\\pi(f)|_{V_{\\pi}^{\\ell}}= \\frac{\\tr(\\pi(f))}{2\\ell+1}\\cdot \\id_{V_{\\pi}^{\\ell}}\n\\qquad\\text{and}\\qquad\\pi(f)|_{V_{\\pi}^{\\ell,\\perp}}=0.\n\\end{equation}\nIn particular, for $\\pi\\in\\Gtemp(\\tau_{\\ell})$ and $f\\in\\mcH(\\tau_{\\ell})$,\n\\begin{equation}\\label{eq:HSfromtr}\n{\\|\\pi(f)\\|}_{\\mathrm{HS}}^2 = \\tr(\\pi(f)\\pi(f)^*)= \\frac{|\\tr(\\pi(f))|^2}{2\\ell+1}.\n\\end{equation}\nFor $(\\nu,p)\\in i\\RR\\times\\frac12\\ZZ$, the condition $\\pi_{\\nu,p}\\in\\Gtemp(\\tau_{\\ell})$ is equivalent to $|p|\\leq\\ell$ and $p\\equiv\\ell\\pmod{1}$. Moreover, for $f\\in L^1(G)\\cap\\mcH(\\tau_{\\ell})$, the trace of $\\pi_{\\nu,p}(f)$ can be expressed\nin terms of the \\emph{$\\tau_\\ell$-spherical trace function}\n\\begin{equation}\\label{eq:def-spherical-function}\n\\varphi_{\\nu,p}^{\\ell}(g):=\\tr(\\pi_{\\nu,p}(\\ov{\\chi_\\ell})\\pi_{\\nu,p}(g)\\pi_{\\nu,p}(\\ov{\\chi_\\ell}))\n\\end{equation}\nas (cf.\\ \\eqref{eq:pif} and \\eqref{eq:tau-projection})\n\\begin{equation}\\label{eq:trpif}\n\\widehat{f}(\\nu,p):=\\tr(\\pi_{\\nu,p}(f)) = \\int_G f(g)\\,\\varphi_{\\nu,p}^{\\ell}(g)\\,\\dd g.\n\\end{equation}\n\nThe function $\\varphi_{\\nu,p}^{\\ell}:G\\to\\CC$ vanishes unless $|p|\\leq\\ell$ and $p\\equiv\\ell\\pmod{1}$, for else $\\tau_{\\ell}$ does not appear in $\\pi_{\\nu,p}$, and $\\varphi_{\\nu,p}^{\\ell}(\\id)=2\\ell+1$ in this latter case. Moreover, we have the integral representation of Harish-Chandra \\cite[Cor.~6.2.2.3]{Warner1972}:\n\\[\\varphi_{\\nu,p}^{\\ell}(g)=\\int_K\\left(\\chi_\\ell\\star\\eta_p\\right)(\\kappa(k^{-1}gk))\\,\ne^{(\\nu-1)\\rho(H(gk))}\\,\\dd k.\\]\nHere, $\\eta_p:M\\simeq S^1\\to\\CC^{\\times}$ is the unitary character $\\eta_p(z)=z^{-2p}$, the convolution is over $M$, and $\\kappa$, $\\rho$, and $H$ are as in \\S\\ref{SL2C-subsec}. For computational purposes, we spell out the $\\chi_\\ell\\star\\eta_p$ term explicitly, cf.~\\eqref{matrix-coeff}, \\cite[(10) \\& Lemma~3.2]{Wang1974}, \\cite[Th.~29.18]{HewittRoss}:\n\\[\\begin{aligned}\n\\left(\\chi_\\ell\\star\\eta_p\\right)(k[\\alpha,\\beta])\n&=(2\\ell+1)\\Phi_{p,p}^{\\ell}(k[\\alpha,\\beta])\\\\\n&=(2\\ell+1)\\sum_{r=0}^{\\ell-|p|}(-1)^r\\binom{\\ell+p}{r}\\binom{\\ell-p}{r}\n\\alpha^{\\ell-p-r}{\\bar\\alpha}^{\\ell+p-r}|\\beta|^{2r}.\n\\end{aligned}\\]\nWe collect further useful properties of $\\varphi_{\\nu,p}^{\\ell}:G\\to\\CC$ in the next lemma, where we write\n\\[a_h:= \\diag(e^{h\/2}, e^{-h\/2}),\\qquad h\\in\\RR.\\]\n\n\\begin{lemma} The $\\tau_\\ell$-spherical trace function $\\varphi_{\\nu,p}^{\\ell}(g)$ extends holomorphically to $\\nu\\in\\CC$, and it satisfies the bound\n\\begin{equation}\\label{eq:spherical-function-bound}\n\\bigl|\\varphi_{\\sigma+it,p}^{\\ell}(k_1a_hk_2)\\bigr|\\leq(2\\ell+1)\\frac{\\sinh(\\sigma h)}{\\sigma\\sinh(h)},\n\\qquad \\sigma,t,h\\in\\RR,\\quad k_1,k_2\\in K.\n\\end{equation}\n(For $\\sigma=0$ or $h=0$, the fraction on the right-hand side is understood as $1$.)\nThe extended function has the symmetries\n\\begin{equation}\\label{eq:spherical-function-symmetry}\n\\varphi_{\\nu,p}^{\\ell}(g)=\\ov{\\varphi_{-\\ov{\\nu},p}^{\\ell}(g)}=\\varphi_{\\nu,p}^{\\ell}(g^{-1}),\n\\end{equation}\n\\begin{equation}\\label{eq:spherical-function-symmetry-2}\n\\varphi_{\\nu,p}^{\\ell}(g)=\\varphi_{p,\\nu}^{\\ell}(g),\\qquad\\nu\\equiv p\\one,\\quad|\\nu|,|p|\\leq\\ell.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nThe holomorphic extension of $\\varphi_{\\nu,p}^{\\ell}(g)$ and the bound \\eqref{eq:spherical-function-bound} are\na straightforward generalization of \\cite[Prop.~3.4]{Wang1974} and its proof. The identity $\\ov{\\varphi_{-\\ov{\\nu},p}^{\\ell}(g)}=\\varphi_{\\nu,p}^{\\ell}(g^{-1})$ follows from \\eqref{eq:def-spherical-function} and $\\pi(g)^*=\\pi(g^{-1})$ for $\\nu\\in i\\RR$, and then also for $\\nu\\in\\CC$ by the uniqueness of analytic continuation. The identity\n$\\varphi_{\\nu,p}^{\\ell}(g)=\\varphi_{\\nu,p}^{\\ell}(g^{-1})$ is \\cite[Lemma~3.2]{Wang1974}. Finally, the remarkable symmetry \\eqref{eq:spherical-function-symmetry-2} follows from \\cite[Cor.~2]{Wang1974}, or more conceptually from the discussion below \\eqref{eq:tau-ell-isotypical-decomposition-algebraic}.\n\\end{proof}\n\nAs we shall see in Theorem~\\ref{thm:tr-plancherel} below, the \\emph{$\\tau_\\ell$-spherical transform} defined by \\eqref{eq:trpif} is inverted by the following \\emph{inverse $\\tau_\\ell$-spherical transform}. For $h\\in L^1(\\Gtemp(\\tau_{\\ell}))\\cap L^2(\\Gtemp(\\tau_{\\ell}))$ and $g\\in G$, we define\n\\begin{equation}\\label{eq:inverse-tauell-transform}\n\\widecheck{h}(g):=\\frac{1}{(2\\ell+1)\\pi^2}\\sum_{\\substack{|p|\\leq\\ell\\\\p\\equiv\\ell\\one}}\n\\int_{0}^{\\infty} h(it,p)\\,\\varphi_{it,p}^{\\ell}(g^{-1})\\,(t^2+p^2)\\,\\dd t.\n\\end{equation}\n\n\\begin{theorem}\\label{thm:tr-plancherel} The transforms defined by \\eqref{eq:trpif} and \\eqref{eq:inverse-tauell-transform} extend (uniquely) to a pair of inverse Hilbert space isometries\n\\[\\mcH(\\tau_{\\ell})\\longleftrightarrow L^2(\\Gtemp(\\tau_{\\ell})).\\]\nIn particular, for $f\\in \\mcH(\\tau_{\\ell})$, the following Plancherel formula holds:\n\\begin{equation}\\label{eq:tr-plancherel}\n\\int_G |f(g)|^2\\,\\dd g = \\frac{1}{(2\\ell+1)\\pi^2}\n\\sum_{\\substack{|p|\\leq\\ell\\\\p\\equiv\\ell\\one}}\\int_{0}^{\\infty}\n|\\widehat{f}(it,p)|^2\\,(t^2+p^2)\\,\\dd t.\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof} The fact that $\\ \\widehat{}\\ $ extends to a Hilbert space isomorphism $\\mcH(\\tau_{\\ell})\\to L^2(\\Gtemp(\\tau_\\ell))$ follows from Theorem~\\ref{thm:abstract-isomorphism} and our discussion above. In particular, \\eqref{eq:tr-plancherel} is a special case of \\eqref{eq:abstract-plancherel} in the light of \\eqref{eq:concrete-plancherel-measure}, \\eqref{eq:HSfromtr}, \\eqref{eq:trpif}. We are left with proving that $\\ \\widecheck{}\\ $ is the inverse of $\\ \\widehat{}\\ $, and for this it suffices to verify that $\\ \\widecheck{}\\ $ applied after $\\ \\widehat{}\\ $ is the identity on the dense subset $C_c^{\\infty}(G)\\cap\\mcH(\\tau_{\\ell})$ of the Hilbert space $\\mcH(\\tau_{\\ell})$. For $f\\in C_c^{\\infty}(G)\\cap \\mcH(\\tau_{\\ell})$, \\eqref{eq:projectionbychi}, \\eqref{eq:pif}, \\eqref{eq:concrete-plancherel-measure}, \\eqref{eq:general-inversion-formula}, \\eqref{eq:opfromtr}, \\eqref{eq:def-spherical-function}, \\eqref{eq:trpif} yield\n\\begin{align*}f(g)\n&=\\int_{\\Gtemp}\\tr(\\pi(f)\\pi(g^{-1}))\\,\\dd\\mupl=\\frac{1}{2\\ell+1}\\int_{\\Gtemp}\\tr(\\pi(f))\\tr(\\pi(\\ov{\\chi_\\ell})\\pi(g^{-1}))\\,\\dd\\mupl\\\\\n&=\\frac{1}{(2\\ell+1)\\pi^2} \\sum_{\\substack{|p|\\leq \\ell \\\\ p\\equiv \\ell\\one}} \\int_{0}^{\\infty} \\widehat{f}(it,p)\\, \\varphi_{it,p}^{\\ell}(g^{-1})\\,(t^2+p^2)\\,\\dd t.\n\\end{align*}\nThe proof is complete.\n\\end{proof}\n\nWang~\\cite{Wang1974} proved an analogue of the Paley--Wiener theorem for the $\\tau_\\ell$-spherical transform, and in particular characterized the image of $\\mcH(\\tau_{\\ell})\\cap C_c^\\infty(G)$ under the transform. The following is \\cite[Prop.~4.5]{Wang1974} and should be compared to Theorem~\\ref{thm:pws} in the introduction.\n\n\\begin{theorem}[Wang]\\label{thm10}\nLet $f\\in\\mcH(\\tau_{\\ell})$ be a test function, and let $R>0$ be a radius. Then the following two conditions are equivalent.\n\\begin{enumerate}[(a)]\n\\item The function $f(g)$ is smooth, and\n\\[f(k_1 a_h k_2)=0,\\qquad |h|>R,\\quad k_1,k_2\\in K.\\]\n\\item The function $\\widehat{f}(\\nu,p)$ extends holomorphically to $\\CC\\times\\tfrac12\\ZZ$ such that\n\\[\\widehat{f}(\\nu,p)=\\widehat{f}(p,\\nu),\\qquad\\nu\\equiv p\\one,\\quad|\\nu|,|p|\\leq\\ell,\\]\nand for any $C>0$ we have\n\\[\\widehat{f}(\\nu,p)\\ll_C (1+|\\nu|)^{-C}e^{R|\\Re\\nu|},\\qquad \\nu\\in\\CC,\\quad p\\in\\tfrac12\\ZZ.\\]\n\\end{enumerate}\n\\end{theorem}\n\nWe now prove a Schwartz class version of this result as stated in Theorem~\\ref{thm:pws}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:pws}]\nAssume condition \\ref{pws-a}. The holomorphic extension of $\\widehat{f}(\\nu,p)$ follows from \\eqref{eq:spherical-function-bound} coupled with \\eqref{eq:mAbound} for $m=0$, and then \\eqref{eq:symmetry} is immediate from \\eqref{eq:spherical-function-symmetry-2}. In order to derive \\eqref{eq:BCbound}, we use an alternate representation of $\\widehat{f}(\\nu,p)$. We shall assume that $|p|\\leq\\ell$ and $p\\equiv\\ell\\pmod{1}$, for else $\\widehat{f}(\\nu,p)=0$. By the second display on \\cite[p.~621]{Wang1974}, we see that the (unique) holomorphic extension is also provided by\n\\begin{equation}\\label{eq:sphericalviaAbel}\n\\widehat{f}(\\nu,p)=\\frac{2\\ell+1}{2}\\int_{-\\infty}^\\infty\\breve{f}(h,p)\\,e^{\\nu h}\\,\\dd h,\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:Abelnew}\n\\breve{f}(h,p):=e^h\\int_K\\int_N f(ka_hn)\\,\\Phi_{p,p}^{\\ell}(k)\\,\\dd k\\,\\dd n.\n\\end{equation}\nWe claim that, for any $m\\in\\ZZ_{\\geq 0}$ and $A>0$, we have\n\\begin{equation}\\label{eq:Abelbound}\n\\frac{\\partial^m}{\\partial h^m}\\breve{f}(h,p)\\ll_{m,A} e^{-A|h|},\\qquad h\\in\\RR,\\quad |p|\\leq\\ell,\\quad p\\equiv\\ell\\one.\n\\end{equation}\nFor $|h|>1$ this follows by writing $a_hn=k_1a_{h'}k_2$ in \\eqref{eq:Abelnew}, and then combining \\eqref{eq:mAbound} with some calculus to keep track of the dependence of $h'\\in\\RR$ and $k_1,k_2\\in K$ on $h\\in\\RR$. For $|h|\\leq 1$ we proceed similarly for the part of the integral in \\eqref{eq:Abelnew} that corresponds to $n=\\big(\\begin{smallmatrix}1&z\\\\& 1\\end{smallmatrix}\\big)$ with $|z|>1$, while we estimate the ($h$-derivatives of the) remaining integral directly by the smoothness of $f(g)$. With \\eqref{eq:Abelbound} at hand, \\eqref{eq:BCbound} follows from \\eqref{eq:sphericalviaAbel} via integration by parts. We proved that \\ref{pws-a} implies \\ref{pws-b}.\n\nAssume condition \\ref{pws-b}. By Theorem~\\ref{thm:tr-plancherel},\n\\[f(g)=\\frac{1}{(2\\ell+1)\\pi^2} \\sum_{\\substack{|p|\\leq \\ell \\\\ p\\equiv \\ell\\one}} \\int_{0}^{\\infty} \\widehat{f}(it,p)\\, \\varphi_{it,p}^{\\ell}(g^{-1})\\,(t^2+p^2)\\,\\dd t.\\]\nLet us restrict, without loss of generality, to $g=k_1a_hk_2$ with $h>0$. Using the display below \\cite[(29)]{Wang1974}, we infer\n\\[f(g)=\\frac{1}{4\\pi^2\\sinh(h)}\\sum_{\\substack{|p|,|j|\\leq \\ell \\\\ p,j\\equiv\\ell\\one}}\n\\int_{-h}^h \\widetilde{f}(s,p) \\,\n\\Phi_{-p,j}^{\\ell}(v_{\\theta}^{-1})\\Phi_{j,j}^{\\ell}(k_1k_2) \\Phi_{j,-p}^{\\ell}(v_{\\theta'})\n\\,\\dd s,\\]\nwhere $\\Phi_{-p,j}^{\\ell}(v_{\\theta}^{-1})$ and $\\Phi_{j,-p}^{\\ell}(v_{\\theta'})$ can be explicated using \\cite[(5) \\& (28)]{Wang1974}, and\n\\begin{equation}\\label{eq:def-tilde-f-s-p}\n\\widetilde{f}(s,p):=\\int_{-\\infty}^\\infty\\widehat{f}(it,p)\\,e^{-its}\\,(t^2+p^2)\\,\\dd t,\\qquad s\\in\\RR.\n\\end{equation}\nBy \\eqref{eq:BCbound} and Cauchy's theorem, it follows for any $n\\in\\ZZ_{\\geq 0}$ and $D>0$ that\n\\begin{equation}\\label{eq:nDbound}\n\\frac{\\partial^n}{\\partial s^n}\\widetilde{f}(s,p)\\ll_{n,D} e^{-D|s|},\\qquad s\\in\\RR.\n\\end{equation}\nThe smoothness of $f(g)$ is now straightforward, and this automatically verifies \\eqref{eq:mAbound} for $|h|\\leq 1$. From now on we can assume, without loss of generality, that $h>1$. From \\eqref{eq:symmetry}, \\eqref{eq:nDbound}, and the calculation around \\cite[(38)--(41)]{Wang1974}, we see that\n\\[\\sum_{\\substack{|p|,|j|\\leq \\ell \\\\ p,j\\equiv\\ell\\one}}\n\\int_{-\\infty}^\\infty \\widetilde{f}(s,p) \\,\n\\Phi_{-p,j}^{\\ell}(v_{\\theta}^{-1})\\Phi_{j,j}^{\\ell}(k_1k_2) \\Phi_{j,-p}^{\\ell}(v_{\\theta'})\n\\,\\dd s=0,\\]\nhence in fact\n\\[f(g)=\\frac{-1}{4\\pi^2\\sinh(h)}\\sum_{\\substack{|p|,|j|\\leq \\ell \\\\ p,j\\equiv\\ell\\one}}\n\\left(\\int_{-\\infty}^{-h}+\\int_h^\\infty\\right)\\widetilde{f}(s,p) \\,\n\\Phi_{-p,j}^{\\ell}(v_{\\theta}^{-1})\\Phi_{j,j}^{\\ell}(k_1k_2) \\Phi_{j,-p}^{\\ell}(v_{\\theta'})\n\\,\\dd s.\\]\nFrom here it is straightforward to deduce \\eqref{eq:mAbound} for $h>1$, using \\eqref{eq:nDbound} and the remarks above it. We proved that \\ref{pws-b} implies \\ref{pws-a}.\n\\end{proof}\n\nWe shall denote by $\\mcH(\\tau_{\\ell})_{\\infty}$ the set of functions satisfying the equivalent conditions \\ref{pws-a} and \\ref{pws-b} of Theorem~\\ref{thm:pws}. It is clear that $\\mcH(\\tau_{\\ell})_{\\infty}$ is a convolution subalgebra of $L^1(G)\\cap L^2(G)$.\n\n\\begin{remark} The last display combined with the observation $\\widetilde{f}(s,p)=\\widetilde{f}(-s,-p)$ yields the following refinement of \\eqref{eq:mAbound} when $m=0$:\n\\begin{equation}\\label{eq:inverse-spherical-transform-estimate}\n\\bigl|f(k_1a_hk_2)\\bigr|\\leq\\sum_{\\substack{|p|\\leq \\ell \\\\ p\\equiv\\ell\\one}}\n\\int_h^\\infty \\bigl|\\widetilde{f}(s,p)\\bigr|\\,\\dd s,\\qquad h>1,\\quad k_1,k_2\\in K. \n\\end{equation}\n\\end{remark}\n\nWe end this subsection by stating a two-variable version of some of the previous definitions and results. Taking (topological) tensor products of Hilbert spaces, we can identify $\\mcH(\\tau_{\\ell}) \\hat{\\otimes} \\mcH(\\tau_{\\ell})$ with the space of functions $f \\in L^2(G \\times G)$ satisfying\n\\begin{itemize}\n\\item $f(g_1, g_2) = f(k_1g_1k_1^{-1}, k_2g_2k_2^{-1})$ for almost every\n$g_1, g_2 \\in G$ and $k_1, k_2 \\in K$;\n\\item $(\\ov{\\chi_\\ell},\\ov{\\chi_\\ell})  \\star f \\star (\\ov{\\chi_\\ell},\\ov{\\chi_\\ell})$.\n\\end{itemize}\nThis can be seen by projecting the isomorphism between $L^2(G)\\hat{\\otimes} L^2(G)$ and $L^2(G\\times G)$ (see e.g.\\ \\cite[Cor.~4.11.9]{Simon2015}) to $\\mcH(\\tau_{\\ell}) \\hat{\\otimes} \\mcH(\\tau_{\\ell})$ and the (closed) subspace of functions in question. By Theorem~\\ref{thm:tr-plancherel}, this space is isometrically isomorphic to $L^2(\\Gtemp(\\tau_{\\ell})^2)$ via the obvious extension of the map \\eqref{eq:trpif}:\n\\begin{equation}\\label{doubletransform}\n\\widehat{f}(\\nu_1, p_1, \\nu_2, p_2) := \\int_{G_1\\times G_2} f(g_1, g_2) \\,\n\\varphi_{\\nu_1, p_1}^{\\ell}(g_1)\\varphi_{\\nu_2, p_2}^{\\ell}(g_2) \\, \\dd g_1\\dd g_2.\n\\end{equation}\nFor $h\\in L^2(\\Gtemp(\\tau_{\\ell})^2)\\cap L^2(\\Gtemp(\\tau_{\\ell})^2)$, the \ninverse transform is given as in \\eqref{eq:inverse-tauell-transform}:\n\\begin{equation}\\label{inversedoubletransform}\n\\begin{split}\n\\widecheck{h}(g_1, g_2) := \\frac{1}{(2\\ell+ 1)^2\\pi^4}&\n\\sum_{\\substack{|p_1|, |p_2|\\leq\\ell\\\\p_1 \\equiv p_2\\equiv\\ell\\one}}\n\\int_{0}^{\\infty}\\int_{0}^{\\infty}  h(it_1,p_1, it_2, p_2)\\\\\n&\\qquad\\varphi_{it_1,p_1}^{\\ell}(g_1^{-1})\\varphi_{it_2,p_2}^{\\ell}(g_2^{-1})\\,(t_1^2+p_1^2)(t_2^2+p_2^2)\\,\\dd t_1\\,\\dd t_2.\n\\end{split}\n\\end{equation}\nIt is straightforward to adapt the above presented proof of Theorem~\\ref{thm:pws} to obtain the following variant for $\\mcH(\\tau_{\\ell}) \\hat{\\otimes} \\mcH(\\tau_{\\ell})$:\n\\begin{theorem}\\label{thm:pws2} For $f\\in\\mcH(\\tau_{\\ell}) \\hat{\\otimes} \\mcH(\\tau_{\\ell})$, the following two conditions are equivalent (with implied constants depending on $f$).\n\\begin{enumerate}[(a)]\n\\item\\label{pws-a} The function $f(g_1,g_2)$ is smooth, and for any $m\\in\\ZZ_{\\geq 0}$ and $A>0$ we have\n\\[\\frac{\\partial^{2m}}{\\partial h_1^m\\partial h_2^m}f(k_1 a_{h_1} k_2,k_3 a_{h_2} k_4)\\ll_{m,A} e^{-A(|h_1|+|h_2|)},\n\\qquad h_1,h_2\\in\\RR,\\quad k_1,k_2,k_3,k_4\\in K.\\]\n\\item\\label{pws-b} The function $\\widehat{f}(\\nu_1,p_1,\\nu_2,p_2)$ extends holomorphically to $\\CC\\times\\tfrac12\\ZZ\\times\\CC\\times\\tfrac12\\ZZ$ such that\n\\begin{align*}\n\\widehat{f}(\\nu_1,p_1,\\nu_2,p_2) &= \\widehat{f}(p_1,\\nu_1,\\nu_2,p_2),\n\\qquad \\nu_1 \\equiv p_1\\pmod{1},\\quad |\\nu_1|, |p_1| \\leq \\ell,\\\\\n\\widehat{f}(\\nu_1,p_1,\\nu_2,p_2) &= \\widehat{f}(\\nu_1,p_1,p_2,\\nu_2),\n\\qquad \\nu_2 \\equiv p_2\\pmod{1},\\quad |\\nu_2|, |p_2| \\leq \\ell,\n\\end{align*}\nand for any $B,C>0$ we have\n\\[\\widehat{f}(\\nu_1,p_1,\\nu_2,p_2) \\ll_{B, C} (1+|\\nu_1|+|\\nu_2|)^{-C},\n\\qquad |\\Re\\nu_1|,|\\Re\\nu_2|\\leq B,\\quad p_1,p_2\\in\\tfrac{1}{2}\\ZZ.\\]\n\\end{enumerate}\n\\end{theorem}\n\n\n\\subsection{Hecke operators}\nThe arithmetic quotient $\\Gamma\\backslash G$ comes equipped with a rich family of Hecke correspondences, which we now describe, referring to \\cite{BlomerHarcosMilicevic2016} for further details and references. For every $n\\in\\ZZ[i]\\setminus\\{0\\}$, consider the set\n\\[ \\Gamma_n:=\\left\\{\\begin{pmatrix} a&b\\\\c&d\\end{pmatrix}\\in\\MM_2(\\ZZ[i]):ad-bc=n\\right\\}. \\]\nIn particular, $\\Gamma_1=\\Gamma$. Then we may define the Hecke operator $T_n$ acting on functions $\\phi:\\Gamma\\backslash G\\to\\CC$ by\n\\begin{equation}\\label{Hecke-def}\n(T_n\\phi)(g):=\\frac1{|n|}\\sum_{\\gamma\\in\\Gamma\\backslash\\Gamma_n}\\phi\\left(\\frac1{\\sqrt{n}}\\gamma g\\right)=\\frac1{4|n|}\\sum_{ad=n}\\sum_{b\\bmod d}\\phi\\left(\\frac1{\\sqrt{n}}\\begin{pmatrix}a&b\\\\0&d\\end{pmatrix}g\\right),\n\\end{equation}\nwhere the result is independent of the choice of the square-root since $\\pm\\id\\in\\Gamma$. In particular, since $\\Gamma_{-1}=\\Gamma\\cdot\\big(\\begin{smallmatrix}-1&\\\\&1\\end{smallmatrix}\\big)$ and $\\frac{1}{i}\\big(\\begin{smallmatrix}-1&\\\\&1\\end{smallmatrix}\\big)=\\big(\\begin{smallmatrix}i&\\\\&-i\\end{smallmatrix}\\big)\\in\\Gamma$, we have $T_{-1}=T_1=\\id$. We also observe that, as $\\gamma$ ranges through a set of representatives of $\\Gamma\\backslash\\Gamma_n$, $n\\gamma^{-1}$ ranges through a set of representatives of $\\Gamma_n\/\\Gamma$.\n\nThese Hecke operators are self-adjoint on $L^2(\\Gamma\\backslash G)$, commute with each other and the Laplace operator; thus they act by constants $\\lambda_n(V)$ on each irreducible component $V\\subset L^2(\\Gamma\\backslash G)$, with non-zero vectors in each $V$ being joint Hecke--Maa{\\ss} eigenfunctions. They also satisfy the multiplicativity relation\n\\begin{equation}\\label{hecke-mult}\nT_mT_n=\\sum_{(d)\\mid(m,n)}T_{mn\/d^2},\\qquad m,n\\in\\ZZ[i]\\setminus\\{0\\},\n\\end{equation}\nwhere it is clear that the right-hand side does not depend on the choice of the generator $d$. Finally we have the Rankin--Selberg bound\n\\begin{equation}\\label{RS-bound}\n\\sum_{|n|^2\\leq x}|\\lambda_n(V)|^2\\ll_Vx.\n\\end{equation}\n\n\\subsection{Eisenstein series and spectral decomposition}\\label{Eisenstein-subsec}\nIn this subsection, we review the construction and properties of the (not necessarily spherical) Eisenstein series on $\\Gamma\\backslash G$. The quotient $\\Gamma\\backslash G$ has a unique cusp at $\\infty$. For $\\ell\\in\\ZZ_{\\geq 0}$, $p,q\\in\\ZZ$ with $2\\mid p$ and $|p|,|q|\\leq\\ell$, and $\\nu\\in\\CC$ with $\\Re\\nu>1$, we define the Eisenstein series of type $(\\ell,q)$ at $\\infty$ as in \\cite[Def.~3.3.1]{Lokvenec-Guleska2004} by the absolutely and locally uniformly convergent series\n\\begin{equation}\\label{Eisdef}\nE_{\\ell,q}(\\nu,p)(g):=\\sum_{\\gamma\\in\\Gamma_{\\infty}\\backslash\\Gamma}\\phi_{\\ell,q}(\\nu,p)(\\gamma g),\n\\end{equation}\nwhere $\\Gamma_{\\infty}$ is the subgroup of upper-triangular matrices in $\\Gamma$ (the stabilizer of $\\infty$ in $\\Gamma$), and\n\\begin{equation}\\label{eq:phi-ell-q-nu-p}\n\\phi_{\\ell,q}(\\nu,p)\\left(\\begin{pmatrix}r&\\ast\\\\&r^{-1}\\end{pmatrix}k\\right):=r^{2(1+\\nu)}\\Phi_{p,q}^{\\ell}(k),\\qquad r>0,\\quad k\\in K.\n\\end{equation}\nThese Eisenstein series possess a meromorphic continuation to $\\nu\\in\\CC$, which is holomorphic along $i\\RR$ \\cite[\\S5.1]{Lokvenec-Guleska2004}. An easy calculation with \\eqref{Hecke-def} and \\eqref{matrix-coeff} shows that they are also eigenfunctions of the Hecke operators $T_n$ with\n\\begin{equation}\n\\label{Eisenstein-Hecke-eigen}\nT_nE_{\\ell,q}(\\nu,p)=\\lambda_n(E(\\nu,p))E_{\\ell,q}(\\nu,p),\\quad \\lambda_n(E(\\nu,p)):=\\frac{1}{4}\\sum_{n=ad}\\chi_{\\nu,p}(a)\\chi_{-\\nu,-p}(d),\n\\end{equation}\nwhere $\\chi_{\\nu,p}(z):=|z|^{\\nu}(z\/|z|)^{-p}$. In particular,\n\\begin{equation}\\label{eq:hecke-i-eigenvalue-eisenstein}\n\\lambda_{in}(E(\\nu,p))=(-1)^{p\/2}\\lambda_n(E(\\nu,p)).\n\\end{equation}\nWhile $E_{\\ell,q}(\\nu,p)$ for individual $\\nu\\in i\\RR$ (barely) fail to lie in $L^2(\\Gamma\\backslash G)$, their averages against $C_c(i\\RR)$ weights $f(\\nu)$ comfortably do, and upon taking the Hilbert space closure of their span and orthocomplements one obtains the familiar orthogonal decomposition\n\\begin{equation}\\label{eq:spectral-decomposition}\nL^2(\\Gamma\\backslash G)=\\CC\\cdot 1\\oplus L^2(\\Gamma\\backslash G)_{\\cusp}\\oplus L^2(\\Gamma\\backslash G)_{\\Eis}.\n\\end{equation}\n\nLet $H(\\nu,p)$ be the linear span of all $\\phi_{\\ell,q}(\\nu,p)$ with $|p|,|q|\\leq\\ell$. By \\eqref{eq:phi-ell-q-nu-p}, the functions $f\\in H(\\nu,p)$ satisfy\n\\[f\\left(\\begin{pmatrix}z&\\ast\\\\&z^{-1}\\end{pmatrix}g\\right)=|z|^2\\chi_{\\nu,p}(z^2)f(g),\\qquad z\\in\\CC^\\times,\\quad g\\in G,\\]\nand they are determined by their restriction to $K$. In fact $H(\\nu,p)$ as a $(\\mfg,K)$-module is isomorphic to the \n$K$-finite part of $\\pi_{\\nu,p}$ featured in \\eqref{eq:tau-ell-isotypical-decomposition-algebraic}. That is, the appropriate completion of $H(\\nu,p)$ serves as a model of the Fr\u00e9chet\/Hilbert space representation $\\pi_{\\nu,p}$, and we shall denote by \n$H^\\infty(\\nu,p)$ the dense subspace of smooth vectors in this completion.\n\nDenoting by $C^K(\\Gamma\\backslash G)$ the space of $K$-finite smooth functions on $\\Gamma\\backslash G$,  an automorphic representation of type $(\\nu,p)$ for $\\Gamma\\backslash G$ may be realized as a $(\\mfg,K)$-module homomorphism $T:H(\\nu,p)\\to C^K(\\Gamma\\backslash G)$, cf.\\ \\cite[\\S3.4]{Lokvenec-Guleska2004}. Such a $T$ may arise as $T_V$ for a cuspidal consituent $V\\simeq\\pi_{\\nu,p}$ occurring discretely in $L^2(\\Gamma\\backslash G)_{\\cusp}$, or from the Eisenstein series via\n\\[T_{E(\\nu,p)}\\phi_{\\ell,q}(\\nu,p):=E_{\\ell,q}(\\nu,p),\\qquad |p|,|q|\\leq\\ell.\\]\nIndeed, by \\eqref{Eisdef}, the last display defines a $(\\mfg,K)$-module homomorphism for $\\Re\\nu>1$, hence by analytic continuation for all $\\nu\\in\\CC$ where the relevant Eisenstein series have no pole. Following custom, we lighten the notation by denoting a generic automorphic representation of type $(\\nu,p)$, whether of type $T_V$ or $T_{E(\\nu,p)}$, as $V$, and its associated Hecke eigenvalues as $\\lambda_n(V)$. Finally, we shall use that the above $(\\mfg,K)$-module homomorphism extends uniquely to a $G$-module homomorphism $H^\\infty(\\nu,p)\\to C^\\infty(\\Gamma\\backslash G)$, and its image consists of functions of moderate growth.\n\nNow \\eqref{eq:spectral-decomposition} is explicated by the following two spectral identities. For $f$ in the space $C^{\\infty}_0(\\Gamma\\backslash G)$ of smooth complex-valued functions on $\\Gamma\\backslash G$ with all rapidly decaying derivatives, we have\n\\begin{equation}\\label{eq:spectral-decomposition-EGM}\n\\begin{split}\nf= \\frac{\\langle f, 1\\rangle }{\\vol(\\Gamma \\backslash G)}  & + \\sum_{\\text{$V$ cuspidal}} \\sum_{\\substack{q,\\ell\\in \\ZZ \\\\ |p_V|,|q|\\leq \\ell}} \\frac{\\langle f,T_V \\phi_{\\ell,q}(\\nu_V,p_V) \\rangle}{{\\|\\Phi_{p_V,q}^{\\ell}\\|}_K^2}  T_V \\phi_{\\ell,q}(\\nu_V,p_V)\\\\ & + \\frac{1}{\\pi i} \\int_{(0)} \\sum_{p\\in 2\\ZZ} \\sum_{\\substack{q,\\ell\\in \\ZZ \\\\ |p|,|q|\\leq \\ell}} \\frac{\\langle f,E_{\\ell,q}(\\nu,p) \\rangle}{{\\|\\Phi_{p,q}\n^{\\ell}\\|}_K^2} E_{\\ell,q}(\\nu,p) \\,\\dd\\nu,\n\\end{split}\n\\end{equation}\nwith the obvious interpretation of $\\langle f,E_{\\ell,q}(\\nu,p) \\rangle$. For\n$f_1,f_2\\in C^{\\infty}_0(\\Gamma \\backslash G)$, we have with the same interpretation\n\\begin{equation}\\label{eq:spectral-decomposition-plancherel}\n\\begin{split}\n\\langle f_1,f_2 \\rangle = \\frac{ \\langle f_1, 1\\rangle \\langle 1, f_2\\rangle}{\\vol(\\Gamma \\backslash G)} & + \\sum_{\\text{$V$ cuspidal}} \\sum_{\\substack{q,\\ell\\in \\ZZ \\\\ |p_V|,|q|\\leq \\ell}} \\frac{\\langle f_1,T_V \\phi_{\\ell,q}(\\nu_V,p_V) \\rangle \\langle T_V \\phi_{\\ell,q}(\\nu_V,p_V), f_2 \\rangle}{{\\|\\Phi_{p_V,q}^{\\ell}\\|}_K^2}  \\\\ & + \\frac{1}{\\pi i} \\int_{(0)} \\sum_{p\\in 2\\ZZ} \\sum_{\\substack{q,\\ell\\in \\ZZ \\\\ |p|,|q|\\leq \\ell}} \\frac{\\langle f_1,E_{\\ell,q}(\\nu,p) \\rangle \\langle  E_{\\ell,q}(\\nu,p),f_2 \\rangle}{{\\|\\Phi_{p,q}\n^{\\ell}\\|}_K^2} \\,\\dd\\nu.\n\\end{split}\n\\end{equation}\nCompare with \\cite[Ch.~6, Th.~3.4]{ElstrodtGrunewaldMennicke1998} and \\cite[Th.~8.1]{Lokvenec-Guleska2004}.\n\nWe shorten the notation in two ways. First, for an automorphic representation $V$ (cuspidal or Eisenstein) of type $(\\nu,p)$ occurring in $L^2(\\Gamma\\backslash G)$, we write\n\\[\\phi_{\\ell,q}^{V}:=\\frac{T_V\\phi_{\\ell,q}(\\nu,p)}{{\\|\\Phi_{p,q}^{\\ell}\\|}_K},\\qquad |p|,|q|\\leq\\ell.\\] \nIn particular, when at least one of two such $V$ and $V'$ is cuspidal, $\\langle\\phi_{\\ell,q}^V,\\phi_{\\ell',q'}^{V'}\\rangle$ equals $\\delta_{(\\ell,q,V)=(\\ell',q',V')}$. Second, while the decompositions in \\eqref{eq:spectral-decomposition-EGM} and \\eqref{eq:spectral-decomposition-plancherel} are over all automorphic representations $V$ (cuspidal or Eisenstein) occurring in $L^2(\\Gamma\\backslash G)$, keeping in mind the $\\tau_{\\ell}$-spherical transform of \\S\\ref{section:sphericaltransform}, it will be useful to introduce the shorthand notation $\\int_{[\\ell]}\\,\\dd V$ for the sum-integral over those $V$ of type $(\\nu,p)$ such that $\\pi_{\\nu,p}\\in\\widehat{G}(\\tau_{\\ell})$ (that is, with $|p|\\leq\\ell$ as well as $p\\in 2\\ZZ$ for $V$ Eisenstein). Thus, for example, \\eqref{eq:spectral-decomposition-EGM} may be rewritten in the more compact form\n\\begin{equation}\n\\label{compactform}\nf=\\frac{\\langle f,1\\rangle}{\\vol(\\Gamma \\backslash G)}+\\sum_{\\ell\\geq 0}\\int_{[\\ell]}\\sum_{|q|\\leq\\ell}\\langle f,\\phi_{\\ell,q}^{V}\\rangle\\phi_{\\ell,q}^{V}\\,\\dd V.\n\\end{equation}\n\n\\subsection{Rankin--Selberg convolutions}\\label{RSSection}\nIn this subsection, we review briefly the properties of Rankin--Selberg $L$-functions. We shall restrict to automorphic representations for $\\Gamma\\backslash G$ on which the Hecke operator $T_i$ acts trivially, so that they lift to automorphic representations for $\\PGL_2(\\ZZ[i])\\backslash\\PGL_2(\\CC)$. This allows us to refer to the theory of $\\GL_2$.\n\nThe Rankin--Selberg $L$-function of two automorphic representations $V_j$ of type\n$(\\nu_j,p_j)\\in i\\RR\\times \\ZZ$ for $\\Gamma\\backslash G$ is defined by the absolutely convergent series (cf.~\\eqref{RS-bound})\n\\begin{equation}\\label{RSdef}\nL(s, V_1\\times V_2) =  \\frac{1}{4}\\zeta_{\\QQ(i)}(2s)\\sum_{n \\in \\ZZ[i] \\setminus \\{0\\}}\n\\frac{\\lambda_n(V_1)\\lambda_n(V_2)}{(|n|^2)^s}, \\qquad \\Re s > 1.\n\\end{equation}\nThis can be verified by matching the Euler factors on the two sides, using \\cite[Th.~15.1]{Jacquet1972}, \\cite[Prop.~3.5]{JacquetLanglands}, \\cite[(3.1.3)]{Tate1977}, and \\cite[Lemma~1.6.1]{Bump1997}.\nIn particular,\n\\[L(s,V\\times E(\\nu,p))=L(s-\\tfrac{1}{2}\\nu,V\\otimes\\chi_p)L(s+\\tfrac{1}{2}\\nu,V\\otimes\\chi_{-p})\\]\nfor $V$ cuspidal and $(\\nu,p)\\in i\\RR\\times 4\\ZZ$ according to \\eqref{eq:hecke-i-eigenvalue-eisenstein}, as well as\n\\[ L\\big(s,E(\\nu_1,p_1)\\times E(\\nu_2,p_2)\\big)=\\prod_{\\epsilon_1,\\epsilon_2\\in\\{\\pm 1\\}}L\\big(s+\\tfrac{1}{2}(\\epsilon_1\\nu_1+\\epsilon_2\\nu_2),\\chi_{-\\epsilon_1p_1-\\epsilon_2p_2}\\big),\\]\nwith $(\\nu_j,p_j)\\in i\\RR\\times 4\\ZZ$ and $\\chi_p(z):=(z\/|z|)^{-p}$. All $L$-functions are meant over $\\QQ(i)$.\n\nThe Rankin--Selberg $L$-function $L(s,V_1\\times V_2)$ possesses a meromorphic continuation to the entire complex plane with the exception of finitely many possible poles along the line $\\Re s=1$. It is in fact entire except as follows (cf.\\ \\cite[Th.~2.2]{GelbartJacquet}):\n\\begin{itemize}\n\\item If $V_1 = V_2\\,(=V)$ is cuspidal of type $(\\nu,p)$ (that is, $(\\nu_1,p_1)=\\pm(\\nu_2,p_2)$), there is a simple pole at $s=1$ with (strictly) positive residue\n\\begin{equation}\\label{RS-res}\n\\mathop{\\mathrm{res}}_{s=1} L(s, V\\times V) = \\frac{\\pi}{4}\\cdot L(1,\\mathrm{ad^2}V) \\gg_{\\eps} \\big((1 + |p|)(1+ |\\nu|)\\big)^{-\\eps},\n\\end{equation}\nwhere the lower bound follows from \\cite[Prop.~3.2]{Maga2013}.\n\\item If $V_1$ and $V_2$ are both Eisenstein series with $p_1 = \\epsilon p_2$ for some $\\epsilon\\in\\{\\pm 1\\}$, there are simple poles at $s = 1+\\eta(\\nu_1 - \\epsilon\\nu_2)\/2$  for $\\eta\\in\\{\\pm 1\\}$ with residue\n\\begin{equation}\n\\label{RS-res-Eis}\n\\mcL_{\\eta} (V_1,V_2):=\\frac{\\pi}{4}\\cdot\\zeta_{\\QQ(i)}(1+\\eta(\\nu_1-\\epsilon\\nu_2))L(1+\\eta \\nu_1,\\chi_{-2\\eta p_1})L(1-\\eta\\epsilon\\nu_2,\\chi_{2\\eta\\epsilon p_2}),\n\\end{equation}\nunless $\\nu_1=\\pm\\nu_2$ or $\\nu_1=0$ or $\\nu_2=0$, in which case, however, the definition still makes sense as a meromorphic function of $\\nu_1,\\nu_2\\in\\CC$.\n\\end{itemize}\n\nFinally, the associated completed $L$-function satisfies the familiar functional equation\n\\begin{equation}\\label{func-eq}\n\\Lambda(s, V_1 \\times V_2) := 16^sL(s, V_1\\times V_2)L_{\\infty} (s, V_1\\times V_2) = \\Lambda(1-s, V_1 \\times V_2),\n\\end{equation}\nwhere the exponential factor $16^s$ coming from the discriminant of $\\QQ(i)$ is included for convenience, and the factor at infinity is given by\n\\begin{align}\n\\notag L_{\\infty}(s, V_1\\times V_2) = \\Gamma(s,\\vec{\\nu},\\vec{p})\n:&=\\prod_{\\epsilon_1,\\epsilon_2\\in\\{\\pm 1\\}}\nL_\\infty(s,\\chi_{\\epsilon_1\\nu_1,\\epsilon_1p_1}\\cdot\\chi_{\\epsilon_2\\nu_2,\\epsilon_2p_2})\\\\\n\\label{inffactor} &=\\prod_{\\epsilon_1,\\epsilon_2\\in\\{\\pm 1\\}}\n\\Gamma_{\\CC}\\left(s+\\textstyle\\frac{1}{2}(\\epsilon_1\\nu_1+\\epsilon_2\\nu_2)+\\textstyle\\frac{1}{2}|\\epsilon_1p_1+\\epsilon_2p_2|\\right).\n\\end{align}\nHere we used the abbreviations\n\\[\\Gamma_{\\CC}(s):=2(2\\pi)^{-s}\\Gamma(s),\\qquad\\vec{\\nu}:=(\\nu_1,\\nu_2),\\qquad\\vec{p}:=(p_1,p_2).\\]\nIndeed, \\eqref{inffactor} follows from \\cite[Prop.~18.2]{Jacquet1972}, \\cite[\\S 3]{Tate1977}, \\cite[Prop.~6 in \\S VII-2]{Weil1974} and its proof, upon noting that $V_j$ is isomorphic to the principal series representation induced from the pair of characters $(\\chi_{-\\nu_j,-p_j},\\chi_{\\nu_j,p_j})$.\n\n\\begin{lemma}\\label{eis-pos}\nLet $f : i\\RR \\rightarrow \\CC$ be a function decaying as $f(\\nu)\\ll (1+|\\nu|)^{-3}$, and let $p \\in \\ZZ$. Then\n\\[\\int_{(0)}\\int_{(0)}\\sum_{\\eta \\in \\{\\pm 1\\}}\nf(\\nu_1)\\ov{f(\\nu_2)}\\,\\mcL_{\\eta}((\\nu_1,p),(\\nu_2,p))\\,\\frac{\\dd\\nu_1}{\\pi i}\\,\\frac{\\dd\\nu_2}{\\pi i}\\geq 0.\\]\n\\end{lemma}\n\n\\begin{proof} First we note that the $\\eta$-sum cancels the individual poles of $\\mcL_{\\eta}((\\nu_1,  p), (\\nu_2,  p))$ at $\\nu_1 = \\nu_2$. For $\\eps > 0$ and $V_j = (\\nu_j,p)$ with $j\\in\\{1,2\\}$ define\n\\[\\mcL(V_1, V_2, \\eps):=\\sum_{\\eta\\in\\{\\pm 1\\}}\\frac{\\pi}{4}\\zeta_{\\QQ(i)}(1+\\eps+\\eta(\\nu_1-\\nu_2))\nL(1+\\eta\\nu_1,\\chi_{-2\\eta p})L(1-\\eta\\nu_2,\\chi_{2\\eta p})\\]\nand\n\\[\\mcI(\\eps):=\\int_{(0)}\\int_{(0)}\\sum_{\\eta\\in\\{\\pm 1\\}}\nf(\\nu_1)\\ov{f(\\nu_2)}\\,\\mcL_{\\eta}(V_1,V_2,\\eps)\\,\\frac{\\dd\\nu_1}{\\pi i}\\,\\frac{\\dd\\nu_2}{\\pi i}.\\]\nThis function is continuous at $\\eps = 0$, so it suffices to show $\\mcI(\\eps) \\geq 0$ for $\\eps > 0$.\nInserting the definition and opening the Dedekind zeta function, we see that\n\\[\\mcI(\\eps) = \\frac{\\pi}{16} \\sum_{\\eta \\in \\{\\pm 1\\}}   \\sum_{n \\in \\ZZ[i]\\setminus\\{0\\}}\n\\frac{1}{|n|^{2+2\\eps}}\n\\biggl|\\int_{(0)} \\frac{1}{|n|^{2\\eta\\nu}}  L(1 + \\eta \\nu, \\chi_{-2\\eta p}) f(\\nu) \\frac{\\dd\\nu}{\\pi i} \\biggr|^2 \\geq 0\\]\nas desired.\n\\end{proof}\n\n\\subsection{Diagonal detection of Voronoi type}\\label{sec28}\nIn this subsection, we prove a Voronoi-type formula that allows us to detect equality of two automorphic representations occurring in $L^2(\\Gamma\\backslash G)$ in terms of a certain weighted orthogonality relation between their Hecke eigenvalues.\nWe shall use that only tempered representations occur in $L^2(\\Gamma\\backslash G)$, e.g.\\ by \\cite[Ch.~7, Prop.~6.2]{ElstrodtGrunewaldMennicke1998}.\n\n\\begin{lemma}\\label{lemma-vor}\nLet $P \\geq 1$ be a parameter. There exists a function\n\\[W_P:\\RR_{>0}\\times\\CC^2\\times\\ZZ^2\\to\\CC,\\]\ngiven explicitly by \\eqref{defW}, with the following properties.\n\\begin{enumerate}[(a)]\n\\item\\label{vor-a} $W_{P}(x,\\vec{\\nu},\\vec{p})$ is an entire function of $\\vec{\\nu}=(\\nu_1,\\nu_2)\\in\\CC^2$, and it is invariant under\n\\[ (\\nu_j,p_j)\\mapsto (-\\nu_j,-p_j)\\quad\\text{as well as}\\quad (\\nu_j,p_j)\\mapsto (p_j,\\nu_j)\\quad (\\nu_j\\in \\ZZ). \\]\n\\item\\label{vor-b} Let us abbreviate $\\tilde P:=\\bigl(1+|p_1+p_2|\\bigr)\\bigl(1+|p_1-p_2|\\bigr)$. Then for every\n$A>|\\Re\\nu_1|+|\\Re\\nu_2|$ we have\n\\begin{equation}\\label{W1}\nW_{P}(x,\\vec{\\nu},\\vec{p})\\ll_{A,\\Re\\nu_1,\\Re\\nu_2}\\bigl(1+(\\tilde P\/P)^{2A-2}\\bigr)\\bigl(1+|\\nu_1|+|\\nu_2|\\bigr)^{4A}x^{-A}.\n\\end{equation}\n\\item\\label{vor-c} For every two automorphic representations $V_j$ of type $(\\nu_j,p_j)\\in i\\RR\\times\\ZZ$ for $\\Gamma\\backslash G$ we have\n\\begin{equation}\\label{vor-formula}\n\\begin{split}\n&\\sum_{n\\in\\ZZ[i]\\setminus\\{0\\}}W_P\\left(\\frac{|n|}{P},\\vec{\\nu},\\vec{p}\\right)\\lambda_n(V_1)\\lambda_n(V_2)\\\\\n&=\\begin{cases}\n\\frac{\\pi}{4}L(1,\\mathrm{ad^2}V_1)P^2,&V_1=V_2\\text{ cuspidal};\\\\\n\\sum_{\\eta\\in\\{\\pm 1\\}}\\mcL_{\\eta}(V_1,V_2)P^{2+\\eta(\\nu_1-\\epsilon\\nu_2)},&V_1,V_2\\text{ Eisenstein, }p_1=\\epsilon p_2,\\,\\,\\epsilon\\in\\{\\pm 1\\};\\\\\n0,&\\text{otherwise,}\n\\end{cases}\n\\end{split}\n\\end{equation}\nwhere $L(1,\\mathrm{ad^2}V_1)$ and $\\mcL_{\\eta}(V_1,V_2)$ are as in \\eqref{RS-res} and \\eqref{RS-res-Eis}.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet $w:\\RR_{>0}\\to\\CC$ be a smooth function supported inside $[1,2]$, and normalized so that its Mellin transform $\\widehat{w}(s)=\\int_0^{\\infty}w(x)x^s\\,\\dd x\/x$ satisfies $\\widehat{w}(1)=1$. We define\n\\begin{equation}\\label{defW}\nW_P(x,\\vec{\\nu},\\vec{p}):=\\frac1{8\\pi i}\\int_{(2)}\\zeta_{\\QQ(i)}(2s)\\left(\\widehat{w}(s)-P^{2-4s}\\frac{16^{2s-1}\\Gamma(s,\\vec{\\nu},\\vec{p})}{\\Gamma(1-s,\\vec{\\nu},\\vec{p})}\\widehat{w}(1-s)\\right)x^{-2s}\\,\\dd s,\n\\end{equation}\nwhere $\\Gamma(s,\\vec{\\nu},\\vec{p})$ is as in \\eqref{inffactor}.\n\nShifting the contour to the far right, we see that $W_P(x,\\vec{\\nu},\\vec{p})$ is entire in $\\vec{\\nu}$. The symmetry with respect to $(\\nu_j, p_j) \\mapsto (-\\nu_j, -p_j)$ is obvious from \\eqref{inffactor}. For $r\\in\\frac{1}{2}\\ZZ$ we have the equality\n\\[\\frac{\\Gamma(z+r)}{\\Gamma(1-z+r)}\n= \\frac{\\Gamma(z-r)}{\\Gamma(1-z-r)}\\cdot\\frac{\\sin(\\pi(z-r))}{\\sin(\\pi(z+r))}\n= (-1)^{2r}\\frac{\\Gamma(z-r)}{\\Gamma(1-z-r)}\\]\nof meromorphic functions in $z\\in\\CC$. This shows that (cf.\\ \\eqref{inffactor})\n\\begin{align*}\n\\frac{\\Gamma(s,\\vec{\\nu},\\vec{p})}{\\Gamma(1-s,\\vec{\\nu},\\vec{p})}\n&=\\prod_{\\epsilon_1,\\epsilon_2\\in\\{\\pm 1\\}}\n\\frac{\\Gamma_{\\CC}\\left(s+\\frac{1}{2}(\\epsilon_1\\nu_1+\\epsilon_2\\nu_2) + \\frac{1}{2}|\\epsilon_1p_1+\\epsilon_2p_2|\\right)}\n{\\Gamma_{\\CC}\\left(1-s-\\frac{1}{2}(\\epsilon_1\\nu_1+\\epsilon_2\\nu_2)+\\frac{1}{2}|\\epsilon_1p_1+\\epsilon_2p_2|\\right)}\\\\\n&=\\prod_{\\epsilon_1,\\epsilon_2\\in\\{\\pm 1\\}}\n\\frac{\\Gamma_{\\CC}\\left(s+\\frac{1}{2}(\\epsilon_1\\nu_1+\\epsilon_2\\nu_2) + \\frac{1}{2}(\\epsilon_1p_1+\\epsilon_2p_2)\\right)}\n{\\Gamma_{\\CC}\\left(1-s-\\frac{1}{2}(\\epsilon_1\\nu_1+\\epsilon_2\\nu_2)+\\frac{1}{2}(\\epsilon_1p_1+\\epsilon_2p_2)\\right)}\n\\end{align*}\nis symmetric with respect to $(\\nu_j, p_j) \\mapsto (p_j, \\nu_j)$, completing the proof of \\ref{vor-a}.\n\nCombining the first line of the previous display with \\cite[Lemma~3.2]{Harcos2002}, we infer for $\\Re(s)>\\frac{1}{2}|\\Re\\nu_1|+\\frac{1}{2}|\\Re\\nu_2|$ that\n\\begin{align*}\n&\\left|\\frac{\\Gamma(s,\\vec{\\nu},\\vec{p})}{\\Gamma(1-s,\\vec{\\nu},\\vec{p})}\\right|\n=\\prod_{\\epsilon_1,\\epsilon_2\\in\\{\\pm 1\\}}\n\\left|\\frac{\\Gamma_{\\CC}\\left(s+\\frac{1}{2}(\\epsilon_1\\nu_1+\\epsilon_2\\nu_2) + \\frac{1}{2}|\\epsilon_1p_1+\\epsilon_2p_2|\\right)}\n{\\Gamma_{\\CC}\\left(1-\\ov{s}-\\frac{1}{2}(\\epsilon_1\\ov{\\nu_1}+\\epsilon_2\\ov{\\nu_2})+\\frac{1}{2}|\\epsilon_1p_1+\\epsilon_2p_2|\\right)}\\right|\\\\\n&\\qquad\\ll_{\\Re s,\\Re\\nu_1,\\Re\\nu_2}\\prod_{\\epsilon_1,\\epsilon_2\\in\\{\\pm 1\\}}\n\\left|s+\\tfrac12(\\epsilon_1\\nu_1+\\epsilon_2\\nu_2) + \n\\tfrac12|\\epsilon_1p_1+\\epsilon_2p_2|\\right|^{\\Re(2s+\\epsilon_1\\nu_1+\\epsilon_2\\nu_2)-1}\\\\\n&\\qquad\\ll_{\\Re s,\\Re\\nu_1,\\Re\\nu_2}\\prod_{\\epsilon_1,\\epsilon_2\\in\\{\\pm 1\\}}\n\\left(1+|\\epsilon_1p_1+\\epsilon_2p_2|\\right)^{\\Re(2s+\\epsilon_1\\nu_1+\\epsilon_2\\nu_2)-1}\n\\left(|s|+|\\nu_1|+|\\nu_2|\\right)^{\\Re(2s+\\epsilon_1\\nu_1+\\epsilon_2\\nu_2)}\\\\\n&\\qquad=\\left(1+|p_1+p_2|\\right)^{4\\Re s-2}\\left(1+|p_1-p_2|\\right)^{4\\Re s-2}\\left(|s|+|\\nu_1|+|\\nu_2|\\right)^{8\\Re s}.\n\\end{align*}\nTurning back to \\eqref{defW}, the singularity of the integrand at $s = 1\/2$ is removable, so we can shift the contour to $\\Re s = A\/2$. The bound \\eqref{W1} follows upon noting that\nthat\n\\begin{itemize}\n\\setlength\\itemsep{3pt}\n\\item $\\widehat{w}(s) \\ll_{C, \\Re s} (1 + |s|)^{-C}$ for all $C > 0$ and $s\\in\\CC$;\n\\item $\\zeta_{\\QQ(i)}(2s) \\ll (1 + |s|)^2$ for $\\Re s > 0$ and $|2s-1|>1$.\n\\end{itemize}\n\nFinally, to show \\ref{vor-c}, we start from the following identity, a consequence of \\eqref{RSdef}:\n\\[\\frac{1}{16}\\sum_{m,n\\in\\ZZ[i]\\setminus\\{0\\}}w\\left(\\frac{|m|^4|n|^2}{P^2}\\right)\\lambda_n(V_1)\\lambda_n(V_2)\n= \\frac{1}{2\\pi i}\\int_{(2)} L(s, V_1 \\times V_2) \\widehat{w}(s) P^{2s} \\,\\dd s.\\]\nWe shift the contour to $\\Re s = -1$; the contribution of the possible poles (on the line $\\Re s = 1$) is recorded on the right-hand side of \\eqref{vor-formula}. In the remaining integral we apply the functional equation \\eqref{func-eq} and change variables $s \\mapsto 1-s$ getting\n$$\\frac{1}{2\\pi i}\\int_{(2)} L(s, V_1 \\times V_2) P^{2-4s} \\frac{16^s\\Gamma(s,\\vec{\\nu},\\vec{p})}{16^{1-s}\\Gamma(1-s,\\vec{\\nu},\\vec{p})}\\widehat{w}(1-s) P^{2s}\\,\\dd s.$$\nMoving this term   to the other side, we obtain the desired formula \\eqref{vor-formula}, first for $(\\nu_1, p_1) \\neq \\pm (\\nu_2, p_2)$, but then by analytic continuation everywhere.\nThis completes the proof of \\ref{vor-c}.\n\\end{proof}\n\n\\section{Pre-trace formula and amplification}\n\n\\subsection{Amplified pre-trace formula}\\label{sec31}\nIn this subsection, we prove an amplified pre-trace formula based on the theory of Eisenstein series and the spectral decomposition of $L^2(\\Gamma\\backslash G)$ (see \\S\\ref{Eisenstein-subsec}). This is a familiar identity between spectral and geometric data, and its full force will be needed in the proof of Theorem~\\ref{thm3}\\ref{thm3-b}; in fact, as an even more general version, we shall use a double pre-trace formula (see \\S\\ref{sec:double-pre-trace-formula}) in two variables. In many situations, however, all the spectral terms are nonnegative, in which case the pre-trace formula is simply used as an inequality.  This is the case  for the proof of Theorems~\\ref{thm1},~\\ref{thm2}~and~\\ref{thm3}\\ref{thm3-a}. An amplified pre-trace inequality can be derived with a less heavy machine. In the next subsection, we make a point of methodology by deriving these inequalities more directly, in a special case that is sufficient for us.\n\nLet $A$ be a bounded operator on $L^2(\\Gamma\\backslash G)$ preserving the subspace $C_0^{\\infty}(\\Gamma\\backslash G)$ of smooth functions with all rapidly decreasing derivatives. Assume that for the basis forms $\\phi_{\\ell,q}^{V}$, indexed as in \\eqref{compactform} by $V$ occurring in $L^2(\\Gamma\\backslash G)$ (cuspidal or Eisenstein) and $\\ell,q\\in\\ZZ$ satisfying $\\ell\\geq\\max(|p_V|,|q|)$, there are constants $c_{\\ell,q}^{V}(A)\\in\\CC$ such that\n\\begin{equation}\\label{newconstants}\n\\langle A\\psi,\\phi_{\\ell,q}^{V}\\rangle=c_{\\ell,q}^{V}(A)\\langle \\psi,\\phi_{\\ell,q}^{V}\\rangle,\\qquad\n\\psi\\in C^{\\infty}_0(\\Gamma\\backslash G).\n\\end{equation}\nThen \\eqref{eq:spectral-decomposition-plancherel} yields, for every $\\psi\\in C^{\\infty}_0(\\Gamma\\backslash G)$,\n\\begin{equation}\\label{Apsi-psi}\n\\langle A\\psi,\\psi\\rangle=\\frac{\\langle A\\psi,1\\rangle\\langle 1,\\psi\\rangle}{\\vol(\\Gamma\\backslash G)}+\\sum_{\\ell\\geq 0}\\int_{[\\ell]}\\sum_{|q|\\leq\\ell}c_{\\ell,q}^{V}(A)|\\langle\\psi,\\phi_{\\ell,q}^{V}\\rangle|^2\\,\\dd V.\n\\end{equation}\n\nFor $f\\in C_0(G)$ a rapidly decaying continuous function on $G$, and $\\psi\\in L^2(\\Gamma\\backslash G)$, we may consider the function $R(f)\\psi\\in L^2(\\Gamma\\backslash G)$ defined by\n\\begin{align*}\n(R(f)\\psi)(g)&:=\\int_Gf(h)\\psi(gh)\\,\\dd h=\\int_Gf(g^{-1}h)\\psi(h)\\,\\dd h\\\\\n&=\\int_{\\Gamma\\backslash G}k_f(g,h)\\psi(h)\\,\\dd h,\\qquad k_f(g,h):=\\sum_{\\gamma\\in\\Gamma}f(g^{-1}\\gamma h).\n\\end{align*}\nThus $R(f)$ is a bounded integral operator on $L^2(\\Gamma\\backslash G)$ with kernel $k_f$. It is clear that $R(f)$ preserves $C_0^{\\infty}(\\Gamma\\backslash G)$, and its adjoint equals $R(f)^\\ast=R(f^\\ast)$ with\n\\[f^\\ast(g):=\\ov{f(g^{-1})},\\qquad g\\in G.\\]\nFurther, for a finitely supported sequence of complex coefficients $x=(x_n)_{n\\in\\ZZ[i]\\setminus\\{0\\}}$, let $R_{\\fin}(x)$ be the operator on $L^2(\\Gamma\\backslash G)$ given by\n\\begin{equation}\\label{eq:def-amplifier}\nR_{\\fin}(x):=\\sum_{n\\in\\ZZ[i]\\setminus\\{0\\}}x_nT_n.\n\\end{equation}\nThe adjoint of this operator equals $R_{\\fin}(x)^\\ast=R_{\\fin}(\\ov{x})$.\n\nLet us now fix an integer $\\ell\\geq 1$. Let $f\\in\\mcH(\\tau_{\\ell})_{\\infty}$ be such that $f=f^\\ast$, and let $x=(x_n)$ be as above such that $x=\\ov{x}$. Further, let $V$ be a non-identity (cuspidal or Eisenstein) automorphic representation of arbitrary type $(\\nu_V,p_V)$ occurring in $L^2(\\Gamma\\backslash G)$, and let $\\ell',q\\in\\ZZ$ be such that $\\ell'\\geq\\max(|p_V|,|q|)$. For $V$ cuspidal, \\eqref{eq:opfromtr} and \\eqref{eq:trpif} show that\n\\begin{equation}\\label{joint-eigenfunctions}\n\\begin{aligned}\nR(f)\\phi_{\\ell',q}^{V}&=\\delta_{\\ell'=\\ell}\\frac{\\widehat{f}(V)}{2\\ell+1}\\phi_{\\ell',q}^{V}, & & & & & \\widehat{f}(V)&:=\\widehat{f}(\\nu_V,p_V);\\\\\nR_{\\fin}(x)\\phi_{\\ell',q}^{V}&=\\widehat{x}(V)\\phi_{\\ell',q}^{V}, & & & & & \\widehat{x}(V)&:=\\sum_{n\\in\\ZZ[i]\\setminus\\{0\\}}x_n\\lambda_n(V).\n\\end{aligned}\n\\end{equation}\nFor $V$ Eisenstein, these equations are still valid with the obvious extension of $R(f)$ and $R_{\\fin}(x)$ to functions in $C^\\infty(\\Gamma\\backslash G)$ of moderate growth, as follows from \\eqref{Eisenstein-Hecke-eigen} and the discussion between \\eqref{eq:spectral-decomposition} and \\eqref{eq:spectral-decomposition-EGM}. Therefore, following the usual argument that $R(f)$ and $R_{\\fin}(x)$ are self-adjoint, we obtain that $A:=R(f)R_{\\fin}(x)$ satisfies \\eqref{newconstants} with\n\\[ c_{\\ell',q}^{V}(A)=\\delta_{\\ell'=\\ell}\\frac{\\widehat{f}(V)\\widehat{x}(V)}{2\\ell+1}.\\]\nHence \\eqref{Apsi-psi} holds with these coefficients and $\\ell$-summation replaced by $\\ell'$-summation. We note that the coefficients decay rapidly in $\\nu$ by Theorem~\\ref{thm:pws}. Moreover, $A(1)=R(f)(1)$ vanishes by $f=f\\star\\overline{\\chi_{\\ell}}$ and the orthogonality of characters (recalling that $\\ell\\geq 1$).\n\nApplying \\eqref{Apsi-psi} and recalling our observation below \\eqref{Hecke-def} about $n\\gamma^{-1}$ as $\\gamma\\in\\Gamma\\setminus\\Gamma_n$, we obtain for every $\\psi\\in C_0^{\\infty}(\\Gamma\\backslash G)$ that\n\\begin{align*}\n\\int_{[\\ell]}\\sum_{|q|\\leq\\ell}c_{\\ell,q}^{V}(A)|\\langle\\psi,\\phi_{\\ell,q}^{V}\\rangle|^2\\,\\dd V\n&=\\iint_{(\\Gamma\\backslash G)^2}k_f(g,h) \\sum_{n\\in\\ZZ[i]\\setminus\\{0\\}}x_n T_n\\psi(h)\\overline{\\psi(g)}\\,\\dd g\\dd h\\\\\n&=\\iint_{(\\Gamma\\backslash G)^2}\\sum_{n\\in\\ZZ[i]\\setminus\\{0\\}}\\frac{x_n}{|n|}\\sum_{\\gamma\\in\\Gamma_n}f(g^{-1}\\tilde{\\gamma}h)\\overline{\\psi(g)}\\psi(h)\\,\\dd g\\dd h,\n\\end{align*}\nwhere $\\tilde\\gamma$ abbreviates $\\gamma\/\\sqrt{\\det\\gamma}$. Letting $\\psi$ range through smooth, nonnegative, $L^{1}$-normalized functions supported in increasingly small open neighborhoods of a fixed point $\\Gamma g\\in\\Gamma\\backslash G$, and taking limits using the rapid decay of $c_{\\ell,q}^{V}(A)$, we obtain the desired \\emph{amplified pre-trace formula}\n\\begin{equation}\\label{APTF}\n\\int_{[\\ell]}\\frac{\\widehat{f}(V)\\widehat{x}(V)}{2\\ell+1}\\sum_{|q|\\leq\\ell}|\\phi_{\\ell,q}^{V}(g)|^2\\,\\dd V=\\sum_{n\\in\\ZZ[i]\\setminus\\{0\\}}\\frac{x_n}{|n|}\\sum_{\\gamma\\in\\Gamma_n}f(g^{-1}\\tilde{\\gamma}g).\n\\end{equation}\n\nThe pre-trace formula \\eqref{APTF} isolates forms $\\phi_{\\ell,q}^{V}$ with a specific value of $\\ell$ (thus, forms in the \nchosen constituent $V^{\\ell}$ in the decomposition \\eqref{decomp} for various $V$'s), a starting point for a proof of Theorem~\\ref{thm1}. To further isolate eigenforms in the specific constituent $V^{\\ell,q}$ (for a fixed $|q|\\leqslant\\ell$), starting from our earlier $f\\in\\mcH(\\tau_{\\ell})_{\\infty}$ satisfying $f=f^\\ast$, we define a smooth function $f_q\\in C_0(G)$ by\n\\begin{equation}\\label{fqg-average-2}\nf_q(g):=\\frac1{2\\pi}\\int_0^{2\\pi}f\\big(g\\diag(e^{i\\varrho},e^{-i\\varrho})\\big)\\,e^{2qi\\varrho}\\,\\dd \\varrho =\\frac1{2\\pi}\\int_0^{2\\pi}f\\big(\\diag(e^{i\\varrho},e^{-i\\varrho})g\\big)\\,e^{2qi\\varrho}\\,\\dd \\varrho.\n\\end{equation}\nWe note that $f_q=f_q^\\ast$, but $f_q$ need not lie in $\\mcH(\\tau_{\\ell})_{\\infty}$. By the orthogonality of characters on $\\RR\/\\ZZ$, we have\n\\begin{equation}\\label{projectq}\nR(f_q)=R(f)\\Pi_q=\\Pi_q R(f),\n\\end{equation}\nwhere $\\Pi_q$ is the projection onto the closed subspace consisting of $\\psi\\in L^2(\\Gamma \\backslash G)$ such that $\\psi(g\\diag(e^{i\\varrho},e^{-i\\varrho}))=e^{2qi\\varrho}\\psi(g)$. In particular, $R(f_q)$ is a bounded, self-adjoint operator, which preserves $C_0^{\\infty}(\\Gamma\\backslash G)$. Moreover, by \\eqref{joint-eigenfunctions} and the surrounding discussion,\n\\[R(f_q)\\phi_{\\ell',q'}^{V}= \\delta_{(\\ell',q')=(\\ell,q)}\\frac{\\widehat{f}(V)}{2\\ell+1}\\phi_{\\ell',q'}^{V}\\]\nholds for $V$ cuspidal, and also for $V$ Eisenstein with the obvious extension of $R(f_q)$ to functions in $C^\\infty(\\Gamma\\backslash G)$ of moderate growth. Thus, applying as above \\eqref{Apsi-psi} with $A=R(f_q)R_{\\fin}(x)$, we obtain the following amplified pre-trace formula for individual forms:\n\\begin{equation}\\label{APTF-single-form}\n\\int_{[\\ell]}\\frac{\\widehat{f}(V)\\widehat{x}(V)}{2\\ell+1}|\\phi_{\\ell,q}^{V}(g)|^2\\,\\dd V=\\sum_{n\\in\\ZZ[i]\\setminus\\{0\\}}\\frac{x_n}{|n|}\\sum_{\\gamma\\in\\Gamma_n}f_q(g^{-1}\\tilde{\\gamma}g).\n\\end{equation}\n\nWe proved \\eqref{APTF} and \\eqref{APTF-single-form} for every $f\\in\\mcH(\\tau_{\\ell})_{\\infty}$ and finitely supported $x=(x_n)$ under the assumption that $f=f^\\ast$ and $x=\\ov{x}$. In fact \\eqref{APTF} and \\eqref{APTF-single-form} hold without this assumption, because both sides are $\\CC$-linear in $f$ and $x$. Alternatively, one can modify the above proof to work without the self-adjointness assumption, starting with the analogue of \\eqref{joint-eigenfunctions} for $R(f^\\ast)\\phi_{\\ell',q}^{V}$ and $R_\\fin(\\ov{x})\\phi_{\\ell',q}^{V}$.\n\n\\subsection{Positivity and amplified pre-trace inequality}\\label{sec31b}\nIf the coefficients on the left hand side of \\eqref{APTF} and \\eqref{APTF-single-form} are nonnegative, then by dropping \ncertain terms, we obtain useful inequalities. In this subsection, we derive these inequalities in a streamlined way, drawing inspiration from \\cite[\\S 3]{BlomerHarcosMagaMilicevic2020}.\n\nLet $A$ be a positive operator operator on $L^2(\\Gamma\\backslash G)$, and let $\\mfB$ be a finite orthonormal system of eigenfunctions $\\phi$ of $A$ with (not necessarily distinct) eigenvalues $(c_{\\phi}(A))_{\\phi\\in\\mfB}$. Then, $A$ preserves the orthodecomposition\n\\[L^2(\\Gamma\\backslash G)=\\mathrm{Span}(\\mfB)\\oplus\\mathrm{Span}(\\mfB)^{\\perp},\\]\nand for any $\\psi\\in L^2(\\Gamma\\backslash G)$ the corresponding decomposition $\\psi=\\psi_1+\\psi_2$ with\n\\[\\psi_1:=\\sum_{\\phi\\in\\mfB}\\langle\\psi,\\phi\\rangle\\phi\\qquad \\text{and} \\qquad \\psi_2:=\\psi-\\psi_1\\]\ngives\n\\begin{equation}\\label{positivity}\n\\langle A\\psi,\\psi\\rangle=\\langle A\\psi_1,\\psi_1\\rangle+\\langle A\\psi_2,\\psi_2\\rangle\n\\geq\\langle A\\psi_1,\\psi_1\\rangle=\\sum_{\\phi\\in\\mfB}c_{\\phi}(A)|\\langle\\psi,\\phi\\rangle|^2.\n\\end{equation}\n\nWe will apply this positivity argument to the operators $A=R(f) R_{\\fin}(x)$ and $A=R(f_q) R_{\\fin}(x)$, where $f\\in\\mcH(\\tau_{\\ell})_{\\infty}$ and $x=(x_n)$ are as in the previous subsection. Positivity is achieved by making the operators $R(f)$ and $R_{\\fin}(x)$ individually positive, because Hecke operators commute with integral operators, and $\\Pi_q$ in \\eqref{projectq} is a positive operator commuting with $R(f)$. For the positivity $R(f)$, it suffices that\n\\begin{equation}\\label{positivity-assumption}\nf=u\\star u\\qquad\\text{for some $u\\in\\mcH(\\tau_{\\ell})_{\\infty}$ satisfying $u=u^\\ast$}.\n\\end{equation}\nFor the positivity of $R_{\\fin}(x)$, it suffices that\n\\begin{equation}\n\\label{gen-amplifier}\n\\begin{gathered}\nR_{\\fin}(x)=\\Bigl(\\sum_{l\\in P}y_lT_l\\Bigr) \\star \\Bigl(\\sum_{m\\in P}\\ov{y_m}T_m\\Bigr) +\n\\Bigl(\\sum_{l\\in P}z_lT_{l^2}\\Bigr) \\star \\Bigl(\\sum_{m\\in P}\\ov{z_m}T_{m^2}\\Bigr),\\\\[4pt]\nx_n:=\\sum_{\\substack{l,m\\in P\\\\ (d)|(l,m)\\\\ lm\/d^2=n}}y_l\\overline{y_m}+\n\\sum_{\\substack{l,m\\in P\\\\ (d)|(l^2,m^2)\\\\ l^2m^2\/d^2=n}}z_l\\overline{z_m},\n\\end{gathered}\n\\end{equation}\nwhere $(y_l)_{l\\in P}$ and $(z_l)_{l\\in P}$ are arbitrary complex coefficients supported on a finite set\n$P\\subset\\ZZ[i]\\setminus\\{0\\}$. Here we used that each Hecke operator $T_n$ is self-adjoint.\n\nNow, let $V$ be a cuspidal automorphic representation that occurs in $L^2(\\Gamma\\backslash G)$ and contains $\\tau_{\\ell}$-type vectors. Let $\\mfB=\\{ \\phi_q : |q| \\leq \\ell\\}$ be an orthonormal basis of $V^\\ell$, with $\\phi_q\\in V^{\\ell,q}$. As in the previous subsection, we evaluate the left hand side of \\eqref{positivity} geometrically, and then apply a limit  in $\\psi$ to both sides. This way we obtain the following \\emph{amplified pre-trace inequalities} in place of \\eqref{APTF} and \\eqref{APTF-single-form}:\n\\begin{equation}\\label{APTI}\n\\frac{\\widehat{f}(V)\\widehat{x}(V)}{2\\ell+1}\\sum_{\\phi\\in\\mfB}|\\phi(g)|^2\\leq\\sum_{n\\in\\ZZ[i]\\setminus\\{0\\}}\\frac{x_n}{|n|}\\sum_{\\gamma\\in\\Gamma_n}f(g^{-1}\\tilde{\\gamma}g)\n\\end{equation}\nand\n\\begin{equation}\\label{APTI-single-form}\n\\frac{\\widehat{f}(V)\\widehat{x}(V)}{2\\ell+1}|\\phi_q(g)|^2\\leq\\sum_{n\\in\\ZZ[i]\\setminus\\{0\\}}\\frac{x_n}{|n|}\\sum_{\\gamma\\in\\Gamma_n}f_q(g^{-1}\\tilde{\\gamma}g).\n\\end{equation}\n\n\\subsection{Test functions and amplifier}\nThe main idea of the amplified pre-trace inequality \\eqref{APTI} is that it can provide a good upper bound for $\\sum_{\\phi\\in\\mfB}|\\phi(g)|^2$ as long as the test function $f\\in\\mcH(\\tau_{\\ell})_{\\infty}$ and the amplifier $x=(x_n)$ in \\S\\ref{sec31b} are chosen so that $\\widehat{f}(V)$ and $\\widehat{x}(V)$ are sizeable while the right-hand side is not too large. In this subsection, we make these choices.\n\nAs in Theorems~\\ref{thm1},~\\ref{thm2}~and~\\ref{thm3}, let $\\ell\\geq 1$ be an integer, $I\\subset i\\RR$ and $\\Omega\\subset G$ be compact sets. Let $V\\subset L^2(\\Gamma\\backslash G)$ be a cuspidal automorphic representation with minimal $K$-type $\\tau_{\\ell}$ and spectral parameter $\\nu_V\\in I$. Let us introduce the spectral weights\n\\begin{equation}\\label{eq:def-gaussian-spectral-weight}\nh(\\nu,p):=\\begin{cases} e^{(p^2-\\ell^2+\\nu^2)\/2},\\qquad &\\text{$\\nu\\in\\CC$, \\quad $p\\in \\frac1{2}\\ZZ$, \\quad $|p|\\leq\\ell$,}\\\\ 0,&\\text{$\\nu\\in\\CC$, \\quad $p\\in \\frac1{2}\\ZZ$, \\quad $|p|>\\ell$.}\\end{cases}\n\\end{equation}\nAccording to Theorems~\\ref{thm:tr-plancherel}~and~\\ref{thm:pws}, the inverse $\\tau_\\ell$-spherical transform $f:=\\widecheck{h}$ given by \\eqref{eq:inverse-tauell-transform} belongs to $\\mcH(\\tau_{\\ell})_{\\infty}$, and it satisfies $\\widehat{f}=h$. Moreover, if we set $u:=\\widecheck{v}$ with\n\\[v(\\nu,p):=\\begin{cases} (2\\ell+1)^{1\/2}e^{(p^2-\\ell^2+\\nu^2)\/4},\\qquad &\\text{$\\nu\\in\\CC$, \\quad $p\\in \\frac1{2}\\ZZ$, \\quad $|p|\\leq\\ell$,}\\\\ 0,&\\text{$\\nu\\in\\CC$, \\quad $p\\in \\frac1{2}\\ZZ$, \\quad $|p|>\\ell$,}\\end{cases}\\]\nthen $u\\in\\mcH(\\tau_{\\ell})_{\\infty}$, $u=u^\\ast$ by \\eqref{eq:spherical-function-symmetry} and \\eqref{eq:inverse-tauell-transform}, and $\\widehat{f}=\\widehat{u}^2\/(2\\ell+1)=\\widehat{u\\star u}$. This shows that \\eqref{positivity-assumption} is satisfied. Hence $R(f)$ is the kind of positive operator considered in \\S\\ref{sec31b}, and\nby \\eqref{joint-eigenfunctions} we have\n\\begin{equation}\\label{lower-bd-arch}\n\\widehat{f}(V)=h(\\nu_V,\\ell)\\gg_I 1.\n\\end{equation}\nWith the notation \\eqref{eq:def-tilde-f-s-p}, we have\n\\[\n\\widetilde{f}(s,p)=\\sqrt{2\\pi}(p^2+1-s^2)e^{(p^2-\\ell^2-s^2)\/2},\n\\]\nwhence by \\eqref{eq:inverse-spherical-transform-estimate}, \\eqref{eq:inverse-tauell-transform}, and the trivial bound\n$\\bigl|\\varphi_{\\nu,p}^{\\ell}(g^{-1})\\bigr|\\leq 2\\ell+1$, we have\n\\begin{equation}\\label{better-be-bounded}\nf(g) \\ll \\ell^2e^{-\\log^2\\|g\\|}.\n\\end{equation}\nWe shall also use the following supplement, a consequence of \\eqref{eq:spherical-function-symmetry}\nand \\eqref{eq:inverse-tauell-transform}:\n\\begin{equation}\\label{in-the-bulk}\nf(g)\\ll \\ell\\sup_{\\nu\\in i\\RR}\\,\\bigl|\\varphi_{\\nu,\\ell}^{\\ell}(g)\\bigr|+\\ell^{-50}.\n\\end{equation}\n\nWe now choose our amplifier, which we do as in \\cite[\\S5]{BlomerHarcosMilicevic2016}. Let $L\\geq 7$ be a parameter, to be chosen at the very end of the proof of Theorems~\\ref{thm1},~\\ref{thm2}~and~\\ref{thm3}, and set\n\\begin{gather*}\nP(L):=\\left\\{\\text{$l\\in\\ZZ[i]$ prime : $0<\\arg(l)<\\tfrac{\\pi}4$ and $L\\leq |l|^2\\leq 2L$}\\right\\};\\\\\ny_l:=\\sgn(\\lambda_{l}(V)),\\qquad z_l:=\\sgn(\\lambda_{l^2}(V)),\\qquad l\\in P(L).\n\\end{gather*}\nIt follows from the thesis of Erd\\H os~\\cite{Erdos} that $P(L)\\neq\\emptyset$, while\nin \\eqref{eq:def-amplifier} and \\eqref{gen-amplifier} we have\n\\begin{equation}\n\\label{ampl-expand}\nx_n=\\begin{cases}\n\\sum\\nolimits_{l\\in P(L)}(y_l^2+z_l{}^2)\\ll L\/\\log L,&n=1;\\\\\n(1+\\delta_{l_1\\neq l_2})y_{l_1}y_{l_2}+\\delta_{l_1=l_2}z_{l_1}z_{l_2}\\ll 1,&\\text{$n=l_1l_2$ for some $l_1,l_2\\in P(L)$};\\\\\n(1+\\delta_{l_1\\neq l_2})z_{l_1}z_{l_2}\\ll 1,&\n\\text{$n=l_1^2l_2^2$ for some $l_1,l_2\\in P(L)$};\\\\\n0,&\\text{otherwise}.\n\\end{cases}\n\\end{equation}\nThis formula is the analogue of \\cite[(9.16)]{BlomerHarcosMagaMilicevic2020}, except that we forgot to insert the factors $1+\\delta_{l_1\\neq l_2}$ there.\nIn particular, by the inequality $|\\lambda_{l}(V)|+|\\lambda_{l^2}(V)|>1\/2$ that follows from \\eqref{hecke-mult}, we have\n\\begin{equation}\n\\label{lower-bd-non-arch}\n\\widehat{x}(V)=\\Bigl(\\sum_{l\\in P(L)}|\\lambda_{l}(V)|\\Bigr)^2+\n\\Bigl(\\sum_{l\\in P(L)}|\\lambda_{l^2}(V)|\\Bigr)^2\\gg \\frac{L^2}{\\log^2L}.\n\\end{equation}\n\nLet $\\mfB$ be an orthonormal basis of $V^\\ell$. Entering the lower bounds \\eqref{lower-bd-arch} and \\eqref{lower-bd-non-arch} into the amplified pre-trace inequality \\eqref{APTI}, we obtain\n\\begin{equation}\\label{pre-APTI-done}\n\\frac{L^{2-\\eps}}{\\ell}\\sum_{\\phi\\in\\mfB}|\\phi(g)|^2\\ll_{\\eps,I}\\sum_{n\\in\\ZZ[i]\\setminus\\{0\\}}\\frac{|x_n|}{|n|}\\sum_{\\gamma\\in\\Gamma_n}|f(g^{-1}\\tilde\\gamma g)|.\n\\end{equation}\nLet us assume that $g\\in\\Omega$. A straightforward counting combined with the divisor bound shows that\n\\begin{equation}\\label{straightforward}\n\\#\\left\\{\\gamma\\in\\Gamma_n:\\|g^{-1}\\tilde\\gamma g\\|\\leq R\\right\\}\\ll_{\\eps,\\Omega} R^{4+\\eps}|n|^{2+\\eps},\n\\end{equation}\nso that, splitting into dyadic ranges for $\\|g^{-1}\\tilde\\gamma g\\|$ and using \\eqref{better-be-bounded}, we obtain\n\\[ \\sum_{\\substack{\\gamma\\in\\Gamma_n\\\\\\log\\|g^{-1}\\tilde\\gamma g\\|>8\\sqrt{\\log\\ell}}}\n|f(g^{-1}\\tilde\\gamma g)|\n\\ll_{\\eps,\\Omega} \\ell^{-50}|n|^{2+\\eps}. \\]\nThus from \\eqref{in-the-bulk} and \\eqref{pre-APTI-done} we conclude that\n\\begin{equation}\\label{pre-APTI-done-2}\n\\begin{aligned}\n&\\sum_{\\phi\\in\\mfB}|\\phi(g)|^2\\ll_{\\eps,I,\\Omega}L^{-2+\\eps}\\ell^2\\sum_{\\substack{n\\in\\ZZ[i]\\setminus\\{0\\} \\\\ \\gamma\\in\\Gamma_n\\\\\\log\\|g^{-1}\\tilde\\gamma g\\|\\leq 8\\sqrt{\\log\\ell}}}\\frac{|x_n|}{|n|}\n\\sup_{\\nu\\in i\\RR}|\\varphi_{\\nu,\\ell}^{\\ell}(g^{-1}\\tilde\\gamma g)|+L^{2+\\eps}\\ell^{-48}.\n\\end{aligned}\n\\end{equation}\nThe bound \\eqref{pre-APTI-done-2} explicitly reduces the non-spherical sup-norm problem of estimating $\\sum_{\\phi\\in\\mfB}|\\phi(g)|^2$ via the amplification method to two ingredients:\n\\begin{itemize}\n\\item estimates on $\\varphi_{\\nu,\\ell}^{\\ell}(g^{-1}\\tilde\\gamma g)$ for $g^{-1}\\tilde\\gamma g\\in G$ of moderate size;\n\\item counting $\\gamma\\in\\Gamma_n$ according to the size of $\\varphi_{\\nu,\\ell}^{\\ell}(g^{-1}\\tilde\\gamma g)$.\n\\end{itemize}\n\nWe now also derive a version of \\eqref{pre-APTI-done-2} adapted to estimating a single form $|\\phi_q(g)|^2$ for some $|q|\\leq\\ell$. With the specific $f\\in\\mcH(\\tau_{\\ell})_{\\infty}$ provided by \\eqref{eq:inverse-tauell-transform} and \\eqref{eq:def-gaussian-spectral-weight}, we obtain by averaging as in \\eqref{fqg-average-2} the test function\n\\[ f_q(g):=\\frac1{(2\\ell+1)\\pi^2}\\sum_{|p|\\leq\\ell}\\int_{0}^{\\infty}e^{(p^2-\\ell^2-t^2)\/2}\\,\\varphi_{it,p}^{\\ell,q}(g^{-1})\\,(t^2+p^2)\\,\\dd t, \\]\nwhere\n\\[\\varphi_{\\nu,p}^{\\ell,q}(g):=\\frac1{2\\pi}\\int_0^{2\\pi}\n\\varphi_{\\nu,p}^{\\ell}\\bigl(g\\diag(e^{i\\varrho},e^{-i\\varrho})\\bigr)\\,e^{-2qi\\varrho}\\,\\dd \\varrho.\\]\nIn particular, this definition generalizes \\eqref{spherical-averaged}, and\nby \\eqref{eq:spherical-function-symmetry} we have the symmetry\n\\begin{equation}\\label{eq:averaged-spherical-function-symmetry}\n\\varphi_{\\nu,p}^{\\ell,-q}(g)=\\ov{\\varphi_{-\\ov{\\nu},p}^{\\ell,q}(g)}=\\varphi_{\\nu,p}^{\\ell,q}(g^{-1}).\n\\end{equation}\nThe analogues of \\eqref{better-be-bounded}--\\eqref{in-the-bulk} clearly hold for the $\\RR\/\\ZZ$-average $f_q$, hence by \\eqref{APTI-single-form} the following analogue of \\eqref{pre-APTI-done-2} holds as well:\n\\begin{equation}\n\\label{pre-APTI-done-2-single-form}\n\\begin{aligned}\n&|\\phi_q(g)|^2\\ll_{\\eps,I,\\Omega} L^{-2+\\eps}\\ell^2\\sum_{\\substack{n\\in\\ZZ[i]\\setminus\\{0\\} \\\\ \\gamma\\in\\Gamma_n\\\\\\log\\|g^{-1}\\tilde\\gamma g\\|\\leq 8\\sqrt{\\log\\ell}}}\\frac{|x_n|}{|n|}\n\\sup_{\\nu\\in i\\RR}|\\varphi_{\\nu,\\ell}^{\\ell,q}(g^{-1}\\tilde\\gamma g)|+L^{2+\\eps}\\ell^{-48}.\n\\end{aligned}\n\\end{equation}\n\n\\subsection{A double pre-trace formula and a fourth moment}\\label{sec:double-pre-trace-formula}\nLet us fix two integers $\\ell,q\\in\\ZZ$ with $\\ell\\geq\\max(1,|q|)$. Let $n\\in\\ZZ[i]\\setminus\\{0\\}$ and $g\\in G$. By \\eqref{APTF-single-form} and the remarks below it, for any $f\\in\\mcH(\\tau_{\\ell})_{\\infty}$ we have\n\\[\\int_{[\\ell]}\\widehat{f}(V)\\lambda_n(V)|\\phi_{\\ell,q}^{V}(g)|^2\\,\\dd V=\\frac{2\\ell+1}{|n|}\\sum_{\\gamma\\in\\Gamma_n}f_q(g^{-1}\\tilde{\\gamma}g).\\]\nIt is straightforward to adapt the proof of this formula to yield the following two-variable version.\nLet $n_1,n_2\\in\\ZZ[i]\\setminus\\{0\\}$ and $g_1,g_2\\in G$. Then for any $f\\in\\mcH(\\tau_{\\ell}) \\hat{\\otimes} \\mcH(\\tau_{\\ell})$  we have\n\\begin{equation}\\label{two-var}\n\\begin{split}\n&\\int_{[\\ell]}\\int_{[\\ell]} \\widehat{f}(V_1, V_2)\\lambda_{n_1}(V_1)\\lambda_{n_2}(V_2) |\\phi^{V_1}_{\\ell,q}(g_1)|^2|\\phi^{V_2}_{\\ell,q}(g_2)|^2 \\,\\dd V_1 \\,\\dd V_2\\\\\n&=\\frac{(2\\ell+1)^2}{|n_1n_2|}\\sum_{\\gamma_1 \\in \\Gamma_{n_1}}\\sum_{\\gamma_2 \\in \\Gamma_{n_2}}\nf_q(g_1^{-1}\\tilde{\\gamma}_1 g_1, g_2^{-1}\\tilde{\\gamma}_2 g_2),\n\\end{split}\n\\end{equation}\nwhere $\\widehat{f}(V_1, V_2)$ is given by \\eqref{doubletransform} when $V_j$ is of type $(\\nu_j,p_j)\\in i\\RR\\times\\ZZ$, and\n\\[f_{q}(g_1, g_2):=\\frac1{(2\\pi)^2}\\int_0^{2\\pi}\\int_0^{2\\pi}  f\\big(g_1\\diag(e^{i\\varrho_1},e^{-i\\varrho_1}), g_2\\diag(e^{i\\varrho_2},e^{-i\\varrho_2})\\big) \\,e^{2q i(\\varrho_1 + \\varrho_2)}\\,\\dd \\varrho_1 \\,\\dd \\varrho_2.\\]\nIn \\eqref{two-var}, we can restrict to pairs $(V_1,V_2)$ satisfying $\\lambda_i(V_j)=1$ by introducing an averaging over $\\{n_1,in_1\\}\\times\\{n_2,in_2\\}$:\n\\begin{equation}\\label{two-var-PGL}\n\\begin{split}\n&\\int_{[\\ell]'}\\int_{[\\ell]'} \\widehat{f}(V_1, V_2)\\lambda_{n_1}(V_1)\\lambda_{n_2}(V_2) |\\phi^{V_1}_{\\ell,q}(g_1)|^2|\\phi^{V_2}_{\\ell,q}(g_2)|^2 \\,\\dd V_1 \\,\\dd V_2\\\\\n&=\\frac{(2\\ell+1)^2}{4|n_1n_2|}\\sum_{\\gamma_1\\in\\Gamma_{n_1}\\cup\\Gamma_{in_1}}\\sum_{\\gamma_2\\in\\Gamma_{n_2}\\cup\\Gamma_{in_2}}\nf_q(g_1^{-1}\\tilde{\\gamma}_1 g_1, g_2^{-1}\\tilde{\\gamma}_2 g_2).\n\\end{split}\n\\end{equation}\nThe prime symbol in $[\\ell]'$ indicates that we sum-integrate over automorphic representations with a lift to $\\PGL_2(\\ZZ[i])\\backslash\\PGL_2(\\CC)$, so that the results of \\S\\ref{RSSection} and \\S\\ref{sec28} are applicable.\n\nNow we consider, for any $n\\in\\ZZ[i]\\setminus\\{0\\}$, the spectral weights\n\\[H(V_1,V_2;n):=h(\\nu_1,p_1)h(\\nu_2,p_2)W_\\ell\\left(\\frac{|n|}{\\ell},\\vec{\\nu},\\vec{p}\\right),\\]\nwhere $h$ is as in \\eqref{eq:def-gaussian-spectral-weight} and $W_\\ell$ is as in Lemma~\\ref{lemma-vor}. Combining \nthe Hilbert space isomorphism\n\\[\\mcH(\\tau_{\\ell}) \\hat{\\otimes} \\mcH(\\tau_{\\ell})\n\\longleftrightarrow L^2(\\Gtemp(\\tau_{\\ell})\\times \\Gtemp(\\tau_{\\ell}))\\]\ninduced by Theorem~\\ref{thm:tr-plancherel} with Theorem~\\ref{thm:pws2} and parts \\ref{vor-a}--\\ref{vor-b} of Lemma~\\ref{lemma-vor}, we see that the function $(g_1,g_2)\\mapsto\\widecheck{H}(g_1,g_2;n)$ given by \\eqref{inversedoubletransform} belongs to $\\mcH(\\tau_{\\ell}) \\hat{\\otimes} \\mcH(\\tau_{\\ell})$, and its double $\\tau_\\ell$-spherical transform equals $H(V_1,V_2;n)$. Therefore, applying \\eqref{two-var-PGL} for $n_1=n_2=n$ and $g_1=g_2=g$, and then summing up over $n$, we arrive at\n\\begin{equation}\\label{eval}\n\\begin{split}\n&\\sum_{n\\in\\ZZ[i]\\setminus\\{0\\}}\\int_{[\\ell]'}\\int_{[\\ell]'}\nH(V_1,V_2;n)\\lambda_n(V_1)\\lambda_n(V_2)|\\phi^{V_1}_{\\ell,q}(g)|^2|\\phi^{V_2}_{\\ell,q}(g)|^2\\,\\dd V_1\\,\\dd V_2\\\\\n&=\\sum_{n\\in\\ZZ[i]\\setminus\\{0\\}}\\frac{(2\\ell+1)^2}{4|n|^2}\\sum_{\\gamma_1,\\gamma_2\\in\\Gamma_n\\cup\\Gamma_{in}}\n\\widecheck{H}_q(g^{-1}\\tilde{\\gamma}_1 g, g^{-1}\\tilde{\\gamma}_2 g;n).\n\\end{split}\n\\end{equation}\nBy Lemma~\\ref{lemma-vor}\\ref{vor-c}, the left-hand side of \\eqref{eval} equals\n\\[\\frac{\\pi}{4}\\ell^2\\sum_{\\substack{\\text{$V$ cuspidal} \\\\T_i(V)=1,\\ |p_V|\\leq \\ell}}\nh(\\nu_V, p_V)^2 \\,L(1,\\mathrm{ad^2} V)\\,|\\phi^{V}_{\\ell,q}(g)|^4\\ +\\ \\text{Eis},\\]\nwhere the term $\\text{Eis}$ is the contribution of Eisenstein representations:\n\\begin{equation}\\label{eval-cusp}\n\\begin{split}\n\\text{Eis}=\\ell^2\\sum_{\\epsilon,\\eta\\in\\{\\pm 1\\}}\\sum_{\\substack{p\\in 4\\ZZ\\\\|p|\\leq\\ell}}\\int_{(0)}\\int_{(0)}\n&\\ell^{\\eta(\\nu_1-\\epsilon \\nu_2)}\\,h(\\nu_1,\\epsilon p) h(\\nu_2,p)\\,\\mcL_{\\eta}((\\nu_1,\\epsilon p),(\\nu_2,p))\\\\\n&|\\phi^{E(\\nu_1,\\epsilon p)}_{\\ell,q}(g)|^2|\\phi^{E(\\nu_2,p)}_{\\ell,q}(g)|^2\n\\,\\frac{\\dd\\nu_1}{\\pi i}\\,\\frac{\\dd\\nu_2}{\\pi i}.\n\\end{split}\n\\end{equation}\nWe make a change of variable $(\\nu_1,\\nu_2,p)\\mapsto(\\eta\\nu_1,\\eta\\epsilon\\nu_2,\\eta\\epsilon p)$. By invariance, we can replace the resulting pairs $(\\eta\\nu_1,\\eta p)$ and $(\\eta\\epsilon\\nu_2,\\eta\\epsilon p)$ by $(\\nu_1,p)$ and $(\\nu_2,p)$, respectively. In this way we see that\n\\[\\begin{split}\n\\text{Eis}=4\\ell^2\\sum_{\\substack{p\\in 4\\ZZ\\\\|p|\\leq\\ell}}\\int_{(0)}\\int_{(0)}\n&\\ell^{\\nu_1-\\nu_2}\\,h(\\nu_1,p) h(\\nu_2,p)\\,\\mcL_{\\eta}((\\nu_1,p),(\\nu_2,p))\\\\\n&|\\phi^{E(\\nu_1,p)}_{\\ell,q}(g)|^2|\\phi^{E(\\nu_2,p)}_{\\ell,q}(g)|^2\n\\,\\frac{\\dd\\nu_1}{\\pi i}\\,\\frac{\\dd\\nu_2}{\\pi i}.\n\\end{split}\\]\nBy Lemma~\\ref{eis-pos}, we conclude that $\\text{Eis}\\geq 0$. In particular, the right-hand side of \\eqref{eval} is real, and it provides an upper bound for the contribution of each cuspidal $V$ in \\eqref{eval-cusp}:\n\\[h(\\nu_V, p_V)^2 \\,L(1,\\mathrm{ad^2} V)\\,|\\phi^{V}_{\\ell,q}(g)|^4\n\\ll\\sum_{n\\in\\ZZ[i]\\setminus\\{0\\}}\\frac{1}{|n|^2}\\sum_{\\gamma_1,\\gamma_2\\in\\Gamma_n\\cup\\Gamma_{in}}\n\\widecheck{H}_q(g^{-1}\\tilde{\\gamma}_1 g, g^{-1}\\tilde{\\gamma}_2 g;n).\\]\nHere we can restrict the $n$-sum to $|n|\\leq\\ell^{1+\\eps}$ at the cost of an error of $\\OO_\\eps(\\ell^{-50})$. Indeed, the contribution of $|n|>\\ell^{1+\\eps}$ on the two sides of \\eqref{eval} are equal, and this contribution is $\\OO_\\eps(\\ell^{-50})$ thanks to the bound $H(V_1, V_2;n)\\ll_A(|n|\/\\ell)^{-A}$ for any $A > 0$ that follows from Lemma~\\ref{lemma-vor}\\ref{vor-b}.\n\nAs before, let $I\\subset i\\RR$ and $\\Omega\\subset G$ be compact subsets. We fix a cuspidal automorphic representation\n$V\\subset L^2(\\Gamma\\backslash G)$ with $\\nu_V\\in I$, $p_V=\\ell$, $\\lambda_i(V)=1$, and we pick a cusp form $\\phi_q\\in V^{\\ell,q}$ with ${\\|\\phi_q\\|}_2=1$. We shall also assume that $g\\in\\Omega$. By \\eqref{RS-res} and our findings above,\n\\begin{equation}\\label{almostdone}\n|\\phi_q(g)|^4\\ll_{\\eps,I}\\ell^\\eps\\sum_{\\substack{n\\in\\ZZ[i]\\setminus\\{0\\}\\\\|n|\\leq\\ell^{1+\\eps}}}\\frac{1}{|n|^2}\\sum_{\\gamma_1,\\gamma_2\\in\\Gamma_n\\cup\\Gamma_{in}}\\widecheck{H}_q(g^{-1}\\tilde{\\gamma}_1 g, g^{-1}\\tilde{\\gamma}_2 g;n)+\\ell^{-50}.\n\\end{equation}\nWe estimate $\\widecheck{H}$ (hence also $\\widecheck{H}_q$) in terms of Cartan coordinates using the two-dimensional analogue of \\eqref{eq:inverse-spherical-transform-estimate}:\n\\[\\begin{split}\n\\big|\\widecheck{H}(k_1a_{h_1}k_2,k_3a_{h_2}k_4;n)\\big|\\leq\n& \\sum_{|p_1|,|p_2|\\leq\\ell}\\ \\ \\iint\\limits_{\\substack{s_1>h_1\\\\s_2>h_2}}\\,\n\\Bigg|\\ \\iint\\limits_{\\substack{t_1\\in\\RR\\\\t_2\\in\\RR}}\nW_\\ell\\left(\\frac{|n|}{\\ell},(it_1, it_2),(p_1,p_2)\\right)\\\\\ne^{-\\ell^2+(p_1^2+p_2^2)\/2}\\,& e^{-(t_1^2 + t_2^2)\/2}\\,e^{-it_1s_1-it_2s_2}\\,(t_1^2+p_1^2)(t_2^2+p_2^2)\n\\,\\dd t_1\\dd t_2\\,\\Bigg|\\,\\dd s_1\\dd s_2.\n\\end{split}\\]\nThis estimate holds for $k_j\\in K$ and $h_j>1$. We restrict the sum to $|p_1|=|p_2|=\\ell$ at the cost of an error of $\\OO_\\eps(\\ell^{-80})$. Then we utilize Lemma~\\ref{lemma-vor}\\ref{vor-a} to further restrict the sum to $p_1=p_2=\\ell$ at the cost of extending the outer double integral to $|s_1|,|s_2|>\\ell$. We arrive at\n\\[\\begin{split}\n\\widecheck{H}(k_1a_{h_1}k_2,k_3a_{h_2}k_4;n)\\ll\n&\\iint\\limits_{\\substack{|s_1|>h_1\\\\|s_2|>h_2}}\\,\n\\Bigg|\\ \\iint\\limits_{\\substack{t_1\\in\\RR\\\\t_2\\in\\RR}}\nW_\\ell\\left(\\frac{|n|}{\\ell},(it_1, it_2),(\\ell,\\ell)\\right)\\\\\n& e^{-(t_1^2 + t_2^2)\/2}\\,e^{-it_1s_1-it_2s_2}\\,(t_1^2+\\ell^2)(t_2^2+\\ell^2)\n\\,\\dd t_1\\dd t_2\\,\\Bigg|\\,\\dd s_1\\dd s_2 + \\ell^{-80}.\n\\end{split}\\]\nWe consider the inner double integral,\nand assume without loss of generality $|s_1| \\geq |s_2|$, the other case being analogous. Shifting the $t_1$-contour downwards if $s_1 > 0$ and upwards if $s_1 < 0$, we conclude from Lemma~\\ref{lemma-vor}\\ref{vor-b} the bound\n\\[\\ll_{\\eps,B}\n\\ell^4 e^{-B|s_1|} \\left(\\frac{|n|}{\\ell}\\right)^{-B-\\eps} = \n\\ell^4 e^{-B\\max(|s_1|,|s_2|)} \\left(\\frac{|n|}{\\ell}\\right)^{-B-\\eps}\\]\nfor the inner double integral, and so\n\\[\\widecheck{H}(k_1a_{h_1}k_2,k_3a_{h_2}k_4;n) \n\\ll_{\\eps,B} \\ell^4 e^{-B\\max(h_1,h_2)} \\left(\\frac{|n|}{\\ell}\\right)^{-B-\\eps} + \\ell^{-80}\\]\nfor any $\\eps, B > 0$ and $h_1, h_2 > 1$. This estimate remains true for general $h_1, h_2 \\geq 0$, as can be seen by using \\eqref{eq:inverse-tauell-transform} instead of \\eqref{eq:inverse-spherical-transform-estimate} for the respective variable if one or both of $h_1, h_2$ are\nat most $1$. The same bound applies for $\\widecheck{H}_q$, hence in particular\n\\[\\widecheck{H}_q(g_1,g_2;n)\n\\ll_{\\eps,C}\\ell^{4+\\eps}\\left(\\frac{\\sqrt{\\ell\/|n|}}{\\|g_1\\|+\\|g_2\\|}\\right)^C + \\ell^{-80}\\]\nfor any $\\eps, C > 0$ and $g_1,g_2\\in G$. So we can refine \\eqref{almostdone} to\n\\[|\\phi_q(g)|^4\\ll_{\\eps,I,\\Omega}\\ell^\\eps\\sum_{\\substack{n\\in\\ZZ[i]\\setminus\\{0\\}\\\\|n|\\leq\\ell^{1+\\eps}}}\n\\sum_{\\substack{\\gamma_1,\\gamma_2\\in\\Gamma_n\\cup\\Gamma_{in}\\\\\\| g^{-1}\\tilde{\\gamma}_j g\\|\\leq\\ell^{\\eps}\\sqrt{\\ell\/|n|}}}\\widecheck{H}_q(g^{-1}\\tilde{\\gamma}_1 g, g^{-1}\\tilde{\\gamma}_2 g;n)+\\ell^{-50}.\\]\n\nIn the last sum, we estimate the terms more directly by \\eqref{inversedoubletransform}, \\eqref{eq:averaged-spherical-function-symmetry}, and Lemma~\\ref{lemma-vor}\\ref{vor-b}:\n\\[\\widecheck{H}_q(g^{-1}\\tilde{\\gamma}_1 g,g^{-1}\\tilde{\\gamma}_2 g;n)\n\\ll_\\eps\\ell^{2+\\eps} F(\\gamma_1)F(\\gamma_2)+\\ell^{-80},\\]\nwhere we abbreviated (suppressing $g$ and $q$ from the notation)\n\\[F(\\gamma):=\\sup_{\\nu\\in i\\RR}|\\varphi_{\\nu,\\ell}^{\\ell,q}(g^{-1}\\tilde{\\gamma}g)|,\\qquad\\gamma\\in\\GL_2(\\CC).\\]\nRecalling also \\eqref{straightforward}, we obtain an inequality of bilinear type:\n\\[|\\phi_q(g)|^4\\ll_{\\eps,I,\\Omega}\\ell^{2+\\eps}\\sum_{\\substack{n\\in\\ZZ[i]\\setminus\\{0\\}\\\\|n|\\leq\\ell^{1+\\eps}}}\n\\sum_{\\substack{\\gamma_1,\\gamma_2\\in\\Gamma_n\\cup\\Gamma_{in}\\\\\\|g^{-1}\\tilde{\\gamma}_j g\\|\\leq\\ell^{\\eps}\\sqrt{\\ell\/|n|}}}F(\\gamma_1)F(\\gamma_2)+\\ell^{-50}.\\]\nIntroducing the notation\n\\[S(n):=\\sum_{\\substack{\\gamma\\in\\Gamma_n\\\\\\| g^{-1}\\tilde{\\gamma} g\\|\\leq\\ell^{\\eps}\\sqrt{\\ell\/|n|}}}F(\\gamma),\\]\nwe arrive at\n\\begin{alignat*}{3}\n|\\phi_q(g)|^4\n&\\ll_{\\eps,I,\\Omega}\\ &&\\ell^{2+\\eps}\\sum_{\\substack{n\\in\\ZZ[i]\\setminus\\{0\\}\\\\|n|\\leq\\ell^{1+\\eps}}}\n\\bigl(S(n)+S(in)\\bigr)^2&&+\\ell^{-50}\\\\\n&\\ll &&\\ell^{2+\\eps}\\sum_{\\substack{n\\in\\ZZ[i]\\setminus\\{0\\}\\\\|n|\\leq\\ell^{1+\\eps}}}\n\\bigl(S(n)^2+S(in)^2\\bigr)&&+\\ell^{-50}.\n\\end{alignat*}\nOf course, the contributions of $S(n)^2$ and $S(in)^2$ are the same. In the end, we conclude\n\\begin{equation}\\label{conclude-KS}\n|\\phi_q(g)|^4\\ll_{\\eps,I,\\Omega}\\ell^{2+\\eps}\\sum_{\\substack{n\\in\\ZZ[i]\\setminus\\{0\\}\\\\|n|\\leq\\ell^{1+\\eps}}}\n\\sum_{\\substack{\\gamma_1,\\gamma_2\\in\\Gamma_n\\\\\\|g^{-1}\\tilde{\\gamma}_j g\\|\\leq\\ell^{\\eps}\\sqrt{\\ell\/|n|}}}\n\\prod_{j=1}^2\\sup_{\\nu\\in i\\RR}|\\varphi_{\\nu,\\ell}^{\\ell,q}(g^{-1}\\tilde{\\gamma}_j g)|+\\ell^{-50},\n\\end{equation}\nwhich serves as an analogue of \\eqref{pre-APTI-done-2-single-form}.\n\n\\subsection{Reduction to Diophantine counting}\\label{reduction}\nIn this subsection, we input into the preliminary estimates \\eqref{pre-APTI-done-2}, \\eqref{pre-APTI-done-2-single-form} and \\eqref{conclude-KS} the results of Theorems~\\ref{thm4},~\\ref{thm6}~and~\\ref{thm5}, which provide the desired estimates on spherical trace functions. We shall assume (as we can) that $\\ell$ is sufficiently large in terms of $\\eps$.\n\nWe begin by explicating the estimate \\eqref{pre-APTI-done-2} using \\eqref{ampl-expand} and Theorem~\\ref{thm4}. For $\\mcL\\geq 1$ and $\\vec{\\delta}=(\\delta_1,\\delta_2)\\in\\RR^{2}_{>0}$, let\n\\[ D(L,\\mcL):=\\left\\{n\\in\\ZZ[i]:\n\\text{$\\mcL\\leq |n|^2\\leq 16\\mcL$, $n=1$ or $n=l_1l_2$ or $n=l_1^2l_2^2$ for some $l_1,l_2\\in P(L)$}\\right\\}, \\]\n\\begin{align*}\nM(g,L,\\mcL,\\vec{\\delta}):=\\sum_{n\\in D(L,\\mcL)}\\#\\bigg\\{\\gamma\\in\\Gamma_n:\\ \n&\\text{$g^{-1}\\tilde{\\gamma}g=k\\begin{pmatrix} z&u\\\\&z^{-1}\\end{pmatrix}k^{-1}$ for some $k\\in K$}\\\\\n&\\text{such that $|z|\\geq 1$, $\\min|z\\pm 1|\\leq\\delta_1$, $|u|\\leq\\delta_2$}\\bigg\\}.\n\\end{align*}\n\nTo each $\\gamma$ occurring in \\eqref{pre-APTI-done-2} we may associate a dyadic vector $\\vec{\\delta}=(\\delta_1,\\delta_2)$ (that is, $\\log_{2}\\delta_{j}\\in\\ZZ$) such that $1\/\\sqrt{\\ell}\\leq\\delta_j\\leq\\ell^\\eps$ and $\\delta_{j}$ are minimal such that $\\gamma$ is counted in the corresponding $M(g,L,\\mcL,\\vec{\\delta})$. Therefore, applying \\eqref{ampl-expand} and the estimates of Theorem~\\ref{thm4} in \\eqref{pre-APTI-done-2} leads to the following result.\n\n\\begin{lemma}\\label{APTI-done-lemma}\nLet $\\ell\\geq 1$ be an integer, $I\\subset i\\RR$ and $\\Omega\\subset G$ be compact sets. Let $V\\subset L^2(\\Gamma\\backslash G)$ be a cuspidal automorphic representation with minimal $K$-type $\\tau_{\\ell}$ and spectral parameter $\\nu_V\\in I$. Let $\\mfB$ be an orthonormal basis of $V^\\ell$, and let $g\\in\\Omega$. Then for any $L\\geq 7$ and $\\eps>0$ we have\n\\begin{align*}\n\\sum_{\\phi\\in\\mfB}|\\phi(g)|^2\n&\\ll_{\\eps,I,\\Omega}\\ell^{3+\\eps}L^{\\eps}\\sum_{\\substack{\\vec{\\delta}\\textnormal{ dyadic}\\\\1\/\\sqrt{\\ell}\\leq\\delta_j\\leq \\ell^{\\eps}}}\\min\\left(\\frac1{\\ell\\delta_1^2},\\frac1{\\sqrt{\\ell}\\delta_2}\\right)\\\\\n&\\qquad\\times\\left(\\frac{M(g,L,1,\\vec{\\delta})}{L}+\\frac{M(g,L,L^2,\\vec{\\delta})}{L^3}+\\frac{M(g,L,L^4,\\vec{\\delta})}{L^4}\\right)+L^{2+\\eps}\\ell^{-48}.\n\\end{align*}\n\\end{lemma}\n\nLemma~\\ref{APTI-done-lemma} is free of any choices of the test function, amplifier, and spherical trace function. It reduces the estimation of $\\sum_{\\phi\\in\\mfB}|\\phi(g)|^2$ to the Diophantine counting problem of estimating $M(g,L,\\mcL,\\vec{\\delta})$ uniformly in $L$, $\\mcL$, and $\\vec{\\delta}$.\n\nNow, we similarly explicate the estimate \\eqref{pre-APTI-done-2-single-form} using \\eqref{ampl-expand} and Theorems~\\ref{thm6}--\\ref{thm5}\\ref{thm5-a}. Recall the sets $\\mcD\\subset G$ and $\\mcS\\subset K\\subset\\mcN\\subset G$ introduced before Theorem~\\ref{thm5}. With $D(L,\\mcL)$ as above, we define for $|q|\\leq\\ell$, $\\mcL\\geq 1$, $\\delta>0$, and $\\vec{\\delta}=(\\delta_1,\\delta_2)\\in\\RR_{>0}^{2}$, the matrix counts\n\\begin{align*}\nM^{\\ast}_0(g,L,\\mcL,\\delta)&:=\\sum_{n\\in D(L,\\mcL)}\\#\\left\\{\\gamma\\in\\Gamma_n:\\dist(g^{-1}\\tilde\\gamma g,\\mcS)\\leq\\delta,\\,\\,\\frac{D(g^{-1}\\tilde\\gamma g)}{\\|g^{-1}\\tilde\\gamma g\\|^2}\\ll\\frac{\\log\\ell}{\\sqrt{\\ell}}\\right\\},\\\\\nM^\\ast(g,L,\\mcL,\\vec{\\delta})&:=\\sum_{n\\in D(L,\\mcL)}\\#\\left\\{\\gamma\\in\\Gamma_n:\\dist\\left(g^{-1}\\tilde\\gamma g,K\\right)\\leq\\delta_1,\\,\\,\\dist(g^{-1}\\tilde\\gamma g,\\mcD)\\leq\\delta_2\\right\\},\n\\end{align*}\nwith a sufficiently large implied constant in the definition of $M^{\\ast}_0(g,L,\\mcL,\\delta)$.\n\nFor $q=0$, we estimate the size of $\\varphi_{\\nu,\\ell}^{\\ell,q}(g^{-1}\\tilde\\gamma g)$ in \\eqref{pre-APTI-done-2-single-form} using Theorem~\\ref{thm5}\\ref{thm5-a}. Since there are at most $\\OO_{\\eps,\\Omega}(\\ell^\\eps|n|^{2+\\eps})$ elements $\\gamma\\in\\Gamma_n$ contributing to the right-hand side of \\eqref{pre-APTI-done-2-single-form}, the total contribution of those elements which fail to satisfy $D(g^{-1}\\tilde\\gamma g)\\ll\\|g^{-1}\\tilde\\gamma g\\|^2(\\log\\ell)\/\\sqrt{\\ell}$ with a sufficiently large implied constant may be absorbed into the existing $\\OO_{\\eps,I,\\Omega}(L^{2+\\eps}\\ell^{-48})$ error term. We may thus restrict to $\\gamma\\in\\Gamma_n$ satisfying these conditions. We associate to each remaining $\\gamma$ in \\eqref{pre-APTI-done-2-single-form} the smallest dyadic $1\/\\sqrt{\\ell}\\leq\\delta\\leq\\ell^{\\eps}$ such that $\\gamma$ is counted in the corresponding $M_0^{\\ast}(g,L,\\mcL,\\delta)$. For a general $|q|\\leq\\ell$, we associate to each $\\gamma$ in \\eqref{pre-APTI-done-2-single-form} the\nlexicographically smallest dyadic vector $\\vec{\\delta}=(\\delta_1,\\delta_2)$ such that $\\delta_j\\leq\\ell^\\eps$ and $\\delta_1^2\\delta_2\\geq 1\/\\sqrt{\\ell}$ and $\\gamma$ is counted in the corresponding $M_q(g,L,\\mcL,\\vec{\\delta})$. Applying \\eqref{ampl-expand} and the estimates of Theorems~\\ref{thm6}--\\ref{thm5}\\ref{thm5-a} in \\eqref{pre-APTI-done-2-single-form} leads to the following result.\n\n\\begin{lemma}\\label{APTI-done-lemma-single-form}\nLet $\\ell\\geq 1$ be an integer, $I\\subset i\\RR$ and $\\Omega\\subset G$ be compact sets. Let $V\\subset L^2(\\Gamma\\backslash G)$ be a cuspidal automorphic representation with minimal $K$-type $\\tau_{\\ell}$ and spectral parameter $\\nu_V\\in I$. Let $\\phi_q\\in V^{\\ell,q}$ such that ${\\|\\phi_q\\|}_2=1$ and let $g\\in\\Omega$. Then for any $L\\geq 7$ and $\\eps>0$ we have\n\\begin{align*}\n|\\phi_0(g)|^2\n&\\ll_{\\eps,I,\\Omega}\\ell^{2+\\eps}L^{\\eps}\\sum_{\\substack{\\delta\\textnormal{ dyadic}\\\\1\/\\sqrt{\\ell}\\leq\\delta\\leq\\ell^{\\eps}}}\\frac1{\\sqrt{\\ell}\\delta}\\\\\n&\\qquad\\times\\left(\\frac{M_0^{\\ast}(g,L,1,\\delta)}{L}+\\frac{M_0^{\\ast}(g,L,L^2,\\delta)}{L^3}+\\frac{M_0^{\\ast}(g,L,L^4,\\delta)}{L^4}\\right)+L^{2+\\eps}\\ell^{-48}.\n\\end{align*}\nMoreover, for $|q|\\leq\\ell$ we have\n\\begin{align*}\n|\\phi_q(g)|^2\n&\\ll_{\\eps,I,\\Omega}\\ell^{2+\\eps}L^{\\eps}\\sum_{\\substack{\\vec{\\delta}\\textnormal{ dyadic},\\,\\,\n\\delta_j\\leq\\ell^{\\eps}\\\\\\delta_1^2\\delta_2\\geq 1\/\\sqrt{\\ell}}}\n \\frac{1}{\\sqrt{\\ell} \\delta_1^2 \\delta_2}\\\\\n&\\qquad\\times\\left(\\frac{M^\\ast(g,L,1,\\vec{\\delta})}{L}+\\frac{M^\\ast(g,L,L^2,\\vec{\\delta})}{L^3}+\\frac{M^\\ast(g,L,L^4,\\vec{\\delta})}{L^4}\\right)+L^{2+\\eps}\\ell^{-48}.\n\\end{align*}\n\\end{lemma}\n\nSimilarly, we explicate \\eqref{conclude-KS} using Theorem~\\ref{thm5}\\ref{thm5-b}. Here we introduce the double matrix count\n\\begin{align*}\nQ(g,L,H_1, H_2):=\\sum_{L \\leq |n|  \\leq 2L} \\#\\Bigg\\{(\\gamma_1, \\gamma_2) \\in\\Gamma_n^2 :\\ \n&\\|g^{-1}\\tilde\\gamma_jg\\|\\leq \\sqrt{\\frac{H_j}{L}},\\\\\n&\\,\\dist(g^{-1}\\tilde\\gamma_jg,\\mcD)\\ll\\sqrt{\\frac{H_j\\log\\ell}{L\\ell}}\\Bigg\\},\n\\end{align*}\nwith a sufficiently large implied constant in the distance condition.\n\n\\begin{lemma}\\label{q=ell-case}\nLet $\\ell\\geq 1$ be an integer,  $I\\subset i\\RR$ and $\\Omega\\subset G$ be compact sets. Let $V\\subset L^2(\\Gamma\\backslash G)$ be a cuspidal automorphic representation with minimal $K$-type $\\tau_{\\ell}$ and spectral parameter $\\nu_V\\in I$. Suppose that $V$ lifts to an automorphic representation for $\\PGL_2(\\ZZ[i])\\backslash\\PGL_2(\\CC)$.\nLet $\\phi_{\\pm \\ell}\\in V^{\\ell,\\pm\\ell}$ such that ${\\|\\phi_{\\pm \\ell}\\|}_2=1$ and let $g\\in\\Omega$. Then for any $\\eps > 0$ we have\n\\[|\\phi_{\\pm \\ell}(g)|^4\\ll_{ \\eps,I,\\Omega}\\ell^{2+ \\eps} \\max_{1 \\leq L, H_1, H_2 \\leq \\ell^{1+\\eps}}  \\frac{Q(g,L,H_1, H_2) }{H_1H_2}+\\ell^{-50}.\\]\n\\end{lemma}\n\n\\section{Proof of Theorem~\\ref{thm4}}\nIn this section, we prove Theorem~\\ref{thm4}. It is clear from the definition \\eqref{eq:def-spherical-function}\nthat we can restrict to $k=1$ without loss of generality, and the first bound holds in the stronger form $|\\varphi_{\\nu,\\ell}^{\\ell}(g)|\\leq 2\\ell+1$. In particular, Theorem~\\ref{thm4} is trivial for $\\ell=1$, hence we shall assume (for notational simplicity) that $\\ell\\geq 2$. In addition, the exponential factor in \\eqref{spherical-def} has absolute value less than $\\|g\\|^2$ thanks to\n\\eqref{eq:kappaH} and the identity\n\\[|ad-bc|^2+|a\\bar b+c\\bar d|^2=(|a|^2+|c|^2)(|b|^2+|d|^2),\\]\nhence it suffices to prove that\n\\begin{equation}\\label{suffices}\n\\int_K  |\\psi_{\\ell}(\\kappa(k^{-1} g k))|\\, \\dd k \\ll_\\eps \\ell^\\eps\n\\min\\left(\\frac{\\|g\\|^4}{|z^2 - 1|^2\\ell}, \\frac{\\|g\\|}{|u|\\sqrt{\\ell}}\\right).\n\\end{equation}\nFinally, we shall use the obvious fact that\n\\begin{equation}\\label{eq:bounded_z_z(-1)_u}\n|u|,|z|,|z^{-1}|\\leq\\|g\\|.\n\\end{equation}\n\nWriting $k=k[\\phi,\\theta,\\psi]$ in Euler angles as in \\eqref{decomp-K}, and setting\n\\[\nx:= (z^2 - 1)\\cos\\theta + i e ^{-2i\\phi} uz \\sin\\theta,\n\\]\none computes\n\\[\nk[\\phi,   \\theta, \\psi]^{-1} g k[\\phi,  \\theta, \\psi] = \\left(\\begin{matrix}  (1 + x \\cos\\theta  )\/z & \\ast\\\\ \\  -ie^{2i\\psi}x \\sin \\theta  \/z& \\ast\\end{matrix}\\right).\n\\]\nOur goal is to estimate then\n\\begin{equation}\\label{eq:thm4_integral_to_bound}\n\\int_0^{\\pi} \\int_0^{\\pi\/2} \\int_{-\\pi}^{\\pi} \\left|\\psi_{\\ell}\\left(\\kappa\\left(\\begin{pmatrix}  (1 + x \\cos\\theta  )\/z & \\ast\\\\ \\  -ie^{2i\\psi}x \\sin \\theta  \/z& \\ast\\end{pmatrix}\\right)\\right)\\right| \\sin 2\\theta\\,\\dd \\psi\\,\\dd \\theta\\,\\dd \\phi.\n\\end{equation}\nWe introduce the notation $\\lambda:=\\sqrt{\\log\\ell}$.\n\n\\subsection{Small values of the integrand}\nFirst we identify a region where $|\\psi_\\ell|$ in the integral \\eqref{eq:thm4_integral_to_bound} is small. Assume that\n\\begin{equation}\\label{tiny}\n\\min\\bigl(\\tan\\theta,|x|\\sin\\theta\\bigr)>\\frac{4\\lambda}{\\sqrt{\\ell}}.\n\\end{equation}\nThen in\n\\[\n\\kappa\\left(\\begin{pmatrix}  (1 + x \\cos\\theta  )\/z & \\ast\\\\ \\  -ie^{2i\\psi}x \\sin \\theta  \/z& \\ast\\end{pmatrix}\\right)=\\begin{pmatrix} \\frac{1 + x \\cos\\theta}{\\sqrt{|1 + x \\cos\\theta|^2+|x\\sin\\theta|^2}}& \\ast \\\\ \\ast & \\ast \\end{pmatrix}\\in K\n\\]\nthe upper left entry has absolute square less than $1-\\lambda^2\/\\ell$, hence\n\\[\\left|\\psi_{\\ell}\\left(\\kappa\\left(\\begin{pmatrix}  (1 + x \\cos\\theta  )\/z & \\ast\\\\ \\  -ie^{2i\\psi}x \\sin \\theta  \/z& \\ast\\end{pmatrix}\\right)\\right)\\right| < \\left(1-\\frac{\\log \\ell}{\\ell}\\right)^{\\ell}<\\frac{1}{\\ell}.\\]\nIn view of \\eqref{eq:bounded_z_z(-1)_u}, this is admissible for \\eqref{suffices}. In the next subsection, we consider the case when \\eqref{tiny} fails.\n\n\\subsection{Large values of the integrand}\nAssume first that $\\tan\\theta\\leq 4\\lambda\/\\sqrt{\\ell}$. Then $\\theta\\leq 4\\lambda\/\\sqrt{\\ell}$, hence the corresponding contribution to \\eqref{eq:thm4_integral_to_bound} is $\\ll\\lambda^2\/\\ell$. This is admissible for \\eqref{suffices} in the light of \\eqref{eq:bounded_z_z(-1)_u}.\n\nNow assume that $|x|\\sin\\theta\\leq 4\\lambda\/\\sqrt{\\ell}$, and decompose the relevant integration domain for $\\theta$ as follows. For any $m,n\\in\\ZZ_{\\geq 0}$ and $\\phi\\in[0,\\pi]$, let\n\\[I(m,n,\\phi):=\\left\\{\\theta\\in\\left(0,\\frac{\\pi}{2}\\right)\\,:\\,\n|x|\\sin\\theta\\leq\\frac{4\\lambda}{\\sqrt{\\ell}},\\\n\\frac{1}{2}<2^m\\sin\\theta\\leq 1,\\ \\frac{1}{2}<2^n\\cos\\theta\\leq 1\\right\\}.\\]\nIf $\\theta\\notin I(m,n,\\phi)$ holds for every $0\\leq m,n\\leq 2\\log\\ell$, then $\\sin 2\\theta=2\\sin\\theta\\cos\\theta\\leq 1\/\\ell$, which is admissible for \\eqref{suffices}. Therefore, by \\eqref{eq:bounded_z_z(-1)_u} and \\eqref{eq:thm4_integral_to_bound}, it suffices to prove the bound\n\\begin{equation}\\label{eq:integral_large_psi}\n\\int_0^{\\pi} \\int_{I(m,n,\\phi)} \\sin 2\\theta\\,\\dd\\theta\\,\\dd\\phi \\ll \\min\\left(\\frac{\\lambda^2}{\\ell|z^2-1|^{2}},\\frac{\\lambda}{\\ell^{1\/2}|uz|}\\right)\n\\end{equation}\nfor every $0\\leq m,n\\leq 2\\log\\ell$. We shall assume that $\\min(m,n)=0$, for otherwise $I(m,n,\\phi)=\\emptyset$.\nWe record also that the Lebesgue measure of $I(m,n,\\phi)$ is $\\OO(2^{-m-n})$, because if $n=0$, then $\\sin\\theta\\asymp \\theta$, while if $m=0$, then $\\cos\\theta\\asymp \\pi\/2-\\theta$. Hence, for any $\\phi\\in[0,\\pi]$, we have\n\\[\\int_{I(m,n,\\phi)} \\sin 2\\theta\\,\\dd \\theta = \\int_{I(m,n,\\phi)} 2\\sin\\theta\\cos\\theta\\,\\dd \\theta \\ll 2^{-2m-2n}.\\]\n\nFirst consider the case when in $x=(z^2 - 1)\\cos\\theta + i e ^{-2i\\phi} uz \\sin\\theta$, whose absolute value does not exceed $2^{m+3}\\lambda\/\\sqrt{\\ell}$, neither of the two summands is large:\n\\[\n|z^2 - 1|2^{-n} \\leq 2^{m+6} \\frac{\\lambda}{\\sqrt{\\ell}},\\qquad |uz|2^{-m}\\leq 2^{m+6} \\frac{\\lambda}{\\sqrt{\\ell}}.\n\\]\nRecalling $\\min(m,n)=0$, the previous two displays imply  for any $\\phi\\in[0,\\pi]$ that\n\\[\\int_{I(m,n,\\phi)}\\sin 2\\theta\\,\\dd\\theta \\ll \\min\\left(\\frac{\\lambda^2}{\\ell|z^2-1|^{2}},\\frac{\\lambda}{\\ell^{1\/2}|uz|}\\right).\\]\nSo in this case \\eqref{eq:integral_large_psi} is clear.\n\nNow consider the case when in $x=(z^2 - 1)\\cos\\theta + i e ^{-2i\\phi} uz \\sin\\theta$,\nwhose absolute value does not exceed $2^{m+3}\\lambda\/\\sqrt{\\ell}$, the two summands are individually large:\n\\begin{equation}\\label{eq:triangle_large_sides}\n|z^2 - 1|2^{-n}> 2^{m+4} \\frac{\\lambda}{\\sqrt{\\ell}},\\qquad |uz|2^{-m}> 2^{m+4} \\frac{\\lambda}{\\sqrt{\\ell}}, \\qquad |z^2-1|2^{-n}\\asymp |uz|2^{-m}.\n\\end{equation}\nWe   claim that this localizes $\\phi$. Indeed, setting\n\\[\n2\\phi_0=\\arg(iuz)-\\arg(z^2-1),\n\\]\nwe see that\n\\[\n|z^2-1|\\cos \\theta + e^{2i(\\phi_0-\\phi)}|uz|\\sin \\theta \\ll 2^m\\frac{\\lambda}{\\sqrt{\\ell}},\n\\]\nand comparing the imaginary parts, we have that\n\\[\n\\sin(2\\phi-2\\phi_0)\\ll\\frac{2^{2m}\\lambda}{|uz|\\sqrt{\\ell}},\\quad\\text{and so}\\quad\n\\phi\\equiv\\phi_0+\\OO\\left(\\frac{2^{2m}\\lambda}{|uz|\\sqrt{\\ell}}\\right)\\!\\!\\!\\pmod{\\pi\/2}.\n\\]\nAlso, $\\theta$ is localized, since\n\\[\n|z^2-1|\\cos \\theta - |uz|\\sin \\theta \\ll 2^m\\frac{\\lambda}{\\sqrt{\\ell}},\n\\]\nand the first term here is monotone decreasing, the second one is monotone increasing in $\\theta$. We see that $\\theta$ is localized to an interval of length $\n\\OO(2^m\\lambda\/|uz|\\sqrt{\\ell})$ for $\\sin\\theta\\leq \\cos\\theta$ (in which case $n=0$), and to an interval of length $\\OO(\\lambda\/|z^2-1|\\sqrt{\\ell})$ for $\\cos\\theta\\leq\\sin\\theta$ (in which case $m=0$).\n\nWe estimate the left-hand side of \\eqref{eq:integral_large_psi} by exploiting the above localizations and all three parts of \\eqref{eq:triangle_large_sides}. If $\\sin\\theta\\leq\\cos\\theta$, then $n=0$ and $\\sin 2\\theta\\leq 2^{1-m}$, so altogether we obtain a contribution to \\eqref{eq:integral_large_psi} of size\n\\[\\ll 2^{-m} \\cdot \\frac{2^m \\lambda}{|uz|\\sqrt{\\ell}}\\cdot \\frac{2^{2m}\\lambda}{|uz|\\sqrt{\\ell}}\n\\ll \\min\\left(\\frac{\\lambda^2}{|z^2-1|^2\\ell},\\frac{\\lambda}{|uz|\\sqrt{\\ell}}\\right).\\]\nSimilarly, if $\\cos\\theta\\leq\\sin\\theta$, then $m=0$ and $\\sin 2\\theta\\leq 2^{1-n}$, so altogether we obtain a contribution to \\eqref{eq:integral_large_psi} of size\n\\[\\ll2^{-n} \\cdot\\frac{\\lambda}{|z^2-1|\\sqrt{\\ell}}\\cdot\\frac{\\lambda}{|uz|\\sqrt{\\ell}}\n\\ll \\min\\left(\\frac{\\lambda^2}{|z^2-1|^2\\ell},\\frac{\\lambda}{|uz|\\sqrt{\\ell}}\\right).\\]\n\nThe proof of Theorem~\\ref{thm4} is complete.\n\n\\section{Proof of Theorems~\\ref{thm6} and \\ref{thm5}}\nIn this section, we prove Theorems~\\ref{thm6}~and~\\ref{thm5}. We recall that the key player is the function\n\\begin{equation}\\label{varphi-def-proof}\n\\varphi_{\\nu,\\ell}^{\\ell,q}(g)  := \\frac{1}{2\\pi} \\int_{0}^{2\\pi} \\varphi_{\\nu,\\ell}^{\\ell}\\big(gk[0, 0, \\varrho]\\big) \\,e^{-2qi\\varrho} \\,\\dd\\varrho,\n\\end{equation}\nwhere\n\\[\\varphi_{\\nu,\\ell}^{\\ell}(g):=(2\\ell+1)\n\\int_K  \\psi_{\\ell}(\\kappa(k^{-1} g k)) \\,e^{(\\nu-1)\\rho(H(gk))}\\,\\dd k.\\]\nThe function $\\psi_\\ell:K\\to\\CC$ was defined in \\eqref{chi-ell}, but for calculational purposes we extend it now to $\\GL_2(\\CC)$:\n\\begin{equation}\\label{psidef}\\psi_{\\ell}\\left( \\begin{pmatrix} \\alpha  &\\beta \\\\ \\gamma & \\delta \\end{pmatrix} \\right) :=\n\\bar{\\alpha}^{2\\ell}, \\qquad \\left( \\begin{matrix} \\alpha  &\\beta \\\\ \\gamma & \\delta \\end{matrix} \\right) \\in\\GL_2(\\CC).\n\\end{equation}\n\n\\subsection{Preliminary computations}\\label{sec:preliminary-computations}\nWe write $g$ in Cartan form\n\\begin{equation}\\label{g}\ng=k[u_1,v_1,w_1]\\begin{pmatrix}r & \\\\ & r^{-1}\\end{pmatrix} k[u_2,v_2,w_2],\n\\end{equation}\nwhere $r\\geq 1$, and we allow $u_j,v_j,w_j\\in\\RR$ to be arbitrary for convenience.\nSpelling out the definitions, and using that the height in the Iwasawa decomposition is left $K$-invariant, we see that $\\varphi_{\\nu,\\ell}^{\\ell,q}(g)$ equals\n\\[\\begin{split}\n\\frac{d_{\\ell}}{4\\pi^3}\\int_{\\substack{0\\leq u\\leq \\pi \\\\ 0\\leq v\\leq \\pi\/2 \\\\ 0\\leq w\\leq 2\\pi \\\\ 0\\leq \\varrho\\leq 2\\pi}} & \\psi_{\\ell}\\left( k[-w,-v,-u]k[u_1,v_1,w_1] \\kappa\\left(\\begin{pmatrix}r&\\\\&r^{-1}\\end{pmatrix} k[u_2,v_2,w_2]k[0,0,\\varrho] k[u,v,w] \\right)\\right)\\\\\n& \\cdot e^{-2iq\\varrho} \\, e^{(\\nu-1)\\rho\\left(H\\left(\\left(\\begin{smallmatrix}r&\\\\&r^{-1}\\end{smallmatrix}\\right) k[u_2,v_2,w_2]k[0,0,\\varrho]k[u,v,w]\\right)\\right)} \\sin 2v \\,\\dd u\\,\\dd v\\,\\dd w\\,\\dd \\varrho.\n\\end{split}\\]\nWith a change of variables $k[u_2,v_2,w_2]k[0,0,\\rho] k[u,v,w] \\mapsto k[u,v,w]$ and dropping the normalized $w$-integration (which is legitimate since the conjugation by $k[0,0,w]$ does not alter the $\\psi_{\\ell}$-value, and the height in the Iwasawa decomposition is also unaffected by right-multiplication by $k[0,0,w]$), we arrive at\n\\[\\begin{split}\n\\frac{d_{\\ell}}{2\\pi^2}\\int_{\\substack{0\\leq u\\leq \\pi \\\\ 0\\leq v\\leq \\pi\/2 \\\\ 0\\leq \\varrho\\leq 2\\pi}} & \\psi_{\\ell}\\left( k[0,-v,-u] k[u_2,v_2,w_2]k[0,0,\\varrho] k[u_1,v_1,w_1] \\kappa\\left(\\begin{pmatrix}r&\\\\&r^{-1}\\end{pmatrix}k[u,v,0]\\right)\\right)\\\\\n& \\cdot e^{-2iq\\varrho} \\, e^{(\\nu-1)\\rho\\left(H\\left(\\left(\\begin{smallmatrix}r&\\\\&r^{-1}\\end{smallmatrix}\\right) k[u,v,0]\\right)\\right)} \\sin 2v   \\,\\dd u\\,\\dd v\\,\\dd\\varrho.\n\\end{split}\\]\nThe sum of absolute squares in the first column of $\\diag(r, r^{-1})k[u,v,0]$ equals\n\\[h(r,v):=r^2\\cos^2v+r^{-2}\\sin^2v,\\]\nhence recalling the definitions \\eqref{eq:kappaH} and \\eqref{psidef}, we can rewrite the integral as\n\\[\\begin{split}\n\\frac{d_{\\ell}}{2\\pi^2}\\int_{\\substack{0\\leq u\\leq \\pi \\\\ 0\\leq v\\leq \\pi\/2 \\\\ 0\\leq \\varrho\\leq 2\\pi}} & \\psi_{\\ell}\\left( k[0,-v,-u] k[u_2,v_2,w_2]k[0,0,\\varrho] k[u_1,v_1,w_1]\n\\begin{pmatrix}r&\\\\&r^{-1}\\end{pmatrix}k[u,v,0]\\right) \\\\\n& \\cdot e^{-2iq\\varrho} \\, h(r,v)^{\\nu-1-\\ell} \\sin 2v \\,\\dd u\\,\\dd v\\,\\dd\\varrho.\n\\end{split}\\]\nReplacing $\\varrho$ by $\\varrho-u_1-w_2$, the integral further simplifies to\n\\[\\begin{split}\n\\frac{d_{\\ell}e^{2iq(u_1+w_2)}}{2\\pi^2}\\int_{\\substack{0\\leq u\\leq \\pi \\\\ 0\\leq v\\leq \\pi\/2 \\\\ 0\\leq \\varrho\\leq 2\\pi}} & \\psi_{\\ell}\\left( \\left(\\begin{matrix} e^{-i\\varrho} I + e^{i\\varrho}J & \\ast \\\\ \\ast & \\ast \\end{matrix}\\right)\\right) e^{-2iq\\varrho} \\, h(r,v)^{\\nu-1-\\ell} \\sin 2v  \\,\\dd u\\,\\dd v\\,\\dd \\varrho,\n\\end{split}\\]\nwhere\n\\begin{align*}\nI:=&\\left(r^{-1}e^{-2iu-iw_1}\\sin v\\cos v_1+re^{iw_1}\\cos v\\sin v_1\\right)\n\\left(e^{2iu-iu_2}\\sin v\\cos v_2-e^{iu_2}\\cos v\\sin v_2\\right),\\\\\nJ:=&\\left(-r^{-1}e^{-2iu-iw_1}\\sin v\\sin v_1+re^{iw_1}\\cos v\\cos v_1\\right)\n\\left(e^{2iu-iu_2}\\sin v\\sin v_2+e^{iu_2}\\cos v\\cos v_2\\right).\\end{align*}\nEvaluating the $\\varrho$-integral, we obtain\n\\begin{equation}\\label{post-expansion}\n\\varphi_{\\nu,\\ell}^{\\ell,q}(g) = \\frac{d_{\\ell}e^{2iq(u_1+w_2)}}{\\pi}\\binom{2\\ell}{\\ell + q}\n\\int_{\\substack{0\\leq u\\leq \\pi \\\\ 0\\leq v\\leq \\pi\/2 } } \\frac{\\sin 2v}{h(r,v)^{\\ell+1-\\nu}}\n\\,\\bar{I}^{\\ell + q} \\bar{J}^{\\ell - q}\\,\\dd u\\,\\dd v.\n\\end{equation}\nTaking the complex conjugate of the right-hand side, and introducing the new variables $t:=r^{-1}\\tan v$ and $\\phi:=2u$, we get\n\\begin{align*}\n\\bigl|\\varphi_{\\nu,\\ell}^{\\ell,q}(g)\\bigr| = \\frac{d_{\\ell}}{\\pi}&\\binom{2\\ell}{\\ell+q}\n\\bigg|\\int_0^{\\infty}\\int_0^{2\\pi} \\frac{t}{(1 + (t\/r)^2)^{\\ell+1+\\nu}(1+(tr)^2)^{\\ell+1-\\nu}}\\\\\n&\\left(e^{-i\\phi-2iw_1}(t\/r)\\cos v_1+\\sin v_1\\right)^{\\ell+q}\n\\left(e^{i\\phi-2iu_2}(tr)\\cos v_2-\\sin v_2\\right)^{\\ell+q}\\\\\n&\\left(e^{-i\\phi-2iw_1}(t\/r)\\sin v_1-\\cos v_1\\right)^{\\ell-q}\n\\left(e^{i\\phi-2iu_2}(tr)\\sin v_2+\\cos v_2\\right)^{\\ell-q}\\,\\dd\\phi\\,\\dd t\\bigg|.\n\\end{align*}\nNow comes the last key step: in the inner $\\phi$-integral, we can remove the $r$'s. This is so because $e^{-i\\phi}$ must be chosen equally many times as $e^{i\\phi}$, and the $r$'s will cancel out in all terms surviving the integration. Another way to see the same thing is to shift the contour as in $\\phi\\mapsto\\phi+i\\log r$ where the boundary terms cancel out by $2\\pi$-periodicity. Either way, using also the opportunity to replace $\\phi\\mapsto\\phi+u_2-w_1$,\nand writing $\\Delta:=u_2+w_1$, we finally obtain\n\\begin{align*}\n\\bigl|\\varphi_{\\nu,\\ell}^{\\ell,q}(g)\\bigr| \\leq \\frac{d_{\\ell}}{\\pi} \\binom{2\\ell}{\\ell+q}\n\\int_{0}^{\\infty}&\\frac{t}{((1 + (t\/r)^2)(1+(tr)^2))^{\\ell+1}}\\\\\n\\times\\int_0^{2\\pi}\n&\\bigl|e^{i\\phi+i\\Delta}t\\cos v_1+\\sin v_1\\bigr|^{\\ell+q}\n\\bigl|e^{i\\phi -i\\Delta}t\\cos v_2-\\sin v_2\\bigr|^{\\ell+q}\\\\\n&\\bigl|e^{i\\phi+i\\Delta}t\\sin v_1-\\cos v_1\\bigr|^{\\ell-q}\n\\bigl|e^{i\\phi -i\\Delta}t\\sin v_2+\\cos v_2\\bigr|^{\\ell-q}\\,\\dd\\phi\\,\\dd t.\n\\end{align*}\n\nWe estimate the inner integrand using the following lemma, which is purely about inequalities. We state it formally so as to clearly separate issues. (In the case $q=\\pm\\ell$, all expressions raised to exponent 0 should simply be omitted.) As in the  previous section, we introduce the notation $\\lambda:=\\sqrt{\\log \\ell}$.\n\n\\begin{lemma}\\label{young}\nLet $\\ell,q\\in\\ZZ$ be such that $\\ell\\geq\\max(1,|q|)$. Let $X>0$ and $\\Lambda>0$.\n\\begin{enumerate}[label=(\\alph*)]\n\\item\\label{part-a}\nIf $A,B\\geq 0$ satisfy $A^2+B^2=X^2$, then\n\\begin{equation}\n\\label{ineq-two-terms}\n\\left(\\frac{2\\ell}{\\ell+q}\\right)^{(\\ell+q)\/2}\\left(\\frac{2\\ell}{\\ell-q}\\right)^{(\\ell-q)\/2}A^{\\ell+q}B^{\\ell-q}\\leq X^{2\\ell}.\n\\end{equation}\nMoreover, the left-hand side is $\\OO_\\Lambda(X^{2\\ell}\\ell^{-\\Lambda})$ unless\n\\begin{equation}\\label{cases-of-eq}\n\\begin{split}\nA^2&=\\frac{\\ell+q}{2\\ell}X^2+\\OO_{\\Lambda}\\left(X^2\\frac{\\lambda^2+\\lambda\\sqrt{\\ell-|q|}}{\\ell}\\right),\\\\\nB^2&=\\frac{\\ell-q}{2\\ell}X^2+\\OO_{\\Lambda}\\left(X^2\\frac{\\lambda^2+\\lambda\\sqrt{\\ell-|q|}}{\\ell}\\right).\n\\end{split}\n\\end{equation}\n\\item\\label{part-b}\nIf $A,B,C,D\\geq 0$ satisfy $A^2+B^2=C^2+D^2=X^2$, then\n\\[ \\binom{2\\ell}{\\ell+q}A^{\\ell+q}B^{\\ell-q}C^{\\ell+q}D^{\\ell-q} \\ll\\frac{X^{4\\ell}}{ 1+\\sqrt{\\ell-|q|}}. \\]\nMoreover, the left-hand side is $\\OO_\\Lambda(X^{4\\ell}\\ell^{-\\Lambda})$ unless \\eqref{cases-of-eq} and the analogous estimates for $C$, $D$ are satisfied.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet us first assume $|q| < \\ell$. We use Young's inequality\n\\[xy \\leq \\frac{x^a}{a}  + \\frac{y^b}{b} , \\qquad \\frac{1}{a} + \\frac{1}{b}=1,\\]\nto conclude with\n\\begin{equation}\\label{xy}\nx := \\left(\\sqrt{\\frac{2\\ell}{\\ell+q}}\\frac{A}{X}\\right)^{\\frac{\\ell+q}{\\ell}}, \\quad\ny := \\left(\\sqrt{\\frac{2\\ell}{\\ell-q}}\\frac{B}{X}\\right)^{\\frac{\\ell-q}{\\ell}}, \\quad\na := \\frac{2\\ell}{\\ell + q}, \\quad\nb := \\frac{2\\ell}{\\ell - q}\n\\end{equation}\nthat\n\\[\n\\left(\\sqrt{\\frac{2\\ell}{\\ell+q}}\\frac{A}{X}\\right)^{\\frac{\\ell+q}{\\ell}}\n\\left(\\sqrt{\\frac{2\\ell}{\\ell-q}}\\frac{B}{X}\\right)^{\\frac{\\ell-q}{\\ell}}\n\\leq \\frac{A^2+B^2}{X^2}=1.\n\\]\nThis is equivalent to \\eqref{ineq-two-terms}. We also conclude (still using the notation \\eqref{xy}) that the left-hand side of \\eqref{ineq-two-terms} is $\\OO_\\Lambda(X^{2\\ell}\\ell^{-\\Lambda})$ unless\n\\begin{equation}\\label{ineq}\nxy>1\/2,\\qquad xy=1+\\OO_\\Lambda(\\delta),\\qquad\\delta:=\\lambda^2\/\\ell.\n\\end{equation}\n\nLet us explore the consequences of \\eqref{ineq}. First, by $x^a\/a+y^b\/b=1$ we have\n\\[1\/3<x,y<3\/2.\\]\n Without loss of generality, $q\\geq 0$ (i.e. $a\\leq b$), and then $x^a<a\\leq 2$. Moreover,\n\\[b\\log x<(b\/a)\\log a<(b\/a)(a-1)=1,\\]\nhence also $-b\\log y<1+\\OO_\\Lambda(b\\delta)$. In particular, $y^b\\gg_\\Lambda 1$ whenever $b\\delta<1$. Now let us consider the function\n\\[F(t):=\\frac{x^a}{a}+\\frac{t^b}{b}-xt.\\]\nNote that $F(y)=1-xy$, and $F(y_0)=F'(y_0)=0$ for $y_0:=x^{a-1}$. Hence, using Lagrange's form for the remainder term in Taylor's theorem, we see that\n\\[\\delta\\gg_\\Lambda F(y)\\geq\\frac{(b-1)}{2}\\min(y_0^{b-2},y^{b-2})\\,(y-y_0)^2.\\]\nHere $y_0^{b-2}=x^{2-a}\\gg 1$. Now let us assume that $y^b>1$ or $b\\delta<1$. Then $y^b\\gg_\\Lambda 1$, whence $y-y_0\\ll_\\Lambda\\sqrt{\\delta\/b}$ by the previous display. From here and \\eqref{ineq} we get the following two approximations for $bxy$:\n\\begin{align*}\nbxy&=bxy_0+\\OO_\\Lambda(\\sqrt{b\\delta})=bx^a+\\OO_\\Lambda(\\sqrt{b\\delta}),\\\\\nbxy&=b+\\OO_\\Lambda(b\\delta)=(b-1)x^a+y^b+\\OO_\\Lambda(b\\delta).\n\\end{align*}\nComparing the right-hand sides, we conclude that\n\\begin{equation}\\label{eq:xayb}\nx^a-y^b\\ll_\\Lambda b\\delta+\\sqrt{b\\delta}.\n\\end{equation}\nIn the remaining case when $y^b\\leq 1$ and $b\\delta\\geq 1$, the inequality \\eqref{eq:xayb} holds automatically in the stronger form $|x^a-y^b|<2\\leq 2b\\delta$.\n\nWe proved that \\eqref{ineq} implies \\eqref{eq:xayb} in all ranges. For our specific set-up \\eqref{xy}, the inequality \\eqref{eq:xayb} says that\n\\[aA^2-bB^2\\ll_\\Lambda X^2(b\\delta+\\sqrt{b\\delta}),\\]\nand this is equivalent to \\eqref{cases-of-eq} in the light of $A^2+B^2=X^2$.\nThis shows \\ref{part-a} under the assumption $0\\leq q<\\ell$, but it is easily seen to continue to hold also for $q = \\ell$ in which case \\eqref{ineq} simply reads $A^2 = X^2 + \\OO_\\Lambda(X^2\\delta)$. The argument for $- \\ell \\leq q < 0$ is identical.\n\nTurning to \\ref{part-b}, we conclude from \\ref{part-a} that\n\\[ \\left(\\frac{2\\ell}{\\ell+q}\\right)^{\\ell+q}\\left(\\frac{2\\ell}{\\ell-q}\\right)^{\\ell-q}A^{\\ell+q}B^{\\ell-q}C^{\\ell+q}D^{\\ell-q}\\leq X^{4\\ell}. \\]\nOn the other hand, using Stirling's formula $n!\\sim(n\/e)^n\\sqrt{2\\pi n}$, we have for $|q|<\\ell$ that\n\\[ \\binom{2\\ell}{\\ell+q}\\asymp\\frac{(2\\ell)^{2\\ell}}{(\\ell+q)^{\\ell+q}(\\ell-q)^{\\ell-q}}\\sqrt{\\frac{2\\ell}{(\\ell+q)( \\ell-q)}}, \\]\nand so combining the two most recent displays we have the announced bound\n \\[ \\binom{2\\ell}{\\ell+q}A^{\\ell+q}B^{\\ell-q}C^{\\ell+q}D^{\\ell-q} \\ll\\frac{X^{4\\ell}}{ 1+\\sqrt{\\ell-|q|}}. \\]\nWe added artificially the $1+$ term in the denominator, so that the inequality also holds for the previously excluded case $|q| = \\ell$ in view of $AC, BD \\leq X^2$ (which follows directly  from $A^2 + C^2 = B^2 +D^2 = X^2$). The claim that the left-hand side is negligible unless \\eqref{cases-of-eq} holds for $(A,B)$ and $(C,D)$ is immediate from \\ref{part-a}.\n\\end{proof}\n\nWe now return to the double integral in the upper bound for $\\varphi_{\\nu,\\ell}^{\\ell,q}(g)$.\nWe estimate\nthe inner integral by writing the integrand as  $A^{\\ell + q}  B^{\\ell - q} C^{\\ell + q} D^{\\ell - q}$ in the obvious way and applying Lemma~\\ref{young}, where\n\\[A^2+B^2=C^2+D^2=X^2=1+t^2,\\]\nand\n\\[ A^2=\\frac{1+t^2}{2}+\\frac{t^2-1}{2}\\cos 2v_1+t\\sin 2v_1\\cos(\\phi+\\Delta), \\]\nwith analogous expressions for $B^2$, $C^2$, and $D^2$. Since\n\\[\\frac{(1 + (t\/r)^2)(1+(tr)^2)}{(1 + t^2)^2} = 1 + \\left(\\frac{r - r^{-1}}{t + t^{-1}}\\right)^2,\\]\nwe conclude that the contribution of the inner integral is $\\OO_\\Lambda(\\ell^{-\\Lambda})$ unless\n\\begin{equation}\\label{r}\n\\min(t,t^{-1})\\ll_\\Lambda\\frac{\\lambda}{(r-1)\\sqrt{\\ell}}.\n\\end{equation}\nFor $r=1$ we treat the right-hand side as infinity. We may then summarize our findings as follows.\n\n\\begin{lemma}\\label{lem1} Let $\\Lambda\\in\\NN$. Let $\\ell,q\\in\\ZZ$ be such that $\\ell\\geq\\max(1,|q|)$, and let $\\nu\\in i\\RR$. Assume that $g\\in\\SL_2(\\CC)$ is given by \\eqref{g}. Let us abbreviate $\\Delta:= u_2 + w_1$ and $\\lambda := \\sqrt{\\log \\ell}$. Let $\\mcM = \\mcM(v_1,v_2,\\Delta,r,\\Lambda)$ be the set of $(\\phi, t) \\in [0, 2\\pi] \\times [0, \\infty)$ satisfying \\eqref{r} as well as\n\\begin{equation}\\label{eq}\n\\begin{split}\n2t\\sin 2v_1\\cos(\\phi+\\Delta)&=(1-t^2)\\cos 2v_1+\\frac{q}{\\ell}(1+t^2)+\\OO_{\\Lambda}\\left((1+t^2)\\frac{\\lambda^2+ \\lambda\\sqrt{\\ell-|q|}}{\\ell} \\right),\\\\\n2t\\sin 2v_2\\cos(\\phi-\\Delta)&=(t^2-1)\\cos 2v_2-\\frac{q}{\\ell}(1+t^2)+\\OO_{\\Lambda}\\left((1+t^2)\\frac{\\lambda^2 + \\lambda\\sqrt{\\ell-|q|}}{\\ell} \\right),\\\\\n\\end{split}\n\\end{equation}\nwith a sufficiently large (but fixed) implied constant depending on $\\Lambda$. Then\n\\begin{equation}\\label{keyupperbound}\n\\varphi_{\\nu,\\ell}^{\\ell,q}(g)  \\ll_\\Lambda  \\frac{\\ell}{1 + \\sqrt{\\ell - |q|}}\n\\int_{\\mcM} \\frac{t}{(1+t^2)^2} \\, \\dd \\phi\\, \\dd t + \\ell^{-\\Lambda}.\n\\end{equation}\n\\end{lemma}\n\n\\subsection{Simplifying assumptions}\\label{simplifying-section}\nFor the proof of Theorems~\\ref{thm6}~and~\\ref{thm5}, we can and we shall assume that $|\\Delta|\\leq\\pi\/4$. Indeed, using the last relation in \\eqref{angleequiv} multiple times, we can choose the coordinates in \\eqref{g} so that this bound is satisfied. Moreover, we can replace $g$ by\n\\[g^{-1}=k\\left[\\frac{\\pi}{2}-w_2,v_2-\\frac{\\pi}{2},u_2+\\frac{\\pi}{2}\\right]\n\\begin{pmatrix}r & \\\\ & r^{-1}\\end{pmatrix}\nk\\left[w_1-\\frac{\\pi}{2},v_1-\\frac{\\pi}{2},\\frac{\\pi}{2}-u_1\\right]\\]\nif needed, because the quantities $\\Delta$, $\\|g\\|$, $D(g)$ do not change under this replacement, \n$\\bigl|\\varphi_{\\nu,\\ell}^{\\ell,q}(g)\\bigr|=\\bigl|\\varphi_{\\nu,\\ell}^{\\ell,q}(g^{-1})\\bigr|$ holds by \\eqref{eq:averaged-spherical-function-symmetry}, and\n\\[\\dist(g,\\mcH)=\\dist(g^{-1},\\mcH),\\qquad\\mcH\\in\\{K,\\mcD,\\mcS\\}\\]\nholds by \\eqref{distinvariance}.\n\nWe shall derive (most of) the bounds in Theorems~\\ref{thm6}~and~\\ref{thm5} from \\eqref{keyupperbound}. In Lemma~\\ref{lem1}, the pair $(\\Delta,r)$ does not change under the above discussed replacement $g\\mapsto g^{-1}$, while\nthe corresponding integration domains $\\mcM$ are related by\n\\[(\\phi,t)\\in\\mcM\\left(v_2-\\frac{\\pi}{2},v_1-\\frac{\\pi}{2},\\Delta,r,\\Lambda\\right)\\quad\\Longleftrightarrow\\quad\n(\\phi,t^{-1})\\in\\mcM(v_1,v_2,\\Delta,r,\\Lambda).\\]\nMoreover, the integrand in \\eqref{keyupperbound} is invariant under $t\\mapsto t^{-1}$, hence we can assume that the contribution of $t\\leq 1$ is not smaller than the contribution of $t>1$. So from now on we restrict $\\mcM$ in \\eqref{keyupperbound} to the corresponding subset of $[0, 2\\pi] \\times [0, 1]$. On this subset we have, by \\eqref{r},\n\\begin{equation}\\label{r1}\nt\\in[0,1]\\qquad\\text{and}\\qquad t \\ll_\\Lambda \\frac{\\lambda}{(r-1)\\sqrt{\\ell}}.\n\\end{equation}\n\n\\subsection{Proof of Theorem~\\ref{thm6}}\nThe bound \\eqref{thm6bound} is trivial for $\\ell\\ll_\\Lambda 1$, hence we shall assume that $\\ell$ is sufficiently large in terms of $\\Lambda$. With the notation\n\\[\\alpha := \\dist(g, K) \\asymp r-1\\qquad\\text{and}\\qquad\\beta := \\dist(g, \\mcD),\\]\nit follows from \\eqref{keyupperbound} and the previous subsection that it suffices to show\n\\begin{equation}\\label{beginThm6}\n\\frac{\\ell}{1 + \\sqrt{\\ell - |q|}}\\int_{\\mcM}  t \\, \\dd \\phi\\, \\dd t \\ll_{\\eps,\\Lambda}\n\\ell^{\\eps}\\min\\left(1,\\frac{\\| g \\|}{\\sqrt{\\ell}\\alpha^2\\beta}\\right),\n\\end{equation}\nwhere $\\mcM$ is now restricted by \\eqref{r1}. In fact our arguments below will show that $\\ell^\\eps$ can be replaced by $(\\log\\ell)^3$.\n\nWe start with the first bound of \\eqref{beginThm6}. With the notation\n\\[\\sigma:=\\lambda^2+\\lambda\\sqrt{\\ell-|q|},\\qquad\n\\mu:=\\frac{q}{\\ell} - \\cos 2v_1,\\qquad \\rho:=\\sin 2 v_1,\\]\nthe first equation in \\eqref{eq} becomes\n\\begin{equation}\\label{tquadratic}\n\\mu t^2 - 2t\\rho\\cos(\\phi+\\Delta) + \\frac{2q}{\\ell}-\\mu+\\OO_\\Lambda\\left(\\frac{\\sigma}{\\ell}\\right)=0.\n\\end{equation}\nWithout loss of generality, $\\mu\\neq 0$, and then we can view \\eqref{tquadratic} as a quadratic equation for $t$. Multiplying by $\\mu$ and completing the square, we obtain the alternative form\n\\begin{equation}\\label{compl}\n\\bigl(\\mu t - \\rho\\cos(\\phi + \\Delta)\\bigr)^2+\\bigl(\\rho\\sin(\\phi + \\Delta)\\bigr)^2\n=1-\\frac{q^2}{\\ell^2}+\\OO_\\Lambda\\left(\\frac{|\\mu|\\sigma}{\\ell}\\right).\n\\end{equation}\nIn particular, the discriminant of \\eqref{tquadratic} equals $4D(\\phi)+\\OO_\\Lambda(|\\mu|\\sigma\/\\ell)$, where\n\\begin{equation}\\label{disc-q}\nD(\\phi) := 1-\\frac{q^2}{\\ell^2}-\\bigl(\\rho\\sin(\\phi + \\Delta)\\bigr)^2\n\\end{equation}\nWe assume first that $|q|\/\\ell\\leq 5\/6$, and decompose $\\mcM$ into two parts $\\mcM^\\pm$ according as\n$|\\rho\\sin(\\phi+\\Delta)|$ exceeds $1\/2$ or not. On $\\mcM^+$, the equation \\eqref{tquadratic} localizes $\\phi$ within $\\ll_\\Lambda\\sigma\/(\\ell t)$ for each given $t\\in[0,1]$. On $\\mcM^-$, we have $D(\\phi)\\geq 1\/18$, hence the equation \\eqref{compl} localizes $t$ within $\\ll_\\Lambda\\sigma\/\\ell$ for each given $\\phi\\in[0,2\\pi]$. This shows that\n\\[\\int_\\mcM t\\,\\dd\\phi\\,\\dd t\\ll_\\Lambda\\int_0^1 t\\,\\frac{\\sigma}{\\ell t}\\,\\dd t+\\int_0^{2\\pi}\\frac{\\sigma}{\\ell}\\,\\dd\\phi\n\\ll\\frac{\\sigma}{\\ell},\\]\nhence the first bound of \\eqref{beginThm6} follows in stronger form. From now on we assume that $|q|\/\\ell>5\/6$. We decompose $\\mcM$ into two parts $\\mcM^\\pm$ according as $D(\\phi)$ is positive or not, and we make two initial observations. First, $\\mcM^+$ is clearly empty when $|q|=\\ell$. Second, $|\\mu|>1\/6$ holds for large $\\ell$, because \\eqref{tquadratic} coupled with $t\\in[0,1]$ yields\n\\[\\frac{2|q|}{\\ell}-|\\mu|-\\OO_\\Lambda\\left(\\frac{\\sigma}{\\ell}\\right)\\leq 2t|\\rho|\n\\leq 2\\sqrt{1 - \\left( \\frac{q}{\\ell} - \\mu\\right)^2}.\\]\nIn order to estimate the contribution of $\\mcM^+$ in \\eqref{beginThm6}, we decompose $\\mcM^+$\ninto pieces\n\\[\\mcM^{+}({\\mtD},\\eta):=\\left\\{(\\phi,t)\\in\\mcM^+:\n\\text{$D(\\phi) \\asymp {\\mtD}$ and $|\\cos(\\phi + \\Delta)| \\asymp \\eta$}\n\\right\\}.\\]\nIf $\\eta \\leq \\ell^{-10}$, we can estimate trivially, so there are only $\\OO(\\log \\ell)$ relevant values for $\\eta$. If $\\rho \\geq \\ell^{-10}$, then by the same argument there are only $\\OO(\\log \\ell)$ relevant values for ${\\mtD}$. If $\\rho < \\ell^{-10}$, then ${\\mtD} > \\ell^{-1}$ by $|q|<\\ell$, hence again\nthere are only $\\OO(\\log \\ell)$ relevant values for ${\\mtD}$. So in all cases it suffices to restrict to $\\OO((\\log \\ell)^2)$ pairs $({\\mtD},\\eta)$. Our current assumptions localize $\\sin(\\phi+\\Delta)$ within\n$\\ll\\sqrt{{\\mtD}}\/|\\rho|$, and hence $\\phi$ within $\\ll\\min(1,\\sqrt{\\mtD}\/|\\rho\\eta|)$, independently of $t$. On the other hand, given $\\phi$, the equation \\eqref{compl} localizes $t$ within $\\ll_\\Lambda\\min((\\sigma\/\\ell){\\mtD}^{-1\/2},\\sqrt{\\sigma\/\\ell})$.\nSuch $t$ are of size $\\ll_\\Lambda |\\rho\\eta| + \\sqrt{\\mtD} + \\sqrt{\\sigma\/\\ell}$, so that\n\\[\\int_{\\mcM^+({\\mtD},\\eta)} t\\,\\dd\\phi\\,\\dd t \\ll_\\Lambda \\left( |\\rho\\eta|+ \\sqrt{\\mtD} + \\sqrt{\\frac{\\sigma}{\\ell}}  \\right)\\min\\left( \\frac{\\sigma}{\\ell \\sqrt{\\mtD} }, \\sqrt{\\frac{\\sigma}{\\ell}}\\right)  \\min\\left(1, \\frac{\\sqrt{\\mtD} }{|\\rho \\eta|}\\right)\n\\ll\\frac{\\sigma}{\\ell}.\\]\nThis contribution is admissible for the first bound of \\eqref{beginThm6}.\nIt remains to estimate the contribution of $\\mcM^-$ in \\eqref{beginThm6}. On this set we have\n\\[0\\leq -D(\\phi)\\ll_\\Lambda\\frac{\\sigma}{\\ell}\\]\nby \\eqref{compl}. The argument is similar as for $\\mcM^+$, in fact simpler as we only need $\\OO(\\log \\ell)$ pieces $\\mcM^{-}(\\eta)$ defined by $|\\cos(\\phi+\\Delta)|\\asymp\\eta$. Initially we localize $\\phi$ within $\\ll\\min(1,|\\rho\\eta|^{-1}\\sqrt{\\sigma\/\\ell})$, independently of $t$. The equation \\eqref{compl} localizes $t$ within $\\ll_\\Lambda\\sqrt{\\sigma\/\\ell}$, and such $t$ are of size $\\ll_\\Lambda |\\rho\\eta| + \\sqrt{\\sigma\/\\ell}$. We obtain altogether\n\\[\\int_{\\mcM^-(\\eta)} t\\,\\dd\\phi\\,\\dd t \\ll_\\Lambda \\left(|\\rho \\eta| + \\sqrt{\\frac{\\sigma}{\\ell}}\\right)\n\\sqrt{\\frac{\\sigma}{\\ell}}\\min\\left(1, \\frac{1}{|\\rho\\eta|}\\sqrt{\\frac{\\sigma}{\\ell}}\\right) \\ll \\frac{\\sigma}{\\ell},\\]\nwhich is again admissible for the first bound of \\eqref{beginThm6}.\n\nWe now turn to the second bound of \\eqref{beginThm6}. We shall assume (as we can) that $\\mcM\\neq\\emptyset$ and $\\sqrt{\\ell}\\alpha^2\\beta>\\| g \\|$. We pick an arbitrary point $(\\phi,t)\\in\\mcM$. Combining \\eqref{eq} and \\eqref{r1}, we get\n\\[\\cos 2v_j = - \\text{sgn}(q) + \\OO(t)\\sin 2v_j + \\OO_\\Lambda\\left(t^2 +\\frac{\\lambda^2+ \\ell - |q|}{\\ell}\\right),\\]\nwhere for $q=0$ we can replace $\\sgn(q)$ by $1$. After squaring and solving for $\\sin 2v_j$, then feeding back the result into the previous display, we get\n\\[\\sin 2v_j=\\OO_\\Lambda\\left(t +\\frac{\\lambda+ \\sqrt{\\ell - |q|}}{\\sqrt{\\ell}}\\right),\\qquad\n\\cos 2v_j = - \\text{sgn}(q) + \\OO_\\Lambda\\left(t^2 +\\frac{\\lambda^2+ \\ell - |q|}{\\ell}\\right).\\]\nRecalling also \\eqref{g}, and using \\eqref{r1} again, we infer that\n\\[\\beta \\ll_\\Lambda\\|g\\|\\left(\\frac{\\lambda}{\\alpha\\sqrt{\\ell}} +\\frac{\\lambda+ \\sqrt{\\ell - |q|}}{\\sqrt{\\ell}}\\right).\\]\nHence we always have\n\\[1 \\ll_\\Lambda \\frac{\\|g\\|\\lambda}{\\alpha\\beta\\sqrt{\\ell}}\n\\qquad \\text{or} \\qquad 1 \\ll_\\Lambda \\|g\\|\\lambda\\frac{1+\\sqrt{\\ell - |q|}}{\\beta\\sqrt{\\ell}}.\\]\nIn either case, for any $c > 0$, the previous display combined with \\eqref{r1} yields that\n\\begin{align*}\n\\frac{\\ell}{1 + \\sqrt{\\ell - |q|}}\\int_{\\mcM}  t \\, \\dd \\phi\\, \\dd t\n&\\ll_{\\Lambda,c}\\frac{\\lambda^2}{(1+\\sqrt{\\ell - |q|})\\alpha^2}\n\\left(\\left(\\frac{\\|g\\|\\lambda}{\\alpha\\beta\\sqrt{\\ell}}\\right)^c+ \\|g \\|\\lambda\\frac{1+\\sqrt{\\ell - |q|}}{\\beta\\sqrt{\\ell}}\\right)\\\\[4pt]\n& \\ll_{\\eps,\\Lambda,c} \\ell^\\eps\\left(\\frac{\\| g \\|^c}{\\ell^{c\/2} \\alpha^{2 + c} \\beta^c}+\\frac{\\| g \\|}{\\sqrt{\\ell} \\alpha^2 \\beta}\\right).\n\\end{align*}\nChoosing $c = 2$, and recalling our initial assumption $\\sqrt{\\ell}\\alpha^2\\beta>\\| g \\|$,\nwe obtain the second bound of \\eqref{beginThm6} in stronger form.\n\nThe proof of Theorem~\\ref{thm6} is complete.\n\n\\subsection{Proof of Theorem~\\ref{thm5}\\ref{thm5-a}}\\label{thm5a-proof-sec}\nThe averaged spherical trace function $\\varphi_{\\nu,\\ell}^{\\ell, q}(g)$ exhibits starkly different behavior depending on the value of $-\\ell\\leq q\\leq\\ell$. Some of these features are already visible along $K=\\SU_2(\\CC)$. From \\eqref{varphi-def-proof} and \\eqref{post-expansion} we can see that, in the notation of \\eqref{decomp-K} and \\eqref{matrix-coeff},\n\\[\\varphi_{\\nu,\\ell}^{\\ell,q}(k[u,v,w])=\\Phi_{q,q}^{\\ell}(k[u,v,w])=\ne^{2\\pi i q(u+w)}(\\cos v)^{2q}P_{\\ell-q}^{(0,2q)}(\\cos 2v).\\]\nThe absolute value of the right-hand side exhibits a primary peak at $v\\in\\pi\\ZZ$ of size 1. For $q=\\pm\\ell$, this is followed by a sharp drop to $\\OO_N(\\ell^{-N})$ after a range of length about $\\ell^{-1\/2}$. For a generic $q$, the drop becomes soft through a highly oscillatory range of magnitude $\\ell^{-1\/2}$ (faster and more oscillatory for smaller $q$) and a secondary, Airy-type peak of size about $\\ell^{-1\/3}$ before the delayed sharp drop. For $q=0$, the secondary peak grows to a full peak of size 1 at $v\\in\\frac12\\pi+\\pi\\ZZ$ (corresponding to skew-diagonal matrices in $K$) and the sharp drop disappears. These varying features, which are illustrated in Figure~\\ref{Jacobi-figure}, become vastly more complicated off $K$, where the hard work in Theorems~\\ref{thm6} and \\ref{thm5} lies. Nevertheless, their traces are visible in the hard localization to $\\mcD$ (but none to $K$!) for $q=\\pm\\ell$ and the hard localization to $\\mcN$ with soft localization to $\\mcS\\subset K\\subset\\mcN$ for $q=0$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.24\\textwidth]{Jacobi1.pdf}\n\\includegraphics[width=0.24\\textwidth]{Jacobi2.pdf}\n\\includegraphics[width=0.24\\textwidth]{Jacobi3.pdf}\n\\includegraphics[width=0.24\\textwidth]{Jacobi4.pdf}\n\\caption{Plots of $(\\cos v)^{2q}P_{\\ell-q}^{(0,2q)}(\\cos 2v)$ for $0\\leq v\\leq\\pi$, $\\ell=120$, $q=120$, $q=100$, $q=20$, and $q=0$.}\n\\label{Jacobi-figure}\n\\end{figure}\n\nIn this subsection, we consider in more detail the case $q = 0$. Then \\eqref{keyupperbound} simplifies to\n\\begin{equation}\\label{keyupperbound:q=0}\n\\varphi_{\\nu,\\ell}^{\\ell,0}(g) \\ll_\\Lambda\\sqrt{\\ell}\n\\int_{\\mcM} \\frac{t}{(1+t^2)^2} \\, \\dd \\phi\\, \\dd t + \\ell^{-\\Lambda},\n\\end{equation}\nwhere by \\eqref{eq} and the last paragraph of \\S \\ref{sec:preliminary-computations}, the set $\\mcM$ can be described by the constraints given in \\eqref{r1} and\n\\begin{equation}\\label{eq0}\n\\begin{split}\n2t\\sin 2v_1\\cos(\\phi+\\Delta)&=(1-t^2)\\cos 2v_1 +\\OO_\\Lambda(\\lambda\/\\sqrt{\\ell}),\\\\\n2t\\sin 2v_2\\cos(\\phi-\\Delta)&=(t^2-1)\\cos 2v_2 +\\OO_\\Lambda(\\lambda\/\\sqrt{\\ell}).\\\\\n\\end{split}\n\\end{equation}\nWe shall use the notations\n\\begin{align*}\nP(\\phi) &:= \\max(|\\sin 2v_1 \\cos(\\phi + \\Delta)|, |\\sin 2v_2 \\cos(\\phi-\\Delta)|),\\\\\nR & := \\max(|\\cos 2v_1|, |\\cos2v_2|),\\\\\nN & := \\max(|\\sin(2v_1+2v_2) \\cos\\Delta|, |\\sin(2v_1-2v_2)\\sin\\Delta|).\n\\end{align*}\nRecall also the earlier notations \\eqref{adbc} and \\eqref{Dgdef}. As\n\\[|a|^2 - |d|^2 = \\frac{r^2-r^{-2}}{2}(\\cos 2v_1 + \\cos 2v_2), \\qquad\n|b|^2 - |c|^2 = \\frac{r^2-r^{-2}}{2}(\\cos 2v_1 - \\cos 2v_2),\\]\nwe can identify $\\mcN$ as the set of matrices with $r=1$ or $\\cos 2v_1 = \\cos 2v_2 = 0$. More precisely,\nby \\eqref{eq0} and \\eqref{r1} we have\n\\[D(g)\\ll r(r-1)R\\ll_\\Lambda r(r-1)\\left(t +\\frac{\\lambda}{\\sqrt{\\ell}}\\right)\\ll_\\Lambda\\frac{r^2\\lambda}{\\sqrt{\\ell}},\\]\nso that unless $D(g)\\ll_\\Lambda\\|g\\|^2\\lambda \\ell^{-1\/2}$, we have $\\mcM=\\emptyset$, yielding $\\varphi^{\\ell, 0}_{\\nu, \\ell}(g)\\ll_\\Lambda\\ell^{-\\Lambda}$. Hence we are left with proving \\eqref{thm5boundq=0}.\n\nIn \\eqref{keyupperbound:q=0}, the contribution of the $t$-integral over the interval $[0,\\ell^{-\\Lambda\/2-1\/4}]$ is negligible, and we split the rest of $\\mcM$ in dyadic ranges $\\mcM(\\delta)$ according to $\\ell^{-\\Lambda\/2-1\/4}<t\\asymp \\delta\\leq 1$. The number of such ranges is $\\OO_{\\Lambda}(\\log \\ell)$. Assume $(\\phi,t)\\in \\mcM(\\delta)$. The discriminants of the two quadratic equations \\eqref{eq0} are $4D_j(\\phi)+\\OO_{\\Lambda}(\\lambda\/\\sqrt{\\ell})$, where\n\\[D_1(\\phi):=1 - \\sin^2 (2v_1) \\sin^2(\\phi+ \\Delta),\\qquad D_2(\\phi):=1 - \\sin^2 (2v_2) \\sin^2(\\phi - \\Delta).\\]\nA simple calculation gives that\n\\begin{equation}\\label{disc-lower-bound}\nD_1(\\phi)+D_2(\\phi)\\geq P(\\phi)^2+R^2.\n\\end{equation}\nIf $|\\sin 2v_1\\sin(\\phi+\\Delta)| > 1\/2$, then for any fixed $t$, \\eqref{eq0} localizes $\\phi$ to a set of measure $\\OO_{\\Lambda}(\\lambda\/\\sqrt{\\ell})$. Otherwise, for any fixed $\\phi$, \\eqref{eq0} localizes $t$ to a set of measure $\\OO_{\\Lambda}(\\lambda\/\\sqrt{\\ell})$. We conclude that\n\\begin{equation}\\label{I1}\n\\meas(\\mcM(\\delta))\\ll_\\Lambda\\lambda\/\\sqrt{\\ell}.\n\\end{equation}\n\nNow we prove the alternative bound\n\\begin{equation}\\label{I2b}\n\\meas(\\mcM(\\delta))\\ll_\\Lambda\\frac{\\lambda^4}{N\\ell}.\n\\end{equation}\nWe shall assume that $N\\ell>1$, for otherwise \\eqref{I2b} follows from \\eqref{I1}. Under this assumption, we have $\\max(|\\sin 2v_1|,|\\sin 2v_2|)\\gg\\ell^{-1}$, which implies that\n\\[\n\\meas(\\{(\\phi,t)\\in \\mcM(\\delta):P(\\phi)\\leq \\ell^{-3}\\}) \\ll \\ell^{-1}.\n\\]\nIndeed, if $\\phi$ changes by at least $\\ell^{-1}$ and at most $\\pi\/4$, then $\\cos(\\phi\\pm\\Delta)$ both change by $\\Omega(\\ell^{-2})$, hence $P(\\phi)$ changes by $\\Omega(\\ell^{-3})$. This implies that $P(\\phi)\\leq \\ell^{-3}$ localizes $\\phi$ to a set of measure $\\OO(\\ell^{-1})$.\nTherefore, the contribution of $\\{(\\phi,t)\\in \\mcM(\\delta):P(\\phi)\\leq \\ell^{-3}\\}$ to the left-hand side of \\eqref{I2b} is\n$\\OO_{\\Lambda}(\\delta\/\\sqrt{\\ell})$, which is admissible by $N\\leq 1$. We decompose the rest of $\\mcM(\\delta)$ into dyadic ranges $\\mcM(\\delta,\\mtP)$ according to $\\ell^{-3}\\leq P(\\phi)\\asymp \\mtP\\leq 1$. The number of such ranges is $\\OO(\\log \\ell)$, hence in order to verify \\eqref{I2b}, it suffices to prove\n\\[\\meas(\\mcM(\\delta,\\mtP))\\ll_\\Lambda\\frac{\\lambda^2}{N\\ell}.\\]\nThe proof of this estimate immediately reduces to the following two localizations:\n\\begin{equation}\\label{phi-loc}\n\\meas(\\{\\phi\\in[0,2\\pi]:(\\phi,t)\\in\\mcM(\\delta,\\mtP)\\text{ for some $t\\asymp \\delta$}\\}) \\ll_{\\Lambda} \\frac{(\\mtP+R)\\lambda}{N\\sqrt{\\ell}},\n\\end{equation}\nand for any $\\phi\\in[0,2\\pi]$,\n\\begin{equation}\\label{t-loc}\n\\meas(\\{t\\in[0,1]:(\\phi,t)\\in\\mcM(\\delta,\\mtP)\\}) \\ll_{\\Lambda} \\frac{\\lambda}{(\\mtP+R)\\sqrt{\\ell}}.\n\\end{equation}\nNow we prove these localizations.\n\nStarting out from \\eqref{eq0}, we execute two eliminations: one to eliminate the main terms of the right-hand sides, and the other one to eliminate the left-hand sides. Introducing\n\\[\nF(\\phi):= \\cos \\phi \\cos \\Delta \\sin(2v_1+2v_2) - \\sin \\phi \\sin \\Delta \\sin(2v_1-2v_2),\n\\]\nthese give\n\\[\ntF(\\phi) \\ll_{\\Lambda} R \\lambda\/\\sqrt{\\ell}\\qquad\\text{and}\\qquad (1-t^2) F(\\phi) \\ll_{\\Lambda} \\mtP \\lambda \/\\sqrt{\\ell}.\n\\]\nIn particular, we obtain both for $t>1\/2$ and $t\\leq 1\/2$ that\n\\begin{equation}\\label{bound-on-Fphi}\nF(\\phi) \\ll_{\\Lambda} (\\mtP+R) \\lambda \/\\sqrt{\\ell}.\n\\end{equation}\nLetting\n\\[\nN':= \\sqrt{\\sin^2(2v_1+2v_2)\\cos^2(\\Delta) + \\sin^2(2v_1-2v_2)\\sin^2(\\Delta)} \\asymp N,\n\\]\nand choosing $\\psi\\in [0,2\\pi)$ such that\n\\[\n\\cos\\psi = \\frac{\\sin(2v_1+2v_2)\\cos\\Delta}{N'},\\qquad\n\\sin\\psi = - \\frac{\\sin(2v_1-2v_2)\\sin \\Delta}{N'},\n\\]\n\\eqref{bound-on-Fphi} gives rise to\n\\[\n\\cos(\\phi-\\psi) \\ll_{\\Lambda} \\frac{(\\mtP+R) \\lambda }{N\\sqrt{\\ell}}.\n\\]\nThis localizes $\\phi$ to a set of measure $\\OO_{\\Lambda}((\\mtP+R)\\lambda\/N\\sqrt{\\ell})$. Indeed, if the right-hand side is very small in terms of the implied constant, then $\\phi-\\psi$ is bounded away from $\\pi\\ZZ$, hence the derivative $\\cos'(\\phi-\\psi)$ is bounded away from zero, while otherwise the claimed localization is trivial. This gives \\eqref{phi-loc}. Fixing $\\phi\\in[0,2\\pi]$, and solving under \\eqref{eq0} the quadratic equation in $t$ of the larger discriminant, we see by \\eqref{disc-lower-bound} that $t$ is localized to a set of measure $\\OO_{\\Lambda}(\\lambda\/(\\mtP+R)\\sqrt{\\ell})$. This gives \\eqref{t-loc}. Altogether, the proof of \\eqref{I2b} is complete.\n\nCombining \\eqref{I1} and \\eqref{I2b}, we obtain\n\\[\\meas(\\mcM(\\delta))\\ll_{\\eps,\\Lambda}\\ell^{\\eps-1}\\mu,\\qquad\\mu:=\\min(\\sqrt{\\ell},N^{-1}).\\]\nWe claim that \n\\begin{equation}\\label{toprovedelta-b}\n\\dist(g,\\mcS)\\ll_{\\Lambda}\\lambda\\delta^{-1}\\mu^{-1},\\qquad \\text{if $\\mcM(\\delta)\\neq\\emptyset$.}\n\\end{equation}\nThis implies the inequality\n\\[\\sqrt{\\ell}\\int_{\\mcM(\\delta)} t\\, \\dd \\phi\\, \\dd t\\ll_{\\eps,\\Lambda}\\ell^{\\eps-1\/2}\\delta\\mu\n\\ll_{\\eps,\\Lambda}\\frac{\\ell^\\eps}{\\sqrt{\\ell}\\dist(g,\\mcS)},\\]\nwhich, summed over the $\\OO(\\log\\ell)$ dyadic ranges for $\\delta$, suffices for the proof of \\eqref{thm5boundq=0}. Note that the bound $\\varphi_{\\nu,\\ell}^{\\ell,q}(g)\\ll_\\eps\\ell^{\\eps}$ is already covered by Theorem~\\ref{thm6}.\n\nTo complete the proof of Theorem~\\ref{thm5}\\ref{thm5-a}, it remains to show \\eqref{toprovedelta-b}. For this final argument, we can and we shall assume that $-\\pi\/8\\leq v_1,v_2\\leq 3\\pi\/8$, because replacing $(u_1,v_1)$ by $(-u_1,v_1+\\pi\/2)$, or $(v_2,w_2)$ by $(v_2+\\pi\/2,-w_2)$, has the effect of multiplying $g$ by $\\left(\\begin{smallmatrix}&i\\\\i&\\end{smallmatrix}\\right)$ from either side without altering $\\Delta$ or the statement \\eqref{toprovedelta-b}. We fix a pair $(\\phi,t)\\in\\mcM(\\delta)$.\n\nNow, $N\\leq \\mu^{-1}$ implies that\n\\begin{equation}\\label{eq:toprovedelta-c}\nv_1+v_2\\in\\frac{\\pi}2\\ZZ+\\OO\\left(\\frac1{\\mu}\\right)\\qquad\\text{and}\\qquad\nv_1-v_2\\in\\frac{\\pi}2\\ZZ+\\OO\\left(\\frac1{\\mu|\\Delta|}\\right).\n\\end{equation}\nLet us introduce the short-hand notation\n\\[m[v]:=\\begin{pmatrix}\\cos v&i\\sin v\\\\i\\sin v&\\cos v\\end{pmatrix},\\qquad v\\in\\RR.\\]\nKeeping \\eqref{decomp-K} and \\eqref{g} in mind, we observe initially that\n\\begin{equation}\\label{initialdistance}\nm[v_1]\\diag\\left(re^{i\\Delta},r^{-1}e^{-i\\Delta}\\right)m[v_2]=m[v_1+v_2]+\\OO\\bigl(r-1+|\\Delta|\\bigr).\n\\end{equation}\nOn the right-hand side, we have $\\dist(m[v_1+v_2],\\mcS)\\ll\\mu^{-1}$ by \\eqref{eq:toprovedelta-c}, and also\n\\begin{equation}\\label{rbound}\nr-1\\ll_\\Lambda\\frac{\\lambda}{t\\sqrt{\\ell}}\\ll\\frac{\\lambda}{\\delta\\mu}\n\\end{equation}\nby \\eqref{r1} and $\\mu\\leq\\sqrt{\\ell}$. Hence \\eqref{toprovedelta-b} follows from \\eqref{initialdistance} as long as $\\Delta\\ll_\\Lambda\\lambda\\delta^{-1}\\mu^{-1}$. In other words, we can and we shall assume that $|\\Delta|\\gg_\\Lambda\\lambda\\delta^{-1}\\mu^{-1}$ holds with a sufficiently large implied constant depending on $\\Lambda$. In particular, we shall assume that the error terms in \\eqref{eq:toprovedelta-c}, and similar error terms for angles in the rest of this subsection, are less than $\\pi\/8$ in size. Under this assumption, \\eqref{eq:toprovedelta-c} breaks into two cases.\n\n\\emph{Case 1:} $v_1,v_2\\ll\\mu^{-1}|\\Delta|^{-1}$ and $v_1+v_2\\ll\\mu^{-1}$. In this case, we refine \\eqref{initialdistance} to\n\\begin{align*}\n&m[v_1]\\diag\\left(re^{i\\Delta},r^{-1}e^{-\\Delta}\\right)m[v_2]\\\\\n&=m[v_1+v_2]+m[v_1]\\diag\\left(re^{i\\Delta}-1,r^{-1}e^{-i\\Delta}-1\\right)m[v_2]\\\\\n&=m[v_1+v_2]+\\diag\\left(re^{i\\Delta}-1,r^{-1}e^{-i\\Delta}-1\\right)+\\OO\\bigl(r-1+\\mu^{-1}\\bigr)\\\\\n&=\\diag\\left(e^{i\\Delta},e^{-i\\Delta}\\right)+\\OO\\bigl(r-1+\\mu^{-1}\\bigr).\n\\end{align*}\nThe main term $\\diag\\left(e^{i\\Delta},e^{-i\\Delta}\\right)$ lies in $\\mcS$, hence \\eqref{toprovedelta-b} follows by \\eqref{rbound}.\n\n\\emph{Case 2:} $v_1,v_2=\\pi\/4+\\OO(\\mu^{-1}|\\Delta|^{-1})$ and $v_1+v_2=\\pi\/2+\\OO(\\mu^{-1})$. As we shall see, this case does not occur. The assumptions imply that $\\sin 2v_1$ and $\\sin 2v_2$ exceed $1\/2$. We multiply the second equation in \\eqref{eq0} by $\\sin 2v_1$, and the first equation in \\eqref{eq0} by $\\sin 2v_2$. Adding and subtracting the resulting two equations, we obtain\n\\begin{align*}\n4t\\sin 2v_1\\sin 2v_2\\cos\\phi\\cos\\Delta&=(t^2-1)\\sin(2v_1-2v_2)+\\OO_\\Lambda(\\lambda\/\\sqrt{\\ell}),\\\\\n4t\\sin 2v_1\\sin 2v_2\\sin\\phi\\sin\\Delta&=(t^2-1)\\sin(2v_1+2v_2)+\\OO_\\Lambda(\\lambda\/\\sqrt{\\ell}).\n\\end{align*}\nWe infer that\n\\[\\delta\\ll|t\\cos\\phi|+|t\\sin\\phi|\n\\ll_\\Lambda|\\sin(2v_1-2v_2)|+\\frac{|\\sin(2v_1+2v_2)|}{|\\Delta|}+\\frac{\\lambda}{\\sqrt{\\ell}|\\Delta|}\n\\ll_\\Lambda\\frac{\\lambda}{\\mu|\\Delta|}.\\]\nThis contradicts our earlier assumption that $\\Delta\\gg_\\Lambda\\lambda\\delta^{-1}\\mu^{-1}$ holds with a sufficiently large implied constant depending on $\\Lambda$.\n\nThe proof of Theorem~\\ref{thm5}\\ref{thm5-a} is complete.\n\n\\subsection{Proof of Theorem~\\ref{thm5}\\ref{thm5-b}}\nWe finally consider the case $q = \\pm \\ell$. By the symmetries \\eqref{distinvariance} and \\eqref{eq:averaged-spherical-function-symmetry}, we can restrict to $q = \\ell$. We have already shown the bound $\\varphi^{\\ell,\\ell}_{\\nu,\\ell}(g)\\ll_\\eps\\ell^{\\eps}$ in greater generality in Theorem~\\ref{thm6}. As a first step, we complement this with a stronger bound for $r \\geq 2$. To this end, we return to \\eqref{post-expansion}. As $q=\\ell$, the binomial coefficient and the $J$-factor disappear. When $\\overline{I}^{2\\ell}$ is expanded, we see a Laurent polynomial of $e^{2iu}$. When we integrate in $u$ from $0$ to $\\pi$, all the terms but the constant one vanish.\nWe calculate the constant term using the binomial theorem and the original product definition of $I$. This way we see that\n\\begin{align*}\n\\varphi_{\\nu,\\ell}^{ \\ell,\\ell}(g) = \\ &  d_{\\ell}  e^{2i\\ell(u_1-u_2-w_1+w_2)} r^{2\\ell}\n\\sum_{m=0}^{2\\ell}\\binom{2\\ell}{m}^2 (r^{-2}e^{2iu_2+2iw_1}\\cos v_1\\cos v_2)^m\\\\\n&(-\\sin v_1\\sin v_2)^{2\\ell-m}\n\\int_0^{\\pi\/2} (\\sin^2 v)^m (\\cos^2 v)^{2\\ell-m} \\frac{\\sin 2v}{h(r,v)^{\\ell+1-\\nu}}\\,\\dd v.\n\\end{align*}\nUsing the variable $x:=\\sin^2 v$, we rewrite this as\n\\begin{align*}\n\\varphi_{\\nu,\\ell}^{ \\ell,\\ell}(g) = \\ &  d_{\\ell} e^{2i\\ell(u_1-u_2-w_1+w_2)} r^{2\\nu-2}\n\\sum_{m=0}^{2\\ell}\\binom{2\\ell}{m}^2\n(r^{-2}e^{2iu_2+2iw_1}\\cos v_1\\cos v_2)^m\\\\\n&(-\\sin v_1\\sin v_2)^{2\\ell-m}\\int_0^1 \\frac{x^m (1-x)^{2\\ell-m}}{(1-x+r^{-4}x)^{\\ell+1-\\nu}}\\,\\dd x.\n\\end{align*}\nWith the short-hand notation\n\\[U:=r^{-1}e^{iu_2+iw_1}\\sqrt{x\\cos v_1\\cos v_2}\\qquad\\text{and}\\qquad V:=i\\sqrt{(1-x)\\sin v_1\\sin v_2},\\]\nwe obtain finally\n\\begin{align}\\label{eq:phi-ell-ell-ell-bound}\n\\begin{split}\n\\bigl|\\varphi_{\\nu,\\ell}^{\\ell,\\ell}(g)\\bigr| \\leq &\\ \\frac{2\\ell+1}{r^2}\\left|\n\\int_0^1\\sum_{m=0}^{2\\ell}\\binom{2\\ell}{m}^2\\frac{U^{2m}V^{4\\ell-2m}}{(1-x+r^{-4}x)^{\\ell+1-\\nu}}\\,\\dd x \\right|\\\\[4pt]\n=&\\ \\frac{2\\ell+1}{r^2} \\left|\\int_0^1\\frac{1}{2\\pi}\\int_0^{2\\pi}\n\\frac{(Ue^{i\\phi}+V)^{2\\ell}(Ue^{-i\\phi}+V)^{2\\ell}}{(1-x+r^{-4}x)^{\\ell+1-\\nu}}\\,\\dd\\phi\\,\\dd x\\right|.\n\\end{split}\n\\end{align}\nUsing that $(Ue^{i\\phi}+V)(Ue^{-i\\phi}+V)=U^2+V^2+2UV\\cos\\phi$ is on the line segment connecting $(U+V)^2$ and $(U-V)^2$, we observe that\n\\[\\frac{|(Ue^{i\\phi}+V)(Ue^{-i\\phi}+V)|^2}{1-x+r^{-4}x}\\leq \\max_\\pm \\frac{|U\\pm V|^4}{1-x+r^{-4}x},\\]\nwhich by the Cauchy--Schwarz inequality can be further upper bounded by\n\\[\\leq\\frac{(1-x+r^{-2}x)^2}{1-x+r^{-4}x}=1-\\frac{x}{1+\\frac{2}{r^2-1}+\\frac{1}{\\left(r^2-1\\right)^2 (1-x)}}.\\]\nHence the contribution to the rightmost expression in \\eqref{eq:phi-ell-ell-ell-bound} of $x\\in[0,1]$ satisfying\n\\[x>\\delta\\left(1+\\frac{2}{r^2-1}+\\frac{1}{\\left(r^2-1\\right)^2 (1-x)}\\right),\\qquad\\delta:=\\frac{\\log\\ell}{\\ell},\\]\nis admissible for \\eqref{thm5boundq=ell}. By $r\\geq 2$, the remaining values $x\\in[0,1]$ satisfy\n\\[x<3\\delta\\qquad\\text{or}\\qquad x(1-x)<\\frac{3\\delta}{(r^2-1)^2},\\]\nhence also $x<3\\delta$ or $1-x<8\\delta\/r^4$. So the remaining contribution is\n\\[\\leq\\frac{2\\ell+1}{r^2}\\int_{[0,3\\delta)\\cup(1-8\\delta\/r^4,1]}\\frac{\\dd x}{1-x+r^{-4}x}\\ll\\frac{\\log\\ell}{r^2},\\]\nwhich is again admissible for \\eqref{thm5boundq=ell}.\n\nBy \\eqref{keyupperbound}, it remains to show that\n\\begin{equation}\\label{toprovedelta-a}\n\\dist(g, \\mcD) \\ll_\\Lambda\\| g \\|\\lambda\/\\sqrt{\\ell},\\qquad \\text{if $\\mcM\\neq\\emptyset$.}\n\\end{equation}\nIn the present case $q=\\ell$, the condition \\eqref{eq} simplifies to\n\\begin{equation}\\label{eqell}\n\\begin{split}\n2t\\sin 2v_1\\cos(\\phi+\\Delta)&=(1-t^2)\\cos 2v_1 + (1 + t^2)+\\OO_\\Lambda(\\lambda^2\/\\ell),\\\\\n2t\\sin 2v_2\\cos(\\phi-\\Delta)&=(t^2-1)\\cos 2v_2 - (1 + t^2)+\\OO_\\Lambda(\\lambda^2\/\\ell),\n\\end{split}\n\\end{equation}\nhence for the proof of \\eqref{toprovedelta-a} we can and we shall assume that $|v_1+v_2|\\leq\\pi\/2$. Indeed, replacing $v_1$ by $v_1+\\pi$ has the effect of replacing $g$ by $-g$ without altering $\\Delta$ or the statement \\eqref{toprovedelta-a}. We fix a pair $(\\phi,t)\\in\\mcM$.\n\nThe two equations in \\eqref{eqell} yield readily that\n\\[(\\sin 2v_j)^2 \\leq 2+2\\cos 2v_j\\ll_\\Lambda t^2+t|\\sin 2v_j|+\\lambda^2\/\\ell.\\]\nHence $\\sin 2v_j \\ll_\\Lambda t+\\lambda\/\\sqrt{\\ell}$, that is,\n\\begin{equation}\\label{pi-coset}\nv_1, v_2 \\in \\frac{\\pi}{2}\\ZZ +\\OO_\\Lambda\\left(t+\\frac{\\lambda}{\\sqrt{\\ell}}\\right).\n\\end{equation}\nCombining \\eqref{eqell} with the Cauchy--Schwarz inequality, we also get\n\\[ (1+t^2)^2+\\OO_\\Lambda(\\lambda^2\/\\ell)\\leq (1-t^2)^2+4t^2\\cos^2(\\phi\\pm\\Delta).\\]\nEquivalently,\n\\[\\sin(\\phi\\pm\\Delta)\\ll_\\Lambda \\frac{\\lambda}{t\\sqrt{\\ell}}.\\]\nUsing also our initial assumption $|\\Delta|\\leq\\pi\/4$, we conclude that\n\\begin{equation}\\label{cos-ess-1}\n\\Delta\\ll_\\Lambda\\frac{\\lambda}{t\\sqrt{\\ell}}\\qquad\\text{and}\n\\qquad\\phi\\in\\pi\\ZZ+\\OO_\\Lambda\\left(\\frac{\\lambda}{t\\sqrt{\\ell}}\\right).\n\\end{equation}\nIn particular, $\\cos(\\phi\\pm\\Delta)=\\epsilon+\\OO_\\Lambda(\\lambda^2\/t^2\\ell)$ for some $\\epsilon\\in\\{\\pm 1\\}$. Plugging this back to \\eqref{eqell}, and using also \\eqref{pi-coset} along with\n\\[t \\left(t+\\frac{\\lambda}{\\sqrt{\\ell}}\\right) \\min\\left(1, \\frac{\\lambda^2}{t^2\\ell}\\right) \\ll \\frac{\\lambda^2}{\\ell},\\]\nwe obtain\n\\begin{equation}\\label{eqell-rewrite}\n\\begin{split}\n2t\\epsilon\\sin 2v_1&=(1-t^2)\\cos 2v_1 + (1 + t^2)+\\OO_\\Lambda(\\lambda^2\/\\ell),\\\\\n2t\\epsilon\\sin 2v_2&=(t^2-1)\\cos 2v_2 - (1 + t^2)+\\OO_\\Lambda(\\lambda^2\/\\ell).\n\\end{split}\n\\end{equation}\nNow consider the following three unit vectors in $\\RR^2$:\n\\[\\mbv_1:=(\\cos 2v_1,\\sin 2v_1),\\qquad \\mbv_2:=(\\cos 2v_2,-\\sin 2v_2),\\qquad\n\\mbt:=\\left(\\frac{t^2-1}{t^2+1},\\frac{2t\\epsilon}{t^2+1}\\right).\\]\nBy \\eqref{eqell-rewrite}, the scalar products $\\mbv_j\\mbt$ are $1+\\OO_\\Lambda(\\lambda^2\/\\ell)$, hence the directed angles $\\arg(\\mbv_j)-\\arg(\\mbt)$ lie in $2\\pi\\ZZ+\\OO_\\Lambda(\\lambda\/\\sqrt{\\ell})$. It follows that\n\\[\\arg(\\mbv_1)-\\arg(\\mbv_2)\\in2\\pi\\ZZ+\\OO_\\Lambda(\\lambda\/\\sqrt{\\ell}),\\]\nand then the assumption $|v_1+v_2|\\leq\\pi\/2$ forces that\n\\begin{equation}\\label{eps}\nv_1+v_2\\ll_\\Lambda\\lambda\/\\sqrt{\\ell}.\n\\end{equation}\n\nWe are now ready to complete the proof of Theorem~\\ref{thm5}\\ref{thm5-b}. By \\eqref{pi-coset} and \\eqref{eps}, there exists a multiple $v$ of $\\pi\/2$ such that\n\\begin{align*}\nm[v_1]&=m[v]+\\OO_\\Lambda\\bigl(t+\\lambda\/\\sqrt{\\ell}\\bigr),\\\\\nm[v_2]&=m[-v]+\\OO_\\Lambda\\bigl(t+\\lambda\/\\sqrt{\\ell}\\bigr),\\\\\nm[v_1+v_2]&=\\id+\\OO_\\Lambda\\bigl(\\lambda\/\\sqrt{\\ell}\\bigr).\n\\end{align*}\nTherefore, using also \\eqref{r1} and \\eqref{cos-ess-1}, we conclude that\n\\begin{align*}\n&m[v_1]\\diag\\left(re^{i\\Delta},r^{-1}e^{-\\Delta}\\right)m[v_2]\\\\\n&=m[v_1+v_2]+m[v_1]\\diag\\left(re^{i\\Delta}-1,r^{-1}e^{-i\\Delta}-1\\right)m[v_2]\\\\\n&=m[v_1+v_2]+m[v]\\diag\\left(re^{i\\Delta}-1,r^{-1}e^{-i\\Delta}-1\\right)m[-v]\n+\\OO_\\Lambda\\bigl(r\\lambda\/\\sqrt{\\ell}\\bigr)\\\\\n&=m[v]\\diag\\left(re^{i\\Delta},r^{-1}e^{-i\\Delta}\\right)m[-v]\n+\\OO_\\Lambda\\bigl(r\\lambda\/\\sqrt{\\ell}\\bigr).\n\\end{align*}\nThe main term $m[v]\\diag\\left(re^{i\\Delta},r^{-1}e^{-i\\Delta}\\right)m[-v]$ lies in $\\mcD$, hence \\eqref{toprovedelta-a} follows.\n\nThe proof of Theorem~\\ref{thm5}\\ref{thm5-b} is complete.\n\n\\section{Proof of Theorem~\\ref{thm1}}\\label{thm1-proof-sec}\nIn this section, we prove Theorem~\\ref{thm1}. Lemma~\\ref{APTI-done-lemma}, which results from the amplified pre-trace inequality and estimates on the spherical trace function, proves an estimate on $|\\Phi(g)|^2$ for $g\\in\\Omega$ in terms of the Diophantine counts $M(g,L,\\mcL,\\vec{\\delta})$. We begin with the key remaining step of estimating these counts.\n\nWe allow all implied constants within this section to depend on $\\Omega$, and we drop the subscript from notation. Moreover, we adopt the notation $A\\preccurlyeq B$ to mean that $|A|\\ll_{\\eps}\\ell^\\eps L^\\eps B$, where $\\eps>0$ is fixed but may be taken as small as desired at each step, and the implied constant is allowed to depend on $\\eps$. \n\nFor each $\\mcL\\in\\{1,L^2,L^4\\}$ and $\\vec{\\delta}=(\\delta_1,\\delta_2)$ with $0<\\delta_1,\\delta_2\\leq\\ell^\\eps$, we will estimate the count $M(g,L,\\mcL,\\vec{\\delta})$ of matrices\n\\begin{align}\\label{detgamma}\n\\begin{aligned}\n&\\gamma=\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}\\in\\MM_2(\\ZZ[i]),\n\\qquad&&\\det\\gamma=n\\in D(L,\\mcL),\\qquad |n|\\asymp\\mcL^{1\/2},\\\\\n&g^{-1}\\tilde{\\gamma}g=k\\begin{pmatrix} z&u\\\\&z^{-1}\\end{pmatrix}k^{-1}\n&&\\text{for some $k\\in K$ such that}\\\\\n&&&\\text{$|z|\\geq 1$, $\\min|z\\pm 1|\\leq\\delta_1$, $|u|\\leq\\delta_2$,}\n\\end{aligned}\n\\end{align}\nwhere as before $\\tilde{\\gamma} = \\gamma\/\\sqrt{n}$. By the symmetry $\\gamma\\leftrightarrow -\\gamma$, we can and we shall assume that $|z-1|\\leq|z+1|$. Then the conditions imply that both $|z-1|$ and $|z^{-1}-1|$ are at most $\\delta_1$, hence\n\\[\\left|\\frac{a+d}{\\sqrt{n}}-2\\right|=|\\tr\\tilde\\gamma-2|=|z+z^{-1}-2|=|z-1||z^{-1}-1|\\leq\\delta_1^2.\\]\nOn the other hand, since $\\|g\\|\\asymp_{\\Omega}1$, we also have that\n\\[ \\|\\tilde{\\gamma}-\\id\\|=\\left\\|gk\\begin{pmatrix}z-1&u\\\\&z^{-1}-1\\end{pmatrix}k^{-1}g^{-1}\\right\\|\n\\ll\\delta_1+\\delta_2. \\]\nSummarizing, we need to estimate the number of matrices $\\gamma$ as in \\eqref{detgamma} such that\n\\begin{equation}\\label{abcd-conditions}\n\\left|a+d-2\\sqrt{n}\\right|\\leq\\delta_1^2\\sqrt{|n|},\\qquad |a-d|,|b|,|c|\\ll(\\delta_1+\\delta_2)\\sqrt{|n|}.\n\\end{equation}\nIn particular, we have $|a+d|\\preccurlyeq\\sqrt{|n|}$ and\n\\begin{equation}\\label{trace-cor}\n(a-d)^2+4bc=(a+d)^2-4n\\preccurlyeq\\delta_1^2|n|.\n\\end{equation}\n\nAs is often the case, parabolic matrices $\\gamma$ (those with trace $\\pm 2\\sqrt{n}$) play a distinctive role in this counting problem, and we split the count accordingly into the parabolic and non-parabolic subcounts as\n\\[ M(g,L,\\mcL,\\vec{\\delta})=M^{\\pp}(g,L,\\mcL,\\vec{\\delta})+M^{\\np}(g,L,\\mcL,\\vec{\\delta}). \\]\nWe shall prove the following result using \\eqref{detgamma}, \\eqref{abcd-conditions}, and \\eqref{trace-cor}.\n\n\\begin{lemma}\\label{counting-for-thm1}\nLet $\\Omega\\subset G$ be a compact subset, $L\\geq 1$, and $\\mcL\\in\\{1,L^2,L^4\\}$. For $g\\in\\Omega$ and $\\vec{\\delta}=(\\delta_1,\\delta_2)$ with $0<\\delta_1,\\delta_2\\preccurlyeq 1$, we have the following bounds.\n\\begin{align}\n\\label{Mbound1}M(g,L,1,\\vec{\\delta})&\\preccurlyeq_{\\Omega}1,\\\\\n\\label{Mbound2}M^{\\pp}(g,L,\\mcL,\\vec{\\delta})&\\preccurlyeq_{\\Omega} \\mcL^{1\/2}+\\mcL\\delta_2^2,\\\\\n\\label{Mbound3}M^{\\np}(g,L,L^2,\\vec{\\delta})&\\preccurlyeq_{\\Omega} L^4\\delta_1^4(\\delta_1^2+\\delta_2^2),\\\\\n\\label{Mbound4}M^{\\np}(g,L,L^4,\\vec{\\delta})&\\preccurlyeq_{\\Omega} L^6\\delta_1^4(\\delta_1^2+\\delta_2^2).\n\\end{align}\nMoreover,\n\\begin{equation}\\label{otherwise-vanishes}\nM^{\\np}(g,L,\\mcL,\\vec{\\delta})=0\\qquad\\text{unless}\\qquad\\delta_1\\succcurlyeq\\mcL^{-1\/4}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nThe bound \\eqref{Mbound1} is immediate from \\eqref{abcd-conditions}. We turn to the bound \\eqref{Mbound2}, which counts parabolic matrices $\\gamma$. In this case, we have $(a-d)^2+4bc=0$ and $z=1$, hence in particular \\eqref{abcd-conditions} holds with $0$ in place of $\\delta_1$. If $bc\\neq 0$, then there are $\\ll\\mcL^{1\/2}$ choices for $a+d=2\\sqrt{n}$, and $\\ll\\mcL^{1\/2}\\delta_2^2$ choices for $a-d\\neq 0$. The difference $a-d$ determines the product $bc$ uniquely, hence by the divisor bound, there are $\\preccurlyeq 1$ choices for $(b,c)$. This is admissible for \\eqref{Mbound2}. If $bc=0$, then there are $\\ll\\mcL^{1\/2}$ choices for $a=d=\\sqrt{n}$, and $\\ll 1+\\mcL^{1\/2}\\delta_2^2$ choices for $(b,c)$. This is again admissible for \\eqref{Mbound2}.\n\nFrom now on we count non-parabolic matrices $\\gamma$, in which case $(a-d)^2+4bc\\neq 0$.\nThe statement \\eqref{otherwise-vanishes} is immediate from \\eqref{trace-cor}, so we are left with proving \\eqref{Mbound3} and \\eqref{Mbound4}. If $bc\\neq 0$, then there are $\\preccurlyeq\\mcL^{1\/2}$ choices for $a+d$, and $\\ll 1+\\mcL^{1\/2}(\\delta_1^2+\\delta_2^2)$ choices for $a-d$. For given $a-d$, there are $\\preccurlyeq 1+\\mcL\\delta_1^4$ choices for $(b,c)$ by \\eqref{trace-cor} and the divisor bound. This is admissible for \\eqref{Mbound3} in the light of \\eqref{otherwise-vanishes}.\nIf $bc=0$, then there are  $\\preccurlyeq\\mcL^{1\/2}$ choices for $a+d$, $\\preccurlyeq\\mcL^{1\/2}\\delta_1^2$ choices for $a-d\\neq 0$ by \\eqref{trace-cor}, and  $\\ll 1+\\mcL^{1\/2}(\\delta_1^2+\\delta_2^2)$ choices for $(b,c)$. This is again admissible for \\eqref{Mbound3} in the light of \\eqref{otherwise-vanishes}. In the high range $\\mcL=L^4$, this argument gives the bound\n\\[M^{\\np}(g,L,L^4,\\vec{\\delta})\\preccurlyeq_{\\Omega} L^8\\delta_1^4(\\delta_1^2+\\delta_2^2),\\]\nwhich is weaker than \\eqref{Mbound4} by a factor of $L^2$. However, we can save a factor of $L^2=\\mcL^{1\/2}$ as follows. In the high range, $n=l_1^2l_2^2$ is a square, and $(a-d)^2+4bc\\neq 0$ factors as $(a+d+2l_1l_2)(a+d-2l_1l_2)$. Hence the triple $(a-d,b,c)$ determines $a+d$ up to $\\preccurlyeq 1$ possibilities by the divisor bound, while earlier we considered $\\preccurlyeq\\mcL^{1\/2}$ possibilities for $a+d$.\n\\end{proof}\n\nCombining Lemmata~\\ref{APTI-done-lemma} and \\ref{counting-for-thm1}, we obtain that\n\\[\\sum_{\\phi\\in\\mfB}|\\phi(g)|^2\\preccurlyeq_{I,\\Omega}\\ell^3\n\\left(\\frac1L+S^{\\pp}(L)+S^{\\np}(L,L^2)+S^{\\np}(L,L^4)\\right)+L^{2}\\ell^{-48},\\]\nwhere\n\\begin{alignat*}{3}\nS^{\\pp}(L)&:=\\sum_{\\substack{\\vec{\\delta}\\text{ dyadic}\\\\1\/\\sqrt{\\ell}\\leq\\delta_j\\preccurlyeq 1}}\\frac1{\\sqrt{\\ell}\\delta_2}\\left(\\frac{L+L^2\\delta_2^2}{L^3}+\\frac{L^2+L^4\\delta_2^2}{L^4}\\right)&&\\preccurlyeq\\frac1{L^2}+\\frac1{\\sqrt{\\ell}},\\\\\nS^{\\np}(L,L^2)&:=\\sum_{\\substack{\\vec{\\delta}\\text{ dyadic}\\\\1\/\\sqrt{\\ell}\\leq\\delta_j\\preccurlyeq 1}}\n\\frac1{\\ell\\delta_1^2}\\cdot\\frac{L^4\\delta_1^4(\\delta_1^2+\\delta_2^2)}{L^3}&&\\preccurlyeq\\frac{L}{\\ell},\\\\\nS^{\\np}(L,L^4)&:=\\sum_{\\substack{\\vec{\\delta}\\text{ dyadic}\\\\1\/\\sqrt{\\ell}\\leq\\delta_j\\preccurlyeq 1}}\\frac1{\\ell\\delta_1^2}\\cdot\\frac{L^6\\delta_1^4(\\delta_1^2+\\delta_2^2)}{L^4}&&\\preccurlyeq\\frac{L^2}{\\ell}.\n\\end{alignat*}\nPutting everything together, we conclude that\n\\[\\sum_{\\phi\\in\\mfB}|\\phi(g)|^2\\preccurlyeq_{I,\\Omega}\\ell^3\n\\left(\\frac1L+\\frac1{\\sqrt{\\ell}}+\\frac{L^2}{\\ell}\\right)+L^2\\ell^{-48}\\ll\\ell^{8\/3}, \\]\nby making the essentially optimal choice $L:=7\\ell^{1\/3}$ (which satisfies our earlier\ncondition $L\\geq 7$).\n\nThe proof of Theorem~\\ref{thm1} is complete.\n\n\\section{Proof of Theorem~\\ref{thm3}}\\label{thm2proof-sec}\nIn this section, we prove Theorem~\\ref{thm3}. For $q= 0$, Lemma~\\ref{APTI-done-lemma-single-form} provides an estimate on $|\\phi_q(g)|^2$ for $g\\in\\Omega$ in terms of the Diophantine count $M_0^{\\ast}(g,L,\\mcL,\\delta)$, while for $q = \\pm \\ell$ we need to analyze $Q(g, L, H_1,H_2)$ as follows from Lemma~\\ref{q=ell-case}. We begin by estimating these counts. We keep the notational conventions from \\S\\ref{thm1-proof-sec}.\n\n\\subsection{A comparison lemma}\nThe Diophantine counts in Lemmata~\\ref{APTI-done-lemma-single-form} and \\ref{q=ell-case} involve the positioning relative to certain special sets of the matrix $g^{-1}\\tilde{\\gamma}g$, which we now explicate in preparation for a counting argument. Using $g\\in\\Omega$, we may write explicitly\n\\[ g=\\begin{pmatrix} g_1&g_2\\\\g_3&g_4\\end{pmatrix},\\qquad g_j\\ll 1. \\]\nAn explicit calculation shows that\n\\[ g^{-1}\\begin{pmatrix} a&b\\\\c&d\\end{pmatrix}g=\\begin{pmatrix}\\frac{a+d}{2}+L_1&L_2\\\\L_3&\\frac{a+d}{2}-L_1\\end{pmatrix}, \\]\nwhere\n\\begin{equation}\\label{coordinates}\n\\begin{alignedat}{7}\nL_1&=\\hphantom{-}(a-d)\\big(\\tfrac12+g_2g_3\\big)&&+bg_3g_4&&-cg_1g_2, \\\\\nL_2&=\\hphantom{-}(a-d)g_2g_4&&+bg_4^2&&-cg_2^2,\\\\\nL_3&=-(a-d)g_1g_3&&-bg_3^2&&+cg_1^2.\n\\end{alignedat}\n\\end{equation}\nWe record the following simple but effective result, which will be used in both parts of Theorem~\\ref{thm3}.\n\n\\begin{lemma}\\label{l2l3-small}\nLet $\\Omega\\subset G$ be a compact subset, and $g\\in\\Omega$. Let $a,b,c,d\\in\\CC$ and $\\Delta>0$ be such that\n$L_2,L_3\\ll\\Delta$.\n\\begin{enumerate}[(a)]\n\\item\\label{1213-a}\nFor at least one $s\\in\\{a-d,b,c\\}$, we have\n\\[\\begin{bmatrix} a-d&b&c\\end{bmatrix}^{\\top}\n=\\begin{bmatrix}\\lambda_1&\\lambda_2&\\lambda_3\\end{bmatrix}^{\\top}s+\\OO(\\Delta)\\]\nwith $\\lambda_1,\\lambda_2,\\lambda_3\\ll 1$ depending only on $g$.\n\\item\\label{1213-b}\nFor the same choice of $s\\in\\{a-d,b,c\\}$, we have\n\\[ (a-d)^2+4bc=\\mu s^2+\\OO(\\Delta |s|+\\Delta^2), \\]\nwith $\\mu=\\lambda_1^2+4\\lambda_2\\lambda_3\\gg 1$. If additionally $(a-d)^2+4bc=0$, then $a-d,b,c\\ll\\Delta$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWe may write the defining equations for $L_2$ and $L_3$ as $\\begin{bmatrix} L_2&L_3\\end{bmatrix}^{\\top}=M\\begin{bmatrix} a-d&b&c\\end{bmatrix}^{\\top}$ for a $2\\times 3$ matrix $M$ whose $2\\times 2$ minors we compute to be\n\\[ \\begin{vmatrix}g_4^2&-g_2^2\\\\-g_3^2&g_1^2\\end{vmatrix}=g_1g_4+g_2g_3,\\quad \\begin{vmatrix} g_2g_4&-g_2^2\\\\-g_1g_3&g_1^2\\end{vmatrix}=g_1g_2,\\quad \\begin{vmatrix} g_2g_4&g_4^2\\\\-g_1g_3&-g_3^2\\end{vmatrix}=g_3g_4. \\]\nAt least one of these minors exceeds $1\/3$ in absolute value, since\n\\[(g_1g_4+g_2g_3)^2-4g_1g_2g_3g_4=1.\\]\nConsider the case when $|g_1g_4+g_2g_3|>1\/3$. Then we may solve the latter two equations in  \\eqref{coordinates} for $b$, $c$, which yields\n\\[ \\begin{bmatrix} b\\\\c\\end{bmatrix}=\\begin{bmatrix}-g_1g_2\\\\g_3g_4\\end{bmatrix}\\frac{a-d}{g_1g_4+g_2g_3}+\\OO(\\Delta).\\]\nThis settles the first claim in the lemma with $s=a-d$. The second claim follows from\n\\[ (a-d)^2+4bc=\\frac{(a-d)^2}{(g_1g_4+g_2g_3)^2}+\\OO\\big(\\Delta|a-d|+\\Delta^2\\big). \\]\nThe other cases (of which it suffices to consider one) are similar. For example, under $|g_1g_2|>1\/3$ we have\n\\[ \\begin{bmatrix}a-d\\\\c\\end{bmatrix}=\\begin{bmatrix}-g_1g_4-g_2g_3\\\\-g_3g_4\\end{bmatrix}\\frac{b}{g_1g_2}+\\OO(\\Delta),\\quad (a-d)^2+4bc=\\frac{b^2}{(g_1g_2)^2}+\\OO\\big(\\Delta|b|+\\Delta^2\\big), \\]\nfrom which the lemma follows.\n\\end{proof}\n\n\\subsection{Second moment count for $q=\\pm\\ell$}\nWe will now establish an upper bound for the quantity $Q(g,L, H_1, H_2)$ counting pairs of matrices $(\\gamma_1,\\gamma_2)$ such that\n\\begin{equation}\\label{thm2a-KS-conditions}\n\\begin{gathered}\n\\gamma_j=\\begin{pmatrix} a_j&b_j\\\\c_j&d_j\\end{pmatrix}\\in\\MM_2(\\ZZ[i]),\n\\qquad \\det\\gamma_1=\\det\\gamma_2=n,\\qquad L \\leq |n|\\leq 2 L,\\\\\n\\|g^{-1}\\tilde{\\gamma}_jg\\|\\leq\\sqrt{\\frac{H_j}{L}},\\qquad\n\\dist(g^{-1}\\tilde{\\gamma}_jg,\\mcD)\\ll\\sqrt{\\frac{H_j\\log\\ell}{L\\ell}}.\n\\end{gathered}\n\\end{equation}\nWe denote the quantities in \\eqref{coordinates} corresponding to $\\gamma_j$ as\n$L_{1j}$, $L_{2j}$, $L_{3j}$. From \\eqref{coordinates} and \\eqref{thm2a-KS-conditions} we deduce that\n\\begin{equation}\\label{2a-KS-immediate}\n\\|\\gamma_j\\|\\ll\\sqrt{H_j},\\qquad L_{2j},L_{3j}\\preccurlyeq\\sqrt{H_j\/\\ell},\n\\end{equation}\nand\n\\begin{equation}\\label{det-cond}\n(a_1+d_1)^2-(a_2+d_2)^2=(a_1-d_1)^2+4b_1c_1-(a_2-d_2)^2-4b_2c_2.\n\\end{equation}\nWe shall prove the following result using \\eqref{thm2a-KS-conditions}, \\eqref{2a-KS-immediate}, and \\eqref{det-cond}.\n\n\\begin{lemma}\\label{lemma-ell-count}\nLet $\\Omega\\subset G$ be a compact subset and $L\\geq 1$. For $g\\in\\Omega$ and\n$1\\leq H_1, H_2\\preccurlyeq\\ell$, we have\n\\begin{equation}\\label{Qbound}\nQ(g,L,H_1, H_2)\\preccurlyeq_\\Omega H_1H_2.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof} We shall use that the entries $a_j,b_j,c_j,d_j\\in\\ZZ[i]$ of each participating $\\gamma_j$ satisfy the conditions of Lemma~\\ref{l2l3-small} with $\\Delta_j\\preccurlyeq 1$ in the role of $\\Delta$. Indeed, this follows from \\eqref{2a-KS-immediate} and $H_1, H_2\\preccurlyeq\\ell$.\n\nLet $s_j\\in\\{a_j-d_j,b_j,c_j\\}$ be as in Lemma~\\ref{l2l3-small}\\ref{1213-a}. By Lemma~\\ref{l2l3-small}\\ref{1213-a} and \\eqref{2a-KS-immediate}, for a given pair $(s_1,s_2)$, there are $\\preccurlyeq 1$ choices for the two triples $(a_j-d_j,b_j,c_j)$, which then determine both sides of \\eqref{det-cond}. Using this preliminary observation, we do the counting in two steps.\n\nFirst we count $(\\gamma_1,\\gamma_2)$ satisfying \\eqref{thm2a-KS-conditions} and $(a_1+d_1)^2\\neq(a_2+d_2)^2$. By \\eqref{2a-KS-immediate}, there are $\\ll H_1H_2$ choices for the pair $(s_1, s_2)$, hence $\\preccurlyeq H_1H_2$ choices for the two triples $(a_j-d_j,b_j,c_j)$. Given the triples, by \\eqref{2a-KS-immediate}--\\eqref{det-cond} and the divisor bound, there are $\\preccurlyeq 1$ choices for $(a_1+d_1,a_2+d_2)$. This is admissible for \\eqref{Qbound}.\n\nNow we count $(\\gamma_1,\\gamma_2)$ satisfying \\eqref{thm2a-KS-conditions} and $(a_1+d_1)^2=(a_2+d_2)^2$.\nIn this case, Lemma~\\ref{l2l3-small}\\ref{1213-b} coupled with \\eqref{2a-KS-immediate}--\\eqref{det-cond} shows that\n$s_1^2-s_2^2\\preccurlyeq\\sqrt{H_1}+\\sqrt{H_2}$. Hence, by the divisor bound (separating the case when $s_1^2=s_2^2$), there are $\\preccurlyeq\\max(H_1,H_2)$ choices for the pair $(s_1, s_2)$ and same for the two triples $(a_j-d_j,b_j,c_j)$. Independently of the triples, by \\eqref{2a-KS-immediate}, there are $\\ll\\min(H_1,H_2)$ choices for $(a_1+d_1,a_2+d_2)$. This is again admissible for \\eqref{Qbound}.\n\\end{proof}\n\n\\subsection{Interlude: a first moment count}\\label{thm2a-first-moment-sec}\nFor the proof of Theorem~\\ref{thm2} in \\S \\ref{sec-proof2} below, we need a variation of the previous Diophantine argument that is most conveniently stated and proved at this point. \nFor $\\mcL\\in\\{1,L^2,L^4\\}$ and every $0<\\delta\\preccurlyeq 1$, we will establish an upper bound on the quantity\n\\begin{equation}\\label{thm2a-conditions}\nM_\\mcD(g,L,\\mcL,\\eps,\\delta):=\\sum_{n\\in D(L,\\mcL)}\n\\#\\left\\{\\gamma\\in\\Gamma_n:\\|g^{-1}\\tilde\\gamma g\\|\\ll\\ell^{\\eps},\\,\\,\\dist(g^{-1}\\tilde\\gamma g,\\mcD)\\leq\\delta\\right\\},\n\\end{equation}\nwhere the implied constant is absolute. As before, we conclude from the conditions in \\eqref{thm2a-conditions} and the explicit description in \\eqref{coordinates} that\n\\begin{equation}\\label{coeff-bound-2a_l2l3-delta0}\n\\|\\gamma\\|\\preccurlyeq\\mcL^{1\/4}\\qquad\\text{and}\\qquad L_2,L_3\\ll\\mcL^{1\/4}\\delta.\n\\end{equation}\nWe shall prove the following result using \\eqref{coeff-bound-2a_l2l3-delta0} and the identity\n\\begin{equation}\\label{parabolicidentity}\n(a-d)^2+4bc=(a+d)^2-4n.\n\\end{equation}\n\n\\begin{lemma}\\label{first-moment-count-lemma}\nLet $\\Omega\\subset G$ be a compact subset, $L\\geq 1$, and $\\eps>0$. For $g\\in\\Omega$ and $0<\\delta\\preccurlyeq 1$, we have the following bounds.\n\\begin{align}\n\\label{Nbound1}M_{\\mcD}(g,L,1,\\eps,\\delta)&\\preccurlyeq_{\\Omega} 1,\\\\\n\\label{Nbound2}M_{\\mcD}(g,L,L^2,\\eps,\\delta)&\\preccurlyeq_{\\Omega} L^2+L^4\\delta^4,\\\\\n\\label{Nbound3}M_{\\mcD}(g,L,L^4,\\eps,\\delta)&\\preccurlyeq_{\\Omega} L^2+L^6\\delta^4.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nThe bound \\eqref{Nbound1} corresponds to $\\mcL=1$, and it is immediate from \\eqref{coeff-bound-2a_l2l3-delta0}. Hence we focus on the bounds \\eqref{Nbound2}--\\eqref{Nbound3} that correspond to $\\mcL\\in\\{L^2,L^4\\}$. We shall use that the entries $a,b,c,d\\in\\ZZ[i]$ of each participating $\\gamma$ satisfy the conditions of Lemma~\\ref{l2l3-small} with $\\Delta=\\mcL^{1\/4}\\delta$, as follows from \\eqref{coeff-bound-2a_l2l3-delta0}.\n\nFirst we count parabolic matrices $\\gamma$. In this case, we have $(a-d)^2+4bc=0$, hence also $a-d,b,c\\ll\\mcL^{1\/4}\\delta$ by Lemma~\\ref{l2l3-small}\\ref{1213-b}. If $bc\\neq 0$, then there are $\\ll\\mcL^{1\/2}$ choices for $a+d=\\pm 2\\sqrt{n}$, and $\\ll\\mcL^{1\/2}\\delta^2$ choices for $a-d\\neq 0$. The difference $a-d$ determines the product $bc$ uniquely, hence by the divisor bound, there are $\\preccurlyeq 1$ choices for $(b,c)$. This is admissible for \\eqref{Nbound2}--\\eqref{Nbound3}. If $bc=0$, then there are $\\ll\\mcL^{1\/2}$ choices for $a=d=\\pm\\sqrt{n}$, and $\\ll 1+\\mcL^{1\/2}\\delta^2$ choices for $(b,c)$. This is again admissible for \\eqref{Nbound2}--\\eqref{Nbound3}.\n\nNow we count non-parabolic matrices $\\gamma$, in which case $(a-d)^2+4bc\\neq 0$. Let $s\\in\\{a-d,b,c\\}$ be as in Lemma~\\ref{l2l3-small}\\ref{1213-a}. There are $\\preccurlyeq\\mcL^{1\/2}$ choices both for $s$ and for $a+d$. For a given $s$, by Lemma~\\ref{l2l3-small}\\ref{1213-a}, there are $\\ll 1+\\mcL\\delta^4$ choices for the triple $(a-d,b,c)$. This is admissible for \\eqref{Nbound2}. In the high range $\\mcL=L^4$, this argument gives the bound $\\preccurlyeq L^4+L^8\\delta^4$ for the relevant count, which is weaker than \\eqref{Nbound3} by a factor of $L^2$. However, we can save a factor of $L^2=\\mcL^{1\/2}$ as follows. In the high range, $n=l_1^2l_2^2$ is a square, and $(a-d)^2+4bc\\neq 0$ factors as $(a+d+2l_1l_2)(a+d-2l_1l_2)$. Hence the triple $(a-d,b,c)$ determines $a+d$ up to $\\preccurlyeq 1$ possibilities by the divisor bound, while earlier we considered $\\preccurlyeq\\mcL^{1\/2}$ possibilities for $a+d$.\n\\end{proof}\n\n\\subsection{Counting setup for $q=0$}\nFor each $\\mcL\\in\\{1,L^2,L^4\\}$ and $0<\\delta\\preccurlyeq 1$, we will establish an upper bound on the quantity $M_0^{\\ast}(g,L,\\mcL,\\delta)$ consisting of matrices\n\\begin{equation}\\label{thm2b-conditions}\n\\begin{gathered}\n\\gamma=\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}\\in\\MM_2(\\ZZ[i]),\n\\qquad\\det\\gamma=n\\in D(L,\\mcL),\\qquad |n|\\asymp\\mcL^{1\/2}\\\\\n\\dist(g^{-1}\\tilde{\\gamma}g,\\mcS)\\leq\\delta,\\qquad\\frac{D(g^{-1}\\tilde\\gamma g)}{\\|g^{-1}\\tilde\\gamma g\\|^2}\\ll\\frac{\\log\\ell}{\\sqrt{\\ell}}.\n\\end{gathered}\n\\end{equation}\nFrom the first distance condition in \\eqref{thm2b-conditions} we conclude that\n\\begin{equation}\\label{coeff-bound}\na,b,c,d\\preccurlyeq\\mcL^{1\/4}.\n\\end{equation}\nUsing the description in \\eqref{coordinates}, the distance conditions in \\eqref{thm2b-conditions} imply that\n\\begin{gather}\n\\label{cond1} \\left\\{\\begin{aligned} &L_2,L_3\\ll\\delta\\sqrt{|n|}\\\\ &\\left|\\tfrac{a+d}{2}\\pm L_1\\right|=(1+\\OO(\\delta))\\sqrt{|n|}\\end{aligned}\\right. \\qquad\\text{or}\\qquad\n\\left\\{\\begin{aligned} &a+d,L_1\\ll\\delta\\sqrt{|n|}\\\\ &|L_2|,|L_3|=(1+\\OO(\\delta))\\sqrt{|n|};\\end{aligned}\\right.\\\\\n\\label{cond0} \\left|\\tfrac{a+d}{2}+L_1\\right|^2-\\left|\\tfrac{a+d}{2}-L_1\\right|^2\n\\preccurlyeq\\sqrt{\\mcL\/\\ell}\\qquad\\text{and}\\qquad|L_2|^2-|L_3|^2\\preccurlyeq\\sqrt{\\mcL\/\\ell}.\n\\end{gather}\nAs in \\S\\ref{thm1-proof-sec}, we split the count into the parabolic and non-parabolic subcounts as\n\\[ M_0^{\\ast}(g,L,\\mcL,\\delta)=M_0^{\\ast\\pp}(g,L,\\mcL,\\delta)+M_0^{\\ast\\np}(g,L,\\mcL,\\delta). \\]\nWe shall prove the following result using \\eqref{thm2b-conditions}--\\eqref{cond0} and \\eqref{parabolicidentity}.\n\n\\begin{lemma}\\label{counting-for-thm2}\nLet $\\Omega\\subset G$ be a compact subset, $L\\geq 1$, and $\\mcL\\in\\{L^2,L^4\\}$. For $g\\in\\Omega$ and $0<\\delta\\preccurlyeq 1$, we have the following bounds.\n\\begin{align}\n\\label{Rbound1} M_0^{\\ast}(g,L,1,\\delta)&\\preccurlyeq_{\\Omega} 1,\\\\\n\\label{Rbound2} M_0^{\\ast\\pp}(g,L,\\mcL,\\delta)&\\preccurlyeq_{\\Omega}\\mcL^{1\/2}+\\mcL\\delta^2,\\\\\n\\label{Rbound3} M_0^{\\ast}(g,L,L^2,\\delta)&\\preccurlyeq_{\\Omega} L^{3\/2}+L^3\\delta^3+\\frac{L^2+L^{7\/2}\\delta^2}{\\sqrt{\\ell}}+\\frac{L^4\\delta^2}{\\ell},\\\\[4pt]\n\\label{Rbound4} M_0^{\\ast\\np}(g,L,L^4,\\delta)&\\preccurlyeq_{\\Omega} L^3+L^5\\delta^2+\\frac{L^4+L^6\\delta^2}{\\sqrt{\\ell}}.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nThe bound \\eqref{Rbound1} is immediate from \\eqref{coeff-bound}. For the proof of \\eqref{Rbound2}, we observe that, in the parabolic case, \\eqref{cond1} implies $L_2,L_3\\ll\\mcL^{1\/4}\\delta$. Indeed, this is clear when the first half of \\eqref{cond1} holds. Otherwise, the conditions $a+d=\\pm 2\\sqrt{n}$ and $a+d\\ll\\delta\\sqrt{|n|}$ force $\\delta\\gg 1$, so the claimed bound is clear again. Applying Lemma~\\ref{l2l3-small}\\ref{1213-b}, we infer that $a-d,b,c\\ll\\mcL^{1\/4}\\delta$ holds in the parabolic case. From here \\eqref{Rbound2} follows readily, as in the second paragraph of the proof of Lemma~\\ref{first-moment-count-lemma}. Finally, we shall prove \\eqref{Rbound3} and \\eqref{Rbound4} in the next two subsections.\n\\end{proof}\n\n\\subsection{Volume argument}\\label{subsec:volume-argument}\nHere, we present a volume argument that we will use repeatedly to estimate the number of lattice points satisfying \\eqref{thm2b-conditions}--\\eqref{cond0}. The symbol $\\vol$ will refer to the Lebesgue measure in $\\CC^m\\simeq\\RR^{2m}$, with $m$ being clear from the context.\n\nThe explicit expressions for the linear forms in \\eqref{coordinates} may be rewritten as \n\\begin{equation}\\label{Ag3}\n\\begin{bmatrix} L_1&L_2&L_3\\end{bmatrix}^{\\top}=A_0(g)\\begin{bmatrix}a-d&b&c\\end{bmatrix}^{\\top},\n\\end{equation}\nwhere $A_0:\\Omega\\to\\GL_3(\\CC)$ is a continuous function.\nIt is straightforward to verify that $\\det A_0(g)=1\/2$ holds identically. \nWe shall also use the $4$-dimensional variant\n\\begin{equation}\\label{Ag4}\n\\begin{bmatrix} a+d&L_1&L_2&L_3\\end{bmatrix}^{\\top}=\\diag(1,A_0(g))\\begin{bmatrix} a+d&a-d&b&c\\end{bmatrix}^{\\top}.\n\\end{equation}\n\nNow, let $m\\geq 1$ be a fixed integer ($m\\in\\{2,3,4\\}$ in our applications), and let $A:\\Omega\\to\\GL_m(\\CC)$ be a fixed continuous function. As $\\Omega$ is compact, there exists a fixed compact subset $K=K(A,\\Omega)\\subset\\CC^m$ such that each $2m$-dimensional lattice $A(g)\\ZZ[i]^m\\subset\\CC^m$ ($g\\in\\Omega)$ has a fundamental parallelepiped lying in $K$ and of volume $\\asymp 1$. It follows by a standard volume argument that for any compact subset $V\\subset\\CC^m$ and $g\\in\\Omega$ we have\n\\begin{equation}\\label{volume-bound}\n\\#\\bigl(V\\cap A(g)\\ZZ[i]^m\\bigr)\\ll\\vol V^\\bullet\\qquad\\text{where}\\qquad V^\\bullet:=V+K.\n\\end{equation}\n\nWe also record for repeated reference a simple volume computation. For $r,\\Delta>0$, we define the sets\n\\begin{align*}\nW_1(r,\\Delta)&:=\\bigl\\{(z_1,z_2)\\in\\CC^2:|z_1|,|z_2|\\leq r,\\,\\,\\Re(z_1\\overline{z_2})\\leq\\Delta\\bigr\\},\\\\\nW_2(r,\\Delta)&:=\\bigl\\{(z_1,z_2)\\in\\CC^2:|z_1|,|z_2|\\leq r,\\,\\,\\bigl||z_1|^2-|z_2|^2\\bigr|\\leq\\Delta\\bigr\\}.\n\\end{align*}\nCutting these into two parts according to whether $|z_2|\\leq |z_1|$ or $|z_2|>|z_1|$, we obtain readily by Fubini's theorem that\n\\[\\vol W_j(r,\\Delta)\\ll\\min(r^4,r^2\\Delta).\\]\nOn the other hand, we have\n\\[W_j(r,\\Delta)^\\bullet\\subset W_j\\bigl(r+\\OO(1),\\Delta+\\OO(r+1)\\bigr)\\]\nwith implied constants depending only on $A$ and $\\Omega$, hence\n\\begin{equation}\\label{volW}\n\\vol W_j(r,\\Delta)^{\\bullet}\\ll\\min\\bigl((r+1)^4,(r+1)^2(\\Delta+r+1)\\bigr)\\ll 1+r^2\\Delta+r^3.\n\\end{equation}\n\n\\subsection{Middle and high range for $q=0$}\nWe now estimate the count $M_0^{\\ast}(g,L,\\mcL,\\delta)$ in the ``middle range'' $\\mcL=L^2$ and the ``high range'' $\\mcL=L^4$. In the high range, we shall focus on the non-parabolic contribution $M_0^{\\ast\\np}(g,L,L^4,\\delta)$, since \nwe have already proved \\eqref{Rbound2}, and here we shall profit substantially from the fact that $\\det\\gamma$ is a square.\n\n\\subsubsection{Middle range}\nIn the middle range $\\mcL=L^2$, we estimate the number of choices in $M_0^{\\ast}(g,L,L^2,\\delta)$ as follows.\n\nFor the case when the first half of \\eqref{cond1} holds, we introduce the set\n\\begin{align*}\nV_1(\\delta):=\\big\\{(z_0,z_1,z_2,z_3)\\in\\CC^4:\\ &z_0,z_1\\preccurlyeq\\mcL^{1\/4},\\,\\,\n\\Re(z_0\\overline{z_1})\\preccurlyeq\\sqrt{\\mcL\/\\ell},\\\\\n&z_2,z_3\\ll\\mcL^{1\/4}\\delta,\\,\\,|z_2|^2-|z_3|^2\\preccurlyeq \\sqrt{\\mcL\/\\ell}\\big\\},\n\\end{align*}\nsuppressing from notation the dependence implicit in $\\preccurlyeq$. Then we have by \\eqref{volW}\n\\begin{align}\n\\nonumber\\vol V_1(\\delta)^{\\bullet}&\\preccurlyeq\n\\vol W_1(\\mcL^{1\/4},\\sqrt{\\mcL\/\\ell})^{\\bullet}\\cdot \n\\vol W_2(\\mcL^{1\/4}\\delta,\\sqrt{\\mcL\/\\ell})^{\\bullet}\\\\\n\\label{V1bound}&\\ll (\\mcL^{3\/4}+\\mcL\/\\sqrt{\\ell})(1+\\mcL^{3\/4}\\delta^3+\\mcL\\delta^2\/\\sqrt{\\ell}).\n\\end{align}\nFor the case when the second half of \\eqref{cond1} holds, we introduce the set\n\\begin{align*}\nV_2(\\delta)=\\big\\{(z_0,z_1,z_2,z_3)\\in\\CC^4:\\ &z_0,z_1\\ll\\mcL^{1\/4}\\delta,\\,\\,\n\\Re(z_0\\overline{z_1})\\preccurlyeq\\sqrt{\\mcL\/\\ell},\\\\\n&z_2,z_3\\preccurlyeq\\mcL^{1\/4},\\,\\,|z_2|^2-|z_3|^2\\preccurlyeq\\sqrt{\\mcL\/\\ell}\\big\\},\n\\end{align*}\nsuppressing from notation the dependence implicit in $\\preccurlyeq$. Then we have by \\eqref{volW}\n\\begin{align}\n\\nonumber\\vol V_2(\\delta)^{\\bullet}&\\preccurlyeq\n\\vol W_1(\\mcL^{1\/4}\\delta,\\sqrt{\\mcL\/\\ell})^{\\bullet}\\cdot \n\\vol W_2(\\mcL^{1\/4},\\sqrt{\\mcL\/\\ell})^{\\bullet}\\\\\n\\label{V2bound}&\\ll (\\mcL^{3\/4}+\\mcL\/\\sqrt{\\ell})(1+\\mcL^{3\/4}\\delta^3+\\mcL\\delta^2\/\\sqrt{\\ell}).\n\\end{align}\n\nUsing \\eqref{thm2b-conditions}--\\eqref{cond0}, \\eqref{Ag4}--\\eqref{volume-bound}, and \\eqref{V1bound}--\\eqref{V2bound}, we conclude \\eqref{Rbound3} in the form\n\\[ M_0^{\\ast}(g,L,L^2,\\delta)\\preccurlyeq (L^{3\/2}+L^2\/\\sqrt{\\ell})(1+L^{3\/2}\\delta^3+L^2\\delta^2\/\\sqrt{\\ell}). \\]\n\n\\subsubsection{High range}\nAs in the proof of Lemmata~\\ref{counting-for-thm1} and \\ref{first-moment-count-lemma}, in the high range $\\mcL=L^4$, once the triple $(a-d,b,c)$ is determined for a non-parabolic matrix $\\gamma$ (so that \\eqref{parabolicidentity} holds), $a+d$ and along with it $\\gamma$ is determined up to $\\preccurlyeq 1$ choices by the divisor bound, using that $n=l_1^2l_2^2$ is a square. We now estimate the number of choices in $M_0^{\\ast\\np}(g,L,L^4,\\delta)$ as follows.\n\nFor the case when the first half of \\eqref{cond1} holds, we introduce the set\n\\[ V_3(\\delta):=\\big\\{(z_1,z_2,z_3)\\in\\CC^3:\nz_1\\preccurlyeq\\mcL^{1\/4},\\,\\,z_2,z_3\\ll\\mcL^{1\/4}\\delta,\\,\\,|z_2|^2-|z_3|^2\\preccurlyeq\\sqrt{\\mcL\/\\ell}\\big\\}, \\]\nsuppressing from notation the dependence implicit in $\\preccurlyeq$. Then we have by \\eqref{volW}\n\\begin{equation}\\label{V3bound}\n\\vol V_3(\\delta)^{\\bullet}\\preccurlyeq\n\\sqrt{\\mcL}\\cdot\\vol W_2(\\mcL^{1\/4}\\delta,\\sqrt{\\mcL\/\\ell})^{\\bullet}\n\\ll \\sqrt{\\mcL}(1+\\mcL\\delta^2\/\\sqrt{\\ell}+\\mcL^{3\/4}\\delta^3).\n\\end{equation}\nFor the case when the second half of \\eqref{cond1} holds, we introduce the set\n\\[ V_4(\\delta):=\\big\\{(z_1,z_2,z_3)\\in\\CC^3:z_1\\ll\\mcL^{1\/4}\\delta,\\,\\,z_2,z_3\\preccurlyeq\\mcL^{1\/4},\\,\\,|z_2|^2-|z_3|^2\\preccurlyeq\\sqrt{\\mcL\/\\ell}\\big\\}, \\]\nsuppressing from notation the dependence implicit in $\\preccurlyeq$. Then we have by \\eqref{volW}\n\\begin{equation}\\label{V4bound}\n\\vol V_4(\\delta)^{\\bullet}\\preccurlyeq\n(1+\\mcL^{1\/4}\\delta)^2\\cdot\\vol W_2(\\mcL^{1\/4},\\sqrt{\\mcL\/\\ell})^{\\bullet}\n\\ll(1+\\sqrt{\\mcL}\\delta^2)(\\mcL^{3\/4}+\\mcL\/\\sqrt{\\ell}).\n\\end{equation}\n\nUsing \\eqref{parabolicidentity}, \\eqref{thm2b-conditions}--\\eqref{cond0}, \\eqref{Ag3}, \\eqref{volume-bound}, and \\eqref{V3bound}--\\eqref{V4bound}, we conclude \\eqref{Rbound4} in the form\n\\[ M_0^{\\ast\\np}(g,L,L^4,\\delta)\\preccurlyeq\nL^2(1+L^4\\delta^2\/\\sqrt{\\ell}+L^3\\delta^3)+(1+L^2\\delta^2)(L^3+L^4\/\\sqrt{\\ell}). \\]\n\nThe proof of Lemma~\\ref{counting-for-thm2} is complete.\n\n\\subsection{Proof of Theorem~\\ref{thm3}}\n\nIn the case $q=0$, we combine Lemmata~\\ref{APTI-done-lemma-single-form} and \\ref{counting-for-thm2} to see that\n\\[ |\\phi_0(g)|^2\\preccurlyeq_{I,\\Omega}\\ell^2\n\\left(\\frac1L+S^{\\ast}_0(L,L^2)+S^{\\ast}_0(L,L^4)\\right)+L^{2}\\ell^{-48}, \\]\nwhere\n\\begin{alignat*}{3}\nS_0^{\\ast}(L,L^2)&:=\\sum_{\\substack{\\delta\\text{ dyadic}\\\\1\/\\sqrt{\\ell}\\leq\\delta\\preccurlyeq 1}}\n\\frac1{\\sqrt{\\ell}\\delta L^3}\n\\left(L^{3\/2}+L^3\\delta^3+\\frac{L^2+L^{7\/2}\\delta^2}{\\sqrt{\\ell}}+\\frac{L^4\\delta^2}{\\ell}\\right)\n&&\\preccurlyeq\\frac1{L^{3\/2}}+\\frac1{\\sqrt{\\ell}}+\\frac{L}{\\ell^{3\/2}},\\\\\nS_0^{\\ast}(L,L^4)&:=\\sum_{\\substack{\\delta\\text{ dyadic}\\\\1\/\\sqrt{\\ell}\\leq\\delta\\preccurlyeq 1}}\n\\frac1{\\sqrt{\\ell}\\delta L^4}\n\\left(L^3+L^5\\delta^2+\\frac{L^4+L^6\\delta^2}{\\sqrt{\\ell}}\\right)\n&&\\preccurlyeq\\frac1L+\\frac{L}{\\sqrt{\\ell}}+\\frac{L^2}{\\ell}.\n\\end{alignat*}\nPutting everything together, we conclude that\n\\[ |\\phi_0(g)|^2\\preccurlyeq_{I,\\Omega}\\ell^2\\left(\\frac1L+\\frac{L}{\\sqrt{\\ell}}+\\frac{L^2}{\\ell}\\right)+L^2\\ell^{-48}\\ll \\ell^{7\/4}, \\]\nby making the essentially optimal choice $L:=7\\ell^{1\/4}$ (which satisfies our earlier\ncondition $L\\geq 7$).\n\nThe case $q = \\pm \\ell$ is immediate from Lemmata~\\ref{q=ell-case} and \\ref{lemma-ell-count}, hence the proof of Theorem~\\ref{thm3} is complete.\n\n\\section{Proof of Theorem~\\ref{thm2}}\\label{sec-proof2}\nIn this section, we prove Theorem~\\ref{thm2}. Here we take the aim of the softest possible proof based on the localization properties of the averaged spherical trace function (proved in Theorem~\\ref{thm6} and then encoded in the form of the amplified pre-trace inequality in Lemma~\\ref{APTI-done-lemma-single-form}) and the already available ingredients for the counting problem.\n\nFor each $\\mcL\\in\\{1,L^2,L^4\\}$ and $\\vec{\\delta}=(\\delta_1,\\delta_2)$ with $0<\\delta_1,\\delta_2\\leq\\ell^\\eps$, the count $M^\\ast(g,L,\\mcL,\\vec{\\delta})$ in Lemma~\\ref{APTI-done-lemma-single-form} may be estimated in a split fashion as\n\\[ M^\\ast(g,L,\\mcL,\\vec{\\delta})\\leq \\min\\big(M_K(g,L,\\mcL,\\delta_1),M_\\mcD(g,L,\\mcL,\\eps,\\delta_2)\\big), \\]\nwhere\n\\[M_K(g,L,\\mcL,\\delta):=\\sum_{n\\in D(L,\\mcL)}\\#\\left\\{\\gamma\\in\\Gamma_n:\\dist\\left(g^{-1}\\tilde\\gamma g,K\\right)\\leq\\delta\\right\\},\\]\nand $M_{\\mcD}(g,L,\\mcL,\\eps,\\delta)$ is as in \\eqref{thm2a-conditions}. The quantity $M_K(g,L,\\mcL,\\delta)$ is the classical Diophantine count in the spherical sup-norm problem in the eigenvalue aspect, which in the present context\nwas treated in detail in \\cite{BlomerHarcosMilicevic2016}. In the notation of that paper, we have:\n\\begin{itemize}\n\\item $u(\\tilde\\gamma gK,gK)\\asymp\\dist(g^{-1}\\tilde\\gamma g,K)^2$ in \\cite[(5.3)]{BlomerHarcosMilicevic2016};\n\\item $N=1$, and $r\\asymp_{\\Omega}1$ for $g\\in\\Omega$, in \\cite[(6.2)]{BlomerHarcosMilicevic2016}.\n\\end{itemize}\nThus the count $M_K(g,L,\\mcL,\\delta_1)$ agrees with $M(gK,L,\\mcL,\\OO(\\delta_1^2))$ in \\cite[(5.17)--(5.18)]{BlomerHarcosMilicevic2016}. Importing estimates \\cite[(7.1), (7.2), (7.5), (11.1), (11.6)]{BlomerHarcosMilicevic2016}, we conclude that\n\\[M_K(g,L,1,\\delta_1)\\preccurlyeq_{\\Omega} 1,\\quad\nM_K(g,L,L^2,\\delta_1)\\preccurlyeq_{\\Omega} L^2+L^4\\delta_1,\\quad\nM_K(g,L,L^4,\\delta_1)\\preccurlyeq_{\\Omega} L^3+L^6\\delta_1.\\]\n\nThe count $M_{\\mcD}(g,L,\\mcL,\\eps,\\delta)$ was estimated in Lemma~\\ref{first-moment-count-lemma}. Combining everything, we obtain the following lemma.\n\n\\begin{lemma}\\label{soft-counting-for-general-q}\nFor $g\\in\\Omega$, $L>0$, and arbitrary $\\eps>0$ and $\\vec{\\delta}=(\\delta_1,\\delta_2)$ with $0<\\delta_{j}\\preccurlyeq 1$, the quantity $M^\\ast(g,L,\\mcL,\\vec{\\delta})$ in Lemma~\\ref{APTI-done-lemma-single-form} satisfies\n\\begin{align*}\nM^\\ast(g,L,1,\\vec{\\delta})&\\preccurlyeq_{\\Omega} 1,\\\\\nM^\\ast(g,L,L^2,\\vec{\\delta})&\\preccurlyeq_{\\Omega}\\min\\big(L^2+L^4\\delta_1,L^2+L^4\\delta_2^4\\big),\\\\\nM^\\ast(g,L,L^4,\\vec{\\delta})&\\preccurlyeq_{\\Omega}\\min\\big(L^3+L^6\\delta_1,L^2+L^6\\delta_2^4\\big).\n\\end{align*}\n\\end{lemma}\n\nWe are now ready for the proof of Theorem~\\ref{thm2}. From Lemma~\\ref{soft-counting-for-general-q}, we have  for every pair $\\vec{\\delta}=(\\delta_1,\\delta_2)$ with $0<\\delta_1,\\delta_2\\leq\\ell^{\\eps}$ that\n\\[\\frac{M^\\ast(g,L,1,\\vec{\\delta})}{L}+\\frac{M^\\ast(g,L,L^2,\\vec{\\delta})}{L^3}+\\frac{M^\\ast(g,L,L^4,\\vec{\\delta})}{L^4}\n\\preccurlyeq_{\\Omega} \\left(\\frac1L+L^2\\min\\left(\\delta_1,\\delta_2^4\\right)\\right).\\]\nInserting this into Lemma~\\ref{APTI-done-lemma-single-form}, we find that\n\\begin{align*}\n|\\phi_q(g)|^2&\\preccurlyeq_{I,\\Omega}\\ell^{2}\n\\sum_{\\substack{\\vec{\\delta}\\textnormal{ dyadic},\\,\\,\\delta_j\\preccurlyeq 1\\\\\\delta_1^2\\delta_2\\geq 1\/\\sqrt{\\ell}}}\n\\frac{1}{\\sqrt{\\ell}\\delta_1^2\\delta_2}\n\\left(\\frac1L+ L^2 \\min\\left(\\delta_1,\\delta_2^4\\right)\\right)+L^{2}\\ell^{-48}\\\\\n&\\preccurlyeq \\ell^{2} \\Biggl(\\frac1L+\n\\sum_{\\substack{\\vec{\\delta}\\textnormal{ dyadic},\\,\\,\\delta_j\\preccurlyeq 1\\\\\\delta_1^2\\delta_2\\geq 1\/\\sqrt{\\ell}}}\nL^2 \\min\\left(\\frac{1}{\\sqrt{\\ell}\\delta_1\\delta_2}, \\delta_1, \\delta_2^4\\right)\\Biggr)\n\\preccurlyeq \\ell^{2} \\left(\\frac1L + \\frac{L^2}{\\ell^{2\/9}}\\right),\n\\end{align*}\nwhere we used $\\min(A, B, C) \\leq A^{4\/9}B^{4\/9} C^{1\/9}$ in the last step.\nThe choice $L:=7\\ell^{2\/27}$ is optimal up to a constant, and it satisfies our earlier\ncondition $L\\geq 7$, hence we obtain Theorem~\\ref{thm2} in the form\n\\[{\\|\\phi_q|_{\\Omega}\\|}_\\infty\\preccurlyeq_{I,\\Omega}\\ell^{26\/27}.\\qedhere \\]\n\nThe proof of Theorem~\\ref{thm2} is complete.\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nFrom AlexNet \\citep{krizhevsky2012imagenet}, VGG \\citep{simonyan2014very}, ResNet \\citep{he2016deep}, to SENet \\citep{hu2018squeeze}, tremendous research effort has been put in efficient deep learning model designs, leading to state-of-the-art performance for various machine learning tasks, including image recognition, object detection, and image segmentation \\citep{ren2015faster, long2015fully}. The success of deep learning models relies on sufficient training on a vast amount of data with correct labels, which however are difficult to acquire in practice. In particular, the privacy issue, which has attracted much attention recently, makes data collection from clients much more challenging \\citep{pmlr-v54-mcmahan17a}.\n\nTo train deep learning models adequately while protecting the privacies of clients, federated learning has come up as the major driving force to enhance the data security of deep learning methods \\citep{pmlr-v54-mcmahan17a}. Instead of training models using collected data from clients, federated learning needs no data collection. Instead, each client trains the machine learning model on its local edge device and then uploads the model parameters to the central server. Since only machine learning models are exchanged in the air, the risk of the client's data leakage becomes much smaller \\citep{truex2019hybrid}. Although straightforward as it may seem, integrating machine learning training methods into the framework of federated learning while with indistinguishable performance loss is of no simple matter. This has triggered recent theoretical research in both machine learning and optimizations \\cite{li2020on}. \n\nTherefore, in federated learning, it is crucial to think about how to use automated machine learning to search for neural architectures directly from the clients' data.\nThanks to gradient-based neural architecture search \\citep{liu2018darts, cai2018proxylessnas}, the model search can be represented as updating architecture parameters. \nSome NAS methods search for backbone cells on proxy data to trim down computational cost \\citep{liu2018darts, xu2020pcdarts}. \nHowever, these proxy strategies do not guarantee that the backbone cells have optimal performance on the target data\\citep{yang2019evaluation}. \nMore importantly, in federated learning, the difference in distribution between proxy data and target data will be larger due to the presence of non-iid data.\nTherefore, there are enough reasons to propose a non-proxy, gradient-based Federated Direct Neural Architecture Search (\\textbf{FDNAS}), with a feature of no data exchange. The first contribution of this paper is to propose such a scheme. \n\nOn the other hand, due to prevalent human biases, preferences, habits, etc., clients are divided into different groups, in each of which the clients are similar to each other in terms of both data and hardware.\nTherefore, instead of looking for an architecture that is too dense to suit each client, we expect that FDNAS should allow each client to use a lightweight architecture that fits their individual tasks.\nTo achieve this goal, We treat different clients' models as a large ensemble, in which the models are \\textit{highly diverse and client-specific}.\nThen, our \\textbf{primary question} is: how can we effectively search for these clusters' diverse models that are near-optimal on their respective tasks in a federated manner, while their weights are entangled? To exploit the model diversity in such a complex ensemble, we resort to the fundamental idea of meta-learning. \n\nParticularly, in meta-learning, the model's weights that have been already meta-trained can be very efficient when they are adapted to different tasks via meta-test \\citep{chen2019closerfewshot}.\nTherefore, we propose to use the SuperNet in a meta-test-like manner in order to obtain all client-specific neural architectures in federated learning. In the order of ``meta-train\" to ``meta-test\", the SuperNet trained on all clients in the first phase are considered as the meta-training model for the next ``meta-test\" client-specific adaptation. Following this idea, we propose the Cluster Federated Direct Neural Architecture Search (\\textbf{CFDNAS}) that divides all clients into groups by data similarity, and each group is trained using the SuperNet from the previous phase. In turn, each group utilizes the previous SuperNet and can adjust the architecture to fit their own client data after only a few rounds of updates like a meta-test. Consequently, CFDNAS can quickly generate a specific architecture for each client's data in parallel, under the framework of federated learning.\nThe contributions of our FDNAS are summarized below: \n\\begin{itemize}\n\\item[1.] \nWe have integrated federated learning with gradient-based and proxy-less NAS. This allows federated learning not only to train weights but also to search model architectures directly from the clients' data.\n\\item[2.] \nInspired by meta-learning, we extended FDNAS to CFDNAS to exploit the model diversity so that client-specific models can be quickly searched at very low computational cost.\n\\end{itemize}\n\n\\section{Related Work}\n\\textbf{Efficient neural architecture} designing is important for the practical deployment.\nMobileNetV2 \\citep{sandler2018mobilenetv2} proposes MBconv blocks, which largely reduce the model's FLOPs. Besides, efficient neural architecture search has recently gained increasing attention.\nTo speed up the NAS, the one-shot NAS approaches treat all normal nets as different subnets of the SuperNet and share weights among the operation candidates \\citep{brock2018smash}.\nENAS \\citep{pham2018efficient} uses the RNN controller to sample subnets in SuperNet and uses reinforce method to obtain approximate gradients of architecture.\nDARTS \\citep{liu2018darts} improves the search efficiency by representing each edge as a mixture of candidate operations and optimizing the weights of operation candidates in continuous relaxations.\nRecently, hardware-aware NAS methods like ProxylessNAS, incorporate the latency feedback in search as the joint optimization task without any expensive manual attempts. \\citep{wu2019fbnet, wan2020fbnetv2, cai2018proxylessnas}.\nAt the same time, some NAS methods search for the best backbone cells and transfer them to other target tasks by stacking them layer by layer \\citep{liu2018darts, xu2020pcdarts}. \nHowever, there is a big difference between the proxy and the target here: typically, the distribution of proxy data differs from the target data, and the best block searched by the proxy method differs from the optimal normal net after stacked \\citep{ yang2019evaluation}.\nTheir motivations and approaches are very different from our FDNAS, where our goal is to search the neural architecture without the gap between proxy task and clients' data, but directly on clients' data in a privacy-preserving way and automatically design a variety of client-specific models that satisfy clients' diversity.\n\n\\noindent \\textbf{Federated learning} commonly deploys predefined neural architectures on the client. They then use FedAvg as a generic algorithm to optimize the model \\citep{pmlr-v54-mcmahan17a}.\nThere are some fundamental challenges associated with the research topic of federated learning \\citep{li2019federated}: communication overhead, statistical heterogeneity of data(non-iid), client privacy.\nThe communication between the central server and the client is a bottleneck due to frequent weights exchange. So some studies aim to design more efficient communication strategies \\citep{konevcny2016federated, 45672, fedpaq19}.\nIn practice, the data is usually non-iid distributed. So the hyper-parameter settings of FedAvg (e.g., learning rate decay) are analyzed to study their impacts on non-iid data \\citep{li2020on}. In addition, a global data-sharing strategy is proposed to improve the accuracy of the algorithm on non-iid data \\citep{zhao2018federated}.\nOther research efforts have focused on privacy security \\citep{agarwal2018cpsgd}. For example, differential privacy is applied to federated training, thus preserving the privacy of the client \\citep{article17eth, 9069945}. Recently, FedNAS searches out the cell architecture and stacks it \\citep{he2020fednas}.\nNote that all of the above techniques are deploying pre-defined network architectures, or only searching for backbone cells for stacking using proxy strategies. However, our FDNAS can search the complete model architecture directly from clients' data and allow the model to better adapt to the clients' data distribution while protecting privacy. Also, with CFDNAS, it is able to provide multiple suitable networks for diverse clients at a very low computational cost.\n\\section{Preliminary}\nTo motivate the idea of this paper, we briefly review the basics of federated learning and ProxylessNAS in this section. \n\\subsection{Federated Learning}\n\\label{sec:FedAvg}\n\nAs an emerging privacy-preserving technology, federated learning (FedAvg) enables edge devices to collaboratively train a shared global model without uploading their private data to a central server \\citep{pmlr-v54-mcmahan17a}. In particular, in the training round $t+1$, an edge device  $k\\in S$ downloads the shared machine learning model  $\\mathbf w^{k}_{t}$ (e.g., a CNN model) from the central server, and utilizes its local data to update the model parameters. Then, each edge device sends its updated model $\\{\\mathbf w^{k}_{t} \\}_{k \\in S}$ to the central server for aggregation. \n\n\n\n\\subsection{ProxylessNAS}\n\\label{sec:plnas}\n\nIn ProxylessNAS, a SuperNet (over-parameterized net) is firstly constructed, which is denoted by a directed acyclic graph (DAG) with $N$ nodes. Each node $x^{(i)}$ represents a latent representation (e.g., a feature map), and each directed edge $e^{(i,j )}$ that connects node  $x^{(i)}$ and  $x^{(j)}$ defines the following operation:\n\\begin{align}\nx^{(j)} =  \\sum_{n=1}^N b_n^{(i \\rightarrow j)} o_n \\left[ x^{(i)} \\rightarrow x^{(j)} \\right],\n\\end{align}\nwhere $o_n\\left[x^{(i)} \\rightarrow x^{(j)}\\right] \\in \\mathcal O$ denotes a operation candidate (e.g., convolution, pooling, identity, ect.) that transforms $x^{(i)}$to $x^{(j)}$, and vector $\\mathbf b^{(i \\rightarrow j)} = [b_1^{(i \\rightarrow j)}, \\cdots, b_N^{(i \\rightarrow j)}]$ is a binary gate that takes values as one hot vector which only set $b_n^{(i \\rightarrow j)}=1$ with a probability $p_n^{(i \\rightarrow j)}$ and other elements are 0 in forwarding.\nRather than computing all the operations in the operation set $\\mathcal O$ \\citep{liu2018darts}, there is only one operation $o_n \\left[ x^{(i)} \\rightarrow x^{(j)}\\right]$ that is utilized to transform each node $x^{(i)}$ to one of its neighbors $x^{(j)}$ with a probability $p_n^{(i \\rightarrow j)}$ at each run-time, thus greatly saving the memory and computations for training.\nSo it opens the door to directly learn the optimal architecture from large-scale datasets without resorting to proxy tasks. Furthermore, it applies the incorporation of hardware latency loss term into the NAS for reducing the inference latency on devices.\n\n\n\n\n\\section{Federated Neural Architecture Search}\n\\subsection{Motivation and Problem Formulation}\nIn the last section, ProxylessNAS is introduced as an efficient framework of great value for real-world applications since it is more efficient to use models directly searched from diverse daily data.\nHowever, the training of ProxylessNAS needs to collect all target data in advance, which inevitably limits its practical use due to the privacy concern of clients. In contrast, FedAvg trains a neural network without exchanging any data and thus can protect the data privacy of clients. However, it usually does not take the NAS into account, and therefore requires cumbersome manual architecture design (or hyperparameter tuning) for the best performance. \n\nTo integrate the merits from ProxylessNAS and FedAvg while overcoming their shortcomings, we develop a federated ProxylessNAS algorithm in this section. In particular, we propose a novel problem formulation as follows:\n\\begin{align}\n    \\min_{\\pmb {\\alpha}} \\quad & \\sum_{k=1}^K\\mathcal{L}_{val}^{k}(\\mathbf w^*(\\pmb {\\alpha}), \\pmb {\\alpha}) \\label{eq:outerfednasFormulation}\\\\\n    \\text{s.t.} \\quad &\\mathbf w^*(\\pmb {\\alpha}) = \\mathrm{argmin}_{\\mathbf w} \\enskip \\sum_{k=1}^K\\mathcal{L}_{train}^{k}(\\mathbf w, \\pmb {\\alpha}), \\label{eq:innerfednasFormulation}\n\\end{align}\nwhere $\\mathcal{L}_{val}^{k}(\\cdot, \\cdot)$ denotes the validation loss function of client $k$,   and $\\mathcal{L}_{train}^{k}(\\cdot, \\cdot)$ denotes its training loss function. Vector $\\pmb {\\alpha}$ collects all architecture parameters $\\{\\alpha_n^{i \\rightarrow j}, \\forall i \\rightarrow j, \\forall n\\}$ and  $\\mathbf w$ collects all weights $\\{\\mathbf w^{i\\rightarrow j}, \\forall i\\rightarrow j\\}$.  This nested bilevel optimization formulation is inspired by those proposed in ProxylessNAS \\citep{cai2018proxylessnas} and FedAvg \\citep{pmlr-v54-mcmahan17a}. More specifically, for each client, the goal is to search the optimal architecture $\\pmb {\\alpha}$ that gives the best performance on its local validation dataset, while with the optimal model weights $\\mathbf w^*(\\pmb {\\alpha})$ learnt from its local training dataset.\n\n\n\\subsection{Federated Algorithm Development}\n\\begin{figure*}[htbp]\n    \\centering\n    \\includegraphics[width=0.825\\linewidth]{illustration-fed-search.pdf}\n    \\caption{Illustration of FDNAS among clients.}\n    \\label{fig:fdnas}\n\\end{figure*}\n\nTo optimize $\\pmb {\\alpha}$ and $\\mathbf w$ while protecting the privacy of client data, the FDNAS is developed in this subsection. \n\nFollowing the set-up of federated learning, assume that there are $K$ clients in the set $S$ and one central server. In communication round $t$, each client downloads global parameters  $\\pmb {\\alpha}^g_t$ and $\\mathbf w^g_t$ from the server, and update these parameters using its local validation dataset and training dataset, respectively.  With global  parameters $\\pmb {\\alpha}^g_t$ and $\\mathbf w^g_t$ as initial values, the updates follow the steps in {\\bf Algorithm \\ref{alg:fdnas}}, i.e., \n\\begin{align}\n    \\mathbf w_{t+1}^k, \\pmb \\alpha_{t+1}^k \\leftarrow \\text{ProxylessNAS}( \\mathbf w_t^{g}, \\pmb \\alpha_t^{g}), \\forall k \\in S. \\label{eq:w-alpha-from-single-client}\n\\end{align}\nThen, each edge device sends their updated parameters $\\{\\mathbf w^{k}_{t+1},\\pmb \\alpha^{k}_{t+1} \\}_{k \\in S}$ to the central server for aggregation. Central server aggregates these parameters to update global parameters $\\pmb {\\alpha}^g_{t+1}$ and $\\mathbf w^g_{t+1}$:\n\\begin{align}\n    \\mathbf w_{t+1}^{g}, \\pmb  \\alpha_{t+1}^{g} \\leftarrow \\sum_{k=1}^K \\frac{N_k}{N} \\mathbf w_{t+1}^k, \\sum_{k=1}^K \\frac{N_k}{N} \\pmb \\alpha_{t+1}^k, \\label{eq:sum-params-from-clients}\n\\end{align}\nwhere $N_k$ is the size of local dataset in client $k$, and $N$ is the sum of all client's data size (i.e.,  $N = \\sum_{k=1}^K N_k)$. \n\nAfter $T$ communication rounds, each client downloads the learnt global parameters $\\pmb {\\alpha}^g_t$ and $\\mathbf w^g_t$ from the server, from which the optimized neural network architecture and model parameters can be obtained, as introduced in Section Preliminary. Furthermore, with the learnt neural network architecture, the weights $\\mathbf w^k$ in each client could be further refined by the conventional FedAvg algorithm for better validation performance. The proposed algorithm is summarized in {\\bf Algorithm \\ref{alg:cfdnas}} and illustrated in Figure \\ref{fig:fdnas}.\n\n\n\\begin{algorithm}[!h]\n\\caption{FDNAS: Federated Direct Neural Architecture Search}\n\\label{alg:fdnas}\n\n\\begin{algorithmic}\n\\SUB{Central server:}\n   \\STATE Initialize $\\mathbf w_0^{g}$ and $\\pmb \\alpha_0^{g}$.\n   \\FOR{each communication round $t = 1, 2, \\dots, T$}\n   \n   \n     \\FOR{each client $k \\in S$ \\textbf{in parallel}}\n       \\STATE $\\mathbf w_{t+1}^k, \\pmb \\alpha_{t+1}^k \\leftarrow \\textbf{ClientUpdate}(\\mathbf w_t^{g}, \\pmb \\alpha_t^{g})$\n     \\ENDFOR\n     \\STATE $\\mathbf w_{t+1}^{g}, \\pmb \\alpha_{t+1}^{g} \\leftarrow \\sum_{k=1}^K \\frac{N_k}{N} \\mathbf w_{t+1}^k, \\sum_{k=1}^K \\frac{N_k}{N} \\pmb \\alpha_{t+1}^k$\n   \n   \\ENDFOR\n   \\STATE\n   \n \\SUB{ClientUpdate($\\mathbf w, \\pmb \\alpha$):} \/\/ \\emph{On client platform.}\n   \\FOR{each local epoch $i$ from $1$ to $E$}\n    \\STATE update  $\\mathbf w, \\pmb \\alpha$ by ProxylessNAS\n  \\ENDFOR\n   \\STATE return $\\mathbf w, \\pmb \\alpha$ to server\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Further Improvement and Insights}\n\\subsubsection{Clustering-aided model compression}\n\\begin{algorithm}[tb]\n\\caption{CFDNAS: Cluster Federated Direct Neural Architecture Search}\n\\label{alg:cfdnas}\n\n\\begin{algorithmic}\n\\SUB{Cluster server:}\n   \\STATE load $\\mathbf w_0^{g}, \\pmb \\alpha_0^{g}$ by central server.\n   \\STATE $\\{S_{1},\\cdots,S_{P}\\} \\leftarrow$ (split $S_{all}$ into clusters by users' tag.)\n   \\FOR {cluster $S_{tag} \\in \\{S_{1},\\cdots,S_{P}\\}$ \\textbf{in parallel}}\n   \\FOR{each round $t = 1, 2, \\dots$}\n   \n   \n     \\FOR{each sampled client $k \\in S_{tag}$ \\textbf{in parallel}}\n       \\STATE $\\mathbf w_{t+1}^k, \\pmb \\alpha_{t+1}^k \\leftarrow \\text{ClientUpdate}_k(\\mathbf w_t^{g}, \\pmb \\alpha_t^{g})$ \n     \\ENDFOR\n     \\STATE $\\mathbf w_{t+1}^{g}, \\pmb \\alpha_{t+1}^{g} \\leftarrow \\sum_{k=1}^K \\frac{n_k}{n_{all}} \\mathbf w_{t+1}^k, \\sum_{k=1}^K \\frac{n_k}{n_{all}} \\pmb \\alpha_{t+1}^k$\n   \\ENDFOR\n   \\ENDFOR\n   \\STATE\n\n\\SUB{ClientUpdate($\\mathbf w, \\pmb \\alpha$):} \/\/ \\emph{On client platform.}\n  \\FOR{each local epoch $i$ from $1$ to $E$}\n    \\STATE update  $\\mathbf w, \\pmb \\alpha$ by ProxylessNAS\n \\ENDFOR\n \\STATE return $\\mathbf w, \\pmb \\alpha$ to server\n\\end{algorithmic}\n\\end{algorithm}\n\nAt the end of the proposed algorithm (i.e., {\\bf Algorithm \\ref{alg:cfdnas}}), the neural network architecture is learnt from the datasets of all the clients in a federated way. Although it enables the the knowledge transfer, accounting for every piece of information might result in structure redundancy for a particular group of clients. More specifically,  assume that 10 clients are collaborated together to train a model for image classification. Some clients are with images about \\textit{birds, cats, and deer} and thus can form a ``animal\" group, while other clients in the ``transportation\" group are with images about \\textit{airplanes, cars, ships, and trucks}. Although these images all contribute to the neural network structure learning in  {\\bf Algorithm \\ref{alg:cfdnas}}, the operations tailored for ``animal\" group might not help the knowledge extraction from the ``transportation\" group significantly. Therefore, an immediate idea is: could we further refine the model architecture by utilizing the clustering information of clients?\n\nTo achieve this, we further propose a clustering-aided refinement scheme into the proposed FDNAS. In particular, after $\\pmb \\alpha^{g}$converges, each client could send a tag about its data to the server, e.g.,  \\textit{animal, transportation}, which are not sensitive about its privacy. Then the server gets the clustering information about the clients: which ones are with similar data distributions, based on which the clients' set $S$ can be divided into several groups:\n\\begin{align}\n    S \\rightarrow \\{S_{1},\\cdots,S_{P}\\}.\n    \\label{eq:s-to-si}\n\\end{align}\nFinally, the clients in the same group could further refine their SuperNet by re-executing the proposed FDNAS algorithm. The proposed clustering-aided refinement scheme labeled as \\textbf{CFDNAS} is summarized in {\\bf Algorithm \\ref{alg:cfdnas}}.  \n\n\\subsubsection{Hardware-tailored model compression}\nClients might deploy models in different devices, such as mobile phone, CPU, and GPU. Previous studies show that taking hardware-ware loss into account could guide the NAS to the most efficient one in terms of the inference speed. \nFor example, ProxylessNAS uses hardware-aware loss to \\textit{avoid} $\\pmb\\alpha$ convergence to the heaviest operations at each layer.\nThe proposed FDNAS and CFDNAS could naturally integrate the hardware information into its search scheme. In particular, we could let each device send their hardware information such as \\textit{GPU, CPU} to the server. Then, the server divides these clients into several groups according to their hardware type. For the clients in the same group, a hardware-ware loss term is added to the training loss. By taking this hardware-ware loss term into the training process, the proposed algorithm will drive the SuperNet to a compact one that gives the fastest inference speed for the particular hardware-platform. The details of hardware-ware loss term refer to ProxylessNAS.\n\n\\begin{table*}[!htbp]\n\\centering\n\\begin{tabular}{lcccc}\n\\toprule\n\\textbf{\\multirow{2}{*}{Architecture}} &\\textbf{\\multirow{2}{*}{Test Acc. (\\%)}}   & \\textbf{Params} & \\textbf{Search Cost} & \\textbf{\\multirow{2}{*}{Method}} \\\\\n\\cmidrule(lr){3-4} &\n & \\textbf{(M)} & \\textbf{(GPU hours)} &\\\\\n\\hline\nDenseNet-BC~ \\citep{densenet}                       & 94.81  & 15.3 & -    & manual \\\\\nMobileNetV2  ~ \\citep{sandler2018mobilenetv2}                                   & 96.05  & 2.5   & -  & manual \\\\\n\\hline\nNASNet-A ~ \\citep{zoph2018learning}                 & 97.35      & 3.3  & 43.2K & RL      \\\\\nAmoebaNet-A ~ \\citep{amoebanet}           & 96.66      & 3.2  & 75.6K & evolution \\\\\nHireachical Evolution~ \\citep{liu2018hierarchical}          & 96.25   & 15.7 & 7.2K  & evolution \\\\\nPNAS~ \\citep{liu2018progressive}                            & 96.59    & 3.2  & 5.4K  & SMBO \\\\\nENAS~ \\citep{pham2018efficient}                    & 97.11     & 4.6  & 12  & RL \\\\\n\\hline\nDARTS~ \\citep{liu2018darts}          & 97.24 & 3.3 & 72 & gradient \\\\\nSNAS ~ \\citep{xie2018snas}        & 97.02 & 2.9  & 36  & gradient \\\\\nP-DARTS C10~ \\citep{chen2019progressive}                                & 97.50 & 3.4  & 7.2 & gradient \\\\\nP-DARTS C100~ \\citep{chen2019progressive}                                & 97.38 & 3.6  & 7.2 & gradient \\\\\nPC-DARTS ~ \\cite{xu2020pcdarts} & 97.39 &3.6 &2.5 &  gradient\\\\\nGDAS  ~ \\citep{dong2019searching}                                   & 97.07 & 3.4  & 5 & gradient \\\\\n\\hline\n\\textbf{FDNAS(ours)}                                  & \\textbf{78.75$^*$\/97.25$\\dag$}  & \\textbf{3.4}  & \\textbf{59} & \\textbf{gradient \\& federated} \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{\\textbf{CIFAR-10 performance.} $^*$: The federated averaged model's accuracy. $\\dag$: mean local accuracy, i.e., Each client completes training local epochs after downloading the server model, and then averages the local accuracy obtained by testing inference on the local test data.}\n\\label{tab_ev_cifar}\n\\end{table*}\n\n\\section{Experiments}\n\n\\subsection{Implementation Details}\nWe use PyTorch \\citep{paszke2019pytorch} to implement FDNAS and CFDNAS.\nWe searched on CIFAR-10 \\citep{cifar10} and then trained normal networks from scratch. CIFAR-10 has 50K training images, 10K test images, and 10 classes of labels. To simulate a federation scenario, we set up 10 clients. The first 3 classes of images are randomly and evenly assigned to the first 3 clients. Then, the middle 3 classes of images were randomly assigned to the middle 3 clients, and the last 4 classes of images were randomly and evenly assigned to the last 4 clients. \nEach client has 4500 images as a training set for learning $\\mathbf w$ and 500 images as a validation set for learning $\\pmb \\alpha$.\nBesides, to test the transferability of the architecture searched out by FDNAS, we used it to train on ImageNet\\citep{5206848}.\n\n\\subsection{Image Classification on CIFAR-10}\n\\subsubsection{Training Settings}\nWe use a total batch size of 256 and set the initial learning rate to $0.05$. Then, we use the cosine rule as a decay strategy for the learning rate. When training SuperNet, we use Adam to optimize $\\pmb \\alpha$ and momentum SGD to optimize $\\mathbf w$. The weight decay of $\\mathbf w$ is $3e-4$, while we do not use weight decay on $\\pmb \\alpha$.\nThe SuperNet has 19 searchable layers, each consisting of MBconv blocks, the same as ProxylessNAS \\citep{cai2018proxylessnas}.\nWe train the FDNAS SuperNet with 10 clients for a total of 125 rounds. Meanwhile, the local epoch of each client is 5. \nWe assume that all clients can always be online during the training procedure.\nAfter that, CFDNAS clusters clients-0, 1, and 2 into the GPU group to train CFDNAS-G SuperNet, and clients-3, 4, and 5 into the CPU group to train CFDNAS-C SuperNet. CFDNAS SuperNets are all searched separately for 25 rounds.\nAfter training the SuperNet, we derive the normal net from the SuperNet and then run 250 rounds of scratch training on clients via FedAvg.\nGPU latency is measured on a TITAN Xp GPU with a batch size of 128 in order to avoid severe underutilization of the GPU due to small batches. CPU latency is measured on two 2.20GHz Intel(R) Xeon(R) E5-2650 v4 servers with a batch size of 128.\nBesides, the random number seeds were set to the same value for all experiments to ensure that the data allocation remained consistent for each training.\n\n\n\\subsubsection{Results}\nSince our FDNAS is based on FedAvg and protects the privacy of clients' data, we put the federated averaged model's accuracy and clients' models mean local accuracy in the Table~\\ref{tab_ev_cifar}.\nAs demonstrated, our FDNAS normal net achieves $78.75\\%$ federated averaged accuracy and $97.25\\%$ mean local accuracy. \nIt costs 59 GPU hours. However, the federated learning framework is naturally suited to distributed training. During training, all clients can be trained simultaneously. \nOur FDNAS outperforms both evolution-based NAS and gradient-based DARTS in terms of search time cost, and our clients'local accuracy is higher than their central accuracy.\nIn addition, we use MobileNetV2 as a predefined, hand-crafted model trained under FedAvg for a fairer comparison with FDNAS. In Table~\\ref{ablation1}, our FDNAS outperforms MobileNetV2 in terms of federated averaged accuracy and mean local accuracy. See ablation study~\\ref{alation-sec} subsection for more analysis and CFDNAS's performance.\n\n\n\\subsection{Image Classification on ImageNet}\n\\begin{table*} [!t]\n\\centering\n\\begin{tabular}{lcccccc}\n\\toprule\n\\textbf{\\multirow{2}{*}{Architecture}} & \\multicolumn{2}{c}{\\textbf{Test Acc. (\\%)}} & \\textbf{Params} & $\\times+$ & \\textbf{Search Cost} & \\textbf{\\multirow{2}{*}{Search Method}} \\\\\n\\cmidrule(lr){2-3}\n& \\textbf{top-1} & \\textbf{top-5} & \\textbf{(M)} & \\textbf{(M)} & \\textbf{(GPU hours)} &\\\\\n\\hline\nMobileNet~ \\citep{howard2017mobilenets}         & 70.6 & 89.5 & 4.2 & 569  & -    & manual \\\\\nShuffleNet 2$\\times$ (v2)~ \\citep{ma2018shufflenet}    & 74.9 & - & $\\sim$5  & 591  & -    & manual \\\\\nMobileNetV2~ \\citep{sandler2018mobilenetv2}         & 72.0 & 90.4 & 3.4 & 300  & -    & manual \\\\\n\n\\hline\nNASNet-A~ \\citep{zoph2018learning}              & 74.0 & 91.6  & 5.3 & 564  & 43.2K & RL \\\\\nNASNet-B~ \\citep{zoph2018learning}              & 72.8 & 91.3  & 5.3 & 488  & 43.2K & RL \\\\\nAmoebaNet-A~ \\citep{amoebanet}        & 74.5 & 92.0  & 5.1 & 555  & 75.6K & evolution \\\\\nAmoebaNet-B~ \\citep{amoebanet}        & 74.0 & 91.5  & 5.3 & 555  & 75.6K & evolution \\\\\nPNAS~ \\citep{liu2018progressive}                & 74.2 & 91.9  & 5.1 & 588  & 5.4K  & SMBO \\\\\nMnasNet~ \\citep{tan2019mnasnet}              & 74.8 & 92.0  & 4.4 & 388  & -    & RL \\\\\n\\hline\nDARTS ~ \\citep{liu2018darts}      & 73.3 & 91.3  & 4.7 & 574  & 96    & gradient \\\\\nSNAS ~ \\citep{xie2018snas}     & 72.7 & 90.8  & 4.3 & 522  & 36  & gradient \\\\\nProxylessNAS ~ \\citep{cai2018proxylessnas}       & 75.1 & 92.5  & 7.1 & 465  & 200  & gradient \\\\\nP-DARTS-C10~ \\citep{chen2019progressive}                 & 75.6 & 92.6  & 4.9 & 557  & 7.2  & gradient \\\\\nP-DARTS-C100 ~ \\citep{chen2019progressive}             & 75.3 & 92.5  & 5.1 & 577  & 7.2  & gradient \\\\\nGDAS ~ \\citep{dong2019searching}           & 74.0 & 91.5 & 5.3 & 581  & 5  & gradient \\\\\\hline\n\n\\textbf{FDNAS(ours)}             & \\textbf{75.3} & \\textbf{92.9}  & \\textbf{5.1} & \\textbf{388}  & \\textbf{59}  & \\textbf{gradient~ \\& federated} \\\\\n\n\\bottomrule\n\\end{tabular}\n\\caption{\\textbf{ImageNet performance.} $\\times+$ denotes the number of multiply-add operations(FLOPs).}\n\\label{ev_imagenet}\n\\end{table*}\n\\subsubsection{Training Settings}\nTo test the generality on larger image classification tasks, we moved the FDNAS general net to ImageNet for training. Following the general mobile setting \\citep{liu2018darts}, we set the input image size to $224\\times224$ and the model's FLOPs were constrained to below 600M.\nWe use an SGD optimizer with a momentum of 0.9. The initial learning rate is 0.4 and decays to 0 by the cosine decay rule. Then the weight decay is 4e-5. The dropout rate is 0.2.\nIn order to fit the FDNAS net to ImageNet's image size, Layer 1, 3, 6, 8, and 16 are set to the downsampling layers.\n\n\\subsubsection{Results}\nAs shown in Table~\\ref{ev_imagenet}, we achieved SOTA performance compared to other methods. FDNAS test accuracy is $75.3\\%$, which is better than GDAS, ProxylessNAS, SNAS, and AmoebaNet. Besides, our FLOPs are 388M, which is also smaller than them. The search cost is only 59 GPU hours, which is smaller than ProxylessNAS and makes sense in real-world deployments.\nAlso, for a fairer comparison, we compare FDNAS to MobileNetV2 since they are both composed of MBconv blocks.\nOur FDNAS normal net's $75.3\\%$ accuracy outperforms MobileNetV2 by $3.3\\%$. Moreover, the MobileNetV2(1.4)'s FLOPs is 585M which is quite more dense than FDNAS 388M. but our FDNAS's accuracy still outperforms its $0.6\\%$. \nSumming up the above analysis, the model searched by our FDNAS in a privacy-preserving manner is highly transferable and it has an outstanding trade-off between accuracy and FLOPs. These show that learning the neural architecture from the data can do away with the bias caused by human effort and attain better efficiency.\n\n\n\\subsection{Ablation study}\n\\label{alation-sec}\n\\begin{table*}[!t]\n\\centering\n\\begin{tabular}{lcccccc}\n\\toprule\n\\textbf{\\multirow{2}{*}{Architecture}} & \\multicolumn{2}{c}{\\textbf{Latency (ms})} & \\textbf{Params}  & \\textbf{$\\times+$} & \\textbf{Search Cost} & \\textbf{\\multirow{2}{*}{Test Acc.(\\%)}} \\\\ \n\\cmidrule(lr){2-3}\n& \\textbf{GPU} & \\textbf{CPU} & \\textbf{(M)} & \\textbf{(M)} & \\textbf{(GPU hours)} &\\\\ \\hline\n MobileNetV2       & 52.31  &  890.69  &2.5 &296.5  & - &  68.45$\\ddag$\/96.51$\\dag$\\\\ \\hline\n\n FDNAS           & 52.78  &  600.17 &  3.4 &346.6  &  59.00 &  78.75$\\ddag$\/97.25$\\dag$\\\\ \\hline\n CFDNAS-G        & 40.33 & 463.86 &  3.3  &318.4 &  3.53 &  73.60$\\ddag$\/ 98.93$\\dag$\\\\\n CFDNAS-C        & 31.00 & 186.52  &  2.0 &169.3 &  3.46 &  71.29$\\ddag$\/ 93.01$\\dag$\\\\\n\\bottomrule\n\\end{tabular}\n\\caption{\\textbf{A comparison between MobileNetV2, FDNAS and CFDNAS on CIFAR-10}: \n$\\dag$ and $\\ddag$ is explained in Table~\\ref{tab_ev_cifar}.}\n\n\\label{ablation1}\n\\end{table*}\n\\begin{table*}[!t]\n\\centering\n\\begin{tabular}{lccccccc}\n\\toprule\n\\textbf{\\multirow{2}{*}{Architecture}} &\\multirow{2}{*}{\\textbf{Client ID}} & \\textbf{Params}  & \\textbf{$\\times+$} &\\textbf{Search Cost}  &{\\textbf{Client Local}} &\\textbf{Mean Local} \\\\ \n&       &                     \\textbf{(M)} & \\textbf{(M)} &\\textbf{(GPU hours)}&\\textbf{Acc.(\\%)}   & \\textbf{Acc.(\\%)} &\\\\ \\hline\n\nFDNAS \n        &0,1,2    &  3.38     & 346.64   &59 & 98.14\/99.35\/98.90    & 98.79   \\\\ \n                       \nnaive-CFDNAS-G  \n   &0,1,2    & 3.70      & 356.83   &18 &97.56\/98.11\/97.36  &97.67  \\\\ \n\\textbf{CFDNAS-G}  \n   &0,1,2    & \\textbf{3.33}      & \\textbf{318.44}   &\\textbf{3.53} &\\textbf{98.39\/99.02\/99.38}  &\\textbf{98.93}  \\\\ \\hline\nFDNAS  \n                        &3,4,5    &3.38 &346.64 &59 & 94.20\/91.81\/93.04    & 93.01   \\\\ \n                       \nnaive-CFDNAS-C  \n   &3,4,5    & 1.92     & 187.89   &18 &88.61\/89.88\/90.25  &89.58  \\\\ \n\\textbf{CFDNAS-C} \n   &3,4,5    & \\textbf{2.03}      & \\textbf{169.35}   &\\textbf{3.46} &\\textbf{93.21\/92.73\/93.12}  &\\textbf{93.02}  \\\\\n\\bottomrule\n\n\\end{tabular}\n\\caption{\\textbf{Enhancement by CFDNAS}. ``naive-CFDNAS'' means that SuperNet for CFDNAS is not inherited from FDNAS, but is searched directly in the cluster group from scratch.}\n\\label{ablation3}\n\\end{table*}\n\\begin{table*}[!t]\n\\centering\n\\begin{tabular}{lccccc}\n\\toprule\n\\textbf{\\multirow{2}{*}{Architecture}} & \\textbf{Params}  & \\textbf{$\\times+$} &\\textbf{Search Cost}  & \\textbf{Test Acc.} \\\\ \n&   \\textbf{(M)} & \\textbf{(M)} &\\textbf{(GPU hours)}& \\textbf{(\\%)}  \\\\ \\hline\nDNAS(on client-0)       & 2.54      & 264.35   &6.4 & 95.83        \\\\ \nDNAS(on client-1)       & 3.50      & 330.58   &6.5 & 97.04        \\\\ \nDNAS(on client-2)       & 2.85      & 269.76   &8.4 & 97.55        \\\\ \nDNAS(on client-3)       & 2.77      & 258.90   &9.9 & 87.33        \\\\ \nDNAS(on client-4)       & 2.28      & 233.99   &6.7 & 88.38        \\\\ \nDNAS(on client-5)       & 3.27      & 299.63   &7.5 & 87.53        \\\\ \nDNAS(on client-6)       & 2.78      & 297.03   &7.7 & 97.83        \\\\ \nDNAS(on client-7)       & 3.25      & 336.20   &7.1 & 96.83        \\\\ \nDNAS(on client-8)       & 2.68      & 263.89   &8.8 & 97.38        \\\\ \nDNAS(on client-9)       & 2.80      & 286.46   &7.8 & 97.47        \\\\ \nmean           & 2.87      & 284.08   &7.7 & 94.31        \\\\ \\hline\n\nDNAS(on collected CIFAR-10)           & 5.02      & 500.71   &24.5 & 96.71        \\\\ \\hline\nFDNAS(on all clients)          & 3.38      & 346.64   &59.0 & 78.75$\\ddag$\/97.25$\\dag$\\\\ \\hline\nCFDNAS-G(on client-0, 1, 2)        & 3.33      & 318.44   &3.53 & 73.60$\\ddag$\/98.93$\\dag$\\\\\nCFDNAS-C(on client-3, 4, 5)       & 2.03      & 169.35   &3.46 & 71.29$\\ddag$\/93.01$\\dag$\\\\\n\\bottomrule\n\\end{tabular}\n\\caption{\\textbf{Compared with conventional DNAS.} $\\dag$ and $\\ddag$ is explained in Table~\\ref{tab_ev_cifar}. The DNAS search algorithm is ProxylessNAS like FDNAS, but uses well-collected data rather than federated learning.\n}\n\\label{ablation2}\n\\end{table*}\n\n\\subsubsection{Effectiveness of CFDNAS}\nFrom the Table~\\ref{ablation1}, we demonstrate CFDNAS for comparison.\nCFDNAS-G (GPU platform) and CFDNAS-C (CPU platform) are trained for 25 epochs each based on the inherited FDNAS SuperNet.\nBoth CFDNAS normal nets have smaller FLOPs than FDNAS. Both nets also have lower GPU\/CPU inference latencies than FDNAS and the hand-crafted MobileNetV2.\nWe then study the improvement in accuracy by CFDNAS in Table~\\ref{ablation3}.\nBenefiting from the clustering approach, for the GPU group (including clients-0, 1, and 2), our CFDNAS-G search model achieves $98.92\\%$ accuracy. It is more accurate than the original FDNAS on clients of the same GPU group and requires only 3.53 GPU hours of SuperNet adaptation, which is a negligible additional cost.\nThe ``naive-CFDNAS'' has no inheritance parameters ($\\mathbf w$ and $\\pmb \\alpha$) and no SuperNet's ``meta-test'' adaptation. As a result, the convergence time of naive-CFDNAS takes 18 GPU hours, which is 5.1 times that of CFDNAS. In the same GPU group, our CFDNAS-G is $1.26\\%$ more accurate than naive-CFDNAS, while its FLOPs are smaller.\nFor the CPU group (including clients-3, 4, and 5), our CFDNAS-C also outperforms FDNAS and naive-CFDNAS-G in terms of accuracy and FLOPs.\nCPU group's data is harder than others, but our CFDNAS-C is still more accurate than naive-CFDNAS-C and more stable than FDNAS.\n\nCompared to FDNAS, CFDNAS gets a better trade-off between accuracy and latency due to the meta-learning mechanism.\nCompared to naive-CFDNAS, CFDNAS gets better performance because it inherits $\\mathbf w$ and $\\pmb \\alpha$ from the FDNAS SuperNet, which is fully trained with data from all clients.\nAlso, because the FDNAS-based ``meta-trained\" SuperNet performs the search, it helps meta-adaptation cost less than naive-CFDNAS.\n\n\\subsubsection{Contributions of federated mechanism}\nIn Table~\\ref{ablation2}, we show the effects of using a traditional DNAS with collected data (including all data) and a single-client DNAS (including only local data, with no federated training).\nDNAS searches the model directly on the clients' data but requires data collection in advance, and it achieves a central accuracy of $96.71\\%$. However, our FDNAS still has higher local accuracy than DNAS while protecting data privacy.\nBesides, our FDNAS has smaller FLOPs than single-client conventional DNAS results. Our federated averaged accuracy is $78.75\\%$, and local accuracy is $97.25\\%$ which is $2.94\\%$ higher than single-client local average accuracy. Thanks to the federated mechanism, FDNAS can use data from a wide range of clients to search for more efficient models. At the same time, it trains models with higher accuracy. Privacy protection and efficiency will be beneficial for the social impacts and effectiveness of actual machine learning deployments.\n\n\\subsubsection{Contributions of directly search}\nFor a fairer comparison, we use MobileNetV2, which is also composed of MBconv blocks, as a predefined hand-crafted neural architecture trained in FedAvg.\nWe train the normal net of FDNAS with the collected data from all clients and obtain an accuracy of $96.03\\%$ in a centralized way, a negligible difference compared to the centralized results of MobileNetV2.\nThen we present the FedAvg results for FDNAS and MobileNetV2 in Table~\\ref{ablation1}. \nOur FDNAS federated averaged accuracy and local accuracy are $6.63\\%$ and $1.4\\%$ higher than MobileNetV2, respectively.\nAlso, although both have similar GPU latency, FDNAS can be faster than MobileNetV2 on the CPU platform. So compared to the FDNAS, MobileNetV2 is not optimal for diverse clients in federated scenarios.\nThe FDNAS search architecture from the dataset is superior to hand-crafted models, and it demonstrates that the NAS approach can provide a significant improvement over human design in federated learning.\n\n\\section{Discussion and Future Work}\nWe propose FDNAS, a privacy-preserving neural architecture search scheme that directly searches models from clients' data under the framework of federated learning. Different from previous federated learning approaches, the complete neural architecture is sought from clients' data automatically without any manual attempts. Extensive numerical results have shown that our FDNAS greatly improves the model's performance and efficiency compared to predefined hand-crafted neural architectures in federated learning.\nOn the other hand, our FDNAS model achieves state-of-the-art results on ImageNet in a mobile setting, which demonstrates the transferability of FDNAS. Moreover, inspired by meta-learning, CFDNAS, an extension to FDNAS, can discover diverse high-accuracy and low-latency models adapted from a SuperNet of FDNAS at a very low computational cost. In future work, we will extend our FDNAS to search for different tasks such as object detection, semantic segmentation, model compression, etc., as well as larger-scale datasets with more categories.\n\\newpage\n\\section{Ethical Impact}\nAutoML has been widely used in many fields, such as searching deep neural networks and tuning hyper-parameters in computer vision and natural language processing. \nIn general, NAS brings some important implications for the neural network design, and here we focus on the use of FDNAS to provide a NAS solution in a better  privacy-preserving way and the use of CFDNAS to provide a suitable neural network architecture for each client. There are many benefits of the schemes, such as exploiting the diversity of models, respecting the diversity of clients, and reducing risks of privacy and security. \n\nNowadays, there is a lot of research about federated learning for improving deep learning's fairness, safety, and trustworthiness. \nTo liberate expensive manual attempts in federated learning, we believe that the impacts of FDNAS in real-world scenarios are worthy of attention. In addition, privacy leak from the gradient update is still an open problem in machine learning. \nIt is worth exploring a more secure AutoML system combining with some potential privacy protection methods such as differential privacy, homomorphic encryption, and secure multi-party computing.\n\\input{ref.bbl}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe sources of majority of cosmic rays (CRs) are believed to be supernovae\n(SNe) and supernova remnants (SNRs), which are capable of accelerating\nparticles to multi-PeV energies. As CRs propagate through the interstellar\nmedium (ISM) they scatter off the magnetic turbulences in a process that on large\nscales can be described with a diffusive transport equation\n\\citep[][]{1964ocr..book.....G,1990acr..book.....B}. The spectrum of\nturbulence controls the value of the diffusion coefficient and its energy\ndependence because the CR particles most efficiently scatter on turbulence that is\ncomparable in size to their gyroradii. The turbulence is\nassumed to initially form at large scales and then cascade down to smaller\nscales thus affecting particles at all rigidities. The power-law shape of the\nenergy spectrum of turbulence\n\\citep{1941DoSSR..30..301K,1964SvA.....7..566I,1965PhFl....8.1385K} translates\ninto a power-law rigidity dependence of the diffusion coefficient with the\nindex taking a value between 0.3 and 0.6. Observations of the ratios of stable secondary to\nprimary species in CRs and the abundances of radioactive secondaries are\nusually employed to determine both the power-law index and normalization of\nthe diffusion coefficient averaged over a significant volume of the\ninterstellar space surrounding the Solar system\n\\citep[e.g.,][]{1998ApJ...509..212S,2007ARNPS..57..285S,JohannessonEtAl:2016},\ntypically several kpc in radius. Recent estimates of the power-law index are\nclustered around 0.35 with a normalization of about $4\\times10^{28}$ cm$^2$\ns$^{-1}$ at a rigidity of 4 GV \\citep[e.g.,][]{BoschiniEtAl:2018b}.\n\nPulsars are rapidly spinning and strongly magnetized neutron stars that are at\nthe final stage of the stellar evolution. They are formed in SN explosions\nand can often be found inside their associated SNR. \nPulsars represent a class of CR sources that has not been considered as extensively as the more usual scenario of acceleration in SNR shocks, but the fact that they may produce CRs is well-known \\citep{1981IAUS...94..175A,1987ICRC....2...92H,1989ApJ...342..807B}. However,\nrecent measurements of positrons in CRs \n\\citep{AdrianiEtAl:2009, AckermannEtAl:2012, AguilarEtAl:2014}\nin excess of predictions of propagation models\n\\citep{1982ApJ...254..391P,1998ApJ...493..694M}, made under an assumption of\ntheir entirely secondary production in the ISM, elevated\npulsars to be one of the primary candidate sources responsible for this excess\n\\citep[e.g.,][]{2009JCAP...01..025H,2009PhRvL.103e1101Y,2009PhRvD..80f3005M}.\nThe magnetospheres of rapidly\nrotating neutron stars are capable of producing electrons and positrons in\nsignificant numbers and accelerating them to very high energies resulting in a\npulsar wind nebula (PWN) that is observable from radio to high-energy $\\gamma$-ray{s} \\citep{GaenslerSlane:2006}; an archetypical example of such a source is the Crab pulsar and its PWN. The accelerated particles eventually escape from the PWN into the ISM and some of them can reach the Solar system. Therefore, PWNe can\nbe natural candidates responsible for the puzzling excess in CR positrons\nobserved by several experiments.\n\nRecent observations of the extended TeV emission around Geminga and PSR~B0656+14\nPWN by the High Altitude Water Cherenkov (HAWC) telescope constrain the\ndiffusion coefficient in their vicinities to be about two orders of\nmagnitude smaller than the average value derived from observations of CRs\n\\citep{AbeysekaraEtAl:2017}. Application of such slow diffusion to the local\nMilky Way results in a contradiction with other CR observations, in particular\nobservations of high-energy CR electrons and positrons.  Fast energy losses of\nTeV particles through inverse Compton (IC) scattering and synchrotron emission\nlimits their lifetime to $\\sim$100~ky \\citep{1998ApJ...509..212S}.\nIf such slow diffusion is representative for the ISM\nwithin about a few hundred pc of the Solar system, the sources of the TeV particles detected at Earth also need to be within a few 10s of pc.\n\\citet{ProfumoEtAl:2018} highlighted that such nearby sources have\nnot been identified and proposed a\ntwo-zone diffusion model with the slow diffusion confined to a small region\naround the PWN. Other authors have considered similar scenarios with varying\ndetails \\citep{TangPiran:2018,FangEtAl:2018,EvoliEtAl:2018}. Interestingly, a\nsimilar suppression of the CR diffusion was observed in the Large Magellanic\nCloud around the 30 Doradus star-forming region, where an analysis of combined\n$\\gamma$-ray{} \nand radio observations \nyielded a diffusion coefficient, averaged over a region with radius\n200--300 pc, an order of magnitude smaller than the typical value in the Milky\nWay \\citep{2012ApJ...750..126M}. Meanwhile, a strong suppression of the\ndiffusion coefficient around a SNR due to excitation of magnetic\nturbulence by escaping CRs\nwas predicted some time before the HAWC observations were reported \\citep[e.g.,][see also references therein]{2008AdSpR..42..486P,MalkovEtAl:2013,2016PhRvD..94h3003D}. A similar mechanism may also be at work in PWN.\n\nIn this paper the two-zone diffusion model is explored using the latest\nversion of the GALPROP\\footnote{\\url{http:\/\/galprop.stanford.edu} \\label{url}}\npropagation code \\citep{PorterEtAl:2017,JohannessonEtAl:2018}.  The results indicate that such a model is a viable interpretation for the HAWC observations and confirm similar conclusions made by other authors. Predictions for lower-energy $\\gamma$-ray{} emission that can be observed with the {\\it Fermi} Large Area Telescope\n({\\em Fermi}-LAT) are made and the contribution of energetic positrons coming from Geminga to the\nobserved CR positron flux in different scenarios is studied. Effects of the\nsize of the slower diffusion zone (SDZ) and the properties of the accelerated electrons\/positrons are taken into\naccount as is the effect of the proper motion of Geminga. Unless\nboth Geminga and PSR~B0656+14 are special cases there are expectations for more regions of\nslower diffusion around other PWNe in the Milky Way.  The implications that\nsuch inhomogeneity of the diffusion in the ISM can have on the CR distribution\nthroughout the Milky Way is also explored.\n\n\\section{A Model for Geminga}\n\n\\subsection{Physical Setup}\n\nIt is assumed that the Geminga pulsar is injecting accelerated electrons and\npositrons into the ISM in equal numbers with a fraction $\\eta$ of its\nspin-down power converted to pairs.   After injection the particles\npropagate via a diffusive process.  The pulsar\nparameters used for this paper are identical to those from\n\\citet{AbeysekaraEtAl:2017} and the\nenergy distribution of the injected electrons\/positrons is described with a\nsmoothly joined broken power-law\n\\begin{equation}\n  \\frac{dn}{dp} \\propto E_k^{-\\gamma_0}\\left[ 1 + \\left( \\frac{E_k}{E_b}\n  \\right)^\\frac{\\gamma_1-\\gamma_0}{s} \\right]^{-s}.\n  \\label{eq:injectionSpectrum}\n\\end{equation}\nHere $n$ is the number density of electrons\/positrons, $p$ is the particle\nmomentum, $E_k$ is the particle kinetic energy, and $\\gamma_1$ is a\npower-law index at high energies.  The smoothness parameter $s=0.5$ is assumed\nconstant and so is the low-energy index $\\gamma_0 = -1$ and the break energy\n$E_b=10$~GeV. This low-energy break effectively truncates the injected particle spectrum that is not expected to extend unbroken to lower energies \\citep[e.g.,][]{Amato:2014}.  The break is required to keep the value of $\\eta$ below 1 for the largest values of $\\gamma_0$ considered.  The truncation occurs at energies below that explored in this paper and has no effect on the results.\nThe injection spectrum is normalized so that the total power injected is given by the expression\n\\begin{equation}\n  L(t) = \\eta \\dot{E}_0 \\left( 1 + \\frac{t}{\\tau_0} \\right)^{-2},\n  \\label{eq:PulsarPower}\n\\end{equation}\nwhere $\\dot{E}_0$ is the initial spin-down power of the pulsar, and $\\tau_0 =\n13$~kyr.  The initial spin-down power is calculated using the current\nspin-down power of $\\dot{E}=3.26\\times10^{34}$ erg s$^{-1}$ assuming that the\npulsar age is $T_p = 340$~kyr.  The distance to Geminga has been determined to\nbe 250~pc \\citep{FahertyEtAl:2007}. The spatial grid for the propagation\ncalculations with GALPROP is right handed with the Galactic centre (GC) at the origin, the Sun\nat $(x,y,z)=(8.5, 0, 0)$~kpc, and the $z$-axis oriented toward the Galactic\nnorth pole. This coordinate system places Geminga at $(8.7407, 0.0651, 0.0186)$~kpc at the current epoch.\n\nPrevious interpretations of the HAWC observations have assumed that the Geminga\npulsar is a stationary object\n\\citep{ProfumoEtAl:2018,TangPiran:2018,FangEtAl:2018,EvoliEtAl:2018}.  Only\n\\citet{TangPiran:2018} discussed the effect of the proper motion of Geminga,\nbut concluded that it has little effect on electrons generating the observed\nTeV $\\gamma$-ray{} emission because of their short cooling timescale.\n\\citet{FahertyEtAl:2007} measured the proper motion of Geminga to be 107.5 mas\nyr$^{-1}$ in right ascension and 142.1 mas yr$^{-1}$ in declination.  Transforming this into\nGalactic coordinates leads to $-80.0$ mas yr$^{-1}$ in longitude and 156.0 mas\nyr$^{-1}$ in latitude.\nAssuming a constant proper velocity that is currently\nperpendicular to the line-of-sight, the corresponding velocity in the spatial\ngrid coordinate system is $(v_x, v_y, v_z) = (24.3, -89.9,\n182.6)$~km s$^{-1}$ with Geminga originally located at $(8.7320, 0.0963, -0.0449)$~kpc.  \nIn this case, Geminga was born in a SN explosion of a star that\nhas travelled from the direction of the Orion OB1a association\n\\citep{SmithEtAl:1994}. While the TeV $\\gamma$-ray{} emission is not significantly\naffected by the proper motion, the large traveled distance and proximity of\nGeminga to the Sun should produce a ``trailing'' tail of CRs whose $\\gamma$-ray{} emissions may be observable\nat lower energies with {\\em Fermi}-LAT.\n\nFor the two-zone diffusion model, the diffusion coefficient in a confined\nregion around the pulsar is assumed to be lower than that in the ISM due to\nthe increased turbulence of the magnetic field.\nThis region will hereafter be called a ``slower diffusion zone'' (SDZ). \nIt is also assumed that the stronger turbulence does not change the power\nspectrum and hence the rigidity dependence of the diffusion coefficient is the\nsame for the SDZ and the ISM. Let $r$ be the distance from the center of the\nSDZ, then the spatial dependence of the diffusion coefficient is given by\n\\begin{equation}\n  D\\!=\\!\\beta \\left( \\frac{R}{R_0} \\right)^\\delta\n  \\begin{cases}\n    \\displaystyle D_z, & r < r_z, \\\\\n    \\displaystyle D_z \\left[ \\frac{D_0}{D_z} \\right]^{\\frac{r-r_z}{r_t-r_z}}, & r_z \\le r \\le r_t, \\\\\n    D_0, & r > r_t. \n  \\end{cases} \n  \\label{eq:diffusion}\n\\end{equation}\nHere $\\beta$ is the particle velocity in units of the speed of light,\n$D_0$ is the normalization of the diffusion coefficient in the general ISM,\n$D_z$ is the normalization of the diffusion coefficient within the SDZ\nwith radius $r_z$, $R$ is the particle rigidity, and $R_0=4$ GV is the normalization (reference) rigidity. In the transitional layer between $r_z$ and $r_t$, the\nnormalization of the diffusion coefficient increases exponentially with $r$\nfrom $D_z$ to the interstellar value $D_0$. Depending on the exact origin of\nthe SDZ, its radius can be time dependent as can its location. To be\nconsistent with the HAWC observations, both $r_z$ and $r_t$ have to be of the\norder of a few tens of pc at the current time.\n\nThe effect of the SDZ around the PWN on the spectra and distribution of\ninjected electrons\/positrons and their emission may depend on its origin. \nTwo general categories distinguished by the origin of the SDZ are\nconsidered. For the first category the SDZ is due to events external to the PWN itself  (external SDZ) that may be related to the\nprogenitor SNR, or surrounding environment.  It is assumed that the\nparticle propagation time in the zone is much longer than the time necessary\nto generate such a zone itself (``instant'' generation) and its location is\nfixed.  For the second category, the SDZ is assumed to be associated with and\ngenerated by the PWN itself (PWN SDZ).\nTherefore, the SDZ is moving with the Geminga pulsar and its size increases\nproportionally to the square root of time. This is in qualitative\nagreement with the evolution of a PWN as determined from simulations\n\\citep[e.g.][]{vanderSwaluw:2003}.\n\nFor the PWN SDZ, the expectation is that there may be a link\nbetween the pulsar that is generating the magnetic turbulence and the surrounding\ninterstellar plasma. In this case, the pulsar would transfer a momentum to the medium within the SDZ.\nTo estimate the evolution of the velocity of Geminga a simple model is put forward. It is assumed that\nthe radius of the SDZ region $r_z$ around Geminga is increasing in a diffusive manner so that $r_z(t) = \\mu \\sqrt{t}$, where $\\mu$ is a constant. It is further assumed that a fraction $f$ of the ISM within\n$r_z$ is swept by the PWN and sped up to the velocity $v$ of the pulsar. Using conservation of momentum $dp = -v dM$ gives\n$v=v_0 (M_0\/M)^2$, where $v_0$ and $M_0\\approx M_\\odot$ are the initial velocity and mass of the system, respectively.\nSubstitution of $r_z(t)$ and $v(M)$ into the mass conservation formula $dM = f \\pi \\rho r_z^2 v dt$, yields the time evolution of the velocity of Geminga:\n\\begin{equation}\n  v = \\frac{v_0}{\\left(3 A_d v_0 t^2 \/2 + 1\\right)^{2\/3}}.\n  \\label{eq:vDiffusion}\n\\end{equation}\nHere $A_d = f \\pi \\rho \\mu^2 \/ M_0$ is the drag coefficient and $\\rho \\approx\n0.03$~$M_\\odot$~pc$^{-3}$ is the mass density of the ISM.\nEq.~(\\ref{eq:vDiffusion}) can be integrated to get the full traveled distance:\n\\begin{equation}\n \\lambda(t) = v_0 t\\ _2F_1\\left( \\frac{1}{2}, \\frac{2}{3}; \\frac{3}{2}; -\\frac{3 A_d v_0 t^2}{2} \\right),\n  \\label{eq:lDiffusion}\n\\end{equation}\nwhere $_2F_1$ is the hypergeometric function. Further assuming that $r_z(T_p)\n\\approx 30$~pc and that $f\\approx 10^{-3}$, the distance traveled by Geminga\nis $\\lambda(T_p) \\approx 330$~pc and its initial velocity is $v_0 \\approx\n4300$~km s$^{-1}$. This places the birth of Geminga at the location of Orion\nOB1a, which is at a distance of 330~pc from its current location. \nThis is a very intriguing possibility, but in this case $v_0$ is much higher\nthan the observed velocities of any other neutron stars that reach only up to\n1000~km s$^{-1}$ \\citep[e.g.,][]{HobbsEtAl:2005}.  Assuming $f\\approx 10^{-4}$\nresults in a more reasonable speed of $v_0 \\approx 400$~km s$^{-1}$ and a\ntotal distance traveled of $\\lambda(T_p) \\approx 100$~pc.  A moderate slow\ndown agrees with observations of older pulsars having, on average, smaller velocities than young pulsars \\citep{HobbsEtAl:2005}.\n\n\n\\subsection{Calculation Setup}\n\nThe calculations are made using the latest release of the GALPROP\ncode\\textsuperscript{\\ref{url}} \\citep{PorterEtAl:2017,\nJohannessonEtAl:2018}.  The GALPROP code solves the diffusion-advection\nequation in three spatial dimensions allowing for diffusive re-acceleration in\nthe ISM -- see the website and GALPROP team papers for full details.  Of major\nrelevance for this paper, the code fully accounts for energy losses due to synchrotron\nradiation and inverse Compton (IC) scattering. The resulting synchrotron and\nIC emission is calculated for an observer located at the position of the Solar\nsystem.  The IC calculations \\citep{2000ApJ...528..357M} take into account the\nanisotropy of the interstellar radiation field (ISRF). The current\ncalculations employ the magnetic field model of \\citet{SunEtAl:2008} described\nby their Eq.~(7) and the R12 model for the ISRF developed by\n\\citet{PorterEtAl:2017}. While the turbulent magnetic field\nis expected to be larger in the SDZ, the magnetic field model is not updated\nto account for this. The expected increase in the synchrotron cooling rate and\ncorresponding electromagnetic emissions will not significantly affect the results presented below.\n\nThe code has also been enhanced to use\nnon-equidistant grids to allow for increased resolution in particular areas of\nthe Milky Way, in this case around Geminga.\nThis improvement to {\\it GALPROP}\\ is inspired by the the Pencil Code\\footnote{See Section 5.4 of\n  \\url{http:\/\/pencil-code.nordita.org\/doc\/manual.pdf}}\n\\citep{BrandenburgDobler:2002}, where the usage of analytic functions can have advantages in terms of speed and memory usage compared to purely numerical implementations for non-uniform grid spacing.\nThe current run uses the grid function\n\\begin{equation}\n  z(\\zeta) = \\frac{\\epsilon}{a}\\tan\\left[ a\\left( \\zeta - \\zeta_0 \\right)  \\right] + z_0\n  \\label{eq:gridFunction}\n\\end{equation}\nfor all spatial coordinates $\\zeta = x, y, z$, where $\\epsilon, a, \\zeta_0, z_0$\nare parameters. This function maps from the linear grid $\\zeta$ to the\nnon-linear grid in each of the spatial coordinates $x, y, z$. The equations are solved on the $\\zeta$ grid after the\ndifferential equations have been updated to account for the change in first\nand second derivatives.\n\nThe parameters of the grid function are chosen such that the resolution is 2\npc at a central location that is defined below for each calculation, but goes\nup to $0.1$\\,kpc at a distance of $700$\\,pc from the grid center.  This setup\nprovides about 0.2 degree resolution on the sky for objects\nlocated at a distance of 250~pc along the line of sight towards the center of the\ngrid.  To minimize artificial asymmetry, the grid\nhas the current location of Geminga close to the center of\na pixel and close enough to the center of the non-equidistant grid so that\nthere is little distortion due to the variable pixel size.  For the chosen\nparameters the grid size is approximately uniform within a box having a width of\n$\\sim60$~pc, which is not enough to enclose the entire evolution of the\nGeminga PWN.  The resolution and size of the grid is bounded by computation costs; the selected parameters are a result of a compromise\nbetween accuracy, memory requirements, and computation speed.  This\nnecessarily means that the start and (or) the end of the evolution of Geminga are (is) not fully resolved.  Because the diffusion process smooths the distribution of particles as time evolves, the grid is chosen such that the current location of Geminga, and hence the TeV $\\gamma$-ray{} emission, is\nalways well resolved, but sometimes this may come with a small decrease of spatial resolution at its birth place.\n\nThe calculations are performed in a square box with a width of 8~kpc.\nThis is wide enough so that the CR propagation calculations in the Solar neighborhood are not affected by the boundary conditions. The non-equidistant grid allows the boundaries to be extended this far without imposing large computational\ncosts.  A fixed timestep of 50 years is\nused for the calculations.  This is small enough to capture the propagation and\nenergy losses near the upper boundary of the energy grid, which is 1~PeV. The\nupper energy boundary is chosen to be well above that for the particles\nproducing the HAWC data.  The lower energy\nboundary is set at 100 MeV, much lower than the cutoff in the injection\nspectrum.  The energy grid is logarithmic with 16 bins per decade.\n\nThe values of $D_0 = 4.5 \\times 10^{28}$ cm$^2$ s$^{-1}$ and $\\delta=0.35$ are\nchosen to match the latest AMS-02 data on secondary CR species (see Section~\\ref{sec:MW}). The SDZ diffusion coefficient of $D_z = 1.3 \\times 10^{26}$ cm$^2$ s$^{-1}$ at $R_0=4$ GV corresponds to the value derived from the HAWC observations at higher energies \\citet{AbeysekaraEtAl:2017}.  No attempt is made to independently fit the latter to the HAWC observations because each propagation calculation is\ncomputationally expensive, taking $\\sim$ 24\nhours to complete on a modern 40 core CPU. The calculations also include\ndiffusive re-acceleration with an Alfv{\\'e}n speed $v_A = 17$~km s$^{-1}$, as determined from the fit to the secondary-to-primary data.  \nTo save CPU time the calculations are made using electrons only,\nbecause the energy losses and propagation of positrons and electrons are\nidentical.  When comparing to the measured positron flux at Earth, the\ncalculated particle flux is simply divided by two.  The IC emission is\ncalculated on a HEALPix \\citep{GorskiEtAl:2005} grid having an order of 9\ngiving a resolution of about 0.1 degrees.  The IC emission is evaluated on a\nlogarithmic grid in energy from 10~GeV to 40~TeV with 32 energy planes.\n\n\\subsection{Results}\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{fig1a}\\\\\n  \\includegraphics[width=0.48\\textwidth]{fig1b}\n  \\caption{Surface brightness of the modelled IC emission shown as a function\n    of angular distance from the current location of Geminga. The models shown\n    here assume Geminga is stationary, $\\gamma_1=2.0$, and the SDZ is\n    stationary and centered at Geminga.  The top panel shows the energy range\n    8~TeV to 40~TeV compared to the observations of HAWC\n    \\citep{AbeysekaraEtAl:2017}. The bottom panel shows the model predictions\n  for the energy range 10~GeV to 30~GeV. The different curves represent\ndifferent assumptions about the size of the SDZ, encoded as SDZ($r_z$, $r_t$)\nin the legend.}\n  \\label{fig:StationarySB}\n\\end{figure}\n\n\nFor a comparison with results of previous studies the first set of calculations is performed\nassuming that there is no proper motion of Geminga.  The index of the injection spectrum is\nfixed to $\\gamma_1 = 2.0$ and the efficiency parameter is $\\eta=0.26$. The SDZ\nis centered at the current location of Geminga $(l_G, b_G) = (195.\\!\\!^\\circ14, 4.\\!\\!^\\circ26)$ and is static in size. The spatial grid is also centered\nat the same location. Several combinations of $(r_z,r_t)$ are used,\ncovering a range \nfrom 30~pc to 50~pc for $r_z$ and from 50~pc to 70~pc for $r_t$. \nCalculations are also made using a model with $(r_z,r_t) =\n(\\infty, \\infty)$ for a comparison with the results from\n\\citet{AbeysekaraEtAl:2017}. Figure~\\ref{fig:StationarySB} (top panel) shows the angular\nprofile of the surface brightness of the modeled IC emission evaluated over the\nenergy range from 8~TeV to 40~TeV and compared with the data from HAWC.  The\nresulting profile is clearly independent of the size of the SDZ,\nbecause the cooling time limits the diffusion length at the corresponding CR\nelectron energies ($\\gtrsim 100$~TeV) to be $\\lesssim 10$~pc.  The same cannot be said for the IC emission evaluated\nfor the energy range from 10~GeV to 30~GeV, which is also shown in\nFigure~\\ref{fig:StationarySB} (bottom panel).  Because of the longer cooling timescales the\nemission profile is highly sensitive to the size of the SDZ, via both $r_z$\nand $r_t$. A smaller SDZ\nsize leads to a correspondingly lower number density of CR electrons\/positrons within the zone, which produces a flatter profile and fainter emission in the GeV energy\nrange.\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{fig2a}\\\\\n  \\includegraphics[width=0.48\\textwidth]{fig2b}\n  \\caption{Spectrum of IC emission averaged over a $10^\\circ$ wide region\n    around the current location of Geminga. The models shown here assume\n    Geminga is stationary and that the SDZ is stationary and centered at\n    Geminga.  The top panel shows the spectrum for different values of\n    $\\gamma_1$ as indicated in the legend, but using fixed values of $(r_z,\n    r_t)=(30,50)$~pc. The bottom panel shows the spectrum for models with\n    different pairs of $(r_z, r_t)$, but fixed value of $\\gamma_1=2.0$. The\n    green shaded region corresponds to HAWC observations assuming a diffusion\n    profile for the spatial distribution \\citep{AbeysekaraEtAl:2017} and the\n  magenta arrow is the upper limit from observations with the MAGIC telescope\n  corrected for the diffusion profile \\citep{TangPiran:2018}.}\n  \\label{fig:StationarySpectrum}\n\\end{figure}\n\nCalculations with different values of $\\gamma_1=1.8$ and 2.2 \nare made using $(r_z, r_t)=(30,50)$~pc.  The conversion efficiency, $\\eta$, is\ncorrespondingly updated for these models to provide consistency with the HAWC data. The values are\n$\\eta = 0.18$ and $\\eta = 0.75$ for $\\gamma_1=1.8$ and $\\gamma_1=2.2$,\nrespectively.  The average spectrum of the IC emission over a circular region\nof $10^\\circ$ in radius around the current location of Geminga is shown in\nFigure~\\ref{fig:StationarySpectrum}.  The models are all tuned to agree\nwith the HAWC data and, therefore, the predicted profiles differ significantly at lower energies.\nAlso shown is the prediction made using the isotropic assumption for the IC\ncross section (dotted curve in upper panel), which is a commonly used approximation that ignores the angular dependence of the ISRF.\nUsing the isotropic approximation under-predicts the emission by $\\sim$10\\% at 10~TeV, and $\\sim25$\\% at 10~GeV.  The discrepancy between the IC\nemission calculated with the realistic angular distribution of the ISRF and\nwith the isotropic distribution depends on the adopted electron injection spectrum\nand the size of the SDZ, with softer injection spectra and larger SDZs producing larger differences. \n\nKinematically, the energy dependence of the discrepancy can be understood where the very highest energy $\\gamma$-ray{s} are produced in head-on collisions from back-scattered soft photons, where the ultrarelativistic electrons ``see'' the angular distribution of soft photons concentrated in a narrow (head on) beam. Meanwhile, for lower electron energies the angular distribution of the background photons is considerably more important and more significantly affects the energy of the upscattered $\\gamma$-ray{s}.  \nThe softer injection spectra have more low-energy electrons and there is more confinement for low-energy electrons for larger sizes of the SDZ. \nThe surface brightness profile is also slightly affected\nby the isotropic assumption of the IC emission as shown in Figure~\\ref{fig:StationarySB} (bottom panel).\nThe profile is more peaked\nunder the isotropic assumption; the under-prediction in the wings of the\nprofile is $\\sim$ 10\\% deeper than near the center.\nConsequently, using the averaging made for the isotropic approximation has some effect that can produce an incorrect determination of the diffusion coefficient from the shape of the profile.\n\nAlso shown in\nFigure~\\ref{fig:StationarySpectrum} is the effect of the size of the SDZ on\nthe average spectrum.  A larger SDZ size results in a larger electron number density that, in turn, leads to an increased emission in the GeV energy range. The observations made with the MAGIC telescope\n(\\citealp{AhnenEtAl:2016}, updated to match the diffusion profile by\n\\citealp{TangPiran:2018}) constrain both the particle injection spectrum and\nthe size of the SDZ. A smaller SDZ and\/or harder particle injection spectrum\nis required by the observations.\nThe perceived degeneracy between the size of the SDZ and the\ninjected spectrum of the particles from this Figure is broken when the angular\nprofile of the low-energy emission is taken into account. Changing the injection spectrum\nwill not affect the shape of the angular profile while changing the SDZ size does. The angular extent of the emission should therefore\nbe a good indicator of the size of the SDZ.\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{fig3a}\\\\\n  \\includegraphics[width=0.48\\textwidth]{fig3b}\n  \\caption{\nPredicted flux of positrons at Earth. The models shown here assume Geminga is\nstationary and that the SDZ is stationary and centered at the current location\nof Geminga.  The top panel shows the spectrum for different values of\n$\\gamma_1$ as indicated in the legend, but using a fixed SDZ size $(r_z,\nr_t)=(30,50)$~pc. The bottom panel shows the spectrum for models with different\npairs of $(r_z, r_t)$, but fixed value of $\\gamma_1=2.0$. The points are\nAMS-02 data \\citep{AguilarEtAl:2014}.}\n  \\label{fig:StationaryPositronSpectrum}\n\\end{figure}\n\nOne of the main conclusions of the study by \\citet{AbeysekaraEtAl:2017} is\nthat the unexpected rise in the positron fraction as measured by PAMELA\n\\citep{AdrianiEtAl:2009}, {\\em Fermi}-LAT{} \\citep{AckermannEtAl:2012}, and AMS-02\n\\citep{2013PhRvL.110n1102A,AguilarEtAl:2014,2014PhRvL.113l1101A} cannot be due\nto the positrons accelerated by the Geminga PWN. This conclusion is based on a\none-zone diffusion model constructed to agree with the HAWC data.  For a two-zone diffusion model the\nconclusion can be quite different. \\citet{ProfumoEtAl:2018} and\n\\citet{FangEtAl:2018} found that the positron excess could easily be\nreproduced with such a model using a power-law injection spectrum for\npositrons (and electrons) with index of $2.34$ and $2.2$, respectively.\n\\citet{TangPiran:2018} also used data from the MAGIC telescope to\nconstrain the model and found\nthat a harder particle injection spectrum below a break energy of 30~TeV was\nnecessary, which \nresulted in a calculated positron flux at Earth that did not fully explain the rise in the positron fraction.\n\nFigure~\\ref{fig:StationaryPositronSpectrum} shows the positron flux at Earth as predicted by the models considered in this paper,\nwhere the top and bottom panels illustrate models with different SDZ sizes, and the effect of changing $\\gamma_1$, respectively. The results clearly show a strong\nvariation in the expected positron flux at Earth.  A smaller SDZ\nleads to larger flux of positrons at Earth because of the larger\n\\emph{effective} diffusion coefficient. Only if the SDZ extends all the way to\nthe Solar system are the results of \\citet{AbeysekaraEtAl:2017} reproduced.  Larger\nvalues of $\\gamma_1$ also results in a larger positron flux at Earth in the observed\nenergy range, meanwhile values larger than $\\gamma_1 \\approx 2.2$ are excluded\nas the predicted positron flux would exceed the data for a \nSDZ of size $(r_z,r_t)=(30,50)$~pc.\n\nSo far the calculations have been made considering a\nstationary source located at the current position of Geminga $(l_G, b_G)$.\nHowever, such an approximation is not supported by its observed large proper motion.\nIn the following analysis, the proper motion is taken into account, but different\nassumptions are made on the origin of the SDZ and the value of\nthe drag coefficient $A_d$ introduced in Eq.~(\\ref{eq:vDiffusion}).  The\ndifferent assumptions are referred to as scenarios A to D and detailed below.\nFor scenarios A and B, the SDZ is assumed to be stationary and is unrelated to\nGeminga.  In both scenarios, $A_d=0$ and the pulsar velocity is constant.\nScenario A assumes that the SDZ is centered on the current position of\nGeminga by chance,\nand that the parameters of the zone are:\n$(r_z,r_t)=(30,50)$~pc.  Scenario B assumes that the SDZ is centered at the\nbirth place of Geminga. In this scenario\nthe SDZ has to be much larger to extend from the birth place of Geminga to its\ncurrent location, $(r_z,r_t)=(90,110)$~pc. The\ncenter of the spatial grid is placed at $(8.7358, 0.0962,\n-0.0124)$~kpc for both these scenarios. This is about half-way between the birth place and the current\nlocation of Geminga for the $x$- and $y$-axes, but only a quarter of the way\nfor the $z$-axis.  As described earlier, the location of the center of the grid is set to give the current location of Geminga preference over its birth location.\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{fig4}\n  \\caption{\nSurface brightness of the modeled IC emission for scenarios A through D shown as a function of the angular distance from the current location of Geminga. Also shown is the profile for the model with a stationary Geminga PWN with $(r_z,r_t)=(30,50)$~pc and $\\gamma_1=2.0$.  The data points show the profile observed by HAWC \\citep{AbeysekaraEtAl:2017}.}\n  \\label{fig:ProfilesScenarios}\n\\end{figure}\n\n\\begin{figure*}[tb]\n  \\centering\n  \\includegraphics[width=0.49\\textwidth]{fig5a}\n  \\includegraphics[width=0.49\\textwidth]{fig5b}\\\\\n  \\includegraphics[width=0.49\\textwidth]{fig5c}\n  \\includegraphics[width=0.49\\textwidth]{fig5d}\n  \\caption{\nIC intensity maps evaluated at 10~GeV around the current location of Geminga.  Shown are maps for scenarios A (top left), B (top right), C (bottom left), and D (bottom right).  The maps are displayed using the colormap bgyw from the colorcet package \\citep{Kovesi:2015}.\n  }\n  \\label{fig:Maps10GeV}\n\\end{figure*}\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{fig6}\n  \\caption{\nPredicted positron fluxes at Earth for scenarios A through D. Also shown are the results for a model where Geminga is stationary with $(r_z,r_t)=(30,50)$~pc and $\\gamma_1=2.0$.  The points are AMS-02 data \\citep{AguilarEtAl:2014}.}\n  \\label{fig:PositronsScenarios}\n\\end{figure}\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{fig7}\n  \\caption{\nSynchrotron intensity averaged over a region around the current location of Geminga with a radius of $1^\\circ$ for scenarios A through D. Also shown are the\nresults for a model where Geminga is stationary with $(r_z,r_t)=(30,50)$~pc\nand $\\gamma_1=2.0$.}\n  \\label{fig:SynchrotronScenarios}\n\\end{figure}\n\nFor scenarios C and D, the center of the SDZ follows the location of Geminga\nand its size increases proportionally to the square root of time, normalized such that the final size of the SDZ is\n$(r_z,r_t)=(30,50)$~pc.  The difference between these two scenarios is the value\nof the drag coefficient, $A_d$.  In scenario C, $A_d=0$ and the velocity is\nconstant over time, while in scenario D,\n$A_d=2.636\\times10^{-8}$~pc$^{-1}$~yr$^{-1}$ giving Geminga an initial\nvelocity of $410$~km~s$^{-1}$ at a\nposition of $(8.72775, 0.113017, -0.0787255)$~kpc.\nFor numerical stability the gradient of the diffusion coefficient is limited to be less than one order of magnitude per grid pixel.\nBecause the ratio $D_0\/D_z \\sim 300$, it is necessary to increase $D_z$ initially for scenarios C and D to fulfill this condition\nuntil the difference between $r_t$ and $r_z$ corresponds to the size of 2 pixels in all directions of the grid.  Correspondingly, $D_z$\nis increased by up to an order of magnitude for the first few thousand years\nonly, but this stability requirement does not significantly affect the results.\nThe center of the spatial grid in scenario C is the same as that in\nscenarios A and B, while in scenario D the center of the grid is at $(8.7300,\n0.1075, -0.0213)$~kpc to better resolve the entire evolution of Geminga.\nAgain the center is about half-way between the origin\nand current location of Geminga for $x$- and $y$-axes, but only a quarter of\nthe way for the $z$-axis.  For scenarios A through D the variables $(r_t,\nr_z)$ are fixed to the values provided above and $\\gamma_1=2.0$. \n\nFigure~\\ref{fig:ProfilesScenarios} shows the angular profile of the surface\nbrightness in the energy range 8~TeV to 40~TeV compared to the HAWC\nobservations \\citep{AbeysekaraEtAl:2017}.  For comparison the results of the\nmodel with stationary Geminga, $(r_z,r_t)=(30,50)$~pc and $\\gamma_1=2.0$ are\nalso shown.  As expected, all models agree well with the data and\nsignificantly overlap. Though it is not seen in the Figure, the emission is\nstill reasonably symmetric around the current location of Geminga.  To\n  quantify the symmetry, the standard deviation of the distribution of\nintensities of pixels in the calculated HEALPix maps within each angular bin\nis calculated. The\nstandard deviation in each angular bin is of the order of 5\\% to 10\\% only.\nMeanwhile, some differences between models are also noticeable: the largest\nstandard deviation of the emission core is seen in scenario D because of the\nlarger proper velocity of the PWN, while the largest deviation in the tail is\nseen in scenario B with the largest SDZ size.\n\nIn contrast, at lower energies all models produce quite different\ndistributions of the surface brightness. In particular, considerable asymmetry in the intensity maps around 10 GeV is seen in\nFigure~\\ref{fig:Maps10GeV}, especially for scenario B with the largest SDZ.\nAll scenarios where Geminga is moving show a distinctive tail in the low\nenergy range.  In the extreme case of scenario B the tail is exceptionally\nbroad and the emission of the tail is, in fact, much brighter than that at the\ncurrent location of Geminga.  The differences between other scenarios are small\nand hard to distinguish in these maps.  The tail in scenario C is slightly\nmore extended than in scenarios A and D.  Observations of the Geminga PWN at\n10 GeV will thus hardly be able to distinguish between these scenarios,\nbut may be able to verify the presence of the tail in Scenario B.\n\nDespite their similar $\\gamma$-ray{} emission maps (with exception of\nscenario B), all scenarios predict different positron fluxes at Earth\n(Figure~\\ref{fig:PositronsScenarios}). For energies below 1 TeV, the positron\nflux in scenario A is about a factor of 5 larger than that predicted for\nscenarios C and D.  The difference between scenarios C and D is much smaller\nand mostly at the highest energies where scenario D produces a higher flux.\nThis is due to the faster movement of the diffusion zone in scenario D that\nallows the CR particles to escape quickly once the diffusion zone has left\nthem behind. This is somewhat idealized and it is more likely that the SDZ\nwill have a shape elongated along the direction of motion. For such a case\nthe results would become closer to scenario B, with a longer IC emission tail\nat low energies and smaller flux of positrons at low and high energies, but\nwith a larger peak at around 300~GeV.\n\nIn addition to IC emission, GALPROP can calculate the predicted synchrotron emission from the models.  In a magnetic field with a strength of several $\\mu$G, as is expected in the vicinity of Geminga, the particles responsible for the TeV $\\gamma$-ray-emission will also emit synchrotron photons with energies of $\\sim$ keV.  Because of the shape of the electron\/positron spectrum of the injected particles from Geminga, the emission in radio and mm wavelengths is significantly dimmer than that from CR electrons from the Milky Way at large.  At keV energies the synchrotron emission follows a profile similar to that of the TeV IC emission, being peaked at the current location of Geminga with a half-width of about a degree.  Figure~\\ref{fig:SynchrotronScenarios} shows the spectrum of the average intensity of the synchrotron emission within a degree of the current location of Geminga.  All the models considered predict a similar intensity level of the emission that approximately follows a power-law $I\\propto E^{-1.8}$.  The calculations are cut off at few 10s of keV because the electron spectrum can only be reliably determined up to the energies corresponding to the HAWC observations, anything beyond that is an extrapolation.  This energy is reached already at a few keV in the synchrotron spectrum.  The synchrotron emission is close to the level of the diffuse X-ray emission observed in the Milky Way ridge \\citep{2000ApJ...534..277V}, but significantly less intense than that observed from the Geminga pulsar and the apparent nearby tails \\citep{2010ApJ...715...66P}.\n\n\n\n\\section{Implications for Propagation Throughout the Milky Way}\n\\label{sec:MW}\n\nIf the result for the two PWNe reported in \\citet{AbeysekaraEtAl:2017} are not special cases, then it is likely that there are similar pockets of slow diffusion around many CR sources elsewhere in the Milky Way. The effective diffusion coefficient would thus be smaller in regions where the number density of CR sources is higher. \nSecondary CR species with different inelastic production cross sections (e.g., $\\bar{p}$ vs.\\ B) probe different propagation distances.  Interpretations of their data would yield different average diffusion coefficients as was indeed found by \\citet{JohannessonEtAl:2016}. \nStarburst galaxies with large star formation rate are expected to have very slow diffusion and should exhibit an energy-loss-dominated spectrum of $\\gamma$-ray{} emission that is much flatter than that observed from galaxies where the leakage of CRs dominates the energy losses. Interestingly, exactly such evolution of the spectral shape is observed when galaxies with different star formation rate, such as the Magellanic Clouds, Milky Way, M31 vs.\\ NGC 253, NGC 4945, M82, NGC 1068, are compared \\citep[][and references therein]{2012ApJ...755..164A}.\n\nHere it is assumed that the distribution of SDZs follows that of the CR sources, while the exact origin of these SDZs is not essential. With current CR propagation codes and reasonable resources resolving the entire Milky Way with a few pc grid size on short time scales is intractable.\nA simpler approach must, therefore, be taken to explore the effects such SDZs have on the propagation of CRs throughout the Milky Way. Assuming that the properties of CR propagation and the distribution of CR sources vary on-average very little over the residence time of CRs in the Milky Way, a steady-state model can be used.  Also assuming that the residence time of CRs in the Milky Way is larger than the active injection time of the CR sources, the CR source distribution can be assumed smooth with injected power that approximates as constant with time.\nThese two approximations have been extensively used in the past and are the starting point of almost all studies on CR propagation across the Milky Way.\n\nAs was shown in the previous Section, the effect of an SDZ is a local increase of the density of CRs for a certain period of time, leading to an increase of CR reaction rates with the ISM contained within the SDZ. CRs thus spend more time in the vicinity of their sources compared to a model with a homogeneous diffusion.  This is equivalent to an effective decrease of the diffusion coefficient averaged over a larger volume. Even though, such an approximation does not account for the detailed spatial distribution of the individual SDZs, it does account for the increased interaction rate with the ISM, such as cooling and generation of secondary CR particles. Therefore, this approximation should still provide for the correct spectra and abundances of CR species. \n\nAssuming that CRs first propagate through a SDZ with diffusion coefficient $D_z$ and\nradius $x_z$ that is embedded in a region with diffusion\ncoefficient $D_0$ and radius $x_0 \\gg x_z$, their residence time in the total volume can be estimated as\n\\begin{equation}\n  \\tau \\sim \\frac{x_0^2}{D_0} + \\frac{x_z^2}{D_z}.\n  \\label{eq:resTimeTwoZone}\n\\end{equation}\nAssuming a single average diffusion coefficient $D$, the residence time can also be expressed as \n\\begin{equation}\n  \\tau \\sim \\frac{x_0^2}{D}.\n  \\label{eq:resTimeAv}\n\\end{equation}\nCombining the two equations results in\n\\begin{equation}\n  D \\approx D_0 \\left[ 1 + \\left( \\frac{x_z}{x_0} \\right)^{2} \\frac{D_0}{D_z}\\right]^{-1}.\n  \\label{eq:avDiffusion}\n\\end{equation}\n\nAssuming that there is a SDZ around each CR source, the density of SDZs is\nthe same as the CR source number density, $q(\\vec{x})$, where $\\vec{x}$ is the spatial coordinate. Let $V_z$ be the volume of each SDZ and $\\Delta V$ be the volume of an element in the spatial grid used in the calculation. Then for each volume element\n\\begin{equation}\n  \\frac{x_z}{x_0} \\approx \\left[\\frac{V_z}{\\Delta V} \\int_{\\Delta V} q(\\vec{x}) dV\\right]^{1\/3} \\approx \\left[q_x V_z\\right]^{1\/3},\n  \\label{eq:xzx0approx}\n\\end{equation}\nwhere $q_x$ is the CR source number density at the center of the volume element. The value $ Q=\\int_V q(\\vec{x}) dV$ is the total number of active CR sources in the Galaxy at any given time, and $QV_z$ is then the combined volume of all SDZs in the Galaxy. \nAssuming that the SN rate is 0.01 year$^{-1}$, and each source continuously accelerates particles for $\\sim$$10^5$ years, there are $\\sim$$10^3$ active CR sources at any given time. Assuming a radius of $x_z\\sim 30$~pc for each source, the combined volume of all SDZs is $QV_z \\approx 0.1$~kpc$^3$.\nThis value is in good agreement with estimates from \\citet{HooperEtAl:2017} and\n\\citet{ProfumoEtAl:2018}.\nThe ratio $x_z\/x_0$ in Eq.~(\\ref{eq:xzx0approx}) depends on the product of the number of SDZs and their sizes.\nBecause of this, equivalent results can be obtained also for, e.g., fewer SDZs of larger sizes, provided that the total occupied volume is the same and their distribution follows $q(\\vec{x})$.\nEq.~(\\ref{eq:avDiffusion}) also depends on the value of $D_0\/D_z$ which is set to 300 in the following  calculations, which is similar to the value used in the Geminga calculations. The results described below, therefore, are valid across different model assumptions provided they are consistent with Eq.~(\\ref{eq:avDiffusion}).\n\n\\begin{figure*}[tb]\n  \\centering\n  \\includegraphics[width=0.49\\textwidth]{fig8a}\n  \\includegraphics[width=0.49\\textwidth]{fig8b}\n  \\caption{Diffusion coefficient evaluated in the Galactic plane at the normalization rigidity $R_0=4$~GV for a model where the effective diffusion coefficient is given by Eq.~(\\ref{D_distr}). The left panel shows the distribution of the diffusion coefficient for the SA0 model while the right panel for the SA50 model.  The location of the Sun is marked as a white star and the GC is at (0,0). The maps are displayed using the colormap bgyw from the colorcet package \\citep{Kovesi:2015}.\n  }\n  \\label{fig:DiffusionPlane}\n\\end{figure*}\n\nThe modified diffusion described above has been implemented in the\nGALPROP code to test its effect on the propagation of CR species in the Milky Way.\nWith a switch in the configuration file it is now possible to change to a diffusion configuration such that\n\\begin{equation}\n    D(\\vec{x}) = D_0 \\left\\{1 + \\left[q\\left( \\vec{x} \\right) V_z \\right]^{2\/3} \\frac{D_0}{D_z} \\right\\}^{-1}.\n    \\label{D_distr}\n\\end{equation}\nTherefore, even with $D_0$ and $D_z$ fixed the {\\it effective} diffusion coefficient is still spatially varying, where the total volume of SDZs in the Milky Way is a normalization parameter $QV_z\\approx 0.1$ kpc$^{3}$, as explained above. In principle, the number distribution of CR sources can be different from the distribution of their injected power into CRs in the Milky Way, but here they are assumed identical.\nTwo models SA0 and SA50 are used for the CR source density distribution\n\\citep{PorterEtAl:2017}.  The SA0 model is a 2D axisymmetric pulsar-like\ndistribution \\citep{2004A&A...422..545Y} for the CR source density in the disk. The\nSA50 model has half of the injected CR luminosity distributed as the SA0\ndensity and the other half distributed as the spiral arms for the R12 ISRF\nmodel.  Each CR source model is\npaired with both a constant homogeneous diffusion and the modified diffusion\ndescribed above resulting in a total of four models.  The model calculations are performed on a 3D spatial grid that now includes the whole Milky Way using the grid function defined in Eq.~(\\ref{eq:gridFunction}).  The parameters of the grid function are chosen such that the resolution in the $x$- and $y$-coordinates is 200~pc at the GC, increasing to 1 kpc at a distance of 20 kpc which is also the boundary of the grid.  At the distance of the Solar system the resolution is about 350~pc in the $x$-direction and the Sun is located at the center of a volume element.  In the $z$-direction the resolution is 50~pc in the plane, increasing to 200~pc at the boundary of the grid at $|z|=6$~kpc.  The energy grid is logarithmic, ranging from 3~MeV to 30~TeV with 112 energy planes.\n\nThe distribution of the effective diffusion coefficient $D(\\vec{x})$ in the Galactic plane is shown in Figure~\\ref{fig:DiffusionPlane}, where the numbers correspond to its value at the normalization rigidity $R_0=4$~GV.\nThe spiral arm structure of the SA50 model is clearly visible.\nThe maximum change in the effective diffusion coefficient is about a factor of 3, which corresponds to a peak value of $x_z\/x_0 \\sim 10^{-3}$.\nAll models employ the R12 ISRF from \\citet{PorterEtAl:2017} and the 3D gas distributions from \\citet{JohannessonEtAl:2018}.\nThe same standard procedure for parameter adjustment to match\nthe recent observations of CR species listed in Table~\\ref{tab:CRdata} is followed for all source density models.\nSolar modulation is treated using the force-field approximation \\citep{1968ApJ...154.1011G}, one\nmodulation potential value for each observational period. The use of the latter is justified by the available resources and the main objective of this paper, i.e.\\ to study the effect of the modified CR diffusion in a self-consistent manner, rather than to update the local interstellar spectra of CR species or to accurately determine the propagation parameters\\footnote{For a detailed treatment of heliospheric propagation of CRs, see, e.g.,\n  \\citet{BoschiniEtAl:2017,BoschiniEtAl:2018b,BoschiniEtAl:2018a} and references therein.}.\nThe procedure for parameter tuning is the same as used by \\citet{PorterEtAl:2017} and \\citet{JohannessonEtAl:2018}.\nThe propagation parameters are first determined by fitting the models to the observed spectra of CR species, Be, B, C, O, Mg, Ne, and Si. These are then\nkept fixed and the injection spectra for electrons, protons, and He fitted\nseparately. To reduce the number of parameters, only the most relevant primary abundances are fitted simultaneously with the propagation parameters while the rest are kept fixed. Because the injection spectra of all CR species are normalized relative to the proton spectrum at normalization energy 100 GeV\/nucleon the procedure is repeated to ensure consistency. One iteration provides a satisfactory accuracy in all cases. The only difference from the procedure described in \\citet{PorterEtAl:2017} and \\citet{JohannessonEtAl:2018} is the inclusion of elemental spectra of Be, C, and O from AMS-02 and the elemental spectra from ACE\/CRIS, while the data from HEAO-C3 and PAMELA has been removed.\n\n\\begin{deluxetable}{lll}[tb!]\n\\tablecolumns{3}\n\\tablewidth{0pc}\n\\tablecaption{Datasets used to derive propagation parameters\n   \\label{tab:CRdata}}\n\\tablehead{\n\\multicolumn{1}{l}{Instrument} &\n\\multicolumn{1}{l}{Datasets} &\n\\colhead{Refs.\\tablenotemark{a}}\n}\n\\startdata\nAMS-02 (2011-2016) & Be, B, Be\/B, Be\/C, B\/C, Be\/O, B\/O & I \\\\\nAMS-02 (2011-2016) & C, O, C\/O & II \\\\\nAMS-02 (2011-2013) & e$^-$ & III \\\\\nAMS-02 (2011-2013) & H & IV \\\\\nAMS-02 (2011-2013) & He & V \\\\\nACE\/CRIS (1997-1998) & B, C, O, Ne, Mg, Si & VI \\\\\nVoyager 1 (2012-2015) & H, He, Be, B, C, O, Ne, Mg, Si & VII \n\\enddata\n\\tablenotetext{a}{I: \\citet{AguilarEtAl:2018}, II: \\citet{AguilarEtAl:2017}, III:\n\\citet{AguilarEtAl:2014}, IV: \\citet{AguilarEtAl:2015a}, V:\n\\citet{AguilarEtAl:2015b}, VI: \\citet{GeorgeEtAl:2009}, VII:\n\\citet{CummingsEtAl:2016}}\n\\end{deluxetable}\n\n\\begin{deluxetable}{lcccc}[tb!]\n\\tablecolumns{5}\n\\tablewidth{0pc}\n\\tablecaption{Final propagation model parameters. \\label{tab:CRparameters} }\n\\tablehead{\n   & \\multicolumn{2}{l}{Homogeneous diffusion} & \\multicolumn{2}{l}{Modified diffusion} \\\\\n\\multicolumn{1}{l}{Parameter} & \n\\colhead{SA0} & \n\\colhead{SA50} &\n\\colhead{SA0} & \n\\colhead{SA50} \n}\n\\startdata\n\\tablenotemark{a}$D_{0,xx}$ [$10^{28}$cm$^2$\\,s$^{-1}$] & $4.36$            & $4.55$            \n                                                       & $4.41$            & $4.78$            \\\\\n\\tablenotemark{a}$\\delta$                              & $0.354$           & $0.344$           \n                                                       & $0.358$           & $0.360$           \\\\\n$v_{A}$ [km s$^{-1}$]                                  & $17.8$            & $18.1$            \n                                                       & $15.7$            & $17.6$            \\smallskip\\\\\n\\tablenotemark{b}$\\gamma_0$                            & $1.33$            & $1.43$            \n                                                       & $1.40$            & $1.47$            \\\\\n\\tablenotemark{b}$\\gamma_1$                            & $2.377$           & $2.399$           \n                                                       & $2.403$           & $2.381$           \\\\\n\\tablenotemark{b}$R_1$ [GV]                            & $3.16$            & $3.44$            \n                                                       & $3.80$            & $3.82$            \\\\\n\\tablenotemark{b}$\\gamma_{0,p}$                        & $1.96$            & $1.99$            \n                                                       & $1.92$            & $1.93$            \\\\\n\\tablenotemark{b}$\\gamma_{1,p}$                        & $2.450$           & $2.466$           \n                                                       & $2.469$           & $2.453$           \\\\\n\\tablenotemark{b}$\\gamma_{2,p}$                        & $2.391$           & $2.355$           \n                                                       & $2.359$           & $2.321$           \\\\\n\\tablenotemark{b}$R_{1,p}$ [GV]                        & $12.0$            & $12.2$            \n                                                       & $11.3$            & $11.3$            \\\\\n\\tablenotemark{b}$R_{2,p}$ [GV]                        & $202$             & $266$             \n                                                       & $213$             & $371$             \\\\\n$\\Delta_{\\rm He}$                                          & $0.033$           & $0.035$           \n                                                       & $0.039$           & $0.032$           \\smallskip\\\\\n\\tablenotemark{b} $\\gamma_{0,e}$                       & $1.62$            & $1.49$            \n                                                       & $1.46$            & $1.46$            \\\\\n\\tablenotemark{b} $\\gamma_{1,e}$                        & $2.843$           & $2.766$           \n                                                       & $2.787$           & $2.762$           \\\\\n\\tablenotemark{b} $\\gamma_{2,e}$                       & $2.494$           & $2.470$           \n                                                       & $2.506$           & $2.480$           \\\\\n\\tablenotemark{b} $R_{1,e}$ [GV]                        & $6.72$            & $5.14$            \n                                                       & $5.03$            & $5.13$            \\\\\n\\tablenotemark{b} $R_{2,e}$ [GV]                        & $52$              & $69$            \n                                                       & $69$              & $69$              \\smallskip\\\\\n\\tablenotemark{c}$J_p$  & $4.096$ & $4.113$ \n                                                       & $4.102$           & $4.099$           \\\\\n\\tablenotemark{c}$J_e$  & $2.386$ & $2.362$ \n                                                       & $2.288$           & $2.345$           \\smallskip\\\\\n\\tablenotemark{d}$q_{0,^{4}\\rm He} \\hfill [10^{-6}]\\qquad\\qquad$          & $ 92495$          & $ 91918$          \n                                         & $ 92094$          & $ 93452$          \\\\\n\\tablenotemark{d}$q_{0,^{12}\\rm C} \\hfill [10^{-6}]\\qquad\\qquad$           & $  2978$          & $  2915$          \n                                         & $  2912$          & $  2986$          \\\\\n\\tablenotemark{d}$q_{0,^{16}\\rm O} \\hfill [10^{-6}]\\qquad\\qquad$           & $  3951$          & $  3852$          \n                                                               & $  3842$          & $  3956$          \\\\\n\\tablenotemark{d}$q_{0,^{20}\\rm Ne} \\hfill [10^{-6}]\\qquad\\qquad$          & $   358$          & $   327$          \n                                          & $   322$          & $   359$          \\\\\n\\tablenotemark{d}$q_{0,^{24}\\rm Mg} \\hfill [10^{-6}]\\qquad\\qquad$          & $   690$          & $   704$          \n                                          & $   681$          & $   744$          \\\\\n\\tablenotemark{d}$q_{0,^{28}\\rm Si} \\hfill [10^{-6}]\\qquad\\qquad$          & $   801$          & $   786$          \n                                                               & $   762$          & $   833$          \\smallskip\\\\\n\\tablenotemark{e}$\\Phi_{\\rm AMS,I}$ [MV]                  & $ 729$            & $ 741$            \n                                                       & $ 709$            & $ 729$            \\\\\n\\tablenotemark{e}$\\Phi_{\\rm AMS,II}$ [MV]                 & $ 709$            & $ 729$            \n                                                       & $ 696$            & $ 729$            \\\\\n\\tablenotemark{e}$\\Phi_{\\rm ACE\/CRIS}$ [MV]               & $ 359$            & $ 370$            \n                                                       & $ 345$            & $ 354$           \n\\enddata\n\\tablenotetext{a}{$D(R) \\propto \\beta R^{\\delta}$, $D_0$ is the normalization at $R_0=4$~GV. \n}\n\\tablenotetext{b}{The injection spectrum is parameterized as $q(R) \\propto R^{-\\gamma_0}$ for $R \\le R_1$, $q(R) \\propto R^{-\\gamma_1}$ for $R_1 < R \\le R_2$, and $q(r) \\propto R^{-\\gamma_2}$ for $R > R_2$.  The spectral shape of the injection spectrum is\nthe same for all species except CR $p$ and He. $R_1$, and $R_2$ are the same for $p$\nand He and $\\gamma_{i,\\rm He} = \\gamma_{i,p}-\\Delta_{\\rm He}$}\n\\tablenotetext{c}{The CR $p$ and e$^-$ fluxes are normalized at the Solar\nlocation at a kinetic energy of 100~GeV.  $J_p$ is in units of $10^{-9}$\ncm$^{-2}$ s$^{-1}$ sr$^{-1}$ MeV$^{-1}$ and $J_e$ is in units of $10^{-11}$ cm$^{-2}$ s$^{-1}$ sr$^{-1}$ MeV$^{-1}$.}\n\\tablenotetext{d}{The injection spectra for isotopes are normalized relative to the proton injection spectrum at 100~GeV\/nuc.  \nThe normalization constants for isotopes not listed here are the same as given in \\citet{JohannessonEtAl:2016}.}\n\\tablenotetext{e}{The force-field approximation is used for calculations of the solar\nmodulation and is determined independently for each model and each\nobserving period. $\\Phi_{\\rm AMS,I}$ and $\\Phi_{\\rm AMS,II}$ correspond\nto the 2011-2016 and 2011-2013 observing periods for the AMS-02\ninstrument, respectively.}\n\\end{deluxetable}\n\nTable~\\ref{tab:CRparameters} lists the best-fit parameters for the four models\nconsidered.  The parameters are similar for all models and the modified\ndiffusion and different source distributions only slightly affect the best-fit\nvalues.\nMost interestingly, despite the fact that the value of the effective diffusion coefficient exhibits strong spatial variations in the Galactic plane including at the Solar system location, where it is about a factor of 2 smaller compared to that in the model with homogeneous diffusion (Fig.~\\ref{fig:DiffusionPlane}), the value of $D_0$ is not significantly different from that obtained for the homogeneous model.\nThis is connected with the relatively small volume affected by the modified diffusion regions because they are associated with the CR sources, which have a relatively narrow distribution about the Galactic plane with exponential $z$ scale-height 200~pc.\nThe modified diffusion slightly affects the low-energy part of the spectrum, resulting in smaller\nmodulation potentials and Alfv{\\'e}n speeds.\nOverall the addition of the SDZs does not significantly alter the global diffusive properties of the ISM.\n\n\\begin{figure*}[tb]\n  \\centering\n  \\includegraphics[width=0.49\\textwidth]{fig9a}\n  \\includegraphics[width=0.49\\textwidth]{fig9b}\\\\\n  \\includegraphics[width=0.49\\textwidth]{fig9c}\n  \\includegraphics[width=0.49\\textwidth]{fig9d}\\\\\n  \\includegraphics[width=0.49\\textwidth]{fig9e}\n  \\includegraphics[width=0.49\\textwidth]{fig9f}\\\\\n  \\caption{A comparison between the best-fit model predictions and observation of CR species\nnear the Solar system. The bottom panel of each subfigure shows the data subtracted from the model in units of data flux. References to the data are provided in Table~\\ref{tab:CRdata}. For uniformity of the presentation, AMS-02 data has been converted from rigidity to kinetic energy per nucleon units using isotopic composition of each element as measured by ACE\/CRIS, but the likelihood in the fitting procedure is calculated using rigidity. Different curves represent the four different models considered.  SA0 and SA50 cases with the homogeneous diffusion are shown as solid and dotted lines, respectively, while SA0 and SA50 cases with the modified diffusion are show as dashed and dash-dotted lines, respectively. Note that there is significant overlap between the lines for the different models for the top panels of the subfigures.\n  }\n  \\label{fig:CRelements}\n\\end{figure*}\n\nA comparison between the calculated spectra of CR species and data is shown in Figure~\\ref{fig:CRelements}.  The\nmodels agree reasonably well with the data and deviate less than 10\\% from the\nmeasurements for most elements.  The largest systematic deviations\nshown are at around 10~GeV for B and C where the relative residuals are as large as\n$\\sim$20\\%.  The residuals for Be and O are similar (not shown).  These\nare likely caused by the lack of freedom in the injection spectra for the\nprimaries, because the deviations are hardly visible in the secondary-to-primary \nratios of which B\/C is shown in the Figure for illustration.  If the deviations were caused by propagation\neffects they would be stronger in the secondary component and be\ncorrespondingly more\nprominent in the secondary-to-primary ratios.  Deviations from the H\nand He data are smaller and the feature at 10~GeV is not visible.\nFigure~\\ref{fig:CRelements} also shows that the models agree with each other to\nwithin 5\\% for all elements, and this further illustrates that the addition of the SDZ pockets does not strongly affect predictions for the local spectra of CR species.\n\n\n\\section{Discussion}\n\nObservations of $\\gamma$-ray{} emission from two PWNe made by HAWC \\citep{AbeysekaraEtAl:2017} provide a direct insight into the diffusive properties of the ISM on small scales. Combined with direct measurements of the spectra of CR species in the Solar neighborhood, the HAWC observations point to quite different diffusion coefficients in space surrounding the Geminga and PSR~B0656+14 PWNe compared to the ISM at large, with that associated with the former (SDZs) about two orders of magnitude smaller than the latter. The size of the SDZs is constrained by the upper limits of $\\gamma$-ray{} emission at lower energies obtained by the MAGIC telescope to be $\\lesssim 100$~pc \\citep{AhnenEtAl:2016, TangPiran:2018}.  Observations by the {\\em Fermi}-LAT{} may be used to further constrain the properties of the SDZs, although it may be difficult to break a degeneracy between the size of the diffusion zone and the shape of the injection spectra of CR electrons and positrons.  This is particularly true for PWNe with a large proper motion, like the case of Geminga, which significantly affects the shape of the IC emission in the energy range of the {\\em Fermi}-LAT.\n\nOne of the main claims of \\citet{AbeysekaraEtAl:2017} is that positrons produced by Geminga provide a negligible fraction of the positrons observed by the AMS-02 instrument \\citep{AguilarEtAl:2014}.  This statement is correct only if the diffusion coefficient is small in the entire region between Geminga and the Solar system.  Using a two-zone model consistent with both HAWC and MAGIC observations of Geminga, the predicted positron flux can vary over a wide range from a small fraction to almost the entire observed positron flux, dependent on the assumed properties of the SDZ and the injection spectrum.  Even if the $\\gamma$-ray{} emission can be constrained with the {\\em Fermi}-LAT{} observations, changing the SDZ properties can result in a variation of the predicted flux by a factor of few.  This is illustrated by Scenarios A and C shown in Figures~\\ref{fig:Maps10GeV} and~\\ref{fig:PositronsScenarios}: while the predicted $\\gamma$-ray{} emission differs only marginally, the predicted positron flux differs by about a factor of 5.\n\nIf the result for the two PWNe reported in \\citet{AbeysekaraEtAl:2017} are not special cases, then it is likely that there are similar pockets of slow diffusion around many CR sources elsewhere in the Milky Way. In the likely case that the distribution of the SDZs follows the distribution of CR sources, CRs spend more time in the inner Milky Way and generally in the plane than in the outer Milky Way or its halo.\nDespite this the predicted fluxes of secondary CR species near the Solar system are not strongly affected (see Figure~\\ref{fig:CRelements}), and the propagation model parameters obtained with SDZs included are very close to those determined for models using homogeneous diffusion (see Table~\\ref{tab:CRparameters}). This is a non-trivial result because the production rate of secondary CR species may also increase in the same regions of slow diffusion provided there is enough interstellar gas there. Compared to a model with homogeneous diffusion, the density of CRs should increase towards the inner Milky Way where the distribution of CR sources peaks. Correspondingly, the interstellar high-energy $\\gamma$-ray{} emission (IEM) should also be brighter in the direction of the inner Milky Way. At the same time, this differs from the results of the analysis of the {\\em Fermi}-LAT{} data which indicate that the gradient is even smaller than predicted by models with two-dimensional axisymmetric geometry and homogeneous diffusion \\citep[e.g.][]{AbdoEtAl:2010,AckermannEtAl:2011}. An updated analysis that uses three-dimensional spatial models and includes localized propagation effects may lead to a new interpretation of the distribution of the diffuse emission and of the {\\em Fermi}-LAT{} data.\n\nIn this study the GALPROP framework has been used, which is fully capable of calculating the spectrum and distribution of the expected interstellar diffuse $\\gamma$-ray{} emission, but because of the assumptions incorporated into the modified diffusion model, the predictions of the diffuse emission would be impractical. The sources of CRs are transient in nature and spatially localized and so are the potential SDZs associated with them. Depending on the exact nature of the SDZs, CR particles  may be confined within these regions for a significant fraction of their residence time in the Milky Way, contrary to the assumption made in Eqs.~(\\ref{eq:avDiffusion}) and (\\ref{eq:xzx0approx}). This may lead to the incorrect brightness distribution of the predicted $\\gamma$-ray{} emission, which is sensitive to the details of CR distribution throughout the Milky Way. Properly accounting for the transient nature of CR sources in the entire Milky Way is beyond the scope of this work, but will be addressed  by a forthcoming paper.\n\nOne plausible explanation for the generation of SDZs near CR sources is the self-excitation of Alfv{\\'e}n waves by the CRs themselves as they stream out into the ISM. Such streaming instabilities have been discussed since the early 70s \\citep{Skilling:1971} and are reviewed in \\citet{AmatoBlasi:2018}. The size and magnitude of the effect of streaming depends on the gradient of the CR distribution and the properties of the ISM, in particular the number density of neutrals and the Alfv{\\'e}n speed. Numerical studies of CR diffusion near SNRs reveal that lower number densities lead to smaller sizes for the SDZ, but also smaller change in the diffusion constant \\citep{NavaEtAl:2016,NavaEtAl:2019}. It is also expected that the injection spectrum is time dependent and may lead to an energy dependence of the diffusion coefficient in SDZs that is quite different from that in the ISM. Such models for the SDZ around Geminga would be in agreement with the HAWC observation that only constrains the diffusion at the highest energies, but produce a wide range of predictions for the expected flux of CR positrons near the Solar system as well as for the expected spectrum of IC emission at low energies. In turn, this would result in an increased CR flux in the inner Galaxy that is dominated by the high-energy particles, leading to a hardening of the spectrum that has been indeed observed by the {\\em Fermi}-LAT{} \\citep{SeligEtAl:2015,AjelloEtAl:2016}. Exploration of these effects is deferred to future work.\n\nTo summarize, observations of the SDZs made by HAWC provide an interesting opportunity to get insight into the fairly complex details of the CR propagation phenomenon. Extension of these observation onto the whole Milky Way \\emph{in specialibus generalia quaerimus} is, however, a non-trivial task \nand further understanding of the origin and properties of the SDZs are necessary to get a correct picture. \n\n\n\\acknowledgements\nT.A.P. and I.V.M. acknowledge partial support from NASA grant NNX17AB48G.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzbobe b/data_all_eng_slimpj/shuffled/split2/finalzzbobe
new file mode 100644
index 0000000000000000000000000000000000000000..0651fb6243dd2421d324211efe43ef4fe4f9bfe7
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzbobe
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\nSolitary waves are an important phenomenon in nonlinear physics and applied mathematics.  Solitary waves have been studied in a diverse array of physical models including water waves~\\cite{Russell,KdV,Ben:67}, quantum electronic devices (Josephson junctions)~\\cite{BEMS:71}, and cosmology~\\cite{AOM:91,BV:01}  One of the most important applications is to nonlinear optical communications where solitary waves have been proposed as information bits in optical fiber transmission systems~\\cite{HT} and produced experimentally about 25 years ago~\\cite{MSG:80}.  Other solitary wave phenomena in nonlinear optics include gap solitons in Bragg gratings~\\cite{AW,CJ} and dispersion managed solitons~\\cite{dmsx,dmnls}, which hold promise for eliminating the timing jitter associated with soliton transmission systems.\n\nA single solitary wave propagating through a uniform medium appears particle-like in its coherence and steady propagation.  Of great interest are the interaction of multiple solitary waves and the behavior of solitary waves propagating through non-uniform media.  Solitary waves of completely integrable equations are known as solitons, and their interactions can be described completely, using multiple-soliton formulas derived via the inverse scattering transform~\\cite{AS:81}.  The infinite set of conservation laws in integrable systems severely constrain the dynamics: collisions are elastic, and the solitons will re-emerge from a collision propagating with their initial amplitudes and speeds intact, although their positions will have undergone a finite jump.  Solitary wave collisions in non-integrable wave equations, can usually not be found in closed form, and show a much richer variety of behaviors: the waves may attract or repel each other and, upon collision, the solitary waves may lose their coherence and break apart, merge into a single localized structure, or even oscillate about one another.~\\cite{KM:89,TY00,TY:01,TY:01a,CPS:86,CP:86,CSW:83,PC:83}.  \n\nIn a soliton-based communications system, the bits are represented by solitons.  In the simplest scenario, the presence of a soliton in a given timing window codes a one, and its absence codes a zero.  \nCollisions between solitons, coupled with random noise in fiber characteristics, can lead to large perturbations in the solitons polarizations and to timing jitter~\\cite{MGH:95}.  A bit that arrives at the wrong time may be interpreted incorrectly by a receiver, as would a soliton that splits in half or two solitons that merge. Ueda and Kath show such behaviors are possible and cite several additional numerical studies of solitons collisions not included here.  We an approach to the modeling and analysis of these phenomena that, while highly idealized, leads to new insights into these collisions.\n\nInteracting pairs of solitary waves from several distinct (non-integrable) physical models have shown an interesting behavior in common.  At high speeds, the solitary waves move right past each other, hardly interacting, while at speeds below some critical velocity, the solitary waves interact strongly and may merge into a single localized state.  Interspersed among the initial velocities that lead to this capture are ``resonance windows'', for which the two waves approach each other, interact with each other for a finite time, and then move apart again; see the second and third graph in figure~\\ref{fig:tanyang}. This has been explored by Tan and Yang in a system of coupled nonlinear Schr\\\"odinger (CNLS) equations that model nonlinear propagation of light in birefringent optical fibers~\\cite{TY00,TY:01,TY:01a}, and by Cambell and collaborators in kink-antikink interaction in the $\\phi^4$ equations and several other nonlinear Klein-Gordon models~\\cite{CPS:86,CP:86,CSW:83,PC:83}.  These windows form a complicated fractal structure that has been described qualitatively and even quantitatively, but for which the underlying mechanism has been poorly understood.\n\nThe same phenomenon was also observed by Fei, Kivshar, and V\\'azquez in the interaction of traveling kink solutions of the sine-Gordon and $\\phi^4$ equations with weak localized defects~\\cite{FKV:92a,FKV:92,KFV:91}.  Instead of two solitary waves merging, in this case the soliton could be captured, or pinned, at the location of the defect.  Almost all of the described models have been studied using the so-called variational approximation, in which the complex dynamics of the full PDE are modeled by a small, finite-dimensional system of ordinary differential equations.\n\nThe sine-Gordon equation with defect and the birefringent fiber-optic model discussed above feature a small parameter measuring the ``non-integrability'' of the system.  In a recent publication, Goodman and Haberman~\\cite{GH:04}, exploited this small parameter to construct approximate solutions to the system of  ODEs  for the sine-Gordon model derived in~\\cite{FKV:92}.  We calculated the critical velocity for defect-induced soliton capture via an energy calculation involving separatrix crossing, and the location of the resonance windows using a quantization condition that occurs in the asymptotic expansion.  In the current paper, we apply the same method to derive similar quantitative features in Ueda and Kath's ODE model of solitary wave collision in coupled nonlinear Schr\\\"odinger equations, and to explain the structure underlying the fractal structure of resonance windows.\n\nIn section~\\ref{sec:theproblem}, we\nintroduce the physical model---a coupled system of Nonlinear Schr\\\"odinger\nequations---and describe previous results in which the ``two pass resonance''\nphenomenon has been observed. In section~\\ref{sec:themodel}, we introduce Ueda\nand Kath's finite-dimensional model system that captures the observed\ndynamics, and introduce a simplified model which partially linearizes the\nsystem and renders it amenable to our analysis.  We show numerically that this\nsimplification does not qualitatively alter the dynamics.  In section~\\ref{sec:DeltaE},\nwe set up the calculation as a singular perturbation problem, and describe the\nunperturbed dynamics.  We determine the critical velocity by calculating the\nenergy that is lost to vibrations as the solitons pass each other, employing a\nMelnikov integral.  We generalize this calculation slightly for subsequent\ninteractions.  In section~\\ref{sec:matching}, we construct approximate\nsolutions using matched asymptotic approximations, incorporating the previously calculated energy\nchanges.  Section~\\ref{sec:nonlinearity} contains a\ndiscussion of the differences between the original model and its\nsimplification and presents a weakly nonlinear theory to account for them.  We\nconclude in section~\\ref{sec:conclusion} with a physical summary and a discussion on\nthe applicability of these results to other systems displaying similar\nbehaviors. \n \n\\section{Physical problem and prior results}\n\\label{sec:theproblem}\nFollowing the previously cited~\\cite{UK:90, TY:01}, we consider the model of polarized light propagation in a  optical fiber, given by the system of coupled nonlinear Schr\\\"odinger equations \n\\begin{equation} \n\\begin{split} \n\\label{eq:CNLS}\n i \\partial_t A + \\partial_z^2 A + \\left(\\abs{A}^2 + \\beta \\abs{B}^2 \\right) A &=0 \\\\\n i \\partial_t B + \\partial_z^2 B + \\left(\\abs{B}^2 + \\beta \\abs{A}^2 \\right) B &=0.\n\\end{split}\n\\end{equation}\nThis system replaces the more familiar scalar Schr\\\"odinger equation when polarization is taken into effect~\\cite{Agrawal}. The equations may be derived using the slowly varying envelope approximation\nto Maxwell's equations in an optical fiber waveguide.  The variables  $A$ and $B$ describe the envelopes of wave\npackets in the two polarization directions and $\\beta$ is the nonlinear cross-phase modulation (XPM) coefficient that arises due to cubic ($\\chi^{(3)}$) terms in the dielectric response of the glass.  Here we use $z$ as a space-like variable and $t$ as a time-like variable.  Of course, in the optics interpretation,\nthe labels $z$ and $t$ are switched, as the signal is defined as a function of\ntime at $z=0$ and the evolution occurs as the pulse moves down the length of\nthe fiber.  For mathematical simplicity, we will use $t$ as the evolution\nvariable. \n\nOur interest is in the interaction of solitary waves in the above system.  In the cases $\\beta=0$ and $\\beta=1$, system~\\eqref{eq:CNLS} is completely integrable~\\cite{ZS, Manakov}.  In the first case it reduces to a pair of uncoupled NLS equations; in the second it is known as the Manakov system.  For other values of $\\beta$, the equations are not integrable.  Of special interest is the case $\\beta= \\frac{2}{3}$, which corresponds to linear fiber birefringence.  For very small values of $\\beta$, this system models light propagation in a two-mode optical fiber~\\cite{CDP}. In the case $\\beta=0$, the equations are simply a pair of focusing nonlinear Schr\\\"odinger equations, with well known soliton solutions, first suggested as carriers of optical signals by Hasegawa and Tappert~\\cite{HT}.  When $\\beta$ takes any other value, the equations are non-integrable.  Yang~\\cite{Yang97} studied these equations in great detail, enumerating several families of solitary waves and determining their stability.  Of these, the only stable solitary waves come from a family of symmetric single-humped solutions.  \n\nThe simplest solutions of interest to~\\eqref{eq:CNLS} consist of an exponentially-localized soliton in the first component, $A$, and zero in the second component, $B$, or vice versa.  A single soliton propagates at constant speed with a fixed spatial profile.  An important problem is the interaction of two such solitons upon collision, as interactions between two such solitons may lead to errors in a soliton-based transmission system.\n\nTan and Yang numerically studied the interaction of two solitons initialized in orthogonal channels with identical amplitude, headed toward each other with exactly opposite initial velocities~\\cite{TY00,TY:01,TY:01a}.  \nFor small values of $\\beta \\approx 0.05$, their simulations show that waves traveling above a critical velocity $\\vc$, the solitons pass by each other, losing a little bit of speed, but not otherwise showing a complicated interaction.  At initial velocities below $\\vc$, the solitons capture each other and merge into a stationary state near their point of collision.   See figure~\\ref{fig:tanyang}, which shows three graphs from~\\cite{TY:01} in which the input velocity of the solitons is plotted on the $x$-axis, and the solutions followed until they separate, and their outgoing velocity plotted on the $y$-axis, and assigned the value zero if they merge.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3in]{fig1a}\n\\includegraphics[width=3in]{fig1b}\n\\includegraphics[width=3in]{fig1c}\n\\caption{The exit velocity as a function of the input velocity for $\\beta=0.05$, $\\beta=0.2$, and $\\beta=0.6$, from Tan and Yang~\\cite{TY:01}, original authors' annotations removed.}\n\\label{fig:tanyang}\n\\end{center}\n\\end{figure}\n\nFor somewhat larger values of $\\beta \\approx 0.2$, they find that in addition to the above behavior, that the capture region is interrupted by a sequence of ``resonant reflection windows.''  Solitons with initial velocities in these resonance windows are reflected instead of being captured.  The numerical simulations show that the solitons pass each other once, undergo a finite number of width oscillations, then pass each other a second time.  Thus they call this the ``two-pass'' resonance.  \n\nFor larger values of $\\beta \\approx 0.6$, they find not only reflection windows, but an intricate fractal-like structure of both reflection and transmission windows.  Certain portions of the structure, when properly scaled, look like copies, in come cases even reflected copies, of other portions of the structure, and such features are seen at many different scales.  \n\nThe two-bounce resonance in kink-antikink interactions was explained qualitatively in the first papers of the Campbell group~\\cite{CSW:83,PC:83}.  As the kinks approach each other, they begin to interact,  and, at time $t_1$,  and energy is transferred into a secondary mode of vibration, with some characteristic frequency $\\omega$.  If the initial velocity is below a critical value, the kinks no longer have enough energy to escape each other's orbit, and turn around to interact a second time $t_2$.  They show numerically that a resonant reflection occurs if $t_2 - t_1 \\approx 2\\pi n\/\\omega +\\delta$. The parameter $\\delta$ is found by a least squares fit with numerical data. This relation is used to estimate the resonant initial velocities.  This reasoning has subsequently been adapted in studies of sine-Gordon kink-defect interactions~\\cite{FKV:92,KFV:91} and of vector soliton collisions~\\cite{TY00,TY:01,TY:01a} which are the focus of this paper.\n\n\\section{The model equations}\n\\label{sec:themodel}\nIn order to gain further insight into the resonance phenomenon, Tan and Yang examine a model system\nderived by Ueda and Kath~\\cite{UK:90} using the variational method.  In the variational method, the solution is assumed to take a certain functional form $A(\\vec p(t))$, $B(\\vec p(t))$, dependent on parameters $\\vec p(t)$ that are allowed to vary as a function of time.  This ansatz is then substituted into the Lagrangian functional for the PDE, which is integrated in space to yield a finite-dimensional effective Lagrangian,\n$$\nL_{\\rm eff} = \\int_{-\\infty}^{\\infty} \\cL(A,A^*,B,B^*) dz,\n$$\nwhose Euler-Lagrange equations describe the evolution of the time-dependent parameters.Equation~\\eqref{eq:CNLS} has Lagrangian density.\n\\begin{equation}\n\\label{eq:CNLS_Laga}\n\\cL = i(A A_z^* - A_z A^*) + i( B B_z^* - B_z B^*) + (\\abs{A_t}^2 -\\abs{A}^4)+( \\abs{B_t}^2-\\abs{B}^4) -2 \\beta\\abs{A}^2\\abs{B}^2\n\\end{equation}\nMany examples using this method for PDE's arising as Euler-Lagrange equations are given in a recent review by Malomed~\\cite{M:02}.\n\nFollowing~\\cite{UK:90}, we take an ansatz corresponding to two solitons at distance $2X$ of equal magnitude heading toward each other with equal speed, \n\\begin{equation}\n\\begin{split}\n\\label{eq:ansatz}\nA & = \\eta \\sech{\\frac{z-X}{w}} \\exp {i \\left(v (z-X) + \\frac{b}{2w}(z-X)^2 +\\sigma\\right)}, \\\\\nB & = \\eta \\sech{\\frac{z+X}{w}} \\exp {i \\left(-v (z+X) + \\frac{b}{2w}(z+X)^2 +\\sigma\\right)}\n\\end{split}\n\\end{equation}\nwhere $\\eta$, $X$, $w$, $v$, $b$, and $\\sigma$ are time-dependent parameters for the amplitude, position, width, velocity, chirp, and phase, whose evolution remains to be determined.  The variational procedure yields a conserved quantity $K = \\eta^2 w$,\nrelated to the conservation of the $L^2$ norm in CNLS, as well as the relations $\\diff{X}{t}=v$ and $\\diff{w}{t}=b$.\nThe evolution is described by the Euler-Lagrange equations\n\\begin{subequations}\n\\begin{align}\n\\diffn{X}{t}{2}&= \\frac{16K\\beta}{w^2}\\frac{d}{d\\alpha} F(\\alpha)  \\label{eq:TY1}\\\\\n\\diffn{w}{t}{2}&=\\frac{16}{\\pi^2 w^2}\\left(\\frac{1}{w}-K \n- 3 \\beta K \\frac{d}{d\\alpha}\\left(\\alpha F(\\alpha)\\right)\\right) \\label{eq:TY2}\n\\end{align}\n\\label{eq:TanYang}\n\\end{subequations}\nwhere $\\alpha= X\/w$ and the potential and coupling terms are given by \n\\begin{equation}\n\\label{eq:F}\nF(x) = \\frac{x \\cosh{x} - \\sinh{x}}{\\sinh^3{x}}.\n\\end{equation}\nNote that $F$, actually $-F$, is a potential term, not a force.  We keep this notation for continuity with previous studies.\n\nNumerical simulations show that for small $\\beta$, a solution to~\\eqref{eq:CNLS} with an initial condition of the form given in ansatz~\\eqref{eq:ansatz} will remain close to that form, i.\\ e.\\ the solution will continue to consist of two nearly orthogonally polarized solitons, at least until they merge into a single bound state.  Using the symmetries of equation~\\eqref{eq:TanYang}, we may set $K=1$ without loss of generality. Equivalently, the PDE symmetry may be used to set $K=1$ in the ansatz used by the variational method.\n\nThese equations display the two-bounce resonance phenomenon, as shown by Tan and Yang.\nConsider the initial value problem, with ``initial'' conditions describing the behavior as $t \\to -\\infty$.\n$$\nX \\to -\\infty;\\; \n\\diff{X}{t} \\to \\vin>0;\\;\nw \\to 1; \\; \n\\diff{w}{t} \\to 0\n$$\nThis does not strictly determine a unique solution, since the solution is invariant to time translation.  We  plot $\\vout$ as a function of $\\vin$ with $\\beta=0.05$ in figure~\\ref{fig:R003}.  These and all other ODE simulations were performed using routines from ODEPACK~\\cite{ODEPACK}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3in,angle=-90]{fig2}\n\\caption{The input vs.\\ output velocity of a pair of orthogonally polarized solitons with $\\beta=0.05$}\n\\label{fig:R003}\n\\end{center}\n\\end{figure}\nCompare this figure to the three plots of figure~\\ref{fig:tanyang}.  There are key similarities and differences between this graph and the exit velocity graphs of the full PDE simulations.  The critical velocity in this figure is about $\\vc=0.19$, close to the value $\\vc=0.1715$ found in~\\cite{TY:01}.\nA noteworthy difference is the complex behavior of solutions with initial velocity below $\\vc$---no such behavior, not even the two-pass windows, was seen in the very careful simulations of Tan and Yang.  This should not be surprising, as system~\\eqref{eq:TanYang} is Hamiltonian, and the set of initial conditions leading to unbounded trajectories in backwards time and bounded trajectories in forward time has measure zero, by reasoning similar to Poincar\\'e recurrence, as shown in Proposition~1 of~\\cite{GHW:02}.  Localized solutions to~\\eqref{eq:CNLS} may lose energy to radiation modes, a dissipation mechanism not present in the ODE model.  As a further result of the dissipation, the output speeds of the reflected solutions are much smaller than the input speeds in the PDE solutions, whereas at the very center of the ODE windows, the output speed exactly matches the input speed.\nA more interesting difference can be seen in the presence of the wide reflection windows, which were not found in the PDE simulations with this value of $\\beta$, summarized in figure~\\ref{fig:tanyang}.\n\nIn figure~\\ref{fig:R012}, the exit velocity graph of~\\eqref{eq:TanYang} shows that even at $\\beta=0.2$, the ODE dynamics display a complex fractal-like structure in addition to the reflection windows, which are not seen in the PDE dynamics for such small values of $\\beta$.  The numerical value of the crititcal velocity is $\\vc=0.86$, close to the value $\\vc=0.936$ found in~\\cite{TY:01}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3in]{fig3a}\n\\includegraphics[width=3in]{fig3b}\n\\caption{Left:  The exit velocity graph for equation~\\eqref{eq:TanYang} for $\\beta=0.2$, showing reflection windows and a variety of more complex fractal-like structures.  Right: The same figure with all but the main resonant reflections removed.}\n\\label{fig:R012}\n\\end{center}\n\\end{figure}\n\nThe numerical solutions of~\\eqref{eq:TanYang} qualitatively explain the resonance windows.  In figure~\\ref{fig:BT009}, the $w(t)$ components of the solutions with initial velocity $v$ at the center of the first two resonance windows (actually the points tangents to the line $\\vout=-\\vin$).  In the leftmost window, the oscillator $w(t)$ is excited, oscillates about 5 times and then is de-excited.  In the next window, $w(t)$ oscillates 6 times.  In each of the successive windows, $w(t)$ oscillates one more time before it is extinguished.  We will refer to the first window as the 2-5 window and the second window as the 2-6 resonance window. Recall that no such windows have been found in the PDE dynamics for this value of $\\beta$, but such windows have been found the in the ODE dynamics for all values of $\\beta$.   Tan and Yang demonstrated a width oscillation in the PDE solutions in analogy with that shown here.  The minimum value of $n$ in the 2-$n$ resonance decreases with increasing $\\beta$.  There does exist a 2-1 resonance with velocity $v=0.649$ in the ODE dynamics shown in figure~\\ref{fig:R012}, while the first resonance window found in the PDE simulations is the 2-$2$ resonance at about $v=0.9$ in figure~\\ref{fig:tanyang}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=4in]{fig4.eps}\n\\caption{Plots of the $w(t)$ component of~\\eqref{eq:TanYang} with initial velocity $v= 0.09988$ (top) and $v=0.13464$ (bottom) and $\\beta=0.05$, showing the 2-5 and 2-6 resonances.}\n\\label{fig:BT009}\n\\end{center}\n\\end{figure}\n\n\\subsection{A further simplified model}\n\nThe model~\\eqref{eq:TanYang}  bears a striking resemblance to the system derived in~\\cite{FKV:92} to study the two-pass resonance in the sine-Gordon equation with defect, and analyzed in~\\cite{GH:04}.  In that case, however, the situation is much simpler: the term equivalent to $w$ in~\\eqref{eq:TanYang} occurs only linearly, and the potential and coupling terms, equivalent to $F(X\/w)$ here, are functions of $X$ alone. This allows us to solve the analog of~\\eqref{eq:TY2} by variation of parameters to solve for this term and then insert it into the equivalent of equation~\\eqref{eq:TY1}, a critical step in our analysis.  In our numerically-computed solutions displaying the two-bounce resonance for small values of $\\beta$, the width $w$ undergoes only a small oscillation about its initial width $w=1$.  Therefore, we may partially linearize system~\\eqref{eq:TanYang}, which allows us to proceed in the same manner as we have for the sine-Gordon system.  We find reasonable agreement, with a few notable differences, between the two ODE systems.  We will discuss the linearized theory first and then discuss corrections due to the nonlinearity.\n\nAllowing $w=1+W$, where $W$ is considered small, expand all the terms in $W$, and keep only leading-order terms.  We arrive at the reduced system:\n\\begin{subequations}\n\\begin{align}\n\\diffn{X}{t}{2}&= 16\\beta\\left(F'(X)+G'(X)W\\right); \\label{eq:simplified_a}\\\\\n\\diffn{W}{t}{2}+ \\frac{16}{\\pi^2}W&=\\frac{48\\beta}{\\pi^2}G(X), \\label{eq:simplified_b}\n\\end{align}\n\\label{eq:simplified}\n\\end{subequations}\nwhere\n$$\nG(X)=-\\left(X F(X) \\right)'.\n$$\nFigure~\\ref{fig:vc} shows that this simplified equation gives an accurate estimate of the critical velocity for small values of $\\beta$ based on numerical simulation.  Figure~\\ref{fig:SR002} shows the equivalent of figure~\\ref{fig:R003} with the same value of $\\beta=0.05$ for the simplified equations.   It shows that the qualitative picture, chaotic scattering interupted by resonance windows for $v<\\vc$, is the same, while the actual location of those windows varies greatly.  In the case $\\beta=0.05$, the simplified equation has a 2-4 window, while the full equation's first resonance is 2-5.  As $\\beta$ was decreased further, the agreement between the two systems improved.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3in]{fig5}\n\\caption{The critical velocity for capture as a function of the coupling $\\beta$ for the fully nonlinear system~\\eqref{eq:TanYang} (solid) and the simplified system~\\eqref{eq:simplified} (dashed).}\n\\label{fig:vc}\n\\end{center}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3in,angle=-90]{fig6}\n\\caption{The exit velocity graph for the simplified system~\\eqref{eq:simplified}, showing qualitative agreement with figure~\\ref{fig:R003}.}\n\\label{fig:SR002}\n\\end{center}\n\\end{figure}\nWe  rescale time by allowing $t \\to 4 \\sqrt{\\beta} t$, transforming equations~\\eqref{eq:TanYang}  to\n\\begin{subequations}\n\\begin{align}\n\\ddot X&= F'(X)+G'(X)W;  \\label{eq:scaled_a}\\\\\n\\ddot W+\\lambda^2 W&=\\frac{3}{\\pi^2}G(X), \\label{eq:scaled_b}\n\\end{align}\n\\label{eq:scaled}\n\\end{subequations}\nwith fast frequency $\\lambda$ given by\n\\begin{equation}\n\\label{eq:lambda}\n\\lambda=\\frac{1}{\\pi\\sqrt{\\beta}}.\n\\end{equation}\nThe dot notation will be used for derivatives with respect to the scaled time.  The conditions in backward time as $t \\to -\\infty$ become:\n\\begin{equation}\nX \\to -\\infty;\\; \n\\dot X \\to \\Vin>0;\\;\nW \\to 0 \\\n\\dot W \\to 0. \\label{eq:init}\n\\end{equation}\nWe will  use capital $V$ to represent velocities in the scaled time $t$ and lower-case $v$ for velocities in the physical time.\n\n\\section{Determination of energy change and critical velocity}\\label{sec:DeltaE}\n\\label{section:DeltaE}\n\\subsection{Setup of Melnikov Integral for $\\Delta E$}\nFirst, note that if $W$ is held equal to zero, equation~\\eqref{eq:scaled_a} has the phase space shown in figure~\\ref{fig:phasespace}, showing three distinct types of orbits: closed orbits, corresponding to a pair of solitons bound together as a breather, unbounded orbits, corresponding to two solitons passing each other by, and orbits heteroclinic to degenerate saddle points at $(X,\\dot X)=(\\pm \\infty,0)$---separatrices---that form a boundary between  the two regimes.  These orbits correspond to level sets, where the energy \n\\begin{equation}\n\\label{eq:energy}\nE= \\half \\dot X^2 - F(X)\n\\end{equation}\nis negative, positive, or zero, respectively. As $W$ is allowed to vary, solutions may cross the separatrices.   We will show below that $W$ remains $O(\\sqrt{\\beta})$ by variation of parameters~\\eqref{eq:varparams} below and, thus, that perturbation methods are applicable.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=4in]{fig7}\n\\caption{The $X$ phase plane, showing trapped (dashed), untrapped (dash-dot), and separatrix (thin solid) orbits, corresponding to level sets of~\\eqref{eq:energy}. Superimposed is the $X-\\dot X$ projection of the 2-6 resonant solution to the fully nonlinear equations~\\eqref{eq:TanYang} with $\\beta=0.05$ (thick solid line).}\n\\label{fig:phasespace}\n\\end{center}\n\\end{figure}\n\nWe wish to asymptotically analyze orbits near the separatrix (see figure~\\ref{fig:phasespace}), since two solitons are initially captured when they cross the separatrix and are reflected or transmitted when they cross it a second time and may escape. We first determine the energy loss as a soliton goes from $X=-\\infty$ to $X=+\\infty$ by computing an energy integral called a Melnikov integral~\\cite{M:63}.  A Melnikov integral is a perturbative device for measuring the change of energy in a given system.  It is simply the integral of the time rate-of-change of the energy along some trajectory in the unperturbed problem.  A zero of the Melnikov integral is commonly a necessary condition for chaos in low-dimensional dynamical systems~\\cite{GH:83}.  In our case, we simply wish to calculate a change in energy.\n\nThe calculation has been simplified significantly from that given in~\\cite{GH:04}, in a manner that yields additional insight into the form of the energy loss.  In particular, we do not need to keep track of whether certain functions possess even or odd symmetry, and we find in an elementary way that the change of energy is negative.  First, we note that the separatrix is given by the level set $E=0$, therefore, along the separatrix, equation~\\eqref{eq:energy} may be solved for $X(t)$, giving\n\\begin{equation}\n\\diff{\\Xs}{t}= \\sqrt{2 F(\\Xs)}.\n\\label{eq:deltadot}\n\\end{equation}\nGiven the function $F$ in~\\eqref{eq:F}, it is not possible to find the separatrix orbit $\\Xs(t)$ in closed form.  The time-dependent energy exactly satisfies the differential equation\n\\begin{equation}\n\\diff{E}{t} = \\left(\\ddot X - F(X) \\right) \\dot X =  \\dot X G'(X) W= \\left( \\diff{}{t} G(X(t))\\right) W,\n\\label{eq:DeltaESetup}\n\\end{equation}\nwhere we have used equations~\\eqref{eq:scaled_a}.  We approximate the change in energy for one nearly heteroclinic orbit along the separatrix (from one saddle at infinity to the next saddle approach) by approximating $X(t)$ in~\\eqref{eq:DeltaESetup} with the known separatrix solution $\\Xs(t)$.  We integrate~\\eqref{eq:DeltaESetup} along the length of the orbit and integrate by parts to find the total change in energy:\n\\begin{equation*}\n\\begin{split}\n\\Delta E & = \\intinf   \\left( \\diff{}{t} G(\\Xs(t))\\right) W\\, dt   \\\\\n&= - \\intinf G(\\Xs(t)) \\diff{W(t)}{t}\\, dt,\n\\end{split}\n\\end{equation*}\nwhere we have integrated by parts.  Given the initial condition~\\eqref{eq:init}, with $V=0$ for the separatrix case, we may solve equation~\\eqref{eq:simplified_b} for $W(t)$ using variation of parameters:\n\\begin{equation}\nW(t) = \\frac{-3}{\\pi^2\\lambda} \\cos{\\lambda t} \\int_{-\\infty}^{t} G(\\Xs(\\tau)) \\sin{\\lambda \\tau} \\, d\\tau\n+\\frac{3}{\\pi^2\\lambda} \\sin{\\lambda t} \\int_{-\\infty}^{t} G(\\Xs(\\tau)) \\cos{\\lambda \\tau} \\, d\\tau,\n\\label{eq:varparams}\n\\end{equation}\n(again approximating $X(t)$ by $\\Xs(t)$) and\n$$\n\\diff{W(t)}{t} = \\frac{3}{\\pi^2} \\sin{\\lambda t} \\int_{-\\infty}^{t} G(\\Xs(\\tau)) \\sin{\\lambda \\tau} \\, d\\tau\n+\\frac{3}{\\pi^2} \\cos{\\lambda t} \\int_{-\\infty}^{t} G(\\Xs(\\tau)) \\cos{\\lambda \\tau} \\, d\\tau\n$$\nSetting \n$I_{\\rm s}(t) = \\int_{-\\infty}^{t} G(\\Xs(\\tau)) \\sin{\\lambda \\tau} \\, d\\tau$ and\n$I_{\\rm c}(t) = \\int_{-\\infty}^{t} G(\\Xs(\\tau)) \\cos{\\lambda \\tau} \\, d\\tau$, we find that \n\\begin{equation}\n\\begin{split}\n\\Delta E  &= - \\frac{3}{\\pi^2}\\intinf    I_{\\rm s}(t)  I_{\\rm s}'(t) dt    \n- \\frac{3}{\\pi^2}\\intinf    I_{\\rm c}(t)  I_{\\rm c}'(t) dt   \\\\\n&= -\\frac{3}{2\\pi^2}(   I_{\\rm s}^2(\\infty) +  I_{\\rm c}^2(\\infty)).\n\\end{split}\n\\end{equation}\nThis may be integrated by a standard substitution to yield\n\\begin{equation}\n\\begin{split}\n\\Delta E  &= - \\frac{3}{2 \\pi^2} \\left(\n\\left(\\intinf G(\\Xs(\\tau)) \\sin{\\lambda \\tau} \\ d\\tau\\right)^2 +\n\\left(\\intinf G(\\Xs(\\tau)) \\cos{\\lambda \\tau} \\ d\\tau\\right)^2  \\right) \\\\\n& =  - \\frac{3}{2\\pi^2} \\left \\lvert\n\\intinf G(\\Xs(\\tau)) e^{i \\lambda \\tau} \\ d\\tau\\right \\rvert^2.\n\\end{split}\n\\label{eq:DeltaEfinal}\n\\end{equation}\nThus, the problem is reduced to to calculating the integral~\\eqref{eq:DeltaEfinal}.  In fact, because in this case $G(\\Xs(t))$ is an even function, \n\\begin{equation}\n\\label{eq:DeltaEFinal2}\n\\Delta E= - \\frac{3}{2\\pi^2} I_{\\rm c}(\\infty)^2.\n\\end{equation}\n  Note that this shows the change in energy is generically negative when we assume $W \\to 0$ as $t \\to -\\infty$.  In fact, it must be negative, as the system conserves an energy that is positive-definite as $\\abs{X} \\to \\infty$, and no energy resides in the width oscillation initially.  Using $\\Delta E = -\\frac{\\vc^2}{2}$, we find\n$$\n\\vc= \\frac{\\sqrt{3}}{\\pi} I_{{\\rm c},\\infty}\n$$\nwhere $ I_{{\\rm c},\\infty}=  I_{\\rm c}(\\infty)$.  The integral in equation~\\eqref{eq:DeltaEFinal2} may be solved numerically by converting it into a differential equation, which may be integrated simultaneously with equation~\\eqref{eq:deltadot}.  Alternatively, we derive closed-form approximations to $\\Delta E $ $\\vc$ in section~\\ref{sec:complexvariables} below using complex analysis.\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=3in]{fig8}\n\\caption{The critical velocity of the ODE system~\\eqref{eq:simplified} computed via direct numerical simulation (solid line), the first-order asymptotic approximation (dots) and the second-order asymptotic approximation (dashed line) from~\\eqref{eq:vseries}, and via numerical evaluation of integral~\\eqref{eq:DeltaEfinal}.}  \n\\label{fig:vseries}\n\\end{center}\n\\end{figure}\n\n\\subsection{A Generalizaton of the calculation}\nNext, we briefly mention two generalizations of the Melnikov calculation above that will be useful later.  First, suppose that instead of approaching zero as $t \\to -\\infty$, \n$$W \\sim \\frac{3}{\\pi^2\\lambda} I_{{\\rm c},\\infty} \\cW \\sin{\\lambda(t-\\phi)}, $$\nThen the change of energy will be given by \n$$\n\\Delta E = \\frac{3 I_{{\\rm c}, \\infty}^2}{\\pi^2}\\left(-\\half + \\cW \\cos{\\lambda\\phi}\\right).\n$$\nAs $X$ traverses the heteroclinic orbit in the reverse direction, the sign of $G'(\\Xs (t))$  in~\\eqref{eq:DeltaESetup} is reversed, which leads to:\n\\begin{equation}\n\\label{eq:DeltaEreturn}\n\\Delta E = \\frac{3 I_{{\\rm c},\\infty}^2}{\\pi^2}\\left(-\\half - \\cW \\cos{\\lambda\\phi}\\right).\n\\end{equation}\nFor a resonance to occur, the change of energy calculated in the first Melnikov integral must cancel with the energy jump on the return trip.  Assume the forward heteroclinic orbit has ``symmetry time'' $t_1$ at which $X=0$, with symmetry time $t_2$ on the return trip.  Then, by~\\eqref{eq:varparams}, as $t\\to\\infty$ on the forward heteroclinic orbit, \n\\begin{equation}\nW(t) \\sim \\frac{3}{\\pi^2\\lambda} I_{{\\rm c},\\infty} \\sin{\\lambda(t-t_1)}.\n\\label{eq:Delta_E_general}\n\\end{equation}\nFor an exact resonance to occur, the energy change along the two heteroclinics must cancel, leading to the condition\n$$\n\\Delta E_1  + \\Delta E_2 =- \\frac{3 I_{{\\rm c},\\infty}^2}{2\\pi^2} +\n \\frac{3 I_{{\\rm c},\\infty}^2}{\\pi^2}\\left(-\\half - \\cos{\\lambda(t_2-t_1)}\\right)=0,\n$$\nobtained by combining~\\eqref{eq:DeltaEfinal} and~\\eqref{eq:DeltaEreturn} with $\\cW=1$ and $\\phi=t_2-t_1$. Thus $\\cos{\\lambda(t_2-t_1)}=-1$, or \n\\begin{equation}\n\\label{eq:t2mt1}\nt_2-t_1 = \\frac{(2n+1)\\pi}{\\lambda}.\n\\end{equation}\nThis differs from the equivalent resonance condition in~\\cite{GH:04}, in which $t_2-t_1 = 2\\pi n\/\\lambda$.  The difference arises because in that system the equivalent term to $G(X)$ was an odd function, whereas here $G$ is even.\n\nMany analyses of two-bounce and two-pass resonance phenomena have been based on the assumption that $t_2-t_1= \\frac{2\\pi n}{\\lambda} + \\delta$ for some undetermined $\\delta$, a phase shift that accounts for unidentified physical processes that have not been modeled.  Equation~\\eqref{eq:t2mt1} shows that in this case $\\delta = \\frac{\\pi}{\\lambda}$.  It is worth computing a linear fit of $t_2-t_1$ vs.\\ $n$ for comparison with earlier studies and, we will see, for comparison with the analogous result for the fully nonlinear ODE~\\eqref{eq:TanYang}.   At $\\beta=0.05$, we find the linear fit $t_2-t_1= 4.935\\left(n + \\half\\right) - 0.011$ and for $\\beta=0.2$,  we find $t_2-t_1\\approx4.931 \\left(n + \\half\\right)+ 0.139$, whereas $\\frac{2\\pi}{\\lambda}\\approx4.9348$.  Therefore, we see that to leading order, relation~\\eqref{eq:t2mt1} holds, and that the agreement improves with decreasing $\\beta$.\n\n\\subsection{Evaluation of critical velocity using complex analysis}\n\\label{sec:complexvariables}\n\nSince $\\lambda$ is large and $G(X)$ is analytic, the calculated change of energy~\\eqref{eq:DeltaEfinal} is exponentially small.  In a calculation for a similar system, we were able to calculate the analogous integrals explicitly because $\\Xs(t)$ was known in a simple closed form~\\cite{GH:04}.  In the present case, we are forced instead to expand the integrand of~\\eqref{eq:DeltaEfinal} about a certain (branch) pole.  Given the form of the potential~\\eqref{eq:F}, $F$ has a pole whenever $\\sinh{X}=0$ and the numerator of $F$ is nonzero or has a zero of order less than 3.  The nearest pole to $X=0$ occurs at $X=i\\pi$.  Along the separatrix\n\\begin{equation}\n\\label{eq:sep}\n \\diff{X}{t} = \\sqrt{2 F(X)},\n \\end{equation}\nso we let $t^*$ be chosen such that $X(t^*)=0$ given the initial condition $X(0)=0$.  This gives the formula:\n\\begin{equation}\n\\label{eq:tstar}\niT \\equiv t^* = \\int_{0}^{t^*} dt =   \\int_{0}^{i\\pi}\\sqrt{\\frac{1}{2F(X)}} \\, dX =\n\\frac{i}{\\sqrt{2}} \\int_{0}^{\\pi} \\sqrt{\\frac{\\sin^3 y}{\\sin y - y\\cos y} } dy\n \\approx 2.10392\\, i.\n\\end{equation}\nWe expand~\\eqref{eq:sep} about $X=i\\pi$ and $t=t^*$ and find\n\\begin{align*}\n\\left(\\frac{(-1)^{-1\/4}}{\\sqrt{2\\pi}} (X-i\\pi)^{3\/2} +O((X-i\\pi)^{9\/2})\\right)\\,dX& =  \\ dt \\\\\n\\frac{(-1)^{-1\/4}\\sqrt{2}}{5\\sqrt{\\pi}} (X-i\\pi)^{5\/2} +O((X-i\\pi)^{11\/2}) & =  t-t^* \n\\end{align*}\nwhich may be inverted to form\n$$\nX-i \\pi = (-1)^{1\/10} \\pi^{1\/5}2^{-1\/5} 5^{-2\/5} (t-t^*)^{2\/5} + O((t-t^*)^{8\/5}).\n$$\nBased on the expansion\n$$\nF(X)=i \\pi (X-i\\pi)^{-3} + O(1)\n$$\nwe compute the two leading order terms of the integrand of~\\eqref{eq:DeltaEfinal}\n$$\nG(\\Xs(t))= \\frac{(-1)^{3\/5} \\pi^{6\/5}2^{4\/5}3}{5^{8\/5}} (t-t^*)^{-8\/5} +\n \\frac{(-1)^{1\/5} \\pi^{2\/5}2^{8\/5}} {5^{6\/5}} (t-t^*)^{-6\/5} + O( (t-t^*)^{-2\/5} ).\n $$\n Therefore~\\eqref{eq:DeltaEfinal} involves integrals of the type\n $$\nI(\\lambda,T,p)= \\intinf e^{i \\lambda t} (t-i T)^p \\, dt\n $$ \n with $\\lambda>0$, $T>0$, and $p<0$. Here $iT$ is the branch pole, from which a branch line extends vertically to $i\\infty$.  By a shift of contour and a change of variables to $z=i \\lambda(t-iT)$, this can be replaced by an integral over the Hankel contour $\\gamma$, which starts at $-\\infty$ below the real axis, circles zero once in the positive direction, and returns to $-\\infty$ along (and above) the real axis~\\cite{CKP}.\n $$\nI(\\lambda,T,p)= \\frac{(-i)^{p+1}}{\\lambda^{p+1}} e^{-\\lambda T} \n\\int_\\gamma e^z z^p dz\n$$\nwhich forms part of a familiar representation of Euler's gamma function and yields the exponentially small term\n\\begin{equation}\nI(\\lambda,T,p)= (-1)^{-p\/2} 2 \\sin{\\left((p+1)\\pi\\right)} \\G{(p+1)} e^{-\\lambda T}\\lambda^{-(p+1)}.\n\\label{eq:Intlambda}\n\\end{equation}\n\nUsing~\\eqref{eq:Intlambda} and standard trigonometric and gamma function identities, we evaluate the integral in~\\eqref{eq:DeltaEfinal} \n\\begin{equation}\n\\intinf G(\\Xs(\\tau)) e^{i \\lambda \\tau} \\ d\\tau =\n\\left(\n\\frac{(-1)^{7\/5}  2^{4\/5} \\pi^{6\/5}}{5^{3\/5}} \\Gamma\\left(\\frac{2}{5}\\right)\\sin{\\frac{2\\pi}{5}}\\lambda^{3\/5}  +\n\\frac{(-1)^{4\/5}  2^{8\/5} \\pi^{2\/5}}{5^{1\/5}} \\Gamma\\left(\\frac{4}{5}\\right)\\sin{\\frac{4\\pi}{5}}\\lambda^{1\/5}  + \nO(\\lambda^{-3\/5}) \\right) e^{-\\lambda T}.\n\\end{equation}\nAs the integrand is real, we choose the branch $(-1)^{1\/5} = -1$ above.  Using that $\\Delta E =  \\frac{\\vc^2}{2}$ and the scaling relation given before~\\eqref{eq:scaled}, we arrive at the expansion for the critical velocity in physical variables:\n\\begin{equation}\n\\vc = \\frac{8\\sqrt{3}}{5} e^{-T\/\\pi\\sqrt{\\beta}}\\left(\\theta(2\/5)\\alpha \\beta^{1\/5} - \\theta(4\/5)\\alpha^2 \\beta^{2\/5} + \\ldots \\right),\n\\label{eq:vseries}\n\\end{equation}\nwhere $\\theta(x)= \\sin{\\pi x}\\G{(x)}$ and $\\alpha= \\pi^{-2\/5} 2^{4\/5} 5^{2\/5}$. using equations~\\eqref{eq:lambda},   \\eqref{eq:energy}, as well as the two integrals above.  \nFigure~\\ref{fig:vseries} shows that the critical velocity is poorly predicted by the first term in this series, but well-predicted up to about $\\beta=0.1$ when the second term is added.  The series expansion of the integrand of~\\eqref{eq:DeltaEfinal} about $t=t^*$ contains one more integrable  term which does not lead to a visible improvement of the approximation to $\\vc$.   In order to improve the approximation, one would have to calculate expansions about the additional singularities of $G(X_S(t))$ further off the imaginary $t$-axis.\n\n\\section{Matched asymptotic construction of solutions}\n\\label{sec:matching}\n\\subsection{The expansion framework}\nIf $\\Vin >\\Vc$, then $\\dot X$ remains positive for all time and $X \\to +\\infty$ monotonically.  We can call this a one-pass transmitted solution.  A ``pass'' will occur each time $X=0$, when the two solitons pass each other and energy is transferred between the translation and vibration modes.  If $\\Vin < \\Vc$, then the energy is negative after one pass, and the solitons reverse direction, setting up the second pass.  On the first pass, the change of energy was shown in~\\eqref{eq:DeltaEfinal} to be negative, but on subsequent passes, it may take either sign, by~\\eqref{eq:DeltaEreturn}.  On the second, and subsequent, passes the solitons may escape if the energy is positive, or may be reversed again.  We will focus primarily on the case that the solitons interact twice before escaping.\n\nFollowing~\\cite{GH:04}, we construct 2-pass solutions by a matched asymptotic expansion.  The solution consists of sequences of nearly heteroclinic orbits connected to near saddle approaches at $X=\\pm\\infty$.  The change in energy from one saddle approach to the next is approximated by the Melnikov integral calculated in section~\\ref{sec:DeltaE}.\nThe 2-bounce solution can be constructed from the following 5 pieces:\n\\begin{enumerate}\n\\item A near saddle approach to $X=-\\infty$ with energy $E_0=\\half \\Vin^2$, such that $\\dot X \\to \\Vin < \\Vc$,  as $t \\searrow -\\infty $ ;\n\\item a heteroclinic orbit with $dX\/dt>0$ such that $X(t_1)=0$, with energy change $\\Delta E_1$ given by~\\eqref{eq:DeltaEFinal2};\n\\item a near saddle approach to $X=+\\infty$ with negative energy $E=-\\half M^2$, such that $X$ achieves its maximum at $t=t^{*}$;\n\\item a heteroclinic orbit with $dX\/dt <0$ such that $X(t_2)=0$,  with energy change $\\Delta E_2$ given by~\\eqref{eq:DeltaEreturn};\n\\item and a near saddle approach to $X=-\\infty$ with positive energy $E=\\half \\Vout^2$,  such $\\dot X \\to -\\Vout$, as $t \\nearrow \\infty$.\n\\end{enumerate}\nThe times $t_1$, $t_2$, and $t^*$, as well as the energy levels, remain to be determined below.  In the language of matched asymptotics, the approximations at steps 1, 3, and 5 are the ``outer solutions'' and steps 2 and 4 are the ``inner solutions.''\n\nA comment about the last step is in order.  For general initial velocity $\\Vin$, the energy at step 5 will not match the energy at step 1.  If these two energies match exactly, then we say the solution is a 2-pass resonance.  If the energy at step 5 is positive but less than $\\Vin^2\/2$, then the solution is in the 2-pass window, and may be called an incomplete resonance.  Physically, the solitons reflect off each other, but with reduced speed and with significant energy remaining in their width oscillation.  The outer edges of the window will be given by velocities where the energy at step 5 is identically zero.  This defines the width of the windows. If at step 5, the energy is instead negative, then the solution remains trapped for another step, alternating between negative energy near-saddle approaches to $X=-\\infty$ and $X=\\infty$ until enough energy is returned to $X$ such that $E=\\Vout^2\/2$, and $X\\to \\pm \\infty$.  Non-resonant solutions and higher resonances are explained in section~\\ref{sec:generalized}.\n\nFor the simpler sine-Gordon system, we wrote down a general asymptotic formula for $n$-pass solutions, calculated the location of 3-pass windows, and calculated the widths of the 2-bounce windows~\\cite{GH:04}.  Analogous results are possible in the present situation and are discussed below in section~\\ref{sec:generalized}, although in less detail than in the previous paper.\n\nWe use the method of matched asymptotic expansions, as in~\\cite{DH:02,DH:03, GH:04}.  The heteroclinic orbits along the separatrix are matched (forward and backward in time) to the finite time singularities associated with the near-saddle approaches. We will not make use of the two positive energy expansions, so we will not compute them.  They enter the analysis  when the energy change calculated over the heteroclinic orbits in the above section will then be used to connect the positive and negative energy expansions. \n\n\\subsection{Asymptotic description of heteroclinic orbit for large $X$}\n\nWe first construct an expansion of the ``inner solution,'' given by the heteroclinic orbit. Along the heteroclinic orbit, ${\\dot X}^2\/2 = F(X)$.  Setting $X=0$ at $t=t_1$, the trajectory is given as the solution to\n\\begin{equation}\n\\label{eq:zero_energy}\n\\int_{t_1}^t dt' = \\int_0^X \\frac{dX'}{\\sqrt{2F(X')}} = \\int_0^{X_0} \\frac{dX'}{\\sqrt{2F(X')}} z+ \\int_{X_0}^X \\frac{dX'}{\\sqrt{2F(X')}} \n\\end{equation}\nfor an arbitrary $O(1)$ constant $X_0<X$.  The first integral is $O(1)$ and will be asymptotically dominated by the second.  For $X\\gg 1$, we may approximate the potential by \n\\begin{equation}\n\\label{eq:Fasympt}\nF(X) \\sim  4 (X-1) e^{-2X}.\n\\end{equation}\nIf we let $Z=X-1$, then \n\\begin{equation}\n\\label{eq:XtoZ}\nF(X) \\sim  (4 e^{-2}) Z e^{-2Z},\n\\end{equation}\nwhich we substitute into~\\eqref{eq:zero_energy} and get\n$$\nt-t_1= \\frac{e}{\\sqrt{8}}  \\int  \\frac{e^{Z}}{\\sqrt{Z}} dZ +O(1).\n$$\nWe make the substitution $Z=C+y$, where $C\\gg 1$ and $y=O(1)$, as motivated in the next section, and expand the integral in powers of $C^{-1}$, yielding\n\\begin{align}\n\\label{eq:expand_heterocline}\nt-t_1=\\frac{e^{1+C}}{\\sqrt{8C}}\\left(e^y + \\frac{1}{2C} (1-y)e^y + O\\left(\\frac{1}{C^2}\\right) \\right) + O(1)\n\\end{align}\n\n\\subsection{Asymptotic description of the saddle approach near $X= \\infty$}\nAs the width perturbation $W(t)$ remains small, we may construct approximate solutions from solutions to equation~\\eqref{eq:scaled_a} with $W=0$.  Since $F(X)\\to 0$ as $\\abs{X}\\to\\infty$, $(\\infty,0)$ is a degenerate fixed point and is of saddle-type.\nFirst we compute the near-saddle approach at $X=+\\infty$, under the approximation that the solution has small constant energy given by $$ E = -\\frac{M^2}{2}$$\nwith $M \\ll 1$. \nSo that, using~\\eqref{eq:XtoZ}, the near-saddle expansion satisfies:\n$$\n\\half \\dot Z^2 - 4 e^{-2} Z e^{-2Z} = -\\frac{M^2}{2}.\n$$\nWe make the expansion $Z= C + y$, where $C \\gg 1$ is determined in~\\eqref{Cn} and show $y=O(1)$. Then \n$$\n\\half \\dot y^2 - 4 e^{-2}(C+y) e^{-2C}e^{-2y} = -\\frac{M^2}{2}.\n $$\nWe define $C$ by\n\\begin{equation}\n\\label{Cn}\n8 C e^{-2}e^{-2  C} = M^2\n\\end{equation}\nand let $T = M t$, which gives the simplified equation\n\\begin{equation}\n\\label{eq:scaledNSA}\n\\left(\\frac{dy}{dT}\\right)^2 - \\left(1+\\frac{y}{C}\\right)e^{-2y} = -1.\n\\end{equation}\n\n\\subsection{The matching procedure for near and exact 2-pass resonances}\n\\subsubsection*{The simplest asymptotic approximation}\nIgnoring the term $\\frac{y}{C}$ in~\\eqref{eq:scaledNSA}, as $C\\gg1$, the near saddle approach takes the form\n$$\ne^y = \\cos{(T-T^{*})}\n$$\nwhere $T^{*}$ is the ``center time'' at which the near saddle approach comes\nclosest to the degenerate saddle at $(X,\\dot X)=(\\infty, 0)$.  This has\nfinite-time singularities forward and backward in time. Backwards in time, this is singular as $T-T^{*} \\searrow -\\frac{\\pi}{2}$, and\nmay be asymptotically expanded as \n\\begin{equation}\n\\label{eq:expansion}\ne^y = T-T^{*} + \\frac{\\pi}{2}.\n\\end{equation}\nFor large $X$, the  heteroclinic orbit in~\\eqref{eq:expand_heterocline} is\nasymptotically approximated using~\\eqref{Cn}, to leading order in $C^{-1}$, by \n $$\ne^y = T-T_1 \n$$\nwith a similar expression for $T_2-T$ along the return heteroclinic.  The algebraic growth of the heteroclinic orbit matches to the finite time singularities of the near-saddle approach only if\n$$\nT^{*}-T_1 = \\frac{\\pi}{2},\n$$\nwhich, combined with a similar relation for $T_2-T^{*}$, yields\n$T_2-T_1 = \\pi$ as in our analysis of the sine-Gordon model, or, in the unscaled time variable\n\\begin{equation}\n\\label{t2-t1_a}\nt_2 - t_1  = \\frac{\\pi}{M},\n\\end{equation}\nwhich shows the energy dependence of the period.  The energies at steps 1 and 3 are related by \n$\\frac{\\Vin^2}{2} + \\Delta E_1 = -\\frac{M^2}{2}.$\n\n\\subsubsection*{Exact Resonance Condition:} \nTo this point, the calculation has been valid for general $\\Vin<\\Vc$.  We now specialize to the case of exact 2-pass resonance. In a resonant solution the second energy jump must balance the first $\\Delta E_2 = -\\Delta E_1$, a condition for which is given in~\\eqref{eq:t2mt1}, implying $\\lambda\/(2n+1)$.  Thus the resonant initial velocity $V_n$ solves\n\\begin{equation}\n\\label{eq:VnDeltaEMn}\n\\frac{V_n^2}{2} + \\Delta E_1 = -\\frac{M_n^2}{2}.\n\\end{equation}\nSolving this for $V_n$, using that $\\Delta E_1 = -\\Vc^2\/2$,\n\\begin{equation}\nv_n=\\sqrt{\\vc^2 - \\frac{16}{\\pi^2 (2n+1)^2}},\n\\label{eq:vn_comp1}\n\\end{equation}\nwhere scaling~\\eqref{eq:lambda} has been used to convert this result back to the physical variables.\nAlso $n_{\\rm min}(\\beta)$ is given by the smallest integer $n$ that makes $v_n$ a real number in equation~\\eqref{eq:vn_comp1}. In figure~\\ref{fig:vn_computed}, we see that this does a relatively poor job at predicting resonant velocities.\n\n\\subsubsection*{An Improved Approximation for Resonance}\nThe above calculation showed clearly the procedure used to find the resonant velocities.  Here it is refined slightly to improve its accuracy.  After some rearrangement, equation~\\eqref{eq:scaledNSA} becomes\n\\begin{equation}\n\\label{eq:period_integral}\n\\frac{dy}{\\sqrt{\\left(1+\\frac{y}{C_n}\\right)e^{-2y} -1}}=dt\n\\end{equation}\nScaling the time variable as before, we  expand this integral in powers of $\\frac{1}{C_n}$ and keep the first two terms. This gives \n\\begin{equation}\n\\label{eq:scaledNSA_2}\n\\frac{dy}{\\sqrt{e^{-2y}-1} }+ \\frac{1}{2 C_n}\\frac{y e^{-2y} dy}{(-1+e^{-2y})^{3\/2}} = dT\n\\end{equation}\nfor the near saddle approach, which has solution:\n\\begin{equation}\n\\label{scaled_NSA_2_sol}\ne^y - \\sin{\\left( \\frac{1}{2 C_n +1} \\frac{y}{\\sqrt{e^{-2y}-1}} \\right)} \n= \\cos{\\left(\\frac{T-T^{*}}{1+ \\frac{1}{2C_n}}\\right)}.\n\\end{equation}\nAs  $T-T^{*} \\searrow \\left(1+ \\frac{1}{2C_n}\\right) \\frac{\\pi}{2}$ on the right-hand side, we find $y\\to-\\infty$.  \nRearranging the expansion of the heteroclinic orbit, we find~\\eqref{eq:expand_heterocline}\n$$\n\\left(1-\\frac{1}{2 C_n +1}y\\right) e^y= \\frac{T-T_1}{1+ \\frac{1}{2{C_n}}}.\n$$\nMatching these two approximations gives $T_2-T_1 = \\left(1+\\frac{1}{2 C_n}\\right)\\pi$ or\n\\begin{equation}\n\\label{t2-t1_b}\nt_2 - t_1  = \\frac{\\left(1+\\frac{1}{2C_n}\\right)\\pi}{M_n},\n\\end{equation}\na more accurate approximation to the period.  Combining this with~\\eqref{eq:t2mt1} yields:\n$$\n \\frac{\\left(1+\\frac{1}{2C_n}\\right)\\pi}{M_n} = \\frac{(2n+1)\\pi}{\\lambda}.\n$$\nThis is still not a closed equation as $C_n$ has yet to be specified.  We may eliminate $M_n$ from this equation and~\\eqref{Cn} to obtain an implicit relation that defines $C_n$:\n\\begin{equation}\n\\label{Cn_implicit}\n32\\pi^2 \\beta (2n+1)^2 C_n^3 - (2C_n+1)^2 e^{2C_n+2} = 0.\n\\end{equation}\nThis has exactly two positive roots as long as $2\\pi^2e^{-2}\\beta(2n+1)^2 > \\frac{3+2\\sqrt{3}}{9}e^{\\sqrt{3}}$, with the larger root relevant. Thus, we come to the revised estimate of the resonant velocities\n\\begin{equation}\n\\label{eq:vn_comp2}\nv_n=\\sqrt{\\vc^2 - \\frac{16\\left(1+\\frac{1}{2C_n}\\right)}{\\pi^2 (2n+1)^2}}.\n\\end{equation}\nFigure~\\ref{fig:vn_computed} shows that this does better than our first estimate.  In a similar computation, we found that this analysis in a neighborhood of $X=\\infty$ was enough to determine the resonant velocities.  We find in the next section that we can do better with a numerical criterion based on~\\eqref{eq:t2mt1}.\n\n\\subsubsection*{A Numerical Condition for Resonance}\nIn all situations where heteroclinic or homoclinic orbits are matched to near-saddle approaches $t_2-t_1$ equals half the period of~\\eqref{eq:scaled_a} with $W$ set to zero,\n\\begin{equation}\n\\label{eq:XnoW}\n\\ddot X - F(X) =0.\n\\end{equation}  \nLet $P(E)$ be the period of the closed orbit of~\\eqref{eq:XnoW} with energy $E<0$.  We may solve for $P(E)$ by evaluating the definite integral\n\\begin{equation}\n\\label{eq:Period}\nP = 2 \\int_{-\\Xmax(E)}^{\\Xmax(E)} \\frac{dX}{\\sqrt{2}\\sqrt{F(X)+E}}\n\\end{equation}\nwhere $\\Xmax(E)$ is the positive root of $F(X)+E=0$. The period $P$ cannot be computed in closed form given the particular potential $-F(X)$ in this problem.  Alternately, one can compute the period simply by integrating equation~\\eqref{eq:XnoW} with initial conditions $X=\\Xmax(E)$, $\\dot X = 0$ until reaching the termination condition $X=-\\Xmax(E)$ at the time $P(E)\/2$.\n\nIn either case, the above calculation must be inverted numerically to yield the energy as a function of the period, using the secant method or some variant.  In the scaled variables, we have, from equation~\\eqref{eq:t2mt1}, \n\\begin{equation}\n\\frac{P(E_n)}{2} = \\frac{(2n+1)\\pi}{\\lambda}\n\\label{eq:nlresonance}\n\\end{equation}\nThis is solved for $E_n = -\\frac{M_n^2}{2}$, and the resonant velocity is found using~\\eqref{eq:VnDeltaEMn}.  This, and the other two approximations are shown in figure~\\ref{fig:vn_computed} \nAll solutions with positive $n$ up to $n=14$ are shown in the figure.  Approximations~\\eqref{eq:vn_comp1} and~\\eqref{eq:vn_comp2} both predict the existence of a 2-3 window, while the numerical calculation~\\eqref{eq:nlresonance} does not, and no such window is found by direct numerical simulation.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=4in]{fig9}\n\\caption{The resonant velocities, indexed by the number of complete oscillations of $w(t)$, with $\\beta=0.05$.  The thick solid curve at the bottom is the result of direct numerical simulation.  From top to bottom, the other curves are the asymptotic results of equations~\\eqref{eq:vn_comp1} and~\\eqref{eq:vn_comp2}, and the value involving numerical calculation of the energy level, given the resonant period~\\eqref{eq:nlresonance}}\n\\label{fig:vn_computed}\n\\end{center}\n\\end{figure}\n\n\\subsection{Generalizaton to near-resonances and higher resonances}\n\\label{sec:generalized}\nThe 2-pass resonant solutions are a countable, and thus measure-zero, family of initial conditions.  Each 2-pass window has finite width whose left and right edges can be found by imposing the conditions that $\\Delta E_2 = M^2\/2$, so that the output energy is identically zero.  It can be shown that the window widths scale as $n^{-3}$ for large $n$.\n\nIn between the 2-pass windows there is a complicated structure consisting of many narrower windows.  These include 3-pass windows, which can be found as follows.  A three-pass resonant solution has three energy jumps.  Just as $W(t)$ and $X(t)$ are even functions about $t^*$ in two-pass solutions, in 3-pass solutions, $W(t)$ and $X(t)$ are odd functions about their center time.  We can place the three ``center times'' at $t=-t_0$, $t=0$, and $t=t_0$, and notice that if the solution is odd, then $\\Delta E=0$ at $t=0$.   The change of energy at the second jump is $\\Delta E = -3 I_{\\rm{c},\\infty}^2\/\\pi^2 (\\frac{1}{2} + \\lambda t_0)$, which implies \n$$\nt_0 = \\frac{\\left(2n+1 \\pm \\frac{1}{3}\\right) \\pi}{\\lambda}.\n$$\nand gives 3-pass resonant solutions with \n$$\nv_{3,n\\pm}  = \\sqrt{\\vc^2 - \\frac{16}{\\pi^2 (2n+1\\pm\\frac{1}{3})^2}}.\n$$\nA corrected formula, as in equation~\\eqref{eq:vn_comp2}, and a more accurate numerical condition, as in equation~\\eqref{eq:nlresonance}, may also be derived.  A general formula for the locations of higher complete resonances can be derived as in~\\cite{GH:04}, but this equation must be solved numerically. \n\n\\section{The effect of coupling to a weakly nonlinear oscillator}\n\\label{sec:nonlinearity}\nWe briefly discuss the discrepancies between the full ODE model~\\eqref{eq:TanYang} and the simplified model~\\eqref{eq:simplified}, in order to account for the marked difference between the window locations between figures~\\ref{fig:R003} and~\\ref{fig:SR002}.  The most obvious, seen by comparing the two graphs of figure~\\ref{fig:w_vs_W}, where the component $w(t)$ and $1+W(t)$ are both plotted for the 2-5 resonance with $\\beta=0.2$.  It is clear that $w(t)$, the fully nonlinear oscillation, has a larger amplitude, a larger period, and the mean about which it oscillates is displaced upward.  As done following equation~\\eqref{eq:t2mt1}, we fit $t_2-t_1$ from our numerical simulations and find $t_2-t_1 \\approx 5.00\\left(n+\\half\\right)+0.541$ when $\\beta=0.05$.  For the case $\\beta=0.2$, careful examination shows that $t_2-t_1$ is not approximated that well by a linear fit, with the growth in $t_2-t_1$ slowing as $n$ increased, and $t_2-t_1\\approx 6.38\\left(n+\\half\\right) + 0.327$ when the first 10 resonances are used, and $t_2-t_1\\approx 6.20 \\left(n+\\half\\right)+ 1.93$ when resonances 11 through 20 are used, and the error in this fit is much larger than in the simplified model, especially when the leftmost windows are included.  We see then that, in addition to a large correction to the oscillation frequency, a significant phase shift appears in the fully nonlinear dynamics.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3in]{fig10a}\n\\includegraphics[width=3in]{fig10b}\n\\caption{The solutions components of the 2-5 resonance of \\textbf{(a)} $1+W(t)$ of the simplified equation~\\eqref{eq:simplified} and \\textbf{(b)} $w(t)$ of the full ODE~\\eqref{eq:TanYang}.}\n\\label{fig:w_vs_W}\n\\end{center}\n\\end{figure}\n\nThis shift in amplitude, frequency, and mean value can be explained using a strained coordinate or Poincar\\'e-Lindstedt expansion~\\cite{KC:96}.  We assume that $w=W+1$ satisfies the homogeneous part of~\\eqref{eq:TY2}\n\\begin{equation}\n\\label{eq:nonlin}\n\\diffn{w}{t}{2}=\\frac{\\lambda^2}{w^2}\\left(\\frac{1}{w}-1\\right)\n\\end{equation}\nwhere we have used scaling~\\eqref{eq:lambda} and have again set $K=1$.  Expanding this as a power series in $W$, we find\n\\begin{equation}\n\\label{eq:W_expansion}\n\\diffn{W}{t}{2} +\\lambda^2 W = \\lambda^2\\left(3 W^2 - 6 W^3 + O(W^4)\\right)\n\\end{equation}\n We look for periodic solutions with initial conditions\n \\begin{equation}\n W(0)=\\epsilon \\text{ and } \\dot W (0)=0,\n \\label{eq:W_ic}\n \\end{equation}\n noting that these initial conditions are somewhat arbitrary, and leaving $\\epsilon$ positive and small, but for now undefined.  We expand both the function $W$ and the frequency of oscillations $\\Omega$ in powers of $\\epsilon$\n\\begin{equation}\n\\label{eq:W_PL}\n\\begin{split}\nW &= \\sum_{k=1}^{\\infty} \\epsilon^k W_{k-1}(T) \\\\\n\\Omega &=  \\sum_{k=0}^{\\infty} \\epsilon^k \\Omega_k \n\\end{split}\n\\end{equation}\nwhere $T=\\Omega t$, and assume that the solution has period $2\\pi$ in $T$. The equation is satisfied at  each order in $\\epsilon$, with the $\\Omega_k$ chosen to suppress secular growth terms.  We find that \n$$\n\\Omega_0 = \\lambda ; \\; \\Omega_1=0;\\;  \\Omega_2=-\\frac{3}{2}\\lambda\n$$\nand\n\\begin{equation}\n\\label{eq:W_k}\n\\begin{split}\nW_0 & = \\cos{\\Omega t}\\\\\nW_1 & = \\frac{3}{2} - \\cos {\\Omega t} -\\half \\cos{2\\Omega t} \\\\\nW_2 &=  -3 + \\frac{13}{8} \\cos{\\Omega t} +  \\cos{2\\Omega t} + \\frac{3}{8}\\cos{3\\Omega t} \n\\end{split}\n\\end{equation}\nThus the period of oscillation is decreased at larger amplitudes, as found from the least squares fits.\n\nIt remains to determine a suitable value of $\\epsilon$ in~\\eqref{eq:W_ic} and its effect on the resonance.  The full ODE model~\\eqref{eq:TanYang} conserves the Hamiltonian \n\\begin{equation}\n\\label{eq:NLEnergy}\nH = \\half{\\dot X}^2 +\\frac{2\\pi^2}{3} \\dot{w}^2  + \\frac{8}{3}\\left(1+ \\frac{1}{w^2}-\\frac{2}{w}\\right) \n- \\frac {16\\beta}{w} F\\left(\\frac{X}{w}\\right),\n\\end{equation}\nnot written here in canonical variables.\nWe see from figure~\\ref{fig:vc} that $\\vc$, and hence $\\Delta E$ is approximately the same for the full and simplified ODE systems.  Expanding this in powers of $W=w-1$, we obtain the approximate Hamiltonian\n\\begin{equation}\n\\label{eq:Energy_expansion}\nH \\approx  \\half{\\dot X}^2 +\\frac{2\\pi^2}{3} \\dot{W}^2  + \\frac{8}{3}\\left( W^2 - 2W^3 + 3 W^4+\\ldots\\right) \n- 16\\beta F(X) - 16\\beta G(X)W + \\ldots.  \n\\end{equation}\n\nAs $t\\to -\\infty$, all of the energy is stored as kinetic energy in the soliton modes $H=\\half v_0^2$.  At the symmetry time, which we can set to $t^{*}=0$, $\\dot X=0$ and $\\dot w=0$, and $X=\\Xmax$ is given, if $G(X) W$ is small enough to be ignored, by the solution to $\\half v_0^2 - \\half \\vc^2  = -16 \\beta F(\\Xmax)$.\nPlugging this back into the energy~\\eqref{eq:NLEnergy} and using the expansion~\\eqref{eq:Energy_expansion} only for the coupling $F(X\/w)$ term, we obtain\n$$\n\\half \\vc^2 \\approx  \\frac{8}{3}\\left(1+ \\frac{1}{(1+W)^2}-\\frac{2}{1+W}\\right) - 16 \\beta G(\\Xmax)W.\n$$\nFor resonant velocities sufficiently close to $\\vc$ or for small enough $\\beta$, $16\\beta G(\\Xmax)$ is negligibly small, and we can solve the resulting quadratic equation for $W(0)$ and obtain  \n$$\nW(0)= \\frac {\\pm \\frac{\\sqrt 3}{4}\\vc}{1\\pm \\frac{\\sqrt 3}{4}\\vc}\\approx\\pm \\frac{\\sqrt 3}{4} \\vc.\n$$\nFor larger values of $16\\beta G(\\Xmax)$, the equation is cubic in $W$ and the roots may be found by a perturbation expansion around the previously found roots.  We use this value of $W(0)$ as our value of $\\epsilon$.  For $\\beta=0.05$, it is sufficient to use $\\epsilon=\\sqrt{3}\\vc\/4$, which gives a period $5.00$, as was found from the linear fit.  For $\\beta=0.2$, we find that weakly nonlinear theory is not useful as the first several terms of the expansion of the Hamiltonian in $W$ of~\\eqref{eq:Energy_expansion} are all found numerically to about the same order, so that ignoring the $W^4$ and $W^5$ term in the Poincar\\'e-Lindstedt expansion is invalid.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nWe have explained many of the phenomena seen in the collision of vector solitons in CNLS.  First we derived and justified a simplified version of a model derived by Ueda and Kath. Using a Melnikov integral, we estimated the critical velocity, and using matched asymptotic expansions near separatrices, we explained how to connect subsequent passes to construct an approximate solution.  Imposing the condition that the total energy change after two passes is zero allowed us to find the locations of the exact two-pass resonant velocities, the centers of the two-pass windows.  It remains to be seen if this phenomenon can be produced in physical experiments, but the experimental setup would appear to be fairly simple.\n\nMore importantly, we have elucidated the mechanism underlying two-pass and two-bounce resonance phenomena in general.  The important elements of an ODE model are the following:\n\\begin{itemize}\n\\item a ``position'' mode $Z(t)$ that moves in a potential well $V(Z)$, which is localized near $Z=0$, so that the force approaches zero at large distances,\n\\item a secondary oscillator mode $W(t)$ that acts as a temporary energy reservoir;\n\\item a term $C(Z,W)$ that couples the two modes together, also localized near $Z=0$, so that coupling decays at large distances.\n\\end{itemize}\nTrapping takes place on the initial interaction if enough energy leaks from the position mode ($Z$) to the energy reservoir ($W$). In that case, the first mode crosses a separatrix curve in its unperturbed phase space.  The  energy change on subsequent interactions may be positive or negative, depending sensitively on the phase of $W(t)$ at the interaction time, even though $W$ may remain exponentially small. Eventually, in any Hamiltonian model, enough energy will eventually be transferred back to the position mode that it returns to the unbounded portions of phase space and escapes.  It is transmitted if it escapes to $+\\infty$ and reflected if it escapes to $-\\infty$.  All the models we have seen are Hamiltonian, but it should be possible to carry through much of the analysis in the presence of a simple dissipative term.\n\nWe believe this mechanism to be present in all the systems which have displayed two-bounce resonance phenomena, including the foundational papers on kink-antikink interactions in nonlinear Klein-Gordon equations~\\cite{CP:86,CPS:86,CSW:83,PC:83}.  A finite dimensional model for kink-antikink interactions in the $\\phi^4$ model is presented by Anninos et al.~\\cite{AOM:91}.  The model they derive is essentially of the form described, but the potential and coupling terms are much more complicated than~\\eqref{eq:TanYang}, and it would be quite difficult even to find expansions about poles in the solution, as was done in section~\\ref{sec:DeltaE} of the current papers.  What's more there is no natural small parameter measuring the coupling and the difference in time scales of the two modes, as we have here.  The topology is slightly different in the $\\phi^4$ kink-antikink problem: the separatrix is given by an orbit homoclinic to infinity, rather than heteroclinic.  This is indeed why the interaction is a ``bounce'' rather than a ``pass.''  We have developed a simple model that displays the same topology as in~\\cite{AOM:91} and are in the process of analyzing it as a next step in understanding the two-bounce case.\n\n\\section*{Acknowledgements}\nWe thank Jianke Yang for useful discussions and use of his figures.  Roy Goodman was supported by NSF-DMS 0204881 and by an NJIT SBR grant.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction}\nAnomaly detection is a binary classification problem to determine whether an input contains an anomaly.\nDetecting anomalies is a critical and long-standing problem faced by the manufacturing and financial industries.\nAnomaly detection is usually formulated as a one-class classification because abnormal examples are either inaccessible or insufficient to model distribution during the training.\nWhen concentrating on image data, detected anomalies can also be localized, and anomaly segmentation problem is to localize the anomalies at the pixel level.\nIn this study, we tackle the problems of image anomaly detection and segmentation.\n\nOne-class support vector machine (OC-SVM)~\\cite{ocsvm} and support vector data description (SVDD)~\\cite{svdd} are classic algorithms used for one-class classification.\nGiven a kernel function, OC-SVM seeks a max-margin hyperplane from the origin in the kernel space.\nLikewise, SVDD searches for a data-enclosing hypersphere in the kernel space.\nThese approaches are closely related, and Vert et al.~\\cite{ocsvm_consistency} showed their equivalence in the case of a Gaussian kernel.\nRuff et al.~\\cite{deepSVDD} proposed a deep learning variant of SVDD, Deep SVDD, by deploying a deep neural network in the place of the kernel function.\nThe neural network was trained to extract a data-dependent representation, removing the need to choose an appropriate kernel function by hand.\nFurthermore, Ruff et al.~\\cite{deep_sad} re-interpreted Deep SVDD in an information-theoretic perspective and applied to semi-supervised scenarios.\n\n\nIn this paper, we extend Deep SVDD to a patch-wise detection method, thereby proposing Patch SVDD.\nThis extension is rendered nontrivial by the relatively high level of intra-class variation of the patches and is facilitated by self-supervised learning.\nThe proposed method enables anomaly segmentation and improves anomaly detection performance.\nFig.~\\ref{fig:example} shows an example of the localized anomalies using the proposed method.\nIn addition, the results in previous studies~\\cite{patch_location,lens} show that the features of a randomly initialized encoder might be used to distinguish anomalies.\nWe detail the more in-depth behavior of random encoders and investigate the source of separability in the random features.\n\n\\input{txt\/1_fig_example}\n\n\\section{Background}\n\\subsection{Anomaly detection and segmentation}\n\\subsubsection{Problem formulation} \nAnomaly detection is a problem to make a binary decision whether an input is an anomaly or not.\nThe definition of \\textit{anomaly} ranges from a tiny defect to an out-of-distribution image.\nWe focus here on detecting a defect in an image.\nA typical detection method involves training a scoring function, $\\mathcal{A}_{\\theta}$, which measures the abnormality of an input.\nAt test time, inputs with high $\\mathcal{A}_{\\theta}(\\mathbf{x})$ values are deemed to be an anomaly.\nA \\textit{de facto} standard metric for the scoring function is the area under the receiver operating characteristic curve (AUROC), as expressed in Eq.~\\ref{eq:auroc}~\\cite{auroc}.\n\n\\begin{equation} \\label{eq:auroc}\n    \\text{AUROC}\\left [ \\mathcal{A}_{\\theta} \\right ] = \\mathbb{P} \\left [ \\mathcal{A}_{\\theta}(\\mathbf{X}_{\\text{normal}}) < \\mathcal{A}_{\\theta}(\\mathbf{X}_{\\text{abnormal}}) \\right ].\n\\end{equation}\nA good scoring function is, thus, one that assigns a low anomaly score to normal data and a high anomaly score to abnormal data.\nAnomaly segmentation problem is similarly formulated, with the generation of an anomaly score for every pixel (i.e., an anomaly map) and the measurement of AUROC with the pixels.\n\n\\subsubsection{Auto encoder-based methods}\nEarly deep learning approaches to anomaly detection used auto encoders~\\cite{vae_ad,recon_and_detect,ocgan}.\nThese auto encoders were trained with the normal training data and did not provide accurate reconstruction of abnormal images.\nTherefore, the difference between the reconstruction and the input indicated abnormality.\nFurther variants have been proposed to utilize structural similarity indices~\\cite{ssim_ae}, adversarial training~\\cite{recon_and_detect}, negative mining~\\cite{ocgan}, and iterative projection~\\cite{iterative_project}.\nCertain previous works utilized the learned latent feature of the auto encoder for anomaly detection.\nAkcay et al.~\\cite{ganomaly} defined the reconstruction loss of the latent feature as an anomaly score, and Yarlagadda et al.~\\cite{satellite} trained OC-SVM~\\cite{ocsvm} using the latent features.\nMore recently, several methods have made use of factors other than reconstruction loss, such as restoration loss~\\cite{itae} and an attention map~\\cite{ve_vae}.\n\n\n\\input{txt\/3_fig_big_picture.tex}\n\n\n\\subsubsection{Classifier-based methods}\nAfter the work of Golan et al.~\\cite{geom}, discriminative approaches have been proposed for anomaly detection.\nThese methods exploit an observation that classifiers lose their confidence~\\cite{oodbaseline} for the abnormal input images.\nGiven an unlabeled dataset, a classifier is trained to predict synthetic labels.\nFor example, Golan et al.~\\cite{geom} randomly flip, rotate, and translate an image, and the classifier is trained to predict the particular type of transformation performed.\nIf the classifier does not provide a confident and correct prediction, the input image is deemed to be abnormal.\nWang et al.~\\cite{e3outlier} proved that such an approach could be extended to an unsupervised scenario, where the training data also contains a few anomalies.\nBergman et al.~\\cite{goad} adopted an open-set classification method and generalized the method to include non-image data.\n\n\n\\subsubsection{SVDD-based methods}\nSVDD~\\cite{svdd} is a classic one-class classification algorithm.\nIt maps all the normal training data into a predefined kernel space and seeks the smallest hypersphere that encloses the data in the kernel space.\nThe anomalies are expected to be located outside the learned hypersphere.\nAs a kernel function determines the kernel space, the training procedure is merely deciding the radius and center of the hypersphere.\n\nRuff et al.~\\cite{deepSVDD} improved this approach using a deep neural network.\nThey adopted the neural network in place of the kernel function and trained it along with the radius of the hypersphere.\nThis modification allows the encoder to learn a data-dependent transformation, thus enhancing detection performance on high-dimensional and structured data.\nTo avoid a trivial solution (i.e., the encoder outputs a constant), they removed the bias terms in the network.\nRuff et al.~\\cite{deep_sad} further applied this method to a semi-supervised scenario.\n\n\\input{txt\/3_fig_map_comparison}\n\n\\subsection{Self-supervised representation learning}\nLearning a representation of an image is a core problem of computer vision.\nA series of methods have been proposed to learn a representation of an image without annotation.\nOne branch of research suggests training the encoder by learning with a \\textit{pretext task}, which is a self-labeled task to provide synthetic learning signals.\nWhen a network is trained to solve the pretext task well, the network is expected to be able to extract useful features.\nThe pretext tasks include predicting relative patch location~\\cite{patch_location}, solving a jigsaw puzzle~\\cite{jigsaw}, colorizing images~\\cite{colorization}, counting objects~\\cite{count}, and predicting rotations~\\cite{rotation}.\n\n\n\\section{Methods}\n\\subsection{Patch-wise Deep SVDD} \\label{sec:patch_svdd}\n\nDeep SVDD~\\cite{deepSVDD} trains an encoder that maps the entire training data to features lying within a small hypersphere in the feature space.\nThe encoder, $f_\\theta$, is trained to minimize the Euclidean distances between the features and the center of the hypersphere using the following loss function:\n\n\\begin{equation} \\label{eq:deep_svdd}\n    \\mathcal{L}_{\\text{SVDD}} = \\sum_i \\left \\| f_\\theta (\\mathbf{x}_i) - \\mathbf{c} \\right \\| _2,\n\\end{equation}\nwhere $\\mathbf{x}$ is an input image.\nAt test time, the distance between the representation of the input and the center is used as an anomaly score.\nThe center $\\mathbf{c}$ is calculated in advance of the training, as shown in Eq.~\\ref{eq:deep_svdd_center}, where $N$ denotes the number of the training data.\nTherefore, the training pushes the features around a single center.\n\n\\begin{equation} \\label{eq:deep_svdd_center}\n    \\mathbf{c} \\doteq \\frac{1}{N} \\sum_i^N  f_\\theta (\\mathbf{x}_i).\n\\end{equation}\n\nIn this study, we extend this approach to patches; the encoder encodes each patch, not the entire image, as illustrated in Fig.~\\ref{fig:svdd_comparison}.\nAccordingly, inspection is performed for each patch.\nPatch-wise inspection has several advantages.\nFirst, the inspection result is available at each position, and hence we can localize the positions of defects.\nSecond, such fine-grained examination improves overall detection performance.\n\n\nA direct extension of Deep SVDD~\\cite{deepSVDD} to a patch-wise inspection is straightforward.\nA patch encoder, $f_\\theta$, is trained using $\\mathcal{L}_{\\text{SVDD}}$ with $\\mathbf{x}$ replaced with a patch, $\\mathbf{p}$.\nThe anomaly score is defined accordingly, and the examples of the resulting anomaly maps are provided in Fig.~\\ref{fig:ablation_loss_map}.\nUnfortunately, the detection performance is poor for the images with high complexity.\nThis is because patches have high intra-class variation; some patches correspond to the background, while the others contain the object.\nAs a result, mapping all the features of dissimilar patches to a single center and imposing a uni-modal cluster weaken the connection between the representation and the content.\nTherefore, using a single center $\\mathbf{c}$ is inappropriate, yet deciding on the appropriate number of multiple centers and the allocation of patches to each center are cumbersome.\n\n\\input{txt\/3_fig_self_supervision}\n\nTo bypass the above issues, we do not explicitly define the center and allocate the patches.\nInstead, we train the encoder to gather semantically similar patches by itself.\nThe semantically similar patches are obtained by sampling spatially adjacent patches, and the encoder is trained to minimize the distances between their features using the following loss function:\n\n\\begin{equation} \\label{eq:patch_svdd}\n    \\mathcal{L}_{\\text{SVDD'}} = \\sum_{i,i'} \\left \\| f_\\theta (\\mathbf{p}_i) - f_\\theta (\\mathbf{p}_{i'}) \\right \\| _2,\n\\end{equation}\nwhere $\\mathbf{p}_{i'}$ is a patch near $\\mathbf{p}_{i}$.\nFurthermore, to enforce the representation to capture the semantics of the patch, we append the following self-supervised learning.\n\n\\subsection{Self-supervised learning} \\label{sec:self_supervised}\nDoersch et al.~\\cite{patch_location} trained an encoder and classifier pair to predict the relative positions of two patches, as depicted in Fig.~\\ref{fig:self_supervision}.\nA well-performing pair implies that the trained encoder extracts useful features for location prediction.\nAside from this particular task, previous research~\\cite{jigsaw,rotation,revisit_ssl} reported that the self-supervised encoder functions as a powerful feature extractor for downstream tasks.\n\nFor a randomly sampled patch $\\mathbf{p}_1$, Doersch et al.~\\cite{patch_location} sampled another patch $\\mathbf{p}_2$ from one of its eight neighborhoods in a 3 $\\times$ 3 grid.\nIf we let the true relative position be $y \\in \\{0, ..., 7\\}$, the classifier $C_{\\phi}$ is trained to predict $y=C_\\phi(f_\\theta(\\mathbf{p}_1), f_\\theta(\\mathbf{p}_2))$ correctly.\nThe size of the patch is the same as the receptive field of the encoder.\nTo prevent the classifier from exploiting shortcuts~\\cite{lens} (e.g., color aberration), we randomly perturb the RGB channels of the patches.\nFollowing the approach by Doersch et al.~\\cite{patch_location}, we add a self-supervised learning signal by adding the following loss term:\n\\begin{equation} \\label{eq:ssl}\n    \\mathcal{L}_{\\text{SSL}} = \\texttt{Cross-entropy} \\left (y, C_\\phi \\left ( f_\\theta(\\mathbf{p}_1), f_\\theta(\\mathbf{p}_2) \\right ) \\right ).\n\\end{equation}\nAs a result, the encoder is trained using a combination of two losses with the scaling hyperparameter $\\lambda$, as presented in Eq.~\\ref{eq:patch_svdd_final}.\nThis optimization is performed using stochastic gradient descent and Adam optimizer~\\cite{adam}.\n\n\\begin{equation} \\label{eq:patch_svdd_final}\n    \\mathcal{L}_{\\text{Patch SVDD}} = \\lambda \\mathcal{L}_{\\text{SVDD'}} + \\mathcal{L}_{\\text{SSL}}.\n\\end{equation}\n\n\n\\input{txt\/3_fig_hierarchical}\n\\subsection{Hierarchical encoding} \\label{sec:hierarchical}\nAs anomalies vary in size, deploying multiple encoders with various receptive fields helps in dealing with variation in size.\nThe experimental results in Section~\\ref{sec:hierarchical_results} show that enforcing a hierarchical structure on the encoder boosts anomaly detection performance as well.\nFor this reason, we employ a hierarchical encoder that embodies a smaller encoder; the hierarchical encoder is defined as\n\\begin{equation} \\label{eq:hierarchical}\n    f_{\\text{big}}(\\mathbf{p})=g_{\\text{big}}(f_{\\text{small}}(\\mathbf{p})).\n\\end{equation}\n\nAn input patch $\\mathbf{p}$ is divided into a 2 $\\times$ 2 grid, and their features are aggregated to constitute the feature of $\\mathbf{p}$, as shown in Fig.~\\ref{fig:hierarchical}.\nEach encoder with receptive field size $K$ is trained with the self-supervised task of patch size $K$.\nThroughout the experiment, the receptive field sizes of the large and small encoders are 64 and 32, respectively.\n\n\n\\input{txt\/3_fig_overall_method}\n\n\n\\subsection{Generating anomaly maps}\nAfter training the encoders, the representations from the encoder are used to detect the anomalies.\nFirst, the representation of every normal train patch, $\\left \\{f_\\theta (\\mathbf{p}_{\\text{normal}}) | \\mathbf{p}_{\\text{normal}}  \\right \\}$, is calculated and stored.\nGiven a query image $\\mathbf{x}$, for every patch $\\mathbf{p}$ with a stride $S$ within $\\mathbf{x}$, the L2 distance to the nearest normal patch in the feature space is then defined to be its anomaly score using Eq.~\\ref{eq:anomaly_score_patch}.\nTo mitigate the computational cost of the nearest neighbor search, we adopt its approximate algorithm\\footnote{\\url{https:\/\/github.com\/yahoojapan\/NGT}}.\nAs a result, the inspection of a single image from MVTec AD~\\cite{mvtecad} dataset for example, requires approximately 0.48 second.\n\n\\begin{equation} \\label{eq:anomaly_score_patch}\n    \\mathcal{A}_{\\theta}^{\\text{patch}}(\\mathbf{p}) \\doteq \\min_{\\mathbf{p}_{\\text{normal}}} \\left \\| f_\\theta (\\mathbf{p}) - f_\\theta (\\mathbf{p}_{\\text{normal}}) \\right \\| _2.\n\\end{equation}\n\nPatch-wise calculated anomaly scores are then distributed to the pixels.\nAs a consequence, pixels receive the average anomaly scores of every patch to which they belong, and we denote the resulting anomaly map as $\\mathcal{M}$.\n\nThe multiple encoders discussed in Section~\\ref{sec:hierarchical} constitute multiple feature spaces, thereby yielding multiple anomaly maps.\nWe aggregate the multiple maps using element-wise multiplication, and the resulting anomaly map, $\\mathcal{M}_{\\text{multi}}$, provides the answer to the problem of anomaly segmentation:\n\n\\begin{equation} \\label{eq:anomaly_map_multi}\n    \\mathcal{M}_{\\text{multi}} \\doteq \\mathcal{M}_{\\text{small}} \\odot \\mathcal{M}_{\\text{big}},\n\\end{equation}\nwhere $\\mathcal{M}_{\\text{small}}$ and $\\mathcal{M}_{\\text{big}}$ are the generated anomaly maps using $f_{\\text{small}}$ and $f_{\\text{big}}$, respectively.\nThe pixels with high anomaly scores in the map $\\mathcal{M}_{\\text{multi}}$ are deemed to contain defects.\n\nIt is straightforward to address the problem of anomaly detection.\nThe maximum anomaly score of the pixels in an image is its anomaly score, as expressed in Eq.~\\ref{eq:anomaly_score_image}.\nFig.~\\ref{fig:overall_method} illustrates the overall flow of the proposed method, and its pseudo-code is provided in Appendix~\\ref{sec:appendix_pseudo_code}.\n\n\\begin{equation} \\label{eq:anomaly_score_image}\n    \\mathcal{A}_{\\theta}^{\\text{image}}(\\mathbf{x}) \\doteq \\max_{i,j} \\mathcal{M}_{\\text{multi}}(\\mathbf{x})_{ij}.\n\\end{equation}\n\n\\section{Results and Discussion}\nTo verify the validity of the proposed method, we applied it to MVTec AD~\\cite{mvtecad} dataset.\nThe dataset consists of 15-class industrial images, each class categorized as either an \\textit{object} or \\textit{texture}.\nTen \\textit{object} classes contain regularly positioned objects, whereas the \\textit{texture} classes contain repetitive patterns.\nThe implementation details used throughout the study are provided in Appendix~\\ref{sec:appendix_implementation}, and please refer to \\cite{mvtecad} for more details on the dataset.\n\n\n\\input{txt\/4_table_auroc.tex}\n\n\\subsection{Anomaly detection and segmentation results}\nFig.~\\ref{fig:anomaly_maps} shows anomaly maps generated using the proposed method, indicating that the defects are properly localized, regardless of their size.\nTable~\\ref{table:anomaly_det_seg} shows the detection and segmentation performances for MVTec AD~\\cite{mvtecad} dataset compared with state-of-the-art baselines in AUROC.\nPatch SVDD provides state-of-the-art performance over the powerful baselines including auto encoder-based and classifier-based methods and outperforms Deep SVDD~\\cite{deepSVDD} by 55.6\\% improvement.\nMore numerical results are provided in Appendix~\\ref{sec:appendix_numerical}.\n\n\n\\input{txt\/4_fig_TSNE_Embedding.tex}\n\n\\subsection{Detailed analysis}\n\\subsubsection{t-SNE visualization}\nFig.~\\ref{fig:tsne} shows t-SNE visualizations~\\cite{tsne} of the learned features of multiple train images.\nPatches located at the points shown in Fig.~\\ref{fig:tsne}(b) are mapped to the points with the same color and size in Fig.~\\ref{fig:tsne}(a) and Fig.~\\ref{fig:tsne}(c).\nIn Fig.~\\ref{fig:tsne}(a), the points with similar color and size form clusters in the feature space.\nSince the images in the cable class are regularly positioned, the patches from the same position have similar content, even if they are from different images.\nLikewise, for the regularly positioned \\textit{object} classes, the points with similar color and size in t-SNE visualization (i.e., the patches with similar positions) can be regarded to be semantically similar.\nBy contrast, features of the leather class in Fig.~\\ref{fig:tsne}(c) show the opposite tendency.\nThis is because the patches in \\textit{texture} classes are analogous, regardless of their position in the image; the positions of the patches are not quite related to their semantics for the \\textit{texture} images.\n\n\n\\input{txt\/4_fig_ablation_loss.tex}\n\\input{txt\/4_fig_ablation_loss_tsne.tex}\n\n\\subsubsection{Effect of self-supervised learning} \\label{sec:ssl_effect}\nPatch SVDD trains an encoder using two losses: $\\mathcal{L}_{\\text{SVDD'}}$ and $\\mathcal{L}_{\\text{SSL}}$, where $\\mathcal{L}_{\\text{SVDD'}}$ is a variant of $\\mathcal{L}_{\\text{SVDD}}$.\nTo compare the roles of the proposed loss terms, we conduct an ablation study.\nTable~\\ref{table:ablation_loss} suggests that the modification of $\\mathcal{L}_{\\text{SVDD}}$ to $\\mathcal{L}_{\\text{SVDD'}}$ and the adoption of $\\mathcal{L}_{\\text{SSL}}$ improve the anomaly detection and segmentation performances.\nFig.~\\ref{fig:ablation_loss} shows that the effects of the proposed loss terms vary among classes.\nSpecifically, the \\textit{texture} classes (e.g. tile and wood) are less sensitive to the choice of loss, whereas the \\textit{object} classes, including cable and transistor, benefit significantly from $\\mathcal{L}_{\\text{SSL}}$.\n\nTo investigate the reason behind these observations, we provide (in Fig.~\\ref{fig:ablation_loss_tsne}) t-SNE visualizations of the features of an \\textit{object} class (the transistor) for the encoders trained with $\\mathcal{L}_{\\text{SVDD}}$, $\\mathcal{L}_{\\text{SVDD'}}$, and $\\mathcal{L}_{\\text{SVDD'}} + \\mathcal{L}_{\\text{SSL}}$.\nWhen training is performed with $\\mathcal{L}_{\\text{SVDD}}$ (Fig.~\\ref{fig:ablation_loss_tsne}(a)) or $\\mathcal{L}_{\\text{SVDD'}}$ (Fig.~\\ref{fig:ablation_loss_tsne}(b)), the features form a uni-modal cluster.\nIn contrast, $\\mathcal{L}_{\\text{SSL}}$ results in multi-modal feature clusters on the basis of their semantics (i.e., color and size), as shown in Fig.~\\ref{fig:ablation_loss_tsne}(c).\nThe multi-modal property of the features is particularly beneficial to the \\textit{object} classes, which have high intra-class variation among the patches.\nFeatures of the patches with dissimilar semantics are separated, and hence anomaly inspection using those features becomes more deliberate and accurate.\n\n\\input{txt\/4_fig_ablation_together}\n\nThe intrinsic dimensions (ID)~\\cite{intrinsic_dimension} of the features also indicate the effectiveness of $\\mathcal{L}_{\\text{SSL}}$.\nThe ID is the minimal number of coordinates required to describe the points without significant information loss~\\cite{intrinsic_dimension2}.\nA larger ID denotes that the points are spreaded in every direction, while a smaller ID indicates that the points lie on low-dimensional manifolds with high separability.\nIn Fig.~\\ref{fig:ablation_loss_id}, we show the average IDs of features in each class trained with three different losses.\nIf the encoder is trained with the proposed $\\mathcal{L}_{\\text{Patch SVDD}}$, features with the lowest ID are yielded, implying that these features are neatly distributed.\n\n\n\\subsubsection{Hierarchical encoding} \\label{sec:hierarchical_results}\nIn Section~\\ref{sec:hierarchical}, we proposed the use of hierarchical encoders.\nFig.~\\ref{fig:hierarchical_helps} shows that aggregating multi-scale results from multiple encoders improves the inspection performances.\nIn addition, an ablation study with a non-hierarchical encoder shows that the hierarchical structure itself also boosts performance.\nWe postulate that the hierarchical architecture provides regularization for the feature extraction.\nNote that the non-hierarchical encoder has a number of parameters similar to that of the hierarchical counterpart.\n\nWe provide an example of multi-scale inspection results, together with an aggregated anomaly map, in Fig.~\\ref{fig:hierarchical_maps}.\nThe anomaly maps from various scales provide complementary inspection results; the encoder with a large receptive field coarsely locates the defect, whereas the one with a smaller receptive field refines the result.\nTherefore, an element-wise multiplication of the two maps localizes the accurate position of the defect.\n\n\\input{txt\/4_fig_hierarchical_map.tex}\n\n\n\\subsubsection{Hyperparameters}\nAs shown in Eq.~\\ref{eq:patch_svdd_final}, the hyperparameter $\\lambda$ balances $\\mathcal{L}_{\\text{SVDD'}}$ and $\\mathcal{L}_{\\text{SSL}}$.\nA large $\\lambda$ emphasizes gathering of the features, while a small $\\lambda$ promotes their informativeness.\nInterestingly, the most favorable value of $\\lambda$ varies among the classes.\nAnomalies in the \\textit{object} classes are well detected under a smaller $\\lambda$, while the \\textit{texture} classes are well detected with a larger $\\lambda$.\nFig.~\\ref{fig:ablation_lambda} shows an example of this difference; the anomaly detection performance for the cable class (\\textit{object}) improves as $\\lambda$ decreases, while the wood class (\\textit{texture}) shows the opposite trend.\nAs discussed in the previous sections, this occurs because the self-supervised learning is more helpful when the patches show high intra-class variation, which is the case for the \\textit{object} classes.\nThe result coincides with that shown in Fig.~\\ref{fig:ablation_loss} because using $\\mathcal{L}_{\\text{SVDD'}}$ as a loss is equivalent to using $\\mathcal{L}_{\\text{Patch SVDD}}$ with $\\lambda >> 1$.\n\n\nThe number of feature dimensions, $D$, is another hyperparameter of the encoder.\nThe anomaly inspection performance for varying $D$ is depicted in Fig.~\\ref{fig:ablation_D}(a).\nA larger $D$ signifies improved performance---a trend that has been discussed in a self-supervised learning venue~\\cite{revisit_ssl}.\nFig.~\\ref{fig:ablation_D}(b) indicates that the ID of the resulting features increases with increasing $D$.\nThe black dashed line represents the $y=x$ graph, and it is the upper bound of ID.\nThe average ID of features among the classes saturates as $D=64$; therefore, we used a value of $D=64$ throughout our study.\n\n\\input{txt\/4_fig_ablation_D.tex}\n\n\n\\subsubsection{Random encoder}\n\\input{txt\/4_table_random_encoder.tex}\nDoersch et al.~\\cite{patch_location} showed that randomly initialized encoders perform reasonably well in image retrieval; given an image, the nearest images in the random feature space look similar to humans as well.\nInspired by this observation, we examined the anomaly detection performance of the random encoders and provided the results in Table~\\ref{table:random_enc}.\nAs in Eq.~\\ref{eq:anomaly_score_patch}, the anomaly score is defined to be the distance to the nearest normal patch, but in the random feature space.\nIn the case of certain classes, the features of the random encoder are effective in distinguishing between normal and abnormal images.\nSome results even outperform the trained deep neural network model (L2-AE).\n\nHere, we investigate the reason for the high separability of the random features.\nFor simplicity, let us assume the encoder to be a one-layered convolutional layer parametrized by a weight $W \\neq \\mathbf{0}$ and a bias $b$ followed by a nonlinearity, $\\sigma$.\nGiven two patches $\\mathbf{p}_1$ and $\\mathbf{p}_2$, their features $h_1$ and $h_2$ are provided by Eq.~\\ref{eq:random_encoder_define}, where $*$ denotes a convolution operation.\n\n\\begin{equation} \\label{eq:random_encoder_define}\n\\begin{split}\n    & h_1 = \\sigma (W * \\mathbf{p}_1 + b) \\\\\n    & h_2 = \\sigma (W * \\mathbf{p}_2 + b).\n\\end{split}\n\\end{equation}\n\nAs suggested by Eq.~\\ref{eq:random_encoder}, when the features are close, so are the patches, and vice versa.\nTherefore, retrieving the nearest patch in the feature space is analogous to doing so in the image space.\n\n\n\\begin{equation} \\label{eq:random_encoder}\n\\begin{split}\n    \\left \\| h_1 - h_2 \\right \\| _2 \\approx 0 & \\Leftrightarrow   (W * \\mathbf{p}_1 + b) - (W * \\mathbf{p}_2 + b) \\approx 0  \\\\\n    & \\Leftrightarrow  W * (\\mathbf{p}_1 - \\mathbf{p}_2) \\approx 0 \\\\\n    & \\Leftrightarrow  \\left \\| \\mathbf{p}_1 - \\mathbf{p}_2 \\right \\| _2 \\approx 0.\n\\end{split}\n\\end{equation}\n\n\nIn Table~\\ref{table:random_enc}, we also provide the results for anomaly detection task using the nearest neighbor algorithm using the raw patches (i.e., $f_\\theta(\\mathbf{p})=\\mathbf{p}$ in Eq.~\\ref{eq:anomaly_score_patch}).\nFor certain classes, the raw patch nearest neighbor algorithm works surprisingly well.\nThe effectiveness of the raw patches for anomaly detection can be attributed to the high similarity among the normal images.\n\nFurthermore, the well-separated classes provided by the random encoder are well-separated by the raw patch nearest neighbor algorithm, and vice versa.\nTogether with the conclusion of Eq.~\\ref{eq:random_encoder}, this observation implies the strong relationship between the raw image patch and its random feature.\nTo summarize, the random features of anomalies are easily separable because they are alike the raw patches, and the raw patches are easily separable.\n\n\\section{Conclusion}\n\nIn this paper, we proposed Patch SVDD, a method for image anomaly detection and segmentation.\nUnlike Deep SVDD~\\cite{deepSVDD}, we inspect the image at the patch level, and hence we can also localize defects.\nMoreover, additional self-supervised learning improves detection performance.\nAs a result, the proposed method achieved state-of-the-art performance on MVTec AD~\\cite{mvtecad} industrial anomaly detection dataset.\n\nIn previous studies~\\cite{jigsaw,colorization}, images were featurized prior to the subsequent downstream tasks because of their high-dimensional and structured nature.\nHowever, the results in our analysis suggest that a nearest neighbor algorithm with a raw patch often discriminates anomalies surprisingly well.\nMoreover, since the distances in random feature space are closely related to those in the raw image space, random features can provide distinguishable signals.\n\n\n\\section{Pseudo code} \\label{sec:appendix_pseudo_code}\n\\input{txt2\/1_pseudo_code.tex}\n\\input{txt2\/1_pseudo_code2.tex}\nAlgorithm~\\ref{algo:pseudo} trains a hierarchical encoder using $\\mathcal{L}_{\\text{Patch SVDD}}$.\nAfter the training, sets of features of normal patches are extracted using the trained multi-scale encoders.\nThe outputs of Algorithm~\\ref{algo:pseudo} are the sets of normal features and trained encoders.\nAlgorithm~\\ref{algo:pseudo2} performs inspection on a query image and outputs the anomaly map and anomaly score.\n\\newpage\n\n\n\\section{Results}\n\\subsection{Numerical results} \\label{sec:appendix_numerical}\n\\input{txt2\/2_table_auroc_full.tex}\n\\input{txt2\/2_table_auroc_hier.tex}\n\\newpage\n\n\\subsection{Anomaly maps}\n\\input{txt2\/2_fig_anomaly_maps1.tex}\n\\clearpage\n\\input{txt2\/2_fig_anomaly_maps2.tex}\n\\clearpage\n\\input{txt2\/2_fig_anomaly_maps3.tex}\n\\clearpage\n\n\\section{Implementation details} \\label{sec:appendix_implementation}\n\\subsection{Dataset}\nThe dataset in the study, MVTec AD~\\cite{mvtecad}, consists of 15-class industrial images.\nEach class is categorized as either an \\textit{object}\\footnote{bottle, cable, capsule, hazelnut, metal\\_nut, pill, screw, toothbrush, transistor, and zipper} or \\textit{texture}\\footnote{carpet, grid, leather, tile, and wood}.\nEach class contains 60 to 390 normal train images and 40 to 167 test images.\nTest images include both normal and abnormal examples, and the defects of the abnormal images are annotated at the pixel level in the form of binary masks.\nWe downsampled every image to a resolution of 256 $\\times$ 256.\nGray-scale images are converted to RGB images by replicating the single channel to three.\nNo data augmentation method (e.g., horizontal flip, rotation) was used for the training.\n\n\\subsection{Networks}\nTwo neural networks are used throughout the study: an encoder and a classifier.\nThe encoder is composed of convolutional layers only.\nThe classifier is a two-layered MLP model having 128 hidden units per layer, and the input to the classifier is a subtraction of the features of the two patches.\nThe activation function for both networks is a LeakyReLU~\\cite{relu} with a $\\alpha=0.1$.\nPlease refer to the code\\footnote{\\url{https:\/\/github.com\/nuclearboy95\/Anomaly-Detection-PatchSVDD-PyTorch}} for the detailed architecture of the networks.\n\nAs proposed in Section 3.3 of the main paper, the encoder has a hierarchical structure.\nThe receptive field of the encoder is $K=64$, and that of the embedded smaller encoder is $K=32$.\nPatch SVDD divides the images into patches with a size $K$ and a stride $S$.\nThe values for the strides are $S=16$ and $S=4$ for the encoders with $K=64$ and $K=32$, respectively.\n\n\\subsection{Environments}\n\nThe experiments throughout the study were conducted on a machine equipped with an Intel i7-5930K CPU and an NVIDIA GeForce RTX 2080 Ti GPU.\nThe code is implemented in python 3.7 and PyTorch~\\cite{pytorch}.\n\n\n\n\n\n\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 Blazars (Flat Spectrum Radio Quasars and BL Lac objects) are the most extreme type of Active Galactic Nuclei (AGN), with\nrelativistic jets pointing toward us. Their emission is non-thermal and it is amplified due to the relativistic bulk motion of the jets.\nBlazar Spectral Energy Distribution (SED) consists of two broad ``humps'', one spanning from radio to optical-UV (and occasionally X-ray) bands and another one\nthat extends from X-rays to multi-GeV and occasionally to TeV $\\gamma$-rays. The low-frequency component is believed to be due to synchrotron emission by non-thermal electrons, while the higher one is attributed to inverse Compton (IC) scattering of the relativistic electrons on\nsynchrotron or external photons. Relativistic plasma motion and radiative cooling affect the temporal variability of these sources, which helps define the emitting region's size. \n Certain patterns in the spectral features became apparent and are now known as the ``Blazar\nSequence\" \\cite{Foss98}. Blazars become redder with increasing bolometric luminosity\n$L_{\\rm bol}$, in that their synchrotron peak frequency $\\nu_{\\rm pk}^{\\rm syn}$ decreases as\n$L_{\\rm bol}$ increases; at the same time, their Compton dominance (CD; i.e., the ratio of their IC to synchrotron luminosities) increases and so do their $\\gamma$-ray spectral indices, i.e. their spectral index becomes steeper.\nThe Blazar Sequence, established originally with 132 objects out of which only 33 were detected in high\nenergy $\\gamma$-rays, was supplemented with the launch of \\emph{Fermi} and the discovery\nof more than 5000 $\\gamma$-ray Blazars as recorded in the  4th Fermi Blazar Catalog\n \\cite{4th}. This\ncompilation provided novel correlations that replaced those of the original Blazar Sequence. This result is implying that the underlying physics is probably related to fundamental parameters of the AGN phenomenology. \\\\\n\\indent The optically thin Blazar GeV emission suggests its location to be far from the\naccreting black hole (BH), possibly out to a distance as large as  $10^6R_S~\\sim$  10 pc \\cite{Marscher10} (where $R_S$ is the Schwarzschild radius ). Furthermore, the AGN torii (dusty, molecular structures of height\/radius ratios\n$z\/D \\simeq 1$)  invoked in the unification of the radio-quiet or radio-loud AGN subclasses \\cite{AntonMill}, are of similar scales and, as we argue, play a significant role in Blazar physics. To reconcile the discrepancy of the torii geometry expected, \\cite{KK94} proposed that these torii are, in fact, MHD accretion disk winds \\cite{BKM19, CL94}; these are launched across the entire disk, from the BH\nvicinity of a few $R_S$ to the BH influence radius $D$  $\\sim~(c\/\\sigma)^2 R_S \\sim 10^6 R_S \\sim 10 pc$ (for $M_{BH}\\simeq 10^8M_{\\odot}$). Furthermore, the discovery of Warm Absorbers (WA, blue-shifted absorption features) and their successful modeling as photoionized MHD\nwinds that extend to $r \\sim 10^6R_S$ \\cite{Behar09,FKCB}, established the combined AGN WA-torii as a\nsingle entity.\nFinally, modeling the absorbers of the Galactic BH GRO 1655-40 with the same type of winds \\cite{FKCB17}\nsuggests the possibility of their presence in any accreting BH. \\\\\n\\indent The physical properties of such MHD winds depend on a few parameters and their presence allowed us to reproduce a theoretical Blazar Sequence by varying only one parameter, namely the mass accretion rate, \\cite{BKM19,20BMK}. Here we will show the results of a two-zone emission model based on the extension of our previous works. However, now we assume that electrons are accelerated into a zone close to the central engine and escape and radiate into a larger region, which will refer as the cooling zone.\n\\vspace{-0.2cm}\n\\section{Emission model-Scalings to mass accretion rate}\n\n\\indent The broader morphology of the non-thermal Blazar SED depends\non the ratio of the magnetic to photon energy densities. To calculate the former we assume some sort of equipartition with the accreting matter, the energy density. By assuming the power of the accretion $P_{\\rm acc}= \\dot{m}{\\cal M}L_{\\rm Edd}$, with  $\\dot{m}$ the mass accretion rate normalized to the Eddington one and \n${\\cal M}= M_{\\rm BH}\/M_{\\rm \\odot}$, where\n$M_{\\rm BH}$ is the mass of the black hole,  one can calculate the magnetic field energy density at position z:\n\\begin{equation}\\label{B}\n U_{\\rm B} \\propto \\eta_{\\rm b} \\dot{m}{\\cal{M}}^{-1},\n\\end{equation}\nwhere $\\eta_{\\rm b}$ is a proportionality constant.\nWe assume for the external photon field that is related to photons which are scattered on accretion disk wind particles \\cite{BKM19} and thus the external photon field density $U_{ext}$ in the jet frame has the form: \n \\begin{equation}\\label{ext}\n  U_{\\rm ext}= \\Gamma^2 {\\rm U}_{\\rm sc}\\propto \\Gamma^2 \\epsilon \\dot{m}^{\\alpha+1}{\\cal M}^{-1}~~(\\alpha =1 ~{\\rm{for}}~\\dot{m}\\geq 0.1 ~~{\\rm{and}}~ \\alpha=2 ~{\\rm{for}}~\\dot{m}<0.1),\n \\end{equation}\n where $\\epsilon$ is  the  efficiency  of  the  conversion\nof  the  accreting  mass  into  radiation and  $\\Gamma$ is the bulk Lorentz factor of the source. We have assumed that the disk emits like a black body characterized by a temperature $T_{\\rm disk} $ in order to estimate the spectrum of the scattered photons. As we pointed in \\cite{BKM19}, all  input parameters  required for the calculation of the spectrum are scaled with $\\dot{m}$ and ${\\cal{M}}$. Using the above definitions for the physical properties of the source, we obtain the Blazar SED, by solving the coupled integro-differential kinetic equations of electrons and photons as described in \\cite{MK95}.\n We emphasize that according to relations \\ref{B},\\ref{ext} the basic parameters of the system of equations depends basically only on the mass accretion rate, $\\dot{m}$.\n \n \n Therefore, we assume a blob of plasma of radius $R$ where particles are accelerated with a characteristic timescale $t_{acc}$, (e.g. \\cite{KRM98}). In the case of the first order Fermi acceleration we have \\begin{equation}\\label{tacc}\n    t_{{acc}_{FI}}\\geq 6\\left(\\frac{c}{u_s}\\right)^2\\frac{\\lambda}{c}\\simeq 6 \\frac{r_g c}{u_s^2},\n\\end{equation}\nwhere \n\\begin{equation}\n    r_g=\\frac{\\gamma m c^2}{eB}.\n\\end{equation}\n\n\\begin{figure}[!htbp]\n\\begin{subfigure}{0.6\\textwidth}\n\\includegraphics[width=\\linewidth]{acc.pdf}\n \\label{fig:a}\n \\end{subfigure}\\hspace*{\\fill}\n\\begin{subfigure}{0.5\\textwidth}\n\\includegraphics[width=\\linewidth]{BS_acc.pdf}\n\\label{fig:b}\n\\end{subfigure}\n\\caption{\\textbf{Left:} The dependece of the acceleration timescale to the normalized mass accretion rate. \\textbf{Right:} The calculated Spectral Energy Distribution of BL Lac objects in the case for various values of the mass accretion rate.  } \\label{fig:1a}\n\\end{figure}\nIn our study, we re-write the Equation \\Ref{tacc} as $t_{acc}\\propto \\gamma A_{acc}(\\dot{m})$ taking into account the dependence on the mass accretion rate. By solving the set of equations we find there is an almost linear dependence between $t_{acc}$ and $\\dot{m}$ in order to explain the high peaked synchrotron blazars, see Figure~\\ref{fig:1a}.\n\n\\section{Particles escape: A two-zone emission model}\nAs a second step, we study a two zone model: particles  accelerate in a zone closer to the central engine, then they escape to a larger volume further away and cool due to synchrotron and inverse Compton losses. \n\\begin{figure}[!htbp]\n    \\centering\n    \\includegraphics[width=0.5\\linewidth]{model_2zone.png}\n    \\caption{Sketch of the two-zone model. Particles accelerate and radiate in the acceleration zone (red). While, a fraction of the accelerated particles escapes into a larger volume(blue), where they cool and radiate.}\n    \\label{fig:model_2zone}\n\\end{figure}\nThe kinetic equation of electrons in the first zone (or acceleration zone) has the form: \n \\begin{equation}\n     \\frac{\\partial n_{e_I}(\\gamma,t)}{\\partial t}+\\frac{n_{e_I}(\\gamma,t)}{t_{{esc}_I}(\\dot{m},\\gamma)}+\\frac{\\partial}{\\partial \\gamma}\\left[\\frac{\\gamma}{t_{acc}(\\dot{m},\\gamma)}n_{e_I}(\\gamma,t)\\right]=\\mathcal{L}_{syn}(\\gamma,t) + \\mathcal{L}_{ICS}(\\gamma,t).\n \\end{equation}\nThe kinetic equation of relativistic electrons in the second zone (or cooling zone) has the form: \n \\begin{equation}\n     \\frac{\\partial n_{e_{II}}(\\gamma,t)}{\\partial t}+\\frac{n_{e_{II}}(\\gamma,t)}{t_{{esc}_{II}}}=Q_{inj}+\\mathcal{L}_{syn}(\\gamma,t) + \\mathcal{L}_{ICS}(\\gamma,t),\n \\end{equation}\nwhere $n_{e_I}, ~n_{e_{II}}$ the electron differential distribution function in the first and second zone respectively, $t_{{esc}_I},~t_{{esc}_{II}}$ the electron escape timescale from the first and the second zone respectively, the term $Q_{inj}=\\frac{n_{e_I}}{tesc_I}$ refers to the relativistic electrons of the acceleration zone that escape and inject to the cooling zone, (e.g. \\cite{1998KRM}). \nTo calculate synchrotron losses, we assume that the magnetic field decreases with the distance $z$ from the central engine as $B\\propto 1\/z$. Furthermore, the energy density of the external photon field $U_{ext}$ is constant with the distance z, \\cite{BKM19}. According to our assumption, the two classes of blazars show a very distinct behavior: \n\\begin{itemize}\n\\item BL Lac objects emission is dominated by the first zone. This results from the fact that the energy density of the magnetic field dominates in both zones. The reason is that in the acceleration zone, we have $U_{B_I}>>U_{ext}$ as BL Lac objects have a weak external photon field. In the radiation zone\\footnote{The size of the second zone is related to the electrons cooling timescale, in order the losses to be Compton dominated.} even the fact that the magnetic energy density $U_B$ decreases with z (see above), \nwe still have that $U_{B_{II}} >> U_{ext}$ everywhere, where $U_{B_{II}}$ the magnetic field strength in the second zone, see Figure \\Ref{fig:sed_2}. As a result, the spectrum is dominated by the emission of the acceleration zone. \n\\item FSRQs synchrotrom emission is mainly produced by the first zone, while the second zone is dominated by the inverse Compton scattering. For the parameter set that we study in the acceleration zone, the magnetic energy density is again more significant than the energy density of the external photon field. However, the second zone is further from the central engine, where the magnetic field decreases, but the external photons' energy density remains constant \\cite{BKM19}. Now we have $U_{ext}>>U_{B_{II}}$ and the contribution of the external inverse Compton scattering is more significant than this of the first zone. As a result, in the total flux, the low-frequency component is related to the synchrotron radiation from the acceleration zone and the high one to the inverse Compton scattering losses from the cooling zone. \n\\end{itemize}\nThe above can be illustrated by Figure \\Ref{fig:sed_2} which depicts our calculations for the two zones in the cases of FSRQs and BL Lac objects. \n\\begin{figure}[!htbp]\n    \\centering\n    \\includegraphics[width=0.75\\linewidth]{BS_comp.pdf}\n    \\caption{Results for FSRQ and BL Lac objects according to the two-zone model. Straight lines depict the emission from the acceleration zone and dotted lines the emission from the cooling zone.}\n    \\label{fig:sed_2}\n\\end{figure}\n\\begin{figure}[!htbp]\n    \\centering\n    \\includegraphics[width=0.75\\linewidth]{BS_2zone.pdf}\n    \\caption{The theoretical Blazar Sequence according to the superposition of the two-zone emission by varying only the mass accretion rate, see Table \\Ref{tab3} for the values of the input parameters. The acceleration zone is at a distance of $z=0.01~pc$ from the central black hole which is assumed to have  a mass $\\mathcal{M}=10^8$. The external photon field is produced from the isotropic scattering of disk photons on the wind particles between radii $R_1=9\\cdot 10^{14}~\\rm{cm}$ and $R_2=3\\cdot 10^{18}~\\rm{cm}$. The efficiencies of the magnetic field and the external photon field are $\\eta_{\\rm b}=0.01$, and $\\epsilon=0.05$, respectively. The number of the injected electrons in the acceleration process, which is assumed to be of Type I Fermi, depends linearly on the mass accretion rate. The bulk Lorentz factor is $\\Gamma=30$ and the Doppler factor is $\\delta=15$. The characteristic temperature of the accretion disk is $T_{disk}=3 \\cdot 10^3~$K.}\n    \\label{fig:sed_2zone}\n\\end{figure}\n\\begin{figure}[!htbp]\n    \\centering\n    \\includegraphics[width=0.75\\linewidth]{3C273.pdf}\n    \\caption{An application of the two-zone model in the case of FSRQ 3C273. The acceleration zone is at distance $z=0.01pc$.  The external photon field is produced from the isotropic scattering of disk photons on the wind particles between radii   $R_1=9\\cdot 10^{14}~\\rm{cm}$ and $R_2=3\\cdot 10^{18}~\\rm{cm}$.  The magnetic field strength in the acceleration zone is $B=1~G$, its radius is $R=5\\cdot 10^{15}$ cm, and the energy density of the external photon field is $U_{ext}=2.5\\cdot10^{-3} ~\\frac{erg}{sec}$. The bulk Lorentz factor is $\\Gamma=30$ and the Doppler factor is $\\delta=15$. The characteristic temparature of the accretion disk is $T_{\\rm disk}=3 \\cdot 10^4$ K. Data are reproduced from \\cite{GIommi12P}.}\n    \\label{fig:3c273}\n\\end{figure}\n\\begin{table} \n\\begin{center}\n{\n  \\begin{tabular}{ |c|c | c | c | c |}\n    \\hline\n    $ \\dot{m}$ & $ B({\\rm G}) $ & $U_{\\rm ext}$ $\\left(\\frac{\\rm erg}{\\rm{cm}^3}\\right)$ & $A_{\\rm acc}$  & Blazar Class\\\\ \\hline\n         -0.5 & 1  & -2.6 &-4& FSRQ\\\\ \\hline\n     -1.5 & 0 & -5.6 & -5& LBL \\\\ \\hline\n    -2.5 & -1  & -8.6 &-6&  HBL \\\\    \n    \\hline\n  \\end{tabular}} \n\\end{center}\n\\caption{The values of the  input parameters for different mass accretion rates when particle acceleration is included in the numerical code. All the values are in a logarithmic scale. }\\label{tab3}\n\\end{table} \n\nIn Figure \\Ref{fig:sed_2zone} we present our results for a theoretical Blazar Sequence in the case of the two-zone model, see Table \\ref{tab3} for the values of the input parameters and the dependence of $\\dot{m}$. In both classes of blazars particles escape from the acceleration zone. However, FSRQs present a  characteristic signature in $\\gamma$-rays, because particles inject into the cooling zone and interact with a strong external photon field. On the contrary, in BL Lac objects, when particles escape, they do not interact with a strong external photon field and as a result, the cooling zone has a lower contribution to the total flux. \n\nIn Figure \\Ref{fig:3c273} we show our results in applying a two-zone model in the case of 3C273, an FSRQ object. We zoom on the high-energy part of the spectrum (X-rays, $\\gamma$-rays). X-rays are produced by SSC of the acceleration zone and $\\gamma$-rays by the external Compton of the cooling zone.\n\n\\section{Conclusion}\nIn this work, we reproduce the theoretical Blazar Sequence based on the model of \\cite{BKM19} by varying only the mass accretion rate that seems to explain the Blazar phenomenology. We solve self-consistently the electron and photon kinetic equations, by assuming that electrons accelerate into a small region and lose energy through synchrotron and inverse Compton scattering. While, a part escapes to a larger volume where they lose energy through the same physical processes. Under this assumption, we produce the theoretical Blazar Sequence by adding up the fluxes of the two zones in both the cases of FSRQs and BL Lac objects. \n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Shortest Common Superstring (SCS) problem is a classical combinatorial problem on strings with applications in many domains, e.g.~DNA fragment assembly, data compression, etc. (see \\cite{GePi14} for a recent survey). It consists, given a finite set of strings $S$ over an alphabet $\\Sigma$, in finding a shortest string  containing as factors (substrings) all the strings in $S$. The decision version of the problem is known to be NP-complete \\cite{MaSt77,GallantPhD,GaMaSt80}, even under several restrictions on the structure of $S$ (see again \\cite{GePi14}). However, a particularly simple greedy algorithm introduced by Gallant in his Ph.D. thesis \\cite{GallantPhD} is widely used in applications since it has very good performance in practice (see for instance \\cite{Ma09} and references therein). It consists in repetitively replacing a pair of strings with maximum overlap with the string obtained by overlapping the two strings, until one string remains. The greedy algorithm can be implemented using Aho-Corasick automaton in $\\Oh(n)$ randomized time (with hashing on the symbols of the alphabet) or $\\Oh(n \\min(\\log m,\\log |\\Sigma|))$ deterministic time (see \\cite{Uk90}), where  $n$ is the sum of the lengths of the strings in $S$ and $m$ its cardinality.\n\nThe approximation of the greedy algorithm is usually measured in two different ways: one consists in taking into account the \\emph{approximation ratio} (also known as the \\emph{length ratio}) $k_g\/k_{min}$, where $k_g$ is the length of the output string of greedy and $k_{min}$ the length of a shortest superstring, the other consists in taking into account the \\emph{compression ratio} $(n-k_g)\/(n-k_{min})$. \n\nFor the approximation ratio, Turner \\cite{Tu89} proved that there is no constant $c<2$ such that $k_g\/k_{min}\\leq c$. The \\emph{greedy conjecture} states that  this approximation ratio is in fact $2$ \\cite{BlJiLiTrYa94}. The best bound currently known is 3.5 due to Kaplan and Shafrir \\cite{KaSh05}. Algorithms with better approximation ratio are known; the best one is due to Mucha, with an approximation ratio of $2\\frac{11}{23}$ \\cite{Mu13}. \n\nFor the compression ratio, Tarhio and Ukkonen \\cite{TaUk88} proved that $(n-k_g)\/(n-k_{min})\\geq \\frac12$ and this bound is tight, since it is achieved for the set $S=\\{ab^h,b^ha,b^{h+1}\\}$ when greedy makes the first choice merging the first two strings together.\n\nLet us formally state the SCS problem:\n\n\\bigskip\n\n  \\defdsproblem{Shortest Common Superstring ($\\mathit{SCS}$)}{\n    \\textbf{Input:} strings $S=\\{s_1,\\ldots,s_m\\}$ of total length $n$.\\\\\n    \\textbf{Output:} a shortest string $u$ that contains $s_i$ for each $i=1,\\ldots,m$ as a factor.\n  }\n \n\n  \\bigskip\n  \n    Several variations of SCS have been considered in literature. For example, shortest common superstring problem with reverse complements was considered in\n  \\cite{DBLP:journals\/algorithmica\/KececiogluM95}. In this setting the alphabet is $\\Sigma=\\{\\mathtt{a,t,g,c}\\}$ and the complement of a string $s$ is $\\bar{s}^{R}$, where $\\, \\bar{}\\, $ is defined by\n  $\\bar{\\mathtt{a}}=\\mathtt{t}$,\n  $\\bar{\\mathtt{t}}=\\mathtt{a}$,\n  $\\bar{\\mathtt{g}}=\\mathtt{c}$,\n  $\\bar{\\mathtt{c}}=\\mathtt{g}$,\n  and $t^R$ denotes the reversal of $t$, that is the string obtained reading $t$ backwards.   In particular, this problem was shown to be NP-complete.\n  \n   Other variations of the SCS problem can be found in \\cite{Jiang94,cpm,swap,rest}.   \n   \n    In this paper, we address the problem of searching for a string $u$ of minimal length such that for every $s_i\\in S$,  $u$ contains as a factor $s_i$ \\emph{or} its reversal $s_i^{R}$.\n  \n    \\bigskip\n    \n  \\defdsproblem{Shortest Common Superstring with Reversals ($\\mathit{SCS\\mbox{-}R}$)}{\n    \\textbf{Input:} strings $S=\\{s_1,\\ldots,s_m\\}$ of total length $n$.\\\\\n    \\textbf{Output:} a shortest string $u$ that contains for each $i=1,\\ldots,m$ at least one of the\n    strings $s_i$ or $s^{R}_i$ as a factor.\n  }\n  \n    \\bigskip\nFor example, if $S=\\{aabb, aaac, abbb\\}$, then a solution of SCS-R for $S$ is $caaabbb$. Notice that a shortest superstring with reversals can be much shorter than a classical shortest superstring. An extremal example is given by an input set of the form $S=\\{ab^h,cb^h\\}$.\n\n\n  The SCS-R problem was already considered by Jiang et al.\\ \\cite{DBLP:journals\/ipl\/JiangLD92}, who observed (not giving any proof) that the problem is still NP-hard. We provide a proof at the end of the paper.\n  \n  In \\cite{DBLP:journals\/ipl\/JiangLD92}, the authors proposed a greedy 4-approximation algorithm. Here, we show that an adaptation of the classical greedy algorithm can be used for solving the SCS-R problem with an (optimal) compression ratio $\\frac12$, and that this algorithm can be implemented in linear time with respect to the total size of the input set.\n\n  \n\n\\section{Basics and Notation}\n\n\nLet $\\Sigma$ be a finite alphabet. We assume that $\\Sigma$ is linearly sortable, e.g., $\\Sigma=\\{0,\\ldots,n^{\\Oh(1)}\\}$.  The \\emph{length} of a string $s$ over $\\Sigma$ is denoted by $|s|$. The \\emph{empty string}, denoted by $\\epsilon$, is the unique string of length zero.\nA string $t$ \\emph{occurs} in a string $s$ if $s=vtz$ for some strings $v,z$. In this case we say that $t$ is a \\emph{factor} of $s$. \nIn particular, we say that $t$ is a \\emph{prefix} of $s$ when $v=\\epsilon$ and a \\emph{suffix} of $s$ when $z=\\epsilon$.\nWe say that a factor $t$ is \\emph{proper} if $s \\ne t$.\n\nThe string $s^{R}$ obtained by reading $s$ from right to left is called the \\emph{reversal} (or \\emph{mirror image}) of $s$.\nGiven a set of strings $S=\\{s_1,\\ldots,s_m\\}$, we define the set $S^R = \\{s_1^{R},\\ldots,s_m^{R}\\}$ and the set $\\widetilde{S}=S \\cup S^R$.\n\nGiven two strings $u,v$, we define the (maximum) overlap between $u$ and $v$, denoted by $\\mathit{ov}(u,v)$, as the length of the longest suffix of $u$ that is also a prefix of $v$. Sometimes we abuse the notation and also say that the suffix of $u$ of length $\\mathit{ov}(u,v)$ is the overlap of $u$ and $v$.\nIn general $\\mathit{ov}(u,v)$ is not equal to $\\mathit{ov}(v,u)$, but it is readily verified that $\\mathit{ov}(u,v) = \\mathit{ov}(v^{R},u^{R})$.\nAdditionally, we define $\\pr(u,v)$ as the prefix of $u$ obtained by removing the suffix of length $\\mathit{ov}(u,v)$ and denote $u \\otimes v=\\pr(u,v) v$.\nNote that the $\\otimes$ operation is in general neither symmetric nor associative.\n\nA set of strings $S$ is called \\emph{factor-free} if no string in $S$ is a factor of another string in $S$. We say that $S$ is \\emph{reverse-factor-free} if there are no distinct strings $u,v\\in S$ such that $u$ is a factor of $v$ or~$v^R$.\n\n\nGiven a factor-free set of strings  $S=\\{s_1,\\ldots,s_m\\}$, the SCS problem for $S$ is known to be equivalent to that of finding a maximum-weight Hamiltonian path $\\pi$ in the \\emph{overlap graph} $G_S$, which is a directed weighted graph $(S,E,w)$ with arcs $E=\\{(s_i,s_j)\\mid i\\neq j\\}$ \nof weights $w(s_i,s_j)=\\mathit{ov}(s_i,s_j)$ (cf.\\ Theorem 2.3 in~\\cite{TaUk88}). \nIn this setting, a path $\\pi=s_{i_1}, \\ldots, s_{i_k}$ corresponds to a string $\\mathit{str}(\\pi):=\\pr(s_{i_1},s_{i_2})\\cdots \\pr(s_{i_{k-1}},s_{i_k})s_{i_k}$.\nBy $\\mathit{ov}(\\pi)$ we denote the total weight of arcs in the path $\\pi$.\n\n\nTo accommodate reversals we extend the notion of an overlap graph to $\\barG{S}=(V,E,w)$.\nHere $V=\\{v_s\\,:\\,s \\in S\\} \\cup \\{v^R_s\\,:\\,s \\in S\\}$ so every $s\\in S$ corresponds to exactly two vertices,\n$v_s$ and $v^R_s$. We define $\\mathit{str}(v_s)=s$ and $\\mathit{str}(v^R_s)=s^R$.\nFor a vertex $\\alpha\\in \\barG{S}$ we define $\\alpha^R$ as $v^R_s$ if $\\alpha=v_s$ for some $s$\nor as $v_s$ if $\\alpha = v^R_s$ for some $s$. Note that $\\mathit{str}(\\alpha^R)=\\mathit{str}(\\alpha)^R$.\nFor every $\\alpha,\\beta \\in V$, $\\alpha \\ne \\beta$, we introduce an arc from $\\alpha$ to $\\beta$ with weight $\\mathit{ov}(\\mathit{str}(\\alpha),\\mathit{str}(\\beta))$.\nFor an arc $e=(\\alpha,\\beta)$ we define $e^R = (\\beta^R,\\alpha^R)$. Note that the weight of $e^R$ is the same as the weight of $e$.\n\nFor paths $\\pi$ in $\\barG{S}$ we also use the notions of $\\mathit{str}(\\pi)$ and $\\mathit{ov}(\\pi)$.\nWe say that a path $\\pi$ in $\\barG{S}$ is \\emph{semi-Hamiltonian} if $\\pi$ contains, for every vertex $\\alpha\\in \\barG{S}$, exactly\none of the two vertices $\\alpha$, $\\alpha^R$.\nObserve that a solution to SCS-R problem for a reverse-factor-free set $S$ corresponds to a maximum-weight semi-Hamiltonian path $\\pi$ in the overlap graph $\\barG{S}$.\n\n\\section{Greedy Algorithm and its Linear-Time Implementation}\nWe define an auxiliary procedure \\textsl{\\textsc{Make-Reverse-Factor-Free}}$(S)$ that\nremoves from $S$ all strings $u$ which are contained as a factor in $v$ or $v^R$ for some $v \\in S$, $v \\ne u$.\nNote that the resulting set $S'$ is reverse-factor-free and, moreover, a string is a common superstring with reversals for $S'$ if and only if it is a common superstring with reversals for $S$.\n\n\n\\begin{example}\n  Let $S=\\{ab,aaa,aab,baa\\}$. Then \\textsl{\\textsc{Make-Reverse-Factor-Free}}$(S)$ produces $S'=\\{aaa,aab\\}$ or $S'=\\{aaa,baa\\}$.\n\\end{example}\n\nThe Greedy-R algorithm works as follows:\nwhile $|S|>1$, choose $u,v \\in \\widetilde{S}$ (excluding $u=v$ and $u=v^{R}$) \nwith largest overlap, insert into $S$ the string $u \\otimes v$,\nand remove from $S$ all strings among $u,v,u^{R},v^{R}$ that belong to $S$;\nsee the pseudocode.\n\n\n\\DontPrintSemicolon\n\\begin{algorithm}\n  \\KwSty{Algorithm} \\textsl{Greedy-R}($S$)\\\\\n  \\KwIn{a non-empty set of strings $S$}\n  \\KwOut{a superstring of $S$ that approximates a solution of SCS-R problem for $S$}\n  \\Begin{\n    $S$ := \\textsl{\\textsc{Make-Reverse-Factor-Free}}($S$) \\;\n    \\While{$|S|>1$}{\n      $P:=\\{ (u,v) : u,v\\in \\widetilde{S}, u\\notin \\{v,v^R\\}\\}$ \\;\n      $\\{$ $S$ is reverse-factor-free and $|S|>1$, so $|P|\\ge 1$ $\\}$ \\\\\n      take $(u,v)\\in P$ with the maximal value of $\\mathit{ov}(u,v)$ \\;\n      $S:= S \\cup \\{u \\otimes v$\\} \\;\n      $S:=S \\setminus \\{u,v,u^{R},v^{R}\\}$\\; \n    }\n    \\Return{the only element of $S$}\n  }\n  \\textbf{end}\n\\end{algorithm}\n\nLet us state two properties of this algorithm useful for its efficient implementation.\n\n\\begin{lemma}\\label{lem:free}\nThe set $S$ stays reverse-factor-free after each iteration of the \\textbf{while} loop.\n\\end{lemma}\n\\begin{proof}\nSuppose that at some point $S$ ceases to be reverse-factor-free.\nThis might only be due to the fact that, when $w=u\\otimes v$ is introduced to $S$, $\\widetilde{S}$ contains a string\n$w' \\notin \\{u,u^R,v,v^R\\}$ such that $w'$ is a factor of $w$ but not of $u$ or $v$.\nThe latter, however, implies $\\mathit{ov}(u,w')>\\mathit{ov}(u,v)$ and $\\mathit{ov}(w',v)>\\mathit{ov}(u,v)$.\nThat contradicts the choice\nof $(u,v)\\in P$ maximizing $\\mathit{ov}(u,v)$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:stays}\nBefore $u\\otimes v$ is inserted to $S$, we have $\\mathit{ov}(w, u\\otimes v)=\\mathit{ov}(w,u)$ and $\\mathit{ov}(u\\otimes v,w)=\\mathit{ov}(v,w)$\nfor every $w\\in \\widetilde{S}$.\n\\end{lemma}\n\\begin{proof}\nClearly, $\\mathit{ov}(w, u\\otimes v)\\ge \\mathit{ov}(w,u)$ and $\\mathit{ov}(u\\otimes v,w)\\ge \\mathit{ov}(v,w)$. \nMoreover, one of these inequalities might be strict only if $\\mathit{ov}(w, u\\otimes v)>|u|$ or $\\mathit{ov}(u\\otimes v, w)>|v|$, in particular, only if $w$ contains respectively $u$ or $v$ as a proper factor.\nThis, however, contradicts Lemma~\\ref{lem:free}.\n\\end{proof}\n\n\n\\subsection{Interpretation on the Overlap Graph}\n\n\nTarhio and Ukkonen~\\cite{TaUk88} work with the following interpretation of the greedy algorithm on the overlap graph\n$G_S$: They maintain a set of arcs $F\\subseteq E(G_S)$ forming a collection of disjoint paths and\nat each step they add to $F$ an arc $e\\in E(G_S)\\setminus F$ of maximum weight so that the resulting set still forms a collection of disjoint paths.\nHere, paths correspond to strings in the collection $S$ maintained by the original implementation. \nInsertion of an arc $(s_i,s_j)$ to $F$ results in merging two paths into one and this corresponds to replacing strings $u,v\\in S$ with $u\\otimes v$.\nIt turns out that $\\mathit{ov}(u,v)=\\mathit{ov}(s_i,s_j)$ and thus \nboth implementations of the greedy algorithm are equivalent.\n\n\n\\DontPrintSemicolon\n\\begin{algorithm}[h]\n  \\KwSty{Algorithm} \\textsl{Greedy-R2}($S$)\\\\\n  \\KwIn{a non-empty set of strings $S$}\n  \\KwOut{a superstring of $S$ that approximates a solution of SCS-R problem for $S$}\n  \\Begin{\n    $S$ := \\textsl{\\textsc{Make-Reverse-Factor-Free}}($S$) \\;\n    Construct $\\barG{S}$\\;\n    $F := \\emptyset$\\;\n    \\For{$i := 1$ \\KwSty{to} $|S|-1$}{\n      $P:=\\{e \\in E(\\barG{S})\\setminus F : F\\cup\\{e,e^R\\}\\text{ forms a collection of disjoint paths in }\\barG{S}\\}$ \\;\n      take $e\\in P$ with the maximal weight $w(e)$ \\;\n      $F := F\\cup\\{e,e^R\\}$\\; \n    }\n    \\Return{$\\mathit{str}(\\pi)$ for one of the maximal paths $\\pi$ formed by $F$ in $\\barG{S}$}\n  }\n  \\textbf{end}\n\\end{algorithm}\n\nHere, we provide an analogous interpretation of Greedy-R on the overlap graph $\\barG{S}$\nand, for completeness, explicitly prove its equivalence to the original implementation.\nWe also maintain a set of arcs $F\\subseteq E(\\barG{S})$ forming a \ncollection of disjoint paths.  In each step we extend $F$ with a pair of arcs $\\{e,e^R\\}$ of maximum (common)\nweight so that the resulting set still forms a collection of disjoint paths; see the pseudocode of algorithm Greedy-R2.\nObserve that this way paths formed by $F$ come in pairs $\\pi,\\pi^R$ such that $\\pi=\\alpha_1,\\ldots,\\alpha_p$\nand $\\pi^R=\\alpha_p^R,\\ldots,\\alpha_1^R$ (in particular, $\\mathit{str}(\\pi^R)=\\mathit{str}(\\pi)^R$).\nWe claim that these pairs of paths correspond to strings $s\\in S$ of the original implementation (with $s=\\mathit{str}(\\pi)$ or $s=\\mathit{str}(\\pi^R)$).\nIn particular, each string $s\\in \\widetilde{S}$ corresponds to a single path formed by $F$ unless\n$s=s^R$ when it corresponds to two reverse paths. \n The claim is certainly true at the beginning of the algorithm, so let us argue\nthat single iterations of the main loops in both algorithms perform analogous operations.\n\nFirst, consider an arc $e=(\\alpha,\\beta)$ such that there exists a path $\\pi$ that ends at $\\alpha$ and a path $\\pi'$\nthat starts at $\\beta$. (Otherwise, $F\\cup\\{e,e^R\\}$ does not form a collection of disjoint paths).\nObserve that strings $u=\\mathit{str}(\\pi)$ and $v=\\mathit{str}(\\pi')$ belong to $\\widetilde{S}$ in Greedy-R and that $u\\in \\{v,v^R\\}$\nif and only if $F\\cup\\{e,e^R\\}$ yields a cycle. Thus, both algorithms consider\nessentially the same set of possibilities \n(the only caveat is that if $u$ or $v$ is a palindrome, then several arcs $e$ correspond to the same pair of strings from $\\widetilde{S}$). \nBy Lemma~\\ref{lem:stays} the weight of an arc $(\\alpha,\\beta)$ is $\\mathit{ov}(\\mathit{str}(\\alpha),\\mathit{str}(\\beta))=\\mathit{ov}(u,v)$. Thus, \nthe maximum-weight arcs correspond to pairs $(u,v)\\in P$ with largest overlap.\n\nClearly, setting $F:= F\\cup\\{e,e^R\\}$ results in merging $\\pi'$ with $\\pi$\nand $\\pi^R$ with $\\pi'^R$. These paths represent $u\\otimes v$ and $v^R \\otimes u^R$.\nSince the set $S$ in Greedy-R stays reverse-factor-free due to Lemma~\\ref{lem:free} and, in particular,\n$S$ does not contain any pair of strings $v,v^R$ such that $v$ is not a palindrome,\nthe ``$S:=S\\setminus \\{u,u^R,v,v^R\\}$'' instruction always removes exactly two elements of $S$.\nThis means that the bijection between $S$ and pairs of paths formed by $F$ is preserved.\n\nFinally, observe that after exactly $|S|-1$ steps $F$ forms two semi-Hamiltonian paths.\nEither of them can be returned as a solution.\n\n\\subsection{Linear-Time Implementation}\n\nUkkonen \\cite{Uk90} showed that the Greedy algorithm for the original SCS problem\ncan be implemented in linear time based on the overlap-graph interpretation.\nOur linear-time implementation of the Greedy-R2 algorithm is quite similar.\n\nFirst, let us show how to efficiently implement the \\textsl{\\textsc{Make-Reverse-Factor-Free}}\\ operation.\nIt is actually slightly easier to compute it for $\\widetilde{S}$ instead of $S$.\nHowever, this is not an issue: if we substitute $S$ with $\\widetilde{S}$ in the very beginning of the greedy algorithm,\nthen the overlap graph will stay the same.\n\n\\begin{lemma}\\label{lem:mrff}\n  The result of \\textsl{\\textsc{Make-Reverse-Factor-Free}}$(\\widetilde{S})$ can be computed in time linear in the total length of strings in $S$.\n\\end{lemma}\n\n\\begin{proof}\n  Let us introduce an auxiliary procedure \\textsl{\\textsc{Make-Factor-Free}}$(X)$ \n  that removes from $X$ all strings $u$ for which there exists\n  a string $v$ in $X$ such that $u$ is a proper factor of $v$.\n  \n    Ukkonen \\cite{Uk90} applied the following result for the preprocessing phase of the greedy algorithm for the ordinary SCS problem.\n\n  \\begin{claim}[\\cite{Uk90}]\n    \\textsl{\\textsc{Make-Factor-Free}}$(X)$ can be implemented in time linear in the total length of the strings in $X$.\n  \\end{claim}\n  \n  Observe that in order to compute \\textsl{\\textsc{Make-Reverse-Factor-Free}}$(\\widetilde{S})$ it suffices to determine\n  $S' = \\,$\\textsl{\\textsc{Make-Factor-Free}}$(\\widetilde{S})$ and then for every pair of strings $u,u^R\\in S'$\n  leave exactly one of these strings, e.g., the lexicographically smaller one. \n  Note that $u\\in S'$ if and only if $u^R \\in S'$, so for the latter\n  it suffices to iterate through $S'$ and report a string $u$ if and only if $u \\le u^R$.\n  \n  The whole procedure works in linear time in the total length of strings in $\\widetilde{S}$,\n  which is at most twice the total length of strings in $S$.\n  \\end{proof}\n\nNow, we show the main result.\n\n\\begin{theorem}\n  Greedy-R algorithm can be implemented in time linear in the total length of strings in $S$.\n\\end{theorem}\n\\begin{proof}\n  By Lemma~\\ref{lem:mrff}, we can make the input set reverse-factor-free in linear time.\n  Let $S=\\{s_1,\\ldots,s_m\\}$ be the set of remaining strings and $n$ the total length of strings in $S$.\n  \n  We actually implement the equivalent algorithm Greedy-R2.\n  We cannot store the whole graph $\\barG{S}$, since this would take too much space.\n  Therefore we only store its vertex set, whereas we will be considering the edges of the graph in an indirect way.\n  \n  Denote by $\\Pref(X)$ the set of all different prefixes of strings in $X$.\n  Each element of this set can be represented as a state of the Aho-Corasick automaton constructed for $X$, thus\n  using $\\Oh(1)$ space per element.\n  Further, given a string $w$, denote by $\\PrefSet(w,X)$, $\\SufSet(w,X)$ the sets (represented as lists of identifiers) of strings\n  in $X$ having $w$ as a prefix, suffix, respectively.\n  Ukkonen \\cite{Uk90} applied the Aho-Corasick automaton for $X$ to show the following fact:\n\n  \\begin{claim}[\\cite{Uk90}]\n    $\\PrefSet(w,X)$, $\\SufSet(w,X)$ for all $w \\in \\Pref(X)$ can be computed in time\n    linear in the total length of the strings in $X$.\n  \\end{claim}\n  In the implementation we use sets of the form $\\Pref(\\widetilde{S})$, $\\PrefSet(w,\\widetilde{S})$ and $\\SufSet(w,\\widetilde{S})$.\n  Our implementation actually requires $\\PrefSet$ and $\\SufSet$ sets to consist of vertices $\\alpha\\in V(\\barG{S})$\n  instead of strings $\\mathit{str}(\\alpha)$. Thus, we replace every string identifier with one or two vertex identifiers\n  if the string is not a palindrome or is a palindrome, respectively.\n\n  Instead of simulating the main loop directly, we will consider all overlaps $w$ of $\\mathit{str}(\\alpha)$ and $\\mathit{str}(\\beta)$\n  to find the maximum-weight arc $e=(\\alpha,\\beta)\\in E(\\barG{S})$.  \n  Observe that at subsequent iterations of the loop, the weight $w(e)$ may only decrease.\n  Hence, we iterate over all  $w \\in \\Pref(\\widetilde{S})$ in decreasing length order\n  and for each $w$ check if there exists an appropriate arc $e=(\\alpha,\\beta)$ such that $F\\cup\\{e,e^R\\}$\n  forms a collection of disjoint paths. More formally, we seek vertices $\\alpha,\\beta\\in V(\\barG{S})$ such that:\n  \\begin{enumerate}[(a)]\n    \\item a path formed by $F$ ends in $\\alpha$ and a \\emph{different} path formed by $F$ starts at $\\beta$,\n    \\item $\\beta \\ne \\alpha^R$, and\n    \\item $w$ is the overlap of $\\mathit{str}(\\alpha)$ and $\\mathit{str}(\\beta)$.\n  \\end{enumerate}\n  Once we find such an arc $e$, we set $F:=F\\cup\\{e,e^R\\}$. \n\n  Let us explain how this approach can be implemented efficiently.  \n  To iterate through all $w \\in \\Pref(\\widetilde{S})$ in decreasing length order we simply traverse all the states of the Aho-Corasick automaton\n  in reverse-BFS order, breaking ties arbitrarily.\n  To check the conditions (a)-(c), we could iterate through pairs of elements\n  $\\beta \\in \\PrefSet(w,\\widetilde{S})$ and $\\alpha \\in \\SufSet(w,\\widetilde{S})$ and verify if they satisfy the conditions.\n  However, to avoid repetitively scanning \\emph{redundant} elements, we remove $\\beta\\in \\PrefSet(w,\\widetilde{S})$\n  and $\\alpha\\in \\SufSet(w,\\widetilde{S})$ if no path starts in $\\beta$ and no path ends in $\\alpha$, respectively.\n  For each vertex we will remember if a path starts or ends there, and, if so, what is the other endpoint of the path.\n  Observe that for each $\\alpha \\in \\SufSet(w,\\widetilde{S})$ there are at most two non-redundant elements $\\beta \\in \\PrefSet(w,\\widetilde{S})$\n  for which the arc $e=(\\alpha,\\beta)$ is not valid.\n  Indeed, these might only be $\\alpha^R$ and the starting vertex of the path ending at $\\alpha$. \n  \n  Hence, with amortized constant-time overhead, either an arc $e=(\\alpha,\\beta)$ satisfying (a)-(c) is found,\n  or it can be verified that no such arc exists for this particular string $w$ and then\n  we can continue iterating through strings in $\\Pref(\\widetilde{S})$.\n  If an arc is found, we start the next search with the same string $w$, since\n  there could still be arcs satisfying conditions (a)-(c) for the same string $w$.\n  \n\n  To conclude: in every step of the simulation, in amortized constant time we either find a pair of arcs to be introduced to $F$\n  or discard the given candidate $w \\in \\Pref(\\widetilde{S})$.\n  The former situation takes place at most $m-1$ times and the latter happens at most $n$ times.\n  The whole algorithm thus works in $\\Oh(n)$ time.\n \\end{proof}\n\n\n\\section{Compression Ratio}\n\nIn this section we prove that the compression ratio of the Greedy-R algorithm is always at least $\\frac12$, and that this value is effectively achieved, so the bound is tight.\n\nLet $S=\\{s_1,\\ldots,s_m\\}$ be the input set of strings.\nWe assume that $S$ is already reverse-factor-free.\nLet $\\opt(S)$ be the length of a longest semi-Hamiltonian path in $G=\\barG{S}$.\nLet $\\mathit{pgreedy}(S)$ denote the length of the semi-Hamiltonian path produced by the Greedy-R algorithm for $S$.\nWe will show that $\\mathit{pgreedy}(S) \\ge \\frac12 \\opt(S)$.\n\nIn the proof we use as a tool the following fact from \\cite{TaUk88} (see Lemma~3.1 in \\cite{TaUk88}):\n\\begin{lemma}\\label{lem:4vert}\n  If strings $x_1,x_2,x_3,x_4$ satisfy\n  $$\\max(\\mathit{ov}(x_1,x_4),\\mathit{ov}(x_2,x_3)) \\le \\mathit{ov}(x_1,x_3),$$\n  then\n  $$\\mathit{ov}(x_1,x_4)+\\mathit{ov}(x_2,x_3) \\le \\mathit{ov}(x_1,x_3)+\\mathit{ov}(x_2,x_4).$$\n\\end{lemma}\n\nWe proceed with the following crucial lemma.\n\n\\begin{lemma}\\label{lem:compression_ratio}\n  Let $S$ be a set of strings and let $u,v \\in \\widetilde{S}$ be two elements for which $\\mathit{ov}(u,v)$ is maximal\n  ($u \\notin \\{v,v^R\\}$).\n  Set $\\mathit{OV}=\\mathit{ov}(u,v)$.\n  Let $\\opt(S)$ be the length of a longest semi-Hamiltonian path in $G$\n  and let $\\opt'(S)$ be the length of a longest semi-Hamiltonian path in $G$ that contains the arc $(u,v)$.\n  Then:\n  $$\\opt'(S) \\ge \\opt(S)-\\mathit{OV}.$$\n\\end{lemma}\n\n\\begin{proof}\n  We consider the path $\\pi$ corresponding to $\\opt(S)$ and show how it can be modified\n  without losing its semi-Hamiltonicity so that the arc $(u,v)$ occurs in the path and the length of the path decreases by at most $\\mathit{OV}$.\n\n  Obviously, if $\\pi$ already contains the arc $(u,v)$, nothing is to be done.\n  If both $u$ and $v$ occur in $\\pi$, we perform transformations as in the proof of a similar fact\n  from \\cite{TaUk88} related to the ordinary SCS problem.\n  If $u$ occurs in $\\pi$ before $v$ then we select $\\pi'$ as in Fig.~\\ref{fig:case_a} and:\n  $$\\mathit{ov}(\\pi') \\ge \\mathit{ov}(\\pi)-\\mathit{ov}(u,b)-\\mathit{ov}(c,v)+\\mathit{ov}(u,v) \\ge \\mathit{ov}(\\pi)-\\mathit{ov}(u,b) \\ge \\mathit{ov}(\\pi)-\\mathit{OV}.$$\n  Note that both nodes $b, c$ exist (they could be the same node, though).\n  If any of the remaining nodes does not exist, it is simply skipped on the path.\n\n  \\begin{figure}[htpb]\n  \\begin{center}\n      \\tikzstyle{mynode} = [circle,draw]\n      \\begin{tikzpicture}[scale=0.85]\n        \\draw (-1,0) node {$\\pi$};\n        \\foreach \\x\/\\c in {0.1\/s,3\/a,4\/u,5\/b,8\/c,9\/v,10\/d,13\/t}\n          \\draw (\\x,0) node[mynode] (\\c) {\\it\\scriptsize \\c};\n        \\foreach \\x\/\\y in {s\/a,b\/c,d\/t}\n          \\draw[decorate, decoration={snake},-latex] (\\x) -- (\\y);\n        \\foreach \\x\/\\y in {a\/u,u\/b,c\/v,v\/d}\n          \\draw[-latex] (\\x) -- (\\y);\n\n        \\begin{scope}[yshift=-1cm]\n        \\draw (-1,0) node {$\\pi'$};\n        \\foreach \\x\/\\c in {0.1\/s,3\/a,4\/u,5\/b,8\/c,9\/v,10\/d,13\/t}\n          \\draw (\\x,0) node[mynode] (\\c) {\\it\\scriptsize \\c};\n        \\foreach \\x\/\\y in {s\/a,b\/c,d\/t}\n          \\draw[decorate, decoration={snake},-latex] (\\x) -- (\\y);\n        \\foreach \\x\/\\y in {a\/u,v\/d}\n          \\draw[-latex] (\\x) -- (\\y);\n        \\draw[-latex] (c) .. controls (5,-2) and (3,-2) .. (s); %\n        \\draw[-latex] (u) .. controls (5.5,-1.5) and (7.5,-1.5) .. (v); %\n        \\end{scope}\n      \\end{tikzpicture}\n  \\end{center}\n  \\vspace*{-0.4cm}\n  \\caption{\\label{fig:case_a}Proof of Lemma~\\ref{lem:compression_ratio}, first case: $u$ occurs in $\\pi$ before $v$.}\n\\end{figure}\n\n  If $v$ occurs before $u$, then by applying the inequality of Lemma~\\ref{lem:4vert}\n  (with $x_1=u$, $x_2=a$, $x_3=v$, $x_4=d$)\n  we have \n  $$\\mathit{ov}(u,d)+\\mathit{ov}(a,v) \\le \\mathit{ov}(u,v)+\\mathit{ov}(a,d).$$\n  By this inequality, for the path $\\pi'$ defined in Fig.~\\ref{fig:case_b} we have:\n  \\begin{align*}\n    \\mathit{ov}(\\pi') &\\ge \\mathit{ov}(\\pi)-\\mathit{ov}(a,v)-\\mathit{ov}(u,d)+\\mathit{ov}(u,v)+\\mathit{ov}(a,d)-\\mathit{ov}(v,b)\\\\\n              &\\ge \\mathit{ov}(\\pi)-\\mathit{ov}(v,b) \\ge \\mathit{ov}(\\pi)-\\mathit{OV}.\n  \\end{align*}\n  As before, if any of the depicted nodes does not exist, we simply skip the corresponding part of the path.\n  In particular, if any of the nodes $u,v$ is an endpoint of the path $\\pi$, we do not need to use the aforementioned\n  inequality to show that $\\mathit{ov}(\\pi') \\ge \\mathit{ov}(\\pi)-\\mathit{OV}$.\n\n  \\begin{figure}[htpb]\n  \\begin{center}\n      \\tikzstyle{mynode} = [circle,draw]\n      \\begin{tikzpicture}[scale=0.85]\n        \\draw (-1,0) node {$\\pi$};\n        \\foreach \\x\/\\c in {0.1\/s,3\/a,4\/v,5\/b,8\/c,9\/u,10\/d,13\/t}\n          \\draw (\\x,0) node[mynode] (\\c) {\\it\\scriptsize \\c};\n        \\foreach \\x\/\\y in {s\/a,b\/c,d\/t}\n          \\draw[decorate, decoration={snake},-latex] (\\x) -- (\\y);\n        \\foreach \\x\/\\y in {a\/v,v\/b,c\/u,u\/d}\n          \\draw[-latex] (\\x) -- (\\y);\n\n        \\begin{scope}[yshift=-1.5cm]\n        \\draw (-1,0) node {$\\pi'$};\n        \\foreach \\x\/\\c in {0.1\/s,3\/a,4\/v,5\/b,8\/c,9\/u,10\/d,13\/t}\n          \\draw (\\x,0) node[mynode] (\\c) {\\it\\scriptsize \\c};\n        \\foreach \\x\/\\y in {s\/a,b\/c,d\/t}\n          \\draw[decorate, decoration={snake},-latex] (\\x) -- (\\y);\n        \\foreach \\x\/\\y in {c\/u}\n          \\draw[-latex] (\\x) -- (\\y);\n        \\draw[-latex] (a) .. controls (5,-1.5) and (8,-1.5) .. (d); %\n        \\draw[-latex] (t) .. controls (10,-2) and (8,-2) .. (b); %\n        \\draw[-latex] (u) .. controls (7.5,1.5) and (5.5,1.5) .. (v); %\n        \\end{scope}\n      \\end{tikzpicture}\n  \\end{center}\n  \\vspace*{-0.4cm}\n  \\caption{\\label{fig:case_b}Proof of Lemma~\\ref{lem:compression_ratio}, second case: $u$ occurs in $\\pi$ after $v$.}\n\\end{figure}\n\n  Differently from the original SCS problem considered in \\cite{TaUk88}, it\n  might not be the case that $u$ and $v$ are in $\\pi$.\n  If none of them is, then $\\pi$ contains both $u^R$ and $v^R$ and by reversing $\\pi$ (that is, taking the path $\\pi^R$)\n  we obtain a semi-Hamiltonian path that contains both $u$ and $v$,\n  which was the case considered before.\n  Thus, we can assume that $u$ and $v^{R}$ occur in $\\pi$ (the case of $u^{R}$ and $v$ is symmetric).\n  Again, we have two cases, depending on which of the two nodes comes first in $\\pi$.\n  If $u$ occurs before $v^{R}$ in $\\pi$, then we have (note that $\\mathit{ov}(c,v^{R})=\\mathit{ov}(v,c^{R})$):\n  $$\\mathit{ov}(\\pi') \\ge \\mathit{ov}(\\pi)-\\mathit{ov}(u,b)-\\mathit{ov}(v^{R},d)+\\mathit{ov}(u,v) \\ge \\mathit{ov}(\\pi)-\\mathit{ov}(u,b) \\ge \\mathit{ov}(\\pi)-\\mathit{OV},$$\n  see also Fig.~\\ref{fig:case_c}.\n \nFinally, if $v^{R}$ occurs before $u$, then (see Fig.~\\ref{fig:case_d}):\n  $$\\mathit{ov}(\\pi') \\ge \\mathit{ov}(\\pi)-\\mathit{ov}(v^{R},b)-\\mathit{ov}(u,d)+\\mathit{ov}(u,v) \\ge \\mathit{ov}(\\pi)-\\mathit{ov}(v^{R},b) \\ge \\mathit{ov}(\\pi)-\\mathit{OV}.$$\n\n\\begin{figure}\n  \\begin{center}\n      \\tikzstyle{mynode} = [circle,draw]\n      \\begin{tikzpicture}[scale=0.85]\n        \\draw (-1,0) node {$\\pi$};\n        \\draw (9,0) node[mynode,inner sep=0pt,minimum size=0.58cm] (vr) {\\scriptsize $v^{R}$};\n        \\foreach \\x\/\\c in {0.1\/s,3\/a,4\/u,5\/b,8\/c,10\/d,13\/t}\n          \\draw (\\x,0) node[mynode] (\\c) {\\it\\scriptsize \\c};\n        \\foreach \\x\/\\y in {s\/a,b\/c,d\/t}\n          \\draw[decorate, decoration={snake},-latex] (\\x) -- (\\y);\n        \\foreach \\x\/\\y in {a\/u,u\/b,c\/vr,vr\/d}\n          \\draw[-latex] (\\x) -- (\\y);\n\n        \\begin{scope}[yshift=-2cm]\n        \\draw (-1,0) node {$\\pi'$};\n        \\draw (5,0) node[mynode,inner sep=0pt,minimum size=0.58cm] (br) {\\scriptsize $b^{R}$};\n        \\draw (8,0) node[mynode,inner sep=0pt,minimum size=0.58cm] (cr) {\\scriptsize $c^{R}$};\n        \\foreach \\x\/\\c in {0.1\/s,3\/a,4\/u,9\/v,10\/d,13\/t}\n          \\draw (\\x,0) node[mynode] (\\c) {\\it\\scriptsize \\c};\n        \\foreach \\x\/\\y in {s\/a,cr\/br,d\/t}\n          \\draw[decorate, decoration={snake},-latex] (\\x) -- (\\y);\n        \\foreach \\x\/\\y in {a\/u,v\/cr}\n          \\draw[-latex] (\\x) -- (\\y);\n        \\draw[-latex] (u) .. controls (5.5,-1.5) and (7.5,-1.5) .. (v); %\n        \\draw[-latex] (br) .. controls (6.5,1.5) and (8.5,1.5) .. (d); %\n        \\end{scope}\n      \\end{tikzpicture}\n  \\end{center}\n  \\vspace*{-0.4cm}\n  \\caption{\\label{fig:case_c}Proof of Lemma~\\ref{lem:compression_ratio}, third case: $u$ occurs in $\\pi$ before $v^R$.}\n\\end{figure}\n\n\\begin{figure}\n\\bigskip\n  \\begin{center}\n      \\tikzstyle{mynode} = [circle,draw]\n      \\begin{tikzpicture}[scale=0.85]\n        \\draw (-1,0) node {$\\pi$};\n        \\draw (4,0) node[mynode,inner sep=0pt,minimum size=0.56cm] (vr) {\\scriptsize $v^{R}$};\n        \\foreach \\x\/\\c in {0.1\/s,3\/a,5\/b,8\/c,9\/u,10\/d,13\/t}\n          \\draw (\\x,0) node[mynode] (\\c) {\\it\\scriptsize \\c};\n        \\foreach \\x\/\\y in {s\/a,b\/c,d\/t}\n          \\draw[decorate, decoration={snake},-latex] (\\x) -- (\\y);\n        \\foreach \\x\/\\y in {a\/vr,vr\/b,c\/u,u\/d}\n          \\draw[-latex] (\\x) -- (\\y);\n\n        \\begin{scope}[yshift=-2cm]\n        \\draw (-1,0) node {$\\pi'$};\n        \\draw (0.1,0) node[mynode,inner sep=0pt,minimum size=0.56cm] (sr) {\\scriptsize $s^{R}$};\n        \\draw (3,0) node[mynode,inner sep=0pt,minimum size=0.56cm] (ar) {\\scriptsize $a^{R}$};\n        \\foreach \\x\/\\c in {4\/v,5\/b,8\/c,9\/u,10\/d,13\/t}\n          \\draw (\\x,0) node[mynode] (\\c) {\\it\\scriptsize \\c};\n        \\foreach \\x\/\\y in {ar\/sr,b\/c,d\/t}\n          \\draw[decorate, decoration={snake},-latex] (\\x) -- (\\y);\n        \\foreach \\x\/\\y in {v\/ar,c\/u}\n          \\draw[-latex] (\\x) -- (\\y);\n        \\draw[-latex] (u) .. controls (7.5,1.5) and (5.5,1.5) .. (v); %\n        \\draw[-latex] (sr) .. controls (3,-2) and (7,-2) .. (d); %\n        \\end{scope}\n      \\end{tikzpicture}\n  \\end{center}\n  \\vspace*{-0.4cm}\n  \\caption{\\label{fig:case_d}Proof of Lemma~\\ref{lem:compression_ratio}, third case: $u$ occurs in $\\pi$ after $v^R$.}\n\\end{figure}\n\nThis completes the proof of the lemma.\n \\end{proof}\n\nTo conclude the proof of the compression ratio of the Greedy-R algorithm we use an inductive argument.\nAssume that $\\mathit{pgreedy}(S') \\ge \\frac12 \\opt(S')$ for all $S'$ such that $|S'|<|S|$.\nBy Lemma~\\ref{lem:compression_ratio}, for $S'=S \\setminus \\{u,v,u^R,v^R\\} \\cup \\{u \\otimes v\\}$:\n$$\\opt(S)-\\mathit{OV} \\le \\opt'(S) \\le \\opt(S')+\\mathit{OV},$$\nso that $\\opt(S')+2\\mathit{OV} \\ge \\opt(S)$.\nMoreover, $\\mathit{pgreedy}(S)=\\mathit{pgreedy}(S')+\\mathit{OV}$ and, by the inductive hypothesis, $\\mathit{pgreedy}(S') \\ge \\frac12 \\opt(S')$.\nConsequently:\n$$\\mathit{pgreedy}(S) = \\mathit{pgreedy}(S')+\\mathit{OV} \\ge \\frac12 (\\opt(S')+2\\mathit{OV}) \\ge \\frac12 \\opt(S).$$\nWe arrive at the following theorem.\n\n\\begin{theorem}\n  Greedy-R has compression ratio $\\frac12$.\n\\end{theorem}\n\nIt turns out that the bound on the compression ratio of the Greedy-R algorithm is tight.\nIt suffices to consider the set of strings $\\{ab^h,b^hc,b^{h+1}\\}$\n(this is actually the same example as from the analysis of the Greedy algorithm \\cite{BlJiLiTrYa94}).\nThe output of Greedy-R can be the string $ab^hcb^{h+1}$ with total overlap $h$,\nwhereas an optimal solution to SCS-R is $ab^{h+1}c$ of total overlap $2h$.\n\n\n\\section{NP-completeness of the SCS-R Problem}\n\n \n  Let $\\Gamma=\\Sigma\\cup\\{\\$,\\#\\}$, where $ \\$,\\# \\not \\in \\Sigma$.\n  Let $h$ be the morphism from $\\Sigma^*$ to $\\Gamma^*$  defined by $h(c)=\\$\\#\\,c$, for every $c \\in \\Sigma$.\n  Given $k\\geq 1$, we also define the morphism $g_k(c)=c^k$, for every $c \\in \\Sigma$. \n\n  \\begin{observation}\\label{obs:small_ov}\n    For every nonempty strings $u,v$, the strings $h(u)$ and $h(v)^{R}$ have an overlap of length at most one.\n  \\end{observation}\n\n  \\begin{observation}\\label{obs:multiple}\n    For every nonempty strings $u,v$, one has\n    $$|h(g_k(u))|=3k|u| \\quad\\mbox{and }\\quad \\mathit{ov}(h(g_k(u)),h(g_k(v))) = 3k\\cdot\\mathit{ov}(u,v).$$\n  \\end{observation}\n\n  In the decision version of the SCS-R problem we need to check if a given set of strings\n  admits a common superstring with reversals of length at most $\\ell$, where $\\ell$ is\n  an additional input parameter.\n\n  \\begin{proposition}\\label{NP}\n    The decision version of the SCS-R problem is NP-complete.\n  \\end{proposition}\n  \\begin{proof}\n    Let $s_1,\\ldots,s_m$ be an instance of the SCS problem.\n    Set  $y_i=h(g_m(s_i))$, for $i=1,\\ldots,m$.\n    We have the following claim.\n\n    \\begin{claim}\n      There exists a common superstring of $s_1,\\ldots,s_m$ of length at most $\\ell$\n      if and only if\n      there exists a common superstring with reversal of $y_1,\\ldots,y_m$ of length at most $3m\\ell$.\n    \\end{claim}\n    \\begin{proof}\n      $(\\Rightarrow)$\n      If $u$ is a common superstring of $s_1,\\ldots,s_m$,\n      then $h(g_m(u))$ is a common superstring of $y_1,\\ldots,y_m$.\n      Hence, $h(g_m(u))$ is also a common superstring with reversals of $y_1,\\ldots,y_m$.\n      If $|u|=\\ell$, then $|h(g_m(u))|=3m\\ell$.\n\n      $(\\Leftarrow)$\n      Let $u$ be a common superstring with reversals of $y_1,\\ldots,y_m$.\n      We will show that, thanks to the special form of the strings, there exists\n      a common superstring \\emph{without} reversals of $y_1,\\ldots,y_m$ of length not much greater than $|u|$.\n\n      Let $\\pi=z_{i_1},\\ldots,z_{i_m}$ be the sequence of nodes on the path in the overlap graph $G$\n      that corresponds to $u$;\n      here $\\{i_1,\\ldots,i_m\\}=\\{1,\\ldots,m\\}$ and each $z_i$ is either $y_i$ or $y^R_i$. Let us construct a new sequence of nodes $\\pi'$, that first contains all nodes from $\\pi$\n      of the form $y_i$ in the same order as in $\\pi$, and then all nodes from $\\pi$\n      of the form $y_i^R$, but given in the reverse order and taken without reversal.\n      Let $u'$ be the common superstring corresponding to $\\pi'$.\n      By Observation~\\ref{obs:small_ov}, $|u'| \\le |u|+m-1$.\n      Note that $u'$ is an ordinary common superstring of $y_1,\\ldots,y_m$.\n\n      By Observation~\\ref{obs:multiple}, $|u'|$ is a multiple of $3m$ and $u'$\n      corresponds to a common superstring $v$ of $s_1,\\ldots,s_m$\n      of length $|u'|\/(3m)$.\n      If $|u|\\le 3m\\ell$, then\n      $$|v|=\\frac{|u'|}{3m} \\le \\left\\lfloor\\frac{|u|+m-1}{3m}\\right\\rfloor \\le \\ell.$$\n    \n    \\vspace{-0.5cm}\n    \\end{proof}\n    \n    The claim provides a reduction of the decision version of the SCS problem to the decision version of the SCS-R problem.\n    This shows that the latter is NP-hard, hence NP-complete, as it is obviously in NP.\n \\end{proof}\n\n\n\n\\section{Acknowledgments}\n\nThe authors thank anonymous referees for a number of helpful comments and remarks.\n\nThis work started during a visit of Gabriele Fici to the University of Warsaw funded by the Warsaw Center of Mathematics and Computer Science. \n\nGabriele Fici is supported by the PRIN 2010\/2011 project ``Automi e Linguaggi Formali: Aspetti Matematici e Applicativi'' of the Italian Ministry of Education (MIUR).\nTomasz Kociumaka is supported by Polish budget funds for science in 2013-2017 as a research project under the `Diamond Grant' program.\nJakub Radoszewski, Wojciech Rytter and Tomasz Wale\\'n are supported by the Polish National Science Center, grant no 2014\/13\/B\/ST6\/00770.\n\n\n\\bibliographystyle{plain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Introduction}\nThe solution of constitutive equations for viscoelastic fluids involves some important considerations, as for instance, the theoretical issues concerning the existence results~\\cite{Chupin2018,Ervin2003,Lukakova2017,Renardy1991}, and the development of numerical schemes for solving complex fluid flows~\\cite{Dellar2014,Han2000,Lee2004}. \n\\par\nSome forms of viscoelastic constitutive equations can be constructed considering the\nupper-convected time derivative or Oldroyd derivative~\\cite{Oldroyd1950}, which is defined as\n\\begin{equation}\n\t\\uctd{\\zeta} \\coloneqq \\frac{\\partial {\\zeta}}{\\partial t} + \\left( {u} \\cdot \\nabla  \\right) {\\zeta} - (\\nabla {u}) {\\zeta} - {\\zeta}(\\nabla {u})^\\top,\n\t\\label{der}\n\t\\end{equation}\nwhere $u(x,t) \\in \\mathbb{R}^d$ is the velocity field of the flow and ${\\zeta}(x,t) \\in \\mathbb{R}^{d\\times d}_{\\rm sym}$ is a tensor to represent the non-Newtonian contribution for $d=(1,) 2, 3$.\nRoughly speaking, the derivative form of~\\eqref{der} is generally used for describing responses of viscoelastic fluids, as for instance, the deformation induced by the rate of strain. Therefore, the upper-convected time derivative~\\eqref{der} is employed to formulate the constitutive equations of the \nmost popular models, as for instance the Oldroyd-B, Phan-Tien--Tanner~(PTT), Giesekus, etc~\\cite{MalPruSkrSul-2018,RenardyBook}. \n\\par\nIn particular, we are interested in the numerical approximations for model equations based on the classical differential constitutive equation for the Oldroyd-B fluid in a dimensionless form:\n\t\\begin{equation}\n\t{\\zeta} + Wi \\, \\uctd{\\zeta} = 2\\left( 1 - \\beta \\right) {D}({u}),\n\t\\label{eq:OB}\n\t\\end{equation}\nwhere ${D}({u}) = {[\\nabla {u} + (\\nabla {u})^\\top]\/{2}}$ is the strain-rate tensor, and the non-dimensional positive parameters $Wi$ and $\\beta$ are respectively the Weissenberg number and the viscosity ratio~$(\\beta \\in (0,1))$.\n\\par\nThe Weissenberg number \\cite{White:1964} is a parameter related to the memory of the fluid, i.e., for a viscoelastic material, the $Wi$ is a dimensionless number which can represent the relaxation time of the fluid. From a rheological point-of-view, the Weissenberg number can be interpreted as a number which can be used to measure the competition between elastic and viscous forces present in the concept of the viscoelasticity. A naive form to interprete the mathematical effect of this non-dimensional number is considering if $Wi=0$ in Eq. (1.2), and in this case, the stress, represented here by $\\zeta$, is given by an explicit relation with the strain-rate tensor $D(u)$. Otherwise, for $Wi\\neq 0$, the relation between the stress and the velocity gradient (rate-of-strain) can be modeled by a differential model, as for instance Eq. (1.2). Notice that increasing the value of the Weissenberg number in Eq. (1.2), the convected time derivative assumes a more significant effect in the equation, and therefore, the numerical treatment of this term needs to be improved in order to obtain a correct approximation of the solution. More details concerning the effect of the Weissenberg number on the partial differential equations whose describe viscoelastic fluid flows can be found in the works of Renardy \\cite{RenardyAnnual,RenardyBook}.\n\\par\nFrom a numerical point of view, in order to preserve the stability of the solutions, Eulerian frameworks for solving equation~\\eqref{eq:OB} need to apply a high-order spatial discretization for treating the convective terms in~\\eqref{der}. Generally, the methods for dealing with convection-dominant terms of the upper-convected time derivative are based on the explicit and implicit upwind methodologies~\\cite{Baba1981,Harten1984,Schlichting2017}. Considering explicit upwind strategies, many numerical approaches have been proposed in the literature for solving constitutive equations of viscoelastic models based on Eq.~\\eqref{eq:OB}, e.g. the Eulerian schemes using Finite-Element~(FE) \\cite{Castillo,Fortin,Hulsen,Sandri}, Finite-Volume~(FV) \\cite{Alves2003,Darwish,Oliveira,Pimenta}, Finite-Difference~(FD)~\\cite{Franca,Martins2015,Tome}, etc. It is worth to notice that the main drawback of the explicit upwind schemes is the severe time step limitations, and the application of implicit time integrators has been used for developing more robust frameworks \\cite{Breuss2006,Schlichting2017,Yee1996}, where a typical example is the so-called CFL condition. However, the construction of fully implicit upwind algorithms is complex resulting in general in high-cost computational schemes due to the solution of large systems. An additional drawback of implicit upwind schemes for solving convection-dominant problems is the excessive numerical diffusion. \n\\par\nIn a different framework, Lagrangian methods combined with the method of characteristics~\\cite{Benitez2012,BermejoGalanSaavedra2012,Douglas1982,NT-2016-M2AN,Sul-1988} for solving viscoelastic fluid flows have been proposed by~\\cite{Baranger1997,Basombrio1991,Hadj1990,Notsu2015,LMNT-Peterlin_Oseen_Part_I,LMNT-Peterlin_Oseen_Part_II,Machmoum2001}. In these schemes, the Eulerian discretization of the convective term in~\\eqref{der}, i.e., $(u\\cdot\\nabla) \\zeta$, is avoided by using a Lagrangian discretization of the material derivative, i.e., $\\partial\\zeta\/\\partial t + (u\\cdot\\nabla) \\zeta$, with the idea of the method of characteristics.\nThe idea is to consider the trajectory of a fluid particle and discretize the material derivative along the trajectory.\nSince it is natural from a physical viewpoint and such Lagrangian schemes have advantages, e.g., the symmetry of resulting coefficient matrices of the system of linear equations in the implicit framework, no artificial parameters and no need of the so-called CFL condition, they are useful for flow problems appearing in the field of scientific computing.\n\\par\nA different approach for avoiding numerical instabilities and to obtain accurate solutions of Eq.~\\eqref{eq:OB} is mathematically rooted on the concept of the generalized Lie derivatives~(GLD)~\\cite{Lee2006,LeeXuZhang2011,LeeBook2011} which modifies the definition of Eq.~\\eqref{der}. In particular, this elegant methodology was firstly presented by Lee and Xu~\\cite{Lee2006} (see also a similar idea proposed in~\\cite{Petera2002}). In that pioneer work, the authors reformulated Eq.~\\eqref{eq:OB} using some mathematical properties to define generalized Ricatti equations in terms of GLD. In summary, the upper-convected time derivative~\\eqref{der} was re-written using the concept of the transition matrix. This idea was adopted in the context of the finite element discretization in Lee et al. \\cite{LeeBook2011} to numerically solve the Poiseuille flow between two parallel plates around a cylinder while in~\\cite{Lee2006} the authors presented theoretical results concerning the discretized version of the formulation proposed in~\\cite{LeeBook2011}.\n\\par\nIn spite of the good stability properties observed in the numerical results and the sophisticated theoretical analysis of the works in~\\cite{Lee2006,LeeBook2011}, to the best knowledge of the authors, the application of the GLD for solving equations in the form of~\\eqref{eq:OB} is limited for finite element discretization resulting in schemes of (mainly) first-order in time.\nIn~\\cite{Lee2006}, two finite element schemes of second-order in time are presented based on the Crank--Nicolson or the Adams--Bashforth method along the trajectory of fluid particle. There are, however, no truncation error analysis of second-order in time and no numerical results yet, while numerical results by a GLD-based finite element scheme of first-order in time are given in~\\cite{LeeBook2011}.\nTherefore, main contributions of this work can be summarized as follows: i)~the combination of the GLD strategy with the method of characteristics to develop temporal second-order finite difference schemes for treating the upper-convected time derivative~\\eqref{der}, and ii)~the application of simple stable algorithms avoiding the need to solve large systems as commonly occur for implicit upwind schemes.\n\\par\nIn this paper, we present finite difference approximations of the upper-convected time derivative~\\eqref{der} based on GLD, and apply them to simple models.\nThe approximations are of second-order in time, where the truncation error of second-order in time is proved in Theorem~\\ref{thm:sec_order}, and a practical form is given in Corollary~\\ref{cor:sec_order}.\nTo the best knowledge of the authors, it is noted that the form, cf.~\\eqref{eq:sec_order_n}, in the corollary is new and that there are no proofs of truncation error of second-order in time for time-discretized approximations using GLD-approach.\nCombining the approximation with the (bi)linear~($p=1$) and (bi)quadratic~($p=2$) Lagrange interpolations, we present full discretizations of the upper-convective time derivative of second-order in time and $p$-th order in space, i.e., $O(\\Delta t^2+h^p)$, which are proved in Theorem~\\ref{prop:sec_order_p_order}.\nWe present two numerical schemes for simple models in $d$-dimensional spaces~($d=1,2$), cf.~\\eqref{scheme}, which are both explicit.\nThe difference of the schemes is the accuracy in space, i.e., one is of first-order~($p=1$) and the other is of second-order~($p=2$) in space as (bi)linear and (bi)quadratic Lagrange interpolation operators have been employed, respectively.\nAfter the presentation of the schemes, numerical experiments for simple models in $d$-dimensional spaces~($d=1,2$) are presented.\nThey are consistent with the theoretical accuracies shown in Theorem~\\ref{prop:sec_order_p_order}.\n\\par\nIn the case of Lagrangian finite element methods (often called Lagrange--Galerkin methods), a numerical integration is often employed in real computation for an integration of a composite function, since it is not easy to compute the integration of a composite function exactly.\nIn fact, a rough numerical integration may cause instability, cf.~\\cite{Tab-2007,TabFuj-2006}, where a robustness of a scheme of second-order in time with a choice of $\\Delta t$ depending on $h$ is discussed in the papers.\nOn the other hand, a quadrature-free scheme is proposed by using a mass-lumping technique in~\\cite{PirTab-2010}, and schemes with the exact integration of a composite function are proposed by introducing a linear interpolation of the velocity and implemented in two-dimensional numerical experiments in~\\cite{TabUch-2016-CD,Tabata2018}.\nIn these quadrature-free schemes, there is no discrepancy between the theory and real computation.\nBesides them, to the best of our knowledge, it is still a standard technique for the integration of a composite function to employ a high-order quadrature rule, cf., e.g., \\cite{BermejoSaavedra2012,ColeraCarpioBermejo2021,HNY-2016,NT-2015-JSC,NT-2016-M2AN}, whose computation cost depends mainly on the number of quadrature points.\nIn the end, we need to choose a suitable high-order quadrature rule by considering the computation cost and the \\emph{error} depending on the (expected) solution, $\\Delta t$, $h$ and so on.\nIn the case of Lagrangian finite difference method, however, there is no need to choose a quadrature rule as no integration is used. \nThis is an advantage of the Lagrangian finite difference method, cf~\\cite{NRT-2013}.\nThe GLD-type Lagrangian finite difference schemes which will be presented in this paper also have this advantage.\n\\par\nThe paper is organized as follows.\nIn Section~\\ref{sec:Preliminaries}, basic concepts for the for flow map and the upper-convected time derivative in the framework of the generalized Lie derivative and a simple model to be dealt in this paper are introduced.\nIn Section~\\ref{sec:FD_discretizations}, finite difference discretizations  of the upper-convected time derivative are presented, where truncation errors are proved.\nIn Section~\\ref{nm}, GLD-type numerical schemes of second-order in time and $p$-th order in space for the simple model and their algorithms are presented.\nIn Section~\\ref{numerics}, numerical results by our schemes are presented to see the experimental orders of convergence.\nIn Section~\\ref{sec:conclusions}, conclusions are given.\nIn Appendix, properties of GLD introduced in Section~\\ref{sec:Preliminaries} are proved, and the main algorithms of the work are described in details.\n\\section{Preliminaries}\n\\label{sec:Preliminaries}\nIn this section, we present some basic concepts concerning the flow map and the ideas of the generalized Lie derivatives. For these purposes, we need to consider some mathematical statements.\n\\par\nLet $\\Omega \\subset \\mathbb{R}^d~(d=1, 2, 3)$ be a bounded domain and $T$ be a positive constant.\nLet $u:\\Omega\\times (0,T) \\to \\mathbb{R}^d$ be a given velocity with the following hypothesis:\n\\begin{Hyp}\\label{hyp:u}\n\tThe velocity $u$ is sufficiently smooth and satisfies $u_{|\\partial\\Omega}=0$.\n\\end{Hyp}\nLet $\\Delta t>0$ be a time increment, $N_T \\coloneqq \\lfloor T\/\\Delta t \\rfloor$ the total number of time steps, and $t^n \\coloneqq n\\Delta t~(n\\in\\mathbb{Z})$.\nFor a function~$f$ defined in $\\Omega\\times (0,T)$, let $f^n \\coloneqq f(\\cdot,t^n)$ be the function at $n$-th time step.\nWe define two mappings $X_1, \\tilde{X}_1: \\Omega\\times (0,T)\\to \\mathbb{R}^d$ by\n\\[\nX_1(x,t)  \\coloneqq x-\\Delta t\\, u(x,t), \\qquad \\tilde{X}_1(x,t)  \\coloneqq x-2\\Delta t\\, u(x,t),\n\\]\nwhich are upwind points of $x$ with respect to $u(x,t)$.\n{We introduce a symbol ``$\\circ$'' to represent a composition of functions defined by}\n\\[\n(g \\circ X_1^n) (x) \\coloneqq g( X_1^n (x) ),\n\\]\nfor a function~$g$ defined in~$\\Omega$, where $X_1^n(x) = X_1(x,t^n) = x - \\Delta t\\,u^n(x)$.\nWe prepare a hypothesis for $\\Delta t$:\n\\begin{Hyp}\\label{hyp:dt}\n\tThe time increment $\\Delta t$ satisfies\n\t$\\displaystyle \\Delta t |u|_{C^0([0,T];W^{1,\\infty}(\\Omega)^d)} \\le 1\/8$.\n\\end{Hyp}\n\\begin{Rmk}\\label{rmk:upwind_points}\n\tHypotheses~\\ref{hyp:u} and~\\ref{hyp:dt} ensure that $X_1(\\Omega, t) = \\tilde{X}_1(\\Omega, t) = \\Omega$, and that {\\rm Jacobian}s of the mappings~$X_1(\\cdot,t)$ and $\\tilde{X}_1(\\cdot,t)$ are greater than or equal to $1\/2$, for $t\\in [0,T]$, cf.~\\cite{RuiTab-2002,Tabata2018}.\n\tWe note that Hypothesis~\\ref{hyp:dt} has no relation with the so-called CFL condition as any spatial mesh size is not included in it.\n\\end{Rmk}\n\\subsection{Lagrangian framework and the generalized Lie derivative}\n\\par\nFor a fixed $(x,t)\\in \\bar\\Omega\\times [0,T]$, let $X(x,t; s) \\in \\mathbb{R}^d$ be a solution of the following ordinary differential equation with an initial condition:\n\t%\n\t\\begin{subequations}\\label{eqns:ode}\n\t\t\\begin{align}\n\t\t\\prz{}{s} X(x,t; s) & = u(X(x,t; s), s),\\quad s \\in (0, T),\\label{eq:ode1}\\\\\n\t\tX(x,t; t) & = x, \\label{eq:ode2}\n\t\t\\end{align}\n\t\\end{subequations}\n\t%\n\tfor $(x,t) \\in \\Omega\\times (0,T)$.\n\tPhysically, $X(x,t; s)$ gives the position of fluid particle at time~$s$ whose position at time~$t$ is $x$. It is known as a flow map and an illustration of this concept can be seen in Fig.~\\ref{fig:flowmap}.\n\t%\n\t\\begin{figure}[!htbp]\n\t\t\\centering\n\t\n\t\t\\begin{tikzpicture}[thick,scale=3, every node\/.style={scale=1.7}]\n\t\n\t\n\t\t\\draw[thick,->] (-1.0,-1.0) -- (1.1,-1.0);\n\t\t\\draw[thick,->] (-1.0,-1.0) -- (-1.0,1.1);\n\t\t\n\t\n\t\t\\draw[dashed] (-1.0,-0.25) -- (1.0,-0.25);\n\t\t\\draw[dashed] (-1.0,0.5) -- (1.0,0.5);\n\t\t\n\t\t\\draw[blue]   (-0.75,-1.0) to[out=90,in=-100] (-0.25,1.0);\n\t\t\\draw[blue]   (-0.125,-1.0) to[out=90,in=-100] (0.375,1.0);\n\t\t\\draw[blue]   (0.5,-1.0) to[out=90,in=-100] (1.0,1.0);\n\t\t\n\t\n\t\t\\filldraw[fill] (-0.125,-1.0) circle (0.02cm)\n\t\t(-0.01,-0.25) circle (0.02cm)\n\t\t(0.25,0.5) circle (0.02cm)\n\t\t(-0.75,-1.0) circle (0.02cm) \n\t\t(-0.64,-0.25) circle (0.02cm)\n\t\t(-0.375,0.5) circle (0.02cm)\n\t\t;\n\t\t\n\t\n\t\t\\draw (-0.15,-1.2) node{$(x,t)$};\n\t\t\\draw (-0.75,-1.2) node{\\scriptsize $(\\tilde{x},t)$};\n\t\t\\draw (1.25,-1.2) node{$\\mathds{R}$};\n\t\t\\draw (-1.15,-1.0) node{$t$};\n\t\t\\draw (-1.15,-0.25) node{$s$};\n\t\t\\draw (-1.15,0.5) node{$\\tilde{t}$};\n\t\t\\draw (-1.25,1.2) node{$time$};\n\t\t\\draw (-0.6,-0.15) node{\\scriptsize $X(\\tilde{x},t;s)$};\n\t\t\\draw (0.425,-0.15) node{$X(x,t;s)$};\n\t\t\\draw (-0.35,0.6) node{\\scriptsize $X(\\tilde{x},t;\\tilde{t})$};\n\t\t\\draw (0.66,0.6) node{$X(x,t;\\tilde{t})$};\n\t\t\n\t\t%\n\t\t\\end{tikzpicture}\n\t\n\t\t\\caption{Sketch of the flow map for $X(x,t; s)$.}\n\t\t\\label{fig:flowmap}\n\t\\end{figure}\n\t%\n\t\\par\n\tFor $(x,t) \\in \\Omega\\times (0,T)$, let us introduce a matrix valued function~$L(x,t; \\cdot, \\cdot): (0,T)\\times (0,T) \\to \\mathbb{R}^{d\\times d}$ defined by\n\t\\begin{eqnarray}\\label{def:L}\n\tL_{ij} (x,t; t_1, t_2) \\coloneqq \\Bigl[ \\prz{}{z_j} X_i(z,t_1; t_2) \\Bigr]_{{\\displaystyle |}z=X(x,t; t_1)},\\quad i,j = 1, \\ldots, d,\n\t\\end{eqnarray}\n\t%\n\twhich is the so-called deformation gradient.\n\tIt is known that the function~$L$ has the following properties:\n\t%\n\t\\begin{subequations}\\label{eqns:L}\n\t\t\\begin{align}\n\t\tL(x,t;t_1,t_2) L(x,t;t_2,t_1) & = L(x,t;t_1,t_1) = I,\n\t\t\\label{eqns:L1} \\\\\n\t\t\\prz{}{s} L(x,t;t_1,s) & = (\\nabla u) \\bigl( X(x,t; s), s \\bigr) L(x,t;t_1,s), \n\t\t\\label{eqns:L2} \\\\\n\t\t\\prz{}{s} L(x,t;s,t_1) & = - L(x,t;s,t_1) (\\nabla u) \\bigl( X(x,t; s), s \\bigr), \n\t\t\\label{eqns:L3}\n\t\t\\end{align}\n\t\\end{subequations}\n\t%\n\tfor $t_1, t_2 \\in [0, T]$, where $I \\in \\mathbb{R}^{d\\times d}_{\\rm sym}$ is the identity matrix.\n\tAlthough the proofs can be found in, e.g., \\cite{LeeBook2011}, we give the proofs again in Appendix~\\ref{A.subsec:L} under the assumption of unique existence of smooth regular~$L$.\n\t%\n\t\\par\n\tLet $D\/Dt$ be the material derivation defined by\n\t\\[\n\t\\frac{D}{Dt} \\coloneqq \\prz{}{t} + u\\cdot\\nabla.\n\t\\]\n\tFor a function~$\\zeta:\\Omega\\times(0,T)\\to\\mathbb{R}^{d\\times d}$, it is well-known that the material derivative of~$\\zeta$ can be written as\n\t%\n\t\\begin{equation}\n\t\\frac{D\\zeta}{Dt} (x,t) = \\Bigl[ \\prz{\\zeta}{t} + (u\\cdot\\nabla) \\zeta \\Bigr] (x,t) = \\frac{\\partial}{\\partial s} \\zeta \\bigl( X(x,t; s), s \\bigr)_{{\\displaystyle |}s=t}.\n\t\\end{equation}\n\t%\n\tHere, we define the so-called generalized Lie derivative~$\\mathcal{L}_u\\zeta$ by\n\t%\n\t\\begin{align}\n\t(\\mathcal{L}_u\\zeta) \\bigl( X(x,t; s), s \\bigr) \n\t& \\coloneqq\n\tL(x,t;t,s) \\prz{}{s}\\Bigl[ L(x,t;s,t) \\zeta \\bigl( X(x,t; s), s \\bigr) L(x,t;s,t)^\\top \\Bigr] L(x,t;t,s)^\\top.\n\t\\label{def:GLD}\n\t\\end{align}\n\t%\n\tFrom~\\eqref{eqns:L}, the upper-convected time derivative can be rewritten by using $\\mathcal{L}_u\\zeta$, i.e.,\n\t%\n\t\\begin{equation}\n\t\\uctd{\\zeta}(x,t) \n\t= (\\mathcal{L}_u\\zeta) (x,t) = (\\mathcal{L}_u\\zeta)\\bigl( X(x,t; s), s \\bigr)_{|s=t},\n\t\\label{eq:UCM_GLD}\n\t\\end{equation}\n\t%\n\twhich is shown in Appendix~\\ref{A.subsec:UCM_GLD}. \n\\subsection{The model equation}\n\\label{sce}\nBased on the above description, we consider a simplified model equation in order to present the application of finite difference schemes for dealing with the generalized Lie derivative. Particularly, based on the Oldroyd-B constitutive equation (\\ref{eq:OB}), the problem is to find $\\zeta: \\Omega\\times (0, T) \\to \\mathbb{R}^{d\\times d}_{\\rm sym}$ such that\n\t%\n\t\\begin{subequations}\\label{eqns:prob}\n\t\t\\begin{eqnarray}\n\t\t\\uctd{\\zeta} & = & F \\quad\\ \\mbox{in}\\ \\Omega\\times (0, T), \n\t\t\\label{eq:prob1}\\\\\n\t\t\\zeta & = & \\zeta_\\mathrm{in} \\quad \\mbox{on}\\ \\Gamma_\\mathrm{in}\\times (0, T), \n\t\t\\label{eq:prob2}\\\\\n\t\t\\zeta & = & \\zeta^0 \\quad\\ \\mbox{in}\\ \\Omega,\\ \\mbox{at}\\ t=0, \n\t\t\\end{eqnarray}\n\t\\end{subequations}\n\t%\n\twhere $\\Gamma_\\mathrm{in}$ is an inflow boundary defined by $\\Gamma_\\mathrm{in} \\coloneqq \\{ x\\in\\partial\\Omega;\\ u(x,t)\\cdot n(x) < 0 \\}$ for the outward unit normal vector~$n: \\partial\\Omega\\to\\mathbb{R}^d$, and $F: \\Omega\\times (0,T) \\to \\mathbb{R}^{d\\times d}_{\\rm sym}$, $\\zeta_\\mathrm{in}: \\Gamma_{\\rm in}\\times (0, T) \\to \\mathbb{R}^{d\\times d}_{\\rm sym}$ and $\\zeta^0: \\Omega \\to \\mathbb{R}^{d\\times d}_{\\rm sym}$ are given functions.\n\t\\begin{Rmk}\n\t\t(i)~From~\\eqref{eq:UCM_GLD}, Eq.~\\eqref{eq:prob1} can be reformulated using the generalized {\\rm Lie} derivative resulting in:\n\t\\begin{equation}\n\t\\mathcal{L}_u\\zeta\n\t= F \\quad \\mbox{in}\\ \\Omega\\times (0, T).\n\t\\label{eq:prob1nf}\n\t\\end{equation}\n\t(ii)~In general, the inflow boundary~$\\Gamma_\\mathrm{in}$ depends on time~$t$, i.e., $\\Gamma_\\mathrm{in} = \\Gamma_\\mathrm{in}(t)$, while $\\Gamma_\\mathrm{in}$ is the empty set under Hypothesis~\\ref{hyp:u}.\n\tThroughout the paper, we deal with the inflow boundary~$\\Gamma_\\mathrm{in}$ independent of time~$t~(\\in (0,T))$.\n\t\\end{Rmk}\n\\section{Finite difference discretizations}\n\\label{sec:FD_discretizations}\nIn this section, we present descriptions concerning the spatial and temporal discretizations. The main results related to the numerical analysis of the schemes are also described in details.\n\\subsection{Space discretizations and interpolation operators}\nIn this subsection, we introduce spatial discretizations and interpolation operators in one- and two-dimensions.\nBefore starting them, for an integer~$i$ and a positive number~$\\delta$, we prepare two functions $\\eta_i^{(1)}(\\,\\cdot\\,; \\delta)$ and $\\eta_i^{(2)}(\\,\\cdot\\,; \\delta):~\\mathbb{R} \\to \\mathbb{R}$.\nThe former, $\\eta_i^{(1)}(\\,\\cdot\\,; \\delta)$, is defined by\n\\begin{align*}\n\\displaystyle\n\\eta_i^{(1)}(s; \\delta) & \\coloneqq\n\\left\\{\n\\begin{aligned}\n& \\frac{s}{\\delta} -i + 1 && \\bigl( s\\in [(i-1)\\delta, i\\delta ) \\bigr),\\\\\n& i+1 - \\frac{s}{\\delta} && \\bigl( s\\in [i\\delta, (i+1)\\delta] \\bigr),\\\\\n& \\ \\ \\ \\ \\   0 && \\bigl( {\\rm otherwise} \\bigr),\n\\end{aligned}\n\\right.\n\\end{align*}\nand the latter, $\\eta_i^{(2)}(\\,\\cdot\\,;\\delta)$, is defined by\n\\begin{align*}\n\\intertext{(i) $i$ : even number}\n\\displaystyle\n\\eta_i^{(2)}(s;\\delta) & \\coloneqq\n\\left\\{\n\\begin{aligned}\n& \\Bigl( \\frac{s}{\\delta} - i + 1 \\Bigr) \\Bigl( \\frac{s}{2\\delta} - \\frac{i}{2} + 1 \\Bigr) && \\bigl( s\\in [(i-2)\\delta, i\\delta ) \\bigr),\\\\\n&\\Bigl( i+1 - \\frac{s}{\\delta} \\Bigr) \\Bigl( \\frac{i}{2} + 1 - \\frac{s}{2\\delta} \\Bigr)  && \\bigl( s\\in [i\\delta, (i+2)\\delta] \\bigr),\\\\\n& \\qquad\\qquad\\ \\ \\  0 && \\bigl( {\\rm otherwise} \\bigr),\n\\end{aligned}\n\\right.\n\\intertext{(ii) $i$ : odd number}\n\\eta_i^{(2)}(s;\\delta) & \\coloneqq\n\\left\\{\n\\begin{aligned}\n& \\Bigl( \\frac{s}{\\delta} - i + 1\\Bigr) \\Bigl( i+1 - \\frac{s}{\\delta} \\Bigr) && \\bigl( s\\in [(i-1)\\delta, (i+1)\\delta] \\bigr),\\\\\n& \\qquad\\quad\\quad \\ \\ \\  0 && \\bigl( {\\rm otherwise} \\bigr).\n\\end{aligned}\n\\right.\n\\end{align*}\nThe functions~$\\eta_i^{(1)}(\\,\\cdot\\,; \\delta)$ and~$\\eta_i^{(2)}(\\,\\cdot\\,; \\delta)$ are used below for the definitions of (bi)linear and (bi)quadratic interpolation operators $\\Pi_h^{(1)}$ and $\\Pi_h^{(2)}$, respectively.\n\\subsubsection{One-dimensional case $(d=1)$}\nInitially, we consider one spatial dimension, i.e., $d = 1$.\nFor the sake of simplicity, we assume $\\Omega = (0, a)$ for a positive number~$a$. \nLet $N\\in\\mathbb{N}$ be a number, $h\\coloneqq a\/N$ a mesh size, and $x_i \\coloneqq ih~(i\\in\\mathbb{Z})$ lattice points.\nWe define a set of lattice points~$\\bar\\Omega_h$ and a discrete function space~$V_h$ restrict to the number $N$, by\n\\begin{align*}\n\\bar\\Omega_h & \n\\coloneqq \\{ x_i \\in \\bar\\Omega;\\ i = 0,\\ldots,N \\} \\ (\\subset \\bar\\Omega \\subset \\mathbb{R}^d = \\mathbb{R}), \\\\\nV_h & \\coloneqq \\{ v_h: \\bar\\Omega_h \\to \\mathbb{R}^{d\\times d}_{\\rm sym} \\} = \\{ v_h: \\bar\\Omega_h \\to \\mathbb{R} \\}.\n\\end{align*}\nWe introduce a set of basis functions~$\\{\\varphi_i^{(1)}: \\bar\\Omega \\to \\mathbb{R}; \\ i=0, \\ldots, N \\}$ defined by\n\\[\n\\varphi_i^{(1)}(x) \\coloneqq \\eta_i^{(1)}(x; h),\n\\qquad\ni=0,\\ldots,N.\n\\]\nThe functions~$\\varphi_0^{(1)}$ and $\\varphi_N^{(1)}$ are simplified to\n\\begin{align*}\n\\displaystyle\n\\varphi_0^{(1)}(x) & \\coloneqq\n\\left\\{\n\\begin{aligned}\n& 1 - \\frac{x}{h} && \\bigl( x\\in [x_0, x_1] \\bigr) \\\\\n& \\ \\ \\   0 && \\bigl( {\\rm otherwise} \\bigr)\n\\end{aligned}\n\\right.\n=\n\\left\\{\n\\begin{aligned}\n&\\frac{x_1-x}{h} && \\bigl( x\\in [ x_0, x_1] \\bigr) \\\\\n& \\ \\ \\ \\   0 && \\bigl( {\\rm otherwise} \\bigr)\n\\end{aligned}\n\\right.,\n\\\\\n\\varphi_N^{(1)}(x) & \\coloneqq\n\\left\\{\n\\begin{aligned}\n& \\frac{x}{h} - N + 1 && \\bigl( x\\in [x_{N-1}, x_N ] \\bigr) \\\\\n& \\ \\ \\ \\ \\ \\   0 && \\bigl( {\\rm otherwise} \\bigr)\n\\end{aligned}\n\\right.\n=\n\\left\\{\n\\begin{aligned}\n&\\frac{x-x_{N-1}}{h} && \\bigl( x\\in [ x_{N-1}, x_N ] \\bigr) \\\\\n& \\ \\ \\ \\ \\ \\   0 && \\bigl( {\\rm otherwise} \\bigr)\n\\end{aligned}\n\\right.,\n\\end{align*}\nas defined in $\\bar\\Omega = [x_0, x_N] = [0, a]$.\nLet $\\Pi_h^{(1)}: V_h \\to C^0(\\bar\\Omega)$ be the linear interpolation operator defined by\n\\[\n\\bigl( \\Pi_h^{(1)} v_h \\bigr) (x) \\coloneqq \\sum_{i=0}^N v_h(x_i)\\varphi_i^{(1)}(x).\n\\]\n\\par\nWe describe the ideas for using a quadratic interpolation. Let $N\\in\\mathbb{N}$ be an even number, and $M\\coloneqq N\/2\\in\\mathbb{N}$. For the definition of the quadratic interpolation operator $\\Pi_h^{(2)}$, we define a set of basis functions~$\\{\\varphi_i^{(2)}: \\bar\\Omega \\to \\mathbb{R}; \\ i=0, \\ldots, N \\}$ by\n\\[\n\\varphi_i^{(2)}(x) \\coloneqq \\eta_i^{(2)}(x; h),\n\\qquad\ni=0,\\ldots,N,\n\\]\nwhere $\\varphi_0^{(2)}$ and $\\varphi_N^{(2)} \\ (=\\varphi_{2M}^{(2)})$ are reduced to\n\\begin{align*}\n\\displaystyle\n\\varphi_0^{(2)}(x) & =\n\\left\\{\n\\begin{aligned}\n&\\frac{(x_1-x)(x_2-x)}{2h^2} && \\bigl( x\\in [x_0, x_2] \\bigr),\\\\\n& \\qquad\\quad \\  0 && \\bigl( {\\rm otherwise} \\bigr),\n\\end{aligned}\n\\right.\n\\\\\n\\varphi_N^{(2)}(x) & =\n\\left\\{\n\\begin{aligned}\n&\\frac{(x-x_{N-1})(x-x_{N-2})}{2h^2} && \\bigl( x\\in [x_{N-2}, x_N ] \\bigr),\\\\\n& \\qquad\\quad\\ \\ \\ \\ \\  0 && \\bigl( {\\rm otherwise} \\bigr).\n\\end{aligned}\n\\right.\n\\end{align*}\nLet $\\Pi_h^{(2)}: V_h \\to C^0(\\bar\\Omega)$ be the quadratic interpolation operator defined by\n\\[\n\\bigl( \\Pi_h^{(2)} v_h \\bigr) (x) \\coloneqq \\sum_{i=0}^N v_h(x_i)\\varphi_i^{(2)}(x).\n\\]\n\\begin{Rmk}\\label{rmk:upwind_cell_1d}\n\tFor $\\alpha$, $\\beta \\in \\mathbb{R}~(\\alpha<\\beta)$, and $N_0\\in\\mathbb{N}$ with $\\delta_0 = \\delta_0(\\alpha,\\beta,N_0) \\coloneqq (\\beta-\\alpha)\/N_0 > 0$, let $\\mathcal{I}(\\cdot; \\alpha, \\beta, N_0) : \\mathbb{R} \\to \\{0,\\ldots,N_0\\}$ be an integer-valued index indicator function defined by\n\t\\begin{align}\n\t\\label{def:index_func}\n\t\\mathcal{I} (s; \\alpha, \\beta, N_0) \\coloneqq\n\t\\left\\{\n\t\\begin{aligned}\n\t& \\left\\lfloor \\frac{s-\\alpha}{\\delta_0} \\right\\rfloor && \\bigl( s \\in (\\alpha,\\beta) \\bigr), \\\\\n\t& \\quad\\ \\ 0 && (s \\le \\alpha), \\\\\n\t& \\quad\\ N_0 && (s \\ge \\beta).\n\t\\end{aligned}\n\t\\right.\n\t\\end{align}\n\n\tWe note that the integer~$i_0 = \\mathcal{I}(s; \\alpha, \\beta, N_0)$ satisfies $i_0\\delta_0+\\alpha \\le s < (i_0+1)\\delta_0+\\alpha$ for $s\\in (\\alpha, \\beta)$, and that, for an even number~$N_0$ with $M_0 = N_0\/2 \\in\\mathbb{N}$, the integer~$k_0 = \\mathcal{I}(s; \\alpha, \\beta, M_0)$ satisfies $2k_0\\delta_0+\\alpha \\le s < 2(k_0+1)\\delta_0+\\alpha$ for $s\\in (\\alpha, \\beta)$ as $\\delta_0 = (\\beta - \\alpha)\/M_0 = 2 (\\beta - \\alpha)\/N_0$. \n\n\t\\par\n\tFor $d=1$, we introduce two notations of intervals,\n\t\\begin{align*}\n\t&&&& K_{i+1\/2}^{(1)} & \\coloneqq [x_i, x_{i+1}], & i & \\in\\{0,\\ldots,N-1\\}, &&&& \\\\\n\t&&&& K_{2k+1}^{(2)} & \\coloneqq [x_{2k}, x_{2k+2}], & k & \\in\\{0,\\ldots,M-1\\}, &&&&\n\t\\end{align*}\n\twhose measures are $h$ and $2h$, respectively.\n\tLet $x \\in \\mathbb{R}$ be given arbitrarily.\n\tThen, the following are practically useful in computation: \\smallskip\\\\\n\t$(i)$~Let $i_0 \\coloneqq \\mathcal{I}(x; 0, a, N)\\in\\{0,\\ldots,N\\}$.\n\tWhen $x\\in\\Omega$, the integer $i_0$ satisfies $x\\in K_{i_0+1\/2}^{(1)} = [x_{i_0}, x_{{i_0}+1}]$, and we have two-points representation of~$( \\Pi_h^{(1)} v_h) (x)$,\n\t%\n\t\\begin{equation}\n\t\\bigl( \\Pi_h^{(1)} v_h \\bigr) (x) = v_{i_0} \\varphi_{i_0}^{(1)}(x) + v_{i_0+1} \\varphi_{{i_0}+1}^{(1)}(x),\n\t\\label{int1}\n\t\\end{equation}\n\n\twhere we have used a notation~$v_i = v_h(x_i)$. \\smallskip\\\\\n\t%\n\t$(ii)$~Let $k_0 \\coloneqq \\mathcal{I}(x; 0, a, M)\\in\\{0,\\ldots,M\\}$. \n\tWhen $x\\in\\Omega$, the integer $k_0$ satisfies $x\\in K_{2k_0+1}^{(2)} = [x_{2k_0}, x_{2{k_0}+2}]$, and we have three-points representation of~$( \\Pi_h^{(2)} v_h) (x)$,\n\n\t\\begin{equation}\n\t\\bigl( \\Pi_h^{(2)} v_h \\bigr) (x) = v_{2k_0} \\varphi_{2k_0}^{(2)}(x) + v_{2k_0+1} \\varphi_{2k_0+1}^{(2)}(x) + v_{2k_0+2} \\varphi_{2k_0+2}^{(2)}(x),\n\t\\label{int2}\n\t\\end{equation}\n\t%\n\tfor $v_i = v_h(x_i)$. \\smallskip\\\\\n\t%\n\t$(iii)$~If the value~$(\\Pi_h^{(p)} v_h)(x)~(p=1,2)$ is needed for $x\\notin\\Omega$, we can employ, instead of it, the closest end value of $v_h$, i.e., $v_0 = v_h(0)~(x \\le 0)$ or $v_N = v_h(a)~(x \\ge a)$, while the value~$v_0$ or~$v_N$ should be given by using~$\\zeta_{\\rm in}$ as $x$ corresponds to an upwind point and $x\\notin\\bar\\Omega$ means the high possibility of existence of ``inflow'' boundary near $x$.\n\tThe function~$\\mathcal{I}(\\cdot;\\alpha,\\beta,N)$ is, therefore, also useful for $x\\notin\\Omega$ in the sense that $\\mathcal{I}(x;\\alpha,\\beta,N)$ provides the closest index of lattice point.\n\\end{Rmk}\n\\subsubsection{Two-dimensional case $(d=2)$}\n\nWe consider two spatial dimensions, i.e., $d=2$.\nFor the sake of simplicity, we assume $\\Omega = (0, a_1)\\times (0, a_2)$ for positive numbers~$a_1$ and $a_2$.\nLet $N_i \\in\\mathbb{N}~(i=1,2)$ be numbers, $h_i\\coloneqq a_i\/N_i~(i=1,2)$ mesh sizes in $x_i$-direction, $h_{\\min} \\coloneqq \\min\\{h_i;\\ i=1,\\ldots,d\\}$ and $h = h_{\\max} \\coloneqq \\max\\{h_i;\\ i=1,\\ldots,d\\}$ minimum and maximum mesh sizes, and $x_{i,j} \\coloneqq (ih_1,jh_2)^\\top~(i,j\\in\\mathbb{Z})$ lattice points.\nWe assume a family of meshes satisfying the next hypothesis:\n\\begin{Hyp}\\label{hyp:mesh}\nThere exist positive constants $h_0$, $\\gamma_1$ and $\\gamma_2$ such that\n\\[\nh \\in (0,h_0], \\quad \\mbox{and} \\quad \\gamma_1 \\le \\frac{h}{h_{\\min}} \\le \\gamma_2.\n\\]\n\\end{Hyp}\n\\begin{Rmk}\nThe hypothesis is set for $d=2$ essentially, as it always holds for $d=1$ with $\\gamma_1 = \\gamma_2 = 1$.\n\\end{Rmk}\n\\par\nWe define a set of lattice points~$\\bar\\Omega_h$ and a discrete function space~$V_h$ restrict to the numbers $N_i \\in\\mathbb{N}~(i=1,2)$, by\n\\begin{align*}\n\\bar\\Omega_h & \n\\coloneqq \\{ x_{i,j} \\in \\bar\\Omega;\\ i = 0,\\ldots,N_1,\\ j = 0,\\ldots,N_2 \\}, \\\\\nV_h &\\coloneqq \\{ v_h: \\bar\\Omega_h \\to \\mathbb{R}^{d\\times d}_{\\rm sym} \\},\n\\end{align*}\nwhere it is noted that $\\bar\\Omega_h \\subset \\bar\\Omega \\subset \\mathbb{R}^d~(= \\mathbb{R}^2)$.\nUsing~$\\eta_i^{(1)}(\\,\\cdot\\,; \\delta)$, we introduce a set of basis functions~$\\{\\varphi_{i,j}^{(1)}: \\bar\\Omega \\to \\mathbb{R}; \\ x_{i,j}\\in\\bar\\Omega_h,\\ i,j\\in\\mathbb{Z} \\}$ defined by\n\\begin{displaymath}\n\\displaystyle\n\\varphi_{i,j}^{(1)}(x) = \\varphi_{i,j}^{(1)}(x_1,x_2) \\coloneqq \n\\eta_i^{(1)}(x_1;h_1) \\eta_j^{(1)}(x_2;h_2).\n\\end{displaymath}\nLet $\\Pi_h^{(1)}: V_h \\to C^0(\\bar\\Omega)$ be the bilinear interpolation operator defined by\n\\[\n\\bigl( \\Pi_h^{(1)} v_h \\bigr) (x) \\coloneqq \\sum_{x_{i,j}\\in \\bar\\Omega_h} v_h(x_{i,j})\\varphi_{i,j}^{(1)}(x).\n\\]\n\\par\nThe extension of the above interpolation using the biquadratic interpolation strategy can be defined as follows. Let $N_1, N_2 \\in\\mathbb{N}$ be even numbers, and $M_i\\coloneqq N_i\/2\\in\\mathbb{N}$ for $i=1,2$.\nFor the definition of the biquadratic interpolation operator $\\Pi_h^{(2)}$, we introduce basis functions~$\\{\\varphi_{i,j}^{(2)}: \\bar\\Omega \\to \\mathbb{R}; \\ x_{i,j}\\in\\bar\\Omega_h \\}$ defined by\n\\[\n\\varphi_{i,j}^{(2)}(x) = \\varphi_{i,j}^{(2)}(x_1,x_2) \\coloneqq \\eta_i^{(2)}(x_1;h_1) \\eta_j^{(2)}(x_2;h_2).\n\\]\nLet $\\Pi_h^{(2)}: V_h \\to C^0(\\bar\\Omega)$ be the biquadratic interpolation operator defined by\n\\[\n\\bigl( \\Pi_h^{(2)} v_h \\bigr) (x) \\coloneqq \\sum_{x_{i,j}\\in\\bar\\Omega_h} v_h(x_{i,j})\\varphi_{i,j}^{(2)}(x).\n\\]\n\\begin{Rmk}\\label{rmk:upwind_cell_2d}\nFor $d=2$, we introduce two notations of boxes (cells),\n\\begin{align*}\nK_{i+1\/2,j+1\/2}^{(1)} & \\coloneqq [ih_1, (i+1)h_1] \\times [jh_2, (j+1) h_2], \n&\n(i,j) & \\in \\{0,\\ldots,N_1-1\\} \\times \\{0,\\ldots,N_2-1\\}, \\\\\nK_{2k+1,2l+1}^{(2)} & \\coloneqq [2kh_1, (2k+2)h_1] \\times [2l h_2, (2l+2)h_2], \n& (k,l) & \\in \\{0,\\ldots,M_1-1\\} \\times \\{0,\\ldots,M_2-1\\},\n\\end{align*}\nwhose measures are $h_1h_2$ and $4h_1h_2$, respectively.\nLet $x \\in \\mathbb{R}^2$ be given arbitrarily.\nThen, the following are practically useful in computation: \\smallskip\\\\\n$(i)$~Let $i_0 \\coloneqq \\mathcal{I}(x_1; 0, a_1, N_1)\\in\\{0,\\ldots,N_1\\}$ and $j_0 \\coloneqq \\mathcal{I}(x_2; 0, a_2, N_2)\\in\\{0,\\ldots,N_2\\}$.\nWhen $x\\in\\Omega$, the set of integers $(i_0,j_0)$ satisfies $x\\in K_{i_0+1\/2,j_0+1\/2}^{(1)} = [i_0h_1, ({i_0}+1)h_1] \\times [j_0h_2, ({j_0}+1)h_2]$, and we have four-points representation of~$( \\Pi_h^{(1)} v_h) (x)$,\n\\begin{align}\n\\label{int3}\n\\bigl( \\Pi_h^{(1)} v_h \\bigr) (x) = \n\\sum_{m,n=0,1} v_{i_0+m,j_0+n} \\, \\varphi_{i_0+m,j_0+n}^{(1)}(x),\n\\end{align}\nwhere we have used a simplified notation~$v_{i,j} = v_h(x_{i,j})$. \\smallskip\\\\\n$(ii)$~Let $k_0 \\coloneqq \\mathcal{I}(x_1; 0, a_1, M_1)\\in\\{0,\\ldots,M_1\\}$ and $l_0 \\coloneqq \\mathcal{I}(x_2; 0, a_2, M_2)\\in\\{0,\\ldots,M_2\\}$.\nWhen $x\\in\\Omega$, the integer $k_0$ satisfies $x\\in K_{2k_0+1}^{(2)} = [x_{2k_0}, x_{2{k_0}+2}]$, and we have nine-points representation of~$( \\Pi_h^{(2)} v_h) (x)$,\n\\begin{equation}\n\\label{int4}\n\t\\bigl( \\Pi_h^{(2)} v_h \\bigr) (x) = \\sum_{m,n=0,1,2} v_{2k_0+m,2l_0+n} \\varphi_{2k_0+m,2l_0+n}^{(2)}(x).\n\\end{equation}\n\t$(iii)$~If the value~$(\\Pi_h^{(p)} v_h)(x)~(p=1,2)$ is needed for $x\\notin\\Omega$, we can employ, instead of it, the closest end value of $v_h$, i.e., one of the values of $v_h(x_{i,j})~(x_{i,j} \\in \\bar\\Omega_h\\cap\\partial\\Omega)$, while the value should be given by using~$\\zeta_{\\rm in}$ as $x$ corresponds to an upwind point.\n\\end{Rmk}\n\\begin{Rmk}\\label{rmk:3D}\nWe omit the extension of the interpolation operators~$\\Pi_h^{(p)}~(p=1, 2)$ to the three-dimensional case, i.e., $d=3$, since it is naturally defined by introducing basis functions~$\\varphi_{i,j,k}^{(p)}(x) = \\varphi_{i,j,k}^{(p)}(x_1,x_2,x_3) \\coloneqq \n\\eta_i^{(p)}(x_1;h_1) \\eta_j^{(p)}(x_2;h_2) \\eta_k^{(p)}(x_3;h_3)$ for $p = 1, 2$ in a similar manner.\n\\end{Rmk}\n\\subsection{Time discretization: truncation error analysis}\\label{subsec:time_discretization}\nFor the velocity~$u$, let $L_1,\\ \\tilde{L}_1: \\Omega \\times (0,T) \\to \\mathbb{R}^{d\\times d}$ be matrices defined by\n\\begin{align}\nL_1 (x,t) \\coloneqq I + \\Delta t (\\nabla u)(x,t),\n\\quad\n\\tilde{L}_1 (x,t) \\coloneqq I + 2\\Delta t (\\nabla u)(x,t),\n\\label{defs:L1_tilde_L1}\n\\end{align}\nwhich are approximations of $L(x,t;\\, t-\\Delta t,t)$ and $L(x,t;\\, t-2\\Delta t,t)$, respectively, cf.~Lemma~\\ref{lem:L_with_k} below.\nNow, we present a theorem which provides an approximation of the upper-convected time derivative of second-order in time.\n\\begin{Thm}\\label{thm:sec_order}\n\tSuppose that Hypotheses~\\ref{hyp:u} and~\\ref{hyp:dt} hold true.\n\tLet $\\zeta: \\bar\\Omega\\times [0,T] \\to \\mathbb{R}^{d\\times d}$ be a sufficiently smooth function.\n\tThen, for any $x\\in\\bar\\Omega$ and $t\\in [2\\Delta t, T]$, we have\n\t%\n\t\\begin{align}\n\t\\uctd{\\zeta} (x,t) \n\t= \\frac{1}{2\\Delta t}\\Bigl[\n\t3\\zeta(x,t) \\notag\n\t-4 L_1 (x,t) \\zeta(X_1(x,t),t-\\Delta t) L_1 (x,t)^\\top \\\\\n\t+ \\tilde{L}_1 (x,t) \\zeta(\\tilde{X}_1(x,t),t-2\\Delta t) \\tilde{L}_1 (x,t)^\\top\n\t\\Bigr] + O(\\Delta t^2).\n\t\\label{eq:sec_order}\n\t\\end{align}\n\\end{Thm}\nWe give the proof of Theorem~\\ref{thm:sec_order} after giving a remark and preparing two lemmas.\n\\begin{Rmk}\n(i)~Let us consider $(x,t) \\in \\bar\\Omega\\times [2\\Delta t,T]$ as a fixed point and employ simple notations~$X = X(x,t;\\,\\cdot\\,)$ and~$L(\\,\\cdot\\,,\\,\\cdot\\,) = L(x,t;\\,\\cdot\\,,\\,\\cdot\\,)$.\nThen, an approximation of~$\\uctd{\\zeta} (x,t)$ of first-order in time is obtained as follows:\n\\begin{align*}\n\\uctd{\\zeta}(x,t) & = (\\mathcal{L}_u\\zeta)\\bigl( X(s), s \\bigr)_{{\\displaystyle |}s=t} \n\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad\n \\mbox{\\rm (by~\\eqref{eq:UCM_GLD})} \\\\\n& = L(t,s) \\prz{}{s}\\Bigl[ L(s,t) \\zeta \\bigl( X(s), s \\bigr) L(s,t)^\\top \\Bigr] L(t,s)^\\top_{{\\displaystyle |}s=t} \n\\qquad\\ \\, \\mbox{\\rm (by definition~\\eqref{def:GLD})} \\\\\n& = L(t,s) \\frac{1}{\\Delta t} \\Bigl[ L(s,t) \\zeta \\bigl( X(s), s \\bigr) L(s,t)^\\top \\\\\n& \\qquad - L(s-\\Delta t,t) \\zeta \\bigl( X(s-\\Delta t), s-\\Delta t \\bigr) L(s-\\Delta t,t)^\\top \\Bigr] L(t,s)^\\top_{{\\displaystyle |}s=t} + O(\\Delta t) \\\\\n& \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad\n \\mbox{\\rm (by the {\\rm Euler} method with respect to~$s$)} \\\\\n& = \\frac{1}{\\Delta t} \\Bigl[ \\zeta \\bigl( X(t), t \\bigr) - L(t-\\Delta t,t) \\zeta \\bigl( X(t-\\Delta t), t-\\Delta t \\bigr) L(t-\\Delta t,t)^\\top \\Bigr] + O(\\Delta t) \\\\\n& \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad\n \\mbox{\\rm (by substituting $t$ into $s$ and~\\eqref{eqns:L1})} \\\\\n& = \\frac{1}{\\Delta t} \\Bigl[ \\zeta (x,t) - L_1(x,t) \\zeta \\bigl( X_1(x,t), t-\\Delta t \\bigr) L_1(x,t)^\\top \\Bigr] + O(\\Delta t),\n\\end{align*}\nwhere the last equality holds true from the initial condition~\\eqref{eq:ode2} for $X$, i.e., $X(t)=x$, and the relations,\n\\begin{align*}\nL_1(x,t) &= L(t-\\Delta t,t)+O(\\Delta t^2), &\nX_1(x,t) &= X(t-\\Delta t) + O(\\Delta t^2),\n\\end{align*}\nwhich will be shown in Lemmas~\\ref{lem:L_with_k} and~\\ref{lem:phi_g_with_k} with $k=1$ below, respectively.\n\\smallskip\\\\\n(ii)~Theorem~\\ref{thm:sec_order} presents an approximation of~$\\uctd{\\zeta} (x,t)$ of second-order in time based on the two-step {\\rm Adams--Bashforth} method, i.e., for a smooth function $f: \\mathbb{R}\\to\\mathbb{R}$,\n\\[\nf^\\prime(t) = \\frac{d}{ds}f(s)_{{\\displaystyle |}s=t} = \\frac{1}{2\\Delta t}[ 3f(t) - 4f(t-\\Delta t) + f(t-2\\Delta t)] + O(\\Delta t^2),\n\\]\nin place of the {\\rm Euler} method in~(i).\n\\end{Rmk}\n\\begin{Lem}\\label{lem:L_with_k}\n\tSuppose that Hypotheses~\\ref{hyp:u} and~\\ref{hyp:dt} hold true.\n\tLet $k = 1$ or $2$ be fixed.\n\tThen, for any $x\\in\\bar\\Omega$ and $t\\in [k\\Delta t, T]$, we have\n\t%\n\t\\begin{equation}\n\tL(x,t;t-k\\Delta t, t) = I + k \\Delta t (\\nabla u)(x,t) + \\frac{(k\\Delta t)^2}{2} U(x,t) + O(\\Delta t^3),\n\t\\label{eq:L_with_k}\n\t\\end{equation}\n\t%\n\twhere $U: \\Omega\\times (0, T) \\to \\mathbb{R}^{d\\times d}$ is a function defined by\n\t%\n\t\\[\n\tU \\coloneqq (\\nabla u)^2 - \\frac{D (\\nabla u)}{Dt}.\n\t\\]\n\t%\n\\end{Lem}\n\\begin{proof}\n\tFrom the Taylor expansion, we have\n\t\\begin{align}\n\t& L(x,t; t-k\\Delta t, t) \n\t= L(x,t; s-k\\Delta t, t)_{|s=t} \\notag\\\\\n\t& = \\Bigl[ L(x,t; s, t) - k\\Delta t \\prz{}{s} L(x,t; s, t) + \\frac{(k\\Delta t)^2}{2} \\prz{^2}{s^2} L(x,t; s, t) \\Bigr]_{{\\displaystyle |}s=t} + O(\\Delta t^3) \\notag\\\\\n\t& = \\Bigl[ L(x,t; s, t) - k\\Delta t \\bigl[ - L(x,t; s, t) (\\nabla u)\\bigl( X(x,t; s), s \\bigr) \\bigr] + \\notag\\\\\n\t& \\quad + \\frac{(k\\Delta t)^2}{2} \\prz{}{s} \\bigl[ - L(x,t; s, t) (\\nabla u)\\bigl( X(x,t; s), s \\bigr) \\bigr] \\Bigr]_{{\\displaystyle |}s=t} + O(\\Delta t^3) \\qquad \\mbox{(by~\\eqref{eqns:L3})}\\notag\\\\\n\t& = I + k\\Delta t (\\nabla u) (x,t) \n\t- \\frac{(k\\Delta t)^2}{2} \\prz{}{s} \\bigl[ L(x,t; s, t) (\\nabla u)\\bigl( X(x,t; s), s \\bigr) \\bigr]_{{\\displaystyle |}s=t} + O(\\Delta t^3).\n\t\\label{eq:proof_lem_L_with_k_1}\n\t\\end{align}\n\tWe evaluate $\\prz{}{s} \\bigl[ L(x,t; s, t) (\\nabla u)\\bigl( X(x,t; s), s \\bigr) \\bigr]_{|s=t}$ as follows:\n\t%\n\t\\begin{align}\n\t& \\prz{}{s} \\bigl[ L(x,t; s, t) (\\nabla u)\\bigl( X(x,t; s), s \\bigr) \\bigr]_{{\\displaystyle |}s=t} \\notag\\\\\n\t& = \\Bigl[\n\t\\Bigl( \\prz{}{s} L(x,t; s, t) \\Bigr) (\\nabla u)\\bigl( X(x,t; s), s \\bigr) \n\t+ L(x,t; s, t) \\Bigl( \\prz{}{s} (\\nabla u)\\bigl( X(x,t; s), s \\bigr) \\Bigr)\n\t\\Bigr]_{{\\displaystyle |}s=t} \\notag\\\\\n\t& = \\Bigl[\n\t- L(x,t; s, t) (\\nabla u)^2 \\bigl( X(x,t; s), s \\bigr) \n\t+ L(x,t; s, t) \\frac{D (\\nabla u)}{Dt} \\bigl( X(x,t; s), s \\bigr)\n\t\\Bigr]_{{\\displaystyle |}s=t} \\notag\\\\\n\t& = \n\t- (\\nabla u)^2 (x,t)\n\t+ \\frac{D (\\nabla u)}{Dt} (x,t)\n\t= - U(x,t).\n\t\\label{eq:proof_lem_L_with_k_2}\n\t\\end{align}\n\t%\n\tCombining~\\eqref{eq:proof_lem_L_with_k_2} with~\\eqref{eq:proof_lem_L_with_k_1}, we obtain~\\eqref{eq:L_with_k}.\n\t%\n\\end{proof}\n\\begin{Lem}\\label{lem:phi_g_with_k}\n\tSuppose that Hypotheses~\\ref{hyp:u} and~\\ref{hyp:dt} hold true.\n\tLet $k = 1$ or $2$ be fixed.\n\tThen, for any $x\\in\\bar\\Omega$ and $t\\in [k\\Delta t, T]$, we have the following:\\\\\n\t%\n\t(i)~It holds that\n\t%\n\t\\[\n\tX(x,t; t-k\\Delta t) = x - k \\Delta t u(x,t) + \\frac{(k\\Delta t)^2}{2}\\frac{Du}{Dt} (x,t) + O(\\Delta t^3).\n\t\\]\n\t(ii)~Let $\\zeta: \\Omega\\times (0,T) \\to \\mathbb{R}^{d\\times d}$ be a sufficiently smooth function.\n\tIt holds that\n\t%\n\t\\[\n\t\\zeta \\bigl( X(x,t; t-k\\Delta t), t-k\\Delta t \\bigr)\n\t = \\zeta \\bigl( x - k \\Delta t u(x,t), t-k\\Delta t \\bigr) + \\frac{(k\\Delta t)^2}{2} Z(x,t) + O(\\Delta t^3),\n\t\\]\n\t%\n\twhere $Z: \\Omega\\times (0,T) \\to \\mathbb{R}^{d\\times d}$ is a function defined by\n\t%\n\t\\[\n\tZ \\coloneqq \\Bigl( \\frac{Du}{Dt} \\cdot \\nabla \\Bigr) \\zeta.\n\t\\]\n\t%\n\\end{Lem}\n\\begin{proof}\n\tWe prove~(i).\nRecalling that $X(x,t;s)$ is a solution to~\\eqref{eqns:ode} and noting that the following identity,\n\\[\nX(x,t; t-k\\Delta t) = x - \\int_{t-k\\Delta t}^t u\\bigl( X(x,t; s), s \\bigr) ds,\n\\]\nholds true, we have\n\t\\begin{align*}\n\t& X(x,t; t-k\\Delta t) - [ x - k\\Delta t u(x,t)]  \\\\\n\t& = x - \\int_{t-k\\Delta t}^t u\\bigl( X(x,t; s), s \\bigr) ds - \\Bigl[ x - \\int_{t-k\\Delta t}^t u\\bigl( X(x,t; t), t \\bigr) ds \\Bigr] \\\\\n\t& = \\int_{t-k\\Delta t}^t \\Bigl[ u \\bigl( X(x,t; t), t \\bigr) - u\\bigl( X(x,t; s), s \\bigr) \\Bigr] ds \n\t= \\int_{t-k\\Delta t}^t ds \\Bigl[ u \\bigl( X(x,t; s_1), s_1 \\bigr) \\Bigr]_{s_1=s}^t \\\\\n\t& = \\int_{t-k\\Delta t}^t ds \\int_s^t \\frac{Du}{Dt} \\bigl( X(x,t; s_1), s_1 \\bigr) ds_1 \n\t= \\int_{t-k\\Delta t}^t ds \\int_s^t \\Bigl( \\frac{Du}{Dt} (x,t) + O(\\Delta t) \\Bigr) ds_1 \\\\\n\t& = \\frac{(k\\Delta t)^2}{2} \\frac{Du}{Dt} (x,t) + O(\\Delta t^3),\n\t\\end{align*}\n\t%\n\twhich completes the proof of~(i).\n\t\\par\n\tWe prove~(ii).\n\tFrom (i) and the Taylor expansion, we have\n\t%\n\t\\begin{align*}\n\t& \\zeta \\bigl( X(x,t; t-k\\Delta t), t-k\\Delta t \\bigr) \\\\\n\t& = \\zeta \\biggl( x - k \\Delta t u(x,t) + \\frac{(k\\Delta t)^2}{2}\\frac{Du}{Dt} (x,t), t-k\\Delta t \\biggr) + O(\\Delta t^3) \\\\\n\t& = \\zeta \\bigl( x - k \\Delta t u(x,t), t-k\\Delta t \\bigr) + \\frac{(k\\Delta t)^2}{2} \\Bigl[ \\Bigl( \\frac{Du}{Dt} (x,t) \\cdot \\nabla \\Bigr) \\zeta \\Bigr] \\bigl( x - k \\Delta t u(x,t), t-k\\Delta t \\bigr) + O(\\Delta t^3) \\\\\n\t& = \\zeta \\bigl( x - k \\Delta t u(x,t), t-k\\Delta t \\bigr) + \\frac{(k\\Delta t)^2}{2} Z (x,t) + O(\\Delta t^3),\n\t\\end{align*}\n\t%\n\twhere we have used the relation,\n\t\\[\n\t\\Bigl[ \\Bigl( \\frac{Du}{Dt} (x,t) \\cdot \\nabla \\Bigr) \\zeta \\Bigr] \\bigl( x - k \\Delta t u(x,t), t-k\\Delta t \\bigr) = Z(x,t) + O(\\Delta t),\n\t\\]\n\tfor the last equality.\n\t%\n\\end{proof}\n\\begin{proof}[Proof of Theorem~\\ref{thm:sec_order}]\n\tIn the proof, we often employ simple notations, $L(\\cdot,\\cdot) = L(x,t;\\,\\cdot\\,, \\cdot\\,)$ and $X = X(x,t; \\,\\cdot\\,)$, if there is no confusion, since $(x,t)$ is considered as a fixed position in space and time.\n\tFrom the Adams--Bashforth method, i.e., for a smooth function~$g$ defined in $\\mathbb{R}$, $g^\\prime (s) = \\frac{1}{2\\Delta t} [ 3g(s)-4g(s-\\Delta t)+g(s-2\\Delta t) ] + O(\\Delta t^2)$, we have\n\t\\begin{align}\n\t\\uctd{\\zeta} (x,t) \n\t& = (\\mathcal{L}_u\\zeta) (x,t)\n\t= (\\mathcal{L}_u\\zeta)\\bigl( X(s), s \\bigr)_{|s=t} \n\t= L(t,s) \\prz{}{s}\\Bigl[ L(s,t) \\, \\zeta \\bigl( X(s), s \\bigr) L(s,t)^{ \\top } \\Bigr] L(t,s)^\\top {}_{{\\displaystyle |}s=t} \\notag\\\\\n\t& = L(t,s) \\frac{1}{2\\Delta t}\\Bigl[\n\t3 L(s,t) \\, \\zeta \\bigl( X(s), s \\bigr) L(s,t)^{ \\top } \n\t-4 L(s-\\Delta t,t) \\, \\zeta \\bigl( X(s-\\Delta t), s-\\Delta t \\bigr) L(s-\\Delta t,t)^{ \\top } \\notag\\\\\n\t& \\quad + L(s-2\\Delta t,t) \\, \\zeta \\bigl( X(s-2\\Delta t), s-2\\Delta t \\bigr) L(s-2\\Delta t,t)^{ \\top } \\Bigr] L(t,s)^{ \\top } {}_{{\\displaystyle |}s=t} + O(\\Delta t^2) \\notag\\\\\n\t& = \\frac{1}{2\\Delta t}\\Bigl[\n\t3 \\zeta \\bigl( X(x,t; s), s \\bigr) \n\t-4 L(t,s) L(s-\\Delta t,t) \\, \\zeta \\bigl( X(s-\\Delta t), s-\\Delta t \\bigr) L(s-\\Delta t,t)^{ \\top } L(t,s)^{ \\top }  \\notag\\\\\n\t& \\quad + L(t,s)L(s-2\\Delta t,t) \\, \\zeta \\bigl( X(s-2\\Delta t), s-2\\Delta t \\bigr) L(s-2\\Delta t,t)^{ \\top } L(t,s)^{ \\top }\n\t\\Bigr] {}_{{\\displaystyle |}s=t} + O(\\Delta t^2) \\quad \\mbox{(by \\eqref{eqns:L1})} \\notag\\\\\n\t& = \\frac{1}{2\\Delta t}\\Bigl[\n\t3 \\zeta (x,t) \n\t-4 L(t-\\Delta t,t) \\, \\zeta \\bigl( X(t-\\Delta t), t-\\Delta t \\bigr) L(t-\\Delta t,t)^{ \\top }  \\notag\\\\\n\t& \\quad + L(t-2\\Delta t,t) \\, \\zeta \\bigl( X(t-2\\Delta t), t-2\\Delta t \\bigr) L(t-2\\Delta t,t)^{ \\top }\n\t\\Bigr] + O(\\Delta t^2) \\quad \\mbox{(by \\eqref{eq:ode2} and \\eqref{eqns:L1})} \\notag\\\\\n\t& = \\frac{1}{2\\Delta t}\\biggl[\n\t3 \\zeta (x,t) -4 \\Bigl[ L_1 + \\frac{\\Delta t^2}{2} U \\Bigr] (x,t) \\, \\zeta \\bigl( X(t-\\Delta t), t-\\Delta t \\bigr) \\Bigl[ L_1 + \\frac{\\Delta t^2}{2} U \\Bigr]^{ \\top }(x,t)  \\notag\\\\\n\t& \\quad + \\bigl[ \\tilde{L}_1 + 2 \\Delta t^2 U \\bigr](x,t) \\, \\zeta \\bigl( X(t-2\\Delta t), t-2\\Delta t \\bigr) \\bigl[ \\tilde{L}_1 + 2 \\Delta t^2 U \\bigr]^{ \\top } (x,t)\n\t\\biggr] + O(\\Delta t^2) \\notag\\\\\n\t& \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\mbox{(by Lem.~\\ref{lem:L_with_k} with definitions of $L_1$ and $\\tilde{L}_1$)} \\notag\\\\\n\t& = \\frac{1}{2\\Delta t}\\Bigl[\n\t3 \\zeta (x,t) -4 L_1 (x,t) \\, \\zeta \\bigl( X(t-\\Delta t), t-\\Delta t \\bigr) L_1^\\top (x,t)  \\notag\\\\\n\t& \\quad + \\tilde{L}_1 (x,t) \\, \\zeta \\bigl( X(t-2\\Delta t), t-2\\Delta t \\bigr) \\tilde{L}_1^\\top (x,t) \\notag\\\\\n\t& \\quad - 2\\Delta t^2 \\bigl[ \\zeta \\bigl( X(t-\\Delta t), t-\\Delta t \\bigr) - \\zeta \\bigl( X(t-2\\Delta t), t-2\\Delta t \\bigr) \\bigr] U^{ \\top }(x,t) \\notag\\\\\n\t& \\quad - 2\\Delta t^2 U(x,t) \\bigl[ \\zeta \\bigl( X(t-\\Delta t), t-\\Delta t \\bigr) - \\zeta \\bigl( X(t-2\\Delta t), t-2\\Delta t \\bigr) \\bigr]\n\t\\Bigr] + O(\\Delta t^2) \\notag\\\\\n\t& = \\frac{1}{2\\Delta t}\\Bigl[\n\t3 \\zeta (x,t) - 4 L_1 (x,t) \\, \\zeta \\bigl( X(t-\\Delta t), t-\\Delta t \\bigr) L_1^\\top(x,t) \\notag\\\\\n\t& \\quad + \\tilde{L}_1(x,t) \\, \\zeta \\bigl( X(t-2\\Delta t), t-2\\Delta t \\bigr) \\tilde{L}_1^\\top (x,t)\n\t\\Bigr] + O(\\Delta t^2),\n\t\\label{eq:proof1}\n\t\\end{align}\n\t%\n\twhere the relation,\n\t\\[\n\t\\zeta \\bigl( X(t-\\Delta t), t-\\Delta t \\bigr) - \\zeta \\bigl( X(t-2\\Delta t), t-2\\Delta t \\bigr) = O(\\Delta t),\n\t\\]\n\thas been employed for the last equality.\n\tCombining~Lemma~\\ref{lem:phi_g_with_k}-(ii) with~\\eqref{eq:proof1} and recalling $x-\\Delta t u(x,t) = X_1(x,t)$ and $x-2\\Delta t u(x,t) = \\tilde{X}_1(x,t)$, we obtain\n\t%\n\t\\begin{align*}\n\t\\uctd{\\zeta} (x,t) = \\frac{1}{2\\Delta t}\\Bigl[\n\t3 \\zeta (x,t) \n\t-4 L_1 (x,t) \\zeta \\bigl( X_1(x,t), t-\\Delta t \\bigr) L_1^\\top(x,t) \\\\\n\t+ \\tilde{L}_1 (x,t) \\zeta \\bigl( \\tilde{X}_1(x,t), t-2\\Delta t \\bigr) \\tilde{L}_1^\\top (x,t)\n\t\\Bigr] + O(\\Delta t^2),\n\t\\end{align*}\n\t%\n\twhich completes the proof.\n\\end{proof}\nSubstituting $t^n$ into $t$ in~\\eqref{eq:sec_order}, the discrete form of second-order in time for the upper-convected time derivative is given as follows.\n\\begin{Cor}\\label{cor:sec_order}\nUnder the same assumptions of Theorem~\\ref{thm:sec_order}, we have\n\\begin{align}\n\t\\uctd{\\zeta} (x, t^n) \n\t= \\frac{1}{2\\Delta t}\\Bigl[\n\t3\\zeta^n(x) -4 L_1^n (x) \\bigl( \\zeta^{n-1}\\circ X_1^n \\bigr)(x) L_1^n (x)^\\top \n\t+ \\tilde{L}_1^n (x) \\bigl( \\zeta^{n-2}\\circ \\tilde{X}_1^n \\bigr)(x) \\tilde{L}_1^n (x)^\\top \\Bigr] + O(\\Delta t^2) \n\t\\label{eq:sec_order_n}\n\t\\end{align}\nfor $n = 2, \\ldots, N_T$.\n\\end{Cor}\n\\begin{Rmk}\nAlthough the approximation~\\eqref{eq:sec_order_n} of $\\uctd{\\zeta}(x,t^n)$ of second-order in time is combined with the finite difference method in this paper below, one can combine it with other methods, e.g., the finite element method and the finite volume method.\n\\end{Rmk}\n\\subsection{Full discretizations of the upper-convected time derivative}\nSuppose that $\\zeta \\in C([0,T];C(\\bar\\Omega;\\mathbb{R}^{d\\times d}_{\\rm sym}))$ and $\\zeta_h = \\{\\zeta_h^n\\}_{n=0}^{N_T}\\subset V_h$ are given.\nFor $n \\in \\{1, \\ldots, N_T\\}$ and $p\\in\\{1, 2\\}$, let $\\mathcal{A}^n \\zeta: \\bar\\Omega \\to \\mathbb{R}^{d\\times d}_{\\rm sym}$ and $\\mathcal{A}_h^{n,(p)} \\zeta_h: \\bar\\Omega_h \\to \\mathbb{R}^{d\\times d}_{\\rm sym}$ be functions defined by\n\\begin{align}\n\t[\\mathcal{A}^n \\zeta] (x)\n\t& \\coloneqq \n\t\\left\\{\n\t\\begin{aligned}\n\t\\frac{1}{2\\Delta t}\\Bigl[\n\t3\\zeta^n(x) \n\t-4 L_1^n (x) \\bigl( \\zeta^{n-1}\\circ X_1^n \\bigr)(x) L_1^n (x)^\\top \\qquad \\\\\n\t+ \\tilde{L}_1^n (x) \\bigl( \\zeta^{n-2}\\circ \\tilde{X}_1^n \\bigr)(x) \\tilde{L}_1^n (x)^\\top \\Bigr] \\quad (n \\ge 2), \\\\\n\t\\frac{1}{\\Delta t} \\Bigl[\n\t\\zeta^1(x) \n\t- L_1^1 (x) \\bigl( \\zeta^0 \\circ X_1^1 \\bigr) (x) L_1^1 (x)^\\top \\Bigr] \\quad (n = 1),\n\t\\end{aligned}\n\t\\right. \\notag \\\\\n\t\\label{def:An_h}\n\t[\\mathcal{A}_h^{n,(p)} \\zeta_h] (x) \n\t& \\coloneqq \\left\\{\n\t\\begin{aligned}\n\t\\frac{1}{2\\Delta t}  \\Bigl[\n\t3\\zeta_h^n (x)\n\t-4 L_1^n (x) \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta_h^{n-1}\\bigr) \\circ X_1^n \\bigr] (x) L_1^n (x)^\\top \\qquad \\\\\n\t+ \\tilde{L}_1^n (x) \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta_h^{n-2} \\bigr) \\circ \\tilde{X}_1^n \\bigr] (x) \\tilde{L}_1^n (x)^\\top\n\t\\Bigr] \\quad (n \\ge 2), \\\\\n\t\\frac{1}{\\Delta t} \\Bigl[\n\t\\zeta_h^1 (x)\n\t- L_1^1 (x) \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta_h^0 \\bigr) \\circ X_1^1 \\bigr] (x) L_1^1 (x)^\\top \\Bigr] \\quad (n = 1),\n\t\\end{aligned}\n\t\\right.\n\\end{align}\nrespectively.\nUsing the notation $\\mathcal{A}^n \\zeta$, we can write Eq.~\\eqref{eq:sec_order_n} as, for $n=\\{2,\\ldots,N_T\\}$,\n\\[\n\t\\uctd{\\zeta} (x, t^n) = [\\mathcal{A}^n \\zeta] (x) + O(\\Delta t^2).\n\\]\n\\par\nNow, we present a theorem on the truncation error of our finite difference approximations of the upper-convected time derivative, where the function~$\\mathcal{A}_h^{n,(p)} \\zeta: \\bar\\Omega_h \\to \\mathbb{R}^{d\\times d}_{\\rm sym}$ to be used in the theorem has meaning since $\\zeta \\in C([0,T];C(\\bar\\Omega;\\mathbb{R}^{d\\times d}_{\\rm sym}))$ can be considered as a series of functions in $V_h$, i.e., $\\zeta = \\{\\zeta^n\\}_{n=0}^{N_T} \\subset V_h$.\n\\begin{Thm}\\label{prop:sec_order_p_order}\n\tSuppose that Hypotheses~\\ref{hyp:u},~\\ref{hyp:dt} and~\\ref{hyp:mesh} hold true.\n\tLet $\\zeta: \\bar\\Omega\\times [0,T] \\to \\mathbb{R}^{d\\times d}$ be a sufficiently smooth function.\n\tThen, we have\n\t%\n\t\\begin{equation}\n\t\\label{eq:sec_order_p_order}\n\t\\uctd{\\zeta} (x, t^n) = [\\mathcal{A}_h^{n,(p)} \\zeta] (x) + O(\\Delta t^2 + h^p)\n\t\\end{equation}\n\tfor $x\\in\\bar\\Omega_h$, $n \\in \\{2, \\ldots, N_T\\}$ and $p=1, 2$.\n\\end{Thm}\n\\begin{proof}\nSince for $x\\in\\bar\\Omega_h$ we have\n\\begin{align}\n\t\\uctd{\\zeta} (x, t^n)\n\t& = [\\mathcal{A}^n \\zeta] (x) + O(\\Delta t^2)  \\notag\\\\\n\t& = [\\mathcal{A}_h^{n,(p)} \\zeta] (x) - \\Bigl( [\\mathcal{A}_h^{n,(p)} \\zeta] (x) - [\\mathcal{A}^n \\zeta] (x) \\Bigr) + O(\\Delta t^2) \\notag\\\\\n\t& = [\\mathcal{A}_h^{n,(p)} \\zeta] (x) + \\frac{2}{\\Delta t} L_1^n (x) \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta^{n-1} - \\zeta^{n-1}\\bigr) \\circ X_1^n \\bigr] (x) L_1^n (x)^\\top \\notag \\\\\n\t& \\quad - \\frac{1}{2\\Delta t} \\tilde{L}_1^n (x) \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta^{n-2} - \\zeta^{n-2} \\bigr) \\circ \\tilde{X}_1^n \\bigr] (x) \\tilde{L}_1^n (x)^\\top + O(\\Delta t^2)\n\t\\label{eq:prop2_proof0}\n\\end{align}\nfrom Corollary~\\ref{cor:sec_order}, it is enough for the proof to show the following estimates,\n\\begin{subequations}\n\\label{eq:prop2_proof1}\n\\begin{align}\n\\frac{2}{\\Delta t}\\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta^{n-1} - \\zeta^{n-1}\\bigr) \\circ X_1^n \\bigr] (x) & = O(h^p), \\\\\n\\frac{1}{2\\Delta t}\\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta^{n-2} - \\zeta^{n-2} \\bigr) \\circ \\tilde{X}_1^n \\bigr] (x)\n& = O(h^p),\n\\end{align}\n\\end{subequations}\nwhere simple estimates~\\eqref{eqns:prop2_proof1_easy} are easily obtained as shown in Remark~\\ref{rmk:prop2_proof1_easy_interpolation} later and the key issue is to eliminate the negative order in $\\Delta t$ from~\\eqref{eqns:prop2_proof1_easy} and get~\\eqref{eq:prop2_proof1}.\nWe prove the former equality of~\\eqref{eq:prop2_proof1} for $d=2$ only, as the equality for $d=1$ is simpler and the latter one is proved similarly.\nLet $x = x_{i,j} \\in \\bar\\Omega_h$ and $y^n \\coloneqq X_1^n(x) = x - u^n(x)\\Delta t$.\nTo simplify notations, we omit superscripts ${}^{n-1}$ and ${}^n$ from $\\zeta^{n-1}$ and $y^n$ in the rest of proof, respectively, if there is no confusion.\n\\par\nLet us start with $p=1$.\nFrom Hypotheses~\\ref{hyp:u} and~\\ref{hyp:dt}, we have $y \\in\\bar\\Omega$ and there exists a pair of indexes~$(i_0, j_0)$ such that $y \\in K^{(1)}_{i_0+1\/2,j_0+1\/2} \\, (= [i_0h_1, (i_0+1)h_1] \\times [j_0h_2, (j_0+1)h_2])$.\nLet $\\Lambda^{(1)}(y)$ be a set of pairs of indexes of lattice points near $y$ defined by $\\Lambda^{(1)}(y) \\coloneqq \\{ (i_0,j_0), (i_0+1,j_0), (i_0,j_0+1), (i_0+1,j_0+1) \\}$.\nLet $a=(a_1,a_2)^{ \\top } \\coloneqq y-x_{i_0,j_0} = ((i-i_0)h_1-u^n_1(x_{i,j})\\Delta t, (j-j_0)h_2-u^n_2(x_{i,j})\\Delta t)$ and $\\tilde{a}=(\\tilde{a}_1,\\tilde{a}_2)^{ \\top } \\coloneqq x_{i_0+1,j_0+1}-y$.\nWithout loss of generality, we can assume that $u^n_k(x_{i,j}) \\ge 0~(k=1,2)$, $i_0 < i$, $j_0 < j$ and $a_k, \\tilde{a}_k \\ge 0~(k=1,2)$, cf. Fig.~\\ref{fig:a1a2}.\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=0.6\\linewidth]{a1a2}\n\\caption{Notations in the proof of Theorem~\\ref{prop:sec_order_p_order}}\n\\label{fig:a1a2}\n\\end{figure}\nThen, we have\n\\begin{align}\n& \\Bigl[ \\bigl( \\Pi_h^{(1)}\\zeta - \\zeta \\bigr) \\circ X_1^n \\Bigr] (x)\n= (\\Pi_h^{(1)}\\zeta) (y) - \\zeta (y) \\notag\\\\\n& = \\sum_{(k,l) \\in \\Lambda^{(1)}(y)} \\bigl[ \\zeta (x_{k,l}) - \\zeta (y) \\bigr] \\varphi^{(1)}_{k,l} (y) \n\\qquad \\mbox{(by~$\\sum_{(k,l)\\in\\Lambda^{(1)}(y)} \\varphi^{(1)}_{k,l} (y) = 1$)} \\notag\\\\\n& = \\sum_{(k,l) \\in \\Lambda^{(1)}(y)} \\Bigl[ \\zeta \\bigl( y + s(x_{k,l}-y) \\bigr) \\Bigr]_{s=0}^1 \\, \\varphi^{(1)}_{k,l} (y) \\notag\\\\\n& = \\sum_{(k,l) \\in \\Lambda^{(1)}(y)} \\int_0^1 \\Bigl( [ (x_{k,l}-y)\\cdot\\nabla ] \\zeta \\Bigr) \\bigl( y + s_1(x_{k,l}-y) \\bigr) ds_1 \\, \\varphi^{(1)}_{k,l} (y) \\notag\\\\\n\\label{eq:prop2_proof2}\n& = \\sum_{(k,l) \\in \\Lambda^{(1)}(y)} \\int_0^1ds_1\\int_0^{s_1} \\Bigl( [ (x_{k,l}-y)\\cdot\\nabla ]^2 \\zeta \\Bigr) \\bigl( y + s_2(x_{k,l}-y) \\bigr) ds_2 \\, \\varphi^{(1)}_{k,l} (y) \\\\\n& \\qquad\\qquad\\qquad\\qquad\\qquad \\mbox{(by~$\\sum_{(k,l) \\in \\Lambda^{(1)}(y)} ( [ (x_{k,l}-y)\\cdot\\nabla ] \\zeta ) ( y ) \\varphi^{(1)}_{k,l} (y) = 0$),} \\notag\n\\end{align}\nand, for $(k,l)=(i_0,j_0)$,\n\\begin{align}\n& \\biggl| \\int_0^1ds_1\\int_0^{s_1} \\Bigl( [ (x_{k,l}-y)\\cdot\\nabla ]^2 \\zeta \\Bigr) \\bigl( y + s_2(x_{k,l}-y) \\bigr) ds_2 \\, \\varphi^{(1)}_{k,l} (y) \\biggr| \n= \\biggl| \\int_0^1ds_1\\int_0^{s_1} \\bigl( [ a \\cdot\\nabla ]^2 \\zeta \\bigr) ( y - s_2 a ) ds_2 \\, \\frac{\\tilde{a}_1\\tilde{a}_2}{h_1h_2} \\biggr| \\notag\\\\\n& \\le c_1 (a_1+a_2)^2 \\|\\zeta^{n-1}\\|_{C^2(K^{(1)}_{i_0+1\/2,j_0+1\/2};\\mathbb{R}^{d\\times d}_{\\rm sym})} \\frac{\\tilde{a}_1\\tilde{a}_2}{h_1h_2} \n\\le c_1^\\prime (a_1\\tilde{a}_1+a_2\\tilde{a}_2) \\|\\zeta^{n-1}\\|_{C^2(\\bar\\Omega;\\mathbb{R}^{d\\times d}_{\\rm sym})} \\notag\\\\ \n& \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n\\mbox{(by $a_k, \\tilde{a}_k \\le h_k,\\ k=1, 2$, and Hyp.~\\ref{hyp:mesh})}\n\\label{eq:prop2_proof3}\n\\end{align}\nfor positive constants~$c_1$ and $c_1^\\prime$ independent of $h$ and $\\Delta t$.\n\\par\nWe evaluate $a_1\\tilde{a}_1$.\nLet $U^\\infty \\coloneqq \\|u\\|_{C([0,T];C(\\bar\\Omega;\\mathbb{R}^d))} = \\max\\{ |u_k(x,t)|; x \\in \\bar\\Omega, t \\in [0,T], k=1,2\\}$.\nFrom $y_1 = [x_{i,j} - u^n(x_{i,j})\\Delta t ]_1 \\in [i_0h_1, (i_0+1)h_1]$, it holds that\n\\begin{displaymath}\n (i-i_0-1)h_1 \\le u^n_1(x_{i,j})\\Delta t \\le (i-i_0)h_1.\n\\end{displaymath}\nIn the case of $i-i_0-1 \\in \\mathbb{N}$, from $h_1 \\le \\frac{u^n_1(x_{i,j})\\Delta t}{i-i_0-1} \\le U^\\infty\\Delta t$, we have $a_1\\tilde{a}_1 \\le h_1^2 \\le h_1 U^\\infty \\Delta t$.\nIn the case of $i-i_0-1 = 0$, from $a_1 \\le h_1$ and $\\tilde{a}_1 = u^n_1(x_{i,j})\\Delta t \\le U^\\infty\\Delta t$, we have $a_1\\tilde{a}_1 \\le h_1 U^\\infty \\Delta t$.\nHence, it holds that, for any case,\n\\[\na_1\\tilde{a}_1 \\le h_1 U^\\infty \\Delta t.\n\\]\nSince it holds that $a_2\\tilde{a}_2 \\le h_2 U^\\infty \\Delta t$ similarly, we obtain\n\\begin{equation}\n\\label{eq:prop2_proof4}\n a_1\\tilde{a}_1 + a_2\\tilde{a}_2 \\le 2 h U^\\infty \\Delta t,\n\\end{equation}\nwhere this estimate holds also for $(k,l)=(i_0+1,j_0), (i_0,j_0+1), (i_0+1,j_0+1)$ similarly.\nCombining~\\eqref{eq:prop2_proof3} and~\\eqref{eq:prop2_proof4} with~\\eqref{eq:prop2_proof2}, we have, for a positive constant~$c_2$ independent of $h$ and~$\\Delta t$,\n\\[\n \\frac{2}{\\Delta t}\\Bigl[ \\bigl( \\Pi_h^{(1)}\\zeta - \\zeta \\bigr) \\circ X_1^n \\Bigr] (x) \n  \\le c_2 U^\\infty h  \\|\\zeta\\|_{C([0,T];C^2(\\bar\\Omega;\\mathbb{R}^{d\\times d}_{\\rm sym}))} = O(h),\n\\]\nwhich implies the former equality in~\\eqref{eq:prop2_proof1} with $p=1$, and the latter is obtained similarly.\nThus, we get~\\eqref{eq:sec_order_p_order} with $p=1$.\n\\par\nIn the case of $p=2$, the result, i.e., \\eqref{eq:sec_order_p_order} with $p=2$, are obtained similarly by taking into account the next identity,\n\\begin{align*}\n\\Bigl[ \\bigl( \\Pi_h^{(2)}\\zeta - \\zeta \\bigr) \\circ X_1^n \\Bigr] (x) \n= \\sum_{(k,l) \\in \\Lambda^{(2)}(y)} \\int_0^1ds_1 \\int_0^{s_1}ds_2 \\int_0^{s_2} \\Bigl( [ (x_{k,l}-y)\\cdot\\nabla ]^3 \\zeta \\Bigr) \\bigl( y + s_3(x_{k,l}-y) \\bigr) ds_3 \\, \\varphi^{(2)}_{k,l} (y),\n\\end{align*}\nwhere $\\Lambda^{(2)}(y) \\coloneqq \\{ (2i_{\\ast}+p,2j_{\\ast}+q);\\ p, q = 0, 1, 2\\}$ for $i_{\\ast}\\in \\{ 0, \\ldots, M_1\\}$ and $j_{\\ast} \\in \\{ 0, \\ldots, M_2\\}$ satisfying $y\\in [2i_{\\ast}h_1, 2(i_{\\ast}+1)h_1] \\times [2j_{\\ast}h_2, 2(j_{\\ast}+1)h_2]$.\n\\end{proof}\n\\begin{Rmk}\n\\label{rmk:prop2_proof1_easy_interpolation}\nIt is obvious that\n\\begin{align}\n\\uctd{\\zeta} (x, t^n) = [\\mathcal{A}_h^{n,(p)} \\zeta] (x) + O\\Bigl(\\Delta t^2 + \\frac{h^{p+1}}{\\Delta t}\\Bigr)\n\\label{eq:dt2_h_to_p_plus_1_over_dt}\n\\end{align}\nfor $x\\in\\bar\\Omega_h$, $n \\in \\{2, \\ldots, N_T\\}$ and $p=1, 2$, since $\\Pi_h^{(p)}\\zeta$ has an accuracy of $O(h^{p+1})$.\nIn fact, from the approximation property of $\\Pi_h^{(p)}\\zeta$, we have\n\\begin{subequations}\n\\label{eqns:prop2_proof1_easy}\n\\begin{align}\n\\frac{2}{\\Delta t}\\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta^{n-1} - \\zeta^{n-1}\\bigr) \\circ X_1^n \\bigr] (x) & = O \\Bigl( \\frac{h^{p+1}}{\\Delta t} \\Bigr), \\\\\n\\frac{1}{2\\Delta t}\\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta^{n-2} - \\zeta^{n-2} \\bigr) \\circ \\tilde{X}_1^n \\bigr] (x)\n& = O \\Bigl( \\frac{h^{p+1}}{\\Delta t} \\Bigr),\n\\end{align}\n\\end{subequations}\nand the relation~\\eqref{eq:dt2_h_to_p_plus_1_over_dt} is obtained by combining~\\eqref{eqns:prop2_proof1_easy} with~\\eqref{eq:prop2_proof0}.\nTheorem~\\ref{prop:sec_order_p_order} eliminates the negative order in $\\Delta t$ from~\\eqref{eq:dt2_h_to_p_plus_1_over_dt} and ensures that we can take small~$\\Delta t$ even for a fixed mesh size from a view point of accuracy.\n\\end{Rmk}\n\\section{Numerical schemes}\n\\label{nm}\nIn this section, we present finite difference schemes of second-order in time and of first- and second-order in space for problem~\\eqref{eqns:prob} by using the ideas of discretizations given in Section~\\ref{sec:FD_discretizations}.\n\\par\nSuppose that $u\\in C^0([0,T];C^1(\\bar\\Omega;\\mathbb{R}^d))$ and $\\zeta^0\\in C^0(\\bar\\Omega;\\mathbb{R}^{d\\times d}_{\\rm sym})$ are given, and that Hypotheses~\\ref{hyp:u}, \\ref{hyp:dt} and~\\ref{hyp:mesh} hold true.\nOur schemes are written in a unified form for $d=1, 2(, 3)$ and $p=1, 2$; find $\\{\\zeta_h^n \\in V_h;\\ n=1, \\ldots, N_T\\}$ such that\n\\begin{subequations}\\label{scheme}\n\t\\begin{align}\n\t\\label{scheme_An_h:general_step}\n\t&&&& [\\mathcal{A}_h^{n,(p)} \\zeta_h] (x) & = F^n(x), & x &\\in \\bar\\Omega_h, \\ n\\ge 1, &&&& \\\\\n\t\\label{scheme_An_h:initial_step}\n\t&&&& \\zeta_h^0(x) & = \\zeta^0(x), & x &\\in \\bar\\Omega_h, &&&&\n\t\\end{align}\n\\end{subequations}\nwhich are equivalent to\n\\begin{subequations}\\label{scheme_detail}\n\t\\begin{align}\n\t\\frac{1}{2\\Delta t}  \\Bigl[\n\t3\\zeta_h^n(x)\n\t-4 L_1^n (x)  \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta_h^{n-1}\\bigr) \\circ X_1^n \\bigr] (x) & L_1^n (x)^\\top \\notag\\\\\n\t+ \\tilde{L}_1^n (x) \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta_h^{n-2} \\bigr) \\circ \\tilde{X}_1^n \\bigr] (x) \\tilde{L}_1^n (x)^\\top\n\t\\Bigr] & = F^n(x), \n\t& x & \\in\\bar\\Omega_h, \\ n\\ge 2, \n\t\\label{scheme:general_step} \\\\\n\t%\n\t\\label{scheme:first_step}\n\t\\frac{1}{\\Delta t} \\Bigl[\n\t\\zeta_h^1(x)\n\t- L_1^1 (x)  \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta_h^0 \\bigr) \\circ X_1^1 \\bigr] (x) L_1^1 (x)^\\top \\Bigr] \n\t& = F^1(x), & x & \\in\\bar\\Omega_h, \\\\\n\t%\n\t\\label{scheme:initial_step}\n\t\\zeta_h^0(x) & = \\zeta^0(x), & x & \\in\\bar\\Omega_h.\n\t\\end{align}\n\\end{subequations}\nThe unified scheme~\\eqref{scheme} (equivalent to~\\eqref{scheme_detail}) includes four schemes, i.e., $p=1$ and $2$ correspond to schemes of first- and second-order in space, respectively, and the spatial dimension~$d~(= 1, 2)$ is implicitly dealt in the symbols~$\\bar\\Omega_h$ and~$V_h$.\nAn approximate initial value~$\\zeta_h^0 \\in V_h$ is given by~\\eqref{scheme:initial_step}.\nWe find $\\zeta_h^1\\in V_h$ from~\\eqref{scheme:first_step} and  $\\zeta_h^n\\in V_h$ for $n \\ge 2$ from~\\eqref{scheme:general_step}.\nHere, we additionally provide practical form of~\\eqref{scheme}:\n\\begin{subequations}\\label{scheme_explicit}\n\t\\begin{align}\n\t\\zeta_h^n(x) & =\n\t\\frac{4}{3} L_1^n (x) \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta_h^{n-1} \\bigr) \\circ X_1^n \\bigr](x) L_1^n (x)^\\top \\notag\\\\\n\t& \\quad \n\t- \\frac{1}{3} \\tilde{L}_1^n (x) \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta_h^{n-2} \\bigr) \\circ \\tilde{X}_1^n \\bigr] (x) \\tilde{L}_1^n (x)^\\top\n\t+ \\frac{2\\Delta t}{3} F^n(x), & x & \\in \\bar\\Omega_h, \\  n \\ge 2, \\label{scheme_explicit:general_step} \\\\\n\t%\n\t\\label{scheme_explicit:first_step}\n\t\\zeta_h^1(x) & =\n\tL_1^1 (x) \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta_h^0 \\bigr) \\circ X_1^1 \\bigr] (x) L_1^1 (x)^\\top + \\Delta t F^1(x), &\n\tx & \\in \\bar\\Omega_h, \\\\\n\t%\n\t\\label{scheme_explicit:initial_step}\n\t\\zeta^0_h(x) & = \\zeta^0(x), &\n\tx & \\in \\bar\\Omega_h,\n\t\\end{align}\n\\end{subequations}\nwhich imply that scheme~\\eqref{scheme} is explicit.\n\\begin{Rmk}\nFrom Hypotheses~\\ref{hyp:u} and~\\ref{hyp:dt} and Remark~\\ref{rmk:upwind_points}, we have $\\Gamma_{\\rm in} = \\emptyset$ and $X_1(\\Omega,t)=\\tilde{X}_1(\\Omega,t)=\\Omega~(t\\in [0, T])$, i.e., all of upwind points are in $\\bar\\Omega$.\nHence, the functions~$(\\Pi_h^{(p)}\\zeta_h^{n-1}) \\circ X_1^n~(n \\ge 1)$ and $(\\Pi_h^{(p)}\\zeta_h^{n-2}) \\circ \\tilde{X}_1^n~(n \\ge 2)$ are well defined in $\\bar\\Omega$ for $p=1,2$.\n\\end{Rmk}\n\\begin{Rmk}\nIn scheme~\\eqref{scheme}, we employ the backward {\\rm Euler} method~\\eqref{scheme:first_step} of first-order in time once to find $\\zeta_h^1$ needed in~\\eqref{scheme:general_step} with $n=2$. It is expected that there is no influence on the second-order convergence in time, cf. \\cite{Notsu2016}.\n\\end{Rmk}\n\\begin{Rmk}\\label{rmk:symmetry}\nSuppose that Hypotheses~\\ref{hyp:u} and~\\ref{hyp:dt} hold true.\nThen, under $F \\in C(\\bar\\Omega\\times [0,T]; \\mathbb{R}^{d\\times d}_{\\rm sym})$ and $\\zeta^0 \\in C(\\bar\\Omega; \\mathbb{R}^{d\\times d}_{\\rm sym})$, the scheme~\\eqref{scheme} preserves the symmetry, i.e., $\\zeta_h^n(x)^{ \\top } = \\zeta_h^n(x)~(x\\in\\bar\\Omega_h,\\ n=0,\\ldots,N_T)$ from the following.\nFor $d=1$, it is obvious, and let us consider $d = 2(, 3)$.\n$\\zeta_h^0(x)~(x\\in\\bar\\Omega_h)$ is symmetric from the symmetry of~$\\zeta^0$.\nWe show the symmetry of $\\zeta^1_h(x)~(x\\in\\bar\\Omega_h)$.\nNoting~\\eqref{scheme_explicit:first_step} and letting $A(x) = L_1^1(x)$, $B(x) = \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta_h^0 \\bigr) \\circ X_1^1 \\bigr] (x)$, and $C(x) = \\Delta t F^1(x)$, we have\n\t\\begin{align*}\n\t\\zeta^1_h(x)^{ \\top }\n\t& = \\bigl[ A(x) B(x) A(x)^{ \\top } + C(x) \\bigr]^{ \\top } \n\t= A(x) B(x)^{ \\top } A(x)^{ \\top } + C(x)^{ \\top } \\\\\n\t& = A(x) B(x) A(x)^{ \\top } + C(x)\n\t= \\zeta^1_h(x),\n\t\\end{align*}\nwhich implies symmetry of~$\\zeta^1_h(x)$ for~$x\\in\\bar\\Omega_h$, where we have used the fact that $B(x)$ and $C(x)$ are symmetric for the second equality from the last.\nFor~$n\\ge 2$, the symmetry of~$\\zeta_h^n(x)$ is obtained similarly from~\\eqref{scheme:general_step}.\n\\end{Rmk}\n\\subsection{Schemes in one-dimensional space~$(d=1)$}\nIn this subsection, we rewrite the finite difference scheme~\\eqref{scheme} in a unified form for $d=1$ and $p=1, 2$.\nWe introduce simplified notations, $\\zeta^n_i \\coloneqq \\zeta_h^n(x_i)$, $u^n_i \\coloneqq u^n(x_i)$, $\\nabla u^n_i \\coloneqq (\\nabla u^n)(x_i)$, $F^n_i \\coloneqq F(x_{i},t^{n})$, $\\Lambda_\\Omega \\coloneqq \\{0,\\ldots,N\\}$, and $\\Lambda_T \\coloneqq \\{1,\\ldots,N_T\\}$.\nThe schemes are to find $\\{\\zeta^n_i \\in \\mathbb{R};\\ i \\in \\Lambda_\\Omega,\\ n \\in \\Lambda_T \\}$ such that\n\\begin{subequations}\\label{scheme_1d}\n\t\\begin{align}\n\t\\zeta^n_i & = \\frac{4}{3} ( 1 + \\Delta t \\nabla u^n_i )^2 \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta_h^{n-1}\\bigr) \\circ X_1^n \\bigr] (x_i) \n\t\\qquad\\qquad\\ \\notag\\\\\n\t& \\quad -\\frac{1}{3} ( 1 + 2\\Delta t \\nabla u^n_i )^2 \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta_h^{n-2} \\bigr) \\circ \\tilde{X}_1^n \\bigr] (x_i) + \\frac{2\\Delta t}{3} F^n_{i}, \n\t& i & \\in \\Lambda_\\Omega, \\ n \\ge 2, \n\t\\label{scheme_1d:general_step} \\\\\n\t%\n\t\\zeta^1_i & = ( 1 + \\Delta t \\nabla u^1_i )^2 \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta_h^0 \\bigr) \\circ X_1^1 \\bigr] (x_i) + \\Delta t F^1_i, & i &\\in \\Lambda_\\Omega, \n\t\\label{scheme_1d:first_step} \\\\\n\t%\n\t\\zeta^0_i & = \\zeta^0(x_i), & i & \\in \\Lambda_\\Omega.\n\t\\label{scheme_1d:initial_step}\n\t\\end{align}\n\\end{subequations}\n\\par\nWe give the algorithm as follows:\n\\smallskip\n\\par\n\\textbf{Algorithm~1~($d=1$).}\nSet $\\bar\\Omega_h = \\{ x_i \\in\\bar\\Omega;\\ i \\in \\Lambda_\\Omega \\}$ with $h=a\/N$, and $\\{ \\zeta^0_i;\\ i\\in\\Lambda_\\Omega\\}$ by~\\eqref{scheme_1d:initial_step} to get $\\zeta_h^0\\in V_h$, where $N$ is an even number and $M = N\/2$ for $p=2$.\n\t\\smallskip\\\\\n\tSet~$n=1$. \\smallskip\\\\\n\t\\quad\n\tFor each $i \\in \\Lambda_\\Omega$ do:\n\t\t\\begin{enumerate}\n\t\t\\item Compute $F^1_i$, $u^1_i$, $\\nabla u^1_i$, and $y^1_i \\coloneqq X_1^{1} (x_i) = x_{i} - \\Delta t\\,u^1_i$.\n\t\t\\item Compute $Z_i^{1,(p)} \\coloneqq [ ( \\Pi_h^{(p)}\\zeta_h^0) \\circ X_1^1 ] (x_i) = ( \\Pi_h^{(p)}\\zeta_h^0) (y_i^1)$ according to~\\eqref{int1} with $i_0 = \\mathcal{I}(y_i^1; 0, a, N)$ for $p=1$, or \\eqref{int2} with $k_0 = \\mathcal{I}(y_i^1; 0, a, M)$ for $p=2$.\n\t\t\\item Compute $\\zeta^1_i$ by~\\eqref{scheme_1d:first_step}, which is equivalent to\n\t\t\\[\n\t\t\\zeta^1_i = ( 1 + \\Delta t \\nabla u^1_i )^2 \\, Z_i^{1,(p)} + \\Delta t F^1_i.\n\t\t\\]\n\t\t\\end{enumerate}\n\t(Here, computation of $\\zeta_h^1 \\in V_h$ is completed.) \\\\\n\tSet~$n=2$. \\smallskip\\\\\n\t\tWhile $n \\le N_T$ do:\\\\\n\t\t\\quad\n\t\tFor each $i\\in \\Lambda_\\Omega$ do:\n\\begin{enumerate}\n\t\\item Compute $F^n_i$, $u^n_i$, $\\nabla u^n_i$, $y^n_i \\coloneqq X_1^{n} (x_i) = x_{i} - \\Delta t\\,u^n_i$, and $\\tilde{y}^n_i \\coloneqq \\tilde{X}_1^{n} (x_i) = x_{i} - 2\\Delta t\\,u^n_i$.\n\t\t\\item Compute $Z_i^{n,(p)} \\coloneqq [ ( \\Pi_h^{(p)}\\zeta_h^{n-1}) \\circ X_1^n ] (x_i) = ( \\Pi_h^{(p)}\\zeta_h^{n-1}) (y_i^n)$ according to~\\eqref{int1} with $i_0 = \\mathcal{I}(y_i^n; 0, a, N)$ for $p=1$, or \\eqref{int2} with $k_0 = \\mathcal{I}(y_i^n; 0, a, M)$ for $p=2$.\n\t\tSimilarly, compute $\\tilde{Z}_i^{n,(p)} \\coloneqq [ ( \\Pi_h^{(p)}\\zeta_h^{n-2}) \\circ \\tilde{X}_1^n ] (x_i) = ( \\Pi_h^{(p)}\\zeta_h^{n-2}) (\\tilde{y}_i^n)$.\n\t\t\\item Compute $\\zeta^n_i$ by~\\eqref{scheme_1d:general_step}, which is equivalent to\n\t\t\\[\n\t\t\\zeta^n_i = \\frac{4}{3}( 1 + \\Delta t \\nabla u^n_i )^2 \\, z_i^{n,(p)} - \\frac{1}{3}( 1 + 2\\Delta t \\nabla u^n_i )^2 \\, \\tilde{Z}_i^{n,(p)} + \\frac{2\\Delta t}{3} F^n_i.\n\t\t\\]\n\\end{enumerate}\t\t\n\t\\quad\n\t(Computation of $\\zeta_h^n \\in V_h$ is completed.) \\\\\n\t\\quad\n\tSet $n=n+1$.\n\\subsection{Schemes in two-dimensional space~($d=2$)}\\label{nm2}\nSimilarly to the previous subsection, we rewrite the unified finite difference scheme~\\eqref{scheme} for $d=2$ and $p=1,2$.\n\\par\nLet us introduce simplified notations, $\\zeta^n_{i,j} \\coloneqq \\zeta_h^n(x_{i,j})$, $u^n_{i,j} \\coloneqq u^n(x_{i,j})$, $\\nabla u^n_{i,j} \\coloneqq (\\nabla u^n)(x_{i,j})$, $F^n_{i,j} \\coloneqq F(x_{i,j},t^{n})$, and $\\Lambda_\\Omega \\coloneqq \\{(i,j);\\ i = 0,\\ldots,N_1,\\ j=0,\\ldots,N_2\\}$.\nThe schemes are to find $\\{\\zeta^n_{i,j} \\in \\mathbb{R}^{2\\times2}_{\\rm sym};\\ (i,j) \\in \\Lambda_\\Omega,\\ n \\in \\Lambda_T \\}$ such that\n\t\\begin{subequations}\\label{scheme_2d}\n\t\t\\begin{align}\n\t\t\\zeta_{i,j}^n & =\n\t\t\\frac{4}{3} \\bigl[I + \\Delta t (\\nabla u^n_{i,j}) \\bigr] \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta_h^{n-1} \\bigr) \\circ X_1^n \\bigr](x_{i,j}) \\bigl[I + \\Delta t (\\nabla u^n_{i,j}) \\bigr]^{ \\top } \\notag\\\\\n\t\t& \\quad - \\frac{1}{3} \\bigl[I + 2\\Delta t (\\nabla u^n_{i,j}) \\bigr] \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta_h^{n-2} \\bigr) \\circ \\tilde{X}_1^n \\bigr] (x_{i,j}) \\bigl[I + 2\\Delta t (\\nabla u^n_{i,j}) \\bigr]^{ \\top } + \\frac{2\\Delta t}{3} F^n_{i,j}, &\n\t\t(i,j) & \\in \\Lambda_\\Omega, \\ n\\ge 2, \n\t\t\\label{scheme_2d:general_step} \\\\\n\t\t%\n\t\t\\zeta_{i,j}^1 & =\n\t\t\\bigl[I + \\Delta t (\\nabla u^1_{i,j}) \\bigr] \\bigl[ \\bigl( \\Pi_h^{(p)}\\zeta_h^0 \\bigr) \\circ X_1^1 \\bigr] (x_{i,j}) \\bigl[I + \\Delta t (\\nabla u^1_{i,j}) \\bigr]^{ \\top } + \\Delta t F^1_{i,j}, & (i,j) & \\in\\Lambda_\\Omega, \\label{scheme_2d:first_step} \\\\\n\t\t%\n\t\t\\zeta^0_{i,j} & = \\zeta^0(x_{i,j}), & (i,j) & \\in \\Lambda_\\Omega.\n\t\t\\label{scheme_2d:initial_step}\n\t\t\\end{align}\n\t\\end{subequations}\n\t%\n\t%\n\t%\n\t%\n\t%\n\\par\nWe give an algorithm of schemes~\\eqref{scheme_2d} for $d=2$ and $p=1,2$, while the construction is analogous to Algorithm~1 for $d=1$.\n\t\\smallskip \\par\n\t\\textbf{Algorithm~2~($d=2$).}\nSet $\\bar\\Omega_h = \\{ x_{i,j} \\in\\bar\\Omega;\\ (i,j) \\in \\Lambda_\\Omega \\}$ with $h_i=a_i\/N_i~(i=1,2)$, and $\\{ \\zeta^0_{i,j};\\ (i,j)\\in\\Lambda_\\Omega\\}$ by~\\eqref{scheme_2d:initial_step} to get $\\zeta_h^0\\in V_h$, where $N_i~(i=1,2)$ are even numbers and $M_i = N_i\/2~(i=1,2)$ for $p=2$.\n\t\\smallskip\\\\\n\tSet~$n=1$. \\smallskip\\\\\n\t\\quad\n\tFor each $(i,j) \\in \\Lambda_\\Omega$ do:\n\t\t\\begin{enumerate}\n\t\t\\item Compute $F^1_{i,j}$, $u^1_{i,j}$, $\\nabla u^1_{i,j}$, and $y^1_{i,j} \\coloneqq X_1^{1} (x_{i,j}) = x_{i,j} - \\Delta t\\,u^1_{i,j}$.\n\t\t\\item Compute $Z_{i,j}^{1,(p)} \\coloneqq [ ( \\Pi_h^{(p)}\\zeta_h^0) \\circ X_1^1 ] (x_{i,j}) = ( \\Pi_h^{(p)}\\zeta_h^0) (y_{i,j}^1)$ according to~\\eqref{int3} with $i_0 = \\mathcal{I}((y_{i,j}^1)_1;\\ 0, a_1, N_1)$ and $j_0 = \\mathcal{I}((y_{i,j}^1)_2;\\ 0, a_2, N_2)$ for $p=1$, or \\eqref{int4} with $k_0 = \\mathcal{I}((y_{i,j}^1)_1;\\ 0, a_1, M_1)$ and $l_0 = \\mathcal{I}((y_{i,j}^1)_2;\\ 0, a_2, M_2)$ for $p=2$.\n\t\t\\item Compute $\\zeta^1_{i,j}$ by~\\eqref{scheme_2d:first_step}, which is equivalent to\n\t\t\\[\n\t\t\\zeta_{i,j}^1 =\n\t\t\\bigl[I + \\Delta t (\\nabla u^1_{i,j}) \\bigr] Z_{i,j}^{1,(p)} \\bigl[I + \\Delta t (\\nabla u^1_{i,j}) \\bigr]^{ \\top } + \\Delta t F^1_{i,j}.\n\t\t\\]\n\t\t\\end{enumerate}\n\t(Here, computation of $\\zeta_h^1 \\in V_h$ is completed.) \\\\\n\tSet~$n=2$. \\smallskip\\\\\n\t\tWhile $n \\le N_T$ do:\\\\\n\t\t\\quad\n\t\tFor each $(i,j) \\in \\Lambda_\\Omega$ do:\n\\begin{enumerate}\n\t\\item Compute $F^n_{i,j}$, $u^n_{i,j}$, $\\nabla u^n_{i,j}$, $y^n_{i,j} \\coloneqq X_1^{n} (x_{i,j}) = x_{i,j} - \\Delta t\\,u^n_{i,j}$, and $\\tilde{y}^n_{i,j} \\coloneqq \\tilde{X}_1^{n} (x_{i,j}) = x_{i,j} - 2\\Delta t\\,u^n_{i,j}$.\n\t\t\\item Compute $Z_{i,j}^{n,(p)} \\coloneqq [ ( \\Pi_h^{(p)}\\zeta_h^{n-1}) \\circ X_1^n ] (x_{i,j}) = ( \\Pi_h^{(p)}\\zeta_h^{n-1}) (y_{i,j}^n)$ according to~\\eqref{int3} with $i_0 = \\mathcal{I}((y_{i,j}^n)_1;\\ 0, a_1, N_1)$ and $j_0 = \\mathcal{I}((y_{i,j}^n)_2;\\ 0, a_2, N_2)$ for $p=1$, or \\eqref{int4} with $k_0 = \\mathcal{I}((y_{i,j}^n)_1; 0, a_1, M_1)$ and $l_0 = \\mathcal{I}((y_{i,j}^n)_2; 0, a_2, M_2)$ for $p=2$.\n\t\tSimilarly, compute $\\tilde{Z}_{i,j}^{n,(p)} \\coloneqq [ ( \\Pi_h^{(p)}\\zeta_h^{n-2}) \\circ \\tilde{X}_1^n ] (x_{i,j}) = ( \\Pi_h^{(p)}\\zeta_h^{n-2}) (\\tilde{y}_{i,j}^n)$.\n\t\t\\item Compute $\\zeta^n_{i,j}$ by~\\eqref{scheme_2d:general_step}, which is equivalent to\n\t\t\\begin{align*}\n\t\t\\zeta_{i,j}^n & =\n\t\t\\frac{4}{3} \\bigl[I + \\Delta t (\\nabla u^n_{i,j}) \\bigr] Z_{i,j}^{n,(p)} \\bigl[I + \\Delta t (\\nabla u^n_{i,j}) \\bigr]^\\top - \\frac{1}{3} \\bigl[I + 2\\Delta t (\\nabla u^n_{i,j}) \\bigr] \\tilde{Z}_{i,j}^{n,(p)} \\bigl[I + 2\\Delta t (\\nabla u^n_{i,j}) \\bigr]^{ \\top } + \\frac{2\\Delta t}{3} F^n_{i,j}.\n\t\t\\end{align*}\n\\end{enumerate}\t\t\n\t\\quad\n\t(Computation of $\\zeta_h^n \\in V_h$ is completed.) \\\\\n\t\\quad\n\tSet $n=n+1$.\n\\section{Numerical results}\n\\label{numerics}\nIn this section, numerical results for problems with manufactured solutions are presented to observe experimental convergence orders of proposed schemes.\nIn the following, we denote scheme~\\eqref{scheme} with $p=1$ and $p=2$ by (S1) and (S2), respectively.\nFrom Theorem~\\ref{prop:sec_order_p_order}, the expected orders of convergence are of $O(\\Delta t^2+h^p)$ for $p=1,2$.\nTo see the experimental orders of convergence, the efficient choices of $\\Delta t$ for (S1) and (S2) are respectively $\\Delta t = c\\sqrt{h}$ and $\\Delta t = c^\\prime h$, for positive constants~$c$ and $c^\\prime$.\nThe choices of $\\Delta t$ for (S1) and (S2) lead to an expected order of convergence of $O(\\Delta t^2)\\ (=O(h^p))$.\nIn the computations below, as mentioned in Remark~\\ref{rmk:upwind_cell_1d}-(iii) and Remark~\\ref{rmk:upwind_cell_2d}-(iii), we employ a value of $\\zeta_{\\rm in}$ at closest lattice point to an upwind point~$X_1^n(x)$ or $\\tilde{X}_1^n(x)$ for $x=x_{i}~(d=1)$ or $x_{i,j}~(d=2)$ when the upwind point is outside the domain, where the integer-valued index indicator function~$\\mathcal{I}$ given by~\\eqref{def:index_func} is used.\n\\par\nFor $\\psi_h:\\bar\\Omega_h\\to\\mathbb{R}$ and $\\phi_h = \\{\\phi_h^n:\\bar\\Omega_h\\to\\mathbb{R};\\ n=1,\\ldots,N_T \\}$, let $\\|\\cdot\\|_{\\ell^\\infty(\\bar\\Omega_h)}$ and $\\|\\cdot\\|_{\\ell^\\infty(\\ell^\\infty)}$ be norms defined by\n\\begin{align*}\n\\| \\psi_h \\|_{\\ell^\\infty(\\bar\\Omega_h)} & = \\| \\psi_h \\|_{\\ell^\\infty(\\bar\\Omega_h;\\mathbb{R})} \\coloneqq \\max \\bigl\\{ | \\psi_h(x) | ;\\ x\\in \\bar\\Omega_h \\bigr\\}, \\\\\n\\| \\phi_h \\|_{\\ell^\\infty(\\ell^\\infty)} & \\coloneqq \\max \\bigl\\{ \\| \\phi_h^n \\|_{\\ell^\\infty(\\bar{\\Omega}_h)};\\ n = 1,\\ldots, N_T \\bigr\\}.\n\\end{align*}\nLet $E_{ij} = E_{ij}(\\Delta t, h),\\ i,j=1,\\ldots,d,$ be errors between a numerical solution~$\\zeta_h = \\{ \\zeta_h^n \\}_{n=1}^{N_T} \\subset V_h$ and a corresponding exact solution~$\\zeta \\in C([0,T];C(\\bar\\Omega;\\mathbb{R}^{d\\times d}_{\\rm sym}))$ defined by\n\\[\n E_{ij}  = E_{ij}(\\Delta t, h) \\coloneqq \\bigl\\| [\\zeta_h]_{ij}-\\zeta_{ij} \\bigr\\|_{\\ell^\\infty(\\ell^\\infty)}, \\quad i, j = 1, \\ldots, d,\n\\]\nand $E_{11}$ is denoted by $E$ simply when $d=1$.\n\\begin{Rmk}\nTo solve the problems proposed in this Section, we are assuming a defined source term and a prescribed velocity field. In addition, we need to establish at least one initial condition and a wall condition where we can call the flow inlet. The initial condition $\\zeta_{h}^{0}(x)$ is directly derived from the exact solution $\\zeta_{\\rm exact}(x,0)$. The boundary condition is computed assuming a Dirichlet-type condition, i.e., we use the exact solution $\\zeta_{\\rm in} = \\zeta_{\\rm exact}^{n}(x_{0})$ at the first point of the boundary for a positive velocity field (if $u^n(x)<0$ then the inlet of the domain is located on the opposite side, making us consider $\\zeta_{\\rm exact}^{n}(x_{N})$). Therefore, when we have the case described in Fig.~\\ref{fig:BC1}, the interpolated point $X^n_1(x_0)$ at previous time is outside the domain; thus we have imposed the boundary condition $\\zeta_{\\rm in}^{n}(x)=\\zeta_{\\rm exact}^{n}(x)$. \n\n\tFor the opposite side of the domain as represented by Fig.~\\ref{fig:NeumannBC1}, we do not impose any wall conditions, since our method can also be used to update the value of unknown function $\\zeta_{\\rm in}^{n}(x_N)$ on the outflow wall. In addition, it is also possible to assume a Neumann boundary condition on this wall and then we apply the method until $x_{N-1}$ and update the last point as in an explicit scheme $\\zeta_{\\rm in}^{n}(x_N) = \\zeta_{\\rm in}^{n}(x_{N-1})$.\n\n\tMore details about the implementation of these strategies can be found in Appendix~\\ref{A.subsec:PseudoCode}.\t \n\n\n\n\n\t\t\\begin{figure}[htbp]\n\t\t\t\\centering\n\t\t\n\t\t\t\\begin{tikzpicture}[thick,scale=1.7, every node\/.style={scale=1.2}]\n\t\t\t\\draw[thick,->] (-1.0,-1.0) -- (1.25,-1.0);\n\t\t\t\\draw[thick,->] (-1.0,-1.0) -- (-1.0,1.1);\n\t\t\t\\draw[thick] (1.0,-1.05) -- (1.0,-0.95);\n\t\t\t\n\t\t\t\\draw (1.5,-0.5) node{$t^{n-1}$};\n\t\t\t\\draw (1.5,0.0) node{$t^{n}$};\n\t\t\t\\draw (-1.05,-1.25) node{$x_0=0$};\n\t\t\t\\draw (0.95,-1.25) node{$x_N=a$};\n\t\t\t\n\t\t\n\t\t\t\\draw[dashed] (-1.0,-0.5) -- (1.0,-0.5);\n\t\t\t\\draw[dashed] (-1.0,0.0) -- (1.0,0.0);\n\t\t\t\\draw[dashed,blue] (-1.5,-1.0) -- (-0.5,1.0);\n\t\t\t\\draw (0.3,1.0) node{$characteristic$};\n\t\t\n\t\t\t\\filldraw[fill] (-1.0,0.0) circle (0.02cm)\n\t\t\t(-1.0,-0.5) circle (0.02cm)\n\t\t\t(-1.25,-0.5) circle (0.02cm);\n\t\t\n\t\t\t\\draw (-0.7,0.2) node{$\\zeta^n_{\\rm in}$};\n\t\t\t\\draw (-0.7,-0.3) node{$\\zeta^{n-1}_{\\rm in}$};\n\t\t\t\\draw (-1.75,-0.4) node{$X^{n}_{1}(x_0)$};\n\t\t\t\n\t\t\t\\end{tikzpicture}\n\t\t\n\t\t\t\\caption{Sketch of the wall treatment for unknown boundary condition.}\n\t\t\t\\label{fig:BC1}\n\t\t\\end{figure}\n\n\t\t\\begin{figure}[htbp]\n\t\t\t\\centering\n\t\t\n\t\t\t\\begin{tikzpicture}[thick,scale=1.7, every node\/.style={scale=1.7}]\n\t\t\n\t\t\t\\draw[thick,->] (-1.0,-1.0) -- (1.25,-1.0);\n\t\t\t\\draw[thick,->] (-1.0,-1.1) -- (-1.0,1.1);\n\t\t\t\\draw[thick] (1.0,-1.1) -- (1.0,-0.9);\n\t\t\t\n\t\t\t\\draw (-1.25,1.0) node{$t$};\n\t\t\t\\draw (1.35,-1.25) node{$x$};\n\t\t\t\\draw (1.75,0.25) node{$\\zeta^n_{i}=\\zeta^n_{i-1}$};\n\t\t\t\\draw (0.45,0.25) node{$\\zeta^n_{i-1}$};\n\t\t\t\n\t\t\n\t\t\t\\draw[dashed] (-1.0,-0.5) -- (1.0,-0.5);\n\t\t\t\\draw[dashed] (-1.0,0.0) -- (1.0,0.0);\n\t\t\t\\draw[dashed,blue] (0.1,-1.0) -- (0.85,0.5);\n\t\t\t\\draw[dashed,blue] (0.5,-1.0) -- (1.0,0.0);\n\t\t\t\n\t\t\n\t\t\t\\draw (-1.0,-1.2) node{$0$};\n\t\t\t\\draw (0.9,-1.2) node{$x_N$};\n\t\t\n\t\t\n\t\t\t\\filldraw[fill] (0.35,-0.5) circle (0.02cm)\n\t\t\t(0.6,0.0) circle (0.02cm)\n\t\t\t(1.0,0.0) circle (0.02cm);\n\t\t\n\t\t\n\t\t\t\\draw (-1.75,0.1) node{$t^n$};\n\t\t\t\\draw (-1.5,-0.4) node{$t^{n-1}$};\n\t\t\t\n\t\t\t\\end{tikzpicture}\n\t\t\n\t\t\t\\caption{Sketch of the wall treatment for Neumann boundary condition.}\n\t\t\t\\label{fig:NeumannBC1}\n\t\t\\end{figure}\n\n\\end{Rmk}\n\\subsection{Examples in one-dimensional space~$(d=1)$}\nWe consider the next example in one-dimensional space.\n\\begin{Ex}[$d=1$]\\label{ex:1d}\nIn problem~\\eqref{eqns:prob}, let $d=1$, $\\Omega = (0,1)$ and $T=1$.\nWe consider three functions for the velocity:\n\\[\n (i)~u(x,t)  = t, \\qquad\n (ii)~u(x,t)  = x+t,  \\qquad\n (iii)~u(x,t)  = \\sin(x+t),\n\\]\nwhich imply $\\Gamma_{\\rm in} = \\{ 0 \\}~(t\\in (0,T])$.\nThe functions~$F$, $\\zeta_{\\rm in}$ and $\\zeta^0$ are given so that the exact solution is\n\\[\n \\zeta(x,t) = \\sin (x+t) + 2.\n\\]\n\\end{Ex}\n\\par\nWe solve Example~\\ref{ex:1d} by (S1) with $\\Delta t = c\\sqrt{h}$ for $c = 1\/50$ and (S2) with $\\Delta t = c^\\prime h$ for $c^\\prime = 1$, where the mesh is constructed for $h=1\/N$ with $N = 10, 20, 40, 80, 160$ and $320$, the constants $c$ and $c^\\prime$ are as larger as possible in order to numerically verify the convergence order of the temporal discretizations.\nTables~\\ref{table:ex1_s1_dt_sqrth} and \\ref{table:ex1_s2_dt_h} show the values of error $E$ and their slopes in~$\\Delta t$.\nAccording to the results in the tables,\twe can confirm that (S1) and (S2) are of second-order in $\\Delta t$ for the three cases of velocity, $(i)$, $(ii)$ and $(iii)$.\nThese results are consistent with the theoretical results in Theorem~\\ref{prop:sec_order_p_order}.\n\t\\begin{table}[!htbp]\n\t\t\\centering\n\t\t\\caption{Example~\\ref{ex:1d} by {\\rm (S1)} with $\\Delta t = c \\sqrt{h}$~$(c=1\/50)$: Values of~$E$ and their slopes in $\\Delta t$.}\n\t\t\\begin{tabular}{rrrcrrcrr}\n\t\t\t\\toprule\n\t\t\t& \\multicolumn{2}{c}{$(i)$} && \\multicolumn{2}{c}{$(ii)$} && \\multicolumn{2}{c}{$(iii)$} \\\\ \\cline{2-3}\\cline{5-6}\\cline{8-9}\n\t\t\t\\multicolumn{1}{c}{$N$} & \\multicolumn{1}{c}{$E$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E$} & \\multicolumn{1}{c}{Slope} \\\\\n\t\t\t\\hline\n\t\t\t$10$ & $1.54 \\times 10^{-2}$ & -- && $3.45 \\times 10^{-2}$ & -- && $2.11 \\times 10^{-2}$ & -- \\\\\n\t\t\t$20$ & $8.07 \\times 10^{-3}$ & $1.86$ && $1.83 \\times 10^{-2}$ & $1.87$ && $1.11 \\times 10^{-2}$ & $1.86$ \\\\\n\t\t\t$40$ & $4.15 \\times 10^{-3}$ & $1.92$ && $9.38 \\times 10^{-3}$ & $1.92$ && $5.69 \\times 10^{-3}$ & $1.92$ \\\\\n\t\t\t$80$ & $2.10 \\times 10^{-3}$ & $1.96$ && $4.75 \\times 10^{-3}$ & $1.96$ && $2.88 \\times 10^{-3}$ & $1.96$ \\\\\n\t\t\t$160$ & $1.06 \\times 10^{-3}$ & $1.98$ && $2.39 \\times 10^{-3}$ & $1.98$ && $1.45 \\times 10^{-3}$ & $1.98$ \\\\\n\t\t\t$320$ & $5.31 \\times 10^{-4}$ & $1.99$ && $1.13 \\times 10^{-3}$ & $2.16$ && $7.27 \\times 10^{-4}$ & $1.99$ \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\t\\label{table:ex1_s1_dt_sqrth}\n\t\\end{table}\n\t\\begin{table}[!htbp]\n\t\t\\centering\n\t\t\\caption{Example~\\ref{ex:1d} by {\\rm (S2)} with $\\Delta t = h$~$(c^\\prime = 1)$: Values of~$E$ and their slopes in $\\Delta t$.}\n\t\t\\begin{tabular}{rrrcrrcrr}\n\t\t\t\\toprule\n\t\t\t& \\multicolumn{2}{c}{$(i)$} && \\multicolumn{2}{c}{$(ii)$} && \\multicolumn{2}{c}{$(iii)$} \\\\ \\cline{2-3}\\cline{5-6}\\cline{8-9}\n\t\t\t\\multicolumn{1}{c}{$N$} & \\multicolumn{1}{c}{$E$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E$} & \\multicolumn{1}{c}{Slope} \\\\\n\t\t\t\\hline\n\t\t\t$10$ & $4.65 \\times 10^{-3}$ & -- && $8.05 \\times 10^{-2}$ & -- && $1.65 \\times 10^{-2}$ & -- \\\\\n\t\t\t$20$ & $1.11 \\times 10^{-3}$ & $2.07$ && $2.19 \\times 10^{-2}$ & $1.88$ && $5.45 \\times 10^{-3}$ & $1.60$ \\\\\n\t\t\t$40$ & $2.68 \\times 10^{-4}$ & $2.04$ && $5.63 \\times 10^{-3}$ & $1.96$ && $1.53 \\times 10^{-3}$ & $1.84$ \\\\\n\t\t\t$80$ & $6.59 \\times 10^{-5}$ & $2.03$ && $1.42 \\times 10^{-3}$ & $1.98$ && $4.02 \\times 10^{-4}$ & $1.93$ \\\\\n\t\t\t$160$ & $1.63 \\times 10^{-5}$ & $2.01$ && $3.58 \\times 10^{-4}$ & $1.99$ && $1.03 \\times 10^{-4}$ & $1.97$ \\\\\n\t\t\t$ 320 $ & $ 4.06 \\times 10^{-6}$ & $ 2.01 $ && $8.96 \\times 10^{-5}$ & $2.00 $ && $ 2.61 \\times 10^{-5}$ & $1.98 $ \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\t\\label{table:ex1_s2_dt_h}\n\t\\end{table}\t\n\nIn order to numerically verify that our methodology is stable for small time-steps, we have fixed a coarse mesh $h=1\/40$ and the finest mesh $h=1\/320$ simulating the reduction of the time-step as $\\Delta t(k)=\\frac{\\sqrt{h}\/50}{2^{k}}$ for the first-order scheme and as $\\Delta t(k)=\\frac{h}{2^{k}}$ for the second-order method. Results for (S1) in Table \\ref{tableX1} while in Table \\ref{tableX2} we have described the results for (S2). \n\t\nAccording to these tables, we can confirm that our methodologies, first- and second-order spatial discretization schemes, are unconditionally stable since the errors are decreasing as $\\Delta t$ is reduced. It is important to highlight that error for the smallest time-step in Table \\ref{tableX2} for $h=1\/40$ is approximately two order smaller than the error of the largest time-step, confirming the good stability property of the second-order scheme.\n\n\t\\begin{table}[ht!]\n\t\\centering\n\t\\caption{Example~\\ref{ex:1d} by {\\rm (S1)}: reducing the time-step as $\\Delta t(k)=\\frac{\\sqrt{h}\/50}{2^{k}}$.}\n\t\\label{tableX1}\n\t\\begin{tabular}{rrr}\n\t\t\\hline\n\t\t\\multicolumn{3}{c}{$h = 0.025$} \\\\ \\hline\n\t\t\\multicolumn{1}{c}{$k$} & \\multicolumn{1}{c}{$\\Delta t$} & \\multicolumn{1}{c}{$Error$}  \\\\\n\t\t\\hline\n\t\t$0$ & $3.16 \\times 10^{-3}$ & $1.18375 \\times 10^{-2}$   \\\\\n\t\t$1$ & $1.58 \\times 10^{-3}$ & $1.08387 \\times 10^{-2}$   \\\\\n\t\t$2$ & $7.91 \\times 10^{-4}$ & $1.03445 \\times 10^{-2}$   \\\\\n\t\t$3$ & $3.95 \\times 10^{-4}$ & $1.01025 \\times 10^{-2}$   \\\\\n\t\t$4$ & $1.98 \\times 10^{-4}$ & $9.98198 \\times 10^{-3}$   \\\\\n\t\t$5$ & $9.88 \\times 10^{-5}$ & $9.92182 \\times 10^{-3}$   \\\\\n\t\t$6$ & $4.94 \\times 10^{-5}$ & $9.89128 \\times 10^{-3}$   \\\\\n\t\n\t\n\t\t\\hline\n\t\t\\multicolumn{3}{c}{$h = 0.003125$} \\\\ \\hline\n\t\t\\multicolumn{1}{c}{$k$} & \\multicolumn{1}{c}{$\\Delta t$} & \\multicolumn{1}{c}{$Error$}  \\\\\n\t\t\\hline\n\t\t$0$ & $1.12 \\times 10^{-3}$ & $1.94633 \\times 10^{-3}$   \\\\\n\t\t$1$ & $5.59 \\times 10^{-4}$ & $1.57935 \\times 10^{-3}$   \\\\\n\t\t$2$ & $2.80 \\times 10^{-4}$ & $1.39709 \\times 10^{-3}$   \\\\\n\t\t$3$ & $1.40 \\times 10^{-4}$ & $1.30627 \\times 10^{-3}$   \\\\\n\t\t$4$ & $6.99 \\times 10^{-5}$ & $1.26087 \\times 10^{-3}$   \\\\\n\t\t$5$ & $3.49 \\times 10^{-5}$ & $1.23823 \\times 10^{-3}$   \\\\\n\t\t$6$ & $1.75 \\times 10^{-5}$ & $1.22691 \\times 10^{-3}$   \\\\\n\t\n\t\n\t\t\\hline\n\t\\end{tabular}\n\n\\end{table}\n\n\n\n\\begin{table}[ht!]\n\t\\centering\n\t\\caption{Example~\\ref{ex:1d} by {\\rm (S2)}: reducing the time-step as  $\\Delta t(k)=\\frac{h}{2^{k}}$.}\n\t\\label{tableX2}\n\t\\begin{tabular}{rrr}\n\t\t\\hline\n\t\t\\multicolumn{3}{c}{$h = 0.025$} \\\\ \\hline\n\t\t\\multicolumn{1}{c}{$k$} & \\multicolumn{1}{c}{$\\Delta t$} & \\multicolumn{1}{c}{$Error$}  \\\\\n\t\t\\hline\n\t\t$0$ & $2.50 \\times 10^{-2}$ & $5.63 \\times 10^{-3}$   \\\\\n\t\t$1$ & $1.25 \\times 10^{-2}$ & $1.50 \\times 10^{-3}$   \\\\\n\t\t$2$ & $6.25 \\times 10^{-3}$ & $4.30 \\times 10^{-4}$   \\\\\n\t\t$3$ & $3.13 \\times 10^{-3}$ & $1.58 \\times 10^{-4}$   \\\\\n\t\t$4$ & $1.56 \\times 10^{-3}$ & $8.97 \\times 10^{-5}$   \\\\\n\t\t$5$ & $7.81 \\times 10^{-4}$ & $7.27 \\times 10^{-5}$   \\\\\n\t\t$6$ & $3.91 \\times 10^{-4}$ & $6.84 \\times 10^{-5}$   \\\\\n\t\n\t\n\t\t\\hline\n\t\t\\multicolumn{3}{c}{$h = 0.003125$} \\\\ \\hline\n\t\t\\multicolumn{1}{c}{$k$} & \\multicolumn{1}{c}{$\\Delta t$} & \\multicolumn{1}{c}{$Error$}  \\\\\n\t\t\\hline\n\t\t$0$ & $3.13 \\times 10^{-3}$ & $8.96 \\times 10^{-5}$   \\\\\n\t\t$1$ & $1.56 \\times 10^{-3}$ & $2.34 \\times 10^{-5}$   \\\\\n\t\t$2$ & $7.81 \\times 10^{-4}$ & $6.64 \\times 10^{-6}$   \\\\\n\t\t$3$ & $3.91 \\times 10^{-4}$ & $2.41 \\times 10^{-6}$   \\\\\n\t\t$4$ & $1.95 \\times 10^{-4}$ & $1.36 \\times 10^{-6}$   \\\\\n\t\t$5$ & $9.77 \\times 10^{-5}$ & $1.10 \\times 10^{-6}$   \\\\\n\t\t$6$ & $4.88 \\times 10^{-5}$ & $1.03 \\times 10^{-6}$   \\\\\n\t\n\t\n\t\t\\hline\n\t\\end{tabular}\n\n\\end{table}\n\n\n\\subsection{Examples for the two-dimensional case~$(d=2)$}\nWe set the next example in two-dimensional space.\n\\begin{Ex}[$d=2$]\\label{ex:2d}\n\tIn problem~\\eqref{eqns:prob}, let $d=2$, $\\Omega = (0,1)^d$ and $T=1$.\n\tWe consider three functions for the velocity:\n\\begin{eqnarray*}\n & (i)~u(x,t) = (t,t)^{ \\top }, \\quad\n (ii)~u(x,t) = (x_1+t,x_2+t)^{ \\top }, \\\\\n & (iii)~u(x,t) = (\\sin(x_1+x_2+t), \\sin(x_1+x_2+t))^{ \\top },\n\\end{eqnarray*}\n\twhich imply $\\Gamma_\\mathrm{in} = \\{ (s,0)^{ \\top }\\in\\partial\\Omega;\\ s\\in [0, 1] \\} \\cup \\{ (0,s)^{ \\top }\\in\\partial\\Omega;\\ s\\in [0, 1] \\}$~$(t\\in (0,T])$.\n\tThe functions~$F$, $\\zeta_\\mathrm{in}$ and $\\zeta^0$ are given so that the exact solution is\n\t\\[\n\t \\zeta(x,t) = \n\t \\begin{bmatrix}\n\t  \\sin(x_1 + x_2 + t) + 2 & \\sin(x_1 + x_2 + t) \\\\\n\t  \\sin(x_1 + x_2 + t) & \\sin(x_1 + x_2 + t) + 2\n\t \\end{bmatrix}.\n\t\\]\n\\end{Ex}\n\\par\nWe solve Example~\\ref{ex:2d} by (S1) with $\\Delta t = c\\sqrt{h}$ for $c = 1\/20$ and (S2) with $\\Delta t = c^\\prime h$ for $c^\\prime = 1\/10$, where the mesh is constructed for $h_1=h_2=h=1\/N$, i.e., $N_1=N_2=N$, with $N = 10, 20, 40$ and $80$.\nTables~\\ref{table:ex2_s1_dt_sqrth} and \\ref{table:ex2_s2_dt_h} show the values of error $E_{11}$ and their slopes in~$\\Delta t$. Slope results for~$E_{12}$ and~$E_{22}$ adopting different velocity fields $(i)$, $(ii)$ and $(iii)$ are very similar to those obtained for $E_{11}$; thus they are omitted here in order to save space.\nWe can confirm that (S1) and (S2) are of second-order in $\\Delta t$ in two-dimensional space for the three cases of velocity, $(i)$, $(ii)$ and $(iii)$.\nThese results are consistent with the theoretical results in Theorem~\\ref{prop:sec_order_p_order}.\n\\begin{table}[!htbp]\n\t\\centering\n\t\\caption{{Example~\\ref{ex:2d} by {\\rm (S1)} with $\\Delta t = c\\sqrt{h}$~$(c=1\/20)$:} Values of~$E_{11}$ and their slopes in $\\Delta t$.}\n\t\\begin{tabular}{rrrcrrcrr}\n\t\\toprule\n\t& \\multicolumn{2}{c}{$(i)$} && \\multicolumn{2}{c}{$(ii)$} && \\multicolumn{2}{c}{$(iii)$} \\\\ \\cline{2-3}\\cline{5-6}\\cline{8-9}\n\t\\multicolumn{1}{c}{$N$} & \\multicolumn{1}{c}{$E_{11}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{11}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{11}$} & \\multicolumn{1}{c}{Slope} \\\\\n\t\\hline\n\t$10$ & $3.87 \\times 10^{-2}$ & -- && $3.84 \\times 10^{-2}$ & -- && $3.87 \\times 10^{-2}$ & -- \\\\\n\t$20$ & $1.98 \\times 10^{-2}$ & $1.94$ && $1.96 \\times 10^{-2}$ & $1.94$ && $1.98 \\times 10^{-2}$ & $1.94$  \\\\\n\t$40$ & $9.99 \\times 10^{-3}$ & $1.97$ && $9.94 \\times 10^{-3}$ & $1.97$ && $9.99 \\times 10^{-3}$ & $1.97$ \\\\\n\t$80$ & $5.03 \\times 10^{-3}$ & $1.98$ && $5.01 \\times 10^{-3}$ & $1.98$ && $5.03 \\times 10^{-3}$ & $1.98$    \\\\ \n\t\\bottomrule\n\t\\end{tabular}\n\t\\label{table:ex2_s1_dt_sqrth}\n\\end{table}\n\\begin{table}[!htbp]\n\t\\centering\n\t\\caption{{Example~\\ref{ex:2d} by {\\rm (S2)} with $\\Delta t = c^\\prime h$~$(c^\\prime=1\/10)$:} Values of~$E_{11}$ and their slopes in $\\Delta t$.}\n\t\\begin{tabular}{rrrcrrcrr}\n\t\\toprule\n\t& \\multicolumn{2}{c}{$(i)$} && \\multicolumn{2}{c}{$(ii)$} && \\multicolumn{2}{c}{$(iii)$} \\\\ \\cline{2-3}\\cline{5-6}\\cline{8-9}\n\t\\multicolumn{1}{c}{$N$} & \\multicolumn{1}{c}{$E_{11}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{11}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{11}$} & \\multicolumn{1}{c}{Slope} \\\\\n\t\\hline\n\t$10$ & $2.07 \\times 10^{-4}$ & -- && $2.18 \\times 10^{-3}$ & -- && $9.79 \\times 10^{-4}$ & -- \\\\\n\t$20$ & $5.10 \\times 10^{-5}$ & $2.02$ && $5.35 \\times 10^{-4}$ & $2.02$ && $2.53 \\times 10^{-4}$ & $1.95$ \\\\\n\t$40$ & $1.27 \\times 10^{-5}$ & $2.00$ && $1.32 \\times 10^{-4}$ & $2.02$ && $6.39 \\times 10^{-5}$ & $1.98$ \\\\\n\t$80$ & $3.17 \\times 10^{-6}$ & $2.00$ && $3.27 \\times 10^{-5}$ & $2.01$ && $1.61 \\times 10^{-5}$ & $1.99$ \\\\\n\t\\bottomrule\n\t\\end{tabular}\n\t\\label{table:ex2_s2_dt_h}\n\\end{table}\n\\subsection{The Oldroyd-B constitutive equation in two-dimensional space}\nWe apply our approximations of the upper-convected time derivative of second-order in time~\\eqref{eq:sec_order_p_order} in Theorem~\\ref{prop:sec_order_p_order} for solving a problem governed by the Oldroyd-B constitutive equation in two-dimensional space; find $\\zeta: \\Omega\\times (0, T) \\to \\mathbb{R}^{d\\times d}_{\\rm sym}$ such that\n\t%\n\t\\begin{subequations}\\label{eqns:prob_ob}\n\t\t\\begin{align}\n\t\t&&&& \\zeta + Wi \\, \\uctd{\\zeta} & = 2 ( 1 - \\beta ) D(u) + F && \\mbox{in}\\ \\Omega\\times (0, T), &&&&\n\t\t\\label{eq:prob_ob1}\\\\\n\t\t&&&& \\zeta & = \\zeta_\\mathrm{in} && \\mbox{on}\\ \\Gamma_\\mathrm{in}\\times (0, T), &&&&\n\t\t\\label{eq:prob_ob2}\\\\\n\t\t&&&& \\zeta & = \\zeta^0 && \\mbox{in}\\ \\Omega,\\ \\mbox{at}\\ t=0. &&&&\n\t\t\\end{align}\n\t\\end{subequations}\n\\par\nThe scheme to solve problem~\\eqref{eqns:prob_ob} is to find $\\{\\zeta_h^n \\in V_h;\\ n=1, \\ldots, N_T\\}$ such that\n\\begin{subequations}\\label{scheme_ob}\n\t\\begin{align}\n\t\\label{scheme_ob:general_step}\n\t\\zeta_h^n(x) + Wi \\, [\\mathcal{A}_h^{n,(p)} \\zeta_h] (x)\n\t& = 2(1-\\beta)D(u^n)(x) + F^n(x), & x & \\in \\bar\\Omega_h,\\ n  \\ge 1, \\\\\n\t\\label{scheme_ob:initial_step}\n\t\\zeta_h^0(x) & = \\zeta^0(x), & x & \\in \\bar\\Omega_h,\n\t\\end{align}\n\\end{subequations}\nwhere $\\mathcal{A}_h^{n,(p)} \\zeta_h:\\bar\\Omega_h\\to\\mathbb{R}^{d\\times d}_{\\rm sym}$ is the function defined already by~\\eqref{def:An_h}.\nWhen an upwind point is outside the domain, we employ a value of $\\zeta_\\mathrm{in}$ at closest lattice point to the upwind point similarly to the case of scheme~\\eqref{scheme} as mentioned in Remark~\\ref{rmk:upwind_cell_2d}-(iii).\nIn the following, scheme~\\eqref{scheme_ob} with $p=1$ and $p=2$ for problem~\\eqref{eqns:prob_ob} are called $({\\rm S}1)^\\prime$ and $({\\rm S}2)^\\prime$, respectively.\n\\par\nWe set two examples below:\n\\begin{Ex}[$d=2$]\\label{ex:OldB2D}\n\tIn problem~\\eqref{eqns:prob_ob}, let $d=2$, $\\Omega = (0,1)^d$, $T=1$ and $\\beta = 1\/9$.\n\tWe consider six values of the {\\rm Weissenberg} number $Wi$,\n\t\\[\n\t\tWi = 0.025, 1, 5, 10, 50, 100,\n\t\\]\n\tand the following function for the velocity field:\n\\begin{align*}\n u(x,t) & = (\\sin(x_1+x_2+t), \\sin(x_1+x_2+t))^{ \\top },\n\\end{align*}\n\twhich implies $\\Gamma_\\mathrm{in} = \\{ (s,0)^{ \\top }\\in\\partial\\Omega;\\ s\\in [0, 1] \\} \\cup \\{ (0,s)^{ \\top }\\in\\partial\\Omega;\\ s\\in [0, 1] \\}$.\n\tThe functions~$F$, $\\zeta_{\\rm in}$ and $\\zeta^0$ are given so that the exact solution is \n\t\\[\n\t \\zeta(x,t) = \n\t \\begin{bmatrix}\n\t  \\sin(x_1 + x_2 + t) + 2 & \\sin(x_1 + x_2 + t) \\\\\n\t  \\sin(x_1 + x_2 + t) & -\\sin(x_1 + x_2 + t) + 2\n\t \\end{bmatrix}.\n\t\\]\n\\end{Ex}\n\\begin{Ex}[$d=2$, {\\cite{Venkatesan2017}}]\\label{ex:Venkatesan2017}\n\tIn problem~\\eqref{eqns:prob_ob}, let $d=2$, $\\Omega = (0,1)^d$, $T=0.5$, $\\beta = 0.75$ and $Wi = 0.25$.\n\tWe consider the following function for the velocity field:\n\\[\nu(x, t) = (\\exp(-0.1t)\\sin(\\pi x_1), -\\pi\\exp(-0.1t)x_2\\cos(\\pi x_1))^{ \\top },\n\\]\n\twhich implies $\\Gamma_{\\rm in} = \\{ (s,0)^{ \\top }\\in\\partial\\Omega;\\ s\\in [0, 1] \\} \\cup \\{ (0,s)^{ \\top }\\in\\partial\\Omega;\\ s\\in [0, 1] \\}$.\n\tThe functions~$F$, $\\zeta_{\\rm in}$ and $\\zeta^0$ are given so that the exact solution is\n\\[\n\\zeta(x,t) = \n\\begin{bmatrix}\n\t\\exp(-0.1t)\\sin(\\pi x_1) & -\\pi\\exp(-0.1t)x_2\\cos(\\pi x_1) \\\\ \n\t-\\pi\\exp(-0.1t)x_2\\cos(\\pi x_1) & \\exp(-0.1t)\\sin(\\pi x_1)\\cos(\\pi x_2) \\end{bmatrix}.\n\\]\n\\end{Ex}\n\\par\nWe solve Example~\\ref{ex:OldB2D} by $({\\rm S}1)^\\prime$ with $\\Delta t = c\\sqrt{h}$ for $c = 1\/50$ and $({\\rm S}2)^\\prime$ with $\\Delta t = c^\\prime h$ for $c^\\prime = 1\/5$, where the mesh is constructed for $h_1=h_2=h=1\/N$, i.e., $N_1=N_2=N$, with $N = 10, 20, 40$ and $80$.\nIn order to further investigate the errors and the orders of convergence of the schemes for solving problem~\\eqref{eqns:prob_ob}, we give the results for the three different components $\\zeta_{11}$, $\\zeta_{12}$ and $\\zeta_{22}$.\nTables~\\ref{table:ErrorOB2D1stOrder} and~\\ref{table:ErrorOB2D2stOrder} show the results by $({\\rm S}1)^\\prime$ and $({\\rm S}2)^\\prime$, respectively, for~$Wi = 0.025$.\nFrom a quantitative point of view, the results are consistent with the theoretical results in Theorem~\\ref{prop:sec_order_p_order}.\n\\begin{table}[htbp!]\n\t\t\\centering\n\t\t\\caption{Example~\\ref{ex:OldB2D} by $({\\rm S}1)^\\prime$ with $\\Delta t = c\\sqrt{h}$~$(c=1\/50)$: Values of each tensor entry ~$E_{11},E_{12},E_{22}$ and their slopes in $\\Delta t$ for $Wi = 0.025$.}\n\t\t\\begin{tabular}{rrrcrrcrr}\n\t\t\t\\toprule\n\t\t\t\\multicolumn{1}{c}{$N$} & \\multicolumn{1}{c}{$E_{11}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{12}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{22}$} & \\multicolumn{1}{c}{Slope} \\\\\n\t\t\\hline\n\t\t\t$10$ & $2.03 \\times 10^{-3}$ & -- && $2.03 \\times 10^{-3}$ & -- && $2.03 \\times 10^{-3}$ & -- \\\\\n\t\t\t$20$ & $1.02 \\times 10^{-3}$ & $1.99 $ && $1.02 \\times 10^{-3}$ & $1.99 $ && $ 1.02 \\times 10^{-3}$ & $1.99 $ \\\\\n\t\t\t$40$ & $5.11 \\times 10^{-4}$ & $ 1.99$ && $5.11 \\times 10^{-4}$ & $1.99 $ && $ 5.11 \\times 10^{-4}$ & $ 1.99$ \\\\\n\t\t\t$80$ & $2.56 \\times 10^{-4}$ & $1.99 $ && $2.56 \\times 10^{-4}$ & $1.99 $ && $2.56 \\times 10^{-4}$ & $ 1.99$ \\\\\n\t\t\t\\bottomrule\n\t\\end{tabular}\n\t\\label{table:ErrorOB2D1stOrder}\n\\end{table}\n\\begin{table}[htbp!]\n\t\\centering\n\t\\caption{Example~\\ref{ex:OldB2D} by $({\\rm S}2)^\\prime$ with $\\Delta t = c^\\prime h$~$(c^\\prime=1\/5)$: Values of each tensor entry~$E_{11},E_{12},E_{22}$ and their slopes in $\\Delta t$ for $Wi = 0.025$.}\n\t\\begin{tabular}{rrrcrrcrr}\n\t\t\\toprule\n\t\t\\multicolumn{1}{c}{$N$} & \\multicolumn{1}{c}{$E_{11}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{12}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{22}$} & \\multicolumn{1}{c}{Slope} \\\\\n\t\t\\hline\n\t\t$10$ & $7.62 \\times 10^{-5}$ & -- && $7.24 \\times 10^{-5}$ & -- && $7.62 \\times 10^{-5}$ & -- \\\\\n\t\t$20$ & $1.89 \\times 10^{-6}$ & $2.02 $ && $1.80 \\times 10^{-6}$ & $2.01 $ && $1.89 \\times 10^{-5}$ & $2.02 $ \\\\\n\t\t$40$ & $4.75 \\times 10^{-6}$ & $1.99 $ && $4.57 \\times 10^{-6}$ & $ 1.98$ && $4.75 \\times 10^{-6}$ & $ 1.99$ \\\\\n\t\t$80$ & $1.21 \\times 10^{-6}$ & $ 1.97$ && $1.17 \\times 10^{-6}$ & $ 1.96$ && $ 1.21 \\times 10^{-6}$ & $1.97 $ \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\label{table:ErrorOB2D2stOrder}\n\\end{table}\n\\par\n\nA computational challenge in viscoelastic fluid flows is the application of high values of the Weissenberg number, i.e., $Wi>1$. In fact, the \\textit{infamous} High Weissenberg Number Problem \\cite{Fattal:2004,Hulsen,Martins2015} depends of some particular factors on viscoelastic flows, as for instance, domain geometry, boundary conditons, fluid type, mesh size, etc. In summary, this instability is related to the unbounded values of the stress tensor during the transient solution resulting in the fail of the numerical methods. It is important to highlight that some classical methods, i.e. without stabilization techniques, have failed for $Wi= O(1)$ exhibiting numerical oscillations of the solution. Rougly speaking, the High Weissenberg Number Problem can be interpreted according to a critical value of the Weissenberg number, $Wi_{crit}$, which the numerical solution is boundly maintained during the simulation of the classical constitutive formulations. For example, considering the traditional Oldroyd-B model, Fattal and Kupferman \\cite{Fattal:2004} described  $Wi_{crit} \\approx 0.5$ for the cavity flow while Oliveira and Miranda \\cite{OliveiraMiranda} pointed out $Wi_{crit} \\approx 1$ for unsteady viscoelastic flow past bounded cylinders. Moreover, Walters and Webster \\cite{Walters2003} presented results for the $4:1$ contraction problem with the critical Weissenberg number near to 3. Therefore, there is an effort of the researchers to circumvent the High Weissenberg Number Problem developing new formulations that can be stable in simulations with $Wi > Wi_{crit}$.\n\nIt is important to highlight that the schemes presented in this current work can deal with high values of $Wi$ without the need to employ stabilization strategies. To test the accuracy of $({\\rm S}2)^\\prime$, we vary the values of Weissenberg number as $Wi=1,5,10,50,100$ in Example~\\ref{ex:OldB2D} and the results are presented in Table \\ref{table:ErrorOB2D2nd_Wi_varied1}. The main focus for varying the Weissenberg number is to verify the ability of $({\\rm S}2)^\\prime$ for dealing with the Oldroyd-B constitutive equation defined on the context of high elasticity. From the results presented in Table \\ref{table:ErrorOB2D2nd_Wi_varied1}, we can notice that the numerical order of convergence of $({\\rm S}2)^\\prime$ is of second-order in both time and space, and that the effect of varying the Weissenberg number is not significant for this example.\n\n\t\\begin{table}[ht!]\n\t\\centering\n\t\\caption{Example~\\ref{ex:OldB2D} by $({\\rm S}2)^\\prime$ with $\\Delta t = c^\\prime h$~$(c^\\prime=1\/5)$ and different values of $Wi$ number.}\n\t\\label{table:ErrorOB2D2nd_Wi_varied1}\n\t\\begin{tabular}{rrrcrrcrr}\n\t\t\\hline\n\t\t\\multicolumn{9}{c}{$Wi=1.0$} \\\\ \\hline\n\t\t\\multicolumn{1}{c}{$N$} & \\multicolumn{1}{c}{$E_{11}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{12}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{22}$} & \\multicolumn{1}{c}{Slope} \\\\ \\hline\n\t\t$10$ & $1.55 \\times 10^{-3}$ & -- && $1.06 \\times 10^{-3}$ & -- && $5.54 \\times 10^{-4}$ & -- \\\\\n\t\t$20$ & $4.23 \\times 10^{-4}$ & $1.88$ && $2.93 \\times 10^{-4}$ & $1.85$ && $1.48 \\times 10^{-4}$ & $1.91$ \\\\\n\t\t$40$ & $1.09 \\times 10^{-4}$ & $1.95$ && $7.65 \\times 10^{-5}$ & $1.94$ && $3.79 \\times 10^{-5}$ & $1.96$ \\\\\n\t\t$80$ & $2.77 \\times 10^{-5}$ & $1.98$ && $1.95 \\times 10^{-5}$ & $1.98$ && $9.58 \\times 10^{-6}$ & $1.99$ \\\\\n\t\n\t\n\t\t\\hline\n\t\t\\multicolumn{9}{c}{$Wi=5$} \\\\ \\hline\n\t\t\\multicolumn{1}{c}{$N$} & \\multicolumn{1}{c}{$E_{11}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{12}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{22}$} & \\multicolumn{1}{c}{Slope} \\\\ \\hline\n\t\t$10$ & $1.97 \\times 10^{-3}$ & -- && $1.37 \\times 10^{-3}$ & -- && $7.13 \\times 10^{-4}$ & -- \\\\\n\t\t$20$ & $5.36 \\times 10^{-4}$ & $1.87$ && $3.80 \\times 10^{-4}$ & $1.85$ && $1.97 \\times 10^{-4}$ & $1.86$ \\\\\n\t\t$40$ & $1.39 \\times 10^{-4}$ & $1.95$ && $9.90 \\times 10^{-5}$ & $1.94$ && $5.14 \\times 10^{-5}$ & $1.94$ \\\\\n\t\t$80$ & $3.51 \\times 10^{-5}$ & $1.98$ && $2.52 \\times 10^{-5}$ & $1.98$ && $1.31 \\times 10^{-5}$ & $1.97$ \\\\\n\t\n\t\n\t\t\\hline\n\t\t\\multicolumn{9}{c}{$Wi=10$} \\\\ \\hline\n\t\t\\multicolumn{1}{c}{$N$} & \\multicolumn{1}{c}{$E_{11}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{12}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{22}$} & \\multicolumn{1}{c}{Slope} \\\\ \\hline\n\t\t$10$ & $2.03 \\times 10^{-3}$ & -- && $1.42 \\times 10^{-3}$ & -- && $7.38 \\times 10^{-4}$ & -- \\\\\n\t\t$20$ & $5.54 \\times 10^{-4}$ & $1.87$ && $3.93 \\times 10^{-4}$ & $1.85$ && $2.04 \\times 10^{-4}$ & $1.85$ \\\\\n\t\t$40$ & $1.43 \\times 10^{-4}$ & $1.94$ && $1.03 \\times 10^{-4}$ & $1.94$ && $5.35 \\times 10^{-5}$ & $1.93$ \\\\\n\t\t$80$ & $3.63 \\times 10^{-5}$ & $1.98$ && $2.61 \\times 10^{-5}$ & $1.98$ && $1.36 \\times 10^{-5}$ & $1.97$ \\\\\n\t\n\t\t\\hline\n\t\t\\multicolumn{9}{c}{$Wi=50$} \\\\ \\hline\n\t\t\\multicolumn{1}{c}{$N$} & \\multicolumn{1}{c}{$E_{11}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{12}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{22}$} & \\multicolumn{1}{c}{Slope} \\\\ \\hline\n\t\t$10$ & $2.08 \\times 10^{-3}$ & -- && $1.46 \\times 10^{-3}$ & -- && $7.59 \\times 10^{-4}$ & -- \\\\\n\t\t$20$ & $5.69 \\times 10^{-4}$ & $1.87$ && $4.05 \\times 10^{-4}$ & $1.85$ && $2.11 \\times 10^{-4}$ & $1.85$ \\\\\n\t\t$40$ & $1.47 \\times 10^{-4}$ & $1.95$ && $1.06 \\times 10^{-4}$ & $1.94$ && $5.53 \\times 10^{-5}$ & $1.93$ \\\\\n\t\t$80$ & $3.72 \\times 10^{-5}$ & $1.98$ && $2.68 \\times 10^{-5}$ & $1.98$ && $1.41 \\times 10^{-5}$ & $1.97$ \\\\\n\t\n\t\t\\hline\n\t\t\\multicolumn{9}{c}{$Wi=100$} \\\\ \\hline\n\t\t\\multicolumn{1}{c}{$N$} & \\multicolumn{1}{c}{$E_{11}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{12}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{22}$} & \\multicolumn{1}{c}{Slope} \\\\ \\hline\n\t\t$10$ & $2.09 \\times 10^{-3}$ & -- && $1.46 \\times 10^{-3}$ & -- && $7.62 \\times 10^{-4}$ & -- \\\\\n\t\t$20$ & $5.71 \\times 10^{-4}$ & $1.87$ && $4.06 \\times 10^{-4}$ & $1.85$ && $2.12 \\times 10^{-4}$ & $1.85$ \\\\\n\t\t$40$ & $1.48 \\times 10^{-4}$ & $1.95$ && $1.06 \\times 10^{-4}$ & $1.94$ && $5.55 \\times 10^{-5}$ & $1.93$ \\\\\n\t\t$80$ & $3.74 \\times 10^{-5}$ & $1.98$ && $2.69 \\times 10^{-5}$ & $1.98$ && $1.42 \\times 10^{-5}$ & $1.97$ \\\\\n\t\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\\par\nFinally, example~\\ref{ex:Venkatesan2017} employs the manufactured solution used by Venkatesan and Ganesan~\\cite{Venkatesan2017}.\nNotice that in this study we are investigating the numerical behavior of the schemes for non-homogeneous boundary conditions in parts of the domain.\nTable~\\ref{table:ErrorOB2_Venkatesan2017} describes the results for Example~\\ref{ex:Venkatesan2017} by $({\\rm S}2)^\\prime$ with $\\Delta t = c^\\prime h$ for $c^\\prime = 1\/10$, where the mesh is constructed for $h_1=h_2=h=1\/N$, i.e., $N_1=N_2=N$, with $N = 10, 20, 40$ and $80$.\nFrom this table we can see that these results are consistent with our truncation error analysis in Theorem~\\ref{prop:sec_order_p_order}.\n\\begin{table}[htbp!]\n\t\\centering\n\t\\caption{Example~\\ref{ex:Venkatesan2017} by $({\\rm S}2)^\\prime$ with $\\Delta t = c^\\prime h$~$(c^\\prime=1\/10)$: Values of each tensor entry~$E_{11},E_{12},E_{22}$ and their slopes in $\\Delta t$ for $Wi = 0.25$ and $\\beta = 0.75$.}\n\t\\begin{tabular}{rrrcrrcrr}\n\t\t\\toprule\n\t\t\\multicolumn{1}{c}{$N$} & \\multicolumn{1}{c}{$E_{11}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{12}$} & \\multicolumn{1}{c}{Slope} && \\multicolumn{1}{c}{$E_{22}$} & \\multicolumn{1}{c}{Slope} \\\\\n\t\t\\hline\n\t\t$10$ & $4.10 \\times 10^{-3}$ & -- && $7.64 \\times 10^{-2}$ & -- && $1.98 \\times 10^{-2}$ & -- \\\\\n\t\t$20$ & $1.02 \\times 10^{-3}$ & $2.01$ && $2.11 \\times 10^{-3}$ & $1.86$ && $5.19 \\times 10^{-3}$ & $1.93$ \\\\\n\t\t$40$ & $2.82 \\times 10^{-4}$ & $1.86$ && $5.83 \\times 10^{-4}$ & $1.85$ && $1.32 \\times 10^{-3}$ & $1.97$ \\\\\n\t\t$80$ & $7.47 \\times 10^{-5}$ & $1.91$ && $1.54 \\times 10^{-4}$ & $1.92$ && $3.30 \\times 10^{-4}$ & $2.00$ \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\label{table:ErrorOB2_Venkatesan2017}\n\\end{table}\n\\section{Conclusions}\n\\label{sec:conclusions}\nThe application of the generalized Lie derivative~(GLD) for constructing schemes to deal with the upper-convected time derivative is an alternative form in the numerical solution of constitutive equations.\nIn spite of the success of this strategy firstly proposed by Lee and Xu~\\cite{Lee2006}, to the best knowledge of the authors, the methodology was only applied in the context of finite elements.\nIn this work, we have combined a Lagrangian framework with GLD to develop new second-order finite difference approximations for the upper-convected time derivative.\nParticularly, the schemes are constructed based on bilinear and biquadratic interpolation operators for solving a simple model in one- and two-dimensional spaces.\nThe schemes are explicit and no CFL condition is required as the Lagrangian framework is employed.\nTruncation errors of~$O(\\Delta t^2 + h^p)$ $(p=1,2)$ for the finite difference approximations of the the upper-convected time derivative have been proved.\nA numerical integration of composite functions may cause an instability in the case of Lagrangian finite element method, our schemes, however, do not have such instability since there is no numerical integration thanks to the finite difference method.\nAccording to our numerical results for simplified model equations, the new finite difference schemes can reach second-order of accuracy in time and {space~$(p=2)$} corroborating with the theoretical analysis. Moreover, the proposed strategy has been also applied to solve a two-dimensional Oldroyd-B constitutive equation subjected to a prescribed velocity field. The results have been very satisfactory since the increasing of the Weissenberg number did not influence the good properties of accuracy and stability of the finite difference approximations. As a future work, we intend to extend our schemes for solving viscoelastic fluid flows governed by different constitutive equations at high Weissenberg numbers.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzcklg b/data_all_eng_slimpj/shuffled/split2/finalzzcklg
new file mode 100644
index 0000000000000000000000000000000000000000..f99d91b31ee5b3262acf064f24f31d74e690931c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzcklg
@@ -0,0 +1,5 @@
+{"text":"\n\\section{Introduction}\n\\input{sections\/introduction}\n\n\\section{Model}\n\\label{sec:model}\n\\input{sections\/model}\n\n\\section{Non-adaptive Optimization}\n\\label{sec:adaptivity}\n\\input{sections\/adaptivity}\n\n\\section{Algorithms}\n\\label{sec:algorithms}\n\\input{sections\/algorithms}\n\n\\section{Experiments}\n\\label{sec:experiments}\n\\input{sections\/experiments}\n\n\\section{Related work}\n\\input{sections\/related}\n\n\\section*{Acknowledgement}\nThis research is supported in part by a Google Research Grant and NSF grant CCF-1301976.\n\n\\bibliographystyle{abbrv}\n\\balance\n\n\n\\subsection{Adaptivity Gap}\\label{sec:gap}\n\nWe will now justify the use of non-adaptive strategies by showing that the\noptimal solution for this form of non-adaptive strategies yields a higher value\nthan adaptive ones.  For brevity, given a probability vector $\\pi\\in[0,1]^m$ we\nwrite:\n\\begin{equation}\\label{eq:multi}\n        F(\\pi) \\equiv\n        \\sum_{R\\subseteq\\neigh{X}}\\left(\\prod_{u\\in\n        R}\\pi_u\\prod_{u\\in\\neigh{X}\\setminus R}(1-\\pi_u)\\right)\n    f(R)\n\\end{equation}\nas well as $\\textbf{p}\\otimes \\textbf{q}$ to denote the component-wise\nmultiplication between vectors $\\textbf{p}$ and $\\textbf{q}$.  Finally, we write\n$\\mathcal{F}_{A} \\equiv \\{S \\subseteq X : |S|\\leq k\\}$, and $\\mathcal{F}_{NA}\n\\equiv\\{(S,\\textbf{q}), |S|+\\textbf{p}^T\\textbf{q} \\leq k, q_u \\leq\n\\mathbf{1}_{\\{u\\in\\neigh{S}\\}}\\}$ to denote the feasible regions of the\nadaptive and non-adaptive problems, respectively.\n\n\\begin{proposition}\\label{prop:gap}\nFor additive functions given by \\eqref{eq:voter}, the value of the optimal\nadaptive policy is upper bounded by the optimal non-adaptive policy:\n    \\begin{displaymath}\n        \\begin{aligned}[t]\n            &\\max_{S\\subseteq X} \\sum_{R\\subseteq\\neigh{S}} p_R\n            \\max_{\\substack{T\\subseteq R\\\\|T|\\leq k-|S|}}f(T)\\\\\n            &\\text{s.t. }S \\in \\mathcal{F}_{A}\n        \\end{aligned}\n        \\leq\n        \\begin{aligned}[t]\n            &\\max_{\\substack{S\\subseteq X\\\\\\textbf{q}\\in[0,1]^n}}\n            F(\\mathbf{p}\\otimes\\mathbf{q})\\\\\n            &\\text{s.t. } (S,\\textbf{q}) \\in \\mathcal{F}_{NA}\n                    \\end{aligned}\n    \\end{displaymath}\n\\end{proposition}\n\n\nThe proof of this proposition can be found in Appendix~\\ref{sec:ad-proofs} and\nrelies on the following fact: the optimal adaptive policy can be written as\na feasible non-adaptive policy, hence it provides a lower bound on the value of\nthe optimal non-adaptive policy.\n\n\\subsection{From Non-Adaptive to Adaptive Solutions}\\label{sec:round}\n\nFrom the above proposition we now know that optimal non-adaptive solutions have\nhigher values than adaptive solutions. Given a non-adaptive solution\n$(S,\\mathbf{q})$, a possible scheme would be to use $S$ as an adaptive\nsolution.  But since $(S, \\mathbf{q})$ is a solution to the non-adaptive\nproblem, Proposition~\\ref{prop:gap} does not provide any guarantee on how well\n$S$ performs as an adaptive solution.\n\nHowever, we show that from a non-adaptive solution $(S,\\mathbf{q})$, we can\nobtain a lower bound on the adaptive value of $S$, that is, the expected\ninfluence attainable in expectation over all possible arrivals of neighbors of\n$S$. Starting from $S$, in every realization of neighbors $R$, sample every\nnode $u \\in R \\cap \\mathcal{N}(S)$ with probability $q_{u}$, to obtain a random\nset of nodes $I_R \\subseteq R \\cap S$. $(S, \\mathbf{q})$ being a non-adaptive\nsolution, it could be that selecting $I_R$ exceeds our budget. Indeed, the only\nguarantee that we have is that $|S| + \\mathbb{E}\\big[|I_R|\\big]\\leq k$. As\na consequence, an adaptive solution starting from $S$ might not be able to\nselect $I_R$ on the second stage.\n\n\n\n \n\n\nFortunately, the probability of exceeding the budget is small enough and with\nhigh probability $I_R$ will be feasible. This is exploited in \\cite{vondrak} to\ndesign a randomized rounding method with approximation guarantees. These\nrounding methods are called \\emph{contention resolution schemes}.\nTheorem~1.3 of this paper gives us a contention resolution scheme which will\ncompute from $\\mathbf{q}$ and for any realization $R$ a \\emph{feasible} set\n$\\tilde{I}_R$, such that:\n\\begin{equation}\n    \\label{eq:cr}\n    \\mathbb{E}_R\\big[f(\\tilde{I}_R)\\big] \\geq (1-\\varepsilon) F(\\mathbf{q})\n\\end{equation}\nWhat this means is that starting from a non-adaptive solution\u00a0$(S,\\mathbf{q})$,\nthere is a way to construct a random \\emph{feasible} subset on the second stage\nsuch that in expectation, this set attains almost the same influence value as\nthe non-adaptive solution. Since the adaptive solution starting from $S$ will\nselect optimally from the realizations $R\\subseteq\\neigh{S}$,\n$\\mathbb{E}_R[f(\\tilde{I}_R)]$ provides a lower bound on the adaptive value of\n$S$ that we denote by $A(S)$.\n\nMore precisely, denoting by $\\text{OPT}_A$ the optimal value of the adaptive\nproblem~\\eqref{eq:problem}, we have the following proposition whose proof can\nbe found in Appendix~\\ref{sec:ad-proofs}.\n\\begin{proposition}\\label{prop:cr}\n    Let $(S,\\textbf{q})$ be an $\\alpha$-approximate solution to the\n    non-adaptive problem \\eqref{eq:relaxed}, then $\\mathrm{A}(S) \\geq \\alpha\n    \\mathrm{OPT}_A$.\n\\end{proposition}  \n\n\n\\subsection{An LP-Based Approach}\n\\label{sec:lp}\nNote that due to linearity of expectation, for a linear function $f$ of the\nform given by \\eqref{eq:voter} we have:\n\\begin{equation}\\label{eq:multi-voter}\n    \\begin{split}\n        F(\\textbf{p}) \n        &=\\mathbb{E}_{R}\\big[f(R)\\big]\n        =\\mathbb{E}_{R}\\Bigg[\\sum_{u\\in\\neigh{X}}w_u\\mathbf{1}_{\\{u\\in\n        R\\}}\\Bigg]\\\\\n        &=\\sum_{u\\in\\neigh{X}}w_u\\mathbb{P}[u\\in R]\n        =\\sum_{u\\in\\neigh{X}}p_uw_u\n    \\end{split}\n\\end{equation}\n\nThus, the non-adaptive optimization problem \\eqref{eq:relaxed} can be written as:\n\\begin{displaymath}\n    \\begin{split}\n        \\max_{\\substack{S\\subseteq X\\\\\\mathbf{q}\\in[0,1]^n} } \n        & \\sum_{u\\in\\neigh{X}}p_uq_uw_u\\\\\n        \\text{s.t. } & |S|+ \\textbf{p}^T\\textbf{q} \\leq k,\\;\n                     q_u \\leq \\mathbf{1}\\{u\\in\\neigh{S}\\}\n    \\end{split}\n\\end{displaymath}\n\nThe choice of the set $S$ can be relaxed by introducing a variable\n$\\lambda_v\\in[0,1]$ for each $v\\in X$. We obtain the following\nLP for the adaptive seeding problem:\n\\begin{equation}\\label{eq:lp}\n    \\begin{split}\n        \\max_{\\substack{\\mathbf{q}\\in[0,1]^n\\\\\\boldsymbol\\lambda\\in[0,1]^m}}\n        & \\;\\sum_{u\\in\\neigh{X}}p_uq_uw_u\\\\\n        \\text{s.t. } & \\sum_{v\\in X}\\lambda_v+\\textbf{p}^T\\textbf{q} \\leq k,\\;\n        q_u \\leq \\sum_{v\\in\\neigh{u}} \\lambda_v\n\\end{split}\n\\end{equation}\n\nAn optimal solution to the above problem can be found in polynomial time using\nstandard LP-solvers.  The solution returned by the LP is \\emph{fractional}, and\nrequires a rounding procedure to return a feasible solution to our problem,\nwhere $S$ is integral.  To round the solution we use the pipage rounding\nmethod~\\cite{pipage}.  We defer the details to Appendix~\\ref{sec:lp-proofs}.\n\n\\begin{lemma}\n    For \\mbox{\\textsc{AdaptiveSeeding-LP}} defined in \\eqref{eq:lp}, any fractional solution $(\\boldsymbol\\lambda, \\mathbf{q})\\in[0,1]^m\\times[0,1]^n$ can be rounded to an integral solution $\\bar{\\boldsymbol\\lambda} \\in \\{0,1\\}^{m}$ s.t. $(1-1\/e) F(\\mathbf{p}\\circ\\mathbf{q}) \\leq A(\\bar{\\lambda})$ in $O(m + n)$ steps.\n\\end{lemma}\n\n\\subsection{A Combinatorial Algorithm}\n\\label{sec:comb}\n\nIn this section, we introduce a combinatorial algorithm with an identical\napproximation guarantee to the LP-based approach. However, its running time,\nstated in Proposition~\\ref{prop:running_time} can be better than the one given\nby LP solvers depending on the relative sizes of the budget and the number of\nnodes in the graph. Furthermore, as we discuss at the end of this section, this\nalgorithm is amenable to parallelization. \n\nThe main idea is to reduce the problem to a monotone submodular maximization\nproblem and apply a variant of the celebrated greedy\nalgorithm~\\cite{nemhauser}.  \nIn contrast to standard influence maximization, the submodularity of the\nnon-adaptive seeding problem is not simply a consequence of properties of the\ninfluence function; it also strongly relies on the combinatorial structure of\nthe two-stage optimization. \n\nIntuitively, we can think of our problem as trying to find a set $S$ in the\nfirst stage, for which the nodes that can be seeded on the second stage have\nthe largest possible value.  To formalize this, for a budget $b\\in\\mathbf{R}^+$\nused in the second stage and a set of neighbors $T\\subseteq\\mathcal{N}(X)$, we\nwill use $\\mathcal{O}(T,b)$ to denote the solution to:\n\\begin{equation}\\label{eq:knap}\n    \\begin{split}\n        \\mathcal{O}(T,b)\\equiv\n        \\max_{\\textbf{q}\\in[0,1]^n} & \\sum_{u\\in\\neigh{X} \\cap T} p_uq_uw_u\\\\\n                            \\text{s.t. } & \\mathbf{p}^T\\mathbf{q}\\leq b\n                           \n\\end{split}\n\\end{equation}\n\nThe optimization problem \\eqref{eq:relaxed} for\nnon-adaptive policies can now be written as:\n\\begin{equation}\\label{eq:sub}\n        \\max_{S\\subseteq X} \\; \\mathcal{O}\\big(\\neigh{S},k-|S|\\big)\n        \\quad \\text{s.t. } |S|\\leq k\n\\end{equation}\n\nWe start by proving in Proposition~\\ref{prop:sub} that for fixed $t$,\n$\\mathcal{O}(\\neigh{\\cdot}, t)$ is submodular. This proposition relies on\nlemmas~\\ref{lemma:nd} and~\\ref{lemma:sub} about the properties of\n$\\mathcal{O}(T,b)$.\n\n\\begin{lemma}\\label{lemma:nd}\n    Let $T \\subseteq \\mathcal{N}(X)$ and $x \\in \\mathcal{N}(X)$, then\n    $\\mathcal{O}(T\\cup\\{x\\},b)-\\mathcal{O}(T,b)$ is\n    non-decreasing in $b$.\n\\end{lemma}\n\nThe proof of this lemma can be found in Appendix~\\ref{sec:comb-proofs}. The main\nidea consists in writing:\n\\begin{multline*}\n    \\mathcal{O}(T\\cup\\{x\\},c)-\\mathcal{O}(T\\cup\\{x\\},b)=\\int_b^c\\partial_+\\mathcal{O}_{T\\cup\\{x\\}}(t)dt\n\\end{multline*}\nwhere $\\partial_+\\mathcal{O}_T$ denotes the right derivative of\n$\\mathcal{O}(T,\\cdot)$. For a fixed $T$ and $b$, $\\mathcal{O}(T,b)$ defines\na fractional Knapsack problem over the set $T$. Knowing the form of the optimal\nfractional solution, we can verify that\n$\\partial_+\\mathcal{O}_{T\\cup\\{x\\}}\\geq\\partial_+\\mathcal{O}_T$ and obtain:\n\\begin{multline*}\n    \\mathcal{O}(T\\cup\\{x\\},c)-\\mathcal{O}(T\\cup\\{x\\},b)\\geq \n    \\mathcal{O}(T,c)-\\mathcal{O}(T,b)\n\\end{multline*}\n\n\\begin{lemma}\\label{lemma:sub}\n    For any $b\\in\\mathbf{R}^+$, $\\mathcal{O}(T,b)$ is submodular in $T$, $T\\subseteq\\neigh{X}$.\n\\end{lemma}\n\nThe proof of this lemma is more technical.\u00a0For $T\\subseteq\\neigh{X}$ and $x,\ny\\in\\neigh{X}\\setminus T$, we need to show that:\n    \\begin{displaymath}\n        \\mathcal{O}(T\\cup\\{x\\},b)-\\mathcal{O}(T,b)\\geq\n        \\mathcal{O}(T\\cup\\{y, x\\},b)-\\mathcal{O}(T\\cup\\{y\\},b)\n    \\end{displaymath}\n    This can be done by partitioning the set $T$ into ``high value\n    items'' (those with weight greater than $w_x$) and ``low value items'' and\n    carefully applying Lemma~\\ref{lemma:nd} to the associated subproblems.\n    The proof is in Appendix~\\ref{sec:comb-proofs}.\n\nFinally, Lemma~\\ref{lemma:sub} can be used to show Proposition~\\ref{prop:sub}\nwhose proof can be found in Appendix~\\ref{sec:comb-proofs}.\n\n\\begin{proposition}\\label{prop:sub}\n    Let $b\\in\\mathbf{R}^+$, then $\\mathcal{O}(\\neigh{S},b)$ is monotone and\n    submodular in $S$, $S\\subseteq X$.\n\\end{proposition}\n\nWe can now use Proposition~\\ref{prop:sub} to reduce \\eqref{eq:sub} to a monotone submodular maximization problem. First, we note that~\\eqref{eq:sub} can be rewritten:\n\\begin{equation}\\label{eq:sub-mod}\n    \\max_{\\substack{S\\subseteq X\\\\ t \\in \\mathbb{N}}} \\; \\mathcal{O}\\big(\\neigh{S},t\\big)\n        \\quad\\text{s.t. } |S| + t\\leq k\n\\end{equation}\n\nIntuitively, we fix $t$ arbitrarily so that the maximization above becomes a submodular maximization problem with fixed budget $t$. We then optimize over the value of $t$. Combining this observation with the greedy algorithm for monotone submodular maximization~\\cite{nemhauser}, we obtain Algorithm~\\ref{alg:comb}, whose performance guarantee is summarized in Proposition~\\ref{prop:main_result}.\n\n\\begin{algorithm}\n    \\caption{Combinatorial algorithm}\n    \\label{alg:comb}\n    \\algsetup{indent=2em}\n    \\begin{algorithmic}[1]\n        \\STATE $S\\leftarrow \\emptyset$\n        \\FOR{$t=1$ \\TO $k-1$}\n            \\STATE $S_t\\leftarrow \\emptyset$\n            \\FOR{$i=1$ \\TO $k-t$}\n                \\STATE $x^*\\leftarrow\\argmax_{x\\in\n                X\\setminus S_t}\\mathcal{O}(\\neigh{S_t\\cup\\{x\\}},t)\n                -\\mathcal{O}(\\neigh{S_t},t)$\\label{line:argmax}\n                \\STATE $S_t\\leftarrow S_t\\cup\\{x^*\\}$\n            \\ENDFOR\n            \\IF{$\\mathcal{O}(\\neigh{S_t},t)>\\mathcal{O}(\\neigh{S},k-|S|)$}\n                \\STATE $S\\leftarrow S_t$\n            \\ENDIF\n        \\ENDFOR\n        \\RETURN $S$\n    \\end{algorithmic}\n\\end{algorithm}\n\n\\begin{proposition}\\label{prop:main_result}\n    Let $S$ be the set computed by Algorithm~\\ref{alg:comb} and let us denote\n    by $\\mathrm{A}(S)$ the value of the adaptive policy selecting $S$ on the first\n    stage. Then $\\mathrm{A}(S) \\geq (1-1\/e)\\mathrm{OPT}_A$.\n\\end{proposition}\n\n\n\n\\noindent\\textbf{Parallelization.}  The algorithm described above considers all\npossible ways to split the seeding budget between the first and the second\nstage.  For each possible split $\\{(t,k-t)\\}_{t=1\\ldots,k-1}$, the algorithm\ncomputes an approximation to the optimal non adaptive solution that uses $k-t$\nnodes in the first stage and $t$ nodes in the second stage, and returns the\nsolution for the split with the highest value (breaking ties arbitrarily).\nThis process can be trivially parallelized across $k-1$ machines, each\nperforming a computation of a single split.  With slightly more effort, for any\n$\\epsilon>0$ one can parallelize over $\\log_{1+\\epsilon}n$ machines at the cost\nof losing a factor of $\\epsilon$ in the approximation guarantee (see Appendix~\\ref{sec:para} for details).\\newline\n\n\\noindent \\textbf{Implementation in MapReduce.} While the previous paragraph\ndescribes how to parallelize the outer \\texttt{for} loop of\nAlgorithm~\\ref{alg:comb}, we note that its inner loop can also be parallelized\nin the MapReduce framework. Indeed, it corresponds to the greedy algorithm\napplied to the function $\\mathcal{O}\\left(\\neigh{\\cdot}, t\\right)$. The\n\\textsc{Sample\\&Prune} approach successfully applied in \\cite{mr} to obtain\nMapReduce algorithms for various submodular maximizations can also be applied\nto Algorithm~\\ref{alg:comb} to cast it in the MapReduce framework. The details\nof the algorithm can be found in Appendix~\\ref{sec:mr}.\n\\newline\n\n\n\n\\noindent \\textbf{Algorithmic speedups.}  To implement Algorithm~\\ref{alg:comb} efficiently, the computation of the $\\argmax$ on line 5 must be dealt with carefully. $\\mathcal{O}(\\neigh{S_t\\cup\\{x\\}},t)$ is the optimal solution to the fractional Knapsack problem~\\eqref{eq:knap} with budget $t$ and can be computed in time $\\min(\\frac{t}{p_\\text{min}},n)$ by iterating over the list of nodes in $\\neigh{S_t\\cup\\{x\\}}$ in decreasing order of the degrees. This decreasing order of $\\neigh{S_t}$ can be maintained throughout the greedy construction of $S_t$ by:\n\\begin{itemize}\n    \\item ordering the list of neighbors of nodes in $X$ by decreasing order of the degrees when initially constructing the graph. This is responsible for a $O(n\\log n)$ pre-processing time.\n\\item when adding node $x$ to $S_t$, observe that $\\neigh{S_t\\cup\\{x\\}} = \\neigh{S_t}\\cup\\neigh{\\{x\\}}$. Hence, if $\\neigh{S_t}$ and $\\neigh{\\{x\\}}$ are sorted lists, then $\\mathcal{O}(\\neigh{S_t\\cup\\{x\\}},t)$ can be computed in a single iteration of length $\\min(\\frac{t}{p_\\text{min}},n)$ where the two sorted lists are merged on the fly.\n\\end{itemize}\nAs a consequence, the running time of line 5 is bounded from above by $m\\min(\\frac{t}{p_\\text{min}},n)$. The two nested \\textsf{for} loops are responsible for the additional $k^2$ factor. The running time of Algorithm~\\ref{alg:comb} is summarized in Proposition~\\ref{prop:running_time}.\n\n\\begin{proposition}\\label{prop:running_time}\n    Let $p_\\text{min} =\\min\\{p_u,u\\in\\neigh{X}\\}$, then Algorithm~\\ref{alg:comb} runs in time \n    ${O\\big(n\\log n + k^2 m \\min(\\frac{k}{p_\\text{min}},n)\\big)}$.\n\\end{proposition}\n\n\\section{Adaptivity proofs}\n\\label{sec:ad-proofs}\n\\begin{proof}[of Proposition~\\ref{prop:gap}]\n    We will first show that the optimal adaptive policy can be interpreted as\n    a non-adaptive policy. Let $S$ be the optimal adaptive\n    solution and define $\\delta_R:\\neigh{X}\\mapsto \\{0,1\\}$:\n    \\begin{displaymath}\n        \\delta_R(u) \\equiv \\begin{cases}\n            1&\\text{if } u\\in\\argmax\\big\\{f(T);\\; T\\subseteq R,\\; |T|\\leq\n        k-|S|\\big\\} \\\\\n            0&\\text{otherwise}\n        \\end{cases},\n    \\end{displaymath}\n    one can write\n    \\begin{displaymath}\n    \\begin{split}\n        \\sum_{R\\subseteq\\neigh{S}} p_R\n        \\max_{\\substack{T\\subseteq R\\\\|T|\\leq k-|S|}}f(T)\n        &=\n        \\sum_{R\\subseteq\\neigh{S}} p_R\n        \\sum_{u\\in\\neigh{X}}\\delta_R(u)w_u\\\\\n        &=\n        \\sum_{u\\in\\neigh{X}}w_u\\sum_{R\\subseteq\\neigh{S}}p_R\\delta_R(u).\n    \\end{split}\n    \\end{displaymath}\n\n    Let us now define for $u\\in\\neigh{X}$:\n    \\begin{displaymath}\n        q_u \\equiv \\begin{cases}\n            \\sum_{R\\subseteq\\neigh{S}}\\frac{p_R}{p_u}\\delta_R(u)\n            &\\text{if }p_u\\neq 0\\\\\n            0&\\text{otherwise}\n        \\end{cases}.\n    \\end{displaymath}\n    This allows us to write:\n    \\begin{displaymath}\n            \\sum_{R\\subseteq\\neigh{S}} p_R\n            \\max_{\\substack{T\\subseteq R\\\\|T|\\leq k-|S|}}f(T)\n            = \\sum_{u\\in\\neigh{X}}p_uq_uw_u = F(\\mathbf{p}\\circ\\mathbf{q})\n    \\end{displaymath}\n    where the last equality is obtained from \\eqref{eq:multi} by successively using the linearity of the expectation and the linearity of $f$.\n\n    Furthermore, observe that $q_u\\in[0,1]$, $q_u=0$ if $u\\notin\\neigh{S}$ and:\n    \\begin{displaymath}\n        \\begin{split}\n        |S|+\\sum_{u\\in\\neigh{X}}p_uq_u\n        &= |S|+\\sum_{R\\subseteq\\neigh{S}}p_R\\sum_{u\\in\\neigh{X}}\\delta_R(u)\\\\\n        &\\leq |S| + \\sum_{R\\subseteq\\neigh{S}}p_R(k-|S|)\\leq k\n    \\end{split}\n    \\end{displaymath}\n\n    Hence, $(S,\\mathbf{q})\\in\\mathcal{F}_{NA}$. In other words, we have written the optimal adaptive solution as a relaxed\n    non-adaptive solution. This conclude the proof of the proposition.\n\\end{proof}\n\n\\vspace{0.5em}\n\n\\begin{proof}[of Proposition~\\ref{prop:cr}]\n    Using the definition of $\\mathrm{A}(S)$, one can write:\n    \\begin{displaymath}\n        \\mathrm{A}(S) = \\sum_{R\\subseteq\\neigh{S}} p_R\n        \\max_{\\substack{T\\subseteq R\\\\|T|\\leq k-|S|}}f(T)\n        \\geq \\sum_{R\\subseteq\\neigh{S}} p_R \\mathbf{E}\\big[f(I)\\big]\n    \\end{displaymath}\n    where the inequality comes from the fact that $I$ is a feasible random set: $|I|\\leq k-|S|$, hence the expected value of $f(I)$ is bounded by the maximum of $f$ over feasible sets.\n\nEquation~\\eqref{eq:cr} then implies:\n\\begin{equation}\\label{eq:tmp}\n    \\mathrm{A}(S) \n    \\geq (1-\\varepsilon)\\sum_{R\\subseteq\\neigh{S}} p_R F(\\mathbf{q})\n    = (1-\\varepsilon)F(\\mathbf{p}\\circ\\mathbf{q}).\n\\end{equation}\n\nEquation~\\eqref{eq:tmp} holds for any $\\varepsilon\\geq 0$. In particular, for $\\varepsilon$ smaller than $\\inf_{S\\neq T} |A(S)-A(T)|$, we obtain that $\\mathrm{A}(S)\\geq F(\\mathbf{p}\\circ\\mathbf{q})$. Note that such a $\\varepsilon$ is at most polynomially small in the size of the instance.\n$(S, \\mathbf{q})$ is an $\\alpha$-approximate non adaptive solution, hence $F(\\mathbf{p}\\circ\\mathbf{q}) \\geq \\alpha\\mathrm{OPT}_{NA}$. We can then conclude by applying Proposition~\\ref{prop:gap}. \n\\end{proof}\n\n\\section{Algorithms Proofs}\n\\label{sec:alg-proofs}\nWe first discuss the NP-hardness of the problem.\n\n\\noindent \\textbf{\\textsc{NP}-Hardness.} In contrast to standard influence maximization, adaptive seeding is already \\textsc{NP}-Hard even for the simplest cases.  In the case when $f(S)=|S|$ and all probabilities equal one, the decision problem is whether given a budget $k$ and target value $\\ell$ there exists a subset of $X$ of size $k-t$ which yields a solution with expected value of $\\ell$ using $t$ nodes in $\\mathcal{N}(X)$.  This is equivalent to deciding whether there are $k-t$ nodes in $X$ that have $t$ neighbors in $\\mathcal{N}(X)$.  To see this is \\textsc{NP}-hard, consider reducing from \\textsc{Set-Cover} where there is one node $i$ for each input set $T_i$, $1\\leq i\\leq n$, with $\\neigh{i}= T_i$ and integers $k,\\ell$, and the output is ``yes'' if there is a family of $k$ sets in the input which cover at least $\\ell$ elements, and ``no'' otherwise.\n\n\n\\subsection{LP-based approach}\n\\label{sec:lp-proofs}\nIn the LP-based approach we rounded the solution using the pipage rounding method.  We discuss this with greater detail here.\n\n\\noindent \\textbf{Pipage Rounding.}\nThe pipage rounding method~\\cite{pipage} is a deterministic rounding method that can be applied to a variety of problems.  In particular, it can be applied to LP-relaxations of the \\textsc{Max-K-Cover} problem where we are given a family of sets that cover elements of a universe and the goal is to find $k$ subsets whose union has the maximal cardinality.  The LP-relaxation is a fractional solution over subsets, and the pipage rounding procedure then rounds the allocation in linear time, and the integral solution is guaranteed to be within a factor of $(1-1\/e)$ of the fractional solution. \nWe make the following key observation: for any given $\\textbf{q}$, one can remove all elements in $\\mathcal{N}(X)$ for which $q_{u}=0$, without changing the value of any solution $(\\boldsymbol\\lambda,\\textbf{q})$.\nOur rounding procedure can therefore be described as follows: given a solution $(\\boldsymbol\\lambda,\\textbf{q})$ we remove all nodes $u \\in \\mathcal{N}(X)$ for which $q_{u}=0$, which leaves us with a fractional solution to a (weighted) version of the \\textsc{Max-K-Cover} problem where nodes in $X$ are the sets and the universe is the set of weighted nodes in $\\mathcal{N}(X)$ that were not removed.  We can therefore apply pipage rounding and lose only a factor of $(1-1\/e)$ in quality of the solution.\n\n\\subsection{Combinatorial Algorithm}\n\\label{sec:comb-proofs}\nWe include the missing proofs from the combinatorial algorithm section.  The scalability and implementation in MapReduce are discussed in this section as well.\n\n\\begin{proof}[of Lemma~\\ref{lemma:nd}]\n\\emph{W.l.o.g} we can rename and order the pairs in $T$ so that $w_1\\geq w_2\\geq\\ldots\\geq w_m $.\nThen, $\\mathcal{O}(T,b)$ has the following simple piecewise linear expression:\n\\begin{displaymath}\\label{eq:pw}\n    \\mathcal{O}(T,b) = \n    \\begin{cases}\n        b w_1&\\text{if }0\\leq b<p_1\\\\\n        \\displaystyle\n        \\sum_{k=1}^{i-1}p_k(w_k-w_i)\n        + b w_i\n        &\\displaystyle\\text{if } 0\\leq b - \\sum_{k=1}^{i-1}p_k< p_i\\\\\n        \\displaystyle\n        \\sum_{k=1}^m p_kw_k\n        &\\displaystyle\\text{if } b\\geq\\sum_{i=1}^m p_k\n\n    \\end{cases}\n\\end{displaymath}\n\nLet us define for $t\\in\\mathbf{R}^+$, $n(t)\\equiv \\inf\\Big\\{i\\text{ s.t.\n} \\sum_{k=1}^i p_k > t\\Big \\}$ with $n(t)=+\\infty$ when the set is empty. In\nparticular, note that $x\\mapsto n(t)$ is non-decreasing. Denoting\n$\\partial_+\\mathcal{O}_T$ the right derivative of $\\mathcal{O}(T,\\cdot)$, one\ncan write $\\partial_+\\mathcal{O}_T(t)=w_{n(t)}$, with the convention that\n$w_\\infty = 0$.\n\nWriting $i \\equiv \\sup\\Big\\{j\\text{ s.t.\n} w_j\\geq w_x\\Big\\}$, it is easy to see that\n$\\partial_+\\mathcal{O}_{T\\cup\\{x\\}}\\geq\\partial_+\\mathcal{O}_T$. Indeed:\n\\begin{enumerate}\n    \\item if $n(t)\\leq i$ then $\\partial_+\\mathcal{O}_{T\\cup\\{x\\}}(t)\n        = \\partial_+\\mathcal{O}_T(t)= w_{n(t)}$.\n    \\item if $n(t)\\geq i+1$ and $n(t-c)\\leq i$ then $\\partial_+\\mathcal{O}_{T\\cup\\{x\\}}(t)\n        = w_x\\geq w_{n(t)}= \\partial_+\\mathcal{O}_T(t)$.\n    \\item if $n(t-c)\\geq i+1$, then $\\partial_+\\mathcal{O}_{T\\cup\\{x\\}}\n        = w_{n(t-c)}\\geq w_{n(t)}=\\partial_+\\mathcal{O}_T(t)$.\n\\end{enumerate}\n\nLet us now consider $b$ and $c$ such that $b\\leq c$. Then, using the integral\nrepresentation of $\\mathcal{O}(T\\cup\\{x\\},\\cdot)$ and $\\mathcal{O}(T,\\cdot)$, we get:\n\\begin{multline*}\n    \\mathcal{O}(T\\cup\\{x\\},c)-\\mathcal{O}(T\\cup\\{x\\},b)=\\int_b^c\\partial_+\\mathcal{O}_{T\\cup\\{x\\}}(t)dt\\\\\n    \\geq\\int_b^c\\partial_+\\mathcal{O}_T(t)dt=\\mathcal{O}(T,c)-\\mathcal{O}(T,b)\n\\end{multline*}\nRe-ordering the terms, $\\mathcal{O}(T\\cup\\{x\\},c)-\\mathcal{O}(T,c)\\geq\n\\mathcal{O}(T\\cup\\{x\\},b)-\\mathcal{O}(T,b)$\nwhich concludes the proof of the lemma.\n\\end{proof}\n\n\\vspace{0.5em}\n\n\\begin{proof}[of Lemma~\\ref{lemma:sub}]\n    Let $T\\subseteq\\neigh{X}$ and $x, y\\in\\neigh{X}\\setminus T$. Using the\n    second-order characterization of submodular functions, it suffices to show\n    that:\n    \\begin{displaymath}\\label{eq:so}\n        \\mathcal{O}(T\\cup\\{x\\},b)-\\mathcal{O}(T,b)\\geq\n        \\mathcal{O}(T\\cup\\{y, x\\},b)-\\mathcal{O}(T\\cup\\{y\\},b)\n    \\end{displaymath}\n\n    We distinguish two cases based on the relative position of $w_x$ and $w_y$.\n    The following notations will be useful: $S_T^x \\equiv \\big\\{u\\in\n    T\\text{ s.t.  }w_x\\leq w_u\\big\\}$ and $P_T^x\\equiv\n    T\\setminus S_T^x$.\n\n    \\textbf{Case 1:} If $w_y\\geq w_x$, then one can\n    write:\n    \\begin{gather*}\n        \\mathcal{O}(T\\cup\\{y,x\\},b) = \\mathcal{O}(P_T^y\\cup\\{y\\},b_1)+\n        \\mathcal{O}(S_T^y\\cup\\{x\\},b_2)\\\\\n        \\mathcal{O}(T\\cup\\{y\\},b) = \\mathcal{O}(P_T^y\\cup\\{y\\},b_1)\n        + \\mathcal{O}(S_T^y,b_2)\n    \\end{gather*}\n    where $b_1$ is the fraction of the budget $b$ spent on $P_T^y\\cup\\{y\\}$ and\n    $b_2=b-b_1$.\n    \n    Similarly:\n    \\begin{gather*}\n        \\mathcal{O}(T\\cup\\{x\\},b) = \\mathcal{O}(P_T^y, c_1) + \\mathcal{O}(S_T^y\\cup\\{x\\},c_2)\\\\\n        \\mathcal{O}(T, b) = \\mathcal{O}(P_T^y, c_1) + \\mathcal{O}(S_T^y,c_2)\n    \\end{gather*}\n    where $c_1$ is the fraction of the budget $b$ spent on $P_T^y$ and $c_2\n    = b - c_1$. \n\n    Note that $b_1\\geq c_1$: an optimal solution will first spent as much\n    budget as possible on $P_T^y\\cup\\{y\\}$ before adding elements in\n    $S_T^y\\cup\\{x\\}$.\n\n    In this case:\n    \\begin{displaymath}\n        \\begin{split}\n            \\mathcal{O}(T\\cup\\{x\\},b)-\\mathcal{O}(T,b)&=\n        \\mathcal{O}(S_T^y\\cup\\{x\\},c_2)+\\mathcal{O}(S_T^y,c_2)\\\\\n        &\\geq \\mathcal{O}(S_T^y\\cup\\{x\\},b_2)+\\mathcal{O}(S_T^y,b_2)\\\\\n        & = \\mathcal{O}(T\\cup\\{y, x\\},b)-\\mathcal{O}(T\\cup\\{y\\},b)\n    \\end{split}\n    \\end{displaymath}\n    where the inequality comes from Lemma~\\ref{lemma:nd} and \n    $c_2\\geq b_2$.\n\n    \\textbf{Case 2:} If $w_x > w_y$, we now decompose\n    the solution on $P_T^x$ and $S_T^x$:\n    \\begin{gather*}\n        \\mathcal{O}(T\\cup\\{x\\},b) = \\mathcal{O}(P_T^x\\cup\\{x\\},b_1)\n        + \\mathcal{O}(S_T^x,b_2)\\\\\n        \\mathcal{O}(T,b) = \\mathcal{O}(P_T^x,c_1)+\\mathcal{O}(S_T^x,c_2)\\\\\n       \n        \\mathcal{O}(T\\cup\\{y, x\\},b) = \\mathcal{O}(P_T^x\\cup\\{x\\},b_1)\n        + \\mathcal{O}(S_T^x\\cup\\{y\\},b_2)\\\\\n        \\mathcal{O}(T\\cup\\{y\\},b) = \\mathcal{O}(P_T^x,c_1)+\\mathcal{O}(S_T^x\\cup\\{y\\},c_2)\n    \\end{gather*}\n    with $b_1+b_2=b$, $c_1+c_2=b$ and $b_2\\leq c_2$. \n\n    In this case again:\n        \\begin{multline*}\n            \\mathcal{O}(T\\cup\\{x\\},b)-\\mathcal{O}(T,b)=\n        \\mathcal{O}(S_T^x,b_2)-\\mathcal{O}(S_T^x,c_2)\\\\\n        \\geq \\mathcal{O}(S_T^x\\cup\\{y\\},b_2)-\\mathcal{O}(S_T^x\\cup\\{y\\},c_2)\\\\\n        = \\mathcal{O}(T\\cup\\{y, x\\},b)-\\mathcal{O}(T\\cup\\{y\\},b)\n    \\end{multline*}\n    where the inequality uses Lemma~\\ref{lemma:nd} and $c_2\\geq b_2$.\n\n    In both cases, we were able to obtain the second-order characterization of submodularity. This concludes the proof of the lemma.\n\\end{proof}\n\n\\vspace{0.5em}\n\n\\begin{proof}[of Proposition~\\ref{prop:sub}]\n    Let us consider $S$ and $T$ such that $S\\subseteq T\\subseteq X$ and $x\\in\n    X\\setminus T$. In particular, note that $\\neigh{S}\\subseteq\\neigh{T}$. \n\n    Let us write $\\neigh{S\\cup\\{x\\}}=\\neigh{S}\\cup R$ with $\\neigh{S}\\cap\n    R=\\emptyset$ and similarly, $\\neigh{T\\cup\\{x\\}}=\\neigh{T}\\cup R'$ with\n    $\\neigh{T}\\cap R'=\\emptyset$. It is clear that $R'\\subseteq R$. Writing $R'=\\{u_1,\\ldots,u_k\\}$:\n    \\begin{multline*}\n            \\mathcal{O}(\\neigh{T\\cup\\{x\\}},b)- \\mathcal{O}(\\neigh{T},b)\\\\\n            =\\sum_{i=1}^k\\mathcal{O}(\\neigh{T}\\cup\\{u_1,\\ldots u_i\\},b)\n            -\\mathcal{O}(\\neigh{T}\\cup\\{u_1,\\ldots u_{i-1}\\},b)\\\\\n            \\leq \\sum_{i=1}^k\\mathcal{O}(\\neigh{S}\\cup\\{u_1,\\ldots u_i\\},b)\n            -\\mathcal{O}(\\neigh{S}\\cup\\{u_1,\\ldots u_{i-1}\\},b)\\\\\n            =\\mathcal{O}(\\neigh{S}\\cup R',b)-\\mathcal{O}(\\neigh{S},b)\n    \\end{multline*}\n    where the inequality comes from the submodularity of $\\mathcal{O}(\\cdot,b)$ proved in Lemma~\\ref{lemma:sub}. This same function is also obviously set-increasing, hence:\n    \\begin{multline*}\n            \\mathcal{O}(\\neigh{S}\\cup R',b)-\\mathcal{O}(\\neigh{S},b)\\\\\n            \\leq \\mathcal{O}(\\neigh{S}\\cup R,b)-\\mathcal{O}(\\neigh{S},b)\\\\\n            =\\mathcal{O}(\\neigh{S\\cup\\{x\\}},b)-\\mathcal{O}(\\neigh{S},b)\n    \\end{multline*}\n    This concludes the proof of the proposition.\n\\end{proof}\n\n\n\\begin{proof}[of Proposition~\\ref{prop:main_result}]\n    We simply note that the content of the outer \\textsf{for} loop on line 2 of Algorithm~\\ref{alg:comb} is the greedy submodular maximization algorithm of \\cite{nemhauser}. Since $\\mathcal{O}(\\neigh{\\cdot}, t)$ is submodular (Proposition~\\ref{prop:sub}), this solves the inner $\\max$ in \\eqref{eq:sub-mod} with an approximation ratio of $(1-1\/e)$. The outer \\textsf{for} loop then computes the outer $\\max$ of \\eqref{eq:sub-mod}.\n\n    As a consequence, Algorithm~\\ref{alg:comb} computes a $(1-1\/e)$-approximate non-adaptive solution. We conclude by applying Proposition~\\ref{prop:cr}.\n\\end{proof}\n\n\\subsection{Parallelization}\n\\label{sec:para}\nAs discussed in the body of the paper, the algorithm can be parallelized across $k$ different machines, each one computing an approximation for a fixed budget $k-t$ in the first stage and $t$ in the second.\nA slightly more sophisticated approach is to consider only $\\log n$ splits: $(1,k-1),(2,k-2),\\ldots,(2^{\\lfloor \\log n \\rfloor},1)$ and then select the best solution from this set.  It is not hard to see that in comparison to the previous approach, this would reduce the approximation guarantee by a factor of at most 2: if the optimal solution is obtained by spending $t$ on the first stage and $k-t$ in the second stage, then since $t \\leq 2\\cdot2^{\\lfloor \\log t \\rfloor}$ the solution computed for $(2^{\\lfloor \\log t \\rfloor}, k - 2^{\\lfloor \\log t \\rfloor})$ will have at least half that value.  \nMore generally, for any $\\epsilon>0$ one can parallelize over $\\log_{1+\\epsilon}n$ machines at the cost of losing a factor of $(1+\\epsilon)$ in the approximation guarantee.\n\n\\subsection{Implementation in MapReduce}\n\\label{sec:mr}\n\nAs noted in Section~\\ref{sec:comb}, lines 4 to 7 of Algorithm~\\ref{alg:comb}\ncorrespond to the greedy heuristic of \\cite{nemhauser} applied to the\nsubmodular function\u00a0$f_t(S) \\equiv \\mathcal{O}\\big(\\neigh{S}, t\\big)$.\nA variant of this heuristic, namely the $\\varepsilon$-greedy heuristic,\ncombined with the \\textsc{Sample\\&Prune} method of \\cite{mr} allows us to write\na MapReduce version of Algorithm~\\ref{alg:comb}. The resulting algorithm is\ndescribed in Algorithm~\\ref{alg:combmr}\n\n\\begin{algorithm}\n    \\caption{Combinatorial algorithm, MapReduce}\n    \\label{alg:combmr}\n    \\algsetup{indent=2em}\n    \\begin{algorithmic}[1]\n        \\STATE $S\\leftarrow \\emptyset$\n        \\FOR{$t=1$ \\TO $k-1$}\n            \\STATE $S_t\\leftarrow \\emptyset$\n            \\FOR{$i=1$ \\TO $\\log_{1+\\varepsilon}\\Delta$}\n                \\STATE $U\\leftarrow X$, $S'\\leftarrow \\emptyset$\n                \\WHILE{$|U|>0$}\n                    \\STATE $R\\leftarrow$ sample from $U$ w.p. $\\min\\left(1,\n                    \\frac{\\ell}{|U|}\\right)$\n                    \\WHILE{$|R|>0$ \\OR $|S_t\\cup S'|< k$}\n                        \\STATE $x\\leftarrow$ some element from $R$\n                        \\IF{$\\nabla f_t(S_t\\cup S', x)\\geq\\frac{\\Delta}{(1+\\varepsilon)^i}$}\n                            \\STATE $S'\\leftarrow S'\\cup\\{x\\}$\n                        \\ENDIF\n                        \\STATE $R\\leftarrow R\\setminus\\{x\\}$\n                    \\ENDWHILE\n                    \\STATE $S_t\\leftarrow S_t\\cup S'$\n                    \\STATE $U\\leftarrow\\{x\\in U\\,|\\, \\nabla f_t(S_t,\n                    x)\\geq\\frac{\\Delta}{(1+\\varepsilon)^i}\\}$\n                \\ENDWHILE\n            \\ENDFOR\n            \\IF{$\\mathcal{O}(\\neigh{S_t},t)>\\mathcal{O}(\\neigh{S},k-|S|)$}\n                \\STATE $S\\leftarrow S_t$\n            \\ENDIF\n        \\ENDFOR\n        \\RETURN $S$\n    \\end{algorithmic}\n\\end{algorithm}\n\nWe denoted by $\\nabla f_t(S, x)$ the marginal increment of $x$ to the set $S$\nfor the function $f_t$, $\\nabla f_t(S, x) =  f_t(S\\cup\\{x\\}) - f_t(S)$.\n$\\Delta$ is an upper bound on the marginal contribution of any element. In our\ncase, $\\Delta = \\max_{u\\in\\neigh{X}} w_u$ provides such an upper bound. The\nsampling in line 7 selects a small enough number of elements that the\n\\texttt{while} loop from lines 8 to 14 can be executed on a single machine.\nFurthermore, lines 7 and 16 can be implemented in one round of MapReduce each.\n\nThe approximation ratio of Algorithm~\\ref{alg:combmr} is\n$1-\\frac{1}{e}-\\varepsilon$. The proof of this result as well as the optimal\nchoice of $\\ell$ follow from Theorem 10 in \\cite{mr}.\n\n\\subsection{Experimental setup}\nWe tested our algorithms on three types of datasets. Each of them allows us to\nexperiment on a different aspect of the adaptive seeding problem. The Facebook\nPages dataset that we collected ourselves has a central place in our\nexperiments since it is the one which is closet to actual applications of\nadaptive seeding.\n\n\\textbf{Synthetic networks.} Using standard models of social networks we\ngenerated large-scale graphs to model the social network.  To emulate the\nprocess of users following a topic (the core set $X$) we sampled subsets of\nnodes at random, and applied our algorithms on the sample and their neighbors.\nThe main advantage of these data sets is that they allow us to generate graphs\nof arbitrary sizes and experiment with various parameters that govern the\nstructure of the graph.  The disadvantages are that users who follow a topic\nare not necessarily random samples, and that social networks often have\nstructural properties that are not captured in generative models.\n\n\\textbf{Real networks.}  We used publicly available data sets of real social\nnetworks available at \\cite{snapnets}. As for synthetic networks, we used\na random sample of nodes to emulate users who follow a topic, which is the main\ndisadvantage of this approach.  The advantage however, is that such datasets\ncontain an entire network which allows testing different propagation\nparameters.   \n  \n\\textbf{Facebook Pages.}  We collected data from several Facebook Pages, each\nassociated with a commercial entity that uses the Facebook page to communicate\nwith its followers. For each page, we selected a post and then collected data\nabout the users who expressed interest (``liked'') the post and their friends.\nThe advantage of this data set is that it is highly representative of the\nscenario we study here. Campaigns run on a social network will primarily target\nusers who have already expressed interests in the topic being promoted. The\nmain disadvantage of this method is that such data is extremely difficult to\ncollect due to the crawling restrictions that Facebook applies and gives us\nonly the 2-hop neighborhood around a post. This makes it difficult to\nexperiment with different propagation parameters. Fortunately, as we soon\ndiscuss, we were able to circumvent some of the crawling restrictions and\ncollect large networks, and the properties of the voter influence model are\nsuch that these datasets suffice to accurately account for influence\npropagation in the graph.\\newline\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{images\/para.pdf}\n    \\caption{Comparison of the average degree of the core set users and the\n    average degree of their friends.}\n    \\label{fig:paradox}\n    \\vspace{-10pt}\n\\end{figure}\n\n\\noindent\\textbf{Data collection.}\nWe selected Facebook Pages in different verticals (topics). Each page is\noperated by an institution or an entity whose associated Facebook Page is\nregularly used for promotional posts related to this topic. On each of these\npages, we selected a recent post (posted no later than January 2014) with\napproximately 1,000 \\emph{likes}. The set of users who liked those posts\nconstitute our core set.  We then crawled the social network of\nthese sets: for each user, we collected her list of friends, and the degrees\n(number of friends) of these friends.\\newline\n\n\\noindent\\textbf{Data description.} Among the several verticals we collected,\nwe select eight of them for which we will report our results. We obtained\nsimilar results for the other ones.  Table~\\ref{tab:data} summarizes statistics\nabout the selected verticals.  We note that depending on the privacy settings\nof the core set users, it was not always possible to access their list of\nfriends.  We decided to remove these users since their ability to spread\ninformation could not be readily determined. This effect, combined with various\nerrors encountered during the data collection, accounts for an approximate 15\\%\nreduction between the users who liked a post and the number of users in the\ndatasets we used.  Following our discussion in the introduction, we observe\nthat on average, the degrees of core set users is much lower than the degrees of\ntheir friends. This is highlighted on Figure~\\ref{fig:paradox} and justifies\nour approach.\n\n\\begin{table}[t]\n    \\small\n    \\centering\n    \\setlength{\\tabcolsep}{3pt}\n    \\begin{tabular}{llrr}\n    \\toprule\n    Vertical & Page & $m$ & $n$ \\\\%& $S$ & $F$\\\\\n    \\midrule\n    Charity & Kiva & 978 & 131334 \\\\%& 134.29 & 1036.26\\\\\n    Travel & Lonely Planet & 753 & 113250 \\\\%& 150.40 & 898.50\\\\\n   \n    Fashion & GAP & 996 & 115524 \\\\%& 115.99 & 681.98\\\\\n    Events & Coachella & 826 & 102291 \\\\%& 123.84 & 870.16\\\\\n    Politics & Green Party & 1044 & 83490 \\\\%& 79.97 & 1053.25\\\\\n    Technology & Google Nexus & 895 & 137995 \\\\%& 154.19 & 827.84\\\\\n    News & The New York Times & 894 & 156222 \\\\%& 174.74 & 1033.94 \\\\\n   \n    Entertainment & HBO & 828 & 108699 \\\\%& 131.28 & 924.09\\\\\n    \\bottomrule\n\\end{tabular}\n\\caption{Dataset statistics. $m$: number of users in the core set, $n$: number\nof friends of core set users.}\n\\label{tab:data}\n    \\vspace{-10pt}\n\\end{table}\n\n\\subsection{Performance of Adaptive Seeding}\n\\label{sec:performance} For a given problem instance with a budget of $k$ we\napplied the adaptive seeding algorithm (the combinatorial version). Recall from\nSection~\\ref{sec:model} that performance is defined as the expected influence\nthat the seeder can obtain by optimally selecting users on the second stage,\nwhere \\emph{influence} is defined as the sum of the degrees of the selected\nusers. We tested our algorithm against the following benchmarks:\n\n\\begin{itemize}\n    \\item \\emph{Random Node} (\\textsf{RN}): we randomly select $k$ users from\n        the core set.  This is a typical benchmark in comparing influence\n        maximization algorithms~\\cite{KKT03}.\n   \\item \\emph{Influence Maximization} (\\textsf{IM}): we apply the optimal\n       influence maximization algorithm on the core set.  This is the naive\n       application of influence maximization. For the voter model, when the\n       propagation time is polynomially large in the network size, the optimal\n       solution is to simply take the $k$ highest degree nodes~\\cite{even-dar}.\n       We study the case of bounded time horizons in Section~\\ref{sec:inf}.\n    \\item \\emph{Random Friend} (\\textsf{RF}): we implement a naive two-stage approach:\n        randomly select $k\/2$ nodes from the core set, and for each\n        node select a random neighbor (hence spending the budget of $k$ rewards\n        overall).  This method was recently shown to outperform standard\n        influence maximization when the core set is random~\\cite{LS13}.\n\\end{itemize}\n\n\n\\begin{figure*}[t]\n    \\centerline{\\includegraphics[width=0.99\\textwidth]{images\/perf2.pdf}}\n    \\vspace{-5pt}\n    \\caption{\\small{Performance of adaptive seeding compared to other influence\n    maximization approaches. The horizontal axis represents the budget used as\na fraction of the size of the core set. The vertical axis is the\nexpected influence reachable by optimally selecting nodes on the second stage.}}\n    \\label{fig:performance}\n    \\vspace{-10pt}\n\\end{figure*}\n\n\\subsection{Performance on Facebook Pages} Figure~\\ref{fig:performance}\ncompares the performance of \\emph{adaptive seeding}, our own approach, to the\nafore-mentioned approaches for all the verticals we collected.  In this first\nexperiment we made simplifying assumptions about the parameters of the model.\nThe first assumption is that all probabilities in the adaptive seeding model\nare equal to one. This implicitly assumes that every friend of a user who\nfollowed a certain topic is interested in promoting the topic given a reward.\nAlthough this is a strong assumption that we will revisit, we note that the\nprobabilities can be controlled to some extent by the social networking service\non which the campaign is being run by showing prominently the campaign material\n(sponsored links, fund-raising banners, etc.).  The second assumption is that\nthe measure of influence is the sum of the degrees of the selected set.  This\nmeasure is an appealing proxy as it is known that in the voter model, after\npolynomially many time steps, the influence of each node is proportional to its\ndegree with high probability~\\cite{even-dar}. Since the influence process\ncannot be controlled by the designer, the assumption is often that the\ninfluence process runs until it stabilizes (in linear thresholds and\nindependent cascades for example, the process terminates after a linear number\nof steps~\\cite{KKT03}).  We perform a set of experiments for different time\nhorizons in Section~\\ref{sec:inf}.\n\n \n\n\nIt is striking to see how well adaptive seeding does in comparison to other\nmethods.  Even when using a small budget (0.1 fraction of the core set, which\nin these cases is about 100 nodes), adaptive seeding improves influence by\na factor of at least 10, across all verticals.  To confirm this, we plot the\nrelative improvements of \\emph{adaptive seeding} over \\textsf{IM} in aggregate\nover the different pages.  The results are shown in Figure~\\ref{fig:compare}.\nThis dramatic improvement is largely due to the friendship paradox phenomenon\nthat adaptive seeding leverages.  Returning to Figure~\\ref{fig:performance}, it\nis also interesting to note that the \\textsf{RF} heuristic significantly\noutperforms the standard \\textsf{IM} benchmark.  Using the same budget, the\ndegree gain induced by moving from the core set to its neighborhood is such\nthat selecting at random among the core set users' friends already does better\nthan the best heuristic restricted only on the core set. Using \\emph{adaptive\nseeding} to optimize the choice of core set users based on their friends'\ndegrees then results in an order of magnitude increase over \\textsf{RF},\nconsistently for all the pages.\\newline\n\n\\begin{figure}[t]\n    \\vspace{-10pt}\n    \\centerline{ \\includegraphics[width=0.4\\textwidth]{images\/comp2.pdf} }\n    \\vspace{-10pt}\n    \\caption{Ratio of the performance of adaptive seeding to \\textsf{IM}. Bars represents the mean improvement across all verticals, and the ``error bar'' represents the range of improvement across verticals.}\n    \\label{fig:compare}\n    \\vspace{-15pt}\n\\end{figure}\n\n\\subsection{The Effect of the Probabilistic Model}\n\\label{sec:robustness}\n\nThe results presented in Section~\\ref{sec:performance} were computed assuming\nthe probabilities in the adaptive seeding model are one.  We now describe\nseveral experiments we performed with the Facebook Pages data set that test the\nadvantages of adaptive seeding under different probability models.\\newline \n\n\n\\noindent\\textbf{Impact of the Bernouilli parameter.} \nFigure~\\ref{fig:prob} shows the impact of the probability of nodes realizing in\nthe second stage.  We computed the performance of \\emph{adaptive seeding} when\neach friend of a seeded user in the core set joins during the second stage\nindependently with probability $p$, using different values of $p$.  We call $p$\nthe \\emph{Bernouilli} parameter, since the event that a given user joins on the\nsecond stage of adaptive seeding is governed by a Bernouilli variable of\nparameter $p$. We see that even with $p=0.01$, \\emph{adaptive seeding} still\noutperforms \\textsf{IM}.  As $p$ increases, the performance of \\emph{adaptive\nseeding} quickly increases and reaches $80\\%$ of the values of\nFigure~\\ref{fig:performance} at $p=0.5$.\\newline\n\n\\begin{figure}[t]\n    \\begin{subfigure}[t]{0.23\\textwidth}\n    \\includegraphics[scale=0.48]{images\/prob.pdf}\n    \\vspace{-10pt}\n    \\caption{}\n    \\label{fig:prob}\n\\end{subfigure}\n\\hspace{1pt}\n\\begin{subfigure}[t]{0.23\\textwidth}\n    \\includegraphics[scale=0.48]{images\/hbo_likes.pdf}\n    \\caption{}\n    \\label{fig:killer}\n     \\end{subfigure}\n \\vspace{-5pt}\n \\caption{\\small{(a) Performance of adaptive seeding for various propagation\n     probabilities. (b) Performance of \\emph{adaptive seeding} when restricted\n to the subgraph of users who \\emph{liked} HBO (red line).}}\n \\vspace{-20pt}\n\\end{figure}\n\n\\noindent\\textbf{Coarse estimation of probabilities.}  \nIn practice, the probability a user may be interested in promoting a campaign\nher friend is promoting may vary. However, for those who have already expressed\ninterest in the promoted content, we can expect this probability to be close to\none.  We therefore conducted the following experiment.  We chose a page (HBO)\nand trimmed the social graph we collected by only keeping on the second stage\nusers who indicated this page (HBO) in their list of interests.  This is\na coarse estimation of the probabilities as it assumes that if a friend follows\nHBO she will be willing to promote with probability 1 (given a reward), and\notherwise the probability of her promoting anything for HBO is 0.\nFigure~\\ref{fig:killer} shows that even on this very restricted set of users,\n\\emph{adaptive seeding} still outperforms \\textsf{IM} and reaches approximately\n$50\\%$ of the unrestricted adaptive seeding.\\newline\n\n\\noindent\\textbf{Impact of the probability distribution.} In order to test\nscenarios where users have a rich spectrum of probabilities of realizing on the\nsecond stage. We consider a setting where the Bernouilli parameter $p$ is drawn\nfrom a distribution. We considered four different distributions; for each\ndistribution for fixed values of the budget and the parameter $p$, we tuned the\nparameters of the distribution so that its mean is exactly $p$. We then plotted\nthe performance as a function of the budget and mean $p$. \n\nFor the Beta distribution, we fixed $\\beta=5$ and tuned the $\\alpha$ parameter\nto obtain a mean of $p$, thus obtaining a unimodal distribution. For the normal\ndistribution, we chose a standard deviation of $0.01$ to obtain a distribution\nmore concentrated around its mean than the Beta distribution. Finally, for the\ninverse degree distribution, we took the probability of a node joining on\nthe second stage to be proportional to the inverse of its degree (scaled so that on\naverage, nodes join with probability $p$).  The results are shown in\nFigure~\\ref{fig:bernouilli}.\n\nWe observe that the results are comparable to the one we obtained in the\nuniform case in Figure~\\ref{fig:prob} except in the case of the inverse degree\ndistribution for which the performance is roughly halved. Remember that the\nvalue of a user $v$ on the second stage of adaptive seeding is given by $p_v\nd_v$ where $d_v$ is its degree and $p_v$ is the its probability of realizing on\nthe second stage.  Choosing $p_v$ to be proportional to ${1}\/{d_v}$ has the\neffect of normalizing the nodes on the second stage and is a strong\nperturbation of the original degree distribution of the nodes available on the\nsecond stage.\n\n\\begin{figure*}[t!]\n\\centering\n\\begin{subfigure}[b]{0.25\\textwidth}\n\\includegraphics[width=\\textwidth]{images\/beta.pdf}\n\\caption{Beta distribution}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.25\\textwidth}\n\\includegraphics[width=\\textwidth]{images\/gauss.pdf}\n\\caption{Normal Distribution}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.24\\textwidth}\n\\includegraphics[width=\\textwidth]{images\/power.pdf}\n\\caption{Power-law distribution}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.24\\textwidth}\n\\includegraphics[width=\\textwidth]{images\/deg.pdf}\n\\caption{Inverse degree}\n\\end{subfigure}\n\\caption{Performance of adaptive seeding as a function of the budget and the\nmean of the distribution from which the Bernouilli parameters are drawn. The\ndetails of the parameters for each distribution can be found in\nSection~\\ref{sec:robustness}.}.\n\\label{fig:bernouilli}\n\\vspace{-10pt}\n\\end{figure*}\n\n\\subsection{Impact of the Influence Model}\n\\label{sec:inf}\n\nThe Facebook Pages data set we collected is limited in that we only have access\nto the 2-hop neighborhood around the seed users and we use the degree of the\nsecond stage users as a proxy for their influence. As proved in\n\\cite{even-dar}, in the voter model, the influence of nodes converges to their\ndegree with high probability when the number of time steps become polynomially\nlarge in the network size.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{images\/voter.pdf}\n    \\vspace{-20pt}\n    \\caption{Performance of adaptive seeding compared to \\textsf{IM} for the voter\n    influence model with $t$ steps.}\n    \\vspace{-10pt}\n    \\label{fig:voter}\n\\end{figure}\n\nIn order to analyze the expected number of nodes influenced according to the\nvoter model that terminates after some fixed number of time steps, we use\npublicly available data sets from \\cite{snapnets} where the entire network is\nat our disposal.  As discussed above, we sample nodes uniformly at random to\nmodel the core set. We then run the voter model for $t$ time steps\nto compute the influence of the second stage users. Figure~\\ref{fig:voter}\nshows the performance of adaptive seeding as a function of $t$ compared to the\nperformance of the \\textsf{IM} benchmark. In this experiment, the budget was\nset to half the size of the core set.\n\nWe see that the performance of adaptive seeding quickly converges (5 time steps\nfor \\textsf{Slashdot}, 15 time steps for \\textsf{Epinions}). In practice, the\nvoter model converges much faster than the theoretical guarantee of\n\\cite{even-dar}, which justifies using the degree of the second stage users as\nmeasure of influence as we did for the Facebook Pages data sets.\nFurthermore, we see that similarly to the Facebook data sets, adaptive seeding\nsignificantly outperforms \\textsf{IM}.\n\n\n\\subsection{Performance on Synthetic Networks} \n\nIn order to analyze the impact of topological variations we generated synthetic\ngraphs using standard network models. All the generated graphs have $100,000$\nvertices, for each model, we tuned the generative parameters to obtain when\npossible a degree distribution (or graph density otherwise) similar to what we\nobserved in the Facebook Pages data sets.\n\n\\begin{itemize}\n    \\item \\emph{Barab\u00e1si-Albert:} this well-known model is often used to model\n        social graphs because its degree distribution is a power law. We took\n        10 initial vertices and added 10 vertices at each step, using the\n        preferential attachment model, until we reached 100,000 vertices.\n    \\item \\emph{Small-World:} also known as the Watts-Strogatz model. This\n        model was one of the first models proposed for social networks. Its\n        diameter and clustering coefficient are more representative of a social\n        network than what one would get with the Erd\u0151s\u2013R\u00e9nyi model. We started\n        from a regular lattice of degree 200 and rewired each edge with\n        probability 0.3.\n    \\item \\emph{Kronecker:} Kronecker graphs were more recently introduced in\n        \\cite{kronecker} as a scalable and easy-to-fit model for social\n        networks. We started from a star graph with 4 vertices and computed\n        Kronecker products until we reached 100,000 nodes.\n    \\item \\emph{Configuration model:} The configuration model allows us to\n        construct a graph with a given degree distribution. We chose a page\n        (GAP) and generated a graph with the same degree distribution using the\n        configuration model.\n\\end{itemize}\nThe performance of adaptive seeding compared to our benchmarks can be found in\nFigure~\\ref{fig:synth}. We note that the improvement obtained by adaptive\nseeding is comparable to the one we had on real data except for the\n\\emph{Small-World} model. This is explained by the nature of the model:\nstarting from a regular lattice, some edges are re-wired at random. This model\nhas similar properties to a random graph where the friendship paradox does not\nhold~\\cite{LS13}. Since adaptive seeding is designed to leverage the friendship\nparadox, such graphs are not amenable to this approach.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{images\/perf_synth.pdf}\n    \\vspace{-15pt}\n    \\caption{Performance of adaptive seeding on synthetic networks.}\n    \\label{fig:synth}\n    \\vspace{-10pt}\n\\end{figure}\n\n\\subsection{Scalability}\\label{sec:scalability}\nTo test the scalability of adaptive seeding we were guided by two central\nquestions.  First, we were interested to witness the benefit our non-sampling\napproach has over the standard SAA method. Secondly, we wanted to understand\nwhen one should prefer to use the LP-based approach from Section~\\ref{sec:lp}\nover the combinatorial one from Section~\\ref{sec:comb}. The computations in\nthis section were run on Intel Core i5 CPU 4x2.40Ghz. For each computation, we\nplot the time and number of CPU cycles it took.\\newline\n\n\\noindent\\textbf{Comparison with SAA.} The objective function of the\nnon-adaptive problem \\eqref{eq:relaxed} is an expectation over exponentially\nmany sets, all possible realizations of the neighbors in the second stage.\nFollowing the sampling-based approach introduced in \\cite{singer}, this\nexpectation can be computed by averaging the values obtained in\n$O\\left(n^2\\right)$ independent sample realizations of the second stage users\n($n$ is the number of neighbors of core set users).  One important aspect of\nthe algorithms introduced in this paper is that in the additive case, this\nexpectation can be computed exactly without sampling, thus significantly\nimproving the theoretical complexity.\n\nIn Figure~\\ref{fig:sampling}, we compare the running time of our combinatorial\nalgorithm to the same algorithm where the expectation is computed via sampling.\nWe note that this sampling-based algorithm is still simpler than the algorithm\nintroduced in \\cite{singer} for general influence models.  However, we observe\na significant gap between its running time and the one of the combinatorial\nalgorithm. Since each sample takes linear time to compute, this gap is in fact\n$O(n^3)$, quickly leading to impracticable running times as the size of the\ngraph increases. This highlights the importance of the \\emph{sans-sampling}\napproach underlying the algorithms we introduced.\\newline\n\n\\begin{figure}[t]\n    \\centerline{ \\includegraphics[width=0.48\\textwidth]{images\/sampling.pdf} }\n    \\vspace{-10pt}\n    \\caption{Running time and number of CPU cycles used by the sampling-based\n    algorithm and the combinatorial adaptive seeding algorithm for different\nsizes of the core set.}\n    \\label{fig:sampling}\n    \\vspace{-10pt}\n\\end{figure}\n\n\\noindent\\textbf{Combinatorial vs. LP algorithm.}\nWe now compare the running time of the LP-based approach and the combinatorial\napproach for different instance sizes.\n\nFigure~\\ref{fig:time} shows the running time and number of CPU cycles used by\nthe LP algorithm and the combinatorial algorithm as a function of the network\nsize $n$. The varying size of the network was obtained by randomly sampling\na varying fraction of core set users and then trimming the social graph by only\nkeeping friends of this random sample on the second stage.  The LP solver used\nwas CLP~\\cite{clp}.\n\n\\begin{figure}[t]\n    \\centerline{ \\includegraphics[scale=0.9]{images\/time.pdf} }\n    \\vspace{-10pt}\n    \\caption{Running time and number of CPU cycles of the combinatorial algorithm and the LP algorithm as a function of the number of nodes $n$. First row with budget $k=100$, second row with budget $k=500$.}\n    \\label{fig:time}\n    \\vspace{-10pt}\n\\end{figure}\n\nWe observe that for a \\emph{small} value of the budget $k$ (first row\nof Figure~\\ref{fig:time}), the combinatorial algorithm outperforms the LP\nalgorithm. When $k$ becomes \\emph{large} (second row of\nFigure~\\ref{fig:time}), the LP algorithm becomes faster. This can be explained\nby the $k^2$ factor in the running time of the combinatorial\nalgorithm (see Proposition~\\ref{prop:running_time}). Even though the asymptotic\nguarantee of the combinatorial algorithm should theoretically\noutperform the LP-based approach for large $n$, we were not able to observe it\nfor our instance sizes. In practice, one can choose which of the two algorithms\nto apply depending on the relative sizes of $k$ and $n$.\\newline\n\n\n\n\n\\subsection{A scalable approach}\n\n\\noindent \\textbf{Scalability.}  One of the challenges in adaptive seeding is\nscalability.  This is largely due to the stochastic nature of the problem\nderived from uncertainty about arrival of neighbors.  The main result\nin~\\cite{singer} is a constant factor approximation algorithm for well-studied\ninfluence models such as independent cascade and linear threshold which is, at\nlarge, a theoretical triumph.  These algorithms rely on various forms of\nsampling, which lead to a significant blowup in the input size.  While such\ntechniques provide strong theoretical guarantees, for social network data sets\nwhich are often either large or massive, such approaches are inapplicable.  The\nmain technical challenge we address in this work is how to design scalable\nadaptive optimization techniques for influence maximization which do not\nrequire sampling.\\newline\n\n\\noindent \\textbf{Beyond random users.}  The motivation for the adaptive\napproach hinges on the friendship paradox, but what if the core set is not\na random sample?  The results in~\\cite{LS13} hold when the core set of users is\nrandom but since users who follow a particular topic are not a random sample of\nthe network, we must somehow evaluate adaptive seeding on representative data\nsets.  The experimental challenge is to estimate the prominence of high degree\nneighbors in settings that are typical of viral marketing campaigns.\nFigure~\\ref{fig:para} is a foreshadowing of the experimental methods we used to\nshow that an effect similar to the friendship paradox exists in such cases as\nwell.  \\newline\n\n   \n\n\n \n\n\n\n\n\n\n\n\n\n\n\\noindent \\textbf{Main results.}\nOur main results in this paper show that adaptive seeding is a scalable\napproach which can dramatically improve upon standard approaches of influence\nmaximization.  We present a general method that enables designing adaptive\nseeding algorithms in a manner that avoids sampling, and thus makes adaptive\nseeding scalable to large size graphs.  We use this approach as a basis for\ndesigning two algorithms, both achieving an approximation ratio of $(1-1\/e)$\nfor the adaptive problem.  The first algorithm is implemented through a linear\nprogram, which proves to be extremely efficient over instances where there is\na large budget.  The second approach is a combinatorial algorithm with the same\napproximation guarantee which can be easily parallelized, has good theoretical\nguarantees on its running time and does well on instances with smaller budgets.\nThe guarantees of our algorithms hold for linear models of influence,\n\\emph{i.e.} models for which the influence of a set can be expressed as the sum\nof the influence of its members.  While this class does not include models such\nas the independent cascade and the linear threshold model, it includes the\nwell-studied \\emph{voter model}~\\cite{holley1975ergodic,even-dar} and measures\nsuch as node degree, click-through-rate or retweet measures of users which\nserve as natural proxies of influence in many settings~\\cite{ZHGS10}.  In\ncomparison to submodular influence functions, the relative simplicity of linear\nmodels allows making substantial progress on this challenging problem.\n\nWe then use these algorithms to conduct a series of experiments to show the\npotential of adaptive approaches for influence maximization both on synthetic\nand real social networks. The main component of the experiments involved\ncollecting publicly available data from Facebook on users who expressed\ninterest (``liked'') a certain post from a topic they follow and data on their\nfriends.  The premise here is that such users mimic potential participants in\na viral marketing campaign.  The results on these data sets suggest that\nadaptive seeding can have dramatic improvements over standard influence\nmaximization methods.\\newline\n\n\\noindent \\textbf{Paper organization.}\nWe begin by formally describing the model and the assumptions we make in the\nfollowing section.  In Section~\\ref{sec:adaptivity} we describe the reduction\nof the adaptive seeding problem to a non-adaptive relaxation.  In\nSection~\\ref{sec:algorithms} we describe our non-adaptive algorithms for\nadaptive seeding. In Section~\\ref{sec:experiments} we describe our\nexperiments, and conclude with a brief discussion on related work.\n \n\n\\subsection{Problem and notations}\nGiven a graph $G=(V,E)$, for a node $v\\in V$ we denote by $\\neigh{v}$\nthe neighborhood of $v$. By extension, for any subset of nodes $S\\subseteq V$,\n$\\neigh{S}\\equiv \\bigcup_{v\\in S}\\neigh{v}$ will denote the neighborhood of\n$S$. The notion of influence in the graph is captured by a function\n$f:2^{|V|}\\rightarrow \\mathbf{R}_+$ mapping a subset of nodes to a non-negative\ninfluence value.\\newline\n\n\\noindent \\textbf{The adaptive seeding model.} \nThe input of the \\emph{adaptive seeding} problem is a \\emph{core set} of nodes\n$X\\subseteq V$ and for any node $u\\in\\neigh{X}$ a probability $p_u$ that $u$\nrealizes if one of its neighbor in $X$ is seeded. We will write $m=|X|$ and\n$n=|\\neigh{X}|$ the parameters controlling the input size. The seeding process\nis the following:\n\\begin{enumerate}\n    \\item \\emph{Seeding:} the seeder selects a subset of nodes $S\\subseteq\n    X$ in the core set.\n    \\item \\emph{Realization of the neighbors:} every node $u\\in\\neigh{S}$\n    realizes independently with probability $p_u$. We denote by\n    $R\\subseteq\\neigh{S}$ the subset of nodes that is realized during this\n    stage.\n    \\item \\emph{Influence maximization:} the seeder selects the set of nodes\n    $T\\subseteq R$ that maximizes the influence function $f$.\n\\end{enumerate}\n\nThere is a budget constraint $k$ on the total number of nodes that can be\nselected: $S$ and $T$ must satisfy $|S|+|T|\\leq k$. The seeder chooses the set\n$S$ before observing the realization $R$ and thus wishes to select optimally in\nexpectation over all such possible realizations. Formally, the objective can be\nstated as:\n\\begin{equation}\\label{eq:problem}\n    \\begin{split}\n        &\\max_{S\\subseteq X} \\sum_{R\\subseteq\\neigh{S}} p_R\n        \\max_{\\substack{T\\subseteq R\\\\|T|\\leq k-|S|}}f(T)\\\\\n        &\\text{s.t. }|S|\\leq k\n    \\end{split}\n\\end{equation}\nwhere $p_R$ is the probability that the set $R$ realizes,\n$    p_R \\equiv \\prod_{u\\in R}p_u\\prod_{u\\in\\neigh{S}\\setminus R}(1-p_u)$.\n\nIt is important to note that the process through which nodes arrive in the\nsecond stage is \\emph{not} an influence process.  The nodes in the second stage\narrive if they are willing to spread information in exchange for a unit of the\nbudget. Only when they have arrived does the influence process occur. This\nprocess is encoded in the influence function and occurs after the influence\nmaximization stage without incentivizing nodes along the propagation path.  In\ngeneral, the idea of a two-stage (or in general, multi-stage) approach is to\nuse the nodes who arrive in the first stage to recruit influential users who\ncan be incentivized to spread information. In standard influence maximization,\nthe nodes who are not in the core set do not receive incentives to propagate\ninformation, and cascades tend to die off\nquickly~\\cite{YC10,BHMW11,GWG12,CADKL14}.\\newline\n\n\n\\noindent \\textbf{Influence functions.}\nIn this paper we focus on \\emph{linear} (or additive) influence models:\nin these models the value of a subset of nodes can be expressed as\na weighted sum of their individual influence.  One important example of such\nmodels is the \\emph{voter model} \\cite{richardson} used to represent the spread\nof opinions in a social network: at each time step, a node adopts an opinion\nwith a probability equal to the fraction of its neighbors sharing this opinion\nat the previous time step. Formally, this can be written as a discrete-time\nMarkov chain over opinion configurations of the network.  In this model\ninfluence maximization amounts to ``converting'' the optimal subset of nodes to\na given opinion at the initial time so as to maximize the number of converts\nafter a given period of time.  Remarkably, a simple analysis shows that under\nthis model, the influence function $f$ is additive:\n\\begin{equation}\\label{eq:voter}\n    \\forall S\\subseteq V,\\; f(S) = \\sum_{u\\in S} w_u\n\\end{equation}\nwhere $w_u, u\\in V$ are weights which can be easily computed from the powers of\nthe transition matrix of the Markov chain. This observation led to the\ndevelopment of fast algorithms for influence maximization under the voter\nmodel~\\cite{even-dar}.\\newline\n\n\\noindent \\textbf{\\textsc{NP}-Hardness.} In contrast to standard influence maximization, adaptive seeding is already \\textsc{NP}-Hard even for the simplest influence functions such as $f(S) = |S|$ and when all probabilities are one.  We discuss this in Appendix~\\ref{sec:alg-proofs}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\nIn their seminal work on spectral properties of the distance matrix $D$ of a tree $T$, Graham and Lov\\'asz \\cite{GL78} showed that, for a tree $T$ and $c_k(T)$ denoting the coefficient of $x^k$ in $\\text{det}(D(T)-xI)=(-1)^np_{D(G)}(x)$, the quantity $d_k(T)=(-1)^{n-1}c_k(T)\/2^{n-k-2}$  is determined as a fixed linear combination of the number of certain subtrees in $T$. The values $d_k(T)$ are called the \\emph{normalized coefficients} of the distance characteristic polynomial of $T$. \n\nA sequence $a_0,a_1,a_2,\\ldots, a_n$ of real numbers is {\\em unimodal} if there is a $k$ such that $a_{i-1}\\leq a_{i}$ for $i\\leq k$ and $a_{i}\\geq a_{i+1}$ for $i\\geq k$. Graham and Lov\\'asz \\cite{GL78} conjectured that the sequence $d_0(T),\\ldots,d_{n-2}(T)$ of normalized coefficients of the distance characteristic polynomial of a tree is unimodal with the maximum value occurring at $\\lfloor\\frac{n}{2}\\rfloor$ for a tree $T$ of order $n$.\nLittle progress on this problem is known. Collins \\cite{C89} confirmed the conjecture for stars, and also showed that for a paths the sequence is unimodal with a maximum value at $(1-\\frac{1}{\\sqrt{5}})n$. Thus, Graham and Lov\\'asz conjecture was reformulated as follows:\n\n\\begin{con}\\cite{GL78,C89}\\label{con:coefficientstree}\nThe normalized coefficients of the distance characteristic polynomial of any tree with $n$ vertices are unimodal with peak between $\\lfloor \\frac{n}{2} \\rfloor$ and $\\lceil(1-\\frac{1}{\\sqrt{5}})n\\rceil$.\n\\end{con}\n\nAalipour et al. \\cite{GRWC2015} confirmed the unimodality part of Conjecture \\ref{con:coefficientstree} by proving that the sequence of normalized coefficients is\nindeed always unimodal; in fact, they proved the stronger statement that this sequence is log-concave. \n\n\n\n\nA natural and widely used generalization of trees are block graphs (also known as clique trees). A connected graph is a \\emph{block graph} or \\emph{clique tree} if its blocks ($2$-connected components) are cliques. Graham, Hoffman, and Hosoya \\cite{GHH77} showed that the determinant of the distance matrix of any graph only depends on the determinant of its $2$-connected components.\nBapat and  Sivasubramanian \\cite{BS} extended the result of the determinant of the distance matrix of a tree by Graham and Lov\\'asz from \\cite{GL78} to block graphs, and Das and Mohanty \\cite{DM} did the same for multi-block graphs. Also the distance eigenvalues of a block graph have received some attention \\cite{DLbook, LLL,JGZ21,XLS2020}. The study of distance matrix has a long history, and many results concerning the distance matrix and distance eigenvalues are reported in the literature, for a survey we refer the reader to the survey papers by Aouchiche and Hansen \\cite{AH} and Hogben and Reinhart \\cite{HR21}. \n\n\nIn this article we consider the sequence of coefficients $c_0,c_1,\\dots,c_n$ of the distance characteristic polynomial of a block graph. Motivated by Conjecture \\ref{con:coefficientstree}, we investigate the following question:\n \n \n\\begin{ques} \\label{que:coefficientsblock}\n Is the sequence of coefficients $c_0,c_1,\\dots,c_n$ of the distance characteristic polynomial of a block graph unimodal, and where does it peak?\n\\end{ques}\n\n\nThis paper is structured as follows. In Section \\ref{preliminaries}, we start by recalling some definitions and preliminary results. In Section \\ref{sec:emb} we develop a general theory regarding the eigenvalues of metric spaces. We use this in Section \\ref{sec:unimodality} to show the unimodality part of Question \\ref{que:coefficientsblock} for block graphs, which extends the corresponding result for trees by Aalipour {\\it et al.}~\\cite{GRWC2015}. In Section \\ref{sec:peak} we prove that the peak part of Question~\\ref{que:coefficientsblock} holds for several extremal classes of block graphs with small diameter. As a corollary of our results we obtain Collins' result for stars \\cite{C89}. Although we show that the peak can move quite a bit when considering block graphs, our results provide evidence that Conjecture~\\ref{con:coefficientstree} might still hold in a more general setting.\n\n\n\\section{Preliminaries}\\label{preliminaries}\n\nThroughout this paper, $G=(V,E)$ denotes an undirected, simple, connected and loopless graph with $n$ vertices. The \\emph{distance matrix} $D(G)$ of a connected graph $G$ is the matrix indexed by the vertices of $G$ whose $(i,j)$-entry equals the distance  between the vertices $v_i$ and $v_j$, i.e., the length of a shortest path between $v_i$ and $v_j$. Dependence on $G$ may be omitted when it is clear from the context. The characteristic polynomial of $D$ is defined by $p_{D}(x)=\\det(xI-D)$ and is called the \\emph{distance characteristic polynomial} of $G$. Since $D$ is a real symmetric matrix, all of  the roots of the distance characteristic polynomial are real. \n\nA metric space can be attached to any connected graph $G =(V,E)$ in the following way. The \\emph{path metric} of $G$, denoted $d_{G}$, is the metric where for all\nvertices $u,v\\in V$, $d_{G}(u,v)$ is the distance between $u$ and $v$ in $G$. Then, $(V,d_{G})$ is a metric space, called the \\emph{graphic metric space}\nassociated with $G$.\n\nA \\emph{cut vertex} of a graph $G$ is a vertex whose deletion increases the number of connected components of $G$. A \\emph{block} of $G$ is a maximal connected subgraph of $G$ which has no cut vertices. Thus, a block is either a maximal\n2-connected subgraph, or a cut-edge, or an isolated vertex, and every such\nsubgraph is a block. Two blocks of $G$ can overlap in at most one vertex, which is a cut-vertex; hence, every edge of $G$ lies in a unique block, and $G$ is the\nunion of its blocks. A connected graph $G$  is called a \\emph{block graph} if all of its blocks are cliques. In\nparticular, we say that $G$ is a \\emph{$t$-uniform block graph} if all of its cliques have size $t$.\n\nGiven two graphs $G$ and $H$, their \\emph{Cartesian product} is the graph $G\\square H$ whose vertex\nset is $V(G)\\times V(H)$ and whose edges are the pairs $((a,x),(b,y))$ with $a,b\\in V(G)$, $x,y\\in V(H)$ and either $(a,b)\\in E(G)$ and $x=y$, or $a=b$ and $(x,y)\\in E(H)$.\nThe Cartesian product $H_{1}\\square\\cdots\\square H_{k}$ of graphs $H_{1},\\ldots,H_{k}$ is also denoted $\\prod_{h=1}^{k}H_{h}$.\n\nFor integers $d\\geq2$ and $n\\geq2$, the \\emph{Hamming graph} $H(d,n)$ is a graph whose vertex set is the $d$-tuples with elements from $\\{0,1,2,\\ldots,n-1\\}$, where two vertices are adjacent if and only if their corresponding $d$-tuples differ only in one coordinate. Equivalently, $H(d,n)$ is the Cartesian product of $d$ copies of $K_n$. For a positive integer $d$, the \\emph{hypercube} or the \\emph{$d$-cube} is the graph $H(d,2)$.\n\nA metric space $(X_1,d_1)$ is \\emph{isometrically embeddable} in a metric space $(X_2,d_2)$ if there exists a mapping $\\sigma:X_1\\rightarrow X_2$ such that $d_2(\\sigma(a), \\sigma(b))=d_1(a,b)$. If $(X_1,d_1)$ and $(X_2,d_2)$ are graphic metric spaces associated with graphs $G$ and $H$ respectively, then $G$ is called an \\emph{isometric subgraph} of $H$.\nThe cases when the graph $H$ is a hypercube, a Hamming graph, or\na Cartesian product of cliques of different sizes are of much importance. For other definitions and notations related to embeddability of metric spaces, we refer the reader to~\\cite{TD1987, DLbook}.\n\nAs mentioned earlier, Graham, Hoffman, and Hosoya \\cite{GHH77} proved the following elegant formula to calculate the determinant of the distance matrix of a block graph. Note that this formula depends only on the sizes of the blocks and not on the graph structure.\n\\begin{thm}[\\cite{GHH77}]\\label{GHH}\nIf $G$ is a block graph with blocks $G_{1},~G_2,~\\ldots,~G_t$, then\n\\begin{eqnarray*}\n\\textnormal{cof}~D(G)&=&\\prod\\limits_{i=1}^{t}\\textnormal{cof}~D(G_i),\\\\\n\\det D(G)&=&\\sum\\limits_{i=1}^{t}\\det D(G_i)\\prod\\limits_{j\\neq i}\\textnormal{cof}~D(G_j).\n\\end{eqnarray*}\n\\end{thm}\n\nThe \\emph{coefficient sequence} of a real polynomial $p(x)=a_nx_n+\\cdots+a_1x+a_0$ is the sequence $a_0,a_1,a_2,\\ldots,a_n$. The polynomial $p$ is \\emph{real-rooted} if all roots of $p$ are real (by convention, constant polynomials are considered real-rooted). A sequence $a_0,a_1,a_2,\\ldots, a_n$ of real numbers is {\\em unimodal} if there is a $k$ such that $a_{i-1}\\leq a_{i}$ for $i\\leq k$ and $a_{i}\\geq a_{i+1}$ for $i\\geq k$, and the sequence is {\\em log-concave} if $a_j^2\\geq a_{j-1}a_{j+1}$ for all $j=1,\\dots,  n-1$.\n\n\n\nFinally, we will make use of the following well-known result (see for instance [\\cite{2}, Lemma 1.1] and [\\cite{5}, Theorem B, p. 270]) with the additional assumption that the polynomial coefficients are nonnegative, but it is straightforward to remove that assumption.\n\n\\begin{lema}\\label{knownunimodalitylemma}\n\t\\leavevmode\n\\begin{description}\n\\item[$(i)$] If $p(x)=a_nx_n+\\cdots+a_1x+a_0$ is a real-rooted polynomial, then the coefficient sequence $a_i$ of $p$ is log-concave.\n\\item[$(ii)$] If the sequence $a_0,a_1,a_2,\\ldots,a_n$ is positive and log-concave, then it is unimodal.\n\\end{description}\n\\end{lema}\n\n\n\n\\section{On $\\ell_1$-embeddability of metric spaces}\\label{sec:emb}\n\n\n\nMetric spaces whose distance matrix has exactly one positive eigenvalue have received much attention since the work of Deza and Laurent~\\cite{DLbook}. Metric properties of regular graphs have also been investigated by Koolen \\cite{Koolenmscthesis}.\n\nIn this section we develop a general theory regarding the eigenvalues of metric spaces. The main result of this section, Theorem~\\ref{thmfinal}, will be used to show the unimodality part of Question \\ref{que:coefficientsblock}.\n\n\t\nWe say that a metric space $(X,d)$ is {\\it of negative type} if for all weight functions $w:X\\to\\mathbb{Z}$ with $\\sum\\limits_{x\\in X} w(x)=0$ we have \n\\begin{equation}\\label{neg-type-ineq}\n\\sum\\limits_{x\\in X}\\sum\\limits_{y\\in X}w(x)w(y)d(x,y)\\leq0.\n\\end{equation} \nIf the same inequality holds for all weight functions with $\\sum\\limits_{x\\in X} w(x)=1$, then we say that $(X,d)$ is {\\it hypermetric}. It is fairly easy to show that $(X,d)$ being hypermetric implies it is of negative type. An alternative way to define metric spaces of negative type is by using Schoenberg's Theorem~\\cite{Sch38}: a metric space $(X,d)$ is of negative type if and only if $(X,\\sqrt{d})$ is isometrically embeddable in the Eucledian space.\n\t\nA metric space $(X,d)$ is said to be \\textit{$\\ell_1$-embeddable} if it can be embedded isometrically into the $\\ell_{1}$-space ($\\mathbb{R}^m, d_{\\ell_{1}}$) for some integer $m\\geq1$. Here, $d_{\\ell_{1}}$ denotes the $\\ell_{1}$-distance defined by\n\\begin{equation*}\nd_{\\ell_1}(x,y):=\\sum\\limits_{1\\leq i\\leq m}|x_i-y_i|~~\\textrm{for}~\nx,y\\in \\mathbb{R}^m.\n\\end{equation*}\nOne of the basic results of $\\ell_{1}$-embeddable metric spaces is a characterization in terms of  cut semimetrics. Given a subset $S$ of the $n$-set $V_n:= \\{1,\\ldots,n\\}$, the \\emph{cut semimetric} $\\delta_S$ is the distance on $V_n$ defined as\n\\begin{equation*}\n\\delta_S(i,j)=\\left\\{\n\\begin{array}{ll}\n1, & \\hbox{$i\\in S,~j\\in V_n\\setminus S$,} \\\\\n0, & \\hbox{$i, j \\in S$, \\textrm{or} $i,j\\in V_n\\setminus S$.}\n\\end{array}\n\\right.\n\\end{equation*}\nNote that every cut semimetric is clearly $\\ell_{1}$-embeddable. In fact, a distance $d$ is $\\ell_{1}$-\\emph{embeddable} if and only if it can be decomposed as a nonnegative linear combination of cut semimetrics. We also note that $\\ell_1$-embeddability of $(X,d)$ implies it is hypermetric~\\cite{TD1987,Kelly67}.\n\t\n\t\nLet $G$ be a graph and let $(X,d)$ be the graphic metric space and associated with~$G$.\nWe have the following implications:\n \n\\begin{gather*}\n(X,d) \\text{ is $\\ell_1$-embeddable} \\\\\n\t\t\\Downarrow \\\\\n(X,d) \\text{ is hypermetric} \\\\\n\t\t\\Downarrow \\\\\n(X,d) \\text{ is of negative type} \\\\\n\t\t\\Downarrow \\\\\n\\text{The distance matrix of $G$ has exactly one positive eigenvalue}\n\\end{gather*}\n\n\n\n\\medskip\n\nFor more details on the metric hierarchy we refer the reader to the book by Deza and Laurent \\cite{DLbook}.\n\t\n\t\nLet ($X_1,d_1$) and ($X_2,d_2$) be two metric spaces. Their direct product is the metric space $(X_1\\times X_2, d_1\\otimes d_2)$ where, for $x_1,y_1\\in X_1$, $x_2,y_2\\in X_2$,\n\\begin{equation}\\label{product-metric}\nd_1\\otimes d_2\\big((x_1,x_2),(y_1,y_2)\\big)=d_1(x_1,y_1)+d_2(x_2,y_2).\n\\end{equation}\nNote that for path metrics, the direct product operation corresponds to the Cartesian product of graphs. Namely, if $G$ and $H$ are two connected graphs, then the direct product of their path metrics coincides with the path metric of the Cartesian product of $G$ and $H$. The following lemmas provide a relation between the metric hierarchy and the direct product of respective metric spaces.\n\\begin{lema}\\label{l1-embed-spaces}\nLet $d_i$ be a distance on the set $X_i$, for $i=1,2$. Then $(X_1\\times X_2, d_1\\otimes d_2)$ is $\\ell_{1}$-embeddable if and only if both $(X_1,d_1)$ and $(X_2,d_2)$ are $\\ell_{1}$-embeddable.\n\\end{lema}\n\\begin{proof}\n\t\t\nSince both $d_1$ and $d_2$ are $\\ell_{1}$-embeddable, they can be decomposed as a nonnegative linear combination of cut semimetrics. Therefore, if $d_1=\\sum\\limits_{S\\subseteq X_1}a_{S}\\delta_{S}$ and $d_2=\\sum\\limits_{T\\subseteq X_2}b_{T}\\delta_{T}$, then by (\\ref{product-metric}) we obtain that\n\\begin{equation*}\nd_1\\otimes d_2=\\sum\\limits_{S\\subseteq X_1}a_{S}\\delta_{S\\times X_2}+\\sum\\limits_{T\\subseteq X_2}b_{T}\\delta_{X_1\\times T}.\n\\end{equation*}\nThis implies that $d_1\\otimes d_2$ is decomposed into nonnegative linear combination of cut semimetrics. Thus $d_1\\otimes d_2$ is $\\ell_{1}$-embeddable.\n\\qedhere\n\\end{proof}\n\t\n\\begin{lema}\\label{l2-embed-spaces}\nLet $d_i$ be a distance on the set $X_i$, for $i=1,2$. Then $(X_1\\times X_2, d_1\\otimes d_2)$ is hypermetric (resp. of negative type) if and only if both ($X_1,d_1$) and ($X_2,d_2$) are hypermetric (resp. of negative type).\n\\end{lema}\n\\begin{proof}\nSimilar to Lemma~\\ref{l1-embed-spaces}, it suffices to show that the inequality~(\\ref{neg-type-ineq}) holds for $d_1\\otimes d_2$ assuming $(X,d_1)$ and $(X,d_2)$ are hypermetric (resp. of negative type). By definition we have\n\\begin{align*}\n&\\sum\\limits_{(x_1,x_2)\\in X_1\\times X_2}\\sum\\limits_{(y_1,y_2)\\in X_1\\times X_2} w(x_1,x_2)w(y_1,y_2)(d_1\\otimes d_2)\\left((x_1,x_2),(y_1,y_2)\\right) = \\\\\n&= \\sum\\limits_{x_1\\in X_1}\\sum\\limits_{x_2\\in X_2}\\sum\\limits_{y_1\\in X_1}\\sum\\limits_{y_2\\in X_2} w(x_1,x_2)w(y_1,y_2)\\left(d_1(x_1,y_1)+d_2(x_2,y_2)\\right) = \\\\\n&= \\sum\\limits_{x_1\\in X_1}\\sum\\limits_{y_1\\in X_1}\\left(\\sum\\limits_{x_2\\in X_2}w(x_1,x_2)\\right)\\left(\\sum\\limits_{y_2\\in X_2}w(y_1,y_2)\\right)d_1(x_1,y_1)+ \\\\\n&+ \\sum\\limits_{x_2\\in X_2}\\sum\\limits_{y_2\\in X_2}\\left(\\sum\\limits_{x_1\\in X_1}w(x_1,x_2)\\right)\\left(\\sum\\limits_{y_1\\in X_1}w(y_1,y_2)\\right)d_2(x_2,y_2).\n\\end{align*}\nFor a weight function $w:X_1\\times X_2\\to \\mathbb{Z}$ such that \n\n$$\\sum\\limits_{(x_1,x_2)\\in X_1\\times X_2} w(x_1,x_2)=1\\text{ (resp. $0$),}$$  we define $w_1:X_1\\to\\mathbb{Z}$ and $w_2:X_2\\to\\mathbb{Z}$ such that \n$$w_1(x)=\\sum\\limits_{x_2\\in X_2} w(x,x_2)$$ and $$w_2(x)=\\sum\\limits_{x_1\\in X_1} w(x_1,x).$$ Note that $$\\sum\\limits_{x_1\\in X_1} w_1(x_1)=\\sum\\limits_{(x_1,x_2)\\in X_1\\times X_2}w(x_1,x_2)=1\\text{ (resp. $0$),}$$ and thus for $(X_1,d_1)$ being hypermetric (resp. of negative type) and the weight function $w_1$ we have $$\\sum\\limits_{x_1\\in X_1}\\sum\\limits_{y_1\\in X_1} w_1(x_1) w_1(y_1)d_1(x_1,y_1)\\leq 0.$$ Similarly, we have $$\\sum\\limits_{x_2\\in X_2}\\sum\\limits_{y_2\\in X_2} w_2(x_2)w_2(y_2)d_2(x_2,y_2)\\leq 0.$$ The inequality~(\\ref{neg-type-ineq}) for $d_1\\otimes d_2$ then follows.\n\\end{proof}\n\n\\begin{thm}\\label{thmfinal}\nLet $G$ be the a graph whose $2$-connected components are of negative type. If $D(G)$ is the distance matrix of $G$, then $D(G)$ has exactly one positive eigenvalue.\n\\end{thm}\n\\begin{proof}\nLemma \\ref{l2-embed-spaces} ensures us that the Cartesian product of the $2$-connected components of $G$ is of negative type. Since $G$ is an isometric subgraph of the Cartesian product, the result follows immediately.\n\\end{proof}\n\n \nTerwilliger and Deza \\cite{TD1987} investigated finite distance spaces having integral distances: a finite set $X$ and a map $d:X^2\\rightarrow \\mathbb{Z}$. The relation $d=1$ is assumed to be connected.  They provided the following classification of hypermetric spaces and metric spaces of negative type: $(X,d)$ has negative type if and only if it is metrically embeddable in a Euclidean space and generates a root lattice (direct sum of lattices of types $A,D,E$). Moreover, $(X,d)$ is hypermetric if and only if it is isomorphic to a subspace of the (complete) Cartesian products of the half-cubes, the CP-graphs, and the Johnson, Schl\u00e4fli and Gosset graphs. These graphs correspond to the minimum vectors in the lattices dual to the root lattices mentioned above. Terwilliger and Deza conclude by describing how a given hypermetric $(X,d)$ may be embedded into a complete one. We should note that the results in \\cite{TD1987} provide an alternative way to show Theorem~\\ref{thmfinal} for hypermetric graphs. \n\n\nNext we observe that the Cartesian product of graphs does not preserve the one positive distance eigenvalue property. \n\n\\begin{rem}\nThe Cartesian product of two graphs having one positive distance eigenvalues does not necessarily have one positive distance eigenvalue; see, e.g., Figure  \\ref{fig:2evproduct}. In fact, if a vertex of $C_5$ is identified with any 4-degree vertex in the graph in Figure \\ref{fig:2evproduct}, right, then the resulting 11-vertex graph has the following distance spectrum:\n$$\\{-7.83, -2.62, -2, -2, -1.38, -1, -1.0000, -0.38, -0.35, 0.08, 18.48\\}.$$\n\n\\begin{figure}[ht!]\n\\centering\n\\resizebox{7cm}{!}{\n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\n\\draw   (121.78,141.11) -- (38.49,140.8) -- (13.05,61.5) -- (80.6,12.79) -- (147.8,61.99) -- cycle ;\n\\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (35.53,140.8) .. controls (35.53,139.16) and (36.86,137.84) .. (38.49,137.84) .. controls (40.13,137.84) and (41.46,139.16) .. (41.46,140.8) .. controls (41.46,142.44) and (40.13,143.76) .. (38.49,143.76) .. controls (36.86,143.76) and (35.53,142.44) .. (35.53,140.8) -- cycle ;\n\\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (118.81,141.11) .. controls (118.81,139.47) and (120.14,138.14) .. (121.78,138.14) .. controls (123.41,138.14) and (124.74,139.47) .. (124.74,141.11) .. controls (124.74,142.74) and (123.41,144.07) .. (121.78,144.07) .. controls (120.14,144.07) and (118.81,142.74) .. (118.81,141.11) -- cycle ;\n\\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (10.08,61.5) .. controls (10.08,59.86) and (11.41,58.53) .. (13.05,58.53) .. controls (14.68,58.53) and (16.01,59.86) .. (16.01,61.5) .. controls (16.01,63.14) and (14.68,64.46) .. (13.05,64.46) .. controls (11.41,64.46) and (10.08,63.14) .. (10.08,61.5) -- cycle ;\n\\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (144.84,61.99) .. controls (144.84,60.35) and (146.17,59.03) .. (147.8,59.03) .. controls (149.44,59.03) and (150.77,60.35) .. (150.77,61.99) .. controls (150.77,63.63) and (149.44,64.96) .. (147.8,64.96) .. controls (146.17,64.96) and (144.84,63.63) .. (144.84,61.99) -- cycle ;\n\\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (77.64,12.79) .. controls (77.64,11.15) and (78.97,9.83) .. (80.6,9.83) .. controls (82.24,9.83) and (83.57,11.15) .. (83.57,12.79) .. controls (83.57,14.43) and (82.24,15.76) .. (80.6,15.76) .. controls (78.97,15.76) and (77.64,14.43) .. (77.64,12.79) -- cycle ;\n\\draw   (305.74,12.29) -- (358.27,90.93) -- (253.22,90.93) -- cycle ;\n\\draw   (358.27,141.93) -- (253.22,90.93) -- (358.27,90.93) -- cycle ;\n\\draw   (253.27,141.93) -- (358.27,90.93) -- (253.27,90.93) -- cycle ;\n\\draw    (305.74,12.29) -- (305.74,116.43) ;\n\\draw   (320.27,55.08) -- (305.74,115.58) -- (305.74,12.29) -- cycle ;\n\\draw    (320.27,55.08) -- (253.22,90.93) ;\n\\draw    (320.27,55.08) -- (253.27,141.93) ;\n\\draw    (358.27,90.93) -- (320.27,55.08) ;\n\\draw    (320.27,55.08) -- (358.27,141.93) ;\n\\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (302.78,12.29) .. controls (302.78,10.65) and (304.1,9.33) .. (305.74,9.33) .. controls (307.38,9.33) and (308.71,10.65) .. (308.71,12.29) .. controls (308.71,13.93) and (307.38,15.26) .. (305.74,15.26) .. controls (304.1,15.26) and (302.78,13.93) .. (302.78,12.29) -- cycle ;\n\\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (317.3,55.08) .. controls (317.3,53.44) and (318.63,52.11) .. (320.27,52.11) .. controls (321.9,52.11) and (323.23,53.44) .. (323.23,55.08) .. controls (323.23,56.71) and (321.9,58.04) .. (320.27,58.04) .. controls (318.63,58.04) and (317.3,56.71) .. (317.3,55.08) -- cycle ;\n\\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (355.3,90.93) .. controls (355.3,89.3) and (356.63,87.97) .. (358.27,87.97) .. controls (359.9,87.97) and (361.23,89.3) .. (361.23,90.93) .. controls (361.23,92.57) and (359.9,93.9) .. (358.27,93.9) .. controls (356.63,93.9) and (355.3,92.57) .. (355.3,90.93) -- cycle ;\n\\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (250.25,90.93) .. controls (250.25,89.3) and (251.58,87.97) .. (253.22,87.97) .. controls (254.85,87.97) and (256.18,89.3) .. (256.18,90.93) .. controls (256.18,92.57) and (254.85,93.9) .. (253.22,93.9) .. controls (251.58,93.9) and (250.25,92.57) .. (250.25,90.93) -- cycle ;\n\\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (302.8,116.43) .. controls (302.8,114.8) and (304.13,113.47) .. (305.77,113.47) .. controls (307.4,113.47) and (308.73,114.8) .. (308.73,116.43) .. controls (308.73,118.07) and (307.4,119.4) .. (305.77,119.4) .. controls (304.13,119.4) and (302.8,118.07) .. (302.8,116.43) -- cycle ;\n\\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (355.3,141.93) .. controls (355.3,140.3) and (356.63,138.97) .. (358.27,138.97) .. controls (359.9,138.97) and (361.23,140.3) .. (361.23,141.93) .. controls (361.23,143.57) and (359.9,144.9) .. (358.27,144.9) .. controls (356.63,144.9) and (355.3,143.57) .. (355.3,141.93) -- cycle ;\n\\draw  [fill={rgb, 255:red, 0; green, 0; blue, 0 }  ,fill opacity=1 ] (250.3,141.93) .. controls (250.3,140.3) and (251.63,138.97) .. (253.27,138.97) .. controls (254.9,138.97) and (256.23,140.3) .. (256.23,141.93) .. controls (256.23,143.57) and (254.9,144.9) .. (253.27,144.9) .. controls (251.63,144.9) and (250.3,143.57) .. (250.3,141.93) -- cycle ;\n\n\n\n\n\\end{tikzpicture}\n\n}\n\\caption{Two graphs whose distance matrices have exactly one positive eigenvalue, but the graph resulting from joining them in one vertex has two positive distance eigenvalues.}\n\\label{fig:2evproduct}\n\\end{figure}\nMoreover, from Zhang's and Godsil's result \\cite[Theorem 3.3]{zg2013} it follows that if the distinguishing one vertex does not preserve the one distance eigenvalues property, then the Cartesian product does not preserve it either.\n\\end{rem}\n \n\\begin{rem} If  all the $2$-connected components of a graph $G$ have a full rank distance matrix, it does not necessarily follow that $D(G)$ has full rank. For example, consider the graph in Figure~\\ref{fig:fullrank} with two biconnected components: $G'$ on vertices $\\{0,1,\\dots,7\\}$ and $G''$ on vertices $\\{0,8,9\\}$. The distance matrices of $G$ and  $G'$ are:\n\n{\\tiny{\n\t\\[\n\tD(G)=\\left(\n\t\\begin{array}{cccccccccc}\n\t\t0 & 1 & 2 & 1 & 2 & 2 & 2 & 2 & 1 & 1 \\\\\n\t\t1 & 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 \\\\\n\t\t2 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 3 & 3 \\\\\n\t\t1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 2 & 2 \\\\\n\t\t2 & 2 & 1 & 1 & 0 & 1 & 2 & 2 & 3 & 3 \\\\\n\t\t2 & 2 & 1 & 1 & 1 & 0 & 2 & 2 & 3 & 3 \\\\\n\t\t2 & 2 & 1 & 1 & 2 & 2 & 0 & 2 & 3 & 3 \\\\\n\t\t2 & 2 & 1 & 1 & 2 & 2 & 2 & 0 & 3 & 3 \\\\\n\t\t1 & 2 & 3 & 2 & 3 & 3 & 3 & 3 & 0 & 1 \\\\\n\t\t1 & 2 & 3 & 2 & 3 & 3 & 3 & 3 & 1 & 0 \\\\\n\t\\end{array}\n\t\\right), \\;\\;\\;\n\tD(G')=\\left(\n\t\\begin{array}{cccccccc}\n\t\t0 & 1 & 2 & 1 & 2 & 2 & 2 & 2 \\\\\n\t\t1 & 0 & 1 & 1 & 2 & 2 & 2 & 2 \\\\\n\t\t2 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \\\\\n\t\t1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\\\\n\t\t2 & 2 & 1 & 1 & 0 & 1 & 2 & 2 \\\\\n\t\t2 & 2 & 1 & 1 & 1 & 0 & 2 & 2 \\\\\n\t\t2 & 2 & 1 & 1 & 2 & 2 & 0 & 2 \\\\\n\t\t2 & 2 & 1 & 1 & 2 & 2 & 2 & 0 \\\\\n\t\\end{array}\n\t\\right),\n\t\\]\n\t}}\nand the ranks of $D(G')$ and $D(G'')$ are $8$ and $3$ respectively, so they are full rank matrices, whereas the rank of $D(G)$ is $9$, so it is not a full-rank matrix.\n\\end{rem}\n \n\n\\begin{figure}[ht!]\n\t\\centering\n\t\\resizebox{4cm}{!}{\n\t\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\t\n\t\t\n\t\n\t\t\\draw    (141.87,52.67) -- (111.33,81) ;\n\t\t\\draw [shift={(111.33,81)}, rotate = 137.14] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\\draw [shift={(141.87,52.67)}, rotate = 137.14] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\n\t\t\\draw    (111.33,81) -- (141.33,111) ;\n\t\t\\draw [shift={(141.33,111)}, rotate = 45] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\\draw [shift={(111.33,81)}, rotate = 45] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\n\t\t\\draw    (141.87,52.67) -- (141.33,111) ;\n\t\t\\draw [shift={(141.33,111)}, rotate = 90.52] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\\draw [shift={(141.87,52.67)}, rotate = 90.52] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\n\t\t\\draw    (111.33,81) -- (81.33,51.67) ;\n\t\t\\draw [shift={(81.33,51.67)}, rotate = 224.36] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\\draw [shift={(111.33,81)}, rotate = 224.36] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\n\t\t\\draw    (81.33,111.67) -- (81.33,51.67) ;\n\t\t\\draw [shift={(81.33,51.67)}, rotate = 270] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\\draw [shift={(81.33,111.67)}, rotate = 270] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\n\t\t\\draw    (111.33,81) -- (81.33,111.67) ;\n\t\t\\draw [shift={(81.33,111.67)}, rotate = 134.37] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\\draw [shift={(111.33,81)}, rotate = 134.37] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\n\t\t\\draw    (81.33,51.67) -- (51.33,82.33) ;\n\t\t\\draw [shift={(51.33,82.33)}, rotate = 134.37] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\\draw [shift={(81.33,51.67)}, rotate = 134.37] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\n\t\t\\draw    (81.33,111.67) -- (51.33,82.33) ;\n\t\t\\draw [shift={(51.33,82.33)}, rotate = 224.36] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\\draw [shift={(81.33,111.67)}, rotate = 224.36] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\n\t\t\\draw    (81.33,51.67) -- (51.33,22.33) ;\n\t\t\\draw [shift={(51.33,22.33)}, rotate = 224.36] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\\draw [shift={(81.33,51.67)}, rotate = 224.36] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\n\t\t\\draw    (51.33,82.33) -- (51.33,22.33) ;\n\t\t\\draw [shift={(51.33,22.33)}, rotate = 270] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\\draw [shift={(51.33,82.33)}, rotate = 270] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\n\t\t\\draw    (51.33,22.33) -- (21.33,53) ;\n\t\t\\draw [shift={(21.33,53)}, rotate = 134.37] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\\draw [shift={(51.33,22.33)}, rotate = 134.37] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\n\t\t\\draw    (51.33,82.33) -- (21.33,53) ;\n\t\t\\draw [shift={(21.33,53)}, rotate = 224.36] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\\draw [shift={(51.33,82.33)}, rotate = 224.36] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\n\t\t\\draw    (81.33,51.67) -- (21.33,53) ;\n\t\t\\draw [shift={(21.33,53)}, rotate = 178.73] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\\draw [shift={(81.33,51.67)}, rotate = 178.73] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\n\t\t\\draw    (20.67,82.33) -- (51.33,82.33) ;\n\t\t\\draw [shift={(51.33,82.33)}, rotate = 0] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\\draw [shift={(20.67,82.33)}, rotate = 0] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\n\t\t\\draw    (20.67,82.33) -- (81.33,51.67) ;\n\t\t\\draw [shift={(81.33,51.67)}, rotate = 333.18] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\\draw [shift={(20.67,82.33)}, rotate = 333.18] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\n\t\t\\draw    (81.33,21) -- (51.33,82.33) ;\n\t\t\\draw [shift={(51.33,82.33)}, rotate = 116.06] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\\draw [shift={(81.33,21)}, rotate = 116.06] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\n\t\t\\draw    (81.33,21) -- (81.33,51.67) ;\n\t\t\\draw [shift={(81.33,51.67)}, rotate = 90] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\\draw [shift={(81.33,21)}, rotate = 90] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][fill={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]      (0, 0) circle [x radius= 1, y radius= 1]   ;\n\t\t\n\t\t\n\t\n\t\t\\draw (106,63.07) node [anchor=north west][inner sep=0.75pt] {$0$};\n\t\n\t\t\\draw (86,43.73) node [anchor=north west][inner sep=0.75pt]  {$1$};\n\t\n\t\t\\draw (47.33,84.4) node [anchor=north west][inner sep=0.75pt] {$2$};\n\t\n\t\t\\draw (68,109.07) node [anchor=north west][inner sep=0.75pt]  {$3$};\n\t\n\t\t\\draw (9.33,38.4) node [anchor=north west][inner sep=0.75pt]  {$4$};\n\t\n\t\t\\draw (37.33,12.4) node [anchor=north west][inner sep=0.75pt]  {$5$};\n\t\n\t\t\\draw (8.67,68.4) node [anchor=north west][inner sep=0.75pt]  {$6$};\n\t\n\t\t\\draw (85.33,10.4) node [anchor=north west][inner sep=0.75pt]  {$7$};\n\t\n\t\t\\draw (145.67,45.73) node [anchor=north west][inner sep=0.75pt]  {$8$};\n\t\n\t\t\\draw (145.33,106.4) node [anchor=north west][inner sep=0.75pt]  {$9$};\n\t\t\n\t\t\n\t\\end{tikzpicture}\n\t}\n\t\\caption{A graph $G$ whose biconnected components have full rank distance matrices but its distance matrix $D(G)$ does not have full rank.}\n\t\\label{fig:fullrank}\n\\end{figure}\n\n\n\\section{Extension of Graham and Lov\\'{a}sz conjecture to block graphs}\\label{sec:conjectureforblockgraphs}\n\nRecall that, for a graph $G$, $c_k(G)$ denotes the coefficient of $x^k$ in $$\\text{det}(D(G)-xI)=(-1)^np_{D(G)}(x).$$ The coefficients of the distance polynomial of a tree all have a common factor of $(-1)^{n-1}2^{n-k-2}$ due to a result of Graham and Lov\\'{a}sz~\\cite{GL78}. We say that the quantities $d_k(T)=(-1)^{n-1}c_k(T)\/2^{n-k-2}$ are the \\emph{normalized coefficients} of the distance characteristic polynomial of a tree $T$. Due to such common factor in  the coefficients of trees \\cite{EGG76}, Conjecture~\\ref{con:coefficientstree} uses the normalized coefficients. We should note that we will not use the normalized coefficients to investigate Question \\ref{que:coefficientsblock}.\n\nIn this section we show that the sequence of coefficients of the distance characteristic polynomial of a block graph is unimodal, and we establish the peak for several extremal classes of block graphs with small diameter. \n\n\\subsection{Unimodality}\\label{sec:unimodality}\n\nTo answer the unimodality part of Question \\ref{que:coefficientsblock}, we will follow a similar approach as it was done in \\cite{GRWC2015} to show the unimodality of trees. The main idea relies on the fact that the distance matrix of a block graph on $n$ vertices has one positive and $n-1$ negative eigenvalues. We begin with a preliminary result, which extends the known result for trees by Edelberg, Garey and Graham \\cite{EGG76}. \n\n\\begin{lema}\\label{claim}\nThe coefficients of the distance characteristic polynomial of a block graph $G$ satisfy\n$$(-1)^{n-1}c_{k}(G)>0 \\quad \\text{for }0\\leq k \\leq n-2.$$\n\\end{lema}\n\\begin{proof}\nIt was shown in~\\cite[Theorem~3.2]{LLL} that the distance matrix $D(G)$ of a block graph $G$ has one positive and $n-1$ negative eigenvalues.\nWe now extend  the argument given in ~\\cite[Theorem 2.3]{EGG76} that $(-1)^{n-1}c_{k}(T)>0$ for $0\\leq k\\leq n-2$ for a tree $T$, given that its distance matrix has one positive  and $n-1$  negative eigenvalues. Let the eigenvalues of the distance matrix $D(G)$ of a block graph $G$ be denoted by $\\lambda_1, -\\lambda_2, \\dots, -\\lambda_n$, where $\\lambda_i>0$ for~$i=1,\\dots,n$. Then the distance characteristic polynomial is\n\\begin{align*}\n\t\\det(D(G)-xI) &= (-1)^n(x-\\lambda_1)(x+\\lambda_2)\\cdots(x+\\lambda_n)= \\\\\n\t&= (-1)^n(x-\\lambda_1)\\sum\\limits_{k=0}^{n-1} g_{n-1-k} x^{k}= \\\\\n\t&= (-1)^n\\left(x^n+\\sum\\limits_{k=1}^{n-1}(g_{n-k}-\\lambda_1g_{n-k-1})x^{k}-\\lambda_1g_{n-1}\\right),\n\\end{align*}\nwhere $g_k$ is the sum of all $k$-fold products of $\\lambda_2,\\dots,\\lambda_n$. Then, $c_{n-1}(G)=g_1-\\lambda_1$, but also $c_{n-1}(G)=-c_n(G)\\text{tr}(D)=0$; thus, $g_1=\\lambda_1$. Then, since $g_{n-k}-\\lambda_1g_{n-k-1}=g_{n-k}-g_1g_{n-k-1}<0$ for $k=1,\\dots,n-2$, \nand since for $k=0$, $-\\lambda_1g_{n-1}=-g_1g_{n-1}=-\\prod\\limits_{i=1}^{n}\\lambda_i<0$, it follows that $(-1)^{n-1}c_{k}(G)>0$ for $0\\leq k\\leq n-2$.\n\\end{proof}\n\n\n\n\n\n\\begin{thm}\\label{theo:unimodality}\nFor a block graph $G$, the sequence of coefficients of the distance characteristic polynomial $(-1)^{n-1}c_0(G),\\ldots,(-1)^{n-1}c_{n-2}(G)$ is unimodal.\n\\end{thm}\n\n\\begin{proof}\nFirst, it follows from Lemma \\ref{claim} that if $G$ is a block graph, then the coefficients of the distance characteristic polynomial satisfy $(-1)^{n-1}c_{k}(G)>0 \\quad \\text{for }0\\leq k \\leq n-2$.\n\nSince the distance matrix $D$ is a real symmetric matrix, the distance characteristic polynomial is real-rooted. From Lemma \\ref{knownunimodalitylemma}$(i)$, it  follows that the sequence is log-concave.\n\nMoreover, since\n$$(-1)^{n-1}c_{k}(G)>0 \\quad \\text{for }0\\leq k \\leq n-2,$$\nthen Lemma \\ref{knownunimodalitylemma}(ii) implies that the sequence of coefficients of the distance characteristic polynomial is unimodal.\n\\end{proof}\n\n\n\\subsection{Peak location}\\label{sec:peak}\n\n\n \n\nIn this section we answer the peak location part of Question~\\ref{que:coefficientsblock} for several extremal families of uniform block graphs with small diameter. \n\nThe idea is to derive an explicit formula for the coefficients and use the unimodality to find the peak. However, while the method for obtaining the peak for stars and paths relies on the algebraic properties of the corresponding distance matrix \\cite{C89}, for block graphs we will exploit several of its spectral properties.\n\nConsider the \\emph{windmill graph} $W(k,t)$, which is a block graph formed by joining $k$ cliques of size $t$ at a shared universal vertex. The following result uses Stirling's approximation to prove an estimate for the peak location of a windmill graph for large $k$. \n\n\\begin{thm}\\label{thm:friend.gen.approx} Consider a windmill graph $W(k,t)$ with $k\\geq 2$ and $t\\geq 3$, so that $n=|V|=k(t-1)+1$. Then, the sequence of coefficients is unimodal, and as $k$ approaches infinity the peak of the sequence occurs at $\\frac{kt(t-1)}{2(t+1)}  + O(\\log k)$.\n\\end{thm}\n\n\n\n\\begin{proof} The distance polynomial of the graph is\n{\\small{\n$$p_D(x)=(x+1)^{(t-2)k}(x+t)^{k-1}(x^2-(t-2+2(t-1)(k-1))x-k(t-1)).$$\n}}\nTo locate the peak of $p_D(x)$, it is sufficient to consider $(x+1)^{(t-2)k}(x+t)^{k-1}$.\n\tWe know $(x+1)^{(t-2)k}$ peaks at $\\frac{k(t-2)}2$. The coefficients of $(x+t)^{k-1}$ are defined by the formula $f_i = {k-1 \\choose i}t^{k-1-i}$. By calculating explicitly $f_{\\frac{k}{t+1}-2},f_{\\frac{k}{t+1}-1},f_{\\frac{k}{t+1}},f_{\\frac{k}{t+1}+1}$ and using unimodality of binomial coefficients we conclude the sequence $f_i$ peaks at $\\frac{k}{t+1}$.\n\t\n\tDefine $$p = \\frac{k(t-2)}2+\\frac{k}{t+1}= \\frac{kt(t-1)}{2(t+1)},$$ and let $d_i$ be the coefficients of $(x+1)^{(t-2)k}(x+t)^{k-1}$. Then  $$d_i=\\sum\\limits_{j=0}^i {(t-2)k \\choose i-j} {k-1 \\choose j} t^{k-1-j}.$$ We define $m_i = \\max\\limits_{j} {(t-2)k \\choose i-j} {k-1 \\choose j} t^{k-1-j}$, the maximal term in the sum. Then we have $d_i \\leq k\\cdot m_i$ for all $i$. On the other hand, $d_p$ has $${(t-2)k \\choose (t-2)k\/2} {k-1 \\choose k\/(t+1)} t^{k-1-k\/(t+1)}$$ as one of the terms in its sum, so $d_p$ is greater than that. The idea is to show that $k m_i \\leq {(t-2)k \\choose (t-2)k\/2} {k-1 \\choose k\/(t+1)} t^{k-1-k\/(t+1)}$ for all $i\\leq k-1$, which implies $d_i \\leq d_p$.\n\t\n\tTo find $m_i$, we apply Stirling's formula to ${(t-2)k \\choose i-j} {k-1 \\choose j} t^{k-1-j}$ and find its derivative with respect to $j$. Note that for $n\\to\\infty$, we have $$\\log n! = n\\log n-n+\\frac12\\log(2\\pi n)+O(\\log n),\\text{ or }$$\n$$n!-\\left(\\frac{n}{e}\\right)^n\\sqrt{2\\pi}=n^{O(1)},$$\n\tmeaning that $n!$ and $\\frac{n^n}{e^n} \\sqrt{2\\pi n}$ are asymptotically equivalent. Also observe that since the sequence is unimodal it is sufficient to find a local peak among the coefficients with high values of $k$ and $n-k$. Hence, for large enough $n$, $k$, and $n-k$ we may derive\n\t{\\small{\n\t$${n \\choose k}= \\frac{n!}{k!(n-k)!}\\sim \\frac{\\sqrt{2\\pi n} n^n e^{n-k} e^k}{2\\pi \\sqrt{k(n-k)} k^k (n-k)^{n-k} e^n} =\\frac{n^n\\sqrt{n}}{ \\sqrt{2\\pi k(n-k)} k^k (n-k)^{n-k}}.$$\n\t}}\n\t\n\tThen, we have\n\t{\\small{\\begin{align*}\n\t    &\\left.{(t-2)k \\choose i-j} {k-1 \\choose j} t^{k-1-j}\\right. \\sim \\\\ \n\t    &\\sim \\frac{((t-2)k)^{k(t-2)+1\/2} (k-1)^{k-1+1\/2}}{2\\pi (i-j)^{i-j+1\/2} ((t-2)k-i+j)^{(t-2)k-i+j+1\/2} j^{j+1\/2} (k-1-j)^{k-1-j+1\/2}},\n\t\\end{align*}}}\n\tand its derivative is then equal to $0$ if and only if\n\t{\\small \\begin{align*}\n\t\t\\log\\left(\\frac{(i-j)(k-1-j)}{j (k(t-2)-i+j)}\\right) &= \\frac1{2j}-\\frac1{2(i-j)}-\\frac1{2(k-1-j)}+\\frac1{2(k(t-2)-i+j)}, \n\t\\end{align*}}\t\t\n\t{\\small \\begin{align*}\t\t\n\t\t\\frac{(i-j)(k-1-j)}{j (k(t-2)-i+j)} &= \\sqrt{\\frac{e^{1\/j} e^{1\/(k(t-2)-i+j)}}{e^{1\/(i-j)} e^{1\/(k-1-j)}}.} \n\t\\end{align*}}\nSince $\\sqrt{e^{1\/z}}$ and $1$ are asymptotically equivalent for $z\\to\\infty$, we may assume\n\\begin{align*}\n\t\\frac{(i-j)(k-1-j)}{j(k(t-2)-i+j)} &\\sim 1, \\\\\n\t\\frac{i(k-1)}{k(t-1)-1} &\\sim j \n\\end{align*}\nFor $k\\to\\infty$ we may assume $j= \\frac{i}{t-1}$ and then the inequality we want to prove is\n\n{\\footnotesize{\n$$k {(t-2)k \\choose i-i\/(t-1)} {k-1 \\choose i\/(t-1)} t^{k-1-i\/(t-1)} \\leq {(t-2)k \\choose (t-2)k\/2} {k-1 \\choose k\/(t+1)} t^{k-1-k\/(t+1)}.$$\n}}\nThis follows for the two inequalities:\n{\\small{\n$$k {(t-2)k \\choose i-i\/(t-1)} t^{k\/(t+1)} \\leq {(t-2)k \\choose (t-2)k\/2} t^{i\/(t-1)} \\text{ and } {k-1 \\choose i\/(t-1)} \\leq {k-1 \\choose k\/(t+1)}.$$\n}}\nThe latter inequality follows from $\\frac{i}{t-1}\\leq\\frac{k}{t+1}$ for $k\\to\\infty$. The former inequality can be shown using Stirling approximation.\n\\end{proof}\n\n\\begin{rem}\nOne may wonder on the extension to a windmill graph in which the cliques are not all of the same size. Consider a block graph with one universal vertex and $k_i$ cliques of size $t_i$ for $i\\in\\{1,\\dots,l\\}$ and some $l$, all sizes $t_1,\\dots,t_l$ are distinct. Then the distance characteristic polynomial takes form\n$$p_D(x) = (x+1)^{n-k-1} \\prod\\limits_{i=1}^l (x+t_i)^{k_i-1} p_B(x),$$\nwhere $p_B(x)$ is the characteristic polynomial of the quotient matrix corresponding to the coarsest partition into $l+1$ vertex subsets $X_0,X_1,\\dots X_l$: $X_0$ only contains the universal vertex, and $X_i$ has all vertices (except for the universal one) from all $k_i$ cliques of size $t_i$. Then the problem of finding $m_i$ as it is defined above is equivalent to a problem of finding a peak of an $l$-dimensional function, which complicates the application of the approximation approach even for small $l$.\n\\end{rem}\n\n\n\nThe \\emph{friendship graph} (or \\emph{Dutch windmill graph}), denoted $F_{2k+1}$, is the graph $W(k,3)$ obtained by joining $k$ copies of $K_3$ by a common vertex so that $n=|V|=2k+1$. As a direct consequence of Theorem~\\ref{thm:friend.gen.approx} we obtain the following corollary.\n\n\\begin{cor}\nLet $F_{2k+1}$ be the friendship graph on $n=|V|=2k+1$ vertices. Then, the sequence of coefficients is unimodal, and as $k$ approaches infinity the peak of the sequence occurs at $3k\/4 + O(\\log k)$.\n\\end{cor}\n\nUsing a different and non-asymptotic approach, next we show that the windmill graph with 2 cliques, $W(2,t)$, has distance characteristic polynomial coefficients with peak exactly at $\\lfloor\\frac{n}{2}\\rfloor$. \n\n\\begin{thm}\nLet $G$ be a graph obtained by adding a universal vertex to the disjoint union of two cliques $K_t$, and let $n=|V(G)|=2t+1$. Then the sequence of coefficients of the distance characteristic polynomial of $G$ is unimodal with peak at $\\lfloor\\frac{n}{2}\\rfloor$.\n\\end{thm}\n\\begin{proof}\nConsider an equitable partition of $G$ into three subsets, one is the universal vertex and the other two correspond to cliques $K_t$. Using the quotient matrix of this partition, we can compute the distance characteristic polynomial of $G$ to be\n\\begin{eqnarray*}\np_{D}(x)&=&(x+1)^{2t-2}(-(t+1)-x)(-2t-(3t-1)x+x^2) =\\\\\n&=&(x+1)^{2t-2}(ax^3 + bx^2 + cx + d),\n\\end{eqnarray*}\nwhere $a=-1$, $b=2t-2$, $c=3t^2+4t-1$, and $d=2t^2+2t$.\nMultiplying the binomial expansion of $(x+1)^{2t-2}$ by $(ax^3 + bx^2 + cx + d)$ and combining the coefficients of terms with the same power, we obtain\n{\\allowdisplaybreaks{\n{\\small{\n\\begin{align*}\np_{D}(x)&=\\left[d{2t-2\\choose 0}\\right]x^0 +\\\\\n&+\\left[c{2t-2\\choose 0}+d{2t-2\\choose 1}\\right]x^1+\\\\\n&+\\left[b{2t-2\\choose0}+c{2t-2\\choose 1}+d{2t-2\\choose 2}\\right]x^2+\\\\\n&+\\sum_{i=0}^{2t-5}\\left[a{2t-2\\choose i} +b{2t-2\\choose i+1}+c{2t-2\\choose i+2}+d{2t-2\\choose i+3}\\right]x^{i+3}+\\\\\n&+\\left[a{2t-2\\choose 2t-4} + b{2t-2\\choose 2t-3} + c{2t-2\\choose 2t-2}\\right]x^{2t-1}+\\\\\n&+\\left[a{2t-2\\choose 2t-3}+b{2t-2\\choose 2t-2}\\right]x^{2t}+\\\\\n&+\\left[a{2t-2 \\choose 2t-2}\\right]x^{2t+1}.\n\\end{align*}\n}}\n}}\nLet $c_i$ be the coefficient of $x^i$ in $p_{D}(x)$. Then, for $t\\geq 4$ and $j\\geq 0$,\n\\begin{equation}\n\\label{beq1}\nc_{t-j}=a{2t-2\\choose t-3-j} +b{2t-2\\choose t-2-j}+c{2t-2\\choose t-1-j}+d{2t-2\\choose t-j}.\n\\end{equation}\nUsing the formula ${n\\choose k} = \\frac{(n - k + 1)}{k} { n\\choose k - 1}$, we can rewrite \\eqref{beq1} as\n\n\\begin{align*}\nc_{t-j} &= \\binom{2 t-2}{t-3-j}\n\\left(\na\n+b\\cdot\\frac{j+t+1}{t-j-2}\n+c\\cdot\\frac{j+t+1}{t-j-2}\\cdot\\frac{j+t}{t-j-1}\\right. +\\\\\n &+\\left. d\\cdot\\frac{j+t-1}{t-j-2}\\cdot\\frac{j+t}{t-j-1}\\cdot\\frac{j+t+1}{t-j}\\right) = \\\\\n&= \\binom{2 t-2}{t-3-j} \\cdot f(t,j),\n\\end{align*}\nwhere $f(t,j)=\\frac{t (j+t) \\left(t \\left(j^2+j (3-4 t)-t (5 t+11)+2\\right)+2\\right)}{(j-t) (j-t+1) (j-t+2)}$.\nThen, $c_{t-j}\\geq c_{t-(j+1)}$ if and only if\n$$\\binom{2 t-2}{t-3-j} \\cdot f(t,j)\\geq \\binom{2 t-2}{t-3-(j+1)} \\cdot f(t,j+1),$$\nwhich is equivalent to\n\\begin{equation}\n\\label{beq2}\n\\frac{f(t,j)}{t-3-j}\\geq \\frac{f(t,j+1)}{t+j+2}.\n\\end{equation}\nIt can be verified using a software with symbolic algebra for rational functions that \\eqref{beq2} holds for all integers $j$ and $t$ with $t\\geq 4$ and $0\\leq j < t-3$; for $t\\leq 3$, it can be verified that $c_{t-j}\\geq c_{t-(j+1)}$ by explicitly computing the distance characteristic polynomial. Similarly as above, it can be shown that $c_{t+j}\\geq c_{t+(j+1)}$ for all $t$ and $j\\geq 0$. Thus, it follows that $c_0\\leq \\ldots\\leq c_t\\geq \\ldots \\geq c_{2t+1}$, and hence the distance characteristic polynomial of $G$ is unimodal with peak  at $t = \\lfloor \\frac{2t+1}{2}\\rfloor=\\lfloor \\frac{n}{2}\\rfloor$.\n\\end{proof}\n\nConsider now the class of \\textit{barbell graphs} $B(t,\\ell)$, which are obtained by connecting two cliques $K_t$ with a path on $\\ell$ vertices by identifying the leaves of the path with one of the vertices in each clique. We also consider \\textit{lollipop graphs} $L(t,\\ell)$, which are obtained by adding a path on $\\ell$ vertices to a single clique $K_t$ so that one of the leaves of the path is also a vertex of the clique. We begin by locating the peak of the distance  characteristic polynomial for a barbell graph with $\\ell\\in\\{2,3,4,5\\}$.\n\n\\begin{thm}\\label{thm.barbell}\t\n\tLet $G$ be a barbell graph $B(t,\\ell)$, so that $|V(G)|=n=2t+\\ell-2$. Then the sequence of coefficients of the distance characteristic polynomial of $G$ is unimodal with peak at $t-1$ if $\\ell=2$ and at $t$ if $\\ell\\in\\{3,4,5\\}$.\n\\end{thm}\n\n\\begin{proof}\nWe prove the claim for $\\ell=2$; the cases $\\ell\\in\\{3,4,5\\}$ can be shown analogously.\nConsider a quotient partition into $4$ vertex sets: $2$ of them are the two vertices of the path of length $2$ connecting the cliques and the other $2$ correspond to the remaining $t-1$ vertices of each clique. We then have the quotient matrix\n$$B=\\left(\n\t\\begin{array}{cccc}\n\t\tt-2 & 1 & 2 & 3 (t-1) \\\\\n\t\tt-1 & 0 & 1 & 2 (t-1) \\\\\n\t\t2 (t-1) & 1 & 0 & t-1 \\\\\n\t\t3 (t-1) & 2 & 1 & t-2 \\\\\n\t\\end{array}\n\t\\right)$$\n\twith characteristic polynomial $$p_B(x)=-x^4+(2 t-4) x^3+\\left(8 t^2-4 t-4\\right) x^2+\\left(14 t^2-12t\\right) x+5 t^2-4 t.$$ \n\tFrom \\cite[Theorem 3.3]{HR21}, it follows that the distance characteristic polynomial of $G$ is \n\t{\\footnotesize{\n\t$$p_D(x)=(x+1)^{2t-4}\\left(-x^4+(2 t-4) x^3+\\left(8 t^2-4 t-4\\right) x^2+\\left(14 t^2-12t\\right) x+5 t^2-4 t\\right).$$\n\t}}\n\t\nLet $a=-1$, $b=2t-4$, $c=8t^2-4t-4$, $d=14t^2-12t$, and $e=5t^2-4t$. We can use $$(x+1)^{2t-4}=\\sum\\limits_{k=0}^{2t-4}\\binom{2t-4}{k} x^k$$ to write down a formula in the case $5\\leq k\\leq 2t-4$:\n\t$$c_{k}=a\\binom{2t-4}{k-4}+b\\binom{2t-4}{k-3}+c\\binom{2t-4}{k-2}+d\\binom{2t-4}{k-1}+e\\binom{2t-4}{k}.$$\n\tUsing the identity $\\binom{n}{k}=\\frac{n-k+1}k\\binom{n}{k-1}$ we obtain\n\n {\\small{\n\\begin{align*}\n\tc_{k}&=\\binom{2t-4}{k-4}\\left(a+\\frac{2t-k}{k-3}\\cdot b+\\frac{2t-k-1}{k-2}\\cdot\\frac{2t-k}{k-3}\\cdot c+\\right. \\\\\n\t&+\\left.\\frac{2t-k-2}{k-1}\\cdot\\frac{2t-k-1}{k-2}\\cdot\\frac{2t-k}{k-3}\\cdot d\\right. + \\\\ &+\\left.\\frac{2t-k-3}{k}\\cdot\\frac{2t-k-2}{k-1}\\cdot\\frac{2t-k-1}{k-2}\\cdot\\frac{2t-k}{k-3}\\cdot e \\right) = \\\\\n\t&= \\binom{2t-4}{k-4}f(t,k).\n\\end{align*}\n }}\n\tThen, $c_k\\geq c_{k-1}$ if and only if\n\t$$\\binom{2t-4}{k-4}f(t,k)\\geq \\binom{2t-4}{k-5}f(t,k-1),$$\n\twhich by applying the identity $\\binom{n}{k}=\\frac{n-k+1}k\\binom{n}{k-1}$ leads to\n\t$$\\frac{f(t,k)}{k-4}\\geq \\frac{f(t,k-1)}{2t-k+1}.$$\n\tIf $k=t$ then the simplified form of the above inequality is\n\t$$\\frac{21 t^4-35 t^3+t^2+3 t+6}{t^4-8 t^3+17 t^2+2 t-24}\\leq 0,$$\n\tand for $k=t-1$ we have $$\\frac{33 t^5+41 t^4-183 t^3+7 t^2+114 t-24}{(t-5) (t-4) (t-3)\n\t\t(t-2) (t+2)}\\geq 0.$$\n\tIt is straightforward to verify that if $t\\geq 6$, the inequality for the $k=t-1$ case holds, but the one for $k=t$ does not, meaning $c_{t-2}\\leq c_{t-1}> c_t$. Since the sequence of coefficients is unimodal by Theorem~\\ref{theo:unimodality}, this implies that $t-1$ is indeed the peak location. The cases $t\\leq5$ can be checked by straightforward calculation.\n\\end{proof}\n\nAn analogous argument as in the proof of Theorem \\ref{thm.barbell}\t can also be used to show the peak location of lollipop graphs with small~$\\ell$.\n\n\\begin{cor} Let $G$ be a lollipop graph $L(t,\\ell)$, so that $|V(G)|=n=t+\\ell-1$. Let $\\ell\\in\\{2,3,4,5\\}$. Then the sequence of coefficients of the distance characteristic polynomial of $G$ is unimodal with peak at $\\lfloor\\frac{n-1}2\\rfloor$.\n\\end{cor}\n\n\n\n\\section{Concluding remarks}\n\nWe end up by discussing another extremal case of block graphs, namely, block paths. According to SageMath simulations, the peak of a block path seems to be located between $\\lfloor\\frac{n}{3}\\rfloor$ and $\\lfloor\\frac{n}{2}\\rfloor$. A natural approach to extend the result from paths to block paths would be to generalize Collins proof for the distance characteristic polynomial of paths~\\cite{C89} and calculate the distance characteristic polynomial coefficients of block paths using the formula for edge-weighted block graphs (see Theorem~\\ref{GHH}). In general, the coefficient formula is given by\n$$c_{n-k}=(-1)^{n-k}\\sum\\limits_{1\\leq i_1\\leq\\dots\\leq i_k\\leq n}\\det D[v_{i_1},\\dots,v_{i_k}],$$\nwhere $D[v_{i_1},\\dots,v_{i_k}]$ is a $k\\times k$ submatrix of a $D$ whose rows and columns are indexed by vertices $v_{i_1},\\dots,v_{i_k}$. The sum is over all possible ways to choose $k$ out of $n$ vertices.\nThe main idea of Collins' approach is to interpret $D[v_{i_1},\\dots,v_{i_k}]$ as a distance matrix of an edge-weighted block graph, and then use Theorem~\\ref{GHH} to calculate its determinant. However, for the case when the blocks within the path have size at least $3$, the description of blocks of $D[v_{i_1},\\dots,v_{i_k}]$ can get very irregular: depending on a particular choice of $k$ vertices, they can be anything from $2$- to $k$-cliques with weights of edges seemingly following no pattern. Thus, for the more general setting of block graphs, it seems hopeless to use an analogous approach as Collins does for trees. \n\n\nOur computational results suggest the following question:\n\n\\begin{ques}\nAre the coefficients of the distance characteristic polynomial of any block graph with $n$ vertices unimodal with peak between $\\lfloor\\frac{n}{3}\\rfloor$ and $\\lfloor\\frac{n}{2}\\rfloor$?\n\\end{ques}\n\n\n We conclude by observing that the formula for the normalized coefficients of the characteristic polynomial of a tree \\cite[page 81]{GL78} probably also holds for uniform block graphs. If this was the case, then we could define the normalized coefficients for a $t$-uniform block graph (the distance eigenvalues of Hamming graphs \\cite[Section 3.1]{GRWC2015v2} suggest that there exists such a common factor). \n\n\n\\subsection*{Acknowledgements}\n\nAida Abiad is partially supported by the FWO (Research Foundation Flanders), grant number 1285921N. Antonina P. Khramova is supported by the NWO (Dutch Science Foundation), grant number OCENW.KLEIN.475. Jack H.~Koolen is partially supported by the National Natural Science Foundation of China (No. 12071454), Anhui Initiative in Quantum Information Technologies (No. AHY150000) and the National Key R and D Program of China (No. 2020YFA0713100). \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Optimal Risk-Aware LQR Controllers\n}\\label{Section_Analysis}\n\nThe analysis of the risk-aware dynamic program~\\eqref{FOR_EQN_LQR_constrained} consists of the following steps. First, we ensure the well-definiteness of~\\eqref{FOR_EQN_LQR_constrained}, also showing that~\\eqref{FOR_EQN_LQR_constrained} can be equivalently reexpressed as a \\textit{sequential variational Quadratically Contrained Quadratic Program (QCQP)}, or, more precisely, as a \\textit{Quadratically Constrained LQR (QC-LQR)} problem (Proposition~\\ref{ANA_PROP_Reformulation}). Then, we exploit Lagrangian duality (Theorem~\\ref{KKT}) to solve~\\eqref{FOR_EQN_LQR_constrained} \\textit{exactly and in closed form}. More specifically, we first derive an explicit expression for the optimal risk-aware controller~(Theorem~\\ref{ANA_THM_Optimal_Input}), given an arbitrary but fixed Lagrange multiplier. Then, we show how an optimal Lagrange multiplier may be efficiently discovered via trivial bisection~(Theorem~\\ref{OPTIMAL} and Proposition \\ref{ANA_PROP_Risk_Evaluation}).\n\\begin{proposition}[\\textbf{QCQP Reformulation}]\\label{ANA_PROP_Reformulation}\n\tLet Assumption~\\ref{FOR_ASS_noise} be in effect, and define the higher-order weighted statistics\n\t\\begin{align}\n\tM_3&\\triangleq \\ensuremath{\\mathbb{E}}\\set{(w_{i}-\\bar{w})(w_{i}-\\bar{w})'Q(w_{i}-\\bar{w})}\\in\\mathbb{R}^n\n\t\\,\\,\\, \\textrm{and}\\nonumber\\\\\n\tm_4&\\triangleq \\ensuremath{\\mathbb{E}}\\Neg{-.5}\\big\\{\\Neg{1.5}\\left[(w_{i}-\\bar{w})'Q(w_{i}-\\bar{w})-\\Tr (WQ)\\right]^2\\Neg{1.5}\\big\\}\\label{ANA_EQN_fourth_moment}\\ge0.\\nonumber\n\t\\end{align}\nThen, the risk-constrained LQR problem~\\eqref{FOR_EQN_LQR_constrained} is well-defined and equivalent to the sequential variational QCQP\n\\begin{equation}\\label{ANA_EQN_Reformulation}\n\\begin{aligned}\n    \\min_{u} &\\hspace{-6pt}& J(u) \\triangleq\\,\\,&  \\ensuremath{\\mathbb{E}}\\set{ x'_NQ x_N+\\sum_{t=0}^{N-1} x'_tQx_t+u'_tRu_t}\\\\\n\\mathrm{s.t.} &\\hspace{-6pt}& J_R(u) \\triangleq\\,\\,& \\ensuremath{\\mathbb{E}}\\set{\\sum_{t=1}^{N}4x'_tQWQx_t+4x_t'QM_3}\\le \\bar{\\epsilon}\\\\\n&\\hspace{-6pt}& & x_{t+1}=Ax_t+Bu_t+w_{t+1}\\\\\n&\\hspace{-6pt}& & u_t\\in \\ensuremath{\\mathcal{L}}_{2}(\\ensuremath{\\mathscr{F}}_t),\\quad t=0,\\dots N-1,\n\\end{aligned}\\,\\,,\n\\end{equation}\nwhere $\\bar{\\epsilon}\\triangleq\\epsilon-Nm_4+4N\\Tr\\set{(WQ)^2}$.\n\\end{proposition}\nThe proof can be found in the Appendix. Proposition~\\ref{ANA_PROP_Reformulation}\nis critical because it shows that our risk constraint is quadratic with respect to the state and control inputs. This enables us to apply duality theory, as discussed next.\n\\subsection{Lagrangian Duality}\nTo tackle problem~\\eqref{FOR_EQN_LQR_constrained}, we now consider the \\textit{variational Lagrangian} $\\ensuremath{{\\mathpzc{L}_{\\hspace{.1pt}}}} : \\ensuremath{\\mathcal{L}}_{2}(\\ensuremath{\\mathscr{F}}_0) \\times \\dots \\times \\ensuremath{\\mathcal{L}}_{2}(\\ensuremath{\\mathscr{F}}_{N-1}) \\times \\mathbb{R}_+ \\rightarrow \\mathbb{R}$ of the sequential QCQP~\\eqref{ANA_EQN_Reformulation}, defined as\n\\begin{align}\\label{ANA_EQN_Lagrangian_Original}\n&\\ensuremath{{\\mathpzc{L}_{\\hspace{.1pt}}}}(u,\\mu)\\triangleq J(u)+\\mu J_R(u)-\\mu \\bar{\\epsilon},\n\\end{align}\nwhere $\\mu\\in\\mathbb{R}_+$ is a multiplier associated with the variational risk constraint of \\eqref{ANA_EQN_Reformulation}. Hereafter, problem \\eqref{ANA_EQN_Reformulation} will be called the \\textit{primal problem}.\nAccordingly, the \\textit{dual function} $D\\hspace{-0.5pt}\\hspace{-0.5pt}:\\hspace{-0.5pt}\\hspace{-0.5pt}\\mathbb{R}_{+}\\hspace{-0.5pt}\\hspace{-0.5pt}\\rightarrow\\hspace{-0.5pt}\\hspace{-0.5pt}\\hspace{-0.5pt}[-\\infty,\\infty)$\nis additionally defined as\n\\begin{equation}\\label{FDUAL}\nD(\\mu)\\triangleq\\inf_{u\\in{\\cal U}_0}\\mathpzc{L}(u,\\mu),\n\\end{equation}\nwhere the \\textit{implicit feasible set} $\\mathcal{U}_0$ obeys ($k\\le N-1$)\n\\begin{equation}\n\\mathcal{U}_k \\Neg{1} \\triangleq \\Neg{1.5} \\set{\\Neg{.5} u_{k:N-1} \\Neg{1}\\in\\Neg{1} \\prod_{t=k}^{N-1} \\ensuremath{\\mathcal{L}}_{2}(\\ensuremath{\\mathscr{F}}_t) \\Neg{-1}\\Bigg|\\Neg{-1} x_{t+1}\\Neg{1}=\\Neg{1}Ax_t\\Neg{1}+\\Neg{1}Bu_t\\Neg{1}+\\Neg{1}w_{t+1} \\Neg{.5}}\\Neg{1}\\Neg{.5},\\Neg{1}\\nonumber\n\\end{equation}\nand contains the constraints of \\eqref{ANA_EQN_Reformulation} that have not been dualized in the construction of the Lagrangian in \\eqref{ANA_EQN_Lagrangian_Original}.\nNote that it is always the case that $D\\le J^{*}$ on $\\mathbb{R}_{+}$, where  $J^{*}\\hspace{-1pt}\\in\\hspace{-1pt}[0,\\infty]$\ndenotes the optimal value of the primal problem \\eqref{ANA_EQN_Reformulation}.\nThen, the optimal value of the always concave \\textit{dual problem}\n\\begin{equation}\\label{DUAL}\n\\sup_{\\mu\\ge0} D(\\mu) \\equiv \\sup_{\\mu\\ge0}\\inf_{u\\in{\\cal U}_0}\\mathpzc{L}(u,\\mu),\n\\end{equation}\n${D}^{*}\\hspace{-2pt}\\triangleq\\hspace{-.5pt}\\sup_{\\mu\\ge0}{D}(\\mu)\\hspace{-1pt}\\in\\hspace{-1pt}[-\\infty,\\hspace{-0.5pt}\\infty]$,\nis the tightest under-estimate of ${J}^{*}$, when knowing\nonly ${D}$.\n\nLeveraging Lagrangian duality, we may now state the following result, which provides sufficient optimality conditions for the QCQP \\eqref{ANA_EQN_Reformulation}. The proof is omitted, as it follows as direct application of \\cite[Theorem 4.10]{ruszczynski2011nonlinear} \n.\n\n\\begin{theorem}[\\textbf{Optimality Conditions}]\\label{KKT}\nLet Assumption \\ref{FOR_ASS_noise} be in effect. Suppose that there exists a feasible policy-multiplier pair $(u^*,\\mu^*) \\in \\mathcal{U}_0 \\times \\mathbb{R}_+$ such that\n\\begin{enumerate}\n    \\item $\\ensuremath{{\\mathpzc{L}_{\\hspace{.1pt}}}}(u^*(\\mu^*),\\mu^*)=\\min_{u\\in\\mathcal{U}_0}\\ensuremath{{\\mathpzc{L}_{\\hspace{.1pt}}}}(u,\\mu^*)=D(\\mu^*)$;\n    \n    \\item $J_R(u^*)\\le\\bar{\\epsilon}$, i.e., the dualized variational risk constraint of \\eqref{ANA_EQN_Reformulation} is satisfied by control policy $u^*$;\n    \n    \\item $\\mu^*\\Neg{.5}(J_R(u^*)-\\bar{\\epsilon})\\Neg{2.1}=\\Neg{1.8}0$, i.e., complementary slackness holds.\n\\end{enumerate}\nThen, $u^*$ is optimal for both the primal problem \\eqref{ANA_EQN_Reformulation} and the initial problem \\eqref{FOR_EQN_LQR_constrained}, $\\mu^*$ is optimal for the dual problem \\eqref{DUAL}, and \\eqref{ANA_EQN_Reformulation} exhibits zero duality gap, that is, $D^*\\equiv P^*<\\infty$.\n\\end{theorem}\n\n\n\nTheorem \\ref{KKT} will be serving as the backbone of our analysis towards the solution to problem \\eqref{ANA_EQN_Reformulation}, presented \nas follows. \n\n\n\n\\subsection{Optimal Risk-Aware Control Policies}\nLet $\\mu\\ge0$ be arbitrary but fixed. First, we may simplify the form of the Lagrangian $\\ensuremath{{\\mathpzc{L}_{\\hspace{.1pt}}}}$ and express it within a canonical dynamic programming framework. In this respect, we have the following, almost obvious, but important result.\n\\begin{lemma}[\\textbf{Lagrangian Reformulation}]\\label{ANA_THM_Lagrangian}\nLet Assumption \\ref{FOR_ASS_noise} be in effect and define the inflated state penalty matrix\n\\begin{equation}\n    Q_{\\mu} \\triangleq Q+4\\mu QWQ.\\nonumber\n\\end{equation}\nThen, for every $u_t\\in\\ensuremath{\\mathcal{L}}_2(\\ensuremath{\\mathscr{F}}_t)$, $t\\le N-1$, the Lagrangian function $\\ensuremath{{\\mathpzc{L}_{\\hspace{.1pt}}}}$ can be expressed as\n\t\\begin{equation}\\label{ANA_EQN_Lagrangian}\n\t\\ensuremath{\\mathcal{L}}(u,\\mu)\\hspace{-1pt}=\\hspace{-1pt}  \\ensuremath{\\mathbb{E}}\\hspace{-1pt}\\set{g_{N}(x_N,\\mu)\\hspace{-1pt}+\\hspace{-1pt}\\sum_{t=0}^{N-1}g_{t}(x_t,u_t,\\mu)\\hspace{-1pt}}\\hspace{-.5pt}+g(\\mu),\\hspace{-1pt}\n\t\\end{equation}\n\twhere, in the notation of Proposition~\\ref{ANA_PROP_Reformulation},\n\t\\begin{align*}\n\t\tg_N(x_N,\\mu)&\\triangleq x_N'Q_{\\mu}x_N+4\\mu M_3'Qx_N,\\\\\n\tg_{t}(x_t,u_t,\\mu)&\\triangleq x_t'Q_{\\mu}x_t+4\\mu M_3'Qx_t+u_t'Ru_t, \\, t\\le N-1,\\Neg{-1.5} \\\\\\\n\t\\textrm{and}\\quad g(\\mu)&\\triangleq -\\mu \\bar{\\epsilon}-4\\mu x_0'QWQ x_0-4\\mu M_3'Qx_0.\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\nIt follows from Proposition~\\ref{ANA_PROP_Reformulation} and the form of $\\ensuremath{{\\mathpzc{L}_{\\hspace{.1pt}}}}$.\n\\end{proof}\n\\begin{remark}[\\textit{Relation to generalized LQR with tracking}]\\label{ANA_REM_LQR}\nThe Lagrangian~\\eqref{ANA_EQN_Lagrangian} has the structure of a generalized LQR problem with a tracking objective. By completing the squares we can rewrite the stage cost $g_t(x_t,u_t,\\mu)$ as\n\\begin{align*}\n    g_{t}(x_t,u_t,\\mu)=(x_t+2\\mu M_3)'Q(x_t+2\\mu M_3)\\Neg{-12}\\\\\n+x'_t(4\\mu QWQ)x_t+u_t'Ru_t-4\\mu^2 M_3'QM_3,\n\\end{align*}\ni.e., the state penalty is quadratic and consists of two distinct terms. The first one, i.e., $(x_t+2\\mu M_3)'Q(x_t+2\\mu M_3)$ is a tracking error term that forces the state to be close to the static target $-2\\mu M_3$. Informally, in the case of skewed noise, by tracking $-2\\mu M_3$ we pre-compensate for directions in which the distribution of the noise has heavy tails. This decreases the statistical variability of the predicted stage cost.  The second term, $x_t'(4\\mu QWQ)x_t$, is a standard quadratic penalty term;  notice that, contrary to the risk-neutral case, the covariance of the noise $W$ now affects the penalty term. Informally, this term penalizes state directions which not only lead to high cost but are also more sensitive to noise, as captured by the product $QWQ$. \nHence, the risk-neutral LQR framework can exhibit inherent risk-averse properties, provided that its parameters are selected in a principled way. Of course, selecting those parameters \\emph{a priori} is not trivial.\n\\end{remark}\n\nThe structure of the Lagrangian as suggested by Lemma~\\ref{ANA_THM_Lagrangian} enables us to derive both a closed-form expression for its minimum \nand an explicit optimal control policy. To this end, define the \\textit{optimal cost-to-go} at stage $k\\le N-1$ as\n\\begin{equation}\n\\begin{aligned}\n& \\Neg{6}\\ensuremath{{\\mathpzc{L}_{\\hspace{.1pt}}}}^*_k(x_k,\\mu) \\\\\n& \\Neg{1.5}\\triangleq\\Neg{1.5}\\inf_{u_{k:N-1} \\in \\Neg{-1} \\mathcal{U}_k} \\ensuremath{\\mathbb{E}}\\Neg{1} \\set{g_N(x_N,\\mu)\\Neg{1}+\\Neg{1}\\sum_{t=k}^{N-1}g_t(x_t,u_t,\\mu)\\Bigg|\\ensuremath{\\mathscr{F}}_k},\\nonumber\n\\end{aligned}\n\\end{equation}\nwhere we omit the constant components of the Lagrangian. Under this definition, it is true that\n\\[\nD(\\mu)\\equiv\\inf_{u\\in\\mathcal{U}_0} \\ensuremath{{\\mathpzc{L}_{\\hspace{.1pt}}}}(u,\\mu)=\\ensuremath{{\\mathpzc{L}_{\\hspace{.1pt}}}}^*_0(x_0,\\mu)+g(\\mu).\n\\]\nWe may now derive the complete solution to \\eqref{FDUAL}, which is one of the main results of this paper, and provides optimal risk-aware control policies for every fixed multiplier $\\mu\\ge0$.\n\\begin{theorem}[\\textbf{Optimal Risk-Aware Controls}]\\label{ANA_THM_Optimal_Input}\n    Let Assumption~\\ref{FOR_ASS_noise} be in effect, choose $\\mu\\ge0$, and adopt the notation of Lemma \\ref{ANA_THM_Lagrangian}. For $t\\le N-1$, the optimal cost-to-go $\\ensuremath{{\\mathpzc{L}_{\\hspace{.1pt}}}}^*_t(x_t,\\mu)$ may be expressed as\n\\begin{align}\\label{FOR_EQN_optimal_cost_to_go}\n\\ensuremath{{\\mathpzc{L}_{\\hspace{.1pt}}}}^*_t(x_t,\\mu)= x'_{t}V_{t}x_{t} +4\\mu M'_3S'_{t}x_{t}+2\\bar{w}'T'_t x_t+c_{t},\n\\end{align}\nwhere the quantities $V_t$, $S_t$, $T_t$ and $c_t$ are evaluated through the backward recursions\n\t\\begin{align}\n\\label{ANA_EQN_DEF_Riccati_Matrix}\nV_{t-1}&\\Neg{1}=\\Neg{1}A'V_{t}A\\hspace{-.5bp}+\\hspace{-.5bp}Q_{\\mu}\\hspace{-1.5bp}-\\hspace{-1.5bp}A'V_{t}B(B'V_{t}B\\hspace{-.5bp}+\\hspace{-.5bp}R)^{-1}B'V_{t}A,\\\\\nK_{t-1}&\\Neg{1}=\\Neg{1}-(B'V_{t}B+R)^{-1}B'V_{t}A,\\\\\nS_{t-1}\t&\\Neg{1}=\\Neg{1}(A+BK_{t-1})'S_t+Q,\\\\\nT_{t-1}\t&\\Neg{1}=\\Neg{1}(A+BK_{t-1})'(T_t+V_t),\\\\\nl_{t-1}&\\Neg{1}=\\Neg{1}-2\\mu(B'V_{t}B+R)^{-1}B' S_t  M_3,\\\\\nh_{t-1}&\\Neg{1}=\\Neg{1}-(B'V_{t}B+R)^{-1}B'(V_t+T_t)\\bar{w}\\quad\\textrm{and}\\\\\nc_{t-1}&\\Neg{1}=\\Neg{0.5}c_t+\\Tr (WV_{t})+\\bar{w}'(2T_{t}'+V_t)\\bar{w}+4\\mu M_3'S_N'\\bar{w}\\nonumber \\\\\n&\\Neg{-6}-(l_{t-1}+h_{t-1})'(B'V_tB+R)(l_{t-1}+h_{t-1}),\n\\end{align}\nwith terminal values $V_N=Q_{\\mu}$, $S_N=Q$, $T_N=0$ and $c_N=0$.\nAdditionally, an optimal control policy \nthat achieves the dual value in \\eqref{FDUAL} may be expressed as\n\t\t\\begin{align}\\label{ANA_EQN_Optimal_Input}\nu^*_{t}(\\mu)=K_{t}x_{t}+l_{t}+h_t \\in  \\ensuremath{\\mathcal{L}}_2(\\ensuremath{\\mathscr{F}}_t),\\,\\,\\,\\forall t\\le N-1,\n\t\\end{align}\nand is unique up to sets of probability measure zero. \n\\end{theorem}\n\\begin{proof}\nBy using dynamic programming and assuming (temporarily) that involved measurability issues are resolved (\\cite{bertsekas2012approximate}, Appendix A),  we have, for every $k\\le N-1$, the recursive optimality condition (i.e., the Bellman equation)\n\\begin{align}\n\\Neg{8.5}\\ensuremath{{\\mathpzc{L}_{\\hspace{.1pt}}}}^*_k\\Neg{0.5}(x_k,\\Neg{0.5}\\mu)&\\Neg{1.5}=\\Neg{1.5}\\inf_{u_k} \\Neg{0.5}\\ensuremath{\\mathbb{E}}\\Neg{1.8}\\set{g_{k}\\Neg{0.5}(x_k,\\Neg{0.5}u_k,\\Neg{0.5}\\mu)\\Neg{2}+\\Neg{2}\\ensuremath{{\\mathpzc{L}_{\\hspace{.1pt}}}}^*_{k+1}\\Neg{0.8}(x_{k+1},\\Neg{0.5}\\mu)\\big|\\ensuremath{\\mathscr{F}}_{k}\\Neg{0.6}}\n\\Neg{1.5},\\Neg{6}\\nonumber\n\\end{align}\nwhere minimization is taken \\textit{pointwise} (i.e., over constants) over the control space $\\mathbb{R}^p$, and with $\\ensuremath{{\\mathpzc{L}_{\\hspace{.1pt}}}}^*_N(x_N,\\mu)=x'_NQx_N$.\nWe will prove the result by induction. The base case $k=N$ is immediate. Assume that~\\eqref{FOR_EQN_optimal_cost_to_go} is true for $k=t+1$;\nwe will show that this implies the same for $k=t$.\nBy Lemma~\\ref{ANA_THM_Lagrangian}, and after standard algebraic manipulations, we obtain\n\\begin{align}\n&\\ensuremath{\\mathbb{E}}\\Neg{1.5}\\set{g_{t}(x_t,u_t,\\mu)\\Neg{1.5}+\\Neg{1.5}\\ensuremath{{\\mathpzc{L}_{\\hspace{.1pt}}}}^*_{t+1}(x_{t+1},\\mu)\\big|\\ensuremath{\\mathscr{F}}_{t}}\n\\Neg{2}=\\Neg{1}u_{t}'(B'V_{t+1}B\\Neg{1}+\\Neg{1}R) u_{t}\\nonumber\\\\&+2\\set{x'_t A' V'_{t+1}+\\bar{w}'(V_{t+1}+T'_{t+1})+2\\mu M'_3S'_{t+1}}Bu_{t}\\nonumber\\\\\n&+x_{t}'(A'V_{t+1}A +\\bar{Q})x_{t}+c_{t+1}\\nonumber\\\\&+2\\set{2\\mu M'_3S'_{t+1}+\\bar{w}'(V_{t+1}+T_{t+1}')}Ax_{t}\\nonumber\\\\\n&+\\bar{w}'(2T_{t}'+V_t)\\bar{w}+4\\mu M_3'S_N'\\bar{w}+\\Tr (WV_{t+1}),\\label{ANA_EQN_Cost_To_Go_Form}\n\\end{align}\nwhere we have exploited the identities $x_{t+1}=Ax_t+Bu_t+w_{t+1}$,  $\\ensuremath{\\mathbb{E}}(w_{t+1}|\\ensuremath{\\mathscr{F}}_{t})=\\bar{w}$ and $\\ensuremath{\\mathbb{E}} (w_{t+1}-\\bar{w})(w_{t+1}-\\bar{w})'=W$. The reader may also verify that all measurability issues are now resolved in a recursive way, retrospectively.\nThe unique stationary point of the convex quadratic \\eqref{ANA_EQN_Cost_To_Go_Form} is\n\\begin{align*}\nu^*_t=K_{t}x_t+l_t+h_t,\n\\end{align*}\nwhich may be readily verified to lie in $\\ensuremath{\\mathcal{L}}_2(\\ensuremath{\\mathscr{F}}_t)$, as well.\nReplacing $u^*_t$ into~\\eqref{ANA_EQN_Cost_To_Go_Form} yields the optimal cost-to-go~\\eqref{FOR_EQN_optimal_cost_to_go}.\n\\end{proof}\nAs suggested by Remark~\\ref{ANA_REM_LQR}, it turns out that the optimal controller~\\eqref{ANA_EQN_Optimal_Input} is affine with respect to the state. The noise-dependent term $\\ell_t$ forces the state to track the reference $-2\\mu M_3$, which points away from \nheavy-tailed regions of the noise distribution.\nMeanwhile, the state-feedback term accounts for the internal dynamics. Similar to the risk-neutral case, the controller's behavior is governed the Riccati difference equation~\\eqref{ANA_EQN_DEF_Riccati_Matrix}. However, we now have an inflated stage cost matrix $\\bar{Q}=Q+4\\mu QWQ$, instead of the original.  As suggested by the product $QWQ$, the risk-aware control gain becomes more strict in directions that are simultaneously more costly and prone to noise, as captured by the covariance $W$. Finally, the term $h_t$ acts against the mean value of the noise--such a term also appears in risk-neutral LQR.\n\nAs a corollary, from standard LQR theory, we immediately obtain that for any $\\mu\\ge 0$, the optimal controller~\\eqref{ANA_EQN_Optimal_Input} will be internally stable, i.e., the spectral radius will be bounded $\\rho(A+BK_t)<1$, as the horizon $N$ grows to infinity.\n\\begin{corollary}[\\textbf{Internal Stability}]\\label{STABILITY}\nLet Assumptions \\ref{FOR_ASS_noise} and \\ref{FOR_ASS_LQR} be in effect, and adopt the notation of Lemma \\ref{ANA_THM_Lagrangian}.\nFor fixed $\\mu\\ge0$, consider the control policy $u^*(\\mu)$, as defined in~\\eqref{ANA_EQN_Optimal_Input}. As $N\\rightarrow \\infty$, $V_t$ converges exponentially fast to the unique stabilizing solution\\footnote{A stabilizing solution renders $A+BK$ stable.} of the algebraic Riccati equation\n\\[\nV=A'VA+Q_{\\mu}- A'VB(B'VB+R)^{-1}B'VA.\n\\] \nAs a result, for every $t\\ge 0$, it is true that, as $N\\rightarrow \\infty$, \n\\begin{align*}\n    K_t&\\rightarrow K\\triangleq -(B'VB+R)^{-1}B'VA,\\\\\n    S_{t}&\\rightarrow S \\triangleq (I-(A+BK)')^{-1}Q,\\\\\n    T_t&\\rightarrow T \\triangleq (I-(A+BK)')^{-1}(A+BK)'V,\\\\\n    l_t&\\rightarrow l \\triangleq -2\\mu(B'VB+R)^{-1}B'SM_3\\quad \\textrm{and}\\\\\n    h_t&\\rightarrow h \\triangleq -(B'VB+R)^{-1}B'(V+T)\\bar{w},\n\\end{align*}\nexponentially fast, and the closed-loop matrix $A+BK$ is stable (spectral radius $\\rho(A+BK)<1$).\n\n\n\n\\end{corollary}\n\\begin{proof}\nSince $Q_{\\mu}\\succeq Q$ and $(A,Q^{1\/2})$ is detectable, the pair $(A,Q_{\\mu}^{1\/2})$ is also detectable.\nSince $(A,B)$ is stabilizable, $(A,Q_{\\mu}^{1\/2})$ is detectable, and $R\\succ 0$, the exponential convergence of $V_t$ and $K_t$ to $V$ and $K$ respectively, and the stability of $A+BK$ follow from standard LQR theory~\\cite[Chapter 4]{anderson2005optimal}.  The proof of the convergence of the remaining terms follows similar steps.\n\\end{proof}\nUp to now we have discussed the properties of the optimal controller given a fixed $\\mu\\ge0$. In what follows, we show how to  compute an optimal multiplier $\\mu^*$, which will also satisfy the sufficient conditions for optimality suggested by Theorem \\ref{KKT}, as stated previously.\n\n\n\\subsection{Recovery of Primal-Optimal Solutions}\nFrom Theorem~\\ref{ANA_THM_Optimal_Input}, we know how to compute the relaxed optimal input $u^*(\\mu)$, for any given multiplier $\\mu\\ge0$.\nBut the risk constraint of the primal problem \\eqref{ANA_EQN_Reformulation} is the only one that has been dualized in the construction of the Lagrangian in~\\eqref{ANA_EQN_Lagrangian_Original}. Then, it turns out that we can also compute an optimal multiplier $\\mu^*$ via bisection, thus providing a complete solution to the primal problem.\nWe exploit the fact that, under the relaxed optimal policy $u^*(\\cdot)$, both the LQR cost $J(u^*(\\cdot))$ and the risk  functional $J_R(u^*(\\cdot))$ are monotone functions.\n\\begin{theorem}[\\textbf{Primal-Optimal Solution}]\\label{OPTIMAL} Let Assumption \\ref{FOR_ASS_noise} be in effect, and consider\nthe control policy $u^*(\\mu), \\mu\\ge0$, as defined in~\\eqref{ANA_EQN_Optimal_Input}. Then, the following statements are true:\n\\begin{enumerate}\n    \\item The LQR cost $J(u^*(\\mu))$ is increasing with $\\mu\\ge0$, while the risk constraint functional $J_R(u^*(\\mu))$ is decreasing.\n    \\item\n    Define the multiplier  \n    \\begin{align}\\label{ANA_EQN_Optimal_Multiplier}\n    \\mu^* \\triangleq \\inf\\set{\\mu \\ge 0:\\:J_R(u^*(\\mu))\\le \\bar{\\epsilon}}.\n\\end{align}\nIf $\\mu^*$ is finite, then the policy $u^*(\\mu^*)$ is optimal for the primal problem~\\eqref{ANA_EQN_Reformulation}, and this is the case as long as \n\\eqref{ANA_EQN_Reformulation} \n satisfies\nSlater's condition.\n\\end{enumerate} \n\\end{theorem}\n\\begin{proof}\nTo prove part 1), let $\\mu_2> \\mu_1\\ge 0$. From the definition of the Lagrangian and optimality of the controller $u^*(\\mu)$, we obtain the inequalities\n\\begin{align*}\n    J(u^*(\\mu_1))+\\mu_1 J_R(u^*(\\mu_1))&\\le  J(u^*(\\mu_2))+\\mu_1 J_R(u^*(\\mu_2))\\\\\n    J(u^*(\\mu_1))+\\mu_2 J_R(u^*(\\mu_1))&\\ge J(u^*(\\mu_2))+\\mu_2 J_R(u^*(\\mu_2))  .\n\\end{align*}\nBy subtracting, we get\n\\[\n(\\mu_2-\\mu_1)\\set{J_R(u^*(\\mu_1))-J_R(u^*(\\mu_2))}\\ge 0,\n\\]\nwhich shows that $J_R(u^*(\\mu_1))\\ge J_R(u^*(\\mu_2))$. The proof of\n$J(u^*(\\mu_1))\\le J(u^*(\\mu_2))$ is similar.\n\nTo prove part 2), we first show that, whenever $\\mu^*<\\infty$, $\\mu^*\\Neg{0.6}(J_R(u^*\\Neg{0.6}(\\mu^*))\\Neg{1}-\\Neg{1}\\bar{\\epsilon})\\Neg{2.2}=\\Neg{2}0$, i.e., complementary slackness holds. \nWe have two cases: either $\\mu^*=0$, where complementary slackness is satisfied trivially; or $\\mu^*>0$, $J_R(u^*(\\mu^*))\\le \\bar{\\epsilon}$. Therefore, it will be sufficient to show that in the latter case we can only have $J_R(u^*(\\mu^*))= \\bar{\\epsilon}$.\nSince $\\mu^*>0$, it is true that $J_R(u^*(0))>\\bar{\\epsilon}$. From Theorem~\\ref{ANA_THM_Optimal_Input} and Proposition~\\ref{ANA_PROP_Risk_Evaluation}, it also follows that the function $J_R(u^*(\\mu))$ is continuous with respect to $\\mu$ (all matrix inverses in~\\eqref{ANA_EQN_DEF_Riccati_Matrix} are continuous since $R\\succ 0$). Now, assume that $J_R(u^*(\\mu^*))< \\bar{\\epsilon}$. Then by continuity, there exists a $0<\\bar{\\mu}<\\mu^*$ such that $J_R(u^*(\\bar{\\mu}))=\\bar{\\epsilon}$, contradicting the definition of $\\mu^*$. Hence, we can only have $J_R(u^*(\\mu^*))= \\bar{\\epsilon}$, which shows that complementary slackness is satisfied. \n\nNow, complementary slackness, along with the trivial fact that $J_R(u^*(\\mu^*))\\le \\bar{\\epsilon}$ imply that the policy-multiplier pair $(u^*(\\mu^*),\\mu^*) \\in \\mathcal{U}_0 \\times \\mathbb{R}_+$ satisfies the sufficient conditions for optimality provided by Theorem \\ref{KKT}. Enough said. \n\nTo prove the last claim of part 2), suppose that \\eqref{ANA_EQN_Reformulation} \n satisfies\nSlater's condition, i.e., that there is an admissible policy $u^\\dagger$ in $\\mathcal{U}_0$ such that $J_R(u^\\dagger)-\\bar{\\epsilon}<0$. For every $\\mu\\ge0$, we have\n\\begin{equation}\n\\begin{aligned}\n    D(\\mu) &\\le J(u^\\dagger) + \\mu (J_R(u^\\dagger)-\\bar{\\epsilon}) \\nonumber\\\\\n\\implies D(\\mu) - \\mu (J_R(u^\\dagger)-\\bar{\\epsilon}) &\\le J(u^\\dagger)<\\infty. \\nonumber\n\\end{aligned}\n\\end{equation}\nNext, suppose that, for every $\\mu\\ge0$, $J_R(u^*(\\mu))-\\bar{\\epsilon}\\ge0$. Because $J(u^*(\\cdot))$ is increasing on $\\mathbb{R}_+$, it must be true that\n\\begin{equation}\n\\begin{aligned}\n    J(u^\\dagger) \\Neg{2}&\\ge\\Neg{1} \\sup_{\\mu\\ge0} D(\\mu) \\Neg{1}-\\Neg{1} \\mu (J_R(u^\\dagger)\\Neg{1}-\\Neg{1}\\bar{\\epsilon}) \\nonumber\\\\\n    & =\\Neg{1} \\sup_{\\mu\\ge0} J(u^*(\\mu)) \\Neg{1}+\\Neg{1} \\mu (J_R(u^*(\\mu))\\Neg{1}-\\Neg{1}\\bar{\\epsilon}) \\Neg{1}-\\Neg{1} \\mu (J_R(u^\\dagger)\\Neg{1}-\\Neg{1}\\bar{\\epsilon}) \\nonumber\\\\\n    & = \\Neg{1}\\infty,\n\\end{aligned}\n\\end{equation}\nwhich contradicts the fact that $J(u^\\dagger)<\\infty$. Therefore, there must exist $\\mu^\\dagger \\ge0$, such that $J_R(u^*(\\mu^\\dagger))-\\bar{\\epsilon}<0$. But $J_R(u^*(\\cdot))$ is decreasing on $\\mathbb{R}_+$ and, consequently, it must be the case that $\\mu^*\\in[0,\\mu^\\dagger)$. The proof is now complete.\n\\end{proof}\n\n\n\nTheorem \\ref{OPTIMAL} implies that we can find an optimal  multiplier satisfying the optimality conditions on Theorem \\ref{KKT} from~\\eqref{ANA_EQN_Optimal_Multiplier}, by performing simple bisection on $\\mu$.\nOf course, this requires evaluating the risk constraint functional $J_R(u^*(\\mu))$ for different values $\\mu\\ge0$. The evaluation may be performed in a recursive fashion, as the following result suggests.\n\\begin{proposition}[\\textbf{Risk Functional Evaluation}]\\label{ANA_PROP_Risk_Evaluation} Let Assumption \\ref{FOR_ASS_noise} be in effect, and adopt the notation of Lemma \\ref{ANA_THM_Lagrangian}.\nFor fixed $\\mu\\ge0$, consider the control policy $u^*(\\mu)$, as defined in~\\eqref{ANA_EQN_Optimal_Input}.\nWith terminal values $P_N=4QWQ$, $z_N=4M_3'Q$, $r_N=0$, consider the backward recursions\n\\begin{align}\n&P_{t-1}=(A+BK_{t-1})'P_t(A+BK_{t-1})+4QWQ,\\nonumber\\\\\n&z_{t-1}=(A+BK_{t-1})'z_t+4QM_3\\nonumber\\\\\n&\\,\\,\\,+2(A+BK_{t-1})'P_t\\paren{Bh_{t-1}+Bl_{t-1}+\\bar{w}}\\,\\,\\,\\, \\textrm{and}\\nonumber\\\\\n&r_{t-1}=r_t+\\Tr(P_tW)+z_t'(\\bar{w}+Bl_{t-1}+Bh_{t-1})\\nonumber\\\\\n&\\,\\,\\,+(Bl_{t-1}+Bh_{t-1}+\\bar{w})'P_{t}(Bl_{t-1}+Bh_{t-1}+\\bar{w}).\\nonumber\n\\end{align}\nThen, the risk constraint in problem \\eqref{ANA_EQN_Reformulation} may be evaluated by\n\\begin{equation}\nJ_R(u^*(\\mu))=x'_0P_0x_0+z'_0x_0+r_0.\\nonumber\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nOmitted; it is similar to that of Theorem~\\ref{ANA_THM_Optimal_Input}. \n\\end{proof}\n\n\n\\section*{Appendix: Proof of Proposition~\\ref{ANA_PROP_Reformulation}}\nLet $\\Delta_t\\triangleq x'_tQ x_t-\\ensuremath{\\mathbb{E}}\\paren{x'_tQx_t|\\ensuremath{\\mathscr{F}}_{t-1}}$ be the prediction error of the stage penalty at time $t$ given $\\ensuremath{\\mathscr{F}}_{t-1}$.\nWe proceed in two steps.\n\tFirst, we show that  $\\Delta_t$ is well-defined and belongs to $\\ensuremath{\\mathcal{L}}_2(\\ensuremath{\\mathscr{F}}_{t})$. Second, we obtain the closed form expression for the expected predictive variance $\\ensuremath{\\mathbb{E}} \\set{\\Delta^2_t}$. \n\t\n\t\\textit{Step a).} The state $x_t$ of the system depends linearly on past inputs $u_k$ as well as past noises $w_{k+1}$, for $k\\le t-1$. Under the constraint $u_k\\in\\ensuremath{\\mathcal{L}}_2(\\ensuremath{\\mathscr{F}}_k)$, and since by Assumption~\\ref{FOR_ASS_noise} $w_k\\in\\ensuremath{\\mathcal{L}}_2(\\ensuremath{\\mathscr{F}}_k)$, it also follows that \n\t$x_t\\in \\ensuremath{\\mathcal{L}}_2(\\ensuremath{\\mathscr{F}}_t)$, for all $t\\le N-1$. As a result, the expectation of $x_t'Qx_t$ exists and any conditional expectation $\\ensuremath{\\mathbb{E}}\\paren{x'_tQ x_t|\\ensuremath{\\mathscr{F}}_{t-1}}$ is well-defined and finite almost everywhere, for all $t\\le N-1$.\n\tDefine\n\t\\begin{align}\n\t\\hat{x}_{t}&\\triangleq \\ensuremath{\\mathbb{E}}(x_t|\\ensuremath{\\mathscr{F}}_{t-1})=Ax_{t-1}+Bu_{t-1}+\\bar{w}\\quad\\textrm{and}\\\\\n\t\\delta_t&\\triangleq w_t-\\bar{w}.\n\t\\end{align}\n\tNote that $\\hat{x}_t$ is well-defined since $x_{t}\\in \\ensuremath{\\mathcal{L}}_2(\\ensuremath{\\mathscr{F}}_t)$.\n\tReplacing $x_t$ with $\\hat{x}_{t}+\\delta_{t}$, we obtain the representation\n\t\\begin{align*}\n\tx'_tQ x_t&=\\hat{x}'_{t}Q\\hat{x}_{t}+2\\hat{x}'_{t}Q\\delta_t+\\delta_t'Q\\delta_t.\n\t\\end{align*}\n\tAll of the terms above are integrable since $\\hat{x}_t$, $\\delta_t$ are square-integrable.\n\tSince $\\hat{x}_{t}$ is measurable with respect to $\\ensuremath{\\mathscr{F}}_{t-1}$, the expectation of $x'_tQ x_t$ conditioned on $\\ensuremath{\\mathscr{F}}_{t-1}$ is\n\t\t\\begin{align*}\n\t\\ensuremath{\\mathbb{E}}\\paren{x'_tQ x_t|\\ensuremath{\\mathscr{F}}_{t-1}}&=\\hat{x}'_{t}Q\\hat{x}_{t}+\\Tr (WQ).\n\t\\end{align*}\nThen, the difference of the above quantities is\n\t\\begin{align*}\n\\Delta_t=\\delta_t'Q\\delta_t -\\Tr (WQ)+2\\hat{x}'_{t}Q\\delta_{t}.\n\t\\end{align*}\n\tComputing the squares of both sides leads to the expression\n\t\\begin{equation}\\label{ANA_EQN_Prediction_Difference}\n\t\\begin{aligned}\n\\Delta^2_t&=(\\delta_t'Q\\delta_t -\\Tr (WQ))^2+4\\hat{x}'_{t}Q\\delta_{t}\\delta'_{t}Q\\hat{x}_t\\\\&\\quad\\,+4\\hat{x}'_{t}Q\\delta_{t}(\\delta_t'Q\\delta_t -\\Tr (WQ)).\n\t\\end{aligned}\n\t\\end{equation}\n\tNote that all of the above terms are integrable, hence $\\ensuremath{\\mathbb{E}}\\set{\\Delta^2_t}$ is well-defined and finite. Integrability of the first term follows from Assumption~\\ref{FOR_ASS_noise}. Integrability of the second term comes from the fact that $\\hat{x}_t$ and $\\delta_t$ are square-integrable and independent of each other. Similarly, itegrability of the last term follows from integrability of $\\hat{x}_t$, Assumption~\\ref{FOR_ASS_noise} \n\tand independence of $\\hat{x}_t,\\delta_t$. \n\t\n\t\\textit{Step b).}\nFrom~\\eqref{ANA_EQN_Prediction_Difference}, it is true that\n\t\\begin{align*}\n\t&\\ensuremath{\\mathbb{E}}\\set{\\Delta_t^2|\\ensuremath{\\mathscr{F}}_{t-1}}=\n\t4\\hat{x}'_{t}QWQ\\hat{x}_{t}+m_4+4\\hat{x}'_{t}QM_3.\n\t\\end{align*}\n  Taking expectation again gives\n  \\begin{align*}\n\t\\ensuremath{\\mathbb{E}}\\set{\\Delta_t^2}=m_4+\\ensuremath{\\mathbb{E}}(4\\hat{x}'_{t}QWQ\\hat{x}_{t}+4\\hat{x}'_{t}QM_3).\n  \\end{align*}\nBy orthogonality of $\\hat{x}_t$, $\\delta_t$, and since $\\ensuremath{\\mathbb{E}} \\delta_t=0$, $\\ensuremath{\\mathbb{E}} \\delta_t\\delta'_t=W$, \n  \\begin{align*}\n\t\\ensuremath{\\mathbb{E}}\\hspace{-1pt}\\set{\\Delta_t^2}&\\hspace{-2pt}=\\hspace{-1pt}\\ensuremath{\\mathbb{E}}(4x_{t}'QWQx_{t}\\hspace{-1pt}+\\hspace{-1pt}4x_{t}'QM_3)\\hspace{-1pt}+\\hspace{-1pt}m_4\\hspace{-1pt}-\\hspace{-1pt}4\\Tr\\Neg{1}\\set{(WQ)^2}\\hspace{-1pt}.\n  \\end{align*}\n  The result follows if we replace $\\ensuremath{\\mathbb{E}}\\set{\\Delta^2_t}$ with the  right-hand side above in the risk constraint\n $\n  \\sum_{t=1}^{N}\\ensuremath{\\mathbb{E}}\\set{\\Delta^2_t}\\le \\epsilon.\n$\n\\hfill $\\qedsymbol$\n\\section{Conclusion and Future Work}\\label{Section_Extensions}\nWe presented a new risk-aware reformulation of the classical LQR problem, where we introduce a new risk measure to be used as an explicit and tunable risk constraint, along with the standard LQR objective. By restricting the expected cumulative predictive variance of the state penalties, we can decrease the variability of the state at will, protecting the system against uncommon but strong random disturbances. The optimal controller enjoys a simple closed-form expression with clear interpretation, is always stable and is easy to tune.\nOur scheme works for arbitrary noise process distributions, as long as the corresponding  fourth-order moments are finite.\n\n\nMoving forward, our framework opens up many directions for extensions and future research. First, we would like to note that our analysis does not depend on the constraints\nhaving the same matrix $Q$ as in the cost. In fact, we can define our risk constraint as\n\\[\n\\ensuremath{\\mathbb{E}} \\set{\\sum_{t=1}^{N} \\left[ x'_tQ_c x_t-\\ensuremath{\\mathbb{E}}\\paren{x'_tQ_cx_t|\\ensuremath{\\mathscr{F}}_{t-1}} \\right]^2}\\le \\epsilon,\n\\]\nwhere $Q_c$ is a design choice. Proposition~\\ref{ANA_PROP_Reformulation} still holds, in the sense that the constraint can be rewritten as a quadratic one.\nThis implies that, thanks to its simplicity, the predictive variance constraint can be easily incorporated in more general problems, e.g., MPC or classical constrained-LQR, adding a risk-aware flavor to them.\nAnother possibility is to employ stage-wise constraints of the form\n\\[\n\\ensuremath{\\mathbb{E}}\\left[ x'_tQ_t x_t-\\ensuremath{\\mathbb{E}}\\paren{x'_tQ_tx_t|\\ensuremath{\\mathscr{F}}_{t-1}} \\right]^2\\le \\epsilon_t,\\quad t\\le N.\n\\]\nIn this case, the optimal controller will be similar to~\\eqref{ANA_EQN_Optimal_Input}, but will depend on multiple Lagrange multipliers that can be optimized with primal-dual algorithms.\nLastly, our predictive variance constraint is based on one-step-ahead prediction. In some cases, careless selection of $Q_c$ \nmight make our controller more myopic. At the same time, though, increasing the prediction horizon  might not always preserve the quadratic form of the constraint. In future work, we would also like to address this issue.\n\n\n\\section{Risk-Constrained LQR Formulation} \\label{Section_Formulation}\n\nConsider a discrete-time linear system in state-space form evolving according to the stochastic difference equation\n\\begin{equation}\\label{FOR_EQN_system}\nx_{t+1}=Ax_{t}+Bu_{t}+w_{t+1},\n\\end{equation}\nwhere $x_t\\in \\ensuremath{\\mathbb{R}}^n$ is the state, $u_t\\in \\ensuremath{\\mathbb{R}}^p$ is an exogenous control signal, $A$ is the state transition matrix, and $B$ is the input matrix. We assume that the initial value $x_0$ is deterministic and fixed. Signal $w_{t}\\in \\ensuremath{\\mathbb{R}}^n$ is a random process noise (not necessarily Gaussian) and is assumed to be \\textit{i.i.d}. \nFor $t\\ge0$, let $\\ensuremath{\\mathscr{F}}_t=\\sigma\\paren{x_{0:t},u_{0:t}}$ be the $\\sigma$-algebra generated by all observables up to time $t$, and let $\\ensuremath{\\mathscr{F}}_{-1}$ be the trivial $\\sigma$-algebra. Based on this notation, $x_t,u_t,w_t$ are $\\ensuremath{\\mathscr{F}}_t$-measurable, while $w_{t+1}$ is independent of $\\ensuremath{\\mathscr{F}}_t$.\nWe also make an additional assumption on the process noise, as follows.\n\\begin{assumption}[\\textbf{Noise Regularity}]\\label{FOR_ASS_noise}\nThe process $w_t$ has finite fourth-order moment, i.e., for every $t\\in \\mathbb{N}$, $\\ensuremath{\\mathbb{E}}\\norm{w_{t}}^4_2<\\infty$.\n\\end{assumption}\n\\noindent The above condition is mild and satisfied by very general noise distributions, including many heavy-tailed ones.\nDenote the mean of the noise by $\\bar{w}\\triangleq\\ensuremath{\\mathbb{E}} w_k$ and its variance by $W\\triangleq \\ensuremath{\\mathbb{E}}(w_k-\\bar{w})(w_k-\\bar{w})'$.\n\n In the classical risk-neutral formulation of the LQR problem, one is interested in the multistage stochastic program\n \\begin{equation}\\label{FOR_EQN_Risk_Neutral_LQR}\n\\begin{aligned}\n\\min_{u}&\\quad \\ensuremath{\\mathbb{E}}\\set{ x'_NQ x_N+\\sum_{t=0}^{N-1} x'_tQx_t+u'_tRu_t}\\\\\n\\mathrm{s.t.}&\\quad x_{t+1}=Ax_t+Bu_t+w_{t+1}\\\\\n&\\quad u_t\\in \\ensuremath{\\mathcal{L}}_{2}(\\ensuremath{\\mathscr{F}}_t),\\,t=0,\\dots N-1\n\\end{aligned},\n\\end{equation}\nwhere $u=u_{0:N-1}$ are the inputs from time $0$ up to time $N-1$, for some horizon $N\\in \\mathbb{N}$. For each $t$, the \\textit{causality constraint} on $u_t$ restricts the inputs to the space of square-integrable $\\ensuremath{\\mathscr{F}}_t$-measurable vector-valued random elements of appropriate dimension,\ndenoted as $\\ensuremath{\\mathcal{L}}_2(\\ensuremath{\\mathscr{F}}_t)$. It also guarantees that the optimization problem is well-defined and with finite cost. In order for the optimal LQR controller to be well-behaved and stable as the horizon $N$ grows, we also make the following standard assumption.\n\\begin{assumption}[\\textbf{LQR}]\\label{FOR_ASS_LQR}\nThe pair $(A,B)$ is stabilizable, the pair $(A,Q^{1\/2})$ is detectable, matrix $Q\\succeq 0$ is positive semi-definite and matrix $R\\succ 0$ is positive definite.\n\\end{assumption}\nAs mentioned above, the classical LQR problem is risk-neutral, since it optimizes performance only on average ~\\cite{bertsekas2017dynamic}. \nStill, even if the average performance is good, the state can grow arbitrarily large under less probable, yet extreme events.\nIn other words, the state can exhibit large variability.\nTo deal with this issue, we propose a risk-constrained formulation of the LQR problem, posed as\n\\begin{equation}\\label{FOR_EQN_LQR_constrained}\n\\begin{aligned}\n\\min_{u} &\\quad \\ensuremath{\\mathbb{E}}\\set{ x'_NQ x_N+\\sum_{t=0}^{N-1} x'_tQx_t+u'_tRu_t}\\\\\n\\mathrm{s.t.} &\\quad \\ensuremath{\\mathbb{E}} \\set{\\sum_{t=1}^{N} \\left[ x'_tQ x_t-\\ensuremath{\\mathbb{E}}\\paren{x'_tQx_t|\\ensuremath{\\mathscr{F}}_{t-1}} \\right]^2}\\le \\epsilon\\\\\n&\\quad x_{t+1}=Ax_t+Bu_t+w_{t+1}\\\\ \n&\\quad u_t\\in \\ensuremath{\\mathcal{L}}_{2}(\\ensuremath{\\mathscr{F}}_t),\\,t=0,\\dots, N-1\n\\end{aligned}\\,.\n\\end{equation}\n\\noindent  Here, the risk measure adopted is the cumulative expected predictive variance of the state cost. The predictive variance incorporates information about the tail \\textit{and} skewness of the penalty $x_t'Qx_t$. This forces the controller to take higher-order noise statistics into account, mitigating the effect of rare though large noise values.\nHence, our risk-aware LQR formulation not only forces the state $x_t$ to be close to zero, but also explicitly restricts its variability.\n\nProblem \\eqref{FOR_EQN_LQR_constrained} offers a simple and interpretable way to control the trade-off between average performance and risk. By simply decreasing $\\epsilon$, we increase risk-awareness. Inspired by standard risk-aware formulations, in the above optimization problem our risk definition is tied to the specific state penalty $x'_tQx_t$ of the LQR. However, all of our results are still valid if we employ the predictive variance of a different quadratic form, e.g., the norm of the state, $\\norm{x_t}^2$, in the constraint---see Section~\\ref{Section_Extensions}. \nLastly, note that the initial state is fixed (for simplicity), so there is no associated risk term for $t=0$.\n\nIn the next section, we show that the risk constraint of \\eqref{FOR_EQN_LQR_constrained} can be rewritten in quadratic form. This will allow us to solve problem \\eqref{FOR_EQN_LQR_constrained} using duality theory and obtain a closed-form solution, exploiting higher-order noise moments.\n\n\n\n\\section{Introduction}\\label{Section_Introduction}\nAchieving good performance in expectation is often insufficient in \nthe design of stochastic control systems, \nespecially when dealing with modern, critical applications. Examples\nappear naturally in many areas, including wireless industrial control \\cite{Ahlen2019}, energy \\cite{Bruno2016,Moazeni2015}, finance \\cite{Markowitz1952,Follmer2002,Shang2018}, robotics \\cite{Kim2019,Pereira2013}, networking \\cite{Ma2018}, and safety \\cite{samuelson2018safety,chapman2019cvar}, to name a few.  Indeed, occurrence of less probable, non-typical or unexpected events might lead the underlying dynamical system to experience shocks with possibly catastrophic consequences, e.g., a drone diverging too much from a given trajectory in a hostile environment, or an autonomous vehicle crashing onto a wall or hitting a pedestrian. In such situations, design of effective \\emph{risk-aware} control policies is highly desirable, systematically compensating for those extreme events, at the cost of slightly sacrificing average performance under nominal conditions.\n\n\nTo highlight the usefulness of a risk-aware control policy, let us consider the following simple, motivating example. Let $x_{k+1}=x_{k}+u_k+w_{k+1}$ model an aerial robot, moving along a line. Assume that the process noise $w_k$ is \\textit{i.i.d.} Bernoulli, taking the values $\\beta>2$ with probability $1\/\\beta$ and $0$ with probability $1-1\/\\beta$. This noise represents shocks, e.g., wind gusts, that can occur with some small probability. We would then like to minimize the LQR cost $\\ensuremath{\\mathbb{E}} \\sum_{t=0}^{N} \\{x^2_t\\}$, i.e., the total displacement of the robot over a horizon of $N$ time steps. In this case, the LQR optimal controller is $u^{\\mathrm{LQR}}_k=-x_k-1$, where $-1\\equiv-\\mathbb{E}w_k$ cancels the mean of the process noise. We see that the LQR solution is \\textit{risk-neutral}, as it does not account for the fact that the shock $\\beta$ could be arbitrarily large.  On the other hand, the risk-aware LQR formulation proposed in this work results in a \\textit{family} of optimal controllers of the form\n\\[\nu^{*}_{t}(\\mu)=-x_t-1-\\frac{\\mu}{1+2\\mu}(\\beta-2),\\quad\\mu\\ge0,\n\\]\nwhere $\\mu$ controls the trade-off between average performance and risk. As $\\mu$ increases, we move from the risk-neutral to the \\textit{maximally risk-aware} controller $u^{*}_{t}(\\infty)=-x_t-\\beta\/2$, which treats the noise as adversarial---see Fig.~\\ref{fig:toy_example}.\n\\begin{figure}[t]\n\\vspace{6.5bp}\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{toy_figure.pdf}\\vspace{-4bp}\n\t\\caption{Comparison between risk-neutral and risk-aware control performance, when the system experiences rare but large shocks---here the shock occurs at time $6$. By sacrificing average behavior, the risk-aware controllers push the state away from the direction of the shock. \n}\n\t\\label{fig:toy_example}\n \t\\vspace{-13bp}\n\\end{figure}\n\nIn both classical and recent literature in linear-quadratic problems, risk awareness in estimation and control is typically achieved by replacing the respective random cost with its exponentiation~\\cite{jacobson1973optimal,Whittle1981,bacsar2000risk,pham2012linear,roulet2019convergence,Speyer1992,Dey1999,Moore1997,Dey1997,Bauerle2014more}. Yet, the resulting stochastic control problem might not be well-defined for general classes of noise distributions, as it requires the moment generating function of the cost to be finite. Thus, heavy-tailed or skewed distributions, which are precisely those exhibiting high risk, are naturally excluded. Also, even if the expectation of the exponential cost is finite, it does not lead to a general, closed-form and interpretable solution. \nA notable exception is that of a\nGaussian process noise, also known as the Linear Exponential Quadratic Gaussian (LEQG) problem, which does enjoy a simple closed-form solution~\\cite{Whittle1981}. \nApparently though, the Gaussian assumption is unable to capture distributions with asymmetric (skewed) structure, as in the above example.\n\nIn this paper, we propose a new risk-aware reformulation of the LQR problem, in which the standard LQR objective is minimized \\textit{subject to} an explicit and tunable risk constraint. Our contributions are as follows.\n\n\\noindent\\textbf{--New Risk Measure.} We introduce  the cumulative expected one-step predictive variance of the associated state penalty as a new risk measure for LQR control.\nIn this way, our risk-constraint formulation ensures not only a small LQR cost, but also \\textit{guaranteed} statistical variability of the state penalty.\n\n\n\\noindent\n\\textbf{--Optimal Risk-Aware Controls \\& Stability.}\nWe show that our new risk-constrained formulation results in a quadratically constrained LQR problem (Proposition~\\ref{ANA_PROP_Reformulation}), which admits an explicit closed-form solution with a natural interpretation. The optimal risk-aware feedback controller is affine with respect to the system state (Theorems~\\ref{ANA_THM_Optimal_Input} \\& \\ref{OPTIMAL}). \nThe affine component of the control law \npushes the state away from directions where the noise exhibits (skewed) heavy tails. Meanwhile, the state feedback gain satisfies a new risk-aware Riccati recursion, in which the state penalty is inflated in riskier directions, where both the noise covariance and state penalty are simultaneously larger. Further, we show that our optimal risk-aware controller is always stable, under standard LQR conditions (Corollary \\ref{STABILITY}).\n\n\\noindent\\textbf{--Arbitrary Noise Model.}\nContrary to the LEQG approach, our results are valid for arbitrary noise distributions, provided the associated fourth-order moments are finite; thus, heavy-tailed or skewed noises are supported within our framework.\n\n\\noindent\\textbf{--Relation to LQR with Tracking.}\nWe show that, \\textit{by appropriate re-parameterization}, our risk-aware LQR problem is equivalent to a generalized risk-neutral LQR problem with a tracking objective. Essentially, this implies that risk-neutral LQR formulations can provide inherent risk-averse behavior, as long as the involved parameters are selected in a principled way, as presented herein.\n\n\\textbf{\\textit{Related Work:}}\nRisk-aware optimization has been studied in a wide variety of decision making contexts \\cite{A.2018,Cardoso2019,W.Huang2017,Jiang2017,Kalogerias2018b,Tamar2017,Vitt2018,Zhou2018,Ruszczynski2010,SOPASAKIS2019281,chapman2019cvar}. The basic idea is to replace expectations by more general functionals, called risk measures\\cite{ShapiroLectures_2ND}, purposed to effectively quantify the statistical volatility of the involved random cost function, in addition to mean performance. Typical examples are mean-variance functionals \\cite{Markowitz1952,ShapiroLectures_2ND},\nmean-semideviations \\cite{Kalogerias2018b}, and Conditional Value-at-Risk\n(CVaR) \\cite{Rockafellar1997}.\n\n \nIn the case of control systems, apart form the aforementioned exponential approach, CVaR optimization techniques have also been considered for risk-aware constraint satisfaction~\\cite{chapman2019cvar}. Although CVaR captures variability and tail events well, CVaR optimization problems rarely enjoy closed-form expressions. Approximations are usually required to make computations tractable, e.g., process noise and controls are assumed to be finite-valued~\\cite{chapman2019cvar}, thus excluding the LQR setting. Predictive variance constraints have also been used as a measure of risk in portfolio optimization~\\cite{abeille2016lqg}. Different from our paper, the noise is assumed to be Gaussian and the variance is with respect to linear stage costs.\n\nAnother related concept is that of robust control, where the system model is unknown. The objective is to optimally control the true system under worst case model uncertainty. When there is no model uncertainty, in the case of stochastic process noise, robust controllers usually reduce to their risk-neutral LQR counterparts~\\cite{tzortzis2016robust,dean2017sample}. On the contrary, in risk-aware control, extreme noise events are part of the system model. Even if the system is exactly modeled, we would still need to consider risk-aware control if the process noise is heavy-tailed or highly variable. From this point of view, robustness and risk are orthogonal concepts.\nInterestingly, in the case of adversarial noise, there is a connection between robust and maximally risk-aware controllers~\\cite{glover1988state}.\n\n\n\\textit{\\textbf{Notation:}}\nThe transpose operation is denoted by $(\\cdot)'$. If $x_{k},\\dots,x_{t}$ is a sequence of vectors, then $x_{k:t}$ denotes the batch vector of all $x_i$ for $k\\le i\\le t$. The $\\sigma$-algebra generated by a random vector $x$ is denoted by $\\sigma(x)$.\n\n\n\\section{Simulations And Discussion}\\label{Section_Simulations}\n\\begin{figure}[t]\n\t\\vspace{3.5bp}\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{xQx_figure.pdf}\\vspace{-4bp}\n\t\\caption{Evolution of the state penalties $x'_kQx_k$, over the first $50$ steps. Notice that our risk-aware LQR controller indeed limits the variability of $x'_kQx_k$. In fact, it sacrifices performance under small wind forces, but protects the system against large wind gusts, for example at time $5-10$.}\n\t\\vspace{-12bp}\n\t\\label{fig:xQx}\n\\end{figure}\nConsider a flying robot that moves on a horizontal plane, i.e., the Euclidean space $\\mathbb{R}^2$. We assume that its linearized dynamics can be abstracted by a double integrator as\n\\begin{equation}\\label{SIM_eq:integrator}\n    x_{k+1}=\\matr{{cccc}1&T_s&0&0\\\\0&1&0&0\\\\0&0&1&T_s\\\\0&0&0&1}x_k+\\matr{{cc}\\tfrac{T_s^2}{2}&0\\\\T_s&0\\\\0&\\tfrac{T_s^2}{2}\\\\0&T_s}(v_k+d_k),\n\\end{equation}\nwhere  $T_s=0.5$ is the sampling time, $x_{k,1}$, $x_{k,3}$ are the position coordinates, $x_{k,2}$, $x_{k,4}$ the respective velocities and $v_k$ is the acceleration input. Let $d_k$ be a wind disturbance force that acts on the robot, which is modeled as follows: We assume that $d_{k,1}$ constitutes the dominant wind direction with non-zero mean and large variability, while the orthogonal direction $d_{k,2}$ is a weak wind direction with zero mean and small variability.\nWe model $d_{k,1}$ as a mixture of two gaussians $\\mathcal{N}(30,30)$, $\\mathcal{N}(80,60)$ with weights $0.8$ and $0.2$, respectively. This bimodal distribution models the presence of infrequent but large wind gusts.  The weak direction $d_{k,2}$ is modeled as zero-mean Gaussian~$\\mathcal{N}(0,5)$.\nIf we cancel the mean of $d_k$ by applying $v_k=u_k-\\ensuremath{\\mathbb{E}} d_k$, then~\\eqref{SIM_eq:integrator}, can be written in terms of~\\eqref{FOR_EQN_system}, where $w_k=B(d_k-\\ensuremath{\\mathbb{E}} d_k)$ is now a zero-mean disturbance. \n\n\nConsider now the LQR problem with parameters \n\\[\nQ = \\textrm{diag}(1,0.1,2,0.1)\n\\Neg{2}\\quad\\textrm{and}\\quad R=I,\n\\] \nand a horizon of length $N=5000$. We primarily compare our risk-aware LQR formulation with the classical, risk-neutral LQR via simulations. To tune our controller, we vary $\\mu$ in~\\eqref{ANA_EQN_Optimal_Input} directly instead of varying $\\epsilon$. We also (heuristically) compare our controller with the exponential (LEQG) method, even though the noise is not Gaussian, by plugging in the second order statistics $W$. Let the tuning parameter of LEQG be $\\theta$. Note that the exponential problem is well defined only if $\\theta<0.001276$ (roughly), where the ``neurotic breakdown\" occurs~\\cite{Whittle1981}. For the purpose of comparison, we simulate all schemes under the same noise sequence $w_{0:N}$. \n\n\nIn Fig.~\\ref{fig:xQx}, we see the evolution of the state penalty terms $x_k'Qx_k$, for the first $50$ time steps, under the different control schemes.  By slightly sacrificing performance under small wind forces, our risk-aware LQR controller forces the state to have less variability and, thus, protects the robot against large gusts. On the other hand, the state of the robot state can grow very large under the risk-neutral \\textit{and} LEQG schemes. This behavior is illustrated  even more clearly in Fig.~\\ref{fig:cdf}, where we present the time-empirical cumulative distribution of the state penalties for all $N$ time steps. The time-empirical \"probability\" of suffering large state penalties is drastically smaller compared to LQR or LEQG.\n\n\\begin{figure}[t]\n\\vspace{3.5bp}\n\t\\centering{}\n\t\\includegraphics[width=\\columnwidth]{cdf_figure.pdf}\\vspace{-4bp}\n\t\\caption{The time-empirical cdf for the state penalties $x_k'Qx_k$, $k\\le N$, for the LQR (risk-neutral), our method, and LEQG. Our method sacrifices some average performance but exhibits much smaller variability for the state penalties. It also protects the system against rare but large wind gusts.}\n\t\\vspace{-12bp}\n\t\\label{fig:cdf}\n\\end{figure}\n\\begin{figure}[t]\n\\vspace{3.5bp}\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{x1_u1_figure.pdf}\\vspace{-4bp}\n\t\\caption{Evolution of the state $x_{k,1}$, and the input $u_{k,1}$ over the first $50$ steps. The controller pushes the state away from the direction of the large gusts, which helps the robot to avoid extreme perturbations. Meanwhile, by inflating the state penalty with the $\\mu QWQ$, we force the state-feedback component to be more cautious with the state. Naturally, being more cautious with the state requires extra control effort.}\n\t\\vspace{-13bp}\n\t\\label{fig:x1}\n\\end{figure}\nTo better illustrate how the proposed risk-aware controller works, we also discuss the evolution of the position $x_{k,1}$ and the input $u_{k,1}$, as shown in Fig. \\ref{fig:x1}, for the first $50$ steps. First, we observe that the controller pushes the state $x_{k,1}$ towards negative values, away from the direction of the large gusts.\nSecond, notice that we penalize $x_{k,3}$ more in $Q$. In fact, the risk-neutral LQR results in the steady state gains $K_{\\mathrm{LQR},11}=-0.697$, $K_{\\mathrm{LQR},12}=-1.201$, $K_{\\mathrm{LQR},23}=-0.925$, $K_{\\mathrm{LQR},24}=-1.376$, i.e., it is stricter with direction $x_{k,3}$. However, $x_{k,1}$ exhibits more variability due to the strong wind direction. In contrast, our risk-aware scheme adapts to the noise in a principled way. Due to the inflation term $\\mu QWQ$, our scheme returns the steady-state gains $K_{11}=-2.1008$, $K_{12}=-2.2132$, $K_{23}=-1.1161$, $K_{24}=-1.5131$, which means that the risky direction $x_{k,1}$ is controlled more strictly. Naturally, being more cautious with the state leads to higher control effort. Lastly, although the LEQG controller is also more state-cautious, it is agnostic to the heavy tails of the wind distribution. Hence, it still suffers from large perturbation due to the wind gusts.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the recent work,  \\cite{Noi} \\cite{Raf}, the problem of the minimum  time optimal control of a two level quantum system ({\\it qubit}) was solved. In the model considered, the system was  subject to a drift and a control field orthogonal to the drift and with bounded norm (cf. section \\ref{Rev} for a precise statement of the optimal control problem). Two level quantum systems are of paramount importance in quantum mechanics and, in particular, in  quantum computation (see, e.g., \\cite{NC}). In this context, evolution in minimum time  is a natural requirement when it is desired to minimize the effect of the environment or to increase the speed of implementation of a given quantum computation. In fact, the circuit model of quantum computation (cf. \\cite{NC}) requires a cascade of simple evolutions on elementary quantum systems which therefore have to occur in very short time in order to maximize the speed of the overall computation. The solution of the minimum time problem given in \\cite{Noi} \\cite{Raf} is quite simple and explicit, requiring only very elementary numerical work, something very rare for optimal control problems.\n\nWhen trying to control $N \\geq 2$  qubits {\\it simultaneously} with independent controls in minimum time, one could argue that the above results could simply be applied $N$ times to obtain  the optimal control. Let us assume, for simplicity of exposition, $N=2$, and let $X_{f1}$ and $X_{f2}$ the desired final condition for system $S_1$ and system $S_2$,  respectively. Let us denote by $T_1$ and $T_2$ the optimal times to reach $X_{f1}$ and $X_{f2}$ for system $S_1$ and $S_2$, respectively. If $T_1=T_2$, then the two optimal controls designed with the techniques of \\cite{Noi} \\cite{Raf},  will drive the two systems to the desired final conditions, in minimum time. However if $T_1 \\not= T_2$, then there is a {\\it slow} system and a {\\it fast} system and the solution is not simply to slow down the fast system to synchronize it with the slow system. The problem is that for a (bilinear) system with drift, such as the ones considered here, the fact that a certain evolution can be performed  at (minimum) time  $T_1$ does not ensure that the same evolution can be achieved at a later time $T_2 > T_1$. Therefore, this problem requires a careful analysis of the {\\it reachable sets} ${\\cal R}(T)$,  for the systems under consideration, that is, the set of evolutions or states which can be reached at exactly time $T$.\n\n\nThe goal of this paper is twofold. On one hand we want to solve the above mentioned time optimal and synchronization control problem for $N$ qubits to any desired final condition. On the other hand we want to describe the structure of the reachable sets for the dynamics of two level quantum systems. In doing so we will determine the features of the dynamics of quantum bits which make the time optimal control problem amenable of such a simple solution as described in \\cite{Noi}, \\cite{Raf}. In fact the underlying geometric structure of the problem, which facilitates its solution, can be found in other problems as well.\n\n\n\n\nThe paper is organized as follows. In section \\ref{Rev} we describe the problem of minimum time control for a two level quantum system which mathematically amounts to a time optimal control problem for a right invariant bilinear system on the Lie group $SU(2)$. We shall consider and review the main results and the approach of \\cite{Noi}, \\cite{Raf}. We will see that the time optimal synthesis can be visualized in the unit disk where all the geometric analysis can be performed. A special trajectory called the {\\it critical trajectory} plays a special role in that time optimal trajectories loose optimality intersecting this curve. In section \\ref{Contin}, we further analyze the optimal synthesis for two level quantum systems by studying the continuity properties of the minimum time function as a function of the bound on the controls or of  the final point. This analysis sheds further light on the time optimal synthesis for this model. It shows that the minimum time function is continuous except for points on the critical trajectory where it presents a right discontinuity. This discontinuity is a manifestation of the fact that the reachable sets for this model do not monotonically grow with time.  The analysis of reachable sets is performed in section \\ref{Reachsets}. Here using a change of coordinates we reduce the problem to the  study of the reachable sets for a driftless system. We use monotonicity of the reachable sets in this case and the relation between the geometry of reachable sets and time optimal control trajectories. In this section, we try to present the results in a way that can be applied to more general systems on Lie groups highlighting the features of the system which allow the treatment to go through.  Finally, the description of the reachable sets is used in section \\ref{synchro} to give  an algorithm for the design  of the  synchronous time optimal control for $N$ qubits.\n\n\\section{The time optimal control problem for a two level quantum system}\\label{Rev}\n\nLet us consider the {\\it Schr\\\"odinger operator equation} (see, e.g., \\cite{Sakurai}) for a spin $\\frac{1}{2}$ particle in a magnetic field with time varying components  (controls) in the $x$ and $y$ direction, $u_x$ and $u_y$. The equation is written as\n\\be{basicmodel}\n\\dot X=\\tilde \\sigma_z X + u_x \\tilde \\sigma_x X+u_y \\tilde \\sigma_y X, \\qquad X(0)={\\bf 1},\n\\end{equation}\nwhere $X \\in SU(2)$, $X(0)={\\bf 1}$ the identity,  and $\\tilde \\sigma_{x,y,z}$ are proportional to the {\\it Pauli matrices},  $\\sigma_{x,y,z}$, and  form a basis of the Lie algebra $su(2)$. They are defined as\n\\be{Paulimat}\n\\tilde \\sigma_x:=\\frac{i}{2}\\sigma_x= \\frac{1}{2}\\begin{pmatrix} 0 & i \\cr i & 0\\end{pmatrix}, \\qquad\n\\tilde \\sigma_y:=\\frac{-i}{2}\\sigma_y= \\frac{1}{2} \\begin{pmatrix} 0 & -1 \\cr 1 & 0 \\end{pmatrix},\n\\qquad \\tilde \\sigma_z:=\\frac{i}{2} \\sigma_z= \\frac{1}{2}\\begin{pmatrix} i & 0 \\cr 0 & -i\\end{pmatrix}.\n\\end{equation}\nThe Lie algebra $su(2)$ is equipped with an inner product between  matrices, $\\langle \\cdot, \\cdot \\rangle$, defined as $\\langle A, B \\rangle:=Tr(A B^\\dagger)$, so that the associated norm is $\\|A\\|:=\\sqrt{\\langle A, A \\rangle}$. The coefficient of $\\tilde \\sigma_z$ in (\\ref{basicmodel}) (which is  called {\\it Larmor frequency} in NMR applications see, e.g., \\cite{Abragam}) is taken equal to $1$, in the appropriate units, without loss of generality, since to this situation we can always reduce ourselves with an appropriate re-scaling and-or reversing  of the time variable and-or a redefinition of the bound on the control\\footnote{This is true unless the Larmor frequency is zero in which case we have a driftless system which will be considered in detail in the following (see section \\ref{Reachsets}).} (cf. Remark 1.1 in \\cite{Noi})\n\n\nThe problem considered in \\cite{Noi} and \\cite{Raf} is, given a desired final condition $X_f \\in SU(2)$, to find the control functions $u_x,u_y$, that steer the state $X$ of system (\\ref{basicmodel}) from the identity, ${\\bf 1}$, to $X_f$ in minimum time, under  the constraint  that $u_x^2(t)+u_y^2(t) \\leq \\gamma^2$, for every $t$. We shall later generalize this problem (cf. section \\ref{synchro}) to the minimum time {\\it simultaneous} control of $N$ two level quantum systems.\n\n\n\n\nThe classical approach to the solution of this type of problems is to apply the {\\it Pontryagin Maximum Principle PMP} (see, e.g., \\cite{Agrachev}, \\cite{Flerish}, and,  in particular  \\cite{Mikobook}, \\cite{Tesi}  for applications to quantum systems.), which gives the necessary conditions for optimality. This results in a set of candidate optimal control functions (more or less explicit) which are typically parametrized by some real values. Then these controls are placed back in the dynamics which is (numerically) integrated. The parameters are then chosen so that the final condition is met and the time of transfer is the minimum one. In essence, the PMP allows one to reduce a search over a space of functions (the controls) to a search over a finite dimensional space (the space of the parameters). In the case of system (\\ref{basicmodel}) application of the PMP\\footnote{Along with a result showing the non-optimality of singular extremals (cf. \\cite{Noi}).} shows that the optimal candidate controls (extremals)  are of the type  \\cite{Noi}:\n\\be{NSE}\nu_x=\\gamma \\sin(\\omega \\tau + \\tilde \\phi), \\qquad u_y=-\\gamma \\cos(\\omega \\tau +\\tilde \\phi),\n\\end{equation}\nwhere $\\tau$ denotes the time variable and $\\omega$ and $\\tilde \\phi$ are two parameters  (frequency and phase) to be tuned in order to reach the desired final condition while minimizing the time.\n\n\n By plugging $u_x$ and $u_y$ in (\\ref{NSE}) into (\\ref{basicmodel}), the corresponding differential equation {\\it can be explicitly integrated}.\nThe solution is given by\n\\be{soluzexpli}\nX(\\tau,\\omega,\\tilde \\phi):=\\begin{pmatrix} e^{i \\omega t}(\\cos(a t)+ i \\frac{b}{a} \\sin(a t)) & e^{i (\\omega t + \\tilde \\phi)} \\frac{\\gamma}{a} \\sin(a t) \\cr  - e^{-i(\\omega t + \\tilde \\phi)} \\frac{\\gamma}{a} \\sin(a t)  & e^{-i \\omega t}(\\cos(a t)-i \\frac{b}{a} \\sin(a t)) \\end{pmatrix},\n\\end{equation}\nfor $t:=\\frac{\\tau}{2}$, $b:=1-\\omega$, $a:=\\sqrt{\\gamma^2 + b^2}$. Direct inspection of formula (\\ref{soluzexpli}) shows that the $\\tilde \\phi$ only affects the phase of the off-diagonal element. This means that the minimum time only depends on the $(1,1)$ element of the matrix giving the desired final condition.\\footnote{This can also be proved without using the explicit form of the optimal controls (cf. Proposition 2.1 in \\cite{Noi}).} We can use an arbitrary phase (for example $\\tilde \\phi=0$) and study the trajectory of the $(1,1)$ element which belongs to the unit disk. Once the frequency $\\omega$ corresponding to the time optimal control steering to the desired point of the unit disk has been found, then the phase $\\tilde \\phi$ is chosen to match the phase of the off diagonal element of the desired final condition.\n\nLet $x=x(t)$ and $y=y(t)$ be the real and the imaginary part of  the (1,1) element in  (\\ref{soluzexpli}), which, parametrized by $\\omega$, is given by:\n\\be{curvex}\nx(t):=x_{\\omega}(t)=\\cos(\\omega t)\\cos(at)-\\frac{b}{a}\\sin(\\omega t)\\sin(at),\n\\end{equation}\n\\be{curvey}\ny(t):=y_\\omega(t)= \\sin(\\omega t)\\cos(at)+\\frac{b}{a}\\cos(\\omega t)\\sin(at).\n\\end{equation}\nIn \\cite{Noi} the optimal synthesis was described in the unit disk.\nThis was done for values of $\\gamma$, $\\frac{1}{\\sqrt{3}} \\leq \\gamma \\leq 1$. A typical picture for the optimal trajectories is in Figure \\ref{Fig1} to which we shall refer in the following discussion. In general, the values of $\\omega$ for which the controls in (\\ref{NSE}) are optimal are $-\\infty < \\omega \\leq 1+\\gamma^2:=\\omega_c$. The limit value $\\omega_c:=1+\\gamma^2$ called the {\\it critical frequency} corresponds to a trajectory, also called the {\\bf critical trajectory} which has a cuspid at the time $T_c=\\frac{\\pi}{2 \\gamma \\sqrt{1+\\gamma^2}}$. The critical trajectory is optimal until time $T_c$ and then looses optimality. It is the curve in blue in Figure \\ref{Fig1}. Among the other trajectories, another important one is the one corresponding to $\\omega=\\omega^*:=\\frac{1+\\gamma^2}{2}$ which corresponds to a circle (in red in Figure \\ref{Fig1}) centered at   $\\left( \\frac{\\gamma^2}{1+\\gamma^2}, 0\\right)$ and with radius $\\frac{1}{1+\\gamma^2}$. This trajectory was called the {\\it separatrix} since optimal trajectories corresponding to $-\\infty < \\omega < \\omega^*$ entirely lay outside of the region bounded by it until they loose optimality upon reaching the boundary of the unit disk. Also, trajectories with $\\omega^* < \\omega \\leq \\omega_c$ remain {\\it inside} the separatrix until reaching the critical trajectory and loosing optimality there\\footnote{The loss of optimality is of different type at the boundary of the unit disk and on the critical trajectory. At the boundary of the unit disk the trajectory (with corresponding $\\omega$) is optimal until an {\\it including} the point on the unit disk. However,  for a curve intersecting the critical trajectory, this curve is optimal until {\\it but not including}, the point on the critical trajectory.}(sample trajectories are drawn in black in Figure \\ref{Fig1}).\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{OS1}\n\\caption{Optimal synthesis for $\\gamma=\\frac{1}{\\sqrt{2}}$. Reported here are the separatrix in red corresponding to $\\omega=\\omega^*=\\frac{3}{4}$ and the critical curve in blue corresponding to $\\omega=2\\omega^*=\\omega_c=\\frac{3}{2}$. Moreover in black are the optimal trajectories corresponding to $\\omega=-3$,  $\\omega=-1$, $\\omega=0$, $\\omega=0.2\\omega^*$, $\\omega=0.5\\omega^*$, $\\omega=0.7\\omega^*$, $\\omega=0.9\\omega^*$, outside  the separatrix, and $\\omega=1.3\\omega^*$, $\\omega=1.45\\omega^*$, $\\omega=1.6\\omega^*$, $\\omega=1.8\\omega^*$.}\n\\label{Fig1}\n\\end{figure}\n\n\n\nGiven, the qualitative picture of the optimal synthesis as in Figure \\ref{Fig1} it is  straightforward to find the time optimal control to steer to a desired final condition. Let $X_f$ be the desired final condition and $P_f$ the point in the unit disk representing the $(1,1)$ entry of this matrix. Then one finds $\\omega$ such that the corresponding trajectory contains $P_f$ (this can be done for example using a bisection algorithm and the graphics of the trajectories). Once $\\omega$ has been found one determines the minimum time. This can be done for example solving  a static optimization problem, minimizing (in $t$) the distance of the trajectory with fixed $\\omega$ from the point $P_f$. Finally, one determines the phase $\\tilde \\phi$ in (\\ref{NSE})  to match (in (\\ref{soluzexpli})) the phase of the off diagonal entry of the desired final condition.\n\n\n\\vspace{0.25cm}\n\nThe assumption  of $\\gamma \\leq 1$ was used  in \\cite{Noi} to guarantee that the minimum time to reach points on the boundary of the unit disk set is an increasing function of the phase of the point. Without this  feature the optimal synthesis becomes more complicated. The assumption of $\\frac{1}{\\sqrt{3}} \\leq \\gamma$ was used to render  the synthesis inside the separatrix analytically tractable. As $\\gamma \\rightarrow 0$ the critical trajectory become longer and longer and resembles a long spiral filling the whole disk bounded by the separatrix. Simulations show that the optimal trajectories follow this trajectory before going around its endpoint (the one corresponding to $T_c=\\frac{\\pi}{2 \\gamma \\sqrt{1+\\gamma^2}}$) and intersecting the critical trajectory. This behavior observed with numerical simulations is however difficult to describe analytically.\n\nThese difficulties were overcome in \\cite{Raf} where the author reconsidered the curves (\\ref{curvex}), (\\ref{curvey}) but this time keeping $t$ fixed and considering $\\omega$ as a variable. If this is done, the curves (\\ref{curvex}) and (\\ref{curvey}) represent, for $\\omega$ in a certain interval, part of the boundary of the reachable set at that time $t$. For every $\\omega$ in that interval the corresponding point is the endpoint of an optimal trajectory. The points where the curve in (\\ref{curvex}) (\\ref{curvey}) are not endpoints of optimal trajectories anymore are the ones where the curve corresponding to time $t$, say ${\\cal F}_t$\\footnote{This curve is called the {\\it optimal frontline} in \\cite{Raf}.} intersects the one  at time $t+dt$, ${\\cal F}_{t+dt}$. The curve of all these intersections with varying $t$ {\\it coincides with the critical trajectory above discussed} which is therefore the {\\it envelope} of these curves. As $\\gamma \\rightarrow \\infty$ the critical trajectory becomes shorter and shorter and disappears at the limit which corresponds to a driftless system\\footnote{Recall that in (\\ref{basicmodel}) we have normalized the Larmor frequency to $1$ and the value of $\\gamma$ is in fact $\\gamma:=\\frac{\\gamma^{'}}{\\omega_0}$ where $\\gamma^{'}$ is the `physical' bound on the norm of the control and $\\omega_0$ is the value of the Larmor frequency which tends to  zero for a driftless system.} At the other end, when $\\gamma \\rightarrow 0$ the critical trajectory becomes very long. Besides giving an  interpretation of the critical trajectory, the analysis of \\cite{Raf} provides an alternative and general method to find the minimum time control. One consider the frontlines ${\\cal F}_t$ with evolving $t$ and bounded on one end by the boundary of the unit disk and on the other end by the critical trajectory (i.e., (\\ref{curvex}), (\\ref{curvey}) with $\\omega=\\omega_c$) and look for the smallest $t$ such that the desired final point $P_f$ is in ${\\cal F}_t$. This idea will be used later in this paper. We refer to \\cite{Raf} for details.\n\nIn the next two sections we shall further elaborate on these results and investigate  the geometric nature of the minimum time problem on $SU(2)$ and its relation with the geometry of the reachable sets.\n\n\n\n\\section{Properties of the minimum time function}\\label{Contin}\n\nWe now study the continuity  and monotonicity properties of the minimum time function for the above problem for a two level quantum system. This is a function of the final state $X_f \\in SU(2)$  fixed and of the bound on the control  $\\gamma$. As discussed above  $X_f$ is represented by a point $P_f$ in the unit disc. For any $\\gamma>0$, and for any final condition $P_f=(x_f,y_f)$, with $0\\leq x_f^2+y_f^2\\leq 1$, we denote by\n $t_{P_f}=t_{P_f}({\\gamma})$ the optimal time to reach the fixed final condition, and by\n$\\omega_{P_f}=\\omega_{P_f}({\\gamma})$   the corresponding frequency $\\omega$ of the optimal control (cf. (\\ref{NSE}))\n\nUsing (\\ref{curvex}) (\\ref{curvey}), we know that\n\\be{curva}\n\\begin{array}{ccl}\nx_f&= &x_{\\omega_{P_f}({\\gamma})}(t_{P_f}({\\gamma}))  \\\\\ny_f&= &y_{\\omega_{P_f}({\\gamma})}(t_{P_f}({\\gamma}))\n\\end{array} \\end{equation}\nas in equations (\\ref{curvex}), (\\ref{curvey}). And we let\n$b_{P_f}({\\gamma}):=1-\\omega_{P_f}({\\gamma})$ and $a_{P_f}({\\gamma})=\\sqrt{(b_{P_f}({\\gamma}))^2+\\gamma^2}$.\n\n\nThe function $t_{P_f}(\\gamma)$ is monotonic non increasing since all controls available for $\\gamma=\\gamma_1$ are also available for any $\\gamma \\geq \\gamma_1$.\n\n\n\nWe consider first the final condition {\\it in the interior} of  the unit circle, so let $P_f=(x_f,y_f)$ be a fixed point such that $0\\leq x_f^2+y_f^2<1$. We give  a bound on the value of the optimal frequency $\\omega$ to be used in (\\ref{NSE}). Define\n$K_{P_f}:=\\sqrt{\\frac{x_f^2+y_f^2}{1-(x_f^2+y_f^2)}}$.\n\n\\bp{ovvia}\nIf $\\omega\\in \\RR$ and $t>0$ is such that  $x_{\\omega}(t)=x_f$ and $y_{\\omega}(t)=y_f$, then\n\\be{ovviae}\n1-\\gamma K_{P_f} \\leq \\omega \\leq 1+\\gamma K_{P_f}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nIf (\\ref{ovviae}) does not hold, then $b^2=(1-\\omega)^2>\\gamma^2K_{P_f}^2$. This implies (using $a^2:=\\gamma^2+b^2$)\n\\[\n\\frac{\\gamma^2}{a^2}\\sin^2(at)<\\frac{1}{1+K_{P_f}^2}.\n\\]\nBy using equations (\\ref{curvex}) and (\\ref{curvey}), we have:\n\\[\nx_f^2+y_f^2=1-\\frac{\\gamma^2}{a^2}\\sin^2(at)> \\frac{K_{P_f}^2}{1+K_{P_f}^2}=x_f^2+y_f^2,\n\\]\nwhich is a contradiction.\n\\end{proof}\nThe above simple proposition gives bounds on the frequencies to be used for a given desired final condition $P_f$. This can be used in the search of the optimal control described in the previous section. In particular, if $P_f=(0,0)$, which corresponds to SWAP-like final conditions, $X_f:=\\begin{pmatrix}0 & e^{i\\phi}\\cr e^{-i\\phi} & 0 \\end{pmatrix}$, then $K_{P_f}=0$, so the only admissible $\\omega$ is $\\omega=\\omega_{P_f}(\\gamma)=1$, independently of $\\gamma$ (these are the resonant controls considered for example in \\cite{OptRes}).\n\nWe now study the limit of $t_{P_f}(\\gamma)$ as $\\gamma$ goes to zero.\n\\bp{seconda}\nIf $P_f$ is in the interior of the unit disc, then\n\\be{t-gamma}\n\\lim_{\\gamma \\to 0^+}  t_{P_f}({\\gamma})=+\\infty.\n\\end{equation}\nIf $P_f$ is on the boundary of the unit disc and corresponds to a phase $\\psi_f$, i.e., $P_f=e^{i \\psi_f}$, then\n\\be{T-circonferenza}\n\\lim_{\\gamma \\to 0^+}  t_{P_f}({\\gamma})=\\psi_f.\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\n\nConsider first the case where $P_f$ is inside the unit circle, i.e., the case of (\\ref{t-gamma}).\nUsing equations (\\ref{curvex}) and (\\ref{curvey}),\nwe have that:\n\\be{relazione}\n\\frac{\\gamma^2}{(b_{P_f})^2+\\gamma^2}\\sin^2\\left(a_{P_f}\\,t_{P_f}\\right)\n=1- x_f^2+y_f^2>0.\n\\end{equation}\n\nFirst we prove that\n\\be{omega-gamma}\n\\lim_{\\gamma \\to 0^+} |b_{P_f}({\\gamma})|=0.\n\\end{equation}\n\n\n\nAssume, by the way of contradiction, that (\\ref{omega-gamma}) does not hold. Then we have:\n\\[\n\\exists \\,\\epsilon >0,  \\text{ such that }\\forall \\ n>0 \\ \\exists \\ 0<\\gamma_n<\\frac{1}{n} \\text{ with }\n(b_{P_f}({\\gamma_n}))^2 >\\epsilon.\n\\]\nThis implies\n\\[\n0\\leq \\frac{\\gamma_n^2}{(b_{P_f}^{\\gamma_n})^2+\\gamma_n^2}\\leq \\frac{\\gamma_n^2}{\\epsilon+\\gamma_n^2}\n\\]\nThus we have:\n\\[\n\\lim_{n\\to+\\infty} \\frac{\\gamma_n^2}{(b_{P_f}({\\gamma_n}))^2+\\gamma_n^2}\n\\sin^2\\left(\\sqrt{((b_{P_f}({\\gamma_n}))^2+\\gamma_n^2)}\\,t_{P_f}(\\gamma_n) \\right)=0\n\\]\nsince the $\\sin$ function is bounded, and $\\gamma_n \\to 0$. This contradicts equation (\\ref{relazione}),\nthus (\\ref{omega-gamma}) holds.\n\nNow we use (\\ref{omega-gamma}) to prove (\\ref{t-gamma}).\nAssume, by the way of contradiction, that (\\ref{t-gamma}) does not hold. As done before, this would imply that, for some $K>0$ there exists a sequence $\\gamma_j$  ($j>0$) with $0<\\gamma_j<\\frac{1}{j}$, such that $0<t_{P_f}({\\gamma_j})<K$.\n\nThus using (\\ref{omega-gamma}):\n\\[\n\\lim_{j\\to +\\infty} \\sqrt{(b_{P_f}({\\gamma_j}))^2+\\gamma_j^2}\\,t_{P_f}({\\gamma_j}) =0.\n\\]\nSince $\\frac{\\gamma^2}{(b_{P_f}^{\\gamma})^2+\\gamma^2}<1$, the previous equation would imply that\n\\[\n\\lim_{j\\to+\\infty} \\frac{\\gamma_j^2}{(b_{P_f}({\\gamma_j}))^2+\\gamma_j^2}\n\\sin^2\\left(\\sqrt{(b_{P_f}({\\gamma_j}))^2+\\gamma_j^2}\\,t_{P_f}({\\gamma_j}) \\right)=0\n\\]\nand this, again, contradicts equation (\\ref{relazione}).\nThus (\\ref{t-gamma}) holds.\n\nFinally for points on the unit circle (\\ref{T-circonferenza}) follows from the explicit expression of the minimum time for a point $P_f=e^{i\\psi_f}$ which was\nderived in \\cite{Noi} under the assumption $\\gamma \\leq 1$. This is\n\\be{T-circonferenza-bis}\nt_{P_f}(\\gamma)=\\frac{\\psi_f (2 \\pi - \\psi_f)}{\\pi - \\psi_f +\\sqrt{\\pi^2 + \\gamma^2 \\psi_f (2\\pi - \\psi_f)}}.\n\\end{equation}\nTaking the limit as $\\gamma \\rightarrow 0$ we obtain (\\ref{T-circonferenza}).\n\\end{proof}\n\n\\vspace{0.25cm}\n\nIn order to study the continuity of the function $t_{P_f}(\\gamma)$ as a function of $\\gamma$ (or as a function of $P_f$) we study, more generally, the continuity of the `joint' function $T(P_f, \\gamma):=T(x_f,y_f,\\gamma):=t_{P_f}(\\gamma)$. This is a function defined on a half infinite cylinder ${\\cal C}:=\\{ x_f, y_f, \\gamma \\, | \\, 0 \\leq x_f^2+y_f^2 \\leq 1, \\, \\gamma >0\\}$. We shall restrict ourselves to study this function in the interior of ${\\cal C}$, i.e., ${\\texttt{int} \\cal C}:= \\{ x_f, y_f, \\gamma \\, | \\, 0 \\leq x_f^2+y_f^2 <  1, \\, \\gamma >0\\}$.\n\n\\vspace{0.25cm}\n\n\\bp{continuita}\nThe function $T:=T(x_f,y_f,\\gamma)$ is continuous at all points in $(P_f, \\gamma) \\in \\texttt{int} {\\cal C}$ such that $P_f$ is not on the critical trajectory corresponding to $\\gamma$.\n\\end{proposition}\n\nRecall from the results of \\cite{Noi}, \\cite{Raf} reviewed in the previous section that the critical trajectory is given by (\\ref{curvex}) (\\ref{curvey}) with $\\omega:=\\omega_c=1+\\gamma^2$, and $0 \\leq t \\leq \\frac{\\pi}{2 a_c}:=\\frac{\\pi}{2 \\gamma \\sqrt{1+\\gamma^2}}$. The locus of points of discontinuity is a surface in ${\\cal C}$ whose intersection with the planes $\\gamma=\\texttt{const}$ is the critical trajectory at that $\\gamma$, which is a spiral-like curve, which is very short for large $\\gamma$ and very long for small $\\gamma$.\n\n\\begin{proof}\n For a given  $P_f$ and $\\bar{\\gamma}$, with $P_f$ not in the critical trajectory of $\\bar{\\gamma}$,\nconsider the corresponding values  $t_{P_f}(\\bar{\\gamma})$ and $\\omega_{P_f}(\\bar{\\gamma})$ which are optimal. Then the two equations\n\\be{zero}\n\\begin{array}{llll}\nF_1(t,\\omega,x,y,\\gamma):=x(t,\\omega,\\gamma)-x=0,\\\\\nF_2(t,\\omega,x,y,\\gamma):=y(t,\\omega,\\gamma)-y=0,\n \\end{array}\n \\end{equation}\nhold at $t_{P_f}(\\bar{\\gamma}),\\omega_{P_f}(\\bar{\\gamma}), x_f,y_f,\\bar{\\gamma}$. Moreover,\nfrom the Implicit Mapping Theorem, they define in an open neighborhood $N$ of $(x_f,y_f,\\bar \\gamma)$ two continuous functions $t:=t(x,y,{\\gamma})$, $\\omega=\\omega(x,y,\\gamma)$, as long as the Jacobian with respect to the variables $t$ and $\\omega$ is different from zero. We calculate using (\\ref{curvex}) (\\ref{curvey}),\n\\[\n\\text{Det } \\left( \\begin{array}{cc}\n  \\frac{\\partial}{\\partial t}F_1 & \\frac{\\partial}{\\partial t}F_2 \\\\\n  & \\\\\n\n  \\frac{\\partial}{\\partial \\omega}F_1 & \\frac{\\partial}{\\partial \\omega}F_2\n  \\end{array} \\right) = \\frac{\\gamma^2(\\gamma^2+1-\\omega)}{a^4} \\sin( a t) \\left[\\sin( at)-at \\cos (at)\\right]\n  \\]\nThis expression is zero if and only if $t=0$, $t=\\pi\/a$ or $\\omega=1+\\gamma^2$.\nSince the optimal time $t_{P_f}({\\bar{\\gamma}})$ is always positive and strictly less then $\\frac{\\pi}{a}$,\\footnote{This follows from the results of \\cite{Noi} \\cite{Raf}. At $t=\\frac{\\pi}{a}$ every optimal candidate  trajectory is at the boundary of the unit circle where it looses optimality (if it had not lost it before by intersecting the critical trajectory).} and  the point $P_f$ does not belong to the critical trajectory, i.e., $\\omega_{P_f}(\\bar{\\gamma})\\neq {1+\\bar{\\gamma}^2}$, the Jacobian is not zero, so the two continuous functions are defined.\nWe know that\n $t(x_f,y_f,\\bar \\gamma)=t_{P_f}(\\bar \\gamma)$ and $\\omega(x_f,y_f,\\bar \\gamma)=\\omega_{P_f}(\\bar \\gamma)$. Let $V=\\left(t(N),\\omega(N)\\right)$, this is a neighborhood of $\\left(t_{P_f}(\\bar \\gamma), \\omega_{P_f}(\\bar \\gamma)\\right)$, moreover if $(t,\\omega,x,y,\\gamma)\\in V \\times N$, and satisfies the equations in (\\ref{zero}), then necessarily $t=t(x,y, \\gamma)$ and $\\omega=\\omega(x,y, \\gamma)$.\n\n  To show continuity  of the map $T$ at $(P_f,{\\bar{\\gamma}})$,i.e., of the time optimal function,\n  we prove that there exists a\n  neighborhood $W\\subseteq N$, such that   if\n $ (x,y,\\gamma)\\in W$ then $T(x,y,\\gamma)$ coincides with the implicit map $t(x,y,{\\gamma})$, whose continuity is guaranteed by the Implicit Mapping  Theorem.\n\n It is obvious by definition that $t(x,y,{\\gamma})\\geq T(x,y,\\gamma)$. Assume, by the way of contradiction, that\n the statement is false, then for all $n$ sufficiently large, there exists a sequence $\\{P_n=(x_n,y_n)\\}$ and a sequence $\\{\\gamma_n\\}$, such that\n the sequence $P_n$ goes to $P_f$,  the sequence $\\gamma_n$ goes to $\\bar{\\gamma}$, and $t(x_n,y_n,\\gamma_n)>T(x_n,y_n,\\gamma_n)$. On the other hand, we have:\n \\[\n x_n=x\\left(t(x_n,y_n,\\gamma_n), \\omega(x_n,y_n,\\gamma_n),\\gamma_n\\right)=\n x\\left(T(x_n,y_n,\\gamma_n), \\omega_{(x_n,y_n)}(\\gamma_n),\\gamma_n\\right),\n \\]\n \\[\n y_n=y\\left(t(x_n,y_n,\\gamma_n), \\omega(x_n,y_n,\\gamma_n),\\gamma_n\\right)=\n y\\left(T(x_n,y_n,\\gamma_n), \\omega_{(x_n,y_n)}(\\gamma_n),\\gamma_n\\right),\n \\]\nwhere $\\omega_{(x_n,y_n)}(\\gamma_n)$ are the optimal values. Since $T(x_n,y_n,\\gamma_n)$ and $\\omega_{(x_n,y_n)}(\\gamma_n)$ belong to compact sets (for the values $\\omega$ see Proposition \\ref{ovvia}), we may assume, after passing if necessary to a subsequence, that:\n\\[\n\\omega_{(x_n,y_n)}(\\gamma_n) \\to \\bar{\\omega},\n\\ \\ T(x_n,y_n,\\gamma_n) \\to \\bar{t}.\\]\nSince by continuity\n\\[\n x\\left(T(x_n,y_n,\\gamma_n), \\omega_{(x_n,y_n)}(\\gamma_n),\\gamma_n\\right) \\to x(\\bar{t},\\bar{\\omega},\\bar{\\gamma}), \\  \\   y\\left(T(x_n,y_n,\\gamma_n), \\omega_{(x_n,y_n)}(\\gamma_n),\\gamma_n\\right) \\to y(\\bar{t},\\bar{\\omega},\\bar{\\gamma}),\n \\]\n we must have\n\\[\nx_f=  x(\\bar{t},\\bar{\\omega},\\bar{\\gamma}), \\  \\  y_f=  y(\\bar{t},\\bar{\\omega},\\bar{\\gamma}).\n\\]\nFrom the fact that  $t(x_n,y_n,\\gamma_n)>T(x_n,y_n,\\gamma_n)$, we have  $t_{P_f}(\\bar \\gamma)=t(x_f,y_f,\\bar \\gamma)\\geq \\bar{t}$, since $t_{P_f}(\\bar \\gamma)$ is optimal we conclude\n$t_{P_f}(\\bar \\gamma)=t(x_f,y_f,\\bar \\gamma)= \\bar{t}$. This, in turn, implies that also\n$\\bar{\\omega}=\\omega_{P_f}(\\bar \\gamma)$, since the optimal values is unique.\\footnote{see \\cite{Noi} or alternatively the geometric analysis of next section which shows that the value $\\omega$ is the value of the parameter at the intersection of two parametric curves which is uniquely defined.}\nThus  for $n$ sufficiently large $\\left(T(x_n,y_n,\\gamma_n), \\omega_{(x_n,y_n)}(\\gamma_n)\\right)$  belongs to $V$, thus we must have $T(x_n,y_n,\\gamma_n)=t(x_n,y_n,\\gamma_n)$ since the\n Implicit Map Theorem guarantees  uniqueness of  the function $t$, which contradicts $t(x_n,y_n,\\gamma_n)>T(x_n,y_n,\\gamma_n)$.\n\n\n\\end{proof}\n\n\nWe now are interested in studying the discontinuity of the function $t_{P_f}=t_{P_f}(\\gamma)$ at the points on the critical trajectory. The following result summarizes the continuity properties of this function.\n\n\\bp{discontin}\nThe function  $t_{P_f}=t_{P_f}(\\gamma)$ is continuous for every $\\gamma$ except for the $\\gamma$'s such that $P_f$ is in the interior of  the critical trajectory. On these points it presents a discontinuity on the left and it is right continuous.\n\\end{proposition}\n\n\\begin{proof}\n\nFirst we first prove right continuity everywhere.\n\nBy monotonicity of the function $t_{P_f}$,  for a fixed value $\\bar{\\gamma}$, we know that:\n\\be{limitedestro}\n\\lim_{\\gamma\\to\\bar{\\gamma}^+} t_{P_f}({\\gamma})=\\sup_{\\gamma>\\bar{\\gamma}}t_{P_f}({\\gamma})=\nl_+({\\bar{\\gamma}})\\leq t_{P_f}({\\bar{\\gamma}}),\n\\end{equation}\nWe will prove  that $l_+({\\bar{\\gamma}})= t_{P_f}({\\bar{\\gamma}})$.\n\nBy definition of $l_+({\\bar{\\gamma}})$, we know that for each $n\\geq 1$ there exists $\\bar{\\gamma}<\\gamma_n<\\bar{\\gamma}+\\frac{1}{n}$, such that $l_+({\\bar{\\gamma}})-\\frac{1}{n}<t_{P_f}({{\\gamma_n}})<l_+({\\bar{\\gamma}})$.\nThen for each $n\\geq 1$ consider the corresponding optimal control value $\\omega_{P_f}({{\\gamma_n}})$.\nBy Proposition \\ref{ovvia}, these control values are bounded, thus they admit a converging subsequence $\\omega_{P_f}({{\\gamma_{n_k}}})$. Along this subsequence we have that\n$t_{P_f}({{\\gamma_{n_k}}})\\to l_+({\\bar{\\gamma}})$, $\\omega_{P_f}({{\\gamma_{n_k}}})\\to \\bar{\\omega}$, and $\\gamma_{n_k}\\to \\bar{\\gamma}$.  By\nusing equations (\\ref{curvex}) and (\\ref{curvey}) we have\n\\[\nx_f=x(\\bar{\\omega}, l_+^{\\bar{\\gamma}}) \\   \\  \\text{ and } \\  \\ y_f=y(\\bar{\\omega},  l_+^{\\bar{\\gamma}}).\n\\]\nThus $ l_+({\\bar{\\gamma}})=t_{P_f}({\\bar{\\gamma}})$, otherwise $t_{P_f}({\\bar{\\gamma}})$ would not be the optimal time, since we have found a control that reaches $P_f=(x_f,y_f)$ in time $l_+({\\bar{\\gamma}})$.\n\nContinuity of the function $t_{P_f}=t_{P_f}(\\gamma)$,  for every $\\gamma$ except for the $\\gamma$'s such that $P_f$ is on the critical trajectory, follows from Proposition \\ref{continuita}, where it is proved that the function is indeed continuous as a function of both variables $P_f$ and $\\gamma$.\n\nAssume now that $P_f$ is a point such that for $\\gamma=\\bar{\\gamma}$ it lies in the interior of the critical trajectory. Therefore $t_{P_f}(\\bar{\\gamma})=\\frac{ \\alpha \\pi}{2\\bar{\\gamma}\\sqrt{(1+\\bar{\\gamma}^2)}}$ with $0< \\alpha<1$. However, from the analysis in \\cite{Noi}, \\cite{Raf}, it follows that, for $\\epsilon > 0$ sufficiently small, $t_{P_f}(\\bar \\gamma -\\epsilon)> \\frac{\\pi}{2 (\\bar \\gamma - \\epsilon) \\sqrt{1+(\\bar \\gamma -\\epsilon)^2}}$. This gives\n \\be{discontinuita77}\n \\lim_{\\gamma \\to \\bar \\gamma^-} t_{P_f}(\\gamma)=\\lim_{\\epsilon \\to 0+}  t_{P_f}(\\bar \\gamma -\\epsilon) \\geq\n \\frac{\\pi}{2 \\bar \\gamma \\sqrt{1+ \\bar \\gamma^2}} >  \\alpha \\frac{\\pi}{2 \\bar \\gamma \\sqrt{1+ \\bar \\gamma^2}}. \\end{equation}\nThe analysis of \\cite{Noi} \\cite{Raf} also shows left continuity of $t_{P_f}$ if $P_f$ is exactly at the endpoint of the critical trajectory.\\footnote{For $\\gamma$ smaller than $\\bar \\gamma$ optimal trajectories reaching $P_f$ travel around the end point of the critical trajectory corresponding to that $\\gamma$ before reaching $P_f$. The time to reach $P_f$ is therefore greater than the maximum time on the critical trajectory. However as $\\gamma \\to \\bar \\gamma^-$ the two points and the two times coincide.}\n\n\n\\end{proof}\n\n\\vspace{0.25cm}\n\n\n\\br{Swapetal}\nIf $P_f$ corresponds to a SWAP like operator, then, since $P_f:=(0,0)$ is not on any critical trajectory,  $t_{P_f}$ is a continuous function of $\\gamma$. For other points $P_f$ however in the interior of the unit disc, there are infinitely many values of $\\gamma$ such that $P_f$ is on the corresponding critical trajectory.\\footnote{They have a limit point at zero} At each of these values,  the function $t_{P_f}$ has a jump on the left. If we are trying to reach a point at exactly a time $T$ which is possibly  larger than the minimum time corresponding to a bound $\\bar \\gamma$, we cannot always  use the time optimal control for a $\\gamma$ smaller than $\\bar \\gamma$. The time $T$ might not be in the range of the function $t_{P_f}$ even though such a function tends to $+\\infty$ as $\\gamma \\rightarrow 0$ as shown in proposition \\ref{seconda}. A characterization of the reachable sets for every $T$ will allow us to know exactly at what times we can reach a given state and how.\n\\end{remark}\n\n\\section{Geometry of the reachable sets} \\label{Reachsets}\n\nWe now give a description of the reachable sets for system (\\ref{basicmodel}), which will then be used in the solution of the minimum time  synchronization problem in the next section \\ref{synchro}.  As the method has more general validity, we shall first describe it for general, bilinear,  right invariant systems on Lie groups and then specialize to system (\\ref{basicmodel}).\\footnote{Some of the concepts and ideas we shall describe are valid for more general families of vector fields. Restricting ourselves to bilinear right invariant vector fields ensure us that the solution of the associated initial value problems exists for every time $t$. For concreteness, our notation refers to {\\it matrix} Lie groups, although we could have extended the discussion to abstract general Lie group simply replacing the exponential of a matrix with the exponential map.}\n\n\n\n\n\\subsection{General method}\n\n\nConsider a system\n\\be{gensys2}\n\\dot X=AX+\\sum_{j=1}^m u_jB_j X, \\qquad X(0)=X_0,\n\\end{equation}\nwhere $A,B_1,\\ldots,B_m$ are matrices in a matrix Lie algebra ${\\cal L}$, $X$ belongs to the corresponding Lie group $e^{\\cal L}$, $X_0$ is the given initial condition and $u_j$, $j=1,\\ldots,m$  are the controls, which are assumed to belong to a set ${\\cal U}$ of functions of time. The {\\bf reachable set} at time $T$, ${\\cal R}(T)$ is the set of states $X_f$ in $e^{\\cal L}$ such that there exist functions $u_1,\\ldots,u_m$ in ${\\cal U}$ defined in $[0,T]$ so that the solution $X$ of  (\\ref{gensys2}) with these controls satisfies  $X(T)=X_f$. ${\\cal R}(0)=\\{ X_0\\}$  by definition and the reachable set ${\\cal R}(\\leq T)$ is defined as\n\\be{reaches}\n{\\cal R}(\\leq T):=\\bigcup_{0 \\leq t \\leq T} {\\cal R}(t).\n\\end{equation}\nIt follows obviously from the definition that the reachable sets ${\\cal R}(\\leq T)$ are non decreasing with $T$, i.e., ${\\cal R}(\\leq T_1) \\subseteq {\\cal R}(\\leq T_2)$ if $T_1 \\leq T_2$, a property not necessarily true for the reachable sets ${\\cal R}(T)$.\n\\vspace{0.25cm}\nTo study reachable sets we can make a, possibly time varying, change of variables  in system (\\ref{gensys2}) and study the reachable sets for the resulting system. The reachable sets for the original system are obtained by mapping back the reachable sets for the new system. In the case of system (\\ref{gensys2}) we define\\footnote{This is called `{\\it passage to the interaction picture}' in the physics literature.}\n\\be{intepic}\nU(t):=e^{-At}X(t).\n\\end{equation} From (\\ref{gensys2}), we obtain the differential equation for $U$,\n\\be{gensys3}\n\\dot U=\\left(\\sum_{j=1}^m u_j e^{-At} B_j e^{At} \\right) U, \\qquad U(0)=X_0.\n\\end{equation}\nLet ${\\cal I}$ the smallest subspace of ${\\cal L}$ which contains $\\{B_1,\\ldots,B_m\\}$ and is invariant under Lie bracket with $A$, and let $\\{ C_1,C_2,\\ldots, C_l \\}$ be a basis of ${\\cal I}$. Then there are analytic functions $\\gamma_{j,i}:=\\gamma_{j,i}(t)$ such that\n\\be{llo2}\ne^{-At} B_j e^{At}:=\\sum_{i=1}^l \\gamma_{j,i}(t)C_i.\n\\end{equation}\nBy replacing this in (\\ref{gensys3}) and defining\n\\be{Vi}\nv_i:=\\sum_{j=1}^m u_j \\gamma_{j,i},\n\\end{equation}\nwe obtain\n\\be{L3e}\n\\dot U=\\left( \\sum_{i=1}^l v_i C_i \\right) U, \\qquad U(0)=X_0.\n\\end{equation}\nIf ${\\cal V}$ denotes the image of the set of control functions, ${\\cal U}$, under the map\n(\\ref{Vi}), then we can study the reachable sets for (\\ref{L3e}) under the set of controls ${\\cal V}$, denote them by ${\\cal R}_U( T)$ and ${\\cal R}_U(\\leq T)$, respectively, and the reachable\nsets for the original system (\\ref{gensys2}), are recovered from (cf. (\\ref{intepic}))\n\\be{recovering}\n{\\cal R}(T)=e^{AT} {\\cal R}_U(T),\\qquad {\\cal R}(\\leq T)=\\bigcup_{0 \\leq t \\leq T} e^{At} {\\cal R}_U(t).\n\\end{equation}\nThis method is particularly useful when the set of controls ${\\cal V}$ has the {\\it scalability property}, that is, $\\vec v \\in {\\cal V}$ implies $L_\\alpha({\\vec v}):= \\alpha \\vec v (\\alpha t) \\in {\\cal V}$ for every $\\alpha \\in [0,1]$. In this case, the reachable sets ${\\cal R}_U(T)$ are increasing with $T$, so that ${\\cal R}_U(T)={\\cal R}_U(\\leq T)$ for every $T$. This is seen with a standard argument for driftless systems such as (\\ref{L3e}). Let $\\vec v$ a control in ${\\cal V}$ steering the initial condition $X_0$ to $U_f$ in time $T_1$, so that $U_f \\in {\\cal R}_U(T_1)$. Let $T_2 > T_1$ and consider the control $L_{\\frac{T_1}{T_2}} \\vec v(t)=\\frac{T_1}{T_2} \\vec v(\\frac{T_1}{T_2}t)$. With this control the function $\\tilde U(t):=U(\\frac{T_1}{T_2}t)$ solves (\\ref{L3e}) and it is such that $\\tilde U(T_2)=U_f$, so that $U_f \\in {\\cal R}_U(T_2)$.\n\n\nIn general, studying ${\\cal R}(\\leq T)$ is easier than studying ${\\cal R}(T)$ because this set is related to the solution of the time optimal control problem. It is a well known fact in geometric control theory that if $X_f$ is a final point of a time optimal trajectory at time $T$, then $X_f$ is on the boundary of the reachable ${\\cal R}(\\leq T)$. On the other hand, by definition, the minimum time to reach $X_f$, is the smallest time $T$ such that $X_f \\in {\\cal R}(T)$. In the above described situation, the minimum time $T$ is the smallest time $t$ such that\n\\be{L4r}\nX_f \\in {\\cal R}(t)=e^{At} {\\cal R}_U(t),\n\\end{equation}\nand, if we have a description of ${\\cal R}_U(t)(={\\cal R}_U(\\leq t))$ this gives an alternative  way to find the minimum time and control.\n\n\\subsection{Reachable sets for systems on $SU(2)$}\n\nWe now apply the strategy outlined above to the case of system (\\ref{basicmodel}). In this case,  the space of controls, ${\\cal U}$ is the space of Lebesgue measurable functions $\\vec u:=(u_x,u_y)$ with Euclidean norm $\\| \\vec u \\|:=\\sqrt{u^2_x+  u^2_y} \\leq \\gamma$. Specializing (\\ref{llo2}), we obtain\n\\be{L1X}\ne^{-\\tilde \\sigma_z \\tau} \\tilde \\sigma_x e^{\\tilde \\sigma_z \\tau}=\\cos(\\tau) \\tilde \\sigma_x -\\sin(\\tau)\\tilde \\sigma_y,\n\\end{equation}\nand\n\\be{L1Y}\ne^{-\\tilde \\sigma_z \\tau} \\tilde \\sigma_y e^{\\tilde \\sigma_z \\tau}=\\sin(\\tau) \\tilde \\sigma_x+ \\cos(\\tau) \\tilde \\sigma_y.\n\\end{equation}\nTherefore, the equation corresponding to (\\ref{L3e}) is\n\\be{L5bis}\n\\dot U=(v_x \\tilde \\sigma_x + v_y \\sigma_y )U, \\qquad U(0)= {\\bf 1}\n\\end{equation}\nwith $v_x:=\\cos(\\tau) u_x+ \\sin(\\tau) u_y$ and $v_y=-\\sin(\\tau) u_x + \\cos(\\tau) u_y$. Therefore, if $\\vec u:=[u_x,u_y]^T$ and $\\vec v:=[v_x,v_y]^T$, we have\n\\be{relat32}\n\\vec v=\\begin{pmatrix}  \\cos(\\tau) & \\sin(\\tau) \\cr -\\sin(\\tau) & \\cos(\\tau) \\end{pmatrix} \\vec u.\n\\end{equation}\n Formula (\\ref{relat32}) gives a one to one correspondence between Lebesgue measurable functions $\\vec u$ with norm bounded by $\\gamma$ and  Lebesgue measurable functions $\\vec v$ with norm bounded by $\\gamma$. That is, in this case, the space of controls ${\\cal V}$ coincides with ${\\cal U}$. Moreover, ${\\cal V}$ has the scalability property, and therefore ${\\cal R}_U(\\leq T)={\\cal R}_U(T)$ for every $T\\geq 0$.\n\n\n\n The following theorem describes ${\\cal R}_U(T)$ for system (\\ref{L5bis}), from which, in the following corollary, we obtain the reachable sets for the original system (\\ref{basicmodel}). As we have already done in the previous section, we scale the time variable as $t:=\\frac{\\tau}{2}$ and we we refer to `time' (as for example for $T$ in ${\\cal R}_U(T)$) we shall refer to the time $t$ defined this way.\n\n\\bt{Reachset1} The reachable set ${\\cal R}_U(T)$ for the system (\\ref{L5bis}) is given by the set of matrices in $SU(2)$ with the $x_{1,1}:=x+iy$ entry in the region of the unit disc bounded by the parametric curve ${\\cal F}_T$ defined as.\\footnote{These are the optimal frontlines studies in \\cite{Raf}.}\n\\be{CurveX9}\nx=x(\\omega)=\\cos(\\omega T) \\cos(aT)+\\frac{\\omega}{a} \\sin(\\omega T) \\sin(aT),\n\\end{equation}\n\\be{CurveY9}\ny=y(\\omega)=\\sin(\\omega T) \\cos(aT)-\\frac{\\omega}{a} \\cos(\\omega T) \\sin(aT),\n\\end{equation}\nwhere $a:=\\sqrt{\\omega^2+\\gamma^2}$ and the parameter $\\omega \\in \\left[ -\\sqrt{\\frac{\\pi^2}{T^2}-\\gamma^2}, \\sqrt{\\frac{\\pi^2}{T^2}-\\gamma^2} \\right]$.\n\\end{theorem}\nThe region of the unit disc representing ${\\cal R}_U(T)$ is at the right of the curve ${\\cal F}_T$ which, for every $T$, connects two points on the boundary of the unit disc and is symmetric with respect to the $x$-axis. The region grows with $T$, and when $T=\\frac{\\pi}{\\gamma}$, the curve ${\\cal F}_T$ collapses to the point $(-1,0)$ and the reachable set ${\\cal R}_U(T)$ becomes all of $SU(2)$. Figure \\ref{Fig2} shows the typical behavior for the curves ${\\cal F}_T$ for various values of $T$.\n\n\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.7\\textwidth, height=0.45\\textheight]{OS2}\n\\caption{Boundaries of reachable sets ${\\cal R}_U(\\leq T)={\\cal R}_U(T)$ for various values of $T$. The parameter $\\gamma$ is chosen equal to $1$. In red, we have the boundary (curves (\\ref{CurveX9}), (\\ref{CurveY9})) for $T=1$, in green for $T=1.1$, in blue for $T=2$, in purple for $T=3$. For $T=\\frac{\\pi}{\\gamma}=\\pi$ the boundary of the reachable set collapses to the single point $(-1,0)$ and the reachable set becomes the whole unit disc and therefore the whole $SU(2)$. }\n\\label{Fig2}\n\\end{figure}\n\nSpecializing (\\ref{recovering}) we obtain\n\\bc{special}\n\\be{cf}\n{\\cal R}(T)=e^{2 \\tilde \\sigma_z T} {\\cal R}_U(T).\n\\end{equation}\nIn other terms, $X_f \\in {\\cal R}(T)$ if and only if, denoted by $P_f$ the $(1,1)$ entry of $X_f$, we have that    $e^{-iT} P_f$ is in the region described in Theorem \\ref{Reachset1}.\n\\end{corollary}\n\\br{explanationdisc}\nThe above corollary gives an alternative geometric explanation of the discontinuity we described in the previous section (cf. Remark \\ref{Swapetal}). Consider the desired point $P_f$, in the unit disc. As $T$ increases $e^{-iT} P_f$ describes  a circle inside the unit disc. At the same time, as $T$ increases the curve ${\\cal F}_T$ moves towards left with a speed which decreases with decreasing $\\gamma$. The optimal time is the minimum time where these two curves intersect for the first time. If this intersection happens at a point of tangency, a small decrease in $\\gamma$ implies that $e^{-iT} P_f$ will have to go around almost an entire circle again before intersecting ${\\cal F}_T$, which explains the discontinuity at that $\\gamma$. Notice that this problem does not occur at the origin (corresponding to SWAP-like operators) since in this case the circle $e^{-iTP_f}$ reduces to a single point.\n\\end{remark}\n\n\n\nTo prove Theorem \\ref{Reachset1} we shall need the following facts about the parametric curves ${\\cal F}_T$ (with parameter $\\omega$ and fixed $T$), and its extension to  values (in (\\ref{CurveX9}), (\\ref{CurveY9})) of ${\\omega} \\in (-\\infty, -\\sqrt{\\frac{\\pi^2}{T^2}-\\gamma^2}) \\bigcup (\\sqrt{\\frac{\\pi^2}{T^2}-\\gamma^2},  +\\infty)$,  which we denote by ${\\cal S}_T$.  It can be inferred by plotting  the curves for different values of $T$, but we present an analytic proof in the appendix.\n\n\n\\newpage\n\n\\bl{SIO}\nLet ${\\cal F}_t$  and ${\\cal S}_t$ the previous  defined curves, then:\n\\begin{enumerate}\n\\item If $t_1\\neq t_2$ then  ${\\cal F}_{t_1}\\cap {\\cal F}_{t_2}=\\emptyset$\n\\item The curve ${\\cal F}_t$ does not have self intersections.\n\\item For any $0<t< \\frac{\\pi}{\\gamma}$, we have ${\\cal F}_{t} \\cap {\\cal S}_t=\\emptyset$.\n\\end{enumerate}\n\\end{lemma}\n\n\n\\begin{proof} (Proof of Theorem \\ref{Reachset1}) Consider the problem of time optimal control subject to $v_x^2+v_y^2 \\leq \\gamma^2$. Analogously to the case of the system with drift considered in \\cite{Noi} \\cite{Raf} and summarized in section \\ref{Rev}, the candidate optimal controls have the form (cf. (\\ref{NSE}))\\footnote{No singular extremals of the form $v_x=v_y\\equiv0$ are admissible in this case since they would mean zero dynamics and therefore they are incompatible with the minimum time requirement.}\n\\be{NSE1}\nv_x=\\gamma \\sin(\\omega \\tau + \\tilde \\phi), \\qquad v_y=-\\gamma \\cos(\\omega \\tau +\\tilde \\phi).\n\\end{equation}\nAs in the case of system with drift of section \\ref{Rev}, with this control, the differential equation (\\ref{L5bis}) can be explicitly integrated. The result is (cf. (\\ref{soluzexpli}))\n\\be{soluzexpli3}\nU(\\tau,\\omega,\\tilde \\phi):=\\begin{pmatrix} e^{i \\omega t}(\\cos(a t)- i \\frac{\\omega}{a} \\sin(a t)) & e^{i (\\omega t + \\tilde \\phi)} \\frac{\\gamma}{a} \\sin(a t) \\cr  - e^{-i(\\omega t + \\tilde \\phi)} \\frac{\\gamma}{a} \\sin(a t)  & e^{-i \\omega t}(\\cos(a t)+i \\frac{\\omega}{a} \\sin(a t)) \\end{pmatrix},\n\\end{equation}\nwith $a:=\\sqrt{\\omega^2+\\gamma^2}$.\nFrom this expression, it follows that, if a final condition is reached optimally in time $T$, the same is true (by simply changing the phase $\\tilde \\phi$) for any state which differs from it only by the phase of the off diagonal element. Therefore, we only have to consider the (1,1) entry of the desired final condition. This is a point inside the unit disc with coordinates (cf. (\\ref{curvex}), (\\ref{curvey}) and (\\ref{CurveX9}) (\\ref{CurveY9}))\n\\be{coordi1}\nx=x_{\\omega}(t)=\\cos(\\omega t) \\cos(at)+\\frac{\\omega}{a} \\sin(\\omega t) \\sin(at),\n\\end{equation}\n\\be{coordi2}\ny=y_{\\omega}(t)=\\sin(\\omega t) \\cos(at)-\\frac{\\omega}{a} \\cos(\\omega t) \\sin(at).\n\\end{equation}\n\n\\vspace{0.25cm}\n\n\nConsider now the curve (\\ref{coordi1}) (\\ref{coordi2}) corresponding to $t$, and with $\\omega \\in \\left[ - \\sqrt{\\frac{\\pi^2}{t^2}-\\gamma^2}, \\sqrt{\\frac{\\pi^2}{t^2}-\\gamma^2} \\right]$, which connects two symmetric points on the unit circle, i.e. ${\\cal F}_t$.\n\n\n\n\nFrom Lemma \\ref{SIO}, we know that ${\\cal F}_{t_1}$ and ${\\cal F}_{t_2}$ do not intersect for $t_1 < t_2$. Moreover by comparing the $x$ values of the endpoints of ${\\cal F}_{t_1}$ and ${\\cal F}_{t_2}$ for $t_1$ and $t_2$, which are $x_1=-\\cos(\\sqrt{\\pi^2 -\\gamma^2 t_1^2})$ and $x_2=-\\cos(\\sqrt{\\pi^2 -\\gamma^2 t_2^2})$, it follows that $x_2 < x_1$. So, if $t_1 < t_2$ ${\\cal F}_{t_1}$ lies entirely to the right of ${\\cal F}_{t_2}$.   Now we want to show that ${\\cal S}_{t_1}$, also is entirely to the right of ${\\cal F}_{t_2}$. In fact, since $\\lim_{\\omega \\to \\pm \\infty} x_\\omega(t)=1$ and $\\lim_{\\omega \\to \\pm \\infty} y_\\omega(t)=0$,  ${\\cal S}_{t_1}$ eventually belongs to the region on the right of  ${\\cal F}_{t_1}$. If ${\\cal S}_{t_1}$ where to intersect ${\\cal F}_{t_2}$, then would have to `go back' to the region on the right of ${\\cal F}_{t_1}$ and therefore intersect ${\\cal F}_{t_1}$, something which is excluded by Lemma \\ref{SIO}.\n\nIn view of this fact, for every $(x_0,y_0) \\in {\\cal F}_T$,  $T$ is the smallest time such that $(x_0, y_0)$ can be reached with an extremal control, and therefore it is the minimum time to reach $(x_0,y_0)$. As a consequence, $(x_0,y_0)$ is on the boundary of the reachable set ${\\cal R}_U(\\leq T)={\\cal R}_U( T)$.\\footnote{or, more properly, the projection of the reachable set onto the unit disc where the actual reachable set in $SU(2)$ is the set of all the matrices whose (1,1) entry is in the region bounded by the curve ${\\cal F}_T$.}\n\n\\end{proof}\n\n\nUsing Theorem \\ref{Reachset1} we obtain an alternative (yet related) method to the ones in \\cite{Noi}, \\cite{Raf} to find the minimum time and the optimal control to reach a final condition $X_f$ for system (\\ref{basicmodel}) represented by a point $P_f$ in the unit disc. The minimum time is the minimum $t$ such that\n\\be{formmintim}\ne^{-it} P_f \\in {{\\cal  R}_U(t)}.\n\\end{equation}\n\n\\vspace{0.25cm}\n\nThe optimal control problem with the method from formula (\\ref{formmintim}) can be solved graphically or numerically by finding for increasing values of $T$ the distance from the point $e^{-iT}P_f$ and the curve ${\\cal F}_t$ until such a distance becomes zero. The first time $T$ where this distance is zero is the minimum time. The value of $\\omega$ of the intersection gives the frequency to be used in the controls, $v_x$ and $v_y$. Inverting the transformation (\\ref{relat32}),  we obtain the control for the original system (\\ref{basicmodel}), which is of the form (\\ref{NSE}).\\footnote{Notice that the frequency $\\omega$ for the driftless system and the frequency $\\omega$ for the original system with drift are not the same. We should have used two different symbols for $\\omega$ (as well as for $a$) but we have not done it to keep the notations simple.} Finally the phase is $\\tilde{\\phi}$ is found in (\\ref{NSE}) to match the desired final condition in $SU(2)$ according to (\\ref{soluzexpli}).\n\n\n\n\n\\section{Solution of the minimum time synchronization problem for $N$ qubits}\\label{synchro}\n\nWith the above description of the reachable set for system (\\ref{basicmodel}) we can now solve the problem to drive simultaneously to the desired final conditions, $N$ qubits in minimum time. Let us assume the bounds on the norm of the controls $\\sqrt{u_x^2+u_y^2}$ are $\\gamma_1,\\ldots,\\gamma_N$ for system $1,\\ldots,N$,\\footnote{This takes into account possible differences in the Larmor frequencies of spin $\\frac{1}{2}$ particles since the equations are normalized.}. Let $X_{f,1},\\ldots,X_{f,N}$ be the desired  final conditions for system $1,\\ldots,N$, respectively. We also denote by ${\\cal R}_j(t)$ the reachable set at time $t$  for system $j$.\n\n\nThe first step to find the time optimal synchronous control is to solve independently the time optimal control problems for the systems $1$ through $N$, and to find therefore the minimum times $T_1$,...,$T_N$. This can be done using the methods of \\cite{Noi}, \\cite{Raf}, or the method based on the evolution of the reachable set in the previous section. Let us consider now the maximum among $T_1,...,T_N$; Assume it is $T_1$. Then for $j=2,\\ldots,N$ we check whether $X_{f,j} \\in {\\cal R}_j(T_1)$. If that is the case, then the minimum time is $T_1$ and we can simply drive system $1$, time optimally and (possibly) slow down systems $2$ through $N$ in order to obtain the desired final conditions $X_{f,2}$,...,$X_{f,N}$ in time $T_1$. To slow down a system, we can apply the optimal control but with a smaller value of $\\gamma$. Graphically $e^{-i T} P_f$ is in the region bounded by ${\\cal F}_T$. If we consider a smaller $\\gamma$ the boundary of this region is moved towards the right. We can do that, until the boundary includes the point $e^{-iT}P_f$ and this gives us the correct $\\gamma$ to use.\n\nAssume instead that there exists a $\\bar j \\in \\{2,\\ldots,N\\}$ such that $X_{f,\\bar j}\\notin {\\cal R}_{\\bar j}(T_1)$. Then the minimum time for synchronous control is greater than $T_1$ and greater than or equal to the smallest $T> T_1$ such that $X_{f,\\bar j} \\in {\\cal R}_{\\bar j}(T)$. Find such a $T$ and then check that for all other $j=1,\\ldots,N$, $j \\not= \\bar j$, $X_{f,j} \\in {\\cal R}_j(T)$. If that is the case, the algorithm stops otherwise it continues going back and replacing $T_1$ with $T$. The algorithms ends because for $T \\geq \\frac{\\pi}{\\gamma_{min}}$, where $\\gamma_{min}:=\\min\\{\\gamma_1,\\ldots,\\gamma_N\\}$, ${\\cal R}_j(T)=SU(2)$, for every $j$.\n\n\\vspace{0.25cm}\n\nWe summarize the procedure in the following formal algorithm.\n\n\\vspace{0.25cm}\n\n\n{\\scshape\n\n\\vspace{0.25cm}\n\n{\\bf ALGORITHM}\n\n\\vspace{0.25cm}\n\\begin{enumerate}\n\n\\item Solve the Time Optimal Control Problem for systems 1 through N and find the minimum\ntimes $T_1,\\ldots,T_N$.\n\nSet $T_{curr}:=\\max \\{T_1,\\ldots,T_N\\}$.\n\nSet $k_{curr}$ a value of $j$ such that $T_j=T_{curr}$.\n\n\\item  For $j \\not=k_{curr}$ check that\n\n  $X_{f,j}  \\in {\\cal R}_j(T_{curr})$\n\n  If this is the case then STOP.\n\n\\item Choose a $\\bar j \\in \\{1,\\ldots,N\\}$  such that\n\n  $X_{f,\\bar j}  \\notin {\\cal R}_{\\bar j}(T_{curr})$\n\n\\item Find the smallest $T > T_{curr}$ such that\n\n  $X_{f,\\bar j}  \\in {\\cal R}_{\\bar j}(T)$\n\n\\item Set $T_{curr}=T$, $k_{curr}=\\bar j$\n\n\\item  Go back to step 2.\n\n\\end{enumerate}\n}\n\n\n\n\n\n\n\\section*{Acknowledgement} Domenico D'Alessandro's  research was  supported by ARO MURI grant W911NF-11-1-0268. He also would like to thank Dr. R. Romano for useful discussions. The authors\n would like to thank Prof. B. Jakubczyk for helpful comments which went as far as providing meaningful examples for  some of the properties we wanted to prove in this paper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nIn this paper, we introduce and solve a new exactly solvable nonlinear partial integro-differential equation which is a natural spin generalization of the Benjamin-Ono (BO) equation \\cite{benjamin1967,ono1975} and which we therefore call the {\\em spin BO (sBO) equation}. We present arguments that the sBO equation is not only interesting from a mathematical point of view but also for physics. We also introduce, discuss, and present results about a closely related spin generalization of the non-chiral intermediate long-wave (ncILW) equation recently introduced and studied by us in \\cite{berntson2020,berntsonlangmann2020,berntsonlangmann2021}.\n\nThe sBO equation describes the time evolution of a square matrix-valued function $\\mathsf{U}=\\mathsf{U}(x,t)$ depending on position and time variables $x\\in{\\mathbb R}$ and $t\\in{\\mathbb R}$, respectively, and it is given by \n\\begin{equation} \n\\label{eq:sBO} \n\\boxed{ \\mathsf{U}_t + \\{\\mathsf{U},\\mathsf{U}_x\\} +  H\\mathsf{U}_{xx}  +\\mathrm{i} [\\mathsf{U},H\\mathsf{U}_x]=0 }\n\\end{equation} \nwhere $\\mathsf{U}_t$ is short for $\\frac{\\partial}{\\partial t}\\mathsf{U}$ etc., $H$ is the usual Hilbert transform (which, for functions $f$ of $x\\in{\\mathbb R}$ is given by \n\\begin{equation}\n\\label{eq:H}  \n(Hf)(x)\\coloneqq \\frac1{\\pi}\\Xint-_{{\\mathbb R}} \\frac1{x'-x} f(x')\\,\\mathrm{d}{x'}\n\\end{equation} \nwhere $\\Xint-$ is the principal value integral), $\\mathrm{i}\\coloneqq\\sqrt{-1}$, and $[\\cdot,\\cdot]$ and $\\{\\cdot,\\cdot\\}$ denote the commutator and anti-commutator of square matrices, respectively (the matrix dimension $d\\times d$ is arbitrary, with $d=1$ corresponding to the standard BO equation). \n The sBO equation is interesting for several reasons:  (i) It is exactly solvable, and it includes well-known soliton equations as limiting cases; in particular, the sBO equation \\eqref{eq:sBO} not only generalizes the BO equation but also the half-wave maps (HWM) equation \\cite{zhou2015,lenzmann2018,lenzmann2018b}. \n(ii) It describes interesting physics beyond what has previously been described by exactly solvable equations; in particular, the sBO equation provides a hydrodynamic description of interacting particles with internal spin degrees of freedom, and it determines the evolution of the nonlinearly coupled charge- and spin-densities of this particle system. As will be explained in more detail, integrable systems of this kind are relevant for the quantum Hall effect in situations where the electron spin cannot be ignored, a topic which has received considerable interest in the physics literature in recent years;  see e.g.\\ \\cite{senthil1999,mccann2006,bernevig2006}. \n(iii) The sBO equation has multi-soliton solutions constructed via a simple spin-pole ansatz where the time evolution of the spins and poles is determined by the spin generalization  of the $A$-type Calogero-Moser (CM) system due to Gibbons and Hermsen \\cite{gibbons1984}, which we call the spin CM (sCM) system; see also \\cite{wojciechowski1987}. (iv) It provides another example of a correspondence between Calogero-Moser-Sutherland type systems and soliton equations; in particular, it is known that the BO equation and the ncILW equation are naturally related to $A$-type Calogero-Moser-Sutherland systems in several different ways \\cite{berntson2020}, and while \\eqref{eq:sBO} is a spin generalization of the BO equation, we also present a spin generalization of the ncILW equation and thus propose a correspondence between soliton equations and sCM systems in all four cases: rational (I), trigonometric (II), hyperbolic (III), elliptic (IV) (see, e.g., \\cite{olshanetsky1981} for basic facts about CM systems).\nWe substantiate this proposal in cases I--III; case IV is technically more demanding and thus left for future work. \n(v) Our results suggest that this equation is exactly solvable in the same strong sense as the BO and HWM equations, and this opens up possibilities for several future research projects. \n\nIn the following, we explain the above points (i)--(v) in more detail; we emphasize that these points are complementary and, depending on the interests of the reader, some of them can be ignored without loss of continuity. \n\n\\paragraph{(i) Special cases.} \nThe BO equation is an exactly solvable nonlinear partial integro-differential equation that describes one-dimensional internal waves in deep water. In particular, physically relevant $N$-soliton solutions of the BO equation can be obtained by a simple pole ansatz where the dynamics of the poles is governed by an $A_{N-1}$ CM system \\cite{chen1979}. \nThus, the BO equation on the real line is related to the rational CM system, while the BO equation with periodic boundary conditions is related to the trigonometric CM system.\n\nThe HWM equation is a nonlinear partial integro-differential equation that describes a spin density, represented by an $S^2$-valued function, propagating in one dimension. As two of us found recently in collaboration with Klabbers  \\cite{berntsonklabbers2020}, the HWM equation is similar to the BO equation in that it has $N$-soliton solutions obtained via a spin-pole ansatz where the dynamics of the poles and of the spin degrees of freedom are determined by the $A_{N-1}$ sCM system and where, again, the real-line and periodic problems for the HWM equation correspond to the rational and trigonometric cases of this sCM system, respectively; see also \\cite{ matsuno2022}. \n It is known that the BO equation is related to a hydrodynamic description of the $A_{N-1}$ CM system \\cite{abanov2009}, and that the HWM equation can be derived as a continuum limit of a spin chain which corresponds to a limiting case of the $A_{N-1}$ sCM system \\cite{zhou2015}. This suggested to us that there should exist a more general soliton equation which includes the BO and HMW equations as limiting cases. As shown in the next two paragraphs, the sBO equation \\eqref{eq:sBO} is such an equation. We found the sBO equation by generalizing the logic in \\cite{abanov2009} to the sCM system, making use of a known B\\\"acklund transformation of $A$-type sCM systems \\cite{gibbons1983}.  \n\nThe BO equation is given by \n\\begin{equation} \n\\label{eq:BO} \nu_t+2uu_x+Hu_{xx}=0 \n\\end{equation} \nfor an ${\\mathbb R}$-valued function $u=u(x,t)$. We make the ansatz $\\mathsf{U}(x,t)=u(x,t)\\mathsf{P}$ with $\\mathsf{P}$ a non-zero hermitian matrix satisfying $\\mathsf{P}^2=\\mathsf{P}$; then the sBO equation \\eqref{eq:sBO} is satisfied if and only if $u(x,t)$ satisfies the BO equation \\eqref{eq:BO}.  It is also interesting to note that by restricting $\\mathsf{U}$ to diagonal matrices, the sBO equation is reduced to $d$ decoupled BO equations: $\\mathsf{U}=\\mathrm{diag}(u_1,\\ldots,u_d)$ satisfies \\eqref{eq:sBO} if and only if $u_\\mu=u_\\mu(x,t)$ solves the BO equation \\eqref{eq:BO} for $\\mu=1,\\ldots,d$. \n\nThe HWM equation describes the time evolution of an ${\\mathbb R}^3$-valued function $\\mathbf{m}=(m^1,m^2,m^3)=\\mathbf{m}(x,t)$ satisfying the constraint $\\mathbf{m}^2\\coloneqq (m^1)^2+(m^2)^2+(m^3)^2=1$ as follows, \n\\begin{equation}\n\\label{eq:HWM}  \n\\mathbf{m}_t = \\mathbf{m}\\wedge H \\mathbf{m}_x \n\\end{equation} \nwhere $\\mathbf{m}\\wedge\\mathbf{n} \\coloneqq (m^2n^3-m^3n^2,m^3n^1-m^1n^3,m^1n^2-m^2n^1)$ is the usual wedge product of three-vectors. By scaling $\\mathsf{U}(x,t)\\to \\lambda\\mathsf{U}(x,2\\lambda t)$ and changing variables $2\\lambda t\\to t$ with $\\lambda>0$ a scaling parameter, the sBO equation \\eqref{eq:sBO} becomes \n\\begin{equation} \n\\label{eq:sBO1} \n\\mathsf{U}_t + \\frac12 \\{\\mathsf{U},\\mathsf{U}_x\\} + \\frac1{2\\lambda}H\\mathsf{U}_{xx}+\\frac{\\mathrm{i}}{2}[\\mathsf{U},H\\mathsf{U}_x] =0 . \n\\end{equation}   \nThis reduces to a generalization of the HWM equation in the limit $\\lambda\\to\\infty$ if we  impose the constraint $\\mathsf{U}^2=I$, where $I$ denotes the identity matrix.  \nIndeed, $\\mathsf{U}^2=I$ implies $ \\{\\mathsf{U},\\mathsf{U}_x\\}=0$, and by specializing $\\mathsf{U}$ to $2\\times 2$ traceless hermitian matrices using the parametrization \n\\begin{equation} \n\\mathsf{U} = \\mathbf{m}\\cdot\\boldsymbol{\\sigma} = \\begin{pmatrix} m^3 & m^1-\\mathrm{i} m^2\\\\ m^1+\\mathrm{i} m^2 & -m^3 \\end{pmatrix} \n\\end{equation} \nwith $\\boldsymbol{\\sigma}=(\\sigma_1,\\sigma_2,\\sigma_3)$ the Pauli matrices,  \\eqref{eq:HWM} is obtained from \\eqref{eq:sBO1} in the limit $\\lambda\\to\\infty$, while $\\mathsf{U}^2=I$ is equivalent to $\\mathbf{m}^2=1$. We note that, while this reduction of the sBO equation to the HWM equation is mathematically simple, there is another similar but more complicated reduction explained in (ii) below which is more interesting from a physics point of view. \n\n\\paragraph{(ii) Physics applications.} While hydrodynamics was initially developed to describe the propagation of fluids, recent developments in physics have established that hydrodynamic equations can provide a powerful tool to compute transport properties of strongly correlated electron systems like the cuprates or graphene; see e.g.\\ \\cite{hartnoll2007,andreev2011,svintsov2013}. Moreover, there exists a variety of topological such systems where the physical behavior is independent of model details and, in such a situation,  one can expect a successful description by integrable hydrodynamic equations (well-known arguments to explain this relation between universality and integrability in the context of soliton equations were given by Calogero \\cite{calogero1991}); as an example, we mention the use of the BO equation to describe nonlinear waves at the boundary of fractional quantum Hall effect systems \\cite{bettelheim2006,wiegmann2012}. \n\nReal electrons have spin and, for this reason, standard hydrodynamic equations describing the time evolution of a scalar density can only account for situations where the electron spin can be ignored. While this is the case for many conventional fractional quantum Hall effect systems, there also exist interesting such systems where the electron spin is important \\cite{senthil1999,mccann2006,bernevig2006}. For such a system, one is interested in a hydrodynamic description by a soliton equation describing the time evolution of a fluid of particles carrying spin. The sBO equation \\eqref{eq:sBO}, in the simplest non-trivial case when $\\mathsf{U}$ is a hermitian $2\\times 2$ matrix, is a natural candidate for such an equation: as shown below, the sBO equation in this case can be written as a coupled system describing the time evolution of a charge- and a spin-density. Thus, we believe that it would be interesting to investigate if (a quantum version of) the sBO equation can be used to describe phenomena observed in quantum Hall effect systems where spin cannot be ignored, in generalization of results in \\cite{bettelheim2006,wiegmann2012}. We recently proposed that the ncILW equation is relevant for parity invariant fractional Hall effect systems \\cite{berntson2020}, and this suggests to us that its spin generalization presented in this paper (see \\eqref{eq:sncILW}) will find applications in the context of the quantum spin Hall effect \\cite{bernevig2006}.  \n\nAs another possible application of the sBO equation in physics, we mention the relation of the BO equation to conformal field theory of spin-less fermions \\cite{abanov2005}. While spin-less fermions\\footnote{To be more precise: spin-less chiral fermions in $1+1$ spacetime dimensions.} are among the simplest examples of a conformal field theory, there are conformal field theories that are natural spin generalizations of these models known as Wess-Zumino-Witten models \\cite{difrancesco1997} which can take electron spin (and more complicated internal degrees of freedom) into account. We  expect that, in a similar way as the BO equation is related to spin-less fermions  \\cite{abanov2005}, the sBO equation can be related to Wess-Zumino-Witten models. It would be interesting to substantiate this expectation. \n\nWe now rewrite the sBO equation in the case where $\\mathsf{U}$ is a hermitian $2\\times 2$ matrix as a coupled system describing the time evolution of a charge density $u$ and a spin density $\\mathbf{m}$. For that, we parametrize $\\mathsf{U}$ as follows, \n\\begin{equation} \n\\label{eq:mUansatz}\n\\mathsf{U} = \\frac{u}{2}\\left(  I + \\mathbf{m}\\cdot\\boldsymbol{\\sigma}\\right)  = \\frac{u}{2} \\begin{pmatrix} 1 +m^3 & m^1-\\mathrm{i} m^2\\\\ m^1+\\mathrm{i} m^2 & 1-m^3 \\end{pmatrix} \n\\end{equation} \nwhere $u=u(x,t)$ and $\\mathbf{m}=(m^1,m^2,m^3)=\\mathbf{m}(x,t)$ are ${\\mathbb R}$- and ${\\mathbb R}^3$-valued functions, respectively. \nWith this parametrization, we find after some computations that \\eqref{eq:sBO} is equivalent to\n\\begin{equation} \n\\label{eq:utbmt} \n\\begin{split} \nu_t + (1+\\mathbf{m}^2)uu_x +\\mathbf{m}\\cdot\\mathbf{m}_x u^2 +Hu_{xx} &= 0 ,\n\\\\\n\\mathbf{m}_t + u_x\\mathbf{m}(1-\\mathbf{m}^2)+u[\\mathbf{m}_x-\\mathbf{m}(\\mathbf{m}\\cdot\\mathbf{m}_x)] +\\frac1u[H(u\\mathbf{m})_{xx} -\\mathbf{m} Hu_{xx}]   -\\mathbf{m}\\wedge H(u\\mathbf{m})_{x} \n &=\\mathbf{0} .\n\\end{split} \n\\end{equation}\nThis system combines and generalizes the physics of the BO equation and of the HWM equation in a non-trivial way, preserving the exact solvability. Indeed, setting $\\mathbf{m}(x,t)=\\mathbf{m}_0$ (constant) such that $\\mathbf{m}_0^2=1$, the first equation in \\eqref{eq:utbmt} becomes the BO equation \\eqref{eq:BO}, while the second equation is trivially fulfilled. On the other hand, setting $u(x,t)=u_0$ (constant) and transforming $\\mathbf{m}(x,t)\\to \\mathbf{m}(x-u_0t,u_0t)$, the first equation in  \\eqref{eq:utbmt} is satisfied if we impose the condition $\\mathbf{m}^2=1$ and, with that, the second equations becomes the HWM equation \\eqref{eq:HWM} in the limit $u_0\\to \\infty$. This suggests that \\eqref{eq:utbmt} can be well approximated by the BO equation if the variation of the spin density in space and time can be ignored, while the HWM equation is a good approximation to \\eqref{eq:utbmt} (on an appropriate time scale) if the charge density is large and only deviates slightly from a constant background $u_0$. \n\n\n\\paragraph{(iii) Multi-soliton solutions.} Following Ref.~\\cite{gibbons1984}, we use the Dirac bra-ket notation \\cite{dirac1939} and denote by $|e\\rangle$ and $\\langle f|$ vectors in some $d$-dimensional complex vector space $\\mathcal{V}$ and its dual $\\mathcal{V}^*$,  respectively; in particular, $|e\\rangle\\langle f|$ represents a $d\\times d$ matrix with complex entries, and $|e\\rangle\\langle f|^\\dag = |f\\rangle\\langle e|$ is the hermitian conjugate of this matrix. Moreover, $*$ is complex conjugation. (Readers not familiar with this notation can identify $|e\\rangle\\in \\mathcal{V}$ with $(e_{\\mu})_{\\mu=1}^d\\in{\\mathbb C}^d$, $\\langle f|\\in \\mathcal{V}^*$ with  $(f^*_{\\mu})_{\\mu=1}^d\\in{\\mathbb C}^d$, $\\langle f|e \\rangle$ with the scalar product $\\sum_{\\mu=1}^d f_\\mu^*e^{\\phantom*}_\\mu$, and $|e\\rangle\\langle f|$ with the matrix $(e^{\\phantom*}_\\mu f_\\nu^*)_{\\mu,\\nu=1}^d$.) \n\nWe show in Section~\\ref{sec:sBO} that, for arbitrary integer $N\\geq 1$ and $x\\in{\\mathbb R}$,  \n\\begin{equation}\n\\label{eq:mUintro}  \n\\mathsf{U}(x,t) = \\mathrm{i} \\sum_{j=1}^N |e_j(t)\\rangle \\langle f_j(t)| \\frac{1}{x-a_j(t)} - \\mathrm{i} \\sum_{j=1}^N |f_j(t)\\rangle \\langle e_j(t)| \\frac{1}{x-a_j(t)^*}\n\\end{equation} \nis a solution of the sBO equation \\eqref{eq:sBO} provided that the variables $a_j=a_j(t)\\in{\\mathbb C}$, $|e_j\\rangle=|e_j(t)\\rangle\\in\\mathcal{V}$ and $\\langle f_j|=\\langle f_j(t)|\\in\\mathcal{V}^*$ satisfy the following time evolution equations,\\footnote{$\\sum_{k\\neq j}^N$ is short for $\\sum_{k=1,k\\neq j}^N$.} \n\\begin{equation} \n\\label{eq:sCMintro} \n\\begin{split} \n\\frac{\\mathrm{d}^2}{\\mathrm{d} t^2} a_j = & \\, 8\\sum_{k\\neq j}^N\\frac{\\langle f_j|e_k\\rangle\\langle f_k|e_j\\rangle }{(a_j-a_k)^3},\\\\\n\\frac{\\mathrm{d}}{\\mathrm{d} t}|e_j\\rangle = &\\, 2\\mathrm{i}\\sum_{k\\neq j}^N \\frac{|e_k\\rangle\\langle f_k|e_j\\rangle}{(a_j-a_k)^2},\\\\ \n\\frac{\\mathrm{d}}{\\mathrm{d} t}\\langle f_j|  = &\\,  -2\\mathrm{i}\\sum_{k\\neq j}^N\\frac{\\langle f_j|e_k\\rangle \\langle f_k|}{(a_j-a_k)^2} \n\\end{split} \n\\end{equation} \nfor $j=1,\\ldots,N$, with initial conditions that satisfy certain constraints; see Theorem~\\ref{thm:sBO} for a precise formulation. \nMoreover, we show that initial conditions satisfying the pertinent constraints can be constructed by solving a linear algebra problem involving a $N d\\times N d$ hermitian matrix, and that the solution \\eqref{eq:mUintro} depends on $N d$ complex parameters; see Section~\\ref{sec:NsolitonssBO} for details. \n\nIt is important to note that, up to a rescaling of time, the time evolution equations \\eqref{eq:sCMintro} and one of the above-mentioned constraints on the initial conditions (given in \\eqref{eq:fjej}) defines the sCM model discovered by Gibbons and Hermsen \\cite{gibbons1984}, and another constraint (given in \\eqref{eq:BTt=0}) is a known B\\\"acklund transformation of the sCM model \\cite{gibbons1983}.\nMoreover, for $d=1$, the solution above reduces to the $N$-soliton solution of the BO equation found in \\cite{chen1979} that relates the BO equation to the rational $A_{N-1}$ CM model. Thus, in the same sense as the sCM model is a natural generalization of the simplest non-trivial CM model, the sBO equation is a natural generalization of the BO equation. \n\nAs shown by Gibbons and Hermsen in their original paper \\cite[Section 6]{gibbons1984}, the sCM model describes the time evolution of poles and spins of rational solutions of the boomeron equation, which is a soliton equation introduced and studied in \\cite{calogero1976coupled}. One might wonder if there is a relation between this fact and our work. We cannot exclude this possibility: it is conceivable to us that it is possible to derive the boomeron equation as a local limit $\\delta\\to\\infty $ of the sncILW equation \\eqref{eq:sncILW} or the sILW equation \\eqref{eq:sILW} introduced below and, in this way, there could be an indirect relation. However, even if this is the case, we expect that it would be challenging to make this relation precise. Indeed, while the boomeron equation is similar to \\eqref{eq:utbmt} in that it is a system describing the time evolution of a vector coupled to a scalar, it is different in other important ways; in particular, \\eqref{eq:utbmt} is rotation invariant (and this is also the case for the certain local limits of these equations that we derive, see Section~\\ref{sec:locallimit} for details), while the boomeron equation is not.\n\n\\paragraph{(iv) Spin generalization of ncILW equation and soliton-CM correspondence.} For $\\delta>0$, we define the following integral operators acting on functions $f$ of $x\\in{\\mathbb R}$, \n\\begin{equation} \n\\label{eq:TT}\n\\begin{split} \n(Tf)(x) &= \\frac1{2\\delta}\\Xint-_{{\\mathbb R}}\\coth\\left(\\frac{\\pi}{2\\delta}(x'-x)\\right)f(x')\\,\\mathrm{d}{x'}, \\\\\n(\\tilde{T} f)(x) &= \\frac1{2\\delta}\\int_{{\\mathbb R}}\\tanh\\left(\\frac{\\pi}{2\\delta}(x'-x)\\right)f(x')\\,\\mathrm{d}{x'}.\n\\end{split} \n\\end{equation} \nThe ncILW equation was introduced in \\cite{berntson2020} as a non-chiral version of the ILW equation and is given for two scalar-valued functions $u(x,t)$ and $v(x,t)$ by\n\\begin{equation} \n\\label{eq:ncILW} \n\\begin{split} \n&u_t + 2 u u_x + Tu_{xx}+\\tilde{T}v_{xx}=0,\\\\\n&v_t - 2 v v_x - Tv_{xx}-\\tilde{T}u_{xx}=0.\n\\end{split} \n\\end{equation} \nHere, we introduce the following spin generalization of the ncILW equation given for two square matrix-valued functions $\\mathsf{U}=\\mathsf{U}(x,t)$ and $\\mathsf{V}=\\mathsf{V}(x,t)$ by\n\\begin{equation} \n\\label{eq:sncILW} \n\\boxed{\n\\begin{aligned} \n\\mathsf{U}_t &+ \\{\\mathsf{U},\\mathsf{U}_x\\} + T\\mathsf{U}_{xx}+\\tilde{T} \\mathsf{V}_{xx} +\\mathrm{i} [\\mathsf{U},T\\mathsf{U}_x]+\\mathrm{i} [\\mathsf{U},\\tilde{T} \\mathsf{V}_x] =0,\\\\\n\\mathsf{V}_t &- \\{\\mathsf{V},\\mathsf{V}_x\\} - T\\mathsf{V}_{xx}-\\tilde{T} \\mathsf{U}_{xx} +\\mathrm{i} [\\mathsf{V},T\\mathsf{V}_x]+\\mathrm{i}[\\mathsf{V},\\tilde{T} \\mathsf{U}_x] =0\n\\end{aligned} \n}\n\\end{equation} \n(again, the matrix size $d\\times d$ of $\\mathsf{U}$ and $\\mathsf{V}$ is arbitrary; $d=1$ corresponds to the ncILW equation \\eqref{eq:ncILW}).\nWe refer to \\eqref{eq:sncILW} as the {\\em spin ncILW (sncILW) equation}. \nOur main result on the sncILW equation is a construction of $N$-soliton solutions obtained via a spin-pole ansatz where the dynamics of the spins and poles are determined by the $A_{N-1}$ sCM model in the hyperbolic case (III), in natural generalization of a known result about the ncILW equation \\cite{berntson2020}; see Theorem~\\ref{thm:sncILW}. \n\nNote that $\\lim_{\\delta\\to+\\infty}\\tilde{T} =0$ \\cite{berntson2020} and thus, clearly, the two equations in \\eqref{eq:sncILW} decouple in the limit $\\delta\\to\\infty$. \nMoreover, since the limit $\\delta\\to\\infty$ of $T$ coincides with the Hilbert transform $H$ given in \\eqref{eq:H} \\cite{kodama1981} (see also \\cite{berntson2020}), the first of these decoupled equations is identical to the sBO equation \\eqref{eq:sBO}, and the second equation is obtained from the sBO equation  \\eqref{eq:sBO} by the parity transformation $\\mathsf{U}(x,t)\\to \\mathsf{V}(x,t)=\\mathsf{U}(-x,t)$. Since the sBO equation and the one obtained from it by this parity transformation are different, the sBO equation   \\eqref{eq:sBO} is chiral, and we therefore regard \\eqref{eq:sncILW} as a non-chiral generalization of the sBO equation where the two chiral degrees of freedom, $\\mathsf{U}$ and $\\mathsf{V}$, are coupled by the operator $\\tilde{T}$.\n\nWe also introduce periodic versions of the sBO and sncILW equations: the former is given by \\eqref{eq:sBO} for an $L$-periodic, square matrix-valued function $\\mathsf{U}$, $\\mathsf{U}(x+L,t)=\\mathsf{U}(x,t)$ with $L>0$ a fixed parameter, and the Hilbert transform\n\\begin{equation}\n\\label{eq:Hp}  \n(Hf)(x)\\coloneqq \\frac1{L}\\Xint-_{-L\/2}^{L\/2} \\cot\\left(\\frac{\\pi}{L}(x'-x) \\right)f(x')\\,\\mathrm{d}{x'}; \n\\end{equation} \nthe latter is given by \\eqref{eq:sncILW} for $L$-periodic, square matrix-valued functions $\\mathsf{U}$ and $\\mathsf{V}$ and the integral operators \n\\begin{equation} \n\\label{eq:TTe}\n\\begin{split} \n(Tf)(x) &= \\frac1{\\pi}\\Xint-_{L\/2}^{L\/2}\\zeta_1(x'-x)f(x')\\,\\mathrm{d}{x'}, \\\\\n(\\tilde{T} f)(x) &= \\frac1{\\pi}\\int_{-L\/2}^{L\/2}\\zeta_1(x'-x+\\mathrm{i}\\delta)f(x')\\,\\mathrm{d}{x'},\n\\end{split} \n\\end{equation} \nwhere \n\\begin{equation}\n\\label{eq:zeta1} \n\\zeta_{1}(z)\\coloneqq  \\zeta(z)-\\frac{\\eta_{1}}{\\omega_{1}}z\n\\end{equation} \nwith $\\zeta(z)$ the Weierstrass $\\zeta$-function with half-periods $(\\omega_1,\\omega_2)=(L\/2,\\mathrm{i}\\delta)$ and $\\eta_1=\\zeta(\\omega_1)$ \\cite{DLMF}. \nSimilarly as in the real-line case, the periodic sBO equation is chiral, and the periodic sncILW equation reduces to two decoupled periodic sBO equations of opposite chirality in the limit $\\delta\\to\\infty$ (details can be found in \\cite{berntson2020}).  \n\nTo explain the relation between the soliton equations discussed above and CM systems, we recall that each CM system comes in four versions which can be distinguished by the special function\n\\begin{equation} \n\\label{eq:alpha} \n\\alpha(z)\\coloneqq \\begin{cases} 1\/z & \\text{(I: rational case)}\\\\\n(\\pi\/L)\\cot(\\pi z\/L) & \\text{(II: trigonometric case)}\\\\\n(\\pi\/2\\delta)\\coth(\\pi z\/2\\delta) & \\text{(III: hyperbolic case)}\n\\end{cases} \\quad (z\\in{\\mathbb C}) \n\\end{equation} \nwith $L>0$ and $\\delta>0$ fixed parameters;  the fourth version corresponds to \n\\begin{equation} \n\\alpha(z)\\coloneqq \\zeta_1(z) \\quad  \\text{(IV: elliptic case)}, \n\\end{equation} \nbut we do not include it in \\eqref{eq:alpha} since it is more complicated and, for this reason, our results in this paper are restricted to cases I--III.  \nThis function $\\alpha(z)$ is important since, for example, it determines the corresponding CM interaction potential as $V(z)=-\\alpha'(z)$ \\cite{olshanetsky1981}. \nMoreover, using these special functions, one can define integral operators\n\\begin{equation} \n\\label{eq:TTalpha}\n\\begin{split} \n(Tf)(x) &\\coloneqq \\frac1\\pi \\Xint- \\alpha(x'-x)f(x')\\,\\mathrm{d}{x'} \\quad \\text{(cases I--IV)}, \\\\\n(\\tilde{T} f)(x)&\\coloneqq \\frac1\\pi \\Xint- \\alpha(x'-x+\\mathrm{i}\\delta)f(x')\\,\\mathrm{d}{x'} \\quad \\text{(cases III and IV)}, \n\\end{split} \n\\end{equation} \nwhere the integrations are over ${\\mathbb R}$ in cases I and III and over $[-L\/2,L\/2]$ in cases II and IV. \nWe note that, in cases I and II, $T$ is identical to the Hilbert transform $H$ in \\eqref{eq:H} and \\eqref{eq:Hp}, respectively, suggesting that the real-line and periodic versions of the sBO equation are related to the $A$-type sCM system in the rational and trigonometric cases, respectively. Similarly, in cases III and IV, the operators $T$ and $\\tilde{T}$ in \\eqref{eq:TTalpha} are identical to the operators in \\eqref{eq:TT}  and \\eqref{eq:TTe}, suggesting that the real-line and periodic versions of the sncILW equation are related to the $A$-type sCM system in the hyperbolic and elliptic cases, respectively. The $N$-soliton solutions of these equations obtained in this paper confirm these expectations in cases I--III, and we conjecture that this result can be generalized to case IV. Thus, the equations proposed in the present paper extend the relation between soliton equations and CM systems proposed in \\cite{berntson2020} to the spin setting. \n\n\nAs shown by two of us in collaboration with Klabbers \\cite{berntsonklabbers2021}, there exists a non-chiral variant of an intermediate generalization of the Heisenberg ferromagnetic equation which generalizes the HWM equation and which has soliton solutions given by a spin-pole ansatz governed by the hyperbolic sCM model, in generalization of a result in \\cite{berntsonklabbers2020} for the HWM equation mentioned above; this {\\em non-chiral intermediate Heisenberg ferromagnetic equation} is given by \n\\begin{equation} \n\\begin{split} \n\\mathbf{m}_t & =\\, + \\mathbf{m}\\wedge T\\mathbf{m}_{x} - \\mathbf{m}\\wedge \\tilde{T}\\mathbf{n}_{x},\\\\\n\\mathbf{n}_t & =\\, - \\mathbf{n}\\wedge T\\mathbf{n}_{x} + \\mathbf{n}\\wedge \\tilde{T}\\mathbf{m}_{x},\\\\\n\\end{split} \n\\end{equation} \nfor two ${\\mathbb R}^3$-valued functions $\\mathbf{m}=\\mathbf{m}(x,t)$ and $\\mathbf{n}=\\mathbf{n}(x,t)$ satisfying $\\mathbf{m}^2=\\mathbf{n}^2=1$  \\cite{berntsonklabbers2021}, and we checked that it is obtained from the sncILW equation \\eqref{eq:sncILW} in a similar way as the HWM equation is obtained from the sBO equation \\eqref{eq:sBO} (as explained in the paragraph containing \\eqref{eq:sBO1}), after changing the sign of $\\mathsf{V}$ (the latter is merely a convention).\n\nOur results in the present paper, together with results in the literature on the standard (chiral) ILW equation \\cite{kodama1981} and on the non-chiral intermediate Heisenberg ferromagnet equation \\cite{berntsonklabbers2021}, suggest that the following is also an integrable generalization of the sBO equation, \n\\begin{equation} \n\\label{eq:sILW} \n\\boxed{ \n\\mathsf{U}_t +  \\{\\mathsf{U},\\mathsf{U}_x\\} + \\frac1\\delta\\mathsf{U}_x  + T\\mathsf{U}_{xx}  + \\mathrm{i} [\\mathsf{U},T\\mathsf{U}_x] =0\n}\n\\end{equation} \n(we added the term $\\mathsf{U}_x\/\\delta$ for convenience; note that this term can be removed by a Galilean transformation $x\\to x-t\/\\delta$). We refer to \\eqref{eq:sILW}  as the {\\em spin ILW (sILW) equation}. Note that, in the limit $\\delta\\to\\infty$, it reduces to the sBO equation \\eqref{eq:sBO}. We show that, in generalization of the well-known fact that the standard ILW equation reduces to the Korteweg-de Vries  (KdV) equation in a limit $\\delta\\downarrow 0$ (see e.g. \\cite{scoufis2005}), \n \\eqref{eq:sILW}  reduces to the following matrix generalization of the KdV equation in a certain $\\delta\\downarrow 0$ limit, \n\\begin{equation} \n\\label{eq:mKdV} \n\\mathsf{U}_t +  \\{\\mathsf{U},\\mathsf{U}_x\\} + \\mathsf{U}_{xxx}  = 0  \n\\end{equation} \n(see Section~\\ref{sec:locallimit}). \nThis so-called {\\em matrix KdV equation} was introduced by Lax \\cite{lax1968}, and its multisoliton solutions were constructed in \\cite{goncharenko2001}. \nMoreover, the sILW equation allows for another limit $\\delta\\downarrow 0$ leading to the following generalization of the Heisenberg ferromagnet (HF) equation \n\\begin{equation} \n\\label{eq:sHF} \n\\mathsf{U}_t + \\mathrm{i} [\\mathsf{U},\\mathsf{U}_{xx}]=0\n\\end{equation} \nwith the constraint $\\mathsf{U}^2=I$, where $I$ denotes the identity matrix (see Section~\\ref{sec:locallimit} for details); note that \\eqref{eq:sHF} reduces to the standard HF equation if $\\mathsf{U}$ is restricted to lie in the class of $2\\times 2$ hermitian traceless matrices. \nThus, the sILW equation is an interpolation between the matrix KdV equation, the HF equation, and the sBO equation. \n \n We call the equations introduced in the present paper the {\\em spin} (rather than the {\\em matrix}) BO and (nc)ILW equations since, in contrast to the matrix KdV equation \\eqref{eq:mKdV}, they  incorporate the nonlinear physics of charge- and spin-densities in a single equation; as discussed above, this property is lost in the local limit $\\delta\\to\\infty$ (since, depending on the scaling, either the Heisenberg-term $\\mathrm{i} [\\mathsf{U},\\mathsf{U}_{xx}]$ or the KdV-terms $\\{\\mathsf{U},\\mathsf{U}\\}_x+\\mathsf{U}_{xxx}$ disappear in that limit).\n\n\\paragraph{(v) Summary and open problems.} \nIn this paper, we introduce a family of equations containing well-known soliton equations as special and limiting cases. This family of equations consists of the sBO equation \\eqref{eq:sBO}, the sncILW equation \\eqref{eq:sncILW}, and the sILW equation \\eqref{eq:sILW} for different matrix sizes $d\\in{\\mathbb Z}_{\\geq 2}$; for $d=1$, these equations reduce to the known  BO, ncILW, and ILW equations, respectively.  \nWe show that the sBO and sncILW equations are exactly solvable and, by that, establish a relation to sCM models.  While we do not give any result about the sILW equation for $0<\\delta<\\infty$, we believe that the arguments we present about this equation strongly suggest that it is integrable.\nObviously, it would be interesting to generalize other known results about the special case $d=1$ (e.g., Hirota bilinear forms, Lax pairs, B\\\"acklund transformations, and inverse scattering transforms) to $d>1$.\n\nFor the ncILW equation, we were able to generalize the multi-soliton solutions in the real-line case to the periodic case in \\cite{berntson2020}. \nHowever, for the sncILW equation \\eqref{eq:sncILW}, we do not have this result. We found that the construction of soliton solutions of the sncILW equation in the periodic case is more challenging for $d>1$ than for $d=1$ for the following reasons: first, the generalization of Proposition~\\ref{prop:BT} to the elliptic case (IV) is more difficult (for $d=1$, this generalization is known), and second, the commutator terms in \\eqref{eq:sncILW} lead to severe complications. We thus believe that the construction of soliton solutions of the periodic sncILW equation is an interesting problem requiring new ideas.\n\nWe were inspired to search for the soliton equations presented in this paper by heuristic arguments suggested to us by the known relation between the quantum version of the BO equation and the Calogero-Sutherland model\\footnote{This is the quantum version of the trigonometric $A$-type CM model.} \\cite{abanov2005}, together with a generalization of this relation to the corresponding elliptic Calogero-Sutherland model that lead us to the ncILW equation \\cite{berntson2020}. \nIt would be interesting to promote these heuristic arguments to precise results by constructing a second quantization of the quantum versions of the trigonometric sCM models, in generalization of results in \\cite{carey1999}. We believe that this can open a way towards finding a precise relation between the sBO equation and Wess-Zumino-Witten models.\n\nAs already discussed in (ii) above, there are several systems in the real world motivating the development of hydrodynamic descriptions of identical particles with spin; however, hydrodynamics for coupled charge- and spin-densities is not a fully developed subject. As one specific example,  we mention recent work deriving such hydrodynamic equations using the known hydrodynamic description of the standard CM model as a guide \\cite{kulkarni2009}; see also Xing's PhD thesis \\cite{xing2015} aiming in this direction.\nWe hope that the present paper opens up a way to push these results further towards a hydrodynamic description of the sCM model which can serve as a prototype of spinful hydrodynamics.\n\n\\paragraph{Plan of paper.} We state and prove the B\\\"acklund transformations for the sCM model in the rational, trigonometric and hyperbolic cases in Section~\\ref{sec:sCM}. In Section~\\ref{sec:sBO} and \\ref{sec:sncILW}, we state and prove our multi-soliton solutions of the sBO and sncILW equation, respectively; we note that while the results in Section~\\ref{sec:sBO} on the sBO equation are for the real-line and periodic cases, the results in Section~\\ref{sec:sncILW} on the sncILW equation only apply to the real-line case.  Section~\\ref{sec:results} contains the Hamiltonian formulations of the sBO and sncILW equations (Section~\\ref{sec:Hamiltonian}), details on the local limit $\\delta\\to\\infty$ of the sILW equation (Section~\\ref{sec:locallimit}), and the spin generalization of the bidirectional BO equation that led us to the results in the present paper (Section~\\ref{sec:2sBO}). We also include three appendices with functional identities that we need (Appendix~\\ref{app:alphaV}), and the non-hermitian solutions of the sBO equation (Appendix~\\ref{app:sBOsolutions}) and the sncILW equation (Appendix~\\ref{app:sncILWsolutions}), respectively.  \n\n\n\\section{Spin Calogero-Moser systems}\n\\label{sec:sCM}  \nIn this section, we collect results about the $A$-type sCM systems due to Gibbons and Hermsen \\cite{gibbons1984} that we need. In particular, we define the $A$-type sCM systems in Section~\\ref{sec:sCM_def}, and, following  \\cite{gibbons1983}, we state and prove a B\\\"acklund transformation for these systems in Sections~\\ref{sec:sCM_BT} and \\ref{sec:BT_proof}. For simplicity, we restrict our discussion to cases I--III; we expect a similar result for the elliptic case IV, but this case is more complicated and thus left to future work. \n\nWe believe that the results in this section are of interest in their own right: to our knowledge, the proof of the relevant B\\\"acklund transformation (see \\eqref{eq:BT}) in the literature has previously been restricted to the rational case (I) and $M=N$ \\cite{gibbons1983}. We mention in passing that, while the original paper introduced and solved the sCM model only in the rational case \\cite{gibbons1984}, the integrability of the sCM model in all cases I--IV was proved in \\cite{hikami1993}, and explicit solutions of the sCM model in the elliptic case (IV) can be found in  \\cite{krichever1995}. \n\n\\subsection{Definition}\n\\label{sec:sCM_def}\nLet $N\\in{\\mathbb Z}_{\\geq 1}$ be arbitrary and let\n\\begin{equation} \n\\label{eq:V} \nV(z)\\coloneqq \\begin{cases} 1\/z^2 & \\text{(I: rational case)}\\\\\n(\\pi\/L)^2\/\\sin^2(\\pi z\/L) & \\text{(II: trigonometric case)}\\\\\n(\\pi\/2\\delta)^2\/\\sinh^2(\\pi z\/2\\delta) & \\text{(III: hyperbolic case).}\\\\\n\\end{cases} \\quad (z\\in{\\mathbb C}) \n\\end{equation} \nFor each case I--III, the corresponding $A_{N-1}$ sCM system is a dynamical system of $N$ particles moving in the complex plane and with internal degrees of freedom described by a $d$-dimensional vector space $\\mathcal{V}$ and its dual $\\mathcal{V}^*$, with $d\\in{\\mathbb Z}_{\\geq 1}$ arbitrary. \nDenoting the position of the $j$th particle at time $t\\in{\\mathbb R}$ by $a_j=a_j(t)\\in{\\mathbb C}$ and its internal degrees of freedom by vectors $|e_j\\rangle=|e_j(t)\\rangle\\in \\mathcal{V}$ and $\\langle f_j|=\\langle f_j(t)|\\in \\mathcal{V}^*$, this system can be defined by the time evolution equations\n\\begin{equation} \n\\label{eq:sCM1a} \n\\ddot a_j = -4\\sum_{k\\neq j}^N \\langle f_j|e_k\\rangle\\langle f_k|e_j\\rangle V'(a_j-a_k) \\quad (j=1,\\ldots,N) \n\\end{equation} \nand\n\\begin{equation} \n\\begin{split} \n\\label{eq:sCM1b} \n|\\dot e_j\\rangle &= 2\\mathrm{i}\\sum_{k\\neq j}^N |e_k\\rangle\\langle f_k|e_j\\rangle V(a_j-a_k),\\\\\n\\langle\\dot f_j| & = -2\\mathrm{i}\\sum_{k\\neq j}^N\\langle f_j|e_k\\rangle \\langle f_k| V(a_j-a_k)\n\\end{split} \\quad (j=1,\\ldots,N) \n\\end{equation} \n(the dot indicates differentiation with respect to $t$ and the prime indicates differentiation with respect to the argument of the respective function), together with the following constraints, \n\\begin{equation} \n\\label{eq:sCM1c} \n\\langle f_j|e_j\\rangle = 1 \\quad  (j=1,\\ldots,N) \n\\end{equation}  \n(our notation is explained in the paragraph above \\eqref{eq:mUintro}).\nObserve that the constraints \\eqref{eq:sCM1c} are preserved under the equations of motion \\eqref{eq:sCM1b}.\n\n\n\\subsection{B\\\"acklund transformations}\n\\label{sec:sCM_BT}\nWe consider the sCM system \\eqref{eq:sCM1a}--\\eqref{eq:sCM1c}, together with another such system involving $M\\in{\\mathbb Z}_{\\geq 0}$ particles located at the positions $b_j=b_j(t)\\in{\\mathbb C}$, $j = 1, \\dots, M$, together with spin degrees of freedom $|g_j\\rangle=|g_j(t)\\rangle\\in \\mathcal{V}$ and $\\langle h_j|=\\langle h_j(t)|\\in \\mathcal{V}^*$ (note that the vector spaces $\\mathcal{V}$ and $\\mathcal{V}^*$ are the same as for the first system; while $M=N$ is an important special case, also the cases $M\\neq N$ and, in particular, $M=0$ are interesting). \nMore specifically, the second system is given by the time evolution equations\n\\begin{equation} \n\\label{eq:sCM2a} \n\\ddot b_j = -4\\sum_{k\\neq j}^M \\langle h_j|g_k\\rangle\\langle h_k|g_j\\rangle V'(b_j-b_k) \\quad (j=1,\\ldots,M) \n\\end{equation} \nand \n\\begin{equation} \n\\begin{split} \n\\label{eq:sCM2b} \n|\\dot g_j\\rangle &= 2\\mathrm{i}\\sum_{k\\neq j}^M |g_k\\rangle\\langle h_k|g_j\\rangle V(b_j-b_k),\\\\\n\\langle\\dot h_j| & = -2\\mathrm{i}\\sum_{k\\neq j}^M\\langle h_j|g_k\\rangle \\langle h_k| V(b_j-b_k), \n\\end{split} \\quad (j=1,\\ldots,M) \n\\end{equation} \nand the constraints \n\\begin{equation} \n\\label{eq:sCM2c} \n\\langle h_j|g_j\\rangle = 1 \\quad (j=1,\\ldots,M) .\n\\end{equation}  \n\nAs shown in \\cite{gibbons1983} in a special case, two such sCM systems \\eqref{eq:sCM1a}--\\eqref{eq:sCM1c} and  \\eqref{eq:sCM2a}--\\eqref{eq:sCM2c} are connected by a B\\\"acklund transformation as follows, \n\\begin{subequations} \n\\label{eq:BT} \n\\begin{align} \n\\label{eq:BTa} \n\\dot a_j \\langle f_j| &= 2\\mathrm{i}\\sum_{k\\neq j}^N  \\langle f_j|e_k\\rangle \\langle f_k|\\alpha(a_j-a_k) -2\\mathrm{i}\\sum_{k=1}^M  \\langle f_j|g_k\\rangle\\langle h_k|\\alpha(a_j-b_k)\\quad &(j=1,\\ldots,N), \\\\\n\\label{eq:BTb} \n\\dot b_j |g_j\\rangle &=-2\\mathrm{i}\\sum_{k\\neq j}^M |g_k\\rangle\\langle h_k|g_j\\rangle\\alpha(b_j-b_k)+2\\mathrm{i}\\sum_{k=1}^N|e_k\\rangle\\langle f_k|g_j\\rangle\\alpha(b_j-a_k) \\quad &(j=1,\\ldots,M),    \n\\end{align} \n\\end{subequations} \nwith the function $\\alpha(z)$ given by \\eqref{eq:alpha}. The precise statement is given below. \n\n\\begin{proposition}[B\\\"acklund transformation for sCM system]\n\\label{prop:BT} \nIn each case I--III, the first order equations  \\eqref{eq:sCM1b}, \\eqref{eq:sCM2b} and \\eqref{eq:BT}, together with the constraints \\eqref{eq:sCM1c} and  \\eqref{eq:sCM2c}, imply the second order equations \\eqref{eq:sCM1a} and \\eqref{eq:sCM2a}.  \n\\end{proposition} \nA self-contained proof of Proposition \\ref{prop:BT} can be found in Section~\\ref{sec:BT_proof}.\n\n\n\\begin{remark} \nAs already mentioned, the special case $M=N$ in the rational case (I) was stated and proved in  \\cite{gibbons1983}, More specifically, in this special case, the equations \\eqref{eq:BTa} and \\eqref{eq:BTb} reduce to the second and first equations in \\cite[Eq.~(17)]{gibbons1983}, respectively, using the transformation $t\\to -2t$ and the identifications \n\\begin{equation} \n (a_j,\\dot a_j,|e_j\\rangle,\\langle f_j|,b_j,\\dot b_j,|g_j\\rangle,\\langle h_j|) \\to x^+_j,p^+_j,|e^+_j\\rangle,\\langle f^+_j|,x_j,p_j,|e_j\\rangle,\\langle f_j|) \\quad (j=1\\ldots,N).\n\\end{equation}  \n\\end{remark} \n\nIt is interesting to note that Proposition~\\ref{prop:BT} has the following consistent reduction when $N=M$,\n\\begin{equation} \n\\label{eq:reduction} \nb_j=a_j^*,\\quad |g_j\\rangle = \\langle f_j|^\\dag = |f_j\\rangle,\\quad \\langle h_j| = |e_j\\rangle^\\dag = \\langle e_j|\\quad (j=1,\\ldots,N)\n\\end{equation} \nwhere $*$ and $\\dag$ indicate complex and hermitian conjugation, respectively. \nIndeed, by imposing these conditions, \\eqref{eq:sCM2a}--\\eqref{eq:sCM2c}  and \\eqref{eq:BTb} become the hermitian conjugate of \\eqref{eq:sCM1a}--\\eqref{eq:sCM1c} and \\eqref{eq:BTa}, respectively, and Proposition~\\ref{prop:BT} simplifies as follows. \n\n\\begin{corollary}\n\\label{cor:BT} \nIn each case I--III, the first order equations \\eqref{eq:sCM1b} and \n\\begin{equation}\n\\label{eq:BThermitian} \n\\dot a_j \\langle f_j| = 2\\mathrm{i}\\sum_{k\\neq j}^N  \\langle f_j|e_k\\rangle \\langle f_k|\\alpha(a_j-a_k) -2\\mathrm{i}\\sum_{k=1}^N  \\langle f_j|f_k\\rangle\\langle e_k|\\alpha(a_j-a^*_k)\\quad (j=1,\\ldots,N), \n\\end{equation} \ntogether with the constraints \\eqref{eq:sCM1c}, imply the  second order equations \\eqref{eq:sCM1a}. \n\\end{corollary} \n\n\\subsection{Proof of Proposition~\\ref{prop:BT}} \n\\label{sec:BT_proof} \nWe note that, in the system of equations \\eqref{eq:sCM1a}--\\eqref{eq:BT}, the sets of variables $\\{a_j,|e_j\\rangle,\\langle f_j|\\}_{j=1}^N$ and $\\{b_j,\\langle h_j|, |g_j\\rangle\\}_{j=1}^M$ can be swapped by hermitian conjugation and renaming $a_j^*\\to a_j$, $b_j^*\\to b_j$. Due to this symmetry, it suffices to verify the claim for the first set of variables, i.e., it is enough to show that \\eqref{eq:sCM1a} follows from \\eqref{eq:sCM1b}, \\eqref{eq:sCM2b}, \\eqref{eq:BT}, subject to \\eqref{eq:sCM1c} and \\eqref{eq:sCM2c}.\n \nWe introduce the shorthand notation \n\\begin{equation} \n\\label{eq:shorthandBT} \n(a_j,|e_j\\rangle,\\langle f_j|,r_j) = \\begin{cases} (a_j,|e_j\\rangle,\\langle f_j|,+1) & (j=1,\\ldots,N) \\\\ (b_{j-N},|g_{j-N}\\rangle,\\langle h_{j-N}|,-1) & (j=N+1,\\ldots,\\mathcal {N}) , \\end{cases} \\quad \n\\mathcal {N}\\coloneqq N+M, \n\\end{equation}\nand\n\\begin{equation} \n\\label{eq:mBj} \n \\mathsf{P}_j\\coloneqq |e_j\\rangle\\langle f_j| , \\quad \\mathsf{B}_j\\coloneqq \\mathrm{i}\\sum_{k\\neq j}^{\\mathcal {N}} r_k \\mathsf{P}_k  \\alpha(a_j-a_k),\\quad (j=1,\\ldots,\\mathcal {N}) \n\\end{equation}  \nto write \\eqref{eq:BT} as  \n\\begin{equation}\n\\label{eq:BTshort} \n\\begin{split}  \n\\dot a_j\\langle f_j| = &\\;  2\\langle f_j|\\mathsf{B}_j\\quad (j=1,\\ldots,N),      \\\\\n\\dot a_j|e_j\\rangle = &\\; 2\\mathsf{B}_j|e_j\\rangle \\quad (j=N+1,\\ldots,\\mathcal {N}).\n\\end{split}\n\\end{equation} \n\nMoreover, this notation allows us to write the two sets of equations \\eqref{eq:sCM1a}--\\eqref{eq:sCM1c} and \\eqref{eq:sCM2a}--\\eqref{eq:sCM2c} as one: \n\\begin{equation} \n\\label{eq:sCM3a} \n\\ddot a_j = -2\\sum_{k\\neq j}^\\mathcal {N} (1+r_jr_k)\\langle f_j|e_k\\rangle\\langle f_k|e_j\\rangle V'(a_j-a_k) \\quad (j=1,\\ldots,\\mathcal {N}) \n\\end{equation} \nand \n\\begin{equation} \n\\begin{split} \n\\label{eq:sCM3b} \n|\\dot e_j\\rangle &= \\mathrm{i}\\sum_{k\\neq j}^\\mathcal {N} (1+r_jr_k) |e_k\\rangle\\langle f_k|e_j\\rangle V(a_j-a_k),\\\\\n\\langle\\dot f_j| & = -\\mathrm{i}\\sum_{k\\neq j}^\\mathcal {N}(1+r_jr_k)\\langle f_j|e_k\\rangle \\langle f_k| V(a_j-a_k), \n\\end{split} \\quad (j=1,\\ldots,\\mathcal {N}) , \n\\end{equation} \ntogether with\n\\begin{equation} \n\\label{eq:sCM3c} \n\\langle f_j|e_j\\rangle = 1 \\quad (j=1,\\ldots,\\mathcal {N}).  \n\\end{equation}  \n\nBy differentiating the first set of equations in \\eqref{eq:BTshort} with respect to time, we obtain \n\\begin{equation} \n\\label{eq:ddotajfj1}\n\\ddot a_j\\langle f_j| =   \\langle \\dot f_j|(2\\mathsf{B}_j-\\dot a_j) + 2\\langle f_j|\\dot \\mathsf{B}_j\n\\end{equation} \nwhere, here and below in this section, $j=1,\\ldots,N$. \nWe compute, using \\eqref{eq:sCM3b} and $|e_k\\rangle\\langle f_k|=\\mathsf{P}_k$ (note that $r_j=+1$), \n\\begin{equation} \n\\label{eq:dotfj} \n\\begin{split} \n\\langle \\dot f_j|(2\\mathsf{B}_j-\\dot a_j) = & -\\mathrm{i}\\sum_{k\\neq j}^\\mathcal {N}(1+r_k) \\langle f_j|e_k\\rangle\\langle f_k |(2\\mathsf{B}_j-\\dot a_j)V(a_j-a_k) \\\\\n=  & -\\mathrm{i}\\sum_{k\\neq j}^\\mathcal {N}(1+r_k)\\langle f_j|( 2\\mathsf{P}_k \\mathsf{B}_j-2\\mathsf{B}_j\\mathsf{P}_k)V(a_j-a_k)  \\\\\n=  & -2\\mathrm{i}\\sum_{k\\neq j}^\\mathcal {N}(1+r_k)\\langle f_j|[\\mathsf{P}_k, \\mathsf{B}_j]V(a_j-a_k), \n\\end{split} \n\\end{equation} \ninserting \\eqref{eq:BTshort} in the second step. Moreover, the definition \\eqref{eq:mBj} of $\\mathsf{B}_j$ and the relation $\\alpha'(z)=-V(z)$  imply\n\\begin{equation} \n2\\langle f_j|\\dot \\mathsf{B}_j = 2\\mathrm{i}\\sum_{k\\neq j}^\\mathcal {N}  r_k \\langle f_j|\\dot\\mathsf{P}_k \\alpha(a_j-a_k) - 2\\mathrm{i}\\sum_{k\\neq j}^\\mathcal {N}  r_k \\langle f_j|\\mathsf{P}_k(\\dot a_j-\\dot a_k)V(a_j-a_k), \n\\end{equation} \nand by using \\eqref{eq:BTshort} we compute\n\\begin{equation} \n\\label{eq:fjPkdotajdotak}\n\\begin{split} \n\\langle f_j|\\mathsf{P}_k(\\dot a_j-\\dot a_k)  = &\\; \\langle f_j|e_k\\rangle\\langle f_k| (\\dot a_j-\\dot a_k) \\\\\n= & \\; \\dot a_j\\langle f_j|e_k\\rangle\\langle f_k|  - \\frac12(1+r_k)\\langle f_j|e_k\\rangle\\dot a_k \\langle f_k| - \\frac12(1-r_k)\\langle f_j|e_k\\rangle \\dot a_k\\langle f_k|\\\\\n= &\\;  2 \\langle f_j|\\mathsf{B}_j\\mathsf{P}_k - (1+r_k)\\langle f_j|\\mathsf{P}_k\\mathsf{B}_k -(1-r_k)\\langle f_j|\\mathsf{B}_k\\mathsf{P}_k\\\\\n= & \\; \\langle f_j|\\big(\\{\\mathsf{P}_k,\\mathsf{B}_j-\\mathsf{B}_k\\} + r_k[\\mathsf{P}_k,\\mathsf{B}_j-\\mathsf{B}_k] -  (1+r_k)[\\mathsf{P}_k,\\mathsf{B}_j]\\big) . \n\\end{split} \n\\end{equation} \nInserting the results in \\eqref{eq:dotfj}--\\eqref{eq:fjPkdotajdotak} into \\eqref{eq:ddotajfj1} and using  $r_k^2=1$,  we find that $ \\langle \\dot f_j|(2\\mathsf{B}_j-\\dot a_j)$ is canceled by the part of  $2\\langle f_j|\\dot \\mathsf{B}_j$ involving the term $-(1+r_k)\\langle f_j[\\mathsf{P}_k,\\mathsf{B}_j]$ in \\eqref{eq:fjPkdotajdotak}, and we obtain  \n\\begin{equation} \n\\label{eq:ddotajfj2}\n\\begin{split} \n\\ddot a_j\\langle f_j| = &\\;   2\\mathrm{i}\\sum_{k\\neq j}^\\mathcal {N}  r_k \\langle f_j|\\dot\\mathsf{P}_k \\alpha(a_j-a_k) \\\\\n& -2\\mathrm{i} \\sum_{k\\neq j}^\\mathcal {N} \\langle f_j|\\big( r_k \\{\\mathsf{P}_k,\\mathsf{B}_j-\\mathsf{B}_k\\} + [\\mathsf{P}_k,\\mathsf{B}_j-\\mathsf{B}_k] \\big)V(a_j-a_k) . \n\\end{split} \n\\end{equation} \nTo proceed, we use $\\mathsf{P}_k=|e_k\\rangle\\langle f_k|$ and \\eqref{eq:sCM3b} to compute \n\\begin{equation} \n\\label{eq:dotPk} \n\\begin{split} \n\\dot \\mathsf{P}_k = &\\; |\\dot e_k\\rangle\\langle f_k| + |e_k\\rangle\\langle \\dot f_k| \\\\\n= &\\; \\mathrm{i}\\sum_{l\\neq k}^{\\mathcal {N}}(1+r_kr_l) ( |e_l\\rangle\\langle f_l | e_k\\rangle\\langle f_k| -  |e_k\\rangle\\langle f_k|e_l\\rangle\\langle f_l|)V(a_k-a_l)  \\\\\n=  & -\\mathrm{i}\\sum_{l\\neq k}^{\\mathcal {N}}(1+r_kr_l) [\\mathsf{P}_k,\\mathsf{P}_l]V(a_k-a_l).\n\\end{split} \n\\end{equation} \nFor future use, it is convenient to rewrite \\eqref{eq:dotPk} as \n\\begin{equation}\n\\label{eq:dotPk2}\n\\dot \\mathsf{P}_k=- \\mathrm{i}(1+r_k)[\\mathsf{P}_k,\\mathsf{P}_j]V(a_j-a_k)-\\mathrm{i}\\sum_{l\\neq j,k}^{\\mathcal {N}}(1+r_kr_l) [\\mathsf{P}_k,\\mathsf{P}_l]V(a_k-a_l),\n\\end{equation}\nusing $r_j=1$ and that $V(z)$ is even in the first term. \nNext, by the definition of $\\mathsf{B}_j$ \\eqref{eq:mBj}, we have, for $k = 1, \\dots, \\mathcal{N}$ with $k \\neq j$,\n\\begin{equation}\n\\begin{split}  \n\\label{eq:mBjmBk} \n(\\mathsf{B}_j-\\mathsf{B}_k)V(a_j-a_k) = &\\; \\mathrm{i} \\Big( \\sum_{l\\neq j}^{\\mathcal {N}} r_l \\mathsf{P}_l \\alpha(a_j-a_l)-  \\sum_{l\\neq k}^{\\mathcal {N}} r_l \\mathsf{P}_l \\alpha(a_k-a_l)\\Bigr)V(a_j-a_k) \\\\\n = &\\; \\mathrm{i}(r_k\\mathsf{P}_k+\\mathsf{P}_j)\\alpha(a_j-a_k)V(a_j-a_k) \\\\ & + \\mathrm{i}\\sum_{l\\neq j,k}^{\\mathcal {N}} r_l\\mathsf{P}_l V(a_j-a_k)\\big( \\alpha(a_j-a_l) - \\alpha(a_k-a_l) \\big), \n\\end{split} \n\\end{equation} \nusing $r_j=1$ and that $\\alpha(z)$ is odd to simplify $r_k\\mathsf{P}_k\\alpha(a_j-a_k)- r_j\\mathsf{P}_j\\alpha(a_k-a_j)=(r_k\\mathsf{P}_k+\\mathsf{P}_j)\\alpha(a_j-a_k)$. \nBy inserting \\eqref{eq:dotPk2} and \\eqref{eq:mBjmBk} into \\eqref{eq:ddotajfj2} and simplifying, we obtain \n\\begin{equation} \n\\label{eq:ddotajfj3}\n\\begin{split} \n\\ddot a_j\\langle f_j|  = & 2\\sum_{k\\neq j}^\\mathcal {N} \\sum_{l\\neq j,k}^{\\mathcal {N}}(r_k+r_l)  \\langle f_j| [\\mathsf{P}_k,\\mathsf{P}_l]\\alpha(a_j-a_k)V(a_k-a_l) \\\\\n& +2\\sum_{k\\neq j}^\\mathcal {N} \\langle f_j|\\big( 2\\mathsf{P}_k+ r_k \\{\\mathsf{P}_k,\\mathsf{P}_j\\} +(2+r_k) [\\mathsf{P}_k,\\mathsf{P}_j] \\big)\\alpha(a_j-a_k)V(a_j-a_k) \\\\ \n& + 2\\sum_{k\\neq j}^\\mathcal {N} \\sum_{l\\neq j,k}^{\\mathcal {N}} \\langle f_j|\\big( r_kr_l \\{\\mathsf{P}_k,\\mathsf{P}_l\\} +r_l[\\mathsf{P}_k,\\mathsf{P}_l]) \n\\big( \\alpha(a_j-a_l) - \\alpha(a_k-a_l) \\big)V(a_j-a_k),   \n\\end{split} \n\\end{equation} \nusing $r_k^2=1$, $r_k\\{\\mathsf{P}_k,r_k\\mathsf{P}_k+\\mathsf{P}_j\\}=2\\mathsf{P}_k^2+r_k\\{\\mathsf{P}_k,\\mathsf{P}_j\\}$, $[\\mathsf{P}_k,r_k\\mathsf{P}_k+\\mathsf{P}_j]=[\\mathsf{P}_k,\\mathsf{P}_j]$, and \n\\begin{equation} \n\\label{eq:mPsquare} \n\\mathsf{P}_k^2 = |e_k\\rangle\\langle f_k|e_k\\rangle\\langle f_k| =  |e_k\\rangle\\langle f_k| =  \\mathsf{P}_k\n\\end{equation}  \nby \\eqref{eq:sCM3c}. \nSince $\\langle f_j|\\mathsf{P}_j = \\langle f_j|e_j\\rangle\\langle f_j|=\\langle f_j|$ by \\eqref{eq:sCM3c}, we can simplify further: \n\\begin{equation}\n\\label{eq:ddotajfj3A}\n \\langle f_j|\\big( 2\\mathsf{P}_k+ r_k \\{\\mathsf{P}_k,\\mathsf{P}_j\\} +(2+r_k) [\\mathsf{P}_k,\\mathsf{P}_j] \\big) \n=  2(1+r_k)\\langle f_j| \\mathsf{P}_k\\mathsf{P}_j.\n\\end{equation} \nMoreover, since $V(z)$ is even, \n\\begin{multline} \\label{eq:ddotajfj3B}\n2\\sum_{k\\neq j}^\\mathcal {N} \\sum_{l\\neq j,k}^{\\mathcal {N}}(r_k+r_l)  \\langle f_j| [\\mathsf{P}_k,\\mathsf{P}_l]\\alpha(a_j-a_k)V(a_k-a_l) \\\\\n= \n\\sum_{k\\neq j}^\\mathcal {N} \\sum_{l\\neq j,k}^{\\mathcal {N}}(r_k+r_l)  \\langle f_j| [\\mathsf{P}_k,\\mathsf{P}_l]\\big( \\alpha(a_j-a_k)-  \\alpha(a_j-a_l)\\big)V(a_k-a_l) . \n\\end{multline} \nTo proceed, we need the identities \n\\begin{equation}\n\\label{eq:IdalphaV}\n\\alpha(z)V(z)=-\\frac12V'(z)\n\\end{equation}\nand  \n\\begin{equation} \n\\label{eq:IdalalV} \n\\big( \\alpha(a_j-a_l) - \\alpha(a_k-a_l) \\big)V(a_j-a_k)  \n= -\\big( \\alpha(a_j-a_k) - \\alpha(a_j-a_l) \\big)V(a_k-a_l).  \n\\end{equation} \nThe first identity \\eqref{eq:IdalphaV} can be obtained by differentiating \\eqref{eq:IdV} with respect to $z$ while the second identity \\eqref{eq:IdalalV} can be obtained by differentiating \\eqref{eq:Idmain} with respect to $b$ and setting $a=a_j$, $b=a_k$, and $c=a_l$. Thus, inserting \\eqref{eq:ddotajfj3A}, \\eqref{eq:ddotajfj3B} with \\eqref{eq:IdalalV}, and \\eqref{eq:IdalphaV}, \\eqref{eq:ddotajfj3} becomes  \n\\begin{equation} \n\\label{eq:ddotajfj4}\n\\begin{split} \n\\ddot a_j\\langle f_j|  = & \\sum_{k\\neq j}^\\mathcal {N} \\sum_{l\\neq j,k}^{\\mathcal {N}}(r_k-r_l)  \\langle f_j| [\\mathsf{P}_k,\\mathsf{P}_l]\\big( \\alpha(a_j-a_k) - \\alpha(a_j-a_l)\\bigl)V(a_k-a_l) \\\\\n& -2\\sum_{k\\neq j}^\\mathcal {N} (1+r_k) \\langle f_j| \\mathsf{P}_k\\mathsf{P}_j   V'(a_j-a_k)\\\\\n& - 2\\sum_{k\\neq j}^\\mathcal {N} \\sum_{l\\neq j,k}^{\\mathcal {N}} r_kr_l\\langle f_j| \\{\\mathsf{P}_k,\\mathsf{P}_l\\}\\big( \\alpha(a_j-a_k) - \\alpha(a_j-a_l) \\big)V(a_k-a_l).\n\\end{split} \n\\end{equation} \nSince $V(z)$ is even, the double sums in the second and third lines in \\eqref{eq:ddotajfj4} both vanish by symmetry, and using that $(1+r_k)=2$ for $k=1,\\ldots,N$ and $0$ otherwise, \nwe get \n\\begin{equation} \n \\ddot a_j\\langle f_j|  =  - 4\\sum_{k\\neq j}^N \\langle f_j|\\mathsf{P}_k\\mathsf{P}_j V'(a_j-a_k)\\quad (j=1,\\ldots,N). \n\\end{equation} \nBy multiplying this from the right with $|e_j\\rangle$ and using $\\mathsf{P}_j|e_j\\rangle=|e_j\\rangle$ and $\\langle f_j|\\mathsf{P}_k|e_j\\rangle = \\langle f_j|e_k\\rangle\\langle f_k|e_j\\rangle$, we obtain \\eqref{eq:sCM1a}. \n\n\\section{Multi-soliton solutions of the sBO equation} \n\\label{sec:sBO} \nIn this section, we present and derive multi-soliton solutions of the sBO equation \\eqref{eq:sBO}, both in the real-line case and the  $L$-periodic case. \nThe special functions $\\alpha(z)$ and $V(z)$ are given in \\eqref{eq:alpha} and \\eqref{eq:V}, with the real-line case corresponding to I (rational case) and the  $L$-periodic case corresponding to II (trigonometric case), respectively, throughout this section. All our results hold true in both cases. \n\n\\subsection{Result}\n\\label{sec:sBOresult}\nWe fix $d\\in{\\mathbb Z}_{\\geq 1}$ and consider the case where $\\mathsf{U}$ is a $d\\times d$ matrix-valued function;  \n for $d=1$, \\eqref{eq:sBO} reduces to the standard BO equation, and our result below is well-known in this case \\cite{chen1979}.  \n \nThe following theorem, whose proof is given in Section \\ref{sec:sBOproof}, is our main result about the sBO equation.\n\n\\begin{theorem} \n\\label{thm:sBO} \nLet $\\{a_j(t),|e_j(t)\\rangle,\\langle f_j(t)|\\}_{j=1}^N$ be a solution of the time evolution equations \\eqref{eq:sCM1a}--\\eqref{eq:sCM1b} with initial conditions \n\\begin{equation} \na_j(0)=a_{j,0},\\quad \\dot a_j(0)=v_j,\\quad |e_j(0)\\rangle =  |e_{j,0}\\rangle,\\quad \\langle f_j(0)|=\\langle f_{j,0}|\\quad (j=1,\\ldots,N)\n\\end{equation} \nsatisfying the constraints \n\\begin{equation} \n\\label{eq:imajt=0}\n\\mathrm{Im} (a_{j,0})<0 \\quad (j=1,\\ldots,N), \n\\end{equation} \n\\begin{equation} \n\\label{eq:fjej}\n\\langle f_{j,0}|e_{j,0}\\rangle =1 \\quad (j=1,\\ldots,N), \n\\end{equation} \nand \n\\begin{equation}\n\\label{eq:BTt=0} \n\\begin{split} \nv_j \\langle f_{j,0}| = &\\; 2\\mathrm{i}\\sum_{k\\neq j}^N  \\langle f_{j,0}|e_{k,0}\\rangle \\langle f_{k,0}|\\alpha(a_{j,0}-a_{k,0}) \\\\\n&  -2\\mathrm{i}\\sum_{k=1}^N  \\langle f_{j,0}|f_{k,0}\\rangle\\langle e_{k,0}|\\alpha(a_{j,0}-a^*_{k,0})\\quad (j=1,\\ldots,N). \n\\end{split} \n\\end{equation} \nThen,\n\\begin{equation} \n\\label{eq:ansatz} \n\\mathsf{U}(x,t)= \\mathrm{i}\\sum_{j=1}^N |e_j(t)\\rangle\\langle f_j(t)|\\alpha(x-a_j(t)) -  \\mathrm{i}\\sum_{j=1}^N |f_j(t)\\rangle\\langle e_j(t)|\\alpha(x-a^*_j(t))  \n\\end{equation} \nis a solution of the sBO equation \\eqref{eq:sBO} for all times $t$ provided that the following condition holds true,\n\\begin{equation} \n\\label{eq:imaj}\n\\mathrm{Im} (a_j(t))<0 \\quad (j=1,\\ldots,N).\n\\end{equation} \n\\end{theorem} \n\n\\begin{remark} \nWe expect that the condition in \\eqref{eq:imaj} is automatically fulfilled for all times $t\\in{\\mathbb R}$ under the stated assumptions, and thus can be dropped. \nIt would be interesting to prove, or falsify, this expectation. Similar remarks apply to Theorems~\\ref{thm:sncILW}, \\ref{thm:sBOgen} and \\ref{thm:sncILWgen}.\n\\end{remark} \n\n\\subsubsection{One-soliton solutions}\nFor $N=1$, the time evolution equations \\eqref{eq:sCM1a}--\\eqref{eq:sCM1b} simplify to $\\ddot a_1=0$, $|\\dot e_1\\rangle=0$, and $\\langle \\dot f_1|=0$. Moreover, the general solution of the constraints \\eqref{eq:imajt=0}--\\eqref{eq:BTt=0} is $|e_{1,0}\\rangle = |f_{1,0}\\rangle\/\\langle f_{1,0}|f_{1,0}\\rangle$ with $\\langle f_{1,0}|\\in\\mathcal{V}^*$ an arbitrary non-zero vector, together with\n\\begin{equation} \nv_1 = -2\\mathrm{i}\\alpha(a_{1,0}^{\\phantom *}-a_{1,0}^*) = \\begin{cases} -1\/\\aI_1 & \\text{(I)}\\\\ - (2\\pi\/L)\\coth(2\\pi\\aI_1\/L) & \\text{(II)}  \\end{cases} ,  \n\\end{equation} \nwhere $a_{1,0}=a^\\mathrm{R}_1+\\mathrm{i}\\aI_1$ with $a^\\mathrm{R}_1\\in{\\mathbb R}$ and $\\aI_1<0$. Thus, $a_1(t)=a^\\mathrm{R}_1+\\mathrm{i}\\aI_1+v_1 t$, $|e_1(t)\\rangle\\langle f_1(t)|=|f_{1,0}\\rangle\\langle f_{1,0}|\/\\langle f_{1,0}|f_{1,0}\\rangle$ independent of $t$,\nand Theorem~\\ref{thm:sBO} implies the following explicit formula for the one-soliton solutions of the sBO equation \\eqref{eq:sBO}, \n\\begin{equation} \n\\label{eq:onesoliton} \n\\mathsf{U}(x,t)= \\frac{|f_{1,0}\\rangle\\langle f_{1,0}|}{\\langle f_{1,0}|f_{1,0}\\rangle}\n\\big(\\mathrm{i}\\alpha(x-a^\\mathrm{R}_1-\\mathrm{i}\\aI_1 - v_1t) - \\mathrm{i}\\alpha(x-a^\\mathrm{R}_1+\\mathrm{i}\\aI_1 - v_1t) \\big) .\n\\end{equation} \nThus, a soliton of the sBO equation is characterized by $d$ complex parameters: the pole $a_{j,0}$ at time $t=0$ (one complex parameter), and a non-zero vector $\\langle f_{1,0}|\\in\\mathcal{V}^*$ modulo the transformations $\\langle f_{1,0}| \\to c\\langle f_{1,0}|$, $c\\in{\\mathbb C}\\setminus\\{0\\}$ arbitrary, which leave \\eqref{eq:onesoliton} invariant ($d-1$ complex parameters).\n It is interesting to note that only $v_1>0$ is allowed: the sBO equation is chiral in the sense that its solitons can only move to the right (note that the center of the soliton is located at the position $x=a^\\mathrm{R}_1+v_1t$).\n \n\\subsubsection{Non-interacting multi-soliton solutions}\nLet $N\\leq d$, and pick $N$ non-zero orthogonal vectors $\\langle f_{j,0}|\\in\\mathcal{V}^*$: \n\\begin{equation} \n\\langle f_{j,0}|f_{k,0}\\rangle=\\delta_{j,k}\\langle f_{j,0}|f_{j,0}\\rangle\\quad (j,k=1,\\ldots,N). \n\\end{equation} \nOne can check that $a_j(t)=a^\\mathrm{R}_j+\\mathrm{i}\\aI_j+v_j t$ and $|e_j(t)\\rangle\\langle f_j(t)|=|f_{j,0}\\rangle\\langle f_{j,0}|\/\\langle f_{j,0}|f_{j,0}\\rangle$\nis a solution of the time evolution equations \\eqref{eq:sCM1a}--\\eqref{eq:sCM1b} satisfying the constraints \\eqref{eq:BTt=0}--\\eqref{eq:imaj} provided that $a^\\mathrm{R}_j\\in{\\mathbb R}$, $\\aI_j<0$ and \n \\begin{equation} \nv_j = \\begin{cases} -1\/\\aI_j & \\text{(I)}\\\\ - (2\\pi\/L)\\coth(2\\pi\\aI_j\/L) & \\text{(II)}  \\end{cases} \n\\end{equation} \nfor $j=1,\\ldots,N$. Thus, by Theorem~\\ref{eq:sBO}, \n\\begin{equation} \n\\mathsf{U}(x,t)= \\sum_{j=1}^N \\frac{|f_{j,0}\\rangle\\langle f_{j,0}|}{\\langle f_{j,0}|f_{j,0}\\rangle}\n\\big(\\mathrm{i}\\alpha(x-a^\\mathrm{R}_j-\\mathrm{i}\\aI_j - v_jt) - \\mathrm{i}\\alpha(x-a^\\mathrm{R}_j+\\mathrm{i}\\aI_j - v_jt) \\big) \n\\end{equation} \nis an exact solution of the sBO equation \\eqref{eq:sBO}. Clearly, this solution describes $N$ non-interacting one-solitons. \n\n\\subsubsection{Generic multi-soliton solutions}\n\\label{sec:NsolitonssBO}\nThe initial data for the $N$-soliton in Theorem \\ref{thm:sBO} is specified in terms of the $2N$ complex numbers $a_{j,0}$ and $v_j$,\nas well as the $2N$ vectors $|e_{j,0}\\rangle \\in \\mathcal{V}$ and $\\langle f_{j,0}| \\in \\mathcal{V}^*$ ($j=1,\\ldots,N$). However, the constraints \\eqref{eq:fjej} and \\eqref{eq:BTt=0} imply that these parameters cannot all be independently specified. Moreover, some choices of these parameters give rise to the same soliton solution $\\mathsf{U}$. In what follows, we show that, in the generic case when certain matrices are invertible, the constraints \\eqref{eq:fjej} and \\eqref{eq:BTt=0} can be solved explicitly in terms of only linear operations. As a result, we will see that if $\\mathcal{V}$ is (complex) $d$-dimensional, then the family of $N$-solitons of the sBO equation generically depends on $Nd$ complex parameters. Moreover, we give a recipe for the construction of the $N$-soliton solution in terms of these $Nd$ parameters.\n\nFor each $j = 1, \\dots, N$, let us identify $|e_{j,0}\\rangle \\in \\mathcal{V}$ with the vector $\\mathbf{e}_j \\in {\\mathbb C}^d$ whose components $(\\mathbf{e}_j)_{\\mu}$, ${\\mu} = 1, \\dots, d$, are the components of $|e_{j,0}\\rangle$ with respect to some given basis of $\\mathcal{V}$.\nNext, let us identify the collection of $N$ vectors $\\mathbf{e}_j$, $j=1,\\ldots,N$, with the single vector $\\mathbf{e} \\in {\\mathbb C}^{Nd}$ whose components $\\mathbf{e}_{j,\\mu} := (\\mathbf{e}_j)_{\\mu}$ are indexed by $j=1,\\ldots,N$ and ${\\mu}=1,\\ldots,d$.\nSimilarly, let us identify the three collections of vectors $\\langle e_{j,0}|$, $\\langle f_{j,0}|$, and $|f_{j,0}\\rangle$, $j=1,\\ldots,N$, with the vectors $\\mathbf{e}^*=(e^*_{j,\\mu})\\in {\\mathbb C}^{Nd}$, $\\mathbf{f}^*=(f^*_{j,\\mu}) \\in {\\mathbb C}^{Nd}$, and $\\mathbf{f}=(f_{j,\\mu})\\in {\\mathbb C}^{Nd}$, respectively. (Since all considerations here will concern the constraints at time $t = 0$, we do not include the subscript $0$ in our notation for simplicity.)\nWith this notation, we can write the constraint \\eqref{eq:BTt=0} as \n\\begin{equation} \n\\label{eq:system}\n\\mathsf{A}\\mathbf{e}^*+ \\mathsf{B}\\mathbf{e}  = \\mathsf{C} \\mathbf{f}^*  \n\\end{equation} \nwhere the $Nd \\times Nd$ matrices $\\mathsf{A},\\mathsf{B}$, and $\\mathsf{C}$ are given by \n\\begin{equation} \n\\label{ABdvdef}\n\\begin{split} \nA_{j,\\mu;k,\\nu} &= -2\\mathrm{i}  \\langle f_j|f_k\\rangle\\delta_{\\mu,\\nu}\\alpha(a^{\\phantom*}_{j,0}-a_{k,0}^*),\\\\\nB_{j,\\mu;k,\\nu} &= 2\\mathrm{i} (1-\\delta_{j,k})f^*_{k,{\\mu}}f^*_{j,{\\nu}}\\alpha(a_{j,0}-a_{k,0}), \\\\\nC_{j,\\mu;k,\\nu} &= v_j\\delta_{j,k}\\delta_{\\mu,\\nu}. \n\\end{split} \n\\end{equation} \nWe write \\eqref{eq:system} and the negative of its complex conjugate as the linear system,\\footnote{Note that the star in $\\mathsf{A}^*$ etc.\\ means complex conjugation, i.e., $\\mathsf{A}^*$ is given by the matrix elements $(A_{j,\\mu;k,\\nu})^*$ where $A_{j,\\mu;k,\\nu}$ are the matrix elements of $\\mathsf{A}$.} \n\\begin{equation} \n\\label{eq:system1}\n \\begin{pmatrix} \\mathsf{A} & \\mathsf{B} \\\\ -\\mathsf{B}^* & -\\mathsf{A}^* \\end{pmatrix}\\begin{pmatrix}\n\\mathbf{e}^*\\\\ \\mathbf{e} \\end{pmatrix} = \\begin{pmatrix} \\mathsf{C}\\mathbf{f}^*\\\\ -\\mathsf{C}^* \\mathbf{f} \\end{pmatrix}, \n\\end{equation} \nand note that the $2Nd\\times 2Nd$ matrix in \\eqref{eq:system1} is hermitian. Thus, restricting ourselves to the generic case when this $2Nd\\times 2Nd$ matrix is invertible, we obtain\n\\begin{equation} \n\\label{eq:solve} \n\\begin{pmatrix}\n\\mathbf{e}^*\\\\ \\mathbf{e} \\end{pmatrix} \n= \\begin{pmatrix} \\mathsf{A} & \\mathsf{B} \\\\ -\\mathsf{B}^* & -\\mathsf{A}^* \\end{pmatrix}^{-1} \n\\begin{pmatrix} \\mathsf{C}\\mathbf{f}^*\\\\ - \\mathsf{C}^* \\mathbf{f} \\end{pmatrix}. \n\\end{equation} \nSubstitution of these expressions for $\\mathbf{e}$ and $\\mathbf{e}^*$ into the constraint \\eqref{eq:fjej} and its complex conjugate gives $2N$ linear equations which can be solved uniquely for the $2N$ initial velocities $v_j, v_j^* \\in {\\mathbb C}$, $j = 1, \\dots, N$, provided that the relevant determinant is nonzero (this is the generic case).\nGiven these expressions for $v_j$, the vectors $\\mathbf{e}$ and $\\mathbf{e}^*$ can be found from (\\ref{eq:solve}). This means that all the constraints can be solved in terms of $\\{a_{j,0}\\}_{j=1}^N$ and $\\mathbf{f}$.\nSo in the Hermitian case considered here, the class of $N$-soliton solutions of the sBO equation is parametrized by the $N$ complex parameters $\\{a_{j,0}\\}_{j=1}^N$ as well as $Nd$ further complex parameters needed to determine $\\mathbf{f}$. However, some of these parameter configurations yield the same $N$-soliton solution $\\mathsf{U}(x,t)$ via \\eqref{eq:ansatz}. Indeed, the equations \\eqref{eq:sCM1a}--\\eqref{eq:sCM1b} and \\eqref{eq:imajt=0}--\\eqref{eq:ansatz} are invariant under the replacements\n\\begin{equation} \n|e_j\\rangle \\to c_j |e_j\\rangle, \\qquad \\langle f_j | \\to \\frac{1}{c_j} \\langle f_j |,\n\\end{equation} \nwhere $\\{c_j\\}_{j=1}^N$ are nonzero complex constants. This means that the family of $N$-solitons generically is $Nd$ complex dimensional; in other words, each soliton is specified by $d$ complex parameters, which is consistent with the result for the one-soliton obtained above.\n\n\\subsection{Proof of Theorem~\\ref{thm:sBO}}\n\\label{sec:sBOproof}\nWe start with the spin-pole ansatz \n\\begin{equation} \n\\label{eq:ansatzh} \n\\mathsf{U}(x,t)=\\mathrm{i} \\sum_{j=1}^N \\mathsf{P}_j(t)\\alpha(x-a_j(t)) - \\mathrm{i} \\sum_{j=1}^N \\mathsf{P}^\\dag_j(t)\\alpha(x-a^*_j(t))  \n\\end{equation}  \nwhere $\\mathsf{P}_j=\\mathsf{P}_j(t)$ are $d\\times d$ matrices and $a_j=a_j(t)\\in{\\mathbb C}$; the $\\mathsf{P}_j$ and $a_j$ correspond to the spin degrees of freedom and poles, respectively. \n\n\\begin{proposition} \n\\label{prop:sBO1} \nThe function $\\mathsf{U}(x,t)$ in \\eqref{eq:ansatzh} satisfies the sBO equation \\eqref{eq:sBO} provided \\eqref{eq:imaj} and the following equations hold true,\n\\begin{equation} \n\\label{eq:BT1a} \n\\dot a_{j} \\mathsf{P}_{j} = 2\\mathrm{i}\\sum_{k\\neq j}^N  \\mathsf{P}_{j}\\mathsf{P}_{k}\\alpha(a_{j}-a_{k}) -2\\mathrm{i}\\sum_{k=1}^N \\mathsf{P}_{j}\\mathsf{P}^\\dag _{k}\\alpha(a_{j}-a^*_{k})\\quad (j=1,\\ldots,N), \\\\\n\\end{equation}\n\\begin{equation} \n\\label{eq:BT2a} \n\\dot \\mathsf{P}_j  = -2\\mathrm{i}\\sum_{k \\neq j}^N[\\mathsf{P}_j,\\mathsf{P}_k]V(a_j-a_k) \\quad (j=1,\\ldots,N),\n\\end{equation} \nand \n\\begin{equation} \n\\label{eq:BT3a} \n\\mathsf{P}_j^2=\\mathsf{P}_j  \\quad (j=1,\\ldots, N).\n\\end{equation} \n\\end{proposition} \n\n\\begin{proof} \nWe use the short-hand notation\n\\begin{equation} \n\\label{eq:shorthand} \n(a_j,\\mathsf{P}_j,r_j)\\coloneqq \\begin{cases} (a_j,\\mathsf{P}_j,+1) & (j=1,\\ldots,N) \\\\ (a^*_{j-N},\\mathsf{P}^\\dag_{j-N},-1) & (j=N+1,\\ldots,\\mathcal {N}) \\end{cases} , \\quad \\mathcal {N}=2N\n\\end{equation} \nto write \\eqref{eq:ansatzh} as  \n\\begin{equation} \n\\label{eq:U} \n\\mathsf{U}= \\mathrm{i}\\sum_{j=1}^{\\mathcal {N}} r_j \\mathsf{P}_j\\alpha(x-a_j) .\n\\end{equation} \nThe proof will follow by inserting \\eqref{eq:U} into the sBO equation \\eqref{eq:sBO} and performing long but straightforward computations.\n\nWe compute each term in \\eqref{eq:sBO}. We start with \n\\begin{equation} \n\\label{eq:Udot}\n\\mathsf{U}_t =  \\sum_{j=1}^\\mathcal {N} \\Big( \\mathrm{i} r_j \\dot \\mathsf{P}_j\\alpha(x-a_j) - \\mathrm{i} r_j \\mathsf{P}_j \\dot a_j \\alpha'(x-a_j)\\Big)  . \n\\end{equation} \nNext, we compute\n\\begin{align}\n\\label{eq:UUx1}\n\\{\\mathsf{U},\\mathsf{U}_x\\} =&\\; -\\sum_{j=1}^{\\mathcal {N}}\\sum_{k=1}^{\\mathcal {N}}  r_j r_k \\{\\mathsf{P}_j, \\mathsf{P}_k\\} \\alpha(x-a_j)\\alpha'(x-a_k)            \\nonumber  \\\\\n=&\\; -2\\sum_{j=1}^{\\mathcal {N}}  \\mathsf{P}_j^2\\alpha(x-a_j)\\alpha'(x-a_j)  -\\sum_{j=1}^{\\mathcal {N}}\\sum_{k\\neq j}^{\\mathcal {N}} r_jr_k \\{\\mathsf{P}_j,\\mathsf{P}_k\\} \\alpha(x-a_j)\\alpha'(x-a_k)   \\nonumber \\\\\n=&\\; \\sum_{j=1}^{\\mathcal {N}} \\mathsf{P}_j^2 \\alpha''(x-a_j) + \\sum_{j=1}^{\\mathcal {N}}\\sum_{k\\neq j}^{\\mathcal {N}} r_j r_k \\{\\mathsf{P}_j,\\mathsf{P}_k\\}\\alpha(a_j-a_k)\\alpha'(x-a_k) \\nonumber \\\\\n&\\; +\\sum_{j=1}^{\\mathcal {N}}\\sum_{k\\neq j}^{\\mathcal {N}}r_j r_k \\{\\mathsf{P}_j,\\mathsf{P}_k\\}V(a_j-a_k)\\big(\\alpha(x-a_j)-\\alpha(x-a_k)\\big),\n\\end{align}\nwhere we have used the identities \n\\begin{equation} \n\\label{eq:Id1} \n2\\alpha(x-a_j)\\alpha'(x-a_j)=-\\alpha''(x-a_j)\n\\end{equation} \nand\\footnote{We write the following identity in a seemingly strange way, mixing $\\alpha'$ and $V=-\\alpha'$, to emphasize the similarity with a corresponding identity \\eqref{eq:Aidentity2} used in the sncILW case.} \n\\begin{equation}\n\\label{eq:Id2}  \n\\alpha(x-a_j)\\alpha'(x-a_k)=- \\alpha(a_j-a_k)\\alpha'(x-a_k) -V(a_j-a_k)\\big(\\alpha(x-a_j)-\\alpha(x-a_k)\\big).\n\\end{equation} \nThe first identity \\eqref{eq:Id1} can be obtained by differentiating \\eqref{eq:IdV} with respect to $z$ and setting $z=x-a_j$ while the second identity \\eqref{eq:Id2} can be obtained by differentiating \\eqref{eq:Idmain} with respect to $c$ and setting $a=x$, $b=a_j$, and $c=a_k$.\n\nThe final sum in \\eqref{eq:UUx1} vanishes because the summand is antisymmetric under the interchange $j\\leftrightarrow k$. Hence,  after re-labelling summation indices  $j\\leftrightarrow k$ in the second sum in \\eqref{eq:UUx1} using \\eqref{eq:Id4}, we are left with\n\\begin{equation}\n\\label{eq:UUx2}\n\\{\\mathsf{U},\\mathsf{U}_x\\} = \\sum_{j=1}^{\\mathcal {N}} \\mathsf{P}_j^2 \\alpha''(x-a_j)-\\sum_{j=1}^{\\mathcal {N}}\\sum_{k\\neq j}^{\\mathcal {N}} r_jr_k \\{\\mathsf{P}_j,\\mathsf{P}_k\\}\\alpha(a_j-a_k)\\alpha'(x-a_j).\n\\end{equation}\n\nTo compute terms in \\eqref{eq:sBO} involving the Hilbert transform $H$, we use\n\\begin{equation}\n\\label{eq:Id3} \n\\begin{split}\n(H\\alpha'(\\cdot-a_j))(x) = \\mathrm{i} r_j\\alpha'(x-a_j) \\\\\n\\end{split} \n\\end{equation}\n(this follows from the well-known facts (i) $\\alpha(x-a)$ is an eigenfunction of $H$ with eigenvalue $+\\mathrm{i}$ for $\\mathrm{Im} (a)<0$ and  $-\\mathrm{i}$ for $\\mathrm{Im} (a)>0$ \\cite{chen1979} and (ii) $H$ commutes with differentiation \\cite[Chapter~4.8]{king2009}). Hence,\n\\begin{equation}\n\\label{eq:HUx}\nH\\mathsf{U}_x= -\\sum_{j=1}^{\\mathcal {N}} \\mathsf{P}_j \\alpha'(x-a_j),\\qquad H\\mathsf{U}_{xx}= -\\sum_{j=1}^{\\mathcal {N}} \\mathsf{P}_j \\alpha''(x-a_j),\n\\end{equation}\nwhere we have again used the fact that $H$ commutes with differentiation to derive the second equation from the first.\nFrom \\eqref{eq:U} and the first equation in \\eqref{eq:HUx}, we compute\n\\begin{equation}\n\\label{eq:UHUx1}\n\\begin{split} \n\\mathrm{i} [\\mathsf{U},H\\mathsf{U}_x]=&\\; \\sum_{j=1}^{\\mathcal {N}}\\sum_{k\\neq j}^{\\mathcal {N}} r_j [\\mathsf{P}_j,\\mathsf{P}_k] \\alpha(x-a_j)\\alpha'(x-a_k) \\\\\n=&\\; -\\sum_{j=1}^{\\mathcal {N}} \\sum_{k\\neq j}^{\\mathcal {N}} r_j[\\mathsf{P}_j,\\mathsf{P}_k] \\alpha(a_j-a_k)\\alpha'(x-a_k)\\\\\n & -\\sum_{j=1}^{\\mathcal {N}}\\sum_{k\\neq j}^{\\mathcal {N}} r_j[\\mathsf{P}_j,\\mathsf{P}_k] V(a_j-a_k)\\big(\\alpha(x-a_j)-\\alpha(x-a_k)\\big), \n\\end{split} \n\\end{equation}\ninserting \\eqref{eq:Id2} \nin the second step. We can rewrite the second sum as follows, \n\\begin{multline}\n\\sum_{j=1}^{\\mathcal {N}} \\sum_{k\\neq j}^{\\mathcal {N}} r_j[\\mathsf{P}_j,\\mathsf{P}_k] V(a_j-a_k)\\big(\\alpha(x-a_j)-\\alpha(x-a_k)\\big) \\\\\n=  \\frac12 \\sum_{j=1}^{\\mathcal {N}} \\sum_{k\\neq j}^{\\mathcal {N}} (r_j+r_k)[\\mathsf{P}_j,\\mathsf{P}_k] V(a_j-a_k)\\big(\\alpha(x-a_j)-\\alpha(x-a_k)\\big) \\\\\n=\\sum_{j=1}^{\\mathcal {N}} \\sum_{k\\neq j}^{\\mathcal {N}} (r_j+r_k)[\\mathsf{P}_j,\\mathsf{P}_k] V(a_j-a_k) \\alpha(x-a_j)\n\\end{multline}\nsince $V(z)$ is an even function.  \nAlso changing variables $j\\leftrightarrow k$ in the first sum in \\eqref{eq:UHUx1} using that $\\alpha(z)$ is odd, we arrive at\n\\begin{equation}\n\\label{eq:UHUx2}\n\\begin{split} \n\\mathrm{i} [\\mathsf{U},H\\mathsf{U}_x]=&\\; -\\sum_{j=1}^{\\mathcal {N}}\\sum_{k\\neq j}^{\\mathcal {N}} r_k[\\mathsf{P}_j,\\mathsf{P}_k] \\alpha(a_j-a_k)\\alpha'(x-a_j)\n\\\\ & -\\sum_{j=1}^{\\mathcal {N}}\\sum_{k\\neq j}^{\\mathcal {N}} (r_j+r_k)[\\mathsf{P}_j,\\mathsf{P}_k]V(a_j-a_k)\\alpha(x-a_j).\n\\end{split} \n\\end{equation}\nInserting \\eqref{eq:Udot}, \\eqref{eq:UUx2}, the second equation in \\eqref{eq:HUx}, and \\eqref{eq:UHUx2} into \\eqref{eq:sBO} gives\n\\begin{equation} \n\\begin{split}\n0=&\\; \\sum_{j=1}^{\\mathcal {N}} \\Bigg(\\mathrm{i} r_j \\dot{\\mathsf{P}}_j- \\sum_{k\\neq j}^{\\mathcal {N}} (r_j+r_k)[\\mathsf{P}_j,\\mathsf{P}_k]V(a_j-a_k)\\Bigg) \\alpha(x-a_j) \\\\\n&\\; +\\sum_{j=1}^{\\mathcal {N}} \\Bigg(-\\mathrm{i} r_j \\mathsf{P}_j\\dot{a}_j -\\sum_{k\\neq j}^{\\mathcal {N}} r_k \\big( r_j\\{\\mathsf{P}_j,\\mathsf{P}_k\\}    +[\\mathsf{P}_j,\\mathsf{P}_k]  \\big)\\alpha(a_j-a_k)\\Bigg)\\alpha'(x-a_j) \\\\\n&\\;  + \\sum_{j=1}^{\\mathcal {N}} \\big( \\mathsf{P}_j^2 -\\mathsf{P}_j \\big) \\alpha''(x-a_j).\n\\end{split}\n\\end{equation} \nThus, the function $\\mathsf{U}$ defined in \\eqref{eq:U} satisfies the sBO equation \\eqref{eq:sBO} if and only if the following conditions are fulfilled,  \n\\begin{equation} \n\\label{eq:BT1} \n\\mathsf{P}_j\\dot a_j =  \\mathrm{i} \\sum_{k\\neq j}^{\\mathcal {N}} r_k \\bigl(  \\{\\mathsf{P}_j,\\mathsf{P}_k\\} + r_j[\\mathsf{P}_j,\\mathsf{P}_k]  \\big) \\alpha(a_j-a_k), \\\\\n\\end{equation} \n\\begin{equation} \n\\label{eq:BT2} \n\\dot \\mathsf{P}_j =  -\\mathrm{i} \\sum_{k\\neq j}^{\\mathcal {N}}(1+r_jr_k)[\\mathsf{P}_j,\\mathsf{P}_k]V(a_j-a_k) ,\\\\\n\\end{equation} \n\\begin{equation} \n\\label{eq:BT3} \n\\mathsf{P}_j^2 = \\mathsf{P}_j \n\\end{equation} \nfor $j=1,\\ldots,\\mathcal {N}$. Recalling \\eqref{eq:shorthand}, one can check that \\eqref{eq:BT1}, \\eqref{eq:BT2}, and \\eqref{eq:BT3} are equivalent to \\eqref{eq:BT1a}, \\eqref{eq:BT2a} and \\eqref{eq:BT3a}, respectively. \n\\end{proof} \n\nWe now make the ansatz \n\\begin{equation}\n\\label{eq:mPj} \n\\mathsf{P}_j= |e_j\\rangle\\langle f_j| \\quad (j=1,\\ldots,N) \n\\end{equation} \nwith vectors $|e_j\\rangle\\in\\mathcal{V}$ and $\\langle f_j|\\in\\mathcal{V}^*$. \nThen the function $\\mathsf{U}(x,t)$ defined in \\eqref{eq:ansatzh} satisfies the equations in Proposition~\\ref{prop:sBO1} whenever $\\{a_j,|e_j\\rangle,\\langle f_j|\\}_{j=1}^N$ satisfy the equations defining the B\\\"acklund transformations of the sCM system discussed in Section~\\ref{sec:sCM_BT}. The precise statement is as follows.\n\n\\begin{lemma}\n\\label{lem:sBOlemma}\nSuppose that $\\{a_j,|e_j\\rangle,\\langle f_j|\\}_{j=1}^N$ satisfy the equations \\eqref{eq:sCM1b}, \\eqref{eq:sCM1c} and \\eqref{eq:BThermitian}, and $\\mathsf{P}_j$ is given by \\eqref{eq:mPj}. \nThen $\\mathsf{U}$ \\eqref{eq:ansatzh} satisfies \\eqref{eq:BT1a}--\\eqref{eq:BT3a}.  \n\\end{lemma} \n\n\\begin{proof} \nBy multiplying \\eqref{eq:BThermitian} from the left by $|e_j\\rangle$, one gets \n\\begin{equation*}\n\\dot a_j |e_j\\rangle\\langle f_j| = 2\\mathrm{i}\\sum_{k\\neq j}^N  |e_j\\rangle\\langle f_j|e_k\\rangle \\langle f_k|\\alpha(a_j-a_k) -2\\mathrm{i}\\sum_{k=1}^M  |e_j\\rangle\\langle f_j|f_k\\rangle\\langle e_k|\\alpha(a_j-a_k^*)\n\\end{equation*} \nwhich is \\eqref{eq:BT1a} (note that $\\mathsf{P}_j^\\dag = |f_j\\rangle\\langle e_j|$). \n\nUsing \\eqref{eq:sCM1b}, we compute \n\\begin{equation} \n\\begin{split} \n\\dot\\mathsf{P}_j = & |e_j\\rangle\\langle \\dot f_j|+|\\dot e_j\\rangle\\langle f_j|  \\\\\n=  & -2\\mathrm{i}\\sum_{k\\neq j}^N \\big(  |e_j\\rangle\\langle f_j|e_k\\rangle \\langle f_k| -  |e_k\\rangle\\langle f_k|e_j\\rangle\\langle f_j|  \\big) \nV(a_j-a_k) \\\\\n= &  -2\\mathrm{i}\\sum_{k\\neq j}^N \\big( \\mathsf{P}_j\\mathsf{P}_k-\\mathsf{P}_k\\mathsf{P}_j \\big) V(a_j-a_k) , \n\\end{split} \n\\end{equation}    \nwhich is \\eqref{eq:BT2a}.\n\nFinally, using \\eqref{eq:sCM1c}, \n\\begin{equation} \n\\mathsf{P}_j^2 = |e_j\\rangle\\langle f_j|e_j\\rangle\\langle f_j| =  |e_j\\rangle\\langle f_j|=\\mathsf{P}_j, \n\\end{equation}  \nwhich is \\eqref{eq:BT3a}. \n\\end{proof} \n\nThen, the theorem is implied by Proposition~\\ref{prop:BT}.\n\n\\section{Multi-soliton solutions of the sncILW equation}\n\\label{sec:sncILW}\nIn this section, we present results for the sncILW equation on the real line in analogy with those for the sBO equation in the previous section. \nWe use the hyperbolic case (III) special functions $\\alpha(z)=(\\pi\/2\\delta)\\coth(\\pi z\/2\\delta)$ and \n$V(z)=(\\pi\/2\\delta)^2\/\\sinh^2(\\pi z\/2\\delta)$ with $\\delta>0$; see \\eqref{eq:alpha} and \\eqref{eq:V}.\n\n\\subsection{Result}\nWe use the same conventions as described in Section~\\ref{sec:sBOresult}. In the case $d=1$, \\eqref{eq:sncILW} reduces to the standard ncILW equation, whose soliton solutions were constructed in \\cite{berntson2020}. Our main result about the sncILW equation, stated below, generalizes the construction in \\cite{berntson2020}.\n\n\\begin{theorem} \n\\label{thm:sncILW} \nLet $\\{a_j(t),|e_j(t)\\rangle,\\langle f_j(t)|\\}_{j=1}^N$ be a solution of the time evolution equations \\eqref{eq:sCM1a}--\\eqref{eq:sCM1b} with initial conditions \n\\begin{equation} \na_j(0)=a_{j,0},\\quad \\dot a_j(0)=v_j,\\quad |e_j(0)\\rangle =  |e_{j,0}\\rangle,\\quad \\langle f_j(0)|=\\langle f_{j,0}|\\quad (j=1,\\ldots,N)\n\\end{equation} \nsatisfying the constraints \n\\begin{equation} \n\\label{eq:imajtsncILW=0}\n-\\frac{3\\delta}{2}<\\mathrm{Im} (a_{j,0})< -\\frac{\\delta}{2}   \\quad (j=1,\\ldots,N), \n\\end{equation} \n\\begin{equation} \n\\label{eq:fjejsncILW}\n\\langle f_{j,0}|e_{j,0}\\rangle =1 \\quad (j=1,\\ldots,N), \n\\end{equation} \nand \n\\begin{equation}\n\\label{eq:BTt=0_sncILW} \n\\begin{split} \nv_j \\langle f_{j,0}| = & 2\\mathrm{i}\\sum_{k\\neq j}^N  \\langle f_{j,0}|e_{k,0}\\rangle \\langle f_{k,0}|\\alpha(a_{j,0}-a_{k,0}) \\\\\n&  -2\\mathrm{i}\\sum_{k=1}^N  \\langle f_{j,0}|f_{k,0}\\rangle\\langle e_{k,0}|\\alpha(a_{j,0}-a^*_{k,0}+\\mathrm{i}\\delta)\\quad (j=1,\\ldots,N). \n\\end{split} \n\\end{equation} \nThen,\n\\begin{equation} \n\\label{eq:mUmV} \n\\begin{split}\n\\mathsf{U}(x,t)=&\\;  \\mathrm{i}\\sum_{j=1}^N |e_j(t)\\rangle\\langle f_j(t)|\\alpha(x-a_j(t) -\\mathrm{i}\\delta\/2) -  \\mathrm{i}\\sum_{j=1}^N |f_j(t)\\rangle\\langle e_j(t)|\\alpha(x-a^*_j(t)+\\mathrm{i}\\delta\/2),  \\\\\n\\mathsf{V}(x,t)=&\\;  -\\mathrm{i}\\sum_{j=1}^N |e_j(t)\\rangle\\langle f_j(t)|\\alpha(x-a_j(t)+\\mathrm{i}\\delta\/2) +  \\mathrm{i}\\sum_{j=1}^N |f_j(t)\\rangle\\langle e_j(t)|\\alpha(x-a^*_j(t)-\\mathrm{i}\\delta\/2) \n\\end{split}\n\\end{equation} \nis a solution of the sncILW equation \\eqref{eq:sncILW} for all times $t$ provided that the following condition holds true,\n\\begin{equation} \n\\label{eq:imaj_sncILW}\n-\\frac{3\\delta}{2}<\\mathrm{Im} ( a_j(t)) <-\\frac{\\delta}{2} \\quad (j=1,\\ldots,N).\n\\end{equation} \n\\end{theorem} \n\n\\subsection{Proof of Theorem~\\ref{thm:sncILW}}\nSince details of the proof are very similar to the ones of Theorem~\\ref{thm:sBO}, we only explain the key differences. \n\nThe proof is facilitated by introducing the notation\n\\begin{equation}\n\\label{eq:circ} \n\\left(\\begin{array}{c} F_1 \\\\ F_2  \\end{array}\\right)\\circ \\left(\\begin{array}{c} G_1 \\\\ G_2 \\end{array}\\right)\\coloneqq  \\left(\\begin{array}{c}F_1G_1 \\\\ -F_2G_2 \\end{array}\\right)\n\\end{equation}\nfor ${\\mathbb C}$-valued functions $F_j$, $G_j$ ($j=1,2$), and the operator\n\\begin{equation}\n\\label{eq:cT} \n\\mathcal{T}\\coloneqq \\left(\\begin{array}{cc} T & \\tilde{T} \\\\ -\\tilde{T} & -T     \\end{array}\\right)\n\\end{equation}\nto be interpreted as linear operator acting on vector-valued functions, see \\cite{berntson2020}. \nIn the present paper, we use the product $\\circ$ defined in  \\eqref{eq:circ} \nalso for vectors $\\mathcal{F}$, $\\mathcal{G}$ whose components $F_j$, $G_j$ are complex $d\\times d$ matrices, and we let\n\\begin{equation}\n\\{\\mathcal{F}\\,\\overset{\\circ}{,}\\, \\mathcal{G}\\}\\coloneqq \\mathcal{F}\\circ\\mathcal{G}+\\mathcal{G}\\circ \\mathcal{F},\\qquad [\\mathcal{F}\\,\\overset{\\circ}{,}\\, \\mathcal{G}]\\coloneqq \\mathcal{F}\\circ\\mathcal{G}- \\mathcal{G}\\circ \\mathcal{F} \n\\end{equation}\nbe the corresponding generalizations of the commutator and anticommutator, respectively. \n\nUsing this notation, the system \\eqref{eq:sncILW} can be written in terms of the vector\n\\begin{equation}\n\\label{eq:cU} \n\\mathcal{U}(x,t)\\coloneqq \\left(\\begin{array}{c}  \\mathsf{U}(x,t) \\\\ \\mathsf{V}(x,t) \\end{array}\\right)\n\\end{equation}\nas\n\\begin{equation}\n\\label{eq:sncILW2}\n\\mathcal{U}_t+ \\{\\mathcal{U}\\,\\overset{\\circ}{,}\\, \\mathcal{U}_x\\} +\\mathcal{T}\\mathcal{U}_{xx}+\\mathrm{i} [\\mathcal{U}\\,\\overset{\\circ}{,}\\, \\mathcal{T}\\mathcal{U}_x]=0.\n\\end{equation}\nTo solve \\eqref{eq:sncILW2}, we use the spin-pole ansatz\n\\begin{equation}\n\\label{eq:ansatzh_sncILW}\n\\mathcal{U}(x,t)=\\mathrm{i} \\sum_{j=1}^{N} \\mathsf{P}_j(t) \\mathcal{A}_+(x-a_j(t))-\\mathrm{i}\\sum_{j=1}^N \\mathsf{P}_j^{\\dag}(t)\\mathcal{A}_-(x-a_j^*(t)),\n\\end{equation}\nwhere $\\mathsf{P}_j=\\mathsf{P}_j(t)$ are $d\\times d$ matrices, $a_j=a_j(t)\\in{\\mathbb C}$, and\n\\begin{equation}\n\\label{eq:cA}\n\\mathcal{A}_{\\pm}(z)\\coloneqq \\left(\\begin{array}{c} +\\alpha(z \\mp \\mathrm{i}\\delta\/2) \\\\ -\\alpha(z\\pm\\mathrm{i}\\delta\/2) \\end{array}\\right)\n\\end{equation}\nfor all $z\\in{\\mathbb C}$ such that $\\delta\/2<|\\mathrm{Im} (z)|<3\\delta\/2$. \nUsing the shorthand notation \\eqref{eq:shorthand}, we can write \\eqref{eq:ansatzh_sncILW} as\n\\begin{equation}\n\\label{eq:ansatz_sncILW}\n\\mathcal{U}=\\mathrm{i}\\sum_{j=1}^{\\mathcal {N}} r_j \\mathsf{P}_j \\mathcal{A}_{r_j}(x-a_j).\n\\end{equation}\nWith this notation in place, one can prove Theorem~\\ref{thm:sncILW} by the very same computations we presented in our proof of Theorem~\\ref{thm:sBO} in Section~\\ref{sec:sBOproof}. The reason for this is that the vector-valued function $\\mathcal{A}(z)$ defined in \\eqref{eq:cA} satisfies the following identities which are natural generalizations of the identities \\eqref{eq:Id1}, \\eqref{eq:Id2} and \\eqref{eq:Id3}, respectively ($j,k=1,\\ldots,\\mathcal {N}$ and $x\\in{\\mathbb R}$):  \n\\begin{equation}\n\\label{eq:Aidentity1}\n2\\mathcal{A}_{r_j}(x-a_j)\\circ \\mathcal{A}_{r_j}'(x-a_j)=- \\mathcal{A}_{r_j}''(x-a_j) \n\\end{equation}\n(this is implied by \\eqref{eq:Id1} and definitions),\n\\begin{align}\n\\label{eq:Aidentity2}\n\\mathcal{A}_{r_j}(x-a_j) & \\circ \\mathcal{A}_{r_k}'(x-a_k)= -\\alpha(a_j-a_k+\\mathrm{i}(r_j-r_k)\\delta\/2) \\mathcal{A}_{r_k}'(x-a_k) \\nonumber \\\\\n&\\; -V(a_j-a_k+\\mathrm{i}(r_j-r_k)\\delta\/2)\\big(\\mathcal{A}_{r_j}(x-a_j)-\\mathcal{A}_{r_k}(x-a_k)\\big) \n\\end{align}\n(this is implied by \\eqref{eq:Id2} and definitions, together with $\\alpha'(z)=-V(z)$ and the fact that $\\alpha(z)$ and $V(z)$ both are $2\\mathrm{i}\\delta$-periodic), \nand\n\\begin{equation} \n\\label{eq:Aidentity3}\n(\\mathcal{T}\\mathcal{A}_{r_j}'(\\cdot -a_j))(x)=\\mathrm{i} r_j\\mathcal{A}_{r_j}'(x-a_j) \n\\end{equation} \n(this follows from the result, proved in \\cite{berntson2020}, that for the vector-valued functions $\\mathcal{A}_\\pm(z)$ in \\eqref{eq:cA}, $\\mathcal{A}_+'(x-a)$  is an eigenfunction of the matrix operator $\\mathcal{T}$ with eigenvalue $+\\mathrm{i}$ provided that $-3\\delta\/2 < \\mathrm{Im} (a) < -\\delta\/2$, and $\\mathcal{A}_-'(x-a)$ is an eigenfunction of $\\mathcal{T}$ with eigenvalue $-\\mathrm{i}$ provided that $\\delta\/2 < \\mathrm{Im} (a) < 3\\delta\/2$; see \\cite[Appendix A.b]{berntson2020}).\n\nThe interested reader can find full details on how to prove Theorem~\\ref{thm:sncILW} in Appendix~\\ref{app:sncILWsolutions}. There, we prove a more general result for solutions that are not necessarily hermitian. The hermitian case of Theorem~\\ref{thm:sncILW} can be established by specializing to \\eqref{eq:ansatzh} at all points in the proof. \n\n\\section{Further results} \n\\label{sec:results}\nWe briefly summarize further basic results about the sBO and sncILW equations. \n\n\\subsection{Hamiltonian structure}\n\\label{sec:Hamiltonian}\nThe sBO equation \\eqref{eq:sBO} can be obtained as a Hamiltonian equation $\\mathsf{U}_t=\\{\\cH_{\\rm sBO},\\mathsf{U}\\}_{\\mathrm{P.B.}}$ from the Hamiltonian ($\\mathsf{U}$ below is short for $\\mathsf{U}(x)$)\n\\begin{equation} \n\\cH_{\\rm sBO} = \\int \\mathrm{tr}\\left( \\frac13 \\mathsf{U}^3 + \\frac12 \\mathsf{U} H\\mathsf{U}_x \\right)\\,\\mathrm{d}{x} \n\\end{equation} \nand the Poisson brackets\\footnote{The subscript $\\mathrm{P.B.}$ is used to avoid confusion with the anti-commutator.} \n\\begin{equation} \n\\{ U_{{\\mu},{\\nu}}(x), U_{{\\mu}',{\\nu}'}(x')\\}_{\\mathrm{P.B.}} = \\mathrm{i}\\delta(x-x')\\big(\\delta_{{\\nu},{\\mu}'}U_{{\\mu},{\\nu}'}(x)-\\delta_{{\\mu},{\\nu}'}U_{{\\mu}',{\\nu}}(x)\\big) + \\delta'(x-x')\\delta_{{\\nu},{\\mu}'}\\delta_{{\\mu},{\\nu}'}  \n\\end{equation} \nwhere $U_{{\\mu},{\\nu}}(x)$ are the matrix elements of $\\mathsf{U}(x)$ and ${\\mu},{\\nu},{\\mu}',{\\nu}'=1,\\ldots,d$; \nthe integration is over ${\\mathbb R}$ and $[-L\/2,L\/2]$ in the real-line and periodic cases, respectively, and $\\mathrm{tr}$ is the usual matrix trace (sum of diagonal elements). \n\nSimilarly, the sncILW equation \\eqref{eq:sncILW} arises from the Hamiltonian \n\\begin{equation} \n\\cH_{\\rm sncILW} = \\int \\mathrm{tr}\\left( \\frac13 \\mathsf{U}^3 +\\frac13\\mathsf{V}^3 + \\frac12 \\mathsf{U} T\\mathsf{U}_x +  \\frac12 \\mathsf{V} T\\mathsf{V}_x + \\frac12 \\mathsf{V} \\tilde{T}\\mathsf{U}_x + \\frac12 \\mathsf{U} \\tilde{T}\\mathsf{V}_x \\right)\\,\\mathrm{d}{x} \n\\end{equation} \nand the Poisson brackets \n\\begin{equation} \n\\begin{split} \n\\{ U_{{\\mu},{\\nu}}(x), U_{{\\mu}',{\\nu}'}(x')\\}_{\\mathrm{P.B.}} &= \\mathrm{i}\\delta(x-x')\\big(\\delta_{{\\nu},{\\mu}'}U_{{\\mu},{\\nu}'}(x)-\\delta_{{\\mu},{\\nu}'}U_{{\\mu}',{\\nu}}(x)\\big) + \\delta'(x-x')\\delta_{{\\nu},{\\mu}'}\\delta_{{\\mu},{\\nu}'}  ,\\\\\n\\{ V_{{\\mu},{\\nu}}(x), V_{{\\mu}',{\\nu}'}(x')\\}_{\\mathrm{P.B.}} &= \\mathrm{i}\\delta(x-x')\\big(\\delta_{{\\nu},{\\mu}'}V_{{\\mu},{\\nu}'}(x)-\\delta_{{\\mu},{\\nu}'}V_{{\\mu}',{\\nu}}(x)\\big) - \\delta'(x-x')\\delta_{{\\nu},{\\mu}'}\\delta_{{\\mu},{\\nu}'},\\\\\n\\{ U_{{\\mu},{\\nu}}(x), V_{{\\mu}',{\\nu}'}(x')\\}_{\\mathrm{P.B.}} &= 0\n\\end{split} \n\\end{equation} \nin the real-line and periodic cases. \n\n(The verifications of these results are by straightforward computations which we omit.) \n\n\\subsection{Local limits of the sILW equation}\n\\label{sec:locallimit}\nWe give details on how the matrix KdV equation \\eqref{eq:mKdV} and the (generalization of the) HF equation \\eqref{eq:sHF} are obtained as limits $\\delta\\downarrow 0$ from the sILW equation \\eqref{eq:sILW}; note that, while the latter equation is non-local, the former equations both are local.\n\nFor simplicity, we only consider the real-line case with $T$ given by \\eqref{eq:TT}. \nWe recall that\n\\begin{equation} \n\\label{eq:Texpand}\n(Tf_x)(x) = -\\frac1\\delta f(x)+\\frac{\\delta}3 f_{xx}(x) + O(\\delta^3)  \n\\end{equation} \nas $\\delta\\downarrow 0$ uniformly for $x\\in{\\mathbb R}$ \\cite[Appendix~A]{scoufis2005}. \nWe scale \n\\begin{equation} \n\\label{eq:scaling1} \n\\mathsf{U}(x,t)\\to \\frac{\\delta}{3}\\mathsf{U}\\left(x,\\frac{\\delta}{3}t\\right) \n\\end{equation} \nin \\eqref{eq:sILW} (and change variables accordingly) to obtain\n\\begin{equation} \n\\mathsf{U}_t + \\{\\mathsf{U},\\mathsf{U}_x\\} +\\frac{3}{\\delta^2}\\mathsf{U}_x + \\frac{3}{\\delta}T\\mathsf{U}_{xx} + \\mathrm{i}[\\mathsf{U},T\\mathsf{U}_x]=0. \n\\end{equation} \nExpanding this in powers of $\\delta$ using \\eqref{eq:Texpand} yields \n\\begin{equation} \n\\mathsf{U}_t+\\{\\mathsf{U},\\mathsf{U}_x\\}+\\mathsf{U}_{xxx} +\\frac{\\delta}3\\mathrm{i}[\\mathsf{U},\\mathsf{U}_x] + O(\\delta^2)=0, \n\\end{equation} \nwhich converges to the matrix KdV equation \\eqref{eq:mKdV} in the limit $\\delta\\downarrow 0$. \n\nSimilarly, by scaling \n\\begin{equation} \n\\label{eq:scaling2} \n\\mathsf{U}(x,t)\\to \\frac1\\delta \\mathsf{U}\\left(x,\\frac{t}{3} \\right), \n\\end{equation} \n\\eqref{eq:sILW} is transformed to\n\\begin{equation} \n\\mathsf{U}_t +\\frac{3}{\\delta}\\{\\mathsf{U},\\mathsf{U}_x\\} + \\frac{3}{\\delta}\\mathsf{U}_x + 3T\\mathsf{U}_{xx}+\\frac{3}{\\delta}\\mathrm{i}[\\mathsf{U},T\\mathsf{U}_x]=0. \n\\end{equation} \nBy inserting this expansion in \\eqref{eq:Texpand}, we obtain\n \\begin{equation} \n \\mathsf{U}_t + \\frac{3}{\\delta}\\{\\mathsf{U},\\mathsf{U}_x\\}+ \\delta \\mathsf{U}_{xxx}+\\mathrm{i}[\\mathsf{U},\\mathsf{U}_{xx}]+O(\\delta^2)=0, \n \\end{equation}  \nyielding the  generalized HF equation \\eqref{eq:sHF} in the limit $\\delta\\downarrow 0$ provided $\\{\\mathsf{U},\\mathsf{U}_x\\}=0$; the latter is achieved by imposing the condition $\\mathsf{U}^2=I$ (note that this condition is preserved under the time evolution \\eqref{eq:sHF}). \n\n\n\\subsection{Spin generalization of the bidirectional Benjamin-Ono equation}\n\\label{sec:2sBO} \nThe bidirectional BO equation was introduced by Abanov, Bettelheim, and Wiegmann \\cite{abanov2009} as a powerful tool to derive a hydrodynamic description of the $A$-type  trigonometric CM system (to mention just one of several interesting applications). In this section, we present a spin generalization of this equation. \n\n\nLet $\\alpha(z)$ \\eqref{eq:alpha} and $V(z)$ \\eqref{eq:V} in II (trigonometric case).  \nWe assume that the variables $a_j\\in{\\mathbb C}$, $|e_j\\rangle\\in\\mathcal{V}$, $\\langle f_j|\\in\\mathcal{V}^*$ for $j=1,\\ldots,N$ and $b_j\\in{\\mathbb C}$, $|g_j\\rangle\\in\\mathcal{V}$, $\\langle h_j|\\in\\mathcal{V}^*$ for $j=1,\\ldots,N$ satisfying the first order equations  \\eqref{eq:sCM1b}, \\eqref{eq:sCM2b} and \\eqref{eq:BT}, together with the constraints \\eqref{eq:sCM1c} and  \\eqref{eq:sCM2c}; note that these are exactly the equations defining the B\\\"acklund transformation of the $A$-type sCM system in the trigonometric case (see Proposition~\\ref{prop:BT}). Using that, we define the functions\n\\begin{equation} \n\\mathsf{U}_0(z,t)\\coloneqq  -\\mathrm{i}\\sum_{j=1}^M |g_j(t)\\rangle\\langle h_j(t)|\\alpha(z-b_j(t)), \\quad \\mathsf{U}_1(z,t)\\coloneqq  \\mathrm{i}\\sum_{j=1}^N |e_j(t)\\rangle\\langle f_j(t)|\\alpha(z-a_j(t)) \n\\end{equation} \nof the position variable $z\\in{\\mathbb C}$ and time $t\\in{\\mathbb R}$, and \n\\begin{equation} \n\\mathsf{U}\\coloneqq \\mathsf{U}_0+\\mathsf{U}_1,\\quad \\tilde{\\mathsf{U}}\\coloneqq \\mathsf{U}_0-\\mathsf{U}_1. \n\\end{equation} \nOne then can verify, by arguments similar to the ones we use to prove Propositions~\\ref{prop:sBO1gen}, \nthat the following equation is satisfied, \n\\begin{equation} \n\\label{eq:bsBO}\n\\mathsf{U}_t + \\{\\mathsf{U},\\mathsf{U}_z\\} +\\mathrm{i}\\tilde{\\mathsf{U}}_{zz}-[\\mathsf{U},\\tilde{\\mathsf{U}}_z]=0. \n\\end{equation} \nThis is the spin generalization of the bidirectional BO equation. It would be interesting to generalize the arguments in  \\cite{abanov2009} to derive a hydrodynamic description of the sCM systems from \\eqref{eq:bsBO}. \n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzcypn b/data_all_eng_slimpj/shuffled/split2/finalzzcypn
new file mode 100644
index 0000000000000000000000000000000000000000..696c277b5409202a9436fc45db7d428971eeb4ab
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzcypn
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\nAt the extremely high temperatures reached in heavy-ion collisions, a phase-transition occurs from ordinary nuclear matter to a QGP state in which quarks and gluons are not confined into hadrons. The quark formation time during the collision is proportional to the inverse of the quark mass \\cite{MassDep}. Therefore, heavy quarks are generated early during the collision and can experience the full evolution of the medium \\cite{qgptime}. The quarks lose energy while moving through the medium by collisional and radiative processes. This energy loss is expected to depend on the path length, the QGP density,the parton colour charge (Casimir factor), and the quark mass (dead-cone effect) \\cite{deadcone,charmbeauty}. Because of this, the following energy loss hierarchy is expected: $\\Delta E_\\mathrm{loss}$(g) $>$ $\\Delta E_\\mathrm{loss}$(u,d) $>$ $\\Delta E_\\mathrm{loss}$(c) $>$ $\\Delta E_\\mathrm{loss}$(b). \n\nThe nuclear modification factor ($R_\\mathrm{AA}$) quantifies the medium effects that affect the heavy quarks when they traverse the medium. This factor, defined as $$R_\\mathrm{AA} = \\frac{1}{\\langle N^\\mathrm{AA}_{coll}\\rangle} \\frac{\\mathrm{d}N^\\mathrm{AA} \/ \\mathrm{d}p_\\mathrm{T}}{\\mathrm{d}N^\\mathrm{pp} \/ \\mathrm{d}p_\\mathrm{T}},$$ is obtained from the ratio of the transverse-momentum-differential yields measured in Pb\u2014Pb and pp collisions. The scaling factor $\\langle N^\\mathrm{AA}_{coll}\\rangle$ represents the average number of binary nucleon-nucleon collisions in Pb--Pb collisions for a given centrality interval. If heavy quarks do not lose energy in the medium $R_\\mathrm{AA} = 1$, while it drops below unity if they do. Heavy quarks are also expected to be affected by the collective motion of the medium. This gives rise to an anisotropic flow usually described by the components of a Fourier expansion of the azimuthal distribution of the outgoing particles. The second coefficient of this expansion is called elliptic flow ($v_2$). \n\n\\begin{figure}[t!]\n\\begin{minipage}[b]{6cm}\n\\centerline{\n\\includegraphics[width=\\textwidth]{1.pdf}}\n\\end{minipage}\n\\begin{minipage}[b]{6cm}\n\\centerline{\n\\includegraphics[width=\\textwidth]{2.pdf}}\n\\end{minipage}\n\\caption{Left: $R_\\mathrm{AA}$ of non-strange D mesons in central Pb--Pb collisions compared with theoretical calculations. Right: Ratio of $R_\\mathrm{AA}$ of non-prompt D$^0$ mesons over the $R_\\mathrm{AA}$ of prompt D$^0$ mesons. The data is compared with models with different energy loss for charm and beauty. Copyright CERN, reused with permission.}\n\\label{Fig:1}\n\\end{figure}\n\n\\section{Open heavy flavour}\nThe left panel in Fig. \\ref{Fig:1} shows a comparison of the $R_\\mathrm{AA}$ of non-strange D-mesons in central Pb--Pb collisions with theoretical calculations. The low momentum reach in central collisions allows setting stringent constraints on energy-loss models for central Pb--Pb collisions. Models without shadowing, like the BAMPS model \\cite{bamps}, overestimate the $R_\\mathrm{AA}$ spectrum at low $p_\\mathrm{T}$. \n\nThe models can be tested more rigorously by requiring a description of multiple observables, like $R_\\mathrm{AA}$ and $v_2$, at the same time, over a wide momentum range, and in different centrality intervals \\cite{dmesonRAA, dmesonFlow}. This shows that accurate modeling of data requires a combination of collisional and radiative energy loss, hadronization via coalescence, cold-nuclear-matter effects, and a realistic description of the medium evolution. \n\nThe right panel shows the ratio of the $R_\\mathrm{AA}$ of non-prompt D$^0$-mesons over the $R_\\mathrm{AA}$ for prompt D$^0$-mesons. Prompt D$^0$-mesons, which come directly from the charm quarks produced in the initial collision, and non-prompt D$^0$-mesons, which are produced later by the decay of beauty hadrons, show a different $R_\\mathrm{AA}$ at intermediate $p_\\mathrm{T}$. Models with different energy loss for charm and beauty can describe within uncertainties  the ratio of non-prompt over prompt D$^0$-meson $R_\\mathrm{AA}$. This is an indication that energy loss depends on the quark mass. \n\n\\begin{figure}[t!]\n\\begin{minipage}[b]{6cm}\n\\centerline{\n\\includegraphics[width=\\textwidth]{3.pdf}}\n\\end{minipage}\n\\begin{minipage}[b]{6cm}\n\\centerline{\n\\includegraphics[width=\\textwidth]{4.pdf}}\n\\end{minipage}\n\\caption{Left: $R_\\mathrm{AA}$ in central Pb--Pb collisions for multiple types of particle species. Right: $\\Lambda_c^+$  \/ D$^0$ ratio as a function of multiplicity for several $p_\\mathrm{T}$ intervals.  Copyright CERN, reused with permission.}\n\\label{Fig:2}\n\\end{figure}\n\nThe left panel in Fig. \\ref{Fig:2} shows the $R_\\mathrm{AA}$ for different particle species with a hierarchy that is consistent with the expected difference in energy loss for charm versus light-flavour and gluons. Strange D-mesons and $\\Lambda_{c}$ baryons show a hint of lower suppression, compared to non-strange D-mesons, that may point at recombination effects. \"Models that include hadronization via coalescence reproduce D$_\\mathrm{S}$ data within uncertainties. \n\nThe right panel in Fig. \\ref{Fig:2} shows the $\\Lambda_c^+$ \/ D$^0$ ratio as a function of multiplicity in pp, p--Pb, and Pb--Pb collisions for several $p_\\mathrm{T}$ intervals. This ratio shows an enhancement at low $p_\\mathrm{T}$ compared to e$^{+}$e$^{-}$ collider measurements in which $\\Lambda_c^+$ \/ D$^0$ $\\approx 0.1$ \\cite{epref}. The multiplicity dependence of the $\\Lambda_c^+$  \/ D$^0$ ratio shows that the enhancement remains higher than electron-positron collider measurements even for low-multiplicity pp collisions, suggesting that charm-quark recombination with quarks from the surrounding hadronic environment may already occur in small systems.\n\n\n\\begin{figure}[t!]\n\\begin{minipage}[b]{6cm}\n\\centerline{\n\\includegraphics[width=\\textwidth]{5.pdf}}\n\\end{minipage}\n\\begin{minipage}[b]{6cm}\n\\centerline{\n\\includegraphics[width=\\textwidth]{6.pdf}}\n\\end{minipage}\n\\caption{Left: $R_\\mathrm{AA}$ as a function of multiplicity for inclusive J\/$\\psi$ in two rapidity intervals. Right: $R_\\mathrm{AA}$ as a function of $\\langle N_\\mathrm{part} \\rangle$ for two $\\Upsilon$ states along with model predictions. Copyright CERN, reused with permission.}\n\\label{Fig:3}\n\\end{figure}\n\n\n\\section{Quarkonium}\nAt high temperatures colour screening in the QGP results in the suppression of quarkonium production \\cite{quarkonium}. Different quarkonium states have different binding energies, which results in the expectation of a sequential melting of states when colliding nuclei at higher energies \\cite{melting}. On the other hand, the c$\\bar{\\mathrm{c}}$ multiplicity increases at higher collision energies. This leads to the expectation of an enhancement of quarkonia production via recombination at hadronization.\n\nThe left panel of Fig. \\ref{Fig:3} shows the $R_\\mathrm{AA}$ as a function of multiplicity for inclusive J\/$\\psi$-mesons in two rapidity intervals. This $R_\\mathrm{AA}$ measurement has a significantly improved precision and $p_\\mathrm{T}$ reach compared to previous measurements \\cite{improved}. At higher multiplicities the $R_\\mathrm{AA}$ at midrapidity is higher than at forward rapidity. This observation may suggest that recombination effects are stronger at midrapidity, where the charm-quark density is higher.\n\nThe centrality dependence of the $R_\\mathrm{AA}$ is shown in the right panel of Fig. \\ref{Fig:3}. The data show a slight bottomonium centrality dependence and match well with the model predictions \\cite{du}. A stronger suppression of $\\Upsilon$(2S) than $\\Upsilon$(1S) is observed.\n\nFor J\/$\\psi$-mesons, measurements show a positive $v_2$ in a large $p_\\mathrm{T}$ range at forward rapidity. This is illustrated in the left panel of Fig. \\ref{Fig:4}. The bottomonium $v_2$ is consistent with zero, however more data are needed for a conclusive interpretation on the difference between J\/$\\psi$ and bottomonium $v_2$\n\n\\begin{figure}[t!]\n\\begin{minipage}[b]{6cm}\n\\centerline{\n\\includegraphics[width=\\textwidth]{7.pdf}}\n\\end{minipage}\n\\begin{minipage}[b]{6cm}\n\\centerline{\n\\includegraphics[width=\\textwidth]{8.pdf}}\n\\end{minipage}\n\\caption{Left: $v_2$ as a function of $p_\\mathrm{T}$ for inclusive J\/$\\psi$. Right: $\\Upsilon$(1S) $v_2$ as a function of $p_\\mathrm{T}$ compared with inclusive J\/$\\psi$ $v_2$ and different models \\cite{Yflow}. Copyright CERN, reused with permission.}\n\\label{Fig:4}\n\\end{figure}\n\n\n\\section{Heavy-flavour jets}\nJets originate from hard parton-parton interactions. In ALICE heavy-flavour tagged jets are measured down to low jet $p_\\mathrm{T}$ (5 GeV\/$c$). The study of jets provides experimental data for gluon-to-hadron fragmentation functions and gluon PDF at low $x$. The study of jet quenching provides additional information to further characterise parton energy loss in the QGP.\n\nFig. \\ref{Fig:5} shows the first measurement of the $\\Lambda_c^+$ probability density distribution of the parallel jet momentum fraction (z$_{||}^{ch}$) compared to data. The Pythia 8 SoftQCD model has the best agreement with data.\n\n Jets with beauty hadrons were reconstructed exploiting the displaced impact parameter of b-hadron decay tracks to the primary vertex. The observed yields are consistent with POWHEG. The nuclear modification factor in p--Pb ($R_\\mathrm{pPb}$) for B-tagged jets is shown in the right panel of Fig. \\ref{Fig:5}. No cold-nuclear-matter effects are observed within uncertainties using B-tagged jets.\n\n\\begin{figure}[t!]\n\\begin{minipage}[b]{6cm}\n\\centerline{\n\\includegraphics[width=\\textwidth]{9.pdf}}\n\\end{minipage}\n\\hspace{0.1cm}\n\\begin{minipage}[b]{6cm}\n\\centerline{\n\\includegraphics[width=\\textwidth]{10.pdf}}\n\\end{minipage}\n\\caption{Left: probability density distribution of the parallel jet momentum fraction (z$_{||}^{ch}$) for $\\Lambda_c^+$-tagged jets compared to expectations from Monte Carlo generators. Right: $R_\\mathrm{pPb}$ for B-tagged jets with a comparison of measurements by ALICE and CMS \\cite{CMSbjet}. Copyright CERN, reused with permission.}\n\\label{Fig:5}\n\\end{figure}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{sec:intro}\n\nThe cosmic microwave background (CMB) anisotropies observed by the\nWilkinson Microwave Anisotropy Probe (WMAP) have revealed that the\nprimordial density fluctuations are almost adiabatic and scale invariant\n\\cite{Komatsu:2008hk,Dunkley:2008ie}.  Inflationary cosmology\n\\cite{inflation} is currently the favored scenario to explain such\nprimordial fluctuations. According to it, density perturbations are\nproduced as quantum vacuum fluctuations on sub-Hubble scales and then\nstretched to super-Hubble scales during the phase of accelerated\nexpansion of space. However, inflationary cosmology is not without its\nproblems (see e.g.  \\cite{RHBrev}),\\footnote{One of the problems in the\nusually considered mechanism, where quantum fluctuations are the origin\nof the present cosmic fluctuations, is the transition from quantum\nquantities to classical observables. However, in the mechanism proposed\nin this paper, such a problem does not exist because they are classical\nfluctuations from the beginning.} and thus it is important to study\nscenarios alternative to inflation. In the 1980s, topological defect\nmodels such as those based on cosmic strings were investigated intensely\nas a possible alternative to generate primordial density fluctuations\n\\cite{topologicaldefects}. However, the fluctuations induced by defects\nin an expanding universe are isocurvature and, even if they might mimic\nthe inflationary predictions for the temperature-temperature (TT)\ncorrelation of the CMB \\cite{Turok:1996wa}, observations of the\nanti-correlation of the temperature and the E-mode polarization (TE),\nprecisely measured by WMAP, confirmed that such fluctuations could not\nbe the dominant source of CMB anisotropies\n\\cite{Peiris:2003ff,Komatsu:2008hk}. Thus, causal scaling seed models\nare ruled out as a main component of primordial density fluctuations.\n\nOn the other hand, in recent years other types of scenarios alternative\nto inflation motivated by developments in string theory have been\nproposed. Examples are the Pre-Big-Bang model (PBB)\n\\cite{Gasperini:1992em} and the Cyclic\/Ekpyrotic scenario\n\\cite{Khoury:2001wf,Steinhardt:2001st}.\\footnote{Another model is string\ngas cosmology \\cite{BV,NBV}, in which even the dimensionality of the\nobservable universe may be determined dynamically.}  The common feature\nof these models is that the universe begins in a contracting phase\nbefore emerging into the expanding phase of Standard Big Bang cosmology\nafter a cosmological bounce. In the contracting phase, comoving scales\nexit the Hubble radius unless the contraction is too rapid. Previous\nstudies have considered quantum mechanical vacuum fluctuations of a\nscalar matter field evaluated when the matter scales exit the Hubble\nradius during the contracting phase. Of course, once they are produced\nin the contracting phase, the fluctuations must be coupled to\nfluctuations in the expanding phase after the bounce. The propagation of\nfluctuations through the bounce phase depends on the details of the\nbounce \\cite{Durrer}.  There are models which yield an almost scale\ninvariant spectrum (see e.g. \\cite{Tolley}) after the bounce.\n\nIn this paper, we suggest the possibility that primordial density\nfluctuations are produced by causal seeds such as cosmic strings in the\ncontracting phase, and show that they could generate adiabatic, almost\nscale invariant, and super-Hubble curvature fluctuations in the\nexpanding universe.\\footnote{Another attempt to produce adiabatic\nfluctuations from cosmic strings is discussed in the context of\ntwo-metric theories of gravity \\cite{Avelino:2000iy}.} One simple\npossibility to realize cosmic strings in the contracting universe is to\nembed our model into a so-called cyclic universe, in which cosmic\nstrings are formed in the usual way during a phase transition in the\nexpanding phase, if the matter Lagrangian admits cosmic strings. In our\nscenario, different from the cyclic scenario of Steinhardt and Turok\n\\cite{Steinhardt:2001st} where quantum fluctuations source the\nprimordial density fluctuations, here cosmic strings seed the\nperturbations. Of course, in the cyclic scenario, topological defects\nmay be dangerous because they may dominate the energy density of the\nuniverse, as pointed out by Avelino et al. in\nRefs.~\\cite{Avelino:2002hx,Avelino:2002xy}. However, Avelino et al. also\ngive a solution to that problem: they point out that a relatively long\nperiod of cosmic acceleration at low energies (late period of one cycle)\ncan dilute topological defects in order that they do not overdominate\nthe universe. A second possibility is to consider the birth of the\nuniverse in the contracting universe (not cyclic). If the universe has\nfinite birth time, the correlated region at the birth of the universe\ndoes not necessarily cover the whole volume of the universe. Then, the\nrandomness of the values of the underling field beyond the correlation\nlength leads to the formation of topological defects.\n\nDensity fluctuations produced by causal seed models naturally become\nsuper-Hubble in the contracting phase. More specifically, the key point\nis that the defect-seeded perturbations which are initially isocurvature\nin nature seed a growing adiabatic mode. At the time when the symmetry\n(whose breaking yields the topological defects) is restored, the seed\nterm disappears and the fluctuations become frozen-in super-Hubble scale\nadiabatic perturbations. As long as no dominant isocurvature\nfluctuations are produced in the expanding phase, the fluctuations in\nthe expanding phase will be adiabatic and thus able to explain the TE\nanti-correlation observed in the CMB. In the following we consider\ncosmic strings as a concrete example and investigate the nature of the\ndensity fluctuations in detail.\n\n\\section{Adiabatic fluctuations from cosmic strings in a contracting\nuniverse}\n\nThe evolution of cosmic strings in a contracting universe was\ninvestigated in Refs. \\cite{Avelino:2002hx,Avelino:2002xy}. As in these\nreferences, we will assume that the distribution of strings on\nsuper-Hubble scale is like a random walk. We make the simplest\nassumption that the universe is initially matter and then radiation\ndominated in the contracting phase. In this context, it was shown that\ncosmic strings obey the scaling solution asymptotically both in the\nradiation and matter dominated epochs.  Specifically, the correlation\nlength $L$ is proportional to $a^{2} \\ln a \\propto (-t) \\ln (-t)$ in the\nradiation era ($a$ is the scale factor and $t(<0)$ is cosmic time, with\nthe bounce time taken as $t=0$). If we take the string loop chopping\nefficiency $\\tilde{c}$ to be a non-zero constant, then the ratio of the\nenergy density in cosmic strings to the total one stays almost\nconstant.\\footnote{There remains the logarithmic dependence in the\nradiation dominated era, which can lead to a small deviation from scale\ninvariance of the final curvature perturbations.}  Therefore, the\ndensity fluctuations produced by cosmic strings are almost scale\ninvariant at least initially at the Hubble radius exit.\n\nOn super-Hubble scales, the dynamics of the defect network\nsets up isocurvature fluctuations which in turn act as a\nseed for growing curvature perturbations.\nAs the universe contracts, the temperature of radiation\nincreases, and eventually leads to symmetry \nrestoration and the disappearance of the\ntopological defects. Thus, there will no longer\nbe any isocurvature fluctuations, and the source on\nsuper-Hubble scales in the differential equation for the\nadiabatic mode vanishes. Hence, the fluctuations become\nfrozen in as adiabatic ones.\n\nOn super-Hubble scales, the equation for the evolution of the curvature\nperturbation on uniform total density hypersurfaces, $\\zeta$, is given\nby\\footnote{\n  Although the density fluctuations of cosmic strings can be large and\n  of the order of unity, the linear perturbation theory applies\n  because the metric perturbations remain small as long as the energy\n  density of the cosmic string is subdominant.\n}  \\cite{Wands:2000dp}\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n \\dot{\\zeta} = - \\frac{H}{\\rho+P} \\delta P_{\\rm nad},\n \\label{eq:zetaevo}\n\\eeq\nwhere $H \\equiv \\dot{a}\/a$ is the Hubble parameter, $\\rho$ and $P$ are\nthe total energy density and pressure, respectively. A dot represents\na derivative with respect to the cosmic time.  The non-adiabatic pressure\nperturbation $\\delta P_{\\rm nad}$ is defined as $\\delta P_{\\rm nad}\n\\equiv \\delta P - c_s^2 \\delta \\rho$ with $\\delta \\rho$ being the\ntotal density fluctuation and $c_s^2 = \\dot{P}\/\\dot{\\rho}$ being the\nadiabatic sound speed. In the multi-fluid case, the total\nnon-adiabatic pressure perturbation consists of two parts. The first\npart comes from intrinsic entropy perturbations which vanish for a\nbarotropic fluid.  Therefore, the intrinsic entropy perturbations of\nmatter and radiation vanish. On the other hand, it is non-trivial that\nthe intrinsic isocurvature perturbation for cosmic strings also\nvanishes as strings can be modeled as a simple fluid with equation of state \n$w_{st} \\equiv P_{st} \/ \\rho_{st} = - 1\/3$. However, we expect that \nthe intrinsic entropy perturbation of cosmic strings is negligible and\nwill assume this in the following.  The second part $\\delta P_{\\rm rel}$ \ncomes from the relative entropy perturbation between different fluids\n$S_{\\alpha\\beta}$ \\cite{Malik:2002jb},\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  \\delta P_{\\rm rel} \\equiv \\frac{1}{6H\\dot{\\rho}} \\sum_{\\alpha,\\beta}\n     \\dot{\\rho}_{\\alpha} \\dot{\\rho}_{\\beta} (c_{\\alpha}^2-c_{\\beta}^2)  \n     S_{\\alpha\\beta},\n  \\label{eq:relnonad}\n\\eeq\nwhere the relative entropy perturbation between different fluids\n$S_{\\alpha\\beta}$ is given by\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  S_{\\alpha\\beta} = - 3H \n           \\left(}  \\newcommand{\\rmk}{\\right)  \\frac{\\delta\\rho_{\\alpha}}{\\dot{\\rho_{\\alpha}}}\n               - \\frac{\\delta\\rho_{\\beta}}{\\dot{\\rho_{\\beta}}} \\rmk,\n  \\label{eq:relentropy}\n\\eeq  \nand the adiabatic sound speed for each component, $c_\\alpha^2$, is\ngiven by $\\dot{P_\\alpha}\/\\dot{\\rho_\\alpha}$ with $\\rho_\\alpha$ and\n$P_\\alpha$ being the energy density and pressure of the component.\n\nNow, let us estimate the amplitude of the curvature perturbation $\\zeta$\nfor a comoving scale $k$. First, we consider a scale $k$ which exits the\nHubble radius during a radiation dominated era. We neglect the curvature\nperturbations which are generated when the corresponding scale is\nsub-Hubble.\\footnote{This can be justified by assuming that string loop\ndistribution is subdominant and the initial value of the curvature\nperturbation at $t \\rightarrow - \\infty$ vanishes. In fact, we expect\nthat the string loop distribution will be subdominant in a contracting\nuniverse compared to an expanding universe since comoving scales are\nexiting rather than entering the Hubble radius. Loops exit the Hubble\nradius before they collapse through emission of gravitational\nwaves. This effect reduces the loop chopping efficiency $\\tilde{c}$, and\nhence the number of produced loops is smaller in a contracting phase.}\nThus, we just follow the evolution of the curvature perturbation from\nthe epoch $t_H(k)$ when a comoving scale $k$ exits the Hubble radius\nuntil the time $t_{cs}$ when the symmetry is restored and the strings\ndisappear.\n\nThe relative entropy perturbation between\nradiation and cosmic strings is \n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  S_{rs} \\simeq - 3H \\frac{\\delta \\rho_{\\rm st}}{\\dot{\\rho}_{\\rm st}},\n  \\label{eq:entropyrs}\n\\eeq\nwhere we have used the fact that \n$\\dot{\\rho}_{\\rm st}\/\\dot{\\rho}_{\\rm rad}$ \nis almost constant and much smaller than unity, and \n$|\\delta \\rho_{\\rm rad}|$ is at most comparable to \n$|\\delta \\rho_{\\rm st}|$. From the scaling solution, it follows that \ncosmic strings can be modeled as a random walk on scales\nlarger than the Hubble radius with step length comparable to\nthe Hubble radius. Then, the density fluctuations of cosmic strings\nfor a super-Hubble scale can be easily estimated as\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  \\delta \\rho_{st}(k) \\simeq N |H| \\mu k_{\\rm phys},\n  \\label{eq:deltarhosr}\n\\eeq\nwhere $\\mu$ is the mass per unit length of a string, $N={\\cal O}} \\newcommand{\\CP}{{\\cal P}(1)$ is\nthe number of long strings crossing any given Hubble volume, and $k_{\\rm\nphys} \\equiv k\/a$. Notice that $N$ can be different in radiation and\nmatter dominated epochs although the difference is expected to be at\nmost ${\\cal O}} \\newcommand{\\CP}{{\\cal P}(1)$.  Inserting these equations to Eq. (\\ref{eq:zetaevo})\nyields\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n \\dot{\\zeta} \\simeq N G \\mu k_{\\rm phys}.\n\\eeq\n\nAs stated above, cosmic strings disappear at the time\n$t_{\\rm cs}$ due to symmetry restoration. Once\ncosmic strings disappear, the curvature perturbation is conserved at\nleast until the bounce. Thus, the final curvature perturbation before\nthe bounce is estimated as\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  \\zeta = \\int_{t_H(k)}^{t_{\\rm cs}} dt \\dot{\\zeta}\n        \\sim  N_{\\rm r} G \\mu k (-t_H(k))^{\\frac 12} \n        \\sim N_{\\rm r} G \\mu.\n\\eeq\nHere we have made use of $k (-t_H(k))^{\\frac 12} = 1\/2$ (in the\nradiation era), and $N_{\\rm r}$ is the value of $N$ (in the\nradiation epoch) which is ${\\cal O}} \\newcommand{\\CP}{{\\cal P}(1)$. Thus,\nthe curvature perturbations are independent of the comoving scale $k$ and\nhence scale invariant at least before the bounce. In the same way, the\ncurvature perturbation for comoving scales whose physical scales exit\nthe Hubble radius during the matter era is estimated as\n\\begin{equation}}  \\newcommand{\\eeq}{\\end{equation}\n  \\zeta = \\int_{t_H(k)}^{t_{\\rm eq}} dt \\dot{\\zeta} +\n          \\int_{t_{\\rm eq}}^{t_{\\rm cs}} dt \\dot{\\zeta}\n        \\sim  N_{\\rm m} G \\mu k (-t_H(k))^{\\frac 13}   \n        \\sim  N_{\\rm m} G \\mu,\n\\eeq\nwhere $t_{\\rm eq}$ is the matter-radiation equality time in the\ncontracting phase, we have made use of $k (-t_H(k))^{\\frac 13}\n= 2\/3$ (in the matter era), and $N_{\\rm m}$ is the value of $N$ in\nthe matter epoch.  Therefore, the curvature perturbation is\nscale invariant at least before the bounce also on these scales.  \n\nAccording to the often-used Hwang-Vishniac \\cite{HV} (Deruelle-Mukhanov\n\\cite{DM}) matching conditions for fluctuations across a space-like\nhypersurface, the curvature perturbation is conserved across the\nbounce. If we apply these matching condition, we conclude that the final\ncurvature perturbations in the expanding phase are almost scale\ninvariant and hence could be responsible for the present density\nfluctuations if $G \\mu \\sim 10^{-5}$. As emphasized in \\cite{Durrer},\nthere are problems with blindly applying these matching\nconditions. Subsequent studies have shown that the actual transfer of\nthe fluctuations depends quite sensitively on the details of the\nbounce. There are cases where the curvature perturbation is conserved\n(see e.g. \\cite{Tsujikawa:2002qc,Copeland:2006tn,Bozza}), but there are\nother examples where this does not hold\n\\cite{Hassan,Tirtho,Cai}. However, if the bounce time is short compared\nto the time scale of the fluctuations of interest, it can be rather\nrigorously shown that the spectrum of $\\zeta$ is maintained through the\nbounce. This can be shown \\cite{Cai,Cai2} by modeling the background\ncosmology with three phases: the initial contracting radiation phase,\nthe ``bounce phase'' during which $H = \\alpha t$, where $\\alpha$ is some\nconstant, and the expanding radiation phase. The matching conditions at\nthe two hypersurfaces between these phases can be consistently applied\n(since the background also satisfies the matching conditions, unlike\nwhat happens in the single matching between contracting and expanding\nphase which has been applied in the case of the singular Ekpyrotic\nbounce).\nHowever, we would like to point out that, even if the\ncurvature perturbation is {\\it not} conserved through the bounce, the\nscale invariance of the final curvature perturbation still holds true as\nlong as its change in the amplitude of the fluctuations across the\nbounce is independent of the comoving scale. This can be reasonably\nexpected for modes we are interested in because their momenta are much\nsmaller than the maximal value of $|H|$ around the bounce point\n(assuming that the bounce is smooth), the only energy scale which can be\nset by the bounce.\\footnote{\n  Even in the case when the change through the bounce depends on the\n  comoving scale, a scale invariant spectrum may be realized by\n  considering the time varying tension of cosmic strings discussed in\n  Refs. \\cite{Yamaguchi:2005gp,Ichikawa:2006rw,Takahashi:2006yc}, which\n  compensates for the variation of the curvature perturbation.\n}\n\nFinally, we comment on some subtleties. First of all, cosmic strings may\nbe formed again in the expanding phase. In this case, cosmic strings\nagain produce isocurvature fluctuations in the expanding phase, which\nshould be suppressed to less than $10\\%$ of the total curvature\nperturbations \\cite{Pogosian:2003mz,Endo:2003fr}. Such a suppression may\nbe realized in the case when the constant $N$ for cosmic strings in the\ncontracting phase is much larger than that in the expanding phase. The\nfact that the loop chopping efficiency $\\tilde{c}$ is smaller implies\nthat the constant $N$ might be larger. Therefore, it is plausible that\nthe constant $N$ for cosmic strings in the contracting phase is larger\nthan that in the expanding phase. Another possibility is that the\ncurvature perturbation sourced by fluctuations of cosmic strings in a\ncontracting phase is amplified at the bounce. Another solution is to\nsimply assume that cosmic string are not produced in the expanding phase\nbecause the symmetry breaking patterns of scalar fields are not\nnecessarily the same in the contracting and the expanding phases.\n\nAnother issue is that cosmic strings generate not only density\nperturbations but also vector and tensor perturbations (gravitational\nwaves). Gravitational waves are produced by oscillations of loops as\nwell as long strings. As explained before, the radius of a loop could\nbecome larger than the Hubble before there has been a significant amount\nof gravitational radiation. Therefore, we expect that the relative\namplitude of gravitational waves to scalar metric fluctuations will be\nsmaller in the contracting phase than the standard scenario of cosmic\nstrings in an expanding universe.  Regarding the vector mode, it has\nbeen shown that vector perturbations exhibit growing mode solutions in\nthe contracting phase \\cite{Battefeld:2004cd}. In particular, the metric\nperturbations always grow while the matter perturbations stay constant\nin the radiation dominated era. However, the vector perturbations will\ndecrease in the expanding phase. If vector fluctuations are suppressed\n(even slightly) across the bounce (or the scalar fluctuations enhanced),\nthen the vector modes will be sufficiently small today not to destroy\nthe successful agreement with the CMB angular power spectrum. Small but\nnot negligible vector fields could, in fact, be useful for generating\nthe observed large scale magnetic fields, as pointed out in\nRef. \\cite{Battefeld:2004cd}. As a final remark, we mention possible\nnon-Gaussianities in the scenario. Since the distribution of cosmic\nstring is highly non-Gaussian, the produced density fluctuations may\ngive large non-Gaussianity, though the bispectrum for a simulated string\nmodel in the expanding universe is shown not to be so large for the\ndiagonal contribution \\cite{Gangui:2001fr}. All these topics are worth\ninvestigating.\n\n\\section{Summary}\n\nWe have shown that adiabatic, super-Hubble, and almost scale invariant\ndensity fluctuations can be produced by cosmic strings in a contracting\nuniverse. Although cosmic strings can only generate isocurvature\nfluctuations in an expanding universe, they can produce adiabatic\nfluctuations by considering them in a contracting universe because\ncosmic strings disappear due to symmetry restoration. Our findings open\nthe possibility that topological defects could be resurrected as the\nmain source of the current cosmic density fluctuations.\n\n\n\\ack\nM.Y is grateful to M. Kawasaki for useful\n  discussions. T.T. and M.Y would like to thank R. H. Brandenberger for\n  kind hospitality at McGill University where this work was finished.\n  This work is supported in part by a Canadian NSERC Discovery Grant and\n  by the Canada Research Chair program (R.B.), and by the Sumitomo\n  Foundation (T.T.), and by Grant-in-Aid for Scientific Research from\n  the Ministry of Education, Science, Sports, and Culture of Japan\n  No.\\,19740145 (T.T.), No.\\,18740157, and No.\\,19340054 (M.Y.).\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nAs artificial intelligence (AI) technologies are playing key roles in our daily lives, developing intelligent systems which can work with humans more effectively (instead of replacing them) is becoming a central research theme \\cite{9153877,peng2022investigations,russell2021human}.  Such theme is mostly pronounced as \\textsl{hybrid intelligence}, aiming to benefit from the strengths of both humans and the machine intelligence in solving problems. Developing  systems of such capability demands fundamentally novel approaches to major research problems in AI: state-of-the-art systems outperform humans in many cognitive tasks from  playing video games \\cite{hester_deep_2017} to pattern recognition \\cite{liu2019comparison}, however they fall short when it comes to other tasks such as common sense reasoning, performing causal discovery, and behavioural human capabilities such as explaining its own decisions, adapting to different environments, collaborating with others, etc. A particular challenge in developing such systems relies on making them more interpretable \\cite{9153877,tjoa2020survey,TIDDI2022103627} which is the main focus of this paper. \t\t\n\nAn obvious medium in making such systems interpretable relies on employing an existing knowledge representation  formalism which is inherently tailored towards expressing human knowledge. One such type of human knowledge that is relevant in problem solving is captured by the notion of \\textit{interrogative agenda} (also called research agenda \\cite{enqvist2012modelling}) of an epistemic agent (which will be explained further in detail in Section ~\\ref{prelim: imterrogative agenda}). Intuitively, given a context, an interrogative agenda abstracts a set of features that an epistemic agent is interested in. In order to express interrogative agendas we employ the knowledge representation formalism of \\textit{formal concept analysis}. \n\nFormal concept analysis (FCA) is an influential foundational theory in knowledge representation and reasoning \\cite{priss2006formal, qadi2010formal, poelmans2010formal, valtchev2004formal, poelmans2013formal, ganter2012formal, wille1996formal} which provides a framework for categorizing objects w.r.t.~a given set of features.\nThe set of features used in the categorization (formal context in FCA) can be identified as its agenda, and different agendas will correspond to different categorizations. The agenda used to categorize a set of objects may be chosen on several factors like the availability and precision of the data, the categorization methodology, and the purpose of the categorization.\\footnote{A logical framework for studying these different categorizations obtained from different agendas and their interaction was developed in our earlier work \\cite{FLexiblecat2022} and applied to auditing domain.} In this paper, we focus on obtaining concept lattices (possibly fuzzy) corresponding to different agendas (possibly non-crisp)\nHowever, in many  applications, it is unclear which interrogative agenda (Sec.~\\ref{prelim: imterrogative agenda}) is best suited to obtain a categorization that can be useful in dealing with a given problem. Thus, in this work, we focus on the task of using a machine learning algorithm to learn such agendas, and hence a ``good categorization'' for the problem at hand. In particular, we will address the task of classification and outlier detection.\n\nIn the realm of machine learning, formal concept analysis has been used in the past for classification, outlier detection, rare concept mining and identification of rare patterns (Sec.~\\ref{Sec:Classification and outlier detection using concept lattice}). However, to the best of our knowledge, all these methods use a single concept lattice  (or its sublattice) to deal with the problems mentioned above. That is, the agenda of the categorization is fixed beforehand. The main difficulty in using such techniques relies on the fact that there are exponentially many subsets of features (and weights) one has to take into account. \nOn the other hand, since some features may not be relevant for a given classification task, removing them can reduce the data collection cost, its complexity, and may even improve the accuracy for some tasks. However, determining the set of relevant features can be difficult, and it is an important part of the preprocessing phase for many such algorithms. \n\nIn this paper, we propose a meta-learning algorithm to identify the best-suited agenda (and hence categorization). That is, to estimate the significance of different sets of features for the given task. The incorporation of such outer-loop on top of an existing classification or outlier detection algorithm can potentially increase its generalising power and the performance. Another major advantage of such method is that the learned agendas provide us an estimation of the importance of different sets of features for the given task, making our results more explainable. \n\n\\paragraph{Structure of paper.} In Section  \\ref{sec:preliminaries}, we provide the relevant preliminaries. In Section \\ref{Sec:Classification and outlier detection using concept lattice}, we give an overview of FCA-based classification and outlier detection algorithms. In Section \\ref{sec:Learning interrogative agendas}, we describe the framework for learning agendas and provide a generic learning algorithm. In Section \\ref{sec:Conclusion}, we conclude and give some directions for future research. \n\\section{Preliminaries}\\label{sec:preliminaries}\n\\subsection{Formal concept analysis}\nA {\\em formal context} \\cite{ganter2012formal}  is a structure $\\mathbb{P} = (A, X, I)$ such that $A$ and $X$ are sets of {\\em objects} and {\\em features}, respectively, and $I\\subseteq A\\times X$ is the so-called {\\em incidence relation} which records whether a given object has a given feature. That is, for any object $a$ and feature $x$, $a I x$ iff $a$ has feature $x$. \nFormal contexts can be thought of as abstract representations of e.g., databases, tabular data and such.\nEvery formal context as above induces maps $I^{(1)}: \\mathcal{P}(A)\\to \\mathcal{P}(X)$ and $I^{(0)}: \\mathcal{P}(X)\\to \\mathcal{P}(A)$, respectively defined by the assignments \n\\begin{equation}\n  I^{(1)}[B]: = \n\\{x\\in X\\mid \\forall a(a\\in B\\Rightarrow aIx)\\},\\quad \n I^{(0)}[Y] = \n\\{a\\in A\\mid \\forall x(x\\in Y\\Rightarrow aIx)\\}.\n\\end{equation}\nA {\\em formal concept} of $\\mathbb{P}$ is a pair \n$c = (\\val{c}, \\descr{c})$ such that $\\val{c}\\subseteq A$, $\\descr{c}\\subseteq X$, and $I^{(1)}[\\val{c}] = \\descr{c}$ and $I^{(0)}[\\descr{c}] = \\val{c}$. \nA subset $B \\subseteq A$ (resp.\\ $Y\\subseteq X$) is said to be {\\em closed} or {\\em Galois-stable} if $Cl(B)=I^{(0)}[I^{(1)}[B]]=B$ (resp.\\ $Cl(Y)=I^{(1)}[I^{(0)}[Y]]=Y$).\nThe set of objects $\\val{c}$ is intuitively understood as the {\\em extension} of the concept $c$, while  the set of features $ \\descr{c}$ is understood as its {\\em intension}. \nThe set   of the all formal concepts of $\\mathbb{P}$ (denoted by $L(\\mathbb{P})$) can be partially ordered as follows: for any $c, d\\in L(\\mathbb{P})$, \n\\begin{equation}\nc\\leq d\\quad \\mbox{ iff }\\quad \\val{c}\\subseteq \\val{d} \\quad \\mbox{ iff }\\quad \\descr{d}\\subseteq \\descr{c}.\n\\end{equation}\nWith this order, $L(\\mathbb{P})$ is a complete lattice, the {\\em concept lattice} $\\mathbb{P}^+$ of $\\mathbb{P}$. \n\\subsection{Interrogative agendas}\\label{prelim: imterrogative agenda} \n\nIn epistemology and formal philosophy, interrogative agenda (or research agenda \\cite{enqvist2012modelling}) of an epistemic agent (or group of agents e.g.,~users) indicates the set of questions they are interested in, or what they want to know relative to a certain circumstance. \nIntuitively, in any context, interrogative agendas act as cognitive filters that block content which is deemed irrelevant by the agent. Only the information the agent considers relevant is used\ne.g.~in the formation of their beliefs, or actions, etc.  Deliberation  and negotiation processes can be described as whether or how agents succeed and interact in shaping their interrogative agendas, and the outcomes of these processes can be described in terms of the aggregated (or \"common ground\") agenda.\nAlso, phenomena such as polarization \\cite{myers1976group}, echo chambers \\cite{sunstein2001republic} and self-fulfilling prophecies \\cite{merton1948self} can be described in terms of the formation and dynamics of interrogative agendas among networks of agents. \n\n\n\n\nDealing with a classification or outlier detection problem, we may have different agendas for different aims. For example, the agenda for the classification of consumers for a grocery store based on their buying preferences is very different from the agenda of a political analyst trying to classify the same set of people based on their political inclinations. Thus, interrogative agendas play an important role in determining natural or useful categorization for a specific purpose. \n\n\\subsection{Interrogative agendas and flexible categorization}\n\\label{ssec:Interrogative agendas and flexible categorization}\nLet $\\mathbb{P}=(A,X,I)$ be a formal context. For a set of features $Y \\subseteq X$, the formal context induced by $Y$ from $\\mathbb{P}$ is $(A,X,I \\cap A \\times Y)$. Given the set of all the features $X$, the (non-crisp) interrogative agenda of an agent can be described by a mass function on $\\mathcal{P}(X)$. For an agenda represented by $m:\\mathcal{P}(X) \\to [0,1]$, and any $Y \\subseteq X$, $m(Y)$ represents the importance (or intensity of the preference) of the set of features $Y$ according to the agenda given by $m$. We assume that mass functions are normalized, that is, \n\\begin{equation}\n\\sum_{Y \\subseteq X} m(Y)=1.\n\\end{equation}\nAny such mass function induces a probability or preference function $p_m: \\mathcal{R} \\to [0,1]$ such that $p_m((A,X,I \\cap A \\times Y))= m(Y)$, where  $\\mathcal{R}$ is the set of all the formal contexts corresponding to the crisp agendas induced by subsets of $X$ (i.e. the formal contexts corresponding to each $Y \\subseteq X$).\n\nThe agendas of different agents can be aggregated using different Dempster-Shafer rules \\cite{shafer1992dempster, sentz2002combination, denoeux2006cautious} to obtain a categorization corresponding to aggregated agendas. A logical framework for deliberation between different agents having different agendas is developed in \\cite{FLexiblecat2022}. This framework can be applied to study categorizations when different agents with different interests interact with each other for communication or joint decision making, as it is the case in auditing, community analysis, linguistics, etc. We also describe a method to approximate the importance of individual features from mass functions describing agendas by plausibility transform \\cite{cobb2006plausibility} or pignistic transformation \\cite{smets2005decision}, methods used in Dempster-Shafer theory to transform Dempster-Shafer mass functions to probability functions. These importance values of individual features can be useful in several  different applications like feature analysis, clustering, etc. \n\\label{ssec:interrogativeag}\n\n\\section{Classification and outlier detection using concept lattices} \\label{Sec:Classification and outlier detection using concept lattice}\nIn this section, we give an overview of different classification  \nand outlier detection techniques using concept lattices.\n\\subsection{Classification using concept lattices}\nDifferent algorithms have been applied to classify objects using formal concept analysis, that is, using concept lattices. \n Fu et al. \\cite{fu2004comparative} provide a comparison between  different FCA-based classification algorithms, such as  LEGAL \\cite{liquiere1990legal}, GALOIS \\cite{carpineto1993galois}, RULEARNER \\cite{sahami1995learning}, CLNN and CLNB \\cite{xie2002concept}. Prokasheva et al. \\cite{prokasheva2013classification} describe different classification algorithms using FCA and challenges to such methods. \n \n In \\cite{kuznetsov2004machine}, Kuznetsov describes a classification algorithm that uses the JSM-method \\cite{finn1989generalized,FINN1983351}. He proposes to use  concept lattices and training examples to form hypotheses as follows. Let $(A, X, I)$ be a formal context for the set of objects $A$ and the set of features $X$. We add an additional target feature $x \\not\\in X$ for denoting a class of an object. This partitions $A$ into three sets of objects $A_+$, $A_-$, and $A_\\tau$ consisting of objects known to have feature $x$, objects known to not have feature $x$, and objects for which it is unknown whether or not they have it, respectively. Positive hypotheses for the JSM-method based on this formal context are given by the sets of features that are shared by a set of positive examples but not by any negative example. That is,  a set $H \\subseteq X$ is a positive hypothesis iff $I^{(0)}[H] \\cap A_+ \\neq \\emptyset$ and $H \\not\\subseteq  I^{(1)}[\\{a\\}] $ for any $a \\in A_-$. Negative hypotheses are defined analogously. For any object $b$, it will be classified positively (resp. negatively) if $I^{(1)}[\\{b\\}]$ contains a positive (resp. negative) hypothesis but no negative (resp. positive) hypotheses. In case  $I^{(1)}[\\{b\\}]$ contains both or neither,  the classification is undetermined or some other method like majority voting can be used to classify $b$. The method sketched above has been used with different modifications in many FCA-based classification algorithms \\cite{ganter2000formalizing,kuznetsov2013fitting,onishchenko2012classification}. Some classification algorithms based on FCA  use concept lattices to augment other classifiers like SVM \\cite{carpineto2009concept},  Naive Bayes classifier and Nearest neighbour classifier \\cite{xie2002concept} in preprocessing or feature selection. Other FCA-based classification methods include biclustering \\cite{onishchenko2012classification}, and cover-based classification \\cite{maddouri2004towards}. \n\\subsection{Outlier detection using concept lattices}\nOutlier detection can be considered as a special case of binary classification where the classes are outlier and non-outliers. Thus, any of the above-mentioned algorithms can be used for outlier detection using concept lattices. Some other methods or algorithms based on formal concept analysis have also been studied specifically for outlier detection or similar tasks like mining rare concepts or patterns  \\cite{sugiyama2013semi,okubo2010algorithm,zhang2014outlier}. The simplest method to define the outlier degree of an element from a concept lattice is by using the size of its closure (i.e. the smallest category containing the element). Smaller size of closure of an object indicates that there are a small number of elements which have the same features as the object and thus it is likely to be an outlier. Sugiyama \\cite{sugiyama2013semi} suggests  that the outlierness of an object in a concept lattice should not depend on the size of its closure but one must consider the number of concepts it creates. He suggests to define the outlierness score of a set of objects $B \\subseteq A$ as\n\\begin{equation}\nq(B): = |\\{ (G,Y) \\in \\mathbb{P}^+ \\mid B \\subseteq G \\, \\text{or}\\, I^{(1)}[B] \\subseteq Y \\}|.\n\\end{equation}\nThis definition is more suited to detect outliers that belong to a densely agglomerated cluster which locates sparsely if we overview the whole set of  objects. Zhang et al.~\\cite{zhang2014outlier} propose an outlier mining algorithm based on  constrained concept lattices to  detect local outliers using a sparsity-based method. One of the key advantages of using formal concept analysis in classification or outlier detection over other algorithms is that FCA can be used to deal with both continuous and discrete attributes simultaneously, through the discretization of continuous attributes by conceptual scaling (Sec. \\ref{ssec:COnceptual scaling}).\n\nOne of the major issues in applications of formal concept analysis is the complexity of the algorithms involved. The fundamental reason behind the high complexity is that in the worst-case scenario the number of categories in a concept lattice grows exponentially with the number of objects and features involved. Several techniques have been devised in past to overcome this complexity problem \\cite{cole1999scalability,dias2010reducing,singh2017concepts}. \n\n\\subsection{Discretization of continuous attributes and conceptual scaling} \\label{ssec:COnceptual scaling}\nIn order to apply formal concept analysis on attributes with continuous values, we  need to discretize them.  The process of converting many-valued (possibly continuous-valued) attributes into binary attributes or features for FCA is known as conceptual scaling \\cite{ganter1989conceptual}. Scaling is an important part of most FCA-based techniques and has been studied extensively \\cite{ganter1989conceptual,prediger1997logical,prediger1999lattice}. Choosing the correct scaling method depends on the specific task the concept lattice is used for. \n\\section{Learning interrogative agendas} \\label{sec:Learning interrogative agendas}\nFormal concept analysis categorizes a given set of objects w.r.t~a given set of features. Thus, the outlier detection (or the classification) task at hand depends on the features (or attributes) under consideration. However, in many applications it is hard to estimate which features are of importance and how important they are, that is the correct agenda, for a given task. Here we describe a machine learning framework that tries to solve this problem by using machine learning to learn a ``good'' agenda for the given task. This provides a way to improve the performance of FCA-based classification or outlier detection algorithms  by choosing the correct agenda. This also makes results more explainable by providing the importance value of each set of features. \n\\subsection{Space of possible agendas}\nAs discussed in Section \\ref{ssec:Interrogative agendas and flexible categorization}, an (non-crisp) interrogative agenda on a given set of features $X$ is given by a mass function $m:\\mathcal{P}(X) \\to [0,1]$, where for any $Y \\subseteq X$, $m(Y)$ denotes the importance of the set of features $Y$ in the categorization.\nThe mass function $m$ induces a probability function $p_m:\\mathcal{R} \\to [0,1]$, where $\\mathcal{R}$ is the set of all the (crisp) formal contexts induced from $\\mathbb{P}=(A,X,I)$ by different crisp agendas i.e. subsets of $X$. For any categorization (formal context) $\\mathbb{P} \\in \\mathcal{R}$, $p_m(\\mathbb{P})$ denotes the likelihood assigned or preference given to the categorization  $\\mathbb{P}$ by the agenda $m$. Thus, the set of all  possible non-crisp categorizations  (resp. non-crisp agendas)  induced from a context $\\mathbb{P}$ is given by the set of all the probability functions on $\\mathcal{R}$ (resp. the set of all the possible mass functions on $\\mathcal{P}(X)$). As discussed in the introduction,  we want to learn  a ``good'' agenda that\nleads to a categorization that can be used to complete a given task effectively. This corresponds to learning  a probability function $p$ on $\\mathcal{R}$ which represents a suitable categorization for the given task. That is, we use machine learning to search for a ``good'' function in the space of probability functions on $\\mathcal{R}$.\nFor the sake of computational and notational convenience, here we propose the following simplifications.\n\n\nLet  $\\mathbb{R}$ be the set of real numbers. Let $f:\\mathcal{R} \\to \\mathbb{R}$ be a map assigning weight $w_\\mathbb{L} \\in \\mathbb{R}$ for every $\\mathbb{P} \\in \\mathcal{R}$. For any $\\mathbb{P} \\in \\mathcal{R}$, $f(\\mathbb{P})$ denotes the importance (or preference) assigned to the context $\\mathbb{P}$ or  to the corresponding set of features $Y$, where $\\mathbb{P}=(A,X, I \\cap A \\times Y)$. We call any such function $f$ a non-crisp agenda as it gives weights (representing importance) to different sets of features. Any such function can be seen as a real-valued vector of dimension $|\\mathcal{R}|$. Thus, the set of all such functions is isomorphic to the space $\\mathbb{R}^{|\\mathcal{R}|}$.  As this space is linear, the shift from  probability functions on $\\mathcal{R}$ to real-valued functions simplifies the task of learning an agenda  (weight function) that minimizes loss using a simple gradient descent method. \n\nThe weights  assigned to lattices can be interpreted as probabilities on $\\mathcal{R}$, (and hence  mass functions on $\\mathcal{P}(X)$) via normalization when all the weights are non-negative. The negative weights suggest that the corresponding   categorization is opposite to the preferred categorization for the task at hand. For example, suppose we are interested in detecting elements  with a value  of feature $f_1$ being abnormally  high, while the outlier detection method used finds \noutliers with value  of $f_1$ low. Then the learning algorithm is likely to assign a negative weight to the  agenda $\\{f_1\\}$. \n\nAs discussed earlier, one of the major problems in applications of formal concept analysis is the complexity of the algorithms involved.  Here, we are proposing to consider priority (or weight) functions on a set of different concept lattices corresponding to different agendas. As the number of different (crisp) agendas induced from a set $X$ of features is exponential in $|X|$, this may add another exponential factor to the complexity of the algorithm. In many applications where the number of features is large, this may make the problem computationally infeasible. Thus, in most applications we  need to choose a smaller set of concept lattices or (crisp) agendas as a basis, that is set of (crisp) concept lattices on which the weight functions  are defined. We propose the following strategies for this choice. \n\n\\begin{enumerate}\n    \\item \\textbf{Choosing agendas that consist of a small number of features} In this strategy, we choose the (crisp) agendas consisting of  $\\alpha$ or a smaller number of features to construct basis concept lattices for some fixed  $\\alpha\\ll |X|$. This is based on the idea that tasks like classification or outlier detection can be performed with good accuracy by considering only a small number of features together. This is especially the case with tasks involving epistemic components as humans use a limited number of features in combination for basic tasks like comparison and classification. As these agendas  consist of a small number of features, the number of concepts in these concept lattices is small. This makes the computational complexity low for most algorithms operating on concept lattices. Thus, this method can be applied for finding agendas when the algorithms may have high computational complexity for lattices with a large number of concepts.  In some situations, it may also be useful to add the full concept lattice (lattice corresponding to the full feature set $|X|$) to the set of basis lattices. This allows us to consider the full concept lattice with all available information for the task at hand while having the possibility of giving higher or lower (compared to other features) importance to some small subsets of features. For example, if the weights attached to all the lattices except those given by agendas $\\{f_1\\}$ and $X$ are close to $0$ and the weights assigned to these agendas are similar, it corresponds to the agenda in which the set of all features and $\\{f_1\\}$ are the only important sets of features. Thus, the concept lattice based on $f_1$ alone would be of high significance.  \n    \n    \\item  \\textbf{Choosing important  agendas based on prior or expert knowledge} For some tasks, we may have prior or expert knowledge  assigning different importance or priority to some lattices or agendas. In such cases, these lattices are taken as the set of basis lattices.  This provides us a way to incorporate prior or expert knowledge with other algorithms using formal concept analysis. \n    \\item  \\textbf{Choosing agendas adaptively} In this strategy, we start with a set of agendas given by all the sets  consisting of less than $\\alpha$ features for some small $\\alpha$ (usually taken as 1). We use machine learning to learn weights assigned to them, and then drop all the oness which get assigned a very low weight (after normalization).  We then consider agendas consisting of any  set of features that is a subset of the union of agendas that are not removed in the first step. Choosing  these agendas can be interpreted as considering combinations of features that are deemed important in the first learning step.  We then repeat the learning process with this new set of lattices. We keep repeating this process until all the agendas (lattices) added in the last step get assigned low weights or we reach $|X|$ (full concept lattice). In this way, we recursively check the possible combinations of  agendas  deemed to be important so far in the next recursive step. This method works on assumption that if a feature is not important on its own, then it is unlikely to be part of a set of features that is important. However, this assumption may fail in several situations. In such cases, this method should not be used to choose a basis. \n    \\end{enumerate}\n    There can be other effective strategies for choosing basis lattices for different tasks and algorithms. \n    \\subsection{Learning algorithm}\n    Once the set  of possible agendas (or concept lattices) is chosen, we apply some \n    classification or outlier detection algorithm on each of these. For every lattice $\\mathbb{L} \\in \\mathcal{R}$, we start assigning it a random weight $w \\in \\mathbb{R}$. Let $Alg$ be any algorithm which performs classification or outlier detection for a fixed concept lattice.\n    \n    Suppose $Alg$ is a classification (resp. outlier detection) algorithm classifying a set $A$ of objects into $n$ classes using concept lattices. For any object $a$ and a class $k$, let $Alg_k(a, \\mathbb{L})$ (resp. $Alg(a, \\mathbb{L})$ denote the membership of  the object $a$ into the class $k$ (resp. outlier degree) according to the classification algorithm $Alg$ acting on the lattice $ \\mathbb{L}$. Notice that we allow for our classifiers (resp. outlier detection algorithms)  to be interpreted as fuzzy or probabilistic such that membership value (resp. outlier degree) of $a$ belongs to $[0,1]$. For an algorithm $Alg$ with crisp output, the value  $Alg_k(a, \\mathbb{L})$ (resp. $Alg(a, \\mathbb{L})$) will be either $0$ or $1$.  For a given weight function $w: \\mathcal{R} \\to \\mathbb{R}$, we say that the membership of $a$ in the class $k$ (resp. outlier degree of $a$) assigned by the algorithm $Alg$ acting on a non-crisp categorization described by $w$ is \n    \\begin{equation}\n    \\label{eqn:outputs}\n    out_k(a,w) = \\frac{\\sum_{\\mathbb{L} \\in \\mathcal{R} } w(\\mathbb{L})Alg_k(a, \\mathbb{L})}{\\sum_{\\mathbb{L} \\in \\mathcal{R} }w(\\mathbb{L})}.\n    \\end{equation}\n    Intuitively, this corresponds to taking the weighted sum of the result given by $Alg$ on lattices with weights provided by the agenda $w$. Let $loss$ be a loss function for a given classification task, and let $loss(out)$   be the total loss for the classification (resp. outlier detection) when classes (outlier degrees) are assigned by $ Alg_k(a,w) $ (resp. $ Alg(a,w) $). We use a gradient descent method to learn the agenda $f_0$ that minimizes the loss. We then use the learnt agenda $f_0$ to assign a class to an object that is for any test object $b$, its predicted membership in class $k$ (resp. outlier degree) is $ Alg_k(b,f_0) $ (resp. $Alg(b, f_0)$). \n    \n\\begin{algorithm}\n\\footnotesize\n\\caption{Meta-Learning Algorithm for Interrogative Agendas}\n\\hspace*{\\algorithmicindent} \\textbf{Input:} a set of objects $A$, a set of features $X$, a training set $T\\subseteq X$, and a map $y:T \\to C$ representing the labels on the training set, an algorithm $Alg$ that takes in input some object and a concept lattice in $\\mathcal{R}$, and outputs an element in $\\mathbb{R}^C$ representing its prediction for each class; a loss function $loss$ that compares two classifications and outputs a real number, and a number of training epochs $M$.\\\\\n\\hspace*{\\algorithmicindent} \\textbf{Output} A model that classifies objects in $X$.\n\\begin{algorithmic}[1]\n\\Procedure{Train}{$A$, $X$, $T$, $y$, $Alg$, $loss$,  $M$}\n    \\State $\\mathbb{L}_1,\\ldots,\\mathbb{L}_n \\leftarrow $ \\textbf{compute} the concept lattices of $\\mathcal{R}$\n    \\State \\textbf{let} $predictions$ be an empty map from $A$ to $\\mathbb{R}^C$\n    \\State \\textbf{let} $w$ be an array of weights of length $n$ initialized with random values in $\\mathbb{R}$\n    \\For{$e = 1, \\ldots, M$ } \n        \\For{$a \\in X$, $k \\in C$}\n            \\State $predictions[a][k] \\leftarrow \\frac{\\sum_{i = 1}^n w(\\mathbb{L}_i)Alg_k(a, \\mathbb{L}_i)}{\\sum_{i = 1}^n w(\\mathbb{L}_i)}$\n        \\EndFor\n        \\State \\textbf{update} $w$ with an iteration of gradient descent using $loss(predictions)$\n    \\EndFor\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\nA generic  algorithm for outlier detection can be given in a similar manner. \n\n\\subsection{Example} \nLet us consider the following toy data table providing some information on different types of apples. It contains information on the color, size, sweetness, and origin of the apples. We assume that all apples under consideration are either green or red. For conceptual scaling,  we divide sweetness, price, and volume into low, medium, and high. This converts these continuous-valued attributes into discrete-valued. The set of features is obtained by considering each value of attributes as a different feature. For example, High volume, red color, Medium price are a few of them. \n\n\\begin{table*}[h]\n    \\label{Table:data table}\n    \\centering\n    \\begin{tabular}{c c c c c c}\n       \\hline\n       \\textbf{Type}& \\textbf{Color}&\\textbf{ Volume}&\\textbf{ Sweetness} &\\textbf{ Local}& \\textbf{Price}\\\\\n       \\hline\n       1 & red & High  & High & Yes & Medium\\\\\n       2 & green & High & High & Yes & Medium \\\\\n       3 & red & Medium & Medium & Yes& Medium\\\\\n       4 & green & Low & High & No& Medium\\\\\n       5 & green & High & Medium & No & Low\\\\\n       6 & red & Medium & Low & Yes& Low \\\\\n       7 & green & High & Medium & Yes &Low \\\\\n       8 & green  & High & Medium & Yes& High\\\\\n       \\hline\n    \\end{tabular}\n     \\caption{Exemplary data table containing the information on different types of apples w.r.t. attributes such as color, volume, sweetness, et cetera. }\n\\end{table*}\n\nLet $A$ and $X$ be the set of all types of apples and features respectively. The (non-crisp) agendas of interest to us are the ones assigning mass to an attribute and not to an individual feature. That is, we consider basis lattices corresponding to feature sets that contain all the values for a given many-valued attribute. As an example, if a $Y \\subseteq X$ in the agendas corresponding to a  basis lattice   contains the feature high volume, then it must also contain the features low and medium volume. We use volume to denote the set of features \\{high volume, low volume, medium volume\\}. A similar convention is used for the other attributes as well. \n\n\nLet  $I \\subseteq A \\times X$ be the incidence relation and\nlet $P$ be a customer. Suppose we are interested in classifying apples into types customer $P$ likes (class 1) and  does not like (class 2).  Given a formal context (concept lattice) $\\mathbb{P}= (A, X, I \\cap A \\times Y)$, describing a categorization of these types of apples for a given agenda of interest $Y$,  we use the following simple algorithm to predict the class for a new type of apple. \nLet $A_+$  and $A_-$ be the set  of apples known to be in class 1 and  class 2  respectively (from the training set). A set of features $H \\subseteq Y$ is said to be a positive (resp. negative) hypothesis in w.r.t.~a lattice $(A, X, I \\cap A \\times Y)$ iff $H$ is Galois-stable,  $I^{(0)}[H]$ is non-empty,  and $I^{(0)}[H] \\cap A_-=\\emptyset$ (resp.~$I^{(0)}[H] \\cap A_+=\\emptyset$). For any new element $t$, \nwe put it in class 1 (resp. class 2) if the category $ I^{(1)}[\\{t\\}]$  contains only positive (resp. negative) hypothesis. The algorithm is inconclusive when it contains neither type of  hypothesis (no information) or contains both type of hypotheses (inconsistent information). \n\n Suppose the classification of apples of types 1-8 into classes 1 and 2 for customer $P$ are as $Class \\, 1=\\{1,2,4,5, 7\\}$ and $Class\\,  2=\\{3,6,8\\}$ and suppose also that we use the full concept lattice (that is,  agenda $Y=X$). Let $t_0$ be a new type of apple that is green, has high volume, high sweetness, is local, and has a high  price. Consider   hypotheses $H_1=$ \\{High sweetness\\} and  $H_2=$ \\{Green, local\\} which are both contained in $ I^{(1)}[\\{t_0\\}]$. The hypothesis $H_1$ is positive while $H_2$ is negative. Thus, the above classification algorithm can not classify this object as the available information is inconsistent.  However, in many cases, some subsets of features are of much more importance to a  customer than others. For example, from the above classification, it is hinted that the customer $P$   considers Sweetness and Price as more important features than color or location. Our algorithm for learning agenda can help to  recognize this  difference in importance of different sets of features and allow us to classify such elements.\n \n Suppose we use our method to find the best categorization (or agenda) for the completion of this task using the above classification algorithm with the basis lattice consisting of lattices given by agendas comprising of one attribute (as discussed earlier, one attribute can correspond to multiple features due to scaling). We start with  random weights assigned to each of these lattices. We then use the classification algorithm described above to classify new types of apples into classes 1 and 2 using each of these lattices. We then sum over the weights of lattices in which elements are assigned to either class. The new object is assigned to the class which has a higher mass (the algorithm is indecisive if such a class does not exist). We use machine learning (gradient descent) to train the algorithm to find the best weights for this classification task. \n \nDuring the training phase, our algorithm can (generally) learn that the attribute  (or set of features) \\{sweetness\\} matters much more than other features to the customer $P$. Thus,  a high weight will be attached to the lattice with agenda \\{sweetness\\}. Thus, the above algorithm in combination with our method assigns  $t_0$ to class 1. \nAdding this method on top of a classification algorithm may give a  better classification (that is, more elements classified correctly with a given amount of training samples) given  our learnt information  `sweetness is much more important for $P$ in decision-making' is true. \n\nSimilarly, higher (resp. lower) masses attached to   agendas consisting of different sets of a single attribute are helpful in better categorization when this attribute is  more (resp. less) important for the customer. Thus, using machine learning techniques  (for example, gradient descent when possible)  to learn the best possible agenda to provide categorization to complement the classification algorithm can improve its accuracy with less training. Considering more basis lattices may further improve the accuracy and sample complexity. For example, it can be seen that the types 5 and 7, which have medium sweetness and low price, belong to class 1. This provides us with another likely hypothesis that the customer likes apples that are of medium size (not necessarily high) but have a low price. This hints to us  that the agenda \\{size,  price\\} may be of significant importance to the customer. In case this agenda is indeed more important to the agent, the learning   would  assign it a high weight during training and  thus allows us to make  more accurate predictions with a fewer number of samples. However, an increasing number of basis lattices may increase computational complexity significantly, meaning that such a decision needs to be made judiciously. \n\nThis simple example shows that the classification  algorithm described above can be improved in terms of accuracy, sample complexity, and explainability by adding a learning step for finding out the best agenda for categorization. Adding this step to the  different algorithms discussed in Section \\ref{Sec:Classification and outlier detection using concept lattice}, used for classification and outlier detection using concept lattices can improve these algorithms in a similar manner.  This is especially the case for the tasks in which the importance of different features may be hard to estimate beforehand. The obtained  agendas can be defined formally using the   logical framework described in \\cite{FLexiblecat2022}. In that paper, a logical model was used to represent deliberation between agents with different agendas. This framework can also be used to model deliberation or interaction between different learning algorithms by aggregating learnt agendas using different techniques described in \\cite{FLexiblecat2022}. The agendas inferred by our learning  algorithm can be used for  further tasks  like aggregation from different sources. For example, if for two different classifiers the agendas learned are given by the mass functions $m_1$ and $m_2$ on $\\mathcal{P}(X)$, then a combined classifier that takes both into account can be obtained by choosing the agenda $F(m_1, m_2)$, where $F$ is a suitable Dempster-Shafer combination rule \\cite{sentz2002combination,smets1993belief,denoeux2006cautious}, and then applying  the classification algorithm to the resulting lattice. \n\n\\section{Conclusion and future directions} \\label{sec:Conclusion}\nIn this paper, targeting the explainability line of hybrid intelligence research~\\cite{9153877}, we proposed a meta-learning algorithm to learn a \"good\" (interrogative) agenda for categorization  (which is used by a potential FCA-based classification or outlier detection algorithm).  Adding such a learning step to a given algorithm  allows us to improve the accuracy and sample complexity of the procedure while also making it more explainable. On the empirical side, a performance evaluation and the ablation study on the results of employing different FCA-based classification and outlier detection algorithms is an avenue of future research. Another investigation line is the transferability analysis of  \"good\" agendas e.g., how much knowledge do we transfer and how good the data efficiency is when such an agenda is used on previously unseen environments\/categorizations.  Noteworthy is extending this methodology towards other interesting application domains such as knowledge discovery, data visualization, information retrieval, etc.\n\nOn the theoretical side, this framework can be used to  model deliberation between agendas learnt from different algorithms, providing us a way to study their interaction, comparison, or combination. Within the interpretation of taking the concept lattice as expert knowledge,  the learnt agendas can also be aggregated or compared with agendas of different experts allowing us to incorporate learning and expert knowledge in categorization. From a multiagent systems perspective, it is especially useful to model subjective categorizations involving multiple agents (human experts and algorithms) with different agendas or goals interacting with each other.  In future work, we are considering investigation in a variety of directions e.g., investigating desirable properties of various aggregation mechanisms, representational power such as proportionality and fairness of induced agendas for multiple parties,  convergence and robustness guarantees for the \"good\" agendas, computational complexity analysis on hard and easy cases for (non-)crisp agendas, and extending our method on a more general framework in order to tackle the problem of features selection in an uniform way.\n\nThe meta-algorithm described in the present paper is currently employed in the development of an outlier detection algorithm with good results. Currently, it has been tested on the datasets from the ELKI toolkit \\cite{Campos2016} and it has been compared against the algorithms discussed in it. A detailed report of the results will be available in the future.\n\n\n\n\\begin{acknowledgments}\n Erman Acar is generously supported by the Hybrid Intelligence Project which is financed by the Dutch Ministry of Education, Culture and Science with project number 024.004.022. Krishna Manoorkar is supported by the NWO grant KIVI.2019.001 awarded to Alessandra Palmigiano.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction}\nRecently the Collider Detector at Fermilab (CDF) Collaboration has measured the mass of $W$ boson to be $80.4335\\pm 0.0094~\\mathrm{GeV}$~\\cite{CDF:2022hxs}, which is deviated from Standard Model (SM) prediction of $80.357\\pm 0.006~\\mathrm{GeV}$~\\cite{ParticleDataGroup:2020ssz} and which seems to indicate new physics beyond SM. And there are lots of works appeared to discuss this topic~\\cite{Cirigliano:2022qdm,Borah:2022obi,Chowdhury:2022moc,Arcadi:2022dmt,Zhang:2022nnh,Mondal:2022xdy,Nagao:2022oin,Kanemura:2022ahw,Kawamura:2022uft,Peli:2022ybi,Ghoshal:2022vzo,Perez:2022uil,Zheng:2022irz,Ahn:2022xeq,Heo:2022dey,Crivellin:2022fdf,Endo:2022kiw,Du:2022brr,Cheung:2022zsb,DiLuzio:2022ziu,Balkin:2022glu,Biekotter:2022abc,Krasnikov:2022xsi,Paul:2022dds,Babu:2022pdn,DiLuzio:2022xns,Bagnaschi:2022whn,Heckman:2022the,Lee:2022nqz,Cheng:2022jyi,Bahl:2022xzi,Song:2022xts,Asadi:2022xiy,Athron:2022isz,Sakurai:2022hwh,Fan:2022yly,Zhu:2022scj,Arias-Aragon:2022ats,Cacciapaglia:2022xih,Blennow:2022yfm,Strumia:2022qkt,Athron:2022qpo,Yang:2022gvz,deBlas:2022hdk,Tang:2022pxh,Du:2022pbp,Campagnari:2022vzx,Zhu:2022tpr,Fan:2022dck}.\nIn this work we will explore physics beyond SM which can give the observed mass of the $W$  boson at tree level. \n\nIn the SM, the mass of the $W$ boson and the $Z$ boson are given by the Higgs mechanism. Since $Z$ is combination of the $B$ boson and the $W^{3}$ boson, which is a component of the gauge triplet $W^{i}$, the mass of the $W$ boson and the $Z$ boson are connected. And it is difficult to change the mass of the $W$ boson solely. One way to alter the mass of the $W$ boson is to mix the $Z$ boson with an other vector boson. Mix the $Z$ boson with another boson will inevitably alter the mass expression of the $Z$ boson which may alter the value of the $\\mathrm{SU}(2)_L$ gauge coupling and thus the mass of the $W$ boson. There are usually two kinds of mixing: direct mixing in mass matrix and kinetic mixing. Though the normalization of the kinetic mixing terms will result in mass mixing, we will consider two models in this work: Derivative Portal Dark Matter (DPDM) model~\\cite{Zeng:2022llh} and the U(1) model~\\cite{Holdom:1990xp,Lao:2020inc}. In these two models, the extra gauge boson are connects to the SM through kinetic mixing to $Z$ boson and $B$ boson respectively. The kinetic mixing will alter the mass expression of the $Z$ boson and thus the mass of the $W$ boson at tree level. Since electroweak oblique parameters have a strong constraint on electroweak physics, we will also consider the electroweak oblique parameters constraints to these models. For the DPDM model, we also consider constraints from the observed Dark Matter (DM) relic density. \n\nThis work is organized as follows: In Sec.~\\ref{sec:general} we generally discuss the mechanism that the mixing between extra boson and $Z$ boson will change the mass of the $W$ boson. In Sec.~\\ref{sec:bsm} we explore two models and discuss their capability of altering the $W$ mass. As well as explore constraints from electroweak oblique parameters and DM relic density. And we conclude in Sec.~\\ref{sec:con}   \n\\section{general discussion of prediction of the mass of $W$  boson}%\n\\label{sec:general}\nIn this section we  will discuss in general how can an extra boson mix with the $Z$ boson will change the mass of $W$ boson. To see this we first write down the mass of the $W$ boson $m_{W}$  and the mass of the $Z$ boson $m_{Z}$  given by SM:\n\\begin{eqnarray}\n    m_{W}^2=\\frac{1}{4}g^2v^2,\\ m_{Z}^2=\\frac{1}{4}(g^2+g^{\\prime 2})v^2.\n\\end{eqnarray}\nWhere $g$ and $g^{\\prime}$ are the gauge couplings of $\\mathrm{SU}(2)_\\mathrm{L}$ and $\\mathrm{U}(1)_\\mathrm{Y}$. And $v$ is the vacuum expectation value (vev) of the Higgs boson. When choosing the Fermi coupling constant $G_{F}$, the $Z$ boson mass $m_{Z}$ and the fine-structure constant $\\alpha$ as input parameter, the $W$ boson mass will then be determined. Because \n\\begin{eqnarray}\n    G_{F}=\\frac{1}{\\sqrt{2}v^2 },\\ e=\\sqrt{4\\pi\\alpha} =\\frac{gg^{\\prime}}{\\sqrt{g^2+g^{\\prime 2}} }.\n\\end{eqnarray}\nGoing beyond the SM, we will mix the $Z$ boson with another vector boson. After that the real mass of the $Z$ will be the square root of one of the eigenvalues of the following mass matrix:\n\\begin{eqnarray}\n    \\begin{pmatrix}\n\t\\frac{1}{4}(g^2+g^{\\prime 2})v^2&b\\\\\n\tb&a\n    \\end{pmatrix}.\n\\end{eqnarray}\nWhere we have used $a$ and $b$ to denote some general mass term. The eigenvalues of the mass matrix can be written as: \n\\begin{eqnarray}\n    m_{Z,Z^{\\prime}}^2=\\frac{1}{2}\\left( \\frac{1}{4}(g^2+g^{\\prime 2})v^2+a\\pm \\sqrt{\\left( \\frac{1}{4}(g^2+g^{\\prime 2})v^2+a \\right)^2-a(g^2+g^{\\prime 2})v^2+4b^2}  \\right) \\label{mass}.\n   \n\\end{eqnarray}\nDefine $c=\\frac{1}{4}(g^2+g^{\\prime 2})v^2$, we have a compact form of $m_{Z,Z^{\\prime}}^2=\\frac{1}{2}\\left( a+c\\pm\\sqrt{(a-c)^2+4b^2}  \\right)$. \nAnd we can see the heavier mass of $m_{Z,Z^{\\prime}}$ will be bigger than both $a$ and $c$, and the lighter mass of $m_{Z,Z^{\\prime}}$ will be smaller than both $a$ and $c$. Therefore in order to have a bigger $c$, since observation of the $W$ mass indicate larger $g$, the value of $a$ must be lager than $c$. And the mass of $Z$ boson should corresponds to the minus sign in Eq.~\\eqref{mass}.  \nAdopting the input parameters as $G_{F}=1.1663787*10^{-5}~\\mathrm{GeV}^{-2},\\ m_{Z}=91.1876~\\mathrm{GeV},\\ \\alpha\\approx 1\/128$~\\cite{ParticleDataGroup:2020ssz}, we can draw a blue band which saturate the observed mass of the $W$ boson in $3\\sigma$ confidence level in Fig.~\\ref{fig:abband}.\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{abband}\n    \\caption{Band which gives the $W$ mass between $80.4053$ and $80.4617$.  }\n    \\label{fig:abband}\n\\end{figure}\n\nActually we can calculate the analytic relation between $a$ and $b$ by taking the mass of $W$ boson $m_{W}$ as an input parameter. From Eq.~\\eqref{mass} we can write:\n\\begin{eqnarray}\n    b^2&&=c(a-m_{Z}^2)+m_{Z}^{4}-m_{Z}^2a\\nonumber\\\\\n   \n      \n\t&&=\\frac{4m_{W}^4}{4m_{W}^2-e^2v^2}(a-m_{Z}^2)+m_{Z}^{4}-m_{Z}^2a\\label{abconstranit}.\n\\end{eqnarray}\nThen we can given constraint to models beyond SM according to Eq.~\\eqref{abconstranit}. Actually the above discussion does not take loop corrections from SM into consideration. Considering the loop corrections from SM we should replace $m_{W}$ in Eq.~\\eqref{abconstranit} with $m_{W}-\\delta m_{W} $, where $\\delta m_{W}$ represents the loop corrections to $m_{W}$ from SM.   \n\\section{models beyond SM}%\n\\label{sec:bsm}\nIn this section we will explore two models beyond SM which mix the $Z$ boson with an extra vector boson and might give the observed $W$ boson mass. We also consider other constraints like electroweak oblique parameters constraint and DM relic density constraint. \n\\subsection{Derivative Portal Dark Matter}\n\\label{sub:u_1_model}\nThe DPDM model extends the SM with an extra vector boson which links the dark sector and the SM through its kinetic mixing with the $Z$ boson. The relevant Lagrangian of the DPDM model can be written as~\\cite{Zeng:2022llh}:\n\\begin{eqnarray}\n    \\mathcal{L}=&&-\\frac{1}{4}Z^{\\mu\\nu}Z_{\\mu\\nu}-\\frac{1}{4}Z^{\\prime\\mu\\nu}Z^{\\prime}_{\\mu\\nu}-\\frac{\\epsilon}{2} Z^{\\mu\\nu}Z_{\\mu\\nu}^{\\prime}\\\\\n\t&&+\\sum\\limits_{f}  Z_{\\mu}\\bar{f}\\gamma^{\\mu}(g_{V}-g_{A}\\gamma^{5})f+g_{\\chi}Z_{\\mu}^{\\prime}\\bar{\\chi}\\gamma^{\\mu}\\chi\\nonumber\\\\\n\t&&+\\frac{1}{2}m_{Z}^2Z_{\\mu}Z^{\\mu}+\\frac{1}{2}m_{Z^{\\prime}}^2Z_{\\mu}^{\\prime}Z^{\\prime\\mu}-m_{\\chi}\\bar{\\chi}\\chi\\nonumber.\n\\end{eqnarray}\nAfter normalization of the kinetic terms, the kinetic mixing between $Z$ and $Z^{\\prime}$ actually result in mass mixing between them. The kinetic mixing term of the Lagrangian can normalized by:\n\\begin{eqnarray}\n   K=\\begin{pmatrix}\n       -k_1&k_2\\\\\n       k_1&k_2\n   \\end{pmatrix},\n\\end{eqnarray}\nwhere $k_1=1\/\\sqrt{2-2\\epsilon} $ and $k_2=1\/\\sqrt{2+2\\epsilon} $. And this operation will result in the following mass matrix between the two vector bosons: \n\\begin{eqnarray}\n\t    \\begin{pmatrix}\n\t    k_1^2M_1&k_1k_2M_2\\\\\n\t    k_1k_2M_2&k_2^2M_1\n\t    \\end{pmatrix}\n\\end{eqnarray}\nwhere $M_1=m_{Z}^2+m_{Z^{\\prime}}^2$ and $M_2=m_{Z^{\\prime}}^2-m_{Z}^2$. \nOne can use an orthogonal matrix $O$ to diagonalize the mass matrix, and $O$ can be defined as \n\\begin{eqnarray}\n\t    O=\\begin{pmatrix}\n\t\t\\cos \\theta&\\sin \\theta\\\\\n\t\t-\\sin \\theta& \\cos \\theta\n\t    \\end{pmatrix},\\ \\text{with} \\tan 2\\theta=\\frac{2k_1k_2M_2}{(k_2^2-k_1^2)M_1}.\n\\end{eqnarray}\nTherefore according to Eq.~\\eqref{abconstranit} we can give constraint to $m_{Z^{\\prime}}$ and $\\epsilon$ as:\n\\begin{eqnarray}\n    k_1k_2M_2=\\sqrt{(k_1^2M_1-m_{Z}^{exp\\ 2})(k_2^2M_1-m_{Z}^{exp\\ 2})}\\label{DPDMcons}.  \n\\end{eqnarray}\nWhere we have used $m_{Z}^{exp}$ to represent the experiment observed mass of $Z$ boson, which is meant to distinguish from $m_{Z}$. \n\nApart from giving mass to the $W$ boson, we will also calculate the tree level $S,T,U$ constraints to this model. The neutral-current coupling between $Z$ boson and SM fermions in the DPDM model can be written as:\n\\begin{eqnarray}\n    L_{NC,Zff}&&=\\sum\\limits_{f}  (-k_2\\sin \\theta -k_1\\cos \\theta) \\hat{Z}_{\\mu}\\bar{f}\\gamma^{\\mu}(g_{V}-g_{A}\\gamma^{5})f\\\\\n\t      &&=\\sum\\limits_{f}  (-k_2\\sin \\theta -k_1\\cos \\theta) \\hat{Z}_{\\mu}\\bar{f}\\gamma^{\\mu}\\frac{e}{s_{w}c_{w} }(T^{3}_{f}\\frac{1-\\gamma^{5}}{2}-Q_{f}s_{w}^2 )f,\n\\end{eqnarray}\nwhere  $\\hat{Z}_{\\mu} $ is the mass eigenstate of the $Z$ boson. The form of the charged-current in the DPDM is the same as the SM. \nUsing the effective-lagrangian techniques given by ~\\cite{burgess1994model}:\n\\begin{eqnarray}\n    &&\\mathcal{L}_{CC, Wff}=-\\frac{e}{\\sqrt{2} \\hat{s}_{w}}(1-\\frac{\\alpha S}{4(\\hat{c}_{w}^2-\\hat{s}_{w}^2)}+\\frac{\\hat{c}_{w}^2\\alpha T}{2(\\hat{c}_{w}^2-\\hat{s}_{w}^2)}+\\frac{\\alpha U}{8\\hat{s}_{w}^2})\\sum\\limits_{ij}V_{ij}\\bar{f}_{i}\\gamma^{\\mu}\\gamma_{L}f_{j}W_{\\mu}^{\\dagger}+\\mathrm{c.c.}.\\\\  \n&&    \\mathcal{L}_{NC, Zff}=\\frac{e}{\\hat{s}_{w}\\hat{c}_{w}}(1+\\frac{\\alpha T}{2})\\sum\\limits_{f}\\bar{f}\\gamma^{\\mu}[T^{3}_{f}\\frac{1-\\gamma^{5}}{2}-Q_{f}(\\hat{s}_{w}^2+\\frac{\\alpha S}{4(\\hat{c}_{w}^2-\\hat{s}_{w}^2)}-\\frac{\\hat{c}_{w}^2\\hat{s}_{w}^2\\alpha T}{\\hat{c}_{w}^2-\\hat{s}_{w}^2})]fZ_{\\mu},\n\\end{eqnarray}\nwhere $\\hat{s}_{w}\\hat{c}_{w}m_{\\hat{Z}}=s_{w}c_{w}\\frac{1}{2}\\sqrt{ g^2+g^{\\prime 2}}v=s_{w}c_{w}m_{Z}$, we can write $S$ and $T$ in the DPDM model as\n\\begin{eqnarray}\n    \\alpha T=2(\\frac{\\hat{s}_{w}\\hat{c}_{w}}{s_{w}c_{w}}(-k_2\\sin \\theta-k_1\\cos \\theta)-1)\\\\\n    \\alpha S=4\\hat{c}_{w}^2\\hat{s}_{w}^2\\alpha T+4(\\hat{c}_{w}^2-\\hat{s}_{w}^2)(s_{w}^2-\\hat{s}_{w}^2)\\\\\n    \\alpha U=8\\hat{s}_{w}^2(\\frac{\\hat{s}_{w}}{s_{w}}-1+\\frac{\\alpha S}{4(\\hat{c}_{w}^2-\\hat{s}_{w}^2)}-\\frac{\\hat{c}_{w}^2\\alpha T}{2(\\hat{c}_{w}^2-\\hat{s}_{w}^2)})\n\\end{eqnarray}\nAnd we constrain the DPDM model with global fit results given by table five of ~\\cite{deBlas:2022hdk}:\n\\begin{eqnarray}\n    S=0.005\\pm 0.097,\\ T=0.04\\pm 0.12,\\ U=0.134\\pm 0.087,\n\\end{eqnarray}\nwith the correlation coefficient $\\rho_{ST}=0.91,\\ \\rho_{SU}=-0.65,\\ \\rho_{TU}=-0.88$.  \n\nThe DPDM model can naturally escape stringent constraint from Dark Matter direct detection due to a cancellation mechanism~\\cite{Cai:2021evx}, and in Fig.~\\ref{fig:DPDMres} we have draw the constraints from observed DM relic density, the observed $W$ mass and the electroweak oblique parameters. \n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{DPDMres}\n    \\caption{The lightblue area are excluded by Planck experiment~\\cite{Planck:2018vyg}. The red line gives the observed $W$ at tree level. The green line has taken the SM model loop corrections into consideration and gives the observed $W$ mass, with the dashed green lines correspond to the $3\\sigma$ upper and lower deviation.  And the blue line gives the observed DM relic density. The magenta contour gives the constraints from electroweak oblique parameters at 95\\% C.L., with the red star gives the best fit. }\n    \\label{fig:DPDMres}\n\\end{figure}\nWhere the red line give the observed $W$ boson mass solely. And the green line give the observed $W$ boson mass with SM loop corrections taken into consideration. The dashed green lines correspond to the $3\\sigma$ mass deviated from the $W$ boson mass. And the blue line saturate the observed DM relic density, while the light blue area are excluded by Planck experiment~\\cite{Planck:2018vyg}. The DM relic density are calculated in settings $m_{\\chi}=60~\\mathrm{GeV},\\ g_{\\chi}=0.1$ by numerical tools: \\texttt{FeynRules~2}~\\cite{Alloul:2013bka}, \\texttt{MadGraph}~\\cite{Alwall:2014hca}, and \\texttt{MadDM}~\\cite{Ambrogi:2018jqj}. The magenta contour corresponds to the 95\\% C.L. constraints from electroweak oblique parameters. And the red star represents the best fit of $STU$: $m_{Z^{\\prime}}\\approx 116~\\mathrm{GeV},\\ \\epsilon\\approx 0.0250$. We see that this point also meets the direct calculation of $m_W$ and observed DM relic density. Also from Fig.~\\ref{fig:DPDMres} we see that the magenta area is consistent with the green lines and is also consistent with the physics that $STU$ not only contains the fitting of the observed $W$ boson mass, but also contains the constraints from electroweak couplings. To give the observed $W$ boson mass $m_{Z^{\\prime}}$ should satisfy $105~\\mathrm{GeV}<m_{Z^{\\prime}}<140~\\mathrm{GeV}$. To make area where $m_{Z^{\\prime}}<118~\\mathrm{GeV}$ and $m_{Z^{\\prime}}>130~\\mathrm{GeV}$ not excluded by Planck experiment, one can change the DM mass $m_{\\chi}$  and thus the annihilation resonance area will move accordingly. On the other hand, one can increase the extra gauge coupling $g_{\\chi}$ or simply not introduce dark matter in this model\n\n\\subsection{U(1) model}\n\\label{sub:u_1_model}\nIn the U(1) model there is a gauge boson of an extra $\\mathrm{U}(1)_\\mathrm{X}$ gauge symmetry which connects to the gauge boson of SM $\\mathrm{U}(1)_\\mathrm{Y}$ symmetry through kinetic mixing. In this section we will adopt the same model setting as~\\cite{Lao:2020inc}. Then the kinetic mixing terms can be written as:\n\\begin{eqnarray}\n    \\mathcal{L}_{\\mathrm{K}}=-\\frac{1}{4}B^{\\mu\\nu}B_{\\mu\\nu}-\\frac{1}{4}X^{\\mu\\nu}X_{\\mu\\nu}-\\frac{\\epsilon}{2}B^{\\mu\\nu}X_{\\mu\\nu},\n\\end{eqnarray}\nwhere $B_{\\mu}$ and $X_{\\mu}$ are the gauge fields of $\\mathrm{U}(1)_\\mathrm{Y}$ and $\\mathrm{U}(1)_\\mathrm{X}$ gauge symmetry. And there will be mass mixing term between $B_{\\mu}$ and $W^{3}_{\\mu}$ after Higgs getting its vev. Therefore the matrix of $(W_{\\mu}^{3},\\ B_{\\mu},\\ X_{\\mu}  )$ can be denoted as:\n\\begin{eqnarray}\n   &&\\frac{1}{2}\\begin{pmatrix}\n W^{3\\mu}&B^{\\mu}&X^{\\mu}      \n   \\end{pmatrix}\n    \\begin{pmatrix}\n\tg^2v^2\/4&-gg^{\\prime}v^2\/4&0\\\\\n\t-gg^{\\prime}v^2\/4&g^{\\prime 2}v^2\/4&0\\\\\n\t0&0&g_{x}^2v_{s}^2\n    \\end{pmatrix}\n   \\begin{pmatrix}\n W_{\\mu}^{3}\\\\\n B_{\\mu}\\\\\n X_{\\mu}      \n   \\end{pmatrix}\\nonumber\\\\\n    =&&\\frac{1}{2}\\begin{pmatrix}\n W^{3\\mu}&B^{\\mu}&X^{\\mu}      \n   \\end{pmatrix}K^{-1T}OO^TK^{T}\n    \\begin{pmatrix}\n\tg^2v^2\/4&-gg^{\\prime}v^2\/4&0\\\\\n\t-gg^{\\prime}v^2\/4&g^{\\prime 2}v^2\/4&0\\\\\n\t0&0&g_{x}^2v_{s}^2\n    \\end{pmatrix}KOO^T K^{-1}\n   \\begin{pmatrix}\n W_{\\mu}^{3}\\\\\n B_{\\mu}\\\\\n X_{\\mu}      \n   \\end{pmatrix}\\\\\n    =&&\\frac{1}{2}\\begin{pmatrix}\n A^{\\mu}&Z^{\\mu}&Z^{\\prime\\mu}      \n   \\end{pmatrix}\n    \\begin{pmatrix}\n\t0&0&0\\\\\n\t0&m_{Z}^2&0\\\\\n\t0&0&m_{Z^{\\prime}}^2\n    \\end{pmatrix}\n   \\begin{pmatrix}\n A_{\\mu}\\\\\n Z_{\\mu}\\\\\n Z^{\\prime}_{\\mu}      \n   \\end{pmatrix}\\nonumber\\label{massmatrix}\n\\end{eqnarray}\nWhere we have used $K$ to normalize the kinetic terms of $B_{\\mu}$ and $X_{\\mu}$ and used $O$ to diagonalize the mass matrix and transform the fields to their mass eigenstates. The masses of the two massive vector boson $Z$ and $Z^{\\prime}$ will be:\n\\begin{eqnarray}\n    m_{Z,Z^{\\prime}}^2=\n    \\frac{1}{8} (\n    g^2 v^2+g^{\\prime 2} k1^2 v^2+g^{\\prime 2} k2^2 v^2+4 g_x^2 k1^2 v_s^2+4 g_x^2 k2^2 v_s^2\\\\\n\\pm\\sqrt{\\left(g^2 v^2+\\left(k1^2+k2^2\\right) \\left(g^{\\prime 2} v^2+4 g_x^2 v_s^2\\right)\\right)^2-16 g_x^2 v^2 v_s^2 \\left(g^2 \\left(k1^2+k2^2\\right)+4 g^{\\prime 2} k1^2 k2^2\\right)})\\nonumber\n\\end{eqnarray}\nNote that the kinetic mixing between $B_{\\mu}$ and $X_{\\mu}$ will not change the form of the electric charge $e$. The definition of electric can be extracted from couplings between Photon and the Higgs doublet. Which in this model will be:\n\\begin{eqnarray}\n    e=g[KO]_{11}=g^{\\prime}[KO]_{21}=\\frac{2g^{\\prime}k_2}{\\sqrt{1+\\frac{4g^{\\prime 2}k_2^2}{g^2}+\\frac{k_2^2}{k_1^2}} }=\\frac{gg^{\\prime}}{\\sqrt{ g^2+g^{\\prime 2}}}.\n\\end{eqnarray}\nWhere $[KO]_{ij}$ represents the element which lies in the $i^{th}$ row and the $j^{th}$ column of matrix $KO$. \nThe neutral-current coupling between $Z$ boson and SM fermions in the U(1) model can be written as:\n\\begin{eqnarray}\n    L_{NC,Zff}&&=\\sum\\limits_{f}Z_{\\mu}\\bar{f}\\gamma^{\\mu}(g_{V}-g_{A}\\gamma^{5})f,\\\\\n    \\mathrm{with}\\ g_{V}&&=g_{A}+g^{\\prime}[KO]_{22}Q_{f},\\ g_{A}=\\frac{T_{f}^3}{2}(-g^{\\prime}[KO]_{22}+g[KO]_{12}).\n\\end{eqnarray}\nAnd we can read $S,T,U$ as:\n\\begin{eqnarray}\n    \\alpha T=\\frac{2\\hat{s}_{w}\\hat{c}_{w}(-g^{\\prime}[KO]_{22}+g[KO]_{12})}{e}-2\\\\\n    \\alpha S=\\frac{-4g^{\\prime}[KO]_{22}(\\hat{c}_{w}^2-\\hat{s}_{w}^2)}{-g^{\\prime}[KO]_{22}+g[KO]_{12}}-4\\hat{s}_{w}^2(\\hat{c}_{w}^2-\\hat{s}_{w}^2)+4\\hat{c}_{w}^2\\hat{s}_{w}^2\\alpha T\\\\\n    \\alpha U=8\\hat{s}_{w}^2(\\frac{\\hat{s}_{w}}{s_{w}}-1+\\frac{\\alpha S}{4(\\hat{c}_{w}^2-\\hat{s}_{w}^2)}-\\frac{\\hat{c}_{w}^2\\alpha T}{2(\\hat{c}_{w}^2-\\hat{s}_{w}^2)})\n\\end{eqnarray}\n\n\nNow we can give a line which predict the observed $W$ mass in this model in Fig.~\\ref{fig:u1res}.\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{u1res}\n    \\caption{The red line gives the observed $W$ at tree level. The green line has taken the SM model loop corrections into consideration and gives the observed $W$ mass, with the dashed green lines correspond to the $3\\sigma$ upper and lower deviation. The magenta contour gives the constraints from electroweak oblique parameters at 99\\% C.L., with the red star gives the best fit.  }\n    \\label{fig:u1res}\n\\end{figure}\nWhere we also use red dashed line to show that the U(1) model can solely give the observed mass of $W$ boson. And the green line takes the SM loop corrections into consideration and gives observed $W$ boson mass, with the dashed green lines correspond to the $3\\sigma$ upper and lower deviation. The magenta contour stands for the 99\\% C.L. constraints from the electroweak oblique parameters constraints, with the red star being the best fit: $m_{Z^{\\prime}}=139.9~\\mathrm{GeV},\\ \\epsilon=0.068$. From Fig.~\\ref{fig:u1res} we see that the electroweak oblique parameters constraints are in accordance with the direct calculation of $m_{W}$. And there are large area in the U(1) model that can give the observed $W$ boson mass. The lower mass limit and the lower $\\epsilon$ limit is about $m_{Z^{\\prime}}\\approx 109.8~\\mathrm{GeV}$ and $\\epsilon\\approx 0.036$.     \n\\section{\\label{sec:con}Conclusion}\nIn this work we have explored the possibility of altering the $W$ boson mas at tree level through mixing between an extra gauge boson and the $Z$ boson. We first give general discussion of the effects from mixing extra boson with $Z$ boson, then explored two realistic models: the DPDM model and the U(1) model. In the DPDM model the extra gauge boson mixes with the $Z$ boson through the kinetic mixing between extra boson and the $Z$ boson, while in the U(1) model the extra gauge boson mixes with the $Z$ boson through the kinetic mixing between extra boson and the $B$ boson. Apart from giving the $W$ boson mass, we also discussed the electroweak oblique parameters constraints for both model, and also explored DM relic density constraints for the DPDM model. We find that there are area in both model which can give the observed $W$ boson mass at tree level, and the best fit value for the extra vector boson mass is around $120~\\mathrm{GeV}$.  \n\n\\begin{acknowledgments}\n    This work is supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 11875327 and 11905300, the China Postdoctoral Science Foundation under Grant\nNo. 2018M643282, the Fundamental Research Funds for the Central Universities, and the Sun Yat-Sen University Science Foundation.\n\\end{acknowledgments}\n\\section*{Note added}\nDuring the finalizing of this manuscript, we noticed that ~\\cite{Zhang:2022nnh} appeared on arxiv. ~\\cite{Zhang:2022nnh} discusses explaination of the $W$ boson mass with U(1) dark matter model as well as several phenomenology constraints on DM. Our work discusses models with an extra gauge boson which can explain the $W$ boson mass. Apart from the DPDM model, we also discussed the U(1) model, but in different scenarios.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n  One extension of Malliavin calculus from the Brownian motion to general L\\'{e}vy processes was made using the It\\^{o} chaos \n  decomposition on the $L_2$-space over the L\\'evy space. This approach was used for\n   instance by Nualart and Vives \\cite{nualart-vives}, Privault \\cite{privault_extension}, Benth, Di Nunno, L{\\o}kka, {\\O}ksendal and Proske \n   \\cite{benth-dinunno-lokka-oksendal-proske}, Lee and Shih \\cite{lee-shih_product_formula},  Sol\\'e, Utzet and Vives\n   \\cite{sole-utzet-vives} and Applebaum\n  \\cite{applebaum2}. \n  \n  The wide interest in Malliavin calculus for L\\'{e}vy processes in stochastics and \n  applications motivates the study of an accessible characterization\n  of differentiability and fractional differentiability.\n  Fractional differentiability is defined by real interpolation between the Malliavin Sobolev space $\\mathbbm{D}_{1,2}$ and $L_2({\\mathbbm{P}})$ and\n  we recall the definition in Section \\ref{section:fractional} of this paper. \n  Geiss and Geiss \\cite{geiss-geiss} and Geiss and Hujo \\cite{geiss-hujo} have shown that Malliavin differentiability and \n  fractional differentiability \n  are in a close connection to discrete-time approximation of certain stochastic integrals when the underlying process is a (geometric)\n  Brownian motion. Geiss et al. \\cite{geiss-geiss-laukkarinen} proved that this applies also to L\\'{e}vy processes with jumps.\n  These works assert that knowing the parameters of \n  fractional smoothness allow to design discretization time-nets\n  such that the optimal approximation rate can be achieved. \n  For details, see \\cite{geiss-geiss}, \\cite{geiss-hujo} and \\cite{geiss-geiss-laukkarinen}.\n  \n\n  \n  Steinicke \\cite{steinicke} and Geiss and Steinicke \\cite{geiss-steinicke} take advantage of the fact that any random variable $Y$ on \n  the L\\'evy space can be represented as a functional $Y = F(X)$ of the L\\'evy process $X$, where $F$ is a real valued measurable mapping \n  on the Skorohod space of right continuous functions. Let us restrict to the case that $F(X)$ only depends on the jump part of $X$. \n  Using the corresponding result from \n  Sol\\'e, Utzet and Vives \\cite{sole-utzet-vives} and Al\\`os, Le\\'on and Vives \\cite{alos-leon-vives} on the canonical space, Geiss and Steinicke\n  \\cite{geiss-steinicke} show that the condition $F(X)\\in\\mathbbm{D}_{1,2}$ is equivalent with\n  $$\\iint_{\\mathbbm{R}_+\\times\\mathbbm{R}}\\mathbbm{E}\\left[ \\left(F(X+x\\mathbbm{1}_{[t,\\infty)}) - F(X) \\right)^2 \\right] {\\mathrm{d}} t\\nu({\\mathrm{d}} x) < \\infty, $$\n  where $\\nu$ is the L\\'evy measure of $X$.\n  On the other hand one gets from Mecke's formula \\cite{mecke} that\n  %\n  $$\\iint_A \\mathbbm{E}[F( X+x\\mathbbm{1}_{[t,\\infty)} )]{\\mathrm{d}} t\\nu({\\mathrm{d}} x) = \\mathbbm{E}[N(A)F(X)]$$\n  %\n  for any nonnegative measurable $F$ and any $A\\in\\mathcal{B}([0,\\infty)\\times\\mathbbm{R}\\setminus\\{0\\})$, \n  where $N$ is the Poisson random measure associated with $X$  as in Section \\ref{section:preliminaries}. \n  These results raise the following questions: when can Malliavin \n  differentiability be described using a weight function such as $N(A)$, and is there a weight function for fractional differentiability?\n  \n\n In this paper  we search for weight functions $\\Lambda$ and measurability conditions on $Y$ such that the criteria  \n   \\begin{equation}\\|Y \\Lambda\\|_{L_2({\\mathbbm{P}})} < \\infty  \\label{equation:weight_criteria}\\end{equation}\n  describes the smoothness of $Y$. We begin by recalling the orthogonal It\\^{o} \n  chaos decomposition\n $$Y = \\sum_{n=0}^\\infty I_n(f_n)$$\n on $L_2({\\mathbbm{P}})$ and the Malliavin Sobolev space\n $$\\mathbbm{D}_{1,2} = \\left\\{ Y\\in L_2({\\mathbbm{P}}): \\|Y\\|_{\\mathbbm{D}_{1,2}} = \\sum_{n=0}^\\infty (n+1) \\|I_n(f_n)\\|_{L_2({\\mathbbm{P}})}^2 < \\infty \\right\\}$$ \n %\n in Section \\ref{section:preliminaries}.\n Then,  in Section \\ref{section:differentiability}, we obtain an equivalent condition for Malliavin differentiability. The assertion is that \n $$Y\\in\\mathbbm{D}_{1,2} \\text{ if and only if } \\normP{ Y \\sqrt{N(A) +1}} < \\infty,$$\n  whenever $Y$ is measurable with respect to ${\\cal F}_A$, the\n completion of the sigma-algebra generated by $\\left\\{ N(B) : B\\subseteq A, \\,B\\in\\mathcal{B}([0,\\infty)\\times\\mathbbm{R})\\right\\}$ \n  and the set $A\\in\\mathcal{B}([0,\\infty)\\times\\mathbbm{R}\\setminus\\{0\\})$ satisfies $\\mathbbm{E}[N(A)]<\\infty$.\n\n Section \\ref{section:fractional} treats fractional differentiability and our aim is to adjust the weight function $\\Lambda$  so that the \n condition \\eqref{equation:weight_criteria} describes \n a given degree of smoothness. We recall the $K$-method of real interpolation which we \n use to determine the interpolation spaces $(L_2({\\mathbbm{P}}),\\mathbbm{D}_{1,2})_{\\theta,q}$ for $\\theta\\in(0,1)$ and $q\\in[1,\\infty]$.  These spaces are intermediate\n between $\\mathbbm{D}_{1,2}$ and $L_2({\\mathbbm{P}})$.\n We show that when $Y$ is ${\\cal F}_A$-measurable and $\\mathbbm{E}[N(A)]<\\infty$, then $Y$ has fractional differentiability of order $\\theta$ for $q=2$ \n if and only if\n$$\\normP{ Y\\sqrt{ N(A) +1}^{\\,\\theta} } < \\infty.$$\n \n \n\n\n\\section{Preliminaries} \\label{section:preliminaries}\n\nConsider a L\\'evy process $X = (X_t)_{t\\geq0}$ with c\\`{a}dl\\`{a}g paths on a\ncomplete probability space $({\\Omega},{\\cal F},\\mathbbm{P})$, where ${\\cal F}$ is the completion of the sigma-algebra generated by $X$.\nThe L\\'{e}vy-It\\^{o} decomposition states that there exist $\\gamma\\in\\mathbbm{R}$, $\\sigma\\geq0$, a standard Brownian motion $W$ and \na Poisson random measure $N$ on $\\mathcal{B}([0,\\infty)\\times\\mathbbm{R})$ such that\n\\[X_t=\\gamma t + \\sigma W_t + \\iint_{(0,t]\\times \\{ |x|>1\\}} x N(\\mathrm{d} s,\\mathrm{d} x) \n+  \\iint_{(0,t]\\times\\left\\{0<|x|\\leq1\\right\\}}x\\tilde{N}(\\mathrm{d} s,\\mathrm{d} x)\\]\n holds for all $t\\geq0$ a.s.\nHere $\\tilde{N}(\\mathrm{d} s,\\mathrm{d} x) = N(\\mathrm{d} s,\\mathrm{d} x)-\\mathrm{d} s\\nu(\\mathrm{d} x)$ is the compensated Poisson random measure and \n$\\nu:\\mathcal{B}(\\mathbbm{R})\\to[0,\\infty]$ is the L\\'{e}vy measure of $X$ satisfying $\\nu(\\{0\\})=0$,\n$\\int_\\mathbbm{R} (x^2\\wedge1)\\nu(\\mathrm{d} x)<\\infty$ and $\\nu(B)=\\mathbbm{E} \\left[ N((0,1]\\times B) \\right]$ when $0\\not\\in \\bar{B}$.\nThe triplet $(\\gamma,\\sigma,\\nu)$ is called the L\\'evy triplet.\n\n\nLet us recall the It\\^{o} chaos decomposition from \\cite{ito}:\nDenote $\\mathbbm{R}_+ := [0,\\infty)$. We consider the following measure $\\mathbbm{m}$ defined as \n\\begin{eqnarray*}\n\n \\mathbbm{m}:\\mathcal{B}(\\mathbbm{R}_+\\times\\mathbbm{R})\\to[0,\\infty],& \n & \\mathbbm{m}(\\mathrm{d} s,\\mathrm{d} x):= \\mathrm{d} s \\left[ {\\sigma}^2 \\delta_0(\\mathrm{d} x)  + x^2 \\nu(\\mathrm{d} x) \\right].\n\\end{eqnarray*}\nFor sets $B\\in\\mathcal{B}(\\mathbbm{R}_+\\times\\mathbbm{R})$ such that $\\mathbbm{m}(B) < \\infty$, a random measure $M$ is defined by\n\\[M(B) := \\sigma \\int_{\\left\\{ s\\in\\mathbbm{R}_+:(s,0)\\in B\\right\\}} \\mathrm{d} W_s + \\lim_{n\\to\\infty}\\iint_{\\left\\{(s,x)\\in B: \\frac{1}{n} < |x| < n\\right\\}} x\\ \\tilde{N}(\\mathrm{d} s,\\mathrm{d} x),\\]\nwhere the convergence is taken in $L_2(\\mathbbm{P}):=L_2({\\Omega},{\\cal F},\\mathbbm{P})$.\nThe random measure $M$ is independently scattered and it holds that $\\mathbbm{E}\nM(B_1) M(B_2) = \\mathbbm{m}(B_1 \\cap B_2)$ for all $B_1,B_2\\in\\mathcal{B}(\\mathbbm{R}_+\\times\\mathbbm{R})$ with\n$\\mathbbm{m}(B_1)<\\infty$ and $\\mathbbm{m}(B_2)<\\infty$.\n\n\n\nFor $n=1,2,\\ldots$ write \n\\[ {L_2\\left( \\mm^{\\otimes n} \\right)} = L_2 \\left((\\mathbbm{R}_+\\times\\mathbbm{R})^n, \\mathcal{B}(\\mathbbm{R}_+\\times\\mathbbm{R})^{\\otimes n}, \\mathbbm{m}^{\\otimes n}\\right)\\] \nand set $L_2 \\left(\\mathbbm{m}^{\\otimes 0} \\right):=\\mathbbm{R}$. A function $f_n:(\\mathbbm{R}_+\\times\\mathbbm{R})^n\\to\\mathbbm{R}$ is said to\nbe symmetric, if it coincides with its symmetrization\n$\\tilde{f}_n$,\n\\[\\tilde{f}_n((s_1,x_1),\\ldots,(s_n,x_n))=\\frac{1}{n!}\\sum_{\\pi}f_n\\left(  \\left( s_{\\pi(1)},x_{\\pi(1)} \\right),\\ldots, \\left( s_{\\pi(n)},x_{\\pi(n)} \\right) \\right),\\]\nwhere the sum is taken over all permutations\n$\\pi:\\{1,\\ldots,n\\}\\to\\{1,\\ldots,n\\}$.\n\nWe let $I_n$ denote the multiple integral of order $n$ defined by It\\^{o} \\cite{ito} and shortly recall the definition. \nFor pairwise disjoint \n$B_1,\\ldots, B_n\\in\\mathcal{B}(\\mathbbm{R}_+\\times\\mathbbm{R})$ with $\\mathbbm{m}(B_i)<\\infty$ the \nintegral of $\\mathbbm{1}_{B_1}\\otimes\\cdots\\otimes\\mathbbm{1}_{B_n}$ \nis defined by\n\\begin{equation} I_n\\left( \\mathbbm{1}_{B_1}\\otimes\\cdots\\otimes\\mathbbm{1}_{B_n}\\right) := M(B_1)\\cdots M(B_n). \\label{equation:multiple_integral}\\end{equation}\nIt is then extended to a linear and continuous operator $I_n:{L_2\\left( \\mm^{\\otimes n} \\right)}\\to L_2(\\mathbbm{P})$. We let $I_0(f_0):=f_0$ for $f_0\\in\\mathbbm{R}$. \nFor the multiple integral we have\n\\begin{equation}\\label{equation:inner_product_L_2}\nI_n(f_n)=I_n(\\tilde{f}_n) \\text{ and } \\mathbbm{E} \\left[ I_n(f_n)I_k(g_k) \\right] \n=\\begin{cases}\n        0, & \\text{ if }n\\neq k\\\\\n       n! \\left( \\tilde{f_n},\\tilde{g_n}\\right)_{L_2(\\mathbbm{m}^{\\otimes n})}, & \\text{ if }n=k\n       \\end{cases}\n\\end{equation}\nfor all $f_n\\in {L_2\\left( \\mm^{\\otimes n} \\right)}$ and $g_k\\in L_2\\left(\\mathbbm{m}^{\\otimes k}\\right)$. \n\nAccording to \\cite[Theorem 2]{ito}, for any $Y\\in L_2({\\mathbbm{P}})$ there exist functions $f_n\\in {L_2\\left( \\mm^{\\otimes n} \\right)}$,\n$n=0,1,2,\\ldots,$ such that\n\\[ Y=\\sum_{n=0}^\\infty\nI_n(f_n) \\quad \\text{ in } L_2(\\mathbbm{P})\\]\nand the functions $f_n$ are unique  in $L_2(\\mathbbm{m}^{\\otimes n})$ when they are chosen to be\nsymmetric. We have\n\\[ \\norm{Y}^2_{L_2(\\mathbbm{P})} = \\sum_{n=0}^\\infty n! \\norm{\\tilde{f}_n}^2_{L_2\\left( \\mm^{\\otimes n} \\right)}.\\]\n\n\n\n\n\n\nWe recall the definition of the Malliavin Sobolev space $\\mathbbm{D}_{1,2}$ based on the\nIt\\^{o} chaos decomposition. We denote by $\\mathbbm{D}_{1,2}$ \nthe space of all $Y = \\sum_{n=0}^\\infty\nI_n(f_n) \\in L_2(\\mathbbm{P})$ such that\n\\[\\|Y\\|^2_{\\mathbbm{D}_{1,2}} := \\sum_{n=0}^\\infty (n+1)! \\norm{\\tilde{f}_n}^2_{L_2\\left( \\mm^{\\otimes n} \\right)} < \\infty.\\]\nLet us write $L_2(\\mathbbm{m}\\otimes\\mathbbm{P}) := L_2(\\mathbbm{R}_+\\times\\mathbbm{R}\\times{\\Omega},\n\\mathcal{B}(\\mathbbm{R}_+\\times\\mathbbm{R})\\otimes{\\cal F}, \\mathbbm{m}\\otimes\\mathbbm{P})$ and define the\nMalliavin derivative $D:\\mathbbm{D}_{1,2}\\to L_2(\\mathbbm{m}\\otimes\\mathbbm{P})$  in the following way.\nFor $B_1,\\ldots,B_n \\in \\mathcal{B}(\\mathbbm{R}_+\\times\\mathbbm{R})$, which are pairwise disjoint and such that $\\mathbbm{m}(B_i)< \\infty$ for all $i=1,\\ldots,n$, we let\n\\begin{align*}D_{t,x} I_n\\left( \\mathbbm{1}_{B_1}\\otimes\\cdots\\otimes\\mathbbm{1}_{B_n}\\right) \n&= nI_{n-1}\\left( \\tilde{\\mathbbm{1}_{B_1}\\otimes\\cdots\\otimes\\mathbbm{1}_{B_n}}(\\cdot,(t,x))\\right)\\\\\n&:= \\sum_{i=1}^n \\prod_{j\\neq i} M(B_j) \\mathbbm{1}_{B_i}(t,x). \n\\end{align*}\nIt holds $\\normmP{DI_n\\left( \\mathbbm{1}_{B_1}\\otimes\\cdots\\otimes\\mathbbm{1}_{B_n}\\right)} \n= \\sqrt{n}\\normP{I_n\\left( \\mathbbm{1}_{B_1}\\otimes\\cdots\\otimes\\mathbbm{1}_{B_n}\\right) }$ and\nthe operator is extended to $\\left\\{ I_n(f_n): f_n\\in L_2(\\mathbbm{m}^{\\otimes n})  \\right\\}$ by linearity and continuity. For \n$Y = \\sum_{n=0}^\\infty I_n(f_n) \\in \\mathbbm{D}_{1,2}$ it then holds that\n\\[D_{t,x}Y := \\sum_{n=1}^\\infty n I_{n-1} \\left(\\tilde{f}_n(\\cdot,(t,x))\\right) \\] \nconverges in $L_2(\\mathbbm{m}\\otimes\\mathbbm{P})$.\n\n\n\\begin{remark}\\label{remark:inner_product_DD}\nNote that also for any $u\\in L_2(\\mathbbm{m}\\otimes{\\mathbbm{P}})$ one finds a chaos representasion $u=\\sum_{n=0}^\\infty I_n(g_{n+1})$, \nwhere the functions $g_{n+1} \\in L_2\\left(\\mathbbm{m}^{\\otimes(n+1)}\\right)$\nare symmetric in the first $n$ variables. For $u,v\\in L_2(\\mathbbm{m}\\otimes{\\mathbbm{P}})$ with \n$u=\\sum_{n=0}^\\infty I_n(g_{n+1})$ and $v=\\sum_{n=0}^\\infty I_n(h_{n+1})$ it then holds\n\\begin{equation}\\label{equation:inner_product_DD}\n (u,v)_{L_2(\\mathbbm{m}\\otimes{\\mathbbm{P}})} = \\sum_{n=0}^\\infty n!  \\left(g_{n+1},h_{n+1} \\right)_{L_2\\left(\\mathbbm{m}^{\\otimes (n+1)}\\right)}.\n \\end{equation}\n\\end{remark}\nFor more information, see for example \\cite{nualart-vives}, \\cite{privault_extension}, \\cite{benth-dinunno-lokka-oksendal-proske}, \n\\cite{lee-shih_product_formula}, \\cite{sole-utzet-vives} and\n  \\cite{applebaum2}. \n\n\\section{Differentiability}\n\\label{section:differentiability}\n\nWe shall use the notation $\\mathbbm{R}_0 = \\mathbbm{R}\\setminus\\{0\\}$. For $A\\in\\mathcal{B}(\\mathbbm{R}_+\\times\\mathbbm{R}_0)$ we denote by\n${\\cal F}_A$ the completion of the sigma-algebra $ \\sigma \\left( N(B) : B\\subseteq A \\text{ and } B\\in\\mathcal{B}(\\mathbbm{R}_+\\times\\mathbbm{R}) \\right) $. \nThe following theorem implies that if $Y\\in L_2({\\mathbbm{P}})$ is ${\\cal F}_A$-measurable and $\\mathbbm{E}[N(A)]<\\infty$, then \n$Y\\in\\mathbbm{D}_{1,2}$ if and only if $\\mathbbm{E}[Y^2 N(A)]<\\infty$.\n\n\\begin{theorem}\\label{theorem:second_double_inequality}\n Let $A\\in\\mathcal{B}(\\mathbbm{R}_+\\times\\mathbbm{R}_0)$ be such that $ \\mathbbm{E}\\left[N(A)\\right] =  ({\\mathrm{d}} t \\otimes \\nu)(A) < \\infty$ and \n $Y\\in L_2({\\mathbbm{P}})$. \n \\begin{enumerate}\n  \\item If $Y\\in\\mathbbm{D}_{1,2}$, then $Y\\sqrt{N(A)}\\in L_2({\\mathbbm{P}})$ and\n        \\begin{align}\\label{equation:second_double_inequality1}\n  \\left| \\normP{Y\\sqrt{N(A)}} - \\normP{Y} \\sqrt{\\mathbbm{E}\\left[ N(A)\\right]}  \\right| \n   \\leq \\normmP{DY\\mathbbm{1}_A}.\\end{align}\n   \\item If $Y\\sqrt{N(A)}\\in L_2({\\mathbbm{P}})$ and $Y$ is ${\\cal F}_A$-measurable, then $Y\\in\\mathbbm{D}_{1,2}$ and\n     \\begin{align}\\label{equation:second_double_inequality2}\n    \\normmP{DY}\n  \\leq \\normP{Y\\sqrt{N(A)}} + \\normP{Y}\\sqrt{\\mathbbm{E}\\left[ N(A)\\right]}.\n \\end{align}\n \\end{enumerate}\n\\end{theorem}\n\n\n\n\n\nWe denote by $\\mathcal{S}$ the set of random variables $Y$ such that there exists $m\\geq 1$, $f\\in C_c^\\infty(\\mathbbm{R}^m)$ and  \n$0\\leq t_0 < t_1 < \\cdots t_m < \\infty$ such that\n$$Y= f\\left( X_{t_1} - X_{t_0},\\ldots,X_{t_m} - X_{t_{m-1}}  \\right).$$\n\n\n\\begin{lemma}[Theorem 4.1, Corollaries 4.1 and 3.1 in \\cite{geiss-laukkarinen}]\\label{lemma:smooths_dense_in}\\hfill\n\n\n\\begin{itemize}\n \\itemm{a} $\\mathcal{S}$ is dense in $\\mathbbm{D}_{1,2}$ and $L_2({\\mathbbm{P}})$.\\label{lemma:smooths_dense_in_a}\n \\itemm{b} For $Y,Z\\in\\mathcal{S}$ it holds\n           $D_{t,x}(YZ) = Y D_{t,x}Z + Z D_{t,x}Y + x D_{t,x} Y D_{t,x} Z$ $\\mathbbm{m}\\otimes{\\mathbbm{P}}$-a.e.\n\\end{itemize}\n\n\\end{lemma}\n\n\n\n\n\\begin{proposition}\\label{proposition:first_double_inequality}\n Let $Y = \\sum_{n=0}^\\infty I_n(f_n)$ be bounded and $A\\in\\mathcal{B}(\\mathbbm{R}_+\\times\\mathbbm{R}_0)$ be such that $ \\mathbbm{E}\\left[N(A)\\right] = \u00a0({\\mathrm{d}} t \\otimes \\nu)(A) < \\infty$.\n Then $\\sum_{n=1}^\\infty nI_{n-1}\\left( \\tilde{f_n}(\\cdot,{ * })\\right) \\mathbbm{1}_A({ *})$ converges in $L_2(\\mathbbm{m}\\otimes{\\mathbbm{P}})$ and\n \\begin{align}\\label{equation:first_double_inequality}\n  &\\left| \\normP{Y\\sqrt{N(A)}} - \\normP{Y} \\sqrt{\\mathbbm{E}\\left[ N(A)\\right]}  \\right| \n   \\leq \\normmP{\\sum_{n=1}^\\infty  \\left(   nI_{n-1}\\!\\left( \\tilde{f_n} \\right) \\mathbbm{1}_A  \\right) } \\nonumber \\\\*\n  &\\qquad\\qquad\\leq \\normP{Y\\sqrt{N(A)}} + \\normP{Y}\\sqrt{\\mathbbm{E}\\left[ N(A)\\right]} .\n \\end{align}\n\\end{proposition}\n\\begin{proof}\n Assume first that $Y\\in\\mathcal{S}$. Then also $Y^2 = \\sum_{n=0}^\\infty I_n(g_n) \\in\\mathcal{S}$. \n %\nLetting $h(t,x) = \\frac{1}{x}\\mathbbm{1}_A(t,x)$ we have $I_1(h) = N(A) - \\mathbbm{E}\\left[ N(A) \\right]$ and we get using\n\\eqref{equation:inner_product_L_2} and \\eqref{equation:inner_product_DD} that\n %\n \\begin{align*}\n \\mathbbm{E}\\left[ Y^2  N(A)\\right] - \\mathbbm{E}\\left[ Y^2 \\right] \\mathbbm{E}\\left[ N(A)\\right]\n & = \\mathbbm{E}\\left[ Y^2 I_1(h) \\right]   \n  = (g_1,h)_{L_2(\\mathbbm{m})}\\\\\n & = (DY^2,h{\\mathbbm{1}_{{\\Omega}}})_{L_2(\\mathbbm{m}\\otimes{\\mathbbm{P}})}.\n\\end{align*}\nFrom Lemma \\ref{lemma:smooths_dense_in} (b) we obtain\n\\begin{align*}\n    \\mathbbm{E}\\left[ Y^2  N(A)\\right]\n & = \\mathbbm{E}\\left[ Y^2 \\right] \\mathbbm{E}\\left[ N(A)\\right] + (DY^2,h)_{L_2(\\mathbbm{m}\\otimes{\\mathbbm{P}})}\\\\\n & = \\mathbbm{E}\\left[ Y^2 \\right] \\mathbbm{E}\\left[ N(A)\\right] + 2  \\iint_A \\mathbbm{E}\\left[ YD_{t,x}Y \\right] x{\\mathrm{d}} t \\nu({\\mathrm{d}} x)\\\\*\n &\\qquad + \\iint_A \\mathbbm{E}\\left[ \\left( D_{t,x} Y\\right)^2\\right]\\mathbbm{m}({\\mathrm{d}} t,{\\mathrm{d}} x).\n \\end{align*}\n Using H\\\"{o}lder's inequality we get\n %\n $$\\left| 2  \\iint_A \\mathbbm{E}\\left[ YD_{t,x}Y \\right] x{\\mathrm{d}} t \\nu({\\mathrm{d}} x)\\right| \\leq 2\\normP{Y}\\sqrt{\\mathbbm{E}\\left[ N(A)\\right]} \\normmP{DY\\mathbbm{1}_A},$$\n so that\n %\n \\begin{align*}\n   &  \\left( -\\normP{Y}\\sqrt{\\mathbbm{E}\\left[ N(A)\\right]} + \\normmP{DY\\mathbbm{1}_A} \\right)^2\n    \\leq \\mathbbm{E} \\left[ Y^2 N(A)\\right]\\\\*\n   & \\qquad\\qquad\\leq \\left( \\normP{Y}\\sqrt{\\mathbbm{E}\\left[ N(A)\\right]} + \\normmP{DY\\mathbbm{1}_A} \\right)^2.\\end{align*}\n Taking the square root yields to the double inequality \\eqref{equation:first_double_inequality}.\n\n Using Lemma \\ref{lemma:smooths_dense_in} (a) we find for any bounded $Y$ a uniformly bounded sequence $(Y_k)\\subset \\mathcal{S}$ such that \n $Y_k\\to Y$ a.s.  Since inequality \\eqref{equation:first_double_inequality} holds for all random variables \n $Y_k-Y_m \\in \\mathcal{S}$, they are uniformly bounded  and $Y_k-Y_m\\to 0$ a.s. as $k,m\\to\\infty$, we have by dominated convergence that\n  \\begin{align*}\n   & \\normmP{D(Y_k-Y_m) \\mathbbm{1}_A}\\\\\n   &\\leq \\normP{(Y_k-Y_m)\\sqrt{N(A)}} + \\normP{Y_k-Y_m}\\sqrt{\\mathbbm{E}\\left[ N(A)\\right]} \\\\ \n   &\\to 0  \n \\end{align*}\n as $k,m\\to\\infty$.\n Thus the sequence $(DY_{k}\\mathbbm{1}_A)_{k=1}^\\infty$ converges\n in $L_2(\\mathbbm{m}\\otimes{\\mathbbm{P}})$ to some mapping $u\\in L_2(\\mathbbm{m}\\otimes{\\mathbbm{P}})$. Write $Y_k=\\sum_{n=0}^\\infty I_n \\left( \\tilde{f}_n^{(k)}  \\right)$.\n The mapping $u$ has a representasion $u = \\sum_{n=0}^\\infty I_n(h_{n+1})$\n (see Remark \\ref{remark:inner_product_DD}), where for all $n\\geq0$ we have that\n %\n \\begin{align*}\n  \\left\\| n\\tilde{f_n}\\mathbbm{1}_A - h_n \\right\\|_{L_2({\\mathbbm{m}^{\\otimes n}})}  \n & \\leq  \\left\\| n\\tilde{f_n}\\mathbbm{1}_A - n\\tilde{f}_n^{(k)}\\mathbbm{1}_A \\right\\|_{L_2({\\mathbbm{m}^{\\otimes n}})} \n    \\!+ \\left\\| n\\tilde{f}_n^{(k)}\\mathbbm{1}_A - h_n \\right\\|_{L_2({\\mathbbm{m}^{\\otimes n}})} \\\\*\n    & \\to 0 \n  \\end{align*} \n   as $k\\to\\infty$.\n  We obtain \\eqref{equation:first_double_inequality} for the random variable $Y$ using dominated convergence, the convergence\n  $DY_k\\mathbbm{1}_A \\to  \\sum_{n=0}^\\infty \\left( D I_n(f_n) \\mathbbm{1}_A \\right)$ in $L_2(\\mathbbm{m}\\otimes{\\mathbbm{P}})$ and the fact that \n  \\eqref{equation:first_double_inequality} holds for all random variables $Y_{k}$. \n\\end{proof}\n\n\n\\begin{lemma}\\label{lemma:Lipschitz}\n If $Y=\\sum_{n=0}^\\infty I_n(f_n\\mathbbm{1}_{\\mathbbm{R}_+\\times\\mathbbm{R}_0}^{\\otimes n}) \\in \\mathbbm{D}_{1,2}$ and $g:\\mathbbm{R}\\to\\mathbbm{R}$ is Lipschitz-continuous, then\n $g(Y)\\in\\mathbbm{D}_{1,2}$ and\n $$D_{t,x}g(Y) = \\frac{g(Y + x D_{t,x}Y) - g(Y)}{x} \\quad \\text{ in } L_2(\\mathbbm{m}\\otimes{\\mathbbm{P}}).$$ \n\\end{lemma}\n\\begin{proof}\nThe lemma is an immediate consequence of \\cite[Lemma 5.1 (b)]{geiss-laukkarinen}.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lemma:measurability}\nLet $Y = \\sum_{n=0}^\\infty I_n(f_n)\\in L_2({\\mathbbm{P}})$ and $A\\in\\mathcal{B}(\\mathbbm{R}_+\\times\\mathbbm{R})$. Then\n$$\\mathbbm{E} \\left[ Y | {\\cal F}_A \\right] = \\sum_{n=0}^\\infty I_n \\left( f_n\\mathbbm{1}_A^{\\otimes n} \\right) \\text{ in }L_2({\\mathbbm{P}}).$$\n\\end{lemma}\n\\begin{proof}\nThe equality can be shown via the construction of the chaos analogously to the proof of \\cite[Lemma 1.2.4]{nualartv1}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{theorem:second_double_inequality}]\n{ \\emph{1.}} Assume $Y \\in\\mathbbm{D}_{1,2}$ and define $g_m(x)= (-m \\vee x)\\wedge m$ for $m\\geq 1$. From Lemma \\ref{lemma:Lipschitz} we get \n$g_m(Y)\\in\\mathbbm{D}_{1,2}$ and $|Dg_m(Y)|\\leq |DY|$.  Then, using monotone convergence and Proposition\n\\ref{proposition:first_double_inequality}, we obtain\n\\begin{align*}\n& \\left| \\normP{Y\\sqrt{N(A)}} - \\normP{Y} \\sqrt{\\mathbbm{E}\\left[ N(A)\\right]}  \\right| \\\\\n& = \\lim_{m\\to\\infty} \\left| \\normP{g_m(Y)\\sqrt{N(A)}} - \\normP{g_m(Y)} \\sqrt{\\mathbbm{E}\\left[ N(A)\\right]}  \\right| \\\\\n&\\leq \\limsup_{m\\to\\infty} \\normmP{Dg_m(Y)\\mathbbm{1}_A}\\\\\n& \\leq  \\normmP{DY\\mathbbm{1}_A} < \\infty.\n\\end{align*}\nHence $Y\\sqrt{N(A)}\\in L_2({\\mathbbm{P}})$.\n\n{ \\emph{2.}} Assume $\\| Y\\sqrt{N(A)}\\| < \\infty$ and define $g_m(Y)$ as above. \nWrite $Y = \\sum_{n=0}^\\infty I_n \\left( f_n \\right)$ and $g_m(Y) = \\sum_{n=0}^\\infty I_n ( f_n^{(m)})$. \nSince $g_m(Y)\\to Y$ in $L_2({\\mathbbm{P}})$, it holds \n$\\| \\tilde{f}_n^{(m)} \\|^2_{L_2(\\mathbbm{m}^{\\otimes n})}\\to \\| \\tilde{f_n} \\|^2_{L_2(\\mathbbm{m}^{\\otimes n})}$ as $m\\to\\infty$.\n Since $g_m(Y)$ is ${\\cal F}_A$-measurable, we have $\\tilde{f}_n^{(m)} = \\tilde{f}_n^{(m)}\\mathbbm{1}_A^{\\otimes n}$ $\\mathbbm{m}^{\\otimes n}$-a.e. \nby Lemma \\ref{lemma:measurability} for all $m\\geq 1$. \n By Fatou's Lemma, \nProposition \\ref{proposition:first_double_inequality} and monotone convergence we get \n\\begin{align*}\n  & \\sqrt{ \\sum_{n=1}^\\infty nn! \\left\\| \\tilde{f_n} \\right\\|^2_{L_2(\\mathbbm{m}^{\\otimes n})}}  \\\\*\n   &\\leq \\liminf_{m\\to\\infty}\\sqrt{\\sum_{n=1}^\\infty nn! \\left\\| \\tilde{f}_n^{(m)} \\right\\|^2_{L_2(\\mathbbm{m}^{\\otimes n})}} \\\\*\n   &\\leq \\liminf_{m\\to\\infty} \\left( \\normP{g_m(Y)\\sqrt{N(A)}} + \\normP{g_m(Y)} \\sqrt{\\mathbbm{E}\\left[ N(A)\\right]}  \\right)\\\\*\n  & = \\normP{Y\\sqrt{N(A)}} + \\normP{Y} \\sqrt{\\mathbbm{E}\\left[ N(A)\\right]} < \\infty.\n\\end{align*}\nThus $Y\\in\\mathbbm{D}_{1,2}$.\n\\end{proof}\n\nWe use the notation $ \\alpha \\sim_c  \\beta$  for $\\frac{1}{c}  \\beta  \\leq  \\alpha \\leq c \\beta$ for $c\\geq 1$ and \n$\\alpha, \\beta\\in[0,\\infty]$.\n \n\\begin{corollary}\\label{corollary:norm_equivalence}\nLet $A\\in\\mathcal{B}(\\mathbbm{R}_+\\times\\mathbbm{R}_0)$ be such that $ \\mathbbm{E} \\left[ N(A)\\right]  < \\infty$ and assume that \n$Y= \\sum_{n=0}^\\infty I_n(f_n)\\in L_2({\\mathbbm{P}})$\n is ${\\cal F}_A$-measurable.  Then\n$$  \\|Y\\|_{\\mathbbm{D}_{1,2}} \\sim_{\\sqrt{2}\\left( \\sqrt{ \\mathbbm{E} \\left[ N(A) \\right]} +1  \\right)}  \n\\left\\|   Y  \\sqrt{N(A) +1 }  \\right\\|_{L_2({\\mathbbm{P}})}, $$\n where the norms may be infinite. \n\\end{corollary}\n\\begin{proof}\nThe inequalities \\eqref{equation:second_double_inequality1} and \\eqref{equation:second_double_inequality2} give  the relation\n$$\\left \\| Y \\sqrt{N(A)} \\right\\|_{L_2({\\mathbbm{P}})} + \\|Y\\|_{L_2({\\mathbbm{P}})} \\sim_{  \\sqrt{ \\mathbbm{E} \\left[ N(A) \\right]} +1 } \\|Y\\|_{L_2({\\mathbbm{P}})} \n+ \\|DY\\|_{L_2(\\mathbbm{m}\\otimes{\\mathbbm{P}})}. $$ \nThe claim follows from $\\|Y\\|_{\\mathbbm{D}_{1,2}} \\leq \\|Y\\|_{L_2({\\mathbbm{P}})} + \\|DY\\|_{L_2(\\mathbbm{m}\\otimes{\\mathbbm{P}})} \\leq \\sqrt{2}\\|Y\\|_{\\mathbbm{D}_{1,2}} $ and\n\\begin{align*}\n       { \\normP{Y\\sqrt{N(A)+1}} }\n & \\leq \\left\\| Y \\left( \\sqrt{N(A)} + 1 \\right)    \\right\\|_{L_2({\\mathbbm{P}})} \\\\\n & \\leq \\left\\| Y \\sqrt{N(A)}  \\right\\|_{L_2({\\mathbbm{P}})} + \\|Y\\|_{L_2({\\mathbbm{P}})}\\\\\n & \\leq { \\sqrt{ 2 \\left( \\left\\| Y \\sqrt{N(A)}  \\right\\|^2_{L_2({\\mathbbm{P}})} + \\|Y\\|^2_{L_2({\\mathbbm{P}})} \\right) } }\\\\\n & = { \\sqrt{2} \\left\\|  Y  \\sqrt{N(A) +1 }   \\right\\|_{L_2({\\mathbbm{P}})}. } \n\\end{align*}\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Fractional differentiability}\n\\label{section:fractional}\n\n\nWe consider fractional smoothness in the sense of real interpolation spaces between $L_2({\\mathbbm{P}})$ and $\\mathbbm{D}_{1,2}$. \nFor parameters $\\theta\\in(0,1)$ and $q\\in[1,\\infty]$ the interpolation space $(L_2({\\mathbbm{P}}),\\mathbbm{D}_{1,2})_{\\theta,q}$\nis a Banach space, intermediate between $L_2({\\mathbbm{P}})$ and $\\mathbbm{D}_{1,2}$.\n\n \n\n\nWe shortly recall the $K$-method of real interpolation. \nThe K-functional of $Y\\in L_2({\\mathbbm{P}})$ is the mapping\n$K(Y,\\cdot; L_2({\\mathbbm{P}}),\\mathbbm{D}_{1,2}): (0,\\infty) \\to [0,\\infty)$ defined by \n\\begin{align*} \n&K(Y,s; L_2({\\mathbbm{P}}),\\mathbbm{D}_{1,2})\\\\ & := \\inf\\{ \\|Y_0\\|_{L_2({\\mathbbm{P}})} + s\\|Y_1\\|_{\\mathbbm{D}_{1,2}}:\\ Y=Y_0+Y_1,\\,Y_0\\in L_2({\\mathbbm{P}}),\\, Y_1\\in \\mathbbm{D}_{1,2} \\}\n\\end{align*}\nand we shall use the abbreviation $K(Y,s)$ for $K(Y,s; L_2({\\mathbbm{P}}),\\mathbbm{D}_{1,2})$.\nLet $\\theta\\in(0,1)$ and $q\\in[1,\\infty]$. The space $(L_2({\\mathbbm{P}}),\\mathbbm{D}_{1,2})_{\\theta,q}$ consists of all $Y\\in L_2({\\mathbbm{P}})$\nsuch that\n\\[\n\\|Y\\|_{(L_2({\\mathbbm{P}}),\\mathbbm{D}_{1,2})_{\\theta,q}} \n= \\begin{cases}\n  \\left[ \\int_0^\\infty \\left| s^{-\\theta} K(Y,s) \\right|^q \\frac{\\mathrm{d}s}{s} \\right]^{\\frac{1}{q}}, & q\\in[1,\\infty)\\\\\n  \\sup_{s>0} s^{-\\theta} K(Y,s), & q=\\infty\n                               \\end{cases}\n\\]\nis finite.\n\n\nThe interpolation spaces are nested in a lexicographical order:\n\\[\n\\mathbbm{D}_{1,2} \\subset (L_2({\\mathbbm{P}}),\\mathbbm{D}_{1,2})_{\\eta,p} \\subset (L_2({\\mathbbm{P}}),\\mathbbm{D}_{1,2})_{\\theta,q} \\subseteq (L_2({\\mathbbm{P}}),\\mathbbm{D}_{1,2})_{\\theta,p} \\subset L_2({\\mathbbm{P}})\n\\]\n for $ 1 \\leq q \\leq p \\leq \\infty$ and\n $ 0 < \\theta < \\eta < 1$.\nFor further properties of interpolation we refer to \\cite{bennet-sharpley} and \\cite{triebel}.\n\n\n\\begin{theorem} \\label{theorem:fractional}\n Let $\\theta\\in(0,1)$, $A\\in\\mathcal{B}(\\mathbbm{R}_+\\times\\mathbbm{R}_0)$ be such that $ \\mathbbm{E}\\left[ N(A) \\right]< \\infty$ \n and $Y\\in L_2({\\mathbbm{P}})$\n be ${\\cal F}_A$-measurable. Then \n$$Y \\in (L_2({\\mathbbm{P}}),\\mathbbm{D}_{1,2})_{\\theta,2}\\text{ if and only if }  \\mathbbm{E}\\left[ Y^2 N(A)^\\theta  \\right] < \\infty.$$ If\n $Y \\in (L_2({\\mathbbm{P}}),\\mathbbm{D}_{1,2})_{\\theta,2}$, then\n %\n$$ \\|Y\\|_{(L_2({\\mathbbm{P}}),\\mathbbm{D}_{1,2})_{\\theta,2}}\n \\sim_{\\sqrt{2}\\frac{  \\sqrt{ \\mathbbm{E}\\left[ N(A) \\right] } +1}{\\sqrt{\\theta(1-\\theta)}}}  \\left\\| Y  \\sqrt{N(A) +1}^{\\,\\theta}  \\right\\|_{L_2({\\mathbbm{P}})}.$$\n\\end{theorem}\n\\begin{proof}\nWe first show that\n \\begin{equation} \\label{equation:Kfunctional}\n  K(Y,s)    \n\\sim_{ 2\\left(  \\sqrt{\\mathbbm{E}\\left[ N(A)\\right]} +1  \\right)} \\left\\|  Y \\min\\left\\{1, s  \\sqrt{N(A) +1}  \\right\\}  \\right \\|_{L_2({\\mathbbm{P}})}.\n  \\end{equation}\nFrom Lemma \\ref{lemma:measurability} we obtain the inequalities $  \\| \\mathbbm{E} \\left[ Y_0 | {\\cal F}_A \\right] \\|_{L_2({\\mathbbm{P}})} \\leq \\|Y_0\\|_{L_2({\\mathbbm{P}})} $\nand  $\\| \\mathbbm{E} \\left[ Y_1 | {\\cal F}_A \\right] \\|_{\\mathbbm{D}_{1,2}} \\leq \\|Y_1\\|_{\\mathbbm{D}_{1,2}} $ for any $Y_0\\in L_2({\\mathbbm{P}})$ and $Y_1\\in\\mathbbm{D}_{1,2}$. Hence\n\\begin{align}\\label{align:K_relation_one}\n     & K(Y,s) \\nonumber\\\\*\n& =  \\inf\\left\\{ \\|Y_0\\|_{L_2({\\mathbbm{P}})} + s\\|Y_1\\|_{\\mathbbm{D}_{1,2}}:\\ Y_0+Y_1 = Y,\\, Y_0\\in L_2({\\mathbbm{P}}),\\, Y_1\\in \\mathbbm{D}_{1,2} \\right\\}\\nonumber \\\\\n& = \\inf\\left\\{ \\| \\mathbbm{E} \\left[ Y_0 | {\\cal F}_A \\right] \\|_{L_2({\\mathbbm{P}})} + s\\| \\mathbbm{E} \\left[ Y_1 | {\\cal F}_A \\right] \\|_{\\mathbbm{D}_{1,2}}: Y_0+Y_1 = Y,\\, Y_1\\in \\mathbbm{D}_{1,2} \\right\\}\\nonumber \\\\*\n& \\sim_{c} \n   \\inf\\left \\{ \\|Y_0\\|_{L_2({\\mathbbm{P}})} +  s \\left\\|  Y_1   \\sqrt{N(A)+1}   \\right\\|_{L_2({\\mathbbm{P}})}  : \n     Y_0 + Y_1 = Y, Y_1\\in\\mathbbm{D}_{1,2} \\right \\}\n\\end{align}\nfor $c = \\sqrt{2}\\left( \\sqrt{\\mathbbm{E}\\left[ N(A) \\right]} +1\\right) $ by Corollary \\ref{corollary:norm_equivalence}. \nNext we approximate the $K$-functional from above with the choice $Y_0 = Y\\mathbbm{1}_{\\left\\{  \\sqrt{N(A)+1}  > \\frac{1}{s}\\right\\}} $ \nand get from \\eqref{align:K_relation_one}  that\n\\begin{align*}\n&     \\frac{1}{c}  K(Y,s)\\\\\n& \\leq   \\left( \\left\\|Y \\mathbbm{1}_{ \\left\\{{ \\sqrt{N(A) +1}} > \\frac{1}{s} \\right\\}} \\right\\|_{L_2({\\mathbbm{P}})} \n       + s \\left\\| Y{ \\sqrt{N(A) +1} }  \\mathbbm{1}_{ \\left\\{ { \\sqrt{N(A) +1 }} \\leq \\frac{1}{s} \\right\\}} \\right\\|_{L_2({\\mathbbm{P}})} \\right) \\\\\n& \\leq \\sqrt{2}  \\left\\| Y \\min\\left\\{ 1, s { \\sqrt{N(A) +1 }} \\right\\}  \\right\\|_{L_2({\\mathbbm{P}})}.\n\\end{align*}\nUsing the triangle inequality and the fact that $$|Y({\\omega}) -y| + |y|a \\geq |Y({\\omega})| \\min\\{1,a\\}$$ for all ${\\omega}\\in{\\Omega}$, \n$y\\in\\mathbbm{R}$ and $a\\geq 0$  we  obtain from \\eqref{align:K_relation_one} the lower bound \n\\begin{align*}\n  &   c  K(Y,s)\\\\*\n& \\geq \\inf \\left\\{ \\left\\| |Y_0| + |Y_1|s { \\sqrt{N(A) +1} }   \\right\\|_{L_2({\\mathbbm{P}})} : Y = Y_0+Y_1,\\, Y_1\\in\\mathbbm{D}_{1,2}  \\right\\}\\\\*\n& \\geq \\left\\| Y \\min \\left\\{ 1, s { \\sqrt{N(A) +1}}  \\right\\}  \\right\\|_{L_2({\\mathbbm{P}})}.\n\\end{align*}\nWe have shown that \\eqref{equation:Kfunctional} holds.\nFrom \\eqref{equation:Kfunctional} we get \n\\begin{align*}\n& \\|Y\\|_{(L_2({\\mathbbm{P}}),\\mathbbm{D}_{1,2})_{\\theta,2}} \\\\*\n\n&\\sim_{2\\left( \\sqrt{\\mathbbm{E} \\left[ N(A) \\right]} +1 \\right)} \n\\left( \\int_0^\\infty \\left| s^{-\\theta} \\normP{ Y \\min\\left\\{ 1, s  { \\sqrt{N(A) +1 } }  \\right\\}  } \\right|^2 \n\\frac{{\\mathrm{d}} s}{s} \\right)^{\\frac{1}{2}}.\n\\end{align*}\nWe finish the proof by computing the integral using first Fubini's theorem. We get\n\\begin{align*}\n&   \\int_0^\\infty \\left| s^{-\\theta} \\normP{ Y \\min\\left\\{ 1, s { \\sqrt{N(A) +1 } }  \\right\\} } \\right|^2 \\frac{{\\mathrm{d}} s}{s}\\\\*\n& =  \\mathbbm{E} \\left[ Y^2 \\int_0^\\infty s^{-2\\theta}  \\min\\left\\{ 1, s^2 {\\left( N(A) +1 \\right) } \\right\\}\\frac{{\\mathrm{d}} s}{s}  \\right]  \\\\*\n& =  \\mathbbm{E} \\left[ Y^2 \\frac{1}{2\\theta(1-\\theta)} { \\left( N(A) + 1 \\right)^{\\theta} } \\right].\n\\end{align*}\n\\end{proof}\n\n\n\n\n\n\\section{Concluding remarks}\n\n\n\n\nFrom Theorem \\ref{theorem:second_double_inequality} assertion \\emph{2.} we can conclude that a higher integrability than square integrability\ncan imply Malliavin differentiability. For example,\nall the spaces $L_p({\\Omega},{\\cal F}_A, {\\mathbbm{P}})$ are subspaces of $\\mathbbm{D}_{1,2}$ when $p>2$ and $\\mathbbm{E}[N(A)]< \\infty$ as we can deduce from the following corollary.\n\n\n\n\\begin{corollary} Let $A\\in\\mathcal{B}(\\mathbbm{R}_+\\times\\mathbbm{R}_0)$ be such that $\\lambda:=\\mathbbm{E}[N(A)] \\in (0, \\infty)$ so that $N(A)\\sim$ Poisson($\\lambda$). \nThen for the space\n$$L_2 \\log^+ L_2({\\Omega},{\\cal F}_A,{\\mathbbm{P}}) : = \\left\\{ Y \\in L_2({\\Omega},{\\cal F}_A,{\\mathbbm{P}}): \\mathbbm{E} \\left[ Y^2 \\ln^+ Y^2  \\right] < \\infty \\right\\},$$\nwhere $\\ln^+x = \\max\\{\\ln x , 0\\}$, it holds that\n$$L_2 \\log^+ L_2({\\Omega},{\\cal F}_A,{\\mathbbm{P}})\\subsetneq \\mathbbm{D}_{1,2} \\cap L_2({\\Omega},{\\cal F}_A,{\\mathbbm{P}}).$$\n\\end{corollary}\n\\begin{proof}\nSuppose $\\mathbbm{E} \\left[ Y^2 \\ln^+ Y^2  \\right] < \\infty$  and let ${\\varphi}(y)=\\ln(y+1)$. The functions $\\Phi$ and $\\Phi^\\star$ with\n$$\\Phi(x) = \\int_0^x {\\varphi}(y) {\\mathrm{d}} y = (x+1)\\ln(x+1) -x  \\leq 1 + x\\ln^+ x$$ \nand\n$$\\Phi^\\star(x)= \\int_0^x {\\varphi}^{-1}(y) {\\mathrm{d}} y =e^x - x -1$$\nare a complementary pair of Young functions. They satisfy the\nYoung inequality $xy \\leq \\Phi(x) + \\Phi^\\star(y)$ for all $x,y\\geq0$ and we get\n\\begin{align*}\n       \\mathbbm{E} \\left[ Y^2 N(A)  \\right] \n& \\leq \\mathbbm{E} \\left[ \\Phi\\left( Y^2 \\right) \\right] + \\mathbbm{E} \\left[ \\Phi^\\star(N(A)) \\right]   \\\\*\n& \\leq \\mathbbm{E} \\left[  Y^2  \\ln^+\\left( Y^2 \\right)   \\right] + e^{(e-1){\\lambda} } - {\\lambda}   \\\\*\n &< \\infty. \\end{align*}\n %\nHence $Y\\in\\mathbbm{D}_{1,2}$ by Theorem \\ref{theorem:second_double_inequality}.\n\nTo see that the inclusion is strict, let $a\\in(1,2]$ and choose a Borel function $f:\\mathbbm{R}\\to\\mathbbm{R}$ such that\n$f(0)=f(1)=0$ and\n$$f(n) = \\sqrt{e^\\lambda \\frac{n!}{\\lambda^n} \\frac{1}{n^2 \\ln^a n}} \\quad \\text{ for }n = 2,3,\\ldots.$$\nThen, since\n $\\ln n!  = \\sum_{k=2}^n \\ln k \\geq \\int_1^n \\ln x\\, {\\mathrm{d}} x  = n\\ln n - n  + 1 $ for $n\\geq 2$ and $a\\leq 2$, we have\n\\begin{align*}\n  \\mathbbm{E}\\left[f^2(N(A))\\ln^+ f^2(N(A))\\right] \n& = \\sum_{n=2}^\\infty  \\frac{1}{n^2 \\ln^a n}\\ln \\left( e^\\lambda \\frac{n!}{\\lambda^n} \\frac{1}{n^2 \\ln^a n}  \\right)\\\\\n& = \\sum_{n=2}^\\infty  \\frac{\\ln n!}{n^2 \\ln^a n}\n    + \\sum_{n=2}^\\infty  \\frac{1}{n^2 \\ln^a n}\\ln \\left(e^\\lambda\\frac{1}{\\lambda^n} \\frac{1}{n^2 \\ln^a n}  \\right) \\\\\n& = \\infty,\n\\end{align*}\nbut\n$$\\mathbbm{E}\\left[N(A) f^2(N(A))\\right] = \\sum_{n=2}^\\infty nf^2(n) e^{-\\lambda} \\frac{\\lambda^n}{n!} = \\sum_{n=2}^\\infty \\frac{1}{n\\ln^a n} < \\infty $$\nso that $f(N(A)) \\in \\mathbbm{D}_{1,2}$ by Theorem \\ref{theorem:second_double_inequality}.\n\\end{proof}\n\n\n\n\n\n\n\\begin{remark}\\label{remark:compound_differentiability}\n Suppose ${\\sigma}=0$ and $\\nu(\\mathbbm{R})<\\infty$, which means that $X$ is a compound Poisson process (with drift) and\n  $$X_t = \\beta t + \\int_{(0,t]\\times\\mathbbm{R}_0} x N({\\mathrm{d}} s, {\\mathrm{d}} x)\\quad  \\text{ for all }t\\geq 0 \\text{ a.s.}$$\n for some $\\beta\\in\\mathbbm{R}$. The process $(N_t)_{t\\geq0}$, with $N_t = N((0,t]\\times\\mathbbm{R}_0)$ a.s., is the Poisson process associated to $X$.\n Let $T\\in(0,\\infty)$ and ${\\cal F}_T$ be the completion of the sigma-algebra generated by  $(X_t)_{t\\in[0,T]}$. Then\n  ${\\cal F}_T = {\\cal F}_{[0,T]\\times\\mathbbm{R}}$ and by Theorems \\ref{theorem:second_double_inequality} and \\ref{theorem:fractional} for\n any\n ${\\cal F}_T$-measurable random variable $Y$ and any $\\theta\\in(0,1)$ it holds that\n \\begin{itemize} \\item[(a)] $Y\\in\\mathbbm{D}_{1,2}$ if and only if\n $\\normP{ Y\\sqrt{N_T+1}}  < \\infty$ and\n \\item[(b)] $Y\\in(L_2({\\mathbbm{P}}),\\mathbbm{D}_{1,2})_{\\theta,2}$ if and only if $\\normP{ Y\\sqrt{N_T+1}^{\\,\\theta}}  < \\infty$.\n \\end{itemize}\n\\end{remark}\n\n\n\n\n\n\n\n{\\bf Acknowledgements.} The author is grateful to Christel Geiss and Stefan Geiss for several\nvaluable ideas and suggestions regarding this work.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzdstu b/data_all_eng_slimpj/shuffled/split2/finalzzdstu
new file mode 100644
index 0000000000000000000000000000000000000000..b63935a537128944dfb246a67ecdbc93100e5646
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzdstu
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\nThe question whether solutions of the two-dimensional Euler equation in vorticity form\n\\begin{align}\\label{eqEuler}\n\\omega_t + u \\cdot \\nabla \\omega &= 0\n\\end{align}\ncan exhibit strong gradient growth in time is a topic of ongoing interest. The best known upper bound predicts\ndouble-exponential growth in time:\n$$\n\\|\\nabla \\omega\\|_\\infty \\leq C_1 \\exp(C_2 \\exp(C_3 t))\n$$\nwith constants $C_i$ depending on the initial data. A natural and important question is: \nAre there flows for which this upper bound is attained? The problem can be considered in bounded domains \nwith no-flow boundary conditions or in domains without a natural boundary (e.g. on the torus). \nFor domains with boundary, a recent breakthrough by A. Kiselev and V. \\v Sver\\'ak \n\\cite{KiselevSverak} answers the question affirmatively. For smooth solutions on the torus, the best known result \nso far was given by S. Denisov.  In \\cite{Denisov1}, he shows that at least superlinear gradient growth is possible\nand in \\cite{Denisov2} he provides an example of double-exponential growth for an arbitrarily long, but finite\ntime interval. In the recent paper \\cite{Zlatos}, A. Zlat\\v os constructs initial data leading to exponential gradient growth, \nhis solution is however in $C^{1, \\gamma}$ for some $\\gamma \\in (0, 1)$ and not in $C^2$. \n\nIn \\cite{KiselevSverak} the construction is based on imposing certain symmetries on the solution\nleading to a \\emph{hyperbolic flow scenario}.  The presence of a boundary and the hyperbolic flow work nicely\ntogether, allowing the construction of examples with double-exponential gradient growth.\nConsidering double-odd solutions, i.e.\n\\begin{align}\\label{doubleOddSym}\n\\omega(-x_1, x_2)& = - \\omega(x_1, x_2), ~\\omega(x_1, -x_2) = - \\omega(x_1, x_2),\n\\end{align}\nis a possible, natural way to  replace the physical wall from \\cite{KiselevSverak}\nby the $x_1$-axis in order to try to create strong gradient growth in the bulk. This \nconstruction was employed in \\cite{Zlatos}. In \\cite{Denisov2}, a perturbation argument\nstarting from a non-smooth double-odd stationary solution (see \\cite{Bahouri}) was used. Creating infinite-time \n\\emph{double-exponential} growth away from the boundary, however, is met with considerable \ndifficulties.\n\nIt is interesting to notice that the result \\cite{KiselevSverak} is in some sense analogous to the still open blowup problem \nfor for the more singular surface quasigeostrophic equation. In SQG blowup means that the solution becomes singular in finite time whereas for the 2d Euler \nequation ``blowup'' would mean maximal (double-exponential) gradient growth on an infinite time interval. \nThere are important conditional regularity results for the SQG equation such as \\cite{Cordoba}, \\cite{CordobaFefferman}, where \none studies a certain blowup scenario, in order to finally exclude it.\nAn analogous ``conditional regularity result'' for 2d Euler equation would be to show that in certain scenarios maximal gradient growth does not occur. Since \nthe possible motions of fluids are various and in general very complicated, studying scenarios is an invaluable method to gain insight into \nregularity problems of fluid mechanics.\n\nOur goal in this paper is to prove such a conditional regularity result in the sense that \na hyperbolic flow cannot create maximal gradient growth near the origin by itself when we start with double-odd $C^2$ initial data,\nprovided a certain ``upstream\" control is assumed on the flow.\nThis is an important step into understanding the double-odd hyperbolic scenario since we rule out the most promising \ncandidate for a mechanism creating maximal gradient growth, i.e. the local hyperbolic compression. Our result does not imply\nimpossibility of double-exponential growth in general, but makes the construction of examples much harder.  \n\nIn some sense, the scenario considered here is complementary to the one considered by D. Cordoba for the SQG\nequation in \\cite{Cordoba}, where a closing hyperbolic saddle is considered. There the solution stays smooth except for\nthe possible closing of the saddle. In our scenario for 2d Euler, the hyperbolic saddle is fixed due to the symmetry\n($\\omega=0$ on the coordinate axes), and we are asking if blowup can happen in another way.\n\nFinally, we would like to mention the recent preprint \\cite{Katz}, where a different approach is proposed\nto study whether double-exponential gradient growth can occur at an interior point (see also\nT. Tao's blog \\cite{Tao} for a related discussion).\n\n\n\n\\subsection{Main result.}\nWe consider \\eqref{eqEuler} on $\\mathbb{T}=[-1,1)^2$ with periodic boundary conditions and double-odd $C^2$ initial data $\\omega_0$. The double-odd \nsymmetry is preserved by the evolution and \\eqref{doubleOddSym} implies that the origin is a stagnant point of the flow field for all times. Moreover, the flow on each coordinate axis is always directed along that axis. When considering smooth solutions $\\omega\\in C^1([0, \\infty), C^2(\\mathbb{T}))$,\n\\eqref{doubleOddSym} also implies \n$$\\omega = 0$$ \non the coordinate axes.\n\nWe will studying the flow in boxes of the form\n\\begin{align}\nD&=(0,\\delta_1)\\times(0,\\delta_2),~~\\widehat{D}=(0,\\delta_1+\\delta_3)\\times(0,\\delta_2),\n\\end{align}\nwhere $\\delta_j$ are positive, but small and  \n\\begin{align}\\label{defDeltas}\n0<\\delta_1< \\delta_2 < \\delta_1+\\delta_3. \n\\end{align}\nIn a hyperbolic flow, which we will explain in detail in section \\ref{secHyperbolic}, fluid particles are supposed to constanty enter the box\n$D$ from the right and leave on the top. Therefore we call $\\widehat{D}\\setminus D$ \\emph{feeding zone}. The following definition\nformalizes the control we assume on the solution in the feeding zone. The meaning of the parameter $\\alpha$\nwill become clear later.\n\\begin{definition} Let $\\alpha\\in(0,\\frac{1}{4})$.\nThe box $\\widehat{D}$ is said to satisfy the conditions of {\\em controlled feeding}, with feeding parameter $R \\geq 0$ if \n\\begin{align}\\label{eqfeeding1}\n|\\d_{x_1}\\omega(x, t)|\\leq R x_2^{1-\\alpha}, ~~|\\d_{x_2}\\omega(x, t)|\\leq R \\quad (x\\in \\widehat{D}\\setminus D)\n\\end{align} \nfor all times $t \\geq 0$.\n\\end{definition}\nWe can think of the first inequality in \\eqref{eqfeeding1} as a H\\\"older-version of a bound on $\\d_{x_2, x_1}\\omega$, keeping in mind that $\\d_{x_1}\\omega(x_1, 0, t)=0$\nfor all times. The concept of controlled feeding conditions allows us to study the evolution of $\\omega$\nin $D$ independent of the remaining flow.\n\nOur main result is the following theorem. \n\\begin{thm} \\label{main} Fix $0< \\alpha < \\frac{1}{4}, 0<\\delta_3$. Let $\\omega$ be a smooth, double-odd solution of the Euler equation, and  \nsuppose the flow is hyperbolic near the origin.  Let $R> 0$ be given. \nThere exist small $\\delta_1, \\delta_2>0$, such that if $\\widehat{D}$ satisfies the controlled feeding conditions with parameter $R$,\nthen  \n\\begin{equation}\n\\|\\nabla \\omega\\|_{D, \\infty} \\leq C_1 \\exp( C_2 t)\\quad (t\\in [0, \\infty))\n\\end{equation}\nfor some $C_1, C_2 > 0$.\n\\end{thm}\nThis means that in this situation for maximal gradient growth near the origin one cannot rely on the hyperbolic compression alone but rather has \nto create in some other way a scenario where the feeding conditions are violated, i.e. there has to be a compression in\n$x_2$-direction in the feeding zone.\n\n\n\\subsection{The hyperbolic scenario.} \\label{secHyperbolic}\n\nIn order to give a definition of hyperbolic flow suitable for our purposes, we introduce the following\nimportant quantity. Let $\\alpha\\in (0, \\frac{1}{4})$ be fixed. For a smooth, periodic function $\\omega$ we set\n\\begin{equation}\nM(x, t) := \\max_{0\\le y_1, y_2\\le \\max\\{x_1, x_2\\}}\\left\\{\\left|y_1^\\alpha \\frac{\\d \\omega}{\\d x_1}(y, t)\\right|, \\left|y_2^\\alpha \\frac{\\d \\omega}{\\d x_2}(y, t)\\right|\\right\\}.\n\\end{equation}\nNote that $M(x,t)$ also depends on $\\omega$ and $\\alpha$.\nThe velocity field $u(x, t) := \\nabla^\\bot (-\\Delta)^{-1}\\omega$ for double-odd $\\omega$ \n($\\omega$ with mean zero over $\\mathbb{T}$) can be written in the form\n\\begin{equation}\\label{velStruc}\nu_1(x, t) = - x_1 Q_1(x, t), ~u_2(x, t) = x_2 Q_2(x, t)\n\\end{equation}\nwhere $Q_1, Q_2$ are scalar fields given by certain integral operators (see \\eqref{defQ}) acting on $\\omega$.\nThe following definition says we regard the flow as hyperbolic if both $Q_1$ and $Q_2$ essentially have a positive\nlower bound, up to a term controlled by the quantity $M(x,t)$. \n\\begin{definition}\nLet $\\omega$ be a smooth solution of the Euler equation, and let $\\alpha\\in (0, \\frac{1}{4})$ be fixed.\nWe say that the flow is hyperbolic near the origin if there are constants $\\rho, A, \\beta_0>0$ \nfor which the following condition is satisfied\n\\begin{equation}\\label{Hyperbolicity}\nQ_i(x, t) + A |x|^{1-\\alpha} M(x, t) \\geq \\beta_0 > 0 \\qquad(0\\le x_1, x_2 \\le \\rho)\n\\end{equation}\nwhere $i=1, 2$, and for all $t \\in [0, \\infty)$. \n\\end{definition}\nBy choosing the initial data $\\omega_0$  suitably, we can ensure hyperbolic flow.\nOne possible choice is, for example, choosing $\\omega_0$ to be nonnegative in $[0, 1]^2$ and such that $\\omega_0 = 1$ on a set of sufficiently large measure, as it\nwas done in \\cite{KiselevSverak}, \\cite{Zlatos}. This creates \na situation where \\eqref{Hyperbolicity} is satisfied. The proof will be given in section \\ref{PotentialTheory}.\nPhysically, we then have compression of the fluid in the $x_1$-direction and expansion of the fluid in $x_2$-direction.\n\n\\section{Gradient growth in the hyperbolic scenario}\nBefore describing our approach, let us explain first why at first sight the hyperbolic scenario seems to be a good candidate \nfor double-exponential growth. Namely, for $Q_1, Q_2$ we have the upper bounds\n\\begin{equation}\nQ_1(x, t), Q_2(x, t) \\lesssim \\|\\omega\\|_{\\infty} |\\log(x_1^2 + x_2^2)|.\n\\end{equation}\nIf it were possible to create a situation where a \\emph{lower bound} of roughly the same order holds, i.e.\n$Q_1 \\geq C |\\log(x_1^2 + x_2^2)|$ over an infinitely long time interval, then for the particle trajectories \nlying on the $x_1$-axis (i.e. $X_2=0$) \n$$\nX_1(t) \\le \\exp( - C_1 \\exp( C_2 t))\n$$\nwould hold, as seen by solving the ODE $\\dot X_1 = -X_1 Q_1$. If, moreover one could arrange for the initial data $\\omega_0$ \nto have suitable nontrivial values on the $x_1$-axis, then this would create double exponential gradient growth. \nHowever, and the simultanenous requirements of smoothness and double-odd symmetry of $\\omega$, \nnecessarily imply $\\omega=0$ on the axes. Moreover, it is highly unclear how a such strong lower bound on $Q_1$ could be achieved. \nAs we shall see later, a certain amount of smoothness of $\\omega$ and the vanishing of $\\omega$ on the axes \nlead to a better upper bound, {\\em without} the logarithmic behavior which is crucial for the double-exponential growth. \n\nAnother way one might hope to get double exponential growth is to consider a ``projectile'', i.e. to track the movement\nof a small domain close to the origin on which $\\omega = 1$, as it was done in \\cite{KiselevSverak}. There the self-interaction of the projectile\nwas able to create enough growth in the values of $Q_1$ to allow double-exponential growth. Namely, while the projectile\napproaches the origin, the values of $Q_1$ on it get larger, this fact being connected to a certain logarithmically divergent integral. \nOur Theorem \\ref{main} shows that in general this is not possible for double-odd solutions, unless there is some compression in \n$x_2$-direction in the feeding zone. Thus a scenario with maximal gradient growth must be much more complicated than \njust using the self-interaction of the projectile.\n\nIn fact, provided the feeding condition holds, the steady fluid compression guaranteed by \\eqref{Hyperbolicity} will turn out \nto stabilize the flow in the neighborhood of the origin.  That is, the hyperbolicity condition \\eqref{Hyperbolicity}\n- essentially a lower bound on $Q_i$ - is converted in the proof of Theorem \\ref{main}\ninto an upper bound for $Q_i$. This is what finally leads to a bound on the gradient growth in $D$. \n\n\\subsection{Heuristic considerations.}\nWe now present an intuitive discussion of our result. Fluid particles carried by the hyperbolic flow will\n constantly enter the box $D$ from the right and leave on\nthe top (see figure \\ref{fig1}). All particles except for those moving on the axes spend a finite time\nin the box. As for the particles on the $x_1$-axis, these move towards the\nleft, approaching the origin asymptotically as $t\\to \\infty$. Particle trajectories\n$t\\mapsto \\mathbf{X}(t)=(X_1(t), X_2(t))$ for which $X_2(0)$ is small approximate the\nstraight trajectories of the particles on the $x_1$-axis, before going steeply\nupward. The time a particle spends in $D$ goes to infinity as $X_2(0)\\to 0$.\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=1]{figure1.pdf}%\n\\end{center}\n\\caption{Illustration of a hyperbolic flow.} \\label{fig1}\n\\end{figure} \nWe now consider the trajectory of a particle $\\mathbf{X}$. The particle may \nhave started inside $D$ at time $t=0$, or may have entered the box at some\ntime $T_0 > 0$, in which case $\\mathbf{X}(T_0)\\in \\d D$. Also, assume that the particle\nexits the box $D$ at some time $T_e$, i.e. $X_2(T_e)=\\delta_2$. \nThe evolution of the gradient of $\\omega$ along the trajectory is given by an ODE of the form\n\\begin{align}\\label{ode1}\n\\frac{d}{d t}\\nabla \\omega(\\mathbf{X}(t), t) = (-\\nabla u)^T(\\mathbf{X}(t), t) \\nabla \\omega(\\mathbf{X}(t), t)\n\\end{align}\nwhere $\\nabla u$ is the velocity gradient. The relation \\eqref{ode1} is simply obtained by\ndifferentiating the Euler equation. The key is now to use the structure \\eqref{velStruc}\nof the velocity field. Combining with \\eqref{ode1}, we obtain\n\\begin{align}\\label{ode2}\n\\frac{d}{d t}\\nabla \\omega(\\mathbf{X}(t), t) = \\begin{pmatrix}\nQ_1 + x_1 \\frac{\\d Q_1}{\\d x_1}& -x_2\\frac{\\d Q_2}{\\d x_1}\\\\\nx_1 \\frac{\\d Q_1}{\\d x_2} & - Q_2 - x_2 \\frac{\\d Q_2}{\\d x_2}\n\\end{pmatrix}\n\\nabla \\omega(\\mathbf{X}(t), t)\n\\end{align}\nWe write the matrix in \\eqref{ode2} as \n\\begin{equation*}\n\\begin{pmatrix}\na(t) & c(t) \\\\\nb(t) & -a(t)\n\\end{pmatrix}\n\\end{equation*}\nevaluating all matrix entries along the given trajectory $\\mathbf{X}$ (note that the matrix has trace zero, since the\nvelocity field $u$ is divergence free). Since in a sufficiently small box\n$x_1 \\frac{\\d Q_1}{\\d x_1}, x_2 \\frac{\\d Q_2}{\\d x_2}$ should be rather ``small\" (due to the prefactors $x_1, x_2$), \n$a$ should be positive and bounded away from zero along the hyperbolic trajectory. Roughly speaking, the the form of \\eqref{ode2} implies that $\\omega_{x_1}$ grows in time like $e^{\\int_{T_0}^t a(\\mathbf{X}(s)) ds}$ whereas\n$\\omega_{x_2}$ should decay in time like $e^{-\\int_{T_0}^t a(\\mathbf{X}(s)) ds}$.\nThis would be exactly true if \\eqref{ode2} were a diagonal system.\n\nTo gain some insight, we consider the case of a particle moving close to the $x_1$-axis, i.e. with small $X_2(T_0) > 0$. We expect that\n$c = x_2 \\frac{\\d Q_2}{\\d x_1}, b = -x_1 \\frac{\\d Q_1}{\\d x_2}$ are ``small\". This suggest to neglect $b, c$ and set $b, c=0$ in \\eqref{ode2},\nso that we have a diagonal system. Denoting $\\boldsymbol{\\xi}(t) = \\nabla \\omega(\\mathbf{X}(t))$\nthe solution would be given by\n\\begin{align}\\label{eqSol}\n\\xi_1(t) = e^{A(t)} \\xi_1(T_0), ~~\\xi_2(t) = e^{-A(t)} \\xi_2(T_0).\n\\end{align}\nwhere $A(t) = \\int_{T_0}^t a(\\mathbf{X}(s)) ds$. \\eqref{eqSol} shows that, in general,\nthe gradient in $x_1$-direction grows along the particle trajectory. However, there is\nan effect which allows us to cancel the growing factor $e^A$. Assume for the sake of the discussion that the following stronger feeding conditions hold:\n\\begin{equation}\\label{eqfeeding2}\n|\\d_{x_1} \\omega(x, t)|\\leq R x_2, ~~|\\d_{x_2} \\omega(x, t)|\\leq R\n\\end{equation}\nThese imply\n\\begin{equation}\\label{eqIntro1}\n|\\xi_1(t)|\\leq R e^A(t) X_2(T_0).\n\\end{equation}\nNow we observe that\n\\begin{align}\\label{eqIntro3}\nA(t) \\approx \\int_{T_0}^t Q_2(s)~ds\n\\end{align}\ntemporarily neglecting the term $x_2 \\frac{\\d Q_2}{\\d x_2}$. Now from \\eqref{velStruc} we have the differential equation $\\dot X_2 = X_2 Q_2$, so that \n$X_2(t) = X_2(T_0) \\expo{\\int_{T_0}^t Q_2(\\mathbf{X}(s)) ds}$\nand hence\n\\begin{equation}\\label{eqIntro2}\nX_2(T_0) = X_2(T_e) \\expo{- \\int_{T_0}^{T_e} Q_2(\\mathbf{X}(s)) ds}\\leq \\delta_2\\expo{- \\int_{T_0}^{T_e} Q_2(\\mathbf{X}(s)) ds}\n\\end{equation}\nCombining \\eqref{eqIntro2}, \\eqref{eqIntro1} and \\eqref{eqIntro3}, we get\n\\begin{equation}\n|\\xi_1(t)|\\leq \\delta_2 R \\expo{ - \\int_{t}^{T_e} Q_2(\\mathbf{X}(s))~ds} \\leq \\delta_2 R\n\\end{equation}\n(we assume $Q_2\\geq 0$ for this heuristic discussion), suggesting that the gradient\nin $x_1$-direction \\emph{does not grow at all in time}. Our rigorous result does not\ngive such a strong conclusion, but we will be able to prove that the gradient \n\\emph{grows at most exponentially in time}. In Remark \\ref{remAlpha} we explain why we actually do not use \n\\eqref{eqfeeding2}. \n\nThe heuristics appear deceivingly simple, but in order to make the argument rigorous, we have to overcome a number of \nformidable technical difficulties. To begin with, the coefficients of \\eqref{ode2} depend on the solution $\\omega$ through the integral operators $Q_1, Q_2$. \nThe derivatives $\\frac{\\d Q_1}{\\d x_1}, \\frac{\\d Q_2}{\\d x_2}$ are given by singular integral operators. These can be controlled if one has \ncontrol over the first derivatives $\\frac{\\d \\omega}{\\d x_1}, \\frac{\\d \\omega}{\\d x_2}$\nof $\\omega$ inside the box, and thus one has a certain control over the coefficients of the ODE system \\eqref{ode2}. \n\nOf course, none of the coefficients may be neglected, and we have to produce sufficiently good estimates on the solutions of the full ODE system \\eqref{ode2}. A major obstacle in getting good estimates, however, is caused by the unstable nature of \\eqref{ode2}. This may be seen, e.g. by setting $c=0$, but keeping $b$, so\nthat we get a supposedly better approximation than the diagonal system. In this model, the solutions can be calculated explictly, and we get\n\\begin{align*}\n\\xi_1(t) = e^A(t) \\xi_1(T_0), ~~\\xi_2(t) = e^{-A(t)} \\left[ \\xi_2(T_0)+ \\xi_1(T_0)\\int_0^t b(s) e^{2 A(s)}~ds\\right].\n\\end{align*}\nThis shows that not only the derivative in $x_1$-direction but also the derivative in $x_2$-direction of $\\omega$ may potentially \ngrow in time (due to the contribution $e^{-A(t)}\\int_0^t b(s) e^{2 A(s)}~ds$),\nmaking things worse, since a possible strong growth in $\\frac{\\d\\omega}{\\d x_2}$ is coupled back into the coefficients of\nthe ODE \\eqref{ode2} via our estimates on $\\frac{\\d Q_1}{\\d x_1}, \\frac{\\d Q_2}{\\d x_2}$. On the other hand,\nthe factor $\\xi_1(T_0)$ may help as before, via the feeding condition \\eqref{eqfeeding2}. \nWe need therefore to proceed with extreme care, looking to cancel the growing factor $e^A$ with the decaying factor $e^{-A}$ whenever possible.\n\n\n\\section{Notation}\n\\subsection{Euler velocity field.}\nFor $x=(x_1,x_2)$ we write $\\til x = (-x_1,x_2)$ and $\\bar x=(x_1,-x_2)$. The velocity field for the Euler equation is\n\\begin{align}\\label{eqVelEuler}\nu(x, t) = \\frac{1}{2\\pi}\\int_{\\ensuremath{\\mathbb{R}}^2} \\frac{(y-x)^\\bot}{|y-x|^2} \\omega(y, t)~dy\n\\end{align}\nwhere $\\omega\\in C^2(\\mathbb{T})$ is periodically extended to all of $\\ensuremath{\\mathbb{R}}^2$. In the calculation of the integral a limit in the\nmean (sequence of unboundedly growing domains) is understood. Note that the velocity field is $\\nabla^\\bot (-\\Delta)^{-1}\\omega$,\nwhere $-\\Delta$ is the periodic Laplacian on the Torus $\\mathbb{T}$. A simple calculation using the double-odd symmetry of $\\omega$ leads to\n\\begin{align*}\nu_1(x, t) = -x_1 Q_1(x, t), ~~u_2(x, t) = x_2 Q_2(x, t)\n\\end{align*}\nwhere $Q_1, Q_2$ are the following integral operators \n\\begin{align}\\label{defQ}\\begin{split}\nQ_1(x, t)&=c_0\\int_{[0, 1]^2}[G_1^1(x,y)+G_1^2(x,y)]\\omega(y)~dy+Q_1^r(x, t)\\\\\nQ_2(x, t)&=c_0\\int_{[0, 1]^2}[G_2^1(x,y)+G_2^2(x,y)]\\omega(y)~dy+Q_2^r(x, t)\n\\end{split}\n\\end{align} \nwith kernels\n\\begin{align*}\n&G_1^1(x,y)=\\frac{y_1(y_2-x_2)}{|y-x|^2|y-\\til x|^2},\n&G_1^2(x,y)=\\frac{y_1(y_2+x_2)}{|y+x|^2|y-\\bar x|^2},\\\\\n&G_2^1(x,y)=\\frac{y_2(y_1+x_1)}{|y+x|^2|y-\\til x|^2},\n&G_2^2(x,y)=\\frac{y_2(y_1-x_1)}{|y-x|^2|y-\\bar x|^2},\n\\end{align*}\nwhere $c_0$ denotes the right constant.\nThe expression $Q_1^r$ is given by the following (limit in the mean) integral\n\\begin{align*}\nc_0 \\int_{\\ensuremath{\\mathbb{R}}^2 \\setminus {[0, 1]^2}}[G_1^1(x,y)+G_1^2(x,y)]\\omega(y)~dy,\n\\end{align*}\na similar formula holding for $Q_2^r$.\n\\subsection{Convention for estimates.} The notation $f \\lesssim g$ means\n$$\nf \\le C g,\n$$\nwhere $C$ may depend on $\\alpha, \\beta, \\|\\omega\\|_\\infty$ and on universal constants,\ne.g. geometrical characteristics of the domain $\\mathbb{T}$.  $C$ does not depend on $\\delta_1, \\delta_2, \\delta_3$.\nWhen using this notation, we shall always imply that $C<\\infty$ for all $\\alpha\\in (0, \\frac{1}{4})$.\n\n\\section{Potential theory of $Q_1, Q_2$} \\label{PotentialTheory}\n\n\\subsection{Sufficient conditions for hyperbolic flow}\n\nWe will be working with boxes of the form\n\\begin{align}\nD&=(0,\\delta_1)\\times(0,\\delta_2)\\\\\n\\widehat{D}&=(0,\\delta_1+\\delta_3)\\times(0,\\delta_2) \\label{defD},\n\\end{align}\nwith the following restriction:\n\\begin{equation}\\label{condDelta}\n0<\\delta_1 < \\delta_2 < \\delta_1+\\delta_3.\n\\end{equation}\nand $\\delta_j$ so small that $\\widehat{D}\\subset [0, 1]^2$. We also write\n\\begin{align}\nd(x) = \\delta_2 - x_2\n\\end{align}\nwhich is the distance of the point $x$ to the top of the box. We write $\\boldsymbol{\\delta}=(\\delta_1, \\delta_2), |\\boldsymbol{\\delta}|^2=\\delta_1^2+\\delta_2^2$.\n\nWe define\n\\begin{align}\nM_{D}(t) := \\max_{y\\in D}\\left\\{\\left|y_1^\\alpha \\frac{\\d \\omega}{\\d x_1}(y,t)\\right|, \\left|y_2^\\alpha \\frac{\\d \\omega}{\\d x_2}(y,t)\\right|\\right\\}\n\\end{align}\nand $M_{\\widehat{D}}$ for the analogous quantity, but where the maximum over $D$ is replaced by a maximum over $\\widehat{D}$. \nNote that $M_{D}$ and $M_{\\widehat{D}}$ depend on $\\omega$ and $\\alpha$. \n\n\n\nAs mentioned before, the flow near the origin can be made hyperbolic, with compression \nin the $x_1$-direction and expansion in $x_2$-direction by choosing the initial data such that\n$\\omega_0 \\geq 0$ on ${[0, 1]^2}$ and such that\n\\begin{equation}\\label{mes}\nm := |\\{ x : \\omega_0(x) = \\|\\omega_0\\|_\\infty \\}|\n\\end{equation}\nis sufficiently large. This is a consequence of theorem \\ref{thmLowerboundQ2}.\n\\begin{remark}\nThe periodicity and double-oddness of $\\omega(\\cdot, t)$ imply also the reflection symmetries\n\\begin{align*}\n\\omega(1+x_1, x_2, t) = -\\omega(1-x_1, x_2, t), ~\\omega(x_1, 1+x_2)=-\\omega(x_1, 1-x_2).\n\\end{align*}\nConsequently, the four corner points of $[-1, 1]\\times [-1, 1]$ are also stagnant points\nof the flow, the flow being confined in ${[0, 1]^2}$. Hence $\\omega_0 \\ge 0$ on ${[0, 1]^2}$ implies $\\omega(x, t)\\ge 0$ on ${[0, 1]^2}$ for all times,\na fact we shall use below.\n\\end{remark}\n\n\\begin{thm}\\label{thmLowerboundQ2}\nSuppose $\\omega_0(x)\\ge 0$ on ${[0, 1]^2}$. There exists a universal $0<m_0<1, 0 < K$ such that if $m_0<m<1$, there are \n$\\beta_0 > 0, A>0$ such that the following estimate holds for all times\n\\begin{align}\\label{ineqthmLowerboundQ2}\n\\begin{split}\nQ_2(x, t) + A M(x, t) |x|^{1-\\alpha} \\geq \\beta_0 \\\\\nQ_1(x, t) + A M(x, t) |x|^{1-\\alpha} \\geq \\beta_0  \n\\end{split}\n\\end{align}\nfor $|x| \\leq K (1-m)$, i.e. the flow is hyperbolic near the origin.\n\\end{thm}\n\nTo prove this, we need the following lemma, which is an adaption of a result in \\cite{Zlatos}.\n\\begin{lem}\nLet $\\Omega(2x) := [2 x_1, 1]\\times[2 x_2, 1]$. Suppose $\\omega(x)\\ge 0$ for $x\\in {[0, 1]^2}$. Then the estimate\n\\begin{equation}\nQ_i(x) \\geq c_0 \\int_{\\Omega(2x)} \\frac{y_1 y_2}{|y|^4}\\omega(y)~dy  - M(x,t) |x|^{1-\\alpha}-C_2 \\|\\omega\\|_{\\infty} \\quad (x\\in D,~i=1,2)\n\\end{equation}\nholds, with universal $C_2 > 0$.\n\\end{lem}\n\n\\begin{proof}\nWe write $G_2 = G_2^1+G_2^2$ and prove the result for $Q_2$. The proof for $Q_1$ is similar. We have\n\\begin{align*}\nQ_2(x) \\geq\t&c_0 \\int_{\\Omega(2x)} \\frac{y_1 y_2}{|y|^4}\\omega(y)~dy + \\int_{\\Omega(2x)} \\left[G_2^2(x, y) - c_0 \\frac{y_1 y_2}{|y|^4}\\right] \\omega(y)~dy\\\\\n& + \\int_{{[0, 1]^2}\\setminus \\Omega(2x)} G_2^2(x, y) \\omega(y)~dy - C_1 \\|\\omega\\|_{\\infty},\n\\end{align*}\nthrowing away the nonnegative contribution from $G_2^1$ and estimating $Q_2^r$ by $C_1  \\|\\omega\\|_{\\infty}$.\nFirst, note that straightforward calculations and estimations give\n\\begin{equation}\n\\left|G_2(x, y)-c_0  \\frac{y_1 y_2}{|y|^4}\\right|\\lesssim \\frac{(|y-x||y-\\bar x|+|y|^2)(|x|^2+|x||y|)}{|y|^2|y-x|^2|y-\\bar x|^2}.\n\\end{equation}\n$y \\in  \\Omega(2 x)$ implies that $|y-x|\\geq \\frac{1}{2}|y|, |y-\\bar x|\\geq \\frac{1}{2}|y|$ so that\n\\begin{equation}\n\\left|G_2(x, y)-c_0  \\frac{y_1 y_2}{|y|^4}\\right|\\lesssim (|y|^{-4}+|y|^{-2})(|x|^2+|x||y|)\n\\end{equation}\nand hence the integral over $\\Omega(2 x)$ is bounded in absolute value by\n\\begin{align*}\n|x|^2 \\int_{2 \\geq |y|\\geq 2|x|}(|y|^{-4}+|y|^{-2})~dy + |x| \\int_{2\\geq |y|\\geq 2|x|}(|y|^{-3}+|y|^{-1})~dy\\\\\n\\leq C|x|^2(|x|^{-2}+\\log|x|)+C|x|(|x|^{-1}+1)\\leq C.\n\\end{align*}\nFor the estimation of the integral with domain of integration ${[0, 1]^2}\\setminus \\Omega(2x)$, we distinguish two cases. \nThe more difficult case is given by the condition $x_2 \\leq x_1$, and we split the domain of integration up into\nthe three parts $[2 x_1, 1]\\times [0, 2x_2], [0, 2 x_1]\\times[2 x_1, 1]$ and $[0, 2 x_1]\\times [0, 2 x_1]$. \nFor the integral over $[2 x_1, 1]\\times [0, 2x_2]$, estimate $\\omega$ by its $L^\\infty$-norm and in the remaining integral \nwe substitute $y_j = x_j + z_j$. \n\\begin{align*}\n&\\int_{x_1}^1 \\int_{-x_2}^{x_2} \\frac{z_1(x_2+z_2)}{(z_1^2+z_2^2)(z_1^2+(2x_2+z_2)^2)} ~dz\n\\leq \\int_0^1 \\int_{-x_2}^{x_2} \\frac{2 z_1 x_2}{(z_1^2+z_2^2)(z_1^2+x_2^2)} ~dz_2~dz_1\\\\\n\\,\\,&\\leq C \\int_0^1 \\frac{z_1 x_2}{(z_1^2+x_2^2)} \\frac{1}{z_1}\\arctan(x_2\/z_1)~ d z_1 \\leq C \\arctan(1\/x_2) \\leq C.\n\\end{align*}\n The same strategy for the integral over $[0, 2 x_1]\\times[2 x_1, 1]$ leads to\n\\begin{align*}\n\\int\\limits_{-x_1}^{x_1}\\int\\limits_{x_1}^{1-x_2}  \\frac{|z_1|(x_2+z_2)}{(z_1^2+z_2^2)(z_1^2+(2x_2+z_2)^2)}~dz\n&\\leq 2  \\int\\limits_{0}^{x_1}\\int\\limits_{x_1}^{1} \\frac{z_1(x_2+z_2)}{(z_1^2+z_2^2)(z_1^2+(2x_2+z_2)^2)}~dz.\n\\end{align*}\nNoting\n\\begin{align*}\n&\\int\\limits_{0}^{x_1}\\int\\limits_{x_1}^{1} \\frac{z_1 x_2}{(z_1^2+z_2^2)(z_1^2+(2x_2+z_2)^2)}~dz\n\\leq \\int\\limits_{0}^{x_1}\\int\\limits_{x_1}^{1} \\frac{z_1 x_2}{(z_1^2+z_2^2)(z_1^2+x_2^2)}~dz_2 dz_1\\\\\n\\,\\,&\\leq C \\int\\limits_{0}^{x_1} \\frac{x_2}{z_1^2+x_2^2}~d z_1 \\leq C \\arctan(x_1\/x_2)\\leq C\n\\end{align*}\nand \n\\begin{align*}\n\\int\\limits_{0}^{x_1}\\int\\limits_{x_1}^{1} \\frac{z_1 z_2}{(z_1^2+z_2^2)(z_1^2+(2x_2+z_2)^2)}~dz &\\leq\n\\int\\limits_{0}^{x_1}\\int\\limits_{x_1}^{1} \\frac{z_1 z_2}{(z_1^2+z_2^2)^2} \\leq C\\log(2 x_1^2\/x_2^2)\n\\end{align*}\nwe can estimate the integral in question by $C\\|\\omega\\|_\\infty$.\n\nIt remains to estimate the integral over $[0, 2 x_1]\\times [0, 2 x_1]$. First note that\n\\begin{align*}\n&\\int_{[0, 2 x_1]\\times [0, 2 x_1]} G_2^2(x, y)\\omega(y)~d y\\ge\\int_{[0,  x_1]\\times [0, 2 x_1]} G_2^2(x, y)\\omega(y)~d y.\n\\end{align*}\nsince $\\omega \\ge 0$ and $G_2^2(x, y) \\ge 0$ if $y_1 \\le x_1$.\nWe will estimate the integral over $[0,  x_1]\\times [0, 2 x_1]$ in absolute value, splitting it again into\n$[0, x_1]\\times [0, x_1]$ and $[0, x_1]\\times [x_1, 2 x_1]$. First, writing $M = M(x,t)$,\n\\begin{align*}\n&\\left|\\int_{[0, x_1]\\times [0, x_1]} G_2^2(x, y)\\omega(y)~d y \\right|\\leq \\int_{[0, x_1]\\times [0, x_1]} \\frac{M y_2^{1-\\alpha}}{|y-x|\n|y-\\bar x|}\\\\\n \\leq &\\int_{[0, x_1]\\times [0, x_1]} \\frac{M |y-\\bar x|^{1-\\alpha}}{|y-x|\n|y-\\bar x|} \\le \\int_{[0, x_1]\\times [0, x_1]} M |y-x|^{-1-\\alpha}~dy\\\\\n\\leq &\\int_{B(x, r)} M |y-x|^{-1-\\alpha}~dy \\leq M r^{1-\\alpha}\n\\end{align*}\nwhere $B(x, r)$ is the smallest ball around $x$ containing $[0, 2 x_1]\\times [0, 2 x_1]$. Clearly $r \\lesssim x_1$,\nso the integral is $\\lesssim M x_1^{1-\\alpha}$.\n\nNext, for the remaining part over $[0, x_1]\\times [x_1, 2 x_1]$, we estimate $\\omega$ by $\\|\\omega\\|_{\\infty}$. We need to bound\n\\begin{align*}\n&\\int_{[0, x_1]\\times [x_1, 2 x_1]} |G_2^2(x, y)| dy = \\int_{-x_1}^0 \\int_{x_1-x_2}^{2 x_1-x_2} \\frac{|z_1| z_2}{(z_1^2+z_2^2)(z_1^2+(2x_2+z_2)^2)}dz\\\\\n&+  \\int_{-x_1}^0 \\int_{x_1-x_2}^{2 x_1-x_2} \\frac{|z_1| x_2}{(z_1^2+z_2^2)(z_1^2+(2x_2+z_2)^2)} dz.\n\\end{align*}\nFor the integral containing $|z_1| z_2$ we distinguish two cases.\nIn case $x_2 \\le \\frac{1}{2} x_1$, we use $z_1^2+(2x_2+z_2)^2\\ge z_1^2+z_2^2$, leading to a bound on the form $\\log(1+\\frac{x_1}{x_1-x_2})\\le C$.\nIf $x_2 \\ge \\frac{1}{2} x_1$, we use $z_1^2+(2x_2+z_2)^2\\ge (x_2+z_2)^2$ in the denominator and $z_2 \\le (z_2+x_2)$\nin the nominator and get the bound $C x_2^{-1} x_1 \\le C$. The integral with $|z_1|x_2$ is estimated as before.\n\nIf $x_1\\leq x_2$, we split ${[0, 1]^2} \\setminus \\Omega(2 x)$ into $[0, 1]\\times [0, x_2], [0, 2 x_1]\\times[2 x_2, 1]$\nand perform similar calculations. In this case, we do not need to use $M(x,t)$. \n\\end{proof}\n\\begin{proof}(of theorem \\ref{thmLowerboundQ2})\nFollowing \\cite{KiselevSverak}, \\cite{Zlatos} we observe that the integral $\\int_{\\Omega(2x)} y_1 y_2|y|^{-4}\\omega(y, t)~dy$ can be bounded\naway from zero by an expression of the form $C_1\\|\\omega\\|_\\infty |\\log(1-m)|$, for $|x|\\leq K (1-m)$.\nwith universal $C_1, K>0$. Hence we obtain \\eqref{ineqthmLowerboundQ2}.\n\\end{proof}\n\n\\subsection{Upper bounds}\nThe following lemma gives an upper bound on $Q_1, Q_2$, in terms of $M_{\\widehat{D}}(t)$. Recall that $d(x)$ is the \ndistance to the top of the box, so the upper bound given blows up close to the top of the box. This is, however\nnot a problem, since we mostly have to integrate $Q_1, Q_2$ along particle trajectories (see the proof \nTheorem \\ref{mainTechnicalThm}).\n\\begin{lem}\\label{lemUpperboundQ}\nFor $x\\in D$, \n\\begin{align}\nQ_i(x) \\lesssim C \\|\\omega\\|_{\\infty}(1+|\\log d(x)|) +  M_{\\widehat{D}} (t) |\\boldsymbol{\\delta}|^{1-\\alpha}\\quad (i=1, 2)\n\\end{align}\n\\end{lem}\n\\begin{proof}\nWe bound $Q_2$, the calculation for $Q_1$ is analogous. First we note\n$$\n|G_2^k| \\lesssim |x-y|^{-1}|y-\\bar x|^{-1} \\quad (k=1, 2)\n$$\nfor $y, x\\in {[0, 1]^2}$. We write $M=M_{\\widehat{D}}(t)$, and split the integral in question\ninto two parts:\n\\begin{align*}\n\\int_{{[0, 1]^2}} G_2^k (x, y) \\omega(y)~dy = \\int_{\\widehat{D}}\\ldots +   \\int_{{[0, 1]^2} \\setminus \\widehat{D}}\\ldots\n\\end{align*} \nSince $|\\omega(y)|\\lesssim M y_2^{1-\\alpha}$,\n\\begin{align*}\n&\\left|\\int_{\\widehat{D}} G_2^k(x, y)\\omega(y)~dy \\right|\\lesssim M \\int_{\\widehat{D}} y_2^{1-\\alpha}  |x-y|^{-1}|y-\\bar x|^{-1}~dy \\\\\n&\\,\\,\\lesssim M \\int_{\\widehat{D}} |x-y|^{-1}|y-\\bar x|^{-\\alpha} ~dy \\lesssim M \\int_{\\widehat{D}} |x-y|^{-1-\\alpha} ~dy\\\\\n&\\,\\,\\leq M \\int_{B(x, r)} |x-y|^{-1-\\alpha} ~dy \\leq M r^{1-\\alpha} \n\\end{align*}\nwhere $B(x, r)$ is the smallest ball centered at $x$ containing $\\widehat{D}$. Obviously $r \\lesssim |\\boldsymbol{\\delta}|$, so the part \nover $\\widehat{D}$ is dominated by $M |\\boldsymbol{\\delta}|^{1-\\alpha}$.\n\nFor the part over ${[0, 1]^2} \\setminus \\widehat{D}$, we have\n\\begin{align*}\n&\\left|\\int_{{[0, 1]^2}\\setminus \\widehat{D}} G_2^k (x, y) \\omega(y)~dy\\right| \\lesssim \\|\\omega\\|_{\\infty} \\int_{{[0, 1]^2}\\setminus \\widehat{D}} |y - x|^{-2} dy \\\\\n&\\,\\,\\lesssim \\|\\omega\\|_{\\infty} \\int\\limits_{B(x, 10)\\setminus B(x, d(x))} |y - x|^{-2} dy \n\\lesssim \\|\\omega\\|_{\\infty}|\\log d(x)|\n\\end{align*} \nwhere we have used $|G_2^k| \\lesssim |x-y|^{-1}|y-\\bar x|^{-1}$ and $|y - \\bar x|\\geq |y - x|$ for $x, y\\in {[0, 1]^2}$. Note also that for $x\\in D$,\n${[0, 1]^2} \\setminus \\widehat{D}$ is completely contained in $B(x, 10)\\setminus B(x, d(x))$ because of \\eqref{condDelta}.\n\\end{proof}\nThe following important lemma allows us to control the coefficients of the ODE system \\eqref{ode2} in terms of\nthe quantity $M_{\\widehat{D}}$. Recall that $d(x)$ is the distance from $x\\in D$ to the top of the box.\n\\begin{lem}\\label{lemFor_c_b}\nWe have the following estimates for $x\\in D$:\n\\begin{align}\n|c(x)|& \\le C(\\alpha) M_{\\widehat{D}}x_2^{1-\\alpha} \n+ C(\\alpha, \\gamma_1, \\gamma_2) x_2^{1-\\gamma_1-\\gamma_2} x_1^{\\gamma_2} d(x)^{-1+\\gamma_1+\\gamma_2},\\label{estc}\\\\\n|b(x)| &\\le C(\\alpha) M_{\\widehat{D}}x_1^{1-\\alpha}(1+|\\log d(x)|) + C(\\alpha, \\gamma) x_1^{1-\\gamma} d(x)^{-1+\\gamma},\\label{estb}\\\\\n\\left|x_i \\frac{\\d Q_i(x)}{\\d x_i} \\right|&\\le C(\\alpha) M_{\\widehat{D}}  x_i^{1-\\alpha}(1+|\\log d(x)|) + C(\\alpha, \\gamma) x_i^{1-\\gamma} d(x)^{-1+\\gamma}\\label{estxdQ}\n\\end{align}\nwhere $\\gamma, \\gamma_1, \\gamma_2\\in (0, 1), \\gamma_1+\\gamma_2 = 1$, $i=1, 2$ and the constants do not depend on $\\delta_1, \\delta_2, \\delta_3$.\n\\end{lem}\n\\begin{proof}\nThis is a consequence of Proposition \\ref{prop:A4} (see appendix) and the definition of $c, b, x_i \\d_{x_i} Q_i(x)$. Note that we have\n\\begin{align*}\n\\left|\\frac{\\d Q_i^r(x)}{\\d x_j}\\right|\\leq C \\|\\omega\\|_{\\infty} \n\\end{align*}\nfor $x\\in D$.\n\\end{proof}\n\n\\begin{remark}\\label{remAlpha}\nIt is not possible to set $\\alpha = 0$ in the estimates of Lemma \\ref{lemFor_c_b}, i.e. if we replace $M_{\\widehat{D}}$ by $\\|\\nabla \\omega\\|_{D, \\infty}$, then\ne.g. the first term on the right-hand side of the estimate for $c$ would contain a logarithmic expression\n$$\n\\|\\nabla \\omega\\|_{D, \\infty} x_2 |\\log x_2|.\n$$  \nThis is the main reason why we do not adopt the stronger feeding condition \\eqref{eqfeeding2}, since we do not know how to deal with\nthe logarithmic terms in our main argument.\n\\end{remark}\n\n\n\\section{Perturbation theory for an ordinary differential equation}\nIn this section we derive estimates for an ODE system of the form\n\\begin{align}\n\\dot\\boldsymbol{\\xi}(t) = \\left(\\begin{array}{cc} a(t) & c(t) \\\\ b(t) & -a(t)\\end{array}\\right) \\boldsymbol{\\xi}(t)\n\\end{align}\nwhere $a, b, c$ are given smooth functions on a time interval $[T_0, T_e]$. For simplicity of notation, we set $T_0=0$.\nThis part is independent of the actual structure of $a, b, c$ from the ODE \\eqref{ode2}. \n\nThe idea will be to perturb from the system with $c\\equiv 0$. We write\n\\begin{align}\n P(t):=\\left(\\begin{array}{cc} a(t) & 0 \\\\ b(t) & -a(t)\\end{array}\\right),  ~~S(t):=\\left(\\begin{array}{cc} 0 & c(t) \\\\ 0 & 0\\end{array}\\right)\n\\end{align}\n\\begin{definition}\nLet the integral operators $\\widehat{P}, \\widehat{S}$ be given by\n\\begin{align}\\label{ode_def1}\n(\\widehat{P} \\boldsymbol{\\xi})(t) = \\int_0^t P(\\tau)\\boldsymbol{\\xi}(\\tau)~d\\tau, ~~(\\widehat{S} \\boldsymbol{\\xi})(t)=\\int_0^t S(\\tau) \\boldsymbol{\\xi}(\\tau) ~d\\tau\n\\end{align}\n\\end{definition}\nRecall that $A(t)=\\int_0^t a(s) ds$. It is convenient to introduce the following operators: \n\\begin{equation}\\label{defF}\n\\begin{split}\n(F^+ g)(t) &= g(t) + e^A \\int_0^t a e^{-A} g(s) ds\\\\\n(F^- g)(t) &= g(t) - e^{-A} \\int_0^t a e^{A} g(s) ds.\n\\end{split}\n\\end{equation}\n\\begin{prop}\n\\begin{enumerate}\n\\item[(a)] The operator $(I-\\widehat{P})$ is bounded and bijective as an operator from $C[0, T]$ into $C[0, T]$.\n\\item[(b)] Consider the Volterra integral equation \n\\begin{align}\\label{ode_eq1}\n\\phi = \\widehat{P} \\phi + g\n\\end{align}\nwith given $g\\in C([0, T], \\ensuremath{\\mathbb{R}}^2)$. The solution  $\\phi = (I-\\widehat{P})^{-1} g$ is given by\n\\begin{equation}\\label{ode_eq3}\n\\begin{split}\n\\phi_1(t) &=  F^+ g_1\\\\\n\\phi_2(t) &= F^- g_2 + e^{-A} \\int_0^t e^{A} b (F^+ g_1)(s) ds\n\\end{split}\n\\end{equation}\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nThe statement (a) is standard. Statement\n(b) is an easy calculation, noting that \\eqref{ode_eq1} is equivalent to the ODE system $\\dot \\boldsymbol{\\xi} = P\\boldsymbol{\\xi} + \\dot g$\nfor $g\\in C^1$.\n\\end{proof}\nThe initial value problem for the system \n\\begin{align*}\n\\dot \\boldsymbol{\\xi} = (P+S)\\boldsymbol{\\xi}, \\,\\, \\boldsymbol{\\xi}(0)~\\text{given}\n\\end{align*}\nis equivalent to the Volterra integral equation\n\\begin{align}\\label{ode_perturbedproblem}\n\\boldsymbol{\\xi} = (\\widehat{P}+\\widehat{S})\\boldsymbol{\\xi} + \\boldsymbol{\\xi}(0).\n\\end{align}\nWe can write $\\boldsymbol{\\xi} = (I-\\widehat{P})^{-1} \\mathbf{w}$ for some $\\mathbf{w}\\in C[0, T]$. This leads to\n\\begin{align}\\label{ode_eq2}\n\\mathbf{w} = \\widehat{S} (I-\\widehat{P})^{-1}\\mathbf{w} + \\boldsymbol{\\xi}(0).\n\\end{align}\nThe following proposition gives a representation of the solution $\\boldsymbol{\\xi}$ in terms of $\\mathbf{w}$:\n\\begin{prop}\nLet $\\boldsymbol{\\xi}\\in C[0, T]$ solve the integral equation \\eqref{ode_perturbedproblem} with given $\\boldsymbol{\\xi}(0)$. Then\n\\begin{equation}\\label{relWXi}\n\\begin{split}\nw_1(t) &= \\xi_1(0)+\\xi_2(0) \\int_0^t e^{-A} c ds + \\int_0^t e^{-A} c \\int_0^s e^{A} b (F^+ w_1)(\\tau) d\\tau,\\\\\nw_2(t) &= \\xi_2(0),\\\\\n\\xi_1(t) &= (F^+ w_1)(t), ~~\\xi_2(t) = \\xi_2(0)e^{-A} + e^{-A} \\int_0^t e^{A} b (F^+ w_1) ds.\n\\end{split}\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nFirst note that \n\\begin{align}\\label{ode_eq6}\n\\widehat{S} (I-\\widehat{P})^{-1} \\mathbf{w} = \\widehat{S} \\boldsymbol{\\xi} = (\\int_0^t c(s) \\xi_2~ ds, 0)\n\\end{align}\nand hence by \\eqref{ode_eq2}, $w_2(t) = \\xi_2(0)$ (the second line of \\eqref{relWXi}). \nRecalling $\\boldsymbol{\\xi} = (I-\\widehat{P})^{-1} \\mathbf{w}$ and using \\eqref{ode_eq3}, we get\nthe following relation:\n\\begin{align}\n\\notag\\xi_2(t) = \\xi_2(0)[1 - e^{-A} \\int_0^t a e^{A} ds]  +e^{-A} \\left[ \\int_0^t e^{A} b w_1(s) ds + \\int_0^t e^{2 A} b \\int_0^s a e^{-A} w_1 d\\tau \\right]&\\\\\n\\label{ode_eq4}=e^{-A} \\left[ \\xi_2(0) +  \\int_0^t e^{A} b w_1(s) ds + \\int_0^t e^{2 A} b \\int_0^s a e^{-A} w_1 d\\tau \\right]&\n\\end{align}\n\\eqref{ode_eq6} and \\eqref{ode_eq2} together give,  \n$$\nw_1(t) = \\xi_1(0)+\\int_0^t c(s) \\xi_2(s) ds.\n$$\nBy inserting \\eqref{ode_eq4}, we get the relation\n\\begin{align*}\nw_1(t) &= \\xi_2(0) \\xi_1(0)+\\int_0^t e^{-A} c\\\\\n&\\,\\,+ \\int_0^t e^{-A} c \\int_0^s e^{A} b w_1 + \\int_0^t e^{-A} c \\int_0^s e^{2 A} b \\int_0^{\\tau} a e^{-A} w_1,\n\\end{align*}\nwhich is the first line of \\eqref{relWXi}.\n\\end{proof}\nWe will need the following Gronwall-type inequality by Wilett \\cite{Wilett}:\n\\begin{lem}\\label{lem_Willet}\nLet $z, f_0, f_1, f_2, v_1, v_2, v_3$ are nonnegative, integrable functions on $[0, T]$ and suppose $z$ satisfies the following\nintegral inequality:\n\\begin{equation}\\label{ode_ineq1}\nz(t) \\leq f_0(t) + f_1(t) \\int_0^t v_1 z + f_2(t) \\int_0^t v_2 z. \n\\end{equation}\nThen $z \\leq H f_0 $, where $H$ is the following functional \n\\begin{align}\\label{eq_H}\n(H f_0)(t) = & f_0 + f_1 \\expo{\\int_0^t v_1 f_1}\\int_0^t v_1 f_0\\\\\n&+\\left[f_2(t)+f_1(t)\\expo{\\int_0^t v_1 f_1}\\int_0^t{v_1 f_2}\\right]\\notag\\\\\n&\\times \\expo{\\int_0^t v_2\\left[f_2(s)+f_1(s)\\expo{\\int_0^s v_1 f_1}\\int_0^s{v_1 f_2}\\right]}\\notag\\\\\n&\\times \\int_0^t v_2 \\left[f_0(s)+f_1(s)\\expo{\\int_0^s v_1 f_1}\\int_0^s{v_1 f_0}\\right]\\notag\n\\end{align}\nWe write $H f_0$ to emphasize the linear dependency on $f_0$.\n\\end{lem}\n\\begin{proof}\nWe give the proof for reference. Recall first the following form basic of Gronwall's integral inequality: suppose\n$z, r, f_1, v_1$ are nonnegative functions on $[0, T]$ satisfying the integral inequality\n\\begin{equation}\\label{ode_ineq2}\nz(t) \\leq r(t) + f_1(t)\\int_0^t v_1 z,\n\\end{equation}\nthen\n\\begin{equation}\\label{ode_eq5}\nz(t) \\leq r(t)+f_1(t) \\expo{\\int_0^t v_1 f_1}\\int_0^t v_1 r~~\\quad (t\\in [0, T]).\n\\end{equation}\nSet $r = f_0 + f_2 \\int_0^t v_2 z$ and apply \\eqref{ode_eq5}. This leads to the following bound for $z$:\n\\begin{align}\\label{ode_ineq3}\nz(t) \\leq f_0 + f_2 \\int_0^t v_2 z +f_1(t) \\expo{\\int_0^t v_1 f_1}\\int_0^t v_1 \\left[f_0 + f_2 \\int_0^s v_2 z\\right].\n\\end{align}\nNote that \n\\begin{align*}\n\\int_0^t v_1 f_2 \\int_0^s v_2 z &= - \\int_0^t \\left(\\int_0^s v_1 f_2 \\right)v_2 u + \\left(\\int_0^t v_1 f_2\\right)\\int_0^t v_2 z \\\\\n&\\leq \\left(\\int_0^t v_1 f_2\\right)\\int_0^t v_2 z\n\\end{align*}\nsince $v_1, f_1, z, v_2 \\geq 0$. Thus \\eqref{ode_ineq3} implies\n\\begin{align*}\\label{ode_ineq4}\nz(t) &\\leq f_0 + f_1 \\expo{\\int_0^t v_1 f_1} \\int_0^t v_1 f_0 + \\\\\n& +\\left[f_2(t)+ f_1(t) \\expo{\\int_0^t v_1 f_1}\\left(\\int_0^t v_1 f_2\\right)\\right] \\int_0^t v_2 z. \\notag\n\\end{align*}\nApplying \\eqref{ode_eq5} again, this time with $r = f_0 + f_1 \\expo{\\int_0^t v_1 f_1}\\int_0^t v_1 f_0$, yields\nthe result \\eqref{eq_H}.\n\\end{proof}\n\\begin{lem}\nLet $\\boldsymbol{\\xi}\\in C[0, T]$ solve the integral equation \\eqref{ode_perturbedproblem} with given $\\boldsymbol{\\xi}(0)$. Then\nthe estimates \n\\begin{equation}\\label{ode_ineq5}\n\\begin{split}\n|\\xi_1(t)| &\\leq (H f_0)(t)+ e^{A} \\int_0^t |a| e^{-A} H f_0\\\\\n|\\xi_2(t)| &\\leq |e^{-A} \\xi_2(0)| + e^{-A} \\left[ \\int_0^t e^{A} |b| H f_0 + \\int_0^t e^{2 A} |b| \\int_0^s |a| e^{-A} H f_0\\right],\n\\end{split}\n\\end{equation}\nhold, where $H$ is the functional \\eqref{eq_H} and where\n\\begin{align*}\n&f_1(t) = \\int_0^t e^{-A} |c| \\int_0^s e^{2 A} |b|,\n&f_2(t)= \\int_0^t e^{-A} |c|,\\\\\n&f_0(t) = |\\xi_1(0)|+f_2(t) |\\xi_2(0)|,\n&v_1(t) = |a(t)| e^{-A},\\\\\n&v_2(t) = |b(t)| e^{A}.\n\\end{align*}\n\\end{lem}\n\\begin{proof}\nUsing obvious estimations, we get from \\eqref{relWXi} the following integral inequality for $|w_1|$:\n\\begin{align*}\n|w_1(t)|&\\leq |\\xi_1(0)| + |\\xi_2(0)| \\int_0^t e^{-A} |c|+\\int_0^t e^{-A} |c| ds ~ \\int_0^t e^{A} |b| |w_1| ds\\\\\n&+  \\int_0^t e^{-A} |c| \\int_0^s e^{2 A} |b| d\\tau ds~ \\int_0^t |a|e^{-A} |w_1|\\\\\n&= f_0(t) + f_1(t) \\int_0^t v_1 |w_1|ds + f_2(t) \\int_0^t v_2 |w_1|ds, \n\\end{align*}\nwhere the expressions $f_0, f_1, f_2, v_1, v_2$ are given as in the statement of the lemma.\nNow using lemma \\ref{lem_Willet}, we obtain $|w_1(t)|\\leq H f_0$ on $[0, T]$. The inequalities \n\\eqref{ode_ineq5} follow from $\\boldsymbol{\\xi} = (I-\\widehat{P})^{-1} \\mathbf{w}$ and the formulas \\eqref{ode_eq3}.\n\\end{proof}\n\n\\section{Main argument}\n\n\\subsection{The main technical result}\nIn order to formulate our main technical result, we introduce a notion of {\\em harmless nonlinear bound}.\n\n\\begin{definition}\nA function  $\\mathcal{N} = \\mathcal{N}(R, \\beta, \\alpha, \\boldsymbol{\\delta}, M)$ where all arguments are nonnegative numbers\n is a {\\em harmless nonlinear function} if for fixed $\\alpha\\in (0, 1), \\beta>0, \\delta_3 > 0$ the following holds: \nfor any given $R > 0$, there exists  $\\bar \\delta_2(R) > 0$ and a number $\\bar \\delta_1=\\bar \\delta_1(R, \\delta_2)$\nsuch that for all $\\delta_2 \\leq \\bar \\delta_2, \\delta_1 \\leq \\bar \\delta_1(\\delta_2)$ the inequality\n\\begin{align}\n\\mathcal{N}(R, \\beta, \\alpha, \\boldsymbol{\\delta}, R) < R\n\\end{align} \nholds.\n\\end{definition}\n\nRecall the box $\\widehat{D}$ is said to satisfy the conditions of {\\em controlled feeding} if there is a $R \\geq 0$ with\n\\begin{align}\n|\\d_{x_1}\\omega(x, t)|\\leq R x_2^{1-\\alpha}, ~~|\\d_{x_2}\\omega(x, t)|\\leq R \\quad (x\\in \\widehat{D}\\setminus D)\n\\end{align} \nfor all times $t \\geq 0$. $R$ is called \\emph{feeding parameter}. For convenience, we introduce the following definition.\n\\begin{definition}\nLet $T > 0, \\beta > 0$. We say that the flow is $\\beta$-hyperbolic in the box $D$ on $[0, T]$ if \n\\begin{equation}\nQ_i(x, t) \\geq \\beta \\quad (x\\in D, \\,t\\in [0, T], i=1, 2).\n\\end{equation}\n\\end{definition}\n\\begin{thm}\\label{mainTechnicalThm}\nLet $0 < \\alpha < 1\/4$. There exists a harmless nonlinear function $\\mathcal{N} = \\mathcal{N}(R, \\beta, \\alpha, \\boldsymbol{\\delta}, M)$ with the following properties.\nIf $\\omega$ is a solution of the Euler equation, $\\widehat{D}$ a box defined by \\eqref{defD} with parameters $\\delta_1, \\delta_2, \\delta_3 > 0$\nsatisfying \\eqref{condDelta} and $T > 0$ is such that \n\\begin{enumerate}\n\\item[(i)] the flow is $\\beta$-hyperbolic in the box $D$ on the time interval $[0, T]$, \n\\item[(ii)] the box $\\widehat{D}$ is satisfies the conditions of {\\em controlled feeding} with parameter $R>0$\n\\item[(iii)] for the initial data,\n\\begin{equation}\nM_{D}(0) < R,~~\\left|\\frac{\\d \\omega_0}{\\d x_1}(x)\\right| \\le R x_2^{1-\\alpha},~~\\left|\\frac{\\d \\omega_0}{\\d x_2}(x)\\right|\\leq R\\qquad (x \\in D),\n\\end{equation}\n\\item[(iv)] there exists a number $K$ such that\n$$\nM_{D}(t) \\leq K  \\quad (t\\in [0, T]), \n$$\n\\end{enumerate}\nthen \n\\begin{equation}\nM_{D}(t) \\leq \\mathcal{N}(R, \\alpha, \\beta, \\boldsymbol{\\delta}, K)\n\\end{equation}\nholds.\n\\end{thm}\n\\subsection{Estimates along particle trajectories}\nWe now begin the proof of our technical main result, theorem \\ref{mainTechnicalThm}. Therefore, let $\\omega$ be a given double-odd solution \nof the Euler equation that is in $C^1([0, \\infty), C^2(\\mathbb{T}))$. \nMoreover, let $\\widehat{D}$ be a box depending on the parameters $\\delta_1, \\delta_2, \\delta_3 > 0$ satisfying the conditions \\eqref{condDelta}.\n\nSuppose also that for the remainder of this section, (i)-(iv) from theorem \\ref{mainTechnicalThm} are satisfied.\nFor abbreviation, we write in the following\n\\begin{align*}\nM := \\max\\{K, R\\}.\n\\end{align*}\nWe observe the following important fact: since $\\delta_1, \\delta_2, \\delta_1 \\leq 1$,\n\\begin{equation}\\label{ineq_M}\nM_{\\widehat{D}}(t) \\leq M\n\\end{equation} \nholds.\n\nWe consider associated particle trajectories, which are the solutions of \n\\begin{equation}\\label{partODE}\n\\dot X_1 = - X_1 Q_1, \\,\\,\\,\\dot X_2 = X_2 Q_2.\n\\end{equation}\nMore precisely, we define the particle trajectories as follows: for any $(x_0, t_0)\\in \\ba D\\times [0, \\infty)$ we take the maximal\nsolution of $t\\mapsto \\mathbf{X}(t)$ of \\eqref{partODE} which passes through $(x_0, t_0)$, and lies $\\ba D$. \n$\\mathbf{X}$ is defined on an interval $[T_0, T_e]$ such that \n\\begin{enumerate}\n\\item[(i)] $\\mathbf{X}(t)\\in \\ba D$ for all $T_0 \\le t \\le T_e$,\n\\item[(ii)] either $T_0 = 0$ or $T_0 > 0$, in which case necessarily $\\mathbf{X}(T_0)\\in \\d D$,\n\\item[(iii)] $\\mathbf{X}(T_e) \\in \\d D$.\n\\end{enumerate}\nObserve that $\\mathbf{X}$ is given by\n\\begin{align}\\label{X2}\n\\begin{split}\nX_1(t) &= X_1(T_0) \\exp\\left(- \\int_{T_0}^t Q_1(\\mathbf{X}(t), s)~ds\\right) \\\\\nX_2(t) &= X_2(T_0) \\exp\\left(\\int_{T_0}^t Q_2(\\mathbf{X}(t), s)~ds\\right).\n\\end{split}\n\\end{align}\nWe call $T_0$ the entry time and $T_e$ the exit time of a particle trajectory. \n$T_0=0$ if the particle starts in $\\ba D$ for $t=0$.\n\nThe next proposition gives a upper bound for the time a particle can spend in the upper\nhalf of the box $D$, provided the flow is $\\beta$-hyperbolic.\n\\begin{prop}\\label{propTime}\nSuppose that the flow is $\\beta$-hyperbolic in the box $D$ on the time interval $[0, T]$. Let $\\mathbf{X}$ be a particle trajectory\nwhose entry time $T_0$ is $< T$.Then if \n\\begin{enumerate}\n\\item[(i)] $X_2(T_0)\\neq 0$,\n\\item[(ii)] $T_0 < T_e$,\n\\end{enumerate}\nthere is a either time $T_e > T_1\\geq T_0$ such that\n$$\nX_2(t) \\geq \\frac{1}{2} \\delta_2 \\quad (t\\in [T_1, T])\n$$\nor \n$$\nX_2(t) \\leq \\frac{1}{2} \\delta_2 \\quad (t\\in [T_0, T]).\n$$\nIf $T_1$ exists, we have the estimate\n\\begin{equation}\nT_e - T_1 \\leq {\\beta}^{-1} \\log(2).  \n\\end{equation}\n\\end{prop}\n\n\\begin{definition}\nWe call a function $g=g(\\alpha, \\beta, \\boldsymbol{\\delta}, M)$ \\emph{harmless generic factor} it has the following property:\nthere exists a $p > 0$ such that for fixed $\\alpha, \\beta, M$ \n\\begin{align*}\ng(\\alpha, \\beta, \\delta_2^p, \\delta_2, M)\n\\end{align*}\nis bounded as $\\delta_2\\to 0$.\n\\end{definition}\nFor example, a function of the form\n\\begin{align*}\ng=C(\\alpha, \\beta) \\left[\\delta_2^{\\gamma_3} M(1+|\\log \\delta_2|)+\\delta_1^{\\gamma_1}{\\delta_2}^{-\\gamma_2}+1\\right]^{\\gamma_4}\n\\end{align*}\n($\\gamma_j > 0$) is a harmless generic factor, and $e^g$ is also a harmless generic factor if $g$ is one.\nWhen performing estimations, we shall often absorb harmless generic factors into one\nanother, so the actual meaning of $g$ may change from line to line.\n\nOur goal will be to obtain estimates for the quantities $f_0, f_1, f_2, v_1, v_2$ along a single particle trajectory,\nup to the given time $T$, so that we can apply our ODE estimates.\nThe crucial point is that our bounds depend \\emph{not directly} on $\\omega,~T,~T_e$ but \nonly on $\\beta, \\alpha, \\mathbf{X}(T_0)$.  \nFor the estimations below we often refer to a fixed particle trajectory with entry time $T_0$, along\nwhich we evaluate integrals over time of the quantities $Q_1, Q_2, c$ etc. To make the notation more compact, \nwe often skip $\\mathbf{X}$ in the arguments of the integrands, e.g. we write\n$$\n\\int_{T_0}^t |c| e^{-A} ds=\\int_{T_0}^t |c| e^{-A(\\mathbf{X}(t),t)} ds = \\int_{T_0}^t |c(\\mathbf{X}(s), s)| \\exp{\\int_{T_0}^s a(\\mathbf{X}(\\tau), \\tau ) d\\tau} ~ds.\n$$\n\\begin{lem}\\label{keyLemma}\nFor any $T^*\\le T_e$,\n\\begin{equation}\nX_2(T_0) \\le \\delta_2 \\expo{-\\int_{T_0}^{T^*} Q_2(\\mathbf{X}) ds}.\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nSince the particle trajectory lies in $D$ for $t\\in [T_0, T_e]$, \n$$\\delta_2 \\ge X_2(T^*) = X_2(T_0) \\expo{\\int_{T_0}^{T^*} Q_2(\\mathbf{X}) ds}$$ holds.\n\\end{proof}\nLet $\\phi: [0, \\infty) \\to [0, \\infty)$ be a function with the properties\n\\begin{equation}\n\\label{phiEst}\\phi(s) \\leq 1-e^{-s}\n\\end{equation}\nand $\\phi$ monotone nondecreasing on $[0, \\infty)$, \n$\\phi$ linear on $[0, s^*]$ and $\\phi$ constant on $[s^*, \\infty)$ for some $s^*$. We fix such a function $\\phi$ for the following.\n\\begin{prop}\\label{propSomeEst}\nAlong a particle trajectory in a $\\beta$-hyperbolic flow in $D$, we have the following\nfor $t\\in [T_0, \\min \\{T_e,T\\}]$:\n\\begin{enumerate}\n\\item[(i)]\\begin{align*}\nX_1(t) &\\leq \\delta_1 \\expo{-\\beta (t-T_0)}\\\\\nX_2(t) &\\leq \\delta_2 \\expo{-\\beta (\\min \\{T_e,T\\}-t)},\n\\end{align*}\n\\item[(ii)]\\begin{align*\nd(\\mathbf{X}(t)) \\geq \\delta_2 \\phi\\left(\\int_{t}^{T_e} Q_2 ~ds\\right) \\geq \\delta_2 \\phi(\\beta(\\min\\{T, T_e\\}-t)),\n\\end{align*}\n\\item[(iii)]For any $\\gamma\\in (0,1)$, $t \\in [T_1, \\min\\{T_e, T\\}]$,\n\\begin{align*}\n\\int_{T_1}^{t} d(\\mathbf{X}(s))^{-1+\\gamma}~ds \\leq C(\\gamma, \\beta) \\delta_2^{-1+\\gamma},\\\\\n\\int_{T_1}^{t} |\\log d(\\mathbf{X}(s))|~ds \\leq C(\\gamma, \\beta) |\\log\\delta_2|\n\\end{align*}\nwith a $C(\\gamma, \\beta)$ independent of the trajectory.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nFor (i), recall that under the assumption of $\\beta$-hyperbolic flow, \n$Q_2 \\geq \\beta$. From \\eqref{X2}, we get\n\\begin{align}\n&X_2(t) = X_2(T_0)  \\expo{\\int_{T_0}^{\\min \\{T_e,T\\}} Q_2~ds\n- \\int_{t}^{\\min \\{T_e,T\\}} Q_2~ds}\\notag\\\\ \n&\\,\\,\\,= X_2(\\min \\{T_e,T\\})\\expo{- \\int_{t}^{\\min \\{T_e,T\\}} Q_2~ds}\\label{propSomeEstEq1}\\\\\n &\\,\\,\\,\\leq \\delta_2 \\expo{-\\beta (\\min \\{T_e,T\\}-t)}\\notag,\n\\end{align}\nnoting that $X_2(\\min \\{T_e,T\\})\\leq \\delta_2$. The bound for $X_1$ is analogous.\n\nNow we show (ii). Recall that $d(\\mathbf{X}) = \\delta_2 - X_2(t)$. Hence by \\eqref{propSomeEstEq1}\n$$\n\\delta_2 - X_2(t) \\geq \\delta_2\\left(1- \\expo{-\\int_t^{\\min \\{T_e,T\\}} Q_2 ds}\\right) = \\delta_2 \\phi\\left(\\int_t^{\\min \\{T_e,T\\}} Q_2 ds\\right).\n$$\n\n(iii) We split the integrals by introducing the time \n$T^*$ defined as follows: $T^*$ is the maximum of all $T_1 \\leq t$ such that \n$$\\phi(\\expo{\\beta(\\min\\{T_e, T\\}-s)})= \\phi(s^*).$$ \nIf there are no such $t$, we set $T^*= T_1$. Thus we split \nas follows:\n$$\n\\int_{T_1}^{t}= \\int_{T_1}^{T^*} \\ldots +  \\int_{T^*}^{t}\\ldots \n$$\nif $t \\geq T^*$, otherwise we have only one integral from $T_1$ to $t$.\nWe calculcate \n\\begin{align*}\n\\int_{T_1}^{T^*} d(\\mathbf{X}(s))^{-1+\\gamma}~ds &\\leq  \n\\delta_2^{-1+\\gamma} \\int_{T_1}^{T^*} \\phi(\\beta(\\min\\{T_e, T\\}-s))^{-1+\\gamma}~ds\\\\\n&\\leq \\delta_2^{-1+\\gamma} (T_e-T_1) \\phi(s^*)^{-1+\\gamma}~ds \\leq C(\\beta, \\gamma)\\delta_2^{-1+\\gamma}\\\\\n\\int_{T^*}^{t} d(\\mathbf{X}(s))^{-1+\\gamma}~ds &\\leq \n\\delta_2^{-1+\\gamma} \\int_{T^*}^{t} \\phi(\\beta(\\min\\{T_e, T\\}-s))^{-1+\\gamma}~ds\\\\\n&\\lesssim \\delta_2^{-1+\\gamma} \\beta^{-1+\\gamma}\\int_{T^*}^{t} (\\min\\{T_e, T\\}-s)^{-1+\\gamma}~ds\\\\\n&\\lesssim \\delta_2^{-1+\\gamma} \\beta^{-1+\\gamma}\\int_{T_1}^{\\min\\{T_e, T\\}} (\\min\\{T_e, T\\}-s)^{-1+\\gamma}~ds\\\\\n&\\lesssim \\delta_2^{-1+\\gamma} \\beta^{-1+\\gamma}\\int_{0}^{T_e-T_1} z^{-1+\\gamma}~dz\n\\lesssim \\delta_2^{-1+\\gamma} C(\\beta, \\gamma).\n\\end{align*}\nusing (ii), Proposition \\ref{propTime} to estimate $T_e-T_1$ and the fact that $\\phi$ is linear on $[0, s^*]$.\n The second integral is treated analogously.\n\\end{proof}\n\\begin{lem}\\label{lemExponential}\nAlong a particle trajectory, we have, for $T_0\\leq t\\leq \\min\\{T, T_e\\}$,\n\\begin{align*}\ne^{\\pm A(t)} &\\leq g(\\alpha, \\beta, \\boldsymbol{\\delta}, M) \\expo{\\pm \\int_{T_0}^t Q_1~ds}, \\\\\ne^{\\pm \\int_{T_0}^t Q_1 ds} &\\leq g(\\alpha, \\beta, \\boldsymbol{\\delta}, M) e^{\\pm \\int_{T_0}^t Q_2 ds}\n\\end{align*}\nwhere $g(\\alpha, \\beta, \\boldsymbol{\\delta}, M)$ are harmless factors depending only on the quantities indicated.\n\\end{lem}\n\\begin{proof}\nWe prove the second inequality of the lemma, the other ones being analogous.\nRecall $a(t) = Q_2(t) + X_2 \\d_{x_2} Q_2(t)$ and thus\n$$\n\\pm A \\leq \\pm \\int_{T_0}^t Q_2(s) ds  + \\int_{T_0}^t |X_2(s) \\d_{x_2} Q_2(s)|~ds.\n$$\nWe now use lemma \\ref{lemFor_c_b}:\n\\begin{align*}\n\\int_{T_0}^t |X_2(s) \\d_{x_2} Q_2(s)|~ds \\lesssim  M \\int_{T_0}^{\\min\\{T, T_e\\}} X_2^{1-\\alpha}(1+|\\log d(\\mathbf{X})|)  ds\\\\\n  + C(\\gamma_1, \\gamma_2) \\int_{T_0}^{\\min\\{T, T_e\\}}  X_2^{1-\\gamma_1-\\gamma_2} X_1^{\\gamma_2} d(\\mathbf{X})^{-1+\\gamma_1}~ds\n\\end{align*}\n(note that the interval of integration has been enlarged). \nWe split the interval of integration into $[T_0, T_1]$ and $[T_1,  \\min\\{T, T_e\\}]$ provided $\\min\\{T, T_e\\} \\geq T_1$. \nThe case $\\min\\{T, T_e\\} < T_1$ is analogous.\n\nIn the part over $[T_0, T_1]$, while $d(\\mathbf{X})\\geq \\frac{1}{2} \\delta_2$, we cannot control the length of the time interval, so we estimate as follows:\n\\begin{align*}\n\\int_{T_0}^{T_1}X_2^{1-\\alpha}(1+|\\log d(\\mathbf{X})|) &\\leq  \\delta_2^{1-\\alpha}\\int_{T_0}^{T_1} e^{-(1-\\alpha)\\beta (\\min\\{T, T_e\\}-s)} (C+|\\log \\delta_2|)~ds\\\\\n&\\lesssim   \\delta_2^{1-\\alpha}  |\\log \\delta_2| \\int_{0}^{\\infty} e^{-(1-\\alpha)\\beta z}~dz\\\\\n&\\lesssim C(\\alpha, \\beta) \\delta_2^{1-\\alpha}|\\log \\delta_2|,\n\\end{align*}\nusing part (i) of Proposition \\ref{propSomeEst} and $d(\\mathbf{X}(s))\\geq \\frac{1}{2} \\delta_2$ for $s\\in [T_0, T_1]$, and $\\delta_2$ sufficiently small.\nIn the part over $[T_1, \\min\\{T, T_e\\}]$ the length of the time interval is bounded but\n$d(\\mathbf{X})$ is unbounded, so we proceed differently:\n\\begin{align*}\n\\int_{T_1}^{\\min\\{T, T_e\\}} X_2^{1-\\alpha}(1+|\\log d(\\mathbf{X})|) &\\lesssim \\delta_2^{1-\\alpha} \\int_{T_1}^{\\min\\{T, T_e\\}}|\\log d(\\mathbf{X})|~ds\\\\\n&\\lesssim \\delta_2^{1-\\alpha}|\\log\\delta_2|.\n\\end{align*}\nusing statement (iii) of Proposition \\ref{propSomeEst} and $X_2\\leq \\delta_2$.\n\nFor the second integral involving $X_2^{1-\\gamma_1+\\gamma_2} X_1^{\\gamma_2} d(\\mathbf{X})^{-1+\\gamma_1}$, we note \n\\begin{align*}\n\\int_{T_0}^{T_1} X_2^{1-\\gamma_1-\\gamma_2} X_1^{\\gamma_2} d(\\mathbf{X})^{-1+\\gamma_1}&\\lesssim  \\delta_1^{\\gamma_2} \\delta_2^{1-\\gamma_1-\\gamma_2} \\delta_2^{-1+\\gamma_1} \\int_{T_0}^{T_1} \ne^{-(1-\\gamma_1-\\gamma_2)\\beta(\\min\\{T, T_e\\}-s)} ds\\\\\n&\\lesssim  \\delta_1^{\\gamma_2} \\delta_2^{-\\gamma_2} \\\\\n\\int_{T_1}^{\\min\\{T, T_e\\}} X_2^{1-\\gamma_1-\\gamma_2} X_1^{\\gamma_2} d(\\mathbf{X})^{-1+\\gamma_1} &\\lesssim \\delta_1^{\\gamma_2}\\delta_2^{-\\gamma_2}\n\\end{align*}\nby Proposition \\ref{propSomeEst}, (i) and (iii) and moreover using $X_1\\leq \\delta_1, X_2\\leq \\delta_2$.\nThis yields finally\n$$\n\\int_{T_0}^t |X_2(s) \\d_{x_2} Q_2(s)|~ds \\leq C(\\alpha, \\beta)[M \\delta_2^{1-\\alpha}|\\log \\delta_2|+(\\delta_1\/\\delta_2)^{\\gamma_2}] \n$$\nimplying the result, since the factor in square brackets is a harmless generic factor. \n\\end{proof}\n\\begin{lem}\\label{lemf2f0}\nThe following estimates hold for $T_0\\leq t \\leq \\min\\{T, T_e\\}$:\n\\begin{align}\nf_2(t)  &\\leq  g(\\alpha, \\beta, \\boldsymbol{\\delta}, M)  X_2(T_0)^{1-\\alpha}\\left[ M+\\delta_1 ^{\\alpha\/2} \\delta_2^{-1+\\alpha\/2} \\right], \\label{lemf2f0_est1}\\\\\nf_0(t) &\\leq R\\, g(\\alpha, \\beta, \\boldsymbol{\\delta}, M)  X_2(T_0)^{1-\\alpha}\\left[ M+\\delta_1 ^{\\alpha\/2} \\delta_2^{-1+\\alpha\/2} \\right].\n\\end{align}\n\\end{lem}\n\\begin{proof} We write $g = g(\\alpha, \\beta, \\boldsymbol{\\delta}, M)$ for any occuring harmless\nfactor. Using Lemma \\ref{lemFor_c_b}, \n\\begin{align*}\nf_2(t) &= \\int_{T_0}^t e^{-A}|c| \\lesssim M \\int_{T_0}^{\\min\\{T, T_e\\}} e^{-A} X_2^{1-\\alpha}~ ds\\\\\n&\\,\\,+ C(\\alpha)\\int_{T_0}^{\\min\\{T, T_e\\}} e^{-A} X_2^{1-\\alpha}X_1^{\\alpha\/2} d(\\mathbf{X})^{-1+\\alpha\/2} ds\n\\end{align*}\n(choose $\\gamma_1=\\gamma_2= \\frac{1}{2} \\alpha$).  Observe first that we can use Lemma \\ref{lemExponential} to replace\n$e^{-A}$ by $\\expo{- \\int_{T_0}^t Q_2 ds}$ and to get the estimate\n\\begin{align}\\label{lem_eq1}\ne^{-A} X_2(s)^{1-\\alpha} \n&\\leq g X_2(T_0)^{1-\\alpha} \\expo{-\\alpha\\int_{T_0}^t Q_2 ~d\\tau}\\notag\\\\\n&\\leq g X_2(T_0)^{1-\\alpha} \\expo{-\\alpha \\beta (s - T_0)} \n\\end{align}\n(we use again $Q_2\\geq \\beta$).\n\nUse \\eqref{lem_eq1} to estimate the integral containing $e^{-A} X_2^{1-\\alpha}$: \n\\begin{align*}\n\\int_{T_0}^{\\min\\{T, T_e\\}} e^{-A} X_2^{1-\\alpha} ds\n&\\leq g\\,X_2(T_0)^{1-\\alpha}  \\int_{T_0}^{\\infty} e^{-\\alpha \\beta(s-T_0)} ds \\\\\n&\\leq g\\,X_2(T_0)^{1-\\alpha} C(\\alpha, \\beta).\n\\end{align*}\nFor integral containing $e^{-A} X_2^{1-\\alpha}X_1^{\\alpha\/2} d(\\mathbf{X})^{1-\\alpha\/2}$, we use \\eqref{lem_eq1} again and estimate\n\\begin{align*}\n&\\int_{T_0}^{\\min\\{T, T_e\\}} e^{-A} X_2^{1-\\alpha}X_1^{\\alpha\/2} d(\\mathbf{X})^{1-\\alpha\/2} ds \\le \\\\\n&\\,\\,\\,g X_2(0)^{1-\\alpha} \\delta_1^{\\alpha\/2} \\int_{T_0}^{\\min\\{T, T_e\\}} e^{-\\alpha \\beta(s-T_0)}d(\\mathbf{X})^{-1+\\alpha\/2}~ds.\n\\end{align*}\nAs in the proof of Lemma \\ref{lemExponential}, we split the interval of integration into $[T_0, T_1]$ and $[T_1, \\min\\{T, T_e\\}]$ in case $T_1 \\leq \\min\\{T, T_e\\}$,\nobtaining\n\\begin{align*}\n\\int_{T_0}^{T_1} e^{-\\alpha \\beta(s-T_0)}d(\\mathbf{X})^{-1+\\alpha\/2}~ds  &\\lesssim \\delta_2^{-1+\\alpha\/2} C(\\alpha, \\beta),\\\\\n\\int_{T_1}^{\\min\\{T, T_e\\}} e^{-\\alpha \\beta(s-T_0)}d(\\mathbf{X})^{-1+\\alpha\/2}~ds &\\lesssim \\int_{T_1}^{\\min\\{T, T_e\\}} d(\\mathbf{X})^{-1+\\alpha\/2}~ds\\\\\n &\\lesssim \\delta_2^{-1+\\alpha\/2}\n\\end{align*}\nwhere we have used $d(\\mathbf{X})\\ge \\frac{1}{2} \\delta_2, e^{-\\alpha \\beta(s-T_0)}\\leq 1$ and \nProposition \\ref{propSomeEst}. In the case $T_1 \\geq \\min\\{T, T_e\\}$, we are left with only integral\nand deal with it in the same way.\n\nTo estimate $f_0$, we use that the feeding condition holds and that assumption (iii) from Theorem \\ref{mainTechnicalThm} holds. This gives \n\\begin{align*}\n|\\xi_1(T_0)| &= |\\d_{x_1}\\omega(\\mathbf{X}(T_0), T_0)| \\leq R X_2(T_0)^{1-\\alpha},\\\\\n|\\xi_2(T_0)|&=|\\d_{x_2}\\omega(\\mathbf{X}(T_0), T_0)|\\leq R\n\\end{align*}\nfor both of the cases $T_0=0$ (particle starts in $D$) and $T_0>0$ (particle starts in feeding zone). \nNow use the definition of $f_0$ and the estimate \\eqref{lemf2f0_est1} for $f_2$.\n\\end{proof}\n\\begin{lem}\\label{lemf1}\nFor $T_0\\leq t\\leq \\min\\{T, T_e\\}$, \n\\begin{align}\nf_1(t) &\\leq g(\\alpha, \\beta, \\boldsymbol{\\delta}, M) \\delta_1^{1-\\alpha} \\delta_2^{1-2\\alpha} e^{ \\alpha \\int_{T_0}^t Q_2 ds} \n\\end{align}\nwith a universal factor $g$ depending on the quantities indicated. \n\\end{lem}\n\\begin{proof}\nWe abbreviate again $g = g(\\alpha, \\beta, \\boldsymbol{\\delta}, M)$. First we claim that \n\\begin{equation}\n\\label{lemf1_claim}\n\\begin{split}\n\\int_{T_0}^t e^{2 A} |b|~ds &\\le g\\, C(\\alpha, \\beta) X_1(0)^{1-\\alpha}\n\\expo{(1+\\alpha)\\int_{T_0}^t Q_1~ds}\\\\\n&\\,\\,\\times\\left[M(1+|\\log \\delta_2|)+\\delta_1^{2\\alpha}{\\delta_2}^{-1+2\\alpha}\\right].\n\\end{split}\n\\end{equation}\nWe treat the case $T_1 \\leq t \\leq \\min\\{T, T_e\\}$. Using Lemma \\ref{lemFor_c_b}\n(recall $M_{\\omega, \\widehat{D}}\\leq M$), with $\\gamma= 2\\alpha$,  and Lemma \\ref{lemExponential} we get\n\\begin{align*}\ne^{2 A} |b| &\\leq e^{2 A} X_1^{1-\\alpha}[M(1+|\\log d(\\mathbf{X})|)+X_1^{2\\alpha} d(\\mathbf{X})^{-1+2\\alpha}]\\\\\n &\\leq g \\,e^{(1+\\alpha)\\int_{T_0}^t Q_1 ds} \\,X_1(0)^{1-\\alpha}[M(1+|\\log d(\\mathbf{X})|)+\\delta_1^{2\\alpha} d(\\mathbf{X})^{-1+2\\alpha}].\n\\end{align*}\nWe integrate this bound from $T_0$ to $t$ and split into two integrals from $T_0$ to $T_1$ and $T_1$ to $t$:\n\\begin{align*}\ng\\,X_1(0)^{1-\\alpha}[M(1+|\\log \\delta_2|)+\\delta_1^{2\\alpha} \\delta_2^{-1+2\\alpha}]\\int_{T_0}^{T_1} e^{(1+\\alpha)\\int_{T_0}^{s} Q_1 d\\tau}~ds\n\\end{align*}\nObserve for the integral:\n\\begin{align*}\n&\\int_{T_0}^{T_1}  e^{(1+\\alpha)\\int_{T_0}^{s} Q_1 d\\tau}~ds = \\int_{T_0}^{T_1}  \\frac{Q_1}{Q_1}  e^{(1+\\alpha)\\int_{T_0}^{s} Q_1 d\\tau}~ds\\\\\n&\\,\\,\\leq \\beta^{-1} (1+\\alpha)^{-1}  \\left.e^{(1+\\alpha)\\int_{T_0}^{s} Q_1 d\\tau}\\right|_{s=T_0}^{s=T_1} \\lesssim e^{(1+\\alpha)\\int_{T_0}^{T_1} Q_1 d\\tau}.\n\\end{align*}\nHence\n\\begin{align*}\n \\int_{T_0}^{T_1} e^{2 A} |b|~ds  &\\lesssim g\\, X_1(0)^{1-\\alpha}[M(1+|\\log \\delta_2|)+\\delta_1^{2\\alpha} \\delta_2^{-1+2\\alpha}] e^{(1+\\alpha)\\int_{T_0}^{T_1} Q_1}\n\\end{align*}\nFor the remaining part $\\int_{T_1}^{t} e^{2 A} |b|~ds$, we use Proposition \\ref{propSomeEst} again, and find the bound\n\\begin{align*}\n&g\\,e^{(1+\\alpha)\\int_{T_0}^t Q_1 ds} X_1(0)^{1-\\alpha} \n\\int_{T_1}^{t}[M(1+|\\log d(\\mathbf{X})|)+\\delta_1^{2\\alpha} d(\\mathbf{X})^{-1+2\\alpha}]~ds\\\\\n&\\,\\,\\lesssim g\\, X_1(0)^{1-\\alpha} \n[M|\\log \\delta_2|+\\delta_1^{2\\alpha} \\delta_2^{-1+2\\alpha}] e^{(1+\\alpha)\\int_{T_0}^t Q_1 ds}.\n\\end{align*}\nThe claim follows for the case $T_1\\leq t\\leq \\min\\{T, T_e\\}$. The calculation for $t\\leq T_1$ is similar (and slightly simpler).\n Next, using again Lemma \\ref{lemFor_c_b}, with $\\gamma= 2\\alpha$,\n\\begin{align*}\ne^{-A} |c|&\\leq e^{-A}X_2^{1-\\alpha} [M + X_2^{\\alpha\/2} X_1^{\\alpha\/2} d(\\mathbf{X})^{-1+\\alpha\/2}]\\\\\n&\\leq g\\, e^{-\\alpha \\int_{T_0}^t Q_2~ds} X_2(0)^{1-\\alpha} \\left[M + \\delta_2^{\\alpha\/2} \\delta_1^{\\alpha\/2} d(\\mathbf{X})^{-1+\\alpha\/2}\\right].\n\\end{align*}\nHence\n\\begin{align*}\n&\\int_{T_0}^t e^{-A} |c|\\int_{T_0}^s e^{2 A} |b| \\lesssim g\\, X_2(0)^{1-\\alpha} X_1(0)^{1-\\alpha}\n\\left[M(1+|\\log \\delta_2|)+\\delta_1^{\\alpha\/2}{\\delta_2}^{-1+\\alpha\/2}\\right]\\\\\n&\\,\\,\\times \\int_{T_0}^t e^{-\\alpha \\int_{T_0}^s Q_2~d\\tau} e^{(1+\\alpha)\\int_{T_0}^s Q_1~d\\tau} [M + \\delta_2^{\\alpha\/2} \\delta_1^{\\alpha\/2} d(\\mathbf{X})^{-1+\\alpha\/2}]~ ds.\n\\end{align*}\nContinuing the estimation, we have\n\\begin{align*}\n\\int_{T_0}^t e^{-\\alpha \\int_{T_0}^s Q_2 d\\tau} e^{(1+\\alpha)\\int_{T_0}^s Q_1 d\\tau} [M + \\delta_2^{\\alpha\/2} \\delta_1^{\\alpha\/2} d(\\mathbf{X})^{-1+\\alpha\/2}]~ ds\\\\\n\\leq g \\int_{T_0}^t e^{-\\alpha \\int_{T_0}^s Q_2 ds} e^{(1+\\alpha)\\int_{T_0}^s Q_2d\\tau} [M + \\delta_2^{\\alpha\/2} \\delta_1^{\\alpha\/2} d(\\mathbf{X})^{-1+\\alpha\/2}]~ ds\\\\\n\\leq g \\int_{T_0}^t e^{ \\int_{T_0}^s Q_2 d\\tau} [M + \\delta_2^{\\alpha\/2} \\delta_1^{\\alpha\/2} d(\\mathbf{X})^{-1+\\alpha\/2}]~ ds\\\\\n\\leq g \\,C(\\alpha, \\beta) e^{ \\int_{T_0}^t Q_2 ds} [M + \\delta_2^{\\alpha\/2} \\delta_1^{\\alpha\/2} \\delta_2^{-1+\\alpha\/2}]\n\\end{align*}\nwhere we have used Lemma \\ref{lemExponential} in the second line to exchange\n$e^{- \\int_{T_0}^s Q_1 d\\tau}$ for $e^{- \\int_{T_0}^s Q_2 d\\tau}$ and used the familiar splitting at $T_1$ to \nestimate the integral. Thus, finally we get \n\\begin{align*}\n\\int_{T_0}^t e^{-A} |c|\\int_{T_0}^s e^{2 A} |b|&\\lesssim g\\, X_1(0)^{1-\\alpha}\n\\left[M(1+|\\log \\delta_2|)+\\delta_1^{\\alpha\/2}{\\delta_2}^{-1+\\delta_2\/2}\\right]\\\\\n&\\,\\,\\,\\times [M + \\delta_1^{\\alpha\/2} \\delta_2^{-1+\\alpha}] e^{ \\int_{T_0}^t Q_2 ds} X_2(0)^{1-\\alpha}.\n\\end{align*}\nIt remains to apply key Lemma \\ref{keyLemma} to estimate the factor $e^{ \\int_{T_0}^{t} Q_2 ds} X_2(0)^{1-\\alpha}$,\nwhich is less than\n\\begin{align}\n&\\delta_2^{1-\\alpha} e^{ \\alpha \\int_{T_0}^t Q_2 ds}\ne^{(1-\\alpha) \\int_{T_0}^{t} Q_2 ds - (1-\\alpha) \\int_{T_0}^{\\min\\{T, T_e\\}} Q_2 ds}\n\\le \\delta_2^{1-\\alpha} e^{ \\alpha \\int_{T_0}^t Q_2 ds}\\label{lemf1_eq1}.\n\\end{align}\nNow observe that the factor $\\delta_2^{1-\\alpha}$ on the right-hand side of \\eqref{lemf1_eq1} can be combined with the expression\n$\\left[M(1+|\\log \\delta_2|)+\\delta_1^{\\alpha\/2}{\\delta_2}^{-1+\\delta_2\/2}\\right]$, giving \n$$\n\\delta_2^{1-2\\alpha} g.\n$$\n\\end{proof}\n\n\n\\begin{lem}\\label{lem11}\n\\begin{align*}\n&v_1 \\lesssim g \\left[Q_2 + M X_2^{1-\\alpha}(1+|\\log d(\\mathbf{X})|) + X_2^{1-\\alpha} d(\\mathbf{X})^{1-\\alpha}\\right] e^{-\\int_{T_0}^t Q_2 ds}\\\\\n&v_2(t) \\leq g X_1(T_0)^{1-\\alpha} \\left[M(1+|\\log d(\\mathbf{X})|)+\\delta_1^{\\alpha\/2} d(\\mathbf{X})^{-1+\\alpha\/2}\\right]  e^{\\alpha \\int_{T_0}^t Q_1 ds},\\\\\n&\\int_{T_0}^{\\min\\{T, T_e\\}} v_1 \\le g [M(1+|\\log \\delta_2|) + 1],\\\\\n&\\int_{T_0}^{\\min\\{T, T_e\\}} v_2 e^{\\alpha \\int_{T_0}^t Q_1 ds} \\le g e^{2 \\alpha \\int_{T_0}^t Q_2 ds} \\\\\n&\\int_{T_0}^{\\min\\{T, T_e\\}} v_1 f_1 \\lesssim g \\delta_1^{1-\\alpha}\\delta_2^{1-\\alpha\/2},\\\\\n&\\int_{T_0}^t v_2 f_1 \\lesssim g \\delta_1^{1-\\alpha}\\delta_2^{1-\\alpha} X_1(T_0)^{1-\\alpha} e^{2\\alpha\\int_{T_0}^t Q_2 ds}.\\\\\n&\\int_{T_0}^{\\min\\{T, T_e\\}} v_1 f_2 \\leq g X_2(0)^{1-\\alpha},\n\\end{align*}\n$g = g(\\alpha, \\beta, \\boldsymbol{\\delta}, M)$ a harmless factor.\n\\end{lem}\n\\begin{proof}\nThe estimates for $v_1$ and $v_2$ follow from Lemma \\ref{lemFor_c_b} and Lemma \\ref{lemExponential}.\nBy Proposition \\ref{propSomeEst} and the usual splitting of the interval of integration,\n\\begin{align*}\n\\int_{T_0}^{\\min\\{T, T_e\\}} v_1 &\\le g [M(1+|\\log \\delta_2|) + 1].\n\\end{align*}\nThe integral $\\int_{T_0}^{\\min\\{T, T_e\\}} v_1 e^{\\alpha \\int_{T_0}^t Q_1 ds}$ is estimated similarly.\n\nUsing Lemma \\ref{lemf1} and Lemma \\ref{lemFor_c_b} we get\n\\begin{align*}\nv_1 f_1 &= |a| e^{-A} f_1 \\leq  g \\delta_1^{1-\\alpha}\\delta_2^{1-\\alpha\/2} e^{\\alpha \\int_{T_0}^t Q_2 ds} e^{-A} \\\\\n&\\,\\,\\,\\,\\,\\times \\left[ Q_2 + M X_2^{1-\\alpha}(1+|\\log d(\\mathbf{X})|) + X_2^{1-\\alpha} d(\\mathbf{X})^{1-\\alpha}\\right]\\\\\n&\\,\\,\\lesssim g \\delta_1^{1-\\alpha}\\delta_2^{1-\\alpha\/2}  e^{(-1+\\alpha) \\int_{T_0}^t Q_2 ds} \\\\\n&\\,\\,\\,\\,\\,\\times\\left[ Q_2 + M \\delta_2^{1-\\alpha}(1+|\\log d(\\mathbf{X})|) + \n\\delta_2^{1-\\alpha} d(\\mathbf{X})^{-1+\\alpha}\\right].\n\\end{align*} \nNote how the growing factor $e^{\\alpha \\int_{T_0}^t Q_2 ds}$ was cancelled by $e^{-A}$.\nBy integrating, we can bound $\\int_{T_0}^{\\min\\{T, T_e\\}} v_1 f_1$: \n\\begin{align*}\n&\\lesssim g \\delta_1^{1-\\alpha}\\delta_2^{1-2\\alpha} \\int\\limits_{T_0}^{\\min\\{T, T_e\\}} e^{(-1+\\alpha) \\int_{T_0}^t Q_2 ds} \\\\\n&\\,\\,\\times\\left[ Q_2 + M \\delta_2^{1-2\\alpha}(1+|\\log d(\\mathbf{X})|) + \\delta_2^{1-\\alpha}d(\\mathbf{X})^{-1+\\alpha}\\right] \\\\\n&\\lesssim g \\delta_1^{1-\\alpha}\\delta_2^{1-2 \\alpha}\\left[M \\delta_2^{1-\\alpha}|\\log \\delta_2| +1 \\right]\n\\end{align*}\nyielding the desired estimate for the integral with integrand $v_2 f_1$.\n\nProceeding analogously, we bound $v_2 f_1 = e^{A} |b| f_1$ by\n\\begin{align*}\n&g \\delta_1^{1-\\alpha} \\delta_2^{1-\\alpha\/2} e^{(1+\\alpha)\\int_{T_0}^t Q_2 ds} X_1^{1-\\alpha}\\left[\nM(1+|\\log d(\\mathbf{X})|) + X_1^{\\alpha\/2} d(\\mathbf{X})^{1-\\alpha\/2}\\right]\\\\\n&\\leq g \\delta_1^{1-\\alpha} \\delta_2^{1-\\alpha\/2} X_1(T_0)^{1-\\alpha} e^{2\\alpha\\int_{T_0}^t Q_2 ds} \\left[\nM(1+|\\log d(\\mathbf{X})|) + \\delta_1^{\\alpha\/2} d(\\mathbf{X})^{1-\\alpha\/2}\\right]\n\\end{align*}\nwhere we have used $e^{(1+\\alpha)\\int_{T_0}^t Q_2 ds} X_1^{1-\\alpha}\\leq g X_1(T_0)^{1-\\alpha} e^{(1+\\alpha)\\int_{T_0}^t Q_2 ds} e^{\n-(1-\\alpha)\\int_0^t Q_1 ds}$ and used Lemma \\ref{lemExponential} to replace $Q_1$ by $Q_2$.\nIntegration now yields \n\\begin{align*}\n\\int_{T_0}^t v_2 f_1 \\leq g \\delta_1^{1-\\alpha} \\delta_2^{1-\\alpha\/2} X_1(T_0)^{1-\\alpha} e^{2\\alpha\\int_{T_0}^t Q_2 ds} \\left[\nM|\\log \\delta_2| + \\delta_1^{\\alpha\/2} \\delta_2^{1-\\alpha\/2}\\right].\n\\end{align*}\n\\end{proof}\n\\begin{lem}\\label{lemEstH}\nAlong a particle trajectory, for $T_0\\leq t \\leq \\min\\{T, T_e\\}$,\n\\begin{align}\\label{lemEstH_eq3}\n(H f_0)(t)&\\leq g \\|f_0\\|_{\\infty} \\left[1+ e^{\\alpha \\int_0^t Q_2 ds}\\right]\n\\end{align}\nholds, where $\\|f_0\\|_{\\infty} = \\|f_0\\|_{\\infty, [T_0, t]}$.\n\\end{lem}\n\\begin{proof}\nFirst we estimate the expression $f_0+f_1 \\expo{\\int_{T_0}^t v_1 f_1}\\int_{T_0}^t v_1 f_0$. Using Lemmas \\ref{lemf1}, \\ref{lem11}, we get the following bound\n\\begin{align}\n&f_0(t) +  g \\delta_1^{1-\\alpha}\\delta_2^{1-2\\alpha} e^{\\alpha \\int_0^t Q_2 ds} \\expo{g \\delta_1^{1-\\alpha}\\delta_2^{1-2\\alpha}} \\|f_0\\|_{\\infty}\\int_{T_0}^{T_e} v_1 ds \\le\\notag\\\\ \n&\\,\\,\\|f_0\\|_{\\infty} (1+g \\delta_1^{1-\\alpha}\\delta_2^{1-2\\alpha} e^{\\alpha \\int_0^t Q_2 ds} M(1+|\\log \\delta_2|))\\le\\notag\\\\\n&\\,\\,\\|f_0\\|_{\\infty} (1+g \\delta_1^{1-\\alpha}\\delta_2^{1-3\\alpha} e^{\\alpha \\int_0^t Q_2 ds}) \\le g e^{\\alpha \\int_0^t Q_2 ds} \\label{lemEstH_eq1}\n\\end{align}\nwhere we have absorbed the harmless factors $\\expo{g \\delta_1^{1-\\alpha}\\delta_2^{1-2\\alpha}}, \n\\delta_2^{\\alpha}M(1+|\\log \\delta_2|)$ into $g$.\nNext we consider \n$$f_2+f_1 \\expo{\\int_{T_0}^t v_1 f_1}\\int_{T_0}^t v_1 f_2,$$ \nfor which we get the following bound (Lemmas \\ref{lemf2f0}, \\ref{lemf1}) \n\\begin{align*}\n&g X_2(T_0)^{1-\\alpha} + g \\delta_1^{1-\\alpha} \\delta_2^{1-2\\alpha} e^{\\alpha \\int_{T_0}^t Q_2 ds} X_2(T_0)^{1-\\alpha} \\\\\n&\\,\\,\\,\\le g X_2(T_0)^{1-\\alpha} + g \\delta_1^{1-\\alpha} \\delta_2^{1-2 \\alpha}  X_2(T_0)^{1-2 \\alpha}\\\\\n&\\,\\,\\,\\le g (\\delta_2^{\\alpha}+\\delta_1^{1-\\alpha}\\delta_2^{1-2\\alpha}) X_2(T_0)^{1-2 \\alpha} \\le g X_2(T_0)^{1-2\\alpha} \\le g X_2(T_0)^{1-2\\alpha}\n\\end{align*}\nusing, in the first step, the key Lemma \\ref{keyLemma} to cancel of $e^{\\alpha \\int_{T_0}^t Q_2 ds}$ using the factor $X_2(T_0)^{\\alpha}$. \n\nSo for $v_2 \\left[f_2+f_1 \\expo{\\int_{T_0}^t v_1 f_1}\\int_{T_0}^t v_1 f_2\\right]$ we obtain the upper bound\n\\begin{align*}\nv_2 g  X_2(T_0)^{1-2 \\alpha} &\\le g e^{\\alpha \\int_{T_0}^t Q_1} X_2(T_0)^{1-2 \\alpha} \\left[M(1+|\\log d(\\mathbf{X})|)+\\delta_1^{\\alpha\/2} d(\\mathbf{X})^{-1+\\alpha\/2}\\right] \\\\\n&\\,\\,\\,\\leq g \\delta_2^\\alpha X_2(T_0)^{1-3 \\alpha} \\left[M(1+|\\log d(\\mathbf{X})|)+\\delta_1^{\\alpha\/2} d(\\mathbf{X})^{-1+\\alpha\/2}\\right] \n\\end{align*}\nusing the key lemma \\ref{keyLemma} again to cancel $e^{\\alpha \\int_{T_0}^t Q_1}$. Thus we see that \n$$\n\\expo{\\int_{T_0}^{\\min\\{T, T_e\\}} v_2 \\left[f_2+f_1 \\expo{\\int_{T_0}^s v_1 f_1} \\int_{T_0}^s v_1 f_2\\right]} \\le g\n$$\nFinally, by \\eqref{lemEstH_eq1} and Lemma \\ref{lem11}\n\\begin{align*}\n&\\int_{T_0}^t v_2 \\left[f_0+f_1 \\expo{\\int_{T_0}^s v_1 f_1}\\int_{T_0}^s v_1 f_0\\right] \\le\\\\\n&\\,\\,g \\|f_0\\|_{\\infty} \\int_{T_0}^t v_2 e^{\\alpha \\int_0^t Q_2 ds} ds \\le g \\|f_0\\|_{\\infty} e^{\\alpha \\int_0^t Q_2 ds}. \n\\end{align*} \nThus, in total we get\n\\begin{equation}\\label{lemEstH_eq2}\n\\begin{split}\n&f_0+f_1 \\expo{\\int_{T_0}^t v_1 f_1}\\int_{T_0}^t v_1 f_0 \\le g \\|f_0\\|_{\\infty} e^{\\alpha \\int_0^t Q_2 ds}\\\\\n&f_2+f_1 \\expo{\\int_{T_0}^t v_1 f_1}\\int_{T_0}^t v_1 f_2 \\le g X_2(T_0)^{1-2\\alpha}\\\\\n&\\expo{\\int_{T_0}^{\\min\\{T, T_e\\}} v_2 \\left[f_2+f_1 \\expo{\\int_{T_0}^s v_1 f_1} \\int_{T_0}^s v_1 f_2\\right]} \\le g\\\\\n&\\int_{T_0}^t v_2 \\left[f_0+f_1 \\expo{\\int_{T_0}^s v_1 f_1}\\int_{T_0}^s v_1 f_0\\right] \\le g \\|f_0\\|_{\\infty} e^{\\alpha \\int_0^t Q_2 ds}.\n\\end{split}\n\\end{equation}\nCombining the inequalities in \\eqref{lemEstH_eq2},\n\\begin{align*}\n\\left(f_2+f_1 \\expo{\\int_{T_0}^t v_1 f_1}\\int_{T_0}^t v_1 f_2\\right) \\\\\n\\times \\expo{\\int_{T_0}^{T_e} v_2 \\left[f_2+f_1 \\expo{\\int_{T_0}^s v_1 f_1}\\int_{T_0}^t v_1 f_2\\right]} \\\\\n \\times \\int_{T_0}^t v_2 \\left[f_0+f_1 \\expo{\\int_{T_0}^s v_1 f_1}\\int_{T_0}^t v_1 f_0\\right]\\\\\n\\le g X_2(T_0)^{1-4 \\alpha} \\|f_0\\|_{\\infty}  \\le g \\|f_0\\|_\\infty\n\\end{align*}\nusing again the key lemma to get rid of the factor $e^{2\\alpha \\int_{T_0}^t Q_2}$, and in the\nvery last step we used $\\alpha\\in (0, 1\/4)$.\nIn view of \\eqref{eq_H}, \\eqref{lemEstH_eq3} now follows.\n\\end{proof}\nWe can now complete the proof of our main technical result, Theorem \\ref{mainTechnicalThm}. At time $t=T$, any\n$x\\in D$ is occupied by a particle, i.e. $x = \\mathbf{X}(T)$ for some particle trajectory. Let us write \n$$\n\\d_{x_j} \\omega(\\mathbf{X}(t), t) = \\xi_j(t) \n$$\nalong that particle trajectory, and so by \\eqref{ode_ineq5},\n\\begin{align}\\label{proofMainTechnicalThmEq1}\n|\\xi_1(t)|&\\leq (H f_0)(t) + e^{A} \\int_{T_0}^t |a| e^{-A} (H f_0)(s) ds\n\\end{align}\nFirst note that by Lemmas \\ref{lemFor_c_b}, \\ref{lemEstH}, \\ref{lemf2f0}, \\ref{lem11}\n\\begin{align}\n&e^{A} \\int_{T_0}^t v_1(s) (H f_0)(s) ds \\lesssim e^{A} g \\|f_0\\|_{\\infty} \\int_{T_0}^t v_1~ds\\notag\\\\  \n&\\,\\,\\, \\le g \\|f_0\\|_{\\infty} (M |\\log\\delta_2|+1) e^{A} \n\\le  g R X_2(T_0)^{1-\\alpha} e^{A} (M |\\log\\delta_2|+1) \\label{proofMainTechnicalThmEq2}\n\\end{align} \nNote that the growing exponential factor $e^{\\alpha \\int_{T_0}^s Q_2 d\\tau}$ is cancelled by the decaying one $e^{-\\int_{T_0}^t Q_2 ds}$\nand that the integral can be estimated using the familiar splitting technique.\nMoreover again by Lemmas \\ref{lemEstH}, \\ref{lemf2f0}\n\\begin{align*}\n(H f_0)(t) \\leq  g R X_2(T_0)^{1-\\alpha} \\left[1+e^{\\alpha \\int_{T_0}^t Q_2 ds}\\right]\n\\end{align*}\nand so in view of \\eqref{proofMainTechnicalThmEq1},\n\\begin{align}\n|\\xi_1(t)|X_1(t)^\\alpha &\\leq g R X_2(T_0)^{1-\\alpha} \\left[1+e^{\\alpha \\int_0^t Q_2 ds}\\right]\\delta_1^{\\alpha} e^{-\\alpha \\int_{T_0}^t Q_1 ds}\\notag\\\\\n&+ g R (M |\\log\\delta_2|+1) e^{A} X_2(T_0)^{1-\\alpha}  \\delta_1^{\\alpha} e^{-\\alpha \\int_{T_0}^t Q_1 ds}\\delta_1^{\\alpha}\\notag\\\\\n&\\le g R \\delta_1^{\\alpha}\\label{proofMainTechnicalThmEq5} \n\\end{align}\nwhere we used the key Lemma \\ref{keyLemma} to cancel $e^A$\n$$e^{A}  X_2(T_0)^{1-\\alpha}e^{-\\alpha \\int_{T_0}^t Q_1 ds}\\le g \\delta_2^{1-\\alpha}$$\nand combined the factor $\\delta_2^{1-\\alpha}$ with $(M |\\log\\delta_2|+1)$ to get a\nharmless generic factor. In fact, this was the most critical estimate in the whole proof, since the dangerous factor $e^{A}$ was barely \ncancelled. \n\nWe now derive a similar estimate for $|\\xi_2(t)|X_2(t)^\\alpha$. From the second line of \\eqref{ode_ineq5}, \n\\begin{align}\\label{proofMainTechnicalThmEq3}\n|\\xi_2(t)| &\\leq R e^{-A} + e^{-A} \\| H f_0\\|_\\infty \\int_{T_0}^t e^{A}|b| ds+ e^{-A} \\int_{T_0}^t e^{2 A} |b| \\int_{T_0}^s |a| e^{-A} H f_0. \n\\end{align}\nFirst we observe that by Lemmas \\ref{lemEstH}, \\ref{lemf2f0}, \\ref{lem11} \n\\begin{align*}\n&\\|H f_0\\|_\\infty e^{-A} \\int_{T_0}^t e^{A}|b| ds  \\leq \\|H f_0\\|_{\\infty} e^{-A} \\int_{T_0}^t v_2 ds\\\\\n &\\,\\,\\leq g \\delta_1^{1-\\alpha} R X_2(T_0)^{1-\\alpha} \\left[1+e^{\\alpha \\int_{T_0}^t Q_2 ds}\\right] e^{-\\int_{T_0}^t Q_2 ds} e^{\\alpha\\int_{T_0}^t Q_2 ds}\\\\\n&\\,\\,\\leq  g \\delta_1^{1-\\alpha} R X_2(T_0)^{1-\\alpha}.\n\\end{align*} \n\nMoreover, using \\eqref{proofMainTechnicalThmEq2} and Lemma \\ref{lemf1}\n\\begin{align*}\n&e^{-A} \\int_{T_0}^t e^{2 A} |b| \\int_{T_0}^s |a| e^{-A} H f_0 \\leq g R X_2(T_0)^{1-\\alpha}(M|\\log \\delta_2|+1) e^{-A} \\int_{T_0}^t e^{2A}|b|\\\\\n&\\,\\,= g R X_2(T_0)^{1-\\alpha} (M|\\log \\delta_2|+1) \\delta_1^{1-\\alpha} \\delta_2^{1-2\\alpha} e^{-A} e^{(1+\\alpha) \\int_{T_0}^t Q_2 ds}\\\\\n&\\,\\,\\leq  g R \\delta_1^{1-\\alpha} X_2(T_0)^{1-\\alpha} e^{\\alpha \\int_{T_0}^t Q_2 ds}\\\\\n &\\,\\,\\leq g R \\delta_1^{1-\\alpha} X_2(T_0)^{1-2\\alpha}.\n\\end{align*}\nHence from \\eqref{proofMainTechnicalThmEq3},\n\\begin{align}\\label{proofMainTechnicalThmEq6}\n|\\xi_2(t)| \\leq g R\n\\end{align}\nand thus\n\\begin{align}\\label{proofMainTechnicalThmEq4}\n|\\xi_2(t)|X_2^{\\alpha} \\leq g R \\delta_2^{\\alpha}.\n\\end{align} follows. Inequalities \\eqref{proofMainTechnicalThmEq4} and \\eqref{proofMainTechnicalThmEq5} imply\n\\begin{align*}\nM_{D}(T) \\leq g(\\alpha, \\beta, \\boldsymbol{\\delta}, M) R \\delta_2^{\\alpha} =: \\mathcal{N}(R, \\alpha, \\beta, \\boldsymbol{\\delta}, M).\n\\end{align*}\nIt remains to show that $\\mathcal{N}$ is a harmless nonlinearity. Therefore, let $\\alpha, \\beta, R$ be given.\nRecall that $g$ has the property that $g(\\alpha, \\beta, \\delta_2^p, \\delta_2, R)$ is bounded as $\\delta_2 \\to 0$,\nwith some $p > 0$. Hence\n\\begin{align*}\ng(\\alpha, \\beta, \\delta_2^p, \\delta_2, R) R \\delta_2^{\\alpha} < R\n\\end{align*}\n for sufficiently small $\\delta_2 > 0$.\n\n\n\\subsection{Proof of the main result}\nWe are now ready to prove Theorem \\ref{main}. So let \n$\\alpha\\in (0, \\frac{1}{4})$ and $R \\geq 0$ be a given nonnegative number.\nLet $\\mathcal{N}$ be the harmless nonlinear function from theorem \\ref{mainTechnicalThm}. \nFix small positive $\\delta_1, \\delta_2$ such that the following set of inequalities hold true:\n\\begin{align}\n\\delta_1, \\delta_2 \\leq \\rho, ~\\beta_0 - A |\\boldsymbol{\\delta}|^{1-\\alpha} R \\ge \\frac{1}{2} \\beta_0. \\label{mainThm_ineq3}\\\\\nM_{D}(0) < R, ~\\left|\\frac{\\d \\omega_0}{\\d x_1}\\right|\\le R x_2^{1-\\alpha}, ~\\left|\\frac{\\d \\omega_0}{\\d x_2}\\right|\\le R, \\label{mainThm_ineq4}\\\\\n\\mathcal{N}(R, \\alpha, \\frac{1}{2}\\beta_0, \\delta_1, \\delta_2, R) < R, \\label{mainThm_ineq2}\n\\end{align}\nwhere $A, \\beta_0, \\rho$ are the numbers from the definition of the hyperbolicity of the flow.\nNote that the box can be chosen so small that that \\eqref{mainThm_ineq4} holds. This a \nconsequence of $\\frac{\\d \\omega_0}{\\d x_2}(0, x_2)=0$ and the\n$C^2$-smoothness of $\\omega_0$.\n\nWe claim now that if the box $\\widehat{D}$ satisfies the controlled feeding conditions with parameter $R$,\nthen we have the bound\n\\begin{align}\\label{mainThm_ineq1}\nM_{D}(t) \\leq R\\quad (t\\in [0, \\infty))\n\\end{align}\nfor all times. Assume \\eqref{mainThm_ineq1} is not true for all times.\nSince $M_{D}(0) < R$, and the solution $\\omega$ is sufficiently smooth in time by assumption,\nthere exists a time $T > 0$ such that $M_{D}(t) < R$ holds\non $[0, T)$ and $M_{D}(T)=R$. Moreover, by \\eqref{mainThm_ineq3},\nthe flow is $\\frac{1}{2} \\beta_0$-hyperbolic in the box $D$ on the time interval $[0, T]$.\nObserve that because of \\eqref{condDelta} and the feeding conditions,\n $M(x,t)\\le M_{\\widehat{D}}(t)\\le R$ for all $x\\in D$ and $t\\in[0,T]$.\n\n\\eqref{mainThm_ineq4} implies that (ii) in the formulation of Theorem \\ref{mainTechnicalThm} holds.\nAlso, on $[0, T]$, we have $M_{D}(t) \\le R$.\nApplying Theorem \\ref{mainTechnicalThm} (choose $K=R$) and \\eqref{mainThm_ineq2}, we get\n$$\nM_{D}(T) \\leq \\mathcal{N}(R, \\alpha, \\frac{1}{2}\\beta_0, \\delta_1, \\delta_2, R) < R.\n$$\nIn combination with \\eqref{mainThm_ineq2}, this gives\n$$\nM_{D}(T) < R,\n$$\na contradiction. This proves \\eqref{mainThm_ineq1}. Now we prove the exponential bound on the gradient growth.\nAt an arbitrary time $t \\geq 0$, each $x \\in D$ is occupied by a particle $\\mathbf{X}(t)$ that has entered the box at some earlier\ntime $T_0$. The same calculation leading to \\eqref{proofMainTechnicalThmEq5}, using \\eqref{proofMainTechnicalThmEq1}\nand \\eqref{proofMainTechnicalThmEq2} yields \n\\begin{align*}\n\\left|\\frac{\\d \\omega}{\\d x_1}(X(t), t)\\right|&\\leq g(\\alpha, \\beta, \\boldsymbol{\\delta}, R) R \\delta_2^{1-\\alpha}\\left[1+e^{\\alpha \\int_{T_0}^t Q_1 ds}\\right].\n\\end{align*} \nfor all $t\\in [T_0, T_e]$ on account of \\eqref{mainThm_ineq1}.\n\nWe apply now Lemma \\ref{lemUpperboundQ}\n\\begin{align*}\n\\int_{T_0}^t Q_1 ds &\\leq \\int_{T_0}^t (C\\|\\omega\\|_{\\infty}+ R|\\boldsymbol{\\delta}|^{1-\\alpha}) (t-T_0) + \\|\\omega\\|_{\\infty} \\int_{T_0}^t |\\log d(\\mathbf{X})| ds  \n\\end{align*}\nThe integral containing the logarithmic term can be estimated using the familiar splitting at $T_1$ and gives\n\\begin{align*}\n\\int_{T_0}^t |\\log d(\\mathbf{X})|~ds &\\leq C (t - T_0)|\\log \\delta_2| + C(\\alpha, \\beta). \n\\end{align*}\nThus, finally, \n\\begin{align*}\n\\left|\\frac{\\d \\omega}{\\d x_1}(\\mathbf{X}(t), t)\\right| \\leq g(\\alpha, \\beta, \\boldsymbol{\\delta}, R) R \\delta_2^{1-\\alpha} e^{\\alpha(C\\|\\omega\\|_{\\infty}+ R|\\boldsymbol{\\delta}|^{1-\\alpha}) t}.\n\\end{align*}\nThe derivative in $x_2$-direction is bounded by \\eqref{proofMainTechnicalThmEq6}.\nThis concludes the proof of Theorem \\ref{main}.\n\n\\section{Acknowlegdements}\nThe authors cordially thank A. Kiselev for suggesting the problem and a great number of helpful discussions. \nVH would like to express his gratitude to the Deutsche Forschungsgemeinschaft (German Research Foundation),\nwithout whose financial support (FOR HO 5156\/1-1) the present research could not have been undertaken.\n\n\\section{Appendix}\n\n\n\\subsection{Appendix A}\n\n\\begin{prop}\\label{prop:A1}\nFor all $x,y\\in{[0, 1]^2},~x\\neq y$ the following estimates hold.\n\\begin{align}\n\\left|G_i^k(x,y)\\right|&\\lesssim|y-x|^{-1}x_i^{-1} \\\\\n\\left|\\frac{\\d G_i^k}{\\d x_j}(x,y)\\right|&\\lesssim|y-x|^{-3}\n\\end{align}\n$(i, k = 1, 2)$.\n\\end{prop}\nThe proofs are straightforward calculations based on the identities in Appendix B, and the reflection identities:\n\\begin{align*}\n|y-\\til x| \\geq |y-x|, |y-\\bar x| \\geq |y-x|, |y+x| \\geq |y-\\bar x| \n\\end{align*}\nholding for $x, y\\in {[0, 1]^2}$.\nAlso, use the obvious inequalities\n\\begin{align*}\nx_2 \\leq |y-\\bar x|, x_1 \\le |y-\\til x|.\n\\end{align*}\n\nWe observe some useful relations for the kernels $G_i^k$ and their derivatives. Let $G$ stand for any $G_i^k$ and let\n$$\n\\Omega_x = (-x_1, 1-x_1)\\times(-x_2, 1-x_2).\n$$\n$G$ has the form $G(x, y) = \\widetilde G(y-x, x, y)$, where $\\widetilde G(z, \\eta, \\mu)$ is \nsmooth provided $\\eta, \\mu\\in (0,1)^2, z\\in \\Omega_x \\setminus \\{0\\}$. For example, if $G=G_1^1$ then\n$$\n\\widetilde G(z, \\eta, \\mu)=\\frac{\\mu_1 z_1}{|z|^2 |\\mu-\\til \\eta|^2}.\n$$\nNote that for $x\\neq y, x, y\\in (0, 1)^2$,\n\\begin{align}\n(\\d_{x_j} G)(x, y) = (\\d_{\\eta_j}\\widetilde G)(y-x, x, y) - (\\d_{z_j}\\widetilde G)(y-x, x, y)\\label{dev_eq4}\\\\\n(\\d_{y_j} G)(x, y) = (\\d_{\\mu_j}\\widetilde G)(y-x, x, y) +(\\d_{z_j}\\widetilde G)(y-x, x, y)\\notag\n\\end{align}\nso that\n\\begin{equation}\\label{relDevXY}\n(\\d_{x_j} G)(x, y) = -(\\d_{y_j} G)(x, y) +(\\d_{\\eta_j}\\widetilde G)(y-x, x, y) + (\\d_{\\mu_j}\\widetilde G)(y-x, x, y).\n\\end{equation}\nMoreover, we always have\n\\begin{align}\\label{dev_ineq1}\n|\\widetilde G(z, x, y)|, \\left|\\frac{\\d \\widetilde G}{\\d \\eta_j}(z, x, x+z)\\right|, \\left|\\frac{\\d \\widetilde G}{\\d \\mu_j}(z, x, x+z)\\right|\\leq C(\\eta)|z|^{-1}.\n\\end{align}\nwhere $C(\\eta)$ is uniformly bounded if $\\eta$ varies in a compact subset of $(0, 1)^2$.\n\n\\begin{prop}\\label{prop:A7}\n\\begin{equation}\n\\frac{\\d G_i^k}{\\d x_j}=-\\frac{\\d G_i^k}{\\d y_j}+x_i^{-2}\\delta_{ij}\\mathcal{O}(|y-x|^{-1})\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nThis is a straighforward calculation using \\eqref{relDevXY}. \n\\end{proof}\n\n\n\n\n\\begin{prop}[Derivatives of $Q_i$]\\label{prop:A2}\n\\begin{align*}\n\\frac{\\d Q_i}{\\d x_j}&=\nc_0P.V.\\!\\!\\int_{[0,1]^2}\\left[\\frac{\\d G_i^1}{\\d x_j}+\\frac{\\d G_i^2}{\\d x_j}\\right] \\omega(y)~dy\\\\\n& - \\omega(x)\\lim_{\\delta \\to 0^+}\\int_{\\d B(\\delta,x)} G_i^i\\cdot\\nu_j~d\\sigma+\\frac{\\d Q_i^3}{\\d x_j}\n\\end{align*}\n\\end{prop}\n\n\\begin{proof}\nWrite  $(G_i^1+G_i^2)(x, y):=G(x, y)$. $G$ has again the form $G(x, y) = \\widetilde G(y-x, x, y)$, where $\\widetilde G(z, \\eta, \\mu)$ is \nsmooth provided $\\eta, \\mu\\in (0,1)^2, z\\in \\Omega_x \\setminus \\{0\\}$. Also \\eqref{dev_eq4}, \\eqref{dev_ineq1} hold for $\\widetilde G$.\nNow \n\\begin{align}\n \\label{dev_eq3}\\frac{\\d}{\\d x_j} \\int_{\\Omega_x} \\widetilde G(z, x, x+z) \\omega(x+z)~dz= \\int_{\\Omega_x} \\widetilde G(z, x, x+z) \\frac{\\d \\omega}{\\d z_j}(x+z)~dz\\\\\n+\\int_{\\Omega_x} \\d_{x_j}(\\widetilde G(z, x, x+z)) \\omega(x+z)~dz\\notag \\\\\n- \\int_{\\d \\Omega_x} \\widetilde G(z, x, x+z) \\omega(x+z)\\nu_j~d\\sigma\\notag\n\\end{align}\nwhere $\\nu_j$ denotes the $j$-th component of the unit outer normal. This is a standard differentiation result (note the\nbounds \\eqref{dev_ineq1}).\n\nNow consider the integral in the line \\eqref{dev_eq3}, exclude the singularity and integrate by parts:\n\\begin{align}\\label{dev_eq5}\n\\int_{\\Omega_x} \\widetilde G(z, x, x+z) \\frac{\\d \\omega}{\\d z_j}(x+z)~dz =- \\int_{\\Omega_x\\setminus B(0, \\delta)} \\d_{z_j}(\\widetilde G(z, x, x+z)) \\omega(x+z)~dz\\\\\n+ \\int_{\\d \\Omega_x} \\widetilde G(z, x, x+z) \\omega(x+z)\\nu_j~d\\sigma \\notag\\\\\n- \\int_{\\d B(0, \\delta)} \\widetilde G(z, x, x+z) \\omega(x+z)\\nu_j~d\\sigma\\notag\n\\end{align}\nObserve that by \\eqref{dev_eq4},\n\\begin{align}\\notag\n-\\d_{z_j}(\\widetilde G(z, x, x+z))+\\d_{x_j}(G(z, x, x+z)) = (\\d_{x_j}G)(z, x, x+z).\n\\end{align}\nSo combining \\eqref{dev_eq3} and \\eqref{dev_eq5}, we finally get\n\\begin{align*}\n\\frac{\\d}{\\d x_j} \\int_{\\Omega_x} \\widetilde G(z, x, x+z) \\omega(x+z)~dz =  -\\int_{\\d B(0, \\delta)} \\widetilde G(z, x, x+z) \\omega(x+z)\\nu_j~d\\sigma\\\\\n+ \\int_{\\Omega_x}  (\\d_{x_j}G)(z, x, x+z) \\omega(x+z)~dz + \\int_{B(0, \\delta)} \\d_{x_j}(\\widetilde G(z, x, x+z)) \\omega(x+z)~dz.\n\\end{align*}\nReplacing $x+z$ by $y$ and sending $\\delta\\to 0$ yields the statement.\n\\end{proof}\n\nRecall that \n\\begin{equation}\nd_1(x) = \\min\\{x_1, x_2\\}\n\\end{equation}\nis the distance of the point $x$ to the coordinate axes.\nObserve also that\n\\begin{align}\\label{relXY}\n\\frac{1}{2} x_r \\le y_r \\le \\frac{3}{2}x_r \n\\end{align}\nfor $y\\in B(\\frac{1}{2} d_1(x), x), r=1, 2$. For the entire appendix, we shall write that\n$M = M_{\\widehat{D}}$, i.e.\n\\begin{equation}\n\\left|\\frac{\\d\\omega}{\\d x_j}(x)\\right| \\le M x_j^{-\\alpha}\\quad (x \\in \\widehat{D}, j=1, 2)\n\\end{equation}\nholds, implying also the inequalities\n\\begin{equation}\\label{estOm}\n|\\omega(x)| \\lesssim M x_j^{1-\\alpha} \\quad (x\\in \\widehat{D}, j=1, 2)\n\\end{equation}\n(by the fact that $\\omega$ vanishes identically on the coordinate axes).\nFigure \\ref{fig2} illustrates the domains we need in the proof of the following propositions. \n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=1]{figure2.pdf}%\n\\end{center}\n\\caption{Domains of integration in Proposition \\ref{prop:A3}.} \\label{fig2}\n\\end{figure}\n\\begin{prop}\\label{prop:A3} Let $I=B(\\frac{1}{2} d_1(x),x)\\cap \\widehat{D}$. Then\n\\begin{equation}\n\\left|P.V.\\!\\!\\int_{I}\\frac{\\d(G_i^1+G_i^2)}{\\d x_j}\\omega(y)~dy\\right| \\lesssim M x_i^{-\\alpha}(1+\\delta_{j2}|\\log d(x)|)\\quad (i\\neq j)\n\\end{equation}\n\\end{prop}\n\n\\begin{proof}\nFirst let $i\\neq j$ and $0<\\delta<\\frac{1}{2} d_1(x)$. By Proposition \\ref{prop:A7} and integration by parts,\n\\begin{equation}\\label{proofOfPropA3eq1}\n\\begin{split}\n&\\left|\\int_{I\\setminus B(\\delta,x)}\\frac{\\d G_i^k}{\\d x_j}\\omega(y)~dy\\right|=\\left|\\int_{I\\setminus B(\\delta,x)}\\frac{\\d G_i^k}{\\d y_j}\\omega(y)~dy\\right|\\\\\n&\\lesssim\\left|\\int_{I\\setminus B(\\delta,x)}G_i^k\\frac{\\d \\omega}{\\d y_j}(y)~dy\\right|+ \\left|\\int_{\\d I} G_i^k  \\omega(y)\\nu_j~d\\sigma \\right| \n+ \\left|\\int_{\\d B(\\delta,x)}G_i^k \\omega(y)\\nu_j~d\\sigma\\right|\n\\end{split}\n\\end{equation}\nwhere $\\nu=(\\nu_1,\\nu_2)$ is the unit outward pointing normal on $\\d I$. We first take care of the integral over $\\d I$.\nObserve that for $x\\in D$, $\\d I$ is either a full circle is the union of a part of a circle and a flat part $\\Sigma$. Hence\n\\begin{align*}\n\\left|\\int_{\\d I} G_i^k \\nu_j~d\\sigma\\right|&\\leq \\int_{\\Sigma} |G_i^k| \\delta_{2j} ~d\\sigma +\\int_{\\d B(0, \\frac{1}{2} d_1(x))} |G_i^k(\\varphi)|~d\\sigma\n\\end{align*}\nFor all sufficiently small $\\varepsilon>0$,\n\\begin{align*}\n\\int_{\\Sigma} |G_i^k|~d\\sigma &\\le\\int_{\\Sigma\\cap \\{|y_1-x_1|\\le \\varepsilon\\}} |G_i^k|~d\\sigma+\\int_{\\Sigma\\cap \\{|y_1-x_1|\\ge \\varepsilon\\}} |G_i^k|~dy_1\\\\\n&\\lesssim\\int_{\\Sigma\\cap \\{|y_1-x_1|\\le \\varepsilon\\}} \\frac{x_i^{-1}}{|y-x|}~dy_1\n+ \\int_{\\Sigma\\cap \\{|y_1-x_1|\\ge \\varepsilon\\}} \\frac{x_i^{-1}}{|y-x|}~dy_1\\\\\n&\\lesssim x_i^{-1}\\int_{x_1-\\varepsilon}^{x_1+\\varepsilon} \\frac{1}{|x_2-\\delta_2|}~dy_1\n+ x_i^{-1}\\int_{1>|x_1-y_1|>\\varepsilon} \\frac{1}{|y_1-x_1|}~dy_1\\\\\n&\\lesssim \\frac{x_i^{-1}\\varepsilon}{|x_2-\\delta_2|}+ x_i^{-1}\\int_{\\varepsilon}^1 \\frac{1}{y_1}~dy_1\\\\\n&\\lesssim \\frac{x_i^{-1}\\varepsilon}{|x_2-\\delta_2|}+ x_i^{-1}|\\log\\varepsilon|.\n\\end{align*}\nHere we used proposition \\ref{prop:A1} again.\nChoosing $\\varepsilon=|x_2-\\delta_2|=d(x)$ we get\n\\begin{align*}\n\\int_{\\Sigma} |G_i^k|~d\\sigma\n&\\lesssim x_i^{-1}(1+|\\log d(x)|).\n\\end{align*}\nThe other part is estimated by (using proposition \\ref{prop:A1} again)\n\\begin{align*}\n\\int_{\\d B(0, \\frac{1}{2} d_1(x))} |G_i^k(\\varphi)|~d\\sigma\n&\\lesssim\nx_i^{-1}\\int_{0}^{2\\pi} |y-x|^{-1} d_1(x) ~d\\varphi\n\\lesssim x_i^{-1}.\n\\end{align*}\nTherefore we get for the integral over $\\d I$, using \\eqref{estOm} and \\eqref{relXY}:\n\\begin{align*}\n\\left|\\int_{\\d I} G_i^k \\omega(y)\\nu_j~d\\sigma \\right|&\\lesssim \\int_{\\d I}  \\left| G_i^k\\nu_j \\right|  \\left|\\omega(y) \\right|~d\\sigma\n\\lesssim M \\int_{\\d I}  \\left|G_i^k\\nu_j \\right|  y_i^{1-\\alpha}~d\\sigma\\\\\n&\\lesssim M x_i^{1-\\alpha}\\int_{\\d I}  \\left|G_i^k\\nu_j \\right| ~d\\sigma\n\\lesssim  M x_i^{1-\\alpha}x_i^{-1}(1+\\delta_{j2}|\\log d(x)|).\n\\end{align*}\nSimilar estimates yield that the contribution from the integral over \n$\\d B(\\delta,x)$ is $\\lesssim M x_i^{-\\alpha}$, with universal constants independent of $\\delta$.\nFor the remaining integral we have, using \\eqref{relXY}:\n\\begin{align*}\n\\left|\\int_{I\\setminus B(\\delta,x)}G_i^k\\frac{\\d \\omega}{\\d y_j}(y)~dy\\right|&\\lesssim M x_i^{-\\alpha}\\int_{I\\setminus B(\\delta,x)}\\left|G_i^k\\right|~dy\n\\lesssim M x_i^{-\\alpha} \\int_{I\\setminus B(\\delta,x)}\\left|y-x\\right|^{-1}x_i^{-1}~dy\\\\\n&\\lesssim M x_i^{-\\alpha} x_i^{-1}\\int_{\\delta}^{d_1(x)}\\frac{1}{\\rho}\\rho ~d\\rho \\lesssim M x_i^{-\\alpha}x_i^{-1}\\int_{\\delta}^{d_1(x)}\\frac{1}{\\rho}\\rho ~d\\rho\\\\\n&\\lesssim M x_i^{-\\alpha} x_i^{-1}d_1(x).\n\\end{align*}\nSince $d_1(x)\\le x_j$ we get:\n\\bn\n\\left|\\int_{I\\setminus B(\\delta,x)}G_i^k\\frac{\\d \\omega}{\\d y_j}(y)~dy\\right|\\lesssim M x_i^{-\\alpha}.\n\\end{equation*}\nThis concludes the case $i\\neq j$, since all the estimates are independent of $\\delta$.\n\nNow let $i=j$. We argue as before, but in \\eqref{proofOfPropA3eq1} now an additional term of the\nform\n$$\nx_i^{-2} \\int_{I\\setminus B(\\delta, x)} \\mathcal{O}(|y-x|^{-1}) |\\omega(y)| dy\n$$\nappears, for which we obtain the estimate $\\lesssim M x_i^{-\\alpha}$ by the same methods.\n\\end{proof}\n\\begin{lem}\\label{lem:A2} Let $\\gamma\\in (0,1)$ and $x_1\\geq 0$. Then\n\\begin{equation}\ny_1^\\gamma\\le|y_1-x_1|^\\gamma+x_1^\\gamma\\quad (y_1\\geq 0).\\notag\n\\end{equation}\n\\end{lem}\n\\begin{proof} If $y_1\\le x_1$, the inequality is obvious. For $y_1>x_1$ we have $y_1\\ge y_1-x_1>0$ and hence \n$\\gamma y_1^{\\gamma-1}\\le\\gamma(y_1-x_1)^{\\gamma-1}$ so that\n$$\ny_1^\\gamma-x_1^\\gamma\\le\\gamma\\int_{x_1}^{y_1}s^{\\gamma-1}~ds\\le\\gamma\\int_{x_1}^{y_1}(s-x_1)^{\\gamma-1}~ds=(y_1-x_1)^\\gamma.\n$$\n\\end{proof}\n\\begin{prop}\\label{prop:A8} Let $II=\\widehat{D} \\setminus I$ with $I$ as in proposition \\ref{prop:A3}. Then\n\\begin{equation}\n\\left| \\int_{II}\\frac{\\d G_i^k}{\\d x_j}\\omega(y)~dy\\right| \\lesssim M x_i^{-\\alpha}\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nUsing proposition \\ref{prop:A1}, \\eqref{estOm} and lemma \\ref{lem:A2} we have\n\\begin{align*}\n&\\left| \\int_{II}\\frac{\\d G_i^k}{\\d x_i}\\omega(y)~dy\\right| \\lesssim \\int_{II} \\left|\\frac{\\d G_i^k}{\\d x_i}\\right||\\omega(y)|~dy\\\\\n&\\lesssim \\int_{II} |y-x|^{-3}y_i^{1-\\alpha}~dy\\lesssim \\int_{II} |y-x|^{-3}(|y_i-x_i|^{1-\\alpha}+x_i^{1-\\alpha})\\\\\n&\\lesssim \\int_{II} |y-x|^{-2-\\alpha}~dy+x_i^{1-\\alpha}\\int_{II} |y-x|^{-3}\\\\\n&\\lesssim \\int_{\\frac{1}{2} d_1(x)}^\\infty\\frac{1}{\\rho^{2+\\alpha}}~\\rho~d\\rho\n+ x_i^{1-\\alpha}\\int_{\\frac{1}{2} d_1(x)}^\\infty\\frac{1}{\\rho^{3}}~\\rho~d\\rho\\\\\n&\\lesssim d_1(x)^{-\\alpha}+x_i^{1-\\alpha}d_1(x)^{-1}\\lesssim x_i^{-\\alpha}+x_i^{1-\\alpha}x_i^{-1}\\lesssim x_i^{-\\alpha}.\n\\end{align*}\n\\end{proof}\n\\begin{prop}\\label{prop:A6}\nFor $i\\neq j$, we have\n\\begin{equation}\n\\left|\\frac{\\d (G_{i}^1+G_{i}^2)}{\\d x_j}\\right|  \\lesssim  x_i^{-\\gamma_1-\\gamma_2} x_j^{\\gamma_2} |y-x|^{-(3 - \\gamma_1)}.\n\\end{equation}  \nwhere $\\gamma_1, \\gamma_2\\in [0, 1], \\gamma_1+\\gamma_2\\leq 1$.\n\\end{prop}\n\\begin{proof}\nThe proof of the proposition is based on a cancellation property of the kernels $G_2^1$ and $G_2^2$ and \nrequires quite tedious computations. First calculate the sum of $\\d_{x_1} G_2^2$ and $\\d_{x_1} G_2^1$ and see that\nit can be grouped into three the expressions\n\\begin{align*}\n\\frac{y_2(y_1-x_1)^2}{|y-x|^4|y-\\bar x|^2}-\\frac{y_2(y_1+x_1)^2}{|y+x|^4|y-\\til x|^2} &= (A)\\\\\n\\frac{y_2(y_1-x_1)^2}{|y-x|^2|y-\\bar x|^4}-\\frac{y_2(y_1+x_1)^2}{|y-\\til x|^2|y + x|^4} &= (B)\\\\\n\\frac{y_2}{|y - \\til x|^2|y+x|^2}-\\frac{y_2}{|y-x|^2|y-\\bar x|^2} &= (C)\n\\end{align*}\nFor convenience, these can be further written as\n\\begin{align*}\n\\frac{y_2(y_1-x_1)^2}{|y-\\bar x|^2}[|y-x|^{-4}-|y-\\til x|^{-4}]+\\frac{y_2}{|y-\\til x|^4}\\left(\n\\frac{(y_1-x_1)^2}{|y-\\bar x|^2} - \\frac{(y_1+x_1)^2}{|y+x|^2}\\right) \\\\= (1)+(2)\\\\\n\\frac{y_2}{|y-\\bar x|^4}\\left[\\frac{(y_1-x_1)^2}{|y-x|^2}-\\frac{(y_1+x_1)^2}{|y-\\til x|^2} \\right]\n+\\frac{y_2}{|y-\\til x|^2}\\left[\\frac{(y_1+x_2)^2}{|y-\\bar x|^4}+\\frac{(y_1+x_1)^2}{|y+x|^4}\\right] \\\\= (3)+(4)\\\\\n\\frac{y_2}{|y+x|^2}\\left[|y-\\til x|^{-2}-|y - x|^{-2}\\right]+\\frac{y_2}{|y-x|^2}[|y+ x|^{-2}-|y - \\bar x|^{-2}]\n\\\\= (5)+(6)\n\\end{align*}\nLet us estimate expression (1). Using $|y-\\til x|^2 - |y-x|^2 = 2 x_1 y_1$ and the relations\n$y_2 \\leq |y-\\bar x|, (y_1-x_1)^2\\leq |y-x|^2, y_1\\leq (y_1+x_1)$, we arrive at\n$$\n|(1)| \\leq \\frac{x_1}{|y-\\bar x||y-\\til x|}[|y-x|^{-2}+|y-\\til x|^{-2}].\n$$\nWrite $\\gamma = \\gamma_1+\\gamma_2$ and noting that $|y-\\bar x|\\geq x_2^{\\gamma}|y-\\bar x|^{1-\\gamma}, |y-\\til x|\\geq x_1^{1-\\gamma_2}|y-\\til x|$ \nand the reflection relations $|y-\\til x|, |y - \\bar x|\\geq |y-x|$ for $y\\in {[0, 1]^2}$, we arrive at\n$$\n|(1)| \\leq x_2^{-\\gamma} x_1^{\\gamma_2} |y-x|^{-(3 - \\gamma + \\gamma_2)}.\n$$\n\nTo estimate (2), we use the relation\n$$\n|y+x|^2 (y_1 - x_1)^2 - |y-\\bar x|^2 (y_1+x_1)^2 = - 4 x_1 y_1 (y_2+x_2)\n$$\nand similar estimations as above to arrive at\n\\begin{align*}\n|(2)| \\lesssim \\frac{y_2 y_1 x_1 (y_2+x_2)^2}{|y-\\til x|^4 |y-\\bar x|^2 |y+x|^2} \\leq \\frac{x_1}{|y-\\bar x||y-\\til x|^3}\\\\\n\\leq x_2^{-\\gamma} x_1^{\\gamma_1} |y-x|^{-3 - \\gamma_2 + \\gamma}.\n\\end{align*}\n\n(3) is similar to the above. (4) is slightly different, since expressions containing $x_1^2$ may appear. In (4),\nwe use\n$$\n|y+x|^4-|y-\\bar x|^4 \\lesssim [|y_1-x_1|x_1+x_1^2][|y+x|^2+|y-\\bar x|^2]\n$$\nto get\n\\begin{align*}\n|(4)|\\leq \\frac{|y_1-x_1|x_1}{|y-\\til x||y-\\bar x|^3|y+x|}+\\frac{x_1^2}{|y-\\til x||y-\\bar x|^3|y+x|}\\\\\n+\\frac{|y_1-x_1|x_1}{|y-\\til x||y-\\bar x||y+x|^3}+\\frac{x_1^2}{|y-\\til x||y-\\bar x ||y+x^3|}.\n\\end{align*}\nThe terms not containing $x_1^2$ are handled in a familar manner. Concerning the others, we have for example\n\\begin{align*}\n&\\frac{x_1^2}{|y-\\til x||y-\\bar x|^3|y+x|}\n\\leq x_2^{-\\gamma}x_1^{\\gamma_2} \\frac{(x_1+y_1)}{|y-\\til x|^\\gamma_2 |y-\\bar x|^{3-\\gamma}|y+x|}\\\\\n&\\leq x_2^{-\\gamma}x_1^{\\gamma_2}|y-x|^{-3+\\gamma-\\gamma_2}.\n\\end{align*} \nThe treatment of (5), (6) parallels the above.\n\\end{proof}\n\n\n\\begin{prop}\\label{prop:A9}\nFor $i\\neq j$,\n\\begin{align*}\n\\left| \\int_{{[0, 1]^2}\\setminus \\widehat{D}}\\left[\\frac{\\d G_i^1}{\\d x_j}+\\frac{\\d G_i^2}{\\d x_j}\\right]\\omega(y)~dy\\right| \n\\leq ~~C(\\gamma_1,\\gamma_2)x_i^{-(\\gamma_1+\\gamma_2)}x_j^{\\gamma_2} d(x)^{-1+\\gamma_1}\n\\end{align*}\nwhere $\\gamma_1\\in (0, 1), \\gamma_2\\in [0, 1), \\gamma_1+\\gamma_2 < 1$. Also,\n\\begin{align*}\n\\left| \\int_{{[0, 1]^2}\\setminus \\widehat{D}}\\left[\\frac{\\d G_i^i}{\\d x_1}+\\frac{\\d G_i^2}{\\d x_i}\\right]\\omega(y)~dy\\right| \n\\leq ~~C(\\gamma)x_i^{-\\gamma} d(x)^{-1+\\gamma_1}\n\\end{align*}\n\\end{prop}\n\\begin{proof}\nAs a preparation, we note that for $x\\in D$, $0<\\gamma_1< 1$,\n\\begin{equation}\\label{prop_a9_ineq1}\n\\int_{{[0, 1]^2}\\setminus \\widehat{D}} |y-x|^{-3 + \\gamma_1} \\lesssim  d(x)^{-1+\\gamma_1}.\n\\end{equation}\nThis follows from\n\\begin{align*}\n\\int_{{[0, 1]^2}\\setminus \\widehat{D}} |y-x|^{-3 + \\gamma_1} &\\leq\\int_{{[0, 1]^2} \\setminus B(x, d(x))} |y-x|^{-3 + \\gamma_1}\\\\\n&\\leq \\int_{B(x, 10)\\setminus B(x, d(x))}|y-x|^{-3 + \\gamma_1},\n\\end{align*}\nsince ${[0, 1]^2}\\setminus \\widehat{D}$ is contained in  ${[0, 1]^2} \\setminus B(x, d(x))$ because of $\\delta_2 < \\delta_3, \\delta_1 < \\delta_2$.\n\nFrom Proposition \\ref{prop:A6} we get in case $i\\neq j$\n\\begin{align*}\n\\left| \\int_{{[0, 1]^2}\\setminus \\widehat{D}}\\left[\\frac{\\d G_i^1}{\\d x_1}+\\frac{\\d G_i^2}{\\d x_j}\\right]\\omega(y)~dy\\right| &\\leq\nx_i^{-(\\gamma_1+\\gamma_2)} x_j^{\\gamma_2} \\int_{{[0, 1]^2}\\setminus \\widehat{D}} |y-x|^{-1+\\gamma_1} \\\\\n&\\leq x_i^{-(\\gamma_1+\\gamma_2)} x_j^{\\gamma_2} d(x)^{-1+\\gamma_1},\n\\end{align*}\naccording to \\eqref{prop_a9_ineq1}.\n\nFor the second inequality of the Proposition, we note that (see Proposition \\ref{prop:A1})\n\\begin{align*}\n\\left|\\frac{\\d G_i^k}{\\d x_i}\\right|&\\lesssim x_i^{-\\gamma} |y-x|^{-3+\\gamma},\n\\end{align*}\nand use \\eqref{prop_a9_ineq1}.\n\\end{proof}\n\n\n\\begin{prop}\\label{prop:A4}For $x\\in D$,\n\\begin{align*}\n\\left|P.V.\\!\\!\\int_{{[0, 1]^2}}\\left[\\frac{\\d G_i^1}{\\d x_j}+\\frac{\\d G_i^2}{\\d x_j}\\right] \\omega(y)~dy\\right|&\\leq\n M x_i^{-1} x_j^{1-\\alpha}(1+\\delta_{j2}|\\log d(x)|)\\\\\n&\\,\\, +  C(\\gamma_1, \\gamma_2) x_i^{-(\\gamma_1+\\gamma_2)}x_j^{\\gamma_2} d(x)^{-1+\\gamma_1}  \\quad (i\\neq j)\\\\\n\\left|P.V.\\!\\!\\int_{{[0, 1]^2}}\\left[\\frac{\\d G_i^1}{\\d x_i}+\\frac{\\d G_i^2}{\\d x_i}\\right] \\omega(y)~dy\\right|&\\leq \n M x_i^{-\\alpha}(1+\\delta_{i2}|\\log d(x)|) + C(\\gamma_1) x_i^{-\\gamma_1}d(x)^{-1+\\gamma_1}\n\\end{align*}\nwith $\\gamma,\\gamma_1\\in (0, 1), \\gamma_2\\in [0, 1), \\gamma_1+\\gamma_2 < 1$. \n\\end{prop}\n\\begin{proof}\nWe split the integral into a principal value integral over $\\widehat{D}$ and a convergent integral over ${[0, 1]^2} \\setminus \\widehat{D}$. The integral over $\\widehat{D}$ is \nfurther split in to integrals over the domains $I=B(x, \\frac{1}{2} d_1(x))$ and $II= \\widehat{D}\\setminus I$, which are estimated by Propositions\n\\ref{prop:A3} and \\ref{prop:A8}. The part over ${[0, 1]^2} \\setminus \\widehat{D}$ is estimated by Proposition \\ref{prop:A9}.\n\\end{proof}\n\n\n\\subsection{Appendix B\n\n\\begin{prop}\\label{propAllG} The following relations hold:\n\\begin{align*}\n\\frac{\\d G_1^1}{\\d x_1}&=-\\frac{2y_1(y_1+x_1)(y_2-x_2)}{|y-x|^2|y-\\til x|^4}\n+\\frac{2y_1(y_1-x_1)(y_2-x_2)}{|y-x|^4|y-\\til x|^2}\\\\\n\\frac{\\d G_1^1}{\\d x_2}&=\\frac{2y_1(y_2-x_2)^2}{|y-x|^2|y-\\til x|^4}\n+\\frac{2y_1(y_2-x_2)^2}{|y-x|^4|y-\\til x|^2}-\\frac{y_1}{|y-x|^2|y-\\til x|^2}\\\\\n\\frac{\\d G_1^1}{\\d y_1}&=-\\frac{2y_1(y_1+x_1)(y_2-x_2)}{|y-x|^2|y-\\til x|^4}\n-\\frac{2y_1(y_1-x_1)(y_2-x_2)}{|y-x|^4|y-\\til x|^2}+\\frac{y_2-x_2}{|y-x|^2|y-\\til x|^2}\\\\\n\\frac{\\d G_1^1}{\\d y_2}&=-\\frac{2 y_1(y_2-x_2)^2}{|y-x|^2|y-\\til x|^4}\n-\\frac{2y_1(y_2-x_2)^2}{|y-x|^4|y-\\til x|^2}+\\frac{y_1}{|y-x|^2|y-\\til x|^2}\n\\end{align*}\n\\begin{align*}\n\\frac{\\d G_1^2}{\\d x_1}&=-\\frac{2y_1(y_1+x_1)(y_2+x_2)}{|y+x|^4|y-\\bar x|^2}\n+\\frac{2y_1(y_1-x_1)(y_2+x_2)}{|y+x|^2|y-\\bar x|^4}\\\\\n\\frac{\\d G_1^2}{\\d x_2}&=-\\frac{2y_1(y_2+x_2)^2}{|y+x|^4|y-\\bar x|^2}\n-\\frac{2y_1(y_2+x_2)^2}{|y+x|^2|y-\\bar x|^4}+\\frac{y_1}{|y+x|^2|y-\\bar x|^2}\\\\\n\\frac{\\d G_1^2}{\\d y_1}&=-\\frac{2y_1(y_1+x_1)(y_2+x_2)}{|y+x|^4|y-\\bar x|^2}\n-\\frac{2y_1(y_1-x_1)(y_2+x_2)}{|y+x|^2|y-\\bar x|^4}+\\frac{y_2+x_2}{|y+x|^2|y-\\bar x|^2}\\\\\n\\frac{\\d G_1^2}{\\d y_2}&=-\\frac{2y_1(y_2+x_2)^2}{|y+x|^4|y-\\bar x|^2}\n-\\frac{2y_1(y_2+x_2)^2}{|y+x|^2|y-\\bar x|^4}+\\frac{y_1}{|y+x|^2|y-\\bar x|^2}\n\\end{align*}\n\\begin{align*}\n\\frac{\\d G_2^1}{\\d x_1}&=-\\frac{2y_2(y_1+x_1)^2}{|y+x|^4|y-\\til x|^2}\n-\\frac{2y_2(y_1+x_1)^2}{|y+x|^2|y-\\til x|^4}+\\frac{y_2}{|y+x|^2|y-\\til x|^2}\\\\\n\\frac{\\d G_2^1}{\\d x_2}&=-\\frac{2y_2(y_1+x_1)(y_2+x_2)}{|y+x|^4|y-\\til x|^2}\n+\\frac{2y_2(y_1+x_1)(y_2-x_2)}{|y+x|^2|y-\\til x|^4}\\\\\n\\frac{\\d G_2^1}{\\d y_1}&=-\\frac{2y_2(y_1+x_1)^2}{|y+x|^4|y-\\til x|^2}\n-\\frac{2y_2(y_1+x_1)^2}{|y+x|^2|y-\\til x|^4}+\\frac{y_2}{|y+x|^2|y-\\til x|^2}\\\\\n\\frac{\\d G_2^1}{\\d y_2}&=-\\frac{2y_2(y_1+x_1)(y_2+x_2)}{|y+x|^4|y-\\til x|^2}\n-\\frac{2y_2(y_1+x_1)(y_2-x_2)}{|y+x|^2|y-\\til x|^4}+\\frac{y_1+x_1}{|y+x|^2|y-\\til x|^2}\n\\end{align*}\n\\begin{align*}\n\\frac{\\d G_2^2}{\\d x_1}&=\\frac{2y_2(y_1-x_1)^2}{|y-x|^2|y-\\bar x|^4}\n+\\frac{2y_2(y_1-x_1)^2}{|y-x|^4|y-\\bar x|^2}-\\frac{y_2}{|y-x|^2|y-\\bar x|^2},\\\\\n\\frac{\\d G_2^2}{\\d x_2}&=-\\frac{2y_2(y_1-x_1)(y_2+x_2)}{|y-x|^2|y-\\bar x|^4}\n+\\frac{2y_2(y_1-x_1)(y_2-x_2)}{|y-x|^4|y-\\bar x|^2},\\\\\n\\frac{\\d G_2^2}{\\d y_1}&=-\\frac{2y_2(y_1-x_1)^2}{|y-x|^2|y-\\bar x|^4}\n-\\frac{2y_2(y_1-x_1)^2}{|y-x|^4|y-\\bar x|^2}+\\frac{y_2}{|y-x|^2|y-\\bar x|^2},\\\\\n\\frac{\\d G_2^2}{\\d y_2}&=-\\frac{2y_2(y_1-x_1)(y_2+x_2)}{|y-x|^2|y-\\bar x|^4}\n-\\frac{2y_2(y_1-x_1)(y_2-x_2)}{|y-x|^4|y-\\bar x|^2}+\\frac{y_1-x_1}{|y-x|^2|y-\\bar x|^2}.\n\\end{align*}\n\\end{prop}\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\n\n\n\n\n\nThis paper gives a method for renormalising  a class of quantum field theories.  The field theories we are going to consider have space of fields of the form $\\mathscr{E} = \\Gamma(M,E)$, where $M$ is a compact manifold and $E$ is a super vector bundle on $M$.    We work within the Batalin-Vilkovisky formalism, so that $\\mathscr{E}$ is equipped with an odd symplectic pairing $\\mathscr{E} \\otimes \\mathscr{E} \\to \\mathbb C$ satisfying a certain non-degeneracy condition\\footnote{Here, and throughout, $\\otimes$ refers to the completed projective tensor product, so that $\\mathscr{E} \\otimes \\mathscr{E} = \\Gamma (M^2, E \\boxtimes E)$.}.  $\\mathscr{E}$ will also be equipped with a differential $Q$, which is an odd differential operator $Q : \\mathscr{E} \\to \\mathscr{E}$ which is skew self adjoint and of square zero.  The action functionals in our quantum field theory will be of the form\n$$\n\\frac{1}{2} \\ip{e, Q e } + S(e )\n$$\nwhere $S$ is a local functional on the space of fields $\\mathscr{E}$, which is at least cubic.  \n\nThe functional integrals we will renormalise are of the form\n$$\nZ(S, \\hbar, a) = \\int_{x \\in L} \\exp \\left( \\frac{1}{2  \\hbar} \\ip{x, Q x } + \\frac{1}{\\hbar} S(x + a )  \\right) \\d x\n$$\nwhere $a \\in \\mathscr{E}$ and $L \\subset \\mathscr{E}$ is an isotropic linear subspace, such that the map $Q : L \\to \\Im Q$ is an isomorphism.   Such an $L$ is known as a \\emph{gauge fixing condition}.  $Z(S,a)$ is a formal functional of the variable $a \\in \\mathscr{E}$, and can be viewed as a generating function for certain Green's functions of the theory.\n\nA fairly wide class of theories can be put in the form we use, including pure Yang-Mills theory in dimension 4, and Chern-Simons theory in any dimension.\n\nThis introduction will give a sketch of the results and of the underlying philosophy, without worrying too much about technical details. \n\n\\subsection{}\nOur gauge fixing conditions are always of the form\n$$\nL = \\Im Q^{GF}\n$$\nwhere $Q^{GF} : \\mathscr{E} \\to \\mathscr{E}$ is an odd self adjoint differential operator of square zero, with the property that the super-commutator\n$$\nH \\overset{\\text{def}}{=} [Q,Q^{GF}]\n$$\n is a positive elliptic operator of second order.  The facts that $Q^{GF}$ is self-adjoint with respect to the symplectic pairing on $\\mathscr{E}$, and that $Q^{GF}$ is of square zero, imply that $L = \\Im Q^{GF}$ is an isotropic subspace.\n\nOnly certain theories admit gauge fixing conditions of this form (this is the main restriction on the kind of theories the techniques from this paper can treat).  In many examples, $Q$ is a first order elliptic operator, and $Q^{GF}$ is a Hermitian adjoint of $Q$.    \n\nA basic example to bear in mind is Chern-Simons theory in dimension $3$.  If $M$ is a compact oriented $3$-manifold, and $\\mathfrak g$ is a Lie algebra with an invariant non-degenerate pairing, then \n$$\n\\mathscr{E} = \\Omega^\\ast(M) \\otimes \\mathfrak g[1].\n$$\n$[1]$ denotes parity shift.  The symplectic pairing on $\\mathscr{E}$ arises from the Poincar\\'e pairing on $\\Omega^\\ast(M)$ and the pairing on $\\mathfrak g$.  The operator $Q$ is simply the de Rham differential $\\d_{DR}$, and $S$ is the cubic term in the standard Chern-Simons action.  The choice of a metric on $M$ leads to a gauge fixing condition $Q^{GF} = \\d_{DR}^\\ast$.  Further examples, including Chern-Simons theory in other dimensions, will be discussed later.\n\n\\subsection{}\nLet us write \n$$\nP(\\varepsilon,T) =  \\int_{\\varepsilon}^T (Q^{GF} \\otimes 1) K_t \\d t \\in \\mathscr{E} \\otimes \\mathscr{E}\n$$\nwhere $K_t \\in \\mathscr{E} \\otimes \\mathscr{E}$ is the heat kernel for $H  = [Q, Q^{GF}]$. Here, $\\mathscr{E} \\otimes \\mathscr{E}$ denotes the completed projective tensor product, \n$$\n\\mathscr{E} \\otimes \\mathscr{E} = C^{\\infty}(M\\times M, E \\boxtimes E ). \n$$\nThe propagator of our theory is \n$$\nP (0,\\infty) = \\int_{0}^\\infty (Q^{GF} \\otimes 1) K_t \\d t. \n$$\nThis is not an element of $\\mathscr{E} \\otimes \\mathscr{E}$, because of the singularities in the heat kernel at $t = 0$. Instead, $P(0,\\infty)$ is an element of a distributional completion of $\\mathscr{E} \\otimes \\mathscr{E}$. \n\nLet $\\mscr O(\\mathscr{E})$ denote the algebra of functions on $\\mathscr{E}$,\n$$\n\\mscr O(\\mathscr{E}) = \\prod_{n \\ge 0} \\Hom ( \\mathscr{E}^{\\otimes n}, \\mathbb C )_{S_n}\n$$\nwhere, as above, the tensor products are completed projective tensor products, and $\\Hom$ denotes continuous linear maps. \n\nAny element $P \\in \\mathscr{E} \\otimes \\mathscr{E}$ gives rise to an order two differential operator $\\partial_P$ on $\\mscr O(\\mathscr{E})$ in a standard way.   \n\n\nUp to a constant factor (the determinant of $Q$), one can write our functional integral as\n$$\nZ(S, \\hbar, a) =  \\lim_{\\varepsilon \\to 0} \\left( e^{\\hbar \\partial_{P(\\varepsilon,\\infty ) } } e^{S \/ \\hbar } \\right) (a).\n$$\nThe right hand side is the exponential of a differential operator applied to a function on $\\mathscr{E}$, yielding a function on $\\mathscr{E}$. This identity is rather formal; in finite dimensions, it is a simple consequence of integration by parts. In infinite dimensions we take it as an attempt at a definition. \n\n When $\\varepsilon > 0$, the right hand side of this equation is well-defined. However, the $\\varepsilon \\to 0$ limit is singular, because $P(0,\\infty)$ is not an element of $\\mathscr{E} \\otimes \\mathscr{E}$, but has singularities  along the diagonal in $M^2$. \n\n\nLet us use the notation\n$$\n\\Gamma ( P(\\varepsilon,T) , S  )  =  \\hbar \\log \\left ( e^{\\hbar \\partial_{P(\\varepsilon,\\infty )} } e^{S \/ \\hbar }\\right). \n$$\nThis is an $\\hbar$ dependent function on $\\mathscr{E}$, that is, an element of $\\mscr O(\\mathscr{E}) [[\\hbar]]$.  We will typically omit the variables $a \\in \\mathscr{E}$ and $\\hbar$ from the notation.\n\nThe expression we would like to make sense of is \n$$\\hbar \\log Z(S, \\hbar, a) =  \\lim_{\\varepsilon \\to 0} \\Gamma( P(\\varepsilon,\\infty), S).$$\n\n\\subsection{}\n An effective action\\footnote{ What I mean by effective action is related to the Wilsonian effective action.  Ignoring for the moment considerations of renormalisation, the Wilsonian effective action is obtained by integrating out all the high-energy fields, i.e.\\ all the eigenspaces of $H$ with high eigenvalues.      The effective action considered here is obtained by averaging over all interactions occurring at small scales.  More precisely, the effective action at scale $\\varepsilon$ is the sum over all Feynman graphs of the theory, using the propagator $P(0,\\varepsilon)$. Using the propagator $P(0,\\varepsilon)$ amounts to allowing particles to propagate for a distance of between $0$ and $\\varepsilon$.   } at scale $\\varepsilon$ is a function $f \\in \\mscr O(\\mathscr{E}) [[\\hbar]]$ which describes all interactions occurring at a scale below $\\varepsilon$.   One can reconstruct the effective action at any other scale using the effective action at scale $\\varepsilon$ and the propagator.  The map $f \\mapsto \\Gamma ( P(\\varepsilon,T) , f)$ is the operation taking an effective action at scale $\\varepsilon$ to the corresponding effective action at scale $T$.  One can imagine that the scale $T$ effective action $\\Gamma(P(\\varepsilon,T), f)$ is obtained from the scale $\\varepsilon$ effective action $f$ by allowing particles to interact according to $f$, and to propagate a distance between $\\varepsilon$ and $T$. \n\n$\\Gamma ( P(\\varepsilon,T) , f)$ is the renormalisation group flow from scale $\\varepsilon$ to scale $T$ applied to $f$. This is well-defined for all those $f \\in \\mscr O(\\mathscr{E})[[\\hbar]]$ which are at least cubic modulo $\\hbar$, as long as $0 < \\varepsilon < T \\le \\infty$.     We have the semi-group law\n$$\n\\Gamma ( P(T_2,T_3 ) , \\Gamma ( P(T_1,T_2) , f) ) = \\Gamma ( P(T_1,T_3) , f).\n$$\nThis equation is  a version of the exact renormalisation group equation.  The operation $f \\to \\Gamma( P(\\varepsilon,T), f)$ is invertible, so that if we know the effective action at any scale, we know it at all other scales. \n\nThe only part of this renormalisation group flow that is problematic is taking an effective action at scale $0$ and turning it into an effective action at any positive scale $\\varepsilon$. This part of the procedure needs to be renormalised.   This is to be expected: an effective action at scale $0$ would describe interactions occurring at infinitely high energy.  \n\nOne way to phrase the main result we prove is that there is a bijection between functionals $S \\in \\mscr O(\\mathscr{E}) [[\\hbar]]$, satisfying a locality axiom, and systems $\\{ S^{eff}(T) \\mid T \\in \\mbb R_{> 0} \\}$ of effective actions at all scales $T > 0$, related by the renormalisation group equation. The effective action $S^{eff}(T)$ must also satisfy a locality condition as $T \\to 0$.  The original action $S$ plays the role of the scale $0$ effective action, and the positive scale effective actions $S^{eff}(T)$ are obtained by renormalising the expression $\\Gamma ( P( 0,T) , S)$.  Every such system of effective actions $S^{eff}(T)$ arises from a unique local functional $S$ in this way.\n\n\\subsection{}\nIn order to renormalise the expression $\\lim_{\\varepsilon \\to 0} \\Gamma ( P(\\varepsilon, T ) , S)$, and thus construct the scale $T$ effective action, we will need a way to extract the ``singular part'' of expressions of the form $\\Gamma ( P(\\varepsilon, T ) , S)$. This will rely on some results about the small $\\varepsilon$ asymptotic expansion of $\\Gamma ( P(\\varepsilon, T ) , S)$. \n\n\nIt's convenient to represent $\\Gamma ( P(\\varepsilon, T ) , S)$ as \n$$\n\\Gamma ( P(\\varepsilon, T ) , S)= \\sum_{i,k \\ge 0} \\hbar^i \\Gamma_{i,k} ( P(\\varepsilon, T ) , S)\n$$\nwhere $\\Gamma_{i,k} ( P(\\varepsilon, T ) , S)$ is homogeneous of degree $k$ as a function of $a \\in \\mathscr{E}$.  This expression is just the Taylor expansion of $\\Gamma ( P(\\varepsilon, T), S)$ as a function of $\\hbar$ and $a \\in \\mathscr{E}$.   In terms of Feynman graphs, $\\Gamma_{i,k}(P(\\varepsilon,T),S)$ is the sum over all graphs with first Betti number $i$  and $k$ external edges. \n\nLet $\\operatorname{An} ( (0,\\infty) )$ be the algebra of analytic functions on $(0,\\infty)$, where $\\varepsilon$ is the coordinate on $(0,\\infty)$.  \n\\begin{thmA}\nThere exists a subalgebra $\\mathscr{A} \\subset \\operatorname{An} ( (0,\\infty) )$ with a countable basis, such that for all local functionals $S \\in \\mscr O(\\mathscr{E}) [[\\hbar]]$, there exists \na small $\\epsilon$ asymptotic expansion\n$$\n\\Gamma_{i,k} ( P(\\varepsilon, T ) , S) \\simeq  \\sum f_r(\\epsilon) \\otimes \\Phi_{i,k,r}(T,a) \n$$\nwhere $f_r \\in \\mathscr{A}$, and $\\Phi_{i,k,r}$ are certain functions of the variables $T \\in (0,\\infty)$ and $a \\in \\mathscr{E}$. \n\\end{thmA}\nA functional $S \\in \\mscr O(\\mathscr{E}) [[\\hbar]]$ is \\emph{local} if, roughly, its homogeneous components $S_{i,k} \\in \\Hom(\\mathscr{E}^{\\otimes k}, \\mathbb C)_{S_k}$, which are distributions on the vector bundle $E^{\\boxtimes k}$ on $M^k$, are supported on the small diagonal, and non-singular in the diagonal directions.  We will denote the space of local functionals by \n$$\n\\mscr O_l(\\mathscr{E}) \\subset \\mscr O(\\mathscr{E}) . \n$$\n\nLet $\\mathscr{A}_{\\ge 0} \\subset \\mathscr{A}$ be the subspace of those functions whose $\\epsilon \\to 0$ limit exists.  In order to extract the singular part of functions in $\\mathscr{A}$, we need to pick a complementary subspace to $\\mathscr{A}_{\\ge 0}$.\n\\begin{definition}\nA \\emph{renormalisation scheme} is a subspace $\\mathscr{A}_{< 0}\\subset \\mathscr{A}$ such that \n$$\n\\mathscr{A}  = \\mathscr{A}_{\\ge 0} \\oplus \\mathscr{A}_{< 0} . \n$$\n\\end{definition}\nLet us fix a renormalisation scheme $\\mathscr{A}_{< 0}$.  Later we will see that things are independent in a certain sense of the choice of renormalisation scheme.  \n\n\\begin{remark}\nThe functions in $\\mathscr{A}$ are quite explicit integrals of multi-variable rational functions.   The algebra $\\mathscr{A}$ only depends on the dimension of the manifold $M$, and not on the details of the particular theory we are working with. \n\\end{remark}\n\\begin{remark}\nInstead of using the algebra $\\mathscr{A}$, one could use any larger algebra of functions on $(0,\\infty)$, and obtain the same results.  It is technically easier to use an algebra with a countable basis. \n\\end{remark}\n\n\\begin{remark}\nAn alternative regularisation scheme, which Jack Morava suggested to me, would be to use the propagator $\\int_0^\\infty t^z (Q^{GF} \\otimes 1)K_t \\d t$, where $z$ is a complex parameter.  If we use this propagator, then integrals attached to Feynman graphs converge if $\\Re z \\gg 0$.  The expressions should admit an analytic continuation to $\\mathbb C$ with poles on $\\tfrac{1}{2} \\mathbb Z$.  Unfortunately, I wasn't able to prove the existence of the analytic continuation. \n\\end{remark}\n\n\n\\subsection{}\nThe first main result of this paper is the following.\n\\begin{thmB}\nFix a renormalisation scheme $\\mathscr{A}_{< 0} \\subset \\mathscr{A}$. \nThen there exists a unique series of counter-terms\n$$\nS^{CT} (\\hbar, \\epsilon ,a ) = \\sum_{i > 0, k\\ge 0} \\hbar^i S^{CT}_{i,k} (\\epsilon, a )\n$$\nwhere \n\\begin{enumerate}\n\\item\neach $S^{CT}_{i,k}(\\epsilon,a)$ is a formal local functional of $a \\in \\mathscr{E}$, homogeneous of order $k$ as a function of $a$, with values in $\\mathscr{A}_{< 0}$ Thus,\n$$\nS^{CT}_{i,k} \\in \\mscr O_l(\\mathscr{E}) \\otimes \\mathscr{A}_{< 0}\n$$ \nwhere $\\mscr O_l(\\mathscr{E}) \\subset \\mscr O(\\mathscr{E})$ is the space of local functionals on $\\mathscr{E}$.\n\\item\nthe limit \n$$\\lim_{\\epsilon \\to 0} \\Gamma ( P(\\varepsilon, T ) , S - S^{CT})$$ \nexists. \n\\end{enumerate}\nThese counter-terms are independent of $T$. \n\\end{thmB}\n\n\nLet me give a brief sketch of the (surprisingly simple) proof of this theorem. As before, let us write \n$$\n\\Gamma ( P(\\varepsilon, T ) , S)= \\sum_{i,k \\ge 0} \\hbar^i \\Gamma_{i,k} ( P(\\varepsilon, T ) , S).\n$$\nThe $\\Gamma_{0,k} ( P(\\varepsilon, T ) , S)$ all have well-defined $\\epsilon \\to 0$ limits.  So the first counter-term we need to construct is $S^{CT}_{1,1}$.  Our choice of renormalisation scheme allows us to extract the singular part of any function of $f(\\epsilon) \\in \\mathscr{A}$; this singular part is simply the projection of $f$ onto $\\mathscr{A}_{< 0}$.  We define\n$$\nS^{CT}_{1,1}(\\epsilon,a) = \\text{singular part of the small $\\epsilon$ expansion of } \\Gamma_{1,1} ( P(\\varepsilon, T ) , S)\\in \\mathscr{A}_{< 0}. \n$$\nIt is easy to see that $\\Gamma_{1,1} ( P(\\varepsilon, T ) , S- \\hbar S^{CT}_{1,1})$ is non-singular as $\\epsilon \\to 0$.  \n\nThe next step is to replace $S$ by $S - \\hbar S^{CT}_{1,1}$, and use this new action to produce the next counter-term, $S^{CT}_{1,2}$.  That is, we define\n$$\nS^{CT}_{1,2}= \\text{singular part of } \\Gamma_{1,2} ( P(\\varepsilon, T ) , S - \\hbar S_{1,1}^{CT})\n$$\nwhere we understand ``singular part'' in the same way as before. We continue like this, to define $S^{CT}_{1,k}$ for all $k \\ge 1$.\n\nThe next step is to define\n$$\nS^{CT}_{2,0} = \\text{singular part of } \\Gamma_{2,0} ( P(\\varepsilon, T ) , S - \\hbar \\sum_{k = 0}^\\infty S_{1,k}^{CT})\n$$\nContinuing in this manner defines all the $S^{CT}_{i,k}$. \n\nThe difficult part of the proof is the verification that the counter-terms $S^{CT}_{i,k}$ are local functionals.  Locality is desirable for many physical and mathematical reasons. More practically, we need $S^{CT}_{i,k}$ to be local in order to apply the procedure at the next inductive step. We only know the existence of a small $\\varepsilon$ asymptotic expansion of the $\\Gamma_{I,K}(P(\\varepsilon,T) , S - \\sum \\hbar^i S^{CT}_{i,k} )$  when the counter-terms $S^{CT}_{i,k}$ are local. \n\nThe main step in the proof of locality is showing that the $S^{CS}_{i,k}$ are independent of $T$.  Once we know they are independent of $T$, we can take $T \\to 0$.  $\\Gamma_{i,k}( P(\\varepsilon, T) , S)$ is concentrated near the diagonal in $M^k$, if $\\varepsilon < T$ and both $\\varepsilon, T$ are very small.   Thus, the counter-terms become supported on the diagonal, and local. \n\n\nThis theorem allows one to define unambiguously the renormalised scale $T$ effective action\n$$\n\\Gamma^R ( P(0,T) , S ) =  \\lim_{\\varepsilon \\to 0} \\Gamma( P(\\varepsilon,T) , S - S^{CT} ) .\n$$\nThus, we can define the renormalised functional integral\n\\begin{multline*}\nZ^R =  \\exp  ( \\Gamma^R ( P(0,\\infty), S)  \/ \\hbar ) = \\text { renormalisation of }  \\\\\n\\int_{x \\in L} \\exp \\left( \\frac{1}{2  \\hbar} \\ip{x, Q x } + \\frac{1}{\\hbar} S(x + a )  \\right) \\d x.\n\\end{multline*}\nThis renormalised partition function is an element of $\\mscr O(\\mathscr{E})((\\hbar))$, that is, a  non-local formal function on the space $\\mathscr{E}$ of fields.\n\n\n\n\\subsection{Independence of choice of renormalisation scheme}\nWe should interpret the expression $\\Gamma^R ( P(0,T) , S ) $ constructed using Theorem B  as the scale $T$ renormalised effective action.  The renormalisation group equation holds:\n$$\n\\Gamma ( P(T_1,T_2 ) , \\Gamma^R ( P(0,T_1) , S) ) = \\Gamma^R ( P(0,T_2) , S).\n$$\n\\begin{definition}\nA \\emph{system of effective actions} on the space of fields $\\mathscr{E}$ is given by an effective action\n$$\nS^{eff}(T) \\in \\mscr O(\\mathscr{E}) [[\\hbar]] \n$$\nfor all $T \\in \\mbb R_{> 0}$, varying smoothly with $T$, \nsuch that\n\\begin{enumerate}\n\\item\nEach $S^{eff}(T)$ is at least cubic modulo $\\hbar$. \n\\item\nThe renormalisation group equation is satisfied,\n$$\nS^{eff}(T_2) =  \\Gamma ( P(T_1,T_2) , S^{eff}(T_1)).\n$$\n\\item\nAs $T \\to 0$, $S^{eff}(T)$ must become local, in the following sense.  There must exist some $T$-dependent local functional $\\Phi(T)$ such that $\\lim_{T \\to 0} \\left( S^{eff}(T) - \\Phi(T) \\right) = 0$. (The $T \\to 0$ limit of $S^{eff}(T)$ itself will generally not exist). \n\\end{enumerate}\n\\end{definition}\nThe renormalised effective actions $\\Gamma^R (  P ( 0,T)  , S)$ constructed from a local functional $S$ satisfy these two axioms. Thus, for any renormalisation scheme $\\mathscr{A}_{< 0}$, theorem B provides a map\n\\begin{align*}\n\\text{local functionals } S \\in \\mscr O_l(\\mathscr{E}) [[\\hbar]] & \\to \\text{ systems of effective actions } \\\\\nS & \\mapsto \\{ \\Gamma^R ( P(0,T), S ) \\mid T \\in \\mbb R_{> 0} \\} .\n\\end{align*}\n(The local functionals $S$ must be at least cubic modulo $\\hbar$, as must the effective actions $S^{eff}(T)$. ) .\n\\begin{thmC}\nFor any renormalisation scheme $\\mathscr{A}_{< 0}$, this map is a bijection.\n\\end{thmC}\nThis set of systems of effective actions on $\\mathscr{E}$ is a canonical object associated to $(\\mathscr{E},Q, Q^{GF})$, independent of the choice of renormalisation scheme.    Renormalisation and regularisation techniques other than those considered should lead to different ways of parametrising the same set of systems of effective actions.  For instance, if one could make sense of dimensional regularisation on general manifolds, one would hope to get simply a different parametrisation of this set.\n\nFrom this point of view, the formalism of counter-terms is simply a convenient way to describe this set of systems of effective actions.   The counter-terms themselves, and the original action $S$, are not really meaningful in themselves.   The main point of introducing counter-terms is that it is otherwise difficult to produce systems of effective actions.   \\emph{A priori}, it is not obvious that there are any non-zero systems of effective actions at all.\n\nThis proposition makes clear in what sense renormalisation is independent of the choice of renormalisation scheme.  To any renormalisation scheme $\\mathscr{A}_{< 0}$ and local funct\\-ional $S \\in  \\mscr O_l(\\mathscr{E}) [[\\hbar]]$ corresponds a ``theory'', i.e.\\  a system of effective actions.  If $\\mathscr{A}'_{< 0}$ is another renormalisation scheme, there is a unique local functional $S'$ such that $(S', \\mathscr{A}'_{< 0})$ gives the same theory as $(S, \\mathscr{A}_{< 0})$.   \n\nThis statement can be expressed more formally as follows. Let $\\operatorname{RS}$ denote the space of renormalisation schemes, and let $\\operatorname{EA}$ denote the set of systems of effective actions.  Theorem C implies gives an isomorphism of fibre bundles on $\\operatorname{RS}$\n$$\n\\mscr O_l(\\mathscr{E}) \\times \\operatorname{RS} \\to \\operatorname{EA} \\times \\operatorname{RS} .\n$$\nGive the right hand side the trivial flat connection; this pulls back to a non-trivial (and non-linear) flat connection on $\\mscr O_l(\\mathscr{E})$.   This flat connection is uniquely characterised by the property that for any flat section $S : \\operatorname{RS} \\to \\mscr O_l(\\mathscr{E}) \\times \\operatorname{RS}$, the system of effective actions $\\{\\Gamma^R ( P(0,T) , S (\\mathscr{A}_{< 0} ) ) \\}$ associated to the functional $S(\\mathscr{A}_{< 0} ) \\in \\mscr O_l(\\mathscr{E})$ is independent of the point $\\mathscr{A}_{< 0}  \\in \\operatorname{RS}$.  \n\nWe will fix once and for all a renormalisation scheme $\\mathscr{A}_{< 0}$. This allows us to always talk about local functionals, as a convenient proxy for the set of systems of effective actions. The choice of $\\mathscr{A}_{< 0}$ is analogous to the choice of a basis in a vector space. \n\nAll statements about a theory should be expressed in terms of the effective actions $\\Gamma^R(P(0,T), S)$, and not directly in terms of the local functional $S$.  This will ensure everything is independent of the choice of renormalisation scheme.\n\n\\subsection{}\n\n Bogoliubov and Parasiuk \\cite{BogPar57}, Hepp \\cite{Hep66} and Zimmerman \\cite{Zim69}   have given an algorithm for the renormalisation of certain quantum field theories.   Their algorithm is based on combinatorial manipulations of Feynman graphs.   Recently, Connes and Kreimer \\cite{ConKre00} have given a beautiful interpretation of the BPHZ algorithm, in terms of the Birkhoff factorisation of loops in a certain Hopf algebra. \n  \nIn the approach used in this paper, no graph combinatorics are needed; all we use is a very simple inductive argument, sketched above.  The reason that things become so simple seems to be the particular kind of functional integrals we consider, which are always of the form\n$$\nZ(S, \\hbar, a) = \\int_{x \\in \\Im Q^{GF}} \\exp \\left( \\frac{1}{2  \\hbar} \\ip{x, Q x } + \\frac{1}{\\hbar} S(x + a )  \\right) \\d x.\n$$\nThus, the moral of this paper is that if we consider functional integrals of this form,\nthen the problem of renormalisation becomes quite simple, and the counter-terms are unique and automatically local.      As we will see shortly, the particular functional integrals we use also play an important role in the Batalin-Vilkovisky formalism. \n\n\nAnother difference between the approach to renormalisation described in this paper and that of Connes-Kreimer and BPHZ is that we do not give finite values to individual Feynman  graphs, but only to the sum over all graphs with a fixed number of loops and external edges. From the point of view of the effective action,  the expression attached to an individual graph has no meaning. \n\n\n\nIf we tried to renormalise different classes of functional integrals the procedure would not be so simple.  For example, if we try to simply renormalise the integral \n$$\\int_{x \\in \\Im Q^{GF}} \\exp \\left( \\frac{1}{2  \\hbar} \\ip{x, Q x } + \\frac{1}{\\hbar} S(x )  \\right) \\d x,$$\nwithout using the variable $a$, then there are many possible choices of counter-terms.  \n\nIf we try to renormalise the functional integral $$\\int_{x \\in \\Im Q^{GF}} \\exp \\left( \\frac{1}{2  \\hbar} \\ip{x, Q x } + \\frac{1}{\\hbar} S(x) + \\frac{1}{\\hbar} \\ip{x,a}  \\right) \\d x, $$  then the terms in the Feynman graph expansion don't fit together in the right way to produce local counter-terms.   \n\nThis type of functional integral is one which appears more commonly in the literature.  A simple change of variables allows one to express this type of functional integral in terms of the kind considered here, but not conversely.  Indeed, if $a =- Q^{-1} \\pi b$, where $\\pi$ is the projection onto $\\Im Q^{GF}$ and $Q^{-1} :  \\Im Q \\to \\Im Q^{GF}$ is the inverse to $Q$, we can write\n\\begin{multline*}\n\\int_{x \\in  \\Im Q^{GF} } \\exp \\left( \\frac{1}{2  \\hbar} \\ip{x, Q x } + \\frac{1}{\\hbar} S(x + a)  \\right) \\d x \\\\\n\\shoveright{ =\\int_{x \\in  \\Im Q^{GF} } \\exp \\left( \\frac{1}{2  \\hbar} \\ip{x- a, Q (x-a) } + \\frac{1}{\\hbar} S(x )  \\right) \\d x  \\hspace{41pt}} \\\\\n=  e^{- \\ip{a, b } \/\\hbar } \\int_{x \\in  \\Im Q^{GF} } \\exp \\left( \\frac{1}{2  \\hbar} \\ip{x, Q x } + \\frac{1}{\\hbar} S(x) + \\frac{1}{\\hbar} \\ip{x,b}  \\right) \\d x \n\\end{multline*}\n\n\n\\subsection{Batalin-Vilkovisky formalism}\nThe Batalin-Vilkovisky quantum master equation is the quantum expression of gauge symmetry.  It takes the form\n$$\n(Q + \\hbar \\Delta ) \\exp ( S \/ \\hbar ) = 0\n$$\nwhere $\\Delta$ is a certain order $2$ differential operator acting on the space of functionals on $\\mathscr{E}$.  This expression makes perfect sense in the simple situation when the space of fields $\\mathscr{E}$ is  finite dimensional (i.e.\\ the underlying manifold $M$ is of dimension $0$).  In the situation we work in, however, this expression is infinite. This is because  $\\Delta S$ involves the multiplication of singular distributions, and thus has the same kind of singularities as appear in one-loop Feynman diagrams.  \n\nThis paper gives a definition of a renormalised quantum master equation that works in the infinite dimensional situation.  There are regularised BV operators $\\Delta_t$, for $t > 0$, defined by \n$$\n\\Delta_t = - \\partial_{K_t}.\n$$\nRecall that $K_t \\in \\mathscr{E} \\otimes \\mathscr{E}$ is the heat kernel for $H = [Q, Q^{GF}]$, and $\\partial_{K_t}$ is the order two differential operator on the algebra $\\mscr O(\\mathscr{E})$ of functions on $\\mathscr{E}$, associated to $K_t$.  The operators $\\Delta_t$ are thus differential operators on $\\mscr O(\\mathscr{E})$, for all $t > 0$.   The ``physical'' BV operator is $\\Delta_0$, which is ill-defined.\n\n\\begin{lemma}\t\nA function $f \\in \\mscr O(\\mathscr{E})[[\\hbar]]$ satisfies the $\\Delta_\\varepsilon$ quantum master equation\n$$\n(Q + \\hbar \\Delta_\\varepsilon)e^{f \/ \\hbar} = 0\n$$\n if and only if \n$$\n\\Gamma ( P(\\varepsilon, T ) , f)\n$$\nsatisfies the $\\Delta_T$ quantum master equation.  \n\\end{lemma}\nThis follows from the fact that\n$$\nQ P(\\varepsilon, T) = -K_T + K_\\varepsilon\n$$\nso that the differential operator $\\partial_{P(\\varepsilon,T)}$ is a chain homotopy between $\\Delta_\\varepsilon$ and $\\Delta_T$.\n\nThis lemma motivates the following definition.\n\\begin{definition}\nA local functional $S \\in \\mscr O_l(\\mathscr{E})[[\\hbar]]$ satisfies the renormalised quantum master equation if the renormalised expression \n$$\\Gamma^R ( P(0,T) , S)  = \\lim_{\\varepsilon \\to 0} \\Gamma (P(\\varepsilon,T), S - S^{CT} )$$\nsatisfies the $\\Delta_T$ quantum master equation, for some (or equivalently, all) $T > 0$. \n\\end{definition}\nThus, the renormalised QME says that the scale $T$ effective action associated to $S$ satisfies the scale $T$ quantum master equation.\n\nOne peculiarity of this renormalised quantum master equation is that, unlike in the finite dimensional situation, it depends on the gauge fixing condition $Q^{GF}$.    We will see shortly that this dependence is very weak.  \n\nIn fact,  the equation depends also on the renormalisation scheme $\\mathscr{A}_{< 0}$, but only in an artificial way.  Recall that theorem C says that the choice of a renormalisation scheme sets up a bijection between local functionals $S \\in \\mscr O_l(\\mathscr{E})[[\\hbar]]$, and systems of effective actions.       The system of effective actions is given by $\\{\\Gamma^R(P(0,T), S) \\mid T \\in \\mbb R_{> 0} \\}$.   The quantum master equation as a statement about the system of effective actions is independent of the choice of renormalisation scheme.   \n\nWe will fix once and for all a renormalisation scheme, and use it to parametrise the set of systems of effective actions.  If we use a different renormalisation scheme, everything works the same, except we are parametrising the set of systems of effective actions in a slightly different way. \n\n\\subsection{Homotopies of solutions to the renormalised quantum master equation}\nLet's consider the finite dimensional situation again for a moment, with the further assumption that the finite dimensional space of fields $V$ has trivial $Q$ cohomology.  Then the subspace $L \\subset V$ is a Lagrangian, and not just isotropic; we have a direct sum decomposition \n$$\nV =  L \\oplus \\Im Q.\n$$\nThe importance of the Batalin-Vilkovisky quantum master equation comes from the fact that in this situation, if $S$ satisfies the Batalin-Vilkovisky quantum master equation, then the partition function $Z(S,\\hbar, a = 0)$  remains unchanged under continuous variations of $L$.  \n\n\nWe would like to prove a version of this in the infinite dimensional situation, for the renormalised quantum master equation.  However, things are more delicate in this situation; the renormalised QME itself depends on the choice of a gauge fixing condition.  What we will show is that if $Q^{GF}(t)$ is a one-parameter family of gauge fixing conditions, corresponding to the family of isotropic subspaces $\\Im Q^{GF}(t)$,  then the set of homotopy classes of solutions to the renormalised QME using $Q^{GF}(0)$ is isomorphic to the corresponding set using $Q^{GF}(1)$.  This result is a corollary of a result about certain simplicial sets of gauge fixing conditions and of solutions to the renormalised QME.  \n\n\nLet me explain this picture in more detail.  One of the axioms we need for our gauge fixing conditions is that there is a direct sum decomposition\n$$\n\\mathscr{E} = \\Im Q^{GF} \\oplus \\Ker H \\oplus \\Im Q\n$$ \nwhere $H = [Q^{GF}, Q]$ so that $\\Ker H$ is the space of harmonic elements of $\\mathscr{E}$.  \nThis leads to the identification\n$$\nH^\\ast(\\mathscr{E}, Q) = \\Ker H.\n$$\nThis cohomology group is finite dimensional.\n\nThere is a notion of homotopy of solutions of the quantum master equation. Briefly, two solutions $S_0, S_1$ of the quantum master equation are homotopic if there exists $S_t + \\d t S'_t$, for $t \\in [0,1]$, such that \n$$\n\\left(Q + \\d t \\frac{\\d}{\\d t} + \\hbar \\Delta \\right) e^{(S_t + \\d t S'_t )\/\\hbar } = 0. \n$$\nIn a similar way, one can define a notion of homotopy of solutions of the renormalised quantum master equation.\n\n\n\\begin{utheorem}\n\\begin{enumerate}\n\\item\nIf $S$ satisfies the renormalised quantum master equation, then the restriction of $\\Gamma^R (P(0,\\infty), S)$ to $H^\\ast(\\mathscr{E}, Q)$ satisfies the finite dimensional quantum master equation.   The map $S \\to \\Gamma^R (P(0,\\infty), S)$ respects homotopies. \n\\item\nLet $Q^{GF}(t)  \\subset \\mathscr{E}$ be a smooth one-parameter family of  gauge fixing conditions, for $t \\in [0,1]$.    Then there exists a natural bijection between the set of homotopy classes of solutions of the renormalised quantum master equation using the gauge fixing condition $Q^{GF}(0)$  and the corresponding set using $Q^{GF}(1)$.\n\\item\nLet $S_0$, $S_1$ be solutions of the renormalised quantum master equations using $Q^{GF}(0)$ and $Q^{GF}(1)$ respectively, which correspond up to homotopy under the bijection coming from the family $Q^{GF}(t)$.  Then $\\Gamma^R (P(0,\\infty), S_0)$ and $\\Gamma^R(P(0,\\infty), S_1)$ (defined using $Q^{GF}(0)$ and $Q^{GF}(1)$ respectively) are homotopic solutions of the quantum master equation on $H^\\ast(\\mathscr{E}, Q)$.  \n\\end{enumerate}\n\n\\end{utheorem}\nIn fact, this result will be a corollary of a more abstract result concerning simplicial sets of solutions of the quantum master equation.\n\\begin{utheorem}\nThere exist simplicial sets\n\\begin{enumerate}\n\\item $\\mathbf{GF}(\\mathscr{E},Q)$ of gauge fixing conditions \n\\item $\\mathbf{BV}( \\mathscr{E}, Q )$ of solutions of the renormalised quantum master equation on $\\mathscr{E}$ \n\\item $\\mathbf{BV}( H^\\ast(\\mathscr{E}, Q) )$ of solutions to the finite-dimensional quantum master equation on $H^\\ast(\\mathscr{E},Q)$\n\\end{enumerate}\nwhich fit  into a diagram of maps of simplicial sets\n$$\n\\xymatrix{ \\mathbf{BV}( \\mathscr{E}, Q )  \\ar[rr]^(.43){\\Gamma^R(P(0,\\infty), - ) } \\ar[d]^{\\pi} & &   \\mathbf{BV}( H^\\ast(\\mathscr{E}, Q) )  \\\\ \n\\mathbf{GF}( \\mathscr{E}, Q)  &  }\n$$\nwhere the vertical arrow  $\\pi$ is a fibration of simplicial sets.  \n\\end{utheorem}\nA solution of the renormalised quantum master equation must be with respect to some gauge fixing condition; this defines the vertical arrow $\\pi : \\mathbf{BV}( \\mathscr{E}, Q ) \\to \\mathbf{GF}(\\mathscr{E},Q)$.   The map $\\mathbf{BV}( \\mathscr{E}, Q ) \\to   \\mathbf{BV}( H^\\ast(\\mathscr{E}, Q) )$ is the simplicial version of the map discussed earlier, which takes a solution $S$ of the renormalised quantum master equation to $\\Gamma^R(P(0,\\infty), S) \\vert_{H^\\ast(\\mathscr{E},Q)}$.  This is a solution of the quantum master equation on $H^\\ast(\\mathscr{E},Q)$.\n\nOne can deduce the previous more concrete result from this abstract statement about simplicial sets using the simplicial analogs of path and homotopy lifting properties for fibrations.  \n\n\\subsection{}\nWe have seen that the set of systems of effective actions is a canonical object associated to $(\\mathscr{E},Q, Q^{GF})$, defined with out reference to a renormalisation scheme.  In a similar way,  we could say that the simplicial set $\\mathbf{BV}(\\mathscr{E},Q)$ is a canonical object associated to $(\\mathscr{E},Q)$ without reference to the choice of gauge fixing condition.  A choice of a renormalisation scheme and a gauge fixing condition gives us a convenient parametrisation of this simplicial set, as a set of local functionals  satisfying the renormalised quantum master equation.  However, if we choose a different renormalisation scheme, we get something canonically isomorphic; and if we choose a different gauge fixing condition we something canonically homotopy equivalent. (At least, this is true as long as the space of natural gauge fixing conditions is contractible, which it always seems to be in examples). \n\nIf we have a classical action $S_0$ which solves the classical master equation, there is a simplicial set $\\mathbf{BV}(\\mathscr{E},Q, S_0)$ of quantisations of $S_0$, i.e.\\ solutions to the renormalised quantum master of the form $S_0+ \\hbar S_1 + \\cdots$.    If the space of natural gauge fixing conditions is contractible, then this simplicial set is canonically associated to the classical theory $(\\mathscr{E},Q,S_0)$, up to canonical homotopy equivalences. \n\nThus, there is a homotopy action of the group of symmetries of the classical theory on the simplicial set $\\mathbf{BV}(\\mathscr{E},Q, S_0)$ of quantisations.  One would like to quantise a given classical theory in a way preserving as many symmetries as possible. \n\n\n\n\n\n\n\n\n\n\n\\subsection{Quantisation of Chern-Simons theory}\nThe results of this paper allow one to renormalise a wide variety of quantum field theories, for example Chern-Simons theory on a compact oriented manifold, or pure Yang-Mills theory on a compact $4$-dimen\\-sional manifold with a conformal class of metrics.  By ``renormalisation'' I simply mean the construction of counter-terms. \n\n\n\nI would like to distinguish between renormalisation and quantisation.  By \\emph{quantisation} I mean the replacement of an action $S_0$ by an action $S = S_0 + \\sum_{i > 0} \\hbar^i S_i$ which satisfies the renormalised quantum master equation.   The only non-trivial theory we succeed in quantising in this paper is Chern-Simons theory.  In fact, we only quantise Chern-Simons theory modulo the constant term (constant as a function on the space of fields). \n\nThe quantisation of Chern-Simons theory is based on a general local-to-global principle, which allows one to construct global solutions to the renormalised QME from local ones.  This local-to-global result is stated and proved for a general class of theories in the body of the paper.  To keep things simple, I'll only discuss Chern-Simons theory in this introduction. \n\nChern-Simons theory is a perturbative gauge theory associated to a compact oriented manifold $M$ and a flat bundle $\\mathfrak g$ of super Lie algebras on $M$, with an invariant pairing of parity opposite to that of $\\dim M$. \n\nIn this situation,  let\n$$\n\\mathscr{E} = \\Omega^\\ast(M , \\mathfrak g ) [1].\n$$\n$\\mathscr{E}$ is the Batalin-Vilkovisky odd symplectic manifold associated to the Chern-Simons gauge theory of connections with values in $\\mathfrak g$.  \n\nA gauge fixing condition on $\\mathscr{E}$ is given by a choice of a metric on $M$.  The space of metrics is of course contractible,.   As we have seen above, the spaces of homotopy classes of solutions to the renormalised QME for different gauge fixing conditions are homotopy equivalent.  Thus, one can speak about homotopy classes of solutions to the renormalised QME without reference to a metric. \n\\begin{utheorem}\nLet $M$, $\\mathfrak g$ be as above.  Then there is a canonical (up to a contractible choice) quantisation $S$ of the Chern-Simons action $S_0$ on $\\mathscr{E}$, modulo constant terms. \n\nThat is, there is a canonical up to homotopy functional $S = S_0 + \\sum_{i > 0} \\hbar^i  S_i$ on $\\mathscr{E}$, where each $S_i$ is defined modulo constants (as a function on $\\mathscr{E}$), which satisfies the renormalised quantum master equation. \n\\end{utheorem}\nI should emphasise that the specific form the $S_i$ will take is dependent on both the metric and the renormalisation scheme we choose to work with. If we use a different renormalisation scheme, then the $S_i$ will change, but the effective action $\\Gamma^R ( P (0,T) , \\sum \\hbar^i S_i)$ remains unchanged.  If we use a different metric, then this effective action changes by a homotopy. \n\n\\begin{ucorollary}\nThere is a canonical, up to homotopy and modulo constants, function $\\Phi^{CS}$ on \n$$H^\\ast(\\mathscr{E}) = H^\\ast(M, \\mathfrak g ) [1]$$\nwhich satisfies the quantum master equation.     \n\\end{ucorollary}\nAs $\\Phi^{CS}$ is well-defined modulo constants, $\\Phi^{CS}$ is an element of \n$$\n\\Phi^{CS}  \\in \\left( \\prod_{k > 0}  \\Sym^k H^\\ast(\\mathscr{E})^{\\vee}  \\right)[[\\hbar]] .\n$$\nThe quantum master equation is\n$$\n\\{ \\Phi^{CS}, \\Phi^{CS} \\} + \\hbar \\Delta \\Phi^{CS} = 0\n$$\nwhich must hold modulo constants, that is, modulo $\\mathbb C[[\\hbar]]$.  \n\nOne can write this identity in more explicit terms.  The Hamiltonian vector field associated to $\\Phi^{CS}$ has Taylor components which are linear maps\n$$\n\\sum \\hbar^i l_{i,k} : H^\\ast(M, \\mathfrak g)^{\\otimes k} \\to H^\\ast(M, \\mathfrak g) [[\\hbar]]. \n$$\nwhere $l_{i,k}$ is independent of $\\hbar$.   The $l_{i,k}$ are defined for all $i, k \\ge 0$.  The $l_{0,k}$ are the usual $L_\\infty$ operations, which arise via the homological perturbation lemma.  The $l_{i,k}$ for $i  > 0$ are the new structure.  The quantum master equation is encoded in a sequence of identities of the form\n\\begin{multline*}\n\\sum_{\\substack{i_1+i_2 = i \\\\ k_1 + k_2 = k-1  }} \\pm l_{i_1,k_1}  ( x_1,\\ldots, x_j,  l_{i_2,k_2}  \n(x_{j+1}, \\ldots, x_{j+i_1}   )  , x_{j + i_1 + 1}, \\ldots, x_{i}       )  \\\\\n+ \\sum \\pm  l_{i-1,k+2} ( x_1, \\ldots, x_{j'}, \\delta', x_{j'+1}, \\ldots, x_{j''}, \\delta'', x_{j''+1}, \\ldots, x_i  ) = 0. \n\\end{multline*}\nfor all $i,k$. \nHere, the $x_a \\in H^\\ast(M , \\mathfrak g)$, and $\\sum \\delta' \\otimes \\delta'' \\in H^\\ast(M, \\mathfrak g)^{\\otimes 2}$ is the tensor inverse to the pairing.    The first term in this identity is the expression appearing in the usual $L_\\infty$ equation.  This algebraic structure is sometimes called a ``quantum $L_\\infty$ algebra''. \n\nAs we have seen, the $l_{0,k}$ give the usual $L_\\infty$ structure on $H^\\ast(M, \\mathfrak g)$. These $L_\\infty$ algebras, for varying $\\mathfrak g$,  encode a great deal of the rational homotopy type of $M$. Thus, the structure defined by the  the $l_{i,k}$ for $i > 0$ can be viewed as a kind of quantisation of the rational homotopy type. \n\nIn the case when $H^\\ast(\\mathscr{E})  = 0$,  Kontsevich \\cite{Kon94} and Axelrod-Singer \\cite{AxeSin92,AxeSin94} (when $\\dim M = 3$) have already constructed the perturbative Chern-Simons invariants.  In some sense, their construction is orthogonal to the construction in this paper.   Because we work modulo constants, the construction in this paper doesn't give anything in the case when $H^\\ast(\\mathscr{E}) = 0$.     On the other hand, their constructions don't apply in the situations where our construction gives something non-trivial.    \n\nThere seems to be no fundamental reason why  a generalisation of the construction to this paper, including the constant term,  should not exist.  Such a generalisation would also generalise the results of Kontsevich and Axelrod-Singer.    However, the problem of constructing such a generalisation does not seem to be amenable to the  techniques used in this paper.\n\nA theory related to the Chern-Simons theory considered here has been treated in the very interesting recent paper \\cite{Mne06}. The BF theory used by Mnev is the same as Chern-Simons theory with a special kind of Lie algebra, of the following form.  Let $\\mathfrak g$ be any finite dimensional Lie algebra.  Then $\\mathfrak g \\oplus \\mathfrak g^\\vee$ is a Lie algebra with an even invariant pairing, and  $\\mathfrak g \\oplus \\mathfrak g^\\vee[1]$ is a Lie algebra with an odd invariant pairing.  The Lie algebra structure arises from the fact that $\\mathfrak g^\\vee$ carries a $\\mathfrak g$ action.  Chern-Simons theory with Lie algebra $\\mathfrak g \\oplus \\mathfrak g^\\vee$ (when $\\dim M$ is odd) or $\\mathfrak g \\oplus \\mathfrak g^\\vee[1]$  (when $\\dim M$ is even) is the same as the BF theory considered by Mnev. \n\n\\subsection{Construction of the quantisation of Chern-Simons theory}\n\n\nLet me sketch the proof of the theorem on quantisation of Chern-Simons theory.  The proof uses the homotopical algebra of simplicial presheaves to glue together local solutions to the renormalised quantum master equation to find a global solution.\n\nRecall that a simplicial presheaf $G$ on $M$ is a presheaf of simplicial sets on $M$; thus, for each open set $U \\subset M$ and each integer $n$, we have a set $G(U,n)$ of $n$-simplices of the simplicial set $G(U)$. \n\nLet $\\mathbf{Met}$ be the simplicial presheaf such that $\\mathbf{Met}(U,n)$ is the set of smooth families of Riemannian metrics on $U$, parametrised by $\\Delta^n$.  Let $\\mathbf{FMet} \\subset \\mathbf{Met}$ be the sub-simplicial presheaf given by families of flat metrics. \n\n It turns out that whether or not an action functional $S$ satisfies the renormalised quantum master equation is a local property.   Further,  the statement that $S$ satisfies the renormalised QME on an open set $U$ depends only on the metric $g$ on $U$. This is true also in families, parametrised by simplices.  Thus, we can arrange the solutions of the renormalised QME into a simplicial presheaf on $M$.\n \n\\begin{utheorem}\nThere is a simplicial presheaf $\\mathbf{BV}$ on $M$, with a map $\\mathbf{BV} \\to \\mathbf{Met}$, whose  $0$-simplices $\\mathbf{BV}(U,0)$ are given by\n\\begin{enumerate}\n\\item\nmetrics $g$ on $U$ \n\\item\na quantisation $S$ of the Chern-Simons action on $U$, modulo constants.  That is, $S$ is a solution of the renormalised quantum master equation (modulo constants) associated to $g$, which modulo $\\hbar$ is the Chern-Simons action $S_0$.\n\\end{enumerate}\nThe one-simplices $\\mathbf{BV} (U,1)$ are homotopies of this data, and so on.\n\\end{utheorem}\n\n\nOur ultimate aim is to construct a canonical (up to contractible choice) point of $\\Gamma(M, \\mathbf{BV})$.   Such a point will consist of a metric $g$ on $M$  and a quantisation of the Chern-Simons action on $\\mathscr{E}$ to a solution of the renormalised quantum master equation associated to $g$, as always modulo constants. \n\nWe can always find a solution to the renormalised QME locally. \n\\begin{uproposition}\nSuppose an open subset $U\\subset M$ is equipped with a flat Riemannian metric.  Then the original Chern-Simons action satisfies the renormalised quantum master equation.\n\\end{uproposition}\n\\begin{remark}\nThis proposition is the only result in this subsection which is really special to Chern-Simons theory.  The proof of this proposition relies heavily on the work of Kontsevich \\cite{Kon94, Kon03a}.    In particular, we use the compactifications of configuration spaces used in these papers.  The quantum master equation is deduced from a theorem about the vanishing of certain integrals on configuration spaces proved by Kontsevich in \\cite{Kon94} when $\\dim M > 2$ and in \\cite{Kon03a} when $\\dim M = 2$.     \n\\end{remark}\n\n\n\nThe last proposition shows that we have a map \n$$\n\\mathbf{FMet} \\to \\mathbf{BV}\n$$\nof simplicial presheaves on $M$.\n\n\n\nIf $G$ is a simplicial presheaf on $M$, one can construct its derived global sections $\\mbb R \\Gamma(M,G)$, which is a simplicial set.  We use a \\v{C}ech definition of $\\mbb R \\Gamma(M,G)$.  If $G_1 \\to G_2$ is a map of simplicial presheaves which induces a  a weak equivalence of simplicial sets $G_1(U) \\to G_2(U)$, for sufficiently small open balls $U$ in $M$, then the map $\\mbb R \\Gamma(M,G_1) \\to \\mbb R \\Gamma(M,G_2)$ is a weak equivalence.  \n\\begin{ulemma}\nFor sufficiently small open balls $U$ in $M$, $\\mathbf{FMet}(U)$ is contractible.\n\\end{ulemma}\nIf $U$ is a ball, then $\\mathbf{FMet}(U)$ is the simplicial set associated to the space of flat metrics on $U$, which one can show is contractible using a simple rescaling argument.\n\nIt follows that $\\mbb R \\Gamma(M,\\mathbf{FMet})$ is contractible. \n\\begin{thmD}\nThe map\n$$\n\\Gamma(M, \\mathbf{BV}) \\to \\mbb R \\Gamma(M,\\mathbf{BV})\n$$\nis a weak equivalence.\n\\end{thmD}\nThis theorem is the heart of the ``local-to-global'' principle; it allows one to glue together local solutions to the renormalised QME to give global ones.  This result is stated and proved for a general class of theories in the body of the paper.  \n\n\nWe have a diagram\n$$\n\\mbb R \\Gamma(M,\\mathbf{FMet}) \\to \\mbb R \\Gamma(M,\\mathbf{BV}) \\xleftarrow{\\simeq} \\Gamma(M, \\mathbf{BV}) \n$$\nwhere the first arrow comes from the map of simplicial presheaves $\\mathbf{FMet} \\to \\mathbf{Met}$ constructed earlier.  The space on the left is contractible, and theorem D says that the arrow on the right is a weak equivalence.  Thus, we get the required point of $\\Gamma(M, \\mathbf{BV}) $ up to contractible choice. \n\n\n\n\n\\subsection{Acknowledgements}\nThis paper benefited a great deal from conversations with many people.  I'd like to thank Dennis Sullivan for inviting me to present these results in his seminar, and for many helpful conversations.   Ezra Getzler's help with heat kernels and simplicial methods was invaluable, and made the paper much clearer.    A conversation with Jack Morava was very helpful early on.   Arthur Greenspoon's comments greatly improved the text.    I'm also grateful to Mohammed Abouzaid, John Baez, Alberto Cattaneo,  Paul Goerss, Dmitry Kaledin, Pavel Mnev, David Nadler, Vasily Pestun, Jared Wunsch and Eric Zaslow, for their help with various aspects of this work.    \n\n\n\\section{A crash course in  the Batalin-Vilkovisky formalism}\n\\label{section intro bv}\n\nThis section should probably be skipped by experts; it consists of an informal introduction to the Batalin-Vilkovisky approach to quantising gauge theories.    The only thing which may not be standard is a discussion of a version of the BV formalism where one integrates over isotropic instead of Lagrangian subspaces.  \n\nIn this section, our vector spaces will \\emph{always} be finite dimensional.  Of course, none of the difficulties of renormalisation are present in this simple case.  Many of the expressions we write in the finite dimensional case are ill-defined in the infinite dimensional case. \n\nLet us suppose we have a finite dimensional vector space $V$ of fields, a group $G$ acting on $V$ in a possibly non-linear way, and a $G$-invariant function $f$ on $V$ such that $0$ is a critical point of $f$.\n\nOne is interested in making sense of functional integrals of the form\n$$\\int_{ V \/ G } e^{f\/ \\hbar }$$\nover the quotient space $V \/ G$.   The starting point in the Batalin-Vilkovisky formalism is the BRST construction, which says one should try to interpret this quotient in a homological fashion.  This means we should consider the  supermanifold\n$$\n\\mathfrak g [ 1 ] \\oplus V\n$$\nwhere $[1]$ refers to a change of degree, so $\\mathfrak g$ is in degree $-1$. The space of functions on this super-manifold is \n$$\n\\mscr O ( \\mathfrak g [ 1 ] \\oplus V ) = \\wedge^\\ast \\mathfrak {g}^\\vee \\otimes \\mscr O (V)\n$$\nwhich is the super vector space underlying the Chevalley-Eilenberg Lie algebra co\\-chain complex for $\\mathfrak g$ with coefficients in the $\\mathfrak g$-module $\\mscr O(V)$.  The Chevalley-Eilenberg differential gives an odd derivation of $\\mscr O(\\mathfrak g [1] \\oplus V)$, which can be thought of as an odd vector field on $\\mathfrak g [1] \\oplus V$.  Let us denote this odd vector field by $X$. \n\n Recall that this Lie algebra cochain complex computes the homotopy invariants for the action of $\\mathfrak g$ on $\\mscr O(V)$.  Thus, we can view $\\mscr O(\\mathfrak g[1] \\oplus V)$, with this differential, as a ``derived'' version of the algebra of functions on the quotient of $V$ by the action of $G$ \n(at least, in a formal neighbourhood of the origin, which is all we really care about).   The BRST construction says one should replace the integral over $V \/ G$ by an integral of the form\n$$\\int_{ \\mathfrak {g} [1 ] \\oplus  V } e^{f \/ \\hbar }.$$\n\nThis leaves us in a better situation than before, as we are in a linear space, and we can attempt to make sense of the integral perturbatively.   However,  we still have problems; the quadratic part of the functional $f$ is highly degenerate on $\\mathfrak {g} [1 ] \\oplus  V$.  Indeed, $f$ is independent of $\\mathfrak g[1]$ and is constant on $G$-orbits on $V$.  Thus,  we cannot compute the integral above by a perturbation expansion around the critical points of the quadratic part of $f$. \n\nThis is where the Batalin-Vilkovisky formalism comes in.    Let $E$ denote the odd cotangent bundle of $\\mathfrak{g} [1] \\oplus V$, so that\n$$\nE = \\mathfrak{g} [1] \\oplus V \\oplus V^\\vee [-1] \\oplus \\mathfrak{g}^\\vee [-2].\n$$\nThe various summands of $E$ are usually given the following names: $\\mathfrak{g} [1]$ is the space of ghosts, $V$ is the space of fields,  $V^\\vee [-1]$ is the space of antifields and $\\mathfrak{g}^\\vee [-2]$ is the space of antighosts. \n\nThe function $f$ on $\\mathfrak{g} [1] \\oplus V$ pulls back to a function on $E$, via the projection $E \\to \\mathfrak{g} [1] \\oplus V$; we continue to call this function $f$.  By naturality, the vector field $X$ on $\\mathfrak{g} [1] \\oplus V$ induces one on $E$, which we continue to call $X$.  As $[X, X] = 0$ on $\\mathfrak{g} [1] \\oplus V$, the same identity holds on $E$.  As $X$ preserves $f$ on $\\mathfrak{g} [1] \\oplus V$, it continues to preserve $f$ on $E$.\n\n$E$ is an odd symplectic manifold, and $X$ is an odd vector field preserving the symplectic form.  Thus, there exists a unique function $h_X$ on $E$ whose Hamiltonian vector field is $X$, and which vanishes at zero.  As $X$ is odd, $h_X$ is an even function.\n\nAs $E$ is odd symplectic, the space of functions on $E$ has an odd Poisson bracket.  The statement that $[X,X] = 0$ translates into the equation $\\{h_X, h_X\\} = 0$. The statement that $X f = 0$ becomes $\\{h_X, f\\} = 0$.   And, as $f$ is pulled back from $\\mathfrak{g} [1] \\oplus V$, it automatically satisfies $\\{f,f\\} = 0$.  These identities together tell us that the function $f + h_X$ satisfies the \\emph{Batalin-Vilkovisky classical master equation}, \n$$\n\\{ f + h_X, f + h_X \\} = 0 . \n$$\n\nLet us write \n$$\nf (e)  + h_X ( e ) = \\frac{1}{2} \\ip{e, Q e }  + S(e)\n$$\nwhere $Q : V \\to V$ is an odd linear map, skew self adjoint for the pairing $\\ip{\\quad}$, and $S$ is a function which is at least cubic.  The fact that $f + h_X$ satisfies the classical master equation implies that $Q^2 = 0$.  Also, the identity \n$$\nQ S + \\frac{1}{2} \\{ S , S \\} = 0\n$$ \nholds as a consequence of the classical master equation for $f + h _X$.  \n\n\nLet $L \\subset E$ be a small, generic, Lagrangian perturbation of the zero section $\\mathfrak {g} [1] \\oplus V \\subset E$.  The Batalin-Vilkovisky formalism tells us to consider the functional integral\n\\begin{equation*}\n\\int_{e \\in L} \\exp  (  f(e) \/ \\hbar + h_X (e)\/ \\hbar  )  = \\int_{e \\in L} \\exp  ( \\frac{1}{2 \\hbar} \\ip{e, Q e }  + \\frac{1}{\\hbar} S(e) ) \n\\end{equation*}\nAs $L$ is generic, the pairing $\\ip{e, Q e}$ will have very little degeneracy on $L$.  In fact, if the complex $(E,Q)$ has zero cohomology, then the pairing $\\ip{e, Q e}$ is non-degenerate on a generic Lagrangian $L$.     This means we can perform the above integral perturbatively, around the critical point $0 \\in L$. \n\n\\subsection{Quantum master equation}\nLet us now turn to a more general situation, where $E$ is a finite dimensional vector space with an odd symplectic pairing, and $Q : E \\to E$ is an odd operator of square zero which is skew self adjoint for the pairing.   $E$ is not necessarily of the form constructed above.\n\nLet $x_i, \\eta_i$ be a Darboux basis for $E$, so that $x_i$ are even, $\\eta_i$ are odd, and $\\ip{x_i, \\eta_i} = 1$.  Let $\\Delta$ be the order two differential operator on $E$ given by the formula\n$$\n\\Delta = \\sum \\partial_{x_i} \\partial_{\\eta_i}.\n$$\nThis operator is independent of the choice of basis of $E$. \n\nLet $S \\in \\mscr O(E ) [[\\hbar]]$ be an $\\hbar$-dependent function on $E$, which modulo $\\hbar$ is at least cubic. The function $S$  satisfies the \\emph{quantum master equation} if \n$$\n(Q + \\hbar \\Delta) e^{S\/  \\hbar }  = 0 . \n$$\nThis equation is equivalent to the equation\n$$\nQ S +  \\frac{1}{2} \\{ S , S \\} + \\hbar \\Delta S = 0 . \n$$\n\nThe key lemma in the Batalin-Vilkovisky formalism is the following.\n\\begin{lemma}\nLet  $L \\subset E$ be a Lagrangian on which the pairing $\\ip{e,Q e }$ is non-degenerate. (Such a Lagrangian exists if and only if $H^\\ast(E, Q ) = 0$).   Suppose that $S$ satisfies the quantum master equation. Then the integral\n$$\n\\int_{e \\in L} \\exp  ( \\frac{1}{2 \\hbar} \\ip{e, Q e }  + \\frac{1}{\\hbar} S(e) ) \n$$\nis unchanged under deformations of $L$.\n\\end{lemma}\nThe non-degeneracy of the inner product on $L$, and the fact that $S$ is at least cubic modulo $\\hbar$,   means that one can compute this integral perturbatively.\n\nSuppose  $E, Q, \\ip{\\  , \\ }, S$ are obtained as before from a gauge theory.  Then $S$ automatically satisfies the classical master equation $Q S + \\frac{1}{2} \\{S, S \\} = 0$.  If, in addition, $\\Delta S = 0$, then $S$ satisfies the quantum master equation.  Thus, we see that we can quantise the gauge theory in a way independent of the choice of $L$ as long as $S$ satisfies the equation $\\Delta S = 0$. When $S$ does not satisfy this equation, one looks to replace $S$ by a series $S ' = S + \\sum_{ i > 0} \\hbar^i S_i$ which does satisfy the quantum master equation $Q S' +  \\frac{1}{2} \\{ S' , S' \\} + \\hbar \\Delta S' = 0 $.\n\n\\subsection{Geometric interpretation of the quantum master equation}\nThe quantum master equation has a geometric interpretation, first described by Albert Schwarz \\cite{Sch93}.  I will give a very brief summary; the reader should refer to this paper for more details.\n\nLet $\\mu$ denote the unique up to scale translation invariant ``measure'' on $E$, that is, section of the Berezinian.    The operator $\\Delta$ can be interpreted as a kind of divergence associated to the measure $\\mu$, as follows.  As $E$ is an odd symplectic manifold,  the algebra $\\mscr O(E)$ has an odd Poisson bracket.  Every function $S \\in \\mscr O(E)$ has an associated vector field $X_S$, defined by the formula $X_S f = \\{ S, f \\}$.  \n\nThe operator $\\Delta$ satisfies the identity\n$$\n\\mathcal L_{X_S} \\mu = (\\Delta S) \\mu\n$$\nwhere $\\mathcal L_{X_S}$ refers to the Lie derivative.  In other words, $\\Delta S$ is the infinitesimal change in volume associated to the vector field $X_S$.  \n\nThus, the two equations\n\\begin{align*}\n\\{S,S\\} &= 0 \\\\\n\\Delta S &= 0\n\\end{align*}\nsay that the vector field $X_S$ has square zero and is measure preserving.\n\nThis gives an interpretation of the quantum master equation in the case when $S$ is independent of $\\hbar$.  When $S$ depends on $\\hbar$, the two terms of the quantum master equation do not necessarily hold independently.  In this situation, we can interpret the quantum master equation as follows.  Let $\\mu_S$ be the measure on $E$ defined by the formula\n$$\n\\mu_S = e^{S \/ \\hbar} \\mu.\n$$\nWe can define an operator $\\Delta_S$ on $\\mscr O(E)$ by the formula\n$$\n\\mathcal L_{X_f} \\mu_S = (\\Delta_S f ) \\mu_S.\n$$\nThis is the divergence operator associated to the measure $\\mu_S$, in the same way that $\\Delta$ is the divergence operator associated to the translation invariant measure $\\mu$.  \n\nThen, a slightly weaker version of the quantum master equation is  equivalent to the statement\n$$\n\\Delta_S^2 = 0.\n$$\nIndeed, one can compute that\n$$\n\\hbar \\Delta_S f = \\{S, f \\} + \\hbar \\Delta f \n$$\nso that \n$$\n\\hbar^2 \\Delta_S^2 f = \\tfrac{1}{2} \\{  \\{S, S\\} , f \\} +  \\hbar \\{ \\Delta S , f \\} . \n$$\nThus, $\\Delta_S^2 = 0$ if and only if $\\tfrac{1}{2} \\{S,S\\} +\\hbar \\Delta S$ is in the centre of the Poisson bracket, that is, is constant. \n\nThis discussion shows that the quantum master equation is the statement that the measure $e^{S \/ \\hbar } \\mu$ is compatible in a certain sense with the odd symplectic structure on $E$. \n\n\\begin{remark}\nIn fact it is better to use half-densities rather than densities in this picture.  A solution of the quantum master equation is then given by a half-density which is compatible in a certain sense with the odd symplectic form. As all of our odd symplectic manifolds are linear, we can ignore this subtlety. \n\\end{remark}\n\n \\subsection{Integrating over isotropic subspaces}\n As we have described it, the BV formalism only has a chance to work when $H^\\ast(E, Q ) = 0$.   This is because one cannot make sense of the relevant integrals perturbatively otherwise.   However, there is a generalisation of the BV formalism which works when $H^\\ast(E,Q ) \\neq 0$.  In this situation, let $L \\subset E$ be an isotropic subspace such that $Q : L \\to \\Im Q$ is an isomorphism.    Let $\\operatorname{Ann}(L) \\subset E$ be the set of vectors which pair to zero with any element of $L$. Then we can identify \n$$\nH^\\ast(E, Q) = \\operatorname{Ann}(L) \\cap \\Ker Q . \n$$\nWe thus have a direct sum decomposition \n$$\nE = L \\oplus H^\\ast(E,Q) \\oplus \\Im Q . \n$$\nNote that $H^\\ast(E,Q)$ acquires an odd symplectic pairing from that on $E$. Thus, there is a BV operator $\\Delta_{H^\\ast(E,Q) }$ on functions on $H^\\ast(E,Q)$.  We say a function $f$ on $H^\\ast(E,Q)$ satisfies the quantum master equation if $\\Delta e^{f \/ \\hbar  } = 0$. \n\nThe analog of the ``key lemma'' of the Batalin-Vilkovisky formalism is the following.  This lemma is well known to experts in the area.  \n\\begin{lemma}\nLet $S \\in \\mscr O(E) [[\\hbar]] $ be an $\\hbar$-dependent function on $E$ which satisfies the quantum master equation.  Then the function on $H^\\ast(E,Q)$ defined by\n$$\na \\mapsto \\hbar \\log \\left( \\int_{e \\in L} \\exp  ( \\frac{1}{2 \\hbar} \\ip{e, Q e }  + \\frac{1}{\\hbar} S(e+  a ) )  \\right) \n$$\n(where we think of $H^\\ast(E,Q)$ as a subspace of $E$) satisfies the quantum master equation. \n\nFurther, if we perturb the isotropic subspace $L$ a small amount, then this solution of the QME on $H^\\ast( E, Q)$ is changed to a homotopic solution of the QME. \n\\end{lemma}\nNote that since $Q a = 0$, we can write the exponential in the integrand in the equivalent way $( \\frac{1}{2 \\hbar} \\ip{e+a, Q (e+a) }  + \\frac{1}{\\hbar} S(e+  a ) )  $.  Thus, there is no real need here to separate out quadratic and higher terms. However, in the infinite dimensional situation we will discuss later, it will be essential to write the integrand as $( \\frac{1}{2 \\hbar} \\ip{e, Q e }  + \\frac{1}{\\hbar} S(e+  a ) )  $, because we will take a field $a$ which is not closed. \n\n\nThis integral is an explicit way of writing the homological perturbation lemma for BV algebras, which transfers a solution of the quantum master equation at chain level to a corresponding solution on cohomology.    From this observation it's clear (at least philosophically) why the lemma should be true; the choice of the Lagrangian $L$ is essentially the same as the choice of symplectic homotopy equivalence between $E$ and its cohomology.  I'll omit a formal proof for now, as a proof of a more general statement will be given later. \n\nLet me explain what a ``homotopy'' of a solution of the QME is.  There is a general concept of homotopy equivalence of algebraic objects, which I learned from the work of Deligne, Griffiths, Morgan and Sullivan \\cite{DelGriMor75}.  Two algebraic objects are homotopic if they are connected by a family of such objects parametrised by the commutative differential algebra $\\Omega^\\ast ([0,1])$.     In our context, this means that two solutions $f_0, f_1$  to the quantum master equation on $H^\\ast(E,Q)$ are homotopic if there exists an element $F \\in \\mscr O(H^\\ast(E,Q) ) \\otimes \\Omega^\\ast([0,1] ) [[\\hbar]]$ which satisfies the quantum master equation\n$$\n (  \\d_{DR} + \\hbar \\Delta_{H^\\ast(E,Q) } ) e^{F \/ \\hbar } = 0\n$$\nand which restricts to $f_0$ and $f_1$ when we evaluate at $0$ and $1$.    Here $\\d _{DR}$ refers to the de Rham differential on $\\Omega^\\ast([0,1])$. \n\nThe quantum master equation imposed on $F$ is equivalent to the equation\n$$\n\\d_{DR} F + \\frac{1}{2} \\{ F , F \\} + \\hbar \\Delta F = 0 . \n$$\nIf we write $F(t, \\d t) =   A(t ) + \\d t B(t)$, then the QME imposed on $F$ becomes the system of equations\n\\begin{align*}\n\\frac{1}{2} \\{ A(t) , A(t) \\} + \\hbar \\Delta A (t) &= 0 \\\\\n\\frac{\\d}  { \\d t} A(t) + \\{ A(t) , B(t) \\} +  \\hbar \\Delta B(t) &= 0 . \n\\end{align*}\nThe first equation says that $A(t)$ satisfies the ordinary QME for all $t$, and the second says that the family $A(t)$ is tangent at every point to an orbit of a certain ``gauge group'' acting on the space of solutions to the QME.  \n\n\\section{Example : Chern-Simons theory}\nI want to discuss a class of quantum field theories in the Batalin-Vilkovisky formalism.  The general definition of the kind of quantum field theory I want to consider is a little technical, so I will start by discussing a simple example in detail, namely Chern-Simons theory on a $3$-manifold.  (Later we will discuss Chern-Simons theory on a manifold of any dimension).    Most of the features of more general theories are already evident in this example.  \n\nLet $M$ be a compact oriented $3$-manifold.  Let $\\mathfrak g$ be a flat bundle of complex\\footnote{Everything works if we take a real Lie algebra and work over $\\mbb R$ throughout.} Lie algebras with a complex valued invariant pairing  on $M$.  For example, $\\mathfrak g$ could be the adjoint bundle associated to a flat principal $SL(n,\\mathbb C)$ bundle, equipped with the Killing form. \n\nWe want to quantise the Chern-Simons gauge theory associated to $\\mathfrak g$. This is the theory whose space of fields $\\mathscr{V}$ is the space $\\Omega^1(M, \\mathfrak g)$, which we think of as being the space of $\\mathfrak g$-valued connections.   The Lie algebra of the gauge group is\n$$\\mathscr{G} = \\Omega^0 (M, \\mathfrak g ).$$    This Lie algebra acts on the space of fields in the usual way that infinitesimal bundle automorphisms act on connections.  \n\nThe Chern-Simons action functional is the function on $\\mathscr{V}$ given by\n$$\n\\frac{1}{2}\\ip{v, \\d v}  + \\frac{1}{3} \\ip{ v, [v,v] } \n$$\nwhere $\\d : \\Omega^1 (\\mathfrak g) \\to \\Omega^2 (\\mathfrak g)$ is the operator obtained by coupling the flat connection on $\\mathfrak g$ to the de Rham differential.\n\nApplying the Batalin-Vilkovisky construction, as described in section \\ref{section intro bv}, yields an odd symplectic vector space \n\\begin{align*}\n\\mathscr{E} = \\mathscr{G} [1] \\oplus \\mathscr{V} \\oplus \\mathscr{V}^\\vee [-1] \\oplus \\mathscr{G}^\\vee [-2].\n\\end{align*}\nIf we interpret the duals of the infinite dimensional vector spaces appropriately, we find that \n$$\n\\mathscr{E} = \\Omega^\\ast (M, \\mathfrak g) [1] . \n$$\nThis is simply because $\\Omega^2 (M,\\mathfrak{g})$ has a natural non-degenerate pairing with $ \\mathscr{V} = \\Omega^1 (M,\\mathfrak{g} )$, and $\\Omega^3 (M,\\mathfrak{g})$ has a natural non-degenerate pairing with $\\Omega^0(M,\\mathfrak{g}) = \\mathscr{G}$.  \n\nThe differential $Q : \\mathscr{E} \\to \\mathscr{E}$ (constructed as in section \\ref{section intro bv}) is simply the de Rham differential, coupled to the flat connection on $\\mathfrak{g}$.     The odd pairing on $\\mathscr{E}$ is given  by\n$$\n\\ip{ \\alpha \\otimes X, \\alpha' \\otimes X'  } = (-1)^{\\abs{\\alpha} }\\int_M \\alpha\\wedge \\alpha' \\ip{X, X' }_{\\mathfrak g}\n$$\nwhere $\\alpha,\\alpha' \\in \\Omega^\\ast(M)$ and $X,X' \\in \\mathfrak{g}$.  The notation $\\ip{\\ , \\ }_{\\mathfrak g}$ refers to the invariant pairing on $\\mathfrak g$. \n\nThe action functional $S$ is given by the formula\n$$\nS( \\sum_i \\alpha_i \\otimes X_i ) = \\sum_{i,j,k} (-1)^{(\\abs{\\alpha_i} + 1) (\\abs{\\alpha_j} + 1 )} \\int_M \\alpha_i \\wedge \\alpha_j \\wedge \\alpha_k \\ip{ X_i, [X_j, X_k ]  }.\n$$\n\n\n\n\n\\subsection{Gauge fixing and functional integrals}\nAs we saw in section \\ref{section intro bv}, the next step in the Batalin-Vilkovisky formalism is the choice of a gauge fixing condition.  This is an isotropic subspace $L \\subset \\mathscr{E}$ such that $Q : L \\to \\Im Q$ is an isomorphism.   \n\nPicking a metric on $M$ gives an operator $\\d^\\ast  : \\Omega^i (M) \\to \\Omega^{i-1} (M)$.  This extends to an operator $Q^{GF} : \\mathscr{E} \\to \\mathscr{E}$, defined (locally) by\n$$\nQ^{GF} ( \\omega \\otimes A)  = (\\d^\\ast \\omega) \\otimes A\n$$\nwhere $A$ is a flat section of the bundle $\\mathfrak g$.  Our gauge fixing condition will be\n$$\nL = \\Im Q^{GF} . \n$$\nThe functional integral we will construct is\n$$\n \\int_{e \\in \\Im Q^{GF}} \\exp  ( \\frac{1}{2 \\hbar} \\ip{e, Q e }  + \\frac{1}{\\hbar} S(e+  a ) ) .\n$$\n\nThus, the Chern-Simons gauge theory in the BV formalism is encoded in the following data.\n\\begin{enumerate}\n\\item\nA vector space $\\mathscr{E}$, which is the space of global sections of a super vector bundle on a compact manifold $M$.\n\\item\nAn odd anti-symmetric  pairing on $\\mathscr{E}$, satisfying a certain non-degeneracy condition.\n\\item\nA skew self adjoint operator $Q : \\mathscr{E} \\to \\mathscr{E}$, of square zero.\n\\item\nAn even functional $S : \\mathscr{E} \\to \\mathbb C$ which is ``local'', meaning roughly that it has a Taylor expansion in terms of continuous linear maps $\\mathscr{E}^{\\otimes k} \\to \\mathbb C$ which are distributions on $M^k$, supported on the small diagonal, and non-singular in the diagonal directions.   (A precise definition will be given soon).\n\\item\nAn auxiliary operator $Q^{GF} : \\mathscr{E} \\to \\mathscr{E}$, which is a self adjoint elliptic operator of square zero.\n\\item\nThe super-commutator $[Q,Q^{GF}]$ must be an elliptic operator of order $2$ with some positivity conditions.\n\\end{enumerate}\n\n\n\\section{Batalin-Vilkovisky formalism in infinite dimensions}\n\\label{section infinite bv}\n\nNow we will define the type of quantum field theory we will consider.  The definition consists essentially of abstracting the data we encountered above in the case of Chern-Simons theory. \n\n\n\\subsection{Functionals}\n If $M,N$ are smooth manifolds and $F,G$ are super vector bundles on $M,N$ respectively, we will use the notation\n$$\n\\Gamma(M,F) \\otimes \\Gamma(N,G)\n$$\nto denote the space $\\Gamma(M \\times N, F \\boxtimes G)$ of smooth sections of $F \\boxtimes G$.  In other words, $\\otimes$ refers to the completed projective tensor product where appropriate.  \n\nLet $E$ be a super vector bundle over $\\mathbb C$ on a compact manifold $M$.  Let $\\mathscr{E}$ denote the space of global sections of $E$.   \n\\begin{definition}\nLet $\\mscr O(\\mathscr{E})$ be the algebra \n$$\n\\mscr O(\\mathscr{E}) = \\prod_{n \\ge 0} \\Hom ( \\mathscr{E}^{\\otimes n}, \\mathbb C ) _{S_n}\n$$\nwhere $\\Hom$ denotes the space of continuous linear maps, and the subscript $S_n$ denotes taking $S_n$ coinvariants.  \n\nDirect product of distributions makes $\\mscr O(\\mathscr{E})$ into an algebra.\n\\end{definition}\nWe can view  $\\mscr O(\\mathscr{E})$ as the algebra  of formal functions at $0 \\in \\mathscr{E}$ which have Taylor expansions of the form\n$$\nf(e) = \\sum f_i(e^{\\otimes i} ) \n$$\nwhere each \n$$\nf_i : \\mathscr{E}^{\\otimes i} \\to \\mathbb C\n$$\nis a continuous linear map (i.e. a distribution).  \n\\begin{definition}\nLet $X$ be an auxiliary manifold.  Define\n$$\n\\mscr O(\\mathscr{E}, C^{\\infty}(X)) = \\prod_{n \\ge 0} \\Hom ( \\mathscr{E}^{\\otimes n}, C^{\\infty}(X) ) _{S_n}\n$$\nwhere, as before, we take the space of continuous linear maps.  \n\\end{definition}\nThus, $\\mscr O(\\mathscr{E},C^{\\infty}(X))$ is a certain completed tensor product of $\\mscr O(\\mathscr{E})$ and $C^{\\infty}(X)$. \nLet $\\hbar$ be a formal parameter.  Let \n$$\n\\mscr O(\\mathscr{E}, \\mathbb C[[\\hbar]] ) =  \\varprojlim  \\mscr O(\\mathscr{E}) \\otimes \\mathbb C[\\hbar]\/\\hbar^n.\n$$\nSimilarly, let $\\mscr O(\\mathscr{E}, C^{\\infty}(X) \\otimes \\mathbb C[[\\hbar]])$ be the inverse limit $ \\varprojlim  \\mscr O(\\mathscr{E}, C^{\\infty}(X) ) \\otimes \\mathbb C[\\hbar] \/ \\hbar^n$. \n\n\\subsection{Local functionals}\n\\begin{definition}\n\nLet $\\operatorname{Diff}(E,E')$ denote the infinite rank vector bundle on $M$ of differential operators between two vector bundles $E$ and $E'$ on $M$.  Let \n$$\n\\operatorname{Poly Diff} ( \\Gamma(E)^{\\otimes n}, \\Gamma(E') ) = \\Gamma ( M, \\operatorname{Diff}(E, \\mathbb C)^{\\otimes n} \\otimes E'  ) \\subset \\operatorname{Hom} (\\Gamma(E)^{\\otimes n} , \\Gamma(E'))\n$$\nwhere $\\mathbb C$ denotes the trivial vector bundle of rank $1$.  All tensor products in this expression are fibrewise tensor products of vector bundles on $M$. \n\\end{definition}\nIt is clear that $\\operatorname{Poly Diff}( \\Gamma(E)^{\\otimes n}, \\Gamma(E'))$ is the space of sections of an infinite rank vector bundle on $M$.     If $F$ is a super vector bundle on another manifold $N$, let \n$$\n\\operatorname{PolyDiff}  ( \\Gamma(E)^{\\otimes n}, \\Gamma(E') ) \\otimes \\Gamma(F)\n$$\ndenote the completed projective tensor product, as usual, so that \n$$\n\\operatorname{PolyDiff}  ( \\Gamma(E)^{\\otimes n}, \\Gamma(E') ) \\otimes \\Gamma(F)  =  \\Gamma ( M \\times N,  (\\operatorname{Diff}(E, \\mathbb C)^{\\otimes n} \\otimes E') \\boxtimes F  ) .\n$$\nIf $X$ is a manifold, we can think of $\\operatorname{PolyDiff}  ( \\Gamma(E)^{\\otimes n}, \\Gamma(E') ) \\otimes C^{\\infty}(X)$ as the space of smooth families of polydifferential operators parametrised by $X$. \n\nOne can give an equivalent definition of polydifferential operators in terms of local trivialisations $\\{e_i\\}, \\{e'_j\\}$ of $E$ and $E'$,  and local coordinates $y_1,\\ldots, y_l$ on $M$.  \nA  map  $\\Gamma(E)^{\\otimes n} \\to \\Gamma(E')$ is a polydifferential operator if, locally, it is a finite sum of operators of the form \n$$\nf_1 e_{i_1} \\otimes \\cdots \\otimes f_n e_{i_n} \\mapsto \\sum _{j} e'_j  \\Phi^{j}_{i_1 \\ldots i_n} (y_1,\\ldots,y_l)  ( D_{I_{i_1}}  f_1 ) \\cdots ( D_{I_{i_n}} f_n ) \n$$\nwhere $\\Phi^{j}_{i_1 \\ldots i_n} (y_1,\\ldots,y_l)$ are smooth functions of the $y_i$, the $I_k$ are multi-indices, and the operators $D_{I_k}$ are the corresponding partial derivatives with respect to the $y_i$.  \n\n\n\n\n\\begin{definition}\nLet \n$$\\mscr O_l(\\mathscr{E}) \\subset \\mscr O(\\mathscr{E}) =  \\prod_{n \\ge 0} \\Hom ( \\mathscr{E}^{\\otimes n}, \\mathbb C ) _{S_n}$$ be the space of formal functionals on $\\mathscr{E}$, each of whose Taylor series terms $f_n : \\mathscr{E}^{\\otimes n} \\to \\mathbb C$ factors as \n$$\n\\mathscr{E}^{\\otimes n} \\to \\dens(M) \\xrightarrow{\\int_M} \\mathbb C\n$$\nwhere the first map is a polydifferential operator.    \n\nElements of $\\mscr O_l(\\mathscr{E})$ will be called \\emph{local functionals} on $\\mathscr{E}$.\n\nIf $X$ is another manifold (possibly with corners), and $F$ is a super vector bundle on $X$, let \n$$\n\\mscr O_l( \\mathscr{E}, \\Gamma(X,F) ) \\subset \\mscr O(\\mathscr{E}, \\Gamma(X,F)) = \\prod_{n \\ge 0} \\Hom ( \\mathscr{E}^{\\otimes n}, \\Gamma(X,F) ) _{S_n}\n$$\nbe the subspace of those functions each of whose terms factors as \n$$\n\\mathscr{E}^{\\otimes n} \\to \\dens(M) \\otimes \\Gamma(X,F) \\xrightarrow{\\int_M} \\Gamma(X,F)\n$$\nwhere the first map is in $\\operatorname{Poly Diff} ( \\mathscr{E}^{\\otimes n}, \\dens(M) ) \\otimes \\Gamma(X,F)$.\n\\end{definition}\nElements of $\\mscr O_l(\\mathscr{E}, C^{\\infty}(X))$ are smooth families of local functionals parametrised by a manifold $X$.    We will use a similar notation when we want to take functionals with values in $\\mathbb C[[\\hbar]]$.  Let\n$$\n\\mscr O_l(\\mathscr{E}, \\mathbb C[[\\hbar]] ) =  \\varprojlim  \\mscr O_l(\\mathscr{E}) \\otimes \\mathbb C[\\hbar]\/\\hbar^n.\n$$\nWe can define $\\mscr O_l(\\mathscr{E}, C^{\\infty}(X) \\otimes \\mathbb C[[\\hbar]])$ as an inverse limit in the same way. \n\n\\begin{remark}\nThe space $\\mscr O_l(\\mathscr{E})$ is \\emph{not} an algebra; the product of two local functionals is not local.\n\\end{remark}\n\n\nIf $f \\in \\mscr O_l(\\mathscr{E})$ is a local functional, then the $i$-th term of its Taylor expansion $f_i : \\mathscr{E}^{\\otimes i} \\to \\mathbb C$ is assumed to factor through a map $\\mathscr{E}^{\\otimes i} \\to \\dens(M)$.\nNote that there will in general exist many such factorisations. However, we have\n\\begin{lemma}\nIf \n$$\\Phi : \\mathscr{E}^{\\otimes n} \\to \\mathbb C$$  is a map which factorises through a polydifferential operator $\\mathscr{E}^{\\otimes n} \\to \\dens(M)$,  there is a unique polydifferential operator\n$$\n\\Psi : \\mathscr{E}^{\\otimes n-1} \\to \\Gamma(E^\\vee) \\otimes_{C^{\\infty}(M)} \\dens(M)\n$$\nsuch that \n$$\n\\Phi ( e_1 \\otimes \\cdots \\otimes e_n ) = \\int_M \\ip{ \\Psi ( e_1 \\otimes \\cdots \\otimes e_{n-1} ) , e_n }.\n$$\n\\label{lemma local functional vector field}\n\\end{lemma}\n\\begin{proof}\nThis is an easy calculation in local coordinates.\n\\end{proof} \n\n\n\n\n\\subsection{Batalin-Vilkovisky formalism}\n\\begin{definition}\nAn odd symplectic structure on $\\mathscr{E}$ is a map\n$$\n\\ip{\\ , \\ }_M : E \\otimes E \\to \\dens_M\n$$\nof graded vector bundles on $M$, which is odd, antisymmetric, and non-degenerate, in the sense that the associated map $E \\to E^{\\vee} \\otimes \\dens_M$ is an isomorphism.  \n\\end{definition}\nSuch an odd symplectic structure  induces an odd pairing \n$$\n\\ip{\\ , \\ } : \\mathscr{E} \\otimes_\\mathbb C \\mathscr{E} \\to \\mathbb C\n$$\non $\\mathscr{E}$.  \n\n\n\n\\begin{lemma}\nAn odd symplectic structure on $\\mathscr{E}$ induces an odd Poisson bracket on the space $\\mscr O_l(\\mathscr{E})$ of local functionals.  Further, if $f \\in \\mscr O_l(\\mathscr{E})$ is local and $g \\in \\mscr O(\\mathscr{E})$ is possibly non-local, the Poisson bracket $\\{f,g\\}$ is well-defined.\n\\end{lemma}\n\\begin{proof}\nThis is an immediate corollary of Lemma \\ref{lemma local functional vector field}.  Indeed, this lemma shows that any local functional on $\\mathscr{E}$ can be replaced by something which has a Taylor expansion  in terms of polydifferential operators $\\mathscr{E}^{\\otimes n} \\to \\mathscr{E}$.   Here we are using the identification of $\\mathscr{E}$ with $\\Gamma(E^\\vee )\\otimes_{C^{\\infty}(M)} \\dens(M)$ given by the odd symplectic form.  \n\nSomething with a Taylor expansion in terms of maps $\\mathscr{E}^{\\otimes n} \\to \\mathscr{E}$ can be considered to be a formal vector field on $\\mathscr{E}$.    This yields the Hamiltonian vector field on $\\mathscr{E}$ associated to a local functional.  The usual formula for the action of vector fields on functions defines the Poisson bracket $\\{f,g\\}$ if $f \\in \\mscr O_l(\\mathscr{E})$ and $g \\in \\mscr O(\\mathscr{E})$.   Since this formula amounts to inserting the output of the map $\\mathscr{E}^{\\otimes n} \\to \\mathscr{E}$ into the input of a map $\\mathscr{E}^{\\otimes k} \\to \\mathbb C$, everything is well-defined.\n\\end{proof} \nIn general, the Poisson bracket of two non-local functionals may be ill-defined. \n\n\n\nLet us fix, for the rest of the paper, the following data:\n\\begin{enumerate}\n\\item\nA compact manifold $M$ with a complex super vector bundle $E$ on $M$, whose algebra of global sections is denoted (as above) by $\\mathscr{E}$.\n\\item\nAn odd linear elliptic differential operator $Q : \\mathscr{E} \\to \\mathscr{E}$, which is self adjoint and satisfies $[Q,Q]  = 0$. $Q$ induces a differential on all the spaces associated to $\\mathscr{E}$, such as $\\mscr O_l(\\mathscr{E})$ and $\\mscr O(\\mathscr{E})$.\n\\end{enumerate}\nA  local functional $S \\in \\mscr O_l(\\mathscr{E})$, which is at least cubic,  satisfies the classical master equation if\n$$\nQ S + \\frac{1}{2}\\{S,S\\} = 0.\n$$\nWe will try to quantise, within the Batalin-Vilkovisky formalism, field theories whose action functionals are of the form\n$$\n\\frac{1}{2} \\ip{e, Q e } + S(e)\n$$\nwhere $S \\in \\mscr O_l(\\mathscr{E} )$ satisfies the classical master equation. A wide class of quantum field theories of physical and mathematical interest, including for instance pure Yang-Mills theory, appear in this way.\n\n\n\n\n\n\n\n \n\n\\begin{definition}\nLet $(M,\\mathscr{E},Q,\\ip{\\ , \\ })$ be as above.  A gauge fixing condition is an odd linear differential operator \n$$\nQ^{GF} :  \\mathscr{E} \\to \\mathscr{E}\n$$\nsuch that\n\\begin{enumerate}\n\\item\n$Q^{GF}$ is skew self adjoint with respect to the odd symplectic pairing on $\\mathscr{E}$.\n\\item\n$Q^{GF}$ is an operator of order $\\le 1$. \n\\item\n$[Q^{GF},Q^{GF}] = 0.$\n\\item\nthe operator \n$$\nH := [Q,Q^{GF}]\n$$\nis a second order elliptic operator, which is a generalised Laplacian in the sense of \\cite{BerGetVer92}. This means that the symbol \n$$\n\\sigma(H) \\in \\Gamma( M, \\Sym^2 TM \\otimes \\End E) \n$$\nis a positive definite metric on $T^\\ast M$, tensored with the identity map $E\\to E$.\n\\item\nThere is a direct sum decomposition\n$$\n\\mathscr{E} = \\Im Q \\oplus \\Im Q^{GF} \\oplus \\Ker H . \n$$\n\\end{enumerate}\n\\end{definition}\nThe fourth condition is a little restrictive. Even so, many interesting quantum field theories admit a natural collection of such gauge fixing conditions.  This condition is imposed because I need to use the asymptotic expansion of the heat kernel of $H$ proved in  \\cite{BerGetVer92}.  \n\n\\section{Examples}\n\\label{section examples}\n\\subsection{$\\phi^4$ theory}\nThe $\\phi^4$ theory is a standard simple example of a quantum field theory. The Batalin-Vilkovisky formalism does not say anything interesting about the $\\phi^4$ theory, because it is a simple bosonic field theory with no gauge symmetries.  \n\nIn our setting, the $\\phi^4$ theory works as follows.  We take $M$ to be a compact Riemannian manifold.  Let \n$$\n\\mathscr{E} = \\mathscr{E}^0 \\oplus  \\mathscr{E}^1 =  C^{\\infty}(M,\\mathbb C) \\oplus \\Pi C^{\\infty}(M,\\mathbb C)\n$$ \nwhere $\\Pi$ refers to parity reversal.   Let $e_0 \\in \\mathscr{E}^0$, $e_1 \\in \\mathscr{E}^1$ be the elements which correspond to the identity function on $M$.   The odd symplectic pairing on $\\mathscr{E}$ is defined by \n$$\n\\ip{ \\phi e_0, \\psi e_1   } = \\int_M \\phi \\psi \n$$\nwhere $\\phi, \\psi \\in C^{\\infty}(M,\\mathbb C)$ and the integration is taken using the measure associated to the Riemannian metric.\n\nThe operator $Q$ is defined by \n\\begin{align*}\nQ (\\phi e_0 ) &= ( \\Delta \\phi + m^2 \\phi ) e_1 \\\\\nQ ( \\phi e_1 ) &= 0 \n\\end{align*}\nwhere $m \\in \\mbb R_{> 0}$ is a ``mass'' parameter, and $\\Delta$ is the Laplacian on $C^{\\infty}(M)$ associated to the Riemannian metric.  \n\nThe action $S$ is defined by\n$$\nS ( \\phi e_0 + \\psi e_1 ) = \\int_M \\phi^4 .\n$$\nThen \n$$\n\\frac{1}{2} \\ip{ \\phi e_0 , Q \\phi e_0  }  + S ( \\phi ) = \\int_M \\frac{1}{2} \\phi \\Delta \\phi + \\frac{1}{2} m^2 \\phi^2 + \\phi^4 \n$$\nwhich is the usual action for the $\\phi^4$ theory.\n\nThe gauge fixing operator $Q^{GF}$ on $\\mathscr{E}$ is defined by\n\\begin{align*}\nQ^{GF} (\\phi e_1 ) &= \\phi e_0 \\\\\nQ^{GF} ( \\phi e_0 ) &= 0 .\n\\end{align*}\nThe operator $H = [Q^{GF}, Q ]$ is given by\n$$\nH ( \\phi e_i) = ( \\Delta \\phi + m^2 \\phi  ) e_i\n$$\nfor $i = 0,1$.   As required, this is a generalised Laplacian.  Also, \n$$\\mathscr{E} = \\Ker H \\oplus \\Im Q^{GF} \\oplus \\Im Q . $$  \nNote that these axioms would not be satisfied if $m = 0$, because $\\Ker H \\cap \\Im Q^{GF}$ would not be zero. \n\nThe functional integral we will construct is \n$$\n\\int_{x \\in C^{\\infty}(M)} \\exp \\left(  \\hbar^{-1}  \\int_M \\frac{1}{2} \\phi \\Delta \\phi + \\frac{1}{2} m^2 \\phi^2 + \\phi^4     \\right).\n$$\nOf course, this construction works if we use any polynomial of $\\phi$ which is at least cubic in place of $\\phi^4$.  \n\nBecause the space of fields is concentrated in degrees $0$ and $1$, any local functional which is homogeneous of degree $0$ satisfies the renormalised quantum master equation.   \n\n\\subsection{Yang-Mills theory} \nWe will use a certain first order formulation of Yang-Mills theory, well-known in the physics literature \\cite{MarZen96, Wit04}.  This was studied in the Batalin-Vilkovisky formalism in \\cite{Cos06a}.  I don't see how to fit the standard formulation of Yang-Mills theory into the framework of this paper; the problem is that I don't see how to construct gauge fixing conditions of the type we need.     I will refer to \\cite{Cos06a} for details of this construction. \n\nLet $M$ be a compact oriented $4$ manifold, and let $V$ be a vector bundle on $M$ with a connection $A$ satisfying the Yang-Mills equation\n$$\n\\d_A F(A)_+ = 0. \n$$\nIn \\cite{Cos06a} a certain differential graded algebra $\\mathscr{B}$ was constructed from  $(M,V,A)$, which looks like\n$$\n\\xymatrix{   \\Omega^{0} (\\End(V)) \\ar[dd]_{\\d_{A}} \\ar[ddrr]^{-[F(A)_{+},\\quad]} &  &  &  \\text{ degree } 0  \\\\ \n & & & \\\\\n \\Omega^{1}(\\End(V)) \\ar[dd]_{\\d_{A+}}   \\ar'[dr]|{-[F(A)_{+}, \\quad]} [ddrr] & \\oplus &  \\Omega^{2}_{+} (\\End(V)) \\ar[dd]^{- \\d_{A}} \\ar[ddll]|(.3){\\operatorname{Id}} &  \\text{ degree } 1    \\\\\n & & & \\\\\n  \\Omega^{2}_{+}(\\End(V))  \\ar[ddrr]^{-[F(A)_{+},\\quad]} & \\oplus & \\Omega^{3} (\\End(V)) \\ar[dd]^{-\\d_{A}} &  \\text{ degree } 2  \\\\\n  & & & \\\\\n   &  & \\Omega^{4} (\\End(V)) &  \\text{ degree } 3  }\n$$\nThe arrows describe the differential $Q$ in $\\mathscr{B}$.   The algebra $\\mathscr{B}$ also has a trace of degree $-3$, which comes from trace over $V$ and integration over $M$.  This trace induces an odd symmetric pairing $\\Tr ab$ on $\\mathscr{B}$. \n\nLet $\\mathscr{E} = \\mathscr{B}[-1]$ be $\\mathscr{B}$ shifted by one, with differential $Q$ inherited from $\\mathscr{B}$.  The pairing on $\\mathscr{B}$ induces a symplectic pairing on $\\mathscr{E}$.   Define a functional $S$ on $\\mathscr{E}$ by \n$$\nS(e) = \\Tr a^3.\n$$\nIt was observed in section 3.3 of \\cite{Cos06a} that $\\mathscr{E}$, with this action functional,  is the Batalin-Vilkovisky odd symplectic space associated with the first order formulation of pure Yang-Mills gauge theory, as studied in for instance \\cite{MarZen96,Wit04}.  Indeed, $\\mathscr{E}^{-1}= \\mathscr{B}^0$ is the space of ghosts, $\\mathscr{E}^0 = \\mathscr{B}^1$ is the space of fields, $\\mathscr{E}^1 = \\mathscr{B}^2$ is the space of antifields, and $\\mathscr{E}^2 = \\mathscr{B}^3$ is the space of antighosts.  \n\nThe choice of a metric on $M$ and a Hermitian metric on the vector bundle $V$ leads to a Hermitian metric on $\\mathscr{B}$.  The operator $Q^{GF}$ is the Hermitian adjoint to $Q$. This operator satisfies the axioms of a gauge fixing condition.   \n\n\\subsection{Holomorphic Chern-Simons}\n\nLet $M$ be a Calabi-Yau manifold of odd complex dimension. Fix a holomorphic  volume form $\\Omega_{Vol}$ on $M$.  Let $\\mathfrak g$ be a complex Lie algebra with a non-degenerate , symmetric, invariant pairing $\\ip{\\ , \\ }$.   Let\n$$\n\\mathscr{E} = \\Omega^{0,\\ast}(M) \\otimes_\\mathbb C \\mathfrak g. \n$$\n$\\mathscr{E}$ has an odd symplectic pairing, defined by\n$$\n\\ip{\\alpha \\otimes X, \\alpha' \\otimes X' } = (-1)^{\\abs{\\alpha} }\\int_M \\Omega_{Vol} \\wedge \\alpha \\wedge \\alpha' \\ip{X, X' } \n$$\nwhere $\\alpha \\in \\Omega^{0,\\ast}(M)$ and $X \\in \\mathfrak g$. \n\nDefine $S \\in \\mscr O_l(\\mathscr{E})$ by\n$$\nS( \\sum_i \\alpha_i \\otimes X_i ) = \\sum_{i,j,k} (-1)^{(\\abs{\\alpha_i} + 1) (\\abs{\\alpha_j} + 1 )} \\int_M \\Omega_{Vol} \\wedge \\alpha_i \\wedge \\alpha_j \\wedge \\alpha_k \\ip{ X_i, [X_j, X_k ]  }.\n$$\nThis is the holomorphic Chern-Simons action functional.\n\nPicking a metric on $M$ leads to an operator $\\br{\\partial}^\\star$ on $\\Omega^{0,\\ast}$, and so on $\\mathscr{E}$.  This is the gauge fixing operator.  The operator $H= [\\br{\\partial},\\br{\\partial}^\\star]$ is a generalised Laplacian.\n\nThe functional integral we will renormalise is\n$$\n\\int_{x \\in \\Im \\br{\\partial}^\\star} \\exp\\left({\\frac{1}{2\\hbar} \\ip{ x, \\br{\\partial} x }  + \\frac{1}{\\hbar} S(x + a)}\\right) .\n$$\n\nThis theory can be generalised in several directions.  Firstly, we can take $\\mathfrak g$ to be a non-trivial holomorphic vector bundle on $M$ with a Lie algebra structure (i.e.\\ a sheaf of Lie algebras over the sheaf of holomorphic functions on $M$).  In that case, to get a gauge fixing condition would require the choice of both a metric on $M$ and a Hermitian metric on the vector bundle.  \n\nIf $M$ is of even complex dimension, one can take $\\mathfrak g$ to be a super-Lie algebra with an odd  invariant pairing (see \\cite{AleKonSch97}).  Of course, there is a version of this where we take a sheaf of such super-Lie algebras.\n\nYet another direction is to use $L_\\infty$ rather than Lie algebras. In that case the action $S$ is no longer purely cubic, but has higher-order contributions from the higher $L_\\infty$ operations.  Such theories are also discussed in \\cite{AleKonSch97}.\n\n\\subsection{Higher dimensional Chern-Simons}\nWe have already discussed Chern-Simons theory in dimension three.  One can readily generalise this theory to manifolds of arbitrary dimension, just as with holomorphic Chern-Simons theory.  \n\nFor odd-dimensional manifolds, one  takes a complex Lie algebra $\\mathfrak g$ with an invariant pairing, and sets $\\mathscr{E} = \\Omega^\\ast(M)\\otimes \\mathfrak g [1]$. The Chern-Simons action is given by essentially the same formula as in the $3$-dimensional case:   \n$$\nS( \\sum_i \\alpha_i \\otimes X_i ) = \\sum_{i,j,k} (-1)^{(-1)^{(\\abs{\\alpha_i} + 1) (\\abs{\\alpha_j} + 1 )}} \\int_M\\alpha_i \\wedge \\alpha_j \\wedge \\alpha_k \\ip{ X_i, [X_j, X_k ]  }\n$$\nwhere $\\alpha_i \\in \\Omega^\\ast(M)$ and $X_i \\in \\mathfrak g$.  The operator $Q$ is defined to be the de Rham differential $\\d$.  If we pick a metric on $M$ then we can define $Q^{GF}$ to be the adjoint operator $\\d^\\star$. \n\nAs with holomorphic Chern-Simons theory, this can be generalised in various directions.  Firstly, we  could replace $\\mathfrak g$ by a locally trivial sheaf of Lie algebras.  Or, we could use a sheaf of $L_\\infty$ algebras instead Lie algebras.\n\nIn the case when $M$ is even-dimensional, we can take $\\mathfrak g$ to be an $L_\\infty$ algebra with an odd invariant pairing, or a locally trivial sheaf of such.\n\nA particularly interesting case occurs when $\\dim M = 2$.  Two dimensional Chern-Simons theory (introduced in \\cite{AleKonSch97}) is essentially the same as the Poisson sigma model.   This theory is used in the proof of Kontsevich's formality theorem \\cite{Kon03a,CatFel01}.  \n\nIf $\\mathfrak g$ is an $L_\\infty$ algebra with an odd invariant pairing, then we can identify $\\mathfrak g^{even} = (\\mathfrak g^{odd})^{\\vee} [1]$. Thus, if we let $V = \\mathfrak g^{odd} [1]$, so that $V$ is a purely even vector space, then \n$$\n\\mathfrak g [1]  = \\Pi T^\\ast V . \n$$\nThe $L_\\infty$ structure on $\\mathfrak g$ is described by a function $f$ on $\\mathfrak g[1] = \\Pi T^\\ast V$, satisfying the classical master equation $\\{f,f\\} = 0$. \n\nWe can identify functions on $\\Pi T^\\ast V$ with polyvector fields on $V$, that is, \n$$\n\\mscr O(\\Pi T^\\ast V) = \\oplus_i \\wedge^i T V\\otimes \\mscr O(V). \n$$\nThe Poisson bracket on $\\mscr O(\\Pi T^\\ast V)$ corresponds to the Schouten bracket on polyvector fields. \n\nThus, the $L_\\infty$ structure on $\\mathfrak g$ defines a generalised Poisson structure on $V$.  If this $L_\\infty$ structure is purely quadratic in the $\\mathfrak g^{ev}[1]$ variables, then we find a Poisson structure in the classical sense on $V$.\n\nThe space\n$$\n\\mathscr{E} = \\Omega^\\ast( M ) \\otimes \\mathfrak g [ 1]\n$$\ncan be identified with the  supermanifold of maps $\\Pi T M \\to \\Pi T^\\ast V$, leading to the sigma model interpretation of the theory. \n\n\n\\section{Differential operators}\n\\label{section differential operators}\n\n\n\\subsection{Functor of points approach to formal super geometry}\n\n\nIn order to clarify sign issues, we will use the  ``functor of points'' approach to formal super geometry. A formal super-space is viewed as a functor  from the category of nilpotent local commutative super-algebras to the category of sets.    For example, the formal super-space $\\mathbb{A}^{m,n}$ sends $R \\mapsto \\left( R^{ev} \\right)^{\\oplus m}  \\oplus  \\left( R^{odd} \\right)^{\\oplus n}$.   \n\nIf \n$$\nV = V^{ev} \\oplus V^{odd}\n$$\nis a $\\mathbb Z\/2$-graded vector space,  the formal super-space $V$ sends $R \\mapsto (V \\otimes R)^{ev}$.    \n\nAn even formal function on $V$ is then a natural transformation of functors $V \\to \\mathbb{A}^{1,0}$,  and an odd formal function on $V$ is a natural transformation to $V \\to \\mathbb{A}^{0,1}$.   More explicitly, an even  formal function $f$ on $V$ is a collection of maps \n$$f_R  : (V \\otimes R)^{ev} \\to R^{ev},$$\nnatural with respect to morphisms $R \\to R'$ of nilpotent local commutative super-algebras.   \n\nLet us denote by $\\mscr O'(V)$ the $\\mathbb Z\/2$-graded algebra of all formal functions on $V$. We can identify this algebra with the inverse limit \n$$\\mscr O'(V) =  \\varprojlim  \\Sym^\\ast V^\\vee \/ \\Sym^{\\ge n} V^\\vee. $$\n\nIn particular, if $\\mathscr{E}$ as above is the global sections of some super vector bundle $E$ on $M$, we can think of $\\mathscr{E}$ as an infinite dimensional formal super-space.  The algebra $\\mscr O(\\mathscr{E})$ is a subalgebra of the algebra $\\mscr O'(\\mathscr{E})$.  An element of $\\mscr O'(\\mathscr{E})$ is in $\\mscr O(\\mathscr{E})$ whenever its Taylor expansion is in terms of continuous linear maps on $\\mathscr{E}^{\\otimes n}$.  \n\nIn the case when $M$ is a point, so that $\\mathscr{E}$ is simply a finite dimensional super vector space, then $\\mscr O(\\mathscr{E})$ and $\\mscr O'(\\mathscr{E})$ coincide.  \n\n\\subsection{Derivations}\nThe functor of points way of looking at things is useful (for instance) when thinking about derivations of the algebra $\\mscr O(\\mathscr{E})$.  Everything we do  could also be written more explicitly, but the advantage of the functor of points formalism is that it allows one to essentially forget about signs.  \n\nLet $\\epsilon$ be an odd parameter.  One can define the derivation $Q$ of $\\mscr O(\\mathscr{E})$ by the formula\n$$\nf ( x + \\epsilon Q x  ) = f( x ) + \\epsilon (Q f ) (x) . \n$$\nThe left hand side is defined using the language of the functor of points, as follows.  For each  nilpotent local commutative superalgebra $R$ as above, $f$ gives a map \n$$f_{R \\otimes \\mathbb C[\\epsilon]} :  (\\mathscr{E} \\otimes R \\otimes \\mathbb C[\\epsilon])^{ev} \\to\n\\begin{cases}\n (R \\otimes \\mathbb C[\\epsilon] )^{ev} \\text{ if } f \\text{ is even} \\\\ \n  (R \\otimes \\mathbb C[\\epsilon] )^{odd} \\text{ if } f \\text{ is odd} . \n\\end{cases}\n$$\nIf $x \\in (\\mathscr{E} \\otimes R)^{ev}$, then $x + \\varepsilon Q x$ is in $( \\mathscr{E} \\otimes R \\otimes \\mathbb C[\\varepsilon] )^{ev}$.  Thus,  $f ( x + \\varepsilon Q x ) \\in (R \\otimes \\mathbb C[\\varepsilon])$.  The coefficient of $\\epsilon$ in $f( x + \\epsilon Q x ) $ gives a map $(\\mathscr{E} \\otimes R)^{ev}$ to $R^{ev}$ (if $f$ is odd), or to $R^{odd}$ (if $f$ is even), and so a formal functional on $\\mathscr{E}$ of parity opposite to that of $f$.    This is defined to be $Q f$.   \n\n\n\nA priori, this construction only yields a derivation of $\\mscr O'(\\mathscr{E})$, but one can check easily that this derivation preserves the subalgebra $\\mscr O(\\mathscr{E})$.   In more explicit terms, if we identify\n$$\n\\mscr O(\\mathscr{E}) = \\prod \\Hom ( \\mathscr{E}^{\\otimes n}, \\mathbb C ) _{S_n} ,\n$$\nthen the derivation $Q$ preserves each factor $\\Hom(\\mathscr{E}^{\\otimes n} , \\mathbb C)_{S_n}$.  On each such factor, $Q$ is simply the usual tensor product differential on each $\\mathscr{E}^{\\otimes n}$, which we then transfer to the dual space and to the space of $S_n$ coinvariants.    \n\n\n To compute the commutator of derivations in terms of the functor of points is very simple.  Let  $D_1,D_2$ be derivations of $\\mscr O'(V)$, where $D_i$ is given by an automorphism of $\\mscr O'(V) \\otimes \\mathbb C[\\epsilon_i] \/ (\\epsilon_i^2)$, which modulo $\\epsilon_i$ is the identity. The parameter $\\epsilon_i$ has the same parity as $D_i$.   We can compute the commutator by \n$$\n(1 + \\epsilon_1 D_1 ) ( 1 + \\epsilon_2 D_2 ) ( 1 - \\epsilon_1 D_1 )(1 - \\epsilon_2 D_2 )  = 1 +  \\epsilon_1 \\epsilon_2 [D_1,D_2] .\n$$\n\nLet $e \\in \\mathscr{E}$.  Differentiation with respect to $e$ is an operator $\\partial_e  : \\mscr O'(\\mathscr{E}) \\to \\mscr O'(\\mathscr{E})$ defined by the formula \n$$\nf ( x + \\epsilon e )  = f ( x ) + \\epsilon \\partial_e f \n$$\nwhere $\\epsilon$ is an auxiliary parameter of the same parity as $e$, and of square $0$.  \n\nIt is easy to see that this derivation preserves $\\mscr O(\\mathscr{E})$.  Indeed, if we identify $\\mscr O(\\mathscr{E}) = \\prod \\Hom ( \\mathscr{E}^{\\otimes n}, \\mathbb C ) _{S_n}$, then the derivation $\\partial_e$ is (up to sign) the map \n$$ \\Hom(\\mathscr{E}^{\\otimes n}, \\mathbb C)_{S_n} \\to \\Hom(\\mathscr{E}^{\\otimes n-1}, \\mathbb C)_{S_{n-1}}$$ \ngiven by contraction with $e$. \n\n\nOne can compute easily that \n$$\n[ \\partial_e , Q] = \\partial_{Q e} .\n$$\n\n\\subsection{Convolution operators}\n\nIf $K \\in \\mathscr{E} \\otimes \\mathscr{E}$, define a convolution map $K \\star : \\mathscr{E} \\to \\mathscr{E}$ by the formula\n$$\nK \\star e = (-1)^{\\abs e} \\sum K' \\otimes \\ip{K'', e}\n$$\nwhere $K = \\sum K' \\otimes K''$.    The reason for the choice of sign in the definition of the convolution is that \n$$\n( Q  K ) \\star e  =   [Q, K\\star ]  e.\n$$\nThe first $Q$ in this formula denotes the tensor product differential on $\\mathscr{E} \\otimes \\mathscr{E}$.\n\n\n\\begin{lemma}\nAn element $K \\in \\mathscr{E} \\otimes \\mathscr{E}$ is symmetric if and only if the map $K \\star $ is self adjoint, and is antisymmetric if and only if $K \\star $ is skew self adjoint. \n\\end{lemma}\n\n\n\\subsection{Order two differential operators}\nLet\n$$\n\\phi = \\sum \\phi' \\otimes \\phi'' \\in \\Sym^2 \\mathscr{E}.\n$$\nAssociated to $\\phi$ we have an order two differential operator $\\partial_{\\phi}$ on $\\mscr O(\\mathscr{E})$, given by the formula\n$$\n\\partial_{\\phi} =  \\frac{1}{2} \\sum \\partial_{\\phi''} \\partial_{\\phi'} =  \\frac{1}{2} \\sum  (-1)^{\\abs{\\phi'} \\abs{\\phi'' } }\\partial_{\\phi'} \\partial_{\\phi''}.\n$$\nEven though the sum may be infinite, this expression is well defined; up to sign, $\\partial_\\phi$ comes from the map \n$$ \\Hom(\\mathscr{E}^{\\otimes n}, \\mathbb C)_{S_n} \\to \\Hom(\\mathscr{E}^{\\otimes n-2}, \\mathbb C)_{S_{n-2}}$$ \ngiven by contraction with $\\phi$.  \n\nThe differential operators $\\partial_\\phi$ mutually super-commute, and \n$$\n[\\partial_\\phi, Q] = \\partial_{Q \\phi} . \n$$\n\nIf $e \\in \\mathscr{E}$, let $e^\\vee : \\mathscr{E} \\to \\mathbb C$ be the continuous linear functional given by $e^\\vee( e' ) = \\ip{e',e}$.   Note that $e^\\vee \\in \\mscr O(\\mathscr{E})$, so there is an associated left multiplication map $\\mscr O(\\mathscr{E}) \\to \\mscr O(\\mathscr{E})$.\nThe following lemma will be useful later.\n\\begin{lemma}\nIf $\\phi \\in \\Sym^2 \\mathscr{E}$ is even, then\n$$\n[\\partial_\\phi, e^\\vee ] =   -\\partial_{\\phi \\star e } .\n$$\n\\end{lemma}\n\\begin{proof}\nIn the language we are using for formal functions, if $R$ is an auxiliary nilpotent ring, and $r \\in R$, $\\eta \\in \\mathscr{E}$ are elements of the same parity, then by definition\n$$\ne^\\vee (r \\eta ) = \\ip{r \\eta , e } = r \\ip{\\eta , e} .\n$$\nTherefore, in particular, \n$$\n\\partial_{e'} e^\\vee = \\ip{ e', e } \n$$\nfor all $e' \\in \\mathscr{E}$.    \n\nWe can assume\n$$\n\\phi = \\phi' \\otimes \\phi'' + (-1)^{\\abs{\\phi'} \\abs{\\phi''}  } \\phi'' \\otimes \\phi'.\n$$\nThen\n$$\n\\partial_\\phi = \\partial_{\\phi''} \\partial_{\\phi'}.\n$$\nThen\n$$\n[\\partial_\\phi, e^\\vee ]  = \\ip{\\phi',e}   \\partial_{\\phi''}  + (-1)^{\\abs{\\phi'} (\\abs{e}+1) } \\ip{\\phi'',e}   \\partial_{\\phi''} \n$$\nwhereas\n$$\n\\phi \\star e = (-1)^{\\abs{e} } \\ip{\\phi'', e}  \\phi' + (-1)^{\\abs{e} + \\abs{\\phi''} \\abs{\\phi' }} \\ip{\\phi'', e}  \\phi'.\n$$\nThe lemma follows from these two equations. \n\\end{proof} \n\n\n\n\n\\subsection{Heat kernels}\nA heat kernel for the operator $H : \\mathscr{E} \\to \\mathscr{E}$ is an element \n$$K_t \\in\\mathscr{E}^{\\otimes 2} = \\Gamma(M^2,E \\boxtimes E),$$ defined for $t \\in \\mbb R_{> 0}$, such that \n$$\nK_t \\star e = e^{- t H} e \n$$\nfor all $e \\in \\mathscr{E}$.  Because $H$ is a generalised Laplacian, the results of \\cite{BerGetVer92} imply that it admits a unique heat kernel $K_t$, which is also smooth as a function of $t$.\n\nLet \n$$L_t = (Q^{GF} \\otimes 1) K_t$$ \nso that $L_t$ is the kernel representing the operator $Q^{GF} e^{-t H}$. This is a smoothing operator for $0< t \\le \\infty$, so that $L_t \\in \\mathscr{E} \\otimes \\mathscr{E}$ for such $t$.  \n\n\nObserve that \n\\begin{align*}\n(Q \\otimes 1 + 1 \\otimes Q) K_t &= 0 \\\\\n\\frac { \\d } { \\d t } K_t  + ( Q \\otimes 1 + 1 \\otimes Q) L_t &= 0 .\n\\end{align*}\nThese formulae together say that the expression\n$$\nK_t + \\d t L_t \n$$\nis a closed element of $\\mathscr{E} \\otimes \\mathscr{E} \\otimes \\Omega^\\ast(\\mbb R_{> 0})$, where we give this space the tensor product differential.  \n\nLet\n$$\nP(\\varepsilon,T)= \\int_{\\epsilon}^T L_t \\d t = \\int_{\\varepsilon}^{T} (Q^{GF} \\otimes 1)K_t \\d t .\n$$\nNote that for $0 < \\epsilon < T \\le \\infty$, $P(\\epsilon,T)$ is in $\\mathscr{E} \\otimes \\mathscr{E}$.    This is clear if $0 < \\epsilon < T < \\infty$.  If $T = \\infty$, one needs to check that \n$$\n\\alpha \\mapsto \\int_\\epsilon^\\infty Q^{GF} e^{-t H} \\alpha \\d t\n$$\nis a smoothing operator for any $t > 0$. The only problems that could occur would be on the $0$ eigenspace of $H$, but $Q^{GF}$ annihilates this eigenspace.  This is one of the axioms of a gauge fixing condition. \n\nThe reason the kernels $P(\\epsilon,T)$ are important comes from the following lemma.\n\\begin{lemma}\nThe operator\n\\begin{align*}\n\\Im Q &\\to \\Im Q^{GF} \\\\\n\\alpha &\\mapsto \\int_0^\\infty Q^{GF} e^{-t H} \\alpha \\d t\n\\end{align*}\nis the inverse to the isomorphism $Q : \\Im Q^{GF} \\to \\Im Q$.\n\\end{lemma}\nThus, the singular kernel $P(0,\\infty)$ represents $Q^{-1}$.\n\\begin{proof}\nIndeed,\n$$\n\\int_0^\\infty Q^{GF} e^{-t H} Q \\alpha \\d t= - Q \\int_0^\\infty Q^{GF} e^{-t H} \\alpha \\d t + \\int_0^\\infty H e^{-t H} \\alpha \\d t . \n$$\nSince $(Q^{GF})^2 = 0$, and $\\alpha \\in \\Im Q^{GF}$, the first term on the right hand side is zero.  Thus, \n$$\n\\int_0^\\infty Q^{GF} e^{-t H} Q \\alpha \\d t =  - \\int_0^\\infty \\frac{\\d } {\\d t }  e^{-t H} \\alpha \\d t  = \\alpha - \\pi \\alpha\n$$ \nwhere $\\pi : \\mathscr{E} \\to \\Ker H$ is the projection onto the zero eigenspace of $H$.  \n\nSince $\\Ker H \\cap \\Im Q^{GF} = 0$, and $\\Im Q^{GF}$ is a direct sum of $H$ eigenspaces, we see that $\\pi \\alpha = 0$, so that\n$$\n\\int_0^\\infty Q^{GF} e^{-t H} Q \\alpha \\d t = \\alpha \n$$\nas desired.  \n\\end{proof} \n\n\\subsection{Functional integrals in terms of differential operators}\nWhen $M$ is a point, we will be able to express certain integrals over the finite dimensional vector space $\\mathscr{E}$ in terms of the differential operators on $\\mscr O(\\mathscr{E})$ introduced above.  When $\\dim M > 0$, so that $\\mathscr{E}$ is infinite dimensional, we will attempt to take this as a definition of the functional integral.\n\n\\begin{definition}\nLet  $\\Phi \\in \\Sym^2 \\mathscr{E}$ be any even element, and  $S \\in \\mscr O(\\mathscr{E}, \\mathbb C [[\\hbar]])$ be an even element, which modulo $\\hbar$ is at least cubic.  Define\n$$\n\\Gamma( \\Phi, S ) = \\hbar \\log\\left( \\exp  (\\hbar \\partial_\\Phi ) \\exp (  S \/ \\hbar )   \\right) \\in \\mscr O(\\mathscr{E}, \\mathbb C [[\\hbar ]] ) . \n$$\n\\end{definition}\nIt is easy to check that this expression is well defined. However, if $S$ was not at least cubic, but contained a quadratic term, this expression would contain some non-convergent infinite sums. \n\nLet us assume for a moment that $M$ is a point.\nLet $\\left(\\Im Q^{GF}\\right)_\\mbb R$ be a real slice of $\\Im Q^{GF} \\subset \\mathscr{E}$, with the property that the quadratic form $\\ip{x,Qx}$ is negative definite on $\\left(\\Im Q^{GF}\\right)_\\mbb R$.  Then the integral\n$$\n\\int_{x \\in \\left(\\Im Q^{GF}\\right)_\\mbb R}  \\exp\\left( \\tfrac{1}{2} \\ip{x,Q x} \/ \\hbar + S(x + a) \/ \\hbar \\right) \\d \\mu\n$$\nis well defined as a formal series in the variables $\\hbar$ and  $a \\in \\mathscr{E}$.  We use an $\\hbar$-dependent Lebesgue measure $\\d \\mu$ normalised so that \n$$\n\\int_{x \\in \\left(\\Im Q^{GF}\\right)_\\mbb R}  \\exp\\left( \\tfrac{1}{2} \\ip{x,Q x}   \/ \\hbar \\right) \\d \\mu = 1 . \n$$\n\\begin{lemma} If $M$ is a point,\n\\label{lemma integral differential}\n$$  \\Gamma(P(0,\\infty), S) (a)  =  \\hbar \\log \\int_{x \\in \\left(\\Im Q^{GF}\\right)_\\mbb R}  \\exp\\left( \\tfrac{1}{2} \\ip{x,Q x} \/ \\hbar + S(x + a) \/ \\hbar \\right) \\d \\mu .  $$\n\\label{lemma integral differential general}\n\\end{lemma}\n\\begin{proof}\nIt suffices to show that for any polynomial  $f \\in \\mscr O(\\mathscr{E})$\n$$\n \\exp ( \\hbar \\partial_{P(0,\\infty)} ) f =  \\int_{\\left(\\Im Q^{GF}\\right)_\\mbb R}  \\exp \\left( \\tfrac{1}{2} \\ip{x,Q x} \/\\hbar \\right)  f (x+a).\n$$\nBoth sides are functions of $a \\in \\mathscr{E}$.  \n\nLet $v \\in \\left(\\Im Q^{GF}\\right)_\\mbb R \\subset \\mathscr{E}$. Define a linear function $l_v = (Q v)^\\vee$ on $\\left(\\Im Q^{GF}\\right)_\\mbb R$, so that\n$$\nl_v(x)  = \\ip{x,Q v}.\n$$\nThen\n$$\n\\partial_v \\ip{x, Q x} = 2l_v (x). \n$$\nWe can integrate by parts, to find\n\\begin{multline*}\n\\int_{\\left(\\Im Q^{GF}\\right)_\\mbb R}  \\exp \\left( \\tfrac{1}{2} \\ip{x,Q x} \/\\hbar \\right) l_v (x + a) f(x+a) \\\\  =   \\hbar \\int_{\\left(\\Im Q^{GF}\\right)_\\mbb R} \\left(\\partial_v  \\exp \\left( \\tfrac{1}{2} \\ip{x,Q x} \/\\hbar \\right)  \\right) f (x+a) \n\\\\ \\shoveright{ +  l_v(a) \\int_{\\left(\\Im Q^{GF}\\right)_\\mbb R}  \\exp \\left( \\tfrac{1}{2} \\ip{x,Q x} \/\\hbar \\right)   f (x+a ))  } \\\\\n= - \\hbar \\int_{\\left(\\Im Q^{GF}\\right)_\\mbb R}  \\exp \\left( \\tfrac{1}{2} \\ip{x,Q x} \/\\hbar \\right) \\partial_v f (x+a)\\\\\n +   l_v(a) \\int_{\\left(\\Im Q^{GF}\\right)_\\mbb R}  \\exp \\left( \\tfrac{1}{2} \\ip{x,Q x} \/\\hbar \\right)   f (x+a ))  \n\\end{multline*}\nA similar identity holds for $\\exp(\\hbar \\partial_{P(0,\\infty)}) f$,  namely\n$$\n \\exp( \\hbar \\partial_{P(0,\\infty)}) l_v  f = - \\hbar   \\exp (\\hbar \\partial_{P(0,\\infty)}) \\partial_v f + l_v(a)  \\exp( \\hbar \\partial_{P(0,\\infty)})  f .\n$$\nThis follows from the equation\n$$\n[\\partial_{P(0,\\infty)}, l_v ] = [\\partial_{P(0,\\infty)}, (Q v)^\\vee ] =   - \\partial_{ P(0,\\infty) \\star (Q v)  }  = - \\partial_{ v } .\n$$\nThe last equation holds because $P(0,\\infty) \\star$ is the operator $Q^{-1} : \\Im Q \\to \\Im Q^{GF}$.  \n\nThese identities allow us to use induction to reduce to the case when $f$ is constant.  The normalisation in the measure on  $\\left(\\Im Q^{GF}\\right)_\\mbb R$ takes care of this initial case.  \n\n\n\n\\end{proof} \n\n\\section{Regularisation}\n\n\n\n\\subsection{Regularisation}\n\nOne can write the formal equality\n$$\n \\hbar \\log \\int_{e \\in (\\Im Q^{GF})_\\mbb R} \\exp(  \\tfrac{1}{2} \\ip{e,Qe}  \/ \\hbar +   S ( e + a) \/ \\hbar  ) =  \\lim_{ \\epsilon\\to 0}  \\Gamma  (P(\\epsilon,\\infty), S)\n$$\nwhere as above,\n$$\n \\Gamma  (P(\\epsilon,\\infty), S) = \\hbar \\log \\left( \\exp (\\hbar \\partial_{P(\\epsilon,\\infty)}) \n  \\exp ( S \/ \\hbar) \\right)  . \n$$\nWhen $\\dim M = 0$, this equality was proved in Lemma \\ref{lemma integral differential}.  When $\\dim M > 0$, we will take this equality as an attempt to define the functional integral over $(\\Im Q^{GF})_\\mbb R$.  \n\nWhen $\\dim M > 0$,  although the expression $\\Gamma(P(\\epsilon,\\infty), S)$ is well defined for all $\\epsilon > 0$, the limit  $\\lim_{\\epsilon \\to 0} \\Gamma(P(\\epsilon,\\infty), S)$ is divergent.   This is because $P(0,\\infty)$ is a distributional section of the vector bundle $E \\boxtimes E$ on $M^2$,  with singularities on the diagonal.  The expression $\\exp (\\hbar \\partial_{P(0,\\infty)} )  \\exp ( S \/ \\hbar)$ involves multiplication of the distribution $P(0,\\infty)$ with distributions with support along the diagonal.    In other words, it involves integrals over products of $M$ where the integrand has singularities on the diagonal.      This is the problem of ultraviolet singularities.\n\nAs I explained in the introduction, we will renormalise the limit by subtracting certain counter-terms from the action $S$.  In order to do this, we need some control over the small $\\epsilon$ asymptotics of $\\Gamma(P(\\varepsilon,\\infty), S)$. \n\nLet us write\n$$\n\\Gamma(P(\\epsilon,T),S) = \\sum_{i\\ge 0,k \\ge 0} \\hbar^i \\Gamma_{i,k} (P(\\epsilon,T),S)\n$$\nwhere $\\Gamma_{i,k}(P(\\varepsilon,T),S)$ is homogeneous of order $k$ as a formal functional of $e \\in \\mathscr{E}$.   This expression is the Taylor expansion of $\\Gamma$ in the variables $\\hbar$ and $e \\in \\mathscr{E}$. \n\nWe will construct a small $\\epsilon$ asymptotic expansion for each $\\Gamma_{i,k}(P(\\epsilon,T), S)$.   This expansion will take values in a certain subalgebra $\\mathscr{A}$ of the algebra of analytic functions on $\\epsilon \\in (0,\\infty)$.   \n\\begin{definition}\nLet $\\mathscr{A} \\subset C^{\\infty}( (0,\\infty) )$ be the subalgebra spanned over $\\mathbb C$ by  functions of $\\epsilon$ of the form\n$$\nf(\\epsilon )  = \\int_{U \\subset (\\epsilon,1)^n } \\frac{  F(t_1,\\ldots,t_n)^{1\/2} } { G(t_1,\\cdots, t_n)^{1\/2} } \\d t_1 \\ldots \\d t_n \n$$\nand functions of the form\n$$\nf(\\epsilon )  = \\int_{U \\subset (\\epsilon,1)^{n-1} } \\frac{  F(t_1,\\ldots,t_n = \\varepsilon)^{1\/2} } { G(t_1,\\ldots, t_n = \\varepsilon)^{1\/2} } \\d t_1 \\cdots \\d t_{n-1}\n$$\nwhere \n\\begin{enumerate}\n\\item\n$F, G \\in \\mathbb Z_{\\ge 0} [t_1,\\ldots, t_n] \\setminus \\{0\\}$; $n$ can take on any value. \n\\item\nthe region of integration $U$ is an open subset cut out by finitely many equations of the form $t_i^l > t_j$, for $l \\in \\mathbb Z$.  \n\\end{enumerate}\n\\end{definition}\n``Spanned'' means in the non-topological sense; we only allow finite sums.  Thus, $\\mathscr{A}$ has a countable basis, and every element is written as a finite sum of basis elements. We give $\\mathscr{A}$ the trivial topology, where all subspaces are closed. \n\nThe details in the definition of $\\mathscr{A}$ aren't all that important; we could always use a larger algebra containing $\\mathscr{A}$. \n\n\\begin{thmA}\n\\begin{enumerate}\n\\item\nFor each $i$ and $k$, there exist functions $f_r \\in \\mathscr{A}$, $r \\in \\mathbb Z_{\\ge 0}$, such that there is a small $\\varepsilon$ asymptotic expansion of the form \n$$\n\\Gamma_{i,k}(P(\\epsilon,T), S(\\epsilon, \\hbar) ) (e) \\simeq  \\sum_{r \\ge 0} f_r (\\epsilon) \\Phi_r (T,e) \n$$\nwhere $e \\in \\mathscr{E}$, and each $\\Phi_r$ is in $\\mscr O(\\mathscr{E}, C^{\\infty}((0,\\infty)))$. \n\\item\nThe $\\Phi_r(T,e)$ have a small $T$ asymptotic expansion\n$$\n\\Phi_r(T,e) = \\sum g_q(T) \\Psi_{q,r} (e)\n$$ where the $\\Psi_{q,r} \\in \\mscr O_l(\\mathscr{E})$ are local functionals of $e$, and $g_q(T)$ are certain smooth functions of $T$. \n\\item\nIf $k > 0$, so $\\Psi_{q,r}$ is a non-constant function on $\\mathscr{E}$,  then we can speak of the germ of each $\\Psi_{q,r}$ near a point $x \\in M$.  This germ only depends on the germ of the data $(E, \\ip{\\ , \\ }, Q, Q^{GF}, S )$ near $x$.   \n\\end{enumerate}\n\\label{theorem_analytic_continuation} \n\\end{thmA}\n\n\\begin{remark}\nThe last point is a little delicate.  For instance, for any $t > 0$, the germ of the heat kernel $K_t$ near a point $(x,x)$ in $M^2$ depends on the \\emph{global} behaviour of the elliptic operator $H$.  Only the $t \\to 0$ asymptotics of the germ of $K_t$ depends on the germ of $H$.\n\\end{remark}\nThe appendix contains a proof of a more general version of this theorem, as well \nas a precise statement of what I mean by small $\\varepsilon$ asymptotic expansion. \n\n\n\\section{Renormalisation}\nThe next step in constructing the quantum field theory is \\emph{renormalisation}.  This amounts to replacing our action functional $S \\in \\mscr O_l(\\mathscr{E}, \\mathbb C[[\\hbar]])$ by a series\n$$\nS^R(\\hbar,\\epsilon) = S(\\hbar) - S^{CT}(\\hbar,\\epsilon) = S - \\sum_{i > 0, k \\ge 0} \\hbar^i S^{CT}_{i,k}(\\epsilon).\n$$\nThe $S_{i,k}^{CT}$ are known as counter-terms; they are $\\varepsilon$-dependent local functionals on $\\mathscr{E}$, homogeneous of degree $k$.   This renormalised action functional $S^R$ is required to be such that the $\\epsilon \\to 0$ limit of $\\Gamma( P(\\varepsilon,T), S^R (\\varepsilon) )$  exists.  Recall that \n$$\nP(\\varepsilon,T)= \\int_{\\varepsilon}^T  (Q^{GF} \\otimes 1) K_t \\d t\n$$\nso that $\\lim_{\\varepsilon \\to 0}\\Gamma(P(\\varepsilon,T), S^R(\\varepsilon) )$ will be our renormalised effective action.\n\nIn order to perform this renormalisation, we need to pick a renormalisation scheme. Let $\\mathscr{A}_{\\ge 0} \\subset \\mathscr{A}$ be the subspace of those functions $f$ such that $\\lim_{\\varepsilon \\to 0} f$ exists. \n\\begin{definition}\nA \\emph{renormalisation scheme} is a subspace $\\mathscr{A}_{< 0}$ to $\\mathscr{A}_{\\ge 0}$ such that \n$$\n\\mathscr{A} = \\mathscr{A}_{< 0} \\oplus \\mathscr{A}_{\\ge 0} . \n$$\n\\end{definition}\nNote that a renormalisation scheme exists, simply because $\\mathscr{A}$ is a vector space with a countable basis.\n\nThe choice of a renormalisation scheme allows us to extract the singular part of functions $f \\in \\mathscr{A}$.  The singular part of $f$ is simply the projection of $f$ onto $\\mathscr{A}_{< 0}$.  \n\n\n\n\n\n\\begin{thmB}\nLet $S \\in \\mscr O_l(\\mathscr{E}, \\mathbb C[[\\hbar]])$ be an even functional which, modulo $\\hbar$, is at least cubic.   Then there exists a unique series\n$$S^{CT}(\\varepsilon,\\hbar) =   \\sum_{i \\ge 1, k \\ge 0} \\hbar^i S_{i,k}^{CT}(s)  $$\nsuch that \n\\begin{enumerate}\n\\item\neach $S_{i,k}^{CT}(\\varepsilon)$ is an element of $\\mscr O_l(\\mathscr{E}, \\mathscr{A}_{< 0})$\nwhich is homogeneous of degree $k$ as a function on $\\mathscr{E}$; \n\\item\nthe limit $\\lim_{\\epsilon \\to 0} \\Gamma(P(\\varepsilon,T),S - S^{CT})$ exists.  \n\\end{enumerate}\n\nIf $k > 0$, the germ of the counter-term $S_{i,k}^{CT}$ at $x \\in M$ depends only on the germ of the data  $(E, \\ip{\\ , \\ }, Q, Q^{GF},  S )$ near $x$.  \n\\label{theorem renormalisation}\n\\end{thmB}\nWe use the notation \n$$\\Gamma^R(P(0,T), S) =  \\lim_{\\varepsilon \\to 0} \\Gamma(P(\\varepsilon,T),S - S^{CT}). $$ $\\Gamma^R(P(0,\\infty), S) $ is a renormalised functional integral.\n\n\\begin{proof}\n\nBefore I begin the proof, let us introduce some notation.   Let us give the set $\\mathbb Z_{\\ge 0} \\times \\mathbb Z_{\\ge 0}$ the lexicographic ordering, so that $(i,k) < (i',k')$ if $i < i'$, or if $i = i'$ and $k < k'$. \n\nLet $A \\in \\mscr O(\\mathscr{E}, \\mathbb C[[\\hbar]])$ be any (not necessarily local) functional.    As before, we will write \n$$\nA = \\sum \\hbar^i A_{i,k} \n$$\nwhere $A_{i,k} \\in \\mscr O(\\mathscr{E})$ is a homogeneous functional of degree $k$.  Let us write \n$$\nA_{\\le (I,K)} = \\sum_{(i,k) \\le (I,K) } \\hbar^i A_{i , k}.\n$$\n\nThe proof of this theorem is actually very simple; no graph combinatorics are required.  All we need to do is to construct the $S_{i,k}^{CT}$ inductively using the lexicographic ordering on $\\mathbb Z_{\\ge 0} \\times \\mathbb Z_{\\ge 0}$. \n\n\nSo, let us suppose, by induction, that we have constructed \n$$S_{i,k}^{CT}(\\varepsilon, T) \\in \\mscr O_l(\\mathscr{E}, \\mathscr{A}_{< 0} \\otimes C^{\\infty}(\\mbb R_{> 0}))$$ \nfor all $(i,k) < (I,K)$,  which are homogeneous of degree $k$ as a function of $\\alpha \\in \\mathscr{E}$.   Here, $\\mbb R_{> 0}$ has coordinate $T$.     \n\nLet \n$$\nS^{CT}_{<(I,K)}  = \\sum_{(i,k) < (I,K)} \\hbar^i S^{CT}_{i,k}\n$$\nWe can write\n$$\n\\Gamma ( P(\\varepsilon,T),  S - S^{CT}_{<(I,K)} )= \\sum_{i,k \\ge 0} \\hbar^i  \\Gamma_{i,k}( P(\\varepsilon,T), S - S^{CT}_{<(I,K)} )\n$$\nLet us make the following further induction assumptions : \n\\begin{enumerate}\n\\item\nthe limit  $\\lim_{\\varepsilon \\to 0} \\Gamma_{i,k}( P(\\varepsilon,T), S - S^{CT}_{<(I,K)} )$ exists, for all $(i,k) < (I,K)$. \n\\item\neach $S^{CT}_{i,k}$ is independent of $T$ if $(i,k) <(I,K)$. \n\\end{enumerate}\n\nNow simply let\n$$\nS^{CT}_{I,K} = \\text{Singular part of } \\Gamma_{I,K}( P(\\varepsilon,T), S - S^{CT}_{<(I,K)} ).\n$$\nBy ``singular part'' I mean the following.  We take the small $\\varepsilon$ expansion of the right hand side, of the form $\\sum f_r(\\varepsilon) \\Phi( T, a)$ where $f_r(\\varepsilon) \\in \\mathscr{A}$.  The singular part is obtained by replacing each $f_r$ by its projection onto $\\mathscr{A}_{< 0}$. \n\nNote that \n$$\n \\Gamma_{i,k}( P(\\varepsilon,T), S -  S^{CT}_{<(I,K)} - S^{CT}_{I,K})  =  \\Gamma_{i,k}( P(\\varepsilon,T), S -  S^{CT}_{<(I,K)} )  - \\delta_{i,I} \\delta_{k,K} S^{CT}_{I,K} \n$$\nif $(i,k) \\le (I,K)$.   It follows that  $\\Gamma_{i,k}( P(\\varepsilon,T), S - S^{CT}_{<(I,K)} - S^{CT}_{I,K}) $ is non-singular (i.e.\\ has a well-defined $\\varepsilon \\to 0$ limit)  if $(i,k) \\le (I,K)$. \n\nTo prove the result, it remains to prove the following. \n\\begin{enumerate}\n\\item\n$S^{CT}_{I,K}$ is independent of $T$.\n\\item\n$S^{CT}_{I,K}$ is a local functional.  \n\\item\n$S^{CT}_{0,K} = 0$ (that is, there are no tree-level counter-terms).\n\\end{enumerate}\n\n\n\\begin{lemma}\n$S^{CT}_{I,K}$ is independent of $T$.  \n\\label{lemma counterterms independent T}\n\\end{lemma}\n\\begin{proof}\nObserve that \n$$\nP(\\varepsilon,T') - P(\\varepsilon,T)= P(T,T')   =  \\int_{T}^{T'}  ( Q^{GF} \\otimes 1) K_t  \\d t\n$$\nis in $\\mathscr{E} \\otimes \\mathscr{E}$ (that is, it has no singularities).    Let \n$$\nA(\\varepsilon,\\hbar) \\in \\mscr O(\\mathscr{E}, \\mathscr{A}_{\\ge 0} \\otimes \\mathbb C[[\\hbar]])\n$$\nbe any (not necessarily local) functional.  Non-singularity of $P(T,T')$ implies that \n$$\n\\Gamma_{\\le (I,K)} ( P(T,T'), A(\\varepsilon,\\hbar))\n$$\nis non-singular (i.e., has a well-defined $\\varepsilon \\to 0$ limit). \n\nWe know by induction that $ \\Gamma_{\\le (I,K)}(P(\\varepsilon,T), S  - S^{CT}_{\\le(I,K)}(\\varepsilon,T) ) $ is non-singular.  It follows that\n$$\n\\Gamma_{\\le(I,K)} \\left(  P(T,T') ,  \\Gamma_{\\le (I,K)}\\left( P(\\varepsilon,T), S - S^{CT}_{\\le(I,K)}(\\varepsilon,T) \\right) \\right) \n$$\nis non-singular.   But,\n\\begin{multline*}\n\\Gamma_{\\le(I,K)} \\left( P(T,T') ,  \\Gamma_{\\le(I,K)}\\left( P(\\varepsilon,T), S - S^{CT}_{\\le(I,K)}(\\varepsilon,T) \\right) \\right)   \\\\ \n\\begin{split}\n& =   \\Gamma_{\\le(I,K)}\\left(P(\\varepsilon,T'), S- S^{CT}_{\\le(I,K)} (\\varepsilon,T)\\right) \\\\ \n& =  \\Gamma_{\\le(I,K)}\\left(P(\\varepsilon,T'), S - S^{CT}_{<(I,K)} (\\varepsilon,T') \\right) - S^{CT}_{(I,K)} (\\varepsilon,T)\n \\end{split}\n\\end{multline*}\n(where we are using the induction assumption that $S^{CT}_{<(I,K)} (\\varepsilon,T') = S^{CT}_{<(I,K)} (\\varepsilon,T)$).\n\nThis makes it clear that\n\\begin{align*}\nS^{CT}_{(I,K)} (\\varepsilon,T) &= \n\\text{Singular part of } \\Gamma_{\\le(I,K)}\\left(P(\\varepsilon,T'), S - S^{CT}_{<(I,K)} (\\varepsilon,T') \\right)\\\\\n&= S^{CT}_{(I,K)} (\\varepsilon,T')\n\\end{align*}\nas desired. \n\\end{proof}\n\n\n\\begin{lemma}\n$S^{CT}_{I,K}$ is local, so that \n$$S^{CT}_{I,K} \\in \\mscr O_l ( \\mathscr{E}, \\mathscr{A}_{< 0} ).$$  \n\\end{lemma}\n\\begin{proof}\nThis follows immediately from the fact that $S^{CT}_{I,K}$ is independent of $T$, and from the $T \\to 0$ asymptotic expansion of the singular part of $\\Gamma_{I,K}( P(\\varepsilon,T), S - S^{CT}_{<(I,K)} )$ proved in theorem A.\n\\end{proof}\n\n\\begin{lemma}\n$S_{0,k}^{CT} = 0$.\n\\end{lemma}\n\\begin{proof}\nAll we need to show is that  $\\Gamma_{0,k} (P(\\varepsilon,T), S ) $ is regular, for all $k$. This is an easy corollary of Lemma \\ref{lemma local functional vector field}.  Namely, if we write $S = \\sum \\hbar^i S_{i,k}$ as usual, then Lemma \\ref{lemma local functional vector field} implies that the functionals $S_{0,k}$ are of the form\n$$\nS_{0,k}(\\alpha) = \\ip { \\Psi_k ( \\alpha^{\\otimes k-1}) ,\\alpha } \n$$\nfor some polydifferential operator\n$$\n\\Psi_k : \\mathscr{E}^{\\otimes k-1} \\to \\mathscr{E}\n$$\nand $\\alpha \\in \\mathscr{E}$.  The tree-level terms $\\Gamma_{0,k}(P(\\varepsilon,T),S)$ of $\\Gamma(P(\\varepsilon,T),S)$ are obtained by composing these polydifferential operators $\\Psi_k$ with each other and with the operator \n$$\nP(\\varepsilon,T) : \\mathscr{E} \\to \\mathscr{E}\n$$\nconstructed by convolution with the kernel $P(\\varepsilon,T)$.  Note that \n$$\nP(\\varepsilon,T)\\star \\alpha = \\int_{\\varepsilon}^T  Q^{GF} e^{-t H} \\alpha \n$$\nThis operator is a map $\\mathscr{E} \\to \\mathscr{E}$,  even when $\\varepsilon = 0$. That is, $\\int_0^T e^{-t H}$ takes smooth sections of the vector bundle $E$ on $M$ to smooth sections of $E$. This implies that all tree-level operators are non-singular, as desired.  \n\\end{proof}\nThis completes the proof. \n\\end{proof}\n\n\n\\section{Renormalisation group flow and converse to theorem B}\nLet $S \\in \\mscr O_l(\\mathscr{E})[[\\hbar]]$ be a local functional. The expression $\\Gamma^R ( P(0,T) , S ) $ constructed using theorem B should be interpreted as the scale $T$ renormalised effective action.  \n\\begin{lemma}\nThe renormalisation group equation holds:\n$$\n\\Gamma^R (   P(0,T') , S  )  = \\Gamma ( P(T,T') , \\Gamma^R ( P(0,T) , S )  ) .\n$$ \n\\end{lemma}\n\\begin{proof}\nThis follows from the fact that the counter-terms $S_{i,k}^{CT}$ are independent of $T$, and the identity\n$$\n\\Gamma ( P(T,T') , \\Gamma ( P(\\varepsilon,T) , S  - S^{CT} )  )  = \\Gamma ( P(\\varepsilon,T'), S  -  S^{CT} ) .\n$$\n\\end{proof} \n\nThus we have seen that to any local functional $S \\in \\mscr O(\\mathscr{E})[[\\hbar]]$, we can associate a system of effective actions $\\Gamma^R ( P (0,T), S)$ satisfying the renormalisation group equation.  In a moment we will see that a converse holds: all such systems of effective actions satisfying a certain locality condition arise in this way.\n\n  \n\\begin{definition}\nA \\emph{system of effective actions} on the space of fields $\\mathscr{E}$ is given by an effective action\n$$\nS^{eff} (T)\\in \\mscr O(\\mathscr{E} , \\mathbb C[[\\hbar]]  )\n$$\nfor each $T \\in \\mbb R_{> 0}$, which are all at least cubic modulo $\\hbar$, and\nsuch that\n\\begin{enumerate}\n\\item\nThe renormalisation group equation is satisfied,\n$$\nS^{eff}(T_2) =  \\Gamma ( P(T_1,T_2) , S^{eff}(T_1)).\n$$\n\\item\nAs $T \\to 0$, $S^{eff}(T)$ must become local, in the following sense.  There must exist some $T$-dependent local functional \n$$\\Phi \\in \\mscr O_l( \\mathscr{E} , C^{\\infty}(0,\\infty) \\otimes \\mathbb C[[\\hbar]]  ) $$ \nwhere $T$ is the coordinate on $(0,\\infty)$,\nsuch that \n$$\\lim_{T \\to 0} \\left( S^{eff}(T) - \\Phi(T) \\right)= 0.$$ \n(The $T \\to 0$ limit of $S^{eff}(T)$ itself will generally not exist). \n\\end{enumerate}\n\\end{definition}\nThe effective actions defined by $\\Gamma^R ( P(0,T) , S)$ for $S \\in \\mscr O_l(\\mathscr{E}, \\mathbb C[[\\hbar]] )$ satisfy these axioms. \n\n\nFix any renormalisation scheme $\\mathscr{A}_{< 0}$. Then theorem B provides a map\n\\begin{align*}\nS \\in \\mscr O_l(\\mathscr{E}) [[\\hbar]] \\text{ which is at least cubic modulo }\\hbar& \\to \\text{ systems of effective actions } \\\\\nS & \\mapsto \\{ \\Gamma^R ( P(0,T), S ) \\mid T \\in \\mbb R_{> 0} \\} \n\\end{align*}\n\\begin{thmC}\n\\label{proposition rge local functional}\nFor any renormalisation scheme $\\mathscr{A}_{< 0}$, this map is a bijection.\n\\end{thmC}\nThis set of systems of effective actions is a canonical object associated to $(\\mathscr{E},Q, Q^{GF})$, independent of the choice of renormalisation scheme.    Renormalisation and regularisation techniques other than those considered should lead to different ways of parametrising the same set of systems of effective actions.  For instance, if one could make sense of dimensional regularisation on general manifolds, one would hope to get simply a different parametrisation of this set.\n\nFrom this point of view, the formalism of counter-terms is simply a convenient way to describe this set of systems of effective actions.   The counter-terms themselves, and the original action $S$, are not really meaningful in themselves.   \n\n\\begin{proof}[Proof of theorem C]\nThe proof is very simple.  Let $\\{S^{eff}(T)\\}$ be a system of effective actions.  We want to construct a corresponding local functional $S \\in \\mscr O_l(\\mathscr{E}) [[\\hbar]]$ such that \n$$\nS^{eff} (T) = \\Gamma^R ( P(0,T) , S ) .\n$$\nSuppose, by induction, that we have constructed $S_{<(I,K)}$ such that \n$$\nS_{(i,k)} ^{eff} (T) = \\Gamma_{(i,k)} ^R ( P(0,T) , S_{<(I,K)} ) \n$$\nfor all $(i,k) < (I,K)$.    The initial case is when $i = 0$ and $k = 2$. Then,  $S^{eff}_{(0,2)} (T) = 0 = S_{(0,2)}$,  so the identity automatically holds. \n\nAssume $(I,K) \\ge (0,3)$.  The renormalisation group equation says that \n\\begin{align*}\nS_{(I,K) }^{eff} (T) &= \\Gamma_{(I,K)}(P(\\varepsilon, T) , S^{eff}_{\\le (I,K)} (\\varepsilon )) \\\\\n&=  S_{(I,K)}^{eff}(\\varepsilon) +   \\Gamma_{(I,K)} ( P(\\varepsilon, T) , S_{<(I,K)}^{eff} (\\varepsilon ) ) \\\\\n&=  S_{(I,K)}^{eff}(\\varepsilon) +   \\Gamma_{(I,K)} ( P(\\varepsilon, T) , \\Gamma_{<(I,K)}^R (P(0,\\varepsilon ), S_{<(I,K)} ) ) \\\\\n&= S_{(I,K)}^{eff}(\\varepsilon) +   \\Gamma_{(I,K)} ( P(\\varepsilon, T) , \\Gamma_{\\le (I,K)}^R (P(0,\\varepsilon ), S_{<(I,K)} ) )  -   \\Gamma^R_{(I,K)} ( P(0, \\varepsilon) , S_{<(I,K)} ) ) \\\\\n&=  S_{(I,K)}^{eff}(\\varepsilon) +   \\Gamma^R_{(I,K)} ( P(0, T) , S_{<(I,K)} ) )  -  \\Gamma^R_{(I,K)} ( P(0, \\varepsilon) , S_{<(I,K)} ) ).\n\\end{align*}\nThe left hand side  is independent of $\\varepsilon$, so the right hand side is also.  Thus, \n$$\n S_{(I,K)}^{eff}(\\varepsilon)  -  \\Gamma^R_{(I,K)} ( P(0, \\varepsilon) , S_{<(I,K)} ) )\n$$\nis independent of $\\varepsilon$.  This allows us to define\n$$\nS_{(I,K)} =  S_{(I,K)}^{eff}(\\varepsilon)  -  \\Gamma^R_{(I,K)} ( P(0, \\varepsilon) , S_{<(I,K)} ) ).\n$$\nWith this definition, we automatically have\n$$\nS_{(I,K)}^{eff}(T) = \\Gamma^R  ( P(0,T) , S_{\\le (I,K)}   ) .\n$$\nIt remains to show that $S_{(I,K)}$ is local.   This is an immediate consequence of the locality axiom for $S^{eff}(\\varepsilon)$.  There exists $\\Phi(\\varepsilon)$, an $\\varepsilon$ dependent local functional, such that $\\lim_{\\varepsilon \\to 0} S^{eff}(\\varepsilon ) - \\Phi(\\varepsilon ) = 0$.  Thus, \n\\begin{align*}\nS_{(I,K)} &=  S_{(I,K)}^{eff}(\\varepsilon)  -  \\Gamma^R_{(I,K)} ( P(0, \\varepsilon) , S_{<(I,K)} ) ) \\\\\n& =  S_{(I,K)}^{eff}(\\varepsilon) -  \\Phi(\\varepsilon) + \\Phi(\\varepsilon) -   \\Gamma^R_{(I,K)} ( P(0, \\varepsilon) , S_{<(I,K)} ) )\\\\\n&= \\lim_{\\varepsilon \\to 0}  \\left( S_{(I,K)}^{eff}(\\varepsilon) -  \\Phi(\\varepsilon) \\right) + \\lim_{\\varepsilon \\to 0} \\left( \\Phi(\\varepsilon) -   \\Gamma^R_{(I,K)} ( P(0, \\varepsilon) , S_{<(I,K)} ) ) \\right) \\\\\n&=  \\lim_{\\varepsilon \\to 0} \\left( \\Phi(\\varepsilon) -   \\Gamma^R_{(I,K)} ( P(0, \\varepsilon) , S_{<(I,K)} ) ) \\right) .\n\\end{align*}\nThis last quantity is local, because $\\Phi(\\varepsilon)$ is local and $\\Gamma^R_{(I,K)} ( P(0,\\varepsilon), S_{< (I,K)} )$ has a small $\\varepsilon$ asymptotic expansion in terms of local functionals. \n\n\\end{proof} \n\n\nAs I explained in the introduction, this result elucidates how the renormalisation procedure depends on the choice of renormalisation scheme.    For the rest of the paper, we will fix one renormalisation scheme $\\mathscr{A}_{< 0}$.  When we talk about local functionals, we are in some sense really talking about systems of effective actions, but using our fixed renormalisation scheme to identify these two sets.  Thus, as long as all the statements we make are about the effective action $\\Gamma^R(P(0,T), S)$, and not directly about the local functional $S$, everything we do is independent of the choice of renormalisation scheme. \n\n\\section{Quantum master equation}\n\n\n\n\n\\subsection{Quantum master equation in finite dimensions}\nRecall that to each element $\\Phi \\in \\Sym^2 \\mathscr{E}$ we have associated (in section \\ref{section differential operators}) an order two differential operator $\\partial_{\\Phi}$ on $\\mscr O(\\mathscr{E})$.  Let $$\\Delta_T =  -\\partial_{K_T}.$$    When $T > 0$, this is a differential operator on $\\mscr O(\\mathscr{E})$. When $T = 0$, because $K_0$ is no longer an element of $\\Sym^2 \\mathscr{E}$ but of some distributional completion, the operator $\\Delta_0$ is  ill-defined.   The operators $\\Delta_T$ are odd, second-order differential operators on $\\mscr O(\\mathscr{E})$, which satisfy the equation\n$$\n\\Delta_T^2 = 0.\n$$ \nThus, each $\\Delta_T$ endows the algebra $\\mscr O(\\mathscr{E})$ with the structure of a Batalin-Vilkovisky algebra.  The ill-defined operator $\\Delta_0$ is the ``physical'' Batalin-Vilkovisky operator.\n\nLet us assume for a moment that $M$ is a point, so that $\\mathscr{E}$ is simply a finite dimensional super vector space.  Then, the operator $\\Delta_0$ is perfectly well defined.  In this situation, $\\Delta_0$ is the Batalin-Vilkovisky odd Laplacian, as described in section \\ref{section intro bv}.   The operator $K_0$ is simply the kernel representing the identity map on $\\mathscr{E}$. In terms of a Darboux basis $x_i, \\xi_i$ of $V$, where $x_i$ are even, $\\xi_i$ are odd, and $\\ip{x_i,\\xi_i} = \\delta_{i j}$, then\n$$\nK_0 =  -  \\sum_i  \\left(  x_i \\otimes \\xi_i + \\xi_i \\otimes x_i  \\right)\n$$\nand\n$$\n\\Delta_0 = - \\partial_{K_0} = \\sum \\partial_{x_i} \\partial_{\\xi_i}.\n$$\n\nStill in the situation when $\\dim M = 0$, the Poisson bracket on the algebra $\\mscr O(\\mathscr{E})$ can be written in terms of $\\Delta_0$, using the equation\n$$\n \\{ f , g \\} = \\Delta_0(f g) - \\Delta_0 (f) g - (-1)^{\\abs f} f \\Delta_0 (g) .\n$$\nWhen $\\dim M > 0$, this Poisson bracket is ill-defined on $\\mscr O(\\mathscr{E})$. However, as we have seen, the Poisson bracket of an element of $\\mscr O_l(\\mathscr{E})$ with an element of $\\mscr O(\\mathscr{E})$ is well-defined. \n\nA functional $S \\in \\mscr O(\\mathscr{E}, \\mathbb C[[\\hbar]])$ satisfies the BV quantum master equation if \n$$\n(Q + \\hbar \\Delta_0 ) \\exp ( S \/ \\hbar ) = 0.\n$$\nAgain, this makes sense when $\\dim M = 0$, but the left hand side of this equation is ill-defined  when $\\dim M > 0$.  This equation can be re-expressed as \n$$\nQ S + \\tfrac{1}{2} \\{ S, S \\} + \\hbar \\Delta_0 S = 0.\n$$\n\n\n\\subsection{The quantum master equation and the renormalisation group flow}\nLet us return to the situation where $\\dim M \\ge 0$.  \n\nRecall that \n$$\nP(\\varepsilon,T)= \\int_{\\epsilon}^T L_t \\d t = \\int_{\\varepsilon}^T (Q^{GF} \\otimes 1) K_t \\d t.\n$$\n\\begin{lemma}\n$$\nQ P _\\epsilon^T = - K_T + K _\\epsilon. \n$$\nwhere $Q$ denotes the tensor product differential on $\\mathscr{E} \\otimes \\mathscr{E}$.\n\\end{lemma}\n\\begin{proof}\nIndeed, our sign conventions are such that $Q P(\\epsilon,T)$ is the kernel for the operator \n\\begin{align*}\n\\alpha &\\mapsto Q \\int_\\epsilon^T Q^{GF} e^{-t H }  \\alpha + \\int_\\epsilon^T Q^{GF} e^{-t H }  Q \\alpha \\\\\n&= \\int_\\epsilon^T H e^{-t H } \\alpha \\\\\n&= -e^{-T H }\\alpha + e^{- \\epsilon H } \\alpha .\n\\end{align*}\nThus, the operator associated to $Q P(\\epsilon,T)$ is the same as that associated to $-K_T + K_\\epsilon$.\n\\end{proof} \n\n\\begin{lemma}\nLet $S \\in \\mscr O(\\mathscr{E}, \\mathbb C[[\\hbar]])$.  Then $S$ satisfies the $\\Delta_{T_1}$ quantum master equation\n$$\n(Q + \\hbar  \\Delta_{T_1} ) \\exp ( S \/ \\hbar ) = 0\n$$\nif and only if $\\Gamma( P(T_1,T_2) , S)$ satisfies the $\\Delta_{T_2}$ quantum master equation\n$$\n(Q + \\hbar \\Delta_{T_2} )  \\exp (\\Gamma( P(T_1,T_2) , S) \/ \\hbar  )  = 0. \n$$\n\\label{lemma propagator homotopy}\n\\end{lemma}\n\\begin{proof}\nIndeed,\n$$\n[ \\partial_{P(T_1,T_2)}  , Q ] = \\partial_{Q P(T_1,T_2) } = \\partial_{K_{T_1} } - \\partial_{K_{T_2} } = \\Delta_{T_2} - \\Delta_{T_1} . \n$$\nThis implies that \n$$\n[ Q , \\exp ( \\partial_{P(T_1,T_2)}  )  ] = \\exp (\\partial_{P(T_1,T_2)}  ) ( \\Delta_{T_1} - \\Delta_{T_2} ) . \n$$\nThus,\n\\begin{align*}\n(Q + \\hbar \\Delta_{T_2 } )  \\exp (\\Gamma( P(T_1,T_2) , S) \/ \\hbar  )  &=  (Q + \\hbar \\Delta_{T_2 } ) \\exp ( \\hbar \\partial_{P(T_1,T_2)} ) \\exp ( S \/ \\hbar  )\\\\ \n&= \\exp ( \\hbar \\partial_{P(T_1,T_2)}  ) ( Q + \\hbar \\Delta_{T_1 }  )  \\exp ( S \/ \\hbar  ) .\n\\end{align*}\nThe converse follows because the operator $\\exp ( \\hbar \\partial_{P(T_1,T_2)}  )$ is invertible. \n\n\n\\end{proof} \n\nSuppose we have an effective action $S^{eff}(T_1) \\in \\mscr O(\\mathscr{E}, \\mathbb C[[\\hbar]])$ at scale $T_1$.   Then $\\Gamma( P(T_1,T_2), S^{eff}(T_1))$  is the effective action at scale $T_2$.      What we see is that if the effective action $S^{eff}(T_1)$ satisfies the scale $T_1$ quantum master equation, then the effective action $S^{eff}(T_2)$ at scale $T_2$ satisfies the scale $T_2$ quantum master equation, and conversely.\n\n\n\n\\subsection{Renormalised quantum master equation}\n\nWe know, using the renormalisation techniques we have discussed so far, how to make sense of the expression $\\Gamma(P(0,T) , S ) $ if $S \\in \\mscr O_l(\\mathscr{E}, \\mathbb C[[\\hbar]])$ is a local functional which is at least cubic modulo $\\hbar$.  The  renormalised scale $T$ effective action is\n$$\\Gamma^R ( P(0,T) , S ) \\overset{\\text{def}}{=} \\lim_{\\varepsilon \\to 0} \\Gamma(P(\\varepsilon,T), S^R )$$ \nwhere $S^R = S - S^{CT}$.  \n\nWe would like to define a renormalised quantum master equation, which is a replacement for the ill-defined equation\n$$\n(Q + \\hbar \\Delta_0 ) e^{S \/ \\hbar }= 0.\n$$\nWhen $\\dim M = 0$, so $\\mathscr{E}$ is finite dimensional, this equation is well-defined and is equivalent to the equation\n$$\n(Q + \\hbar \\Delta_T )\\exp \\left( \\hbar^{-1} \\Gamma ( P(0,T) , S ) \\right)= 0.\n$$\nIn the infinite dimensional situation, we will take this second equation as a definition, where we use the renormalised version $\\Gamma^R ( P(0,T), S )$ of $\\Gamma(P(0,T), S)$.  Thus, \n\\begin{definition}\n$S \\in \\mscr O_l(\\mathscr{E}, \\mathbb C[[\\hbar]])$ satisfies the renormalised quantum master equation if\n$$\n(Q + \\hbar \\Delta_T ) \\exp \\left( \\hbar^{-1}  \\Gamma^R (P(0,T), S ) \\right) = 0 .\n$$\n\\end{definition}\nTheorems B and C show that there is a bijection between the set of local functionals $S \\in \\mscr O_l(\\mathscr{E}, \\mathbb C[[\\hbar]])$ and the set of systems of effective actions, sending $S$ to $\\{\\Gamma^R ( P(0,T) , S) \\mid T \\in \\mbb R_{> 0} \\}$ .  The renormalised QME is thus just saying that the scale $T$ effective action satisfies the scale $T$ QME.  \n\n\nIt is immediate from Lemma \\ref{lemma propagator homotopy} that this condition is independent of $T$.    That is, if $S$ satisfies the renormalised QME for one value of $T$, then it does so for all other values of $T$. \n\nIt seems to me that a successful quantisation of the quantum field theory with classical action $S_0$ consists of replacing $S_0$ by a solution $S = S_0 + \\sum_{i > 0} \\hbar^i S_i$ of the renormalised quantum master equation. Equivalently, a quantisation of a classical action $S_0$ is given by replacing  $S_0$ by a system of effective actions $\\{S^{eff}(T) \\mid T \\in \\mbb R_{> 0} \\}$, which satisfy the quantum master equation, and which, modulo $\\hbar$, converge to $S_0$ as $T \\to 0$. \n\n\\begin{lemma}\nSuppose that $S \\in \\mscr O_l(\\mathscr{E}, \\mathbb C[[\\hbar]])$ satisfies the renormalised quantum master equation.  Then \n$$\n\\Gamma^R ( P(0,\\infty), S ) \\mid_{\\Ker H} \n$$\nsatisfies the quantum master equation on the finite dimensional vector space \n$$\n\\Ker H = H^\\ast(\\mathscr{E}, Q) . \n$$\n\\end{lemma}\n\\begin{proof}\nThis is immediate from the fact that the heat kernel $K_\\infty$ is the element of $\\Ker H \\otimes \\Ker H \\subset \\mathscr{E} \\otimes \\mathscr{E}$ which (as a kernel on $\\Ker H$) represents the identity map $\\Ker H \\to \\Ker H$. \n\\end{proof} \nRecall that\n$\\Gamma^R ( P(0,\\infty), S )(a)$ is a renormalised version of the functional integral\n$$\n\\hbar \\log \\int_{x \\in (\\Im Q^{GF} ) _\\mbb R }  \\exp\\left( \\tfrac{1}{2\\hbar}  \\ip{x,Q x}  + \\tfrac{1}{\\hbar} S ( x + a ) \\right) .\n$$\nIf we take $a$ to be in $\\Ker H = H^\\ast(\\mathscr{E},Q)$, then this functional integral is the Wilsonian effective action in the Batalin-Vilkovisky formalism, obtained by integrating over a Lagrangian $\\Im Q^{GF}$ in the space $\\Im Q^{GF} \\oplus \\Im Q$ of positive eigenvalues of the Hamiltonian $H$.   \n\n\n\\section{Homotopies of solutions of the quantum master equation}\n\n\\label{section simplicial sets}\nIn the usual finite-dimensional Batalin-Vilkovisky formalism \\cite{Sch93, AleKonSch97}, one reason the quantum master equation is so important is that it implies that the functional integral is independent of the choice of gauge fixing condition (usually, a Lagrangian or isotropic subspace in the space of fields).    \n\nWe would like to prove a version of this in the infinite dimensional situation, for the renormalised quantum master equation.  However, things are more delicate in this situation; the renormalised QME itself depends on the choice of a gauge fixing condition.  What we will show is that if $Q^{GF}(t)$ is a one-parameter family of gauge fixing conditions, then the set of homotopy classes of solutions to the renormalised QME using $Q^{GF}(0)$ is isomorphic to the corresponding set using $Q^{GF}(1)$.  This result is a corollary of a result about certain simplicial sets of gauge fixing conditions and of solutions to the renormalised QME.\n\nAs I explained earlier, we are fixing a renormalisation scheme throughout  the paper, which allows us to identify the set of local functionals $S \\in \\mscr O_l(\\mathscr{E}) [[\\hbar]]$ with the set of systems of effective actions.  Everything we say should be in terms of the effective action $\\Gamma^R(P(0,T), S)$ and not directly in terms of the local functional $S$.  As long as we stick to this, everything we do will be independent of the choice of renormalisation scheme. \n\\subsection{Families and simplicial sets of gauge fixing conditions}\n\nLet us denote the $n$-simplex by $\\Delta^n$.\n\n\\begin{definition}\n A family of gauge fixing conditions parametrised by $\\Delta^n$ is a first order differential  operator \n$$\nQ^{GF} : \\mathscr{E} \\otimes C^{\\infty}(\\Delta^n) \\to \\mathscr{E} \\otimes C^{\\infty} (\\Delta^n)\n$$\nwhich is $C^{\\infty}(\\Delta^n)$ linear, and which satisfies a family version of the axioms for a gauge fixing condition.  Namely, \n\\begin{enumerate}\n\\item\nthe operator $H_0 = [Q \\otimes 1, Q^{GF}]$ must be a smooth family of generalised Laplacians in the sense of \\cite{BerGetVer92}, section 9.5.\n\\item\nthere is a direct sum decomposition (of $C^{\\infty}(\\Delta^n)$ modules)\n$$\n\\mathscr{E} \\otimes C^{\\infty}(\\Delta^n) = \\Im (Q\\otimes 1) \\oplus \\Im Q^{GF} \\oplus \\Ker H_0 .\n$$\n\\end{enumerate}\n\\end{definition}\nWe will normally just write $Q$ for the operator $Q \\otimes 1$  on $\\mathscr{E} \\otimes \\Omega^\\ast(\\Delta^n)$, and similarly we will write $\\d_{DR}$ for $ 1 \\otimes \\d_{DR}$.  The operator $Q^{GF}$ extends in a unique $\\Omega^\\ast(\\Delta^n)$-linear way to $\\mathscr{E} \\otimes \\Omega^\\ast(\\Delta^n)$.  Let us use the notation\n$$\nH = [ Q + \\d_{DR} , Q^{GF} ]  : \\mathscr{E} \\otimes \\Omega^\\ast (\\Delta^n) \\to  \\mathscr{E} \\otimes \\Omega^\\ast (\\Delta^n) . \n$$\nIt is easy to see that $H$ is $\\Omega^\\ast(\\Delta^n)$-linear.  If $H_1 = [\\d_{DR}, Q^{GF}]$, then $H = H_0 + H_1$ and $H_1$ is an order $1$ differential operator.  Thus, the symbol of $H$ is the same as that of $H_0$, and, by assumption, the symbol of $H_0$ is given by a smooth family of metrics on $M$, times the identity on the vector bundle $E$ on $M$.    Thus, we can think of $H$ as a smooth family of generalised Laplacians on $\\mathscr{E}$ parametrised by the supermanifold $\\Delta^n \\times \\mbb R^{0,n}$.  The results of \\cite{BerGetVer92}  imply that there exists a unique heat kernel for $H$, \n$$\nK_t \\in \\mathscr{E} \\otimes \\mathscr{E} \\otimes \\Omega^\\ast(\\Delta^n) .\n$$\n\n\nLet us denote by $\\mathbf{GF} [n]$ the set of families of gauge fixing conditions parametrised by $\\Delta^n$.  If $\\phi : \\Delta^n \\to \\Delta^m$ is a smooth map (for instance, the inclusion of a face) then there is an induced map $\\phi^\\ast : \\mathbf{GF}[m]   \\to \\mathbf{GF}[n]$.   In this way, the sets $\\mathbf{GF}[n]$ form a simplicial set, which we denote $\\mathbf{GF}$.  \n\n\n\\subsection{Simplicial sets of solutions to the renormalised QME}\n\nLet $$S \\in \\mscr O_l(\\mathscr{E}, \\Omega^\\ast(\\Delta^n) [[\\hbar]] )$$ be a family of functionals on $\\mathscr{E}$, which, as always, is at least cubic modulo $\\hbar$.   As before, if $P \\in \\mathscr{E} \\otimes \\mathscr{E} \\otimes \\Omega^\\ast (\\Delta^n)$, one can define\n$$\n\\Gamma ( P , S ) = \\hbar \\log \\left( \\exp ( \\hbar \\partial_P ) \\exp ( S  \/ \\hbar ) \\right)  \\in \\mscr O ( \\mathscr{E} , \\Omega^\\ast(\\Delta^n) [[\\hbar]]  ) .\n$$\n\nLet us suppose, as above, that we have a smooth family of gauge fixing conditions $Q^{GF}$ parametrised by $\\Delta^n$. Let $H$ and $K_t$ be as above.  This allows us to construct the propagator we use, \n$$\nP(\\varepsilon,T)= \\int_{\\epsilon}^T  (Q^{GF} \\otimes 1) K_t \\d t .\n$$\nThus, one can define the family version of the renormalisation group flow, which sends\n$$\nf \\in \\mscr O(\\mathscr{E}, \\Omega^\\ast(\\Delta^n) [[\\hbar]] ) \\to \\Gamma(P(\\varepsilon,T) ,f ) \\in \\mscr O(\\mathscr{E}, \\Omega^\\ast(\\Delta^n) [[\\hbar]] ).\n$$\n\\begin{definition}\nA family of systems of effective actions on $\\mathscr{E}$ parametrised by the supermanifold $\\Delta^n \\times \\mbb R^{0,n}$ is given by an effective action\n$$\nS^{eff}(T) \\in \\mscr O(\\mathscr{E}, \\Omega^\\ast(\\Delta^n) [[\\hbar]] )\n$$\nfor all $T \\in \\mbb R_{> 0}$, \nsuch that\n\\begin{enumerate}\n\\item\nThe renormalisation group equation is satisfied,\n$$\nS^{eff}(T_2) =  \\Gamma ( P(T_1,T_2) , S^{eff}(T_1)).\n$$\n\\item\nAs $T \\to 0$, $S^{eff}(T)$ must become local, in the following sense.  There must exist some $T$-dependent local functional \n$$\\Phi \\in \\mscr O_l( \\mathscr{E} , \\Omega^\\ast(\\Delta^n) \\otimes C^{\\infty}(0,\\infty) \\otimes \\mathbb C[[\\hbar]]  ) $$ \nwhere $T$ is the coordinate on $(0,\\infty)$,\nsuch that \n$$\\lim_{T \\to 0} \\left( S^{eff}(T) - \\Phi(T) \\right) = 0.$$ \n(The $T \\to 0$ limit of $S^{eff}(T)$ itself will generally not exist). \n\\end{enumerate}\n\\end{definition}\n\n\n\nTheorems A, B and C apply in this situation. Indeed, theorem A is stated and proved in this generality in the appendix, and the proofs of the family version of theorems  B and C is identical to the proof we gave earlier.    Thus, we can define \n$$\n\\Gamma^R(P(0,T), S ) = \\lim_{\\varepsilon \\to 0} \\Gamma(P(\\varepsilon,T), S - S^{CT} ) \\in \\mscr O ( \\mathscr{E} , \\Omega^\\ast(\\Delta^n) \\otimes \\mathbb C [[\\hbar]]  ) . \n$$\nThis map gives us a  bijection between the set of local functionals $S \\in \\mscr O_l ( \\mathscr{E}, \\Omega^\\ast(\\Delta^n) \\otimes \\mathbb C[[\\hbar]] )$, and the set of families of systems of effective actions parametrised by $\\Delta^n \\times \\mbb R^{0,n}$. This bijection depends on the choice of renormalisation scheme, which we fix throughout the paper.\n\nWe say that $S$ satisfies the renormalised quantum master equation if\n$$\n( \\d_{DR} + Q + \\hbar \\Delta_T) \\exp ( \\hbar^{-1} \\Gamma^R (P(0,T), S )  ) = 0 . \n$$\nThis condition, as before, is independent of $T$.   \n\\begin{definition}\nDefine a simplicial set $\\mathbf{BV}_\\mathscr{E}$ by saying that $\\mathbf{BV}_\\mathscr{E}[n]$ is the set of triples \n\\begin{enumerate}\n\\item\n$Q^{GF} \\in \\mathbf{GF}[n]$\n\\item $S \\in \\mscr O_l(\\mathscr{E}, \\Omega^\\ast(\\Delta^n) [[\\hbar]] )$ which is at least cubic modulo $\\hbar$, and which satisfies the renormalised quantum master equation corresponding to $Q^{GF}$.\n\\end{enumerate}\n\nSimilarly, define a simplicial set $\\mathbf{BV}_{H^\\ast(\\mathscr{E},Q) }$, by saying $\\mathbf{BV}_{H^\\ast(\\mathscr{E},Q) }[n]$  is the set of functions $S \\in \\mscr O (H^\\ast(\\mathscr{E}, Q)  ) \\otimes \\Omega^\\ast(\\Delta^n) [[ \\hbar ]]$ which satisfy the quantum master equation $$ ( \\d_{DR} + \\hbar \\Delta_{H^\\ast(\\mathscr{E}, Q) }  ) e^{S \/ \\hbar } = 0.$$  \n\\end{definition}\n\nAn equivalent (and more natural) definition of the simplicial set $\\mathbf{BV}_\\mathscr{E}$ would be to say that $\\mathbf{BV}_\\mathscr{E}[n]$ is the set of families of systems of effective actions, parametrised by $\\Delta^n \\times \\mbb R^{0,n}$, such that the scale $T$ effective action $S^{eff}(T)$ satisfies the scale $T$ QME, \n$$\n(Q + \\d_{DR} + \\hbar \\Delta_T ) e^{ S^{eff}(T) \/ \\hbar } = 0 . \n$$\nOf course, the choice of renormalisation scheme (which we are fixing throughout the paper) leads to an equivalence between the two definitions.   \n\n\n\n\\begin{lemma}\nThere is a map\n$$\n\\mathbf{BV}_\\mathscr{E}[n] \\to \\mathbf{BV}_{H^\\ast(\\mathscr{E},Q)} [n]\n$$\nof simplicial sets, which sends\n$$\nS \\mapsto \\Gamma^R ( P(0,\\infty) , S )\\mid_{ H^\\ast(\\mathscr{E}, Q)  } .\n$$\nThis extends to a map of simplicial sets\n$$\n\\mathbf{BV}_\\mathscr{E}[n] \\to \\mathbf{BV}_{H^\\ast(\\mathscr{E},Q)} [n]\n$$\nin the natural way.\n\\end{lemma}\n\n\\subsection{Fibration property}\nLet $Q^{GF}$ be a gauge fixing condition on $\\mathscr{E}$.  Let $\\mathbf{BV}_{\\mathscr{E}, Q^{GF}}$ denote the simplicial set of solutions to the renormalised quantum master equation with fixed gauge fixing condition $Q^{GF}$.  In other words, $\\mathbf{BV}_{\\mathscr{E}, Q^{GF}}$ is the fibre of $\\mathbf{BV}$ over the point $Q^{GF} \\in  \\mathbf{GF} [0]$, under the natural map $\\mathbf{BV}_{\\mathscr{E}} \\to \\mathbf{GF} $. \n\n\n The set $\\pi_0 \\left( \\mathbf{BV}_{\\mathscr{E}, Q^{GF}} \\right)$ is the set of homotopy classes of solutions to the renormalised quantum master equation.  The fact that the map $\\mathbf{BV}_{\\mathscr{E}, Q^{GF}} \\to \\mathbf{BV}_{H^\\ast(\\mathscr{E},Q)}$ is a map of simplicial sets tells us that there is an induced map\n\\begin{equation}\n\\pi_0 \\left( \\mathbf{BV}_{\\mathscr{E}, Q^{GF}} \\right)  \\to  \\pi_ 0 \\left( \\mathbf{BV}_{H^\\ast(\\mathscr{E},Q)}  \\right). \\tag{$\\dagger$}  \\label{map to qme cohomology} \n\\end{equation}\nWe would like to show that the set $\\pi_0 ( \\mathbf{BV}_{\\mathscr{E}, Q^{GF}}) $, and the map \\eqref{map to qme cohomology}, are in some sense independent of $Q^{GF}$..  This will follow from an abstract result.\n\n\\begin{theorem}\nThe map\n$$\n\\mathbf{BV}_\\mathscr{E} \\to \\mathbf{GF} \n$$\nis a fibration of simplicial sets.  \n\\label{theorem fibration}\n\\end{theorem}\nWe will prove this later.\n\n\\begin{corollary}\nLet $Q^{GF}(t)$ is a  smooth families of gauge fixing conditions  parametrised by $[0,1]$.  Then the sets $\\pi_0 ( \\mathbf{BV}_{\\mathscr{E}, Q^{GF}(0)} ) $, $\\pi_0 ( \\mathbf{BV}_{\\mathscr{E}, Q^{GF}(1) } ) $ are canonically isomorphic. Further, the diagram\n$$\n\\xymatrix{ \\pi_0 \\left( \\mathbf{BV}_{\\mathscr{E}, Q^{GF}(0)} \\right)  \\ar[r] \\ar[d] &  \\pi_0 \\left( \\mathbf{BV}_{\\mathscr{E}, Q^{GF}(1)}  \\right) \\ar[dl] \\\\\n\\pi_ 0 \\left(\\mathbf{BV}_{H^\\ast(\\mathscr{E}, Q )    } \\right) } \n$$\ncommutes.  \n\\end{corollary}\n\\begin{proof}\nThis follows immediately from the path and homotopy lifting properties for fibrations of simplicial sets.  \n\\end{proof} \n\n\n\\section{A local obstruction to solving the renormalised quantum master equation} \nWe will prove Theorem \\ref{theorem fibration} using obstruction theory.  In this section we will construct  the required obstructions to solving the renormalised QME. \n\nLet us fix a smooth family $Q^{GF}$ of gauge fixing conditions  parametrised by $\\Delta^n$. Let $S \\in \\mscr O_l(\\mathscr{E}, \\Omega^\\ast(\\Delta^n) \\otimes \\mathbb C[[\\hbar]])$ be, as always, at least cubic modulo $\\hbar$.   One way to write the renormalised quantum master equation for $S$ is the equation\n$$\n\\lim_{\\varepsilon \\to 0}    (Q +  \\d_{DR} + \\Delta_T ) \\exp ( \\hbar  \\partial_{P(\\varepsilon,T)} ) \\exp ( S^R \/ \\hbar )   = 0. \n$$\n\n\nOne can re-express this using the  following identity: \n\\begin{multline}\n\\Gamma \\left( P(\\varepsilon,T)+ \\delta \\Delta_\\varepsilon ,  S^R + \\delta Q S^R + \\delta \\d_{DR} S^R \\right)  = \\\\ \\hbar \\log \\left( (1 + \\delta Q + \\delta \\d_{DR} + \\delta  \\Delta_T ) \\exp ( \\hbar  \\partial_{P(\\varepsilon,T)} ) \\exp ( S^R \/ \\hbar )  \\right)  \\label{equation Q inside gamma}\n\\end{multline}\nwhere $\\delta$ is an odd parameter.  This identity is a corollary of the fact that \n$$\n[\\partial_{P(\\varepsilon,T)} , Q + \\d_{DR}] = \\Delta_T -\\Delta_\\varepsilon. \n$$\nThus, we  have an equivalent expression of the renormalised quantum master equation; the renormalised QME holds if and only if \n\\begin{equation*}\n\\lim_{\\varepsilon \\to 0} \\frac{\\d}{\\d \\delta} \\left<   \\Gamma \\left( P(\\varepsilon,T)+ \\delta \\Delta_\\varepsilon,  S^R + \\delta Q S^R + \\delta \\d_{DR} S^R \\right)  \\right> = 0 .\n \\tag{$\\dagger$} \\label{rqme} \n\\end{equation*}\n\nRecall that our choice  of renormalisation scheme gives us a subspace\n$$\n\\mathscr{A}_{< 0} \\subset \\mathscr{A}\n$$\nwhich is complementary to the subspace $\\mathscr{A}_{\\ge 0}$ .\nLet\n$$\n\\mathscr{A}_{\\le 0} = \\mathscr{A}_{< 0} \\oplus \\mathbb C \\subset \\mathscr{A}.\n$$\nIf we denote  by $\\mathscr{A}_{> 0}$ the space of functions whose $\\varepsilon \\to 0$ limit exists and is equal to zero, we have a direct sum decomposition\n$$\n\\mathscr{A} = \\mathscr{A}_{\\le 0} \\oplus \\mathscr{A}_{> 0} . \n$$\n\\begin{proposition}\n\\label{proposition obstruction}\n\\begin{enumerate}\n\\item\nLet $S \\in \\mscr O_l(\\mathscr{E}, \\Omega^\\ast(\\Delta^n) \\otimes \\mathbb C[[\\hbar]] )$ be an even element.  Then\nthere exists a unique odd element \n$$O(S) \\in \\mscr O_l(\\mathscr{E},  \\mathscr{A}_{\\le 0}\\otimes \\Omega^\\ast(\\Delta^n) \\otimes \\mathbb C[[\\hbar]] )$$ such that \n$$\\lim_{\\varepsilon \\to 0} \\frac{\\d}{\\d \\delta} \\left<   \\Gamma \\left( P(\\varepsilon,T)+ \\delta \\Delta_\\varepsilon,  S^R + \\delta Q S^R + \\delta \\d_{DR} S^R + \\delta O(S)   \\right)  \\right>    $$ \nexists, and has value $0$. \n\\item\n$O_{(I,K)}(S)$ depends only on $S_{\\le (I,K)}$, and \n$$\nO_{(I,K)}(S_{<(I,K)} + \\hbar^I S_{(I,K)} ) =   \\hbar^I Q S_{(I,K)} + \\hbar^I \\d_{DR} S_{(I,K)} + O_{(I,K)}(S_{<(I,K)} ). \n$$\n\\item\nFurther,  if $k > 0$, the germ of $O_{i,k}(S)$ at $x \\in M$ depends only on the germ of the data $(\\mathscr{E}, \\ip{\\ , \\ }, Q , Q^{GF} , \\{ S_{i,k} \\mid k > 0 \\} )$ near $x\\times \\Delta^n$.  \n\\end{enumerate}\n\n\\end{proposition}\nThus, $S$ satisfies the renormalised quantum master equation if and only if $O(S) = 0$.\n\\begin{proof}\nThe proof is very similar to that of theorem B.  \n\nSuppose we've constructed $O_{(i,k)}(S) \\in \\mscr O_l(\\mathscr{E}, \\Omega^\\ast(\\Delta^n) \\otimes \\mathscr{A}_{\\le 0})$ for all $(i,k) < (I,K)$, which are independent of $T$, and which are such that \n$$\\frac{\\d}{\\d \\delta} \\left<   \\Gamma_{< (I,K)} \\left( P(\\varepsilon,T)+ \\delta \\Delta_\\varepsilon,  S^R + \\delta Q S^R + \\delta \\d_{DR} S^R + \\delta O_{<(I,K)}(S)   \\right)  \\right>  $$\nis regular at $\\varepsilon = 0$ with value $0$.  \n\nDefine\n\\begin{multline*}\nO_{(I,K)}(S,T) = -\\text{projection onto $\\mathscr{A}_{\\le 0}$ of } \\\\   \\frac{\\d}{\\d \\delta} \\left<   \\Gamma_{(I,K)} \\left( P(\\varepsilon,T)+ \\delta \\Delta_\\varepsilon,  S^R + \\delta Q S^R + \\delta \\d_{DR} S^R + \\delta O_{<(I,K)}(S)   \\right)  \\right>.  \n\\end{multline*}\nThe second clause of the proposition is immediate from this definition.  What  remains to be shown is that  $O_{(I,K)}(S,T)$ is independent of $T$, and so (by the same argument as in the proof of theorem B) local.\n\nIt follows from the definition of $O_{(I,K)}(S,T)$ that\n$$\n\\lim_{\\varepsilon \\to 0}\\Gamma_{\\le (I,K)} \\left( P(\\varepsilon,T)+ \\delta \\Delta_\\varepsilon,   S^R \n+ \\delta Q S^R + \\delta \\d_{DR} S^R  +\n \\delta O_{< (I,K)}(S) + \\hbar^I \\delta O_{(I,K)}(S,T)   \\right)  \n$$\nexists, and this limit is independent of $\\delta$. Thus, the same holds for \n\\begin{multline*}\n\\Gamma_{\\le(I,K)} \\left( P(T,T')  ,  \\Gamma_{\\le(I,K)}\\left( P(\\varepsilon,T)+ \\delta \\Delta_\\varepsilon, S^R + \\delta Q S^R  \\delta \\d_{DR} S^R \n\\right. \\right.\n\\\\ \n \\left. \\left.\n+ \\delta O_{< (I,K)}(S) +  \\hbar^I \\delta O_{(I,K)}(S,T) \\right) \\right) . \n\\end{multline*}\n\nObserve that\n\\begin{multline*}\n\\Gamma_{\\le(I,K)} \\left( P(T,T')  ,  \\Gamma_{\\le(I,K)}\\left( P(\\varepsilon,T)+ \\delta \\Delta_\\varepsilon, S^R + \\delta Q S^R + \\delta \\d_{DR} S^R  \\right. \\right. \\\\\n\\shoveright {\\left. \\left.\n + \\delta O_{< (I,K)}(S) +  \\hbar^I  \\delta O_{(I,K)}(S,T) \\right) \\right)   }\n\\\\ =   \\Gamma_{\\le(I,K)}\\left(P(\\varepsilon,T') + \\delta \\Delta_\\varepsilon, S^R + \\delta Q S^R \n + \\delta \\d_{DR} S^R \n \\right. \\\\\n\\shoveright{ \\left.\n + \\delta O_{< (I,K)}(S)) +  \\hbar^I \\delta O_{(I,K)}(S,T) \\right) }\n\\\\ =  \\Gamma_{\\le(I,K)}\\left(P(\\varepsilon,T') + \\delta \\Delta_\\varepsilon, S^R + \\delta Q S^R + \\delta \\d_{DR} S^R  \n \\right. \\\\\n\\left.\n+ \\delta O_{< (I,K)} (S)  \\right) + \\hbar^I \\delta O_{ (I,K)}(S,T).\n\\end{multline*}\nAs we have seen,  the first line in this expression has an $\\varepsilon \\to 0$ limit which is independent of $\\delta$.  The same is therefore true for the last line; it follows that $O_{(I,K)}(S,T)  = O_{(I,K)} (S,T')$, and so $O_{(I,K)}(S) = O_{(I,K)}(S,T)$ is local as desired.  \n\n\\end{proof} \n\nWe would like to use an inductive, term-by-term method to construct solutions to the renormalised QME.    The following lemma allows us to do so.\n\\begin{lemma}\nLet $S \\in \\mscr O_l ( \\mathscr{E}, \\Omega^\\ast(\\Delta^n) \\otimes \\mathbb C[[\\hbar]] )$ be a local action functional such that $O_{< (I,K)} (S) = 0$.  (In other words, $S$ satisfies the QME up to order $(I,K)$).  Then:\n\\begin{enumerate}\n\\item\n$$\\lim_{\\varepsilon \\to 0} O_{(I,K)}(S)$$ exists. This implies that $O_{(I,K)}(S)$ is independent of $\\varepsilon$, and so is an element\n$$\nO_{(I,K)}(S) \\in  \\mscr O_l(\\mathscr{E}, \\Omega^\\ast(\\Delta^n)  ).\n$$\n\\item\n$$(Q + \\d_{DR}) O_{(I,K)}(S)  = 0. $$ \n\\end{enumerate}\n\\label{closed local obstruction}\n\\end{lemma} \n\\begin{remark}\nThis  lemma implies that if $S$ solves the $<(I,K)$ renormalised QME, that is, $O_{<(I,K)}(S)= 0$, then lifting $S$ to a solution to the $\\le(I,K)$ renormalised QME amounts to finding some $S'\\in \\mscr O_l(\\mathscr{E}, \\Omega^\\ast(\\Delta^n))$, homogeneous of degree $K$ as a function on $\\mathscr{E}$, with  $(Q+ \\d_{DR}) S' = -  O_{(I,K)}(S)$.  The desired solution of the renormalised QME will be $S + \\hbar^I S'$. \n\\end{remark}\n\n\n\\begin{proof}\nBy definition,\n\\begin{multline*}\nO_{(I,K)}(S) =   -\\text{projection onto $\\mathscr{A}_{\\le 0}$ of }   \\\\ \\frac{\\d}{\\d \\delta} \\left<   \\Gamma_{(I,K)} \\left( P(\\varepsilon,T)+ \\Delta_\\varepsilon,  S^R + \\delta Q S^R + \\delta \\d_{DR} S^R + \\delta O_{<(I,K)}(S)   \\right)  \\right>.  \n\\end{multline*}\nEquation \\eqref{equation Q inside gamma} says that\n\\begin{multline*}\n  \\Gamma \\left( P(\\varepsilon,T)+ \\Delta_\\varepsilon,  S^R   + \\delta Q  S^R  + \\delta \\d_{DR} S^R  \\right)  \\\\ \n  = \\hbar \\log \\left(  (1 + \\delta Q + \\delta \\d_{DR} + \\delta  \\Delta_T ) \\exp ( \\hbar  \\partial_{P(\\varepsilon,T)} ) \\exp ( S^R \/ \\hbar ) \\right) .\n\\end{multline*}\nSince $O_{<(I,K)}(S) = 0$, it follows that \n\\begin{multline*}\nO_{(I,K)}(S) =   -\\text{projection onto $\\mathscr{A}_{\\le 0}$ of }  \\\\ \\hbar \\log \\left(  (1 + \\delta Q + \\delta \\d_{DR} + \\delta  \\Delta_T ) \\exp ( \\hbar  \\partial_{P(\\varepsilon,T)} ) \\exp ( S^R \/ \\hbar  ) \\right) .\n\\end{multline*}\nIt follows that the $\\varepsilon \\to 0$ limit of $O_{(I,K)}(S)$ exists.  This implies that $O_{(I,K)}(S)$ is constant as a function of $\\varepsilon$. \n\nThe equation $O_{<(I,K)}(S) = 0$ implies that \n\\begin{multline*}\n\\hbar^I O_{I,K} (S) = - \\text{leading term of }  \\\\ \\lim_{\\varepsilon \\to 0} \\frac{ \\d}{\\d \\delta}  \\left< \\hbar \\log \\left(  (1 + \\delta Q + \\delta \\d_{DR}  +\\delta T \\Delta_T ) \\exp ( \\hbar  \\partial_{P(\\varepsilon,T)} ) \\exp ( S^R \/ \\hbar ) \\right) \\right>\n\\end{multline*}\nwhere ``leading term'' means the first non-zero term in the expansion labelled by $(i,k)$, where the $(i,k)$ term, as always, is $\\hbar^i$ times a homogeneous function of degree $k$ on $\\mathscr{E}$.  We use the  lexicographical ordering on the labels $(i,k)$.   \n\n\nIt follows that\n\\begin{multline*}\n\\hbar^I O_{I,K}(S) = - \\text{leading term of }  \\\\ ( \\delta Q + \\delta \\d_{DR} +  T \\Delta_T )  \\lim_{\\varepsilon \\to 0}  \\left< \\exp ( \\hbar  \\partial_{P(\\varepsilon,T)} ) \\exp ( S^R \/ \\hbar  ) \\right>\n\\end{multline*}\nso that\n\\begin{align*}\n\\hbar^I  (Q +  \\d_{DR})  O_{I,K}(S) =& -\\text{leading term of } \\\\  & (\\delta Q + \\delta \\d_{DR} + T \\Delta_T )^2 \\lim_{\\varepsilon \\to 0}  \\left< \\exp ( \\hbar  \\partial_{P(\\varepsilon,T)} ) \\exp ( S^R \/ \\hbar  ) \\right> \n\\\\ \n  =& 0  \n\\end{align*}\nas desired.\n\n\\end{proof} \n\n\n\\section{Proof of Theorem \\ref{theorem fibration}}\nNow we can prove Theorem \\ref{theorem fibration}, that the map $\\mathbf{BV}_\\mathscr{E} \\to \\mathbf{GF}$ is a fibration of simplicial sets.\n\\subsection{Preliminary definitions}\nSending $\\Delta^n$ to  $\\Omega^\\ast(\\Delta^n)$ defines a simplicial algebra, which we denote $\\Omega^\\ast_\\Delta$.  If $X$ is a simplicial set, let $\\Omega^\\ast(X)$ denote the space of maps $X \\to \\Omega^\\ast_\\Delta$ of simplicial sets.  Concretely, an element of $\\Omega^\\ast(X)$ is something which assigns to an $n$-simplex of $X$ an element of $\\Omega^\\ast(\\Delta^n)$, in a way compatible with the natural maps between simplices. \n\n\nThe spaces \n$$\n\\mscr O_l(\\mathscr{E}, \\Omega^\\ast(\\Delta^n) )\n$$\nassemble into a simplicial object in the category of chain complexes, which we denote $\\mscr O_l(\\mathscr{E}, \\Omega^\\ast_{\\Delta}) $.  For any simplicial set $X$, let $\\mscr O_l(\\mathscr{E}, \\Omega^\\ast(X) )$ denote the space of maps $X \\to \\mscr O_l(\\mathscr{E}, \\Omega^\\ast_{\\Delta}) $.    More concretely, we can view an element of $\\mscr O_l(\\mathscr{E}, \\Omega^\\ast(X) )$ as being an element of $\\mscr O_l(\\mathscr{E}, \\Omega^\\ast(\\Delta^n) )$ for each $n$-simplex of $X$, compatible with the natural maps between simplices.  \n\n\nEarlier we defined a map \n\\begin{align*}\n\\mscr O_l(\\mathscr{E}, \\Omega^\\ast(\\Delta^n) \\otimes \\mathbb C [[\\hbar]] )  &\\to \\mscr O_l(\\mathscr{E}, \\mathscr{A}_{\\le 0} \\otimes \\Omega^\\ast(\\Delta^n)\\otimes \\mathbb C [[\\hbar]] ) \\\\\nS &\\mapsto O(S) .\n\\end{align*}\nThis gives a map of simplicial sets\n$$\n\\mscr O_l(\\mathscr{E}, \\Omega^\\ast_{\\Delta} \\otimes \\mathbb C [[\\hbar]])  \\to \\mscr O_l(\\mathscr{E}, \\mathscr{A}_{\\le 0}\\otimes \\Omega^\\ast_{\\Delta}\\otimes \\mathbb C [[\\hbar]] ).\n$$\nFor each $X$, we get an obstruction  map\n$$\nO : \\mscr O_l(\\mathscr{E}, \\Omega^\\ast(X)\\otimes \\mathbb C [[\\hbar]] )  \\to \\mscr O_l(\\mathscr{E}, \\mathscr{A}_{\\le 0} \\otimes  \\Omega^\\ast(X)\\otimes \\mathbb C [[\\hbar]]).\n$$\n$S \\in \\mscr O_l(\\mathscr{E}, \\Omega^\\ast(X) \\otimes \\mathbb C[[\\hbar]] )$ satisfies the renormalised QME if and only if $O(S) = 0$. \n\n\n$\\mathbf{BV}_{\\mathscr{E}}$ is the simplicial set whose $n$-simplices are elements $S \\in \\mscr O_l(\\mathscr{E}, \\Omega^\\ast(\\Delta^n)  ) $ such that $O(S) = 0$.  \n\n\\begin{definition}\nLet  $\\mathbf{BV}_{\\mathscr{E}, \\le(I,K)}$ be the simplicial set whose $n$-simplices are given by pairs\n$$\n(Q^{GF} ,S ) \n$$ \nwhere $Q^{GF}$ are smooth families of gauge fixing conditions, and \n$$S \\in \\mscr O_l(\\mathscr{E}, \\Omega^\\ast(\\Delta^n)  )$$ \nis such that $S_{(i,k)} = 0$ if $(i,k) > (I,K)$ and $O_{\\le (I,K)} (S) = 0$.  \n\nSimilarly, define $\\mathbf{BV}_{\\mathscr{E}, <(I,K)}$ to be the simplicial set whose $n$-simplices are pairs $(Q^{GF},  S)$ where $Q^{GF}$ are as before, and \n$S \\in \\mscr O_l(\\mathscr{E}, \\Omega^\\ast(\\Delta^n)  ) $ is such that $S_{(i,k)} = 0$ if $(i,k) \\ge (I,K)$ and $O_{< (I,K)} (S) = 0$.  \n\\end{definition}  \nSuppose $X$ is equipped with a map $X \\to \\mathbf{GF}$.  Then a lift of this to a map  $X \\to \\mathbf{BV}_{\\mathscr{E}, \\le(I,K)}$ is given by $S \\in \\mscr O_l(\\mathscr{E}, \\Omega^\\ast(X) )$ such that $S_{>(I,K)} = 0$ and $O_{\\le(I,K)} (S) = 0$.   A similar description holds for maps to $\\mathbf{BV}_{\\mathscr{E}, <(I,K)}$.  \n\nThe following lemma is an immediate corollary of Lemma \\ref{closed local obstruction}. \n\\begin{lemma}\nLet $f : X \\to \\mathbf{BV}_{\\mathscr{E}, <(I,K)}$ be a map.  Then the obstruction $O_{(I,K)}(X)$ to lifting $f$ to a map $X \\to \\mathbf{BV}_{\\mathscr{E}, \\le (I,K) }$ is independent of the parameter $\\varepsilon$, and so is an element of  $\\mscr O_l(\\mathscr{E}, \\Omega^\\ast(X) )$. This is closed, $(Q + \\d_{DR}) O_{(I,K)}(X) = 0$.  \n\nA lift of $f$ to a map $ X \\to \\mathbf{BV}_{\\mathscr{E}, \\le (I,K)}$ is given by an element $S' \\in \\mscr O_l(\\mathscr{E}, \\Omega^\\ast(X) )$, homogeneous of degree $K$ as a function on $\\mathscr{E}$, with \n$$\n(Q + \\d_{DR}) S' = - O_{(I,K)}(X) . \n$$\n\\end{lemma}\n\n\n\n\\subsection{Completion of proof} \nNote that $\\mathbf{BV}_{\\mathscr{E}, \\le (0,0) }$ is $\\mathbf{GF}$.  Thus, to prove Theorem \\ref{theorem fibration},  it suffices to show that\n\\begin{proposition}\nThe map\n$$\\mathbf{BV}_{\\mathscr{E}, \\le (I,K) } \\to \\mathbf{BV}_{\\mathscr{E}, < (I,K) }$$\nis a fibration.  \n\\label{proposition fibration}\n\\end{proposition}\n\\begin{proof}\n\n\n\n\n\nLet $\\Lambda^n \\subset \\Delta^n$ denote an $n$-horn, obtained by removing any face from $\\partial \\Delta^n$.  Suppose we have a map $\\Lambda^n \\to \\mathbf{BV}_{\\mathscr{E}, \\le (I,K) }$, and an extension of this to a map $\\Delta^n \\to \\mathbf{BV}_{\\mathscr{E}, <(I,K) }$.   We need to show that there exists a map $\\Delta^n \\to \\mathbf{BV}_{\\mathscr{E}, \\le (I,K) }$ such that the diagram\n$$\n\\xymatrix{\n\\Lambda^n \\ar[r] \\ar[d] & \\mathbf{BV}_{\\mathscr{E}, \\le (I,K) } \\ar[d]  \\\\\n\\Delta^n \\ar@{-->}[ur] \\ar[r] & \\mathbf{BV}_{\\mathscr{E},  < (I,K) }\n}\n$$\ncommutes.  There is an obstruction \n$$\nO_{I,K} ( \\Delta^n ) \\in \\mscr O_l(\\mathscr{E},  \\Omega^\\ast(\\Delta^n ) ) .\n$$\nto the existence of this lift. \n\nThere is a map $\\Lambda^n \\to \\mathbf{BV}_{\\mathscr{E}, <(I,K) }$.  The obstruction to the lift of this to a map $\\Lambda^n \\to \\mathbf{BV}_{\\mathscr{E}, \\le (I,K) }$ is an element\n$$\nO_{I,K}( \\Lambda^n ) \\in \\mscr O_l ( \\mathscr{E}, \\Omega^\\ast(\\Lambda^n) ) .\n$$\nBecause we are given a lift $\\Lambda^n \\to \\mathbf{BV}_{\\mathscr{E}, \\le (I,K) }$, we are given an element\n$$\nL_{I,K}( \\Lambda^n ) \\in \\mscr O_l ( \\mathscr{E}, \\Omega^\\ast(\\Lambda^n) ) .\n$$\nsuch that $(Q + \\d_{DR}) L_{I,K} (\\Lambda^n) = O_{I,K} (\\Lambda^n)$. \n\n\nUnder the natural map\n$$\n\\mscr O_l ( \\mathscr{E}, \\Omega^\\ast(\\Delta^n) )  \\to \\mscr O_l ( \\mathscr{E}, \\Omega^\\ast(\\Lambda^n) )\n$$\nthe restriction of $O_{I,K}(\\Delta^n)$ is $O_{I,K}(\\Lambda^n)$. \n\nThe spaces $\\mscr O_l ( \\mathscr{E}, \\Omega^\\ast(\\Delta^n))$ form a simplicial abelian group.  All simplicial groups are Kan complexes; see  \\cite{May92}.  This implies that this restriction map is surjective.  It is also a quasi-isomorphism.  (This argument was explained to me by Ezra Getzler).\n\n It follows immediately that there exists an $L_{I,K}(\\Delta^n) \\in \\mscr O_l ( \\mathscr{E}, \\Omega^\\ast(\\Delta^n) )$ which satisfies $(Q + \\d_{DR}) L_{I,K}(\\Delta^n) = O_{I,K}(\\Delta^n)$, and which restricts to $L_{I,K}(\\Lambda^n)$.  The choice of such an $L_{I,K}(\\Delta^n)$ is precisely the choice of a lift of $\\Delta^n \\to \\mathbf{BV}_{\\mathscr{E},  < (I,K) }$ to $\\mathbf{BV}_{\\mathscr{E},  \\le (I,K) }$.  The fact that $L_{I,K}(\\Delta^n)$ restricts to $L_{I,K}(\\Lambda^n)$ implies that the above diagram commutes.  \n\n\n\\end{proof} \n\n\n\\section{A local to global principle} \n\n\\label{section local global}\n\\subsection{Simplicial presheaves and the \\v{C}ech complex}\n\nLet $\\operatorname{sSets}$ denote the category of simplicial sets.   Recall that a simplicial presheaf on a topological space is simply a presheaf of simplicial sets.   Let $\\mathcal U$ be an open cover of $M$, so that $\\mathcal U$ is a topological space with a surjective open map $\\mathcal U \\to M$ which, on every connected component of $\\mathcal U$, is an open embedding.   Let \n$$\n\\mathcal U_n = \\mathcal U \\times_M  \\cdots \\times_M \\mathcal U \n$$\nwhere $\\mathcal U$ appears $n+1$ times on the right hand side. As usual, we can associate to $\\mathcal U$ a simplicial space, whose space of $n$-simplices is $\\mathcal U_n$.\n\nEvery connected component of $\\mathcal U_n$ is an open subset of $X$.  Let $G$ be a simplicial presheaf on $M$, and let \n$$G(\\mathcal U_n) = \\prod_{V \\text{ a connected component of } \\mathcal U_n} G(V) . $$\nThe simplicial sets $G(\\mathcal U_n)$ form a cosimplicial simplicial set. In other words, if $\\Delta$ denotes the category of totally ordered finite sets, there is a functor \n\\begin{align*}\n\\check{G}_\\mathcal U : \\Delta &\\to \\operatorname{sSets} \\\\\n\\check{G} _\\mathcal U[n] &= G(\\mathcal U_n) \n\\end{align*}\nwhere $[n] \\in \\operatorname{Ob} \\Delta$ is the set $\\{0,1,\\ldots,n\\}$. \n\nNext we will define a simplicial set $\\check{C}( \\mathcal U, G )$ which is the \\v{C}ech complex for the open cover $\\mathcal U$, with coefficients in $G$.  \n\nFor any simplicial sets $X,Y$, let us denote by\n$$\n\\Hom^{\\Delta}( X, Y)\n$$\nthe simplicial set of maps $X \\to Y$,  so that \n$$\n\\Hom^{\\Delta}( X, Y) [n] = \\Hom( X \\times \\Delta^n, Y) \n$$\n\nFor each face map $\\phi : \\Delta_{n-1} \\to \\Delta^n$, there are maps\n$$\n\\Hom^\\Delta (\\Delta^n , G( \\mathcal U_n ) ) \\xrightarrow{\\phi^\\ast} \\Hom^\\Delta (\\Delta_{n-1} , G( \\mathcal U_n ) ) \\xleftarrow{\\phi_\\ast} \\Hom^\\Delta (\\Delta_{n-1}  , G( \\mathcal U_{n-1} ) ).\n$$\nThe first arrow is simply composition with $\\phi \\times \\operatorname{Id}$, and the second arrow comes from the cosimplicial set structure on the simplicial sets $G(\\mathcal U_n)$.  \n\nLet \n$$\n\\check{C}( \\mathcal U, G ) \\subset  \\prod_n \\Hom^\\Delta (\\Delta^n , G( \\mathcal U_n ) )\n$$\nbe the sub-simplicial set of those elements which are compatible with the face maps.  More precisely, an element $f \\in \\check{C}( \\mathcal U, G ) [m] $ is a sequence of elements    \n$$f_n \\in \\Hom (\\Delta^n \\times  \\Delta^m , G( \\mathcal U_n ) )$$ for each $n \\ge 0$, such that  for each face map $\\phi : \\Delta_{n-1} \\to \\Delta^n$, \n$$\n\\phi^\\ast f_n = \\phi_\\ast f_{n-1} \\in  \\Hom ( \\Delta_{n-1} \\times \\Delta^m , G( \\mathcal U_n ) ).\n$$\nNote that we \\emph{do not} require compatibility with degeneracy maps.  The simplicial set $\\check{C}(\\mathcal U, G)$ is a version of the total complex of the cosimplicial simplicial set defined by $G(\\mathcal U_n)$ (see \\cite{BouKan72}).\n\nSimilarly, let \n$$\n\\check{C}_{\\le N}( \\mathcal U, G ) \\subset \\prod_{n = 0}^N \\Hom^\\Delta (\\Delta^n , G( \\mathcal U_n ) )\n$$\nbe the sub-simplicial set of those elements compatible with face maps $\\Delta_{n-1} \\to \\Delta_{n}$, for all $0 \\le   n \\le N$.  Note that \n$$\n\\check{C}( \\mathcal U, G ) =  \\varprojlim _N \\check{C}_{\\le N}( \\mathcal U, G )\n$$\n\n\nThe simplicial set $\\check{C}( \\mathcal U, G )$ is the non-linear \\v{C}ech complex for the open cover $\\mathcal U$, with values in the simplicial presheaf $G$.  As with ordinary \\v{C}ech cohomology, we will define the simplicial set of derived global sections of $G$ as a limit over all open covers.  That is, let \n$$\n\\mbb R \\Gamma (M , G ) = \\operatorname{colim}_{\\mathcal U} \\check{C} ( \\mathcal U,G)  .  \n$$\n\\begin{definition}\nA simplicial presheaf $G$ is \\emph{locally fibrant} if for there exists an open cover $\\{U_i\\}$ of $M$, such that for all $i$ and all open sets $W \\subset U_i$, $G(W)$ is a Kan complex (i.e.\\ a fibrant simplicial set). \n\\end{definition}\n\nWe will now see that our definition of $\\mbb R \\Gamma(M,G)$ is well-behaved as long as $G$ is locally fibrant.\n\\begin{lemma}\nIf $G$ is locally fibrant, then the map\n$$\n\\check{C}_{\\le N}( \\mathcal U, G ) \\to \\check{C}_{\\le N-1}( \\mathcal U, G )\n$$\nis a fibration for sufficiently fine open covers $\\mathcal U$.\n\\end{lemma}\n\n\\begin{proof}\nWe  can assume that $G(W)$ is fibrant for all open sets $W$ contained inside some element of the cover $\\mathcal U$. \n\nSuppose we have a map $\\Delta^n \\to \\check{C}_{\\le N-1}( \\mathcal U, G )$, and a map $\\Lambda^n \\to \\check{C}_{\\le N}( \\mathcal U, G )$ lifting the composition $\\Lambda^n \\to \\Delta^n \\to \\check{C}_{\\le N-1}( \\mathcal U, G )$.  We need to construct a lift $\\Delta^n \\to \\check{C}_{\\le N}( \\mathcal U, G )$.\n\nSuch a lift is given by a map $\\Delta^n \\to \\Hom^\\Delta(\\Delta_N , G ( \\mathcal U_N ) )$, in other words, a map $\\Delta_N \\times \\Delta^n \\to G( \\mathcal U_N)$.    This map is known on $\\partial \\Delta_N \\times \\Delta^n$ and on $\\Delta_N \\times \\Lambda^n$.  Indeed, the definition of $\\check{C}_{\\le N}( \\mathcal U, G )$ implies that for each face $\\Delta_{N-1} \\to \\Delta_N$, this map must restrict to the given  map $\\Delta^n \\to \\Hom^\\Delta( \\Delta_{N-1}, G( \\mathcal U_{N-1})$.  On $\\Delta_N \\times \\Lambda^n$, this map must restrict to the given map $\\Lambda^n \\to \\Hom^\\Delta(\\Delta_N , G ( \\mathcal U_N ) )$.  \n\nThus, we are left with finding an extension of a map \n$$\n\\left( \\partial \\Delta_N \\times \\Delta^n \\right) \\amalg_{\\partial \\Delta_N \\times \\Lambda^n} \\Delta_N \\times \\Lambda^n \\to G ( \\mathcal U_N ) \n$$\nto a map \n$$\n\\Delta_N \\times \\Delta^n \\to G ( \\mathcal U_N ) . \n$$\nThe map\n$$\n\\left( \\partial \\Delta_N \\times \\Delta^n \\right) \\amalg_{\\partial \\Delta_N \\times \\Lambda^n} \\Delta_N \\times \\Lambda^n  \\to \\Delta_N \\times \\Delta^n \n$$\nis a cofibration and a weak equivalence.    Since $G ( \\mathcal U_N ) $ is fibrant, the desired extension exists.  \n\\end{proof} \n\n\\begin{lemma}\nLet $G, G'$ be a locally fibrant simplicial presheaves on $M$, and let $\\mathcal U$ be a sufficiently fine open cover of $M$.  Let $G \\to G'$ be a map of simplicial presheaves such that the maps $G(\\mathcal U_n) \\to G'(\\mathcal U_n)$ are all weak equivalence. \n\nThen the map\n$$\n \\check{C} ( \\mathcal U,G)   \\to  \\check{C} ( \\mathcal U,G')    \n$$\nis a weak equivalence. \n\n\\label{lemma local weak equivalence}\n\\end{lemma}\n\\begin{remark}\nThis lemma is standard in the theory of cosimplicial spaces, see \\cite{BouKan72}.\n\\end{remark}\n\n\\begin{proof}\nThe long exact sequence for the homotopy groups of a fibration implies that the maps $\\check{C}_{\\le N} ( \\mathcal U,G)   \\to  \\check{C}_{\\le N} ( \\mathcal U,G')$ are weak equivalences for all $N$.  The result follows easily from this. \n\\end{proof} \n\\begin{lemma}\nLet $G, G'$ be locally fibrant simplicial presheaves on $M$.  Let $G \\to G'$ be a map of simplicial presheaves such that, for all sufficiently small open balls $B$ in $M$, the map $G (B) \\to G'(B)$ is a weak equivalence.\n\nThen the map $\\mbb R \\Gamma(M, G) \\to \\mbb R \\Gamma(M,G') $ is a weak equivalence.\n\\end{lemma}\n\\begin{proof}\nRecall that a \\emph{good cover} of $M$ is a cover all of whose iterated intersections are open balls.  Every cover of $M$ has a refinement by a good cover.  \n\nIf $\\mathcal U \\to M$ is a sufficiently fine good cover, then $\\check{C}(\\mathcal U, G ) \\to \\check{C}(\\mathcal U, G')$ is a weak equivalence, by the previous lemma. \n\nThe colimit over open covers $\\mathcal U$ we use to define $\\mbb R \\Gamma(M,G)$ is a filtered colimit.  Homotopy groups commute with filtered colimits.    It follows immediately that the map $\\mbb R \\Gamma(M,G) \\to \\mbb R \\Gamma(M,G')$ is a weak equivalence.  \n\\end{proof} \n\n\\begin{corollary}\nLet $G$ be a simplicial presheaf such that for all sufficiently small open balls $U$ in $M$, $G(U)$ is contractible.  Then $\\mbb R \\Gamma(M,G)$ is contractible.\n\\label{corollary locally contractible}\n\\end{corollary}\n\\begin{proof}\nThe hypothesis, together with Lemma \\ref{lemma local weak equivalence}, implies that the map $\\mbb R \\Gamma(M,G) \\to \\mbb R \\Gamma(M,\\ast)$ is a weak equivalence, where $\\ast$ is the constant simplicial presheaf on $M$ with value a point.  For any open cover $\\mathcal U$ of $M$, $\\check{C}(\\mathcal U, \\ast) = \\ast$, so $\\mbb R \\Gamma(\\mathcal U, \\ast) = \\ast$.\n\\end{proof} \n\n\\subsection{Simplicial presheaf of gauge fixing conditions}\nIn order to construct a simplicial presheaf of gauge fixing conditions, we need some extra data, which is typically present in applications.    \n\nThe extra data consists of:\n\\begin{enumerate}\n\\item\nA smooth, locally trivial fibre bundle $F$ over $M$, whose fibres are diffeomorphic to $\\mbb R^k$ for some $k$.\n\\item\nA map $D : F \\to \\operatorname{Diff}^{\\le 1} (E)$ of fibre bundles over $M$.\n\\item\nIf $f : U \\to  F\\mid_U$ is any local section, defined over some open set $U$ in $M$, then \n$$\nD(f) : \\Gamma_c(U, E) \\to \\Gamma_c(U,E)\n$$\nmust be an odd operator of square zero, which is self adjoint with respect to the symplectic pairing, and which is such that $[Q, D(f)]$ is a generalised Laplacian.  Here $\\Gamma_c$ denotes sections with compact support.\n\\item\nIf $f : M \\to F$ is a global section of $F$, then $D(f)$ satisfies the axioms of a gauge fixing condition: in addition to the properties already mentioned, this means that there is a direct sum decomposition\n$$\n\\mathscr{E} = \\Gamma(M,E) = \\Im Q \\oplus \\Ker [ D(f), Q ] \\oplus \\Im D(f) . \n$$\n\\end{enumerate}\n\nIt is a little \\emph{ad hoc} to require the existence of such a bundle $F$.  It would be better to construct a bundle of local gauge fixing conditions directly from the data $(\\mathscr{E},Q, \\ip{\\ , \\ })$.    However, I couldn't see a clean way of doing this.   In all examples, one is naturally given an $F$ satisfying these properties.     \n\nFor example, in Chern-Simons theory, $F$ will be the bundle of metrics on the tangent bundle of $M$.  This satisfies these conditions.  Once we fix some metric, we can identify the fibre of the bundle of metrics on the tangent bundle is $GL(n) \/ SO(n)$, which is diffeomorphic to $\\mbb R^{n(n+1)\/2}$.  \n\nAssociated to the bundle $F$ we have a simplicial presheaf $\\mathbf{F}$  on $M$, such that \n$$\\mathbf{F}(U)[n] = \\text{smooth sections of }  F \\vert_U \\times \\Delta^n \\to U \\times \\Delta^n.$$\nThere is a map\n$$\n\\Gamma(M,\\mathbf{F} )  \\to \\mathbf{GF} .\n$$\nThe simplicial presheaf $\\mathbf{F}$ has the following properties.\n\\begin{lemma}\n\\label{lemma properties sheaf gf}\n\\begin{enumerate}\n\\item\n$\\mathbf{F}$ is locally fibrant. \n\\item\nBoth $\\Gamma(M, \\mathbf{F})$ and $\\mbb R \\Gamma(M, \\mathbf{F})$ are contractible.\n\\item\nThe sheaves of sets $U \\mapsto \\mathbf{F}(U)[n]$ are soft. \n\\end{enumerate}\n\\end{lemma}\n\nRecall that a soft sheaf $G$ is one such that for all closed sets $C \\subset M$, the map $G(M) \\to G(C)$ is surjective. \n\n\\begin{proof}\nI will give a proof of the first statement, that $\\mathbf{F}$ is locally fibrant.  The other assertions are straightforward to prove. It suffices to show that for all open subsets $U \\subset M$ on which the bundle $F$ is trivial, the simplicial set $\\mathbf{F}(U)$ is fibrant.\n\nTrivialising $F$ on $U$ gives a diffeomorphism $F \\mid_U \\cong U \\times \\mbb R^{k}$.   Thus, the simplicial presheaf $\\mathbf{F}$ is isomorphic to the simplicial presheaf whose $n$-simplices, on an open set $U$, are smooth maps $U \\times \\Delta^n \\to \\mbb R^k$.  Therefore the simplicial set $\\mathbf{F}(U)$ admits the structure of simplicial abelian groups.  All simplicial abelian groups are Kan complexes, thus $\\mathbf{F}(U)$ is a Kan complex.\n\n\n\\end{proof} \n\n\n\n\\subsection{The sheaf of local functionals}\nWe want to talk about sheaves of solutions to the renormalised quantum master equation.  In order to do this, we need to first define sheaves of local functionals.  Throughout this section, we will work with functions \\emph{modulo constants}.  \n\n\\begin{definition}\nLet $\\mscr O(E)$ denote the presheaf on $M$ whose value on an open set $U$ is \n$$\n\\Gamma(U, \\mscr O(E)) = \\prod_{n \\ge 1} \\Hom ( \\Gamma_c(U,E)^{\\otimes n}, \\mathbb C ) _{S_n}. \n$$\nHere, we work with functions modulo constants, so we take $n \\ge 1$. $\\Hom$ denotes continuous linear maps,  the subscript $S_n$ denotes coinvariants, and the subscript $c$ denotes sections with compact support.\n\\end{definition}\nThus,\n$$\n\\Gamma(M, \\mscr O(E)) = \\mscr O(\\mathscr{E})  \/ \\mathbb C . \n$$\nIn a similar way, if $X$ is an auxiliary manifold, there is a presheaf $\\mscr O(E, C^{\\infty}(X) )$  on $M$,  such that\n$$\n\\Gamma(U, \\mscr O(E,  C^{\\infty}(X)) = \\prod_{n \\ge 1} \\Hom ( \\Gamma_c(U,E)^{\\otimes n}, C^{\\infty}(X) ) _{S_n}. \n$$\n\nRecall that  polydifferential operators $\\mathscr{E}^{\\otimes n} \\to \\dens(M)$ can be identified with global sections of the infinite rank vector bundle\n$$\n\\Diff(E,\\mathbb C)^{\\otimes n} \\otimes \\dens(M)\n$$\non $M$. Here $\\mathbb C$ denotes the trivial bundle, $\\Diff$ is the sheaf of differential operators between two vector bundles, and tensor products are fibrewise tensor products of vector bundles.  \n\\begin{definition}\nLet $\\mscr O_l (E)\\subset \\mscr O(E)$ be the sub-presheaf given locally by the image of \n$$\n\\prod_{n \\ge 1} \\Gamma(U,  \\Diff(E,\\mathbb C)^{\\otimes n} \\otimes \\dens_M ) \\xrightarrow{\\int}  \\prod_{n \\ge 1} \\Hom ( \\Gamma_c(U,E)^{\\otimes n}, C^{\\infty}(X) ) _{S_n} .\n$$\n\\end{definition}\nThus, \n$$\n\\Gamma(M , \\mscr O_l (E) ) = \\mscr O_l(\\mathscr{E}) \/ \\mathbb C .\n$$\nIn a similar way, we can define the sheaf $\\mscr O_l (E ,  C^{\\infty}(X) )$ of local functionals with values in $C^{\\infty}(X)$, or in $\\Omega^\\ast(X)$, if $X$ is an auxiliary manifold (with corners).\n\\begin{lemma}\nThe presheaf $\\mscr O_l (E, \\Omega^\\ast(X))$  is a soft sheaf. \n\\end{lemma}\n\n\\subsection{Simplicial presheaf of solutions to the renormalised QME}\n\nProposition \\ref{proposition obstruction} shows that if we have $S \\in \\mscr O_l(\\mathscr{E}, \\Omega^\\ast(\\Delta^n)\\otimes \\mathbb C [[\\hbar]])$, and a smooth family of gauge fixing conditions $Q^{GF}$, parametrised by $\\Delta^n$, then there are obstructions \n$$O_{(I,K)}(S, Q^{GF} )\\in \\mscr O_l(\\mathscr{E}, \\mathscr{A}_{\\le 0} \\otimes \\Omega^\\ast(\\Delta^n))$$\n to $S$ satisfying the renormalised quantum master equation. These obstructions are homogeneous of order $K$; thus, if $K > 0$, they are elements of $\\Gamma(M, \\mscr O_l (E, \\mathscr{A}_{\\le 0 } \\otimes \\Omega^\\ast(\\Delta^n) ))$.  It is easy to see that the $O_{(I,K)}$, when $K > 0$, don't depend on the constant term of $S$.  Thus, the obstructions give us maps \n$$\nO_{(I,K)}^{global} : \\Gamma(M, \\mscr O_l(E, \\Omega^\\ast(\\Delta^n )   \\otimes \\mathbb C [[\\hbar]] ) \\times \\mathbf{F} [n] ) \\to \\Gamma(M, \\mscr O_l(E, \\mathscr{A}_{\\le 0}\\otimes \\Omega^\\ast(\\Delta^n) ) ).\n$$ \n\\begin{lemma}\nFor all $I \\ge 0$, $K > 0$ and $n \\ge 0$, there are unique maps of simplicial presheaves \n$$O_{(I,K)} :  \\mscr O_l(E, \\Omega^\\ast(\\Delta^n )   \\otimes \\mathbb C [[\\hbar]] )  \\times \\mathbf{F}  [n] \\ \\to \\mscr O_l(E, \\mathscr{A}_{\\le 0} \\otimes \\Omega^\\ast(\\Delta^n) )  $$ such that the diagram\n$$\n\\xymatrix{  \\Gamma(M,  \\mscr O_l(E, \\Omega^\\ast(\\Delta^n )   \\otimes \\mathbb C [[\\hbar]] ) \\times \\mathbf{F} [n]  )    \\ar[r]^>>>>>{O_{(I,K)}^{global}} \\ar[d] &   \\Gamma(M, \\mscr O_l(E, \\mathscr{A}_{\\le 0} \\otimes \\Omega^\\ast(\\Delta^n) ) )  \\ar[d] \\\\ \n\\Gamma(U,  \\mscr O_l(E, \\Omega^\\ast(\\Delta^n )   \\otimes \\mathbb C [[\\hbar]] )   \\times \\mathbf{F} [n])    \\ar[r]^>>>>>{O_{(I,K)}(U)}  &   \\Gamma(U, \\mscr O_l(E, \\mathscr{A}_{\\le 0}\\otimes \\Omega^\\ast(\\Delta^n) ) )  }.\n$$\ncommutes, for all open sets $U \\subset M$. \n\\end{lemma}\n\\begin{proof}\nThe sheaf $ \\mscr O_l(E, \\Omega^\\ast(\\Delta^n )   \\otimes \\mathbb C [[\\hbar]] )  \\times \\mathbf{F}  [n]$ is soft.   This implies that for all open sets $U \\subset M$, and all sections $(S,f) \\in \\Gamma(U,  \\mscr O_l(E, \\Omega^\\ast(\\Delta^n )   \\otimes \\mathbb C [[\\hbar]] )  \\times \\mathbf{F}  [n] \\ ) $, there exists an open cover of $U$ by sets $\\{V_i\\}$, and global sections $(S_i,f_i )$, such that $(S_i,f_i) \\mid_{V_i} = (S,f) \\mid_{V_i}$.   \n\nThis implies uniqueness.   For existence, we need to show that $O_{(I,K)}^{global}(S_i,f_i) \\mid_{V_i}$ only depends on $(S_i,f_i) \\mid {V_i}$. This follows from the third clause of Proposition \\ref{proposition obstruction}. \n\n\\end{proof} \n\n\n\n\\begin{lemma}\n\n\nLet $S \\in \\Gamma(U,   \\mscr O_l(E, \\Omega^\\ast(\\Delta^n )   \\otimes \\mathbb C [[\\hbar]] ) \\times \\mathbf{F}  [n])$ be such that $O_{(i,k)}(S)=  0$ for all $(i,k) < (I,K)$.  Then \\begin{enumerate}\n\\item\n$O_{(I,K)}(S)$ is independent of $\\varepsilon$, so it is a section of $\\mscr O_l(E, \\Omega^\\ast(\\Delta^n ) )$. \n\\item\n$$\n(Q + \\d_{DR} )  O_{(I,K)}(S) = 0. \n$$\n\\item\nIf $S'\\in \\Gamma(U, \\mscr O_l(E, \\Omega^\\ast(\\Delta^n )  )$ is homogeneous of degree $K$, then \n$$\nO_{(I,K)} ( S + \\hbar^I S' ) = O_{(I,K)} (S) + (Q + \\d_{DR} ) S'.\n$$\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nThe proof is identical to that of Lemma \\ref{closed local obstruction}. \n\\end{proof} \n\n\\begin{definition}\nLet $\\mathbf{BV}$ be the simplicial presheaf on $M$ such that   $\\Gamma(U, \\mathbf{BV}[n] )$ is the set of $$(S,f) \\in \\Gamma(U,  \\mscr O_l(E, \\Omega^\\ast(\\Delta^n )   \\otimes \\mathbb C [[\\hbar]] )  \\times \\mathbf{F}[n])$$  such that $$O_{(I,K)}(S,f) = 0$$ for all $I \\ge 0,K > 0$.\n\nLet $\\mathbf{BV}_{\\le (I,K)}$  be the simplicial presheaf such that $\\Gamma(U, \\mathbf{BV}[n] )$ is the set  \n$$(S,f) \\in \\Gamma(U,  \\mscr O_l(E, \\Omega^\\ast(\\Delta^n )   \\otimes \\mathbb C [[\\hbar]] )  \\times \\mathbf{F}[n]  )$$ \nsuch that\n\\begin{enumerate}\n\\item\n$S_{(i,k)} = 0$ if $(i,k) > (I,K)$, where, as usual, $S_{(i,k)}$ is the part homogeneous of degree $(i,k)$ as a function of $(\\hbar, e \\in \\Gamma_c(U,E) )$.   \n\\item\n$O_{(i,k)}(S,f) = 0$ for $(i,k) \\le (I,K)$.\n\\end{enumerate}\n\nLet $\\mathbf{BV}_{< (I,K)}$  be the simplicial presheaf such that $\\Gamma(U, \\mathbf{BV}[n] )$ is the set  \n$$(S,f) \\in \\Gamma(U, \\mscr O_l(E, \\Omega^\\ast(\\Delta^n )   \\otimes \\mathbb C [[\\hbar]] )  \\times \\mathbf{F}[n] )$$ \nsuch that\n\\begin{enumerate}\n\\item\n$S_{(i,k)} = 0$ if $(i,k) \\ge (I,K)$.\n\\item\n$O_{(i,k)}(S,f) = 0$ for $(i,k) < (I,K)$.\n\\end{enumerate}\n\n\\end{definition}\n\n\n\n\\begin{lemma}\n$\\mathbf{BV}$ is a locally fibrant simplicial presheaf.\n\\end{lemma}\n\\begin{proof}\nIt suffices to show that  $\\Gamma ( U , \\mathbf{BV}) $ is a fibrant simplicial set, for all open subsets $U$ of $M$ on which the bundle $F$ is trivial. Since we know $\\Gamma(U, \\mathbf{F})$ is fibrant, it suffices to show that the map $\\Gamma(U, \\mathbf{BV} ) \\to \\Gamma(U, \\mathbf{F})$ is a fibration. The proof is identical to that of Theorem \\ref{theorem fibration}.\n\\end{proof} \n\nThis implies that our definition of $\\mbb R \\Gamma (M ,\\mathbf{BV})$ is well-behaved. \n\n\n\\begin{definition}\nThe operator $Q$ on the sheaf of sections of $E$ is called \\emph{triangular} if \n$E$ admits a decomposition as a direct sum of vector bundles  \n$$E = E_0 \\oplus E_1 \\oplus \\cdots \\oplus E_n$$ such that the operator $Q$ takes sections of $E_i$ to sections of $\\oplus_{j > i} E_j$.\n\\end{definition}\n\n\\begin{thmD}\nSuppose $Q$ is triangular.\nThen the natural map\n$$\n\\Gamma(M, \\mathbf{BV} ) \\to \\mbb R \\Gamma(M, \\mathbf{BV} ) \n$$\nis a weak equivalence. \n\\end{thmD}\nThis theorem I will call the ``local to global principle''.  This result allows one to construct global solutions to the quantum master equation from local local solutions. The local solutions will typically be constructed in explicit model cases, like $\\mbb R^n$ with the flat metric, as we will see below for the case of Chern-Simons theory. \n\n\n\\subsection{Proof of theorem D} \nThere is a simplicial presheaf on $M$ whose presheaf of $n$-simplices is\n$$\n\\Gamma(U, \\mscr O_l(E, \\Omega^\\ast(\\Delta^n ) ) .\n$$\nLet us denote this simplicial presheaf by $\\mscr O_l^{\\Delta}(E)$.  Note that\n$$\n\\mathbf{BV} \\subset \\mscr O_l^{\\Delta}(E) \\times \\mathbf{F} \n$$\nis a sub-simplicial presheaf. \n\n\\begin{lemma}\nLet $X$ be a simplicial set, and let $S_X : X \\to \\mbb R \\Gamma(M, \\mathbf{BV}_{< (I,K)})$ be a map. Then there is an obstruction \n$$\nO(S_X) \\in \\operatorname{Hom} ( X, \\mbb R \\Gamma(M, \\mscr O_l^{\\Delta} (E) ) )\n$$\nwhich is closed, $(Q + \\d_{DR} )O(S_X) = 0$, and homogeneous of degree $K$.  The simplicial set of lifts $X \\to \\mbb R \\Gamma(M, \\mathbf{BV}_{\\le (I,K)})$ is isomorphic to the presheaf of maps $S_X' : X \\to \\mbb R \\Gamma(M, \\mscr O_l^{\\Delta}(E))$, homogeneous of degree $K$, satisfying\n$$\n(Q + \\d_{DR} ) S'_X = O(X) . \n$$\n\\end{lemma}\n\\begin{proof}\nFix an open cover $\\mathcal U$ of $M$.  First, we will prove the corresponding statement for lifts of maps\n$$\nS_X : X \\to \\check{C}(\\mathcal U, \\mathbf{BV}_{<(I,K)}).\n$$\nSuch a map is given by maps\n$$\nS_X^n : X \\times \\Delta^n \\to \\Gamma(\\mathcal U_n,  \\mathbf{BV}_{<(I,K)} )\n$$\nfor all $n$, which are compatible with the face maps $\\Delta^m \\to \\Delta^n$, $\\mathcal U_m \\to \\mathcal U_n$, for $m < n$. \n\nThe obstructions to lifting $S_X^n$ to $\\mathbf{BV}_{\\le (I,K)}$ are elements\n$$\nO(S_X, \\mathcal U_n ) : X \\times \\Delta^n \\to \\Gamma(\\mathcal U_n, \\mscr O_l^\\Delta(E) ).\n$$\nNaturality of these obstructions implies that they fit together into an element\n$$\nO(S_X, \\mathcal U) : X \\to \\check{C}(\\mathcal U, \\mscr O_l^\\Delta(E) ).\n$$\n\nThere is a bijection between lifts of each $S_X^n$ and ways of making the obstruction $O(S_X, \\mathcal U_n)$ exact.  Thus, there is a corresponding bijection between lifts of $S_X$ and ways of making the total obstruction $O(S_X, \\mathcal U)$ exact. Of course, the map $S' : X \\to \\check{C}(\\mathcal U, \\mscr O_l^\\Delta(E) )$ making $O(S_X, \\mathcal U)$ exact must be homogeneous of degree $K$.\n\nThe obstruction $O(S_X, \\mathcal U)$ is natural with respect to restrictions to finer open covers $\\mathcal V$.  Thus, taking the direct limit, we get an element\n$$\nO(S_X) = \\varinjlim_{\\mathcal U} O(S_X, \\mathcal U) \\in \\varinjlim_{\\mathcal U} \\check{C}(\\mathcal U, \\mscr O_l^\\Delta(E) ) = \\mbb R \\Gamma ( M , \\mscr O_l^\\Delta(E) ).\n$$\nAgain, naturality of the obstruction implies that lifts $X \\to \\mbb R \\Gamma(M, \\mathbf{BF}_{\\le (I,K)})$ correspond to ways of making $O(S_X)$ exact.\n\\end{proof} \n\\begin{proposition}\nThe map\n$$\n\\mbb R \\Gamma (M, \\mathbf{BV}_{\\le (I,K)} ) \\to \\mbb R \\Gamma ( M, \\mathbf{BV}_{< (I,K)} )\n$$\nis a fibration.\n\\end{proposition}\n\\begin{proof}\nThe proof is exactly the same as that of Proposition \\ref{proposition fibration},\nusing  the obstruction to lifting maps  to $ \\mbb R \\Gamma ( M, \\mathbf{BV}_{< (I,K)} )$ to maps to  $\\mbb R \\Gamma (M, \\mathbf{BV}_{\\le (I,K)} )$, \\end{proof} \n\nNow we will prove the theorem by induction. The starting point in the induction is the statement that the map\n$$\n\\Gamma(M, \\mathbf{BV}_{(0,0)} ) \\to \\mbb R \\Gamma (M, \\mathbf{BV}_{(0,0)})\n$$\nis a weak equivalence. Note that $\\mathbf{BV}_{(0,0)} = \\mathbf{F}$.  Lemma \\ref{lemma properties sheaf gf} implies that the map $\\Gamma(M, \\mathbf{F}) \\to \\mbb R \\Gamma(M, \\mathbf{F} )$ is a weak equivalence. \n\nWe will assume by induction that the map\n$$\n\\Gamma(M, \\mathbf{BV}_{< (I,K)} )\\to \\mbb R \\Gamma (M, \\mathbf{BV}_{<(I,K)} )\n$$\nis a weak equivalence.\n\nConsider the diagram\n$$\n\\xymatrix{  \\Gamma (M, \\mathbf{BV}_{\\le (I,K)} ) \\ar[r] \\ar[d]^{\\pi} &\\mbb R \\Gamma ( M, \\mathbf{BV}_{\\le (I,K)} )  \\ar[d]^{\\mbb R \\pi} \\\\ \n \\Gamma (M, \\mathbf{BV}_{< (I,K)} ) \\ar[r] &  \\mbb R \\Gamma ( M, \\mathbf{BV}_{< (I,K)} ) .  } \n$$\nBy Proposition \\ref{proposition fibration}, \tthe map $\\pi$ is a fibration, and we have just seen that $\\mbb R \\pi$ is also a fibration.  By induction, the bottom horizontal arrow is a weak equivalence.    By considering the long exact sequence of homotopy groups of a fibration, it suffices to show that the fibres of $\\pi$ and of $\\mbb R \\pi$ are weakly homotopy equivalent. \n\nThus, fix a base point\n$$\nx_0 \\in  \\Gamma (M, \\mathbf{BV}_{< (I,K)} )[0] .\n$$\nIt remains to show that the map\n$$\n\\pi^{-1}(x_0) \\to \\mbb R \\pi^{-1} (x_0)\n$$\nis a weak equivalence. \n\n\n\n\n\nThere is an obstruction\n$$\nO_{(I,K)}(x_0) \\in \\Gamma(M, \\mscr O_l(E) )\n$$\nto lifting $x_0$ to $\\Gamma (M, \\mathbf{BV}_{\\le (I,K)} ) $.  This is a closed element, $Q O_{(I,K)}(x_0) = 0$. \n\nWe can identify $\\pi^{-1}(x_0)$ with the simplicial set whose $n$-simplices are elements of \n$$\\phi \\in \\Gamma(M, \\mscr O_l^{\\Delta}(E) ) [n] = \\Gamma(M, \\mscr O_l(E, \\Omega^\\ast(\\Delta^n) ) )$$\nwhich are homogeneous of degree $K$ as functionals on $\\Gamma(M,E)$, and which satisfy\n$$\n(Q + \\d_{DR}) \\phi = O_{(I,K)}(x_0).\n$$ \n\nSimilarly, we can identify $\\mbb R \\pi^{-1}(x_0)$ with the simplicial set whose $n$-simplices are\n$$\n\\phi \\in  \\mbb R \\Gamma(M, \\mscr O_l^{\\Delta}(E)  ) [n]\n$$\nwhich are homogeneous of degree $K$ and satisfy $(Q + \\d_{DR}) \\phi = O_{(I,K)}(x_0)$.    \n\n\n\\begin{lemma}\nThe natural map of chain complexes\n$$\n(\\Gamma(M, \\mscr O_l(E) ) , Q)  \\to  ( \\mbb R \\Gamma(M, \\mscr O_l^{\\Delta}(E) )   [0] , Q + \\d _{DR} )  \n$$\nis a quasi-isomorphism. \n\\label{lemma linear cech}\n\\end{lemma}\n\n\\begin{proof}\nIt suffices to show that for all open covers $\\mathcal U$ of $M$, the natural map \n$$\n( \\Gamma(M, \\mscr O_l(E) ) , Q) \\to ( \\check{C}(\\mathcal U, \\mscr O_l^{\\Delta} (E) ) [0] , Q + \\d_{DR} )\n$$\nis an isomorphism.\n\n The triangular nature of the differential $Q$ on $E$ implies that the complex of sheaves $\\mscr O_l(E)$ on $M$ admits a grading,  labelled by non-negative integers, such that $Q$ is of strictly negative degree.   This grading is inherited by all auxiliary complexes, such as $\\check{C}(\\mathcal U, \\mscr O_l^{\\Delta} (E) ) [0]$ and $\\Gamma(M, \\mscr O_l(E) ) $.  It follows from a spectral sequence argument that it suffices to show that the map of complexes\n$$\n( \\Gamma(M, \\mscr O_l(E) ) , 0) \\to ( \\check{C}(\\mathcal U, \\mscr O_l^{\\Delta} (E) ) [0] ,  \\d_{DR} )\n$$\nis a quasi-isomorphism, where the complex on the left hand side now has zero differential.\n\nThe complex $\\check{C}(\\mathcal U, \\mscr O_l^{\\Delta} (E) ) [0]$, with the differential $\\d_{DR}$, is quasi-isomorphic to the ordinary  \\v{C}ech complex with coefficients in sheaf $\\mscr O_l(E)$ (with zero differential).  This sheaf is soft. Thus, its  \\v Cech cohomology is the same as its global sections, which is the statement that the map above is a quasi-isomorphism.\n\n\\end{proof} \n\n\n\n\n\\begin{corollary}\n$\\pi^{-1}(x_0)$ is empty if and only if $\\mbb R \\pi^{-1}(x_0)$ is empty.\n\\end{corollary}\n\\begin{proof}\nIndeed, $\\pi^{-1}(x_0)$ is empty if and only if the cohomology class\n$$\n[\\phi ] \\in H^\\ast (   \\Gamma(M, \\mscr O_l(E) ), Q ) \n$$\nis non-zero, and similarly for $\\mbb R \\pi^{-1}(x_0)$. \n\\end{proof}\n\nLet us use the temporary notation $\\mscr O_{l,K}(E) \\subset \\mscr O_{l}(E)$ for the sub-sheaf of functionals which are homogeneous of degree $K$. Also let $\\mscr O_{l,K}^\\Delta(E) \\subset \\mscr O_{l}^\\Delta(E)$  be the sub-simplicial presheaf of $\\mscr O_{l}^\\Delta(E)$ consisting of those functionals  which are homogeneous of degree $K$. Thus, a section of the presheaf of $n$-simplices of $\\mscr O_{l,K}^\\Delta(E)$ is given by an element of $\\Gamma(U, \\mscr O_{l,K}(E, \\Omega^\\ast(\\Delta^n)) )$.   $\\mscr O_{l,K}^{\\Delta}(E)$ is a simplicial presheaf of $\\mathbb Z\/2$-graded complexes, with differential $Q + \\d_{DR}$. \n\n\nLet us suppose that $\\pi^{-1}(x_0)$ is non-empty.  (Otherwise, we are done, as $\\mbb R \\pi^{-1}(x_0)$ will also be empty).  Then by picking a base point in $\\pi^{-1}(x_0)$, we can identify $\\pi^{-1}(x_0)$ with the simplicial set whose $n$-simplices are closed, even elements of $\\Gamma(M, \\mscr O_{l,K}^\\Delta(E) )[n]$.   Let us denote this simplicial set by $\\Gamma(M, \\mscr O_{l,K}^\\Delta(E) )^{closed, even}$.\n\nBy taking the corresponding base point in $\\mbb R \\pi^{-1}(x_0)$, we can identify $\\mbb R \\pi^{-1}(x_0)$ with the simplicial set whose $n$-simplices are closed even elements of $\\mbb R \\Gamma(M, \\mscr O_{l,K}^\\Delta(E) )[n]$.  Let us denote this simplicial set by $\\mbb R \\Gamma(M, \\mscr O_{l,K}^\\Delta(E) )^{closed, even}$.\n\n\nThus, to complete the proof of the theorem, it suffices to show that the map\n$$\n\\Gamma(M, \\mscr O_{l,K}^\\Delta(E) )^{closed, even} \\to \\mbb R \\Gamma(M, \\mscr O_{l,K}^\\Delta(E) )^{closed, even}\n$$\nis a weak equivalence.  This will follow from the following lemma, combined with Lemma \\ref{lemma linear cech}.\n\\begin{lemma}\nThe homotopy groups of the simplicial sets $\\Gamma(M, \\mscr O_{l,K}^\\Delta(E) )^{closed, even}$ and $\\mbb R \\Gamma(M, \\mscr O_{l,K}^\\Delta(E) )^{closed, even}$ are given by\n\\begin{align*}\n\\pi_n \\left(  \\Gamma(M, \\mscr O_{l,K}^\\Delta(E) )^{closed, even} \\right) &=\nH^{\\abs{n}}   (\\Gamma(M, \\mscr O_{l,K} ), Q ) \\\\\n\\pi_n \\left(  \\mbb R \\Gamma(M, \\mscr O_{l,K}^\\Delta(E) )^{closed, even} \\right)  &= H^{\\abs{n}}   (\\mbb R \\Gamma(M, \\mscr O_{l,K}^{\\Delta} )[0], Q + \\d_{DR} ).\n\\end{align*}\nOn the right hand side we are using $\\mathbb Z\/2$-graded cohomology groups. These equations are true for all $n \\ge 0$. \n\\end{lemma}\nThis is just a $\\mathbb Z\/2$-graded version of the Dold-Kan correspondence. \n\\begin{proof}\n\nThe $n$-simplices of $\\Gamma(M, \\mscr O_{l,K}^\\Delta(E) )^{closed, even}$ are closed and even elements of the complex  $\\Omega^\\ast(\\Delta^n)\\otimes \\Gamma(M, \\mscr O_{l,k}(E) )$, where we are using a completed tensor product.  Similarly, the $n$-simplices of $\\mbb R \\Gamma(M, \\mscr O_{l,K}^\\Delta(E) )^{closed, even}$ are given by closed even elements of $\\Omega^\\ast(\\Delta^n) \\otimes \\mbb R \\Gamma ( M , \\mscr O_{(l,K) } ^\\Delta ) [0]$, where we are using, again, an appropriate completed tensor product.\n\nLet $(V, \\d_V)$ be one of these two $\\mathbb Z\/2$-graded complexes, so\n$$\n(V, \\d_V) = \\begin{cases}\n( \\Gamma(M, \\mscr O_{l,k}(E), Q )  &  \\\\\n( \\mbb R \\Gamma ( M , \\mscr O_{(l,K) } ^\\Delta ) [0], Q + \\d_{DR} ) &\n\\end{cases}\n$$\nLet $V^\\Delta$ be the simplicial set whose $n$-simplices are closed even elements of $V \\otimes \\Omega^\\ast(\\Delta^n)$, where we use an appropriate completed tensor product.  We need to calculate the homotopy groups of $V^\\Delta$.\n\nAs $V^\\Delta$ is a Kan complex, we can calculate these homotopy groups as homotopy classes of maps from spheres to $V^\\Delta$.  Let us take $0$ as a base point in $V^\\Delta$.  Let $S^n$ be a simplicial set representing the $n$ sphere, and let $p \\in S^n$ be a  base point.    A base point preserving map $f : S^n \\to V^\\Delta$ is given by an even element \n$$\n\\omega_f \\in V \\otimes \\Omega^\\ast(S^n , p ) .\n$$\nwhich is closed, $(\\d_V + \\d_{DR} ) \\omega = 0$.\n\nThus, $\\omega_f$ gives a class \n$$[\\omega_f] \\in H^{0 }  (V \\otimes \\Omega^\\ast(S^n , p )) = H^{\\abs{n} } (V) .$$\n(Here we are using $\\mathbb Z\/2$-graded cohomology groups).\n\nWe need to show that if the map $f : S^n \\to V^\\Delta$ changes by a homotopy, then the class $[\\omega_f]$ remains unchanged, and conversely, if we change $\\omega_f$ to something cohomologous, then the corresponding map $S^n \\to V^\\Delta$ changes by a homotopy.\n\nIf $f,g : S^n \\to V^\\Delta$ are base point preserving maps, a base point preserving homotopy between them is given by an element $\\eta \\in V \\otimes \\Omega^\\ast ( S^n \\times [0,1], p \\times [0,1] )$, which is even and closed, and which restricts to $\\omega_f$ and $\\omega_g$ at $0$ and $1$.\n\nSince \n$$H^\\ast (V \\otimes \\Omega^\\ast ( S^n \\times [0,1], p \\times [0,1] ) ) = H^{\\abs{n}}(V) $$\nthe existence of such a homotopy implies that $[\\omega_f] = [ \\omega_g]$.\n\nConversely, suppose $\\phi \\in V \\otimes \\Omega^\\ast(S^n , p )$ is an odd element such that \n$$\n(\\d_V + \\d_{DR}) \\phi = \\omega_f - \\omega_g . \n$$\nThen if we let \n$$\n\\eta = \\omega_f + t (\\omega_g - \\omega_f ) + \\phi \\d t \\in V \\otimes \\Omega^\\ast ( S^n \\times [0,1], p \\times [0,1] )\n$$\nthen $\\eta$ is an even element satisfying $(\\d_V + \\d_{DR}) \\eta = 0$, so $\\eta$ defines a homotopy between $f$ and $g$. \n\n\\end{proof} \nThis completes the proof of theorem D. \n\n\\section{Local computations in Chern-Simons theory }\nWe will apply the theory developed in the previous section to construct a canonical up to homotopy solution of the renormalised quantum master equation in the case of Chern-Simons theory.   \n\nIn this section, therefore, let $M$ be a manifold of dimension \n$$\\dim M = n \\ge 2.$$ \nLet $\\mathfrak{g}$ be a locally trivial sheaf of Lie algebras \\footnote{We could also use  $L_\\infty$ algebras, and the construction works in more or less the same way. }  over $\\mathbb C$  on $M$, with a non-degenerate invariant symmetric pairing of the opposite parity to $n$, with values in the constant sheaf $\\mathbb C$.\n\n\nLet\n$$\n\\mathscr{E}  = \\Omega^\\ast(M , \\mathfrak g )[1] . \n$$\nLet $E$ denote the corresponding vector bundle on $M$, so $\\Gamma(M,E)  = \\mathscr{E}$.    Let $S_{CS} \\in \\mscr O_l(\\mathscr{E})$ denote the Chern-Simons action.  \n\n\\begin{theorem}\nThere is a canonical, up to a contractible choice, element \n$$S \\in \\mscr O_l(\\mathscr{E} , \\mathbb C[[\\hbar]] )\/ \\mathbb C[[\\hbar]]$$ \nwhich solves the renormalised quantum master equation modulo constants, and is of the form\n$$\nS = S_{CS } + \\hbar S^{(1)} + \\hbar^2 S^{(2)} +\\cdots\n$$\n\\label{theorem chern simons}\n\\end{theorem}\nThe proof was sketched in the introduction; the idea is to show that locally, in a flat metric, $S_{CS}$ satisfies the renormalised QME.  These local solutions together will give a global solution in a curved metric. \n\n\\subsection{Simplicial presheaves of metrics}\nThe bundle on $M$ playing the role of $F$ in section \\ref{section local global} is the bundle $\\operatorname{Met} \\to M$, whose fibres are metrics on the tangent space of $M$.  As in section \\ref{section local global}, we have a simplicial presheaf associated to $\\operatorname{Met}$,  defined by saying $\\Gamma(U, \\mathbf{Met} [d])$ is the set of smooth family of metrics on $U$ parametrised by $\\Delta^d$.   \n\nLet $\\mathbf{FMet} \\subset \\mathbf{Met}$ be the sub-simplicial presheaf given by families of flat metrics on $U$.  \n\\begin{lemma}\nIf $U$ is a ball in $M$, then $\\Gamma(U, \\mathbf{FMet} )$ is contractible. \\label{lemma flat metrics locally contractible}\n\\end{lemma}\n\\begin{proof}\nPick an isomorphism between $U$ and $\\{x \\in \\mbb R^{n} \\mid \\norm{x} < 1 \\}$.  The standard flat metric on the ball in $\\mbb R^{n}$ gives a metric on $U$, which we call $g^0$. \n\nLet $x_1,\\ldots, x_{n}$ denote our coordinates on $U$.  If $g$ is a smooth family of flat metrics on $U$, parametrised by $\\Delta^d$, then, in these coordinates, we can write $g$ as \n$$\n\\sum_{1 \\le i,j \\le n} g_{ij}  ( x , \\sigma ) \\d x_i \\otimes \\d x_j .\n$$\nHere $g_{ij}(x,\\sigma)$ is smooth in the $\\sigma$ variables, and smooth in the $x$ variables.\nTo show that the simplicial set of such $g$ is contractible, it suffices to construct a simplicial map \n$$\\Gamma(U, \\mathbf{FMet} ) \\times \\Delta_1 \\to \\Gamma(U, \\mathbf{FMet} ),$$ which is a simplicial homotopy between the identity on $\\Gamma(U, \\mathbf{FMet} )$ and the projection onto the flat metric $g^0 = \\sum \\d x_i \\otimes \\d x_i$.   A $d$-simplex in $\\Gamma(U, \\mathbf{FMet} ) \\times \\Delta_1$ is of course a $d$-simplex in $\\Gamma(U, \\mathbf{FMet} )$  and a $d$-simplex on $\\Delta_1$.  A $d$-simplex in $\\Delta_1$ is given by a non-decreasing map of sets $\\{0,1,\\ldots,d\\} \\to \\{0,1\\}$.   Such a $d$-simplex yields an affine linear map $\\Delta^d \\to \\Delta_1$.  \n\nLet $A \\subset \\Gamma(U, \\mathbf{FMet} )$ denote the simplicial set of constant metrics, i.e. those for which $g_{ij}(x,\\sigma)$ is independent of $x \\in U$.  Of course, this is just the simplicial set whose $0$-simplices are positive-definite inner products on the vector space $\\mbb R^{n}$, and whose $d$-simplices are smooth families of such.\n\nFirst we construct a deformation retraction of $\\Gamma(U, \\mathbf{FMet} )$ onto $A$. This is given by a map $\\Phi : \\Gamma(U, \\mathbf{FMet} ) \\times \\Delta_1 \\to \\Gamma(U, \\mathbf{FMet} )$, defined as follows.  Suppose we have a $d$-simplex in $\\Gamma(U, \\mathbf{FMet} ) \\times \\Delta_1$, corresponding to $g \\in \\Gamma(U, \\mathbf{FMet} )[d]$ and an affine linear map $p : \\Delta^d \\to \\Delta_1$. If, as above, $g = \\sum g_{ij}(x,\\sigma) \\d x_i \\otimes \\d x_j$,   we define $\\Phi(g,p)$ by\n$$\n\\Phi(g,p ) = \\sum g_{ij} ( p(\\sigma) x, \\sigma  ) \\d x_i \\otimes \\d x_j . \n$$\n$\\Phi$ is easily seen to give the required homotopy; when we restrict to $\\Gamma(U, \\mathbf{FMet} ) \\times \\{1\\} \\subset \\Gamma(U, \\mathbf{FMet} ) \\times \\Delta_1$, $\\Phi$ is the identity, and when we restrict to $\\Gamma(U, \\mathbf{FMet} )\\times \\{0\\}$, $\\Phi$ is a projection onto $A$.\n\nNext we need to show that the simplicial set $A$ is contractible.  Note that the space of positive-definite inner products on $\\mbb R^n$ is $\\operatorname{GL}(n,\\mbb R) \/ \\operatorname{O}(n.\\mbb R)$.  A simplex in $A$ is a continuous map $\\Delta^d \\to \\operatorname{GL}(n,\\mbb R) \/ \\operatorname{O}(n,\\mbb R)$, which is smooth on the top simplices of some barycentric subdivision of $\\Delta^d$.  Thus, to show $A$ is contractible, it suffices to construct a  smooth homotopy equivalence between $\\operatorname{GL}(n,\\mbb R) \/ \\operatorname{O}(n,\\mbb R)$ and a point.  This is easy to do using (for instance) the Gram-Schmidt procedure.  \n\n\n\\end{proof} \n\nThe simplicial presheaf $\\mathbf{FMet}$ is not necessarily locally fibrant.  Thus, our definition of $\\mbb R \\Gamma$ is not well-behaved when applied to $\\mathbf{FMet}$.  Therefore, we will instead consider $\\mbb R \\Gamma$ applied to a modification of $\\mathbf{FMet}$. \n\nLet $\\operatorname{Ex}^\\infty$ denote Kan's fibrant replacement functor for simplicial sets.  If $X$ is a simplicial set, then $\\operatorname{Ex}^\\infty(X)$ is a fibrant simplicial set equipped with a map to $X$ which is a weak equivalence.  Define $\\operatorname{Ex}^\\infty \\mathbf{FMet}$ by saying that \n$$\\Gamma (U, \\operatorname{Ex}^\\infty \\mathbf{FMet } ) = \\operatorname{Ex}^\\infty \\Gamma ( U, \\mathbf{FMet }).$$\nSince $\\operatorname{Ex}^\\infty \\mathbf{FMet}$ is locally fibrant, $\\mbb R \\Gamma ( M, \\operatorname{Ex}^\\infty \\mathbf{FMet})$ is well-behaved. \nThis is the correct version of derived global sections of $\\mathbf{FMet}$.  \n\n\\begin{corollary}\n$\\mbb R \\Gamma (M , \\operatorname{Ex}^\\infty \\mathbf{FMet} ) $ is contractible.\n\\end{corollary}\nThis is immediate from Corollary \\ref{corollary locally contractible} and Lemma \\ref{lemma flat metrics locally contractible}.\n\nWe apply our local to global principle as follows.  We would like to construct a homotopy point of $\\Gamma(M, \\mathbf{BV})$.  Any map $\\mathbf{FMet} \\to \\mathbf{BV}$ of simplicial presheaves would yield such a homotopy point; indeed, such a map induces a map\n$$\n\\mbb R \\Gamma(M, \\operatorname{Ex}^\\infty \\mathbf{FMet}) \\to \\mbb R \\Gamma(M,  \\operatorname{Ex}^\\infty \\mathbf{BV}) .\n$$\nWe know that $\\mbb R \\Gamma(M, \\operatorname{Ex}^\\infty \\mathbf{FMet}) $ is contractible. Also, $\\Gamma(M,\\mathbf{BV})$ is weakly equivalent to $\\mbb R \\Gamma(M, \\mathbf{BV})$, and therefore to $\\mbb R \\Gamma ( M ,\\operatorname{Ex}^\\infty \\mathbf{BF} )$.     Thus, a map $\\mathbf{FMet} \\to \\mathbf{BV}$ yields a point (up to contractible choice) of $\\Gamma(M, \\mathbf{BV})$.  \n\n\\begin{theorem}\nLet $U \\subset M$ be an open subset, and let\n$$\ng \\in \\Gamma(U, \\mathbf{FMet}[d] ) \n$$\nbe a smooth family of flat metrics on $U$ parametrised by the $d$-simplex $\\Delta^d$.  Then the usual Chern-Simons action $S_{CS}$ satisfies the renormalised quantum master equation on $U$, and so defines an element\n$$\nS_{CS} \\in \\Gamma(U , \\mathbf{BV}[d] ) .\n$$\n\\label{theorem chern simons local calculation}\n\\end{theorem}\nThis theorem gives a map \n$$\n\\mathbf{FMet} \\to \\mathbf{BV}\n$$\n of simplicial presheaves.  Thus, once we have proved this result, we will have proved Theorem \\ref{theorem chern simons}. \n\\subsection{Proof of Theorem \\ref{theorem chern simons local calculation}}\nWe want to show that, locally in a flat metric, all the obstructions $O_{(I,K)}(S_{CS})$ vanish, for $I \\ge 0, K > 0$. \n\nSince everything is local, it suffices to work with small open sets $U$ in $M$, with a smooth family of flat metrics parametrised by $\\Delta^d$. \n\n\nLet us trivialise our flat bundle of Lie algebras $\\mathfrak g$ on $U$.  The space of fields is the $\\Omega^\\ast(\\Delta^d)$-module\n$$\n\\Omega^\\ast(U) \\otimes \\Omega^\\ast(\\Delta^d) \\otimes \\mathfrak g\n$$\nwith the differential $\\d_{DR}^U + \\d_{DR}^{\\Delta^d}$, and the gauge fixing condition $\\d^\\ast$.  The operator $\\d^\\ast$ depends on the coordinates in $\\Delta^d$.   The Hamiltonian $H$ is\n$$\nH = [\\d_{DR}^U + \\d_{DR}^{\\Delta^d}, \\d^\\ast ] .\n$$\n\n\n\n\n\nRecall that $n = \\dim M$. Let us give $\\mbb R^n$ the standard flat metric.  It is easy to check that one can find an open subset $V \\subset \\mbb R^n\\times \\Delta^d$, and an isomorphism $U \\times \\Delta^d \\to V$, such that the diagram\n$$\n\\xymatrix{ U \\times \\Delta^d \\ar[r] \\ar[d] & V \\ar[dl] \\\\ \n\\Delta^d &   }\n$$\ncommutes, and which is compatible with the metrics along the fibres of the projection maps to $\\Delta^d$. \n\n\n \nThe isomorphism chosen above gives an isomorphism\n$$\n\\Omega^\\ast(U) \\otimes \\Omega^\\ast(\\Delta^d) \\otimes \\mathfrak g \\cong \\Omega^\\ast(V)\n$$\nof differential graded $\\Omega^\\ast(\\Delta^d)$-modules.  The differential on both sides is simply the de Rham differential on $U \\times \\Delta^d$, or on $V$. Because the metric along the fibres of the map $U\\times \\Delta^d \\to \\Delta^d$ corresponds to that along the fibres of the map $V \\to \\Delta^d$, this isomorphism also takes the operator $\\d^\\ast$ on the left hand side to the corresponding one on the right hand side.\n\nThe propagator is constructed from the de Rham differential on $U \\times \\Delta^d$, and the operator $\\d^\\ast$. Thus,  it suffices to show that the obstructions $O_{(I,K)}(S_{CS})$ vanish when we work with $V$ instead of $U \\times \\Delta^d$. Since the obstruction is local, it suffices to work in $\\mbb R^n \\times \\Delta^d$.\n\nWe have reduced to the case of $\\mbb R^n$ with the constant family of flat metrics.   Thus, it suffices to show the following.\n\\begin{theorem}\nLet $S_{CS}$ be the usual Chern-Simons action on $\\mbb R^n$ with the flat metric.  Then\n\\begin{enumerate}\n\\item\nThere are no counter-terms.\n\\item\n$S_{CS}$ satisfies the renormalised quantum master equation.\n\\end{enumerate}\n\\end{theorem}\nThis result implies, in particular, that the construction is independent of the choice of renormalisation scheme.  The choice of renormalisation scheme is only involved in constructing the bijection between the set of local functionals $S\\in \\mscr O_l(\\mathscr{E}, \\mathbb C[[\\hbar]])$ and the set of systems of effective actions satisfying the renormalisation group equation. However, because there are no counter-terms, the system of effective actions $\\{\\Gamma ( P (0,T) , S_{CS} )\\}$ associated to the Chern-Simons action $S_{CS}$ is independent of the choice of renormalisation scheme.\n\n\nThe proof of this theorem will follow closely the results of Kontsevich \\cite{Kon94, Kon03a}.  In particular, we make use of the compactifications of configuration spaces used in these papers, and we rely on a certain vanishing theorem proved by Kontsevich in \\cite{Kon94} when $n \\ge 3$, and in \\cite{Kon03a} when $n = 2$. \n\n\nLet $\\operatorname{Conf}_m(\\mbb R^n)$ denote the space of $m$ distinct ordered points in $\\mbb R^n$.    There is a partial compactification $\\overline{\\operatorname{Conf}}_m(\\mbb R^n)$ of $\\operatorname{Conf}_m(\\mbb R^n)$ with the following properties.  \n\\begin{enumerate}\n\\item\n$\\overline{\\operatorname{Conf}}_2(\\mbb R^n)$ is the real blow-up along the diagonal of $\\mbb R^n \\times \\mbb R^n$.\n\\item\n$\\overline{\\operatorname{Conf}}_m(\\mbb R^n)$ admits a proper map to $\\mbb R^{nm}$ such that the diagram\n$$\n\\xymatrix{ \\operatorname{Conf}_m(\\mbb R^n) \\ar[r] \\ar[d] & \\overline{\\operatorname{Conf}}_m(\\mbb R^n)  \\ar[dl] \\\\ \n\\mbb R^{nm} &  }\n$$\ncommutes.\n\\item\nFor any  $i,j$ with $1 \\le i,j \\le m$ and $i \\neq j$,  there is a projection map\n$$\n\\pi_{i,j} : \\overline{\\operatorname{Conf}}_m(\\mbb R^n) \\to \\mbb R^{2n}\n$$\ngiven by forgetting all factors except $i$ and $j$. This map lifts to $\\overline{\\operatorname{Conf}}_2(\\mbb R^n)$.\n\\end{enumerate}\nThe partial compactification we use was constructed in \\cite{Kon94,Kon03a}. This is the real version of the Fulton-MacPherson compactification of configuration spaces of algebraic varieties \\cite{FulMac94}. \n\n\nLet\n$$\n\\mathscr{E} = \\Omega^\\ast(\\mbb R^n) \\otimes \\mathfrak g [1]\n$$\nbe the space of fields, and let\n$$\n\\mathscr{E}_c = \\Omega^\\ast_c(\\mbb R^n) \\otimes \\mathfrak g [1]\n$$\nbe the space of compactly supported fields.\n\nLet\n$$\nP(\\varepsilon,T ) = \\int_{\\varepsilon}^T \\d^\\ast K_t \\d t \\in \\mathscr{E} \\otimes \\mathscr{E} .\n$$\n\\begin{lemma}\nUp to an overall constant,\n$$\nP(\\varepsilon,T) = \\left ( \\int_{ \\norm{x - y}^2 \/ 4 T }^{\\norm{x - y}^2 \/ 4 \\varepsilon }  u^{n\/2 - 1 } e^{- u }  \\d u \\right)   \\pi^\\ast \\operatorname{Vol}_{S^{n-1}} \\otimes  I_{\\mathfrak g} \n$$\nwhere\n\\begin{enumerate}\n\\item\n$$\n\\pi : \\operatorname{Conf}_2(\\mbb R^n) \\to S^{n-1}\n$$\nis the projection \n$$(x,y) \\to (x - y)  \/ \\norm{x - y} .$$\n\\item\n$ \\operatorname{Vol}_{S^{n-1}}$ is the standard volume form on $S^{n-1}$, given by the formula\n$$\n \\operatorname{Vol}_{S^{n-1}} = \\norm{z}^{-n} \\sum_{i = 1}^{n} (-1)^{i}  z_i  \\d z_1 \\cdots \\widehat{ \\d z_i } \\cdots  \\d z_n .\n$$\n\\item\n$$\nI_{\\mathfrak g} \\in \\mathfrak g \\otimes \\mathfrak g\n$$\nis the tensor dual to the pairing on $\\mathfrak g$. \n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nThe proof is an explicit calculation.  The sign conventions we are using imply that the heat kernel is\n$$\nK_t =  C t^{-n\/2 }  e^{- \\norm{x-y}^2 \/ 4t } \\d (x_1 - y_1) \\wedge \\cdots \\wedge \\d (x_n - y_n) \\otimes I_{\\mathfrak g}  \n$$\nwhere $C$ is a certain normalising constant. \nIt follows that \n\\begin{multline*}\n\\d^\\ast K_t  =  C' t^{-n\/2 - 1 } e^{- \\norm{x-y}^2 \/ 4t } \\sum_{i = 1}^{n} (-1)^{i}  (x_i - y_i) \\d (x_1 - y_1)  \\cdots  \\widehat{ \\d (x_i -  y_i) } \\\\  \\cdots \\d (x_n - y_n )  \\otimes  I_{\\mathfrak g} \n\\end{multline*}\nwhere $C'$ is a constant.\nNow, \n$$\n\\int_\\varepsilon^T  t^{-n\/2 - 1 } e^{- \\norm{x-y}^2 \/ 4t } \\d t = C'' \\norm{x - y}^{-n}  \\int_{ \\norm{x - y}^2 \/ 4 T }^{\\norm{x - y}^2 \/ 4 \\varepsilon }  u^{n\/2 - 1 } e^{- u }  \\d u \n$$\nfor some constant $C''$.   This follows from the change of variables $u = \\norm{x-y}^2 \/ 4t$.   \n\\end{proof} \n\\begin{corollary}\nThe form $P(0,T) \\in \\Omega^\\ast(\\operatorname{Conf}_2(\\mbb R^n) ) \\otimes \\mathfrak g^{\\otimes 2}$ extends to a smooth form in $ \\Omega^\\ast(\\overline{\\operatorname{Conf}}_2(\\mbb R^n)) \\otimes \\mathfrak g^{\\otimes 2} $. \n\\end{corollary}\n\\begin{proof}\nThe map $\\pi : \\operatorname{Conf}_2(\\mbb R^n) \\to S^{n-1}$ extends to a map $\\overline{\\operatorname{Conf}}_2(\\mbb R^n) \\to S^{n-1}$, which implies that the form $ \\pi^\\ast \\operatorname{Vol}_{S^{n-1}}$ extends to a smooth form on $\\overline{\\operatorname{Conf}}_2(\\mbb R^n)$.  It remains to show that the $\\varepsilon \\to 0$ limit of $\\int_{ \\norm{x - y}^2 \/ 4 T }^{\\norm{x - y}^2 \/ 4 \\varepsilon }  u^{n\/2 - 1 } e^{- u }  \\d u$ is a smooth function on $\\overline{\\operatorname{Conf}}_2(\\mbb R^n)$.   Note that this $\\varepsilon \\to 0$ limit is the incomplete gamma function\n$$\n\\int_{ \\norm{x - y}^2 \/ 4 T }^{\\infty }  u^{n\/2 - 1 } e^{- u }  \\d u  = \\Gamma (n\/2, \\norm{x - y}^2 \/ 4 T ) .\n$$\n The function $t \\to \\Gamma(n\/2, t^2)$ is smooth, and the function $\\norm{x - y} : \\overline{\\operatorname{Conf}}_2(\\mbb R^n) \\to \\mbb R$ is smooth, which implies that $\\Gamma( n\/2, \\norm{x- y}^2 \/ 4T)$ is smooth.  \n\\end{proof} \n\n\\begin{lemma}\nThe differential of $P(0,T)$ on $\\overline{\\operatorname{Conf}}_2 (\\mbb R^n)$ is $-K_T$.\n\\end{lemma}\n\\begin{proof}\nIf we think of $P(0,T)$ as a de Rham current on $\\mbb R^n \\times \\mbb R^n$, we know its differential is the delta current on the diagonal, minus $K_T$.  Since $P(0,T)$ is a smooth form on $\\overline{\\operatorname{Conf}}_2(\\mbb R^n)$, its differential must be smooth, and is determined by its restriction to $\\operatorname{Conf}_2(\\mbb R^n)$, where it is equal to $-K_T$.   \n\\end{proof} \n\nAs before, we can attempt to define\n$$\n\\Gamma( P(\\varepsilon,T), S_{CS} ) \n$$\nas an element of $\\mscr O(\\mathscr{E}_c, \\mathbb C[[\\hbar]])$, the space of functionals on the compactly supported fields.  It is not completely obvious that this is well defined. However,\n\\begin{proposition}\nFix $0 < T < \\infty$ and $i \\ge 0, k > 0$. \nThen $\\Gamma_{(i,k)}(P(\\varepsilon,T), S_{CS})$  is a well-defined continuous linear functional $\\mathscr{E}_c^{\\otimes k} \\to \\mathbb C$. Further, the limit $\\lim_{\\varepsilon \\to 0} \\Gamma_{(i,k)}(P(\\varepsilon,T), S_{CS})$ exists.\n\\label{prop cs convergence}\n\\end{proposition}\nThis proposition implies that all the counter-terms vanish. \n\\begin{proof}\nLet $\\alpha \\in \\mathscr{E}_c^{\\otimes k}$.\nStandard Feynman graph techniques allow one to write\n$$\n\\Gamma_{(i,k)}(P(\\varepsilon,T) , S_{CS} )(\\alpha) = \\sum_{\\gamma} \\frac{1}{\\Aut(\\gamma)} w_\\gamma(\\varepsilon, T, \\alpha)\n$$\nwhere the sum is over connected trivalent graphs $\\gamma$, whose first Betti number is $i$, with $k$ external edges.   The weight $w_\\gamma$ attached to each graph is a certain integral;  we will show that these integrals converge absolutely, even when we set $\\varepsilon =0$. \n\nLet $\\gamma$ be such a trivalent graph, and let $V(\\gamma), E(\\gamma)$ denote the sets of vertices and edges of $\\gamma$. Vertices are all trivalent; the end points of the external edges are not considered vertices.  Also, the external edges are not elements of $E(\\gamma)$.\n\nWe will define a differential form $\\omega_\\gamma(\\varepsilon, T, \\alpha)$ on the space $\\overline{\\operatorname{Conf}}_{V(\\gamma)}(\\mbb R^n)$.  I'll ignore signs, as we only want to show convergence. (Technically, the form $\\omega_\\gamma$ is associated to a trivalent graph with a certain orientation).  \n\nOnly graphs with no loops (i.e.\\ edges joining a vertex to itself) appear in the sum.  The weights of graphs with loops vanish, because the propagator $P(\\varepsilon,T)$  (which is form on $\\mbb R^n \\times \\mbb R^n$)  is zero when restricted to the diagonal.  \n\nFor each edge $e \\in E(\\gamma)$, let\n$$\n\\pi_e : \\overline{\\operatorname{Conf}}_{V(\\gamma)}(\\mbb R^n) \\to \\overline{\\operatorname{Conf}}_2(\\mbb R^n)\n$$\nbe the projection correspond to the two vertices attached to $e$.  \n\nLet \n$$\\phi : \\overline{\\operatorname{Conf}}_{V(\\gamma)}(\\mbb R^n) \\to \\mbb R^{n k}$$\nbe the projection corresponding to the $k$ vertices of $\\gamma$ which are attached to external edges. \n\nThe form attached to the graph is\n$$\n\\omega_\\gamma (\\varepsilon,T , \\alpha) = \\otimes_{v \\in V(\\gamma)} \\Tr^{\\mathfrak g}_v \\left(  \\phi^\\ast \\alpha \\wedge_{e \\in E(\\gamma)} \\pi_e^\\ast P(\\varepsilon,T) \\right).\n$$\nLet me explain this notation.  The expression inside the bracket is an element  \n$$ \\phi^\\ast \\alpha \\wedge_{e \\in E(\\gamma)} \\pi_e^\\ast P(\\varepsilon,T) \\in \\Omega^\\ast(\\overline{\\operatorname{Conf}}_{V(\\gamma)} (\\mbb R^n) ) \\otimes \\mathfrak g^{\\otimes H(\\gamma)}, $$ where $H(\\gamma)$ is the set of half-edges (or germs of edges) of $\\gamma$. Half-edges include external edges. Let $H(v)$ denote the set of $3$ half-edges at a vertex $v$.   For each $v$,  there is a trace map \n$$\n\\Tr_v^{\\mathfrak g} : \\mathfrak g^{\\otimes H(v) } \\to \\mathbb C\n$$\ndefined by $\\ip{X, [Y,Z]}$.  Thus, \n$$\n\\otimes_{v \\in V(\\gamma)} \\Tr_v^{\\mathfrak g} : \\mathfrak g^{\\otimes H(\\gamma)} \\to \\mathbb C \n$$\nso that\n$$\n\\omega_\\gamma (\\varepsilon,T , \\alpha) \\in \\Omega^\\ast(\\overline{\\operatorname{Conf}}_{V(\\gamma)} (\\mbb R^n) ) .\n$$\n\nThe weight attached to $\\gamma$ is\n$$\nw_\\gamma(\\varepsilon,T, \\alpha) = \\int_{\\overline{\\operatorname{Conf}}_{V(\\gamma)}} \\omega_\\gamma(\\varepsilon,T,\\alpha).\n$$\nWe have to show that this integral converges absolutely, for all $\\varepsilon \\ge 0$ and all $0 < T < \\infty$.  \n\nThe problems that could occur would be  if some of the points in the configuration space went to $\\infty$. However, we are putting a compactly supported form at the vertices which are attached to external edges, and (since $k > 0$) there is at least one such vertex.     If the two points in $\\mbb R^{n}$ attached to the end points of an edge $e$ become very far apart, then $\\pi_e^\\ast P(\\varepsilon,T)$ decays exponentially.   If the point in $\\mbb R^n$ attached to one of the vertices goes to $\\infty$, and another point is constrained to lie in a compact set because of the compactly supported form, then we must have an edge $e$ whose end points are far apart.  Thus, if any of the points in $\\mbb R^n$ attached to vertices tend to $\\infty$, the integrand decays exponentially. This implies that the integral converges absolutely.\n\\end{proof} \nThis proposition implies that\n$$\n\\Gamma(P(0,T), S_{CS} ) \\in \\mscr O( \\mathscr{E}_c, \\mathbb C[[\\hbar]] ) \/ \\mathbb C[[\\hbar]] \n$$\nis well-defined. Note that we are working modulo the ring of constants. \n\nTo prove our result, it remains to show that the appropriate quantum master equation is satisfied. \n\\begin{proposition}\n$$\n(\\d_{DR}  +\\hbar \\Delta_T ) \\exp ( \\Gamma(P(0,T), S_{CS} ) \/ \\hbar )  = 0\n$$\nmodulo constants.\n\\end{proposition}\n\\begin{proof}\nWe can re-express the quantum master equation as \n$$\n\\d_{DR} \\Gamma(P(0,T), S_{CS}  ) +  \\{\\Gamma(P(0,T), S_{CS}  ), \\Gamma(P(0,T), S_{CS}  )\\}_T   +\\hbar \\Delta_T \\Gamma(P(0,T), S_{CS}  ) = 0\n$$\nwhere $\\{\\quad\\}_T$ is a certain bracket on the space of functionals.   Up to sign, if $f, g \\in \\mscr O(\\mathscr{E})$ are functionals, and $K_T = \\sum \\phi' \\otimes \\phi''$, then\n$$\n\\{f,g\\}_T = \\sum \\frac{\\partial f }{\\partial \\phi'} \\frac{\\partial g }{\\partial \\phi''} .\n$$\nWorking modulo constants amounts to ignoring the terms $\\hbar \\Delta_T \\Gamma_{(i,2)}(P(0,T), S_{CS} )$ for any $i$,  and $\\{ \\Gamma_{(i,1)}(P(0,T), S_{CS} ) , \\Gamma_{(j,1)}(P(0,T), S_{CS} ) \\}_T$ for any $i,j$. \n\nIn the proof of Proposition \\ref{prop cs convergence} we showed how to express $\\Gamma_{(i,k)}(P(0,T) , S_{CS} ) (\\alpha)$ as a sum over trivalent graphs, with $k$ external edges, and first Betti number $i$.  The weight attached to each trivalent graph $\\gamma$  is\n$$\nw_\\gamma(0,T,\\alpha) = \\int_{\\overline{\\operatorname{Conf}}_{V(\\gamma)}} \\otimes_{v \\in V(\\gamma)} \\Tr^{\\mathfrak g}_v \\left(  \\phi^\\ast \\alpha \\bigwedge_{e \\in E(\\gamma)} \\pi_e^\\ast P(0,T) \\right)\n$$\nHere $\\alpha \\in \\mathscr{E}_c^{\\otimes k}$ is the input, which we put at the external edges.\n\nBy definition, \n$$\\d_{DR} \\Gamma_{(i,k)}(P(0,T), S_{CS})(\\alpha) =  \\Gamma_{(i,k)}(P(0,T), S_{CS})(\\d_{DR} \\alpha). $$\nStokes' theorem implies that\n\\begin{multline*}\nw_\\gamma(0,T, \\d_{DR} \\alpha) = - \\int_{\\partial \\overline{\\operatorname{Conf}}_{V(\\gamma)}} \\otimes_{v \\in V(\\gamma)} \\Tr^{\\mathfrak g}_v \\left(  \\phi^\\ast  \\alpha \\bigwedge_{e \\in E(\\gamma)} \\pi_e^\\ast P(0,T) \\right) \\\\\n+ \\sum_{e \\in E(\\gamma)} \\pm  \\int_{ \\overline{\\operatorname{Conf}}_{V(\\gamma)}} \\otimes_{v \\in V(\\gamma)} \\Tr^{\\mathfrak g}_v \\left(  \\phi^\\ast  \\alpha \\wedge \\pi_e^\\ast (K_T) \\bigwedge_{e' \\neq e} \\pi_e^\\ast P(0,T) \\right).\n\\end{multline*}\nIn the second line, we are using the fact that $\\d_{DR} P(0,T) = -K_T$. \n\nThe terms in the sum\n$$\n\\sum_{e \\in E(\\gamma)} \\pm  \\int_{ \\overline{\\operatorname{Conf}}_{V(\\gamma)}} \\otimes_{v \\in V(\\gamma)} \\Tr^{\\mathfrak g}_v \\left(  \\phi^\\ast  \\alpha \\wedge \\pi_e^\\ast (K_T) \\bigwedge_{e' \\neq e} \\pi_e^\\ast P(0,T) \\right)\n$$\nwhere $e$ is a separating edge cancel with terms in the graphical expansion of $$ \\{\\Gamma(P(0,T), S_{CS}  ), \\Gamma(P(0,T), S_{CS}  )\\}_T .$$  The terms where $e$ is a non-separating edge cancel with terms in the graphical expansion of $\\hbar \\Delta_T \\Gamma(P(0,T), S_{CS}  )$.  \n\nThus, it remains to show that \n$$\n\\sum_{\\gamma} \\pm \\int_{\\partial \\overline{\\operatorname{Conf}}_{V(\\gamma)}} \\otimes_{v \\in V(\\gamma)} \\Tr^{\\mathfrak g}_v \\left(  \\phi^\\ast  \\alpha \\bigwedge_{e \\in E(\\gamma)} \\pi_e^\\ast P(0,T) \\right)  = 0 . \n$$\nThis has been proved by Kontsevich in \\cite{Kon94} when $n \\ge 3$, and in  \\cite{Kon03a} when $n = 2$.   Indeed, lemma 2.1 of \\cite{Kon94} implies that when $n \\ge 3$, the only boundary strata of $\\overline{\\operatorname{Conf}}_{V(\\gamma)}$ which could contribute are those where precisely two points collide.  The first lemma in section 6 of \\cite{Kon03a} proves the same statement when $n = 2$.  The boundary strata where only two points collide are taken care of by the Jacobi identity.\n\\end{proof} \n\n\\section*{Appendix}\n\n\n\n\n\nThis appendix contains the proof of a generalised version of theorem A. \n\\subsection{Statement of results}\nLet $E$ be a super vector bundle on a compact manifold $M$, whose space of global sections is denoted by $\\mathscr{E}$. As always, suppose $\\mathscr{E}$ has an odd symplectic structure.  \nLet\n$$\nH_0 : \\mathscr{E} \\otimes C^{\\infty}(\\Delta^d) \\to \\mathscr{E} \\otimes C^{\\infty}(\\Delta^d) \n$$\nbe a smooth family of generalised Laplacians, parametrised by $\\Delta^d$. Thus, $H_0$ is a $C^{\\infty}(\\Delta^d)$ linear map, which is an order two differential operator with respect to $M$.  The symbol of $H_0$ is a section\n$$\n\\sigma(H_0 ) : \\Gamma( T^\\ast M \\times \\Delta^d, \\End E) .\n$$\nThe statement that $H_0$ is a family of generalised Laplacians says that $\\sigma(H_0)$ is a smooth family of metrics on $T^\\ast M$, parametrised by $\\Delta^d$, times the identity in $\\End E$.\n\nLet \n$$\nH_1 : \\mathscr{E} \\otimes \\Omega^\\ast(\\Delta^d) \\to \\mathscr{E} \\otimes \\Omega^\\ast(\\Delta^d)\n$$\nbe an even $\\Omega^\\ast(\\Delta^d)$ linear map, which is a first order differential operator with respect to $M$.  Then\n$$\nH = H_0 + H_1\n$$\nis a smooth family of generalised Laplacians parametrised by the super-manifold $\\Delta^d \\times \\mbb R^{0,d}$. \n\nThe results of \\cite{BerGetVer92}, appendix to chapter 9, imply that there is a unique heat kernel\n$$K_t \\in \\mathscr{E} \\otimes \\mathscr{E} \\otimes \\Omega^\\ast(\\Delta^d)$$ for the operator $H$.\n\n\n\nLet\n$$\nD : \\mathscr{E} \\otimes \\Omega^\\ast(\\Delta^d) \\to \\mathscr{E} \\otimes \\Omega^\\ast(\\Delta^d)\n$$\nbe any odd $\\Omega^\\ast(\\Delta^d)$ linear differential operator, commuting with $H$.\n\nAs before, let\n$$\nP(\\varepsilon,T) = \\int_\\varepsilon^T (D \\otimes 1) K_t \\d t \\in \\mathscr{E} \\otimes  \\mathscr{E} \\otimes \\Omega^\\ast(\\Delta^d) \n$$\nbe the propagator.\n\\begin{theorem}\nLet \n$$S \\in \\mscr O_l(\\mathscr{E},  \\Omega^\\ast(\\Delta^d) \\otimes \\mathscr{A} \\otimes \\mathbb C[[\\hbar]] )$$ be a function which, modulo $\\hbar$, is at least cubic.    More explicitly, $S$ has a Taylor series expansion \n$$\nS = \\sum_{i,k \\ge 0} \\hbar^i S_{i,k}  \\phi_{i,k}(\\varepsilon)\n$$\nwhere $S_{i,k} : \\mathscr{E} \\to \\Omega^\\ast(\\Delta^d)$ is a local functional, homogeneous of degree $k$, and $\\phi_{i,k}(\\varepsilon) \\in \\mathscr{A}$.  \n\nThen we can form\n$$\n\\Gamma\\left( P (\\epsilon,T) , S  \\right)  = \\sum_{i\\ge 0,k \\ge 0} \\hbar^i \\Gamma_{i,k} (P (\\epsilon, T),  S)\n$$\nas before.  \n\n\\begin{enumerate}\n\\item\nThere exist functions $f_r \\in \\mathscr{A}$ and $\\Phi_r  \\in \\mscr O(\\mathscr{E}, \\Omega^\\ast(\\Delta^d) \\otimes C^{\\infty}(0,\\infty))$, for   $r \\in \\mathbb Z_{\\ge 0}$, such that there is a small $\\varepsilon$ asymptotic expansion \n$$\n\\Gamma_{i,k}(P(\\epsilon,T),S ) (e)    \\simeq \\sum f_r(\\varepsilon) \\Phi_r  (e,T)\n$$\nfor all $e \\in \\mathscr{E}$.\n\\item\nEach $\\Phi_r(e,T)$ has a small $T$ asymptotic expansion \n$$\n\\Phi_r(e,T) \\simeq \\sum \\Phi_{r,s} (e) g_s(T)\n$$\nwhere the functions $\\Phi_{r,s}$ are local functionals of $e$, that is, $$\\Phi_{r,s} \\in \\mscr O_l(\\mathscr{E},\\Omega^\\ast(\\Delta^d)).$$ The $g_s(T)$ are certain smooth functions of $T \\in (0,\\infty)$.  \n\\item\nWe can view the coefficients $\\Phi_{r,s}$ as linear maps $\\mathscr{E}^{\\otimes k} \\to \\Omega^\\ast(\\Delta^d)$.  Thus, we can speak of the germ of $\\Phi_{r,s}$ near a point $x$.  This germ only depends on the germ of the operators $H,D, S$ near $x \\times \\Delta^d$.  \n\\end{enumerate}\n\\end{theorem}\n\\begin{remark}\nAt one stage in the paper, we need a slight generalisation of this result, which involves a propagator $\\delta K_\\varepsilon$ where $\\delta$ is an odd parameter of square zero.  The proof given below incorporates this case also. \n\\end{remark}\n\n\\begin{remark}\nRecall that  $\\mathscr{A} \\subset C^{\\infty}((0,\\infty))$ is the sub-algebra spanned over $\\mathbb C$ by  functions of $\\epsilon$ of the form\n$$\nf(\\epsilon )  = \\int_{U \\subset (\\epsilon,1)^n } \\frac{  F(t_1,\\ldots,t_n)^{1\/2} } { G(t_1,\\ldots, t_n)^{1\/2} } \\d t_1 \\cdots \\d t_n\n$$\nand functions of the form\n$$\nf(\\epsilon )  = \\int_{U \\subset (\\epsilon,1)^{n-1} } \\frac{  F(t_1, \\ldots,t_n = \\varepsilon)^{1\/2} } { G(t_1,\\ldots, t_n = \\varepsilon)^{1\/2} } \\d t_1 \\cdots \\d t_{n-1}\n$$\nwhere \n\\begin{enumerate}\n\\item\n$F, G \\in \\mathbb Z_{\\ge 0} [t_1,\\ldots, t_n] \\setminus \\{0\\}$; $n$ can take on any value. \n\\item\nthe region of integration $U$ is an open subset cut out by finitely many equations of the form $t_i^l > t_j$, for $l \\in \\mathbb Z$.  \n\\end{enumerate}\n\n\\end{remark}\n\n\n\n\n\nLet me explain more precisely what I mean by small $\\varepsilon$ asymptotic expansion. Let $\\norm{\\cdot}_{l}^{\\Delta^d}$  be the $C^l$ norm on the space $\\Omega^\\ast(\\Delta^d)$, so that $\\norm{\\omega}_{l}^{\\Delta^d}$ is the supremum over $\\Delta^d$ of the sum of all order $\\le l$ derivatives of $\\omega$. \n\nLet us consider $  \\Gamma_{i,k}(P(\\varepsilon,T),S ) $ as a linear map $\\mathscr{E}^{\\otimes k} \\to \\Omega^\\ast(\\Delta^d) \\otimes C^{\\infty}( \\{ 0 < \\varepsilon <  T \\}  ) $.   The precise statement is that for all $R , l\\in \\mathbb Z_{\\ge 0}$ and compact subsets $K \\subset (0,\\infty)$, there exists $m \\in \\mathbb Z_{\\ge 0}$ such that \n$$\n\\sup_{T \\in K}   \\norm{   \\Gamma_{i,k}(P(\\varepsilon,T),S ) (\\alpha)  -  \\sum_{r = 0}^R f_r(\\varepsilon) \\Phi_r (T,\\alpha)   }_{l}^{\\Delta^d}  < \\varepsilon^{R+1} \\norm{\\alpha}_m\n$$\nfor all $T$ sufficiently small.  Here $\\norm{\\alpha}_m$ denotes the $C^m$ norm on the space $\\mathscr{E}^{\\otimes k}$. \n\nThe small $T$ asymptotic expansion in part (2) has a similar definition. \n\n\n\n\n\n\n\n\\subsection{Expressions in terms of integrals attached to graphs}\n\n\n\nStandard Feynman graph techniques allow one to represent each $\\Gamma_{i,k}(P(\\epsilon,T),S ) $ as a finite sum \n$$\n\\Gamma_{i,k}(P(\\epsilon,T),S )  (\\alpha)  = \\sum_\\gamma \\frac{1}{\\operatorname{Aut} \\gamma} w_\\gamma (\\alpha)\n$$\nwhere $\\alpha \\in \\mathscr{E}^{\\otimes k}$,  and the sum is over certain graphs $\\gamma$.   The graphs $\\gamma$ that appear in the sum are connected graphs, with $k$ legs (or external lines).  Each vertex is assigned a degree in $\\mathbb Z_{\\ge 0}$, corresponding to the power of $\\hbar$ attached to the vertex.    Degree zero vertices are at least trivalent, and the sum of the degrees of the vertices, plus the first Betti number of the graph, must be equal to $i$.   There are a finite number of such graphs.  To each graph is attached a certain integral, whose value is $w_\\gamma(\\alpha)$.   \n\nI won't describe in detail the formula for the particular graph integrals appearing in the expansion of $\\Gamma_{(i,k)}(P(\\varepsilon,T),S)$.  Instead, I will describe a very general class of graph integrals, which include those appearing in the sum above.   The theorem will  be proved for this general class of graph integrals. \n\n\n\n\n\n\n\\subsection{Asymptotics of graph integrals}\n\n\\begin{definition}\nA labelled graph is a connected graph $\\gamma$,  with some number of legs (or external edges). \n\nFor each vertex $v$ of $\\gamma$, let $H(v)$ denote the set of half edges (or germs of edges) emanating from $v$; a leg attached at $v$ counts as a half edge.\n\nAlso, $\\gamma$ has an ordering on the sets of vertices, edges, legs and on the set of half edges  attached to each vertex.  \n\nEach vertex $v$ of $\\gamma$ is labelled by an element\n$$S_v \\in  \\operatorname{PolyDiff}( \\mathscr{E}^{\\otimes H(v)}, \\operatorname{Dens}(M) ) \\otimes \\Omega^\\ast(\\Delta^d).$$\n$S_v$ is a smooth family of polydifferential operators parametrised by $\\Delta^d$.  Let $O(v)$ denote the order of $S_v$.\n\\end{definition}\nLet $L(\\gamma)$ denote the set of legs of $\\gamma$.\nFix $\\alpha \\in \\mathscr{E}^{\\otimes L(\\gamma)}$, and fix $t_e \\in (0,\\infty)$ for each $e \\in E(\\gamma)$.  Define a function $f_\\gamma (t_e, \\alpha)$ as follows.\n\nLet $H(\\gamma)$ denote the set of half edges of $\\gamma$, so $H(\\gamma) = \\cup_{v \\in V(\\gamma)} H(v)$.  By putting $K_{t_e}$ at each edge of $\\gamma$, and $\\alpha$ at the legs, we get an element\n$$\n\\alpha \\otimes_{e \\in E(\\gamma)} K_{t_e}   \\in \\mathscr{E}^{\\otimes H(\\gamma)} .\n$$\nOn the other hand, the polydifferential operators $S_v$ at the vertices define a map\n$$\n\\int_{M^{V(\\gamma) } } \\otimes S_v :   \\mathscr{E}^{\\otimes H(\\gamma) }   \\to \\operatorname{Dens}(M)^{\\otimes V(\\gamma) } \\otimes \\Omega^\\ast(\\Delta^d) \\xrightarrow{\\int_{M^{V(\\gamma) } } }      \\Omega^\\ast(\\Delta^d).\n$$\nLet \n$$\nf_\\gamma(t_e,\\alpha) = \\int_{M^{V(\\gamma) } } \\otimes S_v \\left( \\alpha \\otimes_{e \\in E(\\gamma)} K_{t_e}  \\right)  \\in \\Omega^\\ast(\\Delta^d).\n$$\n\n\nThe graph integrals $w_\\gamma$ which appear in the expansion of $\\Gamma_{i,k}(P(\\varepsilon,T), S)$   can be realised as finite sums of functions of the form \n$$f_\\gamma (t_e, \\alpha, S_v )$$\nfor certain local functionals $S_v$.   Terms in the sum can be multiplied by elements of $ \\mathscr{A}$. \n\n\n\n\\begin{theorem}\n\\label{graph integral expansion}\nThe integral\n$$\nF_\\gamma(\\epsilon, T , \\alpha )  = \\int_{t_e \\in (\\epsilon,T)^{E(\\gamma)} } f_\\gamma (t_e, \\alpha  )  \\prod \\d t_e\n$$\nhas an asymptotic expansion as $\\epsilon \\to 0$ of the form\n$$\nF_\\gamma(\\epsilon, T , \\alpha  ) \\simeq \\sum f_r(\\varepsilon) \\Psi_r( T,\\alpha)\n$$\nwhere $f_r \\in \\mathscr{A}$, and the $\\Psi_r$ are continuous linear  maps \n$$\n\\mathscr{E}^{\\otimes L(\\gamma ) } \\to  C^{\\infty}((0,\\infty) ) \\otimes \\Omega^\\ast(\\Delta^d) \n$$\nwhere $T$ is the coordinate on $(0,\\infty)$.\n\nFurther, each $\\Psi_r(T,\\alpha)$ has a small $T$ asymptotic expansion\n$$\n\\Psi_r(T,\\alpha) \\simeq \\sum g_r(T) \\int_M \\Phi_{r,k} (\\alpha)\n$$\nwhere \n$$\n\\Phi_{r,k} \\in \\operatorname{PolyDiff}( \\mathscr{E}^{\\otimes L(\\gamma)}, \\dens(M) ) \\otimes \\Omega^\\ast(\\Delta^d). \n$$\nand $g_r$ are smooth functions of $T \\in (0,\\infty)$. \n\\end{theorem}\nThe phrase ``asymptotic expansion'' is to be interpreted in the same sense as before. \n\n\\begin{remark}\n\n\n\nThere is a variant of this result, which incorporates a propagator of the form $P(\\varepsilon,T) + \\delta K_\\varepsilon$, where $\\delta$ is an odd parameter of square zero.  In this case, the graphs we use may have one special edge, on which we put $K_\\varepsilon$ instead of $K_{t_e}$.  Thus, instead of integrating over the parameter $t_e$ for this edge, we specialise to $\\varepsilon$.  \n\\end{remark}\n\\subsection{Asymptotics of heat kernels}\nThe proof is based on the asymptotic expansion of the heat kernels of generalised Laplacians, proved in \\cite{BerGetVer92}.  The operator\n$$\nH  : \\mathscr{E} \\otimes \\Omega^\\ast(\\Delta^d) \\to \\mathscr{E} \\otimes \\Omega^\\ast(\\Delta^d)\n$$\nis a smooth family of generalised Laplacians parametrised by the supermanifold with corners $\\Delta^d \\times \\mbb R^{0,d}$.     A generalised Laplacian on the vector bundle $E$ on $M$ (whose global sections is $\\mathscr{E}$) is specified by a metric on $M$, a connection on $E$ and a potential $F \\in \\Gamma(M, \\End(E))$.    A smooth family of generalised Laplacians is a family where this data varies smoothly; this is equivalent to saying that the operator varies smoothly. We are dealing with a smooth family parametrised by the super-manifold $\\Delta^d \\times\\mbb R^{0,d}$. The metric on $M$ will be independent of the odd coordinates $\\mbb R^{0,d}$, but the parameters for the connection and the potential will both involve Grassmann variables.  (I'm very grateful to Ezra Getzler for explaining this point of view to me.)\n\nLet $x \\in M$.   Let $U \\subset M \\times \\Delta^d$ denote the open subset of points $(y,\\sigma)$ where $d_\\sigma(x,y) < \\varepsilon$.  Normal coordinates on $U$ gives an isomorphism \n$$\nU \\cong B_\\varepsilon^n \\times \\Delta^d\n$$\nwhere $B_\\varepsilon^n$ is the ball of radius $\\varepsilon$ in $\\mbb R^n$, and $n = \\dim M$.  \n\nThus, we get an isomorphism \n$$\nC^{\\infty}( U \\times \\mbb R^{0,d} ) = C^{\\infty}(U) \\otimes \\mathbb C [ \\d t_1 ,\\ldots, \\d t_d ]  \\cong C^{\\infty}(B_\\varepsilon^n) \\otimes \\Omega^\\ast(\\Delta^d)\n$$ \nof $\\Omega^\\ast(\\Delta^d)$ algebras.    \n\nThe vector bundle $E$ becomes a vector bundle (still called $E$) on $B_{\\varepsilon}^{n}$, and we find\n$$\n\\Gamma( U, E) \\otimes \\mathbb C[\\d t_1, \\ldots, \\d t_d ] \\cong \\Gamma(B_\\varepsilon^n, E ) \\otimes \\Omega^\\ast(\\Delta^d) .   \n$$\nThe following is a variant of a result proved in \\cite{BerGetVer92}, following \\cite{MinPle49, BerGauMaz71,McKSin67}. \n\\begin{theorem}\nThere exists a small $t$ asymptotic expansion of $K_t$ which, in these coordinates, is of the form\n$$\nK_t \\simeq  t^{-\\operatorname{dim} M  \/ 2} e^{-\\norm{x-y}^2\/t }  \\sum_{i \\ge 0} t^i \\phi_i\n$$\nwhere $x,y$ denote coordinates on the two copies of $B_\\varepsilon^n$,  and\n$$\n\\phi_i \\in  \\Gamma(B_\\varepsilon^n,E) \\otimes \\Gamma(B_\\varepsilon^n,E) \\otimes \\Omega^\\ast(\\Delta^d).\n$$\nIf we denote\n$$\nK_t^N = t^{-\\operatorname{dim} M  \/ 2}  e^{-\\norm{x-y}^2\/t }  \\sum_{i =0}^N t^i \\phi_i\n$$\nthen for all $l \\in \\mathbb Z_{\\ge 0}$\n$$\n\\norm{ K_t - K_t^N }_l  = O(t^{N - \\dim M \/ 2 - l}).\n$$\nHere $\\norm{\\cdot}_l$ denotes the $C^l$ norm on the space $C^{\\infty}(B_\\varepsilon^n) \\otimes C^{\\infty}(B_\\varepsilon^n) \\otimes \\Omega^\\ast(\\Delta^d)$.  \n\\end{theorem}\nThis precise statement is not proved in \\cite{BerGetVer92}, as they do not use Grassmann parameters.  But, as Ezra Getzler explained to me, the proof in \\cite{BerGetVer92} goes through \\emph{mutatis mutandis}. \n\n\\subsection{Proof of Theorem \\ref{graph integral expansion}}\nIn this proof, I will often avoid mention of the parameter space $\\Delta^d \\times \\mbb R^{0,d}$.  Thus, if $f$ is some expression which depends on this parameter space, I will often abuse notation and write $\\abs{ f }$ for the $C^l$ norm of $f$ as a function of the $\\Delta^d \\times \\mbb R^{0,d}$ variables, for some $l$.  \n\nLet us enumerate the edges of $\\gamma$ as $e_1,\\ldots, e_k$.  Let $t_i = t_{e_i}$, and let us consider the region where $t_1 > t_2 > \\cdots > t_k$.    (Of course we have to consider all orderings of the edges of $\\gamma$).\n\nFor a function $I : E(\\gamma) = \\{1,\\ldots, k\\} \\to \\mathbb Z_{\\ge 0}$, let $\\abs{I} = \\sum I(i)$.    Let $t^I = \\prod t_i^{I(i)}$.  Similarly, if $n \\in \\mathbb Z$, let $t^n = \\prod t_i^n$.\n\nLet \n$$\nO(\\gamma) = \\sum_{v \\in V(\\gamma)} O(v) .\n$$\n\nFor $R > 1$ let \n$$\nA_{R,T} \\subset (0,T)^{E(\\gamma)}\n$$\nbe the region where $t_i^R < t_{j}$ for all $i,j$.  This means that the $t_i$ are all of a similar size.\n\\begin{proposition}\n\\label{prop wicks lemma expansion}\nFor all $r \\ge 0$, there exist\n\\begin{enumerate}\n\\item\n $F_i,G_i \\in \\mathbb Z_{\\ge 0} [ t_1,\\ldots, t_k ] \\setminus \\{0\\} $ for  $1 \\le i \\le c_r$,\n \\item\n polydifferential operators\n$$\\Psi_{i} \\in \\operatorname{Poly Diff} ( \\mathscr{E}^{\\otimes L(\\gamma)}, \\operatorname{Dens}(M) ) \\otimes \\Omega^\\ast(\\Delta^d)     $$ \nfor $1 \\le i \\le c_r$,\n\\end{enumerate}\nsuch that\n$$\n\\abs{ f_\\gamma(t_1,\\ldots, t_k) -  \\sum_{i = 1}^{c_r} \\frac{ F_i (t_1,\\ldots, t_k )^{1\/2}} { G_i (t_1,\\ldots, t_k)^{1\/2} } \\int_M \\Psi_i  (\\alpha  ) } \\le \\norm{\\alpha}_{r + 1 - \\chi(\\gamma) \\dim M \/ 2 + O(\\gamma)(\\abs{E(\\gamma)} + 1) } t_1^{r+1} \n$$\nfor all $\\{t_1,\\ldots, t_k\\} \\in A_{R,T}$ with $t_1 > t_2 > \\cdots  > t_k$ and  $t_1$ sufficiently small. In this expression, $\\chi(\\gamma)$ is the Euler characteristic of the graph.\n\\end{proposition}\n\n\\begin{proof}\nAs above, let\n$$\nK^N_{t}(x,y) =  t^{-\\operatorname{dim} M  \/ 2} \\Psi(x,y) e^{-\\norm{x-y}^2\/t }  \\sum_{i = 0}^N t^i \\phi_i(x,y)\n$$\nbe the  approximation to the heat kernel to order $t^N$.  This expression is written in normal coordinates near the diagonal in $M^2$.  The $\\phi_i(x,y)$ are sections of $E \\boxtimes E$ defined near the diagonal on $M^2$. $\\Psi(x,y)$ is a cut-off function, which is $1$ when $\\norm{x-y} < \\epsilon$ and $0$ when $\\norm{x-y} > 2\\epsilon$. \n\n\n\nWe have the bound\n$$\n\\norm{  K_{t}(x,y) - K^N_{t}(x,y) }_l = O ( t^{N  - \\operatorname{dim} M  \/2 -  l    }  ) .\n$$\n\nThe first step is to replace each $K_{t_i}$ by $K^{N}_{t_i}$ on each edge of the graph.   Thus, let $f_\\gamma^N(t_i,\\alpha)$ be the function constructed like $f_\\gamma(t_i,\\alpha)$ except using $K_{t_i}^N$ in place of $K_{t_i}$. \n\nEach time we replace $K_{t_i}$ by $K_{t_i}^{N}$, we get a contribution of $t_i^{N  - O(\\gamma) - \\operatorname{dim} M \/ 2}$ from the edge $e_i$, times the $O(\\gamma)$ norm of the contribution of the remaining edges, times $\\norm{\\alpha}_{O(\\gamma)}$. \n\nThe $O(\\gamma)$ norm of the contribution of the remaining edges is $\\prod_{j \\neq i} t_j^{ -  O(\\gamma) - \\operatorname{dim} M \/ 2}$.  We are thus left with the bound \n$$\n\\abs {  f_\\gamma(t_i,\\alpha ) -  f_\\gamma^N (t_i,\\alpha )   } <  C    t^{- O(\\gamma) - \\dim M \/ 2  }    t_1^{N} \\norm{\\alpha}_{O(\\gamma) } \n$$\nwhere $C$ is a constant.   (Recall our notation : $t^n$ denotes $\\prod t_i^n$). \n\nIn particular, if the $t_i$ are in $A_{R,T}$, we find that \n$$ \\abs { f_\\gamma(t_i, \\alpha  ) -   f_\\gamma^N(t_i,\\alpha) }   <   \\norm{\\alpha}_{O(\\gamma)} t_1^{N -  \\abs{E(\\gamma)} ( O(\\gamma) + \\dim M \/ 2 )  R + 1 }  $$\nif $t_i \\in A_{R,T}$ and $t_i$ are all sufficiently small.\n\n\n\n\nNext, we construct a  small $t_i$ asymptotic expansion of $f_\\gamma^N(t_i,\\alpha)$.   Recall that $f_\\gamma^N(t_i,\\alpha)$ is defined as an integral over a small neighbourhood of the small diagonal in $M^{V(\\gamma)}$.  Let \n$$n = \\operatorname{dim} M .$$  \nBy using a partition of unity we can consider $f_\\gamma^N(t_i,\\alpha)$ as an integral over a small neighbourhood of zero in $\\mbb R^{n V(\\gamma)}$.  This allows us to express  $f_\\gamma^N(t_i,\\alpha)$ as a finite sum of integrals over $\\mbb R^{n V(\\gamma)}$, of the following form.\n\nFor each vertex $v$ of $\\gamma$, we have a coordinate map $x_v : \\mbb R^{n \\abs{V(\\gamma)}} \\to \\mbb R^n$.  Fix any $\\varepsilon> 0$, and let $\\chi : [0,\\infty) \\to [0,1]$ be a smooth function with $\\chi(x) = 1$ if $x < \\epsilon$, and $\\chi(x) = 0$ if $x > 2 \\epsilon$.  Let us define a cut-off function $\\psi$ on $\\mbb R^{n V(\\gamma)}$ by the formula\n$$\n\\psi = \\chi (  \\norm{\\sum x_v}^2  ) \\chi \\left( \\sum_{v' \\in V(\\gamma)}  \\norm{ x_{v'} -  \\abs{V(\\gamma)}^{-1} \\sum_{v \\in V(\\gamma)} x_v }^2  \\right). \n$$\nThus, $\\psi$ is zero unless all points $x_v$ are near their centre of mass $\\abs{V(\\gamma)}^{-1}\\sum x_v$, and $\\psi$ is zero when this centre of mass is too far from the origin.  \n\n For each $1 \\le i \\le k$, let $Q_i$ be the quadratic for on $\\mbb R^{n \\abs{V(\\gamma)}}$ defined by\n$$\nQ_i ( x) = \n\\begin{cases} \n0 & \\text{ if the edge } e_i \\text{ is a loop, i.e.\\ is attached to only one vertex } \\\\\n\\norm{ x_{v_1} - x_{v_2} }^2 & \\text{ if } v_1, v_2 \\text{ are the vertices attached to the edge } e_i \n\\end{cases}\n$$\n\nWe can write $f_\\gamma^N$ as a finite sum of integrals of the form\n$$\n\\int_{\\mbb R^{n V(\\gamma)} } \\psi    e^{- \\sum Q_i  \/ t_i } \\sum_{ I, K} t^{ I -  \\dim M \/ 2 - O(\\gamma) } \\Phi_{I,K} \\partial_{K,x} \\alpha.\n$$\nIn this expression, \n\\begin{itemize}\n\\item\nThe sum is over $I : E(\\gamma) \\to \\mathbb Z_{\\ge 0}$, with all $I(e) \\le N + O(\\gamma) + 1$, and multi-indices \n$K : V(\\gamma) \\times \\{1,\\ldots,n\\} \\to \\mathbb Z_{\\ge 0}$, with $\\sum K(v,i) \\le O(\\gamma)$.  The notation $\\partial_{K,x}$ denotes \n$$\\partial_{K,x}=  \\prod_{v \\in V(\\gamma), 1 \\le i \\le n}  \\frac {\\partial}{\\partial x_{v,i}^{K(v,i)} }.$$\n\nIn this notation, we are pretending (by trivialising the vector bundle $E$ on some small open sets in $M$) that $\\alpha$ is a function on $\\mbb R^{n V(\\gamma)}$. \n\\item\nThe $\\Phi_{I,K}$ are smooth functions on $\\mbb R^{n V(\\gamma)}$.\n\\end{itemize}\n\nNext, we will use Wick's lemma to compute the asymptotics of this integral.   Let $c = \\left(1\/\\abs{V(\\gamma)} \\right)\\sum x_v$ be the centre of mass function $\\mbb R^{n V(\\gamma)}\\to \\mbb R^n$.       We can perform the integral in two steps, first integrating over the coordinates $y_v =  x_v -  c$,  and secondly by integrating over the variable $c$.  (Of course, there are $\\abs{V(\\gamma)} - 1$ independent $y_v$ coordinates). The quadratic form $\\sum Q_i \/ t_i$ on $\\mbb R^{n V(\\gamma)}$ is non-degenerate on the subspace of $\\mbb R^{n V(\\gamma)}$ of vectors with a fixed centre of mass, for all $t_i \\in (0,\\infty)$.  Thus, the integral over the variables $y_v$ can be approximated with the help of Wick's lemma.  \n\n\nLet us order the set $V(\\gamma)$ of vertices as $v_1,v_2, \\ldots, v_m$.  We will use the coordinates $y_1,\\ldots,y_{m-1}$, and $c$ on $\\mbb R^{n V(\\gamma)}$.  Then  $f^N_\\gamma$  is a finite sum of integrals of the form\n$$\n\\int_{w \\in \\mbb R^n} \\chi ( \\abs{w}^2 ) \\int_{y_1,\\ldots,y_{m-1} \\in \\mbb R^n} \\chi ( \\sum \\norm{y_v}^2 )  e^{- \\sum Q_i(y) \/ t_i} \\sum t^{I - O(\\gamma)  - \\dim M \/ 2} \\Phi_{I,K} \\partial_{K, w,y_i} \\alpha.\n$$\nHere we are using the same notation as before, in these new coordinates.    \n\nTo get an approximation to the inner integral, we take a Taylor expansion of the functions $\\alpha$ and $\\Phi_{I,K}$ around the point $\\{y_i = 0, w\\}$, only expanding in the variables $y_i$.  We take the expansion to order $N$.  We find, as an approximation to the inner integral, an expression of the form\n$$\n\\int_{y_1,\\ldots,y_{m-1} \\in \\mbb R^n} e^{- \\sum Q_i(y) \/ t_i} \\sum t^{I -  \\dim M \/ 2 - O(\\gamma)} y^K c_{K,I,L} \\left( \\prod \\frac{\\partial^{L_i}} {\\partial^{L_i} y_i} \\frac{\\partial ^{L_w}} {\\partial^{L_w} w}  \\alpha  \\right)_{y_i = 0}\n$$\nwhere the sum is over a finite number of multi-indices $I,K,L$, and $c_{K,I,L}$ are constants.\n\nWe can calculate each such integral by Wick's lemma.   The application of Wick's lemma involves inverting the quadratic form $\\sum Q_i(y) \/ t_i$.    Let $A = A(t_i)$ denote the matrix of the quadratic form $\\sum Q_i(y) \/ t_i$; this is a square matrix of size $(\\dim M) \\abs{V(\\gamma)}$, whose entries are sums of $t_{i}^{-1}$.   Note that $\\left( \\prod_{i = 1}^{k} t_i \\right)  A$ has polynomial entries.  \n\nLet \n$$\nP_\\gamma (t_i) = \\det \\left( \\left( \\prod_{i = 1}^{k}t_i \\right) A \\right). \n$$\nThis is the graph polynomial associated to $\\gamma$ (see \\cite{BloEsnKre06}).   One important property of $P_\\gamma$ is that it is a sum of monomials, each with a non-negative integer coefficient. \n\nWe can write\n$$\nA^{-1} = P_\\gamma^{-1} B\n$$\nwhere the entries of $B$ are polynomial in the $t_i$.   Note also that \n$$\n\\det A = P_\\gamma t^{-(\\dim M)(\\abs{V(\\gamma)} - 1)  \/ 2 } .\n$$\n(The expansion from Wick's lemma gives an overall factor of $(\\det A)^{-1\/2}$).\n\nThus, we find, using Wick's lemma, an approximation of the form \n$$\nf^N_\\gamma(t_i) \\simeq P_\\gamma^{-1\/2} \\sum_{l \\ge 0, I: E(\\gamma) \\to \\tfrac{1}{2}\\mathbb Z_{\\ge 0}} P_\\gamma^{-l} t^{I - \\dim M \/ 2 - O(\\gamma) } \\int_M \\Psi_{l,I}(\\alpha)\n$$\nwhere the $\\Psi_{l,I}$ are polydifferential operators\n$$\n\\Psi_{l,I} : \\mathscr{E}^{\\otimes L(\\gamma)} \\to \\dens(M)\n$$\n and the sum is finite (i.e.\\ all but finitely many of the $\\Psi_{l,I}$ are zero).    \n \nThis expansion is of the desired form; it remains to bound the error term.\n\\begin{lemma}\nThe error term in this expansion is bounded by $$t_1^{R (N+1 + (\\dim M) \\chi(\\gamma)\/2 - \\abs{E(\\gamma)} O(\\gamma) ) } \\norm{\\alpha}_{N + 1 + O(\\gamma)}$$ for $N$ sufficiently large and $t_1$ sufficiently small.  Here $\\chi(\\gamma)$ is the Euler characteristic of the graph. \n\\end{lemma}\n\\begin{proof}\n\n\nThe error in this expansion arises from the error in the Taylor expansion of the functions $\\alpha, \\Phi_{I,K}$ around $0$, and from the fact that we are neglecting the cut-off function.  Thus, if $N$ is sufficiently large, the magnitude of the error in the expansion can be bounded by an expression of the form\n$$\nt^{- \\dim M \/ 2 - O(\\gamma) }\\int_{w \\in \\mbb R^n} \\chi(w)   \\int_{y_1,\\ldots, y_{m-1} \\in  \\mbb R^{n}  } \\abs{ \\prod_e e^{-\\sum Q_i(y)\/t_i} \\sum_{K} \\phi_K \\partial_{K,w,y_j} \\alpha  } . \n$$\nHere, the sum is over multi-indices $K$ for the variables $y_i,w$, each $\\phi_K$ is homogeneous of order $N+1$ as a function of the variables $y_i$,  and $\\abs{K} \\le N+1+O(\\gamma)$, so we are differentiating $\\alpha$ at most this number of times.\n\nThis integral only decreases if we  decrease each $t_i$.  Since $t_i > t_1^{R}$, we find we can bound the integral by the corresponding integral using the quadratic form $\\sum Q_i(y) \/ t_1^{R}$. Thus, we find a bound of   \n$$\nt_1^{- R \\abs{E(\\gamma)} ( \\dim M \/ 2 + O(\\gamma) )}\\int_{w \\in \\mbb R^n} \\chi(w)   \\int_{y_1,\\ldots, y_{m-1} \\in  \\mbb R^{n}  } \\abs{ \\prod_e e^{-\\sum Q_i(y)\/t_1^R} \\sum_{K} \\phi_K \\partial_{K,w,y_j} \\alpha  } . \n$$\nWick's lemma bounds this integral by\n$$\n\\operatorname{det}(\\sum Q_i(y)\/ t_1^{R})^{-1\/2} t_1^{R (N+1 - \\abs{E(\\gamma)}(\\dim M \/  2 + O(\\gamma)) )} \\norm{\\alpha}_{N+1 + O(\\gamma) }\n$$\nAlso, \n$$\n\\operatorname{det}(\\sum Q_i(y)\/ t_1^{R})^{-1\/2} = C t_1^{  R\\operatorname{dim} M  \\abs{V(\\gamma)} \/2    }\n$$\nwhere $C$ is a constant, to yield the desired bound. \n\\end{proof} \n\n\\end{proof} \n\nA \\emph{subgraph} $\\gamma'$ of a graph $\\gamma$ is given by the set  of edges $E(\\gamma') \\subset E(\\gamma)$.   The vertices of the subgraph $\\gamma'$  are the ones that adjoin edges in $E(\\gamma')$.     The legs of $\\gamma'$ are the half-edges of $\\gamma$, which adjoint vertices of $\\gamma'$, but which are not part of any edge of $\\gamma'$.  \n\nLet us fix a proper subgraph $\\gamma'$, and let us enumerate the edges of $\\gamma$ as $e_1,\\ldots, e_k$,  where $e_1,\\ldots, e_l \\in E(\\gamma')$ and $e_{l+1},\\ldots, e_k \\in E(\\gamma) \\setminus E(\\gamma')$.  Let $t_i = t_{e_i}$.  \n\n\nLet \n$$ \\phi_{ \\gamma,\\gamma' } (t_{l+1}, \\ldots, t_k ) =  \\int_{ t_1,\\ldots, t_l \\in (t_{l+1}^{1\/R} , T)   }  f_\\gamma(t_1,\\ldots, t_k) \\d t_1 \\cdots \\d t_l  $$\n\\begin{lemma}\nLet $R \\gg 0$ be sufficiently large. Then for all $r > 0$,  there exist $m_r \\in \\mathbb Z_{\\ge 0}$,  a finite number of $g_i \\in \\mathscr{A}$, $F_i, G_i \\in \\mathbb Z_{\\ge 0 }[t_{l+1},\\ldots,t_k]\\setminus\\{0\\}$, and continuous linear maps $\\Psi_i : \\mathscr{E}^{\\otimes L(\\gamma)} \\to \\Omega^\\ast(\\Delta^d) \\otimes C^{\\infty}((0,\\infty))$ such that \n$$\n\\abs{ \\phi_{\\gamma,\\gamma'} (t_{l+1},\\ldots, t_k ) -  \\sum g_i(t_{l+1} )\\frac{ F_i (t_{l+1}, \\ldots, t_k )^{1\/2} } {G_i(t_{l+1}, \\ldots, t_k)^{1\/2} }  \\Psi_i ( T,\\alpha)     } < \\norm{\\alpha}_{m_r} t_{l+1}^{r+1}\n$$\nfor all $t_{l+1} >  \\cdots >  t_k > 0$ with $t_{l+1}^R < t_k$, and $t_{l+1}$ sufficiently small.\n\nFurther, the $\\Psi_i$ admit small $T$ asymptotic expansions \n$$\n\\Psi_i(T,\\alpha) \\simeq \\sum \\eta_{i,k}(T)\\int_M \\Phi_{i,k}(\\alpha)\n$$\nwhere \n$$\\Phi_{i,k}  \\in \\operatorname{Poly Diff}(\\mathscr{E}^{\\otimes L(\\gamma)}, \\dens(M) ) \\otimes \\Omega^\\ast(\\Delta^d) $$\nand $\\eta_{i,k}(T)$ are smooth functions of $T$.\n\\label{lemma subgraph}\n\\end{lemma}\n\n\\begin{proof}\nWe can write  \n$$f_\\gamma(t_1,\\ldots, t_k, \\alpha)  = f_{\\gamma'}(t_{l+1}, \\ldots, t_k, \\alpha \\otimes K_{t_1} \\otimes \\cdots \\otimes K_{t_l}). $$ The right hand side of this equation denotes the graph integral for $\\gamma'$ with inputs being tensor products of the heat kernels $K_{t_i}$, for $1 \\le i \\le l$, and $\\alpha$.\n\nThe starting point of the proof is Proposition \\ref{prop wicks lemma expansion} applied to each connected component of the graph $\\gamma'$.  This proposition implies that we can approximate the contribution of each such connected component by a local functional applied to the legs of that connected component.  If  we approximate the contribution of each connected component in this way, we are left with a graph integral $\\Psi_{\\gamma\/\\gamma'}(t_1,\\ldots, t_l, \\alpha)$ for the quotient graph $\\gamma \/ \\gamma'$, with certain local functionals at the vertices of $\\gamma \/ \\gamma'$.   \n\nMore formally, Proposition \\ref{prop wicks lemma expansion} implies that  there exists a finite number of $F_i, G_i \\in \\mathbb Z_{\\ge 0}[t_{l+1},\\ldots,t_k]\\setminus\\{0\\}$, and $\\Psi^i_{\\gamma \/ \\gamma' } (t_1,\\ldots, t_l , \\alpha )$, which are graph integrals for $\\gamma \/ \\gamma'$,  such that \n\\begin{multline*}\n\\abs{f_{\\gamma}(t_{1}, \\ldots, t_k, \\alpha)  -  \\sum \\frac{ F_i (t_{l+1}, \\ldots, t_k )^{1\/2} } {G_i(t_{l+1}, \\ldots, t_k)^{1\/2} }    \\Psi^i_{\\gamma \/ \\gamma' } (t_1,\\ldots, t_l , \\alpha )  } \\\\ < t_{l+1}^{r+1} \\norm{\\alpha \\otimes K_{t_1} \\otimes \\cdots \\otimes K_{t_l}}_{r + 1 -\\chi(\\gamma) \\dim M \/ 2 + O(\\gamma)(\\abs{E(\\gamma)} + 1) }  \n\\end{multline*}\nfor all $t_1,\\ldots,t_k$ with $t_{l+1} > \\ldots > t_k$, $t_{l+1}^R < t_k$, and $t_{l+1}$ sufficiently small.\n\nWe are only interested in the region where $t_i^R > t_{l+1}$ if $1 \\le i \\le l$.  Note that \n\\begin{align*}\n\\norm{K_{t_i}}_{r + 1 -\\chi(\\gamma) \\dim M \/ 2 + O(\\gamma) (\\abs{E(\\gamma)}+1)} &= O(t_i^{(\\chi(\\gamma) -1)\\dim M\/2    - r -1 - O(\\gamma)(\\abs{E(\\gamma)}+1) }  )\\\\ & = O( t_{l+1}^{R^{-1}\\left(( \\chi(\\gamma) -1)\\dim M\/2    - r -1 - O(\\gamma)(\\abs{E(\\gamma)}+1)  \\right) }   )\n\\end{align*}\nSince we can take $R$ as large as we like, this contribution is small, and we find, \n$$\n\\abs{f_{\\gamma}(t_{1}, \\ldots, t_k, \\alpha)  -  \\sum \\frac{ F_i (t_{l+1}, \\ldots, t_k )^{1\/2} } {G_i(t_{l+1}, \\ldots, t_k)^{1\/2} }    \\Psi^i_{\\gamma \/ \\gamma' } (t_1,\\ldots, t_l , \\alpha )  } \\\\ < t_{l+1}^{r ( 1 - 1 \/ R ) } \\norm{\\alpha}_{m_r}  \n$$\nfor  some $m_r \\gg 0$.  \n\nWe want to integrate over the variables $t_1,\\ldots, t_l$, in the region $(t_{l+1}^{1\/R} , T)$.  The integral \n$$\n\\int_{t_1,\\ldots, t_l \\in (t_{l+1}^{1\/R}, T) } \\Psi^i_{\\gamma \/ \\gamma' } (t_1,\\ldots, t_l , \\alpha )   \\d t_1 \\cdots \\d t_l\n$$\nis approximated (by induction) using Theorem \\ref{graph integral expansion}, applied to the graph $\\gamma \/ \\gamma'$, with $t_{l+1}^{1\/R}$ playing the role of $\\varepsilon$.   It is easy to see that this yields the desired approximation of $\\phi_{\\gamma , \\gamma ' }(t_{l+1},\\ldots, t_k,\\alpha)$. \n\\end{proof}\n\n\nLet $\\gamma' \\subset \\gamma$ be a subgraph.  \nLet $A_{R,T}^{\\gamma'} \\subset (0,T)^{E(\\gamma)}$ be the open subset where \n\\begin{align*}\nt_e^R > t_{e'} & \\text{ if } e \\in E(\\gamma) \\setminus E(\\gamma') \\text{ and } e' \\in E(\\gamma') \\\\\nt_e^R < t_{e'} & \\text{ if } e,e' \\in E(\\gamma') .\n\\end{align*} \nThis means that the lengths of the edges of the subgraph $\\gamma'$ are all around the same size, and are all much smaller than the lengths of the other edges. \n\n\\begin{lemma}\nFix $R \\gg 0$, and $0 < T < \\infty$.  Then the closures of the regions\n $A_{R^{2^k},T}^{\\gamma'} $, \nwhere $0 \\le k \\le \\abs{E(\\gamma)}$ and $\\gamma' \\subset \\gamma$ is non-empty, cover $(0,T)^{E(\\gamma)}$. (The regions $A_{R^{2k},T}$ appear as $A_{R^{2k}, T}^{\\gamma}$, where $\\gamma$ is considered as a subgraph of itself). \n\\end{lemma}\n\\begin{proof}\nLet $\\{t_e\\} \\in (0,T)^{E(\\gamma)}$.  As before, label the elements of $E(\\gamma)$ by $\\{1,2,\\ldots, k \\}$,  in such a way that $t_1 \\ge t_2 \\ge \\cdots \\ge t_{k}$.  \n\nEither $t_j^R  \\ge t_{k}$ for all $j < k$, or, there is a smallest $i_1 < k$ such that $t_{i_1}^{R} \\le t_{k}$.  In the first case, we are done, as then $\\{t_e\\} \\in \\overline{ A_{R,T}^{\\gamma'}} $ where $\\gamma'$ is the subgraph with the single edge corresponding to $t_{k}$.  \n\nSuppose the second possibility holds.  Then either for all $j < i_1$,  $t_j^{R} \\ge t_{i_1}$,   or there exists a smallest $i_2 < i_1$ with $t_{i_2}^R \\le t_{i_1}$.  In the first case, we're done, as we are in $\\overline{ A_{R,T}^{\\gamma'} }$ where $\\gamma'$ is the subgraph whose edges correspond to $t_{i_1}, t_{i_1+1}, \\ldots, t_{k}$.  \n\nAgain, let's suppose the second possibility holds.  Then $t_{i_2}^{R^2} \\le t_{i_1}^R \\le t_{k}$.    Either, for all $j < i_2$, $t_j^{R^2} \\ge t_{i_2}$,  and then we are in $\\overline{ A_{R^2,T}^{\\gamma'} }$ where $\\gamma'$ is the subgraph whose edges correspond to $t_{i_2}, t_{i_2+1} , \\ldots, t_k$.\n\nOtherwise, there is some smallest $i_3 < i_2$ with $t_{i_3}^{R^2} \\le t_{i_2}$.  Then $t_{i_3}^{R^4} \\le t_{k}$.   And so forth.\n\nWe eventually end up either finding ourselves in one of the regions $\\overline { A_{R^{2^k}, T}^{\\gamma'} }$, for some non-empty proper subgraph $\\gamma' \\subset \\gamma$, or we find some $i_{k+1} = 1$, so we are in  $\\overline{ A_{R^{2^{k}}, T}^{\\gamma} } = \\overline{ A_{R^{2^{k}}, T} }$.  \n\\end{proof} \n\\begin{definition}\nAn open subset $U$ of $(0,T)^{E(\\gamma)}$ is called \\emph{good} if it is a subset of some  $A_{R,T}^{\\gamma'}$ which is cut out by a finite number of inequalities $t_e^{R^n} > t_{e'}$, where $l \\in \\mathbb Z_{\\ge 0}$ and both $e,e' \\in E(\\gamma')$. \n\\end{definition}\n\\begin{lemma}\nThe intersection of any two good regions is good. \n\\end{lemma}\nNow, we have seen that $(\\varepsilon,T)^{E(\\gamma)}$ is covered by the closures of a finite number of good regions.  Thus, we can write \n$$F_\\gamma (\\varepsilon , T, \\alpha ) =  \\int_{(\\varepsilon,T)^{E(\\gamma)} } f_\\gamma ( t_e ,\\alpha ) \\prod_{e \\in E(\\gamma)} \\d t_e $$ as an alternating sum of integrals of $f_\\gamma(t_e)$ over  $U \\cap (\\varepsilon,T)^{E(\\gamma)}$, where $U$ is a good subset of $(0,T)^{E(\\gamma)}$. Thus, in order to understand the small $\\varepsilon$ asymptotic expansions of $F_\\gamma(\\varepsilon, T,\\alpha)$, it suffices to consider the integrals of $f_\\gamma$ over such regions.\n\n\n\\begin{lemma}\nLet us fix $R$ to be a (sufficiently large) integer.\n\nLet $U \\subset (0,T)^{E(\\gamma)}$ be a good subset.  Then the integral\n$$\nF_{\\gamma,U}(\\varepsilon,T,\\alpha) = \\int_{U \\cap (\\varepsilon,T)^{E(\\gamma)} } f_\\gamma ( t_e ,\\alpha ) \\prod_{e \\in E(\\gamma)} \\d t_e\n$$\nadmits a small $\\varepsilon$ asymptotic expansion\n$$F_{\\gamma,U}(\\varepsilon,T,\\alpha)  \\simeq \\sum \\phi_r(\\varepsilon) \\Psi_r( T,\\alpha)$$\nwhere $\\phi_r \\in \\mathscr{A}$ and $\\Psi_r : \\mathscr{E}^{\\otimes L(\\gamma)} \\to \\Omega^\\ast(\\Delta^d)\\otimes C^{\\infty}((0,T))$ are continuous maps.\n\nEach $\\Psi_r(T,\\alpha)$ admits a small $T$ asymptotic expansion\n$$\n\\Psi_r(T,\\alpha) \\simeq \\sum g_k(T)\\int_M \\Phi_{r,k}(\\alpha)\n$$\nwhere $g_k$ is a smooth function of $T$, and $\\Phi_{r,k} \\in \\operatorname{Poly Diff}(\\mathscr{E}^{\\otimes L(\\gamma)}, \\dens(M) ) \\otimes \\Omega^\\ast(\\Delta^d)$. \n\\end{lemma}\n\\begin{proof}\nThis  follows from Proposition \\ref{prop wicks lemma expansion} and Lemma \\ref{lemma subgraph}.  \n\nIndeed, we can assume (without loss of generality) that \n$$\nU \\subset  \\{ t_1,\\ldots, t_k \\mid t_i^N > t_{l+1} \\text{ for all } 1 \\le i \\le l  , \\quad t_{l+1} >  t_{l+2} > \\cdots > t_k, \\quad t_{l+1}^N < t_k     \\} \n$$\nis an open subset cut out by a finite number of inequalities of the form $t_i^{a} > t_j$, where $a \\in \\mathbb Z_{> 0}$ and $l+1 \\le i,j \\le k$.\n\nThen let\n$$\n\\eta(t_{l+1}, T, \\alpha)  = \\int_{(t_1,\\ldots, t_k) \\in U} f_\\gamma( t_1,\\ldots, t_k , \\alpha) \\d t_1 \\cdots \\widehat{\\d t_{l+1} } \\cdots \\d t_k .\n$$\nHere, we're integrating over all variables except $t_{l+1}$.\n\nThen  Proposition  \\ref{prop wicks lemma expansion} (if $l = 0$) or Lemma \\ref{lemma subgraph} (if $l > 0$) imply that $\\eta(t_{l+1}, T,\\alpha)$ has a nice $t_{l+1}$ asymptotic expansion. More precisely, for all $r \\in \\mathbb Z_{> 0}$, there exist $m_r \\in \\mathbb Z_{> 0}$, and a finite number of functions $g_i(t_{l+1})$, $\\psi_i(T,\\alpha)$, such that\n\\begin{enumerate}\n\\item\n$$\\abs{ \\eta(t_{l+1}, T, \\alpha) -  \\sum g_i(t_{l+1}) \\psi_i(T,\\alpha)  } < t_{l+1}^{r+1} \\norm{\\alpha}_{m_r}   $$\n\\item\nthe $\\psi_i$ are continuous linear maps $\\psi_i : \\mathscr{E}^{\\otimes L(\\gamma)} \\to \\Omega^\\ast(\\Delta^d) \\otimes C^{\\infty}((0,\\infty))$, which admit a small $T$ asymptotic expansion \n$$\n\\psi_i(T,\\alpha) \\sim \\sum \\zeta_{i,k}(T) \\int_M \\phi_{i,k}(\\alpha)\n$$ where \n$$\\phi_{i,k}  \\in \\operatorname{Poly Diff}(\\mathscr{E}^{\\otimes L(\\gamma)}, \\dens(M) ) \\otimes \\Omega^\\ast(\\Delta^d) $$\nand $\\zeta_{i,k}(T)$ are smooth functions of $T$.\n\\item\n$$\n\\int_{\\varepsilon}^1 g_i(t_{l+1}) \\d t_{l+1} \\in \\mathscr{A}\n$$\n\\end{enumerate}\n(The third part follows from the particular form of the terms of the expansions proved in Proposition \\ref{prop wicks lemma expansion} and Lemma \\ref{lemma subgraph}.)\n\nNow, it remains to integrate out the variable $t_{l+1}$.  Let\n$$\n\\eta_r (t_{l+1}, T, \\alpha) =  \\sum g_i(t_{l+1}) \\psi_i(T,\\alpha)\n$$\nbe the approximation to order $t_{l+1}^{r+1}$ to $\\eta(t_{l+1}, T,\\alpha)$.  \n\nThen\n$$\n\\abs{ \\int_\\varepsilon^T \\eta \\d t_{l+1}  - \\left( \\int_0^T (\\eta - \\eta_r) \\d t_{l+1}   + \\int_1^T \\eta_r \\d t_{l+1} +    \\int_\\varepsilon^1 \\eta_r \\d t_{l+1}  \\right) } < \\varepsilon^{r+2} \\norm{\\alpha}_{m_r} .\n$$\nThus, \n$$\\int_0^T (\\eta - \\eta_r) \\d t_{l+1}   + \\int_1^T \\eta_r \\d t_{l+1} +    \\int_\\varepsilon^1 \\eta_r \\d t_{l+1} $$\ngives the desired small $\\varepsilon$ asymptotic expansion.  Note that the integral in the first term converges.   The first two terms are independent of $\\varepsilon$, and the third term is in $\\mathscr{A} \\otimes \\Hom ( \\mathscr{E}^{\\otimes L(\\gamma)}, \\Omega^\\ast(\\Delta^d) \\otimes C^{\\infty}((0,\\infty)) )$, as desired. \n\nIt's easy to check that the small $T$ asymptotic expansion of the approximation above is in terms of local functionals of $\\alpha$, as desired. \n\n\n\n\\end{proof} \nThis completes the proof of Theorem \\ref{graph integral expansion}.    \n\nThe proof of the variant result, when we include a propagator of the form $P(\\varepsilon,T) + \\delta K_\\varepsilon$, is identical, except that instead of integrating over the smallest variable $t_k$, we specialise $t_k = \\varepsilon$.  In our definition of the algebra $\\mathscr{A}$, we included functions of $\\varepsilon$ of the form\n$$\nf(\\epsilon )  = \\int_{U \\subset (\\epsilon,1)^{k-1} } \\frac{  F(t_1,\\ldots,t_k = \\varepsilon)^{1\/2} } { G(t_1,\\ldots, t_k = \\varepsilon)^{1\/2} } \\d t_1 \\cdots \\d t_{k-1}\n$$\nwhere $F, G \\in \\mathbb Z_{\\ge 0}[ t_1, \\ldots, t_k]$ and $U$ is cut out by equations of the form $t_i^l > t_j$, for $i,j < k$.  Functions of this kind arise when we have a $\\delta K_\\varepsilon$ term in the propagator. \n\n\n\n\n\\def$'${$'$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLight carrying OAM, which corresponds to beams with a helical twist of phase, has gained interest due to its applications in various fields of optics. Generally termed as `Optical Vortex' in classical optics, light having phase singularity has been used in the generation of bright optical solitons \\cite{swartzlander1, tikhonenko}, study of optical chronograph \\cite{swartzlander2, berkhout} and trapping of particles \\cite{gahagan}. The transverse intensity distribution of an optical vortex mode is radially symmetric in nature that originates from the azimuthal dependence of the phase. Among such classes of phase singular modes, Laguerre-Gaussian (LG) modes are most commonly used in practical applications due to their profile stability upon free space propagation. Since the size of a normal optical vortex strongly depends on its topological charge \\cite{reddydivergence}, they have their limitations in applications involving transmission of OAM modes through optical fibers \\cite{li, gregg} in communication \\cite{willner}. Due to this, projective measurements based on `phase-flattening' technique become difficult for higher-order orbital angular momentum (OAM) modes \\cite{qassim}. Bessel-Gaussian (BG) modes are another class of structured light modes carrying OAMs, which come from the non-diffracting solution of paraxial wave equation \\cite{gori1987bessel, mcgloin2005bessel}. The transverse field distribution of BG modes is described by a Bessel function \\cite{korenev2002bessel}. They exhibit non-diffracting nature \\cite{durnin1987diffraction, cruz2012observation} as well as self-healing property \\cite{litvin2009conical, mclaren2014self} when propagated in space. In most of the practical cases, Bessel-Gaussian modes are generated using axicons \\cite{arlt2000generation} and spatial light modulators (SLM) \\cite{leach2006generation}. Fourier transformation of BG modes gives a class of size invariant modes known as Perfect optical vortex (POV) \\cite{vaity2015}. They are a new class of ring-shaped beams carrying OAM. The size of a POV modes does not change with respect to the change in OAM values.\n\nThe concept of OAM has been well established in quantum information too. Access to higher dimensions in Hilbert space makes OAM suitable for encoding higher amounts of information per photon. SPDC is the most common workhorse for the generation of single photons carrying OAM \\cite{boydbook, mair}. The paired photons generated in SPDC are inherently correlated in their OAM values \\cite{walborn2004entanglement}. It has been experimentally verified that the amplitude, as well as the helical phase of an optical vortex pump, gets transferred to SPDC photons \\cite{vicuna, anwar2018direct}. Based on the OAM selection rule in SPDC process, single photons carrying OAM are generated in `heralding' configuration by pumping an optical vortex beam into a nonlinear crystal and projecting one photon from the generated pair to `zero OAM' (Gaussian) mode \\cite{lal2020photon}. Single-photons carrying OAM find potential applications in quantum information \\cite{erhard}, quantum gates \\cite{babazadeh}, quantum memories \\cite{nicolas} etc. \n\nApart from the Gaussian modes, OAM correlations among the SPDC photon-pairs are explored using structured pump modes \\cite{romero}. Such systems can be used to generate versatile high-dimensional quantum states by careful shaping of pump beam having additional spiral modes using different pump engineering techniques \\cite{liu2018coherent, kovlakov2018quantum, anwar2020selective}. Also, it was shown that the use of BG modes in photon-pair detection using OAM projection improved the dimensionality of the OAM state generated. However, the potential of the class of such modes in further enhancement of the heralded single-photon detection and dimensionality of the quantum state is yet unexplored. In this article, we discuss the use of POV modes as the pump in a parametric down-conversion process to generate heralded single photons carrying OAM. By exploiting the OAM independence on the size of POV modes, we show that the heralding efficiency of the single photons with POV pump is higher than that with a NOV pump. We also show that a configuration with POV pump beams with generated SPDC photon-pairs projected to BG modes give improved fidelity than the configuration with an LG mode projection.\n\n\\section{Helical modes and their generations} \\label{gen-non-diffr-size-inv-modes}\nDepending on the transverse intensity distribution, there are three main classes of optical modes carrying azhimuthal phase. All those modes have phase singularity at the center where the intensity is zero. Here we introduce different helical modes and an experimental method to generate them.\n\\begin{figure}[h]\n\t\\centering\n    \\includegraphics[width=0.46\\textwidth]{POV-generation-scheme.pdf}\n\t\\caption{Experimental schemes to generate NOV, then BGV, and then POV: (a) Conversion of a Gaussian beam to helical modes Using spiral phase plate, axicon lens and a spherical lens. (b) Generation of helical modes by diffraction of fork grating hologram imprinted on a SLM.} \\label{pump_prep_POV}\n\\end{figure}\n\n\\subsection{Size-variant: Normal optical vortex (NOV)}\nA typical field distribution of a normal optical vortex (NOV) mode of topological charge $\\ell$ in polar coordinates $(r,\\theta)$, is given by\n\\begin{equation}\nE_{\\text{NOV}}^{\\ell}(r,\\theta)=\\sqrt{\\frac{2^{\\abs{\\ell}+1}}{\\pi w^2|\\ell|!}}\\left(\\frac{r}{w}\\right)^{|\\ell|}\\exp\\left(-\\frac{r^2}{w^2}\\right)\\exp(i\\ell\\theta),\n\\label{NOV}\n\\end{equation}\nwhere $w$ is the radius of the mode. As shown in Fig. \\ref{pumpmodes}, the size of NOV modes increases with OAM due to the OAM-dependant radial term $r^\\abs{\\ell}$ in the above expression.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.46\\textwidth]{pumpmodes.pdf}\n  \\caption{(a) Theoretical (top) and experimental (bottom) intensity distributions of NOV and POV beams of order varying from $\\ell=1$ to $25$. (b) and (c) compares the intensity distribution\nof a NOV and POV with $\\ell=1$ and $\\ell=25$.}\\label{pumpmodes}\n\\end{figure}\n\n\\subsection{Size-invariant: Perfect optical vortex (POV)}\nA new class of optical vortex beam, termed as `perfect optical vortex', was introduced by Ostrovsky \\textit{et. al.} that solves size effects of a normal vortex \\cite{ostrovskygeneration}. Conventionally, POV of order $\\ell$ are formed as a Fourier transform of Bessel-Gaussian vortex (BGV) of order $\\ell$ \\cite{vaity2015}. The normalized expression for a BGV mode of azhimuthal order $\\ell$ is written as\n\\begin{equation}\nE_{\\rm{BG}}^{\\ell}(r,\\theta)=\\sqrt{\\frac{2e^{\\sfrac{1}{4}}}{\\pi w^2I_\\ell(\\sfrac{1}{4})}}J_\\ell(k_rr)\\exp\\left(-\\frac{r^2}{w^2}\\right)\\exp(i\\ell\\theta).\n\\label{BG}\n\\end{equation}\nThe Fourier transform of the above equation leads to the field of a typical perfect optical vortex of order $\\ell$ is given as\n\\begin{equation}\nE_{\\rm{POV}}^{\\ell}(r,\\theta)=i^{\\ell-1}\\frac{w_g}{w_o}\\exp(i\\ell\\theta)\\exp\\left(-\\frac{(r^2+r_r^2)}{w_o^2}\\right)I_\\ell\\left(\\frac{2r_rr}{w_o^2}\\right),\n\\label{POV-expression}\n\\end{equation}\nwhere $w_g$ is the waist radius of the initial Gaussian beam, $w_o=2f\/kw_g$ is half of the ring width and $r_r$ is radius of the ring. Here $f$ is the focal length of the Fourier lens and $k$ is the magnitude of the wave-vector of the light beam. \n\n\\subsection{Experimental generation of NOV and POV modes}\nFigure \\ref{pump_prep_POV}(a) shows a simple method to generate POV beams using a spiral phase plate (SPP), an axicon, and a lens. A Gaussian beam passing through an SPP of topological order $\\ell$ generates an NOV of OAM $\\ell$. This beam then passing through the axicon becomes a BGV, and Fourier transformation of the BGV using lens generates a POV beam. Alternatively, these helical modes can also be generated from the diffraction of a laser beam using a fork-grating pattern imprinted onto SLM, as shown in Fig. \\ref{pump_prep_POV}(b). The first diffracted order is imaged in the far-field plane will contain the desired helical mode, or NOV modes. A Gaussian light incident on a fork-grating hologram obtained through interference converts into BG modes in the immediate plane after SLM, and the corresponding POV mode is generated at the far-field plane.\n\nIn Fig \\ref{pumpmodes}(a), the experimental intensity distribution of normal optical vortex pump is compared with the perfect optical vortex pump of different orders $\\ell$=1, 5, 10, 15, 20 and 25. For a NOV pump, Eqn.1 shows an $\\ell$ dependence on the radial terms. So, the size of the OAM modes increases with $\\ell$. However, for a POV pump, radial terms are independent of $\\ell$ (\\eqref{BG}), and thus the size of modes remains the same for higher OAM values. The spatial profiles are shown in Fig \\ref{pumpmodes}(b) and (c) for NOV and POV, respectively. From the intensity profile, one can observe that the spatial profile change considerably for a NOV, while remains unchanged for POVs.\n\n\\section{Parametric down conversion of helical modes} \\label{PDC}\nFirst, we will discuss the theory of SPDC with NOV and POV pump beams. A comparative study of down-converted photons with these pump modes is important to understand their two-photon modal spectra.\n\n\\subsection{Theory} \\label{PDC:theory}\nIn the perturbative treatment of spontaneous parametric down-conversion process, interaction of pump ($p$), signal ($s$) and idler ($i$) modes in a medium (Non-linear $\\chi^{(2)}$ crystal) is represented by an interaction Hamiltonian $\\mathcal{H}_I$. The initial state is a vacuum state $\\vert0\\rangle_s\\vert0\\rangle_i$, therefore the output state of SPDC is approximated as\n\\begin{equation}\n\\vert\\Phi\\rangle\\approx\\left(1-\\frac{i}{\\hbar}\\int_{0}^{\\tau}\\mathcal{H}_I(t)dt\\right)\\vert0\\rangle_s\\vert0\\rangle_i.\n\\label{biphotonstate}\n\\end{equation}\nThe biphoton mode function of the generated twin photons in transverse momentum coordinates ($\\mathbf{k}$) is obtained as\n\\begin{equation}\n\\Phi(\\mathbf{k}^{\\perp})=\\langle\\mathbf{k}^{\\perp}_s\\vert\\langle-\\mathbf{k}^{\\perp}_i\n\\vert\\Phi\\rangle,\n\\label{biphotonmode}\n\\end{equation}\nwhere $\\mathbf{k}^{\\perp}_s$ and $-\\mathbf{k}^{\\perp}_i$ represents the transverse position in the momentum coordinates for signal and idler respectively. On simplification, the biphoton mode \nfunction in transverse momentum coordinates is given by\n\\begin{equation}\n\\Phi(\\mathbf{k}_s^{\\perp},\\mathbf{k}_i^{\\perp},\\Delta k)=E_p(\\mathbf{k}_p^{\\perp})L\\text{sinc}\\left(\\dfrac{\\Delta kL}{2}\\right)\\exp\\left(i\\dfrac{\\Delta kL}{2}\\right),\n\\label{modefna}\n\\end{equation}\nwhere $E_p(\\mathbf{k}_p^{\\perp})$ represents the pump transverse amplitude distribution, $\\mathbf{k}_p^{\\perp} (= \\mathbf{k}^{\\perp}_s+\\mathbf{k}^{\\perp}_i)$ is the angular coordinates of the pump, $L$ is the thickness of the crystal and the exponential factor in the Eqn. \\ref{modefna} is a global phase term. $\\Delta k$ is the longitudinal phase mismatch given by\n\\begin{equation}\n    \\Delta k(\\mathbf{k}_s^{\\perp},\\mathbf{k}_i^{\\perp}) = k_p^z(\\mathbf{k}_s^{\\perp},\\mathbf{k}_i^{\\perp})-k_s^z(\\mathbf{k}_s^{\\perp})-k_i^z(\\mathbf{k}_i^{\\perp}).\n    \\label{phase_mismatch}\n\\end{equation}\n\nConsider a Type-0\/I SPDC using a $\\chi^{(2)}$ crystal of thickness $L$. The magnitude of the wave-vector of the interacting fields under phase-matching condition are given by\n\\begin{align}\nk_{p,s,i}^z(\\mathbf{k}_{p,s,i}^{\\perp})&=\\sqrt{k_{p,s,i}^2-\\vert \\mathbf{k}_{p,s,i}^{\\perp}\\vert^2}.\n\\end{align}\nHere $k_j$=$n_j\\omega_j\/c$ $(j=p,s,i)$ are the magnitudes of wave-vectors of the fields.  Here,  $n_j\\equiv n_j(\\omega_j)$ $(j=p,s,i)$ are the refractive indices and $\\omega_j$ are the frequencies of the pump, signal and idler respectively. $c$ is the speed to light in vacuum. For near-collinear SPDC, the z-component of the wave-vector is approximated by $k-|\\mathbf{k}^{\\perp}|^2\/2k$. So, \\eqref{phase_mismatch} becomes\n\\begin{align}\n    \\Delta k(\\mathbf{k}_s^{\\perp},\\mathbf{k}_i^{\\perp}) = \\frac{|\\mathbf{k}_s^{\\perp}-\\mathbf{k}_i^{\\perp}|^2}{2k_p}.\n    \\label{phase-mismatch-approx}\n\\end{align}\nAlso, an additional term of $-2\\pi\/\\Lambda$ will be added to the phase-mismatch in \\eqref{phase-mismatch-approx}, in case of a quasi phase-matched crystal, where $\\Lambda$ is the grating period.\n\n\\subsection{Angular spectrum of SPDC} \\label{PDC:experiment}\nThe generated signal and idler photons in SPDC propagate in space according to the phase-matching condition of SPDC. The locus of all points in space that satisfies phase-matching conditions comes out to be an annular ring distribution where the signal and idler photons in a pair are present. The angular spectrum of the down-converted signal photons for frequency $\\omega_s$ is obtained by tracing the biphoton mode function overall idler photons \\cite{yasser}\n\\begin{equation}\n    R_s(\\mathbf{k}^{\\perp}_s)=\\int \n    d\\mathbf{k}_i^{\\perp}\\vert\\Phi(\\mathbf{k}_s^{\\perp},\\mathbf{k}_i^{\\perp},\\Delta k)\\vert^2.\n    \\label{SPDCAS}\n\\end{equation}\n\n\\begin{figure}[b]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{SPDC_imaging_setup.pdf}\n  \\caption{Experimental setup to record the angular spectrum of SPDC photons generated by pumping a non-linear crystal with different pump beams. Figure in inset is the SPDC annular distribution in the angular spectrum. Two diametrically opposite points (circled in red) on the distribution are chosen to collect the signal-idler pairs.}\n  \\label{exptalsetup_imaging}\n\\end{figure}\n\nThe experimental setup to record the angular spectra of SPDC photons with different helical pump modes is given in Fig. \\ref{exptalsetup_imaging}. A pump beam of wavelength 405$\\pm$2~nm from a continuous-wave diode laser (TOPTICA iBeam Smart) of 50~mW power is incident on a $\\chi^{(2)}$ crystal. The dashed box corresponds to the case-by-case method to prepare NOV or POV helical beams using the method given in Fig. \\ref{pump_prep_POV}(a). A half-wave plate (HWP) is used to orient the pump polarization along the optic axis of the crystal. As discussed earlier, we use Type-I BBO crystal to generate SPDC photon-pairs. The down-converted photons, signal \\& idler, are generated in a non-collinear fashion at diametrically opposite points of the SPDC ring. A bandpass filter (BPF) of width 10~nm centered at 810~nm is used to filter down-converted photons and block the unconverted pump beam after the crystal. A $2f$-imaging configuration with a plano-convex lens of focal length 50~mm is used to image the SPDC in $k$-space. The angular spectrum of SPDC is recorded using an electron-multiplying CCD (EMCCD) camera with a gain $\\times$100 and the addition of 100 frames, each having an exposure time of 0.5~s. The EMCCD camera has an imaging area of 512$\\times$512 pixels with a pixel size of 16~$\\mu$m.\n\nThe numerical angular spectra of SPDC with different pump OAM modes with similar experimental conditions obtained from Eqn. \\eqref{SPDCAS} is shown in Fig. \\ref{SPDC-angular-spectra}(top). This is well-matched with their corresponding recorded intensity distributions of SPDC photons with NOV and POV pumps of different OAMs, as shown in Fig. \\ref{SPDC-angular-spectra}(bottom). The asymmetric intensity distribution for all the angular spectra is mainly due to the spatial walk-off of the thick crystal used. For a NOV pump, the SPDC annular distribution broadens with an increase in the topological order. However, the signature of 'size-invariance' of POV pump beams is observed in their corresponding angular spectra, where the intensity distribution remains the same for higher OAM values. \n\n\\begin{figure}[h]\n\t\\centering\n    \\includegraphics[width=0.48\\textwidth]{SPDC_AS.pdf}\n\t\\caption{Numerical (top) and experimental (bottom) angular spectra of SPDC generated using NOV, and POV pump modes of various orders.} \\label{SPDC-angular-spectra}\n\\end{figure}\n\nFor a non-collinear down-conversion, the signal and idler are located at diametrically opposite points of the annular distribution (Fig. \\ref{exptalsetup_imaging}). If one chooses to place a collection optics setup each at such two points and detect the photons, then we could observe a decrease in the number of detected photons with the increase in OAM of an NOV pump. There will not be a significant change in the number of detected photons in the case of a POV pump. To achieve the full advantage of our study, we preferred the pump generation method illustrated in Fig.\\ref{pump_prep_POV}(a) over Fig.\\ref{pump_prep_POV}(b) due to the limitation of spiral-phase plates of order more than 3. As the low down-conversion rates in BBO crystal restricted us to investigate further the size effects on photon-pair detection, we choose periodically poled KTP (ppKTP) crystal, which not only provides higher photon-pairs production rate, however also removes the asymmetry in the SPDC distribution.\nPumping the crystal with a beam carrying OAM $l_p$ and the collection of idler photon through a single-mode fiber leads to a conditional detection of signal photon (heralded) carrying an OAM identical to that of the pump, which is discussed below.\n\n\\subsection{Heralded twisted single photons}\n\nEquation \\eqref{modefna} gives the two-photon modal distribution for a pump with electric field $E_p(\\mathbf{k}_p^{\\perp})$. According to conservation of OAM in SPDC process \\cite{walborn2004entanglement}, for a pump carrying an OAM $l_p$ and the idler photon projected to a zero-OAM mode ($l_i=0$), the corresponding heralded signal photon will have the same OAM as that of the pump ($l_s=l_p$). This heralded single-photon carrying OAM can be experimentally generated by coupling one photon in a pair to a single-mode fiber and the other one to a multimode fiber. So, the coincidence counts can be calculated from the modal distribution as\n\\begin{equation}\n    C \\propto \\int_0^{a_s}\\int_0^{a_i}\\Phi(\\mathbf{k}_s^{\\perp},\\mathbf{k}_i^{\\perp},\\Delta k)\\xi_s^*(\\mathbf{k}_s^{\\perp})\\xi_i^*(\\mathbf{k}_i^{\\perp})d\\mathbf{k}_s^{\\perp}\\mathbf{k}_i^{\\perp},\n    \\label{coinc_counts_herladed}\n\\end{equation}\nwhere $\\xi_s$ and $\\xi_i$ are the characteristic functions of spatial mode projectors in signal and idler arms respectively. The fibers in each arm is represented by a Gaussian function with mode-field diameters $2a_s$ and $2a_i$, given by\n\\begin{align}\n    \\xi_{s,i}^*(\\mathbf{k}_{s,i}^{\\perp})=\\sqrt{\\frac{a_{s,i}^2}{2\\pi}}\\text{exp}\\left(-\\frac{a_{s,i}^2}{4}|\\mathbf{k}_{s,i}^{\\perp}|^2\\right).\n    \\label{FiberExpression}\n\\end{align}\n\nThe experimental setup to generate heralded twisted single-photons from SPDC is given in Fig. \\ref{exptalsetup_her_twist}. Here, we have used a blue diode laser (TopMode) of wavelength 405~nm and power 10~mW with a spectral bandwidth of 0.1~nm, as the pump beam. NOV and POV beams are generated by illuminating the Gaussian laser mode onto the corresponding grating holograms imprinted on a SLM (Hamamatsu), and the first order far-field diffracted beam after Fourier transformation using a 750~mm lens is incident on a Type-0 ppKTP crystal of thickness 30mm and transverse dimensions of 1~mm$\\times$2~mm. The HWP allows us to vary the pump beam polarization along the crystal axis. \n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{SPDC_twist_photons_expt_setup.pdf}\n  \\caption{Experimental setup for generating heralded twisted single photons. BPF - Band pass filter; PM - Prism mirror; SMF - Single mode fiber; MMF - Multimode fiber; FC - Fiber coupler; SPCM - Single photon counting module; CC - Coincidence counter.}\\label{exptalsetup_her_twist}\n\\end{figure}\n\nThe down-converted photons (signal \\& idler) of wavelength 810 nm each (degenerate) are generated in a non-collinear fashion at diametrically opposite points of the SPDC ring. To measure the number of generated photon pairs, two diametrically opposite portions of the SPDC ring at a given plane were selected using apertures (not shown in the setup) and the photons coming out of each aperture were collected using fiber collimators (CFC-2X-B, Thorlabs), of focal lengths 2 mm each. The fiber collimator in the idler arm is attached to a SMF (P1-780A-FC-2, Thorlabs) having a numerical aperture of 0.13 and a mode field diameter of $5\\pm0.5$ $\\mu$m, and that in the signal arm is attached to a multi-mode fiber (M43L02, Thorlabs). The fibers are connected to the single-photon detectors SPCMs (SPCM-AQRH-16-FC, Excelitas). The detectors have a timing resolution of 350 ps with 25 dark counts per second. To count the correlated photon-pairs, the two detectors are connected to a coincidence counter (IDQuantique-ID800), having a time resolution of 81 ps.\n\nWe recorded the coincidences of signal and idler for NOV and POV pumps starting from $\\ell=1$ to 25. Figure \\ref{plot1}(a) shows the coincidence counts for NOV and POV pump beams with different OAM. For a pump OAM up to 10, the coincidence counts of SPDC photon pairs is considerably higher for a POV pump than that with a NOV pump beam. \n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{plot2.png}\n  \\caption{Plot of (a) measured coincidence counts for different values of OAM of pump, and (b) corresponding heralding efficiency, for NOV and POV pumps.}\\label{plot1}\n\\end{figure}\n\nThe difference in coincidences corresponding to the two pump modes of the same order decreases for higher orders. The robustness of the source with POV pump mode can be more understood by comparing the heralding efficiencies. Figure \\ref{plot1}(b) shows the variation of heralding efficiency of SPDC single photons with pump OAM for NOV and POV pumps. From the graph, it is clear that, for up to a pump OAM of 15, the efficiency of heralded single photons carrying OAM with POV pump is considerably greater than that corresponding to the NOV pump. Ideally, the coincidences for POV pump should be independent of the order. However, we observe a drop in their coincidences for higher OAM values. This is due to the fact that the ring diameter of the POV mode gradually increases with order and the difference is more exposed for orders starting from $l=15$, which causes a considerable drop in counts.\n\n\\section{Two-photon OAM states using POV pump} \\label{high-dim-OAM-states}\n\\subsection{Theory}\n\nParametric down-converted photons generated using a pump beam of any spatial profile will have an incoherent sum of different OAMs. Based on the OAM conservation in SPDC process \\cite{walborn2004entanglement}, the photon-pairs represent a joint signal-idler OAM state in the allowed OAM subspace, given as\n\\begin{equation}\n    \\ket{\\psi}=\\sum_{\\ell_i=-\\infty}^\\infty C_{\\ell_p-\\ell_i,\\ell_i}\\ket{\\ell_p-\\ell_i}_a\\ket{\\ell_i}_b,\n    \\label{SPDC_single_sum}\n\\end{equation}\nwhere $l_i$ is the OAM of idler photon, $C_{l_p-l_i,l_i}$ is the probability amplitude for occurrence of the state $\\ket{l_p-l_i}_a\\ket{l_i}_b$. The subscripts $a$ \\& $b$ represents signal and idler modes respectively. In the experiment, the probability amplitude corresponds to the coincidence counts obtained by projecting conjugate azimuthal modes in signal and idler. The coincidence counts representing each projection is given by\n\\begin{equation}\n    C_{\\ell_s,\\ell_i}\\propto {}^{}_{a}\\langle \\ell_s^{\\text{(proj)}}\\vert {}^{}_{b}\\langle \\ell_i^{\\text{(proj)}}\\vert \\psi\\rangle.\n    \\label{prob_ampl}\n\\end{equation}\nHere, $\\ell_s^{\\text{(proj)}}$ and $\\ell_i^{\\text{(proj)}}$ are the OAM values of azimuthal modes considered in projective measurement. \n\nFor a nearly collinear phase-matched SPDC, the probability amplitude in Eqn. (\\ref{prob_ampl}) is rewritten in terms of an integral in polar coordinates representing the field overlap of pump, projected signal and idler modes\n\\begin{equation}\n    C_{\\ell_s,\\ell_i}\\propto\\int_0^{2\\pi}d\\theta\\int_0^\\infty rE_p(r,\\theta)E_s^*(r,\\theta)E_i^*(r,\\theta)dr,\n    \\label{prob_ampl_num}\n\\end{equation}\nwhere $E_p$ is the pump mode and $E_s$ \\& $E_i$ are the projected signal and idler modes respectively. Here, we consider a POV pump represented by \\eqref{POV-expression} and NOV or BGV modes in both signal and idler arms. \n\n\\subsection{Experiment}\nThe schematic of the experimental setup to measure in higher dimension is shown in Fig. \\ref{OAMentlsetup}. In this case, the combination of a SLM and a SMF was used as the detection system, instead of the heralding configuration using fibres as discussed earlier. The combination was added in both the signal \\& idler arms. The SLM is used to generate either the NOV or POV modes at the plane of the crystal. The crystal is then imaged on the two SLMs using two lenses and then imaged again to the SMF. The photon is then sent to the APDs, and the signal is analysed using the coincidence logic (CC). \n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{OAMentlsetup.pdf}\n  \\caption{Experimental setup for generating OAM entangled photon-pairs with different helical pump modes.}\\label{OAMentlsetup}\n\\end{figure}\n\nWe measured OAM spectra of SPDC photon-pairs with a POV pump and estimated their bandwidths using LG and BG mode projections in signal\/idler. The OAM bandwidth is quantified as Schmidt number, which estimates the dimensionality of the state generated \\cite{pors2008shannon, pires2010, straupe}. Fig. \\ref{OAMspectra} shows the OAM spectra of photon-pairs for POV pump of order 0 \\& 1. \n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{oamspectrum-highdim-states.pdf}\n  \\caption{OAM spectrum of SPDC photon-pairs generated using POV pump of OAM (a) $\\ell_p=0$ and (b) $\\ell_p=1$.}\\label{OAMspectra}\n\\end{figure}\nFor $\\ell_p=0$ (Fig. \\ref{OAMspectra}(a)), the Schmidt numbers are 3.8 \\& 8.7 for LG \\& BG mode projections respectively. For $\\ell_p=1$ (Fig. \\ref{OAMspectra}(b)), the Schmidt numbers are 6.3 \\& 11.6 for respective LG and BG mode projections. This shows that projecting the photon-pairs generated from a POV pump to a BG mode will effectively bring down the 'size effects' that causes a reduction in OAM bandwidth. Studies had already shown that the bandwidth could be further increased by increasing the radial wavenumber $k_r$, which broadens the spectrum further. \\cite{mclaren2012entangled}.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{densitymatrix-2doamstates.pdf}\n  \\caption{The density matrices of the two-photon OAM entangled states. (a)-(c) correspond to the estimation by projecting photon-pairs on LG basis, and (d)-(f) are the estimations in BG projection. The holograms or the true state and the fidelity of the estimated state are given in the inset of each density matrix. (a) \\& (d) corresponds to a POV pump of $\\ell_p=0$ and the other density matrices correspond to a POV pump of $\\ell_p=1$. }\\label{densitymatrix-2doamstates}\n\\end{figure}\n\nWe also studied the effect of zero-order mode projection in idler on the fidelity of generated quantum states using POV pump. For simplicity, we consider two-dimensional OAM states in the form of Bell states $(\\ket{\\ell_s}\\ket{\\ell_i}+\\ket{\\ell_i}\\ket{\\ell_s})\/\\sqrt{2}$, with different $\\ell_s$ and $\\ell_i$ values. Figure \\ref{densitymatrix-2doamstates}(a)-(c) show the density matrices of OAM states estimated with LG mode projection and Fig. \\ref{densitymatrix-2doamstates}(d)-(f) gives the corresponding density matrices with BG mode projections. In all these cases, the pump OAM $\\ell_p$ is the sum of OAMs of signal and idler, $\\ell_p=\\ell_s+\\ell_i$. Here, we observe that the state fidelity is lower for the case of a zero-OAM mode in signal or idler, which is the setting for the generation of heralded twisted single photons, as discussed in the earlier section. So, there is a trade-off in using a two-photon SPDC system with complete mode projection capabilities and using heralding with the fiber projective systems. \n\nWe also observed that the fidelity of the state is improved with the use of BG projection. Efficient OAM detection based on phase-flattening of POV modes in classical light source has been demonstrated \\cite{pinnell2019quantitative, pinnell2019perfect}. Thus, a robust source of high dimensional quantum states in OAM degree of freedom can be implemented by using a size-invariant twisted optical mode like a perfect optical vortex as a pump, as well as by performing projection of photon-pairs in Bessel-Gaussian modes.\n\n\\section{Conclusion} \\label{Conclusion}\nIn conclusion, we have experimentally demonstrated that the conditional coupling efficiency of the heralded twisted single photons for higher OAM values can be improved by using a perfect optical vortex beam as the pump. We also showed that the dimensionality of the two-photon OAM states is increased with the use of POV modes in the pump, as well as projective measurements using Bessel-Gaussian vortex modes that give POV, instead of the Laguerre-Gaussian modes. The presented results may be utilized for the practical realization of efficient higher dimensional OAM entangled photon-pair sources. \n\n\\section*{Disclosures}\nThe authors declare that there are no conflicts of interest related to this article.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\nLet $L$ be a link in the 3-dimensional Euclidean space $\\mathbb{R}^3$. The \\textit{unknotting number} $u(L)$ is the minimal number of crossing changes\\ (Fig.\\ \\ref{cc}) from $L$ to a trivial link. The \\textit{crossing number} $c(L)$ is the minimal number of crossing points among all regular diagrams of $L$. It is well-known that $u(L)$ is less than or equal to half of $c(L)$\\ (see for example\\ \\cite{unbound}). In\\ \\cite{unbound} Taniyama characterized the links which satisfy the equality as follows. \n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=0.12\\linewidth]{cc1.eps}\n\\caption{}\n\\label{cc}\n\\end{figure}\n\n\\begin{thm} \\cite[Theorem $1.5\\,(2)$] {unbound}\nLet L be a $\\mu-$component link that satisfies the equality $u(L) =\\dfrac{c(L)}{2}$. Then $L$ has a diagram $D = \\gamma_1 \\cup \\cdots \\cup \\gamma_{\\mu}$ such that each $\\gamma_i$ is a simple closed curve on $\\mathbb{R}^2$ and for each pair $i,\\ j$, the subdiagram $\\gamma_i \\cup \\gamma_j$ is an alternating diagram or a diagram without crossings.\n\\end{thm}\n\nIn\\ \\cite{uncr} Taniyama and the author showed that this inequality is not extended to spatial embeddings of planar graphs and this inequality is extended to spatial embeddings of trivializable planar graphs. Namely for any spatial embedding $f$ of a trivializable planar graph, $u(f)$ is less than or equal to half of $c(f)$. For example, a handcuff-graph and a theta-curve as illustrated in Fig.\\ \\ref{hanthe} are trivializable. We characterize the spatial embeddings of a handcuff-graph or a theta curve which satisfy the equality as follows.\n\n\\begin{figure}[H]\n      \\begin{tabular}{c}\n      \\begin{minipage}[b]{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.3\\linewidth]{hand.eps}\\\\\n          \\ \\\\\n          handcuff-graph\n        \\end{center}\n      \\end{minipage}\n      \\begin{minipage}[b]{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.3\\linewidth]{theta.eps}\\\\\n          \\ \\\\\n          theta curve\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{hanthe}\n\\end{figure}\n\n\\begin{thm}\\label{handcuff}\nLet $G$ be a handcuff-graph and let $f$ be a spatial embedding of $G$. Then $f$ satisfies the equality $u(f)=\\dfrac{c(f)}{2}$ if and only if $f$ has a diagram $D$ with the following conditions {\\rm:} \\\\\n$(1)\\,$Each edge of $D$ has no self-crossings. \\\\\n$(2)\\,$All crossings of $D$ are crossings between two loops.\\\\\n$(3)\\,$Two loops of $D$ form an alternating diagram or a diagram without crossings.\n\\end{thm}\n\n\\begin{thm}\\label{theta}\nLet $G$ be a theta curve and let $f$ be a spatial embedding of $G$. Then $f$ satisfies the equality $u(f)=\\dfrac{c(f)}{2}$ if and only if $f$ is trivial.\n\\end{thm}\n\nWe note that unknotting numbers of spatial embeddings of a theta curve is studied in \\cite{ob}.\n\nA \\textit{handlebody-knot} is an embedded handlebody in the 3-dimensional Euclidean space $\\mathbb{R}^3$, which is introduced by Ishii in \\cite{handle}. Two handlebody-knots $H_1$ and $H_2$ are equivalent if there is an\norientation-preserving homeomorphism $h$ of $\\mathbb{R}^3$ with $h(H_1) = H_2$. A \\textit{spine} of a handlebody-knot $H$ is a spatial graph whose regular neightborhood is $H$. In this paper, we assume that spines have no degree 1 verticies. Any handlebody-knot $H$ can be represented by a spatial trivalent graph that is a spine of $H$. In particular, genus 2 handlebody-knot can be represented by a spatial embedding of a handcuff-graph or a theta curve. A \\textit{crossing change} of a handlebody-knot $H$ is that of a spatial trivalent graph representing $H$. In\\ \\cite{unqc} Iwaliri showed that a crossing change of a handlebody-knot is an unknotting operation and give lower bounds of the unknotting numbers for handlebody-knots by the numbers of some finite Alexander quandle colorings. \n\nWe have the following well-known relation between unknotting number and crossing number of classical knots.\n\\begin{prop} \\label{12c1k}\nLet $K$ be a nontrivial knot. Then $u(K) \\leq \\dfrac{c(K)-1}{2}$.\n\\end{prop}\nIn \\cite{unbound} Taniyama characterized the knots which satisfy the equality as follows.\n\\begin{thm} \\cite[Theorem1.4\\,(2)]{unbound} \\label{12c2k}\nLet $K$ be a nontrivial knot that satisfies the equality $u(K)=\\dfrac{c(K)-1}{2}$.   Then $K$ is a $(2,\\ p)$-torus knot for some odd number $p \\neq \\pm 1$.\n\\end{thm}\n\nIn this paper, as an extension of Proposition \\ref{12c1k}, we show the following theorem.\n\n\\begin{thm}\\label{12c1}\nLet $H$ be a non-trivial handlebody-knot.\\ Then $u(H) \\leq \\dfrac{c(H)-1}{2}$.\n\\end{thm}\nThe spine of genus 1 handlebody-knot is a classical knot. Therefore Theorem \\ref{12c1} is an extention of Proposition \\ref{12c1k}. It follows from Theorem \\ref{handcuff} and Theorem \\ref{theta} that for any non-trivial handlebody-knot $H$ with genus 2 twice the unknotting number of $H$ is less than or equal to the crossing number of $H$ minus one \\ (see section 4). \n\nIt follows from Theorem \\ref{12c2k} that genus 1 handlebody-knot $H$ with $u(H) = \\dfrac{c(H)-1}{2}$ is a regular neighborhood of a $(2,\\ p)-$torus knot. We also characterize genus $n \\geq 2$ handlebody-knots which satisfy the equality as follows.\n\\begin{thm}\\label{12c2}\nLet $n\\geq 2$ and let $H$ be a nontrivial genus $n$ handlebody-knot that satisfies the equality $u(H)=\\dfrac{c(H)-1}{2}$.\\,Then $H$ is a handlebody-knot represented by $D_3$ or $D_{-3}$ illustrated in Fig.\\ \\ref{d3}.\n\\end{thm}\n\\begin{figure}[H]\n      \\begin{tabular}{c}\n      \\begin{minipage}{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.6\\linewidth]{d311.eps}\n        \\end{center}\n      \\end{minipage}\n      \\begin{minipage}{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.6\\linewidth]{d312.eps}\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{d3}\n\\end{figure}\n\nThis paper consists of five sections. In section 2 we review trivializability of planar graphs and  inequalities between unknotting numbers and crossing numbers of spatial embeddings of planar graphs. In section 3 we introduce unknotting number of handlebody-knots. In section 4 we give proofs of Theorem \\ref{handcuff} and Theorem \\ref{theta}. In section 5 we give proofs of Theorem \\ref{12c1} and Theorem \\ref{12c2}.\n\n\\section{UNKNOTTING NUMBERS AND CROSSING NUMBERS OF SPATIAL EMBEDDINGS OF PLANAR GRAPHS}\n\nLet $G$ be a planar graph. A \\textit{spatial embedding} of $G$ is an embedding $f : G\\rightarrow \\mathbb{R}^3$. Its image $f(G)$ is said to be a \\textit{spatial graph}. Let $\\pi: \\mathbb{R}^3 \\rightarrow \\mathbb{R}^2$ be a natural projection defined by $\\pi(x,y, z) = (x, y)$. Let $SE(G)$ be the set of all spatial embeddings of $G$. A \\textit{regular projection} of $G$ is a continuous map $\\tilde{f} : G \\rightarrow \\mathbb{R}^2$ whose double points are only finitely many transversal double points. Such a double point is said to be a \\textit{crossing point} or simply a \\textit{crossing}.  If we give over\/under informations at each crossing points of a regular projection $\\tilde{f}$ of $G$, then $\\tilde{f}$ together with the over\/under informations represents a spatial embedding $f:G \\rightarrow \\mathbb{R}^3$ such that $\\tilde{f} = \\pi \\circ f$. Such a regular projection together with the over\/under informations is said to be a \\textit{diagram} of $f(G)$. Then we say that $f$ is \\textit{obtained from} $\\tilde{f}$. We also call $\\tilde{f}$ a \\textit{regular projection} of $f(G)$. For a diagram $D$ of a spatial embedding, the set of all crossings of $D$ is denoted by $\\textit{C}(D)$. The number of crossings of $D$ is denoted by $c(D) = |\\textit{C}(D)|$.\n\nAn element $f \\in SE(G)$ is said to be \\textit{trivial}, if it is ambient isotopic to  $t \\in SE(G)$ such that $t(G) \\subset \\mathbb{R}^2$.\nAny spatial embedding of a planar graph can be transformed into trivial one by crossing changes. Therefore unknotting number is naturally extended to spatial embeddings of planar graphs as follows. For $f \\in SE(G)$, the unknotting number $u(f)$ is defined to be the minimal number of crossing changes from $f$ to a trivial embedding of $G$. The \\textit{crossing number} $c(f)$ is defined to be the minimal number of crossing points among all diagrams of spatial embeddings that are ambient isotopic to $f$. \n\nFor any link $L$, $L$ satisfies the inequality $u(L) \\leq \\dfrac{c(L)}{2}$. But this is not extended for spatial embeddings of planar graph, namely there are a planar graph $G$ and a spatial embedding $f$ of $G$ such that $u(f) > \\dfrac{c(f)}{2}$. Let $P_3$ the cube graph and $f_3 \\in SE(P_3)$ a spatial embedding of $P_3$ as illustrated in Fig.\\ \\ref{f3p3}. The spatial graph $f_3(P_3)$ contains three Hopf-links and one crossing change of edges of $f_3(P_3)$ unknot at most two of them\\ (See Fig.\\ \\ref{hopf3}). Then $u(f_3) \\geq 2$. Since $f_3(P_3)$ contains a trefoil whose crossing number is 3, $c(f_3)=3$ and $u(f_3) > \\dfrac{c(f_3)}{2}$ \\cite{uncr}. \n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{f3p3.eps}\\\\\n\\caption{}\n\\label{f3p3}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.55\\linewidth]{hopf3.eps}\n\\caption{}\n\\label{hopf3}\n\\end{figure}\n\nNow we review the reason why it happens for some planar graphs. The key point of the proof of $u(L) \\leq \\dfrac{c(L)}{2}$ for a link $L$ is that any link diagram can be transformed into a trivial link diagram by changing over\/under informations at some crossings of the diagram. Let $D$ be a minimal crossing diagram of $L$. Let $A$ be a subset of $C(D)$ such that changing over\/under informations at all crossings in $A$ turns $D$ to a diagram $T_1$ of a trivial link. Let $T_2$ be a diagram that is obtained from $T_1$ by changing over\/under informations at all crossings. A mirror image of a trivial link is also  trivial.  Thus $T_2$ is a diagram of a trivial link. Note that $T_2$ is obtained from $D$ by changing over\/under informations at all crossings in $C(D)-A$. Therefore we have\n$$u(L) \\leq u(D) \\leq \\min  \\{|A|,|C(D)-A|\\} \\leq \\dfrac{c(D)}{2}=\\dfrac{c(L)}{2}$$\n\nOn the other hand, all diagrams obtained from $\\pi \\circ f_3(P_3)$ (Fig.\\ \\ref{knotted}) represent non-trivial spatial graphs since each of the spatial graphs obtained from these diagrams contains at least one Hopf-link. A regular projection $\\tilde{f}$ of a planar graph $G$ is said to be a \\textit{knotted projection} \\cite{knotted}, if all spatial embeddings of $G$ which can be obtained from $\\tilde{f}$ are non-trivial. \n\n\n\n\\begin{figure}[H]\n        \\begin{center}\n          \\includegraphics[height=0.18\\linewidth]{knotted.eps}\\\\\n          \\ \\\\\n          $\\pi \\circ f_3(P_3)$\n        \\end{center}\n\\caption{}\n\\label{knotted}\n\\end{figure}\n\nA planar graph is said to be \\textit{trivializable} if it has no knotted projections.  In \\cite{knotted} Taniyama gave a class of trivializable graphs. In \\cite{class} Sugiura and Suzuki extended the class. In \\cite{tamura} Tamura gave another class of trivializable graphs.\n\nFor a spatial embedding of a trivializable planar graph, the same argument as\nfor a link works, and we have the following proposition.\n\n\\begin{prop}\\label{tri12} \\cite{uncr}\nLet $G$ be a trivializable planar graph and $f: G \\rightarrow \\mathbb{R}^3$ a spatial embedding of $G$. Then $u(f) \\leq \\dfrac{c(f)}{2}$.\n\\end{prop}\n\n\\newpage\n\n\\section{UNKNOTTING NUMBERS AND CROSSING NUMBERS OF HANDLEBODY-KNOTS}\n\nWe review that crossing change of a handlebody-knot is an unknotting operation \\cite{unqc}.\n\nA \\textit{diagram} of a handlebody-knot $H$ is that of a spatial trivalent graph representing $H$. In \\cite{handle}, Ishii gave a list of fundamental moves among diagrams of handlebody-knots, which is called R1-6 moves illustrated in Fig.\\ \\ref{reide}. Ishii showed that two handlebody-knots are equivalent if and only if their representing diagrams are related by a finite sequence of R1-6 moves. Note that R6-move is also called \\textit{IH-move}. \n\n\\begin{figure}[H]\n      \\begin{tabular}{c}\n      \\begin{minipage}{0.3\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.4\\linewidth]{R1.eps}\n        \\end{center}\n      \\end{minipage}\n      \\begin{minipage}{0.25\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.48\\linewidth]{R2.eps}\n        \\end{center}\n      \\end{minipage}\n      \\begin{minipage}{0.25\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.48\\linewidth]{R3.eps}\n        \\end{center}\n      \\end{minipage}\n         \\end{tabular}\\\\\n\\ \\vspace{0.5cm} \\\\\n       \\begin{tabular}{c}\n      \\begin{minipage}{0.36\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.333333\\linewidth]{R4.eps}\n        \\end{center}\n      \\end{minipage}\n      \\begin{minipage}{0.36\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.333333\\linewidth]{R5.eps}\n        \\end{center}\n      \\end{minipage}\n      \\begin{minipage}{0.2\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.6\\linewidth]{R6.eps}\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{reide}\n\\end{figure}\n\nA crossing change of a handlebody-knot $H$ is that of a spatial trivalent graph representing $H$. This move can be realized by switching two tubes illustrated in Fig.\\ \\ref{cch}. A genus $n$ handlebody-knot is \\textit{trivial} if it is equivalent to a handlebody-knot represented by a diagram  illustrated in Fig.\\ \\ref{tg}.\n\n\n\\begin{figure}[H]\n\\centering\n     \n     \n      \n          \\includegraphics[height=0.12\\linewidth]{cc2.eps}\n       \n     \n     \n       \n        \n       \n     \n      \n         \\caption{}\n         \\label{cch}\n\\end{figure}\n\n\n\\begin{figure}[H]\n      \\centering\n      \\includegraphics[width=0.4\\linewidth]{tg.eps}\n         \\caption{}\n         \\label{tg}\n\\end{figure}\n\n\nLet $T_n$ be the trivalent graph whose image is illustrated in Fig.\\ \\ref{tg}. Any handlebody-knot is represented by a diagram of a spatial embedding of $T_n$ since a genus $n$ handlebody has $T_n$ as a spine. Note that $T_n$ is a trivializable graph \\cite{class}. Namely, any diagram $D$ of a spatial embedding of $T_n$ can be changed to a trivial spatial graph diagram by changing over\/under informations at some crossings of $D$. Then we have the following proposition.\n\n\\begin{prop}\\label{unop} \\cite[Proposition\\ 2.1]{unqc} Any handlebody-knot can be transformed into trivial one by crossing changes.\n\\end{prop}\n\nTherefore unknotting number is naturally extended to handlebody-knots as follows. For a handlebody-knot $H$, the unknotting number $u(H)$ is the minimal number of crossing changes needed to obtain a trivial handlebody-knot from $H$. The crossing number $c(H)$ is the minimal number of crossing points among all diagrams of handlebody-knots that are equivalent to $H$. \n\nBy the proof of \\cite[Proposition\\ 3.1]{forbidden} we see that any diagram $D$ of a spatial graph can be transformed into a diagram of a spatial graph whose neighborhood are ambient isotopic to a neighborhood of a trivial bouquet by changing over\/under informations at some crossings of $D$. Therefore, in \\cite{unqc}, Iwakiri also showed that Proposition\\ \\ref{unop} can be refined to the strong statement as follows.\n\n\\begin{prop}\\label{dunop} \\cite{unqc} Any handlebody-knot diagram can be transformed into a trivial handlebody-knot diagram by changing over\/under informations at some crossings of the diagram.\n\\end{prop}\n\n\n\nFor a handlebody-knot diagram $D$, the unknotting number $u(D)$ is the minimal number of changing over\/under informations at crossings of $D$ needed to obtain a trivial handlebody-knot diagram. Same as Proposition\\ \\ref{tri12} we have $u(D) \\leq \\dfrac{c(D)}{2}$ and $u(H) \\leq \\dfrac{c(H)}{2}$.\n\nIn section 5, we show that a handlebody-knot $H$ satisfies $u(H)=\\dfrac{c(H)}{2}$ if and only if $H$ is trivial\\ (Theorem\\ \\ref{12c1}). Then it is natural to ask when handlebody-knots satisfy the equality $u(H) =\\dfrac{c(H)-1}{2}$. Let $H$,\\ $H_1$ and $H_2$ be handlebody-knots in $\\mathbb{R}^3$ and let $S$ be a $2-$sphere in $\\mathbb{R}^3$. Suppose that $H \\cap S=H_1 \\cap H_2$ is a $2-$disk and $H=H_1 \\cup H_2$. Then $H$ is said to be a \\textit{disk sum} of $H_1$ and $H_2$ and denoted by $H=H_1 \\# H_2$. In \\cite{unbound} Taniyama showed that if a classical knot $K$ satisfies $u(K) = \\dfrac{c(K)-1}{2}$ then $K$ is a $(2,\\ p)$-torus knot for some odd number $p \\neq \\pm 1$\\ (Theorem\\ \\ref{12c2k}). Therefore the handlebody-knots illustrated in Fig.\\ \\ref{d2n1} may satisfy the equality. But by the following proposition only two of these handlebody-knots satisfy the equality.\n\n\\begin{prop}\\label{2br}\nLet $n\\geq 2$ and let $H$ be a genus $n$ handlebody-knot such that $H=K\\ \\#\\ O_{n-1}$, where $K$ is a genus $1$ handlebody-knot whose spine is a $2-$bridge knot and $O_{n-1}$ is a genus $n-1$ trivial handlebody-knot. Then $u(H)=1$.\n\\end{prop}\n\n\\begin{figure}[H]\n\\begin{tabular}{c}\n\n      \\begin{minipage}[t]{0.24\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.6\\linewidth]{d311.eps}\\\\\n          $D_3$\n        \\end{center}\n      \\end{minipage}\n      \n      \\begin{minipage}[t]{0.24\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.6\\linewidth]{d51.eps}\\\\\n          $D_5$\n        \\end{center}\n      \\end{minipage}\n      \n        \\begin{minipage}[t]{0.32\\hsize}\n         \\begin{center}\n          \\includegraphics[height=0.45\\linewidth]{d71.eps}\\\\\n          $D_7$\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\\\\\n      \\ \\vspace{0.5cm} \\\\\n\\begin{tabular}{c}\n\n      \\begin{minipage}[t]{0.24\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.6\\linewidth]{d312.eps}\\\\\n          $D_{-3}$\n        \\end{center}\n      \\end{minipage}\n      \n      \\begin{minipage}[t]{0.24\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.6\\linewidth]{d52.eps}\\\\\n          $D_{-5}$\n        \\end{center}\n      \\end{minipage}\n      \n        \\begin{minipage}[t]{0.32\\hsize}\n         \\begin{center}\n          \\includegraphics[height=0.45\\linewidth]{d72.eps}\\\\\n          $D_{-7}$\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{d2n1}\n\\end{figure}\n\n\n\n\\begin{proof}\nLet $K'$ be the spine of $K$. Let $H'$ be the handlebody-knot obtained from $K\\# O_{n-1}$ by one crossing change as illustrated in the left of Fig.\\ \\ref{dut}. By \\cite[Proposition\\ 3.1]{tunnel} we see that the tunnel $\\tau$ for $K'$ as illustrated in the right of Fig.\\ \\ref{dut} is an unknotting tunnel. Therefore the genus 2 handlebody-knot represented by the right of Fig.\\ \\ref{dut} is trivial. Since a disk sum of two trivial handlebody-knots is trivial,  $H'$ is also trivial.\n\n\\begin{figure}[H]\n      \\begin{tabular}{c}\n      \\begin{minipage}[t]{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.95\\linewidth]{2briuntn.eps} \\vspace{0.5cm}\\\\\n          \\hspace{0.2\\linewidth}$b_1,\\ b_2,\\ \\cdots b_n:\\ 2$-braids\n        \\end{center}\n        \n      \\end{minipage}\n      \\begin{minipage}[t]{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.9\\linewidth]{2briunt.eps}\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{dut}\n\\end{figure}\n\n\\end{proof}\n\n\n\n\n\\section{PROOFS OF THEOREM\\ \\ref{handcuff} AND THEOREM\\ \\ref{theta}}\nLet $D$ be a diagram of a spatial graph $f(G)$ and let $H$ be a subgraph of $G$. Then the diagram of $f(H)$ that is contained in $D$ is said to be a \\textit{subdiagram} of $D$. For subdiagrams $A,\\ B$ of a diagram $D$, let $c(A)$ be the number of all crossings on $A$ among the crossings of $D$ and let $c(A,\\,B)$ be the number of all crossings between  $A$ and $B$. \n\n\\begin{lemma}\\label{self}\nLet $G$ be a trivializable graph and let $f$ be a spatial embedding of $G$. Let $D$ be a diagram of $f(G)$. If $D$ has a self-crossing, then $u(D) \\leq \\dfrac{c(D)-1}{2}$.\n\\end{lemma}\n\\noindent\n\\begin{proof}\nLet $P$ be a self-crossing of $D$. By smoothing $D$ at $P$, we have a diagram $D'$ such that one of the components of $D'$ represents a knot\\ (see Fig.\\ \\ref{selfc}). Let $\\gamma_1$ be a component of $D'$ that represents a knot and let $\\gamma_2$ be the other component of $D'$. \n\n\\begin{figure}[H]\n      \\begin{tabular}{c}\n      \\begin{minipage}[b]{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.5\\linewidth]{selfc1.eps} \\vspace{0.35cm} \\\\\n          \\hspace{0.08\\linewidth}$D$\n        \\end{center}\n        \n      \\end{minipage}\n      \\begin{minipage}[b]{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.5\\linewidth]{selfc2.eps}\\\\\n          \\hspace{0.08\\linewidth}$D'$\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{selfc}\n\\end{figure}\n\nIf we change some crossings on $\\gamma_1$ so that the part $\\gamma_1$ is over other component of $D$ and itself unknotted then we have a spatial embedding that has a diagram $\\gamma_2$. Also we may change some crossings on $\\gamma_1$ so that the part $\\gamma_1$ is under the other component of $D$ and itself unknotted. Note that we can choose these two crossing changes complementary on the crossings on $\\gamma_1$. We choose one of them that have no more crossing changes than the other. Thus by changing no more than $\\dfrac{c(D)-c(\\gamma_2)-1}{2}$ crossings of $D$ we have a spatial embedding that has a diagram $\\gamma_2$. Note that the key point here is that we do not need to change the crossing $c$. Since $\\gamma_2$ is also a diagram of a spatial embedding of a trivializable graph, we have $u(\\gamma_2) \\leq \\dfrac{c(\\gamma_2)}{2}$. Therefore we have \n$$u(D) \\leq u(\\gamma_2) + \\dfrac{c(D)-c(\\gamma_2)-1}{2} \\leq \\dfrac{c(\\gamma_2) +(\\ c(D)-c(\\gamma_2)-1\\ )}{2}=\\dfrac{c(D)-1}{2}$$\n\\end{proof}\n\n\\begin{lemma}\\label{mini}\nLet $G$ be a trivializable planar graph and let $f$ be a spatial embedding of $G$ such that $u(f)=\\dfrac{c(f)}{2}$. Let $D$ be a minimal crossing diagram of $f(G)$. Then $u(D)=\\dfrac{c(D)}{2}$.\n\\end{lemma}\n\n\\begin{proof} It is sufficient to show that $u(D) \\geq \\dfrac{c(D)}{2}$. Since $u(f) \\leq u(D)$ and $c(f)=c(D)$ we have\n$$u(D) \\geq u(f)=\\dfrac{c(f)}{2}=\\dfrac{c(D)}{2}$$\n\\end{proof}\n\n\\begin{lemma}\\label{dhandcuff}\nLet $D$ be a diagram of a spatial embedding of a handcuff-graph such that $u(D)=\\dfrac{c(D)}{2}$. Then $D$ satisfies the following conditions {\\rm:} \\\\\n$(1)\\,$Each edge of $D$ has no self-crossings. \\\\\n$(2)\\,$All crossings of $D$ are crossings between two loops.\\\\\n$(3)\\,$Two loops of $D$ form an alternating diagram or a diagram without crossings.\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{self} $D$ satisfies $(1)$. In the following we show that $D$ satisfies $(2)$ and $(3)$. \n\nLet $\\gamma_1$ and $\\gamma_2$ be two loops of $D$ and let $e$ be the edge of $D$ that is not $\\gamma_i$\\ ($i=1,\\ 2$).  If we change some crossings on $\\gamma_2$ so that the part $\\gamma_2$ is over $D-\\gamma_2$ of $D$ then we have a diagram of a trivial spatial embedding of $G$ since $\\gamma_i$ is a simple closed curve on $\\mathbb{R}^2$\\ ($i=1,\\ 2$). See for example Fig.\\ \\ref{alc}.  Also we may change some crossings on $\\gamma_2$ that the part $\\gamma_2$ is under $D-\\gamma_2$ of D and itself unknotted. Note that these two crossing changes are complementary on the crossings on $\\gamma_2$. We choose one of them that have no more  crossing changes than the other. Thus by changing no more than $\\dfrac{c(D)-c(\\gamma_1,\\ e)}{2}$ crossings of $D$ we have a trivial diagram and $u(D) \\leq \\dfrac{c(D)-c(\\gamma_1,\\ e)}{2}$. The key point here is that we do not need to change crossings between $\\gamma_1$ and $e$.  Since $u(D) = \\dfrac{c(D)}{2}$ we have $c(\\gamma_1,\\ e)=0$. Similarly we have $c(\\gamma_2,\\ e)=0$. Therefore $D$ satisfies $(2)$. \n\n\n\\begin{figure}[H]\n      \\begin{tabular}{c}\n      \\begin{minipage}[b]{0.3\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.7\\linewidth]{lemma431.eps}\n        \\end{center}\n        \n      \\end{minipage}\n      \\begin{minipage}[b]{0.7\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.3\\linewidth]{lemma432.eps}\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{alc}\n\\end{figure}\n\nSuppose that $\\gamma_1 \\cup \\gamma_2$ is not an alternating diagram. Then we may suppose without loss of generality that there is an arc $\\alpha$ of $\\gamma_1$ disjoint from $e$ such that $\\alpha \\cap \\gamma_2=\\partial \\alpha=\\{c_1,\\ c_2\\}$ and $\\gamma_1$ is over $\\gamma_2$ at $c_1$ and $c_2$. See Fig.\\ \\ref{disarc} .\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.25\\linewidth]{disarc.eps}\n\\caption{}\n\\label{disarc}\n\\end{figure}\n\nLet $A$ be the set of all crossings of $D$ at which $\\gamma_1$ is under $\\gamma_2$. Let $B=C(D)\\backslash (A \\cup \\{c_1.\\ c_2\\})$. Then by the height function argument first used in \\cite{knotted} we see that changing all crossings in $A$ (resp. $B$) produce a trivial spatial embedding. See Fig. \\ref{althand}.\n\n\\begin{figure}[H]\n\\begin{tabular}{c}\n      \\begin{minipage}[b]{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.8\\linewidth]{althandover.eps} \\vspace{0.55cm}\n\n        \\end{center}\n        \n      \\end{minipage}\n      \\begin{minipage}[b]{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.75\\linewidth]{althandunder.eps} \n\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{althand}\n\\end{figure}\n\nTherefore we have\n$$u(D) \\leq \\min \\{|A|,\\ |B|\\} \\leq \\dfrac{c(D)-2}{2}$$\nThis is contradicts to the equation $u(D) = \\dfrac{c(D)}{2}$. Thus $D$ satisfies $(3)$ as desired. \n\\end{proof}\n\n\n\\noindent\n\\textit{Proof of Theorem\\ref{handcuff}}\\\\\nFirst, we show that if there exists a diagram $D$ of $f(G)$ satisfying $(1)$, $(2)$ and $(3)$, then $u(f) = \\dfrac{c(f)}{2}$. We may suppose that $c(D)>0$. Let $L=l_1 \\cup l_2$ be a $2-$component link represented by two loops of $D$. See for example Fig.\\ \\ref{altlink}. Since the diagram of $L$ consists of two simple closed curves and it is alternating, we see that twice the absolute value of the linking number $2|\\,lk(l_1,l_2)\\,|$ is equal to $C(D)$. \n Therefore we have \n$$u(f) \\geq u(L) \\geq |\\,lk(l_1,l_2)\\,| = \\dfrac{c(D)}{2}=\\dfrac{c(f)}{2}$$\nBy Proposition \\ref{tri12} we have $u(f)=\\dfrac{c(f)}{2}$. \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.25\\linewidth]{altlink.eps}\n\\caption{}\n\\label{altlink}\n\\end{figure}\n\nLet $f$ be a spatial embedding of $G$ such that $u(f)=\\dfrac{c(f)}{2}$ and let $D$ be a minimal crossing diagram of $f(G)$. By Lemma\\ \\ref{mini} we have $u(D)=\\dfrac{c(D)}{2}$. By Lemma \\ref{dhandcuff} $D$ satisfies $(1)$, $(2)$ and $(3)$ as desired. \\qed\n\n\\begin{lemma}\\label{dtheta}\nLet $G$ be a theta curve. Let $D$ be a diagram of a spatial embedding of $G$ such that $u(D)=\\dfrac{c(D)}{2}$. Then $c(D)=0$.\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{self} we may suppose that each edge of $D$ has no self-crossings. Suppose that $c(D)>0$. Then there exists a crossing $c$ on $D$ between two edges. Let $\\tilde{f}:G \\rightarrow \\mathbb{R}^2$ be a regular projection of $G$ where $D$ is obtained from $\\tilde{f}(G)$. Let $v$ and $u$ be two vertices of $G$. Let $G'$ be the graph obtained by adding $2$ verticies $v_1,\\ v_1'$ to $G$ such that $\\tilde{f}(v_1)=\\tilde{f}(v_1')=c$ and $v_1$\\ (resp.\\ $v_1'$) is contained in the over-arc\\ (resp.\\ the under-arc) at $c$. Let $P$ be the path from $v$ to $u$ that contains $v_1$. We fix a spanning tree $T$ of $G'$ that contains $P$\\ (see for example Fig.\\ \\ref{subdthe}). Let $h: G' \\rightarrow \\mathbb{R}$ be a continuous function with the following properties : \\\\\n$(1)\\,$For each vertex $t$ of $G'$, $h(t)=-d_{T}(t,\\ v)$. Here $d_{T}(t,v)$ be the number of edges of the path in $T$ joining $t$ and $v$.\\\\\n$(2)\\,h|_e$ is injective for each edge $e$ of $G'$\\\\\n\\begin{figure}[H]\n      \\begin{tabular}{c}\n      \\begin{minipage}[b]{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.45\\linewidth]{kino.eps} \\vspace{0.3cm}\\\\\n          $D$\n        \\end{center}\n        \n      \\end{minipage}\n      \\begin{minipage}[b]{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.7\\linewidth]{kinospa.eps} \\vspace{0.3cm}\\\\\n          $G'$\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{subdthe}\n\\end{figure}\nWe can deform $h$ slightly so that $h(v_1)>h(v_1')$ since $d_{T}(v,v_1)=d_{T}(v,v_1')=1$. Then we give over\/under information to $\\tilde{f}$ to produce a spatial embedding $f : G \\rightarrow \\mathbb{R}^3 = \\mathbb{R}^2 \\times \\mathbb{R}$ such that $p_1 \\circ f=\\tilde{f}$ and $p_2 \\circ f =h$, where $p_1$\\ (resp. $p_2)$ denotes the projection of $\\mathbb{R}^3$ to the first factor (respectively to the second factor) of $\\mathbb{R}^2 \\times \\mathbb{R}$. Let $\\Pi:\\mathbb{R}^3 \\rightarrow \\mathbb{R}^2$ be a projection defined by $\\Pi(x, y, z) = (x, z)$. We deform $f$ slightly by an ambient isotopy if necessary so that $\\Pi \\circ f$ is a regular projection. Then we can eliminate all crossings of $\\Pi \\circ f$ by  eliminating the crossing nearest to $v$ repeatedly (see Fig.\\ \\ref{heithe}). Therefore $f$ is trivial.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.4\\linewidth]{kinohei.eps}\n      \\caption{}\n      \\label{heithe}\n\\end{figure}\nLet $D'$ be the diagram of $f(G)$ where $D'$ is obtained from $\\tilde{f}(G)$. We note that $D$ and $D'$ are deformed  into each other by changing over\/under informations of all crossing points without changing over\/under informations of $c$. Let $D''$ be the diagram that is obtained from $D'$ by changing over\/under informations of all crossing points with the exception of  $c$\\ (see for example Fig.\\ \\ref{dandd}). Let $h':G' \\rightarrow \\mathbb{R}$ be a continuous function such that $h'=-h$. We can deform $h'$ slightly so that $h'(v_1)>h'(v_1')$. Then $D''$ is the diagram of a spatial embedding of $f' : G \\rightarrow \\mathbb{R}^3 = \\mathbb{R}^2 \\times \\mathbb{R}$ such that $p_1 \\circ f'=\\tilde{f}$ and $p_2 \\circ f'=h'$. Same as the case of $f(G)$, $f'$ is also trivial.\n\\begin{figure}[H]\n      \\begin{tabular}{c}\n      \\begin{minipage}[b]{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.45\\linewidth]{kinod1.eps} \\vspace{0.3cm}\\\\\n          $D'$\n        \\end{center}\n        \n      \\end{minipage}\n      \\begin{minipage}[b]{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.45\\linewidth]{kinod2.eps} \\vspace{0.3cm}\\\\\n          $D''$\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{dandd}\n\\end{figure}\n\nLet $A$ be a subset of $C(D)$ such that changing all crossings in $A$ turns $D$ to $D'$. We note that changing all crossings in $(\\ C(D)-\\{c\\}\\ )-A$ turns $D$ to $D''$. Therefore we have\n$$u(D) \\leq \\min \\{|A|,|(\\ C(D)-\\{c\\}\\ )-A|\\} \\leq \\dfrac{c(D)-1}{2}$$ \n\nThis is contradicts to the equation $u(D) = \\dfrac{c(D)}{2}$. Therefore we have $c(D)=0$ and $D$ is a diagram of a trivial theta curve. \n\\end{proof}\n\n\n\n\n\\noindent\n\\textit{Proof of Theorem\\ref{theta}}\\\\\nLet $f$ be a spatial embedding of $G$ such that $u(f)=\\dfrac{c(f)}{2}$ and let $D$ be a minimal crossing diagram of $f(G)$. By Lemma\\ \\ref{mini} we have $u(D)=\\dfrac{c(D)}{2}$. By Lemma \\ref{dtheta} we see that $f$ is trivial.\\qed\n\n\\begin{remark}\\label{ge2} {\\rm We can prove Theorem\\ \\ref{12c1} in the case of genus 2  by observing Lemma \\ref{dhandcuff} and Lemma \\ref{dtheta}. Let $D$ be a minimal crossing diagram of a non-trivial genus 2 handlebody-knot $H$. Then $D$ is also a diagram of a spatial embedding of a handcuff-graph or  a theta curve. \n\nIn the case $D$ is a diagram of a spatial handcuff-graph, by Lemma \\ref{dhandcuff} all crossings of $D$ are between two loops or $u(D) \\leq \\dfrac{c(D)-1}{2}$. In the case all crossings of $D$ are between two loops, by one IH-move on the edge that is not a loop we have a diagram $D'$ of $H$ such that $c(D')=c(D)=c(H)$ and $D'$ is also a diagram of a spatial theta curve (see Fig.\\ \\ref{wh}). By Lemma \\ref{dtheta} we have $u(D') \\leq \\dfrac{c(D')-1}{2}$.  \n\nIn the case $D$ is a diagram of a spatial theta curve, by Lemma\\ \\ref{dtheta} we have $u(D) \\leq \\dfrac{c(D)-1}{2}$. In the both cases we have $u(H) \\leq \\dfrac{c(H)-1}{2}$.}\n\\end{remark}\n\n\\begin{figure}[H]\n      \\begin{tabular}{c}\n      \\begin{minipage}[b]{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.35\\linewidth]{nselfhand.eps}\\vspace{0.5cm} \\\\\n          $D$\n        \\end{center}\n        \n      \\end{minipage}\n      \\begin{minipage}[b]{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.35\\linewidth]{nselftheta.eps}\\vspace{0.5cm} \\\\\n          $D'$\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{wh}\n\\end{figure}\n\n\n\\section{PROOFS OF THEOREM\\ \\ref{12c1} AND THEOREM\\ \\ref{12c2}}\nIn this section we prove Theorem \\ref{12c1} and Theorem \\ref{12c2}. In the following we give an inequality between unknotting number and crossing number by an observation of subdivided graph. \n\nLet $\\tilde{f}:G \\rightarrow \\mathbb{R}^2$ be a regular projection of a graph $G$. Let $c_1,\\ c_2,\\ \\cdots,\\ c_k$ be crossing points of $\\tilde{f}(G)$. A \\textit{subdivided graph} of $G$ at $\\{c_1,\\ c_2,\\ \\cdots,\\ c_k\\}$ is a graph obtained by adding $2k$ vertices $v_1,\\ v_1',\\ v_2,\\ v_2',\\ \\cdots ,v_k,\\ v_k'$ to $G$ such that $\\tilde{f}(v_i)=\\tilde{f}(v_i')=c_i$ and $v_i$\\ (resp.\\ $v_i'$) is contained in the over-arc\\ (resp.\\ the under-arc) at $c_i$. Then we say that $v_i$\\ (resp.\\ $v_i')$ is an \\textit{over-vertex}\\ (resp.\\ \\textit{under-vertex}) at $c_i$. Let $T$ be a spanning tree of $G'$. For any two vertices $v$ and $u$ of $G'$, let $d_{T}(v,u)$ be the number of edges of the path in $T$ joining $v$ and $u$.\n\n\\begin{lemma}\\label{31}\nLet $D$ be a diagram of a nontrivial handlebody-knot $H$. Let $\\tilde{f}:G \\rightarrow \\mathbb{R}^2$ be a regular projection of a connected trivalent graph $G$ where $D$ is obtained from $\\tilde{f}(G)$. Let $c_1,\\ c_2,\\ \\cdots,\\ c_k$ be crossing points of $\\tilde{f}(G)$. Let $G'$ be the subdivided graph of $G$ at $\\{c_1,\\ c_2,\\ \\cdots,\\ c_k\\}$. Let $v_1,\\ v_1',\\ v_2,\\ v_2',\\ \\cdots ,v_k,\\ v_k'$ be vertices of $G'$ such that $v_i$\\ (resp.\\ $v_i'$) is an over-vertex\\ (resp.\\ under-vertex) at $c_i\\ (i=1,2,\\cdots ,k)$. If there exists a vertex $v$ of $G'$ and a spanning tree $T$ of $G'$ such that $d_{T}(v,v_i)=d_{T}(v,v_i')$ for all $i \\in \\{1,2,\\cdots,k\\}$, then $u(D) \\leq \\dfrac{c(D)-k}{2}$. \n\\end{lemma}\n\\begin{proof}\nThe proof is analogous to the proof of \\cite[Proposition\\ 3.2]{forbidden}. We fix a vertex $v$ of $G'$ and a spanning tree $T$ of $G'$ such that $d_{T}(v,v_i)=d_{T}(v,v_i')$ for all $i \\in \\{1,2,\\cdots,k\\}$\\ (see Fig.\\ \\ref{311}). Let $h:G' \\rightarrow \\mathbb{R}$ be a continuous function with the following properties : \n\\begin{figure}[H]\n      \\begin{tabular}{c}\n      \\begin{minipage}{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.5\\linewidth]{41c2.eps}\n        \\end{center}\n      \\end{minipage}\n      \\begin{minipage}{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.5\\linewidth]{41spa.eps}\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{311}\n\\end{figure}\n\\noindent\n$(1)\\,$For each vertex $u$ of $T$, $h|_T(u)=-d_{T}(v,u)$.\\\\\n$(2)\\,h|_e$ is injective for each edge $e$ of $T$.\\\\\n$(3)\\,$Each edge of $G'-T$ has exactly one minimum point of $h$. \\\\\nWe can deform $h$ slightly so that $h(v_i)>h(v_i')\\ (i=1,2,\\cdots,k)$ since $d_{T}(v,v_i)=d_{T}(v,v_i')\\ (i=1,2,\\cdots,k)$. Then we give over\/under informations to $\\tilde{f}$ to produce a spatial embedding $f : G \\rightarrow \\mathbb{R}^3 = \\mathbb{R}^2 \\times \\mathbb{R}$ such that $p_1 \\circ f=\\tilde{f}$ and $p_2 \\circ f =h$, where $p_1$\\ (respectively $p_2)$ denotes the projection of $\\mathbb{R}^3$ to the first factor (respectively to the second factor) of $\\mathbb{R}^2 \\times \\mathbb{R}$. \n\n\nLet $D'$ be the diagram of $f(G)$ where $D'$ is obtained from $\\tilde{f}(G)$. We note that $D$ and $D'$ are deformed  into each other by changing over\/under informations of crossing points without changing over\/under informations of $c_1,\\ c_2,\\ \\cdots,\\ c_k$\\ (see for example Fig.\\ \\ref{312}). Since we obtain a bouquet as in Fig.\\ \\ref{313} which is trivial by contracting spatial edges of $f(T)$, $D'$ is a diagram of a trivial handlebody-knot. \n\n\\begin{figure}[H]\n      \\begin{tabular}{c}\n      \\begin{minipage}{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.6\\linewidth]{41d1hei.eps}\n        \\end{center}\n      \\end{minipage}\n      \\begin{minipage}{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.5\\linewidth]{41d1.eps}\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{312}\n\\end{figure}\n\n\\begin{figure}[H]\n      \\begin{tabular}{c}\n      \\begin{minipage}{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.4\\linewidth]{41ec.eps}\n        \\end{center}\n      \\end{minipage}\n      \\begin{minipage}{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.4\\linewidth]{41braid.eps}\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{313}\n\\end{figure}\n\n\nLet $D''$ be the diagram that is obtained from $D'$ by changing over\/under informations of all crossing points with the exception of  $c_1,\\ c_2,\\ \\cdots,\\ c_k$. Let $h':G' \\rightarrow \\mathbb{R}$ be a continuous function such that $h'=-h$. We can deform $h'$ slightly so that $h'(v_i)>h'(v_i')\\ (i=1,2,\\cdots,k)$\\  (see for example Fig.\\ \\ref{314}). Then $D''$ is the diagram of a spatial embedding of $f' : G \\rightarrow \\mathbb{R}^3 = \\mathbb{R}^2 \\times \\mathbb{R}$ such that $p_1 \\circ f'=\\tilde{f}$ and $p_2 \\circ f'=h'$. Thus we obtain a trivial bouquet by contracting spatial edges of $f'(T)$ and $D''$ is a diagram of a trivial handlebody-knot. \n\n\\begin{figure}[H]\n      \\begin{tabular}{c}\n      \\begin{minipage}{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.5\\linewidth]{41d2.eps}\n        \\end{center}\n      \\end{minipage}\n      \\begin{minipage}{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.6\\linewidth]{41d2hei.eps}\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{314}\n\\end{figure}\n\nLet $A$ be a subset of $C(D)$ such that changing all crossings in $A$ turns $D$ to $D'$. We note that changing all crossings in $(\\ C(D)-\\{c_1,\\ c_2,\\ \\cdots,\\ c_k\\}\\ )-A$ turns $D$ to $D''$. Therefore we have\n$$u(D) \\leq \\min \\{|A|,|(\\ C(D)-\\{c_1,\\ c_2,\\ \\cdots,\\ c_k\\}\\ )-A|\\} \\leq \\dfrac{c(D)-k}{2}$$ \n\\end{proof}\n\\noindent\n\\textit{Proof of Theorem \\ref{12c1}}\\\\\nLet $D$ be a minimal crossing diagram of a nontrivial handlebody-knot $H$. Let $\\tilde{f}:G \\rightarrow \\mathbb{R}^2$ be a regular projection of a connected trivalent graph $G$ where $D$ is obtained from $\\tilde{f}(G)$. Let $c_1$ be a crossing point of $\\tilde{f}(G)$. Let $G'$ be the subdivided graph of $G$ at $\\{c_1\\}$. Let $v_1$\\ (resp.\\ $v_1'$) be a vertex of $G'$ such that $v_1\\ (resp.\\ v_1')$ is over-vertex (resp.\\ under-vertex) at $c_1$. Let $T$ be a spanning tree of $G'$ containing $v_1$ and $v_1'$. \\\\\n\\begin{figure}[H]\n      \\begin{tabular}{c}\n      \\begin{minipage}{0.45\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.5\\linewidth]{52d.eps}\n        \\end{center}\n      \\end{minipage}\n      \\begin{minipage}{0.45\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.5\\linewidth]{theta52.eps}\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{141}\n\\end{figure}\nBy subdividing $T$ if necessary, we can choose a vertex $v$ of $T$ such that $d_{T}(v,v_1)=d_{T}(v,v_1')$ since there exists a path in $T$ joining $v_1$ and $v_1'$\\ (see for example Fig.\\ \\ref{141}). Then by Lemma \\ref{31} we have $u(D) \\leq \\dfrac{c(D)-1}{2}$. Since $u(H) \\leq u(D)$ and $c(D)=c(H)$ we have $u(H) \\leq \\dfrac{c(H)-1}{2}$. \\qed\n\n\\begin{lemma}\\label{d12c2}\nLet $D$ be a minimal crossing diagram of a nontrivial handlebody-knot $H$ that satisfies the equality $u(D)=\\dfrac{c(D)-1}{2}$. Let $\\gamma$ be a cycle of $D$ that has at least one crossing of $D$. Then the following $(1)$ and $(2)$ holds.\\\\\n$(1)\\,$All crossings of $D$ are self-crossings of $\\gamma$.\\\\\n$(2)\\,$There exists an odd number $p \\neq \\pm1$ such that $\\gamma$ is a reduced alternating diagram of a $(2,\\ p)$-torus knot.\n\\end{lemma}\n\\begin{proof}\nSuppose that $\\gamma$ has just one crossing. Suppose that $\\gamma$ itself is a simple closed curve on $\\mathbb{R}^2$. Then we have a diagram $D'$ of $H$ as illustrated in Fig.\\ \\ref{cl1} such that $c(D')=c(D)-1$. Suppose that $\\gamma$ is not a simple closed curve on $\\mathbb{R}^2$ and $\\gamma$ has exactly one crossing of $D$. Then by a similar deformation we have a diagram $D'$ of $H$ with $c(D')=c(D)-1$. This contradicts that $D$ is a minimal crossing diagram. Thus $\\gamma$ has at least two crossings of $D$. \\\\\n\\begin{figure}[h]\n      \\begin{tabular}{c}\n     \n      \\begin{minipage}{0.3\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.8\\linewidth]{cl11.eps}\\\\\n          $D$\n        \\end{center}\n      \\end{minipage}\n\n     \n      \\begin{minipage}{0.3\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.8\\linewidth]{cl131.eps}\\\\\n          \\ \\ \n        \\end{center}\n      \\end{minipage}\n\n     \n      \\begin{minipage}{0.3\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.8\\linewidth]{cl14.eps}\\\\\n          $D'$\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{cl1}\n\\end{figure}\\\\\nLet $\\tilde{f}$ be a regular projection of a trivalent graph $G$ where $D$ is obtained from $\\tilde{f}(G)$. First, we show that if $(1)$ does not hold, then $u(D)\\leq \\dfrac{c(D)-2}{2}$. We will show this claim step by step as follows.\\\\\n\\ \\\\\n\\textbf{Subclaim 1.}\\ \\textit{If one of the crossings on $\\gamma$, say $c_1$, is a crossing  between $\\gamma$ and $D-\\gamma$ and another crossing on $\\gamma$, say $c_2$, is a self-crossing of $\\gamma$, then $u(D) \\leq \\dfrac{c(D)-2}{2}$.}\n\\begin{proof}\nLet $G'$ be the subdivided graph of $G$ at $\\{c_1,\\ c_2\\}$. Let $v_i$\\ (resp.\\ $v_i')$ be the over-vertex\\ (resp.\\ under-vertex) at $c_i$\\ $(i=1,\\ 2)$. Then $G'$ is the graph as illustrated in Fig.\\ \\ref{slmu}\\ $(a)$ or $(b)$. \n\n\n\\begin{figure}[H]\n      \\begin{tabular}{c}\n      \\begin{minipage}{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.4\\linewidth]{slmu1.eps}\n        \\end{center}\n        \\subcaption{}\n      \\end{minipage}\n      \\begin{minipage}{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.4\\linewidth]{slmu2.eps}\n        \\end{center}\n        \\subcaption{}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{slmu}\n\\end{figure}\n\n\n\\begin{figure}[H]\n        \\begin{center}\n          \\includegraphics[width=0.2\\linewidth]{slmu1a.eps}\n        \\end{center}     \n      \\caption{}\n      \\label{slmua}\n\\end{figure}\n\nBy subdividing if necessary, we can choose a spanning tree $T$ of $G'$ and a vertex $v$ of $T$ such that $d_T(v,v_i)=d_T(v,v_i')\\ (i=1,2)$. A choice of $T$ and  $v$ for the case of Fig.\\ \\ref{slmu}\\ $(a)$ is illustrated in Fig.\\ \\ref{slmua}. By Lemma \\ref{31} we have $u(D)\\leq \\dfrac{c(D)-2}{2}$. \n\\end{proof}\n\\noindent\n\\textbf{Subclaim 2.}\\ \\textit{If two of crossings on $\\gamma$, say $c_1$ and $c_2$, are  crossings  between $\\gamma$ and $D-\\gamma$ then $u(D) \\leq \\dfrac{c(D)-2}{2}$.} \n\\begin{proof}\nLet $G'$ be the subdivided graph of $G$ at $\\{c_1,\\ c_2\\}$. Let $v_i$\\ (resp.\\ $v_i')$ be the over-vertex\\ (resp.\\ under-vertex) at $c_i$\\ $(i=1,\\ 2)$. Then $G'$ is one of the graphs as illustrated in Fig.\\ \\ref{mumu}. \n\n\\begin{figure}[H]\n      \\begin{tabular}{c}\n      \\begin{minipage}{0.24\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.8\\linewidth]{mumu3.eps}\n        \\end{center}\n        \\subcaption{}\n      \\end{minipage}\n      \\begin{minipage}{0.24\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.8\\linewidth]{mumu2.eps}\n        \\end{center}\n        \\subcaption{}\n        \\end{minipage}\n        \\begin{minipage}{0.24\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.8\\linewidth]{mumu1.eps}\n        \\end{center}\n        \\subcaption{}\n      \\end{minipage}\n      \\begin{minipage}{0.24\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.8\\linewidth]{mumu4.eps}\n        \\end{center}\n        \\subcaption{}\n      \\end{minipage}\n      \n      \\end{tabular}\n      \n      \\caption{}\n      \\label{mumu}\n\\end{figure}\n\n\\begin{figure}[H]\n        \\begin{center}\n          \\includegraphics[width=0.25\\linewidth]{mumu31.eps}\n        \\end{center}     \n      \\caption{}\n      \\label{mumua}\n\\end{figure}\n\nBy subdividing if necessary, we can choose a spanning tree $T$ of $G'$ containing $v_1,\\,v_1',\\,v_2,\\,v_2'$ and a vertex $v$ of $T$ such that $d_T(v,v_i)=d_T(v,v_i')\\ (i=1,2)$. A choice of $T$ and $v$ for the case of Fig.\\ \\ref{mumu}\\ $(a)$ is illustrated in Fig.\\ \\ref{mumua}. By Lemma \\ref{31} we have $u(D)\\leq \\dfrac{c(D)-2}{2}$. \n\\end{proof}\n\\noindent\n\\textbf{Subclaim 3.}\\ \\textit{If there exists a self-crossing of $D-\\gamma$, say $c_1$,  then $u(D) \\leq \\dfrac{c(D)-2}{2}$.} \n\\begin{proof}\nBy Subclaim 2 we may assume that $\\gamma$ has a self-crossing, say $c_2$. Let $G'$ be the subdivided graph of $G$ at $\\{c_1,\\ c_2\\}$. Let $v_i$\\ (resp.\\ $v_i')$ be the over-vertex\\ (resp.\\ under-vertex) at $c_i$\\ $(i=1,\\ 2)$. Then $G'$ is one of the graphs as illustrated in Fig.\\ \\ref{sese}. \n\n\\begin{figure}[H]\n      \\begin{tabular}{c}\n      \\begin{minipage}{0.24\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.42\\linewidth]{sese4.eps}\n        \\end{center}\n        \\subcaption{}\n      \\end{minipage}\n      \\begin{minipage}{0.24\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.42\\linewidth]{sese1.eps}\n        \\end{center}\n        \\subcaption{}\n        \\end{minipage}\n        \\begin{minipage}{0.24\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.42\\linewidth]{sese2.eps}\n        \\end{center}\n        \\subcaption{}\n      \\end{minipage}\n      \\begin{minipage}{0.24\\hsize}\n        \\begin{center}\n          \\includegraphics[height=0.42\\linewidth]{sese3.eps}\n        \\end{center}\n        \\subcaption{}\n      \\end{minipage}\n      \n      \\end{tabular}\n      \\caption{}\n      \\label{sese}\n\\end{figure}\n\n\n\\begin{figure}[H]\n        \\begin{center}\n          \\includegraphics[width=0.25\\linewidth]{sese41.eps}\n        \\end{center}     \n      \\caption{}\n      \\label{sesea}\n\\end{figure}\n\nBy subdividing if necessary, we can choose a spanning tree $T$ of $G'$ containing $v_1,\\,v_1',\\,v_2,\\,v_2'$ and a vertex $v$ of $T$ such that $d_T(v,v_i)=d_T(v,v_i')\\ (i=1,2)$.  A choice of $T$ and $v$ for the case Fig.\\ \\ref{sese}\\ $(a)$ is illustrated in Fig.\\ \\ref{sesea}.  By Lemma \\ref{31} we have $u(D)\\leq \\dfrac{c(D)-2}{2}$. \n\\end{proof}\n \nFrom the above we see that $\\gamma$ satisfies $(1)$. \\\\\n\\ \\\\\n\\textbf{Subclaim 4.}\\ \\textit{If $\\gamma$ is not obtained from a standard projection of a $(2,\\ p)$-torus knot for any odd number $p>1$, then $u(D) \\leq \\dfrac{c(D)-2}{2}$.}\n\n\\begin{proof}\n\nLet $G'$ be the subdivided graph of $G$ at $C(D)$ and let $\\Gamma$ be a cycle of $G'$ such that $\\gamma$ is obtained from $\\tilde{f}(\\Gamma)$. Note that if $\\tilde{f}(\\Gamma)$ is a standard projection of a $(2,\\ p)$-torus knot for some odd number $p\\neq \\pm 1$ as the case $p=5$ is illustrated in the left of Fig.\\ \\ref{pro25}, then any pair of crossings on $\\gamma$ are not parallel. Namely $\\Gamma$ is as illustrated in the right of Fig.\\ \\ref{pro25}. It follows from \\cite[Theorem\\ 1]{proje} that the converse is also true (see also \\cite[Proof of Theorem\\ 1.11]{pseudo}). \n\n\n\\begin{figure}[h]\n      \\begin{tabular}{c}\n      \n     \n      \\begin{minipage}[b]{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.4\\linewidth]{pro25.eps}\n        \\end{center}\n      \\end{minipage}\n      \n     \n      \\begin{minipage}[b]{0.5\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.4\\linewidth]{pro25cd.eps}\n        \\end{center}\n      \\end{minipage}\n      \n      \\end{tabular}\n      \\caption{}\n      \\label{pro25}\n\\end{figure}\n\n\nTherefore there are two crossings $c_1$, $c_2$ of $\\gamma$ such that $c_1$ and $c_2$ are parallel, namely $\\Gamma$ is illustrated as the left of Fig.\\ \\ref{para}. Let $v_i$\\ (resp.\\ $v_i')$ be the over-vertex\\ (resp.\\ under-vertex) at $c_i$\\ $(i=1,\\ 2)$. By subdividing if necessary, we can choose a spanning tree $T$ of $G'$ and a vertex $v$ of $T$ such that $d_T(v,v_i)=d_T(v,v_i')\\ (i=1,2)$\\ (see the right of Fig.\\ \\ref{para}). By Lemma \\ref{31} we have $u(D)\\leq \\dfrac{c(D)-2}{2}$. \n\\begin{figure}[h]\n      \\begin{tabular}{c}\n     \n      \\begin{minipage}[t]{0.45\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.4\\linewidth]{paracd.eps}\n        \\end{center}\n      \\end{minipage}\n\n     \n      \\begin{minipage}[t]{0.45\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.4\\linewidth]{paraspa.eps}\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{para}\n\\end{figure}\n\n\\end{proof}\n\nFinally, we show that if $u(D)=\\dfrac{c(D)-1}{2}$, then $u(\\gamma)=u(D)$ and $\\gamma$ is a reduced alternating diagram of a $(2,\\ p)-$ torus knot. \n\nLet $\\Gamma$ be a cycle of $G$ such that $\\gamma$ is obtained from $\\tilde{f}(\\Gamma)$. From the above $\\tilde{f}(\\Gamma)$ is a standard projection of a $(2,\\ p)-$torus knot as the case $p=5$ is illustrated in the left of Fig.\\ \\ref{pro25} and $c(\\gamma)=c(D)=p$. If we can join two components of $\\tilde{f}(\\Gamma) \\backslash C(\\tilde{f}(G))$ by a path $P$ of $\\tilde{f}(G)$ as illustrated in the left of Fig.\\ \\ref{pro2}, then there exists a cycle $\\gamma'$ of $D$ that has a crossing between $\\gamma$ and $D-\\gamma$ as illustrated in the right of Fig.\\ \\ref{pro2}. This is contradict to Lemma\\ \\ref{d12c2}\\ $(1)$. \n\n\\begin{figure}[h]\n      \\begin{tabular}{c}\n     \n      \\begin{minipage}[t]{0.45\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.5\\linewidth]{pro25p.eps}\n        \\end{center}\n      \\end{minipage}\n\n     \n      \\begin{minipage}[t]{0.45\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.5\\linewidth]{pro25p2.eps}\n        \\end{center}\n      \\end{minipage}\n      \\end{tabular}\n      \\caption{}\n      \\label{pro2}\n\\end{figure}\n\n\nTherefore we may assume that $\\tilde{f}(G)$ has no paths as illustrated in the left of Fig.\\ \\ref{pro2}, namely $\\tilde{f}(G)$ is a projection as illustrated in Fig.\\ \\ref{pro3}. Then by changing over\/under informations at $u(\\gamma)$ crossings on $D$ we can obtain a diagram of a trivial handlebody-knot. Since $c(\\gamma)=c(D),\\ u(D)=\\dfrac{c(D)-1}{2}$ and $u(D) \\leq u(\\gamma)$, we have $u(\\gamma)=\\dfrac{c(\\gamma)-1}{2}$. By Theorem \\ref{12c2k}, $\\gamma$ is a reduced alternating diagram of a $(2,\\ p)$-torus knot for some odd number $p \\neq \\pm 1$. \n\n\\begin{figure}[h]\n\\centering\n  \\includegraphics[width=0.25\\linewidth]{pro25g.eps}\n  \\caption{}\n      \\label{pro3}\n\\end{figure}\n\n\\end{proof}\n\n\\noindent\n\\textit{Proof of Thm \\ref{12c2}}\\\\\nLet $H$ be a nontrivial handlebody-knot that satisfies the equality $u(H) = \\dfrac{c(H) -1}{2}$. Let $D$ be a minimal crossing diagram of $H$. Since $u(H) \\leq u(D)$ and $\\dfrac{c(D)-1}{2} =\\dfrac{c(H)-1}{2}$ we have $u(D) \\geq \\dfrac{c(D)-1}{2}$. Thus by the proof of Theorem\\ \\ref{12c1} we have $u(D) = \\dfrac{c(D)-1}{2}$. Then by Lemma\\ \\ref{d12c2} we see that $H$ is a handlebody-knot represented by one of diagrams illustrated in Fig.\\ \\ref{d2n1}. We note that the unknotting number of handlebody-knot represented by $D_{2n-1}(n \\ne 0,1)$ are one by Proposition \\ref{2br}. Therefore $H$ is a handlebody-knot represented by $D_3$ or $D_{-3}$ as desired. \\qed \n\n\n\n\\section*{Acknowledgements}\nThe author would like to thank Professor Tomo Murao for his helpful comments. He is particularly grateful to Professor Kouki Taniyama for invaluable advice and\nhis suggestions.\n\n\\bibliographystyle{myplain2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nThe study of turbulence in a collisionless plasma is an extremely challenging problem to face because it is a strongly nonlinear process involving many decades of scales that extend from fluid magnetohydrodynamic (MHD) scales to ion kinetic and electron kinetic scales that are associated with different physical regimes. No general theory is currently capable of describing the full turbulent cascade process in a plasma. On the other hand, different reduced models have been formulated to describe the properties of the turbulent system in a limited range of spatial and temporal scales and in special physical conditions such as in the presence of a strong magnetic field that makes the plasma anisotropic (see, e.g., \\citet{00,01,02} and references therein). Thus, the properties of plasma turbulence can be studied in detail only by means of numerical simulations, within the limits of the currently available computational resources \\citep{03a,03,04}. Numerical studies are inspired and guided by in situ satellite measurements taken in the solar wind and in the terrestrial magnetosphere. Space plasmas represent natural laboratories for the study of plasma turbulence through extremely accurate spatial and temporal satellite data \\citep{06}. It is worth noting that today, space is the only environment where measurements down to electron scales are accessible, as in the case of the Magnetospheric Multiscale (MMS) space mission \\citep{07}, and even ion-scale measurements are far more accurate than in the laboratory.\n\nSolar wind studies focusing on the formation of the turbulent cascade at MHD scales have unambiguously demonstrated the fundamental role of low-frequency Alfv\u00e9n waves in nonlinearly building up the turbulent spectrum \\citep{02}. On the other hand, the properties of the turbulent cascade at kinetic scale are not yet fully understood. Energy transfers at sub-ion scales are thought to be driven by nonlinear interaction between relatively high-frequency modes such as kinetic Alfv\u00e9n waves and whistler waves \\citep{08}. Recent studies instead suggest that the development of turbulence at small scales is closely related to magnetic reconnection phenomena developing inside the current sheets that are spontaneously generated by the turbulent MHD dynamics, which create small-scale coherent structures where energy is thought to be dissipated \\citep{04,09,09a}. Understanding the nature of kinetic-scale turbulence in plasmas is therefore an open problem.\n\nObservations of reconnection driven by turbulence have been reported in space plasmas \\citep{10a,10b,10c}, and recently, satellite measurements of the MMS mission in the turbulent magnetosheath of Earth have given evidence of unusual reconnection events driven only by electrons, while ions were found to be decoupled from the magnetic field \\citep{10}. In particular, satellite data show electron-scale current sheets in which divergent bidirectional electron jets were not accompanied by any ion outflow. This situation is quite different from the standard reconnection picture, in which an electron-scale diffusion region is embedded within a wider ion-scale current sheet. For these reasons, these new phenomena were dubbed \"electron-only reconnection events\" (e-rec from now on). This discovery stimulated great interest first of all because it is not trivial to determine how electron-scale current sheets undergoing e-rec may form in a large-scale turbulent environment. For instance, in the terrestrial magnetosheath, energy is typically first transferred in a continuous way from large MHD scales down to ion kinetic scales (or directly injected by reconnection at ion kinetic scales) and finally to the electron kinetic scale. A fundamental question to answer is therefore how e-rec can be triggered by the turbulent motion of a plasma. This problem has recently been addressed by Califano et al. \\citep{11}, who showed using 2D-3V dimensional Eulerian hybrid Vlasov-Maxwell simulations \\citep{12a,12} that if the scale of injection of energy in a turbulent plasma is close to the ion kinetic scale,  ions decouple from the magnetic field and reconnection processes taking place in the system are driven exclusively by electrons, showing the same features of the e-rec events detected by MMS. The transition from standard ion-coupled reconnection to e-rec has recently been studied in detail by Pyakurel et al. \\citep{12b} using 2D-3V dimensional particle-in-cell simulations of laminar reconnection with conditions appropriate for the magnetosheath. By gradually increasing the size of the simulation box from a few ion inertial lengths to several tens of ion inertial lengths, they observed a smooth transition from the e-rec regime, where ions are decoupled from the reconnection dynamics, to the more familiar ion-coupled reconnection. \n\nAnother important aspect concerning the relationship between e-rec and turbulence is to understand whether and how this new reconnection process in turn affects the development of the turbulent energy cascade and its statistical properties, and if there are any differences with respect to the turbulence associated with standard reconnection. In this context, the magnetosheath data collected by the MMS satellites were recently analyzed by Stawarz et al. \\citep{12c}, who showed that the statistical distribution of the turbulent magnetic field fluctuations associated with e-rec and their spectral properties are analogous to those observed in other turbulent plasmas, such as the solar wind, and in numerical simulations of plasma turbulence. \n\nIn this paper we present a study of the statistical properties of fluctuations developing in a simulation of freely decaying plasma turbulence in which e-rec occurs. The results obtained from this simulation are then compared to those of a different simulation of plasma turbulence where standard reconnection takes place. We aim at finding possible differences between the statistical features of these two turbulent systems by taking advantage of the different dynamics of the ions  associated with the reconnection structures in the two simulations. In particular, we investigate if there is any specific signature of e-rec in the turbulence statistics. Our study is based on the analysis of the structure functions (hereafter, SFs) of turbulent fields. SFs have been used extensively to analyze numerical simulations \\citep{12d,12d2} and observational data \\citep{12e}, showing that the turbulent magnetic field undergoes a transition from an intermittent dynamics to a self-similar one at sub-ion scales \\citep{12d,12e}. Here we extend the SFs analysis to ion velocity fluctuations as well in order to characterize the behavior of this species, which has a very different role in the reconnection dynamics of the two simulations. Our main finding is that the turbulent fluctuations associated with e-rec show the same statistical properties as the turbulent fluctuations associated with standard ion-coupled reconnection. The structure of the turbulent cascade is also examined. In particular, the properties of the magnetic field dissipation range of a collisionless turbulent plasma are discussed, and we claim that only electrons contribute to its formation.\n\nThe paper is structured as follows: the numerical model implemented in our simulations is discussed in section 2. In section 3 we describe the specific setup adopted for the two simulations considered here, which are the same as were analyzed by Califano et al. (2020). The method of analysis based on the study of SFs is introduced in section 4, and our results are presented in section 5. Our conclusions are finally discussed in the last section.\n\n\\section{Numerical Model} \n\nThe two simulations analyzed in this paper were performed using an Eulerian hybrid Vlasov-Maxwell (HVM) 2D-3V dimensional code that advances the Vlasov equation for ions in time \\citep{12a}, coupled with an isothermal fluid model with finite mass for the electrons \\citep{12}. The electron response is described by the generalized Ohm law that includes electron inertia terms, allowing the complete decoupling of the magnetic field at electron scales \\citep{12},\\\\\n\\begin{gather}\n\\textbf{E}-\\frac{d^2_e}{n}\\nabla^2\\textbf{E}=\\frac{1}{n}(\\textbf{J}\\times\\textbf{B})-(\\textbf{u}\\times\\textbf{B})-\\frac{1}{n}\\nabla P_e + \\notag \\\\\n\\\\\n+\\frac{d^2_e}{n} \\nabla \\cdot \\left(\\textbf{u}\\textbf{J}+\\textbf{J}\\textbf{u}- \\frac{\\textbf{J}\\textbf{J}}{n} \\right), \\notag\n\\end{gather}\n\\\\where $\\textbf{E}$ and $\\textbf{B}$ are the electric and magnetic fields, respectively, $\\textbf{J}\\!=\\!\\nabla \\times \\textbf{B}$ is the current density (we neglect the displacement current), $\\textbf{u}$ and $P_i$ are the ion velocity and pressure, respectively, $P_e\\!=\\!nT_e$ is the electron isothermal pressure, and $d_e$ is the electron inertial length. Furthermore, quasi-neutrality is assumed so that ion and electron densities are equal $n_e\\!=\\!n_i\\!=\\!n$. Finally, the evolution of the magnetic field is described by the Faraday equation. All equations were normalized and transformed in dimensionless units using the ion mass $m_i$, charge $+e$, inertial length $d_i$ , and cyclotron frequency $\\Omega_i$ (see \\citet{12}). For the sake of numerical stability, a numerical filter that smooths out the electromagnetic fields at high wave numbers was used \\citep{13}.\n\n\\section{Simulations Setup}\n\nWe report two simulations that were identical in every aspect except for the spectrum of modes that was used to initially drive turbulence: the first simulation was initialized with ion-scale fluctuations so that e-rec can take place, with ions that do not participate significantly, whereas the second simulation has only large-scale perturbations and reconnection occurs in the usual ion-coupled regime.\n\nThe equations of the HVM model were integrated on a 2D-3V domain (bidimensional in real space and tridimensional in velocity space). In both simulations we took a square spatial domain of size $L\\!=\\!20 \\pi d_i$ covered by a uniform grid consisting of $1024^2$ mesh points, while the velocity domain was cubic with sides spanning from $-5v_{th,i}$ to $+5v_{th,i}$ in each direction (where $v_{th,i}$ is the ion thermal velocity) and sampled by a uniform grid consisting of $51^3$ mesh points. The ion-to-electron mass ratio was $m_i\/m_e\\!=\\!144,$ which implies $d_i\/d_e\\!=\\!\\sqrt[]{m_i\/m_e}\\!=\\!12$, the electron temperature was set to $T_e\\!=\\!0.5,$ and the plasma beta was $\\beta\\!=\\!1$, corresponding to $v_{th,i}\\!=\\!\\sqrt[]{\\beta\/2}\\!=\\!\\sqrt[]{1\/2}$ (in Alfv\u00e9n speed units). Both simulations were initialized with an isotropic Maxwellian distribution for ions and an homogeneous out-of-plane guide field $\\textbf{B}_0$ along the $z$ -axis. Turbulence was triggered by adding to the guide field some large-scale, random phase, isotropic magnetic field sinusoidal perturbations $\\delta \\textbf{B}$. For the first simulation (hereafter, sim.1), we took perturbations with wavenumber $k$ in the range $0.1 \\leqslant k d_i \\leqslant 0.6$, mean amplitude $\\left| \\delta\\textbf{B} \\right|_{rms}\/B_0\\!\\simeq\\!0.2,$ and maximum amplitude $\\left| \\delta\\textbf{B} \\right|_{max}\/B_0\\!\\simeq\\!0.5$. The scales of the largest wavenumbers of these perturbations were close to ion kinetic scales in order for the ions to be nearly decoupled from the magnetic field dynamics from the beginning of the simulation and therefore to drive a turbulent environment in which e-rec occurs \\citep{11}. In the second simulation (hereafter, sim.2), the system was perturbed by fewer modes with wavenumber $k$ in the range $0.1 \\leqslant k d_i \\leqslant 0.3$, all being far larger than ion kinetic scales, mean amplitude $\\left| \\delta\\textbf{B} \\right|_{rms}\/B_0\\!\\simeq\\!0.25,$ and maximum amplitude $\\left| \\delta\\textbf{B} \\right|_{max}\/B_0\\!\\simeq\\!0.5$. In this way, ions were magnetized at the beginning of the simulation, eventually leading to a turbulent environment in which standard reconnection occurs. The time step used for both simulations was $\\Delta t\\!=\\!0.005 \\,\\Omega^{-1}_i$ in order to accurately resolve phenomena with frequencies between the electron cyclotron frequency $\\Omega_e$ and the ion cyclotron frequency $\\Omega_i$. This choice is consistent with the limits of our HVM model, where ions are kinetic, while electrons, with mass, are taken as fluid but adopting the Ohm law corresponding to an electron magnetohydrodynamics (EMHD) dynamics \\citep{11}. It is worth noting that in our simulations, with the spatial resolution we chose, we have only two points to resolve the electron inertial length $d_e$ , but nonetheless this was sufficient to distinguish the EMHD invariant $F\\!\\doteq\\!\\psi-d_{\\rm e}^2\\nabla^2\\psi$ from the flux function $\\psi$ \\citep{23}, in other words, it allowed us to accurately resolve the electron physics at sub-ion scales (see \\citet{11} for a detailed discussion of this point).\n\nWe did not include any external forcing term in our model (in this case, we talk about freely decaying turbulence simulations). This means that when the plasma is in a turbulent regime, the energy dissipated at small scales  is not replaced by any large-scale energy source and the system will never reach the statistically stationary state corresponding to a fully developed turbulence \\citep{14}. However, there is a time interval during which 2D freely decaying turbulence reaches a peak of activity that shows statistical properties that are very similar to those of homogeneous and isotropic fully developed turbulence. This time interval corresponds to a period in which the out-of-plane mean square current $\\langle J^2_z \\rangle$ reaches and maintains a roughly constant peak value \\citep{03a,04,12d} that corresponds to an intense small-scale activity \\citep{16}. For this reason, the analysis of the turbulence statistics was carried out at a fixed time close to the peak of $\\langle J^2_z \\rangle$ in both simulations.\n\n\\section{Structure functions and intermittency} \\label{sec04}\n\nOne way to characterize a turbulent process, regardless of its nature, is to analyze the statistical features of the fluctuations of the physical quantities at different scales \\citep{01,14}. For a plasma, these quantities could be for example the magnetic field $\\textbf{B}$ or the ion velocity $\\textbf{u}$. Given a generic vector quantity $\\textbf{q}(\\textbf{x})$, its fluctuations in the direction of $\\textbf{r}$ at scale $r\\!=\\!|\\textbf{r}|$ can be defined as \\citep{14}\\\\\n\\begin{equation}\n\\Delta q_\\parallel(\\textbf{x},\\textbf{r})\\!=\\![\\textbf{q}(\\textbf{x}+\\textbf{r})\\!-\\!\\textbf{q}(\\textbf{x})]\\!\\cdot\\!\\frac{\\textbf{r}}{r}\n.\\end{equation}\n\\\\The moments of order $p$ of such fluctuations are given by\\\\\n\\begin{equation}\nS_p(\\textbf{r})\\!=\\!\\langle |\\Delta q_\\parallel(\\textbf{x},\\textbf{r})|^p \\rangle \\label{sf}\n\\end{equation}\n\\\\and are known as (longitudinal) structure functions of the variable $\\textbf{q}(\\textbf{x})$, where the symbol $\\langle \\cdot \\rangle$ indicates the average on a suitable statistical ensemble. For a homogeneous and isotropic system, SFs depend solely on $r$ and the ensemble average can be replaced by an average over the real space. \n\nThe importance of SFs in analyzing turbulence lies in the fact that in many turbulent processes, they take the form of a power law,\\\\\n\\begin{equation}\nS_p(r)\\sim r^{\\xi(p)} \\label{power}\n,\\end{equation}  \n\\\\where $\\xi(p)$ is called the scaling exponent of the process. This exponent contains important information about the spatial distribution of fluctuations. It is possible to prove that if $\\xi(p)\\!=\\!ph$ (with $h$ being a constant), the fluctuations are self-similar, that is, they are uniformly distributed in the system at all scales. On the other hand, if $\\xi(p)$ is nonlinear in $p,$ the fluctuations are intermittent, which means that they become increasingly less homogeneous with decreasing scale length, and they tend to be concentrated only in some portions of the system \\citep{14}. Therefore SFs represent a powerful analysis tool that allows us to identify some key properties of a turbulent process.\n\nSometimes the SFs of finite systems where turbulence is not fully developed do not take the form of the power law of eq. (\\ref{power}). Nevertheless, the turbulent flow can still be characterized using a set of scaling exponents $\\xi(p)$ if by plotting SFs of different order one against the other, the following scaling is obtained:\\\\\n\\begin{equation}\nS_p(r) \\sim S_q(r)^{\\beta(p,q)}\n,\\end{equation}\n\\\\where $\\beta(p,q)\\!=\\!\\xi(p)\/\\xi(q)$. In this case, we talk about extended self-similarity (ESS), which has been observed in many experimental turbulent systems as well as in numerical simulations \\citep{17,18,19}. In the case of ESS, it is not possible to calculate all the $\\xi(p)$ separately because they appear in the form of a fraction in the scaling exponents $\\beta(p,q)$. However, the knowledge of $\\beta(p,q)$ alone is sufficient to determine whether the turbulent cascade is self-similar or intermittent. In the case of self-similarity, $\\xi(p)\\!=\\!ph$ and so $\\beta(p,q)\\!=\\!p\/q$, while in the case of intermittency, $\\beta(p,q)\\!\\neq\\!p\/q$ \\citep{12d}.\n\nIn our case, the SFs were calculated by assuming homogeneity and isotropy in both simulations. In this way, eq. (\\ref{sf}) reduces to\\\\\n\\begin{gather}\nS_p(r)=\\langle |q_x(x+r,y)-q_x(x,y)|^p \\rangle=\\notag \\\\\n\\\\\n\\langle |q_y(x,y+r)-q_y(x,y)|^p \\rangle, \\notag\n\\end{gather}\n\\\\where the ensemble average is replaced by the average over real space. The assumption of homogeneity and isotropy was confirmed by comparing the SFs calculated using $q_x$ with the SFs calculated using $q_y$ , and we found only very little difference between them for all quantities we considered in the two simulations. SFs higher than $p\\!=\\!4$ were not considered here because calculating them requires a larger simulation grid with many more points in the real space domain than we used \\citep{20a,20}. This problem is related to the fact that the calculation of high-order moments of a quantity strongly depends on the tails of its distribution, which are often associated with low probability. As a result, when the ensemble average is replaced with the real space average, it is necessary to ensure that the number of sampled grid points is large enough to include the tail events. \n\n\\section{Results}\n\nThe statistical analysis of the turbulent fluctuations in the two simulations was carried out at a fixed time when the turbulent activity was at its maximum. For sim.1, where e-rec is observed, this time corresponds to $t_1\\! = \\!131.7 \\,\\Omega^{-1}_i$ , while for sim.2, where magnetic reconnection develops according to the standard picture, this time corresponds to $t_2\\! = \\!147.5 \\,\\Omega^{-1}_i$.\n\n\\begin{figure}[t]\n\\centering\n\\subfloat{\n\\begin{tikzpicture}\n\\node[] at (0,0) { \\includegraphics[width=.48\\linewidth]{e-rec_Jz} };\n\\node[] at (-4.2,3) { \\Large{(a)} };\n\\end{tikzpicture}\n}\n\\subfloat{\n\\begin{tikzpicture}\n\\node[] at (0,0) { \\includegraphics[width=.48\\linewidth]{n-rec_Jz} };\n\\node[] at (-4.2,3) { \\Large{(b)} };\n\\end{tikzpicture}\n}\n\\\\\n\\subfloat{\n\\begin{tikzpicture}\n\\node[] at (0,0) { \\includegraphics[width=.48\\linewidth]{e-rec_jdote} };\n\\node[] at (-4.2,3) { \\Large{(c)} };\n\\end{tikzpicture}\n}\n\\subfloat{\n\\begin{tikzpicture}\n\\node[] at (0,0) { \\includegraphics[width=.48\\linewidth]{n-rec_jdote} };\n\\node[] at (-4.2,3) { \\Large{(d)} };\n\\end{tikzpicture}\n}\n\\caption{Top panels: Shaded contour plots of the out-of-plane current $J_z$ (colored) and contour lines of the flux function $\\Psi$ (black lines) of sim.1 at $t_1\\!=\\!131.7 \\,\\Omega^{-1}_i$ (a) and of sim.2 at $t_2\\!=\\!147.5 \\,\\Omega^{-1}_i$ (b). Bottom panels: Shaded contour plots of $\\textbf{J}\\cdot\\textbf{E}$ (colored) and contour lines of $\\Psi$ (black lines) of sim.1 at $t_1\\!=\\!131.7 \\,\\Omega^{-1}_i$ (c) and of sim.2 at $t_2\\!=\\!147.5 \\,\\Omega^{-1}_i$ (d).}\n\\label{fig_Jz_e-rec}\n\\end{figure}\n\nIn the top panels of figure \\ref{fig_Jz_e-rec} we show for both simulations the shaded contour plots of the out-of-plane current $J_z$ together with the contour lines of the flux function $\\Psi$, related to the in-plane magnetic field by $\\textbf{B}_\\perp\\!=\\!\\nabla\\Psi\\times\\textbf{e}_z$ (with $\\textbf{e}_z$ being the out-of-plane unit vector). In both simulations we see that the magnetic configuration of the system is characterized by a large number of island-like magnetic structures of various sizes and shapes, produced by the nonlinear evolution of the initial perturbation. The process of the formation of reconnection sites and the development of an intermittent turbulent cascade of magnetic energy can be understood in terms of the nonlinear interaction between these magnetic islands, which attract one another when the associated central current $J_z$ is of the same sign (and vice versa in the case of opposite sign). In particular, as two islands with central $J_z$ of the same sign approach each other, the magnetic field lines of opposite sign between them are pushed against each other, and this leads to the formation of a thin current sheet where reconnection occurs and magnetic energy is dissipated. Thus, as a result of this dynamics, reconnecting current sheets are not uniformly distributed in a turbulent plasma, they tend to be concentrated between merging magnetic islands, and therefore the dissipation of magnetic energy is nonuniform, that is, the turbulent cascade of magnetic energy is intermittent. The relation between the formation of localized reconnecting current sheets and the development of an intermittent turbulent cascade is highlighted by the contour plots of $\\textbf{J}\\cdot\\textbf{E}$ shown in figure \\ref{fig_Jz_e-rec} (the flux function $\\Psi$ is overplotted), bottom panels, made at the same time instants and for the same runs as the corresponding contour plots of $J_z$ in the top panels. The quantity $\\textbf{J}\\cdot\\textbf{E}$, representing the energy exchange between the electromagnetic field and the plasma, is significantly nonzero only in correspondence to the intense current structures, thus marking the strong correlation between reconnection and the intermittent dissipation of magnetic energy.\n\nThe characteristic size of the magnetic islands depends on the wavelength of the initial fluctuations, that is, on the injection scale, and therefore magnetic islands in sim.1 are smaller than those in sim.2. As a result, the characteristic thickness and length of the current sheets in the two simulations are different as well, and this affects the ion magnetization and consequently the dynamics of magnetic reconnection \\citep{12b}. It has been shown in Califano et al. (2020) that in sim.1, ions are (and remain) decoupled from the magnetic field on the scale of the current sheets and because of this, e-rec develops. Conversely, the current sheets of sim.2 are large enough to let the ions participate in the magnetic field dynamics, hence reconnection proceeds according to the standard ion-coupled reconnection model. A statistical analysis of the characteristic widths and lengths of the current structures of the two simulations here considered has been carried out by Califano et al. (2020),   who showed that the reconnecting current sheets of sim.1 are shorter than those of sim.2, while their characteristic width is about the same in the two simulations. In particular, in sim.2 the characteristic length of the current sheets was found to be at least about $10 d_i$ and to vary up to scales of some tens of $d_i$. On the other hand, in sim.1, all the reconnecting current sheets have about the same length, which is about a few $d_i$.\n\nIn summary, the turbulent magnetic fluctuations of sim.1 and sim.2 have a significantly different local dynamics. We now determine whether there is a difference in their statistical features, in particular by analyzing the SFs of the magnetic field.\n\n\\begin{figure}[t]\n\\centering\n\\subfloat{\n\\begin{tikzpicture}\n\\node[] at (0,0) { \\includegraphics[width=.48\\linewidth]{e-rec_SFBs} };\n\\node[] at (-4.1,1.7) { \\Large{(a)} };\n\\node[] at (-1.2,-1.4) { \\scriptsize{Range I} };\n\\node[] at (0.6,-1.4) { \\scriptsize{Range II} };\n\\end{tikzpicture}\n}\n\\subfloat{\n\\begin{tikzpicture}\n\\node[] at (0,0) { \\includegraphics[width=.48\\linewidth]{n-rec_SFBs} };\n\\node[] at (-4.1,1.7) { \\Large{(b)} };\n\\node[] at (-1.2,-1.4) { \\scriptsize{Range I} };\n\\node[] at (0.6,-1.4) { \\scriptsize{Range II} };\n\\end{tikzpicture}\n}\n\\\\\n\\subfloat{\n\\begin{tikzpicture}\n\\node[] at (0,0) { \\includegraphics[width=.48\\linewidth]{e-rec_ESS2} };\n\\node[] at (-4.1,1.7) { \\Large{(c)} };\n\\end{tikzpicture}\n}\n\\subfloat{\n\\begin{tikzpicture}\n\\node[] at (0,0) { \\includegraphics[width=.48\\linewidth]{n-rec_ESS2} };\n\\node[] at (-4.1,1.7) { \\Large{(d)} };\n\\end{tikzpicture}\n}\n\\\\\n\\subfloat{\n\\begin{tikzpicture}\n\\node[] at (0,0) { \\includegraphics[width=.48\\linewidth]{e-rec_beta1} };\n\\node[] at (-4.1,1.7) { \\Large{(e)} };\n\\end{tikzpicture}\n}\n\\subfloat{\n\\begin{tikzpicture}\n\\node[] at (0,0) { \\includegraphics[width=.48\\linewidth]{n-rec_beta1} };\n\\node[] at (-4.1,1.7) { \\Large{(f)} };\n\\end{tikzpicture}\n}\n\\caption{Top panels: Magnetic field structure functions $S_{B,p}$ (log-scale) of sim.1 at $t_1\\!=\\!131.7 \\,\\Omega^{-1}_i$ (a) and of sim.2 at $t_2\\!=\\!147.5 \\,\\Omega^{-1}_i$ (b); dashed straight lines represent the power laws $r^p$ , and vertical dash-dotted lines delimit ranges I and II. Middle panels: $S_{B,4}$ vs. $S_{B,2}$ (filled black dots, in log-scale) for $r\\!<\\!10\\,d_i$ from sim.1 at $t_1\\!=\\!131.7 \\,\\Omega^{-1}_i$ (c) and sim.2 at $t_2\\!=\\!147.5 \\,\\Omega^{-1}_i$ (d); ranges I and II were fit separately with straight lines (blue and red lines, respectively); vertical dash-dotted lines separate range I from range II. Bottom panels: Magnetic field scaling exponents $\\beta_B(p,1)$ within range I (blue) and range II (red) from sim.1 at $t_1\\!=\\!131.7 \\,\\Omega^{-1}_i$ (e) and sim.2 at $t_2\\!=\\!147.5 \\,\\Omega^{-1}_i$ (f); the dashed straight line, representing the self-similar scaling $\\beta(p,1)\\!=\\!p$, is given as reference.}\n\\label{fig02}\n\\end{figure}\n\nIn panels (a) and (b) of figure \\ref{fig02} we compare the first four magnetic field structure functions $S_{B,p}$ (in logarithmic scale) of sim.1 and sim.2, respectively. These SFs were calculated using $B_x$. The same results were obtained using $B_y$ (not shown here), which means that magnetic field turbulence is isotropic in our simulations. Figure \\ref{fig02} shows that all magnetic field SFs of both simulations have the same behavior over the range of scales we considered and that there are no significant differences between sim.1 and sim.2. In particular, we see that for $r\\!>\\!10\\,d_i$ , all SFs start to saturate, while for $r\\!<\\!10\\, d_i$ , it is possible to distinguish two ranges that correspond to two different scalings. The first range, hereafter called range I, extends from $r\\!\\simeq\\!0.06\\, d_i$ to $r\\!\\simeq\\!0.3\\, d_i$. Here the SFs follow the power law of eq. (\\ref{power}). The second range, hereafter called range II, reaches from $r\\!\\simeq\\!0.3\\, d_i$ to $r\\!\\simeq\\!10\\, d_i$. In this range, $log(S_{B,p})$ is nonlinear in $log(r),$ which means that the SFs do not take the form of a power law.\n\n\\begin{figure}[t]\n\\centering\n\\subfloat{\n\\begin{tikzpicture}\n\\node[] at (0,0) { \\includegraphics[width=.48\\linewidth]{e-rec_SFus} };\n\\node[] at (-4.1,1.7) { \\Large{(a)} };\n\\end{tikzpicture}\n}\n\\subfloat{\n\\begin{tikzpicture}\n\\node[] at (0,0) { \\includegraphics[width=.48\\linewidth]{n-rec_SFus} };\n\\node[] at (-4.1,1.7) { \\Large{(b)} };\n\\end{tikzpicture}\n}\n\\\\\n\\subfloat{\n\\begin{tikzpicture}\n\\node[] at (0,0) { \\includegraphics[width=.48\\linewidth]{e-rec_ESSu2} };\n\\node[] at (-4.1,1.7) { \\Large{(c)} };\n\\end{tikzpicture}\n}\n\\subfloat{\n\\begin{tikzpicture}\n\\node[] at (0,0) { \\includegraphics[width=.48\\linewidth]{n-rec_ESSu2} };\n\\node[] at (-4.1,1.7) { \\Large{(d)} };\n\\end{tikzpicture}\n}\n\\\\\n\\subfloat{\n\\begin{tikzpicture}\n\\node[] at (0,0) { \\includegraphics[width=.48\\linewidth]{e-rec_betau1} };\n\\node[] at (-4.1,1.7) { \\Large{(e)} };\n\\end{tikzpicture}\n}\n\\subfloat{\n\\begin{tikzpicture}\n\\node[] at (0,0) { \\includegraphics[width=.48\\linewidth]{n-rec_betau1} };\n\\node[] at (-4.1,1.7) { \\Large{(f)} };\n\\end{tikzpicture}\n}\n\\caption{Top panels: Ion velocity structure functions $S_{u,p}$ (in log-scale) of sim.1 at $t_1\\!=\\!131.7 \\,\\Omega^{-1}_i$ (a) and of sim.2 at $t_2\\!=\\!147.5 \\,\\Omega^{-1}_i$ (b); dashed straight lines represent the power laws $r^p$ , and the vertical dash-dotted lines separate the power law-like region from the saturation region. Middle panels: $S_{u,4}$ vs. $S_{u,2}$ (filled black dots, in log-scale) for $r\\!<\\!7\\,d_i$ from sim.1 at $t_1\\!=\\!131.7 \\,\\Omega^{-1}_i$ (c) and sim.2 at $t_2\\!=\\!147.5 \\,\\Omega^{-1}_i$ (d); these curves were fit with a straight line (in magenta). Bottom panels: Ion velocity scaling exponents $\\beta_u(p,1)$ in range $r\\!<\\!7\\,d_i$ from sim.1 at $t_1\\!=\\!131.7 \\,\\Omega^{-1}_i$ (e) and sim.2 at $t_2\\!=\\!147.5 \\,\\Omega^{-1}_i$ (f); the dashed straight line, representing the self-similar scaling $\\beta(p,1)\\!=\\!p$, is given as reference.}\n\\label{fig03}\n\\end{figure}\n\nThe large-scale behavior observed for $r\\!>\\!10\\,d_i$ is expected because we used periodic boundary conditions in a finite box, which causes the SFs to become periodic and even in $r$ \\citep{21}. Because of these properties, all SFs here considered tend to grow for $r\\!>\\!0$ and start to saturate around $r\\!=\\!L\/2$ (where $L$ is the box size), while for $r\\!>\\!L\/2,$ they decrease symmetrically with respect to $r\\!=\\!L\/2$. We did not analyze the SFs for $r\\!>\\!10\\,d_i$ because the statistics there would be affected by these finite-box effects.\n\nThe small-scale power law behavior observed in range I is expected as well \\citep{18,22} because of the dissipation effects that become important at small $r$ and tend to smooth out magnetic field fluctuations. This implies that in the dissipation range $B_x(x+r,y)-B_x(x,y)\\!=\\!\\Delta B_x(r)\\!\\sim\\!r$ and consequently, the magnetic field SFs take the form of the power law $S_{B,p}\\!\\sim\\!r^p$. This appears evident in the first two panels of figure \\ref{fig02}, where in range I all SFs overlap almost perfectly with their corresponding smooth scaling power law $r^p$ in both simulations. Thus, range I can be identified as the magnetic field dissipation range.\n\nAs discussed in section \\ref{sec04}, even if the SFs do not scale as a power law, as in range II, it can still be possible to characterize the turbulent fluctuations with a set of scaling exponents $\\beta(p,q)$ if ESS is observed. Thus, we tested range II for ESS by analyzing all combinations of magnetic field SFs of different order plotted against each other. Panels (c) and (d) of figure \\ref{fig02} show an example of two magnetic field SFs of different orders plotted against each other for $r\\!<\\!10\\,d_i$ from sim.1 and sim.2, respectively. Each of these curves was fit separately in range I and range II using two straight lines, and we find that in both simulations, they are linear in range I (blue line) and range II (red line), but with different slopes. This means that ESS holds well in range II but with a different scaling exponent than in range I. The same behavior was found for any other combination of magnetic field SFs of different order in both simulations (not shown here). As ESS is observed, we proceed by evaluating the magnetic field scaling exponents $\\beta_B(p,q)$. Panels (e) and (f) of figure \\ref{fig02} show $\\beta_B(p,q)$ as a function of $p$ at fixed $q\\!=\\!1$ within range I (blue curve) and range II (red curve) for sim.1 and sim.2, respectively. These exponents were calculated for both ranges separately by taking the gradients of the linear fits of all possible combinations of $log(S_{B,p})$ versus $log(S_{B,q})$. In both simulations, $\\beta_B(p,1)$ is linear in $p$ within range I and becomes nonlinear in range II. It is worth noting that a very small deviation from the self-similar scaling is observed in range I as p increases because the calculation of SFs performed by averaging over the simulation grid becomes increasingly less accurate with increasing $p,$ as pointed out in section \\ref{sec04}. The same behavior with $\\beta_B(p,q)$ being linear in range I and nonlinear in range II was observed for any other value of $q$  in sim.1 and sim.2 (not shown here). This result suggests that in both simulations, the magnetic field fluctuations are intermittent for $r\\!>\\!0.3\\,d_i,$ and they become self-similar for $r\\!<\\!0.3\\,d_i$.\n\nThis small-scale transition from an intermittent inertial range to a self-similar magnetic field dissipation range has previously been observed in numerical simulations \\citep{12d} and is consistent with the Cluster satellite measurements \\citep{12e}. However, in our case, it takes place at scales of about a few electron inertial lengths (around $r\\!\\simeq\\!0.3\\,d_i\\!\\simeq\\!3.6\\,d_e$) rather than at $r\\!\\simeq\\!1\\,d_i$. Furthermore, no relevant differences between the magnetic field statistics of sim.1 and sim.2 are detected, suggesting that the statistical features of the turbulent cascade of magnetic energy, and in particular, the formation of the dissipation range, are independent of the specific reconnection mechanism associated with the evolution of magnetic field fluctuations. These results agree with recent MMS measurements in the magnetosheath of Earth, showing that the statistical properties of turbulent magnetic fluctuations associated with e-rec are analogous to those of other turbulent plasmas where standard reconnection occurs \\citep{12c}.\n\nAs the magnetic field statistics do not show any significant difference between sim.1 and sim.2, we analyzed the SFs of the ion fluid velocity $\\textbf{u}$ to determine whether they show any signature of e-rec because the ions do play a very different role in the reconnection dynamics of the two simulations. In panels (a) and (b) of figure \\ref{fig03} we compare the first four ion velocity structure functions $S_{u,p}$ (in logarithmic scale) of sim.1 and sim.2, respectively. All the ion velocity SFs we considered were calculated using $u_x$ , but the same results were obtained using $u_y$ (not shown here). This again implies that ion turbulence is essentially isotropic in our simulations. Surprisingly, as in the case of the magnetic field SFs, figure \\ref{fig03} shows that the ion velocity SFs of both simulations shows the same features in the range of scales we considered, and there are no noticeable differences between sim.1 and sim.2. On the other hand, their behavior is significantly different from that of the magnetic field SFs because we do not observe any sub-ion scale transition such as the one between range I and II that characterizes the magnetic field statistics (see the top panels of figure \\ref{fig02}). In particular, we see that for $r\\!>\\!7\\, d_i$ all SFs start to saturate, while for $r\\!<\\!7\\, d_i$ , they behave like a power law, although the transition between these two regions is not sharp and introduces some curvature between about $2\\, d_i$ and $7\\, d_i$.\n\nThe large-scale saturation observed for $r\\!>\\!7\\, d_i$ is caused, as in the case of the magnetic field SFs, by the use of periodic boundary conditions in a finite simulation box. We did not analyze the ion velocity SFs in this range because their properties here are significantly affected by these finite box effects.\n\nAs concerning the ion velocity SFs behavior for $r\\!<\\!7\\,d_i$, we already said that SFs are usually expected to take the form of the power law $r^p$ at small $r$ because of dissipation that tends to smooth out fluctuations on small scales. However, the first two panels of figure \\ref{fig03} show that in both simulations, all ion velocity SFs are well approximated by their corresponding $r^p$ power law for $r\\!<\\!2\\,d_i$, a range that is much wider than the dissipation range of the magnetic field SFs that was identified with range I. This means that the ion velocity fluctuations are smooth on a wider range than the magnetic field fluctuations. However, the formation of this extended ion dissipation range observed in the ion velocity SFs must have a different origin than the magnetic field dissipation range as it covers a range of scales that far exceeds range I and extends to ion scales. A possible explanation is that the development of the ion dissipation range is related to ions being decoupled from the magnetic field at scales of about the ion Larmor radius $\\rho_i$ (which is on the same order as $d_i$ for $\\beta\\!=\\!1$, as in our simulations) where ion thermal effects become important. It is reasonable to assume that if the system develops magnetic fluctuations at scales on the same order of $\\rho_i$ or smaller, then ions  are unable to follow the rapid magnetic field variations in space, so they will decouple from it and no ion structures will be formed at those scales. Therefore, as an effect of ions decoupling, ion velocity becomes smooth at scales smaller than some $\\rho_i\\!\\simeq\\!d_i$. On the other hand, even if ions are decoupled, the intermittent cascade of magnetic energy proceeds toward smaller scales, supported by the electrons that remain coupled to the magnetic field. However, when electron scales are reached, even the electron dynamics decouples from the magnetic field and the magnetic dissipation range is formed. Thus we claim that only the electrons play a role in the formation of the magnetic field dissipation range as the ions decouple from the magnetic field dynamics long before the formation of the magnetic dissipation range.\n\nFurthermore, as all ion velocity SFs exhibit some curvature between $2\\,d_i$ and $7\\,d_i$, we verified that ESS holds by analyzing all combinations of ion velocity SFs of different order plotted against each other. Panels (c) and (d) of figure \\ref{fig03} show an example of two ion velocity SFs of different order plotted against other other for $r\\!<\\!7\\,d_i$ from sim.1 and sim.2, respectively. These curves were fit using a single straight line over the whole range, and we find that in both simulations, they are linear for $r\\!<\\!7\\,d_i$ without any change in slope between the region where all SFs behave like $r^p$ and the region where they show some curvature. This means that ESS holds and that the whole range $r\\!<\\!7\\,d_i$ is characterized by a single scaling exponent. The same behavior was observed for every other combination of ion velocity SFs of different order in both simulations (not shown here). Finally, as ESS is observed, we calculated the ion velocity scaling exponents $\\beta_u(p,q)$. Panels (e) and (f) of figure \\ref{fig03} show $\\beta_u(p,q)$ as a function of $p$ at fixed $q\\!=\\!1$ for sim.1 and sim.2, respectively. These exponents were calculated taking the gradients of the linear fits of all possible combinations of $log(S_{u,p})$ versus $log(S_{u,q})$. We find that $\\beta_u(p,1)$ is linear in $p$ in range $r\\!<\\!7\\,d_i$ , and the same behavior was observed for every other value of $q$  in sim.1 and sim.2 (not shown here). This result suggests that in both simulations, ion velocity fluctuations are self-similar at scales smaller than about $7\\, d_i$, even in the region where all SFs show some curvature. The ion velocity fluctuations are therefore likely to be smooth over the whole $r\\!<\\!7\\,d_i$ range.\n\nThus, the analysis of ion velocity SFs clearly shows that the ion statistics is also not influenced by the specific reconnection mechanism associated with the evolution of magnetic field fluctuations. No signature of e-rec is present because we do not see any difference between the statistical features of the ions in sim.1 and sim.2. \n\n\\section{Conclusions}\n\nBy combining the information obtained from the magnetic field and the ion velocity SFs, we find that the turbulent cascade associated with e-rec has the same statistical properties of the turbulent cascade associated with standard reconnection. This result is consistent with a recent analysis of turbulent magnetic fluctuations associated with e-rec, measured in the terrestrial magnetosheath by the satellites of the MMS space mission \\citep{12c}.\n\nFurthermore, our analysis suggests that in both simulations, it is possible to identify two dynamical regimes. The first is the ion-decoupled regime, associated with scales in the range $4\\,d_e\\!<\\!r\\!<\\!7\\,d_i$, where magnetic field fluctuations are intermittent while ion velocity fluctuations are self-similar and smooth as this species is strongly decoupled from the magnetic field. The second regime is the dissipative one, associated with scales in the range $r\\!<\\!4\\,d_e$, where both magnetic field and ion velocity fluctuations are self-similar and smooth because of small-scale dissipation. This result is consistent with the analysis of Pyakurel et al. \\citep{12b}, according to which ions decouple from the magnetic field at scales of about $10\\, d_i\\!\\simeq\\!10\\, \\rho_i$ and no ion structures are formed at these scales or smaller. In addition, we claim that the formation of the self-similar magnetic field dissipation range is only guided by the small-scale electron dynamics and that it is independent of the ion dynamics as these particles are decoupled from the magnetic field in this range. These results suggest that the statistical features of the turbulent cascade in a collisionless magnetized plasma depend solely on the coupling between the magnetic field and the different particle species present in the system, but they are independent of the specific process that is responsible for the decoupling of these particles (whether it is e-rec or standard magnetic reconnection, e.g.). In other words, this means that e-rec dissipates the turbulent magnetic energy in the same way as standard ion-coupled reconnection does, and this happens because turbulent dissipation is guided by electrons whose dynamics remains unaltered from standard reconnection to e-rec. This seems to be a robust and universal feature of turbulent magnetized plasmas, independent of the reconnection dynamics, and this result has a potential impact on the formulation of new theoretical models of plasma turbulence. In addition, in this context, the SFs proved to be a useful tool for investigating the coupling between particles and the magnetic field, and their use may be extended to the analysis of satellite data as well.\n\nAdditional studies are necessary to better characterize the transition between the ion-decoupled regime and the dissipative regime. The spatial grid spacing of our simulations is on the same order as the electron inertial length $d_e$ , and because of this, it is not possible to accurately resolve the small-scale electron dynamics in the dissipation range. Moreover, even if our hybrid model is computationally very efficient and able to highlight the different roles of ions and electrons, it is still too simplified to completely describe the small-scale electron physics. Thus, simulations with higher resolution and including electron kinetic effects are required for a much more detailed study of the formation of the magnetic field dissipation range.\n\nFinally, the natural extension of our work will be to perform full 3D-3V simulations of plasma turbulence to study three dimensional effects on the transition between the different physical regimes that characterize the turbulent energy cascade. \n\n\\section*{\\normalsize{Acknowledgments}}\n\nThis paper has received funding from the European Unions Horizon 2020 research and innovation programme under grant agreement No 776262 (AIDA, www.aida-space.eu). Numerical simulations have been performed on Marconi at CINECA (Italy) under the ISCRA initiative. FC thank Dr.~M.~Guarrasi (CINECA, Italy) for his essential contribution for his contribution on code implementation on Marconi. We acknowledge dr. S. Cerri, M. Sisti, F. Finelli and S. Fadanelli for very useful discussions.\n\n\\section*{\\normalsize{Data Availability}}\n\nThe simulation dataset (UNIPI e-rec) is available at Cineca on the AIDA-DB. In order to access the meta-information and the link to the raw data, look at the tutorial at http:\/\/aida-space.eu\/AIDAdb-iRODS.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzehri b/data_all_eng_slimpj/shuffled/split2/finalzzehri
new file mode 100644
index 0000000000000000000000000000000000000000..8fe5e62c20cfee20a6e5573d40272d282b6a09c8
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzehri
@@ -0,0 +1,5 @@
+{"text":"\\section{INTRODUCTION}\n\n\nUltracold Fermi gas \\cite{giorgini_rmp2008},\nwith tunable atom-atom interaction through Feshbach resonance \\cite{chin_rmp2010}, has been an ideal platform for the study of the crossover physics from weak-coupling Bardeen-Cooper-Schrieffer (BCS) pairing to a Bose-Einstein condensate (BEC) of bound pairs \\cite{regal_prl2004,zwierlein_prl2004,bartenstein_prl2004}.\nRecently synthetic gauge fields \\cite{lin_nat2009} and spin-orbit (SO) coupling \\cite{lin_nat2011, ji_nphys2014, chunlei_pra2013, pengjun_prl2012, cheuk_prl2012} were realized in experiments,\nopening up a completely new avenue of research in superfluid Fermi gases.\nThe interplay of Zeeman fields and SO coupling leads to many novel phenomena at both zero\n\\cite{gong_prl2011,han_pra2012,seo_pra2012,jiang_pra2011,zengqiang_prl2011,zhoupra2013}\nand finite temperatures \\cite{lindong_njp2013,hu_njp2013},\nfrom mixed singlet-triplet pairing to topological phase transition\n\\cite{gong_prl2011,xiaji_pra2012,wu_prl2013,zheng_pra2013,chunlei_natc2013,weizhang_natc2013}.\n\nMany of the interesting physics of SO-coupled Fermi gas can be captured by mean-field theory. However, it is also true that mean-field theory may fail under many circumstances. For example, mean-field theory, which does not include the effect of noncondensed pairs,\nfails to describe accurately the phase transition from a superfluid to a normal gas, particularly for systems with strong interaction.\nBeyond-mean-field theoretical methods have been proposed  \\cite{nozieres1985,melo_prl1993,ohashi_prl2002,machida_pra2006,xiaji_pra2005}.\nHere we present a theoretical investigation under the framework of the T-matrix scheme to address the superfluid properties of a Rashba SO coupled Fermi gas over the entire BCS-BEC crossover regime \\cite{qijin_prl1998,maly1999,loktev_pr2001,perali_prb2007,qijin_pr2005,kinnunen,huihu_prl2010,qijin,kinast, stajic,bauer_prl2014, haoguo_arxiv2013,lianyi_pra2013,zhangjing}.\nIn the absence of the Zeeman field, it was shown that the SO coupling enhances superfluid pairing \\cite{renyuan_prl2012,zengqiang_prl2011,lianyi_prl2012,lianyi_pra2013}. At the mean-field level, we know that\nthe presence of both the SO coupling and a perpendicular (out-of-plane) Zeeman field give rise to effective $p$-wave pairing \\cite{tewari_prl2007,chuanwei_prl2012}.\nMeanwhile introducing an in-plane Zeeman component creates an anisotropic Fermi surface which favors finite-momentum pairing,\ngiving rise to a Fulde-Ferrell superfluid \\cite{wu_prl2013,zheng_pra2013,chunlei_natc2013,weizhang_natc2013,ff1,ff4}.\nThese previous studies motivated our present work, in which we investigate the thermodynamic properties of an SO coupled Fermi gas subject to a Zeeman field. We will focus our calculation on two important quantities that are measurable in experiment: the superfluid to normal transition temperature and the isothermal compressibility.\n\nWe organize the paper as follows.\nIn Sec.~\\ref{sec_model_hamiltonian}, we give an introduction to the Rashba SO coupled model\nand briefly describe the T-matrix scheme used in the calculation.\nThen in Sec.~\\ref{sec_zero-temperature_properties},\nwe briefly review the zero-temperature properties of the system.\nThe superfluid transition temperature $T_c$ in the BEC-BCS crossover is investigated in Sec.~\\ref{sec_superfluid_transition_temperature}, and its\ndependence on the Zeeman field is emphasized. We present the numerical results of compressibility in Sec.~\\ref{sec_isothermal_compressibility} before we provide a summary in Sec~\\ref{summary}. We show that both the superfluid transition temperature and the compressibility have distinct dependence on the out-of-plane and the in-plane Zeeman fields.\nIn the Appendix we present technical details of the T-matrix formalism.\n\n\\section{Model}\n\\label{sec_model_hamiltonian}\n\nWe consider a three-dimensional two-component degenerate Fermi gas with Rashba SO coupling together with effective Zeeman fields.\nThis system can be described by the following Hamiltonian:\n\\begin{eqnarray}\n\\mathcal{H} &=& \\int d{\\bf r}\\,\n\\psi^\\dag ( \\mathcal{H}_0 + \\mathcal{H}_\\mathrm{so} + h_z\\sigma_z +h_x\\sigma_x) \\psi({\\bf r}) \\nonumber\\\\\n&& + U \\int d{\\bf r}\\, \\psi^\\dag_\\uparrow({\\bf r}) \\psi^\\dag_\\downarrow({\\bf r})\n\\psi_\\downarrow({\\bf r}) \\psi_\\uparrow({\\bf r})~,\n\\end{eqnarray}\nwhere $\\psi({\\bf r})=(\\psi_\\uparrow \\,,\\psi_\\downarrow)^T$ represents the fermionic field operator and $\\mathcal{H}_0 = -\\nabla^2\/(2m)-\\mu$ represents the kinetic energy\nwith $\\mu$ being the chemical potential.\nThe Rashba SO-coupling term takes the form\n$\\mathcal{H}_\\mathrm{so} = \\alpha ( k_y\\sigma_x - k_x\\sigma_y )$ in the $xy$-plane, with the parameter $\\alpha$ characterizing the strength of the SO coupling.\nWe consider both an out-of-plane Zeeman field $h_z$ and an in-plane Zeeman field $h_x$.\nThe quantity $U$ represents the bare two-body interaction constant and in the calculation will be replaced by the $s$-wave scattering length $a_s$ through the standard regularization scheme:\n$1\/U = m\/(4\\pi\\hbar^2a_s)-\\sum_k m\/k^2$.\n\nIn the mean-field BCS theory, we introduce an order parameter\n$\\Delta=U\\sum_{\\bm{k}}\\langle \\psi_{\\bm{Q}+\\bm{k},\\uparrow}\\psi_{\\bm{Q}-\\bm{k},\\downarrow}\\rangle$ to characterize the property of the superfluid. Here $\\bm{Q}$ represents the center-of-mass momentum of the pairs. A finite ${\\bm{Q}}$ arises from the presence of the in-plane Zeeman field \\cite{wu_prl2013,zheng_pra2013,chunlei_natc2013,ff1,lindong_njp2013,hu_njp2013,ff4}.\nAt finite temperature in a T-matrix scheme, the order parameter $\\Delta(T)$ can be divided into two parts:\n$\\Delta^2 = \\Delta_\\mathrm{sc}^2 + \\Delta_\\mathrm{pg}^2$ \\cite{lianyi_pra2013}. Here $\\Delta_{\\rm sc}$ is the superfluid gap arising from the condensed pairs and vanishes above the superfluid transition temperature $T_c$.\nThe pseudo-gap $\\Delta^2_\\mathrm{pg}\\sim\\langle\\Delta^2(T)\\rangle - \\langle\\Delta(T)\\rangle^2$ describes the thermodynamic fluctuation of non-condensed pairs \\cite{qijin_prl1998,kosztin_prb2000}.\nBelow the superfluid transition temperature $T_c$, the thermodynamic quantities are determined\nby the Thouless criterion \\cite{thouless_1960} within the T-matrix scheme instead of the BCS formalism in the mean-field model.\nThis is because at finite temperature thermodynamic fluctuation plays a critical role with a tendency towards destroying the pairing condensation.\nThe Thouless criterion for finite pairing momentum $\\bm{Q}$ takes the form:\n\\begin{equation}\nU^{-1}+\\chi(0,\\bm{Q})=0 ~,\n\\end{equation}\nwhere $\\chi(0,\\bm{Q})$ is the spin symmetrized pair susceptibility.\nMore technical details are given in the Appendix.\n\nThe T-matrix scheme adopted here was first developed in the context of high-$T_c$ cuprates \\cite{qijin_prl1998,maly1999,loktev_pr2001,perali_prb2007}. It was later applied to study the BEC-BCS crossover phenomenon in ultracold atomic Fermi gases \\cite{qijin_pr2005}. The T-matrix theory was found to be reasonably successful in providing theoretical support for the measured radio-frequency (rf) spectrum \\cite{kinnunen,huihu_prl2010,qijin}, density profiles \\cite{stajic}, and thermodynamic properties such as heat capacity \\cite{kinast}, pressure \\cite{bauer_prl2014}, and isothermal compressibility \\cite{haoguo_arxiv2013} of ultracold Fermi gases. More recently, this theory was also applied to Fermi gases with spin-orbit coupling \\cite{lianyi_pra2013, zhangjing}. In Ref.~\\cite{zhangjing}, it was found that the theoretical calculation qualitatively agrees with the measured rf spectrum of a spin-orbit-coupled Fermi gas on the BEC side of the resonance. Given these past successes, we are confident that the T-matrix scheme indeed represents a vast improvement over the simple mean-field theory and should be at least qualitatively valid over the whole BEC-BCS crossover regime.   \n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.35\\textwidth]{fig1.eps}\n\\caption{(Color online) Thermodynamic quantities at zero temperature,\n(a) order parameter, and (b) pairing momentum, as functions of interaction strength characterized by $1\/a_s K_F$ for different $\\theta_h$. Here the SO coupling strength is\n$\\alpha K_F=2.0E_F$, and the effective Zeeman field strength is $h=0.5E_F$. $\\theta_h$ is the angle between the effective Zeeman field and the $z$-axis, such that $h_z = h \\cos \\theta_h$ and $h_x = h \\sin \\theta_h$. Therefore $\\theta_h=0$ represents a pure out-of-plane Zeeman field, while $\\theta_h=\\pi\/2$ represents a pure in-plane Zeeman field. $K_F$ and $E_F$ are the Fermi wave number and Fermi energy of the ideal Fermi gas, respectively.}\n\\label{crossover_para}\n\\end{figure}\n\n\n\\section{Zero-temperature properties}\n\\label{sec_zero-temperature_properties}\n\nWe first briefly review the main features of the system at zero temperature. Note that the pseudo-gap tends to zero in the low-temperature limit as $\\Delta_\\mathrm{pg}\\sim T^{3\/4}$ [see Eq.~(\\ref{delta_pg})] and vanishes at $T=0$. Hence, the T-matrix scheme we adopted here reduces to the mean-field theory at zero temperature. The Rashba SO coupling and the Zeeman field have opposite effects on the magnitude of the gap parameter: the former tends to enhance the gap, while the latter tends to reduce the gap. Fig.~\\ref{crossover_para}(a) displays the gap parameter as a function of interaction strength for several different Zeeman fields. The in-plane Zeeman field is known to break the symmetry of the band structure and results in Cooper pairs with finite momentum, a signature of the Fulde-Ferrell superfluid. In Fig.~\\ref{crossover_para}(b), we show how the magnitude of the momentum of Cooper pairs $Q=|\\bm{Q}|$ ($\\bm{Q}$ is along the $y$-axis \\cite{zheng_pra2013}) changes in the BEC-BCS crossover. $Q$ decreases quickly on the BEC side of the resonance as the two-body $s$-wave interaction, which favors zero-momentum pairing, dominates in that regime.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.35\\textwidth]{fig2.eps}\n\\caption{(Color online) Superfluid gap at $T=0$ as a function of the Zeeman field strength for (a) $1\/a_sK_F= -0.5$, (b) 0, and (c) 0.5. The solid, dashed, and dot-dashed lines correspond to $\\theta_h= 0$, $\\pi\/4$, and $\\pi\/2$, respectively. The SO-coupling strength is\n$\\alpha K_F=2.0E_F$. The vertical arrows indicate the critical Zeeman field strength at which the system becomes gapless.}\n\\label{deltah}\n\\end{figure}\n\nIn Fig.~\\ref{deltah}, we show how the zero-temperature superfluid gap varies as the Zeeman field strength changes. For a weak Zeeman field, the gap is insensitive to the orientation of the field. In general, $\\Delta$ decreases as $h=\\sqrt{h_z^2+h_x^2}$ increases due to the pair-breaking effect of the Zeeman field. However, as $h$ exceeds some threshold value, the decrease of the gap becomes sensitive to the orientation of the field: The larger the in-plane Zeeman field component is, the steeper the decrease of the gap becomes. As we will show below, this threshold value corresponds to the field strength at which the quasi-particle excitation gap vanishes.\n\nAnother important feature induced by the Zeeman field is that it closes the bulk quasi-particle excitation gap at a critical value. Fig.~\\ref{phase}(a) represents a phase diagram in which we plot the critical Zeeman field at which the system changes from gapped to gapless. The region below each line represents the gapped phase. As the interaction strength varies from the BCS side to the BEC side, the order parameter increases [see Fig.~\\ref{crossover_para}(a)], and correspondingly the critical field strength increases and the gapped region enlarges. In the absence of the in-plane Zeeman field (i.e., when $h_x=0$), the system becomes gapless due to the presence of the discrete Fermi points \\cite{gong_prl2011} located along the $k_z$-axis in momentum space at $k_z=\\pm\\sqrt{2m(\\mu\\pm\\sqrt{h^2-\\Delta^2})}$. These Fermi points are topological defects. Hence the transition from the gapped to the gapless region in this case also represents a transition from a topologically trivial to a topologically nontrivial quantum phase. For finite $h_x$, by contrast, the gapless region features a nodal surface in momentum space on which the single particle excitation gap vanishes \\cite{lindong_njp2013,zhoupra2013,yong_prl2014}. An example of such a nodal surface is plotted in Fig.~\\ref{phase}(b). The vertical arrows in Fig.~\\ref{deltah} indicate the critical Zeeman field strength at which the system becomes gapless. One can see that the sharp drop of the superfluid gap is correlated with the appearance of the nodal surface induced by a large in-plane Zeeman field. As we shall show below, the presence of the nodal surface also has dramatic effects on the thermodynamic properties of the system.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.32\\textwidth]{fig3.eps}\n\\caption{(Color online) (a) The critical magnetic field that separates the gapped region and the gapless region. The region below the lines is gapped. Here the SO-coupling strength is\n$\\alpha K_F=2.0E_F$. (b) The nodal surface in the gapless region for $h_z=0.0$, $h_x=1.0E_F$, $\\alpha K_F=2.0E_F$, and $1\/a_sK_F=0.0$. The range of the plot is from $-2 K_F$ to $2K_F$ along each direction.}\n\\label{phase}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{fig4.eps}\n\\caption{(Color online) (a) The total gap $\\Delta$, pseudo-gap $\\Delta_{\\rm pg}$ and superfluid gap $\\Delta_{\\rm sc}$ as functions of the temperature.\n(b) The corresponding pairing momentum $Q$ as a function of the temperature.\nThe parameters used are\n$1\/a_sK_F=0.0$, $\\alpha K_F=2.0E_F$, $h_z=0.0$, and $h_x=0.5E_F$. In this example, the green vertical dashed line indicates the critical temperature $T_c=0.210E_F$.}\n\\label{delta}\n\\end{figure}\n\n\\section{Superfluid Transition Temperature}\n\\label{sec_superfluid_transition_temperature}\n\nWe now turn our attention to the finite-temperature properties of the system. The first quantity we want to address is the superfluid transition temperature $T_c$, which is identified as the lowest temperature at which the superfluid gap $\\Delta_{\\rm sc}$ vanishes.\nFig.~\\ref{delta}(a) shows an example of how the gap varies as the temperature increases. As temperature increases, the superfluid gap $\\Delta_{\\rm sc}$ decreases monotonically and vanishes at $T_c$.\nIn contrast, the pseudo-gap $\\Delta_{\\rm pg}$ is a monotonically increasing function $\\Delta_{\\rm pg}\\sim T^{3\/4}$ below $T_c$ and decreases above $T_c$.\nOn the other hand, Fig.~\\ref{delta}(b) shows that the center-of-mass momentum $Q$ of the Cooper pairs is quite insensitive to the temperature for $T<T_c$, and slowly decreases for $T>T_c$. This indicates that the in-plane Zeeman field not only induces a Fulde-Ferrell superfluid below $T_c$, but also induces an exotic normal state above $T_c$ featuring a finite-momentum pseudo-gap \\cite{ff4}.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.36\\textwidth]{fig5.eps}\n\\caption{(Color online) Critical temperature $T_c$ in the BEC-BCS crossover for (a) different spin-orbit couplings (the values of the SO-coupling strength $\\alpha$, in units of $E_F\/K_F$, are shown in the legends) with no Zeeman field;\nand for (b) different ($h$, $\\theta_h$) with $\\alpha K_F=2.0E_F$. $h$ is in units of $E_F$.}\n\\label{crossover}\n\\end{figure}\n\nIn Fig. \\ref{crossover}, we show the superfluid transition temperature $T_c$ in the BEC-BCS crossover. In Fig. \\ref{crossover}(a), we plot $T_c$ as a function of the interaction strength for several different values of the SO-coupling strength without the Zeeman field. For comparison, the mean-field result is also included. For weak interaction, where $1\/(a_sK_F)$ is large and negative, the mean-field result agrees with the T-matrix result. As the interaction increases towards the BEC limit, the mean-field theory predicts an unphysically large critical temperature, a clear indication of its breakdown. In the BEC limit where the two-body attractive interaction is dominant, the system behaves like a condensation of weakly interacting tightly bound bosonic dimers with effective mass $2m$, regardless of the SO coupling strength \\cite{jiang_pra2011,zengqiang_prl2011,hu_prl2011}. The BEC transition temperature for this system is 0.218$E_F$ \\cite{melo_prl1993,ohashi_prl2002,machida_pra2006}.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.35\\textwidth]{fig6.eps}\n\\caption{(Color online) Critical temperature $T_c$ as functions of the Zeeman field strength for (a) $1\/a_sK_F= -0.5$, (b) 0, and (c) 0.5. The solid, dashed, and dot-dashed lines correspond to $\\theta_h= 0$, $\\pi\/4$, and $\\pi\/2$, respectively. The SO-coupling strength is\n$\\alpha K_F=2.0E_F$. The vertical arrows indicate the value of the Zeeman field strength at which the system become gapless.}\n\\label{htc}\n\\end{figure}\n\nIn Fig. \\ref{crossover}(b), we examine how the Zeeman field affects $T_c$. In the BEC limit, we have again $T_c=0.218 E_F$, insensitive to either the SO-coupling strength or the Zeeman field strength \\cite{jiang_pra2011}. On the BCS side, however, the Zeeman field tends to decrease $T_c$. This effect is much more pronounced with the in-plane Zeeman field than with the out-of-plane Zeeman field, indicating that the finite-momentum Fulde-Ferrell pairing is less robust than the zero-momentum Cooper pairing.\n\nTo show the effect of the Zeeman field on the transition temperature more clearly, we plot in Fig.~\\ref{htc} $T_c$ as a function of the Zeeman field strength at different orientations. Across the BEC-BCS crossover, $T_c$ is not very sensitive to the out-of-plane Zeeman field over a large range of Zeeman field strengths. By contrast, when the in-plane Zeeman field is present, $T_c$ starts to drop rather sharply near the critical Zeeman field strength where the system becomes gapless. This steep down turn of $T_c$ is particularly pronounced on the BEC side of the resonance. We attribute this steep drop of $T_c$ and the corresponding steep drop of the zero-temperature superfluid gap (see Fig.~\\ref{deltah}) to the enhanced fluctuation as a result of the emergence of the nodal surface induced by a large in-plane Zeeman field.\n\n\n\n\n\\section{Isothermal Compressibility}\n\\label{sec_isothermal_compressibility}\n\nThe isothermal compressibility, defined as $\\kappa = \\frac{1}{n} \\left. \\frac{\\partial n}{\\partial P} \\right|_{T}$, measures the change in the density $n$ in response to the change of the pressure $P$. A recent experiment measured the compressibility of a Fermi gas across the superfluid phase transition \\cite{ku_sci2012}, which is found to be in reasonable agreement with the T-matrix theory \\cite{haoguo_arxiv2013}. In the superfluid regime, it is found that $\\kappa$ increases with temperature, i.e., the gas becomes more compressible as temperature increases. This can be intuitively understood as a lower temperature yields a larger superfluid gap, hence a gas less likely to be compressed. Here we want to examine how SO coupling affects the behavior of $\\kappa$.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{fig7.eps}\n\\caption{(Color online) Compressibility in the superfluid regime as a function of temperature for different Zeeman field strengths and scattering lengths. The scattering lengths are: $1\/K_Fa_s = -0.5$ for upper panels (a1) and (b1), $1\/K_Fa_s = 0.$ for middle panels (a2) and (b2), and $1\/K_Fa_s = 0.5$ for lower panels (a3) and (b3). For left panels (a1), (a2) and (a3), $h_x=0$; for right panels (b1), (b2) and (b3), $h_z=0$. The SO-coupling strength is\n$\\alpha K_F=2.0E_F$. The compressibility is normalized by $\\kappa_0=\\frac{3}{2}\\frac{1}{N E_F}$, the compressibility of an ideal Fermi gas, and the temperature is normalized to the superfluid transition temperature $T_c$ for the given set of parameters. In the legends, we also indicate whether the quasiparticle excitations of the corresponding system are gapped or gapless. In (a2), we show $\\kappa$ in the absence of SO couplings and Zeeman fields (the dotted line) by taking $\\kappa=h=0$. In this limit, our calculation recovers the result reported in Ref. \\cite{haoguo_arxiv2013}.}\n\\label{compressibility}\n\\end{figure}\n\nFor our system, $\\kappa$ can be expressed as \\cite{seo_arxiv2011, seo_pra2013, haoguo_arxiv2013}\n\\begin{eqnarray}\n\\kappa &=& \\dfrac{1}{N^2}\\Big[\n\\Big(\\dfrac{\\partial N}{\\partial \\mu}\\Big)_{T,\\Delta,Q}\n+ \\Big(\\dfrac{\\partial N}{\\partial \\Delta}\\Big)_{T,\\mu,Q}  \\Big(\\dfrac{\\partial \\Delta}{\\partial \\mu}\\Big)_{T,Q} \\notag\\\\\n&& + \\Big(\\dfrac{\\partial N}{\\partial Q}\\Big)_{T,\\mu,\\Delta}  \\Big(\\dfrac{\\partial Q}{\\partial \\mu}\\Big)_{T,\\Delta}\n\\Big] ~,\n\\end{eqnarray}\nwhere\n\\[\n\\Big(\\dfrac{\\partial \\Delta}{\\partial \\mu}\\Big)_{T,Q} =\n\\Big(\\dfrac{\\partial N}{\\partial \\Delta}\\Big)_{T,\\mu,Q} \\Big\/ \\Big(\\dfrac{\\partial^2 \\Omega}{\\partial \\Delta^2}\\Big)_{T,\\mu,Q} ~,\n\\]\n\\[\n\\Big(\\dfrac{\\partial Q}{\\partial \\mu}\\Big)_{T,\\Delta} =\n\\Big(\\dfrac{\\partial N}{\\partial Q}\\Big)_{T,\\mu,\\Delta} \\Big\/ \\Big(\\dfrac{\\partial^2 \\Omega}{\\partial Q^2}\\Big)_{T,\\mu,\\Delta} ~.\n\\]\nIn Fig. \\ref{compressibility}, we show the compressibility $\\kappa$ as a function of temperature $T$ in the superfluid regime on both sides of the Feshbach resonance. In the left panels, we set the in-plane Zeeman field $h_x=0$. Across the Feshbach resonance, we see that regardless of the strength of the out-of-plane Zeeman field $h_z$, $\\kappa$ increases smoothly as the temperature increases from 0 to $T_c$, just like in a superfluid Fermi gas without SO coupling \\cite{ku_sci2012, haoguo_arxiv2013}. By contrast, the presence of the in-plane Zeeman field gives rise to some surprising effects. In the right panels of Fig.~\\ref{compressibility}, we set $h_z=0$. One can see that for small $h_x$ such that the system is gapped, $\\kappa$ is a monotonically increasing function of $T$. However, once $h_x$ exceeds the critical value such that the system becomes gapless, $\\kappa$ becomes a non-monotonic function of temperature. In certain regimes, $\\kappa$ even decreases as $T$ increases.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.32\\textwidth]{fig8.eps}\n\\caption{(Color online) Compressibility at $T=0$ as a function of the Zeeman field strength. The dashed line corresponds to the in-plane Zeeman field ($\\theta_h=\\pi\/2$), and the solid line corresponds to the out-of-plane Zeeman field ($\\theta_h=0$). The dashed and solid arrows indicate the critical field strength at which the system becomes gapless in the presence of an in-plane Zeeman field and an out-of-plane Zeeman field, respectively. The SO coupling strength is\n$\\alpha K_F=2.0E_F$, and the scatter length is set at $1\/K_Fa_s=0$.}\n\\label{comph}\n\\end{figure}\n\nThe drastically different effects on compressibility by the in-plane and the out-of-plane Zeeman field can also be seen in Fig.~\\ref{comph}, where we plot $\\kappa$ at the unitarity limit at zero temperature as a function of the Zeeman field strength. For the out-of-plane Zeeman field, $\\kappa$ decreases smoothly as the field strength increases. In particular, it does not exhibit any distinctive feature when the system changes from gapped to gapless. This is consistent with the fact that the gapped to gapless transition in the presence of a pure out-of-plane Zeeman field represents a topological phase transition which does not leave a trace in thermodynamic quantities. On the other hand, when the Zeeman field is in plane with its strength increasing from zero, $\\kappa$ first decreases and then rises sharply near the critical field when the system becomes gapless. Hence the in-plane Zeeman-field-induced gapless superfluid, featuring a Fermi nodal surface, is highly compressible.\n\nThe nodal surface resulting from the in-plane Zeeman field is not unique to Rashba SO coupling. For example, in the experimentally realized equal-weight Rashba-Dresselhaus coupling, a large in-plane Zeeman field can also induce a nodal surface \\cite{nodal1,nodal2}. We have checked that for that system, the Zeeman-field dependence of the compressibility exhibits very similar behavior.\nAs it is known, using the standard form of the fluctuation-dissipation\ntheorem for a balanced system, the isothermal compressibility can be rewritten as\n$\\kappa= \\frac{V}{T}( \\frac{\\langle\\hat{N}^2\\rangle-N^2}{N^2} )$ \\cite{seo_arxiv2011}, i.e., $\\kappa$ is directly proportional to number fluctuation of the system. The increase in $\\kappa$ can therefore be interpreted as a consequence of enhanced number fluctuation induced by the nodal surface.\n\n\n\n\n\\section{SUMMARY}\n\\label{summary}\n\nIn summary,\nwe have investigated the effect of the pairing fluctuation\non the thermodynamic properties of a Rashba SO coupled superfluid Fermi gas by adopting a T-matrix scheme. We focus on the effect of the Zeeman field. In particular, the in-plane Zeeman component leads to finite-momentum Cooper pairing and, when its magnitude becomes sufficiently large, it induces a nodal Fermi surface on which the quasi-particle excitation gap vanishes. The presence of the nodal surface has dramatic effects on the superfluid properties: it greatly suppresses the superfluid transition temperature and increases the isothermal compressibility. Both phenomena can be attributed to the enhanced fluctuation due to the presence of the nodal surface. By stark contrast, when only the out-of-plane Zeeman field is present, both $T_c$ and $\\kappa$ exhibit smooth behavior when the Zeeman field strength is increased, even when the system becomes gapless at large Zeeman field strength. The key difference here is that a large out-of-plane Zeeman field, unlike its in-plane counterpart, does not give rise to a nodal surface, but only discrete Fermi points along the $k_z$-axile.\nThese Fermi points are topological defects, and their appearance does not manifest on any thermodynamic quantities. In addition,\nwe also find an unconventional pseudo-gap state above the superfluid transition temperature, in which the non-condensed pairs possess nonzero center-of-mass momentum.\nWe attribute it to the anisotropic Fermi surface induced by the in-plane Zeeman field, which is independent of temperature.\n\n\n\n\n\\section*{ACKNOWLEDGEMENTS}\nZ.Z., X.Z. and G.G. are supported by the National Natural\nScience Foundation of China (Grants No. 11074244\nand No. 11274295), and the National 973 Fundamental Research Program (2011cba00200).\nH.P. acknowledges support from the NSF, and\nthe Welch Foundation (Grant No. C-1669).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Counting of branched covers}\n\n\nLet us consider a connected compact surface without boundary $\\Omega$ and a branched covering $f:\\Sigma\\rightarrow\\Omega$\nby a connected or non-connected surface $\\Sigma$. We will consider a covering $f$ of the degree $d$. It means that the\npreimage $f^{-1}(z)$ consists of $d$ points $z\\in\\Omega$ except some finite number of points. This points are called\n\\textit{critical values of $f$}.\n\nConsider the preimage $f^{-1}(z)=\\{p_1,\\dots,p_{\\ell}\\}$ of $z\\in\\Omega$. Denote by $\\delta_i$ the degree of $f$ at $p_i$. It\nmeans that in the neighborhood of $p_i$ the function $f$ is homeomorphic to $x\\mapsto x^{\\delta_i}$. The set \n$\\Delta=(\\delta_1\\dots,\\delta_{\\ell})$\nis the partition of $d$, that is called \\textit{topological type of $z$}.\n\n\nFor a partition $\\Delta$ of a number $d=|\\Delta|$ denote by $\\ell(\\Delta)$ the number of the non-vanishing parts ($|\\Delta|$ and\n$\\ell(\\Delta)$ are called the weight and the length of $\\Delta$, respectively). We denote a partition and its Young diagram by \nthe same letter. Denote by $(\\delta_1,\\dots,\\delta_{\\ell})$ the Young diagram with rows of length $\\delta_1,\\dots,\\delta_{\\ell}$\nand corresponding partition of $d=\\sum \\delta_i$. \n\n\nFix now points $z_1,\\dots,z_{\\textsc{f}}$ and partitions $\\Delta^{(1)},\\dots,\\Delta^{(\\textsc{f})}$ of $d$. Denote by\n\\[\\widetilde{C}_{\\Omega (z_1\\dots,z_{\\textsc{f}})} (d;\\Delta^{(1)},\\dots,\\Delta^{(\\textsc{f})})\\]\nthe set of all branched covering $f:\\Sigma\\rightarrow\\Omega$ with critical points $z_1,\\dots,z_{\\textsc{f}}$ of topological\ntypes  $\\Delta^{(1)},\\dots,\\Delta^{(\\textsc{f})}$.\n\nCoverings $f_1:\\Sigma_1\\rightarrow\\Omega$ and $f_2:\\Sigma_2\\rightarrow\\Omega$ are called isomorphic if there exists an\nhomeomorphism $\\varphi:\\Sigma_1\\rightarrow\\Sigma_2$ such that $f_1=f_2\\varphi$. Denote by $\\texttt{Aut}(f)$  the group of\nautomorphisms of the covering $f$. Isomorphic coverings have isomorphic groups of automorphisms of degree $|\\texttt{Aut}(f)|$.\n\nConsider now the set $C_{\\Omega (z_1\\dots,z_{\\textsc{f}})} (d;\\Delta^{(1)},\\dots,\\Delta^{(\\textsc{f})})$ of isomorphic classes\nin $\\widetilde{C}_{\\Omega (z_1\\dots,z_{\\textsc{f}})} (d;\\Delta^{(1)},\\dots,\\Delta^{(\\textsc{f})})$. This is a finite set.\nThe sum\n\\begin{equation}\\label{Hurwitz-number-geom-definition}\nH^{\\textsc{e},\\textsc{f}}(d;\\Delta^{(1)},\\dots,\\Delta^{(\\textsc{f})})=\n\\sum\\limits_{f\\in C_{\\Omega (z_1\\dots,z_{\\textsc{f}})}(d;\\Delta^{(1)},\\dots,\n\\Delta^{(\\textsc{f})})}\\frac{1} {|\\texttt{Aut}(f)|}\\quad,\n\\end{equation}\ndon't depend on the location of the points $z_1\\dots,z_{\\textsc{f}}$ and is called \\textit{Hurwitz number}.\nHere $\\textsc{f}$ denotes the number of the branch points, and $\\textsc{e}$ is the Euler characteristic of the base surface.\n\nIn case it will not produce a confusion we admit 'trivial' profiles $(1^d)$ among $\\Delta^1,\\dots,\\Delta^{\\textsc{f}}$ in\n(\\ref{Hurwitz-number-geom-definition})\nkeeping the notation $H^{\\textsc{e},\\textsc{f}}$ though the number of critical points now is less than $\\textsc{f}$.\n\n\nIn case we count only connected covers $\\Sigma$ we get the \\textit{connected} Hurwitz numbers \n$H_{\\rm con}^{\\textsc{e},\\textsc{f}}(d;\\Delta^{(1)},\\dots,\\Delta^{(\\textsc{f})})$.\n\n\n\\vspace{1ex}\n\nThe Hurwitz numbers arise in different fields of mathematics: from algebraic geometry to integrable systems. They are well\nstudied for orientable $\\Omega$. In this case the Hurwitz number coincides with the weighted number of holomorphic branched\ncoverings of a Riemann surface $\\Omega$ by other Riemann surfaces, having critical points $z_1,\\dots,z_\\textsc{f}\\in\\Omega$ of\nthe topological types $\\Delta^{(1)},\\dots,\\Delta^{(\\textsc{f})}$ respectively. The well known isomorphism between Riemann\nsurfaces and complex algebraic curves gives the interpretation of the Hurwitz numbers as the numbers of morphisms of\ncomplex algebraic curves.\n\nSimilarly, the Hurwitz number for a non-orientable surface $\\Omega$ coincides with the weighted number of the dianalytic\nbranched coverings of the Klein surface without boundary by another Klein surface and coincides with the weighted number\nof morphisms of real algebraic curves without real points \\cite{AG,N90,N2004}. An extension of the theory to all Klein surfaces\nand all real algebraic curves leads to Hurwitz numbers for surfaces\nwith boundaries may be found in \\cite{AN,N}.\n\n\nRiemann-Hurwitz formula related the Euler characteristic of the base surface $\\textsc{e}$ and the Euler characteristic of\nthe $d$-fold cover $\\textsc{e}'$ as follows:\n\\begin{equation}\\label{RH}\n \\textsc{e}'= d\\textsc{e}+\\sum_{i=1}^{\\textsc{f}}\\left(\\ell(\\Delta^{(i)})-d\\right)=0\n\\end{equation}\nwhere the sum ranges over all branch points $z_i\\,,i=1,2,\\dots$ with ramification profiles given by partitions $\\Delta^i\\,,i=1,2,\\dots$\nrespectively, and $\\ell(\\Delta^{(i)})$ denotes the length of the partition $\\Delta^{(i)}$ which is equal to the number of\nthe preimages $f^{-1}(z_i)$ of the point $z_i$.\n\n\\vspace{1ex}\n{\\bf Example 1}.\nLet $f:\\Sigma\\rightarrow\\mathbb{CP}^1$ be a covering without critical points.\nThen, each $d$-fold cover is a union of $d$ Riemann spheres: $\\mathbb{CP}^1 \\coprod \\cdots \\coprod \\mathbb{CP}^1$, then\n$\\deg f =d!$ and $H^{2,0}(d)=\\frac{1}{d!}$\n\n\n\\vspace{1ex}\n{\\bf Example 2}.\nLet $f:\\Sigma\\rightarrow\\mathbb{CP}^1$ be a $d$-fold covering with two critical points with the profiles \n$\\Delta^{(1)}=\\Delta^{(2)}=(d)$.\n(One may think of $f=x^d$). Then $H^{2,2}(d;(d),(d))=\\frac 1d$. Let us note that $\\Sigma$ is connected in this case \n(therefore $H^{2,2}(d;(d),(d))=H_{\\rm con}^{2,2}(d;(d),(d)) $)\nand its Euler\ncharacteristic $\\textsc{e}'=2$.\n\n\n\\vspace{1ex}\n{\\bf Example 3}.\n The generating\nfunction for the Hurwitz numbers $H^{2,2}(d;(d),(d))$ from the previous Example may be writen as\n$$\nF(h^{-1}\\mathbf{p}^{(1)},h^{-1}\\mathbf{p}^{(2)}):=\\, h^{-2}\\sum_{d>0}\\, H_{\\rm con}^{2,2}(d;(d),(d)) p_d^{(1)}p_d^{(2)}=\nh^{-2}\\sum_{d>0} \\frac 1d p_d^{(1)}p_d^{(2)}\n$$ \nHere $\\mathbf{p}^{(i)}=(p_1^{(i)},p_2^{(i)},\\dots),\\,i=1,2$ are two sets of formal parameters. The powers of the auxilary\nparameter $\\frac 1h$ count the Euler characteristic of the cover $\\textsc{e}'$ which is 2 in our example.\nThen thanks to the known general statement about the link between generating functions of \"connected\" and \"disconnected\"\nHurwitz numbers (see for instance \\cite{ZL}) one can write down the generating function for the Hurwitz numbers for covers with\ntwo critical points,\n$H^{2,2}(d;\\Delta^{(1)},\\Delta^{(2)})$, as follows:\n\\[\n\\tau(h^{-1}\\mathbf{p}^{(1)},h^{-1}\\mathbf{p}^{(2)})=e^{F(h^{-1}\\mathbf{p}^{(1)},h^{-1}\\mathbf{p}^{(2)}) } =\n\\]\n\\begin{equation}\\label{E=2,F=2Hurwitz}\ne^{h^{-2}\\sum_{d>0} \\frac 1d p_d^{(1)}p_d^{(2)}}\\,\n= \\,\n\\sum_{d\\ge 0} \\sum_{\\Delta^{(1)},\\Delta^{(2)}}\n H^{2,2}(d;\\Delta^{(1)},\\Delta^{(2)}) \\,h^{-\\textsc{e}'} \\mathbf{p}^{(1)}_{\\Delta^{(1)}}\\mathbf{p}^{(2)}_{\\Delta^{(2)}}\n\\end{equation}\nwhere $\\mathbf{p}^{(i)}_{\\Delta^{(i)}}:=p^{(i)}_{\\delta^{(i)}_1}p^{(i)}_{\\delta^{(i)}_2}p^{(i)}_{\\delta^{(i)}_3}\\cdots$, $i=1,2$\nand where $\\textsc{e}'=\\ell(\\Delta^{(1)}) + \\ell(\\Delta^{(2)})$ in agreement with (\\ref{RH}) where we put $\\textsc{f}=2$.\nFrom (\\ref{E=2,F=2Hurwitz}) it follows that the profiles of both critical points\ncoincide, otherwise the Hurwitz number vanishes. Let us denote this profile $\\Delta$, \nand $|\\Delta|=d$ and from the last equality we get\n$$\nH^{2,2}(d;\\Delta,\\Delta) = \\frac {1}{z_{\\Delta}}\n$$\nHere\n\\begin{equation}\nz_\\Delta\\,=\\,\\prod_{i=1}^\\infty \\,i^{m_i}\\,m_i!\n\\end{equation}\nwhere $m_i$ denotes the number of parts equal to $i$ of the partition $\\Delta$ (then the partition $\\Delta$ is often\ndenoted by $(1^{m_1}2^{m_2}\\cdots)$).\n\n\n\\vspace{1ex}\n{\\bf Example 4}.\nLet $f:\\Sigma\\rightarrow\\mathbb{RP}^2$ be a covering without critical points.\nThen, if $\\Sigma$ is connected, then $\\Sigma=\\mathbb{RP}^2$,\n$\\deg f=1$\\quad or $\\Sigma=S^2$, $\\deg f=2$. Next, if $d=3$, then\n$\\Sigma=\\mathbb{RP}^2\\coprod\\mathbb{RP}^2\\coprod\\mathbb{RP}^2$ or $\\Sigma=\\mathbb{RP}^2\\coprod S^2$.\nThus $H^{1,0}(3)=\\frac{1}{3!}+\\frac{1}{2!}=\\frac{2}{3}$.\n\n\n\n\n\\vspace{1ex}\n{\\bf Example 5}.\nLet $f:\\Sigma\\rightarrow\\mathbb{RP}^2$ be a covering with a single critical point with profile $\\Delta$, and $\\Sigma$ \nis connected.\n Note that due to (\\ref{RH}) the Euler\ncharacteristic of $\\Sigma$ is $\\textsc{e}'=\\ell(\\Delta)$. \n(One may think of $f=z^d$ defined in the unit disc where we identify $z$ and $-z$ if $|z|=1$).\nIn case we cover the Riemann sphere by the Riemann sphere $z\\to z^m$ we get\ntwo critical points with the same profiles. However we cover $\\mathbb{RP}^2$ by the Riemann sphere, then we have the \ncomposition of the\nmapping $z\\to z^{m}$ on the\nRiemann sphere and the factorization by antipodal involution $z\\to - \\frac{1}{\\bar z}$. Thus we have the ramification \nprofile $(m,m)$\nat the single critical point $0$ of $\\mathbb{RP}^2$.\nThe automorphism group is the dihedral group of the order $2m$ which consists of rotations on $\\frac{2\\pi }{m}$ and \nantipodal involution\n$z\\to -\\frac{1}{\\bar z}$.\nThus we get that \n$$\nH_{\\rm con}^{1,1}\\left(2m;(m,m)\\right)=\\frac{1}{2m}\n$$\nFrom (\\ref{RH}) we see that $1=\\ell(\\Delta)$ in this case.\nNow let us cover $\\mathbb{RP}^2$ by $\\mathbb{RP}^2$ via $z\\to z^d$. From (\\ref{RH}) we see that $\\ell(\\Delta)=1$.\nFor even $d$ we have the critical point\n$0$, in addition each point of the unit\ncircle $|z|=1$ is critical (a folding), while from the beginning we restrict our consideration only on isolated critical points.\nFor odd $d=2m-1$ there is\nthe single critical point $0$, the automorphism group consists of rotations on the angle $\\frac{2\\pi}{2m-1}$. Thus in this case\n$$\nH_{\\rm con}^{1,1}\\left(2m-1;(2m-1)\\right)=\\frac{1}{2m-1}\n$$\n\n\\vspace{1ex}\n{\\bf Example 6}.\nThe generating series of the connected Hurwitz numbers with a single critical point from the previous Example  is\n\\[\nF(h^{-1}\\mathbf{p})=\n \\frac{1}{h^2}\\sum_{m>0} p_m^2 H_{\\rm con}^{1,1}\\left(2m;(m,m)\\right) +\n \\frac{1}{h} \\sum_{m>0} p_{2m-1} H_{\\rm con}^{1,1}\\left(2m-1;(2m-1)\\right)\n\\]\nwhere $H_{{\\rm con}}^{1,1}$ describes $d$-fold covering either by the Riemann\nsphere ($d=2m$) or by the projective plane ($d=2m-1$). \nWe get the generating function for  Hurwitz numbers with a single critical point\n\\[\n\\tau(h^{-1}\\mathbf{p})=e^{F(h^{-1}\\mathbf{p} ) } =\n\\]\n\\begin{equation}\\label{single-branch-point'}\ne^{\\frac {1}{h^2}\\sum_{m>0} \\frac {1}{2m}p_m^2  +\\frac 1h\\sum_{m {\\rm odd}} \\frac 1m p_m }=\n\\sum_{d>0} \n\\sum_{\\Delta\\atop |\\Delta|=d} h^{-\\ell(\\Delta)} \\mathbf{p}_\\Delta\nH^{1,a}(d;\\Delta)\n\\end{equation}\nwhere $a=0$ and if $\\Delta=(1^d)$, and where $a=1$  and\n otherwise. Then $H^{1,1}(d;\\Delta)$ is the Hurwitz number \ndescribing $d$-fold covering of $\\mathbb{RP}^2$ with a single\nbranch point of type $\\Delta=(d_1,\\dots,d_l),\\,|\\Delta|=d$ by a (not necessarily connected) Klein surface of\nEuler characteristic $\\textsc{e}'=\\ell(\\Delta)$. For instance, for $d=3$, $\\textsc{e}'=1$ we get\n$H^{1,1}(3;\\Delta)=\\frac 13\\delta_{\\Delta,(3)}$.\nFor unbranched coverings (that is for $a=0$, $\\textsc{e}'=d$) we get formula (\\ref{unbranched}).\n\n\\paragraph{Tau functions.}\n\nLet us note that the expression presented in (\\ref{E=2,F=2Hurwitz}), namely,\n\\begin{equation}\\label{2KP-tau-vac-Schur}\n \\tau^{\\rm 2KP}_1(h^{-1}\\mathbf{p}^{(1)},h^{-1}\\mathbf{p}^{(2)}) = \ne^{h^{-2}\\sum_{d>0} \\frac 1d p_d^{(1)}p_d^{(2)}}\n\\end{equation}\ncoincides with the simplest two-component KP tau function\nwith two sets of higher times $h^{-1}\\mathbf{p}^{(i)},\\, i=1,2$, while (\\ref{single-branch-point'}) may be recognized\nas the simplest non-trivial tau function of the BKP hierarchy of Kac and van de Leur \\cite{KvdLbispec} \n\\begin{equation}\n \\tau^{\\rm BKP}_1(h^{-1}\\mathbf{p}) =\ne^{\\frac {1}{h^2}\\sum_{m>0} \\frac {1}{2m}p_m^2  +\\frac 1h\\sum_{m {\\rm odd}} \\frac 1m p_m }\n\\end{equation}\nwritten down in\n\\cite{OST-I}. In (\\ref{E=2,F=2Hurwitz}) and in (\\ref{single-branch-point'}) the higher times are rescaled as\n$p_m\\to h^{-1}p_m,\\,m>0$ as it is common in the study of the integrable dispersionless equations where only\nthe top power of the 'Plank constant' $h$ is taken into account. For instance, see \\cite{NZ} where the counting\nof coverings of the Riemann sphere by Riemann spheres was related to the so-called Laplacian growth problem \\cite{MWZ},\n\\cite{Z}.\nAbout the quasiclassical limit of the DKP hierarchy see \\cite{ATZ}.\nThe rescaling is also common for tau functions used in two-dimensional gravity where the powers of $h^{-\\textsc{e}}$ \ngroup contributuions of surfaces of Euler characteristic $\\textsc{e}$ to the 2D gravity partition function \\cite{BrezinKazakov}.\nIn the context of the links between Hurwitz numbers and integrable hierarchies the rescaling $\\mathbf{p} \\to h^{-1}\\mathbf{p}$ was\nconsidered in \\cite{Harnad-overview-2015} and in \\cite{NO-LMP}. In our case the role similar to $h$ plays $N^{-1}$, \nwhere $N$ is the size of matrices in matrix integrals.\n\nWith the help of these tau functions we shall construct integral over matrices. To do this we present the variables\n$\\mathbf{p}^{(i)},\\, i=1,2$ and $\\mathbf{p}$ as traces of a matrix we are interested in. We write $\\mathbf{p}(X)=\\left(p_1(X),p_2(X),\\dots  \\right)$,\nwhere\n\\begin{equation}\\label{p_m(X)}\np_m(X) = \\mathrm {tr} X^m = \\sum_{i=1}^N x_i^m\n\\end{equation}\nand where $x_1,\\dots,x_N$ are eigenvalues of $X$. \n\n\nIn this case we use non-bold capital letters for the matrix argument and our tau functions\nare tau functions of the matrix argument:\n\\begin{equation}\\label{tau_1}\n\\tau_1^{\\rm 2KP}(X,\\mathbf{p}):=\\tau_1^{\\rm 2KP}(\\mathbf{p}(X),\\mathbf{p})=\n\\sum_\\lambda s_\\lambda(X)s_\\lambda(\\mathbf{p}) =e^{\\mathrm {tr} V(X,\\mathbf{p})}=\n\\prod_{i=1}^N e^{\\sum_{m=1}^\\infty \\frac{1}{m}x_i^mp_m}\n\\end{equation}\nwhere $x_i$ are eigenvalues of $X$,\nwhere $\\mathbf{p}=(p_1,p_2,\\dots)$ is a semi-infinite set of parameters,\nand\n\\begin{equation}\\label{tau_1^B}\n \\tau_1^{\\rm BKP}(X): =\\tau_1^{\\rm BKP}(\\mathbf{p}(X)) =\n \\sum_\\lambda s_\\lambda(X) =\\prod_{i=1}^N (1-x_i)^{-1}\\prod_{i<j} (1-x_ix_j)^{-1}\n\\end{equation}\nHere $s_\\lambda$ denotes the Schur function, see Section \\ref{Partitions-and-Schur-functions} in Appendix. We recall the fact\n\\cite{Mac} we shall need: if $X$ is $N\\times N$ matrix, then\n\\begin{equation}\\label{ell<N}\ns_\\lambda(X)=0,\\qquad {\\rm if}\\quad \\ell(\\lambda)>N\n\\end{equation}\nwhere $\\ell(\\lambda)$ is the length of a partition $\\lambda=(\\lambda_1,\\dots,\\lambda_\\ell),\\,\\lambda_\\ell >0$.\n\nFor further purposes we need the following spectral invariants\nof a matrix $X$:\n\\begin{equation}\\label{spectral-invariant}\n{\\bf P}_\\Delta(X):=\\prod_{i=1}^\\ell p_{\\delta_i}(X)\n\\end{equation}\nwhere $\\Delta=(\\delta_1,\\dots, \\delta_\\ell)$ is a partition and each $p_{\\delta_i}$ is defined by (\\ref{p_m(X)})\n\n\nIn our notation one can write\n\\begin{equation}\\label{tau_1-XY}\n\\tau_1^{\\rm 2KP}(X,Y)=\\tau_1^{\\rm 2KP}(\\mathbf{p}(X),\\mathbf{p}(Y))=\n\\sum_{\\Delta}\\frac{1}{z_\\Delta} {\\bf P}_\\Delta(X){\\bf P}_\\Delta(Y)\n\\end{equation}\n\n\\paragraph{Combinatorial approach.} The study of the homomorphisms between the fundemental group of the base Riemann sufrace \nof genus $g$ (the Euler characterisic is resectively $\\textsc{e}=2-2g$)\nwith \n$\\textsc{f}$ marked points and the symmetric group in the context of the counting of the non-equivalent $d$-fold covering with \ngiven profiles \n$\\Delta^{i},\\,i=1,\\dots,\\textsc{f}$ results to the following equation (for instance, for the details, see Appendix A written by Zagier for the \nRussian edition of \\cite{ZL} or works \\cite{M1}, \\cite{GARETH.A.JONES})\n\\begin{equation}\\label{Hom-pi-S_d-Riemann}\n\\prod_{j=1}^g a_jb_ja_j^{-1}b_j^{-1}X_1\\cdots X_{\\textsc{f}} =1\n\\end{equation}\nwhere $a_j,b_j,X_i\\in S_d$ and where each $X_i$ belongs to the cycle class $C_{\\Delta^i}$. Then the Hurwitz number \n$H^{2-2g,\\textsc{f}}(d;\\Delta^1,\\dots,\\Delta^\\textsc{f})$ is equal to the number of solutions of equation (\\ref{Hom-pi-S_d-Riemann})\ndivided by the order of symmetric group $S_d$ (to exclude the equivalent solutions obtained by the conjugation of all factors in\n(\\ref{Hom-pi-S_d-Riemann}) by elements of the group. In the geometrical approach each conjugation means the re-enumaration of $d$ sheets\nof the cover).\n\nFor instance Example 3 considered above counts non-equivalent solutions of the equation $X_1X_2=1$ with given cycle classes \n$C_{\\Delta^1}$ and $C_{\\Delta^2}$. Solutions of this equation consist of all elements of class $C_{\\Delta^1}$ and inverse elements, so \n$\\Delta^2=\\Delta^1=:\\Delta$. The number of elements of any class $C_\\Delta$ (the cardinality of $|C_\\Delta|$) divided by $|\\Delta|!$ \nis $1 \\over z_\\Delta$ as we got in the Example 3.\n\nFor Klein surfaces (see \\cite{M2},\\cite{GARETH.A.JONES}) instead of (\\ref{Hom-pi-S_d-Riemann}) we get \n\\begin{equation}\\label{Hom-pi-S_d-Klein}\n\\prod_{j=1}^g R_j^2 X_1\\cdots X_{\\textsc{f}} =1\n\\end{equation}\nwhere $R_j,X_i\\in S_d$ and where each $X_i$ belongs to the cycle class $C_{\\Delta^i}$. In (\\ref{Hom-pi-S_d-Klein}),\n$g$ is the so-called genus of non-orientable surface which is related to its Eular chatacteristic $\\textsc{e}$ as \n$\\textsc{e}=1-g$. For the projective plane ($\\textsc{e}=1$) we have $g=0$, for the Klein bottle ($\\textsc{e}=1$) $g=1$.\n\nConsider unbranched covers of the torus (equation (\\ref{Hom-pi-S_d-Riemann})), projective plane and Klein bottle\n(\\ref{Hom-pi-S_d-Klein})). In this we put each $X_i=1$ in (\\ref{Hom-pi-S_d-Riemann})) and (\\ref{Hom-pi-S_d-Klein})).\nHere we present three pictures, for the torus ($\\textsc{e}=0$), the real projective plane ($\\textsc{e}=1$) and Klein bottle \n($\\textsc{e}=0$) which may be obtained by the identification of square's edges. We get $aba^{-1}b^{-1}=1$ for torus,\n$abab=1$ for the projective plane and $abab^{-1}=1$ for the Klein bottle.\n\n\n3 pictures.\n\nConsider unbranched coverings ($\\textsc{f}=0$).\nFor the real projective plane we have $g=1$ in (\\ref{Hom-pi-S_d-Klein}) only one $R_1=ab$. If we treat the projective plane as the unit disk\nwith identfied opposit points of the boarder $|z|=1$, then $R$ is related to the path from $z$ to $-z$.\nFor the Klein bottle ($g=2$ in (\\ref{Hom-pi-S_d-Klein})) there are $R_1=ab$ and $R_2=b^{-1}$.\n\nTo avoid confisions in what follows  we will use the notion of genus and the notations $g$ only for Rieamnn surfaces, while\nthe notion of the Euler characteristic $\\textsc{e}$ we shall use both for orientable and non-orientable surfaces.\n\n \n \n\\section{Random matrices. Complex Ginibre ensemble.}\nOn this subject there is an extensive literature, for instance see \\cite{Ak1}, \\cite{Ak2}, \\cite{AkStrahov}, \n\\cite{S1}, \\cite{S2}.\n\nWe will consider integrals over complex matrices $Z_1,\\dots,Z_n$ where the measure is defined as\n\\begin{equation}\nd\\Omega(Z_1,\\dots,Z_n)= \\prod_{\\alpha=1}^n d\\mu(Z_\\alpha)=c \n\\prod_{\\alpha=1}^n\\prod_{i,j=1}^N d\\Re (Z_\\alpha)_{ij}d\\Im (Z_\\alpha)_{ij}e^{-|(Z_\\alpha)_{ij}|^2}\n\\end{equation}\nwhere the integration range is $\\mathbb{C}^{N^2}\\times \\cdots \\times\\mathbb{C}^{N^2}$ and where $c$ is the normalization \nconstant defined via $\\int d \\Omega(Z_1,\\dots,Z_n)=1$. \n\n\nWe treat this measure as the probability measure. The related ensemble is called the ensemble of $n$ independent \ncomplex Ginibre enesembles. \nThe expectation of a quantity\n$f$ which depends on entries of the matrices $Z_1,\\dots,Z_n$ is defined by\n$$\n\\mathbb{E}(f)=\\int f(Z_1,\\dots,Z_n) d\\Omega(Z_1,\\dots,Z_n).\n$$\n\nLet us introduce\nthe following products\n\\begin{eqnarray}\\label{Z}\nX&:=&(Z_1 C_1) \\cdots (Z_n C_n)\\\\ \n\\label{tildeZ^*}\nY_{t}&:=& Z^\\dag_n Z^\\dag_{n-1} \\cdots Z^\\dag_{t+1} Z^\\dag_1Z^\\dag_2  \\cdots Z^\\dag_{t} ,\\qquad 0<  t < n\n\\end{eqnarray}\nwhere $Z_\\alpha^\\dag$ is the Hermitian conjugate of $Z_\\alpha$. \nWe are interested in correlation functions of spectral invariants of matrices $X$ and $Y_t$.\n\nWe denote by $x_1,\\dots,x_N$ and by $y_1,\\dots,y_N$  the eigenvalues of the matrices $X$ and $Y_t$, respectively.\nGiven partitions $\\lambda=(\\lambda_1,\\dots,\\lambda_l)$, $\\mu=(\\mu_1,\\dots,\\mu_k)$, $l,k\\le N$. Let us introduce the following \nspectral invariants\n\\begin{equation}\n{\\bf P}_\\lambda (X)=p_{\\lambda_1}(X)\\cdots p_{\\lambda_l} (X),\\qquad\n{\\bf P}_\\mu(Y_t)=p_{\\mu_1}(Y_t)\\cdots p_{\\mu_k}(Y_t)\n\\end{equation}\nwhere each $p_m(X)$ is defined via (\\ref{p_m(X)}).\n\nFor a given partition $\\lambda$, such that $d:=|\\lambda|\\le N$, let us consider the spectral invariant \n${\\bf P}_\\lambda$ of the matrix $XY_{t}$ \n(see (\\ref{spectral-invariant})). We have \n\n\n\\begin{Theorem}\\label{Theorem1}\n $X$ and $Y_t$ are defined by (\\ref{Z})-(\\ref{tildeZ^*}).\n Denote $\\textsc{e}=2-2g$. \n \n(A)   Let $n > t=2g \\ge 0$. \nThen\n \\[\n \\mathbb{E}\\left({\\bf P}_\\lambda (X Y_{2g})\\right)=\n \\]\n \\begin{equation}\n z_\\lambda \\sum_{\\Delta^1,\\dots,\\Delta^{n-2g+1}\\atop |\\lambda|=|\\Delta^{j}| =d,\\, j\\le n-2g+1}\nH^{2-2g,n+2-2g}(d;\\lambda,\\Delta^{1},\\dots,\\Delta^{n-2g+1})P_{\\Delta^{n-2g+1}}(C' C'')\n\\prod_{i=1}^{n-2g}  P_{\\Delta^i}(C_{2g+i})\n \\end{equation}\n where\n \\begin{equation}\\label{C'C''2g}\n C' = C_1 \\cdots C_{2g-1} ,\n\\qquad\nC''=C_2C_4\\cdots C_{2g}\n\\end{equation} \n\n\n(B)   Let $n > t=2g+1 \\ge 1$. \nThen\n \\[\n \\mathbb{E}\\left({\\bf P}_\\lambda (X Y_{2g+1})\\right)=\n \\] \n \\begin{equation}\n z_\\lambda \\sum_{\\Delta^1,\\dots,\\Delta^{n-2g+1}\\atop |\\lambda|= |\\Delta^{j}|=d,\\,j\\le n-2g+1}\nH^{2-2g,n+2-2g}(d;\\lambda,\\Delta^{1},\\dots,\\Delta^{n-2g+1})P_{\\Delta^{n-2g}}(C')P_{\\Delta^{n-2g+1}}(C'')\n\\prod_{i=1}^{n-2g-1}  P_{\\Delta^i}(C_{2g+1+i})\n \\end{equation}\n where\n \\begin{equation}\\label{C'C''2g+1}\nC'= C_1C_3 \\cdots C_{2g+1} ,\n\\qquad\nC''=C_2C_4\\cdots C_{2g}\n\\end{equation} \n \n\\end{Theorem}\n\n\n\\begin{Corollary}\nLet  $|\\lambda|=d\\le N$ as before, and let each $C_i=I_N$ ($N\\times N$ unity matrix).\nThen\n \\begin{equation}\n \\frac{1}{z_\\lambda}\\mathbb{E}\\left({\\bf P}_\\lambda (X Y_{2g})\\right)=\n \\frac{1}{z_\\lambda}\\mathbb{E}\\left({\\bf P}_\\lambda (X Y_{2g+1})\\right)=\n  N^{nd-\\ell(\\lambda)} \\sum_{\\textsc{e}'} N^{\\textsc{e}'}\n S^{\\textsc{e}'}_{\\textsc{e}}(\\lambda)\n \\end{equation}\n where $\\textsc{e}=2-2g$ and where\n \\begin{equation}\n S^{\\textsc{e}'}_{\\textsc{e}}(\\lambda) :=\n\\sum_{\\Delta^1,\\dots,\\Delta^{n+\\textsc{e}-1}\\atop \n\\sum_{i=1}^{n+\\textsc{e}-1}\\ell(\\Delta^{i}) =L}\nH^{\\textsc{e},n+\\textsc{e}}(d;\\lambda,\\Delta^{1},\\dots,\\Delta^{n+\\textsc{e}-1}),\\quad L=-\\ell(\\lambda)+ nd +\\textsc{e}' \n \\end{equation}\nis the sum of Hurwitz numbers counting all $d$-fold coverings with the following properties:  \n\n(i) the Euler characteristic of the base surface is $\\textsc{e}$ \n\n(ii) the Euler characteristic of the cover is $\\textsc{e}'$ \n\n(iii) there are at most $\\textsc{f}=n+\\textsc{e}$ critical points\n\n\n\\end{Corollary}\n\nThe item (ii) in the Corollary follows from the equality ${\\bf P}_\\Delta(I_N)=N^{\\ell(\\Delta)}$ \n(see (\\ref{spectral-invariant}) and (\\ref{p_m(X)})) and\nfrom the Riemann-Hurwitz relation which relates Euler characteristics of a base and a cover via branch points\nprofile's lengths (see (\\ref{RH})):\n$$\n\\sum_{i=1}^{n+\\textsc{e}-1}\\ell(\\Delta^{i}) =-\\ell(\\lambda) +(\\textsc{f}- \\textsc{e})d +\\textsc{e}'\n$$ \nIn our case $\\textsc{f}- \\textsc{e}=n$.\n\n\n\n\\begin{Theorem} \\label{Theorem2}\n$X$ and $Y_t$ are defined by (\\ref{Z})-(\\ref{tildeZ^*}).\n  \n  (A)  If $|\\lambda|\\neq |\\mu|$ then\n $\\mathbb{E}\\left({\\bf P}_\\lambda (X) {\\bf P}_\\mu(Y_t)\\right)=0$.\n\n \n (B)    Let $|\\lambda| = |\\mu| =d$ and   $n-1 > t=2g+1 \\ge 1$.\n \n\nThen\n \\[\n \\mathbb{E}\\left({\\bf P}_\\lambda (X) {\\bf P}_\\mu(Y_{2g+1})\\right)=\n \\]\n \\begin{equation}\n z_\\lambda z_\\mu \n  \\sum_{\\Delta^1,\\dots,\\Delta^{n-2g}\\atop |\\lambda|= |\\Delta^{j}|=d,\\,j\\le n-2g}\nH^{2-2g,n+2-2g}(d;\\lambda,\\mu,\\Delta^{1},\\dots,\\Delta^{n-2g})P_{\\Delta^{n-2g-1}}(C')P_{\\Delta^{n-2g}}(C_{n}C'')\n\\prod_{i=1}^{n-2g-2}  P_{\\Delta^i}(C_{2g+1+i}) \n \\end{equation} \n where $C'$ and $C''$ are given by (\\ref{C'C''2g}).\n \n (C) \n Let $|\\lambda| = |\\mu|$ $n > t=2g \\ge 0$ . \n\nThen\n \\[\n \\mathbb{E}\\left({\\bf P}_\\lambda (X) {\\bf P}_\\mu(Y_{2g})\\right)=\n \\]\n \\begin{equation}\n z_\\lambda z_\\mu \n \\sum_{\\Delta^1,\\dots,\\Delta^{n-2g}\\atop |\\lambda|=|\\Delta^{j}| =d,\\, j\\le n-2g}\nH^{2-2g,n+2-2g}(d;\\lambda,\\mu,\\Delta^{1},\\dots,\\Delta^{n-2g})P_{\\Delta^{n-2g+1}}(C'C_n C'')\n\\prod_{i=1}^{n-2g}  P_{\\Delta^i}(C_{2g+i})\n \\end{equation}\n where $C'$ and $C''$ are given by (\\ref{C'C''2g+1}).\n \n \n\\end{Theorem}\n\n\n\\begin{Corollary}\n Let  $|\\lambda|=d\\le N$ as before, and let each $C_i=I_N$.\nThen\n \\begin{equation}\n \\frac{1}{z_\\lambda z_\\mu}\\mathbb{E}\\left({\\bf P}_\\lambda (X){\\bf P}_\\lambda(Y_{2g})\\right)=\n \\frac{1}{z_\\lambda z_\\mu}\\mathbb{E}\\left({\\bf P}_\\lambda (X){\\bf P}_\\lambda(Y_{2g+1})\\right)\n =\n \\frac{1}{z_\\lambda}\\mathbb{E}\\left({\\bf P}_\\lambda (X Y_{2g})\\right)=\n \\frac{1}{z_\\lambda}\\mathbb{E}\\left({\\bf P}_\\lambda (X Y_{2g+1})\\right)\n \\end{equation}\n\n\n\n\\end{Corollary}\n\n\n\n\\begin{Theorem} \n\\label{Theorem3}\n$X$ and $Y_t$ are defined by (\\ref{Z})-(\\ref{tildeZ^*}).\n  \n \n (A)    Let   $n-1 > t=2g+1 \\ge 0$.\n \n\nThen\n \\[\n \\mathbb{E}\\left({\\bf P}_\\lambda (X) \\tau^{\\rm BKP}_1(Y_{2g+1})\\right)=\n \\]\n \\begin{equation}\n z_\\lambda \n  \\sum_{\\Delta^1,\\dots,\\Delta^{n-2g}\\atop |\\lambda|= |\\Delta^{j}|=d,\\,j\\le n-2g}\nH^{1-2g,n+1-2g}(d;\\lambda,\\Delta^{1},\\dots,\\Delta^{n-2g})P_{\\Delta^{n-2g-1}}(C')P_{\\Delta^{n-2g}}(C_{n}C'')\n\\prod_{i=1}^{n-2g-2}  P_{\\Delta^i}(C_{2g+1+i}) \n \\end{equation} \n where $C'$ and $C''$ are given by (\\ref{C'C''2g}).\n \n (B) \n Let $n > t=2g \\ge 0$ . \n\nThen\n \\[\n \\mathbb{E}\\left({\\bf P}_\\lambda (X) \\tau^{\\rm BKP}_1(Y_{2g})\\right)=\n \\]\n \\begin{equation}\n z_\\lambda z_\\mu \n \\sum_{\\Delta^1,\\dots,\\Delta^{n-2g}\\atop |\\lambda|=|\\Delta^{j}| =d,\\, j\\le n-2g}\nH^{1-2g,n+1-2g}(d;\\lambda,\\Delta^{1},\\dots,\\Delta^{n-2g})P_{\\Delta^{n+1-2g}}(C'C_n C'')\n\\prod_{i=1}^{n-2g}  P_{\\Delta^i}(C_{2g+i})\n \\end{equation}\n where $C'$ and $C''$ are given by (\\ref{C'C''2g+1}).\n \n \n\\end{Theorem}\n\nThe sketch of proof.\n \nThe character Frobenius-type formula by Mednykh-Pozdnyakova-Jones \\cite{M2},\\cite{GARETH.A.JONES} \n\\begin{equation}\\label{Hurwitz-number}\nH^{\\textsc{e},k}(d;\\Delta^1,\\dots,\\Delta^{k})=\\sum_{\\lambda\\atop |\\lambda|=d}\n\\left(\\frac{{\\rm dim}\\lambda}{d!} \\right)^{\\textsc{e}}\\varphi_\\lambda(\\Delta^1)\\cdots \n\\varphi_\\lambda(\\Delta^k)\n\\end{equation}\nwhere \n${\\rm dim}\\lambda$ is the dimension of the irreducible representation of $S_d$, and\n\\begin{equation}\n\\label{varphi}\n\\varphi_\\lambda(\\Delta^{(i)}) := |C_{\\Delta^{(i)}}|\\,\\,\\frac{\\chi_\\lambda(\\Delta^{(i)})}{{\\rm dim}\\lambda} ,\n\\quad {\\rm dim}\\lambda:=\\chi_\\lambda\\left((1^d)\\right)\n\\end{equation}\n$\\chi_\\lambda(\\Delta)$ is the character of the symmetric group $S_d$ evaluated at a cycle type $\\Delta$,\nand $\\chi_\\lambda$ ranges over the irreducible complex characters of $S_d$ (they are\nlabeled by partitions $\\lambda=(\\lambda_1,\\dots,\\lambda_{\\ell})$ of a given weight $d=|\\lambda|$). It \nis supposed that $d=|\\lambda|=|\\Delta^{1}|=\\cdots =|\\Delta^{k}|$.\n$|C_\\Delta |$ is the cardinality of the cycle\nclass $C_\\Delta$ in $S_d$.\n\nThen we use the characteristic map relation\n\\cite{Mac}:\n\\begin{equation}\\label{Schur-char-map}\ns_\\lambda(\\mathbf{p})=\\frac{{\\rm dim}\\lambda}{d!}\\left(p_1^d+\\sum_{\\Delta\\atop |\\Delta|=d } \\varphi_\\lambda(\\Delta)\\mathbf{p}_{\\Delta}\\right)\n\\end{equation}\nwhere $\\mathbf{p}_\\Delta=p_{\\Delta_1}\\cdots p_{\\Delta_{\\ell}}$ and where $\\Delta=(\\Delta_1,\\dots,\\Delta_\\ell)$ is a partition whose weight\ncoinsides with the weight of $\\lambda$: $|\\lambda|=|\\Delta|$. Here \n\\begin{equation}\n{\\rm dim}\\lambda =d!s_\\lambda(\\mathbf{p}_\\infty),\\qquad \\mathbf{p}_\\infty = (1,0,0,\\dots)\n\\end{equation}\nis the dimension of the irreducable representation of the symmetric group $S_d$. We imply that \n$\\varphi_\\lambda(\\Delta)=0$ if $|\\Delta|\\neq |\\lambda|$.\n\nThen we know how to evaluate the integral with the Schur function via Lemma\n used in \\cite{O-2004-New} and \\cite{NO-2014}, \\cite{NO-LMP}\n(for instance see \\cite{Mac} for the derivation). \n\\begin{Lemma} \\label{useful-relations}\nLet $A$ and $B$ be normal  matrices (i.e. matrices diagonalizable by unitary transformations). Then\nBelow $p_{\\infty}=(1,0,0,\\dots)$. \n\\begin{equation}\\label{sAZBZ^+'}\n\\int_{\\mathbb{C}^{n^2}} s_\\lambda(AZBZ^+)e^{-\\textrm{Tr}\nZZ^+}\\prod_{i,j=1}^n d^2Z=\n\\frac{s_\\lambda(A)s_\\lambda(B)}{s_\\lambda(p_{\\infty})}\n\\end{equation}\nand\n\\begin{equation}\\label{sAZZ^+B'}\n\\int_{\\mathbb{C}^{n^2}} s_\\mu(AZ)s_\\lambda(Z^+B) e^{-\\textrm{Tr}\nZZ^+}\\prod_{i,j=1}^nd^2Z= \\frac{s_\\lambda(AB)}{s_\\lambda(p_{\\infty})}\\delta_{\\mu,\\lambda}\\,.\n\\end{equation}\n\\end{Lemma}\nTo prove Theorem \\ref{Theorem1} we use that we can equate the integral over $E(\\tau^{\\rm 2KP}(XY_y))$ using this Lemma\nand (\\ref{2KP-tau-vac-Schur}) and then compare it to the same integral where now we use (\\ref{tau_1}).\nTo prove Theorem \\ref{Theorem2} in the similar way we equate $E(\\tau^{\\rm 2KP}(X)\\tau^{\\rm 2KP}(Y_y))$.\nTo prove Theorem \\ref{Theorem3} we similarly $E(\\tau^{\\rm 2KP}(X)\\tau^{\\rm 2KP}(Y_y))$ in the same way taking into acount\nalso (\\ref{tau_1^B}).\n\n\n\n\n\\section*{Acknowledgements}\n\nThe work has been funded by the RAS Program ``Fundemental problems of nonlinear mechanics'' and by the Russian Academic Excellence Project '5-100'.\nI thank A. Odziyevich and university of Bialystok for warm hospitality which allowed\nit is accurate to write down this work. I am grateful to S. Natanzon, A. Odziyevich, J. Harnad, A. Mironov (ITEP) and\nto van de Ler for various remarks concerning the questions connected with this work. Special gratitude\nto E. Strakhov for the fact that he drew my attention to the works on quantum chaos devoted to the products \nof random matrices and for fruitful discussions.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Gross-Pitaevskii (GP) equation was independently derived by L.P.~Pitaevskii \\cite{Pitaevskii61} and E.P.~Gross \\cite{Gross61} in 1961. It describes a superfluid gas of weakly interacting bosons at zero temperature. The solution of the equation is a complex function $\\Psi=|\\Psi| \\exp i \\varphi$, whose  modulus squared represents the particle density, $n=|\\Psi|^2$, and the gradient of the phase gives the local velocity of the fluid, ${\\bf v}= (\\hbar\/m)\\nabla \\varphi$, where $m$ is the particle mass.  In the derivation by L.P. Pitaevskii, the GP equation emerges as a generalization of Bogoliubov's theory \\cite{Bogolyubov47} to a spatially inhomogeneous superfluid \\cite{Ginzburg58}.  A quantized vortex can exist as a stationary solution of the GP equation where all particles circulate with the same angular momentum $\\hbar$ around a line where the density vanishes; the solution has the form $\\sqrt{n(r)} \\exp i \\varphi$, where now $\\varphi$ is the angle around the vortex axis and $r$ is the distance from the axis in cylindrical coordinates.  The density $n(r)$ is a smooth function which increases from $0$ to a constant asymptotic value $n_0$ over a length scale characterized by $\\xi$, known as the {\\it healing length}, determined by $n_0$ and the strength of the interaction. \n\nQuantized vortices have been extensively studied in superfluid $^4$He \\cite{Donnelly91}, which is a strongly correlated liquid. The core of the vortex in $^4$He is only qualitatively captured by the GP equation and more refined theories are needed to account for the atom-atom interactions and many-body effects \\cite{Dalfovo92,Ortiz95,Vitiello96,Giorgini96,Galli14}. A direct comparison between theory and experiment for the structure of the vortex core is not available, and is likely unrealistic, the main reason being that the core size in $^4$He is expected to be of the same order as the atom size. The only way to observe such a vortex thus consists of looking at its effects on the motion of impurities that may be attached to it. Electrons \\cite{Yarmchuk79,Yarmchuk82,Maris09,Maris11}, solid hydrogen particles \\cite{Bewley06,Bewley08,Paoletti10,Paoletti11,Fonda14}, and $^4$He$_2^*$ excimer molecules \\cite{Zmeev13} have been used for this purpose. These impurities act as tracers for the position of vortex filaments in order to infer their motion on a macroscopic scale, but the fine structure of the core remains inaccessible. Furthermore, impurities may themselves affect the dynamics of the vortex filaments \\cite{Barenghi09}. \n\nIn dilute ultracold atomic gases the situation is more favorable. On the one hand, the GP theory furnishes a very accurate description of the system in regimes of temperature and diluteness that are attainable in typical experiments with trapped Bose-Einstein condensates (BECs) \\cite{Dalfovo99,PSbook16}. On the other hand, beginning with a series of seminal experiments \n\\cite{Matthews99,Madison00,Anderson2000,Anderson2001,Haljan01,AboShaeer01,Hodby02},\nquantized vortices are routinely produced and observed with different techniques (see \\cite{Fetter09} for a review). \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{vortex-core-fig1.png}\n\\caption{Experimental absorption images of a condensate with $7 \\times 10^6$ atoms after $120$~ms of free expansion. The small blue ellipse at the center of (a) represents the shape of the trapped condensate before the expansion, which is an elongated ellipsoid with the long axis in the $x$-direction. The expansion is faster in the transverse direction, so that the aspect ratio is inverted and the atomic distribution aquires a pancake shape. (a) Column density along a transverse direction. The faint vertical stripe is a signature of the presence of a vortex, and its shape is an interference pattern originating from the anisotropic velocity field around the vortex and the velocity field of the expansion. The  field of view is $1.3 \\times 3$~mm. (b) Column density along the axial direction. The vortex is almost invisible. The field of view is $3 \\times 3$~mm. (c) Residual column density. From the previous image we subtract spurious interference fringes, due to imperfections in the optical imaging, and the background density, using a Thomas-Fermi fit (see text). The result is an image of the residual column density which neatly reveals a vortex filament. (d)-(g) Other examples of vortex filaments shown by the residual column density for different condensates with one or more vortices. Note that even though the {\\it in-situ} condensate is always isotropic in the $y$-$z$ plane it becomes slightly elliptic after a long expansion due to a residual curvature of the magnetic field used to levitate the condensate against gravity.}\n\\label{fig:expt-images}\n\\end{figure*}\n\nDespite such an abundance of work, it may sound surprising that no detailed quantitative comparison between theory and experiment for the\nstructure of the vortex core in three-dimensional (3D) condensates has yet been performed. A reason is that the healing length $\\xi$ in typical trapped BECs, though much larger than in liquid $^4$He, is still smaller than the optical resolution, which is limited by the wavelength of the laser beams used for imaging. Another reason is that, when illuminating the atomic cloud with light, the result is the optical density, which is determined by an integral of the density along the imaging axis (column density); thus, a vortex filament has a strong contrast only if it is rectilinear and aligned along the imaging axis. One can overcome the first limitation by switching off the confining potential, letting the condensate freely expand. The vortex core expands as well, at least as fast as the condensate radius \\cite{Lundh98,Dalfovo00,Hosten03,Teles13a}, so that it can become visible after a reasonable expansion time.  Concerning vortex alignment, one can strongly confine a BEC along one spatial direction, squeezing it to within a width of several $\\xi$. In such a geometry, vortices orient themselves along the short direction, thus behaving as point-like topological defects in a quasi-2D system rather than filaments in a 3D fluid (a recent discussion about the structure of the vortex core in expanding quasi-2D condensates can be found in \\cite{Sang13}). Conversely, if the condensate width is significantly larger than $\\xi$ in all directions, the vortex filaments can easily bend \\cite{Aftalion01,Garcia01,Aftalion03,Modugno03}, with a consequent reduction of their visibility in the column density. Bent vortex filaments have indeed been observed in \\cite{Raman01,Rosenbusch02,Bretin03,Donadello14}.\nBending and optical resolution particularly limit the quality of comparisons between theory and experiment for the structure of the vortex core (see Fig.~14.10 in \\cite{PSbook16}). \n\nIn this work, we show that 3D vortex filaments can be optically observed with enough accuracy to permit a direct comparison with the predictions of the GP theory. In our experiment, we produce large condensates of sodium atoms in an elongated axially symmetric harmonic trap and we image each condensate, in both the axial and a transverse direction, after free expansion. When a vortex filament is present, it produces a visible modification of the column density distribution of the atoms. \nWe use numerical GP simulations, as well as scaling laws which are valid for the expansion of large condensates, to make direct comparisons with our experimental observations and find good agreement. \n\n\\section{Experiment}\n\nWe produce ultracold samples of sodium atoms in the internal state $|3S_{1\/2},F=1,m_{\\mathrm{F}}=-1\\rangle$ in a cigar-shaped harmonic magnetic trap with trap frequencies $\\omega_x\/2 \\pi=9.3$~Hz and $\\omega_\\perp\/2 \\pi=93$~Hz.  The thermal gas is cooled via forced evaporative cooling and pure BECs of typically around $10^7$ atoms are finally obtained with negligible thermal component. The evaporation ramp in the vicinity of the BEC phase transition is performed at different rates: slow quenches eventually produce condensates which are almost in their ground state, while faster quenches lead to the formation of quantized vortices in the condensate as a result of the Kibble-Zurek mechanism \\cite{Lamporesi13,Donadello16}.  The quench rate can be chosen in such a way to obtain condensates with one vortex on average. \n\nThe trapped condensate has a radial width on the order of $30 \\ \\mu$m and an axial width that is $10$ times larger. The healing length in the center of the condensate is about $0.2\\ \\mu$m, smaller than the optical resolution. It is also about two orders of magnitude smaller than the radial width of the condensate, which means that, as far as the density distribution is concerned, a vortex is a thin filament living in a 3D superfluid background with smoothly varying density, and the local properties of the vortex core are hence almost unaffected by boundary conditions. However, boundaries are still important for the superfluid velocity field. In fact, the ellipsoidal shape of the condensate causes a preferential alignment of the vortex filament along a (randomly chosen) radial direction so as to minimize its energy. Moreover, this geometry makes the flow around the vortex line anisotropic, meaning that on the larger scale of the entire condensate a vortex behaves as an almost planar localized object. For this reason, such vortices in elongated condensates are also known as solitonic-vortices \\cite{Donadello14,Brand02,Komineas03,Ku14}. For our purposes, such localization is an advantage since it significantly reduces the bending of the vortex filaments, while at the same time keeping their local core structure three dimensional. \n\nObservations are performed by releasing the atoms from the trap and taking simultaneous absorption images of the full atomic distribution along the radial and axial directions after a sufficiently long expansion in free space, so that the vortex core becomes larger than the imaging resolution \\cite{Donadello14,Donadello16}. The presence of a levitating magnetic field gradient makes it possible to achieve long expansion times preventing the BEC  from falling. Typical images are shown in Fig.~\\ref{fig:expt-images}.  In the radial direction (panel {\\it a}), the vortex is seen as a dark stripe. This soliton-like character is due to the interference of the two halves (ends) of the elongated condensate which, on the large length scale of the entire condensate, have approximately a $\\pi$ phase difference \\cite{Donadello14,Ku14,Tylutki15}.\nIf a vortex filament is parallel to the imaging direction, the dark stripe exhibits a central dip, corresponding to the vortex core seen along its axis, and a twist due to the anisotropic quantized circulation. The $2\\pi$ phase winding around the vortex core was also detected in the same setup \\cite{Donadello14} by means of an interferometric technique based on a sequence of Bragg pulses. In the axial direction (panel {\\it b}), the soliton-like character is integrated out and the vortex filament is only a faint (and almost invisible) perturbation in the column density. However, by subtracting the background represented by a condensate without any vortex, the filament clearly emerges in the residual density distribution (panel {\\it c}). In the following we show how this signal can be used to extract quantitative information on the vortex structure after expansion, and how this is related to the shape of the vortex core in the condensate {\\it in-situ}, before the expansion. \n\n\\section{Theory}\n\nThe GP equation for the macroscopic wave function $\\Psi({\\bf r,t})$ for a BEC of weakly interacting bosons of mass $m$ at zero temperature is \\cite{Pitaevskii61,Gross61,Dalfovo99,PSbook16}\n\\begin{equation}\ni \\hbar \\frac{\\partial \\Psi}{\\partial t} = \\left( -\\frac{\\hbar^2 \\nabla^2}{2m} + V_{\\rm ext} + g |\\Psi|^2 \\right) \\Psi ,\n\\end{equation}\nwhere $V_{\\rm ext}$ is the external potential and $t$ is time. The quantity $g$ is a coupling constant characterizing the interaction between the atoms, which is positive for our condensates.  The stationary version of the GP equation is obtained by choosing $\\Psi({\\bf r},t)= \\psi({\\rm r}) \\exp (-i\\mu t\/\\hbar)$, so that\n\\begin{equation}\n\\left( -\\frac{\\hbar^2 \\nabla^2}{2m} + V_{\\rm ext} + g |\\psi|^2 \\right) \\psi = \\mu \\psi\n\\label{eq:stationaryGP}\n\\end{equation}\nwhere $\\mu$ is the chemical potential and  $n=|\\psi|^2$ is the density. In our case, we use the stationary GP equation to describe the condensate confined by the axially symmetric harmonic potential $V_{\\rm ext}=(m\/2)[\\omega_x^2 x^2+\\omega_\\perp^2 (y^2+z^2)]$, with the aspect ratio $\\lambda=\\omega_x\/\\omega_\\perp=0.1$, as in the experiment.  Then we simulate the expansion by using this solution as the $t=0$, starting condition for the solution of the time dependent GP equation with $V_{\\rm ext}=0$. We simulate condensates with and without a vortex. In the former case, the vortex is rectilinear, passing through the center and aligned along the $z$-axis. The need to accurately describe the dynamics of the system on both the scale of the healing length $\\xi$ and the scale of the width of the entire expanding condensate poses severe computational constraints. With this in mind, we are only able to perform simulations up to values of the chemical potential on the order of $10\\hbar \\omega_\\perp$, which are smaller than the experimental values, ranging from about $15$ to $30\\hbar \\omega_\\perp$. Experiments can also be performed for smaller values of $N$, and hence smaller $\\mu$, but fluctuations in the density distribution become relatively larger with decreasing $N$, and the signal-to-noise ratio for the visibility of vortices in axial imaging becomes too small. The comparison between theory and experiments hence requires an extrapolation of the GP results to larger $\\mu$ and this is possible thanks to scaling laws which are valid for large condensates.  \n\nIf $\\mu$ is significantly larger than both $\\hbar \\omega_\\perp$ and  $\\hbar \\omega_x$, then the ground state of the condensate, i.e., the lowest energy stationary solution of the GP equation, is well approximated by the Thomas-Fermi (TF) approximation, which corresponds to neglecting the first term in the parenthesis of Eq.~(\\ref{eq:stationaryGP}), so that the density becomes  \\cite{Dalfovo99,PSbook16}\n\\begin{equation}\nn_{\\rm TF}(x,y,z) =  \\frac{1}{g} \\left[\\mu -\\frac{1}{2}m \\omega_x^2 x^2 -\\frac{1}{2}m \\omega_\\perp^2 (y^2+ z^2) \\right] \n\\label{eq:TF}\n\\end{equation}\nwithin the central region where $n_{\\rm TF}$ is positive, and is $0$ elsewhere.\nWe can then define the boundary TF radii $R_x = (2\\mu\/m\\omega_x^2)^{1\/2}$ and  $R_\\perp = (2\\mu\/m\\omega_\\perp^2)^{1\/2}$, the central density $n_0=\\mu\/g$, and the rescaled coordinates $\\tilde{x}=x\/R_x$, $\\tilde{y}=y\/R_\\perp$ and $\\tilde{z}=z\/R_\\perp$, and rewrite the density in the form \n\\begin{equation}\nn_{\\rm TF}(\\tilde{x},\\tilde{y},\\tilde{z}) =  n_0 ( 1 - \\tilde{x}^2 - \\tilde{y}^2 - \\tilde{z}^2 ) \\, .\n\\label{eq:rescaledTF}\n\\end{equation}\nThis inverted parabola is a very good approximation for the density profiles of our condensates except in a narrow region near the condensate boundaries  \\cite{Dalfovo96b}. \n\nIn the regime where the TF approximation is valid, the free expansion is governed by simple scaling laws \\cite{Castin96,Kagan96,Dalfovo97}. In particular, one can prove that the condensate preserves its shape with a rescaling of the TF radii in time according to $R_x(t)=b_x(t)R_x(0)$ and $R_\\perp(t)=b_\\perp(t)R_\\perp(0)$, where the scaling parameters $b_x$ and $b_\\perp$ are solutions of the coupled differential equations\n$\\ddot{b}_\\perp -  \\omega_{\\perp}^2\/(b_xb_\\perp^3)  =  0 $ and $\\ddot{b}_x -  \\omega_{x}^2\/(b_x^2b_\\perp^2)   = 0$,\nwith initial conditions $b_x=b_\\perp=1$ and $\\dot{b}_x=\\dot{b}_\\perp=0$ at $t=0$.\nBy using the aspect ratio $\\lambda$ and introducing the dimensionless time $\\tau=\\omega_{\\perp}t$, one can rewrite the same equations as \n\\begin{equation}\n\\frac{d^2 b_\\perp}{d \\tau^2}  -  \\frac{1}{b_xb_\\perp^3} =  0 \\ \\ \\ , \\ \\ \\ \n\\frac{d^2 b_x}{d \\tau^2} -  \\frac{\\lambda^2}{b_x^2b_\\perp^2}  =  0  \\, . \n\\label{eq:b}\n\\end{equation}\nAnalytic solutions exist in the limit $\\lambda \\ll 1$, that is, for a very elongated ellipsoid, for which one finds \\cite{Castin96}\n\\begin{eqnarray}\nb_\\perp (\\tau) & = &  \\sqrt{1+\\tau^2} \\nonumber \\\\\nb_x (\\tau) & = & 1 + \\lambda^2 [ \\tau \\ {\\rm arctan} \\tau - \\ln \\sqrt{1+\\tau^2} \\ ] \\, .\n\\label{eq:banalytic} \n\\end{eqnarray}\nThe correction proportional to $\\lambda^2$ becomes vanishingly small in the limit of the infinite cylinder, where the condensate is known to follow a scaling behavior that preserves its radial shape, even in regimes where the TF approximation does not apply \\cite{Pitaevskii97}.\n\n\\begin{figure}[]\n\\includegraphics[width=0.9\\linewidth]{vortex-core-fig2.pdf}\n\\caption{Residual column density  (\\ref{eq:nres}) calculated for a GP simulation of an expanding condensate with $\\mu=9.7 \\hbar \\omega_\\perp$ and with a vortex aligned along $z$, passing through the origin. Curves are plotted for different values of the expansion time, $\\tau=\\omega_\\perp t$, and are normalized to the value $n^{\\rm TF}_{\\rm col} (0,\\tau)$, which is the maximum of the fitted TF column density at the same time. The coordinate $\\tilde{y}=y\/R_\\perp$ is the distance from the vortex axis in units of the transverse TF radius obtained from the same fit. The spatial range is limited to half the TF radius in order to highlight the print of the vortex in the column density; the effects of the condensate boundaries are almost negligible in this range.      }\n\\label{fig:restheo}\n\\end{figure}\n\n\\begin{figure}[]\n\\includegraphics[width=0.9\\linewidth]{vortex-core-fig3.pdf}\n\\caption{Time evolution of the depth (top) and width (bottom) of the depletion produced by a vortex in the residual column density of expanding condensates with different chemical potentials $\\mu$.\nDepth and width are defined as the amplitude and the width $\\sigma$ of a Gaussian fit, respectively.\nAs in Fig.~\\ref{fig:restheo}, these parameters are normalized by the central TF column density and the transverse TF radius.\nNote that to be consistent with our experiments, for the purpose of improving the fit quality, prior to fitting we average  $\\delta n (\\tilde{y},\\tilde{z},\\tau)\/n^{\\rm TF}_{\\rm col} (0,\\tilde{z},\\tau)$ over different $z$ values within the interval [$-R_{\\perp}\/3,R_{\\perp}\/3$]. At very early times, $\\tau\\lesssim 3$, the dip in the residual is too small for the fit to quantitatively represent the vortex's characteristics. The dashed line is the prediction (\\ref{eq:maxresidualvortex}) of the empty core model.}\n\\label{fig:depthwidth}\n\\end{figure}\n\nThe TF density profile (\\ref{eq:rescaledTF}) is not only an accurate fitting function of the GP density distribution during the free expansion of an elongated condensate with $\\mu \\sim 10 \\hbar \\omega_{\\perp}$, but the TF radii extracted from the fit also agree with the scaling solutions of (\\ref{eq:b}), as well as with the analytic expressions  (\\ref{eq:banalytic}), the discrepancy being less than $2$\\% in all our simulations, even for long expansion times. The agreement is expected to be even better for larger values of $\\mu$. This justifies the use of a TF fit to extract the residual density both in the experiments and in the GP simulations. The fit also provides the values of the TF radii and $n_0$ at any given time $t$, which can be used to rescale the coordinates and the density.  \n\n For comparison with experiments, the key quantity is the column density, that is, the integral of the density along the imaging axis. Let us consider a cut of the density in the $z=0$  plane and define $n_{\\rm col}(\\tilde{y},t) =  \\int d\\tilde{x} \\ n(\\tilde{x},\\tilde{y},0,t)$, where the integral is restricted to the region where the density is positive. Using the analytic TF density, one finds \n\\begin{equation}\nn^{\\rm TF}_{\\rm col}(\\tilde{y},t)=n^{\\rm TF}_{\\rm col} (0,t)(1 - \\tilde{y}^2)^{3\/2} \n\\label{eq:ncolaxial}\n\\end{equation}\nand we can finally define the residual column density as\n\\begin{equation}\n\\delta n (\\tilde{y},t) =  n_{\\rm col} (\\tilde{y},t) - n^{\\rm TF}_{\\rm col}(\\tilde{y},t) \\,  .\n\\label{eq:nres}\n\\end{equation}\nAn example is shown in Fig.~\\ref{fig:restheo}, where we plot $\\delta n$  obtained in the GP simulation of the expansion for a condensate with $\\mu= 9.7 \\hbar \\omega_\\perp$. The figure shows that, as expected, a vortex produces a (column) density depletion whose depth is very small, i.e., only a few percent of the central column density of the condensate. It also shows that the depth increases in time during the expansion, while the width seems to remain almost constant.   In Fig.~\\ref{fig:depthwidth} we show results for the depth and the width obtained in simulations of condensates with different chemical potentials,  plotted as a function of the expansion time.\n\nThese results can be qualitatively understood by using a simplified model where the GP vortex core in the initial condensate is modelled by an empty cylinder of radius $r_v = c  \\xi_0$, where  $c$ is a number of order $1$ and $\\xi_0$ is the healing length of a uniform condensate with density $n_0$, which is given by $\\xi_0= \\hbar\/\\sqrt{2mgn_0} = \\hbar\/\\sqrt{2m\\mu}$. The rescaled radius is $\\tilde{r}_v=r_v\/R_\\perp= c \\xi_0\/R_\\perp= c \\hbar\\omega_\\perp\/2\\mu$. Then, let us assume that the initial expansion of the condensate is dominated by the mean-field interaction in the following sense: a segment of vortex filament near the center of the condensate expands as if it were in a uniform condensate, preserving its shape, but adiabatically following the time variation of the density of the medium around it. Hence, the vortex radius grows because the density decreases and the healing length is inversely proportional to $\\sqrt{n_0}$. Meanwhile, the transverse and axial TF radii $R_\\perp$ and $R_x$ grow, but with different scaling laws; such a difference is precisely the origin of the increased visibility of the vortex.  The empty-cylinder model allows us to calculate the column density, analytically taking into account all of these effects. In particular, using the scaling law (\\ref{eq:banalytic}) and neglecting the $\\lambda^2$ term, one can easily prove that $\\tilde{r}_v$ is constant during the expansion, while the residual column density takes the form \n\\begin{equation}\n\\delta n (\\tilde{y},\\tau) =  - \\frac{3 \\lambda \\tilde{r}_v}{2}  n^{\\rm TF}_{\\rm col} (0;\\tau)  \\sqrt{1+\\tau^2} \n(1 - \\tilde{y}^2) \\left( 1 - \\frac{\\tilde{y}^2}{\\tilde{r}_v^2} \\right)^{\\frac{1}{2}}   \\ ,\n\\end{equation}\nand the normalized depth can be written as    \n\\begin{equation}\n\\frac{|\\delta n (0,\\tau)|}{n^{\\rm TF}_{\\rm col} (0,\\tau)} =   \\frac{3}{2} \\lambda \\tilde{r}_v  \\sqrt{1+\\tau^2}  \\, .\n\\label{eq:maxresidualvortex}\n\\end{equation}\nThe dashed line in Fig.~\\ref{fig:depthwidth} corresponds to this prediction when $c=1.6$ and $\\mu= 9.7 \\hbar \\omega_\\perp$. With the same parameters, the rescaled width of the empty cylinder is $\\tilde{r}_v \\sim 0.08$, which is in qualitative agreement with the data in the bottom panel of the same figure. However, the assumption of adiabaticity is expected to be valid only at short times, when the density of the expanding condensate remains sufficiently large. As the expansion proceeds, the mean-field interactions lose their strength and the velocity field gradually assumes the characteristics of a ballistic expansion \\cite{Lundh98,Dalfovo00}.\nThe crossover from mean-field to ballistic expansion is smooth and,\nfor reference, we note that a spherically trapped condensate is expected to decouple at around $\\tau_{\\rm dec} \\sim \\sqrt{2\\mu\/\\hbar\\omega}$ \\cite{Lundh98}, which, for $\\mu = 9.7\\hbar\\omega$, would correspond to $\\tau_{\\rm dec} \\sim 4$ in Fig.~\\ref{fig:depthwidth}.\nThe full GP simulations show that the width remains approximately constant throughout the simulation, while the depth significantly deviates from the $\\sqrt{1+\\tau^2}$ law and saturates to a constant value deep in the ballistic regime. \n\n\\section{Experiment \\lowercase{vs.} Theory}\n\nIn this section, we compare the results of the experiments with the predictions of the GP theory for the overall shape, width and depth of the vortex in the residual column density.  \n\nThe depth and the width after a given expansion time $t$ are shown in Fig.~\\ref{fig:mu_scaling} as a function of $1\/\\mu$.  The two quantities are extracted from Gaussian fits, and normalized  by the central TF column density and the transverse TF radius as in Fig.~\\ref{fig:depthwidth}. In the case of experimental data, we first select condensates exhibiting a rectilinear vortex filament near their center, at an axial distance smaller than $R_\\perp\/3$. We then fit the column density with the analytic TF profile, but excluding points lying within a few healing lengths of the filament. From the fit we obtain the chemical potential and the TF radii of the ``background\" condensate and, by subtracting this background from the column density, we get the residual $\\delta n (\\tilde{y})$, where $\\tilde{y}$ is taken to be orthogonal to the filament. In order to increase the signal-to-noise ratio we average the normalized depth $\\delta n(\\tilde{y})\/n_{\\rm col}^{\\rm TF}(0)$ over different $z$ values within the interval [$-R_{\\perp}\/3,R_{\\perp}\/3$]. \nMoreover, if a vortex line is displaced from the center by a distance $\\tilde \\rho = \\sqrt{\\tilde x ^ 2 +\\tilde y ^2 + \\tilde z ^2}$, its core structure is that of a vortex in a background condensate with a density $(1 - \\tilde \\rho^2)$ times lower than the central density; we thus assign to the vortex a value of $\\mu$ corrected by the same factor.\nFinally, for long expansion times the residual external field makes the condensate slightly elliptic in the radial plane. For this reason, we use both $R_y$ and $R_z$ as independent TF radii and then we define $R_\\perp=\\sqrt{R_yR_z}$.  The same fitting procedure is applied to the GP density distributions, for which the condensate radius is always axially symmetric and the vortex is centered by construction. The experimental points correspond to four independent sets of data, where the cooling, evaporation, and imaging procedures are optimized for condensates with different atom numbers: red and orange points correspond to the largest condensates in our laboratory ($\\mu \\sim 30 \\hbar\\omega_\\perp$, $t=150$~ms and $120$~ms), blue points are the smallest condensates in which vortices are still observable ($\\mu \\sim 15 \\hbar\\omega_\\perp$, $t=100$~ms), while green points represent an old data set \\cite{Bisset17} for intermediate condensates ($\\mu \\sim 20 \\hbar\\omega_\\perp$, $t=120$~ms). Error bars account for statistical noise in the residual column density and for the uncertainties in the fit.   \n\nThe GP results clearly show that the rescaled width $\\sigma\/R_\\perp$ scales linearly with $1\/\\mu$. This is consistent with the fact that, in the elongated geometry of our condensates, the rescaled width remains almost constant during the expansion.\nAnother way to understand this is to note that the in-trap width is proportional to $\\xi_0 \/R_\\perp$, and hence to $1\/\\mu$, and this scaling survives after long expansion times, even deep within the ballistic regime where length ratios become frozen.\nThe dashed line is a linear fit to the GP points, including the limiting case of an infinite condensate at $1\/\\mu=0$. Figure~\\ref{fig:mu_scaling} shows that the experimental data are in good agreement with the GP predictions, especially for the largest condensates, where the vortex signal-to-noise ratio is the largest.  \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\linewidth]{vortex-core-fig4.pdf}\n\\caption{Depth (top) and width (bottom) of the depletion produced by a vortex in the residual column density for condensates of different $\\mu$. The black $+$ symbols are obtained from GP simulations for an expansion time $\\tau=\\omega_\\perp t= 70$, corresponding to $120$~ms; the point at $1\/\\mu=0$ is the limit of an infinitely large condensate, where both quantities must vanish.  The dashed line in the bottom panel is the linear law $\\sigma\/R_\\perp \\sim \\xi_0 \/R_\\perp \\propto 1\/\\mu$ predicted by GP theory in  the TF scaling regime.  Points with error bars are the experimental data. The expansion time is $t=150$~ms (red), $t=120$~ms (green and orange) and $t=100$~ms (blue); varying $t$ in this range would change the vertical position of the experimental data by a negligible amount of the order of $1\\%$.  The depth and width are calculated from Gaussian fits to both GP and experimental distributions of the residual column density by using the same procedure.   }\n\\label{fig:mu_scaling}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\linewidth]{vortex-core-fig5.pdf}\n\\caption{Residual column density after $150$~ms of free expansion for a condensate with $2 \\times 10^7$ atoms and $\\mu=33 \\hbar \\omega_\\perp$, containing a vortex. The inset shows the full residual column density in the $y$-$z$ plane. The quantity  $\\delta n (\\tilde{y},\\tilde{z})\/n^{\\rm TF}_{\\rm col} (0,\\tilde{z})$ is averaged in the direction $z$ within the rectangular box and the resulting values (blue points) are plotted in the main panel as a function of the rescaled coordinate $\\tilde{y}=y\/R_\\perp$, with $\\tilde{y}=0$ at the vortex position. The solid line is the same quantity, obtained with the same fitting procedure applied to the GP residual column density of a condensate with $\\mu=9.7 \\hbar \\omega_\\perp$, after linearly rescaling its width according to  the dashed line of Fig.~\\ref{fig:mu_scaling}, and reducing its depth to match the experimental value. }\n\\label{fig:profile}\n\\end{figure}\n\nFor the case of vortex depth, the GP theory does not provide any simple scaling law to compare with the experimental results considered here. The reason is that, as discussed in the previous section, the visibility of the vortex in the residual column density exhibits a nontrivial dependence on the expansion time, associated with the crossover from the mean-field dominated early stages of expansion to the later ballistic expansion dynamics. Eventually, for large $t$, the normalized depth saturates at a value weakly dependent on $\\mu$ (see Fig.~\\ref{fig:depthwidth}). The experimental points lie in a range fully compatible with a smooth interpolation from the GP results down to the infinite condensate limit, in the sense that any reasonable interpolating function would clearly pass through most of the experimental points, within the experimental uncertainties. \n\nIn Fig.~\\ref{fig:profile}, we show an example of vortex profile in a condensate with $2 \\times 10^7$ atoms and chemical potential $\\mu_{\\rm expt}=33 \\hbar \\omega_\\perp$, after an expansion time $t=150$~ms. The full residual column density $\\delta n (\\tilde{y},\\tilde{z})$ is plotted in the inset. The quantity $\\delta n (\\tilde{y},\\tilde{z})\/n^{\\rm TF}_{\\rm col} (0,\\tilde{z})$ is averaged in the $z$ direction within the rectangular box, and the resulting $\\delta n (\\tilde{y})\/n^{\\rm TF}_{\\rm col} (0)$ is shown in the main panel of the figure as a function of $\\tilde{y}$. In order to compare the experimental data with GP theory we proceed as follows. We first check that the shape of the vortex core in the residual column density of GP simulations with different values of $\\mu$ is the same up to a rescaling of the width and the depth as in Fig.~\\ref{fig:mu_scaling}, except for small fluctuations in the tails, which are expected to become negligible for large $\\mu$. This implies that the GP profile of $\\delta n (\\tilde{y})\/n^{\\rm TF}_{\\rm col} (0)$ for the experimental chemical potential $\\mu_{\\rm expt}=33 \\hbar \\omega_\\perp$ should be the same as for the GP simulation for $\\mu_{\\rm GP}=9.7 \\hbar \\omega_\\perp$, after rescaling the width linearly with $\\mu$ (dashed line in Fig.~\\ref{fig:mu_scaling}). The solid line in Fig.~\\ref{fig:profile} is the resulting GP profile, where we fixed the depth to the experimental value.\nThere is good agreement between theory and experiment for the overall shape, including quantitative agreement for the width. The depth has good qualitative agreement if one considers that the experimental value lies within a range between the GP results for smaller $\\mu$ and the trivial limit for $\\mu \\to \\infty$, in a way that is compatible with any reasonable smooth interpolation as already shown in the top panel of Fig.~\\ref{fig:mu_scaling}.\n \nIt is worth noticing that the optical resolution in our experiments is not limiting the comparison with theory. To check this, we convolve the GP profile with a Gaussian having a width in the range $\\sigma_{\\rm res} \\sim 2 - 3\\ \\mu$m, corresponding to our optical resolution, and we find that the effects on the points in Figs.~\\ref{fig:mu_scaling}  and \\ref{fig:profile} are negligible (note that the vortex core in Fig.~\\ref{fig:profile} has a width $\\sigma \\sim 30\\ \\mu {\\rm m} \\gg \\sigma_{\\rm res}$). The fluctuations in the experimental data, which contribute to the error bars in Fig.~\\ref{fig:mu_scaling}, are dominated by photon shot-noise in the absorption images and by systematic spurious optical fringes which are not completely filtered out.   \n\nFinally, we note that thermal atoms are not visible in our samples, which means that the temperature of the condensates is significantly smaller than the critical temperature for Bose-Einstein condensation. Nevertheless, a certain number of thermal atoms is still expected to be present in the trapped condensate, and some of them can be confined within the vortex core \\cite{Coddington04}. These atoms should not be present in the vortex core after the expansion, since their kinetic energy is sufficient to separate them from the expanding condensate, leaving an empty vortex core.\nIn any case, our observations suggest that the effect of thermal atoms on the {\\it in situ} vortex core is limited.\nIn fact, the good agreement that we find with GP theory (valid at zero temperature) is an indication that, if thermal atoms are present, their effects on the shape, width and depth of the vortex are negligible within the uncertainties of our experiments. \n\n\\section{Conclusion}\n\nIn summary, we have shown that quantized vortex filaments can be observed by optical means in 3D Bose-Einstein condensates of weakly interacting ultracold atoms, at a level of accuracy which is enough to allow for a direct comparison with the predictions of the Gross-Pitaevskii theory for the width, depth, and overall shape of the vortex core. We found good agreement between theory and experiment. We have performed experiments with large condensates of sodium atoms and compared the results to those obtained in numerical simulations. In order to make the vortex visible we let the condensate expand for a long time. The expansion dynamics were included in the numerical simulations. We have shown that Thomas-Fermi scaling laws, valid for large elongated condensates, can be efficiently used to relate the observed features after expansion to the structure of the vortex core in the initially trapped condensate.   \\\\\n\n\\bigskip\n\n{\\bf Acknowledgments:}\nWe dedicate this paper to Lev P. Pitaevskii in celebration of his 85th birthday.\nNo words can express our gratitude for the times spent working alongside him and, of course, for his pioneering contributions to physics itself.\nThis work is supported by Provincia Autonoma di Trento and by QuantERA ERA-NET cofund project NAQUAS. \\\\\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nKeeping information secure has become a major concern with the advancement in technology. In this work, the information theory aspect of security is analyzed, as entropies are used to measure security. The system also incorporates some traditional ideas surrounding cryptography, namely Shannon's cipher system and adversarial attackers in the form of eavesdroppers. In cryptographic systems, there is usually a message in plaintext that needs to be sent to a receiver. In order to secure it, the plaintext is encrypted so as to prevent eavesdroppers from reading its contents. This ciphertext is then transmitted to the receiver. Shannon's cipher system (mentioned by Yamamoto\\cite{shannon1_yamamoto}) incorporates this idea. The definition of Shannon's cipher system has been discussed by Hanawal and Sundaresan~\\cite{hanawal_shannon}. In Yamamoto's~\\cite{shannon1_yamamoto} development on this model, a correlated source approach is introduced. This gives an interesting view of the problem, and is depicted in Figure~\\ref{fig:yamamoto_shannoncipher}. Correlated source coding incorporates the lossless compression of two or more correlated data streams. Correlated sources have the ability to decrease the bandwidth required to transmit and receive messages because a syndrome (compressed form of the original message) is sent across the communication links instead of the original message. A compressed message has more information per bit, and therefore has a higher entropy because the transmitted information is more unpredictable. The unpredictability of the compressed message is also beneficial in terms of securing the information. \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics [scale = 0.7]{yamamoto_shannoncipher.pdf}\n\\caption{Yamamoto's development of the Shannon Cipher System}\n\\label{fig:yamamoto_shannoncipher}\n\\end{figure}\n\nThe source sends information for the correlated sources, $X$ and $Y$ along the main transmission channel. A key $W_k$, is produced and used by the encoder when producing the ciphertext. The wiretapper has access to the transmitted codeword, $W$. The decoded codewords are represented by $\\widehat{X}$ and $\\widehat{Y}$. In Yamamoto's scheme the security level was also focused on and found to be $\\frac{1}{K} H(X^K,Y^K|W)$ (i.e. the joint entropy of $X$ and $Y$ given $W$, where $K$ is the length of $X$ and $Y$) when $X$ and $Y$ have equal importance, which is in accordance with traditional Shannon systems where the security is measured by the equivocation. When one source is more important than the other then the security level is measured by the pair of the individual uncertainties $(\\frac{1}{K} H(X^K|W), \\frac{1}{K} H(Y^K|W))$. \n\nIn practical communication systems links are prone to eavesdropping and as such this work incorporates wiretapped channels, i.e. channels where an eavesdropper is present.\n\nThere are specific kinds of wiretapped channels that have been developed. The mathematical model for this Wiretap Channel is given by Rouayheb \\textit{et al.}~\\cite{ref12_rouayheb_soljanin}, and can be explained as follows: the channel between a transmitter and receiver  is error-free and can transmit $n$ symbols $Y=(y_1,\\ldots,y_n)$ from which $\\mu$ bits can be observed by the eavesdropper and the maximum secure rate can be shown to equal $n-\\mu$ bits. The security aspect of wiretap networks have been looked at in various ways by Cheng \\textit{et al.} \\cite{ref21_cheng_yeung}, and Cai and Yeung \\cite{ref11_cai_yeung}, emphasising that it is of concern to secure these type of channels. \n\nVillard and Piantanida \\cite{pablo_secure_multiterminal} also look at correlated sources and wireap networks: A source sends information to the receiver and an eavesdropper has access to information correlated to the source, which is used as side information. There is a second encoder that sends a compressed version of its own correlation observation of the source privately to the receiver. Here, the authors show that the use of correlation decreases the required communication rate and increases secrecy. Villard \\textit{et al.} \\cite{pablo_secure_transmission_receivers} explore this side information concept further where security using side information at the receiver and eavesdropper is investigated. Side information is generally used to assist the decoder to determine the transmitted message. An earlier work involving side information is that by Yang \\textit{et al.}~\\cite{feedback_yang}. The concept can be considered to be generalised in that the side information could represent a source. It is an interesting problem when one source is more important and Hayashi and Yamamoto\\cite{Hayashi_coding} consider it in another scheme, where only $X$ is secure against wiretappers and $Y$ must be transmitted to a legitimate receiver. They develop a security criterion based on the number of correct guesses of a wiretapper to retrieve a message. In an extension of the Shannon cipher system, Yamamoto \\cite{coding_yamamoto} investigated the secret sharing communication system. \n\nIn this case, we generalise a model for correlated sources across a channel with an eavesdropper and the security aspect is explored by quantifying the information leakage and reducing the key lengths when incorporating Shannon's cipher system. \n\nThis paper initially describes a two correlated source model across wiretapped links, which is detailed in Section II. In Section III, the information leakage is investigated and proven for this two correlated source model. The information leakage is quantified to be the equivocation subtracted from the total obtained uncertainty. In Section IV the two correlated sources model is looked at according to Shannon's cipher system. The notation contained in the tables will be clarified in the following sections. The proofs for this Shannon cipher system aspect are detailed in Section V. Section VI details the extension of the two correlated source model where multiple correlated sources in a network scenario is investigated. There are two subsections here; one quantifying information leakage for the Slepian-Wolf scenario and the other incorporating Shannon's cipher system where key lengths are minimized and a masking method to save on keys is presented. Section VII explains how the models detailed in this paper are a generalised model of Yamamoto's~\\cite{shannon1_yamamoto} model, and further offers comparison to other models. The future work for this research is detailed in Section VIII and the paper is concluded in Section IX.\n\n\n\n\n\n\\section{Model}\n\nThe independent, identically distributed (i.i.d.) sources $X$ and $Y$ are mutually correlated random variables, depicted in Figure~\\ref{fig:new_model}. The alphabet sets for sources $X$ and $Y$ are represented by $\\mathcal{X}$ and $\\mathcal{Y}$ respectively. Assume that ($X^K$, $Y^K$) are encoded into two syndromes ($T_{X}$ and $T_{Y}$). We can write $T_X = (V_X, V_{CX})$ and $T_Y = (V_Y, V_{CY})$ where $T_X$ and $T_Y$ are the syndromes of $X$ and $Y$. Here, $T_X$ and $T_Y$ are characterised by $(V_X, V_{CX})$ and $(V_Y, V_{CY})$ respectively. The Venn diagram in Figure \\ref{fig:new_venn2}  easily illustrates this idea where it is shown that $V_X$ and $V_Y$ represent the private information of sources $X$ and $Y$ respectively and $V_{CX}$ and $V_{CY}$ represent the common information between $X^K$ and $Y^K$ generated by $X^K$ and $Y^K$ respectively. \n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics [scale = 0.7]{new_model.pdf}\n\\caption{Correlated source coding for two sources}\n\\label{fig:new_model}\n\\end{figure}\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics [scale = 0.7]{new_venn2.pdf}\n\\caption{The relation between private and common information}\n\\label{fig:new_venn2}\n\\end{figure}\n\nThe correlated sources $X$ and $Y$ transmit messages (in the form of syndromes) to the receiver along wiretapped links. The decoder determines $X$ and $Y$ only after receiving all of $T_X$ and $T_Y$. The common information between the sources are transmitted through the portions $V_{CX}$ and $V_{CY}$. In order to decode a transmitted message, a source's private information and both common information portions are necessary. This aids in security as it is not possible to determine, for example $X$ by wiretapping all the contents transmitted along $X$'s channel only. This is different to Yamamoto's~\\cite{shannon1_yamamoto} model as here the common information consists of two portions.  The aim is to keep the system as secure as possible and these following sections show how it is achieved by this new model. \n\nWe assume that the function $F$ is a one-to-one process with high probability, which means based on $T_X$ and $T_Y$ we can retrieve $X^K$ and $Y^K$ with minimal error. Furthermore, it reaches the Slepian-Wolf bound, $H(T_X, T_Y)=H(X^K,Y^K)$. Here, we note that the lengths of $T_X$ and $T_Y$ are not fixed, as it depends on the encoding process and nature of the Slepian-Wolf codes. The process is therefore not ideally one-to-one and reversible and is another difference between our model and Yamamoto's~\\cite{shannon1_yamamoto} model.\n\nThe code described in this section satisfies the following inequalities for $\\delta > 0$ and sufficiently large $K$.\n\n\\begin{eqnarray}\nPr \\{X^K \\neq G(V_X, V_{CX}, V_{CY})\\} \\le \\delta\n\\label{x_prob}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nPr \\{Y^K \\neq G(V_Y, V_{CX}, V_{CY})\\} \\le \\delta\n\\label{y_prob}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nH(V_X, V_{CX}, V_{CY})\\le H(X^K) + \\delta \n\\label{x_entropy}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nH(V_Y, V_{CX}, V_{CY})\\le H(Y^K) + \\delta \n\\label{y_entropy}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nH(V_X, V_Y, V_{CX}, V_{CY})\\le H(X^K,Y^K) + \\delta \n\\label{xy_entropy}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nH(X^K|V_X, V_Y) \\geq H(V_{CX}) + H(V_{CY}) - \\delta \n\\label{H_inequality1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nH(X^K|V_{CX}, V_{CY}) \\geq H(V_X) + H(V_{CY}) - \\delta \n\\label{H_inequality2}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nH(X^K|V_{CX}, V_{CY}, V_Y) \\geq H(V_X) + H(V_{CY}) - \\delta \n\\label{H_inequality3}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nH(V_{CX}) + H(V_X) - \\delta \\le H(X^K|V_{CY}, V_{Y})  \\nonumber\n\\\\ \\le H(X) - H(V_{CY}) + \\delta\n\\label{H_inequality4}\n\\end{eqnarray}\n\\\\\nwhere $G$ is a function to define the decoding process at the receiver. It can intuitively be seen from \\eqref{x_entropy} and \\eqref{y_entropy} that $X$ and $Y$ are recovered from the corresponding private information and the common information produced by $X^K$ and $Y^K$. Equations \\eqref{x_entropy}, \\eqref{y_entropy} and \\eqref{xy_entropy} show that the private information and common information produced by each source should contain no redundancy. \nIt is also seen from \\eqref{H_inequality2} and \\eqref{H_inequality3} that $V_Y$ is independent of $X^K$ asymptotically. Here, $V_X$, $V_Y$, $V_{CX}$ and $V_{CY}$ are disjoint, which ensures that there is no redundant information sent to the decoder. \n\nTo recover $X$ the following components are necessary: $V_X$, $V_{CX}$ and $V_{CY}$. This comes from the property that $X^K$ cannot be derived from $V_X$ and $V_{CX}$ only and part of the common information between $X^K$ and $Y^K$ is produced by $Y^K$.\n\nYamamoto~\\cite{shannon1_yamamoto} proved that a common information between $X^K$ and $Y^K$ is represented by the mutual information $I(X;Y)$. Yamamoto~\\cite{shannon1_yamamoto} also defined two kinds of common information. The first common information is defined as the rate of the attainable minimum core $V_C$ (i.e. $V_{CX}, V_{CY}$ in this model) by removing each private information, which is independent of the other information, from ($X^K$, $Y^K$) as much as possible. The second common information is defined as the rate of the attainable maximum core $V_C$ such that if we lose $V_C$ then the uncertainty of $X$ and $Y$ becomes $H(V_C)$. Here, we consider the common information that $V_{CX}$ and $V_{CY}$ represent.\n\nWe begin demonstrating the relationship between the common information portions by constructing the prototype code ($W_X$, $W_Y$, $W_{CX}$, $W_{CY}$) as per Lemma 1. \n\n\\textit{Lemma 1: For any $\\epsilon_0 \\geq 0$ and sufficiently large $K$, there exits a code $W_X = F_X(X^K)$, $W_Y = F_Y(Y^K)$, $W_{CX} = F_{CX}(X^K)$, $W_{CY} = F_{CY}(Y^K)$, $\\widehat{X}^K,\\widehat{Y}^K = G(W_X, W_Y, W_{CX}, W_{CY})$, where $W_X \\in I_{M_X}$, $W_Y \\in I_{M_Y}$, $W_{CX} \\in I_{M_{CX}}$, $W_{CY} \\in I_{M_{CY}}$ for $I_{M_{\\alpha}}$, which is defined as $\\{0, 1, \\ldots, M_{\\alpha} - 1\\}$, that satisfies},\n\n\\begin{eqnarray}\nPr\\{\\widehat{X}^K, \\widehat{Y}^K \\neq X^K, Y^K\\} \\le \\epsilon\n\\label{lemma1_1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nH(X|Y) - \\epsilon_0 \\le \\frac{1}{K} H(W_X) \\le \\frac{1}{K} \\log M_X \\le H(X|Y) + \\epsilon_0\n\\label{lemma1_2}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nH(Y|X) - \\epsilon_0 \\le \\frac{1}{K} H(W_Y) \\le \\frac{1}{K} \\log M_Y \\le H(Y|X) + \\epsilon_0\n\\label{lemma1_3}\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n& & I(X;Y) - \\epsilon_0 \\le \\frac{1}{K} (H(W_{CX}) + H(W_{CY})) \\nonumber \\\\\n    & \\le & \\frac{1}{K} (\\log M_{CX} + \\log M_{CY}) \\le I(X;Y) + \\epsilon_0\n\\label{lemma1_4}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K|W_Y) \\geq H(X) - \\epsilon_0\n\\label{lemma1_5}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} H(Y^K|W_X) \\geq H(Y) - \\epsilon_0\n\\label{lemma1_6}\n\\end{eqnarray}\n\nWe can see that \\eqref{lemma1_2} - \\eqref{lemma1_4} mean\n\\begin{eqnarray}\n&& H(X,Y) - 3\\epsilon_0 \\le \\frac{1}{K} (H(W_X) + H(W_Y) + H(W_{CX}) \\nonumber \\\\\n& + & H(W_{CY})) \\nonumber \\\\\n& \\le & H(X,Y) + 3\\epsilon_0\n\\label{lemma1_7}\n\\end{eqnarray}\n\nHence from \\eqref{lemma1_1}, \\eqref{lemma1_7} and the ordinary source coding theorem, ($W_X$, $W_Y$, $W_{CX}$, $W_{CY}$) have no redundancy for sufficiently small $\\epsilon_0 \\geq 0$. It can also be seen that $W_X$ and $W_Y$ are independent of $Y^K$ and $X^K$ respectively. \n\n\\begin{proof}[Proof of Lemma 1]\n\nAs seen by Slepian and Wolf, mentioned by Yamamoto\\cite{shannon1_yamamoto} there exist $M_X$ codes for the $P_{Y|X}(y|x)$ DMC (discrete memoryless channel) and $M_Y$ codes for the $P_{X|Y}(x|y)$ DMC. The codeword sets exist as $C^X_i$ and $C^Y_j$, where $C^X_i$ is a subset of the typical sequence of $X^K$ and $C^Y_j$ is a subset of the typical sequence of $Y^K$.\nThe encoding functions are similar, but we have created one decoding function as there is one decoder at the receiver:\n\n\\begin{eqnarray}\nf_{Xi}:I_{M_{CX}} \\rightarrow C^X_i\n\\label{lemma1proof_1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nf_{Yj}:I_{M_{CY}} \\rightarrow C^Y_j\n\\label{lemma1proof_1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\ng: X^K, Y^K \\rightarrow I_{M_{CX}} \\times I_{M_{CY}}\n\\label{lemma1proof_2}\n\\end{eqnarray}\n\nThe relations for $M_X$, $M_Y$ and the common information remain the same as per Yamamoto's and will therefore not be proven here. \n\nIn this scheme, we use the average $(V_{CX}, V_{X},V_{CY}, V_{Y})$ transmitted for many codewords from $X$ and $Y$. Thus, at any time either $V_{CX}$ or $V_{CY}$ is transmitted. Over time, the split between which common information portion is transmitted is determined and the protocol is prearranged accordingly. Therefore all the common information is either transmitted as $l$ or $m$, and as such Yamamoto's encoding and decoding method may be used. \n\nAs per Yamamoto's method the code does exist and that $W_X$ and $W_Y$ are independent of $Y$ and $X$ respectively, as shown by Yamamoto\\cite{shannon1_yamamoto}. \n\n\n\\end{proof}\n\nThe common information is important in this model as the sum of $V_{CX}$ and $V_{CY}$ represent a common information between the sources. The following theorem holds for this common information:\n\\\\\n\\textit{Theorem 1:}\n\\begin{eqnarray}\n\\frac{1}{K} [H (V_{CX}) + H (V_{CY})] = I (X;Y) \n\\label{theorem1}\n\\end{eqnarray}\n\nwhere $V_{CX}$ is the common portion between $X$ and $Y$ produced by $X^K$ and $V_{CY}$ is the common portion between $X$ and $Y$ produced by $Y^K$. It is noted that the \\eqref{theorem1} holds asymptotically, and does not hold with equality when $K$ is finite. Here, we show the approximation when $K$ is infinitely large.\nThe private portions for $X^K$ and $Y^K$ are represented as $V_X$ and $V_Y$ respectively. As explained in Yamamoto's~\\cite{shannon1_yamamoto} Theorem 1, two types of common information exist (the first is represented by $I(X;Y)$ and the second by $\\text{min} (H(X^K), H(Y^K))$. We will develop part of this idea to show that the sum of the common information portions produced by $X^K$ and $Y^K$ in this new model is represented by the mutual information between the sources. \n\n\\begin{proof}[Proof of Theorem 1]\nThe first part is to prove that $H(V_{CX}) + H(V_{CY}) \\geq I(X;Y)$, and is done as follows. \nWe weaken the conditions \\eqref{x_prob} and \\eqref{y_prob} to\n\\begin{eqnarray}\n\\text{Pr }\\{X^K,Y^K \\neq G_{XY} (V_X, V_Y, V_{CX}, V_{CY}\\}) \\le \\delta_1\n\\label{weakenedxy_prob}\n\\end{eqnarray}\n\nFor any ($V_X$,$V_Y$, $V_{CX}$, $V_{CY}$) $\\in C(3\\epsilon_0)$ (which can be seen from \\eqref{lemma1_7}), we have from \\eqref{weakenedxy_prob} and the ordinary source coding theorem that\n\n\\begin{eqnarray}\nH(X^K,Y^K) - \\delta_1 &\\le & \\frac{1}{K} H(V_X, V_Y, V_{CX}, V_{CY}) \\nonumber \\\\\n\t& \\le & \\frac{1}{K} [H(V_X) + H(V_Y) + H (V_{CX})  \\nonumber \\\\\n\t& + & H(V_{CY})]\n\\label{theorem1proof_1}\n\\end{eqnarray}\n\nwhere $\\delta_1 \\rightarrow 0$ as $\\delta \\rightarrow 0$. From Lemma 1,\n\\begin{eqnarray}\n\\frac{1}{K} H(V_Y|X^K) \\geq \\frac{1}{K} H(V_Y) - \\delta\n\\label{theorem1proof_2}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} H(V_X|Y^K) \\geq \\frac{1}{K} H(V_X) - \\delta\n\\label{theorem1proof_3}\n\\end{eqnarray}\n\nFrom \\eqref{theorem1proof_1} - \\eqref{theorem1proof_3},\n\\begin{eqnarray}\n\\frac{1}{K} [H(V_{CX}) + H(V_{CY})] &\\ge & H(X,Y) - \\frac{1}{K} H(V_X)  \\nonumber \\\\\n& - & \\frac{1}{K} H(V_Y)  - \\delta_1\t\t\t\t\t\t\t\\nonumber \\\\\n& \\geq & H(X,Y) - \\frac{1}{K} H(V_X|Y)  \t\t\\nonumber \\\\\n& - & \\frac{1}{K} H(V_Y|X) - \\delta_1 - 2\\delta\n\\label{theorem1proof_4}\n\\end{eqnarray}\n\nOn the other hand, we can see that\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K, V_Y) \\le  H(X,Y) + \\delta\n\\label{theorem1proof_5}\n\\end{eqnarray}\n\nThis implies that \n\\begin{eqnarray}\n\\frac{1}{K} H(V_Y|X^K) \\le H(Y|X) + \\delta  \n\\label{theorem1proof_6}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n\\frac{1}{K} H(V_X|Y^K) \\le H(X|Y) + \\delta \n\\label{theorem1proof_7}\n\\end{eqnarray}\n\nFrom \\eqref{theorem1proof_4}, \\eqref{theorem1proof_6} and \\eqref{theorem1proof_7} we get\n\\begin{eqnarray}\n\\frac{1}{K} [H(V_{CX}) + H(V_{CY})] & \\ge & H(X,Y) - H(X|Y) - H(Y|X)  \\nonumber \\\\\n& - & \\delta_1 - 4\\delta\t\\nonumber \\\\\n& = & I(X;Y) - \\delta_1 - 4\\delta\n\\label{theorem1proof_8}\n\\end{eqnarray}\n\nIt is possible to see from \\eqref{lemma1_4} that $H(V_{CX}) +  H(V_{CY}) \\le I(X;Y)$. From this result,  \\eqref{lemma1proof_2} and \\eqref{theorem1proof_8}, and as $\\delta_1 \\rightarrow 0$ and $\\delta \\rightarrow 0$ it can be seen that\n\\begin{eqnarray}\n\\frac{1}{K} [H (V_{CX} + H(V_{CY})] = I(X;Y)\n\\end{eqnarray}\n\\end{proof}\n \nThis model can cater for a scenario where a particular source, say $X$ needs to be more secure than $Y$ (possibly because of eavesdropping on the $X$ channel). In such a case, the $\\frac{1}{K} H(V_{CX})$ term in \\eqref{theorem1proof_8} needs to be as high as possible. When this uncertainty is increased then the security of $X$ is increased. Another security measure that this model incorporates is that $X$ cannot be determined from wiretapping only $X$'s link. \n \n\n\\section{Information Leakage}\nIn order to determine the security of the system, a measure for the amount of information leaked has been developed. This is a new notation and quantification, which emphasizes the novelty of this work. The obtained information and total uncertainty are used to determine the leaked information. Information leakage is indicated by $L_{\\mathcal{Q}}^\\mathcal{P}$. Here $\\mathcal{P}$ indicates the source\/s for which information leakage is being quantified, $\\mathcal{P} = \\{S_1, \\ldots, S_n\\}$ where $n$ is the number of sources (in this case, $n = 2$). Further, $\\mathcal{Q}$ indicates the syndrome portion that has been wiretapped, $\\mathcal{Q} = \\{V_1, \\ldots, V_m\\}$ where $m$ is the number of codewords (in this case, $m = 4$).\n\nThe information leakage bounds are as follows:\n\\begin{eqnarray}\nL_{V_X,V_Y}^{X^K} \\le H(X^K) - H(V_{CX}) - H(V_{CY}) + \\delta\n\\label{L_inequality1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nL_{V_{CX},V_{CY}}^{X^K} \\le H(X^K) - H(V_X) - H(V_{CY}) + \\delta\n\\label{L_inequality2}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nL_{V_{CX},V_{CY},V_Y}^{X^K} \\le H(X^K) - H(V_X) - H(V_{CY}) + \\delta\n\\label{L_inequality3}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&& H(V_{CY}) - \\delta \\le L_{V_{Y},V_{CY}}^{X^K} \\nonumber\n\\\\ & \\le & H(X^K) - H(V_{CX}) - H(V_X) + \\delta \n\\label{L_inequality4}\n\\end{eqnarray}\n \nHere, $V_Y$ is private information of source $Y^K$ and is independent of $X^K$\nand therefore does not leak any information about $X^K$, shown\nin \\eqref{L_inequality2} and \\eqref{L_inequality3}. Equation \\eqref{L_inequality4} gives an indication of the minimum and maximum amount of leaked information for the interesting case where a syndrome has been wiretapped and its information leakage on the alternate source is quantified. The outstanding common information component is the maximum information that can be leaked. For this case, the common information $V_{CX}$ and $V_{CY}$ can thus consist of added\nprotection to reduce the amount of information leaked. These bounds developed in \\eqref{L_inequality1} - \\eqref{L_inequality4} are proven in the next section.\n\nThe proofs for the above mentioned information leakage inequalities are now detailed. First, the inequalities in \\eqref{H_inequality1} - \\eqref{H_inequality4} will be proven, so as to prove that the information leakage equations hold. \\\\\n\\begin{proof}[Lemma 2]\nThe code ($V_X$, $V_{CX}$, $V_{CY}$, $V_Y$) defined at the beginning of Section I, describing the model and \\eqref{x_prob} - \\eqref{xy_entropy} satisfy \\eqref{H_inequality1} - \\eqref{H_inequality4}. Then the information leakage bounds are given by \\eqref{L_inequality1} - \\eqref{L_inequality4}.\n\\\\\\\\ \\textit{Proof for \\eqref{H_inequality1}}:\n\\begin{eqnarray}\n& & \\frac{1}{K} H(X^K|V_X,V_Y) \t\t\t\t\t\t\t\t\t\t\t\t\\nonumber\\\\\n& = & \\frac{1}{K} [H(X^K,V_X,V_Y) - H(V_X,V_Y)] \t\t\t\t\t\t\t\\nonumber\\\\\n& = & \\frac{1}{K} [H(X^K,V_Y) - H(V_X,V_Y)] \t\t\t\t\t\t\t\t\\label{lemma2_ref1}\\\\\n& = & \\frac{1}{K} [H(X^K|V_Y) + I(X^K;V_Y) + H(V_Y|X^K)] \t\t\t\t\t\t\t\\nonumber\\\\\n& & - \\frac{1}{K} [H(V_X|V_Y) + I(V_X;V_Y) + H(V_Y|V_X)] \t\t\t\t\t\t\\nonumber\\\\\n& = & \\frac{1}{K} [H(X^K|V_Y) + H(V_Y|X^K) - H(V_X|V_Y)\t\t\t\t\t\t\t\\nonumber\\\\\n& & - H(V_Y|V_X)]\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\nonumber\\\\\n& = & \\frac{1}{K} [H(X^K) + H(V_Y) - H(V_X) - H(V_Y)]\t\t\t\t\t\t\t\\label{lemma2_ref2}\\\\ \n& = & \\frac{1}{K} [H(X^K) - H(V_X)]\t\t\t\t\t\t\t\t\t\t\t\\nonumber\\\\\n& \\geq & \\frac{1}{K} [H(V_X) + H(V_{CX}) + H(V_{CY}) - H(V_X)]\t- \\delta\t\t\t\\nonumber\\\\\n& = & \\frac{1}{K} [H(V_{CX}) + H(V_{CY})] -\\delta\t\t\t\t\t \t\t\t\n\\label{lemma2_part1}\n\\end{eqnarray}\n\nwhere \\eqref{lemma2_ref1} holds because $V_X$ is a function of $X$ and \\eqref{lemma2_ref2} holds because $X$ is independent of $V_Y$ asymptotically and $V_X$ is independent of $V_Y$ asymptotically.\n\nFor the proofs of \\eqref{H_inequality2} and \\eqref{H_inequality3}, the following simplification for $H(X|V_{CY})$ is used:\n\\begin{eqnarray}\nH(X^K|V_{CY}) & = & H(X^K,Y^K) - H(V_{CY}) \\nonumber \\\\\n& = & H(X^K) + H(V_{CY}) - I(X; V_{CY}) - H(V_{CY}) \\nonumber \\\\\n& = & H(X^K) + H(V_{CY}) - H(V_{CY}) - H(V_{CY}) \\nonumber \\\\\n& + & {\\delta}_1 \\label{new_55} \\\\\n& = & H(X^K) - H(V_{CY}) = {\\delta}_1\n\\label{simpli}\n\\end{eqnarray}\n\nwhere $I(X; V_{CY})$ approximately equal to $H(V_{CY})$ in \\eqref{new_55} can be seen intuitively from the Venn diagram in Figure \\ref{fig:new_venn2}. Since it is an approximation, ${\\delta}_1$, which is smaller than $\\delta$ in the proofs below has been added to cater for the tolerance. \n\\\\\\\\  \\textit{Proof for \\eqref{H_inequality2}}:\n\\begin{eqnarray}\n& & \\frac{1}{K} H(X^K|V_{CX},V_{CY})\t\t\t\t\t\\nonumber\\\\\n& = & \\frac{1}{K} [H(X^K,V_{CX},V_{CY}) - H(V_{CX},V_{CY})]\t\t\t\t\t\t\\nonumber\\\\\n& = & \\frac{1}{K} [H(X^K,V_{CY}) - H(V_{CX},V_{CY})] \t\t\t\t\t\t\t\t\\label{lemma2_ref3}\\\\\n& = & \\frac{1}{K} [H(X^K) - H(V_{CY}) + I(X;V_{CY}) + H(V_{CY}|X^K)] \t\t\t\t\\nonumber\\\\\n& & - \\frac{1}{K} [H(V_{CX}|V_{CY}) + I(V_{CX};V_{CY}) + H(V_{CY}|V_{CX})] \\nonumber \\\\\n& + & \\delta_1\t\t\\nonumber\\\\\t\n& = & \\frac{1}{K} [H(X^K) - H(V_{CY}) + H(V_{CY})- H(V_{CX}) - H(V_{CY})]\t \\nonumber \\\\\n& + & \\delta_1\t\t\t\t\t\\label{lemma2_ref4} \\\\\n& = & \\frac{1}{K} [H(X^K) - H(V_{CY}) - H(V_{CX})]\t\t\t\t\t\t\t\t+ \\delta_1\t\t\t\\nonumber\\\\\n& \\geq & \\frac{1}{K} [H(V_X) + H(V_{CX}) + H(V_{CY}) - H(V_{CY}) - H(V_{CX})] -\\delta\t\t\t\\nonumber\\\\\n& = & \\frac{1}{K} H(V_X) + \\delta_1 -\\delta\n\\label{lemma2_part2}\n\\end{eqnarray}\n\nwhere \\eqref{lemma2_ref3} holds because $V_{CX}$ is a function of $X^K$ and \\eqref{lemma2_ref4} holds because $X$ is independent of $V_{CY}$ asymptotically and $V_{CX}$ is independent of $V_{CY}$ asymptotically. \n\nThe proof for $H(X|V_{CX},V_{CY},V_Y)$ is similar to that for $H(X|V_{CX},V_{CY})$, because $V_Y$ is independent of $X$.\n\\\\\\\\ \\textit{Proof for \\eqref{H_inequality3}}:\n\\begin{eqnarray}\n& & \\frac{1}{K} H(X^K|V_{CX},V_{CY},V_Y)\t\t\t\t\t\t\t\t\t\t\\nonumber\\\\\n& = & \\frac{1}{K} H(X^K|V_{CX},V_{CY})\n\\label{lemma2_ref5}\\\\\n& = & \\frac{1}{K} [H(X^K,V_{CX},V_{CY}) - H(V_{CX},V_{CY})]\t\t\t\t\t\t\\nonumber\\\\\n& = & \\frac{1}{K} [H(X^K,V_{CY}) - H(V_{CX},V_{CY})] \t\t\t\t\t\t\t\t\\label{lemma2_ref6}\\\\\n& = & \\frac{1}{K} [H(X^K) - H(V_{CY}) + I(X;V_{CY}) + H(V_{CY}|X^K)] \t\t\t\t\t\\nonumber\\\\\n& & - \\frac{1}{K} [H(V_{CX}|V_{CY}) + I(V_{CX};V_{CY}) + H(V_{CY}|V_{CX})]  \\nonumber \\\\\n& + & \\delta_1\t\t\\nonumber\\\\\t\n& = & \\frac{1}{K} [H(X^K) - H(V_{CY}) + H(V_{CY})- H(V_{CX}) \t\t\\nonumber\\\\\n& - &  H(V_{CY})] + \\delta_1\t\t\t\t\\label{lemma2_ref7}\\\\\n& = & \\frac{1}{K} [H(X^K) - H(V_{CY}) - H(V_{CX})]\t\t\t\t\t\t\t\t+ \\delta_1\t\t\t\\nonumber\\\\\n& \\geq & \\frac{1}{K} [H(V_X) + H(V_{CX}) + H(V_{CY}) - H(V_{CY}) \\nonumber\\\\\n& - & - H(V_{CX})]  - \\delta\t\t+ \\delta_1\t\\nonumber\\\\\n& = & \\frac{1}{K} H(V_X) -\\delta + \\delta_1\n\\label{lemma2_part3}\n\\end{eqnarray}\n\nwhere \\eqref{lemma2_ref5} holds because $V_Y$ and $X^K$ are independent, \\eqref{lemma2_ref6} holds because $V_{CX}$ is a function of $X^K$ and \\eqref{lemma2_ref7} holds because $X^K$ is independent of $V_{CY}$ asymptotically and $V_{CX}$ is independent of $V_{CY}$ asymptotically. \n\nFor the proof of \\eqref{H_inequality4}, we look at the following probabilities:\n\\begin{eqnarray}\n\\text{Pr} \\{V_X,V_{CX} \\neq G(T_X)\\} \\le \\delta\n\\label{lemma2_eqn1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\text{Pr} \\{V_Y,V_{CY} \\neq G(T_Y)\\} \\le \\delta\n\\label{lemma2_eqn2}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n& & \\frac{1}{K} H(X^K|T_Y)\t\t\t\t\t\t\t\t\t\t\t\\nonumber\\\\\n& \\le & \\frac{1}{K} H(X^K, V_{CY},V_Y)] + \\delta\t\t\t\t\t\\label{lemma2_ref8}\\\\\n& = & \\frac{1}{K} [H(X^K, V_{CY},V_{Y}) - H(V_{CY},V_{Y})] + \\delta\t\t\t\t\t\\nonumber\\\\\n& = & \\frac{1}{K} [H(X^K, V_{Y}) - H(V_{CY},V_{Y})] + \\delta\t\t\t\t\t\t\\label{lemma2_ref9}\\\\\n& = & \\frac{1}{K} [H(X^K|V_{Y}) + I(X^K;V_{Y}) + H(V_{Y}|X^K)] \t\t\t\t\t\\nonumber\\\\\n& & - \\frac{1}{K} [H(V_{CY}|V_{Y}) + I(V_{CY};V_{Y}) + H(V_{Y}|V_{CY})]  + \\delta\t\t\\nonumber\\\\\t\n& = & \\frac{1}{K} [H(X^K) + H(V_{Y})- H(V_{CY}) - H(V_{Y})]+ \\delta\t\\label{lemma2_ref10}\\\\\n& = & \\frac{1}{K} [H(X^K) - H(V_{CY})] + \\delta\t\t\t\t\t\t\t\t\t\t\t\n\\label{lemma2_part4.1}\n\\end{eqnarray}\n\nwhere \\eqref{lemma2_ref8} holds from \\eqref{lemma2_eqn2}, \\eqref{lemma2_ref9} holds because $V_{CY}$ and $V_Y$ are asymptotically independent. Furthermore, \\eqref{lemma2_ref10} holds because $V_{CY}$ and $V_{Y}$ are asymptotically independent and $X^K$ and $V_{Y}$ are asymptotically independent.\n\nFollowing a similar proof to those done above in this section, another bound for $H(X^K|V_{CY},V_Y)$ can be found as follows:\n\\begin{eqnarray}\n& & \\frac{1}{K} H(X^K|V_{CY},V_Y)\t\t\t\t\t\t\t\t\t\t\t\\nonumber\\\\\n& = & \\frac{1}{K} [H(X^K,V_{CY},V_{Y}) - H(V_{CY},V_{Y})]\t\t\t\t\t\\nonumber\\\\\n& = & \\frac{1}{K} [H(X^K,V_{Y}) - H(V_{CY},V_{Y})] \t\t\t\t\t\t\\label{lemma2_ref11}\\\\\n& = & \\frac{1}{K} [H(X^K|V_{Y}) + I(X^K;V_{Y}) + H(V_{Y}|X)] \t\t\t\t\t\\nonumber\\\\\n& & - \\frac{1}{K} [H(V_{CY}|V_{Y}) + I(V_{CY};V_{Y}) + H(V_{Y}|V_{CY})] \t\t\\nonumber\\\\\t\n& = & \\frac{1}{K} [H(X^K) + H(V_{Y})- H(V_{CY}) - H(V_{Y})]\t\t\t\t\\label{lemma2_ref12}\\\\\n& = & \\frac{1}{K} [H(X^K) - H(V_{CY})]\t\\nonumber\\\\\n& \\geq & \\frac{1}{K} [H(V_X) + H(V_{CX}) + H(V_{CY}) - H(V_{CY})]\t- \\delta \\nonumber\\\\\n& = & \\frac{1}{K} [H(V_X) + H(V_{CX})]\t- \\delta\n\\label{lemma2_part4.2}\n\\end{eqnarray}\n\nwhere \\eqref{lemma2_ref11} and \\eqref{lemma2_ref12} hold for the same reason as \\eqref{lemma2_ref9} and \\eqref{lemma2_ref10} respectively. \n\nSince we consider the information leakage as the total information obtained subtracted from the total uncertainty, the following hold for the four cases considered in this section:\n\n\\begin{eqnarray}\nL_{V_X,V_Y}^{X^K} & = & H(X^K) - H(X^K|V_X,V_Y) \t\\nonumber\\\\ \n& \\le & H(X^K) - H(V_{CX}) - H(V_{CY}) + \\delta\n\\label{Lemma2_proof_inequality1}\n\\end{eqnarray}\nwhich proves \\eqref{L_inequality1}.\n\n\\begin{eqnarray}\nL_{V_{CX},V_{CY}}^{X^K} & = & H(X^K) - H(X^K|V_{CX},V_{CY}) \\nonumber\n\\\\ & \\le & H(X^K) - H(V_X) + \\delta\n\\label{Lemma2_proof_inequality2}\n\\end{eqnarray}\nwhich proves \\eqref{L_inequality2}.\n\n\\begin{eqnarray}\nL_{V_{CX},V_{CY},V_Y}^{X^K} & = & H(X^K) - H(X^K|V_{CX},V_{CY},V_Y)  \\nonumber\n\\\\ & \\le & H(X^K) - H(V_X) + \\delta\n\\label{Lemma2_proof_inequality3}\n\\end{eqnarray}\nwhich proves \\eqref{L_inequality3}.\n\nThe two bounds for $H(V_{CY},V_Y)$ are given by \\eqref{lemma2_part4.1} and \\eqref{lemma2_part4.2}. \nFrom \\eqref{lemma2_part4.1}:\n\n\\begin{eqnarray}\nL_{V_{Y},V_{CY}}^{X^K} & \\geq & H(X^K) - [H(X) - H(V_{CY}) + \\delta] \\nonumber \\\\\n& \\geq & H(V_{CY}) - \\delta\n\\label{Lemma2_proof_inequality4.1}\n\\end{eqnarray}\n\nand from \\eqref{lemma2_part4.2}:\n\\begin{eqnarray}\nL_{V_{Y},V_{CY}}^{X^K} & \\le & H(X^K) - \\left (H(V_X) + H(V_{CX})  - \\delta \\right)  \\nonumber \\\\\n& \\le & H(X^K) - H(V_X) - H(V_{CX}) + \\delta \n\\label{Lemma2_proof_inequality4.2}\n\\end{eqnarray}\n\n\nCombining these results from \\eqref{Lemma2_proof_inequality4.1} and \\eqref{Lemma2_proof_inequality4.2} gives \\eqref{L_inequality4}.\n\\end{proof}\n\n\\section{Shannon's Cipher System}\n\nHere, we discuss Shannon's cipher system for two independent correlated sources (depicted in Figure \\ref{fig:shannon_cipher_2sources}). The two source outputs are i.i.d random variables $X$ and $Y$, taking on values in the finite sets $\\mathcal{X}$ and $\\mathcal{Y}$. Both the transmitter and receiver have access to the key, a random variable, independent of $X^K$ and $Y^K$ and taking values in $I_{M_k} = \\{0, 1, 2, \\ldots ,M_{k} - 1\\}$. The sources $X^K$ and $Y^K$ compute the ciphertexts $X^{'}$ and $Y^{'}$, which are the result of specific encryption functions on the plaintext from $X$ and $Y$ respectively. The encryption functions are invertible, thus knowing $X^{'}$ and the key, $X^K$ can be retrieved. \n\nThe mutual information between the plaintext and ciphertext should be small so that the wiretapper cannot gain much information about the plaintext. For perfect secrecy, this mutual information should be zero, then the length of the key should be at least the length of the plaintext.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics [scale = 0.7]{shannon_2sources.pdf}\n\\caption{Shannon cipher system for two correlated sources}\n\\label{fig:shannon_cipher_2sources}\n\\end{figure}\n\nThe encoder functions for $X$ and $Y$, ($E_X$ and $E_Y$ respectively) are given as:\n\n\\begin{eqnarray}\nE_X : \\mathcal{X}^K \\times I_{M_{kX}} & \\rightarrow & I_{M_X'} =  \\{0, 1, \\ldots, M_X' - 1\\} \\nonumber \n\\\\ && I_{M_{CX}'} =  \\{0, 1, \\ldots, M_{CX}' - 1\\}\n\\label{xencoder_fcn}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nE_Y : \\mathcal{Y}^K \\times I_{M_{kY}} & \\rightarrow & I_{M_Y'} =  \\{0, 1, \\ldots, M_Y' - 1\\} \\nonumber \n\\\\ && I_{M_{CY}'} =  \\{0, 1, \\ldots, M_{CY}' - 1\\}\n\\label{yencoder_fcn}\n\\end{eqnarray}\n\nThe decoder is defined as:\n\n\\begin{eqnarray}\nD_{XY} : (I_{M'_X}, I_{M'_Y}, I_{M'_{CX}},I_{M'_{CY}})  & \\times &  I_{M_{kX}}, I_{M_{kY}} \\nonumber \\\\\n& \\rightarrow & \\mathcal{X}^K \\times \\mathcal{Y}^K\n\\end{eqnarray}\n\nThe encoder and decoder mappings are below:\n\\begin{eqnarray}\nW_1 = F_{E_X} (X^K, W_{kX})\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_2 = F_{E_Y} (Y^K, W_{kY})\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\widehat{X}^K = F_{D_X} (W_1, W_2, W_{kX})\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\widehat{Y}^K = F_{D_Y} (W_1, W_2, W_{kY})\n\\end{eqnarray}\n\nor \n\n\\begin{eqnarray}\n(\\widehat{X}^K, \\widehat{Y}^K) = F_{D_{XY}} (W_1, W_2, W_{kX}, W_{kY})\n\\end{eqnarray}\n\n\nThe following conditions should be satisfied for cases 1- 4:\n\n\\begin{eqnarray}\n\\frac{1}{K}\\log M_X \\le R_X +\\epsilon\n\\label{cond1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K}\\log M_Y \\le R_Y +\\epsilon\n\\label{cond2}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K}\\log M_{kX} \\le R_{kX} +\\epsilon\n\\label{cond3}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K}\\log M_{kY} \\le R_{{kY}} +\\epsilon\n\\label{cond4}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\text {Pr} \\{\\widehat{X}^K \\neq X^K\\} \\le \\epsilon\n\\label{cond5}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\text{Pr} \\{ \\widehat{Y}^K \\neq Y^K\\} \\le \\epsilon\n\\label{cond6}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K|W_1) \\le h_X + \\epsilon\n\\label{cond7}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} H(Y^K|W_2) \\le h_Y + \\epsilon\n\\label{cond8}\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K,Y^K|W_1) \\le h_{XY} + \\epsilon\n\\label{cond8.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K,Y^K|W_2) \\le h_{XY} + \\epsilon\n\\label{cond9}\n\\end{eqnarray}\n\nwhere $R_X$ is the the rate of source $X$'s channel and $R_Y$ is the the rate of source $Y$'s channel. Here, $R_{kX}$ is the rate of the key channel at $X^K$ and $R_{kY}$ is the rate of the key channel at $Y^K$. The security levels, which are measured by the total and individual uncertainties are $h_{XY}$ and $(h_X, h_Y)$ respectively. \n\\\\\\\\\nThe cases 1 - 5 are:\n\\\\ \\textit{Case 1:} When $T_X$ and $T_Y$ are leaked and both $X^K$ and $Y^K$ need to be kept secret.\n\\\\ \\textit{Case 2:} When $T_X$ and $T_Y$ are leaked and $X^K$ needs to be kept secret.\n\\\\ \\textit{Case 3:} When $T_X$ is leaked and both $X^K$ and $Y^K$ need to be kept secret.\n\\\\ \\textit{Case 4:} When $T_X$ is leaked and $Y^K$ needs to be kept secret.\n\\\\ \\textit{Case 5:} When $T_X$ is leaked and $X^K$ needs to be kept secret.\n\\\\ where $T_X$ is the syndrome produced by $X$, containing $V_{CX}$ and $V_X$ and $T_Y$ is the syndrome produced by $Y$, containing $V_{CY}$ and $V_X$ .\n\\\\\\\\\nThe admissible rate region for each case is defined as follows:\n\\\\ \\textit{Definition 1a:} ($R_X$, $R_Y$, $R_{kX}$, $R_{kY}$, $h_{XY}$) is admissible for case 1 if there exists a code ($F_{E_{X}}$, $F_{D_{XY}}$) and ($F_{E_{Y}}$, $F_{D_{XY}}$) such that \\eqref{cond1} - \\eqref{cond6} and \\eqref{cond9} hold for any $\\epsilon \\rightarrow 0$ and sufficiently large $K$.\n\\\\ \\textit{Definition 1b:} ($R_X$, $R_Y$, $R_{kX}$, $R_{kY}$, $h_{X}$) is admissible for case 2 if there exists a code ($F_{E_{X}}$, $F_{D_{XY}}$) such that \\eqref{cond1} - \\eqref{cond7} hold for any $\\epsilon \\rightarrow 0$ and sufficiently large $K$.\n\\\\ \\textit{Definition 1c:} ($R_X$, $R_Y$, $R_{kX}$, $R_{kY}$, $h_{X}$, $h_{Y}$) is admissible for case 3 if there exists a code ($F_{E_{X}}$, $F_{D_{XY}}$) and ($F_{E_{Y}}$, $F_{D_{XY}}$) such that \\eqref{cond1} - \\eqref{cond6} and \\eqref{cond8}, \\eqref{cond9} hold for any $\\epsilon \\rightarrow 0$ and sufficiently large $K$.\n\\\\ \\textit{Definition 1d:} ($R_X$, $R_Y$, $R_{kX}$, $R_{kY}$, $h_{Y}$) is admissible for case 4 if there exists a code ($F_{E_{X}}$, $F_{D_{XY}}$) such that \\eqref{cond1} - \\eqref{cond6} and \\eqref{cond8} hold for any $\\epsilon \\rightarrow 0$ and sufficiently large $K$.\n\\\\ \\textit{Definition 1e:} ($R_X$, $R_Y$, $R_{kX}$, $R_{kY}$, $h_{X}$) is admissible for case 5 if there exists a code ($F_{E_{X}}$, $F_{D_{XY}}$) such that \\eqref{cond1} - \\eqref{cond6} and \\eqref{cond7} hold for any $\\epsilon \\rightarrow 0$ and sufficiently large $K$.\n\\\\ \\textit{Definition 2:} The admissible rate regions of $\\mathcal{R}_j$ and of $\\mathcal{R}_k$ are defined as:\n\n\\begin{eqnarray}\n\\mathcal{R}_1(h_{XY}) = \\{(R_X, R_Y, R_{kX}, R_{kY}):\t\t\t\\nonumber\n\\\\(R_X, R_Y, R_{kX}, R_{kY}, h_{XY} ) \\text{ is admissible for case 1} \\}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{R}_2(h_{X}) = \\{(R_X, R_Y, R_{kX}, R_{kY}):\t\t\t\\nonumber\n\\\\ (R_X, R_Y, R_{kX}, R_{kY}, h_{X} ) \\text{ is admissible for case 2} \\}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{R}_3(h_X, h_Y) = \\{(R_X, R_Y, R_{kX}, R_{kY}):\t\t\t\\nonumber\n\\\\ (R_X, R_Y, R_{kX}, R_{kY}, h_{X}, h_{Y} ) \\text{ is admissible for case 3} \\}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{R}_4(h_{Y}) = \\{(R_X, R_Y, R_{kX}, R_{kY}):\t\t\t\\nonumber\n\\\\(R_X, R_Y, R_{kX}, R_{kY}, h_{Y} ) \\text{ is admissible for case 4} \\}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{R}_5(h_{X}) = \\{(R_X, R_Y, R_{kX}, R_{kY}):\t\t\t\\nonumber\n\\\\(R_X, R_Y, R_{kX}, R_{kY}, h_{X} ) \\text{ is admissible for case 5} \\}\n\\end{eqnarray}\n\nTheorems for these regions have been developed:\n\n\\textit{Theorem 2:} For $0 \\le h_{XY} \\le H(X,Y)$,\n\\begin{eqnarray}\n&& \\mathcal{R}_1(h_{XY}) = \\{(R_X, R_Y, R_{kX},R_{kY}): \t\t\\nonumber\n\\\\ && R_X \\geq H(X|Y), \t\t\t\t\\nonumber\n\\\\ && R_Y \\geq H(Y|X),\t\t\\nonumber\n\\\\ && R_X + R_Y\t\\geq H(X,Y)\t\t\t\t\t\t\\nonumber\n\\\\ && R_{kX} \\geq h_{XY} \\text{ and } R_{kY} \\geq h_{XY} \\}\t\t\t\n\\label{theorem2}\n\\end{eqnarray}\n\n\\textit{Theorem 3:} For $0 \\le h_{X} \\le H(X)$,\n\\begin{eqnarray}\n&& \\mathcal{R}_2(h_{X}) = \\{(R_X, R_Y, R_{kX},R_{kY}): \t\t\\nonumber\n\\\\ && R_X \\geq H(X|Y), \t\t\t\t\\nonumber\n\\\\ && R_Y \\geq H(Y|X),\t\t\\nonumber\n\\\\ && R_X + R_Y\t\\geq H(X,Y)\t\t\t\t\t\t\\nonumber\n\\\\ && R_{kX} \\geq h_X \\text{ and } R_{kY} \\geq h_Y \\}\t\t\t\n\\label{theorem3}\n\\end{eqnarray}\n\n\\textit{Theorem 4:} For $0 \\le h_{X} \\le H(X)$ and $0 \\le h_{Y} \\le H(Y)$,\n\\begin{eqnarray}\n&& \\mathcal{R}_3(h_{X}, h_{Y}) = \\{(R_X, R_Y, R_{kX},R_{kY}): \t\t\\nonumber\n\\\\ && R_X \\geq H(X|Y), \t\t\t\t\\nonumber\n\\\\ && R_Y \\geq H(Y|X),\t\t\\nonumber\n\\\\ && R_X + R_Y\t\\geq H(X,Y)\t\t\t\t\t\t\\nonumber\n\\\\ && R_{kX} \\geq h_{X} \\text{ and } R_{kY} \\geq h_{Y} \\}\n\\label{theorem4}\n\\end{eqnarray}\n\n\\textit{Theorem 5:} For $0 \\le h_{X} \\le H(X)$,\n\\begin{eqnarray}\n&& \\mathcal{R}_5(h_{X}, h_{Y}) = \\{(R_X, R_Y, R_{kX},R_{kY}): \t\t\\nonumber\n\\\\ && R_X \\geq H(X|Y), \t\t\t\t\\nonumber\n\\\\ && R_Y \\geq H(Y|X),\t\t\\nonumber\n\\\\ && R_X + R_Y\t\\geq H(X,Y)\t\t\t\t\t\t\\nonumber\n\\\\ && R_{kX} \\geq h_{X} \\text{ and } R_{kY} \\geq 0 \\}\n\\label{theorem5}\n\\end{eqnarray}\n\nWhen $h_X = 0$ then case $5$ can be reduced to that depicted in \\eqref{theorem4}.  \nHence, Corollary 1 follows:\n\\\\ \\textit{Corollary 1:} $\\mathcal{R}_4(h_{Y}) = \\mathcal{R}_3(0, h_Y)$\n\nThe security levels, which are measured by the total and individual uncertainties $h_{XY}$ and $(h_X, h_Y)$ respectively give an indication of the level of uncertainty in knowing certain information. When the uncertainty increases then less information is known to an eavesdropper and there is a higher level of security. \n\n\n\\section{Proof of Theorems 2 - 5}\nThis section initially proves the direct parts of Theorems 2 - 5 and thereafter the converse parts.\n\n\\subsection{Direct parts}\nAll the channel rates in the theorems above are in accordance with Slepian-Wolf's theorem, hence there is no need to prove them. \nWe construct a code based on the prototype code ($W_X, W_Y, W_{CX}, W_{CY}$) in Lemma 1. In order to include a key in the prototype code, $W_X$ is divided into two parts as per the method used by Yamamoto \\cite{shannon1_yamamoto}:\n\\begin{eqnarray}\nW_{X1} = W_X \\text{ mod } M_{X1} \\in I_{M_{X1}} = \\{0, 1, 2, \\ldots, M_{X1} - 1\\}\n\\label{theorems2-4_eq_1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{X2} = \\frac{W_X - W_{X1}}{M_{X1}} \\in I_{M_{X2}} = \\{0, 1, 2, \\ldots, M_{X2} - 1\\}\n\\label{theorems2-4_eq_2}\n\\end{eqnarray}\n\nwhere $M_{X1}$ is a given integer and $M_{X2}$ is the ceiling of $M_X\/M_{X1}$. The $M_X\/M_{X1}$ is considered an integer for simplicity, because the difference between the ceiling value and the actual value can be ignored when $K$ is sufficiently large. In the same way, $W_Y$ is divided:\n\n\\begin{eqnarray}\nW_{Y1} = W_Y \\text{ mod } M_{Y1} \\in I_{M_{Y1}} = \\{0, 1, 2, \\ldots, M_{Y1} - 1\\}\n\\label{theorems2-4_eq_3}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{Y2} = \\frac{W_Y - W_{Y1}}{M_{Y1}} \\in I_{M_{Y2}} = \\{0, 1, 2, \\ldots, M_{Y2} - 1\\}\n\\label{theorems2-4_eq_4}\n\\end{eqnarray}\n\nThe common information components $W_{CX}$ and $W_{CY}$ are already portions and are not divided further. \nIt can be shown that when some of the codewords are wiretapped the uncertainties of $X^K$ and $Y^K$ are bounded as follows:\n\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K|W_{X2},W_Y) \\geq I(X;Y) + \\frac{1}{K} \\log M_{X1} - \\epsilon_{0}^{'}\n\\label{theorems2-4_ineq_1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} H(Y^K|W_{X},W_{Y2}) \\geq I(X;Y) + \\frac{1}{K} \\log M_{Y1} - \\epsilon_{0}^{'}\n\\label{theorems2-4_ineq_2}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K|W_{X},W_{Y2}) \\geq I(X;Y) - \\epsilon_{0}^{'}\n\\label{theorems2-4_ineq_3}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K|W_{X},W_Y, W_{CY}) \\geq \\frac{1}{K} \\log M_{CX} - \\epsilon_{0}^{'}\n\\label{theorems2-4_ineq_4}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} H(Y^K|W_{X},W_Y, W_{CY}) \\geq \\frac{1}{K} \\log M_{CX} - \\epsilon_{0}^{'}\n\\label{theorems2-4_ineq_5}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K|W_Y, W_{CY}) \\geq H(X|Y) + \\frac{1}{K} \\log M_{CX} - \\epsilon_{0}^{'}\n\\label{theorems2-4_ineq_6}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} H(Y^K|W_Y, W_{CY}) \\geq \\frac{1}{K} \\log M_{CX} - \\epsilon_{0}^{'}\n\\label{theorems2-4_ineq_7}\n\\end{eqnarray}\n\nwhere $\\epsilon_{0}^{'} \\rightarrow 0$ as  $\\epsilon_{0} \\rightarrow 0$.\nThe proofs for \\eqref{theorems2-4_ineq_1} - \\eqref{theorems2-4_ineq_7} are the same as per Yamamoto's\\cite{shannon1_yamamoto} proof in Lemma A1. The difference is that $W_{CX}$, $W_{CY}$, $M_{CX}$ and $M_{CY}$ are described as $W_{C1}$, $W_{C2}$, $M_{C1}$ and $M_{C2}$ respectively by Yamamoto. Here, we consider that $W_{CX}$ and $W_{CY}$ are represented by Yamamoto's $W_{C1}$ and $W_{C2}$ respectively. In addition there are some more inequalities considered here:\n\\begin{eqnarray}\n \\frac{1}{K} H(Y^K|W_X, W_{CX}, W_{CY}, W_{Y2}) & \\geq & \\frac{1}{K} \\log M_{Y1}  \\nonumber\n \\\\ & - & \\epsilon_{0}^{'}\n\\label{theorems2-4_ineq_8}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n \\frac{1}{K} H(Y^K|W_X, W_{CX}, W_{CY}) & \\geq & \\frac{1}{K} \\log M_{Y1}  \\nonumber\n\\\\ & + & \\frac{1}{K} \\log M_{Y2} - \\epsilon_{0}^{'}\n\\label{theorems2-4_ineq_9}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K|W_{X2}, W_{CY}) & \\geq & \\frac{1}{K} \\log M_{X1} \t\\nonumber\n\\\\ & + & \\frac{1}{K} \\log M_{CX} - \\epsilon_{0}^{'}\n\\label{theorems2-4_ineq_10}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} H(Y^K|W_{X2}, W_{CY}) & \\geq & \\frac{1}{K} \\log M_{Y1} \t\\nonumber\n\\\\ & + & \\frac{1}{K} \\log M_{Y2} + \\frac{1}{K} \\log M_{CX} \t\t\t\\nonumber\n\\\\ & - & \\epsilon_{0}^{'}\n\\label{theorems2-4_ineq_11}\n\\end{eqnarray}\n\nThe inequalities \\eqref{theorems2-4_ineq_8} and \\eqref{theorems2-4_ineq_9} can be proved in the same way as per Yamamoto's\\cite{shannon1_yamamoto} Lemma A2, and  \\eqref{theorems2-4_ineq_10} and \\eqref{theorems2-4_ineq_11} can be proved in the same way as per Yamamoto's\\cite{shannon1_yamamoto} Lemma A1. \n\nFor each proof we consider cases where a key already exists for either $V_{CX}$ or $V_{CY}$ and the encrypted common information portion is then used to mask the other portions (either $V_{CX}$ or $V_{CY}$ and the private information portions). There are two cases considered for each; firstly, when the common information portion entropy is greater than the entropy of the portion that needs to be masked, and secondly when the common information portion entropy is less than the entropy of the portion to be masked. For the latter case, a smaller key will need to be added so as to cover the portion entirely. This has the effect of reducing the required key length, which is explained in greater detail in Section VII.\n\n\\begin{proof}[Proof of Theorem 2]\nSuppose that ($R_X$, $R_Y$, $R_{KX}$, $R_{KY}$) $\\in$ \n$\\mathcal{R}_1$ for $h_{XY} \\le H(X,Y)$. Without loss of generality, we assume that $h_X \\le h_Y$. Then, from \\eqref{theorem2} \n\\begin{eqnarray}\n&& R_X \\geq H(X^K|Y^K)  \t\t\t\t\\nonumber\n\\\\&& R_Y  \\geq H(Y^K|X^K) \t\t\t\t\\nonumber\n\\\\&& R_X + R_Y  \\geq H(X^K, Y^K)\n\\label{theorem2_proof_1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nR_{kX} \\geq h_{XY}, R_{kY} \\geq h_{XY} \n\\label{theorem2_proof_2}\n\\end{eqnarray}\n\nAssuming a key exists for $V_{CY}$. For the first case, consider the following: $H(V_{CY}) \\geq H(V_X)$, $H(V_{CY}) \\geq H(V_Y)$ and $H(V_{CY}) \\geq H(V_{CX})$.\n\n\\begin{eqnarray}\nM_{CY} = 2^{K h_{XY}}\n\\label{theorem2_proof_6}\n\\end{eqnarray}\n\nThe codewords $W_X$ and $W_Y$ and their keys $W_{kX}$ and $W_{kY}$ are now defined:\n\n\\begin{eqnarray}\nW_X = (W_{X1} \\oplus W_{kCY}, W_{X2}  \\oplus W_{kCY}, W_{CX}  \\oplus W_{kCY})\n\\label{theorem2_proof_7}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_Y = (W_{Y1} \\oplus W_{kCY}, W_{Y2}  \\oplus W_{kCY}, W_{CY})\n\\label{theorem2_proof_8}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{kY} = (W_{kCY})\n\\label{theorem2_proof_10}\n\\end{eqnarray}\n\nwhere $W_\\alpha \\in I_{M_\\alpha} = \\{0, 1, \\ldots, M_\\alpha - 1\\}$. The wiretapper will not know $W_{X1}$, $W_{X2}$ and $W_{CX}$ from $W_X$ and $W_{Y1}$, $W_{Y2}$ and $W_{CY}$ from $W_Y$ as these are protected by the key ($W_{kCY}$.\n\nIn this case, $R_X$, $R_Y$, $R_{kX}$ and $R_{kY}$ satisfy from \\eqref{lemma1_2} - \\eqref{lemma1_4} and \\eqref{theorem2_proof_1} - \\eqref{theorem2_proof_6}, that\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_X + \\frac{1}{K} \\log M_Y  & = & \\frac{1}{K} (\\log M_{X1} + \\log M_{X2}  \\nonumber\n\\\\ & + &  \\log M_{CX}) + \\frac{1}{K} (\\log M_{Y1}  \\nonumber\n\\\\ & + &  \\log M_{Y2} +  \\log M_{CY}) \\nonumber \n\\\\ & \\le & H(X|Y) + H(Y|X) \t\t\t\t\\nonumber\n\\\\ & + & I(X;Y) + \\epsilon_0 \\nonumber\n\\\\ & = & H(X,Y)  \t\t\\nonumber\n\\\\ & \\le & R_X + R_Y\n\\label{theorem2_proof_11}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{kX} & = & \\frac{1}{K} \\log M_{CY}\t\\nonumber\n\\\\ & = & h_{XY}\t\t\t\\label{num3}\n\\\\ & \\le & R_{kX}\n\\label{theorem2_proof_13}\n\\end{eqnarray}\n\nwhere \\eqref{num3} comes from \\eqref{theorem2_proof_6}.\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{kY} & = & \\frac{1}{K} \\log M_{CY}\t\\nonumber\n\\\\ & = & h_{XY}\t\t\t\\label{num4}\n\\\\ & \\le & R_{kY}\n\\label{theorem2_proof_14}\n\\end{eqnarray}\nwhere \\eqref{num4} comes from \\eqref{theorem2_proof_6}.\n\nThe security levels thus result:\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K|W_X, W_Y) & = & \\frac{1}{K} H(X|W_{X1} \\oplus W_{kCY}, \\nonumber\n\\\\ && W_{X2} \\oplus W_{kCY}, W_{CX} \\oplus W_{kCY},\t\t\t\\nonumber\n\\\\ && W_{Y1} \\oplus W_{kCY}, W_{Y2} \\oplus W_{kCY},\t\\nonumber\n\\\\ &&  W_{CY}) \t\t\t\t\\nonumber\n\\\\ & = & H(X^K)\t\\label{num5}\n\\\\ & \\ge & h_X - \\epsilon^{'}_0\n\\label{theorem2_proof_16}\n\\end{eqnarray}\n\nwhere \\eqref{num5} holds because $W_{X1}$, $W_{X2}$, $W_{CX}$, $W_{Y1}$, $W_{Y2}$ are covered by key $W_{CY}$ and $W_{CY}$ is covered by an existing random number key. Equations \\eqref{lemma1_1} - \\eqref{lemma1_7} imply that $W_{X1}$, $W_{X2}$, $W_{Y1}$ and $W_{Y2}$ have almost no redundancy and they are mutually independent.\n\nSimilarly, \n\\begin{eqnarray}\n\\frac{1}{K} H(Y^K|W_X, W_Y) \\geq h_Y - \\epsilon^{'}_0\n\\label{theorem2_proof_17}\n\\end{eqnarray}\n\nTherefore ($R_X$, $R_Y$, $R_{kX}$, $R_{kY}$, $h_{XY}$, $h_{XY}$) is admissible from \\eqref{theorem2_proof_11} -  \\eqref{theorem2_proof_17}.\n\nNext the case where: $H(V_{CY}) < H(V_X)$, $H(V_{CY}) < H(V_Y)$ and $H(V_{CY}) < H(V_{CX})$ is considered. Here, there are shorter length keys used in addition to the key provided by $W_{CY}$ in order to make the key lengths required by the individual portions. For example the key $W_{k1}$ comprises $W_{kCY}$ and a short key $W_1$, which together provide the length of $W_{X1}$.\nThe codewords $W_X$ and $W_Y$ and their keys $W_{kX}$ and $W_{kY}$ are now defined:\n\n\\begin{eqnarray}\nW_X = (W_{X1} \\oplus W_{k1}, W_{X2}  \\oplus W_{k2}, W_{CX}  \\oplus W_{k3})\n\\label{theorem2_proof_7.1.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_Y = (W_{Y1} \\oplus W_{k4}, W_{Y2}  \\oplus W_{k5}, W_{CY})\n\\label{theorem2_proof_8.1.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{kX} = (W_{k1}, W_{k2}, W_{k3})\n\\label{theorem2_proof_10.1.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{kY} = (W_{k4}, W_{k5})\n\\label{theorem2_proof_10.1.2}\n\\end{eqnarray}\n\nwhere $W_\\alpha \\in I_{M_\\alpha} = \\{0, 1, \\ldots, M_\\alpha - 1\\}$. The wiretapper will not know $W_{X1}$, $W_{X2}$ and $W_{CX}$ from $W_X$ and $W_{Y1}$, $W_{Y2}$ and $W_{CY}$ from $W_Y$ as these are protected by the key ($W_{kCY}$.\n\nIn this case, $R_X$, $R_Y$, $R_{kX}$ and $R_{kY}$ satisfy that\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_X + \\frac{1}{K} \\log M_Y  & = & \\frac{1}{K} (\\log M_{X1} + \\log M_{X2}  \\nonumber\n\\\\ & + &  \\log M_{CX}) + \\frac{1}{K} (\\log M_{Y1}  \\nonumber\n\\\\ & + &  \\log M_{Y2} +  \\log M_{CY}) \\nonumber \n\\\\ & \\le & H(X|Y) + H(Y|X) \t\t\t\t\\nonumber\n\\\\ & + & I(X;Y) + \\epsilon_0 \\nonumber\n\\\\ & = & H(X,Y)  \t\t\\nonumber\n\\\\ & \\le & R_X + R_Y\n\\label{theorem2_proof_11.1.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{kX} & = & \\frac{1}{K} [\\log M_{k1} + \\log M_{k2} + \\log M_{k3}]\t\\nonumber\n\\\\ & = & \\log M_{kCY} + \\log M_{1} \\nonumber\n\\\\ & + & \\log M_{kCY} + \\log M_{2} \\nonumber\n\\\\ & + & \\log M_{kCY} + \\log M_{3} \\nonumber\n\\\\ & = & 3 \\log M_{kCY} + \\log M_{1} \\nonumber\n\\\\ & + & \\log M_{2} + \\log M_{3} \\nonumber\n\\\\ & \\geq & 3 h_{XY} - \\epsilon_0  \\label{num333.1}\n\\\\ & \\geq & h_{XY}\n\\label{theorem2_proof_13.1.1}\n\\end{eqnarray}\n\nwhere \\eqref{num333.1} results from \\eqref{theorem2_proof_6}.\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{kX} & = & \\frac{1}{K} [\\log M_{k3} + \\log M_{k4} + \\log M_{kCY}]\t\\nonumber\n\\\\ & = & \\log M_{kCY} + \\log M_{3} +\t\\log M_{kCY} \\nonumber\n\\\\ & + & \\log M_{4} + \\log M_{kCY} \t\t\\nonumber\n\\\\ & = & 3 \\log M_{kCY} + \\log M_{3} + \\log M_{4}\n\\\\ & \\geq & 3 h_{XY} - \\epsilon_0 \\label{num333}\n\\\\ & \\geq & h_{XY}\n\\label{theorem2_proof_14.1.1}\n\\end{eqnarray}\n\n\nwhere \\eqref{num333} results from \\eqref{theorem2_proof_6}.\n\nThe security levels thus result:\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K|W_X, W_Y) & = & \\frac{1}{K} H(X|W_{X1} \\oplus W_{k1}, W_{X2} \\oplus W_{k2}, \\nonumber\n\\\\ && W_{CX} \\oplus W_{k3},\t\t\t\\nonumber\n\\\\ && W_{Y1} \\oplus W_{k4}, W_{Y2} \\oplus W_{k5},\t\\nonumber\n\\\\ &&  W_{CY}) \t\t\t\t\\nonumber\n\\\\ & = & H(X^K)\t\\label{num5.1.1}\n\\\\ & \\ge & h_X - \\epsilon^{'}_0\n\\label{theorem2_proof_16.1.1}\n\\end{eqnarray}\n\nwhere \\eqref{num5} holds because $W_{X1}$, $W_{X2}$, $W_{CX}$, $W_{Y1}$, $W_{Y2}$ are covered by key $W_{CY}$ and some shorter length key and $W_{CY}$ is covered by an existing random number key. \n\nSimilarly, \n\\begin{eqnarray}\n\\frac{1}{K} H(Y^K|W_X, W_Y) \\geq h_Y - \\epsilon^{'}_0\n\\label{theorem2_proof_17.1.1}\n\\end{eqnarray}\n\nTherefore ($R_X$, $R_Y$, $R_{kX}$, $R_{kY}$, $h_{XY}$, $h_{XY}$) is admissible from \\eqref{theorem2_proof_11.1.1} -  \\eqref{theorem2_proof_17.1.1}.\n\n\\end{proof}\n\nTheorem 3 - 5 are proven in the same way with varying codewords and keys. The proofs follow:\n\n\\begin{proof}[Theorem 3 proof]\n\nThe consideration for the security levels is that $h_Y \\geq h_X$ because $Y$ contains the key the is used for masking.\nSuppose that ($R_X$, $R_Y$, $R_{KX}$, $R_{KY}$) $\\in$ \n$\\mathcal{R}_2$. From \\eqref{theorem3} \n\\begin{eqnarray}\n&& R_X \\geq H(X^K|Y^K)  \t\t\t\t\\nonumber\n\\\\&& R_Y  \\geq H(Y^K|X^K) \t\t\t\t\\nonumber\n\\\\&& R_X + R_Y  \\geq H(X^K, Y^K)\n\\label{theorem3_proof_1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nR_{kX} \\geq h_{X}, R_{kY} \\geq h_{Y} \n\\label{theorem3_proof_2}\n\\end{eqnarray}\n\nAssuming a key exists for $V_{CY}$. For the first case, consider the following: $H(V_{CY}) \\geq H(V_X)$, $H(V_{CY}) \\geq H(V_Y)$ and $H(V_{CY}) \\geq H(V_{CX})$.\n\n\\begin{eqnarray}\nM_{CY} = 2^{K h_{Y}}\n\\label{theorem3_proof_6}\n\\end{eqnarray}\n\nThe codewords $W_X$ and $W_Y$ and their keys $W_{kX}$ and $W_{kY}$ are now defined:\n\n\\begin{eqnarray}\nW_X = (W_{X1} \\oplus W_{kCY}, W_{X2}  \\oplus W_{kCY}, W_{CX}  \\oplus W_{kCY})\n\\label{theorem3_proof_7}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_Y = (W_{Y1} \\oplus W_{kCY}, W_{Y2}  \\oplus W_{kCY}, W_{CY})\n\\label{theorem3_proof_8}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{kY} = (W_{kCY})\n\\label{theorem3_proof_10}\n\\end{eqnarray}\n\nwhere $W_\\alpha \\in I_{M_\\alpha} = \\{0, 1, \\ldots, M_\\alpha - 1\\}$. The wiretapper will not know $W_{X1}$, $W_{X2}$ and $W_{CX}$ from $W_X$ and $W_{Y1}$, $W_{Y2}$ and $W_{CY}$ from $W_Y$ as these are protected by the key ($W_{kCY}$.\n\nIn this case, $R_X$, $R_Y$, $R_{kX}$ and $R_{kY}$ satisfy from \\eqref{lemma1_2} - \\eqref{lemma1_4} and \\eqref{theorem3_proof_1} - \\eqref{theorem3_proof_6}, that\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_X + \\frac{1}{K} \\log M_Y  & = & \\frac{1}{K} (\\log M_{X1} + \\log M_{X2}  \\nonumber\n\\\\ & + &  \\log M_{CX}) + \\frac{1}{K} (\\log M_{Y1}  \\nonumber\n\\\\ & + &  \\log M_{Y2} +  \\log M_{CY}) \\nonumber \n\\\\ & \\le & H(X|Y) + H(Y|X) \t\t\t\t\\nonumber\n\\\\ & + & I(X;Y) + \\epsilon_0 \\nonumber\n\\\\ & = & H(X,Y)  \t\t\\nonumber\n\\\\ & \\le & R_X + R_Y\n\\label{theorem3_proof_11}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{kX} & = & \\frac{1}{K} \\log M_{CY}\t\\nonumber\n\\\\ & = & h_{Y}\t\t\t\\label{num3.2}\n\\\\ & \\geq & h_X - \\epsilon_0  \\label{num3.3}\n\\\\ &  & R_{kX}\n\\label{theorem3_proof_13}\n\\end{eqnarray}\n\nwhere \\eqref{num3.2} comes from \\eqref{theorem3_proof_6} and \\eqref{num3.3} comes form the consideration stated at the beginning of this proof.\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{kY} & = & \\frac{1}{K} \\log M_{CY}\t\\nonumber\n\\\\ & = & h_{XY}\t\t\t\\label{num4.1}\n\\\\ & \\le & R_{kY}\n\\label{theorem3_proof_14}\n\\end{eqnarray}\nwhere \\eqref{num4.1} comes from \\eqref{theorem3_proof_6}.\n\nThe security levels thus result:\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K|W_X, W_Y) & = & \\frac{1}{K} H(X|W_{X1} \\oplus W_{kCY}, \\nonumber\n\\\\ && W_{X2} \\oplus W_{kCY}, W_{CX} \\oplus W_{kCY},\t\t\t\\nonumber\n\\\\ && W_{Y1} \\oplus W_{kCY}, W_{Y2} \\oplus W_{kCY},\t\\nonumber\n\\\\ &&  W_{CY}) \t\t\t\t\\nonumber\n\\\\ & = & H(X^K)\t\\label{num5.1}\n\\\\ & \\ge & h_X - \\epsilon^{'}_0\n\\label{theorem3_proof_16}\n\\end{eqnarray}\n\nwhere \\eqref{num5.1} holds because $W_{X1}$, $W_{X2}$, $W_{CX}$, $W_{Y1}$, $W_{Y2}$ are covered by key $W_{CY}$ and $W_{CY}$ is covered by an existing random number key. Equations \\eqref{lemma1_1} - \\eqref{lemma1_7} imply that $W_{X1}$, $W_{X2}$, $W_{Y1}$ and $W_{Y2}$ have almost no redundancy and they are mutually independent.\n\nSimilarly, \n\\begin{eqnarray}\n\\frac{1}{K} H(Y^K|W_X, W_Y) \\geq h_Y - \\epsilon^{'}_0\n\\label{theorem3_proof_17}\n\\end{eqnarray}\n\nTherefore ($R_X$, $R_Y$, $R_{kX}$, $R_{kY}$, $h_{XY}$, $h_{XY}$) is admissible from \\eqref{theorem2_proof_11} -  \\eqref{theorem2_proof_17}.\n\nNext the case where: $H(V_{CY}) < H(V_X)$, $H(V_{CY}) < H(V_Y)$ and $H(V_{CY}) < H(V_{CX})$ is considered. Here, there are shorter length keys used in addition to the key provided by $W_{CY}$ in order to make the key lengths required by the individual portions. For example the key $W_{k1}$ comprises $W_{kCY}$ and a short key $W_1$, which together provide the length of $W_{X1}$.\nThe codewords $W_X$ and $W_Y$ and their keys $W_{kX}$ and $W_{kY}$ are now defined:\n\n\\begin{eqnarray}\nW_X = (W_{X1} \\oplus W_{k1}, W_{X2}  \\oplus W_{k2}, W_{CX}  \\oplus W_{k3})\n\\label{theorem3_proof_7.1.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_Y = (W_{Y1} \\oplus W_{k4}, W_{Y2}  \\oplus W_{k5}, W_{CY})\n\\label{theorem3_proof_8.1.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{kX} = (W_{k1}, W_{k2}, W_{k3})\n\\label{theorem3_proof_10.1.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{kY} = (W_{k4}, W_{k5})\n\\label{theorem3_proof_10.1.2}\n\\end{eqnarray}\n\nwhere $W_\\alpha \\in I_{M_\\alpha} = \\{0, 1, \\ldots, M_\\alpha - 1\\}$. The wiretapper will not know $W_{X1}$, $W_{X2}$ and $W_{CX}$ from $W_X$ and $W_{Y1}$, $W_{Y2}$ and $W_{CY}$ from $W_Y$ as these are protected by the key ($W_{kCY}$.\n\nIn this case, $R_X$, $R_Y$, $R_{kX}$ and $R_{kY}$ satisfy that\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_X + \\frac{1}{K} \\log M_Y  & = & \\frac{1}{K} (\\log M_{X1} + \\log M_{X2}  \\nonumber\n\\\\ & + &  \\log M_{CX}) + \\frac{1}{K} (\\log M_{Y1}  \\nonumber\n\\\\ & + &  \\log M_{Y2} +  \\log M_{CY}) \\nonumber \n\\\\ & \\le & H(X|Y) + H(Y|X) \t\t\t\t\\nonumber\n\\\\ & + & I(X;Y) + \\epsilon_0 \\nonumber\n\\\\ & = & H(X,Y)  \t\t\\nonumber\n\\\\ & \\le & R_X + R_Y\n\\label{theorem3_proof_11.1.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{kX} & = & \\frac{1}{K} [\\log M_{k1} + \\log M_{k2} + \\log M_{k3}]\t\\nonumber\n\\\\ & = & \\log M_{kCY} + \\log M_{1} +\t\\log M_{kCY}  \\nonumber\n\\\\ & + &  \\log M_{2} + \\log M_{kCY} + \\log M_{3} \\nonumber\n\\\\ & = & 3 \\log M_{kCY} + \\log M_{1} + \\log M_{2} + \\log M_{3} \\nonumber\n\\\\ & \\geq & 3 h_{Y} - \\epsilon_0  \\label{num334.1}\n\\\\ & \\geq & 3 h_{X} - \\epsilon_0 \\nonumber\n\\\\ & \\geq & h_{X}\n\\label{theorem3_proof_13.1.1}\n\\end{eqnarray}\n\nwhere \\eqref{num334.1} results from \\eqref{theorem3_proof_6} and the result is from the consideration at the beginning of this proof.\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{kY} & = & \\frac{1}{K} [\\log M_{k3} + \\log M_{k4} + \\log M_{kCY}]\t\\nonumber\n\\\\ & = & \\log M_{kCY} + \\log M_{3} +\t\\log M_{kCY}  \\nonumber\n\\\\ & + & \\log M_{4} + \\log M_{kCY}  \\nonumber\n\\\\ & = & 3 \\log M_{kCY} + \\log M_{3} + \\log M_{4} \\nonumber\n\\\\ & \\geq & 3 h_{Y} - \\epsilon_0 \\label{num333.5}\n\\\\ & \\geq & h_{Y}\n\\label{theorem3_proof_14.1.1}\n\\end{eqnarray}\n\n\nwhere \\eqref{num333.5} results from \\eqref{theorem3_proof_6}.\n\nThe security levels thus result:\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K|W_X, W_Y) & = & \\frac{1}{K} H(X|W_{X1} \\oplus W_{k1}, \\nonumber \n\\\\ && W_{X2} \\oplus W_{k2}, W_{CX} \\oplus W_{k3},\t\t\t\\nonumber\n\\\\ && W_{Y1} \\oplus W_{k4}, W_{Y2} \\oplus W_{k5},\t\\nonumber\n\\\\ &&  W_{CY}) \t\t\t\t\\nonumber\n\\\\ & = & H(X^K)\t\\label{num5.1.1.1}\n\\\\ & \\ge & h_X - \\epsilon^{'}_0\n\\label{theorem3_proof_16.1.1}\n\\end{eqnarray}\n\nwhere \\eqref{num5.1.1.1} holds because $W_{X1}$, $W_{X2}$, $W_{CX}$, $W_{Y1}$, $W_{Y2}$ are covered by key $W_{CY}$ and some shorter length key and $W_{CY}$ is covered by an existing random number key. \n\nSimilarly, \n\\begin{eqnarray}\n\\frac{1}{K} H(Y^K|W_X, W_Y) \\geq h_Y - \\epsilon^{'}_0\n\\label{theorem3_proof_17.1.1}\n\\end{eqnarray}\n\nTherefore ($R_X$, $R_Y$, $R_{kX}$, $R_{kY}$, $h_{XY}$, $h_{XY}$) is admissible from \\eqref{theorem2_proof_11.1.1} -  \\eqref{theorem2_proof_17.1.1}.\n\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem 4]\n\n\nAgain, the consideration for the security levels is that $h_Y \\geq h_X$ because $Y$ contains the key the is used for masking.\nSuppose that ($R_X$, $R_Y$, $R_{KX}$, $R_{KY}$) $\\in$ \n$\\mathcal{R}_3$. From \\eqref{theorem3} \n\\begin{eqnarray}\n&& R_X \\geq H(X^K|Y^K)  \t\t\t\t\\nonumber\n\\\\&& R_Y  \\geq H(Y^K|X^K) \t\t\t\t\\nonumber\n\\\\&& R_X + R_Y  \\geq H(X^K, Y^K)\n\\label{theorem4_proof_1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nR_{kX} \\geq h_{X}, R_{kY} \\geq h_{Y} \n\\label{theorem4_proof_2}\n\\end{eqnarray}\n\nAssuming a key exists for $V_{CY}$. For the first case, consider the following: $H(V_{CY}) \\geq H(V_X)$, $H(V_{CY}) \\geq H(V_Y)$ and $H(V_{CY}) \\geq H(V_{CX})$.\n \n\\begin{eqnarray}\nM_{CY} = 2^{K h_{Y}}\n\\label{theorem4_proof_6}\n\\end{eqnarray}\n\nIn the same way as theorem 2 and 3, the codewords $W_X$ and $W_Y$ and their keys $W_{kX}$ and $W_{kY}$ are now defined:\n\n\\begin{eqnarray}\nW_X = (W_{X1} \\oplus W_{kCY}, W_{X2}  \\oplus W_{kCY}, W_{CX}  \\oplus W_{kCY})\n\\label{theorem4_proof_7}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_Y = (W_{Y1} \\oplus W_{kCY}, W_{Y2}  \\oplus W_{kCY}, W_{CY})\n\\label{theorem4_proof_8}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{kY} = (W_{kCY})\n\\label{theorem4_proof_10}\n\\end{eqnarray}\n\nwhere $W_\\alpha \\in I_{M_\\alpha} = \\{0, 1, \\ldots, M_\\alpha - 1\\}$. The wiretapper will not know $W_{X1}$, $W_{X2}$ and $W_{CX}$ from $W_X$ and $W_{Y1}$, $W_{Y2}$ and $W_{CY}$ from $W_Y$ as these are protected by the key ($W_{kCY}$.\n\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_X + \\frac{1}{K} \\log M_Y  & = & \\frac{1}{K} (\\log M_{X1} + \\log M_{X2}  \\nonumber\n\\\\ & + &  \\log M_{CX}) + \\frac{1}{K} (\\log M_{Y1}  \\nonumber\n\\\\ & + &  \\log M_{Y2} +  \\log M_{CY}) \\nonumber \n\\\\ & \\le & H(X|Y) + H(Y|X) \t\t\t\t\\nonumber\n\\\\ & + & I(X;Y) + \\epsilon_0 \\nonumber\n\\\\ & = & H(X,Y)  \t\t\\nonumber\n\\\\ & \\le & R_X + R_Y\n\\label{theorem4_proof_11}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{kX} & = & \\frac{1}{K} \\log M_{CY}\t\\nonumber\n\\\\ & = & h_{Y}\t\t\t\\label{num4.2}\n\\\\ & \\geq & h_X - \\epsilon_0  \\label{num4.3}\n\\\\ &  & R_{kX}\n\\label{theorem4_proof_13}\n\\end{eqnarray}\n\nwhere \\eqref{num4.2} comes from \\eqref{theorem4_proof_6} and \\eqref{num4.3} comes form the consideration stated at the beginning of this proof.\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{kY} & = & \\frac{1}{K} \\log M_{CY}\t\\nonumber\n\\\\ & = & h_{XY}\t\t\t\\label{num5.1}\n\\\\ & \\le & R_{kY}\n\\label{theorem4_proof_14}\n\\end{eqnarray}\nwhere \\eqref{num5.1} comes from \\eqref{theorem4_proof_6}.\n\nThe security levels thus result:\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K|W_X, W_Y) & = & \\frac{1}{K} H(X|W_{X1} \\oplus W_{kCY}, \n\\\\ && W_{X2} \\oplus W_{kCY}, W_{CX} \\oplus W_{kCY},\t\t\t\\nonumber\n\\\\ && W_{Y1} \\oplus W_{kCY}, W_{Y2} \\oplus W_{kCY},\t\\nonumber\n\\\\ &&  W_{CY}) \t\t\t\t\\nonumber\n\\\\ & = & H(X^K)\t\\label{num6.1}\n\\\\ & \\ge & h_X - \\epsilon^{'}_0\n\\label{theorem4_proof_16}\n\\end{eqnarray}\n\nwhere \\eqref{num6.1} holds because $W_{X1}$, $W_{X2}$, $W_{CX}$, $W_{Y1}$, $W_{Y2}$ are covered by key $W_{CY}$ and $W_{CY}$ is covered by an existing random number key. Equations \\eqref{lemma1_1} - \\eqref{lemma1_7} imply that $W_{X1}$, $W_{X2}$, $W_{Y1}$ and $W_{Y2}$ have almost no redundancy and they are mutually independent.\n\nSimilarly, \n\\begin{eqnarray}\n\\frac{1}{K} H(Y^K|W_X, W_Y) \\geq h_Y - \\epsilon^{'}_0\n\\label{theorem4_proof_17}\n\\end{eqnarray}\n\nTherefore ($R_X$, $R_Y$, $R_{kX}$, $R_{kY}$, $h_{XY}$, $h_{XY}$) is admissible from \\eqref{theorem2_proof_11} -  \\eqref{theorem2_proof_17}.\n\nNext the case where: $H(V_{CY}) < H(V_X)$, $H(V_{CY}) < H(V_Y)$ and $H(V_{CY}) < H(V_{CX})$ is considered. Here, there are shorter length keys used in addition to the key provided by $W_{CY}$ in order to make the key lengths required by the individual portions. For example the key $W_{k1}$ comprises $W_{kCY}$ and a short key $W_1$, which together provide the length of $W_{X1}$.\nThe codewords $W_X$ and $W_Y$ and their keys $W_{kX}$ and $W_{kY}$ are now defined:\n\n\\begin{eqnarray}\nW_X = (W_{X1} \\oplus W_{k1}, W_{X2}  \\oplus W_{k2}, W_{CX}  \\oplus W_{k3})\n\\label{theorem4_proof_7.1.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_Y = (W_{Y1} \\oplus W_{k4}, W_{Y2}  \\oplus W_{k5}, W_{CY})\n\\label{theorem4_proof_8.1.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{kX} = (W_{k1}, W_{k2}, W_{k3})\n\\label{theorem4_proof_10.1.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{kY} = (W_{k4}, W_{k5})\n\\label{theorem4_proof_10.1.2}\n\\end{eqnarray}\n\nwhere $W_\\alpha \\in I_{M_\\alpha} = \\{0, 1, \\ldots, M_\\alpha - 1\\}$. The wiretapper will not know $W_{X1}$, $W_{X2}$ and $W_{CX}$ from $W_X$ and $W_{Y1}$, $W_{Y2}$ and $W_{CY}$ from $W_Y$ as these are protected by the key ($W_{kCY}$.\n\nIn this case, $R_X$, $R_Y$, $R_{kX}$ and $R_{kY}$ satisfy that\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_X + \\frac{1}{K} \\log M_Y  & = & \\frac{1}{K} (\\log M_{X1} \\nonumber\n\\\\ & + & \\log M_{X2} +  \\log M_{CX}) \\nonumber\n\\\\ & + & \\frac{1}{K} (\\log M_{Y1} + \\log M_{Y2} +  \\log M_{CY}) \\nonumber\n\\\\ & \\le & H(X|Y) + H(Y|X) + I(X;Y) + \\epsilon_0\t\t\t\t\\nonumber\n\\\\ & = & H(X,Y)  \t\t\\nonumber\n\\\\ & \\le & R_X + R_Y\n\\label{theorem4_proof_11.1.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{kX} & = & \\frac{1}{K} [\\log M_{k1} + \\log M_{k2} + \\log M_{k3}]\t\\nonumber\n\\\\ & = & \\log M_{kCY} + \\log M_{1} +\t\\log M_{kCY} \\nonumber\n\\\\ & + & \\log M_{2} + \\log M_{kCY} + \\log M_{3} \\nonumber\n\\\\ & = & 3 \\log M_{kCY} + \\log M_{1} + \\log M_{2} + \\log M_{3} \\nonumber\n\\\\ & \\geq & 3 h_{Y} - \\epsilon_0  \\label{num335.1}\n\\\\ & \\geq & 3 h_{X} - \\epsilon_0 \\nonumber\n\\\\ & \\geq & h_{X}\n\\label{theorem4_proof_13.1.1}\n\\end{eqnarray}\n\nwhere \\eqref{num335.1} results from \\eqref{theorem4_proof_6} and the result is from the consideration at the beginning of this proof.\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{kY} & = & \\frac{1}{K} [\\log M_{k3} + \\log M_{k4} + \\log M_{kCY}]\t\\nonumber\n\\\\ & = & \\log M_{kCY} + \\log M_{3} +\t\\log M_{kCY} \\nonumber\n\\\\ & + &  \\log M_{4} + \\log M_{kCY}  \\nonumber\n\\\\ & = & 3 \\log M_{kCY} + \\log M_{3} + \\log M_{4} \\nonumber\n\\\\ & \\geq & 3 h_{Y} - \\epsilon_0 \\label{num336.5}\n\\\\ & \\geq & h_{Y}\n\\label{theorem4_proof_14.1.1}\n\\end{eqnarray}\n\n\nwhere \\eqref{num336.5} results from \\eqref{theorem4_proof_6}.\n\nThe security levels thus result:\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K|W_X, W_Y) & = & \\frac{1}{K} H(X|W_{X1} \\oplus W_{k1}, \n\\\\ && W_{X2} \\oplus W_{k2}, W_{CX} \\oplus W_{k3},\t\t\t\\nonumber\n\\\\ && W_{Y1} \\oplus W_{k4}, W_{Y2} \\oplus W_{k5},\t\\nonumber\n\\\\ &&  W_{CY}) \t\t\t\t\\nonumber\n\\\\ & = & H(X^K)\t\\label{num6.1.1.1}\n\\\\ & \\ge & h_X - \\epsilon^{'}_0\n\\label{theorem4_proof_16.1.1}\n\\end{eqnarray}\n\nwhere \\eqref{num6.1.1.1} holds because $W_{X1}$, $W_{X2}$, $W_{CX}$, $W_{Y1}$, $W_{Y2}$ are covered by key $W_{CY}$ and some shorter length key and $W_{CY}$ is covered by an existing random number key. \n\nSimilarly, \n\\begin{eqnarray}\n\\frac{1}{K} H(Y^K|W_X, W_Y) \\geq h_Y - \\epsilon^{'}_0\n\\label{theorem4_proof_17.1.1}\n\\end{eqnarray}\n\nTherefore ($R_X$, $R_Y$, $R_{kX}$, $R_{kY}$, $h_{XY}$, $h_{XY}$) is admissible from \\eqref{theorem4_proof_11.1.1} -  \\eqref{theorem4_proof_17.1.1}.\n\nThe region indicated for $\\mathcal{R_4}$ is derived from this region for $\\mathcal{R_3}$, when $h_X = 0$.\n\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem 5]\nAs before, $V_{CY}$ may be used as a key, however here we use $V_{CX}$ as the key in this proof to show some variation. \n\nNow the consideration for the security levels is that $h_X \\geq h_Y$ because $X$ contains the key that is used for masking.\nSuppose that ($R_X$, $R_Y$, $R_{KX}$, $R_{KY}$) $\\in$ \n$\\mathcal{R}_5$. From \\eqref{theorem3} \n\\begin{eqnarray}\n&& R_X \\geq H(X^K|Y^K)  \t\t\t\t\\nonumber\n\\\\&& R_Y  \\geq H(Y^K|X^K) \t\t\t\t\\nonumber\n\\\\&& R_X + R_Y  \\geq H(X^K, Y^K)\n\\label{theorem5_proof_1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nR_{kX} \\geq h_{X}, R_{kY} \\geq h_{Y} \n\\label{theorem5_proof_2}\n\\end{eqnarray}\n\nAssuming a key exists for $V_{CX}$. For the first case, consider the following: $H(V_{CX}) \\geq H(V_X)$, $H(V_{CX}) \\geq H(V_Y)$ and $H(V_{CX}) \\geq H(V_{CX})$.\n \n\\begin{eqnarray}\nM_{CX} = 2^{K h_{X}}\n\\label{theorem5_proof_6}\n\\end{eqnarray}\n\nThe codewords $W_X$ and $W_Y$ and their keys $W_{kX}$ and $W_{kY}$ are now defined:\n\n\\begin{eqnarray}\nW_X = (W_{X1} \\oplus W_{kCX}, W_{X2}  \\oplus W_{kCX}, W_{CX})\n\\label{theorem5_proof_7}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_Y = (W_{Y1} \\oplus W_{kCX}, W_{Y2}  \\oplus W_{kCX}, W_{CY}  \\oplus W_{kCX})\n\\label{theorem5_proof_8}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{kX} = (W_{kCX})\n\\label{theorem5_proof_10}\n\\end{eqnarray}\n\nwhere $W_\\alpha \\in I_{M_\\alpha} = \\{0, 1, \\ldots, M_\\alpha - 1\\}$. The wiretapper will not know $W_{X1}$, $W_{X2}$ and $W_{CX}$ from $W_X$ and $W_{Y1}$, $W_{Y2}$ and $W_{CY}$ from $W_Y$ as these are protected by the key $W_{kCX}$ and $W_{kCX}$ is protected by a random number key.\n\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_X + \\frac{1}{K} \\log M_Y  & = & \\frac{1}{K} (\\log M_{X1} + \\log M_{X2}  \\nonumber\n\\\\ & + &  \\log M_{CX}) + \\frac{1}{K} (\\log M_{Y1}  \\nonumber\n\\\\ & + &  \\log M_{Y2} +  \\log M_{CY}) \\nonumber \n\\\\ & \\le & H(X|Y) + H(Y|X) \t\t\t\t\\nonumber\n\\\\ & + & I(X;Y) + \\epsilon_0 \\nonumber\n\\\\ & = & H(X,Y)  \t\t\\nonumber\n\\\\ & \\le & R_X + R_Y\n\\label{theorem5_proof_11}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{kX} & = & \\frac{1}{K} \\log M_{CX}\t\\nonumber\n\\\\ & \\geq & h_X - \\epsilon_0  \\label{num6.31}\n\\\\ &  & R_{kX}\n\\label{theorem5_proof_13}\n\\end{eqnarray}\n\nwhere \\eqref{num6.31} comes from \\eqref{theorem5_proof_6}.\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{kY} & = & \\frac{1}{K} \\log M_{CX}\t\\nonumber\n\\\\ & = & h_{X}\t\t\t\\label{num6.32}\n\\\\ & \\geq & h_Y \\label{num6.32}\n\\\\ & \\le & R_{kY}\n\\label{theorem5_proof_14}\n\\end{eqnarray}\nwhere \\eqref{num6.32} comes from \\eqref{theorem5_proof_6} and \\eqref{num6.32} comes form the consideration stated at the beginning of this proof.\n\nThe security levels thus result:\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K|W_X, W_Y) & = & \\frac{1}{K} H(X|W_{X1} \\oplus W_{kCX}, \\nonumber\n\\\\ && W_{X2} \\oplus W_{kCX}, W_{CX}, W_{CY} \\oplus W_{kCX},\t\t\t\\nonumber\n\\\\ && W_{Y1} \\oplus W_{kCX}, W_{Y2} \\oplus W_{kCX}) \t\t\t\t\\nonumber\n\\\\ & = & H(X^K)\t\\label{num16.1}\n\\\\ & \\ge & h_X - \\epsilon^{'}_0\n\\label{theorem5_proof_16}\n\\end{eqnarray}\n\nwhere \\eqref{num6.1} holds because $W_{X1}$, $W_{X2}$, $W_{CX}$, $W_{Y1}$, $W_{Y2}$ are covered by key $W_{CY}$ and $W_{CY}$ is covered by an existing random number key. Equations \\eqref{lemma1_1} - \\eqref{lemma1_7} imply that $W_{X1}$, $W_{X2}$, $W_{Y1}$ and $W_{Y2}$ have almost no redundancy and they are mutually independent.\n\nSimilarly, \n\\begin{eqnarray}\n\\frac{1}{K} H(Y^K|W_X, W_Y) \\geq h_Y - \\epsilon^{'}_0\n\\label{theorem5_proof_17}\n\\end{eqnarray}\n\nTherefore ($R_X$, $R_Y$, $R_{kX}$, $R_{kY}$, $h_{XY}$, $h_{XY}$) is admissible from \\eqref{theorem2_proof_11} -  \\eqref{theorem2_proof_17}.\n\nNext the case where: $H(V_{CX}) < H(V_X)$, $H(V_{CX}) < H(V_Y)$ and $H(V_{CX}) < H(V_{CX})$ is considered. Here, there are shorter length keys used in addition to the key provided by $W_{CX}$ in order to make the key lengths required by the individual portions. For example the key $W_{k1}$ comprises $W_{kCX}$ and a short key $W_1$, which together are the length of $W_{X1}$.\nThe codewords $W_X$ and $W_Y$ and their keys $W_{kX}$ and $W_{kY}$ are now defined:\n\n\\begin{eqnarray}\nW_X = (W_{X1} \\oplus W_{k1}, W_{X2}  \\oplus W_{k2}, W_{CX})\n\\label{theorem5_proof_7.1.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_Y = (W_{CY} \\oplus W_{k3}, W_{Y1} \\oplus W_{k4}, W_{Y2}  \\oplus W_{k5})\n\\label{theorem5_proof_8.1.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{kX} = (W_{k1}, W_{k2}, W_{k3})\n\\label{theorem5_proof_10.1.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{kY} = (W_{k4}, W_{k5})\n\\label{theorem5_proof_10.1.2}\n\\end{eqnarray}\n\nwhere $W_\\alpha \\in I_{M_\\alpha} = \\{0, 1, \\ldots, M_\\alpha - 1\\}$. The wiretapper will not know $W_{X1}$, $W_{X2}$ and $W_{CX}$ from $W_X$ and $W_{Y1}$, $W_{Y2}$ and $W_{CY}$ from $W_Y$ as these are protected by the key ($W_{kCY}$.\n\nIn this case, $R_X$, $R_Y$, $R_{kX}$ and $R_{kY}$ satisfy that\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_X + \\frac{1}{K} \\log M_Y  & = & \\frac{1}{K} (\\log M_{X1} \\nonumber\n\\\\ & + & \\log M_{X2} +  \\log M_{CX}) \\nonumber\n\\\\ & + & \\frac{1}{K} (\\log M_{Y1} + \\log M_{Y2} \\nonumber\n\\\\ & + &  \\log M_{CY}) \\nonumber\n\\\\ & \\le & H(X|Y) + H(Y|X) + I(X;Y)  \\nonumber\n\\\\ & + & \\epsilon_0\t\t\t\t\\nonumber\n\\\\ & = & H(X,Y)  \t\t\\nonumber\n\\\\ & \\le & R_X + R_Y\n\\label{theorem5_proof_11.1.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{kX} & = & \\frac{1}{K} [\\log M_{k1} + \\log M_{k2} + \\log M_{k3}]\t\\nonumber\n\\\\ & = & \\log M_{kCX} + \\log M_{1} +\t\\log M_{kCX}  \\nonumber\n\\\\ & + & \\log M_{2} + \\log M_{kCX} + \\log M_{3} \\nonumber\n\\\\ & = & 3 \\log M_{kCX} + \\log M_{1} + \\log M_{2} + \\log M_{3} \\nonumber\n\\\\ & \\geq & 3 h_{X} - \\epsilon_0  \\label{num3335.1}\n\\\\ & \\geq & h_{X}\n\\label{theorem5_proof_13.1.1.1}\n\\end{eqnarray}\n\nwhere \\eqref{num3335.1} results from \\eqref{theorem5_proof_6}.\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{kY} & = & \\frac{1}{K} [\\log M_{k3} + \\log M_{k4} + \\log M_{kCY}]\t\\nonumber\n\\\\ & = & \\log M_{kCX} + \\log M_{3} +\t\\log M_{kCX} \\nonumber\n\\\\ & + & \\log M_{4} + \\log M_{kCX} \\nonumber\n\\\\ & = & 3 \\log M_{kCX} + \\log M_{3} + \\log M_{4} \\nonumber\n\\\\ & \\geq & 3 h_{X} - \\epsilon_0  \\label{num33335.1}\n\\\\ & \\geq & 3 h_{Y} - \\epsilon_0 \\label{num336.5}\n\\\\ & \\geq & h_{Y}\n\\label{theorem5_proof_14.1.1}\n\\end{eqnarray}\n\nwhere \\eqref{num33335.1} results from \\eqref{theorem5_proof_6} and \\eqref{num336.5} results from the consideration at the beginning of this proof.\n\n\nThe security levels thus result:\n\\begin{eqnarray}\n\\frac{1}{K} H(X^K|W_X, W_Y) & = & \\frac{1}{K} H(X|W_{X1} \\oplus W_{k1}, W_{X2} \\oplus W_{k2}, \\nonumber\n\\\\ && W_{CX}, W_{CY} \\oplus W_{k3},\t\t\t\\nonumber\n\\\\ && W_{Y1} \\oplus W_{k4}, W_{Y2} \\oplus W_{k5},\t\\nonumber\n\\\\ &&  W_{CY}) \t\t\t\t\\nonumber\n\\\\ & = & H(X^K)\t\\label{num6.1.1.1.1}\n\\\\ & \\ge & h_X - \\epsilon^{'}_0\n\\label{theorem5_proof_16.1.1.1}\n\\end{eqnarray}\n\nwhere \\eqref{num6.1.1.1.1} holds because $W_{X1}$, $W_{X2}$, $W_{CY}$, $W_{Y1}$, $W_{Y2}$ are covered by key $W_{CX}$ and some shorter length key and $W_{CX}$ is covered by an existing random number key. \n\nSimilarly, \n\\begin{eqnarray}\n\\frac{1}{K} H(Y^K|W_X, W_Y) \\geq h_Y - \\epsilon^{'}_0\n\\label{theorem5_proof_17.1.1}\n\\end{eqnarray}\n\nTherefore ($R_X$, $R_Y$, $R_{kX}$, $R_{kY}$, $h_{XY}$, $h_{XY}$) is admissible from \\eqref{theorem5_proof_11.1.1} -  \\eqref{theorem5_proof_17.1.1}.\n\n\\end{proof}\n\n\n\\subsection{Converse parts}\nFrom Slepian-Wolf's theorem we know that the channel rate must satisfy $R_X \\geq H(X|Y)$, $R_Y \\geq H(Y|X)$ and $R_X + R_Y \\geq H(X,Y)$ to achieve a low error probability when decoding.\nHence, the key rates are considered in this subsection. \n\\\\\n\\textit{Converse part of Theorem 2:}\n\\begin{eqnarray}\nR_{kX} & \\geq & \\frac{1}{K} log M_{kX} - \\epsilon\t\t\\nonumber\n\\\\ & \\geq & \\frac{1}{K} H(W_{kX}) - \\epsilon\t\t\t\\nonumber\n\\\\ & \\geq & \\frac{1}{K} H(W_{kX|W_1}) - \\epsilon\t\t\t\\nonumber\n\\\\ & = & \\frac{1}{K} [H(W_{kX}) - I(W_{kX}; W_1)] - \\epsilon\t\t\\nonumber\n\\\\ & = & \\frac{1}{K} [H((W_{kX|X, Y, W_1}) + I(W_{kX}; W_1) \t\t\\nonumber\n\\\\ & + & I(W_{kX};X|Y, W_1) + I(X, Y, W_{kX}|W_1) \t\t\\nonumber\n\\\\ & + & I(Y, W_{kX}|X, W_1) - I(W_{kX}; W_1)] - \\epsilon\t\t\\nonumber\n\\\\ & = & \\frac{1}{K} [H(X, Y|W_1) - H(X,Y|W_1, W_{kX})] - \\epsilon\t\t\\nonumber\n\\\\ & \\geq & h_{XY} - \\frac{1}{K} H(X,Y|W_1, W_{kX})  - \\epsilon  \\label{conv_1}\n\\\\ & = & h_{XY} - H(V_{CY})  - \\epsilon  \t\t\\nonumber\n\\\\ & = & h_{XY}  - \\epsilon\n\\end{eqnarray}\n\nwhere \\eqref{conv_1} results from equation \\eqref{cond8.1}. Here, we consider the extremes of $H(V_{CY})$ in order to determine the limit for $R_{kX}$. When this quantity is minimum then we are able to achieve the maximum bound of $h_{XY}$.\n\n\\begin{eqnarray}\nR_{kY} & \\geq & \\frac{1}{K} log M_{kY} - \\epsilon\t\t\\nonumber\n\\\\ & \\geq & \\frac{1}{K} H(W_{kY}) - \\epsilon\t\t\t\\nonumber\n\\\\ & \\geq & \\frac{1}{K} H(W_{kY|W_2}) - \\epsilon\t\t\\nonumber\n\\\\ & = & \\frac{1}{K} [H(W_{kY}) - I(W_{kY}; W_2)] - \\epsilon\t\t\\nonumber\n\\\\ & = & \\frac{1}{K} [H((W_{kY|X, Y, W_2}) + I(W_{kY}; W_2) \t\t\\nonumber\n\\\\ & + & I(W_{kY};X|Y, W_2) + I(X, Y, W_{kY}|W_2) \t\t\t\t\\nonumber\n\\\\ & + & I(Y, W_{kY}|X, W_2) - I(W_{kY}; W_2)] - \\epsilon\t\t\t\\nonumber\n\\\\ & = & \\frac{1}{K} [H(X, Y|W_2) - H(X,Y|W_2, W_{kY})] - \\epsilon\t\t\\nonumber\n\\\\ & \\geq & h_{XY} - \\frac{1}{K} H(X,Y|W_2, W_{kY})  - \\epsilon  \\label{conv_2}\n\\\\ & = & h_{XY} - H(V_{CX})  - \\epsilon   \t\\nonumber\n\\\\ & = & h_{XY}  - \\epsilon\n\\end{eqnarray}\n\n\nwhere \\eqref{conv_2} results from equation \\eqref{cond9}. Here, we consider the extremes of $H(V_{CX})$ in order to determine the limit for $R_{kY}$. When this quantity is minimum then we are able to achieve the maximum bound of $h_{XY}$.\n\n\n\\textit{Converse part of Theorem 3:}\n\\begin{eqnarray}\nR_{kX} & \\geq & \\frac{1}{K} log M_{kX} - \\epsilon\t\t\\nonumber\n\\\\ & \\geq & \\frac{1}{K} H(W_{kX}) - \\epsilon\t\t\\nonumber\n\\\\ & \\geq & \\frac{1}{K} H(W_{kX|W_1}) - \\epsilon\t\t\\nonumber\n\\\\ & = & \\frac{1}{K} [H(W_{kX}) - I(W_{kX}; W_1)] - \\epsilon\t\t\\nonumber\n\\\\ & = & \\frac{1}{K} [H((W_{kX|X, W_1}) + I(W_{kX}; W_1) \t\t\\nonumber\n\\\\ & + & I(X, W_{kX}|W_1) - I(W_{kX}; W_1)] - \\epsilon\t\t\\nonumber\n\\\\ & \\geq & \\frac{1}{K} I(X, W_{kX}|W_1) - \\epsilon\t\t\\nonumber\n\\\\ & = & \\frac{1}{K} [H(X|W_1) - H(X|W_1, W_{kX})] - \\epsilon\t\\nonumber\n\\\\ & \\geq & h_{X} - H(V_{CY})  - \\epsilon  \\label{conv_3}\n\\\\ & = & h_{X}  - \\epsilon\n\\end{eqnarray}\n\n\nwhere \\eqref{conv_3} results from \\eqref{cond7}. Here, we consider the extremes of $H(V_{CY})$ in order to determine the limit for $R_{kX}$. When this quantity is minimum then we are able to achieve the maximum bound of $h_{X}$.\n\n\\begin{eqnarray}\nR_{kY} & \\geq & \\frac{1}{K} log M_{kY} - \\epsilon\t\\nonumber\n\\\\ & \\geq & \\frac{1}{K} H(W_{kY}) - \\epsilon\t\t\\nonumber\n\\\\ & \\geq & \\frac{1}{K} H(W_{kY|W_2}) - \\epsilon\t\t\\nonumber\n\\\\ & = & \\frac{1}{K} [H(W_{kY}) - I(W_{kY}; W_2)] - \\epsilon\t\t\\nonumber\n\\\\ & = & \\frac{1}{K} [H((W_{kY|Y, W_2}) + I(W_{kY}; W_2) \t\t\\nonumber\n\\\\ & + & I(X, W_{kY}|W_2) - I(W_{kY}; W_2)] - \\epsilon\t\t\\nonumber\n\\\\ & \\geq & \\frac{1}{K} I(Y, W_{kY}|W_2) - \\epsilon\t\t\t\t\\nonumber\n\\\\ & = & \\frac{1}{K} [H(Y|W_2) - H(Y|W_2, W_{kY})] - \\epsilon\t\t\\nonumber\n\\\\ & \\geq & h_{Y} - H(V_{CX})  - \\epsilon  \\label{conv_4}\n\\\\ & = & h_{Y}  - \\epsilon\n\\end{eqnarray}\n\nwhere \\eqref{conv_3} results from \\eqref{cond8}. Here, we consider the extremes of $H(V_{CX})$ in order to determine the limit for $R_{kY}$. When this quantity is minimum then we are able to achieve the maximum bound of $h_{Y}$.\n\nSince theorems 4-5 also have key rates of $h_X$ and $h_Y$ for $X$ and $Y$ respectively we can use the same methods to prove the converse. \n\n\n\\section{Scheme for multiple sources}\n\nThe two correlated source model presented in Section II is generalised even further, and now concentrates on multiple correlated sources transmitting syndromes across multiple wiretapped links. This new approach represents a network scenario where there are many sources and one receiver. We consider the information leakage for this model for Slepian-Wolf coding and thereafter consider the Shannon's cipher system representation.\n\n\\subsection{Information leakage using Slepian-Wolf coding}\n\nHere, Figure~\\ref{fig:extended} gives a pictorial view of the new extended model for multiple correlated sources.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics [scale = 0.7]{extended.pdf}\n\\caption{Extended generalised model}\n\\label{fig:extended}\n\\end{figure}\n\n\nConsider a situation where there are many sources, which are part of the ${\\bf S}$ set:\n \n\\begin{eqnarray}\n{\\bf S} = \\{S_{1}, S_{2}, \\ldots, S_{n}\\} \\nonumber\n\\end{eqnarray}\nwhere $i$ represents the $i$th source ($i = 1,\\ldots, n$) and there are $n$ sources in total. Each source may have some correlation between some other source and all sources are part of a binary alphabet. There is one receiver that is responsible for performing decoding. The syndrome for a source $S_i$ is represented by $T_{S_i}$, which is part of the same alphabet as the sources. \n\nThe entropy of a source is given by a combination of a specific conditional entropy and mutual information. In order to present the entropy we first define the following sets:\n\n\n\n\\begin{itemize}\n\\item [-] The set, ${\\bf S}$ that contains all sources: ${\\bf S} = \\{S_1, S_2,\\ldots, S_n\\}$. \n\\item [-] The set, ${\\bf S}_t$ that contains $t$ unique elements from ${\\bf S}$ and ${\\bf S}_t$ $\\subseteq$ $\\bf{S}$, ${S}_i \\in {\\bf S}_t$, ${\\bf S}_t \\cup {\\bf S}_t^c$ $=$ $\\bf{S}$  and $|{\\bf S}_t|$ $= t$ \n\\end{itemize}\n\n\nHere, $H(S_i)$ is obtained as follows:\n\n\\begin{eqnarray}\nH(S_i) = H(S_i|{\\bf S}_{\\backslash S_i}) + \\displaystyle\\sum_{t=2}^{n} (-1)^{t-1} \\displaystyle\\sum_{\\text{all possible ${\\bf S}_t$}}^{} I({\\bf S}_t|{\\bf S}_t^c)\n\\label{entropy}\n\\end{eqnarray}\n\n\nHere, $n$ is the number of sources, $ H(S_i|{\\bf S}_{\\backslash S_i})$ denotes the conditional entropy of the source $S_i$ given $S_i$ subtracted from the set ${\\bf S}$ and $I({\\bf S}_t|{\\bf S}_t^c)$ denotes the mutual information between all sources in the subset ${\\bf S}_t$ given the complement of ${\\bf S}_t$.\nIn the same way as for two sources, the generalised probabilities and entropies can be developed. It is then possible to decode the source message for source $S_i$ by receiving all components related to $S_i$. This gives rise to the following inequality for $H(S_i)$ in terms of the sources:\n\\begin{eqnarray}\nH(S_i|{ {\\bf S}_{\\backslash S_i}}) & + & \\displaystyle\\sum_{t=2}^{n} (-1)^{t-1}\t\\nonumber \\displaystyle\\sum_{\\text{all possible ${\\bf S}_t$}}^{} I({\\bf S}_t|{\\bf S}_t^c) \n\\\\ & \\le & H(S_i) + \\delta\n\\label{3source_2}\n\\end{eqnarray}\n\nIn this type of model information from multiple links need to be gathered in order to determine the transmitted information for one source. Here, the common information between sources is represented by the $I({\\bf S}_t|{\\bf S}_t^c)$ term. The portions of common information sent by each source can be determined upfront and is an arbitrary allocation in our case. For example in a three source model where $X$, $Y$ and $Z$ are the correlated sources, the common information shared with $X$ and the other sources is represented as: $I(X;Y|Z)$ and $I(X;Z|Y)$. Each common information portion is divided such that the sources having access to it are able to produce a portion of it themselves. The common information $I(X;Y|Z)$ is divided into $V_{CX1}$ and $V_{CY1}$ where the former is the common information between $X$ and $Y$, produced by $X$ and the latter is the common information between $X$ and $Y$, produced by $Y$. Similarly, $I(X;Z|Y)$ consists of two common information portions, $V_{CX2}$ and $V_{CZ1}$ produced by $X$ and $Z$ respectively. \n\nAs with the previous model for two correlated sources, since wiretapping is possible there is a need to develop the information leakage for the model. The information leakages for this multiple source model is indicated in \\eqref{remark1} and \\eqref{remark2}. \n\n\\textit{Remark 1:} The leaked information for a source $S_i$ given the transmitted codewords $T_{S_i}$, is given by:\n\\begin{eqnarray}\nL^{S_i}_{T_{S_i}} = I(S_i ; T_{S_i})\n\\label{remark1}\n\\end{eqnarray}\n\nSince we use the notion that the information leakage is the conditional entropy of the source given the transmitted information subtracted from the source's uncertainty (i.e $H(S_i) - H(S_i| T_{S_i})$), the proof for \\eqref{remark1} is trivial. Here, we note that the common information is the minimum amount of information leaked. Each source is responsible for transmitting its own private information and there is a possibility that this private information may also be leaked. The maximum leakage for this case is thus the uncertainty of the source itself, $H(S_i)$.\n\nWe also consider the information leakage for a source $S_i$ when another source $S_{j_(j \\neq i)}$ has transmitted information. This gives rise to Remark 2.\n\\\\\n\\textit{Remark 2:} The leaked information for a source $S_i$ given the transmitted codewords $T_{S_j}$, where $i \\neq j$ is:\n\\begin{eqnarray}\nL^{S_i}_{T_{S_j}} & = & H(S_i) - H(S_i|T_{S_j})\t\\nonumber\n\\\\ & = & H(S_i) - [H(S_i) - I(S_i; T_{S_j})]\t\\nonumber\n\\\\ & = & I(S_i ; T_{S_j})\n\\label{remark2}\n\\end{eqnarray}\n\nThe information leakage for a source is determined based on the information transmitted from any other channel using the common information between them. The private information is not considered as it is transmitted by each source itself and can therefore not be obtained from an alternate channel. Remark 2 therefore gives an indication of the maximum amount of information leaked for source $S_i$, with knowledge of the syndrome $T_{S_j}$. \n\nThese remarks show that the common information can be used to quantify the leaked information. The common information provides information for more than one source and is therefore susceptible to leaking information about more than one source should it be compromised. This subsection gives an indication of the information leakage for the new generalised multiple correlated sources model when a source's syndrome and other syndromes are wiretapped.\n\n\\subsection{Information leakage for Shannon's cipher system}\n\nThis subsection details a novel masking method to minimize the key length and thereafter builds this multiple correlated source model on Shannon's cipher system. \n\nThe new masking method encompasses masking the conditional entropy portion with a mutual information portion. By masking, certain information is hidden and it becomes more difficult to obtain the information that has been masked. Masking can typically be done using random numbers, however we eliminate the need for random numbers that represent keys and rather use a common information to mask with. \n\nWe make the following assumptions:\n\\begin{itemize}\n\\item\nThe capacity of each link cannot be exhausted using this method.\n\\item\nA common information is used to mask certain private information and can be used to mask multiple times. Further, private information that needs to be masked always exists in this method.\n\\end{itemize}\nThe allocation of common information for transmission are done on an arbitrary basis. The objective of this subsection is to minimize the key lengths while achieving perfect secrecy. \n\nThe private information for source $i$ is given by $H(S_i|{ {\\bf S}_{\\backslash S_i}})$ according to \\eqref{entropy}, which is called $W_{S_i}$ and the common information associated with source $S_i$ is given by $W_{CS_i}$. First, choose a common information with which to mask. Then we take a part of  $W_{S_i}$, i.e.  ${W_{S_i}}^{'}$, that has entropy equal to $H(W_{CS_i})$, and mask as follows: \n\\begin{eqnarray}\nW_{S_i}^{'} \\oplus W_{CS_i}\n\\label{masking}\n\\end{eqnarray}\n\nWhen the two sequences are xor'ed the result is a single sequence that may look different to the originals. We then transmit the masked portion instead of the $W_{S_i}^{'}$ portion when transmitting $W_{S_i}$, thus providing added security. \nThis brings in the interesting factor that there are many possibilities for a specific mutual information to mask conditional entropy portions. For example when considering  three sources as before, it is possible to mask the private information for $X$, $Y$ and $Z$ with the common portion $I(X;Y;Z)$. If $Y$ is secure then this common information can be transmitted along $Y$'s channel, ensuring the information is kept secure. The ability to mask using the common information is a unique and interesting feature of this new model for multiple correlated sources. The underlying principle is that the secure link should transmit more common information after transmitting their private information. \n\nWe find that the lower bound for the channel rate when the masking approach is used is given by:\n\\begin{eqnarray}\nR_i^M \\geq H(S_1, \\ldots, S_n) - \\displaystyle\\sum_{t=2}^{n} \\displaystyle\\sum_{\\text{all possible ${\\bf S}_t$}} (t -1)  I({\\bf S}_t|{\\bf S}_t^c)\n\\end{eqnarray}\nwhere $R_i^M$ is the $i$th channel rate when masking is used. \n\nThe method works theoretically but may result in some concern practically as there may be a security compromise when common information is sent across non secure links. We see that if the $ W_{CS_i}$ component used for masking has been compromised then the private portion it has masked will also be compromised. A method to overcome this involves using two common information parts for masking. Equation \\eqref{masking} representing the masking would become:\n\\begin{eqnarray}\nW_{S_i}^{'} \\oplus W_{CS_i} \\oplus W_{CS_j}\n\\label{masking2}\n\\end{eqnarray}\n\nwhere $i \\neq j$ and both $W_{CS_i}$ and $W_{CS_j}$ are common information associated with source $S_i$. This way, if only $W_{CS_j}$ is compromised then $W_{S_i}$ is not compromised as it is still protected by $W_{CS_i}$. Here, combinations of common information are used to increase the security. The advantage with \\eqref{masking2} is that keys may be reused because common information may be shared by more than one source. Further, the method will not result in an increase in key length. \n\n\nThe Shannon's cipher system for this multiple source model is now presented in order to determine the rate regions for perfect secrecy. The multiple sources each have their own encoder and there is a universal decoder. Each source has an encoder represented by:\n\\begin{eqnarray}\nE_i : \\mathcal{S} \\times I_{W_{S_i}} & \\rightarrow & I_{W_{CS_i}} =  \\{0, 1, \\ldots, W_{S_i} - 1\\} \\nonumber \n\\\\ && I_{W_{CS_i}} =  \\{0, 1, \\ldots, W_{CS_i} - 1\\}\n\\label{ms_xencoder_fcn}\n\\end{eqnarray}\nwhere $I_{MPi}$ is the alphabet representing the private portion for source $S_i$ and $I_{MCi}$ is the alphabet representing the common information for source $S_i$.\nThe decoder at the receiver is defined as:\n\\begin{eqnarray}\nD : (I_{W_{S_i}}, I_{W_{CS_i}})  & \\times &  I_{Mk} \\rightarrow \\mathcal{S}\n\\end{eqnarray}\n\nThe encoder and decoder mappings are below:\n\\begin{eqnarray}\nW_i = F_{E_i} (S_i, W_{ki})\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\widehat{S_i} = F_{D_i} (W_i, W_{ki}, W_{\\{{kp}\\}})\n\\end{eqnarray}\n\nwhere $p = 1, \\ldots, n$, $p \\neq i$ and $W_{\\{{kp}\\}}$ represents the set of common information required to find $S_i$, and $\\widehat{S_i}$ is the decoded output. \n\nThe following conditions should be satisfied for the general cases:\n\n\\begin{eqnarray}\n\\frac{1}{K}\\log W_{S_i} \\le R_i +\\epsilon\n\\label{ms_cond1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K}\\log M_{ki} \\le R_{ki} +\\epsilon\n\\label{ms_cond2}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\text {Pr} \\{\\widehat{S_i} \\neq S_i\\} \\le \\epsilon\n\\label{ms_cond3}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} H(S_i|W_i) \\le h_i - \\epsilon\n\\label{ms_cond7}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} H(S_j|W_i) \\le h_j - \\epsilon\n\\label{ms_cond8}\n\\end{eqnarray}\n\nwhere $R_i$ is the the rate of source $S_i$'s channel and $R_{k_{i}}$ is the key rate of $S_i$. The security levels, for source $i$ and any other source $j$ are measured uncertainties $h_{i}$ and $h_j$ respectively. \n\\\\\\\\\nThe general cases considered are:\n\\\\ \\textit{Case 1:} When $T_{S_i}$ is leaked and $S_i$ needs to be kept secret.\n\\\\ \\textit{Case 2:} When $T_{S_i}$ is leaked and $S_i$ and\/or $S_j$ needs to be kept secret.\n\\\\\\\\\nThe admissible rate region for each case is defined as follows:\n\\\\ \\textit{Definition 1a:} ($R_i$, $R_{ki}$, $h_{i}$) is admissible for case 1 if there exists a code ($F_{E_{i}}$, $F_{D}$) such that \\eqref{ms_cond1} - \\eqref{ms_cond7} hold for any $\\epsilon \\rightarrow 0$ and sufficiently large $K$.\n\\\\ \\textit{Definition 1b:} ($R_i$, $R_{ki}$, $R_j$, $R_{kj}$, $h_{j}$) is admissible for case 2 if there exists a code ($F_{E_{i}}$, $F_{D}$) such that \\eqref{ms_cond1} - \\eqref{ms_cond3} and \\eqref{ms_cond8} hold for any $\\epsilon \\rightarrow 0$ and sufficiently large $K$.\n\\\\ \\textit{Definition 2:} The admissible rate regions are defined as:\n\n\\begin{eqnarray}\n\\mathcal{R}(h_{i}) = \\{(R_i, R_{ki}):\t\t\t\\nonumber\n\\\\(R_i, R_{ki}, h_{i} ) \\text{ is admissible for case 1} \\}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathcal{R}(h_{i}, h_{j}) = \\{(R_i, R_{ki}, R_j, R_{kj}):\t\t\t\\nonumber\n\\\\(R_i, R_{ki}, R_j, R_{kj}, h_{j} ) \\text{ is admissible for case 2} \\}\n\\end{eqnarray}\n\nThe theorems developed for these regions follow:\n\n\\textit{Theorem 6:} For $0 \\le h_{i} \\le I(S_i;S_n|S_n^c)$,\n\\begin{eqnarray}\n&& \\mathcal{R}_1(h_{i}) = \\{(R_i, R_{ki}): \t\t\\nonumber\n\\\\ && R_i \\geq H(S_i), \t\t\t\t\\nonumber\n\\\\ && R_{k_i} \\geq I({\\bf S}_t|{\\bf S}_t^c) \\}\t\t\t\n\\label{theorem6}\n\\end{eqnarray}\n\n\\textit{Theorem 7:} For $0 \\le h_{j} \\le H(S_i, S_j)$,\n\\begin{eqnarray}\n&& \\mathcal{R}_2(h_{i}, h_{j}) = \\{(R_i, R_{ki}, R_j, R_{kj}): \t\t\\nonumber\n\\\\ && R_i \\geq H(S_i, S_j), R_j \\geq H(S_i, S_j), \t\t\t\t\t\t\t\\nonumber\n\\\\ && R_{ki} \\geq I(S_i;S_j) \\text{ and } R_{kj} \\geq I(S_i;S_j) \\}\t\t\t\n\\label{theorem7}\n\\end{eqnarray}\n\nThe proofs for these theorems follow. The source information components are first identified. Assume the private portions of source $i$ and $j$ are given by $W_i$ and $W_j$ respectively.  \n\n\\begin{proof}[Theorem 6 proof]\nHere, $R_i \\geq H(S_i)$, $R_{ki} \\geq I({\\bf S}_t|{\\bf S}_t^c)$. For the case where $h_i > I({\\bf S}_t|{\\bf S}_t^c)$, the definitions for $W_{CS_i}$, $W_i$ and $W_{ki}$ follow:\n\n\\begin{eqnarray}\nW_{CS_i} = 2^{K I({\\bf S}_t|{\\bf S}_t^c)}\n\\label{theorem6_proof_41}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_i = (W_{Pi}, W_{kCi})\n\\label{theorem6_proof_43}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{ki} = W_{Ci}\n\\label{theorem6_proof_44}\n\\end{eqnarray}\n\nThe keys and uncertainties are calculated as follows:\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{i} & = & \\frac{1}{K} (\\log W_{S_i} + \\log W_{CS_i}) \\nonumber\n\\\\ & \\le & H(S_i|{\\bf S}_{\\backslash {S_i}}) + \\frac{1}{K} W_{CS_i} + \\epsilon_0 \t\\nonumber\n\\\\ & = & H(S_i|{\\bf S}_{\\backslash {S_i}}) + I({\\bf S}_t|{\\bf S}_t^c)  + \\epsilon_0 \\nonumber \n\\\\ & = & \\frac{1}{K} H(S_i) + \\epsilon_0\t\t\t\\nonumber\n\\\\ & \\le & R_i + \\epsilon_0\n\\label{theorem6_proof_45.0}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&& \\frac{1}{K} \\log M_{ki} \t\t\\nonumber\n\\\\ & = & \\frac{1}{K} \\log W_{CS_i}\t\\nonumber\n\\\\ & = & I({\\bf S}_t|{\\bf S}_t^c) \n\\\\ & \\le & R_{ki} + \\epsilon_0\n\\label{theorem6_proof_45.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&& \\frac{1}{K} H(S_i|W_{Pi}, W_{Ci}) \t\t\\nonumber\n\\\\ & \\geq & \\frac{1}{K} H(S_i) - \\epsilon_0^{'} \t\t\\nonumber\n\\\\ & = & h_i - \\epsilon_0^{'}\t\t\n\\label{theorem6_proof_47}\n\\end{eqnarray}\n\nFrom \\eqref{theorem6_proof_45.0} - \\eqref{theorem6_proof_47}, ($R_i$, $R_{ki}$, $h_i$) is admissible for $h_i > I({\\bf S}_t|{\\bf S}_t^c))$. We now consider the case where $h_i \\le I({\\bf S}_t|{\\bf S}_t^c))$ and define $W_{CS_i}$, $W_i$ and $W_{ki}$ as follows:\n\n\n\\begin{eqnarray}\nW_{CS_i} = 2^{K I({\\bf S}_t|{\\bf S}_t^c)}\n\\label{theorem6_proof_49}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{i} = (W_{Pi}, W_{kCi})\n\\label{theorem6_proof_53}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{ki} = W_{Ci}\n\\label{theorem6_proof_54}\n\\end{eqnarray}\n\nThe keys and uncertainties are calculated as follows:\n\\begin{eqnarray}\n&& \\frac{1}{K} \\log M_{ki} \t\t\\nonumber\n\\\\ & = & \\frac{1}{K} \\log W_{CS_i}\t\\nonumber\n\\\\ & = & I({\\bf S}_t|{\\bf S}_t^c)) \n\\\\ & \\le & R_{ki} + \\epsilon_0\n\\label{theorem6_proof_55}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&& \\frac{1}{K} H(S_i|W_{Pi}, W_{Ci}) \t\t\\nonumber\n\\\\ & \\geq & \\frac{1}{K} H(S_i|W_{Ci}) + I({\\bf S}_t|{\\bf S}_t^c) - \\epsilon_0^{'} \t\t\\nonumber\n\\\\ & = & H(S_i) - \\epsilon_0^{'} \\nonumber\n\\\\ & = & h_i - \\epsilon_0^{'}\n\\label{theorem6_proof_57}\n\\end{eqnarray}\n\nFrom \\eqref{theorem6_proof_55} - \\eqref{theorem6_proof_57} it is seen that ($R_i$, $R_{ki}$, $h_i$) is admissible for $h_i \\le I({\\bf S}_t|{\\bf S}_t^c))$.\n\\end{proof}\n\nTheorem 7 is proven in a similar manner. \n\n\\begin{proof}[Theorem 7 proof]\nHere, $R_i \\geq H(S_i, S_j)$, $R_j \\geq H(S_i, S_j)$, $R_{ki} \\geq I(S_i;S_j)$ and $R_{kj} \\geq I(S_i;S_j)$. For the case where $h_j \\le H(S_i;S_j)$, the definitions for $W_{CS_i}$, $M_{Cj}$ $W_i$, $W_{ki}$, $W_j$ and $W_{kj}$ follow:\n\n\\begin{eqnarray}\nW_{CS_i} = 2^{K I({\\bf S}_t|{\\bf S}_t^c)}\n\\label{theorem7_proof_40.9}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nM_{Cj} = 2^{K I({\\bf S}_t|{\\bf S}_t^c)}\n\\label{theorem7_proof_40.10}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_i = (W_{Pi}, W_{kCi})\n\\label{theorem7_proof_41}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{ki} = W_{Ci}\n\\label{theorem7_proof_42}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_j = (W_{Pj}, W_{kCj})\n\\label{theorem7_proof_43}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{kj} = W_{Cj}\n\\label{theorem7_proof_44}\n\\end{eqnarray}\n\nThe keys and uncertainties are calculated as follows:\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{i} & = & \\frac{1}{K} (\\log W_{S_i} + \\log W_{CS_i}) \\nonumber\n\\\\ & \\le & \\frac{1}{K} H(S_i|{\\bf S}_{\\backslash S_i}) + \\frac{1}{K} W_{CS_i} + \\epsilon_0 \t\\nonumber\n\\\\ & = & \\frac{1}{K} H(S_i|{\\bf S}_{\\backslash S_i}) + I({\\bf S}_t|{\\bf S}_t^c)  + \\epsilon_0 \t\n\\\\ & = & \\frac{1}{K} H(S_i) + \\epsilon_0\t\t\t\\nonumber\n\\\\ & \\le & R_i + \\epsilon_0\n\\label{theorem7_proof_45.0}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\frac{1}{K} \\log M_{j} & = & \\frac{1}{K} (\\log M_{Pj} + \\log M_{Cj}) \\nonumber\n\\\\ & \\le & \\frac{1}{K} H(S_j|{\\bf S}_{\\backslash S_j}) + \\frac{1}{K} M_{Cj} + \\epsilon_0 \t\\nonumber\n\\\\ & = & \\frac{1}{K} H(S_j|{\\bf S}_{\\backslash S_j}) + I(S_j;{\\bf S}_t|{\\bf S}_t^c)  + \\epsilon_0 \t\n\\\\ & = & \\frac{1}{K} H(S_j) + \\epsilon_0\t\t\t\\nonumber\n\\\\ & \\le & R_j + \\epsilon_0\n\\label{theorem7_proof_45.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&& \\frac{1}{K} \\log M_{ki} \t\t\\nonumber\n\\\\ & = & \\frac{1}{K} \\log W_{CS_i}\t\\nonumber\n\\\\ & = & I({\\bf S}_t|{\\bf S}_t^c)\n\\\\ & \\le & R_{ki} + \\epsilon_0\n\\label{theorem7_proof_46}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&& \\frac{1}{K} \\log M_{kj} \t\t\\nonumber\n\\\\ & = & \\frac{1}{K} \\log M_{Cj}\t\\nonumber\n\\\\ & = & I(S_i; S_j) + \\epsilon_0\n\\\\ & \\le & R_{ki} + \\epsilon_0\n\\label{theorem7_proof_46.1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&& \\frac{1}{K} H(S_j|W_{Pi}, W_{Ci}) \t\t\\nonumber\n\\\\ & \\geq & H(S_j) - H(S_i) - \\epsilon_0^{'} \t\t\\nonumber\n\\\\ & = & H(S_i, S_j) - H(S_i) - \\epsilon_0^{'} \\nonumber\n\\\\ & \\geq & h_j - H(S_i)\n\\\\ & = & h_j - h_i - \\epsilon_0^{'}\t\t\n\\label{theorem7_proof_47}\n\\end{eqnarray}\n\nFrom \\eqref{theorem7_proof_45.0} - \\eqref{theorem7_proof_47}, ($R_i$, $R_{ki}$, $R_j$, $R_{kj}$, $h_j$) is admissible for $h_j \\le H(S_i, S_j)$. We now consider the case where $h_j > H(S_i,S_j)$, and define $W_{CS_i}$, $W_i$ and $W_{ki}$ as follows:\n\n\\begin{eqnarray}\nW_{CS_i} = 2^{K I({\\bf S}_t|{\\bf S}_t^c)}\n\\label{theorem7_proof_40.91}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nM_{Cj} = 2^{K I({\\bf S}_t|{\\bf S}_t^c)}\n\\label{theorem7_proof_40.11}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_i = (W_{Pi}, W_{kCi})\n\\label{theorem7_proof_48}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{ki} = W_{Ci}\n\\label{theorem7_proof_49}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_j = (W_{Pj}, W_{kCj})\n\\label{theorem7_proof_50}\n\\end{eqnarray}\n\n\\begin{eqnarray}\nW_{kj} = W_{Cj}\n\\label{theorem7_proof_51}\n\\end{eqnarray}\n\nThe keys and uncertainties are calculated as follows:\n\n\\begin{eqnarray}\n&& \\frac{1}{K} \\log M_{ki} \t\t\\nonumber\n\\\\ & = & \\frac{1}{K} \\log W_{CS_i}\t\\nonumber\n\\\\ & = & I({\\bf S}_t|{\\bf S}_t^c)\n\\\\ & \\le & R_{ki} + \\epsilon_0\n\\label{theorem7_proof_52}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&& \\frac{1}{K} \\log M_{kj} \t\t\\nonumber\n\\\\ & \\le & I(S_i; S_j) + \\epsilon_0\n\\\\ & \\le & R_{ki} + \\epsilon_0\n\\label{theorem7_proof_53}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&& \\frac{1}{K} H(S_j|W_{Pi}, W_{Ci}) \t\t\\nonumber\n\\\\ & \\le & H(S_j) - H(S_i) + \\epsilon_0^{'} \t\t\\nonumber\n\\\\ & = & H(S_i, S_j) - H(S_i) + \\epsilon_0^{'} \\nonumber\n\\\\ & \\le & h_j - H(S_i)\n\\\\ & = & h_j - h_i - \\epsilon_0^{'}\t\t\n\\label{theorem7_proof_54}\n\\end{eqnarray}\n\nFrom \\eqref{theorem7_proof_52} - \\eqref{theorem6_proof_54} it is seen that ($R_i$, $R_{ki}$, $R_j$, $R_{kj}$, $h_j$) is admissible for $h_j \\le H(S_i,S_j)$.\n\\end{proof}\n\nThese theorems demonstrate the necessary rates required for perfect secrecy. The goal of the Shannon's cipher aspect was to reduce the key lengths. The masking method explained in this section is able to use common information as keys and therefore minimise the key rates for the general cases. \n\nThe information leakage described in the Slepian-Wolf aspect indicates the common information that should be given added protection to ensure that even less information will be leaked. \nThe new extended model presented here also incorporates a multiple correlated sources approach using Shannon's cipher system, which is more practical than looking at two sources. \n\n\n\\section{Comparison to other models}\nThe two correlated sources model across a channel with an eavesdropper is a more generalised approach of Yamamoto's~\\cite{shannon1_yamamoto} model. If we were to combine the links into one link, we would have the same situation as per Yamamoto's ~\\cite{shannon1_yamamoto}. From Section VI it is evident that the model can be implemented for multiple sources with Sahnnon's cipher system. Due to the unique scenario incorporating multiple sources and multiple links, the new model is more secure as private information and common information from other link\/s are required for decoding.\n\nFurther, information at the sources may be more secure in the new model because if one source is compromised then only one source's information is known. In Yamamoto's ~\\cite{shannon1_yamamoto} method both source's information is contained at one station and when that source is compromised then information about both sources are known. The information transmitted along the channels (i.e. the syndromes) do not have a fixed length as per Yamamoto's ~\\cite{shannon1_yamamoto} method. Here, the syndrome length may vary depending on the encoding procedure and nature of Slepian-Wolf codes, which is another feature of this generalised model. \n\nThe generalised model also has the advantage that varying amounts of the common information $V_{CX}$ and $V_{CY}$ (in the case of two sources) may be transmitted depending on the security of the transmission link and\/or sources. For example, for two correlated sources, if $Y$'s channel is not secure we can specify that more of the common information is transmitted from $X$. In this way we're able to make better use of the transmission link's security. For Yamamoto's ~\\cite{shannon1_yamamoto} method the common information was transmitted as one portion, $V_{C}$.\n\nIn this model, information from more than one link is required in order to determine the information for one source. This gives rise to added security as even if one link is wiretapped it is not possible to determine the contents of a particular source. This is attributed to the fact that this model has separate common information portions, which is different to Yamamoto's model. \n\nAnother major feature is that private information can be hidden using common information. Here, common information produced by a source may be used to mask its private codeword thus saving on key length. The key allocation is specified by general rules presented in Section VI. The multiple correlated sources model presents a combination masking scheme where more than one common information is used to protect a private information, which is a practical approach. This is an added feature developed in order to protect the system.  This approach has not been considered in the other models mentioned in this section. \n \nThe work by Yang \\textit{et al.}~\\cite{feedback_yang} uses the concept of side information to assist the decoder in determining the transmitted message. The side information could be considered to be a source and is related to this work when the side information is considered as correlated information. Similar work with side information that incorporates wiretappers, by Villard and Piantanida \\cite{pablo_secure_multiterminal} and Villard \\textit{et al.} \\cite{pablo_secure_transmission_receivers} may be generalised in the sense that side information can be considered to be a source, however this new model is distinguishable as syndromes, which are independent of one another are transmitted across an error free channel in the new model. Further, to the author's knowledge Shannon's cipher system has not been incorporated into these models by Villard and Piantanida \\cite{pablo_secure_multiterminal} and Villard \\textit{et al.} \\cite{pablo_secure_transmission_receivers}.  \n\n\n\n\\section{Future work}\nThis work has room for expansion and future work. It would be interesting to consider the case where the channel capacity has certain constraints (according to the assumptions in Section VI the channel capacity is enough at all times). In the new model, the channels are either protected by keys or not however this is limited and a real case scenario where there are varying security levels for the channels is an avenue for future work. Another aspect for expansion is to investigate the allocation of common information as keys to minimize additional keys with links having varying security levels and limited capacity. \n\n\n\n\\section{Conclusion}\nThe information leakage for two correlated sources across a channel with an eavesdropper was initially considered. Knowing which components contribute most to information leakage aids in keeping the system more secure, as these terms can be made more secure. The information leakage for the two correlated source model was quantified and proven. Shannon's cipher system was also incorporated for this model and channel and key rates to achieve perfect secrecy have been provided. The two correlated sources model has been extended for the network scenario where we consider multiple sources transmitting information across multiple links. The information leakage for this extended model is detailed. The channel and key rates are also considered for the multiple correlated source model when Shannon's chipher system is implemented. A masking method is further presented to minimize key lengths and a combination masking method is presented to address its practical shortcoming.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe $n$-vector model, introduced by Stanley in 1968 \\cite{S68} is described by the Hamiltonian $${\\mathcal H}(d,n) = -J\\sum_{\\langle i,j \\rangle} {\\bf s_i }\\cdot{\\bf  s_j} ,$$\nwhere $d$ denotes the dimensionality of the lattice, and ${\\bf s_i}$\nis an $n$-dimensional vector of magnitude $\\sqrt{n}$. When $n=1$ this Hamiltonian\ndescribes the Ising model, when $n=2$ it describes the classical\nXY model, and in the limit $n \\to 0,$ one recovers the self-avoiding walk (SAW) model, as first pointed out by de Gennes \\cite{dG72}. The $n$-vector model has been shown to be equivalent to a loop model with a weight $n$ assigned to each closed loop \\cite{DMNS81} and weight $x$ to each edge of the loop. The two-dimensional O($n$) model on a honeycomb lattice, which is the focus of this paper, is a particular case of this equivalence. The partition function of the loop model can be written as\n\\begin{equation}\nZ(x)=\\sum_{G}x^{l(G)}n^{c(G)},\n\\end{equation}\nwhere $G$ is a configuration of loops, $l(G)$ is the number of loop segments and $c(G)$ is the number of closed loops. The parameter $x$ is defined by the high-temperature expansion of the $O(n)$ model partition function and is related to the coupling $J$, the temperature $T$ and Boltzmann's constant $k$ by \n\\begin{equation}\n{\\rm e}^{ \\frac{J}{kT} {\\bf s_i }\\cdot{\\bf  s_j} } \\approx 1+ x{\\bf s_i }\\cdot{\\bf  s_j}.\n\\end{equation}\n\nIn 1982 Nienhuis \\cite{N82} showed that, for $n \\in [-2,2],$ the model on the honeycomb lattice could be mapped onto a solid-on-solid model, from which he was able to derive the critical points and critical exponents, subject to some plausible assumptions. These results agreed with the known exponents and critical point for the Ising model, and predicted exact values for those models corresponding to other values of the spin dimensionality $n.$ In particular, for $n=0$ the critical point for the honeycomb lattice SAW model was predicted to be $x_{\\rm c}=1\/\\sqrt{2+\\sqrt{2}},$ a result finally proved 28 years later by Duminil-Copin and Smirnov \\cite{DC-S10}. \n\nThe proof of Duminil-Copin and Smirnov involves the use of a non-local \\emph{parafermionic observable} $F(z)$ where $z$ is the (complex) coordinate of the plane. This function can be thought of as a complex function with the ``parafermionic'' property $F({\\rm e}^{2\\pi{\\rm i}} z) = {\\rm e}^{-2\\pi{\\rm i} \\sigma} F(z)$ where the real-valued parameter $\\sigma$ is called the $\\emph{spin}$. For special values of $\\sigma$, this observable satisfies a discrete analogue of (one half of) the Cauchy-Riemann equations. This \\emph{discrete holomorphic} or \\emph{preholomorphic} property allowed Smirnov and Duminil-Copin to derive an important identity for self-avoiding walks on the honeycomb lattice and, consequently, the Nienhuis prediction for $x_{\\rm c}$.\n\nSmirnov~\\cite{Smirnov10} has also derived such an identity for the general honeycomb $O(n)$ model with $n \\in [-2,2]$. This identity provides an alternative way of predicting the value of the critical point $x_{\\rm c}(n)=1\/\\sqrt{2+\\sqrt{2-n}}$ as conjectured by Nienhuis for values of $n$ other than $n=0$.\n\nThis paper contains two new results. We first present an off-critical deformation of the discrete Cauchy-Riemann equations, by relaxing the preholomorphicity condition, which allows us to consider critical exponents near criticality. Indeed, this deformation gives rise to an identity between bulk and boundary generating functions, and we utilize this identity in Section~\\ref{ssec:wedge} to determine, based on some assumptions, the asymptotic form of the winding angle distribution function for SAWs on the half-plane and in a wedge in terms of boundary critical exponents. It is important to note that up to this point the only assumptions we make are the existence of the critical exponents and the value of the critical point. We will not rely on Coulomb gas techniques or conformal invariance. We find perfect agreement with the conjectured winding angle distribution function on the cylinder predicted by Duplantier and Saleur \\cite{DS88} in terms of bulk critical exponents. Finally we conjecture the values of the wedge critical exponents as a function of the wedge angle for $n\\in [-2, 2)$.\n\n\\section{Off-critical identity for the honeycomb O($n$) model}\n\\subsection{Smirnov's observable on the honeycomb lattice}\nWe briefly review an important result of Smirnov for self-avoiding walks on the honeycomb lattice. \n\nFirstly, a $\\emph{mid-edge}$ is defined to be a point equidistant from two adjacent vertices on a lattice. A \\emph{domain} $\\Omega$ is a simply connected collection of mid-edges on the half-plane honeycomb lattice. The set of vertices of the half-plane honeycomb lattice is denoted $V(\\Omega)$. Those mid-edges of $\\Omega$ which are adjacent to only one vertex in $V(\\Omega)$ form $\\partial\\Omega$. \n\\begin{figure}\n\\centering\n\\begin{picture}(150,180)\n\\put(0,180){\\includegraphics[scale=0.5, angle=270]{exampleF}}\n\\put(-10,85){$a$}\n\\put(15,85){$v_a$}\n\\put(48,122){$z$}\n\\put(48,143){$v$}\n\\end{picture}\n\\caption{A configuration $\\gamma$ on a finite domain. Point $a$ is a boundary mid-edge, point $z$ is another mid-edge, and $v_a$ and $v$ are corresponding vertices. The contribution of $\\gamma$ to $F(z)$ is ${\\rm e}^{-2{\\rm i}\\sigma\\pi\/3}x^{30} n$.\n} \\label{fig:exampleF}\n\\end{figure}\nLet $\\gamma$ be a configuration on a domain $\\Omega$ comprising a single self-avoiding walk and a number (possibly zero) of closed loops. We denote by $\\ell(\\gamma)$ the number of vertices occupied by $\\gamma$ and $c(\\gamma)$ the number of closed loops. Furthermore let $W(\\gamma)$ be the winding angle of the self-avoiding walk component. Define the following observable.\n\\begin{definition}[Preholomorphic observable]\n\\label{def:Fdef}\n\\mbox{}\\newline\n\\begin{itemize}\n\\item For $a\\in\\partial\\Omega, z\\in\\Omega$, set\n\\begin{equation}\nF(\\Omega,z;x,n,\\sigma):=F(z) = \\sum_{\\gamma: a\\rightarrow z} {\\rm e}^{-{\\rm i} \\sigma W(\\gamma)} x^{\\ell(\\gamma)} n^{c(\\gamma)},\n\\label{eq:Fmidedge}\n\\end{equation}\nwhere the sum is over all configurations $\\gamma$ for which the SAW component runs from the mid-edge $a$ to a mid-edge $z$ (we say that $\\gamma$ ends at $z$).\n\\item Let $F(p;v)$ only include configurations where there is a walk terminating at the mid-edge $p$ adjacent to the vertex $v$ and the other two mid-edges adjacent to $v$ are not occupied by a loop segment. For $v_a, v\\in V(\\Omega)$ and $p, q$ and $r$ mid-edges adjacent to $v$, set\n\\begin{align}\n\\overline{F}(V(\\Omega),v;x,n,\\sigma):=\\overline{F}(v) = (p-v)F(p;v)+(q-v)F(q;v)+(r-v)F(r;v),\n\\label{eq:Fvertex}\n\\end{align}\nSince this is a function involving walks that terminate at mid-edges adjacent to the vertex $v$ we consider this as a function defined at the vertices of the lattice.\n\\end{itemize}\nSee Fig.~\\ref{fig:exampleF} for an example:\n\\end{definition}\nSmirnov \\cite{Smirnov10} proves the following: \n\\begin{lemma}[Smirnov]\n\\label{lem:Slemma4}\nFor $n\\in[-2,2]$, set $n=2\\cos\\phi$ with $\\phi\\in[0,\\pi]$. Then for\n\\begin{align}\n\\sigma &= \\frac{\\pi-3\\phi}{4\\pi},\\qquad x^{-1}=x_{\\rm c}^{-1}=2\\cos\\left(\\frac{\\pi+\\phi}{4}\\right) = \\sqrt{2-\\sqrt{2-n}},\\qquad\\text{or}\n\\label{eq:Slemma_dense}\\\\\n\\sigma &= \\frac{\\pi+3\\phi}{4\\pi},\\qquad x^{-1}=x_{\\rm c}^{-1}=2\\cos\\left(\\frac{\\pi-\\phi}{4}\\right) = \\sqrt{2+\\sqrt{2-n}},\n\\label{eq:Slemma_dilute}\n\\end{align}\nthe observable $F$ satisfies the following relation for every vertex $v\\in V(\\Omega)$:\n\\begin{equation} \\label{eqn:localidentity}\n(p-v)F(p) + (q-v)F(q) + (r-v)F(r)=0,\n\\end{equation}\nwhere $p,q,r$ are the mid-edges adjacent to $v$.\n\\end{lemma}\n\nThe first equation in Lemma~\\ref{lem:Slemma4} corresponds to the larger of the two critical values of the step weight $x$ and thus describes the so-called dense regime as configurations with many loops are favoured. The second equation corresponds to the line of critical points separating the dense and dilute phases \\cite{N82}. Eqn. (\\ref{eqn:localidentity}) can be interpreted as the vanishing of a discrete contour integral, hence the name preholomorphic observable for $F(z)$.\n\\begin{figure}\n\\centering\n\\begin{picture}(250,180)\n\\put(0,180){\\includegraphics[scale=1.1, angle=270]{identity_groups}}\n\\put(48,8){$p$}\n\\put(65, 20){$r$}\n\\put(58, 0){$q$}\n\\end{picture}\n\\caption{The two types of configurations which end at mid-edges $p,q,r$ adjacent to vertex $v$. The first type, on the left, involves configurations which visit all three mid-edges. On the right are those configurations where the self-avoiding walk visits at most two mid-edges.} \n\\label{fig:identity_groups}\n\\end{figure}\n\\begin{proof}\nConsider a vertex $v$ adjacent to a mid-edge $p$. The two other adjacent mid-edges we refer to as $q$ and $r$ and are labelled as shown in Fig.~\\ref{fig:identity_groups}. For a self-avoiding walk entering the vertex $v$ from the mid-edge $p$ and terminating at either $p, q$ or $r$ there are two disjoint sets of configurations to consider, each corresponding to a different external connectivity of the remaining mid-edges $q$ and $r$. These are also shown in Fig. (\\ref{fig:identity_groups}). Since the two sets of configurations are disjoint we can consider the identity (\\ref{eqn:localidentity}) for each case separately.\nIn the following, we define $\\lambda={\\rm e}^{-{\\rm i}\\sigma\\pi\/3}$ (the weight accrued by a walk for each left turn) and $j={\\rm e}^{2{\\rm i}\\pi\/3}$ (the value of $p-v$ when mid-edge $p$ is to the north-west of its adjacent vertex $v$).\n\\begin{enumerate}\n\\item\nIn the first case, we consider all configurations where mid-edges $p$ and $q$ are connected. There are three ways for this to occur: two with the self-avoiding walk visiting all three sites, and one with a closed loop running through $v$. Furthermore, we define $F_L(P;v)$ to be the contribution to $F(p)$ involving only configurations where the walk ends at the point $p$, adjacent to the vertex $v$ and where the two other mid-edges adjacent to $v$ are occupied by a closed loop. Requiring \\eqref{eqn:localidentity} to hold then implies\n\\begin{equation}\n(p-v)F_L(p;v)+(q-v)\\frac{1}{n}\\bar{\\lambda}^4F_L(p;v)+(r-v)\\frac{1}{n}\\lambda^4F_L(p;v)=0.\n\\label{eqn:localidentity2}\n\\end{equation}\nThe factor of $\\frac{1}{n}$ arises from the absence of a closed loop and the complex phase factors arise from the additional winding: the loop makes an additional four left turns to arrive at $q$ and four right turns to arrive at $r$. Substituting these into \\eqref{eqn:localidentity2} we find\n\\begin{equation}\n\\frac{1}{n} (p-v)F_L(p;v)(-n-\\bar{\\lambda}^4 j- \\lambda^4\\bar{j} )=0, \\nonumber\n\\end{equation}\nwhere we have used that $q-v=j(p-v)$ and $r-v=\\bar{j}(p-v)$. Since the choice of $v$ and $p$ was arbitrary this implies\n\\begin{equation}\nn+\\lambda^4 \\bar{j} +\\bar{\\lambda}^{4}j=0. \\nonumber\n\\end{equation}\nFinally, using the parameterisation of $n$ in terms of $\\phi$ and solving for $\\sigma$ we obtain\n\\begin{align}\n\\sigma &= \\frac{\\pi-3\\phi}{4\\pi} \\qquad \\text{for } \\lambda^4 \\bar{j} = -{\\rm e}^{{\\rm i}\\phi},\n\\label{eq:sigmadense}\\\\\n\\sigma &= \\frac{\\pi+3\\phi}{4\\pi} \\qquad \\text{for } \\lambda^4 \\bar{j} = -{\\rm e}^{-{\\rm i}\\phi},\n\\label{eq:sigmadilute}\n\\end{align}\n\\item\nIn the second case only one or two mid-edges are occupied in the configuration and mid-edges $q$ and $r$ are not connected.\nRecalling the definition of $F(p; v)$ in (\\ref{def:Fdef}) and Eqn. \\eqref{eqn:localidentity} we have\n\\begin{equation}\n(p-v)F(p; v)+(q-v)x\\bar{\\lambda} F(p; v)+(r-v)x\\lambda F(p; v)=0,\n\\end{equation}\nwhich simplifies to\n\\begin{equation}\nF(p; v)(-1-x_{\\rm c}\\bar{\\lambda}j-x_{\\rm c}\\lambda\\bar{j})=0.\n\\end{equation}\nAgain, since this equation holds for arbitrary $v$ we obtain\n\\begin{equation}\n1+x_{\\rm c}\\lambda\\bar{j}+x_{\\rm c}\\bar{\\lambda}j=0,\n\\end{equation}\nwhich leads to\n\\begin{equation}\nx_{\\rm c}^{-1}=2\\cos\\left(\\frac\\pi3(\\sigma-1)\\right).\n\\end{equation}\n\\end{enumerate}\nThe two possible values of $\\sigma$ give rise to the corresponding two values for $x_{\\rm c}$.\n\\end{proof}\n\n\\subsection{Off-critical deformation}\nFirst we evaluate the discrete divergence of the second set of configurations in Fig.~\\ref{fig:exampleF} for general $x$, below the critical value. This gives\n\\begin{lemma}[Massive preholomorphicity identity]\n\\label{lem:massiveCR}\nFor a given vertex $v$ with mid-edges $p$, $q$ and $r,$ and $x$ below the critical value $x_{\\rm c}$, the parafermionic observable $F(z)$ satisfies\n\\begin{align}\n\\label{eqn:massiveCR}\n(p-v)F(p)+(q-v)F(q)+(r-v)F(r)=(1-\\frac{x}{x_{\\rm c}}) \\overline{F}(v),\n\\end{align}\nwhere $F(z)$ and $\\overline{F}(v)$ are defined in Definition \\ref{def:Fdef}.\n\\end{lemma}\nWe use the term massive preholomorphic as \\eqref{eqn:massiveCR} is of a similar form to that described in \\cite{MS09} and \\cite{Smirnov10}.\n\\begin{proof}\nSimilar to Lemma~\\ref{lem:Slemma4} the proof splits into two parts. The first part, concerning cancellations of contributions coming from walks depicted in the left-hand side of Fig. \\ref{fig:identity_groups}, is completely analogous, and as before fixes the value of $\\sigma$. The difference is that now we relax the requirement that the contribution from the second set of configurations (shown on the right in Fig.~\\ref{fig:identity_groups}) to the discrete contour integral vanishes. Consequently,  $x$ is no longer fixed to the critical value.\n\nConsider a vertex $v$ with mid-edges labelled $p$, $q$ and $r$ in a counter-clockwise fashion. There are three disjoint sets of configurations, depending on which of the three mid-edges $p$, $q$ or $r$ the walk enters from. These are shown in Fig. \\ref{fig:pqrvertex}. Recall that we denote by $F(p;v)$ the contributions to $F(p)$ that only include configurations where there is a walk terminating at the mid-edge $p$ adjacent to the vertex $v$ and where the two other mid-edges adjacent to $v$ are unoccupied. The contribution to the left-hand side of \\eqref{eqn:massiveCR} from walks entering the vertex from $p$ is the sum of three terms\n\\begin{eqnarray}\n\\label{eqn:venterp}\n&&(p-v)F(p;v)+(q-v)x\\,{\\rm e}^{\\pi\\sigma{\\rm i}\/3}F(p;v)+(r-v)x\\,{\\rm e}^{-\\pi\\sigma{\\rm i}\/3}F(p;v).\n\\end{eqnarray}\nThe first term is simply from walks that enter and terminate at $p$. The second term is from those walks that enter from $p$, make a right turn and terminate at $q$. The final term is  from walks that enter at $p$ and make a left turn to terminate at $r$. The last two terms acquire an additional weight $x$ from the extra step and a phase factor from the turn. We can simplify \\eqref{eqn:venterp} to obtain\n\\begin{eqnarray*}\n&=& (p-v)F(p;v)+(p-v)xj\\bar{\\lambda}F(p;v)+(p-v)x\\lambda \\bar{j}F(p;v) \\\\\n&=& (p-v)F(p;v)(1-xj\\bar{\\lambda} -x\\bar{j}\\lambda)\\\\\n&=& (p-v)F(p;v)(1-\\frac{x}{x_{\\rm c}}),\n\\end{eqnarray*}\nwhere in the first line we have used that \n\\[\nq-v=j(p-v),\\qquad r-v=\\bar{j}(p-v).\n\\]\nFor walks entering from mid-edges $q$ and $r$ similar calculations give contributions\n\\[\n(q-v)F(q;v)(1-\\frac{x}{x_{\\rm c}}) \\text{ and }\n(r-v)F(r;v)(1-\\frac{x}{x_{\\rm c}}).\n\\]\nAdding the three contributions together and using Definition~\\ref{def:Fdef} gives the right-hand side of Eqn. \\eqref{eqn:massiveCR}. \n\\end{proof}\nUsing the above lemma we can now derive the following relationship between generating functions. \\\\\n\\begin{lemma}[Off-critical generating function identity]\n\\label{lem:offcriticalDCS}\n\\begin{align}\n\\sum_{\\gamma : a \\to z\\in\\partial \\Omega\\backslash\\{a\\}} e^{{\\rm i} \\tilde{\\sigma} W(\\gamma)} x^{|\\gamma|}n^{c(\\gamma)}  + (1- x\/x_c)\\sum_{\\gamma : a \\to z\\in\\Omega\\backslash \\partial \\Omega} e^{{\\rm i} \\tilde{\\sigma} W(\\gamma)} x^{|\\gamma|}n^{c(\\gamma)}  = C_{\\Omega}(x),\n\\end{align}\nwhere\n\\begin{equation}\nC_{\\Omega}(x) = \\sum_ {\\gamma : a \\to a} x^{|\\gamma|}n^{c(\\gamma)},\n\\end{equation}\nis the usual generating function of the honeycomb lattice O($n$) model, i.e. closed loops without the SAW component, and $\\tilde{\\sigma} = 1-\\sigma$.\n\\end{lemma}\n\\begin{figure}\n\\centering\n\\begin{picture}(400,100)\n\\put(85,0){\\includegraphics[scale=0.75, angle=0]{pqrvertex2.pdf}}\n\\put(122,58){$p$}\n\\put(207,58){$p$}\n\\put(290,57){$p$}\n\\put(110,38){$q$}\n\\put(194,38){$q$}\n\\put(276,38){$q$}\n\\put(135,37){$r$}\n\\put(219,37){$r$}\n\\put(303,37){$r$}\n\\end{picture}\n\\caption{The three possible ways for a walk to enter a given vertex via each of the three mid-edges, $p$, $q$ and $r$. The discrete divergence is evaluated for all three cases in order to derive the off-critical, or `massive' preholomorphicity condition.}\n\\label{fig:pqrvertex}\n\\end{figure}\n\\begin{proof}\nWe begin by summing Eqn. \\eqref{eqn:massiveCR} over all the vertices of the lattice $\\Omega$. The contribution to the left-hand side of \\eqref{eqn:massiveCR} from those mid-edges that are in the bulk cancels, since each bulk mid-edge is summed over twice but with opposite signs. This leaves only the boundary mid-edges contributing to the left-hand side which can be written as\n\\begin{equation}\n\\sum_{\\gamma : a \\to z \\in\\partial \\Omega} {\\rm e}^{ -{\\rm i} \\sigma W(\\gamma)} x^{|\\gamma|} n^{c(\\gamma)}{\\rm e}^{{\\rm i}\\phi(\\gamma)}, \\nonumber\n\\end{equation}\nwhere ${\\rm e}^{{\\rm i}\\phi(\\gamma)}$ is the complex number that describes the direction from the boundary vertex to the boundary mid-edge. It is easy to check that this equals ${\\rm e}^{{\\rm i} W(\\gamma)}$ for all walks terminating on boundary mid-edges other than the starting mid-edge $a$ and is $-1$ if the walk terminates at $a$ (which we call a $\\emph{zero-length walk}$) . Using $\\tilde{\\sigma}=1-\\sigma$ we then have\n\\begin{equation}\n\\label{eqn:lhssum}\n\\sum_{\\gamma : a \\to z\\in\\partial \\Omega\\backslash\\{a\\}} {\\rm e}^{{\\rm i} \\tilde{\\sigma} W(\\gamma)} x^{|\\gamma|} n^{c(\\gamma)}-\\sum_{\\gamma : a \\to a} x^{|\\gamma|} n^{c(\\gamma)}.\n\\end{equation}\nThis first term arises from all configurations where the walk terminates on a boundary mid-edge other than the starting mid-edge $a$. The second is from all configurations with a zero-length walk, that is one that terminates at $a$. Note that we define the winding angle of a zero-length walk to be $0$.\n\n \nAs for the right-hand side of Eqn. \\eqref{eqn:massiveCR}, using Definition \\ref{def:Fdef} this can be written as\n\\begin{equation}\n\\label{eqn:rhssum}\n\\left (1-\\frac{x}{x_{\\rm c}} \\right ) \\sum_{\\gamma : a \\to z \\in\\Omega\\backslash \\partial \\Omega} \\left [ F(z;v_1(z)) (z-v_1(z))+ F(z;v_2(z)) (z-v_2(z)) \\right ].\n\\end{equation}\nThis is because for a given end point $z$, a walk can be heading towards one of two possible vertices which we call $v_1$ and $v_2$, the labelling being unimportant.  This is illustrated in Fig. \\ref{fig:2vertex}. Equating \\eqref{eqn:lhssum} and \\eqref{eqn:rhssum} we have\n\\begin{align}\n&\\sum_{\\gamma : a \\to z\\in \\partial \\Omega\\backslash \\{a\\}}{\\rm e}^{ {\\rm i} \\tilde{\\sigma} W(\\gamma)} x^{|\\gamma|}n^{c(\\gamma)}-\\sum_{\\gamma : a \\to a}x^{|\\gamma|}n^{c(\\gamma)} \\nonumber\\\\\n= &\\left (1-\\frac{x}{x_{\\rm c}} \\right ) \\sum_{\\gamma : a \\to z \\in\\Omega\\backslash \\partial \\Omega} \\left ( F(z;v_1) (z-v_1(z))+ F(z;v_2) (z-v_2(z)) \\right ) \n\\label{eqn:identity1}\n\\end{align}\nUsing $\\sigma=1-\\tilde{\\sigma}$ and the definition of $F(z;v)$ the summation on the right-hand side becomes \n\\[\n{\\rm e}^{ {\\rm i} \\phi } \\left ( \\sum_{\\gamma : a \\to z \\to v_1} x^{|\\gamma|}n^{c(\\gamma)}{\\rm e}^{ {\\rm i} \\tilde{\\sigma} W(\\gamma)}{\\rm e}^{-{\\rm i} W(\\gamma)} - \\sum_{\\gamma : a \\to z \\to v_2} x^{|\\gamma|}n^{c(\\gamma)}e^{{\\rm i} \\tilde{\\sigma} W(\\gamma)} {\\rm e}^{-{\\rm i} W(\\gamma)}  \\right ), \n\\] \\\\\nwhere $e^{{\\rm i}\\phi}$ is the unit vector from $v_1$ to $z$, which is the negative of the unit vector from $v_2$ to $z$.\n\\begin{figure}\n\\centering\n\\begin{picture}(250,75)\n\\put(50,0){\\includegraphics[scale=0.75, angle=0]{2vertex2.pdf}}\n\\put(37,0){$v_1$}\n\\put(90,35){$v_2$}\n\\put(66, 22){$z$}\n\\put(147,0){$v_1$}\n\\put(200,35){$v_2$}\n\\put(180, 22){$z$}\n\\put(70,0){$e^{i\\phi}$}\n\\end{picture}\n\\caption{A walk terminating at the mid-edge $z$. The mid-edge lies between two vertices $v_1$ and $v_2$ and the unit vector from $v_1$ to $z$ is given by $e^{{\\rm i} \\phi}$. The labelling of the vertices is arbitrary.} \n\\label{fig:2vertex}\n\\end{figure}\n\nA walk that terminates at $z$ and moves in the direction of vertex $v_2$ has winding $W(\\gamma_2)=2\\pi m' + \\phi$ while a walk heading in the direction of the vertex $v_1$ and terminating at $z$ has winding $W(\\gamma_1)=(2m + 1)\\pi + \\phi$ for some $m, m' \\in\\mathbb{Z}$. In each case the angle $\\phi$ from the unit vector is cancelled by the $\\phi$ appearing in the winding angle term $e^{-{\\rm i} W(\\gamma)}$ and this leaves\n\\begin{equation}\n\\label{exp:rhsidentity1}\n-\\sum_{\\gamma : a \\to z \\in \\Omega\\backslash \\partial \\Omega} x^{|\\gamma|}n^{c(\\gamma)}{\\rm e}^{ {\\rm i} \\tilde{\\sigma} W(\\gamma)}. \n\\end{equation}\n\nThe left-hand side \\eqref{eqn:lhssum} is a sum of walks to the boundary and walks of length zero, which is equal to $C_{\\Omega}(x)$. Substituting expression (\\ref{exp:rhsidentity1}) into Eqn. (\\ref{eqn:identity1}) and rearranging we obtain \n\\begin{equation} \\label{eqn:keyidentity}\n\\sum_{\\gamma : a \\to z\\in \\partial \\Omega\\backslash \\{a \\}} {\\rm e}^{{\\rm i}\\tilde{\\sigma} W(\\gamma)} x^{|\\gamma|}n^{c(\\gamma)}  + (1- x\/x_c)\\sum_{\\gamma : a \\to z\\in \\Omega\\backslash \\partial \\Omega} {\\rm e}^{{\\rm i}\\tilde{\\sigma} W(\\gamma)} x^{|\\gamma|}n^{c(\\gamma)}  = C_{\\Omega}(x)\n\\end{equation}\n\\end{proof}\n\\section{Winding angle}\n\\subsection{Generating function definitions}\nLet us now restrict to a particular trapezoidal domain $\\Omega=S_{T,L}$ of width $T$ and left-height $2L$, see Fig.~\\ref{fig:S_boundary}.\nThe winding angle distribution function can be calculated directly from the off-critical generating function identity (\\ref{eqn:keyidentity}). We remind the reader that $\\gamma$ describes a walk along with a configuration of loops. For convenience we use the terms $\\emph{generating function of walks}$ and $\\emph{configuration of walks}$ but it should be understood that these include configurations of closed loops as well. We define the following generating function\n\\[\nG_{\\theta,\\Omega}(x)=\\sum_{\\substack{\\gamma : a \\to z\\in\\Omega\\backslash \\partial \\Omega\\\\W(\\gamma)=\\theta}}x^{|\\gamma|}n^{c(\\gamma)}.\n\\]\n$G_{\\theta,\\Omega}(x)$ contains only those contributions to $G_\\Omega(x) = \\sum_\\theta G_{\\theta,\\Omega}(x)$ where the walk has winding angle $\\theta$. We also define\n\\[\nH_{\\Omega}(x)=\\sum_{\\gamma : a \\to z\\in\\partial \\Omega\\backslash \\{a \\} }{\\rm e}^{{\\rm i}\\tilde{\\sigma} W(\\gamma)}x^{|\\gamma|}n^{c(\\gamma)},\n\\]\nwhich is the generating function describing walks that terminate on the boundary of the domain, and thus have a winding angle associated to that boundary.\n\\begin{figure}[t]\n\\begin{center}\n\\begin{picture}(140,190)\n\\put(0,180){\\includegraphics[height=120pt, angle=270]{honeycomb}}\n\\put(15,77){$\\alpha$}\n\\put(123,76){$\\beta$}\n\\put(55,143){$\\varepsilon$}\n\\put(55,10){$\\bar{\\varepsilon}$}\n\\put(15,89){$a$}\n\\put(-11,77){$2L$}\n\\put(69,182){$T$}\n\\end{picture}\n\\end{center}\n\\caption{Finite patch $S_{5,1}$ of the hexagonal lattice. The SAW component of a loop configuration starts on the central mid-edge of the left boundary (shown as $a$).}\n\\label{fig:S_boundary}\n\\end{figure}\nUsing this notation (\\ref{eqn:keyidentity}) becomes\n\\[\nH_{\\Omega}(x)  + (1- x\/x_c)\\sum_{\\theta} {\\rm e}^{{\\rm i}\\tilde{\\sigma} \\theta}G_{\\theta,\\Omega}(x) = C_{\\Omega}(x).\n\\]\nNow let $H_{\\Omega}^{*}(x)$ and $G_{\\theta,\\Omega}^{*}(x)$ be $H_{\\Omega}(x)\/C_{\\Omega}(x)$ and $G_{\\theta,\\Omega}(x)\/C_{\\Omega}(x)$ respectively. For $x<x_{\\rm c}$ define $H^{*}(x)$ and $G_{\\theta}^{*}(x)$ to be $H_{\\Omega}^{*}(x)$ and $G_{\\theta,\\Omega}^{*}(x)$ respectively in the limit as $\\Omega$ approaches the half plane. Assuming that $x_{\\rm c}$ is the location of the critical point\\footnote{For SAWs ($n=0$) it was proved in \\cite{DC-S10} that $x_{\\rm c}$ is indeed the critical point. Likewise for the $n=1$ case \\cite{H50} and the $n=-2$ case \\cite{BT73} this is rigorously known.}, as we will do in the next section, the limits of $H_{\\Omega}^{*}(x)$ and $G_{\\theta,\\Omega}^{*}(x)$ exist for $x<x_{\\rm c}$ by definition of the critical point.\n\nMoreover, since $H_{\\Omega}^{*}(x)$ converges, the contributions from configurations whose walk ends at the top\/bottom or right-hand boundary are tail terms of a converging series and hence vanish as $\\Omega$ approaches the half-plane. Thus $H^{*}(x)$ only contains walks starting and ending at the surface $\\alpha$, which is the only surface remaining in the domain. In the limit as the strip width becomes infinite we thus obtain\n\n\\begin{equation}\\label{eqn:f*g*}\nH^{*}(x)  + (1- x\/x_c)\\sum_{\\theta} {\\rm e}^{{\\rm i}\\tilde{\\sigma} \\theta}G^{*}_{\\theta}(x) = 1.\n\\end{equation}\n\n\\subsection{Susceptibilities and critical exponents}\nThe first term $H^{*}(x)$ is (up to a normalisation) the generating function of walks that start and end at the $\\alpha$ surface (and have an additional half-step to the left of their starting and ending vertices). This generating function is usually denoted in the literature as $\\chi_{11}(x)$ \\cite{BH72}. One conventionally writes \n\\[\n\\chi_{11}(x) \\sim d_0(x)+d_1(x)(1-x\/x_{\\rm c})^{-\\gamma_{11}},\\qquad x\\lesssim x_{\\rm c},\n\\]\nwhere $d_0$ and $d_1$ are analytic at $x=x_{\\rm c}$ and by $\\sim$ we mean that the ratio of the left and right sides approaches $1$ as $x$ approaches the critical point. In doing this we are assuming the existence of the exponent $\\gamma_{11}$ as well as that the value of the critical exponent is given by $x_{\\rm c}$ as defined in \\eqref{eq:Slemma_dilute}.\n\nSimilarly, the generating function of walks that start at the surface and finish anywhere inside the domain, is usually denoted in the literature as $\\chi_{1}(x).$ One conventionally writes \n\\[\n\\chi_{1}(x) \\sim c(x)(1-x\/x_{\\rm c})^{-\\gamma_{1}}, \\qquad x\\lesssim x_{\\rm c},\n\\]\nwith $c(x_c)$ analytic. This assumes the existence of the exponent $\\gamma_{1}$ and the value of the critical point, but no other assumption is made. We will see later that this assumption for example is not valid at $n=2$. \n\nWith the assumption as to the existence of the exponents for the susceptibilities we have\n\\[\nH^{*}(x)\\propto \\chi_{11}(x) \\sim 1+const \\times(1-x\/x_{\\rm c})^{-\\gamma_{11}},\n\\]\nand\n\\begin{equation}\n\\sum_{\\theta}G_{\\theta}^{*}(x) \\propto \\chi_1(x) \\sim const\\times (1-x\/x_{\\rm c})^{-\\gamma_{1}},\n\\label{eq:Gasymp}\n\\end{equation}\nUsing Eqn. (\\ref{eqn:f*g*}) we also obtain\n\\begin{equation}\\label{eqn:g*}\n\\sum_{\\theta} e^{\\tilde{\\sigma} i \\theta}G^{*}_{\\theta}(x)=(1-H^{*}(x))(1- x\/x_c)^{-1}\\sim const\\times (1-x\/x_{c})^{-\\gamma_{11}-1}.\n\\end{equation}\n\n\\subsection{Asymptotic winding angle distribution}\nWe first state the following result regarding the asymptotic expansion of certain singular functions \\cite{FS09}\n\\begin{thm}\n\\label{thm:singgam}\nLet $\\alpha$ be an arbitrary complex number in $\\mathbb{C}\\backslash\\mathbb{Z}_{\\leq 0}$. The coefficient of $z^n$ in\n\\[\nf(z)=(1-z)^{-\\alpha}\n\\]\nadmits for large $n$ a complete asymptotic expansion in descending powers of $n$,\n\\[\n[z^n]f(z) \\sim \\frac{n^{\\alpha-1}}{\\Gamma(\\alpha)}\\left(1+O \\left ( \\frac{1}{n} \\right ) \\right ).\n\\]\n\\end{thm}\nThis allows one to show the following:\n\\begin{lemma}\n\\label{lem:GFasymp}\nLet $G(x)=\\sum_{j\\ge 0} g_j x^j$. If $G(x)\\sim A (1-x\/x_{\\rm c})^{-\\eta}$ for $x\\lesssim x_{\\rm c}$ and some constant $A$, then\n\\[\ng_j \\sim A x_{\\rm c}^{-j} j^{\\eta-1}\/\\Gamma(\\eta).\n\\]  \n\\end{lemma}\n\nWe denote the number of walks of length $j$ with winding angle exactly $\\theta$ by $a_{\\theta}(j)$, so that we can write\n\\[\nG^{*}_{\\theta}(x)=\\sum_{j=0}^{\\infty}a_{\\theta}(j)x^{j}.\n\\]\nSumming over $\\theta$ and using \\eqref{eq:Gasymp} and Lemma~\\ref{lem:GFasymp}, the total number of walks of length $j$ behaves like\n\\[\n\\sum_{\\theta}a_{\\theta}(j)\\sim const\\times x_{\\rm c}^{-j}j^{\\gamma_{1}-1}.\n\\]\nSimilarly, from (\\ref{eqn:g*}) we have \n\\[\n\\sum_{\\theta} {\\rm e}^{{\\rm i}\\tilde{\\sigma} \\theta}a_{\\theta}(j)\\sim const\\times x_{\\rm c}^{-j}j^{\\gamma_{11}}.\n\\]\n\n\\begin{definition}\nThe probability density function $P(\\theta,j)$ for the winding angle of walks of length $j$ is the fraction of walks of length $j$ with winding angle $\\theta$: \n\\[\nP(\\theta,j) = \\frac{a_{\\theta}(j)}{\\sum_{\\theta}a_{\\theta}(j)}.\n\\]\n\\end{definition}\n\nFrom the reasoning above, the following result follows immediately.\n\\begin{prop}\n\\label{prop:Psurf}\nLet $\\tilde{\\sigma}=1-\\sigma$ where $\\sigma$ is given by \\eqref{eq:sigmadilute}. Then\n\\[\n\\sum_{\\theta} {\\rm e}^{{\\rm i}\\tilde{\\sigma} \\theta}P(\\theta,j)\\sim const\\times j^{\\gamma_{11}-\\gamma_{1}+1}.\n\\]\n\\end{prop}\nThat is to say, the probability density function is characterised by an exponent that can be expressed solely in terms of the half-plane exponents $\\gamma_1$ and $\\gamma_{11}.$\n\\subsection{Winding angle in a wedge}\\label{ssec:wedge}\n\nRather than taking $L\\rightarrow\\infty$ as in the previous sections, we now set $L=0$. In this case the trapezoidal domain $S_{T,0}$ reduces to a wedge with wedge angle $\\alpha=\\pi\/3$. Using similar arguments as before, we take the limit $T\\rightarrow\\infty$, giving\n\\begin{equation}\\label{eqn:f*g*wedge}\nH^{*}_\\alpha(x)  + (1- x\/x_{\\rm c})\\sum_{\\theta} {\\rm e}^{{\\rm i}\\tilde{\\sigma} \\theta}G^{*}_{\\theta,\\alpha}(x) = 1,\n\\end{equation}\nwhere now $H^{*}_\\alpha(x)$ is (again up to a normalisation factor) the generating function of walks that start at $a$ and end at the $\\varepsilon$ or $\\bar{\\varepsilon}$ surface of the wedge with angle $\\alpha=\\pi\/3$. This generating function is usually denoted in the literature $\\chi_{21}(x,\\alpha)$ \\cite{GT84}. It is known that self-avoiding walks ($n=0$) in a wedge have the same connective constant as those in the plane, for arbitrary wedge angle \\cite{HW85}. This is also known to be true in the Ising ($n=1$) case \\cite{BPP84}, and for the O$(-2)$ model, the Gaussian case \\cite{BT73}. We assume that this holds also for all $n\\in[-2,2]$. We thus write \n\\[\n\\chi_{21}(x,\\pi\/3) \\sim \\tilde{d}_0(x)+\\tilde{d}_1(x)(1-x\/x_{\\rm c})^{-\\gamma_{21}(\\pi\/3)},\n\\]\nwhere $\\tilde{d}_0$ and $\\tilde{d}_1$ are analytic near $x=x_{\\rm c}$. This assumes the existence of the exponent $\\gamma_{21}(\\alpha)$ which in general depends on the wedge angle $\\alpha.$ Similarly, the generating function of walks that start at $a$ and finish anywhere inside the domain, is usually denoted in the literature as $\\chi_{2}(x,\\alpha).$ Assuming the existence of the relevant critical exponent $\\gamma_{2}(\\alpha)$, we write \n\\[\n\\chi_{2}(x,\\pi\/3) \\sim \\tilde{c}(x)(1-x\/x_{\\rm c})^{-\\gamma_{2}(\\pi\/3)},\n\\]\nwith $\\tilde{c}(x)$ analytic near $x = x_c.$  \n\nDenote the probability density function for the winding angle of walks of length $j$ in a wedge with angle $\\alpha$ by $P_\\alpha(\\theta,j)$. Using exactly the same reasoning as in the previous section, we find that\n\\begin{equation}\n\\sum_{\\theta} {\\rm e}^{{\\rm i}\\tilde{\\sigma} \\theta}P_{\\alpha=\\pi\/3}(\\theta,j)\\sim const\\times j^{\\gamma_{21}(\\pi\/3)-\\gamma_{2}(\\pi\/3)+1}.\n\\end{equation}\nIn fact, one can approximate every wedge angle by an appropriate stacking of additional boundary vertices and mid-edges. Using the same reasoning as before we can therefore generalise to arbitrary wedge angle and find:\n\\begin{equation}\n\\sum_{\\theta} {\\rm e}^{{\\rm i}\\tilde{\\sigma} \\theta}P_{\\alpha}(\\theta,j)\\sim const\\times j^{\\gamma_{21}(\\alpha)-\\gamma_{2}(\\alpha)+1}.\n\\label{eq:Pwedge}\n\\end{equation}\nThis reduces to the previous result in the special case $\\alpha = \\pi.$ \n\n\\subsection{Exponent inequalities}\nUsing the techniques employed above we can derive rigorous exponent bounds in the following way. Recall\n\\[\nH^{*}(x)\\propto \\chi_{11}(x) \\sim 1+const \\times(1-x\/x_{\\rm c})^{-\\gamma_{11}},\n\\]\nand\n\\[\n\\sum_{\\theta}G_{\\theta}^{*}(x) \\propto \\chi_1(x) \\sim const\\times (1-x\/x_{\\rm c})^{-\\gamma_{1}}.\n\\]\nSince $$\\sum_{\\theta} {\\rm e}^{{\\rm i}\\tilde{\\sigma} \\theta}G_{\\theta}^{*}(x) \\le \\sum_{\\theta}G_{\\theta}^{*}(x) ,$$ it follows from\n\\[\nH^{*}(x)  + (1- x\/x_{\\rm c})\\sum_{\\theta} {\\rm e}^{{\\rm i}\\tilde{\\sigma} \\theta}G^{*}_{\\theta}(x) = 1,\n\\]\nthat\n\\begin{equation} \n\\gamma_1 -\\gamma_{11} \\ge 1.\n\\label{ineq}\n\\end{equation}\nThe only assumptions here are the existence of the critical exponents and the assumption that the critical point is at $x = x_c,$ as given by Lemma \\ref{lem:Slemma4} (except for for $n=0, \\,\\, 1$ and $-2,$ where it is known rigorously \\cite{DC-S10,H50, BT73}). To see how strong the inequality is, one must substitute the conjectured exponent values. For the O$(n=-2)$ model, the inequality is an equality. As $n$ increases, the bound gets progressively weaker. For the O$(n=-1)$ model, the l.h.s. of \\eqref{ineq} is $67\/64,$  for the O$(n=0)$ model, the l.h.s. is $73\/64,$ while for the O$(n=1)$ model it is $11\/8$. The exponents do not exist for the O$(2)$ model.\n\nAs we have shown that $\\gamma_2(\\alpha) -\\gamma_{21}(\\alpha)$ is independent of $\\alpha,$ it follows that the above inequality can be written more generally as \n\\begin{equation} \n\\gamma_2(\\alpha) -\\gamma_{21}(\\alpha) \\ge 1.\n\\label{ineq2}\n\\end{equation}\n\n\\section{Conjectures}\n\\subsection{Winding angle distribution from conformal field theory}\nDuplantier and Saleur \\cite{DS88} conjectured the winding angle distribution function for the general O($n$) model on a cylinder. We use  the parametrisation $n=-2\\cos(4\\pi\/\\kappa)$ whereas Duplantier and Saleur use the symbol $g$ which is related to $\\kappa$ by $g=4\/\\kappa$ ($\\kappa=8\/3$ for SAWs). The parafermionic spin $\\sigma$ is related to $\\kappa$ by $\\sigma=\\frac{3}{\\kappa}-\\frac{1}{2}$. They conjecture, from CFT and Coulomb gas arguments, that\n\\begin{equation}\\label{DS}\nP(x=\\theta) \\propto \\exp\\left(-\\frac{\\theta^2}{2\\kappa\\nu \\log \\ell}\\right) ,\\qquad\\ell\\rightarrow\\infty, \n\\end{equation}\nwhere $\\ell$ is the length of the walk. Here $\\nu$ is the standard critical exponent relating the circumference of the cylinder to the length of the walk, $L\\sim \\ell^{\\nu}$, and is given by\n\\[\n\\nu=\\frac{1}{4-\\kappa},\n\\]\nwhere $\\kappa = 2, \\,\\, 12\/5, \\,\\, 8\/3, \\,\\, 3, \\,\\, 4$ for $n = -2, \\,\\, -1,\\,\\, 0, \\,\\, 1, \\,\\, 2$ respectively. Hence $\\nu$ is not defined for $n=2$. Note that the winding angle distribution \\eqref{DS} is expressed entirely in terms of bulk critical exponents.  \n\nWe recall from the previous sections that\n\\begin{equation}\\label{distrib}\n\\sum_{\\theta} {\\rm e}^{{\\rm i}\\tilde{\\sigma}\\theta} P_{\\alpha}(\\theta,\\ell) \\approx \\int_{-\\infty}^\\infty {\\rm e}^{{\\rm i}\\tilde{\\sigma} \\theta} P_\\alpha(\\theta,\\ell) d\\theta \\propto \\ell^{-\\omega},\n\\end{equation}\nwhere $\\omega=-\\gamma_{21}(\\alpha)+\\gamma_2(\\alpha)-1$. The half plane corresponds to $\\alpha=\\pi$. \nUsing (\\ref{DS}) this is a straightforward integral, and gives\n\\[\n\\omega=\\nu\\kappa\\tilde{\\sigma}^2\/2 = \\frac{\\kappa\\tilde{\\sigma}^2}{2(4-\\kappa)}.\n\\]\nHence we find\n\\begin{equation}\n\\label{eq:surfbulk}\n-\\gamma_{21}(\\alpha)+\\gamma_2(\\alpha)-1= \\omega=\\frac98\\frac{(2-\\kappa)^2}{\\kappa(4-\\kappa)}.\n\\end{equation}\nIn particular we note that $\\omega=-\\gamma_{21}(\\alpha)+\\gamma_2(\\alpha)-1$ is independent of the wedge angle $\\alpha$. This is confirmed in the case of SAW ($n=0$) by the results of \\cite{GT84}, and for the Ising case ($n=1$) by the results of \\cite{BPP84}.\n\nFrom the existing physics literature \\cite{BM77,N82,C84} one can express the conjectured values of the half-plane exponents as follows: \n\\begin{equation}\\label{gam1}\n\\gamma_1=\\frac{\\kappa^2+12\\kappa-12}{8\\kappa(4-\\kappa)},\\qquad \\gamma_{11}= -\\frac{2(3-\\kappa)}{\\kappa(4-\\kappa)},\n\\end{equation}\nand thus\n\\[ \n-\\gamma_{11}+\\gamma_1 -1= \\frac98\\frac{(2-\\kappa)^2}{\\kappa(4-\\kappa)},\n\\]\nin perfect agreement with \\eqref{eq:surfbulk} for $\\alpha=\\pi$.\n\n\\subsection{Wedge exponents}\nThe expected values of the wedge exponents have not previously been derived for general $n$. In \\cite{BD08} this is done for the $n=0$ case with the assumption that $\\mathrm{SLE}_{8\/3}$ describes the scaling limit. However by extrapolating certain special cases we can provide conjectured values for these exponents for $n\\in [-2,2)$. Following the notation of \\cite{GT84}, we write the free energy of a $d$-dimensional wedge-shaped system as $$F=Vf_b + Af_s+Lf_e + \\ldots,$$ where $V$ is the $d$-dimensional `volume' of the system, $A$ is the $(d-1)$-dimensional `area' of a surface, and $L$ is the $(d-2)$-dimensional `length' of the edge formed by the intersection of the two surfaces. In our case, $d=2,$ so the `volume' is an area, the `area' is the length of the boundary, and the `length' corresponds to the point at the apex of a wedge. Using the scaling hypothesis, the singular part of the  corresponding free energies can be written as\n\\begin{eqnarray}\nf_b & \\sim & t^{2-\\alpha}\\psi_b(ht^{-y_0\\nu}) \\\\ \\nonumber\nf_s & \\sim & t^{2-\\alpha_s}\\psi_s(ht^{-y_0\\nu},h_1t^{-y_1\\nu}) \\\\ \\nonumber\nf_e & \\sim & t^{2-\\alpha_e}\\psi_e(ht^{-y_0\\nu},h_1t^{-y_1\\nu},h_2t^{-y_2\\nu}), \n\\end{eqnarray}\nwhere $t$ is the reduced temperature $(T-T_c)\/T_c;$ $y_0,$ $y_1$ and $y_2$ are the bulk, surface and edge scaling indices, from which all the susceptibility critical exponents follow. The reduced magnetic fields in the bulk, surface and edge are denoted, respectively $h,$ $h_1$ and $h_2.$ In particular, the bulk susceptibility is given by $$\\chi = \\partial^2f_b\/\\partial h^2 \\asymp t^{-\\gamma}; \\,\\,\\, \\gamma=\\nu(2y_0-d),$$ the surface susceptibilities are given by\n$$\\chi_{1} = \\partial^2f_s\/\\partial h \\partial h_1 \\asymp t^{-\\gamma_{1}},\\qquad \\gamma_{1}=\\nu(y_0+y_1-d+1),$$\n$$\\chi_{11} = \\partial^2f_s\/\\partial h_1^2  \\asymp t^{-\\gamma_{11}},\\qquad \\gamma_{11}=\\nu(2y_1-d+1),$$\nand the edge susceptibilities are given by\n$$\\chi_{2} = \\partial^2f_e\/\\partial h \\partial h_2 \\asymp t^{-\\gamma_{2}},\\qquad \\gamma_{2}=\\nu(y_0+y_2-d+2),$$\n$$\\chi_{21} = \\partial^2f_e\/\\partial h_1 \\partial h_2  \\asymp t^{-\\gamma_{21}},\\qquad \\gamma_{21}=\\nu(y_1+y_2-d+2).$$\nIn \\cite{GT84} it was shown (non-rigorously) that, for the O$(n=0)$ model, $y_2(\\alpha) = -5\\pi\/8\\alpha,$ where $\\alpha$ is the wedge angle. Similarly, in \\cite{BPP84} it was shown (non-rigorously) that, for the O$(n=1)$ model, $y_2(\\alpha) = -\\pi\/2\\alpha.$ For the Gaussian model, corresponding to the O$(n=-2)$ model, $y_2(\\alpha) = -\\pi\/\\alpha.$  Thus for these three cases, we have $$y_2(\\alpha) = -\\pi\\sigma\/\\alpha.$$ \nIf, as we conjecture, this is true for other values of $n,$ we then find\n$$\\gamma_2 = \\frac{\\kappa^2+8\\kappa+12-24\\pi\/\\alpha+4\\pi \\kappa\/\\alpha}{8\\kappa(4-\\kappa)},$$\nand\n$$\\gamma_{21} = \\frac{3\\kappa-6-6\\pi\/\\alpha+\\pi \\kappa\/\\alpha}{2\\kappa(4-\\kappa)}.$$\nwhere we have used $\\sigma=\\frac{3}{\\kappa}-\\frac{1}{2}$. If we set $\\alpha = \\pi,$ they reduce to (\\ref{gam1}), providing evidence for the validity of our assumption that they hold for all $n \\in [-2,2).$ For $n=2$ the free energy is believed to exhibit an essential singularity, so that critical exponents do not exist. This is signalled in the conjectured exponent values by the divergence at $\\kappa=4,$ corresponding to the O$(n=2)$ model.\n\nWe remark that the scaling indices $y_0,$ $y_1$ and $y_2$ take  particularly simple forms if parameterised in terms of $\\sigma$: \n$$y_0=\\frac{(\\sigma+1)(\\sigma+2)}{2\\sigma+1},\\qquad y_1=1-\\sigma,\\qquad y_2=-\\frac{\\pi \\sigma}{\\alpha}.$$\n\n\\section{Conclusion}\nWe have generalised the identity of Duminil-Copin and Smirnov off criticality, which allows us to make statements about critical exponents. We have proved an inequality for surface and wedge exponents, subject only to their existence.\n\nWe have similarly proved, under the same assumption, a relationship between the surface susceptibility exponents and the winding angle exponent of the O($n$) model for $n\\in[-2,2)$. Previously conjectured values of the surface and winding angle exponents are in agreement with the relationship we have derived for all values of $n\\in[-2,2)$. \n\nA study of the edge exponents that arise when considering the O$(n)$ model in a wedge geometry permits us to conjecture the exact value of the exponents for all wedge angles. \n\nThe off-critical extension of Smirnov's identity that we have obtained seems likely to yield a number of other significant results, and we are currently studying this possibility.\n\n\\section*{Acknowledgment}\nWe are grateful for financial support from the Australian Research Council (ARC). This work was carried out  during the visit of three of the authors to the US Mathematical Sciences Research Institute (MSRI, USA), during the Spring 2012 Random Spatial Processes Program. The authors thank the institute for its hospitality and the NSF (grant DMS-0932078) for its financial support. AJG wishes to thank Neal Madras and Mireille Bousquet-M$\\mathrm{\\acute{e}}$lou for helpful discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzfrgt b/data_all_eng_slimpj/shuffled/split2/finalzzfrgt
new file mode 100644
index 0000000000000000000000000000000000000000..192a08bb4a65a1901d5331694e8ed4fe5ddeabb5
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzfrgt
@@ -0,0 +1,5 @@
+{"text":"\\section{Conclusion}\nStudying the black hole thermodynamics and finding the origin of the black hole entropy is one of the most popular topics in the current era. The purpose of this paper is to present a very simple and efficient method for deriving The Bekenstein-Hawking entropy. And we showed that by maximizing the entropy of two-dimensional space, can reach the Bekenstein-Hawking entropy. On the other hand, our calculations show that the excitation of space quanta at the microscopic scale causes gravity at the macroscopic scale and so, gravity is a quantum mechanics phenomenon. Summarizing, the above calculations show that issues such as the origin of the black hole entropy and the cause of space-time bending have the same origin.\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThermal transport and energy conversion at the scale of micrometers to nanometers is a fascinating topic of research. The carriers of heat and charge as well as their transport mechanisms have characteristic length scales in this regime. Consequently, a multitude of effects can be studied and ultimately exploited\\cite{Pop2010,Shi2012_nanostructures}. Thermoelectric energy conversion, for example, has been predicted and shown to occur at enhanced efficiency in micro- and nanoscale structures\\cite{Hicks1993,Shi2012_nanostructures, Pop2010}. Particularly interesting realizations are axial and radial nanowire heterostructures, combining effects of reduced dimension both in the transport direction and perpendicular to it\\cite{Sistani2018,Swinkels_2018}. The transition from an extended thermoelectric material to a finite length along the transport direction, however, has not yet been fully explored. For example, the influence of interfaces between a thermoelectric material and metallic contacts or the transition between the conventional Peltier effect and thermionic emission may still hold some insights in the experimental realization\\cite{Shakouri1998}.\n\nOne of the challenges in researching this area is the difficulty in performing nanoscopic heat transport measurements\\cite{Pop2010}. Thermal contact resistances dominating at small length scales translate into systematic challenges for measuring the local temperature or heat flux in a structure. The thermal contacts to nanowire samples, for example, have turned out to be a major issue\\cite{Swinkels_2018,Menges_nature}. It has been proposed that this can be mitigated using extended measurement series for example of thermal transport as a function of length, sometimes called transmission line method\\cite{Vakulov2020} or related\\cite{Kim2015}.  Apart from being time-intensive, these solutions rely on other assumptions such as the ability to fabricate reproducible test structures with the same contact resistances to electrodes or thermometers. For semiconducting nanowires this is, oftentimes, a major difficulty\\cite{Wagner2018}.\n\nMotivated by the recent success of using scanning thermal microscopy (SThM)\\cite{gomes2015}, we here describe the development of a method to systematically extract thermal and thermoelectric transport properties from the spatial temperature information. In contrast to transport measurements using external heaters and thermometers, here we analyse the temperature distribution in self-heated nanostructures to extract information through fitting with appropriate models. While thermal conductivity measurements with SThM are often hampered by the thermal resistance between probing tip and sample, we apply periodic self-heating to circumvent this problem and are able to measure the sample temperatures\\cite{Menges_nature}. In this way, we can not only mitigate but also quantify the effect of thermal contact resistances inside and around the device. Thereby, we can address length scales not readily accessible using other methods (such as a segment and its interfaces within a nanowire). Additionally, the temperature distribution is measured in its topographic context by atomic force microscopy (AFM). It is by that possible to align and compare scans under different operating currents. Also, the method is independent on the geometry and material of the device. This means, that there is no need for altering the sample architecture for the thermal measurements. \n\nThere are examples of interpreting the temperature field of nanowire structures using SThM to extract thermal conductivity\\cite{Menges_nature, konemann2019, konemann2020, Shi2009_sthm}. Here, we extend the method to extract the thermal conductivity of a thermoelectric material and the metal leads, the thermal interfaces to a substrate, and the Peltier coefficient from the observation of the temperature fields. The nanowire device consists of a germanium (Ge) segment which is monolithically integrated in a single crystalline aluminium (c-Al) nanowire with atomically sharp interfaces\\cite{sistani2017} and a diamter of 35\\,nm. A dark field scanning transmission electron microscope (DF-STEM) image of the nanowire cross-section, a SEM image and a sketch of the device are shown in Figure \\ref{fig:device}(b-d). Further information on the fabrication, electrical and structural characterization of the device can be found in the supplementary information and in \\cite{Elhajouri2019, Brunbauer_2016, Kral2015}. The thermoelectric figure of merit can thus be calculated from a single measurement collecting the complementary information from the Joule and the Peltier signals. The analysis concentrates on the temperature profile in the lateral direction of the wire for the different operating currents.\n\n\\section{Experimental}\n\n\\begin{figure}[tb] \n\t\\centering \n\t\\includegraphics[width=1.0\\columnwidth]{figure_setupandsample.pdf} \n    \\caption{a) Schematic of the SThM set-up. A slow AC-bias is applied over the device to induce an oscillating sample temperature $T_{\\rm sample}$ which is thermally coupled via the tip to the heated sensor. The signal is de-modulated with a lock-in amplifier at the modulation frequency $f_{\\rm mod}$ and $2f_{\\rm mod}$ for measuring the temperature increase due to the Peltier and Joule effect respectively. The simultaneous measurement of the AC- and DC-signal allows to eliminate the thermal resistance between sample and tip and to infer $T_{\\rm sample}$. b)-d) The single crystalline Al-Ge nanowire is electrically connected over Al electrodes and lies on SiO$_2$ substrate. It is naturally integrated in a back-gated FET set-up. b) shows a dark field scanning transmission electron microscope (DF-STEM) image of the nanowire cross-section, c) a SEM image and d) a sketch of the device.}\n    \\label{fig:device} \n\\end{figure}\n\n\\subsection{Thermal imaging with the SthM}\nThe present SThM setup is operated in a highly shielded lab environment\\cite{NFL} in a high vacuum chamber at a pressure of $10^{-6}$\\,mbar. The measured thermal maps have a spatial resolution below 10\\,nm and mK sensitivity\\cite{Menges_nature}. The technique is based on an AFM, where the tip is operated in contact with the sample. The silicon-based temperature sensor is located at the base of the tip and heated to a temperature $T_{\\rm sensor} = 273\\,^{\\circ}$C, around which the electrical resistance of the sensor depends linearly on the temperature. When the heated tip is put in contact with the sample, the temperature difference induces a heat flow between the sensor and the sample. A change in sample temperature leads to a change in the heat flux that is detected through a small change in sensor temperature. At the same time, the sample temperature is not expected to be influenced beyond the uncertainty of the method as the thermal resistances within the sample are orders of magnitude lower than the thermal resistance between tip and surface. The nanoscopic contact between tip and sample has typically a resistance of more than $ 10^7-10^8$\\,W\/K compared to the cantilever leads with a thermal resistance of $R_{\\rm cl} \\approx 2\\cdot10^{5}$\\,W\/K. This difference prevents the sensor from equilibrating with the sample temperature\\cite{Menges2016_instruments}.\n\nA major difficulty in the determination of the sample temperature $T_{\\rm sample}$ lies in quantifying the thermal resistance of the tip, the tip-sample contact and the spreading into the substrate. They are all summarized in $R_{\\rm ts}$, which then depends on the surface material, the precise shape of the tip and largely on the size of the touching point that changes with the surface granularity and the topography of the sample. To overcome this problem, the nanowire is operated with an electrical AC-bias of frequency $f_{\\rm mod}=1234$\\,Hz that creates a continuously modulated temperature field. $R_{\\rm ts}$ is then eliminated in the equations by simultaneous measurement of the time-averaged sensor signal $\\Delta V_{\\rm DC}$ and the de-modulated sensor signal amplitude $\\Delta V_{\\rm AC}$. An illustration of the set-up is depicted in Figure \\ref{fig:device}a. The sample temperature difference to ambient temperature, $\\Delta T_{\\rm sample}$, for linear devices is then given by\\cite{Menges_nature}\n\\begin{equation}\n\\Delta T_{\\rm sample} = \\Delta T_{\\rm sensor} \\cdot \\frac{\\Delta V_{\\rm AC}}{\\Delta V_{\\rm DC} - \\Delta V_{\\rm AC}}.\n\\label{eq:sampletemp}\n\\end{equation}\nAn additional feature of the lock-in detection is the possibility to separate the temperature change that is caused by Joule heating and the Peltier effect, respectively. Joule heating, on the one hand, is proportional to the dissipated electrical power and the signal appears at $2f_{\\rm mod}$. On the other hand, the Peltier effect is directly proportional to the current and produces a thermal signal at $1f_{\\rm mod}$.\n\n\n\\section{Results and analysis}\n\\begin{figure*}[t] \n\t\\centering\n\t\\includegraphics[width=2.0\\columnwidth]{figure_scans.pdf} \n\t\\caption{Scanning thermal microscopy data of the wire operating with an AC-current of amplitude $I = 26\\,\\mu$A.\n\t\\newline a) AFM topography. b) Thermal signal amplitude taken at 2\\,$f_{\\rm mod}$ in response to Joule heating. c) Phase of the 2\\,$f_{\\rm mod}$ signal. d) Joule thermal map calculated using equation\\,\\ref{eq:sampletemp} and the scans in b) and e). e) DC-thermal measurement proportional to the thermal resistance of the tip-sample contact. f) Thermal signal amplitude taken at 1\\,$f_{\\rm mod}$ in response to Peltier heating\/cooling. g) Phase of the 1\\,$f_{\\rm mod}$ AC signal, where $\\pm$5\\,V equals $\\pm$90$^\\circ$. h) Peltier thermal map calculated using equation\\,\\ref{eq:sampletemp} and the scans in e), f) and g).}\n\t\\label{fig:sthmmaps} \n\\end{figure*}\n\n\\subsection{Temperature maps of self-heated nanowires}\nFigure \\ref{fig:sthmmaps} shows the results from a SthM scan at an operating current of 26\\,$\\mu$A. (a) is the sample topography from the SThM measurement of the operating nanowire with a spatial resolution of 10\\,nm. The inferred temperature fields that are induced by Joule heating $\\Delta T_{\\rm Joule}$ and the Peltier effect $\\Delta T_{\\rm Peltier}$ are represented in (d) and (h). Using equation \\ref{eq:sampletemp}, they are calculated  from the DC thermal signal (e), and the AC signals at $2f_{\\rm mod}$ (b) and $f_{\\rm mod}$ (f), respectively. The phase signals (c, g) are recorded with the lock-in detector and show the expected behavior for an approximately linear resistor.\n\nThe following analysis concentrates on the line-profiles extracted from the two-dimensional temperature maps of Figure \\ref{fig:sthmmaps} (d) and (h) and according ones for other currents between 13 and 26\\,$\\mu$A. The temperature profiles along the long axis of the nanowire are shown in Figure \\ref{fig:SThMProfiles} for both Joule and Peltier signals.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=2.0\\columnwidth]{figure_thermalprofiles_1.pdf}\n    \\caption{Temperature profiles along the wire with the Ge-segment shaded in red. a) The temperature change induced by Joule heating $\\Delta T_{\\rm Joule}$ as a function of the position $x$ along the wire, for the operation under increasing voltage biases. b) The temperature change due to the Peltier effect $\\Delta T_{\\rm Peltier}$ as a function of the position $x$ along the wire operating at different currents $I$ with a zoom on the area of the energy barrier.}\n\t\\label{fig:SThMProfiles} \n\\end{figure*}\n\n\\subsection{Analysis based on a 1D heat diffusion model}\nWe analyze the data through fitting the experimental temperature profiles to a model based on the one-dimensional heat diffusion equation. It consists of the following contributions: (1) a heat diffusion term, (2) a Joule heating term, proportional to the square of the current $I$, (3) a Peltier heating term, linearly proportional to the current, (4) a term for heat-loss to the substrate, proportional to the temperature difference between sample and substrate and (5) a term related to the heat capacity $C_V$: \n\\begin{equation}\n\\kappa A\\frac{\\partial^2T}{\\partial x^2} + I^2 \\cdot \\frac{\\rho}{A} + I\\cdot \\pi - g(T-T_{\\rm ambient}) = C_V A\\frac{\\partial T}{\\partial t},\n\\label{eq:diffusion}\n\\end{equation}\nwhere $T$ and $T_{\\rm ambient}$ are device and ambient temperature, $A$ the nanowire's cross sectional area, $\\kappa$ and $\\rho$ the material dependent thermal and electric conductivities and $g$ the substrate coupling constant counting the heat loss of the wire per unit length. In this context $\\pi$ is the local Peltier coefficient per unit length, as discussed below. The nanowire is slightly tapered on both sides near the metal electrodes. Nevertheless, in the regions of significant temperature rise, the diameter is constant and a position-independent $\\kappa$, $A$ and $g$ is justified. A similar diffusion equation model has been used to analyse SThM data of nanowires previously\\cite{Shi2009_sthm, Menges_nature, konemann2020}. However, there are underlying assumptions that need to be justified for each system under study:\n\nFirstly, the radial temperature distribution can be neglected. This assumption works well for wires with high aspect ratio as in this case. For this composite system we further require that the thermal interface resistance between the core and Al$_2$O$_3$-shell is smaller than that between wire and substrate. While this is a plausible assumption (see below), our study is limited in studying lateral intra-wire effects.\n\nSecondly, the heat spreading into the substrate from a line heat source has no analytical solution. To be able to simplify this into a single thermal conductance term per unit length between a nanowire and the substrate ($g \\Delta T$) requires, that the thermal interface resistance to the substrate is larger than the spreading resistances within the wire and the substrate. As a first indication, we observe a discontinuity in the horizontal temperature profile between wire and substrate. Figure \\ref{fig:fits}a shows that the temperature rise along a line section perpendicular to the wire is approximately constant across the nanowire surface and also five to ten times larger than on the substrate. Note, that the wire appears to have a width of 150\\,nm (instead of the actual 65 \\,nm) due to the well-known convolution effects with the tip shape. In the convolution area and due to the finite diameter of the tip-surface the temperature distribution appears smeared out over some tens of nanometers in this line scan, in contrast to a higher resolution along the nanowire axis.\n\nThirdly, equation \\ref{eq:diffusion} implies that the transport is essentially diffusive. This is a well established assumption for thermal transport in nanowires of these dimensions with one notable exception where reduced dimensions lead to a ballistic regime\\cite{Vakulov2020}. However, in this study the observed temperature profiles within the segments are typical for diffusive transport and cannot be explained using ballistic effects. They can be observed only around the interfaces and boundaries, where the spatial resolution of the scan is higher than the carrier scattering length. Near the Ge-Al interfaces, the thermalization lengths of non-equilibrium charge carriers lead to a spatial distribution of the thermoelectric effect\\cite{konemann2020, Lake1992, Shakouri1998}. The effective Peltier coefficient is then the integration over the spatial distribution of the local Peltier coefficient\n\\begin{equation}\n\\Pi = \\int \\pi(x) dx.\n\\label{eq:petcoefficient}\n\\end{equation} \n\nFinally, the AC-modulation period of the driving current is orders of magnitudes shorter than any thermal time constant of the system. Therefore, the scan is taken at steady state and the last term of equation\\,\\ref{eq:diffusion} is negligible.\n\nFor a system whose electric response is sufficiently linear, the total temperature distribution is disentangled to $T(x,t) = T_{\\rm ambient} + \\Delta T_{\\rm Peltier}(x)\\sin{(2\\pi f t)} + \\Delta T_{\\rm Joule}(x)\\sin{(4\\pi f_{\\rm mod} t)}$ for the current $I(t) = I_0\\sin{(2\\pi f_{\\rm mod} t)}$. The respective thermal profiles are, then, described independently by  \n\\begin{equation}\n\\kappa A\\frac{\\partial^2}{\\partial x^2}\\Delta T_{\\rm Joule} + I_0^2 \\cdot \\frac{\\rho}{A} -  g\\cdot \\Delta T_{\\rm Joule} = 0\n\\label{eq:diffusionJ}\n\\end{equation}\nand\n\\begin{equation}\n\\kappa A\\frac{\\partial^2 }{\\partial x^2}\\Delta T_{\\rm Peltier} + I_0\\cdot \\pi - g\\cdot \\Delta T_{\\rm Peltier} = 0.\n\\label{eq:diffusionP}\n\\end{equation}\n\nAn initial qualitative observation of the line profiles, shown in Figure \\ref{fig:SThMProfiles}, already allows for important conclusions. The Ge and Al sections of the wire exhibit very different characteristics. In the Ge segment, the temperature profile is approximately parabolic, as expected for a diffusive 1D system with uniform Joule dissipation. A heat source term in equation\\ref{eq:diffusion} generally leads to a concave temperature profile. In contrast, the profile in the Al sections is convex, indicative of a situation which is dominated by heat dissipation and loss to the substrate. Here, the approximately exponential decay is expected for a uniform wire without any local Joule dissipation. The Al-Ge interface marks the Joule profiles by a sudden change of slope. When extracting the points of maximum temperature gradient for the Joule signals, their relative distances coincides for all measurements exactly with the length of the Ge segment. Also, after the deflection points there is a transition length of about 40-50\\,nm in the Al-segments. This area corresponds to the phonon-electron thermalization length that is blurred by a combination of spatial resolution and parallel heat transport in the oxide shell (the Figure is in the supplemental information).\n\n\\subsection{Analysis of the aluminium leads}\n\\label{sec:Al}\n\\begin{figure}[tb]\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{figure_fitting.pdf}\n\t\\caption{Visualization of the parameter extraction. a) The horizontal profile of the temperature change with respect to RT that caused by Joule heating $\\Delta T_{\\rm Joule}$. The discontinuity in the temperature profile when going from wire to substrate justifies an analysis in one dimension. b) The exponential fit in the Al-segment allows to extract the ratio between the thermal conductivity $\\kappa$ and the substrate coupling constant $g$. The thermal data is marked with the corresponding uncertainty. c) The thermal conductivity in the Ge-segment is extracted from a hyperbolic fit to the measured temperature change caused by Joule heating in the nanowire. d) Finally, the substrate coupling constant is calculated via the observation, that all heat dissipated in the wire is lost into the substrate. e) Shows the values for $\\frac{\\kappa_{\\rm Al}}{g}$ extracted from the exponential fit for the Peltier and Joule profile for different operating currents. f) presents the values for the thermal substrate coupling $g$ extracted from the more general model as in d) (in red) and the values extracted from the exponential fit (in blue).}\n\t\\label{fig:fits}\n\\end{figure}\nWe turn first to the sections of the line profiles, in which the core is single crystalline Al. In these sections, the heat diffusion equation has no source terms and therefore reduces to  \n\\begin{equation}\n\\frac{\\partial^2}{\\partial x^2} \\Delta T_{\\rm j,p}(x)= \\frac{g}{\\kappa_{\\rm Al}A}\\cdot \\Delta T_{\\rm j,p}(x).\n\\end{equation}\nThe indices j and p denote Joule and Peltier terms, respectively. As the temperature profile reaches room temperature before reaching the electrodes, the analytic solution to this differential equation is given by\n\\begin{equation}\n\\label{eq:exponetial}\n\\Delta T_{\\rm j,p}(x-x_0) = T_0 \\cdot \\exp{-\\sqrt{\\frac{g}{\\kappa_{\\rm Al} A}}\\cdot (x-x_0)},\n\\end{equation}\nwhere $T_0$ is the temperature at point $x_0$. Thus, the exponential fit to the temperature data reveals the relation between the substrate coupling constant $g$ and the thermal conductivity in Al $\\kappa_{\\rm Al}$. The exponential fit is applied to both the Peltier and Joule profiles under different operating currents. It is shown in Figure \\ref{fig:fits}b. The extracted ratios for $\\kappa_{\\rm Al}\/g$ are similar for all measurements, as can be seen in \\ref{fig:fits}e. The values for $\\kappa_{\\rm Al}\/g$ suggest, that the heat loss to the substrate is about $150$ times smaller than the heat conduction to the sides in each infinitesimal element d$x$. The assumption of the uniformity of the temperature distribution over a cross-section is, thereby, in retrospect justified by this result. Furthermore, the high consistency of the values derived from the $1\\times f_{\\rm mod}$ and $2\\times f_{\\rm mod}$-measurements is an experimental confirmation for negligible Joule heating or Peltier heating\/cooling in this section.\n\nThe analysis is pushed further by deriving $\\kappa_{\\rm Al}$ and $g$ independently. It is observed, that the wire temperature of the Joule profiles decays to room temperature before reaching the electrodes. In other words, the entire heat that is generated over electric power dissipation in the wire ($P_{\\rm dis} = I^2 \\rho L_{\\rm Ge} \/ A_{\\rm core}$) is equal to the heat lost to the substrate. By making use of a finite element integral over all the measuring points on the nanowire, the substrate coupling is then the only unknown quantity. The conservation of heat requires\n\\begin{equation}\n\\begin{split}\nP_{\\rm dis} &= \\int\\limits_{\\rm NW} g\\cdot \\Delta T_{\\rm Joule}(x) {\\rm d}x  \\\\\n&\\approx g\\cdot\\Delta x \\cdot \\sum\\limits_{\\rm NW} \\Delta T_{\\rm Joule}(x_i),\n\\end{split}\n\\end{equation}\nwhere $\\Delta x \\approx 10$\\,nm is the spacing between adjacent  measurement points $x_i$, $g$ the substrate coupling, $T$ the sample temperature and $\\Delta T_{\\rm Joule}(x_i)$ the measured temperature increase of the finite element $x_i$ caused by Joule heating with respect to ambient temperature. With this approach we extracted values for $g$ of $2.1\\pm0.3$, $2.1\\pm0.3$ and $2.0\\pm0.4$\\,W\/(mK) for the experiments using 23, 24, and 26\\,$\\mu$A, respectively.\n\nWe note that one reaches the same result by considering continuity of heat flux at the Ge-Al interface in addition to the 1D heat diffusion equation. The heat transported into the Al segment is given by $\\dot{Q}_{\\rm Al} = \\grad T(x_0) \\cdot \\kappa_{\\rm Al} A_{\\rm wire}$. The temperature gradient is then calculated from the exponential fit and extrapolated to the Al-Ge interface. Energy conservation implies:\n\\begin{equation}\nP_{\\rm dis} = 2\\dot{Q}_{\\rm Al} + g\\cdot \\Delta T_{\\rm avg\\, Ge} \\cdot L_{\\rm Ge},\n\\end{equation}\nwhere $L_{\\rm Ge}$ is the length of the Ge segment and $T_{\\rm avg Ge}$ is the average temperature along the Ge segment. Solving this latter equation based on the exponential fit leads to the same results as previously, this is shown in blue on Figure \\ref{fig:fits}f. The good agreement with the result based on the more general result of energy conservation strengthens the diffusion equation based analysis further. As argued above, the value of $g$ is dominated by the thermal interface between the oxide shell and the substrate. The contact width is $(56 \\pm 5)$\\,nm (see Appendix). Therefore, we can calculate a value per unit area of approximately $3.5\\cdot 10^7$\\,W\/(Km$^2$). This is a typical value for a thermal interface between dielectrics.\n\nNext, we turn to the thermal conductivity of the Al segment. Using the extracted values of $g$ and the ratio $\\kappa \/ g$, we arrive at a thermal conductivity for the Al segments. To interpret, we need to consider that the value from the fit is an average over the cross section of both the Al core and the Al$_2$O$_3$ shell which are weighed by the respective thermal conductivities and obtain for the core $\\kappa_{\\rm Al} \\approx 150$\\,W\/(mK). We use the measured cross-sectional areas from the STEM image and the tabulated values for Al$_2$O$_3$. To compare with expected values we apply the Wiedemann Franz law, which results in an electrical resistivity of $\\rho_{\\rm Al} \\approx 50\\cdot 10^{-9} \\, \\Omega$m about twice as high than theoretical predictions for bulk c-Al \\cite{Gall2016,Jain2016} which is expected due to increased surface scattering in nanostructures. Indeed, this value is three times lower than previously determined for pure c-Al wires \\cite{Brunbauer_2016}. However, in an SThM scan of such an operating c-Al wire a dominant local heat source is identified in the center (see the temperature profiles in the supplemental information). This may be expected according to the wire fabrication and indicates a grain boundary. Therefore, the expected values for the electrical conductivity of the c-Al segments without any grain boundary are significantly higher and in good agreement with our experimental value.  This result is shows that the influence of grain boundary scattering on the electrical resistivity exceeds the impact of surface scattering in this device. This is typical for polycrystalline nanowires of these dimensions\\cite{Durkan2000}. Finally, we would like to mention the importance of taking into account the contribution of an oxide shell to the heat transport in a nanowire.\n\n\\subsection{Analysis of the germanium section}\nNext, a value for the thermal conductivity in the Ge-segments is extracted. We note, that the conduction of electrons is limited by the area of the core, whereas the conduction of phonons occurs in both, the Ge and the Al$_2$O$_3$-shell. Equation \\ref{eq:diffusionJ} becomes\n\\begin{equation}\n\\label{eq:GeDGL}\n\\frac{\\partial^2}{\\partial x^2}\\Delta T_{\\rm j} + \\frac{g}{\\kappa_{\\rm Ge}A_{\\rm wire}}\\Delta T_{\\rm j}= - \\frac{I^2 \\rho_{\\rm Ge}}{A_{\\rm core}} \\frac{1}{A_{\\rm wire}\\kappa_{\\rm Ge}}.\n\\end{equation}\nHere, $\\kappa_{\\rm Ge}$ is an average value over core and shell in the Ge segment.\nThis second order differential equation is analytically solved by\n\\begin{equation}\n\\begin{split}\n\\label{eq:GeSolution}\n\\Delta T_{\\rm j}(x) = \\left( \\frac{\\Delta T_1 + \\Delta T_2}{2}- \\frac{q}{g} \\right) \\frac{\\cosh (m(x-x_c))}{ \\cosh (mL\/2)} \\\\ + \\frac{\\Delta T_2 - \\Delta T_1}{2} \\frac{\\sinh (m(x-x_c))}{ \\sinh (mL\/2)} + \\frac{q}{g}\n\\end{split}\n\\end{equation}\nwhere $x_c$ is the center on the nanowire, $\\Delta T_{1,2}$ is the temperature at the boundaries of the fit, the Joule dissipation per unit length $q = I^2 \\rho_{\\rm Ge} \/ A_{\\rm core}$, and $m = \\sqrt{g\/k_{\\rm Ge} A_{\\rm wire}}$. \nThis function is fitted to the temperature profile in the Ge-segment. In the Casimir result, the mean free path of phonons is limited by the diameter of the nanowire, although in this core-shell system the mean free path may be even smaller than that. The heat diffusion equation, equation\\,\\ref{eq:diffusion}, is valid for problems on a larger length scale than the mean free path. Therefore, the points lying closer to the thermal interface than the diameter are excluded for the fitting. Furthermore, to reduce uncertainty of the fit, we fix $g$ from the analysis above and use only $\\kappa_{\\rm Ge}$ and the boundary conditions as a fitting parameter in equation \\ref{eq:GeSolution}.\nThe fit is shown in Figure \\ref{fig:fits}c, and the extracted value for the thermal conductivity $\\kappa_{\\rm Ge}$ is $16.9\\pm2.5$\\,W\/m\/K at a current of 26\\,$\\mu$A.\n\nIt is interesting to note, that in this case of relatively small values of $m$, the temperature profile looks parabolic. For $g=0$, supposing that the temperature profile is dominated by Joule heating and heat evacuation towards the electrodes, the solution to equation \\ref{eq:GeDGL} would indeed be a parabola. Such a fit seems to match the data well. However, the extracted value is given by $\\kappa_{\\rm Ge} = 23.7 \\pm 3.3$\\,W\/m\/K higher by 40\\% and we conclude that the hyperbolic model should not be simplified.\n\nTo interpret the result, we consider that the extracted value for $\\kappa_{\\rm Ge}$ comprises the contributions of both oxide shell and Ge core. The area of the shell is double the area of the core, however, the exact values are measured on the STEM image. Both Al$_2$O$_3$ and Ge crystals have a larger conductivity in bulk, of about 30 and 50 to 60\\,W\/m\/K, respectively. Both values, however, are expected to be significantly reduced, due to the boundary scattering of the phonons at reduced dimensions\\cite{Mingo2003} and the measured value is consistent with expectations.  \n\n\\subsection{Extraction of the Peltier coefficient and thermoelectric properties}\n\\begin{figure}[tb]\n\t\\centering\n\n\t\\includegraphics[width=1.0\\columnwidth]{figure_peltier.pdf}\n\t\\caption[A floating figure]{Illustration of the Peltier effect a) The Fermi energy of electrons in a metal is lower than the average transport energy in a semiconductor. When electrons pass from a metal to a semiconductor they absorb energy from the lattice which creates a local heat sink. On the other side, when electrons pass from the semiconductor to the metal the higher energy electrons thermalize with the lattice which creates a local heat source.\n\tb) The Peltier coefficient is extracted by considering, that the injected heat is equal to the heat flux to the left and the heat flux to the right. The temperature gradients are calculated by help of an exponential fit in the Al-segment and a linear fit in the Ge-barrier.}\n\t\\label{fig:peltier}\n\\end{figure}\n\nAt this length scale, the distinction between the Peltier effect and thermionic emission becomes blurry. Whereas the Peltier effect exists also in bulk materials, the thermionic effect is understood in a microscopic regime around an energy barrier present at the interface of a heterostructure\\cite{thermoelec_book}. This region is marked by the thermalization length of the non-equilibrium charge carriers with the lattice. In this analysis, an effective Peltier coefficient is assigned to the thermionic cooling and heating by charge carriers transported over a potential barrier. The local Peltier coefficient is associated with a length scale $\\lambda$  through $\\pi (x) = (\\Pi \/ \\lambda) \\exp (-(x-x_0)\/\\lambda)$. On the scale of $\\lambda$ the non-equilibrium charge carriers equilibrate with the lattice (i.e. the phonon system) after passing the the metal-semiconductor interface. Indeed, we observe this distance to be larger than the scattering length of charge carriers for electron-phonon scattering of about 19 to 25\\,nm\\cite{haynes2014crc, Kanter1970}. \n\nThe extraction of the Peltier coefficient is based on the conservation of heat flux at steady state. In the Al wires and near the Al-Ge interfaces, there is either a heat sink or a heat source caused by the Peltier effect $\\dot{Q}_{\\rm GeAl}=\\Pi\\cdot I$ or $\\dot{Q}_{\\rm AlGe}=-\\Pi\\cdot I$, respectively, where $\\dot{Q}_{\\rm AlGe}>0$ and $\\dot{Q}_{\\rm GeAl}<0$, and $\\Pi$ is the Peltier coefficient. An illustration is seen in Figure \\ref{fig:peltier}b. The injected heat $\\dot{Q}$ from these regions is then equal to the sum of the heat flux towards the Al-electrodes and the heat flux over the Ge-segment to the other heat source\/sink. (We can again neglect on the short length scale $\\lambda$, as discussed above.) By use of Fourier's law the relation is then written as\n\\begin{equation}\n\\dot{Q}= (-\\kappa_{\\rm Al}\\nabla T_{\\rm Al} - \\kappa_{\\rm Ge}\\nabla T_{\\rm Ge})\\cdot A_{\\rm wire}.\n\\label{eq:injectedHeat}\n\\end{equation}\nIn order to take into account the above described microscopic extension of the Peltier coefficient we consider the temperature profile beyond the length scale away from the Al-Ge interface. Then we can treat the temperature profiles as solutions to equation \\ref{eq:diffusionP} in sections without source terms.\nThe temperature gradient in the Al-segment is the derivative with respect to the position $x$ of the exponential fit to the Peltier profile evaluated at the junction as $\\nabla T_{\\rm Al} = \\sqrt{\\frac{g}{\\kappa A}} \\cdot\\Delta T_{\\rm AlGe}$, see equation\\,\\ref{eq:exponetial}. By extrapolating the exponential fit we conceptually concentrate the extended heat source in a point at the AlGe interface. The temperature gradient in the Ge-segment is obtained from a linear fit to the data between heat source and heat sink. The linearity of the temperature profile within the Ge section is a result from the strong temperature gradient and the $\\kappa \/ g$ ratio which justifies neglecting substrate loss $g$ in this region. \nThe effective Peltier coefficient is then calculated as $\\Pi=267\\pm25$\\,mW\/A, resulting in a Seebeck coefficient of $S=790\\pm95\\,\\mu$V\/K for the temperature at the Al-Ge interface. \n\nIt is interesting to relate the effective Peltier coefficient to the Schottky barrier height to test the validity of the derived value. The cooling power by thermionic emission over a barrier is given by the total current times the average energy of the carriers over the barrier\n\\begin{equation}\n\\dot{Q}=I\\cdot\\bigg(\\phi_C+\\frac{2k_BT}{e}\\bigg).\n\\end{equation}\nHence, the barrier height may be calculated from our measurements. We find $\\phi_{\\rm c}=325\\pm45$\\,meV, which is in agreement with previously determined barrier heights from gating experiments of similar devices\\cite{Kral2015}. Independent of metal type and doping concentrations, metal-germanium junctions form Schottky contacts and exhibit very strong Fermi-level pinning close to the valence band \\cite{THANAILAKIS19731383, Dimoulas2006}.\nFinally, the thermoelectric figure of merit is calculated for the scan taken at a current of $I=26\\,\\mu$A to be $ZT = 0.020\\pm0.005$. This finalizes the thermoelectric characterization of the Al-Ge heterostructure nanowire with a segment length of 168\\,nm from a single measurement.\n\nLastly, we have a closer look at the Ge-Al interface regions. We acknowledge that extracting quantitative results is hampered by the oxide shell, which does not comprise an interface. However, important observations can nevertheless be made. The conjecture that the Peltier source term has an exponential spatial distribution has testable consequences. As a result of the distributed source term (equation \\ref{eq:petcoefficient}), the point of maximum temperature should be shifted with respect to the Al-Ge interface. Using the differential equation with this source term and the material parameters extracted above, we calculate that the maximum temperature should be about 46\\,nm separated from the interface using an equilibration length of 22\\,nm. This shift is a good estimation with the position of maximum temperature extracted from the experimental temperature profile of $58 \\pm 10$\\,nm. \n\n\\section{Conclusion}\nThis study demonstrates the capability to use thermal imaging by SThM for extracting relevant information on material properties and the dynamics of nanoscale devices and structures. Most notably, we were able to obtain the thermal and thermoelectric properties of a model heterostructure from a single measurement. By quantifying the thermal contact resistance between sample and substrate, its effect can be quantified and, in contrast to other techniques, does no longer hamper the measurements. The values are confirmed by independent measurements, such as the derivation of the substrate coupling with two different models and the comparison of the barrier height investigated thermally with previous electrical transport measurements. It appears, there is currently no other method available to extract this set of information from a sample of this size. For thermoelectric applications, the performance of short segments is interesting, and the results create a link between device design and materials properties. \n\n\\section{Acknowledgements}\nThis project has received funding from the European Union's Horizon 2020 research and innovation programme under Grants 482\nNo. 766853 (EFINED) and No. 767187 (QuIET).\n\nContinuous support from Anna Fontcuberta i Morral, Ilario Zardo, \nMarta De Luca, Kirsten Moselund, Heike Riel and Steffen Reidt is gratefully acknowledged.\n\n\n\n\\section{Device Fabrication}\nThe NWs are fabricated by thermally induced substitution of gold assisted vapor-liquid-solid (VLS) grown Ge-NWs by Al. At first, <111> oriented Ge-NWs are grown heteroepitaxially on silicon substrates in a low pressure chemical vapor deposition system by use of a gold-assisted VLS process. The NWs are then coated with 15\\,nm high-\\textit{k} Al$_2$O$_3$ by atomic layer deposition (ALD) and drop-cast on a highly p-doped Si substrate with a 100\\,nm thick dielectric layer of SiO$_2$. Afterwards, the Al-contacts are formed with electron-beam lithography, sputter deposition and lift-off techniques in preparation for the subsequent thermal exchange reaction. Lastly, in order to form the single crystalline Al-Ge-Al heterostructure, the Ge-NWs are thermally annealed at 623\\,K. Under these conditions, the diffusion constants of Ge and Al are $10^{12}$ times higher in Al than in Ge\\cite{Elhajouri2019}. This means, that the Ge can easily diffuse into the Al-contact pats, whereas the Al-atoms are efficiently supplied via fast self-diffusion to take over the released lattice sites. The successive substitution of Ge-atoms with Al has been monitored \\textit{in-situ} in both, scanning electron microscopy (SEM) and scanning transmission electron microscopy (STEM) studies. They revealed atomically sharp interfaces between the Al and Ge during the anneal. Thus, the length $L_{\\rm Ge}$ of the Ge-segment is tuned by varying the annealing time. For sufficiently long baking times the Ge diffuses completely out of the wire into the contact pads. It leaves behind single crystalline Al (c-Al) NWs with a single grain boundary in the center. Furthermore, the Ge-segments are integrated in a back-gated field effect transistor (FET) by construction. The Al-contacts are naturally self-aligned with the wire. More details are found in references \\cite{Kral2015,Elhajouri2019,Brunbauer_2016}.\n\n\\section{NW properties}\nFigure 1 b-d in the paper show a schematic, a STEM image of the cross-section and a SEM of the nanowire heterostructure studied here. More particularly it consists of a single-crystalline Ge-segment of 168\\,nm length and 37\\,nm diameter, as determined using scanning transmission electron microscopy (STEM). The Ge-segment is contacted by two self-aligned c-Al nanowires of the same diameter and around 1.7\\,$\\mu$m length. An aluminium oxide shell (Al$_2$O$_3$) of 15\\,nm thickness deposited by atomic layer deposition (ALD) surrounds the Al-Ge-Al core. The interfaces between the c-Al and Ge are abrupt to the atomic level. The Al-sections of the nanowire are contacted using Al pads patterned with electron-beam lithography on a Si\/SiO$_2$-substrate. \n\nThe electrical properties of pure c-Al nanowires fabricated using the same process revealed a conductivity of $\\sigma = (7.6 \\pm 1.5) \\cdot 10^6 \\, (\\Omega$m$)^{-1}$ for Al\\cite{Brunbauer_2016}. The Al-Ge-Al heterostructures with atomically abrupt interfaces show the non-linear current-voltage (IV) relationship of two back-to-back Schottky diodes in series for Ge-segments of length $L_{\\rm Ge}>45$\\,nm with an overall resistance  proportional to the Ge-segment length\\cite{sistani2017}. From gating experiments it is further concluded that the Ge segments act like p-type semiconductors. Independent of metal type and doping concentrations, metal-germanium junctions form Schottky contacts, exhibiting very strong Fermi-level pinning close to the valence band \\cite{THANAILAKIS19731383, Dimoulas2006}.\n\nOne issue regarding the thermal characterization of current-carrying Al-Ge-Al nanowires are traps for charge carriers located at the Ge\/Ge-oxide interface. In this regard, the protective Al$_2$O$_3$-shell ensures reliable and reproducible measurements by avoiding any influence of adsorbates rather than eliminating charge trapping due to dangling bonds at the Ge\/Ge-oxide interface. These traps lead to hysteretic current voltage relationships. However, the hysteresis decreases with increasing current range, drive speed and operating time. It was found that for currents larger than 21\\,$\\mu$A, the hysteresis is reduced and the current-voltage graph sufficiently linear to assume a constant resistance in the thermal analysis. Consequently, we estimate a contribution to the systematic error to the extracted temperature values for lower currents shown below due to these charging effects.\n\n\\section{Imaging and Geometric Characterization of the Wire}\n\nAfter the thermal scans, a scanning transmission electron microscopy analysis of an Al segment is carried out with a double spherical aberration-corrected JEOL JEM- ARM200F operated at 200 kV. The Al segment was prepared by focused ion beam, using a dual beam FIB Helios NanoLab 660 from FEI. The DF STEM images of the cross-section are shown in figure \\ref{fig:STEM} and reflect the round shape. A sharp interface separates the core and the oxide shell with respective areas of $A_{\\rm core} = 1075\\pm 80 $\\,nm$^2$ and $A_{\\rm shell} = 2450\\pm230 $\\,nm$^2$. The Al-core shows some poly-crystallinity, however, the loss of crystalline order probably occurred during sample preparation in the FIB. Previous more extensive studies suggest crystalline order. Furthermore, the Al$_2$O$_3$-shell is observed to be crystalline in the Fourier image of the cross-section (not shown here). \n\nEnergy Dispersive X-ray Spectroscopy (EDS) chemical maps reveal a well defined Al-core and Al$_2$O$_3$ shell with a sharp transition between the two. No residual Ge is detected beyond the detection limit which does not resolve doping concentrations. They are shown in figure \\ref{fig:eds}. To conclude, all geometric properties of the NW are measured and summarized in table \\ref{tab:geometry}. \n\nFinally, figure \\ref{fig:sem_after} shows the SEM-image taken in the FIB, when the wire was no longer conducting. The location at which the wire broke, lies close to the upper Al-electrode. In most of the experiments the wires break in a similar spot, even though the heating is observed closer to the Ge-segment. The wire is slightly thinner towards the electrodes due to the chemical etching of the oxide shell before the deposition of the electrodes.\n\n\\begin{figure}[tb]\n\t{\\includegraphics[width=1.0\\columnwidth]{figure_stm.pdf}}\n\t\\caption{DF-STEM scans taken after the thermal measurements. The oxide Aluminium Oxide layer shows a certain degree of crystalinity which enhances the thermal conductivity.}\n\t\\label{fig:STEM}\n\\end{figure}\n\\begin{figure*}[tb]\n\t\\centering\n\t\\includegraphics[width=2.0\\columnwidth]{figure_stmanalysis.pdf}\n\t\\caption{Chemical Analysis (EDS) of the nanowire cross-section. a) Along the a line through the center of the wire's cross-section, see the green line on the STEM image in the inset. b), c), d), e) show the number of counts at the energy of the Al, O, Pt and Si respectively. They demonstrate the high purity of the different materials. The platinum was deposited after the thermal measurements, to protect the wire before the FIB cut.}\n\t\\label{fig:eds}\n\\end{figure*}\n\\begin{figure}[tb]\n\t{\\includegraphics[width=0.8\\columnwidth]{semafter.jpg}}\n\t\\caption{The SEM image after the thermal scans when the wire was no longer conducting. The breaking point lies close to the electrodes, even though the thermal scans revealed only little heating in these areas.}\n\t\\label{fig:sem_after}\n\\end{figure}\n\n\n\\begin{table}[tbh]\n\t\\centering\n\t\\begin{tabular}{|c|l|} \n\t\t\\hline\n\t\t$L_{\\rm wire}$ & $(2475\\pm5)$\\,nm\\\\ \\hline\n\t\t$L_{\\rm Ge}$ & $(168\\pm1)$\\,nm \\\\ \\hline\n\t\t$r_{\\rm core}$ & $(18.5\\pm2)$\\,nm \\\\ \\hline\n\t\t$d_{\\rm oxide}$ & $(15\\pm2)$\\,nm \\\\ \\hline\n\t\t$d_{\\rm touch}$ & $(56\\pm5)$\\,nm \\\\ \\hline\n\t\t$A_{\\rm core}$ & $(1075\\pm 80) $\\,nm$^2$ \\\\ \\hline\n\t\t$A_{\\rm shell}$ & $(2450\\pm230 $\\,nm$^2$ \\\\ \\hline\n\t\t$A_{\\rm wire}$ & $(0.0035\\pm0.0003)\\, \\mu $m$^2$ \\\\ \\hline\n\t\\end{tabular}\n\t\\caption{Geometric properties of the NW obtained from SEM and STEM images: $L_{\\rm Ge}$ is the length of the Ge segment, $r_{\\rm core}$ is the radius of the conducting core, $d_{\\rm oxide}$ is the thickness of the surrounding Al$_2$O$_3$ shell, $d_{\\rm touch}$ is the length perpendicular to the wire that is in contact with the substrate. $A_{\\rm core}$, $A_{\\rm shell}$ and $A_{\\rm wire}$ are the cross-sections of the conducting inner core, the passivating Al$_2$O$_3$ shell and the entire wire respectively. The error estimation for the areas follows Gaussian error propagation.}\n\t\\label{tab:geometry}\n\\end{table}\n\n\n\\section{Electrical Characterization of the nanowire}\n\nThis section provides some overview on the electrical behavior of the NW under measuring conditions for the thermal scans. Figure \\ref{fig:ivbeforeheating} displays the I-V relation that is taken just below the threshold voltage at which any heating of the wire is detected. The I-V has two very striking features: First, a pronounced hysteresis appears between ramping the current up and down. Second, the electrical resistance depends highly on the applied voltage. Both observations are well visible on the resistance-voltage (R-V) plot in the logarithmic scale depicted next to the I-V. It reveals a change in resistance of two orders of magnitude. A heating of the wire is observed only if the applied electric field is higher than the electrical breakdown field in pure Ge wires that has been previously observed at $E=1.25\\cdot10^5$\\,V\/cm \\cite{Lugstein_2013}. For a Ge-segment of length $L_{\\rm Ge} = 168$\\,nm this corresponds to an applied voltage bias of $V_{\\rm bias}=2.1$\\,V.\n\nThe wire is integrated in an electrical circuit including a series resistor of similar resistance $R_{\\rm S} = 100$\\,k$\\Omega$ that protects from current spikes. The real time data acquisition system is is also used to measure the IV after each scan. The IV is conducted in the DC-mode with 200 steps distributed over the AC-modulation. Each point is averaged over $20$\\,ms.\n\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=2.0\\columnwidth]{figure_ivs.pdf} \n\t\\caption[A floating figure]{Characterization of the electrical behavior of the nanowire. The data in a) and b) is measured just below a current density $J$ is that is high enough to lead to a detectable change in the temperature for the wire. a) is the current density is represented as a function of the voltage drop over the device, b) the same information shown with the device resistance as a function of the voltage drop over the NW. c) and d) show the I-V and R-V curves measured immediately after each SthM scan up to the same voltage bias. The total applied voltage $v_{\\rm tot}$ drops additionally over a series resistance of $R_{\\rm S}=100$\\,k$\\Omega$.}\n\t\\label{fig:ivbeforeheating} \n\\end{figure*}\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.8\\columnwidth]{rhoge.pdf} \n\t\\caption[A floating figure]{The electrical resistivity $\\rho_{\\rm Ge}$ is shown as a function of the average temperature in the Ge segment.}\n\t\\label{fig:rhoge} \n\\end{figure*}\n\nFigure \\ref{fig:ivbeforeheating} shows a plot of all I-V and R-V curves going up to the maximum voltage bias applied in the respective thermal measurements. The electric field in the Ge-segment is between $E \\approx V_{\\rm device}\/L_{\\rm Ge} = 1.27-1.44\\cdot10^5$\\,V\/cm. Each curves is recorded after one hour of operation time and, therefore, charging of the oxide shell during the thermal scan. The wires have some seconds to cool down after the scan. The very different behavior of the IVs for different maximally applied voltages is, therefore, not caused by changes in the device temperature which should be the same for equal current density. Hence, it might be attributed to the charging effect in the oxide shell. Moreover, the amount of charging depends upon whether the NW is operated under AC- or DC-current. It is observed, that this local gating phenomenon changes the resistance of the device by orders of magnitude for only small changes of the experimental condition. However, the amount of charging tends to a stationary state when the IV is run quickly several times. Therefore, the hysteresis is expected to disappear sufficiently during the scans in the SThM. A difficulty for investigating thermoelectric effects is the simultaneous presence of Joule heating, which, on the other hand, creates a positive feedback phenomenon by decreasing the resistance during heating. It is counteracted only by an increased charge carrier scattering at higher temperature.\n\nAn increase in drain current with the gate voltage is linked to the trapping of negative surface charges in the interband levels due to surface states and bulk impurities. An applied negative gate voltage provokes the accumulation of holes that continuously neutralize the trapped electrons\\cite{sistani2017}. A similar p-type charging has been shown to appear in undoped Si-NWs. In the Si-system, changing the passivation layer from Al-oxide to Si-oxide turns the device into a n-type FET, being OFF for negative gate voltage. Therefore, surface charges originating from interface defects and dangling bonds can generate doping effects in NWs\\cite{Winkler2015}. The use of a high quality thermal oxide reduces the number of charge traps and thereby also the hysteresis that occurs when the gate voltage is sweeped from negative to positive values and vice versa\\cite{sistani2017}. Under a given gate voltage, the neutralization of holes is a rather slow process. It reaches a stationary state after approximately 20 minutes of operation. Such a long time-span is owed to a kinetic limitation by either a diffusion or tunnel barrier in form of a GeO$_x$ layer at the interface between the Ge and the Al$_2$O$_3$. Indeed, such a layer is expected to form during ALD-deposition of the Al$_2$O$_3$ layer. It acts as a local gate and provides a large number of trapping states\\cite{Zhang2015}\\cite{staudinger2018}. To conclude, the nature of the passivation layer highly influences the reliability and reproducible performance of the device under the above described charging mechanism.\n\nThe electrical resistivity in the Ge-segment is determined as follows. It is calculated from the electrical resistance measured in the I-V curves which are taken after each scan. The resistance at the maximal voltage in the I-V is the same as the AC-voltage applied during the scan closest to the conditions during the respective thermal measurement. The contribution of the Al-wire and contacts is subtracted from the overall device resistance. The values are represented as a function of the average temperature in the Ge-segment in figure \\ref{fig:rhoge}. The electrical resistivity depends strongly on the temperature.\n\n\n\n\\section{SThM operation and analysis details}\nThe images of the scans are generated and treated with the help of the open-source software Gwyddion. Apart from the straight forward calculation of the temperature field using equation 1 in the paper, some further data processing is conducted: Firstly, under the observation that the trace and retrace are nearly identical, they are averaged for noise reduction. Secondly, a low-pass scaling is applied to compensate for the signal attenuation that is due to a delay in temperature response of the cantilever. Thirdly, a DC-signal offset is manually corrected to eliminate some remaining non-uniformity in the temperature field which is caused by the artifacts in the DC-field. It is hard to measure the temperature offset of the sensor exactly due to small changes in DC-signal when approaching the tip. However, by eye it is easy to see when the artifacts at the edge of the NW disappear. The procedure might be compared to aligning a microscope. The final heat maps are shown in figure 2.\n\n\nWhen turning the focus to the temperature profile around the Al-Ge interface, the location of the Ge segment on the temperature line has to be determined first. The temperature gradient of the Joule profile is plotted in figure \\ref{fig:gradient}a). A pronounced maximum and\nminimum peak marks the deflection points in the Joule profile. They result from a sudden change of the thermal and electrical conductivities between the Ge- and the Al-parts. Their distance of 169\\,nm corresponds precisely to the expected length of the Ge-segment in a scan with 10\\,nm resolution. The hereby identified location of the Ge-segment is then shaded in red on the profile lines and used as orientation in the following analysis. After the deflection point there is a transition length of about 40-50\\,nm, that is well seen on the zoom-in figure \\ref{fig:gradient}b). This area corresponds to the phonon-electron thermalization that is blurred by a combination of the spatial resolution and a parallel heat transport in the oxide shell.\n\\begin{figure}[tb] \n\t\\centering\n    \\includegraphics[width=1.0\\columnwidth]{figure_gradient.pdf}\n\t\\caption{Determination of the location of the Ge segment is possible due to a discontinuity in the temperature gradient at the thermal interface between the Ge and the Al segments. a) The temperature gradient $\\nabla T_{\\rm Joule}$ as a function of the position $x$ along the wire for the different operating currents. The gradients are normalized by the maximum temperature in the center of the red shaded Ge segment. b) shows a zoom on the gradient at the Ge-Al interface.}\n\t\\label{fig:gradient} \n\\end{figure}\n\nIn general, the temperature profiles show mostly symmetric behavior for the Joule and the Peltier signal. It reflects the good quality of the measurement data and the sample. However for the thermal measurements that were conducted at lower operating currents, the profile curves show a slight asymmetry. This is when the SThM method reaches its limitations due to the underlying assumption of a linear electrical behavior of the sample. This requirement is better fulfilled at higher currents and leads to a mixing of the $1f_{\\rm mod}$ and $2f_{\\rm mod}$ signal at lower currents.\n\\begin{figure}[tb]\n\t\\centering\n    \\includegraphics[width=1.0\\columnwidth]{figure_linearfits.pdf} \n\t\\caption{In a), the maximum temperature increase caused by Joule heating is plotted as a function of the dissipated power $P$. In b), the temperature difference between the two poles in the Peltier scan as a function of the operating current. The errorbars reflect the uncertainties on the temperature measurement.}\n\t\\label{fig:propI2I}\n\\end{figure}\nNonetheless, the comparison of the the maximum temperature in the Joule-scan with the dissipated power and the comparison between the maximum temperature difference in the Peltier-scan and the current reveal linear relationships. These results is expected under the the 1D diffusion equation 2 from the paper. They can be seen in figure \\ref{fig:propI2I}.\nThe noise levels are quite low. No noise reduction is used at any stage in the lateral direction of the wire, to omit altering the shape of the profile. The expected error on the measured temperature is a combination of a constant absolute error and a more important relative contribution that depends on the sample temperature. The absolute error is determined by taking the root mean squared of the temperature on the substrate far from the wire where no signal is expected. The relative error the result of a multiplication with the temperature difference to the sensor according to equation 1 is estimated to be around $5\\%$ of the sample temperature\\cite{konemann2019}. The estimated uncertainty for each temperature measurement along the profile of the NW is then calculated as\n\\begin{equation}\n\\Delta \\bigg(\\Delta T _{\\rm J\/P}\\bigg) = \\frac{1.84  [\\rm{K}]}{\\sqrt{15}}+0.05\\cdot\\Delta T_{\\rm J\/P}\n\\label{eq:error}\n\\end{equation}\nTo conclude, the method allows to resolve the temperature profiles of the samples that are caused by the Joule and Peltier effects down to the thermalization lengths of the heat carriers. In order to extract further information by the most simple means, the analysis is divided in different sectors on the wire.\n\n\n\\section{Joule heating in pure Aluminium nanowires}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{figure_AlJoule.pdf}\n\t\\caption[A floating figure]{Joule heating in an operando c-Al wire. a) shows the heat map measured with the SThM and b) shows the temperature profile along the nanowire axis under different voltage bias.}\n\t\\label{fig:AlJoule} \n\\end{figure}\n\nThermal measurements are also conducted on a c-Al NW. However, the electrical characterization indicates leaking of the device. Nonetheless,\nthe thermal measurements give some insight on where heat is generated. The Joule heat map in figure \\ref{fig:AlJoule} shows heating of the entire wire, compared to the Al-Ge-wire, where it is more localized around the Ge-segment. Even more striking is the heating of the contacts which is not present in the heterostructure. In the profile lines along the wire, also shown in figure A.4, a distinct heat source is identified. Most probably it is located where the Ge-segment disappeared during fabrication, leaving behind some grain boundary or impurities inducing more scattering of electrons during operation. Consequently, the electrical conductivity of the single crystalline Al-parts of the heterostructure wire is expected to be higher than the values determined for the entire c-Al-wires due to the absence of this defect.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThis paper is concerned with the interplay of the symmetric trilinear form $\\mu$ on the second cohomology group $H^{2}(X,\\mathbb{Z})$ \nand the Chern classes $c_{2}(X)$, $c_{3}(X)$ of a Calabi--Yau threefold $X$.  \nIt is an open problem whether or not the number of topological types of Calabi--Yau threefolds is bounded and \nthe original motivation of this work was to investigate topological types of Calabi--Yau threefolds via the trilinear form $\\mu$ on $H^{2}(X,\\mathbb{Z})$. \nThe role that the trilinear form $\\mu$ plays in the geography of $6$-manifolds is indeed prominent as \nC.T.C. Wall proved the following celebrated theorem by using surgery methods and homotopy information associated with these surgeries.\n\\begin{Thm}[C.T.C. Wall \\cite{Wall}] \\label{Wall}\nDiffeomorphism classes of simply-connected, spin, oriented, closed $6$-manifolds $X$\nwith torsion-free cohomology correspond bijectively to isomorphism classes of systems of invariants consisting of \n\\begin{enumerate}\n\\item free Abelian groups  $H^{2}(X,\\mathbb{Z})$ and $H^{3}(X,\\mathbb{Z})$, \n\\item a symmetric trilinear from $\\mu:H^{2}(X,\\mathbb{Z})^{\\otimes 3}\\rightarrow H^{6}(X,\\mathbb{Z})\\cong \\mathbb{Z}$ defined by $\\mu(x,y,z)=x\\cup y \\cup z$, \n\\item a linear map $p_{1}:H^{2}(X,\\mathbb{Z})\\rightarrow  H^{6}(X,\\mathbb{Z})\\cong \\mathbb{Z}$ defined by $p_{1}(x)=p_{1}(X)\\cup x$,\nwhere $p_{1}(X) \\in H^{4}(X,\\mathbb{Z})$ is the first Pontrjagin class of $X$, \n\\end{enumerate}\nsubject to: for any $x,y \\in H = H^{2}(X,\\mathbb{Z})$, \n$$\n\\mu(x,x,y)+\\mu(x,y,y)\\equiv 0 \\pmod{2},\\ \\\n4\\mu(x,x,x) -p_{1}(x)\\equiv 0\\pmod{24}.\n$$\nThe isomorphism $H^{6}(X,\\mathbb{Z})\\cong \\mathbb{Z}$ above is given by pairing the cohomology class with the fundamental class $[X]$ with natural orientation. \n\\end{Thm}\nAt present the classification of trilinear forms, which is as difficult as that of diffeomorphism classes of $6$-manifolds, is unknown.  \nIn the light of the essential role of the K3 lattice in the study of K3 surfaces, we would like to propose the following question:\n\\textit{what kind of trilinear forms $\\mu$ occur on Calabi--Yau threefolds?} \nThe quantized version of the trilinear forms, known as Gromov--Witten invariants or A-model Yukawa couplings, are also of interest to both mathematicians and physicists. \nOne advantage of working with complex threefolds is that we can reduce  our questions to the theory of complex surfaces by considering linear systems of divisors. \nFurthermore, for Calabi--Yau threefolds $X$, \nthe second Chern class $c_{2}(X)$ and the K\\\"{a}hler cone $\\mathcal{K}_{X}$ turn out to encode important information about $\\mu$ (see \\cite{Wil2, Wil4} for details).  \nOne purpose of this paper is to take the first step towards an investigation on \nhow the Calabi--Yau structure affects the trilinear form $\\mu$ and the Chern classes of the underlying manifold. \\\\\n\nIt is worth mentioning some relevant work from elsewhere. Let $(X,H)$ be a polarized Calabi--Yau threefold. \nA bound for the value $c_{2}(X)\\cup H$ in terms of the triple intersection $H^{3}$ is well-known (see for example \\cite{Wil3})  \nand hence there are only finitely many possible Hilbert polynomials $\\chi(X,\\mathcal{O}_{X}(nH))=\\frac{H^{3}}{6}n^{3} +\\frac{c_{2}(X)\\cup H}{12}n$ for such $(X,H)$.   \nBy the footnote below \nand standard Hilbert scheme theory, we know that the Calabi--Yau threefold $X$ belongs to a finite number of families. \nThis implies that once we fix a positive integer $n \\in \\mathbb{N}$, there are only finitely many diffeomorphism classes of polarized Calabi--Yau threefolds $(X,H)$ with $H^{3}=n$,  \nand in particular only finitely many possibilities for the Chern classes $c_{2}(X)$ and $c_{3}(X)$ of $X$.  \nExplicit bounds on the Euler characteristic $c_{3}(X)$ in terms of $H^{3}$ for certain types of Calabi--Yau threefolds are given in \\cite{BH, CK}; \nthe idea of this article is to record the following simple explicit result which holds in general, and which may be  useful for both mathematicians and physicists. \n\\begin{Thm} \\label{MAIN} \nLet $(X,H)$ be a very amply polarized Calabi--Yau threefold, i.e. $x=H$ is a very ample divisor on $X$.  \nThen the following inequality holds:\n$$\n-36\\mu(x,x,x)-80\\le \\frac{c_{3}(X)}{2}=h^{1,1}(X)-h^{2,1}(X)\\le 6\\mu(x,x,x)+40.\n$$\nMoreover, the above inequality can be sharpened by replacing the left hand side by $-80$, $-180$ and right hand side by $28$, $54$ when $\\mu(x,x,x)=1,3$ respectively \n\\footnote{It is shown by K. Oguiso and T. Peternell \\cite{OP} that we can always pass from an ample divisor $H$ on a Calabi--Yau threefold to a very ample one $10H$. }.  \n\\end{Thm}\n\nIn the last section,  we study the cubic form $\\mu(x,x,x):H^{2}(X,\\mathbb{Z})\\rightarrow \\mathbb{Z}$ for a K\\\"{a}hler threefold $X$, assuming that $\\mu(x,x,x)$ has a linear factor over $\\mathbb{R}$.  \nSome properties of the linear form and the residual quadratic form on $H^{2}(X,\\mathbb{R})$ are obtained;  \npossible signatures of the residual quadratic form are determined under a certain condition (for example $X$ is a Calabi--Yau threefold).   \n\n\\section{Bound for $c_{2}(X)\\cup H$} \n\nIn this section, we collect some properties of the trilinear form and the second Chern classes of a Calabi--Yau threefold. \nWe will always work over the field of complex numbers $\\mathbb{C}$. \\\\\n\nLet $X$ be a smooth K\\\"{a}hler threefold. \nThroughout this paper, we write $c_{i}(X)=c_{i}(TX)$ the $i$-th Chern class of the tangent bundle $TX$.  \nK\\\"{a}hler classes constitute an open cone $\\mathcal{K}_{X}\\subset H^{1,1}(X,\\mathbb{C})\\cap H^{2}(X,\\mathbb{R})$, called the  K\\\"{a}hler cone. \nThe closure $\\overline{\\mathcal{K}_{X}}$  then consists of nef classes and hence is called the nef cone. \nThe second Chern class $c_{2}(X) \\in H^{4}(X, \\mathbb{Z})$ defines a linear function on $H^{2}(X,\\mathbb{R})$. \nUnder the assumption that $X$ is minimal (for instance a Calabi--Yau threefold), \nresults of Y. Miyaoka \\cite{Miy} imply  that for any nef class $x \\in \\overline{\\mathcal{K}_{X}}$, we have  $c_{2}(X)\\cup x \\ge 0$. \\\\\n\n\nLet $X$ be a smooth complex threefold. \nWe define a symmetric trilinear form $\\mu:H^{2}(X,\\mathbb{Z})^{\\otimes3} \\rightarrow H^{6}(X,\\mathbb{Z})\\cong \\mathbb{Z}$ \nby setting $\\mu(x,y,z)=x\\cup y \\cup z$ for $x,y,z \\in H^{2}(X,\\mathbb{Z})$. \nBy small abuse of notation we also use $\\mu$ for its scalar extension. \n\n\n\\begin{Def}\nA Calabi--Yau threefold $X$ is a complex projective smooth threefold with trivial canonical bundle $K_{X}$ such that $H^{1}(X, \\mathcal{O}_{X})=0$. \n\\end{Def}\nFor a Calabi--Yau threefold $X$, the exponential exact sequence gives an identification $\\mathrm{Pic}(X)=H^{1}(X, \\mathcal{O}_X ^{\\times}) \\cong H^{2}(X,\\mathbb{Z})$. \nThe divisor class $[D]$ is then identified with the first Chern class $c_{1}(\\mathcal{O}_{X}(D))$ of the associated line bundle $\\mathcal{O}_{X}(D)$. \nIn the following we freely use this identification. \\\\\n\nThe Hirzebruch--Riemann--Roch theorem for a Calabi--Yau threefold $X$ states that $\\chi(X,\\mathcal{O}_{X}(D))=\\frac{1}{6} \\mu(x,x,x)+\\frac{1}{12}c_{2}(X)\\cup x$ for any $x=D \\in H^{2}(X,\\mathbb{Z})$. \nTherefore \n$$\n2\\mu(x,x,x)+c_{2}(X)\\cup x\\equiv0 \\pmod{12}. \n$$\nIn particular, $c_{2}(X)\\cup x$ is an even integer for any $x \\in H^{2}(X,\\mathbb{Z})$. \nIn the case when the cohomology is torsion-free, this also follows from the fact $p_{1}(X)=-2c_{2}(X)$ and  Wall's Theorem \\ref{Wall}. \nThe role played by $p_{1}(X)$ in his theorem is replaced by $c_{2}(X)$ for Calabi--Yau threefolds. \\\\ \n\nFor a compact complex surface $S$, the geometric genus $p_{g}(S)$ is defined by $p_{g}(S)=\\dim_{\\mathbb{C}}H^{0}(S,\\Omega_{S}^{2})$.  \nThe basic strategy we take in the following is to reduce the question on Calabi--Yau threefolds to compact complex surface theory by considering linear systems of divisors. \n\n\\begin{Prop} \\label{NC} \nLet $X$ be a Calabi--Yau threefold. \n\\begin{enumerate}\n\\item For any ample $x=H \\in \\mathcal{K}_{X}\\cap H^{2}(X,\\mathbb{Z})$ with $|H|$ free and $\\dim_{\\mathbb{C}}|H|\\ge 2$, the following inequalities hold. \n$$\n\\frac{1}{2}c_{2}(X)\\cup x  \\le 2\\mu(x,x,x)+C\n$$\nwhere $C=18$ when $\\mu(x,x,x)$ even and $C=15$ otherwise. \n\\item If furthermore the canonical map $\\Phi_{|K_{H}|}:H\\rightarrow \\mathbb{P}^{|K_{H}|}$ (which is given by the restriction of the map $\\Phi_{|H|}$ to $H$) is birational onto its image, the following inequality holds. \n$$\n\\frac{1}{2}c_{2}(X)\\cup x  \\le \\mu(x,x,x)+20 \n$$\n\\item If furthermore the image of the canonical map in (2) is generically an intersection of quadrics, the following inequality holds.\n$$\nc_{2}(X)\\cup x \\le \\mu(x,x,x)+48\n$$\n\\end{enumerate}\n\\end{Prop}\n\n\\begin{proof}\n(1) By Bertini's theorem, a general member of the complete linear system $|H|$ is irreducible and gives us a smooth compact complex surface $S \\subset X$. \nApplying the Hirzebruch--Riemann--Roch theorem and the Kodaira vanishing theorem to the ample line bundle $\\mathcal{O}_{X}(H)$, \nwe can readily show that the geometric genus $p_{g}(S)=\\frac{1}{6}\\mu(x,x,x)+\\frac{1}{12}c_{2}(X)\\cup x-1$. \nSince $K_{S}$ is ample, the surface $S$ is a minimal surface of general type. \nThen the Noether's inequality $\\frac{1}{2}K_{S}^{2}\\ge p_{g}(S)-2$ yields the desired two equalities depending on the parity of $K_{S}^{2}=\\mu(x,x,x)$.\\\\\n(2) The proof is almost identical to the first case. \nSince the surface $S$ obtained above is a minimal canonical surface, i.e. the canonical map $\\Phi_{|K_{S}|}:S\\rightarrow \\mathbb{P}^{|K_{S}|}$ is birational onto  its image,  \nthe Castelnuovo inequality for minimal canonical surfaces $K_{S}^{2}\\ge 3p_{g}(S)-7$ yields the inequality.\\\\\n(3) We say that an irreducible variety $S\\subset \\mathbb{P}^{p_{g}-1}$ is generically an intersection of quadrics \nif $S$ is one component of the intersection of all quadrics through $S$. \nIn this case, M. Reid \\cite{MR} improved the above inequality to $K_{S}^{2}\\ge 4p_{g}(S)+q(S)-12$. \nThe irregularity $q(S)=\\dim_{\\mathbb{C}}H^{1}(S,\\mathcal{O}_{S})=0$ in our case. \n\\end{proof}\n\nIf $x\\in \\mathcal{K}_{X}$ is very ample, the conditions in Proposition \\ref{NC} (1) and (2) are automatically satisfied. \nThe first two inequalities are optimal in the sense that equalities hold for the complete intersection Calabi--Yau threefolds $\\mathbb{P}_{(1^{4},4)}\\cap(8)$ and $\\mathbb{P}^{4}\\cap (5)$. \\\\\n\n\nIt is worth noting that polarized Calabi--Yau threefolds $(X,H)$ with $\\Delta$-genus $\\Delta(X, H)\\le 2$ are classified by K. Oguiso \\cite{O} \nand it is observed by the second author \\cite{Wil3} that the inequality $c_{2}(X)\\cup H \\le 10 H^{3}$ holds for those with $\\Delta(X,H)>2$. \nR. Schimmrigk's experimental observation \\cite{Sc} however conjectures the existence of a better linear upper bound of $c_{2}(X)$  \nfor Calabi--Yau hypersurfaces in weighted projective spaces.  \n\n\n\\begin{Prop}\nThe surface $S$ in the proof of Proposition \\ref{NC} is a minimal surface of general type with non-positive second Segre class $s_{2}(S)$. \n$s_{2}(S)$ is negative if and only if $c_{2}(X)$ is not identically zero. \n\\end{Prop}\n\\begin{proof}\nLet $i:S \\hookrightarrow X$ be the inclusion and we identify $H^{4}(S,\\mathbb{Z})\\cong \\mathbb{Z}$. \nA simple computation shows $c_{1}(S)=-i^{*}(x)$ and $c_{2}(S)=\\mu(x,x,x)+c_{2}(X)\\cup x$. \nSince $x \\in \\mathcal{K}_{X}$, $s_{2}(S)=c_{1}(S)^{2}-c_{2}(S)=-c_{2}(X)\\cup x \\le 0$ by the result of Y. Miyaoka \\cite{Miy}. \nThe second claim follows from the fact that $\\mathcal{K}_{X} \\subset H^{2}(X,\\mathbb{R})$ is an open cone.   \n\\end{proof}\nIf $X$ is a Calabi--Yau threefold and the linear form $c_{2}(X)$ is identically zero, it is well known that $X$ is the quotient of an Abelian threefold by a finite group acting freely on it. \n\n\n\n\\section{Bound for $c_{3}(X)$}\nIn this section, we apply to smooth projective threefolds the Fulton--Lazarsfeld  theory for nef vector bundles developed by J.P. Demailly, T. Peternell and M. Schneider \\cite{DPS}.  \nThis gives us several inequalities among  Chern classes and cup products of certain cohomology classes. \nWhen $X$ is a Calabi--Yau threefold, these inequalities simplify and provide us with effective bounds for the  Chern classes.\\\\\n\nRecall that a vector bundle $E$ on a complex manifold $X$ is called nef if the Serre line bundle $\\mathcal{O}_{\\mathbb{P}(E)}(1)$ on the projectivized bundle $\\mathbb{P}(E)$ is nef. \n\n\\begin{Thm}[J.P. Demailly, T. Peternell, M. Schneider \\cite{DPS}] \\label{DPS}\nLet $E$ be a nef vector bundle over a complex manifold $X$ equipped with a K\\\"{a}hler class $\\omega_{X} \\in \\mathcal{K}_{X}$. \nThen for any Schur polynomial $P_{\\lambda}$ of degree $2r$ and any complex submanifold $Y$ of dimension $d$, we have $\\int_{Y}P_{\\lambda}(c(E))\\wedge \\omega_{X}^{d-r} \\ge 0$.\n\\end{Thm}\nHere we let $\\deg c_{i}(E)=2i$ for $0\\le i \\le \\mathop{\\mathrm{rank}}\\nolimits E$ and the Schur polynomial $P_{\\lambda}(c(E))$ of degree $2r$ is defined by $P_{\\lambda}(c(E))=\\det(c_{\\lambda_{i}-i+j}(E))$ \nfor each partition $\\lambda=(\\lambda_{1},\\lambda_{2},\\dots) \\dashv r$ of a non-negative integer $r\\le \\dim Y$ with $\\lambda_{k}\\ge \\lambda_{k+1}$ for all $k\\in \\mathbb{N}$.  \n\n\\begin{Ex} (\\cite{Laz}, page 118) \nLet $X$ be a complex threefold and $E$ a vector bundle of $\\mathop{\\mathrm{rank}}\\nolimits E=3$, then\n$$\nP_{(1)}(c(E))=c_{1}(E), \\ \\ \nP_{(2)}(c(E))=c_{2}(E), \\ \\ \nP_{(1,1)}(c(E))=c_{1}(E)^{2}-c_{2}(E)\n$$\n$$\nP_{(3)}(c(E))=c_{3}(E), \\ \\ \\ \nP_{(2,1)}(c(E))=c_{1}(E)\\cup c_{2}(E)-c_{3}(E),\n$$\n$$\nP_{(1,1,1)}(c(E))=c_{1}(E)^{3}-2c_{1}(E)\\cup c_{2}(E)+c_{3}(E). \n$$\n\n\\end{Ex}\n\n\\begin{Prop}\nLet $X$ be a smooth projective threefold, $x,y \\in \\mathcal{K}_{X}\\cap H^{2}(X,\\mathbb{Z})$ and assume $x$ is very ample, then the following inequalities hold. \n\\begin{enumerate}\n\\item $8\\mu(x,x,x)+2c_{2}(X)\\cup x \\ge 4\\mu(c_{1}(X),x,x)+c_{3}(X)$\n\\item $64\\mu(x,x,x)+4\\mu(c_{1}(X),c_{1}(X),x)+4c_{2}(X)\\cup x  +c_{3}(X)\n           \\\\ \\ \\ \\ge 32\\mu(c_{1}(X),x,x)+c_{1}(X)\\cup c_{2}(X)$\n\\item $80\\mu(x,x,x)+10\\mu(c_{1}(X),c_{1}(X),x)+2c_{1}(X)\\cup c_{2}(X)\n           \\\\ \\ \\ \\ge 40\\mu(c_{1}(X),x,x)+\\mu(c_{1}(X),c_{1}(X),c_{1}(X))+10c_{2}(X)\\cup x +c_{3}(X)$\n\\item $12\\mu(x,x,y)+c_{2}(X)\\cup y \\ge 4\\mu(c_{1}(X),x,y)$\n\\item $24 \\mu(x,x,y) +\\mu(c_{1}(X),c_{1}(X),y) \\ge 8\\mu(c_{1}(X),x,y)+c_{2}(X)\\cup y$\n\\item $6\\mu(x,y,y)\\ge\\mu(c_{1}(X),y,y)$\n\\end{enumerate}\n\\end{Prop}\n\n\\begin{proof}\nThe very ample divisor $x=H$ gives us an embedding $\\Phi_{|H|}:X\\rightarrow \\mathbb{P}(V)$, where $V=H^{0}(X,\\mathcal{O}_{X}(H))$.  \nUsing the Euler sequence and the Koszul complex, we obtain the following exact sequence of sheaves \n$$\n0 \\longrightarrow \\Omega_{\\mathbb{P}(V)}^{k+1} \\longrightarrow \\bigwedge^{k+1}V \\otimes \\mathcal{O}_{\\mathbb{P}(V)}((-k-1)H) \\longrightarrow  \\Omega_{\\mathbb{P}(V)}^{k}\\longrightarrow 0 \n$$\nfor each $1\\le k \\le \\dim_{\\mathbb{C}}V-1$. \nWe see that $\\Omega_{\\mathbb{P}(V)}(2H)$ is a quotient of $\\mathcal{O}_{\\mathbb{P}(V)}^{\\oplus\\binom{\\dim_{\\mathbb{C}}V}{2}}$. \nThe vector bundle $\\Omega_{X}(2H)$ is then generated by global sections because it is a quotient of the globally generated vector bundle $\\Omega_{\\mathbb{P}(V)}|_{X}(2H)$. \nWe hence conclude that $\\Omega_{X}(2H)$ is a nef vector bundle. \nApplying Theorem \\ref{DPS} (or rather the inequalities derived using the above example) to our nef vector bundle $\\Omega_{X}(2H)$, straightforward computation shows the desired inequalities.  \n\\end{proof}\n\n\nThe above result (with appropriate modification) certainly carries over to complex manifolds of dimension other than $3$.  \n\n\n\\begin{Cor} \\label{FL} \nLet $X$ be a Calabi--Yau threefold, $x,y \\in \\mathcal{K}_{X}\\cap H^{2}(X,\\mathbb{Z})$ and assume $x$ is very ample, then the following inequalities hold. \n\\begin{enumerate}\n\\item $8\\mu(x,x,x)+2c_{2}(X)\\cup x \\ge c_{3}(X)$ \n\\item $64\\mu(x,x,x)+4c_{2}(X)\\cup x +c_{3}(X)\\ge 0$\n\\item $80\\mu(x,x,x)\\ge 10c_{2}(X)\\cup x+c_{3}(X)$\n\\item $24 \\mu(x,x,y)\\ge c_{2}(X)\\cup y$\n\\end{enumerate}\n\\end{Cor}\n\nIn recent literature there has been some interest in finding practical bounds for topological invariants of Calabi--Yau threefolds.  \nAs is mentioned in the introduction, the standard Hilbert scheme theory assures \nthat possible Chern classes of a polarized Calabi--Yau threefold $(X,H)$ are in principle bounded once we fix a triple intersection number $H^{3}=n \\in \\mathbb{N}$, \nbut now that we have effective bounds for the Chern classes (with a bit of extra data for the second Chern class $c_{2}(X)$) as follows.  \nRecall first that it is shown by K. Oguiso and T. Peternell \\cite{OP} that we can always pass from an ample divisor $H$ on a Calabi--Yau threefold to a very ample one $10H$.   \nThen the last inequality in Corollary \\ref{FL} says that once we know the trilinear form $\\mu$ on the ample cone $\\mathcal{K}_{X}$ \nthere are only finitely many possibilities for the linear function $c_{2}(X) : H^{2}(X,\\mathbb{Z}) \\rightarrow \\mathbb{Z}$. \nWe shall now give a simple explicit formula to give a range of the Euler characteristic $c_{3}(X)$ of a Calabi--Yau threefold $X$. \n\n\\begin{Thm} \\label{MAIN} \nLet $(X,H)$ be a very amply polarized Calabi--Yau threefold, i.e. $x=H$ is a very ample divisor on $X$.  \nThen the following inequality holds:\n$$\n-36\\mu(x,x,x)-80\\le \\frac{c_{3}(X)}{2}=h^{1,1}(X)-h^{2,1}(X)\\le 6\\mu(x,x,x)+40.\n$$\nMoreover, the above inequality can be sharpened by replacing the left hand side by $-80$, $-180$ and right hand side by $28$, $54$ when $\\mu(x,x,x)=1,3$ respectively. \n\\end{Thm}\n\\begin{proof}\nThis is readily proved by combining Proposition \\ref{NC} (1), (2) and Corollary \\ref{FL} (1), (2), (4).  \n\\end{proof}\n\nThe smallest and largest known Euler characteristics $c_{3}(X)$ of a Calabi--Yau threefold $X$ are $-960$ and $960$ respectively. \nOur formula may replace the question of finding a range of $c_{3}(X)$ by that of estimating the value $\\mu(x,x,x)$ for an ample class $x \\in \\mathcal{K}_{X} \\cap H^{2}(X,\\mathbb{Z})$.   \n\n\n\n\\section{Quadratic forms associated with special cubic forms}\n\nIn this section we further study the cubic form $\\mu(x,x,x):H^{2}(X,\\mathbb{Z})\\rightarrow \\mathbb{Z}$ for a K\\\"{a}hler threefold $X$, assuming that $\\mu(x,x,x)$ has a linear factor over $\\mathbb{R}$.  \nWe will see that the linear factor and the residual quadratic form are not independent. \nPossible signatures of the residual quadratic form are also determined under a certain condition. \nIf the second Betti number $b_{2}(X)>3$, the residual quadratic form may endow the second cohomology $H^{2}(X,\\mathbb{Z})$ mod torsion with a lattice structure. \\\\\n\nWe start with fixing our notation. \nLet $\\xi:V\\rightarrow \\mathbb{R}$ be a real quadratic form. \nOnce we fix a basis of the $\\mathbb{R}$-vector space $V$, $\\xi$ may be represented as $\\xi(x)=x^{t}A_{\\xi}x$ for some symmetric matrix $A_{\\xi}$. \nThe signature of a quadratic form $\\xi$ is a triple $(s_{+},s_{0},s_{-})$ \nwhere $s_{0}$ is the number of zero eigenvalues of $A_{\\xi}$ and $s_{+}$ $(s_{-})$ is the number of positive (negative) eigenvalues of $A_{\\xi}$. \n$A_{\\xi}$ also defines a linear map $A_{\\xi}:V\\rightarrow V^{\\vee}$ (or a symmetric bilinear form $A_{\\xi}:V^{\\otimes2}\\rightarrow \\mathbb{R}$).  \nThe quadratic form $\\xi$ is called (non-)degenerate if $\\dim_{\\mathbb{R}} Ker(A_{\\xi})>0 \\ (=0)$. \nWe say that $\\xi$ is definite if it is non-degenerate and either $s_{+}$ or $s_{-}$ is zero, and indefinite otherwise. \\\\\n\n\nLet $X$ be a K\\\"{a}hler threefold and assume that its cubic form $\\mu(x,x,x)$ factors as $\\mu(x,x,x)=\\nu(x)\\xi(x)$, where $\\nu$ is linear and $\\xi$ is quadratic map $H^{2}(X,\\mathbb{R}) \\rightarrow \\mathbb{R}$. \nWe can always choose the linear form $\\nu$ so that it is positive on the K\\\"{a}hler cone $\\mathcal{K}_{X}$. \nIt is proven (see the proof of Lemma 4.3 in \\cite{Wil}) that there exists a non-zero point on the quadric $Q_{\\xi}=\\{x\\in H^{2}(X,\\mathbb{R}) \\ | \\ \\xi(x)=0\\}$ and hence $\\xi$ is indefinite \nprovided that the irregularity $q(X)=\\dim_{\\mathbb{C}}H^{1}(X,\\mathcal{O}_{X})=0$ and the second Betti number $b_{2}(X)>3$.  \n\n\\begin{Prop}\nLet $X$ be a K\\\"{a}hler threefold. \nAssume that the trilinear form $\\mu(x,x,x)$ decomposes as $\\nu(x)\\xi(x)$ over $\\mathbb{R}$ (if the quadratic form is not a product of linear forms, then we may work \nover $\\mathbb{Q}$)\nand the linear form $\\nu$ is positive on the K\\\"{a}hler cone $\\mathcal{K}_{X}$. \nThen the following hold. \n\\begin{enumerate}\n\\item $\\dim_{\\mathbb{R}} Ker(A_{\\xi})\\le 1$.   \nIf $\\xi$ is a degenerate quadratic form, its restriction $\\xi|_{H_{\\nu}}$ to the hyperplane $H_{\\nu}=\\{x \\in H^{2}(X,\\mathbb{R})\\ | \\ \\nu(x)=0\\}$ is non-degenerate. \n\\item If the irregularity $q(X)=0$ (for example a Calabi--Yau threefold), then the signature of $\\xi$ is either $(2,0,b_{2}(X)-2)$, $(1,1,b_{2}(X)-2)$ or $(1,0,b_{2}(X)-1)$. \n\\item The above three signatures are realized by some Calabi--Yau threefolds with $b_{2}(X)=2$.  \n\\end{enumerate}\n\\end{Prop}\n\n\\begin{proof}\n(1) Let $\\omega_{X} \\in \\mathcal{K}_{X}$ be a K\\\"{a}hler class. \nThe Hard Lefschetz theorem states that the map $H^{2}(X,\\mathbb{R})\\rightarrow H^{4}(X,\\mathbb{R})$ defined by $\\alpha \\mapsto \\omega_{X} \\cup \\alpha$ is an isomorphism.\nHence the cubic form $\\mu(x,x,x)$ depends on exactly $b_{2}(X)$ variables. \nThen the quadratic form $\\xi$ must depend on at least $b_{2}(X)-1$ variables and thus we have $\\dim_{\\mathbb{R}}(Ker(A_{\\xi}))\\le 1$. \nAssume next that the quadratic form $\\xi$ is degenerate. \nThen the linear form $\\nu$ is not the zero form on $Ker(A_{\\xi})$ (otherwise $\\mu(x,x,x)$ depends on less than $b_{2}(X)$ variables).  \nThe restriction $\\xi|_{H_{\\nu}}$ is non-degenerate because $H^{2}(X,\\mathbb{R})=H_{\\nu} \\oplus Ker(A_{\\xi})$ as a $\\mathbb{R}$-vector space. \\\\\n(2) Let $L_{1} \\in \\mathcal{K}_{X} \\cap H^{2}(X,\\mathbb{R})$ be an ample class such that $\\mu(L_{1},L_{1},L_{1})=1$.  \nSince the K\\\"{a}hler cone $\\mathcal{K}_{X}\\subset H^{2}(X,\\mathbb{R})$ is an open cone, $X$ is projective by the Kodaira embedding theorem. \nThen the Hodge index theorem states \nthat the symmetric bilinear form $b_{\\mu,L_{1}}=\\mu(L_{1},*,**):H^{2}(X,\\mathbb{R})^{\\otimes 2}\\cong (NS(X)\\otimes \\mathbb{R})^{\\otimes 2} \\rightarrow \\mathbb{R}$ has signature $(1,0,b_{2}(X)-1)$, \nwhere $NS(X)$ is the Neron--Severi group of $X$. \nNote that $\\dim_{\\mathbb{R}} (L_{1} ^{\\perp} \\cap H_{\\nu}) \\ge b_{2}(X) -2$, \nwhere $L_1^{\\perp}$ denotes the orthogonal space to $L_{1}$ with respect to the non-degenerate bilinear form $b_{\\mu , L_1}$.   \nWe then have two cases; the first is when $\\dim_{\\mathbb{R}} (L_{1} ^{\\perp} \\cap H_{\\nu})=b_2(X) -1$ (i.e. $L_{1}^{\\perp} = H_{\\nu}$). \nIn this case we can write down a basis $L_{2}, \\ldots , L_{b_{2}(X)}$ for the subspace $H_{\\nu}$ which diagonalizes the quadratic form $b_{\\mu , L_{1}}|_{H_{\\nu}}$,  \nand hence (noting that $L_{1} \\not\\in H_{\\nu})$ the Gramian matrix of $b_{\\mu , L_{1}}$ \nwith respect to the basis $L_{1}, \\ldots , L_{b_{2}(X)}$ of $H^2(X, \\mathbb{R})$ is $A_{b_{\\mu,L_{1}}}=(b_{\\mu,L_{1}}(L_{i},L_{j}))=diag(1,-1,\\dots,-1)$. \nIf $\\dim_{\\mathbb{R}} (L_{1} ^{\\perp} \\cap H_{\\nu})=b_{2}(X)-2$, then we can write down a basis $L_{2} , \\ldots, L_{b_{2}(X)-1}$ \nfor the subspace $L_{1} ^{\\perp} \\cap H_{\\nu}$ which diagonalizes the quadratic form $b_{\\mu , L_{1}}|_{L_{1} ^{\\perp} \\cap H_{\\nu}}$, \nand then extend it to a basis $L_2, \\ldots , L_{b_2(X)}$ of $H_\\nu$.  \nThus in both cases $L_1 , \\ldots ,L_{b_2(X)}$ is a basis for $H^2 (X, \\mathbb{R})$; \nthe corresponding matrix $A_{b_\\mu ,L_1}$ will not be diagonal in this second case, \nbut the first $(b_{2}(X) -1)$-principal minor is, with one $+1$ and $b_{2}(X)-2$ entries $-1$ on the diagonal.\n\nLet us define a new basis $\\{M_{i}\\}_{i=1}^{b_{2}(X)}$ of $H^{2}(X,\\mathbb{R})$ by setting $M_{i}=L_{i}$ for $1\\le i \\le b_{2}(X)-1$ and  \n$$\nM_{b_{2}(X)}=L_{b_{2}(X)}+\\sum_{i=2}^{b_{2}(X)}b_{\\mu,L_{1}}(L_{i},L_{b_{2}(X)})L_{i} \\in H_{\\nu}. \n$$ \nLet $x=\\sum_{i=1}^{b_2{(X)}} a_{i}M_{i}$. \nThen the hyperplane $H_{\\nu}$ is defined by the equation $a_{1}=0$ and the K\\\"{a}hler cone $\\mathcal{K}_{X}$ lies on the side where $a_{1}>0$ by the assumption on $\\nu$.  \nTherefore we have \n$$\n\\mu(x,x,x)=a_{1}(a_{1}^{2}-\\sum_{i=2}^{b_{2}(X)-1}a_{i}^{2}+Ca_{1}a_{b_{2}(X)}+Da_{b_{2}(X)}^{2})\n$$\nfor some (explicit) constants $C,D \\in \\mathbb{R}$.\nSince the quadratic form is positive on the the K\\\"{a}hler cone $\\mathcal{K}_{X}$, there must be at least one positive eigenvalue \nand hence possible signatures are $(2,0,b_{2}(X)-2)$, $(1,1,b_{2}(X)-2)$ and $(1,0,b_{2}(X)-1)$. \\\\\n(3) Consider a Calabi--Yau threefold $X_{7}^{II}(1,1,1,2,2)^{2}_{-186}$ from page 575 \\cite{HLY} \ngiven as a resolution of a degree $7$ hypersurface in the weighted projective space $\\mathbb{P}_{(1,1,1,2,2)}$. \nIts cubic form is given by $a_{1}(14a_{1}^{2} +21a_{1}a_{2}+9a_{2}^{2})$, whose quadratic form has signature $(2,0,0)$. \nThe cubic form of a hypersurface Calabi--Yau threefold $(\\mathbb{P}^{3}\\times \\mathbb{P}^{1})\\cap(4,2)$ is $2a_{1}^{3}+12a_{1}^{2}a_{2}$,  \nwhose quadratic form has signature either $(1,0,1)$ or $(1,1,0)$, depending on its decomposition. \n\\end{proof}\n\n\nThe restriction $\\xi|_{H_{\\nu}}$ may be degenerate if $\\xi$ is non-degenerate. \nThe cubic form of the above Calabi--Yau threefold $(\\mathbb{P}^{3}\\times \\mathbb{P}^{1})\\cap(4,2)$ gives an example of such phenomenon. \nLet $\\nu(a)=2a_{1}$ and $\\xi(a)=a_{1}(a_{1}+6a_{2})$. \nThen $\\xi$ is hyperbolic and non-degenerate, but its restriction to $H_{\\nu}$ is trivial. \\\\\n\n\nLet $X$ be a K\\\"{a}hler threefold. \nIf $b_{2}(X)>3$, the cubic form $\\mu$ cannot consist of three linear factors over $\\mathbb{R}$ and hence if $\\mu$ contains a linear factor it must be rational (see also the comment after Lemma 4.2 \\cite{Wil}).  \nHence an appropriate scalar multiple of $\\xi$ endows the second cohomology $H^{2}(X,\\mathbb{Z})$ mod torsion with a lattice structure. \n\n\\begin{Ex}\nLet $X$ be a K3 surface and $\\iota_{S}$ an involution of $S$ without fixed point (hence the quotient $S\/\\langle \\iota_{S}\\rangle$ is an Enriques surface).  \nLet $E$ be an elliptic curve with the canonical involution $\\iota_{E}$. \nThen we can define a new involution $\\iota$ of $S\\times E$ given by  $\\iota=(\\iota_{S},\\iota_{E})$. \nThe quotient $X=(S\\times E)\/\\langle \\iota \\rangle$ is a Calabi--Yau threefold with $b_{2}(X)=11$. \nThe cubic form $\\mu(x,x,x$) of $X$ has a linear factor (which, we assume, is positive on the  K\\\"{a}hler cone $\\mathcal{K}_{X}$) and the residual quadratic form $\\xi$ has signature $(1,1,9)$. \nMore precisely, the lattice structure on $H^{2}(X,\\mathbb{Z})$ mod torsion associated with appropriate $\\xi$ is given by $U\\oplus E_{8}(-1)\\oplus \\langle 0\\rangle $, \nwhere $U$ is the hyperbolic lattice, $E_{8}(-1)$ is the root lattice of type $E_{8}$ multiplied by $-1$ and $\\langle 0\\rangle$ is a trivial lattice of rank $1$.  \n\\end{Ex}\n\n\n\n\n\\begin{Prop}\nLet $G$ be a finite group acting on a K\\\"{a}hler threefold $X$ and $\\phi:G\\rightarrow GL(H^{2}(X,\\mathbb{Z}))$ the induced representation.  \nAssume that the trilinear form decomposes $\\mu(x,x,x)=\\nu(x)\\xi(x)$ as above. \nThen the image of $\\phi:G\\rightarrow GL(H^{2}(X,\\mathbb{Z}))$ lies in the orthogonal group $O(\\xi)$ associated with the quadratic form $\\xi$.   \n\\end{Prop}\n\\begin{proof}\nSince the cubic form $\\mu:H^{2}(X,\\mathbb{R})\\rightarrow \\mathbb{R}$ is invariant under $G$, it is enough to show that the linear form $\\nu$ is invariant under $G$. \nThere exists $x\\in \\mathcal{K}_{X}$ such that $\\mathbb{R} x$ is a trivial representation of $G$ (by averaging a K\\\"{a}hler class over $G$) \nand then the representation $\\phi$ is a direct sum of two subrepresentations $\\mathbb{R} x\\oplus H_{\\nu}$. \nSince $\\nu$ is a linear form, this shows the invariance of $\\nu$ under $G$.  \n\\end{proof}\n\nThis proposition may be useful to study group actions on the cohomology group $H^{2}(X,\\mathbb{Z})$. \n\n\n\\subsection*{Acknowledgement}\nThe present work was initiated during the the first author's stay at the Workshop on Arithmetic and Geometry of K3 surfaces and Calabi--Yau threefolds, August 16-25, 2011. \nHe is grateful to the organizers of the workshop and Fields Institute for their warm hospitality and partial travel support.  \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nHeavy-tailed distributions are ubiquitous in modern datasets, especially those arising in economy, finance, imaging, biology, see \\cite{woolson2011statistical,sahu2019noising,ibragimov2015heavy,biswas2007statistical,swami2002some,kruczek2020detect} for instance. In contrast to sub-Gaussian distributions, outliers and extreme values appear much more frequently in data with heavy-tailed distributions (referred to as heavy-tailed data), which challenges statistical analysis. In fact, many standard statistical procedures developed for sub-Gaussian data suffer from performance degradation in the heavy-tailed regime. Consequently, the past decade has witnessed considerable interests on     statistical estimation methods that are robust to heavy-tailedness \\cite{catoni2012challenging,fan2017estimation,nemirovskij1983problem,minsker2015geometric,brownlees2015empirical,hsu2016loss,fan2021shrinkage,zhu2021taming,lugosi2019mean}. \n\n\n\n\nIn the era of digital signal processing, quantization is an inevitable process that maps signals to bitstreams so that they can be stored, processed  and transmitted.  In particular, the resolution of quantization should be selected to achieve a trade-off between accuracy and various data processing costs, and in some applications relatively low resolution would be preferable. For instance, in a distributed\/federated learning setting or a MIMO system, the frequent information transmission among multiple parties often result in prohibitive communication cost   \\cite{konevcny2016federated,mo2018limited}, and  quantizing signals or data to fairly low resolution (while preserving satisfying utility) is an effective approach to lower the cost. Under such a big picture, in recent years there appeared growing literature on high-dimensional signal recovery from quantized data (see, e.g., \\cite{boufounos20081,davenport20141,chen2022high,thrampoulidis2020generalized,jung2021quantized,dirksen2021non,dirksen2021covariance} for 1-bit quantization, \\cite{thrampoulidis2020generalized,jung2021quantized,xu2020quantized,jacques2017time} for multi-bit uniform quantization), trying to design appropriate quantization scheme for some fundamental estimation or recovery problems.\n\n\n\n\n\n\n\nIndependently, the heavy-tailed difficulty can be overcome by some robustifying techniques, and also, data quantization under uniform dither is shown to cost very little in some recovery problems. Considering ubiquitousness of heavy-tailed behaviour and quantization, a natural question is to design quantization scheme for heavy-tailed data that only incurs minor information loss. For instance, when applied to statistical estimation problems with heavy-tailed data, an appropriate quantization scheme should at least enable a faithful estimator from the quantized data, and  ideally an estimator achieving the (near) optimal error rate. Nevertheless, to the best of our knowledge, the only existing results that simultaneously take quantization and heavy-tailedness into account are in our earlier work   \\cite{chen2022high}, and these results are essentially sub-optimal. It remains an open question {\\em whether (near) minimax rates can be achieved if only quantized heavy-tailed data are available.} \n\n\nIn this paper, we address this question in the affirmative. We propose to truncate the data prior to a dithered uniform quantizer; under our quantization scheme, (near) minimax estimators are proposed in several estimation problems. We also study covariate quantization and uniform recovery in quantized compressed sensing with heavy-tailed data. \n\n\n\n \\subsection{Related works and our contributions}\n We review the most  relevant works and then explain our main results.\n \\subsubsection{Heavy-tailed data and quantization in estimation problems}\nWe first review   prior works on estimation with heavy-tailed data or under quantization. \n\n\n\n\n\n\n Compared to sub-Gaussian data, heavy-tailed data may contain many outliers and extreme values that are overly influential to traditional estimation methods. Hence, developing estimation methods that are robust to heavy-tailedness has become a recent focus in statistics literature, where heavy-tailed distributions are only assumed to have bounded moments of some order. In particular, significant efforts have been devoted to the fundamental problem of mean-estimation for heavy-tailed distribution. For instance, effective techniques for estimating the mean of heavy-tailed distribution include Catoni's mean estimator \\cite{catoni2012challenging,fan2017estimation}, median of means \\cite{nemirovskij1983problem,minsker2015geometric}, and trimmed mean \\cite{lugosi2021robust,devroye2016sub}. While the strategies to achieve robustness are different, these methods indeed share the same core spirit of making the outliers less influential. To this end, the trimmed method (also referred to as truncation or shrinkage) may be the most intuitive --- it truncates overlarge data to some threshold, the extreme values can thus be well controlled. For more in-depth discussions we refer to the recent survey \\cite{lugosi2019mean}. Furthermore, these robust methods for estimating the mean have been applied to empirical risk minimization \\cite{brownlees2015empirical,hsu2016loss} and various high-dimensional estimation problems \\cite{fan2021shrinkage,zhu2021taming}, achieving near optimal guarantees. For instance, by invoking M-estimators with truncated data, (near) minimax rates can be achieved in linear regression, matrix completion, covariance estimation \\cite{fan2021shrinkage}. \n\n\n\n\nDue to the prominent role of quantization in signal processing and machine learning, quantized compressed sensing (QCS) that studies the interplay between compressed sensing (CS) and  data quantization has become an active research branch in the field. In this work, we focus on memoryless quantization scheme\\footnote{This means the quantization for different measurements are independent. For other quantization schemes we refer to the recent survey \\cite{dirksen2019quantized}.} that embraces simple hardware design. An important QCS model is 1-bit compressed sensing (1-bit CS) where only the sign of the measurement is retained \\cite{boufounos20081,jacques2013robust,plan2012robust,plan2013one}, for instance, the recovery of sparse $\\bm{\\theta^\\star}\\in \\mathbb{R}^d$ from $\\sign(\\bm{X\\theta^\\star})$ under some sensing matrix $\\bm{X}\\in \\mathbb{R}^{n\\times d}$. However, 1-bit CS with this direct quantization suffers from some frustrating limitations, e.g., the loss of norm information, and the identifiability issue under some regular sensing matrix (e.g., under Bernoulli sensing matrix, see \\cite{dirksen2021non})\\footnote{All guarantees for this 1-bit CS model are restricted to standard Gaussian measurement, with a single exception of \\cite{ai2014one}, which is still quite restricted.}. Fortunately, these limitations can be overcome by random dithering prior to the quantization, meaning that the quantized measurements are $\\sign(\\bm{X\\theta^\\star}+\\bm{\\tau})$ for some random dither $\\bm{\\tau}\\in \\mathbb{R}^n$ to be chosen: with Gaussian dither, full reconstruction with norm becomes possible \\cite{knudson2016one}; with uniform dither, recovery with norm can be achieved under general sub-Gaussian sensing matrix  \\cite{dirksen2021non,jung2021quantized,thrampoulidis2020generalized,chen2022high} even with near minimax error rate \\cite{thrampoulidis2020generalized,chen2022high}. \n\n\n\nBesides the 1-bit quantizer that retains the sign, the uniform quantizer maps $a\\in \\mathbb{R}$ to $\\mathcal{Q}_\\Delta(a)= \\Delta\\big(\\lfloor \\frac{a}{\\Delta}\\rfloor +\\frac{1}{2} \\big)$ for some pre-specified $\\Delta>0$ (we refer to $\\Delta$  as the quantization level, and note that smaller $\\Delta$ represents higher resolution). While recovering $\\bm{\\theta^\\star}$ from $\\mathcal{Q}_\\Delta(\\bm{X\\theta^\\star})$ encounters identifiability issue (e.g., under Bernoulli sensing matrix and $\\Delta=1$, $1.1\\bm{e}_1$ and $1.2\\bm{e}_2$ can never be distinguished because of $\\mathcal{Q}_1(1.1\\bm{Xe}_1)=\\mathcal{Q}_1(1.2\\bm{Xe}_1)$), it is again beneficial to use  random dithering and observe $\\mathcal{Q}_\\Delta(\\bm{X\\theta^\\star}+\\bm{\\tau})$. More specifically, by using uniform dither the Lasso estimator \\cite{thrampoulidis2020generalized,sun2022quantized} and projected back projection (PBP) method \\cite{xu2020quantized} achieve   minimax rate. These error bounds demonstrate that the dithered uniform quantization only worsens the multiplicative factor in the error rate. While the above results of QCS with random dither are quite recent, we point out that the technique of dithering in quantization has a long history, and we refer to \\cite{gray1993dithered} for a brief introduction. \n\n\n\n\n\nSeveral other models were also   investigated under dithered 1-bit quantization, including signal parameter estimation, covariance matrix estimation, and matrix completion (MC). Specifically, \\cite{balan2006signal} studied a general estimation problem under dithered 1-bit  quantization in a traditional setting where sample size tends to infinity. Their results indicate that only logarithmic rate loss is incurred by the quantization. By collecting 2 bits per entry from each sample $\\bm{x}_k$, \\cite{dirksen2021covariance} proposed a covariance matrix estimator for sub-Gaussian distribution, which was then extended to high-dimensional case (with sparsity) in \\cite{chen2022high}. Moreover, 1-bit matrix completion (1-bit MC) was first proposed and studied in \\cite{davenport20141} based on    maximum likelihood estimation with a nuclear norm constraint. A series of follow-up works   developed this likelihood approach with various regularizers or constraints, or under multi-bit quantization on a finite alphabet \\cite{cai2013max,lafond2014probabilistic,klopp2015adaptive,cao2015categorical,bhaskar2016probabilistic}. With uniform dither, the method in \\cite{chen2022high}   essentially departs from the standard likelihood approach, representing the first 1-bit MC method robust to pre-quantization noise with unknown distribution.\n\n\n\nIt should be noted that the results we just reviewed are   for heavy-tailed data with full precision, or quantization of sub-Gaussian data. While for quantization of heavy-tailed data, e.g., those corrupted by heavy-tailed noise, the only results we are aware of were presented in \\cite{chen2022high} concerning covariance estimation, CS, MC. Applying truncation and a dithered 1-bit quantizer, however, all rates derived in \\cite{chen2022high} are essentially sub-optimal. To the best of our knowledge, there exists no previous example for heavy-tailed data with quantization  that nearly achieves optimal rate.  \n\n\n\\subsubsection{Quantization of heavy-tailed data and (near) minimax rates}\nWe now introduce our quantization scheme and the  (near) minimax guarantees for covariance (matrix) estimation, CS, MC. \n\n\nOur procedure for quantizing heavy-tailed data contains three steps: {\\it truncation} that shrinks data to some threshold, {\\it dithering} with proper random dither, {\\it uniform quantization}. For sub-Gaussian or sub-exponential data the truncation step is unnecessary, and one can set truncation threshold as $\\infty$ in this case. In fact, we replace the dithered 1-bit quantizer ($\\sign(.)$) in  \\cite{chen2022high} with the less extreme dithered uniform quantizer ($\\mathcal{Q}_\\Delta(.)$). The gain turns out to be significant --- we derive (near) optimal rates that improve the sub-optimal ones in \\cite{chen2022high}. Specifically,  \n  we consider the canonical   examples of covariance estimation (in low dimension or high dimension with sparsity), CS (particularly sparse recovery), and MC. From the quantized data, (near) minimax estimators are constructed in closed form or as solution of convex programs, see Theorems \\ref{thm1}-\\ref{thm9}.  \n  \n\n  The goal of truncation is to achieve a bias-and-variance tradeoff ---  introducing some bias can render significantly reduced   variance, thereby allowing more robust estimator \\cite{fan2021shrinkage}. Some of our results strictly generalize \\cite{fan2021shrinkage} to dithered uniform quantization setting without extra assumption, in the sense that the corresponding ones in \\cite{fan2021shrinkage} can be exactly recovered by setting quantization level to zero. For the effect of dithered uniform quantizaion, a unified conclusion is that   the dithered quantization only results in a slightly worse multiplicative factor in the error rate. For instance, in QCS with (regular) sub-Gaussian covariate but noise with only bounded $(2+\\nu)$-th moment (for some $\\nu>0$), we derive the rate $O\\big(\\mathscr{L}\\sqrt{\\frac{s \\log d}{n}}\\big)$ with $\\mathscr{L}=M^{1\/(2l)} +\\Delta$ (Theorem \\ref{thm4}).  Such unified conclusion generalizes similar ones for QCS in \\cite{thrampoulidis2020generalized,sun2022quantized,xu2020quantized} towards two directions, i.e., to heavy-tailed data, and to more estimation problems (i.e., covariance matrix estimation, MC). From a technical side, many of our analyses on the dithered quantizer are cleaner than prior works  because we make full use of the nice property of a dithered uniform quantizer\\footnote{Prior work that did not fully leveraged the property of dithered uniform quantizer may incur extra technical complication, e.g., the symmetrization and contraction in \\cite[Lemma A.2]{thrampoulidis2020generalized}.}, see section \\ref{prequan}. \n\n\n\n\n \n\n\\subsubsection{Covariate quantization}\nThis work contains some novel QCS results that involve the quantization of covariate.\n\n\nPrior works on QCS focused on quantization of the response, \nbut the consideration on covariate quantization is rather natural because of the direct outcome of lower communication or memory cost (e.g., in distributed machine learning one often encounters some communication constraint on the number of bits). Indeed, for QCS results with quantizd covariate we are only aware of \\cite[Theorems 7-8]{chen2022high} concerning 1-bit quantization of the covariate-response pair in CS. The key downside of these two results is that they require the uncommon sparsity on the covariance matrix of the covariate (denoted $\\bm{\\Sigma^\\star}$) to provide positive definiteness for the covariance matri estimator. See \\cite[Assumption 3]{chen2022high}.   \n\n\n\nIn this work, we present QCS results with covariate quantization free of  non-standard assumption such as sparsity on $\\bm{\\Sigma^\\star}$. Specifically, the quantization scheme is applied to $\\bm{x}_k$ with triangular dither, then,   our covariance matrix estimator is plugged in the framework of regularized M-estimator. While the resulting recovery program is non-convex, we prove (near) minimax error bounds for all local minimizers (Theorems \\ref{thm6}-\\ref{thm7}). To derive guarantee of this kind, our analysis bears resemblance to a line of works on non-convex M-estimator \\cite{loh2011high,loh2017statistical,loh2013regularized}, but also exhibit some essential differences (Remark \\ref{rem4}). As byproducts, we present 1-bit QCS results with quantized covariate that are comparable to \\cite{chen2022high} without assuming sparsity on $\\bm{\\Sigma^\\star}$ (Theorems \\ref{sg1bitqccs}-\\ref{ht1bitqccs}).    \n\n\n\n\n\n\n\\subsubsection{Uniform recovery guarantee}\nWe also derive a uniform error bound for QCS with heavy-tailed data. \n\n\nIt is standard in CS to leverage measurements with randomness, and consequently, a prominent feature of a CS result (e.g., exact recovery, error bound for approximate recovery) is the uniformity. In brief, a uniform result guarantees the recovery of all structured signals with a single usage of the randomness, and CS associated with a uniform guarantee can be referred to as a uniform compressive coding scheme. In contrast, a non-uniform result is only valid for a fixed structured signal, and basically, a new realization of the sensing matrix is required for the sensing of a new signal. \n\n\n\nIt is well-known in linear CS that, the restricted  isometry property (RIP) of the sensing matrix implies uniform recovery of all sparse signals (e.g., \\cite{Foucart2013}). However, when it comes to various nonlinear CS models,   uniform recovery is still eagerly pursued so far. For example, in the specific regime of quantization (1-bit\/uniform quantization with\/without dithering), or a more general single index model $y_k = f\\big(\\bm{x}_k^\\top\\bm{\\theta^\\star}\\big)$ with possibly unknown nonlinearity $f(.)$, most representative results are non-uniform (e.g., \\cite{jacques2013robust,plan2017high,plan2016generalized,thrampoulidis2020generalized,chen2022high,jacques2021importance}, or part of \\cite{xu2020quantized,jung2021quantized}), see the discussions in \\cite{genzel2022unified}. For examples of uniform recovery, we refer to \\cite{plan2013one,dirksen2021non,xu2020quantized,jung2021quantized,genzel2022unified,chen2022uniform}; some of them remain (near) optimal, while others suffer from essential degradation   compared to the non-uniform ones. \n\n\n\nWe improve the non-uniform Theorem \\ref{thm4} to a uniform recovery guarantee  in Theorem \\ref{uniformtheorem}, seemingly the first uniform   result for QCS with heavy-tailed data. It states that, with a single realization of the sub-Gaussian sensing matrix, heavy-tailed noise, uniform dither, all $s$-sparse signals within a $\\ell_2$ ball can be   estimated with the near minimiax $\\ell_2$ norm error ${O}\\big(\\sqrt{\\frac{s\\log \\mathscr{W}}{n}}\\big)$ for some $\\mathscr{W}$ depending on $(n,d,s,\\Delta)$ and other parameters. Our proof builds on an in-depth   concentration result and a covering argument. Some new machinery is also needed to handle the truncation step.\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Outline}\nThe remainder of this paper is structured as follows. We provide the notation and preliminaries in section \\ref{sec2}. We present the first set of   main results, i.e., the (near) optimal guarantees for three quantized estimation problems with heavy-tailed data, in section \\ref{sec3}. Our second set of results for covariate quantization and uniform recovery in QCS is given in section \\ref{qc-qcs}. To corroborate our theory, numerical results on   synthetic data  are reported in section \\ref{sec6}. We give some remarks to conclude the paper in section \\ref{sec7}. All the proofs are postponed to the Appendices.\n\n\n\n\n\\section{Preliminaries}\\label{sec2}\nWe adopt the following conventions throughout the paper:\n\n\n\n 1) We use  capital   letters (e.g., $\\bm{A}$, $\\bm{x}$) to denote matrices and vectors, and regular letters (e.g, $a$, $x$) for scalars. We write $[m] = \\{1,...,m\\}$ for positive integer $m$. We denote the complex unit by $\\bm{i}$. The $i$-th entry for a vector $\\bm{x}$ (likewise, $\\bm{y}$, $\\bm{\\tau}$) is denoted by $x_i$ (likewise, $y_i$, $\\tau_i$).\n \n \n \n 2) Notation with \"$\\star$\" as superscript denotes the desired underlying parameter or signal, e.g., $\\bm{\\Sigma}^\\star$, $\\bm{\\theta}^\\star$.   Moreover, notation marked by a tilde (e.g., $\\bm{\\widetilde{x}}$) and a dot (e.g., $\\bm{\\dot{\\bm{x}}}$)    stands for the truncated data and quantized data, respectively.\n\n\n3) We reserve $d$, $n$ for the problem dimension and sample size, repectively. \n\n\n \n \n \n 4) For vector $\\bm{x} \\in \\mathbb{R}^d$, we work with its transpose $\\bm{x}^\\top$,  $\\ell_p$ norm $\\|\\bm{x}\\|_p = (\\sum_{i\\in [d]} |x_i|^p)^{1\/p}$ ($p\\geq 1$), max norm $\\|\\bm{x}\\|_{\\infty} = \\max_{i\\in [d]}|x_i|$.\nWe define the standard Euclidean sphere as $\\mathbb{S}^{d-1}= \\{\\bm{x}\\in\\mathbb{R}^d:\\|\\bm{x}\\|_2=1\\}$. \n\n\n\n5) For   matrix $\\bm{A}= [a_{ij}]\\in \\mathbb{R}^{m\\times n}$ with singular values $\\sigma_1\\geq \\sigma_2\\geq ...\\geq \\sigma_{\\min\\{m,n\\}}$, recall the operator norm $\\|\\bm{A}\\|_{op}= \\sup_{\\bm{v}\\in \\mathbb{S}^{n-1}}\\|\\bm{Av}\\|_2=\\sigma_1$, Frobenius norm $\\|\\bm{A}\\|_F = (\\sum_{i,j}a_{ij}^2)^{1\/2}$, nuclear norm $\\|\\bm{A}\\|_{nu} = \\sum_{k=1}^{\\min\\{m,n\\}}\\sigma_k$, and max norm $\\|\\bm{A}\\|_{\\infty}= \\max_{i,j}|a_{ij}|$. $\\lambda_{\\min}(\\bm{A})$ (resp. $\\lambda_{\\max}(\\bm{A})$) stands for the minimum eigenvalue (resp. maximum eigenvalue) of a symmetric $\\bm{A}$. \n\n\n\n6) We  denote universal constants by $C$, $c$, $C_i$ and $c_i$, whose value may vary from line to line.   We write $T_1 \\lesssim T_2$ or $T_1 = O(T_2)$ if $T_1\\leq CT_2$. Conversely, if $T_1\\geq CT_2$ we write $T_1 \\gtrsim T_2$ or $T_1 = \\Omega(T_2)$. Also, we write $T_1 \\asymp T_2$ if both $T_1=O(T_2)$ and $T_2 = \\Omega(T_1)$ hold.\n\n\n\n7) We use $\\mathscr{U}(\\Omega)$ to denote the uniform distribution over $\\Omega\\subset \\mathbb{R}^N$, $\\mathcal{N}(\\bm{\\mu},\\bm{\\Sigma})$ to denote Gaussian distribution with mean $\\bm{\\mu}$ and covariance $\\bm{\\Sigma}$, $\\mathsf{t}(\\nu)$ to denote student's t distribution with degrees of freedom $\\nu$. \n\n\n8) $\\mathcal{Q}_\\Delta(.)$ is the uniform quantizer with quantization level  $\\Delta>0$. It applies to scalar $a$ by $\\mathcal{Q}_\\Delta(a)=\\Delta\\big(\\big\\lfloor\\frac{a}{\\Delta}\\big\\rfloor+\\frac{1}{2}\\big)$, and we set $\\mathcal{Q}_0(a) =a$. Given a threshold $\\mu$, the hard thresholding of scalar $a$ is $\\mathcal{T}_\\mu(a)= a\\cdot \\mathbbm{1}(|a|\\geq \\mu)$. Both functions element-wisely apply to vectors or matrices.\n\n\n\n\\subsection{High-dimensional statistics}\n \nLet $X$ be a real random variable, we provide some basic knowledge on the sub-Gaussian, sub-exponential random variable. We also give the formulation of heavy-tailed distribution.\n\n\n\n1) The sub-Gaussian norm is defined as $\\|X\\|_{\\psi_2} = \\inf\\{t>0:\\mathbbm{E}\\exp(\\frac{X^2}{t^2})\\leq 2\\}$. A random variable $X$ with  finite $\\|X\\|_{\\psi_2}$ is said to be sub-Gaussian. Sub-Gaussian variable shares similar properties with the Gaussian variable, e.g., a probability tail with similar decaying rate and the moment constraint up to any order\\begin{equation}\n    \\begin{aligned}\\label{2.2}\n        &\\mathbbm{P}(|X|\\geq t)\\leq 2\\exp(-\\frac{ct^2}{\\|X\\|_{\\psi_2}^2});\\\\\n        &(\\mathbbm{E}|X|^p)^{1\/p}\\leq C\\|X\\|_{\\psi_2}\\sqrt{p},~\\forall~ p\\geq 1.\n    \\end{aligned}\n\\end{equation}    \nNote that,   these two properties can also define $\\|.\\|_{\\psi_2}$ up to multiplicative constant, for instance, $\\|X\\|_{\\psi_2}\\asymp \\sup_{p\\geq 1}\\frac{(\\mathbbm{E}|X|^p)^{1\/p}}{\\sqrt{p}}$, see \\cite[Proposition 2.5.2]{vershynin2018high}. For a $d$-dimensional random vector $\\bm{x}$ we have the sub-Gaussian norm $\\|\\bm{x}\\|_{\\psi_2}=\\sup_{\\bm{v}\\in\\mathbb{S}^{d-1}}\\|\\bm{v}^\\top\\bm{x}\\|_{\\psi_2}$.\n\n\n\n2) The sub-exponential norm is defined as $\\|X\\|_{\\psi_1} = \\inf\\{t>0:\\mathbbm{E}\\exp(\\frac{|X|}{t})\\leq 2\\}$, \nand $X$  is sub-exponential if $\\|X\\|_{\\psi_1}<\\infty$. The sub-exponential $X$ enjoys the properties \n\\begin{equation}\n    \\begin{aligned}\\nonumber\n       & \\mathbbm{P}(|X|\\geq t)\\leq 2\\exp(-\\frac{ct}{\\|X\\|_{\\psi_1}});\\\\\n       & (\\mathbbm{E}|X|^p)^{1\/p}\\leq C\\|X\\|_{\\psi_1}p,~\\forall~p\\geq 1.\n    \\end{aligned}\n\\end{equation}\n To relate $\\|.\\|_{\\psi_1}$ and $\\|.\\|_{\\psi_2}$ one has $\\|XY\\|_{\\psi_2}\\leq \\|X\\|_{\\psi_1}\\|Y\\|_{\\psi_1}$ \\cite[Lemma 2.7.7]{vershynin2018high}. \n  \n  \n  \n\n\n3) In contrast to the moment constaint in (\\ref{2.2}),  the heavy-tailed distributions in this work  only satisfy bounded moments of some order. More precisely, for  a given $l>0$, we assume $\\mathbbm{E}|X|^l\\leq M$ for some $M>0$. \nWhile for random vector $\\bm{x}\\in \\mathbb{R}^d$, we may use an element-wise assumption $\\sup_{i\\in [d]}\\mathbbm{E}|x_i|^l\\leq M$ or a stronger one $\\mathbbm{E}|\\bm{v}^\\top\\bm{x}|^l\\leq M$ towards all directions. Our focus is on quantization of heavy-tailed data drawn from such distributions. \n\n\n\n\\subsection{Dithered uniform quantization}\\label{prequan}\nIn this part, we describe the dithered uniform quantizer and its property in detail. We also specify the choices of random dither in this work.\n\n\n\n1) We first provide the detailed procedure of dithered quantization and     its general property.   Independent of the  input signal $\\bm{x}\\in \\mathbb{R}^N$ with dimension $N\\geq 1$, we generate the random dither $\\bm{\\tau}\\in \\mathbb{R}^N$ with i.i.d. entries from some distribution, and then quantize $\\bm{x}$ to $\\bm{\\dot{x}}= \\mathcal{Q}_\\Delta(\\bm{x}+\\bm{\\tau})$. Following \\cite{gray1993dithered}, we refer to $\\bm{w}:= \\bm{\\dot{x}}- (\\bm{x}+\\bm{\\tau})$ as the quantization error, while $\\bm{\\xi}:= \\bm{\\dot{x}} - \\bm{x}$ as the quantization noise. The principal properties of dithered quantization is provided in  Lemma \\ref{lem1}.\n\n\n\\begin{lem} \\label{lem1}  {\\rm \\scshape\t(Theorems 1-2 in} {\\rm \\cite{gray1993dithered}){\\bf \\sffamily.}}\nConsider the dithered uniform quantization described above for the input signal $\\bm{x} = [x_i]$, with random dither $\\bm{\\tau} = [\\tau_i]$, quantization error $\\bm{w} = [w_i]$ and quantization noise $\\bm{\\xi} = [\\xi_i]$. Let $Y$ be the random variable having the same distributioin as the random dither $w_i$.  \n\n\\noindent{\\rm (a)} {\\rm \\scshape(Quantization Error){\\bf \\sffamily.}} If $f(u):=\\mathbbm{E}(\\exp(\\bm{i} uY))$ satisfies $f\\big(\\frac{2\\pi  l}{\\Delta}\\big)=0$ for all non-zero integer $l$, then $x_i$ and $w_j$ are independent for all $i,j\\in [N]$. Moreover, $\\{w_j:j\\in [N]\\}$ is an i.i.d. sequence of $\\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}])$. \n\n\n\\noindent{\\rm(b)} {\\rm \\scshape(Quantization Noise){\\bf \\sffamily.}} Assume $Z\\sim \\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}])$ is independent of $Y$. Let $g(u):=\\mathbbm{E}(\\exp(\\ii uY))\\mathbbm{E}(\\exp(\\ii u Z))$. Given positive integer $p$, if the $p$-th order derivative $g^{(p)}(u)$ satisfies  $g^{(p)}\\big( \\frac{2\\pi  l}{\\Delta}\\big)=0$ for all non-zero integer $l$, then the $p$-th conditional moment of $\\xi_i$ does not depend on $\\bm{x}$: $\\mathbbm{E}[\\xi_i^p|\\bm{x}] = \\mathbbm{E}(Y+Z)^p$. \n\\end{lem}\n\n\n\n2) We use {\\it uniform dither} for quantization of the response in CS and MC. More specifically, under  $\\Delta>0$, we adopt the uniform dither $\\tau_k\\sim \\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}])$ for the response $y_k\\in \\mathbb{R}$, which is also a common choice in previous works (e.g., \\cite{thrampoulidis2020generalized,xu2020quantized,dirksen2021non,jung2021quantized}).  For $Y\\sim  \\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}])$, it can be calculated that \n\\begin{equation}\\nonumber\n     \\mathbbm{E}(\\exp(\\bm{i}uY))= \\int_{-\\Delta\/2}^{\\Delta\/2}~\\frac{1}{\\Delta}\\big(\\cos (ux) + \\bm{i}\\sin (ux)\\big)~\\mathrm{d}x = \\frac{2}{\\Delta u}\\sin\\big(\\frac{\\Delta u}{2}\\big),\n\\end{equation}\nand hence $\\mathbbm{E}(\\exp(\\bm{i} \\frac{2\\pi l}{\\Delta}  Y))=0$ holds for all non-zero integer $l$.  Therefore, the benefit of using $\\tau_k \\sim \\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}])$\nis that the quantization errors $w_k=\\mathcal{Q}_\\Delta(y_k+\\tau_k)-(y_k+\\tau_k)$ i.i.d. follow $\\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}])$, and are independent of $\\{y_k\\}$.\n\n\n\n3) We use {\\it triangular dither} for quantization of the covariate, i.e., the sample in covariance estimation or the covariate in CS. Particularly, when considering the uniform quantizer $\\mathcal{Q}_\\Delta(.)$  for   the covariate $\\bm{x}_k\\in \\mathbb{R}^d$, we adopt the dither $\\bm{\\tau}_k\\sim \\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}]^d)+\\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}]^d)$, which is  the sum of two independent  $\\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}]^d)$ and referred to as a triangular dither \\cite{gray1993dithered}. Simple calculations verify that the triangular dither respects not only the condition in Lemma \\ref{lem1}(a), but also the one in Lemma \\ref{lem1}(b) with $p=2$. Thus, at the cost of a larger dithering variance (compared to uniform dither), the advantage of triangular dither over uniform one is the additional property of signal-independent quantization noise --- $\\mathbbm{E}(\\xi_{ki}^2)=\\frac{1}{4}\\Delta^2$, where $\\xi_{ki}$ is the $i$-th entry of $\\bm{\\xi}_k = \\mathcal{Q}_\\Delta(\\bm{x}_k+\\bm{\\tau}_k)-(\\bm{x}_k+\\bm{\\tau}_k)$. \n\n\n\nTo the best of our knowledge, the triangular dither is new to the literature of QCS. We will explain its necessity if covariance estimation is involved. This is also complemented by numerical simulation (see Figure \\ref{fig5}(a)).\n\n\n\n \n\n\n\\section{(Near) minimax error rates} \\label{sec3}\nIn this section we derive (near) optimal error rates for several canonical statistical estimation problems, including quantized covariance estimation, quantized compressed sensing, and quantized matrix completion. For convenience, we represent these three models by QCE, QCS, QMC hereafter. The novelty is that the optimality is achieved under the two-fold difficulty of heavy-tailedness and quantization\n\\subsection{Quantized covariance estimation}\nGiven   $\\mathscr{X}:=\\{\\bm{x}_1,...,\\bm{x}_n\\}$ as i.i.d. copies of a zero-mean  random vector $\\bm{x}\\in \\mathbb{R}^d$, one often encounters the covariance (matrix) estimation problem, i.e., to estimate $\\bm{\\Sigma^\\star} = \\mathbbm{E}(\\bm{x} \\bm{x}^\\top)$. This estimation problem is of fundamental importance in multivariate analysis and  has   attracted a mass of research interest (e.g., \\cite{cai2012optimal,cai2010optimal,cai2011adaptive,bickel2008covariance}), whereas the more practical quantized regime remains under-developed, for which we are only aware of the extreme 1-bit quantization results in \\cite{dirksen2021covariance,chen2022high}.\n\nFormally put, the problem of quantized covariance estimation (QCE) investigates {\\it how to design a quantizer   for quantizing $\\bm{x}_k$, together with a fairly faithful estimation procedure only based on quantized data}.\nIn contrast to the conventional sub-Gaussian assumption in most previous works, we only assume that $\\bm{x}$ possesses bounded fourth moments. \n\nUnder such essential weaker assumption, the observed data may contain outliers, \n which we handle by a data truncation step. We first truncate $\\bm{x}_k$ to $\\bm{\\widetilde{x}}_k$, where the core spirit is to make the outliers less influential. Here, we defer the precise definition of $\\bm{\\widetilde{x}}_k$ to the concrete results, because it should be well suited to the error measure and hence can vary in different instances. \n \n \n After the truncation, we dither and quantize $\\bm{\\widetilde{x}}_k$ to $\\bm{\\dot{x}}_k = \\mathcal{Q}_\\Delta(\\bm{\\widetilde{x}}_k+\\bm{\\tau}_k)$ with the triangular dither $\\bm{\\tau}_k\\sim \\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}]^d)+\\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}]^d)$. Different from the uniform dither adopted in the literature (e.g., \\cite{chen2022high,thrampoulidis2020generalized,xu2020quantized,dirksen2021non,jung2021quantized}),  first let us explain our usage of triangular dither. Recall the definitions of quantization noise $\\bm{\\xi}_k:=\\bm{\\dot{x}}_k - \\bm{\\widetilde{x}}_k$ and quantization error $\\bm{w}_k:=\\bm{\\dot{x}}_k - \\bm{\\widetilde{x}}_k - \\bm{\\tau}_k$,\n  then $\\bm{\\xi}_k = \\bm{\\tau}_k+\\bm{w}_k$. If $\\bm{w}_k$ is independent of $\\bm{\\widetilde{x}}_k$ and follows $\\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}]^d)$ (as satisfied by both uniform dither and triangular dither), then we have\n\\begin{equation}\n    \\begin{aligned}\n    \\label{3.1}\n   & \\mathbbm{E}(\\bm{\\dot{x}}_k\\bm{\\dot{x}}_k^\\top) = \\mathbbm{E}\\big((\\bm{\\widetilde{x}}_k + \\bm{\\xi}_k)(\\bm{\\widetilde{x}}_k + \\bm{\\xi}_k)^\\top\\big)\\\\& = \\mathbbm{E}(\\bm{\\widetilde{x}}_k\\bm{\\widetilde{x}}_k^\\top)+ \\mathbbm{E}(\\bm{\\widetilde{x}}_k\\bm{\\xi}_k^\\top) + \\mathbbm{E}(\\bm{\\xi}_k\\bm{\\widetilde{x}}_k^\\top) + \\mathbbm{E}(\\bm{\\xi}_k\\bm{\\xi}_k^\\top)\\\\&\\stackrel{(i)}{=}\\mathbbm{E}(\\bm{\\widetilde{x}}_k\\bm{\\widetilde{x}}_k^\\top)+ \\mathbbm{E}(\\bm{\\xi}_k\\bm{\\xi}_k^\\top).\n    \\end{aligned}\n\\end{equation}\nNote that $(i)$ is because $\\mathbbm{E}(\\bm{\\xi}_k\\bm{\\widetilde{x}}_k^\\top)=\\mathbbm{E}(\\bm{\\tau}_k\\bm{\\widetilde{x}}_k^\\top)+ \\mathbbm{E}(\\bm{w}_k\\bm{\\widetilde{x}}_k^\\top)=\\mathbbm{E}(\\bm{\\tau}_k)\\mathbbm{E}(\\bm{\\widetilde{x}}_k^\\top)+\\mathbbm{E}(\\bm{w}_k)\\mathbbm{E}(\\bm{\\widetilde{x}}_k^\\top)=0$, and here we use the fact that $\\bm{\\tau}_k$, $\\bm{w}_k$ are independent of   $\\bm{\\widetilde{x}}_k$. While with suitable choice of the truncation threshold $\\mathbbm{E}(\\bm{\\widetilde{x}}_k\\bm{\\widetilde{x}}_k^\\top)$ is expected to well approximate $\\bm{\\Sigma^\\star}$, the remaining $\\mathbb{E}(\\bm{\\xi}_k\\bm{\\xi}_k^\\top)$  gives rise to constant deviation. Thus, it must be removed, and the only possibility is that one has full knowledge on $\\mathbbm{E}(\\bm{\\xi}_k\\bm{\\xi}_k^\\top)$, i.e., the covariance matrix of the quantization noise.  For $i\\neq j$, we  can observe that \\begin{equation}\n    \\begin{aligned}\\nonumber\n        \\mathbbm{E}(\\xi_{ki}\\xi_{kj})=\\mathbbm{E}\\big((w_{ki}+\\tau_{ki})(w_{kj}+\\tau_{kj})\\big)=0,\n    \\end{aligned}\n\\end{equation}\nshowing that $\\mathbbm{E}(\\bm{\\xi}_k\\bm{\\xi}_k^\\top)$ is diagonal. Moreover,   under triangular dither the $i$-th diagonal entry is also known as $\\mathbbm{E}|\\xi_{ki}|^2=\\frac{\\Delta^2}{4}$, see section \\ref{prequan}. Now we can conclude that $\\mathbbm{E}(\\bm{\\xi}_k\\bm{\\xi}_k^\\top)=\\frac{\\Delta^2}{4}\\bm{I}_d$; combined with  (\\ref{3.1}) we  propose the  following estimator\n\\begin{equation}\n\\label{3.2}\n    \\bm{\\widehat{\\Sigma}} = \\frac{1}{n} \\sum_{k=1}^n \\bm{\\dot{x}}_k \\bm{\\dot{x}}_k^\\top - \\frac{\\Delta^2}{4}\\bm{I}_d,\n\\end{equation}\nwhich is the sample covariance of the quantized sample $\\dot{\\mathscr{X}}:=\\{\\bm{\\dot{x}}_1,...,\\bm{\\dot{x}}_n\\}$ followed by a correction step. On the other hand, the reason why the standard uniform dither is not suitable for QCE  becomes self-evident --- the diagonal of $\\mathbbm{E}(\\bm{\\xi}_k\\bm{\\xi}_k^\\top)$ remains unknown\\footnote{It depends on the input signal, see \\cite[Page 3]{gray1993dithered}.} and thus the bias cannot be precisely removed.\n\n\n\nWe are now ready to present error bounds for $\\bm{\\widehat{\\Sigma}}$ under max-norm, operator norm. We will also investigate the high-dimensional setting by assuming sparse structure of $\\bm{\\Sigma^\\star}$, for which we propose a thresholding estimator based on $\\bm{\\widehat{\\Sigma}}$ in Theorem \\ref{thm1}. \n\n\nIn an  estimation with error measured by $\\|.\\|_\\infty$, we invoke element-wise truncation to obtain $\\bm{\\widetilde{x}}_k$. The bounded fourth moment assumption is also in an element-wise manner. \n\n\n\n\\begin{theorem}\\label{thm1}\n\\noindent{\\rm\\scshape(Element-wise Error){\\bf \\sffamily.}} Given $\\Delta>0$ and $\\delta > 2$, we consider the problem QCE described above. We suppose that $\\bm{x}_k = [x_{ki}]$ satisfies  $\\mathbbm{E}|x_{ki}|^4\\leq M$, $\\forall ~i\\in [d]$. We adopt the element-wise truncation with threshold $\\zeta>0$: $\\bm{\\widetilde{x}}_k=[\\widetilde{x}_{ki}]$, $\\widetilde{x}_{ki}=\\sign(x_{ki})\\min\\{|x_{ki}|,\\zeta\\}$;  and recall that $\\bm{\\dot{x}}_k  = \\mathcal{Q}_\\Delta(\\bm{\\widetilde{x}}_k+ \\bm{\\tau}_k)$ with triangular dither $\\bm{\\tau}_k$. We set $\\zeta \\asymp \\big(\\frac{nM}{\\delta \\log d}\\big)^{1\/4}$. Then if $n\\gtrsim \\delta \\log d$, the estimator in (\\ref{3.2}) enjoys the bound \n\\begin{equation}\\nonumber\n    \\mathbbm{P}\\Big(\\|\\bm{\\widehat{\\Sigma}}-\\bm{\\Sigma^\\star}\\|_{\\infty}\\geq C\\mathscr{L}\\sqrt{\\frac{\\delta \\log d}{n}}\\Big)\\leq  2d^{2-\\delta},\n\\end{equation}\nwith $\\mathscr{L}:=\\sqrt{M}+\\Delta^2$. \n\\end{theorem}\n\n\n\n\\vspace{2mm}\n\nNotably, despite the heavy-tailedness and quantization, the estimator achieves an element-wise rate $O(\\sqrt{\\frac{\\log d}{n}})$ coincident with the one for sub-Gaussian case. Indeed, after truncation the data are essentially more benign for estimation, while one can clearly position quantization level $\\Delta$ in the   multiplicative factor $\\mathscr{L}=\\sqrt{M}+\\Delta^2$. Thus, the cost of quantization is inessential without affecting the key part of the error rate. These remarks on the (near) optimality and the degradation incurred by quantization remain  valid in our subsequent Theorems. \n\n\n\n\nWhen estimating $\\bm{\\Sigma^\\star}$ under operator norm,  we  truncate $\\bm{x}_k$ regarding $\\ell_4$ norm with threshold $\\zeta$. To distinguish with the truncation in Theorem \\ref{thm1}, we change notation and use $\\bm{\\check{x}}_k$ to denote the truncated date, i.e.,  $\\bm{\\check{x}}_k=\\frac{\\bm{x}_k}{\\|\\bm{x}_k\\|_4} \\min\\{\\|\\bm{x}_k\\|_4,\\zeta\\}$.\nWe also require   bounded fourth moments towards all directions.\nThen, we still define the estimator as (\\ref{3.2}). \n\\begin{theorem}\\label{thm2}\n\\noindent{\\rm\\scshape(Operator Norm Error){\\bf \\sffamily.}} Given $\\Delta>0$ and $\\delta>0$, we consider the problem of QCE described above.  We suppose that $\\bm{x}_k$ satisfies $ \\mathbbm{E}|\\bm{v}^\\top \\bm{x}_k|^4 \\leq M$  for any $\\bm{v}\\in \\mathbb{S}^{d-1}$. We adopt the $\\ell_4$ norm truncation with threshold $\\zeta>0$: $\\bm{\\check{x}}_k=\\frac{\\bm{x}_k}{\\|\\bm{x}_k\\|_4}\\min \\{\\|\\bm{x}_k\\|_4,\\zeta\\}$; also recall that $\\bm{\\dot{x}}_k=\\mathcal{Q}_\\Delta (\\bm{\\check{x}}_k+\\bm{\\tau_k})$ with triangular dither $\\bm{\\tau}_k$. We set $\\zeta \\asymp (M^{1\/4}+\\Delta)\\big(\\frac{n}{\\delta \\log d}\\big)^{1\/4}$.   Then if $n\\gtrsim \\delta d \\log d$, the estimator in (\\ref{3.2}) enjoys the bound \\begin{equation}\\nonumber\n    \\mathbbm{P}\\Big(\\|\\bm{\\widehat{\\Sigma}}-\\bm{\\Sigma^\\star}\\|_{op}\\geq C\\mathscr{L}\\sqrt{\\frac{\\delta d\\log d}{n}}\\Big)\\leq 2d^{-\\delta},\n\\end{equation} \nwith $\\mathscr{L}:=\\sqrt{M}+\\Delta^2$.\n\\end{theorem}\n\nThe operator norm error rate in Theorem \\ref{thm2} is near optimal (e.g., compared to \\cite[Theorem 7]{fan2021shrinkage}), and  the effect of dithered uniform quantization is solely on the factor $\\mathscr{L}$. Nevertheless, one still needs (at least) $n \\gtrsim d$ to achieve small operator norm error.\n\n\nTo handle estimation under operator norm in a high-dimensional regime (where $d$ may even exceed $n$), we resort to additional structure on $\\bm{\\Sigma^\\star}$. We use sparse $\\bm{\\Sigma^\\star}$ as an example, which corresponds to the situations where dependencies among different features are  weak. To promote sparsity, we invoke a  thresholding regularization \\cite{bickel2008covariance,cai2012optimal} to the element-wise estimator in Theorem \\ref{thm1}. \nIn spite of the heavy-tailedness and quantization,  our estimator ($\\bm{\\widehat{\\Sigma}}_s$) achieves minimax rates $O\\big(s\\sqrt{\\frac{\\log d}{n}}\\big)$ under operator norm (e.g., compared to \\cite[Theorem 2]{cai2012optimal}).\n\n\n \n\n\\begin{theorem}\\label{thm3}\n\\noindent{\\rm\\scshape(Sparse Covariance Matrix estimation){\\bf \\sffamily.}} We   assume all columns of $\\bm{\\Sigma}^* = [\\sigma^\\star_{ij}]$ are $s$-sparse. Under conditions and estimator $\\bm{\\widehat{\\Sigma}}$ in Theorem \\ref{thm1},  we set $\\mu=C_1 (\\sqrt{M}+\\Delta^2)\\sqrt{\\frac{\\delta \\log d}{n}}$ for some sufficiently large $C_1$, and consider the thresholding estimator $\\bm{\\widehat{\\Sigma}}_s := \\mathcal{T}_{\\mu}(\\bm{\\widehat{\\Sigma}})$. Then it enjoys the operator norm error bound \\begin{equation}\\nonumber\n    \\mathbbm{P}\\Big(\\|\\bm{\\widehat{\\Sigma}}_s - \\bm{\\Sigma^\\star}\\|_{op} \\leq C\\mathscr{L}s\\sqrt{\\frac{ \\delta \\log d}{n}}\\Big) \\geq 1-\\exp(-0.25\\delta),\n\\end{equation}\nwith $\\mathscr{L}:=\\sqrt{M}+\\Delta^2$.\n\\end{theorem}\n To analyse the thresholding estimator, our proof resembles those developed in prior works\n (e.g., \\cite{cai2012optimal}), but note that additional efforts are needed to bound the   bias term induced by the data truncation. We also point out that the results for full-data regime can be exactly recovered by setting $\\Delta=0$. As a result, Theorems \\ref{thm1}-\\ref{thm2} represent the strict extension of   \\cite[Section 4]{fan2021shrinkage}, while Theorem \\ref{thm3} complements \\cite{fan2021shrinkage} by applying the   truncation technique to estimating sparse $\\bm{\\Sigma^\\star}$. \n\n\n\n \n \n\n \n\\begin{rem}\\label{rem2}\n\\noindent{\\rm\\scshape(Sub-Gaussian case){\\bf \\sffamily.}} \nWhile we concentrate on quantization of heavy-tailed data in this work, our results can be readily adjusted to sub-Gaussian $\\bm{x}_k$, for which the truncation step is inessential and can be removed (i.e., $\\zeta=\\infty$). These results are also new to the literature but will not be presented here.\n\\end{rem}\n\n\\subsection{Quantized compressed sensing}\\label{sec4}\nWe consider the linear CS model   $\\bm{y} = \\bm{X} \\bm{\\theta^\\star} + \\bm{\\epsilon}$, \nwhere $\\bm{\\theta^\\star}\\in\\mathbb{R}^d$, $\\bm{X}\\in \\mathbb{R}^{n\\times d},\\bm{\\epsilon}$ are the desired $s$-sparse signal, sensing matrix and noise vector, respectively; our goal is to recover $\\bm{\\theta^\\star}$ from $(\\bm{X},\\bm{y})$.\n In the area of    QCS, we are interested in {\\it developing quantization scheme (mainly for $\\bm{y}$ in prior works)  that is associated with a faithful  recovery procedure merely based on the quantized data} \\cite{dirksen2019quantized}. \n\n \n Let $\\bm{x}_k$ be the $k$-th row of $\\bm{X}$, in statistical estimation this problem is also termed as sparse linear regression and formulated as \\begin{equation}\\label{csmodel}\n    y_k = \\bm{x}_k^\\top \\bm{\\theta^\\star}+\\epsilon_k,~k=1,...,n.\n\\end{equation}   In this work, we  adopt the more statistical conventions and refer to $\\bm{x}_k$ as the covariate, $y_k$ as the response. Also, we assume   $\\bm{x}_k$ are i.i.d. drawn from some multi-variate distribution,  $\\{\\epsilon_k:k\\in[n]\\}$ are i.i.d. statistical noise independent of the covariate.\n\n\n\n\n\n\n \n\n\n \nAs already mentioned, we will apply a dithered uniform quantizer to $\\bm{y}$.\nNote that, under the above statistical assumptions and dithered quantization of $\\{y_k\\}$,  the near optimal recovery guarantee has been established in \\cite{thrampoulidis2020generalized,jung2021quantized,xu2020quantized}, but restricted to sub-Gaussian regime, i.e., both $\\bm{x}_k$ and $\\epsilon_k$ are drawn from sub-Gaussian distribution. In contrast, our focus is on quantization of heavy-tailed data. In particular, the noise $\\epsilon_k$ is i.i.d. drawn from some heavy-tailed distribution and hence leads to     heavy-tailed response. We will also study a more challenging setting where both $\\bm{x}_k$, $\\epsilon_k$ are heavy-tailed. These two settings will be presented separately because of some essential differences, see Remark \\ref{htcova} below.\n\n\n\nAs before, our strategy to handle heavy-tailed data is the truncation step prior to the dithered quantization. Particularly, with threshold $\\zeta_y$  we truncate $y_k$ to   $\\widetilde{y}_k = \\sign(y_k) \\min\\{|y_k|,\\zeta_y\\}$, which is then quantized to $\\dot{y}_k = \\mathcal{Q}_\\Delta(\\widetilde{y}_k+\\tau_k)$. To estimate the sparse $\\bm{\\theta^\\star}$,  a classical approach is the Lasso \\cite{tibshirani1996regression}  (also encompassed by the framework of regularized M-estimator \\cite{negahban2011estimation,negahban2012unified})\n\\begin{equation}\n    \\begin{aligned}\\nonumber\n    \\mathop{\\arg\\min}\\limits_{\\bm{\\theta}}~\\frac{1}{2n}\\sum_{k=1}^n(y_k-\\bm{x}_k^\\top\\bm{\\theta})^2 + \\lambda \\|\\bm{\\theta}\\|_1,\n    \\end{aligned}\n\\end{equation}\nwhich combines the $\\ell_2$ loss for data fidelity and $\\ell_1$ norm that encourages sparsity. Evidently, the main issue to invoke Lasso under quantization lies in the $\\ell_2$ loss that requires full data. To illustrate our remedy for the issue, we calculate the expected $\\ell_2$ loss: \n$$\\mathbbm{E}(y_k-\\bm{x}_k^\\top\\bm{\\theta})^2\\stackrel{(i)}{=}\\bm{\\theta}^\\top\\big(\\mathbbm{E}\\bm{x}_k\\bm{x}_k^\\top\\big)\\bm{\\theta}-2\\big(\\mathbbm{E}y_k\\bm{x}_k\\big)^\\top\\bm{\\theta}:\\stackrel{(ii)}{=}\\bm{\\theta}^\\top\\bm{\\Sigma^\\star}\\bm{\\theta}-2\\bm{\\Sigma}_{y\\bm{x}}^\\top\\bm{\\theta},$$\nwhere $(i)$ holds up to an inessential constant $\\mathbbm{E}|y_k|^2$, and in $(ii)$ we let $\\bm{\\Sigma^\\star}:= \\mathbbm{E}(\\bm{x}_k\\bm{x}_k^\\top)$, $\\bm{\\Sigma}_{y\\bm{x}}=\\mathbbm{E}(y_k\\bm{x}_k)$. Thus, we can still consider a generalized Lasso as  \n\\begin{equation}\\label{4.3}\n   \\bm{\\widehat{\\theta}} =  \\mathop{\\arg\\min}\\limits_{\\bm{\\theta}\\in \\mathcal{S}}~\\frac{1}{2}\\bm{\\theta}^\\top \\bm{Q}\\bm{\\theta} - \\bm{b}^\\top \\bm{\\theta} +\\lambda\\|\\bm{\\theta}\\|_1.\n\\end{equation}\n  Enlightened by the expected loss, we will look for surrogates for the covariance $\\bm{\\Sigma^\\star}$, $\\bm{\\Sigma}_{y\\bm{x}}$, and then plug them into $\\bm{Q}$, $\\bm{b}$, respectively. We also introduce the constraint $\\bm{\\theta}\\in \\mathcal{S}$ to allow more flexibility.\n\n\nThe next Theorem is concerned with QCS under the conventional sub-Gaussian covariate but heavy-tailed response. Note that the heavy-tailedness of $y_k$ stems from the noise distribution with bounded $2+\\nu$ moment ($\\nu>0$). As \\cite{fan2021shrinkage,chen2022high,zhu2021taming}, it is easier to impose the moment constraint on the response $y_k$.\n\n\n \n   \n \n\n\n\n\\begin{theorem}\\label{thm4}\n\\noindent{\\rm\\scshape(sub-gaussian covariate, heavy-tailed response){\\bf \\sffamily.}} For the model (\\ref{csmodel}) we suppose that   $\\bm{x}_k$ are i.i.d., zero-mean and sub-Gaussian with $\\|\\bm{x}_k\\|_{\\psi_2}\\leq \\sigma$, and that $\\kappa_0\\leq \\lambda_{\\min}(\\bm{\\Sigma^\\star})\\leq \\lambda_{\\max}(\\bm{\\Sigma^\\star})\\leq \\kappa_1$ for some $\\kappa_1>\\kappa_0>0$, where $\\bm{\\Sigma^\\star}=\\mathbbm{E}(\\bm{x}_k\\bm{x}_k^\\top)$, $\\bm{\\theta^\\star}\\in\\mathbb{R}^d$ is   $s$-sparse, \\textcolor{black}{  $\\mathbbm{E}|y_k|^{2l}\\leq M$ for some fixed $l>1$}. Recall our quantization scheme that $\\widetilde{y}_k= \\sign(y_k)\\min\\{|y_k|,\\zeta_y\\}$, $\\dot{y}_k = \\mathcal{Q}_\\Delta(\\widetilde{y}_k+\\tau_k)$ with $\\tau_k\\sim \\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}])$. For recovery, we define the estimator $\\bm{\\widehat{\\theta}}$  as (\\ref{4.3}) with $\\bm{Q} = \\frac{1}{n}\\sum_{k=1}^n\\bm{x}_k\\bm{x}_k^\\top$, $\\bm{b} = \\frac{1}{n}\\sum_{k=1}^n\\dot{y}_k\\bm{x}_k$, $\\mathcal{S} = \\mathbb{R}^d$.  We   set $\\zeta_y\\asymp \\big(\\frac{nM^{1\/l}}{\\delta \\log d}\\big)^{1\/2}$, $\\lambda = C_1 \\frac{\\sigma^2}{\\sqrt{\\kappa_0}}(\\Delta +M^{1\/(2l)})\\sqrt{\\frac{\\delta\\log d}{n}}$ with sufficiently large $C_1$.  If $n \\gtrsim \\delta s\\log d$ for some hidden constant only depending on $(\\kappa_0,\\sigma)$, then with probability at least $1-9d^{1-\\delta}$, the estimation error $\\bm{\\widehat{\\Delta}} = \\bm{\\widehat{\\theta}} - \\bm{\\theta^\\star}$ enjoys the error bounds \\begin{equation}\\nonumber\n    \\|\\bm{\\widehat{\\Delta}}\\|_2\\leq C_3\\mathscr{L} \\sqrt{\\frac{\\delta s\\log d}{n}}~~~\\mathrm{and}~~~\\|\\bm{\\widehat{\\Delta}}\\|_1 \\leq C_4\\mathscr{L}s\\sqrt{\\frac{\\delta\\log d}{n}}~\n\\end{equation}\nwhere $\\mathscr{L} := \\frac{\\sigma^2(\\Delta+M^{1\/(2l)})}{\\kappa_0^{3\/2}}$.\n\\end{theorem}\n\n\n\n \n The rate $O\\big(\\sqrt{\\frac{s\\log d}{n}}\\big)$ for $\\ell_2$ norm error is   minimax optimal up to logarithmic factor (e.g., compared to \\cite{raskutti2011minimax}). Note that a  random noise \n bounded by $\\Delta$   roughly contributes $\\Delta^{2l}$ to $\\mathbbm{E}|y_k|^{2l}$ bounded by $M$; because in the error bound $\\Delta$ and $M^{1\/(2l)}$ almost play the same role, the effect of uniform quantization  can be  readily interpreted as  an additional bounded noise. This generalizes previous findings in \\cite{thrampoulidis2020generalized,jung2021quantized,sun2022quantized} to the heavy-tailed regime.\n \nNext, we switch to the more tricky situation where both \n $\\bm{x}_k$ and $y_k$ are heavy-tailed,   particularly  we assume they both possess bounded fourth moments.  We need to  element-wisely truncate $\\bm{x}_k$ to $\\bm{\\widetilde{x}}_k$ and then set $\\bm{Q}:= \\frac{1}{n}\\bm{\\widetilde{x}}_k\\bm{\\widetilde{x}}_k^\\top$, which can be viewed as a robust covariance estimator and is indeed encompassed by our Theorem \\ref{thm1} (i.e., the special case of $\\Delta=0$).\n\\begin{theorem}\n\\label{thm5}\\noindent{\\rm\\scshape(heavy-tailed covariate and response){\\bf \\sffamily.}} For the model (\\ref{csmodel}) we suppose that $\\bm{x}_k=[x_{ki}]$ are i.i.d. zero-mean with bounded fourth moment $\\mathbbm{E}|x_{ki}|^4\\leq M$,      and that $\\kappa_0\\leq \\lambda_{\\min}(\\bm{\\Sigma^\\star})\\leq \\lambda_{\\max}(\\bm{\\Sigma^\\star})\\leq \\kappa_1$ for some $\\kappa_1>\\kappa_0>0$, where $\\bm{\\Sigma^\\star}=\\mathbbm{E}(\\bm{x}_k\\bm{x}_k^\\top)$, and that $\\mathbbm{E}|y_k|^4\\leq M$. We assume   $\\bm{\\theta^\\star}\\in\\mathbb{R}^d$ is sparse and satisfies $\\|\\bm{\\theta^\\star}\\|_1\\leq R$ for some $R>0$. In the signal processing, with some $\\zeta_x,\\zeta_y$ we truncate $\\bm{x}_k$ to $\\bm{\\widetilde{x}}_k = [\\widetilde{x}_{ki}]$ with $\\widetilde{x}_{ki}:=\\sign(x_{ki})\\min\\{|x_{ki}|,\\zeta_x\\}$, and $y_k$  to $\\widetilde{y}_k:= \\sign(y_k)\\min\\{|y_k|,\\zeta_y\\}$, which is then quantized to  $\\dot{y}_k = \\mathcal{Q}_\\Delta(\\widetilde{y}_k+ \\tau_k)$ with $\\tau_k\\sim\\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}])$. For recovery, we define the estimator $\\bm{\\widehat{\\theta}}$ as (\\ref{4.3}) with $\\bm{Q}=\\frac{1}{n}\\sum_{k=1}^n\\bm{\\widetilde{x}}_k\\bm{\\widetilde{x}}_k^\\top$, $\\bm{b}=\\frac{1}{n}\\sum_{k=1}^n\\dot{y}_k\\bm{\\widetilde{x}}_k$, $\\mathcal{S}=\\mathbb{R}^d$. We  set $\\zeta_x, \\zeta_y\\asymp\\big(\\frac{nM}{\\delta \\log d}\\big)^{1\/4}$, $\\lambda = C_1(R\\sqrt{M}+\\Delta^2)\\sqrt{\\frac{\\delta \\log d}{n}}$ with sufficiently large $C_1$. If $n\\gtrsim \\delta s^2\\log d$ for some hidden constant only depending on $(\\kappa_0,M)$, then with probability at least $1-4d^{2-\\delta}$, the estimation error $\\bm{\\widehat{\\Delta}}:= \\bm{\\widehat{\\theta}} - \\bm{\\theta^\\star}$ enjoys the error bounds \\begin{equation}\\nonumber\n      \\|\\bm{\\widehat{\\Delta}}\\|_2\\leq C_2\\mathscr{L}  \\sqrt{\\frac{s\\log d}{n}}~~~\\mathrm{and}~~~\\|\\bm{\\widehat{\\Delta}}\\|_1 \\leq C_3\\mathscr{L}s\\sqrt{\\frac{\\log d}{n}}~\n\\end{equation} \nwhere $\\mathscr{L}:=\\frac{R\\sqrt{M}+\\Delta^2}{\\kappa_0}$.\n\\end{theorem}\n\n\n\\vspace{1mm}\n\nTheorem \\ref{thm5} generalizes   \\cite[Theorem 2(b)]{fan2021shrinkage} to the uniform  quantization setting. Clearly,  the obtained rate   remains  near minimax optimal if $R$ is of minor scaling (e.g., bounded or logarithmic factors). Nevertheless,   such near optimality in Theorem \\ref{thm5} comes at the cost of more stringent conditions and stronger scaling, see the next remark.\n\n\\begin{rem}\\label{htcova} {\\rm \\scshape (Comparing Theorems \\ref{thm4}-\\ref{thm5}){\\bf \\sffamily.}}\nCompared with $n\\gtrsim s\\log d$ in Theorem \\ref{thm4}, Theorem \\ref{thm5} requires a   sub-optimal sample complexity $n\\gtrsim s^2\\log d$. In fact, $n\\gtrsim s^2\\log d$ also appears in \\cite[Theorem 2(b)]{fan2021shrinkage}; while by explicitly adding the constraint $\\|\\bm{\\theta}\\|_1\\leq R$ to the recovery program and more careful analysis (with the first-order optimality of $\\bm{\\widehat{\\theta}}$), it can be improved to  $n\\gtrsim s\\log d$  and hence the bound is valid in a wider range of $n$ (see Remark \\ref{impro} for detail). Besides,  in Theorem \\ref{thm5} we impose a constraint $\\|\\bm{\\theta^\\star}\\|_1\\leq R$ that is stronger than $\\|\\bm{\\theta^\\star}\\|_2\\lesssim\\frac{M^{1\/(2l)}}{\\sigma}$ derived in the proof of Theorem \\ref{thm4}. \n\\end{rem}\n\n\n\\subsection{Quantized matrix completion}\\label{sec5}\nCompleting a low-rank matrix from only a partial observation of its entries is known as the matrix completion problem, which has found many applications including recommendation system, image inpainting, quantum state tomography   \\cite{chen2022color,davenport2016overview,bennett2007netflix,nguyen2019low,gross2010quantum}, to name just a few. It is currently well understood that, the rather stringent incoherence condition   is required to pursue exact recovery (e.g., \\cite{candes2012exact,recht2011simpler}), whereas a faithful estimation   can be achieved as long as the underlying matrix is not overly spiky (e.g., \\cite{klopp2014noisy,negahban2012restricted}). The latter condition is also known as low spikiness   (e.g., \\cite{chen2022color,klopp2017robust,negahban2012restricted}) and  is indeed necessary for the well-posedness of matrix completion problem (e.g., \\cite{davenport2016overview,negahban2012restricted}). \n\n\n\n\nIn this part, we do not pursue exact recovery but focus on the estimation problem.  \nLet $\\bm{\\Theta^\\star}\\in \\mathbb{R}^{d\\times d}$ be the underlying matrix satisfying $\\rank(\\bm{\\Theta^\\star})\\leq r$, we formulate the model as \n\\begin{equation}\\label{mcmodel}\n    y_k = \\big<\\bm{X}_k, \\bm{\\Theta^\\star}\\big> + \\epsilon_k, ~k=1,2,...,n,\n\\end{equation}\nwhere the design $\\bm{X}_k$ is randomly distributed on $\\mathcal{X}:=\\{\\bm{e}_i\\bm{e}_j^\\top: i,j\\in [d]\\}$ ($\\bm{e}_i$ is the $i$-th column of $\\bm{I}_d$), $\\epsilon_k$ is the noise independent of $\\bm{X}_k$. Note that for $\\bm{X}_k=\\bm{e}_{i(k)}\\bm{e}_{j(k)}^\\top$ one has $\\big<\\bm{X}_k,\\bm{\\Theta^\\star}\\big> = \\theta^\\star_{i(k),j(k)}$; hence, each observation is a noisy entry. For simplicity we consider the uniform sampling scheme $\\bm{X}_k\\sim \\mathscr{U}(\\mathcal{X})$, but with a little bit more work it generalizes to more general sampling scheme  \\cite{klopp2014noisy}.\nIt has been noted that the low-spikiness condition \\cite{negahban2012restricted} can be substituted with the simpler max-norm constraint \\cite{klopp2014noisy,chen2022color,koltchinskii2011nuclear}, which we adopt here and assume $\\|\\bm{\\Theta^\\star}\\|_\\infty\\leq \\alpha$. \n\n\n\n\n Our main interest is on quantized matrix completion (QMC), where our goal is to {\\it design quantizer for the observations $y_k$ that allows accurate post-quantization estimation on $\\bm{\\Theta^\\star}$, which is often shown by faithful estimation procedure from quantized data}. Besides, we   study heavy-tailed $y_k$, which should be truncated  as before. In particular, $y_k$ will be truncated to $\\widetilde{y}_k=\\sign(y_k)\\min\\{|y_k|,\\zeta_y\\}$, which is then processed by a uniform quantizer $\\dot{y}_k=\\mathcal{Q}_\\Delta(\\widetilde{y}_k+\\tau_k)$ with dither $\\tau_k\\sim \\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}])$. From the given data $(\\bm{X}_k,\\dot{y}_k)$, we estimate $\\bm{\\Theta^\\star}$ by the regularized M-estimator \\cite{negahban2011estimation,negahban2012unified}\n\\begin{equation}\\label{5.2}\n    \\bm{\\widehat{\\Theta}} = \\mathop{\\arg\\min}\\limits_{\\|\\bm{\\Theta}\\|_\\infty\\leq \\alpha}~\\frac{1}{2n}\\sum_{k=1}^n \\big(\\dot{y}_k- \\big<\\bm{X}_k,\\bm{\\Theta}\\big>\\big)^2+\\lambda\\|\\bm{\\Theta}\\|_{nu}\n\\end{equation} \nthat combines an $\\ell_2$ loss and nuclear norm regularizer.\n\n\nIt should be pointed out that there is a line of works on 1-bit or multi-bit matrix completion  related to our results to be presented   \\cite{cai2013max,lafond2014probabilistic,klopp2015adaptive,cao2015categorical,bhaskar2016probabilistic}. While the referenced works commonly adopted a likelihood approach, our method  is an essential departure and embraces some advantage, see a precise comparison in Remark \\ref{rem8}. Because of such novelty, we include   the result for sub-exponential $\\epsilon_k$ in Theorem \\ref{thm8}. \n\n\nNote that in the case of  sub-exponential noise, the truncation of $y_k$ becomes unnecessary and so we simply set $\\zeta_y=\\infty$. \n\n\n\n\n\\begin{theorem}\n\\label{thm8}{\\rm \\scshape(QMC with Sub-Exponential Noise){\\bf \\sffamily.}} Given some $\\Delta,\\delta$, we consider the   MC model (\\ref{mcmodel}) described above under i.i.d. zero-mean $\\{\\epsilon_k:k\\in [n]\\}$ satisfying $\\|\\epsilon_k\\|_{\\psi_1}\\leq \\sigma$. In the quantization scheme we set $\\zeta_y=\\infty$ so that $\\widetilde{y}_k=y_k$. We choose $\\lambda =C_1 (\\sigma+\\Delta) \\sqrt{\\frac{\\delta\\log d}{nd}}$ with sufficiently large $C_1$, and define $\\bm{\\widehat{\\Theta}}$ as (\\ref{5.2}). If $d\\log^3 d\\lesssim n\\lesssim r^2d^2 \\log d$, then with probability at least $1-4d^{-\\delta}$, $\\bm{\\widehat{\\Delta}}:=\\bm{\\widehat{\\Theta}} - \\bm{\\Theta^\\star}$ enjoys the error bounds  \\begin{equation}\\nonumber\n    \\frac{\\|\\bm{\\widehat{\\Delta}}\\|_F}{d}\\leq C_2 \\mathscr{L}\\sqrt{\\frac{\\delta r d \\log d}{n}} ~~\\mathrm{and}~~ \\frac{\\|\\bm{\\widehat{\\Delta}}\\|_{nu}}{d} \\leq  C_3\\mathscr{L}r\\sqrt{\\frac{\\delta   d \\log d}{n}}\n\\end{equation}\nwhere $\\mathscr{L}:= \\alpha+\\sigma+\\Delta $. \n\\end{theorem}\n\n \n\nThen, we present our result for the heavy-tailed case where the noise $\\epsilon_k$ possesses bounded second moment. \n\n\n\\begin{theorem}\n\\label{thm9}{\\rm \\scshape(QMC with Heavy-tailed Noise){\\bf \\sffamily.}} Given some $\\Delta,\\delta$, we consider the MC model (\\ref{mcmodel}) described above under     i.i.d., zero-mean $\\{\\epsilon_k:k\\in [n]\\}$   satisfying  $\\mathbbm{E}|\\epsilon_k|^2\\leq M$. In the quantization scheme we  set the threshold $\\zeta_y\\asymp (\\sqrt{M}+\\alpha)\\sqrt{\\frac{n}{\\delta d \\log d}}$. We choose  $\\lambda =C_1 (\\alpha+\\sqrt{M}+\\Delta) \\sqrt{\\frac{\\delta\\log d}{nd}}$ with sufficiently large $C_1$, and define $\\bm{\\widehat{\\Theta}}$ as (\\ref{5.2}). If $d\\log d\\lesssim n\\lesssim r^2d^2 \\log d$, then with probability at least $1-6d^{-\\delta}$, $\\bm{\\widehat{\\Delta}}:=\\bm{\\widehat{\\Theta}} - \\bm{\\Theta^\\star}$ enjoys the error bounds  \\begin{equation}\\nonumber\n    \\frac{\\|\\bm{\\widehat{\\Delta}}\\|_F}{d}\\leq C_2 \\mathscr{L}\\sqrt{\\frac{\\delta r d \\log d}{n}} ~~\\mathrm{and}~~ \\frac{\\|\\bm{\\widehat{\\Delta}}\\|_{nu}}{d} \\leq  C_3\\mathscr{L}r\\sqrt{\\frac{\\delta   d \\log d}{n}}\n\\end{equation}\nwhere $\\mathscr{L}:=\\alpha+\\sqrt{M}+\\Delta$. \n\\end{theorem}\n\n\n\\vspace{1mm}\n\nCompared to the information-theoretic lower bounds in  \\cite{negahban2012restricted,koltchinskii2011nuclear}, the error rates obtained in Theorems \\ref{thm8}-\\ref{thm9} are   minimax optimal up to logarithmic factors. Specifically, Theorem \\ref{thm9} derives near optimal guarantee for QMC with heavy-tailed observations, as the key standpoint of this paper. Note that, the 1-bit quantization counterpart of these two Theorems was derived in our previous work \\cite{chen2022high}; in sharp contrast to the present Theorem \\ref{thm9}, the error rate for the heavy-tailed  regime in \\cite[Theorem 12]{chen2022high} suffers from essential degradation.\n\n\n\nTo close this section, we  give a remark to illustrate the novelty and advantage of our method by a careful comparison with prior works.  \n\n\\begin{rem}\n\\label{rem8} \n  QMC with  1-bit or multi-bit  quantized observations has attracted considerable research interest  \\cite{davenport20141,cai2013max,lafond2014probabilistic,klopp2015adaptive,cao2015categorical,bhaskar2016probabilistic}. Adapted to our notation, these works studied the model $\\dot{y}_k = \\mathcal{Q}(\\big<\\bm{X}_k,\\bm{\\Theta^\\star}\\big>+\\tau_k)$ under general random dither $\\tau_k$ and quantizer $\\mathcal{Q}(.)$, and they commonly adopted regularized (or constrained) maximum likelihood estimation  for estimating $\\bm{\\Theta^\\star}$. By contrast,   with   the random dither and quantizer specialized to \n$\\tau_k\\sim \\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}])$ and $\\mathcal{Q}_\\Delta(.)$ (resp.), we consider $\\dot{y}_k=\\mathcal{Q}_\\Delta(\\mathsf{T}(\\big<\\bm{X}_k,\\bm{\\Theta^\\star}\\big>+\\epsilon_k)+\\tau_k)$ where $\\mathsf{T}(.)$ stands for the truncation under heavy-tailed $\\epsilon_k$. Thus, while suffering from less generality in $(\\tau_k,\\mathcal{Q})$, our method embraces the   advantage of robustness to pre-quantization noise $\\epsilon_k$, whose distribution is unknown and can even be heavy-tailed. Note that such unknown $\\epsilon_k$ evidently forbids the likelihood approach.\n\\end{rem}\n \n\\section{More developments of QCS}\\label{qc-qcs}\nBy now we have presented near optimal results in the context of QCE, QCS and QMC, all under the two-fold challenge of heavy-tailedness and quantization. While positioning such achievable (near) optimality as the primary standpoint of this work, in this section, we further develop our theoretical results for QCS. \n\\subsection{Covariate quantization}\nIn the area of QCS, almost all prior works  merely focused on the quantization of response $y_k$, see the recent survey \\cite{dirksen2019quantized}. Thus, it is of both theoretical and practical interest to  study a setting of \"complete quantization\" --- meaning that the covariate $\\bm{x}_k$ is also quantized. From the practical side, for instance, the immediate benefit of covariate quantization is the lower memory load. Perhaps more prominently, in a federated or distributed learning system,  multiple parties may need to communicate the feature information among themselves, then quantization may become necessary to render lower  communication cost. \n\n\nThe main goal of this part is to study QCS with covariate quantization. \n\n\n\n\n\n\\subsubsection{Multi-bit QCS with quantized covariate}\n\n Since we will also consider the 1-bit quantization, we more precisely refer to the QCS under uniform quantizer (as considered in Theorems \\ref{thm4}-\\ref{thm5}) as multi-bit QCS. We will generalize Theorems \\ref{thm4}-\\ref{thm5} to covariate quantization in the next two theorems. \n\n\n Let $(\\bm{\\dot{x}}_k,\\dot{y}_k)$ be the quantized covariate-response pair, we first quickly sketch the idea of our approach: we stick to the framework of M-estimator in (\\ref{4.3}); as already pointed out, this boils down to looking for accurate surrogates of the covariance $\\bm{\\Sigma^\\star} =\\mathbbm{E}(\\bm{x}_k\\bm{x}_k^\\top)$ and $\\bm{\\Sigma}_{y\\bm{x}}=\\mathbbm{E}(y_k\\bm{x}_k)$ from $(\\bm{\\dot{x}}_k,\\dot{y}_k)$. Fortunately, with triangular dither for $\\bm{x}_k$, this problem has been addressed by our developments of QCE.  \n \n\n \n Now we are in a position to state our quantization scheme as follows:  \n \\begin{itemize}\n     \\item (Response quantization){\\bf \\sffamily.} It is the same as Theorems \\ref{thm4}-\\ref{thm5}. With threshold $\\zeta_y>0$, $y_k$ is truncated to $\\widetilde{y}_k = \\sign(y_k)\\min\\{|y_k|,\\zeta_y\\}$, which is then processed by the uniform quantizer with   uniform dither and some $\\Delta>0$, i.e.,  $\\dot{y}_k = \\mathcal{Q}_\\Delta(\\widetilde{y}_k+ \\phi_k)$ with $\\phi_k\\sim \\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}])$. \n\n     \\item (Covariate quantization){\\bf \\sffamily.} It is the same as Theorem \\ref{thm1}. With threshold $\\zeta_x>0$, $\\bm{x}_k$ is element-wisely truncated to $\\bm{\\widetilde{x}}_k$ by $\\widetilde{x}_{ki} = \\sign(x_{ki})\\min\\{|x_{ki}|,\\zeta_x\\}$, which is then processed by the uniform quantizer with  triangular dither and some $\\bar{\\Delta}>0$, i.e., $\\bm{\\dot{x}}_k = \\mathcal{Q}_{\\bar{\\Delta}}(\\bm{\\widetilde{x}}_k+\\bm{\\tau}_k)$ with $\\bm{\\tau}_k\\sim \\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}]^d)+\\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}]^d)$. \n\n\n     \\item (Notation){\\bf \\sffamily.} We write the quantization noise $\\varphi_k=\\dot{y}_k-\\widetilde{y}_k$, $\\bm{\\xi}_k = \\bm{\\dot{x}}_k- \\bm{x}_k$, and the quantization error $\\vartheta_k= \\dot{y}_k-(\\widetilde{y}_k+\\phi_k)$, $\\bm{w}_k=\\bm{\\dot{x}}_k-(\\bm{\\widetilde{x}}_k+ \\bm{\\tau}_k)$. \n \\end{itemize}\nWe will adopt the above notation in subsequent developments. Based on the quantized covariate-response pair $(\\bm{\\dot{x}}_k,\\dot{y}_k)$, we specify  our estimator by setting $(\\bm{Q},\\bm{b})$ in (\\ref{4.3}) as\\begin{equation}\\label{quanlasso}\n    \\bm{Q}=\\frac{1}{n}\\sum_{k=1}^n \\bm{\\dot{x}}_k\\bm{\\dot{x}}_k^\\top- \\frac{\\bar{\\Delta}^2}{4}\\bm{I}_d~~\\mathrm{and}~~\\bm{b}=\\frac{1}{n}\\sum_{k=1}^n \\dot{y}_k\\bm{\\dot{x}}_k.\n\\end{equation}\nNote that  the choice of $\\bm{Q}$ is due to the estimator in Theorem \\ref{thm1}, while $\\bm{b}$ is because of the relation $\\mathbbm{E}(\\dot{y}_k\\bm{\\dot{x}}_k)=\\mathbbm{E}(\\widetilde{y}_k\\bm{\\widetilde{x}}_k)$. \n\n \n \n However, the issue is that $\\bm{Q} $ is not positive semi-definite.\n  To explain this, note that  of primary interest in CS  is the high-dimensional setting where $d\\gg n$, while evidently, at least $d-n$ eigenvalues of $\\bm{Q}$ are $-\\frac{\\bar{\\Delta}^2}{4}$. We mention that, the lack of positive semi-definiteness of $\\bm{Q}$ is problematic in both statistics and optimization aspects --- the Lemma  \\ref{csframework}  for deriving statistical rates in Theorems \\ref{thm4}-\\ref{thm5} do not apply, and it is in general unclear how to globally optimize a non-convex program. \n \n\n\n\n\n\nMotivated by a previous line of works on non-convex M-estimator \\cite{loh2011high,loh2013regularized,loh2017statistical}, we add a $\\ell_1$ norm constraint to (\\ref{4.3}) by setting $\\mathcal{S}=\\{ \\bm{\\theta}\\in \\mathbb{R}^d:\\|\\bm{\\theta}\\|_1\\leq \\mathcal{R}\\}$, where $\\mathcal{R}$ represents the prior estimation on $\\|\\bm{\\theta^\\star}\\|_1$. Let $\\nabla\\|\\bm{\\theta}_1\\|_1$ be the sub-differential of $\\|\\bm{\\theta}\\|_1$ at $\\bm{\\theta}=\\bm{\\theta}_1$, we consider any local minimizer for the proposed recovery program\\footnote{The existence of local minimizer is guaranteed because of the additional $\\ell_1$ constraint.}, or more generally $\\bm{\\widetilde{\\theta}}\\in \\mathcal{S}$\\footnote{To distinguish the global minimizer in (\\ref{4.3}), we denote  by $\\bm{\\widetilde{\\theta}}$ the estimator in QCS with quantized covariate.} satisfying \\begin{equation}\n    \\big<\\bm{Q}\\bm{\\widetilde{\\theta}}-\\bm{b}+ \\lambda \\cdot \\nabla \\|\\bm{\\widetilde{\\theta}}\\|_1, \\bm{\\theta} - \\bm{\\widetilde{\\theta}}\\big> \\geq 0,~~\\forall~\\bm{\\theta}\\in \\mathcal{S}.\\label{4.17}\n\\end{equation}  \nWe will prove a fairly strong guarantee that all local minimizers enjoy  near minimax error rate.\nWhile   result of this kind bears resemblance to the ones in \\cite{loh2013regularized}, we point out that, \\cite{loh2013regularized} only derived concrete result for sub-Gaussian regime; because of the heavy-tailed data and quantization in our setting,   some essentially different ingredients are required  for the technical analysis (see {Remark} \\ref{rem4}).\n\n\nWe are ready to present our results concerning multi-bit QCS with quantized covariate; the settings of sub-Gaussian $\\bm{x}_k$ and heavy-tailed $\\bm{x}_k$ are presently separately.\n\n\n\n\n\\begin{theorem}\\label{thm6}\n{\\rm \\scshape (Quantized sub-gaussian Covariate){\\bf \\sffamily.}} Given $\\Delta\\geq 0$, $\\bar{\\Delta}\\geq 0$, $\\delta>0$, we consider the same model  as  Theorem \\ref{thm4}, \nwith the additional assumption of $\\|\\bm{\\theta^\\star}\\|_2\\leq R$   for some $R>0$. We consider the quantization scheme described above with $\\zeta_x=\\infty$ so that $\\bm{\\widetilde{x}}_k=\\bm{x}_k$.\nFor recovery, we let $\\bm{Q} = \\frac{1}{n}\\sum_{k=1}^n\\bm{\\dot{x}}_k\\bm{\\dot{x}}_k^\\top -\\frac{\\bar{\\Delta}^2}{4}\\bm{I}_d$, $\\bm{b}= \\frac{1}{n}\\sum_{k=1}^n\\dot{y}_k\\bm{\\dot{x}}_k$, $\\mathcal{S} = \\{\\bm{\\theta}:\\|\\bm{\\theta}\\|_1\\leq R\\sqrt{s}\\}$. We also set $\\zeta_y \\asymp \\big(\\frac{\\sigma M^{1\/l}}{\\sigma+\\Delta}\\frac{n}{\\delta\\log d}\\big)^{1\/2}$, $\\lambda = C_1\\frac{(\\sigma+\\bar{\\Delta})^2}{\\sqrt{\\kappa_0}}(\\Delta+M^{1\/(2l)})\\sqrt{\\frac{\\delta\\log d}{n}}$ with sufficiently large $C_1$. If $n\\gtrsim \\delta s\\log d$ for some hidden constant only depending on $(\\kappa_0,\\sigma,\\Delta,\\bar{\\Delta},M,R)                $,\nwith probability at least $1- 8d^{1-\\delta}-C_2\\exp(-C_3n)$,  all $\\bm{\\widetilde{\\theta}}\\in \\mathcal{S}$ satisfying (\\ref{4.17}) deliver an estimation error $\\bm{\\widetilde{\\Delta}}:=\\bm{\\widetilde{\\theta}}-\\bm{\\theta^\\star}$ bounded by \n\\begin{equation}\\nonumber\n    \\|\\bm{\\widetilde{\\Delta}}\\|_2\\leq C \\mathscr{L}\\sqrt{\\frac{\\delta s\\log d}{n}} ~~\\mathrm{and}~~\\|\\bm{\\widetilde{\\Delta}}\\|_1\\leq C' \\mathscr{L}s\\sqrt{\\frac{\\delta \\log d}{n}}\n\\end{equation}\nwhere $ \\mathscr{L} :=\\frac{(\\sigma+\\bar{\\Delta})^2(\\bm{\\Delta}+M^{1\/(2l)})}{\\kappa_0^{3\/2}}$.\n\\end{theorem}\n \n\n\\vspace{1mm}\n\nThen, we turn to QCS with heavy-tailed covariate studied in Theorem \\ref{thm5}. The difference is that we also quantize the covariate. \n\n\n\\begin{theorem}\\label{thm7}\n{\\rm \\scshape (Quantized heavy-tailed Covariate){\\bf \\sffamily.}} Given $\\Delta\\geq 0$, $\\bar{\\Delta}\\geq 0$, $\\delta>0$, we consider the same model  as  Theorem  \\ref{thm5}, with the additional assumption of $\\|\\bm{\\theta^\\star}\\|_1\\leq R$   for some $R>0$.\nWe adopt the quantization scheme described above.\nFor recovery, we let $\\bm{Q}=\\frac{1}{n}\\sum_{k=1}^n \\bm{\\dot{x}}_k\\bm{\\dot{x}}_k^\\top- \\frac{\\Delta^2}{4}\\bm{I}_d$, $\\bm{b}= \\frac{1}{n}\\sum_{k=1}^n\\dot{y}_k\\bm{\\dot{x}}_k$, $\\mathcal{S}=\\{\\bm{\\theta}:\\|\\bm{\\theta}\\|_1\\leq R\\}$.\nWe also set $\\zeta_x,\\zeta_y\\asymp \\big(\\frac{nM}{\\delta\\log d}\\big)^{1\/4}$, \n$\\lambda =C_1 (R\\sqrt{M}+\\Delta^2+R\\bar{\\Delta}^2)\\sqrt{\\frac{\\delta \\log d}{n}}$ with sufficiently large $C_1$. If $n\\gtrsim \\delta s\\log d$ for some hidden constant only depending on $(\\kappa_0,M)$, with probability at least $1- 8d^{1-\\delta}$,  all $\\bm{\\widetilde{\\theta}}\\in \\mathcal{S}$ satisfying (\\ref{4.17}) have an estimation error $\\bm{\\widetilde{\\Delta}}:=\\bm{\\widetilde{\\theta}}-\\bm{\\theta^\\star}$ bounded by \\begin{equation}\\nonumber\n    \\|\\bm{\\widetilde{\\Delta}}\\|_2\\leq C_3\\mathscr{L}\\sqrt{\\frac{\\delta s\\log d}{n}} ~~\\mathrm{and}~~\\|\\bm{\\widetilde{\\Delta}}\\|_1\\leq C_4\\mathscr{L}s\\sqrt{\\frac{\\delta \\log d}{n}}.\n\\end{equation}  \nwhere $\\mathscr{L}:=\\frac{R\\sqrt{M}+\\Delta^2+R\\bar{\\Delta}^2}{\\kappa_0}$.\n\\end{theorem}\n\n\n\\begin{rem}\\label{rem4}\n{\\rm \\scshape (Comparing Our Analyses with \\cite{loh2013regularized}){\\bf \\sffamily.}} The above results are motivated by a line of works on nonconvex M-estimator \\cite{loh2011high,loh2013regularized,loh2017statistical}, and our guarantee for the whole set of stationary points (\\ref{4.17}) resembles \\cite{loh2013regularized} most. While the main strategy for proving Theorem \\ref{thm6} is adjusted from \\cite{loh2013regularized},   the proof of Theorem \\ref{thm7} does involve an essentially different RSC condition, see our (\\ref{4.23}). In particular, compared with  \\cite[equation (4)]{loh2013regularized}, the leading factor of $\\|\\bm{\\widetilde{\\Delta}}\\|_1^2$ in (\\ref{4.23}) degrades from $O\\big(\\frac{\\log d}{n}\\big)$ to $O\\big(\\sqrt{\\frac{\\log d}{n}}\\big)$. To retain near optimal rate we need to impose a stronger scaling $\\|\\bm{\\theta^\\star}\\|_1\\leq R$  with proper changes in the proof. Although Theorem \\ref{thm7} is presented for a   concrete setting, it sheds light on   extension of \\cite{loh2013regularized} to a weaker RSC condition that can possibly accommodate covariate with heavier tail. Such extension is formally presented as a deterministic framework in Proposition \\ref{framework}.  \n\\end{rem}\n\\begin{pro}\n\\label{framework}\nSuppose that the $s$-sparse $\\bm{\\theta^\\star}\\in \\mathbb{R}^d$  satisfies $\\|\\bm{\\theta^\\star}\\|_1\\leq R$, and that   $\\bm{\\Sigma^\\star}=\\mathbbm{E}(\\bm{x}_k\\bm{x}_k^\\top)$ admits $\\lambda_{\\min}(\\bm{\\Sigma^\\star})\\geq \\kappa_0$. If  \\begin{equation}\\label{4.26}\n    \\lambda \\geq C_1\\max\\big\\{\\|\\bm{Q\\theta^\\star} - \\bm{b}\\|_\\infty,~ R\\cdot\\|\\bm{Q}-\\bm{\\Sigma^\\star}\\|_\\infty\\big\\}\n\\end{equation}\nholds for sufficiently large $C$. Then, all $\\bm{\\widetilde{\\theta}}$ satisfying (\\ref{4.17}) with $\\mathcal{S} = \\{\\bm{\\theta}\\in \\mathbb{R}^d:\\|\\bm{\\theta}\\|_1\\leq R\\}$ has estimation error $\\bm{\\widetilde{\\Delta}}:=\\bm{\\widetilde{\\theta}}-\\bm{\\theta^\\star}$ bounded by \n\\begin{equation}\n    \\begin{aligned} \\nonumber\n      \\|\\bm{\\widetilde{\\Delta}}\\|_2\\leq C_2 \\frac{\\sqrt{s}\\lambda}{\\kappa_0} ~~\\mathrm{and}~~\\|\\bm{\\widetilde{\\Delta}}\\|_1\\leq C_3 \\frac{s\\lambda}{\\kappa_0}~.\n    \\end{aligned}\n\\end{equation} \n\\end{pro}\nWe then compare Theorem \\ref{thm7} with Theorem \\ref{thm5} where full $\\bm{x}_k$ is available.  \n\\begin{rem}\n\\label{rem5} \\label{impro}By setting $\\bar{\\Delta}=0$, Theorem \\ref{thm7} produces a result (with convex program) for the setting of Theorem \\ref{thm5}.  Interestingly, with the additional $\\ell_1$ constraint, a notable improvement is that the sub-optimal $n\\gtrsim s^2\\log d$ in Theorem \\ref{thm5} is sharpened to the near optimal one in Theorem \\ref{thm7}.  \nMore concretely, this is because (ii) in (\\ref{4.8}) can be  tightened  to $(ii)$ of     (\\ref{4.24}). Going back to the full-date regime,   Theorem \\ref{thm7} with $\\Delta=\\bar{\\Delta}=0$ recovers \\cite[Theorem 2(b)]{fan2021shrinkage} with improved sample complexity requirement. \n\n\\end{rem}\n   \n \n \n  \n\\subsubsection{1-bit QCS with quantized covariate} \\label{sec4.3}\nOur consideration of covariate quantization in QCS seems fairly new to the literature. To the best of our knowledge, the only related results are  \\cite[Theorems 7-8]{chen2022high} for QCS with 1-bit quantized covariate and response. The assumption there, however, is quite restrictive. Specifically, it is assumed that $\\bm{\\Sigma^\\star}=\\mathbbm{E}(\\bm{x}_k\\bm{x}_k^\\top)$ has sparse columns (see \\cite[Assumption 3]{chen2022high}), which may not be available and is non-standard in CS\\footnote{While it is conventional to use isotropic sensing vector in CS (i.e., $\\bm{\\Sigma^\\star}=\\bm{I}_d$), most results in the literature do not really rely on the sparsity of $\\bm{\\Sigma^\\star}$.}.    \n\n\nIn this part, we slightly deviate from the focus of dithered uniform quantization. Rather, our   goal is to study QCS under dithered 1-bit quantization. Specifically, as byproducts of this work, we will apply Proposition \\ref{framework} to derive results comparable to \\cite[Theorems 7-8]{chen2022high} without resorting to the sparsity of $\\bm{\\Sigma^\\star}$.  \n \nWe first give some remarks on Proposition \\ref{framework}. Compared with the framework \\cite[Theorem 1]{loh2013regularized}, the key strength of   Proposition \\ref{framework} is that it does not explicitly assume the  RSC condition on the loss function that is hard to verify without assuming sub-Gaussian covariate. \nInstead, the role of the RSC assumption in \\cite{loh2013regularized} is now played by $\\lambda\\gtrsim R\\|\\bm{Q}-\\bm{\\Sigma^\\star}\\|_\\infty$,  which immediately yields a kind of  RSC condition by simple argument as (\\ref{4.24}). Although this RSC condition is often essentially weaker than the conventional one  in terms of the leading factor of $\\|\\bm{\\widetilde{\\Delta}}\\|_1^2$ (see   Remark \\ref{rem4}), along this line one can still   derive non-trivial (or even near optimal) error rate. The gain of replacing RSC assumption with $\\lambda\\gtrsim R\\|\\bm{Q}-\\bm{\\Sigma^\\star}\\|_\\infty$ is that the latter amounts  to \nconstructing element-wise estimator for $\\bm{\\Sigma^\\star}$, which is often much easier for heavy-tailed covariate (e.g., due to many existing robust covariance estimator).\n\n\n\nWe first review the 1-bit quantization scheme developed in \\cite{chen2022high}: \n\\begin{itemize}\n    \\item (Response quantization){\\bf \\sffamily.} With threshold $\\zeta_y>0$, $y_k$ is truncated to $\\widetilde{y}_k = \\sign(y_k)\\min\\{|y_k|,\\zeta_y\\}$, which is then quantized to $\\dot{y}_k = \\sign(\\widetilde{y}_k+\\phi_k)$ with $\\phi_k\\sim \\mathscr{U}([-\\gamma_y,\\gamma_y])$.\n\n\n    \n    \\item (Covariate quantization){\\bf \\sffamily.} With threshold $\\zeta_x>0$, $\\bm{x}_k$ is element-wisely truncated to $\\bm{\\widetilde{x}}_k$ by $\\widetilde{x}_{ki} = \\sign(x_{ki})\\min\\{|x_{ki}|,\\zeta_x\\}$, which is then quantized to $\\bm{\\dot{x}}_{k1}= \\sign(\\bm{\\widetilde{x}}_k+\\bm{\\tau}_{k1})$, $\\bm{\\dot{x}}_{k2}= \\sign(\\bm{\\widetilde{x}}_k+\\bm{\\tau}_{k2})$, with $\\bm{\\tau}_{k1},\\bm{\\tau}_{k2}\\sim \\mathscr{U}([-\\gamma_x,\\gamma_x])$. (Note that we collect two bits for each entry).\n\n   \n   \n\\end{itemize}\n\n\n\n\n\nAs arranged in  \\cite{chen2022high}, we divide it into sub-Gaussian case (Theorem \\ref{sg1bitqccs}) and heavy-tailed case (Theorem \\ref{ht1bitqccs}). \n\n\\begin{theorem}\\label{sg1bitqccs}\n{\\rm \\scshape (1-bit QCS with sub-gaussian Data){\\bf \\sffamily.}} For the model $y_k = \\bm{x}_k^\\top\\bm{\\theta^\\star}+\\epsilon_k$ with $s$-sparse $\\bm{\\theta^\\star}$ that admits $\\|\\bm{\\theta^\\star}\\|_1\\leq R$, and with i.i.d. zero-mean, sub-Gaussian\n $\\bm{x}_k, \\epsilon_k$; for simplicity we assume $\\|\\bm{x}_k\\|_{\\psi_2}\\leq \\sigma,\\|y_k\\|_{\\psi_2}\\leq \\sigma$; also suppose that  $\\lambda_{\\min}\\big(\\bm{\\Sigma^\\star}\\big)\\geq \\kappa_0$ holds for some $\\kappa_0>0$, where $\\bm{\\Sigma^\\star}=\\mathbbm{E}(\\bm{x}_k\\bm{x}_k^\\top)$. In the above quantization scheme we set $\\zeta_x=\\zeta_y=\\infty$, $\\gamma_x,\\gamma_y\\asymp \\sigma \\sqrt{\\log\\big(\\frac{n}{2\\delta \\log d}\\big)}$.  For the recovery we let $\\bm{Q}:=\\frac{\\gamma_x^2}{2n}\\sum_{k=1}^n\\big(\\bm{\\dot{x}}_{k1}\\bm{\\dot{x}}_{k2}^\\top+\\bm{\\dot{x}}_{k2}\\bm{\\dot{x}}_{k1}^\\top\\big)$, $\\bm{b}:=\\frac{\\gamma_x\\gamma_y}{n}\\sum_{k=1}^n \\dot{y}_k \\bm{\\dot{x}}_{k1}$, $\\mathcal{S}=\\{\\bm{\\theta}:\\|\\bm{\\theta}\\|_1\\leq R\\}$ and set   $\\lambda = C_1\\sigma^2R \\sqrt{\\frac{\\delta \\log d(\\log n)^2}{n}}$   with sufficiently large $C_1$. Then if $n\\gtrsim \\delta s\\log d(\\log n)^2$, with probability at least $1-4d^{2-\\delta}$, all local minimizers $\\bm{\\widetilde{\\theta}}$ satisfying (\\ref{4.17}) have   estimation error $\\bm{\\widetilde{\\Delta}}:=\\bm{\\widetilde{\\theta}}-\\bm{\\theta^\\star}$ bounded by \\begin{equation}\\nonumber\n    \\|\\bm{\\widetilde{\\Delta}}\\|_2\\leq C_2 \\frac{\\sigma^2}{\\kappa_0}\\cdot R\\sqrt{\\frac{\\delta s\\log d (\\log n)^2}{n}} ~~\\mathrm{and}~~\\|\\bm{\\widetilde{\\Delta}}\\|_1\\leq  C_3\\frac{\\sigma^2}{\\kappa_0}\\cdot Rs\\sqrt{\\frac{\\delta \\log d (\\log n)^2}{n}}.\n\\end{equation}\n\\end{theorem}\n\n\\begin{theorem}\\label{ht1bitqccs}\n{\\rm \\scshape (1-bit QCS with heavy-tailed Data){\\bf \\sffamily.}} For the model $y_k = \\bm{x}_k^\\top\\bm{\\theta^\\star}+\\epsilon_k$ with $s$-sparse $\\bm{\\theta^\\star}$ that admits $\\|\\bm{\\theta^\\star}\\|_1\\leq R$, and with i.i.d. zero-mean, heavy-tailed  $\\bm{x}_k, \\epsilon_k$; for simplicity we assume $\\sup_{\\bm{v}\\in \\mathbb{S}^{d-1}}\\mathbbm{E}|\\bm{v}^\\top \\bm{x}_k|^4\\leq M$, $\\mathbbm{E}|y_k|^4\\leq M$;  also suppose that  $\\lambda_{\\min}\\big(\\bm{\\Sigma^\\star}\\big)\\geq \\kappa_0$ holds for some $\\kappa_0>0$, where $\\bm{\\Sigma^\\star}=\\mathbbm{E}(\\bm{x}_k\\bm{x}_k^\\top)$. In the above quantization scheme we set $\\zeta_x,\\zeta_y,\\gamma_x,\\gamma_y \\asymp \\big(\\frac{nM^2}{\\delta \\log d}\\big)^{1\/8}$ that satisfy $\\zeta_x <\\gamma_x$ and $\\zeta_y<\\gamma_y$. For the recovery we let $\\bm{Q}:=\\frac{\\gamma_x^2}{2n}\\sum_{k=1}^n\\big(\\bm{\\dot{x}}_{k1}\\bm{\\dot{x}}_{k2}^\\top+\\bm{\\dot{x}}_{k2}\\bm{\\dot{x}}_{k1}^\\top\\big)$, $\\bm{b}:=\\frac{\\gamma_x\\gamma_y}{n}\\sum_{k=1}^n \\dot{y}_k \\bm{\\dot{x}}_{k1}$, $\\mathcal{S}=\\{\\bm{\\theta}:\\|\\bm{\\theta}\\|_1\\leq R\\}$ and set $\\lambda = C_1\\sqrt{M}R\\big(\\frac{\\delta \\log d}{n}\\big)^{1\/4}$ with sufficiently large $C_1$. Then if $n\\gtrsim \\delta s^2\\log d$, with probability at least $1-4d^{2-\\delta}$,\nall local minimizers $\\bm{\\widetilde{\\theta}}$ satisfying (\\ref{4.17}) have estimation error $\\bm{\\widetilde{\\Delta}}:=\\bm{\\widetilde{\\theta}}-\\bm{\\theta^\\star}$ bounded by \\begin{equation}\\nonumber\n    \\|\\bm{\\widetilde{\\Delta}}\\|_2\\leq C_2\\frac{\\sqrt{M}}{\\kappa_0}\\cdot R \\Big(\\frac{\\delta s^2\\log d}{n}\\Big)^{1\/4} ~~\\mathrm{and}~~\\|\\bm{\\widetilde{\\Delta}}\\|_1\\leq C_3\\frac{\\sqrt{M}}{\\kappa_0}\\cdot Rs\\Big(\\frac{\\delta \\log d}{n}\\Big)^{1\/4}.\n\\end{equation}\n\\end{theorem}\n\\subsection{Uniform recovery guarantee}\n The main goal of this part is to improve the previous non-uniform  Theorem  \\ref{thm4} to uniform reconstruction guarantee. The proof requires significantly more work and deeper technical tools. Some techniques are inspired by previous works, e.g., we handle the discontinuity of the dithered uniform quantizer via covering argument as \\cite{xu2020quantized}, and similar to \\cite{genzel2022unified}, a key tool is   the powerful concentration inequality for product process due to Mendelson \\cite{mendelson2016upper}. \n\n\n\nHowever, while previous works concentrated on sub-Gaussian data\\footnote{More precisely, \\cite{xu2020quantized} considered RIP sensing matrix without noise; \\cite{chen2022uniform} assumed standard Gaussian complex Gaussian sensing matrix.},   we study  quantization of heavy-tailed data that involves additional data processing steps (i.e., truncation and dithered quantization). As a consequence, it is necessary to develop some new machinery. For instance, we consider a setting without covariate quantization ($\\bar{\\Delta}=0$) and  let $\\mathscr{S}_{\\zeta_y}(a)= \\sign(a)\\cdot\\min\\{|a|,\\zeta_y\\}$ be the truncation operator; then following similar lines in the proof of Theorem \\ref{thm4}, we will need to bound the term $\\mathscr{T}'=\\sup_{\\bm{v},\\bm{\\theta}} \\sum_{k=1}^n\\big(\\widetilde{y}_k\\bm{x}_k-\\mathbbm{E}(\\widetilde{y}_k\\bm{x}_k)\\big)^\\top \\bm{v}$, where $\\bm{v}$ takes value in the range of the normalized estimation error, and the supremum of $\\bm{\\theta}$ is taken over all sparse signals within an $\\ell_2$ ball, as we are pursuing a uniform guarantee and $\\widetilde{y}_k=\\mathscr{S}_{\\zeta_y}(\\bm{x}_k^\\top\\bm{\\theta^\\star}+ \\epsilon_k)$ does hinge on $\\bm{\\theta}$. Our main tool to bound the product process $\\mathscr{T}'$ is Lemma \\ref{productpro}, but the issue is that $\\widetilde{y}_k$ does not possess $O(1)$ sub-Gaussian norm (due to $\\epsilon_k$), and using the trivial sub-Gaussianity\n$\\|\\widetilde{y}_k\\|_{\\psi_2}=O(\\zeta_y)$ definitely leads to sub-optimal rate. To address the issue, our main idea is to introduce $\\mathscr{S}_{\\zeta_y}(\\epsilon_k)$ and $\\widetilde{z}_k= \\widetilde{y}_k-\\mathscr{S}_{\\zeta_y}(\\epsilon_k)$, then we can write \n\\begin{equation}\\label{n4.9}\n    \\mathscr{T}'\\leq \\underbrace{\\sup_{\\bm{v},\\bm{\\theta}}~ \\sum_{k=1}^n\\big(\\widetilde{z}_k\\bm{x}_k-\\mathbbm{E}(\\widetilde{z}_k\\bm{x}_k)\\big)^\\top\\bm{v}}_{\\mathscr{T}'_1}+ \\underbrace{\\sup_{ \\bm{v}}~ \\sum_{k=1}^n\\big(\\mathscr{S}_{\\zeta_y}(\\epsilon_k)\\bm{x}_k-\\mathbbm{E}(\\mathscr{S}_{\\zeta_y}(\\epsilon_k)\\bm{x}_k)\\big)^\\top\\bm{v}}_{\\mathscr{T}'_2}.\n\\end{equation}\nAfter this crucial workaround, Lemma \\ref{productpro} applies to $\\mathscr{T}'_1$\nsince $\\widetilde{z}_k$ is sub-Gaussian and admits sub-Gaussian increments; also, $\\mathscr{T}_2'$ is now independent of $\\bm{\\theta}$ and hence essentially easier. In addition, compared to \\cite[Proposition 6.1]{xu2020quantized}, {\\sffamily step 2.4} in the proof involves some   technical changes, as will be explicitly pointed out there.\n\n\n\n\n\n\nIn the following we present the uniform recovery guarantee. We adopt most assumptions in Theorem \\ref{thm4}, but specify the range of $\\bm{\\theta^\\star}$ as $\\bm{\\theta^\\star}\\in \\Sigma_{s,R_0}$ and impose the moment constraint on $\\epsilon_k$.   For a more straightforward analysis we consider constrined Lasso instead of (\\ref{4.3}) as prior works.\nThe proof is provided in the appendix.    \n\n\\begin{theorem}\n    \\label{uniformtheorem}\n    {\\rm \\scshape (Uniform version of Theorem \\ref{thm4}){\\bf \\sffamily.}} We suppose in $y_k = \\bm{x}_k^\\top\\bm{\\theta^\\star}+\\epsilon_k$ that $\\bm{x}_k$ are i.i.d., zero-mean, and sub-Gaussian with $\\|\\bm{x}_k\\|_{\\psi_2}\\leq \\sigma$; that     $\\kappa_0\\leq \\lambda_{\\min}(\\bm{\\Sigma^\\star})\\leq \\lambda_{\\max}(\\bm{\\Sigma^\\star})\\leq \\kappa_1$ $(\\bm{\\Sigma^\\star}=\\mathbbm{E}(\\bm{x}_k\\bm{x}_k^\\top))$ for some $\\kappa_1\\geq \\kappa_0>0$, $\\bm{\\theta^\\star}\\in \\Sigma_{s,R_0}:= \\Sigma_s \\cap \\{\\bm{\\theta}:\\|\\bm{\\theta}\\|_2\\leq R_0\\}$ for some absolute constant $R_0$ (this is unnecessary and only for less parameter); and that $\\epsilon_k$ are i.i.d. noise satisfying $\\mathbbm{E}|\\epsilon_k|^{2l}\\leq M$ for some fixed $l>1$.     Recall our quantization scheme that $\\widetilde{y}_k= \\sign(y_k)\\min\\{|y_k|,\\zeta_y\\}$, $\\dot{y}_k = \\mathcal{Q}_\\Delta(\\widetilde{y}_k+\\tau_k)$ with $\\tau_k\\sim \\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}])$, and we set $\\zeta_y\\asymp \\big(\\frac{n(M^{1\/l}+\\sigma^2)}{\\delta \\log d}\\big)^{1\/2}$. For recovery, we define the estimator $\\bm{\\widehat{\\theta}}$ as the solution to constrained Lasso \\begin{equation}\\nonumber\n        \\bm{\\widehat{\\theta}}= \\mathop{\\arg\\min}\\limits_{\\|\\bm{\\theta}\\|_{1}\\leq \\|\\bm{\\theta^\\star}\\|_1}~\\frac{1}{2n}\\sum_{k=1}^n(\\dot{y}_k-\\bm{x}_k^\\top\\bm{\\theta})^2 \n    \\end{equation}\n    If $n \\gtrsim \\delta s\\log \\mathscr{W}$ for $\\mathscr{W}= \\frac{\\kappa_1d^2n^3}{\\Delta^2s^5\\delta^3}$ and some hidden constant depending on $(\\kappa_0,\\sigma)$,     with probability at least $1-Cd^{1-\\delta}$ on a single random draw of $(\\bm{x}_k,\\epsilon_k, \\tau_k)_{k=1}^n$, it holds uniformly for all $\\bm{\\theta^\\star}\\in\\Sigma_{s,R_0}$ that the estimation errors $\\bm{\\widehat{\\Delta}}:=\\bm{\\widehat{\\theta}}-\\bm{\\theta^\\star}$ satisfy \n    \\begin{equation}\\nonumber\n         \\|\\bm{\\widehat{\\Delta}}\\|_2\\leq C_3\\mathscr{L} \\sqrt{\\frac{\\delta s\\log \\mathscr{W}}{n}}~~~\\mathrm{and}~~~\\|\\bm{\\widehat{\\Delta}}\\|_1 \\leq C_4\\mathscr{L}s\\sqrt{\\frac{\\delta\\log \\mathscr{W}}{n}}~\n    \\end{equation}\n    where $\\mathscr{L}:=\\frac{\\sigma(\\sigma+\\Delta+M^{1\/(2l)})}{\\kappa_0}$.\n\\end{theorem}\n\n\n Notably, our uniform guarantee is still minimax optimal up to logarithmic factors. Compared to the clean $\\sqrt{\\log d}$ in previous results,  the worse logarithmic factor $\\sqrt{\\log \\mathscr{W}}$ is because of a covering argument on $\\Sigma_{s,R_0}$. See {\\sffamily step 2.4} in the proof  for more details, and for similar techniques we refer to \\cite{xu2020quantized,chen2022uniform}. Because in the full-data regime $(\\Delta =0)$ we do not need to deal with quantizer noise as {\\sffamily step 2.4},   we can prove the uniform error bound $O\\big(\\sqrt{\\frac{s\\log d}{n}}\\big)$  that is in full agreement with previous results. Such result has its own  interest --- by using constrained Lasso instead of the regularized one, it strengthens \\cite[Theorem 2(a)]{fan2021shrinkage} to a uniform reconstruction guarantee almost with no cost.\n\n\n\n \n\n\n\\section{Numerical Simulations}\\label{sec6}\nIn this section we provide two sets of experimental results to support and demonstrate the previous developments. As the standpoint of this work is to demonstrate near minimax error rates are achievable when applying the proposed scheme to heavy-tailed data, the first set of simulations aim to validate the obtained error rates. Then, the second set of results are presented to illustrate the crucial role played by  the appropriate dither (i.e., triangular dither for covariate, uniform dither for response) before uniform quantization. For the importance of data truncation we refer to the relatively complete simulations in \\cite{fan2021shrinkage}, which includes three estimation problems in this work and contrasts the estimations with or without the data truncation. \n\n\\subsection{(Near) minimax error rates}\nEach data point in our results is set to be the mean value of $50$ or $100$ independent trials. \n\n\n\n\n\\subsubsection{Quantized covariance estimation}\nWe start from covariance matrix estimation, specifically we verify the element-wise rate $\\mathscr{B}_1:=O\\big(\\mathscr{L}\\sqrt{\\frac{\\log d}{n}}\\big)$ and operator norm rate $\\mathscr{B}_2:=O\\big(\\mathscr{L}\\sqrt{\\frac{d\\log d}{n}}\\big)$ in Theorems \\ref{thm1}-\\ref{thm2}.\n\n\nFor estimator in Theorem \\ref{thm1}, we draw $\\bm{x}_k = (x_{ki})$ such that the first two coordinates are i.i.d. from $\\mathsf{t}(4.5)$, $(x_{ki})_{i=3,4}$ are from $\\mathsf{t}(6)$ with covariance $\\mathbbm{E}(x_{k3}x_{k4})=1.2$, and the remaining $d-4$ coordinates are from $\\mathsf{t}(6)$. We test different choices of $(d,\\Delta)$  under $n=80:20:220$, and the log-log plots are shown in Figure \\ref{fig1}(a). Clearly, for each $(d,\\Delta)$ the experimental points roughly exhibit a straight line that is well aligned with the dashed line representing the $n^{-1\/2}$ rate. As predicted by the factor $\\mathscr{L}=\\sqrt{M}+\\Delta^2$, the curves with larger $\\Delta$ are higher, but note that the error decreasing rates remain unchanged. In addition, the curves of $(d,\\Delta)=(100,1),(120,1)$ are extremely close, thereby confirming that the element-wise error only depends on ambient dimension $d$ logarithmically. \n\n\nFor the error bound $\\mathscr{B}_2$, the coordinates of $\\bm{x}_k$ are drawn from a scaled version of $\\mathsf{t}(4.5)$ such that $\\bm{\\Sigma^\\star}=\\mathrm{diag}(2,2,1,...,1)$, and we test different settings of $(d,\\Delta)$ under $n= 200:100:1000$. As shown in Figure \\ref{fig1}(b), the operator norm error decreases with $n$ in the optimal rate $n^{-1\/2}$, and using a coarser dithered quantizer (i.e., larger $\\Delta$) only slightly lifts the curves. Indeed, the effect seems consistent with $\\mathscr{L}$'s quadratic dependence on $\\Delta$. To validate the relative scaling of $n$ and $d$, we try $(d,\\Delta)= (150,1)$ under $n=1.5\\times(200:100:1000)$, and surprisingly the obtained curve coincides with the one for $(d,\\Delta)=(100,1)$. Thus, ignoring the logarithmic factor $\\log d$, the operator norm error can be characterized by $\\mathscr{B}_2$ fairly well. \n\n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[scale = 0.6]{Thm_1_2.eps}\n    \n ~~~   (a) \\hspace{4.4cm} (b) \n    \\caption{(a): Element-wise error (Theorem \\ref{thm1}); (b): operator norm error (Theorem \\ref{thm2}).}\n    \\label{fig1}\n\\end{figure}\n\n\n\n\\subsubsection{Quantized compressed sensing}\n\nWe now switch to QCS with full covariate and aim to verify the $\\ell_2$ norm error rate $\\mathscr{B}_3=O\\big(\\mathscr{L}\\sqrt{\\frac{s\\log d}{n}}\\big)$ obtained in Theorems \\ref{thm4}-\\ref{thm5}. We let the support of the $s$-sparse   $\\bm{\\theta^\\star}\\in \\mathbb{R}^d$ be $[s]$, and then draw the non-zero entries from a uniform distribution over $\\mathbb{S}^{s-1}$ (hence $\\|\\bm{\\theta^\\star}\\|_2=1$). For the setting of Theorem \\ref{thm4} we adopt $\\bm{x}_k\\sim \\mathcal{N}(0,\\bm{I}_d)$ and $\\epsilon_k\\sim \\frac{1}{\\sqrt{6}}\\mathsf{t}(3)$, while $\\bm{x}_{ki}\\stackrel{iid}{\\sim}\\frac{\\sqrt{5}}{3}\\mathsf{t}(4.5)$ and $\\epsilon_k\\sim \\frac{1}{\\sqrt{3}}\\mathsf{t}(4.5)$ for Theorem \\ref{thm5}. We simulate different choices of $(d,s,\\Delta)$ under $n=100:100:1000$, and the proposed convex program (\\ref{4.3}) is solved with the framework of ADMM (we refer to the review \\cite{boyd2011distributed}). Experimental results are shown as   log-log plots in Figure \\ref{fig2}. Consistent with the theoretical bound $\\mathscr{B}_3$, the error in both case decreases in the (near) optimal rate of $n^{-1\/2}$, whereas the effect of uniform quantization is merely on the multiplicative factor $\\mathscr{L}$. Interestingly, it seems that the gaps between $\\Delta=0,0.5$ and $\\Delta = 0.5,1$ are in agreement with the explicit form of $\\mathscr{L}$, i.e., $\\mathscr{L}\\asymp M^{1\/(2l)}+\\Delta$ for Theorem \\ref{thm4}, and $\\mathscr{L}\\asymp \\sqrt{M}+\\Delta^2$ for Theorem \\ref{thm5}. In addition, note that the curves of $(d,s)=(150,5),(180,5)$ are close, whereas increasing $s=8$ suffers from significantly larger error, thus validating the relative scaling of $(n,d,s)$ in $\\mathscr{B}_3$.\n\n\n\n\n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[scale = 0.62]{Thm_4_5.eps}\n    \n ~~~   (a) \\hspace{4.4cm} (b) \n    \\caption{(a): QCS in Theorem \\ref{thm4}; (b): QCS in Theorem \\ref{thm5}.}\n    \\label{fig2}\n\\end{figure}\n\nThen, we simulate the complete quantization setting where both covariate and response are quantized (Theorems \\ref{thm6}-\\ref{thm7}). The simulation details are the same as before except that $\\bm{x}_k$ is also quantized with $\\Delta$ same as $y_k$. We provide  the best $\\ell_1$ norm constraint, i.e., $\\mathcal{S}:=\\{\\bm{\\theta}:\\|\\bm{\\theta}\\|_1\\leq\\|\\bm{\\theta^\\star}\\|_1 \\}$. Then,   composite gradient descent \\cite{loh2011high,loh2013regularized} is invoked to handle the non-convex estimation program. We show the log-log plots in Figure \\ref{fig3}. Note that these results have implications similar to Figure \\ref{fig2}, in terms of the $n^{-1\/2}$ rate, the effect of quantization, and the relative scaling of $(n,d,s)$.\n\n\n\n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[scale = 0.58]{Thm_6_7.eps}\n    \n ~~~   (a) \\hspace{4.4cm} (b) \n    \\caption{(a):  QCS in Theorem \\ref{thm6}; (b):  QCS in Theorem \\ref{thm7}.}\n    \\label{fig3}\n\\end{figure}\n\n\n\n\\subsubsection{Quantized matrix completion}\nFinally, we simulate QMC and demonstrate the   error bound $\\mathscr{B}_4=O\\big(\\mathscr{L}\\sqrt{\\frac{rd\\log d}{n}}\\big)$ for $\\|\\bm{\\widehat{\\Delta}} \\|_F\/d$  in Theorems \\ref{thm8}-\\ref{thm9}. We generate the rank-$r$ $\\bm{\\Theta^\\star}\\in \\mathbb{R}^{d\\times d}$ as follows: we first generate $\\bm{\\Theta}_0\\in \\mathbb{R}^{d\\times r}$ with i.i.d. standard Gaussian entries to obtain the rank-$r$ $\\bm{\\Theta}_1:=\\bm{\\Theta}_0\\bm{\\Theta}_0^\\top$, then we set $\\bm{\\Theta^\\star}:=k_1\\bm{\\Theta}_1$ such that $\\|\\bm{\\Theta^\\star}\\|_F=d$. \nWe use $\\epsilon_k\\sim \\mathcal{N}(0,\\frac{1}{4})$ to simulate the sub-exponential noise in Theorem \\ref{thm8}, while $\\epsilon_k\\sim \\frac{1}{\\sqrt{6}}\\mathsf{t}(3)$ for Theorem \\ref{thm9}. The convex program (\\ref{5.2}) is fed with  $\\alpha=\\|\\bm{\\Theta^\\star}\\|_\\infty$ and optimized by the ADMM algorithm. We test different choices of $(d,r,\\Delta)$ under $n=2000:1000:8000$, with the log-log error plots   displayed in Figure \\ref{fig4}. Firstly, the experimental curves are well aligned with the dashed line that represents the optimal $n^{-1\/2}$ rate. Then, comparing the results for $\\Delta = 0,0.5,1$, we conclude that quantization only affects the multiplicative factor $\\mathscr{L}$ in the estimation error. It should also be noted that, increasing either $d$ or $r$ leads to significantly larger error, which is consistent with the $\\mathscr{B}_4$'s essential  dependence on $d$ and $r$. \n\n\n\n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[scale = 0.72]{Thm_10_11.eps}\n    \n ~~~   (a) \\hspace{4.4cm} (b) \n    \\caption{(a): QMC in Theorem \\ref{thm8}; (b): QMC in Theorem \\ref{thm9}.}\n    \\label{fig4}\n\\end{figure}\n\n\\subsection{Importance of appropriate dithering}\nTo demonstrate the crucial role played by the suitable dither, we   provide the second set of simulations. In order to observe more significant phenomena and then conclude evidently, we may test huge sample size but a rather simple estimation problem under coarse quantization (i.e., large $\\Delta$). \n\n\nSpecifically, for covariance matrix estimation we set $d=1$ and i.i.d. draw $X_1,...,X_n$ from $\\mathcal{N}(0,1)$. Thus, the problem boils down to estimating $\\mathbbm{E}|X_k|^2$, for which the estimators in Theorems \\ref{thm1}-\\ref{thm2} coincide. Since $X_k$ is sub-Gaussian, we do not perform data truncation before dithered quantization. Besides our estimator $\\bm{\\widehat{\\Sigma}}=\\frac{1}{n}\\sum_{k=1}^n\\dot{X}_k^2-\\frac{\\Delta^2}{4}$ where $\\dot{X}_k=\\mathcal{Q}_\\Delta(X_k+\\tau_k)$ and $\\tau_k$ is triangular dither, we invite the following competitors: \n\\begin{itemize}\n    \\item $\\bm{\\widehat{\\Sigma}}_{no}=\\frac{1}{n}\\sum_{k=1}^n(\\dot{X}'_k)^2$, where $\\dot{X}'_k=\\mathcal{Q}_\\Delta(X_k)$ is the direct quantization without dithering;\n    \\item $\\bm{\\widehat{\\Sigma}}_u-\\frac{\\Delta^2}{6}$ and $\\bm{\\widehat{\\Sigma}}_u$, where $\\bm{\\widehat{\\Sigma}}_u=\\frac{1}{n}\\sum_{k=1}^n(\\dot{X}''_k)^2$, and $\\dot{X}''_k=\\mathcal{Q}_\\Delta(X_k+\\tau''_k)$ is quantized under uniform dither $\\tau''_k\\sim \\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}])$.\n\\end{itemize}   To illustrate the choice of $\\bm{\\widehat{\\Sigma}}_u-\\frac{\\Delta^2}{6}$ and $\\bm{\\widehat{\\Sigma}}_u$, we write $\\dot{X}''_k=X_k+\\tau''_k+w_k=X_k+\\xi_k$ with quantization error $w_k\\sim \\mathscr{U}([-\\frac{\\Delta}{2},\\frac{\\Delta}{2}])$ (due to Lemma \\ref{lem1}(a)) and quantization noise $\\xi_k=\\tau''_k+w_k$, then (\\ref{3.1})  gives $\\mathbbm{E}(\\dot{X}''_k)^2=\\mathbbm{E}|X_k|^2 +\\mathbbm{E}|\\xi_k|^2$, while $\\mathbbm{E}|\\xi_k|^2$ remains unknown. Thus, we consider $\\bm{\\widehat{\\Sigma}}_u-\\frac{\\Delta^2}{6}$    because of an unjustified  guess $\\mathbbm{E}|\\xi_k|^2\\approx \\mathbbm{E}|\\tau''_k|^2+\\mathbbm{E}|w_k|^2=\\frac{\\Delta^2}{6}$, while   $\\bm{\\widehat{\\Sigma}}_u$ simply gives up the correction of $\\mathbbm{E}|\\xi_k|^2$. We test $\\Delta =3$ under $n=(2:2:20)\\cdot10^3$. From the results shown in Figure \\ref{fig5}(a), the proposed estimator based on quantized data under triangular dither embraces the lowest estimation errors and the optimal rate of $n^{-1\/2}$, whereas other competitors are not consistent, i.e., they all reach some error floors under a large sample size.\n\n\n\nFor the two remaining signal recovery problems, we simply focus on the quantization of the measurement $y_k$. In particular, we simulate QCS in the setting of Theorem \\ref{thm4}, with $(d,s,\\Delta)=(50,3,2)$ under $n:=(2:2:20)\\cdot 10^3$. Other experimental details are as previously stated. We compare our estimator $\\bm{\\widehat{\\theta}}$ with its counterpart  $\\bm{\\widehat{\\theta}}'$ defined by (\\ref{4.3}) with the same $\\bm{Q},\\mathcal{S}$ but $\\bm{b}'=\\frac{1}{n}\\sum_{k=1}^n\\dot{y}'_k\\bm{x}_k$, where $\\dot{y}_k' = \\mathcal{Q}_\\Delta(\\widetilde{y}_k)$ is a direct uniform quantization with no dither. Evidently, the simulation results   in Figure \\ref{fig5}(b) confirm that the application of a uniform dither significantly lessens the recovery errors. Without dithering, although  our results under Gaussian covariate still exhibit $n^{-1\/2}$ decreasing rate,  identifiability issue does arise under Bernoulli covariate. In that case, the simulation without dithering will evidently deviate from the $n^{-1\/2}$ rate, see  \\cite[Figure 1]{sun2022quantized} for instance.\n\n\nIn analogy, we simulate QMC   (Theorem \\ref{thm8}) with data generated as previous experiments, and specifically we try $(d,r,\\Delta)= (30,5,1.5)$ under $n=(5:5:25)\\cdot 10^3$. While our estimator $\\bm{\\widehat{\\Theta}}$ is defined in (\\ref{5.2}) involving $\\dot{y}_k$ from a dithered quantizer, we simulate the performance of its counterpart without dithering, i.e., $\\bm{\\widehat{\\Theta}}'$ defined in (\\ref{5.2}) with $\\dot{y}_k$ substituted by $\\dot{y}_k'=\\mathcal{Q}_\\Delta(y_k)$. From the experimental results displayed in Figure \\ref{fig5}(c), one shall clearly see that $\\bm{\\widehat{\\Theta}}$ performs much better in terms of the   decreasing rate of $n^{-1\/2}$ and the estimation error; while the curve without dithering even does not decrease.\n\n\n\n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[scale = 0.72]{importance.eps}\n    \n ~~~   (a) \\hspace{4.6cm} (b) \\hspace{4.6cm} (c) \n    \\caption{(a): covariance matrix estimation; (b): QCS in Theorem \\ref{thm4}; (c): QMC in Theorem \\ref{thm8}.}\n    \\label{fig5}\n\\end{figure}\n  \n\\section{Final remarks}\\label{sec7}\nIn digital signal processing and many distributed or federated machine learning problems, data quantization is an indispensable process. On the other hand,  many modern datasets exhibit   heavy tail behaviour, and the past decade has witnessed an increasing interest in statistic method robust to heavy-tailedness.\nIn this work we study the quantization of heavy-tailed data.\nThe proposed scheme is to   truncate the heavy-tailed data prior to  a   uniform quantizer with random dither well suited to the problem. Applying our quantization scheme to covariance   estimation, compressed sensing, and matrix completion, we have proposed (near) optimal estimators from quantized data. Indeed, the quantization only affects the multiplicative factor of theoretical error bounds, which we also confirmed via numerical simulations. We believe that our approach can be applied to more statistical estimation  problems with   heavy-tailed data that need to be quantized. \n\n\nWe also presented more developments for quantized compressed sensing in two respects. Firstly, we study a novel setting   that involves covariate quantization.   Following the strategy of regularized M-estimator, the key is the covariance estimation from quantized data. Still, the resulting recovery program is non-convex, but we proved that all local minimizers enjoy near minimax   rate. At a higher level, this development extends a line of works on non-convex M-estimator \\cite{loh2011high,loh2013regularized,loh2017statistical} to accommodate heavy-tailed covariate, see the deterministic framework Proposition \\ref{framework}. As application, we   derive results for (dithered) 1-bit compressed sensing as byproducts. Secondly,  we established uniform recovery guarantee by more in-depth concentration inequality and covering argument. Notably, with a single realization of $(\\bm{x}_k,\\epsilon_k,\\tau_k)$, all sparse signals within an $\\ell_2$ ball can be uniformly recovered from the quantized data.   \n\n\n \n\n\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{introduction}\n\\hf\\hf Enumeration of singular curves in $\\mathbb{P}^2$ (complex projective space) is a classical \nproblem in algebraic geometry. In our first paper \\cite{BM13}, we used purely topological \nmethods to answer the following question: \n\\begin{que}\nHow many degree $d$-curves are there in $\\mathbb{P}^2$, passing through $(d(d+3)\/2 -k)$ generic points and having  one singularity \nof codimension $k$, where $k$ is at most $7$?  \n\\end{que} \nIn this paper, we extend the methods applied in \\cite{BM13} to enumerate curves with two singular points. More precisely, we obtain an \nexplicit answer for the following question: \n\\begin{que}\nHow many degree $d$-curves are there in $\\mathbb{P}^2$, passing through $(d(d+3)\/2 -(k+1))$ generic points, having one node and one singularity \nof codimension $k$, where $k$ is at most $6$?  \n\\end{que} \n\\hf\\hf Let us denote the space of curves of degree $d$ in $\\mathbb{P}^2$ by $\\mathcal{D}$. It follows that $\\mathcal{D} \\cong \\mathbb{P}^{\\delta_d}$, where $\\delta_d = d(d+3)\/2$. Let $\\gamma_{_{\\mathbb{P}^2}}\\longrightarrow \\mathbb{P}^2$ be the tautological line bundle. \nA homogeneous polynomial $f$, of degree $d$ and in $3$ variables, induces a holomorphic section of the line bundle $\\gamma_{_{\\mathbb{P}^2}}^{*d} \\longrightarrow \\mathbb{P}^2$. \nIf $f$ is non-zero, then we will denote its \\textit{equivalence class} in $\\mathcal{D}$ by $\\tilde{f}$. Similarly, if $p$ is a non-zero vector in $\\mathbb{C}^3$, \nwe will denote its equivalence class in $\\mathbb{P}^2$ by $\\tilde{p}$ \\footnote{In this paper we will use the symbol $\\tilde{A}$ to denote the equivalence class of \n$A$ instead of the standard $[A]$. This will make some of the calculations in section \\ref{closure_of_spaces} easier to read.}.     \n\\begin{defn}\n\\label{singularity_defn}\nLet $\\tilde{f} \\in \\mathcal{D}$ and $\\tilde{p} \\in \\mathbb{P}^2$. A point $\\tilde{p} \\in f^{-1}(0)$ \\textsf{is of singularity type} $\\mathcal{A}_k$,\n$\\mathcal{D}_k$, $\\mathcal{E}_6$, $\\mathcal{E}_7$, $\\mathcal{E}_8$ or $\\mathcal{X}_8$ if there exists a coordinate system\n$(x,y) :(\\mathcal{U},\\tilde{p}) \\longrightarrow (\\mathbb{C}^2,0)$ such that $f^{-1}(0) \\cap \\mathcal{U}$ is given by \n\\begin{align*}\n\\mathcal{A}_k: y^2 + x^{k+1}   &=0  \\qquad k \\geq 0, \\qquad \\mathcal{D}_k: y^2 x + x^{k-1} =0  \\qquad k \\geq 4, \\\\\n\\mathcal{E}_6: y^3+x^4 &=0,  \\qquad \\mathcal{E}_7: y^3+ y x^3=0, \n\\qquad \\mathcal{E}_8: y^3 + x^5=0, \\\\\n\\mathcal{X}_8: x^4 + y^4 &=0.   \n\\end{align*}\n\\end{defn}\nIn more common terminology, $\\tilde{p}$ is a {\\it smooth} point of $f^{-1}(0)$ if \nit is a singularity of type $\\mathcal{A}_0$; a {\\it simple node} if its singularity type is $\\mathcal{A}_1$; a {\\it cusp} if its type is $\\mathcal{A}_2$; a {\\it tacnode} \nif its type is $\\mathcal{A}_3$; a {\\it triple point} if its type is $\\mathcal{D}_4$; and a {\\it quadruple point} if its type is $\\mathcal{X}_8$. \\\\\n\\hf\\hf We have several results (cf. Theorem \\ref{algoa1a1}\\,-\\,\\ref{algope6a1}, section \\ref{algorithm_for_numbers})  \nwhich can be summarized collectively as our main result. Although \\eqref{algoa1a1}-\\eqref{algope6a1} may appear as equalities, \nthe content of each of these equations is a theorem.\n\\begin{mthm}\n\\label{main_result}\nLet $\\mathfrak{X}_k$ be a singularity of type $\\mathcal{A}_k$, $\\mathcal{D}_k$ or $\\mathcal{E}_k$. Denote $\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k,n)$ to be the number of \ndegree $d$ curves in $\\mathbb{P}^2$ that pass through $\\delta_d - (k+1+n)$ generic points having  \none $\\mathcal{A}_1$ node and  one singularity of type $\\mathfrak{X}_k$ at the intersection of $n$ generic lines. \\\\\n\\textup{(i)} There is a formula for $\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k,n)$ if $k+1 \\leq 7$, provided  $d \\geq \\mathcal{C}_{\\mathcal{A}_1\\mathfrak{X}_k}$ where \n\\bgd\n\\mathcal{C}_{\\mathcal{A}_1\\mathcal{A}_k} = k+3, ~~\\mathcal{C}_{\\mathcal{A}_1\\mathcal{D}_k} = k+1, ~~\\mathcal{C}_{\\mathcal{A}_1\\mathcal{E}_6} = 6. \n\\edd\n\\textup{(ii)} There is an algorithm to explicitly compute these numbers.     \n\\end{mthm}\n\n\\begin{rem}\nNote that $\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k,n)$ is zero if $n>2$, since three or more generic lines do not intersect at a common point. Moreover, $\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k,2)$ is the the number of curves, of degree $d$, that pass through $\\delta_d-(k+3)$ generic points having one $\\mathcal{A}_1$-node and one singularity of type $\\mathfrak{X}_k$ lying at a given fixed point (since the intersection of two generic lines is a point).\n\\end{rem}\n\n\nIn \\cite{BM13}, we obtained an explicit formula for \n$\\mathcal{N}(\\mathfrak{X}_k, n)$, the number of degree $d$ \ncurves in $\\mathbb{P}^2$ that pass through $\\delta_d - (k+n)$ generic points having  \none singularity of type $\\mathfrak{X}_k$ at the intersection of $n$ generic lines.\nWe extend the methods applied in \\cite{BM13} to obtain an explicit \nformula for $\\mathcal{N}(\\mathcal{A}_1 \\mathfrak{X}_k, n)$. \\\\[0.1cm]\n\n\\hf \\hf The numbers $\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k,0)$ till \n$k+1\\leq 7$ have also been computed by Maxim Kazarian \\cite{Kaz}   \nusing different methods. Our results for $n=0$ \nagree with his. The bound $d \\geq \\mathcal{C}_{\\mathcal{A}_1\\mathfrak{X}_k}$ is imposed to ensure that the \nrelevant bundle sections are transverse.$\\footnote{However, this bound is not the optimal bound.}$ \nThe formulas for $\\mathcal{N}(\\mathcal{A}_1\\mathcal{A}_1,n)$ and $\\mathcal{N}(\\mathcal{A}_1\\mathcal{A}_2,n)$ also appear in \\cite{Z1}. \nWe extend the methods applied by the author to obtain the remaining formulas. \n\n\n\n\n\\section{Overview} \n\\hf\\hf Our main tool will be the following well known fact from topology (cf. \\cite{BoTu}, Proposition 12.8).\n\\begin{thm} \n\\label{Main_Theorem} \nLet $V\\longrightarrow X$ be a vector bundle over a manifold $X$. Then the following are true: \n\\hspace*{0.5cm}\\textup{(1)} A generic smooth section $s: X\\longrightarrow V$ is transverse to the zero set. \\\\\n\\hspace*{0.5cm}\\textup{(2)} Furthermore, if $V$ and $X$ are oriented with $X$ compact then the zero set of such a section defines an integer homology class  in $X$, \nwhose Poincar\\'{e} dual is the Euler class of $V$. \nIn particular, if the rank of $V$ is same as the dimension of $X$, \nthen the signed cardinality of $s^{-1}(0)$ is the Euler class of $V$, evaluated on the fundamental class of \n$X$, i.e., \n\\bgd\n|\\pm s^{-1}(0)| = \\langle e(V), [X] \\rangle. \n\\edd\n\\end{thm}\n\\begin{rem}\nLet $X$ be a compact, complex manifold, $V$ a holomorphic vector bundle and $s$ a holomorphic section that is transverse to the zero set. If the rank of $V$ is same as the dimension of $X$, \nthen the signed cardinality of $s^{-1}(0)$ is same as its actual cardinality (provided $X$ and $V$ have their natural orientations). \n\\end{rem}\nHowever, for our purposes, the requirement that $X$ is a smooth manifold is too strong. We will typically be dealing with spaces that are \nsmooth but have non-smooth closure. The following result is a stronger  version of Theorem \\ref{Main_Theorem}, that applies to singular spaces, provided \nthe set of singular points is of real codimension two or more.   \n\\begin{thm} \n\\label{Main_Theorem_pseudo_cycle} \nLet $M \\subset \\mathbb{P}^{N}$ be a smooth, compact algebraic variety and $X \\subset M$ a smooth subvariety, not necessarily closed. Let $V \\longrightarrow M$ be an oriented vector bundle, such that the rank of $V$ is same as the dimension of $X$. Then the following are true: \\\\\n\\hspace*{0.5cm}\\textup{(1)} The closure of $X$ is an algebraic variety and defines a homology class.\\\\\n\\hspace*{0.5cm}\\textup{(2)} The zero set of a generic smooth section $s: M \\longrightarrow V$ intersects $X$ transversely and does not intersect $\\overline{X}-X$ anywhere. \\\\\n\\hspace*{0.5cm}\\textup{(3)} The number of zeros of such a section inside $X$, counted with signs, \nis the Euler class of $V$ evaluated on the homology class $[\\overline{X}]$, i.e., \n\\bgd\n|\\pm s^{-1}(0) \\cap \\overline{X}| = |\\pm s^{-1}(0) \\cap X| = \\big\\langle e(V), ~[\\overline{X}] \\big\\rangle.\n\\edd\n\\end{thm}\n\n\\begin{rem}\nAll the subsequent statements we make are true provided $d$ is sufficiently large. The precise bound on $d$ is given in section \\ref{bundle_sections}. Although the results of this paper are an extension of \\cite{BM13}, our aim has been to keep this paper self-contained. Ideally, a reader not familar with \\cite{BM13} should have no difficulty following this paper.\n\\end{rem}\n\n\n\\hf\\hf We will now explain our strategy to compute  $\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k, n)$. \nThe strategy is very similar to that of computing $\\mathcal{N}(\\mathfrak{X}_k, n)$, which \nwas the content of \\cite{BM13}. Let $\\mathrm{X}_1$ and  $\\mathrm{X}_2$ be two subsets of $\\mathcal{D} \\times \\mathbb{P}^2$. Then we define \n\\begin{align*}\n\\mathrm{X}_1 \\circ \\mathrm{X}_2 &:= \\{ (\\tilde{f}, \\tilde{p}_1, \\tilde{p}_2) \\in  \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P}^2: (\\tilde{f}, \\tilde{p}_1) \\in \\mathrm{X}_1, ~~(\\tilde{f}, \\tilde{p}_2) \\in \\mathrm{X}_2, ~~\\tilde{p}_1 \\neq \\tilde{p}_2 \\}. \n\\end{align*}\nNext, given a subset $\\mathrm{X}$ of $\\mathcal{D} \\times \\mathbb{P}^2$ we define \n\\begin{align*}\n\\Delta \\mathrm{X} &:= \\{ (\\tilde{f}, \\tilde{p}, \\tilde{p}) \\in  \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P}^2: (\\tilde{f}, \\tilde{p}) \\in \\mathrm{X} \\}. \n\\end{align*}\nSimilarly, let $\\mathrm{X}_1$ and  $\\mathrm{X}_2$ be two subsets of $\\mathcal{D} \\times \\mathbb{P}^2$ and $\\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2$ respectively.\nThen we define \n\\begin{align*}\n\\mathrm{X}_1 \\circ  \\mathrm{X}_2 &:= \\{ (\\tilde{f}, \\tilde{p}_1, l_{\\tilde{p}_2}) \\in  \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2: (\\tilde{f}, \\tilde{p}_1) \\in \\mathrm{X}_1, ~~(\\tilde{f}, l_{\\tilde{p}_2}) \\in \\mathrm{X}_2, ~~\\tilde{p}_1 \\neq \\tilde{p}_2 \\}. \n\\end{align*}\nFinally, given a subset $\\mathrm{X}$ of $\\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2$ we define \n\\begin{align*}\n\\Delta \\mathrm{X} &:= \\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in  \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T \\mathbb{P}^2: (\\tilde{f}, l_{\\tilde{p}}) \\in \\mathrm{X} \\}. \n\\end{align*}\n\\noindent The following result is clear from the definition of closure.\n\\begin{lmm}\n\\label{closure_obvious}\nWe have the following equality of sets \n\\begin{align*}\n \\overline{\\mathrm{X}_1 \\circ \\mathrm{X}_2} & = \\overline{\\overline{\\mathrm{X}}_1 \\circ \\mathrm{X}_2} = \\overline{\\mathrm{X}_1 \\circ \\overline{\\mathrm{X}}_2} = \\overline{\\overline{\\mathrm{X}}_1 \\circ \\overline{\\mathrm{X}}_2}. \n\\end{align*}\n\\end{lmm}\n\\hf\\hf Given a singularity $\\mathfrak{X}_k$, we also denote by $\\mathfrak{X}_k$, the \\textit{space} of \ncurves of degree $d$ with a marked point $\\tilde{p}$ such that the curve has a singularity of type $\\mathfrak{X}_k$ at $\\tilde{p}$. \nSimilarly, $\\mathcal{A}_1\\circ \\mathfrak{X}_k$ is the the \\textit{space} of degree $d$ curves with two distinct marked points $\\tilde{p}_1$ and $\\tilde{p}_2$ such that the curve has a node at $\\tilde{p}_1$ and a singularity of type $\\mathfrak{X}_k$ at $\\tilde{p}_2$. Note that except when $\\mathfrak{X}_k=\\mathcal{A}_1$, the space $\\mathcal{A}_1\\circ\\mathfrak{X}_k$ {\\it is} the fibre product $\\mathcal{A}_1\\times_\\mathcal{D} \\mathfrak{X}_k$.\\\\\n\\hf\\hf Let $\\tilde{p}_1, \\tilde{p}_2, \\ldots , \\tilde{p}_{\\delta_d -(k+1+n)}$ be \n$\\delta_d-(k+1+n)$ generic points in $\\mathbb{P}^2$ \nand $\\mathrm{L}_1, \\mathrm{L}_2, \\ldots, \\mathrm{L}_n $ be $n$ generic lines \nin $\\mathbb{P}^2$. Define the following sets\n\\begin{align}\n\\label{hyperplane} \n\\Hpl_i& := \\{ \\tilde{f} \\in \\mathcal{D}: f(p_i)=0 \\}, \\qquad \\Hpl_i^* := \\{ \\tilde{f} \\in \\mathcal{D}: f(p_i)=0,  \\nabla f|_{p_i} \\neq 0 \\},  \\nonumber \\\\\n\\hat{\\Hpl}_i& := \\Hpl_i \\times \\mathbb{P}^2 \\times \\mathbb{P}^2, \\qquad \\hat{\\Hpl}_i^* := \\Hpl_i^* \\times \\mathbb{P}^2 \\times \\mathbb{P}^2 \\qquad \\textnormal{and}  \\qquad \\hat{\\mathrm{L}}_i := \\mathcal{D} \\times\\mathbb{P}^2 \\times \\mathrm{L}_i.  \n\\end{align}\nBy definition, our desired number $\\mathcal{N}(\\mathcal{A}_1 \\mathfrak{X}_k,n)$ is the cardinality of \nthe set\n\\bge\n\\label{number_Xk_defn}\n\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k,n) := |\\mathcal{A}_1 \\circ \\mathfrak{X}_k \\cap \\hat{\\Hpl}_1 \\cap \\ldots \\cap \\hat{\\Hpl}_{\\delta_d-(n+1+k)} \\cap \n\\hat{\\mathrm{L}}_1 \\cap \\ldots \\cap \\hat{\\mathrm{L}}_n|.\n\\ede\nLet us now clarify an important point to avoid confusion: as per our notation, \nthe number $\\mathcal{N}(\\mathcal{A}_1 \\mathcal{A}_1,0)$ \nis the number of degree $d$ curves through $\\delta_d-2$ generic points having \ntwo \\textit{ordered} nodes. To find the the corresponding number of curves where the \nnodes are \\textit{unordered}, we have to divide by $2$.\\\\\n\\hf\\hf We will now describe the various steps involved to obtain an explicit formula for $\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k,n)$. \n\n\\begin{step} \nOur first observation is that if $d$ is sufficiently large then $\\mathcal{A}_1 \\circ \\mathfrak{X}_k$ is a smooth algebraic variety and its closure defines a homology class.\n\\begin{lmm}{\\bf (cf. section \\ref{bundle_sections})}\nThe space $\\mathcal{A}_1 \\circ \\mathfrak{X}_k$ is a smooth subvariety of $\\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P}^2$ of dimension $\\delta_d-k$. \n\\end{lmm}\n\\end{step}\n\n\\begin{step}\nNext we observe that if the points and lines are chosen generically, \nthen the corresponding hyperplanes and lines defined in \\eqref{hyperplane} will intersect our space \n$\\mathcal{A}_1 \\circ \\mathfrak{X}_k$ transversely. Moreover, they would not intersect any extra points in the closure. \n\\begin{lmm}\n\\label{gpl2}\nLet $\\tilde{p}_1, \\tilde{p}_2, \\ldots , \\tilde{p}_{\\delta_d -(k+1+n)}$ be \n$\\delta_d-(k+1+n)$ generic points in $\\mathbb{P}^2$ \nand $\\mathrm{L}_1, \\mathrm{L}_2, \\ldots, \\mathrm{L}_n $ be $n$ generic lines \nin $\\mathbb{P}^2$. Let $\\hat{\\Hpl}_i$, $\\hat{\\Hpl}_i^*$  and $\\hat{\\mathrm{L}}_i$ be as defined in \n\\eqref{hyperplane}.  \nThen \n\\bgd\n\\overline{\\mathcal{A}_1 \\circ \\mathfrak{X}}_k \\cap \\hat{\\Hpl}_1 \\cap \\ldots \\cap \\hat{\\Hpl}_{\\delta_d -(k+n+1)} \n\\cap \\hat{\\mathrm{L}}_1 \\cap \\ldots \\cap \\hat{\\mathrm{L}}_n = \n\\mathcal{A}_1 \\circ \\mathfrak{X}_k \\cap \\hat{\\Hpl}_1^* \\cap \\ldots \\cap \\hat{\\Hpl}_{\\delta_d -(k+n+1)}^* \n\\cap \\hat{\\mathrm{L}}_1 \\cap \\ldots \\cap \\hat{\\mathrm{L}}_n  \n\\edd\nand every intersection is transverse. \n\\end{lmm}\nWe omit the details of the proof; it follows from an application of the families transversality theorem and Bertini's theorem. \nThe details of this proof  can be found in \\cite{BM_Detail}. \n \n\\begin{notn}\n\\label{tau_bundle_defn}\nLet $\\gamma_{\\mathcal{D}}\\longrightarrow \\mathcal{D}$ and $\\gamma_{_{\\mathbb{P}^2}}\\longrightarrow\\mathbb{P}^2$ denote the tautological line bundles. If $c_1(V)$ denotes the first Chern class of a vector bundle then we set \n\\bgd\ny : = c_1(\\gamma_{\\mathcal{D}}^*) \\in H^{2}(\\mathcal{D}; \\mathbb{Z}), \\qquad \na := c_1(\\gamma_{_{\\mathbb{P}^2}}^*)  \\in H^{2}(\\mathbb{P}^2; \\mathbb{Z}). \n\\edd\n\\end{notn}\n  \n\\hf\\hf As a consequence of Lemma \\ref{gpl2}  \nwe obtain the following fact: \n\\begin{lmm}\n\\label{gpl}\nThe number $\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k,n)$ is given by~  \n$$ \\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k, n) =\n\\big\\langle (\\pi_{\\mathcal{D}}^*y)^{\\delta_d-(n+k+1)} (\\pi_2^*a)^n, ~[\\overline{\\mathcal{A}_1 \\circ \\mathfrak{X}}_k] \\big\\rangle$$\n where ~$\\pi_{\\mathcal{D}}, \\pi_1, \\pi_2: \\mathcal{D} \\times \\mathbb{P}_1^2 \\times \\mathbb{P}^2_2 \n \\longrightarrow \\mathcal{D}, \\mathbb{P}^2_1, \\mathbb{P}^2_2 $\n are the projection maps.  \n\\end{lmm}\n\\end{step}\n\\noindent \\textbf{Proof:  } This follows from  Theorem \\ref{Main_Theorem_pseudo_cycle} and Lemma \\ref{gpl2}. \\qed \\\\\n\n\\noindent As explained in \\cite{BM13}, \nthe space $\\mathfrak{X}_k$ is not easy to describe directly and hence \ncomputing $\\mathcal{N}(\\mathfrak{X}_k,n)$ \\textit{directly} is not easy. As a result we define another space $\\mathcal{P} \\mathfrak{X}_k \\subset \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2.$ This is the space of curves $\\tilde{f}$, of degree $d$, with a marked point $\\tilde{p} \\in \\mathbb{P}^2$ and a marked direction $l_{\\p} \\in \\mathbb{P} T_{\\tilde{p}}\\mathbb{P}^2$, such that the curve $f$ has a singularity of type $\\mathfrak{X}_k$ at $\\tilde{p}$ and certain directional derivatives \\textit{vanish along $l_{\\p}$}, and certain other derivatives \\textit{do not vanish}. To take a simple example, $\\mathcal{P} \\mathcal{A}_2$ is the space of curves $\\tilde{f}$ with a marked point $\\tilde{p}$ and a marked direction $l_{\\p}$ such that $f$ has an $\\mathcal{A}_2$-node at $\\tilde{p}$ and the Hessian is degenerate along $l_{\\p}$, but the third derivative along $l_{\\p}$ is non-zero. It turns out that this space is much easier to describe. We have defined $\\mathcal{P} \\mathfrak{X}_k$ in section \\ref{definition_of_px}. Similarly, instead of dealing with the space $\\mathcal{A}_1 \\circ \\mathfrak{X}_k$, we deal with the space $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathfrak{X}_k$. \n\n\\begin{step}\nNext we observe that since $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathfrak{X}_k$ is described locally as the vanishing of certain sections that are transverse to the zero set, these are \\text{smooth} algebraic varieties.\n\n\\begin{lmm}{\\bf (cf. section \\ref{bundle_sections})}\n\\label{pr_sp_pseudo}\nThe space \n$\\mathcal{A}_1 \\circ \\mathcal{P} \\mathfrak{X}_k$ is a smooth subvariety \nof $\\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2$ of dimension $\\delta_d-(k+1)$. \n\\end{lmm}\n\n\\begin{notn}\n\\label{tau_bundle_pv_defn}\nLet $\\tilde{\\gamma} \\longrightarrow \\mathbb{P} T\\mathbb{P}^2$ be the tautological line bundle. The first Chern class of the dual will be denoted by $\\lambda := c_1(\\tilde{\\gamma}^*)\\in H^{2}(\\mathbb{P} T\\mathbb{P}^2; \\mathbb{Z})$.   \n\\end{notn}\n\\noindent Lemma \\ref{pr_sp_pseudo} now motivates the following definition: \n \\begin{defn}\n\\label{up_number_defn}\n We define the number \n$\\mathcal{N}(\\mathcal{A}_1 \\circ \\mathcal{P} \\mathfrak{X}_k,n,m)$ as \n\\bge\n\\label{num_proj}\n\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathfrak{X}_k, n,m) := \\big\\langle \\pi_{\\mathcal{D}}^*y^{\\delta_d-(k+n+m+1)} \\pi_2^*a^n \\pi_2^*\\lambda^m, ~[\\overline{\\mathcal{A}_1 \\circ \\mathcal{P} \\mathfrak{X}}_k] \\big\\rangle. \n\\ede\nwhere ~$\\pi_{\\mathcal{D}}, \\pi_1, \\pi_2: \\mathcal{D} \\times \\mathbb{P}_1^2 \\times \\mathbb{P} T\\mathbb{P}^2_2 \\longrightarrow \\mathcal{D}, \\mathbb{P}^2_1, \\mathbb{P} T\\mathbb{P}^2_2 $ are the projection maps. \n\\end{defn}\n\n\\noindent The next Lemma relates the numbers $\\mathcal{N}(\\mathcal{A}_1\\mathcal{P}\\mathfrak{X}_k,n,0)$ and \n$\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k,n)$. \n\\begin{lmm}\n\\label{up_to_down}\nThe projection map $ \\pi: \\mathcal{A}_1 \\circ \\mathcal{P} \\mathfrak{X}_k \\longrightarrow \\mathcal{A}_1 \\circ \\mathfrak{X}_k $ is one to one if $\\mathfrak{X}_k = \\mathcal{A}_k, \\mathcal{D}_k, \\mathcal{E}_6, \\mathcal{E}_7$ or $\\mathcal{E}_8$ except for $\\mathfrak{X}_k = \\mathcal{D}_4$ when it is three to one. In particular, \n\\bge\n\\label{up_down_equation}\n\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k, n)  = \\mathcal{N}(\\mathcal{A}_1 \\mathcal{P} \\mathfrak{X}_k, n,0) \n\\qquad \\textnormal{if} ~~\\mathfrak{X}_k \\neq \\mathcal{D}_4   \\qquad \\textnormal{and} \n\\qquad \\mathcal{N}(\\mathcal{A}_1 \\mathcal{D}_4, n) = \n\\frac{\\mathcal{N}(\\mathcal{A}_1 \\mathcal{P} \\mathcal{D}_4,n,0)}{3}.\n\\ede\n\\end{lmm}\n\\noindent \\textbf{Proof:  } This is identical to the proof of the corresponding lemma in \\cite{BM13}.\\qed\n\\end{step}\n\\noindent To summarize, the \\textit{definition} of $\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k,n)$ is \\eqref{number_Xk_defn}. \nLemma \\ref{gpl} equates this number to a \\textit{topological} computation. \nWe then introduce another number $\\mathcal{N}(\\mathcal{A}_1\\mathcal{P}\\mathfrak{X}_k,n,m)$ in Definition \\ref{up_number_defn} and relate it to \n$\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k,n) $ in Lemma \\ref{up_to_down}. \nIn other words, we do not compute $\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k,n)$ \\textit{directly}; \nwe compute it \\textit{indirectly} by first computing $\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathfrak{X}_k, n,m)$ and then using Lemma \\ref{up_to_down}. \\\\\n\\hf\\hf We now give a brief idea of how to compute these numbers. \nSuppose we want to compute $\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathfrak{X}_k, n, m)$. \nWe first find some singularity $\\mathfrak{X}_l$ for which \n$\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathfrak{X}_l, n,m)$ has been calculated \nand which contains $\\mathfrak{X}_k$ in its closure, i.e., we want $\\mathcal{P} \\mathfrak{X}_k$ to be a subset of $\\overline{\\mathcal{P} \\mathfrak{X}}_{l}$. \nUsually, $l=k-1$ but it is not necessary. Our next task is to \ndescribe the closure of $ \\mathcal{P} \\mathfrak{X}_l$ and   $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathfrak{X}_{l}$ explicitly as\n\\begin{align}\n\\overline{\\mathcal{P} \\mathfrak{X}}_{l}&=  \\mathcal{P} \\mathfrak{X}_l \\sqcup \\overline{\\mathcal{P} \\mathfrak{X}}_k \\cup \\mathcal{B}_1  \\qquad \\textnormal{and} \\label{stratification_general_one_point}  \\\\\n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathfrak{X}}_{l} & = \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathfrak{X}_l \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathfrak{X}}_l- \\mathcal{P} \\mathfrak{X}_l) \\sqcup \\big( \\Delta \\mathcal{B}_2 \\big)  \\nonumber \\\\ \n& = \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathfrak{X}_l \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathfrak{X}}_k \\cup \\mathcal{B}_1) \\sqcup \\big( \\Delta \\mathcal{B}_2 \\big), \\qquad \n\\textnormal{where} \\label{stratification_general} \\\\\n\\Delta\\mathcal{B}_2 & := \\{ (\\tilde{f}, \\tilde{p}_1, l_{\\tilde{p}_2}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathfrak{X}}_l: \\tilde{p}_1 = \\tilde{p}_2 \\}. \\nonumber \n\\end{align}\nNote that $\\overline{\\mathcal{A}_1 \\circ \\mathcal{P} \\mathfrak{X}_{l}} = \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathfrak{X}_l}$.\nThe main content of \\cite{BM13} was to express $\\overline{\\mathcal{P} \\mathfrak{X}}_l$ as in \\eqref{stratification_general_one_point}. \nThe main content of this paper is to \ncompute $\\Delta \\mathcal{B}_2$, i.e    \nexpressing $\\overline{\\mathcal{A}_1 \\circ \\mathcal{P} \\mathfrak{X}}_l$ as in \\eqref{stratification_general}.\nConcretely, computing $\\Delta \\mathcal{B}_2$ means figuring out what happens to a $\\mathfrak{X}_k$ singularity, when it collides with an $\\mathcal{A}_1$-node. \nAs a simple example, when two nodes collide, we get an \n\n\\begin{figure}[h!]\n\\vspace*{0.2cm}\n\\begin{center}\\includegraphics[scale = 0.5]{collide1.eps}\\vspace*{-0.2cm}\\end{center}\n\\caption{Two nodes colliding into a tacnode}\n\\end{figure}\n\n$\\mathcal{A}_3$-node (which is basically the content of Lemma \\ref{cl_two_pt}, statement \\ref{a1a1_up_cl}). \nBy definition \\ref{up_number_defn} and Theorem \\ref{Main_Theorem_pseudo_cycle} \n\\bgd\n\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathfrak{X}_k, n, m) := \\big\\langle e(\\mathbb{W}_{n,m,k}^{1}), ~[\\overline{\\mathcal{A}_1 \\circ \\mathcal{P} \\mathfrak{X}}_k] \\big\\rangle = \\big|\\pm \\mathcal{Q}^{-1}(0) \\cap \\mathcal{A}_1 \\circ \\mathcal{P} \\mathfrak{X}_k\\big|, \n\\edd\nwhere\n\\bge\n\\label{generic_Q}  \n\\mathcal{Q} :\\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2  \\longrightarrow \\mathbb{W}_{n,m,k}^{\\delta}:= \\bigg({\\textstyle \\bigoplus}_{i=1}^{\\delta_d -(n+m+k+\\delta)}\\pi_{\\mathcal{D}}^*\\gamma_{\\mathcal{D}}^*\\bigg)\\oplus\\bigg({\\textstyle \\bigoplus}_{i=1}^{n} \n\\pi_1^*\\gamma_{_{\\mathbb{P}^2}}^*\\bigg)\\oplus\\bigg({\\textstyle \\bigoplus}_{i=1}^{m}\\pi_2^*\\tilde{\\gamma}^*  \\bigg)\n\\ede\nis a generic smooth section. \nIn \\cite{BM13}, we constructed a section $\\Psi_{\\mathcal{P} \\mathfrak{X}_k}$ \nof an appropriate vector bundle\n\\bgd\n\\mathbb{V}_{\\mathcal{P} \\mathfrak{X}_k}\\longrightarrow \\overline{\\mathcal{P} \\mathfrak{X}}_{l} = \\mathcal{P} \\mathfrak{X}_{l} \\cup \\overline{\\mathcal{P}\\mathfrak{X}}_k \\cup \\mathcal{B}_1\n\\edd\nwith the following properties:  the section ~$ \\Psi_{\\mathcal{P} \\mathfrak{X}_k}: \\overline{\\mathcal{P} \\mathfrak{X}}_k \\longrightarrow  \\mathbb{V}_{\\mathcal{P} \\mathfrak{X}_k}$\ndoes not vanish on $\\mathcal{P} \\mathfrak{X}_{l}$ and it vanishes \\textit{transversely} on $\\mathcal{P} \\mathfrak{X}_k$. \nWith a similar reasoning one can show that the induced section \n$$ \\pi_2^*\\Psi_{\\mathcal{P} \\mathfrak{X}_k}: \\overline{\\mathcal{A}_1 \\circ \\mathcal{P} \\mathfrak{X}}_l \\longrightarrow  \\pi_2^*\\mathbb{V}_{\\mathcal{P} \\mathfrak{X}_k} $$\nvanishes \\textit{transversely} on $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathfrak{X}_k$.\\footnote{However the bound on $d$ for which transversality \nis achieved increases.} \nHere $\\pi_2$ is the following projection map \n$$ \\pi_2: \\mathcal{D}\\times \\mathbb{P}_1^2 \\times \\mathbb{P} T\\mathbb{P}^2_2 \\longrightarrow \\mathcal{D} \\times \\mathbb{P} T \\mathbb{P}^2_2.$$\nSince $\\Psi_{\\mathcal{P} \\mathfrak{X}_k}$ does not vanish on  \n$\\mathcal{P} \\mathfrak{X}_{l}$, the section $ \\pi_2^*\\Psi_{\\mathcal{P} \\mathfrak{X}_k}$ does not \nvanish on $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathfrak{X}_l$. \nTherefore, \n\\begin{align}\n\\Big\\langle e(\\pi_2^*\\mathbb{V}_{\\mathcal{P} \\mathfrak{X}_{k}} \\oplus \\mathbb{W}_{n,m,k}^1), ~\\big[\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathfrak{X}}_{l}\\big] \\Big\\rangle \n& = \\mathcal{N}(\\mathcal{A}_1 \\mathcal{P} \\mathfrak{X}_k,n,m) + \\mathcal{C}_{\\mathcal{A}_1 \\circ \\mathcal{B}_1}(\\pi_2^*\\Psi_{\\mathcal{P} \\mathfrak{X}_k} \\oplus \\mathcal{Q}) \\nonumber \\\\  \n& \\,\\,+ \\mathcal{C}_{\\Delta \\mathcal{B}_2}(\\pi_2^*\\Psi_{\\mathcal{P} \\mathfrak{X}_k} \\oplus \\mathcal{Q}) \\label{Euler_equal_number_plus_bdry}\n\\end{align}\nwhere $\\mathcal{C}_{\\mathcal{A}_1 \\circ \\mathcal{B}_1}(\\pi_2^*\\Psi_{\\mathcal{P} \\mathfrak{X}_k} \\oplus \\mathcal{Q})$ and  $\\mathcal{C}_{\\Delta \\mathcal{B}_2}(\\pi_2^*\\Psi_{\\mathcal{P} \\mathfrak{X}_k} \\oplus \\mathcal{Q})$ \nare the contributions of the section\n$\\pi_2^*\\Psi_{\\mathcal{P} \\mathfrak{X}_k} \\oplus \\mathcal{Q}$ to the Euler class from the points of $\\mathcal{A}_1 \\circ \\mathcal{B}_1$ and $\\Delta \\mathcal{B}_2$ \nrespectively.  \nThe number $\\mathcal{C}_{\\mathcal{A}_1 \\circ \\mathcal{B}_1}(\\pi_2^*\\Psi_{\\mathcal{P} \\mathfrak{X}_k} \\oplus \\mathcal{Q})$ was computed \nin \\cite{BM13}. The main content of this paper is to compute $\\mathcal{C}_{\\Delta \\mathcal{B}_2}(\\pi_2^*\\Psi_{\\mathcal{P} \\mathfrak{X}_k} \\oplus \\mathcal{Q})$.   \nOnce we have computed these numbers, we observe that the left hand side of \\eqref{Euler_equal_number_plus_bdry} \nis computable via splitting principle and the \nfact that $\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathfrak{X}_l, n,m)$ is known. Therefore, we get a recursive formula for $\\mathcal{N}(\\mathcal{A}_1 \\mathcal{P} \\mathfrak{X}_k,n,m)$ in terms of $\\mathcal{N}(\\mathcal{A}_1 \\mathcal{P} \\mathfrak{X}_l,n^{\\prime},m^{\\prime})$ and $\\mathcal{N}(\\mathcal{P} \\mathfrak{X}_{k+1}, n, m)$. The main result of  \n\\cite{BM13} was to find an explicit formula for $\\mathcal{N}(\\mathcal{P} \\mathfrak{X}_{k+1}, n, m)$. Using this and iterations, \nwe get an explicit formula for $\\mathcal{N}(\\mathcal{A}_1 \\mathcal{P} \\mathfrak{X}_k,n,m)$. \nFinally, using \nLemma \\ref{up_down_equation}, we get our desired numbers $\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k, n)$.\n\\begin{eg}\nSuppose we wish to compute $\\mathcal{N}(\\mathcal{A}_1\\mathcal{A}_5, n)$. \nThis can be deduced from the knowledge of $\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{A}_5, n,m)$. \nThe obvious singularities which have $\\mathcal{A}_5$-nodes in its closure are $\\mathcal{A}_4$-nodes. \nIn order to analyze the space $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_4$, \nwe infer (cf. Lemma \\ref{cl_two_pt}, statement \\ref{a1_pa5_cl}) that\n\\bgd\n\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}}_4 = \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}_4 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{A}}_4- \\mathcal{P} \\mathcal{A}_4) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_6 \\cup \\Delta \\overline{\\PP \\D_7^{s}} \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{E}}_6   \\Big). \n\\edd\nBy \\cite{BM13} (cf. Lemma \\ref{cl}, statement \\ref{A3cl} we conclude that \n\\bgd\n\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}}_4 = \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}_4 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{A}}_5 \\cup \\overline{\\mathcal{P} \\mathcal{D}}_5 ) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_6 \\cup \\Delta \\overline{\\PP \\D_7^{s}} \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{E}}_6   \\Big). \n\\edd\nThe corresponding line bundle $\\UL_{\\mathcal{P} \\mathcal{A}_5} \\longrightarrow \\overline{\\mathcal{P} \\mathcal{A}}_4$ with a \nsection $\\Psi_{\\mathcal{P} \\mathcal{A}_5}$ that does not vanish on $\\mathcal{P} \\mathcal{A}_4$ and vanishes transversely on \n$\\mathcal{P} \\mathcal{A}_5$ is defined in section \\ref{summary_vector_bundle_definitions}. \nIn section \\ref{bundle_sections}, we indicate that \n\\[ \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_5}: \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_4 \\longrightarrow \\pi_2^* \\UL_{\\mathcal{P} \\mathcal{A}_5} \\] \nvanishes \non $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{A}_5$ transversely. \nLet $\\mathcal{Q}$ be a generic section of the vector  bundle \n\\bgd\n\\mathbb{W}_{n,m,5}^1 \\longrightarrow \\mathcal{D} \\times\\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2.\n\\edd\nBy \\cite{BM13} \n$\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_5} \\oplus \\mathcal{Q} $ vanishes on all points of $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{D}_5$ with a multiplicity \nof $2$. By Corollary \\ref{a1_pak_mult_is_2_Hess_neq_0} and \\ref{a1_pa4_mult_is_5_around_pe6},\n$\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_5} \\oplus \\mathcal{Q} $ vanishes on all the points of $\\Delta \\mathcal{P} \\mathcal{A}_6$ and $\\Delta \\mathcal{P} \\mathcal{E}_6$ \nwith a multiplicity of $2$ and $5$ respectively. \nFurthermore, we also show that $\\Delta \\overline{\\PP \\D_7^{s}}$ is contained inside $\\Delta \\overline{\\mathcal{P} \\mathcal{D}}_7$. \nSince the dimension of $\\Delta \\mathcal{P} \\mathcal{D}_7$ is one less than the rank of $\\pi_2^* \\UL_{\\mathcal{P} \\mathcal{A}_5} \\oplus \\mathbb{W}_{n,m,5}^1$ \nand \n$\\mathcal{Q}$ is generic, $\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_5} \\oplus \\mathcal{Q} $ does \nnot vanish on $\\Delta \\overline{\\mathcal{P} \\mathcal{D}}_7$. Hence, it does not vanish on  $\\Delta \\overline{\\PP \\D_7^{s}}$. \nTherefore, we conclude that\n\\begin{align}\n\\big\\langle e(\\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_5} \\oplus \\mathbb{W}_{n,m,5}^1), ~~[\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_4] \\big\\rangle =  & \\,\\,\\mathcal{N}(\\mathcal{A}_1\\mathcal{P}\\mathcal{A}_5,n,m)\\nonumber\\\\\n& +2 \\mathcal{N}(\\mathcal{P} \\mathcal{D}_5, n, m)+ 2 \\mathcal{N}(\\mathcal{P} \\mathcal{A}_6, n, m)+ 5\\mathcal{N}(\\mathcal{P} \\mathcal{E}_6, n, m).  \\label{sample_computation}\n\\end{align}\nThis gives us a recursive formula for $\\mathcal{N}(\\mathcal{A}_1 \\mathcal{P} \\mathcal{A}_5, n, m)$ \nin terms of $\\mathcal{N}(\\mathcal{A}_1 \\mathcal{P} \\mathcal{A}_4, n^{\\prime}, m^{\\prime})$, $\\mathcal{N}(\\mathcal{P} \\mathcal{A}_6, n, m)$, \n$\\mathcal{N}(\\mathcal{P} \\mathcal{D}_5, n, m)$ and $\\mathcal{N}(\\mathcal{P} \\mathcal{E}_6, n, m)$, \nwhich is \\eqref{algopa4a1} in our algorithm. \n\\end{eg}\n\n\\begin{rem}\nWe remind the reader that $\\mathcal{N}(\\mathcal{P}\\mathfrak{X}_k, n,m)$ has been defined in \\cite{BM13}.\nThe definition \nis analogous to the definition of  \n$\\mathcal{N}(\\mathcal{A}_1 \\mathcal{P} \\mathfrak{X}_k, n, m)$ as given in definition \\ref{up_number_defn} in this paper. \n\\end{rem}\n\n\n\\hf\\hf Now we describe the basic organization of our paper. In section \\ref{algorithm_for_numbers} \nwe state the explicit algorithm to obtain the numbers $\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k, n)$ in our MAIN THEOREM. \nIn section \\ref{summary_notation_def} we recapitulate all the spaces, vector bundles \nand sections of vector bundles we encountered in the process of enumerating curves with one singular point. \nIn section \\ref{bundle_sections} we introduce \na few new notation needed for this paper and \nwrite down the relevant sections that are transverse to the zero set. \nThe proof of why the sections are transverse to the zero set can be found in \\cite{BM_Detail}. \nIn section \\ref{closure_of_spaces} we stratify the space $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathfrak{X}}_k$ as described in \\eqref{stratification_general}. Along the way we also compute the \\textit{order} to which a certain section vanishes around certain points (i.e., the contribution of the section to the Euler class of a bundle). Finally, using the splitting principal, in section \\ref{Euler_class_computation} we compute the \nEuler class of the relevant bundles and obtain the recursive formula similar to \\eqref{sample_computation} above. \n\n\\begin{ack}\\nonumber \n\\textup{One of the crucial results of this paper is to compute the closure of relevant spaces, i.e.,  \nto stratify the space $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathfrak{X}}_k$ as described in \\eqref{stratification_general}. A key step here is to observe that certain sections are transverse to the zero set and utilize them to describe the neighborhood of a point. The second author is indebted to Aleksey Zinger for sharing his understanding of transversality and explaining this crucial idea to him (i.e., how to  describe the neighborhood of a point using transversality of bundle sections). In addition, the second author is also grateful to Aleksey Zinger for suggesting several non trivial low degree checks to verify our formulas. One of those low degree checks proved to be crucial in figuring out a mistake the second author had made earlier.  \\\\ \n\\hf \\hf The authors are grateful to Dennis Sullivan for sharing his perspective on this problem and indicating its connection to other areas of mathematics.}\n\\end{ack}\n\n\n\n\n\n\n\n\n\n\n\\section{Algorithm}  \n\\label{algorithm_for_numbers}\n\\hf\\hf We now give an algorithm to compute the numbers $\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k,n)$. \nEquations \\eqref{algoa1a1}-\\eqref{algope6a1} are recursive formulas for $\\mathcal{N}(\\mathcal{A}_1\\mathcal{P}\\mathfrak{X}_k,n,m)$ in terms of \n$\\mathcal{N}(\\mathcal{A}_1 \\mathcal{P} \\mathfrak{X}_{k-1}, n^{\\prime}, m^{\\prime})$ and  \n$\\mathcal{N}(\\mathcal{P}\\mathfrak{X}_{k+1},n, m)$. In \\cite{BM13} we had obtained\nan explicit formula for $\\mathcal{N}(\\mathcal{P}\\mathfrak{X}_{k+1},n, m)$. Finally, using \nLemma \\ref{up_down_equation}, we get our desired numbers $\\mathcal{N}(\\mathcal{A}_1\\mathfrak{X}_k, n)$. We have implemented this algorithm in a Mathematica program to obtain the final answers. The program is available on our web page \\url{https:\/\/www.sites.google.com\/site\/ritwik371\/home}. We prove the formulas in section \\ref{Euler_class_computation}.  \\\\\n\\hf \\hf First we note that \nusing the ring structure of $H^*(\\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2; \\mathbb{Z})$, it is easy to see that \nfor every singularity  type $\\mathfrak{X}_k$ we have\n\\begin{align}\n\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathfrak{X}_k,n,m)\n& = -3\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathfrak{X}_k,n+1,m-1)\n-3\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathfrak{X}_k,n+2,m-2)   \\qquad\\forall ~~m\\ge2. \\label{ringp}\n\\end{align}\nWe now give recursive formulas for $\\mathcal{N}(\\mathcal{A}_1\\mathcal{A}_1, n)$ \nand  $\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathfrak{X}_k, n, m)$:    \n\\begin{eqnarray}\n\\mathcal{N}(\\mathcal{A}_1\\mathcal{A}_1, n) &=& \\mathcal{N}(\\mathcal{A}_1,0) \\times \\mathcal{N}(\\mathcal{A}_1, n)   \\nonumber \\\\ \n                  & & -\\big( \\mathcal{N}(\\mathcal{A}_1, n) + d \\mathcal{N}(\\mathcal{A}_1, n+1)  + 3 \\mathcal{N}(\\mathcal{A}_2, n)\\big)  \\label{algoa1a1} \\\\\n\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{A}_2, n, 0) & = & 2 \\mathcal{N} (\\mathcal{A}_1\\mathcal{A}_1,n) + 2(d-3) \\mathcal{N}(\\mathcal{A}_1\\mathcal{A}_1,n+1)  \\nonumber  \\\\ \n                         & & - 2\\mathcal{N}(\\mathcal{P} \\mathcal{A}_3, n, 0)\\label{algopa20a1}\\\\\n\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{A}_2, n, 1) & = & \\mathcal{N}(\\mathcal{A}_1\\mathcal{A}_1,n) + (2d-9) \\mathcal{N}(\\mathcal{A}_1\\mathcal{A}_1,n+1) + (d^2-9d+18) \\mathcal{N}(\\mathcal{A}_1\\mathcal{A}_1,n+2) \\nonumber \\\\ \n                     & & - 2\\mathcal{N}(\\mathcal{P} \\mathcal{A}_3, n, 1) - 3 \\mathcal{N}(\\mathcal{D}_4,n)  \\label{algopa21a1}\\\\\n\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{A}_3, n, m) & = & \\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{A}_2, n, m ) +  3\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{A}_2, n, m+1) + d\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{A}_2, n+1, m) \\nonumber \\\\ \n                         & &   -2\\mathcal{N}(\\mathcal{P} \\mathcal{A}_4,n,m)   \\label{algopa3a1}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\mathcal{N}( \\mathcal{A}_1 \\mathcal{P} \\mathcal{A}_4, n, m) & = &  2\\mathcal{N}( \\mathcal{A}_1\\mathcal{P} \\mathcal{A}_3, n, m) + 2\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{A}_3, n, m+1) + (2d-6)\\mathcal{N}( \\mathcal{A}_1\\mathcal{P} \\mathcal{A}_3, n+1, m) \\nonumber \\\\ \n                           &   &  -2\\mathcal{N}(\\mathcal{P} \\mathcal{A}_5, n,m) \\label{algopa4a1} \\\\\n\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{A}_5, n, m) & = & 3\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{A}_4, n, m) + \\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{A}_4, n, m+1) + (3d -12)\\mathcal{N}( \\mathcal{A}_1\\mathcal{P} \\mathcal{A}_4, n+1, m)\\nonumber\\\\\n&     & -2\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{D}_5, n, m) -2\\mathcal{N}(\\mathcal{P} \\mathcal{A}_6, n, m)- 5\\mathcal{N}(\\mathcal{P} \\mathcal{E}_6, n, m) \\label{algopa5a1}\\\\ \n\\mathcal{N}( \\mathcal{A}_1\\mathcal{P}\\mathcal{A}_6, n, m) & = & 4\\mathcal{N}(\\mathcal{A}_1 \\mathcal{P}\\mathcal{A}_5, n, m) +0\\mathcal{N}(\\mathcal{P} \\mathcal{A}_1\\mathcal{A}_5, n, m+1) +  (4d -18)\\mathcal{N}(\\mathcal{P} \\mathcal{A}_1\\mathcal{A}_5, n+1, m) \\nonumber \\\\   \n              &     & -4\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{D}_6, n, m) - 3\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{E}_6, n, m)\\nonumber\\\\\n              &     & -2\\mathcal{N}(\\mathcal{P} \\mathcal{A}_7, n, m) -6\\mathcal{N}(\\mathcal{P} \\mathcal{E}_7, n, m)  \\label{algopa6a1}\\\\\n\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{D}_4, n, 0) & = & \\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{A}_3, n, 0) -2\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{A}_3, n, 1) + (d-6)\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{A}_3, n+1, 0) \\nonumber \\\\\n                         &   & -2\\mathcal{N}(\\mathcal{D}_5,n) \\label{algopd4a1}   \\\\ \n\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{D}_4, n, 1) & = & \\mathcal{N}(\\mathcal{A}_1 \\mathcal{D}_4, n, 0) + (d-9)\\mathcal{N}(\\mathcal{A}_1 \\mathcal{D}_4, n+1, 0)  \\label{algopd4a1_lambda} \\\\\n                        \n\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{D}_5, n, m) & = & \\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{D}_4, n, m) + \\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{D}_4, n, m+1) + (d-3)\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{D}_4, n+1, m) \\nonumber \\\\ \n                         & & -2\\mathcal{N}(\\mathcal{P} \\mathcal{D}_6,n,m) \\label{algopd5a1} \\\\\n\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{D}_6, n, m) & = & \\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{D}_5, n, m) + 4\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{D}_5, n, m+1) + d\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{D}_5, n+1, m)  \\nonumber \\\\  \n                         & &   -2\\mathcal{N}(\\mathcal{P} \\mathcal{D}_7, n, m) - \\mathcal{N}(\\mathcal{P} \\mathcal{E}_7,n,m) \\label{algopd6a1} \\\\\n\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{E}_6, n, m) & = & \\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{D}_5, n, m) -\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{D}_5, n, m+1) + (d-6)\\mathcal{N}(\\mathcal{A}_1\\mathcal{P} \\mathcal{D}_5, n+1, m) \\nonumber  \\\\ \n                         & &   -\\mathcal{N}(\\mathcal{P} \\mathcal{E}_7, n,m) \\label{algope6a1} \n\\end{eqnarray}\n\n\n\\section{Review of definitions and notations for one singular point} \n\\label{summary_notation_def}\n\\hf\\hf We recall a few definitions and notation from \\cite{BM13} so that our paper is self-contained.  \n\\subsection{The vector bundles involved}\\label{summary_vector_bundle_definitions}\n\\hf\\hf The first three of the vector bundles we will encounter, the tautological line bundles, have been defined in notations \\ref{tau_bundle_defn} and \\ref{tau_bundle_pv_defn}. \nLet $\\pi:\\mathcal{D}\\times \\mathbb{P} T\\mathbb{P}^2\\longrightarrow \\mathcal{D}\\times\\mathbb{P}^2$ be the projection map. \n\\begin{rem}\n\\label{an1_again}\nWe will make the abuse of notation of usually omitting the pullback maps \n$\\pi_{\\mathcal{D}}^*$ and $\\pi_{\\mathbb{P}^2}^*$.   \nOur intended meaning should be clear when we say, for instance, $\\gamma_{\\mathcal{D}}^* \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2$. \nHowever, we will not omit to write the pullback via $\\pi^*$. \n\\end{rem}\nWe have the following bundles over $\\mathcal{D}\\times\\mathbb{P}^2$ :\n\\begin{eqnarray*}\n\\mathcal{L}_{\\mathcal{A}_0} &:= & \\gamma_{\\mathcal{D}}^*\\otimes \\gamma_{_{\\mathbb{P}^2}}^{*d} \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2 \\\\\n\\mathcal{V}_{\\mathcal{A}_1} &:= & \\gamma_{\\mathcal{D}}^*\\otimes \\gamma_{_{\\mathbb{P}^2}}^{*d} \\otimes T^*\\mathbb{P}^2 \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2 \\\\\n\\mathcal{L}_{\\mathcal{A}_2} &:= & (\\gamma_{\\mathcal{D}}^* \\otimes \\gamma_{_{\\mathbb{P}^2}}^{*d} \\otimes \\Lambda^2 T^*\\mathbb{P}^2)^{\\otimes 2} \n\\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2 \\\\ \n\\mathcal{V}_{\\mathcal{D}_4} &:= & \\gamma_{\\mathcal{D}}^*\\otimes \\gamma_{_{\\mathbb{P}^2}}^{*d} \\otimes \n\\textnormal{Sym}^2 (T^*\\mathbb{P}^2 \\otimes T^*\\mathbb{P}^2) \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2 \\\\\n\\mathcal{V}_{\\mathcal{X}_8} &:= &  \\gamma_{\\mathcal{D}}^*\\otimes \\gamma_{_{\\mathbb{P}^2}}^{*d} \\otimes \n\\textnormal{Sym}^3 (T^*\\mathbb{P}^2 \\otimes T^*\\mathbb{P}^2 \\otimes T^*\\mathbb{P}^2) \n\\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2 \n\\end{eqnarray*}\nAssociated to the map $\\pi$ there are pullback bundles \n\\begin{eqnarray*}\n\\UL_{\\hat{\\A}_0} &:= &\\pi^* \\mathcal{L}_{\\mathcal{A}_0} \\longrightarrow \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\\\\n\\mathbb{V}_{\\hat{\\A}_1} &:= & \\pi^{*} \\mathcal{V}_{\\mathcal{A}_1} \\longrightarrow \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\\\\n\\mathbb{V}_{\\hat{\\D}_4} &:= & \\pi^{*} \\mathcal{V}_{\\mathcal{D}_4} \\longrightarrow \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\\\\n\\mathbb{V}_{ \\hat{\\XC}_8} &:= & \\pi^{*} \\mathcal{V}_{\\mathcal{X}_8} \\longrightarrow \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\\\\n\\mathbb{V}_{\\mathcal{P} \\mathcal{A}_2} &:= & \\tilde{\\gamma}^*\\otimes \\gamma_{\\mathcal{D}}^*\\otimes \\gamma_{_{\\mathbb{P}^2}}^{*d} \\otimes \\pi^* T^*\\mathbb{P}^2 \n\\longrightarrow  \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\\\\n\\mathbb{V}_{\\mathcal{P} \\mathcal{D}_5} &:= & \\tilde{\\gamma}^{*2}\\otimes \\gamma_{\\mathcal{D}}^*\\otimes \\gamma_{_{\\mathbb{P}^2}}^{*d} \\otimes \\pi^* T^*\\mathbb{P}^2 \n\\longrightarrow  \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2.\n\\end{eqnarray*}\nFinally, we have \n\\begin{align*}\n\\UL_{\\mathcal{P} \\mathcal{D}_4} &:= (T\\mathbb{P}^2\/\\tilde{\\gamma})^{*2} \\otimes \\gamma_{\\mathcal{D}}^* \\otimes \\gamma_{_{\\mathbb{P}^2}}^{*d} \n \\longrightarrow \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2  \\\\ \n \\UL_{\\mathcal{P} \\mathcal{D}_5} &:= \\tilde{\\gamma}^{*2} \\otimes (T\\mathbb{P}^2\/\\tilde{\\gamma})^* \\otimes \n \\gamma_{\\mathcal{D}}^* \\otimes \\gamma_{_{\\mathbb{P}^2}}^{*d} \\longrightarrow \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\\\\n\\UL_{\\mathcal{P} \\mathcal{D}_5^{\\vee}} &:= \\tilde{\\gamma}^{*2} \\otimes (T\\mathbb{P}^2\/\\tilde{\\gamma})^{*4} \\otimes \n \\gamma_{\\mathcal{D}}^{*2} \\otimes \\gamma_{_{\\mathbb{P}^2}}^{*2d} \\longrightarrow \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\\\\n \\UL_{\\mathcal{P} \\mathcal{D}_6^{\\vee}} &:= \\tilde{\\gamma}^{*8} \\otimes (T\\mathbb{P}^2\/\\tilde{\\gamma})^{*4} \\otimes \n \\gamma_{\\mathcal{D}}^{* 5} \\otimes \\gamma_{_{\\mathbb{P}^2}}^{*5d} \\longrightarrow \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\\\\n\\UL_{\\mathcal{P} \\mathcal{E}_6} &:= \\tilde{\\gamma}^{*} \\otimes (T\\mathbb{P}^2\/\\tilde{\\gamma})^{*2} \\otimes \n \\gamma_{\\mathcal{D}}^* \\otimes \\gamma_{_{\\mathbb{P}^2}}^{*d} \\longrightarrow \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2\\\\\n\\UL_{\\mathcal{P} \\mathcal{E}_7} &:=  \\tilde{\\gamma}^{*4} \\otimes \\gamma_{\\mathcal{D}}^* \\otimes \\gamma_{_{\\mathbb{P}^2}}^{*d}\n\\longrightarrow \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\\\\n\\UL_{\\mathcal{P} \\mathcal{E}_8} &:=  \\tilde{\\gamma}^{*3} \\otimes (T\\mathbb{P}^2\/\\tilde{\\gamma})^* \\otimes  \\gamma_{\\mathcal{D}}^* \\otimes \\gamma_{_{\\mathbb{P}^2}}^{*d}\n\\longrightarrow \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\\\\n\\UL_{\\mathcal{P} \\mathcal{X}_8} &:= (T\\mathbb{P}^2\/\\tilde{\\gamma})^{*3} \\otimes \\gamma_{\\mathcal{D}}^* \\otimes \\gamma_{_{\\mathbb{P}^2}}^{*d} \n\\longrightarrow \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\\\\n\\UL_{\\mathcal{J}} &:= \\tilde{\\gamma}^{*9}\\otimes(T\\mathbb{P}^2\/\\tilde{\\gamma})^{*3}\\otimes \\gamma_{\\mathcal{D}}^{*3} \\otimes \\gamma_{_{\\mathbb{P}^2}}^{*d} \n\\longrightarrow \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\\\\nk \\geq 3 \\qquad \\UL_{\\mathcal{P} \\mathcal{A}_k} &:= \\tilde{\\gamma}^{*k} \\otimes (T\\mathbb{P}^2\/\\tilde{\\gamma})^{*(2k-6)} \\otimes \\gamma_{\\mathcal{D}}^{*(k-2)} \\otimes \\gamma_{_{\\mathbb{P}^2}}^{*(d(k+1)-3d)}  \n\\longrightarrow \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\\\  \nk \\geq 6 \\qquad \\UL_{\\mathcal{P} \\mathcal{D}_k} & := \\tilde{\\gamma}^{*(k-2 + \\epsilon_k)} \\otimes (T\\mathbb{P}^2\/\\tilde{\\gamma})^{*(2\\epsilon_k)} \\otimes \\gamma_{\\mathcal{D}}^{*(1+\\epsilon_k)} \\otimes \\gamma_{_{\\mathbb{P}^2}}^{*(d(1+ \\epsilon_k))} \\longrightarrow \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2,\n\\end{align*}\nwhere $\\epsilon_6 =0$, $\\epsilon_7=1$ and $\\epsilon_8 =3$. \n\nThe reason for defining these bundles will become clearer in section \\ref{summary_sections_of_vector_bundle_definitions}, when we define \nsections of these bundles.\\\\\n\\hf \\hf With the abuse of notation as explained in Remark \\ref{an1_again}, the bundle $T\\mathbb{P}^2\/\\tilde{\\gamma}$ is the quotient of the bundles $V$ and $W$, \nwhere $V$ is the pullback of the tangent bundle $T\\mathbb{P}^2\\to \\mathbb{P}^2$ via $\\mathcal{D}\\times\\mathbb{P} T\\mathbb{P}^2\\stackrel{\\pi}{\\rightarrow}\\mathcal{D}\\times \\mathbb{P}^2\\to \\mathbb{P}^2$ and $W$ \nis pullback of $\\tilde{\\gamma}\\to \\mathbb{P} T \\mathbb{P}^2$ via $\\mathcal{D}\\times \\mathbb{P} T \\mathbb{P}^2\\to \\mathbb{P} T \\mathbb{P}^2$.\n\n\\subsection{Sections of Vector Bundles}\\label{summary_sections_of_vector_bundle_definitions}\n\\hf\\hf Let us recall the definition of \\textit{vertical derivative}. \n\\begin{defn}\n\\label{vertical_derivative_defn}\n\\noindent Let $\\pi:V\\longrightarrow M$ be a holomorphic vector bundle of rank $k$ and $s:M\\longrightarrow V$ be a holomorphic section. Suppose $h: V|_{\\mathcal{U}} \\longrightarrow \\mathcal{U}\\times \\mathbb{C}^{k}$ is a holomorphic trivialization of $V$ and $\\pi_{1}, \\pi_{2}: \\mathcal{U} \\times \\mathbb{C}^{k} \\longrightarrow \\mathcal{U}, \\mathbb{C}^{k}$ the projection maps.\nLet \n\\begin{align}\n\\label{section_local_coordinate}\n\\hat{s}&:= \\pi_{2} \\circ h \\circ s.\n\\end{align}\nFor $q\\in \\mathcal{U}$, we define the {\\it vertical derivative} of $s$ to be the $\\mathbb{C}$-linear map  \n\\begin{align*}\n\\nabla s|_{q}: T_{q}M \\longrightarrow V_{q}, \\qquad  \n\\nabla s|_{q} & := (\\pi_{2} \\circ h)|_{V_{q}}^{-1} \\circ d \\hat{s}|_q,\n\\end{align*}\nwhere $V_{q} = \\pi^{-1}(q)$, the fibre at $q$. In particular, if $v \\in T_q M$  \nis given by a \nholomorphic map $\\gamma: \\mathrm{B}_{\\epsilon}(0) \\longrightarrow M$ such that $\\gamma (0)=q$ and $\\frac{\\partial \\gamma}{\\partial z}\\big|_{z=0} = v$, then\n\\begin{align*} \n\\nabla s|_q (v) & := (\\pi_{2} \\circ h)|^{-1}_{V_{q}} \\circ \n\\frac{\\partial \\hat{s}(\\gamma(z))}{\\partial z}\\bigg|_{z=0}\n\\end{align*}\nwere $\\mathrm{B}_{\\epsilon}$ is an open $\\epsilon$-ball in $\\mathbb{C}$ around the origin.$\\footnote{Not every tangent vector is given by a holomorphic map; \nhowever combined with \nthe fact that $\\nabla s|_p $ is $\\mathbb{C}$-linear, this definition determines $\\nabla s|_p$ completely.} $ \nFinally, if $v, w \\in T_q M$  are tangent vectors such that  \nthere exists a family of complex curves  $\\gamma: \\mathrm{B}_\\epsilon\\times \\mathrm{B}_\\epsilon \\longrightarrow M$ such that \n\\bgd\n\\gamma (0,0) =q, \\qquad \n \\frac{\\partial \\gamma(x,y)}{\\partial x}\\bigg|_{(0,0)}= v, \\qquad \n\\qquad \\frac{\\partial \\gamma(x,y)}{\\partial y}\\bigg|_{(0,0)} = w\n\\edd\nthen  \n\\begin{align}\n\\label{verder}\n\\nabla^{i+j} s|_q \n(\\underbrace{v,\\cdots v}_{\\textnormal{$i$ times}}, \\underbrace{w,\\cdots w}_{\\textnormal{$j$ times}}) & :=  \n(\\pi_{2} \\circ h)\\mid^{-1}_{V_{q}} \\circ\\left[\n\\frac{ \\partial^{i+j} \\hat{s}(\\gamma(x,y))}{\\partial^i x \\partial^j y}\\right]\\bigg|_{(0,0)}.\n\\end{align}\n\\end{defn}\n\\begin{rem}\nIn general the quantity in \\eqref{verder} is not well defined, i.e., it depends on the \ntrivialization and the curve $\\gamma$. In \\cite{BM_Detail} we explain on what subspace \nthis quantity is well defined. \n\\end{rem}\n\\begin{rem}\n\\label{transverse_local}\nThe section $ s: M \\longrightarrow V $  is transverse to the zero set if and only if the induced map\n\\begin{align} \n\\label{section_local_coordinate_calculus}\n\\hatii{s} & := \\hat{s} \\circ \\varphi_{\\mathcal{U}}^{-1} : \\mathbb{C}^m \\longrightarrow \\mathbb{C}^k \n\\end{align}\nis transverse to the zero set in the usual calculus sense, where $\\varphi_{\\mathcal{U}}: \\mathcal{U} \\longrightarrow \\mathbb{C}^m $ is a coordinate chart and $\\hat{s}$ is as defined in \\eqref{section_local_coordinate}.\n\\end{rem}\n\\hf\\hf Let $f:\\mathbb{P}^2 \\longrightarrow \\gamma_{_{\\mathbb{P}^2}}^{*d}$ be a section and $\\tilde{p} \\in \\mathbb{P}^2$. \nWe can think of $p$ as a non-zero vector in $\\gamma_{_{\\mathbb{P}^2}}$ and $p^{\\otimes d}$ a non-zero vector \nin $\\gamma_{_{\\mathbb{P}^2}}^{\\otimes d}$ $\\footnote{Remember that $p$ is an element of $\\mathbb{C}^3-0$ while $\\tilde{p}$ is the corresponding equivalence class in $\\mathbb{P}^2$.}$. \nThe quantity $\\nabla f|_{\\tilde{p}}$ acts on a vector in  $\\gamma_{_{\\mathbb{P}^2}}^{d}|_{\\tilde{p}}$ and produces an element of $T^*_{\\tilde{p}}\\mathbb{P}^2$ . Let us denote this quantity as $\\nabla f|_p$, i.e., \n\\begin{align}\n\\nabla f|_p &:= \\{\\nabla f|_{\\tilde{p}}\\}(p^{\\otimes d}) \\in T^*_{\\tilde{p}}\\mathbb{P}^2.   \n\\end{align}\nNotice that  $\\nabla f|_{\\tilde{p}}$ is an element of the fibre of $T^*\\mathbb{P}^{2} \\otimes \\gamma_{_{\\mathbb{P}^2}}^{*d}$ at $\\tilde{p}$ while $\\nabla f|_{p}$ is an element of $T^*_{\\tilde{p}}\\mathbb{P}^{2}$. \\\\\n\\hf\\hf Now observe that $\\pi^{*} T\\mathbb{P}^2 \\cong \\tilde{\\gamma} \\oplus \\pi^*T\\mathbb{P}^2\/\\tilde{\\gamma} \\longrightarrow \\mathbb{P} T\\mathbb{P}^2$, where \n$\\pi: \\mathbb{P} T\\mathbb{P}^2 \\longrightarrow \\mathbb{P}^2~$ is the projection map. Let us denote a vector in $\\tilde{\\gamma}$ by $v$ and  a vector \nin $\\pi^*T \\mathbb{P}^2\/\\tilde{\\gamma}$ by $\\tilde{w}$. \nGiven $\\tilde{f} \\in \\mathcal{D}$ and $\\tilde{p} \\in \\mathbb{P}^2$, let \n\\begin{align}\n\\label{abbreviation}\nf_{ij} & := \\nabla^{i+j} f|_p \n(\\underbrace{v,\\cdots v}_{\\textnormal{$i$ times}}, \\underbrace{w,\\cdots w}_{\\textnormal{$j$ times}}).\n\\end{align}\nNote that $f_{ij}$ is a \\textit{number}. \nIn general $f_{ij}$ is not well defined; it depends on the trivialization and  the curve. \nMoreover it is also not well defined on the quotient space. \nSince our sections are not defined on the whole space, \nwe will use the notation $ s:M \\dashrightarrow V$ to indicate that $s$ is defined only on \na subspace of $M$. \\\\\n\\hf \\hf With this terminology, we now explicitly define the sections that we will \nencounter.\n\\begin{eqnarray*}\n\\psi_{\\mathcal{A}_0}: \\mathcal{D} \\times \\mathbb{P}^2 \\longrightarrow \\mathcal{L}_{\\mathcal{A}_0}, & & \\{\\psi_{\\mathcal{A}_0}(\\tilde{f}, \\tilde{p})\\} (f \\otimes p^{\\otimes d}):= f(p) \\\\\n\\psi_{\\mathcal{A}_1}: \\mathcal{D} \\times \\mathbb{P}^2 \\dashrightarrow \\mathcal{V}_{\\mathcal{A}_1}, & & \\{\\psi_{\\mathcal{A}_1}(\\tilde{f}, \\tilde{p})\\}(f \\otimes p^{\\otimes d}):= \\nabla f|_p  \\\\\n\\psi_{\\mathcal{D}_4}: \\mathcal{D} \\times \\mathbb{P}^2 \\dashrightarrow \\mathcal{V}_{\\mathcal{D}_4}, & & \\{\\psi_{\\mathcal{D}_4}(\\tilde{f}, \\tilde{p})\\}(f\\otimes p^{\\otimes d}):= \\nabla^2 f|_p  \\\\ \n\\psi_{\\mathcal{X}_8}: \\mathcal{D} \\times \\mathbb{P}^2 \\dashrightarrow \\mathcal{V}_{\\mathcal{X}_8}, & & \\{\\psi_{\\mathcal{X}_8}(\\tilde{f}, \\tilde{p})\\}(f \\otimes p^{\\otimes d}):= \\nabla^3 f|_p  \\\\ \n\\psi_{\\mathcal{A}_2} : \\mathcal{D} \\times \\mathbb{P}^2 \\dashrightarrow \\mathcal{L}_{\\mathcal{A}_2}, & & \\{\\psi_{\\mathcal{A}_2}(\\tilde{f}, \\tilde{p})\\}(f \\otimes p^{\\otimes d}):= \\textnormal{det}\\, \\nabla^2 f|_p \\\\\n\\Psi_{\\hat{\\A}_0}:  \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\dashrightarrow \\UL_{\\hat{\\A}_0}, & & \\Psi_{\\hat{\\A}_0}(\\tilde{f}, l_{\\p}):= \\psi_{\\mathcal{A}_0}(\\tilde{f},\\tilde{p})  \\\\\n\\Psi_{\\hat{\\A}_1}: \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\dashrightarrow \\mathbb{V}_{\\hat{\\A}_1}, & & \\Psi_{\\hat{\\A}_1}(\\tilde{f},l_{\\p}):= \\psi_{\\mathcal{A}_1}(\\tilde{f}, \\tilde{p}) \\\\\n\\Psi_{\\hat{\\D}_4}: \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\dashrightarrow \\mathbb{V}_{\\hat{\\D}_4}, & & \\Psi_{\\hat{\\D}_4}(\\tilde{f},l_{\\p}):= \\psi_{\\mathcal{D}_4}(\\tilde{f}, \\tilde{p}) \\\\\n\\Psi_{ \\hat{\\XC}_8}: \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\dashrightarrow \\mathbb{V}_{ \\hat{\\XC}_8}, & & \\Psi_{ \\hat{\\XC}_8}(\\tilde{f},l_{\\p}):= \\psi_{\\mathcal{X}_8}(\\tilde{f}, \\tilde{p}).\n\\end{eqnarray*}\nWe also have\n\\begin{eqnarray*}\n\\Psi_{\\mathcal{P} \\mathcal{A}_2}: \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\dashrightarrow \\mathbb{V}_{\\mathcal{P} \\mathcal{A}_2}, & & \\{\\Psi_{\\mathcal{P} \\mathcal{A}_2}(\\tilde{f},l_{\\p})\\}(f \\otimes p^{\\otimes d} \\otimes v):= \\nabla^2 f|_p (v,\\cdot) \\\\  \n\\Psi_{\\mathcal{P} \\mathcal{D}_5}^{\\mathbb{V}}: \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\dashrightarrow \\mathbb{V}_{\\mathcal{P} \\mathcal{D}_5}, & & \\{\\Psi_{\\mathcal{P} \\mathcal{D}_5}^{\\mathbb{V}}(\\tilde{f},l_{\\p})\\}(f \\otimes p^{\\otimes d} \\otimes v^{\\otimes 2}):= \\nabla^3 f|_p (v,v,\\cdot)  \\\\\n\\Psi_{\\mathcal{P} \\mathcal{D}_4}: \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\dashrightarrow \\UL_{\\mathcal{P} \\mathcal{D}_4}, & &  \\{\\Psi_{\\PP D_4}(\\tilde{f},l_{\\p})\\}(f \\otimes p^{\\otimes d} \\otimes w^{\\otimes 2}):= f_{02} \\\\\n\\Psi_{\\mathcal{P} \\mathcal{D}_5}^{\\UL}: \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\dashrightarrow \\UL_{\\mathcal{P} \\mathcal{D}_5}, & & \\{\\Psi_{\\mathcal{P} \\mathcal{D}_5}^{\\UL}(\\tilde{f},l_{\\p})\\}(f \\otimes p^{\\otimes d} \\otimes v^{\\otimes 2}\\otimes w):= f_{21} \\\\\n\\Psi_{\\mathcal{P} \\mathcal{E}_6}: \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\dashrightarrow \\UL_{\\mathcal{P} \\mathcal{E}_6}, & & \\{\\Psi_{\\mathcal{P} \\mathcal{E}_6}(\\tilde{f},l_{\\p})\\}(f \\otimes p^{\\otimes d} \\otimes v\\otimes w^{\\otimes 2}):= f_{12} \\\\ \n\\Psi_{\\mathcal{P} \\mathcal{E}_7}: \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\dashrightarrow \\UL_{\\mathcal{P} \\mathcal{E}_7}, & & \\{\\Psi_{\\mathcal{P} \\mathcal{E}_7}(\\tilde{f},l_{\\p})\\}(f \\otimes p^{\\otimes d} \\otimes v^{\\otimes 4}):= f_{40} \\\\\n\\Psi_{\\mathcal{P} \\mathcal{E}_8}: \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\dashrightarrow \\UL_{\\mathcal{P} \\mathcal{E}_8}, & & \\{\\Psi_{\\mathcal{P} \\mathcal{E}_8}(\\tilde{f},l_{\\p})\\}(f \\otimes p^{\\otimes d} \\otimes v^{\\otimes 3} \\otimes w ):= f_{31} \\\\\n\\Psi_{\\mathcal{P} \\mathcal{X}_8}: \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\dashrightarrow \\UL_{\\mathcal{P} \\mathcal{X}_8}, & & \\{\\Psi_{\\mathcal{P} \\mathcal{X}_8}(f,l_{\\p})\\}(f \\otimes p^{\\otimes d} \\otimes w^{\\otimes 3}):= f_{03}.\n\\end{eqnarray*}\nWe also have sections of the following bundles:  $\\Psi_{\\mathcal{P} \\mathcal{D}_5^{\\vee}}: \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\dashrightarrow \\UL_{\\mathcal{P} \\mathcal{D}_5^{\\vee}}$ given by\n\\bge\n\\label{psi_d5_dual}\n\\{\\Psi_{\\mathcal{P} \\mathcal{D}_5^{\\vee}}(\\tilde{f},l_{\\p})\\}( f^{\\otimes 2} \\otimes p^{\\otimes 2 d} \n\\otimes v^{\\otimes 2} \\otimes w^{\\otimes 4}) := 3 f_{12}^2 - 4 f_{21} f_{03}, \n\\ede\nand  $\\Psi_{\\mathcal{P} \\mathcal{D}_6^{\\vee}}: \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\dashrightarrow \\UL_{\\mathcal{P} \\mathcal{D}_6^{\\vee}}$ at $(\\tilde{f},l_{\\p})$ is given by\n\\bge\n\\label{psi_d6_dual}\n(f^{\\otimes 5} \\otimes p^{\\otimes 5 d} \n\\otimes v^{\\otimes 8} \\otimes w^{\\otimes 4})\\mapsto \\big(f_{12}^4 f_{40} - 8f_{12}^3 f_{21} f_{31} + 24 f_{12}^2 f_{21}^2 f_{22}-32 f_{12} f_{21}^3 f_{13} + 16 f_{21}^4 f_{04} \\big),\n\\ede\nand $\\Psi_{\\mathcal{J}}: \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\dashrightarrow \\UL_{\\mathcal{J}}$ given by\n\\bge\n\\label{J_psi}\n\\{\\Psi_{\\mathcal{J}}(\\tilde{f},l_{\\p})\\}( f^{\\otimes 3} \\otimes p^{\\otimes d} \\otimes v^{\\otimes 9} \\otimes w^{\\otimes 3}) := \\Big(- \\frac{f_{31}^3}{8 } + \\frac{3 f_{22} f_{31} f_{40}}{16 } - \\frac{f_{13} f_{40}^2}{16}  \\Big).\n\\ede\nWhen $k\\geq 3$ we have $\\Psi_{\\mathcal{P} \\mathcal{A}_k}: \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\dashrightarrow \\UL_{\\mathcal{P} \\mathcal{A}_k}$ given by\n\\bgd\n\\{\\Psi_{\\mathcal{P} \\mathcal{A}_k}(\\tilde{f},l_{\\p}) \\}\\big(f^{\\otimes (k-2)} \\otimes p^{\\otimes d} \\otimes v^{\\otimes k} \\otimes w^{\\otimes (2k-6)}\\big):= f_{02}^{k-3} \\mathcal{A}^f_k.\n\\edd\nSimilarly, when $k \\geq 6$ we have $\\Psi_{\\mathcal{P} \\mathcal{D}_k}: \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 \\dashrightarrow \\UL_{\\mathcal{P} \\mathcal{D}_k}$ given by\n\\bgd\n\\{\\Psi_{\\mathcal{P} \\mathcal{D}_k}(\\tilde{f},l_p)\\}\\big(f^{\\otimes (1+ \\epsilon_k)} \\otimes p^{\\otimes d(1+\\epsilon_k)} \\otimes v^{\\otimes (k-2+\\epsilon_k)}\\otimes w^{\\otimes (2\\epsilon_k)}\\big):= f_{12}^{\\epsilon_k} \\mathcal{D}^f_k,\n\\edd\nwhere, \n$\\epsilon_6 = 0$, $\\epsilon_7 =1$ and \n$\\epsilon_8=3$. The expressions for $\\mathcal{A}^f_k$ (resp. $\\mathcal{D}^f_k$) are given below explicitly in \n\\eqref{Formula_Ak} (resp. \\eqref{Formula_Dk}), till $k=7$ (resp. till $k=8$). \\\\\n\\hf\\hf Here is an explicit  formula for  \n$\\mathcal{A}^f_k$ till $k=7$:  \n\\begin{align}\n\\label{Formula_Ak}\n\\mathcal{A}^f_3&= f_{30},\\qquad\n\\mathcal{A}^f_4 = f_{40}-\\frac{3 f_{21}^2}{f_{02}}, \\qquad\n\\mathcal{A}^f_5= f_{50} -\\frac{10 f_{21} f_{31}}{f_{02}} + \n\\frac{15 f_{12} f_{21}^2}{f_{02}^2} \\nonumber \\\\\n\\mathcal{A}^f_6 &= f_{60}- \\frac{ 15 f_{21} f_{41}}{f_{02}}-\\frac{10 f_{31}^2}{f_{02}} + \\frac{60 f_{12} f_{21} f_{31}}{f_{02}^2}\n   +\n   \\frac{45 f_{21}^2 f_{22}}{f_{02}^2} - \\frac{15 f_{03} f_{21}^3}{f_{02}^3}\n   -\\frac{90 f_{12}^2 f_{21}^2}{f_{02}^3} \\nonumber \\\\  \n\\mathcal{A}^f_7 &= f_{70} - \\frac{21 f_{21} f_{51}}{f_{02}} \n- \\frac{35 f_{31} f_{41}}{f_{02}} + \\frac{105 f_{12} f_{21} f_{41}}{f_{02}^2} + \\frac{105 f_{21}^2 f_{32}}{f_{02}^2} + \n\\frac{70 f_{12} f_{31}^2}{f_{02}^2}+ \\frac{210 f_{21}f_{22}f_{31}}{f_{02}^2} \\nonumber \\\\\n&\n-\\frac{105 f_{03} f_{21}^2 f_{31}}{ f_{02}^3}\n-\\frac{420 f_{12}^2 f_{21} f_{31}}{f_{02}^3}\n-\\frac{630 f_{12}f_{21}^2 f_{22}}{f_{02}^3}\n-\\frac{105 f_{13} f_{21}^3}{f_{02}^3}\n+ \\frac{315 f_{03} f_{12} f_{21}^3}{f_{02}^4}\n+ \\frac{630 f_{12}^3 f_{21}^2}{f_{02}^4}.\n\\end{align} \nHere is an explicit formula for \n$\\mathcal{D}^f_k$  till $k=8$:\n\\begin{align}\n\\label{Formula_Dk}\n\\mathcal{D}^f_6 &=  f_{40},\\,\\,\n\\mathcal{D}^f_7 =   f_{50} -\\frac{5 f_{31}^2}{3 f_{12}},\\,\\,\n\\mathcal{D}^f_8 = f_{60} + \n\\frac{5 f_{03} f_{31} f_{50}}{3 f_{12}^2} \n-\\frac{5 f_{31} f_{41}}{f_{12}} - \\frac{10 f_{03} f_{31}^3}{3 f_{12}^3} \n+ \\frac{5 f_{22} f_{31}^2}{f_{12}^2}.\n\\end{align}  \n\n\\subsection{The spaces involved}\n\\hf\\hf We begin by explaining a terminology. If $l_{\\tilde{p}} \\in \\mathbb{P} T_{\\tilde{p}}\\mathbb{P}^2$, then we say that $v \\in l_{\\tilde{p}}$ if \n$v$ is a tangent vector in $T_{\\tilde{p}}\\mathbb{P}^2$ and lies over the fibre of $l_{\\tilde{p}}$.\nWe now define the spaces that we will encounter.\n\\label{definition_of_px}\n\\begin{align*}\n\\mathfrak{X}_k &:= \\{ ( \\tilde{f},\\tilde{p}) \\in \\mathcal{D} \\times \\mathbb{P}^2~~~~~: \\textnormal{$f$ has a singularity of type $\\mathfrak{X}_k$ at $\\tilde{p}$} \\} \\\\\n\\hat{\\mathfrak{X}}_k &:= \\{ (\\tilde{f},l_{\\p}) \\in \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2: \\textnormal{$f$ has a singularity of type $\\mathfrak{X}_k$ at $\\tilde{p}$} \\} ~= \\pi^{-1}(\\mathfrak{X}_k) \\\\\n\\textnormal{if $~k>1$ } \\quad \n\\mathcal{P} \\mathcal{A}_k &:= \\{ (\\tilde{f},l_{\\p}) \\in \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2: \n\\textnormal{$f$ has a singularity of type $\\mathcal{A}_k$ at $\\tilde{p}$},\\\\\n& \\qquad\\qquad\\qquad\\qquad \\qquad  \\qquad \\nabla^2 f|_p(v, \\cdot) =0\\,\\,\\textup{if}\\,\\,v \\in  l_{\\tilde{p}}\\}\\\\ \n\\mathcal{P} \\mathcal{D}_4 &:= \\{ (\\tilde{f},l_{\\p}) \\in \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2: \n\\textnormal{$f$ has a singularity of \ntype $\\mathcal{D}_4$ at $\\tilde{p}$}, \\\\ \n& \\qquad \\qquad \\qquad \\qquad \\qquad  \\qquad \\nabla^3 f|_p(v,v,v) =0\\,\\,\\textup{if}\\,\\,v  \\in  l_{\\tilde{p}}\\} \\\\\n\\textnormal{if $~k>4$} \\qquad \n\\mathcal{P} \\mathcal{D}_k &:= \\{ (\\tilde{f},l_{\\p}) \\in \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2: \\textnormal{$f$ has a singularity of \ntype $\\mathcal{D}_k$ at $\\tilde{p}$} \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad\\qquad \\nabla^3 f|_p(v, v, \\cdot) =0\\,\\,\\textup{if}\\,\\,v \\in l_{\\tilde{p}}\\} \\\\\n\\textnormal{if $k=6, 7$ or  $8$}\\qquad \n\\mathcal{P} \\mathcal{E}_k &:= \\{ (\\tilde{f},l_{\\p}) \\in \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2: \\textnormal{$f$ has a singularity of \ntype $\\mathcal{E}_k$ at $\\tilde{p}$}\\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad\\qquad \\nabla^3 f|_p(v, v, \\cdot) =0\\,\\,\\textup{if}\\,\\,v\\in l_{\\tilde{p}}\\} \\\\\n\\textnormal{if $~k>4$} \\qquad \\mathcal{P}\\mathcal{D}_k^{\\vee} &:= \\{ (\\tilde{f},l_{\\p}) \\in \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2: \n\\textnormal{$f$ has a singularity of \ntype $\\mathcal{D}_k$ at $\\tilde{p}$}, \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad  \\qquad \\nabla^3 f|_p(v,v,v) =0, ~~\\nabla^3 f|_p(v,v,w) \\neq 0 \\\\ \n& \\qquad \\qquad \\qquad \\qquad \\qquad  \\qquad  \\,\\,\\textup{if}\\,\\,v  \\in  l_{\\tilde{p}} ~~\\textup{and} ~~w \\in (T_{\\tilde{p}}\\mathbb{P}^2)\/l_{\\tilde{p}}\\} \n\\end{align*}\n\\noindent We also need the definitions for a few other spaces which will make our computations convenient. \n\\begin{align*}\n\\hat{\\mathcal{A}}_1^{\\#} := \\{ (\\tilde{f}, l_p) \\in \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 &: \n f(p) =0, \\nabla f|_p =0,  \\nabla^2 f|_p(v, \\cdot) \\neq 0, \\forall ~v\\neq 0 \\in l_{\\tilde{p}}  \\}  \\\\\n\\hat{\\mathcal{D}}_4^{\\#} := \\{ (\\tilde{f}, l_{\\p}) \\in \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 &: \nf(p) =0, \\nabla f|_p =0, \\nabla^2 f|_p \\equiv 0, \\nabla^3 f|_p (v,v,v) \\neq 0, \\forall ~v \\neq 0 \\in l_{\\tilde{p}} \\} \\\\\n\\hat{\\mathcal{D}}_k^{\\#\\flat} := \\{ (\\tilde{f}, l_{\\p}) \\in \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 &: \\textnormal{$f$ has a $\\mathcal{D}_k$ singularity at $\\tilde{p}$},  ~~\\nabla^3 f|_p (v,v,v) \\neq 0, \\forall ~v \\neq 0 \\in l_{\\tilde{p}},\\,\\,k\\geq 4\\} \\\\ \n\\hat{\\mathcal{X}}_8^{\\#} := \\{ (\\tilde{f}, l_{\\p}) \\in \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 &: \nf(p) =0, \\nabla f|_p =0, \\nabla^2 f|_p \\equiv 0, \\nabla^3 f|_p =0,  \\\\ \n                & \\qquad \\nabla^4 f|_p (v,v,v, v) \\neq 0 ~\\forall ~v \\neq 0  \\in l_{\\tilde{p}} \\}    \\\\\n \\hat{\\XC}_{8}^{\\# \\flat} :=  \\{ (\\tilde{f}, l_{\\p}) \\in \\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2 &: (\\tilde{f}, l_{\\p}) \\in \\hat{\\mathcal{X}}_8^{\\#}, \\Psi_{\\mathcal{J}}(\\tilde{f}, l_{\\p})\\neq 0, \n\\textnormal{where $\\Psi_{\\mathcal{J}}$ is defined in \\eqref{J_psi}}\\}.                        \n\\end{align*}\n\n\n\n\n\n\\section{Transversality} \n\\label{bundle_sections}\n\n\\hf\\hf In this section we list down all the relevant bundle sections that are transverse to the zero set. We set up our notation first. Let us define the following projection maps: \n\\begin{align*}\n\\pi_{1}&:\\mathcal{D} \\times \\mathbb{P}^2_1 \\times \\mathbb{P}^2_2  \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2_1, \\\\\n\\pi_{2}&:\\mathcal{D} \\times \\mathbb{P}^2_1 \\times \\mathbb{P}^2_2  \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2_2, \\\\\n\\pi_{1}&:\\mathcal{D} \\times \\mathbb{P}^2_1 \\times \\mathbb{P} T \\mathbb{P}^2_2  \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2_1, \\\\\n\\pi_{2}&:\\mathcal{D} \\times \\mathbb{P}^2_1 \\times \\mathbb{P} T \\mathbb{P}^2_2  \\longrightarrow \\mathcal{D} \\times \\mathbb{P} T \\mathbb{P}^2_2. \n\\end{align*}\nHence, given a vector bundle over $\\mathcal{D} \\times \\mathbb{P}^2$ or $\\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2$  we obtain a bundle over \n$\\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P}^2$ and $\\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2$ respectively, via the pullback maps.  \\\\\n\\hf\\hf A section of a bundle over $\\mathcal{D} \\times \\mathbb{P}^2$ or $\\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2$ induces a section over \nthe corresponding bundle over  $\\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P}^2$ and $\\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2$ respectively, via the pullback maps.\n\n\\begin{rem}\n To describe bundles over $\\mathcal{D}  \\times  \\mathbb{P}^2$ or $\\mathcal{D} \\times \\mathbb{P} T\\mathbb{P}^2$, we follow the abuse of notation \nof omitting pullback maps (as mentioned in Remark \\ref{an1_again}). \nHowever, to describe bundles over $\\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P}^2$ or $\\mathcal{D} \\times\\mathbb{P}^2 \\times  \\mathbb{P} T\\mathbb{P}^2$ we do  write the pullback maps. \n\\end{rem}\n\\begin{lmm}\n\\label{tube_lemma}\nLet $\\pi:E \\longrightarrow M$ be a fibre bundle with compact fibers. Let $X\\subseteq E$ and $Y\\subseteq M$. Then \n\\begin{align}\n \\pi (\\overline{X}) & = \\overline{\\pi (X)}  \\label{tube_lemma_X}\\\\\n\\pi^{-1} (\\overline{Y}) &= \\overline{\\pi^{-1}(Y)}. \\label{tube_lemma_Y}\n\\end{align}\n\\end{lmm}\n\\noindent \\textbf{Proof:  } Since the fibers are compact, $\\pi$ is a closed map (Tube Lemma). Combined with the fact that $\\pi$ is continuous \n\\eqref{tube_lemma_X} follows. Secondly, equation \\eqref{tube_lemma_Y} holds for the trivial bundle, hence it also for \nan arbitrary fibre bundle, since this is a local statement. \\qed  \n\\begin{prp}\n\\label{ift_ml}\nThe sections of the vector bundles\n\\begin{align*}\n\\pi_2^*\\psi_{\\mathcal{A}_0}:\\overline{\\mathcal{A}}_1 \\times \\mathbb{P}^2 - \\Delta \\overline{\\mathcal{A}}_1   \\longrightarrow \\pi_2^* \\mathcal{L}_{\\mathcal{A}_0}, \\qquad \\pi_2^*\\psi_{\\mathcal{A}_1}: \\pi_2^*\\psi_{\\mathcal{A}_0}^{-1}(0) \\longrightarrow \\pi_2^*\\mathcal{V}_{\\mathcal{A}_1}  \n\\end{align*}\nare transverse to the zero set if $d \\geq 3$. \n\\end{prp}\n\n\\begin{prp}\n\\label{A2_Condition_prp}\nThe section of the vector bundle $\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_2}:  \\overline{\\mathcal{A}}_1 \\circ \\overline{\\hat{\\mathcal{A}}}_1 \\longrightarrow \\pi_2^*\\mathbb{V}_{\\mathcal{P} \\mathcal{A}_2}$ is transverse to the zero set, provided $d \\geq 4$.  \n\\end{prp}\n\n\\begin{prp}\n\\label{D4_Condition_prp}\nThe sections of the vector bundles \n\\begin{align*}\n\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_3}: \\overline{\\mathcal{A}}_1 \\circ \\overline{\\mathcal{P} \\mathcal{A}}_2 \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_3}, \\quad \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4}: \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_3}^{-1} (0) \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{D}_4},  \n\\quad \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_5}^{\\UL}: \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4}^{-1} (0) \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{D}_5} \n\\end{align*} \nare transverse to the zero set provided $d \\geq 5$.\n\\end{prp}\n\n\\begin{prp}\n\\label{A3_Condition_prp}\nIf $i \\geq 4$, then the sections of the vector bundles \n\\begin{align*}\n\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_3}&:\\overline{\\mathcal{A}}_1 \\circ \\overline{\\mathcal{P} \\mathcal{A}}_2 \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_3}, \n\\qquad \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_4}:\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_{3}}^{-1} (0) - \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4}^{-1}(0) \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_4},  \\ldots, \\\\ \n\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_{i}}&:\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_{i-1}}^{-1} (0) - \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4}^{-1}(0) \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_{i}} \n\\end{align*} \nare transverse to the zero set provided $d\\geq i+2$.\n\\end{prp}\n\n\\begin{prp}\n\\label{D6_Condition_prp}\nIf $i \\geq 6$, then the sections of the vector bundles \n\\begin{align*}\n\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_6}&:\\overline{\\mathcal{A}}_1 \\circ \\overline{\\mathcal{P} \\mathcal{D}}_5 \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{D}_6}, \n\\qquad \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_7}:\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_{6}}^{-1} (0) - \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{E}_6}^{-1}(0) \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{D}_7},  \\ldots, \\\\ \n\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_{i}}&:\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_{i-1}}^{-1} (0) - \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{E}_6}^{-1}(0) \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{D}_{i}} \n\\end{align*} \nare transverse to the zero set provided $d\\geq i+2$.\n\\end{prp}\n\n\\begin{prp}\n\\label{E6_Condition_prp}\nThe section of the vector bundle $\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{E}_6}:\\overline{\\mathcal{A}}_1 \\circ \\overline{\\mathcal{P} \\mathcal{D}}_5 \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{E}_6}$ is transverse to the zero set provided $d\\geq 5$.\n\\end{prp}\n\n\\begin{prp}\n\\label{PD4_Condition_prp}\nThe section of the vector bundle $\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_3}:\\overline{\\mathcal{A}}_1 \\circ \\overline{\\hat{\\mathcal{D}}}_4 \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_3}$ is transverse to the zero set provided $d\\geq 5$.\n\\end{prp}\n\n\\noindent \\textbf{Proof:  }\nWe have omitted the proofs here; they can be found in \\cite{BM_Detail}. \nThey are similar to the way we prove transversality of bundle sections in \\cite{BM13}.\n\\qed  \n\n\\section{Closure and Euler class contribution} \n\\label{closure_of_spaces}\n\n\\hf\\hf In this section we compute closure of a singularity with one $\\mathcal{A}_1$-node. Along the way \nwe also compute how much a certain section contributes to the Euler class of a bundle. But \nfirst, let us recapitulate what we know about the closure of one singular point \nwhich was proved in \\cite{BM13}.\n\\begin{lmm}\n\\label{cl}\nLet $\\mathfrak{X}_k$ be a singularity of type $\\mathcal{A}_k$, $\\mathcal{D}_k$, \n$\\mathcal{E}_k$ or $\\mathcal{X}_8$. Then the closures are given by :\n\\begin{enumerate} \n\\item \\label{A0cl} $\\overline{\\mathcal{A}}_0 = \\mathcal{A}_0 \\cup \\overline{\\mathcal{A}}_1$ \n\\qquad if  $d \\geq 3$.\n\\item \\label{A1cl}\n$\\overline{\\hat{\\mathcal{A}}_1} = \\overline{\\hat{\\mathcal{A}}^{\\#}_1} = \\hat{\\mathcal{A}}_1^{\\#} \\cup \n\\overline{\\mathcal{P} \\mathcal{A}}_2$ \\qquad if $d \\geq 3$.\n\\item \\label{D4_cl_no_direction}$ \\overline{\\hat \\mathcal{D}^{\\#}_4} = \\hat \\mathcal{D}^{\\#}_4 \\cup \\overline{\\mathcal{P} \\mathcal{D}}_4 $ \\qquad if $d \\geq 3$.\n\\item \\label{D4cl}$ \\overline{\\mathcal{P} \\mathcal{D}}_4 = \\mathcal{P} \\mathcal{D}_4 \\cup \\overline{\\mathcal{P} \\mathcal{D}}_5 \\cup \\overline{\\mathcal{P} \\mathcal{D}_5^{\\vee}}$ \n\\qquad if $d \\geq 4$. \n\\item \\label{E6cl}$\\overline{\\mathcal{P} \\mathcal{E}}_6 = \n\\mathcal{P} \\mathcal{E}_6 \\cup \\overline{\\mathcal{P} \\mathcal{E}}_7 \\cup \\overline{\\hat{\\mathcal{X}}^{\\#}_8}$ \n\\qquad if $d \\geq 4$. \n\\item \\label{D5cl}$\\overline{\\mathcal{P} \\mathcal{D}}_5 = \\mathcal{P} \\mathcal{D}_5 \\cup \\overline{\\mathcal{P} \\mathcal{D}}_6 \\cup \\overline{\\mathcal{P} \\mathcal{E}}_6$ \\qquad if $d \\geq 4$.\n\\item \\label{D6cl}$\\overline{\\mathcal{P} \\mathcal{D}}_6 = \\mathcal{P} \\mathcal{D}_6 \\cup \\overline{\\mathcal{P} \\mathcal{D}}_7 \\cup \\overline{\\mathcal{P} \\mathcal{E}}_7$ \\qquad if $d \\geq 5$.\n\\item \\label{A2cl}$\\overline{\\mathcal{P} \\mathcal{A}}_2 = \\mathcal{P} \\mathcal{A}_2 \\cup \n\\overline{\\mathcal{P} \\mathcal{A}}_3 \\cup \\overline{\\hat{\\mathcal{D}}_4^{\\#}} $ \n\\qquad if $d \\geq 4$.\n\\item \\label{A3cl}$\\overline{\\mathcal{P} \\mathcal{A}}_3 = \\mathcal{P} \\mathcal{A}_3 \\cup  \\overline{\\mathcal{P} \\mathcal{A}}_4 \\cup \\overline{\\mathcal{P} \\mathcal{D}}_4$ \\qquad if $d \\geq 5$.\n\\item \\label{A4cl}$\\overline{\\mathcal{P} \\mathcal{A}}_4 = \\mathcal{P} \\mathcal{A}_4 \\cup  \\overline{\\mathcal{P} \\mathcal{A}}_5 \\cup \\overline{\\mathcal{P} \\mathcal{D}}_5$ \\qquad if $d \\geq 6$.\n\\item \\label{A5cl}$\\overline{\\mathcal{P} \\mathcal{A}}_5 = \\mathcal{P} \\mathcal{A}_5 \\cup  \\overline{\\mathcal{P} \\mathcal{A}}_6 \\cup \\overline{\\mathcal{P} \\mathcal{D}}_6 \\cup \\overline{\\mathcal{P} \\mathcal{E}}_6 $ \\qquad if $d \\geq 7$.\n\\item \\label{A6cl}$\\overline{\\mathcal{P} \\mathcal{A}}_6 = \\mathcal{P} \\mathcal{A}_6 \\cup  \\overline{\\mathcal{P} \\mathcal{A}}_7 \\cup \n\\overline{\\mathcal{P} \\mathcal{D}}_7 \\cup \\overline{\\mathcal{P} \\mathcal{E}}_7 \\cup \\overline{\\hat{\\mathcal{X}}_8^{\\# \\flat}}$  \n \\qquad if $d \\geq 8$.\n\\end{enumerate}\n\\end{lmm}\n\n\\noindent Let us now state a few facts about the closure of one singular point that will be required in this paper. \nThese facts were not explicitly stated in \\cite{BM13} because it was not needed in that paper.  \n\n\\begin{lmm}\n\\label{Dk_sharp_closure}\nWe have the following equality (or inclusion) of sets \n\\begin{enumerate}\n\\item \\label{a1_cl_new} $\\overline{\\mathcal{A}}_1 = \\mathcal{A}_1 \\cup \\overline{\\mathcal{A}}_2$ \\qquad if ~$d \\geq 2$. \\vspace*{-0.1cm} \n \\item \\label{d4_new} $\\overline{\\hat{\\mathcal{D}}^{\\#}_4} = \\overline{\\hat{\\mathcal{D}}}_4$ \\qquad if ~$d \\geq 3$. \\vspace*{-0.1cm}\n \\item \\label{dk_new} $\\overline{\\hat{\\mathcal{D}}^{\\#\\flat}_k} = \\overline{\\hat{\\mathcal{D}}}_k$ \\qquad if ~$k\\geq 4$ ~~ and ~~$d \\geq 3$. \n \\vspace*{-0.1cm}\n \\item \\label{d5_pa3_zero} $\\big\\{ (\\tilde{f}, l_{\\p}) \\in \\overline{\\hat{\\mathcal{D}}}_5: \\Psi_{\\mathcal{P} \\mathcal{A}_3}(\\tilde{f}, l_{\\p}) =0 \\big\\} \n = \\overline{\\mathcal{P} \\mathcal{D}}_5 \\cup \\overline{\\mathcal{P} \\mathcal{D}^{\\vee}_5}$  \\qquad if ~$d \\geq 3$. \\vspace*{-0.1cm} \n \\item \\label{pd6_pd7_pd8_closure} \n $\\big\\{(\\tilde{f}, l_{\\p}) \\in  \\overline{\\mathcal{P} \\mathcal{D}}_6: \\Psi_{\\mathcal{P} \\mathcal{E}_6}(\\tilde{f}, l_{\\p}) \\neq 0 \\big\\} \n =\\mathcal{P} \\mathcal{D}_6 \\cup \\mathcal{P} \\mathcal{D}_7\\cup \\big\\{(\\tilde{f}, l_{\\p}) \\in \\overline{\\mathcal{P} \\mathcal{D}}_8: \\Psi_{\\mathcal{P} \\mathcal{E}_6}(\\tilde{f}, l_{\\p}) \\neq 0 \\big\\}$\n \\item \\label{pe6_subset_of_cl_pd5_dual} $\\overline{\\mathcal{P} \\mathcal{E}}_6  \\subset \\overline{\\mathcal{P} \\mathcal{D}_5^{\\vee}}$ \\qquad if ~$d \\geq 3$. \n\\end{enumerate}\n\\end{lmm}\nThe proofs are straightforward; the details can be found in \\cite{BM_Detail}. \nHence we stated Lemma \\ref{Dk_sharp_closure}, statements \\ref{d4_new} and \\ref{dk_new} separately to avoid confusion.\n\n\\hf \\hf Before stating the main results of this section, \nlet us define three more spaces which will be required \nwhile formulating some of the Lemmas: \n\\begin{align}\n\\Delta \\PP \\D_7^{s} &:= \\{ (\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_4: \n\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4}(\\tilde{f}, \\tilde{p}, l_{\\p}) =0, \n~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{E}_6}(\\tilde{f}, \\tilde{p}, l_{\\p}) \\neq 0  \\}, \\nonumber \\\\\n\\Delta \\PP \\D_8^{s} &:= \\{ (\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_5: \n\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4}(\\tilde{f}, \\tilde{p}, l_{\\p}) =0, \\nonumber \n~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{E}_6}(\\tilde{f}, \\tilde{p}, l_{\\p}) \\neq 0  \\}, \\\\\n\\Delta \\PP \\D_6^{\\vee s} & := \\{ (\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_4: \n\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_5}(\\tilde{f}, \\tilde{p}, l_{\\p}) \\neq 0  \\}.\\label{new_defn_delta}\n\\end{align} \nWe are now ready to state the main Lemmas. \n\n\\begin{lmm}\n\\label{cl_two_pt}\nLet $\\mathfrak{X}_k$ be a singularity of type $\\mathcal{A}_k$, $\\mathcal{D}_k$, \n$\\mathcal{E}_k$. Then their closures with one $\\mathcal{A}_1$-node are given by :\n\\begin{enumerate}\n\\item \\label{a1a_minus_1_cl} $\\overline{\\overline{\\mathcal{A}}_1 \\circ (\\mathcal{D} \\times \\mathbb{P}^2)} = \\overline{\\mathcal{A}}_1 \\circ (\\mathcal{D} \\times \\mathbb{P}^2) \\sqcup \\Delta \\overline{\\mathcal{A}}_1 $ \n\\qquad if  $d \\geq 1$. \n\\item \\label{a1a1_up_cl} $\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\hat{\\mathcal{A}}^{\\#}_1} = \\overline{\\mathcal{A}}_1\\circ \\hat{\\mathcal{A}}^{\\#}_1 \\sqcup  \\overline{\\mathcal{A}}_1 \\circ  (\\overline{\\hat{\\mathcal{A}}_1^{\\#}}- \\hat{\\mathcal{A}}_1^{\\#}  ) \\sqcup \\Delta \\overline{\\hat{\\mathcal{A}}}_3$ \n\\qquad if  $d \\geq 3$. \n\\item \\label{a1_pa2_cl} $\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}}_2 = \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}_2 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{A}}_2- \\mathcal{P} \\mathcal{A}_2) \\sqcup \\Big(\\Delta \\overline{\\mathcal{P} \\mathcal{A}}_4 \\cup \n\\Delta \\overline{\\hat{\\mathcal{D}}^{\\#\\flat}_5} \\Big )$  \\qquad if  $d \\geq 4$.\n\\item \\label{a1_pa3_cl} $\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}}_3 = \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}_3 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{A}}_3- \\mathcal{P} \\mathcal{A}_3) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_5 \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{D}^{\\vee}_5} \\Big)$ \\qquad if  $d \\geq 5$.  \n\\item \\label{a1_pa4_cl} $\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}}_4 = \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}_4 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{A}}_4- \\mathcal{P} \\mathcal{A}_4) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_6 \\cup  \n\\Delta \\overline{\\PP \\D_7^{s}} \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{E}}_6  \\Big) $ \\qquad if  $d \\geq 6$. \n\\item \\label{a1_pa5_cl} $\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}}_5 = \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}_5 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{A}}_5- \\mathcal{P} \\mathcal{A}_5) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_7 \\cup \\Delta  \\overline{\\PP \\D_8^{s}} \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{E}}_7 \\Big)$ \\qquad if  $d \\geq 7$. \n\\item \\label{a1_pd4_cl} $\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{D}}_4 = \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{D}_4 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{D}}_4- \\mathcal{P} \\mathcal{D}_4) \\sqcup \n\\Big( \\Delta \\overline{\\PP \\D_6^{\\vee s}} \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{D}}_6 \\Big) $  \\qquad if  $d \\geq 4$.\n\\item \\label{a1_d4_cl} $\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\hat{\\mathcal{D}}_4} = \\overline{\\mathcal{A}}_1 \\circ  \\hat{\\mathcal{D}}_4 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\hat{\\mathcal{D}}}_4- \\hat{\\mathcal{D}}_4) \\sqcup   \n\\Big( \\Delta \\overline{\\hat{\\mathcal{D}}}_6 \\Big) $  \\qquad if  $d \\geq 4$.\n\\item \\label{a1_pd5_cl} $\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{D}}_5 = \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{D}_5 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{D}}_5- \\mathcal{P} \\mathcal{D}_5) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{D}}_7 \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{E}}_7 \\Big)$ \\qquad if  $d \\geq 5$.  \n\\end{enumerate}\n\\end{lmm}\n\n\\begin{rem} Although the statement of \nLemma \\ref{cl_two_pt} (\\ref{a1a_minus_1_cl}) is trivial we give an elaborate proof for two reasons. \nFirstly, along the way, we prove a few other statements that will be required later. \nSecondly, as a corollary we also compute the contribution of certain sections to the Euler class of relevant bundles. \n\\end{rem}\n\n\n\\textbf{Proof of Lemma \\ref{cl_two_pt} (\\ref{a1a_minus_1_cl}):} It suffices to show  that \n\\begin{align}\n\\{ (\\tilde{f}, \\tilde{p}, \\tilde{p}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ (\\mathcal{D} \\times \\mathbb{P}^2)}\\} &= \\Delta \\overline{\\mathcal{A}}_{1}. \\label{a1_a0_is_a1}  \n\\end{align} \nClearly the lhs\\footnote{We shall use {\\it lhs} to denote {\\it left hand side} and {\\it rhs} to denote {\\it right hand side} of an equation.} of \\eqref{a1_a0_is_a1} is a subset of its rhs. To show the converse we will prove the \nfollowing two claims simultaneously: \n\\begin{align}\n \\overline{\\overline{\\mathcal{A}}_1 \\circ (\\mathcal{D} \\times \\mathbb{P}^2)} & \\supset \\Delta (\\mathcal{A}_1 \\sqcup \\mathcal{A}_2),  \\label{a1_du_a2_is_subset_of_a1_a0}\\\\\n \\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{A}}_1  \\cap \\Delta (\\mathcal{A}_1 \\sqcup \\mathcal{A}_2)  & = \\varnothing. \\label{a1_du_a2_intersect_a1_a1_is_empty} \n\\end{align}\nSince $\\overline{\\overline{\\mathcal{A}}_1 \\circ (\\mathcal{D} \\times \\mathbb{P}^2)}$ \nis a closed set, \\eqref{a1_du_a2_is_subset_of_a1_a0} \nimplies that the rhs of \\eqref{a1_a0_is_a1} is a subset of its lhs.\\footnote{In fact\nthe full strength of \\eqref{a1_du_a2_is_subset_of_a1_a0} is not really needed; \nwe simply need that $\\overline{\\overline{\\mathcal{A}}_1 \\circ (\\mathcal{D} \\times \\mathbb{P}^2)} \\supset \\Delta \\mathcal{A}_1$.}\nMoreover, \\eqref{a1_du_a2_intersect_a1_a1_is_empty} is not required at all \nfor the proof of Lemma \\ref{cl_two_pt} \\eqref{a1a_minus_1_cl}. \nHowever, these statements will be required in the proofs of  \nLemma \\ref{cl_two_pt} (\\ref{a1a1_up_cl}).\nFurthermore, in the process of proving \\eqref{a1_du_a2_is_subset_of_a1_a0} and \\eqref{a1_du_a2_intersect_a1_a1_is_empty}, \nwe will also be computing the contribution of certain sections to the Euler class of relevant bundles as a corollary. \n\n\n\\begin{claim}\n\\label{a1_a1_closure_intersect_a1_or_a2_empty_equations_claim}\nLet $(\\tilde{f},\\tilde{p}, \\tilde{p}) \\in \\Delta (\\mathcal{A}_1 \\cup \\mathcal{A}_{2})$.\nThen there exist solutions \n$$(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)  ) \\in \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P}^2 $$ \nsufficiently close to $(\\tilde{f},\\tilde{p}, \\tilde{p})$ to the set of equations\n\\begin{align}\n\\pi_1^*\\psi_{\\mathcal{A}_0}( \\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)) & = 0, ~~\\pi_1^*\\psi_{\\mathcal{A}_1}( \\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1) ) = 0, ~~\\tilde{p}(t_1, t_2) \\neq \\tilde{p}(t_1).  \\label{a1_a1_sharp_closure_functional_eqn_a1_plus_a2} \n\\end{align}\nFurthermore, any such solution sufficiently close to $(\\tilde{f}, \\tilde{p}, \\tilde{p})$ satisfies \n\\begin{align}\n\\Big( \\pi_2^*\\psi_{\\mathcal{A}_0}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)), ~\\pi_2^*\\psi_{\\mathcal{A}_1}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)) \\Big) \\neq (0,0). \\label{a1_a1_closure_intersect_a1_is_empty_equation}\n\\end{align} \nIn particular, $(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1))$ does not lie in $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{A}}_1$.\n\\end{claim}\nIt is easy to see that claim \\ref{a1_a1_closure_intersect_a1_or_a2_empty_equations_claim} proves statements \\eqref{a1_du_a2_is_subset_of_a1_a0} and \n\\eqref{a1_du_a2_intersect_a1_a1_is_empty} \nsimultaneously. \\\\   \n\n\\noindent \\textbf{Proof:  }  Choose homogeneous coordinates $[\\mathrm{X}: \\mathrm{Y}: \\mathrm{Z}]$ \nso that $\\tilde{p} = [0:0:1]$ and  let $\\mathcal{U}_{\\tilde{p}}$ be a sufficiently \nsmall neighbourhood of $\\tilde{p}$ inside $\\mathbb{P}^2$.\nDenote   \n$\\pi_{x}, \\pi_y : \\mathcal{U}_{\\tilde{p}} \\longrightarrow \\mathbb{C} $ \n to be the projection maps given by \n$$\\pi_{x}([\\mathrm{X}: \\mathrm{Y}:\\mathrm{Z}]) := \\mathrm{X}\/\\mathrm{Z} \\qquad \\textnormal{and} \\qquad \\pi_{y}([\\mathrm{X}:\\mathrm{Y}:\\mathrm{Z}]) := \\mathrm{Y}\/\\mathrm{Z},$$ \nand $v, w: \\mathcal{U}_{\\tilde{p}} \\longrightarrow T \\mathbb{P}^2$ the vector fields dual to the one forms $d\\pi_x$ and $d\\pi_y $ respectively. Let $ (\\tilde{f} (t_1, t_2),  \\tilde{p}(t_1)) \\in \\mathcal{D} \\times \\mathbb{P}^2 $ be an arbitrary point that is close to \n$( \\tilde{f}, \\tilde{p})$ and let $\\tilde{p}(t_1, t_2)$ be a point in $\\mathbb{P}^2$ that is close to $\\tilde{p}(t_1)$. Let \n\\begin{align*}\n\\tilde{p}(t_1) := [x_{t_1}:y_{t_1}:1] \\in \\mathbb{P}^2,  \\qquad p(t_1) &:= (x_{t_1},y_{t_1},1) \\in \\mathbb{C}^3,\\\\ \n\\tilde{p}(t_1, t_2) := [x_{t_1} + x_{t_2}: ~y_{t_1}+y_{t_2}: 1] \\in \\mathbb{P}^2, \\qquad p(t_1, t_2) &:= (x_{t_1} + x_{t_2}, ~y_{t_1}+y_{t_2}, 1) \\in \\mathbb{C}^3, \\\\\n\\tilde{f}(t_1, t_2)  \\in \\mathbb{P}^{\\delta_d}, \\qquad f(t_1, t_2) & \\in \\mathbb{C}^{\\delta_d +1}.  \n\\end{align*}\nDefine the following numbers: \n\\begin{align*}\nf_{ij}(t_1, t_2) & := \\{ \\nabla^{i+j} f(t_1, t_2)|_{\\tilde{p}(t_1)} \n(\\underbrace{v,\\cdots v}_{\\textnormal{$i$ times}}, \\underbrace{w,\\cdots w}_{\\textnormal{$j$ times}}) \\}(p(t_1)^{\\otimes d}),\\\\\n\\mathrm{F}   &:= f_{00}(t_1, t_2) + f_{10}(t_1, t_2) \\xt + f_{01}(t_1, t_2) y_{t_2} +  \\sum_{i+j=2}\\big({\\textstyle \\frac{f_{ij}(t_1, t_2)}{i ! j !}} x_{t_2}^i y_{t_2}^j\\big) + \\ldots,  \\\\ \n\\mathrm{F}_{x_{t_2}} &:= f_{10}(t_1, t_2)  +  f_{20}(t_1, t_2) \\xt + f_{11}(t_1, t_2) y_{t_2} + \\ldots, \\\\\n\\mathrm{F}_{y_{t_2}} &:= f_{01}(t_1, t_2) + f_{11}(t_1, t_2) x_{t_2} + f_{02}(t_1, t_2) y_{t_2} + \\ldots.  \n\\end{align*}\nIt is easy to see that  \n\\begin{align*}\n\\{ \\pi_1^* \\psi_{\\mathcal{A}_0}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  \\tilde{p}(t_1)) \\} (f(t_1, t_2) \\otimes p(t_1, t_2)^{\\otimes d} ) & = \\mathrm{F}, \\\\\n\\{ \\pi_1^* \\psi_{\\mathcal{A}_1}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  \\tilde{p}(t_1)) \\} (f(t_1, t_2) \\otimes p(t_1, t_2)^{\\otimes d} \\otimes v) & = \\mathrm{F}_{x_{t_2}}, \\\\ \n\\{ \\pi_1^* \\psi_{\\mathcal{A}_1}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  \\tilde{p}(t_1)) \\} (f(t_1, t_2) \\otimes p(t_1, t_2)^{\\otimes d} \\otimes w) &=  \\mathrm{F}_{y_{t_2}}.\n\\end{align*}\nWe now observe  that \n\\eqref{a1_a1_sharp_closure_functional_eqn_a1_plus_a2}  \nhas a solution if and only if \nthe following set of equations has a solution \n\\begin{align}\n\\mathrm{F} = 0, \\qquad \\mathrm{F}_{x_{t_2}} = 0, \\qquad \\mathrm{F}_{y_{t_2}} = 0,  \\qquad (x_{t_2}, y_{t_2}) \\neq (0, 0) \\qquad \\textnormal{(but small)}. \\label{eval_f1} \n\\end{align} \nNote that in equation \\eqref{a1_a1_sharp_closure_functional_eqn_a1_plus_a2} the equality holds as \n\\textit{functionals}, while in equation \\eqref{eval_f1}, the equality holds as \\textit{numbers}. \n\\begin{comment}\nIt will be useful to define the following number:\n\\begin{align*}\n\\mathrm{G} &:= \\xt \\mathrm{F}_{x_{t_2}} + y_{t_2} \\mathrm{F}_{y_{t_2}} - \\mathrm{F} \\\\ \n           & =  \\frac{f_{20}(t_1, t_2)}{2} x_{t_2}^2 + f_{11}(t_1, t_2) x_{t_2} y_{t_2} +  \\frac{f_{02}(t_1, t_2)}{2} y_{t_2}^2 + \\ldots.\n\\end{align*}\nHence, \\eqref{eval_f1} has a solution if and only if \n\\begin{align}\n\\mathrm{G} = 0, \\qquad \\mathrm{F}_{x_{t_2}} = 0, \\qquad \\mathrm{F}_{y_{t_2}} = 0,  \\qquad (x_{t_2}, y_{t_2}) \\neq (0, 0) \\qquad \\textnormal{(but small)} \\label{eval_g} \n\\end{align} \nhas a solution. It easy to see \\eqref{eval_g} always has solutions. \nTo see why, let $\\mathrm{L}$ be a solution of \n\\begin{align}\nf_{20} + 2 f_{11}\\mathrm{L} + f_{02}\\mathrm{L}^2 & =0. \\label{equation_for_L}\n\\end{align}\nThen   \n\\begin{align} \n\\xt & = u, \\qquad y_{t_2} = \\mathrm{L} u, \\qquad \\textnormal{$u$ small but non zero}, \\nonumber \\\\\nf_{20}(t_1, t_2) &= -2 f_{11}(t_1, t_2) \\mathrm{L} - f_{02}(t_1, t_2) \\mathrm{L}^2 + \\mathrm{O}(u) + \\ldots \\nonumber \\\\\nf_{10}(t_1, t_2) &= -f_{20}(t_1, t_2) \\xt -f_{11}(t_1, t_2) y_{t_2}  + \\ldots, \\nonumber  \\\\\nf_{01}(t_1, t_2) & = - f_{11}(t_1, t_2) x_{t_2} - f_{02}(t_1, t_2) y_{t_2} + \\ldots  \\label{a1_neighbourhood_inside_a1_a0}\n\\end{align}\nis a solution to \\eqref{eval_g}.\\footnote{If \\eqref{equation_for_L} has no solution then solve the equation  \n$f_{02} + 2 f_{11}\\mathrm{L} + f_{20}\\mathrm{L}^2 =0$ and interchange  \n$\\xt$ and $y_{t_2}$ in \\eqref{a1_neighbourhood_inside_a1_a0}.} \nHence \\eqref{eval_g} always has solutions.\\footnote{However, we are not saying that \nthe solutions constructed in \\eqref{a1_neighbourhood_inside_a1_a0} are the only solutions.}\\\\\n\\end{comment}   \nWe will now show that  \n\\eqref{eval_f1} (and hence \\eqref{a1_a1_sharp_closure_functional_eqn_a1_plus_a2}) \nhas solutions whenever $(\\tilde{f}, \\tilde{p}, \\tilde{p}) \\in \\Delta (\\mathcal{A}_1 \\cup \\mathcal{A}_2)$. \nFurthermore, for all those solutions,  \n\\eqref{a1_a1_closure_intersect_a1_is_empty_equation} holds. \\\\\n\\hf \\hf First let us assume \n$(\\tilde{f}, \\tilde{p}, \\tilde{p}) \\in \\Delta \\mathcal{A}_1$. \nIt is obvious that solutions to \\eqref{eval_f1} exist; we can solve for $f_{10}(t_1, t_2)$ and $f_{01}(t_1, t_2)$ \nusing $\\mathrm{F}_{x_{t_2}} =0$ and $\\mathrm{F}_{y_{t_2}} =0$ and \nand then solve for \n$f_{00}(t_1, t_2)$ using $\\mathrm{F} =0$. \nTo show that \\eqref{a1_a1_closure_intersect_a1_is_empty_equation} holds it suffices to show that \nif $(\\xt, y_{t_2})$ is small but non zero,  then \n\\begin{align}\n\\big(f_{00}(t_1, t_2), ~f_{10}(t_1, t_2), ~f_{01}(t_1, t_2) \\big) &\\neq (0,0,0). \\label{a1_a1_closure_intersect_a1_is_empty_equation_numbers}\n\\end{align}\nObserve that  \n\\eqref{eval_f1}  implies that\n\\begin{equation}\n\\label{f_ij_matrix_equation}\n\\left( \\begin{array}{c} f_{10}(t_1, t_2) \\\\ f_{01}(t_1, t_2) \\end{array} \\right) = \n-\\begin{pmatrix} f_{20}(t_1,t_2) & f_{11}(t_1, t_2)  \\\\ f_{11}(t_1, t_2)  & f_{02}(t_1, t_2) \n\\end{pmatrix} \\left(\\begin{array}{c} x_{t_2} \\\\ y_{t_2} \\end{array} \\right)+ \\left (\\begin{array}{c} \\mathrm{E}_1(x_{t_2}, y_{t_2}) \\\\ \\mathrm{E}_2(x_{t_2}, y_{t_2})   \n\\end{array} \\right) \n\\end{equation}\nwhere $\\mathrm{E}_i(x_{t_2}, y_{t_2})$ are second order in $(x_{t_2}, y_{t_2})$. \nSince $(\\tilde{f},\\tilde{p}, \\tilde{p}) \\in \\Delta \\mathcal{A}_1$, the matrix\n\\begin{equation*}\n\\mathrm{M} := \\left( \\begin{array}{cc}\nf_{20}(t_1,t_2) & f_{11}(t_1, t_2)    \\\\\nf_{11}(t_1, t_2) & f_{02}(t_1, t_2)  \\end{array} \\right)\n\\end{equation*}\nis invertible if $\\tilde{f}(t_1, t_2)$ is sufficiently close to $\\tilde{f}$. \nEquation \\eqref{f_ij_matrix_equation} now implies that  \nthat if $(\\xt, y_{t_2})$ is small but non zero, then  \n$f_{10}(t_1, t_2)$ and $f_{01}(t_1, t_2)$ can not both be zero. \nHence \\eqref{a1_a1_closure_intersect_a1_is_empty_equation_numbers}  holds and hence \\eqref{a1_a1_closure_intersect_a1_is_empty_equation} holds. \\\\\n\\begin{comment}\n\n***************\n\n\n  \n\n\nWe claim in fact that $(f_{10}(t_1, t_2), f_{01}(t_1, t_2)) \\neq (0,0)$. \n\nLet us define \n\\begin{align}\n \\mathrm{G} &:= \\xt \\mathrm{F}_{x_{t_2}} + y_{t_2} \\mathrm{F}_{y_{t_2}} - \\mathrm{F} \\nonumber \\\\ \n            & = \\frac{f_{20}(t_1, t_2)}{2} x_{t_2}^2 + f_{11}(t_1, t_2) x_{t_2} y_{t_2} +  \\frac{f_{02}(t_1, t_2)}{2} y_{t_2}^2 + \\ldots.\\nonumber \n\\end{align}\nNotice that the quadratic term of $\\mathrm{G}$ is same as the quadratic term of $\\mathrm{F}$; moreover it has no first order terms \n(and also no zeroth order term since $f_{00}(t_1, t_2)$ is zero). \nWe observe that \n\\eqref{eval_f1} has a solution if and only if \n\\begin{align}\n\\mathrm{G} = 0, \\qquad \\mathrm{F}_{x_{t_2}} = 0, \\qquad \\mathrm{F}_{y_{t_2}} = 0,  \\qquad (x_{t_2}, y_{t_2}) \\neq (0, 0) \\qquad \\textnormal{(but small)} \\label{eval_f1_g} \n\\end{align} \nhas a solution. Since $(\\tilde{f}, \\tilde{p}, \\tilde{p}) \\in \\Delta \\mathcal{A}_1$ one of the possibility holds; at least one of $f_{20}(t_1, t_2)$  or $f_{02}(t_1, t_2)$ is non zero, \nor $f_{11}(t_1, t_2)$ is non zero. Let us assume $f_{02}(t_1, t_2) \\neq 0$. \nWe claim that there exists a unique holomorphic function $\\mathrm{B}(x_{t_2})$, \nvanishing at the origin  such that \nafter we make a change of coordinates $y_{t_2} = \\hat{y}_{t_2} + \\mathrm{B}(x_{t_2})$, \nthe function $\\mathrm{G}$ is given by \n\\bgd\n\\mathrm{G} = \\hat{\\Z}_0(x_{t_2}) + \\hat{\\Z}_2(x_{t_2}) \\hat{y}_{t_2}^2 + \\hat{\\Z}_3(x_{t_2}) \\hat{y}_{t_2}^3 + \\ldots \n\\edd\nfor some $\\hat \\Z_k(x_{t_2})$ (i.e., $\\hat{\\Z}_1(x_{t_2}) \\equiv 0$). \nTo see why, we note that this is possible if $\\mathrm{B}(x_{t_2})$ satisfies the identity\n\\begin{align}\n\\Z_1(x_{t_2}) + 2 \\Z_2(x_{t_2}) \\mathrm{B} + 3 \\Z_3(x_{t_2}) \\mathrm{B}^2 + \\ldots \\equiv 0. \n\\end{align}\nSince ~$\\Z_2(0) \\neq 0$, $\\mathrm{B}(x_{t_2})$ exists by the Implicit Function Theorem.\nHence, we get that \n\\begin{align}\n\\mathrm{G} & =  \\frac{f_{02}(t_1, t_2)}{2} \\hat{\\hat{y}}_{t_2}^2 + \\frac{1}{2}\\Big( f_{20}(t_1, t_2) - \\frac{f_{11}(t_1, t_2)^2}{f_{02}(t_1, t_2)} \\Big) \\hat{x}_{t_2}^2, \\label{G_modified}\\\\\n\\textnormal{where} ~~~\\hat{\\hat{y}}_{t_2} &:= \\sqrt{\\frac{\\hat{2 \\Z}_2(x_{t_2}) + 2 \\hat{\\Z}_3(x_{t_2}) \\hat{y}_{t_2} + \\ldots}{f_{02}(t_1, t_2)}}, ~~~\\hat{x}_{t_2} := \n\\sqrt{\\frac{2 f_{02}(t_1, t_2)\\hat{\\Z}_0(x_{t_2})}{f_{20}(t_1, t_2)f_{02}(t_1, t_2) - f_{11}(t_1, t_2)^2 }}.  \\nonumber \n\\end{align}\nHence,\\eqref{G_modified} implies that  solutions to \\eqref{eval_f1_g}  are given by \n\\begin{align}\n\\hat{x}_{t_2} &= u, ~~ \\hat{\\hat{y}}_{t_2} = \\alpha u, ~~f_{10}(t_1, t_2) = \\mathrm{O}(u), ~~f_{01}(t_1, t_2) = f_{02}(t_1, t_2) \\alpha u + \\mathrm{O}(u^2) \\nonumber \\\\\n\\textnormal{where} ~~\\alpha &:= \\frac{\\sqrt{f_{20}(t_1, t_2)f_{02}(t_1, t_2) - f_{11}(t_1, t_2)^2 }}{f_{02}(t_1, t_2)}  ~~\\textnormal{a branch of the square root.} \\label{eval_f1_g_soln}\n\\end{align}\nSince $f_{02}(t_1, t_2) \\alpha \\neq 0$, \\eqref{eval_f1_g_soln} implies that \nif $u$ is small but non zero, then \n$f_{01}(t_1, t_2) \\neq 0$.   \nIn other words $(f_{10}, f_{01}) \\neq (0,0) $;\nhence \\eqref{a1_a1_closure_intersect_a1_is_empty_equation} holds. We can use a similar argument if \n$f_{02}(t_1, t_2) = f_{20}(t_1, t_2) =0$, but $f_{11}(t_1, t_2) \\neq 0$. \n\n\n\n\n\nTherefore, we can compute $\\mathrm{B}(x_{t_2})$ \nas a power series using \n\n\nObserve that  \nthe second and third equation of\n\\eqref{eval_f1} \n(namely $\\mathrm{F}_{x_{t_2}} =0$ and $\\mathrm{F}_{y_{t_2}} =0$ ) can be written as \n\\begin{equation}\n\\label{f_ij_matrix_equation}\n\\left( \\begin{array}{c} f_{10}(t_1, t_2) \\\\ f_{01}(t_1, t_2) \\end{array} \\right) = \n-\\begin{pmatrix} f_{20}(t_1,t_2) & f_{11}(t_1, t_2)  \\\\ f_{11}(t_1, t_2)  & f_{02}(t_1, t_2) \n\\end{pmatrix} \\left(\\begin{array}{c} x_{t_2} \\\\ y_{t_2} \\end{array} \\right)+ \\left (\\begin{array}{c} \\mathrm{E}_1(x_{t_2}, y_{t_2}) \\\\ \\mathrm{E}_2(x_{t_2}, y_{t_2})   \n\\end{array} \\right) \n\\end{equation}\nwhere $\\mathrm{E}_i(x_{t_2}, y_{t_2})$ are second order in $(x_{t_2}, y_{t_2})$. \nSince $(\\tilde{f},\\tilde{p}, \\tilde{p}) \\in \\Delta \\mathcal{A}_1$, the matrix\n\\begin{equation}\\label{matrix_M}\n\\mathrm{M} := \\left( \\begin{array}{cc}\nf_{20}(t_1,t_2) & f_{11}(t_1, t_2)    \\\\\nf_{11}(t_1, t_2) & f_{02}(t_1, t_2)  \\end{array} \\right)\n\\end{equation}\nis invertible if $(t_1, t_2)$ is sufficiently small.\nHence, by the Implicit Function Theorem, solution to \n\\eqref{f_ij_matrix_equation} exists;  \nwe can solve for $(x_{t_2}, y_{t_2})$ \nin terms of $f_{ij}(t_1, t_2)$. \nFinally, since $x_{t_2}$ and $y_{t_2}$ \nare not both equal to zero, we can use the first equation of \n\\eqref{eval_f1} (namely $\\mathrm{F} =0$) \nto solve for either $f_{10}(t_1, t_2)$ or \n$f_{01}(t_1, t_2)$. Hence solutions to \n\\eqref{eval_f1} (and hence to \\eqref{a1_a1_sharp_closure_functional_eqn_a1_plus_a2}) \nexist.  \\\\\n\\hf \\hf Next we will show that \\eqref{a1_a1_closure_intersect_a1_is_empty_equation} holds. \nObserve that by \\eqref{f_ij_matrix_equation}  \nif $(x_{t_2}, y_{t_2}) \\neq (0, 0)$, but sufficiently small, \nthen $(f_{10}(t_1, t_2), f_{01}(t_1, t_2) ) \\neq (0,0)$. This is \nbecause $\\mathrm{M}$ is injective and $\\mathrm{E}_i(x_{t_2}, y_{t_2})$ are second order in $(x_{t_2}, y_{t_2})$. \nThat proves that \\eqref{a1_a1_closure_intersect_a1_is_empty_equation} holds (as functionals). \\\\ \n\\end{comment}\n\\hf \\hf Next, let $(\\tilde{f}, \\tilde{p}, \\tilde{p}) \\in \\Delta \\mathcal{A}_2$. \nSince $\\Delta \\mathcal{A}_2 \\subset \\Delta \\overline{\\mathcal{A}}_1$, we conclude that solutions to \n\\eqref{a1_a1_sharp_closure_functional_eqn_a1_plus_a2} exist; we only need to show that \n\\eqref{a1_a1_closure_intersect_a1_is_empty_equation} holds. \nObserve that \n$f_{20}$ and $f_{02}$ can not both be zero; assume $f_{02} \\neq 0$ \n(and hence $f_{02}(t_1, t_2) \\neq 0$). \nWrite $\\mathrm{F}$ as \n\\bgd\n\\mathrm{F} = f_{00}(t_1, t_2) +f_{10}(t_1, t_2) \\xt + f_{01}(t_1, t_2) y_{t_2} +  \\Z_0(x_{t_2}) + \\Z_1(x_{t_2})y_{t_2} + \\Z_2(x_{t_2}) y_{t_2}^2 + \\ldots\n\\edd\nwhere $\\Z_2(0) \\neq 0.$ We claim that there exists a unique holomorphic function $\\mathrm{B}(x_{t_2})$, \nvanishing at the origin,  such that \nafter we make a change of coordinates $y_{t_2} = \\hat{y}_{t_2} + \\mathrm{B}(x_{t_2})$, \nthe function $\\mathrm{F}$ is given by \n\\bgd\n\\mathrm{F} = f_{00}(t_1, t_2) +f_{10}(t_1, t_2) \\xt + f_{01}(t_1, t_2) \\hat{y}_{t_2} +  f_{01}(t_1, t_2)\\mathrm{B}(x_{t_2}) +  \n \\hat{\\Z}_0(x_{t_2}) + \\hat{\\Z}_2(x_{t_2}) \\hat{y}_{t_2}^2 + \\hat{\\Z}_3(x_{t_2}) \\hat{y}_{t_2}^3 + \\ldots \n\\edd\nfor some $\\hat \\Z_k(x_{t_2})$ (i.e., $\\hat{\\Z}_1(x_{t_2}) \\equiv 0$). \nThis is possible if $\\mathrm{B}(x_{t_2})$ satisfies the identity\n\\begin{align}\n\\Z_1(x_{t_2}) + 2 \\Z_2(x_{t_2}) \\mathrm{B}(\\xt) + 3 \\Z_3(x_{t_2}) \\mathrm{B}(\\xt)^2 + \\ldots \\equiv 0.  \\label{psconvgg2}\n\\end{align}\nSince $\\Z_2(0) \\neq 0$, $\\mathrm{B}(x_{t_2})$ exists by the Implicit Function Theorem and \nwe can compute $\\mathrm{B}(x_{t_2})$ explicitly  \nas a power series using \\eqref{psconvgg2} and then\ncompute $\\hat \\Z_{0}(x_{t_2})$. Hence, \n\\begin{align*} \n\\mathrm{F} & = f_{00}(t_1, t_2) +f_{10}(t_1, t_2) x_{t_2} + f_{01}(t_1, t_2) \\hat{y}_{t_2} +  \\varphi(x_{t_2}, \\hat{y}_{t_2})  \\hat{y}_{t_2}^2  \n+f_{01}(t_1, t_2)\\mathrm{B}(x_{t_2}) \\\\ \n    & + \\underbrace{{\\textstyle \\frac{\\mathcal{B}^{f(t_1, t_2)}_2}{2!}x_{t_2}^2 + \\frac{\\mathcal{B}^{f(t_1, t_2)}_3}{3!} x_{t_2}^3 + \\mathrm{O}(x_{t_2}^4)}}_{\\hat \\Z_{0}(x_{t_2})},\n \\end{align*}\n where $\\varphi(0,0) \\neq 0$ and\n\\begin{align*}\n \\mathcal{B}^{f(t_1, t_2)}_2  & := f_{20}(t_1, t_2) - \\frac{f_{11}(t_1, t_2)^2}{f_{02}(t_1, t_2)}, \n\\qquad \\textup{and}\\\\\n\\mathcal{B}^{f(t_1, t_2)}_{3} & :=f_{30}(t_1, t_2)-\\frac{3 f_{11}(t_1, t_2) f_{21}(t_1, t_2)}\n{f_{02}(t_1, t_2)} + \\frac{3 f_{11}(t_1, t_2)^2 f_{12}(t_1, t_2)}{f_{02}(t_1, t_2)^2}- \\frac{f_{11}(t_1, t_2)^3 \nf_{03}(t_1, t_2)}{f_{02}(t_1, t_2)^3}\\neq 0.  \n\\end{align*}\nThe last inequality holds because $(\\tilde{f}, \\tilde{p}, \\tilde{p}) \\in \\Delta \\mathcal{A}_2$.\nIn these new coordinates $\\hat{y}_{t_2}$ \nand $x_{t_2}$, \nequation \\eqref{eval_f1} is equivalent to  \n\\begin{align}\n\\mathrm{F} = f_{00}(t_1, t_2) + f_{10}(t_1, t_2) x_{t_2} + f_{01}(t_1, t_2) \\hat{y}_{t_2} + \\varphi(x_{t_2}, \\hat{y}_{t_2})  \\hat{y}_{t_2}^2 + f_{01}(t_1, t_2)\\mathrm{B}(x_{t_2}) + &   \\nonumber \\\\\n    +\\frac{\\mathcal{B}^{f(t_1, t_2)}_2}{2!}x_{t_2}^2 + \\frac{\\mathcal{B}^{f(t_1, t_2)}_3}{3!} x_{t_2}^3  + \\mathrm{O}(x_{t_2}^4) & = 0,  \\label{closure_a1_a1_is_not_a2_F} \\\\\nf_{10}(t_1, t_2)+ f_{01} (t_1, t_2) \\mathrm{B}^{\\prime}(x_{t_2})+ \\varphi_{x_{t_2}}(x_{t_2}, \\hat{y}_{t_2})  \\hat{y}_{t_2}^2  + \\mathcal{B}^{f(t_1, t_2)}_2 x_{t_2} + \\frac{\\mathcal{B}^{f(t_1, t_2)}_3}{2!} x_{t_2}^2   + \n\\mathrm{O}(x_{t_2}^3) & = 0, \n\\label{closure_a1_a1_is_not_a2_Fx}\\\\\nf_{01}(t_1, t_2)+ 2 \\hat{y}_{t_2}  \\varphi(x_{t_2}, \\hat{y}_{t_2}) +  \\hat{y}_{t_2}^2 \\varphi_{\\hat{y}_{t_2}}(x_{t_2}, \\hat{y}_{t_2}) & =0, \\nonumber \\\\ \n(\\xt, \\hat{y}_{t_2})  \\neq (0,0) \\qquad \\textnormal{but small}.& \\label{closure_a1_a1_is_not_a2_Fy}\n\\end{align}\nLet us clarify a point of confusion: we are claiming that \\eqref{eval_f1} has a solution if and only if the equation \n\\begin{align}\n\\mathrm{F}& = 0, \\qquad \\mathrm{F}_{x_{t_2}} = 0, \\qquad \\mathrm{F}_{\\hat{y}_{t_2}} = 0,  \\qquad (x_{t_2}, \\hat{y}_{t_2}) \\neq (0, 0) \\qquad \\textnormal{(but small)} \\label{eval_f1_hat} \n\\end{align}\nhas a solution. We are \\textit{not} claiming that the partial derivatives in the old \ncoordinates are individually equal to the partial derivatives in the new coordinates.  \nSince $\\varphi(0,0) \\neq 0$ we can use \\eqref{closure_a1_a1_is_not_a2_Fy} to solve for $\\hat{y}_{t_2}$ in terms of $x$ and \n$f_{01}(t_1, t_2)$ to get \n\\begin{align}\n \\hat{y}_{t_2} & = f_{01} (t_1, t_2) \\mathrm{E}(x_{t_2}, f_{01}(t_1, t_2)), \\label{closure_a1_a1_is_not_a2_F_hat_y}\n\\end{align}\nwhere $\\mathrm{E}(x_{t_2}, f_{01}(t_1, t_2))$ is a  holomorphic \nfunction of $(x_{t_2}, f_{01}(t_1, t_2))$.\nUsing \\eqref{closure_a1_a1_is_not_a2_F_hat_y}, \\eqref{closure_a1_a1_is_not_a2_Fx} and \\eqref{closure_a1_a1_is_not_a2_F}  \nwe get (by eliminating $\\mathcal{B}^{f(t_1, t_2)}_2$ and $\\hat{y}_{t_2}$)    \n\\begin{align}\n\\mathrm{F} = -\\frac{\\mathcal{B}^{f(t_1, t_2)}_3}{12} x_{t_2}^3 + \\mathrm{O}(x_{t_2}^4) + f_{00}(t_1, t_2)+f_{10}(t_1, t_2) \\mathrm{E}_{1} (x_{t_2}, f_{10}(t_1, t_2), f_{01}(t_1, t_2))  &  \\nonumber \\\\ \n+f_{01}(t_1, t_2) \\mathrm{E}_{2} (x_{t_2}, f_{10}(t_1, t_2), f_{01}(t_1, t_2))  &  = 0 \\label{closure_a1_a1_is_not_a2_F_eliminated}\n\\end{align}\nwhere $\\mathrm{E}_i (x_{t_2}, f_{10}(t_1, t_2), f_{01}(t_1, t_2))$ is a  holomorphic \nfunction of $(x_{t_2}, f_{10}(t_1, t_2), f_{01}(t_1, t_2))$. \nSince solutions to equation \\eqref{eval_f1}\nsatisfy  \\eqref{closure_a1_a1_is_not_a2_F_eliminated}, we conclude \nthat $f_{00}(t_1, t_2)$,\n$f_{10}(t_1, t_2)$ and $f_{01}(t_1, t_2)$ can not all be zero.  \nIf they were all zero \nthen $\\mathrm{F}$ could not be zero for small but non zero $x_{t_2}$ (by \\eqref{closure_a1_a1_is_not_a2_F_eliminated}), \nsince \n$\\mathcal{B}^{f(t_1, t_2)}_3 \\neq 0$ (this is where we are using $(\\tilde{f}, \\tilde{p}, \\tilde{p}) \\in \\Delta \\mathcal{A}_2$). \nThis implies \\eqref{a1_a1_closure_intersect_a1_is_empty_equation} holds \nas functionals. This completes the proof of Lemma \\ref{cl_two_pt} \\eqref{a1a_minus_1_cl}. \\qed  \\\\\n\n\n\\hf \\hf Before proceeding to the proof of Lemma \\ref{cl_two_pt} (\\ref{a1a1_up_cl}),  \nlet us prove  a corollary which will be needed in the proof of equation \\eqref{algoa1a1} in section \\ref{Euler_class_computation}. The proof of this corollary follows from the setup of the preceding proof, hence we prove it here.\n\\begin{cor}\n\\label{a1_section_contrib_from_a1_and_a2}\nLet $\\mathcal{W} \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P}^2$ \nbe a vector bundle  \nsuch that the rank of $\\mathcal{W}$  is \nsame as the dimension of $\\Delta \\mathcal{A}_2$\\footnote{This dimension is also one less than the dimension of $ \\Delta \\mathcal{A}_{1}$, equal to $\\delta_d-2$.} \nand $\\mathcal{Q}: \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P}^2 \\longrightarrow \\mathcal{W}$ \na generic smooth section.  \nThen the contribution of the section \n\\[ \\pi_2^* \\psi_{\\mathcal{A}_{0}} \\oplus \\pi_2^* \\psi_{\\mathcal{A}_{1}}  \\oplus \\mathcal{Q}: \\overline{\\mathcal{A}}_1 \\times \\mathbb{P}^2 \\longrightarrow \\pi_2^* \\mathcal{L}_{\\mathcal{A}_0} \\oplus  \\pi_2^* \\mathcal{V}_{\\mathcal{A}_1} \\oplus \\mathcal{W} \\]\nto the Euler class from the points of $\\Delta \\mathcal{A}_{1}$ is given by \n\\begin{align}\n\\mathcal{C}_{\\Delta \\mathcal{A}_1} (\\pi_2^* \\psi_{\\mathcal{A}_{0}} \\oplus \\pi_2^* \\psi_{\\mathcal{A}_{1}} \\oplus \\mathcal{Q}) = \\big\\langle e( \\pi_2^* \\mathcal{L}_{\\mathcal{A}_0} \\oplus \\mathcal{W}), ~[\\Delta \\overline{\\mathcal{A}}_1] \\big\\rangle.  \\label{contrib_from_delta_a1_psi_a1}\n\\end{align}\nSecondly, \nif  $(\\tilde{f},\\tilde{p}, \\tilde{p}) \\in \\Delta \\mathcal{A}_{2} \\cap \\mathcal{Q}^{-1}(0)$, \nthen this section vanishes on $(\\tilde{f}, \\tilde{p},  \\tilde{p})$ with a multiplicity of $3$. \n\\end{cor}\n\n\\begin{rem}\nWhen we use the phrase ``number of zeros''  (resp. ``number of solutions'') our intended meaning is number of zeros counted with a sign (resp. the number of solutions counted with a sign).\n\\end{rem}\n\n\\textbf{Proof of Corollary  \\ref{a1_section_contrib_from_a1_and_a2}:} The contribution of $\\pi_2^* \\psi_{\\mathcal{A}_{0}}\\oplus \\pi_2^* \\psi_{\\mathcal{A}_{1}} \\oplus \\mathcal{Q}$ \nto the the Euler class \nfrom the points of $\\Delta\\mathcal{A}_1$\nis\nthe number of solutions of \n\\begin{align}\n\\pi_2^*\\psi_{\\mathcal{A}_0} (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)) &=  \\nu_0(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)), \\nonumber \\\\  \n\\pi_2^*\\psi_{\\mathcal{A}_1} (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)) &=  \\nu_1(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)), \\nonumber  \\\\ \n\\mathcal{Q}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)) &= 0, ~~(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)) \\in \\mathcal{U}_{\\mathcal{K}} \\subset \\overline{\\mathcal{A}}_1 \\times \\mathbb{P}^2, \\label{psi_A1_contribution_from_A1}\n\\end{align}\nwhere $\\mathcal{K}$ is a sufficiently \nlarge compact subset of $\\Delta \\mathcal{A}_1$, $\\mathcal{U}_{\\mathcal{K}}$ is a sufficiently small neighborhood of $\\mathcal{K}$ inside $\\overline{\\mathcal{A}}_1 \\times \\mathbb{P}^2$, and $\\nu_0$ and $\\nu_1$ are generic smooth perturbations. We do not need to perturb $\\mathcal{Q}$ since it is already generic. We will convert the functional equation \\eqref{psi_A1_contribution_from_A1} into an equation that involves equality of numbers. \nLet $\\theta_1, \\theta_2, \\ldots, \\theta_{\\delta_d-2}$ form a basis for $\\mathcal{W}^*$ at the \npoint $(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1))$. \nDefine \n\\begin{align}\n\\xi_{0} (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1), (\\xt, y_{t_2})) &:=  \\{ \\nu_0(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)) \\}(f(t_1, t_2) \\otimes p(t_1)^{\\otimes d}) \\in \\mathbb{C}, \\nonumber \\\\\n\\xi_{1x} (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1), (\\xt, y_{t_2})) &:=  \\{ \\nu_1(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)) \\}(f(t_1, t_2) \\otimes p(t_1)^{\\otimes d} \\otimes v) \\in \\mathbb{C},\\nonumber  \\\\\n\\xi_{1y} (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1), (\\xt, y_{t_2})) &:=  \\{ \\nu_1(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)) \\}(f(t_1, t_2) \\otimes p(t_1)^{\\otimes d} \\otimes w)\\in \\mathbb{C}, \\nonumber \\\\\n\\mathcal{R}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1), (\\xt, y_{t_2})) &:= \\{ \\mathcal{Q}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)) \\}(\\theta_1 \\oplus \\theta_2 \\oplus \\ldots \\oplus \\theta_{\\delta_{d-2}}) \\in \\mathbb{C}^{\\delta_d-2}. \\label{number_defn}\n\\end{align}\nConsider now the following set of equations \n\\begin{align}\n\\xi_0 (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1), (\\xt, y_{t_2})) + \\xt \\xi_{1x} (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1), (\\xt, y_{t_2})) &  \\nonumber \\\\ \n  \\qquad \\qquad + y_{t_2} \\xi_{1y} (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1), (\\xt, y_{t_2})) & \\nonumber \\\\ \n+ \\frac{f_{20}(t_1, t_2)}{2} x_{t_2}^2 + f_{11}(t_1, t_2) x_{t_2} y_{t_2} +  \n\\frac{f_{02}(t_1, t_2)}{2} y_{t_2}^2 + \\ldots &=0  \\label{eval_perturbed_number_again} \\\\ \n\\mathcal{R}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1), (\\xt, y_{t_2})) &=0 \\label{relation_number_generic} \\\\ \n\\xi_{1x} (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1), (\\xt, y_{t_2}))  +  f_{20}(t_1, t_2) \\xt + f_{11}(t_1, t_2) y_{t_2} + \\ldots &=0 \\label{fx_perturbed_number} \\\\ \n\\xi_{1y} (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1), (\\xt, y_{t_2}))  + f_{11}(t_1, t_2) x_{t_2} + f_{02}(t_1, t_2) y_{t_2} + \\ldots &= 0 \\label{fy_perturbed_number} \\\\ \n\\textnormal{$(\\xt, y_{t_2})=$  small}, \\qquad \\big|f_{20}(t_1, t_2) f_{02}(t_1, t_2) - f_{11}(t_1, t_2)^2\\big| & > \\mathrm{C}   \\label{psi_A1_contribution_from_A1_numbers}\n\\end{align}\nwhere $\\mathrm{C}$ is a small positive constant. Observe that the number of solutions \nof \\eqref{psi_A1_contribution_from_A1} is same as the number of solutions of \n\\eqref{eval_perturbed_number_again}-\\eqref{psi_A1_contribution_from_A1_numbers}. \nLet $\\mathrm{N}$ be the number of solutions $(\\tilde{f}, \\tilde{p})$ of     \n\\begin{align}\n\\xi_{0}(\\tilde{f}, \\tilde{p}, 0, 0) =0, ~~\\mathcal{R}(\\tilde{f},\\tilde{p}, 0,0) =0, ~~f_{00}=0, ~~ (f_{10}, ~f_{01}) = (0,0).  \\label{eval_perturbed_number} \n\\end{align}\nObserve that this number is same as\nthe number of solutions \n\\begin{align}\n\\nu_0(\\tilde{f}, \\tilde{p}, \\tilde{p}) =0, ~~\\mathcal{Q}(\\tilde{f}, \\tilde{p}, \\tilde{p}) =0, ~~(\\tilde{f}, \\tilde{p}, \\tilde{p}) \\in \\Delta \\overline{\\mathcal{A}}_1.  \\label{psi_a0_nu_0_Q_c0}\n\\end{align}\nHence \n\\bge\n\\mathrm{N} = \\big\\langle e( \\pi_2^* \\mathcal{L}_{\\mathcal{A}_0} \\oplus \\mathcal{W}), ~[\\Delta \\overline{\\mathcal{A}}_1] \\big\\rangle. \\label{N_equal_Euler}\n\\ede\nSince $\\mathcal{Q}$ is generic, all solutions of \\eqref{psi_a0_nu_0_Q_c0} \n(and hence \\eqref{eval_perturbed_number})\nbelong to $\\Delta \\mathcal{A}_1$, i.e,  \n\\begin{align}\nf_{20}  f_{02} - f_{11}^2 \\neq 0 \\implies \\big|f_{20} f_{02} - f_{11}^2\\big| > \\mathrm{C}. \\label{ift_hypothesis_det_hess}\n\\end{align}\nLet $(\\tilde{f}, \\tilde{p})$ be a solution of \\eqref{eval_perturbed_number}. \nSince the sections \n\\[ (\\pi_2^*\\psi_{\\mathcal{A}_0}+\\nu_0) \\oplus \\mathcal{Q}: \\overline{\\mathcal{A}}_0\\times\\{\\tilde{p}\\} \\longrightarrow \\pi_2^* \\mathcal{L}_{\\mathcal{A}_0} \\oplus \\mathcal{W}, \\qquad  (\\pi_2^*\\psi_{\\mathcal{A}_0} +\\nu_0) \\oplus \\mathcal{Q}: \\overline{\\mathcal{A}}_0 \\times \\mathbb{P}^2 \\longrightarrow \\pi_2^* \\mathcal{L}_{\\mathcal{A}_0} \\oplus \\mathcal{W} \\]\nare transverse to the zero set (for any $\\tilde{p} \\in \\mathbb{P}^2$)\nwe conclude that given an $(\\xt, y_{t_2})$ sufficiently small, there exists a unique \n$(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1)) \\in \\overline{\\mathcal{A}}_0$\nclose to  $(\\tilde{f}, \\tilde{p})$, such that $(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1), (\\xt, y_{t_2}))$ \nsolves \\eqref{eval_perturbed_number_again} and \\eqref{relation_number_generic}.  \nPlugging in this value in \\eqref{fx_perturbed_number} and \\eqref{fy_perturbed_number} and \nusing \\eqref{ift_hypothesis_det_hess}, we conclude \nthere is  a unique $(\\xt, y_{t_2})$ that solves \\eqref{fx_perturbed_number} and \\eqref{fy_perturbed_number} (provided the norm of $\\xi_{1x}$ and $\\xi_{1y}$ is sufficiently small). \nHence, there is a one to one correspondence between the number solutions of  \\eqref{eval_perturbed_number} and the solutions of \n\\eqref{eval_perturbed_number_again}-\\eqref{psi_A1_contribution_from_A1_numbers}. Equation \\eqref{N_equal_Euler} now proves \n\\eqref{contrib_from_delta_a1_psi_a1}. \\\\ \n\\hf \\hf Next, suppose $(\\tilde{f},\\tilde{p}, \\tilde{p}) \\in \\Delta \\mathcal{A}_{2} \\cap \\mathcal{Q}^{-1}(0)$. \nSince $f_{20}$ and $f_{02}$ are not both zero, let us assume $f_{02} \\neq 0$.\nThe contribution of the section \n\\[ \\pi_2^* \\psi_{\\mathcal{A}_{0}} \\oplus \\pi_2^* \\psi_{\\mathcal{A}_{1}} \\oplus \\mathcal{Q}: \\overline{\\mathcal{A}}_1 \\times \\mathbb{P}^2 \\longrightarrow \\pi_2^* \\mathcal{L}_{\\mathcal{A}_0} \\oplus \\pi_2^* \\mathcal{V}_{\\mathcal{A}_1} \\oplus \\mathcal{W} \\]\nto the Euler class is the number of solutions of  \n\\begin{align}\n\\pi_2^*\\psi_{\\mathcal{A}_0} (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1))&=  \\nu_0, ~~\\pi_2^*\\psi_{\\mathcal{A}_1} (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1))=  \\nu_1 \\nonumber \\\\ \n\\mathcal{Q}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)) &= 0, \\qquad (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)) \\in \\mathcal{U}_{(\\tilde{f}, \\tilde{p},\\tilde{p})} \\subset \\overline{\\mathcal{A}}_1 \\times \\mathbb{P}^2 \\label{contrib_psi_a0_plus_a1_from_cusps}\n\\end{align}\nwhere $\\mathcal{U}_{(\\tilde{f}, \\tilde{p},\\tilde{p})}$ is a sufficiently small neighbourhood of $(\\tilde{f}, \\tilde{p}, \\tilde{p})$ inside \n$\\overline{\\mathcal{A}}_1 \\times \\mathbb{P}^2$ and $\\nu_0$ and $\\nu_1$\nare generic smooth perturbations. Let $\\xi_0$, $\\xi_{1x}$, $\\xi_{1y}$ and $\\mathcal{R}$ be defined \nas in \\eqref{number_defn}. \nThe number of solutions of \\eqref{contrib_psi_a0_plus_a1_from_cusps} \n(which is a functional equation), is equal to the number of \nsolutions of \\eqref{eval_perturbed_number_again}-\\eqref{fy_perturbed_number} and \n\\begin{align}\n\\textnormal{$(\\xt, y_{t_2})=$  small}, \\qquad  f_{02}(t_1, t_2)\\mathcal{B}^{f(t_1, t_2)}_2 := f_{02}(t_1, t_2)f_{20}(t_1, t_2) - f_{11}(t_1, t_2) = \\textnormal{small}.   \\label{psi_A1_contribution_from_A2_numbers}\n\\end{align}\nSince the section\n\\[ \\psi_{\\mathcal{A}_2} \\oplus \\mathcal{Q} : \\Delta \\overline{\\mathcal{A}}_1 \\longrightarrow \\mathcal{L}_{\\mathcal{A}_2} \\oplus \\mathcal{W}  \\]\nis transverse to the zero set, we conclude that \nthere is a unique $(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1)) \\in \\Delta \\overline{\\mathcal{A}}_1 \\cap \\mathcal{Q}^{-1}(0)$ \nclose to $(\\tilde{f}, \\tilde{p})$ with a specified value of $\\mathcal{B}^{f(t_1, t_2)}_2$.\\footnote{This is true provided $\\mathcal{B}^{f(t_1, t_2)}_2$ is sufficiently small.} \nIn other words, we can express all the \n$f_{ij}(t_1, t_2)$ in terms of $\\mathcal{B}^{f(t_1, t_2)}_2$. \nPlug in this expression for $f_{ij}(t_1, t_2)$\nin  \n\\eqref{eval_perturbed_number_again}, \\eqref{fx_perturbed_number} and \\eqref{fy_perturbed_number}. \nSince $~(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1))$  \nis close to $\\Delta \\mathcal{A}_2$,   \nwe conclude that after a change of coordinates, the set of equations  \n\\eqref{eval_perturbed_number_again}, \\eqref{fx_perturbed_number} and \\eqref{fy_perturbed_number}\nis equivalent to   \n\\eqref{closure_a1_a1_is_not_a2_F}, \n\\eqref{closure_a1_a1_is_not_a2_Fx} and \\eqref{closure_a1_a1_is_not_a2_Fy}, \nwith  \n\\begin{align}\nf_{00}(t_1, t_2) & = \\xi_{0}, \\qquad  f_{10}(t_1, t_2) = \\xi_{1x}, \\qquad f_{01}(t_1, t_2) = \\xi_{1y}. \\label{psi_a1_contribution_from_a2}\n\\end{align}\nHence, \\eqref{closure_a1_a1_is_not_a2_F_eliminated} holds which combined with \\eqref{psi_a1_contribution_from_a2} \nimplies that the multiplicity is $3$ to one in $x_{t_2}$. \nGiven $x_{t_2}$, we can solve for $\\hat{y}_{t_2}$ uniquely using \\eqref{closure_a1_a1_is_not_a2_F_hat_y}. \nAnd given $x_{t_2}$ and $\\hat{y}_{t_2}$, we can uniquely solve for  $\\mathcal{B}^{f(t_1, t_2)}_2$ \nusing \\eqref{closure_a1_a1_is_not_a2_Fx}, provided $\\xi_{1x}$ and $\\xi_{1y}$ are sufficiently small. \nHence, the total multiplicity is $3$. \\qed \\\\\n\n\n\\begin{comment}\nThe rest of the $f_{ij}$ can be expressed in terms of $x_{t_2}$, $\\hat{y}_{t_2}$ and $\\mathcal{B}^{f(t_1, t_2)}_2$. \nTo see why this is so, first observe that \n$\\mathcal{Q}$ is a generic section; hence the section \n$$ \\pi_2^*\\psi_{\\mathcal{A}_2} \\oplus \\mathcal{Q}: \\overline{\\mathcal{A}}_1 \\circ \\overline{\\mathcal{A}}_1 \\longrightarrow \\pi_2^*\\mathcal{L}_{\\mathcal{A}_2} \\oplus \\mathcal{W}  $$\nis transverse to the zero set. \nBy the Implicit Function Theorem, if $(\\tilde{p}(t_1, t_2), \\tilde{p}(t_1))$ is \nclose to $(\\tilde{p}, \\tilde{p})$ and \n$\\pi_2^*\\psi_{\\mathcal{A}_2}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1))$\nis close to zero, \nthen there exists a unique $\\tilde{f}(t_1, t_2)$ close to \n$\\tilde{f}$ that solves \n$$ \\mathcal{Q}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)) = 0, \\qquad (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1)) \\in \\overline{\\mathcal{A}}_1\\circ \\overline{\\mathcal{A}}_1.$$ \nIn other words, we can express $\\tilde{f}(t_1, t_2)$ as a function of \n$\\tilde{p}(t_1, t_2)$ , $\\tilde{p}(t_1)$ and $\\pi_2^*\\psi_{\\mathcal{A}_2}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), \\tilde{p}(t_1))$.  \nHence $f_{ij}$ can be expressed in terms of $x_{t_2}$, $\\hat{y}_{t_2}$ and $\\mathcal{B}^{f(t_1, t_2)}_2$. \nHence the set of equations \\eqref{psi_a1_contribution_from_a2} is  $3$ to one.  \\qed \n\\end{comment}   \n\n\n\n\\textbf{Proof of Lemma \\ref{cl_two_pt} (\\ref{a1a1_up_cl}):} It suffices to show  that \n\\bge\n\\Big\\{ (\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\hat{\\mathcal{A}}^{\\#}_1} \\Big\\} = \\Delta \\overline{\\hat{\\mathcal{A}}}_{3}. \\label{a1_hat_a1_sharp_is_a3_sharp}  \n\\ede\nBy using Lemma \\ref{tube_lemma} with Lemma \\ref{Dk_sharp_closure} \\eqref{a1_cl_new} we get $\\overline{\\hat{\\mathcal{A}}}_1=\\hat{\\mathcal{A}}_1\\cup\\overline{\\hat{\\mathcal{A}}}_2$. \nNow we use Lemma \\ref{tube_lemma} again with Lemma \\ref{cl} \\eqref{A2cl} to get $\\overline{\\hat\\mathcal{A}}_2=\\hat{\\mathcal{A}}_2\\sqcup \\overline{\\hat\\mathcal{A}}_3 \\cup \\overline {\\hat{\\mathcal{D}}}_4$. \nBy Lemma \\ref{cl} \\eqref{A3cl} and \\ref{tube_lemma}, we conclude that $\\overline{\\hat{\\mathcal{D}}}_4 $ is a subset of $\\overline{\\hat{\\mathcal{A}}}_3$. \nHence $\\overline{\\hat\\mathcal{A}}_2=\\hat{\\mathcal{A}}_2\\sqcup \\overline{\\hat\\mathcal{A}}_3$. \nThis implies that\n\\bgd\n\\Delta \\overline{\\hat{\\mathcal{A}}}_1=\\Delta \\hat{\\mathcal{A}}_1\\sqcup \\Delta\\hat{\\mathcal{A}}_2\\sqcup \\Delta\\overline{\\hat{\\mathcal{A}}}_3.\n\\edd\nFirst we observe that the lhs  of \\eqref{a1_hat_a1_sharp_is_a3_sharp} \nis a subset of its rhs. To see this, observe that  by \\eqref{a1_du_a2_intersect_a1_a1_is_empty}\n\\begin{align*}\n\\{ (\\tilde{f}, \\tilde{p}, \\tilde{p}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\overline{\\mathcal{A}}_1} \\} \\cap \\Delta (\\mathcal{A}_1 \\cup \\mathcal{A}_{2}) & = \\varnothing \\\\\n\\implies \\{ (\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\overline{\\hat{\\mathcal{A}}}_1} \\} \\cap \\Delta (\\hat{\\mathcal{A}}_1 \\cup \\hat{\\mathcal{A}}_{2}) & = \\varnothing \\\\\n\\implies \\{ (\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\hat{\\mathcal{A}}^{\\#}_1} \\} \\cap \\Delta (\\hat{\\mathcal{A}}_1 \\cup \\hat{\\mathcal{A}}_{2}) &= \\varnothing  \\qquad \\textnormal{since} ~\\overline{\\hat{\\mathcal{A}}^{\\#}_1} = \\overline{\\hat{\\mathcal{A}}}_1\\,\\,(\\textup{Lemma}\\, \\,\\ref{cl}\\,\\eqref{A1cl}). \n\\end{align*} \n\n\\noindent Next we will show that the rhs  of \\eqref{a1_hat_a1_sharp_is_a3_sharp} is a \nsubset of its lhs. We will simultaneously prove the following two statements. \n\\begin{align}\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\hat{\\mathcal{A}}_1^{\\#}} \\} &\\supset \\Delta (\\hat{\\mathcal{A}}_{3} \\sqcup \\hat{\\mathcal{D}}_4^{\\#\\flat}),  \\label{a3_is_subset_of_a1_a1_closure_down_stairs}   \\\\\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}_2} \\} \\cap \\Delta (\\hat{\\mathcal{A}}_{3} \\sqcup \\hat{\\mathcal{D}}_4^{\\#\\flat}) &= \\varnothing. \\label{a3_interesct_a1_pa2_is_empty_set_equation}\n\\end{align} \n\n\\noindent Since $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\hat{\\mathcal{A}}_1^{\\#}}$ is a closed set, \\eqref{a3_is_subset_of_a1_a1_closure_down_stairs} implies that \nthe rhs  of \\eqref{a1_hat_a1_sharp_is_a3_sharp} is a subset of its lhs.\\footnote{As before, we do not need the full strength of \n\\eqref{a3_is_subset_of_a1_a1_closure_down_stairs}. We simply need that \n$\\{ (\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\hat{\\mathcal{A}}_1^{\\#}} \\} \\supset \\Delta \\hat{\\mathcal{A}}_{3}$.}\n\n\\begin{claim}\n\\label{claim_a3_subset_of_a1_a1_closure}\nLet $~(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta (\\hat{\\mathcal{A}}_{3} \\sqcup \\hat{\\mathcal{D}}_4^{\\#\\flat})$.\nThen there exist solutions \n$$ (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) \\in \\overline{(\\mathcal{D} \\times \\mathbb{P}^2) \\circ \\hat{\\mathcal{A}}_1^{\\#}}$$ \nclose to  $(\\tilde{f}, \\tilde{p}, l_{\\p})$ to the set of equations \n\\begin{align}\n\\label{a3_is_subset_of_a1_a1_closure_down_stairs_equation}\n\\pi_1^* \\psi_{\\mathcal{A}_0}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, ~\\pi_1^* \\psi_{\\mathcal{A}_1}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)})= 0, ~~\\tilde{p}(t_1, t_2) \\neq \\tilde{p}(t_1). \n\\end{align}\nMoreover, whenever such a solution is sufficiently close to $(\\tilde{f},\\tilde{p}, l_{\\p})$, it lies in $\\overline{\\mathcal{A}}_1 \\circ \\hat{\\mathcal{A}}_1^{\\#}$, i.e., \n\\begin{align}\n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{A}_2}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  \\tilde{p}(t_1)) & \\neq 0. \\label{pa3_intersect_a1_a2_is_empty}\n\\end{align}\nIn particular, $(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  \\tilde{p}(t_1))$ does not lie in $\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}_2$.\n\\end{claim}\nIt is easy to see that claim \\ref{claim_a3_subset_of_a1_a1_closure} proves  \\eqref{a3_is_subset_of_a1_a1_closure_down_stairs} and \\eqref{a3_interesct_a1_pa2_is_empty_set_equation} \nsimultaneously. \\\\\n\n\\noindent \\textbf{Proof:  } Choose homogeneous coordinates $[\\mathrm{X}: \\mathrm{Y}: \\mathrm{Z}]$ so that \n$\\tilde{p} = [0:0:1]$ and let ~$\\mathcal{U}_{\\tilde{p}}$,   \n$\\pi_x$, $\\pi_y$,  $v$, $w$, \n$x_{t_1}$, $y_{t_1}$, $x_{t_2}$, $y_{t_2}$,\n$f_{ij}(t_1, t_2)$\nbe exactly the same as defined in the \nproof of claim \\ref{a1_a1_closure_intersect_a1_or_a2_empty_equations_claim}.\nTake\n\\[(\\tilde{f} (t_1, t_2),  l_{\\tilde{p}(t_1)}) \\in \\overline{\\hat{\\mathcal{A}}^{\\#}_1}\\] \nto be a point that is close to \n$( \\tilde{f}, l_{\\tilde{p}})$ and $l_{\\tilde{p}(t_1, t_2)}$ a point \nin $\\mathbb{P} T\\mathbb{P}^2$ that is close to $l_{\\tilde{p}(t_1)}$.\nWithout loss of generality, we can assume that   \n\\[ v + \\eta w \\in l_{\\tilde{p}}, ~~v + \\eta_{t_1} w \\in l_{\\tilde{p}(t_1)} ~~\\textnormal{and} ~~v + (\\eta_{t_1} + \\eta_{t_2}) w \\in l_{\\tilde{p}(t_1, t_2)} \\] \nfor some complex \nnumbers $\\eta$, $\\eta_{t_1}$ and $\\eta_{t_1} +\\eta_{t_2}$ close to each other. \nLet the numbers \n$\\mathrm{F}$, $\\mathrm{F}_{x_{t_2}}$ and $\\mathrm{F}_{y_{t_2}}$ be the same as in the proof of  claim \\ref{a1_a1_closure_intersect_a1_or_a2_empty_equations_claim}.\nSince  $(\\tilde{f} (t_1, t_2),  l_{\\tilde{p}(t_1)}) \\in \\overline{\\hat{\\mathcal{A}}^{\\#}_1}$, we conclude that \n\\[ f_{00}(t_1, t_2) =f_{10}(t_1, t_2) = f_{01}(t_1, t_2) =0.\\] \nThe functional equation   \n\\eqref{a3_is_subset_of_a1_a1_closure_down_stairs_equation}  \nhas a solution if and only if the following has a numerical solution: \n\\begin{align}\n\\mathrm{F} = 0, \\qquad \\mathrm{F}_{x_{t_2}} = 0, \\qquad \\mathrm{F}_{y_{t_2}} = 0,  \\qquad (x_{t_2}, y_{t_2}) \\neq (0, 0) \\qquad \\textnormal{(but small)}. \\label{eval_f1_again_d4_hat} \n\\end{align} \nFirst let us assume $(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\Delta \\hat{\\mathcal{A}}_3$. \nIt is easy to see that $f_{20}$ and $f_{02}$ can not both be zero; \nlet us assume $f_{02}$ is non zero.  \nFollowing the same argument as in the proof of claim \\ref{a1_a1_closure_intersect_a1_or_a2_empty_equations_claim}, \nwe make a change of coordinates and write $\\mathrm{F}$ as \n\\begin{align*}\n\\mathrm{F} & = \\hat{\\hat{y}}_{t_2}^2 +\\frac{\\mathcal{B}^{f(t_1, t_2)}_{2}}{2!} x_{t_2}^2 + \\frac{\\mathcal{B}^{f(t_1, t_2)}_{3}}{3!} x_{t_2}^3 + \n\\frac{\\mathcal{B}^{f(t_1, t_2)}_{4}}{4!} x_{t_2}^4 + \\mathrm{O}(x_{t_2}^5), \\\\ \n\\textnormal{where} & \\qquad \\hat{\\hat{y}}_{t_2} := \\sqrt{\\varphi ( x_{t_2}, \\hat{y}_{t_2})} \\hat{y}_{t_2}, \\qquad \\mathcal{B}^{f(t_1, t_2)}_2 = f_{20}(t_1, t_2) - \\frac{f_{11}(t_1, t_2)^2}{f_{02}(t_1, t_2)}.  \n\\end{align*}\nIt is easy to see that the only solutions to \\eqref{eval_f1_again_d4_hat}  (in terms of the new coordinates) are\n\\bge\n\\mathcal{B}_{2}^{f(t_1, t_2)}  = \\ {\\textstyle \\frac{\\mathcal{B}_{4}^{f(t_1, t_2)}}{12}} x_{t_2}^2  + \\mathrm{O}(x_{t_2}^3), ~\\mathcal{B}_{3}^{f(t_1, t_2)} = {\\textstyle \\frac{-\\mathcal{B}_{4}^{f(t_1, t_2)}}{2}} x_{t_2}+\\mathrm{O}(x_{t_2}^2), ~\\hat{\\hat{y}}_{t_2} =0, ~x_{t_2} \\neq 0 ~\\textnormal{(but small).} \\label{b20_equation} \n\\ede\nIt remains to show that these solutions satisfy \\eqref{pa3_intersect_a1_a2_is_empty}. \nFirst consider the case when $(\\tilde{f}, \\tilde{p},  l_{\\p}) \\notin \\mathcal{P} \\mathcal{A}_3 $, i.e., \n$\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{A}_2}(\\tilde{f}, \\tilde{p},  l_{\\p})  \\neq 0$.\nThen \\eqref{pa3_intersect_a1_a2_is_empty} is obviously true, since the section $\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{A}_2}$ is \\textit{continuous}. \nNext, consider the case when $(\\tilde{f}, \\tilde{p},  l_{\\p}) \\in \\mathcal{P} \\mathcal{A}_3$. \nDefine the numbers \n\\begin{align}\n\\mathrm{J}_1 &:= f_{20}(t_1, t_2) + \\eta_{t_1} f_{11}(t_1, t_2), ~~ \\mathrm{J}_2:= f_{11}(t_1, t_2) + \\eta_{t_1} f_{02}(t_1, t_2). \\label{b2_can_not_be_zero}\n\\end{align}\nObserve that \n\\begin{align}\n\\{ \\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{A}_2}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} ) \\} (f(t_1, t_2) \\otimes p(t_1)^{\\otimes d} \\otimes (v + \\eta_{t_1} w) \\otimes v) &= \\mathrm{J}_1, \\nonumber  \\\\   \n\\{ \\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{A}_2}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} ) \\} (f(t_1, t_2) \\otimes p(t_1)^{\\otimes d} \\otimes (v + \\eta_{t_1} w) \\otimes w) &= \\mathrm{J}_2.  \n\\label{psi_pa2_number_form}\n\\end{align}\nIf \\eqref{pa3_intersect_a1_a2_is_empty} were false, then $\\mathrm{J}_1$ and $\\mathrm{J}_2$ would vanish (by \\eqref{psi_pa2_number_form}). \nEquations \\eqref{b2_can_not_be_zero} and \\eqref{b20_equation} imply\n\\begin{align}\n\\frac{\\mathcal{B}^{f(t_1, t_2)}_{4}}{12} x_{t_2}^2 + \\mathrm{O}(x_{t_2}^3) &= \\mathrm{J}_1 - \\frac{f_{11}(t_1, t_2)}{f_{02}(t_1, t_2)} \\mathrm{J}_2.   \\label{b2_can_not_be_zero_again}\n\\end{align}\nSince $(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\hat{\\mathcal{A}}_3$, \nwe conclude  $\\mathcal{B}^{f(t_1, t_2)}_{4} \\neq 0$. Hence, \n\\eqref{b2_can_not_be_zero_again} implies that \n$\\mathrm{J}_1$ and $\\mathrm{J}_2$ can not both be zero. \nHence \\eqref{pa3_intersect_a1_a2_is_empty} holds.  \\\\\n\\hf \\hf Next, let us assume that $(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\Delta \\hat{\\mathcal{D}}_4^{\\#\\flat}$. \nDefine the following number \n\\begin{align*}\n\\mathrm{G}&:= x_{t_2} \\mathrm{F}_{x_{t_2}} + y_{t_2} \\mathrm{F}_{y_{t_2}} -2 \\mathrm{F} \\\\\n          & = \\frac{f_{30}(t_1, t_2)}{6} x_{t_2}^3 + \\frac{f_{21}(t_1, t_2)}{2} x_{t_2}^2 y_{t_2} +  \\frac{f_{12}(t_1, t_2)}{2} x_{t_2} y_{t_2}^2 + \\frac{f_{03}(t_1, t_2)}{6} y_{t_2}^3 + \\ldots.\n\\end{align*}\nNote that the cubic term of $\\mathrm{G}$ is same as the cubic term of $\\mathrm{F}$. \nSince $(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\Delta\\hat{\\mathcal{D}}_4$  \nwe conclude, using the same argument as in \\cite{BM13}\nthat there exists a change of coordinates  \n\\begin{align*}\nx_{t_2} & = \\hat{x}_{t_2} +\\mathrm{E}_1 (\\hat{x}_{t_2}, \\hat{y}_{t_2}), \\qquad y_{t_2} = \\hat{y}_{t_2} + \\mathrm{E}_2 (\\hat{x}_{t_2}, \\hat{y}_{t_2}),  \n\\end{align*}\n(where $\\mathrm{E}_i(\\hat{x}_{t_2}, \\hat{y}_{t_2})$ are second order in $\\hat{x}_{t_2}$ and $\\hat{y}_{t_2}$)\nso that  $\\mathrm{G}$ is given by \n\\begin{align}\n\\mathrm{G} & =\\frac{f_{30}(t_1, t_2)}{6} \\hat{x}_{t_2}^3 + \\frac{f_{21}(t_1, t_2)}{2} \\hat{x}_{t_2}^2 \\hat{y}_{t_2} +  \\frac{f_{12}(t_1, t_2)}{2} \\hat{x}_{t_2} \\hat{y}_{t_2}^2 + \n\\frac{f_{03}(t_1, t_2)}{6} \\hat{y}_{t_2}^3.  \\label{new_G}\n          \n\\end{align}\nSince $(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\Delta\\hat{\\mathcal{D}}_4$, \nthere \nare three possibilities to consider;  \n\\begin{align}\nf_{30}(t_1, t_2) & \\neq 0  \\qquad \\textnormal{or} \\qquad f_{03}(t_1, t_2) \\neq 0 ~~ \\textnormal{or} \\nonumber \\\\ \nf_{30}(t_1, t_2) & = f_{03}(t_1, t_2) =0, ~~~~\\textnormal{but} ~~~~f_{21}(t_1, t_2) \\neq 0 ~~\\textnormal{and} ~~f_{12}(t_1, t_2) \\neq 0. \\label{fij_cases_D4_hat}\n\\end{align}\nLet us assume $f_{30}(t_1, t_2)\\neq 0$. \nSince $(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\Delta\\hat{\\mathcal{D}}_4^{\\#\\flat}$, \nequation \\eqref{new_G} now can be written as   \n\\begin{align}\n\\mathrm{G}&=  \n\\frac{f_{30}(t_1,t_2)}{6}(\\hat{x}_{t_2} - \\mathrm{A}_1 \\hat{y}_{t_2}) (\\hat{x}_{t_2} - \\mathrm{A}_2 \\hat{y}_{t_2}) (\\hat{x}_{t_2} - \\mathrm{A}_3 \\hat{y}_{t_2})\n\\end{align}\nwhere $\\mathrm{A}_i$ are complex numbers such that \n\\begin{align}\n \\mathrm{A}_1 & \\neq \\mathrm{A}_2 \\neq \\mathrm{A}_3\\neq \\mathrm{A}_1 ~~\\textnormal{and} ~~\\eta \\neq \\frac{1}{\\mathrm{A}_1}, ~\\frac{1}{\\mathrm{A}_2} ~~\\textnormal{or} ~\\frac{1}{\\mathrm{A}_3}. \n \\label{eta_neq_d4_sharp}\n\\end{align}\nTo see why the last  inequality is true,   \nnote that if $\\eta$ is either $\\frac{1}{\\mathrm{A}_1}$, \n$\\frac{1}{\\mathrm{A}_2}$ or $\\frac{1}{\\mathrm{A}_3}$ then \n\\[ \\nabla^3 f|_{\\tilde{p}} (v+ \\eta w, v + \\eta w, v + \\eta w) =0. \\]\nSince \n$(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\Delta \\hat{\\mathcal{D}}_{4}^{\\#\\flat}$ \nthe \nlast inequality of \\eqref{eta_neq_d4_sharp}  holds. \nHence,   \n\\eqref{a3_is_subset_of_a1_a1_closure_down_stairs_equation}  \nhas a solution if and only if \n\\begin{align}\n\\mathrm{G} = &\\,\\,  \\frac{f_{30}(t_1,t_2)}{6}(\\hat{x}_{t_2} - \\mathrm{A}_1 \\hat{y}_{t_2}) (\\hat{x}_{t_2} - \\mathrm{A}_2 \\hat{y}_{t_2}) (\\hat{x}_{t_2} - \\mathrm{A}_3 \\hat{y}_{t_2}) = 0, \\nonumber \\\\ \n\\mathrm{F}_{x_{t_2}} =&\\,\\,  \\hat{x}_{t_2} f_{20}(t_1, t_2) + \\hat{y}_{t_2} f_{11}(t_1, t_2) \\nonumber \\\\ \n                      & + \\frac{f_{30}(t_1, t_2)}{6} \\Big( 3 \\hat{x}_{t_2}^2 \n                      -2 \\hat{x}_{t_2} \\hat{y}_{t_2}\\big ({\\textstyle \\sum_{i=1}^3\\mathrm{A}_i}\\big) + \n                                                                (\\mathrm{A}_1\\mathrm{A}_2 + \\mathrm{A}_1\\mathrm{A}_3+ \\mathrm{A}_2\\mathrm{A}_3 ) \n                                                                \\hat{y}_{t_2}^2 \\Big)+ \n                                                                \\mathrm{E}_3(\\hat{x}_{t_2}, \\hat{y}_{t_2}) =0, \\nonumber \\\\\n\\mathrm{F}_{y_{t_2}} =&\\,\\, \\hat{x}_{t_2} f_{11}(t_1, t_2) + \\hat{y}_{t_2} f_{02}(t_1, t_2) \\nonumber \\\\ \n                      & -\\frac{f_{30}(t_1, t_2)}{6}\\Big(\\hat{x}_{t_2}^2\\big({\\textstyle\\sum_{i=1}^3  \\mathrm{A}_i}\\big) \n                      -2 \\hat{x}_{t_2} \\hat{y}_{t_2}\\big({\\textstyle \\sum_{i\\neq j}\\mathrm{A}_i\\mathrm{A}_j}\\big)  \n                                                                +3\\hat{y}_{t_2}^2\\mathrm{A}_1\\mathrm{A}_2 \\mathrm{A}_3 \\Big) + \\mathrm{E}_4(\\hat{x}_{t_2}, \\hat{y}_{t_2}) =0 \\nonumber \\\\  \n(\\hat{x}_{t_2}, \\hat{y}_{t_2}) \\neq &\\,\\, (0,0) \\qquad \\textnormal{(but small)}  \\label{hat_D4_neighbourhood_inside_a1_hat_a1}\n\\end{align}\nhas a solution, where $\\mathrm{E}_i(\\hat{x}_{t_2}, \\hat{y}_{t_2})$ are third order in $(\\hat{x}_{t_2}, \\hat{y}_{t_2})$. \\\\  \n\\hf \\hf To avoid confusion let us clarify one point; in the above equation \n$\\mathrm{F}_{x_{t_2}}$ \nand $\\mathrm{F}_{y_{t_2}}$  \nare simply  \nexpressed in terms of the new coordinates \n$\\hat{x}_{t_2}$ and $\\hat{y}_{t_2}$. They are still the partial derivatives of \n$\\mathrm{F}$ with respect to $x_{t_2}$ and $y_{t_2}$;  \nthey are \\textit{not} $\\mathrm{F}_{\\hat{x}_{t_2}}$ \nand $\\mathrm{F}_{\\hat{y}_{t_2}}$, the  partial derivatives of  $\\mathrm{F}$ with respect to \n$\\hat{x}_{t_2}$ and $\\hat{y}_{t_2}$. Now we will construct the solutions to \\eqref{hat_D4_neighbourhood_inside_a1_hat_a1}. \nThere are three solutions; we will just give one of the solutions, the rest are similar.   \nThey are given by: \n\\begin{align}\n\\hat{x}_{t_2}&= \\mathrm{A}_1 \\hat{y}_{t_2}, ~~\\hat{y}_{t_2} \\neq 0  ~~\\textnormal{(but small)}, ~~f_{20}(t_1, t_2) = \\textnormal{small}, \\nonumber \\\\\nf_{02}(t_1, t_2) & = \\frac{f_{30}(t_1, t_2)}{6}\\Big( \n2 \\mathrm{A}_1^3 - 2 \\mathrm{A}_1^2 \\mathrm{A}_2 - \\mathrm{A}_1^2 \\mathrm{A}_3 + 2 \\mathrm{A}_1 \\mathrm{A}_2\\mathrm{A}_3  \\Big) \\hat{y}_{t_2} \n+ \\mathrm{A}_1^2 f_{20}(t_1, t_2) + \n\\mathrm{E}_5(\\hat{y}_{t_2}), \\nonumber \\\\  \nf_{11}(t_1, t_2) & = \\frac{f_{30}(t_1, t_2)}{6}\\Big( -\\mathrm{A}_1^2 +\\mathrm{A}_1 \\mathrm{A}_2 +\\mathrm{A}_1 \\mathrm{A}_3 - \\mathrm{A}_2 \\mathrm{A}_3 \\Big)\\hat{y}_{t_2} \n-\\mathrm{A}_1 f_{20}(t_1, t_2) + \n\\mathrm{E}_6(\\hat{y}_{t_2}), \\label{pa2_section_around_hat_d4_sharp} \n\\end{align}\nwhere $\\mathrm{E}_i(\\hat{y}_{t_2})$ are second order in $\\hat{y}_{t_2}$ and independent of $f_{20}(t_1, t_2)$.  \nIt remains to show that \\eqref{pa3_intersect_a1_a2_is_empty} holds.\nEquation \\eqref{pa2_section_around_hat_d4_sharp} implies that \n\\begin{align}\n & (1 -\\eta_{t_1} \\mathrm{A}_1)\\mathrm{J}_2 + (\\mathrm{A}_1 - \\mathrm{A}_1^2 \\eta_{t_1}) \\mathrm{J}_1 = \\beta \\hat{y}_{t_2} + \\mathrm{O}(\\hat{y}_{t_2}^2), \\label{pa2_section_around_hat_d4_sharp_again}\\\\\n\\textnormal{where} \\qquad  \\beta &:= \\frac{f_{30}(t_1, t_2)}{6}(\\mathrm{A}_1 - \\mathrm{A}_2) (\\mathrm{A}_1 - \\mathrm{A}_3)(-1 + \\mathrm{A}_1 \\eta_{t_1})^2, \\nonumber \n\\end{align}\nand $\\mathrm{J}_1$ and $\\mathrm{J}_2$ are as defined in \\eqref{b2_can_not_be_zero}.\nNote that the rhs of \\eqref{pa2_section_around_hat_d4_sharp_again} \nis independent of $f_{20}(t_1, t_2)$. By \\eqref{eta_neq_d4_sharp}, \n$\\beta \\neq 0$. Hence, by \\eqref{pa2_section_around_hat_d4_sharp_again} $\\mathrm{J}_1$ and $\\mathrm{J}_2$ \ncan not both vanish. As a result, \\eqref{pa3_intersect_a1_a2_is_empty} holds. \nSimilar argument holds for the other two solutions of \\eqref{hat_D4_neighbourhood_inside_a1_hat_a1}. \\\\\n\\hf \\hf A similar argument will go through if \nany of the \nother two cases of \\eqref{fij_cases_D4_hat} holds. \\qed \n\n\\begin{cor}\n\\label{pa2_section_mult_around_pa3}\nLet $\\mathbb{W} \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2$ be a vector bundle such that \nthe rank of $\\mathbb{W}$ is equal to dimension of $ \\Delta \\mathcal{P} \\mathcal{A}_{3}$ and \n$\\mathcal{Q}: \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2 \\longrightarrow \\mathbb{W}$  a \\textit{generic} \nsmooth section. Suppose $(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{A}_3$. \nThen the section \n$$ \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_{2}} \\oplus \\mathcal{Q}: \\overline{\\overline{\\mathcal{A}}_1 \\circ \\hat{\\mathcal{A}}_1^{\\#}}  \\longrightarrow \\pi_2^* \\mathbb{V}_{\\mathcal{P} \\mathcal{A}_2} \\oplus \\mathbb{W}$$ \nvanishes around $(\\tilde{f}, \\tilde{p}, l_{\\p})$ with a multiplicity of $2$.   \n\\end{cor}\n\\noindent \\textbf{Proof:  } Since $\\mathcal{Q}$ is generic, the sections \n\\[ \\Psi_{\\mathcal{P} \\mathcal{A}_2} \\oplus \\mathcal{Q}: \\Delta \\overline{\\hat{\\mathcal{A}}^{\\#}_1} \\longrightarrow \\mathbb{V}_{\\mathcal{P} \\mathcal{A}_2} \\oplus \\mathbb{W}, \n~~ \\Psi_{\\mathcal{P} \\mathcal{A}_3}: \\Psi_{\\mathcal{P} \\mathcal{A}_2}^{-1}(0) \\longrightarrow \\UL_{\\mathcal{P} \\mathcal{A}_3} \\]\nare transverse to the zero set. \nHence, there exists a unique \n$(\\tilde{f}(t_1, t_2), l_{p(t_1)}) \\in  \\Delta \\overline{\\hat{\\mathcal{A}}^{\\#}_1}$ close to $(\\tilde{f}, l_{\\p})$ for a specified \nvalue of $\\mathrm{J}_1$, $\\mathrm{J}_2$ and $f_{30}(t_1, t_2)$. \\footnote{Provided \n$\\mathrm{J}_1$, $\\mathrm{J}_2$ and $f_{30}(t_1, t_2)$ are sufficiently small.} \nIn other words we can express all the $f_{ij}(t_1, t_2)$ in terms of \n$\\mathrm{J}_1$, $\\mathrm{J}_2$ and $f_{30}(t_1, t_2)$. \nSince $\\mathcal{B}^{f(t_1, t_2)}_4 \\neq 0$, \nequation \\eqref{b2_can_not_be_zero_again} implies that \nthe number of \nsolutions to the set of equations \n\\[ \\mathrm{J}_1 = \\xi_1, \\qquad \\mathrm{J}_2 = \\xi_2 \\] \nis $2$, where $\\xi_i$ is a small perturbation. \\qed   \n\n\\begin{cor}\n\\label{pa2_section_mult_around_hat_d4}\nLet $\\mathbb{W} \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2$ be a vector bundle such that \nthe rank of $\\mathbb{W}$ is equal to dimension of $ \\Delta \\hat{\\mathcal{D}}_{4}^{\\#\\flat}$ and \n$\\mathcal{Q}: \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2 \\longrightarrow \\mathbb{W}$  a \\textit{generic} \nsmooth section.  Suppose $(\\tilde{f}, \\tilde{p}, \\tilde{p}) \\in \\Delta \\hat{\\mathcal{D}}_4^{\\#\\flat}$. \nThen the section \n$$ \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_{4}} \\oplus \\mathcal{Q}: \\overline{\\overline{\\mathcal{A}}_1 \\circ \\hat{\\mathcal{A}}_1^{\\#}}  \\longrightarrow \\pi_2^* \\UL_{\\mathcal{P} \\mathcal{D}_4} \\oplus \\mathbb{W}$$ \nvanishes around $(\\tilde{f}, \\tilde{p}, l_{\\p})$ with a multiplicity of $3$.   \n\\end{cor}\n\n\\noindent \\textbf{Proof:  } Since $\\mathcal{Q}$ is generic, the sections \n\\[ \\Psi_{\\mathcal{P} \\mathcal{A}_2} \\oplus \\mathcal{Q}: \\Delta \\overline{\\hat{\\mathcal{A}}^{\\#}_1} \\longrightarrow \\mathbb{V}_{\\mathcal{P} \\mathcal{A}_2} \\oplus \\mathbb{W}, \n~~ \\Psi_{\\mathcal{P} \\mathcal{D}_4}: \\Psi_{\\mathcal{P} \\mathcal{A}_2}^{-1}(0) \\longrightarrow \\UL_{\\mathcal{P} \\mathcal{D}_4} \\]\nare transverse to the zero set. \nHence, there exists a unique \n$(\\tilde{f}(t_1, t_2), l_{p(t_1)}) \\in  \\Delta \\overline{\\hat{\\mathcal{A}}^{\\#}_1}$ close to $(\\tilde{f}, l_{\\p})$ for a specified \nvalue of $\\mathrm{J}_1$, $\\mathrm{J}_2$ and $f_{02}(t_1, t_2)$. \\footnote{Provided \n$\\mathrm{J}_1$, $\\mathrm{J}_2$ and $f_{02}(t_1, t_2)$ are sufficiently small.} \nIn other words we can express all the $f_{ij}(t_1, t_2)$ in terms of \n$\\mathrm{J}_1$, $\\mathrm{J}_2$ and $f_{02}(t_1, t_2)$. Since $\\beta \\neq 0$,  \nequation \n\\eqref{pa2_section_around_hat_d4_sharp_again} \nimplies that \nthe number of \nsolutions to the set of equations \n\\[ \\mathrm{J}_1 = \\xi_1, \\qquad \\mathrm{J}_2 = \\xi_2 \\] \nis $1$, where $\\xi_i$ is a small perturbation.  \nSince there are a total of $3$ solutions to \\eqref{hat_D4_neighbourhood_inside_a1_hat_a1}, \nthe total multiplicity is $3$. \\qed   \\\\\n\n\n\n\n\\textbf{Proof of Lemma \\ref{cl_two_pt} (\\ref{a1_pa2_cl}):} It suffices to prove the following two statements in view of \\eqref{pak2_is_subset_of_a1_and_pak} :  \n\\begin{align}\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\overline{\\overline{\\mathcal{A}}_1\\circ \\mathcal{P} \\mathcal{A}}_2: ~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) \\neq 0  \\} &=  \n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_{4}: ~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) \\neq 0  \\}\n\\label{closure_a1_pa2_f02_not_zero}  \\\\\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_2: ~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) = 0  \\} &=  \\Delta \\overline{\\hat{\\mathcal{D}}^{\\#\\flat}_5}   \\label{a1_pa2_d5}\n \\end{align}\nLet us directly prove a more general version of \\eqref{closure_a1_pa2_f02_not_zero}: \n\\begin{lmm}\n\\label{closure_a1_pak_f02_not_zero}\nIf $k \\geq 2$, then  \n\\begin{align*}\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}}_k: ~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) \\neq 0  \\} &= \n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_{k+2}: ~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) \\neq 0  \\}.  \n\\end{align*}\n\\end{lmm}\nNote that \\eqref{closure_a1_pa2_f02_not_zero} is a special case of Lemma \\ref{closure_a1_pak_f02_not_zero}; take $k=2$.\\\\\n\n\\noindent \\textbf{Proof:  } We will prove the following two facts simultaneously:\n\\begin{align}\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_{k} \\} \\supset \\Delta \\mathcal{P} \\mathcal{A}_{k+2} \\qquad & \\forall ~k \\geq 2, \\label{pak2_is_subset_of_a1_and_pak} \\\\\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_{k+1} \\} \\cap \\Delta \\mathcal{P} \\mathcal{A}_{k+2} = \\varnothing \\qquad  & \\forall ~k \\geq 1.  \\label{pak2_intersect_a1_and_pa1k+1_is_empty}\n\\end{align}\nIt is easy to see that \\eqref{pak2_is_subset_of_a1_and_pak} and \\eqref{pak2_intersect_a1_and_pa1k+1_is_empty} imply \nLemma \\ref{closure_a1_pak_f02_not_zero}. We will now prove the following claim: \n\\begin{claim}\n\\label{claim_a4_closure_simultaneous}\nLet $~(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{A}_{k+2}$ and $ k\\geq 2$.\nThen there exists a solution \n$$ (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} ) \\in \\overline{ (\\mathcal{D} \\times \\mathbb{P}^2) \\circ \\mathcal{P} \\mathcal{A}}_2$$ \n\\textit{near} $(\\tilde{f}, \\tilde{p}, l_{\\p})$ to the set of equations\n\\begin{align}\n\\label{closure_a1_ak_hessian_not_zero}\n\\pi_1^* \\psi_{\\mathcal{A}_0}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, ~\\pi_1^* \\psi_{\\mathcal{A}_1}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) = 0, \\nonumber \\\\\n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{A}_3}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, \\ldots, \\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{A}_k}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) = 0, \\nonumber \\\\\n~\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{D}_4}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & \\neq  0, ~\\tilde{p}(t_1, t_2) \\neq \\tilde{p}(t_1). \n\\end{align}\nMoreover, \\textit{any} solution $(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)})$ sufficiently close to $(\\tilde{f}, \\tilde{p}, l_{\\p})$ \nlies in $ \\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}_k$, i.e.,\n\\begin{align}\n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{A}_{k+1}}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) \\neq  0. \\label{psi_pa_k_plus_1_does_not_vanish}\n\\end{align}\nIn particular $(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)})$ \\textit{does not} lie in $ \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_{k+1} $.\n\\end{claim}\n\\noindent It is easy to see that claim \\ref{claim_a4_closure_simultaneous} implies \\eqref{pak2_is_subset_of_a1_and_pak} and \\eqref{pak2_intersect_a1_and_pa1k+1_is_empty} \nsimultaneously for all $k\\geq 2$.\nThe fact that \\eqref{pak2_intersect_a1_and_pa1k+1_is_empty} holds for $k=1$ follows from \\eqref{a3_interesct_a1_pa2_is_empty_set_equation} \n(since $\\mathcal{P} \\mathcal{A}_3$ is a subset of $\\hat{\\mathcal{A}}_3$.)\\\\\n\n\\noindent \\textbf{Proof:  } Choose homogeneous coordinates $[\\mathrm{X}: \\mathrm{Y}: \\mathrm{Z}]$ so that \n$\\tilde{p} = [0:0:1]$ and let ~$\\mathcal{U}_{\\tilde{p}}$,   \n$\\pi_x$, $\\pi_y$, \n$x_{t_1}$, $y_{t_1}$, $x_{t_2}$, $y_{t_2}$\nbe exactly the same as defined in the \nproof of claim \\ref{a1_a1_closure_intersect_a1_or_a2_empty_equations_claim}.\nLet $v_1, w:\\mathcal{U}_{\\tilde{p}} \\longrightarrow T\\mathbb{P}^2$ be vectors dual to the one \nforms $d\\pi_x$ and $d\\pi_y$ respectively.\nTake\n\\[(\\tilde{f} (t_1, t_2),  l_{\\tilde{p}(t_1)}) \\in \\overline{\\mathcal{P} \\mathcal{A}}_2\\] \nto be a point that is close to \n$( \\tilde{f}, l_{\\tilde{p}})$ and $l_{\\tilde{p}(t_1, t_2)}$ a point \nin $\\mathbb{P} T\\mathbb{P}^2$ that is close to $l_{\\tilde{p}(t_1)}$.\nWithout loss of generality, we can assume that   \n\\[ v:= v_1 + \\eta w \\in l_{\\tilde{p}}, ~~v_1 + \\eta_{t_1} w \\in l_{\\tilde{p}(t_1)} ~~\\textnormal{and} ~~v + (\\eta_{t_1} + \\eta_{t_2}) w \\in l_{\\tilde{p}(t_1, t_2)} \\] \nfor some complex \nnumbers $\\eta$, $\\eta_{t_1}$ and $\\eta_{t_2}$ close to each other. Let \n\\begin{align*}\nf_{ij}(t_1, t_2) & := \\nabla^{i+j} f(t_1, t_2)|_{p(t_1)} \n(\\underbrace{v,\\cdots v}_{\\textnormal{$i$ times}}, \\underbrace{w,\\cdots w}_{\\textnormal{$j$ times}}).\n\\end{align*}\nThe numbers \n$\\mathrm{F}$, $\\mathrm{F}_{x_{t_2}}$ and $\\mathrm{F}_{y_{t_2}}$ are the same as in the proof of  claim \\ref{a1_a1_closure_intersect_a1_or_a2_empty_equations_claim}. \nSince  $(\\tilde{f} (t_1, t_2),  l_{\\tilde{p}(t_1)}) \\in \\overline{\\mathcal{P} \\mathcal{A}}_2$, we conclude that \n\\[ f_{00}(t_1, t_2) = f_{10}(t_1, t_2) = f_{01}(t_1, t_2) =f_{20}(t_1, t_2) = f_{11}(t_1, t_2)=0.\\]\nMoreover, since $(\\tilde{f}, l_{\\p}) \\in \\mathcal{P} \\mathcal{A}_{k+2}$ we conclude that \n$f_{02}$ and $\\mathcal{A}^{f}_{k+3}$ are non zero.\nHence $f_{02}(t_1, t_2)$ and $\\mathcal{A}^{f(t_1, t_2)}_{k+3}$ are non zero  \nif $\\tilde{f}(t_1, t_2)$ is sufficiently close to $\\tilde{f}$.\nSince $f_{02}(t_1, t_2) \\neq 0$ , \nfollowing the same argument as in the proof of claim \\ref{a1_a1_closure_intersect_a1_or_a2_empty_equations_claim}, \nwe can make a change of coordinates to write $\\mathrm{F}$ as   \n\\begin{align*} \n\\mathrm{F}&= \\hat{\\hat{y}}_{t_2}^2 + \\frac{\\mathcal{A}^{f(t_1, t_2)}_3}{3!}x_{t_2}^3 + \\frac{\\mathcal{A}^{f(t_1, t_2)}_4}{4!} x_{t_2}^4 + \\ldots \n\\end{align*}\nThe functional equation   \\eqref{closure_a1_ak_hessian_not_zero}  has a solution if and only if \nthe following set of equations has a solution (as numbers): \n\\begin{align}\n\\label{closure_a1_ak_hessian_not_zero_numbers}\n\\hat{\\hat{y}}_{t_2}^2 + \\frac{\\mathcal{A}^{f(t_1, t_2)}_3}{3!}x_{t_2}^3 + \\frac{\\mathcal{A}^{f(t_1, t_2)}_4}{4!} x_{t_2}^4 + \\ldots &=0, \\qquad 2 \\hat{\\hat{y}}_{t_2} = 0, \\nonumber \\\\\n\\frac{\\mathcal{A}^{f(t_1, t_2)}_3}{2!}x_{t_2}^3 + \\frac{\\mathcal{A}^{f(t_1, t_2)}_4}{3!} x_{t_2}^4 + \\ldots &= 0, \\qquad \\mathcal{A}^{f(t_1, t_2)}_3, \\ldots, \\mathcal{A}^{f(t_1, t_2)}_k = 0, \\nonumber  \\\\\n(\\hat{\\hat{y}}_{t_2}, x_{t_2}) & \\neq (0,0)   \\qquad \\textnormal{(but small).}\n\\end{align}\nIt is easy to see that the solutions to \\eqref{closure_a1_ak_hessian_not_zero_numbers} \nexist given by   \n\\begin{align}\n\\mathcal{A}^{f(t_1, t_2)}_3,& \\ldots, \\mathcal{A}^{f(t_1, t_2)}_k = 0, \\nonumber \\\\\n\\mathcal{A}^{f(t_1, t_2)}_{k+1} &=  \\frac{\\mathcal{A}_{k+3}^{f(t_1, t_2)}}{(k+2)(k+3)} x_{t_2}^2 + \\mathrm{O}(x_{t_2}^3),  \\label{pak+2_inside_a1_and_pak_solution}\\\\ \n\\mathcal{A}^{f(t_1, t_2)}_{k+2} &= -\\frac{2 \\mathcal{A}^{f(t_1, t_2)}_{k+3}}{(k+3)} x_{t_2} + \\mathrm{O}(x_{t_2}^2), \n\\qquad \\hat{\\hat{y}}_{t_2} = 0, \\qquad x_{t_2}  \\neq 0 \\qquad \\textnormal{(but small).} \\nonumber \n\\end{align}\nBy \\eqref{pak+2_inside_a1_and_pak_solution}, it immediately follows that \\eqref{psi_pa_k_plus_1_does_not_vanish} holds (since $\\mathcal{A}^{f(t_1, t_2)}_{k+3} \\neq 0$). \\qed \n\n\\begin{cor}\n\\label{a1_pak_mult_is_2_Hess_neq_0}\nLet $\\mathbb{W} \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2$ be a vector bundle such that \nthe rank of $\\mathbb{W}$ is same as the dimension of $ \\Delta \\mathcal{P} \\mathcal{A}_{k+2}$ and \n$\\mathcal{Q}: \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2 \\longrightarrow \\mathbb{W}$  a \\textit{generic} \nsmooth section. Suppose $(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{A}_{k+2} \\cap \\mathcal{Q}^{-1}(0)$. \nThen the section $$ \\pi_2^\\ast\\Psi_{\\mathcal{P} \\mathcal{A}_{k+1}} \\oplus \\mathcal{Q}: \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_{k} \\longrightarrow \\pi_2^* \\UL_{\\mathcal{P} \\mathcal{A}_{k+1}} \\oplus \\mathbb{W}$$\nvanishes around $(\\tilde{f}, \\tilde{p},  l_{\\p})$ with a multiplicity of $2$.\n\\end{cor}\n\\noindent \\textbf{Proof:  }  This follows from the fact that the sections \n\\begin{align*}\n\\pi_2^\\ast\\Psi_{\\mathcal{P} \\mathcal{A}_{i}}:& \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_{i-1} - \\pi_2^\\ast\\Psi_{\\mathcal{P} \\mathcal{D}_4}^{-1}(0) \\longrightarrow \\pi_2^\\ast \\UL_{\\mathcal{P} \\mathcal{A}_{i}}\n\\end{align*}\nare transverse to the zero set for all $3 \\leq i \\leq k+2$, the fact that \n$\\mathcal{Q}$ is generic and \\eqref{pak+2_inside_a1_and_pak_solution}. \nThe proof is now similar to that of Corollary \\ref{a1_section_contrib_from_a1_and_a2}, \n\\ref{pa2_section_mult_around_pa3} and \\ref{pa2_section_mult_around_hat_d4}. \\qed \\\\\n\n\\hf \\hf Next let us prove \\eqref{a1_pa2_d5}. First we will prove the following two facts: \n\\begin{align}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}_2} \\cap \\mathcal{P} \\mathcal{D}_4 &= \\varnothing,\\label{a1_pa2_intersect_pd4_empty} \\\\ \n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}_3} \\cap \\mathcal{P} \\mathcal{D}_5 &= \\varnothing. \\label{a1_pa3_intersect_pd5_empty} \n\\end{align}\n\\noindent Although \\eqref{a1_pa3_intersect_pd5_empty} is not needed to prove \\eqref{a1_pa2_d5}, we will prove these two \nstatements in one go since their proofs are very similar. \n\n\\begin{claim}\n\\label{a1_pa2_intersect_pd4_empty_claim}\nLet $(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{D}_4$. Then there exist no solutions \n\\[ (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), l_{\\tilde{p}(t_1)}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_2 \\]\nnear $(\\tilde{f}, \\tilde{p}, l_{\\p})$ to the set of equations \n\\begin{align}\n\\pi_1^* \\psi_{\\mathcal{A}_0}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, ~\\pi_1^* \\psi_{\\mathcal{A}_1}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) = 0. \n\\label{a1_pa2_intersect_pd4_empty_equation}\n\\end{align}\nSecondly, let $(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{D}_5$. Then there exist no solutions \n\\[ (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), l_{\\tilde{p}(t_1)}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_3 \\]\nnear $(\\tilde{f}, \\tilde{p}, l_{\\p})$ to the set of equations \n\\begin{align}\n\\pi_1^* \\psi_{\\mathcal{A}_0}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, ~\\pi_1^* \\psi_{\\mathcal{A}_1}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) = 0. \n\\label{a1_pa3_intersect_pd5_empty_equation}\n\\end{align}\n\\end{claim}\n\\noindent It is easy to see that  claim \\ref{a1_pa2_intersect_pd4_empty_claim} proves \\eqref{a1_pa2_intersect_pd4_empty} \nand \\eqref{a1_pa3_intersect_pd5_empty} .\\\\\n\n\\noindent \\textbf{Proof:  } For the first part, choose homogeneous coordinates $[\\mathrm{X}: \\mathrm{Y}: \\mathrm{Z}]$ so that \n$\\tilde{p} = [0:0:1]$ and let ~$\\mathcal{U}_{\\tilde{p}}$,   \n$\\pi_x$, $\\pi_y$,  $v_1$, $w$, $v$, $\\eta$, $\\eta_{t_1}$, $\\eta_{t_2}$, \n$x_{t_1}$, $y_{t_1}$, $x_{t_2}$, $y_{t_2}$, \n$f_{ij}(t_1, t_2)$, $\\mathrm{F}$, $\\mathrm{F}_{x_{t_2}}$ and $\\mathrm{F}_{y_{t_2}}$\nbe exactly the same as defined in the \nproof of claim \\ref{claim_a4_closure_simultaneous}.\nSince $(\\tilde{f}(t_1, t_2), l_{\\tilde{p}(t_1)}) \\in \\overline{\\mathcal{P} \\mathcal{A}}_2$, we conclude that \n\\[ f_{00}(t_1, t_2) = f_{10}(t_1, t_2) = f_{01}(t_1, t_2) =f_{20}(t_1, t_2) = f_{11}(t_1, t_2)=0.\\]\nThe functional equation \\eqref{a1_pa2_intersect_pd4_empty_equation}  has a solution if and only if \nthe following set of equations has a solution (as numbers): \n\\begin{align}\n\\mathrm{F} = 0, \\qquad \\mathrm{F}_{x_{t_2}} = 0, \\qquad \\mathrm{F}_{y_{t_2}} = 0,  \\qquad (x_{t_2}, y_{t_2}) \\neq (0, 0) \\qquad \\textnormal{(but small)}. \\label{eval_f1_pd4} \n\\end{align} \nFor the convenience of the reader, let us rewrite the expression for $\\mathrm{F}$: \n\\begin{align*}\n\\mathrm{F} &: = \\frac{f_{02}(t_1, t_2)}{2} y_{t_2}^2 + \\frac{f_{30}(t_1, t_2)}{6} x_{t_2}^3 + \\frac{f_{21}(t_1, t_2)}{2} x_{t_2}^2 y_{t_2}+ \n\\frac{f_{12}(t_1, t_2)}{2} x_{t_2} y_{t_2}^2 + \\frac{f_{03}(t_1, t_2)}{6} y_{t_2}^3 + \\ldots.\n\\end{align*} \nSince $(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\mathcal{P} \\mathcal{D}_4$ we conclude that \n\\begin{align}\nf_{02} &=0, ~~f_{30} =0, ~~f_{21} \\neq 0, ~~3f_{12}^2 - 4f_{21} f_{03} \\neq 0. \\label{pd4_nv_condition_equation}\n\\end{align}\nTo see why the two non vanishing conditions hold, first notice that since $(\\tilde{f}, l_{\\p}) \\in \\hat{\\mathcal{D}}_4$ we conclude that \nthe cubic term in the Taylor expansion of $f$ has no repeated root. In other words \n\\begin{align*}\n\\beta &:= f_{30}^2 f_{03}^2-6 f_{03} f_{12} f_{21} f_{30} + 4 f_{12}^3 f_{30} + 4 f_{03} f_{21}^3 - 3 f_{12}^2 f_{21}^2 \\neq 0.   \n\\end{align*}\nSince $(\\tilde{f}, l_{\\p}) \\in \\mathcal{P} \\mathcal{D}_4$ we conclude $f_{30} =0$. Hence we get \\eqref{pd4_nv_condition_equation}. \nNow we will show that \\eqref{eval_f1_pd4} has no solutions. First of all we claim that \n$y_{t_2} \\neq 0 $; we will justify that at the end. Assuming that, define \n$\\mathrm{L} := \\frac{x_{t_2}}{y_{t_2}}$. Substituting $x_{t_2} = \\mathrm{L} y_{t_2}$ in $\\mathrm{F}_{x_{t_2}} =0 $ and using $y_{t_2} \\neq 0$ and $f_{21}(t_1, t_2) \\neq 0$ \nwe can solve for $\\mathrm{L}$ using the Implicit Function Theorem. That gives us \n\\begin{align}\n\\mathrm{L} &= -\\frac{f_{12}(t_1, t_2)}{2 f_{21}(t_1, t_2)} + y_{t_2} \\mathrm{E}_1(y_{t_1}, f_{30}(t_1, t_2)) + f_{30}(t_1, t_2) \\mathrm{E}_2(y_{t_1}, f_{30}(t_1, t_2)),  \n\\label{L_value_pd4_again}\n\\end{align}\nwhere $\\mathrm{E}_i(0,0) =0$. Using the value of $\\mathrm{L}$ from \\eqref{L_value_pd4_again}, \nand substituting $x_{t_2} = \\mathrm{L} y_{t_2}$ in $\\mathrm{F} - \\frac{y_{t_2}\\mathrm{F}_{y_{t_2}}}{2} =0$, \nwe conclude that as $(y_{t_2}, f_{30}(t_1, t_2))$ go to zero\n\\begin{align}\n-\\frac{f_{03}}{12} + \\frac{f_{12}^2}{16 f_{21}} &=0. \\label{pd5_dual_eqn}\n\\end{align}\nIt is easy to see that \\eqref{pd5_dual_eqn} contradicts \\eqref{pd4_nv_condition_equation}. \nIt remains to show that $y_{t_2} \\neq 0$. To see why that is so, consider the equation \n$\\mathrm{F}_{y_{t_2}} =0$. It is easy to see that if $y_{t_2} =0$ then $f_{21}(t_1, t_2)$ will go to zero, contradicting \n\\eqref{pd4_nv_condition_equation}. Hence \\eqref{eval_f1_pd4} has no solutions. \\\\ \n\\hf \\hf  For the second part of the claim, we use the same set up except for one difference: we require $(\\tilde{f}, l_{\\p}) \\in \\overline{\\mathcal{P} \\mathcal{A}}_3$. \nHence \n\\[ f_{00}(t_1, t_2) = f_{10}(t_1, t_2) = f_{01}(t_1, t_2) =f_{20}(t_1, t_2) = f_{11}(t_1, t_2)= f_{30}(t_1, t_2)=0.\\]\nThe functional equation \\eqref{a1_pa3_intersect_pd5_empty_equation}  has a solution if and only if \nthe following set of equations has a solution (as numbers): \n\\begin{align}\n\\mathrm{F} = 0, \\qquad \\mathrm{F}_{x_{t_2}} = 0, \\qquad \\mathrm{F}_{y_{t_2}} = 0,  \\qquad (x_{t_2}, y_{t_2}) \\neq (0, 0) \\qquad \\textnormal{(but small)}. \\label{eval_f1_pd5_again} \n\\end{align} \nFor the convenience of the reader, let us rewrite the expression for $\\mathrm{F}$: \n\\begin{align*}\n\\mathrm{F} &: = \\frac{f_{02}(t_1, t_2)}{2} y_{t_2}^2 + \\frac{f_{21}(t_1, t_2)}{2} x_{t_2}^2 y_{t_2}+ \n\\frac{f_{12}(t_1, t_2)}{2} x_{t_2} y_{t_2}^2 + \\frac{f_{03}(t_1, t_2)}{6} y_{t_2}^3 + \\ldots.\n\\end{align*} \nSince $(\\tilde{f}, l_{\\p}) \\in \\mathcal{P} \\mathcal{D}_5$ we conclude that \n\\begin{align}\nf_{02} &=0, ~~f_{30} =0, ~~f_{21} =0, ~~f_{40}\\neq 0, ~~f_{12} \\neq 0.  \\label{pd5_nv_condition_equation_again}\n\\end{align}\nWe will now show that there are no solutions to \\eqref{eval_f1_pd5_again}. First we claim that $y_{t_2} \\neq 0$; we will justify that at the end. \nAssuming that, define $\\mathrm{L} := \\frac{x_{t_2}}{y_{t_2}}$. Substituting $x_{t_2} = \\mathrm{L} y_{t_2}$ in $\\mathrm{F}_{x_{t_2}} =0 $ and using $y_{t_2} \\neq 0$, we conclude that as $y_{t_2}$ and $f_{21}(t_1, t_2)$ \ngo to zero, $f_{12}(t_1, t_2)$ goes to zero, contradicting \\eqref{pd5_nv_condition_equation_again}. It remains to show that $y_{t_2} \\neq 0$. Consider the equation $\\mathrm{F}_{x_{t_2}} =0$. If $y_{t_2} =0$ then $f_{40}(t_1, t_2)$ would go to zero as $x_{t_2}$ goes to zero, \ncontradicting \\eqref{pd5_nv_condition_equation_again}. Hence \\eqref{eval_f1_pd5_again} has no solutions. \\qed \\\\\n\n\\hf \\hf Now we return to the proof of \\eqref{a1_pa2_d5}. First of all we observe that \n\\eqref{a3_interesct_a1_pa2_is_empty_set_equation} and \n\\eqref{a1_pa2_intersect_pd4_empty} imply that \n\\begin{align}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_2 \\cap \\Delta \\hat{\\mathcal{D}}_4 & = \\varnothing.  \\label{a1_pa2_intersect_hat_d4_is_empty}\n\\end{align}\nHence, the lhs of \\eqref{a1_pa2_d5} is a subset of its \nrhs. This is because \n\\begin{align}\n \\overline{\\hat{\\mathcal{D}}}_4 & = \\hat{\\mathcal{D}}_4 \\cup \\overline{\\hat{\\mathcal{D}}}_5 \\qquad \\textnormal{and} \\qquad \\overline{\\hat{\\mathcal{D}}^{\\#\\flat}_5} = \\overline{\\hat{\\mathcal{D}}}_5. \\label{lot_of_closure_claims}\n\\end{align}\nThe first equality follows by applying Lemma \\ref{tube_lemma} twice to Lemma \\ref{cl} \\eqref{D4cl} while the second is covered by Lemma \\ref{Dk_sharp_closure}. To show \nrhs of \\eqref{a1_pa2_d5} is a subset of its lhs, \nit suffices to show that   \n\\begin{align}\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}_2}: ~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) = 0\\}\n& \\supset \\Delta \\hat{\\mathcal{D}}_5^{\\#\\flat}.   \\label{d5_hat+sharp_is_subset_of_a1_pa2}\n\\end{align}\n\\begin{claim}\n\\label{a1_pa2_closure_and_also_pa3}\nLet $(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\Delta \\hat{\\mathcal{D}}_5^{\\#\\flat}$. Then there exists a solution \n$$ (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} ) \\in \\overline{ (\\mathcal{D} \\times \\mathbb{P}^2) \\circ \\mathcal{P} \\mathcal{A}}_2$$ \n\\textit{near} $(\\tilde{f}, \\tilde{p}, l_{\\p})$ to the set of equations\n\\begin{align}\n\\pi_1^* \\psi_{\\mathcal{A}_0}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, ~\\pi_1^* \\psi_{\\mathcal{A}_1}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) = 0, ~~\\tilde{p}(t_1, t_2) \\neq \\tilde{p}(t_1). \\label{a1_pa2_closure_d5_sharp}\n\\end{align}\n\\end{claim}\n\\noindent It is easy to see that claim \\ref{a1_pa2_closure_and_also_pa3} \nimplies \\eqref{d5_hat+sharp_is_subset_of_a1_pa2}.\\\\ \n\n\\noindent \\textbf{Proof:  } Choose homogeneous coordinates $[\\mathrm{X}: \\mathrm{Y}: \\mathrm{Z}]$ so that \n$\\tilde{p} = [0:0:1]$ and let ~$\\mathcal{U}_{\\tilde{p}}$,   \n$\\pi_x$, $\\pi_y$,  $v_1$, $w$, $v$, $\\eta$, $\\eta_{t_1}$, $\\eta_{t_2}$, \n$x_{t_1}$, $y_{t_1}$, $x_{t_2}$, $y_{t_2}$, \n$f_{ij}(t_1, t_2)$, $\\mathrm{F}$, $\\mathrm{F}_{x_{t_2}}$ and $\\mathrm{F}_{y_{t_2}}$\nbe exactly the same as defined in the \nproof of claim \\ref{claim_a4_closure_simultaneous}.\nSince $(\\tilde{f}(t_1, t_2), l_{\\tilde{p}(t_1)}) \\in \\overline{\\mathcal{P} \\mathcal{A}}_2$, we conclude that \n\\[ f_{00}(t_1, t_2) = f_{10}(t_1, t_2) = f_{01}(t_1, t_2) =f_{20}(t_1, t_2) = f_{11}(t_1, t_2)=0.\\]\nThe functional equation \\eqref{a1_pa2_closure_d5_sharp}  has a solution if and only if \nthe following set of equations has a solution (as numbers): \n\\begin{align}\n\\mathrm{F} = 0, \\qquad \\mathrm{F}_{x_{t_2}} = 0, \\qquad \\mathrm{F}_{y_{t_2}} = 0,  \\qquad (x_{t_2}, y_{t_2}) \\neq (0, 0) \\qquad \\textnormal{(but small)}. \\label{eval_f1_d5_sharp} \n\\end{align} \nFor the convenience of the reader, let us rewrite the expression for $\\mathrm{F}$: \n\\begin{align*}\n\\mathrm{F} &: = {\\textstyle \\frac{f_{02}(t_1, t_2)}{2} y_{t_2}^2 + \\frac{f_{30}(t_1, t_2)}{6} x_{t_2}^3 + \\frac{f_{21}(t_1, t_2)}{2} x_{t_2}^2 y_{t_2}+ \n\\frac{f_{12}(t_1, t_2)}{2} x_{t_2} y_{t_2}^2 + \\frac{f_{03}(t_1, t_2)}{6} y_{t_2}^3 + \\ldots}.\n\\end{align*}\nLet us define  \n\\begin{align}\n\\beta_1 &:= f_{21}^2 - f_{12} f_{30},  \n~~\\beta_2^{\\pm}:= -\\frac{f_{03}}{12} -\\frac{f_{21}^3}{6 f_{30}^2} + \\frac{f_{12} f_{21}}{4 f_{30}} \\pm\n\\sqrt{\\beta_1} \\Big( \\frac{f_{21}^2}{6 f_{30}^2} - \\frac{f_{12}}{6 f_{30}} \\Big) \\qquad \\textnormal{and} \\nonumber \\\\\n\\beta_3  & := f_{30}^2 f_{03}^2-6 f_{03} f_{12} f_{21} f_{30} + 4 f_{12}^3 f_{30} + 4 f_{03} f_{21}^3 - 3 f_{12}^2 f_{21}^2 = 4 f_{30}^2 \\beta_{2}^{+} \\beta_2^{-}. \n\\label{beta_3_beta_2_plus_beta_2_minus}\n\\end{align}\nDefine $\\beta_k(t_1, t_2)$ similarly with $f_{ij}$ replaced by $f_{ij}(t_1, t_2)$. \nSince $(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\hat{\\mathcal{D}}_5$, the cubic \n\\[ \\Phi(\\theta):= {\\textstyle \\frac{f_{30}}{6} \\theta^3 + \\frac{f_{21}}{2} \\theta^2+ \n\\frac{f_{12}}{2} \\theta + \\frac{f_{03}}{6}} \\]\nhas a repeated root, \nbut not all the three roots are the same. Hence we conclude  that \n\\begin{align}\n\\beta_3 &=0, ~~\\beta_1 \\neq 0 ~~\\textnormal{and} ~~f_{30} \\neq 0. \n\\end{align}\nThe last inequality follows from the fact that $(\\tilde{f}, \\tilde{p}, l_{\\p})$ belongs to  $\\hat{\\mathcal{D}}_5^{\\#\\flat}$ as opposed to\n$\\hat{\\mathcal{D}}_5$.\nWe will now construct solutions to \\eqref{eval_f1_d5_sharp}.    \nCorresponding to each branch of $\\sqrt{\\beta_1(t_1, t_2)}$, the solutions are: \n\\begin{align}\nx_{t_2} &= \\frac{-f_{21}(t_1, t_2) + \\sqrt{\\beta_1(t_1, t_2)}}{f_{30}(t_1, t_2)} y_{t_2} + \\mathrm{O}(y_{t_2}^2), \n~~f_{02}(t_1, t_2) = \\mathrm{O}(y_{t_2}), ~~\\beta_2^{+}(t_1, t_2) = \\mathrm{O}(y_{t_2}). \\label{x_y_beta_soln_d5}  \n\\end{align}\nLet us explain how we obtained these solutions. To obtain the value of $x_{t_2}$ we used $\\mathrm{F}_{x_{t_2}} =0$. To obtain  the value of \n$f_{02}(t_1, t_2)$ we used $\\mathrm{F}_{y_{t_2}} =0$ and the value of $x_{t_2}$ from the previous equation. Finally we used the fact that \n$2\\mathrm{F} - y_{t_2}\\mathrm{F}_{y_{t_2}} =0$ and the value of $x_{t_2}$ to obtain $\\beta_2^{+}(t_1, t_2)$. We get a similar solution \nfor the other branch of $\\sqrt{\\beta_1(t_1, t_2)}$. \nBy \\eqref{beta_3_beta_2_plus_beta_2_minus} and \\eqref{x_y_beta_soln_d5} , we conclude that \nas $y_{t_2}$ goes to zero, $f_{02}(t_1, t_2)$ and \n$\\beta_3$ go to zero. Hence, the solutions in \\eqref{x_y_beta_soln_d5} lie in \n$\\overline{\\mathcal{A}}_1\\circ\\overline{\\mathcal{P} \\mathcal{A}}_2 $ and converge to a point \n$(\\tilde{f}, \\tilde{p} , l_{\\p})$  in  $\\hat{\\mathcal{D}}^{\\#\\flat}_5$.  \\qed \\\\\n\n\n\n\\textbf{Proof of Lemma \\ref{cl_two_pt} (\\ref{a1_pa3_cl}):} By Lemma \\ref{closure_a1_pak_f02_not_zero} ($k=3$), if we show that\n\\begin{align}\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_3: ~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) = 0  \\} &=  \\Delta \\overline{\\mathcal{P} \\mathcal{D}_5^{\\vee}} \\cup \n\\Delta \\overline{\\mathcal{P} \\mathcal{D}}_6  \\label{a1_pa3_pd5_dual_eqn}\n\\end{align}\nthen we have\n\\bgd\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_3\\}\\subseteq \\Delta\\overline{\\mathcal{P}\\mathcal{A}}_5 \\cup \\Delta\\overline{\\mathcal{P} \\mathcal{D}_5^{\\vee}}\\cup\\Delta\\overline{\\mathcal{P} \\mathcal{D}}_6.\n\\edd\nBy Lemma \\ref{cl} \\eqref{A5cl}, we conclude that \n\\bgd\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_3\\}\\subseteq \\Delta\\overline{\\mathcal{P}\\mathcal{A}}_5 \\cup \\Delta\\overline{\\mathcal{P} \\mathcal{D}_5^{\\vee}}.\n\\edd\nOn the other hand, by \\eqref{pak2_is_subset_of_a1_and_pak} applied with $k=3$ and \\eqref{a1_pa3_pd5_dual_eqn} we conclude that\n\\bgd\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_3\\}\\supseteq \\Delta\\overline{\\mathcal{P}\\mathcal{A}}_5 \\cup \\Delta\\overline{\\mathcal{P} \\mathcal{D}_5^{\\vee}}.\n\\edd\nThus, it suffices to prove \\eqref{a1_pa3_pd5_dual_eqn}. Note that the intersection of the lhs of \\eqref{a1_pa3_pd5_dual_eqn} with $\\hat{\\mathcal{D}}_4$ is empty. This follows from \n\\eqref{a1_pa2_intersect_hat_d4_is_empty} and the fact that $\\mathcal{P} \\mathcal{A}_3$ is a subset of $\\overline{\\mathcal{P} \\mathcal{A}}_2$ \n(see Lemma \\ref{cl}). \nEquation \\eqref{lot_of_closure_claims} \nand Lemma \\ref{Dk_sharp_closure}, statement \\ref{d5_pa3_zero}  now implies that the \nlhs of \\eqref{a1_pa3_pd5_dual_eqn} is a subset of $\\Delta \\overline{\\mathcal{P} \\mathcal{D}_5^{\\vee}} \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{D}}_5$.\nHowever, the intersection of the lhs of \\eqref{a1_pa3_pd5_dual_eqn} with $\\Delta \\mathcal{P} \\mathcal{D}_5$ is also empty by \\eqref{a1_pa3_intersect_pd5_empty}. \nBy Lemma \\ref{cl}, statement \\ref{D5cl} and Lemma \\ref{Dk_sharp_closure}, statement \\ref{pe6_subset_of_cl_pd5_dual} we have that \n\\begin{align}\n\\overline{\\mathcal{P} \\mathcal{D}}_5 & = \\mathcal{P} \\mathcal{D}_5 \\cup \\overline{\\mathcal{P} \\mathcal{D}}_6 \\cup \\overline{\\mathcal{P} \\mathcal{E}}_6 \\qquad \\textnormal{and} \\qquad \\overline{\\mathcal{P} \\mathcal{E}}_6  \\subset \\overline{\\mathcal{P} \\mathcal{D}_5^{\\vee}}.  \\label{lot_of_closure_claims_again}\n\\end{align}\nHence the lhs of \\eqref{a1_pa3_pd5_dual_eqn} is a subset of its rhs. \nTo show the converse, we will first simultaneously prove the following three statements: \n\\begin{align}\n \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_3  & \\supset \\Delta \\mathcal{P} \\mathcal{D}_5^{\\vee},  \\label{pd5_subset_a1_pa3}\\\\ \n \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_4 \\cap \\Delta \\mathcal{P} \\mathcal{D}_5^{\\vee} & = \\varnothing,  \\label{pd5_intersect_a1_pd4_is_empty}\\\\\n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_4 \\cap \\Delta \\mathcal{P} \\mathcal{D}_5^{\\vee} & = \\varnothing.  \\label{pd5_intersect_a1_pa4_is_empty}\n\\end{align}\nAnd then we will prove the following two statements simultaneously:\n\\begin{align}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_3 & \\supset \\Delta \\mathcal{P} \\mathcal{D}_6,   \\label{a1_pa3_is_supsetof_pd6} \\\\  \n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_4 \\cap \\Delta \\mathcal{P} \\mathcal{D}_6 & = \\varnothing. \\label{a1_pa4_intersect_pd6_is_empty}  \n\\end{align}\nNote that \\eqref{pd5_subset_a1_pa3} and \\eqref{a1_pa3_is_supsetof_pd6} imply rhs of \n\\eqref{a1_pa3_pd5_dual_eqn} is a subset of its lhs, since $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_3$ is a closed set.  \n\\begin{claim}\n\\label{claim_pd5_subset_of_a1_pa3}\nLet $(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{D}_5^{\\vee}$.\nThen there exists a solution \n$$ (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} ) \\in \\overline{ (\\mathcal{D} \\times \\mathbb{P}^2) \\circ \\mathcal{P} \\mathcal{A}}_3$$ \n\\textit{near} $(\\tilde{f}, \\tilde{p}, l_{\\p})$ to the set of equations\n\\begin{align}\n\\pi_1^* \\psi_{\\mathcal{A}_0}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, ~\\pi_1^* \\psi_{\\mathcal{A}_1}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) = 0, ~~\\tilde{p}(t_1, t_2) \\neq \\tilde{p}(t_1). \\label{pd5_dual_limit_a1_pa3_functional_eqn}\n\\end{align}\nMoreover, such solution  \nlies in $\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}_3$, i.e., \n\\begin{align}\n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{D}_4}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), l_{\\tilde{p}(t_1)}) & \\neq 0, \\label{psi_pd4_neq_0_a1_pa3_functional} \\\\\n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{A}_4}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), l_{\\tilde{p}(t_1)}) & \\neq 0. \\label{psi_pa4_neq_0_a1_pa3_functional_new}\n\\end{align}\nIn particular, $(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} )$ does not lie in $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_4$ or $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_4$. \n\\end{claim}\n\\noindent Note that claim \\ref{claim_pd5_subset_of_a1_pa3} implies  \\eqref{pd5_subset_a1_pa3}, \\eqref{pd5_intersect_a1_pd4_is_empty} and \n\\eqref{pd5_intersect_a1_pa4_is_empty} simultaneously. \\\\\n\n\\noindent \\textbf{Proof:  } Choose homogeneous coordinates $[\\mathrm{X}: \\mathrm{Y}: \\mathrm{Z}]$ so that \n$\\tilde{p} = [0:0:1]$ and let ~$\\mathcal{U}_{\\tilde{p}}$,   \n$\\pi_x$, $\\pi_y$,  $v_1$, $w$, $v$, $\\eta$, $\\eta_{t_1}$, $\\eta_{t_2}$, \n$x_{t_1}$, $y_{t_1}$, $x_{t_2}$, $y_{t_2}$, \n$f_{ij}(t_1, t_2)$, $\\mathrm{F}$, $\\mathrm{F}_{x_{t_2}}$ and $\\mathrm{F}_{y_{t_2}}$\nbe exactly the same as defined in the \nproof of claim \\ref{claim_a4_closure_simultaneous}, except for one difference:\nwe take $(\\tilde{f}(t_1, t_2), l_{\\tilde{p}(t_1)})$ to be a point in $\\overline{\\mathcal{P} \\mathcal{A}}_3$.\nHence \n\\[ f_{00}(t_1, t_2) = f_{10}(t_1, t_2) = f_{01}(t_1, t_2) =f_{11}(t_1, t_2)= f_{20}(t_1, t_2) = f_{30}(t_1, t_2)=0.\\]\nThe functional equation \\eqref{pd5_dual_limit_a1_pa3_functional_eqn}  \nhas a solution if and only if \nthe following set of equations has a solution (as numbers): \n\\begin{align}\n\\mathrm{F} = 0, \\qquad \\mathrm{F}_{x_{t_2}} = 0, \\qquad \\mathrm{F}_{y_{t_2}} = 0,  \\qquad (x_{t_2}, y_{t_2}) \\neq (0, 0) \\qquad \\textnormal{(but small)}. \\label{eval_f1_d5_dual} \n\\end{align} \nFor the convenience of the reader, let us rewrite the expression for $\\mathrm{F}$: \n\\begin{align*}\n\\mathrm{F} &: = {\\textstyle \\frac{f_{02}(t_1, t_2)}{2} y_{t_2}^2 +\\frac{f_{21}(t_1, t_2)}{2} x_{t_2}^2 y_{t_2}+ \n\\frac{f_{12}(t_1, t_2)}{2} x_{t_2} y_{t_2}^2 + \\frac{f_{03}(t_1, t_2)}{6} y_{t_2}^3 + \\ldots}.\n\\end{align*}\nSince $(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{D}_5^{\\vee}$, we claim that \n\\begin{align}\nf_{21} & \\neq 0, \\qquad \\beta_1 := \\frac{(f_{12} \\partial_x - 2 f_{21} \\partial_y)^3 f}{2 f_{21}^2} = 3 f_{12}^2 -4f_{21} f_{03} =0  \n\\qquad \\textnormal{and} \\label{pd5_dual_nv1} \\\\\n\\beta_2&:= (f_{12} \\partial_x - 2 f_{21} \\partial_y)^4 f = \nf_{12}^4 f_{40} - 8f_{12}^3 f_{21} f_{31} + 24 f_{12}^2 f_{21}^2 f_{22}-32 f_{12} f_{21}^3 f_{13} + 16 f_{21}^4 f_{04} \\neq 0. \n\\label{pd5_dual_nv_2}\n\\end{align}\nLet us justify this. Since $(\\tilde{f}, \\tilde{p}) \\in \\mathcal{D}_5$ there exists a non zero vector $u = m_1v + m_2w$ such that \n\\begin{align}\n\\nabla^3 f|_{\\tilde{p}} (u,u, v) & = m_1^2 f_{30} + 2 m_1 m_2 f_{21} + m_2^2 f_{12} =0,  \\label{nabla_cube_f_1} \\\\\n\\nabla^3 f|_{\\tilde{p}} (u,u, w) &= m_1^2 f_{21} + 2 m_1 m_2 f_{12} + m_2^2 f_{03} =0, \\label{nabla_cube_f_2} \\\\\n\\nabla^4 f|_{\\tilde{p}} (u,u,u,u) & \\neq 0. \\label{nabla_fourth_nv}\n\\end{align}\nSince $(\\tilde{f}, l_{\\p}) \\in \\mathcal{P} \\mathcal{D}_5^{\\vee}$, we conclude \\textit{by definition} that \n\\begin{align}\nf_{30} &=0, \\qquad f_{21} \\neq 0, \\qquad m_2 \\neq 0. \\label{m_pd5_dual} \n\\end{align}\n(If $m_2 =0$ then $f_{21}$ would be zero). \nEquations \\eqref{m_pd5_dual} and \\eqref{nabla_cube_f_1} now imply that \n\\begin{align}\nm_1\/ m_2 &= - f_{12}\/(2 f_{21}). \\label{m1_m2_value}\n\\end{align}\nEquation \\eqref{m1_m2_value} and \\eqref{nabla_cube_f_2} implies \\eqref{pd5_dual_nv1}. \nFinally, \\eqref{nabla_fourth_nv} implies \\eqref{pd5_dual_nv_2}.\\\\\n\\hf \\hf We claim that solutions to \\eqref{eval_f1_d5_dual} are given by  \n\\begin{align}\nx_{t_2} = -\\frac{f_{12}(t_1, t_2)}{2 f_{21}(t_1, t_2)} y_{t_2} + \\mathrm{O}(y_{t_2}^2), \\qquad \n\\beta_1(t_1, t_2) &= -\\frac{\\beta_2(t_1, t_2)}{8 f_{21}(t_1, t_2)^3} y_{t_2} + \\mathrm{O}(y_{t_2}^2) \\qquad \\textnormal{and} \\nonumber \\\\\nf_{02}(t_1, t_2) &= -\\frac{\\beta_2(t_1, t_2)}{192 f_{21}(t_1, t_2)^4} y_{t_2}^2 + \\mathrm{O}(y_{t_2}^3). \\label{d5_dual_multilicity}\n\\end{align}\nLet us explain how we obtained these solutions. Assuming $y_{t_2} \\neq 0$ (to be justified at the end) define $~\\mathrm{L} := \\frac{\\xt}{y_{t_2}}.$\nUsing $x_{t_2} = \\mathrm{L} y_{t_2}$ with $y_{t_2}\\neq 0$ in the equation $\\mathrm{F}_{x_{t_2}}=0$ we can solve for $\\mathrm{L}$ via the Implicit Function Theorem. That gives us  \n\\begin{equation}\n\\mathrm{L} = \\textstyle{\\frac{-f_{12}(t_1, t_2)}{2 f_{21}(t_1, t_2)}}+ \\Big(-\\textstyle{\\frac{f_{13}(t_1, t_2)}{6 f_{21}(t_1, t_2)}\n+\\frac{f_{12}(t_1, t_2) f_{22}(t_1, t_2)}{4 f_{21}(t_1, t_2)^2} - \\frac{f_{12}(t_1, t_2)^2 f_{31}(t_1, t_2)}{8 f_{21}(t_1, t_2)^3} + \n\\frac{f_{12}(t_1, t_2)^3 f_{40}(t_1, t_2)}{48 f_{21}(t_1, t_2)^4}} \\Big) y_{t_2} + \\mathrm{O}(y_{t_2}^2). \\label{L_value_beta}\n\\end{equation}\nNext, using the equation $2\\mathrm{F} - y_{t_2} \\mathrm{F}_{y_{t_2}} =0$ and the fact that \n$\\xt = \\mathrm{L} y_{t_2}$ and \\eqref{L_value_beta}, \nwe obtain the expression for $\\beta_1(t_1, t_2)$ in \\eqref{d5_dual_multilicity}. \nNext, observe that \n\\begin{align}\nf_{30}(t_1, t_2) & = \\frac{3 f_{12}(t_1, t_2)^2- \\beta_1(t_1, t_2)}{4 f_{21}(t_1, t_2)}. \\label{f30_beta1}\n\\end{align}\nFinally, using the equation $\\mathrm{F}_{y_{t_2}}=0$,  the fact that \n$\\xt = \\mathrm{L} y_{t_2}$, \\eqref{L_value_beta}, the expression for $\\beta_1(t_1, t_2)$ in \\eqref{d5_dual_multilicity} and \n\\eqref{f30_beta1} \nwe obtain the expression for $f_{02}(t_1, t_2)$ in \\eqref{d5_dual_multilicity}.  \nIt is now easy to see that \\eqref{psi_pd4_neq_0_a1_pa3_functional} holds. It remains to show that $y_{t_2} \\neq 0$. \nTo see why that is so, suppose $y_{t_2}=0$. Then using the fact that $\\mathrm{F}_{y_{t_2}} =0$, we conclude that \n$f_{21}(t_1, t_2) = \\mathrm{O}(\\xt)$. Hence $f_{21}(t_1, t_2)$ goes to zero as $\\xt$ goes to zero, contradicting \\eqref{pd5_dual_nv1}.\\\\ \n\\hf \\hf Finally, \\eqref{psi_pa4_neq_0_a1_pa3_functional_new} is true because \nthe section $\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_4}$ does not vanish on $\\Delta \\mathcal{P} \\mathcal{D}_5^{\\vee}$. \\qed \n\n\\begin{cor}\n\\label{psi_pd4_section_vanishes_order_two_around_dual_d5}\nLet $\\mathbb{W} \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2$ be a vector bundle such that \nthe rank of $\\mathbb{W}$ is same as the dimension of $ \\Delta \\mathcal{P} \\mathcal{D}_5^{\\vee}$ and \n$\\mathcal{Q}: \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2 \\longrightarrow \\mathbb{W}$  a \\textit{generic} \nsmooth section. Suppose $(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{D}_5^{\\vee}\\cap \\mathcal{Q}^{-1}(0)$. \nThen the section $$ \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4} \\oplus \\mathcal{Q}: \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P}\\mathcal{A}}_3 \\longrightarrow \\pi_2^* (\\UL_{\\mathcal{P} \\mathcal{D}_4}) \\oplus \\mathbb{W}$$\nvanishes around $(\\tilde{f}, \\tilde{p},  l_{\\p})$ with a multiplicity of $2$.\n\\end{cor}\n\\noindent \\textbf{Proof:  } This follows by observing that at $\\Delta \\mathcal{P} \\mathcal{D}_5^{\\vee}$, the sections \ninduced by $f_{02}$ and $\\beta_1$  (the corresponding functionals) \nare transverse to the zero set over $\\overline{\\mathcal{P} \\mathcal{A}}_3$ \n\\footnote{To see why; just take the partial derivative with respect to \n$f_{02}$ and $f_{03}$. Since $f_{21} \\neq 0$, transversality follows.}, \n$\\mathcal{Q}$ is generic and \\eqref{d5_dual_multilicity}. \\qed\n\n\\begin{claim}\n\\label{claim_a1_pa4_intersect_pd6_is_empty}\nLet $(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{D}_6$.\nThen there exist solutions \n$$ (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} ) \\in \\overline{ (\\mathcal{D} \\times \\mathbb{P}^2) \\circ \\mathcal{P} \\mathcal{A}}_3$$ \n\\textit{near} $(\\tilde{f}, \\tilde{p}, l_{\\p})$ to the set of equations\n\\begin{align}\n\\pi_1^* \\psi_{\\mathcal{A}_0}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, ~\\pi_1^* \\psi_{\\mathcal{A}_1}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) = 0, \\nonumber  \\\\ \n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{D}_4}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & \\neq 0,  ~\\tilde{p}(t_1, t_2) \\neq \\tilde{p}(t_1). \\label{pd6_intersect_a1_pa4_is_empty_functional_eqn}\n\\end{align}\nMoreover, any such solution sufficiently close to $(\\tilde{f}, \\tilde{p}, l_{\\p})$ lies in \n$\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}_3$, i.e., \n\\begin{align}\n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{A}_4}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), l_{\\tilde{p}(t_1)}) \\neq 0. \\label{psi_pa4_neq_0_a1_pa3_functional}\n\\end{align}\nIn particular, $(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} )$ does not lie in $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_4$. \n\\end{claim}\n\\noindent It is easy to see that claim \\ref{claim_a1_pa4_intersect_pd6_is_empty} implies \\eqref{a1_pa3_is_supsetof_pd6} and   \n\\eqref{a1_pa4_intersect_pd6_is_empty} simultaneously. \\\\\n\n\\noindent \\textbf{Proof:  } Choose homogeneous coordinates $[\\mathrm{X}: \\mathrm{Y}: \\mathrm{Z}]$ so that \n$\\tilde{p} = [0:0:1]$ and let ~$\\mathcal{U}_{\\tilde{p}}$,   \n$\\pi_x$, $\\pi_y$,  $v_1$, $w$, $v$, $\\eta$, $\\eta_{t_1}$, $\\eta_{t_2}$, \n$x_{t_1}$, $y_{t_1}$, $x_{t_2}$, $y_{t_2}$, \n$f_{ij}(t_1, t_2)$, $\\mathrm{F}$, $\\mathrm{F}_{x_{t_2}}$ and $\\mathrm{F}_{y_{t_2}}$\nbe exactly the same as defined in the \nproof of claim \\ref{claim_pd5_subset_of_a1_pa3}. \nSince $(\\tilde{f}(t_1, t_2), l_{\\tilde{p}(t_1)} ) \\in \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_3$ we conclude that \n\\[ f_{00}(t_1, t_2) = f_{10}(t_1, t_2) = f_{01}(t_1, t_2) =f_{11}(t_1, t_2)= f_{20}(t_1, t_2) = f_{30}(t_1, t_2)=0.\\]\nFurthermore, since $(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{D}_6$, we conclude that \n\\begin{align}\nf_{21}, ~f_{40} &=0 ~~\\textnormal{and} ~~f_{12}, ~\\mathcal{D}^{f}_7 \\neq 0. \\label{pd6_vanish_non_vanish}\n\\end{align}\nThe functional equation \\eqref{pd6_intersect_a1_pa4_is_empty_functional_eqn}has a solution if and only if the following has a numerical solution: \n\\begin{align}\n\\mathrm{F} = 0, \\qquad \\mathrm{F}_{x_{t_2}} = 0, \\qquad \\mathrm{F}_{y_{t_2}} = 0,  \\qquad (x_{t_2}, y_{t_2}) \\neq (0, 0) \\qquad \\textnormal{(but small)}. \\label{eval_f1_pd6} \n\\end{align} \nFor the convenience of the reader, let us rewrite the expression for $\\mathrm{F}$: \n\\begin{align*}\n\\mathrm{F} &: = {\\textstyle \\frac{f_{02}(t_1, t_2)}{2} y_{t_2}^2 + \\frac{f_{21}(t_1, t_2)}{2} x_{t_2}^2 y_{t_2}+ \\frac{f_{12}(t_1, t_2)}{2} x_{t_2} y_{t_2}^2 + \\frac{f_{03}(t_1, t_2)}{6} y_{t_2}^3} + \\ldots.\n\\end{align*}\nWe will now construct solutions to \\eqref{eval_f1_pd6}. \nLet us define $\\mathrm{G}:= \\mathrm{F}-\\frac{y_{t_2} \\mathrm{F}_{y_{t_2}}}{2}-\\frac{x_{t_2} \\mathrm{F}_{x_{t_2}}}{4}$. Then\n\\begin{align*}\n\\mathrm{G}&:= -\\frac{f_{03}(t_1, t_2)}{12} y_{t_2}^3 + y_{t_2}^3\\mathrm{E}_1(y_{t_2})\\\\\n          & ~~~ -\\xt \\big({\\textstyle \\frac{f_{12}(t_1, t_2)}{8} y_{t_2}^2  +\\frac{f_{31}(t_1, t_2)}{8} \\xt^2 y_{t_2} +\\frac{f_{50}(t_1, t_2)}{96} \\xt^4  + y_{t_2}^2 \\mathrm{E}_2(\\xt, y_{t_2})+\\xt^2 y_{t_2} \\mathrm{E}_3(\\xt) + \n\\xt^4 \\mathrm{E}_4(\\xt)} \\big),\n\\end{align*}\nwhere $\\mathrm{E}_i$ is a holomorphic function vanishing at the origin. \\\\\n\\hf \\hf We now make a change of variables by using a holomorphic function $\\mathrm{C}(y_{t_2})$ such that by making the substitution $x_{t_2} = \\hat{x}_{t_2} + \\mathrm{C}(y_{t_2})$, the coefficients of $y_{t_2}^n$ are killed for all $n$ in $\\mathrm{G}$. We may now make a change of coordinates $y_{t_2} = \\hat{y}_{t_2} + \\mathrm{B}(\\hat{x}_{t_2})$ so that the coefficient of $\\hat{x}_{t_2}\\hat{y}_{t_2}^n$ is killed for all $n$ in $\\mathrm{G}$. The existence of these functions follow from an identical argument as in \nin \\cite{BM13}. After these changes, $\\mathrm{G}$ is given by   \n\\begin{align*}\n\\mathrm{G} &=  {\\textstyle -\\frac{f_{12}(t_1, t_2)}{8} \\hat{x}_{t_2} \\Big( \\hat{y}_{t_2}^2 + \\frac{\\mathcal{D}^{f(t_1, t_2)}_7}{60 f_{12}(t_1, t_2)} \\hat{x}_{t_2}^4 + \\mathrm{O}(\\hat{x}_{t_2}^5)\\Big)}. \n\\end{align*}\nHence we are solving, in terms of the new variables $\\hat{x}_{t_2}$ and $\\hat{y}_{t_2}$, for \n\\begin{align}\n \\mathrm{G}&=0, ~~\\mathrm{F}_{x_{t_2}} =0, ~~\\mathrm{F}_{y_{t_2}} =0. \\label{G_eq_0_Fx_eq_0_Fy_eq_0_D6}\n\\end{align}\nNote that $\\mathrm{F}_{x_{t_2}}$ and $\\mathrm{F}_{y_{t_2}}$ are partials with respect to $x_{t_2}$ and $y_{t_2}$ \nexpressed in the new coordinates, they are not the partials with respect to $\\hat{x}_{t_2}$ and $\\hat{y}_{t_2}$. \nLet us write $\\mathrm{C}(y_{t_2})$ and $\\mathrm{B}(\\hat{x}_{t_2})$ to second order: \n\\begin{align*}\n\\mathrm{C}(y_{t_2}) = -& {\\textstyle \\frac{2 f_{03}(t_1, t_2)}{3 f_{12}(t_1, t_2)} y_{t_2}}\\\\ \n                + &  \\big( {\\textstyle -\\frac{f_{04}(t_1, t_2)}{3 f_{12}(t_1, t_2)}  \n+ \\frac{2 f_{03}(t_1, t_2) f_{13} (t_1, t_2) }{3 f_{12}(t_1, t_2)^2 } - \n\\frac{4 f_{03}(t_1, t_2)^2 f_{22} (t_1, t_2) }{9 f_{12}(t_1, t_2)^3 } + \\frac{8 f_{03}(t_1, t_2)^3 f_{31} (t_1, t_2) }{81 f_{12}(t_1, t_2)^4}} \\big) y_{t_2}^2\\\\ \n+ & \\mathrm{O}(y_{t_2}^3), \\\\ \n\\mathrm{B}(\\hat{x}_{t_2}) = - &  {\\textstyle \\frac{f_{31}(t_1, t_2)}{6 f_{12}(t_1, t_2)} \\hat{x}_{t_2}^2 + b_3 \\hat{x}_{t_2}^3 + \\mathrm{O}(\\hat{x}_{t_2}^4)}.\n\\end{align*}\nThe exact value of $b_3$ is not important, \nbut it will play a role in the \nsubsequent calculation. \nThe solutions to \\eqref{G_eq_0_Fx_eq_0_Fy_eq_0_D6} are given by \n\\begin{align}\n\\hat{y}_{t_2} &=  \\alpha \\hat{x}_{t_2}^2 +\\mathrm{O}(\\hat{x}_{t_2}^3) \\label{pd6_hat_y}\\\\\nf_{21}(t_1, t_2) &=\\Big({\\textstyle  - 2 \\alpha f_{02}(t_1, t_2) + \\frac{f_{02}(t_1, t_2) f_{31}(t_1, t_2)}{3 f_{12}(t_1, t_2)}} \\Big) + \\Big[{\\textstyle -2 b_3 f_{02}(t_1, t_2) -\\frac{8 \\alpha^2 f_{02}(t_1, t_2) f_{03}(t_1, t_2)}{3 f_{12}(t_1, t_2)}} \\nonumber\\\\\n&  \\,{\\textstyle -2\\alpha f_{12}(t_1, t_2)+\\frac{8 \\alpha f_{02}(t_1, t_2) f_{03}(t_1, t_2) f_{31}(t_1, t_2)}{9 f_{12}(t_1, t_2)^2} -\\frac{2 f_{02}(t_1, t_2) f_{03}(t_1, t_2) f_{31}(t_1, t_2)^2}{27 f_{12}(t_1, t_2)^3}} \\Big] \\hat{x}_{t_2} \\nonumber\\\\\n& + \\mathrm{O}(\\hat{x}_{t_2}^2)  \\label{pd6_f21}\\\\\nf_{40}(t_1, t_2) &= \\Big( 12 \\alpha^2 f_{02}(t_1, t_2)- \\frac{4 \\alpha f_{02}(t_1, t_2) f_{31}(t_1, t_2)}{f_{12}(t_1, t_2)} + \\frac{f_{02}(t_1, t_2) f_{31}(t_1, t_2)^2}{3 f_{12}(t_1, t_2)^2} \\Big) \\nonumber  \\\\\n                 &+ \\big({\\textstyle 24 \\alpha b_3 f_{02}(t_1, t_2)+\\frac{32 \\alpha^3 f_{02}(t_1, t_2) f_{03}(t_1, t_2)}{f_{12}(t_1, t_2)} + 24 \\alpha^2 f_{12}(t_1, t_2)-4 \\alpha f_{31}(t_1, t_2)} \\nonumber \\\\\n                 &{\\textstyle -\\frac{16 \\alpha^2 f_{02}(t_1, t_2) f_{03}(t_1, t_2) f_{31}(t_1, t_2)}{f_{12}(t_1, t_2)^2}-\\frac{4 b_3 f_{02}(t_1, t_2) f_{31}(t_1, t_2)}{f_{12}(t_1, t_2)}+ \\frac{8 \\alpha f_{02}(t_1, t_2) f_{03}(t_1, t_2) f_{31}(t_1, t_2)^2}{3 f_{12}(t_1, t_2)^3}}  \\nonumber \\\\ \n                 & {\\textstyle - \\frac{4 f_{02}(t_1, t_2) f_{03}(t_1, t_2) f_{31}(t_1, t_2)^3}{27 f_{12}(t_1, t_2)^4}}\\big) \\hat{x}_{t_2} \n    + \\mathrm{O}(\\hat{x}_{t_2}^2), \\label{pd6_f40}\\\\\n\\textnormal{where} ~~\\alpha &:= \\sqrt{-\\frac{\\mathcal{D}^{f(t_1, t_2)}_7}{60 f_{12}(t_1, t_2)}} ~~\\textnormal{is a branch of the square root.}  \\nonumber \n\\end{align}\nNote that each value of $\\alpha$ corresponds to a different solution. Let us now explain how we obtained these solutions. \nFirst of all we claim that $\\hat{x}_{t_2} \\neq 0$; we will justify that at the end. Assuming that, we obtain \n\\eqref{pd6_hat_y} from the fact $\\mathrm{G}=0$. Next, we obtain \\eqref{pd6_f21} from \\eqref{pd6_hat_y} and using the fact that \n$\\mathrm{F}_{y_{t_2}} =0$. Finally, we obtain \\eqref{pd6_f40} from  \\eqref{pd6_hat_y}, \\eqref{pd6_f21} and using the fact that \n$\\mathrm{F}_{x_{t_2}} =0$.   \nObserve that equations \\eqref{pd6_hat_y}, \\eqref{pd6_f21} and \\eqref{pd6_f40} imply that   \n\\begin{align}\nf_{02}(t_1, t_2)\\mathcal{A}^{f(t_1, t_2)}_4 &= \\textstyle{\\frac{1}{5}}\\mathcal{D}^{f(t_1, t_2)}_7 f_{12}(t_1, t_2) \\hat{x}_{t_2}^2 + \\xt^2 \\mathrm{E}(\\xt, f_{02}(t_1, t_2)) \\label{pd6_f02_pa4}\n\\end{align}\nwhere $\\mathrm{E}(0,0) =0$.\nHence, if $f_{02}(t_1, t_2)$ and $\\hat{x}_{t_2}$ are small and non zero, $f_{02}(t_1, t_2)\\mathcal{A}^{f(t_1, t_2)}_4$ is non zero. \nHence \\eqref{psi_pa4_neq_0_a1_pa3_functional} holds.  \\\\\n\\hf \\hf It remains to show that $\\hat{x}_{t_2} \\neq 0$. If $\\hat{x}_{t_2} =0$, then  \nusing the fact that $\\mathrm{F}_{x_{t_2}} =0$ we get  \n\\begin{align*}\nf_{12}(t_1, t_2) & = \\frac{4 f_{03}(t_1, t_2) f_{21}(t_1, t_2)}{3 f_{12}(t_1, t_2)} + \\mathrm{O}(\\hat{y}_{t_2}).\n\\end{align*}\nHence $f_{12}(t_1, t_2)$ goes to zero as $f_{21}(t_1, t_2)$ and $\\hat{y}_{t_2}$ go to zero, which  \ncontradicts \\eqref{pd6_vanish_non_vanish}. \\qed \\\\\n\n\\noindent This proves Lemma \\ref{cl_two_pt} (\\ref{a1_pa3_cl}). Before proceeding further, note that  \n\\eqref{a1_pa3_intersect_pd5_empty} and \\eqref{a1_pa4_intersect_pd6_is_empty} imply that \n\\begin{align}\n\\Delta \\PP \\D_7^{s} & \\subset \\Delta \\overline{\\mathcal{P} \\mathcal{D}}_7. \\label{one_a1_one_pa4_f02_zero_f12_not_zero_is_pd7}\n\\end{align}\n\n \n\n\\textbf{Proof of Lemma \\ref{cl_two_pt} (\\ref{a1_pa4_cl}):} By Lemma \\ref{closure_a1_pdk_f12_not_zero} and \\eqref{pak2_is_subset_of_a1_and_pak} for $k=4$, it suffices to show that \n\\begin{align}\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_4: ~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) = 0  \\} &=  \\Delta \\overline{\\PP \\D_7^{s}} \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{E}}_6 \\label{a1_pa4_pd7s_pe6}\n\\end{align}\nBy the definition of $\\Delta \\PP \\D_7^{s}$, to prove \\eqref{a1_pa4_pd7s_pe6} it suffices to show that \n\\begin{align}\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}})  \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_4& : \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) = 0, ~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{E}_6}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) =0  \\}  = \\Delta \\overline{\\mathcal{P} \\mathcal{E}}_6. \n\\label{one_a1_one_pa4_f02_zero_is_pe6}\n\\end{align}\nIt is clear that the lhs of \\eqref{one_a1_one_pa4_f02_zero_is_pe6} is a subset of its rhs. \nTo prove the converse, let us prove the following three facts simultaneously:\n\\begin{align}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_4 & \\supset \\Delta \\mathcal{P} \\mathcal{E}_6,    \\label{a1_pa4_is_supsetof_pe6} \\\\  \n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_5 \\cap \\Delta \\mathcal{P} \\mathcal{E}_6 & = \\varnothing,  \\label{a1_pa5_intersect_pe6_is_empty} \\\\\n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_4 \\cap \\Delta \\mathcal{P} \\mathcal{E}_6 & = \\varnothing.  \\label{a1_pd4_intersect_pe6_is_empty} \n\\end{align}\nNote that since $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_4$ is a closed set,  \n\\eqref{a1_pa4_is_supsetof_pe6} implies  that the rhs of \\eqref{one_a1_one_pa4_f02_zero_is_pe6}  is a subset of its lhs. \nWe will need \\eqref{a1_pd4_intersect_pe6_is_empty} later, since it follows from the present setup, we prove it here. \n\n\\begin{claim}\n\\label{claim_a1_pa5_intersect_pe6_is_empty}\nLet $(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{E}_6$.\nThen there exist solutions \n$$ (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} ) \\in \\overline{ (\\mathcal{D} \\times \\mathbb{P}^2) \\circ \\mathcal{P} \\mathcal{A}}_3$$ \n\\textit{near} $(\\tilde{f}, \\tilde{p}, l_{\\p})$ to the set of equations\n\\begin{align}\n\\pi_1^* \\psi_{\\mathcal{A}_0}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, ~\\pi_1^* \\psi_{\\mathcal{A}_1}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) = 0, \\nonumber \\\\\n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{A}_4}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, \n~\\tilde{p}(t_1, t_2) \\neq \\tilde{p}(t_1). \\label{pe6_intersect_a1_pa5_is_empty_functional_eqn}\n\\end{align}\nMoreover, any such solution sufficiently close to $(\\tilde{f}, \\tilde{p}, l_{\\p})$ lies in \n$\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}_4$, i.e., \n\\begin{align}\n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{A}_5}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), l_{\\tilde{p}(t_1)}) \\neq 0,  \\label{psi_pa5_neq_0_a1_pa4_functional} \\\\\n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{D}_4}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), l_{\\tilde{p}(t_1)}) \\neq 0. \\label{psi_pd4_neq_0_a1_pa4_functional}\n\\end{align}\nIn particular, $(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} )$ does not lie in $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_5$ or \n$\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_4$. \n\\end{claim}\n\\noindent It is easy to see that claim \\ref{claim_a1_pa5_intersect_pe6_is_empty} implies \\eqref{a1_pa4_is_supsetof_pe6},  \n\\eqref{a1_pa5_intersect_pe6_is_empty} and \\eqref{a1_pd4_intersect_pe6_is_empty} simultaneously. \\\\\n\n\\noindent \\textbf{Proof:  } Choose homogeneous coordinates $[\\mathrm{X}: \\mathrm{Y}: \\mathrm{Z}]$ so that \n$\\tilde{p} = [0:0:1]$ and let ~$\\mathcal{U}_{\\tilde{p}}$,   \n$\\pi_x$, $\\pi_y$,  $v_1$, $w$, $v$, $\\eta$, $\\eta_{t_1}$, $\\eta_{t_2}$, \n$x_{t_1}$, $y_{t_1}$, $x_{t_2}$, $y_{t_2}$, \n$f_{ij}(t_1, t_2)$, $\\mathrm{F}$, $\\mathrm{F}_{x_{t_2}}$ and $\\mathrm{F}_{y_{t_2}}$\nbe exactly the same as defined in the \nproof of claim \\ref{claim_pd5_subset_of_a1_pa3}.\nSince $(\\tilde{f}(t_1, t_2) , l_{\\tilde{p}(t_1)}) \\in  \\overline{\\mathcal{P} \\mathcal{A}}_3$, we conclude that \n\\[ f_{00}(t_1, t_2) = f_{10}(t_1, t_2) = f_{01}(t_1, t_2) =f_{11}(t_1, t_2)= f_{20}(t_1, t_2) = f_{30}(t_1, t_2)=0.\\]\nMoreover, since  $(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{E}_6$, we conclude that \n\\begin{align}\nf_{21}, f_{12} &=0, ~~f_{03}, f_{40} \\neq 0. \\label{pe6_condition_v_and_nv}\n\\end{align}\nThe functional equation \\eqref{pe6_intersect_a1_pa5_is_empty_functional_eqn}  \nhas a solution if and only if \nthe following set of equations has a solution (as numbers): \n\\begin{align}\n\\mathrm{F} = 0, ~~\\mathrm{F}_{x_{t_2}} = 0, ~~\\mathrm{F}_{y_{t_2}} = 0, ~~f_{02}(t_1, t_2)\\mathcal{A}^{f(t_1, t_2)}_4 =0, \\quad (x_{t_2}, y_{t_2}) \\neq (0, 0) \n~~\\textnormal{(small)}. \\label{eval_f1_pe6} \n\\end{align}\nSince $f_{02}(t_1, t_2)\\mathcal{A}^{f(t_1, t_2)}_4 =0$ we conclude that \n$f_{02}(t_1, t_2) = \\frac{3 f_{21}(t_1, t_2)^2}{f_{40}(t_1, t_2)}$. \nHence \n\\begin{align*}\n\\mathrm{F} & = \\frac{3 f_{21}(t_1, t_2)^2}{2 f_{40}(t_1, t_2)} y_{t_2}^2 + \\frac{f_{21}(t_1, t_2)}{2} x_{t_2}^2 y_{t_2}+ \n\\frac{f_{12}(t_1, t_2)}{2} x_{t_2} y_{t_2}^2 + \\frac{f_{03}(t_1, t_2)}{6} y_{t_2}^3 + \\frac{f_{40}(t_1, t_2)}{24} x_{t_2}^4 \\\\ \n& + \\frac{\\mathcal{R}_{50}(x_{t_2})}{120} x_{t_2}^5+ \n\\frac{\\mathcal{R}_{31}(x_{t_2})}{6} x_{t_2}^3 y_{t_2} \n+\\frac{\\mathcal{R}_{22}(x_{t_2})}{4} x_{t_2}^2 y_{t_2}^2 \n+\\frac{\\mathcal{R}_{13}(x_{t_2}, y_{t_2})}{4} x_{t_2} y_{t_2}^2 \n+ \\frac{\\mathcal{R}_{04}(y_{t_2})}{24} y_{t_2}^4.  \n\\end{align*}\nWe will now eliminate $f_{12}(t_1, t_2)$ and $f_{21}(t_1, t_2)$ from \\eqref{eval_f1_pe6}. \nFirst we  make a simple observation: let \n\\begin{align*}\n\\mathrm{A}(\\theta) &:= \\mathrm{A}_0 + \\mathrm{A}_1 \\theta + \\mathrm{A}_2 \\theta^2, \\quad \\mathrm{B}(\\theta) := \\mathrm{B}_0 + \\mathrm{B}_1 \\theta, \n\\quad p_1 := -\\mathrm{A}_2 \\mathrm{B}_1^2 , \\quad p_2 := -\\mathrm{A}_2^2 \\mathrm{B}_0 + \\mathrm{A}_1 \\mathrm{A}_2 \\mathrm{B}_1 +\\mathrm{A}_2^2 \\mathrm{B}_1 \\theta.  \n\\end{align*}\nThen $p_1 \\mathrm{A}(\\theta) + p_2\\mathrm{B}(\\theta)$ is independent of $\\theta$. With this observation we will now proceed to define  \n\\begin{align*}\n\\mathrm{G}_1:=\\mathrm{F} - x_{t_2} \\mathrm{F}_{x_{t_2}}, ~~\\mathrm{G}_2 := \\mathrm{F} - \\frac{y_{t_2} \\mathrm{F}_{y_{t_2}}}{2}, ~~ \\mathrm{G} := \n\\mathrm{P}_1(t_1, t_2) \\mathrm{G}_1 + \\mathrm{P}_2(t_1, t_2) \\mathrm{G}_2\n\\end{align*}\nwhere\n\\begin{align*}\n\\mathrm{P}_1 := - \\frac{3 x^4 y^4}{32 f_{40}}, \\qquad \\mathrm{P}_2 &:= \\frac{3 y^7 f_{03}}{16 f_{40}^2} + \\frac{9 x^2 y^5 f_{21}}{16 f_{40}^2} - \\frac{9 x^4 y^4}{32 f_{40}} + \\frac{3 y^8 \\mathcal{R}_{04}(y)}{32 f_{40}^2}\n+\\frac{3 x y^7 \\mathcal{R}_{13}(x,y)}{16 f_{40}^2}  \\\\\n             &-\\frac{3 x^3 y^5 \\mathcal{R}_{31}(x)}{16 f_{40}^2} - \\frac{3 x^5 y^4 \\mathcal{R}_{50}(x)}{160 f_{40}^2} \n                +\\frac{3 y^9 \\mathcal{R}_{04}^{(1)}(y)}{64 f_{40}^2}+\\frac{3 x y^8 \\mathcal{R}_{13}^{(0,1)}(x,y)}{16 f_{40}^2}.  \n\\end{align*}\nThe quantity $\\mathrm{P}_k(t_1, t_2)$ is similarly defined, with $f_{ij}$, $x$ and $y$ \nreplaced by $f_{ij}(t_1, t_2)$, \n$\\xt$ and $y_{t_2}$.\\\\   \n\\hf \\hf Note that $\\mathrm{G}_1$ and $\\mathrm{G}_2$ are independent of $f_{12}(t_1, t_2)$.  \nSecondly, they are quadratic and linear in $f_{21}(t_1, t_2)$ respectively. \nHence, using our previous observation, $\\mathrm{G}$ is independent of $f_{12}(t_1, t_2)$ and $f_{21}(t_1, t_2)$. \n\\footnote{Replace $\\theta \\longrightarrow f_{21}(t_1, t_2)$,  $\\mathrm{A}(\\theta) \\longrightarrow \\mathrm{G}_1$, $\\mathrm{B}(\\theta) \\longrightarrow \\mathrm{G}_2$, $p_1 \\longrightarrow \\mathrm{P}_1(t_1, t_2)$ and $p_2 \\longrightarrow \\mathrm{P}_2(t_1, t_2)$.}\\\\\n\\hf \\hf\nWe claim that $x_{t_2} \\neq 0$ and $y_{t_2} \\neq 0$; \nwe will justify that at the end. \nAssuming this claim we conclude that  solving \\eqref{eval_f1_pe6} \nis equivalent to solving: \n\\begin{align}\n\\mathrm{G}=0, ~~\\mathrm{G}_2 =0, ~~\\mathrm{F} =0, \n\\quad (x_{t_2}, y_{t_2}) \\neq (0, 0) \n~~\\textnormal{(small).} \\label{eval_f1_pe6_modified}\n\\end{align}\nDefine $\\mathrm{L}:= \\xt^4 \/ y_{t_2}^3$. We will first solve for $\\mathrm{L}$ in terms of $\\xt$ and $y_{t_2}$ and then we will parametrize $(\\xt, y_{t_2})$.  \nNotice that we can rewrite $\\mathrm{G}$ in terms of $\\xt$, $y_{t_2}$  and $\\mathrm{L}$ in such a way that the highest power of \n$\\xt$ is $3$; whenever there is an $\\xt^4$ we replace it with $\\mathrm{L} y_{t_2}^3$.\nHence  \n\\begin{align*}\n\\mathrm{G} &= y_{t_2}^{10}\\Big(- \\frac{f_{03}(t_1, t_2)^2}{64 f_{40}(t_1, t_2)^2} +\\frac{f_{03}(t_1, t_2)}{64 f_{40}(t_1, t_2)} \\mathrm{L}  + \\mathrm{E}(\\xt, y_{t_2}, \\mathrm{L}) \\Big)\n\\end{align*}\nwhere $\\mathrm{E}(0,0, \\mathrm{L}) =0$. Hence $\\mathrm{G} =0$ and $y_{t_2} \\neq 0$ implies that \n\\begin{align*}\n \\Phi (\\xt, y_{t_2}, \\mathrm{L}) &:= - \\frac{f_{03}(t_1, t_2)^2}{64 f_{40}(t_1, t_2)^2} +\\frac{f_{03}(t_1, t_2)}{64 f_{40}(t_1, t_2)} \\mathrm{L}  + \\mathrm{E}(\\xt, y_{t_2}, \\mathrm{L}) =0.\n\\end{align*}\nHence, by the Implicit Function Theorem we conclude that \n\\begin{align*}\n\\mathrm{L}(\\xt, y_{t_2}) = \\frac{f_{03}(t_1, t_2)}{f_{40}(t_1, t_2)} + \\mathrm{E}_2(\\xt, y_{t_2})  \n\\end{align*}\nwhere $\\mathrm{E}_2(0, 0)$ is zero. Hence $(\\xt, y_{t_2})$ is parametrized by \n\\begin{align*}\ny_{t_2} &= u^4, \\qquad \\xt = \\alpha u^3 + \\mathrm{O}(u^4)  \n\\end{align*}\nwhere $\\alpha:= \\sqrt[4]{\\frac{f_{03}(t_1, t_2)}{f_{40}(t_1, t_2)}}$, \na branch of the fourth root. Note that just one branch of the fourth root gives all the solutions, choosing another branch does not give us any more solutions.\\footnote{Observe that a neighbourhood of the origin for the curve $x^4 -y^3 =0$ is just one copy of $\\mathbb{C}$ parametrized by $x = u^3$ and $y = u^4$.} We have\n\\begin{eqnarray}\nf_{02}(t_1, t_2) & = & \\frac{f_{03}(t_1, t_2)}{12} u^4 + \\mathrm{O}(u^5)  \\label{psi_pd4_neq_0_a1_pa4_number}\\\\\nf_{02}(t_1, t_2)^2 \\mathcal{A}^{f(t_1, t_2)}_5 & = &  -\\frac{5 f_{03}(t_1, t_2)^2 f_{40}(t_1, t_2)}{18 \\alpha} u^5 + \\mathrm{O}(u^6) \\label{pe6_multiplicity}\n\\end{eqnarray}\nTo arrive at these, use $\\mathrm{G}_2 =0$ to get\n\\begin{align*}\nf_{21}(t_1, t_2) &=  \\frac{f_{03}(t_1, t_2)}{3 \\alpha^2} u^2 + \\mathrm{O}(u^3).\n\\end{align*}\nWe then use the fact that $f_{02}(t_1, t_2) = \\frac{3 f_{21}(t_1, t_2)^2}{f_{40}(t_1, t_2)}$ to get \\eqref{psi_pd4_neq_0_a1_pa4_number}. Finally using $\\mathrm{F} =0$ we get that \n\\begin{align*}\nf_{12}(t_1, t_2) &= -\\frac{2\\alpha^3 f_{40}(t_1, t_2)}{3} u + \\mathrm{O}(u^2)\n\\end{align*}\nPlugging all this in we get \\eqref{pe6_multiplicity}. Equations \\eqref{psi_pd4_neq_0_a1_pa4_number} and \\eqref{pe6_multiplicity} imply that \\eqref{psi_pa5_neq_0_a1_pa4_functional} and \\eqref{psi_pd4_neq_0_a1_pa4_functional} hold respectively. \\\\\n\\hf \\hf It remains to show that that $x_{t_2} \\neq 0$ and $y_{t_2} \\neq 0$. \nIf $y_{t_2} =0$ then using the fact that $\\mathrm{F} =0$ we get  \nthat \n\\begin{align*}\nf_{40}(t_1, t_2) &= -\\frac{\\xt \\mathcal{R}_{50}(\\xt)}{5}.\n\\end{align*}\nAs $\\xt$ goes to zero, $f_{40}(t_1, t_2)$ goes to zero, contradicting \n\\eqref{pe6_condition_v_and_nv}. \nSimilarly, if $\\xt=0$, then using the fact that \n$\\mathrm{F} - \\frac{y_{t_2} \\mathrm{F}_{y_{t_2}}}{2} =0$ we get that \n\\begin{align*}\nf_{03}(t_1, t_2) &= -\\frac{2 y_{t_2} \\mathcal{R}_{04}(y_{t_2}) + \ny_{t_2}^2 \\mathcal{R}_{04}^{(1)}(y_{t_2})}{4}\n\\end{align*}\nAs $y_{t_2}$ goes to zero, $f_{03}(t_1, t_2)$ goes to zero, contradicting \n\\eqref{pe6_condition_v_and_nv}. \n\n\\begin{cor}\n\\label{a1_pa4_mult_is_5_around_pe6}\nLet $\\mathbb{W} \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2$ be a vector bundle such that \nthe rank of $\\mathbb{W}$ is same as the dimension of $ \\Delta \\mathcal{P} \\mathcal{E}_{6}$ and \n$\\mathcal{Q}: \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2 \\longrightarrow \\mathbb{W}$  a \\textit{generic} \nsmooth section. Suppose $(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{E}_{6} \\cap \\mathcal{Q}^{-1}(0)$. \nThen the section $$ \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_{5}} \\oplus \\mathcal{Q}: \\overline{\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{A}}_{4} \\longrightarrow \\pi_2^* (\\UL_{\\mathcal{P} \\mathcal{A}_{5}}) \\oplus \\mathbb{W}$$\nvanishes around $(\\tilde{f}, \\tilde{p},  l_{\\p})$ with a multiplicity of $5$.\n\\end{cor}\n\\noindent \\textbf{Proof:  } Follows from the fact that the sections \ninduced by $f_{02}$, $f_{21}$ and $f_{12}$ (the corresponding functionals) \nare transverse to the zero set over $\\Delta \\overline{\\mathcal{P} \\mathcal{A}}_3$ \\footnote{To see why, just take the partial derivatives with respect to \n$f_{02}$, $f_{21}$ and $f_{12}$}, the fact that  $\\mathcal{Q}$ is generic and \\eqref{pe6_multiplicity}. \\qed \\\\\n\n\\noindent This completes the proof of  Lemma \\ref{cl_two_pt} (\\ref{a1_pa4_cl}). \\\\\n\n\n\\textbf{Proof of Lemma \\ref{cl_two_pt} (\\ref{a1_pa5_cl}):} We have to show that \n\\bgd\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_5\\} =\\Delta\\overline{\\mathcal{P}\\mathcal{A}}_7\\cup\\Delta\\overline{\\mathcal{P} \\mathcal{D}_8^s}\\cup\\Delta\\overline{\\mathcal{P} \\mathcal{E}}_7.\n\\edd\nBy Lemma \\ref{closure_a1_pak_f02_not_zero} and \\eqref{pak2_is_subset_of_a1_and_pak}, it is equivalent to showing that  \n\\begin{align}\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_5: ~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) = 0  \\} &=  \\Delta \\overline{\\PP \\D_8^{s}} \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{E}}_7. \\label{a1_pa5_pd8s_pe7}\n\\end{align}\nBy the definition of $\\Delta \\PP \\D_8^{s}$, to prove \\eqref{a1_pa5_pd8s_pe7} it suffices to show that \n\\begin{align}\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}})  \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_5 & : \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) = 0, ~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{E}_6}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) =0  \\}  = \\Delta \\overline{\\mathcal{P} \\mathcal{E}}_7. \n\\label{one_a1_one_pa5_f02_zero_is_pe7}\n\\end{align}\nBefore we prove \\eqref{one_a1_one_pa5_f02_zero_is_pe7}, we will first prove the following fact: \n\\begin{align}\n\\Delta \\PP \\D_8^{s} & \\subset \\Delta \\overline{\\mathcal{P} \\mathcal{D}}_8. \\label{one_a1_one_pa5_f02_zero_f12_not_zero_is_pd8}\n\\end{align}\nAlthough \\eqref{one_a1_one_pa5_f02_zero_f12_not_zero_is_pd8} \nis not required for the proof of Lemma \\ref{cl_two_pt} (\\ref{a1_pa5_cl}), it will be needed later. \nTo prove \\eqref{one_a1_one_pa5_f02_zero_f12_not_zero_is_pd8}, it suffices to show that \n\\begin{align}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_5 \\cap \\Delta \\mathcal{P} \\mathcal{D}_7 & = \\varnothing. \\label{a1_pa5_intersect_pd7_is_empty}  \n\\end{align}\nTo see why, note that by \\eqref{a1_pa4_intersect_pd6_is_empty} combined with $\\mathcal{P} \\mathcal{A}_5\\subset \\overline{\\mathcal{P} \\mathcal{A}}_4$ we have \n\\bgd\n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_5 \\cap \\Delta \\mathcal{P} \\mathcal{D}_6=\\varnothing.\n\\edd\nSince $\\Delta\\overline{\\mathcal{P} \\mathcal{D}}_8^s\\subseteq \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_5$,  \nwe conclude that \\eqref{a1_pa5_intersect_pd7_is_empty} implies  \n\\bgd\n\\Delta\\overline{\\mathcal{P} \\mathcal{D}}_8^s \\cap \\big(\\Delta \\mathcal{P} \\mathcal{D}_6\\cup \\Delta \\mathcal{P} \\mathcal{D}_7\\big)=\\varnothing.\n\\edd\nOn the other hand, by Lemma \\ref{cl} (\\ref{A5cl}) we \nknow that $\\Delta\\overline{\\mathcal{P} \\mathcal{D}}_8^s\\subseteq \\Delta \\overline{\\mathcal{P} \\mathcal{D}}_6$. \nLemma \\ref{Dk_sharp_closure} (\\ref{pd6_pd7_pd8_closure}) now \nproves \\eqref{one_a1_one_pa5_f02_zero_f12_not_zero_is_pd8} assuming \\eqref{a1_pa5_intersect_pd7_is_empty}.\n\\begin{claim}\n\\label{claim_a1_pa5_intersect_pd7_is_empty}\nLet $(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{D}_7$. \nThen there are no solutions\n$$ (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} ) \\in \\overline{ (\\mathcal{D} \\times \\mathbb{P}^2) \\circ \\mathcal{P} \\mathcal{A}}_3$$ \n\\textit{near} $(\\tilde{f}, \\tilde{p}, l_{\\p})$ to the set of equations\n\\begin{align}\n\\label{pd7_intersect_a1_pa5_is_empty_functional_eqn}\n\\pi_1^* \\psi_{\\mathcal{A}_0}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, ~\\pi_1^* \\psi_{\\mathcal{A}_1}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) = 0, \\nonumber \\\\ \n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{A}_4}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, ~\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{A}_5}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) = 0,  \\nonumber \\\\ \n\\pi_1^* \\Psi_{\\mathcal{P} \\mathcal{D}_4}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & \\neq 0, ~\\tilde{p}(t_1, t_2) \\neq \\tilde{p}(t_1).\n\\end{align}\nIn particular, if $(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} )$ is sufficiently close to \n$(\\tilde{f},\\tilde{p}, l_{\\p})$, it does not lie in $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_5$. \n\\end{claim}\n\\noindent It is easy to see that claim \\ref{claim_a1_pa5_intersect_pd7_is_empty} implies \\eqref{a1_pa5_intersect_pd7_is_empty}. \\\\\n\n\\noindent \\textbf{Proof:  } Choose homogeneous coordinates $[\\mathrm{X}: \\mathrm{Y}: \\mathrm{Z}]$ so that \n$\\tilde{p} = [0:0:1]$ and let ~$\\mathcal{U}_{\\tilde{p}}$,   \n$\\pi_x$, $\\pi_y$,  $v_1$, $w$, $v$, $\\eta$, $\\eta_{t_1}$, $\\eta_{t_2}$, \n$x_{t_1}$, $y_{t_1}$, $x_{t_2}$, $y_{t_2}$, \n$f_{ij}(t_1, t_2)$, $\\mathrm{F}$, $\\mathrm{F}_{x_{t_2}}$ and $\\mathrm{F}_{y_{t_2}}$\nbe exactly the same as defined in the \nproof of claim \\ref{claim_pd5_subset_of_a1_pa3}. \nSince $(\\tilde{f}(t_1, t_2), l_{\\tilde{p}(t_1)} ) \\in \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_3$ we conclude that \n\\[ f_{00}(t_1, t_2) = f_{10}(t_1, t_2) = f_{01}(t_1, t_2) =f_{11}(t_1, t_2)= f_{20}(t_1, t_2) = f_{30}(t_1, t_2)=0.\\]\nFurthermore, since $(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{D}_7$, we conclude that \n\\begin{align}\nf_{21}, ~f_{40}, ~\\mathcal{D}_7^f &=0 ~~\\textnormal{and} ~~f_{12}, ~\\mathcal{D}^{f}_8 \\neq 0. \\label{pd7_vanish_non_vanish}\n\\end{align}\nThe functional equation \\eqref{pd7_intersect_a1_pa5_is_empty_functional_eqn}  \nhas a solution if and only if \nthe following set of equations has a solution (as numbers): \n\\begin{align}\n\\mathrm{F} & = 0, \\qquad \\mathrm{F}_{x_{t_2}} = 0, \\qquad \\mathrm{F}_{y_{t_2}} = 0, \\qquad \\mathcal{A}^{f(t_1, t_2)}_4 =0, \\qquad \\mathcal{A}^{f(t_1, t_2)}_5 =0, \\nonumber \\\\ \n(x_{t_2}, y_{t_2}) & \\neq (0, 0), \\qquad f_{02}(t_1, t_2) \\neq 0  \\qquad \\textnormal{(but small)}. \\label{eval_f1_pd7} \n\\end{align} \nFor the convenience of the reader, let us rewrite the expression for $\\mathrm{F}$: \n\\bge\n\\mathrm{F} : = {\\textstyle \\frac{f_{02}(t_1, t_2)}{2}} y_{t_2}^2 + {\\textstyle \\frac{f_{21}(t_1, t_2)}{2}} x_{t_2}^2 y_{t_2}+ {\\textstyle \\frac{f_{40}(t_1, t_2)}{24}}\\xt^4 + {\\textstyle \\frac{f_{12}(t_1, t_2)}{2}} x_{t_2} y_{t_2}^2 +{\\textstyle  \\frac{f_{03}(t_1, t_2)}{6}} y_{t_2}^3 + \\ldots. \\label{F_recap_pd8}\n\\ede\nWe will now show that solutions to \n\\eqref{eval_f1_pd7} can not exist.  \nWe will show that if $(f_{ij}(t_1, t_2), \\xt, y_{t_2})$ is a sequence converging \nto $(f,0,0) $\nthat satisfies \\eqref{eval_f1_pd7}, then $\\mathcal{D}^{f(t_1, t_2)}_8$ goes to zero (after passing to a subsequence).  \nThat would contradict \\eqref{pd7_vanish_non_vanish}. \nFirst, we observe that any solution to the set of equations $\\mathcal{A}^{f(t_1, t_2)}_4 =0$ and $\\mathcal{A}^{f(t_1, t_2)}_5 =0$ \nis given by \n\\begin{align}\nf_{21}(t_1, t_2) & = {\\textstyle \\Big( \\frac{f_{31}(t_1, t_2) \\pm v }{ 3 f_{12}(t_1, t_2)} \\Big)f_{02}(t_1, t_2), ~~f_{40}(t_1, t_2) = \\frac{3 f_{21}(t_1, t_2)^2}{f_{02}(t_1, t_2)},~~\\mathcal{D}^{f(t_1, t_2)}_7 = -\\frac{5 v^2}{3 f_{12}(t_1, t_2)}}. \\label{af4_af5_pd8}\n\\end{align}\nLet us choose the $+v$ solution for $f_{21}(t_1, t_2)$, a similar argument will \ngo through for the $-v$ solution. Now, using the value of $f_{40}(t_1, t_2)$ we observe that the first three terms in the expression for \n$\\mathrm{F}$ in \\eqref{F_recap_pd8} form a perfect square. We will now rewrite the remaining part of $\\mathrm{F}$ by making a change of coordinates. \nUsing an identical argument that is given in \nin \\cite{BM13} and using \\eqref{af4_af5_pd8}, we can make a change of coordinates \n\\[ x_{t_2} = \\hat{x}_{t_2} + \\mathrm{G}(y_{t_2}, v), \\qquad y_{t_2} = \\hat{y}_{t_2} + \\mathrm{B}(\\hat{x}_{t_2}, v)  \\]\nso that $\\mathrm{F}$ is given by \n\\begin{align*}\n\\mathrm{F} &= \\frac{u}{2} \\Big( \\hat{y}_{t_2} + \\mathrm{B}(\\hat{x}_{t_2},v) + \\frac{f_{31}(t_1, t_2)}{6 f_{12}(t_1, t_2)} (\\hat{x}_{t_2} + \\mathrm{J} )^2  + \n\\frac{v (\\hat{x}_{t_2} + \\mathrm{J})}{6 f_{12}(t_1, t_2)}\\Big)^2  \\\\ \n& -\\frac{v^2 \\hat{x}_{t_2}^5}{72 f_{12}(t_1, t_2)} + \\frac{f_{12}(t_1, t_2)}{2} \\hat{x}_{t_2} \\hat{y}_{t_2}^2+ \\frac{\\mathcal{D}^{f(t_1, t_2)}_8}{720} \\hat{x}_{t_2}^6 + \n\\beta(\\hat{x}_{t_2}) \\hat{x}_{t_2}^7,\n\\end{align*}\nwhere \n\\[ \\mathrm{J} := \\mathrm{G}(\\hat{y}_{t_2} + \\mathrm{B}(\\hat{x}_{t_2},v), v),\\,\\,\\,u:=f_{02}(t_1,t_2).\\]\nNote that $\\mathrm{B}$ is also a function of  $v$ because the coefficients of $\\xt^n$ may depend   \non $f_{50}(t_1, t_2)$, which is equal to  $\\mathcal{D}^{f(t_1, t_2)}_7 + \\frac{5 f_{31}(t_1, t_2)^2}{3 f_{12}(t_1, t_2)} $. \nLet \n\\[ z_{t_2} := \\hat{y}_{t_2} + \\frac{\\hat{x}_{t_2}^2 v}{6 f_{12}(t_1, t_2)}.\\]\nSince \n\\[\\mathrm{G}(0,v) =0 \\qquad \\textnormal{and} \\qquad  \\mathrm{B}(\\hat{x}_{t_2}, v) + \\frac{f_{31}(t_1, t_2)}{6 f_{12}(t_1, t_2)} \\hat{x}_{t_2}^2 = \\mathrm{O}(\\hat{x}_{t_2}^3) \\]\nwe conclude that in the new coordinates $(\\hat{x}_{t_2}, z_{t_2})$,  $\\mathrm{F}$ is given by  \n\\begin{align}\n\\mathrm{F} &= \\frac{u}{2} \\Big( z_{t_2} + \\alpha(\\hat{x}_{t_2}, v) \\hat{x}_{t_2} ^3 + \\mathrm{E}(\\hat{x}_{t_2}, z_{t_2}, v) z_{t_2} \\Big)^2  \\nonumber \\\\ \n& -\\frac{v \\hat{x}_{t_2}^3 z_{t_2}}{6 f_{12}(t_1, t_2)} + \\frac{f_{12}(t_1, t_2)}{2} \\hat{x}_{t_2} z_{t_2}^2+ \\frac{\\mathcal{D}^{f(t_1, t_2)}_8}{720} \\hat{x}_{t_2}^6 + \n\\beta(\\hat{x}_{t_2}) \\hat{x}_{t_2}^7, \\qquad \\textnormal{where} ~~\\mathrm{E}(0,0,v) \\equiv 0.\\label{FF_new_form_pd8}\n\\end{align}\nHence, \\eqref{eval_f1_pd7} has solutions if and only if the following set of equations has a solution \n\\begin{align}\n\\mathrm{F} &=0, \\qquad \\mathrm{F}_{\\hat{x}_{t_2}} =0, \\qquad \\mathrm{F}_{z_{t_2}} =0, \\qquad (\\hat{x}_{t_2}, z_{t_2}) \\neq (0,0), ~~u \\neq 0, ~~v ~~\\textnormal{small}. \\label{f_eval_pd7_new}\n\\end{align}\nWe will now analyze solutions of \\eqref{f_eval_pd7_new}. \nNotice that we can rewrite \\eqref{f_eval_pd7_new} in the following way \n\\begin{align}\np_0 + p_1 w + p_2 v &=0, ~~q_0 + q_1 w + q_2 v =0, ~~r_0 + r_1 w + r_2 v =0, \\label{p0_q0_etc}\n\\end{align}\nwhere \n\\begin{align}\nw &:= u (z_{t_2} + \\alpha (\\hat{x}_{t_2}, v) \\hat{x}_{t_2}^3 + \\mathrm{E}(\\hat{x}_{t_2}, z_{t_2},  v)), \n~~\\eta := z_{t_2} + \\alpha (\\hat{x}_{t_2}, v) \\hat{x}_{t_2}^3 + \\mathrm{E}(\\hat{x}_{t_2}, z_{t_2},  v), \\nonumber \\\\\np_0 &:=  \\frac{f_{12}(t_1, t_2)}{2} \\hat{x}_{t_2} z_{t_2}^2+ \\frac{\\mathcal{D}^{f(t_1, t_2)}_8}{720} \\hat{x}_{t_2}^6 + \n\\beta(\\hat{x}_{t_2}) \\hat{x}_{t_2}^7, ~~p_1 := \\frac{\\eta}{2}, ~~p_2 := -\\frac{\\hat{x}_{t_2}^3 z_{t_2}}{6}, \\nonumber \\\\ \nq_0 &:= \\frac{f_{12}(t_1, t_2)}{2}  z_{t_2}^2+ \\frac{\\mathcal{D}^{f(t_1, t_2)}_8}{120} \\hat{x}_{t_2}^5 + \n(7 \\beta(\\hat{x}_{t_2}) + \\hat{x}_{t_2} \\beta^{\\prime} (\\hat{x}_{t_2})) \\hat{x}_{t_2}^6, \\nonumber \\\\ \nq_1 & := 3 \\alpha (\\hat{x}_{t_2}) \\hat{x}_{t_2}^2 + \\alpha^{\\prime} (\\hat{x}_{t_2}) \\hat{x}_{t_2}^3 + \\mathrm{E}_{{\\hat{x}_{t_2}}}(\\hat{x}_{t_2}, z_{t_2}, v), \n~~q_2:= -\\frac{\\hat{x}_{t_2}^2 z_{t_2}}{2}, \\nonumber  \\\\ \nr_0& := f_{12}(t_1, t_2) \\hat{x}_{t_2} z_{t_2}  ~~r_1 := 1+ \\mathrm{E}(\\hat{x}_{t_2}, z_{t_2}, v)+ z \\mathrm{E}_{z_{t_2}}(\\hat{x}_{t_2}, z_{t_2}, v)), \n~~r_2 := -\\frac{\\hat{x}_{t_2}^3}{6}. \\label{p0_q0_defn}\n\\end{align}\nSince \\eqref{p0_q0_etc} holds, we conclude \nthat \n\\begin{align}\np_0 q_2 r_1 -p_0 q_1 r_2 + p_2 q_1 r_0 -p_1 q_2 r_0 -p_2q_0 r_1 + p_1 q_0 r_2 =0. \\label{p0_q0_eliminated}  \n\\end{align}\nEquations \\eqref{p0_q0_eliminated} and \\eqref{p0_q0_defn} now imply that  \n\\begin{align}\n\\Phi(\\hat{x}_{t_2}, z_{t_2}) & : = z_{t_2}^3(f_{12}(t_1, t_2) + \\mathrm{E}_2(\\hat{x}_{t_2}, z_{t_2},  v)) \n+ z_{t_2}^2 \\hat{x}_{t_2}^3(f_{12}(t_1, t_2) + \\mathrm{E}_3(\\hat{x}_{t_2}, z_{t_2}, v)) \\nonumber \\\\  \n & ~~~~ +z_{t_2} \\hat{x}_{t_2}^6(f_{12}(t_1, t_2) + \\mathrm{E}_4(\\hat{x}_{t_2}, z_{t_2}, v)) + \n \\hat{x}_{t_2}^9 \\mathcal{R}_5(z_{t_2}, \\hat{x}_{t_2}, v)=0,   \\label{eliminate_u_v_pd8}\n\\end{align}\nwhere $\\mathrm{E}_i(0,0,v) =0$ and $\\mathcal{R}_5(z_{t_2}, \\hat{x}_{t_2}, v)$ is a holomorphic function. \nNow let us define $\\mathrm{L} := z_{t_2}\/ \\hat{x}_{t_2}^3.$ Let $(f(t_1, t_2), \\hat{x}_{t_2}, z_{t_2})$ \nbe a sequence  converging to $(f, 0,0)$ \nthat satisfies \\eqref{f_eval_pd7_new} and such that $(\\hat{x}_{t_2}, z_{t_2}) \\neq (0,0)$.  \nFirst we observe that $\\hat{x}_{t_2} \\neq 0$; this follows from \n\\eqref{eliminate_u_v_pd8} and the fact that $f_{12} \\neq 0$. \nNext, it is easy to see \nusing \\eqref{eliminate_u_v_pd8}, that\n$\\mathrm{L}$ is bounded, since $f_{12}\\neq 0$. Hence, after passing to a subsequence $\\mathrm{L}$ \nconverges. Since $\\mathrm{F} =0$, we can easily see from \\eqref{FF_new_form_pd8}, that \nas $(\\hat{x}_{t_2}, z_{t_2}), u$ and $v$ go to zero, $\\mathcal{D}^{f(t_1, t_2)}_8$ goes to zero. \nThis contradicts \\eqref{pd7_vanish_non_vanish}. \\qed \\\\\n\n\n\n\\noindent Now let us prove \\eqref{one_a1_one_pa5_f02_zero_is_pe7}. It is clear that the lhs of \\eqref{one_a1_one_pa5_f02_zero_is_pe7}  is a subset of its rhs. \nNext, we will prove the \nfollowing two facts simultaneously:\n\\begin{align}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_5 & \\supset \\Delta \\mathcal{P} \\mathcal{E}_7,    \\label{a1_pa5_is_supsetof_pe7} \\\\  \n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_6 \\cap \\Delta \\mathcal{P} \\mathcal{E}_7 & = \\varnothing.  \\label{a1_pa6_intersect_pe7_is_empty}  \n\\end{align}\nSince $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_5$ is a closed set, \\eqref{a1_pa5_is_supsetof_pe7} implies that the \nrhs of \\eqref{one_a1_one_pa5_f02_zero_is_pe7} is a subset of its lhs. \n\n\\begin{claim}\n\\label{claim_a1_pa6_intersect_pe7_is_empty}\nLet $(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{E}_7$.\nThen there exist solutions \n$$ (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} ) \\in \\overline{ (\\mathcal{D} \\times \\mathbb{P}^2) \\circ \\mathcal{P} \\mathcal{A}}_3$$ \n\\textit{near} $(\\tilde{f}, \\tilde{p}, l_{\\p})$ to the set of equations\n\\begin{align}\n\\pi_1^* \\psi_{\\mathcal{A}_0}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, ~\\pi_1^* \\psi_{\\mathcal{A}_1}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) = 0, \\nonumber \\\\\n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{A}_4}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, ~~\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{A}_5}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)})  = 0, \\nonumber \\\\\n ~\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{D}_4}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & \\neq 0, ~~\\tilde{p}(t_1, t_2) \\neq \\tilde{p}(t_1). \\label{pe7_intersect_a1_pa6_is_empty_functional_eqn} \n\\end{align}\nMoreover, any such solution sufficiently close to $(\\tilde{f}, \\tilde{p}, l_{\\p})$ lies in \n$\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}_5$, i.e., \n\\begin{align}\n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{A}_6}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), l_{\\tilde{p}(t_1)}) \\neq 0. \\label{psi_pa6_neq_0_a1_pa5_functional}\n\\end{align}\nIn particular, $(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} )$ does not lie in $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_6$. \n\\end{claim}\n\\noindent It is easy to see that claim \\ref{claim_a1_pa6_intersect_pe7_is_empty} implies \\eqref{a1_pa5_is_supsetof_pe7} and \n\\eqref{a1_pa6_intersect_pe7_is_empty} simultaneously. \\\\ \n   \n\\noindent \\textbf{Proof:  } Choose homogeneous coordinates $[\\mathrm{X}: \\mathrm{Y}: \\mathrm{Z}]$ so that \n$\\tilde{p} = [0:0:1]$ and let ~$\\mathcal{U}_{\\tilde{p}}$,   \n$\\pi_x$, $\\pi_y$,  $v_1$, $w$, $v$, $\\eta$, $\\eta_{t_1}$, $\\eta_{t_2}$, \n$x_{t_1}$, $y_{t_1}$, $x_{t_2}$, $y_{t_2}$, \n$f_{ij}(t_1, t_2)$, $\\mathrm{F}$, $\\mathrm{F}_{x_{t_2}}$ and $\\mathrm{F}_{y_{t_2}}$\nbe exactly the same as defined in the \nproof of claim \\ref{claim_pd5_subset_of_a1_pa3}.\nSince $(\\tilde{f}(t_1, t_2), l_{\\tilde{p}(t_1)}) \\in \\overline{\\mathcal{P} \\mathcal{A}}_3$, we conclude \n\\[ f_{00}(t_1, t_2) = f_{10}(t_1, t_2) = f_{01}(t_1, t_2) =f_{11}(t_1, t_2)= f_{20}(t_1, t_2) = f_{30}(t_1, t_2)=0.\\]\nMoreover, since $(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{E}_7$, we conclude that\n\\begin{align}\nf_{12}, f_{40} &=0, ~~f_{31}, f_{03} \\neq 0. \\label{pe7_nv}\n\\end{align}\nThe functional equation \\eqref{pe6_intersect_a1_pa5_is_empty_functional_eqn}  \nhas a solution if and only if \nthe following set of equations has a solution (as numbers): \n\\begin{align}\n\\mathrm{F} & = 0, ~~\\mathrm{F}_{x_{t_2}} = 0, ~~\\mathrm{F}_{y_{t_2}} = 0, ~~\\mathcal{A}^{f(t_1, t_2)}_4 =0, ~~\\mathcal{A}^{f(t_1, t_2)}_5 =0, \\nonumber \\\\ \nf_{02}(t_1, t_2) &\\neq 0, \\qquad (x_{t_2}, y_{t_2}) \\neq (0, 0) \\qquad \\textnormal{(but small)}.  \\label{eval_f1_pe7} \n\\end{align}\nLet us now define the following quantities: \n\\begin{align*}\n\\mathrm{G}_1 &:= \\mathrm{F} \n-\\frac{\\xt \\mathrm{F}_{\\xt}}{4} -\\frac{y_{t_2} \\mathrm{F}_{y_{t_2}}}{2}, ~~\\mathrm{G}_2 := \\mathrm{F}-\\frac{y_{t_2} \\mathrm{F}_{y_{t_2}}}{2}, \n~~\\mathrm{G} := \\mathrm{F}+ 4 \\mathrm{G}_1 - 2 \\mathrm{G}_2. \n\\end{align*}\nNote that $\\mathrm{G}_1$ depends linearly on $f_{12}(t_1, t_2)$\nand is independent of $f_{40}(t_1, t_2)$, \n$f_{21}(t_1, t_2)$ and $f_{02}(t_1, t_2)$; \n$\\mathrm{G}_2$ depends linearly on $f_{40}(t_1, t_2)$ \nand $f_{21}(t_1, t_2)$\nand is independent of $f_{12}(t_1, t_2)$, \nand $f_{02}(t_1, t_2)$; finally \n$\\mathrm{G}$ depends linearly on $f_{02}(t_1, t_2)$ \nand $f_{40}(t_1, t_2)$\nand is independent of $f_{12}(t_1, t_2)$, \nand $f_{21}(t_1, t_2)$. \nNext, we claim that $\\xt \\neq 0$ and $y_{t_2} \\neq 0$; we will \njustify that at the end. Assuming this claim, we observe that \n\\eqref{eval_f1_pe7} combined with \\eqref{pe7_nv} \nis equivalent to\n\\begin{align}\n\\mathrm{G}_1 & = 0, ~~\\mathrm{G}_2 = 0, ~~\\mathrm{G} = 0, ~~f_{21}(t_1, t_2) = \\frac{5f_{12}(t_1, t_2)f_{40}(t_1, t_2) + f_{02}(t_1, t_2) f_{50}(t_1, t_2)}{10 f_{31}(t_1, t_2)},  \\nonumber \\\\\n\\mathcal{A}^{f(t_1, t_2)}_4 & =0,  ~~f_{02}(t_1, t_2) \\neq 0, ~~ \\xt \\neq 0, ~~y_{t_2} \\neq 0 \\qquad \\textnormal{(but small)}.  \\label{eval_f1_pe7_modified} \n\\end{align}\nWe will now construct solutions for  \n\\eqref{eval_f1_pe7_modified}. \nFirst of all, using $\\mathrm{G} =0$ we can solve for $f_{02}(t_1, t_2)$ as a function of \n$f_{40}(t_1, t_2)$, $\\xt$ and $y_{t_2}$. Next, \nusing that $\\mathrm{G}_1 =0$, we get $f_{12}(t_1, t_2)$ as a function of $\\xt$ and $y_{t_2}$. Finally, using \n$\\mathrm{G}_2 =0$, the value of $f_{02}(t_1, t_2)$, $f_{12}(t_1, t_2)$  \nfrom the previous two equations and the value of $f_{21}(t_1, t_2)$ from \\eqref{eval_f1_pe7_modified}, \nwe get $f_{40}(t_1, t_2)$ in terms of $\\xt$ and $y_{t_2}$. Plugging the expression back in, we get \n$f_{12}(t_1, t_2)$, $f_{21}(t_1, t_2)$, $f_{02}(t_1, t_2)$ and $f_{40}(t_1, t_2)$ in terms of $\\xt$ \nand $y_{t_2}$.\\\\\n\\hf \\hf Next, let us define $\\mathrm{L} := \\xt^3 \/y_{t_2}^2.$ We note that any expression involving $\\xt$ and $y_{t_2}$ can be re written in  terms of $\\xt$, $y_{t_2}$ \nand $\\mathrm{L}$ so that the highest power of $\\xt$ is $2$; replace $\\xt^3$ by $\\mathrm{L} y_{t_2}^2$. \nUsing that fact, \nwe conclude    \n\\begin{align*}\n\\mathcal{A}^{f(t_1, t_2)}_4 \\frac{\\xt^4}{y_{t_2}^3}&= -\\frac{f_{03}(t_1, t_2)^2}{3} + \\frac{f_{03}(t_1, t_2) f_{31}(t_1, t_2)}{3} \\mathrm{L} + \\mathrm{E}_1(\\xt, y_{t_2}, \\mathrm{L}) =0 \n\\end{align*}\nwhere $\\mathrm{E}_1(0,0, \\mathrm{L}) =0$.\nHence, by the Implicit function theorem, we conclude that \n\\begin{align*}\n\\mathrm{L} &= \\frac{f_{03}(t_1, t_2)}{f_{31}(t_1, t_2)} + \\mathrm{E}_2(\\xt, y_{t_2}) \n\\end{align*}\n where $\\mathrm{E}_2(0,0)=0$. \nHence, $\\xt$ and $ y_{t_2}$ are parametrized by \n\\[ y_{t_2} = u^3, ~~\\xt = \\alpha u^2 + \\mathrm{O}(u^3) \\qquad \\textnormal{where} ~~\\alpha := \\sqrt[3]{\\frac{f_{03}(t_1, t_2)}{f_{31}(t_1, t_2)}}, ~~\\textnormal{a branch of the cube root.}  \\]\nPlugging in all this we get \n\\begin{align}\nf_{02}(t_1, t_2) &= \\frac{f_{03}(t_1, t_2)}{3} u^3 + \\mathrm{O}(u^4), \n~~f_{12}(t_1, t_2)= -\\frac{f_{03}(t_1, t_2)}{3 \\alpha } u + \\mathrm{O}(u^2), ~~f_{21}(t_1, t_2) = \\mathrm{O}(u^3), \\nonumber \\\\ \nf_{40}(t_1, t_2) &= \\mathrm{O}(u^2), ~~f_{02}(t_1, t_2)^3 \\mathcal{A}^{f(t_1, t_2)}_6  =  -\\frac{10 f_{03}(t_1, t_2)^2 f_{31}(t_1, t_2)^2 }{9 } u^6 + \\mathrm{O}(u^7). \\label{pe7_multiplicity}\n\\end{align}\nEquation \\eqref{pe7_multiplicity} implies that \\eqref{psi_pa6_neq_0_a1_pa5_functional} holds. \\qed \\\\\n\n\n\n\\begin{cor}\n\\label{a1_pa5_mult_is_5_around_pe7}\nLet $\\mathbb{W} \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2$ be a vector bundle such that \nthe rank of $\\mathbb{W}$ is same as the dimension of $ \\Delta \\mathcal{P} \\mathcal{E}_{7}$ and \n$\\mathcal{Q}: \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2 \\longrightarrow \\mathbb{W}$  a \\textit{generic} \nsmooth section. Suppose $(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{E}_{7} \\cap \\mathcal{Q}^{-1}(0)$. \nThen the section $$ \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_{6}} \\oplus \\mathcal{Q}: \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_{5} \\longrightarrow \\pi_2^* (\\UL_{\\mathcal{P} \\mathcal{A}_{6}}) \\oplus \\mathbb{W}$$\nvanishes around $(\\tilde{f}, \\tilde{p},  l_{\\p})$ with a multiplicity of $6$.\n\\end{cor}\n\n\\noindent \\textbf{Proof:  } Follows from the fact that the sections \ninduced by $f_{02}$, $f_{21}$, $f_{12}$ and $f_{40}$ (the corresponding functionals) \nare transverse to the zero set over $\\Delta\\overline{\\mathcal{P} \\mathcal{A}}_3$ \\footnote{Take partial derivative with respect to \n$f_{02}$, $f_{21}$, $f_{12}$ and $f_{40}$.}, the fact that  $\\mathcal{Q}$ is generic and \\eqref{pe7_multiplicity}. \\qed \\\\\n\n\\noindent This finishes the proof of Lemma \\ref{cl_two_pt} (\\ref{a1_pa5_cl}).\\\\\n\n\n\n\\textbf{Proof of Lemma \\ref{cl_two_pt} (\\ref{a1_pd4_cl}):}  By definition of $\\Delta \\PP \\D_6^{\\vee s}$ in \\eqref{new_defn_delta}, \nit suffices to show  that \n\\begin{align}\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_4: \\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{D}_5}(\\tilde{f}, l_{\\p}) =0 \\} &= \\Delta \\overline{\\mathcal{P} \\mathcal{D}}_{6}. \\label{a1_pd4_is_pd6}  \n\\end{align} \nObserve that \n\\begin{align}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_4 \\cap \\Delta (\\mathcal{P} \\mathcal{D}_4 \\cup \\mathcal{P} \\mathcal{D}_5 \\cup \\mathcal{P} \\mathcal{E}_6) & = \\varnothing. \\label{a1_pd4_intersetc_pd4_is_empty}\n\\end{align}\nThis follows from \\eqref{a1_pa2_intersect_pd4_empty}, \\eqref{a1_pa3_pd5_dual_eqn} and \\eqref{a1_pd4_intersect_pe6_is_empty} \ncombined with \nthe fact that \n\\[ (\\mathcal{P} \\mathcal{D}_4 \\cup \\mathcal{P} \\mathcal{D}_5) \\cap (\\overline{\\mathcal{P} \\mathcal{D}_5^{\\vee}} \\cup \\overline{\\mathcal{P} \\mathcal{D}}_6) = \\varnothing. \\] \nEquation \\eqref{a1_pd4_intersetc_pd4_is_empty} implies that the lhs of \\eqref{a1_pd4_is_pd6} is a subset of its rhs. \nNext, we will simultaneously, prove the following two statements: \n\\begin{align}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_4 & \\supset \\Delta \\mathcal{P} \\mathcal{D}_6, \\label{a1_pa4_supset_pd6}\\\\ \n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_5 \\cap \\Delta \\mathcal{P} \\mathcal{D}_6 & = \\varnothing, \\label{a1_pa5_intersetc_pd6_is_empty} \n\\end{align}\nSince $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_4$ is a closed set,  \\eqref{a1_pa4_supset_pd6} implies that the rhs of \\eqref{a1_pd4_is_pd6} is a subset of its lhs. \n\n\\begin{claim}\n\\label{claim_pd5_actual_subset_of_a1_pa3}\nLet $(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{D}_6$.\nThen there exists a solution \n$$ (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} ) \\in \\overline{ (\\mathcal{D} \\times \\mathbb{P}^2) \\circ \\mathcal{P} \\mathcal{D}}_4$$ \n\\textit{near} $(\\tilde{f}, \\tilde{p}, l_{\\p})$ to the set of equations\n\\begin{align}\n\\pi_1^* \\psi_{\\mathcal{A}_0}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, ~\\pi_1^* \\psi_{\\mathcal{A}_1}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) = 0, \n~\\tilde{p}(t_1, t_2) \\neq \\tilde{p}(t_1). \\label{pd5_actual_limit_a1_pa3_functional_eqn}\n\\end{align}\nMoreover, such a solution  \nlies in $\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}_4$, i.e., \n\\begin{align}\n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{D}_5}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), l_{\\tilde{p}(t_1)}) & \\neq 0. \\label{psi_pa4_neq_0_a1_pa3_functional_new_pd5_actual}\n\\end{align}\nIn particular, $(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} )$ does not lie in $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_5$. \n\\end{claim}\n\\noindent Note that claim \\ref{claim_pd5_actual_subset_of_a1_pa3} implies  \\eqref{a1_pa4_supset_pd6} and  \n\\eqref{a1_pa5_intersetc_pd6_is_empty} simultaneously. \\\\\n\n\\noindent \\textbf{Proof:  } Choose homogeneous coordinates $[\\mathrm{X}: \\mathrm{Y}: \\mathrm{Z}]$ so that \n$\\tilde{p} = [0:0:1]$ and let ~$\\mathcal{U}_{\\tilde{p}}$,   \n$\\pi_x$, $\\pi_y$,  $v_1$, $w$, $v$, $\\eta$, $\\eta_{t_1}$, $\\eta_{t_2}$, \n$x_{t_1}$, $y_{t_1}$, $x_{t_2}$, $y_{t_2}$, \n$f_{ij}(t_1, t_2)$, $\\mathrm{F}$, $\\mathrm{F}_{x_{t_2}}$ and $\\mathrm{F}_{y_{t_2}}$\nbe exactly the same as defined in the \nproof of claim  \n\\ref{claim_a4_closure_simultaneous}, except for one difference:\nwe take $(\\tilde{f}(t_1, t_2), l_{\\tilde{p}(t_1)})$ to be a point in $\\overline{\\mathcal{P} \\mathcal{D}}_4$.\nSince $(\\tilde{f}(t_1, t_2), l_{\\tilde{p}(t_1)}) \\in \\overline{\\mathcal{P} \\mathcal{D}}_4$, \nwe conclude that \n\\[ f_{00}(t_1, t_2) = f_{10}(t_1, t_2) = f_{01}(t_1, t_2) =f_{11}(t_1, t_2)= f_{20}(t_1, t_2)=f_{02}(t_1, t_2) = f_{30}(t_1, t_2)=0.\\]\nThe functional equation \\eqref{pd5_dual_limit_a1_pa3_functional_eqn}  \nhas a solution if and only if \nthe following set of equations has a solution (as numbers): \n\\begin{align}\n\\mathrm{F} = 0, \\qquad \\mathrm{F}_{x_{t_2}} = 0, \\qquad \\mathrm{F}_{y_{t_2}} = 0, \\qquad (x_{t_2}, y_{t_2}) \\neq (0, 0) \\qquad \\textnormal{( but small)}. \\label{eval_f1_d5_actual} \n\\end{align} \nFor the convenience of the reader, let us rewrite the expression for $\\mathrm{F}$: \n\\begin{align*}\n\\mathrm{F} &: = \\frac{f_{21}(t_1, t_2)}{2} x_{t_2}^2 y_{t_2}+ \n\\frac{f_{12}(t_1, t_2)}{2} x_{t_2} y_{t_2}^2 + \\frac{f_{03}(t_1, t_2)}{6} y_{t_2}^3 + \\ldots.\n\\end{align*}\nSince $(\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{D}_6$, we conclude that \n\\begin{align}\nf_{21}, ~f_{40} &= 0 ~~\\textnormal{and} ~~f_{12}, ~\\mathcal{D}^{f}_7 \\neq 0. \n\\end{align}\nA little bit of thought will reveal \nthat the solutions to \n\\eqref{eval_f1_d5_actual}\nare exactly the same as in \\eqref{pd6_hat_y}, \n\\eqref{pd6_f21} and \\eqref{pd6_f40}, with $f_{02}(t_1, t_2) =0$. \nSince $\\alpha \\neq 0$, we conclude that $f_{21}(t_1, t_2) \\neq 0$ \nfor small but non zero $\\hat{x}_{t_2}$. \nHence \\eqref{psi_pa4_neq_0_a1_pa3_functional_new_pd5_actual} holds. \\qed \n\n\\begin{cor}\n\\label{psi_pd5_section_vanishes_order_two_around_pd6}\nLet $\\mathbb{W} \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2$ be a vector bundle such that \nthe rank of $\\mathbb{W}$ is same as the dimension of $ \\Delta \\mathcal{P} \\mathcal{D}_6$ and \n$\\mathcal{Q}: \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2 \\longrightarrow \\mathbb{W}$  a \\textit{generic} \nsmooth section. Suppose $(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{D}_6\\cap \\mathcal{Q}^{-1}(0)$. \nThen the section $$ \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4} \\oplus \\mathcal{Q}: \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P}\\mathcal{D}}_4 \\longrightarrow \\pi_2^* (\\UL_{\\mathcal{P} \\mathcal{D}_4}) \\oplus \\mathbb{W}$$\nvanishes around $(\\tilde{f}, \\tilde{p},  l_{\\p})$ with a multiplicity of $2$.\n\\end{cor}\n\\noindent \\textbf{Proof:  } Follows from the fact that the sections induced by $f_{21}$ and $f_{40}$ are transverse to the zero set over \n$\\Delta \\overline{\\mathcal{P} \\mathcal{D}}_4$, \\footnote{Take partial derivatives with respect to $f_{21}$ and $f_{40}$} \nthe fact that $\\mathcal{Q}$ is generic and \\eqref{pd6_f21} (combined with $f_{02}(t_1, t_2) =0$). Each branch of \n$\\alpha := \\sqrt{\\frac{\\mathcal{D}^{f(t_1, t_2)}_7}{-60 f_{12}(t_1, t_2)}}$  \ncontributes with a multiplicity of $1$. Hence, the total multiplicity is $2$. \\qed \\\\\n\n\\noindent Before proceeding further, observe that \\eqref{pd5_intersect_a1_pd4_is_empty} implies that \n\\begin{align}\n\\Delta \\overline{\\PP \\D_6^{\\vee s}} & \\subset \\Delta \\overline{\\mathcal{P} \\mathcal{D}_6^{\\vee}}. \\label{pd6_dual_s_is_subset_of_pd6_dual} \n\\end{align}\n\n\n\n\\textbf{Proof of Lemma \\ref{cl_two_pt} (\\ref{a1_d4_cl}):}  Follows from Lemma \\ref{cl_two_pt} (\\ref{a1_pd4_cl}), \\eqref{pd6_dual_s_is_subset_of_pd6_dual}, \nLemma \\ref{tube_lemma}, \\eqref{tube_lemma_X} and \n\\eqref{tube_lemma_Y}. \\qed \\\\\n\n\n\\textbf{Proof of Lemma \\ref{cl_two_pt} (\\ref{a1_pd5_cl}):}  It suffices to prove the following two statements:  \n\\begin{align}\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\overline{\\overline{\\mathcal{A}}_1\\circ \\mathcal{P} \\mathcal{D}}_5: ~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{E}_6}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) \\neq 0  \\} &=  \n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\Delta \\overline{\\mathcal{P} \\mathcal{D}}_{7}: ~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{E}_6}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) \\neq 0  \\}\n\\label{closure_a1_pd5_f12_not_zero}  \\\\\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_5: ~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{E}_6}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) = 0  \\} &=  \\Delta \\overline{\\mathcal{P} \\mathcal{E}}_7.   \\label{a1_pd5_pe7}\n \\end{align}\nLet us directly prove a more general version of \\eqref{closure_a1_pd5_f12_not_zero}: \n\\begin{lmm}\n\\label{closure_a1_pdk_f12_not_zero}\nIf $k \\geq 5$ then  \n\\begin{align*}\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{D}}_k: ~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{E}_6}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) \\neq 0  \\} &= \n\\{ (\\tilde{f}, \\tilde{p}, l_{\\tilde{p}}) \\in \\Delta \\overline{\\mathcal{P} \\mathcal{D}}_{k+2}: ~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{E}_6}( \\tilde{f}, \\tilde{p}, l_{\\tilde{p}} ) \\neq 0  \\}.  \n\\end{align*}\n\\end{lmm}\nNote that \\eqref{closure_a1_pd5_f12_not_zero} is a special case of Lemma \\ref{closure_a1_pdk_f12_not_zero}; take $k=5$. \nWe will prove the following two facts simultaneously:\n\\begin{align}\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_{k} \\} \\supset \\Delta \\mathcal{P} \\mathcal{D}_{k+2} \\qquad & \\forall ~k \\geq 5, \\label{pdk2_is_subset_of_a1_and_pdk} \\\\\n\\{ (\\tilde{f}, \\tilde{p}, l_{\\p}) \\in \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_{k+1} \\} \\cap \\Delta \\mathcal{P} \\mathcal{D}_{k+2} = \\varnothing \\qquad  & \\forall ~k \\geq 4.  \\label{pdk2_intersect_a1_and_pd1k+1_is_empty}\n\\end{align}\nIt is easy to see that \\eqref{pdk2_is_subset_of_a1_and_pdk} and \\eqref{pdk2_intersect_a1_and_pd1k+1_is_empty} imply \nLemma \\ref{closure_a1_pdk_f12_not_zero}. We will now prove the following claim:\n\n\\begin{claim}\n\\label{claim_d6_closure_simultaneous}\nLet $~(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{D}_{k+2}$ and $ k\\geq 5$.\nThen there exists a solution \n$$ (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} ) \\in \\overline{ (\\mathcal{D} \\times \\mathbb{P}^2) \\circ \\mathcal{P} \\mathcal{D}}_5$$ \n\\textit{near} $(\\tilde{f}, \\tilde{p}, l_{\\p})$ to the set of equations\n\\begin{align}\n\\label{closure_a1_dk_f12_not_zero}\n\\pi_1^* \\Psi_{\\mathcal{A}_0}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, ~\\pi_1^* \\Psi_{\\mathcal{A}_1}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) = 0, \\nonumber \\\\\n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{D}_6}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, \\ldots, \\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{D}_k}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) = 0, \\nonumber \\\\\n~\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{E}_6}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & \\neq  0, ~\\tilde{p}(t_1, t_2) \\neq \\tilde{p}(t_1). \n\\end{align}\nMoreover, \\textit{any} solution $(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)})$ sufficiently close to $(\\tilde{f}, \\tilde{p}, l_{\\p})$ \nlies in $ \\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}_k$, i.e.,\n\\begin{align}\n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{D}_{k+1}}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) \\neq  0. \\label{psi_pd_k_plus_1_does_not_vanish}\n\\end{align}\nIn particular $(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)})$ \\textit{does not} lie in $ \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_{k+1} $.\n\\end{claim}\n\n\\noindent It is easy to see that claim \\ref{claim_d6_closure_simultaneous} implies \\eqref{pdk2_is_subset_of_a1_and_pdk} and \\eqref{pdk2_intersect_a1_and_pd1k+1_is_empty} \nsimultaneously for all $k\\geq 5$.\nThe fact that \\eqref{pdk2_intersect_a1_and_pd1k+1_is_empty} holds for $k=4$ is the content of \\eqref{a1_pa5_intersetc_pd6_is_empty}. \\\\\n\n\n\\noindent \\textbf{Proof:  } Choose homogeneous coordinates $[\\mathrm{X}: \\mathrm{Y}: \\mathrm{Z}]$ so that \n$\\tilde{p} = [0:0:1]$ and let ~$\\mathcal{U}_{\\tilde{p}}$,   \n$\\pi_x$, $\\pi_y$,  $v_1$, $w$, $v$, $\\eta$, $\\eta_{t_1}$, $\\eta_{t_2}$, \n$x_{t_1}$, $y_{t_1}$, $x_{t_2}$, $y_{t_2}$, \n$f_{ij}(t_1, t_2)$, $\\mathrm{F}$, $\\mathrm{F}_{x_{t_2}}$ and $\\mathrm{F}_{y_{t_2}}$\nbe exactly the same as defined in the \nproof of claim \\ref{claim_pd5_subset_of_a1_pa3}.\nHence \n\\[ f_{10}(t_1, t_2) = f_{01}(t_1, t_2) =f_{11}(t_1, t_2)= f_{20}(t_1, t_2)= f_{02}(t_1, t_2) = f_{30}(t_1, t_2)= f_{21}(t_1, t_2)=0.\\]\nSince $(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{D}_{k+2}$, we conclude that $f_{12}(t_1, t_2) \\neq 0$. \nHence, \nwe can make a change of coordinates to write $\\mathrm{F}$ as   \n\\begin{align*} \n\\mathrm{F}&= \\hat{y}_{t_2}^2 \\hat{x}_{t_2} + \\frac{\\mathcal{D}^{f(t_1, t_2)}_6}{4!} \\hat{x}_{t_2}^4 + \\frac{\\mathcal{D}^{f(t_1, t_2)}_7}{5!} \\hat{x}_{t_2}^5+ \\ldots \n\\end{align*}\nThe functional equation   \\eqref{closure_a1_dk_f12_not_zero}  has a solution if and only if \nthe following set of equations has a solution (as numbers): \n\\begin{align}\n\\label{closure_a1_dk_f12_not_zero_numbers}\n\\hat{y}_{t_2}^2 \\hat{x}_{t_2} + \\frac{\\mathcal{D}^{f(t_1, t_2)}_6}{4!} \\hat{x}_{t_2}^4 + \\frac{\\mathcal{D}^{f(t_1, t_2)}_7}{5!} \\hat{x}_{t_2}^5+ \\ldots &=0, \n\\qquad \\hat{y}_{t_2} \\hat{x}_{t_2} = 0, \\nonumber \\\\\n\\hat{y}_{t_2}^2 + \\frac{\\mathcal{D}^{f(t_1, t_2)}_6}{3!}\\hat{x}_{t_2}^3 + \\frac{\\mathcal{D}^{f(t_1, t_2)}_7}{4!} \\hat{x}_{t_2}^4 + \\ldots &= 0, \\qquad \\mathcal{D}^{f(t_1, t_2)}_6, \\ldots, \\mathcal{D}^{f(t_1, t_2)}_k = 0, \\nonumber  \\\\\n(\\hat{y}_{t_2}, \\hat{x}_{t_2}) & \\neq (0,0)   \\qquad \\textnormal{(but small).}\n\\end{align}\nIt is easy to see that the solutions to \\eqref{closure_a1_dk_f12_not_zero_numbers} exist given by   \n\\begin{align}\n\\mathcal{D}^{f(t_1, t_2)}_6,& \\ldots, \\mathcal{D}^{f(t_1, t_2)}_k = 0, \\nonumber \\\\\n\\mathcal{D}^{f(t_1, t_2)}_{k+1} &=  \\frac{\\mathcal{D}_{k+3}^{f(t_1, t_2)}}{k(k+1)} \\hat{x}_{t_2}^2 + \\mathrm{O}(\\hat{x}_{t_2}^3),   \\label{pdk+2_inside_a1_and_pdk_solution}\\\\ \n\\mathcal{D}^{f(t_1, t_2)}_{k+2} &= -\\frac{2 \\mathcal{D}^{f(t_1, t_2)}_{k+3}}{(k+1)} \\hat{x}_{t_2} + \\mathrm{O}(\\hat{x}_{t_2}^2), \\qquad \\hat{y}_{t_2} = 0, \\qquad \\hat{x}_{t_2}  \\neq 0 \\qquad \\textnormal{(but small).} \\nonumber \n\\end{align}\nBy \\eqref{pdk+2_inside_a1_and_pdk_solution}, it immediately follows that \\eqref{psi_pd_k_plus_1_does_not_vanish} holds.  \\qed  \n\n\n\\begin{cor}\n\\label{a1_pdk_mult_is_2_f12_neq_0}\nLet $\\mathbb{W} \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2$ be a vector bundle such that \nthe rank of $\\mathbb{W}$ is same as the dimension of $ \\Delta \\mathcal{P} \\mathcal{D}_{k+2}$ and \n$\\mathcal{Q}: \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2 \\longrightarrow \\mathbb{W}$  a \\textit{generic} \nsmooth section. Suppose $(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{D}_{k+2} \\cap \\mathcal{Q}^{-1}(0)$. \nThen the section $$ \\pi_2^\\ast\\Psi_{\\mathcal{P} \\mathcal{D}_{k+1}} \\oplus \\mathcal{Q}: \\Delta \\overline{\\mathcal{P} \\mathcal{D}}_{k} \\longrightarrow \\pi_2^* (\\UL_{\\mathcal{P} \\mathcal{D}_{k+1}}) \\oplus \\mathbb{W}$$\nvanishes around $(\\tilde{f}, \\tilde{p},  l_{\\p})$ with a multiplicity of $2$.\n\\end{cor}\n\\noindent \\textbf{Proof:  }  This follows from the fact that the sections \n\\begin{align*}\n\\pi_2^\\ast\\Psi_{\\mathcal{P} \\mathcal{D}_{i}}:& \\Delta \\overline{\\mathcal{P} \\mathcal{D}}_{i-1} - \\pi_2^\\ast\\Psi_{\\mathcal{P} \\mathcal{E}_6}^{-1}(0) \\longrightarrow  \\pi_2^\\ast\\UL_{\\mathcal{P} \\mathcal{D}_{i}}\n\\end{align*}\nare transverse to the zero set for all $6 \\leq i \\leq k+2$, the fact that \n$\\mathcal{Q}$ is generic and \\eqref{pdk+2_inside_a1_and_pdk_solution}. \\qed \\\\ \n\n\n\\noindent Next, we will prove \\eqref{a1_pd5_pe7}. The lhs of \\eqref{a1_pd5_pe7} is a subset of its rhs; this follows from \n\\eqref{a1_pd4_intersect_pe6_is_empty} and the fact that $\\mathcal{P} \\mathcal{D}_5$ is a subset of $\\overline{\\mathcal{P} \\mathcal{D}}_4$. To prove the converse, we will prove the following three statements simultaneously:\n\\begin{align}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_5 & \\supset \\Delta \\mathcal{P} \\mathcal{E}_7, \\label{a1_pd5_supset_pe7}\\\\\n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_6 \\cap \\Delta \\mathcal{P} \\mathcal{E}_7 & = \\varnothing. \\label{a1_pd6_intersect_pe7_empty}\\\\\n\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{E}}_6 \\cap \\Delta \\mathcal{P} \\mathcal{E}_7 & = \\varnothing. \\label{a1_pe6_intersect_pe7_empty}\n\\end{align}\nSince  $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_5$ is a closed set, \\eqref{a1_pd5_supset_pe7} implies that the rhs of \n\\eqref{a1_pd5_pe7} is a subset of its lhs.\n\n\\begin{claim}\n\\label{claim_a1_pd6_intersect_pe7_is_empty}\nLet $(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{E}_7$.\nThen there exist solutions \n$$ (\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} ) \\in \\overline{ (\\mathcal{D} \\times \\mathbb{P}^2) \\circ \\mathcal{P} \\mathcal{D}}_5$$ \n\\textit{near} $(\\tilde{f}, \\tilde{p}, l_{\\p})$ to the set of equations\n\\begin{align}\n\\label{pe7_intersect_a1_pd6_is_empty_functional_eqn}\n\\pi_1^* \\Psi_{\\mathcal{A}_0}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) & = 0, ~\\pi_1^* \\Psi_{\\mathcal{A}_1}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)}) = 0, ~~\\tilde{p}(t_1, t_2) \\neq \\tilde{p}(t_1).\n\\end{align}\nMoreover, any such solution sufficiently close to $(\\tilde{f}, \\tilde{p}, l_{\\p})$ lies in \n$\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}_5$, i.e., \n\\begin{align}\n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{D}_6}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), l_{\\tilde{p}(t_1)}) \\neq 0.\n\\end{align}\nIn particular, $(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} )$ does not lie in $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_6$. Since\n\\begin{align}\n\\pi_2^* \\Psi_{\\mathcal{P} \\mathcal{E}_6}(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2), l_{\\tilde{p}(t_1)}) \\neq 0\n\\end{align}\nthe solution $(\\tilde{f}(t_1, t_2), \\tilde{p}(t_1, t_2),  l_{\\tilde{p}(t_1)} )$ does not lie in $\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{E}}_6$. \n\\end{claim}\n\n\\noindent Note that claim \\ref{claim_a1_pd6_intersect_pe7_is_empty} implies \\eqref{a1_pd5_pe7} and \\eqref{a1_pd6_intersect_pe7_empty} \nsimultaneously. \\\\ \n\n\\noindent \\textbf{Proof:  } Choose homogeneous coordinates $[\\mathrm{X}: \\mathrm{Y}: \\mathrm{Z}]$ so that \n$\\tilde{p} = [0:0:1]$ and let ~$\\mathcal{U}_{\\tilde{p}}$,   \n$\\pi_x$, $\\pi_y$,  $v_1$, $w$, $v$, $\\eta$, $\\eta_{t_1}$, $\\eta_{t_2}$, \n$x_{t_1}$, $y_{t_1}$, $x_{t_2}$, $y_{t_2}$, \n$f_{ij}(t_1, t_2)$, $\\mathrm{F}$, $\\mathrm{F}_{x_{t_2}}$ and $\\mathrm{F}_{y_{t_2}}$\nbe exactly the same as defined in the \nproof of claim \\ref{claim_pd5_subset_of_a1_pa3}\nexcept for one difference:\nwe take $(\\tilde{f}(t_1, t_2), l_{\\tilde{p}(t_1)})$ to be a point in $\\overline{\\mathcal{P} \\mathcal{D}}_5$.\nSince $(\\tilde{f}(t_1, t_2), l_{\\tilde{p}(t_1)}) \\in \\overline{\\mathcal{P} \\mathcal{D}}_5$, we conclude that \n\\[ f_{10}(t_1, t_2) = f_{01}(t_1, t_2) =f_{11}(t_1, t_2)= f_{20}(t_1, t_2)= f_{02}(t_1, t_2) = f_{30}(t_1, t_2)= f_{21}(t_1, t_2)=0.\\]\nSince $(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{E}_7$, we conclude that \n\\begin{align}\nf_{03}, f_{31} &\\neq 0,  ~~f_{12}, f_{40} =0. \\label{pe7_nv_again} \n\\end{align} \nThe functional equation \\eqref{pe7_intersect_a1_pd6_is_empty_functional_eqn}  \nhas a solution if and only if \nthe following set of equations has a solution (as numbers): \n\\begin{align}\n\\mathrm{F} = 0, \\qquad \\mathrm{F}_{x_{t_2}} = 0, \\qquad \\mathrm{F}_{y_{t_2}} = 0,  \\qquad (x_{t_2}, y_{t_2}) \\neq (0, 0) \\qquad \\textnormal{(but small)}. \\label{eval_f1_pe7_again} \n\\end{align} \nLet us now define \n\\begin{align*}\n\\mathrm{G} &:= -8\\mathrm{F} + 2 \\xt \\mathrm{F}_{\\xt} + 3 y_{t_2} \\mathrm{F}_{y_{t_2}}. \n\\end{align*}\nWe claim that $ \\xt \\neq 0$ and $ y_{t_2} \\neq 0$; we will justify that at the end. Assuming that claim we \nconclude that solving \\eqref{eval_f1_pe7_again} is equivalent to solving \n\\begin{align}\n\\mathrm{G} = 0, \\qquad \\mathrm{F}_{x_{t_2}} = 0, \\qquad \\mathrm{F}_{y_{t_2}} = 0,  \\qquad (x_{t_2}, y_{t_2}) \\neq (0, 0) \\qquad \\textnormal{(but small)}. \\label{eval_f1_pe7_again_modified} \n\\end{align}\nNote that $\\mathrm{G}$ is independent of $f_{12}(t_1, t_2)$ and $f_{40}(t_1, t_2)$. Hence, $\\mathrm{G}$ is explicitly given by \n\\begin{equation*}\n\\mathrm{G}= \\frac{\\mathrm{P}_{03}(\\xt,y_{t_2})}{6}y_{t_2}^3+ \\frac{\\mathrm{P}_{31}(\\xt,y_{t_2})}{6}\\xt^3y+\\kappa \\xt^2y_{t_2}^2+\\mathrm{P}_{50}(\\xt)\\xt^5,\n\\end{equation*}\nwhere   $\\mathrm{P}_{03}(0,0) = f_{03}(t_1, t_2)$ and $\\mathrm{P}_{31}(0,0) = f_{31}(t_1, t_2)$.  \nUsing the same argument as in \\cite{BM13}, \nthere exists a holomorphic function $\\mathrm{B}(\\hat{x}_{t_2})$ and   \nconstant $\\eta$  such that if we make the substitution \n$$\\xt = \\hat{x}_{t_2} + \\eta \\hat{y}_{t_2}, \\qquad y_{t_2} = \\hat{y}_{t_2} + \\mathrm{B}(\\hat{x}_{t_2}) \\hat{x}_{t_2}^2 $$ \nthen $\\mathrm{G}$ is given by\n\\begin{align*}\n\\mathrm{G} &= \\frac{\\hat{\\mathrm{P}}_{03}(\\hat{x}_{t_2}, \\hat{y}_{t_2})}{6} \\hat{y}_{t_2}^3 + \\frac{\\hat{\\mathrm{P}}_{31}(\\hat{x}_{t_2}, \\hat{y}_{t_2})}{6} \\hat{x}_{t_2}^3 \\hat{y}_{t_2},\n\\end{align*}\nwhere $\\hat{\\mathrm{P}}_{03} (0,0) = f_{03}(t_1, t_2)$ and $\\hat{\\mathrm{P}}_{31} (0,0) = f_{31}(t_1, t_2)$. \nWe claim that $\\hat{y}_{t_2} \\neq 0$; we will justify that at the end. Assuming that claim, we conclude from $\\mathrm{G} =0$ that \n\\begin{align*}\n\\hat{y}_{t_2} &= u^3, ~~\\hat{x}_{t_2} = \\alpha u^2 + \\mathrm{O}(u^2) \\qquad \\textnormal{where} \\qquad \\alpha := \\sqrt[3]{-\\frac{f_{03}(t_1, t_2)}{f_{31}(t_1, t_2)}} \\qquad \\textnormal{a branch of the cuberoot.} \n\\end{align*}\nUsing this and the remaining two equations of \\eqref{eval_f1_pe7_again_modified} we conclude that\n\\begin{align}\nf_{12}(t_1, t_2) & = - \\frac{f_{03}(t_1, t_2)}{3 \\alpha} u + \\mathrm{O}(u^2), ~~f_{40}(t_1, t_2) = -\\frac{4 f_{31}(t_1, t_2)}{\\alpha} u + \\mathrm{O}(u^2).  \\label{f12_and_f40_mult_around_pe7}\n\\end{align}\nIt remains to show that $\\xt \\neq 0$, $y_{t_2} \\neq 0$ and $\\hat{y}_{t_2} \\neq 0$. If $\\xt =0$, then $\\mathrm{F} =0$ implies that \n$f_{03}(t_1, t_2) = \\mathrm{O}(y_{t_2})$, \ncontradicting \\eqref{pe7_nv_again}. Next, if $y_{t_2} =0$ then $\\mathrm{F}_{y_{t_2}} =0$ implies that $f_{31} (t_1, t_2) = \\mathrm{O}(\\xt)$, contradicting \\eqref{pe7_nv_again}.\nFinally, if $\\hat{y}_{t_2} =0$, then $\\mathrm{F}_{y_{t_2}} =0$ implies that \n\\begin{align*}\nf_{31}(t_1, t_2) &= -6\\mathrm{B}(0) f_{12}(t_1, t_2) + \\mathrm{O}(\\xt).  \n\\end{align*}\nAs $f_{12}(t_1, t_2)$ and $\\xt$ go to zero, $f_{31}(t_1, t_2)$ goes to zero, contradicting \\eqref{pe7_nv_again}. \\qed \n\n\\begin{cor}\n\\label{psi_pe6_and_pd6_section_vanishes_order_one_around_pe7}\nLet $\\mathbb{W} \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2$ be a vector bundle such that \nthe rank of $\\mathbb{W}$ is same as the dimension of $ \\Delta \\mathcal{P} \\mathcal{E}_7$ and \n$\\mathcal{Q}: \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P} T\\mathbb{P}^2 \\longrightarrow \\mathbb{W}$  a \\textit{generic} \nsmooth section. Suppose $(\\tilde{f},\\tilde{p}, l_{\\p}) \\in \\Delta \\mathcal{P} \\mathcal{E}_7\\cap \\mathcal{Q}^{-1}(0)$. \nThen the sections \n\\begin{align*}\n\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_6} \\oplus \\mathcal{Q}: \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P}\\mathcal{D}}_5 \\longrightarrow \\pi_2^* (\\UL_{\\mathcal{P} \\mathcal{D}_6}) \\oplus \\mathbb{W}, \n~~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{E}_6} \\oplus \\mathcal{Q}: \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P}\\mathcal{D}}_5 \\longrightarrow \\pi_2^* (\\UL_{\\mathcal{P} \\mathcal{E}_6}) \\oplus \\mathbb{W}  \n\\end{align*}\nvanish around $(\\tilde{f}, \\tilde{p},  l_{\\p})$ with a multiplicity of $1$.\n\\end{cor}\n\\noindent \\textbf{Proof:  } Follows from the fact \nthat the sections induced by $f_{12}$ and $f_{40}$ are transverse to the zero set over \n$\\Delta \\overline{\\mathcal{P} \\mathcal{D}}_5$, \\footnote{Take partial derivatives with respect to $f_{12}$ and $f_{40}$.}\n$\\mathcal{Q}$ is generic and \\eqref{f12_and_f40_mult_around_pe7}. \\qed \n\n\n\\section{Euler class computation} \n\\label{Euler_class_computation}\n\n\\hf\\hf We are ready to prove the recursive formulas stated in section \\ref{algorithm_for_numbers}.\\\\\n\n\\noindent \\textbf{Proof of \\eqref{algoa1a1}:}  \nLet $\\mathcal{Q}: \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P}^2 \\longrightarrow \\mathcal{W}$ \nbe a generic smooth section to\n\\bgd\n\\mathcal{W} := \\bigg({\\textstyle \\bigoplus}_{i=1}^{\\delta_d -(n+2)} \\pi_{\\mathcal{D}}^*\\gamma_{\\mathcal{D}}^* \n\\bigg)\\oplus\\Big({\\textstyle \\bigoplus}_{i=1}^{n}  \\pi_2^* \\gamma_{_{\\mathbb{P}^2}}^*\\Big) \\longrightarrow \\mathcal{D} \\times \\mathbb{P}^2 \\times \\mathbb{P}^2.\n\\edd\nNote that  \n\\begin{align*}\n\\mathcal{N}(\\mathcal{A}_1 \\mathcal{A}_1, n) = \\big\\langle e(\\mathcal{W}), ~[\\overline{\\mathcal{A}_1\\circ\\mathcal{A}}_1] \\big\\rangle = | \\pm (\\mathcal{A}_1\\circ \\mathcal{A}_1) \\cap  \\mathcal{Q}^{-1}(0)|.\n\\end{align*}\nBy Lemma \\ref{cl_two_pt} (\\ref{a1a_minus_1_cl}) \n\\begin{align*} \n\\overline{\\mathcal{A}}_1 \\times \\mathbb{P}^2 & = \\overline{\\overline{\\mathcal{A}}_1 \\circ (\\mathcal{D} \\times \\mathbb{P}^2)} = \\overline{\\mathcal{A}}_1 \\circ (\\mathcal{D} \\times \\mathbb{P}^2) \\sqcup \\Delta \\overline{\\mathcal{A}}_1.\n\\end{align*} \nThe sections \n\\begin{align}\n \\pi_2^*\\psi_{\\mathcal{A}_0}:\\overline{\\mathcal{A}}_1 \\times \\mathbb{P}^2 - \\Delta \\overline{\\mathcal{A}}_1   \\longrightarrow \\pi_2^* \\mathcal{L}_{\\mathcal{A}_0}, \\qquad \\pi_2^*\\psi_{\\mathcal{A}_1}: \\pi_2^*\\psi_{\\mathcal{A}_0}^{-1}(0) \\longrightarrow \\pi_2^*\\mathcal{V}_{\\mathcal{A}_1}\n\\end{align}\nare transverse to the zero set. \n(cf. Proposition \\ref{ift_ml}). \nHence \n\\begin{align}\n\\big\\langle e(\\pi_2^*\\mathcal{L}_{\\mathcal{A}_0}) e(\\pi_2^*\\mathcal{V}_{\\mathcal{A}_1}) e(\\mathcal{W}), ~[\\overline{\\mathcal{A}}_1 \\times \\mathbb{P}^2] \\big\\rangle & =  \\mathcal{N}(\\mathcal{A}_1 \\mathcal{A}_1, n) + \n\\mathcal{C}_{\\Delta \\overline{\\mathcal{A}}_1}(\\pi_2^*\\psi_{\\mathcal{A}_0} \\oplus \\pi_2^*\\psi_{\\mathcal{A}_1} \\oplus \\mathcal{Q}), \\label{A1A1_Euler_Class}\n\\end{align}\nwhere $\\mathcal{C}_{\\Delta \\overline{\\mathcal{A}}_1}(\\pi_2^*\\psi_{\\mathcal{A}_0} \\oplus \\pi_2^*\\psi_{\\mathcal{A}_1} \\oplus \\mathcal{Q})$ \nis the contribution of the section \n$\\pi_2^*\\psi_{\\mathcal{A}_0} \\oplus \\pi_2^*\\psi_{\\mathcal{A}_1} \\oplus \\mathcal{Q}$ to the Euler class from \n$\\Delta \\overline{\\mathcal{A}}_1$. \nThe lhs of \\eqref{A1A1_Euler_Class}, as computed by splitting principle and a case by case check, is \n\n\\begin{align}\n\\big\\langle e(\\pi_2^*\\mathcal{L}_{\\mathcal{A}_0}) e(\\pi_2^*\\mathcal{V}_{\\mathcal{A}_1}) e(\\mathcal{W}), ~[\\overline{\\mathcal{A}}_1 \\times \\mathbb{P}^2] \\big\\rangle  &= \\mathcal{N}(\\mathcal{A}_1, 0) \\times \\mathcal{N}(\\mathcal{A}_1, n) \\label{a1a1_Euler_class_Main_stratum}.\n\\end{align}\nIn fact, the usual answer one arrives at is\n\\bgd\n\\mathcal{N}(\\mathcal{A}_1,n)+3(d-1)\\mathcal{N}(\\mathcal{A}_1,n+1)+3(d-1)^2\\mathcal{N}(\\mathcal{A}_1,n+2).\n\\edd\nOne then uses a result from \\cite{BM13}:\n\\begin{align} \n\\mathcal{N}(\\mathcal{A}_1,n) = \\begin{cases}\n3(d-1)^{2},&\\textnormal{if}~n=0;\\\\\n3(d-1),&\\textnormal{if}~n=1;\\\\\n1,&\\textnormal{if}~n=2;\\\\\n0,&\\textnormal{otherwise}.\n\\end{cases}\n\\end{align}\n\\noindent Next, we  compute \n$\\mathcal{C}_{\\Delta \\overline{\\mathcal{A}}_1}(\\pi_2^*\\psi_{\\mathcal{A}_0}\\oplus\\pi_2^*\\psi_{\\mathcal{A}_1} \\oplus \\mathcal{Q})$.\nNote that ~$ \\overline{\\mathcal{A}}_1 = \\mathcal{A}_1 \\sqcup \\overline{\\mathcal{A}}_2.$\nBy claim \\ref{a1_section_contrib_from_a1_and_a2} we get that \n\\begin{align}\n\\mathcal{C}_{\\Delta \\mathcal{A}_1}(\\pi_2^*\\psi_{\\mathcal{A}_0}\\oplus\\pi_2^*\\psi_{\\mathcal{A}_1} \\oplus \\mathcal{Q}) & = \\big\\langle e(\\pi_2^* \\mathcal{L}_{\\mathcal{A}_0})e(\\mathcal{W}) , ~[\\Delta \\overline{\\mathcal{A}}_1] \\big\\rangle  = \\mathcal{N}(\\mathcal{A}_1, n) + d \\mathcal{N}(\\mathcal{A}_1, n+1),  \\label{a1a1_a1_contribution}\\\\\n\\mathcal{C}_{\\Delta \\overline{\\mathcal{A}}_2}(\\pi_2^*\\psi_{\\mathcal{A}_0}\\oplus\\pi_2^*\\psi_{\\mathcal{A}_1} \\oplus \\mathcal{Q}) &= 3 \\mathcal{N}(\\mathcal{A}_2, n).  \\label{a1a1_a2_contribution}\n\\end{align}\nIt is easy to see that \\eqref{a1a1_Euler_class_Main_stratum}, \\eqref{a1a1_a1_contribution} and \\eqref{a1a1_a2_contribution} \nprove \\eqref{algoa1a1}. \\qed \n\\begin{rem}\nIn \\cite{Z1} a different method is used to compute $\\mathcal{C}_{\\Delta \\mathcal{A}_1}(\\pi_2^\\ast\\psi_{\\mathcal{A}_0}\\oplus\\pi_2^*\\psi_{\\mathcal{A}_1} \\oplus \\mathcal{Q})$. The author later on pointed out this simpler method to the second author of this paper. \n\\end{rem}\n\n\n\\noindent \\textbf{Proof of \\eqref{algopa20a1} and \\eqref{algopa21a1}:} Let $\\mathbb{W}_{n,m,2}^1$ and $\\mathcal{Q}$ be as \nin \\eqref{generic_Q} with $k=2$. \nBy definition, $~\\mathcal{N}(\\mathcal{A}_1\\mathcal{P}\\mathcal{A}_2, n,m)$ is the signed \ncardinality of the intersection of $\\mathcal{A}_1\\circ \\mathcal{P} \\mathcal{A}_2$ with $\\mathcal{Q}^{-1}(0)$. \nBy Lemma \\ref{cl_two_pt}, statement \\ref{a1a1_up_cl} we gather that\n\\begin{align*}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\hat{\\mathcal{A}}^{\\#}_1}& = \\overline{\\mathcal{A}}_1\\circ \\hat{\\mathcal{A}}^{\\#}_1 \\sqcup  \\overline{\\mathcal{A}}_1 \\circ  (\\overline{\\hat{\\mathcal{A}}_1^{\\#}}- \\hat{\\mathcal{A}}_1^{\\#}  ) \\sqcup \\Delta \\overline{\\hat{\\mathcal{A}}}_3 \\\\ \n                           & =  \\overline{\\mathcal{A}}_1\\circ \\hat{\\mathcal{A}}^{\\#}_1  \\sqcup \\overline{\\mathcal{A}}_1 \\circ  \\overline{\\mathcal{P} \\mathcal{A}}_2 \\sqcup \\Delta \\overline{\\hat{\\mathcal{A}}}_3 \\qquad \n\\textnormal{(by \\cite{BM13} (cf. Lemma \\ref{cl}, statement \\ref{A1cl}).}\n\\end{align*}\nBy Proposition \\ref{A2_Condition_prp}, the section \n$$\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_2} : \\overline{\\overline{\\mathcal{A}}_1 \\circ  \\hat{\\mathcal{A}}^{\\#}_1} \\longrightarrow \\pi_2^*\\mathbb{V}_{\\mathcal{P} \\mathcal{A}_2}$$ \nvanishes on $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{A}_2$ transversely. \nHence, the zeros of the section \n$$ \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_2} \\oplus \\mathcal{Q} : \\overline{\\overline{\\mathcal{A}}_1 \\circ  \\hat{\\mathcal{A}}^{\\#}_1} \\longrightarrow \\pi_2^*\\mathbb{V}_{\\mathcal{P} \\mathcal{A}_2} \\oplus \\mathbb{W}_{n,m,2}^1, $$\nrestricted to $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{A}_2$ counted with a sign, is our desired number. \nIn other words \n\\bgd\n\\Big\\langle e(\\pi_2^*\\mathbb{V}_{\\mathcal{P} \\mathcal{A}_2}) e(\\mathbb{W}_{n,m,2}^1), ~[\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\hat{\\mathcal{A}}^{\\#}_1}] \\Big\\rangle = \\mathcal{N}(\\mathcal{A}_1 \\mathcal{P}\\mathcal{A}_2, n,m) + \\mathcal{C}_{\\Delta \\overline{\\hat{\\mathcal{A}}}_3}(\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_2} \\oplus \\mathcal{Q}) \n\\edd\nwhere $\\mathcal{C}_{\\Delta \\overline{\\hat{\\mathcal{A}}}_3}(\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_2} \\oplus \\mathcal{Q})$ is the contribution of the section to the Euler class from \n$\\Delta \\overline{\\hat{\\mathcal{A}}}_3$. Note that $ \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_2} \\oplus \\mathcal{Q}$ vanishes only on $\\mathcal{P} \\mathcal{A}_3$ and $\\hat{\\mathcal{D}}_4$ and not on the entire $\\overline{\\hat{\\mathcal{A}}}_3$.   \nBy Corollary \\ref{pa2_section_mult_around_pa3} and  \\ref{pa2_section_mult_around_hat_d4}, \nthe contribution from $\\mathcal{P} \\mathcal{A}_3$ and $\\hat{\\mathcal{D}}_4$ are  $2$ and $3$ respectively. This proves the claim.   \n\\begin{rem}\nIn the above proof we are using Lemma \\ref{cohomology_ring_of_pv} with $M:= \\overline{\\overline{\\mathcal{A}}_1 \\circ  \\hat{\\mathcal{A}}^{\\#}_1}$. However, in this case \n$M$ is not a smooth manifold; it is only a pseudocycle. Lemma \\ref{cohomology_ring_of_pv} is actually true even when $M$ happens to be a \npseudocycle. \n\\end{rem}\n\n\\begin{rem}\nA completely different method is used in \\cite{Z1} to compute $\\mathcal{N}(\\mathcal{A}_1 \\mathcal{A}_2, n)$; instead of removing the cusp, the node is removed.\nIn fact, all the numbers $\\mathcal{N}(\\mathcal{A}_1 \\mathfrak{X}_k,n)$ can also be computed by removing the node, instead of removing the $\\mathfrak{X}_k$ singularity. However, in order to obtain a recursive formula for the number of degree $d$ curves \nthrough $\\delta_{d} - (\\delta+k)$ generic points and having $\\delta$-nodes and one singularity of type $\\mathfrak{X}_k$, \nwe have to apply the method employed in this paper \n(i.e., we have to remove the $\\mathfrak{X}_k$-singularity, not the node). \nThis observation is again due to Aleksey Zinger. \n\\end{rem}\n\n\n\n\\noindent \\textbf{Proof of \\eqref{algopa3a1}:} Let $\\mathbb{W}_{n,m,3}^1$ and $\\mathcal{Q}$ be as in \\eqref{generic_Q} with $k=3$.\nBy Lemma \\ref{cl_two_pt}, statement \\ref{a1_pa2_cl} we have \n\\begin{align*}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}}_2& = \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}_2 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{A}}_2- \\mathcal{P} \\mathcal{A}_2) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_4 \\cup \\Delta \\overline{\\hat{\\mathcal{D}}^{\\#}_5}\\Big), \\\\\n                         &= \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}_2 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{A}}_3 \\cup \\overline{\\hat{\\mathcal{D}}^{\\#}_4}) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_4 \\cup \\Delta \\overline{\\hat{\\mathcal{D}}^{\\#}_5}\\Big) \n\\end{align*}\nwhere the last equality follows from \\cite{BM13} (cf. Lemma \\ref{cl}, statement \\ref{A2cl}). By Proposition \\ref{A3_Condition_prp}, the section \n$$ \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_3} \\oplus \\mathcal{Q} : \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_2 \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_3} \\oplus \\mathbb{W}_{n,m,3}^1 $$\nvanishes transversely on $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{A}_3$. By definition, it does not vanish on $\\mathcal{A}_1 \\circ \\hat{\\mathcal{D}}_4^{\\#}$. \nBy Corollary \\ref{a1_pak_mult_is_2_Hess_neq_0}, the contribution to the Euler class from the points of $\\Delta \\mathcal{P} \\mathcal{A}_4$ is $2$. \nFurthermore, by definition the section does not vanish on $\\Delta\\hat{\\mathcal{D}}^{\\#}_5$.\nHence \n\\bgd\n\\Big\\langle e(\\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_3} \\oplus \\mathbb{W}_{n,m,3}^1), ~~[\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_2] \\Big\\rangle = \\mathcal{N}(\\mathcal{A}_1\\mathcal{P}\\mathcal{A}_3,n,m) + 2 \\mathcal{N}(\\mathcal{P} \\mathcal{A}_4, n, m)  \n\\edd\nwhich proves the equation.  \\qed \\\\\n\n\n\n\\noindent \\textbf{Proof of \\eqref{algopa4a1}:} Let $\\mathbb{W}_{n,m,4}^1$ and $\\mathcal{Q}$ be as in \\eqref{generic_Q} with $k=4$.\nBy Lemma \\ref{cl_two_pt}, statement \\ref{a1_pa2_cl} we have \n\\begin{align*}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}}_3 &=  \n\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}_3 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{A}}_3- \\mathcal{P} \\mathcal{A}_3) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_5 \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{D}^{\\vee}_5} \\Big), \\\\\n&= \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}_3 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{A}}_4 \\cup \\overline{\\mathcal{P} \\mathcal{D}}_4) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_5 \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{D}^{\\vee}_5} \\Big) \n\\end{align*}\nwhere the last equality follows from \\cite{BM13} (cf. Lemma \\ref{cl}, statement \\ref{A3cl}). By Proposition \\ref{A3_Condition_prp}, the section \n$$ \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_4} \\oplus \\mathcal{Q} : \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_3 \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_4} \\oplus \\mathbb{W}_{n,m,4}^1 $$\nvanishes transversely on $\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}_4$. It is easy to see that it does not vanish on $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{D}_4$. \nBy Corollary \\ref{a1_pak_mult_is_2_Hess_neq_0}, the contribution to the Euler class from the points of $\\Delta\\mathcal{P} \\mathcal{A}_5$ is $2$. \nMoreover, the section does not vanish on $\\Delta \\mathcal{P} \\mathcal{D}^{\\vee}_5$. \nHence \n\\bgd\n\\Big\\langle e(\\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_4} \\oplus \\mathbb{W}_{n,m,4}^1), ~~[\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_3] \\Big\\rangle = \\mathcal{N}(\\mathcal{A}_1\\mathcal{P}\\mathcal{A}_4,n,m) + 2 \\mathcal{N}(\\mathcal{P} \\mathcal{A}_5, n, m)  \n\\edd\nwhich proves the equation.  \\qed \\\\ \n\n\\noindent \\textbf{Proof of \\eqref{algopa5a1}:} Let $\\mathbb{W}_{n,m,5}^1$ and $\\mathcal{Q}$ be as in \\eqref{generic_Q} with $k=5$.\nBy Lemma \\ref{cl_two_pt}, statement \\ref{a1_pa2_cl} we have \n\\begin{align*}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}}_4 &=  \n\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}_4 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{A}}_4- \\mathcal{P} \\mathcal{A}_4) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_6 \\cup  \n\\Delta \\overline{\\PP \\D_7^{s}} \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{E}}_6  \\Big), \\\\\n&= \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}_4 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{A}}_5 \\cup \\overline{\\mathcal{P} \\mathcal{D}}_5) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_6 \\cup  \n\\Delta \\overline{\\PP \\D_7^{s}} \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{E}}_6  \\Big)  \n\\end{align*}\nwhere the last equality follows from \\cite{BM13} (cf. Lemma \\ref{cl}, statement \\ref{A4cl}). By Proposition \\ref{A3_Condition_prp}, the section \n$$ \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_5} \\oplus \\mathcal{Q} : \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_4 \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_5} \\oplus \\mathbb{W}_{n,m,5}^1 $$\nvanishes transversely on $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{A}_5$. \nBy \\cite{BM13}\nwe conclude that this section  \nvanishes on $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{D}_5$ with a multiplicity of $2$. \nBy Corollary  \\ref{a1_pak_mult_is_2_Hess_neq_0} and \\ref{a1_pa4_mult_is_5_around_pe6}, \nthe contribution to the Euler class from the points of $\\Delta\\mathcal{P} \\mathcal{A}_6$ and $\\Delta\\mathcal{P} \\mathcal{E}_6$ are $2$ and $5$ respectively. \nSince the dimension of $\\mathcal{P} \\mathcal{D}_7$ is one less than the rank of  $\\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_5} \\oplus \\mathbb{W}_{n,m,5}^1$ and $\\mathcal{Q}$ is generic, \nthe section \ndoes not vanish on $\\Delta \\overline{\\mathcal{P} \\mathcal{D}}_7$. Since $\\overline{\\PP \\D_7^{s}}$ is a subset of $\\Delta \\overline{\\mathcal{P} \\mathcal{D}}_7$ \n(by \\eqref{one_a1_one_pa4_f02_zero_f12_not_zero_is_pd7}),  \nthe section \ndoes not vanish on $\\overline{\\PP \\D_7^{s}}$ either. \nHence\n\\begin{align*}\n\\Big\\langle e(\\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_5} \\oplus \\mathbb{W}_{n,m,5}^1), ~[\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_4] \\Big\\rangle & = \\mathcal{N}(\\mathcal{A}_1\\mathcal{P}\\mathcal{A}_5,n,m) +2\\mathcal{N}(\\mathcal{A}_1\\mathcal{P}\\mathcal{D}_5,n,m)\\\\ \n                                                                                    & ~~ + 2 \\mathcal{N}(\\mathcal{P} \\mathcal{A}_6, n, m)+ 5\\mathcal{N}(\\mathcal{P} \\mathcal{E}_6, n ,m)  \n\\end{align*}\nwhich proves the equation.  \\qed \\\\ \n\n\\noindent \\textbf{Proof of \\eqref{algopa6a1}:} Let $\\mathbb{W}_{n,m,6}^1$ and $\\mathcal{Q}$ be as in \\eqref{generic_Q} with $k=6$.\nBy Lemma \\ref{cl_two_pt}, statement \\ref{a1_pa5_cl} we have \n\\begin{align*}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}}_5 & = \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}_5 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{A}}_5- \\mathcal{P} \\mathcal{A}_5) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_7 \\cup \\Delta  \\overline{\\PP \\D_8^{s}} \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{E}}_7 \\Big) \\\\\n&=  \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}_5 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{A}}_6 \\cup \\overline{\\mathcal{P} \\mathcal{D}}_6 \\cup \\overline{\\mathcal{P} \\mathcal{E}}_6 ) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_7 \\cup \\Delta  \\overline{\\PP \\D_8^{s}} \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{E}}_7 \\Big) \n\\end{align*}\nwhere the last equality follows from \\cite{BM13} (cf. Lemma \\ref{cl}, statement \\ref{A5cl}). By Proposition \\ref{A3_Condition_prp}, the section \n$$ \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_6} \\oplus \\mathcal{Q} : \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_5 \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_6} \\oplus \\mathbb{W}_{n,m,6}^1 $$\nvanishes transversely on $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{A}_6$. \nBy \\cite{BM13}\nwe conclude that this section  \nvanishes on $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{D}_6$ with a multiplicity of $4$. \nBy Corollary \\ref{a1_pak_mult_is_2_Hess_neq_0} and \\ref{a1_pa5_mult_is_5_around_pe7}, \nthe contribution to the Euler class from the points of $\\Delta\\mathcal{P} \\mathcal{A}_7$ and $\\Delta\\mathcal{P} \\mathcal{E}_7$ are $2$ and $6$ respectively. \nSince the dimension of $\\mathcal{P} \\mathcal{D}_8$ is one less than the rank of  $\\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_6} \\oplus \\mathbb{W}_{n,m,6}^1$ and $\\mathcal{Q}$ is generic, \nthe section \ndoes not vanish on $\\Delta \\overline{\\mathcal{P} \\mathcal{D}}_8$. Since $\\Delta \\PP \\D_8^{s}$ is a subset of $\\Delta \\overline{\\mathcal{P} \\mathcal{D}}_8$ \n(by \\eqref{one_a1_one_pa5_f02_zero_f12_not_zero_is_pd8}), the section does not vanish \non $\\Delta \\overline{\\PP \\D_8^{s}}$ either. Hence \n\\begin{align*}\n\\Big\\langle e(\\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_6} \\oplus \\mathbb{W}_{n,m,6}^1), ~[\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_5] \\Big\\rangle & = \\mathcal{N}(\\mathcal{A}_1\\mathcal{P}\\mathcal{A}_6,n,m) +4\\mathcal{N}(\\mathcal{A}_1\\mathcal{P}\\mathcal{D}_6,n,m)\\\\ \n                                                                                    & ~~ + 2 \\mathcal{N}(\\mathcal{P} \\mathcal{A}_7, n, m)+ 6\\mathcal{N}(\\mathcal{P} \\mathcal{E}_7, n ,m)  \n\\end{align*}\nwhich proves the equation.  \\qed \\\\ \n\n\\noindent \\textbf{Proof of \\eqref{algopd4a1}:} Let $\\mathbb{W}_{n,0,4}^1$ and $\\mathcal{Q}$ be as in \\eqref{generic_Q} with $k=4$ and $m=0$.\nBy Lemma \\ref{cl_two_pt}, statement \\ref{a1_pa2_cl} we have \n\\begin{align*}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}}_3 &= \n\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}_3 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{A}}_3- \\mathcal{P} \\mathcal{A}_3) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_5 \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{D}^{\\vee}_5} \\Big) \\\\\n&= \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{A}_3 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{A}}_4 \\cup \\overline{\\mathcal{P} \\mathcal{D}}_4) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{A}}_5 \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{D}^{\\vee}_5} \\Big)\n\\end{align*}\nwhere the last equality follows from \\cite{BM13} (cf. Lemma \\ref{cl}, statement \\ref{A3cl}). By Proposition \\ref{A3_Condition_prp}, the section \n$$ \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_4} \\oplus \\mathcal{Q} : \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_3 \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{D}_4} \\oplus \\mathbb{W}_{n,m,4}^1 $$\nvanishes transversely on $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{D}_4$. It is easy to see that the section does not vanish on $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{A}_4$. \nBy Corollary \\ref{a1_pak_mult_is_2_Hess_neq_0} and \\ref{psi_pd4_section_vanishes_order_two_around_dual_d5}, \nthe contribution to the Euler class from the points of $\\Delta\\mathcal{P} \\mathcal{A}_5$ and $\\mathcal{P} \\mathcal{D}_5^{\\vee}$ are $2$ and $2$ respectively. \nHence \n\\begin{align}\n\\Big\\langle e(\\pi_2^*\\UL_{\\mathcal{P} \\mathcal{D}_4} \\oplus \\mathbb{W}_{n,0,4}^1), ~[\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{A}}_3] \\Big\\rangle & = \\mathcal{N}(\\mathcal{A}_1\\mathcal{P}\\mathcal{A}_4,n,0) + 2 \\mathcal{N}(\\mathcal{P} \\mathcal{A}_5, n, 0) \\nonumber \\\\\n&  +2\\big\\langle e(\\mathbb{W}_{n,0,4}^1), ~[\\overline{\\mathcal{P} \\mathcal{D}^{\\vee}_5}] \\big\\rangle. \\label{pd5_dual_prelim_a1_pa4}\n\\end{align}\nSince the map $\\pi: \\mathcal{P} \\mathcal{D}_5^{\\vee} \\longrightarrow \\mathcal{D}_5$ is one to one, we conclude that \n\\begin{align}\n\\big\\langle e(\\mathbb{W}_{n,0,4}^1), ~[\\overline{\\mathcal{P} \\mathcal{D}^{\\vee}_5}] \\big\\rangle &= \\mathcal{N}(\\mathcal{D}_5, n). \\label{pd5_dual_d5_equality_numbers}\n\\end{align}\nThis follows from the same argument as in \\cite{BM13}\nEquations \\eqref{pd5_dual_d5_equality_numbers} and \\eqref{pd5_dual_prelim_a1_pa4} imply \n\\eqref{algopd4a1}.\\\\  \n\\hf \\hf Here is an alternative way to compute the lhs of \\eqref{pd5_dual_d5_equality_numbers}. \nRecall that \n\\[ \\overline{\\mathcal{P} \\mathcal{D}}_4 = \\mathcal{P} \\mathcal{D}_4 \\cup \\overline{\\mathcal{P} \\mathcal{D}}_5 \\cup \\overline{\\mathcal{P} \\mathcal{D}^{\\vee}_5}. \\]\nThe section \n$\\Psi_{\\mathcal{P} \\mathcal{D}_5^{\\vee}}: \\overline{\\mathcal{P} \\mathcal{D}}_4 \\longrightarrow \\UL_{\\mathcal{P} \\mathcal{D}_5^{\\vee}}$ vanishes transversely on $\\mathcal{P} \\mathcal{D}_5^{\\vee}$ and \ndoes not vanish on $\\mathcal{P} \\mathcal{D}_5$. Hence \n\\begin{align*}\n\\big\\langle e(\\UL_{\\mathcal{P} \\mathcal{D}_5^{\\vee}} \\oplus \\mathbb{W}_{n,0,5}^0), ~[\\overline{\\mathcal{P} \\mathcal{D}}_4] \\big\\rangle &= \\big\\langle e(\\mathbb{W}_{n,0,5}^0), ~ [\\overline{\\mathcal{P} \\mathcal{D}^{\\vee}_5}]\\big\\rangle =  \\mathcal{N}(\\mathcal{D}_5, n).  \n\\end{align*}\nIt is easy to check directly that these two methods give the same answer for $\\mathcal{N}(\\mathcal{D}_5, n)$. \\qed \\\\ \n\n\n\n\n\n\n\\noindent \\textbf{Proof of \\eqref{algopd4a1_lambda} :} Let $\\mathbb{W}_{n,1,4}^1$ and $\\mathcal{Q}$ be as in \\eqref{generic_Q} with $k=4$ and $m=1$.\nBy Lemma \\ref{cl_two_pt}, statement \\ref{a1_d4_cl} we have \n\\begin{align}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\hat{\\mathcal{D}}_4} & = \\overline{\\mathcal{A}}_1 \\circ  \\hat{\\mathcal{D}}_4 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\hat{\\mathcal{D}}}_4- \\hat{\\mathcal{D}}_4) \\sqcup \n\\Big( \\Delta \\overline{\\hat{\\mathcal{D}}}_6 \\Big) \\nonumber \\\\\n\\implies \\overline{\\overline{\\mathcal{A}}_1 \\circ  \\hat{\\mathcal{D}}^{\\#}_4} &= \\overline{\\mathcal{A}}_1 \\circ  \\hat{\\mathcal{D}}_4^{\\#} \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\hat{\\mathcal{D}}^{\\#}_4}- \\hat{\\mathcal{D}}_4^{\\#}) \\sqcup \n\\Big( \\Delta \\overline{\\hat{\\mathcal{D}}^{\\# \\flat}_6} \\Big) \\qquad (\\textnormal{since} ~~\\overline{\\hat{\\mathcal{D}}}_4 = \\overline{\\hat{\\mathcal{D}}^{\\#}_4} ~~\\textnormal{and} \n~~\\overline{\\hat{\\mathcal{D}}}_6 = \\overline{\\hat{\\mathcal{D}}^{\\# \\flat}_6}) \\nonumber \\\\\n&= \\overline{\\mathcal{A}}_1 \\circ  \\hat{\\mathcal{D}}_4^{\\#} \\sqcup \\overline{\\mathcal{A}}_1 \\circ \\overline{\\mathcal{P} \\mathcal{D}}_4 \\sqcup \n\\Big( \\Delta \\overline{\\hat{\\mathcal{D}}^{\\# \\flat}_6} \\Big) \\qquad \\textnormal{(by definition).} \\nonumber\n\\end{align}\nBy Proposition \\ref{PD4_Condition_prp}, the section \n$$ \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{A}_3} \\oplus \\mathcal{Q} : \\overline{\\overline{\\mathcal{A}}_1 \\circ  \\hat{\\mathcal{D}}^{\\#}_4} \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_3} \\oplus \\mathbb{W}_{n,m,4}^1 $$\nvanishes transversely on $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{D}_4$. By definition, the section does not vanish on $\\hat{\\mathcal{D}}^{\\# \\flat }_6$. \nHence \n\\begin{align*}\n\\Big\\langle e(\\pi_2^*\\UL_{\\mathcal{P} \\mathcal{A}_3} \\oplus \\mathbb{W}_{n,1,4}^1), ~[\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\hat{\\mathcal{D}}^{\\#}_4}] \\Big\\rangle & = \\mathcal{N}(\\mathcal{A}_1\\mathcal{P}\\mathcal{D}_4,n,1)\n\\end{align*}\nwhich proves the equation. \\qed \\\\ \n\n\\noindent \\textbf{Proof of \\eqref{algopd5a1}:} Let $\\mathbb{W}_{n,m,5}^1$ and $\\mathcal{Q}$ be as in \\eqref{generic_Q} with $k=5$. \nBy Lemma \\ref{cl_two_pt}, statement \\ref{a1_pd4_cl} we have \n\\begin{align*}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{D}}_4 & = \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{D}_4 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{D}}_4- \\mathcal{P} \\mathcal{D}_4) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{D}}_6  \\cup \\Delta \\overline{\\PP \\D_6^{\\vee s}} \\Big)\\\\\n&= \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{D}_4 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{D}}_5 \\cup \\overline{\\mathcal{P} \\mathcal{D}^{\\vee}_5})  \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{D}}_6 \\cup \\Delta \\overline{\\PP \\D_6^{\\vee s}} \\Big)\n\\end{align*}\nwhere the last equality follows from \\cite{BM13} (cf. Lemma \\ref{cl}, statement \\ref{D4cl}). By Proposition \\ref{D4_Condition_prp}, the section \n$$ \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_5}^{\\mathbb{L}} \\oplus \\mathcal{Q} : \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_4 \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{D}_5} \\oplus \\mathbb{W}_{n,m,5}^1 $$\nvanishes transversely on $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{D}_5$. Moreover, it does not vanish on $\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}_5^{\\vee}$ by definition. \nBy Corollary \\ref{psi_pd5_section_vanishes_order_two_around_pd6}, \nthe contribution to the Euler class from the points of $\\Delta\\mathcal{P} \\mathcal{D}_6$ is $2$.  \nThe section does not vanish on $\\Delta \\PP \\D_6^{\\vee s}$ by definition. \nSince the dimension of $\\mathcal{P} \\mathcal{D}_6^{\\vee}$ is same as the dimension of $\\mathcal{P} \\mathcal{D}_6$ \nand $\\mathcal{Q}$ is generic, by \\eqref{pd6_dual_s_is_subset_of_pd6_dual}, the section does not vanish on $\\Delta \\overline{\\PP \\D_6^{\\vee s}}$.  \nHence \n\\begin{align*}\n\\Big\\langle e(\\pi_2^*\\UL_{\\mathcal{P} \\mathcal{D}_5} \\oplus \\mathbb{W}_{n,m,5}^1), ~[\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_4] \\Big\\rangle & = \\mathcal{N}(\\mathcal{A}_1\\mathcal{P}\\mathcal{D}_5,n,m) + 2 \\mathcal{N}(\\mathcal{P} \\mathcal{D}_6, n, m)\n\\end{align*}\nwhich proves the equation. \\qed \\\\\n\n\n\\noindent \\textbf{Proof of \\eqref{algopd6a1} and \\eqref{algope6a1}:} Let $\\mathbb{W}_{n,m,6}^1$ and $\\mathcal{Q}$ be as in \\eqref{generic_Q} with $k=6$. \nBy Lemma \\ref{cl_two_pt}, statement \\ref{a1_pd5_cl} we have \n\\begin{align*}\n\\overline{\\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{D}}_5 & = \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{D}_5 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{D}}_5- \\mathcal{P} \\mathcal{D}_5) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{D}}_7 \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{E}}_7 \\Big) \\\\\n&= \\overline{\\mathcal{A}}_1 \\circ  \\mathcal{P} \\mathcal{D}_5 \\sqcup \\overline{\\mathcal{A}}_1 \\circ (\\overline{\\mathcal{P} \\mathcal{D}}_6 \\cup \\overline{\\mathcal{P} \\mathcal{E}}_6) \\sqcup \n\\Big( \\Delta \\overline{\\mathcal{P} \\mathcal{D}}_7 \\cup \\Delta \\overline{\\mathcal{P} \\mathcal{E}}_7 \\Big) \n\\end{align*}\nwhere the last equality follows from \\cite{BM13} (cf. Lemma \\ref{cl}, statement \\ref{D5cl}). By Propositions \\ref{D6_Condition_prp} and \\ref{E6_Condition_prp}, the sections \n\\[ \\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_6} \\oplus \\mathcal{Q} : \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_5 \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{D}_6} \\oplus \\mathbb{W}_{n,m,6}^1, \n~~\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{E}_6} \\oplus \\mathcal{Q} : \\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_5 \\longrightarrow \\pi_2^*\\UL_{\\mathcal{P} \\mathcal{E}_6} \\oplus \\mathbb{W}_{n,m,6}^1 \\]\nvanishes transversely on $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{D}_6$ and $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{E}_6$ respectively. \nMoreover, they do not vanish on $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{E}_6$ and $\\mathcal{A}_1 \\circ \\mathcal{P} \\mathcal{D}_6$ respectively.  \nBy Corollary \\ref{a1_pdk_mult_is_2_f12_neq_0} and \\ref{psi_pe6_and_pd6_section_vanishes_order_one_around_pe7}\nthe contribution of the section  $\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{D}_6} \\oplus \\mathcal{Q}$ \nto the Euler class from the points of $\\Delta\\mathcal{P} \\mathcal{D}_7$ and $\\Delta \\mathcal{P} \\mathcal{E}_7$ are $2$ and $1$ respectively. \nBy Corollary \\ref{psi_pe6_and_pd6_section_vanishes_order_one_around_pe7}, the contribution of the section    \n$\\pi_2^*\\Psi_{\\mathcal{P} \\mathcal{E}_6} \\oplus \\mathcal{Q}$ from the points of $\\Delta \\mathcal{P} \\mathcal{E}_7$ is $1$; moreover it does not vanish on $\\Delta \\mathcal{P} \\mathcal{D}_7$. \nHence \n\\begin{align*}\n\\Big\\langle e(\\pi_2^*\\UL_{\\mathcal{P} \\mathcal{D}_6} \\oplus \\mathbb{W}_{n,m,6}^1), ~[\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_5] \\Big\\rangle & = \\mathcal{N}(\\mathcal{A}_1\\mathcal{P}\\mathcal{D}_6,n,m) + 2 \\mathcal{N}(\\mathcal{P} \\mathcal{D}_7, n, m)+\\mathcal{N}(\\mathcal{P} \\mathcal{E}_7, n, m), \\\\\n\\Big\\langle e(\\pi_2^*\\UL_{\\mathcal{P} \\mathcal{E}_6} \\oplus \\mathbb{W}_{n,m,6}^1), ~[\\overline{\\overline{\\mathcal{A}}_1 \\circ \\mathcal{P} \\mathcal{D}}_5] \\Big\\rangle & = \\mathcal{N}(\\mathcal{A}_1\\mathcal{P}\\mathcal{E}_6,n,m) + \\mathcal{N}(\\mathcal{P} \\mathcal{E}_7, n, m)\n\\end{align*}\nwhich prove equations \\eqref{algopd6a1} and \\eqref{algope6a1}.  \\qed \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzgdmr b/data_all_eng_slimpj/shuffled/split2/finalzzgdmr
new file mode 100644
index 0000000000000000000000000000000000000000..d9994d2bc8f4453417ee6b6b16325c914a975e0f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzgdmr
@@ -0,0 +1,5 @@
+{"text":"\\section{INTRODUCTION}\nZero-sum games (ZSGs) are mathematical models describing the interaction of mutually adversarial decision makers. \nIn the realm of machine learning, ZSGs  have played a central role in the development of techniques such as generative adversarial networks (GANs) \\cite{goodfellow_2014_generative} and adversarial training \\cite{madry_2017_towards}.\nWithin this context, two solution concepts have been adopted for predicting the outcome of ZSGs: the Nash equilibrium (NE)~\\cite{nash_1950_equilibrium} and the Stackelberg equilibrium (SE) \\cite{Stackelberg-1952}.\nThe NE is a prediction observed under the assumption that both players simultaneously choose their strategies (probability measures over the set of possible actions or decisions).\nOn the other hand, the SE describes the outcome in which one of the players (the leader) publicly and irrevocably commits to use a particular strategy before its opponent (the follower). In such a case, the follower chooses its strategy as a best response to the commitment of the leader. \n\nCommitments are said to be in mixed strategies when the leader is allowed to commit to strategies whose support contains more than one action. In this case, the relevant solution concept is the SE in mixed strategies~\\cite{conitzer_2006_computing,  conitzer_2016_stackelberg,  leonardos_2018_commitment}. \nInterestingly, in finite ZSGs, the payoffs at the NE and the SE in mixed strategies are identical, as shown in~\\cite{von_2010_leadership}.\nThe commitment is said to be in pure strategies when the leader is constrained to commit to play one action with probability one. This is assimilated to the case in which the follower perfectly observes the action played by the leader.\nThe relevant solution concept under these assumptions is the SE in pure strategies~\\cite{Stackelberg-1952, simaan_1973_stackelberg, simaan_1973_additional}.\nThe expected payoff at the SE in pure strategies is equal to the $\\min\\max$ or $\\max\\min$ solution, where the optimization is over the set of actions \\cite{jin_2020_local, bai_2021_sample}.\n\nIn adversarial training, the underlying assumption is that the follower (the attacker or adversary) perfectly observes the action played by the leader (the learner) \\cite{huang_2022_robust, zuo_2021_adversarial, bruckner_2011_stackelberg, gao_2022_achieving, bai_2021_sample}. Similarly, in data integrity attacks, the follower (the learner) perfectly observes the action of the leader (the attacker) \\cite{chivukula_2017_adversarial, liu_2009_game, kantarciouglu_2011_classifier}. \nThat is, adversarial training and data integrity attacks are studied using the SE in pure strategies.\nAlternatively, GANs are modelled by ZSGs in which the relevant solution concept is the NE (or SE in mixed strategies)~\\cite{hsieh_2019_finding, oliehoek_2018_beyond}. Essentially, ZSGs are used to predict game outcomes in terms of mixed strategies (probability measures), instead of actions (pure strategies). \nIn a nutshell, the underlying assumption of the SE in mixed strategies is that the strategy to which the leader commits to is perfectly observed by the follower and the actions are unobservable.  Alternatively, the assumption of the SE in pure strategies is that actions are perfectly observable, which makes the notion of commitment irrelevant. \nThese assumptions are difficult to justify in practice. Often, data is obtained via data acquisition systems that are subject to quantization errors, additive noise, and impairments due to data transmission and storage. These additional impairments are not necessarily due to malicious agents but the nature of the data acquisition and information processing \\cite{cover_elements_2012}. \nIn real system implementations, the observations of actions and commitments, if they occur, are   subject to noise. Nonetheless, the impact of noisy observations in adversarial training, GANs, and most areas of ML relying on ZSGs remains an uncharted territory, in part due to the lack of simple and adapted solution concepts.  This work makes progress in this direction and proposes a game formulation that takes into account noisy observations of both actions and commitments.\n\n\\subsection{Contributions}\n\n For pedagogical purposes, results are presented for $2 \\times 2$ ZSGs. Nonetheless, the results can be readily extended to the case of two-player ZSGs, with finite number of actions. \nThe contributions are presented as follows.\n\nSection~\\ref{SecNOA} introduces a game formulation in which the follower obtains a noisy observation of the action played by the leader, whereas the commitment is assumed to be perfectly observed. Three results are presented: First, the set of best responses of the follower is characterized and the role of the priors formed by the follower with the available information is presented. Second, the set of optimal commitments for the leader is calculated and it is shown that, even subject to noise, observations either benefit or never make a difference to the follower. Third, the equilibrium is shown to always exist. Benefits for the follower are observed at the equilibrium exclusively when the ZSG exhibits a unique NE in mixed strategies. In all other cases, e.g., ZSG exhibiting strategic dominance, unique NE in pure strategies, or infinitely many NEs, the payoffs with and without observations are identical.\n\nSection~\\ref{SecCM} introduces a game formulation in which the follower obtains a noisy observation of both the action played by the leader and the commitment. The commitment mismatch is modelled by a deterministic distortion (affine function) that is assumed to be known by the leader and ignored by the follower.\nCommitment mismatch is shown to be either beneficial or immaterial to the leader. Nonetheless,  beneficial situations for the leader are shown to be not stable, in part because, an equilibrium does not necessarily exist. \nThis phenomenon arises due to the fact that optimal commitments induce infinitely many best responses for the follower and each of them leads to a different payoff.  To overcome this challenge, the leader is subject to commit to suboptimal strategies in order to induce a unique best response that can be predicted.\n\nThe work is finalized with a discussion and final remarks presented in Section~\\ref{SecDiscussion}. The proofs of all results are presented in the supplementary material. \n \n\\subsection*{Notation}\nGiven a finite set $\\set{X}$, the notation $2^{\\set{X}}$ represents \nthe power set of $\\set{X}$. The notation $\\simplex{\\set{X}}$ represents the set of all probability measures that can be defined on the measurable space $\\left( \\set{X}, 2^{\\set{X}}\\right)$. The set of all subsets of $\\simplex{\\set{X}}$ is denoted by $2^{\\simplex{\\set{X}}}$.\nGiven two matrices $\\matx{a}$ and $\\matx{b}$, their Hadamard product is denoted by $\\matx{a} \\circ \\matx{b}$.\n\n\\section{PRELIMINARIES}\\label{SecPrel}\nConsider a two-player two-action zero-sum game in normal form with payoff matrix \n\\begin{IEEEeqnarray}{rcl}\n\\label{EqMatrixU}\n\\matx{u} & = & \n\\begin{pmatrix}\nu_{1,1} & u_{1,2}\\\\\nu_{2,1} & u_{2,2}\n\\end{pmatrix}.\n\\end{IEEEeqnarray}\nLet the elements of the set $\\set{K} \\triangleq \\{1,2\\}$ represent the indices of the players; \nand let the elements of the set $\\set{A}_1 = \\set{A}_2 \\triangleq \\{a_1, a_2\\}$ represent the actions of the players.\nHence, for all $(i,j) \\inCountTwo^2$, when \\Pone plays $a_i$ and \\Ptwo plays $a_j$, the outcome of the game is $u_{i, j}$.\nIn the following, such a game is represented by the tuple \n\\begin{IEEEeqnarray}{rcl}\n\\label{EqTheGame}\n\\gameNF{\\matx{u}} & \\triangleq & \\left(\\set{K}, \\set{A}_1 , \\set{A}_2 , \\matx{u} \\right).\n\\end{IEEEeqnarray} \nThe remaining part of this section relies on the following assumptions:\n$(i)$~The game $\\gameNF{\\matx{u}}$ is repeated infinitely many times; \n$(ii)$~At each repetition, the players are oblivious of all previous repetitions; and \n$(iii)$~actions are  simultaneously chosen at each repetition.\nUnder assumptions $(i) - (iii)$, the average payoff achieved by the players in the \\emph{repeated game} can be expressed in terms of their \\emph{strategies}.\nFor all $k \\in \\set{K}$, the strategy of \\Pkth  is a probability measure denoted by $P_{A_k} \\in \\simplex{\\set{A}_k}$.  \nAt each repetition of the game, players choose their actions by sampling their probability measures (strategies). \nLet the average payoff be represented by the function $u: \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_2} \\to \\reals$ such that, given the strategies $P_{A_1}$ and $P_{A_2}$,  \n\\begin{IEEEeqnarray}{rcl}\n\\label{EqNormalFormU}\nu\\left(P_{A_1}, P_{A_2} \\right) & = &  \\sum_{(i,j) \\in \\{1,2\\}^2} P_{A_1} \\left( a_i \\right)  P_{A_2}\\left(a_j\\right) u_{i,j}. \\squeezeequ\n\\end{IEEEeqnarray}\nNote that the average payoff coincides with the expected payoff under assumptions $(i) - (iii)$.\n\\Pone chooses its strategy $P_{A_1}$ aiming to maximize the expected payoff $u\\left(P_{A_1}, P_{A_2} \\right)$, whereas \\Ptwo  chooses the strategy $P_{A_2}$ to minimize it. \nInterestingly, under assumptions $(i) - (iii)$, players can calculate their optimal strategies before the beginning of the repeated game, as shown in the following sections.  \n\nWhen the repeated game is played without commitments, the relevant solution concept for the ZSG $\\gameNF{\\matx{u}}$ in~\\eqref{EqTheGame} is the NE. The following lemma characterizes the payoff at the NE and shows  that $2\\times2$ ZSGs exhibit either a unique NE or infinitely many NEs. \n\\begin{lemma}\\label{LemmaNE}\nLet the probability measures $P^{\\star}_{A_1} \\in \\simplex{\\set{A}_1}$ and $P^{\\star}_{A_2} \\in \\simplex{\\set{A}_2}$ form a NE of the game $\\gameNF{\\matx{u}}$ in~\\eqref{EqTheGame}. \nIf the entries of the matrix $\\matx{u}$ in~\\eqref{EqMatrixU} satisfy\n\\begin{subequations}\n\\begin{IEEEeqnarray}{lcl}\n\\left( u_{1,1} - u_{1,2} \\right) \\left( u_{2,2} - u_{2,1} \\right)   >   0 & \\mbox{ and } & \\\\\n\\left( u_{1,1} - u_{2,1} \\right) \\left( u_{2,2} - u_{1,2} \\right)   >   0,\n\\end{IEEEeqnarray}\n\\label{EqMixedAssumption}\n\\end{subequations}\nthen, the NE of the game $\\gameNF{\\matx{u}}$ in~\\eqref{EqTheGame} is unique, with \n\\begin{subequations}\\label{EqNEStratExample}\n\\begin{IEEEeqnarray}{rcl}\n\\label{EqPA1StarExample}\nP^{\\star}_{A_1}(a_1) & = &  \\frac{u_{2,2}-u_{2,1}}{u_{1,1} - u_{1,2} - u_{2,1}+u_{2,2}} \\in (0,1)\\mbox{  and  } \\IEEEeqnarraynumspace\\\\\n\\label{EqPA2StarExample}\nP^{\\star}_{A_2}(a_1) & = &\\frac{u_{2,2}-u_{1,2}}{u_{1,1} - u_{1,2} - u_{2,1}+u_{2,2}}\\in (0,1).\n\\end{IEEEeqnarray} \n\\end{subequations}\nMoreover,  the expected payoff at the NE is\n\\begin{IEEEeqnarray}{rcl}\nu(P_{A_1}^{\\star},P_{A_2}^{\\star}) & = & \\frac{u_{1,1}u_{2,2} - u_{1,2}u_{2,1}}{u_{1,1} - u_{1,2} - u_{2,1}+u_{2,2}}.\n\\end{IEEEeqnarray}\nIf the entries of the matrix $\\matx{u}$ in~\\eqref{EqMatrixU}  satisfy\n\\begin{subequations}\n\\begin{IEEEeqnarray}{lcl}\n\\left( u_{1,1} - u_{1,2} \\right) \\left( u_{2,2} - u_{2,1} \\right)   \\leqslant   0 & \\mbox{ or } & \\\\\n\\left( u_{1,1} - u_{2,1} \\right) \\left( u_{2,2} - u_{1,2} \\right)   \\leqslant   0,\n\\end{IEEEeqnarray}\n\\label{EqNotMixedAssumption}\n\\end{subequations}\nthen, there exists either a unique NE or infinitely many NEs. Moreover, all NE strategies lead to the same payoff,\n\\begin{IEEEeqnarray}{rcl}\\label{Equ:f_3}\nu(P_{A_1}^{\\star},P_{A_2}^{\\star}) & = & \\min \\lbrace \\max\\lbrace u_{1,1}, u_{2,1}\\rbrace,  \\max\\lbrace u_{1,2}, u_{2,2}\\rbrace \\rbrace \\squeezeequ\\\\\n& = & \\max \\lbrace \\min\\lbrace u_{1,1}, u_{1,2}\\rbrace,  \\min\\lbrace u_{2,1}, u_{2,2}\\rbrace \\rbrace. \\IEEEeqnarraynumspace \\squeezeequ\n\\end{IEEEeqnarray}\n\\end{lemma}\nA payoff matrix $\\matx{u}$ that satisfies~\\eqref{EqMixedAssumption} represents a ZSG exhibiting a unique NE in strictly mixed strategies. Alternatively, a payoff matrix $\\matx{u}$ that satisfies~\\eqref{EqNotMixedAssumption} represents a ZSG exhibiting \\emph{strategic dominance}, a unique pure NE, or infinitely many NEs. \n\n\n\\section{NOISY OBSERVATIONS OF THE ACTIONS}\\label{SecNOA}\nIn this section, the repeated game is assumed to be played with commitments under the assumptions $(i)$ and $(ii)$ in Section~\\ref{SecPrel}, and a new assumption: $(iv)$~at each repetition, the leader chooses its action and the follower obtains a noisy observation. \nThat is, assumption $(iii)$ is dropped, and the follower chooses a strategy at each repetition, knowing the  commitment and a noisy observation of the action played by the leader.\n\\subsection{Game Formulation}\n\nDenote by $A_1$, $A_2$, and $\\tilde{A}_2$ the random variables representing the actions of \\Pone (the follower), \\Ptwo (the leader), and the noisy observation of the action played by \\Ptwo at each repetition of the game, respectively. \nLet also $P_{A_1 \\tilde{A}_2 A_2} \\in \\simplex{\\set{A}_1 \\times \\set{A}_2 \\times \\set{A}_2}$ be  the probability measure jointly induced by $A_1$,  $\\tilde{A}_2$, and $A_2$, which satisfies for all $\\left( a,  \\tilde{b},b  \\right) \\in \\set{A}_1 \\times \\set{A}_2 \\times \\set{A}_2$, \n\\begin{IEEEeqnarray}{rcl}\n\\label{EqTheJointThing}\nP_{A_1 \\tilde{A}_2 A_2} \\left( a, \\tilde{b},  b \\right) = P_{A_2} \\left( b \\right) P_{\\tilde{A}_2 | A_2 = b} \\left(  \\tilde{b}  \\right) P_{A_1 | \\tilde{A}_2 = \\tilde{b}} \\left( a  \\right), \\IEEEeqnarraynumspace \\supersqueezeequ\n\\end{IEEEeqnarray}\nwhere the probability measure $P_{A_2 } \\in \\simplex{\\set{A}_2}$ is the strategy of \\Ptwo;\nthe pair of  probability measures $\\left( P_{A_1| \\tilde{A}_2 = a_1}, P_{A_1| \\tilde{A}_2 = a_2}\\right) \\in \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_1}$ form the strategy of \\Pone;\nand the pair of measures $\\left( P_{\\tilde{A}_2 | A_2 = a_1}, P_{\\tilde{A}_2 | A_2 = a_2}\\right)$  form a binary channel through which the follower observes the action of the leader.\n\nUsing this notation, the development of the repeated game is described as follows. \nBefore the beginning of the repetittions,  \\Ptwo publicly and irrevocably announces its strategy $P_{A_2}$. \nAt each repetition,  \\Ptwo (the leader) plays the action $b \\in \\set{A}_2$ with probability $P_{A_2}\\left( b \\right)$. \n\\Pone observes action $\\tilde{b} \\in \\set{A}_2$ with probability  $P_{\\tilde{A}_2 | A_2 = b} \\left( \\tilde{b} \\right)$.\nFinally, \\Pone plays the action $a \\in \\set{A}_1$, with probability $P_{A_1| \\tilde{A}_2 = \\tilde{b}}\\left( a \\right)$ and both players obtain their payoffs. \n\nThe expected payoff obtained by the players is determined by the function $v: \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_2} \\to \\reals$, such that given the strategy $\\left( P_{A_1| \\tilde{A}_2 = a_1}, P_{A_1| \\tilde{A}_2 = a_2}\\right)$ of \\Pone, often denoted by $ P_{A_1| \\tilde{A}_2 }$, and the strategy $P_{A_2}$ of \\Ptwo, the expected payoff $v\\left(P_{A_1| \\tilde{A}_2}, P_{A_2} \\right)$ is  \n\\begin{IEEEeqnarray}{rcl}\n\\label{EqTheCostFunction}\nv\\left(P_{A_1| \\tilde{A}_2}, P_{A_2} \\right) & = & \\sum_{(i,j) \\inCountTwo^2} u_{i,j} P_{A_1 A_2} \\left( a_i , a_j\\right),  \\IEEEeqnarraynumspace \\squeezeequ\n\\end{IEEEeqnarray}\nwhere the joint probability measure $P_{A_1 A_2}$ is the marginal probability measure on $A_1$ and $A_2$ of the probability measure $P_{A_1 \\tilde{A}_2 A_2}$ in~\\eqref{EqTheJointThing}.\n\nThe generalization of the game in normal form $\\gameNF{\\matx{u}}$ in~\\eqref{EqTheGame} obtained by including the binary channel formed by the measures $ P_{\\tilde{A}_2 | A_2 =a_{1} }$ and  $P_{\\tilde{A}_2 | A_2 =a_{2} }$ is \ndescribed by  the tuple \n\\begin{IEEEeqnarray}{c}\n\\label{EqTheGameNoise}\n\\game{G}\\left(\\matx{u}, \\matx{w} \\right) = \\left(\\set{K}, \\set{A}_1, \\set{A}_2 , \\matx{u}, \\matx{w} \\right),\n\\end{IEEEeqnarray} \nwhere the  $2\\times2$ matrix $\\matx{w}$ satisfies\n\\begin{IEEEeqnarray}{rcl}\n\\label{EqTheChannel}\n\\matx{w}=\n\\begin{pmatrix}\nP_{\\tilde{A}_2 | A_2 = a_1}(a_1) &P_{\\tilde{A}_2 | A_2 = a_2}(a_1) \\\\\nP_{\\tilde{A}_2 | A_2 = a_1}(a_2) & P_{\\tilde{A}_2 | A_2 = a_2}(a_2)\n\\end{pmatrix}.\n\\end{IEEEeqnarray}\n\n\\subsection{The Set of Best Responses of \\Pone}\nThe set of best responses of \\Pone to a given strategy of \\Ptwo is determined by the correspondence $\\BR_1: \\simplex{\\set{A}_{2}} \\to 2^{\\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_1} }$. That is, given the commitment $P_{A_2}$, it holds that\n\\begin{IEEEeqnarray}{rcl}\\label{Equ:BR_1}\n\\label{EqBR1}\n\\BR_1\\left( P_{A_2} \\right) & = &\\arg\\max_{ (Q_1, Q_2) \\in \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_1}} v(Q_1, Q_2, P_{A_2}),  \\IEEEeqnarraynumspace \\supersqueezeequ\n\\end{IEEEeqnarray}\nwhere the function $v$ is defined in~\\eqref{EqTheCostFunction}.\nIn order to study the set $\\BR_1\\left( P_{A_2} \\right)$ in~\\eqref{EqBR1}, consider the $2\\times2$ matrix\n\\begin{IEEEeqnarray}{rcl}\n\\label{EqMatrixUs}\n\\matx{u}^{(i)} & = & \\matx{u}\n\\begin{pmatrix}\nP_{\\tilde{A}_2 | A_2 = a_1}(a_i) & 0 \\\\\n 0 & P_{\\tilde{A}_2 | A_2 = a_2}(a_i)\n\\end{pmatrix},\n\\end{IEEEeqnarray}\nwith $i \\inCountTwo$, where the matrix $\\matx{u}$ is defined in~\\eqref{EqMatrixU}; and the probability measures $P_{\\tilde{A}_2 | A_2 = a_1}$ and $P_{\\tilde{A}_2 | A_2 = a_2}$ are defined in~\\eqref{EqTheChannel}.\nThe following lemma shows that, given a commitment  $P_{A_2}$,  the set of best responses $\\BR_1(P_{A_2})$ in~\\eqref{EqBR1} is the Cartesian product of two sets that can be described independently. \n\n\\begin{lemma}\\label{CorBR1Separates}\nThe correspondance $\\BR_1$ in~\\eqref{EqBR1} satisfies for all  $P \\in \\simplex{\\set{A_2}}$,\n\\begin{IEEEeqnarray}{rcl}\n\\label{EqBR1X}\n\\BR_1\\left( P \\right) &=& \\BR_{1,1}\\left( P \\right) \\times \\BR_{1,2}\\left( P \\right), \n\\end{IEEEeqnarray}\nwhere for all $i \\inCountTwo$, the correspondence $\\BR_{1,i}:  \\simplex{\\set{A}_{2}} \\to 2^{\\simplex{\\set{A}_1}}$ is such that \n\\begin{IEEEeqnarray}{rcl}\n\\label{EqBR1i}\n\\BR_{1,i}\\left( P \\right) & = & \\arg\\max_{ Q \\in \\simplex{\\set{A}_1}} \n\\begin{pmatrix}\nQ\\left( a_1 \\right)\\\\\nQ\\left( a_2 \\right)\n\\end{pmatrix}^{\\sfT}\n\\matx{u}^{(i)}\n\\begin{pmatrix}\nP\\left(a_1\\right)\\\\\nP\\left(a_2\\right)\n\\end{pmatrix}, \\IEEEeqnarraynumspace \n\\end{IEEEeqnarray}\nwhere the matrix $\\matx{u}^{(i)}$ is in~\\eqref{EqMatrixUs}.\n\\end{lemma}\n %\nA first observation from Lemma~\\ref{CorBR1Separates} is that  in the case in which the matrices $\\matx{u}^{(1)}$ and $\\matx{u}^{(2)}$ in~\\eqref{EqMatrixUs} are identical, \\Pone chooses its actions independently of the noisy observation of the action played by \\Ptwo. In such a case, $\\BR_{1,1}\\left( P_{A_2} \\right) = \\BR_{1,2}\\left( P_{A_2} \\right)$, and thus, the best response of \\Pone depends exclusively on its opponent's strategy~$P_{A_2}$.\n\nFor all $i \\inCountTwo$ and for all $P \\in \\simplex{\\set{A}_2}$, the cardinality of set $\\BR_{1,i}\\left( P \\right)$ is either one or infinite. In the case in which $\\BR_{1,i}\\left( P \\right)$ is unitary, the only element is a pure strategy. Alternatively, when the cardinality is infinity, the set $\\BR_{1,i}\\left( P \\right)$ is identical to the set of all possible probability measures on $\\set{A}_1$, i.e., $\\BR_{1,i}\\left( P \\right) = \\simplex{\\set{A}_1}$. That is, at each repetition of the game, \\Pone chooses its actions either deterministically, i.e,  with probability one; or indifferently. \nThe following lemma formalizes this observation. \n\n\\begin{lemma}\\label{LemmaBreakingBad1}\nGiven a probability measure $P \\in \\simplex{\\set{A}_2}$,  for all $i \\inCountTwo$, the correspondence $\\BR_{1,i}$ in~\\eqref{EqBR1i} satisfies\n\\begin{IEEEeqnarray}{rCl}\\label{Equ:f_1_1}\n&&\\BR_{1,i}(P)=\n\\ \\left\\{ \n \\begin{array}{cl}\n\\hspace{-1.5ex} \\{ Q \\in \\Delta(\\set{A}_1): Q(a_1) = 1 \\}, &  \\hspace{-1ex}\\textnormal{if } s_i> 0,\\\\\n\\hspace{-1.5ex} \\{ Q \\in \\Delta(\\set{A}_1): Q(a_1) = 0 \\}, &  \\hspace{-1ex}\\textnormal{if }  s_i< 0,\\\\\n\\simplex{\\set{A}_1}, &  \\hspace{-1ex}\\textnormal{if }  \ns_i= 0, \n \\end{array}\n \\right. \\IEEEeqnarraynumspace \\supersqueezeequ\n\\end{IEEEeqnarray}\nwhere $s_i \\in \\reals$ is given by \n\\begin{IEEEeqnarray}{rcl} \\label{Eqsi}\n&&s_i \\triangleq \\left(u_{1,1} - u_{2,1}\\right) P\\left( a_1 \\right) P_{\\tilde{A}_2 | A_2 = a_1}\\left( a_i \\right) \\IEEEnonumber \\\\\n&& \\quad + \\left(u_{1,2}-u_{2,2} \\right)P\\left( a_2 \\right) P_{\\tilde{A}_2 | A_2 = a_2}\\left( a_i \\right). \\label{EqBigOne}\n\\end{IEEEeqnarray}\n\\end{lemma}\n\nFor all $\\left( i, j\\right) \\inCountTwo^2$, at a given game repetition, the expected payoff, when \\Pone plays $a_j$, \\Ptwo has committed to $P_{A_2}$, and the noisy observation is $a_i$, is $u_{j,1} P_{A_2}\\left( a_1 \\right) P_{\\tilde{A}_2 | A_2 = a_1}\\left( a_i \\right) +  u_{j,2} P_{A_2}\\left( a_2 \\right) P_{\\tilde{A}_2 | A_2 = a_2}\\left( a_i \\right)$. \nThus, the right-hand side of the equality in~\\eqref{EqBigOne} is the difference between the expected payoff obtained by the players when \\Pone plays $a_1$ and when it plays $a_2$, subject to the observation $a_i$ and the commitment $P_{A_2}$. \nHence, from Lemma~\\ref{LemmaBreakingBad1}, the optimal action to be played at a given repetition by \\Pone is the action that maximizes the expected payoff subject to the noisy observation and the  commitment. When both actions induce the same expected payoff,  \\Pone chooses its actions following any strategy.\n\nThe following lemma presents a different view of the correspondences $\\BR_{1,1}$ and $\\BR_{1,2}$ in~\\eqref{EqBR1i}. It suggests that at each game repetition, \\Pone performs an estimation of the action played by \\Ptwo based on the knowledge of the commitment and the noisy observation.\n\n\\begin{lemma}\\label{LemmaBreakingGrounds}\nGiven a probability measure $P \\in \\simplex{\\set{A}_2}$,  for all $i \\inCountTwo$, the correspondence $\\BR_{1,i}$ in~\\eqref{EqBR1i} satisfies\n\\begin{IEEEeqnarray}{rcl}\n\\BR_{1,i}\\left( P \\right) & = & \\arg\n\\max_{Q \\in \\simplex{\\set{A}_1}} u\\left( Q , P_{A_2 | \\tilde{A}_2 = a_i} \\right),\n\\end{IEEEeqnarray}\nwhere the function $u$ is defined in~\\eqref{EqNormalFormU}; the probability measure $P_{A_2 | \\tilde{A}_2 = a_i}$ satisfies for all $j \\inCountTwo$,\n\\begin{IEEEeqnarray}{rcl}\n\\label{EqPost}\nP_{A_2 | \\tilde{A}_2 = a_i} \\left( a_j \\right) & = & \\frac{P_{\\tilde{A}_2 | A_2 = a_j} \\left( a_i \\right) P\\left( a_j \\right)}{\\displaystyle\\sum_{\\ell \\inCountTwo} P_{\\tilde{A}_2 | A_2 = a_{\\ell}} \\left( a_i \\right) P\\left( a_{\\ell} \\right)},\n\\end{IEEEeqnarray}\nwith the probability measures $P_{\\tilde{A}_2 | A_2 = a_{1}}$ and $P_{\\tilde{A}_2 | A_2 = a_{2}}$ defined in~\\eqref{EqTheChannel}.\n\\end{lemma}\n\nFor all $(i,j) \\inCountTwo^2$,  the likelihood with which \\Ptwo has chosen action $a_j$ given the commitment  $P$ and the noisy observation $a_i$ is $P_{A_2 | \\tilde{A}_2 = a_i} \\left( a_j \\right)$ in~\\eqref{EqPost}. \nHence, from Lemma~\\ref{LemmaBreakingBad1} and Lemma~\\ref{LemmaBreakingGrounds}, at a given repetition of the game $\\game{G}\\left(\\matx{u}, \\matx{w} \\right)$, the optimal action of \\Pone to the observation $a_i$ and the commitment $P$ is identical to the optimal action of a player in the game $\\gameNF{\\matx{u}}$ in~\\eqref{EqTheGame} when its  opponent plays the strategy $P_{A_2 | \\tilde{A}_2 = a_i}$  in~\\eqref{EqPost}. \n   \n\\subsection{The Best Strategies of \\Ptwo}\nLet the function $\\hat{v}: \\simplex{\\set{A}_2} \\rightarrow \\reals$ be such that given  a probability measure $P \\in \\simplex{\\set{A}_2}$,\n\\begin{IEEEeqnarray}{rcl}\n\\label{Eqv}\n\\hat{v}\\left( P \\right) = \\max_{\\left(Q_1, Q_2\\right) \\in  \\BR_{1}\\left( P \\right)} v\\left(Q_1, Q_2, P \\right),\n\\end{IEEEeqnarray}\nwhere the function $v$ is defined in~\\eqref{EqTheCostFunction} and the correspondence $\\BR_{1}$ is defined in~\\eqref{EqBR1}.\nThe set of best strategies for \\Ptwo, under the assumption that \\Pone observes the commitment and obtains a noisy observation of the action played by \\Ptwo, is the set of minimizers of the function $\\hat{v}$ in~\\eqref{Eqv}. \nLet  $P^{(1)}$ and $P^{(2)}$ be two real numbers such that for all $i \\inCountTwo$,\n\\begin{IEEEeqnarray}{rcl}\n\\label{EqPi}\n\\begin{pmatrix}\n1\\\\\n0\n\\end{pmatrix}^{\\sfT}\n\\matx{u}^{(i)}\n\\begin{pmatrix}\nP^{(i)}\\\\\n1- P^{(i)}\n\\end{pmatrix} \n& = &\n\\begin{pmatrix}\n0\\\\\n1\n\\end{pmatrix}^{\\sfT}\n\\matx{u}^{(i)}\n\\begin{pmatrix}\nP^{(i)}\\\\\n1- P^{(i)}\n\\end{pmatrix}.  \\IEEEeqnarraynumspace \n\\end{IEEEeqnarray}\nFrom Lemma~\\ref{LemmaBreakingBad1}, it holds that if $P^{(i)} \\in [0,1]$, for some $i\\inCountTwo$, and \\Ptwo adopts a strategy $P_{A_2} \\in \\simplex{\\set{A}_2}$ such that $P_{A_2}\\left( a_1 \\right) = P^{(i)}$, then $\\BR_{1,i}\\left( P_{A_2} \\right) = \\simplex{\\set{A}_1}$. \n\nLet the function $\\hat{u}: \\simplex{\\set{A}_2} \\to \\reals$ be such that for all $P \\in \\simplex{\\set{A}_2}$, \n\\begin{IEEEeqnarray}{rcl}\n\\label{EqHatu}\n\\hat{u}\\left( P \\right) & = & \\max_{Q \\in \\simplex{\\set{A}_1}} u\\left(Q,  P\\right),\n\\end{IEEEeqnarray}\nwhere the function $u$ is defined in~\\eqref{EqNormalFormU}. \n\nThe set of best strategies of \\Ptwo, under the assumption that \\Pone observes the commitment but not the actual action played by the leader, is formed by the probability measures in $\\simplex{\\set{A}_2}$ that minimize the function $\\hat{u}$ in~\\eqref{EqHatu}. \nUnder this assumption, if the probability measure $P \\in \\simplex{\\set{A}_2}$ is one of the best strategies of \\Ptwo, then it is also a strategy of \\Ptwo in at least one NE. Moreover, $\\hat{u}\\left( P \\right) $ is the payoff at the NE.\n\nFigure~\\ref{FigBellesFigures} depicts the functions  $\\hat{v}$ in~\\eqref{Eqv} and $\\hat{u}$ in~\\eqref{EqHatu}. Note that in all cases, the function $\\hat{v}$ is lower bounded by the function $\\hat{u}$. This implies that, when the follower is granted with an observation of the action played by the leader, even subject to noise, the payoff does not decrease, and in some cases, might significantly increase, as shown by the following lemma. \n\\begin{lemma}\\label{LemmaLowerBounds}\nLet the probability measures $P^{\\star}_{A_1} \\in \\simplex{\\set{A}_1}$ and $P^{\\star}_{A_2} \\in \\simplex{\\set{A}_2}$ form one of the NEs of the game $\\gameNF{\\matx{u}}$ in~\\eqref{EqTheGame}. \nFor all $P \\in \\simplex{\\set{A}_2}$, it holds that\n\\begin{IEEEeqnarray}{rcl}\\label{EqLBTricky}\n u(P_{A_1}^{\\star},P_{A_2}^{\\star})\n & \\leqslant & \\hat{u}(P)\\\\\n\\label{EqLBTrickyA}\n & \\leqslant & \\hat{v}\\left( P \\right)\\\\\n\\label{EqLBTrickyB}\n & \\leqslant &  \\sum_{k\\inCountTwo}P(a_k) \\left( \\max_{i\\inCountTwo}  u_{i,k} \\right),  \\IEEEeqnarraynumspace \\supersqueezeequ\n\\end{IEEEeqnarray}\nwhere the functions $u$, $\\hat{v}$, and $\\hat{u}$ are defined in~\\eqref{EqNormalFormU},~\\eqref{Eqv} and~\\eqref{EqHatu}, respectively.\n\\end{lemma}\nThe inequality in~\\eqref{EqLBTricky} follows from the definition of the function $\\hat{u}$ in~\\eqref{EqHatu}. The inequality in~\\eqref{EqLBTrickyA} shows that for all the strategies that \\Ptwo might adopt,  the payoff with noisy observations is larger than or equal to the payoff without observations. \nHence, even subject to noise, granting the follower with an observation of the action played by the leader is either beneficial or immaterial for the follower. \nThe inequality  in~\\eqref{EqLBTrickyB} holds with equality when \\Pone is always able to best respond to the actual action played by the leader. \nThis is for instance the case when the observation of the action played by the leader is noiseless.\n\n \n\n\\subsection{Equilibria}\n\nThe solution concept for the game  $\\game{G}\\left(\\matx{u}, \\matx{w} \\right)$ in~\\eqref{EqTheGameNoise} is defined hereunder.\n\\begin{definition}[Equilibrium]\\label{DefEquilibrium}\nThe tuple $\\left( P_{A_1| \\tilde{A}_2 = a_1}^{\\dagger}, P_{A_1| \\tilde{A}_2 = a_2}^{\\dagger}, P_{A_2}^{\\dagger}\\right) \\in \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_2}$ is said to form an equilibrium of the game $\\game{G}\\left(\\matx{u}, \\matx{w} \\right)$ in~\\eqref{EqTheGameNoise} if\n\\begin{IEEEeqnarray}{l}\nP_{A_2}^{\\dagger} \\in  \\arg  \\min_{P \\in \\simplex{\\set{A}_2}}\\hat{v}\\left( P \\right) \\supersqueezeequ \\IEEEeqnarraynumspace \\mbox{ and }\\\\\n\\label{EqBigMax}\n\\left( P_{A_1| \\tilde{A}_2 = a_1}^{\\dagger}, P_{A_1| \\tilde{A}_2 = a_2}^{\\dagger} \\right)  \\in  \\BR_1\\left( P_{A_2}^{\\dagger} \\right),\n\\end{IEEEeqnarray}\nwhere the function $\\hat{v}$ is in~\\eqref{Eqv} and the correspondence $ \\BR_1$ is in~\\eqref{EqBR1X}.\n\\end{definition}\n\nThe following theorem ensures the existence of an equilibrium for the game $\\game{G}\\left(\\matx{u}, \\matx{w} \\right)$ in~\\eqref{EqTheGameNoise}.\n\n\\begin{theorem}[Existence]\\label{TheoExistance}\nThe game $\\game{G}\\left(\\matx{u}, \\matx{w} \\right)$ in~\\eqref{EqTheGameNoise} always possesses an equilibrium.\n\\end{theorem}\n\nWhen the game $\\game{G}\\left(\\matx{u}, \\matx{w} \\right)$ in~\\eqref{EqTheGameNoise} possesses several equilibria, the payoff is identical at all equilibria, as shown by the following theorem.\n\n\n\\begin{theorem}[Equilibrium Payoff]\\label{TheoEquilibrium}\nLet the tuple $\\left( P_{A_1| \\tilde{A}_2 = a_1}^{\\dagger}, P_{A_1| \\tilde{A}_2 = a_2}^{\\dagger}, P_{A_2}^{\\dagger}\\right) \\in \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_2}$ form an equilibrium of the game $\\game{G}\\left(\\matx{u}, \\matx{w} \\right)$ in~\\eqref{EqTheGameNoise}. \nIf the matrix $\\matx{u}$ in~\\eqref{EqMatrixU} satisfies~\\eqref{EqMixedAssumption}, then\n\\begin{IEEEeqnarray}{rcl}\\label{Equ:f_1}\n\\hat{v}\\left( P_{A_2}^ \\dagger\\right)  & = & \\min\\lbrace \\hat{v}\\left( P_{1}\\right),\\hat{v}\\left( P_{2}\\right) \\rbrace,\n\\end{IEEEeqnarray}\nwhere, the function $\\hat{v}$ is defined in~\\eqref{Eqv}, and for all $i \\inCountTwo$, the probability measure $P_i \\in \\simplex{\\set{A}_2}$ is such that $P_{i}\\left( a_1 \\right) = P^{(i)}$, with $P^{(i)}$ in~\\eqref{EqPi}.\nAlternatively,  if the entries of the matrix $\\matx{u}$ satisfy~\\eqref{EqNotMixedAssumption}, then\n\\begin{IEEEeqnarray}{rcl}\n\\hat{v}\\left( P_{A_2}^ \\dagger\\right) & = & \\min \\left\\{ \\max \\left\\{ u_{1,1},  u_{2,1} \\right\\}, \\max \\left\\{ u_{1,2},  u_{2,2} \\right\\}\\right\\}. \\squeezeequ \\IEEEeqnarraynumspace\n\\end{IEEEeqnarray}\n\\end{theorem}\n\nWhen the payoff matrix $\\matx{u}$ in~\\eqref{EqMatrixU} satisfies~\\eqref{EqMixedAssumption}, at the equilibrium, \\Ptwo commits to a strategy that renders \\Pone indifferent to play any of its actions for at least one of the noisy observations.  That is, for some $j \\inCountTwo$, when at a given game repetition, \\Pone obtains $a_j$ as the noisy observation, it holds that $\\BR_{1,j}\\left( P_{A_2}^{\\dagger} \\right) = \\simplex{\\set{A}_1}$.\nThe following lemma sheds more light into this particularity.\n\n\n\\begin{lemma}\\label{LemmaEquilibriumEquality}\nLet $\\set{S} \\subseteq \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_2}$ be the set of equilibria of the game $\\game{G}\\left(\\matx{u}, \\matx{w} \\right)$ in~\\eqref{EqTheGameNoise}. \nLet the matrix $\\matx{u}$ in~\\eqref{EqMatrixU} satisfy~\\eqref{EqMixedAssumption}, and let $P^{\\star}_{A_{1}} \\in \\simplex{\\set{A}_1}$ and $P^{\\star}_{A_{2}} \\in \\simplex{\\set{A}_2}$ form a NE in the game $\\game{G}(\\matx{u})$ in~\\eqref{EqTheGame}.\nThen, there exists a tuple $ \\left(Q_1, Q_2, P \\right) \\in \\set{S}$ such that $P(a_1) \\in \\left\\{ P^{(1)}, P^{(2)}\\right\\}$. \nFurthermore, if $P(a_1) = P^{(i)}$, with $i \\in \\{1,2 \\}$, then it holds that  $P_{A_2| \\tilde{A}_2 = a_{i}} \\left( a_{1} \\right)  =  P^{\\star}_{A_{2}}  \\left( a_{1} \\right)$, \nwhere $P_{A_2|\\tilde{A}_2 = a_i}$ is in~\\eqref{EqPost}.\n\\end{lemma}\n\nLemma~\\ref{LemmaBreakingGrounds} and Lemma~\\ref{LemmaEquilibriumEquality} lead to a deeper conclusion. When the payoff matrix $\\matx{u}$ in~\\eqref{EqMatrixU} satisfies~\\eqref{EqMixedAssumption}, at the equilibrium, \\Ptwo (the leader) commits to a strategy such that for at least one $i \\inCountTwo$, the posterior  $P_{A_2 | \\tilde{A}_2 = a_i}$ in~\\eqref{EqPost} is equal to the strategy of \\Ptwo at the (unique) NE.  \n\nFinally, when the payoff matrix $\\matx{u}$ in~\\eqref{EqMatrixU} satisfies~\\eqref{EqNotMixedAssumption}, the payoff at the equilibrium is achieved by committing to a pure strategy. \nMore specifically, the payoff at the equilibrium of the game $\\game{G}\\left(\\matx{u}, \\matx{w}\\right)$ in~\\eqref{EqTheGameNoise} is the same as the payoff at the SE in pure strategies of the game $\\game{G}\\left(\\matx{u}\\right)$ in~\\eqref{EqTheGame}.\n\n\\subsection{Relevance of Noisy Observations}\n\nThe following lemma presents necessary and sufficient conditions under which the follower cannot benefit from the noisy observations.\n\n\\begin{lemma}\\label{LemmaEquilibriumEquality1}\nLet the tuple $\\left( P_{A_1| \\tilde{A}_2 = a_1}^{\\dagger}, P_{A_1| \\tilde{A}_2 = a_2}^{\\dagger}, P_{A_2}^{\\dagger}\\right) \\in \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_2}$ form an equilibrium of the game $\\game{G}\\left(\\matx{u}, \\matx{w} \\right)$ in~\\eqref{EqTheGameNoise}. Let also the tuple $\\left(P^{\\star}_{A_1}, P^{\\star}_{A_2}\\right) \\in \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_2}$ form one of the NEs of the game $\\gameNF{\\matx{u}}$ in~\\eqref{EqTheGame}. Then, \n\\begin{IEEEeqnarray}{rcl}\\label{Equ:f_5}\nv\\left( P_{A_1| \\tilde{A}_2}^ \\dagger, P_{A_2}^ \\dagger\\right)  & = &u(P_{A_1}^{\\star},P_{A_2}^{\\star})  , \n\\end{IEEEeqnarray}\nif and only if, $(a)$ the matrix $\\matx{u}$ in~\\eqref{EqMatrixU} satisfies~\\eqref{EqNotMixedAssumption}; or $(b)$ the matrix $\\matx{u}$ in~\\eqref{EqMatrixU} satisfies~\\eqref{EqMixedAssumption} and the channel $\\matx{w}$ in~\\eqref{EqTheChannel} satisfies $\\det{\\matx{w}} =0$.\n\\end{lemma}\nLemma~\\ref{LemmaEquilibriumEquality1} establishes that  granting the follower with noisy observations of the action played by the leader does not make any difference in two particular scenarios. \nFirst, in ZSGs with strategic dominance, NEs in pure strategies and infinitely many NEs (condition $(a)$).\nSecond, in ZSGs with unique NE in strictly mixed strategies when the observation given to the follower of the action of the leader is independent of the action actually played (condition $(b)$). Note that a channel that satisfies $\\det{\\matx{w}} =0$ is a channel whose mutual information between the channel input and channel output is zero.\n\nLemma~\\ref{LemmaLowerBounds} and Lemma~\\ref{LemmaEquilibriumEquality1} imply that granting the follower with relevant noisy observations of the action played by the leader makes a difference exclusively for ZSGs with a unique NE in mixed strategies. In this case, relevant noisy observations refer to observations obtained through channels with positive mutual information between the channel input and the channel output ($\\det{\\matx{w}}\\neq 0$). \n\nThe following lemma describes the special case of channels with maximum mutual information between the channel input and the channel output. That is, channels whose output is deterministic given the channel input. These channels satisfy the condition $\\abs{\\det\\matx{w}} = 1$.\n\n\\begin{lemma}\\label{LemmaPureEquilibriumEquality}\nLet  $\\left( P_{A_1| \\tilde{A}_2 = a_1}^{\\dagger}, P_{A_1| \\tilde{A}_2 = a_2}^{\\dagger}, P_{A_2}^{\\dagger}\\right) \\in \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_2}$ form an equilibrium of the game $\\game{G}\\left(\\matx{u}, \\matx{w} \\right)$ in~\\eqref{EqTheGameNoise}.  If $\\abs{\\det\\matx{w}} = 1$, then\n\\begin{IEEEeqnarray}{rcl}\\label{Equ:PureEquilibriumEquality}\n\\hat{v}\\left( P_{A_2}^ \\dagger\\right) & = & \\min \\lbrace \\max\\lbrace u_{1,1}, u_{2,1}\\rbrace,  \\max\\lbrace u_{1,2}, u_{2,2}\\rbrace \\rbrace. \\IEEEeqnarraynumspace\n\\end{IEEEeqnarray}\n\\end{lemma}\n %\nLemma~\\ref{LemmaPureEquilibriumEquality}  strengthens the observation that under perfect observations at each repetition, the strategy to which the leader commits to becomes irrelevant. \n\n\\section{COMMITMENT MISMATCH}\\label{SecCM} \nIn this section, the commitment observed by the follower  in the game $\\game{G}\\left(\\matx{u}, \\matx{w} \\right)$ in~\\eqref{EqTheGameNoise} is assumed to be  different from the actual commitment of the leader. This scenario is referred to as \\emph{commitment mismatch}.\n\n\\subsection{Game Formulation}\nLet $\\matx{t}$ be a given $2\\times2$ nonsingular stochastic matrix. \nLet $P_{A_2} \\in \\simplex{\\set{A}_2}$ be the commitment announced by \\Ptwo before the beginning of the game repetitions. The commitment observed by \\Pone is denoted by $\\tilde{P}_{A_2} \\in \\simplex{\\set{A}_2}$ and satisfies,\n\\begin{IEEEeqnarray}{rcl}\n\\label{EqPtilde}\n\\begin{pmatrix}\n\\tilde{P}_{A_2} (a_1)\\\\\n\\tilde{P}_{A_2} (a_2)\n\\end{pmatrix}\n= \n\\matx{t} \n\\begin{pmatrix}\nP_{A_2} (a_1)\\\\\nP_{A_2} (a_2)\n\\end{pmatrix}.\n\\end{IEEEeqnarray}\nThat is, the commitment observed by the follower is a deterministic distortion of the commitment announced by the leader. Note that the leader is not engaged on learning the commitment observed by the follower, nor the follower is engaged in learning the commitment announced by the leader, as in~\\cite{muthukumar_2019_robust}. Here, the follower assumes that $\\tilde{P}_{A_2}$ is the commitment actually announced by the leader, and the leader is aware of this.\nThe extension of the game $\\game{G}\\left(\\matx{u}, \\matx{w} \\right)$ in~\\eqref{EqTheGameNoise} obtained by including the binary channel represented by the stochastic matrix $\\matx{t}$ is described by  the tuple \n\\begin{IEEEeqnarray}{c}\n\\label{EqTheGameNoiseTilde}\n\\game{G}\\left(\\matx{u}, \\matx{w}, \\matx{t} \\right) = \\left(\\set{K}, \\set{A}_1, \\set{A}_2 , \\matx{u}, \\matx{w}, \\matx{t} \\right).\n\\end{IEEEeqnarray} \n\n\\subsection{The Best Strategies of \\Ptwo}\nLet the correspondence $\\tilde{v}: \\simplex{\\set{A}_2} \\rightarrow \\reals$ be such that given the probability measure $P_{A_2} \\in \\simplex{\\set{A}_2}$,\n\\begin{IEEEeqnarray}{rcl}\n\\label{Eqvtilde}\n\\tilde{v}\\left( P_{A_2} \\right) = \\max_{\\left(Q_1, Q_2\\right) \\in  \\BR_{1}\\left( \\tilde{P}_{A_2} \\right)} v\\left(Q_1, Q_2, P_{A_2} \\right),\n\\end{IEEEeqnarray}\nwhere the function $v$ is defined in~\\eqref{EqTheCostFunction}, the correspondence $\\BR_{1}$ is defined in~\\eqref{EqBR1}, and the probability measure $\\tilde{P}_{A_2}$ is in~\\eqref{EqPtilde}.\nThe correspondence $\\tilde{v}$ in~\\eqref{Eqvtilde} determines the payoff achieved by the players. \nThus, the set of best strategies for \\Ptwo, under the assumption that \\Pone observes a distorted commitment, is the set of probability measures that minimize $\\tilde{v}$ in~\\eqref{Eqvtilde}, if such a minimum exists. \nMore specifically, the cardinality of the set $\\BR_{1}\\left( \\tilde{P}_{A_2} \\right)$ in~\\eqref{Eqvtilde} is either one or infinite (Lemma~\\ref{LemmaBreakingBad1}). Hence, there might exist two tuples $\\left(Q_1, Q_2\\right) \\in  \\BR_{1}\\left( \\tilde{P}_{A_2} \\right)$ and $\\left(Q_3, Q_4\\right) \\in  \\BR_{1}\\left( \\tilde{P}_{A_2} \\right)$, for which $v\\left(Q_1, Q_2, P_{A_2} \\right) \\neq v\\left(Q_3, Q_4, P_{A_2} \\right)$. In this case, $\\tilde{v}\\left( P_{A_2} \\right)$ corresponds to a subset of $\\reals$ in which each element is induced by an element of the set  $\\BR_{1}\\left( \\tilde{P}_{A_2} \\right)$.\nFrom this perspective, the minimization of $\\tilde{v}$ might not be a well posed optimization problem.\n\nFor all $i \\inCountTwo$, let  $\\tilde{P}^{(i)} \\in \\reals$ be such that\n\\begin{IEEEeqnarray}{rcl}\n\\label{EqPiTilde}\n\\begin{pmatrix}\n\\tilde{P}^{(i)}\\\\\n1-\\tilde{P}^{(i)}\n\\end{pmatrix}\n=\n\\matx{t}^{-1}\n\\begin{pmatrix}\nP^{(i)}\\\\\n1-P^{(i)}\n\\end{pmatrix},\n\\end{IEEEeqnarray}\nwith $P^{(i)}$ in~\\eqref{EqPi}. \nUsing this notation, the following lemma shows an explicit expression for the correspondence $\\tilde{v}$ in a special case.\n\\begin{lemma}\\label{LemmaPayoffVP2MIsmatch}\nAssume that the matrix $\\matx{u}$ in~\\eqref{EqMatrixU} satisfies~\\eqref{EqMixedAssumption} and $ u_{1,1} - u_{1,2} -u_{2,1} + u_{2,2} >0$. \nAssume also that $\\det{\\matx{w}}>0$ and $\\det{\\matx{t}}>0$, with $\\matx{w}$ in~\\eqref{EqTheChannel} and $\\matx{t}$ in~\\eqref{EqTheGameNoiseTilde}. \nFor all $P  \\in \\simplex{\\set{A}_2}$, it holds that:\nIf $P(a_1) >  \\tilde{P}^{(2)}$, with $\\tilde{P}^{(2)}$ in~\\eqref{EqPiTilde}, then it follows that \n\\begin{IEEEeqnarray}{rcccl}\\label{EqVP2Mismatch_1}\n\\tilde{v}\\left( P \\right)  & = &  u_{1,1} P(a_1)  +  u_{1,2} P(a_2).\n \\end{IEEEeqnarray}\nIf $P(a_1) =  \\tilde{P}^{(2)}$, then it follows that \n\\begin{IEEEeqnarray}{l}\\label{EqVP2Mismatch_2}\n\\nonumber\n\\tilde{v}\\left( P \\right)\n  =  \\bigg\\lbrace   \\left(u_{1,1} P(a_1)+ u_{1,2}P(a_2)\\right) \\beta \\\\\n +  \\left(  \\begin{pmatrix}\n1 \\\\\n1\n \\end{pmatrix}^\\sfT \n \\left(\\matx{u} \\circ \\matx{w} \\right) \n \\begin{pmatrix}\n P(a_1) \\\\\n P(a_2)\n \\end{pmatrix}\n  \\right) \\left( 1 -\\beta \\right) : \\beta \\in [0,1] \\bigg\\rbrace.\\supersqueezeequ \\IEEEeqnarraynumspace \n \\end{IEEEeqnarray} \nIf  $\\tilde{P}^{(1)} <  P(a_1) <  \\tilde{P}^{(2)} $, then  it follows that \n\n \n \\begin{IEEEeqnarray}{l}\n\\tilde{v}\\left( P \\right) \\supersqueezeequ \n = \\left(  \\begin{pmatrix}\n1 \\\\\n1\n \\end{pmatrix}^\\sfT \n \\left(\\matx{u} \\circ \\matx{w} \\right) \n \\begin{pmatrix}\n P(a_1) \\\\\n P(a_2)\n \\end{pmatrix}\n  \\right) . \\label{EqVP2Mismatch_3} \\supersqueezeequ \\IEEEeqnarraynumspace\n\\end{IEEEeqnarray}\n\nIf  $P(a_1) =  \\tilde{P}^{(1)}$, with $\\tilde{P}^{(1)}$ in~\\eqref{EqPiTilde},  then it follows that \n\\begin{IEEEeqnarray}{l}\\label{EqVP2Mismatch_4}\n\\nonumber\n\\tilde{v}\\left( P \\right)\n  =  \\bigg\\lbrace   \\left(u_{2,1} P(a_1)+ u_{2,2}P(a_2)\\right) \\left(1-\\beta\\right) \\\\\n +  \\left(  \\begin{pmatrix}\n1 \\\\\n1\n \\end{pmatrix}^\\sfT \n \\left(\\matx{u} \\circ \\matx{w} \\right) \n \\begin{pmatrix}\n P(a_1) \\\\\n P(a_2)\n \\end{pmatrix}\n  \\right) \\beta  : \\beta \\in [0,1] \\bigg\\rbrace.\\supersqueezeequ \\IEEEeqnarraynumspace \n \\end{IEEEeqnarray} \nIf $P(a_1) <  \\tilde{P}^{(1)}$, then it follows that \n\\begin{IEEEeqnarray}{rcccl}\n\\tilde{v}\\left( P \\right)  & = & u_{2,1} P(a_1) + u_{2,2} P(a_2).\\label{EqVP2Mismatch_5}\n\\end{IEEEeqnarray}\n\\end{lemma}\nFigure~\\ref{FigBellesFigures} depicts the correspondence $\\tilde{v}$ in~\\eqref{Eqvtilde}. \nNote that for all $i \\inCountTwo$, if the commitment of the leader $P_{A_2}$ satisfies \n$P_{A_2}(a_1) = \\tilde{P}^{(i)}$,  it holds that $\\tilde{v}\\left(P_{A_2} \\right)$ is a closed interval. \nFigure~\\ref{FigBellesFigures} also depicts the existence of some commitments of \\Ptwo for which the expected payoff is smaller than the expected payoff achieved when no distortion of the commitment is considered. \nThe following lemma formalizes this observation and shows that even a deterministic distortion of the commitment can benefit the leader in particular cases.\n\\begin{lemma}\\label{LemmaMixMismatch}\nConsider the following assumptions: \n$(a)$~The matrix $\\matx{u}$ in~\\eqref{EqMatrixU} satisfies~\\eqref{EqMixedAssumption}; \n$(b)$~ For all $i \\inCountTwo$, the probability measures $Q_i \\in \\simplex{\\set{A}_2}$ such that $Q_{i}\\left( a_1 \\right) = P^{(i)}$, with $P^{(i)}$ in~\\eqref{EqPi}, satisfy $\\hat{u}\\left( Q_{1}\\right) \\neq \\hat{u}\\left( Q_{2}\\right)$, \nwhere the function $\\hat{u}$ is in~\\eqref{EqHatu}.\nIf $ \\det{\\matx{t}} \\notin \\lbrace 0, 1 \\rbrace$, then, there exists a strategy $ P \\in \\simplex{\\set{A}_2}$ such that $\\tilde{v}(P) < \\hat{v}(P_{A_2}^\\dagger)$, \nwhere the tuple $\\left( P_{A_1| \\tilde{A}_2 = a_1}^{\\dagger}, P_{A_1| \\tilde{A}_2 = a_2}^{\\dagger}, P_{A_2}^{\\dagger}\\right) \\in \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_2}$ is an equilibrium of the game $\\game{G}\\left(\\matx{u}, \\matx{w} \\right)$ in~\\eqref{EqTheGameNoise}. \n\\end{lemma}\n\n\\subsection{Equilibria}\n\nThe solution concept for the game $\\game{G}\\left(\\matx{u}, \\matx{w}, \\matx{t} \\right)$ in~\\eqref{EqTheGameNoiseTilde} is the following.\n\n\\begin{definition}[Equilibrium]\\label{DefEquilibriumTilde}\nThe tuple $\\left( P_{A_1| \\tilde{A}_2 = a_1}, P_{A_1| \\tilde{A}_2 = a_2}, P_{A_2}\\right) \\in \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_1} \\times \\simplex{\\set{A}_2}$ is said to form an equilibrium of the game  $\\game{G}\\left(\\matx{u}, \\matx{w}, \\matx{t} \\right)$ in~\\eqref{EqTheGameNoiseTilde}  if\n\\begin{IEEEeqnarray}{l}\nP_{A_2} \\in  \\arg  \\min_{P \\in \\simplex{\\set{A}_2}}\\tilde{v}\\left( P \\right) \\supersqueezeequ \\IEEEeqnarraynumspace \\mbox{ and }\\\\\n\\label{EqBigMaxTintin}\n\\left( P_{A_1| \\tilde{A}_2 = a_1}, P_{A_1| \\tilde{A}_2 = a_2} \\right)  \\in  \\BR_1\\left( \\tilde{P}_{A_2} \\right),\n\\end{IEEEeqnarray}\nwhere the correspondence $\\tilde{v}$ is in~\\eqref{Eqvtilde}, the correspondence $ \\BR_1$ is in~\\eqref{EqBR1X}, and the probability measures $P_{A_2}$ and $\\tilde{P}_{A_2}$ satisfy~\\eqref{EqPtilde}.\n\\end{definition}\nThe game $\\game{G}\\left(\\matx{u}, \\matx{w}, \\matx{t} \\right)$ in~\\eqref{EqTheGameNoiseTilde} does not necessarily possess an equilibrium.  This is due to the fact that the minimum of $\\tilde{v}$ in~\\eqref{Eqvtilde} does not always exist, as shown hereunder. \n\nFor all $i \\inCountTwo$, let the measure $\\tilde{P}_{i} \\in \\simplex{\\set{A}_2}$ be such that $\\tilde{P}_{i}(a_1) = \\tilde{P}^{(i)}$. \nLet the function $\\omega: \\simplex{\\set{A}_2} \\to \\reals$ be defined for all $P \\in \\simplex{\\set{A}_2}$ as follows:\nIf $P(a_1) >  \\tilde{P}^{(2)}$, with $\\tilde{P}^{(2)}$ in~\\eqref{EqPiTilde}, then\n$\\omega\\left( P \\right)  =   u_{1,1} P(a_1)  +  u_{1,2} P(a_2)$.\nIf  $\\tilde{P}^{(1)} \\leqslant  P_{A_2}(a_1)  \\leqslant  \\tilde{P}^{(2)} $, then\n$\\omega\\left( P \\right)   = \\left( u_{1,1} P_{\\tilde{A}_2 | A_2 = a_1}(a_1) + u_{2,1} P_{\\tilde{A}_2 | A_2 = a_1}(a_2)\\right)P(a_1)  +  \\left( u_{1,2}P_{\\tilde{A}_2 | A_2 = a_2}(a_1) + u_{2,2}P_{\\tilde{A}_2 | A_2 = a_2}(a_2) \\right) P(a_2)$. \nIf $P(a_1) <  \\tilde{P}^{(1)}$, then \n$\\omega\\left( P \\right)  =  u_{2,1} P(a_1) + u_{2,2} P(a_2)$. \n\nIn the example of Figure~\\ref{FigBellesFigures}, for all $P \\in \\simplex{\\set{A}_2}\\setminus\\lbrace \\tilde{P}_{1}, \\tilde{P}_{2} \\rbrace$, it holds that $\\tilde{v}(P) = \\omega(P)$, \nand the function $\\omega$ is discontinuous at $\\tilde{P}_{1}$ and $\\tilde{P}_{2}$ with a minimum at $\\tilde{P}_{2}$ (magenta triangle). \nNote that $\\tilde{v}(\\tilde{P}_1)$ and $\\tilde{v}(\\tilde{P}_2)$ are intervals, and thus, the minimization of  $\\tilde{v}$ does not have a solution. \nEssentially, if \\Ptwo (the leader) commits to play the probability measure $\\tilde{P}_{A_2}$ that minimizes $\\omega$, i.e., $P_{A_2} = \\tilde{P}_{2}$, then the set of the best responses of \\Pone when the noisy observation is $a_{2}$ is  $\\BR_{1,2}(\\tilde{P}_{A_2}) =\\simplex{\\set{A}_1}$, where the probability measures $P_{A_2}$ and $\\tilde{P}_{A_2}$ satisfy~\\eqref{EqPtilde}. \nThat is, for each game repetition in which the output of the channel is $a_{2}$, \\Pone might choose its action by sampling any probability measure in $\\simplex{\\set{A}_1}$ and achieve an expected payoff in the interval $\\tilde{v}(P_{2})$. If such a payoff is larger than $\\omega(P_{A_2})$, then \\Ptwo can deviate and obtain a payoff arbitrarily close to $\\omega(P_{A_2})$. This shows the non existence of an equilibrium in the example of Figure~\\ref{FigBellesFigures}.\n\nNote that the equilibrium exists for the cases in which the strategy of \\Pone is independent of the commitment, e.g., ZSG with strategic dominance.\n\\begin{figure}\n\\centering\n            \\includegraphics[width=\\linewidth]{Figures\/1.pdf}\n\n\n\n\n\\caption{Plots of the functions $\\hat{v}$ in~\\eqref{Eqv} and $\\hat{u}$ in~\\eqref{EqHatu}; and the correspondence $\\tilde{v}$ in~\\eqref{Eqvtilde} as a function of the probability $P_{A_2}(a_1)$, with parameters $\\matx{u} =\\left( -8, 6 ; 2, -2\\right)$, $\\matx{w} = \\left( 0.8,  0.2; 0.2, 0.8\\right)$ and  $\\matx{t} = \\left( 0.9,  0.1; 0.1, 0.9\\right)$.\nThe tuple $(P_{A_1}^{\\star},P_{A_2}^{\\star})$ is the unique NE  in~\\eqref{EqNEStratExample} and for all $i \\inCountTwo$,  $P_i\\left(a_1 \\right)= P^{(i)}$, with $P^{(i)}$ in~\\eqref{EqPi} and $\\tilde{P}_i\\left(a_1 \\right)= \\tilde{P}^{(i)}$, with $\\tilde{P}^{(i)}$ in~\\eqref{EqPiTilde}.\n}\n\\label{FigBellesFigures}\n\\end{figure}\n\\subsection{Equilibrium Refinements}\n \nIn the example in Figure~\\ref{FigBellesFigures}, the function $\\omega$, which can be minimized, is obtained from the correspondence $\\tilde{v}$ in~\\eqref{Eqvtilde} by replacing the closed intervals $\\tilde{v}(\\tilde{P}_1)$ and $\\tilde{v}(\\tilde{P}_2)$ by the real numbers $\\min\\tilde{v}(\\tilde{P}_1)$ and $\\min\\tilde{v}(\\tilde{P}_2)$, respectively. Hence, the correspondence $\\tilde{v}$ and the function $\\omega$ are identical if \\Pone is forced to choose the strategy that minimizes the expected payoff every time it observes that \\Ptwo commits either to the strategies  $\\tilde{P}_1$ or $\\tilde{P}_2$.\nUnder this assumption, the game $\\game{G}\\left(\\matx{u}, \\matx{w}, \\matx{t} \\right)$ in Figure~\\ref{FigBellesFigures} possesses an equilibrium in which \\Ptwo commits to play  the strategy $P_{A_2}$ that satisfies $P_{A_2} = \\tilde{P}_{2}$.\nThis observation is reminiscent of the equilibrium refinements proposed in~\\cite{Leitmann-JOTA-1978} for the SE of bi-matrix games, i.e., the \\emph{strong}-SE. \n\nAnother refinement of the solution concept in Definition~\\ref{DefEquilibriumTilde} can be obtained when the leader commits to a strategy $P_{A_2}$ that satisfies $\\tilde{v}(P_{A_2}) = \\omega(\\tilde{P}_{2}) + \\epsilon$, with $\\epsilon > 0$ arbitrarily small. This refinement is reminiscent of the solution concept known as $\\epsilon$-equilibrium for the NE~\\cite{Fudenberg-Tirole-Book}. In this case, the leader admits to committing to a suboptimal strategy in order to be able to force a unique (and predictable) best response from its opponent. \n \n\\section{FINAL REMARKS AND DISCUSSION}\\label{SecDiscussion}\n\nThe analysis of ZSG with commitments in which the follower is granted with a noisy observation of the action and the commitment of the leader have been studied following a Bayesian approach. This approach relies on the capability of the follower to construct posterior probability measures on the actions of the leader based on the available information. The construction of posteriors is more general than the notion of incomplete information in the extensive form of ZSGs, which is limited to modelling the inability of players to distinguish between elements of the \\emph{information sets} (maximum entropy posteriors) \\cite{von_1947_theory}. \nNote also that noisy observations cannot be modelled using Bayesian games as introduced in \\cite{harsanyi_1967_games1, harsanyi_1968_games2}.\nThis new game formulation is shown to always posses an equilibrium under the assumption that the commitment is observed perfectly.  When this assumption is dropped, the game is shown to have an equilibrium only under strict conditions. Despite that commitment mismatch can significantly benefit the leader, such benefits are not achievable at a stable point. To benefit from commitment mismatch, the leader must admit to commit to a suboptimal strategy in order to be able to unequivocally  predict the best response of its opponent. \n\nThis work relies on the assumption that actions are observed through discrete channels for which the sets of channel inputs and channel outputs are finite and identical.  Nonetheless, these channels fail to model many  typical data processing impairments, which calls for more elaborate channel models. For instance, the effect of erasures is modelled by the \\emph{erasure channel} \\cite{Elias-1956}; and the effect of additive white Gaussian noise (AWGN) is modelled by the AWGN channel \\cite{Shannon-1948, Shannon-1948_2}. Nonetheless, the extension of this work to these and other channel models is not trivial. \nFinally, it is important to highlight that the conclusions of this work hold under the assumption that both players are aware of the existence of a channel through which actions are observed. Moreover, such a channel is assumed to be known by both players. \n\n\\balance\n \\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}\n\n\nAccurate camera relocalization plays an important role in autonomous driving.  Given query images, the camera relocalization task aims at estimating the camera poses for which the images are taken. In recent years, many camera relocalization approaches have been proposed. Generally speaking, they fall into two categories.\n\\begin{enumerate}\n    \\item One solution is to treat relocalization as a matching task. This solution assumes the availability of a database or a map that stores prior information (e.g., 3D point clouds, images, or descriptors) of sample points. Given a query image, the matching model finds the best match between the query and the database based on a similarity score. The estimated camera pose is then inferred from the matched prior information in the database.\n   \n    \\item Another solution does not assume the availability of a database and uses only neural networks to regress the camera poses of the query images. This approach constructs an implicit relation between images and poses, which is called \\emph{camera pose regression} (CPR).\n\\end{enumerate}\nThe prerequisite of a database on the one hand can boost the accuracy of the camera relocalization by storing useful prior information. On the other hand, the computation and storage requirements are proportionate to the number of sample points in the database. To decouple relocalization from the need for a database, there has been a recent surge of research interest in the second category CPR. \n\n\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics[width=0.47\\textwidth]{figures\/neighboring.pdf}\n\\end{center}\n\\caption{Multi-view camera pose regression with neighboring information, without the need for any database. }\n\\label{fig:neighboring}\n\\end{figure}\n\nThe pioneering work PoseNet \\cite{posenet} uses a convolutional neural network (CNN) to extract features from a single image as vector embeddings, and the embeddings are directly regressed to the 6-DoF poses. To further improve the regression performance in driving scenarios, multi-view-based models extend the input from a single image to multi-view images. MapNet \\cite{mapnet} leverages pre-computed visual odometry to post-process the output pose trajectory. GNNMapNet \\cite{gnnmapnet} integrates a graph neural network (GNN) into CNN to make image nodes interact with neighbors.\n\nThe above-mentioned multi-view-based models show promising performance in benign driving environments. To operate well in challenging environments, the model must be robust to environmental perturbations (e.g., changing seasons, weather, illumination, and unstable objects), and effectively leverage neighboring information from spatially or temporally nearby frames of a single vehicle or multi-view images shared from other spatially nearby agents (e.g., using V2X communication) as shown in \\cref{fig:neighboring}. Images sharing such neighboring information are said to be \\emph{covisible}.\n\nRecently, neural Ordinary Differential Equations (ODEs)  \\cite{chen2018neural}  and Partial Differential Equations (PDEs) \\cite{chamberlain2021grand, chamberlain2021blend}  have demonstrated their robustness against input perturbations  \\cite{yan2019robustness,kang2021Neurips}. Moreover, GNNs can effectively aggregate neighborhood information. We thus propose RobustLoc that not only explores the relations between graph neighbors but also utilizes neural differential equations to improve robustness. We test our new multi-view-based model on three challenging autonomous driving datasets and verify that it outperforms existing state-of-the-art (SOTA) CPR methods.\n\nOur main contributions are summarized as follows:\n\\begin{enumerate}\n\n\\item \nWe represent the features extracted from a CNN in a graph and apply graph neural diffusion layers at each stage. I.e., we design feature diffusion blocks at both the feature map extraction and vector embedding stages to achieve robust feature representations. Each diffusion block consists of not only cross-diffusion from node to node in a graph but also self-diffusion within each node. We also propose multi-level training with the branched decoder to better regress the target poses.\n\n\n\\item We conduct experiments in both ideal and challenging noisy autonomous driving datasets to demonstrate the robustness of our proposed method. The experiments verify that our method achieves better performance than the current SOTA CPR methods. \n\n\\item We conduct extensive ablation studies to provide insights into the effectiveness of our design. \n\\end{enumerate}\n\n\n\n\\section{Related Work}\n\n\n\n\\subsection{Image Matching}\nNetVLAD \\cite{netvlad} integrates CNN into image retrieval, where a trainable VLAD layer is proposed to implicitly split images into different clusters. SuperGlue \\cite{sarlin2020superglue}  assumes 2D image key points are available, and point-wise matching is achieved by an attentional graph network and the Sinkhorn algorithm. Pixloc \\cite{pixloc} integrates 3D point clouds into image matching, which estimates the pose by minimizing the reprojection error of the 3D points. \n\n\\subsection{Camera Pose Regression}\nGiven the query images, CPR models directly regress the camera poses of these images without the need for a database. Thus, it does not depend on the scale of the database, which is definitely a born gift compared with those database methods.  \\\\\nPoseNet \\cite{posenet} and GeoPoseNet \\cite{geoposenet2017} propose the simultaneous learning for location and orientation by integrating balance parameters. MapNet \\cite{mapnet} uses visual odometry to serve as the post-processing technique to optimize the regressed poses. LsG \\cite{lsg} and LSTM-PoseNet \\cite{lstmpose} integrates the sequential information by fusing PoseNet and LSTM. AD-PoseNet and AD-MapNet \\cite{adposenet} leverages the semantic masks to drop out the dynamic area in the image. AtLoc \\cite{atloc} introduces the global attention to guide the network to learn better representation. GNNMapNet \\cite{gnnmapnet} expands the feature exploration from a single image to multi-view images using GNN. IRPNet \\cite{irpnet} proposes to use two branches to regress translation and orientation respectively. Coordinet \\cite{coordinet} uses the coordconv \\cite{coordconv} and weighted average pooling \\cite{fc4} to capture spatial relations. \n\n\n\n\\subsection{Neural Differential Equations and Robustness}\n\nThe dynamics of a system are usually described by ordinary or partial differential equations. The paper \\cite{chen2018neural} first proposes trainable neural ODEs by parameterizing the continuous dynamics of hidden units. The hidden state of the ODE network is modeled as:\n\\begin{align}\n\\ddfrac{\\bm{y}(t)}{t}=f_{\\theta}(\\bm{y}(t)) \\label{eq:ode_f}\n\\end{align}\nwhere  $\\bm{y}(t)$ denotes the latent state of the trainable network $f_{\\theta}$ that is parameterized by weights $\\theta$. Recent studies \\cite{yan2019robustness,kang2021Neurips} have demonstrated that neural ODEs are intrinsically more robust against input perturbations compared to vanilla CNNs. \n\n\nIn addition, neural PDEs \\cite{chamberlain2021grand, chamberlain2021blend} have been proposed and applied to GNN, where the diffusion process is modeled on the graph. Furthermore, the stability of the heat semigroup and the heat kernel under perturbations of the Laplace operator (i.e., local perturbation of the manifold) is studied \\cite{SonKanWan:C22}.\n\n\n\\section{Proposed Model}\n\n\nIn this section, we provide a detailed description of our proposed CPR approach.\nWe assume that the input is a set of images $\\{\\bm{I}_{i}\\}_{i\\in [N]}$ that may be covisible (see \\cref{fig:neighboring}).\\footnote{\\emph{Notations:} In this paper, we use $[N]$ to denote the set of integers $\\{1, 2, \\ldots, N\\}$.We use boldfaced lowercase letters like $\\bm{m}$ to denote vectors and boldface capital letters like $\\bm{W}$ to denote matrices.} Our objective is to perform CPR on the input images.\n\n\n\n\n\n\n\n\n\n\\begin{figure*}[ht]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{figures\/architecture.pdf}\n\\end{center}\n\\caption{The main architecture of RobustLoc. Feature diffusion is performed at both the feature map stage and the vector embedding stage. The branched decoder regresses the 6-DoF poses based on the vector embeddings or the pooled feature maps. The details for multi-layer decoding are shown in \\cref{fig:multi-level}. }\n\\label{fig:model}\n\\end{figure*}\n\n\\subsection{RobustLoc Overview}\nWe first summarize our multi-view CPR pipeline, which can be decomposed into three different stages, as follows (see \\cref{fig:model} and \\cref{fig:multi-level}): \n\\begin{enumerate}\n\\item Given $N$ neighboring images, a CNN extracts the feature maps of all these images. Our proposed feature map diffusion block then performs cross-self diffusion on the feature maps.\n\\item After feature map diffusion, a global average pooling module aggregates the feature maps as vector embeddings, which contain global representations of these images. Similarly, those vector embeddings are then diffused by cascaded diffusion blocks. \n\\item Based on the vector embeddings, the branched decoder module regresses the output camera poses. During training, decoding is performed on multiple levels to provide better feature constraints.\n\\end{enumerate}\n\n\n\n\\subsection{Neural Diffusion for Feature Maps}\n The input images $\\{\\bm{I}_{i}\\}_{i\\in [N]}$ are passed through a CNN to obtain the feature maps $ \\{\\bm{m}_{i} \\in \\mathbb{R}^{H \\times W \\times C } \\}_{i\\in [N]}$. Here, $C$ is the channel dimension, while $H$ and $W$ are the dimensions of a feature map. For each feature map $\\bm{m}_i$, we denote its $j$-th element as $\\bm{m}_{i,j}\\in \\mathbb{R}^{C} , j\\in [HW]$. We next describe the feature map diffusion block, where we perform cross-diffusion from node to node in a graph, and self-diffusion within each node. The two diffusion processes update the feature map by leveraging the neighboring information or only using each node's individual information, respectively.\n\\subsubsection{Cross-Diffusion Dynamics.}\nTo support the cross-diffusion over feature maps, we formulate the first graph in our pipeline as:\n\\begin{align}\n\\mathcal{G}^{\\mathrm{feat}}=(\\mathcal{V}^{\\mathrm{feat}}, \\mathcal{E}^{\\mathrm{feat}}), \n\\end{align}\nwhere the node set $\\mathcal{V}^{\\mathrm{\\mathrm{feat}}}=\\{\\bm{m}_{i,j}\\}_{(i,j)\\in [N]\\times[HW]}$ contains element-wise features $\\bm{m}_{i,j}$ and the edge set $\\mathcal{E}^{\\mathrm{\\mathrm{feat}}}$ is defined as the complete graph edges associated with attention weights as discussed below. And the complete graph architecture is demonstrated to be an effective design shown in \\cref{tab:graph}.\n\nTo achieve robust feature interaction, we next define the cross-diffusion process as: \n\\begin{align}\n\\frac{\\partial}{\\partial t} \\bm{x}(t)\n& = f_{\\mathrm{cross}}(\\bm{x}(t)), \\label{eq:pde}\n\\end{align}\nwhere $f_{\\mathrm{cross}}(\\bm{x}(t))$ is a neural network and can be approximately viewed as a neural PDE  with the partial differential operations over a manifold space replaced by the attention modules that we will introduce later. We denote the input to the feature map diffusion module as the initial state at $t=t_{0}$ as $\\bm{x}(t_{0})=\\left\\{\\bm{m}_{i,j}\\right\\}_{(i,j)\\in [N]\\times[HW]}$, where $\\bm{x}(t)=\\left\\{\\bm{m}_{i,j}(t)\\right\\}_{(i,j)\\in [N]\\times[HW]}$ denotes the hidden state of the diffusion.\nThe diffusion process is known to have robustness against local perturbations of the manifold \\cite{chen1998stability} where the local perturbations in our CPR task include challenging weather conditions, dynamic street objects, and unexpected image noise. Therefore, we expect our module \\cref{eq:pde} is simultaneously capable of leveraging the neighboring image information and holding robustness against local perturbations.\n\n\nWe next introduce the computation of attention weights in $f_{\\mathrm{cross}}(\\bm{x}(t))$ for node features at time $t$. \nWe first generate the embedding of each node using multi-head fully connected (FC) layers with learnable parameter matrix $\\bm{W}_{k}$ and bias $\\bm{b}_{k}$ at each head $k=[K]$, where $K$ is the number of heads. The output at each head $k$ can be written as:\n\\begin{align}\n\\bm{m}_{i,j;k}^{\\mathrm{FC}}(t)=\\bm{W}_{k} \\bm{m}_{i,j}(t) + \\bm{b}_{k}.\n\\end{align}\nThe attention weights are then generated by computing the dot product among all the neighboring nodes using the features \n$\\bigl\\{\\bm{m}_{i,j;k}^{\\mathrm{FC}}(t)\\bigr\\}_{(i,j)\\in[N]\\times[HW]}$. We have\n\\begin{align}\n& \\{a_{(i,j),(i',j');k}(t)\\}_{(i',j')\\in \\mathcal{N}_{i,j}} \\nonumber\\\\ \n& =\\mathrm{Softmax}_{(i',j') \\in \\mathcal{N}(i,j)}(\\bm{m}_{i,j;k}^{\\mathrm{FC}}(t) \\cdot  \\bm{m}_{i', j';k}^{\\mathrm{FC}}(t)),\n\\end{align}\nwhere $\\mathcal{N}_{i,j}$ denotes the set of neighbors of node $\\bm{m}_{i,j}$.  \nLet\n\\begin{align}\n\\bm{m}_{i,j;k}^{\\mathrm{weighted}}(t)= \\sum_{(i',j') \\in \\mathcal{N}_{i,j}} a_{(i,j),(i',j');k}(t) \\bm{m}_{i',j';k}^{\\mathrm{FC}}(t).\n\\end{align} Finally, the updated node features are obtained by concatenating the weighted node features from all heads as \n\\begin{align}\nf_{\\mathrm{cross}}(\\bm{x}(t))=\\set*{ \\concat_{k\\in[K]}(\\bm{m}_{i,j;k}^{\\mathrm{weighted}}(t)) }_{(i,j)\\in [N]\\times[HW]}.\n\\end{align}\n\nBased on the above pipeline, the output of the cross-diffusion at time $t=t_{1}$ can be obtained as:\n\n\\begin{align}\n\\bm{x}(t_{1})=F_{\\mathrm{cross}}(\\bm{x}(t_{0})),\n\\end{align}\nwhere $F_{\\mathrm{cross}}(\\cdot)$ denotes the solution of \\cref{eq:pde} integrated from $t=t_{0}$ to $t=t_{1}$. \n\n\\subsubsection{Self-Diffusion Dynamics.} In the next step, we update each node feature independently. The node-wise feature update can be regarded as a rewiring of the complete graph to an edgeless graph, and the node-wise feature update is described as:\n\\begin{align}\n\\ddfrac{\\bm{m}_{i,j}(t)}{t}=f_{\\mathrm{self}}(\\bm{m}_{i,j}(t))=\\mathrm{MLP}(\\bm{m}_{i,j}(t)).\n\\label{eq:NODE}\n\\end{align}%\nAnd the output of self-diffusion can be obtained as:\n\\begin{align}\n\\bm{m}_{i,j}(t_{2})=F_{\\mathrm{self}}(\\bm{m}_{i,j}(t_{1})).\n\\end{align}\nwhere $F_{\\mathrm{self}}(\\cdot)$ denotes the solution of \\cref{eq:NODE} integrated from $t=t_{1}$ to $t=t_{2}$. \nAs neural ODEs are robust against input perturbations  \\cite{yan2019robustness,kang2021Neurips}, we expect the updating of each node feature according to the self-diffusion \\cref{eq:NODE} to be robust against perturbations like challenging weather conditions, dynamic street objects, and image noise.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Vector Embeddings and Diffusion}\nAfter the feature map neural diffusion, we feed the updated feature maps into a global average pooling module to generate the vector embeddings $\\{ \\bm{h}_{i}\\in \\mathbb{R}^{C} \\}_{i\\in [N]}$, where\n\\begin{align}\n\\bm{h}_{i}=\\mathrm{Pooling} (\\bm{m}_{i}) .\n\\end{align}\nEach vector embedding contains rich global representations for the input image together with the information diffused from the neighboring images. To enable diffusion for the global information, we propose to design the vector embedding graph as:\n\\begin{align}\n\\mathcal{G}^{\\mathrm{vect}}=(\\mathcal{V}^{\\mathrm{vect}}, \\mathcal{E}^{\\mathrm{vect}}),  \n\\end{align}\nwhere the node set $\\mathcal{V}^{\\mathrm{vect}}=\\{\\bm{h}_{i}\\}_{i\\in[N]}$ contains image vector embeddings $\\bm{h}_{i}$ and the edge set $\\mathcal{E}^{\\mathrm{vect}}$ is also defined to be the complete graph. Based on this graph $\\mathcal{G}^{\\mathrm{vect}}$, we construct the cascaded diffusion blocks, to perform global information diffusion. Within the cascaded blocks, each basic diffusion block consists of two diffusion layers: a cross-diffusion layer and a self-diffusion layer, similar to the two diffusion schemes introduced at the feature map diffusion phase.\n\n\\subsection{Pose Decoding}\nIn this subsection, we explain the pose decoding operations.\n\n\\subsubsection{Branched Pose Decoder.}\\label{decoder}\n\nEach camera pose $\\bm{p}=\\{\\bm{d},\\bm{r}\\}\\in\\mathbb{R}^{6}$, consists of a 3-dimensional translation $ \\bm{d}\\in\\mathbb{R}^{3}$ and a 3-dimensional rotation $ \\bm{r}\\in \\mathbb{R}^{3}$. Thus CPR can be viewed as a multi-task learning problem. However, since the translation and rotation elements of $\\bm{p}$ do not scale compatibly, the regression converges in different basins. To deal with it, previous methods consider regression for translation and rotation respectively and demonstrate it is an effective way to improve performance \\cite{irpnet}. In our paper, we also follow this insight to design the decoder.\n\nFirstly, the feature embeddings $\\{ \\bm{h}_{\\bm{d}},  \\bm{h}_{\\bm{r}}\\}$ for translation and rotation are extracted from the feature embedding $\\bm{h}$ using different non-linear MLP layers as:\n\n\\begin{align}\n\\bm{h}_{\\bm{d}}=\\mathrm{MLP}_{\\bm{d}}(\\bm{h}),\n\\end{align}\n\\begin{align}\n\\bm{h}_{\\bm{r}}=\\mathrm{MLP}_{\\bm{r}}(\\bm{h}),\n\\end{align}\nThus, the features of translation and rotation are decoupled. Next in the second stage, the pose output can be regressed as:\n\n\\begin{align}\n\\bm{p}=\\bm{W} (\\bm{h}_{\\bm{d}}\\concat\\bm{h}_{\\bm{r}}) + \\bm{b}\n\\end{align}\nwhere $\\bm{W},\\bm{b}$ are learnable parameters. During training, we compute the regression loss of decoded poses from multiple levels, which we will introduce below. During inference, we use the decoded pose from the last layer as the final output pose.\n\n\n\n\n\n\n\n\n\n\\subsubsection{Multi-level Pose Decoding Graph.}  \nTo better regularize the whole regression pipeline, we propose to leverage the feature maps at multiple levels. As shown in \\cref{fig:multi-level}, at the vector embedding stage, we use the vector embeddings to regress the poses, while at the feature map stage, we use the feature maps. Denoting the feature maps at layer $l$ as $\\{ \\bm{m}_{i}^{l} \\in \\mathbb{R}^{H \\times W \\times C} \\}_{i\\in[N]}$, the pose decoding graph at layer $l$ can be formulated as: \n\\begin{align}\n\\mathcal{G}^{\\mathrm{pose},l}=(\\mathcal{V}^{\\mathrm{pose},l},\\mathcal{E}^{\\mathrm{pose},l}),\n\\end{align}\nwhere edge set $\\mathcal{E}^{\\mathrm{pose},l}$ is defined to be connected with two spatially adjacent nodes which can be viewed as the odometry connection, while the node set $\\mathcal{V}^{\\mathrm{pose},l}$ is defined depending on layers since the information used to regress poses is different:\n\\begin{align}\n\\mathcal{V}^{\\mathrm{pose},l}=\n\\left\\{\n\\begin{array}{ll}\n\\{ \\bm{h}_{i} \\}_{i\\in[N]} & \\text{if}\\  l=L, \\\\\n\\{ \\bm{m}_{i}^{l} \\}_{i\\in[N]} & \\text{otherwise}, \\\\\n\\end{array}\n\\right.\n\\end{align}\nwhere $L$ represents the last layer in our network.\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics[width=0.42\\textwidth]{figures\/multi-level.pdf}\n\\end{center}\n\\caption{Multi-level pose decoding. Decoding can be directly applied to vector embeddings. Feature maps are first pooled and then decoded.  }\n\\label{fig:multi-level}\n\\end{figure}\n\nAt the last layer where there are vector embeddings, we can directly apply the pose decoder to generate absolute pose messages. By contrast, at feature map layers,  we first apply a global average pooling module on the feature maps to formulate feature vectors, and pose messages can be obtained using the  pose decoder:\n\\begin{align}\n\\bm{p}^{l}_{i}=\n\\left\\{\n\\begin{array}{l l}\nf_{\\mathrm{decoder}}^{l}( \\bm{h}^{l}_{i}) & \\text{if}\\  l=L, \\\\\nf_{\\mathrm{decoder}}^{l}( \\mathrm{Pooling}(\\bm{m}^{l}_{i})) & \\text{otherwise}. \\\\\n\\end{array}\n\\right.\n\\end{align}\nwhere $f_{\\mathrm{decoder}}^{l}(\\cdot)$ is the pose decoder at layer $l$. Using the simplified relative pose computation technique in \\cite{atloc}, the relative pose messages $\\bm{p}^{l}_{i,i'}$ at layer $l$ can be generated as:\n\\begin{align}\n\\bm{p}^{l}_{i,i'} = \\bm{p}^{l}_{i'} - \\bm{p}^{l}_{i}.\n\\end{align}\nBy leveraging multi-layer information, we expect not only the last layer but also the preceding middle-level layers can directly learn the implicit relation between images and poses, which helps to improve the robustness against perturbations.\n\n\\subsection{Loss Function}\nFollowing the approach in \\cite{atloc}, we use a weighted balance loss for translation and rotation predictions. For the input image $I_{i}$, we denote the translation and rotation targets as $\\bm{d}_{i}^{*} \\in \\mathbb{R}^{3}$ and $\\bm{r}_{i}^{*} \\in \\mathbb{R}^{3}$ respectively. Then the absolute pose loss term $\\mathcal{L}^{l}_{i}$ and the relative pose loss term $\\mathcal{L}^{l}_{i,i^{'}}$ at decoding layer $l$ are computed as: \n\\begin{align}\n&\\ml{\\mathcal{L}^{l}_{i} = \\norm{\\bm{d}_{i}^{l}-\\bm{d}_{i}^{*}} \\exp(-\\alpha^{l}) + \\alpha^{l} \\\\\n+ \\norm{\\bm{r}_{i}^{l}-\\bm{r}_{i}^{*}} \\exp(-\\beta^{l}) + \\beta^{l},}\\\\\n&\\ml{\\mathcal{L}^{l}_{i,i^{'}} = \\norm{\\bm{d}_{i,i^{'}}^{l}-\\bm{d}_{i,i^{'}}^{*}} \\exp(-\\gamma^{l}) + \\gamma^{l} \\\\ \n+ \\norm{\\bm{r}_{i,i^{'}}^{l}-\\bm{r}_{i,i^{'}}^{*}} \\exp(-\\lambda^{l}) + \\lambda^{l}},\n\\end{align}\nwhere $\\bm{d}_{i}^{l},\\bm{r}_{i}^{l},\\bm{d}_{i,i^{'}}^{l},\\bm{r}_{i,i^{'}}^{l}$ are outputs at layer $l$, while $\\alpha^{l}, \\beta^{l}, \\gamma^{l}, \\lambda^{l}$ are all learnable parameters at layer $l$. Finally, the overall loss function can be obtained as:\n\\begin{align}\n\\mathcal{L}=\\sum_{l\\in\\{3,4,L\\}}\\sum_{i\n\\in[N], i^{'}\\in \\mathcal{N}^l_i}\\mathcal{L}^{l}_{i} + \\mathcal{L}^{l}_{i,i^{'}},\n\\end{align}\nwhere $\\mathcal{N}^l_i$ is the neighborhood of node $i$ in $\\mathcal{G}^{\\mathrm{pose},l}$. We use the logarithmic form of the quaternion to represent rotation $\\bm{r}$ as:\n\\begin{align}\n\\bm{r}=\\log \\bm{q}=\\left\\{\n\\begin{array}{ll}\n\\frac{(\\bm{q}_{2},\\bm{q}_{3},\\bm{q}_{4})}{\\norm{(\\bm{q}_{2},\\bm{q}_{3},\\bm{q}_{4})}}\\cos^{-1} \\bm{q}_{1} &\\ \\text{if}\\  \\norm{(\\bm{q}_{2},\\bm{q}_{3},\\bm{q}_{4})} \\not= 0,\\\\\n0 &\\ \\text{otherwise},\n\\end{array}\n\\right.\n\\end{align}\nwhere $\\bm{q} = (\\bm{q}_{1}, \\bm{q}_{2}, \\bm{q}_{3}, \\bm{q}_{4}) \\in \\mathbb{R}^{4}$ represents a quaternion.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Experiments}\nIn this section, we first evaluate our proposed model on three large autonomous driving datasets. We next present an ablation study to demonstrate the effectiveness of our model design.\n\n\\begin{table*}[!htp]\\footnotesize\n\\centering\n\\begin{tabular}{c | l | c  c   c  c     c  c  } \n\\toprule\n& \\multirow{3}{*}{Model} & \\multicolumn{2}{c}{Loop (cross-day)} & \\multicolumn{2}{c}{Loop (within-day)} & \\multicolumn{2}{c}{Full} \\\\\n& & \\multicolumn{1}{c}{Mean} & \\multicolumn{1}{c}{Median} & \\multicolumn{1}{c}{Mean} & \\multicolumn{1}{c}{Median} & \\multicolumn{1}{c}{Mean} & \\multicolumn{1}{c}{Median} \\\\ \n\\midrule\n\\multirow{5}{*}{\\rotatebox{90}{+ Extra Data}}\n& GNNMapNet + \\emph{post.}  & 7.96 \/ \\underlinecloser{2.56}  & - & - & - & 17.35 \/ \\underlinecloser{3.47} & -  \\\\\n\n& ADPoseNet & - & - & - & 6.40 \/ 3.09 & - & 33.82 \/ 6.77 \\\\\n\n& ADMapNet & - & - & - & 6.45 \/ 2.98 & - & 19.18 \/ 4.60 \\\\\n\n& MapNet+  & 8.17 \/ 2.62 & - & - & - & 30.3 \/ 7.8 &  \\\\\n\n& MapNet+ + \\emph{post.} & 6.73 \/ \\textbf{2.23} & - & - & - & 29.5 \/ 7.8 &  -\\\\\n\n\\midrule\n\\multirow{7}{*}{\\rotatebox{90}{CPR Only}}\n& GeoPoseNet &  27.05 \/ 18.54 & 6.34 \/ 2.06  & -  & -  & 125.6 \/ 27.1 &  107.6 \/ 22.5 \\\\\n\n& MapNet & 9.30 \/ 3.71 &  5.35 \/ \\underlinecloser{1.61} & - & -  & 41.4 \/ 12.5 &  17.94 \/ 6.68\\\\\n\n& LsG & 9.08 \/ 3.43   & -  & -  & -  & 31.65 \/ 4.51  &  - \\\\\n\n& AtLoc &   8.74 \/ 4.63 & 5.37 \/ 2.12 & -  & -  & 29.6 \/ 12.4  & 11.1 \/ 5.28 \\\\\n\n& AtLoc+ & \\underlinecloser{7.53} \/ 3.61 &  \\underlinecloser{4.06} \/ 1.98 & -  & -  &  21.0 \/ 6.15 & 6.40 \/ 1.50\\\\\n\n& CoordiNet & -  & -  & \\underlinecloser{4.06}  \/  \\underline{1.44} & \\underline{2.42}  \/  \\underline{0.88} &  \\underline{14.96}  \/  5.74  & \\textbf{3.55}  \/  \\underline{1.14} \\\\\n\n& RobustLoc (ours) & \\textbf{4.68} \/ 2.67  & \\textbf{3.70} \/ \\textbf{1.50} & \\textbf{2.49} \/  \\textbf{1.40}  & \\textbf{1.97}\/ \\textbf{0.84} & \\textbf{9.37} \/ \\textbf{2.47}  & \\underline{5.93} \/ \\textbf{1.06} \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Median and mean translation\/rotation estimation error (m\/$^\\circ$) on the Oxford RobotCar dataset. The best and the second-best results in each metric are highlighted in \\textbf{bold} and \\underlinecloser{underlined} respectively. ``-'' denotes no data provided.\n}\n\n\\label{tab:robotcar}\n\\end{table*}\n\\subsection{Datasets and Implemention Details}\n\\subsubsection{Oxford RobotCar.}\nThe Oxford RobotCar dataset\\cite{robotcar} is a large autonomous driving dataset collected by a car driving along a route in Oxford, UK. It consists of two different routes: 1) Loop with a trajectory area of $8.8\\times10^{4}\\mathrm{m}^{2}$ and length of $10^{3}\\mathrm{m}$,  and 2) Full with a trajectory area of $1.2\\times10^{6}\\mathrm{m}^{2}$ and length of $9\\times10^{3}\\mathrm{m}$.\n\n\\subsubsection{4Seasons.}\nThere are only a few existing methods designed for robust CPR in driving environments, and the experiment on the Oxford dataset is insufficient for comparison. Thus we also conduct experiments on another driving dataset to cover more driving scenarios. The 4Seasons dataset \\cite{4seasons} is a comprehensive dataset for autonomous driving SLAM. It was collected in Munich, Germany, covering varying perceptual conditions. Specifically, it contains different environments including the business area, the residential area, and the town area. In addition, it consists of a wide variety of weather conditions and illuminations. In our experiments, we use 1) Business Campus (business area), 2) Neighborhood (residential area), and 3) Old Town (town area).\n\n\\subsubsection{Perturbed RobotCar.}\nTo further evaluate the performance under challenging environments, we inject noise into the RobotCar Loop dataset and call this the Perturbed RobotCar dataset as shown in \\cref{fig:noisy}. We create three scenarios: 1) Medium (with fog, snow, rain, and spatter on the lens), 2) Hard (with added Gaussian noise), and 3) Hard (+ \\emph{noisy training}) (i.e., training with noisy augmentation).\n\n\n\\begin{figure}[!htp]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{figures\/noisy.pdf}\n\\end{center}\n\\caption{Visualization of the Perturbed RobotCar dataset. Medium is with fog, snow, rain, and spatter on the lens. Hard is with added Gaussian noise.}\n\\label{fig:noisy}\n\\end{figure}\n\n\n\\subsubsection{Implemention}\nWe use ResNet34 as the backbone, which is pre-trained on the ImageNet dataset. We set the maximum number of input images as $11$. We resize the shorter side of each input image to $128$ and set the batch size to $64$. The Adam optimizer with a learning rate $2\\times10^{-4}$ and weight decay $5\\times10^{-4}$ is used to train the network. Data augmentation techniques include random cropping and color jittering. We set the integration times $t_{0}=0$, $t_{1}=1$, and $t_{2}=2$. The number of attention heads is 8. We train our network for $300$ epochs. All of the experiments are conducted on an NVIDIA A5000. \n\n\\subsubsection{Baselines.} The baselines are described in the section ``Related Work''.\n\n\\subsection{Main Results}\\label{subsec:main results}\nOn the Oxford RobotCar dataset, as shown in \\cref{tab:robotcar}, we obtain the best performance in 10 out of 12 metrics. Using the mean error, which is easily influenced by outlier predictions, RobustLoc outperforms the baselines by a significant margin. In the most challenging route Full, to the best of our knowledge, RobustLoc is the first to achieve less than $10\\mathrm{m}$ mean translation error for CPR.\n\nThe 4Seasons dataset consists of more varied driving scenes. As shown in \\cref{tab:4seasons}, RobustLoc achieves the best performance in 11 out of 12 metrics. Again, using the mean error metric, RobustLoc outperforms the baselines by a significant margin.\n\nOn the Perturbed RobotCar dataset, where the images contain more challenging weather conditions and noisy perturbations, RobustLoc achieves the best in all metrics. The superiority of RobustLoc over other baselines is more obvious in \\cref{tab:noisy robotcar}.\n\n\n\n\n\n\n\\begin{table*}[!htp]\\footnotesize\n\\centering\n\\begin{tabular}{l | c c  c c  c c   } \n\\toprule\n\\multirow{2}{*}{Model} & \\multicolumn{2}{c}{Business Campus} & \\multicolumn{2}{c}{Neighborhood}  & \\multicolumn{2}{c}{Old Town}\\\\\n& \\multicolumn{1}{c}{Mean}& \\multicolumn{1}{c}{Median}& \\multicolumn{1}{c}{Mean}& \\multicolumn{1}{c}{Median}& \\multicolumn{1}{c}{Mean}& \\multicolumn{1}{c}{Median}\\\\\n\\midrule\nGeoPoseNet  & 11.04 \/ 5.78 & 5.93 \/ 2.03 & 2.87 \/ 1.30 & 1.92 \/ 0.88   & 64.81 \/ 6.67 & 15.03 \/ 1.57  \\\\\n\nMapNet & 10.35 \/ 3.78  & 5.66 \/ 1.83  & 2.81 \/ 1.05   & 1.89 \/ 0.92   & 46.56 \/ 7.14  & 16.52 \/ 2.12  \\\\\n\nGNNMapNet & \\underlinecloser{7.69} \/ 4.34 & \\underlinecloser{5.52} \/ 2.16 & 3.02 \/ 2.92  & 2.14 \/ 1.45 & \\underlinecloser{41.54} \/ 7.30 & 19.23 \/ 3.26 \\\\\n\n\nAtLoc & 11.53 \/ 4.84 & 5.81 \/ \\underlinecloser{1.50} & 2.80 \/ 1.16 & 1.83 \/ 0.93  & 84.17 \/ 7.81 & 17.10 \/ 1.73 \\\\\n\nAtLoc+   & 13.70 \/ 6.41 & 5.58 \/ 1.94 & 2.33 \/ 1.39 & 1.61 \/ 0.88 & 68.40 \/ 5.51 &  14.52 \/ 1.69 \\\\\n\nIRPNet &  10.95 \/ 5.38  & 5.91 \/ 1.82 & 3.17 \/ 2.85   & 1.98 \/ 0.90 & 55.86 \/ 6.97    &  17.33 \/ 3.11  \\\\\n\nCoordiNet  & 11.52 \/ \\underlinecloser{3.44} & 6.44 \/ \\textbf{1.38} & \\underlinecloser{1.72} \/ \\underlinecloser{0.86} & \\underlinecloser{1.37} \/ \\underlinecloser{0.69} & 43.68 \/ \\underlinecloser{3.58} &  \\underlinecloser{11.83} \/ \\underlinecloser{1.36} \\\\\n\nRobustLoc (ours)  & \\textbf{4.28} \/ \\textbf{2.04} & \\textbf{2.55} \/ \\underlinecloser{1.50} & \\textbf{1.36} \/ \\textbf{0.83} & \\textbf{1.00} \/ \\textbf{0.65} & \\textbf{21.65} \/ \\textbf{2.41} &  \\textbf{5.52} \/ \\textbf{1.05} \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Median and mean translation\/rotation estimation error (m\/$^\\circ$) on the 4Seasons dataset. The best and the second-best results in each metric are highlighted in \\textbf{bold} and \\underlinecloser{underlined} respectively.\n}\n\\label{tab:4seasons}\n\\end{table*}\n\n\n\n\n\n\n\\begin{table*}[!htp]\\footnotesize\n\\centering\n\\begin{tabular}{l |  c c c c  c c } \n\\toprule\n\n\\multirow{2}{*}{Model} & \\multicolumn{2}{c}{Medium} & \\multicolumn{2}{c}{Hard} & \\multicolumn{2}{c}{Hard (+ \\emph{noisy training})} \\\\\n& \\multicolumn{1}{c}{Mean} & \\multicolumn{1}{c}{Median} & \\multicolumn{1}{c}{Mean}& \\multicolumn{1}{c}{Median} \n& \\multicolumn{1}{c}{Mean} & \\multicolumn{1}{c}{Median}  \\\\\n\\midrule\nGeoPoseNet  & 20.47 \/ 8.76  & 8.70 \/ 2.30  & 41.71 \/ 17.63 & 14.02 \/ 3.13  & 24.03 \/ 11.14  &  7.14 \/ 1.70\\\\\nMapNet   & 17.93 \/ 7.01 & 6.89 \/ 2.00 & 49.36 \/ 20.01  & 18.37 \/ \\underlinecloser{2.58}  & 21.22 \/ 8.38  & 6.38 \/ 1.97 \\\\\nGNNMapNet  &  \\underlinecloser{16.17} \/ 7.24    &  8.02 \/ 2.35 &  73.97 \/ 35.57 &  61.47 \/ 19.73  & \\underlinecloser{14.55} \/ \\underlinecloser{7.62}  & 6.69 \/ \\underlinecloser{1.57} \\\\\n\nAtLoc  & 19.92 \/ 7.25 & 7.26 \/ \\underlinecloser{1.74} & 52.56 \/ 23.46 &  15.01 \/ 3.17   &  23.48 \/ 11.43 &  7.42 \/ 2.38\\\\\nAtLoc+  & 17.68 \/ 7.48 & \\underlinecloser{6.19}  \/ 1.80 & \\underlinecloser{37.92} \/ 18.65  & \\underlinecloser{12.17} \/ 2.93   & 22.61 \/ 11.23  & \\underlinecloser{6.21} \/ 1.83 \\\\\nIRPNet  & 16.35 \/ 7.56 & 8.71 \/ 2.28    & 45.72 \/ 21.84  & 17.99 \/ 3.50   &  24.73 \/ 11.20 &  6.73 \/ 1.82\\\\\nCoordiNet   & 17.67 \/ \\underlinecloser{6.66} & 7.63 \/ 1.79 & 44.11 \/ \\underlinecloser{16.42}  & 17.21 \/ 2.70   &  24.06 \/ 12.27 &  6.25 \/ 1.61 \\\\\nRobustLoc (ours)  & \\bf{8.12}  \/  \\bf{3.83}  & \\textbf{5.34} \/ \\textbf{1.53}   & \\textbf{27.75} \/ \\textbf{9.70}  & \\textbf{11.59} \/ \\textbf{2.64}   & \\textbf{10.06} \/ \\textbf{4.95}  & \\textbf{5.18} \/ \\textbf{1.43} \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Median and mean translation\/rotation estimation error (m\/$^\\circ$) on the Perturbed RobotCar dataset. The best and the second-best results in each metric are highlighted with \\textbf{bold} and \\underlinecloser{underline} respectively. RobustLoc achieves the best in \\textbf{all} metrics.\n}\n\\label{tab:noisy robotcar}\n\\end{table*}\n\n\n\n\\subsection{Analysis}\n\\subsubsection{Ablation Study.}\nWe justify our design for RobustLoc by ablating each module. From \\cref{tab:ablation study}, we observe that every module in our design contributes to the final improved estimation. We see that making use of neighboring information from covisible frames and learning robust feature maps contribute to more accurate CPR.\n\n\\begin{table}[!htb]\\footnotesize\n\\centering\n\\begin{tabular}{l  c  c } \n\\toprule\n\\multirow{1}{*}{Method} & \\multicolumn{1}{c}{ Mean Error (m\/$^\\circ$) on Loop (c.)} \\\\\n\\midrule\nbase model  &  8.38 \/ 4.29 \\\\\n+ feature map graph & 7.01 \/ 3.86 \\\\\n+ vector embedding graph & 6.24 \/ 3.21\\\\\n+ diffusion &  5.53 \/ 2.95\\\\\n+ branched decoder &  5.14 \/ 2.79\\\\\n+ multi-level decoding & \\textbf{4.68} \/ \\textbf{2.67} \\\\\n\\midrule\ndiffusion at stage 3 & 5.27 \/ 2.90  \\\\\ndiffusion at stage 3,4 & 4.86 \/ 3.18 \\\\\ndiffusion at stage 4 & \\textbf{4.68} \/ \\textbf{2.67}  \\\\\nmulti-layer concatenation & 5.80 \/ 3.26  \\\\\n\\midrule\nmore augmentation & \\textbf{4.68} \/ \\textbf{2.67}  \\\\\nless augmentation & 5.32 \/ 3.17  \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Ablation study, diffusion design, and augmentation design comparison on the Oxford RobotCar dataset.}\n\\label{tab:ablation study}\n\\end{table}\n\n\n\\subsubsection{Salience Visualization.}\nSalience maps shown in \\cref{fig:salience} suggest that in driving environments, RobustLoc pays more attention to relatively robust features such as the skyline and the road, similar to PixLoc \\cite{pixloc}. In addition, dynamic objects such as vehicles are implicitly suppressed in RobustLoc's regression pipeline.\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{figures\/saliency.pdf}\n\\end{center}\n\\caption{Robust features from RobustLoc. }\n\\label{fig:salience}\n\\end{figure}\n\n\n\\subsubsection{Diffusion and Augmentation.}\nUsing multi-level features is an effective method in dense prediction tasks such as depth estimation \\cite{yan2021cadepth}. To test if this holds in CPR, we use the feature maps from the lower stage 3 (see \\cref{fig:multi-level}), which however does not lead to performance improvement shown in \\cref{tab:ablation study}. We also utilize the multi-level concatenation strategy used in GNNMapNet. This does not lead to significant changes. These experiments demonstrate that CPR benefits more from high-level features with more semantic information than from low-level local texture features. Finally, we test the performance when training with less data augmentation, which leads to worse performance. This suggests that more extensive data augmentation can enhance the model robustness in challenging scenarios, which is consistent with the experimental results on the Perturbed Robotcar dataset in \\cref{tab:noisy robotcar}. \n\n\n\n\n\n\\subsubsection{Graph Design.}\nWe next explore the use of different graph designs for feature map diffusion and vector embedding diffusion. The grid graph stacks an image with two other spatially adjacent images as a cube, and the attention weights are formulated within the 6-neighbor area (for feature maps) or the 2-neighbor area (for vector embeddings). The self-cross graph computes attention weights first within each image and then across different images. From \\cref{tab:graph}, we see that the complete graph has the best performance. This is because, in the complete graph, each node can interact with all other nodes, allowing the aggregation of useful information with appropriate attention weights.\n\n\n\\begin{table}[!htb]\\footnotesize\n\\centering\n\\begin{tabular}{l  c   c} \n\\toprule\n\\multirow{1}{*}{Method} & \\multicolumn{1}{c}{Mean Error (m\/$^\\circ$) on Full} \\\\\n\\midrule\ngrid graph & 15.67 \/ 2.95  \\\\\nself-cross graph & 15.31 \/ 3.28  \\\\\ncomplete graph & \\textbf{9.37} \/ \\textbf{2.47}  \\\\\n\\midrule\n\\multirow{1}{*}{} & \\multicolumn{1}{c}{Mean Error ($^\\circ$) on Business Campus}  \\\\\n\\midrule\nquaternion & 2.23  \\\\\nLie group & 2.20  \\\\\nrotation matrix & 2.25  \\\\\nlog (quaternion) & \\textbf{2.04}  \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Graph design comparison on the Oxford RobotCar dataset and rotation representation comparison on the 4Seasons dataset.}\n\\label{tab:graph}\n\\end{table}\n\n\n\\subsubsection{Rotation Representation.}\nWe compare different representations of rotation in \\cref{tab:graph}, where the log form of the quaternion is the optimal choice. The other three representations, including the vanilla quaternion, the Lie group, and the vanilla rotation matrix, show similar performance.\n\n\n\n\n\\subsubsection{Trajectory Visualization.}\nWe visualize the output pose trajectories as shown in \\cref{fig:trajectory}, where a significant gap can be seen from the comparison. RobostLoc outputs more smooth and globally accurate poses compared with the previous method, which shows the effectiveness of our design.\n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{figures\/trajectory.pdf}\n\\end{center}\n\\caption{Trajectory visualization on the Oxford RobotCar dataset. The ground truth trajectories are shown in bold blue lines, and the estimated trajectories are shown in thin red lines. The stars mark the start of the trajectories.}\n\\label{fig:trajectory}\n\\end{figure}\n\n\n\\subsubsection{Inference Speed.}\nWe finally test the performance using a different number of input frames. The inference speed does not drop significantly when increasing the input frames. And even the slowest one (using $11$ frames) can run $50$ iterations per second and achieve real-time regression.  On the other hand, more frames can bring performance improvement when the input size is small, while further increasing frame size does not bring significant change.\n\n\\begin{table}[!htp]\\footnotesize\n\\centering\n\\begin{tabular}{l  c   c   c c c} \n\\toprule\n\\multirow{1}{*}{\\#frames} & \\multicolumn{1}{c}{3} & \\multicolumn{1}{c}{5} & \\multicolumn{1}{c}{7} & \\multicolumn{1}{c}{9} & \\multicolumn{1}{c}{11} \\\\\n\\midrule\nSpeed (iters\/s)  & \\textbf{56}  & 55  & 53  &  52  &  50\\\\\nMean Error (m)   &  5.28  &  5.09  &  4.96 & \\textbf{4.68}   & 4.72 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{The performance using different numbers of frames on the Oxford RobotCar Loop (cross-day).}\n\\label{tab:additional insight}\n\\end{table}\n\n\n\n\n\n\n\\section{Conclusion}\nWe have proposed and verified the performance of a robust CPR model RobustLoc. The model's robustness derives from the use of information from covisible images and neural graph diffusion to aggregate neighboring information, which is present in challenging driving environments. Extensive experimental results demonstrate that RobustLoc achieves SOTA performance.\n\n\n\\section{Acknowledgments}\nThis work is supported under the RIE2020 Industry Alignment Fund-Industry Collaboration Projects (IAF-ICP) Funding Initiative, as well as cash and in-kind contribution from the industry partner(s), and by the National Research Foundation, Singapore and Infocomm Media Development Authority under its Future Communications Research \\& Development Programme.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe first discuss briefly the problem of dictionary learning and the different choices one has to make to perform this task depending on the problem at hand. We also introduce notations and standard operators.\n\\subsection{Why a Dictionary ?}\n\nIn order to analyze a given finite dataset $X:=\\{x_n \\in \\mathbb{R}^D,n=1,..,N\\}$ different approaches are possible. One of them lies on the assumption that those observations actually come from some latent representation that are mixed together, different mixing leading to different observations but with fixed dictionary $\\Phi_K:=\\{\\phi_k \\in \\mathbb{R}^D,k=1,...,K\\}$ with usually $K \\ll N$. One can thus rewrite \n\\begin{equation}\\label{eq0}\n    x_n = f(\\Phi,x_n).\n\\end{equation}\nIn practice, a linear assumption is used for $f$ and we write the prediction as \n\\begin{equation}\n    \\hat{x}_n=\\hat{f}(\\hat{\\Phi}_K,x_n),\n\\end{equation}\nwhere one estimate the functional and the filter-bank through a reconstruction loss defined as\n\\begin{equation}\n    E_n = ||x_n-\\hat{x}_n ||^2,\n\\end{equation}\nwhere we assume here a squared error but any loss function can be used in general.\nThe linear assumption imposed on $f$ leads to the estimation weighting coefficients we denote by $\\alpha_{n,k}$ weighting for observation $n$ the atom $k$, and those attributes are the new features used to represent the observations.\n\nThis new representation of $x_n$ via its corresponding feature vector $\\bm{\\alpha_n}$ can be used for many tasks s.a. clustering, denoising, data generation, anomaly detection, compression and much more.\nDepending on the constraints one imposes on the feature vectors $\\bm{\\alpha_n}$ and the dictionary $\\Phi_K$, one can come back to standard known frameworks s.a. Principal Component Analysis (PCA)\\cite{jolliffe2002principal}, Independent Component Analysis (ICA)\\cite{hyvarinen2004independent}, Sparse Coding (SC)\\cite{olshausen1997sparse}, (Semi) Nonegative Matrix Factorization (sNMF)\\cite{lee1999learning,ding2010convex},  Gaussian Mixture Model (GMM)\\cite{bilmes1998gentle}, and many more, but all those approaches can be categorized into two main categories: Complete Dictionary Learning and Over-Complete Dictionary Learning.\n\nThe use of a dictionary also extends to standard template matching, the mot popular technique and optimal in the GLRT sense, where new examples are to be mapped to one of the template to be clustered or else.\n\n\n\\subsection{(Over)complete Dictionary Learning}\nAs we saw, the dictionary learning problem finds many formulations but the main difference resides in the properties of the learned dictionary, namely if it is complete or over-complete. \nThe general case of complete or orthogonal basis imposes the following constraint on the filter-bank:\n\\begin{align}\n    <\\phi_j,\\phi_k>=0,\\forall j \\not = k,\\;K=D,\n\\end{align}\nand with sometimes the complementary constraints that $||\\phi_k||=1,\\forall k$ leads to an orthonormal basis. The orthogonality of the atoms allow to have exact reconstruction leading to $E_n=0,\\forall n$ as by definition one has \n\\begin{equation}\\label{eq1}\n    \\hat{x}_n=\\sum_i \\frac{<x_n,\\hat{\\phi}_k>}{||\\hat{\\phi}_k||^2}\\hat{\\phi}_k,\\forall n.\n\\end{equation}\nGiven a dictionary $\\Phi$ this decomposition is unique and thus we have $(\\hat{\\Phi}_K,x_n) \\Rightarrow \\hat{\\bm{\\alpha}}_n,\\forall n$.\nHowever, while through the Gram-Schmidt process existence of such a basis is guaranteed, it is not unique.\n\n\nOn the other hand, $\\Phi_K$ can be an over-complete basis with the main difference coming that now $K>D$ and with the extreme case of $K=N$. Containing more atoms than the dimension of the space leads to interesting properties in practice such as sparse and independent features (ICA), clustering properties (K-means, NMF), biological understanding as is has been empirically shown that visual and auditory cortex of many mammals contains over-complete dictionaries.\nYet, all those benefits coming from the redundancy of the atoms also lead to the non-unique features $\\bm{\\alpha}_n$ even when the dictionary is kept fixed, thus we have that $(\\hat{\\Phi}_K,x_n) \\not \\Rightarrow \\hat{\\bm{\\alpha}}_n$.\nAs a result, one has to solve the additional optimization problem of\n\\begin{equation}\n    \\bm{\\hat{\\alpha}_n}=\\argmin_{\\bm{\\alpha}\\in \\Omega \\subset \\mathbb{R}^K} ||x_n-\\sum_k\\alpha_k\\hat{\\phi}_k ||.\n\\end{equation}\n\nAs a result, whether one is in a complete or over-complete dictionary setting, the optimization problems are always ill-posed by the non-uniqueness of the solutions forcing to impose additional structures or constraints in order to reach well posed problems. For the complete case, the main approach consist in imposing that as few atoms as possible are used leading to PCA, a very powerful approach for dimensionality reduction and compression. For over-complete cases, different sparsity criteria are imposed on the features $\\bm{\\alpha}_n$ such as norm-(0,1,2). For a norm-0 we are in the Matching pursuit case, norm1 is sparse coding and norm2 is ridge regression.\nFor each of those cases many very efficient exact or iterative optimization algorithms have been developed to estimate $\\hat{\\Phi}_k$ and $\\hat{\\bm{\\alpha}}_n$ yet there still exist a conceptual gap between the two concepts and the two approaches are often seen as orthogonal.\n\n\nAs we have seen, both settings lead to different beneficial aspects, compression, easy of projection and reconstruction or sparsity\/clustering but more complex optimization problems, but, at a higher level, the signal processing community has always put a gap between those frameworks. As well put by Meyer in [CITE] one has to choose between encoding and representation.\n\n\nWe thus propose in this paper a novel approach allowing to have an over-complete dictionary yet given an input, a dynamic basis selection reduces it to an optimal complete dictionary without need for optimization in order to reconstruct. The selection is done without optimization in a forward manner leading to an efficient algorithm. This thus allows to inherit from all the properties induced by orthonormal basis while allowing adaptivity for better allowing to learn an over-complete basis with a nonlinear basis selection leading to an orthonormal basis when conditioned on the input. We also provide results from a low-dimensional manifold perspective and show that our approach perform nonparametric orbit estimation.\nWe validate on compression, dictionary learning tasks and clustering.\n\n\n\n\n\n\n\\section{Deep Residual Oja Network}\n\\subsection{Shallow Model}\nIs it possible to learn a dictionary inheriting the benefits or complete and over-complete dictionaries ? We present one solution here. \nWe first motivate the need for such a framework as well as present the general approach and notations. Throughout the next sections, the choice of the Oja name for the algorithm will become blatant for the reader.\n\n\nKeeping the previously defined notation, we aim at learning an over-complete basis with the number of atoms defined by $FK$ with $F>1$ called the increase factor, note that $F=1$ lead to a complete basis, $K=D$ unless otherwise defined. By definition, the following projection-reconstruction scheme\n\\begin{equation}\n    \\hat{x}_n=\\sum_k \\frac{<x_,\\hat{\\phi}_k>}{||\\hat{\\phi}_k||^2}\\hat{\\phi}_k,\n\\end{equation}\ncan not reach an error $E_n<\\epsilon$ and in fact $E_n$ increases with $F$. One way to resolve this issue comes from an optimal basis selection point of view leading to\n\\begin{equation}\\label{loss1}\n    E_n:=||x_n-\\sum_k \\delta_{n,k}\\frac{<x_,\\hat{\\phi}_k>}{||\\hat{\\phi}_k||^2}\\hat{\\phi}_k|| < \\epsilon,\n\\end{equation}\nwith $\\delta{n,k}\\in \\{0,1\\},\\forall n,k$ representing a mask allowing to use a subset of the dictionary $\\hat{\\Phi}_{FK}$ we denote by $\\rho_{\\delta_{n,.}}[\\hat{\\Phi}_{FK}]$. \n\\iffalse\n\\tiny\nAs we aim at finding a systematic way to perform this basis selection, there are few possible approaches: either selecting the top-$\\kappa$ atoms $(S1)_\\kappa$ w.r.t. their activation maps energy, or keep the ones above a given threshold via either soft-thresholding (S2-1) or hard-thresholding (S2-2). If one assumes some kind of concentration among the dictionary, another approach or local winner-takes-all (S3) can be used.\n\nWe first present the conditions to have optimal reconstruction for each of the strategies.\n\\begin{theorem}\nFor an over-complete basis, $(S1)_\\kappa$ is optimal if $\\kappa=1$ and $F\\Rightarrow \\infty$ s.t. $\\forall x \\in \\Omega, \\exists k : \\phi_k=x$, (S2) is optimal if all inputs are made of atoms with always the same energy, (S3) is optimal is the dictionary can be rewritten as blocks s.t. selecting one atom per block lead to a complete basis.  \n\\end{theorem}\nThe strategies are thus\n\\begin{itemize}\n\\item $(S1)_\\kappa$ : $\\delta_{n,k}=1 \\iff Card(\\{j:|<x,\\phi_{j}>|<|<x,\\phi_{k}>|,j=1,..,FK,j\\not = k\\})>FK-\\kappa$\n\\item $(S2-1)$ : for this case in addition of applying a mask, there is a change in the value used to project back the atom, it is defined as \n\\begin{equation}\nE_n:=||x_n-\\sum_k \\delta_{n,k}(<x_n,\\hat{\\phi}_k>-b_k*sgn(<x_n,\\hat{\\phi}_k>))\\hat{\\phi}_k|| < \\epsilon,\n\\end{equation}\nand with $\\delta_{n,k}=1 \\iff $\n\\item $(S2-2)$ : $\\delta_{n,k}=1 \\iff |<x,\\phi_k>|>b_k$ where $b_k$ is a atom dependent threshold value either learned or imposed.\n\\item $(S3)$ : Let first partition $\\Phi_{FK}$ into an union of $D$ sets of $F$ atoms we denote $\\Phi'_{d},d=1,...,D$ as \n\\begin{align}\n    \\Phi_{FK} :=\\big[ \\Phi'_{1}\\vline \\dots \\vline \\Phi'_{D} \\big],\n\\end{align}\nwith \n\\begin{equation}\n    \\Phi'_d=\\big[\\phi_{Fk} \\vline \\dots \\vline \\phi_{F(k+1)} \\big], \\phi_{.}\\in \\mathbb{R}^D,\n\\end{equation}\nlet rewrite $\\delta_{n,d,f}$ be the indicator function applying to the $n^{th}$ example and $f^{th}$ filter of $\\Phi'_d$. We thus have\n\\[\n\\hat{\\delta}_{n,d,f}=1 \\iff f=\\argmax_{\\phi \\in \\hat{\\Phi}'_d} |<x_n,\\phi>|, \\forall d\n\\]\n\\end{itemize}\nWe are interested in this paper into the $(S1)$ strategy.\n\nFor (S1) the assumption of having a very large number of atoms is intractable. Or is it ? We now propose a way to achieve this and present results showing optimality of (S1) over other strategies, yet, we first present the framework allowing to have efficiently a humongous number of atoms.\n\n\\normalsize\n\n\\fi\n\n\n\n\n\\subsubsection{Error Bounds, Learning and Link with Oja Rule}\nWe first provide an error upper-bound for the proposed scheme $(S1)_{\\kappa=1}$.\nTo simplify notations be also define by $\\phi_\\kappa(x_n)$ the atom defined by\n\\begin{equation}\n\\phi_\\kappa(x_n)=\\phi_{k'},k'=\\argmax_k \\frac{|<x_n,\\phi_k>|^2}{||\\phi_k||^2}.\n\\end{equation} \n\\begin{theorem}\nThe error induced by $(S1)_{\\kappa=1}$ is $||x_n||^2-\\frac{|<x,\\phi_\\kappa(x_n)>|^2}{||\\phi_\\kappa(x_n)||^2}$ which is simply the reconstruction error from the best atom as since only one filter is used, nothing else is present.\n\\end{theorem}\n\\begin{proof}\nBy the incomplete basis theorem, there exists a basis s.t. it contains the atom $\\phi_{\\kappa}$, we denote by $\\phi_k$ such a basis, with $k=1,...,D$, we thus have\n\\begin{align}\nE_n=&|| x_n- \\frac{<x,\\phi_\\kappa(x_n)>}{||\\phi_\\kappa(x_n)||^2}\\phi_\\kappa(x_n)||^2\\nonumber\\\\\n=&|| \\sum_k\\frac{<x_n,\\phi_k>}{||\\phi_k||^2}\\phi_k- \\frac{<x,\\phi_\\kappa(x_n)>}{||\\phi_\\kappa(x_n)||^2}\\phi_\\kappa(x_n)||^2 &&\\text{ incomplete basis theorem} \\nonumber \\\\\n=&|| \\sum_{k\\not = \\kappa}\\frac{<x_n,\\phi_k>}{||\\phi_k||^2}\\phi_k||^2\\nonumber\\\\\n=&\\sum_{k\\not = \\kappa}\\frac{|<x_n,\\phi_k>|^2}{||\\phi_k||^2}\\nonumber\\\\\n=&||x||^2-\\frac{|<x_n,\\phi_\\kappa(x_n)>|^2}{||\\phi_\\kappa(x_n)||^2}&&\\text{Parseval's Theorem}\\nonumber\\\\\n=&||x||^2\\Big(1-\\cos(\\theta(x_n,\\phi_\\kappa(x_n))^2\\Big)\\label{en_eq}\n\\end{align}\nAnd we have by definition $E_n\\geq 0, E_n=0 \\iff x_n=\\phi_\\kappa(x_n)$\n\\end{proof}\nThere is first a few comments on the loss and updates. First of all, the loss is closely related to a k-mean objective with cosine similarity and specifically spherical k-means which is the case where the centers and the data points are re-normalized to unit norm and has the objective to minimize\n\\begin{equation}\n    \\sum_n(1-cos(x_n,p_{c(n)})).\n\\end{equation}\n\n\n\\subsubsection{Learning and Oja Rule}\nIn order to learn the filter-bank $\\Phi_K$, a common approach is to use an alternating scheme between finding the cluster belongings and optimizing the atoms w.r.t. this estimate. We first derive a gradient descent scheme to update the atoms and study some of its characteristics.\n\nIf we now denote by $n(k):=\\{n:n=1,...,N|\\kappa(x_n)=k\\}$ be the collection of the sample index in cluster $k$, he resulting loss $E_{n(k)}:=\\frac{\\sum_{n \\in n(k)}E_n}{Card(n(k))}$ we can now derive a gradient descent step as\n\\begin{equation}\n    \\phi_k^{(t+1)}(\\lambda)=\\phi_k^{(t)}-\\lambda \\frac{d E_{n(k)}}{d \\phi_k},\n\\end{equation}\nwith \n\\begin{align}\n    \\frac{d E_{n(k)}}{d \\phi_k}&=\\frac{1}{Card(n(k))}\\sum_{n \\in n(k)} \\frac{2|<x_n,\\phi_k>|}{||\\phi_k||^2}\\Big( \\frac{|<x_n,\\phi_k>|\\phi_k}{||\\phi_k||^2}-(-1)^{1_{<x_n,\\phi_k> <0}}x_n \\Big),\\nonumber\\\\\n    &=\\frac{1}{Card(n(k))}\\sum_{n \\in n(k)} \\frac{2<x_n,\\phi_k>}{||\\phi_k||^2}\\Big( \\frac{<x_n,\\phi_k>\\phi_k}{||\\phi_k||^2}-x_n \\Big).\\label{oja_eq}\n\\end{align}\nOn the other hand, if one adopts an adaptive gradient step $\\lambda$ per atom and point with one of the two strategies $\\lambda_1,\\lambda_2$ defined as\n\\begin{align}\n    \\lambda_1&=\\frac{<x_n,\\phi_k>}{2||x_n||^2}\\\\\n    \\lambda_2&=\\frac{||\\phi_k||^4}{2<x_n,\\phi_k>^2}\n\\end{align}\nthen we have the \n\\begin{align}\n    \\phi_k^{(t+1)}(\\lambda_1)&=\\phi_k^{(t)}-\\frac{1}{\\sum_n \\cos(\\theta(x_n,\\phi_k))^2}\\sum_{n \\in n(k)}\\cos(\\theta(x_n,\\phi_k))^2\\Big( \\frac{<x_n,\\phi_k>\\phi_k}{||\\phi_k||^2}-x_n \\Big),\\label{eq_online1}\\\\\n    \\phi_k^{(t+1)}(\\lambda_2)&=\\frac{1}{Card(n(k))}\\sum_{n \\in n(k)}\\frac{||\\phi_k||^2}{<x_n,\\phi_k>}x_n\\label{eq_online2}\n\\end{align}\nwe thus end up with in the $\\lambda_1$ case to a simple update rule depending on a weighted average of the points in the cluster based on their cosine similarity squared whereas for $\\lambda_2$ we obtain a rule a la convex NMF with is a plain without update combination of the points available.\n\nOn the other hand, it is clear that minimizing $E_n$ from Eq. \\ref{en_eq} is equivalent to maximizing $E^+_n=\\frac{<x_n,\\phi_\\kappa(x_n)>}{||\\phi_\\kappa(x_n)||^2}$. As a result,  one can seize in Eq. \\ref{oja_eq} the Oja rule as we can rewrite a GD update of $\\phi_k$ as\n\\begin{align}\n    \\phi_k^{(t+1)}&=\\phi^{(t)}_k+\\gamma \\frac{d E^+_n}{d \\phi_k}(\\phi^{(t)}_k)\\\\\n    \\phi_k^{(t+1)}&=\\phi^{(t)}_k+\\gamma \\Big( x_n\\frac{<x_n,\\phi_k>}{||\\phi_k||^2}-(\\frac{<x_n,\\phi_k>}{||\\phi_k||^2})^2\\phi_k \\Big)\\label{eq_online3}\n\\end{align}\nknown as the Oja rule.\nSPEAK ABOUT OJA RULE. And in fact, the convergence of Oja rule toward the first eigenvector-eigenvalue is not surprising as $E^+_{n(k)}$ leads explicitly to \n\\begin{equation}\n    \\phi_k=\\argmax_{\\phi}\\frac{1}{Card(n(k))}\\frac{\\phi^TX(k)^TX(k)\\phi}{\\phi^T\\phi},\\label{eq_pca}\n\\end{equation}\nwhich is known as the Rayleigh quotient and is a formulation of PCA leading to a one step global optimum being the greatest eigenvector-eigenvalue.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=5in]{time_mnist.png}\n\\end{figure}\n\n\n\\begin{pseudocode}[doublebox]{Filter-Bank Learning strategy}{X,K}\n\\text{Initialize }\\Phi_K\\\\\n\\WHILE \\text{not converged} \\DO\n\\BEGIN\n\\FOR k \\GETS 1 \\TO K \\DO\n\\BEGIN\n\\text{Compute }n(k) \\text{ with current }\\Phi_K\\\\\n\\text{Update }\\phi_k \\text{with }n(k) \\text{ and }X(k)\\text{ according to Eq. \\ref{eq_pca}}\\\\\n\\END\\\\\n\\END\\\\\n\\RETURN{\\Phi_k}\n\\end{pseudocode}\n\n\\begin{pseudocode}[doublebox]{Online Filter-Bank Learning strategy}{X,K}\n\\text{Initialize }\\Phi_K\\\\\n\\WHILE \\text{not converged} \\DO\n\\BEGIN\n\\FOR n \\GETS 1 \\TO N \\DO\n\\BEGIN\n\\kappa = \\argmax_k \\frac{|<x_n,\\phi_k>|^2}{||\\phi_k||^2||x_n||^2}\\\\\n\\text{Update }\\phi_\\kappa \\text{ according to Eq. \\ref{eq_online1} or Eq.\\ref{eq_online2} or Eq.\\ref{eq_online3}}\\\\\n\\END\\\\\n\\END\\\\\n\\RETURN{\\Phi_k}\n\\end{pseudocode}\n\n\n\n\n\n\n\\begin{theorem}\nIf the distribution of the $x_n$ in the space is uniformly distributed, the optimal over-complete basis for $(S1)_{kappa=1}$ is thus the one corresponding of a quantization of the sphere, it is unique up to a change of sign and global rotations (same applied to each atom). For the $2$-dimensional case it is easy to see that the maximum error for any given point $x_n$ is exactly upper-bounded by $||x||^2\\Big(1-\\cos(\\frac{\\pi}{2FK})^2\\Big)$ if $FK$ growths exponentially .(?? CHECK POWER OF HIGH DIMENION COSINE decompose as union of 2D spaces)\n\\end{theorem}\n\\begin{proof}\nFor 2D we now the upper bound is $||x||^2\\Big(1-\\cos(\\frac{\\pi}{2FK})^2\\Big)$ with $FK$ atoms, we thus rewrite \n\\begin{align*}\n||x-\\hat{x}||^2=\\sum_{d=1}^{D\/2}||x_d-\\hat{x}_d||^2\n\\end{align*}\nand see that in order to have the upper bound for each subspace we need the cartesian product of all the subspace basis $\\Phi_{FK}$ leading to $FK$ atoms. Thus one need to grow exponentially the number of atom w.r.t the dimension to have a loss increasing linearly.\n\\end{proof}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=5in]{error_bound.png}\n\\end{figure}\n\n\n\n\nHowever this pessimistic upper-bound assumes the worst possible scenario: uniform distribution of the data point in the space $\\mathbb{R}^D$ which in general is false. In fact, many dataset have inherent structures and at least lye only in a small subset sometime regular of $\\mathbb{R}^D$.\nIn general, data might be living in unions of subsets and thus providing a general strategy or optimal basis is a priori more complex thus pushing the need to learn the atoms as it is done in general for k-mean applications.\n\\begin{theorem}\\label{th5}\nA sufficient condition for $(S1)_{kappa=1}$ to be optimal is that the data are already clustered along $FD$ lines, or orbits ??? :)\n\\end{theorem}\nWe now present one way to tackle the curse of dimensionality in the next section\n\n\n\n\\subsection{Multiple Atoms}\n\\begin{equation}\n    E_n=|| x_n- \\sum_{k=1}^K\\frac{<x,\\phi^k_\\kappa(x_n)>}{||\\phi^k_\\kappa(x_n)||^2}\\phi^k_\\kappa(x_n)||^2\n\\end{equation}\nFor learning atom after atom a la coordinate ascent we have that \n\\begin{align*}\n    \\hat{\\phi}^{k'}_j=& \\argmin_{\\phi^{k'}_j}\\sum_n E_n\\\\\n    =&\\argmin_{\\phi^{k'}_j}\\sum_{n \\in n(k,j)} || \\Big(x_n- \\sum_{k=1,\\not = k'}^K\\frac{<x,\\phi^k_\\kappa(x_n)>}{||\\phi^k_\\kappa(x_n)||^2}\\Big)-\\frac{<x,\\phi^{k'}_j>}{||\\phi^{k'}_j||^2}\\phi^{k'}_\\kappa(x_n)||^2\n\\end{align*}\nas we showed in the last section we end up with the same update rule but with the input being substracted by the other used atoms. Thus we still perform PCA but on the input minus the other atoms. Note that this ensures orthogonality between the atoms.\n\n\\subsection{From Shallow to Deep Residual for better Generalization Error Bounds and Combinatorially Large Dictionaries}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=5in]{laroue.png}\n\\end{figure}\n\n\n\n\n\n\nWe now consider the analysis of the generalization performance on out of bag observations as well as the problem of having really large dataset.\n\n\\begin{theorem}\nIf we suppose an finite training set and an Oja Network with sufficient filters in order to reach a training error of $0$ then we know that the generalization error is directly lower-bounded by how close the testing and training examples are. In fact\n\\begin{equation}\n    E_{new}\\propto \\cos(\\theta(x_\\kappa,x_{new})),\\kappa=\\argmin_n \\cos(\\theta(x_n,x_{new})).\n\\end{equation}\n\\end{theorem}\n\nThe proof is straightforward as we know the network is able to perfectly reconstruct the training set and only the training set.\nAs a result, if the training set is well and uniformly sampled among the space of possible observations, a shallow Oja Network can be considered as optimal also for the testing set.\nHowever, and especially for computer vision task, it is well known that every observation is very far form each other in term of distance when dealt with in the pixel domain, also, requiring a proper sampling of the space of images is clearly outrageous.\nWe thus now present the result motivating deep architectures in general including the Oja Network.\n\n\\begin{align}\nR^{(l)}_n=&R^{(l-1)}_n-\\frac{<R^{(l-1)}_n,\\phi_\\kappa^{(l)}>}{||\\phi_\\kappa^{(l)}||^2}\\phi_\\kappa^{(l)}, \\kappa = \\argmax_k \\frac{|<R^{(l-1)}_n,\\phi_k^{(l)}>|}{||R^{(l-1)}_n||^2||\\phi_k^{(l)}||^2} \\nonumber \\\\\nR^{(0)}_n=&x_n\n\\end{align}\n\n\nas a result as soon as the input and the template are not orthogonal there is convergence. \n\\begin{theorem}\nSince we have by definition that the selected atom is the one with smaller angle, if it is $0$ it means that the input $R^{(l-1)}$ is orthogonal to all the learned dictionary $\\Phi^{(l)}$\n\\begin{equation}\n\\cos \\Big(\\theta (R^{(l-1)}_n,\\phi^{(l)}_\\kappa) \\Big)^2=0 \\iff R^{(l-1)} indep \\phi^{(l)}_k \\forall k,\n\\end{equation}\nand thus they live in two orthogonal spaces.\nTO PROVE : ALL THE NEXT ONES ARE ALSO 0 \n\\end{theorem}\n\n\\begin{theorem}\nThe residual decreases exponentially w.r.t. the depth of the model.\n\\end{theorem}\n\\begin{proof}\n\\begin{align}\n||R^{(l)}_n||^2=&||R^{(l-1)}_n-\\frac{<R^{(l-1)}_n,\\phi_\\kappa^{(l)}>}{||\\phi_\\kappa^{(l)}||^2}\\phi_\\kappa^{(l)}||^2 \\nonumber \\\\\n=&||R^{(l-1)}_n||^2-\\frac{<R^{(l-1)}_n,\\phi_\\kappa^{(l)}>^2}{||\\phi_\\kappa^{(l)}||^2}\\nonumber \\\\\n=&||R^{(l-1)}_n||^2\\Big(1-\\cos \\Big(\\theta(R^{(l-1)}_n,\\phi_\\kappa^{(l)}) \\Big)^2\\Big)\\\\\n=&||x_n||^2\\prod_{l=1}^l\\Big(1-\\cos \\Big(\\theta(R^{(l-1)}_n,\\phi_\\kappa^{(l)}) \\Big)^2 \\Big)\n\\end{align}\n\\end{proof}\n\nThe final template can be flattened via\n\\begin{align}\nT_n =& \\sum_l \\frac{<R^{(l-1)}_n,\\phi_\\kappa^{(l)}>}{||\\phi_\\kappa^{(l)}||^2}\\phi_\\kappa^{(l)}\\\\\n=&\\sum_l P^{(l)}_n\n\\end{align}\n\n\\subsubsection{Learning}\nComputing the gradient finds a great recursion formula we define as follows:\n\\begin{align}\n    A_{i,j}=\\left\\{ \\begin{matrix}\n         0 \\iff j<i\\\\\n         \\frac{<R^{(i-1)},\\phi^{(i)}_\\kappa>I_d+R^{(i-1)}\\phi^{(i)}_\\kappa}{||\\phi^{(i)}_\\kappa||^2}+\\frac{2<R^{(i-1)},\\phi^{(i)}_\\kappa>\\phi^{(i)}_\\kappa\\phi^{(i)T}_\\kappa}{||\\phi^{(i)}_\\kappa||^4} \\iff i=j\\\\\n         A_{i,j-1}-\\frac{\\phi^{(j)}_\\kappa\\phi^{(j)T}_\\kappa A_{i,j-1}}{||\\phi^{(j)}_\\kappa||^2}\\iff j>i\n    \\end{matrix} \\right.\n\\end{align}\nthus $A_{i,j} \\in \\mathbb{R}^{D \\times D}$ then we have\n\\begin{align}\n    \\mathcal{L}_n=&||R^{(L)}_n||,\\\\\n    \\frac{\\textbf{d} \\mathcal{L}^2}{\\textbf{d} \\phi^{(l)}_\\kappa}=&2R^{(L)}_n\\frac{\\textbf{d} R^{(L)}_n}{\\textbf{d} \\phi^{(l)}_\\kappa}\\\\\n    \\end{align}\nHowever as we will see below we have a nice recursive definition to compute all those derivatives, in fact\n\\begin{equation}\\text{Init. }\n\\begin{cases}\n    \\frac{\\textbf{d} P^{(l)}_n}{\\textbf{d} \\phi^{(l)}_\\kappa}&=\\frac{<R^{(l-1)}_n,\\phi^{(l)}_\\kappa>I_d+R^{(l-1)_n}\\phi^{(l)}_\\kappa}{||\\phi^{(l)}_\\kappa||^2}+\\frac{2<R^{(i-1)}_n,\\phi^{(l)}_\\kappa>\\phi^{(l)}_\\kappa\\phi^{(l)T}_\\kappa}{||\\phi^{(l)}_\\kappa||^4},\\\\\n    \\frac{\\textbf{d} R^{(l)}_n}{\\textbf{d} \\phi^{(l)}_\\kappa}&=-\\frac{\\textbf{d} P^{(l)}_n}{\\textbf{d} \\phi^{(l)}_\\kappa}\n\\end{cases}\n\\end{equation}\n\n\n\\begin{equation}\\text{Recursion }\n\\begin{cases}\n    \\frac{\\textbf{d} P^{(l+1)}_n}{\\textbf{d} \\phi^{(l)}_\\kappa}&=\\frac{\\phi^{(l+1)}_\\kappa\\phi^{(l+1)^T}_\\kappa}{||\\phi^{(l+1)}_\\kappa||^2}\\frac{\\textbf{d} R^{(l)}_n}{\\textbf{d} \\phi_\\kappa^{(l)}},\\\\\n    \\frac{\\textbf{d} R^{(l+1)}_n}{\\textbf{d} \\phi^{(l)}_\\kappa}&=\\frac{\\textbf{d} R^{(l)}_n}{\\textbf{d} \\phi^{(l)}_\\kappa}-\\frac{\\textbf{d} P^{(l+1)}_n}{\\textbf{d} \\phi^{(l)}_\\kappa}\n\\end{cases}\n\\end{equation}\n\n\n\\begin{pseudocode}[doublebox]{Residual Oja Network}{X,K}\nR_n \\GETS X_n, \\forall n \\\\\n\\FOR l \\GETS 1 \\TO L \\DO\n\\BEGIN\n\\text{Initialize }\\Phi^{(l)}_K \\text{ from }R\\\\\n\\WHILE \\text{not converged} \\DO\n\\BEGIN\n\\FOR k \\GETS 1 \\TO K \\DO\n\\BEGIN\n\\text{Compute }n(k) \\text{ with current }\\Phi^{(l)}_K\\\\\n\\text{Update }\\phi^{(l)}_k \\text{with }n(k) \\text{ and }R(k)\\text{ according to Eq. \\ref{eq_pca}}\\\\\n\\END\\\\\n\\END\\\\\nR_n = (R_n-\\frac{<R_n,\\phi^{(l)}_\\kappa>}{||\\phi^{(l)}_\\kappa ||^2}\\phi^{(l)}_\\kappa)\n\\END\\\\\n\\RETURN{\\Phi^{(l)}_k, \\forall l}\n\\end{pseudocode}\n\n\n\\begin{pseudocode}[doublebox]{Online Residual Oja Network}{X,K}\nR_n \\GETS X_n, \\forall n \\\\\n\\FOR l \\GETS 1 \\TO L \\DO\n\\BEGIN\n\\text{Initialize }\\Phi^{(l)}_K \\text{ from }R\\\\\n\\WHILE \\text{not converged} \\DO\n\\BEGIN\n\\FOR n \\GETS 1 \\TO N \\DO\n\\BEGIN\n\\kappa = \\argmax_k \\frac{|<R_n,\\phi^{(l)}_k>|^2}{||\\phi^{(l)}_k||^2||R_n||^2}\\\\\n\\text{Update }\\phi^{(l)}_\\kappa \\text{ according to Eq. \\ref{eq_online1} or Eq.\\ref{eq_online2} or Eq.\\ref{eq_online3}}\\\\\n\\END\\\\\n\\END\\\\\nR_n = (R_n-\\frac{<R_n,\\phi^{(l)}_\\kappa>}{||\\phi^{(l)}_\\kappa ||^2}\\phi^{(l)}_\\kappa)\n\\END\\\\\n\\RETURN{\\Phi^{(l)}_k, \\forall l}\n\\end{pseudocode}\n\n\n\n\\begin{theorem}\nWith a Deep (Oja) Network, the previously presented lower-bound of the generalization error becomes an upper-bound.\n\\end{theorem}\nIn addition of guaranteeing better generalization errors through depth, we also benefit from another gain. The depth as we will see allows for an exponential amount of possible templates to be constructed perfectly with only a linear increase in the number of learned parameters.\n\n\n\\begin{lstlisting}[caption=Input to Mask]\n####################\n#  INPUT: X(N,channels,Ix,Jx),w(n_filters,channels,Iw,Jw)\n####################\nk    = T.nnet.conv.conv2d(x,w,stride=stride,\n        border_mode='valid',flip_filters=False,input_shape=(N,channels,Ix,Jx),\n        filters_shape=(n_filters,channels,Iw,Jw))#(N,n_filters,(Ix-Iw)\/stride+1,(Jx-Jw)\/stride+1)\noutput = ((k>0)*2-1)*T.signal.pool.max_pool_2d_same_size(\n\t\ttheano.tensor.abs_(k).dimshuffle([0,2,3,1]),\n\t\t(1,n_filters)).dimshuffle([0,3,1,2])#(N,n_filters,(Ix-Iw)\/stride+1,(Jx-Jw)\/stride+1)\nmask = T.switch(T.eq(output,0),0,1)#(N,n_filters,(Ix-Iw)\/stride+1,(Jx-Jw)\/stride+1)\n\\end{lstlisting}\n\n\\begin{lstlisting}[caption=Reconstruction]\n####################\n# INPUTS: Z(N,n_filters,Iz,Jz),w(n_filters,channels,Iw,Jw),stride\n####################\ndilated_output = T.set_subtensor(T.zeros((N,n_filters,(Iz-1)*stride),(Iz-1)*stride),\n\t\tdtype='float32')[:,:,::stride,::stride],Z)#(N,n_filters,Ix-Iw+1,Jx-Jw+1)\nrec = T.nnet.conv.conv2d(dilated_Z,w.dimshuffle([1,0,2,3]),stride=1,\n        border_mode='full',flip_filters=False)#(N,channels,Ix,Jx)\n\\end{lstlisting}\n\n\\begin{lstlisting}[caption=Mask to Grad]\n###################\n#  INPUT : rec(N,C,Ix,Jx),mask(N,n_filters,Iz,Jz),Iw,Jw\n###################\nd_W,outputs=theano.scan(fn=lambda acc,i,X,mask:\n        acc+conv2d(rec[i].dimshuffle([0,'x',1,2]),mask[i].dimshuffle([0,'x',1,2]),\n        input_shape=(C,1,Ix,Jx),\n        filter_shape=(n_filters,1,Iz,Jz)).dimshuffle([1,0,2,3]),\n        sequences=[theano.tensor.arange(N,dtype='int32')],\n        non_sequences=[rec,mask],outputs_info = T.zeros((n_filters,C,Iw,Jw),dtype='float32'))\nd_W = d_W[-1]\n\\caption{algo}\n\\end{lstlisting}\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=5in]{error_energy.png}\n\\caption{Top : evolution of the reconstruciton error w.r.t. epochs. Bottom: evolution of the energy captured per level during training. At first stage the last levels capture everything since random initialization makes the global filters almost orthogonal to images, during training global filters learn to capture the low frequencies. Since it is known that natural images have a $1\/f$ decay of energy over frequencies $f$ we can see that the final energy repartition is indeed bigger for low-frequency\/global filters and go down for smaller filters.}\n\\end{figure}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=5in]{rec_example.png}\n\\caption{Example of decomposition and reconstruction of some CIFAR10 images. From right to left is the final residual (reconstruction minus original), the original image, the reconstructed images and then all the decompositions, on the left is the global\/large one. Summing elementwise columns $1$ to $8$ leads to column $9$ the reconstrued input.}\n\\end{figure}\n\n\n\\subsection{Previous Work}\nThe proposedm ethod can be seen as bilinear sparse coding with one-hot latent vector $y$ \\cite{grimes2005bilinear} for the case of only one filter used. There is also a direct link with the probabilistic version of this work, namely mixture of PPCA \\cite{tipping1999probabilistic,tipping1999mixtures}, as here we are in a ''hard clustering'' case similarly to k-means versus GMM. \nBy the selection of the best matching atom, we find some links with matching pursuit \\cite{tropp2007signal,pati1993orthogonal} and also  locality sensitive hashing \\cite{indyk1998approximate,johnson1984extensions} especially in the  cosine similarity distance.\n\nThis problem can also be seen from a best basis selection point of view coupled with dictionary learning. \nPopular examples with varying degrees of computational overhead include convex relaxations such\nas $L1$-norm minimization \\cite{beck2009fast,candes2006robust,tibshirani1996regression},  greedy approaches like orthogonal matching pursuit (OMP)\n\\cite{pati1993orthogonal,tropp2004greed}, and many flavors of iterative hard-thresholding (IHT) \\cite{blumensath2009iterative,blumensath2010normalized}\nVariants of these algorithms find practical relevance in numerous disparate domains, including feature selection \\cite{cotter2002sparse,figueiredo2002adaptive}, outlier removal \\cite{candes2005decoding,ikehata2012robust}, compressive sensing \\cite{baraniuk2007compressive}, and source localization \\cite{baillet2001electromagnetic,model2006signal}\n\n\\section{Conclusion}\nWe presented a hierarchical version of the deterministic mixture of PCA and presented results on CIFAR10 images. We also provided algorithms allowing GPU computation for large scale dataset and speed. The main novelty comes from the deterministic formulate of the probabilistic mixture of PCA which allows easier use as it is known in general that MPPCA is unstable for large scale problems. From this we derived its hierarchical residual version which inherits many benefits and allow for exponentially good reconstruction w.r.t. the depth. We also believe that this residual approach allowing to learn orthogonal spaces will lead to interesting dictionary learning mixing for example residual networks with this approach.\n\\iffalse\n\\section{Validation Results}\n\\subsection{MNIST}\nWe present now reconstruction error on MNIST for different $F$ values as $1,4,8$. Note that for better comparison, we also provided the reconstruction error provided by PCA for the case of taking the best $KF$ projectors. Note that for the proposed approach, we always have $||\\alpha||_0=K$.\n\\begin{table}\n\\begin{tabular}{c|c|c|c|}\\hline\n      &F=1&F=4&F=8\\\\\\hline\nPCA         &0.000245  \\textbf{0.000246} & 6.08e-05  \\textbf{6.16e-05} & 1.92e-05 \\textbf{1.97e-05} \\\\ \\hline\nPROPOSED    &0.000251  \\textbf{0.000252} & 0.000135  \\textbf{0.000136} & 0.000116  \\textbf{0.000117}\\\\  \\hline\n\\end{tabular}\n\\caption{Global Dictionary with $K=32$}\n\\end{table}\n\n\\begin{table}\n\\begin{tabular}{c|c|c|c|}\\hline\n      &F=1&F=4&F=8\\\\\\hline\nPCA         &0.000517 \\textbf{0.000518}&0.000245 \\textbf{0.000246}&0.000133 \\textbf{0.000134} \\\\ \\hline\nPROPOSED    &0.000538 \\textbf{0.000538}&0.000361  \\textbf{0.000362} & 0.000309  \\textbf{0.000310}\\\\  \\hline\n\\end{tabular}\n\\caption{Global Dictionary with $K=8$}\n\\end{table}\n\nWe also provide results for the case of local patches below without overlap (thus the two global and local errors can be compared) :\n\\begin{table}\n\\begin{tabular}{c|c|c|c|}\\hline\n      &F=1&F=4&F=8\\\\\\hline\nPROPOSED (8,8) K=8    &0.000348  \\textbf{0.000347}& 0.000205 \\textbf{0.000205} & 0.000178  \\textbf{0.000177} \\\\  \\hline\nPROPOSED (8,8) K=32    &0.000123  \\textbf{0.000122}&9.68e-05  \\textbf{9.66e-05} & 9.47e-05 \\textbf{9.457e-05}\\\\  \\hline\n\\end{tabular}\n\\caption{Global Dictionary with $K=8,32$}\n\\end{table}\n\n\nFor all the presented tables, the associated filters are in the appendix.\n\n\\subsection{CIFAR10}\n\n\n\n\n\\subsubsection*{Acknowledgments}\n\nUse unnumbered third level headings for the acknowledgments. All\nacknowledgments go at the end of the paper. Do not include\nacknowledgments in the anonymized submission, only in the final paper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe growing number of known exoplanet systems raises the question of\nhow to properly define the habitability zone around a star \\citep{Kasting:1993hw,Barnes:2011tc}. Its\ndefinition depends on the interactions existing\nbetween a planet and its host star, which are gravitational (tidal forces),\nmagnetic (wind-planet interactions, hereafter referred to as SPMI) and\nradiative (\\textit{e.g.}, stellar EUV ionisation flux).\nMagnetized interactions between a star and its orbiting planets have\nrecently been suggested to be at the origin of a possibly enhanced planet detectability\n\\citep{Jardine:2008ec,Fares:2010hq,Miller:2012gq}. In the case of a\nclose-in planet, these interactions may also be at the origin of anomalous stellar\nmagnetic activity \\citep{Cuntz:2000ef,Lanza:2008fn,Donati:2008hw}. It\nwas also suggested that it could affect the star-planet rotational\nevolution\n\\citep{Laine:2008dx,Pont:2009ip,Cohen:2010jm,Vidotto:2010iv,Lanza:2010bo}. Theoretical \nwork is needed to better understand SPMIs.\n\nBased on a pioneering work done in the context of the satellites of Jupiter\n\\citep{Goldreich:1969kf,Kivelson:2004vf}, \\citet{Laine:2011jt} built\nan analytical model describing the various\ncomponents of SPMIs in the case of unmagnetized\nplanets. Pursuing the same goal, \\citet{Lanza:2013gj} also developed\nsemi-analytical models of SPMIs in the context of magnetized\nplanets. However, a systematic numerical validation of those models still\nremains to be properly done \\citep[see][for first\nsteps towards such a validation]{Ip:2004ba,Cohen:2011gg}. \n\nFocusing on close-in planets, the SPMIs include magnetic reconnection,\nmagnetic field  diffusion at\nthe stellar surface and in the planet vicinity,\nradiation and ionisation processes in the planetary magnetosphere and\nmagneto-sonic wave propagation.\nA numerical investigation of SPMI requires a careful description\nof those physical processes although it is generally not possible to\ntreat all of them simultaneously with a unique model. Hence,\nspecific strategies such as dedicated boundary conditions have to be\ndeveloped to study SPMI from a global point of view. We detail in this\nwork how to develop both stellar (section \\ref{sec:star-bound-cond})\nand planetary (section \\ref{sec:plan-bound-cond}) boundary conditions\nto globally model the different SPMI cases,\nwithin the MHD formalism.\n\n\\section{Stellar boundary conditions}\n\\label{sec:star-bound-cond}\n\n\\begin{figure}[b]\n\\begin{center}\n \n  \\includegraphics[width=0.9\\linewidth]{fig1} \n \\caption{\\textbf{(a)} Schematic of the multi-layer boundary condition\n ensuring good conservation properties of the MHD solution as well as\n reactivity to external stimuli. Fixed quantities are forced to the\n Parker wind solution. The subscript $p$ stands for the poloidal\n component $(\\varpi,z)$ of vector in cylindrical\n coordinates. \\textbf{(b)} Typical wind solution \n used for SPMI. The color map represents the logarithmic density, the white lines\n the poloidal magnetic field lines. The slow and fast Alfv\\`en surfaces are labeled\n by the dashed lines, and the arrows show velocity field. The stellar\n surface is labeled by a black quarter of a circle. The axes are in\n stellar radius units. \\textbf{(c)}\n Effective rotation rate as a function of the streamfunction\n $\\psi$ for good (blue dots) and bad (black dots) boundary conditions. The red dashed horizontal line labels the stellar rotation\n rate. Low values of $\\psi$ correspond to open polar field lines and\n larger values of $\\psi$ to closed equatorial field lines. Each dot corresponds to a grid point.}\n   \\label{fig:fig1}\n\\end{center}\n\\end{figure}\n\nWe model stellar winds following numerous previous\nanalytical and numerical studies\n\\citep{Weber:1967kx,Washimi:1993vm,Ustyugova:1999ig,Keppens:2000ea,Matt:2004kd,Matt:2012ib}. We use standard\nMHD theory to numerically model with the PLUTO code\n\\citep{Mignone:2007iw} magnetized steady state flows \nanchored at the surface \nof a rotating star. We model winds driven by the thermal pressure of\nthe stellar corona in a 2D axisymmetric cylindrical geometry \\citep[see][for a more detailed description of the MHD\nmodel we use]{Strugarek:2012th}. \n\nThe steady-state wind solution can depend very sensitively on the\ntype of boundary conditions that are imposed under the stellar\nsurface. Because we want to use our model to study SPMIs, the stellar boundary\nconditions have to be able to both react and adapt to external stimuli\noriginating from the orbiting planet. The design of a boundary\ncondition satisfying those two conditions, and its associated stellar\nwind solution, are displayed in panels (a) and (b) of fig. \\ref{fig:fig1}.\n\n\nWe developed a layered boundary condition over which the stellar wind\ncharacteristics are progressively enforced as we go deeper\nunder the stellar surface. This boundary condition ensures \nvery good conservation properties \\citep{Lovelace:1986kd,Zanni:2009kc}\nalong the magnetic field lines. This \nis exemplified in panel (c) of fig. \\ref{fig:fig1}. We display \nthe effective rotation rate $\\Omega_{\\mbox{eff}} \\equiv \\frac{1}{\\varpi}\\left(v_\\phi\n  -\\frac{v_p}{B_p}B_\\phi \\right)$ as a function of the streamfunction\n$\\psi$ generating the poloidal magnetic field. In a steady-state,\nideal MHD wind, $\\Omega_{\\mbox{eff}}$ should be constant along each\nfield line and equal to $\\Omega_{\\star}$. The blue dots\ncorrespond to the boundary condition described in panel (a), and the\nblack dots to a case where $B_{\\phi}$ is set to 0 at all latitudes in\nthe third boundary level. We observe that the target\nstellar rotation rate (dashed horizontal red line) is\nrecovered only with the correct boundary conditions. Conservation errors\nexist at the open-closed field lines\nboundary ($\\psi \\sim 0.23$), but they remain confined to very few grid\npoints in the simulation domain. Finally, this boundary condition is intrinsically\nable to react to a perturbation by a planet orbiting a star by,\n\\textit{e.g.}, modifying the stellar wind topology.\nWe discuss now the importance of\nplanetary boundary conditions when studying SPMIs.\n\n\\section{Planetary boundary conditions}\n\\label{sec:plan-bound-cond}\n\n\\begin{figure}[b]\n\\begin{center}\n \n  \\includegraphics[width=0.9\\linewidth]{fig2_2}\n \\caption{Zoom on planetary boundary conditions effects for dipolar (upper panels) and\n   unipolar (lower panels) interactions. The color map represents the\n   gas pressure in logarithmic scale,\n   and the white lines the magnetic field lines. The planet surface is\n   labeled by a black circle at $1$ stellar radius. Panels (a) and (b) show the fiducial dipolar case,\n and panel (c) is the unrealistic case of a planet with a very high\n internal pressure. Panel (d) represents a\n Venus-like interaction and panels (e) and (f) two Io-Jupiter-like interactions.}\n   \\label{fig:fig2}\n\\end{center}\n\\end{figure}\n\nSPMIs are generally decomposed in two categories: the so-called\nunipolar and dipolar interactions \\citep{Zarka:2007fo}, which refer to \nthe cases of unmagnetized and magnetized planets.\nBoth interactions can be modeled\nwithin the MHD formalism with an adequate boundary condition design. We\ndetail in this section how to design such boundary conditions. The\nexamples given here were all done for a planet with a radius of $r_p = 0.1\\,\nr_\\star$, a mass of $M_p=0.01\\, M_\\star$, an orbital radius\nof $r_{\\rm{orb}}=3r_\\star$ and a resolution of $0.03\\, r_p$ at the\nplanetary surface.\n\nWe consider the planet itself as a boundary condition. The PLUTO\ncode allows one to define internal domains as boundary conditions\nover which all variables can be altered during the model evolution. In\nall cases, we set the poloidal velocity to zero and the\nazimuthal velocity to the keplerian velocity inside the planet. We\nalso set the density and pressure values inside the planet to \nfiducial values which are consistent with its gravity field. These\nvalue have to be carefully prescribed since they can trigger undesirable\neffects in the vicinity of the planet. We give an example of a\ndipolar case in panels (a) and (b) of fig. \\ref{fig:fig2} (the planetary\nmagnetic field is simply enforced in the planetary interior in this\ncase). A stable configuration is obtained when the magnetic pressure and the\ngas pressure equilibrate at the interface between the planetary\nmagnetosphere and the stellar wind. The ram pressure plays little role\nhere because the planet we consider is in the so called\n\\textit{dead-zone} of the stellar wind, in which the poloidal\nvelocity is negligible. We show in panel (c) the exact same simulation for\nan extreme case where we multiplied the internal pressure of the\nplanet by a factor of $20$. The former pressure balance then fails and\na wind is driven from the planet itself. The planetary dipole opens up and a shock\neventually creates at the interface between the two\n``winds''. Such undesirable effects may also be obtained by varying the\ndensity of the interior of the planet. Hence, any SPMI model must be developed\nto minimize such undesirable effects in the final solution.\n\nModeling a planet in the unipolar case is a bit more complex than in the\ndipolar case. Two classes of unipolar interactions can indeed be\ndistinguished: Venus-like interaction (case V) and Io-Jupiter like\ninteraction (case IJ). Note however that in both cases, we consider a\nplanet located inside the stellar wind dead-zone, at $r_{p}=3\\, R_{\\star}$.\n\nIn case V, the ionisation of the planetary atmosphere \nby the stellar EUV radiation flux allows the creation of a\nionosphere which  acts\nas a barrier between the stellar wind magnetic field and the \nunmagnetized interior of the planet \\citep{Russell:1993jk}. Depending\non the stellar wind conditions around the planet, an induced\nmagnetosphere may then be sustained on secular time scales. We show\ncase V in panel (d) of fig. \\ref{fig:fig2}. The\nionosphere is modeled as a very thin ($< 0.2\\,\nr_{p}$) highly conductive boundary layer under the \nplanetary surface. The wrapping of the magnetic field lines around\nplanet \\citep{Russell:1993jk} is naturally recovered.\n\nIn case IJ, no ionosphere is created and the stellar wind magnetic field\npervades inside the planet. The SPMI then depends on the ratio of\nelectrical conductivities between the planetary interior and the stellar\nsurface where the magnetic field lines are \\textit{a priori}\nanchored. This ratio sets the effective drag the planet is able to\ninduce on the stellar wind magnetic field lines. We use the ability of\nthe PLUTO code to add extra ohmic diffusion in\nthe planet interior to model it and show in\nfigure \\ref{fig:fig2} two extreme cases in which\nmagnetic field lines are dragged (panel e) or diffused (panel f) by\nthe planet. In all cases, we obtain a statistical steady state in\nwhich the SPMI can be analyzed in details. \n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe showed in this work that is it possible to model the global,\nmagnetized and non-linear interactions between a star and\na planet, within the MHD formalism. It requires a careful\ndevelopment of adequate boundary conditions to represent\nthe various interaction cases. We showed that boundary conditions play\na very important role both at the stellar surface and in the planetary\ninterior. Steady state solutions could be found in the dipolar case as\nwell as in both the Venus-like and Io-Jupiter-like unipolar\ncases. \n\nThe SPMI model we developed will be useful for exploring\nstable interaction configurations between a close-in planet and its\nhost star. In addition, it will enable quantitative predictions of\nrotational evolution of star-planet systems due to the\neffective magnetic torques which develop in the context of dipolar and unipolar\ninteractions \\citep{Strugarek:2013uh}. Finally, such models\ncould also be used to study potential SPMI induced emissions, which we\nwill analyze in a future work.\n\n\\acknowledgements\n\nWe thank A. Mignone and his team for making the PLUTO code\nopen-source. We thank A. Cumming, R. Pinto, C. Zanni and P. Zarka\nfor inspiring discussions on star-planet magnetized interactions. This\nwork was supported by the ANR TOUPIES and the ERC project\nSTARS2. We acknowledge access to supercomputers through GENCI project\n1623 and Prace infrastructures. A. Strugarek acknowledges support from\nthe Canada's Natural Sciences and Engineering Research Council.  \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThis article is motivated by two perennial questions of approximation\ntheory: Assume that a finite number of  samples of a continuous function $f$ on some\ncompact set is given,  (i) find a good or optimal approximation of $f$\nfrom these samples and  derive  error estimates,  and   (ii) approximate an\nintegral $\\int f$ from these samples and derive error estimates, in\nother words, find a quadrature rule based on the given samples.\n\n We argue, completely  in line with the tradition of approximation theory, that these\nquestions are best answered by means of \\mz\\ families and  inequalities. \nRoughly speaking, a \\mz\\ family is a double-indexed set of points\n$\\xkn $ such that the sampled $\\ell ^2$-norm of the $n$-th layer $\\sum\n_{k} |p(\\xkn )|^2 $ is an equivalent norm for the space of ``polynomials''\nof degree $n$ with uniform constants.\nOur main result then  shows that the existence of a \\mz\\ family already\nimplies (i) approximation theorems from pointwise samples of a function, and (ii)\nquadrature rules. This is, of course, folkore, and the content of an\nabundance of results in approximation theory and numerical analysis \non many  levels of generality. In the literature \\mz\\ families are  constructed for\nthe purpose of quadrature rules and approximation theorems~\\cite{FM10,MNW01},\nour main insight is that  quadrature rules\nand approximation theorems follow automatically  from  a\n\\mz\\ family.   It is one of our objectives to explain\nthis conceptual  hierarchy:  \\mz\\ families are first, then  quadrature rules and\napproximation theorems come for free. \n\n\nThe main novelty of our contribution is a completely elementary\nderivation of approximation theorems and quadrature rules based on the \nexistence of a \\mz\\ family. This derivation is fairly simple and is  based solely on the\nbasic definitions of \\mz\\ families, orthogonal projections, Sobolev\nspaces, and least square problems. The assumptions are minimal and\nonly require an orthonormal basis $\\{\\phi _k\\}$  and an associated\nnon-decreasing sequence of ``eigenvalues'' $\\{\\lambda _k\\} \\subseteq\n\\bR ^+$. This set-up is similar to the one in~\\cite{FM10,FM11,MM08}. \n\nOur point of view is informed by the theory of non-uniform\nsampling of bandlimited functions and their discrete analogs developed\nin the 1990's by many groups~\\cite{BH90,FGtp94,fgs95,PST01,Sunw02}.\nIndeed, a sampling theorem is simply a \\mz\\ inequality (upper and\nlower) for a fixed function space, and some of the first \\mz\\\ninequalities for scattered points on the torus (or non-uniform sampling points)\nwere derived in this context~\\cite{Gro93c}. The  method  in this paper \nwas essentially developed in~\\cite{Gro99} for the local approximation of\nbandlimited functions by trigonometric polynomials from samples. \n\nSeveral technical aspects deserve special mention.\n\n(i) In general, the error estimates for quadrature rules depend on a\ncovering radius (or mesh size) or on the number of nodes that  arise in a particular construction of a\n\\mz\\ family~\\cite{BCCG14,EGO17}. In our derivation the constants depend only on the\ncondition number of the \\mz\\ family and thus on their definition.\n\n(ii)   Whereas quadrature\nrules are often connected to \\mz\\ inequalities with respect to the\n$L^1$-norm~\\cite{FM10}, we derive such rules from \\mz\\ inequalities in\nthe $L^2$-norm by means of  frame theory.  The frame approach to\nquadrature rules  is \nmotivated by a  question of N.~ Trefethen about convergence of\nthe standard  quadrature rules  after a perturbation of the uniform\ngrid, see~\\cite{TW14,AT17}\n\n(iii)  In several articles on \\mz\\ families \\cite{FM10,MM08,OP12} the polynomial growth of\nthe spectral function $\\sum _{k=1}^n |\\phi _k(x)|^2$ is used\nimplicitly  or as a hypothesis.  In our treatment\n(Lemma~\\ref{critval}),  the growth of the spectral function  is  \nrelated to the critical Sobolev exponent and leads to  explicit and \ntransparent error estimates.  In hindsight the appearance\nof the spectral function is not surprising, as it is the reciprocal of\nthe Christoffel function associated to an orthonormal basis (or to a\nset of orthogonal polynomials), and is thus absolutely fundamental for\npolynomial interpolation and quadrature rules. See~\\cite{Nev86} for an extended survey. \n\n\\vspace{ 2mm} \nThe paper is organized as follows: the end of this introduction\nprovides a brief survey of related literature. In Section~2 we \nintroduce \\mz\\ families for the torus and prove the resulting\napproximation theorems and quadrature rules. In Section~3 we treat the\nsame question in more  generality on a compact space with a given\northonormal basis and corresponding eigenvalues. With  the appropriate\ndefinitions of Sobolev spaces and error terms, the formulation of the\nmain results and the proofs are then identical. In a sense Section~2\nis redundant, but we preferred to separate the proofs from the\nconceptual work. This separation reveals the simplicity of the\narguments more clearly. \n\n\n\n\\subsection{Discussion of related work}\nIt is impossible to  do full justice to the extensive literature on \\mz\\ inequalities,\n approximation theorems, and quadrature rules,  we will therefore\n mention only some aspects and apologize for any  omissions. \n\nThe classical theory of \\mz\\ inequalities deals with the interpolation\nby polynomials and the associated quadrature rules and is surveyed\nbeautifully in~\\cite{Lub98}. An  extended recent  survey with a\ncomplementary point of vie\nis contained in~\\cite{DTT19}. \n\n(i) \\emph{\\mz\\ families on the torus and trigonometric polynomials.} \n\\mz\\ inequalities for scattered nodes were considered in the theory of\nnonuniform sampling~\\cite{FGtp94}. An early example of  \\mz\\\ninequalities  on the torus  is contained in the\nestimates of~\\cite[Thm.~4]{Gro93c}. \\mz\\ with respect to different measures\nmeasures were then studied in~\\cite{Erd99,Lub99,MT00,RS97}.\nA complete   characterization of\n \\mz\\ families on the torus with respect to Lebesgue measure  in terms of suitable\n Beurling densities was given by Ortega-Cerd\\`a\n and Saludes~\\cite{OS07}. \n\n (ii) The next phase concerned \\emph{\\mz\\ inequalities and quadrature\n   on the sphere}: The goal of \\cite{MNW01} is the construction of\n good quadrature rules on the sphere via \\mz\\ inequalities, sufficient\n conditions for \\mz\\ families are obtained in~\\cite{MP14}. \n  Necessary  density conditions for \\mz\\ inequalities on the\n sphere are derived in~\\cite{Mar07}, and \\cite{MOC10} studies the\n connection between \\mz\\ families and Fekete points on the sphere.\n \\cite{HS06} derives worst case errors for quadratures on the  sphere. \n\n (iii) \\emph{\\mz\\ families on metric measure spaces.} The most\n general constructions of \\mz\\ families and quadrature rules  are due to Filbir and Mhaskar\n in a series of papers~\\cite{FM10,FM11,MM08,Mh18}. They work  for metric measure spaces with\n Gaussian estimates for the heat kernel associated to an orthonormal\n basis. This theory includes in particular \\mz\\ families on compact\n Riemannian manifolds. A related theory is contained in~\\cite{FFP16}\n whose goal is the construction of frames for Besov spaces. Again,\n necessary density conditions for \\mz\\ families on compact Riemannian\n manifolds have been derived by Ortega-Cerd\\`a and\n Pridhnani~\\cite{OP12}. \n\n(iv) \\emph{Approximation of functions from samples via least squares:}\nIn general the approximating polynomials do not interpolate, therefore\nthe best approximation of a given function by a ``polynomial'' is\nobtained by solving a least squares problem. A \\mz\\ inequality then\nimplies bounds on the condition number of the underlying matrix. This\nconnection appears, among others,  in~\\cite{Gro99,FM11} and is\nhighlighted in Proposition~\\ref{prop1}.  Modern\nversions use random sampling to generate \\mz\\ inequalities. This\naspect was studied in~\\cite{BG04,SZ04} for random sampling in\nfinite-dimensional subspaces, \\cite{CM17,ABC19,AC19} contain recent studies of the stability of\nleast squares reconstruction. We point out in particular~\\cite{CM17}\nwhere the Christoffel function is identified as the optimal weight for\na given probability measure. Finally we highlight the series of papers\non generalized sampling~\\cite{AH12,AHP13} as an alternative approach to the\napproximation of functions from finitely many linear measurements. In this\ncase the constants in the error estimates are formulated with the angle between\nsubspaces rather than with the condition number of the \\mz\\ family.  \n\n\n\n\n\\section{Approximation of functions on the torus from nonuniform samples}\nAs a  model example where the technique is completely transparent, we\nfirst deal with nonuniform sampling on the torus $\\bT $. On $\\bT$ the\napproximation spaces are  the space of trigonometric polynomials $\\cT\n_n$ of degree $n$, i.e.,\n $p\\in \\cT _n$, if $p(x) = \\sum _{k=-n}^n c_k e^{2\\pi i k x}$. \n\n \\begin{definition} \\label{defmz}\n   Let  $\\cX    = \\{ \\xkn : n\\in \\bN , k=1, \\dots , L_n \\}$ be  a\n   doubly-indexed set  of points in $\\bT \\simeq (-1\/2,1\/2] $ and $\\tau\n   =  \\{ \\tkn : n\\in \\bN , k= 1, \\dots , L_n \\}\\subseteq (0,\\infty ) $ be  a\n   family of non-negative weights. Then $\\cX $ \n   is called a \\mz\\ family, if there\n  exist constants $A,B >0$ such that \n  \\begin{equation}\n    \\label{eq:1}\nA \\|p\\|_2^2 \\leq \\sum _{k=1} ^{L_n} |p(\\xkn )|^2 \\tau _{n,k}    \\leq B\n\\|p \\|_2^2 \\qquad \\text{ \\emph{for all} } p \\in \\cT _n \\, .\n  \\end{equation}\nThe ratio $\\kappa = B\/A$ is the global condition number of the \\mz\\\nfamily, and $\\cX _n =\\{ \\xkn : k=1, \\dots , L_n \\}$ is the $n$-th\nlayer of $\\cX $. \n\\end{definition}\n\nThe point of  Definition~\\ref{defmz} is that the constants are uniform in the\ndegree $n$ and that usually  $\\cX _n$  contains more than\n$\\mathrm{dim}\\, \\cT _n = 2n+1$ points,  so\nthat it is not an interpolating set for $\\cT _n$. Weights are\nomnipresent in the classical theory of \\mz\\ inequalities~\\cite{Lub99},\n  in sampling\ntheory they are used to improve  condition numbers~\\cite{fgs95}, and\nin Fourier sampling they serve as density compensating factors. \nCurrently  they  play an  important role in weighted least squares\nproblem in statistical estimation in~\\cite{CM17,ABC19,AC19}.\n\nGiven the samples $\\{ f(\\xkn ) \\}$ of a continuous  function $f $ on\n$\\bT $ on the $n$-th layer $\\cX _n$, we first need to approximate $f$\nusing only these samples. For this we  solve a sequence of least\nsquares problems with samples taken from the $n$-th layer $\\cX _n$: \n\\begin{equation}\n  \\label{eq:8}\np_n = \\mathrm{argmin} _{p\\in \\cT _n} \\sum _{k=1} ^{L_n} |f(\\xkn ) -\np(\\xkn )|^2 \\tkn \\, .\n\\end{equation}\nThis procedure yields a  sequence of trigonometric\npolynomials for every $f\\in C(\\bT )$. In general,  these polynomials\ndo not interpolate the given $f$ on $\\cX _n$,\nbut they yield the best $\\ell ^2$-approximation of the data $\\{f(\\xkn )\\}$ by a\ntrigonometric polynomial in $\\cT _n$. Therefore  $p_n$ is usally called a\nquasi-interpolant.\n\nThe question is now  how the $p_n$'s approximate  $f$ on all of $\\bT\n$. As always in approximation theory, the answer depends on the\nsmoothness of $f$. For this we use the standard Sobolev spaces $H^\\sigma\n(\\bT )$ with  norm \n\\begin{equation}\n  \\label{eq:2}\n  \\|f\\| _{H^\\sigma } = \\Big(\\sum _{k\\in \\bZ } |\\fhat (k)|^2\n  (1+k^2)^{\\sigma } \\Big)^{1\/2} \\, ,\n\\end{equation}\nwhere  $\\fhat (k) = \\int _0^1 f(x)\ne^{-2\\pi i kx} \\, dx$ is the $k$-th Fourier coefficient of $f$. \n\nOur main theorem asserts the  convergence of the\nquasi-interpolants of $f$. \n\n\\begin{tm} \\label{tm1}\nLet $\\cX $ be a \\mz\\ family with associated weights $\\tau $ and\ncondition number $\\kappa = B\/A$. \n\n(i)   If $f\\in H^\\sigma $ for $\\sigma >1\/2$, then \n  \\begin{equation}\n    \\label{eq:3}\n    \\|f-p_n\\|_2  \\leq  C_\\sigma \\sqrt{1+\\kappa ^2} \\|f\\|_{\\hs }\n    n^{-\\sigma +1\/2} \\, ,\n  \\end{equation}\nwith a constant depending on $\\sigma$ (roughly $C_\\sigma \\approx\n(\\sigma -1\/2)^{-1\/2}$).\n \n(ii) If $f$ extends to an analytic function on an strip  $\\{z\\in \\bC\n: |\\mathrm{Im} z| <  \\rho  _0 \\}$, then the convergence is geometric, i.e.,\n  \\begin{equation}\n    \\label{eq:3a}\n    \\|f-p_n\\|_2 = \\cO (e^{-\\rho n}) \\, .\n  \\end{equation}\n  for every $\\rho <\\rho _0$.\n\\end{tm}\n\n\n\nThe proof starts with   the orthogonal projection  $P_nf(x) = \\sum _{|k|\\leq n}\n\\fhat (k) e^{2\\pi i kx} $  of $f$ onto the trigonometric \npolynomials $\\cT _n$. Note that $P_nf $ is the $n$-th partial sum of\nthe Fourier series of $f$. The proof of Theorem~\\ref{tm1} is based on the orthogonal decomposition\n\\begin{equation}\n  \\label{eq:4}\n  \\|f- p_n \\|_2^2 =   \\|f- P_nf \\|_2^2 +  \\|P_nf - p_n \\|_2^2 \\, .\n\\end{equation}\nThe first term measures how fast the partial sums of the Fourier\nseries converge to $f$, whereas the second, and more interesting, term\ncompares the best $L^2$-approximation of $f$ in $\\cT _n$ with the\napproximation $p_n$ obtained from the samples of $f$ on $\\cX _n$. \n\n\\subsection{Sampling and embeddings in $\\hs $ }\n\nBefore entering the details of the proof, we state some well-known \nfacts about the Sobolev space $\\hs $. \n\\begin{lemma} \\label{lm1}\nAssume that $\\sigma > 1\/2$. \n\n(i) Sobolev embedding: Then $\\hs (\\bT )  $ is continuously embedded in\n$C(\\bT )$.\n  \n(ii) Convergence rate: for all $f\\in \\hs (\\bT ) $\n\\begin{equation}\n  \\label{eq:5}\n  \\|f-P_nf \\|_\\infty  \\leq  \\|f \\|_{\\hs } \\, \\phi _\\sigma (n) \\, ,\n\\end{equation}\nwhere $\\phi _\\sigma (n) = (\\sigma -1\/2)^{-1\/2}\\, n^{-\\sigma\n  +1\/2}$.\n\n(iii) Sampling in $\\hs $: If $\\cX $ satisfies the sampling\ninequalities \\eqref{eq:1} and  $f\\in \\hs $, then \n\\begin{equation}\n  \\label{eq:6}\n\\sum _{k=1}^{L_n} |f(\\xkn )|^2 \\tkn \\leq  B \\|f\\|_\\infty ^2  \\leq B C_\\sigma ^2 \\|f\\|_{\\hs }^2 \\, , \n\\end{equation}\n \\end{lemma}\n\n \\begin{proof}\n(i) and (ii) are standard (and also  follow from  Lemma~\\ref{sobgen}).\n\n(iii)  The sampling inequalities\n\\eqref{eq:1} applied to the constant function $p\\equiv 1$  with\n$\\|p\\|_2=1$ yield\n\\begin{equation}\n  \\label{eq:7}\n  A \\leq \\sum _{k=1}^{L_n} \\tkn \\leq B \\, .\n\\end{equation}\nThe claim follows from \\eqref{eq:7} and the Sobolev embedding.\n \\end{proof}\n\n\n\n\\subsection{Quasi-interpolation versus projection}\nTo estimate the norm $\\| P_n f - p_n\\|_2$, let us introduce the\nvectors and matrices that arise in the explicit solution of the least squares\nproblem \\eqref{eq:8}.\nLet $$\ny_n = (\\tau _{n,1}^{1\/2} f(x_{n,1}), \\dots , \\tau _{n,L_n}^{1\/2}\nf(x_{n,L_n})) \\in \\bC ^{L_n}$$\nbe the\ngiven data vector, and $U_n $ be the $L_n \\times (2n+1)$-matrix (a\nVandermonde matrix) with\nentries \n\\begin{equation}\n  \\label{eq:10}\n(U_n)_{kl} =  \\tkn ^{1\/2} e^{2\\pi i x_{n,k} l} \\qquad k=1,\\dots , L_n, |l| \\leq n\n\\, .  \n\\end{equation}\nWe write \n\\begin{equation}\n  \\label{eq:11}\n  T_n = U_n ^* U_n \\, . \n\\end{equation}\nFor the numerical construction of $p_n$ we  note that $T_n$ is a\nToeplitz matrix and thus accessible to fast algorithms~\\cite{Gro93c,fgs95,PST01}. For our\nanalysis we collect the following facts.\n\n\\begin{lemma} \\label{lem2a}\n  Assume that $\\cX $ is a \\mz\\ family. Then\n\n(i) the spectrum of $T_n$ is contained in the\ninterval $[A,B]$ for all $n\\in \\bN $, and \n\n(ii)  the  solution of the least\nsquares problem \\eqref{eq:8} yields a trigonometric polynomial $p_n =\n\\sum _{|k| \\leq n} a_{n,k} e^{2\\pi i kx}\n\\in \\cT _n$ with a coefficient vector $a_n \\in \\bC ^{2n+1}$ given by\n\\begin{equation}\n  \\label{eq:12}\n  a_n = T_n \\inv U_n ^* y_n \\, .\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n  (i) Note that for $p\\in \\cT _n$  and $p(x) = \\sum _{|l|\\leq n} a_l\n  e^{2\\pi i l x}$ the point evaluation at $\\xkn \\in \\cX _n$ is\n  precisely \n$$\n \\tkn ^{1\/2} p(\\xkn ) = (U_n a )_k    \\, ,\n$$ and  the sampled $2$-norm is \n$$\n\\sum _{k=1} ^{L_n} |p(\\xkn )|^2 \\tau _{n,k}  = \\langle U_n a , U_n\na\\rangle = \\langle T_n a, a \\rangle  \\, .\n$$\nBy \\eqref{eq:1} the spectrum of $T_n$ is contained in the interval\n$[A,B]$. \n\n(ii) This is the standard formula for the solution of a least squares\nproblem by means of the Moore-Penrose pseudo-inverse $U_n ^\\dagger =\n(U_n^* U_n)\\inv U_n ^* = T_n\\inv U_n^*$. \n\\end{proof}\n\n\nHere is  the decisive estimate for the  second term in\n\\eqref{eq:4}. The following lemma relates the solution to the least\nsquares problem~\\eqref{eq:8} to the best approximation of $f$ in $\\cT\n_n$. Compare~\\cite{Gro99,Gro01b} for an early use of this argument.  \n\n\\begin{prop}\\label{prop1}\nLet $p_n$ be the solution of the least squares problem\n\\eqref{eq:8}. Then \n\\begin{equation}\n  \\label{eq:13}\n  \\|P_nf - p_n \\|_2^2 \\leq A^{-2} B \\sum _{k=1}^{L_n} |f(\\xkn ) - P_n\n  f(\\xkn )|^2 \\tkn \\, .\n\\end{equation}\n  \\end{prop}\n\n  \\begin{proof}\n Let $f_n \\in \\bC ^{2n+1}$ be the Fourier coefficients of the\nprojection $P_nf$, i.e, $f_n = (\\fhat (-n), \\fhat (-n+1), \\dots, \\fhat\n(n-1), \\fhat (n))$. Then by Plancherel's theorem\n$$\n\\|P_n f - p_n \\|^2_2 = \\| f_n  - a_n \\|^2_2 \\, ,\n$$\nwhere the norm on the left-hand side is taken in $L^2(\\bT)$ and on the\nright-hand side in $\\bC ^{2n+1}$. Using \\eqref{eq:12} for the solution   of\nthe least squares problem \\eqref{eq:8} and $T_n = U_n^* U_n$, we  obtain\n\\begin{align*}\n   \\| f_n  - a_n \\|^2_2 &= \\|f_n - T_n \\inv U_N^* y_n\\|_2^2 \\\\\n&= \\| T_n \\inv U_n^* (U_n  f_n - y_n)\\|_2^2 \\\\\n& \\leq A^{-2} B \\|U_n f_n - y_n\\|_2^2 \\, , \n\\end{align*}\n      because the operator norm of $T_n\\inv $ is bounded by $A\\inv $\n      and the norm of $U^*_n$ is bounded by $\\|U_n ^*\\| = \\|U_n\\|=\n      \\|U_n^* U_n \\|^{1\/2} = \\|T_n \\|^{1\/2} \\leq B^{1\/2}$. \nFinally, \n$$\n(U_nf_n)_k = \\tkn ^{1\/2} \\sum _{|l| \\leq n} e^{2\\pi i \n  \\xkn l}  \\hat{f}(l) = \\tkn ^{1\/2} P_n\nf(\\xkn )  \\, ,\n$$\nand thus\n$$\n\\|U_n f_n - y\\|_2^2 = \\sum _{k=1} ^{L_n} |P_nf(\\xkn ) - f(\\xkn )|^2\n\\tkn  \\, ,\n$$ \nand the statement is proved. \n  \\end{proof}\n\n\n\\subsection{Proof of Theorem~\\ref{tm1}}\n\\begin{proof}\n(i) We use  the orthogonal  decomposition  $\n\\|f- p_n \\|_2^2 =   \\|f- P_nf \\|_2^2 +  \\|P_nf - p_n \\|_2^2 \\, .\n$\nThen  Lemma \\ref{lm1}(ii) yields \n$$\n  \\|f- P_nf \\|_2^2 \\leq \\|f- P_nf \\|_\\infty^2 \\leq  \\,\\|f\\|_{\\hs } ^2 \\,\n\\phi _\\sigma (n)^2\n  \\, ,\n$$\nand Proposition~\\ref{prop1} yields \n$$\n  \\|P_nf - p_n \\|_2^2 \\leq A^{-2} B \\sum _{k=1}^{L_n} |f(\\xkn ) - P_n\n  f(\\xkn )|^2 \\tkn \\, .\n$$\nWe now apply \\eqref{eq:7} and Lemma~\\ref{lm1}(ii) to $f-P_nf \\in \\hs\n(\\bT )$ and\ncontinue the inequality as \n$$\n\\sum _{k=1}^{L_n} |f(\\xkn ) - P_n\n  f(\\xkn )|^2 \\tkn  \\leq B \\|f-P_nf\\|_{\\infty} ^2 \\leq B \\|f\\|_{\\hs }\n  ^2 \\phi _\\sigma (n)^2 \\, .\n$$\nThe combination of these inequalities yields the final error estimate\n\\begin{equation}\n  \\label{eq:14}\n  \\|f-p_n\\|_2^2 \\leq     \\big( 1 +\n  \\frac{B^2}{A^2} \\big) \\|f\\|_{\\hs } ^2 \\, \\phi _\\sigma (n) ^2 \\, .\n\\end{equation}\nSince $\\phi _\\sigma (n)  = \\cO (n^{-\\sigma +1\/2})$, Theorem~\\ref{tm1} is\nproved. \n\n(ii) If $f$ can be extended to an analytic function of the strip $\\{z\\in \\bC\n: |\\mathrm{Im} z| <  \\rho  _0 \\}$, then its Fourier coefficients decay\nexponentially as $|\\fhat (k) | \\leq c_\\rho e^{-\\rho |k|}$ for every\n$\\rho <\\rho _0$ with an appropriate constant. Consequently\n\\begin{equation}\n  \\label{eq:abc}\n\\|f-P_nf\\|_\\infty \\leq \\sum _{|k|>n} |\\fhat (k)| \\leq  c_\\rho \\sum\n_{|k|>n} e^{-\\rho |k|} = \\frac{2c_\\rho}{e^{\\rho}-1} e^{-\\rho n} \\, ,\n\\end{equation}\nwhich proves the exponential decay.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Quadrature Rules}\nTo deduce a set of quadrature rules, we use frame theory to obtain\nsuitable weights. See~\\cite{Chr16,DS52} for the basic facts. The\nfollowing argument is typical in sampling theory, whereas often the\nderivation of quadrature rules relies on abstract functional analytic\narguments as in~\\cite{MNW01}.\n\nLet $k^{(n)}_x\\in \\cT_n$ be the reproducing kernel of $\\cT _n$ defined by\n$p(x) = \\langle p, k^{(n)}_x \\rangle $ for all $\\pi \\in \\cT _n$ and $x\\in\n\\bT $. In fact, $k^{(n)}_x(y) = \\frac{\\sin (2n+1)\\pi (y-x)}{\\sin \\pi (y-x)}$ is just the Dirichlet kernel for $\\cT\n_n$. In the language of frame theory each inequality of \\eqref{eq:1}\nsimply says that $\\{\\tkn ^{1\/2} \\, k^{(n)}_{\\xkn\n} : k=1, \\dots , L_n\\}$ is a frame for $\\cT _n$ with  frame\nbounds $A,B>0$ independent of $n$. Equivalently, the   associated frame\noperator\n$S_np = \\sum _{k=1}^{L_n} \\tkn  \\langle p, k^{(n)}_{\\xkn } \\rangle k^{(n)}_{\\xkn }\n$\nis invertible on $\\cT _n$ for every $n\\in \\bN $ and we obtain the dual\nframe $e_{n,k} =  S_n \\inv ( \\tkn ^{1\/2}  k^{(n)}_{\\xkn }) $. The factorization\n$S_n\\inv S_n = \\mathrm{I}_{\\cT _n}$ yields the following\nreconstruction formula for all  trigonometric polynomials $p \\in \\cT\n_n$ from their samples:    \n\\begin{equation}\n  \\label{eq:17}\n  p = \\sum _{k=1}^{L_n} \\tkn   \\langle p , k^{(n)}_{\\xkn } \\rangle\n  S\\inv k^{(n)}_{\\xkn } = \\sum _{k=1}^{L_n} \\tkn ^{1\/2} p(\\xkn ) \\,\n  e_{n,k} \\, \n \n\\end{equation}\nFurthermore,  $\\{e_{n,k} : k=1, \\dots, L_n\\}$ is a frame for $\\cT _n$\nwith frame bounds $B\\inv $ and $ A\\inv $, again independent of $n$.\nThis property implies in particular that for the constant \nfunction $1$ with $\\|1\\|_2 = 1$ we have\n\\begin{equation}\n  \\label{eq:a1}\n  \\sum _{k=1} ^{L_n} |\\langle  1, e_{n,k}  \\rangle |^2 \\leq A\\inv \\|1\\|_2\n  = A\\inv \\, .\n\\end{equation}\nWe now define the weights for the quadrature rules by \n\\begin{equation}\n  \\label{eq:18}\n  w_{n,k} = \\tkn ^{1\/2} \\langle  e_{n,k} ,1 \\rangle =  \\tkn ^{1\/2} \\int\n  _{-1\/2} ^{1\/2} e_{n,k}(x) \\, dx \\, , \n\\end{equation}\nand the corresponding quadrature rule by \n\\begin{equation} \\label{eq:18b}\n  I_n(f) = \\sum _{k=1}^{L_n} f(\\xkn ) w_{n,k} \\, .\n\\end{equation}\nWe also  write $I(f) = \\int _{-1\/2} ^{1\/2} f(x) \\, dx $ for the\nintegral of $f$ on $\\bT $. \n\nAs a consequence of the definitions we obtain the following easy\nproperties of this quadrature rule.\n\\begin{lemma} \\label{lem-quad}\n  Let  $w_{n,k}$ and $I_n$ be  defined as in \\eqref{eq:18} and\n  \\eqref{eq:18b}.\n\n  (i) Then the quadrature rule $I_n$ is exact on $\\cT _n$, i.e.,\n  $I_n(p) = I(p)$ for all $p \\in \\cT _n$.\n\n  (ii) For $f\\in \\hs (\\bT )$  we have\n  \\begin{equation}\n    \\label{eq:a2}\n    |I_n(f)|^2 \\leq A\\inv \\sum _{k=1}^{L_n} |f(\\xkn )|^2 \\tkn  \\leq\n    \\tfrac{B}{A}\\|f\\|_\\infty ^2 \\, .\n  \\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n  (i) follows from \\eqref{eq:17}. For (ii) we use\n  \\begin{equation*}\n    |I_n(f)|^2 \\leq \\big( \\sum _{k=1}^{L_n} |f(\\xkn )|^2 \\tkn \\Big) \\,\n\\Big(  \\sum _{k=1}^{L_n} |\\langle  e_{n,k} , 1 \\rangle )|^2  \\Big)\n\\leq \\frac{B}{A} \\|f\\|_\\infty ^2 \\, .\n  \\end{equation*}\n\\end{proof}\n\nAs a consequence of Theorem~\\ref{tm1} we obtain the following\nconvergence theorem.\n\n\\begin{tm}\n  \\label{tm2}\nLet $\\cX $ be a \\mz\\ family with weights $\\tau $ and let $\\{I_n: n \\in \\bN \\}$ be the\nassociated sequence of  quadrature rules.\n\n(i) If $f\\in C(\\bT )$, then \n\\begin{equation}\n  \\label{eq:190}\n  |I(f)  - I_n (f)| \\leq (1+\\sqrt{\\kappa} )  \\inf _{p \\in \\cT _n}\n  \\|f-p\\|_\\infty \\, .  \n\\end{equation}\nConsequently, if $f\\in C^\\sigma (\\bT )$, then $  |I(f)  - I_n (f)| =\n\\cO (n^{-\\sigma })$. \n\n\n(ii) If $f\\in \\hs $ for $\\sigma >1\/2$, then \n\\begin{equation}\n  \\label{eq:19}\n  |I(f)  - I_n (f)| \\leq (1+ \\sqrt{\\kappa })   \\|f\\|_{\\hs } \\phi\n  _\\sigma (n) \\, ,\n\\end{equation}\nwith $\\phi _\\sigma (n) = (\\sigma -1\/2)^{-1\/2} n^{-\\sigma +1\/2}$. \n\n\n \n(iii) If $f$ extends to an analytic function on a strip  $\\{z\\in \\bC\n: |\\mathrm{Im}\\, z| <  \\rho _0 \\}$, then  for $\\rho <\\rho _0$\n$$\n|I(f) - I_n (f)| = \\cO (e^{-\\rho n}) \\, .\n$$\n\\end{tm}\n\n\n\\begin{proof}\n(i) and (ii)  Let  $P_nf$ is the orthogonal projection of $f$\nonto $\\cT _N$ and  $q_n$ be the best approximation of $f$ in $\\cT _n$  with respect\nto $\\| \\cdot \\|_\\infty $. \n \n  Since   $I_n$ is exact on\n  $\\cT _n$, we  have $I(P_nf)  = I_n (P_nf)$ and $I(q_n) =\n  I_n(q_n)$. Then we obtain  with \\eqref{eq:a2} that \n  \\begin{align*}\n    |I(f) - I_n(f)| &\\leq |I(f-q_n)| + |I_n(q_n - f)| \\\\\n                    &\\leq \\|f-q_n\\|_\\infty +   (B\/A)^{1\/2} \\|f-q_n\\|_\\infty \\\\\n                    & =  (1+\\sqrt{\\kappa } ) \\inf _{p\\in \\cT _n}\n                      \\|f-p\\|_\\infty \\, .\n  \\end{align*}\nFor the approximating polynomial $P_nf$ we obtain \n  \\begin{align*}\n    |I(f) - I_n(f)| &\\leq |I(f-P_nf)| + |I_n(P_nf - f)| \\\\\n                    &\\leq \\|f-P_nf\\|_\\infty +   (B\/A)^{1\/2} \\|f-P_nf\\|_\\infty \\\\\n    &\\leq (1+\\sqrt{\\kappa } ) \\,    \\|f\\|_{\\hs } \\, \\phi _\\sigma (n)\n      \\, ,   \\end{align*}\n    with $\\phi _\\sigma (n) = (\\sigma -1\/2)^{-1\/2} n^{-\\sigma +1\/2}$\n  by Lemma~\\ref{lm1}.\n  \n(iii) If $f$ can be extended to the strip $\\{z\\in \\bC\n: |\\mathrm{Im} z| <  \\rho  _0 \\}$, then we use the error estimate\n\\eqref{eq:abc} for $\\|f-P_nf\\|_\\infty$ and obtain \n$$\n  |I(f) - I_n(f)| \\leq (1+ (B\/A)^{1\/2}) \\|f-P_nf\\|_\\infty \\leq c_\\rho '\n  (1+\\sqrt{\\kappa }) \\, e^{-\\rho n}\n$$\nfor every $\\rho <\\rho _0$ with a constant $c_\\rho '$ depending on $\\rho\n$. \n\\end{proof}\n\nTheorem~\\ref{tm2} answers a question of N.\\ Trefethen~\\cite{AT17}\nabout the rate of convergence of the standard quadrature rules for\nscattered nodes.\n\n\\section{General Approximation Theorems}\n Theorem~\\ref{tm1} and the quadrature rule of Theorem~\\ref{tm2}\nrequired hardly any tools, and the proofs  use  only the definitions of  \\mz\\\ninequalities, Sobolev spaces, and the solution formula for least square\nproblems. We will now show that the results for $\\bT $  can be\nextended   significantly  with a mere change of\nnotation.\n\nFor an axiomatic approach to approximation theorems from samples and\nquadrature rules, we assume that $M$ is a\ncompact space and $\\mu $ is a probability measure on $M$. Furthermore, \n\n(i)  $\\{\\phi _k : k\\in \\bN\\}$ is an orthonormal basis for $L^2(M,\\mu\n)$.  In agreement with the notation for Fourier series, we write $\\fhat (k)\n= \\langle f, \\phi _k\\rangle =\n\\int _M f(x) \\overline{\\phi _k(x)} \\, d\\mu (x)$ for the $k$-th\ncoefficient, so that the orthogonal expansion is $f= \\sum _k \\fhat (k)  \\phi _k$. \n\n(ii) Next, let  $\\lambda _k \\geq 0$ be  a non-decreasing  sequence  with\n$\\lim _{k\\to \\infty } \\lambda _k = \\infty $. \nThe associated Sobolev\nspace $\\hs (M)$ is defined by\n\\begin{equation}\n  \\label{eq:20}\n  \\hs (M) = \\{ f\\in L^2(M): \\|f\\|_{\\hs } ^2 = \\sum _{k=1}^\\infty\n  |\\fhat (k)|^2 (1+\\lambda _k ^2)^{\\sigma } < \\infty \\} \\, . \n\\end{equation}\nWith $\\cP _n$ we denote the set of ``polynomials'' of degree $n$ on\n$M$ by \n\\begin{equation}\n  \\label{eq:21}\n  \\cP _n = \\{ p \\in L^2(M): p = \\sum _{k: \\lambda _k \\leq n} \\fhat (k)\n  \\, \\phi _k \\} \\, .\n\\end{equation}\n This space is  finite-dimensional because $\\lim _{k\\to \\infty }\n \\lambda _k = \\infty $.  The definition of $\\cP _n$ encapsulates the\n appropriate  notion of bandlimitedness with respect to the basis\n $\\{ \\phi _n \\}$; in the  terminology of \\cite{FM10}  the functions \n in $\\cP_n $ are called  diffusion polynomials.\n\n Note that both $\\hs (M)$ and $\\cP _n$ depend on the orthonormal basis\n and on the sequence $\\{\\lambda _k\\}$. We may think of $\\{\\phi _k\\}$\n as the set of eigenfunctions of an unbounded positive operator on\n $L^2(M,\\mu )$ with eigenvalues $ \\lambda _k$.  \n\n The following table illustrates the transition from the set-up in\nSection~2 to the general theory. \n\n\n\n\n\\begin{table}[h!]\n  \\begin{center}\n    \\caption{Generalization}\n    \\label{tab:table1}\n    \\begin{tabular}{l|l}\n      \\emph{from torus $\\bT$} & \\emph{to manifold $M$} \\\\\n      \\hline\n      ONB $e^{2\\pi i k x}$ for $L^2(\\bT )$ &   ONB $\\phi _k (x) $ for  $L^2(M,\\mu )$ \\\\\n    Fourier coefficients $\\fhat (k) $ & Fourier coefficients~$ \\langle f, \\phi _k \\rangle $ \\\\\n    Eigenvalues $ k^2$ of $-\\frac{1}{4\\pi ^2}\\tfrac{d^2}{dx^2} $ &  ``Eigenvalues'' $\\lambda _k \\geq 0 $, $\\lambda _k \\to \\infty $ \\\\\n    Sobolev space $\\hs (\\bT)$ &   Sobolev space $\\hs (M) $  \\\\\n    Trigonometric polynomials $\\cT _n$  &   ``Polynomials'' $\\cP _n\n   $, $p = \\sum _{k: \\lambda _k \\leq n } \\fhat (k) \\, \\phi _k  $\n     \\end{tabular}\n  \\end{center}\n\\end{table}\n\n \n \n For meaningful statements we make the following natural assumptions:\n\n (i)  Every basis element $\\phi _k$ is continuous (and thus bounded)\n on $M$ and $\\phi\n_1 \\equiv 1$. Then the point evaluation $p \\to p(x)$ makes sense on\n$\\cP _n$ and $\\int _M f d\\mu = \\langle f, \\phi _1\\rangle$. \n\n(ii) There exists  a critical index\n$\\sigma _{\\mathrm{crit}}$  such that for $\\sigma > \\scrit $ the sum\n\\begin{equation}\n  \\label{eq:a3}\nC_\\sigma ^2 = \\sup _{x\\in M}  \\sum _{k=1}^\\infty |\\phi _k(x)|^2\n(1+\\lambda _k ^2)^{-\\sigma } <\\infty\n  \\, \n\\end{equation}\nconverges\\footnote{In  \\cite{BCCG14} the expression  $\\sum _{k=1}^\\infty \\phi _k(x) \\phi _k(y)\n(1+\\lambda _k ^2)^{-\\sigma }$ is called the Bessel kernel associated\nto the orthonormal basis $\\{\\phi _k\\}$.}.\n\n(iii) The error estimates will be in terms of the remainder function \n\\begin{equation}\n  \\label{eq:26}\n  \\phi _\\sigma  (n) = \\sup _{x\\in M} \\Big(\\sum _{k: \\lambda _k > n} |\\phi _k(x)|^2 (1+ \\lambda _k ^2)^{-\\sigma\n} \\Big)^{1\/2}  \\, .\n\\end{equation}\nBy \\eqref{eq:a3}  $\\phi _\\sigma (n) \\to 0$ as $n\\to \\infty $.  \n\nA \\mz\\ family  $\\cX $ for $M$  is  a doubly-indexed set $\\cX = \\{ \\xkn\n: n\\in \\bN , k=1 , \\dots , L_n\\} \\subseteq\nM$ with associated weights $\\{ \\tkn \\}$, such that \n\\begin{equation}\n  \\label{eq:1a}\nA \\|p\\|_2^2 \\leq \\sum _{k=1} ^{L_n} |p(\\xkn )|^2 \\tau _{n,k} \\leq B\n\\|p \\|_2^2 \\qquad \\text{ for all } p \\in \\cP _n \\, ,\n\\end{equation}\nwith constants $A,B >0$ independent of $n$.\n\n\\subsection{Embeddings}\nWe first prove the versions for approximation and embedding in the\ngeneral context.\n\n\\begin{lemma}\n  \\label{sobgen}\n  (i) If $\\sigma > \\scrit $, then $\\hs (M) \\subseteq C(M)$, in fact,\n  $\\| f \\|_\\infty \\leq C_\\sigma \\, \\|f\\|_{\\hs }$.  \n\n  (ii)\nFor $f\\in C(M)$ \n  \\begin{equation}\n    \\label{eq:a4}\n    \\|f-P_nf \\|_2 \\leq \\|f-P_nf\\|_\\infty \\leq \\|f\\|_{\\hs } \\phi\n    _\\sigma (n) \\, .\n  \\end{equation}\n\n(iii) Assume that $\\cX $ is a \\mz\\ family with weights $\\tau $.  For $f\\in \\hs (M) $ we have \n\\begin{equation}\n  \\label{eq:6a}\n  \\sum _{k=1}^{L_n} |f(\\xkn )|^2 \\tkn \\leq B C_\\sigma ^2 \\|f\\|_{\\hs }^2 \\, .\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n(i) and (ii):\nLet $f = \\sum _{k=1}^\\infty \\fhat (k) \\phi _k$. Then with the\nCauchy-Schwarz inequality, \n\\begin{align*}\n|f(x) | \\leq \\Big(\\sum _{k=1}^\\infty |\\fhat (k)|^2 (1+\\lambda\n          _k ^2)^\\sigma   \\Big)^{1\/2} \\, \\sup _{x\\in M} \\Big(\\sum\n          _{k=1}^\\infty |\\phi _k(x)|^2 (1+\\lambda\n          _k ^2)^{-\\sigma }  \\Big)^{1\/2}  =  C_\\sigma \\|f\\|_{\\hs } \\, .\n\\end{align*}\nThe approximation error is estimated by \n\\begin{align}\n    \\|f-P_nf\\|_2 & \\leq   \\|f - P_n f \\|_\\infty \\notag \\\\\n  &\\leq \\|f \\|_{\\hs }  \\sup _{x\\in M}  \\Big(\\sum _{k: \\lambda _k > n} |\\phi _k(x)|^2 (1+ \\lambda _k )^{-\\sigma\n} \\Big)^{1\/2} = \\|f\\|_{\\hs } \\, \\phi _\\sigma (n) \\, .  \\label{eq:25}\n\\end{align}\nIn the first inequality we have used the fact that $\\mu $ is a\nprobability measure. \n\n(iii) is proved  as in Lemma~\\ref{lm1}.  The sampling inequalities\n\\eqref{eq:1} applied to the constant function $p\\equiv 1$  with\n$\\|p\\|_2=1$ yields\n\\begin{equation}\n  \\label{eq:7a}\n  A \\leq \\sum _{k=1}^{L_n} \\tkn \\leq B \\, .\n\\end{equation}\nConsequently, with the embedding of (i), we have \n$$\n\\sum _{k=1}^{L_n} |f(\\xkn )|^2 \\tkn \\leq  \\|f\\|_\\infty ^2 \\sum\n_{k=1}^{L_n}  \\tkn \\leq B C_\\sigma ^2 \\|f\\|_{\\hs }^2 \\, .\n$$ \n\\end{proof}\n\n\n\\subsection{Quasi-interpolation versus projection}\n\n To produce optimal\napproximations of $f$ in $\\cP _n$ from the samples $\\cX _n= \\{ \\xkn : k=1 , \\dots\n, L_n\\} $ in the $n$-th layer of $\\cX $, we solve the  sequence of least\nsquares problems\n\\begin{equation}\n  \\label{eq:8b}\np_n = \\mathrm{argmin} _{p\\in \\cT _n} \\sum _{k=1} ^{L_n} |f(\\xkn ) -\np(\\xkn )|^2 \\tkn \\, .\n\\end{equation}\nNow let $y_n = (\\tau _{n,1} ^{1\/2} f(x_{n,1}), \\dots , \\tau\n_{n,L_n}^{1\/2} f(x_{n,L_n})) \\in \\bC ^{L_n}$ be the\ngiven data vector, and $U_n $ be the $L_n \\times \\mathrm{dim}\\, \\cP _n$-matrix  with\nentries \n\\begin{equation}\n  \\label{eq:10a}\n(U_n)_{kl} = \\tkn ^{1\/2} \\phi _l (x_{n,k})  \\qquad k=1,\\dots , L_n,\n l = 1, \\dots , \\mathrm{dim}\\, \\cP _n\n\\, , \n\\end{equation}\nand set $T_n = U_n^* U_n $. With this notation the solution to\n\\eqref{eq:8b} is the polynomial   $p_n =\n\\sum _{k: \\lambda _k  \\leq n} a_{n,k} \\phi _k\n\\in \\cP _n$  with  coefficient vector \n\\begin{equation}\n  \\label{eq:12b}\n  a_n = T_n \\inv U_n ^* y_n  \\, .\n\\end{equation}\nAgain, since $\\langle T_nc,c\\rangle  = \\langle U_nc, U_nc  \\rangle =\n\\sum _{k=1}^{L_n} |p(\\xkn )|^2 \\tkn  $,  the spectrum of\nevery $T_n$ is contained in the interval $[A,B]$ and we have uniform\n upper bounds for the norms of $T_n$ and $T_n\\inv $. \n\nWe now have the analogue of Proposition~\\ref{prop1}.\n\n\\begin{prop} \\label{prop1gen}\nLet $\\cX $ be a \\mz\\ family with weights $\\tau $.   Let $p_n$ be the solution of the least squares problem\n\\eqref{eq:8b}. Then \n\\begin{align}\n  \\label{eq:13a}\n  \\|P_nf - p_n \\|_2^2 &\\leq A^{-2} B \\sum _{k=1}^{L_n} |f(\\xkn ) - P_n\n                        f(\\xkn )|^2 \\tkn  \\\\\n  & \\leq \\tfrac{B^2}{A^2} \\|f-P_nf\\|^2_\\infty \\leq\n  \\kappa ^2  \\|f\\|^2_{\\hs } \\phi _\\sigma (n)^2 \\, . \\notag \n\\end{align}\n\\end{prop}\n\nThe proof is identical to the proof of Proposition~\\ref{prop1}, this\ntime combined with Lemma~\\ref{sobgen}.\n\n\\subsection{Approximation of continuous functions from samples}\n\nIn this general context the  analog of Theorem~\\ref{tm1} read as\nfollows. \n\n\\begin{tm}\n  \\label{tm1a}\nAssume that $\\cX = \\{\\cX _n :n\\in \\bN \\}$ is a \\mz\\ family for $M$ with\ncondition number $\\kappa = B\/A$ and associated weights $\\{\\tkn \\}$. \n\n   If $f\\in H^\\sigma (M)$, then \n  \\begin{equation}\n    \\label{eq:3b}\n    \\|f-p_n\\|_2  \\leq \\sqrt{1+\\kappa ^2} \\|f\\|_{\\hs } \\phi _\\sigma (n) \\, .\n  \\end{equation}\n\\end{tm}\n\n\\begin{proof}\n The proof is identical to the proof of\n  Theorem~\\ref{tm1}. This is our main point. We simply use Lemma~\\ref{sobgen} and\n  Proposition~\\ref{prop1gen}, precisely, \\eqref{eq:a4} and\n  \\eqref{eq:13a}  in the  decomposition\n$$\n\\|f- p_n \\|_2^2 =   \\|f- P_nf \\|_2^2 +  \\|P_nf - p_n \\|_2^2 \\, .\n$$\n\\end{proof}\n\nNoting that the error estimates also hold for fixed $n$, we obtain the\nfollowing  useful consequence. \n\\begin{cor}\nAssume that $\\{ y_k : k=1, \\dots ,L\\}\\subseteq M$ is a sampling set for $\\cP _n$\nwith weights $\\tau _k$,\ni.e., for some $A,B>0$ and all $p \\in \\cP _n$\n$$\nA \\|p\\|_2^2 \\leq \\sum _{k=1}^L |p(y_k)|^2 \\tau _k \\leq B \\|p\\|_2^2 \\,\n.\n$$\n For $f\\in C(M)$ solve  $q = \\mathrm{argmin} _{p\\in \\cP _n} \\sum _{k=1}^L\n|f(y_k) - p(y_k)|^2 \\tau _k$.  Then \n  \\begin{align*}\n  \\|f-q\\|_2 &\\leq  \\big(1+\\big(\\tfrac{B}{A}\\big)^2 \\big)^{1\/2} \\|f-P_nf\\|_\\infty  \\, .\n\\end{align*}  \n\\end{cor}\nThe corollary gives a possible answer to  how well a given\nfunction on $M$ can be approximated from samples. Again, the statement\nhighlights the importance of sampling inequalities in approximation\ntheoretic problems. \n\n\n\\subsection{Quadrature rules}\n\\label{sec:Quadrat}\n\nLikewise the derivation and  convergence of the quadrature rules is\ncompletely analogous to Theorem~\\ref{tm2}.  The reproducing kernel for $\\cP _n$ is given by\n$k^{(n)}_x(y) = \\sum _{k : \\lambda _k \\leq n} \\overline{\\phi _k (x)}\n\\phi _k(y)$.\nThen  the \\mz\\ inequalities  \\eqref{eq:1a}  say that every set  $\\{\\tkn\n^{1\/2} \\, k^{(n)}_{\\xkn\n} : k=1, \\dots , L_n\\}$ is a frame for $\\cP _n$ with  uniform frame\nbounds $A,B>0$. We thus  obtain the dual\nframe $e_{n,k} \\in \\cP _n$ such that every  polynomial $p \\in \\cP\n_n$ can be reconstructed from the samples on $\\cX _n$ by     \n\\begin{equation}\n  \\label{eq:17a}\n  p = \\sum _{k=1}^{L_n} \\tkn   \\langle p , k^{(n)}_{\\xkn } \\rangle\n  S\\inv k^{(n)}_{\\xkn } = \\sum _{k=1}^{L_n} \\tkn ^{1\/2} p(\\xkn ) \\,\n  e_{n,k} \\, \n \n\\end{equation}\nSince $1\\in \\cP _n$ for all $n$, we have again\n\\begin{equation}\n  \\label{eq:a1b}\n  \\sum _{k=1} ^{L_n} |\\langle  1, e_{n,k}  \\rangle |^2 \\leq A\\inv \\|1\\|_2\n  = A\\inv \\, .\n\\end{equation}\n The weights for the quadrature rules are defined by \n\\begin{equation}\n  \\label{eq:18a}\n  w_{n,k} = \\tkn ^{1\/2} \\langle  e_{n,k} ,1 \\rangle =  \\tkn ^{1\/2} \\int\n  _M e_{n,k}(x) \\, d\\mu(x) \\, , \n\\end{equation}\nand the corresponding quadrature rule is defined by \n\\begin{equation} \\label{eq:18bb}\n  I_n(f) = \\sum _{k=1}^{L_n} f(\\xkn ) w_{n,k} \\, .\n\\end{equation}\n  Writing $I(f) = \\int _M f(x) \\, d\\mu(x) $,\nthe convergence rules for the quadrature \\eqref{eq:18bb} can now be\nstated as follows. \n\n\n\\begin{tm}\n  \\label{tm2a}\nLet $\\cX $ be a \\mz\\ family on $M$ with weights $\\tau $ and let $\\{I_n: n \\in \\bN \\}$ be the\n associated sequence of quadrature rules. Assume that $\\sigma\n>\\scrit $. \n\n\n(i) If $f\\in C(M )$, then \n\\begin{equation}\n  \\label{eq:190b}\n  |I(f)  - I_n (f)| \\leq (1+\\sqrt{\\kappa })  \\inf _{p \\in \\cP _n}\n  \\|f-p\\|_\\infty  \\, .\n\\end{equation}\n\n\n\n(ii) If $f\\in \\hs $ for $\\sigma >\\scrit $, then \n\\begin{equation}\n  \\label{eq:19b}\n  |I(f)  - I_n (f)| \\leq (1+\\sqrt{\\kappa })  \\|f\\|_{\\hs }\n  \\phi _\\sigma (n)  \\, . \n\\end{equation}\n\\end{tm}\n\n\nError estimates of this type are, of course, \nwell-known, see, e.g. ~\\cite{CGT11,BCCG14,HS06}.  \nThe main difference is in the constants: usually these depend mainly\non  the  mesh size of the points $\\xkn $, whereas Theorems~\\ref{tm1a}\nand \\ref{tm2a}  involve only the bounds of the \\mz\\ inequalities. \n\nTheorems~\\ref{tm1a} and \\ref{tm2a} are  pure formalism. Their main\ninsight is  that\napproximation theorems from samples are a direct consequence of the\nexistence of \\mz\\ families. Thus  the ``real'' and deep question was and still is how to construct a \\mz\\\nfamily for a given $M$ and orthonormal basis. This is precisely what\nis accomplished in \\cite{FM10,FM11} under similar assumptions on an\nabstract metric measure space.  \n\n\\vspace{2mm}\n\n\\noindent \\textbf{Example.}\n  For the torus $\\phi _k (x) = e^{2\\pi i kx}$ and\n$\\lambda _k = k$. Then the error function is \n$$\n\\phi _\\sigma (n)^2  = \\sum _{|k|>n} (1+k^2)^{-\\sigma } \\leq 2 \\int\n_n^\\infty x^{-2\\sigma } \\, dx = \\frac{2}{2\\sigma -1} n^{-2\\sigma +1}\n\\, .\n$$\nConsequently  $\\phi\n_\\sigma (n) = (\\sigma - 1\/2)^{-1\/2} n^{-\\sigma +1\/2} $ in Lemma~\\ref{lm1} and\nTheorem~\\ref{tm1}. \n\n\n\\subsection{The spectral function and \\mz\\ families}\n\nTheorem~\\ref{tm1a}, as formulated, is almost void of content, as the\nerror function $\\phi _\\sigma $ from \\eqref{eq:26} depends on the orthonormal basis and the chosen \n$\\lambda _k$ in a rather intransparent manner. With the interpretation\nof $(\\phi _k,\n\\lambda _k)$ as the eigenvalues and eigenfunctions of a positive\nunbounded operator, the  spectral theory of partial differential\noperators suggests a suitable condition to elaborate  the error\nfunction $\\phi _\\sigma $ further.\n\n\\begin{definition}\n  We say the orthonormal basis $\\{\\phi _k:k\\in \\bN \\}$ and the\n  eigenvalues $\\{\\lambda _k: k\\in \\bN \\}$ satisfy Weyl's law, if there\n  exist constants $d=d_{\\phi,\\lambda }>0$ and $C= C_{\\phi,\\lambda\n  }>0$, such that\n  \\begin{equation}\n    \\label{eq:weyl}\n    \\sum _{k: \\lambda _k \\leq n} |\\phi _k(x)|^2 \\leq C n^d \\, .\n  \\end{equation}\n\\end{definition}\nIntegrating   over $M$, \\eqref{eq:weyl} implies the eigenvalue count\n\\begin{equation}\n  \\label{eq:weyl2}\n\\# \\{ k: \\lambda _k \\leq n\\}  \\leq C n^d \\, .  \n\\end{equation}\nIn the spectral  theory of partial differential operators or pseudodifferential\noperators the function $  \\sum _{k: \\lambda _k \\leq n} |\\phi _k(x)|^2$\nis called the spectral function, and \\eqref{eq:weyl2} is Weyl's law\nfor the count of eigenvalues~\\cite{Gar53,hormander3,Shubin91}. In the theory\nof orthogonal polynomials  the function  $ (\\sum _{k\\leq n} |\\phi\n_k(x)|^2)\\inv $ is called the Christoffel function and  plays a\ncentral role in the investigation of orthogonal\npolynomials~\\cite{Nev86}. In this context assumption \\eqref{eq:weyl}\nsays that  the Christoffel function along a subsequence  determined by\nthe  $\\lambda _k$'s is bounded\npolynomially from below.  \n\nUnder the assumption of Weyl's law we can determine the critical\nexponent and the asymptotics of the error function precisely.  One may\nalso say that Weyl's law implies the correct version of the Sobolev\nembedding. \n\\begin{prop}\n  \\label{critval}\nLet $\\{\\phi _k:k\\in \\bN \\}$ be an orthonormal basis for $L^2(M,\\mu )$\nand $\\lambda _k \\geq 0$ be a non-decreasing sequence with $\\lim _{k\\to\n  \\infty } \\lambda _k = \\infty $. Assume that $(\\phi _k, \\lambda _k)$\nsatisfies Weyl's law \\eqref{eq:weyl} with exponent $d$. Then the critical value is $\\scrit = d\/2$, i.e., if $\\sigma\n  >d\/2$, then\n  $$\n  C_\\sigma ^2 = \\sup _{x\\in M}  \\sum _{k=1}^\\infty |\\phi _k(x)|^2\n  (1+\\lambda _k^2)^{-\\sigma } <\\infty \\, . \n  $$\nMoreover, the error function is\n  \\begin{equation}\n    \\label{eq:a5}\n    \\phi _\\sigma (n) = C_\\sigma  \\, n^{-\\sigma +d\/2}\n  \\end{equation}\n\\end{prop}\n\\begin{proof}\n  We only show \\eqref{eq:a5}. Choose $M \\in \\bN$, such that $2^{M}\n  \\leq n \\leq 2^{M+1}$. We  split the sum defining $\\phi _\\sigma $\n  into dyadic blocks and use \\eqref{eq:weyl} as follows:  \n  \\begin{align*}\n  \\sum _{\\lambda _k >n}^\\infty |\\phi _k(x)|^2\n    (1+\\lambda _k^2)^{-\\sigma }     & \\leq   \\sum _{n=M}^\\infty \\sum _{k: 2^n \\leq\n    \\lambda _k <  2^{n+1}} |\\phi _k(x)|^2   (1+\\lambda _k^2)^{-\\sigma } \\\\\n    &\\leq  \\sum _{n=M}^\\infty 2^{-2n \\sigma } \\sum _{k: 2^n \\leq\n      \\lambda _k <  2^{n+1}} |\\phi _k(x)|^2  \\\\\n    &\\leq   C \\sum _{n=M} ^\\infty 2^{-2n \\sigma } 2^{d(n+1)} \n     =  C 2^d\\sum _{n=M} ^\\infty 2^{n(d-2\\sigma )} \\\\\n    &= C \\frac{2^d}{ (1-2^{d-2\\sigma } )} \\,  2^{M(d-2\\sigma ) }  \\leq\n      C_\\sigma n^{-2\\sigma +d}  \\, ,\n  \\end{align*}\nwith convergence precisely for $\\sigma > d\/2$.   \n\\end{proof}\n\n\n\n\\noindent \\textbf{Example.} Let $-\\Delta $ be the Laplacian on a\nbounded domain $M\\subseteq \\rd $ with $C^\\infty$-boundary and $\\phi\n_k$ be its  eigenfunctions, \ni.e., $-\\Delta \\phi _k =\n\\lambda _k^2 \\phi _k$.\nThen the $(\\phi _k,\n\\lambda _k)$'s satisfy Weyl's law with exponent $d$ and the constant is\nroughly $C= \\mathrm{vol}\\,  (M)$.  Similar statements hold for the\nLaplace-Beltrami operator on a compact Riemannian manifold and for\ngeneral elliptic partial differential operators, see~\\cite{Gar53,hormander68,hormander3,Shubin91}\n\nWeyl's law plays an important role in  the theory and construction of \\mz\\\nfamilies in metric measure spaces by Filbir and\nMhaskar~\\cite{FM10,FM11}. In particular, they show that Weyl's law is\nequivalent to Gaussian estimates for the heat kernel. A weaker form of\nWeyl's law is equivalent to a sampling inequality~\\cite{Pes16}.\n\n\n\n\n\n\n\n\n\n\\vspace{ 2mm}\n\n\\noindent\\textbf{Concluding remarks.} The simplicity of the proofs\nlead to conceptual insights into the role of \\mz\\ families, but the\nresults are certainly limited.\n\n(i) The proofs work only for $p=2$.  \\mz\\ families with respect to\ngeneral $p$-norms seem to require different techniques. \n\n(ii) The weights for the  quadrature rules are not necessarily\npositive. So far, positive weights  are obtained only with special\nconstructions from \\mz\\\nfamilies with $p=1$ and sufficiently dense sampling on each\nlevel~\\cite{FM10}.\n\n(iii) The existence of \\mz\\ families has been established on many\nlevel of generality~\\cite{FM11,FFP16} through the construction of\nsufficiently fine meshes on each level. So far necessary density\nconditions have been found only for \\mz\\ on compact Riemannian\nmanifolds~\\cite{OP12}. It would be interesting to extend the scope of\nthe necessary conditions to the conditions used in Section~3 and to\ncompare the densities of the existing \\mz\\ families to the necessary\ndensity conditions. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzguah b/data_all_eng_slimpj/shuffled/split2/finalzzguah
new file mode 100644
index 0000000000000000000000000000000000000000..dc978679905bd66ea4459a95f62f08b7edcb1055
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzguah
@@ -0,0 +1,5 @@
+{"text":"\\section{\\@startsection {section}{1}{{\\zeta}@}{3.25ex plus 1ex minus\n\n\n\\numberwithin{equation}{section}\n\n\n\n\n\n\\begin{document}\n\n\\title{Backward and forward filtering under the weak H\\\"ormander conditio\n\n\n\n\\author{Andrea Pascucci         \\and\n        Antonello Pesce\n}\n\n\n\\institute{A. Pascucci \\at\n              Department of Mathematics, University of Bologna, Piazza di Porta san Donato, 5 Bologna \\\\\n              Tel.: +39-0512094428\\\\\n              \\email{andrea.pascucci@unibo.it}          \n           \\and\n           A. Pesce \\at\n              Department of Mathematics, University of Bologna, Piazza di Porta san Donato, 5 Bologna \\\\\n              \\email{antonello.pesce2@unibo.it}          \n}\n\n\\date{Received: date \/ Accepted: date}\n\n\n\\maketitle\n\n\\begin{abstract}\nWe derive the forward and backward filtering equations for a class of degenerate partially\nobservable diffusions, satisfying the weak H\\\"ormander condition. Our approach is based on the\nH\\\"older theory for degenerate SPDEs that allows to pursue the direct approaches proposed by N. V.\nKrylov and A. Zatezalo, and A. Yu. Veretennikov, avoiding the use of general results from\nfiltering theory. As a by-product we also provide existence, regularity and estimates for the\nfiltering density. \\keywords{filtering \\and H\\\"older theory of SPDEs \\and Langevin equation \\and\nweak H\\\"ormander condition}\n \\subclass{60G35 \\and 60H15 \\and 60J60 \\and 35H20}\n\\end{abstract}\n\n\n\\section{Introduction}\nThe classical kinetic model\n\\begin{equation}\\label{aaee1}\n  \\begin{cases}\n    dX_{t}=V_{t}dt, \\\\\n    dV_{t}={\\sigma} dW_{t},\\qquad {\\sigma}>0,\n  \\end{cases}\n\\end{equation}\nis a remarkable example of a system of SDEs whose Kolmogorov equation\n\\begin{equation}\\label{aaee1bb}\n \n  \\frac{{\\sigma}^{2}}{2}{probabilit\\`a }_{vv}f+v{probabilit\\`a }_{x}f+{probabilit\\`a }_{t}f=0,\\qquad (t,x,v)\\in{\\mathbb {R}}^{3},\n\\end{equation}\nis hypoelliptic but not uniformly parabolic. Precisely, \\eqref{aaee1bb} satisfies the {\\it weak}\nH\\\"ormander condition in that the drift plays a key role in the noise propagation (see\n\\cite{Kolmogorov2} and the introduction in \\cite{Hormander}). In \\eqref{aaee1} $W$ is a Brownian\nmotion and $X,V$ represent position and velocity of a particle. This type of SDEs arises in\nseveral linear and non-linear models in physics (see, for instance, \\cite{Cercignani},\n\\cite{Lions1},\n\\cite{Desvillettes}, \\cite{MR2130405})\nand in mathematical finance (see, for instance, \\cite{BarucciPolidoroVespri},\n\\cite{Pascucci2011}).\n\nIn this paper we study the filtering problem for \\eqref{aaee1}. To the best of our knowledge, this\nkind of problem was never considered in the literature, possibly because the known results for\nhypoelliptic SPDEs (e.g. \\cite{MR736147}, \\cite{MR705933}, \\cite{Krylov17}, \\cite{MR3839316} and\n\\cite{MR3706782}) do not apply in this case. Here we propose a unified approach for the derivation\nof the {\\it backward and forward filtering equations} based on the H\\\"older theory for degenerate\nSPDEs recently developed in \\cite{PascucciPesce1} and \\cite{pasc:pesc:19} (see also \\cite{Chow94}\nand \\cite{MR1755998} for similar results for uniformly parabolic SPDEs).\nHaving an existence and regularity theory at hand, we can pursue the ``direct'' approaches\nproposed by Krylov and Zatezalo \\cite{MR1795614} and Veretennikov \\cite{Veretennikov}, thus\navoiding the use of general results from filtering theory. In particular, as in \\cite{Veretennikov} we\nderive the backward filtering equation ``by hand'',\nwithout resorting to prior knowledge of the SPDE, in a more direct way compared to the classical\napproach in \\cite{MR553909}, \\cite{MR583435}, \\cite{MR1070361} or \\cite{MR3839316}.\n\nTo be more specific, we consider the following general setup: we assume that the position $X_{t}$\nand the velocity $V_{t}$ of a particle are scalar stochastic processes only partially observable\nthrough some observation process $Y_{t}$. The joint dynamics of $X,V$ and $Y$ is given by the\nsystem of SDEs\n\\begin{equation}\\label{eq1}\n\\begin{cases}\n  dX_t=V_tdt,\\\\\n  dV_t={b(t,X_t,V_t,Y_t)dt}\n  \\sigma^i(t,X_t,V_t,Y_t)dW^i_t,\\\\\n  dY_t=h(t,X_t,V_t,Y_t)dt\n  {_{\\scalebox{.6}{0}}}\\sigma^i(t,Y_t)dW^i_t,\n\\end{cases}\n\\end{equation}\nwhere we adopt the Einstein summation convention, $W_t=(W_t^1,\\cdots, W_t^{n})$ denotes a\n$n$-dimensional Brownian motion, with $n\\ge2$, defined on a complete probability space\n$({\\Omega},\\mathcal{F},P)$ with a filtration $(\\mathcal{F}_t)_{t\\in [0,T]}$ satisfying the usual assumptions. Hereafter,\nfor simplicity we set $Z_{t}=(X_{t},V_{t})$ and denote by $z=(x,v)$ and ${\\zeta}=({\\xi},{\\nu})$ the points in\n${\\mathbb {R}}^{2}$.\n\nLet $\\mathcal{F}_{t,T}^{Y}={\\sigma}(Y_s,t\\le s\\le T)$ define the filtration of observations and let ${\\varphi}$ be a\nbounded and continuous function, ${\\varphi}\\in bC({\\mathbb {R}}^{2})$. The filtering problem consists in finding\nthe best $\\mathcal{F}_{t,T}^{Y}$-measurable least-square estimate of ${\\varphi}(Z_{T})$, that is the conditional\nexpectation $E\\left[{\\varphi}(Z_{T})\\mid \\mathcal{F}_{t,T}^{Y}\\right]$. Our first result, Theorem \\ref{th2}\nshows that\n\\begin{equation}\\label{ae7bis}\n E\\left[{\\varphi}(Z_{T}^{t,z})\\mid {\\mathcal{F}^{Y}_{t,T}}\\right]=\n \\int_{{\\mathbb {R}}^{2}}\\hat{\\mathbf{\\mathbf{\\Gamma}}}(t,z;T,{\\zeta}){\\varphi}({\\zeta})d{\\zeta},\n\\end{equation}\nwhere $\\hat{\\mathbf{\\mathbf{\\Gamma}}}$ is the (normalized) fundamental solution of the {\\it forward filtering\nequation}; the latter is a SPDE of the form\n\\begin{equation}\\label{spde_forwbbb}\n d_{\\mathbf{B}}u_s({\\zeta})=\\mathcal{A}_{s,{\\zeta}}u_s({\\zeta})ds+\\mathcal{G}_{s,{\\zeta}}u_s({\\zeta})dW_s\n\\end{equation}\nwhere $\\mathbf{B}={probabilit\\`a }_s+{\\nu}{probabilit\\`a }_{\\xi}$ and\n\\begin{align}\n \\mathcal{A}_{s,{\\zeta}}u_s({\\zeta})=\\frac{1}{2}\\bar{a}_s({\\zeta}){probabilit\\`a }_{{\\nu}\\n}u_s({\\zeta})+\\text{\\it ``first order terms''},\\qquad\n \\mathcal{G}_{s,{\\zeta}}u_s({\\zeta})=\\bar{{\\sigma}}_s({\\zeta}){probabilit\\`a }_{{\\nu}}u_s({\\zeta})+\\bar{h}_s({\\zeta})u_s({\\zeta}).\n\\end{align}\nThe forward filtering SPDE is precisely formulated in \\eqref{Forward_eq2}. The symbol\n$d_{\\mathbf{B}}$ in \\eqref{spde_forwbbb} indicates that the SPDE is understood in the It\\^o (or\nstrong) sense, that is\n\\begin{equation}\\label{spde_forw1bbb}\n u_{s}\\left(\\gamma^{\\mathbf{B}}_{s-t}({\\zeta})\\right)=u_{t}({\\zeta})+\\int_{t}^{s}{\\mathcal{A}_{{\\tau},\\gamma^{\\mathbf{B}}_{{\\tau}-t}({\\zeta})}}u_{\\tau}(\\gamma^{\\mathbf{B}}_{{\\tau}-t}({\\zeta}))d{\\tau}\n + \\int_{t}^{s}\\mathcal{G}_{{\\tau},\\gamma^{\\mathbf{B}}_{{\\tau}-t}({\\zeta})}u_{\\tau}(\\gamma^{\\mathbf{B}}_{{\\tau}-t}({\\zeta}))dW_{{\\tau}},\\qquad s\\in[t,T],\n\\end{equation}\nwhere $s\\mapsto\\gamma^{\\mathbf{B}}_{s}({\\xi},{\\nu})$ denotes the integral curve, starting from $({\\xi},{\\nu})$, of the\nadvection vector field ${\\nu}{probabilit\\`a }_{{\\xi}}$ or, more explicitly, $\\gamma^{\\mathbf{B}}_{s}({\\xi},{\\nu})=({\\xi}+s{\\nu},{\\nu})$.\n{\\begin{example\nThe prototype of \\eqref{spde_forwbbb} is the Langevin SPDE\n\\begin{equation}\\label{spde_forw_pro}\n d_{\\mathbf{B}}u_s({\\xi},{\\nu})=\\frac{{\\sigma}^{2}}{2}{probabilit\\`a }_{{\\nu}\\n}u_s({\\xi},{\\nu})ds+{\\beta}{probabilit\\`a }_{{\\nu}}u_s({\\xi},{\\nu})dW_s,\n\\end{equation}\nwith ${\\sigma},{\\beta}$ constant parameters. Clearly, if $u_s=u_s({\\xi},{\\nu})$ is a smooth function then\n\\eqref{spde_forw_pro} can be written in the usual It\\^o form\n\\begin{equation}\\label{spde_forw_pro_bis}\n du_s({\\xi},{\\nu})=\\left(\\frac{{\\sigma}^{2}}{2}{probabilit\\`a }_{{\\nu}\\n}u_s({\\xi},{\\nu}){-}{\\nu}{probabilit\\`a }_{{\\xi}}u_s({\\xi},{\\nu})\\right)ds+{\\beta}{probabilit\\`a }_{{\\nu}}u_s({\\xi},{\\nu})dW_s.\n\\end{equation}\nNotice that ${probabilit\\`a }_{{\\xi}}$, being equal to the Lie bracket $[{probabilit\\`a }_{{\\nu}},\\mathbf{B}]$, has to be regarded\nas a {\\it third order derivative} in the intrinsic sense of subelliptic operators (cf.\n\\cite{MR657581}): this motivates the use of the ``Lie stochastic differential'' $d_{\\mathbf{B}}$\ninstead of the standard It\\^o differential in \\eqref{spde_forw}. Notice also that\n\\eqref{spde_forw_pro} reduces to the forward Kolmogorov (or Fokker-Planck) equation for\n\\eqref{aaee1} when ${\\beta}=0$.\n\\end{example}\n\nAnalogously, in Section \\ref{bSPDE} we prove that\n\\begin{equation}\n E\\left[{\\varphi}(Z_{T}^{t,z,y},Y_{T}^{t,z,y})\\mid {\\mathcal{F}^{Y}_{t,T}}\\right]=\n \\int\\limits_{{\\mathbb {R}}^{3}}\\bar{\\mathbf{\\Gamma}}(t,z,y;T,{\\zeta},{\\eta}){\\varphi}({\\zeta},{\\eta})d{\\zeta} d{\\eta}, \\qquad (t,z,y)\\in [0,T]\\times\n {\\mathbb {R}}^2\\times{\\mathbb {R}},\n\\end{equation}\nwhere $\\bar{\\mathbf{\\Gamma}}$ denotes the (normalized) fundamental solution of the {\\it backward filtering\nequation} that is a SPDE of the form\n\\begin{equation}\\label{spde_backbbb}\n -d_{{\\mathbf{B}}}u_t(z,y)=\\tilde{\\mathcal{A}}_{t}\n u_t(z,y)dt+\\tilde{\\mathcal{G}}_{t}u_t(z,y)\\star dW_{t}.\n\\end{equation}\nWe refer to \\eqref{Backward_eq2} for the precise formulation of the backward filtering SPDE. The\nsymbol $\\star$ means that \\eqref{spde_backbbb} is written in terms of the {\\it backward It\\^o\nintegral} whose definition is recalled in Section \\ref{Itoback} for reader's convenience.  We\nshall see that the coefficients of the forward filtering SPDE are random, while the coefficients\nof the backward filtering SPDE are deterministic. Moreover, \\eqref{spde_forwbbb} is posed in\n${\\mathbb {R}}^{3}$ (including the time variable) while \\eqref{spde_backbbb} is posed in ${\\mathbb {R}}^{4}$.\n\nThe rest of the paper is organized as follows. In Section \\ref{sec1} we resume and extend the\nH\\\"older theory for degenerate SPDEs satisfying the weak H\\\"ormander condition, developed in\n\\cite{PascucciPesce1} and \\cite{pasc:pesc:19}. In Section \\ref{sec2}, which is the core of the\npaper, we state the filtering problem and derive the forward and backward filtering SPDEs. Section\n\\ref{proofK} contains the proof of the results about the existence and Gaussian estimates for the\nfundamental solutions of the filtering SPDEs. In Section \\ref{Itoback} we collect the definition\nand some basic result about backward stochastic integration.\n\n\n\\section{Fundamental solution of Langevin-type SPDEs}\\label{sec1}\nWe present the H\\\"older theory for degenerate SPDEs that will be used in the derivation of the\nfiltering equations. We first introduce some general notation and the functional spaces used\nthroughout the paper.\n\nWe denote by $z=(x,v_{1},\\dots,v_{d})$ and ${\\zeta}=({\\xi},{\\nu}_{1},\\dots,{\\nu}_{d})$ the points in\n${\\mathbb {R}}\\times{\\mathbb {R}}^d$. Moreover, for any $k\\in{\\mathbb {N}}$, $0<{\\alpha}<1$ and $0\\le t<T$,\n\\begin{itemize}\n  \\item[i)] $m\\mathscr{B}_{t,T}$  (resp. $b\\mathscr{B}_{t,T}$) is the space of all real-valued (resp. bounded)\n  Borel measurable functions $f=f_{s}(z)$ on $[t,T]\\times {\\mathbb {R}}^{d+1}$;\n  \\item[ii)] $C^{0}_{t,T}$ (resp. $bC^{0}_{t,T}$) is the space of functions $f\\in m\\mathscr{B}_{t,T}$ (resp. $f\\in b\\mathscr{B}_{t,T}$)\n  that are continuous in $z$ and $C^{{\\alpha}}_{t,T}$ (resp. $bC^{{\\alpha}}_{t,T}$) is the space of functions $f\\in m\\mathscr{B}_{t,T}$ (resp. $f\\in b\\mathscr{B}_{t,T}$)\n  that are ${\\alpha}$-H\\\"older continuous in $z$ uniformly with respect to $s$, that is\n  $$\\sup_{s\\in[t,T]\\atop z\\neq {\\zeta}}\\frac{|f_{s}(z)-f_{s}({\\zeta})|}{|z-{\\zeta}|^{{\\alpha}}}<\\infty.$$\n  {We also denote by  $C^{0,1}_{t,T}$ the space of functions $f\\in m\\mathscr{B}_{t,T}$ that are Lipschitz continuous in $z$ uniformly with respect to\n  $s\\in [t,T]$};\n  \\item[iii)] $C^{k+{\\alpha}}_{t,T}$ (resp. $bC^{k+{\\alpha}}_{t,T}$) is the space of functions $f\\in m\\mathscr{B}_{t,T}$ that are $k$-times differentiable with respect to $z$\n  with derivatives in $C^{{\\alpha}}_{t,T}$  (resp. $bC^{{\\alpha}}_{t,T}$).\n\\end{itemize}\nWe use boldface to denote the stochastic version of the previous functional spaces. Let\n$(W_t)_{t\\in [0,T]}$ be a one-dimensional Brownian motion on a complete probability space\n$({\\Omega},\\mathcal{F},P)$, endowed with a filtration $(\\mathcal{F}_t)_{t\\in [0,T]}$ satisfying the usual conditions, and\nlet $\\mathcal{P}_{t,T}$ be the predictable ${\\sigma}$-algebra on $[t,T]\\times{\\Omega}$.\n\\begin{definition}\\label{def1}\nWe denote by $\\mathbf{C}^{k+{\\alpha}}_{t,T}$ the family of functions $f=f_{s}(z,{\\omega})$ on\n$[t,T]\\times{\\mathbb {R}}^{d+1}\\times{\\Omega}$ such that:\n\\begin{itemize}\n \\item[i)] $(s,z)\\mapsto f_{s}(z,{\\omega})\\in C^{k+{\\alpha}}_{t,T}$ for any ${\\omega}\\in{\\Omega}$;\n \\item[ii)] $(s,{\\omega})\\mapsto f_{s}(z,{\\omega})$ is $\\mathcal{P}_{t,T}$-measurable for any $z\\in{\\mathbb {R}}^{d+1}$.\n\\end{itemize}\nSimilarly, we define $\\mathbf{bC}^{k+{\\alpha}}_{t,T}$.\n\\end{definition}\n\nWe consider a class of degenerate SPDEs of the form\n\\begin{equation}\\label{spde_forw}\n d_{\\mathbf{B}}u_s({\\zeta})=\\mathcal{A}_{s,{\\zeta}}u_s({\\zeta})ds+\\mathcal{G}_{s,{\\zeta}}u_s({\\zeta})dW_s\n\\end{equation}\nwhere $\\mathbf{B}={probabilit\\`a }_s+{\\nu}_1{probabilit\\`a }_{\\xi}$ and\n\\begin{align}\n \\mathcal{A}_{s,{\\zeta}}u_s({\\zeta})&:=\\frac{1}{2}a^{ij}_s({\\zeta}){probabilit\\`a }_{{\\nu}_i{\\nu}_j}u_s({\\zeta})+b^i_s({\\zeta}){probabilit\\`a }_{{\\nu}_i}u_s({\\zeta})+c_s({\\zeta})u_s({\\zeta}), \\\\\n \\mathcal{G}_{s,{\\zeta}}u_s({\\zeta})&:={\\sigma}^{i}_s({\\zeta}){probabilit\\`a }_{{\\nu}_i}u_s({\\zeta})+h_s({\\zeta})u_s({\\zeta}).\n\\end{align}\n\\begin{definition}\\label{ad1}\nA solution to \\eqref{spde_forw} on $[t,T]$ is a process $u=u_{s}({\\xi},{\\nu})\\in \\mathbf{C}^{0}_{t,T}$\nthat is twice continuously differentiable in the variables ${\\nu}$ and solves the equation\n\\begin{equation}\\label{spde_forw1}\n u_{s}\\left(\\gamma^{\\mathbf{B}}_{s-t}({\\zeta})\\right)=u_{t}({\\zeta})+\\int_{t}^{s}{\\mathcal{A}_{{\\tau},\\gamma^{\\mathbf{B}}_{{\\tau}-t}({\\zeta})}}u_{\\tau}(\\gamma^{\\mathbf{B}}_{{\\tau}-t}({\\zeta}))d{\\tau}\n + \\int_{t}^{s}\\mathcal{G}_{{\\tau},\\gamma^{\\mathbf{B}}_{{\\tau}-t}({\\zeta})}u_{\\tau}(\\gamma^{\\mathbf{B}}_{{\\tau}-t}({\\zeta}))dW_{{\\tau}},\\qquad s\\in[t,T],\n\\end{equation}\nwhere $s\\mapsto\\gamma^{\\mathbf{B}}_{s}({\\xi},{\\nu})$ denotes the integral curve, starting from $({\\xi},{\\nu})$, of the\nadvection vector field ${\\nu}_1{probabilit\\`a }_{{\\xi}}$, that is $\\gamma^{\\mathbf{B}}_{s}({\\xi},{\\nu})=({\\xi}+s{\\nu}_1,{\\nu})$.\n\\end{definition}\n\\begin{definition}\\label{d1}\nA fundamental solution of the forward SPDE \\eqref{spde_forw} is a stochastic process\n${\\mathbf{\\Gamma}}={\\mathbf{\\Gamma}}(t,z;s,{\\zeta})$, defined for $0\\le t<s\\le T$ and $z,{\\zeta}\\in{\\mathbb {R}}^{d+1}$, such that for any\n$(t,z)\\in[0,T)\\times{\\mathbb {R}}^{d+1}$ and $t_{0}\\in(t,T)$ we have:\n\\begin{itemize}\n  \\item[i)] ${\\mathbf{\\Gamma}}(t,z;\\cdot,\\cdot)$ is a solution to \\eqref{spde_forw} on $[t_{0},T]$;\n  \\item[ii)] for any ${\\varphi}\\in bC({\\mathbb {R}}^{d+1})$ and $z_{0}\\in{\\mathbb {R}}^{d+1}$, we have\n    $$\\lim_{(s,{\\zeta})\\to(t,z_{0})\\atop s>t}\\int_{{\\mathbb {R}}^{2}}\n    {\\mathbf{\\Gamma}}(t,z;s,{\\zeta}){\\varphi}(z)dz={\\varphi}(z_{0}),\\qquad P\\text{-a.s.}$$\n\\end{itemize}\n\\end{definition}\nIn \\cite{pasc:pesc:19}, under suitable assumptions on the coefficients, we proved existence and\nGaussian-type estimates of a fundamental solution for \\eqref{spde_forw} when $b_s\\equiv c_s\\equiv\nh_s\\equiv 0$ and $d=1$. Here we slightly extend those results to an SPDE of the general form\n\\eqref{spde_forw} and to the backward version of it, that is\n\\begin{equation}\\label{spde_back}\n -d_{\\mathbf{B}}u_t(z)=\\mathcal{A}_{t,z}u_t(z)dt+\\mathcal{G}_{t,z}u_t(z)\\star dW_t, \\qquad\n \\mathbf{B}={probabilit\\`a }_t+v_1{probabilit\\`a }_x.\n\\end{equation}\n\nWe denote by $\\cev{\\mathbf{C}}^{k+{\\alpha}}_{t,T}$ (and $\\mathbf{b}\\cev{\\mathbf{C}}^{k+{\\alpha}}_{t,T}$) the\nstochastic H\\\"older spaces formally defined as in Definition \\ref{def1} with $\\mathcal{P}_{t,T}$\nin condition ii) replaced by the backward predictable ${\\sigma}$-algebra $\\cev{\\mathcal{P}}_{t,T}$\ndefined in terms of the backward Brownian filtration (cf. Section \\ref{Itoback}). Again,\n\\eqref{spde_back} is understood in the strong sense:\n\\begin{definition}\\label{ad2}\nA solution to \\eqref{spde_back} on $[0,s]$ is a process $u=u_{t}(x,v)\\in\n\\cev{\\mathbf{C}}^{0}_{0,s}$ that is twice continuously differentiable in the variables $v$ and\nsuch that\n\\begin{equation}\\label{spde_back1}\n u_{t}\\left(\\gamma^{\\mathbf{B}}_{s-t}(z)\\right)=u_{s}(z)+\\int_{t}^{s}{\\mathcal{A}_{{\\tau},\\gamma^{\\mathbf{B}}_{s-{\\tau}}(z)}}u_{\\tau}(\\gamma^{\\mathbf{B}}_{s-{\\tau}}(z))d{\\tau}\n + \\int_{t}^{s}\\mathcal{G}_{{\\tau},\\gamma^{\\mathbf{B}}_{s-{\\tau}}(z)}u_{\\tau}(\\gamma^{\\mathbf{B}}_{s-{\\tau}}(z))\\star dW_{{\\tau}},\\qquad t\\in[0,s].\n\\end{equation}\n\\end{definition}\n\\begin{definition}\\label{d2} A fundamental solution for\nthe backward SPDE \\eqref{spde_back} is a stochastic process $\\cev{\\mathbf{\\Gamma}}=\\cev{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})$\ndefined for $0\\le t<s\\le T$ and $z,{\\zeta}\\in{\\mathbb {R}}^{d+1}$, such that for any $(s,{\\zeta})\\in\n(0,T]\\times{\\mathbb {R}}^{d+1}$ and $t_{0}\\in(0,s)$ we have:\n\\begin{itemize}\n  \\item[i)] $\\cev{\\mathbf{\\Gamma}}(\\cdot,\\cdot;s,{\\zeta})$ is a solution to \\eqref{spde_back} on $[0,t_{0}]$;\n \\item[ii)] for any ${\\varphi}\\in bC({\\mathbb {R}}^{d+1})$ and $z_{0}\\in{\\mathbb {R}}^{d+1}$, we have\n    $$\\lim_{(t,z)\\to(s,z_{0})\\atop t<s}\\int_{{\\mathbb {R}}^{2}}\n    \\cev{\\mathbf{\\Gamma}}(t,z;s,{\\zeta}){\\varphi}({\\zeta})d{\\zeta}={\\varphi}(z_{0}),\\qquad P\\text{-a.s.}$$\n\\end{itemize}\n\\end{definition}\n\nNext we pose the standing assumptions on the coefficients of \\eqref{spde_forw} and\n\\eqref{spde_back}.\n\\begin{assumption}[\\bf Regularity]\\label{ass1}\nFor some ${\\alpha}\\in(0,1)$, we have:\n\\begin{itemize}\n  \\item[i)] $a\\in\\mathbf{bC}^{{\\alpha}}_{0,T}$, ${\\sigma}\\in\\mathbf{bC}^{3+{\\alpha}}_{0,T}$, $b,c\\in\\mathbf{bC}^{0}_{0,T}$, $h\\in\n\\mathbf{bC}^2_{0,T}$ in the forward SPDE \\eqref{spde_forw};\n  \\item[ii)] $a\\in\\mathbf{b}\\,\\cev{\\mathbf{C}}^{{\\alpha}}_{0,T}$, ${\\sigma}\\in\\mathbf{b}\\,\\cev{\\mathbf{C}}^{3+{\\alpha}}_{0,T}$,\n$b,c\\in\\mathbf{b}\\,\\cev{\\mathbf{C}}^{0}_{0,T}$, $h\\in \\mathbf{b}\\,\\cev{\\mathbf{C}}^2_{0,T}$ in the\nbackward SPDE \\eqref{spde_back}.\n\\end{itemize}\n\\end{assumption}\n\\begin{assumption}[\\bf Coercivity]\\label{ass2}\nThere exists a random, finite and positive constant $\\mathbf{m}$ such that\n\\begin{equation}\n \\langle (a_{t}(z)-{\\sigma}_{t}(z){\\sigma}_t^{\\ast}(z)){\\zeta},{\\zeta}\\rangle \\ge \\mathbf{m} |{\\zeta}|^2, \\qquad t\\in [0,T],\\ z,{\\zeta}\\in {\\mathbb {R}}^{d+1},\\ P\\text{-a.s.}\n\\end{equation}\n\\end{assumption}\nIn our analysis we make use of the It\\^o-Wentzell transform. Let $(x,v)\\in{\\mathbb {R}}^{d+1}$. For a fixed\n$t\\in [0,T)$ we consider the SDE in ${\\mathbb {R}}^{d}$\n\\begin{align}\\label{sde_forw}\n \\gamma^{\\text{\\rm\\tiny IW}}_{t,s}(x,v)&=v-\\int_{t}^s{\\sigma}_{\\tau}(x,\\gamma^{\\text{\\rm\\tiny IW}}_{t,{\\tau}}(x,v))dW_{\\tau}, \\qquad s\\in [t,T],\n\\intertext{and, for a fixed $s\\in (0,T]$, the SDE}\n \\theta^{\\text{\\rm\\tiny IW}}_{t,s}(x,v)&=v+\\int_{t}^{s}{\\sigma}_{\\tau}(x,\\theta^{\\text{\\rm\\tiny IW}}_{{\\tau},s}(x,v))\\star dW_{\\tau},\\qquad t\\in[0,s].\\label{sde_back}\n\\end{align}\nAssumption \\ref{ass1} ensures that \\eqref{sde_forw} and \\eqref{sde_back} have strong solutions and\nthe maps $(x,v)\\mapsto \\left(x,\\gamma^{\\text{\\rm\\tiny IW}}_{t,s}(x,v)\\right)$ and $(x,v)\\mapsto\n\\left(x,\\theta^{\\text{\\rm\\tiny IW}}_{t,s}(x,v)\\right)$ define forward and backward flows of diffeomorphisms of ${\\mathbb {R}}^{d+1}$\nrespectively. These changes of coordinates allow to transform the SPDEs \\eqref{spde_forw} and\n\\eqref{sde_back} into PDEs with random coefficients whose properties depend on the gradient of the\nstochastic flow: to have a control on it, we impose the some additional condition. {For any\nsuitably regular function $f=f(w):{\\mathbb {R}}^{N}\\longrightarrow {\\mathbb {R}}$, ${\\varepsilon}>0$ and multi-index ${\\beta}\\in\n{\\mathbb {N}}_{0}^{N}$, we set\n\\begin{equation}\\label{aea1}\n \\langle f\\rangle_{{\\varepsilon},{\\beta}}:=\\sup_{w\\in{\\mathbb {R}}^{N}}(1+|w|^2)^{{\\varepsilon}}|{probabilit\\`a }_{w}^{{\\beta}}f(w)|.\n\\end{equation}\n\\begin{assumption}\\label{ass3}\nThere exist ${\\varepsilon}>0$ and two random variables $M_1\\in L^{p}({\\Omega})$, with\n$p>\\max\\left\\{2,\\frac{1}{{\\varepsilon}}\\right\\}$, and $M_2\\in L^{\\infty}({\\Omega})$ such that with probability one\n\\begin{align\n  \\sup_{t\\in[0,T]}\\left(\\langle {\\sigma}_{t}\\rangle_{{\\varepsilon},{\\beta}}+\\langle {\\sigma}_{t}\\rangle_{1\/2+{\\varepsilon},{\\beta}'}\\right)&\\le M_{1},\\qquad |{\\beta}|=1,\\\n  |{\\beta}'|=2,3,\\\\\n  \\sup_{t\\in[0,T]}\\langle h_{t}\\rangle_{1\/2,{\\beta}}&\\le M_{2},\\qquad |{\\beta}|=1.\n\\end{align}\n\\end{assumption}} Assumption \\ref{ass3} requires that ${\\sigma}_{t}(z)$ and $h_t(z)$ flatten as $z\\to \\infty$. In\nparticular, this condition is clearly satisfied if ${\\sigma}$ and $h$ depend only on $t$ or, more\ngenerally, if the spatial gradients of ${\\sigma}$ and $h$ have compact support.\n\nIn order to state the main result of this section, Theorem \\ref{TH1} below, we need to introduce\nsome additional notation: we consider the Gaussian kernel\n\\begin{equation}\\label{gammal}\n \\Gamma_{\\lambda}(t,x,v)=\\frac{\\lambda}{t^{\\frac{d+3}{2}}}\\exp\\left(-\\frac{1}{2\\lambda}\\left(\\frac{x^{2}}{t^{3}}+\\frac{|v|^{2}}{t}\\right)\\right),\n  \\qquad t>0,\\ (x,v)\\in{\\mathbb {R}}\\times{\\mathbb {R}}^{d},\\ \\lambda>0.\n\\end{equation}\nTo fix ideas, for $d=1$ and up to some renormalization, $\\Gamma_{\\lambda}$ is the fundamental solution\nof the degenerate Langevin equation \\eqref{aaee1bb}. For a recent survey on the theory of this\nkind of ultra-parabolic operators and the related sub-elliptic structure, we refer to\n\\cite{anceschi}.\n\nIn the following statement, we denote by $g^{\\text{\\rm\\tiny IW},-1}$ (and $\\cev{g}^{\\text{\\rm\\tiny IW},-1}$) the inverse of the It\\^o-Wentzell\nstochastic flow $(x,v)\\mapsto g^{\\text{\\rm\\tiny IW}}(x,v):= \\left(x,\\gamma^{\\text{\\rm\\tiny IW}}_{t,s}(x,v)\\right)$ defined by\n\\eqref{sde_forw} (and $(x,v)\\mapsto \\cev{g}^{\\text{\\rm\\tiny IW}}(x,v):= \\left(x,\\theta^{\\text{\\rm\\tiny IW}}_{t,s}(x,v)\\right)$ defined by\n\\eqref{sde_back}, respectively). Moreover, we consider the vector field\n\\begin{align}\\label{flusso2}\n  \\mathbf{Y}_{t,s}(z)&:=\\Big((\\gamma^{\\text{\\rm\\tiny IW}}_{t,s})_1(z),-(\\gamma^{\\text{\\rm\\tiny IW}}_{t,s}(z))_1(\\nabla_v\\gamma^{\\text{\\rm\\tiny IW}}_{t,s})^{-1}(z){{probabilit\\`a }_x\\gamma^{\\text{\\rm\\tiny IW}}_{t,s}(z)}\\Big),\n\\end{align}\nwith $\\nabla_v\\gamma^{\\text{\\rm\\tiny IW}}=({probabilit\\`a }_{v_j}\\gamma^{\\text{\\rm\\tiny IW}}_i)_{i,j=1,\\cdots d}$ and ${probabilit\\`a }_x\\gamma^{\\text{\\rm\\tiny IW}}=({probabilit\\`a }_x\\gamma^{\\text{\\rm\\tiny IW}}_i)_{i=1,\\cdots d}$,\nand define $\\cev{\\mathbf{Y}}_{t,s}$ analogously. Eventually, equation\n  $$\\gamma_{s}^{t,z}=z+\\int_{t}^{s}\\mathbf{Y}_{t,{\\tau}}(\\gamma_{{\\tau}}^{t,z})d{\\tau},\\qquad s\\in[t,T],$$\ndefines the integral curve of $\\mathbf{Y}_{t,s}$ starting from $(t,z)$, and equation\n  $$\\cev{\\gamma}_{t}^{s,{\\zeta}}={\\zeta}+\\int_{t}^{s}\\cev{\\mathbf{Y}}_{{\\tau},s}(\\cev{\\gamma}_{{\\tau}}^{s,{\\zeta}})d{\\tau},\\qquad t\\in[0,s],$$\ndefines the integral curve of $\\cev{\\mathbf{Y}}_{t,s}$ ending at $(s,{\\zeta})$. The main result of this section is\nthe following theorem whose proof is postponed to Section \\ref{proofK}.\n\\begin{theorem}\\label{TH1}\nUnder Assumptions \\ref{ass1}-i), \\ref{ass2} and \\ref{ass3}, the forward SPDE \\eqref{spde_forw} has\na fundamental solution $\\mathbf{\\mathbf{\\Gamma}}$ and there exists a positive random variable $\\lambda$ such\nthat\n\\begin{align}\n \\Gamma_{\\lambda^{-1}}\\left({s-t}, g^{\\text{\\rm\\tiny IW},-1}_{t,s}({\\zeta})-\\gamma_{s}^{t,z} \\right) &\\leq\n \\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta}) \\leq \\Gamma_{\\lambda}\\left({s-t}, g^{\\text{\\rm\\tiny IW},-1}_{t,s}({\\zeta})-\\gamma_{s}^{t,z}\\right), \\label{t_e1}\\\\\n \\left| {probabilit\\`a }_{{\\nu}_i}\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\xi},{\\nu})\\right|&\\leq\n \\frac{1}{\\sqrt{s-t}}\\Gamma_{\\lambda}\\left({s-t}, g^{\\text{\\rm\\tiny IW},-1}_{t,s}({\\xi},{\\nu})-\\gamma_{s}^{t,z}\\right), \\label{t_e2}\\\\\n \\left| {probabilit\\`a }_{{\\nu}_i{\\nu}_j}\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\xi},{\\nu})\\right|\n &\\leq \\frac{1}{s-t}\\Gamma_{\\lambda}\\left({s-t}, g^{\\text{\\rm\\tiny IW},-1}_{t,s}({\\xi},{\\nu})-\\gamma_{s}^{t,z}\\right), \\label{t_e3}\n\\end{align}\nfor every $0\\leq t<s\\leq T$, $z,{\\zeta}=({\\xi},{\\nu})\\in {\\mathbb {R}}^{d+1}$ and $i,j=1,\\cdots d$, with probability\none.\n\nSimilarly, under Assumptions \\ref{ass1}-ii), \\ref{ass2} and \\ref{ass3}, the backward SPDE\n\\eqref{spde_back} has a fundamental solution $\\cev{\\mathbf{\\Gamma}}$ satisfying estimates\n\\begin{align}\n \\Gamma_{\\lambda^{-1}}\\left({s-t}, \\cev{g}^{\\text{\\rm\\tiny IW},-1}_{t,s}(z)-\\cev{\\gamma}_{t}^{s,{\\zeta}}\\right)\n &\\leq \\cev{\\mathbf{\\Gamma}}(t,z;s,{\\zeta}) \\leq \\Gamma_{\\lambda}\\left({s-t}, \\cev{g}^{\\text{\\rm\\tiny IW},-1}_{t,s}(z)-\\cev{\\gamma}_{t}^{s,{\\zeta}}\\right), \\label{t_e4}\\\\\n \\left| {probabilit\\`a }_{v_i}\\cev{\\mathbf{\\Gamma}}(t,x,v;s,{\\zeta})\\right|&\\leq\n \\frac{1}{\\sqrt{s-t}}\\Gamma_{\\lambda}\\left({s-t}, \\cev{g}^{\\text{\\rm\\tiny IW},-1}_{t,s}(x,v)-\\cev{\\gamma}_{t}^{s,{\\zeta}}\\right), \\label{t_e5}\\\\\n \\left| {probabilit\\`a }_{v_iv_j}\\cev{\\mathbf{\\Gamma}}(t,x,v;s,{\\zeta})\\right|\n &\\leq \\frac{1}{s-t}\\Gamma_{\\lambda}\\left({s-t}, \\cev{g}^{\\text{\\rm\\tiny IW},-1}_{t,s}(x,v)-\\cev{\\gamma}_{t}^{s,{\\zeta}}\\right), \\label{t_e6}\n\\end{align}\nfor every $0\\leq t<s\\leq T$, $z=(x,v),{\\zeta}\\in {\\mathbb {R}}^{d+1}$ and $i,j=1,\\dots d$, with probability one.\n\\end{theorem}\n\\begin{remark}\nWe would like to emphasize that Theorem \\ref{TH1} is new even in the deterministic case, i.e. when\n${\\sigma}\\equiv 0$, $h\\equiv 0$ and the coefficients are deterministic functions. In fact, a study of\nKolmogorov PDEs with coefficients measurable in time was only recently proposed in \\cite{brampol}:\nhowever in \\cite{brampol} the coefficients are assumed to be independent of the spatial variables\nthat is a very particular case where the fundamental solution is known explicitly.\n\\end{remark}\nIn the derivation of the forward filtering SPDE, we will use a deterministic backward Kolmogorov\nPDE to which Theorem \\ref{TH1} applies. Precisely, we will use the following\n\\begin{corollary}\\label{L_Cauchy}\nLet Assumption \\ref{ass2} with ${\\sigma}\\equiv 0$ be satisfied and let $a\\in bC^{{\\alpha}}_{0,T}$, $b,c\\in\nbC^{0}_{0,T}$, for some ${\\alpha}\\in(0,1)$, and ${\\varphi}\\in bC({\\mathbb {R}}^{d+1})$. Then there exists a bounded\nsolution of the backward Cauchy problem\n\\begin{equation}\\label{e_Cauchy2}\n  \\begin{cases}\n   -d_{\\mathbf{B}}f(t,z)=\\mathcal{A}_{t,z}f(t,z)dt,\\\\\n   f(T,\\cdot)={\\varphi},\n  \\end{cases}\n\\end{equation}\nin the sense of Definition \\ref{ad2}, that is\n\\begin{equation}\\label{e_Cauchy}\n f\\left(t,\\gamma^{\\mathbf{B}}_{T-t}(z) \\right)={\\varphi}(z)+\\int_t^T \\mathcal{A}_{s,\\gamma^{\\mathbf{B}}_{T-s}(z)} f\\left(s,\\gamma^{\\mathbf{B}}_{T-s}(z)\\right)ds, \\qquad\n (t,z)\\in[0,T]\\times {\\mathbb {R}}^{d+1},\n\\end{equation}\nwhere $\\gamma^{\\mathbf{B}}_{s}(x,v)=(x+sv_1,v)$. Moreover, {if ${\\varphi}\\in bC^{{\\alpha}}({\\mathbb {R}}^{d+1})$ for some ${\\alpha}\\in (0,1)$}\nthen there exists a positive constant $C$ such that,\n\\begin{equation}\\label{coreq}\n {\\sup_{(x,v)\\in {\\mathbb {R}}\\times\n {\\mathbb {R}}^{d}}|{probabilit\\`a }_{v}^{{\\beta}}f(t,x,v)|\\le C(T-t)^{-\\frac{|{\\beta}|-{\\alpha}}{2}},\\qquad 1\\le |{\\beta}|\\le 2.}\n\\end{equation}\n\\end{corollary}\n\n\n\n\\section{The filtering problem}\\label{sec2}\nConsider system \\eqref{eq1} and suppose that $h\\equiv {_{\\scalebox{.6}{0}}}\\sigma\\equiv 0$, that is no observation is available on\nthe solution $Z^{t,z}$ starting from $z$ at time $t$. Then, it is well known that, under suitable\nregularity and non-degeneracy assumptions on\n${\\sigma}$, we have\n\\begin{equation}\\label{aaee10}\n E\\left[{\\varphi}(Z_{T}^{t,z})\\right]=\\int_{{\\mathbb {R}}^2}{\\Gamma}(t,z;T,{\\zeta}){\\varphi}({\\zeta})d{\\zeta},\n\\end{equation}\nwhere\n${\\Gamma}={\\Gamma}(t,z;T,{\\zeta})$ is the fundamental solution of the backward Kolmogorov\noperator\n\\begin{equation}\\label{eai1}\n \\mathcal{K}=\\frac{|{\\sigma}|^{2}}{2}{probabilit\\`a }_{vv}+b{probabilit\\`a }_{v}+v{probabilit\\`a }_x+{probabilit\\`a }_t\n\\end{equation}\nwith respect to the variables $(t,x,v)$ and of its adjoint, the Fokker-Plank operator\n$\\mathcal{K}^{\\ast}$,\nw.r.t the forward variables $(T,{\\xi},{\\nu})$\n\n\nIn this section we study the filtering problem for system \\eqref{eq1} and, assuming that $Y$ is\nnot trivial, we prove a representation formula for $E\\left[{\\varphi}(Z_{T})\\mid \\mathcal{F}_{t,T}^{Y}\\right]$\nthat is analogous to \\eqref{aaee10} in the sense that it is written in terms of the fundamental\nsolution of a backward and a forward SPDE, whose existence is guaranteed by Theorem \\ref{TH1}.\nActually, in filtering theory, the derivation itself of the filtering SPDE is a non-trivial task.\n\nAs already mentioned, in our analysis we will adopt a {\\it direct approach}. However, we should\nacknowledge that there are at least two quite different {\\it direct approaches} proposed in the\nliterature: both of them are meant to avoid the purely probabilistic techniques of the general\nfiltering theory.\n\nThe direct approach by Krylov and Zatezalo \\cite{MR1795614} mimicks the derivation of the standard\nKolmogorov operator \\eqref{eai1}: roughly speaking, {\\it assuming that the filtering SPDE is known\nin advance}, one takes a solution $u_{t}$ (whose existence is guaranteed by Theorem \\ref{TH1}),\napplies the It\\^o formula to $u_{t}(Z_{t})$ and finally takes expectations. This is the approach\nwe follow in Section \\ref{fSPDE} to prove the existence of the {\\it forward filtering density} and\nthe representation of the conditional expectation $E\\left[{\\varphi}(Z_{T})\\mid \\mathcal{F}_{t,T}^{Y}\\right]$ in\nterms of it.\n\nOn the other hand, the direct approach by Veretennikov \\cite{MR1352749}, \\cite{Veretennikov},\nallows to derive the backward filtering SPDE ``by hand'', without knowing the equation in advance:\nthe main tools are the backward It\\^o calculus and the remarkable {\\it backward diffusion SPDE} of\nTheorem \\ref{t1}. We follow this approach in Section \\ref{bSPDE} to derive the backward filtering\nSPDE and the corresponding {\\it filtering density}. Note however that in Section \\ref{bSPDE} we\nonly provide an informal, yet quite detailed, derivation: a full proof is outside the scope of the\npresent paper and would require a generalization of the results of Section \\ref{Itoback} to\ndegenerate diffusions. This is certainly possible but would require some additional effort and is\npostponed to future research.\n\n\n\\medskip\nThroughout this section we assume the following non-degeneracy condition:\nthere exists a positive constant $m$ such that\n\\begin{align}\\label{A1}\n {|{_{\\scalebox{.6}{0}}}\\sigma(t,y)|^{2}\\ge m}, \\qquad\n \\langle\\left(I-\\frac{{_{\\scalebox{.6}{0}}}\\sigma(t,y){_{\\scalebox{.6}{0}}}\\sigma^{\\ast}(t,y)}{|{_{\\scalebox{.6}{0}}}\\sigma(t,y)|^2}\\right)\\sigma(t,z,y),\\sigma(t,z,y)\\rangle \\ge m,\n \\qquad t\\in [0,T],\\ z\\in{\\mathbb {R}}^2,\\ y\\in{\\mathbb {R}}.\n\\end{align}\nEquivalently, $|Q(t,y)\\sigma(t,z,y)|^2 \\ge m$, where $Q$ is the orthogonal projector on ${\\rm\nKer}{_{\\scalebox{.6}{0}}}\\sigma$.\n\nUnder condition \\eqref{A1}, {up to a straightforward transformation (see \\cite{MR3839316},\nSection 6.1),\nsystem \\eqref{eq1}} can be written in the canonical form\n\\begin{equation}\\label{eq2bis}\n \\begin{cases}\n dX_t=V_tdt, \\\\\n dV_t=b(t,X_t,V_t,Y_t)dt+\n {\\sigma}^i(t,X_t,V_t,Y_t)dW^i_t,\\\\\n dY_t=h(t,Z_t,Y_t)dt+{_{\\scalebox{.6}{0}}}\\sigma(t,Y_t)d{W}^{1}_t,\n \\end{cases}\n\\end{equation}\nwhere $W=(W^1,\\cdots, W^{n})$ is a $n$-dimensional Brownian motion. Setting\n$\\hat{\\sigma}:=\\left({\\sigma}^{2},\\dots,{\\sigma}^{n}\\right)$ so that ${\\sigma}=({\\sigma}^{1},\\hat{\\sigma})$, assumption \\eqref{A1}\nbecomes {\n\\begin{assumption}[Coercivity]\\label{A1bis}\nThere exists a positive constant $m$ such that\n\\begin{align}\n {_{\\scalebox{.6}{0}}}\\sigma(t,y)^{2}\\ge m, \\qquad |\\hat{\\sigma}(t,z,y)|^{2}\\ge m, \\qquad t\\in [0,T], \\ z\\in{\\mathbb {R}}^2, \\ y\\in{\\mathbb {R}}\n\\end{align}\n\\end{assumption}}\nMoreover, system \\eqref{eq2bis} can be written more conveniently as\n\\begin{equation}\\label{eq2}\n\\begin{cases}\n dZ_t=B Z_t dt +\\mathbf{e}_2\\left(b(t,Z_t,Y_t)dt+\\sigma^i(t,Z_t,Y_t)dW^i_t\\right),\\\\\n dY_t=h(t,Z_t,Y_t)dt+{_{\\scalebox{.6}{0}}}\\sigma(t,Y_t)dW^1_t,\n\\end{cases}\n\\end{equation}\nwith\n $$B=\\begin{pmatrix}\n   0 & 1 \\\\\n   0 & 0 \\\n \\end{pmatrix},\\qquad \\mathbf{e}_2=\\begin{pmatrix} 0 \\\\ 1\n\\end{pmatrix}.$$\n\n\\subsection{Forward filtering SPDE}\\label{fSPDE}\nWe consider the solution $(Z^{t,z}_{s},Y_{s})_{s\\in[t,T]}$ of system \\eqref{eq2} with initial\ncondition $Z^{t,z}_{t}=z\\in{\\mathbb {R}}^{2}$; we do not impose any initial condition on the $Y$-component.\nWe introduce the stochastic processes\n$${\\begin{aligned}\n  \\sigma_{s}({\\zeta})&:=\\sigma(s,{\\zeta},Y_{s}),\\qquad \n  {_{\\scalebox{.6}{0}}}\\sigma_s:={_{\\scalebox{.6}{0}}}\\sigma(s,Y_{s}),\\qquad\n  b_s({\\zeta}):=b_s({\\zeta},Y_s),\\qquad \\widetilde{h}_s({\\zeta}):=\\frac{h(s,{\\zeta},Y_{s})}{{_{\\scalebox{.6}{0}}}\\sigma(s,Y_{s})},\n\\end{aligned}}$$\nThe {\\it forward filtering SPDE} for system \\eqref{eq2} reads as follows\n\\begin{equation}\\label{Forward_eq2}\n d_{\\mathbf{B}}v_s({\\xi},{\\nu})=\\mathcal{A}^{\\ast}_{s}v_s({\\xi},{\\nu})ds+\\mathcal{G}^{\\ast}_sv_s({\\xi},{\\nu})\\frac{dY_s}{{_{\\scalebox{.6}{0}}}\\sigma_s},\n \\qquad \\mathbf{B}={probabilit\\`a }_s+{\\nu}{probabilit\\`a }_{\\xi},\n\\end{equation}\nwhere $\\mathcal{A}^{\\ast}$ and $\\mathcal{G}^{\\ast}$ are the adjoints of the differential operators (with random\ncoefficients)\n  $$\\mathcal{A}_s:={\\frac{|\\sigma_{s}({\\xi},{\\nu})|^{2}}{2}{probabilit\\`a }_{{\\nu}\\n}}+b_s({\\xi},{\\nu}){probabilit\\`a }_{\\nu},\\qquad \\mathcal{G}_{s}:=\\sigma^{1}_s({\\xi},{\\nu}){probabilit\\`a }_{{\\nu}}+\\widetilde{h}_s({\\xi},{\\nu}),$$\nrespectively.\n\nIn order to apply to \\eqref{Forward_eq2} the general results of Section 1, in particular Theorem\n\\ref{TH1} and Corollary \\ref{L_Cauchy}, we assume the following conditions. We recall notation\n\\eqref{aea1} and that ${\\sigma}=({\\sigma}^{1},\\hat{{\\sigma}})\\equiv({\\sigma}^{1},{\\sigma}^{2},\\dots,{\\sigma}^{n})$.\n\\begin{assumption}[Regularity]\\label{A2} The coefficients of \\eqref{eq2} are such that\n$\\sigma^{1}\\in bC^{3+{\\alpha}}_{0,T}({\\mathbb {R}}^3)$, $\\hat{\\sigma}\\in bC^{2+{\\alpha}}_{0,T}({\\mathbb {R}}^3)$, ${_{\\scalebox{.6}{0}}}\\sigma\\in bC^{{\\alpha}}_{0,T}({\\mathbb {R}})$,\n$b\\in bC^{1}_{0,T}({\\mathbb {R}}^3)$, $h\\in bC^{2}_{0,T}({\\mathbb {R}}^3)$.\n\\end{assumption}\n{\\begin{assumption}[Flattening at infinity]\\label{A3} There exist two positive constants ${\\varepsilon},M$\nsuch that\n\\begin{align}\n \\sup_{t\\in[0,T]\\atop y\\in{\\mathbb {R}}}\\left(\\langle \\sigma^1(t,\\cdot,y)\\rangle_{{\\varepsilon},{\\beta}}+\\langle\n \\sigma^1(t,\\cdot,y)\\rangle_{1\/2+{\\varepsilon},{\\beta}'} +\\langle h(t,\\cdot,y)\\rangle_{1\/2,{\\beta}}\\right)&\\le M\n\\end{align}\nfor $|{\\beta}|=1$ and $|{\\beta}'|=2,3$.\n\\end{assumption}}\n\n\n\n\n\\begin{remark}\\label{ar1\nWith regard to the existence of solutions to \\eqref{Forward_eq2}, let us introduce the process\n $$\\widetilde{W}_{s}:=\\int_t^{s}{_{\\scalebox{.6}{0}}}\\sigma_{\\tau}^{-1}dY_{\\tau}={W^{1}_{s}-W^{1}_{t}}+\\int_t^{s}\\tilde{h}_{\\tau}(Z^{t,z}_{{\\tau}})d{\\tau}, \\qquad s\\in [t,T].$$\nBy Girsanov's theorem, $(\\widetilde{W}_{s})_{s\\in[t,T]}$ is a Brownian motion w.r.t the measure $Q$\ndefined by $dQ=({\\varrho}^{t,z}_T)^{-1}dP$ where\n\\begin{equation}\\label{ae11}\n d{\\varrho}^{t,z}_s=\\tilde{h}_{s}(Z^{t,z}_{s})^2{\\varrho}^{t,z}_sdt+\\tilde{h}_{s}(Z^{t,z}_{s}){\\varrho}^{t,z}_s{dW^{1}_s},\\qquad {\\varrho}^{t,z}_t=1.\n\\end{equation}\nMoreover, $(\\widetilde{W}_{s})_{s\\in[t,T]}$ is adapted to $(\\mathcal{F}^{Y}_{t,s})_{s\\in[t,T]}$.\nThen, equation \\eqref{Forward_eq2} can be written in the equivalent form\n\\begin{equation}\\label{Forward_eq3}\n d_{\\mathbf{B}}v_s({\\zeta})=\\mathcal{A}^{\\ast}_{s}v_s({\\zeta})ds+\\mathcal{G}^{\\ast}_sv_s({\\zeta}){d\\widetilde{W}_s}\n\\end{equation}\nunder $Q$. Under Assumptions \\ref{A1bis}, \\ref{A2} and \\ref{A3}, by Theorem \\ref{TH1} a\nfundamental solution $\\mathbf{\\mathbf{\\Gamma}}=\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})$ for \\eqref{Forward_eq3} exists,\nsatisfies estimates \\eqref{t_e1}, \\eqref{t_e2}, \\eqref{t_e3} and $s\\mapsto\n\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})$ is adapted to $(\\mathcal{F}^{Y}_{t,s})_{s\\in[t,T]}$.\nWe say that the stochastic process\n\\begin{equation}\\label{ae100}\n  \\hat{\\mathbf{\\mathbf{\\Gamma}}}(t,z;s,{\\zeta})=\\frac{\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})}{\\int_{{\\mathbb {R}}^{2}}\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta}_{1})d{\\zeta}_{1}},\\qquad\n  0\\le t<s\\le T,\\ z,{\\zeta}\\in{\\mathbb {R}}^{2},\n\\end{equation}\nis the {\\it forward filtering density} for system \\eqref{eq2}. This definition is motivated by the\nfollowing\n\\end{remark}\n\\begin{theorem}\\label{th2}\nLet $(Z^{t,z}_{s},Y_{s})_{s\\in[t,T]}$ denote the solution of system \\eqref{eq2} with initial\ncondition $Z^{t,z}_{t}=z$. {Under Assumptions \\ref{A1bis}, \\ref{A2} and \\ref{A3}}, for any\n${\\varphi}\\in {bC({\\mathbb {R}}^{2})}$ we have\n\\begin{equation}\\label{ae7}\n E\\left[{\\varphi}(Z_{T}^{t,z})\\mid {\\mathcal{F}^{Y}_{t,T}}\\right]=\n \\int_{{\\mathbb {R}}^{2}}\\hat{\\mathbf{\\mathbf{\\Gamma}}}(t,z;T,{\\zeta}){\\varphi}({\\zeta})d{\\zeta}, \\qquad (t,z)\\in [0,T]\\times {\\mathbb {R}}^2.\n\\end{equation}\n\\end{theorem}\n\\proof By Remark \\ref{ar1}, $\\int_{{\\mathbb {R}}^{2}}\\hat{\\mathbf{\\mathbf{\\Gamma}}}(t,z;T,{\\zeta}){\\varphi}({\\zeta})d{\\zeta}\\in\nm\\mathcal{F}^{Y}_{t,T}$. We prove that, for any {bounded and $\\mathcal{F}^{Y}_{t,T}$-measurable random variable\n$G$,}\nwe have\n\\begin{equation}\\label{ae5}\n  E\\left[G{\\varphi}(Z_{T}^{t,z}\n  \\right]=\n E\\left[G ({\\varrho}_T^{t,z})^{-1} \\int_{{\\mathbb {R}}^{2}}{\\mathbf{\\mathbf{\\Gamma}}}(t,z;T,{\\zeta}){\\varphi}({\\zeta})d{\\zeta}\\right],\n\\end{equation}\nwith ${\\varrho}^{t,z}$ as in \\eqref{ae11}. From\n\\eqref{ae5} with ${\\varphi}\\equiv 1$ it will follow that\n\\begin{equation}\n E\\left[({\\varrho}_T^{t,z})^{-1}\\mid\n \\mathcal{F}^{Y}_{t,T}\\right]=\\left(\\int_{{\\mathbb {R}}^{2}}\\mathbf{\\mathbf{\\Gamma}}(t,z;T,{\\zeta})d{\\zeta}\\right)^{-1}\n\\end{equation}\nand therefore also \\eqref{ae7} will follow from \\eqref{ae5}.\n\nBy a standard approximation argument, it is enough to take ${\\varphi}$ in the class of test functions\nand $G$ of the form $G=e^{-\\int_t^Tc_{s}ds}$\nwhere $c_{s}=c(s,Y_{s})$ with $c=c(s,y)$\nbeing a {smooth, bounded and non-negative function} on $[t,T]\\times{\\mathbb {R}}$. Thus, we are left with the\nproof of the following identity:\n\\begin{equation}\\label{to_prove}\n E\\left[e^{-\\int_t^Tc_{s}ds}{\\varphi}(Z_{T}^{t,z})\\right]=\n E\\left[e^{-\\int_t^Tc_{s}ds}({\\varrho}_T^{t,z})^{-1}\\int_{{\\mathbb {R}}^{2}}{\\mathbf{\\mathbf{\\Gamma}}}(t,z;T,{\\zeta}){\\varphi}({\\zeta})d{\\zeta}\\right].\n\\end{equation}\nTo this end, we consider the deterministic backward Cauchy problem\n\\begin{equation}\\label{Cauchy_pb}\n {f\\left(s,e^{(T-s)B}{\\zeta},y\\right)={\\varphi}({\\zeta})+\\int_s^T(\\tilde{\\mathcal{A}}_{\\tau}-c({\\tau},y))f\\left({\\tau},e^{(T-{\\tau})B}{\\zeta},y\\right)d{\\tau}, \\qquad\n (s,{\\zeta},y)\\in[t,T]\\times {\\mathbb {R}}^{2}\\times{\\mathbb {R}},}\n\\end{equation}\n{where\n\\begin{equation}\\label{fLL}\n \\tilde{\\mathcal{A}}_{{\\tau}}:=\\frac{1}{2}\\left(|\\sigma({\\tau},{\\zeta},y)|^{2}{probabilit\\`a }_{{\\nu}\\n}+2{_{\\scalebox{.6}{0}}}\\sigma({\\tau},y)\\sigma^{1}({\\tau},{\\zeta},y){probabilit\\`a }_{{\\nu} y}+{_{\\scalebox{.6}{0}}}\\sigma^{2}({\\tau},y){probabilit\\`a }_{yy}\\right)\n +b({\\tau},{\\zeta},y){probabilit\\`a }_{{\\nu}}+h({\\tau},{\\zeta},y){probabilit\\`a }_{y}.\n \\end{equation}\n} In differential form, \\eqref{Cauchy_pb} reads as\n  $$\n  \\begin{cases}\n    -d_{{\\mathbf{B}}}f(s,{\\zeta},y)=\\left(\\tilde{\\mathcal{A}}_sf(s,{\\zeta},y)-c(s,y)f(s,{\\zeta},y)\\right)ds, \\\\\n    f(T,{\\zeta},y)={\\varphi}({\\zeta}).\n  \\end{cases}\n  $$\n{Corollary \\ref{L_Cauchy}} ensures existence and estimates of a strong solution $f$ to\n\\eqref{Cauchy_pb}.\n\nNext, we consider the process\n $$M_{s}^{t,z}:=e^{-\\int_{t}^{s} c_{{\\tau}}d{\\tau}} ({\\varrho}_s^{t,z})^{-1}\n \\int_{{\\mathbb {R}}^{2}}{\\mathbf{\\mathbf{\\Gamma}}}(t,z;s,{\\zeta})f(s,{\\zeta},Y_s)d{\\zeta},\\qquad s\\in [t,T].$$\nBy definition, we have\n $$M_T^{t,z}=e^{-\\int_t^Tc_{s}ds}({\\varrho}_T^{t,z})^{-1}\n \\int_{{\\mathbb {R}}^{2}}{\\mathbf{\\mathbf{\\Gamma}}}(t,z;T,{\\zeta}){\\varphi}({\\zeta})d{\\zeta}.$$\nOn the other hand, by the Feynman-Kac theorem\nwe have\n $$M_{t}^{t,z}=f(t,z,Y_t) =E\\left[e^{-\\int_{t}^Tc_{s}ds}{\\varphi}(Z_T^{t,z})\\mid Y_{t}\\right].$$\n\nHence to prove \\eqref{to_prove} it suffices to check that $M=(M_s^{t,z})_{s\\in[t,T]}$ is a\nmartingale: to this end, we prove the representation\n\\begin{align}\\label{SviluppoM}\n  M_T^{t,z}&={M_t^{t,z}+\\int_{t}^{T}G^{t,z}_{s}{dW^{1}_s}},\\\\ \\label{SviluppoM1}\n  G^{t,z}_{s}&=e^{-\\int_{t}^{s} c_{{\\tau}}d{\\tau}} ({\\varrho}_s^{t,z})^{-1}\n \\int_{{\\mathbb {R}}^{2}}{\\mathbf{\\mathbf{\\Gamma}}}(t,z;s,{\\zeta})\\left(\\mathcal{G}_s+{_{\\scalebox{.6}{0}}}\\sigma_s{probabilit\\`a }_y\\right)f(s,{\\zeta},Y_s)d{\\zeta},\\qquad\n s\\in[t,T],\n\\end{align}\nand conclude by showing that\n\\begin{equation}\\label{StimaM}\n   E\\left[\\int_{t}^{T}|G^{t,z}_{s}|^{2}ds\\right]<\\infty.\n\\end{equation}\nWe first compute the stochastic differential $d_{\\mathbf{B}}f(s,{\\zeta},Y_s)$: {by Corollary\n\\ref{L_Cauchy}} we have\n\\begin{align}\n d_{\\mathbf{B}}f(s,{\\zeta},Y_s)&=\\left(-\\tilde{\\mathcal{A}}_s+\\frac{1}{2}{_{\\scalebox{.6}{0}}}\\sigma_s^2{probabilit\\`a }_{yy}+c_s\\right)f(s,{\\zeta},Y_s)ds+{probabilit\\`a }_yf(s,{\\zeta},Y_s)dY_s\\\\\n &=\\left(-\\tilde{\\mathcal{A}}_s+\\frac{1}{2}{_{\\scalebox{.6}{0}}}\\sigma_s^2{probabilit\\`a }_{yy}+{h_s}(Z_s){probabilit\\`a }_y+c_s\\right)f(s,{\\zeta},Y_s)ds+\n {_{\\scalebox{.6}{0}}}\\sigma_s {probabilit\\`a }_yf(s,{\\zeta},Y_s){dW^{1}_s}.\n\\end{align}\nOn the other hand,\nwe have\n\\begin{align}\nd_{\\mathbf{B}}\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})&=\\mathcal{A}^{\\ast}_s\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})ds+\n  \\mathcal{G}^{\\ast}_s\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})\\frac{dY_s}{{_{\\scalebox{.6}{0}}}\\sigma_s}\\\\\n &=\\left(\\mathcal{A}^{\\ast}_s+\\widetilde{h}_s(Z_s)\\mathcal{G}^{\\ast}_s \\right)\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})ds+ \\mathcal{G}^{\\ast}_s\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta}){dW^{1}_s.}\n\\end{align}\nThen, by It\\^o formula we have\n\\begin{align}\n d_{\\mathbf{B}}\\left(f(s,{\\zeta},Y_s)\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})\\right\n=I_{1}(t,z;s,{\\zeta})ds+I_{2}(t,z;s,{\\zeta}){dW^{1}_{s}}\n\\end{align}\nwhere\n\\begin{align}\n I_{1}(t,z;s,{\\zeta})&=f(s,{\\zeta},Y_s) \\left(\\mathcal{A}^{\\ast}_s+\\widetilde{h}_s(Z_s)\\mathcal{G}^{\\ast}_s \\right)\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})\n \\\\ &\\quad+\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})\\left(-\\tilde{\\mathcal{A}}_s+\\frac{1}{2}{_{\\scalebox{.6}{0}}}\\sigma_s^2{probabilit\\`a }_{yy}+h_s(Z_s){probabilit\\`a }_y+c_s\\right)f(s,{\\zeta},Y_s)\n + {_{\\scalebox{.6}{0}}}\\sigma_s \\mathcal{G}^{\\ast}_s\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta}){probabilit\\`a }_yf(s,{\\zeta},Y_s),\\\\\n I_{2}(t,z;s,{\\zeta})&=f(s,{\\zeta},Y_s)\\mathcal{G}^{\\ast}_s\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})+ {_{\\scalebox{.6}{0}}}\\sigma_s \\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})\n {probabilit\\`a }_yf(s,{\\zeta},Y_s).\n\\end{align}\nThis means that for any $s\\in(t,T]$ we have\n\\begin{align}\n f(T,e^{(T-s)B}{\\zeta},Y_T)\\mathbf{\\mathbf{\\Gamma}}(t,z;T,e^{(T-s)B}{\\zeta})&=\n  f(s,{\\zeta},Y_s)\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})\\\\\n  &\\quad+\\int_{s}^{T}I_{1}(t,z;{\\tau},e^{({\\tau}-s)B}{\\zeta})d{\\tau}\n  +\\int_{s}^{T}I_{2}(t,z;{\\tau},e^{({\\tau}-s)B}{\\zeta}){dW^{1}_{{\\tau}}}.\n\\end{align}\nNext, we integrate over ${\\mathbb {R}}^2$ the previous identity and apply the standard and stochastic\nFubini's theorems (see, for instance, \\cite{MR3839316}, Chapter 1) to get\n\\begin{align}\n \\int_{{\\mathbb {R}}^{2}}f(T,e^{(T-s)B}{\\zeta},Y_T)\\mathbf{\\mathbf{\\Gamma}}(t,z;T,e^{(T-s)B}{\\zeta})d{\\zeta}&=\n  \\int_{{\\mathbb {R}}^{2}}f(s,{\\zeta},Y_s)\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})d{\\zeta}\\\\\n  &\\quad+\\int_{s}^{T}\\int_{{\\mathbb {R}}^{2}}I_{1}(t,z;{\\tau},e^{({\\tau}-s)B}{\\zeta})d{\\zeta} d{\\tau}\\\\\n  &\\quad+\\int_{s}^{T}\\int_{{\\mathbb {R}}^{2}}I_{2}(t,z;{\\tau},e^{({\\tau}-s)B}{\\zeta})d{\\zeta} {dW^{1}_{{\\tau}}}.\n\\end{align}\n{By the estimates of the fundamental solution in Theorem \\ref{TH1}, the estimates of the solution\n$f$ and its derivatives in Corollary \\ref{L_Cauchy}, the boundedness of the coefficients and the\nnon-degeneracy condition \\eqref{A1bis}, we have\n\\begin{align}\n\\int_s^T\\int_{{\\mathbb {R}}^2}|I_1(t,z;{\\tau},{\\zeta})|d{\\zeta} d{\\tau} \\le \\int_s^{T}\\frac{C}{(T-{\\tau})^{\\frac\n12}(s-t)}\\int_{{\\mathbb {R}}^2} {\\Gamma}_{\\lambda}({\\tau}-t,z;{\\tau},g^{\\text{\\rm\\tiny IW},-1}_{t,{\\tau}}({\\zeta})-\\gamma_{\\tau}^{t,z})d{\\zeta} d{\\tau} \\le C'\n\\frac{(T-s)^{\\frac 12}}{s-t}, \\intertext{and, analogously}\n\\int_s^T\\left(\\int_{{\\mathbb {R}}^2}|I_2(t,z;{\\tau},{\\zeta})|d{\\zeta} \\right)^2d{\\tau} \\le\n\\int_s^{T}\\left(\\frac{C}{(s-t)^{\\frac 12}}\\int_{{\\mathbb {R}}^2}\n{\\Gamma}_{\\lambda}({\\tau}-t,z;{\\tau},g^{\\text{\\rm\\tiny IW},-1}_{t,{\\tau}}({\\zeta})-\\gamma_{\\tau}^{t,z})d{\\zeta}\\right)^2d{\\tau} \\le \\frac{C'}{s-t},\n\\end{align}\nfor some positive constants $C,C'$. This justifies the use of Fubini's theorems.}\n\nNow, since the Jacobian of the transformation ${\\zeta}'=e^{sB}{\\zeta}$ equals one for any $s$, the previous\nequality yields\n\\begin{align}\n \\int_{{\\mathbb {R}}^{2}}f(T,{\\zeta},Y_T)\\mathbf{\\mathbf{\\Gamma}}(t,z;T,{\\zeta})d{\\zeta}&=\n  \\int_{{\\mathbb {R}}^{2}}f(s,{\\zeta},Y_s)\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})d{\\zeta}\\\\\n  &\\quad+\\int_{s}^{T}\\int_{{\\mathbb {R}}^{2}}I_{1}(t,z;{\\tau},{\\zeta}){\\zeta} d{\\tau}\n  +\\int_{s}^{T}\\int_{{\\mathbb {R}}^{2}}I_{2}(t,z;{\\tau},{\\zeta})d{\\zeta} {dW^{1}_{{\\tau}}}.\n\\end{align}\nIntegrating by parts and using the identity\n\\begin{align}\n &\\int_{{\\mathbb {R}}^2}\\left(f(s,{\\zeta},Y_s)\\mathcal{A}^{\\ast}_s\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})+\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})\\frac{1}{2}{_{\\scalebox{.6}{0}}}\\sigma^2_s{probabilit\\`a }_{yy}f(s,{\\zeta},Y_s)+\n {_{\\scalebox{.6}{0}}}\\sigma_s\\mathcal{G}^{\\ast}_s\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta}){probabilit\\`a }_yf(s,{\\zeta},Y_s)\\right)d{\\zeta} \\\\\n &=\\int_{{\\mathbb {R}}^2}\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})\\left(\\mathcal{A}_s+\\frac{1}{2}{_{\\scalebox{.6}{0}}}\\sigma^2_s{probabilit\\`a }_{yy}+{_{\\scalebox{.6}{0}}}\\sigma_s\\sigma^{1}_s{probabilit\\`a }_{y{\\nu}}+h_s({\\zeta},Y_s){probabilit\\`a }_y\\right)f(s,{\\zeta},Y_s)d{\\zeta} \\\\\n &=\\int_{{\\mathbb {R}}^2}\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})\\tilde{\\mathcal{A}}_sf(s,{\\zeta},Y_s)d{\\zeta},\n\\end{align}\nwe get\n\\begin{align}\n \\int_{{\\mathbb {R}}^{2}}f(T,{\\zeta},Y_T)\\mathbf{\\mathbf{\\Gamma}}(t,z;T,{\\zeta})d{\\zeta}&=\n  \\int_{{\\mathbb {R}}^{2}}f(s,{\\zeta},Y_s)\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})d{\\zeta}\\\\\n &\\quad+\\int_{s}^{T}\\int_{{\\mathbb {R}}^2}\\mathbf{\\mathbf{\\Gamma}}(t,z;{\\tau},{\\zeta})\\left(\\tilde{h}_{\\tau}(Z_{\\tau})\\mathcal{G}_{\\tau}+h_{\\tau}(Z_{\\tau}){probabilit\\`a }_y+c_{\\tau}\\right)f({\\tau},{\\zeta},Y_{\\tau})d{\\zeta}\n d{\\tau}\\\\\n &\\quad+ \\int_{s}^{T}\\int_{{\\mathbb {R}}^2}\\mathbf{\\mathbf{\\Gamma}}(t,z;{\\tau},{\\zeta})\\left(\\mathcal{G}_{\\tau}+{_{\\scalebox{.6}{0}}}\\sigma_{\\tau}{probabilit\\`a }_y\\right)f({\\tau},{\\zeta},Y_{\\tau}) d{\\zeta} {dW^{1}_{\\tau}}.\n\\end{align}\nEventually, we multiply the expression above by $e^{-\\int_{t}^{s} c_{{\\tau}}d{\\tau}}({\\varrho}_s^{t,z})^{-1}$:\nsince\n\\begin{align}\n d \\left(e^{-\\int_{t}^{s} c_{{\\tau}}d{\\tau}}({\\varrho}_s^{t,z})^{-1}\\right)&= e^{-\\int_{t}^{s} c_{{\\tau}}d{\\tau}}({\\varrho}_s^{t,z})^{-1}\n \\left(-c_sds-\\tilde{h}_s(Z_s){dW^{1}_s}\\right),\\\\\n d\\langle e^{-\\int_{t}^{\\cdot}\n  c_{{\\tau}}d{\\tau}}({\\varrho}_{\\cdot}^{t,z})^{-1},\n \\int_{{\\mathbb {R}}^2}f(\\cdot,{\\zeta},Y_{\\cdot})\\mathbf{\\mathbf{\\Gamma}}(t,z;\\cdot,{\\zeta})d{\\zeta}\\rangle_s\n &= -\\int_{{\\mathbb {R}}^2}\\mathbf{\\mathbf{\\Gamma}}(t,z;s,{\\zeta})\\left(\\tilde{h}_s(Z_s)\\mathcal{G}_s+h_s(Z_s){probabilit\\`a }_y\\right)f(s,{\\zeta},Y_s)\n d{\\zeta} ds,\n\\end{align}\nby It\\^o formula, for $s\\in(t,T]$ we have\n\\begin{align}\n M_T^{t,z}&=e^{-\\int_{t}^{T} c_{{\\tau}}d{\\tau}}({\\varrho}_T^{t,z})^{-1}\\int_{{\\mathbb {R}}^{2}}f(T,{\\zeta},Y_T)\\mathbf{\\mathbf{\\Gamma}}(t,z;T,{\\zeta})d{\\zeta} \\\\\n &\n M_s^{t,z}+ \\int_{s}^{T}e^{-\\int_{t}^{{\\tau}} c_{{\\varrho}}d{\\varrho}}({\\varrho}_{\\tau}^{t,z})^{-1} \\int_{{\\mathbb {R}}^2}\n \\mathbf{\\mathbf{\\Gamma}}(t,z;{\\tau},{\\zeta})\\left(\\mathcal{G}_{\\tau}+{_{\\scalebox{.6}{0}}}\\sigma_{\\tau}{probabilit\\`a }_y\\right)f({\\tau},{\\zeta},Y_{\\tau}) d{\\zeta} {dW^{1}_{\\tau}} \\\\\n &=M_s^{t,z}+ \\int_{s}^{T} {G^{t,z}_{{\\tau}}}\\label{ae10}\n{dW^{1}_{\\tau}}.\n\\end{align}\nwith $G^{t,z}_{{\\tau}}$ as in \\eqref{SviluppoM1}. Now, {again} by the estimates of the fundamental\nsolution (cf. Theorem \\ref{TH1}), the estimates of the solution $f$ and its derivatives (cf.\nCorollary \\ref{L_Cauchy}), the boundedness of the coefficients and the non-degeneracy condition\n\\eqref{A1bis}, we deduce the estimate\n\\begin{align}\\label{Uniform_integrability}\n |G^{t,z}_{{\\tau}}|&\\le C({\\varrho}_{\\tau}^{t,z})^{-1}\\int_{{\\mathbb {R}}^2}\\Gamma_\\lambda({\\tau}-t,g^{\\text{\\rm\\tiny IW},-1}_{{\\tau},t}({\\zeta})-\\gamma^{t,z}_{\\tau})d{\\zeta}\\le {C'}\n\\end{align}\nfor some positive constants $C,C'$. This implies \\eqref{StimaM} and concludes the proof.\n\\endproof\n\n\\subsection{Backward filtering SPDE}\\label{bSPDE}\nAs in the previous section, in order to apply the general results of Section 1 to the filtering\nSPDE for system \\eqref{eq2}, we impose the following conditions:\n\\begin{assumption}[Regularity]\\label{A2ii} The coefficients of \\eqref{eq2} are such that\n$\\sigma^{1}\\in bC^{3+{\\alpha}}_{0,T}({\\mathbb {R}}^3)$, $\\hat{\\sigma}\\in bC^{{\\alpha}}_{0,T}({\\mathbb {R}}^3)$, ${_{\\scalebox{.6}{0}}}\\sigma\\in bC^{3+{\\alpha}}_{0,T}({\\mathbb {R}})$,\n$b\\in bC^{0}_{0,T}({\\mathbb {R}}^3)$, $h\\in bC^{2}_{0,T}({\\mathbb {R}}^3)$.\n\\end{assumption}\n\\begin{assumption}[Flattening at infinity]\\label{A3ii} There exist two positive constants\n${\\varepsilon},M$ such that\n\\begin{align\n  \\sup_{t\\in[0,T]}\\left(\\langle \\sigma^1(t,\\cdot,\\cdot)\\rangle_{{\\varepsilon},{\\beta}}+\\langle \\sigma^1(t,\\cdot,\\cdot)\\rangle_{1\/2+{\\varepsilon},{\\beta}'}\n  +\\langle {_{\\scalebox{.6}{0}}}\\sigma(t,\\cdot)\\rangle_{{\\varepsilon},{\\beta}}+\\langle {_{\\scalebox{.6}{0}}}\\sigma(t,\\cdot)\\rangle_{1\/2+{\\varepsilon},{\\beta}'}\n  +\\langle h(t,\\cdot,\\cdot)\\rangle_{1\/2,{\\beta}}\\right)&\\le M\n\\end{align}\nfor $|{\\beta}|=1$ and $|{\\beta}'|=2,3$.\n\\end{assumption}\n\n\n\n\nThe backward filtering SPDE for system \\eqref{eq2} reads\n\\begin{equation}\\label{Backward_eq2}\n -d_{{\\mathbf{B}}}u_t(z,y)=\\tilde{\\mathcal{A}}_{t}\n u_t(z,y)dt+\\tilde{\\mathcal{G}}_{t}u_t(z,y)\\star\\frac{dY_t}{{_{\\scalebox{.6}{0}}}\\sigma(t,y)}, \\qquad\n \\mathbf{B}:={probabilit\\`a }_t+v{probabilit\\`a }_x,\n\\end{equation}\nwhere $z=(x,v)$ and {\\begin{align}\\label{fLL2}\n \\tilde{\\mathcal{A}}_t&:=\\frac{1}{2}\\left(|\\sigma(t,z,y)|^{2}{probabilit\\`a }_{vv}+2{_{\\scalebox{.6}{0}}}\\sigma(t,y)\\sigma^{1}(t,z,y){probabilit\\`a }_{v y}+{_{\\scalebox{.6}{0}}}\\sigma^{2}(t,y){probabilit\\`a }_{yy}\\right)\n +b(t,z,y){probabilit\\`a }_v+h(t,z,y){probabilit\\`a }_{y},\\\\\n  \\label{fGG}\n \\tilde{\\mathcal{G}}_{t}&:=\\sigma^{1}(t,z,y){probabilit\\`a }_v+{_{\\scalebox{.6}{0}}}\\sigma(t,y){probabilit\\`a }_{y}+\\tilde{h}(t,z,y),\\qquad\n \\tilde{h}(t,z,y):=\\frac{h(t,z,y)}{{_{\\scalebox{.6}{0}}}\\sigma(t,y)}.\n\\end{align}\nBefore presenting the main result of this section, we comment on the existence of solutions to\n\\eqref{Backward_eq2}.\nLet $(Z_{s}^{t,z,y},Y_{s}^{t,z,y},{{\\varrho}}^{t,z,y,{\\eta}}_{s})_{s\\in[t,T]}$ be the solution, starting at\ntime $t$ from $(z,y,{\\eta})$, of the system of SDEs\n\\begin{equation}\n{\\begin{cases}\\label{eq4}\n dZ_t=B Z_t dt +\\mathbf{e}_2({b(t,Z_t,Y_t)dt}+\\sigma^{i}(t,Z_t,Y_t)dW^{i}_{t},\\\\\n dY_t=h(t,Z_t,Y_t)dt+{_{\\scalebox{.6}{0}}}\\sigma(t,Y_t)d{W}^{1}_t, \\\\\n d{\\varrho}_t=\\tilde{h}(t,Z_t,Y_t)^2{\\varrho}_tdt+\\tilde{h}(t,Z_t,Y_t){\\varrho}_td{W}^{1}_t.\n\\end{cases}}\n\\end{equation}\n{By Girsanov's theorem, the process\n\\begin{align}\n \\widetilde{W}^{t,z,y}_{s}:\\!\\!&={\\int_{t}^{s}{_{\\scalebox{.6}{0}}}\\sigma^{-1}({\\tau},Y^{t,z,y}_{\\tau})dY^{t,z,y}_{\\tau}}\\\\\n &={W^{1}_{s}-W^{1}_t}+\\int_t^{s}\\tilde{h}({\\tau},Z^{t,z,y}_{{\\tau}},Y^{t,z,y}_{{\\tau}})d{\\tau}, \\qquad s\\in [t,T],\n\\end{align}\nis a Brownian motion w.r.t the measure $Q^{t,z,y}$ defined by\n$dQ^{t,z,y}=({\\varrho}_T^{t,z,y,1})^{-1}dP$.} Notice also that $(\\widetilde{W}^{t,z,y}_{s})_{s\\in[t,T]}$ is\nadapted to $(\\mathcal{F}^{Y}_{t,s})_{s\\in[t,T]}$ where {\\bf $\\mathcal{F}^{Y}_{t,s}={\\sigma}(Y^{t,z,y}_{{\\tau}},\\, t\\le {\\tau}\\le\ns)$}. Then equation \\eqref{Backward_eq2} can be written in the equivalent form\n\\begin{equation}\\label{Backward_eq3}\n -d_{{\\mathbf{B}}}u_s(z,y)=\\tilde{\\mathcal{A}}_{s}\n u_s(z,y)ds+\\tilde{\\mathcal{G}}_{s}u_s(z,y)\\star {d\\widetilde{W}_{s}^{t}\n\\end{equation}\nor, more explicitly,\n\\begin{equation}\\label{spde_back4}\n u_{t}\\left(\\gamma^{\\mathbf{B}}_{T-t}(z,y)\\right)=u_{T}(z,y)+\\int_{t}^{T}\\tilde{\\mathcal{A}}_{s}u_s(\\gamma^{\\mathbf{B}}_{T-s}(z,y))ds\n + \\int_{t}^{T}\\tilde{\\mathcal{G}}_{s}u_s(\\gamma^{\\mathbf{B}}_{T-s}(z,y))\\star d{\\widetilde{W}^{t}_{s}},\\qquad t\\in[0,T],\n\\end{equation}\nwhere {$\\gamma^{\\mathbf{B}}_{s}(z,y)=\\gamma^{\\mathbf{B}}_{s}(x,v,y)=(x+sv,v,y)$}.\nIn \\eqref{Backward_eq3} and \\eqref{spde_back4}, we simply write $\\widetilde{W}^{t}_{s}$ instead of\n$\\widetilde{W}^{t,z,y}_{s}$ because the starting point of the Brownian motion is irrelevant in the\nstochastic integration.\nTheorem \\ref{TH1} guarantees that a fundamental solution $\\cev{\\mathbf{\\Gamma}}=\\cev{\\mathbf{\\Gamma}}(t,z,y;s,{\\zeta},{\\eta})$ for\n\\eqref{Backward_eq3} exists and satisfies estimates \\eqref{t_e4}, \\eqref{t_e5} and \\eqref{t_e6}.\nMoreover, $t\\mapsto \\cev{\\mathbf{\\Gamma}}(t,z,y;T,{\\zeta},{\\eta})$ is adapted to $({\\mathcal{F}^{Y}_{t,T}})_{t\\in[0,T]}$.\nThe main result of this section is the following\n\\begin{theorem}\\label{at1} Let\n$(Z^{t,z,y}_T,Y^{t,z,y}_T)$ denote the solution of system \\eqref{eq2} starting from $(z,y)$ at\ntime $t\\in[0,T)$ and ${\\varphi}\\in bC({\\mathbb {R}}^{3})$. {Under Assumptions \\ref{A1bis}, \\ref{A2ii} and\n\\ref{A3ii}}, we have\n\\begin{equation}\\label{ae2}\n E\\left[{\\varphi}(Z_{T}^{t,z,y},Y_{T}^{t,z,y})\\mid\\mathcal{F}^{Y}_{t,T}\\right]=\\frac{u^{({\\varphi})}_{t}(z,y)}{u^{(1)}_{t}(z,y)},\n \\qquad (t,z,y)\\in [0,T]\\times {\\mathbb {R}}^2\\times {\\mathbb {R}},\n\\end{equation}\nwhere $u^{({\\varphi})}_{t}$ denotes the solution to \\eqref{Backward_eq2} with final datum\n$u^{({\\varphi})}_T={\\varphi}$.\n\\end{theorem}\n\\begin{definition}[\\bf Backward filtering density\nThe normalized process\n\\begin{equation}\\label{ae1}\n  \\bar{\\mathbf{\\Gamma}}(t,z,y;T,{\\zeta},{\\eta})=\\frac{\\cev{\\mathbf{\\Gamma}}(t,z,y;T,{\\zeta},{\\eta})}{\\int\\limits_{{\\mathbb {R}}^{3}}\\cev{\\mathbf{\\Gamma}}(t,z,y;T,{\\zeta}_{1},{\\eta}_{1})d{\\zeta}_{1}\n  d{\\eta}_{1}},\n\\end{equation}\nfor $0\\le t<T$ and $(z,y),({\\zeta},{\\eta})\\in{\\mathbb {R}}^{2}\\times{\\mathbb {R}}$, is called the backward filtering density of\nsystem \\eqref{eq2}. By Theorem\n\\ref{at1}, we have\n\\begin{equation}\\label{form1}\n E\\left[{\\varphi}(Z_{T}^{t,z,y},Y_{T}^{t,z,y})\\mid {\\mathcal{F}^{Y}_{t,T}}\\right]=\n \\int\\limits_{{\\mathbb {R}}^{3}}\\bar{\\mathbf{\\Gamma}}(t,z,y;T,{\\zeta},{\\eta}){\\varphi}({\\zeta},{\\eta})d{\\zeta} d{\\eta}, \\qquad (t,z,y)\\in [0,T]\\times\n {\\mathbb {R}}^2\\times{\\mathbb {R}},\n\\end{equation}\nfor any ${\\varphi}\\in bC({\\mathbb {R}}^{3})$.\n\\end{definition}\n\\begin{remark\nNotice that formulas \\eqref{ae2} and \\eqref{form1} represent the conditional expectation in terms\nof solutions to the Cauchy problem for the backward filtering SPDE. This is not the case for\nformula \\eqref{ae7} in the forward case.\n\\end{remark}\n\n\n\nIn the rest of the section we sketch the proof of Theorem \\ref{at1}. First, notice that under\n$Q^{t,z,y}$ we have\n\\begin{equation}\\label{eq3}\n {\\varrho}^{t,z,y,{\\eta}}_s={\\eta}\\exp\\left(\\int_t^s\\tilde{h}({\\tau},Z^{t,z,y}_{\\tau},Y^{t,z,y}_{\\tau}){d\\widetilde{W}^{t}_{\\tau}}-\n \\frac{1}{2}\\int_t^s\\tilde{h}({\\tau},Z^{t,z,y}_{\\tau},Y^{t,z,y}_{\\tau})^2d{\\tau}\\right), \\qquad s\\in [t,T],\n\\end{equation}\nand system \\eqref{eq4} reads\n\\begin{equation}\\label{ae3}\n\\begin{cases}\ndZ^{t,z,y}_s=\\tilde{B}(s,Z^{t,z,y}_s,Y^{t,z,y}_s)ds +\n\\mathbf{e}_2\\left(\\hat{\\sigma}^i(s,Z^{t,z,y}_s,Y^{t,z,y}_s)\ndW^i_t+\\sigma^{1}(s,Z^{t,z,y}_s,Y^{t,z,y}_s){d\\widetilde{W}^{t}_s}\\right),\\\\\ndY^{t,z,y}_s={_{\\scalebox{.6}{0}}}\\sigma(s,Y^{t,z,y}_s){d\\widetilde{W}^{t}_s},\\\\\nd{\\varrho}^{t,z,y,{\\eta}}_s=\\tilde{h}(s,Z^{t,z,y}_s,Y^{t,z,y}_s){\\varrho}^{t,z,y,{\\eta}}_s{d\\widetilde{W}^{t}_s},\n\\end{cases}\n\\end{equation}\nwhere $\\tilde{B}(s,z,y)=Bz+\\mathbf{e}_2({b(s,z,y)}-\\tilde{h}(s,z,y)\\sigma^{1}(s,z,y))$. Recalling the notation\n$z=(x,v)\\in{\\mathbb {R}}^{2}$ and omitting the arguments of the coefficients for brevity, the correspondent\ncharacteristic operator is\n\\begin{align}\n {\\mathcal{L}}}%{\\tilde{\\tilde{\\mathcal{L}}}=\\frac{1}{2}\\left({|\\sigma|^{2}}{probabilit\\`a }_{vv}+{_{\\scalebox{.6}{0}}}\\sigma^2{probabilit\\`a }_{yy}+{\\eta}^2\\tilde{h}^2{probabilit\\`a }_{{\\eta}\\y}+\n 2\\sigma^{1}{_{\\scalebox{.6}{0}}}\\sigma{probabilit\\`a }_{vy}+2{\\eta}\\sigma^{1}\\tilde{h}{probabilit\\`a }_{v{\\eta}}+2{\\eta}{_{\\scalebox{.6}{0}}}\\sigma\\tilde{h}{probabilit\\`a }_{y{\\eta}}\\right) +\\langle\n \\tilde{B},\\nabla_{z}\\rangle.\n\\end{align}\nWe write the backward diffusion SPDE for system \\eqref{ae3}. {Assuming that ${\\varphi}$ is smooth} and\nletting $V_s(z,y):={\\varphi}(Z_{T}^{s,z,y},Y_{T}^{s,z,y})$, by Corollary \\ref{c1} we have\n\\begin{align}\n -d(V_s(z,y){\\varrho}_{T}^{s,z,y,{\\eta}})&={\\mathcal{L}}}%{\\tilde{\\tilde{\\mathcal{L}}}(V_s(z,y){\\varrho}_{T}^{s,z,y,{\\eta}})ds+{probabilit\\`a }_v(V_s(z,y){\\varrho}_{T}^{s,z,y,{\\eta}})\n\\left(\\hat{\\sigma}^i(s,z,y)\\star dW_s^i+\\sigma^{1}(s,z,y)\\star {d\\widetilde{W}^{t}_s}\\right)\\\\\n &\\quad +{probabilit\\`a }_y(V_s(z,y){\\varrho}_{T}^{s,z,y,{\\eta}}){_{\\scalebox{.6}{0}}}\\sigma(s,z,y)\\star {d\\widetilde{W}^{t}_s}+\n {probabilit\\`a }_{\\eta}(V_s(z,y){\\varrho}_{T}^{s,z,y,{\\eta}}){\\eta}\\tilde{h}(s,z,y)\\star {d\\widetilde{W}^{t}_s}\n\\intertext{(noting that ${probabilit\\`a }_{{\\eta}}Z^{t,z,y}_{T}={probabilit\\`a }_{{\\eta}}Y^{t,z,y}_{T}={probabilit\\`a }_{{\\eta}\\y}{{\\varrho}}^{t,z,y,{\\eta}}_{T}=0$\nand ${\\eta}{probabilit\\`a }_{{\\eta}}{{\\varrho}}^{t,z,y,{\\eta}}_{T}={{\\varrho}}^{t,z,y,{\\eta}}_{T}$)}\n  &=\\frac{1}{2}\\left({{|\\sigma(t,z,y)|^{2}}}{probabilit\\`a }_{vv}+{_{\\scalebox{.6}{0}}}\\sigma^2(t,z,y){probabilit\\`a }_{yy}+2\\sigma^{1}{_{\\scalebox{.6}{0}}}\\sigma(t,z,y){probabilit\\`a }_{vy}\\right)(V_s(z,y){\\varrho}_{T}^{s,z,y,{\\eta}})ds\\\\\n  &\\quad +\\Big(\\tilde{h}(s,z,y)(\\sigma^{1}{probabilit\\`a }_v+{_{\\scalebox{.6}{0}}}\\sigma{probabilit\\`a }_y)+\\langle \\tilde{B}(t,z,y),\\nabla_{z}\\rangle\\Big)(V_s(z,y){\\varrho}_{T}^{s,z,y,{\\eta}})ds\\\\\n &\\quad +\\hat{\\sigma}^i(s,z,y) {probabilit\\`a }_v(V_s(z,y){\\varrho}_{T}^{s,z,y,{\\eta}})\\star {dW_s^i}\\\\\n &\\quad +\\left(\\sigma^{1}(s,z,y){probabilit\\`a }_v+{_{\\scalebox{.6}{0}}}\\sigma(s,z,y){probabilit\\`a }_y+\\tilde{h}(s,z,y)\\right)(V_s(z,y){\\varrho}_{T}^{s,z,y,{\\eta}})\\star {d\\widetilde{W}^{t}_s}\n\\intertext{(noting that $\\tilde{h}(s,z,y)(\\sigma^{1}{probabilit\\`a }_v+{_{\\scalebox{.6}{0}}}\\sigma{{probabilit\\`a }_y})+\\langle\n\\tilde{B}(t,z,y),\\nabla_{z}\\rangle={v{probabilit\\`a }_x+b(t,z,y){probabilit\\`a }_v+h(t,z,y){probabilit\\`a }_y}$)}\n &=\\tilde{\\mathcal{L}}(V_s(z,y){\\varrho}_{T}^{s,z,y,{\\eta}})ds+\\hat{\\sigma}^i(s,z,y) {probabilit\\`a }_v(V_s(z,y){\\varrho}_{T}^{s,z,y,{\\eta}})\\star\n {dW_s^i}\\\\\n &\\quad +\\left(\\sigma^{1}(s,z,y){probabilit\\`a }_v+{_{\\scalebox{.6}{0}}}\\sigma(s,z,y){{probabilit\\`a }_y}+\\tilde{h}(s,z,y)\\right)(V_s(z,y){\\varrho}_{T}^{s,z,y,{\\eta}})\\star\n {d\\widetilde{W}^{t}_s}.\n\\end{align}\nwhere ${\\tilde{\\mathcal{L}}}=\\tilde{\\mathcal{A}}_t+v{probabilit\\`a }_x$, with $\\tilde{\\mathcal{A}}_t$ as in \\eqref{fLL2}, is the infinitesimal generator of\n$(Z_t,Y_t)$. Therefore we have\n\\begin{align}\n{\\varphi}(Z_{T}^{t,z,y},Y_{T}^{t,z,y}){\\varrho}_{T}^{t,z,y,1}-{\\varphi}(z,y)&=\nV_t(z,y){\\varrho}_{T}^{t,z,y,1}-V_{T}(z,y){\\varrho}_{T}^{{T},z,y,1}\\\\ &=\n\\int_t^{T}\\tilde{\\mathcal{L}}(V_s(z,y){\\varrho}_{T}^{s,z,y,{\\eta}})ds+\\int_t^{T}\\hat{\\sigma}^i(s,z,y){probabilit\\`a }_v(V_s(z,y){\\varrho}_{T}^{s,z,y,{\\eta}})\\star\n{d{W}_s^i}\\\\ &\\quad + \\int_t^{T} \\tilde{\\mathcal{G}}_s\n(V_s(z,y){\\varrho}_{T}^{s,z,y,{\\eta}}\n\\star {d\\widetilde{W}^{t}_s}.\\label{ae4}\n\\end{align}\nNow we take the conditional expectation in \\eqref{ae4} and exploit the fact that\n$(W^{2},\\dots,W^{n})$ is independent of $\\mathcal{F}^{Y}_{t,T}$ under $Q^{t,z,y}$ (this follows from the\ncrucial assumption that ${_{\\scalebox{.6}{0}}}\\sigma$ is a function of $t,y$ only): setting\n \\begin{equation}\n u^{({\\varphi})}_{t}(z,y)=E^{Q^{t,z,y}}\\left[V_{t}(z,y){{\\varrho}}^{t,z,y,1}_{T}\\mid\\mathcal{F}^{Y}_{t,T}\\right],\n\\end{equation}\nand applying the standard and stochastic Fubini's theorems, we directly get the filtering equation\n\\begin{align}\n u^{({\\varphi})}_{t}(z,y)&={\\varphi}(z,y)+ \\int_{t}^{T}\\tilde{\\mathcal{L}}_s u^{({\\varphi})}_{s}(z,y)ds +\\int_{t}^{T}\n\n \\tilde{\\mathcal{G}}_s u^{({\\varphi})}_{s}(z,y\n \\star\n \\frac{dY_s^{t,z,y}}{{_{\\scalebox{.6}{0}}}\\sigma(s,y)}\n\\end{align}\nwhich is equivalent to \\eqref{Backward_eq2}.\nAnalogously,\n  $$u^{(1)}_{t}(z,y):=E^{Q^{t,z,y}}\\left[{{\\varrho}}^{t,z,y,1}_{T}\\mid\\mathcal{F}^{Y}_{t,T}\\right]$$\nsolves the same SPDE with terminal condition $u^{(1)}_{T}(z,y)\\equiv 1$. To conclude, it suffices\nrecall the Bayes representation for conditional expectations or the Kallianpur-Striebel's formula\n(cf. \\cite{MR3839316}, Lemma 6.1) according to which we have\n\\begin{equation}\nE\\left[{\\varphi}(Z_{T}^{t,z,y},Y_{T}^{t,z,y})\\mid\\mathcal{F}^{Y}_{t,T}\\right]=\n\\frac{E^{Q^{t,z,y}}\\left[{\\varphi}(Z_{T}^{t,z,y},Y_{T}^{t,z,y}){{\\varrho}}^{t,z,y,1}_{T}\\mid\\mathcal{F}^{Y}_{t,T}\\right]}{E^{Q^{t,z,y}}\n\\left[{{\\varrho}}^{t,z,y,1}_{T}\\mid\\mathcal{F}^{Y}_{t,T}\\right]}.\n\\end{equation}\n\\endproof\n\n\n\n\\section{Proof of Theorem \\ref{TH1}}\\label{proofK}\nAs in \\cite{pasc:pesc:19} the main ingredient in the proof of Theorem \\ref{TH1} is the\nIt\\^o-Wentzell formula that transforms the original SPDE into a PDE with random coefficients. In\nthis section we explain how to tweak the change of variables introduced in \\cite{pasc:pesc:19} to\ndeal with the additional term $h$ and we also consider the backward equation. We set $d=1$ for\nsimplicity.\n\n\\subsection{It\\^o-Wentzell change of variables}\nWe first recall some global estimates, proved in \\cite{pasc:pesc:19}, Section 4, for $\\gamma^{\\text{\\rm\\tiny IW}}$,\n$\\theta^{\\text{\\rm\\tiny IW}}$ in \\eqref{sde_forw}-\\eqref{sde_back} and their derivatives under Assumptions \\ref{ass1},\n\\ref{ass2} and \\ref{ass3}.\n\\begin{lemma}\\label{lemmaIW} For any $\\bar{{\\alpha}}\\in [0,{\\alpha})$, we have ${\\gamma^{\\text{\\rm\\tiny IW}}_{t,\\cdot}}\\in{\\mathbf{C}^{3+\\bar{{\\alpha}}}_{t,T}}$. Moreover there exists\n${\\varepsilon}\\in\\left(0,\\frac{1}{2}\\right)$ and a random, finite constant $\\mathbf{c}$ such that, with\nprobability one,\n\\begin{align}\n |\\gamma^{\\text{\\rm\\tiny IW}}_{t,s}({\\xi},{\\nu})|&\\le \\mathbf{c} \\sqrt{1+{\\xi}^2+{\\nu}^2},\\\\\n e^{-\\mathbf{c}(s-t)^{{\\varepsilon}}}\\le {probabilit\\`a }_{\\nu}\\gamma^{\\text{\\rm\\tiny IW}}_{t,s}({\\xi},{\\nu})&\\le e^{\\mathbf{c}(s-t)^{{\\varepsilon}}},\\\\\n |{probabilit\\`a }_{\\xi} \\gamma^{\\text{\\rm\\tiny IW}}_{t,s}({\\xi},{\\nu})|&\\le \\mathbf{c}(s-t)^{{\\varepsilon}},\\\\\n |{probabilit\\`a }^{{\\beta}}\\gamma^{\\text{\\rm\\tiny IW}}_{t,s}({\\xi},{\\nu})|&\\le \\frac{\\mathbf{c}(s-t)^{{\\varepsilon}}}{\\sqrt{1+{\\xi}^2+{\\nu}^2}},\n\\end{align}\nfor any $({\\xi},{\\nu})\\in{\\mathbb {R}}^{2}$, $0\\le t\\le s\\le T$ and $|{\\beta}|=2$. Analogous estimates hold for\n$\\theta^{\\text{\\rm\\tiny IW}}_{\\cdot,s}\\in\\mathbf{C}^{3,\\bar{{\\alpha}}}_{0,s}$.\n\\end{lemma}\n\nWe introduce the ``hat'' operator which transforms any function $f_{s}({\\xi},{\\nu})$, {$s\\in[t,T]$,}\ninto\n\\begin{equation}\n\\hat{f}_{t,s}({\\xi},{\\nu}):=f_{s}({\\xi},\\gamma^{\\text{\\rm\\tiny IW}}_{t,s}({\\xi},{\\nu})).\n\\end{equation}\nLet $u_{s}({\\xi},{\\nu})$ a solution to \\eqref{spde_forw} {on $[t,T]$}. Then we define\n\\begin{equation}\nv_{t,s}({\\zeta}):=\\hat{{\\varrho}}_{t,s}({\\zeta})\\hat{u}_{t,s}({\\zeta}), \\quad {\\varrho}_{t,s}({\\zeta}):=\\exp\\left(-\\int_{t}^s\n{h}_{{\\tau}}({\\zeta})dW_{\\tau}-\\frac{1}{2}\\int_{t}^s{h}_{{\\tau}}^2({\\zeta})d{\\tau} \\right).\n\\end{equation}\nWe have the following\n\\begin{proposition} \\label{Prop_IW}\n$u_s$ is a solution to the SPDE \\eqref{spde_forw} {on $[t,T]$} if and only if $v_{t,s}$ is a\nsolution on $[t,T]$ to the PDE with random coefficients\n\\begin{equation}\\label{PDE_FORW}\n d_{\\mathbf{\\hat{B}}}v_{t,s}({\\zeta})=\\left(a^{\\ast}_{t,s}({\\zeta}){probabilit\\`a }_{{\\nu}\\n}v_{t,s}+\n b_{t,s}^{\\ast}({\\zeta}){probabilit\\`a }_{{\\nu}}v_{t,s}({\\zeta})+{c}^{\\ast}_{t,s}({\\zeta})v_{t,s}({\\zeta})\\right)ds,\n \\qquad \\mathbf{\\hat{B}}={probabilit\\`a }_s+\\mathbf{Y}_{t,s}\n\\end{equation}\nwhere\n\\begin{equation}\\label{aee13}\n  \\mathbf{Y}_{t,s}=\\mathbf{Y}_{t,s}({\\xi},{\\nu}):=(\\gamma^{\\text{\\rm\\tiny IW}}_{t,s})_1({\\xi},{\\nu}){probabilit\\`a }_{{\\xi}}-(\\gamma^{\\text{\\rm\\tiny IW}}_{t,s}({\\xi},{\\nu}))_1({probabilit\\`a }_{{\\nu}}\\gamma^{\\text{\\rm\\tiny IW}}_{t,s})^{-1}({\\xi},{\\nu}){{probabilit\\`a }_{{\\xi}}\n  \\gamma^{\\text{\\rm\\tiny IW}}_{t,s}({\\xi},{\\nu})}{probabilit\\`a }_{{\\nu}},\n\\end{equation}\nis the first order operator identified with the vector field in \\eqref{flusso2} (with $d=1$) and\nthe coefficients $a^{\\ast}_{t,\\cdot}$, $b^{\\ast}_{t,\\cdot}$, $c^{\\ast}_{t,\\cdot}$ are defined in\n\\eqref{coeff_deterministic} below. Moreover, $a^{\\ast}_{t,\\cdot}\\in \\mathbf{bC}^{{\\alpha}}_{t,T}$,\n$b^{\\ast}_{t,\\cdot}, c^{\\ast}_{t,\\cdot}\\in \\mathbf{bC}^{0}_{t,T}$, {$\\mathbf{Y}_{t,\\cdot}\\in\n{\\mathbf{C}^{0,1}_{t,T}}$}, ${probabilit\\`a }_{\\nu}(\\mathbf{Y}_{t,\\cdot})_1\\in \\mathbf{bC}^{\\bar{{\\alpha}}}_{t,T}$ {for any\n$\\bar{{\\alpha}}\\in [0,{\\alpha})$,} and there exist two random, finite and positive constants $\\mathbf{m}_1$,\n$\\mathbf{m}_2$ such that, for $s\\in [t,T]$, ${\\zeta} \\in {\\mathbb {R}}^2$, we have\n\\begin{align}\\label{Ass5}\n& \\mathbf{m}_1^{-1}\\le a^{\\ast}_{t,s}({\\zeta})\\le \\mathbf{m}_1, \\qquad \\mathbf{m}_2^{-1}\\le {probabilit\\`a }_{\\nu}\n(\\mathbf{Y}_{t,s}({\\zeta}))_1\\le \\mathbf{m}_2,\n\\end{align}\nwith probability one.\n\\end{proposition}\n\\proof\nBy a standard regularization argument, we may assume $u\\in \\mathbf{C}_{t,T}^2$ so that equation\n\\eqref{spde_forw} can be written in the usual It\\^o sense, namely\n$$du_{s}({\\zeta})=(\\mathcal{A}_{s,{\\zeta}}-{\\nu}_1{probabilit\\`a }_{\\xi})u_{s}({\\zeta})ds+\\mathcal{G}_{s,{\\zeta}}u_{s}(s)dW_s. $$ By the standard\nIt\\^o-Wentzell formula (see for instance \\cite{MR3839316}, Theorem 1.17), and the chain rule we\nhave\n\\begin{align}\\label{p_e1}\nd\\hat{u}_{t,s}&=\\left(\\widehat{\\mathcal{A}_{s,{\\zeta}}u}_{t,s}-\\gamma^{\\text{\\rm\\tiny IW}}_{t,s}\\widehat{{probabilit\\`a }_1u}_{t,s}+\\frac{1}{2}\\hat{{\\sigma}}^2_{t,s}\\widehat{{probabilit\\`a }_2^2\nu}_{t,s}-\\widehat{{probabilit\\`a }_2\\mathcal{G}}_{s,{\\zeta}}\\hat{{\\sigma}}_{t,s}\\right)ds+\\hat{h}_{t,s}\\hat{u}_{t,s}dW_s\\\\\n &=\\left(\\mathfrak{L}_{t,s}\n \\mathbf{Y}_{t,s}\\right)\\hat{u}_{t,s}dt+\\hat{h}_{t,s}\\hat{u}_{t,s}dW_s,\n\\end{align}\nwhere ${\\mathfrak{L}_{t,s}}:= \\bar{a}_{t,s}{probabilit\\`a }_{vv}+\\bar{b}_{t,s}{probabilit\\`a }_v+\\bar{c}_{t,s}$ with\n\\begin{equation}\\label{aee14}\n \\begin{split}\n \\bar{a}_{t,s}&=\\frac{1}{2}({probabilit\\`a }_{\\nu}\\gamma^{\\text{\\rm\\tiny IW}}_{t,s})^{-2}(\\hat{a}_{t,s}-\\hat{{\\sigma}}^2_{t,s}),\\\\\n \\bar{b}_{t,s}&=({probabilit\\`a }_{\\nu}\\gamma^{\\text{\\rm\\tiny IW}}_{t,s})^{-1}\\left(\\hat{b}_{t,s}-\\hat{{\\sigma}}_{t,s}\\hat{h}_{t,s}-({probabilit\\`a }_{\\nu}\\gamma^{\\text{\\rm\\tiny IW}}_{t,s})^{-1}{\\hat{{\\sigma}}_{t,s}{probabilit\\`a }_{{\\nu}}\\hat{{\\sigma}}_{t,s}}-\n \\bar{a}_{t,s}{probabilit\\`a }_{{\\nu}\\n}\\gamma^{\\text{\\rm\\tiny IW}}_{t,s}\\right), \\\\\n \\bar{c}_{t,s}&=\\hat{c}_{t,s}-({probabilit\\`a }_{\\nu}\\gamma^{\\text{\\rm\\tiny IW}}_{t,s})^{-1}{\\hat{{\\sigma}}_{t,s}{probabilit\\`a }_{{\\nu}}\\hat{h}_{t,s}}.\n \\end{split}\n\\end{equation}\nNotice that the change of variable is well defined by the estimates of Lemma \\ref{lemmaIW}. Next\nwe compute the product $v_{t,s}=\\hat{{\\varrho}}_{t,s}\\hat{u}_{t,s}$: by the It\\^o formula\n$d\\hat{{\\varrho}}_{t,s}=-\\hat{{\\varrho}}_{t,s}\\hat{h}_{t,s}dW_s$ and therefore\n\\begin{align}\\label{p_e2}\n dv_{t,s}({\\zeta})&=\\hat{{\\varrho}}_{t,s}({\\zeta})d\\hat{u}_{t,s}({\\zeta})+\\hat{u}_{t,s}({\\zeta})d\\hat{{\\varrho}}_{t,s}+ d\\langle\n \\hat{u}_{t,\\cdot}({\\zeta})\\hat{{\\varrho}}_{t,\\cdot}({\\zeta}) \\rangle_s \\\\\n &=\\left(\\hat{{\\varrho}}_{t,s}({\\zeta})\\mathfrak{L}_{t,s}(\\hat{{\\varrho}}^{-1}_{t,s}v_{t,s})({\\zeta})-\n \\hat{{\\varrho}}_{t,s}({\\zeta})\n (\\mathbf{Y}_{t,s}(\\hat{{\\varrho}}^{-1}_{t,s}v_{t,s}))({\\zeta})-\\bar{h}_{t,s}^2({\\zeta})v_{t,s}({\\zeta})\\right)ds.\n\\end{align}\nNow we notice that\n\\begin{align}\n \\hat{{\\varrho}}_{t,s}({\\zeta})(\\mathbf{Y}_{t,s}(\\hat{{\\varrho}}^{-1}_{t,s}v_{t,s}))({\\zeta})&=\n (\\mathbf{Y}_{t,s}v_{t,s})({\\zeta}) + (\\mathbf{Y}_{t,s}\\ln \\hat{{\\varrho}}^{-1}_{t,s})({\\zeta})v_{t,s}({\\zeta}),\n\\end{align}\nand eventually, by a standard application of the Leibniz rule, we get\n $$dv_{t,s}({\\zeta})=\\left(a^{\\ast}_{t,s}({\\zeta}){probabilit\\`a }_{vv}v_{t,s}({\\zeta})-(\\mathbf{Y}_{t,s}\n v_{t,s})({\\zeta})+ b_{t,s}^{\\ast}({\\zeta}){probabilit\\`a }_vv_{t,s}({\\zeta})+c^{\\ast}_{t,s}({\\zeta})v_{t,s}({\\zeta})\\right)ds, $$\nwhere\n\\begin{align}\\label{coeff_deterministic}\n a_{t,s}^{\\ast}&=\\bar{a}_{t,s}=\\frac{1}{2}({probabilit\\`a }_{\\nu}\\gamma^{\\text{\\rm\\tiny IW}}_{t,s})^{-2}(\\hat{a}_{t,s}-\\hat{{\\sigma}}^2_{t,s}),\\\\\n b_{t,s}^{\\ast}&=\\bar{b}_{t,s}+2 \\bar{a}_{t,s}{probabilit\\`a }_{\\nu}\\ln \\hat{{\\varrho}}^{-1}_{t,s}, \\\\\n c^{\\ast}_{t,s}&=\\bar{c}_{t,s}+\\bar{b}_{t,s}{probabilit\\`a }_{\\nu}\\ln \\hat{{\\varrho}}^{-1}_{t,s}+\\bar{a}_{t,s}\\left({probabilit\\`a }_{\\nu}\\ln\n \\hat{{\\varrho}}^{-1}_{t,s}+ {probabilit\\`a }^2_{\\nu}\\ln \\hat{{\\varrho}}^{-1}_{t,s}\\right)+\\mathbf{Y}_{t,s}\\ln\n \\hat{{\\varrho}}^{-1}_{t,s}-{\\hat{h}_{t,s}^2}.\n\\end{align}\nThe regularity of the coefficients and \\eqref{Ass5} follow directly from\n\\eqref{coeff_deterministic}, Assumption \\ref{ass2} and Lemma \\ref{lemmaIW}.\n\\endproof\n\\begin{remark}\\label{Hormander}\nWhen the coefficients are smooth, condition \\eqref{Ass5} ensures the validity of the weak\nH\\\"ormander condition: indeed the vector fields $\\sqrt{a^{\\ast}_{t,\\cdot}}{probabilit\\`a }_{\\nu}$ and\n$\\mathbf{Y}_{t,\\cdot}$, together with their commutator, span ${\\mathbb {R}}^{2}$ at any point. In this case a smooth\nfundamental solution to \\eqref{PDE_FORW} exists by H\\\"ormander's theorem.\n\\end{remark}\n\n\nIn the backward case the computations are completely analogous since it only suffices to reverse\nthe time in equations \\eqref{p_e1} and \\eqref{p_e2}. Precisely, we introduce the ``check''\ntransform\n\\begin{equation}\n \\check{f}_{t,s}(x,v):=f_{t}({\\xi},\\theta^{\\text{\\rm\\tiny IW}}_{t,s}(x,v)),\\qquad t\\in [0,s],\n\\end{equation}\nwith $\\theta^{\\text{\\rm\\tiny IW}}_{t,s}$ as in \\eqref{sde_back}. For a solution $u_{t}=u_{t}(z)$ to \\eqref{spde_back} on\n$[0,s]$, we define\n\\begin{equation}\n v_{t,s}(z):=\\check{{\\varrho}}_{t,s}(z)\\check{u}_{t,s}(z), \\quad {\\varrho}_{t,s}(z):=\\exp\\left(-\\int_{t}^s\n{h}_{{\\tau}}(z)\\star dW_{\\tau}-\\frac{1}{2}\\int_{t}^s{h}_{{\\tau}}^2(z)d{\\tau} \\right),\n\\end{equation}\nwhich solves, on $[0,s]$, the deterministic equation with random coefficients\n\\begin{equation}\\label{PDE_BACK}\n -d_{\\small{\\cev{\\mathbf{B}}}}v_{t,s}(z)=\\left(\\cev{a}^{\\ast}_{t,s}(z){probabilit\\`a }_{vv}v_{t,s}+\n \\cev{b}_{t,s}^{\\ast}(z){probabilit\\`a }_{v}v_{t,s}(z)+\\cev{c}^{\\ast}_{t,s}(z)v_{t,s}(z)\\right)dt,\n \\qquad {\\cev{\\mathbf{B}}}={probabilit\\`a }_t+\\cev{\\mathbf{Y}}_{t,s},\n\\end{equation}\nwhere $\\cev{\\mathbf{Y}}_{t,s}$ and the coefficients are defined similarly to \\eqref{aee13} and\n\\eqref{aee14}, exchanging the {\\it hat}-  and  {\\it check}-transforms  in the definitions. As for\nthe forward case, by Assumption \\ref{ass2} and Lemma \\ref{lemmaIW},\n$a^{\\ast}_{t,\\cdot}\\in \\mathbf{b}\\cev{\\mathbf{C}}^{{\\alpha}}_{t,T}$, $b^{\\ast}_{t,\\cdot},\nc^{\\ast}_{t,\\cdot}\\in \\mathbf{b}\\cev{\\mathbf{C}}^{0}_{t,T}$,\n{$\\cev{\\mathbf{Y}}_{t,\\cdot}\\in\\cev{\\mathbf{C}}^{0,1}_{t,T}$}, ${probabilit\\`a }_{\\nu}(\\cev{\\mathbf{Y}}_{t,\\cdot})_1\\in\n\\mathbf{b}\\cev{\\mathbf{C}}^{\\bar{{\\alpha}}}_{t,T}$, {for any $\\bar{{\\alpha}}\\in [0,{\\alpha})$}, and there exist two\nrandom, finite and positive constant $\\mathbf{m}_1$, $\\mathbf{m}_2$ such that, for $t\\in [0,s]$\nand $z \\in {\\mathbb {R}}^2$, we have\n\\begin{align}\n& \\mathbf{m}_1^{-1}\\le \\cev{a}^{\\ast}_{t,s}(z)\\le \\mathbf{m}_1, \\qquad \\mathbf{m}_2^{-1}\\le {probabilit\\`a }_v\n(\\cev{\\mathbf{Y}}_{t,s}(z))_1\\le \\mathbf{m}_2,\n\\end{align}\nwith probability one, which ensures the weak H\\\"ormander condition to hold.\n\n\\subsection{The parametrix expansion}\nEquations of the form \\eqref{PDE_FORW} have been studied in \\cite{pasc:pesc:19} by means of a\ntime-dependent parametrix expansion which takes into account the unbounded drift $\\mathbf{Y}$. The only\nminor difference here is the presence of a term of order zero $c^{\\ast}$ which, as we shall see,\ndoes not modify the analysis substantially.\n\nIn this section we briefly resume the parametrix construction and show how it works in the\nbackward framework. For the sake of readability, here we reset the notations and rewrite equation\n\\eqref{PDE_FORW} as\n\\begin{equation}\\label{aee16}\n  \\mathcal{A}_{s}v_{s}({\\zeta})-\\mathbf{Y}_{s}v_{s}({\\zeta})-{probabilit\\`a }_{s}v_{s}({\\zeta})=0,\\qquad s\\in(t,T],\\ {\\zeta}\\in{\\mathbb {R}}^{2},\n\\end{equation}\nwhere $\\mathcal{A}_s$ is a second order operator of the form\n  $$\\mathcal{A}_s=a_s{probabilit\\`a }_{{\\nu}\\n} + b_s{probabilit\\`a }_{\\nu} +c_\n  $$\nand\n$\\mathbf{Y}_{s}\n(\\mathbf{Y}_{s})_{1}{probabilit\\`a }_{{\\xi}}+(\\mathbf{Y}_{s})_{2}{probabilit\\`a }_{{\\nu}}$ is the vector field in \\eqref{aee13}. For fixed\n$(t_{0},z_{0})\\in [t,T)\\times {\\mathbb {R}}^2$, we linearize $\\mathbf{Y}_{s}$ by setting\n\\begin{equation}\\label{aee17}\n  \\mathbf{Y}_{s}^{t_{0},z_{0}}({\\zeta})=\\mathbf{Y}_{s}(\\gamma_{s}^{t_{0},z_{0}})+\\left(D\\mathbf{Y}_{s}\\right)(\\gamma_{s}^{t_{0},z_{0}})\n  \\left({\\zeta}-\\gamma_{s}^{t_{0},z_{0}}\\right)\n\\end{equation}\nwhere\n   {$$\\gamma_{s}^{t_{0},z_{0}}=z_{0}+\\int_{t_{0}}^{s}\\mathbf{Y}_{{\\tau}}(\\gamma_{{\\tau}}^{t_{0},z_{0}})d{\\tau},\\qquad s\\in[t_{0},T],$$}\nand $D\\mathbf{Y}_s$ is the reduced Jacobian defined as\n $$D\\mathbf{Y}_s:=\\begin{pmatrix}0 & {probabilit\\`a }_v(\\mathbf{Y}_{s})_1 \\\\ 0 & 0 \\end{pmatrix}.$$\nThen we consider the linearized version of \\eqref{aee16}, that is\n\\begin{equation}\\label{aee16lin}\n\n \\mathcal{A}^{t_{0},z_{0}}_{s}v_{s}({\\zeta})-\\mathbf{Y}_{s}^{t_{0},z_{0}}v_{s}({\\zeta})-{probabilit\\`a }_{s}v_{s}({\\zeta})=0,\\qquad s\\in(t,T],\\ {\\zeta}\\in{\\mathbb {R}}^{2},\n\\end{equation}\nwhere\n\\begin{equation}\\label{linearized_PDE_forw}\n \\mathcal{A}^{t_{0},z_{0}}_s:=a_s(\\gamma_s^{t_{0},z_{0}}){probabilit\\`a }_{{\\nu}\\n}.\n\\end{equation}\nIt turns out (cf. \\cite{pasc:pesc:19}, Section 5) that, for any choice of the parameters\n$(t_{0},z_{0})$, equation \\eqref{linearized_PDE_forw} has a fundamental solution\n$\\mathbf{\\Gamma}^{t_{0},z_{0}}=\\mathbf{\\Gamma}^{t_{0},z_{0}}(t,z;s,{\\zeta})$. Moreover, $\\mathbf{\\Gamma}^{t_{0},z_{0}}$ has an explicit\nGaussian expression and satisfies the following estimates {for some $\\lambda>0$ that depends only on\nthe general constants of Assumptions \\ref{ass1}, \\ref{ass2} and \\ref{ass3}}:\n\\begin{equation}\\label{aee18}\n\\begin{split}\n \\Gamma_{\\lambda^{-1}}\\left(s-t,{\\zeta}-{{\\gamma}}_{t,s}^{t_{0},z_{0}}(z)\\right)\\le\\,\n \\mathbf{\\Gamma}^{t_{0},z_{0}}(t,z;s,{\\zeta})&\\le \\Gamma_{\\lambda}\\left(s-t,{\\zeta}-{{\\gamma}}_{t,s}^{t_{0},z_{0}}(z)\\right), \\\\\n |{probabilit\\`a }_{\\nu} \\mathbf{\\Gamma}^{t_{0},z_{0}}(t,z;s,{\\xi},{\\nu})|&\\le  \\frac{1}{\\sqrt{s-t}}\\Gamma_{\\lambda}\\left(s-t,({\\xi},{\\nu})-{{\\gamma}}_{t,s}^{t_{0},z_{0}}(z)\\right),\\\\\n |{probabilit\\`a }_{{\\nu}\\n}\\mathbf{\\Gamma}^{t_{0},z_{0}}(t,z;s,{\\xi},{\\nu})|&\\le  \\frac{1}{s-t}\\Gamma_{\\lambda}\\left(s-t,({\\xi},{\\nu})-{{\\gamma}}_{t,s}^{t_{0},z_{0}}(z)\\right),\n\\end{split}\n\\end{equation}\nfor $0\\le t<s\\le T$ and $z,{\\zeta}\\in{\\mathbb {R}}^{2}$, with $\\Gamma_{\\lambda}$ as in \\eqref{gammal} and\n$s\\mapsto{\\gamma}_{t,s}^{t_{0},z_{0}}(z)$ defined by\n $${{\\gamma}}_{t,s}^{t_{0},z_{0}}(z)=z+\\int_t^{s}\\mathbf{Y}_{{\\tau}}^{t_{0},z_{0}}({{\\gamma}}_{{\\tau},t}^{t_{0},z_{0}}(z))d{\\tau},\\qquad s\\in[t,T].$$\n\nWe introduce the so-called \\textit{forward parametrix}\n  $$Z(t,z;s,{\\zeta}):=\\mathbf{\\Gamma}^{t,z}(t,z;s,{\\zeta}),\\qquad 0\\le t<s\\le T,\\ z,{\\zeta}\\in{\\mathbb {R}}^{2},$$\nthat will be used as a first approximation of a fundamental solution $\\mathbf{\\Gamma}$ of \\eqref{aee16}.\nOwing to the fact that {${\\gamma}^{t,z}_{t,s}(z)=\\gamma^{t,z}_s$}, $Z$ satisfies estimates \\eqref{aee18}.\n\nNow we set\n\\begin{align}\n H(t,z;s,{\\zeta})&:=\\left(\\mathcal{A}_s-\\mathbf{Y}_{s}-(\\mathcal{A}^{{t,z}}_s-\\mathbf{Y}_{s}^{{t,z}})\\right)Z(t,z;s,{\\zeta})\n\\end{align}\nand notice that\n\\begin{align}\n |H (t,z;s,{\\zeta})|&\\le |a_s({\\zeta})-a_s({\\gamma}_s^{t,z})||{probabilit\\`a }_{{\\nu}\\n}Z(t,z;s,{\\zeta})|\\\\ &\\quad+\n |(\\mathbf{Y}_s-\\mathbf{Y}_s^{t,z})Z(t,z;s,{\\zeta})| +\n |b_s({\\zeta})||{probabilit\\`a }_{{\\nu}}Z(t,z;s,{\\zeta})|+|c_s({\\zeta})||Z(t,z;s,{\\zeta})| \\intertext{(since ${probabilit\\`a }_{\\nu}(\\mathbf{Y}_s)_1$ is\n $\\bar{{\\alpha}}$-H\\\"older continuous by Proposition \\ref{Prop_IW})} &\\le\n \\left(\\frac{|{\\zeta}-{\\gamma}_s^{t,z}|^{{\\alpha}}}{s-t}+\\frac{|{\\zeta}-{\\gamma}_s^{t,z}|^{1+\\bar{{\\alpha}}}}{(s-t)^{3\/2}}\n +\\frac{|{\\zeta}-{\\gamma}_s^{t,z}|^{{\\alpha}}}{(s-t)^{1\/2}}+1\\right)\\mathbf{\\Gamma}_{\\lambda}(s-t,{\\zeta}-{\\gamma}_s^{t,z})\n\\intertext{(for some $\\bar{\\lambda}>\\lambda$)}\n &\\le\n \\frac{1}{(s-t)^{1-\\bar{{\\alpha}}\/2}}\\mathbf{\\Gamma}_{\\bar{\\lambda}}(s-t,{\\zeta}-{\\gamma}_s^{t,z}).\n\\end{align}\nNext, we set\n\\begin{align}\n  \\mathbf{\\Gamma}\\otimes H (t,z;s,{\\zeta})&:=\\int_t^s\\int_{{\\mathbb {R}}^2}H(t,z;{\\tau},w)\\mathbf{\\Gamma}({\\tau},w;s,{\\zeta})dw d{\\tau}.\n\\end{align}\nA recursive application of the Duhamel principle shows that\n\\begin{align}\\label{Expansion1}\n \\mathbf{\\Gamma}(t,z;s,{\\zeta})&=Z(t,z;s,{\\zeta})+\\mathbf{\\Gamma}\\otimes H (t,z;s,{\\zeta})\\\\ &= Z(t,z;s,{\\zeta}) + \\sum_{k=1}^{N-1}Z\\otimes\n H^{\\otimes k} (t,z;s,{\\zeta})+\\mathbf{\\Gamma}\\otimes H^{\\otimes N} (t,z;s,{\\zeta}),\\qquad N\\ge1.\n\\end{align}\nAs $N$ tends to infinity we formally obtain a representation of $\\mathbf{\\Gamma}$ as a series of convolution\nkernels. Unfortunately, as already noticed in \\cite{MR2659772} and \\cite{pasc:pesc:19}, the\npresence of the transport term makes it hard to control the iterated kernels uniformly in $N$,\nas opposed to the classical parametrix method for uniformly parabolic PDEs.\nThus the remainder $\\mathbf{\\Gamma}\\otimes H^{\\otimes N}$ must be handled with a different technique, borrowed\nfrom stochastic control theory: the rest of the proof proceeds exactly in the same way as in\n\\cite{pasc:pesc:19} to which we refer for a detailed explanation.\n\n\\medskip\n\nNext, we consider the backward equation\n\\begin{equation}\\label{aee16b}\n  \\cev{\\mathcal{A}}_{t}u_{t}(z)+\\cev{\\mathbf{Y}}_{t}u_{t}(z)+{probabilit\\`a }_{t}u_{t}(z)=0,\\qquad t\\in[0,s),\\ z=(x,v)\\in{\\mathbb {R}}^{2},\n\\end{equation}\nwhere $\\cev{\\mathcal{A}}_t$ is a second order operator of the form\n  $$\\cev{\\mathcal{A}}_t=\\cev{a}_t{probabilit\\`a }_{vv} + \\cev{b}_t{probabilit\\`a }_{v} +\\cev{c}_t,\\qquad z=(x,v)\\in{\\mathbb {R}}^{2}\n  $$\nand\n$\\cev{\\mathbf{Y}}_{t\n=(\\cev{\\mathbf{Y}}_{t})_{1}{probabilit\\`a }_{x}+(\\cev{\\mathbf{Y}}_{t})_{2}{probabilit\\`a }_{v}.$\nFor a fixed $(s_{0},{\\zeta}_{0})\\in (0,s]\\times{\\mathbb {R}}^{2}$, we define the linearized version of\n\\eqref{aee16b}, that is\n\\begin{equation}\\label{aee16lin_b}\n\n \\cev{\\mathcal{A}}^{s_{0},{\\zeta}_{0}}_{t}u_{t}(z)+\\cev{\\mathbf{Y}}_{t}^{s_{0},{\\zeta}_{0}}u_{t}(z)+{probabilit\\`a }_{t}u_{t}(z)=0,\\qquad t\\in[0,s),\\ z\\in{\\mathbb {R}}^{2},\n\\end{equation}\nwhere the definition of $\\cev{\\mathbf{Y}}_{t}^{s_{0},{\\zeta}_{0}}$ is analogous to that of\n$\\mathbf{Y}_{s}^{t_{0},z_{0}}$ in \\eqref{aee17} and\n\\begin{equation}\\label{linearized_PDE_back}\n \\cev{\\mathcal{A}}^{s_{0},{\\zeta}_{0}}_t:=\\cev{a}_t(\\cev{{\\gamma}}_t^{s_{0},{\\zeta}_{0}}){probabilit\\`a }_{vv},\\qquad\n \\cev{{\\gamma}}_{t}^{s_{0},{\\zeta}_{0}}={\\zeta}_{0}+\\int_{t}^{s_{0}}\\cev{\\mathbf{Y}}_{{\\tau}}(\\cev{{\\gamma}}_{{\\tau}}^{s_{0},{\\zeta}_{0}})d{\\tau},\\qquad t\\in[0,s_{0}].\n\\end{equation}\nEquation \\eqref{aee16lin_b} has an explicit fundamental solution\n$\\cev{\\mathbf{\\Gamma}}^{s_{0},{\\zeta}_{0}}=\\cev{\\mathbf{\\Gamma}}^{s_{0},{\\zeta}_{0}}(t,z;s,{\\zeta})$ of Gaussian type, that satisfies\nestimates analogous to \\eqref{aee18}.\nThe {\\it backward parametrix} for {\\eqref{aee16b}} is defined as\n $$\\cev{Z}(t,z;s,{\\zeta})=\\cev{\\mathbf{\\Gamma}}^{s,{\\zeta}}(t,z;s,{\\zeta}),\\qquad 0\\le t<s\\le T,\\ z,{\\zeta}\\in{\\mathbb {R}}^{2}.$$\nAs in the forward case, Duhamel principle yields the expansion\n\\begin{align}\\label{Expansion2}\n \\cev{\\mathbf{\\Gamma}}(t,z;s,{\\zeta}) = \\cev{Z}(t,z;s,{\\zeta}) + \\sum_{k=1}^{N-1}\\cev{Z}\\otimes \\cev{H}^{\\otimes k}\n (t,z;s,{\\zeta})+\\cev{\\mathbf{\\Gamma}}\\otimes \\cev{H}^{\\otimes N} (t,z;s,{\\zeta}),\\qquad N\\ge 1,\n\\end{align}\nwhere\n$\\cev{H}(t,z;s,{\\zeta})=\\left(\\cev{\\mathcal{A}}_t+\\cev{\\mathbf{Y}}_{t}-\\cev{\\mathcal{A}}^{s_{0},{\\zeta}_{0}}_t-\n\\cev{\\mathbf{Y}}^{s_{0},{\\zeta}_{0}}_{t}\\right)\\cev{Z}(t,z;s,{\\zeta})$ and the rest of the proof proceeds as\nin the forward case. In particular, existence and estimates for the fundamental solutions  of\n\\eqref{PDE_FORW} and \\eqref{PDE_BACK} (in the sense of Definitions \\ref{d1} and \\ref{d2}) follow\nfrom the parametrix expansions \\eqref{Expansion1} and \\eqref{Expansion2}. Eventually, it suffices\nto go back to the original variables to conclude the proof: we refer to \\cite{pasc:pesc:19},\nSection 6, for full details.\n\n\\subsection{Proof of Corollary \\ref{L_Cauchy}}\n{By Theorem \\ref{TH1} there exists a fundamental solution $\\cev{\\mathbf{\\Gamma}}$ of equation\n\\eqref{e_Cauchy}, in the sense of Definition \\ref{d2}. Moreover, since ${\\sigma}\\equiv 0$, $\\cev{\\mathbf{\\Gamma}}$\nsatisfies estimates \\eqref{t_e4}, \\eqref{t_e5} and \\eqref{t_e6} with $\\cev{g}^{\\text{\\rm\\tiny IW},-1}_{t,s}\\equiv {\\rm Id}$\nand $\\cev{\\gamma}_t^{s,{\\zeta}}=\\gamma^{\\mathbf{B}}_{t-s}({\\zeta})$ as in Definition \\ref{ad1}.}\nThen, the function\n\\begin{equation}\n {f_t(z)}:={\\int_{{\\mathbb {R}}^{d+1}}}\\cev{\\mathbf{\\Gamma}}(t,z,T,{\\zeta}){\\varphi}({\\zeta})d{\\zeta},\\qquad (t,z)\\in[0,T]\\times {\\mathbb {R}}^{d+1},\n\\end{equation}\nsolves problem \\eqref{e_Cauchy2}. Since ${\\varphi}\\in bC({\\mathbb {R}}^{d+1})$, we have\n $$\\sup_{z\\in{\\mathbb {R}}^{d+1}}|f_z(z)|\\le \\|{\\varphi}\\|_{\\infty}\\sup_{z\\in{\\mathbb {R}}^{d+1}}\\int_{{\\mathbb {R}}^{d+1}}\\cev{\\mathbf{\\Gamma}}(t,z,T,{\\zeta})d{\\zeta}\\le C$$\nfor a positive constant $C$. Estimate \\eqref{coreq} is proven in \\cite{MR2352998}, Proposition\n3.3\n\\endproof\n\n\n\\section{Backward It\\^o calculus}\\label{Itoback}\nIn this section we collect some basic result about {\\it backward It\\^o integrals} and the {\\it\nbackward diffusion SPDE} (or {\\it Krylov equation} according to \\cite{MR3839316}). This is\nstandard material which resume the original results in \\cite{MR0339338}, \\cite{MR653654},\n\\cite{MR673162}, \\cite{MR665405}, \\cite{MR706230} (see also the monographs \\cite{MR3839316} and\n\\cite{MR1070361}).\n\nLet $W=(W_{t})_{t\\in[0,T]}$ be a $d$-dimensional Brownian motion on $({\\Omega},\\mathcal{F},P,\\mathcal{F}^{W})$ where\n$\\mathcal{F}^{W}$ denotes the standard Brownian filtration satisfying the usual assumptions. We consider\n  $$\\mathcal{F}^{W,t}_{T}={\\sigma}(\\mathcal{G}_{t}\\cup \\mathcal{N}),\\qquad \\mathcal{G}_{t}={\\sigma}(W_{s}-W_{t},\\, t\\le s\\le T),\\qquad t\\in[0,T],$$\nthe augmented  ${\\sigma}$-algebra of Brownian increments between $t$ and $T$. Notice that\n$(\\mathcal{F}^{W,t}_{T})_{0\\le t\\le T}$ is a decreasing family of ${\\sigma}$-algebras. Then the process\n $$\\cev{W}_{t}:=W_{T}-W_{T-t},\\qquad t\\in[0,T],$$\nis a Brownian motion on $({\\Omega},\\mathcal{F},P,\\cev{\\mathcal{F}})$ where\n  $$\\cev{\\mathcal{F}}_{t}:=\\mathcal{F}^{W,T-t}_{T},\\qquad t\\in[0,T],$$\nis the ``backward'' Brownian filtration. The backward stochastic It\\^o integral is defined as\n\\begin{equation}\\label{e2}\n  \\int_{t}^{s}u_{r}\\star dW_{r}:=\\int_{T-s}^{T-t}u_{T-r}d\\cev{W}_{r},\\qquad 0\\le t\\le s\\le T,\n\\end{equation}\nunder the assumptions on $u$ for which the RHS of \\eqref{e2} is defined in the usual It\\^o sense,\nthat is\n\\begin{itemize}\n  \\item[i)] $t\\mapsto u_{T-t}$ is $\\cev{\\mathcal{F}}$-progressively\n  measurable (thus $u_{t}\\in m \\mathcal{F}^{W,t}_{T}$ for any $t\\in[0,T]$);\n\n  \\item[ii)] $u\\in L^{2}([0,T])$ a.s.\n\\end{itemize}\nFor practical purposes, if $u$ is continuous, the backward integral is the limit\n\\begin{equation}\\label{e3}\n  \\int_{t}^{s}u_{r}\\star dW_{r}:=\\lim_{|\\pi|\\to 0^+}\\sum_{k=1}^{n}u_{t_{k}}\\left(W_{t_{k}}-W_{t_{k-1}}\\right)\n\\end{equation}\nin probability, where $\\pi=\\{t=t_{0}<t_{1}<\\cdots<t_{n}=s\\}$ denotes a partition of $[t,s]$.\n\nA backward It\\^o process is a process of the form\n\\begin{equation}\\label{e4}\n  X_{t}=X_{T}+\\int_{t}^{T}b_{s}ds+\\int_{t}^{T}{\\sigma}_{s}\\star dW_{s},\\qquad t\\in[0,T],\n\\end{equation}\nalso written in differential form as\n\\begin{equation}\\label{e5}\n  -d X_{t}=b_{t}dt+{\\sigma}_{t}\\star dW_{t}.\n\\end{equation}\n\\begin{theorem}[\\bf Backward It\\^o formula]\\label{t1}\nLet $v=v(t,x)\\in C^{1,2}({\\mathbb {R}}_{\\ge0}\\times{\\mathbb {R}}^{d})$ and let $X$ be the process in \\eqref{e5}. Then\n\\begin{equation}\\label{e6}\n  -dv(t,X_{t})=\\left(({probabilit\\`a }_{t}v)(t,X_{t})+\\frac{1}{2}({\\sigma}_{t}{\\sigma}_{t}^{\\ast})_{ij}({probabilit\\`a }_{x_{i}x_{j}}v)(t,X_{t})+(b_{t})_{i}({probabilit\\`a }_{x_{i}}v)\n  (t,X_{t})\\right)dt+\n  ({\\sigma}_{t})_{ij}\\left({probabilit\\`a }_{x_{i}} v\\right)(t,X_{t})\\star dW^{j}_{t}.\n\\end{equation}\n\\end{theorem}\nA crucial tool in our analysis is the followin\n\\begin{theorem}[\\bf Backward diffusion SPDE]\\label{t2}\nAssume $b,{\\sigma}\\in bC^{3}({\\mathbb {R}}_{\\ge0}\\times{\\mathbb {R}}^{d})$ and denote by $s\\mapsto X^{t,x}_{s}$ the solution\nof the SDE\n\\begin{equation}\\label{e7}\n  dX^{t,x}_{s}=b(s,X^{t,x}_{s})ds+{\\sigma}(s,X^{t,x}_{s})dW_{s}\n\\end{equation}\nwith initial condition $X^{t,x}_{t}=x$. Then the process $(t,x)\\mapsto X^{t,x}_{T}$ solves the\nbackward SPDE\n\\begin{align}\\label{e8}\n  \\begin{cases}\n  -dX^{t,x}_{T}=\\mathcal{L}X^{t,x}_{T}dt\n +{\\sigma}_{ij}(t,x){probabilit\\`a }_{x_{i}}X^{t,x}_{T}\\star dW^{j}_{t}, \\\\\n    X^{T,x}_{T}=x,\n  \\end{cases}\n\\end{align}\nwhere\n  \\begin{equation}\\label{e11}\n  \\mathcal{L}=\\frac{1}{2}({\\sigma}(t,x){\\sigma}^{\\ast}(t,x))_{ij}{probabilit\\`a }_{x_{j}x_{i}}+b_{i}(t,x){probabilit\\`a }_{x_{i}}\n\\end{equation}\nis the characteristic operator of $X$. More explicitly, in \\eqref{e8} we have\n  $$\\mathcal{L}X^{t,x}_{T}\\equiv \\frac{1}{2}({\\sigma}(t,x){\\sigma}^{\\ast}(t,x))_{ij}{probabilit\\`a }_{x_{j}x_{i}}X^{t,x}_{T}+b_{i}(t,x){probabilit\\`a }_{x_{i}}X^{t,x}_{T}.$$\n\\end{theorem}\n\\begin{remark}\nThe regularity assumption of Theorem \\ref{t2} on the coefficients is by no means optimal:\n\\cite{MR3839316}, Theorem 5.1, proves that $(t,x)\\mapsto X^{t,x}_{T}$ is a generalized (or\nclassical, under non-degeneracy conditions) solution of \\eqref{e8} if $b,{\\sigma}\\in\nbC^{1}({\\mathbb {R}}_{\\ge0}\\times{\\mathbb {R}}^{d})$.\n\\end{remark}\n\\proof For illustrative purposes we only consider the one-dimensional, autonomous case. A general\nproof can be found in \\cite{MR3839316}, Proposition 5.3. Here we follow the ``direct'' approach\nproposed in \\cite{MR706230}. By standard results for  stochastic flows (cf. \\cite{MR1070361}),\n$x\\mapsto X^{t,x}_{T}$ is sufficiently regular to support the derivatives in the classical sense.\nWe use the Taylor expansion for $C^{2}$-functions:\n\\begin{equation}\\label{e12}\n  f({\\delta})-f(0)={\\delta} f'(0)+\\frac{{\\delta}^{2}}{2}f''(\\lambda{\\delta}),\\qquad \\lambda\\in[0,1].\n \n\\end{equation}\nWe have\n\\begin{align}\n X^{t,x}_{T}-x&=X^{t,x}_{T}-X^{T,x}_{T}\\\\\n &=\\sum_{k=1}^{n}\\left(X^{t_{k-1},x}_{T}-X^{t_{k},x}_{T}\\right)=\n\\intertext{(by the flow property)}\n &=\\sum_{k=1}^{n}\\left(X^{t_{k},X^{t_{k-1},x}_{t_{k}}}_{T}-X^{t_{k},x}_{T}\\right)=\n\\intertext{(by \\eqref{e12} with $f({\\delta})=X_{T}^{t_{k},x+{\\delta}}$ and\n ${\\delta}=\\Delta_{k}X:=X^{t_{k-1},x}_{t_{k}}-x$)}\\label{e14}\n &=\\sum_{k=1}^{n}\\left(\\Delta_{k}X{probabilit\\`a }_{x}X^{t_{k},x}_{T}+\n \\frac{(\\Delta_{k}X)^{2}}{2}{probabilit\\`a }_{xx}X^{t_{k},x+\\lambda_{k}\\Delta_{k}X}_{T}\\right)\n\\end{align}\nfor some $\\lambda_{k}=\\lambda_{k}({\\omega})\\in[0,1]$. Now, we have\n  $$\\Delta_{k}X=X^{t_{k-1},x}_{t_{k}}-x=\\int_{t_{k-1}}^{t_{k}}b(X^{t_{k-1},x}_{s})ds+\\int_{t_{k-1}}^{t_{k}}{\\sigma}(X^{t_{k-1},x}_{s})dW_{s}.$$\nThus, setting\n  $$\\Delta_{k}t=t_{k}-t_{k-1},\\qquad \\Delta_{k}W=W_{t_{k}}-W_{t_{k-1}},\\qquad \\widetilde{\\Delta}_{k}X=b(x)\\Delta_{k}t+{\\sigma}(x)\\Delta_{k}W,$$\nby standard estimates for solutions of SDEs, we have\n\\begin{align}\n \\Delta_{k}X-\\widetilde{\\Delta}_{k}X=\n \\int_{t_{k-1}}^{t_{k}}\\left(b(X^{t_{k-1},x}_{s})-b(x)\\right)ds+\\int_{t_{k-1}}^{t_{k}}\\left({\\sigma}(X^{t_{k-1},x}_{s})-{\\sigma}(x)\\right)dW_{s}\n &=\\text{O}(\\Delta_{k}t),\\\\ \\label{e16}\n {probabilit\\`a }_{xx}X^{t_{k},x+\\lambda_{k}\\Delta_{k}X}_{T}-{probabilit\\`a }_{xx}X^{t_{k},x}_{T} &=\\text{O}(\\Delta_{k}t),\n\\end{align}\nin the square mean sense or, more precisely,\n  $\n  E\\left[|\\Delta_{k}X-\\widetilde{\\Delta}_{k}X|^{2}+\\left|{probabilit\\`a }_{xx}X^{t_{k},x+\\lambda_{k}\\Delta_{k}X}_{T}-\n  {probabilit\\`a }_{xx}X^{t_{k},x}_{T}\\right|^{2}\\right]\\le c(1+|x|^{2}\n  (\\Delta_{k}t)^{2}$$\nwith $c$ depending only on $T$ and the Lipschitz constants of $b,{\\sigma}$. From \\eqref{e14} we get\n\\begin{align}\n X^{t,x}_{T}-\n &=\\sum_{k=1}^{n}\\left(\\widetilde{\\Delta}_{k}X{probabilit\\`a }_{x}X^{t_{k},x}_{T}+\n \\frac{(\\widetilde{\\Delta}_{k}X)^{2}}{2}{probabilit\\`a }_{xx}X^{t_{k},x}_{T}\\right)+\\text{O}(\\Delta_{k}t).\n\\end{align}\nNext we recall {\\eqref{e3}} and notice that\n  ${probabilit\\`a }_{x}X^{t_{k},x}_{T},{probabilit\\`a }_{xx}X^{t_{k},x}_{T}\\in m\\mathcal{F}^{W,t_{k}}_{T}.$\nThus, passing to the limit, we have\n\\begin{align}\n \\sum_{k=1}^{n}\\widetilde{\\Delta}_{k}X{probabilit\\`a }_{x}X^{t_{k},x}_{T} &\\longrightarrow\n \\int_{t}^{T}b(t,x){probabilit\\`a }_{x}X^{s,x}_{T}ds+\\int_{t}^{T}{\\sigma}(x){probabilit\\`a }_{x}X^{s,x}_{T}\\star dW_{s},\\\\\n \\sum_{k=1}^{n}(\\widetilde{\\Delta}_{k}X)^{2}{probabilit\\`a }_{xx}X^{t_{k},x}_{T} &\\longrightarrow\n \\int_{t}^{T}{\\sigma}^{2}(x){probabilit\\`a }_{xx}X^{s,x}_{T}ds,\n\\end{align}\nin the square mean sense and this concludes the proof.\n\\endproof\nWe have a useful corollary of Theorem \\ref{t2}.\n\\begin{corollary}[\\bf Invariance of the backward diffusion SPDE]\\label{c1}\nFor $v\\in bC^{2}({\\mathbb {R}}^{d})$ and $X$ as in \\eqref{e7}, let $V^{t,x}_{T}=v(X^{t,x}_{T})$. Then\n$V^{t,x}_{T}$ satisfies the same SPDE \\eqref{e8}, that is\n\\begin{equation}\\label{e8b}\n -dV^{t,x}_{T}=\\mathcal{L}V^{t,x}_{T}dt+\n {\\sigma}_{ij}(t,x){probabilit\\`a }_{x_{i}}V^{t,x}_{T}\\star dW^{j}_{t}\n\\end{equation}\nwith terminal condition $V^{T,x}_{T}=g(x)$.\n\\end{corollary}\n\\begin{proof\nTo fix ideas, we first consider the one-dimensional case: by the backward SPDE \\eqref{e8} and the\nbackward It\\^o formula \\eqref{e6}, we have\n\\begin{align\n  -dv(X^{t,x}_{T})&=\n  \\left(\\frac{{\\sigma}^{2}(t,x)}{2}v''(X^{t,x}_{T})({probabilit\\`a }_{x} X^{t,x}_{T})^{2}+\\frac{{\\sigma}^{2}(t,x)}{2}v'(X^{t,x}_{T}){probabilit\\`a }_{xx}X^{t,x}_{T}\n  +b(t,x)v'(X^{t,x}_{T}){probabilit\\`a }_{x} X^{t,x}_{T}\\right)dt\\\\\n  &\\quad+{\\sigma}(t,x)v'(X^{t,x}_{T}){probabilit\\`a }_{x} X^{t,x}_{T}\\star dW_{t}=\n\\intertext{(using the identities ${probabilit\\`a }_{x} V^{t,x}_{T}=v'(X^{t,x}_{T}){probabilit\\`a }_{x} X^{t,x}_{T}$ and\n${probabilit\\`a }_{xx}V^{t,x}_{T}=v''(X^{t,x}_{T})({probabilit\\`a }_{x}X^{t,x}_{T})^{2}+\n  v'(X^{t,x}_{T}){probabilit\\`a }_{xx}X^{t,x}_{T}$)}\n  &=\\left(\\frac{{\\sigma}^{2}(t,x)}{2}{probabilit\\`a }_{xx}V^{t,x}_{T}+b(t,x){probabilit\\`a }_{x} V^{t,x}_{T}\\right)dt+{\\sigma}(t,x){probabilit\\`a }_{x} V^{t,x}_{T}\\star\n  dW_{t}\n\\end{align}\nand this proves the thesis. In general, we have\n\\begin{equation}\\label{e9}\n\\begin{split}\n  {probabilit\\`a }_{x_{h}} V^{t,x}_{T}&=(\\nabla v)(X^{t,x}_{T}){probabilit\\`a }_{x_{h}} X^{t,x}_{T},\\\\\n  {probabilit\\`a }_{x_{h}x_{k}}V^{t,x}_{T}&\n  ({probabilit\\`a }_{ij} v)(X^{t,x}_{T})({probabilit\\`a }_{x_{h}}X^{t,x}_{T})_{i}({probabilit\\`a }_{x_{k}}X^{t,x}_{T})_{j}+\n  (\\nabla v)(X^{t,x}_{T})({probabilit\\`a }_{x_{h}x_{k}}X^{t,x}_{T}),\n\\end{split}\n\\end{equation}\nand by \\eqref{e8} and \\eqref{e6}\n\\begin{align\n  -dv(X^{t,x}_{T})&=\n  \\left(\\frac{1}{2}\\left((\\nabla X^{t,x}_{T}){\\sigma}(t,x)((\\nabla X^{t,x}_{T}){\\sigma}(t,x))^{\\ast}\\right)_{ij}({probabilit\\`a }_{ij}v)(X^{t,x}_{T})\\right)dt\\\\\n  &\\quad+\\left(\\frac{1}{2}({\\sigma}(t,x){\\sigma}^{\\ast}(t,x))_{ij}{probabilit\\`a }_{x_{j}x_{i}}X^{t,x}_{T}+b(t,x)\\nabla X^{t,x}_{T}\\right)(\\nabla v)\n  (X^{t,x}_{T})dt\\\\\n  &\\quad+(\\nabla v)(X^{t,x}_{T})(\\nabla X^{t,x}_{T}){\\sigma}(t,x)\\star dW_{t}=\n\\intertext{(by \\eqref{e9})}\n  &=\\left(\\frac{1}{2}({\\sigma}(t,x){\\sigma}^{\\ast}(x))_{ij}{probabilit\\`a }_{x_{j}x_{i}}V^{t,x}_{T}+b(t,x)\\nabla V^{t,x}_{T}\\right)dt+\\nabla V^{t,x}_{T}{\\sigma}(t,x)\\star\n  dW_{t}.\n\\end{align}\n\\end{proof}\n\n\n\\bigskip\\noindent\n{\\bf Acknowledgments.} This was supported by the Gruppo Nazionale per l'Analisi Matematica, la\nProbabilit\\`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica\n(INdAM).\n\n\n\\bibliographystyle{spmpsci}     \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\nIn this report, we present the phased array feed (PAF) model equations\nfor two definitions of embedded beam patterns. In Roshi \\& Fisher (2016), \nwe presented the PAF model by defining the embedded beam pattern\nas the beam pattern due to a 1 V excitation to one port and all other\nports short circuited. This embedded beam pattern is referred to as\nvoltage-embedded-beam (VEB). The embedded beam pattern can also be defined as\nthe beam pattern due to a 1 A excitation to one port and all other\nports open circuited. This definition is usually used in engineering\nliterature and we refer to the pattern as current-embedded-beam (CEB). \nHere we derive the relationship between the two embedded\nbeam patterns and present the corresponding model equations.\n\n\\section{Introduction}\n\n\\subsection{Embedded beam pattern}\n\n\\begin{figure}[t]\n\\begin{tabular}{cc}\n\\includegraphics[width=3.0in, height=2.5in, angle =0]{pafmodela.pdf} & \n\\includegraphics[width=3.0in, height=2.5in, angle =0]{pafmodelb.pdf} \n\\end{tabular} \n\\caption{PAF in transmitting mode. The net radiation pattern of the PAF\ncan be expressed as a `weighted' sum of the embedded beam patterns (see\nEq.~\\ref{Evsvembpat} \\& \\ref{Evsiembpat}). The `weights' can be either the\nport voltages along with the VEB, \n$\\boldsymbol{\\vec{\\mathcal{E}}^e}$ (left)\nor port currents along with the CEB,  \n$\\boldsymbol{\\vec{\\mathcal{\\psi}}^e}$ (right).\n}\n\\label{fig1}\n\\end{figure}\n\nIt is convenient to express the radiation pattern of PAF in terms of the {\\em embedded beam\npattern} (see Roshi \\& Fisher 2016). The embedded beam pattern can be defined \nin different ways. A definition \nused in Roshi \\& Fisher (2016) is : the $j^{th}$ embedded beam pattern, $\\vec{\\mathcal{E}}^e_j$ is \nthe beam pattern of the PAF when $j^{th}$ port is excited with\n1 V (i.e. $v_{0_j} = 1$ V) and all other ports are short circuited \n(i.e. $v_{0_i} = 0$ V for $i \\neq j$; see Fig.~\\ref{fig1}a).\nThe source impedance for excitation is considered to be equal to $z_0$, the \ncharacteristic impedance of the transmission line connected to the dipole.\nThere will be $M$ embedded beam patterns for a $M$ element PAF, which \nare represented conveniently as a vector $\\boldsymbol{\\vec{\\mathcal{E}}^e}$,\n\\begin{equation}\n\\boldsymbol{\\vec{\\mathcal{E}}^e}^T = \\left[\\vec{\\mathcal{E}}^e_1, \\vec{\\mathcal{E}}^e_2, ... \\right].\n\\end{equation} \nThese beam patterns are functions of the position vector $\\vec{r}$ with the origin of the\ncoordinate system located at the center of the PAF (see Roshi \\& Fisher 2016). \nThe beam patterns are specified at the far-field \nie $|\\vec{r}| >> \\frac{2 D_{array}^2}{\\lambda}$, where $D_{array}$ is the maximum\nphysical size of the PAF and $\\lambda$ is the wavelength of operation of the PAF.  \nThe radiation pattern when the PAF is excited by an arbitrary set of port voltages\nis obtained by scaling the embedded beam patterns with the\nport voltage and summing them up.\nHence the dimension of the embedded beam pattern in this definition is m$^{-1}$. \nAt the far-field, the beam pattern can be described by an outgoing spherical wave,\n\\begin{equation}\n\\vec{\\mathcal{E}}^e_i(\\vec{r}) =  \\vec{E}^e_i(\\theta, \\phi) \\; \\frac{e^{j\\vec{k}.\\vec{r}}}{r},\n\\label{farf}\n\\end{equation}\nwhere $\\vec{\\mathcal{E}}^e_i$ is the $i^{th}$ embedded beam pattern,\n$r$ and $\\hat{r}$ are the magnitude and the unit vector in the direction \nof $\\vec{r}$ respectively, $\\vec{k} = \\frac{2\\pi}{\\lambda} \\hat{r}$ is \nthe propagation vector. Here $\\vec{E}^e_i$ depends only \non the coordinates $\\theta, \\phi$. The geometric phase due to the location\nof elements (or in other words the excitation current distribution) away\nfrom the co-ordinate center is included in $\\vec{E}^e_i$. From the \ndefinition of embedded pattern it follows that $\\vec{E}^e_i$ is dimensionless. \nThe fields here are harmonic quantities, and for simplicity we omit the term $e^{j\\omega t}$.\nThe radiation pattern of the PAF when excited by a set of arbitrary\nport voltages is \n\\begin{eqnarray} \n\\vec{\\mathcal{E}}(\\vec{r}) & =  & \\sum_{i=1,M} v_{0_i} \\vec{\\mathcal{E}}^e_i(\\vec{r}), \\nonumber \\\\ \n                           & =  & \\bm V_0^T \\boldsymbol{\\vec{\\mathcal{E}}^e},  \n\\label{PAFfpat}\n\\end{eqnarray}\nwhere $\\bm V_0$ is the vector of port voltages $v_{0_i}$ (see Fig.~\\ref{fig1}a). The \nradiation pattern $\\vec{\\mathcal{E}}$ has units V\/m. In the far-field,\nthe ($\\theta, \\phi$) dependence of the radiation pattern can be written\nin a similar fashion,\n\\begin{equation}\n\\vec{E}(\\theta, \\phi) = \\bm V_0^T \\boldsymbol{\\vec{E}^e}.\n\\label{Evsvembpat}\n\\end{equation}\nThe unit of $\\vec{E}$ is V. In this report, we refer to both $\\boldsymbol{\\vec{\\mathcal{E}}^e}$ \nand $\\boldsymbol{\\vec{E}^e}$ as embedded beam pattern VEB. \n\nAnother definition for embedded beam pattern is: the $j^{th}$ embedded beam pattern, \n$\\vec{\\mathcal{\\psi}}^e_j$ is \nthe beam pattern of the PAF when $j^{th}$ port is excited with\n1 A and all other ports are open circuited, i.e.\n\\begin{eqnarray}\n\\mathcal{J}_{0_i} & = & 1\\;\\; \\textrm{A}\\;\\; \\textrm{for}\\; i = j,  \\nonumber\\\\\n\\mathcal{J}_{0_i} & = & 0\\;\\; \\textrm{A}\\;\\; \\textrm{for}\\; i \\neq j,\n\\end{eqnarray}\nwhere $\\mathcal{J}_{0_i}$ are the port currents.\nThe source impedance for excitation is considered to be equal to $z_0$.  \nAs before there are $M$ embedded beam patterns, which are represented conveniently\nas a vector $\\boldsymbol{\\vec{\\mathcal{\\psi}}^e}$,\n\\begin{equation}\n\\boldsymbol{\\vec{\\mathcal{\\psi}}^e}^T = \\left[\\vec{\\mathcal{\\psi}}^e_1, \\vec{\\mathcal{\\psi}}^e_2, ... \\right].\n\\end{equation} \nThe beam pattern at far field can be written as,\n\\begin{equation}\n\\vec{\\mathcal{\\psi}}^e_i(\\vec{r}) =  \\vec{\\Psi}^e_i(\\theta, \\phi) \\; \\frac{e^{j\\vec{k}.\\vec{r}}}{r},\n\\label{ifarf}\n\\end{equation}\nThe radiation pattern of the PAF when excited by a set of arbitrary\nport currents is \n\\begin{eqnarray} \n\\vec{\\mathcal{E}}(\\vec{r}) & =  & \\sum_{i=1,M} \\mathcal{J}_{0_i} \\vec{\\mathcal{\\psi}}^e_i(\\vec{r}), \\nonumber \\\\ \n                           & =  & \\bm I_0^T \\boldsymbol{\\vec{\\mathcal{\\psi}}^e},  \n\\label{PAFfpat2}\n\\end{eqnarray}\nwhere $\\bm I_0$ is the vector of port currents $\\mathcal{J}_{0_i}$ (see Fig.~\\ref{fig1}b). The \nradiation pattern $\\vec{\\mathcal{E}}$ has the unit V\/m and\n$\\vec{\\mathcal{\\psi}}^e_i$ has unit V\/A\/m. As before, \nthe ($\\theta, \\phi$) dependence of the far-field radiation pattern can be written\nas,\n\\begin{equation}\n\\vec{E}(\\theta, \\phi) = \\bm I_0^T \\boldsymbol{\\vec{\\Psi}^e}.\n\\label{Evsiembpat}\n\\end{equation}\nThe unit of $\\vec{E}$ is V and that of $\\vec{\\Psi}^e$ is V\/A. In this report,\nwe refer to both $\\boldsymbol{\\vec{\\mathcal{\\psi}}^e}$ and $\\boldsymbol{\\vec{\\Psi}^e}$ as the \nembedded beam pattern CEB. \n\nThe relationship between the two embedded beam patterns can be obtained\nusing the network relationship between the port voltages and\ncurrents, $\\bm V_0 = \\bm Z \\bm I_0$.\nSubstituting this relationship in Eq.~\\ref{PAFfpat}, we get \n\\begin{equation}\n\\vec{\\mathcal{E}}  =  \\bm V_0^T \\boldsymbol{\\vec{\\mathcal{E}}^e} =  \\bm I_0^T \\bm Z^T \\boldsymbol{\\vec{\\mathcal{E}}^e}.\n\\label{eq11}\n\\end{equation}\nFrom Eq.~\\ref{PAFfpat2} \\& \\ref{eq11} it follows\n\\begin{equation}\n\\boldsymbol{\\vec{\\mathcal{\\psi}}^e} = \\bm Z^T \\boldsymbol{\\vec{\\mathcal{E}}^e}\n\\end{equation}\nFor a reciprocal PAF, $\\bm Z^T = \\bm Z$, and so the above equation \ncan also be written as\n\\begin{equation}\n\\boldsymbol{\\vec{\\mathcal{\\psi}}^e} = \\bm Z \\boldsymbol{\\vec{\\mathcal{E}}^e}\n\\end{equation}\n\n\n\\section{PAF model equations corresponding to the two embedded beam patterns}\n\nThe PAF model equations are somewhat simplified when written in terms  \nof CEB $\\boldsymbol{\\vec{\\mathcal{\\psi}}^e}$. Essentially in almost \nall relevant equations the impedance matrix $\\bm Z$ is absorbed in the\nembedded beam pattern when $\\boldsymbol{\\vec{\\mathcal{\\psi}}^e}$ is used. For example,\nthe open circuit voltage vector (see Eq. 37 in Roshi \\& Fisher 2016)\nat the output of the PAF for VEB, \n$\\boldsymbol{\\vec{\\mathcal{E}}^e}$, and CEB, $\\boldsymbol{\\vec{\\mathcal{\\psi}}^e}$, is given by Eqs.~\\ref{eq14a} \\& \\ref{eq14b}\nrespectively; \n\\begin{eqnarray}\n\\bm V_{oc} &  = & \\bm Z \\int_{A_{free}} \\left(\\boldsymbol{\\vec{\\mathcal{E}}^e}^T\n             \\times \\boldsymbol{\\mathcal{I}} \\vec{\\mathcal{H}_r} -\n             \\boldsymbol{\\mathcal{I}} \\vec{\\mathcal{E}_r} \\times \\boldsymbol{\\vec{\\mathcal{H}}^e}\\right)\n             \\cdot \\hat{n}\\; \\textrm{d}A, \\label{eq14a} \\\\\n           &  = & \\int_{A_{free}} \\left(\\boldsymbol{\\vec{\\mathcal{\\psi}}^e}^T\n             \\times \\boldsymbol{\\mathcal{I}} \\vec{\\mathcal{H}_r} -\n             \\boldsymbol{\\mathcal{I}} \\vec{\\mathcal{E}_r} \\times \\boldsymbol{\\vec{\\mathcal{J}}^e}\\right)\n             \\cdot \\hat{n}\\; \\textrm{d}A. \\label{eq14b}\n\\label{voc}\n\\end{eqnarray} \nHere $\\boldsymbol{\\vec{\\mathcal{H}}^e}$ and $\\boldsymbol{\\vec{\\mathcal{J}}^e}$ are the magnetic \nfield patterns corresponding to the VEB, $\\boldsymbol{\\vec{\\mathcal{E}}^e}$\nand the CEB, $\\boldsymbol{\\vec{\\mathcal{\\psi}}^e}$ respectively, $\\vec{\\mathcal{E}_r}$ and\n$\\vec{\\mathcal{H}_r}$ are the incident electric and magnetic fields on the PAF respectively, \n$\\boldsymbol{\\mathcal{I}}$ is the identify matrix, and the integration is over a region outside the\nPAF (see Fig. 2 in Roshi \\& Fisher 2016). The notation used\nin Eq.~\\ref{voc} is explained in Appendix J of Roshi \\& Fisher (2016).\nA list of model equations corresponding to the VEB, \n$\\boldsymbol{\\vec{\\mathcal{E}}^e}$ (left) and the CEB, $\\boldsymbol{\\vec{\\mathcal{\\psi}}^e}$ (right) is given below.\n\\begin{align}\n\\bm R_{spill} & =  \\frac{4 k_B T_g}{z_f} \\bm Z \\bm C_{Ce1}, \\bm Z^H  \n&\\bm R_{spill}  = & \\frac{4 k_B T_g}{z_f} \\bm C_{C\\psi1},  \\\\\n\\bm R_{signal}& =  \\frac{2 S_{source}}{z_f} \\bm Z \\bm C_{Ie}, \\bm Z^H\n&\\bm R_{signal} = & \\frac{2 S_{source}}{z_f} \\bm C_{I\\psi},  \\\\\nT_{spill}     & =  T_g\n                  \\frac{\\bm w_1^H \\bm Z \\bm C_{Ce1} \\bm Z^H  \\bm w_1}\n                  {\\bm w_1^H \\bm Z \\bm C_{Ce} \\bm Z^H  \\bm w_1},  \n&T_{spill}      = &  T_g\n                  \\frac{\\bm w_1^H \\bm C_{C\\psi1}  \\bm w_1}\n                  {\\bm w_1^H \\bm C_{C\\psi}  \\bm w_1},  \\\\\nT_A           & =  \\frac{S_{source}}{2 k_B} \\frac{\\bm w_1^H \\bm Z \\bm C_{Ie} \\bm Z^H \\bm w_1}\n                  {\\bm w_1^H \\bm Z \\bm C_{Ce} \\bm Z^H   \\bm w_1},\n&T_A            = & \\frac{S_{source}}{2 k_B} \\frac{\\bm w_1^H \\bm C_{I\\psi} \\bm w_1}\n                  {\\bm w_1^H  \\bm C_{C\\psi}  \\bm w_1} ,\\\\\n\\eta_{app}      & =  \\frac{1}{A_{ap}} \\; \\; \\frac{\\bm w_1^H \\bm Z \\bm C_{Ie} \\bm Z^H \\bm w_1}\n                  { \\bm w_1^H \\bm Z \\bm C_{Ce} \\bm Z^H \\bm w_1} ,\n&\\eta_{app}      = & \\frac{1}{A_{ap}} \\; \\; \\frac{\\bm w_1^H \\bm C_{I\\psi} \\bm w_1}\n                  { \\bm w_1^H \\bm C_{C\\psi} \\bm w_1}. \n\\end{align}\nHere $\\bm R_{spill}$, $\\bm R_{signal}$ are the open circuit voltage correlations due to spillover\nnoise and that due to radiation from source respectively, $T_{spill}$ is the spillover\ntemperature and $T_A$ is the antenna temperature due to the source, $\\eta_{app}$ is the\naperture efficiency, $\\bm w_1$ is the weight vector applied on the open circuit voltage correlations\n(see Roshi \\& Fisher 2016),  \n\\begin{eqnarray}\n\\bm C_{Ce1} & \\equiv & \\int_{\\Omega_{spill}} \\boldsymbol{\\vec{E^e}}\\cdot \\boldsymbol{\\vec{E^e}}^H \\textrm{d}\\Omega,\n\\label{eqce1} \\\\\n\\bm C_{C\\psi1} & \\equiv & \\int_{\\Omega_{spill}} \\boldsymbol{\\vec{\\Psi}^e}\\cdot \\boldsymbol{\\vec{\\Psi}^e}^H \\textrm{d}\\Omega, \\label{eqcsi1}\\\\\n\\bm C_{Ce} & \\equiv & \\int_{4\\pi} \\boldsymbol{\\vec{E^e}}\\cdot \\boldsymbol{\\vec{E^e}}^H \\textrm{d}\\Omega, \\\\\n\\bm C_{C\\psi} & \\equiv & \\int_{4\\pi} \\boldsymbol{\\vec{\\Psi}^e}\\cdot \\boldsymbol{\\vec{\\Psi}^e}^H \\textrm{d}\\Omega,\\\\\n\\bm C_{Ie} & \\equiv & \\left(\\int_{A_{pap}} \\boldsymbol{\\vec{\\mathcal{E}}^e_{pap}} \\textrm{d} A \\right) \\cdot\n                  \\left(\\int_{A_{pap}} \\boldsymbol{\\vec{\\mathcal{E}}^e_{pap}} \\textrm{d} A \\right)^H,\n\\label{eqie} \\\\\n\\bm C_{I\\psi} & \\equiv & \\left(\\int_{A_{pap}} \\boldsymbol{\\vec{\\mathcal{\\psi}}^e_{pap}} \\textrm{d} A \\right) \\cdot\n                  \\left(\\int_{A_{pap}} \\boldsymbol{\\vec{\\mathcal{\\psi}}^e_{pap}} \\textrm{d} A \\right)^H, \n\\label{eqisi}\n\\end{eqnarray}\n$k_B$ is the Boltzmann constant, $T_g$ is the ground temperature, $z_f$ is the free\nspace impedance, $S_{source}$ is the flux density of the observed source \nand $\\boldsymbol{\\vec{\\mathcal{E}}^e_{pap}}$ and \n$\\boldsymbol{\\vec{\\mathcal{\\psi}}^e_{pap}}$ are the aperture fields (see Roshi \\& Fisher 2016) due to \nthe VEB, $\\boldsymbol{\\vec{\\mathcal{E}}^e}$ and the CEB, $\\boldsymbol{\\vec{\\mathcal{\\psi}}^e}$ respectively. \nIn Eqs.~\\ref{eqce1} \\& \\ref{eqcsi1} the integration is over the\nparts of the beam solid angle, $\\Omega_{spill}$, seeing the ground radiation field and\nin Eqs.~\\ref{eqie} \\& \\ref{eqisi} the integration is over the aperture\nplane, $A_{pap}$, of physical area $A_{ap}$.\nThe model equations\nthat are not affected by the embedded beam pattern definition are\n\\begin{eqnarray}\n\\bm R_{rec} & = & 4 k_B T_0 \\; \\Big(R_n \\boldsymbol{\\mathcal{I}} + \\sqrt{R_n g_n} \\;\\big(\\rho \\bm Z + \\rho^* \\bm Z^H\\big) + g_n \\bm Z\\bm Z^H \\Big), \\\\\nT_n & =& T_{min} + N T_0 \\; \\frac{\\bm w_1^H (\\bm Z - Z_{opt} \\boldsymbol{\\mathcal{I}})\n   (\\bm Z - Z_{opt} \\boldsymbol{\\mathcal{I}})^H \\bm w_1} {\\mbox{Re}\\{Z_{opt}\\} \\; \\frac{1}{2}\\bm w_1^H (\\bm Z + \\bm Z^H) \\bm w_1}, \\\\\n\\bm R_{cmb} & =& 2 k_B T_{cmb} (\\bm Z + \\bm Z^H), \\\\\n\\bm R_{sky} & \\approx& 2 k_B T_{sky} (\\bm Z + \\bm Z^H).\n\\end{eqnarray}\nHere $\\bm R_{rec}$, $\\bm R_{cmb}$, $\\bm R_{sky}$ are the open circuit voltage correlations\ndue to the amplifier noise, the cosmic microwave background and the sky background\nradiation respectively,\n$T_n$ is the receiver temperature of the PAF, $T_0 = 290 K$, $R_n$, $g_n$ and $\\rho$ \nare the noise parameters of the amplifier,\nwhich can equivalently be expressed in terms of the minimum noise temperature $T_{min}$,\nLange invariance $N$ and optimum impedance $Z_{opt}$ (Pospieszalski 2010); \n$T_{cmb}$ is the cosmic microwave\nbackground temperature and $T_{sky} = T_{cmb} + T_{bg,\\nu_0} \\left(\\frac{\\nu}{\\nu_0}\\right)^{-2.7}$\nis the temperature of the sky background at the observed off-source\nposition, $T_{bg,\\nu_0}$ is the galactic background radiation\ntemperature at $\\nu_0$, and $\\nu$ is the frequency at which $\\bm R_{sky}$\nis computed.\n\n\\section{Embedded beam patterns from the CST far-field patterns}\n\\label{A8}\n\nThe CST (\\verb|https:\/\/www.cst.com\/|) microwave studio provides the\nfar-field pattern $\\vec{E'}_j$ when the $j^{th}$ port is excited and all\nother ports are terminated with the CST port impedance (in our case it is 50 $\\Omega$).\nFrom Eqs.~\\ref{Evsvembpat} \\& \\ref{Evsiembpat} we get\n\\begin{equation}\n\\vec{E'}_j = \\sum_{i=1,M} q_{ij} \\; \\vec{E}^e_i,\n\\label{cstembed}\n\\end{equation}\n\\begin{equation}\n\\vec{E'}_j = \\sum_{i=1,M} \\mathcal{J}_{ij} \\; \\vec{\\Psi}^e_i.\n\\label{cstembed1}\n\\end{equation}\nHere $q_{ij} = v_{0_i}$ is the port voltage and $\\mathcal{J}_{ij}$ is the\nport current. These voltages and currents are computed below. \nThe elements of the wave amplitude vector for the excitation are \n\\begin{eqnarray}\na_i & =  & \\sqrt{2\\,P_{stim}} \\quad \\textrm{for} \\; i=j \\nonumber \\\\\n    & =  & 0 \\quad\\quad\\quad\\quad\\;\\; \\textrm{for}\\; i \\neq j\n\\end{eqnarray}\nwhere $P_{stim} = 0.5$ W, is the RMS excitation power in the CST simulation. \nThe wave amplitude vector $\\bm b$ is then\n\\begin{equation}\n\\bm b = a_j \\begin{bmatrix}\n    S_{1j} \\\\\n    S_{2j} \\\\\n    \\threevdots \\\\\n    S_{jj} \\\\\n    \\threevdots \\\\\n    S_{Mj}\n\\end{bmatrix}\n,\n\\end{equation}\nwhere $a_j$ is the $j^{th}$ element of the vector $\\bm a$,\n$S_{ij}, i = 1$ to $M$ is the $j^{th}$ column of $\\bm S$. The\nport voltages and currents are then\n\\begin{eqnarray}\nq_{ij} & = & \\sqrt{z_0} (a_i + b_i) \\nonumber \\\\\n       & = & \\sqrt{z_0} (1 + S_{jj}) a_j \\quad \\textrm{for}\\;\\;\\; i = j \\nonumber \\\\\n       & = & \\sqrt{z_0} S_{ij} a_j \\quad\\quad\\quad \\textrm{for}\\;\\;\\; i \\neq j \\label{qij}\\\\\n\\mathcal{J}_{ij} & = & \\frac{1}{\\sqrt{z_0}} (a_i - b_i) \\nonumber \\\\\n       & = & \\frac{1}{\\sqrt{z_0}} (1 - S_{jj}) a_j \\quad \\textrm{for}\\;\\;\\; i = j \\nonumber \\\\\n       & = & \\frac{-1}{\\sqrt{z_0}} S_{ij} a_j \\quad\\quad\\quad \\textrm{for}\\;\\;\\; i \\neq j\n\\label{jij}\n\\end{eqnarray}\nThe set of far-field patterns provided by the CST along with\nthe port voltages and currents can be used to obtain the VEB,\n$\\boldsymbol{\\vec{E}^e}$ and the CEB, $\\boldsymbol{\\vec{\\Psi}^e}$. \nEq.~\\ref{cstembed} \\& \\ref{cstembed1} for the set of far-field patterns\ncan be concisely written as\n\\begin{eqnarray}\n\\boldsymbol{\\vec{E}^{'}} & = & \\bm Q \\; \\boldsymbol{\\vec{E}^e}, \\\\\n\\boldsymbol{\\vec{E}^{'}} & = & \\bm J \\; \\boldsymbol{\\vec{\\Psi}^e}.\n\\end{eqnarray}\nwhere the elements of the matrix $\\bm Q$ are $q_{ij}$ and that of the \nmatrix $\\bm J$ are $\\mathcal{J}_{ij}$.\nThis equation is valid for each $\\theta, \\phi$.\nUsing Eqs.~\\ref{qij} \\& \\ref{jij} $\\bm Q$ and $\\bm J$ can be written as\n\\begin{eqnarray}\n\\bm Q & = & \\sqrt{2\\, z_0\\, P_{stim}} \\;\\;(\\boldsymbol{\\mathcal{I}} + \\bm S),\\\\\n\\bm J & = & \\sqrt{\\frac{2\\, P_{stim}}{z_0}} \\;\\;(\\boldsymbol{\\mathcal{I}} - \\bm S).\n\\end{eqnarray}\nThe matrices $\\bm Q$ and $\\bm J$ are also related through the equation \n\\begin{equation}\n\\bm J = \\bm Q \\bm Z^{-1}.\n\\end{equation}\nThe embedded beam patterns are then obtained as\n\\begin{eqnarray}\n\\boldsymbol{\\vec{E}^e} & = & \\bm Q^{-1} \\; \\boldsymbol{\\vec{E}^{'}}, \\\\\n\\boldsymbol{\\vec{\\Psi}^e} & = & \\bm J^{-1} \\; \\boldsymbol{\\vec{E}^{'}}.\n\\end{eqnarray}\n\n\\section{Some sanity checks}\n\n\\subsection{Energy conservation}\nWe verify here whether the computed embedded beam patterns \nsatisfy energy conservation.\nDetails of such a verification for the VEB $\\boldsymbol{\\vec{E}^e}$ are given in\nRoshi \\& Fisher (2016).\nWe consider below the case for CEB, $\\boldsymbol{\\vec{\\mathcal{\\psi}}^e}$. \nFrom the definition of embedded beam pattern $\\vec{\\mathcal{\\psi}}^e_j$  \nthe port currents are\n\\begin{eqnarray}\n\\mathcal{J}_{0_i} & = & 1\\; \\textrm{A} \\quad \\textrm{for}\\;\\;\\; i = j, \\nonumber \\\\\n    & = & 0\\; \\textrm{A} \\quad \\textrm{for}\\;\\;\\; i \\neq j,\n\\end{eqnarray}\nand hence the wave amplitudes are \n\\begin{eqnarray}\n\\frac{1}{\\sqrt{z_0}} (a_i - b_i) & = & 1 \\quad \\textrm{for}\\;\\;\\; i = j, \\nonumber \\\\\n                                 & = & 0 \\quad \\textrm{for}\\;\\;\\; i \\neq j. \n\\end{eqnarray}\nThe vector $\\bm a$ can be written as\n\\begin{equation}\n\\bm a = \\bm b + \\sqrt{z0} \\begin{bmatrix}\n    0 \\\\\n    0 \\\\\n    \\threevdots \\\\\n    1 \\\\\n    \\threevdots \\\\\n    0\n\\end{bmatrix},\n\\end{equation}\nwhere the non-zero element (which is 1) is located at $j^{th}$ row. Substituting\nthis in the equation $\\bm b = \\bm S \\bm a $ \nand re-arranging we get\n\\begin{equation}\n\\bm b = \\sqrt{z0}\\;(\\boldsymbol{\\mathcal{I}} - \\bm S)^{-1} \\begin{bmatrix}\n    S_{1j} \\\\\n    S_{2j} \\\\\n    \\threevdots \\\\\n    S_{jj} \\\\\n    \\threevdots \\\\\n    S_{Mj}\n\\end{bmatrix}.\n\\end{equation}\nPower dissipated at the $j^{th}$ port is\n\\begin{eqnarray}\nP_{dis} & = & \\frac{1}{2} (a_j a_j^* - b_j b_j^*), \\\\\n        & = & \\frac{\\sqrt{z_0}}{2} (\\sqrt{z_0} + (b_j + b_j^*)).\n\\label{pdis1}\n\\end{eqnarray}\nThe far-field beam pattern of the PAF for the above excitation is the\nembedded beam pattern $\\vec{\\mathcal{\\psi}}^e_j$ and hence\nthe radiated power is,  \n\\begin{eqnarray}\nP_{rad} & = & \\frac{1}{2 z_f}\\int_{sphere} \\vec{\\mathcal{\\psi}}^e_j \\cdot \\vec{\\mathcal{\\psi}}^{e*}_j \\; \\textrm{d}A, \\nonumber \\\\\n        &  = & \\frac{1}{2 z_f}\\int_{4\\pi} \\vec{\\Psi}^e_j  \\cdot \\vec{\\Psi}^{e*}_j \\; \\textrm{d}\\Omega.\n\\end{eqnarray}\nFor loss-less PAF $P_{dis} = P_{rad}$. This equality is satisfied in our\nPAF model computation. Further, for a loss-less antenna,\n\\begin{equation}\nP_{rad} = \\frac{1}{2} \\mathcal{J}_{0_j}^2 \\; \\textrm{Re}\\{Z_{pin_j}\\},\n\\end{equation}\nwhere $\\mathcal{J}_{0_j}$ is the current flowing to port $j$, which for the embedded pattern\n$\\vec{\\mathcal{\\psi}}^e_j$ is 1 A and $Z_{pin_j}$ is the input\nimpedance of port $j$ when all other ports are open circuited. The input\nimpedance for this case is given by\n\\begin{equation}\nZ_{pin_j} = z_{jj},\n\\label{embZin}\n\\end{equation}\nwhere $z_{jj}$ is the $j^{th}$ diagonal element of the impedance matrix ${\\bm Z}$.\nThus\n\\begin{equation}\nP_{rad} = \\frac{1}{2} \\textrm{Re}\\{z_{jj}\\}.\n\\label{embpdis}\n\\end{equation}\n\n\\subsection{PAF in a thermal radiation field}\n\nIn this Section, we show that the open circuit voltage correlations $\\bm R_t$,\nobtained from the two embedded beam patterns, when the the PAF is\nembedded in a black body radiation field are equal to the result given by\nTwiss's theorem (Twiss 1955). \nThe correlation $\\bm R_t$ is given by (Roshi \\& Fisher 2016) \n\\begin{eqnarray}\n\\bm R_t & = &\\frac{4 k_B T_0}{z_f} \\bm Z \\left(\n\\int_{4\\pi} \\boldsymbol{\\vec{E}^e}\\cdot \\boldsymbol{\\vec{E}^e}^H \\textrm{d}\\Omega \\right) \\bm Z^H, \\nonumber \\\\\n        & = & \\frac{4 k_B T_0}{z_f} \\bm Z \\bm C_{Ce} \\bm Z^H,\n\\label{thcorr1}\n\\end{eqnarray}\nfor the VEB, $\\boldsymbol{\\vec{\\mathcal{E}}^e}$ and \n\\begin{equation}\n\\bm R_t  =  \\frac{4 k_B T_0}{z_f} \\bm C_{C\\psi},\n\\label{thcorr2}\n\\end{equation}\nfor the CEB, $\\boldsymbol{\\vec{\\mathcal{\\psi}}^e}$.\nFor a loss-less antenna the power dissipated at the ports should be equal to the radiated power,\nwhich can be used to calculate $\\bm C_{Ce}$ and $\\bm C_{C\\psi}$. The \nenergy balance condition gives, \n\\begin{eqnarray}\n\\frac{1}{2} \\left(\\frac{\\bm V_0^H \\bm I_0}{2} + \\frac{\\bm I_0^H \\bm V_0}{2}\\right)  & = & \\frac{1}{2 z_f} \\bm V_0^H \\bm C_{Ce} \\bm V_0, \\nonumber \\\\\n\\frac{1}{4} \\bm V_0^H \\left(\\bm Z^{-1} + \\left(\\bm Z^{-1}\\right)^H \\right) \\bm V_0 & = &\n\\frac{1}{2 z_f} \\bm V_0^H \\bm C_{Ce} \\bm V_0,\n\\label{ce}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\frac{1}{2} \\left(\\frac{\\bm V_0^H \\bm I_0}{2} + \\frac{\\bm I_0^H \\bm V_0}{2}\\right)  & = & \\frac{1}{2 z_f} \\bm I_0^H \\bm C_{C\\psi} \\bm I_0, \\nonumber \\\\\n\\frac{1}{4} \\bm I_0^H \\left(\\bm Z + \\bm Z^H \\right) \\bm I_0 & = &\n\\frac{1}{2 z_f} \\bm I_0^H \\bm C_{C\\psi} \\bm I_0.\n\\label{csi}\n\\end{eqnarray}\nSince Eqs.~\\ref{ce} \\& \\ref{csi} are valid for arbitrary excitations it follows\nthat\n\\begin{eqnarray} \n\\frac{1}{2} \\left(\\bm Z^{-1} + \\left(\\bm Z^{-1}\\right)^H \\right) & = &\n\\frac{1}{z_f} \\bm C_{Ce}, \\label{impbc1} \\\\\n\\frac{1}{2} \\left(\\bm Z + \\bm Z^H \\right) & = &\n\\frac{1}{z_f} \\bm C_{C\\psi}. \n\\label{impbc2}\n\\end{eqnarray}\nSubstituting Eq.~\\ref{impbc1} in Eq.~\\ref{thcorr1} and Eq.~\\ref{impbc2} in \nEq.~\\ref{thcorr2}, we get \n\\begin{equation}\n\\bm R_t = 2 k_B T_0 \\Big(\\bm Z + \\bm Z^H\\Big),\n\\end{equation}\nfrom both Eqs.~\\ref{thcorr1} \\& \\ref{thcorr2},\nwhich is the voltage correlation given by Twiss's theorem (Twiss 1955). \n\n\\section*{Acknowledgment}\n\nI thank Rick Fisher and Bill Shillue for carefully proof reading the report and providing useful\ncomments.\n\n\\section*{References}\n\n\\noindent \nPospieszalski, M. W., 2010, IEEE Microwave Magazine, 11, 61 \n\n\\noindent\nRoshi, D. A., Fisher, J. R., 2016, NRAO, Electronics division internal report, 330. \\\\\n\\url{https:\/\/library.nrao.edu\/public\/memos\/edir\/EDIR_330.pdf}\n\n\\noindent\nTwiss, R. Q., J. Appl. Phys., 1955, 26(5) 599.\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{Definition}\n\n\\begin{definition}[Primitive root of unity]\n\\label{defi:racprim}\n  Let $q$ be a prime power. A matrix\n  $A \\in M_\\ell(\\mathbb{F}_{q^s})$ is called a \\emph{primitive $m$-th root of\n  unity} if\n  \\begin{itemize}\n    \\item $A^m = I_\\ell$,\n    \\item $A^i \\neq I_\\ell$ if $i < m$,\n    \\item $\\det(A^i - A^j) \\neq 0$, whenever $i \\neq j$.\n  \\end{itemize}\n\\end{definition}\n\n\\begin{proposition}\n\\label{prop:racprim}\n  Let $q$ be a prime power and suppose that $q^{s\\ell} - 1 = m$.\n  Then there exists a primitive $m$-th root of unity in\n  $M_{\\ell}(\\mathbb{F}_{q^s})$.\n\\end{proposition}\n\n\\begin{proof}\n  Let $\\alpha \\in \\mathbb{F}_{q^{s\\ell}}$ be a primitive $m$-th root of unity\n  and $A \\in M_{\\ell}(\\mathbb{F}_{q^s})$ be the companion matrix of the\n  irreducible polynomial $f(X) \\in \\mathbb{F}_{q^s}[X]$ of $\\alpha$ over\n  $\\mathbb{F}_{q^s}$. There exists $P \\in \\GL_{\\ell}(\\mathbb{F}_{q^{s\\ell}})$ and an\n  upper triangular matrix $U \\in M_{\\ell}(\\mathbb{F}_{q^{s\\ell}})$ whose\n  diagonal coefficients are the eigenvalues of $A$ such that\n  $A = P^{-1} U P$. The eigenvalues of $A$ are exactly the\n  roots of $f$ and then are primitive $m$-th roots of unity. Therefore\n  $A$ satisfies the three conditions of Definition~\\ref{defi:racprim}.\n\\end{proof}\n\n\\begin{definition}[Block minimum distance]\n  Let $\\mathcal{C}$ be a linear code over $\\mathbb{F}_q$ of length $m\\ell$.\n  We define the \\emph{$\\ell$-block minimum distance} of $\\mathcal{C}$ to be\n  the minimum distance of the folded code of $\\mathcal{C}$.\n\\end{definition}\n\n\\begin{definition}[Left quasi-BCH codes]\n\\label{defi:QBCH}\n  Let $A$ be a primitive $m$-th root of unity in\n  $M_{\\ell}(\\mathbb{F}_{q^s})$ and $\\delta \\leq m$.\n  We define the $\\ell$-quasi-BCH code of length $m\\ell$,\n  with respect to $A$, with designed minimum distance $\\delta$,\n  over $\\mathbb{F}_q$ by\n  \\begin{multline*}\n    \\qbch_q(m,\\ell,\\delta,A) := \\\\\n    \\left\\lbrace\n      (c_1,\\ldots,c_m) \\in (\\mathbb{F}_q^\\ell)^m :\n        \\sum_{j = 0}^{m - 1} A^{ij}c_j = 0\n        \\text{ for } i = 1,\\ldots,\\delta - 1\n    \\right\\rbrace.\n  \\end{multline*}\n  We call the linear map\n  \\begin{equation*}\n    \\begin{array}{rcl}\n      \\mathcal{S}_A : (\\mathbb{F}_q^{\\ell})^m & \\rightarrow & (\\mathbb{F}_{q^s}^{\\ell})^m \\\\\n      x = (x_1,\\ldots,x_m)     & \\mapsto     & \\sum_{j = 0}^{m - 1}\n                                                     A^j x_j\n    \\end{array}\n  \\end{equation*}\n  the \\emph{syndrome} map with respect to $\\qbch(m,\\ell,\\delta,A)$.\n\\end{definition}\n\n\\begin{proposition}\n  Using the notation of Definition~\\ref{defi:QBCH},\n  $\\qbch_q(m,\\ell,\\delta,A)$ has dimension at least\n  $(m - e(\\delta - 1))\\ell$ and $\\ell$-block minimum distance at least\n  $\\delta$. In other words $\\qbch_q(m,\\ell,\\delta,A)$ is an\n  $[m\\ell , \\geq (m - s(\\delta - 1))\\ell , \\geq \\delta]_{\\mathbb{F}_q}$-code.\n\\end{proposition}\n\n\\begin{proof}\n  According to Definition~\\ref{defi:QBCH} we have that\n  \\begin{equation*}\n    H = \\begin{pmatrix}\n      I_{\\ell} & A              & \\cdots & A^{m - 1} \\\\\n      I_{\\ell} & A^2            & \\cdots & A^{2(m - 1)} \\\\\n      \\vdots   & \\vdots         &        & \\vdots \\\\\n      I_{\\ell} & A^{\\delta - 1} & \\cdots & A^{(\\delta - 1)(m - 1)}\n    \\end{pmatrix} \\in M_{(\\delta - 1)\\ell,m\\ell}(\\mathbb{F}_{q^s})\n  \\end{equation*}\n  is a parity check matrix of $\\qbch_q(m,\\ell,\\delta,A)$.\n  Let\n  \\begin{equation*}\n    V = \\begin{pmatrix}\n      I_\\ell & A              & \\cdots & A^{\\delta - 1} \\\\\n      I_\\ell & A^2            & \\cdots & A^{2(m - 1)} \\\\\n      \\vdots & \\vdots         &        & \\vdots \\\\\n      I_\\ell & A^{\\delta - 1} & \\cdots & A^{(\\delta - 1)^2}\n    \\end{pmatrix}.\n  \\end{equation*}\n  Using the Vandermonde matrix trick we find that the determinant $D$\n  of $V$ over $M_{\\ell}(\\mathbb{F}_{q^s})[A]$ is $\\prod_{i < j} (A^i - A^j)$.\n  By the definition of $A$ we have $\\det_{\\mathbb{F}_{q^s}} D \\neq 0$, thus\n  $V$ is invertible over $M_{\\ell}(\\mathbb{F}_{q^s})[A]$ and then, invertible\n  over $\\mathbb{F}_{q^s}$. Therefore $H$ has full rank over $\\mathbb{F}_{q^s}$.\n  \n  Let $i:\\mathbb{F}_q^{m\\ell} \\rightarrow \\mathbb{F}_{q^s}^{m\\ell}$ be the canonical\n  injection and denote by\n  $h:\\mathbb{F}_{q^s}^{m\\ell} \\rightarrow \\mathbb{F}_{q^s}^{(\\delta - 1)\\ell}$\n  the $\\mathbb{F}_q$-linear map given by $H$. Then we have\n  $\\dim_{\\mathbb{F}_q}(\\im h) = e(\\delta - 1)\\ell$. Thus\n  $\\dim_{\\mathbb{F}_{q^s}}(\\im h \\circ i) \\leq (\\delta - 1)\\ell$ and\n  $\\dim_{\\mathbb{F}_q}(\\im h \\circ i) \\leq e(\\delta - 1)\\ell$. Therefore\n  $\\dim_{\\mathbb{F}_q}(\\ker h \\circ i) \\geq m\\ell - e(\\delta - 1)\\ell$.\n  Suppose that there exists a codeword\n  \\mbox{$c = (c_1,\\ldots,c_m) \\in \\mathcal{C} \\setminus \\{0\\}$}\n  with $\\ell$-block weight $b \\leq \\delta - 1$. Note $i_1,\\ldots,i_b$\n  the indexes such that $c_{i_j}\\neq 0$ for $i=1,\\ldots,b$.\n  This implies that the matrix\n  \\begin{equation*}\n    \\begin{pmatrix}\n      A^{i_1}  & A^{i_2}  & \\cdots & A^{i_b} \\\\\n      A^{2i_1} & A^{2i_2} & \\cdots & A^{2i_b} \\\\\n      \\vdots   & \\vdots   &        & \\vdots \\\\\n      A^{(\\delta - 1) i_1} &\n        A^{(\\delta - 1) i_2} &\n          \\cdots &\n            A^{(\\delta - 1) i_b} \\\\\n    \\end{pmatrix}\n  \\end{equation*}\n  has not full rank which is absurd.   \n\\end{proof}\n\n\\begin{example}\n  Consider the $3$-quasi-BCH codes defined by primitive roots in\n  $M_3(\\mathbb{F}_{2^2})$ of length $63$ over $\\mathbb{F}_2$ with designed\n  minimum distance $6$ defined by a $21$-th root of unity in\n  $\\mathbb{F}_{2^2}$. In other words, $q = 2,m = 21,\\ell = 3,s = 2$ and\n  $\\delta = 6$. There are $22$ non-equivalent codes splitting as\n  follows:\n  \\begin{equation*}\n    \\begin{array}{|c|c|}\n      \\hline\n        \\text{Number of codes} & \\text{Parameters} \\\\\n      \\hline\n        2  & [63,33,6]_{\\mathbb{F}_2} \\\\\n      \\hline\n        18 & [63,33,7]_{\\mathbb{F}_2} \\\\\n      \\hline\n        2  & [63,36,6]_{\\mathbb{F}_2} \\\\\n      \\hline\n    \\end{array}\n  \\end{equation*}\n  Notice that their dimension is always at least\n  $(m - e(\\delta - 1))\\ell = 33$ and their minimum distance is\n  at least $\\delta = 6$. All the computations have been performed\n  with the \\textsc{magma} computer algebra system \\cite{magma}.\n\\end{example}\n\n\\begin{example}\n  Let $q = 5,m = 7,\\ell = 3,s = 2$ and $\\delta = 3$. Let\n  $\\omega \\in \\mathbb{F}_{5^2}$ be a primitive $(5^2 - 1)$-th root of unity and\n  \\begin{equation*}\n    A=\\begin{pmatrix}\n      \\omega^9    & \\omega^4    & \\omega^{22} \\\\\n      \\omega^{11} & \\omega^{11} & \\omega^{15} \\\\\n      \\omega^2    & \\omega^{19} & 1 \n    \\end{pmatrix}\n    \\in M_3(\\mathbb{F}_{5^2}).\n  \\end{equation*}\n  Then the left $3$-quasi-BCH code of length $21$ with respect\n  to $A$ with designed minimum distance $3$ over $\\mathbb{F}_5$ has parameters\n  $[21,9,7]_{\\mathbb{F}_5}$. Its generator polynomial is given by\n  \\begin{multline*}\n    g(X) = \n    \\begin{pmatrix} 1 & 4 & 3 \\\\ 3 & 3 & 4 \\\\ 1 & 1 & 4 \\end{pmatrix} X^4 +\n    \\begin{pmatrix} 4 & 0 & 0 \\\\ 4 & 0 & 0 \\\\ 4 & 0 & 4 \\end{pmatrix} X^3 +\n    \\begin{pmatrix} 3 & 0 & 4 \\\\ 0 & 3 & 4 \\\\ 0 & 0 & 0 \\end{pmatrix} X^2 +\\\\\n    \\begin{pmatrix} 2 & 3 & 2 \\\\ 4 & 4 & 4 \\\\ 3 & 1 & 1 \\end{pmatrix} X +\n    \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix}\n  \\in M_3(\\mathbb{F}_5)[X].\n  \\end{multline*}\n\\end{example}\n\n\n\\subsection{The one-to-one correspondence}\n\nIt is well-known \\cite[Theorem~1, page~190]{SloMacWil86} that there\nis a one-to-one correspondence between cyclic codes of length $n$\nover $\\mathbb{F}_q$ and monic factors of $X^n - 1 \\in \\mathbb{F}_q[X]$ \\emph{i.e.}\nideals of $\\mathbb{F}_q[X] \/ (X^n - 1)$.\nIn \\cite{CayChaAbd2010,Chabot2011} the authors start to exhibit\nsuch a correspondence for quasi-cyclic codes. They show that there\nis a correspondence between a subfamily of $\\ell$-quasi-cyclic codes\nof length $m\\ell$ over $\\mathbb{F}_q$ and reversible factors of\n$X^n - 1 \\in M_{\\ell}(\\mathbb{F}_q)[X]$.\n\nThe one-to-one correspondence between $\\ell$-quasi cyclic codes\nand left ideals of $M_{\\ell}(\\mathbb{F}_q)[X] \/ (X^m - 1)$ is a consequence\nof the two following lemmas.\n\n\\begin{lemma}\n\\label{lem:mod_gen}\n  Let $R$ be a commutative principal ring and $M$ be a free left\n  module of finite rank $s$ over $R$. Then every submodule $N$ of $M$\n  can be generated by at most $s$ elements.\n\\end{lemma}\n\n\\begin{proof}\n  It is an easy adaptation of the proof of\n  \\cite[Theorem~7.1, page~146]{Lang2002}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:morita}\n  Let $s$ be a positive integer and $R$ be a commutative principal\n  ring. Then there is a one-to-one correspondence between the\n  submodules of $R^s$ and the left ideals of $M_s(R)$.\n\\end{lemma}\n\n\\begin{proof}\n  Note that this is a particular case of the Morita equivalence\n  for modules. See for example\n  \\cite[n\\textsuperscript{o}4, page~99]{BourbakiAlgCom8}. This\n  particular case can be proved directly.\n  To a submodule $N \\subseteq R^s$, we can build a left ideal\n  of $M_s(R)$ whose elements have rows in $N$. Conversely,\n  to a left ideal $I \\subseteq M_s(R)$ we associate the\n  submodule of $R^s$ generated by all the rows of all the elements\n  of $I$. It is straightforward to check that these maps are\n  inverse to each other.\n\\end{proof}\n\nNote that $M_{\\ell}(\\mathbb{F}_q)[X] \/ (X^m - 1)$ and\n$M_{\\ell}(\\mathbb{F}_q[X] \/ (X^m - 1))$ are isomorphic as rings and that\n$R = \\mathbb{F}_q[X] \/ (X^m - 1)$ is a commutative principal ring. By\nLemma~\\ref{lem:mod_gen} any submodule of $R^{\\ell}$ can be generated\nby at most $\\ell$ elements. Therefore by Lemma~\\ref{lem:morita}\nany left ideal of $M_{\\ell}(R) = M_{\\ell}(\\mathbb{F}_q)[X] \/ (X^m - 1)$ is\nprincipal.\n\n\\begin{theorem}\n\\label{thm:one-to-one}\n  There is a one-to-one correspondence between $\\ell$-quasi-cyclic\n  codes over $\\mathbb{F}_q$ of length $m\\ell$\n  and left ideals of $M_{\\ell}(\\mathbb{F}_q)[X] \/ (X^m - 1)$.\n\\end{theorem}\n\n\\begin{proof}\n  Let $g = (g_{11},\\ldots,g_{1\\ell},g_{21},\\ldots,g_{2\\ell},\\ldots,\n       g_{m1},\\ldots,g_{m\\ell}) \\in \\mathbb{F}_q^{m\\ell}$. We associate\n  to $g$ the element $\\varphi(g) \\in (\\mathbb{F}_q[X] \/ (X^m - 1))^{\\ell}$\n  defined by\n  \\begin{multline*}\n    \\varphi(g) =\n    \\left(\n      g_{11} + g_{21} X + \\cdots + g_{m1} X^{m - 1} ;\n    \\right. \\\\\n    g_{12} + g_{22} X + \\cdots + g_{m2} X^{m - 1} ; \\ldots ; \\\\\n    \\left.\n      g_{1\\ell} + g_{2\\ell} X + \\cdots + g_{m\\ell} X^{m - 1}\n    \\right).\n  \\end{multline*}\n  Then $\\varphi$ induces a one-to-one correspondence between\n  $\\ell$-quasi-cyclic codes of length $m\\ell$ over $\\mathbb{F}_q$\n  and submodules of $(\\mathbb{F}_q[X] \/ (X^m - 1))^{\\ell}$.\n  The theorem follows by Lemma~\\ref{lem:morita}.\n\\end{proof}\n\nLet $\\pr_{i,j}$ be the projection of the $i,i + 1,\\ldots,j$ coordinates:\n\\begin{equation*}\n  \\begin{array}{rcl}\n    \\pr_{i,j} : \\mathbb{F}_q^n & \\longrightarrow & \\mathbb{F}_q^{j - i + 1} \\\\\n    (x_1,\\ldots,x_n)   & \\longmapsto     &\n                      (x_i,x_{i + 1},\\ldots,x_{j - 1},x_j).\n  \\end{array}\n\\end{equation*}\nWe have the following obvious lemma:\n\n\\begin{lemma}\n\\label{LemmaBlockRank}\n  Let $\\mathcal{C}$ be an $\\ell$-quasi-cyclic code over $\\mathbb{F}_q$ of dimension $k$\n  and length $m\\ell$.\n  Then there exists an integer $r$ such that $1 \\leq r \\leq k$ and\n  for any generator matrix $G$ of $\\mathcal{C}$ and $0 \\leq i \\leq m - 1$,\n  the rank of the $i\\ell + 1,i\\ell + 2,\\ldots,(i + 1)\\ell$ columns of\n  $G$ is $r$.\n\\end{lemma}\n\n\n\\begin{definition}[Block rank]\n  Taking the notation of Lemma~\\ref{LemmaBlockRank},\n  we call the integer $r$ the \\emph{block rank} of $\\mathcal{C}$. Note that\n  $r$ depends only on $\\mathcal{C}$ and not on any particular generator matrix\n  of $\\mathcal{C}$.\n\\end{definition}\n\n\\subsection{The generator polynomial of an $\\ell$-quasi-cyclic code}\n\nIn this subsection we fix an $\\ell$-quasi-cyclic code $\\mathcal{C}$ over $\\mathbb{F}_q$.\nIf $\\ell = 1$, then $\\mathcal{C}$ is a cyclic code of length $n$ and a\ngenerator matrix of $\\mathcal{C}$ can be given\n\\cite[Theorem~1,~(e), page~191]{SloMacWil86} by\n\\begin{equation}\n\\label{equ:cyclic-gen}\n  \\begin{pmatrix}\n    g(X)  &       &       & \\\\\n          & Xg(X) &       & \\\\\n          &       & \\dots & \\\\\n          &       &       & X^{n - \\deg g}g(X)\n  \\end{pmatrix},\n\\end{equation}\nwhere $g(X) \\in \\mathbb{F}_q[X]$ is the generator polynomial of $\\mathcal{C}$.\nThe block rank of $\\mathcal{C}$ is $1$ and we see that we can write\na generator matrix of $\\mathcal{C}$ with only $1$ vector and its shifts\n(by $T^{\\ell} = T$). The natural generalization of this result\nfor quasi-cyclic codes is done using the block rank.\n\nLet $r$ be the block rank of $\\mathcal{C}$, the following algorithm computes\na basis of $\\mathcal{C}$ from $r$ vectors of $\\mathcal{C}$ and their shifts.\nWe call the \\emph{first index} of a nonzero vector\n$x = (x_1,\\ldots,x_{m\\ell})$ the least integer $0 \\leq i \\leq m - 1$\nsuch that \\mbox{$(x_{i\\ell + 1},\\ldots,x_{(i + 1)\\ell}) \\neq 0$} and denote\nit by $\\first(x) = \\first(x_1,\\ldots,x_{m\\ell})$.\nLet\n\\begin{equation*}\n  \\begin{array}{rlc}\n  p:\\mathbb{F}_q^{m\\ell} &\\longrightarrow & \\mathbb{F}_q^{\\ell}\\\\\n  x = (x_1,\\ldots,x_{m\\ell}) & \\longmapsto &\n        (x_{i\\ell + 1},\\ldots,x_{(i + 1)\\ell}),\n  \\end{array}\n\\end{equation*}\nwhere $i = \\first(x_1,\\ldots,x_n)$ if $x \\neq 0$ and $p(0) = 0$.\n\n\\begin{algorithm}\n\\label{al:basis-block-rank}\n\\caption{Basis computation with the block rank}\n\\begin{algorithmic}[1]\n\\REQUIRE A generator matrix $G$ of $\\mathcal{C}$.\n\\ENSURE A generator matrix formed by $r$ rows from $G$\n        and some of their shifts.\n\\STATE $G' \\gets $ a row echelon form of $G$.\n\\STATE Denote by $g_1,\\ldots,g_k$ the rows of $G'$.\n\\STATE $M \\gets \\max \\{ \\first(g_i) : i \\in \\{ 0,\\ldots,m - 1 \\} \\}$.\n\\STATE $B_M' \\gets \\emptyset$.\n\\STATE $G_{M + 1} \\gets \\emptyset$.\n\\FOR{$j = M \\to 0$}\n  \\STATE $B_j \\gets$ $\\{ g_i : i \\in \\{ 1,\\ldots,k \\}\n         \\text{ and } \\first(g_i) = j \\}$.\n  \\FOR{each element $x$ of $B_j$}\n    \\IF{$p(B_j') \\cup \\{ p(x) \\}$ are independent}\n      \\STATE $B_j' \\gets B_j' \\cup \\{ x \\}$.\n    \\ENDIF\n  \\ENDFOR\n  \\STATE $G_j \\gets G_{j + 1} \\cup B_j'$.\n  \\STATE $B_{j - 1}' \\gets T^{\\ell}(B_j')$.\n\\ENDFOR\n\\RETURN $G_0$.\n\\end{algorithmic}\n\\end{algorithm}\n\nNote that Algorithm~\\ref{al:basis-block-rank} applied to a cyclic\ncode, \\emph{i.e.} $\\ell = 1$, returns exactly the\nmatrix~\\eqref{equ:cyclic-gen} and we can deduce the generator\npolynomial of $\\mathcal{C}$ at the cost of the computation of a row echelon\nform of any generator matrix of $\\mathcal{C}$.\n\n\\begin{proposition}  \n  Algorithm~\\ref{al:basis-block-rank} works correctly as expected and\n  returns a generator matrix $G$ of $\\mathcal{C}$ made of $r$ linearly\n  independent vectors of $\\mathcal{C}$ and some of their shifts.\n\\end{proposition}\n\n\\begin{proof}\n  We will prove by descending induction on $j$ that:\n  \\begin{enumerate}\n    \\item $B_j' \\supseteq T^{\\ell} (B_{j + 1}')\n                \\supseteq \\dots\n                \\supseteq T^{(M - j)\\ell} (B_M')$.\n    \\item $\\#B_j' \\leq r$.\n    \\item The vectors of $B_j'$ are linearly independent.\n    \\item The vectors of $G_j$ are linearly independent.\n    \\item $\\gen{G_j} = \\gen{g_i : i \\in \\{ 1,\\ldots,k \\}\n                                  \\text{ and } \\first(g_i) \\geq j }$.\n  \\end{enumerate}\n  Let $j = M$. By step~3, we have $B_M \\neq \\emptyset$. Item~1\n  is trivially satisfied. By Lemma~\\ref{LemmaBlockRank},\n  $\\#B_M \\leq r$ and item~2 is satisfied. As\n  $G_{M + 1} = B_M' = \\emptyset$ then\n  $G_M = B_M' = B_M = \\{ g_i : i \\in \\{ 1,\\ldots,k \\}\n                               \\text{ and } \\first(g_i) \\geq M \\}$ and\n  items~3 to~5 are satisfied.\n\n  Suppose that $j < M$ and that items~1 to~5 are satisfied for\n  $i = j + 1,\\ldots,M$. First note that $B_j \\neq \\emptyset$.\n  If we had $B_j = \\emptyset$ then, as $G'$ is in row echelon form,\n  $g_1,\\ldots,g_k,T^{(M - j)\\ell}(g_k)$ would be linearly independent\n  which is a contradiction.\n\n  Items~1 and~3 are satisfied by steps~7, 9 and~10 of the algorithm. By\n  Lemma~\\ref{LemmaBlockRank} and step~9, item~2 is satisfied. For all\n  $x \\in G_{j + 1}$, we have $\\first(x) \\geq j + 1$, thus, by\n  item~3, the elements of $G_j$ are linearly independent and item~4\n  is satisfied. Let $g$ be a vector of $G'$ such that $\\first(g) = j$, then\n  the construction of $B_j'$ implies that we have\n  \\begin{equation*}\n    \\first \\left( g - \\sum_{u \\in B_j'} \\mu_u u \\right) \\geq j + 1\n  \\end{equation*}\n  where $\\mu_u \\in \\mathbb{F}_q$ for $u \\in B_j'$. Then by item~5 of the\n  inductive hypothesis, we have\n  \\begin{equation*}\n    \\left( g - \\sum \\mu_u u \\right) \\in G_{j + 1}.\n  \\end{equation*}\n  Thus we have\n  $\\gen{G_j} = \\gen{ g_i : i \\in \\{1,\\ldots,k\\}\n                           \\text{ and } \\first(g_i) \\geq j }$ and item~5\n  is satisfied.\n\n  As a consequence of the previous induction, $G_0$ is constituted of\n  linearly independent vectors and generates\n  $\\gen{ g_i : i \\in \\{1,\\ldots,k\\} \\text{ and } \\first(g_i) \\geq 0 }\n   = \\mathcal{C}$ by item~5. By Lemma~\\ref{LemmaBlockRank} we must have\n  exactly $r$ vectors $g \\in G_0$ such that $\\first(g) = 0$. Thus\n  by items~1 and~2 we have\n  \\begin{equation*}\n    r = \\#B_0' = \\sum_{\\lambda = 0}^M\n                 \\#\\left(\n                   B_{\\lambda}' \\setminus T^{\\ell} (B_{\\lambda + 1}')\n                 \\right)\n  \\end{equation*}\n  which shows that $G_0$ is constituted of $r$ linearly independent\n  vectors of $\\mathcal{C}$ and some of their shifts.\n\\end{proof}\n\n\\begin{corollary}\n\\label{cor:gen}\n  There exist $g_1,\\ldots,g_r$ linearly independent vectors of $\\mathcal{C}$\n  such that $\\gggg$ span $\\mathcal{C}$.\n  If we denote by $g_{i,j}$ the $j$'th coordinate of $g_i$ and let\n  \\begin{equation*}\n    G_i =\n    \\begin{pmatrix}\n      g_{1,i\\ell + 1} & \\dots & g_{1,(i + 1)\\ell} \\\\\n      \\vdots          &       & \\vdots \\\\\n      g_{r,i\\ell + 1} & \\dots & g_{r,(i + 1)\\ell} \\\\\n                      & 0     &\n    \\end{pmatrix}\n    \\in M_{\\ell}(\\mathbb{F}_q)\n  \\end{equation*}\n  and\n  \\begin{equation*}\n    g(X) = \\frac{1}{X^{\\nu}} \\sum_{i = 0}^{m - 1} G_i X^i\n           \\in M_{\\ell}(\\mathbb{F}_q)[X],\n  \\end{equation*}\n  where $\\nu$ is the least integer such that $G_i \\neq 0$,\n  then $\\mathcal{C}$ corresponds to the left ideal $\\gen{g(X)}$\n  by Theorem~\\ref{thm:one-to-one}.\n\\end{corollary}\n\n\\begin{corollary}\n  Taking the notation of the proof of Theorem~\\ref{thm:one-to-one},\n  the submodule $\\varphi(\\mathcal{C}) \\subseteq (\\mathbb{F}_q[X] \/ (X^m - 1))^{\\ell}$\n  is generated by $r$ elements as an $\\mathbb{F}_q[X] \/ (X^m - 1)$-module\n  but cannot be generated by less that $r$ elements. If $\\mathcal{C}$ is a\n  cyclic code then we have $r = 1$ and we find the classical result\n  about cyclic codes.\n\\end{corollary}\n\n\\begin{definition}[Generator polynomial]\n  The polynomial $g(X) \\in M_{\\ell}(\\mathbb{F}_q)[X]$ from\n  Corollary~\\ref{cor:gen} is called a \\emph{generator polynomial}\n  of $\\mathcal{C}$.\n\\end{definition}\n\n\\begin{example}\n  Let $I = \\gen{P(X),Q(X)} \\subset M_3(\\mathbb{F}_4)[X]\/(X^5-1)$ be a left ideal.\n  The row echelon form generator matrix of the $3$-quasi cyclic code\n  $\\mathcal{C}_I$ associated to the left ideal $I$ is \n  \\begin{equation*}\n    G = \\left( \\begin{array}{ccc|ccc|ccc|ccc|ccc}\n      1 & 0 & \\omega^2 &\n      0 & 0 & 0 &\n      0 & \\omega^2 & \\omega &\n      \\omega & 0 & 1 &\n      0 & 0 & 0 \\\\\n \n      0 & 1 & \\omega^2 &\n      0 & 0 & 0 &\n      0 & 0 & 0 &\n      \\omega & \\omega & 0 &\n      1 & 0 & \\omega^2 \\\\\n      \n      \\hline\n      \n      0 & 0 & 0 &\n      1 & 0 & \\omega^2 &\n      0 & 0 & 0 &\n      0 & \\omega^2 & \\omega &\n      \\omega & 0 & 1 \\\\\n \n      0 & 0 & 0 &\n      0 & 1 & \\omega^2 &\n      0 & \\omega^2 & \\omega &\n      \\omega & 0 & 1 &\n      \\omega & \\omega & 0 \\\\\n \n      \\hline\n \n      0 & 0 & 0 &\n      0 & 0 & 0 &\n      1 & 1 & 0 &\n      \\omega^2 & 0 & \\omega &\n      0 & \\omega^2 & \\omega \n    \\end{array} \\right).\n  \\end{equation*}\n  Algorithm~\\ref{al:basis-block-rank} gives that\n  $(g_4,g_5,T^3(g_4),T^3(g_5),T^{2 \\times 3}(g_5))$ is a basis of\n  $\\mathcal{C}_I$. Moreover\n  \\begin{equation*}\n    g(X) =\n    \\begin{pmatrix}\n    0 & 1 & \\omega^2 \\\\\n    0 & 0 & 0 \\\\\n    0 & 0 & 0\n    \\end{pmatrix} +\n    \\begin{pmatrix}\n    0 & \\omega^2 & \\omega \\\\\n    1 & 1 & 0 \\\\\n    0 & 0 & 0\n    \\end{pmatrix} X +\n    \\begin{pmatrix}\n    \\omega & 0 & 1 \\\\\n    \\omega & 0 & \\omega \\\\\n    0 & 0 & 0\n    \\end{pmatrix} X^2 +\n    \\begin{pmatrix}\n    \\omega & \\omega & 0 \\\\\n    0 & \\omega^2 & \\omega \\\\\n    0 & 0 & 0\n    \\end{pmatrix} X^3\n  \\end{equation*}\n  is a generator polynomial of $\\mathcal{C}_I$ and\n  $I=\\gen{P(X),Q(X)} = \\gen{g(X)}$.\n\\end{example}\n\n\\subsection{A property of generator polynomials}\n\nThe following proposition generalizes\n\\cite[Theorem~1,~(c), page~190]{SloMacWil86} and\n\\cite[Theorem~4, page~196]{SloMacWil86}.\n\n\\begin{proposition}\n  Let $\\mathcal{C}$ be an $\\ell$-quasi-cyclic code of length $m\\ell$ over $\\mathbb{F}_q$.\n  Let $P(X)$ be a generator polynomial of $\\mathcal{C}$ and $Q(X)$ a\n  generator polynomial of its dual.\n  Then \n  \\begin{equation*}\n    P(X) \\left( ^t Q^{\\star}(X) \\right) = 0 \\pmod{X^m - 1}\n  \\end{equation*}\n  where $Q^{\\star}$ denotes the reciprocal polynomial of $Q$ and $^t Q$\n  the polynomial whose coefficients are the transposed matrices of\n  the coefficients of $Q$.\n\\end{proposition}\n\n\\begin{proof}\n  Since $P(X) = \\sum_{i = 0}^{m - 1} P_i X^i$ is a generator\n  polynomial of $\\mathcal{C}$, the rows of the matrix\n  \\begin{equation*}\n    \\begin{pmatrix} P_0 & P_1 & \\ldots & P_{m - 1} \\end{pmatrix}\n  \\end{equation*}\n  and their shifts span $\\mathcal{C}$.\n  Similarly $Q(X) = \\sum_{i = 0}^{m - 1} Q_i X^i$ and the rows of\n  \\begin{equation*}\n    \\begin{pmatrix} Q_0 & Q_1 & \\ldots & Q_{m - 1} \\end{pmatrix}\n  \\end{equation*}\n  and their shifts span $\\mathcal{C}^{\\perp}$.\n  By definition of a dual code, we have\n  \\begin{equation*}\n    \\begin{pmatrix} P_0 & P_1 & \\cdots & P_{m-1} \\end{pmatrix}\n    \\begin{pmatrix} ^t Q_0 \\\\ ^t Q_1 \\\\ \\vdots \\\\ ^t Q_{m-1} \\end{pmatrix}\n    = \\sum_{i=0}^{m-1} P_i \\left( ^t Q_i \\right) = 0.\n  \\end{equation*}\n  As $\\mathcal{C}$ and $\\mathcal{C}^{\\perp}$ are $\\ell$-quasi cyclic codes we also have\n  \\begin{equation*}\n    \\sum_{i = 0}^{m - 1} P_i \\left( ^t Q_{i + j \\mod m} \\right) = 0\n  \\end{equation*}\n  for all $j \\in \\mathbb{Z}$. Therefore \n  \\begin{equation*}\n    P(X) \\left( ^t Q^{\\star}(X) \\right) =\n    \\sum_{j = 0}^{m-1} \\sum_{i = 0}^{m - 1}\n    P_i \\left( ^tQ_{i - j \\mod m} \\right) X^j\n    = 0 \\mod (X^m-1).\n  \\end{equation*}\n  Hence the proposition.\n\\end{proof}\n\n\n\n\\subsection{The key equation}\n\nAs in the scalar case, we exhibit a key equation for quasi-BCH codes.\nIn this subsection, all vectors are considered to be single-column\nmatrices. Consider $\\mathbb{F}_q^{\\ell}$ as a product ring of $\\ell$ copies of\n$\\mathbb{F}_q$. We define a map\n\\begin{equation*}\n  \\begin{array}{rcl}\n    \\Psi : M_{\\ell}(\\mathbb{F}_{q^s})[[X]] \\times \\mathbb{F}_q^{\\ell}[[X]] &\n              \\rightarrow & \\mathbb{F}_{q^s}^{\\ell}[[X]] \\\\\n    (f,g) & \\mapsto & \\sum_{i,j} f_j g_i X^{i + j}\n  \\end{array}\n\\end{equation*}\nwhere the $f_i g_j$ are matrix-vector products. In the sequel we will\ndenote $\\Psi(f,g)$ simply by $f \\mathbin{\\diamond} g$. Note that we have\n$(fh) \\mathbin{\\diamond} g = f \\mathbin{\\diamond} (h \\mathbin{\\diamond} g)$ for any $h \\in M_{\\ell}(\\mathbb{F}_{q^s})$.\n\nLet $c$ be a codeword of $\\mathcal{C}$ sent over a channel,\n$y \\in (\\mathbb{F}_q^{\\ell})^m$ be the received word and let $e$ be\nthe error vector \\emph{i.e.} $e = y - c$ such that\n$\\w(e) = w \\leq \\lfloor (\\delta - 1) \/ 2 \\rfloor$. Let\n$W = \\supp(e) = \\{ i_1,\\ldots,i_w \\}$.\n\n\\begin{definition}[Locator and evaluator polynomials]\n  We define the \\emph{locator polynomial} by\n  \\begin{equation*}\n    \\Lambda(X) := \\prod_{i \\in W} (1- A^i X)\n    \\in M_{\\ell}(\\mathbb{F}_{q^s})\n  \\end{equation*}\n  and the \\emph{evaluator polynomial} by\n  \\begin{equation*}\n    L(X) := \\sum_{i \\in W}\n    \\left(\n      \\prod_{j \\neq i}^{w}\n      A^i (1 - A^j) X\n    \\right) \\diamond y_i\n    \\in \\mathbb{F}_{q^s}^{\\ell}[X].\n  \\end{equation*}\n\\end{definition}\n\n\\begin{lemma}\n  \\label{lem:invertSerie}\n  Let $B \\in M_\\ell(\\mathbb{F}_q)$ be a nonzero matrix, then $1 - BX$ has a\n  left- and right- inverse in $M_\\ell(\\mathbb{F}_q)[[X]]$, both equal to\n  \\begin{equation*}\n    \\sum_{j = 0}^{+\\infty} B^j X^j.\n  \\end{equation*}\n\\end{lemma}\n\nWe see that the locator polynomial $\\Lambda(X)$ is invertible in\nthe power series ring $M_{\\ell}(\\mathbb{F}_{q^s})[[X]]$ and we have\n\n\\begin{align*}\n  \\left( \\Lambda(X)^{-1} \\right) \\diamond L(X) &=\n      \\sum_{i \\in W} \\left(\n        A^i (1 - A^i X)^{-1}\n      \\right) \\mathbin{\\diamond} y_i \\\\\n  &= \\sum_{i \\in W} \\left(\n       \\sum_{j = 0}^{+\\infty} A^{i(j + 1)} X^j\n     \\right) \\mathbin{\\diamond} y_i \\\\\n  &= \\sum_{j = 0}^{+\\infty}\n       \\sum_{i \\in W} A^{i(j + 1)} y_i X^j.\n\\end{align*}\n\nUsing the fact that $y = c + e$ and that, by definition,\n$\\mathcal{S}_{A^i}(y) = \\mathcal{S}_{A^i}(e)$ for any $i = 0,\\ldots,\\delta - 1$\nwe have\n\n\\begin{equation*}\n  \\left( \\Lambda(X)^{-1} \\right) \\diamond L(X) =\n     \\sum_{j = 0}^{+\\infty} \\mathcal{S}_{A^{j + 1}}(e) X^j\n       := S_{\\infty}(X).\n\\end{equation*}\n\n\\begin{proposition}\n\\label{prop:KE}\n  For any error vector $e \\in \\mathbb{F}_q^{m\\ell}$ such that\n  $w(e) \\leq \\lfloor (\\delta - 1) \/ 2 \\rfloor$ we have\n  \\begin{center}\n    \\fbox{$\\Lambda(X) \\mathbin{\\diamond} S_{\\infty}(X) = L(X)$}\n  \\end{center}\n  and therefore\n  \\begin{equation}\n  \\label{equ:KE}\n    \\Lambda(X) \\mathbin{\\diamond} S_{\\infty}(X) \\equiv L(X) \\mod X^{\\delta}.\n  \\end{equation}\n  We will refer to~\\eqref{equ:KE} as the \\emph{key equation}.\n\\end{proposition}\n\n\\subsubsection{Problems solving the key equation}\n\\label{sss:solving_ke}\n\nIn the case of BCH codes, the extended Euclidean and Berlekamp-Massey\nalgorithms can be used to solve the key equation.\nWe denote by $S_{\\delta}(X)$ the polynomial\n$S_{\\infty}(X) \\mod X^{\\delta}$ from~\\eqref{equ:KE} which can\nbe written as\n\n\\begin{equation}\n\\label{equ:lambda_L}\n  \\begin{pmatrix}\n    \\Lambda_0 & \\dots & \\Lambda_{\\delta - 1} &\n        \\vline & L_0 & \\dots & L_{\\delta - 1}\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    S_0 & S_1    & \\dots  & S_{\\delta - 1} \\\\\n        & S_0    &        & \\vdots \\\\\n        &        & \\ddots & \\vdots \\\\\n        &        &        & S_0 \\\\\n    \\hline\n    -1     & 0      & \\dots  & 0 \\\\\n     0     & -1     &        & \\vdots \\\\\n    \\vdots &        & \\ddots & 0 \\\\\n     0     & \\dots  & 0      & -1\n  \\end{pmatrix}\n  = 0.\n\\end{equation}\n\nWhere the $S_i$'s and $L_i$'s are column vectors such that\nthe $S_i$'s are the coefficients of $S_{\\delta}$ in\n$\\mathbb{F}_{q^s}^{\\ell}$ and the $L_i$'s are\nthe coefficients in $\\mathbb{F}_{q^s}^{\\ell}$ of $L(X)$.\nThe $\\Lambda_i$'s are the coefficients of $\\Lambda(X)$ in\n$M_{\\ell}(\\mathbb{F}_{q^s})$.\nThis system of linear equations over $\\mathbb{F}_{q^s}$ has many solutions in\n$\\mathbb{F}_{q^s}$ since there are $\\ell\\delta + \\delta$ unknowns and only\n$\\delta$ equations for each row of\n\\begin{equation*}\n  \\begin{pmatrix}\n    \\Lambda_0 & \\dots & \\Lambda_{\\delta - 1} &\n        \\vline & L_0 & \\dots & L_{\\delta - 1}\n  \\end{pmatrix}.\n\\end{equation*}\nHowever, we are only interested in the solution such that\n$(\\Lambda_0,\\ldots,\\Lambda_{\\delta - 1})$ is an error locator\npolynomial. In other words, if we let $\\mathfrak{B}$ be the\nsolutions of~\\eqref{equ:lambda_L} and\n\\begin{equation*}\n  \\mathfrak{S} = \\left\\{\n    \\prod_{i \\in W} (1- A^i X) \\in M_{\\ell}(\\mathbb{F}_{q^s}) :\n      W \\subset \\{ 1,\\ldots,m \\} \\text{ and }\n      \\#W \\leq \\lfloor (\\delta - 1) \/ 2 \\rfloor\n  \\right\\}\n\\end{equation*}\nbe the set of all possible locator polynomials corresponding\nto errors of weight at most $\\lfloor (\\delta - 1) \/ 2 \\rfloor$,\nwe are interested in the elements of $\\mathfrak{B} \\cap \\mathfrak{S}$.\n\n\\begin{proposition}\n  There exists one and only one solution of\n  equation~\\eqref{equ:lambda_L} in $\\mathfrak{S}$.\n\\end{proposition}\n\n\\begin{proof}\n  Equation~\\ref{equ:KE} ensures that there exists at least one element\n  in $\\mathfrak{B} \\cap \\mathfrak{S}$. If there were more than one\n  solution in $\\mathfrak{S}$ there would exist more than one codeword\n  in a Hamming ball of radius $\\lfloor (\\delta - 1) \/ 2 \\rfloor$ which\n  is absurd.\n\\end{proof}\n\nThe solving of~\\eqref{equ:lambda_L} remains difficult. One\nneeds an exponential (in $\\ell\\delta$) number of arithmetic operations\nin $\\mathbb{F}_{q^s}$ to find the element of $\\mathfrak{B} \\cap \\mathfrak{S}$.\nFor small values of $q$, $\\ell$ and $\\delta$ the solution can be found\nby exhaustive search on the solutions of~\\eqref{equ:lambda_L}.\n\n\\subsubsection{Unambiguous decoding scheme}\n\nIn this subsection, we prove that, as in the BCH case, the roots of\nthe locator polynomial (in $\\mathbb{F}_{q^s}[A]$) give precious\ninformation about the location of errors. The factorization of\npolynomials of $M_\\ell(\\mathbb{F}_{q^s})[X]$ is not unique, all the\nroots of the locator polynomial do not indicate an error position.\n\n\\begin{proposition}\n\\label{prop:equivRacineLoca}\n  Let $e \\in \\mathbb{F}_q^{m\\ell}$ be an error vector such that\n  $w(e) \\leq \\lfloor (\\delta - 1) \/ 2 \\rfloor$ and $\\Lambda(X)$ be\n  the locator polynomial associated to $e$. We have \n  \\begin{equation*}\n    e_i \\neq 0 \\Longleftrightarrow \\Lambda(A^{-i}) = 0.\n  \\end{equation*}\n\\end{proposition}\n\n\\begin{proof}\n  By definition, we have $\\Lambda(A^{-i}) = 0$ if $e_i \\neq 0$.\n  Conversely, if $e_i = 0$ then $A^j A^{-i} \\neq I_{\\ell}$ for\n  $j \\in \\supp(e)$. Thus $1 - A^j A^{-i}$ is a unit in\n  $\\mathbb{F}_{q^s}[A]$ by definition of $A$. Therefore\n  $\\Lambda(A^{-i}) \\neq 0$.\n\\end{proof}\n  \nThese roots can be found by an exhaustive search on the powers of $A$\nin at most $m$ attempts. At this step the support of the error vector\n$e$ is known. The last step to complete the decoding is to find the\nvalue of the error.\n\n\\begin{proposition}\n\\label{prop:errorEvaluation}\n  Let $e \\in \\mathbb{F}_q^{m\\ell}$ be an error such that\n  $w(e) \\leq \\lfloor (\\delta - 1) \/ 2 \\rfloor$, \\mbox{$W=\\supp(e)$}, $\\Lambda(X)$ be the\n  locator and $L(X)$ be the evaluator polynomials associated to\n  $e$. If $A^{-i}$ is a root of $\\Lambda(X)$ for $ i \\in W$,\n  then \n  \\begin{equation*}\n    e_i =  \\prod_{j \\in W \\setminus \\{i\\}}\n                (A^{i} - A^{j})^{-1} L(A^{-i})\n  \\end{equation*}\n  where $L(A^j)$ denotes $\\sum (A^j)^i L_i$.\n\\end{proposition}\n\n\\begin{proof}\n  Let $i_0 \\in W$. We have\n  \\begin{align*}\n    L(A^{-i_0}) &= \\sum_{i = 1}^{w} \\prod_{j \\neq i}^{w}\n                   A_i (1 - A^{-i_0} A_j) y_i\\\\\n                &= \\prod_{j \\in W \\setminus \\{i_0\\}}\n                   A^{i_0} (1 - A^{-i_0} A^j) e_{i_0}\\\\\n                &= \\prod_{j \\in W \\setminus \\{i_0\\}}\n                   (A^{i_0} - A^j) e_{i_0}.\n  \\end{align*}\n  By definition of $A$, $A^{i_0} - A^{j}$ is invertible for all\n  $j \\in W$ hence the result.\n\\end{proof}\n\n\\begin{algorithm}\n\\label{al:DecodingQCBCH}\n\\caption{Decoding algorithm for quasi-BCH codes}\n\\begin{algorithmic}\n  \\REQUIRE{The received word $y = c + e$ where\n           $c \\in \\mathcal{C}$ and $w(e) \\leq \\lfloor (\\delta - 1) \/ 2 \\rfloor$.}\n  \\ENSURE{The codeword $c$, if it exists such that\n          $d(y,c) \\leq \\lfloor (\\delta - 1) \/ 2 \\rfloor$.}\n  \\STATE $S_{\\delta}(X) \\gets$ Syndrome of $y$.\n  \\STATE Compute $\\Lambda(X)$ and $L(X)$\n         (Subsection~\\ref{sss:solving_ke}).\n  \\STATE $\\mathfrak{R} \\gets$ roots of $\\Lambda(X)$ in $\\mathbb{F}_{q^s}[A]$.\n  \\STATE $W \\gets \\{i | A^{-i} \\in \\mathfrak{R} \\}$.\n  \\STATE $\\zeta \\gets (0,\\ldots,0)$.\n  \\FOR{$i \\in W$}\n    \\STATE $\\zeta_i = \\prod_{j \\in W \\setminus \\{i\\}}\n                      (A^{i} - A^{j})^{-1} L(A^{-i})$.\n  \\ENDFOR\n  \\RETURN $y - \\zeta$.\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\n\\section{Introduction}\n\\label{Sec:Intro}\n\n  \\subsection{Context}\n  \\input{Intro}\n\n  \\subsection{First definitions}\n  \\label{Ssec:Rappel}\n  \\input{Rappel}\n\n\\section{Properties of quasi-cyclic codes}\n\\label{Sec:Classification}\n\\input{Classification}\n\n\\section{Quasi-BCH}\n\\label{Sec:BCH}\n\\input{BCH}\n\n\\section{Decoding scheme for quasi-BCH codes}\n\\label{Sec:Key}\n\\input{KE}\n\n\\section{Evaluation codes}\n\\label{Sec:Eval}\n\\input{eval}\n\n\\section{Conclusion}\n\\label{Sec:Conlu}\n\\input{conclusion}\n\n\\section*{Acknowledgments} \n\\input{acknowledgements}\n\n\\bibliographystyle{plain}\n\n\n\\subsection{Definition and parameters}\n\nIn this subsection we generalize evaluation codes. For any ring $R$\nand any positive integer $k$, we denote by $R[X]_{<k}$ the left\n$R$-module of all polynomials of $R[X]$ of degree at most $k - 1$.\n\n\\begin{proposition}\n\\label{FqAField}\n  Let $q$ be a prime power and $\\ell,m$ be positive integers such\n  that $m = q^{\\ell} - 1$. Let $A \\in M_{\\ell}(\\mathbb{F}_q)$ be a primitive\n  $m$-th root of unity. Then $\\mathbb{F}_q[A]$ and $\\mathbb{F}_{q^{\\ell}}$ are\n  isomorphic as rings.\n\\end{proposition}\n\n\\begin{proof}\n  Let $\\mu(X)$ be the minimal polynomial of $A$ of degree at\n  most~$\\ell$. We have $\\mu | X^m - 1$, thus the roots of $\\mu$\n  are all distinct. By Definition~\\ref{defi:racprim}-~(3), the\n  roots of $\\mu$ lie in $\\mathbb{F}_{q^{\\ell}}$ and not in any subfield.\n  Therefore $\\mu$ is irreducible.\n\\end{proof} \n\n\\begin{definition}[Quasi-cyclic evaluation codes]\n\\label{defi:evalcode}\n  Let $\\ell$ be a positive integer and $q$ be a prime power. Let\n  $m = q^{\\ell} - 1$ and $k \\leq m$. Let $A \\in M_\\ell(\\mathbb{F}_q)$ a\n  primitive $m$-th root of unity.\n  Let $\\pi$ be a $\\mathbb{F}_q$-linear map from $\\mathbb{F}_q[A]$\n  into $\\mathbb{F}_q^{\\ell}$. We denote by $C_{A,k,\\pi}$ the image of:\n  \\begin{equation*}\n    \\begin{array}{ccccc}\n      (\\mathbb{F}_q[A])[X]_{<k} & \\stackrel{\\ev_A}{\\longrightarrow} &\n        (\\mathbb{F}_q[A])^m & \\stackrel{\\pi^{\\times m}}{\\longrightarrow} &\n          (\\mathbb{F}_q^\\ell)^m \\\\\n      P(X) & \\longmapsto & \\left(P(A^0),\\ldots,P(A^{m-1})\\right) &\n        \\longmapsto &\n          \\left(\\pi(P(A^0)),\\ldots,\\pi(P(A^{m-1}))\\right).\n    \\end{array}\n  \\end{equation*}\n\\end{definition}\n\n\\begin{proposition}\n  Taking the notation of Definition~\\ref{defi:evalcode},\n  $C_{A,k,\\pi}$ is a $\\ell$-quasi cyclic code over $\\mathbb{F}_q$ of\n  length $m\\ell$ and of dimension over $\\mathbb{F}_q$ at least \n  $k \\ell - \\dim_{\\mathbb{F}_q}(\\ker \\pi^{\\times m})$.\n\\end{proposition}\n\n\\begin{proof}\n  By Proposition~\\ref{FqAField} the statement about the dimension of\n  $C_{A,k,\\pi}$ is obvious. Let\n  \\begin{equation*}\n    P(X) = \\sum_{i = 0}^{k - 1} \\sum_{j = 0}^{m - 1} P_{ij} A^j X^i\n    \\in \\mathbb{F}_q[A][X]_{<k}\n  \\end{equation*}\n  with $P_{ij} \\in \\mathbb{F}_q$.\n  Then\n  \\begin{equation*}\n    Q(X) = \\sum_{i = 0}^{k - 1} \\sum_{j = 0}^{m - 1} P_{ij}\n           A^{j + i} X^i\n    \\in \\mathbb{F}_q[A][X]_{<k}\n  \\end{equation*}\n  is such that $Q(A^i) = P(A^{i + 1})$ for all $i \\in \\mathbb{Z}$\n  and $C_{A,k,\\pi}$ is $\\ell$-quasi cyclic.\n\\end{proof}\n\n\\subsection{New good codes}\n\n\\begin{proposition}\n  Using the notation of Definition~\\ref{defi:evalcode},\n  if $\\pi$ is such that for $B=(b_{ij}) \\in \\mathbb{F}_q[A]$\n  \\begin{itemize}\n    \\item $\\pi(B)=(b_{i1},\\ldots,b_{i\\ell})$ for some $i$,\n    \\item or $\\pi(B)=(b_{1j},\\ldots,b_{\\ell j})$ for some $j$,\n  \\end{itemize}\n  then $\\dim C_{A,k,\\pi} \\geq k \\ell$ and $C_{A,k,\\pi}$ has minimum\n  distance $d \\geq m - k + 1$.\n\\end{proposition}\n\n\\begin{proof}\n  In both cases, it suffices to notice that $\\pi^{\\times m}$ is\n  injective. If $\\pi^{\\times m}(B_1,\\ldots,B_m) = 0$ then\n  $\\det B_i = 0$ for $i = 1,\\ldots,m$. As $\\mathbb{F}_q[A]$ is a field we must\n  have $B_i = 0$ for $i = 1,\\ldots,m$. In fact under the assumptions\n  of the proposition $\\pi^{\\times m}$ is an isomorphism since\n  $\\#((F_q[A])^m) = q^{m \\ell} = \\#((F_q^{\\ell})^m)$.\n\\end{proof}\n\n\\begin{remark}\n\\label{rem:new_codes}\n  \\begin{enumerate}\n    All the computations of the examples below have been performed\n    with the \\textsc{magma} computer algebra system \\cite{magma}.\n    \\item \\label{rem:item:ze_one}\n          For some particular choices of $\\pi$, especially when we\n          decrease the dimension~$k$, we observe that the minimum\n          distance is multiplied by $\\ell - 1$. For example, with\n          \\begin{equation*}\n            A =\n            \\begin{pmatrix}\n              0      & \\omega   & 0 \\\\\n              \\omega & \\omega^2 & \\omega^2 \\\\\n              1      & \\omega^2 & 1\n            \\end{pmatrix} \\in M_3(\\mathbb{F}_4)\n            \\text{ with } \\mathbb{F}_4 = \\mathbb{F}_2[\\omega],\n          \\end{equation*}\n          $k = 4$ and $\\pi((b_{ij}))=(b_{2,1},b_{1,2},b_{2,3})$,\n          we find a $[189,11,125]_{\\mathbb{F}_4}$-code. According to\n          \\cite{codetables}, the previous best known minimum\n          distance was $121$.\n    \\item \\label{rem:item:rs}\n          As for Reed-Solomon codes, we can evaluate polynomials of\n          $(\\mathbb{F}_q[A])[X]_{<k}$ at less than $m = q^{\\ell} - 1$ points.\n          Following this approach, we find the following new good\n          codes listed below together with the corresponding previous\n          best known minimum distances:\n          \\begin{center}\n            $[186, 11, 122]_{\\mathbb{F}_4}$, 120; \\\\\n            $[183, 11, 119]_{\\mathbb{F}_4}$, 117; \\\\\n            $[180, 11, 116]_{\\mathbb{F}_4}$, 114; \\\\\n            $[177, 11, 113]_{\\mathbb{F}_4}$, 112. \\\\\n          \\end{center}\n    \\item Markus Grassl applied different methods to construct new codes\n          from our $[189, 11, 125]_{\\mathbb{F}_4}$ code\n          (item~\\ref{rem:item:ze_one} of Remark~\\ref{rem:new_codes}).\n          For example, he used a puncturing method \\cite{GrasWhi2004}.\n          Some of the codes he obtained have the same parameters as the\n          codes listed in item~\\ref{rem:item:rs} of\n          Remark~\\ref{rem:new_codes}. He found \n          $[186, 11, 122]_{\\mathbb{F}_4}$, $[183, 11, 119]_{\\mathbb{F}_4}$ and\n          $[180, 11, 116]_{\\mathbb{F}_4}$ codes. He also found a\n          $[177, 11, 114]_{\\mathbb{F}_4}$ code while the best known\n          minimum distance was~$112$.\n          The 49 new codes found with the help of Markus Grassl are\n          listed in Table~\\ref{tab:BestCodes}. All the methods used\n          for the construction of these codes are detailed\n          in~\\cite{codetables}.\n          \\begin{table}\n            \\begin{tabular}{|ccccc|}\n              \\hline\n              \\multicolumn{5}{|c|}{New codes over $\\mathbb{F}_{4}$} \\\\\n              \\hline\n              \\hline\n              $[171, 11, 109]_4$ & $[172, 11, 110]_4$ &\n              $[173, 11, 110]_4$ &\n              $[174, 11, 111]_4$ & $[175, 11, 112]_4$ \\\\\n      \n              $[176,11,113]_4$ & $[177,11,114]_4$ &\n              $[178,11,115]_4$ &\n              $[179,11,115]_4$ & $[180,11,116]_4$ \\\\\n          \n              $[181, 11, 117]_4$ & $[182, 11, 118]_4$ &\n              $[183, 11, 119]_4$ &\n              $[184, 10, 121]_4$ & $[184, 11, 120]_4$ \\\\\n          \n              $[185,10,122]_4$ & $[185,11,121]_4$ &\n              $[186,10,123]_4$ &\n              $[186,11,122]_4$ & $[187,10,124]_4$ \\\\\n          \n              $[187,11,123]_4$ & $[188,10,125]_4$ &\n              $[188,11,124]_4$ &\n              $[189,10,126]_4$ & $[189,11,125]_4$ \\\\\n              \n              $[190,10,127]_4$ & $[190,11,126]_4$ &\n              $[191,10,128]_4$ &\n              $[191,11,127]_4$ & $[192,11,128]_4$ \\\\\n          \n              $[193,11,128]_4$ & $[194,11,128]_4$ &\n              $[195,11,128]_4$ &\n              $[196,11,129]_4$ & $[197,11,130]_4$ \\\\\n          \n              $[198,11,130]_4$ & $[199,11,131]_4$ &\n              $[200,11,132]_4$ &\n              $[201,10,133]_4$ & $[201,11,132]_4$ \\\\\n          \n              $[202,10,134]_4$ & $[202,11,132]_4$ &\n              $[203,10,135]_4$ &\n              $[204,10,136]_4$ & $[204,11,133]_4$ \\\\\n          \n              $[205,11,134]_4$ & $[210,11,137]_4$ &\n              $[213,11,139]_4$ & $[214,11,140]_4$ & \\\\ \n          \\hline\n        \\end{tabular}\n        \\caption{\\label{tab:BestCodes}\n          49 new codes over $\\mathbb{F}_4$ which have a larger minimum\n          distance than the previously known ones.}\n        \\end{table}\n  \\end{enumerate}\n\\end{remark}\n\n\\begin{remark}\n  We have proved in Proposition~\\ref{FqAField} that $\\mathbb{F}_q[A]$ is a\n  field such that $[\\mathbb{F}_q[A]:\\mathbb{F}_q] = \\ell$. Thus there is a\n  $\\mathbb{F}_q$-linear isomorphism from $\\mathbb{F}_q[A]$ to $\\mathbb{F}_q^{\\ell}$.\n  Consider the following one:\n  \\begin{equation*}\n    \\begin{array}{ccc}\n      \\mathbb{F}_q[A] & \\stackrel{\\psi}{\\longrightarrow} & \\mathbb{F}_q^{\\ell} \\\\ \n      B=b_0 I_{\\ell} + b_1 A + \\cdots + b_{\\ell - 1} A^{\\ell - 1} &\n          \\longmapsto & (b_0,b_1,\\ldots,b_{\\ell - 1}).\n    \\end{array}\n  \\end{equation*}\n  Then\n  \\begin{equation*}\n    C_{A,k,\\psi}=\\psi^{\\times m}(\\ev_A(\\mathbb{F}_q[A][X]_{<k}))\n  \\end{equation*}\n  is still an $\\ell$-quasi cyclic code of length $m\\ell$ and of\n  dimension $k\\ell$. Let $\\Pi \\in M_{\\ell}(\\mathbb{F}_q)$ and let\n  \\begin{equation*}\n    \\begin{array}{rcl}\n      \\pi : \\mathbb{F}_q^{\\ell} & \\rightarrow & \\mathbb{F}_q^{\\ell} \\\\\n                      x & \\mapsto     & x \\Pi\n    \\end{array}\n  \\end{equation*}\n  for a given $\\Pi \\in M_{\\ell}(\\mathbb{F}_q)$.\n  Then\n  \\begin{equation*}\n    C_{A,k,\\psi,\\pi} =\n      \\pi^{\\times m}(\\psi^{\\times m}(\\ev_A(\\mathbb{F}_q[A][X]_{<k})))\n  \\end{equation*}\n  is an $\\ell$-quasi cyclic code of length $m\\ell$ and dimension\n  $\\geq k \\ell - \\dim(\\ker \\pi)$.\n\n  We notice that there exist matrices $\\Pi$ for which the obtained\n  minimum distance is always greater than $m - k + 1$. For instance,\n  taking $\\ell = 3$, $q = 4$ and the matrix\n  \\begin{equation*}\n    \\Pi =\n    \\begin{pmatrix}\n      1        & \\omega^2 & \\omega \\\\\n      \\omega^2 & \\omega   & 1 \\\\\n      1        & 1        & 1\n    \\end{pmatrix},\n   \\end{equation*}\n   give codes with minimum distance close to $2(m - k + 1)$.\n\\end{remark}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSince the discovery of topological matter in two dimensions \\cite{tsui82,laughlin83} it has become clear that the distinction between fermions, as particles which obey the Pauli exclusion principle, and bosons, as particles which do not, is incomplete on the level of emergent particles. Instead, topological many-body systems can host also quasiparticles, so called anyons with intermediate quantum-statistical behavior.  \\cite{leinaas77,wilczek82}. In many respects, an anyon behaves like the fraction of a particle, and accordingly it possesses fractional quantum numbers. For instance, electronic systems in the fractional quantum Hall regime host quasiparticles whose electric charge is only a fraction of the electron's charge. If two identical anyons are exchanged, their wave function may acquire a $U(1)$ phase, which in contrast to the case of bosons and fermions is not restricted to integer multiples of $\\pi$. An even more exotic type of anyons are the non-Abelian ones \\cite{moore-read}: they have a characteristic number of (quasi-)degenerate ground states, and under particle exchange a state in this manifold can evolve into another one. Importantly, such mixing is not possible under local perturbations, which has triggered the hope for an exciting technological application, namely a robust quantum memory. The quantum information stored in the topologically protected state of the anyons can be processed by the braiding of non-Abelian anyons, possibly allowing for fault-tolerant quantum computing \\cite{nayak08}. The first step to achieve this goal is to gain control over quasiparticles in fractional quantum Hall systems.\n\nThe standard way of creating fractional excitations is by tuning the magnetic field strength and\/or the electrostatic backgate potential. Current schemes for detecting anyonic behavior are based on transport measurements in interferometric devices \\cite{camino05,mcclure12,willett13}. However, there are also different optical techniques which can be used to probe quantum Hall physics beyond electronic transport measurements: Since the early days of quantum Hall physics, the light emission from quantum Hall samples has been measured \\cite{goldberg90,turberfield90} in order to probe the interaction between electrons and holes \\cite{macdonald90,cooper97,wojs00}. In addition to emission spectra, also the elastic and inelastic scattering of light has been detected \\cite{levy16}. Recently, a novel spectroscopic approach with improved energy resolution has been achieved by bringing a GaAs quantum well into a cavity, and detecting polariton resonances via light reflection \\cite{ravets18}. Landau level transitions in graphene have been probed by infrared absorption spectroscopy \\cite{jiang07,orlita08}, and Raman spectroscopy \\cite{goler12,faugeras15}. Photocurrent measurements in graphene have combined optical probing with transport measurements \\cite{nazin10,olivier}. Moreover, using a scanning single-electron transistor \\cite{Ilani2004} or a scanning tunneling microscope \\cite{morgenstern03,matsui2005,hashimoto08,luican-mayer14}, the local density of states has been detected for graphene in the quantum Hall regime. The resolution of these measurements allows for identifying single quasiparticles, and this technique has recently been suggested for the direct imaging also of fractional quasiparticles \\cite{papic18}.\n\nAt the same time, there have also been remarkable advances in optical control and manipulation of synthetic many-body systems. For atomic quantum Hall droplets, it has been proposed to create anyons via AC Stark shift, and to directly observe their dynamical behavior \\cite{paredes01,julia-diaz12,grass12}. In such systems, spectroscopic properties can also reveal the fractional statistics of excitations \\cite{cooper15}. In optical lattices, adiabatic flux insertion is suited to grow fractional quantum Hall states \\cite{grusdt14}, or to create anyonic excitations \\cite{wang18}. Angular-momentum resolved spectroscopy of emitted light has been suggested as a tool to gain microscopic insight to a photonic quantum Hall system \\cite{umucal17}. Exploiting light beams with orbital angular momentum has been proposed for the engineering of polaritonic fractional quantum Hall systems \\cite{ivanov18}. \n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig1_arxiv.eps}\n\\caption{\\label{scheme} \n{\\bf Two schemes to optically prepare quasi-particles in a FQH system.} (a) Synthetic flux insertion: Light with orbital angular momentum couples to a quantum Hall system on a Corbino disk, and shifts (quasi-)particles through the annulus. Since only entire electrons may flow through a wire connecting inner and outer edge,  a fractional quantum Hall system at filling $\\nu=1\/q$ requires $q$ pumping cycles for a measurable signal. (b)  Light-induced potentials: Local light beams create an optical Stark shift which is suited to trap quasiparticles in a FQH system.\n}\n\\end{figure}\n\n\nIn the present paper, we apply a quantum optics toolbox to manipulate electrons in quantum Hall liquids.  A major advantage of optical methods is their versatility. For instance, while manipulating an electronic material with a gate potential requires built-in contacts, optical potentials could have less hardware requirements. Moreover, compared to transport measurements, optical schemes can be suited for local probes, and the position of an optical potential can flexibly be tuned. These properties suggest that optical techniques may become particularly useful for braiding quasiparticles. In this paper, we present two different schemes to create and manipulate quasiparticles in electronic fractional quantum Hall systems, as schematically shown in Fig.~\\ref{scheme}: (a) Synthetic flux insertion creates quasiparticles, and (b) light-induced potentials are able to trap quasiparticles.\n\n\\textbf{(a) Synthetic flux insertion:} In this scheme, presented in Sec. \\ref{pump}, we exploit the orbital angular momentum of light to synthesize the insertion of a magnetic flux, and to create individual quasiholes or quasielectrons. Specifically,  we use a pulsed light field with a non-zero orbital angular momentum, and coherent light-matter interactions to pump the electrons into a state with the angular momentum shifted by the value of the photons' orbital angular momentum. From the conceptual point of view, this process is equivalent to adding\/removing a magnetic flux into\/from the system. Therefore, each light pulse can be designed to exactly produce one quasihole or one quasielectron, if the orbital angular momentum of the light field is $\\pm1$.\n\nDetails of the optical coupling depend on the material:\n(i) For Dirac materials like graphene, we consider a single optical transition from the Landau level at the Fermi surface to an empty Landau level. Such a selective coupling is enabled by the anharmonicity of the relativistic energy spectrum, in contrast to systems with quadratic dispersion, e.g., GaAs.  If the electron and the photon exchange orbital angular momentum, such a transition can be used to change the angular momentum of an electron. By timing the pulse duration, such that it matches the value $\\pi$ (in units given by the inverse Rabi frequency), we can coherently increase (decrease) the angular momentum of all electrons by one, and thereby, produce a quasihole (quasielectron). \n(ii) In systems with quadratic dispersion, we consider a Raman-type coupling between two spin manifolds in the conduction band, and the valence band.\nThis approach requires spin-orbit coupling in the valence band, as found in GaAs. Exploiting a STIRAP-like protocol, cf. Ref. \\onlinecite{vitanov17}, the Raman beams coherently flip the spin of the (spin-polarized) fractional quantum Hall state without producing excitations from the valence band. As in the case of graphene, it is again possible to increase (decrease) the angular momentum of each electron by using light with orbital angular momentum. Both protocols are robust against disorder provided the  timescale for the optical transfer process is fast compared to the disorder potential.\n\nOur optical method is particularly unique as it provides an experimentally applicable and well controlled method to generate quasielectrons in a similar setting as the quasiholes. Despite their apparent similarity, there is a fundamental distinction between the two types of quasiparticles, and this can be understood from the fact that the quasihole state is an eigenstate of the parent Hamiltonian for the Laughlin state in the presence of an additional repulsive potential. In contrast, adding an attractive potential in place of the repulsive one does not generally produce the quasielectron state, due to the fact that the position of the potential might be already occupied by another electron \\cite{Hansson2009,Nielsen2018}. \n\n\\textbf{(b)  Light-induced potentials:} In this scheme, we propose that by using an off-resonance light field, one can generate an AC Stark shift to produce a local potential, as presented in Sec. \\ref{pin}. By choosing the frequency of the light field to be larger than the one of the corresponding Landau level transition, we show that the resulting repulsive potential can stabilize the system with a single quasihole. The optimal size of the trap corresponds to a situation where  the potential width is of the order of the magnetic length. However, within the quantum Hall regime, this length scale is usually smaller than  the wavelength of light, which sets the minimal trap size. Improvements on the potential can be achieved by an alternative sub-wavelength trapping scheme. Specifically, by coupling three electronic levels, e.g., from two spin levels and the valence band, it is possible to engineer optical potentials below the diffraction limit. Furthermore, we  show that by adiabatically moving the optical potential, the electronic wave function acquires a Berry phase which reflects the fractional charge of the excitation. By explicitly simulating the dynamics of a small system, we determine the maximum speed at which the potential can be moved without resulting in non-adiabatic behavior. Our simulation demonstrates that optical traps may become a useful tool for the braiding of anyons. \n\n\n\nAlthough our optical schemes can be applied to different fractional or integer quantum Hall phases, the present paper focuses on systems in the Laughlin phase \\cite{laughlin83}. Despite its relatively simple form, the Laughlin wave function supports fractional quasiparticles, and it captures very well the ground state of electrons at filling 1\/3. In the remainder of this introductory section, we briefly discuss some general properties of the Laughlin wave function and of its excitations, which will be used in this work. \n\nThe Laughlin wave function is given by \n\\begin{align}\n\\Psi_{\\rm L}\\sim\\prod_{i,j} (z_i-z_j)^3 \\exp[-\\sum_i |z_i|^2\/(4 l_{\\rm B})],\n\\end{align}\nwhere $z_j=x_j- i y_j$ are the spatial coordinates of the $j$th electron, with $l_{\\rm B} = \\sqrt{\\hbar\/(eB)}$ being the magnetic length in a field of strength $B$. The wave function assumes that electrons are confined to the lowest Landau level, denoted by a Landau level index $n=0$. We note that within the $n=0$ Landau level, there is no difference between graphene and semiconducting materials, except for an additional valley degree of freedom in graphene. In the Laughlin state, spin and valley degrees of freedom are assumed to be fully polarized.\n\nFor $N$ electrons, the $z$-component of total angular momentum in the Laughlin state is $L_N=\\frac{3}{2}N(N-1)$. In the thermodynamic limit, this quantum number is replaced by the filling factor $\\nu$, that is, by the ratio between the number of electrons, and the number of states within a Landau level. The filling factor is also well-defined in compact geometries such as the torus. The Laughlin wave function corresponds to a filling $\\nu=1\/3$. \n\nOne distinguishes between low-energy excitations at the edge and in the bulk of a Laughlin system. Excitations at the edge are gapless deformations, which increase the angular momentum slightly (by a value $\\ll N \\hbar$). The anyonic quasiparticles which we are interested in here are excitations in the bulk.\nThey are gapped excitations which appear as fractional electrons (``quasielectrons'') or fractional holes (``quasiholes''), that is, as a local increase or decrease of the charge density. The wave function of a quasihole at position $\\xi$ is obtained by multiplying the Laughlin wave function with a prefactor $f_{\\rm qh}^\\xi=\\prod_{i=1}^N (z_i-\\xi)$. From this expression it is seen that a quasihole increases the $z$-component of total angular momentum by $\\mathcal{O}(N)$ above the Laughlin value (in units $\\hbar$). In contrast, for a quasielectron in the lowest Landau level located at a position $\\xi$ the Laughlin wave function should be multiplied by $f_{\\rm qe}^\\xi=\\prod_{i=1}^N (\\partial_{z_i}-\\xi)$, where the derivative does not act on the exponential factor of the Laughlin wave function. Obviously, the quasielectron has the opposite effect on total angular momentum compared to quasihole.\n\nWithin the lowest Landau level, the coordinate $z_i$ can be replaced by the operator $b_i^\\dagger$ which raises the angular momentum of an electron. With this, we can re-write the quasihole state as \n\\begin{align}\n\\label{bqh}\n\\Psi_{\\rm qh}^{\\xi} \\sim  \\prod_{i=1}^N \\left(b_i^\\dagger-\\xi\\right) \\Psi_{\\rm L},\n\\end{align}\nChoosing the quasihole position to be in the center, $\\xi=0$, this expression shows that the quasihole state can be produced by shifting each electron into the next angular momentum orbital. A similar expression can be obtained for the quasielectron by replacing $b_i^\\dagger$ with $b_i$, which shows that in this case the angular momentum of each electron should be decreased by one. This observation outlines the strategy which we will pursue in the following section in order to generate Laughlin quasiparticles.\n\n\n\\section{Synthetic flux insertion}\n\\label{pump}\n\nIn this section, we present two approaches to synthesize the insertion of a flux, i.e. to add quantized angular momenta to the electronic system. In both approaches, we achieve this by applying a light field with orbital angular momentum. Our first approach, presented in Sec. \\ref{graphene}, uses several $\\pi$-pulses, which resonantly couple two Landau levels. This approach is best suited for Dirac materials such as graphene with an anharmonic Landau level spectrum. The schematics of our approach is illustrated in Fig. \\ref{fig:raman}(a). The proposed coupling brings an electron from the Fermi surface into an empty Landau level, so the action of the coupling is to change the  Landau level index by one:  $n \\rightarrow n+1$. Simultaneously, the coupling increases the electron angular momentum by $\\hbar$, i.e. the orbital quantum number within the Landau level increases by one: $m \\rightarrow m+1$. By a proper timing of this coupling (that is, by applying a $\\pi$-pulse), all electrons can coherently be transferred, resulting in a quasihole state according to Eq. (\\ref{bqh}) within a higher Landau level. To remove the Landau level excitation, one can apply a second $\\pi$-pulse at constant angular momentum.  In this Section, we perform a numerical simulation of the system dynamics which shows that decoherence due to electron-electron interactions is small, if the Rabi frequency is on the order of the Coulomb interactions ($\\sim1$~eV). Spontaneous emission from the excited Landau level can then be neglected, as lifetimes on the order of picoseconds are much longer than the duration of a $\\pi$-pulse. \n\nThe second approach, presented in Sec. \\ref{GaAs}, is based on a three-level scheme, as shown in Fig. \\ref{fig:raman}(b). Here, a Raman-type coupling between two spin Landau levels in the conduction band leads to a spin flip. An important ingredient to enable the Raman transition is spin-orbit coupling in the valence band, which can be found in prominent quantum Hall materials, including GaAs. As in our first approach, angular momentum transfer from the photons to the electrons generates the desired orbital shift, $m \\rightarrow m+1$. This approach produces a quasihole state within the spin-reversed Landau level. Since interactions are spin-independent, this scheme is free from decoherence due to interactions. Moreover, lifetimes of spin excitations are extremely long (on the order of nanoseconds) \\cite{kikkawa98,ohno99}, so the final state is sufficiently stable.  Excitation of valence-band electrons can be avoided by applying a detuned STIRAP protocol. \n\nIn Sec. \\ref{sec:corbino}, we discuss an experimental proposal to measure the fractional charge of an anyon. The main idea is that by increasing the angular momentum of each electron by $\\hbar$, a charge $e\/q$ is pumped through the system, with $q=1\/\\nu$, which is set by the filling factor $\\nu$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\textwidth, angle=0]{fig2.eps}\n\\caption{\\label{fig:raman} {\\bf Different schemes for generating quasiholes.} (a) Coupling scheme for graphene: Electrons from the Fermi surface at LL0 are shifted to an empty Landau level (LL1) by a $\\pi$-pulse. If the beam carries orbital angular momentum, it will also act on the orbital quantum number, $m\\rightarrow m+1$. A second $\\pi$-pulse is applied to remove the Landau level excitation, while leaving the angular momentum increased. The final state is a quasihole excitation at the Fermi level. (b) Coupling scheme for GaAs quantum well: A Raman-like coupling, consisting of a $\\pi$- and a $\\sigma$- polarized light beam, couple two spin Landau levels in the conduction band to the valence band. Using the shown STIRAP-like timing of the pulses, we can flip the spin of all conduction band electrons, while avoiding excitations from the valence band. If one of the light beams carries orbital angular momentum, the spin flip is combined with an orbital shift, $m\\rightarrow m+1$. The final state is a quasihole excitation within the spin-excited Landau level.}\n\\end{figure}\n\n\n\\subsection{ $\\pi$-pulse coupling in graphene \\label{graphene}}\nWe consider an optical coupling between the fractionally filled Landau level $n$ at the Fermi surface to an empty Landau level $n'$. Such a selective coupling is possible in Dirac materials such as graphene, as they exhibit a non-equidistant Landau level spectrum. The selection rules for circularly polarized light, $|n| \\leftrightarrow \\pm(|n|\\pm 1)$, determine optically allowed transitions, cf. Ref. \\onlinecite{jiang07}.\nFor concreteness, we will focus on graphene at a fractionally filled $n=0$ level, and consider a coupling to $n'=1$. We note that both Landau levels support Laughlin-like ground states \\cite{areg,toke06,amet15} at filling $\\nu=1\/3$. If the light also carries orbital angular momentum $\\ell$, the selection rule regarding the orbital quantum number $m$ is given by $|\\Delta m|=\\ell$, see Ref. \\onlinecite{gullans17}. \n\nIn the rotating frame, such coupling is described by a time-independent Hamiltonian \\cite{areg}:\n\\begin{align}\n\\label{H0}\n H_0 = \\sum_{m} \\left[ \\hbar \\delta c_{m,1}^\\dagger c_{m,1}  + \\hbar \\frac{\\Omega_m}{2} \\left( c_{m+\\ell,1}^\\dagger c_{m,0}  + {\\rm h.c.} \\right) \\right].\n\\end{align}\nThe operators $c_{m,n}^\\dagger$ ($c_{m,n}$) create (annihilate) electrons in the $m$th orbital of the $n$th Landau level, $\\delta$ is the detuning of the laser field from the Landau level transition frequency, and $\\Omega_m= \\int {\\rm d}^2r \\ \\bra{m+\\ell,1} {\\bf E}({\\bf r}) \\cdot {\\bf r} \\ket{m,0}$ is the Rabi frequency of the optical transition $m \\rightarrow m+\\ell$ in the (3D) electric field ${\\bf E}({\\bf r})$ within the 2D-plane ${\\bf r}$. \n\nIn the following, we will take $\\Omega_m=\\Omega$, that is, a constant for all orbitals $m$. This assumption is not strictly valid for any system size, since in order to carry orbital angular momentum, the light beam must have a vortex line somewhere, and orbitals which are localized near the vortex line will experience a weaker Rabi frequency than others. In a large enough system only few orbitals are affected from the vortex, and our assumption of a $m$-independent Rabi frequency holds approximately. The assumption becomes more rigorous, if one considers a Corbino disk geometry, i.e. a disk with a hole in the center, such that the hole may coincide with the vortex of the light beam.\n\nConsidering the time evolution under $H_0$ in the weakly detuned limit, $\\delta\\rightarrow 0$, the electrons are found to perform Rabi oscillations between orbitals $\\ket{n=0,m}$ and $\\ket{n=1,m+\\ell}$ with period $T=2\\pi\/\\Omega$. That is, if initially all electrons were in the $n=0$ level, they will be flipped into $n=1$ after a time $t=T\/2$.\nA light field which is properly timed, i.e. a $\\pi$-pulse, will therefore modify the initial $N$-electron state, $\\ket{\\Psi(0)}$, in the following way:\n\\begin{align}\n\\label{qhprime}\n\\ket{\\Psi(T\/2)} &= e^{-i \\pi H_0\/\\Omega }\\ket{\\Psi(0)} = \\prod_{i=1}^N [\\tilde a_i^\\dagger (\\tilde b_i^\\dagger)^\\ell] \\ket{\\Psi(0)} \\nonumber \\\\ &\n\\equiv \\ket{\\Psi_{\\rm qh}'}.\n\\end{align}\nHere, $\\tilde a_i^\\dagger$ and $\\tilde b_i^\\dagger$ denote the operators which raise the Landau level index $n$ and the orbital quantum number $m$:\n\\begin{align}\n\\label{btilde}\n \\tilde a^\\dagger \\equiv \\sum_{n,m} \\ket{n+1,m}\\bra{n,m},  \\ \\ \\  \\tilde b^\\dagger \\equiv \\sum_{n,m} \\ket{n,m+1}\\bra{n,m}.\n\\end{align}\nFor $\\ell=1$, the state defined in Eq.~(\\ref{qhprime}) describes a quasihole excitation similar to the one defined in Eq.~(\\ref{bqh}). However, Eq.~(\\ref{bqh}) defines the quasihole by applying ladder operators to the ground state wave function, whereas Eq.~(\\ref{qhprime}) uses projection operators $\\tilde b^\\dagger$. The ladder operators differ from the projection operators by a normalization factor $\\sqrt{m+1}$. The effect of these normalization factors in the ladder operators is minor for small systems and vanishes in the thermodynamic limit, as we show in the appendix. Thus, the orbital shift in Eq.~(\\ref{qhprime}) produces a quasihole excitation. In addition to the orbital shift, the $\\pi$-pulse also increases the Landau level index of each electron. Such a projection of the Laughlin state and its excitations into higher Landau levels is a straightforward generalization which has been discussed in Refs. \\onlinecite{macdonald84,macdonald86}. \n\nHaving established that an idealized light pulse creates a quasihole, we will in the following consider different processes which cause decoherence, and which could reduce the fidelity of our scheme: (a) electron-electron interactions, (b) spontaneous emission from the excited level, or  non-radiative dissipation (heating). \n\n\\paragraph{Electron-electron interactions.} Since interaction in the $n=1$ Landau level also support a Laughlin-like phase, interactions will not cause decoherence once the full population has been transferred from one Landau level into the other, that is to say, the initial and the final state will not be affected by interactions. However, during the transfer both Landau levels are occupied, and inter-level interactions differ significantly from intra-level interactions, cf. Ref. \\onlinecite{areg}. A straightforward strategy to keep the resulting decoherence small is by using short pulses, that is, by applying a strong Rabi frequency. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\textwidth, angle=0]{fig3.eps}\n\\caption{\\label{fig2} {\\bf Fidelity of the quasihole pump.} We consider a system of $N=5$ electrons, initially prepared in the ground state of $V$ within the $n=0$ Landau level at total angular momentum $L_n$. This initial state $\\Psi(0)$ has a large ($>0.99$) overlap with the Laughlin state. We then evolve this state  under $H_0+V_{\\rm C}$, and consider the overlap of the evolved state $\\Psi(t)$ with other trial wave functions, including the one for the quasihole state. In (a), we plot the maximally attained overlap between evolved state $\\Psi(t)$ and quasihole state $\\Psi_{\\rm qh}'$ (defined in Eq. (\\ref{qhprime}) as a function of the detuning $\\delta$ and the Rabi frequency $\\Omega$. In (b), we plot the overlap between $\\Psi(t)$ and different trial wave functions as a functions of time. This includes the overlaps with the initial state $\\Psi(0)$ (blue dashed line), with the quasihole state $\\Psi_{\\rm qh}'$ (red dotted line), and with the model wave function $\\Psi_{\\rm model}(t)$ given in Eq. (\\ref{model}) (green solid line). Here we have chosen coupling parameters $\\Omega=0.2 e^2\/\\epsilon l_{\\rm B}$ and $\\delta=0.04 e^2\/\\epsilon l_{\\rm B}$. Units of time are given as inverts of $\\Omega' = \\sqrt{\\Omega^2+\\delta^2}$.\n}\n\\end{figure}\n\nTo make this assessment more quantitative, we have numerically simulated the time evolution under $H=H_0+V_{\\rm C}$ for $N=5$ electrons, where the single-particle part $H_0$ is defined in Eq. (\\ref{H0}), and $V_{\\rm C}$ denotes Coulomb interactions. In the simulation, we have restricted the Hilbert space to the two coupled Landau levels, and \nthe angular momentum of the initial state fixes the quantum number $\\sum_i (m_i-\\ell n_i)$. For the initial state $\\Psi(0)$, we have chosen the ground state of $V_{\\rm C}$ within the $n=0$ level at fixed total angular momentum $L_N$. This state has large overlap ($\\sim0.99$) with the Laughlin state. We then determine the overlap of the evolved state $\\Psi(t)$ with the state $\\Psi_{\\rm qh}' \\equiv \\prod_i a_i^\\dagger f_{\\rm qh}^0 \\Psi(0) $, that is, a state obtained from the initial state by introducing a quasihole and raising the Landau level index of all electrons. In Fig. \\ref{fig2}(a), we plot the maximally attained overlap during the course of the evolution as a function of the detuning $\\delta$ and Rabi frequency $\\Omega$. As a promising result, we find that the Rabi frequency does not have to be much larger than the many-body gap for the fidelity to reach values close to one. \nWe note that the many-body gap above the Laughlin phase is on the order $0.15 e^2\/\\epsilon l_{\\rm B}$. This value corresponds to 0.3 eV, if we assume a typical magnetic field strength of 10 teslas, and use the permittivity of the vacuum, $\\epsilon=\\epsilon_0$.\nOur numerical simulation also shows that the best choice for the detuning is not at resonance, but at about  $\\delta=0.05~e^2\/\\epsilon l_{\\rm B}$, that is, for an optical frequency below the Landau level resonance. The value of the detuning roughly compensates the interaction energy difference when electrons are pumped into the quasihole state.  Due to a larger total angular momentum in the quasihole state, the Coulomb repulsion in this state is decreased, and the many-body resonance is shifted away from the single-particle value.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\textwidth, angle=0]{fig4.eps}\n\\caption{\\label{fig2qe} {\\bf Fidelity of the quasielectron pump.} The setup in this case is similar to Fig.~\\ref{fig2}, but the pump photons have orbital angular momentum $l=-1$. Moreover, we simulate pumping in the presence of an additional potential in the lowest Landau level acting on $m=0$ state, to initially remove the population of this state (see main text). (a) We plot the maximally attained overlap with the quasielectron state during a pumping cycle. (b) We plot the overlap of the time-evolved state with the quasielectron state (shifted into LL1) $\\Psi_{\\rm qe}'$ (red dotted line), the Laughlin state $\\Psi_{\\rm L}$ (blue dashed line), or a time-dependent model wave function $\\Psi_{\\rm model}(t)$ (green solid line). The coupling parameters are $\\Omega=0.4 e^2\/\\epsilon l_{\\rm B}$ and $\\delta=-0.1 e^2\/\\epsilon l_{\\rm B}$}.\n\\end{figure} \n\n\\paragraph{Spontaneous emission.} Spontaneous emission limits the lifetime of any state above the Fermi level. Therefore, we need to prepare the state of interest in the Landau level at the Fermi energy.  This can be achieved by applying two subsequent $\\pi$-pulses, as shown in Fig. \\ref{fig:raman}: The first pulse, with orbital angular momentum $\\ell=1$, transfers the electrons into an excited Landau level, and  simultaneously shifts the orbital quantum number $m$ to $m+1$, as discussed above. The second pulse with $\\ell=0$ returns the electrons to the original Landau level, without changing orbital states. Using sufficiently large Rabi frequencies, both pulses can operate at large fidelities.  The combination of both pulses then results in a quasihole excitation within the Landau level at the Fermi surface. With this scheme, spontaneous emission can only occur during the pulse duration. To neglect this effect, we have to demand that the lifetime in the excited level is large compared to the duration $t=\\pi\/\\Omega$ of a $\\pi$-pulse. In other words, \nthe coupling has to be fast compared to the emission rate. In summary, large Rabi frequencies (on the order of eV) suppress both decoherence due to interactions or due to spontaneous emission. However, strong Rabi couplings also lead to non-radiative losses. This will set a practical limit to the Rabi strength of the pulse, and thus, to the fidelity of our scheme. A further requirement on the Rabi frequency is that it is large compared to the disorder potential, which ensures that the selection rules for orbital angular momentum are well obeyed.\n\nWithin our simulation, we have also studied how the system evolves from the initial Laughlin-like state (polarized in $n=0$ at $t=0$) into a quasihole state (polarized in $n=1$ at $t=\\pi\/\\Omega'$ with $\\Omega'\\equiv \\sqrt{\\delta^2+\\Omega^2}$). It is found that at intermediate times $0<t<\\pi\/\\Omega'$ the system evolves through a series of edge-like excitations. The most relevant edge states, denoted by $\\Psi^{(s)}$, are of the form:\n\\begin{align}\n \\Psi^{(s)} =  \\frac{1}{\\sqrt{\\binom{N}{s}}} \\sum_{ \\{ k_1,\\dots,k_s\\} } (-1)^{\\sum_{j=1}^s k_j} \\prod_{j=1}^s \\tilde a_{k_j}^\\dagger \\tilde b_{k_j}^\\dagger \\Psi(0).\n\\end{align}\nHere the sum is over all $s$-tuples $\\{k_1,\\dots,k_s\\}$ with $1\\leq k_1 <\\dots<k_s \\leq N$, i.e. all ways of choosing $s$ out of $N$ particles. For $s=0$ and $s=N$, this definition recovers the initial state $\\Psi(0)$ and the quasihole state $\\Psi_{\\rm qh}'$ of Eq. (\\ref{qhprime}). \nGenerally, $s$ specifies the number of electrons in $n=1$, which is equal to the excess of angular momentum quanta with respect to the Laughlin value $L_N$. Thus, the states $\\Psi^{(s)}$ interpolate between edge and quasihole excitations.\nBy definition, a quasihole excitation increases the total angular momentum by $\\mathcal{O}(N)$, while an edge excitation is characterized by an increase of $\\mathcal{O}(1)$. \n\nWe also note that the family $\\Psi^{(s)}$ contains only a selection of edge states, namely those where $s$ electrons are excited by only one angular momentum quantum.\nOther edge states are barely relevant for the dynamics in our system, and we can model with high fidelity the system evolution using only states  $\\Psi^{(s)}$ with $0\\leq s \\leq N$. Therefore, we make the following ansatz:\n\\begin{align}\n\\label{model}\n \\Psi_{\\rm model}(t) = \\sum_{s=0}^N \n {\\cal N}_s\n \\cos(\\frac{1}{2} \\Omega' t)^{N-s} \\sin(\\frac{1}{2} \\Omega' t)^s \\Psi^{(s)},\n\\end{align}\nwhere  ${\\cal N}_s=(-i)^{{\\rm mod}(s,2)}\\sqrt{ \\binom{N}{s} }$. In Fig. \\ref{fig2}(b) we plot, for $N=5$ electrons, the overlap of the exact state with this model wave function as a function of time. Despite our choice of a relatively weak Rabi frequency, $\\Omega=0.2 e^2\/\\epsilon l_{\\rm B}$, the model wave function $\\Psi_{\\rm model}(t)$ captures the evolution rather well.\n\nThis establishes the following picture for our quasihole pump: During a pumping period, the quasihole state is reached via a series of edge excitations.  Therefore, as shown by Eq. (\\ref{model}), for large systems our scheme requires fine-tuning of the pumping period in order to reach the quasihole state. As a function of time, the overlap with the quasihole state behaves like $\\sim\\sin(\\frac{1}{2} \\Omega' t)^N$, so the time window where the evolved state has large overlap with the quasihole state becomes short for large $N$. \n\nWe note that the scenario here is different from another mechanism to produce a quasihole which has been discussed in the context of cold atoms \\cite{paredes01}, and which is based on a local repulsive potential. By adiabatically increasing the potential strength, the Laughlin state is turned into the quasihole state without involving significant contributions from other states. In contrast to our scheme, this approach involves a first-order phase transition which would make the adiabatic evolution prohibitively slow in the thermodynamic limit. On the other hand, in small systems, the method provides an intermediate superposition between Laughlin state and quasihole state which can be used for interferometric measurements. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\textwidth, angle=0]{fig5.eps}\n\\caption{\\label{cases} {\\bf Distinct cases in the STIRAP scheme.} In Case I, the $\\uparrow$-electron (black ball) is transferred into the empty $\\downarrow$ state (empty dotted ball) via a coupling to the filled valence band (VB) level. Under a particle-hole transformation, this process maps onto a single-particle process, flipping the spin of the conduction band (CB) hole, and the standard single-particle STIRAP scenario applies. In Case II, both CB states are empty, but coupled to a filled VB state. To avoid Rabi oscillation of the valence band electron, both couplings $\\Omega_\\pi$ and $\\Omega_\\sigma$ must be sufficiently strongly detuned from the single-photon resonance.\n}\n\\end{figure}\n\nThe optical scheme makes it possible to induce the quasielectron state in a similar manner as for the quasihole. To this end, we consider an optical pump with angular momentum $l=-1$ and follow a similar procedure as outlined above. An important difference in this case is that the pump cannot transfer the electron with angular momentum $m=0$ to the higher Landau level due to the absence of the resonantly coupled state. In contrast, in order to produce the quasielectron wave function, the components of the Laughlin wave function for which the $m=0$ state is occupied should be destroyed by the quasielectron prefactor. Given the strongly correlated nature of the Laughlin ground state, this might appear as a challenge for generating the quasielectron state. \nParticularly, in exact diagonalization calculations with finite number of electrons and using the Laughlin state as the initial state, the highest value of the overlap which we were able to obtain with the quasi-electron state is 87\\% (not shown).\nHowever, the overlap can be significantly increased to 99\\% (Fig.~\\ref{fig2qe}), if we add a repulsive potential acting on the $m=0$ state in the lowest Landau level. Such potential excludes the $m=0$ state from the initial wave function, which thus deviates from Laughlin state (Fig.~\\ref{fig2qe} (b)). The addition of this potential might appear as a mathematical artifact, but it clearly demonstrates that in the absence of a $m=0$ (as on an annulus), the quasielectron wave function can be produced by our pumping scheme. Moreover, optical methods may even allow to directly implement such a potential in real experiments. \nTo this end, one could exploit optical Stark effect (which is discussed in the next section) using an off-resonant coupling from the Fermi level (e.g. $(n=0$) to another Landau level (e.g. $n=-1$). If the light beam has angular momentum $l=1$, it will couple levels $(n=-1,m)$ and $(n=0,m+1)$, and the level $(n=0,m=0)$ remains uncoupled. Thus, the AC Stark effect shifts the energy of all holes in the $n=0$ Landau level, except for the $m=0$ state, and effectively can repel electrons from this state. The large overlap with the quasielectron state which is achieved in exact diagonalization, after we implement such potential, shows the promise of our method for the controlled generation of quasielectrons, and for manipulating such states. As can be seen from Fig.~\\ref{fig2qe} (b), the model wave function (\\ref{model}) with the appropriate change of $\\tilde b_i^\\dagger$ with $\\tilde b_i$ captures the dynamics quite well for this case as well. The maximum overlap is now attained for detuning $\\delta=-0.12~e^2\/\\epsilon l_{\\rm B}$, that is, at opposite sign compared to the quasihole pumping. This is due to the fact that producing a quasielectron reduces the total angular momentum in the system, and thus leads to increased Coulomb repulsion. As a consequence, the many-body resonance is shifted to an optical frequency above the Landau level gap.\n\n\\subsection{STIRAP spin-flip scheme in GaAs \\label{GaAs}}\nWe now consider an alternative coupling scheme which is illustrated in Fig. \\ref{fig:raman}(b).\nThe goal and the general strategy is the same as in the previous subsection, but instead of selectively coupling Landau levels, we now achieve the desired momentum transfer via a Raman spin flip process. This approach avoids the need of an anharmonic Landau level spectrum, and thus it can be applied to non-relativistic materials. On the other hand, coupling different spin manifolds in the conduction band to the same level in the valence band requires the presence of spin-orbit coupling \\cite{imamoglu00,vanderwal11}. Therefore, the approach in this section is well suited for GaAs, but it cannot be applied to graphene. One of the Raman beams shall carry orbital angular momentum, such that the coupling effectively transfers conduction band electrons from $\\ket{n=0,m,\\uparrow}$ into $\\ket{n=0,m+1,\\downarrow}$. As discussed before, the angular momentum transfer creates a quasihole state, but now, as an additional bonus, the final state remains in the $n=0$ Landau level, with only the electron spin being flipped. Given the long spin lifetimes of the order of nanoseconds, measured for GaAs in Refs. \\onlinecite{kikkawa98,ohno99}, the final state is effectively stable. Moreover, since the Coulomb interactions are spin-independent, the final state will not be subject to decoherence due to interactions. We remark that our approach is robust against sample disorder provided the timescale for the optically induced transfer process is much faster than the characteristic frequency of the disorder potential.\n\nIn order to avoid excitations from the valence band, the timing of the light fields may follow a STIRAP protocol (see Ref. \\onlinecite{vitanov17} for a recent review on STIRAP techniques). In the standard STIRAP scenario, a particle is transferred between two stable states, involving two fields which couple these states to a third radiative level. The characteristic feature of STIRAP is the fact that, for properly timed fields, full state transfer is possible without populating the radiative level at any time. \nOur case, though, is different from the standard STIRAP scenario: While we also want to transfer a particle between two (relatively) stable conduction band states, we achieve this via a coupling to a filled valence band level. The scenario is illustrated as Case I in Fig. \\ref{cases}. Although this process involves two electrons, STIRAP can be applied if we view the process as the transfer of a single hole. This particle-hole transformation only requires to interchange pump- and probe-fields, and Coulomb interactions between the electrons simply renormalize the resonance frequencies. Our situation, however, is more complicated through the presence of a second scenario, illustrated as Case II in Fig. \\ref{cases}. This case may occur whenever the many-body state is at fractional filling, such that the couplings also act onto empty orbitals. This scenario may give rise to undesired excitations from the valence band. A natural way to avoid these excitations is by operating far detuned from a single-photon resonance.\n\nIn order to achieve high fidelities, the STIRAP protocol should be characterized by $T \\gg 2\\pi\/\\Omega \\gtrsim 2\\pi\/\\delta$. Here, $T$ denotes the duration of a STIRAP pulse, which needs to be significantly longer than a $\\pi$-pulse. However, due to the spin-independence of Coulomb interactions, there is no need to keep $T$ small compared to the time scale of interactions, set by $e^2\/(\\epsilon l_{\\rm B})$, where $\\epsilon=\\epsilon_0\\epsilon_{\\rm d}$. Note that in GaAs, a dielectric constant $\\epsilon_{\\rm d}\\approx 12$ suppresses Coulomb interactions by a factor of 12 compared to graphene. Thus, at a field strength of about $B\\approx10 \\ {\\rm T}$, the energy scale of Coulomb interactions is on the order of tens of THz, i.e., at femtosecond time scales. In contrast, the lifetimes of spin excitations is on the order of nanoseconds, so it is justified to treat both spin levels as stable levels.\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.88\\textwidth, angle=0]{fig6.eps}\n\\caption{\\label{stirap} {\\bf Simulation of many-body STIRAP scheme.} \n(a) Illustration of the STIRAP scheme for which we have performed a numerical simulation in a minimal system: Each Landau level is truncated at $m=3$ (4 states), and the total number of electrons is six. For simplicity, we have considered a coupling at constant angular momentum, i.e. between $\\ket{m,\\uparrow}$ and $\\ket{m,\\downarrow}$. Bandgap $\\Delta_{\\rm bg}$ and Zeeman gap $\\Delta_{Z}$ are chosen such that in the absence of the coupling, 4 electrons fill the valence band, and the remaining 2 electrons polarize in the spin-up manifold of the conduction band, where they form a $N=2$ ``Laughlin'' state, $\\Psi\\sim(z_1-z_2)^3$. (b) Applied STIRAP pulses, with the pulse duration $T=415 \\epsilon l_{\\rm B}\/e^2$, that is, on the order of hundreds of picoseconds. The pulse maxima are separated by $\\Delta t=250\\epsilon l_{\\rm B}\/e^2$. (c) We simulate the time evolution for different values of Rabi frequency $\\Omega$ and detuning $\\delta$, obtaining the valence band filling (occupation per states), and the overlap with initial state (a $N=2$ Laughlin state in the spin-up manifold) \nand target state ((a $N=2$ Laughlin state in the spin-down manifold), as a function of time. In the upper plot of panel (c), the transfer is poor due to a relatively weak Rabi frequency $\\Omega=0.025 e^2\/(\\epsilon l_{\\rm B})$, and comparably strong detuning $\\delta=0.1 e^2\/(\\epsilon l_{\\rm B})$. The plot in the center is for an increased Rabi frequency $\\Omega=\\delta=0.1e^2\/(\\epsilon l_{\\rm B})$, which allows for good transfer, but excitations from the valence band become relatively large. The lower plot, for $\\Omega=0.1e^2\/(\\epsilon l_{\\rm B})$ and $\\delta=0.3e^2\/(\\epsilon l_{\\rm B})$, leads to good transfer at low excitation rates.\n}\n\\end{figure*}\n\nSince the transfer dynamics now involve three different Landau levels, including the filled valence band Landau level, its numerical simulation is hard. We restrict ourselves to simulating a minimal example of the scheme: As plotted in Fig. \\ref{stirap}(a), we truncate the Landau levels to having only 4 states, and load the system with $N=6$ electrons. Choosing the bandgap and the Zeeman gap sufficiently large compared to the Coulomb interactions, the ground state of this system will be a filled valence band, and two electrons in the spin-up manifold of the conduction band, forming a ``Laughlin'' state of two electrons, i.e. $\\Psi(z_1,z_2) \\sim (z_1-z_2)^3 \\ket{\\uparrow \\uparrow}$. If we wanted to pierce a hole into this state, this would increase the angular momentum by 2, which is not permitted in our truncated Landau levels. Therefore, we simulate only the coherent transfer part of our scheme, that is, a coupling as shown in Fig. \\ref{stirap}(a), leaving the angular momentum constant. Applying the STIRAP protocol shown in Fig. \\ref{stirap}(b), we evaluate the fidelity, that is, the overlap of the time-evolved state $\\Psi(t)$ with the target state,  $\\Psi_{\\rm target}(z_1,z_2) \\sim (z_1-z_2)^3 \\ket{\\downarrow \\downarrow}$. As seen in Fig. \\ref{stirap}(c), this fidelity reaches unity, if the Rabi frequency is sufficiently strong, i.e. $\\Omega T \\gg 2\\pi$. For the pulse duration (defined by the FWHM of a Gaussian pulse), we have chosen $T\\approx 415 \\epsilon l_{\\rm B}\/e^2$, which for typical magnetic field strengths is on the order of hundreds of picoseconds. To avoid excitations from the valence band, the detuning should be larger than the Rabi frequency, $\\delta > \\Omega$.\n\n \n\\subsection{Detecting anyonic properties \\label{sec:corbino}}\nIn the remainder of this section, we briefly discuss possible detection schemes for fractional charge and statistics, which potentially benefit from a method to generate exactly one quasihole by a pulsed light beam.\n\n\\paragraph{Fractional charge.}\nA possible charge measurement can be performed on a Corbino disk. The insertion of flux through a Corbino disk has been discussed in Ref. \\onlinecite{thouless90}. As described in the previous section, our scheme increases the angular momentum of the electrons. This shifts them towards the outer edge in the same way as creation of an additional flux through the inner circle of the Corbino disk would do. The reverse process, which transports charges towards the inner edge can be achieved by decreasing the angular momentum of the electrons. \n\nThe confining potential at the edge makes it energetically favorable for the charge to return to its original position. This leads to transport through a wire connecting the two edges of the Corbino disk. However, considering a fractional quantum Hall system at filling $\\nu=1\/q$, $q$ quasiparticles need to be shifted to the outer edge in order to accumulate a total electronic charge $e$. Thus, $n$ pumping cycles are expected to produce a current of $n\/q$ electrons, and the number of pumping cycles serves as a direct measure of the fractional charge.\n\n\\paragraph{Fractional statistics.}\nThe detection of fractional statistics is possible using interferometers, either of the Fabry-Perot or the Mach-Zehnder type. Such devices are suited to detect Aharonov-Bohm phases, as proposed for instance in Ref. \\onlinecite{chamon97} and realized in Refs. \\onlinecite{camino05,mcclure12,willett13}, by measuring the interference of currents along different paths. In these schemes, the interference pattern is sensitive to changes of the magnetic field, which yields the value of the fractional charge. It is also sensitive to the number of quasiparticle between the different arms of the interferometer, and from this, the statistical angle of the excitations can be deduced. However, to extract both charge and statistical angle from the interference pattern, exact knowledge about the number of excitations is needed. Thus, our scheme may allow for improved measurements as it provides individual control over these excitations.\n\n\n\\section{Light-induced potentials:}\n\\label{pin}\nThe previous section has demonstrated that light with orbital angular momentum can be used to mimic the addition of a flux, and to produce a quasiparticle excitation. In the present section, we will not be concerned with the production of the excitation, but we will be interested in ways to stabilize and control the quasiparticle. Specifically, we will \nconsider an optical potential which locally repels the electrons and thereby traps a quasihole. \n\nA major concern addressed in this section is the finite width of the optical potential, in contrast to $\\delta$-like potentials which have been studied earlier in the context of cold atoms \\cite{paredes01,julia-diaz12}. A numerical investigation shows that a potential with small but finite width is even better suited for trapping quasiholes than a point-like potential. However, the gap above the quasihole state is found to decrease when the potential becomes broader than the magnetic length. Since the optical wavelength is usually larger than the magnetic length, we will present some ideas to achieve subwavelength potentials using a three-level coupling.\n\nGiven the flexibility of optical potentials, they appear to be particularly well suited for moving the quasihole. Thus, an optical trap for quasiholes may provide a tool for braiding anyonic excitations. To demonstrate that ability, we show that, when the potential is moved on a closed contour, the wave function acquires a Berry phase proportional to the fractional charge of the quasihole. \n\nThe calculations and discussions in this section hold for both non-relativistic systems and for Dirac materials.\n\n\\subsection{AC Stark shift}\nThe mechanism which provides the desired optical potential is AC Stark shift. The AC Stark shift is routinely used to trap cold atoms in optical lattices. Recently, it has been suggested to trap Dirac electrons in graphene by exploiting AC Stark shift \\cite{morina18}. In a GaAs quantum well, this shift can be produced by optically pumping below the band gap \\cite{vonlehmen86}. Alternatively, if the system is coupled to a cavity, an enhanced Stark shift can be achieved using a resonance of the cavity \\cite{edo}. In general, the energy shift $\\Delta E$ experienced by the electronic energy in a laser field ${\\bf E}({\\bf r},t)$, is given by $\\Delta E= {\\bf d}\\cdot {\\bf E}({\\bf r},t)$, where ${\\bf d}= \\alpha [ E_x({\\bf r},t),E_y({\\bf r},t)]$ is the dipole moment induced by the field. The polarizability $\\alpha$ is inversely proportional to the detuning $\\Delta$ from the closest resonance. With this, the optical potential reads:\n\\begin{align}\n V_{\\rm opt} \\propto \\frac{I(z)}{\\Delta},\n\\end{align}\nwhere $I(z)$ is the laser intensity in the complex plane, assumed to be constant in time. In the following, we will consider a Gaussian beam, that is, an optical potential $V_{\\rm opt}^{(\\xi,w)}(z) = \\left(\\frac{l_{\\rm B}}{w}\\right)^2 V_{{\\rm opt},0} \\exp[|z-\\xi|^2\/w^2]$, characterized by the position of the beam focus $\\xi$, the width $w$ of the beam, and an potential strength $V_0$. The prefactor $ \\left(\\frac{l_{\\rm B}}{w}\\right)^2$ normalizes the intensity such that $\\lim_{w\\rightarrow 0} V_{\\rm opt}^{(\\xi,w)}(z)= V_{{\\rm opt},0} \\delta(z)$, with $\\delta(z)$ being the Dirac distribution. \n\n For the purpose of trapping a quasihole, it is necessary that the strength of the potential compensates the energy gap above the Laughlin state. Thus, the relevant energy scale is given by the Coulomb energy $e^2\/(\\epsilon l_{\\rm B})$, with the magnetic length $l_{\\rm B}$ representing the typical length scale relevant for the quantum Hall physics.\nThis length scale determines the size of an electronic orbital, but also of defects like  quasiparticles and quasiholes.  If $\\tilde B$ is the magnetic field strength in tesla, $l_{\\rm B} = 26 \\ {\\rm nm}\/\\sqrt{\\tilde B}$. For a magnetic field of 9 T, and with a dielectric constant $\\epsilon_{\\rm d}=12$ (as in GaAs), this energy scale is on the order of 150 meV, and a typical gap will be on the order of 15 meV. An early measurement in GaAs \\cite{vonlehmen86} has obtained an AC shift of 0.2 meV was with a laser intensity of 8 $\\rm MW\/cm^2$. \n\nApart from the energy scale, also the length scale of the potential plays an important role. With the size of a quasihole being on the order of the magnetic length, we expect that the length scale of a trapping potential should not significantly exceed this scale. However, the minimum length scale of an optical potential is limited by the wavelength of the light, which in the visible regime is on the order of hundreds of nanometers. In contrast, for magnetic field strengths on the order of a few teslas, the magnetic length is only a few nanometers. We will thus need to evaluate whether a potential with finite width $w \\gg l_{\\rm B}$ is still suited to trap quasiholes.\n \n \\subsection{$\\delta$-like potentials}\n Before considering the case of finite-width potentials, we verify that a point-like potential ($w=0$) gives rise to the desired excitations. This becomes obvious when we look at the parent Hamiltonian of the Laughlin state, that is, at some model interactions $V_{\\rm parent}$ for which the Laughlin wave function $\\Psi_{\\rm L}$ is the densest  zero-energy eigenstate. Such parent Hamiltonian is given in terms of Haldane pseudopotentials $V_m$, specifying the interaction strength between two electrons at relative angular momentum $\\hbar m$. In the 1\/3-Laughlin state, all pairs of electrons have relative angular momentum $3\\hbar$, so the Laughlin state has zero energy in a model Hamiltonian with $V_m=0$ for $m\\geq 3$. Since for spin-polarized fermions the relative angular momentum cannot be even, a Hamiltonian with only a single non-zero pseudopotential, $V_1$, provides a  parent Hamiltonian for the Laughlin state. It follows that the quasihole state $f_{\\rm qh}^\\xi \\Psi_{\\rm L}$ becomes the densest zero-energy eigenstate of $V_{\\rm parent}+V_0\\delta(\\xi)$, when the potential strength exceeds a critical value. To see this, we note that the quasihole state carries the same anti-correlations between the electrons as the Laughlin state, but at the same time has vanishing density at position $\\xi$. \n \n The scenario of a $\\delta$-potential has been studied before in greater detail in the context of cold atoms \\cite{paredes01,julia-diaz12}. In these systems of neutral particles, which can be brought into the fractional quantum Hall regime by artificial gauge fields, the cyclotron frequency is on the order of the trap frequency ($\\sim 10$ Hz), resulting in a magnetic length $l_{\\rm B} = \\sqrt{\\hbar\/(M\\omega_{\\rm c})}$ on the order of microns, with $M$ being the mass of the atoms. Due to this different length scale, finite-width effects can indeed be neglected in these artificial systems. \n\n \\subsection{Finite-width potentials}\nTo study the role of the finite potential width $w$, we turn to numerical diagonalization methods, by which we obtain the ground state of $V_{\\rm C}+V_{\\rm opt}^{(\\xi,w)}$ for different laser positions $\\xi$ and different beam widths $w$. Generally, we find large overlaps of these states with the Laughlin quasihole state $f_{\\rm qh}^\\xi \\Psi_{\\rm L}$, even when the beam becomes as broad as (or even broader than) the electronic cloud. In our numerics, we have considered both disk and torus geometries which we discuss separately below.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\textwidth, angle=0]{fig7.eps}\n\\caption{\\label{gap} {\\bf Gap above the quasihole state on a torus.} We plot the energy gap above the three degenerate quasihole states on a square torus in the presence of an optical potential $V_{\\rm opt}^{(\\xi,w)}$, as a function of the potential width $w$. The $N$ electrons are confined in the $n=0$ Landau level, generated by the presence of $N_{\\Phi}=3N+1$ magnetic fluxes.}\n\\end{figure}\n\n\\paragraph{Torus.} The torus geometry is convenient because due to its compact nature no trapping potential is required to confine the electrons. Since the torus has no edge, this geometry is well suited for studying the bulk behavior of large systems for which deformations at the edge are irrelevant. Interestingly, on the torus, the ground state of $V_{\\rm C}+V_{\\rm opt}^{(\\xi,w)}$ is almost independent of $w$. The overlap with the exact Laughlin quasihole state \\footnote{The Laughlin quasihole states can be obtained as zero-energy states from the parent Hamiltonian, that is, from pseudopotential interactions with $V_1$ being the only non-zero term, together with a $\\delta$-like potential to create the quasihole. On the torus, one needs to consider the Hilbert space of $N$ electrons and $N_{\\Phi}=3N+1$ magnetic fluxes, which corresponds to LL filling 1\/3 plus one additional flux.} takes large values close to 1, cf. Table \\ref{table1}. While this result shows that the finite potential width does not modify the quasihole state in the bulk, we also find that the energy gap above the quashihole states is quite sensitive to the width of the beam (see Fig. \\ref{gap}). Up to a certain value, of the order of the magnetic length, a finite potential width is found to increase the stability of the quasihole. However, the gap starts to decrease when the beam width exceeds the magnetic length. \nThis result can be understood by noticing that also the quasihole has a finite size of the order of the magnetic length, and the formation of a quasihole reduces the energy due to the optical potential most efficiently when the spatial overlap between quasihole and potential becomes largest. Obviously, in broader potentials a quasihole becomes less efficient for reducing potential energy.\n\n\n\\paragraph{Disk.}\nThe disk, though the most natural geometry to study quantum Hall physics, suffers strongly from finite-size effects. Even the concept of a filling factor is not defined on an infinite disk because each Landau level contains an infinite amount of states. It is necessary to assume a trapping potential which controls the electron density. A realistic trapping potential consists of hard walls, so the potential is flat in the entire system, except for the edge, where the potential energy steeply increases. Effectively, such potential puts a constraint on the Hilbert space, as it restricts the orbitals to those which fit into the flat region. This means that angular momenta beyond a certain value are not available anymore. Since we are interested in the Laughlin state (with angular momentum $L_N$), and in its quasihole excitation (increasing the angular momentum by up to $N$ quanta), we will assume that the trap effectively truncates the Hilbert space at $L_N+N$. Therefore, we perform the exact diagonalization study within a space of Fock states of angular momentum $L_N \\leq L \\leq L_N+N$. This choice yields the quasihole state as the only zero-energy eigenstate if the parent Hamiltonian is applied, that is, for a point-like potential $V_{\\rm opt}^{(\\xi,w=0)}$ and pseudopotential interactions $V_m \\sim \\delta_{1,m}$. \n\n\\begin{table}\n\\begin{tabular}{c|c|c}\n $w\/l_{\\rm B}$ & Overlap on torus & Overlap on disk \\\\\n & $N=7$ & $N=8$ \\\\\n & $N_{\\phi}=22$ & $84\\leq L\/\\hbar \\leq 92$ \\\\ \n \\hline\n 0 & 0.9884 & 0.9450 \\\\\n 3 & 0.9885 & 0.9552 \\\\\n 6 & 0.9851 & 0.9409\n\\end{tabular} \n\\caption{\\label{table1}\nOverlaps between Laughlin quasihole state, and ground state of $V_{\\rm C}+V_{\\rm opt}^{(\\xi,w)}$ on disk and square torus, for different $w$.\nParameters on the torus: $V_0=1,\\xi=0$, and on the disk: $V_0=10,\\xi=2$. On the torus, exhibiting three (quasi)-degenerate ground states, overlap refers to the three (equal) eigenvalues of the 3x3 overlap matrix.\n}\n\\end{table}\n\nImportantly, we find that Coulomb interactions do not significantly alter the scenario. Comparing the exact Laughlin quasihole state and the ground state of $V_{\\rm C}+V_{\\rm opt}^{(\\xi,w)}$, we obtain  an overlap of about 0.95 for $N=8$. Strikingly, the potential width $w$ has only a minor effect on these numbers if the potential is chosen sufficiently strong, see Table \\ref{table1}.\n\nThere is, however, a notable consequence of finite-range interactions appearing in our numerics on the disk: the quasihole position does not exactly coincide with the position of the optical potential anymore, as seen in Fig. \\ref{rxi}. Although this observation seems to be an artifact due to the neglection of the trap, it will be important to take it into account  when determining the quasihole charge, as discussed in the next subsection. To this end, the data in Fig. \\ref{rxi} is needed to calibrate the quasihole position.\n\nEnergetic arguments explain the mismatch between quasihole position and potential minimum: shifting the quasihole towards the center increases the angular momentum, and thereby reduces the energy of long-ranged interactions. A realistic trapping potential would compensate this effect by penalizing the angular momentum increase, but this term is missing in our numerical study. If Coulomb interactions are replaced by the short-ranged pseudopotential model, the quasihole position coincides with the potential minimum.  In this case, the interaction energy is zero, and shifting the quasihole cannot lead to an interaction energy gain.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\textwidth, angle=0]{fig8.eps}\n\\caption{\\label{rxi} {\\bf Calibrating the quasihole position on the disk.} By neglecting the trapping energy, long-range interactions lead to a shift of the radial position $r$ of the quasihole towards the center.  The plotted curve is used to calibrate the true quasihole position $r$ as a function of the parameter $|\\xi|$, specifying the maximum of the optical potential, for $N=8$ electrons.}\n\\end{figure}\n\n\n\\subsection{Realization of sub-diffraction\\xspace potentials}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.48\\textwidth, angle=0]{fig9.eps}\n\\caption{\\label{fig:sub} \n\\textbf{Sub-diffraction potentials.}\n(a) \nEngineering a sub-diffraction\\xspace potential via three-level coupling.\nAn $\\Uparrow$-hole at the Fermi level (empty dotted ball) experiences an attractive potential by coupling to the electrons (black balls) in\nthe $\\downarrow$-level of the conduction band and in the valence band state $\\ket{v}$. A particle-hole transformation relates this process to the standard single-particle EIT scenario, applied to a single hole. Although the laser fields do not induce a direct potential for ${\\uparrow}$-electrons, the attractive potential for  $\\Uparrow$-holes results in an effective repulsive potential for the  ${\\uparrow}$-electrons.\n(b) \nWe show a 1D cut through the potential $V(z)$ and the laser fields $\\Omega_c(z)$ and  $\\Omega_p$. \nEven though $\\Omega_c$ is diffraction limited, we can achieve a sub-diffraction\\xspace $V(z)$ by working with ${\\rm max}[\\Omega_c(z)]\\gg \\Omega_p$.\n}\n\\end{figure}\nIn the previous section, we showed that the manipulation of anyons profits from potentials of width $w\\sim l_B$. This requires a sub-diffraction\\xspace addressability which can be achieved by employing techniques analogous to the ones used in ultra-cold atoms~\\cite{Gorshkov2008a,Wang2018}. \nThe basic idea is to use three energy levels which provides much more flexibility than just the two-level scheme used to induce AC Stark shift. \nAs an example we consider GaAs, for which  we can use the level scheme shown in Fig. \\ref{fig:sub}~(a), in analogy to Fig. \\ref{cases} used for the STIRAP. The scheme consists of two spin-levels in the conduction band and one level in the valence band. We now choose the Fermi energy through the upper spin level ${{\\uparrow}}$, so both the ${\\downarrow}$-level and the valence band are occupied. The two-electron system can be mapped onto a single-particle problem via particle-hole transformation, and a repulsive potential on ${{\\uparrow}}$-electrons will be achieved by engineering an attractive potential for $\\Uparrow$-holes. Therefore, we operate  at the two-photon detuning $\\delta_\\down<0$. \n\nMoreover, we use two laser fields: a strong $\\Omega_c\\xspace(z)$\nwhich is position dependent [for the easiness of presentation we fix it to $\\Omega_c\\xspace(z)^2=\\Omega_0^2(1-\\exp[|z-\\xi|^2\/w^2])$], \nand a weaker $\\Omega_p\\xspace$ which is homogeneous in space. \n  The Hamiltonian reads\n\\begin{equation}\nH_{\\rm \\scriptscriptstyle al}=\n\\left(\n\\begin{array}{ccc}\n\\delta_\\down& 0 & \\Omega _c(z) \\\\\n0&0& \\Omega_p  \\\\\n \\Omega _c(z) & \\Omega_p  & \\Delta\\\\%-i\\gamma \n\\end{array}\n\\right)\\label{HaEIT}\n\\end{equation}\nin the bare hole-state-basis: $\\{\\ket{{\\Downarrow}},\\ket{{\\Uparrow}}, \\ket{v}\\}$.\nFor $|\\delta_\\down| \\ll\\Omega_p\\xspace$ which ensures that we can consider $\\delta_\\down$ perturbatively, and for  an appropriate preparation scheme~\\cite{Wang2018,Lacki2016}, the internal state of a hole\ncan be described using a dark state $\\ket{D}=\\frac{\\Omega_c\\xspace(z)}{\\sqrt{\\Omega_p\\xspace^2+\\Omega_c\\xspace(z)^2}}\\ket{{\\Uparrow}}-\\frac{\\Omega_p\\xspace}{\\sqrt{\\Omega_p\\xspace^2+\\Omega_c\\xspace(z)^2}}\\ket{{\\Downarrow}}$.\nFrom the form of $\\ket{D}$, we see that the hole experiences an attractive potential $V(z)=\\delta_{\\Downarrow}\\frac{\\Omega_p\\xspace^2}{\\Omega_p\\xspace^2+\\Omega_c\\xspace(z)^2}$, which for $\\Omega_0\\gg\\Omega_p\\xspace$ can have sub-diffraction\\xspace width $w_s=w\/s$ characterized by the enhancement factor $s\\sim \\Omega_0\/\\Omega_p\\xspace$ and the depth $V_0=\\delta_\\down$.\n\n\nAssuming that we can describe our system using three levels,\nthe available depth of the trap is mainly limited by: (i)  the validity of the rotating wave approximation, and (ii) the coupling to the short-lived intermediate level.\nThe (i) limitation constrains the strength of $\\Omega_c(z)$  to $\\Omega_0\\ll\\Delta{\\rm \\scriptscriptstyle bg}$.\nTogether with $\\Omega_p\\gg V_0$ and $s=\\Omega_0\/\\Omega_p$, we get that $s\\ll \\Omega_0\/V_0\\ll \\Delta_{\\rm \\scriptscriptstyle bg}\/V_0$. \nFor $\\Delta_{\\rm \\scriptscriptstyle bg}\\sim 1.5$~eV, $\\Omega_0\\sim 0.5$~eV,  and $V_0\\sim15$~meV, we see that enhancement  factors $s$ on the order of $10$ are within a reach.\nThe losses in (ii), lead to the broadening of the trapping potential by $\\gamma_v \\frac{V_0^2}{\\Omega_p^2}\\ll \\gamma_v$, which [compared with the depth of the potential $V_0$] is negligible for the lifetimes $\\tau_v=1\/\\gamma_v$ on the order of  10~ps.\nNote that in contrast to ultra-cold atoms~\\cite{Lacki2016,Jendrzejewski2016,Wang2018,BieniasInPreparation}, the kinetic energy is quenched in a magnetic field, and therefore non-adiabatic corrections to the Born-Oppenheimer potentials~\\cite{Lacki2016,Jendrzejewski2016} are negligible. This relaxes some of the constraints posed on the possible trapping depths. \nWe leave a more detailed analysis, beyond the estimates presented here, for the future work.\n\nFinally, in the case of graphene, we envision similar possibilities: for example, one can use other filled Landau levels as the additional two levels in the ladder or lambda three-level scheme.\n\n\n\\subsection{Moving a quasihole.}\nIn the remaining part of this section, we consider an optical potential which is moved on a closed loop. As a quasihole is trapped by the potential, this procedure is expected to imprint a Berry phase onto the wave function which is proportional to the charge of the quasihole. Thus, by calculating the quasihole charge from the Berry phase we will verify that moving the optical potential is suited to move an excitation. By considering short-range interactions instead of Coulomb interactions, finite-size effects become small, and the fractional charge matches with the value 1\/3, expected for a thermodynamically large Laughlin system. We will also compare an idealized adiabatic evolution, restricted to the ground state Hilbert space, with the true dynamic evolution. This establishes the maximal speed with which the potential should be moved.\n\n\n\\paragraph{Relation between Berry phase and charge.} If the position $\\xi$ of a single charge $q$ is moved, the wave function $\\Psi(\\xi)$ will pick up a Berry phase $\\gamma=\\oint {\\rm d}\\xi \\langle \\Psi(\\xi) | \\bigtriangledown_\\xi | \\Psi(\\xi) \\rangle$, and this phase is proportional to the magnetic flux through the enclosed area $A$ times the value $q$ of the electric charge. This relation is normalized such that the electron charge $e$ acquires a Berry phase $\\gamma=2\\pi$ when it encircles one flux. In a constant magnetic field, with the magnetic length defined such that an area $2 \\pi l_{\\rm B}^2$ contains one flux quantum, we have the relation\n\\begin{align}\n\\label{qe}\n \\frac{q}{e} = \\gamma\\frac{l_{\\rm B}^2}{A}.\n\\end{align}\nThus, by studying the phase of the wave function upon moving the quasihole, we can extract the electric charge of the excitation. \n\n\\paragraph{Results from adiabatic evolution.}\nWe have performed such calculation using the disk geometry with $N=8$ electrons. We considered a Hamiltonian $H=V_{\\rm int}+V_{\\rm opt}^{(\\xi,w)}$, where the interactions $V_{\\rm int}$ are either Coulomb interactions or the parent Hamiltonian of the Laughlin state. For the optical potential $V_{\\rm opt}^{(\\xi,w)}$, we considered a finite width $w$ up to $3l_{\\rm B}$ as well as the limit $w\\rightarrow 0$. Our results are plotted in Fig. \\ref{charge}: Importantly, in case of the ideal interactions from the pseudopotential model, the width of the optical potential has a minor effect on the Berry phase, or, respectively the measured charge of the quasihole. Both, for a point-like potential and for a broad beam ($w=3 l_{\\rm B}$),  the quasihole charge is almost independent of the quasihole position, as it should be in a quantum liquid. Moreover, the value of the charge is close to the expected value $1\/3$. The accuracy of this result despite the small system size is due to the particular choice of interactions. As the pseudopotential model has short range interactions, finite-size effects are much weaker than in the long-range Coulomb case. For Coulomb interactions, the charge as a function of quasihole position is found to be $>0.4e$, that is, it differs significantly from $e$\/3. Surprisingly, in the presence of Coulomb interactions, the results for the finite-width potential are closer to the ideal value 1\/3.\n\nThe important conclusion which we draw from Fig. \\ref{charge} is that the finite width of the optical potential does not appear as a limiting factor for a charge measurement by moving the potential. In other words, the observed behavior suggests that even if the optical trap is much wider than the actual size of a quasihole, the quasihole still follows the contour described by the moving potential, and despite their broad width optical potentials can be used for moving and braiding anyons.\n\nThe results shown in Fig. \\ref{charge} were obtained from an ``adiabatic'' simulation, that is, we actually did not simulate the dynamics of the system while the potential is moved, but we assumed that for any potential position the system remains in its ground state. Thus, we simply determine the ground state $\\Psi_n$ at different quasihole positions $R e^{i\\varphi_n}= R e^{i n \\Delta \\varphi}$, and obtain the phase difference between subsequent states from their overlap. Summing up all phase differences along the contour gives the Berry phase\n\\begin{align}\n\\label{gamma_static}\n\\gamma =  \\sum_n {\\rm Im}\\left( \\langle \\Psi_{n+1} | \\Psi_n \\rangle -1\\right).\n\\end{align}\nIn this approach, it is important to fix the global gauge.  In our case of a non-degenerate ground state, the possible global gauge transformations are $U(1)$ rotation. Since we compare eigenvectors obtained from two different diagonalization procedures, we have to assure that the global gauge remains the same.  This can be done by demanding that a certain component of the state vector is real and positive \\cite{kohmoto}. However, this procedure requires that some assumptions and conditions hold: Of course, any state along the path needs to have a non-zero overlap with this reference state.  Moreover, one must ensure that, after discretization of the parameter space, the global gauge transformation does not remove the Berry phase. We can achieve this by choosing a reference component which does not gain a phase when the potential is moved.  It is easy to find such a component, since we know that the quasihole is described by a wave function of the type $\\prod_i (z_i-\\xi)\\Psi$. This means that the part of the wave function given by $\\prod_i z_i\\Psi$ is not affected by the quasihole position. Any component which has non-zero overlap with this expression can be used as a reference component, that is, any occupied Fock state with angular momentum $L=L_N+N\\hbar$.\n\n \\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\textwidth, angle=0]{fig10.eps}\n\\caption{\\label{charge} {\\bf Charge of Laughlin quasihole.}\nWe plot the charge $q$ of a quasihole in a system of $N=8$ electrons on the disk as a function of radial quasihole position $r$. The total angular momentum is restricted to the Laughlin regime, $L_N\\leq L \\leq L_N+N$, and the Hamiltonian consists of interactions $V_{\\rm int}$ and an optical potential $V_{\\rm opt}^{(\\xi,w)}$, of width $w$ and focused at position $\\xi$.\nWe have tuned the radial position $|\\xi|\/l_{\\rm B}$ of the optical potential from 0.1 to 2, and obtained the corresponding radial position position $r$ of the quasihole. The potential is then moved on a circle around the origin, which leads to a Berry phase which we evaluate using the static method of Eq. (\\ref{gamma_static}) for 200 discrete steps. We consider both Coulomb and pseudopotential interactions, the latter providing a parent Hamiltonian for the Laughlin state. We compare point-like potentials ($w=0$) and finite-width potentials ($w=3l_{\\rm B}$). Independently of the potential width, the system with pseudopotential interactions agrees well with the expected value $q\/e=1\/3$, whereas finite-size effects spoil the numerical value in the system with Coulomb interactions.\n}\n\\end{figure}\n\n\\paragraph{Dynamical evolution.}\nIn the remainder, we compare the ``adiabatic'' approach with a dynamic one. In the dynamic approach, we really simulate the time evolution of the system while the potential is moved, without assuming adiabaticity of the process. Of course, the dynamic approach is much more costly, as it requires full diagonalization of the Hamiltonian, whereas in the static approach only the ground state is needed. But there are some conceptual advantages of the dynamic method: First, this method yields the overlap between initial and final state which provides a measure for the adiabaticity of the process. From this one can also obtain information about how fast the optical potential may be varied. Second, the dynamic method does not require the gauge fixing procedure described above. \n\n\nFor our dynamical simulation we discretize time, and define $H_n$ as the Hamiltonian with the optical potential $V_{\\rm opt}$ at position $R e^{i\\varphi_n}$. We then evolve for short periods $\\Delta t$ under $H_n$, applying the evolution operator $U_n=\\exp\\left(i H_n \\Delta t\\right)$ to the quantum state, and afterwards we quench from $H_n$ to $H_{n+1}$. \n Starting from $\\Psi_0$, the ground state of $H_0$, we reach a final state $\\Psi=\\prod_{n=1}^{n_{\\rm max}} U_n \\Psi_0$, where $n_{\\rm max}=2\\pi\/\\Delta \\varphi-1$. If the process was adiabatic, initial and final state only differ by a phase $\\langle \\Psi | \\Psi_0 \\rangle = e^{i \\phi}$. This phase now consists of a dynamical contribution $\\phi_T$,  determined by the energy of the state, and the Berry phase $\\gamma$. In the particular case of a circular rotation around the origin, the energy does not change, and $\\phi_T=E_0 T$ where $T=n_{\\rm max}\\Delta t$. Thus, the Berry phase is obtained by\n\\begin{align}\n \\label{gamma_dyn}\n \\gamma= {\\rm Im}\\left({\\rm ln}  \\langle \\Psi | \\Psi_0 \\rangle \\right)-E_0 n_{\\rm max}\\Delta t.\n\\end{align}\n\n If the duration of a time step $\\Delta t$ is of the order of $1\/\\Delta E$, where $\\Delta E$ is the energy gap, the dynamic method produces exactly the same result as the adiabatic one. Interestingly, even for much shorter time steps, the system still behaves adiabatic in the sense that its overlap with excited states remains negligible, and initial and final state remain the same up to a phase difference. However, this phase difference acquires some errors, in the sense that it differs from the adiabatic value. This behavior is demonstrated in the data shown in Fig. \\ref{adi} for a system of $N=5$ electrons with Coulombic interactions and an optical potential of width $w=3 l_{\\rm B}$. This finding suggests that a quasihole can be moved relatively fast without energetically exciting the system, but this yet does not guarantee an adiabatic phase evolution.\n \n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\textwidth, angle=0]{fig11.eps}\n\\caption{\\label{adi} {\\bf Deviations from adiabatic process due to finite times.}\nWe compare two types of errors occurring if the quasihole is not moved adiabatically, as a function of the time step duration $\\Delta t$ (in units $\\hbar \\epsilon l_{\\rm B}\/e^2$).\nThe red curve shows the relative phase error, defined as $\\Delta \\gamma\/\\gamma_{\\rm ad}$. Here, $\\Delta \\gamma$ is the difference between the adiabatically obtained Berry phase  $\\gamma_{\\rm ad}$ via Eq. (\\ref{gamma_static}), and the value obtained dynamically using Eq. (\\ref{gamma_dyn}). The blue curve shows the quantum state error, defined as $1-|\\langle {\\rm initial \\ state} | {\\rm final \\ state} \\rangle |$, that is, the amount of norm which becomes excited during the evolution. The state error is low ($<10^{-3}$) for \ntime steps $\\Delta t > 0.5$, while a similar phase error can only be achieved by significantly longer time steps $\\Delta t > 15$.\n}\n\\end{figure}\n \n\n\n\\section{Summary}\nWe have proposed several optical tools which can provide microscopic control over excitations in integer or fractional quantum Hall systems. In Sec. \\ref{pump}, we have \ndeveloped ideas for a quasiparticle pump, based on interactions between electrons and photons with orbital angular momentum. For graphene, empty and filled Landau levels can optically be coupled as discussed in Sec. \\ref{graphene}. For GaAs, a spin-flip Raman coupling  is possible, see Sec. \\ref{GaAs}. We have applied a STIRAP scheme on this many-body scenario, which allows avoiding decoherence.  Our techniques to create individual quasiparticles are robust against disorder and can give rise to novel ways of measuring fractional charge and statistics. A possible application within a Corbino disk geometry is given in Sec. \\ref{sec:corbino}. \n\nIn Sec. \\ref{pin}, we have discussed different strategies for optically trapping a quasihole. We have studied the role played by the potential width for the stability of the trap. Even shallow potentials are found to support quasihole states, but the gap above the quasihole state is largest when the width is on the order of the magnetic length. A simple way of achieving an optical potential is based on the AC Stark shift, but the potential width, limited by the wavelength, exceeds the ideal length scale. Improvements are possible using a three-level coupling scheme, where for the prize of a weaker trap the potential width can be brought below the diffraction limit. We have also simulated the system dynamics in a moving potential, showing that such a process imprints a Berry phase in the electronic wave function according to the fractional charge of the quasihole. The optical potentials thus might become useful for braiding quasiparticles, which is the operation on which future topological quantum computers might be based on.\n\nIn summary, our manuscript advances quantum-optical tools for engineering and manipulating quantum Hall systems. In previous work, optical driving near a Landau level resonance has been suggested as a tool for engineering novel quantum Hall states \\cite{areg}. Here, we have applied similar ideas in order to control bulk excitations of a quantum Hall system. Other interesting aspects of optically coupled Landau levels regard quantized dissipation rates \\cite{tran18}, or optically induced electron localization \\cite{arikawa17}.\n\n\n\\acknowledgments\nWe acknowledge fruitful discussions with Wade de Gottardi, Ze-Pei Cian, and Hwanmun Kim. This research was financially supported by the NSF through the PFC@JQI, AFOSR-MURI  FA95501610323, EFRI-1542863,  CNS-0958379,\nCNS-0855217,  ACI-1126113  and  the  City  University  of\nNew  York  High  Performance  Computing  Center  at  the\nCollege of Staten Island, Sloan Fellowship, YIP-ONR.\nP.B. acknowledges support by AFOSR, NSF PFC@JQI, NSF QIS, ARL CDQI, ARO MURI, and ARO.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn special relativity, the Lorentz transforms supersede their\nclassical equivalent, the Galilean transforms~\\citep{goldstein1980}.\nLorentz transforms operate on four-vectors such as the four-velocity\nor four-potential and are usually operationalised as multiplication by\na $4\\times 4$ matrix.  A Lorentz transform takes the components of an\narbitrary four-vector as observed in one coordinate system and returns\nthe components observed in another system which is moving at constant\nvelocity with respect to the first.\n\nThere are a few existing software tools for working with Lorentz\ntransforms, mostly developed in an educational context.  Early work\nwould include that of \\citet{horwitz1992} who describe {\\tt relLab}, a\nsystem for building a range of {\\em gendanken} experiments in an\ninteractive graphical environment.  The author asserts that it runs on\n``any Macintosh computer with one megabyte of RAM or more'' but it is\nnot clear whether the software is still available.  More modern\ncontributions would include the {\\tt OpenRelativity}\ntoolkit~\\citep{sherin2016} which simulates the effects of special\nrelativity in the {\\tt Unity} game engine.\n\nThe {\\tt lorentz} package~\\cite{hankin2022_lorentz_package} is written\nin the R programming language~\\cite{rcore2022}, providing {\\tt\nR}-centric functionality for the physics of special relativity.  It\ndeals with formal Lorentz boosts, converts between three-velocities\nand four-velocities, and provides computational support for the\ngyrogroup structure of relativistic three-velocity addition.  I\nleverage the power of the R programming language and the package\nitself to search for a gyrodistributive law in appendix A.\n\n\n\\section{The R programming language}\n\nThe R programming language~\\cite{rcore2022} has an emphasis on\nstatistics and data analysis~\\cite{chambers2008}.  However, R is a\ngeneral-purpose tool and is increasingly being used in the physical\nsciences~\\cite{mullen2022}.  For example, functionality for working\nwith general relativity is given\nin~\\cite{hankin2020,hankin2021string}.  R is interpreted, not\ncompiled, giving instant feedback on commands and allowing rapid\ndevelopment.  In this document, the typical cycle is presented as\nfollows:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> 2+2\n\\end{Sinput}\n\\begin{Soutput}\n[1] 4\n\\end{Soutput}\n\\end{Schunk}\n\nWe see the user's query of {\\tt 2+2} is accepted, and the result, {\\tt\n4}, given as a response.  The ``{\\tt [1]}'' indicates that the\nreturned value is a vector of length 1.\n\n\\subsection{The {\\tt lorentz} package: an overview}\n\nR's capabilities are extended through user-created ``packages'', which\noffer specialist additional functionality to base R.  Packages may be\ninstalled independently and cover a wide range of computational\nfacilities.  R packages contain code, data, and documentation in a\nstandardised format that can be installed by users of R, typically via\na centralised software repository such as CRAN.  R packages must\nconform to a strict specification and pass extensive quality control\nchecks which ensure the usability and long-term stability of packages\nfor end users.\n\n\\subsection{Installation of the {\\tt lorentz} package}\n\nThe R system may be downloaded from \\url{https:\/\/cran.r-project.org\/};\nmany users prefer the Rstudio IDE, available from\n\\url{https:\/\/posit.co\/}.  Once R is installed, the {\\tt lorentz}\npackage is easily loaded.  Type:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> install.packages(\"lorentz\")\n\\end{Sinput}\n\\end{Schunk}\n\nat the command line, and this will download the package from CRAN.  To\ninstall the package, use\n\n\\begin{Schunk}\n\\begin{Sinput}\n> library(\"lorentz\")\n\\end{Sinput}\n\\end{Schunk}\n\nand this will make the package functions available to the R session.\n\n\\section{Lorentz transforms: active and passive}\n\nPassive transforms are the usual type of transforms taught and used in\nrelativity.  However, sometimes active transforms are needed and it is\neasy to confuse the two.  Here I will discuss passive and then active\ntransforms, and illustrate both in a computational context.\n\n\\subsection*{Passive transforms}\n\n\\newcommand{\\vvec}[2]{\\begin{pmatrix}#1 \\\\ #2\\end{pmatrix}}\n\\newcommand{\\twomat}[4]{\\begin{pmatrix} #1 & #2 \\\\ #3 & #4\\end{pmatrix}}\n\nConsider the following canonical Lorentz transform in which we have\nmotion in the $x$-direction at speed $v>0$; the motion is from left to\nright.  We consider only the first two components of four-vectors, the\n$y$- and $z$- components being trivial.  A typical physical\ninterpretation is that I am at rest, and my friend is in his spaceship\nmoving at speed $v$ past me; and we are wondering what vectors which I\nmeasure in my own rest frame look like to him.  The (passive) Lorentz\ntransform is:\n\n\\begin{equation*}\n\\twomat{\\gamma}{-\\gamma v}{-\\gamma v}{\\gamma}\n\\end{equation*}\n\nAnd the canonical example of that would be:\n\n\\begin{equation*}\n\\twomat{\\gamma}{-\\gamma v}{-\\gamma v}{\\gamma}\\vvec{1}{0}=\\vvec{\\gamma}{-\\gamma v}\n\\end{equation*}\n\nwhere the vectors are four velocities (recall that $\\vvec{1}{0}$ is\nthe four-velocity of an object at rest).  Operationally, I measure the\nfour-velocity of an object to be $\\vvec{1}{0}$, and he measures the\nsame object as having a four-velocity of $\\vvec{\\gamma}{-\\gamma v}$.\nSo I see the object at rest, and he sees it as moving at speed $-v$;\nthat is, he sees it moving to the left (it moves to the left because\nhe is moving to the right relative to me).  The {\\tt lorentz}\npackage~\\cite{hankin2022_lorentz_package} makes computations easy.\nSuppose $v=0.6c$ in the $x$-direction.\n\n\\begin{Schunk}\n\\begin{Sinput}\n>                           # NB: speed of light = 1 by default\n> u <- as.3vel(c(0.6,0,0))  # coerce to a three-velocity\n> u\n\\end{Sinput}\n\\begin{Soutput}\nA vector of three-velocities (speed of light = 1)\n       x y z\n[1,] 0.6 0 0\n\\end{Soutput}\n\\begin{Sinput}\n> as.4vel(u)                # four-velocity is better for calculations\n\\end{Sinput}\n\\begin{Soutput}\nA vector of four-velocities (speed of light = 1)\n        t    x y z\n[1,] 1.25 0.75 0 0\n\\end{Soutput}\n\\begin{Sinput}\n> (B <- boost(u))           # transformation matrix\n\\end{Sinput}\n\\begin{Soutput}\n      t     x y z\nt  1.25 -0.75 0 0\nx -0.75  1.25 0 0\ny  0.00  0.00 1 0\nz  0.00  0.00 0 1\n\\end{Soutput}\n\\end{Schunk}\n\n(note that element $[1,2]$ of the boost matrix $B$ is negative as we\nhave a passive transform).  Then a four-velocity of $(1,0,0,0)^T$\nwould appear in the moving frame as\n\n\\begin{Schunk}\n\\begin{Sinput}\n> B\n\\end{Sinput}\n\\begin{Soutput}\n   [,1]\nt  1.25\nx -0.75\ny  0.00\nz  0.00\n\\end{Soutput}\n\\end{Schunk}\n\nThis corresponds to a speed of $-0.75\/1.25=-0.6$.  Observe that it is\npossible to transform an arbitrary four-vector:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> B\n\\end{Sinput}\n\\begin{Soutput}\n  [,1]\nt  0.5\nx  4.5\ny -8.0\nz  9.0\n\\end{Soutput}\n\\end{Schunk}\n\n\n\\subsubsection*{Null vectors: light}\n\nLet's try it with light (see section~\\ref{photonsection} for more\ndetails on photons).  Recall that we describe a photon in terms of its\nfour momentum, not four-velocity, which is undefined for a photon.\nSpecifically, we {\\em define} the four-momentum of a photon to be\n\n\\begin{equation*}\n  \\left(\n  \\begin{array}{c}\n    E\/c\\\\Ev_x\/c^2\\\\Ev_y\/c^2\\\\Ev_z\/c^2\n  \\end{array}\n  \\right)\n  \\end{equation*}\n\nSo if we consider unit energy and keep $c=1$ we get $p=\\vvec{1}{1}$\nin our one-dimensional world (for a rightward-moving photon) and the\nLorentz transform is then\n\n\\begin{equation*}\n  \\twomat{\\gamma}{-\\gamma v}{-\\gamma v}{\\gamma}\\vvec{1}{1}=\\vvec{\\gamma-\\gamma v}{\\gamma-\\gamma v}\n  \\end{equation*}\n\n\nSo, in the language used above, I see a photon with unit energy, and\nmy friend sees the photon with energy\n$\\gamma(1-v)=\\sqrt{\\frac{1-v}{1+v}}<1$, provided that $v>0$: the\nphoton has less energy in his frame than mine because of Doppler\nredshifting.  It's worth doing the same analysis with a\nleftward-moving photon:\n\n\\begin{equation*}\n  \\twomat{\\gamma}{-\\gamma v}{-\\gamma v}{\\gamma}\\vvec{1}{-1}=\\vvec{\\gamma(1+v)}{-\\gamma(1+v)}\n  \\end{equation*}\n\nHere the photon has more energy for him than me because of blue\nshifting: he is moving to the right and encounters a photon moving to\nthe left.  The R idiom would be\n\n\\begin{Schunk}\n\\begin{Sinput}\n> B\n\\end{Sinput}\n\\begin{Soutput}\n  [,1]\nt  0.5\nx  0.5\ny  0.0\nz  0.0\n\\end{Soutput}\n\\begin{Sinput}\n> B\n\\end{Sinput}\n\\begin{Soutput}\n  [,1]\nt    2\nx   -2\ny    0\nz    0\n\\end{Soutput}\n\\end{Schunk}\n\nfor the left- and right- moving photons respectively. \n\nThe above analysis uses {\\em passive} transforms: there is a single\nphysical reality, and we describe that one physical reality using two\ndifferent coordinate systems.  One of the coordinate systems uses a\nset of axes that are {\\em boosted} relative to the axes of the other.\n\nThis is why it makes sense to use prime notation as in\n$x\\longrightarrow x'$ and $t\\longrightarrow t'$ for a passive Lorentz\ntransform: the prime denotes measurements made using coordinates\nthat are defined with respect to the boosted system, and we see\nnotation like\n\n\\begin{equation*}\n\\vvec{t'}{x'}=\\twomat{\\gamma}{-\\gamma v}{-\\gamma v}{\\gamma}\\vvec{t}{x}\n\\end{equation*}  \n\nThese are the first two elements of a displacement four-vector.  It is\nthe same four-vector but viewed in two different reference frames.\n\n\\subsection*{Active transforms}\n\nIn the passive view, there is a single physical reality, and we are\njust describing that one physical reality using two different\ncoordinate systems.  Now we will consider active transforms: there are\ntwo physical realities, but one is boosted with respect to another.  \n\nSuppose me and my friend have zero relative velocity, but my friend is\nin a spaceship and I am outside it, in free space, at rest.  He\nconstructs a four-vector in his spaceship; for example, he could fire\nbullets out of a gun which is fixed in the spaceship, and then\ncalculate their four-velocity as it appears to him in his\nspaceship-centric coordinate system.  We both agree on this\nfour-velocity as our reference frames are identical: we have no\nrelative velocity.\n\nNow his spaceship acquires a constant velocity, leaving me stationary.\nMy friend continues to fire bullets out of his gun and sees that their\nfour-velocity, as viewed in his spaceship coordinates, is the same as\nwhen we were together.\n\nNow he wonders what the four-velocity of the bullets is in my\nreference frame.  This is an {\\em active} transform: we have two\ndistinct physical realities, one in the spaceship when it was at rest\nwith respect to me, and one in the spaceship when moving.  And both\nthese realities, by construction, look the same to my friend in the\nspaceship.\n\nSuppose, for example, he sees the bullets at rest in his spaceship;\nthey have a four-velocity of $\\vvec{1}{0}$, and my friend says to\nhimself: ``I see bullets with a four velocity of $\\vvec{1}{0}$, and I\nknow what that means.  The bullets are at rest.  What are the bullets'\nfour velocities in Robin's reference frame?\".  This is an {\\em active}\ntransform:\n\n\\begin{equation*}\n\\twomat{\\gamma}{\\gamma v}{\\gamma v}{\\gamma}\\vvec{1}{0}=\\vvec{\\gamma}{\\gamma v}\n\\end{equation*}\n\n(we again suppose that the spaceship moves at speed $v>0$ from left to\nright).  So he sees a four velocity of $\\vvec{1}{0}$ and I see\n$\\vvec{\\gamma}{\\gamma v}$, that is, with a positive speed: the bullets\nmove from left to right (with the spaceship).  The R idiom would be:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> (B <- boost(as.3vel(c(0.8,0,0))))  # 0.8c left to right\n\\end{Sinput}\n\\begin{Soutput}\n          t         x y z\nt  1.666667 -1.333333 0 0\nx -1.333333  1.666667 0 0\ny  0.000000  0.000000 1 0\nz  0.000000  0.000000 0 1\n\\end{Soutput}\n\\begin{Sinput}\n> solve(B)\n\\end{Sinput}\n\\begin{Soutput}\n      [,1]\nt 1.666667\nx 1.333333\ny 0.000000\nz 0.000000\n\\end{Soutput}\n\\end{Schunk}\n\n\\section{Successive Lorentz transforms}\n\nCoordinate transformation is effected by standard matrix\n multiplication; thus composition of two Lorentz transforms is also\n ordinary matrix multiplication:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> u <- as.3vel(c(0.3,-0.4,+0.8))\n> v <- as.3vel(c(0.4,+0.2,-0.1))\n> L <- boost(u)\n> L\n\\end{Sinput}\n\\begin{Soutput}\n          t          x          y          z\nt  3.256577 -2.2327055  0.5419596 -2.0800479\nx -1.437147  1.6996791 -0.0237489  0.4194255\ny  1.091131 -0.7581795  1.1190282 -0.6029155\nz -2.519789  1.5878378 -0.2023170  2.1879612\n\\end{Soutput}\n\\end{Schunk}\n\nBut observe that the resulting transform is not a pure boost, as the\nspatial components are not symmetrical.  We may decompose the matrix\nproduct $L$ into a pure translation composed with an orthogonal\nmatrix, which represents a coordinate rotation.  The R idiom is\n{\\tt pureboost()} for the pure boost component, and {\\tt orthog()}\nfor the rotation:\n             \n\\begin{Schunk}\n\\begin{Sinput}\n> (P <- pureboost(L))  # pure boost\n\\end{Sinput}\n\\begin{Soutput}\n           t          x          y          z\nt  3.2565770 -2.2327055  0.5419596 -2.0800479\nx -2.2327055  2.1711227 -0.2842745  1.0910491\ny  0.5419596 -0.2842745  1.0690039 -0.2648377\nz -2.0800479  1.0910491 -0.2648377  2.0164504\n\\end{Soutput}\n\\begin{Sinput}\n> P - t(P)   # check for symmetry\n\\end{Sinput}\n\\begin{Soutput}\n  t x y z\nt 0 0 0 0\nx 0 0 0 0\ny 0 0 0 0\nz 0 0 0 0\n\\end{Soutput}\n\\end{Schunk}\n\n\nNow we compute the rotation:\n\\begin{Schunk}\n\\begin{Sinput}\n> (U <- orthog(L))                  # rotation matrix\n\\end{Sinput}\n\\begin{Soutput}\n              t             x            y             z\nt  1.000000e+00 -1.332268e-14 3.219647e-15 -1.509903e-14\nx -1.054712e-14  9.458514e-01 1.592328e-01 -2.828604e-01\ny  8.659740e-15 -1.858476e-01 9.801022e-01 -6.971587e-02\nz -1.953993e-14  2.661311e-01 1.185098e-01  9.566241e-01\n\\end{Soutput}\n\\begin{Sinput}\n> U[2:4,2:4]                        # inspect the spatial components\n\\end{Sinput}\n\\begin{Soutput}\n           x         y           z\nx  0.9458514 0.1592328 -0.28286043\ny -0.1858476 0.9801022 -0.06971587\nz  0.2661311 0.1185098  0.95662410\n\\end{Soutput}\n\\begin{Sinput}\n> round(crossprod(U) - diag(4),10)  # check for orthogonality\n\\end{Sinput}\n\\begin{Soutput}\n  t x y z\nt 0 0 0 0\nx 0 0 0 0\ny 0 0 0 0\nz 0 0 0 0\n\\end{Soutput}\n\\begin{Sinput}\n> ## zero to within numerical uncertainty\n\\end{Sinput}\n\\end{Schunk}\n\n\\section[Units in which c is not 1]{Units in which ${\\mathbf c\\neq 1}$}  \n\nThe preceding material used units in which $c=1$.  Here I show how the\npackage deals with units such as SI in which $c=299792458\\neq 1$.  For\nobvious reasons we cannot have a function called {\\tt c()} so the\npackage gets and sets the speed of light with function {\\tt sol()}:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> sol(299792458)\n\\end{Sinput}\n\\begin{Soutput}\n[1] 299792458\n\\end{Soutput}\n\\begin{Sinput}\n> sol()\n\\end{Sinput}\n\\begin{Soutput}\n[1] 299792458\n\\end{Soutput}\n\\end{Schunk}\n\nThe speed of light is now~$299792458$ until re-set by {\\tt sol()} (an\nempty argument queries the speed of light).  We now consider speeds\nwhich are fast by terrestrial standards but involve only a small\nrelativistic correction to the Galilean result:\n                                                \n\\begin{Schunk}\n\\begin{Sinput}\n> u <- as.3vel(c(100,200,300))\n> as.4vel(u)\n\\end{Sinput}\n\\begin{Soutput}\nA vector of four-velocities (speed of light = 299792458)\n     t   x   y   z\n[1,] 1 100 200 300\n\\end{Soutput}\n\\end{Schunk}\n\nThe gamma correction term $\\gamma$ is only very slightly larger\nthan~$1$ and indeed R's default print method suppresses the\ndifference:\n                    \n\\begin{Schunk}\n\\begin{Sinput}\n> gam(u)\n\\end{Sinput}\n\\begin{Soutput}\n[1] 1\n\\end{Soutput}\n\\end{Schunk}\n\nHowever, we can display more significant figures by subtracting one:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> gam(u)-1\n\\end{Sinput}\n\\begin{Soutput}\n[1] 7.789325e-13\n\\end{Soutput}\n\\end{Schunk}\n\nor alternatively we can use the {\\tt gamm1()} function which\ncalculates $\\gamma-1$ more accurately for speeds $\\ll c$:\n                                                  \n\\begin{Schunk}\n\\begin{Sinput}\n> gamm1(u)\n\\end{Sinput}\n\\begin{Soutput}\n[1] 7.78855e-13\n\\end{Soutput}\n\\end{Schunk}\n\nThe Lorentz boost is again calculated by the {\\tt boost()} function:\n              \n\\begin{Schunk}\n\\begin{Sinput}\n> boost(u)\n\\end{Sinput}\n\\begin{Soutput}\n     t             x            y             z\nt    1 -1.112650e-15 -2.22530e-15 -3.337950e-15\nx -100  1.000000e+00  1.11265e-13  1.668975e-13\ny -200  1.112650e-13  1.00000e+00  3.337950e-13\nz -300  1.668975e-13  3.33795e-13  1.000000e+00\n\\end{Soutput}\n\\end{Schunk}\n\nThe boost matrix is not symmetrical, even though it is a pure boost,\nbecause $c\\neq 1$.  \n\n\nNote how the transform is essentially the Galilean\nresult, which is discussed below.\n\n\\subsection{Changing units}\n\nOften we have a four-vector in SI units and wish to express this in natural units.\n\n\\begin{Schunk}\n\\begin{Sinput}\n> sol(299792458)\n\\end{Sinput}\n\\begin{Soutput}\n[1] 299792458\n\\end{Soutput}\n\\begin{Sinput}\n> disp  <- c(1,1,0,0)\n\\end{Sinput}\n\\end{Schunk}\n\nIf we interpret {\\tt disp} as a four-displacement, it corresponds to\nmoving 1 metre along the x-axis and waiting for one second.  To\nconvert this to natural units we multiply by the passive\ntransformation matrix given by {\\tt ptm()}:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> ptm(to_natural=TRUE)\n\\end{Sinput}\n\\begin{Soutput}\n       [,1]\nt 299792458\nx         1\ny         0\nz         0\n\\end{Soutput}\n\\end{Schunk}\n\nIn the above, see how the same vector is expressed in natural units in\nwhich the speed of light is equal to 1: the unit of time is about\n$3\\times 10^{-9}$ seconds and the unit of distance remains the metre.\nAlternatively, we might decide to keep the unit of time equal to one\nsecond, and use a unit of distance equal to 299792458 metres which\nagain ensures that~$c=1$:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> ptm(to_natural=TRUE,change_time=FALSE)\n\\end{Sinput}\n\\begin{Soutput}\n          [,1]\nt 1.000000e+00\nx 3.335641e-09\ny 0.000000e+00\nz 0.000000e+00\n\\end{Soutput}\n\\end{Schunk}\n\nAs a further check, we can take two boost matrices corresponding to\nthe same coordinate transformation but expressed using different units\nof length and verify that their orthogonal component agrees:\n\n\n\\begin{Schunk}\n\\begin{Sinput}\n> sol(1)\n\\end{Sinput}\n\\begin{Soutput}\n[1] 1\n\\end{Soutput}\n\\begin{Sinput}\n> B1 <- boost((2:4)\/10)\n> orthog(B1)[2:4,2:4]\n\\end{Sinput}\n\\begin{Soutput}\n           x           y          z\nx  0.9832336  0.09752166 0.15408208\ny -0.1020390  0.99454439 0.02166761\nz -0.1511284 -0.03702671 0.98782044\n\\end{Soutput}\n\\end{Schunk}\n\nNow we create {\\tt B2} which is the same physical object but using a\nlength scale of one-tenth of {\\tt B2} (which requires that we\nmultiply the speed of light by a factor of 10):\n\n\\begin{Schunk}\n\\begin{Sinput}\n> sol(10)\n\\end{Sinput}\n\\begin{Soutput}\n[1] 10\n\\end{Soutput}\n\\begin{Sinput}\n> B2 <- boost(2:4)\n> orthog(B2)[2:4,2:4]\n\\end{Sinput}\n\\begin{Soutput}\n           x           y          z\nx  0.9832336  0.09752166 0.15408208\ny -0.1020390  0.99454439 0.02166761\nz -0.1511284 -0.03702671 0.98782044\n\\end{Soutput}\n\\end{Schunk}\n\nso the two matrices agree, as expected.\n\n\n\\section{Infinite speed of light}\n\nIn the previous section considered speeds that were small compared\nwith the speed of light and here we will consider the classical limit\nof infinite $c$:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> sol(Inf)\n\\end{Sinput}\n\\begin{Soutput}\n[1] Inf\n\\end{Soutput}\n\\end{Schunk}\n\nThen the familiar parallelogram law operates:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> u <- as.3vel(1:3)\n> v <- as.3vel(c(-6,8,3))\n> u+v\n\\end{Sinput}\n\\begin{Soutput}\nA vector of three-velocities (speed of light = Inf)\n      x  y z\n[1,] -5 10 6\n\\end{Soutput}\n\\begin{Sinput}\n> v+u\n\\end{Sinput}\n\\begin{Soutput}\nA vector of three-velocities (speed of light = Inf)\n      x  y z\n[1,] -5 10 6\n\\end{Soutput}\n\\end{Schunk}\n\nAbove we see that composition of velocities is commutative, unlike the\nrelativistic case.  The boost matrix is instructive:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> boost(u)\n\\end{Sinput}\n\\begin{Soutput}\n   t x y z\nt  1 0 0 0\nx -1 1 0 0\ny -2 0 1 0\nz -3 0 0 1\n\\end{Soutput}\n\\begin{Sinput}\n> boost(u+v)\n\\end{Sinput}\n\\begin{Soutput}\n    t x y z\nt   1 0 0 0\nx   5 1 0 0\ny -10 0 1 0\nz  -6 0 0 1\n\\end{Soutput}\n\\begin{Sinput}\n> boost(u)\n\\end{Sinput}\n\\begin{Soutput}\n    t x y z\nt   1 0 0 0\nx   5 1 0 0\ny -10 0 1 0\nz  -6 0 0 1\n\\end{Soutput}\n\\end{Schunk}\n\nAbove, see how the boost matrix for the composed velocity of $u+v$\ndoes not have any rotational component, unlike the relativistic case\n[recall that {\\tt boost()} gives a {\\em passive} transform, which is\nwhy the sign of the numbers in the first column is changed].  With an\ninfinite speed of light, even ``large'' speeds have zero relativistic\ncorrection:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> gamm1(1e100)\n\\end{Sinput}\n\\begin{Soutput}\n[1] 0\n\\end{Soutput}\n\\end{Schunk}\n\nFunction {\\tt rboost()} returns a random Lorentz transform matrix,\nwhich is in general a combination of a pure Lorentz boost and an\northogonal rotation.  With an infinite speed of light, it requires a\nspeed:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> set.seed(0)\n> options(digits=3)\n> (B <- rboost(1))  # random boost, speed 1\n\\end{Sinput}\n\\begin{Soutput}\n          t      x       y      z\n[1,]  1.000  0.000  0.0000  0.000\n[2,] -0.411  0.213 -0.9402  0.266\n[3,] -0.279 -0.917 -0.0989  0.385\n[4,]  0.868 -0.336 -0.3260 -0.884\n\\end{Soutput}\n\\end{Schunk}\n\nWe can decompose {\\tt B} into a pure boost and an orthogonal\ntransformation:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> orthog(B)\n\\end{Sinput}\n\\begin{Soutput}\n     [,1]   [,2]    [,3]   [,4]\n[1,]    1  0.000  0.0000  0.000\n[2,]    0  0.213 -0.9402  0.266\n[3,]    0 -0.917 -0.0989  0.385\n[4,]    0 -0.336 -0.3260 -0.884\n\\end{Soutput}\n\\begin{Sinput}\n> pureboost(B)\n\\end{Sinput}\n\\begin{Soutput}\n          t x y z\n[1,]  1.000 0 0 0\n[2,] -0.123 1 0 0\n[3,]  0.131 0 1 0\n[4,] -0.984 0 0 1\n\\end{Soutput}\n\\end{Schunk}\n\n\nBoost matrices can be applied to any four-vector.  Here I show how\npure spatial displacements transform with an infinite light speed.\n\n\\begin{Schunk}\n\\begin{Sinput}\n> (u <- as.3vel(c(10,0,0))) # velocity of 10, parallel to x axis\n\\end{Sinput}\n\\begin{Soutput}\nA vector of three-velocities (speed of light = Inf)\n      x y z\n[1,] 10 0 0\n\\end{Soutput}\n\\begin{Sinput}\n> (B <- boost(u))\n\\end{Sinput}\n\\begin{Soutput}\n    t x y z\nt   1 0 0 0\nx -10 1 0 0\ny   0 0 1 0\nz   0 0 0 1\n\\end{Soutput}\n\\begin{Sinput}\n> d <- c(0,1,0,0) # displacement of distance one, parallel to the x-axis\n> B\n\\end{Sinput}\n\\begin{Soutput}\n  [,1]\nt    0\nx    1\ny    0\nz    0\n\\end{Soutput}\n\\end{Schunk}\n\nAbove we see that a spatial displacement is the same for both\nobservers.  We can similarly apply a boost to a temporal displacement:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> d <- c(1,0,0,0) # displacement of one unit of time, no spatial component\n> B\n\\end{Sinput}\n\\begin{Soutput}\n  [,1]\nt    1\nx  -10\ny    0\nz    0\n\\end{Soutput}\n\\end{Schunk}\n\nAbove we see the result expected from classical mechanics.\n\n\n\n\\section{Vectorization}\n\nHere I discuss vectorized operations (to avoid confusion between boost\nmatrices and their transposes we will use $c=10$).  The issue is\ndifficult because a Lorentz boost is conceptually a matrix product of\na $4\\times 4$ matrix with vector with four elements:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> sol(10)\n\\end{Sinput}\n\\begin{Soutput}\n[1] 10\n\\end{Soutput}\n\\begin{Sinput}\n> u <- as.3vel(c(5,-6,4))\n> (U <- as.4vel(u))\n\\end{Sinput}\n\\begin{Soutput}\nA vector of four-velocities (speed of light = 10)\n        t    x     y    z\n[1,] 2.09 10.4 -12.5 8.34\n\\end{Soutput}\n\\begin{Sinput}\n> B <- boost(U)\n> B\n\\end{Sinput}\n\\begin{Soutput}\n       [,1]\nt  1.00e+00\nx -1.33e-15\ny  4.44e-16\nz  1.78e-15\n\\end{Soutput}\n\\end{Schunk}\n\n(note that the result is the four-velocity of an object at rest, as\nexpected, for we use passive transforms by default).  However, things\nare different if we wish to consider many four-vectors in one R\nobject.  A vector ${\\mathbf V}$ of four-velocities is a matrix: each\n{\\em row} of ${\\mathbf V}$ is a four-velocity.  In the package we\nrepresent this with objects of class {\\tt 4vel}.  Because a vector is\ntreated (almost) as a one-column matrix in R, and the four-velocities\nare rows, we need to take a transpose in some sense.\n\n\\begin{Schunk}\n\\begin{Sinput}\n> u <- 1:7  # speed in the x-direction [c=10]\n> jj <- cbind(gam(u),gam(u)*u,0,0)\n> (U <- as.4vel(jj))\n\\end{Sinput}\n\\begin{Soutput}\nA vector of four-velocities (speed of light = 10)\n        t    x y z\n[1,] 1.01 1.01 0 0\n[2,] 1.02 2.04 0 0\n[3,] 1.05 3.14 0 0\n[4,] 1.09 4.36 0 0\n[5,] 1.15 5.77 0 0\n[6,] 1.25 7.50 0 0\n[7,] 1.40 9.80 0 0\n\\end{Soutput}\n\\end{Schunk}\n\nNow a boost, also in the x-direction:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> (B <- boost(as.3vel(c(6,0,0))))  # 6\n\\end{Sinput}\n\\begin{Soutput}\n      t      x y z\nt  1.25 -0.075 0 0\nx -7.50  1.250 0 0\ny  0.00  0.000 1 0\nz  0.00  0.000 0 1\n\\end{Soutput}\n\\end{Schunk}\n\nNote the asymmetry of $B$, in this case reflecting the speed of light\nbeing 10 (but note that boost matrices are not always symmetrical,\neven if $c=1$).\n\nTo effect a {\\em passive} boost we need to multiply each row of $U$ by\nthe transpose of the boost matrix $B$:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> U\n\\end{Sinput}\n\\begin{Soutput}\n        t     x y z\n[1,] 1.18 -6.28 0 0\n[2,] 1.12 -5.10 0 0\n[3,] 1.07 -3.93 0 0\n[4,] 1.04 -2.73 0 0\n[5,] 1.01 -1.44 0 0\n[6,] 1.00  0.00 0 0\n[7,] 1.02  1.75 0 0\n\\end{Soutput}\n\\end{Schunk}\n\nwe can verify that the above is at least plausible:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> is.consistent.4vel(U\n\\end{Sinput}\n\\begin{Soutput}\n[1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE\n\\end{Soutput}\n\\end{Schunk}\n\nthe above shows that the four velocities $U$, as observed by an\nobserver corresponding to boost $B$, satisfies $U^iU_i=-c^2$.  Anyway,\nin this context we really ought to use {\\tt tcrossprod()}:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> tcrossprod(U,B)\n\\end{Sinput}\n\\begin{Soutput}\n        t     x y z\n[1,] 1.18 -6.28 0 0\n[2,] 1.12 -5.10 0 0\n[3,] 1.07 -3.93 0 0\n[4,] 1.04 -2.73 0 0\n[5,] 1.01 -1.44 0 0\n[6,] 1.00  0.00 0 0\n[7,] 1.02  1.75 0 0\n\\end{Soutput}\n\\end{Schunk}\n\nwhich would be preferable (because this idiom does not require one to\ntake a transpose) although the speed increase is unlikely to matter\nmuch because $B$ is only $4\\times 4$.\n\nThe above transforms were passive: we have some four-vectors measured\nin my rest frame, and we want to see what these are four-vectors as\nmeasured by my friend, who is moving in the positive x direction at\n60\\% of the speed of light (remember that $c=10$).  See how the\nx-component of the transformed four-velocity is negative, because in\nmy friend's rest frame, the four velocities are pointing backwards.\n\nTo effect an {\\em active} transform we need to take the matrix inverse\nof $B$:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> solve(B)\n\\end{Sinput}\n\\begin{Soutput}\n     t     x y z\nt 1.25 0.075 0 0\nx 7.50 1.250 0 0\ny 0.00 0.000 1 0\nz 0.00 0.000 0 1\n\\end{Soutput}\n\\end{Schunk}\n\nand then\n\n\\begin{Schunk}\n\\begin{Sinput}\n> tcrossprod(U,solve(B))\n\\end{Sinput}\n\\begin{Soutput}\n        t     x y z\n[1,] 1.33  8.79 0 0\n[2,] 1.43 10.21 0 0\n[3,] 1.55 11.79 0 0\n[4,] 1.69 13.64 0 0\n[5,] 1.88 15.88 0 0\n[6,] 2.13 18.75 0 0\n[7,] 2.49 22.75 0 0\n\\end{Soutput}\n\\end{Schunk}\n\nIn the above, note how the positive x-component of the four-velocity\nis increased because we have actively boosted it.  We had better check\nthe result for consistency:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> is.consistent.4vel(tcrossprod(U,solve(B)))\n\\end{Sinput}\n\\begin{Soutput}\n[1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE\n\\end{Soutput}\n\\end{Schunk}\n\n\\section{Multiple boosts}\n\n\nIf we are considering multiple boosts, it is important to put them in\nthe correct order.  First we will do some passive boosts.\n\n\\begin{Schunk}\n\\begin{Sinput}\n> sol(100)\n\\end{Sinput}\n\\begin{Soutput}\n[1] 100\n\\end{Soutput}\n\\begin{Sinput}\n> B1 <- boost(r3vel(1))\n> B2 <- boost(r3vel(1))\n> (U <- r4vel(5))\n\\end{Sinput}\n\\begin{Soutput}\nA vector of four-velocities (speed of light = 100)\n        t      x     y      z\n[1,] 1.99 -162.4  46.3  34.32\n[2,] 2.58  149.0 185.0  -3.07\n[3,] 1.75 -110.4 -18.4 -90.19\n[4,] 1.70   55.1 -88.3  90.15\n[5,] 2.62 -205.6  98.2  81.53\n\\end{Soutput}\n\\end{Schunk}\n\nSuccessive boosts are effected by matrix multiplication; there are at\nleast four equivalent R constructions:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> U\n\\end{Sinput}\n\\begin{Soutput}\n         t    x     y    z\n[1,] 11.09 -601  -844 -383\n[2,]  3.00  -70  -166 -218\n[3,] 13.98 -639 -1087 -594\n[4,]  7.75 -239  -697 -220\n[5,] 12.23 -706  -911 -394\n\\end{Soutput}\n\\begin{Sinput}\n> U\n\\end{Sinput}\n\\begin{Soutput}\n         t    x     y    z\n[1,] 11.09 -601  -844 -383\n[2,]  3.00  -70  -166 -218\n[3,] 13.98 -639 -1087 -594\n[4,]  7.75 -239  -697 -220\n[5,] 12.23 -706  -911 -394\n\\end{Soutput}\n\\begin{Sinput}\n> tcrossprod(U, B2\n\\end{Sinput}\n\\begin{Soutput}\n         t    x     y    z\n[1,] 11.09 -601  -844 -383\n[2,]  3.00  -70  -166 -218\n[3,] 13.98 -639 -1087 -594\n[4,]  7.75 -239  -697 -220\n[5,] 12.23 -706  -911 -394\n\\end{Soutput}\n\\begin{Sinput}\n> U\n\\end{Sinput}\n\\begin{Soutput}\n         t    x     y    z\n[1,] 11.09 -601  -844 -383\n[2,]  3.00  -70  -166 -218\n[3,] 13.98 -639 -1087 -594\n[4,]  7.75 -239  -697 -220\n[5,] 12.23 -706  -911 -394\n\\end{Soutput}\n\\end{Schunk}\n\n(in the above, note that the result is the same in each case).\n\n\\subsection*{A warning}\n\nIt is easy to misapply matrix multiplication in this context.  Note\ncarefully that the following natural idiom is {\\bf incorrect}:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> U\n\\end{Sinput}\n\\begin{Soutput}\n         t      x     y      z\n[1,]  1220 -203.2  46.3  34.32\n[2,] -1115  186.1 185.0  -3.07\n[3,]   830 -138.1 -18.4 -90.19\n[4,]  -411   68.7 -88.3  90.15\n[5,]  1545 -257.1  98.2  81.53\n\\end{Soutput}\n\\begin{Sinput}\n> ## The above idiom is incorrect.  See\n> ## https:\/\/www.youtube.com\/watch?v=m7-bMBuVmHo&t=1s\n> ## (in particular @1:08) for a technical explanation of why \n> ## this is a Very Bad Idea (tm).\n\\end{Sinput}\n\\end{Schunk}\n\nIt is not clear to me that the idiom above has any meaning at all.\n\n\\section{The stress-energy tensor}\n\nThe stress-energy tensor (sometimes the energy-momentum tensor) is a\ngeneralization and combination of the classical concepts of density,\nenergy flux, and the classical stress tensor~\\citep{schutz1985}.  It\nis a contravariant tensor of rank two, usually represented as a\nsymmetric $4\\times 4$ matrix.  The {\\tt lorentz} package includes\nfunctionality for applying Lorentz transforms to the stress energy\ntensor.\n\n\\begin{Schunk}\n\\begin{Sinput}\n> sol(1)        # revert to natural units \n\\end{Sinput}\n\\begin{Soutput}\n[1] 1\n\\end{Soutput}\n\\begin{Sinput}\n> D <- dust(1)  # Dust is the simplest nontrivial SET, with \n> D             # only one nonzero component\n\\end{Sinput}\n\\begin{Soutput}\n  t x y z\nt 1 0 0 0\nx 0 0 0 0\ny 0 0 0 0\nz 0 0 0 0\n\\end{Soutput}\n\\end{Schunk}\n\nThe stress-energy tensor is usually written with two upstairs\n(contravariant) indices, as in~$T^{\\alpha\\beta}$; it may be\ntransformed using the {\\tt transform\\_uu()} function: package:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> B <- boost(as.3vel(c(0.0,0.8,0.0)))\n> transform_uu(D,B)\n\\end{Sinput}\n\\begin{Soutput}\n      t x     y z\nt  2.78 0 -2.22 0\nx  0.00 0  0.00 0\ny -2.22 0  1.78 0\nz  0.00 0  0.00 0\n\\end{Soutput}\n\\end{Schunk}\n\nIn this reference frame, the dust is not at rest: the stress-energy\ntensor has components corresponding to nonzero pressure and momentum\ntransfer, and the $[t,t]$ component is greater, at 2.78, than its rest\nvalue of 1.  Note that the $[t,y]$ component is negative as we use\npassive transforms.  If one wants to consider the stress-energy tensor\nwith downstairs indices (here we will use a photon gas), we need to\nuse {\\tt transform\\_dd()}:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> pg <- photongas(3)\n> pg\n\\end{Sinput}\n\\begin{Soutput}\n  t x y z\nt 3 0 0 0\nx 0 1 0 0\ny 0 0 1 0\nz 0 0 0 1\n\\end{Soutput}\n\\begin{Sinput}\n> transform_uu(pg,B)\n\\end{Sinput}\n\\begin{Soutput}\n      t x     y z\nt 10.11 0 -8.89 0\nx  0.00 1  0.00 0\ny -8.89 0  8.11 0\nz  0.00 0  0.00 1\n\\end{Soutput}\n\\end{Schunk}\n\nagain we see that the $[0,0]$ component is larger than its rest value,\nand we see nonzero off-diagonal components which correspond to the\ndynamical behaviour.  As a consistency check we can verify that this\nis the same as transforming the SET with upstairs indices, using the\n{\\tt lower()} and {\\tt raise()} functions:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> raise(transform_dd(lower(pg),lower(B)))\n\\end{Sinput}\n\\begin{Soutput}\n      t x     y z\nt 10.11 0 -8.89 0\nx  0.00 1  0.00 0\ny -8.89 0  8.11 0\nz  0.00 0  0.00 1\n\\end{Soutput}\n\\begin{Sinput}\n> raise(transform_dd(lower(pg),lower(B))) - transform_uu(pg,B) #zero to numerical precision\n\\end{Sinput}\n\\begin{Soutput}\n  t x y z\nt 0 0 0 0\nx 0 0 0 0\ny 0 0 0 0\nz 0 0 0 0\n\\end{Soutput}\n\\end{Schunk}\n\nOne of the calls to {\\tt lower()} is redundant; for a photon gas,\nraising or lowering both indices does not change the components as the\nMinkowski metric is symmetric and orthogonal.\n\n\\subsection{Successive boosts}\n  \nSuccessive boots are represented as ordinary matrix multiplication.\nAgain the {\\tt magrittr} package can be used for more readable idiom.\n \n\\begin{Schunk}\n\\begin{Sinput}\n> B1 <- boost(as.3vel(c(0.5,-0.4,0.6)))\n> B2 <- boost(as.3vel(c(0.1,-0.1,0.3)))\n> pf <- perfectfluid(4,1)\n> pf\n\\end{Sinput}\n\\begin{Soutput}\n  t x y z\nt 4 0 0 0\nx 0 1 0 0\ny 0 0 1 0\nz 0 0 0 1\n\\end{Soutput}\n\\begin{Sinput}\n> pf\n\\end{Sinput}\n\\begin{Soutput}\n      t      x      y     z\nt  38.4 -18.17  15.24 -28.2\nx -18.2   9.38  -7.03  13.0\ny  15.2  -7.03   6.89 -10.9\nz -28.2  12.98 -10.89  21.1\n\\end{Soutput}\n\\begin{Sinput}\n> pf\n\\end{Sinput}\n\\begin{Soutput}\n      t      x      y     z\nt  38.4 -18.17  15.24 -28.2\nx -18.2   9.38  -7.03  13.0\ny  15.2  -7.03   6.89 -10.9\nz -28.2  12.98 -10.89  21.1\n\\end{Soutput}\n\\end{Schunk}\n\nAgain as a consistency check, we may verify that transforming\ndownstairs indices gives the same result:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> lower(pf)\n\\end{Sinput}\n\\begin{Soutput}\n      t      x      y     z\nt  38.4 -18.17  15.24 -28.2\nx -18.2   9.38  -7.03  13.0\ny  15.2  -7.03   6.89 -10.9\nz -28.2  12.98 -10.89  21.1\n\\end{Soutput}\n\\end{Schunk}\n\n(note that the matrix representation of the Lorentz transforms\nrequires that the order of multiplication be reversed for successive\ncovariant transforms, so {\\tt B1} and {\\tt B2} must be swapped).\n\n\\subsection{Speed of light and the stress-energy tensor}\n\nHere I will perform another consistency check, this time with non-unit\nspeed of light, for a perfect fluid:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> sol(10)\n\\end{Sinput}\n\\begin{Soutput}\n[1] 10\n\\end{Soutput}\n\\begin{Sinput}\n> pf_rest <- perfectfluid(1,4)\n> pf_rest\n\\end{Sinput}\n\\begin{Soutput}\n     t    x    y    z\nt 1.04 0.00 0.00 0.00\nx 0.00 0.04 0.00 0.00\ny 0.00 0.00 0.04 0.00\nz 0.00 0.00 0.00 0.04\n\\end{Soutput}\n\\end{Schunk}\n\nThus {\\tt pf\\_rest} is the stress energy for a perfect fluid at rest\nin a particular frame $F$.  We may now consider the same perfect\nfluid, but moving with a three velocity of~$(3,4,5)'$: with respect to\n$F$:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> u <- as.3vel(3:5)\n> pf_moving <- perfectfluid(1,4,u)\n> pf_moving\n\\end{Sinput}\n\\begin{Soutput}\n      t     x     y    z\nt  2.08  6.24  8.32 10.4\nx  6.24 18.76 24.96 31.2\ny  8.32 24.96 33.32 41.6\nz 10.40 31.20 41.60 52.0\n\\end{Soutput}\n\\end{Schunk}\n\nThe consistency check is to verify that transforming to a frame in\nwhich the fluid is at rest will result in a stress-energy tensor that\nmatches {\\tt pf\\_rest}:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> transform_uu(perfectfluid(1,4,u),boost(u))\n\\end{Sinput}\n\\begin{Soutput}\n          t         x         y         z\nt  1.04e+00 -3.01e-16 -1.65e-15  9.04e-16\nx -1.33e-15  4.00e-02 -1.87e-15 -4.95e-15\ny -3.33e-15  7.18e-15  4.00e-02 -1.87e-15\nz -3.55e-15  9.87e-16  1.08e-14  4.00e-02\n\\end{Soutput}\n\\end{Schunk}\n\nthus showing agreement to within numerical precision.\n\n\\section{Photons}\n\\label{photonsection}\nIt is possible to define the four-momentum of photons by specifying\ntheir three-velocity and energy, and using {\\tt as.photon()}:\n \n\\begin{Schunk}\n\\begin{Sinput}\n> sol(1)\n\\end{Sinput}\n\\begin{Soutput}\n[1] 1\n\\end{Soutput}\n\\begin{Sinput}\n> (A <- as.photon(as.3vel(cbind(0.9,1:5\/40,5:1\/40))))\n\\end{Sinput}\n\\begin{Soutput}\n     E   p_x    p_y    p_z\n[1,] 1 0.990 0.0275 0.1375\n[2,] 1 0.992 0.0551 0.1103\n[3,] 1 0.993 0.0828 0.0828\n[4,] 1 0.992 0.1103 0.0551\n[5,] 1 0.990 0.1375 0.0275\n\\end{Soutput}\n\\end{Schunk}\n\nabove, $A$ is a vector of four-momentum of five photons, all of unit\nenergy, each with a null world line.  They are all moving\napproximately parallel to the x-axis.  We can check that this is\nindeed a null vector:\n                                                 \n\\begin{Schunk}\n\\begin{Sinput}\n> inner4(A)\n\\end{Sinput}\n\\begin{Soutput}\n[1]  1.45e-16  2.56e-17 -5.55e-17  2.56e-17  1.45e-16\n\\end{Soutput}\n\\end{Schunk}\n\nshowing that the vectors are indeed null to numerical precision.  What\ndo these photons look like in a frame moving along the $x$-axis at\n$0.7c$?\n  \n\\begin{Schunk}\n\\begin{Sinput}\n> tcrossprod(A,boost(as.3vel(c(0.7,0,0))))\n\\end{Sinput}\n\\begin{Soutput}\n         t     x      y      z\n[1,] 0.430 0.406 0.0275 0.1375\n[2,] 0.428 0.409 0.0551 0.1103\n[3,] 0.427 0.410 0.0828 0.0828\n[4,] 0.428 0.409 0.1103 0.0551\n[5,] 0.430 0.406 0.1375 0.0275\n\\end{Soutput}\n\\end{Schunk}\n\nAbove, see how the photons have lost the majority of their energy due\nto redshifting.  Blue shifting is easy to implement as either a\npassive transform:\n                                            \n\\begin{Schunk}\n\\begin{Sinput}\n> tcrossprod(A,boost(as.3vel(c(-0.7,0,0))))\n\\end{Sinput}\n\\begin{Soutput}\n        t    x      y      z\n[1,] 2.37 2.37 0.0275 0.1375\n[2,] 2.37 2.37 0.0551 0.1103\n[3,] 2.37 2.37 0.0828 0.0828\n[4,] 2.37 2.37 0.1103 0.0551\n[5,] 2.37 2.37 0.1375 0.0275\n\\end{Soutput}\n\\end{Schunk}\n                  \nor an active transform:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> tcrossprod(A,solve(boost(as.3vel(c(0.7,0,0)))))\n\\end{Sinput}\n\\begin{Soutput}\n        t    x      y      z\n[1,] 2.37 2.37 0.0275 0.1375\n[2,] 2.37 2.37 0.0551 0.1103\n[3,] 2.37 2.37 0.0828 0.0828\n[4,] 2.37 2.37 0.1103 0.0551\n[5,] 2.37 2.37 0.1375 0.0275\n\\end{Soutput}\n\\end{Schunk}\n\ngiving the same result.\n\n\n\\subsection{Reflection in mirrors}\n\n\\citet{gjurchinovski2004} discusses reflection of light from a\nuniformly moving mirror and here I show how the {\\tt lorentz} package\ncan illustrate some of his insights.  We are going to take the five\nphotons defined above and reflect them in an oblique mirror which is\nitself moving at half the speed of light along the $x$-axis.  The\nfirst step is to define the mirror {\\tt m}, and the boost {\\tt B}\ncorresponding to its velocity:\n                                                              \n\\begin{Schunk}\n\\begin{Sinput}\n> m <- c(1,1,1)\n> B <- boost(as.3vel(c(0.5,0,0)))\n\\end{Sinput}\n\\end{Schunk}\n\nAbove, the three-vector $m$ is parallel to the normal vector of the\nmirror and $B$ shows the Lorentz boost needed to bring it to rest.  We\nare going to reflect these photons in this mirror.  The R idiom for\nthe reflection is performed using a sequence of transforms.  First,\ntransform the photons' four-momentum to a frame in which the mirror is\nat rest:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> A\n\\end{Sinput}\n\\begin{Soutput}\n     E   p_x    p_y    p_z\n[1,] 1 0.990 0.0275 0.1375\n[2,] 1 0.992 0.0551 0.1103\n[3,] 1 0.993 0.0828 0.0828\n[4,] 1 0.992 0.1103 0.0551\n[5,] 1 0.990 0.1375 0.0275\n\\end{Soutput}\n\\begin{Sinput}\n> (A <- as.4mom(A\n\\end{Sinput}\n\\begin{Soutput}\n         E   p_x    p_y    p_z\n[1,] 0.583 0.566 0.0275 0.1375\n[2,] 0.582 0.569 0.0551 0.1103\n[3,] 0.581 0.569 0.0828 0.0828\n[4,] 0.582 0.569 0.1103 0.0551\n[5,] 0.583 0.566 0.1375 0.0275\n\\end{Soutput}\n\\end{Schunk}\n\nAbove, see how the photons have lost energy because of a redshift (the\n{\\tt as.4mom()} function has no effect other than changing the column\nnames).  Next, reflect the photons in the mirror (which is at rest):\n\n\\begin{Schunk}\n\\begin{Sinput}\n> (A <- reflect(A,m))\n\\end{Sinput}\n\\begin{Soutput}\n         E    p_x    p_y    p_z\n[1,] 0.583 0.0786 -0.460 -0.350\n[2,] 0.582 0.0793 -0.434 -0.379\n[3,] 0.581 0.0795 -0.407 -0.407\n[4,] 0.582 0.0793 -0.379 -0.434\n[5,] 0.583 0.0786 -0.350 -0.460\n\\end{Soutput}\n\\end{Schunk}\n\nAbove, see how the reflected photons have a reduced the x-component of\nmomentum; but have acquired a substantial $y$- and $z$- component.\nFinally, we transform back to the original reference frame.  Observe\nthat this requires an {\\em active} transform which means that we need\nto use the matrix inverse of $B$:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> (A <- as.4mom(A\n\\end{Sinput}\n\\begin{Soutput}\n         E   p_x    p_y    p_z\n[1,] 0.719 0.427 -0.460 -0.350\n[2,] 0.718 0.427 -0.434 -0.379\n[3,] 0.717 0.427 -0.407 -0.407\n[4,] 0.718 0.427 -0.379 -0.434\n[5,] 0.719 0.427 -0.350 -0.460\n\\end{Soutput}\n\\end{Schunk}\n\nThus in the original frame, the photons have lost about a quarter of\ntheir energy as a result of a Doppler effect: the mirror was receding\nfrom the source.  The photons have imparted energy to the mirror as a\nresult of mechanical work.  It is possible to carry out the same\noperations in one line:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> A <- as.photon(as.3vel(cbind(0.9,1:5\/40,5:1\/40)))\n> A\n\\end{Sinput}\n\\begin{Soutput}\n         E   p_x    p_y    p_z\n[1,] 0.719 0.427 -0.460 -0.350\n[2,] 0.718 0.427 -0.434 -0.379\n[3,] 0.717 0.427 -0.407 -0.407\n[4,] 0.718 0.427 -0.379 -0.434\n[5,] 0.719 0.427 -0.350 -0.460\n\\end{Soutput}\n\\end{Schunk}\n\n\\subsection{Disco ball}\n\nIt is easy to define a disco ball, which is a sphere covered in\nmirrors.  For the purposes of exposition, we will use a rather shabby\nball with only 7 mirrors:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> dfun <- function(n){matrix(rnorm(n*3),ncol=3)\n> (disco <- dfun(7))\n\\end{Sinput}\n\\begin{Soutput}\n        [,1]    [,2]    [,3]\n[1,] -0.8255  0.5638 -0.0246\n[2,]  0.4893 -0.0811  0.8683\n[3,] -0.0654  0.4519  0.8897\n[4,] -0.6690 -0.2310 -0.7065\n[5,] -0.7243 -0.6781 -0.1248\n[6,] -0.5078 -0.4407  0.7402\n[7,]  0.5224 -0.4929  0.6958\n\\end{Soutput}\n\\end{Schunk}\n\nThen we define a unit-energy photon moving parallel to the x-axis, and\nreflect it in the disco ball:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> p <- as.photon(c(1,0,0))\n> reflect(p,disco)\n\\end{Sinput}\n\\begin{Soutput}\n  E     p_x     p_y     p_z\nx 1 -0.3630  0.9309 -0.0406\nx 1  0.5212  0.0794 -0.8497\nx 1  0.9914  0.0591  0.1164\nx 1  0.1049 -0.3091 -0.9452\nx 1 -0.0491 -0.9823 -0.1807\nx 1  0.4843 -0.4476  0.7518\nx 1  0.4542  0.5150 -0.7270\n\\end{Soutput}\n\\end{Schunk}\n\n(above, {\\tt p} is a photon moving along the x-axis; standard R\nrecycling rules imply that we effectively have one photon per mirror\nin {\\tt disco}.  See how the photons' energy is unchanged by the\nreflection).  We might ask what percentage of photons are reflected\ntowards the source; but to do this we would need a somewhat more\nstylish disco ball, here with 1000 mirrors:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> table(reflect(p,dfun(1000))[,2]>0) # should be TRUE with probability sqrt(0.5)\n\\end{Sinput}\n\\begin{Soutput}\nFALSE  TRUE \n  294   706 \n\\end{Soutput}\n\\end{Schunk}\n\n(compare the expected value of $1000\/\\sqrt{2}\\simeq 707$).  But it is\nperhaps more fun to consider a relativistic disco in which the mirror\nball moves at 0.5c:\n\n\n\\begin{Schunk}\n\\begin{Sinput}\n> B <- boost(as.3vel(c(0.5,0,0)))\n> p\n\\end{Sinput}\n\\begin{Soutput}\n      t      x       y       z\nx 0.546 0.0913  0.5375 -0.0235\nx 0.840 0.6808  0.0458 -0.4906\nx 0.997 0.9943  0.0341  0.0672\nx 0.702 0.4033 -0.1785 -0.5457\nx 0.650 0.3006 -0.5671 -0.1043\nx 0.828 0.6562 -0.2584  0.4340\nx 0.818 0.6361  0.2973 -0.4197\n\\end{Soutput}\n\\end{Schunk}\n\nAbove, note the high energy of the photon in the third row.  This is\nbecause the third mirror of {\\tt disco} is such that the photon hits\nit with grazing incidence; this means that the receding of the mirror\nis almost immaterial.  Note further that a spinning disco ball would\ngive the same (instantaneous) results.\n\n\\subsection{Mirrors and rotation-boost coupling}\n\nConsider the following situation: we take a bunch of photons which in\na certain reference frame are all moving (almost) parallel to the\n$x$-axis.  Then we reflect the photons from a mirror which is moving\nwith a composition of pure boosts, and examine the reflected light in\ntheir original reference frame.  The R idiom for this would be:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> sol(1)\n\\end{Sinput}\n\\begin{Soutput}\n[1] 1\n\\end{Soutput}\n\\begin{Sinput}\n> light_start <- as.photon(as.3vel(cbind(0.9,1:5\/40,5:1\/40)))\n> m <- c(1,0,0)     # mirror normal to x-axis\n> B1 <- boost(as.3vel(c(-0.5, 0.1, 0.0)))\n> B2 <- boost(as.3vel(c( 0.2, 0.0, 0.0)))\n> B3 <- boost(as.3vel(c( 0.0, 0.0, 0.6)))\n> B <- B1\n> light <- light_start\n> light <- reflect(light,m)\n> light <- as.4mom(light\n> light\n\\end{Sinput}\n\\begin{Soutput}\n        E   p_x   p_y   p_z\n[1,] 2.30 -2.11 0.119 0.920\n[2,] 2.31 -2.13 0.147 0.897\n[3,] 2.32 -2.14 0.175 0.874\n[4,] 2.32 -2.15 0.203 0.849\n[5,] 2.33 -2.16 0.230 0.824\n\\end{Soutput}\n\\end{Schunk}\n\nSee how the photons have picked up momentum in the $y$- and $z$-\ndirection, even though the mirror is oriented perpendicular to the\n$x$-axis (in its own frame).  Again it is arguably preferable to use\npipes:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> light_start\n\\end{Sinput}\n\\begin{Soutput}\n        E   p_x   p_y   p_z\n[1,] 2.30 -2.11 0.119 0.920\n[2,] 2.31 -2.13 0.147 0.897\n[3,] 2.32 -2.14 0.175 0.874\n[4,] 2.32 -2.15 0.203 0.849\n[5,] 2.33 -2.16 0.230 0.824\n\\end{Soutput}\n\\end{Schunk}\n\nCompare when the speed of light is infinite:\n\\begin{Schunk}\n\\begin{Sinput}\n> sol(Inf)\n\\end{Sinput}\n\\begin{Soutput}\n[1] Inf\n\\end{Soutput}\n\\begin{Sinput}\n> light_start <- as.photon(as.3vel(cbind(0.9,1:5\/40,5:1\/40)))\n> B1 <- boost(as.3vel(c(-0.5, 0.1, 0.0)))\n> B2 <- boost(as.3vel(c( 0.2, 0.0, 0.0)))\n> B3 <- boost(as.3vel(c( 0.0, 0.0, 0.6)))\n> B <- B1\n> light_start\n\\end{Sinput}\n\\begin{Soutput}\n     E   p_x    p_y    p_z\n[1,] 0 0.990 0.0275 0.1375\n[2,] 0 0.992 0.0551 0.1103\n[3,] 0 0.993 0.0828 0.0828\n[4,] 0 0.992 0.1103 0.0551\n[5,] 0 0.990 0.1375 0.0275\n\\end{Soutput}\n\\begin{Sinput}\n> light_start\n\\end{Sinput}\n\\begin{Soutput}\n     E    p_x    p_y    p_z\n[1,] 0 -0.990 0.0275 0.1375\n[2,] 0 -0.992 0.0551 0.1103\n[3,] 0 -0.993 0.0828 0.0828\n[4,] 0 -0.992 0.1103 0.0551\n[5,] 0 -0.990 0.1375 0.0275\n\\end{Soutput}\n\\end{Schunk}\n\nNote that, in the infinite light speed case, the energy of the photons\nis zero (photons have zero rest mass); further observe that in this\nclassical case, the effect of the mirror is to multiply the $x$-momentum\nby $-1$ and leave the other components unchanged, as one might expect\nfrom a mirror perpendicular to $(1,0,0)$.\n\n\\section{Three-velocities}\n\nIn contrast to four-velocities, three-velocities do not form a group\nunder composition as the velocity addition law is not\nassociative~\\citep{ungar2006}.  Instead, three-velocity composition\nhas an algebraic structure known as a {\\em gyrogroup} (this\nobservation was the original motivation for the package).\n\\citeauthor{ungar2006} shows that the velocity addition law for\nthree-velocities is\n\n\\begin{equation}\n\\mathbf u\\oplus\\mathbf v=\n\\frac{1}{1+\\mathbf u\\cdot\\mathbf v}\n\\left\\{\n\\mathbf u + \\frac{\\mathbf v}{\\gamma_\\mathbf u} + \\frac{\\gamma_\\mathbf u\n\\left(\\mathbf u\\cdot\\mathbf v\\right)\\mathbf u}{1+\\gamma_\\mathbf u}\n\\right\\}\n\\end{equation}\n   \nwhere~$\\gamma_\\mathbf u=\\left(1-\\mathbf u\\cdot\\mathbf u\\right)^{-1\/2}$ and we are\nassuming~$c=1$.  \\citeauthor{ungar2006} goes on to show that, in\ngeneral, $\\mathbf u\\oplus\\mathbf v\\neq\\mathbf v\\oplus\\mathbf u$\nand~$(\\mathbf u\\oplus\\mathbf v)\\oplus\\mathbf w\\neq\\mathbf u\\oplus(\\mathbf v\\oplus\\mathbf w)$.  He also\ndefines the binary operator~$\\ominus$\nas~$\\mathbf u\\ominus\\mathbf v=\\mathbf u\\oplus\\left(-\\mathbf v\\right)$, and implicitly\ndefines~$\\ominus\\mathbf u\\oplus\\mathbf v$ to be~$\\left(-\\mathbf u\\right)\\oplus\\mathbf v$.  If\nwe have\n\n\\begin{equation}\n\\gyr{u}{v}\\mathbf x=-\\left(\\mathbf u\\oplus\\mathbf v\\right)\\oplus\\left(\\mathbf u\\oplus\\left(\\mathbf v\\oplus\\mathbf x\\right)\\right)\n\\end{equation}\n\nthen\n\n\\begin{eqnarray}\n\\mathbf u\\oplus\\mathbf v &=& \\gyr{u}{v}\\left(\\mathbf v\\oplus\\mathbf u\\right)\\label{noncom}\\\\\n\\gyr{u}{v}\\mathbf x\\cdot\\gyr{u}{v}\\mathbf y &=& \\mathbf x\\cdot\\mathbf y\\label{doteq}\\\\\n\\gyr{u}{v}\\left(\\mathbf x\\oplus\\mathbf y\\right) &=& \\gyr{u}{v}\\mathbf x\\oplus\\gyr{u}{v}\\mathbf y\\\\\n\\left(\\gyr{u}{v}\\right)^{-1} &=& \\left(\\gyr{v}{u}\\right)\\label{gyrinv}\\\\\n\\mathbf u\\oplus\\left(\\mathbf v\\oplus\\mathbf w\\right) &=&\\left(\\mathbf u\\oplus\\mathbf v\\right)\\oplus\\gyr{u}{v}\\mathbf w\\label{nonass1}\\\\\n\\left(\\mathbf u\\oplus\\mathbf v\\right)\\oplus\\mathbf w &=&\\mathbf u\\oplus\\left(\\mathbf v\\oplus\\gyr{v}{u}\\mathbf w\\right)\\label{nonass2}\n\\end{eqnarray}\n\nConsider the following R session:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> sol(1)\n\\end{Sinput}\n\\begin{Soutput}\n[1] 1\n\\end{Soutput}\n\\begin{Sinput}\n> u <- as.3vel(c(-0.7,+0.2,-0.3))\n> v <- as.3vel(c(+0.3,+0.3,+0.4))\n> w <- as.3vel(c(+0.1,+0.3,+0.8))\n> x <- as.3vel(c(-0.2,-0.1,-0.9))\n> u\n\\end{Sinput}\n\\begin{Soutput}\nA vector of three-velocities (speed of light = 1)\n        x   y    z\n[1,] -0.7 0.2 -0.3\n\\end{Soutput}\n\\end{Schunk}\n\nHere we have three-vectors {\\tt u} etc.  We can see that {\\tt u} and\n{\\tt v} do not commute:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> u+v\n\\end{Sinput}\n\\begin{Soutput}\nA vector of three-velocities (speed of light = 1)\n          x     y        z\n[1,] -0.545 0.482 -0.00454\n\\end{Soutput}\n\\begin{Sinput}\n> v+u\n\\end{Sinput}\n\\begin{Soutput}\nA vector of three-velocities (speed of light = 1)\n          x     y     z\n[1,] -0.429 0.572 0.132\n\\end{Soutput}\n\\end{Schunk}\n\n(the results differ).  We can use equation~\\ref{noncom}\n\\begin{Schunk}\n\\begin{Sinput}\n> (u+v)-gyr(u,v,v+u)\n\\end{Sinput}\n\\begin{Soutput}\nA vector of three-velocities (speed of light = 1)\n            x         y        z\n[1,] 1.77e-16 -7.08e-16 1.23e-16\n\\end{Soutput}\n\\end{Schunk}\n\nshowing agreement to within numerical error.  It is also possible to\nuse the functional idiom in which we define {\\tt f()} to be the\nmap~$\\mathbf x\\mapsto\\gyr{u}{v}\\mathbf x$.  In R:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> f <- gyrfun(u,v)\n> (u+v)-f(v+u)    # should be zero\n\\end{Sinput}\n\\begin{Soutput}\nA vector of three-velocities (speed of light = 1)\n            x         y        z\n[1,] 1.77e-16 -7.08e-16 1.23e-16\n\\end{Soutput}\n\\end{Schunk}\n\nFunction {\\tt gyrfun()} is vectorized, which means that it plays\nnicely with (R) vectors.  Consider\n\n\\begin{Schunk}\n\\begin{Sinput}\n> u9 <- r3vel(9)\n> u9\n\\end{Sinput}\n\\begin{Soutput}\nA vector of three-velocities (speed of light = 1)\n            x       y       z\n [1,] -0.6635 -0.1850 -0.0324\n [2,]  0.7232  0.3007  0.0448\n [3,] -0.4816 -0.2698  0.0903\n [4,]  0.7186  0.5215  0.1149\n [5,] -0.4988 -0.5469  0.4781\n [6,] -0.0446  0.7934 -0.4839\n [7,]  0.4225  0.6831 -0.2474\n [8,] -0.5546 -0.1326  0.4232\n [9,]  0.0366  0.0748  0.4407\n\\end{Soutput}\n\\end{Schunk}\n\nThen we can create a vectorized gyrofunction:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> f <- gyrfun(u9,v)\n> f(x)\n\\end{Sinput}\n\\begin{Soutput}\nA vector of three-velocities (speed of light = 1)\n            x         y      z\n [1,] -0.0282 -7.04e-02 -0.924\n [2,] -0.3568 -1.36e-01 -0.845\n [3,] -0.0692 -2.67e-02 -0.924\n [4,] -0.3503 -1.89e-01 -0.838\n [5,]  0.0444  1.59e-01 -0.913\n [6,] -0.2297 -4.52e-01 -0.776\n [7,] -0.3268 -3.10e-01 -0.811\n [8,]  0.0130  3.85e-06 -0.927\n [9,] -0.1409 -5.01e-02 -0.915\n\\end{Soutput}\n\\end{Schunk}\n\nNote that the package vectorization is transparent when using syntactic sugar:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> u9+x\n\\end{Sinput}\n\\begin{Soutput}\nA vector of three-velocities (speed of light = 1)\n            x       y       z\n [1,] -0.7436 -0.2345 -0.5826\n [2,]  0.6411  0.2532 -0.6615\n [3,] -0.6319 -0.3444 -0.6273\n [4,]  0.6860  0.5266 -0.4421\n [5,] -0.6903 -0.6791 -0.0511\n [6,] -0.0952  0.7096 -0.6909\n [7,]  0.3115  0.6168 -0.6976\n [8,] -0.8231 -0.2462 -0.3690\n [9,] -0.2550 -0.0524 -0.7806\n\\end{Soutput}\n\\end{Schunk}\n\n(here, the addition operates using R's standard recycling rules).\n\n\\subsection{Associativity}\n\nThree velocity addition is not associative:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> (u+v)+w\n\\end{Sinput}\n\\begin{Soutput}\nA vector of three-velocities (speed of light = 1)\n          x     y     z\n[1,] -0.465 0.655 0.501\n\\end{Soutput}\n\\begin{Sinput}\n> u+(v+w)\n\\end{Sinput}\n\\begin{Soutput}\nA vector of three-velocities (speed of light = 1)\n          x     y     z\n[1,] -0.549 0.667 0.416\n\\end{Soutput}\n\\end{Schunk}\n\nBut we can use equations~\\ref{nonass1} and~\\ref{nonass2}:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> (u+(v+w)) - ((u+v)+gyr(u,v,w))\n\\end{Sinput}\n\\begin{Soutput}\nA vector of three-velocities (speed of light = 1)\n            x         y         z\n[1,] 6.92e-16 -1.38e-15 -6.92e-16\n\\end{Soutput}\n\\begin{Sinput}\n> ((u+v)+w) - (u+(v+gyr(v,u,w)))\n\\end{Sinput}\n\\begin{Soutput}\nA vector of three-velocities (speed of light = 1)\n     x y        z\n[1,] 0 0 5.35e-16\n\\end{Soutput}\n\\end{Schunk}\n\n\\subsection{Visualization of noncommutativity and nonassociativity of three-velocities}\n\nConsider the following three-velocities:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> u <- as.3vel(c(0.4,0,0))\n> v <- seq(as.3vel(c(0.4,-0.2,0)), as.3vel(c(-0.3,0.9,0)),len=20)\n> w <- as.3vel(c(0.8,-0.4,0))\n\\end{Sinput}\n\\end{Schunk}\n\nObjects~$\\mathbf v$ and $\\mathbf w$ are single three-velocities, and object $\\mathbf v$\nis a vector of three velocities.  We can see the noncommutativity of\nthree velocity addition in figures~\\ref{comfail1} and~\\ref{comfail2},\nand the nonassociativity in figure~\\ref{assfail}.\n\n\\begin{figure}[htbp]\n  \\begin{center}\n\\begin{Schunk}\n\\begin{Sinput}\n> comm_fail1(u=u, v=v)\n\\end{Sinput}\n\\end{Schunk}\n\\includegraphics{lorentz_arxiv-comfail1_fig}\n\\caption{Failure\\label{comfail1} of the commutative law for velocity\n  composition in special relativity.  The arrows show successive\n  velocity boosts of $+\\mathbf u$ (purple), $+\\mathbf v$ (black), $-\\mathbf u$ (red),\n  and~$-\\mathbf v$ (blue) for $\\mathbf u,\\mathbf v$ as defined above.  Velocity $\\mathbf u$ is\n  constant, while $\\mathbf v$ takes a sequence of values.  If velocity\n  addition is commutative, the four boosts form a closed\n  quadrilateral; the thick arrows show a case where the boosts almost\n  close and the boosts nearly form a parallelogram.  The blue dots\n  show the final velocity after four successive boosts; the distance\n  of the blue dot from the origin measures the combined velocity,\n  equal to zero in the classical limit of low speeds.  The discrepancy\n  becomes larger and larger for the faster elements of the sequence\n  $\\mathbf v$}\n  \\end{center}\n\\end{figure}\n\n\\begin{figure}[htbp]\n  \\begin{center}\n\\begin{Schunk}\n\\begin{Sinput}\n> comm_fail2(u=u, v=v)\n\\end{Sinput}\n\\end{Schunk}\n\\includegraphics{lorentz_arxiv-comfail2_fig}\n\n\\caption{Another view of the failure of the commutative\n  law\\label{comfail2} in special relativity.  The black arrows show\n  velocity boosts of $\\mathbf u$ and the blue arrows show velocity boosts of\n  $\\mathbf v$, with $\\mathbf u,\\mathbf v$ as defined above; $\\mathbf u$ is constant while\n  $\\mathbf v$ takes a sequence of values.  If velocity addition is\n  commutative, then $\\mathbf u+\\mathbf v=\\mathbf v+\\mathbf u$ and the two paths end at the\n  same point: the parallelogram is closed.  The red lines show the\n  difference between $\\mathbf u+\\mathbf v$ and $\\mathbf v+\\mathbf u$}\n  \\end{center}\n\\end{figure}\n\n\\begin{figure}[htbp]\n  \\begin{center}\n\\begin{Schunk}\n\\begin{Sinput}\n> ass_fail(u=u, v=v, w=w, bold=10)\n\\end{Sinput}\n\\end{Schunk}\n\\includegraphics{lorentz_arxiv-assfail_fig}\n\\caption{Failure of the associative law \\label{assfail} for velocity\n  composition in special relativity.  The arrows show successive\n  boosts of $\\mathbf u$ followed by $\\mathbf v+\\mathbf w$ (black lines), and $\\mathbf u+\\mathbf v$\n  followed by $\\mathbf w$ (blue lines), for $\\mathbf u$, $\\mathbf v$, $\\mathbf w$ as defined\n  above; $\\mathbf u$ and $\\mathbf w$ are constant while $\\mathbf v$ takes a sequence of\n  values. The mismatch between $\\mathbf u+\\left(\\mathbf v+\\mathbf w\\right)$ and\n  $\\left(\\mathbf u+\\mathbf v\\right)+\\mathbf w$ is shown in red}\n  \\end{center}\n\\end{figure}\n\n\n\\subsection[The magrittr package: pipes]{The {\\tt magrittr} package: pipes}\n\nThree velocities in the {\\tt lorentz} package work nicely with\n{\\tt magrittr}.  If we define\n\n\\begin{Schunk}\n\\begin{Sinput}\n>  u <- as.3vel(c(+0.5,0.1,-0.2))\n>  v <- as.3vel(c(+0.4,0.3,-0.2))\n>  w <- as.3vel(c(-0.3,0.2,+0.2))\n\\end{Sinput}\n\\end{Schunk}\n\nThen pipe notation operates as expected:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> jj1 <- u\n> jj2 <- u+v\n> speed(jj1-jj2)\n\\end{Sinput}\n\\begin{Soutput}\n[1] 2.21e-16\n\\end{Soutput}\n\\end{Schunk}\n\nThe pipe operator is left associative:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> jj1 <- u\n> jj2 <- (u+v)+w\n> speed(jj1-jj2)\n\\end{Sinput}\n\\begin{Soutput}\n[1] 7.39e-17\n\\end{Soutput}\n\\end{Schunk}\n\n\nIf we want right associative addition, the pipe operator needs\nbrackets:\n\n\\begin{Schunk}\n\\begin{Sinput}\n> jj1 <- u\n> jj2 <- u+(v+w)\n> speed(jj1-jj2)\n\\end{Sinput}\n\\begin{Soutput}\n[1] 3.45e-17\n\\end{Soutput}\n\\end{Schunk}\n\n\\subsection{Numerical verification}\n\nHere I provide numerical verification of equations~\\ref{noncom}\nto~\\ref{nonass2}.  If we have\n\n\\begin{Schunk}\n\\begin{Sinput}\n> x <- as.3vel(c(0.7, 0.0, -0.7))\n> y <- as.3vel(c(0.1, 0.3, -0.6))\n> u <- as.3vel(c(0.0, 0.8, +0.1))   # x,y,u: single three-velocities\n> v <- r3vel(5,0.9)\n> w <- r3vel(5,0.8)                 # v,w: vector of three-velocities\n> f <- gyrfun(u,v)\n> g <- gyrfun(v,u)\n\\end{Sinput}\n\\end{Schunk}\n\nThen we can calculate the difference between the left hand side and\nright hand side numerically:\n  \n\\begin{Schunk}\n\\begin{Sinput}\n> max(speed((u+v) - f(v+u)))              # equation 3\n\\end{Sinput}\n\\begin{Soutput}\n[1] 3.49e-13\n\\end{Soutput}\n\\begin{Sinput}\n> max(abs(prod3(f(x),f(y)) - prod3(x,y))) # equation 4\n\\end{Sinput}\n\\begin{Soutput}\n[1] 5.94e-15\n\\end{Soutput}\n\\begin{Sinput}\n> max(speed(f(x+y) - (f(x)+f(y))))        # equation 5\n\\end{Sinput}\n\\begin{Soutput}\n[1] 3.81e-12\n\\end{Soutput}\n\\begin{Sinput}\n> max(speed(f(g(x)) - g(f(x))))           # equation 6\n\\end{Sinput}\n\\begin{Soutput}\n[1] 7.3e-13\n\\end{Soutput}\n\\begin{Sinput}\n> max(speed((u+(v+w)) - ((u+v)+f(w))))    # equation 7\n\\end{Sinput}\n\\begin{Soutput}\n[1] 6.25e-14\n\\end{Soutput}\n\\begin{Sinput}\n> max(speed(((u+v)+w) - (u+(v+g(w)))))    # equation 8\n\\end{Sinput}\n\\begin{Soutput}\n[1] 2.84e-14\n\\end{Soutput}\n\\end{Schunk}\n\n(all zero to numerical precision). \n\n\n\\section{Conclusions}\n\nThe {\\tt lorentz} package furnishes some functionality for\nmanipulating four-vectors and three-velocities in the context of\nspecial relativity.  The R idiom is relatively natural and the package\nhas been used to illustrate different features of relativistic\nkinematics.  The package leverages the powerful R programming language\nto conduct a systematic search for a gyrodistributive law, without\nsuccess.  If such a law exists, it is complicated: it is not one of\nthe 688128 natural forms considered in the search.\n\n\n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzhvxy b/data_all_eng_slimpj/shuffled/split2/finalzzhvxy
new file mode 100644
index 0000000000000000000000000000000000000000..751e475144481f86dec4e401a04c3c624be3f1a6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzhvxy
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nOur present understanding of the primordial Universe relies on the paradigm of inflation \\citep{inflationhist1,inflationhist2,inflationhist3}, introducing a phase of accelerated expansion in the first fractions of a second after the primordial singularity. Such a phenomenon is expected to leave a background of gravitational waves propagating in the primordial plasma during recombination, leaving a permanent mark imprinted in the polarization anisotropies of the cosmic microwave background (CMB): the primordial $B$-modes \\citep{InflationmodesB3,InflationsmodesB1,InflationsmodesB2}. The amplitude of the angular power spectrum of those primordial $B$-modes is characterized by the {tensor-to-scalar ratio} $r$, which is proportional to the energy scale at which inflation occurred \\citep{renergyscale}. Hence, looking for this smoking gun of inflation allows us to test our best theories of fundamental physics in the primordial Universe at energy scales far beyond the reach of particle accelerators. In this scope, it is one of the biggest challenges of cosmology set out for the next decades. The best experimental upper limit on the $r$ parameter so far is $r<0.032$ \\citep[95\\,\\% C.L.,][]{tristram,bicep2021,PlanckandBICEP}.\n\nThe JAXA Lite (Light) satellite, used for the $B$-mode polarization and Inflation from cosmic background Radiation Detection (\\textit{LiteBIRD}{}) mission, is designed to observe the sky at large angular scales in order to constrain this parameter $r$ down to $\\delta r= 10^{-3}$, including all sources of uncertainty \\citep{litebird,LiteBIRDUpdatedDesign}. Exploring this region of the parameter space is critical, because this order of magnitude for the tensor-to-scalar ratio is predicted by numerous physically motivated inflation models (for a review see e.g.,  \\cite{EncyclopediaInflationaris})  \n\nHowever, the success of this mission relies on our ability to treat polarized foreground signals. Indeed various diffuse astrophysical sources emit polarized $B$-mode signals above the primordial ones, the strongest being due to the diffuse polarized emission of our own Galaxy \\citep{PlanckCompoSep}. Even in a diffuse region like the BICEP\/Keck field, the Galactic $B$-modes are at least ten times stronger at 150\\,GHz than the $r=0.01$ tensor $B$-modes targeted by the current CMB experiments \\citep{BICEPKECKGW}.\n\nThe true complexity of polarized foreground emission that the next generation of CMB experiments will face is still mostly unknown today. Underestimation of this complexity can lead to the estimation of a spurious nonzero value of $r$ \\citep[see e.g.,][]{PlanckL,Remazeilles_etal_2016}.\n\nAt high frequencies ($>100$ GHz), the thermal emission of interstellar dust grains is the main source of Galactic foreground contaminating the CMB \\citep{dusthighfreq,PlanckDust2}. The canonical model of the spectral energy distribution (SED) of this thermal emission for intensity and polarization is given by the modified black body (MBB) law \\citep{Desertdustmodel}. This model provides a good fit to the dust polarization SED at the sensitivity of the \\textit{Planck}{} satellite \\citep{PlanckDust2} but it may not fully account for\nit at the sensitivity of future experiments \\citep{HensleyBull}. Furthermore, due to changes of physical conditions across the galaxy, spatial variations of the SEDs are present between and along the lines of sight. The former leads to what is known as \\emph{frequency decorrelation} in the CMB community \\citep[see e.g.][]{tassis,PlanckL,pelgrims2021}. Moreover, both effects lead to averaging MBBs when observing the sky (unavoidable line-of-sight or beam-integration effects). Because of the nonlinearity of the MBB law, those averaging effects will {distort} the SED, leading to deviations from this canonical model \\citep{Chluba}. \n\n\\cite{Chluba} proposed a general framework called {``moment expansion''} of the SED to take into account those distortions, using a Taylor expansion around the MBB with respect to its spectral parameters \\citep[Taylor expansion of foreground SEDs was discussed in previous studies; see e.g.,][]{stolyarov2005}. This method is agnostic: it does not require any assumption on the real complexity of the polarized dust emission. The moment expansion approach thus provides a promising tool with which to model the unanticipatable complexity of the dust emission in real data.\n\n\\cite{Mangilli} generalized this formalism for the sake of CMB data analysis in harmonic space and for cross-angular power spectra and applied it successfully to complex simulations and \\textit{Planck}{} High-Frequency Instrument (HFI) intensity data. This latter work shows that the real complexity of Galactic foregrounds could be higher than expected, encouraging us to follow the path opened by the moment expansion formalism.\n\nIn the present work, we apply the moment expansion in harmonic space to characterize and treat the dust foreground polarized emission of \\textit{LiteBIRD}{} high-frequency simulations, using  dust-emission models of increasing complexity. We discuss the ability of this method to recover an unbiased value for the $r$ parameter, with enough accuracy to achievethe scientific objectives of the \\textit{LiteBIRD} mission.\n\nIn Sect.~\\ref{sec:formalism}, we first review the formalism of moment expansion in map and harmonic domains. We then describe in Sect.~\\ref{sec:sims} how we realize several sets of simulations of the sky as seen by the \\textit{LiteBIRD}{} instrument with varying dust complexity and how we estimate the angular power spectra. In Sect.~\\ref{sec:fit}, we describe how we estimate the moment parameters and the tensor-to-scalar ratio $r$ in those simulations. The results are then presented in Sect.~\\ref{sec:results}. Finally, we discuss those results and the future work that has to be done in the direction opened by moment expansion in Sect.~\\ref{sec:discussion}.\n\n\\section{\\label{sec:formalism}Formalism}\n\n\\subsection{Characterizing the dust SED in real space}\n\n\\subsubsection{Modified black body model \\label{sec:mbb}}\n\nThe canonical way to characterize astrophysical dust-grain emission in every volume element of the Galaxy is given by the modified black body (MBB) function, consisting of multiplying a standard black body SED $B_\\nu(T)$ at a given temperature $T_0$ by a power-law of the frequency $\\nu$ with a spectral index $\\beta_0$. The dust intensity map $I_{\\rm D}(\\nu,\\vec{n})$ observed at a frequency $\\nu$ in every direction with respect to the unit vector $\\vec{n}$, can then be written as:\n\\begin{equation}\n\\label{eq:MBB}\n     I(\\nu,\\vec{n}) =  \\left(\\frac{\\nu}{\\nu_0}\\right)^{\\beta_0} \\frac{B_{\\nu}(T_0)}{B_{\\nu_0}(T_0)} A(\\vec{n}) \n     =\\frac{I_{\\nu}(\\beta_0,T_0)}{I_{\\nu_0}(\\beta_0,T_0)} A(\\vec{n}),\n\\end{equation}\n\n\\noindent where $A(\\vec{n})$ is the dust intensity template at a reference frequency $\\nu_0$\\footnote{Throughout this work, we use $\\nu_0 = 353$\\,GHz.}.\nWe know that the physical conditions (thermodynamic and dust grain properties) change through the interstellar medium across the Galaxy, depending, in an intricate fashion, on the gas velocity and density, the interstellar radiation field, the distance to the Galactic center \\citep[see e.g., ][]{dustacrossMW,dustvarMW,PlanckVardust,PlanckCompoSep,vardustdisk,dustradfield}. This change of physical conditions leads to variations in $\\beta$ and $T$ depending on the direction of observation $\\vec{n}$:\n\n\\begin{equation}\n\\label{eq:MBBn}\n     I(\\nu,\\vec{n}) =  \\frac{I_{\\nu}(\\beta(\\vec{n}),T(\\vec{n}))}{I_{\\nu_0}(\\beta(\\vec{n}),T(\\vec{n}))} A(\\vec{n}).\n\\end{equation}\n\nThe SED amplitude and parameters (temperature and spectral index) are then different for every line of sight. It is therefore clear that, in order to provide a realistic model of the dust emission, the frequency and spatial dependencies may not be trivially separated. \n\n\\subsubsection{\\label{sect:limitsmbb}Limits of the modified black body}\n\nThe dust SED model given by the MBB has proven to be highly accurate \\citep{Planck2014dust,Planck2015dust}. However, it must be kept in mind that this model is empirical and is therefore not expected to give a perfect description of the dust SED in the general case. Indeed, physically motivated dust grain emission models predict deviations from it \\citep[e.g.,][]{modelbeyondmbb}. Surveys tend to show that the dust-emission properties vary across the observed 2D sky and the 3D Galaxy \\citep{PlanckDust2}. Furthermore, in true experimental conditions, one can never directly access the pure SED of a single volume element with specific emission properties and unique spectral parameters. Averages are therefore made over different SEDs emitted from distinct regions with different physical emission properties, in a way that may not be avoided:  along the line of sight; between different lines of sight, inside the beam of the instrument or; when doing a spherical harmonic decomposition to calculate the angular power spectra over large regions of the sky.\n\nThe MBB function is nonlinear, and therefore summing MBBs with different spectral parameters does not return another MBB function and produces \\emph{SED distortions}. For all these reasons, modeling the dust emission with a MBB is intrinsically limited, even when doing so with spatially varying spectral parameters. As a consequence, inaccuracies might appear when modeling the dust contribution to CMB data that will unavoidably impact the final estimation of the cosmological parameters. \n\n\\subsubsection{Moment expansion in pixel space}\n\\label{sec:moment_pixel}\n\nA way to address the limitation of the MBB model in accurately describing the dust emission is given by the {moment expansion} formalism proposed by \\cite{Chluba}. This formalism is designed to take into account the SED distortions due to averaging effects by considering a multidimensional Taylor expansion of the distorted SED $I(\\nu,\\vec{p})$ around the mean values $\\vec{p}_{0}$ of its spectral parameters $\\vec{p} = \\{p_i\\}$. This is the so-called {moment expansion} of the SED, which can be written as\n\\begin{align}\n    I(\\nu,\\vec{p}) = I(\\nu,\\vec{p}_{0}) &+ \\sum_i \\omega_1^{p_i}\\langle\\partial_{p_i}I(\\nu,\\vec{p})\\rangle_{\\vec{p}=\\vec{p}_0}\n    \\nonumber \\\\\n    &+ \\frac{1}{2}\\sum_{i,j} \\omega_2^{p_ip_j}\\langle\\partial_{p_i}\\partial_{p_j}I(\\nu,\\vec{p})\\rangle_{\\vec{p}=\\vec{p}_0}\n    \\nonumber \\\\\n    &+ \\dots \\nonumber\\\\\n    &+\\frac{1}{\\alpha!} \\sum_{i,\\dots,k} \\omega_\\alpha^{p_i\\dots p_k}\\langle\\partial_{p_i}\\dots\\partial_{p_k}I(\\nu,\\vec{p})\\rangle_{\\vec{p}=\\vec{p}_0},\n\\label{eq:momentgeneral}\n\\end{align}\n\nwhere the first term on the right-hand side is the SED without distortion $I(\\nu,\\vec{p}_{0})$ evaluated at $\\vec{p}=\\vec{p}_0$, and the other terms are the so-called {moments} of order $\\alpha$, quantified by the {moment parameters} $\\omega_\\alpha^{p_i\\dots p_k}$ for the expansion with respect to any parameter of $\\vec{p}$. Performing the expansion to increasing order adds increasing complexity to the SED $I(\\nu,\\vec{p}_{0})$. \n\nFor the MBB presented in Sect.~\\ref{sec:mbb}, there are two parameters so that $\\vec{p} = \\{\\beta,T\\}$. Thus the dust moment expansion reads\n\n\\begin{align}\n     I(\\nu,\\vec{n})  = \\frac{I_{\\nu}(\\beta_0,T_0)}{I_{\\nu_0}(\\beta_0,T_0)} \\bigg\\{ & A(\\vec{n}) + \\omega^\\beta_1(\\vec{n}) \\ln\\left(\\frac{\\nu}{\\nu_0}\\right)+ \\frac{1}{2}\\omega^\\beta_2(\\vec{n}) \\ln^2\\left(\\frac{\\nu}{\\nu_0}\\right)\\nonumber \\\\[2mm]\n     &+ \\omega^T_1(\\vec{n})\\Big( \\Theta(\\nu,T_0) - \\Theta(\\nu_0,T_0) \\Big) + \\dots \\bigg\\},\n\\label{eq:momentinttemp}\n\\end{align}\n\n\\noindent where the expansion has been written up to order two in $\\beta$ (with moment expansion parameters $\\omega^\\beta_1$ at order one and $\\omega^\\beta_2$ at order two) and to order one in $T$ (with a moment expansion parameter $\\omega^T_1$ at order one). The following expression has been introduced to simplify the black body derivative with respect to $T$:\n\n\\begin{align}\n    \\Theta(\\nu,T) = \\frac{x}{T}\\frac{e^{x}}{e^{x}-1},\\ {\\rm with}\\ x = \\frac{h \\nu}{k T}.\n\\end{align}\n\nThe moment expansion in pixel space can be used for component separation and possibly crossed with other methods \\citep[see e.g.,][]{RemazeillesmomentsILC,Debabrata_2021}. However, in the present work, we are interested in the modeling of the dust at the $B$-mode angular power spectrum level. Performing the moment expansion at the angular power spectrum level adds some complexity to the SEDs due to the additional averaging occurring when dealing with spherical harmonic coefficients. Indeed, these coefficients are estimated on potentially large fractions of the sky and probe regions with various physical conditions. On the other hand, the expansion at the power spectrum level possibly drastically reduces the parameter space with respect to performing the expansion in every sky pixel.\n\n\\subsection{Characterizing the dust SED in harmonic space}\n\n\\subsubsection{Dust SED in spherical harmonic space \\label{sec:formalismspectra}}\n\nThe expansion presented in Sect.~\\ref{sec:moment_pixel} can be applied in spherical harmonic space using the same logic. The sky emission projection then reads\n\n\\begin{equation}\n        I(\\nu,\\vec{n}) = \\sum_{\\ell = 0}^{\\infty}\\sum_{m=-\\ell}^{\\ell}I^\\nu_{\\ell m}Y_{\\ell m} (\\vec{n}).\n\\end{equation}\n\nApplying the moment expansion to the spherical harmonics coefficients, with respect to $\\beta$ and $T$, as in Eq.~\\ref{eq:momentinttemp}, leads to\n\n\\begin{align}\n     I^\\nu_{\\ell m} = & \\frac{I_{\\nu}(\\beta_0(\\ell),T_0(\\ell))}{I_{\\nu_0}(\\beta_0(\\ell),T_0(\\ell))} \\bigg\\{ A_{\\ell  m} + \\omega^\\beta_{1,\\ell  m} \\ln\\left(\\frac{\\nu}{\\nu_0}\\right) + \\frac{1}{2}\\omega^\\beta_{2,\\ell  m} \\ln^2\\left(\\frac{\\nu}{\\nu_0}\\right) \\nonumber \\\\[2mm]\n     &+ \\omega^T_{1,\\ell  m} \\Big( \\Theta(\\nu,T_0(\\ell)) - \\Theta(\\nu_0,T_0(\\ell)) \\Big) + \\dots \\quad\\bigg\\},\n\\label{eq:momentint2temp}\n\\end{align}\n\n\n\\noindent where this time $\\beta_0(\\ell)$ and $T_0 (\\ell)$ are the averages of $\\beta$ and $T$ at a given multipole $\\ell$ over the sky fraction we are looking at. We note that the moment parameters $\\omega^{p_i}_{\\alpha,\\ell m}$ involved here are different from the $\\omega^{p_i}_i(\\vec{n})$ appearing in Eq.~\\ref{eq:momentinttemp} in the map space because they involve different averaging. In principle, the moment expansion in harmonic space can take into account the three kinds of spatial averages presented in Sect.~\\ref{sect:limitsmbb}.\n\nAs the dust spectral index and temperature are difficult to separate in the frequency range considered for CMB studies \\citep[{i.e.}, Rayleigh-Jeans domain, see e.g.][]{betatcorr}, the moment expansion in harmonic space has  only been applied in the past with respect to $\\beta$, with the temperature being fixed to a reference value $T=T_0$ \\citep{Mangilli,Azzoni}. In the present paper, for the first time, the moment expansion in harmonic space is instead performed with respect to both $\\beta$ and $T$, as it was in real space in \\citet{RemazeillesmomentsILC}.\n\n\\subsubsection{Cross-power spectra}\n\nRelying on the derivation made by \\cite{Mangilli} and Eq.~\\ref{eq:momentint2temp}, we can explicitly write the cross-spectra between two maps $M_{\\nu_i}$ and $M_{\\nu_j}$ at frequencies $\\nu_i$ and $\\nu_j$, using the moment expansion in $\\beta$ and $T$ as follows:\n\n\\begin{align}\n    \\mathcal{D}_\\ell(\\nu_i \\times \\nu_j) &= \\frac{I_{\\nu_i}(\\beta_0(\\ell),T_0(\\ell))I_{\\nu_j}(\\beta_0(\\ell),T_0(\\ell))}{I_{\\nu_0}(\\beta_0(\\ell),T_0(\\ell))^2} \\cdot \\bigg\\{ \\nonumber \\\\[-0.5mm]\n   \n    0^{\\rm th}\\ \\text{order}\\;&\n    \\begin{cases}\n    & \\ \\mathcal{D}_\\ell^{A \\times A}\n    \\end{cases}\n    \\nonumber \\\\[-0.5mm]\n   \n    1^{\\rm st}\\ \\text{order}\\ \\beta\\;&\n    \\begin{cases}\n    &+\\mathcal{D}_\\ell^{A \\times \\omega^{\\beta}_1}\\left[ \\ln\\left(\\frac{\\nu_i}{\\nu_0}\\right) + \\ln\\left(\\frac{\\nu_j}{\\nu_0}\\right) \\right] \\nonumber \\\\\n    &+ \\mathcal{D}_\\ell^{\\omega^{\\beta}_1 \\times \\omega^{\\beta}_1} \\left[\\ln\\left(\\frac{\\nu_i}{\\nu_0}\\right)\\ln\\left(\\frac{\\nu_j}{\\nu_0}\\right) \\right]\\nonumber \\\\  \\end{cases}\\\\[-0.5mm]\n   \n    1^{\\rm st}\\ \\text{order}\\ T \\;&\n    \\begin{cases}\n    &+\\mathcal{D}_\\ell^{A \\times \\omega_1^T} \\left( \\Theta_i +  \\Theta_j -  2\\Theta_0\\right) \\\\\n    &+ \\mathcal{D}_\\ell^{\\omega_1^T \\times \\omega_1^T}\\Big(\\Theta_i - \\Theta_0\\Big)\\left(\\Theta_j - \\Theta_0\\right)\\nonumber\n    \\end{cases}\\\\[-0.5mm]\n   \n    1^{\\rm st}\\ \\text{order}\\ T\\beta \\;&\n    \\begin{cases}\n    &+ \\mathcal{D}_\\ell^{\\omega^{\\beta}_1 \\times \\omega_1^T} \\left[ \\ln\\left(\\frac{\\nu_j}{\\nu_0} \\right)\\Big( \\Theta_i - \\Theta_0\\Big) + \\ln\\left(\\frac{\\nu_i}{\\nu_0} \\right)\\left( \\Theta_j - \\Theta_0\\right)\\right]  \\nonumber \\\\\n    \\end{cases}\\\\[-0.5mm]\n   \n    2^{\\rm nd}\\ \\text{order}\\ \\beta \\;&\n    \\begin{cases}\n    &+ \\frac{1}{2} \\mathcal{D}_{\\ell}^{A \\times\\omega^{\\beta}_{2}}  \\left[ \\ln^2\\left(\\frac{\\nu_i}{\\nu_0}\\right)\n    + \\ln^2\\left(\\frac{\\nu_j}{\\nu_0}\\right)\n    \\right]\n    \\\\[-0.5mm]\n    &+ \\frac{1}{2} \\mathcal{D}_{\\ell}^{\\omega^{\\beta}_{1} \\times \\omega^{\\beta}_{2}}  \\Big[ \\ln \\left(\\frac{\\nu_i}{\\nu_0}\\right)  \\ln^2\\left(\\frac{\\nu_j}{\\nu_0}\\right)  \n    +\\ln\\left(\\frac{\\nu_j}{\\nu_0}\\right)\n    \\ln^2 \\left(\\frac{\\nu_i}{\\nu_0}\\right) \\Big] \n    \\\\[-0.5mm]\n    &+\\frac{1}{4} \\mathcal{D}_{\\ell}^{\\omega^{\\beta}_{2} \\times \\omega^{\\beta}_{2}}  \\left[\\ln^2 \\left(\\frac{\\nu_i}{\\nu_0}\\right)  \\ln^2 \\left(\\frac{\\nu_j}{\\nu_0}\\right) \\right]\n    \\,\n    \\end{cases}\\nonumber\\\\[-0.5mm]\n    &+ \\dots \\bigg\\},\n\\label{eq:moments}\n\\end{align}\n\n\\noindent where we use the following abbreviation: $\\Theta(\\nu_k,T_0(\\ell))\\equiv \\Theta_k$, so that $\\Theta_0=\\Theta(\\nu_0,T_0(\\ell))$, and we defined the moment expansion cross-power spectra between two moments $\\mathcal{M}$ and $\\mathcal{N}$ as\n\n\\begin{equation}\n    \\mathcal{C}_\\ell^{\\mathcal{M}\\times\\mathcal{N}} = \\sum_{m, m'=-\\ell}^{\\ell} \\mathcal{M}_{\\ell m} \\mathcal{N}_{\\ell m'},\\ {\\rm with}\\ (\\mathcal{M},\\mathcal{N})\\in\\left\\{A,\\omega^{\\beta}_1,\\omega^{T}_1, \\omega^{\\beta}_2,\\ \\dots\\right\\}.\n\\label{eq:moments_spectra}\n\\end{equation}\n\nIn the remainder of this article, we use the $\\mathcal{D}_\\ell$ quantity, which is a scaling of the angular power spectra, and is defined as \n\\begin{equation}\n    \\mathcal{D}_\\ell \\equiv \\frac{\\ell(\\ell +1)}{2 \\pi} \\mathcal{C}_\\ell.\n\\label{eq:dl}\n\\end{equation}\n\nEquation~\\ref{eq:moments} has been written using the expansion with respect to $\\beta$ at order two and $T$ at order one, as in Eq.~\\ref{eq:momentint2temp}. Nevertheless, the terms involving power spectra between order two in $\\beta$ and order one in $T$ have been neglected so as to match the needs of the implementation of our method in the following. \n\nHereafter, when we refer to \"order $k$\" at the angular power spectrum level, we are referring to moment expansion terms involving the pixel space moment up to order $k$. For example, $D_\\ell^{A\\times\\omega_1^T}$ and $D_\\ell^{\\omega^\\beta_1\\times\\omega_1^T}$ are order one, while $D_\\ell^{A\\times\\omega^\\beta_2}$, $D_\\ell^{\\omega^\\beta_1\\times\\omega^\\beta_2}$ and $D_\\ell^{\\omega^\\beta_2\\times\\omega^\\beta_2}$ are order two. At order zero, one retrieves the MBB description of the cross-angular power spectra SED $\\mathcal{D}_\\ell(\\nu_i\\times\\nu_j)$ as a function of the frequencies $\\nu_i$ and $\\nu_j$.\n\nThis formalism was originally introduced to analyze the complexity of intensity data in \\cite{Mangilli}. In the present work, we focus on $B$-mode polarization power spectra. This was put forward after analyzing the \\textit{Planck}{} and balloon-borne Large Aperture Submillimeter Telescope for Polarimetry (BLASTPol) data and finding that the polarization fraction appears to be constant in the far-infrared-to-millimetre wavelengths {\\citep{blastpol1,blastpol2}}. This allows us to assume that the same grain population is responsible for the total and polarized foreground emission \\citep{dustmodels}. As a result, intensity and polarization SED complexity may be similar.\nNevertheless, $Q$ and $U$ can have a different SED because of the polarization angle frequency dependence \\citep[see e.g.,][]{tassis,moment_polar} and so can $E$ and $B$. This could be a limitation when analyzing the dust $E$ and $B$ with a single moment expansion, especially when SED variations occur along the line of sight.\nEven when trying to model a single polarization component ---as we do in the present work, dealing only with $B$ modes--- it is not clear whether the distorted SED can be modeled in terms of $\\beta$ and $T$ moments only. Further work needs to be done to assess this question. However, they should not impact the present study in which variations along the line of sight are not simulated.\n\nModeling the complexity of the foreground signals by means of the moment expansion of the $B$-mode angular power spectrum has already been successfully applied to Simons Observatory \\citep{SimonsObservatory} simulated data  \\citep{Azzoni}. However, the approach taken by these latter authors is different from the one presented above. They apply a \\emph{minimal} moment expansion: assumptions are made to keep only the $\\mathcal{D}_\\ell^{\\omega^{\\beta}_1\\times\\omega^{\\beta}_1}$ and $\\mathcal{D}_\\ell^{A\\times\\omega^{\\beta}_2}$ parameters, which are modeled with a power-law scale dependence. These assumptions may not hold for experiments with higher sensitivity and observing wider sky patches. Furthermore, they assume a scale-invariant dust spectral index. In this work, on the other hand, we relax these assumptions in order to characterize the required spectral complexity of the dust emission for \\textit{LiteBIRD}{}.\n\n\\section{\\label{sec:sims}Simulations and cross-spectra estimation}\n\n\\subsection{\\label{sec:LiteBIRD}\\textit{LiteBIRD}{}}\n\n\\textit{LiteBIRD}{} is an international project proposed by the Japanese spatial agency (JAXA), which selected it in May 2019 as a strategic large class mission. The launch is planned for 2029 for a minimal mission duration of 3 years. \\citep{2020SPIE,Ptep}\n\n\\textit{LiteBIRD}{} is designed to realize a full sky survey of the CMB at large angular scales in order to look for the reionization bump of primordial $B$-modes and explore the scalar-to-tensor ratio ($r$) parameter space with a total uncertainty $\\delta r$ below $10^{-3}$, including foreground cleaning and systematic errors.\n\\textit{LiteBIRD}{} is composed of three telescopes observing in different frequency intervals: the Low-, Medium- and High-Frequency Telescopes (LFT, MFT and HFT).\nEach of the telescopes illuminates a focal plane composed of hundreds of polarimetric detectors. The whole instrument will be cooled down to 5\\,K \\citep{LiteBIRDUpdatedDesign} while the focal plane will be cooled down to 100\\,mK \\citep{LBsubK}. In order to mitigate the instrumental systematic effects, the polarization is modulated by a continuously rotating half-wave plate. \\textit{LiteBIRD}{} will observe the sky in 15 frequency bands from 40 to 402\\,GHz. Table~\\ref{tab:litetab} gives the details of the frequency bands and their sensitivities in polarization \\citep[adapted from][see Sect.~\\ref{sec:Instrsim}]{2020SPIE}.\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{cccc}\n        \\multirow{2}{*}{Telescope} & Frequency & Sensitivity  $\\sigma^{\\rm noise}_{Q,U}(\\nu)$ & $\\boldsymbol \\theta_{\\rm FWHM}$  \\\\\n         & [GHz]&  [$\\mu$K$\\cdot$arcmin] & {arcmin}\\\\\n        \\hline\n         LFT & 40.0 & 37.42 & {70.5}     \\\\\n        LFT & 50.0 & 33.46 & {58.5} \\\\\n        LFT & 60.0 & 21.31 & {51.1} \\\\\n        LFT & 68.0 & {19.91\/31.77} & {41.6\/47.1} \\\\\n        LFT & 78.0 & {15.55\/19.13} & {36.9\/43.8} \\\\\n        LFT & 89.0 & {12.28\/28.77} & {33.0\/41.5} \\\\\n     LFT\/MFT & 100.0 & {10.34\/8.48} & {30.2\/37.8} \\\\\n     LFT\/MFT & 119.0 & {7.69\/5.70} & {26.3\/33.6} \\\\\n     LFT\/MFT & 140.0 & {7.25\/6.38} & {23.7\/30.8}  \\\\\n         MFT & 166.0 & 5.57 & {28.9} \\\\\n     MFT\/HFT & 195.0 & {7.05\/10.50} & {28.0\/28.6} \\\\\n         HFT & 235.0 & 10.79 & {24.7}\\\\\n        HFT & 280.0 & 13.8 & {22.5} \\\\\n        HFT & 337.0 & 21.95 & {20.9} \\\\\n        HFT & 402.0 & 47.45 & {17.9} \\\\\n\\end{tabular}\n\\caption{\\footnotesize Instrumental characteristics of \\textit{LiteBIRD}{} used in this study \\citep[adapted from][see Sect.~\\ref{sec:Instrsim}]{2020SPIE}. Some frequency bands are shared by two different telescopes or detector arrays. If so, the two values of polarization sensitivities $\\sigma^{\\rm noise}_{Q,U}(\\nu)$ and instrumental beam full width at half maximum $\\theta_{\\rm FWHM}$ are displayed on the same line.}\n\\label{tab:litetab}\n\\end{center}\n\\end{table}\n\n\\subsection{Components of the simulations}\n\\label{sec:ingredients}\n\nWe build several sets of \\textit{LiteBIRD}{} sky simulations. These multi-frequency sets of polarized sky maps are a mixture of CMB, dust, and instrumental noise. The simulations are made at the nine highest frequencies accessible by the instrument ($\\geq 100$ GHz), where dust is the predominant source of foreground contamination.\nFor every studied scenario, we built $N_{\\rm sim} = 500$ simulations, each composed of a set of $N_{\\rm freq}=9$ pairs of sky maps $(Q,U)$ built using the {\\sc HEALPix} package, with $N_{\\rm side} = 256$ \\citep{healpix}. All the signals will be expressed in $\\mu{\\rm K}_{\\rm CMB}$ units.\n\n\\subsubsection{Cosmic microwave background signal}\n\nTo generate the CMB signal, we use the {\\it Code for Anisotropies in the Microwave Background } \\citep[CAMB,][]{CAMB} to create a fiducial angular power spectrum from the best-fit values of cosmological parameters estimated by the recent \\textit{Planck}{} data analysis \\citep{PlanckOverview}.\n\nFor the $B$-modes, we consider the two different components of the spectrum: lensing-induced and primordial (tensor), so that $\\mathcal{D}_{\\ell}^{BB} =\\mathcal{D}_{\\ell}^{{\\rm lensing}} + r_{\\rm sim} \\cdot \\mathcal{D}_{\\ell}^{{\\rm tensor}}$, where $\\mathcal{D}_{\\ell}^{{\\rm tensor}}$ refers to the tensor $B$-modes for $r=1$ and $r_{\\rm sim}$ labels the input values of the tensor-to-scalar ratio $r$ contained in the simulation. We use two different values throughout this work: $r_{\\rm sim}=0$, which is used in the present work as the reference simulations and $r_{\\rm sim}=10^{-2}$ used for consistency checks when the CMB primordial signal is present.\n\nFor all simulations, we then generate the Stokes $Q$ and $U$ CMB polarization Gaussian realization maps $S^{\\rm CMB}_{\\nu,r_{\\rm sim}}$ from the angular power spectra using the {\\tt synfast} function of {\\sc HEALPix}.\n\n\\subsubsection{Foregrounds: dust}\n\nOur study focuses on high frequencies ($\\geq 100$\\,GHz) only, where thermal dust emission is the main source of polarized foreground as mentioned in Sect.~\\ref{sec:intro}. We make use of two different scenarios of increasing complexity included in the {\\sc PySM} \\citep{Pysm} and one of intermediate complexity not included in the {\\sc PySM}:\n\n\\begin{itemize}\n    \\item {\\tt d0}, included in the {\\sc PySM}: the dust polarization $Q$ and $U$ maps are taken from $S^{Planck}_{\\nu=353}$, the \\textit{Planck}{} 2015 data at 353\\,GHz \\citep{planck_2015_overview}, extrapolated to a frequency $\\nu$ using the MBB given in Eq.~\\ref{eq:MBB} with a temperature $T_0=T_{\\tt d0}=20$\\,K and spectral index $\\beta_0=\\beta_{\\tt d0}=1.54$ constant over the sky:\n    \n    \\begin{equation}S^{\\rm dust}_\\nu=S_\\nu^{\\tt d0}=\\frac{I_\\nu(\\beta_{\\tt d0},T_{\\tt d0})}{I_{\\nu_0}(\\beta_{\\tt d0},T_{\\tt d0})}\\cdot S^{Planck}_{353},\n    \\end{equation}\n    \n    \\item {\\tt d1T}, introduced here: the dust polarization $Q$ and $U$ maps are also taken from \\citet{planck_2015_overview} but they are extrapolated to a frequency $\\nu$ using the MBB given in Eq.~\\ref{eq:MBBn}, with spatially varying  spectral index $\\beta(\\vec{n})$, as in {\\tt d1} and a fixed temperature $T_0=T_{\\tt d1T}=21.9$\\,K, obtained as the mean of the \\textit{Planck}{} {\\sc Commander} dust temperature map  \\citep{planck_2015_commander} on our $f_{\\rm sky}=0.7$ sky mask: \n    \n    \\begin{equation}\n        S^{\\rm dust}_\\nu=S_\\nu^{\\tt d1T}=\\frac{I_\\nu(\\beta(\\vec{n}),T_{\\tt d1T})}{I_{\\nu_0}(\\beta(\\vec{n}),T_{\\tt d1T})}\\cdot S^{Planck}_{353}.\n    \\end{equation}\n    \n    \\item {\\tt d1}, included in the {\\tt PySM}: similar to { \\tt d1T} with both a spatially varying temperature $T(\\vec{n})$ and spectral index $\\beta(\\vec{n})$ obtained from the \\textit{Planck}{} data using the {\\sc Commander} code \\citep{planck_2015_commander}: \n    \n    \\begin{equation}\n        S^{\\rm dust}_\\nu=S_\\nu^{\\tt d1}=\\frac{I_\\nu(\\beta(\\vec{n}),T(\\vec{n}))}{I_{\\nu_0}(\\beta(\\vec{n}),T(\\vec{n}))}\\cdot S^{Planck}_{353}.\n    \\end{equation}\n\n\\end{itemize}\n\n\\subsubsection{ \\label{sec:Instrsim}Instrumental noise}\n\nThe band polarization sensitivities  $\\sigma^{\\rm noise}_{Q,U}(\\nu)$ are derived from the noise equivalent temperature (NET) values converted into $\\mu$K$\\cdot$arcmin for each telescope (LFT, MFT and HFT). As seen in Table~\\ref{tab:litetab}, some frequency bands are overlapping between two telescopes. In this situation, we take the mean value of the two NETs, weighted by the beam full width at half maximum (FWHM) $\\theta$ as:\n\n\\begin{equation}\n    \\sigma^{\\rm noise}_{Q,U}(\\nu_{\\rm overlapping}) = \\sqrt{\\frac{1}{ \\left(\\frac{\\theta_{\\rm min}}{{\\theta_{\\rm max}}}\\sigma^{\\rm noise}_{Q,U}(\\nu_{\\theta_{\\rm min}})\\right)^{-2} + \\left(\\sigma^{\\rm noise}_{Q,U}(\\nu_{\\theta_{\\rm max}})\\right)^{-2} }},\n\\end{equation}\n\n{\\noindent where $\\theta_{\\rm min}$ is the smallest FWHM among the two and $\\theta_{\\rm max}$ the largest. The band polarization sensitivities} are displayed in Table~\\ref{tab:litetab}. For every simulation, the noise component $N_\\nu$ is generated in every pixel of the maps with a Gaussian distribution centered on zero, with standard deviation $\\sigma^{\\rm noise}_{Q,U}(\\nu)$ weighted by the pixel size (and $\\sqrt{2}\\cdot\\sigma^{\\rm noise}_{Q,U}(\\nu)$ for the maps used to compute the auto-power spectra, see Sect.~\\ref{sec:spectra_estimation}).\n\nFor simplicity, we choose to ignore beam effects in our simulations, assuming they can be taken into account  perfectly. Simulations are thus produced at infinite (0\\,arcmin) resolution and no beam effect is corrected for when estimating the angular power spectrum. This is equivalent to convolving the maps by Gaussian beams of finite resolution and correcting the power spectra for the associated Gaussian beam window functions.\n\n\\subsection{Combining signals and building the simulated maps}\n\n\\label{sec:simulations}\n\nThe simulated $(Q,U)$ maps $M_\\nu$, for a given simulation, can be expressed as the sum:\n\n\\begin{equation}\nM_\\nu=S^{\\rm CMB}_{\\nu,r_{\\rm sim}}+S^{\\rm dust}_\\nu+N_\\nu.\n\\label{eq:map}\n\\end{equation}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale = 0.2]{Figures\/simulations.pdf}\n    \\caption{\\footnotesize Mean value over the $N_{\\rm sim}$ simulations of the $B$-mode angular power spectra $\\mathcal{D}_{\\ell}(\\nu_i \\times \\nu_j)$ for the {\\tt d1c} simulation type, with $r_{\\rm sim}=0$. The color bar spans all the $N_{\\rm cross}$ spectra $\\mathcal{D}_{\\ell}(\\nu_i \\times \\nu_j)$, associated to their reduced cross-frequency $\\nu_{\\rm red.}=\\sqrt{\\nu_i \\nu_j}$, from 100\\,GHz (dark red) to 402\\,GHz (dark blue). The input CMB lensing power spectrum is shown as a black dashed line.}\n    \\label{fig:simulations}\n\\end{figure*}\n\n\nCosmic microwave background and noise are simulated stochastically: for each simulation, we generate a new realization of the CMB maps $S^{\\rm CMB}_{\\nu,r_{\\rm sim}}$ and the noise maps $N_\\nu$. The dust map $S^{\\rm dust}_\\nu$ is the same for each simulation, at a given frequency.\n\nHereafter, we use the notation {\\tt d0}, {\\tt d1T,} and {\\tt d1} to refer to simulations containing only dust and \\textit{LiteBIRD}{} noise, {\\tt d0c}, {\\tt d1Tc,} and {\\tt d1c} for simulations including CMB, dust, and \\textit{LiteBIRD}{} noise and, finally, and {\\tt c} for the simulation containing only CMB and \\textit{LiteBIRD}{} noise. The different components present in these different {simulation types} are summarized in Table~\\ref{tab:sims}.\n\n\\begin{table}[t!]\n\\centering\n        \\begin{tabular}{c !{\\color{white}\\vrule width 1pt} c!{\\color{white}\\vrule width 1pt}c!{\\color{white}\\vrule width 1pt}c!{\\color{white}\\vrule width 1pt}c!{\\color{white}\\vrule width 1pt}c}\n        & $S^{\\rm CMB}_{\\nu,r_{\\rm sim}}$ & $S^{\\tt d0}_\\nu$ & $S^{\\tt d1T}_\\nu$ & $S^{\\tt d1}_\\nu$ & $N_\\nu$ \\\\[0.5ex]\\noalign{\\color{white}\\hrule height 1pt}\n        {\\tt c} & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{myred}$\\times$  & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        {\\tt d0} & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;  & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\n        \\noalign{\\color{white}\\hrule height 1pt}\n        {\\tt d1T} & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$  & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\n        \\noalign{\\color{white}\\hrule height 1pt}\n        {\\tt d1} & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$  & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\n        \\noalign{\\color{white}\\hrule height 1pt}\n        {\\tt d0c} & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;  & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\n        \\noalign{\\color{white}\\hrule height 1pt}\n        {\\tt d1Tc} & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{myred}$\\times$  & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\n        \\noalign{\\color{white}\\hrule height 1pt}\n        {\\tt d1c} & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{myred}$\\times$  & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\n    \\end{tabular}\n\n\\caption{\\footnotesize Summary of the different components present in the simulated maps $M_\\nu$ in Eq.~\\ref{eq:map}, for every \\emph{simulation type}. A tick on a green background signifies that the component is present in the simulations, red with a cross symbol shows that it is absent.}\n\\label{tab:sims}\n\\end{table}\n\n\\subsection{ \\label{sect:spectra} Angular power spectra of the simulations}\n\n\\subsubsection{\\label{sect:mask}Mask}\n\nA mask is applied on the simulated maps presented in Sect.~\\ref{sec:simulations} in order to exclude the Galactic plane from the power-spectrum estimation. The mask is created by setting a threshold on the polarized intensity ($P=\\sqrt{Q^2+U^2}$) of the \\textit{Planck}{} 353\\,GHz map \\citep{PlanckOverview} \\footnote{\\url{http:\/\/pla.esac.esa.int\/pla\/}}, smoothed with a $10^{\\circ}$ beam. In order to keep $f_{\\rm sky}= 0.7$,$f_{\\rm sky}= 0.6,$ and $f_{\\rm sky}= 0.5$, the cut is applied at 121\\,$\\mu$K, 80\\,$\\mu$K,  and 53\\,$\\mu$K, respectively. We then realize a C2 apodization of the binary mask with a scale of $5^{\\circ}$ using {\\sc Namaster} \\citep{namaster}. The resulting Galactic masks are displayed in Fig.~\\ref{fig:mask}. These masks are similar to those used in \\citet{PlanckDust2}.\n\n\\subsubsection{Estimation of the angular power spectra}\n\\label{sec:spectra_estimation}\n\nWe use the {\\sc Namaster}\\footnote{\\url{https:\/\/github.com\/LSSTDESC\/NaMaster}} software \\citep{namaster} to compute the angular power spectra of each simulation. {\\sc Namaster} allows us to correct for the $E$ to $B$ leakage bias due to the incomplete sky coverage. Therein we use a \\emph{purification} process to suppress the effect of the $E$ to $B$ leakage in the variance. For every simulation, from the set of maps $M_{\\nu_i}$,  we compute all the possible auto-frequency and cross-frequency spectra $\\mathcal{D}_\\ell({\\nu_i\\times\\nu_j})\\equiv\\mathcal{D}_\\ell({M_{\\nu_i}\\times M_{\\nu_j}})$ with\n\n\\begin{align}\n\\nu_i\\times \\nu_j \\in \\left\\{\\right.&100\\times 100, 100 \\times 119, 100 \\times 140,\\   \\dots, 100\\times402,\\nonumber\\\\\n&119\\times140,\\ \\dots, 119\\times402,\\nonumber\\\\[-2mm]\n&\\qquad\\qquad\\vdots\\nonumber\\\\\n&337\\times337, 337\\times402,\\nonumber\\\\\n&\\left.\\!\\!402 \\times 402 \\right\\},\n\\label{eq:all_cross}\n\\end{align}\n\n\\noindent leading to $N_{\\rm cross}=N_{\\rm freq}\\cdot(N_{\\rm freq}+1)\/2=45$ cross-frequency spectra. These spectra are displayed in Fig.~\\ref{fig:simulations} for the case of the {\\tt d1c} simulation\ntype.\\\\\n\nIn order to avoid noise auto-correlation in the auto-spectra ({i.e.}, $\\mathcal{D}_\\ell(\\nu_i\\times\\nu_j)$ when $i=j$), the latter are estimated in a way that differs slightly from what is presented in Sect.~\\ref{sec:Instrsim}. We simulate two noise-independent data subsets at an observing frequency $\\nu_i$, with a noise amplitude $\\sqrt{2}$ higher than that of the frequency band, and compute the cross-angular power spectrum between those. Thus, $\\mathcal{D}_\\ell(\\nu_i\\times\\nu_i)$ is free from noise auto-correlation bias at the expense of multiplying the noise amplitude in the spectrum by a factor of two. This approach is similar to that commonly used by the \\textit{Planck}{} Collaboration \\citep[see e.g.,][]{PlanckDust,PlanckDust2,tristram}.\n\n\n\nThe spectra are evaluated in the multipole interval $\\ell \\in [1,200]$ in order to be able to focus on the reionization and recombination bumps of the primordial $B$-modes spectra. \nThe spectra are binned in $N_{\\ell}=20$ bins of size $\\Delta \\ell = 10$ using {\\sc Namaster}. The same binning is applied throughout this article such that, in the following, the multipole $\\ell$ denotes the multipole bin of size $\\Delta\\ell=10$ centered on $\\ell$\\footnote{ The $N_\\ell$ multipole bins are centered on the following $\\ell$ values: $[6.5,  16.5,  26.5,  36.5,  46.5,  56.5,  66.5,  76.5,  86.5,\n96.5, 106.5,\\\\ 116.5, 126.5, 136.5, 146.5, 156.5, 166.5, 176.5,\n186.5, 196.5]$}.\n\nFrom the sets of $(Q,U)$ maps, {\\sc Namaster} computes the $\\mathcal{D}_\\ell^{EE}$, $\\mathcal{D}_\\ell^{BB}$, and $\\mathcal{D}_\\ell^{EB}$ angular power spectra; for the sake of the present analysis, we keep only $\\mathcal{D}_\\ell^{BB}$. Hence, when we discuss or analyze power spectra, we are referring to the $B$-mode power spectra  $\\mathcal{D}_\\ell^{BB}$. All spectra are expressed in $(\\mu{\\rm K}_{\\rm CMB})^2$.\n\n\\section{\\label{sec:fit}Best-fit implementation}\n\n\nIn order to characterize the complexity of the dust SED that will be measured by \\textit{LiteBIRD}{}, we modeled the angular power spectra of our simulations described in Sect.~\\ref{sec:sims} over the whole frequency and multipole ranges with the moment expansion formalism introduced in Sect.~\\ref{sec:formalism}.\n\n\\subsection{General implementation}\n\\label{sec:fit_dust}\n\nFor each multipole $\\ell$, we ordered the angular power spectra $\\mathcal{D}_\\ell^{BB}(\\nu_i\\times\\nu_j$) as in Eq.~\\ref{eq:all_cross} in order to build a SED that is a function of both $\\nu_i$ and $\\nu_j$. We fit this SED with models, as in Eq.~\\ref{eq:moments} for example, using a Levenberg-Marquardt $\\chi^2$ minimization with {\\tt mpfit} \\citep{mpfit}\\footnote{\\url{https:\/\/github.com\/segasai\/astrolibpy\/tree\/master\/mpfit}}. All the fits performed with {\\tt mpfit} were also realized with more computationally heavy Monte Carlo Markov Chains (MCMC) with {\\tt emcee} \\citep{emcee}, giving compatible results, well within the error bars.\n\nThe reduced $\\chi^2$ minimization is given by\n\n\\begin{equation}\n    \\chi^2 = \\frac{1}{N_{\\rm d.o.f.}}\\vec{R}^T\\mathbb{C}^{-1}\\vec{R},\n\\label{eq:chi2}\n\\end{equation}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale = 0.3]{Figures\/corr2.png}\n    \\caption{\\footnotesize Correlation matrix (${\\rm Corr}_{\\ell\\ell'} \\equiv \\mathbb{C}_{\\ell\\ell'}\/\\sqrt{\\mathbb{C}_{\\ell\\ell}\\mathbb{C}_{\\ell'\\ell'}}$) for the $N_{\\rm sim}$ simulations in {\\tt d1c}. Every block represents a value of $\\ell$ and contains the ordered $N_{\\rm cross} = 45$ cross-spectra. The red squares represent the truncation of the full covariance matrix applied in the analysis (kept entries in red, other entries set to zero).}\n    \\label{fig:cov}\n\\end{figure}\n\n\\noindent where $N_{\\rm d.o.f.}$ is the number of degrees of freedom and $\\mathbb{C}$ is the covariance matrix of our $N_{\\rm sim}$ simulations, represented in Fig.~\\ref{fig:cov}, of dimension $(N_{\\ell}\\cdot N_{\\rm cross})^2$:\n\\begin{equation}\n    \\mathbb{C}_{\\ell,\\ell'}^{i\\times j,k\\times l} = {\\rm cov}\\left(\\mathcal{D}^{\\rm sim}_\\ell (\\nu_i \\times \\nu_j),\\mathcal{D}^{\\rm sim}_{\\ell'} (\\nu_k \\times \\nu_l)\\right).\n\\label{eq:cov}\n\\end{equation}\n\nThe entire covariance matrix $\\mathbb{C}$ is, in general, not invertible. To avoid this, we kept only the $\\ell=\\ell'$ block-diagonal of $\\mathbb{C}$ with the strongest correlation values\\footnote{${\\rm Corr}_{\\ell\\ell'} \\equiv \\mathbb{C}_{\\ell\\ell'}\/\\sqrt{\\mathbb{C}_{\\ell\\ell}\\mathbb{C}_{\\ell'\\ell'}}$}, as well as the $(\\ell=6.5,\\ell'=16.5)$ off-diagonal blocks showing a significant anti-correlation, as illustrated in Fig.~\\ref{fig:cov}. It was then possible to invert the thus-defined truncated correlation matrix  with the required precision most of the time. \n\nIn the case of the {\\tt d1} simulation type, we experienced a fit convergence issue for $\\sim20$\\,\\% of the simulations, leading to a very large $\\chi^2$. In order to overcome this problem, two options lead to identical results: throwing away the outliers from the analysis or fitting using only the block-diagonal matrix (i.e., the $\\ell=6.5$, $\\ell'=16.5$ block is set to zero). This last option solves the conversion issue while providing sufficient precision. The results presented in the following are using the block-diagonal matrix when the simulation type is {\\tt d1}.\n\nFinally, in Eq.~\\ref{eq:chi2}, $\\vec{R}$ is the residual vector associated with every simulation of size $N_{\\ell}\\times N_{\\rm cross}$:\n\n\\begin{equation}\n\\vec{R} =\n\\begin{pmatrix}\n\\mathcal{R}_{\\ell = 6.5}(100 \\times 100) \\\\\n\\mathcal{R}_{\\ell = 6.5}(100 \\times 119) \\\\\n\\vdots \\\\\n\\mathcal{R}_{\\ell = 16.5}(100 \\times 100)\\\\\n\\vdots \\\\\n\\mathcal{R}_{\\ell = 196.5}(402 \\times 402)\\\\\n\\end{pmatrix}, \n\\end{equation}\n\n\\noindent with $\\mathcal{R}_\\ell (\\nu_i \\times \\nu_j) = \\mathcal{D}^{\\rm sim}_\\ell (\\nu_i \\times \\nu_j) - \\mathcal{D}^{\\rm model}_\\ell(\\nu_i \\times \\nu_j)$.\n\nThe expression used for the model to fit is given by:\n\n\\begin{equation}\n\\begin{split}\n  \\mathcal{D}_{\\ell}^{\\rm model}(\\nu_i \\times \\nu_j) &= \\mathcal{D}_{\\ell}^{{\\rm dust}}\\left(\\beta_0(\\ell), T_0(\\ell), \\mathcal{D}^{\\mathcal{M}\\times\\mathcal{N}}_{\\ell}(\\nu_i\\times\\nu_j)\\right) \\\\ & + A_{\\rm lens} \\cdot \\mathcal{D}_{\\ell}^{{\\rm lensing}} + r \\cdot  \\mathcal{D}_{\\ell}^{{\\rm tensor}}, \n\\label{eq:model}\n\\end{split}\n\\end{equation}\n\n\\noindent  where $A_{\\rm lens}$ is not a free parameter and will remain fixed to zero (when there is no CMB, simulation types {\\tt d0}, {\\tt d1T,} and {\\tt d1}) or one (when the CMB is included, simulation types {\\tt d0c}, {\\tt d1Tc} and {\\tt d1c}). We leave the question of the impact of dust modeling with moments on the lensing measurement for future work. In Eq.~\\ref{eq:model}, the free parameters can thus be $\\beta_{0}(\\ell)$, $T_{0}(\\ell)$, and $\\mathcal{D}^{\\mathcal{M}\\times\\mathcal{N}}_{\\ell}(\\nu_i\\times\\nu_j)$ and the tensor-to-scalar ratio $r$. The estimated value of $r$ is referred to as $\\hat{r}$\n\nNo priors on the parameters are used in order to explore the parameter space with minimal assumptions. Finally, a frequency-dependent conversion factor is included in $\\mathcal{D}_{\\ell}^{{\\rm dust}}$ -- from (MJy$\\cdot$sr$^{-1})^2$ to $(\\mu$K$_{\\rm CMB})^2$ -- to express the dust spectra in ($\\mu$K$_{\\rm CMB})^2$ units. In those units, $\\mathcal{D}_{\\ell}^{{\\rm lensing}}$ and $\\mathcal{D}_{\\ell}^{\\rm tensor}$ are frequency-independent.\n\nTo mitigate the impact of outliers in our simulations, all the final values of the best-fit parameters and $\\chi^2$ distributions are represented by their median and median absolute deviations over $N_{\\rm sim}$ values. For the tensor-to-scalar ratio $\\hat{r}$, we chose to represent all the best-fit values from the $N_{\\rm sim}$ simulations in a histogram and we assume its distribution is normal. Fitting a Gaussian curve on this histogram and getting the mean and standard deviation gives us the final values of $\\hat{r}$ and $\\sigma_{\\hat{r}}$ presented in the paper.\n\n\n\n\\subsection{Implementation for the dust component}\n\\label{sec:fit_general}\n\nFor the dust component, we consider four different \\emph{fitting schemes}, corresponding to four expressions for the dust model $\\mathcal{D}^{\\rm dust}_\\ell$ in Eq.~\\ref{eq:model}, which are referred to as \"MBB\", \"$\\beta$-1\", \"$\\beta$-$T$\", and \"$\\beta$-2\". Each of them corresponds to a truncation of Eq.~\\ref{eq:moments}, keeping only some selected terms of the moment expansion: MBB stands for those of the modified black body, $\\beta$-1 for those of the expansion in $\\beta$ at first order, $\\beta$-2 for the expansion in $\\beta$ at second order, and $\\beta$-$T$ for the expansion in both $\\beta$ and $T$ at first order. We chose the $\\beta$-1 and $\\beta$-2 truncations based on the studies of \\citet{Mangilli} and \\citet{Azzoni}, where the dust SED moment expansion is performed only with respect to $\\beta$. The $\\beta$-$T$ fitting scheme is instead the first-order truncation in both $\\beta$ and $T$, introduced here for the first time at the power spectrum level. The parameters fitted in each of these fitting schemes are summarized in Table~\\ref{tab:parameter}. We note that the $\\beta$-2 and $\\beta$-$T$ fitting schemes share the same number of free parameters. Finally, when we fit $\\hat{r}$ at the same time as the dust parameters, the fitting schemes will be referred to as $r$MBB, $r\\beta$-1, $r\\beta$-$T,$ and $r\\beta$-2.\n\nDifferent physical processes are expected to occur at different angular scales, leading to different SED properties. Thus, we estimate the dust-related parameters with one parameter per multipole bin. As an example, we estimate $\\beta_0 = \\beta_0(\\ell)$ and $T_0 = T_0(\\ell)$ to be able to take into account their scale dependence, at the cost of increasing the number of free parameters in our model. This is also true for the higher order moments. On the other hand, $\\hat{r}$ is not scale dependent and, when it is fitted, we add one single parameter over the whole multipole range.\n\nIn \\cite{Mangilli}, the first-order moment expansion parameter $\\mathcal{D}_\\ell^{A\\times\\omega^\\beta_1}$ is considered to be  the leading order correction to the MBB spectral index. We applied a similar approach in the present work, extending it to the dust temperature when it is fitted. In our pipeline, we proceed iteratively:\n\n\\begin{enumerate}\n\\item(i)  we fit $\\beta_0(\\ell)$ and $T_0(\\ell)$ at order zero (MBB), for each $\\ell$, \n\n\\item (ii) we fix $\\beta_0(\\ell)$ and $T_0(\\ell)$ and fit the higher order parameters, as in Eq.~\\ref{eq:moments}, (iii) we update the $\\beta_0(\\ell)$ to $\\beta_{\\rm corr}(\\ell)$ {(and $T_0(\\ell)$ to $T_{\\rm corr}(\\ell)$ in the case of $\\beta$-$T$)} as:\n\n\\begin{equation}\n    \\beta_{\\rm corr}(\\ell) = \\beta_0(\\ell) + \\frac{\\mathcal{D}^{A \\times \\omega^{\\beta}_1 }_{\\ell}}{\\mathcal{D}^{A \\times A}_{\\ell}},\\quad T_{\\rm corr}(\\ell) = T_0(\\ell) + \\frac{\\mathcal{D}^{A \\times \\omega^{T}_1 }_{\\ell}}{\\mathcal{D}^{A \\times A}_{\\ell}},\n\\label{eq:iteration}\n\\end{equation}\n\\item[iv)] and we iterate from (ii) fixing $\\beta_0(\\ell)=\\beta_{\\rm corr}(\\ell)$, until $\\mathcal{D}_\\ell^{A \\times \\omega^{\\beta}_1}$ converges to be compatible with zero (and $T_0(\\ell)=T_{\\rm corr}(\\ell)$, until $\\mathcal{D}_\\ell^{A \\times \\omega^{T}_1}$ converges to zero in the case of $\\beta$-$T$).\n\\end{enumerate}\nWe used three such iterations, which we found to be sufficient to guarantee the convergence. As the moment expansion is a nonorthogonal and incomplete basis \\citep{Chluba}, this iterative process is performed to ensure that the expansions up to different orders share the same $\\beta_0(\\ell)$ and $T_0(\\ell)$ with $\\mathcal{D}^{A \\times \\omega^{\\beta}_1 }_\\ell=0$ and $\\mathcal{D}^{A \\times \\omega^{T}_1 }_\\ell=0$.\n\n\n\\begin{table}[t!]\n\\centering\n        \\normalsize\n        \\begin{tabular}{c!{\\color{white}\\vrule width 1pt}c!{\\color{white}\\vrule width 1pt}c!{\\color{white}\\vrule width 1pt}c!{\\color{white}\\vrule width 1pt}c}\n        & MBB & $\\beta$-1 & $\\beta$-$T$ & $\\beta$-2\\\\\n        \\noalign{\\color{white}\\hrule height 1pt}\n        $N_{\\rm param.}$ & \\cellcolor{mygrey} $3N_\\ell$& \\cellcolor{mygrey} $2N_\\ell$ & \\cellcolor{mygrey} $5N_\\ell$ & \\cellcolor{mygrey} $5N_\\ell$ \\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $\\beta_0(\\ell)$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;& \\cellcolor{myyellow}$\\circ$ & \\cellcolor{myyellow}$\\circ$ & \\cellcolor{myyellow}$\\circ$\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $T_0(\\ell)$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;& \\cellcolor{myred}$\\times$ & \\cellcolor{myyellow}$\\circ$ & \\cellcolor{myred}$\\times$\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{A\\times A}$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;& \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{A\\times\\omega_1^\\beta}$ & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{\\omega_1^\\beta\\times\\omega_1^\\beta}$ & \\cellcolor{myred}$\\times$& \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{A\\times\\omega_1^T}$ & \\cellcolor{myred}$\\times$& \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; &\\cellcolor{myred}$\\times$\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{\\omega_1^T\\times\\omega_1^T}$ & \\cellcolor{myred}$\\times$& \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{myred}$\\times$\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{\\omega_1^\\beta\\times\\omega_1^T}$ & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{myred}$\\times$\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{A\\times\\omega_2^\\beta}$ & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{\\omega_1^\\beta\\times\\omega_2^\\beta}$ & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{\\omega_2^\\beta\\times\\omega_2^\\beta}$ & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\\noalign{\\color{white}\\hrule height 1pt}\n    \\end{tabular}\n    \\normalsize\n\\caption{\\footnotesize Summary of the fitted parameters in the four dust moment expansion \\emph{fitting schemes} we consider (MBB, $\\beta$-1, $\\beta$-$T,$ and $\\beta$-2), in Eq.~\\ref{eq:moments}. A tick on a green background signifies that the parameter is fitted, red with a cross symbol shows that the parameter is not fitted, and a circle symbol on yellow means that the parameter is fixed and corrected through an iterative process as presented in Sect.~\\ref{sec:fit_general}. $\\mathcal{D}_\\ell^{A\\times A}$ is fixed to the MBB best-fit value in the case of $\\beta$-1, $\\beta$-$T,$ and $\\beta$-2 and all the other moments are set to zero when they are not fitted. When $\\hat{r}$ is fitted at the same time, the fitting schemes are denoted $r$MBB, $r\\beta$-1, $r\\beta$-$T,$ and $r\\beta$-2, and they have one more parameter than the number of parameters reported in the first line.}\n\\label{tab:parameter}\n\\end{table}\n\n\\section{\\label{sec:results} Results}\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/chi2dustgauss_d0.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/chi2dustgauss_d1T.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/chi2dustgauss_d1.pdf}\n    \\caption{\\footnotesize Median of the reduced $\\chi^2$ in every multipole bin $\\ell$, for all the $N_{\\rm sim}$ simulations of {\\tt d0} (top, orange), {\\tt d1T} (middle, green) and {\\tt d1} (bottom, blue), on $f_{\\rm sky}=0.7$. The reduced $\\chi^2$ values are reported for the four different fitting schemes: MBB (circles), $\\beta$-1 (crosses), $\\beta$-$T$ (diamonds) and $\\beta$-2 (triangles). The values for the four fitting schemes are shifted from each others by $\\ell=2$, in order to distinguish them. The black dashed line represents $\\chi^2_{\\rm red}=1$.}\n    \\label{fig:chi2dust}\n\\end{figure}\n\nIn this section, we present our evaluation of the best-fit parameters for the different {fitting schemes} presented in Sect.~\\ref{sec:fit_dust} on the $B$-mode cross-angular power spectra computed from the different {simulation types} presented in Sect.~\\ref{sec:simulations} and on the Galactic mask keeping $f_{\\rm sky}=0.7$, which is defined in Sect.~\\ref{sect:mask}.\nWe first tested the simulation types containing only dust and noise in order to calibrate the dust complexity of our data sets in Sect.~\\ref{sec:results_dust_only}. We then used CMB only plus noise simulations to assess the minimal error on $\\hat{r}$ in Sect.~\\ref{sec:c} and, finally, we explored the dust, CMB, and noise simulation types to assess the impact of the dust complexity on $\\hat{r}$ in Sect.~\\ref{sec:results_dust_CMB}.\n\n\\subsection{Dust only}\n\\label{sec:results_dust_only}\n\nTo evaluate the amplitude of the dust moment parameters contained in the dust simulations in the absence of CMB, we ran the fitting schemes presented in Sect.~\\ref{sec:fit_dust} in the three simulation types {\\tt d0}, ${\\tt d1T,}$ and ${\\tt d1}$ presented in Sect.~\\ref{sec:simulations}. In these cases, $A_{\\rm lens}$ and $r$ in Eq.~\\ref{eq:model} are both fixed to zero and the fitted parameters are given  in Table~\\ref{tab:parameter} for every fitting scheme.\n\n\\subsubsection{{\\tt d0}}\n\\label{sec:d0}\n\nThe {\\tt d0} dust maps presented in Sect.~\\ref{sec:ingredients} extrapolate between frequency bands with a MBB SED with constant parameters over the sky: $\\beta_{\\tt d0} = 1.54$ and $T_{\\tt d0} = 20$\\,K. We performed the fit with the four fitting schemes presented in Sect.~\\ref{sec:fit_dust}.\n\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/betadustgauss.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/Tgauss.pdf}\n    \\caption{\\footnotesize (Top): Median of the best fit values of $\\beta_0(\\ell)$ in {\\tt d0} (orange), {\\tt d1T} (green), and {\\tt d1} (blue) for the MBB (circles). $\\beta_{\\tt d0}$ is marked by the dashed  black line. (Bottom): Same as above but with $T_0(\\ell)$, the black dashed-lines being $T_{\\tt d0}=20$\\,K and $T_{\\tt d1T}=21.9$\\,K.}\n    \\label{fig:betaTdust}\n\\end{figure}\n\n\nIn Fig.~\\ref{fig:chi2dust} the values of the reduced $\\chi^2(\\ell)$ for each fitting scheme are displayed. For every fitting scheme (MBB, $\\beta$-1, $\\beta$-$T$ and $\\beta$-2), the reduced $\\chi^2$ are close to 1 over the whole multipole range (slightly below 1 for the $\\beta$-1, $\\beta$-$T$ and $\\beta$-2 fitting scheme). This indicates that the MBB is a good fit to the cross-angular power spectra computed from the  {\\tt d0} maps with a spatially invariant MBB SED, as expected. Adding additional (higher order) parameters, such as with $\\beta$-1, $\\beta$-$T$ and $\\beta$-2, has no significant effect on the $\\chi^2$.\n\nIn Fig.~\\ref{fig:betaTdust} we can see that the best-fit values of $\\beta_0(\\ell)$ and $T_0(\\ell)$ are compatible with constant values $\\beta_0(\\ell)=\\beta_{\\tt d0}$ and $T_0(\\ell)=T_{\\tt d0}$, as expected for this simulated data set.\n\nThe best-fit values of the dust amplitude and the moment-expansion parameters are presented in Figs.~\\ref{fig:Adust}, \\ref{fig:moments1}, \\ref{fig:moments2}, and \\ref{fig:moments3}, respectively. The amplitude power spectrum is compatible with that of the dust template map used to build {\\tt d0} and the moment-xpansion parameters are compatible with zero for every fitting scheme, as expected with no spatial variation of the SED. \nTherefore, the moment expansion method presented in Sect.~\\ref{sec:formalism} passes the \\emph{null test} in the absence of SED distortions, with the {\\tt d0} simulated data set.  \n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/Adustgauss.pdf}\n    \\caption{\\footnotesize Median of the best-fit values of $\\mathcal{D}_\\ell^{A \\times A}$ for {\\tt d0} (orange), {\\tt d1T} (green), and {\\tt d1} (blue) using the MBB fitting scheme. The values for the three simulation types are shifted with respect to one another by $\\ell=2$ in order to distinguish them. The black dashed line is the amplitude power spectrum of the dust template map used to build the three simulation sets {\\tt d0}, {\\tt d1T,} and  {\\tt d1}.}\n    \\label{fig:Adust}\n\\end{figure}\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/w1w1dustgauss.pdf}\n    \\caption{\\footnotesize Best-fit values of the first-order moment $\\mathcal{D}_\\ell^{\\omega^{\\beta}_1\\times\\omega^{\\beta}_1}$ for {\\tt d0} (orange), {\\tt d1T} (green), and {\\tt d1} (blue), fitting with $\\beta$-1 (crosses), $\\beta$-2 (triangles), and $\\beta$-$T$ (diamonds).}\n    \\label{fig:moments1}\n\\end{figure}\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/Aw2dustgauss.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/w1w2dustgauss.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/w2w2dustgauss.pdf}\n    \\caption{\\footnotesize Best-fit values of the second-order $\\mathcal{D}_\\ell^{A\\times\\omega^{\\beta}_2}$, $\\mathcal{D}_\\ell^{\\omega^{\\beta}_1\\times\\omega^{\\beta}_2}$ and $\\mathcal{D}_\\ell^{\\omega^{\\beta}_2\\times\\omega^{\\beta}_2}$ moment parameters in {\\tt d0} (orange), {\\tt d1T} (green), and {\\tt d1} (blue) for $\\beta$-2 (triangles). }\n    \\label{fig:moments2}\n\\end{figure}\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/b1t1dustgauss.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/t1t1dustgauss.pdf}\n    \\caption{\\footnotesize Best-fit values of the first-order  $\\mathcal{D}_\\ell^{\\omega^{\\beta}_1\\times\\omega^{T}_1}$ and $\\mathcal{D}_\\ell^{\\omega^{T}_1\\times\\omega^{T}_1}$ moment parameters in {\\tt d0} (orange), {\\tt d1T} (green), and {\\tt d1} (blue) for $\\beta$-$T$ (diamonds). }\n    \\label{fig:moments3}\n\\end{figure}\n\n\n\\subsubsection{{\\tt d1T}}\n\\label{sec:d1T}\n\n\nWe now introduce, as a first layer of complexity, the spatial variations of the spectral index associated to a fixed temperature over the sky with the {\\tt d1T} simulation type. The dust temperature was fixed to $T_{\\rm d1T}=21.9$\\,K while the spectral index $\\beta(\\vec{n})$ was allowed to vary between lines of sight. The four different fitting schemes presented in Sect.~\\ref{sec:fit_dust} are fitted over the cross-spectra of our simulations as in Sect.~\\ref{sec:d0}.\n\nThe reduced $\\chi^2(\\ell)$ values for each fitting scheme can be found in Fig.~\\ref{fig:chi2dust}. It can be seen that the MBB no longer provides a good fit for the dust SED, especially at low multipoles. Averaging effects of spatially varying SEDs are more important over large angular scales and thus SED distortions and moments are expected to be more significant at low multipoles. Indeed, the moments added to the fit in $\\beta$-1 are enough to lower the reduced $\\chi^2$ such that it becomes compatible with 1 over almost all of the multipole range. The fitting schemes $\\beta$-$T$ and $\\beta$-2, including more parameters than $\\beta$-1, provide a fit of similar goodness, except in the multipole bin $\\ell=66.5$ where they are closer to 1.\n\nFigure~\\ref{fig:betaTdust} presents the best-fit values of $\\beta_0(\\ell)$ in the case of the MBB fit. For the sake of clarity, the values after iteration (see Sect.~\\ref{sec:fit_dust}) for $\\beta$-1, $\\beta$-$T,$ and $\\beta$-2 are not shown, but they present comparable trends. We can see that the best-fit values of $\\beta_0(\\ell)$ for this {\\tt d1T} simulation type are no longer compatible with a constant. $\\beta_0(\\ell)$ fitted values show a significant increase at low (<100) multipoles, up to $\\beta_0(\\ell=16.5)=1.65$. For $\\ell>100$, $\\beta_0(\\ell)$ is close to a constant of value $\\sim1.53$. This increase towards the low $\\ell$ is correlated to the increase of the MBB $\\chi^2$ discussed in the previous paragraph. However, we note that in the lowest $\\ell$-bin, the $\\beta_0(\\ell)$ value is close to 1.53 and that the $\\chi^2$ of the MBB fit is close to unity.\n\n\nThe best-fit values of $T_0(\\ell)$ are also presented in Fig.~\\ref{fig:betaTdust} in the case of the MBB fit. Here again, the values after iteration for the other fitting schemes are not presented, but are similar. The {\\tt d1T} $T_0(\\ell)$ best-fit values oscillate around $T_{\\tt d1T}=21.9$\\,K, without being strictly compatible with a constant value, as would be expected for this simulation type. This tends to indicate that the SED distortions due to the spectral index spatial variations are affecting the accuracy at which we can recover the correct angular dependence of the sky temperature. \n\nThe amplitude power spectrum is displayed in Fig.~\\ref{fig:Adust} for the MBB fitting scheme. The other fitting scheme results are not presented for clarity and would not be distinguishable from those of the MBB. The fitted $\\mathcal{D}_\\ell^{A\\times A}$ is compatible with the one of the dust template map used to build the simulations. \n\nAll the parameters of the moment expansion with respect to $\\beta$ can be found in Figs.~\\ref{fig:moments1} and \\ref{fig:moments2}, and are now significantly detected, except for $\\mathcal{D}_\\ell^{\\omega^\\beta_2 \\times \\omega^\\beta_2}$. In Fig.~\\ref{fig:moments3}, we can observe that the parameters of the moment expansion with respect to the temperature (only present in the $\\beta$-$T$ fit) remain undetected. The SED distortions due to the spatial variations of $\\beta$ are well detected, while no SED distortion linked to the temperature is seen, as expected for the ${\\tt d1T}$ simulation type.\n\n\n\\subsubsection{{\\tt d1}}\n\\label{sec:d1}\n\nWe now discuss the {\\tt d1} simulations, with the highest complexity in the polarized dust SED. In this more physically relevant simulation type, the dust emission is given by a MBB with variable index $\\beta(\\vec{n})$ and temperature $T(\\vec{n})$ over the sky.\nWe ran the four different fitting schemes on the {\\tt d1} simulation type, as we did in Sect.~\\ref{sec:d0} and \\ref{sec:d1T}.\n\nThe values of the reduced $\\chi^2(\\ell)$ are displayed in Fig.~\\ref{fig:chi2dust}. For the MBB and $\\beta$-1, the reduced $\\chi^2$ are not compatible with unity, especially at low multipole. This indicates that none of them are a good fit anymore for the spatially varying SED with $\\beta(\\vec{n})$ and $T(\\vec{n})$. With $\\beta$-2 and $\\beta$-$T$, the $\\chi^2(\\ell)$ values become compatible with unity, except for the $\\ell=26.5$ bin. We note that $\\beta$-$T$ provides a slightly better fit than $\\beta$-2 in this bin.\n\nLooking at the medians of the best-fit values of $\\beta_0(\\ell)$ for {\\tt d1} in Fig.~\\ref{fig:betaTdust}, we can see that the spectral index is changing with respect to $\\ell$, as discussed in Sect.~\\ref{sec:d1T}, in a similar manner as for the {\\tt d1T} simulation type. The fitted temperature $T_0(\\ell)$ values for {\\tt d1} show an increasing trend from $\\sim17$ to $\\sim20.5$\\,K and from $\\ell=16.5$ to $\\ell\\sim100$. At higher multipoles, $T_0(\\ell)$ is close to a constant temperature of 20.5\\,K. In {\\tt d1}, as for {\\tt d1T}, the angular scales at which we observe strong variations of $\\beta_0(\\ell)$ and $T_0(\\ell)$ are the ones for which we observe a poor $\\chi^2$ for some fitting schemes. Also, as for {\\tt d1T}, the largest angular scale $\\ell$-bin, at $\\ell=6.5,$ shows $\\beta$ and $T$ values close to the constant value at high $\\ell$, which are associated with $\\chi^2$ values closer to unity.\nThe best-fit values of the amplitude $\\mathcal{D}_\\ell^{A\\times A}$ are shown in Fig.~\\ref{fig:Adust}. These are similar to those of the other simulation types. \n\nThe moment-expansion parameters fitted on {\\tt d1} are shown in Figs.~\\ref{fig:moments1}, \\ref{fig:moments2}, and \\ref{fig:moments3}. For this simulation type, the moment parameters are all significantly detected with respect to both $\\beta$ and $T$. This was already the case with the \\textit{Planck}{} intensity simulations, produced in a similar way, as discussed in \\cite{Mangilli}. Their detections quantify the complexity of dust emission and SED distortions from the MBB present in the {\\tt d1} simulation type, due to the spatial variations of $\\beta(\\vec{n})$ and $T(\\vec{n})$. \n\n\n\\subsection{\\label{sec:nodust} CMB only}\n\\label{sec:c}\n\nIn order to calibrate the accuracy at which the $r$ parameter can be constrained with the \\textit{LiteBIRD}{} simulated data sets presented in Sect.~\\ref{sec:simulations}, we tested the simulation type with no dust component, $M_\\nu^{\\tt c}$, and with no tensor modes ($r_{\\rm sim}=0$, only CMB lensing and noise). We fit the expression in Eq.~\\ref{eq:model} with $\\mathcal{D}_\\ell^{\\rm dust}$ fixed to zero and $A_{\\rm lens}$ fixed to one ({i.e.}, $r$ is the only parameter we fit in this case).\nDoing so over the $N_{\\rm sim}$ simulations, we obtain $\\hat{r}= (0.7 \\pm 3.5) \\times 10^{-4}$.\nThis sets the minimal value we can expect to retrieve for $\\hat{r}$ with our assumptions if the dust component is perfectly taken into account.\n\n\n\\subsection{Dust and CMB}\n\\label{sec:results_dust_CMB}\n\nWe now present our analysis of the simulations including dust, CMB (lensing), and noise ({\\tt d0c}, {\\tt d1Tc} and {\\tt d1c}) with no primordial tensor modes ($r_{\\rm sim}=0$). As described above, we applied the four fitting schemes for the dust on the three simulation types, fitting $\\hat{r}$ and fixing $A_{\\rm lens}$ to~one    (namely $r$MBB, $r\\beta$-1, $r\\beta$-$T$ and $r\\beta$-2) simultaneously. \n\nThe best-fit values of $\\beta_0(\\ell)$, $T_0(\\ell)$ and the moment expansion parameters $\\mathcal{D}_\\ell^{\\mathcal{M}\\times\\mathcal{N}}$ derived with the simulation types {\\tt d0c}, {\\tt d1Tc,} and {\\tt d1c} are not discussed further when they are compatible with the ones obtained for the {\\tt d0}, {\\tt d1T,} and {\\tt d1} simulation types and presented in Sect.~\\ref{sec:results_dust_only}.\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/gauss_r_0_d0_SLD_full.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/gauss_r_0_d1T_SLD_full.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/gauss_r_0_d1_SLD_full.pdf}\n    \\caption{\\footnotesize \\emph{(Top panel)}: Posterior on $\\hat{r}$ in the {\\tt d0c} simulation type for the different fitting schemes: $r$MBB (blue, dotted line), $r\\beta$-1 (red, dashed line), $r\\beta$-$T$ (green, solid line), and $r\\beta$-2 (yellow, dash-dotted line). The vertical black dashed line marks the value of $r_{\\rm sim}=0$. \\emph{(Central panel)}: Same, but in the case of the {\\tt d1Tc} simulation type. \\emph{(Bottom panel)}: Same, but in the case of the {\\tt d1c} simulation type.}\n    \\label{fig:rgauss1}\n\\end{figure}\n\n\n\\subsubsection{{\\label{sec:d0c}}{\\tt d0c}}\n\nFor {\\tt d0c}, as for {\\tt d0}, we recover the input constant spectral index and temperature $\\beta_{\\tt d0}$ and $T_{\\tt d0}$ at all angular scales for every fitting scheme. Furthermore, we do not detect any moment, when fitting $r\\beta$-1, $r\\beta$-$T,$ and $r\\beta$-2. This simulation type therefore constitutes our {null-test} when $\\hat{r}$ and the dust parameters are fitted at the same time. The addition of the CMB lensing in the simulations and the addition of $r$ to the fits thus does not lead to the detection of the moment parameters nor biases the recovery of the spectral index and the temperature.\n\nThe posterior distributions of the estimated tensor-to-scalar ratio $\\hat{r}$ are displayed in Fig.~\\ref{fig:rgauss1} and their mean and standard deviations are summarized in Table~\\ref{tab:rvalues}. We note that $\\hat{r}$ is compatible with the input value ($r_{\\rm sim}=0)$ for all the fitting schemes. For $r$MBB and $r\\beta$-1, the dispersion $\\sigma_{\\hat{r}}$ is comparable with the CMB-only scenario discussed in Sect.~\\ref{sec:nodust}. For $r\\beta$-$T$ and $r\\beta$-2, the width of the distribution increases by a factor of $\\sim 2$ and $\\sim 4$,  respectively.\n\n\\subsubsection{{\\tt d1Tc}}\n\\label{sec:d1Tc}\n\nThe posterior distribution of $\\hat{r}$ in the case of the {\\tt d1Tc} simulation type is displayed in Fig.~\\ref{fig:rgauss1} for the four fitting schemes and the mean value and standard deviation of these distributions are summarized in Table~\\ref{tab:rvalues}. We can see that in the case of $r$MBB, we fit $\\hat{r}\\pm\\sigma_{\\hat{r}}=(99.7\\pm6.2)\\times10^{-4}$. In that case, the input tensor-to-scalar ratio $r_{\\rm sim}=0$ is not recovered and we obtain a bias on the central value of $\\hat{r}$ of $\\sim 16\\,\\sigma_{\\hat{r}}$. As discussed in Sect.~\\ref{sect:limitsmbb}, this is expected because we know that the MBB is not a good dust model for a SED with spatially varying spectral index, as we also verify in Sect.~\\ref{sec:d1T} looking at the $\\chi^2$ values.\n\nUsing the $r\\beta$-1 fitting scheme allows us to recover $\\hat{r}=(-8.0\\pm6.4)\\times10^{-4}$, where $r_{\\rm sim}$ is recovered within $\\sim\\,2\\sigma_{\\hat{r}}$, while $r\\beta$-2 and $r\\beta$-$T$ recover the input value within $1\\,\\sigma_{\\hat{r}}$ (with $\\hat{r}=(-3.6\\pm 13.0)\\times10^{-4}$ and $\\hat{r}=(0.7\\pm20.9)\\times10^{-4}$, respectively). As in Sect~\\ref{sec:d0c}, the deviation remains similar between $r$MBB and $r\\beta$-1 and increases by a factor of $\\sim 2$ and $\\sim 4$ from $r\\beta$-1 to $r\\beta$-$T$ and $r\\beta$-2,  respectively.\n\n\\begin{table}[t!]\n\\centering\\scalebox{1}{\n    \\centering\n    \\begin{tabular}{c | ccc}\n        ($\\hat{r} \\pm \\sigma_{\\hat{r}})\\times10^{4}$ & {\\tt d0c} & {\\tt d1Tc} & {\\tt d1c}\\\\\n        \\hline\n        $r$MBB     &  \\cellcolor{mygreen} $0.3 \\pm 3.9$ & \\cellcolor{myred}$99.7 \\pm 6.2$  & \\cellcolor{myred} $125.1 \\pm 5.9$\\\\\n        $r\\beta$-1   & \\cellcolor{mygreen} $0.5 \\pm 4.5$ & \\cellcolor{myyellow} $-8.0 \\pm 6.4$& \\cellcolor{myred} $32.9 \\pm 6.5$\\\\\n        $r\\beta$-$T$   & \\cellcolor{mygreen} $0.3 \\pm 9.5$ & \\cellcolor{mygreen} $-3.6 \\pm 13.0$  & \\cellcolor{mygreen} $-3.3 \\pm 11.7$\\\\\n        $r\\beta$-2   & \\cellcolor{mygreen} $0.7 \\pm 16.4$ & \\cellcolor{mygreen} $0.7 \\pm 20.9$ & \\cellcolor{myyellow} $-37.4 \\pm 19.4$\\\\ \n        \\hline \n    \\end{tabular}}\n\\caption{\\footnotesize Best-fit values of $\\hat{r}$ in units of $10^{-4}$ on $f_{\\rm sky}=0.7$. The green values are compatible with $r_{\\rm sim}=0$ at $1\\,\\sigma_{\\hat{r}}$, the yellow values are compatible with $r_{\\rm sim}=0$ at $2\\,\\sigma_{\\hat{r}}$ and the red values are incompatible with $r_{\\rm sim}=0$ at more than $2\\,\\sigma_{\\hat{r}}$.}\n\\label{tab:rvalues}\n\\end{table}\n\n\\subsubsection{{\\tt d1c}}\n\\label{sec:d1c}\n\nIn the case of the {\\tt d1c} simulation type, as in {\\tt d0c} and {\\tt d1Tc}, we fit $\\hat{r}$ in addition to the dust-related parameters. In that case, dust moment parameters are recovered as for {\\tt d1} (see Sect.~\\ref{sec:d1}), except for the $r\\beta$-2 fitting scheme.\n\nFigure~\\ref{fig:momentsd1d1c} compares the moment parameters between $\\beta$-2 on the {\\tt d1c} simulations type, fitting only the dust-related parameters and $r\\beta$-2 on {\\tt d1c} when jointly fitting the dust parameters and $\\hat{r}$. We observe that $\\mathcal{D}_\\ell^{\\omega^{\\beta}_2 \\times \\omega^{\\beta}_2}$ is not consistently recovered when fitting $\\hat{r}$ in addition to the dust parameters.\n\nA similar comparison can be found in Fig.~\\ref{fig:moments2d1d1c} for the moment parameters between $\\beta$-$T$ and $r\\beta$-$T$ on the {\\tt d1c} simulation type. Using this fitting scheme, we can see that all the moments are correctly recovered when adding $\\hat{r}$ to the fit.\n\n\nThe $\\hat{r}$ posterior distributions in the case of {\\tt d1c} are displayed in  Fig.~\\ref{fig:rgauss1} and summarized in Table~\\ref{tab:rvalues}. As discussed in Sect.~\\ref{sect:limitsmbb} and observed in Sect.~\\ref{sec:d1c}, the $r$MBB fit is highly biased, with $\\hat{r}=(125.1 \\pm 5.9)\\times10^{-4}$ (by more than 21\\,$\\sigma_{\\hat{r}}$). When fitting the $r\\beta$-1, this bias is significantly reduced ($\\hat{r}=(32.9 \\pm 6.5)\\times10^{-4}$, $~5\\,\\sigma_{\\hat{r}}$ away from $r_{\\rm sim}=0$), illustrating the ability of the first-moment parameters to correctly capture part of the SED complexity. However, performing the expansion in both $\\beta$ and $T$ with $r\\beta$-$T$ allows us to recover $r_{\\rm sim}$ without bias ($\\hat{r}=(-3.3\\pm 11.7)\\times10^{-4}$), highlighting the need for the description of the SED complexity in terms of dust temperature for this simulated data set where both $\\beta$ and $T$ vary spatially. On the other hand, for $r\\beta$-2, a negative tension ($1.9\\,\\sigma_{\\hat{r}}$) can be observed: $\\hat{r}=(-37.4\\pm 19.4)\\times10^{-4}$. This tension is discussed in Sect.~\\ref{sec:bias}.  \n\nFor {\\tt d1c}, the $\\hat{r}$ distribution widths roughly meet the foreground cleaning requirements of \\textit{LiteBIRD}{} presented in Sect.~\\ref{sec:LiteBIRD} for $r$MBB and $r\\beta$-1 but are higher for $r\\beta$-$T$ and $r\\beta$-2. We also note that, with the same number of free parameters, all the standard deviations $\\sigma_{\\hat{r}}$ slightly increase compared to the {\\tt d0c} simulation type. This is expected due to the increasing dust complexity. \n\n\\section{\\label{sec:discussion} Discussion}\n\n\n\\subsection{Lessons learnt}\n\nIn Sect.~\\ref{sec:results}, we apply the fitting pipeline introduced in Sect.~\\ref{sec:fit} on \\textit{LiteBIRD}{} simulated data sets on $f_{\\rm sky}=0.7$ and for $r_{\\rm sim}=0$, including the various dust simulation types defined in Sect.~\\ref{sec:ingredients}. We fitted the estimated $B$-mode power-spectra with the four different fitting schemes summarized in Table~\\ref{tab:parameter}. Our main results can be summarized as follows: \n\n\\begin{itemize}\n\\item The MBB fitting scheme provides a good fit for the dust component in the {\\tt d0} and {\\tt d0c} simulation types. \n    However, when the spectral index changes with the angular scale, such as in the {\\tt d1T}, {d1Tc}, {\\tt d1,} and {\\tt d1c} simulations, this approach no longer provides a good fit because of the complexity of the dust SED. As a consequence, in the $r$MBB case, $r_{\\rm sim}$ cannot be recovered without a significant bias. \n\n\\item The $\\beta$-1 fitting scheme  allows us to perform a good fit for the dust complexity using the {\\tt d0} and {\\tt d1T} simulations but not for {\\tt d1}, while the $r\\beta$-1 fitting  scheme yields estimates of $\\hat{r}$ close to $r_{\\rm sim}$ within $1\\,\\sigma_{\\hat{r}}$ for {\\tt d0c}, and within $2\\,\\sigma_{\\hat{r}}$ for {\\tt d1Tc}, but presenting a bias of $\\sim 6\\,\\sigma_{\\hat{r}}$ for {\\tt d1c}. \n\n\\item The $\\beta$-$T$ fitting scheme provides a good fit for every dust model, while using the $r\\beta$-$T$ fitting  scheme allows us to recover $\\hat{r}$  values consistent with $r_{\\rm sim}$ within $1\\,\\sigma_{\\hat{r}}$ for all the simulation types, but is associated with an increase of $\\sigma_{\\hat{r}}$ by a factor $\\sim 2$ compared to the $r\\beta$-1 case.\n\n\\item The $\\beta$-2 fitting scheme also provides a good fit for each dust model, and the $r\\beta$-2 fitting scheme leads to values of $\\hat{r}$ compatible with $r_{\\rm sim}$ within $1\\,\\sigma_{\\hat{r}}$ for all the simulation types but {\\tt d1c}. In this last case, there is a negative tension of $\\sim2\\,\\sigma_{\\hat{r}}$.  \n    For all the simulation types, there is an increase of $\\sigma_{\\hat{r}}$ by a factor of $\\sim 4$ compared to the $r\\beta$-1 case.\n\n\\end{itemize}\n\nThe present analysis shows that the temperature could be a critical parameter for the moment expansion in the context of \\textit{LiteBIRD}{}.\n\nIndeed, for simulations including a dust component with a spectral index and a temperature that both vary spatially, as in {\\tt d1}, the only fitting scheme allowing us to recover $r_{\\rm sim}$ within $1\\,\\sigma_{\\hat{r}}$ is $r\\beta$-$T$, the expansion to first order in both $\\beta$ and $T$. This shows that expanding in $\\beta$ only, without treating $T$, is not satisfactory when looking at such large fractions of the sky. Indeed, when applying the $\\beta$-2 fitting scheme, the $\\mathcal{D}_\\ell^{\\omega^{\\beta}_2 \\times \\omega^{\\beta}_2}$ parameter remains undetected for the {\\tt d1T} simulation type (Sect~\\ref{sec:d1T}), while it is significantly detected using the {\\tt d1} simulation type (Sect~\\ref{sec:d1}). Nevertheless, {\\tt d1T} and {\\tt d1}share the same template of $\\beta(\\vec{n})$ (Sect.~\\ref{sec:ingredients}) and they only differ by the sky temperature (constant for {\\tt d1T} and varying for {\\tt d1}). This suggests that the observed $\\mathcal{D}_\\ell^{\\omega^{\\beta}_2 \\times \\omega^{\\beta}_2}$ with the {\\tt d1} simulations originates from the temperature variations and not those in the spectral index. This observation shows that it is less convenient to use the $\\beta$-2 fitting scheme than the $\\beta$-$T$ one in order to correctly recover the moment-expansion parameters and $\\hat{r}$ when temperature varies spatially.\n\nMoreover, we saw that $ \\sigma_{\\hat{r}}$ is lower when using the fitting scheme $r\\beta$-$T$ instead of $r\\beta$-2 for every simulation type, even if both have the same number of free parameters. This second observation additionally encourages an approach where the SED is expanded with respect to both $\\beta$ and $T$. Nevertheless, the uncertainty on $\\hat{r}$ we obtain in this case ($\\sigma_{\\hat{r}}=1.17\\times10^{-3}$) is larger than the \\textit{LiteBIRD}{} requirements. \n\n\\subsection{Increasing the accuracy on the tensor-to-scalar ratio}\n\\label{sec:Opt}\n\nIn Sect.~\\ref{sec:d1} and Fig.~\\ref{fig:chi2dust}, we see that the MBB and $\\beta$-1 fitting schemes do not provide good fits for the {\\tt d1} dust simulations, especially at low multipoles ($\\ell \\lesssim 100$). Conjointly, in Fig.~\\ref{fig:moments3}, we can see that the $\\beta$-$T$ moment parameters are significantly detected for $\\ell \\lesssim 100$ and compatible with zero above that threshold, suggesting that their corrections to the SED are predominantly required at large angular scales.\n\nThis implies that we can improve the pipeline presented in Sect.~\\ref{sec:fit} to keep only the required parameters in order to recover $\\hat{r}$ compatible with $r_{\\rm sim}$ with a minimal $\\sigma_{\\hat{r}}$. It can be achieved by applying the $r\\beta$-1 fitting sheme over the whole multipole range, while restricting the $r\\beta$-$T$-specific ($\\mathcal{D}_\\ell^{\\omega^\\beta_1\\times\\omega^T_1}$ and $\\mathcal{D}_\\ell^{\\omega^T_1\\times\\omega^T_1}$) moment-expansion parameters fit to the low multipoles range. We note that in order to correct the bias, it is still necessary to keep the $r\\beta$-1 moment parameters even at high multipoles, because the MBB does not provide a good fit even for $\\ell \\in [100,200]$, as we can see in Fig.~\\ref{fig:chi2dust}. We define $\\ell_{\\rm cut}$ as the multipole bin under which we keep all the $r\\beta$-$T$ moment parameters and above which we use the $r\\beta$-1 scheme. \n\n\nThe best-fit values and standard deviations of $\\hat{r}$ for different values of $\\ell_{\\rm cut}$ are displayed in Table~\\ref{tab:rellcut}. We can see that a trade-off has to be found: the smaller the $\\ell_{\\rm cut}$ , the bigger the shift from $r_{\\rm sim}$, and the bigger the $\\ell_{\\rm cut}$, the higher the value of $\\sigma_{\\hat{r}}$. The trade-off point seems to be found for $\\ell_{\\rm cut} \\sim 80$, allowing us to recover $\\hat{r}$ without tension, with $\\sigma_{\\hat{r}} =8.8\\times10^{-4}$. The error on $r$ is thus reduced by more than $\\sim30\\,\\%$ with respect to the nonoptimized fit and meets the \\textit{LiteBIRD}{} requirements.\n \n\n\n\n\n\\subsection{\\label{sec:fsky} Tests with smaller sky fractions}\n\n\\begin{table}[t!]\n\\centering\\scalebox{1}{\n    \\centering\n    \\begin{tabular}{c|c}\n        $\\ell_{\\rm cut}$ & $(\\hat{r} \\pm \\sigma_{\\hat{r}})\\times 10^{4}$\\\\\\hline\n        50 &  \\cellcolor{myred}$12.0 \\pm 7.3$ \\\\\n        60 &  \\cellcolor{mygreen}$7.3 \\pm 7.9$ \\\\\n        70 &  \\cellcolor{mygreen}$4.9 \\pm 8.1$ \\\\\n        80 & \\cellcolor{mygreen}$-0.9 \\pm 8.8$ \\\\\n        90 &  \\cellcolor{mygreen}$-2.1 \\pm 9.9$ \\\\\n    \\end{tabular}}\n\\caption{\\footnotesize Best-fit values of $\\hat{r} \\pm \\sigma_{\\hat{r}}$ in units of $10^{-4}$ for different values of $\\ell_{\\rm cut}$ for the {\\tt d1c} simulations with $f_{\\rm sky}=0.7$, when applying the $r\\beta$-$T$ fitting scheme. The green values are compatible with $r_{\\rm sim}=0$ at $1\\,\\sigma_{\\hat{r}}$.}\n\\label{tab:rellcut}\n\\end{table}\n\n\\begin{table}[t!]\n\\centering\\scalebox{1}{\n    \\centering\n    \\begin{tabular}{c | ccc}\n        $(\\hat{r} \\pm \\sigma_{\\hat{r}})\\times10^4$ & $r_{\\rm sim}=0.01$ & $f_{\\rm sky}=0.5$ & $f_{\\rm sky}=0.6$\\\\\n        \\hline\n        $r$MBB     & \\cellcolor{myred} $204.8 \\pm 7.7$ & \\cellcolor{myred} $47.3 \\pm 5.6$ & \\cellcolor{myred}  $59.2 \\pm 5.4$\\\\\n        $r\\beta$-1  & \\cellcolor{myred} $129.0 \\pm 8.3$ & \\cellcolor{myyellow} $-8.4 \\pm 6.7$ & \\cellcolor{mygreen} $1.8 \\pm 6.2$\\\\\n        $r\\beta$-$T$ & \\cellcolor{mygreen} $94.6 \\pm 15.1$ &\\cellcolor{mygreen} $0.02 \\pm 13.4$ & \\cellcolor{mygreen} $-1.1 \\pm 12.0$\\\\\n        $r\\beta$-2   & \\cellcolor{myyellow} $62.5 \\pm 25.0$ & \\cellcolor{mygreen} $4.3 \\pm 24.2$ & \\cellcolor{mygreen} $-3.2 \\pm 22.4$\\\\\n         \\hline\n    \\end{tabular}}\n\\caption{\\footnotesize Best-fit values of $\\hat{r}$ in units of $10^{-4}$ for an alternative {\\tt d1c} simulation with $r_{\\rm sim}=0.01$ on $f_{\\rm sky}=0.7$, and with $r_{\\rm sim}=0$ but on $f_{\\rm sky}=0.5$ and $f_{\\rm sky}=0.6$. The green values are compatible with $r_{\\rm sim}$ at $1\\,\\sigma_{\\hat{r}}$, the yellow values are compatible with $r_{\\rm sim}$ at $2\\,\\sigma_{\\hat{r}}$ , and the red values are incompatible with $r_{\\rm sim}$ at more than $2\\,\\sigma_{\\hat{r}}$.}\n\\label{tab:rvalues2}\n\\end{table}\n\nIn all the results presented in Sect.~\\ref{sec:results}, we were considering a sky fraction of $f_{\\rm sky}=0.7$. This sky mask keeps a considerable fraction of the brightest Galactic dust emission. To quantify the impact of the sky fraction on our analysis, we ran the pipeline as in Sect.~\\ref{sec:d1c} with the different masks introduced in Sect.~\\ref{sect:mask} ($f_{\\rm sky}=0.5$ and $f_{\\rm sky}=0.6$). This was done with the {\\tt d1c} simulation type. \n\nThe posteriors on $\\hat{r}$ for the different fitting schemes are displayed in Fig.~\\ref{fig:rgauss2} and Table~\\ref{tab:rvalues2}. We can see that, while the $r$MBB fiting scheme always leads to biased estimates, the $r\\beta$-1 case allows us to recover $\\hat{r}$ at $1.25\\,\\sigma_{\\hat{r}}$ for $f_{\\rm sky}=0.5$ and within $1\\,\\sigma_{\\hat{r}}$ for $f_{\\rm sky}=0.6$. In the two situations, the results using the $r\\beta$-$T$ and $\\beta$-2 fitting schemes are both unbiased with estimates of $\\hat{r}$ compatible with $r_{\\rm sim}$ within $1\\,\\sigma_{\\hat{r}}$. The $\\sigma_{\\hat{r}}$ hierarchy between the $r$MBB, $r\\beta$-1, $r\\beta$-$T,$ and $r\\beta$-2 fitting schemes is the same as for $f_{\\rm sky}=0.7$ (see Sect.~\\ref{sec:d1c}). Nevertheless, we observe that $\\sigma_{\\hat{r}}$ increases as the sky fraction decreases, as does the statistical error (cosmic variance of the lensing and noise). The bias, on the other hand, decreases for all the fitting schemes with the sky fraction, which is expected because less dust emission contributes to the angular power spectra. The negative tension observed on the $\\hat{r}$ posterior in Sect.~\\ref{sec:d1c} for the $r\\beta$-2 case is not present when using smaller sky fractions. In Fig.~\\ref{fig:fskydust}, the $r\\beta$-2 moment parameters are displayed. We can see that they are not significantly detected for the $f_{\\rm sky}=0.5$ and 0.6, unlike for $f_{\\rm sky}=0.7$. As we have seen that some of the moments in the $\\beta$-2 fitting scheme failed to model SED distortions coming from temperature, we can suppose that, in our simulations, the temperature variations play a less significant role in the dust SED on the $f_{\\rm sky}=0.5$ and 0.6 masks than they play in the $f_{\\rm sky}=0.7$ one. As a consequence, they have a smaller impact on $r$ when not properly taken into account.\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/gauss_r_0_d1_SLD_full_fsky0.5.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/gauss_r_0_d1_SLD_full_fsky0.6.pdf}\n    \\caption{\\footnotesize \\emph{(Top panel)}: Posterior on $\\hat{r}$ in the {\\tt d1c} simulation type on $f_{\\rm sky}=0.5$ for the different fitting schemes: $r$MBB (blue, dotted line), $r\\beta$-1 (red, dashed line), $r\\beta$-$T$ (green, solid line), and $r\\beta$-2 (yellow, dash-dotted line). The vertical black dashed line marks the value of $r_{\\rm sim}=0$.\\emph{(Bottom panel)}: Same, in the case of the {\\tt d1c} simulation type on $f_{\\rm sky}=0.6$.}\n    \\label{fig:rgauss2}\n\\end{figure}\n\n\\subsection{Tests with nonzero input tensor modes}\n\\label{sec:rnot0}\n\nWe show in Sect.~\\ref{sec:d1c} that the $r\\beta$-$T$ fitting scheme allows us to retrieve $\\hat{r}$ compatible with zero when $r_{\\rm sim}=0$. We now want to assess the potential leakage of $\\hat{r}$ in the moment expansion parameters if $r_{\\rm sim} \\neq 0$. In this case, primordial tensor signals would be incorrectly interpreted as dust complexity. We run the pipeline as described in Sect.~\\ref{sec:d1c} with $r_{\\rm sim}=0.01$, in the ${\\tt d1c}$ simulation type. This value of $r_{\\rm sim}=0.01$ is larger than the value targeted by \\textit{LiteBIRD},{} but given the order of magnitude of the error on $\\hat{r}$ observed in the previous sections, a potential leakage could be left unnoticed using a smaller $r_{\\rm sim}$. \n\nLooking at the final posterior on $\\hat{r}$ (Fig.~\\ref{fig:rgauss3} and Table~\\ref{tab:rvalues2}), we can see that the results are comparable with the $r_{\\rm sim}=0$ case, but centered on the new input value $r_{\\rm sim}=0.01$. The $r$MBB fitting scheme gives a highly biased posterior of $\\hat{r}=(2.048\\pm0.077)\\times10^{-2}$ ; the bias is reduced but still significant when using the $r\\beta$-1 scheme ($\\hat{r}=129.0\\pm 8.3\\times10^{-4}$) ; in the $\\beta$-$T$ case we get an estimate of $\\hat{r}=94.6\\pm 15.1\\times10^{-4}$ compatible with the input value of $r_{\\rm sim}=100\\times10^{-4}$ ; and  finally, the $\\beta$-2 fitting scheme leads to a negative $2\\,\\sigma_{\\hat{r}}$ tension ($\\hat{r}=62.5\\pm25.0\\times10^{-4}$). This demonstrates the robustness of our method and its potential application to component separation. We note that the negative bias at second order is still present in the $r_{\\rm sim }=0.01$ case, illustrating that setting a positive prior on $\\hat{r}$ would not have been a satisfying solution when $r_{\\rm sim}=0$.\n\n\\subsection{\\label{sec:bias} Exploring the correlations between the parameters}\n\nWe now examine the substantial increase in the dispersion on the $\\hat{r}$ posteriors between the $r\\beta$-1 fitting scheme on the one hand and the $r\\beta$-$T$ and $r\\beta$-2 ones on the other. Indeed, in Sect.~\\ref{sec:d1c}, we show that $\\sigma_{\\hat{r}}$ is about two times greater when using the $r\\beta$-$T$ scheme than the $r\\beta$-1 one, and about four times larger in the case of $r\\beta$-2, while the $r\\beta$-$T$ and $r\\beta$-2 schemes share the same number of free parameters. Some other points to clarify are the shift on $\\hat{r}$ appearing for $r\\beta$-2 in the {\\tt d1c} scenario, discussed in Sect.~\\ref{sec:d1c}, and the inability to correctly recover $\\mathcal{D}_\\ell^{\\omega_2^\\beta \\times \\omega_2^\\beta}$ when $\\hat{r}$ is added to the fit illustrated in Fig.~\\ref{fig:momentsd1d1c}. \n\nThe 2D-SED shapes of the parameters $\\mathcal{D}_\\ell^{\\mathcal{N} \\times \\mathcal{M}}(\\nu_i \\times \\nu_j)$ in the $(\\nu_i, \\nu_j)$ space\\footnote{For example, $\\mathcal{S}(\\nu_i,\\nu_j) = \\frac{I_{\\nu_i}(\\beta_0,T_0)I_{\\nu_j}(\\beta_0,T_0)}{I_{\\nu_0}(\\beta_0,T_0)^2}\\cdot \\left[\\ln\\left(\\frac{\\nu_i}{\\nu_0}\\right)\\ln\\left(\\frac{\\nu_j}{\\nu_0}\\right) \\right]$ is associated to the $\\mathcal{D}^{\\omega_1 \\times \\omega_1}_\\ell$ parameter (see Eq.~\\ref{eq:moments}).} are displayed in Fig.~\\ref{fig:momentshapes}. We used the nine frequencies of \\textit{LiteBIRD}{} presented in Sect.~\\ref{sec:Instrsim} and fixed $\\beta_0=1.54$ and $T_0 = 20$\\,K. We also introduce the CMB 2D-SED shape with the black body function:\n\n\\begin{equation}\n    B_{\\rm CMB}(\\nu_i \\times \\nu_j) = \\frac{B_{\\nu_i}(T_{\\rm CMB})B_{\\nu_j}(T_{\\rm CMB})}{B_{\\nu_0}(T_{\\rm CMB})^2},\n\\end{equation}\n\n\\noindent where $T_{\\rm CMB} = 2.726$\\,K. \n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/gauss_r_0.01_d1_SLD_full.pdf}\n    \\caption{\\footnotesize Posterior on $\\hat{r}$ in the {\\tt d1c} simulation type with $r_{\\rm sim}=0.01$ and $f_{\\rm sky}=0.7$ for the different fitting schemes: $r$MBB (blue, dotted line), $r\\beta$-1 (red, dashed line), $r\\beta$-$T$ (green, solid line), and $r\\beta$-2 (yellow, dash-dotted line). The vertical black dashed line marks the value of $r_{\\rm sim}$.}\n    \\label{fig:rgauss3}\n\\end{figure}\n\nThe 2D correlation coefficients between these 2D-SED shapes are displayed in Fig.~\\ref{fig:corr3D}. We present the correlations between the shapes of the parameters in the case of the $r\\beta$-$T$ and $r\\beta$-2 fitting schemes. We can see that all the moment parameters in $\\omega^{\\beta}_2$ are strongly correlated with the CMB SED signal, while the ones in $\\omega^{T}_1$ are not. \n\nWe showed that, when fitting $\\beta$-2 on {\\tt d1c}, the SED distortions due to spatial variations of $T$ are incorrectly detected by the second-order moment parameters with respect to the spectral index $\\beta$. Due to the correlations highlighted above, those spurious moment parameters could then leak into $\\hat{r}$ when adding it to the fit in $r\\beta$-2. This explains both the negative shift on the $\\hat{r}$ posterior using $\\beta$-2 in the {\\tt d1c} simulation type with $f_{\\rm sky}=0.7$ presented in Sect.~\\ref{sec:d1c} and \\ref{sec:rnot0}, and the inability to  correctly recover the  $\\omega^{\\beta}_2 \\times \\omega^{\\beta}_2$ dust moment parameter presented in Fig.~\\ref{fig:momentsd1d1c}. In addition, it gives a natural reason for the surge of $\\sigma_{\\hat{r}}$ when the second-order moments in $\\beta$ are added to the fit. \n\nOn the other hand, the moment parameters in $\\omega^{T}_1$ are strongly correlated with the moments in $\\omega^{\\beta}_1$. This behavior is expected due to the strong correlation between $\\beta$ and $T$  \\citep[see e.g.,][]{betatcorr}. However those moment parameters are less correlated with the CMB signal than the second-order parameters of $\\beta$-2. This points out that the factor of $\\sim 2$ on $\\sigma_{\\hat{r}}$ between $\\beta$-$T$ and $\\beta$-2 is due to this correlation of the 2D-SED shapes.\nAs the parameters in $\\omega^{T}_1$ are highly correlated with one another, we expect them to be highly redundant in the fit. However, repeating the process described in Sect.~\\ref{sec:d1c} using only $\\mathcal{D}^{A \\times \\omega^{T}_1}_\\ell$ for $\\beta$-$T$ ---which is equivalent to applying the $\\beta$-1 fitting scheme with an iterative correction to the temperature $T_0(\\ell)$--- gives a $\\hat{r}$ posterior similar to the one obtained for $\\beta$-1 alone. Taking the other $\\omega^{T}_1$ terms  into account appears to be necessary in order to recover an unbiased distribution of $\\hat{r}$.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/corr3D.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/corr3DLB_betaT.pdf}\n    \\caption{\\footnotesize Correlation matrices of the 2D-SED shapes of the CMB ($B_{\\rm CMB}(\\nu_i\\times\\nu_j)$ and dust moments $\\mathcal{D}_\\ell^{\\mathcal{N} \\times \\mathcal{M}}(\\nu_i \\times \\nu_j)$ in the $(\\nu_i, \\nu_j)$ space). Each element represents the Pearson correlation coefficient between any 2 of these 2D-SED shapes. The correlation matrices are displayed in the case of the $\\beta$-2 fitting scheme (top panel) and the $\\beta$-$T$ one (bottom panel).}\n    \\label{fig:corr3D}\n\\end{figure}\n\n\\subsection{Adding synchrotron to the simulations }\n\\label{sec:synchrotron}\n\nThermal dust is not the only source of polarized foreground that must be considered for CMB studies. Although subdominant at high frequencies ($\\geq 100\\,$GHz), the synchrotron emission due to accelerated electrons in the interstellar medium is still expected to represent a significant fraction of the total polarized signal. \n\nIn order to take one more step towards realistic forecasts for the \\textit{LiteBIRD}{} instrument, we add a synchrotron contribution to the {\\tt d1c} simulations presented in \\ref{sec:simulations} using the {\\tt s1} template included in the {\\sc PySM}. In this scenario, the synchrotron SED for each line of sight is given by a power law of the form  (in antenna temperature units)\n\n\\begin{equation}\n    S_\\nu^{\\tt s1} = A_{\\tt s1}(\\vec{n}) \\left(\\frac{\\nu}{\\nu^{\\tt s1}_0}\\right)^{\\beta_{\\tt s1}(\\vec{n})},\n\\label{eq:s1powerlaw}\n\\end{equation}\n\n\\noindent where the amplitude $A_{\\tt s1}(\\vec{n})$ and the spectral index $\\beta_{\\tt s1}(\\vec{n})$ maps are derived from the combination of the \\textit{WMAP}{} mission 23 GHz map \\cite{WMAPfg} and Haslam 408 GHz map \\cite{Haslam1}. $\\nu^{\\tt s1}_0$ is defined as 23 GHz. The simulations containing synchrotron are referred to as {\\tt d1s1c} below.\n\nIf not treated in the fit, the presence of synchrotron is expected to induce a bias on the $\\hat{r}$ posterior distribution. Regarding the dust MBB discussed in Sect.~\\ref{sect:limitsmbb}, the synchrotron SED is expected to have distortions. However, as the synchrotron polarized emission is significantly lower than that of dust, in the frequency range considered in the present work, we expect the distortions to be small compared to the ones induced by dust and we leave their modeling to a further study. \n\nIn order to minimize the number of free parameters used for fitting the synchrotron emission, we model its power spectrum as a power law of the multipole $\\ell$ \\citep{krachmalnicoff_s-pass_2018}. Therefore, combining with the synchrotron SED in Eq.~\\ref{eq:s1powerlaw}, the synchrotron component of the cross-angular power spectra reads \n\n\\begin{equation}\n    \\mathcal{D}^{\\rm sync}_\\ell (\\nu_i \\times \\nu_j) = A_{\\rm s}  \\left(\\frac{\\nu_i \\nu_j}{\\nu_0}\\right)^{\\beta_{\\rm s}} \\ell^{\\alpha_{\\rm s}},\n\\label{eq:syncromoment}\n\\end{equation}\n\nwhere the amplitude coefficient $A_{\\rm s}$ is treated as a free parameter while we fix $\\beta_{\\rm s}=-3$ (median value of the ${\\tt s1}$ $\\beta_{\\rm s}$ map on our $f_{\\rm sky}=0.7$ mask) and $\\alpha_{\\rm s}=-1$ \\citep{krachmalnicoff_s-pass_2018}. \n\nWhen fitting the {\\tt d1s1c} simulations, we either use the $r\\beta$-$T$ fitting scheme, neglecting the synchrotron component, or we add the synchrotron component in Eq.~\\ref{eq:syncromoment} to the model in Eq.~\\ref{eq:model}. We refer to this latter case as the {\\tt s}$r\\beta$-$T$ fitting scheme.\nIn Fig.~\\ref{fig:synchrotron}, the $\\hat{r}$ posteriors derived from the {\\tt d1s1c} simulations are displayed with $r_{\\rm sim}=0$ and $f_{\\rm sky}=0.7$. \n\nUsing the $r\\beta$-$T$ fitting scheme, we find $\\hat{r} = (143.1 \\pm 13.5)\\times 10^{-4}$. As expected, even at high frequencies, modeling the synchrotron component is critical and cannot be neglected in order to recover an unbiased value of $\\hat{r}$.\nOn the other hand, using {\\tt s}$r\\beta$-$T$ fitting scheme, we recover $\\hat{r} = (-5.4 \\pm 13.2)\\times 10^{-4}$. This result is comparable with the one obtained for the {\\tt d1c} simulations in Sect.~\\ref{sec:d1c}, with a minor increase in $\\sigma_{\\hat{r}}$. We can therefore conclude that a model as simple as that of Eq.~\\ref{eq:syncromoment} is sufficient to take into account the {\\tt s1} component at $\\nu > 100$\\,GHz and the corresponding SED distortions can be neglected in order to recover an unbiased value of $\\hat{r}$. In principle, as we know that the dust-synchrotron spatial correlation is significant at large scales \\citep{PlanckDust2},  Eq.~\\ref{eq:model} should include a dust-synchrotron term \\citep[see e.g.,][]{SOgalactic}. In our study, where we consider cross-spectra from 100 to 402\\,GHz, this dust-synchrotron term is subdominant, but it could be significant when considering cross-spectra between LiteBIRD's extreme frequency bands (e.g., the 40$\\times$402 cross-spectrum). The moment expansion might be more complicated as well in this case, as we could expect some correlation between the dust and synchrotron moment-terms.\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/gauss_r_0_d1s1_SLD_full.pdf}\n    \\caption{\\footnotesize Posterior on $\\hat{r}$ in the {\\tt d1s1c} simulation type with $r_{\\rm sim}=0$ and $f_{\\rm sky}=0.7$ for the different fitting schemes: $r\\beta$-$T$ (green, solid line) and {\\tt s}$r\\beta$-$T$ (orange, dash-dotted line). The vertical black dashed line marks the value $r_{\\rm sim}=0$.}\n    \\label{fig:synchrotron}\n\\end{figure}\n\nThis result shows that a full polarized foreground content can be treated at high frequencies when using a power law SED for the synchrotron coupled with the moment expansion of the MBB up to first order in both $\\beta$ and $T$ for the dust SED. A full study remains to be done in that direction using all the frequency bands of the \\textit{LiteBIRD}{} instrument. Eventually, Eq.~\\ref{eq:syncromoment} will also have to be expanded in moments with respect to its parameters. Doing so, one can expect to recover an unbiased value of $\\hat{r}$ associated with a decrease in $\\sigma_{\\hat{r}}$ down to a value compatible with the full success criterion of the mission.\n\n\\subsection{Limitations of this work and caveats}\n\n{As discussed in Sect.~\\ref{sec:formalismspectra}, we neglected polarization effects through this work by treating the $BB$ signal as an intensity signal. This is not problematic in the present work, because no variations along the lines of sight were present in the simulations. However, this point has to be addressed using complex simulations or real sky data.}\n\nThe choice of reference frequency $\\nu_0$ used for the normalization of the MBB in Eq.~\\ref{eq:MBB}, which is not discussed in this study, can potentially have a significant impact on the moment expansion and, in turn, on the measurement of $\\hat{r}$. Indeed, $\\nu_0$ is the pivot frequency of the moment expansion (moments are equal to zero at $\\nu_0$) and will determine the shape of the SED distortion around it. A poor choice for this reference frequency can have disastrous consequences for the moment fit: for example, if it is chosen far away from the observed bands, all the moments will become degenerated. In our case, the reference frequency (353\\,GHz) is within the observed frequency range (100 to 402\\,GHz), but we have not tried to optimize its position. In addition, the $\\nu_0$ pivot of our moment expansion coincides with the one used to extrapolate the dust template map in the {\\sc PySM} and we have not quantified how much this impacts our results. \n\nFinally, as pointed out several times in this work, the quantitative results depend strongly on the sky model of our simulations. Moreover, we lack dedicated sky models where we can control the complexity of the dust SED, either by directly including moments or by averaging the emission from the 3D structure of the Galaxy. However, both methods are beyond the scope of the present work.\n\n\n\\section{Conclusion}\n\nBeing able to precisely characterize the complexity of the Galactic thermal dust polarized SED has become critical for the measurement of the faint primordial $B$-mode signal of the CMB, especially at the sensitivity targeted by future CMB experiments such as the \\textit{LiteBIRD}{} satellite mission.\n\nIn this work, we applied the moment expansion formalism to the dust emission SED as a component-separation tool to recover the tensor-to-scalar ratio parameter $r$ in \\textit{LiteBIRD}-simulated data. This formalism, proposed by \\citet{Chluba} and implemented in harmonic space by \\citet{Mangilli}, allows us to deal with the spectral complexity of the Galactic dust signal by modeling its deviations from the canonical MBB model at the cross-angular power spectrum level. In the case of the data-driven realistic dust emission model ---we explore ({\\sc PySM} {\\tt d1}) \nhere---, suitably taking into account the dust SED distortions prevents the spurious detection of the primordial $B$-mode signal. \n\nWe show that the dust spectral index $\\beta$ and dust temperature $T$ spatial variations significantly distort the dust cross-power spectrum SED. The MBB is not a good model to describe the data in that case and the estimation of $r$ is dramatically affected. In the case where no primordial signal is included in the simulated data sets, not taking into account the dust SED complexity leads to a highly significant spurious detection of $r$ with \\textit{LiteBIRD}{} (from $\\hat{r}\\simeq5\\times10^{-3}$ to $1.25\\times10^{-2}$, with a 8.4 to 21.2\\,$\\sigma$ significance, from 50 to 70\\,\\% of the sky, respectively). \n\nTo overcome this obstacle, we applied the moment expansion formalism in order to model these SED distortions. We demonstrate that, at \\textit{LiteBIRD}{} sensitivity, the previously studied moment expansion with respect to the dust spectral index $\\beta$  \\citep{Mangilli,Azzoni} does not give satisfactory results. Indeed, expanding in $\\beta$ to first order (following the angular power spectrum definition of the order) leads to a significant bias on 70\\,\\% of the sky ($\\hat{r}=(3.29\\pm0.65)\\times10^{-3}$ when $r_{\\rm sim}=0$ and $\\hat{r}=(1.29\\pm0.08)\\times10^{-2}$, when $r_{\\rm sim}=10^{-2}$). At second order in $\\beta$, we observe a $\\sim$2\\,$\\sigma$ negative tension ($\\hat{r}=(-3.7\\pm1.9)\\times10^{-3}$ when $r_{\\rm sim}=0$ and $\\hat{r}=(6.25\\pm2.50)\\times10^{-3}$, when $r_{\\rm sim}=10^{-2}$). \n\nWe introduce for the first time in this work the expansion of the dust angular cross-power spectra with respect to both $\\beta$ and $T$. We show that by using this expansion up to first order, we correctly model the dust SED distortions due to spatial variations of both $\\beta$ and $T$ at the map level. This allows us to recover $r$ parameter without bias, with $\\hat{r}=(-3.3\\pm11.7)\\times10^{-4}$ if $r_{\\rm sim}=0$ and $\\hat{r}=(0.95\\pm0.15)\\times10^{-2}$ if $r_{\\rm sim}=10^{-2}$. Thus, despite the known degeneracy between the dust spectral index and its temperature in the Rayleigh-Jeans domain, it is important to correctly model the latter in order to accurately retrieve the tensor-to-scalar ratio $r$ at the unprecedented precision reached by experiments such as \\textit{LiteBIRD}{}. \n\nAdding parameters to tackle the dust SED complexity means an increase in the error budget. Given the \\textit{LiteBIRD}{} bands and sensitivities we consider in this work (frequency bands above 100\\,GHz), the ideal sensitivity on $r$ without delensing is $\\sigma_{\\hat{r}}=3.4\\times10^{-4}$. In the ideal case, where the dust $\\beta$ and $T$ are constant over the sky ({\\sc PySM} {\\tt d0}), separating the CMB from dust leads to $\\sigma_{\\hat{r}}=3.9\\times10^{-4}$ on 70\\,\\% of the sky. Adding the expansion to first order in $\\beta$ does not significantly increase the error ($\\sigma_{\\hat{r}}=4.5\\times10^{-4}$), but expanding to first order in both $\\beta$ and $T$ multiplies it by a factor of $\\sim2$ ($\\sigma_{\\hat{r}}=9.5\\times10^{-4}$) and to second order in $\\beta$ by a factor of $\\sim4$ ($\\sigma_{\\hat{r}}=16.4\\times10^{-3}$). We show that the surge of $\\sigma_{\\hat{r}}$ between the two latter cases, sharing the same number of free parameters, is due to strong correlations between the SED of the second-order moments in $\\beta$ and the CMB. This is an important point, as it could lead to some intrinsic limitation for component-separation algorithms based exclusively on the modeling of the SED. Furthermore, when dealing with real data, if the dust SED is complex enough to have significant second-order distortions with respect to $\\beta$, CMB experiments might reach a dilemma: either include the second order in the modeling at the cost of losing sensitivity on $r,$ or neglect it at the cost of a potential spurious detection. Coupling the SED-based separation with methods exploiting the diversity of spatial distribution between components \\citep[e.g.,][]{Wavelets} seems a natural way to overcome this issue.\n\nNevertheless, moment expansion at the cross-angular power spectrum level provides a powerful and agnostic tool, allowing us to analytically recover the actual dust complexity without making any further assumptions. We additionally show that this method is robust, in the sense that it can effectively distinguish the primordial tensor signal from dust when $r_{\\rm sim} \\neq 0$, as in the case of \\textit{LiteBIRD}{} simulations. The dust moments in $\\beta$ and $T$ at first order are needed in order to retrieve a reliable measure of $r$; they are significantly detected for $\\ell\\lesssim100$. We can therefore define a cut in $\\ell$ above which we do not fit for the whole complexity of the dust (we fit only the expansion up to first order in $\\beta$ and not in $\\beta$ and $T$). Doing so, we can reduce the error on $\\hat{r}$ while keeping the bias negligible ($\\hat{r}=(-0.9\\pm8.8)\\times10^{-4}$). We could imagine other ways to reduce the number of free parameters in our model \\cite[e.g., assuming a power-law of $\\ell$ behavior for the moments, as in][]{Azzoni} and hence reduce the error on $r$. However, this optimization really depends on the simulated sky complexity and has not been comprehensively explored in the present work.\n\nThe {\\sc PySM} {\\tt d1} sky simulations, being data-driven, are widely used by the CMB community as they contain some of the real sky complexity. Nevertheless, at high-Galactic latitudes, the dust spectral index and temperature templates from \\textit{Planck}{} are dominated by systematic errors (uncertainty on the assumed zero-level of the \\textit{Planck}{} intensity maps, residual cosmic infrared background (CIB), anisotropies, instrumental noise, etc.). Therefore, some of the complexity we observe far from the Galactic plane in this sky model is not real. On the other hand, the modeled SED of the dust is {exactly} a MBB in each pixel, and line-of-sight averages or more complex dust models are ignored. As a consequence, our method and CMB $B$-mode component-separation algorithms in general need to be confronted with more complex models in order to really assess their performances in a  quantitative manner. \n\nFinally, although we demonstrate that the synchrotron component can be tackled at frequencies above 100\\,GHz with a minimal model under our assumptions, a study over the full \\textit{LiteBIRD}{} frequency bands, including synchrotron and the potential moment expansion of its SED, will be considered as a natural next step for a further application.\n\n\n\\section*{Acknowledgments}\n\n\nThis work is supported in Japan by ISAS\/JAXA for Pre-Phase A2 studies, by the acceleration program of JAXA research and development directorate, by the World Premier International Research Center Initiative (WPI) of MEXT, by the JSPS Core-to-Core Program of A. Advanced Research Networks, and by JSPS KAKENHI Grant Numbers JP15H05891, JP17H01115, and JP17H01125. The Italian \\textit{LiteBIRD}{} phase A contribution is supported by the Italian Space Agency (ASI Grants No. 2020-9-HH.0 and 2016-24-H.1-2018), the National Institute for Nuclear Physics (INFN) and the National Institute for Astrophysics (INAF). The French \\textit{LiteBIRD}{} phase A contribution is supported by the Centre National d'Etudes Spatiale (CNES), by the Centre National de la Recherche Scientifique (CNRS), and by the Commissariat \u00e0 l'Energie Atomique (CEA). The Canadian contribution is supported by the Canadian Space Agency. The US contribution is supported by NASA grant no. 80NSSC18K0132. \nNorwegian participation in \\textit{LiteBIRD}{} is supported by the Research Council of Norway (Grant No. 263011). The Spanish \\textit{LiteBIRD}{} phase A contribution is supported by the Spanish Agencia Estatal de Investigaci\u00f3n (AEI), project refs. PID2019-110610RB-C21 and AYA2017-84185-P. Funds that support the Swedish contributions come from the Swedish National Space Agency (SNSA\/Rymdstyrelsen) and the Swedish Research Council (Reg. no. 2019-03959). The German participation in \\textit{LiteBIRD}{} is supported in part by the Excellence Cluster ORIGINS, which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy (Grant No. EXC-2094 - 390783311). This research used resources of the Central Computing System owned and operated by the Computing Research Center at KEK, as well as resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy.\n\nMR acknowledges funding support from the ERC Consolidator Grant CMBSPEC (No. 725456) under the European Union's Horizon 2020 research and innovation program.\n\nThe authors would like to thank David Alonso, Josquin Errard, Nicoletta Krachmalnicoff and Giuseppe Puglisi, for useful discussions as well as Jens Chluba for insightful comments on earlier version of this work. \n\n\n\n\\bibliographystyle{aa}\n\n\\section{Introduction}\n\\label{sec:intro}\n\nOur present understanding of the primordial Universe relies on the paradigm of inflation \\citep{inflationhist1,inflationhist2,inflationhist3}, introducing a phase of accelerated expansion in the first fractions of a second after the primordial singularity. Such a phenomenon is expected to leave a background of gravitational waves propagating in the primordial plasma during recombination, leaving a permanent mark imprinted in the polarization anisotropies of the cosmic microwave background (CMB): the primordial $B$-modes \\citep{InflationmodesB3,InflationsmodesB1,InflationsmodesB2}. The amplitude of the angular power spectrum of those primordial $B$-modes is characterized by the {tensor-to-scalar ratio} $r$, which is proportional to the energy scale at which inflation occurred \\citep{renergyscale}. Hence, looking for this smoking gun of inflation allows us to test our best theories of fundamental physics in the primordial Universe at energy scales far beyond the reach of particle accelerators. In this scope, it is one of the biggest challenges of cosmology set out for the next decades. The best experimental upper limit on the $r$ parameter so far is $r<0.032$ \\citep[95\\,\\% C.L.,][]{tristram,bicep2021,PlanckandBICEP}.\n\nThe JAXA Lite (Light) satellite, used for the $B$-mode polarization and Inflation from cosmic background Radiation Detection (\\textit{LiteBIRD}{}) mission, is designed to observe the sky at large angular scales in order to constrain this parameter $r$ down to $\\delta r= 10^{-3}$, including all sources of uncertainty \\citep{litebird,LiteBIRDUpdatedDesign}. Exploring this region of the parameter space is critical, because this order of magnitude for the tensor-to-scalar ratio is predicted by numerous physically motivated inflation models (for a review see e.g.,  \\cite{EncyclopediaInflationaris})  \n\nHowever, the success of this mission relies on our ability to treat polarized foreground signals. Indeed various diffuse astrophysical sources emit polarized $B$-mode signals above the primordial ones, the strongest being due to the diffuse polarized emission of our own Galaxy \\citep{PlanckCompoSep}. Even in a diffuse region like the BICEP\/Keck field, the Galactic $B$-modes are at least ten times stronger at 150\\,GHz than the $r=0.01$ tensor $B$-modes targeted by the current CMB experiments \\citep{BICEPKECKGW}.\n\nThe true complexity of polarized foreground emission that the next generation of CMB experiments will face is still mostly unknown today. Underestimation of this complexity can lead to the estimation of a spurious nonzero value of $r$ \\citep[see e.g.,][]{PlanckL,Remazeilles_etal_2016}.\n\nAt high frequencies ($>100$ GHz), the thermal emission of interstellar dust grains is the main source of Galactic foreground contaminating the CMB \\citep{dusthighfreq,PlanckDust2}. The canonical model of the spectral energy distribution (SED) of this thermal emission for intensity and polarization is given by the modified black body (MBB) law \\citep{Desertdustmodel}. This model provides a good fit to the dust polarization SED at the sensitivity of the \\textit{Planck}{} satellite \\citep{PlanckDust2} but it may not fully account for\nit at the sensitivity of future experiments \\citep{HensleyBull}. Furthermore, due to changes of physical conditions across the galaxy, spatial variations of the SEDs are present between and along the lines of sight. The former leads to what is known as \\emph{frequency decorrelation} in the CMB community \\citep[see e.g.][]{tassis,PlanckL,pelgrims2021}. Moreover, both effects lead to averaging MBBs when observing the sky (unavoidable line-of-sight or beam-integration effects). Because of the nonlinearity of the MBB law, those averaging effects will {distort} the SED, leading to deviations from this canonical model \\citep{Chluba}. \n\n\\cite{Chluba} proposed a general framework called {``moment expansion''} of the SED to take into account those distortions, using a Taylor expansion around the MBB with respect to its spectral parameters \\citep[Taylor expansion of foreground SEDs was discussed in previous studies; see e.g.,][]{stolyarov2005}. This method is agnostic: it does not require any assumption on the real complexity of the polarized dust emission. The moment expansion approach thus provides a promising tool with which to model the unanticipatable complexity of the dust emission in real data.\n\n\\cite{Mangilli} generalized this formalism for the sake of CMB data analysis in harmonic space and for cross-angular power spectra and applied it successfully to complex simulations and \\textit{Planck}{} High-Frequency Instrument (HFI) intensity data. This latter work shows that the real complexity of Galactic foregrounds could be higher than expected, encouraging us to follow the path opened by the moment expansion formalism.\n\nIn the present work, we apply the moment expansion in harmonic space to characterize and treat the dust foreground polarized emission of \\textit{LiteBIRD}{} high-frequency simulations, using  dust-emission models of increasing complexity. We discuss the ability of this method to recover an unbiased value for the $r$ parameter, with enough accuracy to achievethe scientific objectives of the \\textit{LiteBIRD} mission.\n\nIn Sect.~\\ref{sec:formalism}, we first review the formalism of moment expansion in map and harmonic domains. We then describe in Sect.~\\ref{sec:sims} how we realize several sets of simulations of the sky as seen by the \\textit{LiteBIRD}{} instrument with varying dust complexity and how we estimate the angular power spectra. In Sect.~\\ref{sec:fit}, we describe how we estimate the moment parameters and the tensor-to-scalar ratio $r$ in those simulations. The results are then presented in Sect.~\\ref{sec:results}. Finally, we discuss those results and the future work that has to be done in the direction opened by moment expansion in Sect.~\\ref{sec:discussion}.\n\n\\section{\\label{sec:formalism}Formalism}\n\n\\subsection{Characterizing the dust SED in real space}\n\n\\subsubsection{Modified black body model \\label{sec:mbb}}\n\nThe canonical way to characterize astrophysical dust-grain emission in every volume element of the Galaxy is given by the modified black body (MBB) function, consisting of multiplying a standard black body SED $B_\\nu(T)$ at a given temperature $T_0$ by a power-law of the frequency $\\nu$ with a spectral index $\\beta_0$. The dust intensity map $I_{\\rm D}(\\nu,\\vec{n})$ observed at a frequency $\\nu$ in every direction with respect to the unit vector $\\vec{n}$, can then be written as:\n\\begin{equation}\n\\label{eq:MBB}\n     I(\\nu,\\vec{n}) =  \\left(\\frac{\\nu}{\\nu_0}\\right)^{\\beta_0} \\frac{B_{\\nu}(T_0)}{B_{\\nu_0}(T_0)} A(\\vec{n}) \n     =\\frac{I_{\\nu}(\\beta_0,T_0)}{I_{\\nu_0}(\\beta_0,T_0)} A(\\vec{n}),\n\\end{equation}\n\n\\noindent where $A(\\vec{n})$ is the dust intensity template at a reference frequency $\\nu_0$\\footnote{Throughout this work, we use $\\nu_0 = 353$\\,GHz.}.\nWe know that the physical conditions (thermodynamic and dust grain properties) change through the interstellar medium across the Galaxy, depending, in an intricate fashion, on the gas velocity and density, the interstellar radiation field, the distance to the Galactic center \\citep[see e.g., ][]{dustacrossMW,dustvarMW,PlanckVardust,PlanckCompoSep,vardustdisk,dustradfield}. This change of physical conditions leads to variations in $\\beta$ and $T$ depending on the direction of observation $\\vec{n}$:\n\n\\begin{equation}\n\\label{eq:MBBn}\n     I(\\nu,\\vec{n}) =  \\frac{I_{\\nu}(\\beta(\\vec{n}),T(\\vec{n}))}{I_{\\nu_0}(\\beta(\\vec{n}),T(\\vec{n}))} A(\\vec{n}).\n\\end{equation}\n\nThe SED amplitude and parameters (temperature and spectral index) are then different for every line of sight. It is therefore clear that, in order to provide a realistic model of the dust emission, the frequency and spatial dependencies may not be trivially separated. \n\n\\subsubsection{\\label{sect:limitsmbb}Limits of the modified black body}\n\nThe dust SED model given by the MBB has proven to be highly accurate \\citep{Planck2014dust,Planck2015dust}. However, it must be kept in mind that this model is empirical and is therefore not expected to give a perfect description of the dust SED in the general case. Indeed, physically motivated dust grain emission models predict deviations from it \\citep[e.g.,][]{modelbeyondmbb}. Surveys tend to show that the dust-emission properties vary across the observed 2D sky and the 3D Galaxy \\citep{PlanckDust2}. Furthermore, in true experimental conditions, one can never directly access the pure SED of a single volume element with specific emission properties and unique spectral parameters. Averages are therefore made over different SEDs emitted from distinct regions with different physical emission properties, in a way that may not be avoided:  along the line of sight; between different lines of sight, inside the beam of the instrument or; when doing a spherical harmonic decomposition to calculate the angular power spectra over large regions of the sky.\n\nThe MBB function is nonlinear, and therefore summing MBBs with different spectral parameters does not return another MBB function and produces \\emph{SED distortions}. For all these reasons, modeling the dust emission with a MBB is intrinsically limited, even when doing so with spatially varying spectral parameters. As a consequence, inaccuracies might appear when modeling the dust contribution to CMB data that will unavoidably impact the final estimation of the cosmological parameters. \n\n\\subsubsection{Moment expansion in pixel space}\n\\label{sec:moment_pixel}\n\nA way to address the limitation of the MBB model in accurately describing the dust emission is given by the {moment expansion} formalism proposed by \\cite{Chluba}. This formalism is designed to take into account the SED distortions due to averaging effects by considering a multidimensional Taylor expansion of the distorted SED $I(\\nu,\\vec{p})$ around the mean values $\\vec{p}_{0}$ of its spectral parameters $\\vec{p} = \\{p_i\\}$. This is the so-called {moment expansion} of the SED, which can be written as\n\\begin{align}\n    I(\\nu,\\vec{p}) = I(\\nu,\\vec{p}_{0}) &+ \\sum_i \\omega_1^{p_i}\\langle\\partial_{p_i}I(\\nu,\\vec{p})\\rangle_{\\vec{p}=\\vec{p}_0}\n    \\nonumber \\\\\n    &+ \\frac{1}{2}\\sum_{i,j} \\omega_2^{p_ip_j}\\langle\\partial_{p_i}\\partial_{p_j}I(\\nu,\\vec{p})\\rangle_{\\vec{p}=\\vec{p}_0}\n    \\nonumber \\\\\n    &+ \\dots \\nonumber\\\\\n    &+\\frac{1}{\\alpha!} \\sum_{i,\\dots,k} \\omega_\\alpha^{p_i\\dots p_k}\\langle\\partial_{p_i}\\dots\\partial_{p_k}I(\\nu,\\vec{p})\\rangle_{\\vec{p}=\\vec{p}_0},\n\\label{eq:momentgeneral}\n\\end{align}\n\nwhere the first term on the right-hand side is the SED without distortion $I(\\nu,\\vec{p}_{0})$ evaluated at $\\vec{p}=\\vec{p}_0$, and the other terms are the so-called {moments} of order $\\alpha$, quantified by the {moment parameters} $\\omega_\\alpha^{p_i\\dots p_k}$ for the expansion with respect to any parameter of $\\vec{p}$. Performing the expansion to increasing order adds increasing complexity to the SED $I(\\nu,\\vec{p}_{0})$. \n\nFor the MBB presented in Sect.~\\ref{sec:mbb}, there are two parameters so that $\\vec{p} = \\{\\beta,T\\}$. Thus the dust moment expansion reads\n\n\\begin{align}\n     I(\\nu,\\vec{n})  = \\frac{I_{\\nu}(\\beta_0,T_0)}{I_{\\nu_0}(\\beta_0,T_0)} \\bigg\\{ & A(\\vec{n}) + \\omega^\\beta_1(\\vec{n}) \\ln\\left(\\frac{\\nu}{\\nu_0}\\right)+ \\frac{1}{2}\\omega^\\beta_2(\\vec{n}) \\ln^2\\left(\\frac{\\nu}{\\nu_0}\\right)\\nonumber \\\\[2mm]\n     &+ \\omega^T_1(\\vec{n})\\Big( \\Theta(\\nu,T_0) - \\Theta(\\nu_0,T_0) \\Big) + \\dots \\bigg\\},\n\\label{eq:momentinttemp}\n\\end{align}\n\n\\noindent where the expansion has been written up to order two in $\\beta$ (with moment expansion parameters $\\omega^\\beta_1$ at order one and $\\omega^\\beta_2$ at order two) and to order one in $T$ (with a moment expansion parameter $\\omega^T_1$ at order one). The following expression has been introduced to simplify the black body derivative with respect to $T$:\n\n\\begin{align}\n    \\Theta(\\nu,T) = \\frac{x}{T}\\frac{e^{x}}{e^{x}-1},\\ {\\rm with}\\ x = \\frac{h \\nu}{k T}.\n\\end{align}\n\nThe moment expansion in pixel space can be used for component separation and possibly crossed with other methods \\citep[see e.g.,][]{RemazeillesmomentsILC,Debabrata_2021}. However, in the present work, we are interested in the modeling of the dust at the $B$-mode angular power spectrum level. Performing the moment expansion at the angular power spectrum level adds some complexity to the SEDs due to the additional averaging occurring when dealing with spherical harmonic coefficients. Indeed, these coefficients are estimated on potentially large fractions of the sky and probe regions with various physical conditions. On the other hand, the expansion at the power spectrum level possibly drastically reduces the parameter space with respect to performing the expansion in every sky pixel.\n\n\\subsection{Characterizing the dust SED in harmonic space}\n\n\\subsubsection{Dust SED in spherical harmonic space \\label{sec:formalismspectra}}\n\nThe expansion presented in Sect.~\\ref{sec:moment_pixel} can be applied in spherical harmonic space using the same logic. The sky emission projection then reads\n\n\\begin{equation}\n        I(\\nu,\\vec{n}) = \\sum_{\\ell = 0}^{\\infty}\\sum_{m=-\\ell}^{\\ell}I^\\nu_{\\ell m}Y_{\\ell m} (\\vec{n}).\n\\end{equation}\n\nApplying the moment expansion to the spherical harmonics coefficients, with respect to $\\beta$ and $T$, as in Eq.~\\ref{eq:momentinttemp}, leads to\n\n\\begin{align}\n     I^\\nu_{\\ell m} = & \\frac{I_{\\nu}(\\beta_0(\\ell),T_0(\\ell))}{I_{\\nu_0}(\\beta_0(\\ell),T_0(\\ell))} \\bigg\\{ A_{\\ell  m} + \\omega^\\beta_{1,\\ell  m} \\ln\\left(\\frac{\\nu}{\\nu_0}\\right) + \\frac{1}{2}\\omega^\\beta_{2,\\ell  m} \\ln^2\\left(\\frac{\\nu}{\\nu_0}\\right) \\nonumber \\\\[2mm]\n     &+ \\omega^T_{1,\\ell  m} \\Big( \\Theta(\\nu,T_0(\\ell)) - \\Theta(\\nu_0,T_0(\\ell)) \\Big) + \\dots \\quad\\bigg\\},\n\\label{eq:momentint2temp}\n\\end{align}\n\n\n\\noindent where this time $\\beta_0(\\ell)$ and $T_0 (\\ell)$ are the averages of $\\beta$ and $T$ at a given multipole $\\ell$ over the sky fraction we are looking at. We note that the moment parameters $\\omega^{p_i}_{\\alpha,\\ell m}$ involved here are different from the $\\omega^{p_i}_i(\\vec{n})$ appearing in Eq.~\\ref{eq:momentinttemp} in the map space because they involve different averaging. In principle, the moment expansion in harmonic space can take into account the three kinds of spatial averages presented in Sect.~\\ref{sect:limitsmbb}.\n\nAs the dust spectral index and temperature are difficult to separate in the frequency range considered for CMB studies \\citep[{i.e.}, Rayleigh-Jeans domain, see e.g.][]{betatcorr}, the moment expansion in harmonic space has  only been applied in the past with respect to $\\beta$, with the temperature being fixed to a reference value $T=T_0$ \\citep{Mangilli,Azzoni}. In the present paper, for the first time, the moment expansion in harmonic space is instead performed with respect to both $\\beta$ and $T$, as it was in real space in \\citet{RemazeillesmomentsILC}.\n\n\\subsubsection{Cross-power spectra}\n\nRelying on the derivation made by \\cite{Mangilli} and Eq.~\\ref{eq:momentint2temp}, we can explicitly write the cross-spectra between two maps $M_{\\nu_i}$ and $M_{\\nu_j}$ at frequencies $\\nu_i$ and $\\nu_j$, using the moment expansion in $\\beta$ and $T$ as follows:\n\n\\begin{align}\n    \\mathcal{D}_\\ell(\\nu_i \\times \\nu_j) &= \\frac{I_{\\nu_i}(\\beta_0(\\ell),T_0(\\ell))I_{\\nu_j}(\\beta_0(\\ell),T_0(\\ell))}{I_{\\nu_0}(\\beta_0(\\ell),T_0(\\ell))^2} \\cdot \\bigg\\{ \\nonumber \\\\[-0.5mm]\n   \n    0^{\\rm th}\\ \\text{order}\\;&\n    \\begin{cases}\n    & \\ \\mathcal{D}_\\ell^{A \\times A}\n    \\end{cases}\n    \\nonumber \\\\[-0.5mm]\n   \n    1^{\\rm st}\\ \\text{order}\\ \\beta\\;&\n    \\begin{cases}\n    &+\\mathcal{D}_\\ell^{A \\times \\omega^{\\beta}_1}\\left[ \\ln\\left(\\frac{\\nu_i}{\\nu_0}\\right) + \\ln\\left(\\frac{\\nu_j}{\\nu_0}\\right) \\right] \\nonumber \\\\\n    &+ \\mathcal{D}_\\ell^{\\omega^{\\beta}_1 \\times \\omega^{\\beta}_1} \\left[\\ln\\left(\\frac{\\nu_i}{\\nu_0}\\right)\\ln\\left(\\frac{\\nu_j}{\\nu_0}\\right) \\right]\\nonumber \\\\  \\end{cases}\\\\[-0.5mm]\n   \n    1^{\\rm st}\\ \\text{order}\\ T \\;&\n    \\begin{cases}\n    &+\\mathcal{D}_\\ell^{A \\times \\omega_1^T} \\left( \\Theta_i +  \\Theta_j -  2\\Theta_0\\right) \\\\\n    &+ \\mathcal{D}_\\ell^{\\omega_1^T \\times \\omega_1^T}\\Big(\\Theta_i - \\Theta_0\\Big)\\left(\\Theta_j - \\Theta_0\\right)\\nonumber\n    \\end{cases}\\\\[-0.5mm]\n   \n    1^{\\rm st}\\ \\text{order}\\ T\\beta \\;&\n    \\begin{cases}\n    &+ \\mathcal{D}_\\ell^{\\omega^{\\beta}_1 \\times \\omega_1^T} \\left[ \\ln\\left(\\frac{\\nu_j}{\\nu_0} \\right)\\Big( \\Theta_i - \\Theta_0\\Big) + \\ln\\left(\\frac{\\nu_i}{\\nu_0} \\right)\\left( \\Theta_j - \\Theta_0\\right)\\right]  \\nonumber \\\\\n    \\end{cases}\\\\[-0.5mm]\n   \n    2^{\\rm nd}\\ \\text{order}\\ \\beta \\;&\n    \\begin{cases}\n    &+ \\frac{1}{2} \\mathcal{D}_{\\ell}^{A \\times\\omega^{\\beta}_{2}}  \\left[ \\ln^2\\left(\\frac{\\nu_i}{\\nu_0}\\right)\n    + \\ln^2\\left(\\frac{\\nu_j}{\\nu_0}\\right)\n    \\right]\n    \\\\[-0.5mm]\n    &+ \\frac{1}{2} \\mathcal{D}_{\\ell}^{\\omega^{\\beta}_{1} \\times \\omega^{\\beta}_{2}}  \\Big[ \\ln \\left(\\frac{\\nu_i}{\\nu_0}\\right)  \\ln^2\\left(\\frac{\\nu_j}{\\nu_0}\\right)  \n    +\\ln\\left(\\frac{\\nu_j}{\\nu_0}\\right)\n    \\ln^2 \\left(\\frac{\\nu_i}{\\nu_0}\\right) \\Big] \n    \\\\[-0.5mm]\n    &+\\frac{1}{4} \\mathcal{D}_{\\ell}^{\\omega^{\\beta}_{2} \\times \\omega^{\\beta}_{2}}  \\left[\\ln^2 \\left(\\frac{\\nu_i}{\\nu_0}\\right)  \\ln^2 \\left(\\frac{\\nu_j}{\\nu_0}\\right) \\right]\n    \\,\n    \\end{cases}\\nonumber\\\\[-0.5mm]\n    &+ \\dots \\bigg\\},\n\\label{eq:moments}\n\\end{align}\n\n\\noindent where we use the following abbreviation: $\\Theta(\\nu_k,T_0(\\ell))\\equiv \\Theta_k$, so that $\\Theta_0=\\Theta(\\nu_0,T_0(\\ell))$, and we defined the moment expansion cross-power spectra between two moments $\\mathcal{M}$ and $\\mathcal{N}$ as\n\n\\begin{equation}\n    \\mathcal{C}_\\ell^{\\mathcal{M}\\times\\mathcal{N}} = \\sum_{m, m'=-\\ell}^{\\ell} \\mathcal{M}_{\\ell m} \\mathcal{N}_{\\ell m'},\\ {\\rm with}\\ (\\mathcal{M},\\mathcal{N})\\in\\left\\{A,\\omega^{\\beta}_1,\\omega^{T}_1, \\omega^{\\beta}_2,\\ \\dots\\right\\}.\n\\label{eq:moments_spectra}\n\\end{equation}\n\nIn the remainder of this article, we use the $\\mathcal{D}_\\ell$ quantity, which is a scaling of the angular power spectra, and is defined as \n\\begin{equation}\n    \\mathcal{D}_\\ell \\equiv \\frac{\\ell(\\ell +1)}{2 \\pi} \\mathcal{C}_\\ell.\n\\label{eq:dl}\n\\end{equation}\n\nEquation~\\ref{eq:moments} has been written using the expansion with respect to $\\beta$ at order two and $T$ at order one, as in Eq.~\\ref{eq:momentint2temp}. Nevertheless, the terms involving power spectra between order two in $\\beta$ and order one in $T$ have been neglected so as to match the needs of the implementation of our method in the following. \n\nHereafter, when we refer to \"order $k$\" at the angular power spectrum level, we are referring to moment expansion terms involving the pixel space moment up to order $k$. For example, $D_\\ell^{A\\times\\omega_1^T}$ and $D_\\ell^{\\omega^\\beta_1\\times\\omega_1^T}$ are order one, while $D_\\ell^{A\\times\\omega^\\beta_2}$, $D_\\ell^{\\omega^\\beta_1\\times\\omega^\\beta_2}$ and $D_\\ell^{\\omega^\\beta_2\\times\\omega^\\beta_2}$ are order two. At order zero, one retrieves the MBB description of the cross-angular power spectra SED $\\mathcal{D}_\\ell(\\nu_i\\times\\nu_j)$ as a function of the frequencies $\\nu_i$ and $\\nu_j$.\n\nThis formalism was originally introduced to analyze the complexity of intensity data in \\cite{Mangilli}. In the present work, we focus on $B$-mode polarization power spectra. This was put forward after analyzing the \\textit{Planck}{} and balloon-borne Large Aperture Submillimeter Telescope for Polarimetry (BLASTPol) data and finding that the polarization fraction appears to be constant in the far-infrared-to-millimetre wavelengths {\\citep{blastpol1,blastpol2}}. This allows us to assume that the same grain population is responsible for the total and polarized foreground emission \\citep{dustmodels}. As a result, intensity and polarization SED complexity may be similar.\nNevertheless, $Q$ and $U$ can have a different SED because of the polarization angle frequency dependence \\citep[see e.g.,][]{tassis,moment_polar} and so can $E$ and $B$. This could be a limitation when analyzing the dust $E$ and $B$ with a single moment expansion, especially when SED variations occur along the line of sight.\nEven when trying to model a single polarization component ---as we do in the present work, dealing only with $B$ modes--- it is not clear whether the distorted SED can be modeled in terms of $\\beta$ and $T$ moments only. Further work needs to be done to assess this question. However, they should not impact the present study in which variations along the line of sight are not simulated.\n\nModeling the complexity of the foreground signals by means of the moment expansion of the $B$-mode angular power spectrum has already been successfully applied to Simons Observatory \\citep{SimonsObservatory} simulated data  \\citep{Azzoni}. However, the approach taken by these latter authors is different from the one presented above. They apply a \\emph{minimal} moment expansion: assumptions are made to keep only the $\\mathcal{D}_\\ell^{\\omega^{\\beta}_1\\times\\omega^{\\beta}_1}$ and $\\mathcal{D}_\\ell^{A\\times\\omega^{\\beta}_2}$ parameters, which are modeled with a power-law scale dependence. These assumptions may not hold for experiments with higher sensitivity and observing wider sky patches. Furthermore, they assume a scale-invariant dust spectral index. In this work, on the other hand, we relax these assumptions in order to characterize the required spectral complexity of the dust emission for \\textit{LiteBIRD}{}.\n\n\\section{\\label{sec:sims}Simulations and cross-spectra estimation}\n\n\\subsection{\\label{sec:LiteBIRD}\\textit{LiteBIRD}{}}\n\n\\textit{LiteBIRD}{} is an international project proposed by the Japanese spatial agency (JAXA), which selected it in May 2019 as a strategic large class mission. The launch is planned for 2029 for a minimal mission duration of 3 years. \\citep{2020SPIE,Ptep}\n\n\\textit{LiteBIRD}{} is designed to realize a full sky survey of the CMB at large angular scales in order to look for the reionization bump of primordial $B$-modes and explore the scalar-to-tensor ratio ($r$) parameter space with a total uncertainty $\\delta r$ below $10^{-3}$, including foreground cleaning and systematic errors.\n\\textit{LiteBIRD}{} is composed of three telescopes observing in different frequency intervals: the Low-, Medium- and High-Frequency Telescopes (LFT, MFT and HFT).\nEach of the telescopes illuminates a focal plane composed of hundreds of polarimetric detectors. The whole instrument will be cooled down to 5\\,K \\citep{LiteBIRDUpdatedDesign} while the focal plane will be cooled down to 100\\,mK \\citep{LBsubK}. In order to mitigate the instrumental systematic effects, the polarization is modulated by a continuously rotating half-wave plate. \\textit{LiteBIRD}{} will observe the sky in 15 frequency bands from 40 to 402\\,GHz. Table~\\ref{tab:litetab} gives the details of the frequency bands and their sensitivities in polarization \\citep[adapted from][see Sect.~\\ref{sec:Instrsim}]{2020SPIE}.\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{cccc}\n        \\multirow{2}{*}{Telescope} & Frequency & Sensitivity  $\\sigma^{\\rm noise}_{Q,U}(\\nu)$ & $\\boldsymbol \\theta_{\\rm FWHM}$  \\\\\n         & [GHz]&  [$\\mu$K$\\cdot$arcmin] & {arcmin}\\\\\n        \\hline\n         LFT & 40.0 & 37.42 & {70.5}     \\\\\n        LFT & 50.0 & 33.46 & {58.5} \\\\\n        LFT & 60.0 & 21.31 & {51.1} \\\\\n        LFT & 68.0 & {19.91\/31.77} & {41.6\/47.1} \\\\\n        LFT & 78.0 & {15.55\/19.13} & {36.9\/43.8} \\\\\n        LFT & 89.0 & {12.28\/28.77} & {33.0\/41.5} \\\\\n     LFT\/MFT & 100.0 & {10.34\/8.48} & {30.2\/37.8} \\\\\n     LFT\/MFT & 119.0 & {7.69\/5.70} & {26.3\/33.6} \\\\\n     LFT\/MFT & 140.0 & {7.25\/6.38} & {23.7\/30.8}  \\\\\n         MFT & 166.0 & 5.57 & {28.9} \\\\\n     MFT\/HFT & 195.0 & {7.05\/10.50} & {28.0\/28.6} \\\\\n         HFT & 235.0 & 10.79 & {24.7}\\\\\n        HFT & 280.0 & 13.8 & {22.5} \\\\\n        HFT & 337.0 & 21.95 & {20.9} \\\\\n        HFT & 402.0 & 47.45 & {17.9} \\\\\n\\end{tabular}\n\\caption{\\footnotesize Instrumental characteristics of \\textit{LiteBIRD}{} used in this study \\citep[adapted from][see Sect.~\\ref{sec:Instrsim}]{2020SPIE}. Some frequency bands are shared by two different telescopes or detector arrays. If so, the two values of polarization sensitivities $\\sigma^{\\rm noise}_{Q,U}(\\nu)$ and instrumental beam full width at half maximum $\\theta_{\\rm FWHM}$ are displayed on the same line.}\n\\label{tab:litetab}\n\\end{center}\n\\end{table}\n\n\\subsection{Components of the simulations}\n\\label{sec:ingredients}\n\nWe build several sets of \\textit{LiteBIRD}{} sky simulations. These multi-frequency sets of polarized sky maps are a mixture of CMB, dust, and instrumental noise. The simulations are made at the nine highest frequencies accessible by the instrument ($\\geq 100$ GHz), where dust is the predominant source of foreground contamination.\nFor every studied scenario, we built $N_{\\rm sim} = 500$ simulations, each composed of a set of $N_{\\rm freq}=9$ pairs of sky maps $(Q,U)$ built using the {\\sc HEALPix} package, with $N_{\\rm side} = 256$ \\citep{healpix}. All the signals will be expressed in $\\mu{\\rm K}_{\\rm CMB}$ units.\n\n\\subsubsection{Cosmic microwave background signal}\n\nTo generate the CMB signal, we use the {\\it Code for Anisotropies in the Microwave Background } \\citep[CAMB,][]{CAMB} to create a fiducial angular power spectrum from the best-fit values of cosmological parameters estimated by the recent \\textit{Planck}{} data analysis \\citep{PlanckOverview}.\n\nFor the $B$-modes, we consider the two different components of the spectrum: lensing-induced and primordial (tensor), so that $\\mathcal{D}_{\\ell}^{BB} =\\mathcal{D}_{\\ell}^{{\\rm lensing}} + r_{\\rm sim} \\cdot \\mathcal{D}_{\\ell}^{{\\rm tensor}}$, where $\\mathcal{D}_{\\ell}^{{\\rm tensor}}$ refers to the tensor $B$-modes for $r=1$ and $r_{\\rm sim}$ labels the input values of the tensor-to-scalar ratio $r$ contained in the simulation. We use two different values throughout this work: $r_{\\rm sim}=0$, which is used in the present work as the reference simulations and $r_{\\rm sim}=10^{-2}$ used for consistency checks when the CMB primordial signal is present.\n\nFor all simulations, we then generate the Stokes $Q$ and $U$ CMB polarization Gaussian realization maps $S^{\\rm CMB}_{\\nu,r_{\\rm sim}}$ from the angular power spectra using the {\\tt synfast} function of {\\sc HEALPix}.\n\n\\subsubsection{Foregrounds: dust}\n\nOur study focuses on high frequencies ($\\geq 100$\\,GHz) only, where thermal dust emission is the main source of polarized foreground as mentioned in Sect.~\\ref{sec:intro}. We make use of two different scenarios of increasing complexity included in the {\\sc PySM} \\citep{Pysm} and one of intermediate complexity not included in the {\\sc PySM}:\n\n\\begin{itemize}\n    \\item {\\tt d0}, included in the {\\sc PySM}: the dust polarization $Q$ and $U$ maps are taken from $S^{Planck}_{\\nu=353}$, the \\textit{Planck}{} 2015 data at 353\\,GHz \\citep{planck_2015_overview}, extrapolated to a frequency $\\nu$ using the MBB given in Eq.~\\ref{eq:MBB} with a temperature $T_0=T_{\\tt d0}=20$\\,K and spectral index $\\beta_0=\\beta_{\\tt d0}=1.54$ constant over the sky:\n    \n    \\begin{equation}S^{\\rm dust}_\\nu=S_\\nu^{\\tt d0}=\\frac{I_\\nu(\\beta_{\\tt d0},T_{\\tt d0})}{I_{\\nu_0}(\\beta_{\\tt d0},T_{\\tt d0})}\\cdot S^{Planck}_{353},\n    \\end{equation}\n    \n    \\item {\\tt d1T}, introduced here: the dust polarization $Q$ and $U$ maps are also taken from \\citet{planck_2015_overview} but they are extrapolated to a frequency $\\nu$ using the MBB given in Eq.~\\ref{eq:MBBn}, with spatially varying  spectral index $\\beta(\\vec{n})$, as in {\\tt d1} and a fixed temperature $T_0=T_{\\tt d1T}=21.9$\\,K, obtained as the mean of the \\textit{Planck}{} {\\sc Commander} dust temperature map  \\citep{planck_2015_commander} on our $f_{\\rm sky}=0.7$ sky mask: \n    \n    \\begin{equation}\n        S^{\\rm dust}_\\nu=S_\\nu^{\\tt d1T}=\\frac{I_\\nu(\\beta(\\vec{n}),T_{\\tt d1T})}{I_{\\nu_0}(\\beta(\\vec{n}),T_{\\tt d1T})}\\cdot S^{Planck}_{353}.\n    \\end{equation}\n    \n    \\item {\\tt d1}, included in the {\\tt PySM}: similar to { \\tt d1T} with both a spatially varying temperature $T(\\vec{n})$ and spectral index $\\beta(\\vec{n})$ obtained from the \\textit{Planck}{} data using the {\\sc Commander} code \\citep{planck_2015_commander}: \n    \n    \\begin{equation}\n        S^{\\rm dust}_\\nu=S_\\nu^{\\tt d1}=\\frac{I_\\nu(\\beta(\\vec{n}),T(\\vec{n}))}{I_{\\nu_0}(\\beta(\\vec{n}),T(\\vec{n}))}\\cdot S^{Planck}_{353}.\n    \\end{equation}\n\n\\end{itemize}\n\n\\subsubsection{ \\label{sec:Instrsim}Instrumental noise}\n\nThe band polarization sensitivities  $\\sigma^{\\rm noise}_{Q,U}(\\nu)$ are derived from the noise equivalent temperature (NET) values converted into $\\mu$K$\\cdot$arcmin for each telescope (LFT, MFT and HFT). As seen in Table~\\ref{tab:litetab}, some frequency bands are overlapping between two telescopes. In this situation, we take the mean value of the two NETs, weighted by the beam full width at half maximum (FWHM) $\\theta$ as:\n\n\\begin{equation}\n    \\sigma^{\\rm noise}_{Q,U}(\\nu_{\\rm overlapping}) = \\sqrt{\\frac{1}{ \\left(\\frac{\\theta_{\\rm min}}{{\\theta_{\\rm max}}}\\sigma^{\\rm noise}_{Q,U}(\\nu_{\\theta_{\\rm min}})\\right)^{-2} + \\left(\\sigma^{\\rm noise}_{Q,U}(\\nu_{\\theta_{\\rm max}})\\right)^{-2} }},\n\\end{equation}\n\n{\\noindent where $\\theta_{\\rm min}$ is the smallest FWHM among the two and $\\theta_{\\rm max}$ the largest. The band polarization sensitivities} are displayed in Table~\\ref{tab:litetab}. For every simulation, the noise component $N_\\nu$ is generated in every pixel of the maps with a Gaussian distribution centered on zero, with standard deviation $\\sigma^{\\rm noise}_{Q,U}(\\nu)$ weighted by the pixel size (and $\\sqrt{2}\\cdot\\sigma^{\\rm noise}_{Q,U}(\\nu)$ for the maps used to compute the auto-power spectra, see Sect.~\\ref{sec:spectra_estimation}).\n\nFor simplicity, we choose to ignore beam effects in our simulations, assuming they can be taken into account  perfectly. Simulations are thus produced at infinite (0\\,arcmin) resolution and no beam effect is corrected for when estimating the angular power spectrum. This is equivalent to convolving the maps by Gaussian beams of finite resolution and correcting the power spectra for the associated Gaussian beam window functions.\n\n\\subsection{Combining signals and building the simulated maps}\n\n\\label{sec:simulations}\n\nThe simulated $(Q,U)$ maps $M_\\nu$, for a given simulation, can be expressed as the sum:\n\n\\begin{equation}\nM_\\nu=S^{\\rm CMB}_{\\nu,r_{\\rm sim}}+S^{\\rm dust}_\\nu+N_\\nu.\n\\label{eq:map}\n\\end{equation}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale = 0.2]{Figures\/simulations.pdf}\n    \\caption{\\footnotesize Mean value over the $N_{\\rm sim}$ simulations of the $B$-mode angular power spectra $\\mathcal{D}_{\\ell}(\\nu_i \\times \\nu_j)$ for the {\\tt d1c} simulation type, with $r_{\\rm sim}=0$. The color bar spans all the $N_{\\rm cross}$ spectra $\\mathcal{D}_{\\ell}(\\nu_i \\times \\nu_j)$, associated to their reduced cross-frequency $\\nu_{\\rm red.}=\\sqrt{\\nu_i \\nu_j}$, from 100\\,GHz (dark red) to 402\\,GHz (dark blue). The input CMB lensing power spectrum is shown as a black dashed line.}\n    \\label{fig:simulations}\n\\end{figure*}\n\n\nCosmic microwave background and noise are simulated stochastically: for each simulation, we generate a new realization of the CMB maps $S^{\\rm CMB}_{\\nu,r_{\\rm sim}}$ and the noise maps $N_\\nu$. The dust map $S^{\\rm dust}_\\nu$ is the same for each simulation, at a given frequency.\n\nHereafter, we use the notation {\\tt d0}, {\\tt d1T,} and {\\tt d1} to refer to simulations containing only dust and \\textit{LiteBIRD}{} noise, {\\tt d0c}, {\\tt d1Tc,} and {\\tt d1c} for simulations including CMB, dust, and \\textit{LiteBIRD}{} noise and, finally, and {\\tt c} for the simulation containing only CMB and \\textit{LiteBIRD}{} noise. The different components present in these different {simulation types} are summarized in Table~\\ref{tab:sims}.\n\n\\begin{table}[t!]\n\\centering\n        \\begin{tabular}{c !{\\color{white}\\vrule width 1pt} c!{\\color{white}\\vrule width 1pt}c!{\\color{white}\\vrule width 1pt}c!{\\color{white}\\vrule width 1pt}c!{\\color{white}\\vrule width 1pt}c}\n        & $S^{\\rm CMB}_{\\nu,r_{\\rm sim}}$ & $S^{\\tt d0}_\\nu$ & $S^{\\tt d1T}_\\nu$ & $S^{\\tt d1}_\\nu$ & $N_\\nu$ \\\\[0.5ex]\\noalign{\\color{white}\\hrule height 1pt}\n        {\\tt c} & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{myred}$\\times$  & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        {\\tt d0} & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;  & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\n        \\noalign{\\color{white}\\hrule height 1pt}\n        {\\tt d1T} & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$  & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\n        \\noalign{\\color{white}\\hrule height 1pt}\n        {\\tt d1} & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$  & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\n        \\noalign{\\color{white}\\hrule height 1pt}\n        {\\tt d0c} & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;  & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\n        \\noalign{\\color{white}\\hrule height 1pt}\n        {\\tt d1Tc} & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{myred}$\\times$  & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\n        \\noalign{\\color{white}\\hrule height 1pt}\n        {\\tt d1c} & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{myred}$\\times$  & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\n    \\end{tabular}\n\n\\caption{\\footnotesize Summary of the different components present in the simulated maps $M_\\nu$ in Eq.~\\ref{eq:map}, for every \\emph{simulation type}. A tick on a green background signifies that the component is present in the simulations, red with a cross symbol shows that it is absent.}\n\\label{tab:sims}\n\\end{table}\n\n\\subsection{ \\label{sect:spectra} Angular power spectra of the simulations}\n\n\\subsubsection{\\label{sect:mask}Mask}\n\nA mask is applied on the simulated maps presented in Sect.~\\ref{sec:simulations} in order to exclude the Galactic plane from the power-spectrum estimation. The mask is created by setting a threshold on the polarized intensity ($P=\\sqrt{Q^2+U^2}$) of the \\textit{Planck}{} 353\\,GHz map \\citep{PlanckOverview} \\footnote{\\url{http:\/\/pla.esac.esa.int\/pla\/}}, smoothed with a $10^{\\circ}$ beam. In order to keep $f_{\\rm sky}= 0.7$,$f_{\\rm sky}= 0.6,$ and $f_{\\rm sky}= 0.5$, the cut is applied at 121\\,$\\mu$K, 80\\,$\\mu$K,  and 53\\,$\\mu$K, respectively. We then realize a C2 apodization of the binary mask with a scale of $5^{\\circ}$ using {\\sc Namaster} \\citep{namaster}. The resulting Galactic masks are displayed in Fig.~\\ref{fig:mask}. These masks are similar to those used in \\citet{PlanckDust2}.\n\n\\subsubsection{Estimation of the angular power spectra}\n\\label{sec:spectra_estimation}\n\nWe use the {\\sc Namaster}\\footnote{\\url{https:\/\/github.com\/LSSTDESC\/NaMaster}} software \\citep{namaster} to compute the angular power spectra of each simulation. {\\sc Namaster} allows us to correct for the $E$ to $B$ leakage bias due to the incomplete sky coverage. Therein we use a \\emph{purification} process to suppress the effect of the $E$ to $B$ leakage in the variance. For every simulation, from the set of maps $M_{\\nu_i}$,  we compute all the possible auto-frequency and cross-frequency spectra $\\mathcal{D}_\\ell({\\nu_i\\times\\nu_j})\\equiv\\mathcal{D}_\\ell({M_{\\nu_i}\\times M_{\\nu_j}})$ with\n\n\\begin{align}\n\\nu_i\\times \\nu_j \\in \\left\\{\\right.&100\\times 100, 100 \\times 119, 100 \\times 140,\\   \\dots, 100\\times402,\\nonumber\\\\\n&119\\times140,\\ \\dots, 119\\times402,\\nonumber\\\\[-2mm]\n&\\qquad\\qquad\\vdots\\nonumber\\\\\n&337\\times337, 337\\times402,\\nonumber\\\\\n&\\left.\\!\\!402 \\times 402 \\right\\},\n\\label{eq:all_cross}\n\\end{align}\n\n\\noindent leading to $N_{\\rm cross}=N_{\\rm freq}\\cdot(N_{\\rm freq}+1)\/2=45$ cross-frequency spectra. These spectra are displayed in Fig.~\\ref{fig:simulations} for the case of the {\\tt d1c} simulation\ntype.\\\\\n\nIn order to avoid noise auto-correlation in the auto-spectra ({i.e.}, $\\mathcal{D}_\\ell(\\nu_i\\times\\nu_j)$ when $i=j$), the latter are estimated in a way that differs slightly from what is presented in Sect.~\\ref{sec:Instrsim}. We simulate two noise-independent data subsets at an observing frequency $\\nu_i$, with a noise amplitude $\\sqrt{2}$ higher than that of the frequency band, and compute the cross-angular power spectrum between those. Thus, $\\mathcal{D}_\\ell(\\nu_i\\times\\nu_i)$ is free from noise auto-correlation bias at the expense of multiplying the noise amplitude in the spectrum by a factor of two. This approach is similar to that commonly used by the \\textit{Planck}{} Collaboration \\citep[see e.g.,][]{PlanckDust,PlanckDust2,tristram}.\n\n\n\nThe spectra are evaluated in the multipole interval $\\ell \\in [1,200]$ in order to be able to focus on the reionization and recombination bumps of the primordial $B$-modes spectra. \nThe spectra are binned in $N_{\\ell}=20$ bins of size $\\Delta \\ell = 10$ using {\\sc Namaster}. The same binning is applied throughout this article such that, in the following, the multipole $\\ell$ denotes the multipole bin of size $\\Delta\\ell=10$ centered on $\\ell$\\footnote{ The $N_\\ell$ multipole bins are centered on the following $\\ell$ values: $[6.5,  16.5,  26.5,  36.5,  46.5,  56.5,  66.5,  76.5,  86.5,\n96.5, 106.5,\\\\ 116.5, 126.5, 136.5, 146.5, 156.5, 166.5, 176.5,\n186.5, 196.5]$}.\n\nFrom the sets of $(Q,U)$ maps, {\\sc Namaster} computes the $\\mathcal{D}_\\ell^{EE}$, $\\mathcal{D}_\\ell^{BB}$, and $\\mathcal{D}_\\ell^{EB}$ angular power spectra; for the sake of the present analysis, we keep only $\\mathcal{D}_\\ell^{BB}$. Hence, when we discuss or analyze power spectra, we are referring to the $B$-mode power spectra  $\\mathcal{D}_\\ell^{BB}$. All spectra are expressed in $(\\mu{\\rm K}_{\\rm CMB})^2$.\n\n\\section{\\label{sec:fit}Best-fit implementation}\n\n\nIn order to characterize the complexity of the dust SED that will be measured by \\textit{LiteBIRD}{}, we modeled the angular power spectra of our simulations described in Sect.~\\ref{sec:sims} over the whole frequency and multipole ranges with the moment expansion formalism introduced in Sect.~\\ref{sec:formalism}.\n\n\\subsection{General implementation}\n\\label{sec:fit_dust}\n\nFor each multipole $\\ell$, we ordered the angular power spectra $\\mathcal{D}_\\ell^{BB}(\\nu_i\\times\\nu_j$) as in Eq.~\\ref{eq:all_cross} in order to build a SED that is a function of both $\\nu_i$ and $\\nu_j$. We fit this SED with models, as in Eq.~\\ref{eq:moments} for example, using a Levenberg-Marquardt $\\chi^2$ minimization with {\\tt mpfit} \\citep{mpfit}\\footnote{\\url{https:\/\/github.com\/segasai\/astrolibpy\/tree\/master\/mpfit}}. All the fits performed with {\\tt mpfit} were also realized with more computationally heavy Monte Carlo Markov Chains (MCMC) with {\\tt emcee} \\citep{emcee}, giving compatible results, well within the error bars.\n\nThe reduced $\\chi^2$ minimization is given by\n\n\\begin{equation}\n    \\chi^2 = \\frac{1}{N_{\\rm d.o.f.}}\\vec{R}^T\\mathbb{C}^{-1}\\vec{R},\n\\label{eq:chi2}\n\\end{equation}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale = 0.3]{Figures\/corr2.png}\n    \\caption{\\footnotesize Correlation matrix (${\\rm Corr}_{\\ell\\ell'} \\equiv \\mathbb{C}_{\\ell\\ell'}\/\\sqrt{\\mathbb{C}_{\\ell\\ell}\\mathbb{C}_{\\ell'\\ell'}}$) for the $N_{\\rm sim}$ simulations in {\\tt d1c}. Every block represents a value of $\\ell$ and contains the ordered $N_{\\rm cross} = 45$ cross-spectra. The red squares represent the truncation of the full covariance matrix applied in the analysis (kept entries in red, other entries set to zero).}\n    \\label{fig:cov}\n\\end{figure}\n\n\\noindent where $N_{\\rm d.o.f.}$ is the number of degrees of freedom and $\\mathbb{C}$ is the covariance matrix of our $N_{\\rm sim}$ simulations, represented in Fig.~\\ref{fig:cov}, of dimension $(N_{\\ell}\\cdot N_{\\rm cross})^2$:\n\\begin{equation}\n    \\mathbb{C}_{\\ell,\\ell'}^{i\\times j,k\\times l} = {\\rm cov}\\left(\\mathcal{D}^{\\rm sim}_\\ell (\\nu_i \\times \\nu_j),\\mathcal{D}^{\\rm sim}_{\\ell'} (\\nu_k \\times \\nu_l)\\right).\n\\label{eq:cov}\n\\end{equation}\n\nThe entire covariance matrix $\\mathbb{C}$ is, in general, not invertible. To avoid this, we kept only the $\\ell=\\ell'$ block-diagonal of $\\mathbb{C}$ with the strongest correlation values\\footnote{${\\rm Corr}_{\\ell\\ell'} \\equiv \\mathbb{C}_{\\ell\\ell'}\/\\sqrt{\\mathbb{C}_{\\ell\\ell}\\mathbb{C}_{\\ell'\\ell'}}$}, as well as the $(\\ell=6.5,\\ell'=16.5)$ off-diagonal blocks showing a significant anti-correlation, as illustrated in Fig.~\\ref{fig:cov}. It was then possible to invert the thus-defined truncated correlation matrix  with the required precision most of the time. \n\nIn the case of the {\\tt d1} simulation type, we experienced a fit convergence issue for $\\sim20$\\,\\% of the simulations, leading to a very large $\\chi^2$. In order to overcome this problem, two options lead to identical results: throwing away the outliers from the analysis or fitting using only the block-diagonal matrix (i.e., the $\\ell=6.5$, $\\ell'=16.5$ block is set to zero). This last option solves the conversion issue while providing sufficient precision. The results presented in the following are using the block-diagonal matrix when the simulation type is {\\tt d1}.\n\nFinally, in Eq.~\\ref{eq:chi2}, $\\vec{R}$ is the residual vector associated with every simulation of size $N_{\\ell}\\times N_{\\rm cross}$:\n\n\\begin{equation}\n\\vec{R} =\n\\begin{pmatrix}\n\\mathcal{R}_{\\ell = 6.5}(100 \\times 100) \\\\\n\\mathcal{R}_{\\ell = 6.5}(100 \\times 119) \\\\\n\\vdots \\\\\n\\mathcal{R}_{\\ell = 16.5}(100 \\times 100)\\\\\n\\vdots \\\\\n\\mathcal{R}_{\\ell = 196.5}(402 \\times 402)\\\\\n\\end{pmatrix}, \n\\end{equation}\n\n\\noindent with $\\mathcal{R}_\\ell (\\nu_i \\times \\nu_j) = \\mathcal{D}^{\\rm sim}_\\ell (\\nu_i \\times \\nu_j) - \\mathcal{D}^{\\rm model}_\\ell(\\nu_i \\times \\nu_j)$.\n\nThe expression used for the model to fit is given by:\n\n\\begin{equation}\n\\begin{split}\n  \\mathcal{D}_{\\ell}^{\\rm model}(\\nu_i \\times \\nu_j) &= \\mathcal{D}_{\\ell}^{{\\rm dust}}\\left(\\beta_0(\\ell), T_0(\\ell), \\mathcal{D}^{\\mathcal{M}\\times\\mathcal{N}}_{\\ell}(\\nu_i\\times\\nu_j)\\right) \\\\ & + A_{\\rm lens} \\cdot \\mathcal{D}_{\\ell}^{{\\rm lensing}} + r \\cdot  \\mathcal{D}_{\\ell}^{{\\rm tensor}}, \n\\label{eq:model}\n\\end{split}\n\\end{equation}\n\n\\noindent  where $A_{\\rm lens}$ is not a free parameter and will remain fixed to zero (when there is no CMB, simulation types {\\tt d0}, {\\tt d1T,} and {\\tt d1}) or one (when the CMB is included, simulation types {\\tt d0c}, {\\tt d1Tc} and {\\tt d1c}). We leave the question of the impact of dust modeling with moments on the lensing measurement for future work. In Eq.~\\ref{eq:model}, the free parameters can thus be $\\beta_{0}(\\ell)$, $T_{0}(\\ell)$, and $\\mathcal{D}^{\\mathcal{M}\\times\\mathcal{N}}_{\\ell}(\\nu_i\\times\\nu_j)$ and the tensor-to-scalar ratio $r$. The estimated value of $r$ is referred to as $\\hat{r}$\n\nNo priors on the parameters are used in order to explore the parameter space with minimal assumptions. Finally, a frequency-dependent conversion factor is included in $\\mathcal{D}_{\\ell}^{{\\rm dust}}$ -- from (MJy$\\cdot$sr$^{-1})^2$ to $(\\mu$K$_{\\rm CMB})^2$ -- to express the dust spectra in ($\\mu$K$_{\\rm CMB})^2$ units. In those units, $\\mathcal{D}_{\\ell}^{{\\rm lensing}}$ and $\\mathcal{D}_{\\ell}^{\\rm tensor}$ are frequency-independent.\n\nTo mitigate the impact of outliers in our simulations, all the final values of the best-fit parameters and $\\chi^2$ distributions are represented by their median and median absolute deviations over $N_{\\rm sim}$ values. For the tensor-to-scalar ratio $\\hat{r}$, we chose to represent all the best-fit values from the $N_{\\rm sim}$ simulations in a histogram and we assume its distribution is normal. Fitting a Gaussian curve on this histogram and getting the mean and standard deviation gives us the final values of $\\hat{r}$ and $\\sigma_{\\hat{r}}$ presented in the paper.\n\n\n\n\\subsection{Implementation for the dust component}\n\\label{sec:fit_general}\n\nFor the dust component, we consider four different \\emph{fitting schemes}, corresponding to four expressions for the dust model $\\mathcal{D}^{\\rm dust}_\\ell$ in Eq.~\\ref{eq:model}, which are referred to as \"MBB\", \"$\\beta$-1\", \"$\\beta$-$T$\", and \"$\\beta$-2\". Each of them corresponds to a truncation of Eq.~\\ref{eq:moments}, keeping only some selected terms of the moment expansion: MBB stands for those of the modified black body, $\\beta$-1 for those of the expansion in $\\beta$ at first order, $\\beta$-2 for the expansion in $\\beta$ at second order, and $\\beta$-$T$ for the expansion in both $\\beta$ and $T$ at first order. We chose the $\\beta$-1 and $\\beta$-2 truncations based on the studies of \\citet{Mangilli} and \\citet{Azzoni}, where the dust SED moment expansion is performed only with respect to $\\beta$. The $\\beta$-$T$ fitting scheme is instead the first-order truncation in both $\\beta$ and $T$, introduced here for the first time at the power spectrum level. The parameters fitted in each of these fitting schemes are summarized in Table~\\ref{tab:parameter}. We note that the $\\beta$-2 and $\\beta$-$T$ fitting schemes share the same number of free parameters. Finally, when we fit $\\hat{r}$ at the same time as the dust parameters, the fitting schemes will be referred to as $r$MBB, $r\\beta$-1, $r\\beta$-$T,$ and $r\\beta$-2.\n\nDifferent physical processes are expected to occur at different angular scales, leading to different SED properties. Thus, we estimate the dust-related parameters with one parameter per multipole bin. As an example, we estimate $\\beta_0 = \\beta_0(\\ell)$ and $T_0 = T_0(\\ell)$ to be able to take into account their scale dependence, at the cost of increasing the number of free parameters in our model. This is also true for the higher order moments. On the other hand, $\\hat{r}$ is not scale dependent and, when it is fitted, we add one single parameter over the whole multipole range.\n\nIn \\cite{Mangilli}, the first-order moment expansion parameter $\\mathcal{D}_\\ell^{A\\times\\omega^\\beta_1}$ is considered to be  the leading order correction to the MBB spectral index. We applied a similar approach in the present work, extending it to the dust temperature when it is fitted. In our pipeline, we proceed iteratively:\n\n\\begin{enumerate}\n\\item(i)  we fit $\\beta_0(\\ell)$ and $T_0(\\ell)$ at order zero (MBB), for each $\\ell$, \n\n\\item (ii) we fix $\\beta_0(\\ell)$ and $T_0(\\ell)$ and fit the higher order parameters, as in Eq.~\\ref{eq:moments}, (iii) we update the $\\beta_0(\\ell)$ to $\\beta_{\\rm corr}(\\ell)$ {(and $T_0(\\ell)$ to $T_{\\rm corr}(\\ell)$ in the case of $\\beta$-$T$)} as:\n\n\\begin{equation}\n    \\beta_{\\rm corr}(\\ell) = \\beta_0(\\ell) + \\frac{\\mathcal{D}^{A \\times \\omega^{\\beta}_1 }_{\\ell}}{\\mathcal{D}^{A \\times A}_{\\ell}},\\quad T_{\\rm corr}(\\ell) = T_0(\\ell) + \\frac{\\mathcal{D}^{A \\times \\omega^{T}_1 }_{\\ell}}{\\mathcal{D}^{A \\times A}_{\\ell}},\n\\label{eq:iteration}\n\\end{equation}\n\\item[iv)] and we iterate from (ii) fixing $\\beta_0(\\ell)=\\beta_{\\rm corr}(\\ell)$, until $\\mathcal{D}_\\ell^{A \\times \\omega^{\\beta}_1}$ converges to be compatible with zero (and $T_0(\\ell)=T_{\\rm corr}(\\ell)$, until $\\mathcal{D}_\\ell^{A \\times \\omega^{T}_1}$ converges to zero in the case of $\\beta$-$T$).\n\\end{enumerate}\nWe used three such iterations, which we found to be sufficient to guarantee the convergence. As the moment expansion is a nonorthogonal and incomplete basis \\citep{Chluba}, this iterative process is performed to ensure that the expansions up to different orders share the same $\\beta_0(\\ell)$ and $T_0(\\ell)$ with $\\mathcal{D}^{A \\times \\omega^{\\beta}_1 }_\\ell=0$ and $\\mathcal{D}^{A \\times \\omega^{T}_1 }_\\ell=0$.\n\n\n\\begin{table}[t!]\n\\centering\n        \\normalsize\n        \\begin{tabular}{c!{\\color{white}\\vrule width 1pt}c!{\\color{white}\\vrule width 1pt}c!{\\color{white}\\vrule width 1pt}c!{\\color{white}\\vrule width 1pt}c}\n        & MBB & $\\beta$-1 & $\\beta$-$T$ & $\\beta$-2\\\\\n        \\noalign{\\color{white}\\hrule height 1pt}\n        $N_{\\rm param.}$ & \\cellcolor{mygrey} $3N_\\ell$& \\cellcolor{mygrey} $2N_\\ell$ & \\cellcolor{mygrey} $5N_\\ell$ & \\cellcolor{mygrey} $5N_\\ell$ \\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $\\beta_0(\\ell)$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;& \\cellcolor{myyellow}$\\circ$ & \\cellcolor{myyellow}$\\circ$ & \\cellcolor{myyellow}$\\circ$\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $T_0(\\ell)$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;& \\cellcolor{myred}$\\times$ & \\cellcolor{myyellow}$\\circ$ & \\cellcolor{myred}$\\times$\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{A\\times A}$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;& \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{A\\times\\omega_1^\\beta}$ & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{\\omega_1^\\beta\\times\\omega_1^\\beta}$ & \\cellcolor{myred}$\\times$& \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{A\\times\\omega_1^T}$ & \\cellcolor{myred}$\\times$& \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; &\\cellcolor{myred}$\\times$\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{\\omega_1^T\\times\\omega_1^T}$ & \\cellcolor{myred}$\\times$& \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{myred}$\\times$\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{\\omega_1^\\beta\\times\\omega_1^T}$ & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \\cellcolor{myred}$\\times$\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{A\\times\\omega_2^\\beta}$ & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{\\omega_1^\\beta\\times\\omega_2^\\beta}$ & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\\noalign{\\color{white}\\hrule height 1pt}\n        $D_\\ell^{\\omega_2^\\beta\\times\\omega_2^\\beta}$ & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{myred}$\\times$ & \\cellcolor{mygreen}\\tikz\\fill[scale=0.3](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\\\\\noalign{\\color{white}\\hrule height 1pt}\n    \\end{tabular}\n    \\normalsize\n\\caption{\\footnotesize Summary of the fitted parameters in the four dust moment expansion \\emph{fitting schemes} we consider (MBB, $\\beta$-1, $\\beta$-$T,$ and $\\beta$-2), in Eq.~\\ref{eq:moments}. A tick on a green background signifies that the parameter is fitted, red with a cross symbol shows that the parameter is not fitted, and a circle symbol on yellow means that the parameter is fixed and corrected through an iterative process as presented in Sect.~\\ref{sec:fit_general}. $\\mathcal{D}_\\ell^{A\\times A}$ is fixed to the MBB best-fit value in the case of $\\beta$-1, $\\beta$-$T,$ and $\\beta$-2 and all the other moments are set to zero when they are not fitted. When $\\hat{r}$ is fitted at the same time, the fitting schemes are denoted $r$MBB, $r\\beta$-1, $r\\beta$-$T,$ and $r\\beta$-2, and they have one more parameter than the number of parameters reported in the first line.}\n\\label{tab:parameter}\n\\end{table}\n\n\\section{\\label{sec:results} Results}\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/chi2dustgauss_d0.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/chi2dustgauss_d1T.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/chi2dustgauss_d1.pdf}\n    \\caption{\\footnotesize Median of the reduced $\\chi^2$ in every multipole bin $\\ell$, for all the $N_{\\rm sim}$ simulations of {\\tt d0} (top, orange), {\\tt d1T} (middle, green) and {\\tt d1} (bottom, blue), on $f_{\\rm sky}=0.7$. The reduced $\\chi^2$ values are reported for the four different fitting schemes: MBB (circles), $\\beta$-1 (crosses), $\\beta$-$T$ (diamonds) and $\\beta$-2 (triangles). The values for the four fitting schemes are shifted from each others by $\\ell=2$, in order to distinguish them. The black dashed line represents $\\chi^2_{\\rm red}=1$.}\n    \\label{fig:chi2dust}\n\\end{figure}\n\nIn this section, we present our evaluation of the best-fit parameters for the different {fitting schemes} presented in Sect.~\\ref{sec:fit_dust} on the $B$-mode cross-angular power spectra computed from the different {simulation types} presented in Sect.~\\ref{sec:simulations} and on the Galactic mask keeping $f_{\\rm sky}=0.7$, which is defined in Sect.~\\ref{sect:mask}.\nWe first tested the simulation types containing only dust and noise in order to calibrate the dust complexity of our data sets in Sect.~\\ref{sec:results_dust_only}. We then used CMB only plus noise simulations to assess the minimal error on $\\hat{r}$ in Sect.~\\ref{sec:c} and, finally, we explored the dust, CMB, and noise simulation types to assess the impact of the dust complexity on $\\hat{r}$ in Sect.~\\ref{sec:results_dust_CMB}.\n\n\\subsection{Dust only}\n\\label{sec:results_dust_only}\n\nTo evaluate the amplitude of the dust moment parameters contained in the dust simulations in the absence of CMB, we ran the fitting schemes presented in Sect.~\\ref{sec:fit_dust} in the three simulation types {\\tt d0}, ${\\tt d1T,}$ and ${\\tt d1}$ presented in Sect.~\\ref{sec:simulations}. In these cases, $A_{\\rm lens}$ and $r$ in Eq.~\\ref{eq:model} are both fixed to zero and the fitted parameters are given  in Table~\\ref{tab:parameter} for every fitting scheme.\n\n\\subsubsection{{\\tt d0}}\n\\label{sec:d0}\n\nThe {\\tt d0} dust maps presented in Sect.~\\ref{sec:ingredients} extrapolate between frequency bands with a MBB SED with constant parameters over the sky: $\\beta_{\\tt d0} = 1.54$ and $T_{\\tt d0} = 20$\\,K. We performed the fit with the four fitting schemes presented in Sect.~\\ref{sec:fit_dust}.\n\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/betadustgauss.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/Tgauss.pdf}\n    \\caption{\\footnotesize (Top): Median of the best fit values of $\\beta_0(\\ell)$ in {\\tt d0} (orange), {\\tt d1T} (green), and {\\tt d1} (blue) for the MBB (circles). $\\beta_{\\tt d0}$ is marked by the dashed  black line. (Bottom): Same as above but with $T_0(\\ell)$, the black dashed-lines being $T_{\\tt d0}=20$\\,K and $T_{\\tt d1T}=21.9$\\,K.}\n    \\label{fig:betaTdust}\n\\end{figure}\n\n\nIn Fig.~\\ref{fig:chi2dust} the values of the reduced $\\chi^2(\\ell)$ for each fitting scheme are displayed. For every fitting scheme (MBB, $\\beta$-1, $\\beta$-$T$ and $\\beta$-2), the reduced $\\chi^2$ are close to 1 over the whole multipole range (slightly below 1 for the $\\beta$-1, $\\beta$-$T$ and $\\beta$-2 fitting scheme). This indicates that the MBB is a good fit to the cross-angular power spectra computed from the  {\\tt d0} maps with a spatially invariant MBB SED, as expected. Adding additional (higher order) parameters, such as with $\\beta$-1, $\\beta$-$T$ and $\\beta$-2, has no significant effect on the $\\chi^2$.\n\nIn Fig.~\\ref{fig:betaTdust} we can see that the best-fit values of $\\beta_0(\\ell)$ and $T_0(\\ell)$ are compatible with constant values $\\beta_0(\\ell)=\\beta_{\\tt d0}$ and $T_0(\\ell)=T_{\\tt d0}$, as expected for this simulated data set.\n\nThe best-fit values of the dust amplitude and the moment-expansion parameters are presented in Figs.~\\ref{fig:Adust}, \\ref{fig:moments1}, \\ref{fig:moments2}, and \\ref{fig:moments3}, respectively. The amplitude power spectrum is compatible with that of the dust template map used to build {\\tt d0} and the moment-xpansion parameters are compatible with zero for every fitting scheme, as expected with no spatial variation of the SED. \nTherefore, the moment expansion method presented in Sect.~\\ref{sec:formalism} passes the \\emph{null test} in the absence of SED distortions, with the {\\tt d0} simulated data set.  \n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/Adustgauss.pdf}\n    \\caption{\\footnotesize Median of the best-fit values of $\\mathcal{D}_\\ell^{A \\times A}$ for {\\tt d0} (orange), {\\tt d1T} (green), and {\\tt d1} (blue) using the MBB fitting scheme. The values for the three simulation types are shifted with respect to one another by $\\ell=2$ in order to distinguish them. The black dashed line is the amplitude power spectrum of the dust template map used to build the three simulation sets {\\tt d0}, {\\tt d1T,} and  {\\tt d1}.}\n    \\label{fig:Adust}\n\\end{figure}\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/w1w1dustgauss.pdf}\n    \\caption{\\footnotesize Best-fit values of the first-order moment $\\mathcal{D}_\\ell^{\\omega^{\\beta}_1\\times\\omega^{\\beta}_1}$ for {\\tt d0} (orange), {\\tt d1T} (green), and {\\tt d1} (blue), fitting with $\\beta$-1 (crosses), $\\beta$-2 (triangles), and $\\beta$-$T$ (diamonds).}\n    \\label{fig:moments1}\n\\end{figure}\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/Aw2dustgauss.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/w1w2dustgauss.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/w2w2dustgauss.pdf}\n    \\caption{\\footnotesize Best-fit values of the second-order $\\mathcal{D}_\\ell^{A\\times\\omega^{\\beta}_2}$, $\\mathcal{D}_\\ell^{\\omega^{\\beta}_1\\times\\omega^{\\beta}_2}$ and $\\mathcal{D}_\\ell^{\\omega^{\\beta}_2\\times\\omega^{\\beta}_2}$ moment parameters in {\\tt d0} (orange), {\\tt d1T} (green), and {\\tt d1} (blue) for $\\beta$-2 (triangles). }\n    \\label{fig:moments2}\n\\end{figure}\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/b1t1dustgauss.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/t1t1dustgauss.pdf}\n    \\caption{\\footnotesize Best-fit values of the first-order  $\\mathcal{D}_\\ell^{\\omega^{\\beta}_1\\times\\omega^{T}_1}$ and $\\mathcal{D}_\\ell^{\\omega^{T}_1\\times\\omega^{T}_1}$ moment parameters in {\\tt d0} (orange), {\\tt d1T} (green), and {\\tt d1} (blue) for $\\beta$-$T$ (diamonds). }\n    \\label{fig:moments3}\n\\end{figure}\n\n\n\\subsubsection{{\\tt d1T}}\n\\label{sec:d1T}\n\n\nWe now introduce, as a first layer of complexity, the spatial variations of the spectral index associated to a fixed temperature over the sky with the {\\tt d1T} simulation type. The dust temperature was fixed to $T_{\\rm d1T}=21.9$\\,K while the spectral index $\\beta(\\vec{n})$ was allowed to vary between lines of sight. The four different fitting schemes presented in Sect.~\\ref{sec:fit_dust} are fitted over the cross-spectra of our simulations as in Sect.~\\ref{sec:d0}.\n\nThe reduced $\\chi^2(\\ell)$ values for each fitting scheme can be found in Fig.~\\ref{fig:chi2dust}. It can be seen that the MBB no longer provides a good fit for the dust SED, especially at low multipoles. Averaging effects of spatially varying SEDs are more important over large angular scales and thus SED distortions and moments are expected to be more significant at low multipoles. Indeed, the moments added to the fit in $\\beta$-1 are enough to lower the reduced $\\chi^2$ such that it becomes compatible with 1 over almost all of the multipole range. The fitting schemes $\\beta$-$T$ and $\\beta$-2, including more parameters than $\\beta$-1, provide a fit of similar goodness, except in the multipole bin $\\ell=66.5$ where they are closer to 1.\n\nFigure~\\ref{fig:betaTdust} presents the best-fit values of $\\beta_0(\\ell)$ in the case of the MBB fit. For the sake of clarity, the values after iteration (see Sect.~\\ref{sec:fit_dust}) for $\\beta$-1, $\\beta$-$T,$ and $\\beta$-2 are not shown, but they present comparable trends. We can see that the best-fit values of $\\beta_0(\\ell)$ for this {\\tt d1T} simulation type are no longer compatible with a constant. $\\beta_0(\\ell)$ fitted values show a significant increase at low (<100) multipoles, up to $\\beta_0(\\ell=16.5)=1.65$. For $\\ell>100$, $\\beta_0(\\ell)$ is close to a constant of value $\\sim1.53$. This increase towards the low $\\ell$ is correlated to the increase of the MBB $\\chi^2$ discussed in the previous paragraph. However, we note that in the lowest $\\ell$-bin, the $\\beta_0(\\ell)$ value is close to 1.53 and that the $\\chi^2$ of the MBB fit is close to unity.\n\n\nThe best-fit values of $T_0(\\ell)$ are also presented in Fig.~\\ref{fig:betaTdust} in the case of the MBB fit. Here again, the values after iteration for the other fitting schemes are not presented, but are similar. The {\\tt d1T} $T_0(\\ell)$ best-fit values oscillate around $T_{\\tt d1T}=21.9$\\,K, without being strictly compatible with a constant value, as would be expected for this simulation type. This tends to indicate that the SED distortions due to the spectral index spatial variations are affecting the accuracy at which we can recover the correct angular dependence of the sky temperature. \n\nThe amplitude power spectrum is displayed in Fig.~\\ref{fig:Adust} for the MBB fitting scheme. The other fitting scheme results are not presented for clarity and would not be distinguishable from those of the MBB. The fitted $\\mathcal{D}_\\ell^{A\\times A}$ is compatible with the one of the dust template map used to build the simulations. \n\nAll the parameters of the moment expansion with respect to $\\beta$ can be found in Figs.~\\ref{fig:moments1} and \\ref{fig:moments2}, and are now significantly detected, except for $\\mathcal{D}_\\ell^{\\omega^\\beta_2 \\times \\omega^\\beta_2}$. In Fig.~\\ref{fig:moments3}, we can observe that the parameters of the moment expansion with respect to the temperature (only present in the $\\beta$-$T$ fit) remain undetected. The SED distortions due to the spatial variations of $\\beta$ are well detected, while no SED distortion linked to the temperature is seen, as expected for the ${\\tt d1T}$ simulation type.\n\n\n\\subsubsection{{\\tt d1}}\n\\label{sec:d1}\n\nWe now discuss the {\\tt d1} simulations, with the highest complexity in the polarized dust SED. In this more physically relevant simulation type, the dust emission is given by a MBB with variable index $\\beta(\\vec{n})$ and temperature $T(\\vec{n})$ over the sky.\nWe ran the four different fitting schemes on the {\\tt d1} simulation type, as we did in Sect.~\\ref{sec:d0} and \\ref{sec:d1T}.\n\nThe values of the reduced $\\chi^2(\\ell)$ are displayed in Fig.~\\ref{fig:chi2dust}. For the MBB and $\\beta$-1, the reduced $\\chi^2$ are not compatible with unity, especially at low multipole. This indicates that none of them are a good fit anymore for the spatially varying SED with $\\beta(\\vec{n})$ and $T(\\vec{n})$. With $\\beta$-2 and $\\beta$-$T$, the $\\chi^2(\\ell)$ values become compatible with unity, except for the $\\ell=26.5$ bin. We note that $\\beta$-$T$ provides a slightly better fit than $\\beta$-2 in this bin.\n\nLooking at the medians of the best-fit values of $\\beta_0(\\ell)$ for {\\tt d1} in Fig.~\\ref{fig:betaTdust}, we can see that the spectral index is changing with respect to $\\ell$, as discussed in Sect.~\\ref{sec:d1T}, in a similar manner as for the {\\tt d1T} simulation type. The fitted temperature $T_0(\\ell)$ values for {\\tt d1} show an increasing trend from $\\sim17$ to $\\sim20.5$\\,K and from $\\ell=16.5$ to $\\ell\\sim100$. At higher multipoles, $T_0(\\ell)$ is close to a constant temperature of 20.5\\,K. In {\\tt d1}, as for {\\tt d1T}, the angular scales at which we observe strong variations of $\\beta_0(\\ell)$ and $T_0(\\ell)$ are the ones for which we observe a poor $\\chi^2$ for some fitting schemes. Also, as for {\\tt d1T}, the largest angular scale $\\ell$-bin, at $\\ell=6.5,$ shows $\\beta$ and $T$ values close to the constant value at high $\\ell$, which are associated with $\\chi^2$ values closer to unity.\nThe best-fit values of the amplitude $\\mathcal{D}_\\ell^{A\\times A}$ are shown in Fig.~\\ref{fig:Adust}. These are similar to those of the other simulation types. \n\nThe moment-expansion parameters fitted on {\\tt d1} are shown in Figs.~\\ref{fig:moments1}, \\ref{fig:moments2}, and \\ref{fig:moments3}. For this simulation type, the moment parameters are all significantly detected with respect to both $\\beta$ and $T$. This was already the case with the \\textit{Planck}{} intensity simulations, produced in a similar way, as discussed in \\cite{Mangilli}. Their detections quantify the complexity of dust emission and SED distortions from the MBB present in the {\\tt d1} simulation type, due to the spatial variations of $\\beta(\\vec{n})$ and $T(\\vec{n})$. \n\n\n\\subsection{\\label{sec:nodust} CMB only}\n\\label{sec:c}\n\nIn order to calibrate the accuracy at which the $r$ parameter can be constrained with the \\textit{LiteBIRD}{} simulated data sets presented in Sect.~\\ref{sec:simulations}, we tested the simulation type with no dust component, $M_\\nu^{\\tt c}$, and with no tensor modes ($r_{\\rm sim}=0$, only CMB lensing and noise). We fit the expression in Eq.~\\ref{eq:model} with $\\mathcal{D}_\\ell^{\\rm dust}$ fixed to zero and $A_{\\rm lens}$ fixed to one ({i.e.}, $r$ is the only parameter we fit in this case).\nDoing so over the $N_{\\rm sim}$ simulations, we obtain $\\hat{r}= (0.7 \\pm 3.5) \\times 10^{-4}$.\nThis sets the minimal value we can expect to retrieve for $\\hat{r}$ with our assumptions if the dust component is perfectly taken into account.\n\n\n\\subsection{Dust and CMB}\n\\label{sec:results_dust_CMB}\n\nWe now present our analysis of the simulations including dust, CMB (lensing), and noise ({\\tt d0c}, {\\tt d1Tc} and {\\tt d1c}) with no primordial tensor modes ($r_{\\rm sim}=0$). As described above, we applied the four fitting schemes for the dust on the three simulation types, fitting $\\hat{r}$ and fixing $A_{\\rm lens}$ to~one    (namely $r$MBB, $r\\beta$-1, $r\\beta$-$T$ and $r\\beta$-2) simultaneously. \n\nThe best-fit values of $\\beta_0(\\ell)$, $T_0(\\ell)$ and the moment expansion parameters $\\mathcal{D}_\\ell^{\\mathcal{M}\\times\\mathcal{N}}$ derived with the simulation types {\\tt d0c}, {\\tt d1Tc,} and {\\tt d1c} are not discussed further when they are compatible with the ones obtained for the {\\tt d0}, {\\tt d1T,} and {\\tt d1} simulation types and presented in Sect.~\\ref{sec:results_dust_only}.\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/gauss_r_0_d0_SLD_full.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/gauss_r_0_d1T_SLD_full.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/gauss_r_0_d1_SLD_full.pdf}\n    \\caption{\\footnotesize \\emph{(Top panel)}: Posterior on $\\hat{r}$ in the {\\tt d0c} simulation type for the different fitting schemes: $r$MBB (blue, dotted line), $r\\beta$-1 (red, dashed line), $r\\beta$-$T$ (green, solid line), and $r\\beta$-2 (yellow, dash-dotted line). The vertical black dashed line marks the value of $r_{\\rm sim}=0$. \\emph{(Central panel)}: Same, but in the case of the {\\tt d1Tc} simulation type. \\emph{(Bottom panel)}: Same, but in the case of the {\\tt d1c} simulation type.}\n    \\label{fig:rgauss1}\n\\end{figure}\n\n\n\\subsubsection{{\\label{sec:d0c}}{\\tt d0c}}\n\nFor {\\tt d0c}, as for {\\tt d0}, we recover the input constant spectral index and temperature $\\beta_{\\tt d0}$ and $T_{\\tt d0}$ at all angular scales for every fitting scheme. Furthermore, we do not detect any moment, when fitting $r\\beta$-1, $r\\beta$-$T,$ and $r\\beta$-2. This simulation type therefore constitutes our {null-test} when $\\hat{r}$ and the dust parameters are fitted at the same time. The addition of the CMB lensing in the simulations and the addition of $r$ to the fits thus does not lead to the detection of the moment parameters nor biases the recovery of the spectral index and the temperature.\n\nThe posterior distributions of the estimated tensor-to-scalar ratio $\\hat{r}$ are displayed in Fig.~\\ref{fig:rgauss1} and their mean and standard deviations are summarized in Table~\\ref{tab:rvalues}. We note that $\\hat{r}$ is compatible with the input value ($r_{\\rm sim}=0)$ for all the fitting schemes. For $r$MBB and $r\\beta$-1, the dispersion $\\sigma_{\\hat{r}}$ is comparable with the CMB-only scenario discussed in Sect.~\\ref{sec:nodust}. For $r\\beta$-$T$ and $r\\beta$-2, the width of the distribution increases by a factor of $\\sim 2$ and $\\sim 4$,  respectively.\n\n\\subsubsection{{\\tt d1Tc}}\n\\label{sec:d1Tc}\n\nThe posterior distribution of $\\hat{r}$ in the case of the {\\tt d1Tc} simulation type is displayed in Fig.~\\ref{fig:rgauss1} for the four fitting schemes and the mean value and standard deviation of these distributions are summarized in Table~\\ref{tab:rvalues}. We can see that in the case of $r$MBB, we fit $\\hat{r}\\pm\\sigma_{\\hat{r}}=(99.7\\pm6.2)\\times10^{-4}$. In that case, the input tensor-to-scalar ratio $r_{\\rm sim}=0$ is not recovered and we obtain a bias on the central value of $\\hat{r}$ of $\\sim 16\\,\\sigma_{\\hat{r}}$. As discussed in Sect.~\\ref{sect:limitsmbb}, this is expected because we know that the MBB is not a good dust model for a SED with spatially varying spectral index, as we also verify in Sect.~\\ref{sec:d1T} looking at the $\\chi^2$ values.\n\nUsing the $r\\beta$-1 fitting scheme allows us to recover $\\hat{r}=(-8.0\\pm6.4)\\times10^{-4}$, where $r_{\\rm sim}$ is recovered within $\\sim\\,2\\sigma_{\\hat{r}}$, while $r\\beta$-2 and $r\\beta$-$T$ recover the input value within $1\\,\\sigma_{\\hat{r}}$ (with $\\hat{r}=(-3.6\\pm 13.0)\\times10^{-4}$ and $\\hat{r}=(0.7\\pm20.9)\\times10^{-4}$, respectively). As in Sect~\\ref{sec:d0c}, the deviation remains similar between $r$MBB and $r\\beta$-1 and increases by a factor of $\\sim 2$ and $\\sim 4$ from $r\\beta$-1 to $r\\beta$-$T$ and $r\\beta$-2,  respectively.\n\n\\begin{table}[t!]\n\\centering\\scalebox{1}{\n    \\centering\n    \\begin{tabular}{c | ccc}\n        ($\\hat{r} \\pm \\sigma_{\\hat{r}})\\times10^{4}$ & {\\tt d0c} & {\\tt d1Tc} & {\\tt d1c}\\\\\n        \\hline\n        $r$MBB     &  \\cellcolor{mygreen} $0.3 \\pm 3.9$ & \\cellcolor{myred}$99.7 \\pm 6.2$  & \\cellcolor{myred} $125.1 \\pm 5.9$\\\\\n        $r\\beta$-1   & \\cellcolor{mygreen} $0.5 \\pm 4.5$ & \\cellcolor{myyellow} $-8.0 \\pm 6.4$& \\cellcolor{myred} $32.9 \\pm 6.5$\\\\\n        $r\\beta$-$T$   & \\cellcolor{mygreen} $0.3 \\pm 9.5$ & \\cellcolor{mygreen} $-3.6 \\pm 13.0$  & \\cellcolor{mygreen} $-3.3 \\pm 11.7$\\\\\n        $r\\beta$-2   & \\cellcolor{mygreen} $0.7 \\pm 16.4$ & \\cellcolor{mygreen} $0.7 \\pm 20.9$ & \\cellcolor{myyellow} $-37.4 \\pm 19.4$\\\\ \n        \\hline \n    \\end{tabular}}\n\\caption{\\footnotesize Best-fit values of $\\hat{r}$ in units of $10^{-4}$ on $f_{\\rm sky}=0.7$. The green values are compatible with $r_{\\rm sim}=0$ at $1\\,\\sigma_{\\hat{r}}$, the yellow values are compatible with $r_{\\rm sim}=0$ at $2\\,\\sigma_{\\hat{r}}$ and the red values are incompatible with $r_{\\rm sim}=0$ at more than $2\\,\\sigma_{\\hat{r}}$.}\n\\label{tab:rvalues}\n\\end{table}\n\n\\subsubsection{{\\tt d1c}}\n\\label{sec:d1c}\n\nIn the case of the {\\tt d1c} simulation type, as in {\\tt d0c} and {\\tt d1Tc}, we fit $\\hat{r}$ in addition to the dust-related parameters. In that case, dust moment parameters are recovered as for {\\tt d1} (see Sect.~\\ref{sec:d1}), except for the $r\\beta$-2 fitting scheme.\n\nFigure~\\ref{fig:momentsd1d1c} compares the moment parameters between $\\beta$-2 on the {\\tt d1c} simulations type, fitting only the dust-related parameters and $r\\beta$-2 on {\\tt d1c} when jointly fitting the dust parameters and $\\hat{r}$. We observe that $\\mathcal{D}_\\ell^{\\omega^{\\beta}_2 \\times \\omega^{\\beta}_2}$ is not consistently recovered when fitting $\\hat{r}$ in addition to the dust parameters.\n\nA similar comparison can be found in Fig.~\\ref{fig:moments2d1d1c} for the moment parameters between $\\beta$-$T$ and $r\\beta$-$T$ on the {\\tt d1c} simulation type. Using this fitting scheme, we can see that all the moments are correctly recovered when adding $\\hat{r}$ to the fit.\n\n\nThe $\\hat{r}$ posterior distributions in the case of {\\tt d1c} are displayed in  Fig.~\\ref{fig:rgauss1} and summarized in Table~\\ref{tab:rvalues}. As discussed in Sect.~\\ref{sect:limitsmbb} and observed in Sect.~\\ref{sec:d1c}, the $r$MBB fit is highly biased, with $\\hat{r}=(125.1 \\pm 5.9)\\times10^{-4}$ (by more than 21\\,$\\sigma_{\\hat{r}}$). When fitting the $r\\beta$-1, this bias is significantly reduced ($\\hat{r}=(32.9 \\pm 6.5)\\times10^{-4}$, $~5\\,\\sigma_{\\hat{r}}$ away from $r_{\\rm sim}=0$), illustrating the ability of the first-moment parameters to correctly capture part of the SED complexity. However, performing the expansion in both $\\beta$ and $T$ with $r\\beta$-$T$ allows us to recover $r_{\\rm sim}$ without bias ($\\hat{r}=(-3.3\\pm 11.7)\\times10^{-4}$), highlighting the need for the description of the SED complexity in terms of dust temperature for this simulated data set where both $\\beta$ and $T$ vary spatially. On the other hand, for $r\\beta$-2, a negative tension ($1.9\\,\\sigma_{\\hat{r}}$) can be observed: $\\hat{r}=(-37.4\\pm 19.4)\\times10^{-4}$. This tension is discussed in Sect.~\\ref{sec:bias}.  \n\nFor {\\tt d1c}, the $\\hat{r}$ distribution widths roughly meet the foreground cleaning requirements of \\textit{LiteBIRD}{} presented in Sect.~\\ref{sec:LiteBIRD} for $r$MBB and $r\\beta$-1 but are higher for $r\\beta$-$T$ and $r\\beta$-2. We also note that, with the same number of free parameters, all the standard deviations $\\sigma_{\\hat{r}}$ slightly increase compared to the {\\tt d0c} simulation type. This is expected due to the increasing dust complexity. \n\n\\section{\\label{sec:discussion} Discussion}\n\n\n\\subsection{Lessons learnt}\n\nIn Sect.~\\ref{sec:results}, we apply the fitting pipeline introduced in Sect.~\\ref{sec:fit} on \\textit{LiteBIRD}{} simulated data sets on $f_{\\rm sky}=0.7$ and for $r_{\\rm sim}=0$, including the various dust simulation types defined in Sect.~\\ref{sec:ingredients}. We fitted the estimated $B$-mode power-spectra with the four different fitting schemes summarized in Table~\\ref{tab:parameter}. Our main results can be summarized as follows: \n\n\\begin{itemize}\n\\item The MBB fitting scheme provides a good fit for the dust component in the {\\tt d0} and {\\tt d0c} simulation types. \n    However, when the spectral index changes with the angular scale, such as in the {\\tt d1T}, {d1Tc}, {\\tt d1,} and {\\tt d1c} simulations, this approach no longer provides a good fit because of the complexity of the dust SED. As a consequence, in the $r$MBB case, $r_{\\rm sim}$ cannot be recovered without a significant bias. \n\n\\item The $\\beta$-1 fitting scheme  allows us to perform a good fit for the dust complexity using the {\\tt d0} and {\\tt d1T} simulations but not for {\\tt d1}, while the $r\\beta$-1 fitting  scheme yields estimates of $\\hat{r}$ close to $r_{\\rm sim}$ within $1\\,\\sigma_{\\hat{r}}$ for {\\tt d0c}, and within $2\\,\\sigma_{\\hat{r}}$ for {\\tt d1Tc}, but presenting a bias of $\\sim 6\\,\\sigma_{\\hat{r}}$ for {\\tt d1c}. \n\n\\item The $\\beta$-$T$ fitting scheme provides a good fit for every dust model, while using the $r\\beta$-$T$ fitting  scheme allows us to recover $\\hat{r}$  values consistent with $r_{\\rm sim}$ within $1\\,\\sigma_{\\hat{r}}$ for all the simulation types, but is associated with an increase of $\\sigma_{\\hat{r}}$ by a factor $\\sim 2$ compared to the $r\\beta$-1 case.\n\n\\item The $\\beta$-2 fitting scheme also provides a good fit for each dust model, and the $r\\beta$-2 fitting scheme leads to values of $\\hat{r}$ compatible with $r_{\\rm sim}$ within $1\\,\\sigma_{\\hat{r}}$ for all the simulation types but {\\tt d1c}. In this last case, there is a negative tension of $\\sim2\\,\\sigma_{\\hat{r}}$.  \n    For all the simulation types, there is an increase of $\\sigma_{\\hat{r}}$ by a factor of $\\sim 4$ compared to the $r\\beta$-1 case.\n\n\\end{itemize}\n\nThe present analysis shows that the temperature could be a critical parameter for the moment expansion in the context of \\textit{LiteBIRD}{}.\n\nIndeed, for simulations including a dust component with a spectral index and a temperature that both vary spatially, as in {\\tt d1}, the only fitting scheme allowing us to recover $r_{\\rm sim}$ within $1\\,\\sigma_{\\hat{r}}$ is $r\\beta$-$T$, the expansion to first order in both $\\beta$ and $T$. This shows that expanding in $\\beta$ only, without treating $T$, is not satisfactory when looking at such large fractions of the sky. Indeed, when applying the $\\beta$-2 fitting scheme, the $\\mathcal{D}_\\ell^{\\omega^{\\beta}_2 \\times \\omega^{\\beta}_2}$ parameter remains undetected for the {\\tt d1T} simulation type (Sect~\\ref{sec:d1T}), while it is significantly detected using the {\\tt d1} simulation type (Sect~\\ref{sec:d1}). Nevertheless, {\\tt d1T} and {\\tt d1}share the same template of $\\beta(\\vec{n})$ (Sect.~\\ref{sec:ingredients}) and they only differ by the sky temperature (constant for {\\tt d1T} and varying for {\\tt d1}). This suggests that the observed $\\mathcal{D}_\\ell^{\\omega^{\\beta}_2 \\times \\omega^{\\beta}_2}$ with the {\\tt d1} simulations originates from the temperature variations and not those in the spectral index. This observation shows that it is less convenient to use the $\\beta$-2 fitting scheme than the $\\beta$-$T$ one in order to correctly recover the moment-expansion parameters and $\\hat{r}$ when temperature varies spatially.\n\nMoreover, we saw that $ \\sigma_{\\hat{r}}$ is lower when using the fitting scheme $r\\beta$-$T$ instead of $r\\beta$-2 for every simulation type, even if both have the same number of free parameters. This second observation additionally encourages an approach where the SED is expanded with respect to both $\\beta$ and $T$. Nevertheless, the uncertainty on $\\hat{r}$ we obtain in this case ($\\sigma_{\\hat{r}}=1.17\\times10^{-3}$) is larger than the \\textit{LiteBIRD}{} requirements. \n\n\\subsection{Increasing the accuracy on the tensor-to-scalar ratio}\n\\label{sec:Opt}\n\nIn Sect.~\\ref{sec:d1} and Fig.~\\ref{fig:chi2dust}, we see that the MBB and $\\beta$-1 fitting schemes do not provide good fits for the {\\tt d1} dust simulations, especially at low multipoles ($\\ell \\lesssim 100$). Conjointly, in Fig.~\\ref{fig:moments3}, we can see that the $\\beta$-$T$ moment parameters are significantly detected for $\\ell \\lesssim 100$ and compatible with zero above that threshold, suggesting that their corrections to the SED are predominantly required at large angular scales.\n\nThis implies that we can improve the pipeline presented in Sect.~\\ref{sec:fit} to keep only the required parameters in order to recover $\\hat{r}$ compatible with $r_{\\rm sim}$ with a minimal $\\sigma_{\\hat{r}}$. It can be achieved by applying the $r\\beta$-1 fitting sheme over the whole multipole range, while restricting the $r\\beta$-$T$-specific ($\\mathcal{D}_\\ell^{\\omega^\\beta_1\\times\\omega^T_1}$ and $\\mathcal{D}_\\ell^{\\omega^T_1\\times\\omega^T_1}$) moment-expansion parameters fit to the low multipoles range. We note that in order to correct the bias, it is still necessary to keep the $r\\beta$-1 moment parameters even at high multipoles, because the MBB does not provide a good fit even for $\\ell \\in [100,200]$, as we can see in Fig.~\\ref{fig:chi2dust}. We define $\\ell_{\\rm cut}$ as the multipole bin under which we keep all the $r\\beta$-$T$ moment parameters and above which we use the $r\\beta$-1 scheme. \n\n\nThe best-fit values and standard deviations of $\\hat{r}$ for different values of $\\ell_{\\rm cut}$ are displayed in Table~\\ref{tab:rellcut}. We can see that a trade-off has to be found: the smaller the $\\ell_{\\rm cut}$ , the bigger the shift from $r_{\\rm sim}$, and the bigger the $\\ell_{\\rm cut}$, the higher the value of $\\sigma_{\\hat{r}}$. The trade-off point seems to be found for $\\ell_{\\rm cut} \\sim 80$, allowing us to recover $\\hat{r}$ without tension, with $\\sigma_{\\hat{r}} =8.8\\times10^{-4}$. The error on $r$ is thus reduced by more than $\\sim30\\,\\%$ with respect to the nonoptimized fit and meets the \\textit{LiteBIRD}{} requirements.\n \n\n\n\n\n\\subsection{\\label{sec:fsky} Tests with smaller sky fractions}\n\n\\begin{table}[t!]\n\\centering\\scalebox{1}{\n    \\centering\n    \\begin{tabular}{c|c}\n        $\\ell_{\\rm cut}$ & $(\\hat{r} \\pm \\sigma_{\\hat{r}})\\times 10^{4}$\\\\\\hline\n        50 &  \\cellcolor{myred}$12.0 \\pm 7.3$ \\\\\n        60 &  \\cellcolor{mygreen}$7.3 \\pm 7.9$ \\\\\n        70 &  \\cellcolor{mygreen}$4.9 \\pm 8.1$ \\\\\n        80 & \\cellcolor{mygreen}$-0.9 \\pm 8.8$ \\\\\n        90 &  \\cellcolor{mygreen}$-2.1 \\pm 9.9$ \\\\\n    \\end{tabular}}\n\\caption{\\footnotesize Best-fit values of $\\hat{r} \\pm \\sigma_{\\hat{r}}$ in units of $10^{-4}$ for different values of $\\ell_{\\rm cut}$ for the {\\tt d1c} simulations with $f_{\\rm sky}=0.7$, when applying the $r\\beta$-$T$ fitting scheme. The green values are compatible with $r_{\\rm sim}=0$ at $1\\,\\sigma_{\\hat{r}}$.}\n\\label{tab:rellcut}\n\\end{table}\n\n\\begin{table}[t!]\n\\centering\\scalebox{1}{\n    \\centering\n    \\begin{tabular}{c | ccc}\n        $(\\hat{r} \\pm \\sigma_{\\hat{r}})\\times10^4$ & $r_{\\rm sim}=0.01$ & $f_{\\rm sky}=0.5$ & $f_{\\rm sky}=0.6$\\\\\n        \\hline\n        $r$MBB     & \\cellcolor{myred} $204.8 \\pm 7.7$ & \\cellcolor{myred} $47.3 \\pm 5.6$ & \\cellcolor{myred}  $59.2 \\pm 5.4$\\\\\n        $r\\beta$-1  & \\cellcolor{myred} $129.0 \\pm 8.3$ & \\cellcolor{myyellow} $-8.4 \\pm 6.7$ & \\cellcolor{mygreen} $1.8 \\pm 6.2$\\\\\n        $r\\beta$-$T$ & \\cellcolor{mygreen} $94.6 \\pm 15.1$ &\\cellcolor{mygreen} $0.02 \\pm 13.4$ & \\cellcolor{mygreen} $-1.1 \\pm 12.0$\\\\\n        $r\\beta$-2   & \\cellcolor{myyellow} $62.5 \\pm 25.0$ & \\cellcolor{mygreen} $4.3 \\pm 24.2$ & \\cellcolor{mygreen} $-3.2 \\pm 22.4$\\\\\n         \\hline\n    \\end{tabular}}\n\\caption{\\footnotesize Best-fit values of $\\hat{r}$ in units of $10^{-4}$ for an alternative {\\tt d1c} simulation with $r_{\\rm sim}=0.01$ on $f_{\\rm sky}=0.7$, and with $r_{\\rm sim}=0$ but on $f_{\\rm sky}=0.5$ and $f_{\\rm sky}=0.6$. The green values are compatible with $r_{\\rm sim}$ at $1\\,\\sigma_{\\hat{r}}$, the yellow values are compatible with $r_{\\rm sim}$ at $2\\,\\sigma_{\\hat{r}}$ , and the red values are incompatible with $r_{\\rm sim}$ at more than $2\\,\\sigma_{\\hat{r}}$.}\n\\label{tab:rvalues2}\n\\end{table}\n\nIn all the results presented in Sect.~\\ref{sec:results}, we were considering a sky fraction of $f_{\\rm sky}=0.7$. This sky mask keeps a considerable fraction of the brightest Galactic dust emission. To quantify the impact of the sky fraction on our analysis, we ran the pipeline as in Sect.~\\ref{sec:d1c} with the different masks introduced in Sect.~\\ref{sect:mask} ($f_{\\rm sky}=0.5$ and $f_{\\rm sky}=0.6$). This was done with the {\\tt d1c} simulation type. \n\nThe posteriors on $\\hat{r}$ for the different fitting schemes are displayed in Fig.~\\ref{fig:rgauss2} and Table~\\ref{tab:rvalues2}. We can see that, while the $r$MBB fiting scheme always leads to biased estimates, the $r\\beta$-1 case allows us to recover $\\hat{r}$ at $1.25\\,\\sigma_{\\hat{r}}$ for $f_{\\rm sky}=0.5$ and within $1\\,\\sigma_{\\hat{r}}$ for $f_{\\rm sky}=0.6$. In the two situations, the results using the $r\\beta$-$T$ and $\\beta$-2 fitting schemes are both unbiased with estimates of $\\hat{r}$ compatible with $r_{\\rm sim}$ within $1\\,\\sigma_{\\hat{r}}$. The $\\sigma_{\\hat{r}}$ hierarchy between the $r$MBB, $r\\beta$-1, $r\\beta$-$T,$ and $r\\beta$-2 fitting schemes is the same as for $f_{\\rm sky}=0.7$ (see Sect.~\\ref{sec:d1c}). Nevertheless, we observe that $\\sigma_{\\hat{r}}$ increases as the sky fraction decreases, as does the statistical error (cosmic variance of the lensing and noise). The bias, on the other hand, decreases for all the fitting schemes with the sky fraction, which is expected because less dust emission contributes to the angular power spectra. The negative tension observed on the $\\hat{r}$ posterior in Sect.~\\ref{sec:d1c} for the $r\\beta$-2 case is not present when using smaller sky fractions. In Fig.~\\ref{fig:fskydust}, the $r\\beta$-2 moment parameters are displayed. We can see that they are not significantly detected for the $f_{\\rm sky}=0.5$ and 0.6, unlike for $f_{\\rm sky}=0.7$. As we have seen that some of the moments in the $\\beta$-2 fitting scheme failed to model SED distortions coming from temperature, we can suppose that, in our simulations, the temperature variations play a less significant role in the dust SED on the $f_{\\rm sky}=0.5$ and 0.6 masks than they play in the $f_{\\rm sky}=0.7$ one. As a consequence, they have a smaller impact on $r$ when not properly taken into account.\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/gauss_r_0_d1_SLD_full_fsky0.5.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/gauss_r_0_d1_SLD_full_fsky0.6.pdf}\n    \\caption{\\footnotesize \\emph{(Top panel)}: Posterior on $\\hat{r}$ in the {\\tt d1c} simulation type on $f_{\\rm sky}=0.5$ for the different fitting schemes: $r$MBB (blue, dotted line), $r\\beta$-1 (red, dashed line), $r\\beta$-$T$ (green, solid line), and $r\\beta$-2 (yellow, dash-dotted line). The vertical black dashed line marks the value of $r_{\\rm sim}=0$.\\emph{(Bottom panel)}: Same, in the case of the {\\tt d1c} simulation type on $f_{\\rm sky}=0.6$.}\n    \\label{fig:rgauss2}\n\\end{figure}\n\n\\subsection{Tests with nonzero input tensor modes}\n\\label{sec:rnot0}\n\nWe show in Sect.~\\ref{sec:d1c} that the $r\\beta$-$T$ fitting scheme allows us to retrieve $\\hat{r}$ compatible with zero when $r_{\\rm sim}=0$. We now want to assess the potential leakage of $\\hat{r}$ in the moment expansion parameters if $r_{\\rm sim} \\neq 0$. In this case, primordial tensor signals would be incorrectly interpreted as dust complexity. We run the pipeline as described in Sect.~\\ref{sec:d1c} with $r_{\\rm sim}=0.01$, in the ${\\tt d1c}$ simulation type. This value of $r_{\\rm sim}=0.01$ is larger than the value targeted by \\textit{LiteBIRD},{} but given the order of magnitude of the error on $\\hat{r}$ observed in the previous sections, a potential leakage could be left unnoticed using a smaller $r_{\\rm sim}$. \n\nLooking at the final posterior on $\\hat{r}$ (Fig.~\\ref{fig:rgauss3} and Table~\\ref{tab:rvalues2}), we can see that the results are comparable with the $r_{\\rm sim}=0$ case, but centered on the new input value $r_{\\rm sim}=0.01$. The $r$MBB fitting scheme gives a highly biased posterior of $\\hat{r}=(2.048\\pm0.077)\\times10^{-2}$ ; the bias is reduced but still significant when using the $r\\beta$-1 scheme ($\\hat{r}=129.0\\pm 8.3\\times10^{-4}$) ; in the $\\beta$-$T$ case we get an estimate of $\\hat{r}=94.6\\pm 15.1\\times10^{-4}$ compatible with the input value of $r_{\\rm sim}=100\\times10^{-4}$ ; and  finally, the $\\beta$-2 fitting scheme leads to a negative $2\\,\\sigma_{\\hat{r}}$ tension ($\\hat{r}=62.5\\pm25.0\\times10^{-4}$). This demonstrates the robustness of our method and its potential application to component separation. We note that the negative bias at second order is still present in the $r_{\\rm sim }=0.01$ case, illustrating that setting a positive prior on $\\hat{r}$ would not have been a satisfying solution when $r_{\\rm sim}=0$.\n\n\\subsection{\\label{sec:bias} Exploring the correlations between the parameters}\n\nWe now examine the substantial increase in the dispersion on the $\\hat{r}$ posteriors between the $r\\beta$-1 fitting scheme on the one hand and the $r\\beta$-$T$ and $r\\beta$-2 ones on the other. Indeed, in Sect.~\\ref{sec:d1c}, we show that $\\sigma_{\\hat{r}}$ is about two times greater when using the $r\\beta$-$T$ scheme than the $r\\beta$-1 one, and about four times larger in the case of $r\\beta$-2, while the $r\\beta$-$T$ and $r\\beta$-2 schemes share the same number of free parameters. Some other points to clarify are the shift on $\\hat{r}$ appearing for $r\\beta$-2 in the {\\tt d1c} scenario, discussed in Sect.~\\ref{sec:d1c}, and the inability to correctly recover $\\mathcal{D}_\\ell^{\\omega_2^\\beta \\times \\omega_2^\\beta}$ when $\\hat{r}$ is added to the fit illustrated in Fig.~\\ref{fig:momentsd1d1c}. \n\nThe 2D-SED shapes of the parameters $\\mathcal{D}_\\ell^{\\mathcal{N} \\times \\mathcal{M}}(\\nu_i \\times \\nu_j)$ in the $(\\nu_i, \\nu_j)$ space\\footnote{For example, $\\mathcal{S}(\\nu_i,\\nu_j) = \\frac{I_{\\nu_i}(\\beta_0,T_0)I_{\\nu_j}(\\beta_0,T_0)}{I_{\\nu_0}(\\beta_0,T_0)^2}\\cdot \\left[\\ln\\left(\\frac{\\nu_i}{\\nu_0}\\right)\\ln\\left(\\frac{\\nu_j}{\\nu_0}\\right) \\right]$ is associated to the $\\mathcal{D}^{\\omega_1 \\times \\omega_1}_\\ell$ parameter (see Eq.~\\ref{eq:moments}).} are displayed in Fig.~\\ref{fig:momentshapes}. We used the nine frequencies of \\textit{LiteBIRD}{} presented in Sect.~\\ref{sec:Instrsim} and fixed $\\beta_0=1.54$ and $T_0 = 20$\\,K. We also introduce the CMB 2D-SED shape with the black body function:\n\n\\begin{equation}\n    B_{\\rm CMB}(\\nu_i \\times \\nu_j) = \\frac{B_{\\nu_i}(T_{\\rm CMB})B_{\\nu_j}(T_{\\rm CMB})}{B_{\\nu_0}(T_{\\rm CMB})^2},\n\\end{equation}\n\n\\noindent where $T_{\\rm CMB} = 2.726$\\,K. \n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/gauss_r_0.01_d1_SLD_full.pdf}\n    \\caption{\\footnotesize Posterior on $\\hat{r}$ in the {\\tt d1c} simulation type with $r_{\\rm sim}=0.01$ and $f_{\\rm sky}=0.7$ for the different fitting schemes: $r$MBB (blue, dotted line), $r\\beta$-1 (red, dashed line), $r\\beta$-$T$ (green, solid line), and $r\\beta$-2 (yellow, dash-dotted line). The vertical black dashed line marks the value of $r_{\\rm sim}$.}\n    \\label{fig:rgauss3}\n\\end{figure}\n\nThe 2D correlation coefficients between these 2D-SED shapes are displayed in Fig.~\\ref{fig:corr3D}. We present the correlations between the shapes of the parameters in the case of the $r\\beta$-$T$ and $r\\beta$-2 fitting schemes. We can see that all the moment parameters in $\\omega^{\\beta}_2$ are strongly correlated with the CMB SED signal, while the ones in $\\omega^{T}_1$ are not. \n\nWe showed that, when fitting $\\beta$-2 on {\\tt d1c}, the SED distortions due to spatial variations of $T$ are incorrectly detected by the second-order moment parameters with respect to the spectral index $\\beta$. Due to the correlations highlighted above, those spurious moment parameters could then leak into $\\hat{r}$ when adding it to the fit in $r\\beta$-2. This explains both the negative shift on the $\\hat{r}$ posterior using $\\beta$-2 in the {\\tt d1c} simulation type with $f_{\\rm sky}=0.7$ presented in Sect.~\\ref{sec:d1c} and \\ref{sec:rnot0}, and the inability to  correctly recover the  $\\omega^{\\beta}_2 \\times \\omega^{\\beta}_2$ dust moment parameter presented in Fig.~\\ref{fig:momentsd1d1c}. In addition, it gives a natural reason for the surge of $\\sigma_{\\hat{r}}$ when the second-order moments in $\\beta$ are added to the fit. \n\nOn the other hand, the moment parameters in $\\omega^{T}_1$ are strongly correlated with the moments in $\\omega^{\\beta}_1$. This behavior is expected due to the strong correlation between $\\beta$ and $T$  \\citep[see e.g.,][]{betatcorr}. However those moment parameters are less correlated with the CMB signal than the second-order parameters of $\\beta$-2. This points out that the factor of $\\sim 2$ on $\\sigma_{\\hat{r}}$ between $\\beta$-$T$ and $\\beta$-2 is due to this correlation of the 2D-SED shapes.\nAs the parameters in $\\omega^{T}_1$ are highly correlated with one another, we expect them to be highly redundant in the fit. However, repeating the process described in Sect.~\\ref{sec:d1c} using only $\\mathcal{D}^{A \\times \\omega^{T}_1}_\\ell$ for $\\beta$-$T$ ---which is equivalent to applying the $\\beta$-1 fitting scheme with an iterative correction to the temperature $T_0(\\ell)$--- gives a $\\hat{r}$ posterior similar to the one obtained for $\\beta$-1 alone. Taking the other $\\omega^{T}_1$ terms  into account appears to be necessary in order to recover an unbiased distribution of $\\hat{r}$.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/corr3D.pdf}\n    \\includegraphics[width=\\columnwidth]{Figures\/corr3DLB_betaT.pdf}\n    \\caption{\\footnotesize Correlation matrices of the 2D-SED shapes of the CMB ($B_{\\rm CMB}(\\nu_i\\times\\nu_j)$ and dust moments $\\mathcal{D}_\\ell^{\\mathcal{N} \\times \\mathcal{M}}(\\nu_i \\times \\nu_j)$ in the $(\\nu_i, \\nu_j)$ space). Each element represents the Pearson correlation coefficient between any 2 of these 2D-SED shapes. The correlation matrices are displayed in the case of the $\\beta$-2 fitting scheme (top panel) and the $\\beta$-$T$ one (bottom panel).}\n    \\label{fig:corr3D}\n\\end{figure}\n\n\\subsection{Adding synchrotron to the simulations }\n\\label{sec:synchrotron}\n\nThermal dust is not the only source of polarized foreground that must be considered for CMB studies. Although subdominant at high frequencies ($\\geq 100\\,$GHz), the synchrotron emission due to accelerated electrons in the interstellar medium is still expected to represent a significant fraction of the total polarized signal. \n\nIn order to take one more step towards realistic forecasts for the \\textit{LiteBIRD}{} instrument, we add a synchrotron contribution to the {\\tt d1c} simulations presented in \\ref{sec:simulations} using the {\\tt s1} template included in the {\\sc PySM}. In this scenario, the synchrotron SED for each line of sight is given by a power law of the form  (in antenna temperature units)\n\n\\begin{equation}\n    S_\\nu^{\\tt s1} = A_{\\tt s1}(\\vec{n}) \\left(\\frac{\\nu}{\\nu^{\\tt s1}_0}\\right)^{\\beta_{\\tt s1}(\\vec{n})},\n\\label{eq:s1powerlaw}\n\\end{equation}\n\n\\noindent where the amplitude $A_{\\tt s1}(\\vec{n})$ and the spectral index $\\beta_{\\tt s1}(\\vec{n})$ maps are derived from the combination of the \\textit{WMAP}{} mission 23 GHz map \\cite{WMAPfg} and Haslam 408 GHz map \\cite{Haslam1}. $\\nu^{\\tt s1}_0$ is defined as 23 GHz. The simulations containing synchrotron are referred to as {\\tt d1s1c} below.\n\nIf not treated in the fit, the presence of synchrotron is expected to induce a bias on the $\\hat{r}$ posterior distribution. Regarding the dust MBB discussed in Sect.~\\ref{sect:limitsmbb}, the synchrotron SED is expected to have distortions. However, as the synchrotron polarized emission is significantly lower than that of dust, in the frequency range considered in the present work, we expect the distortions to be small compared to the ones induced by dust and we leave their modeling to a further study. \n\nIn order to minimize the number of free parameters used for fitting the synchrotron emission, we model its power spectrum as a power law of the multipole $\\ell$ \\citep{krachmalnicoff_s-pass_2018}. Therefore, combining with the synchrotron SED in Eq.~\\ref{eq:s1powerlaw}, the synchrotron component of the cross-angular power spectra reads \n\n\\begin{equation}\n    \\mathcal{D}^{\\rm sync}_\\ell (\\nu_i \\times \\nu_j) = A_{\\rm s}  \\left(\\frac{\\nu_i \\nu_j}{\\nu_0}\\right)^{\\beta_{\\rm s}} \\ell^{\\alpha_{\\rm s}},\n\\label{eq:syncromoment}\n\\end{equation}\n\nwhere the amplitude coefficient $A_{\\rm s}$ is treated as a free parameter while we fix $\\beta_{\\rm s}=-3$ (median value of the ${\\tt s1}$ $\\beta_{\\rm s}$ map on our $f_{\\rm sky}=0.7$ mask) and $\\alpha_{\\rm s}=-1$ \\citep{krachmalnicoff_s-pass_2018}. \n\nWhen fitting the {\\tt d1s1c} simulations, we either use the $r\\beta$-$T$ fitting scheme, neglecting the synchrotron component, or we add the synchrotron component in Eq.~\\ref{eq:syncromoment} to the model in Eq.~\\ref{eq:model}. We refer to this latter case as the {\\tt s}$r\\beta$-$T$ fitting scheme.\nIn Fig.~\\ref{fig:synchrotron}, the $\\hat{r}$ posteriors derived from the {\\tt d1s1c} simulations are displayed with $r_{\\rm sim}=0$ and $f_{\\rm sky}=0.7$. \n\nUsing the $r\\beta$-$T$ fitting scheme, we find $\\hat{r} = (143.1 \\pm 13.5)\\times 10^{-4}$. As expected, even at high frequencies, modeling the synchrotron component is critical and cannot be neglected in order to recover an unbiased value of $\\hat{r}$.\nOn the other hand, using {\\tt s}$r\\beta$-$T$ fitting scheme, we recover $\\hat{r} = (-5.4 \\pm 13.2)\\times 10^{-4}$. This result is comparable with the one obtained for the {\\tt d1c} simulations in Sect.~\\ref{sec:d1c}, with a minor increase in $\\sigma_{\\hat{r}}$. We can therefore conclude that a model as simple as that of Eq.~\\ref{eq:syncromoment} is sufficient to take into account the {\\tt s1} component at $\\nu > 100$\\,GHz and the corresponding SED distortions can be neglected in order to recover an unbiased value of $\\hat{r}$. In principle, as we know that the dust-synchrotron spatial correlation is significant at large scales \\citep{PlanckDust2},  Eq.~\\ref{eq:model} should include a dust-synchrotron term \\citep[see e.g.,][]{SOgalactic}. In our study, where we consider cross-spectra from 100 to 402\\,GHz, this dust-synchrotron term is subdominant, but it could be significant when considering cross-spectra between LiteBIRD's extreme frequency bands (e.g., the 40$\\times$402 cross-spectrum). The moment expansion might be more complicated as well in this case, as we could expect some correlation between the dust and synchrotron moment-terms.\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figures\/gauss_r_0_d1s1_SLD_full.pdf}\n    \\caption{\\footnotesize Posterior on $\\hat{r}$ in the {\\tt d1s1c} simulation type with $r_{\\rm sim}=0$ and $f_{\\rm sky}=0.7$ for the different fitting schemes: $r\\beta$-$T$ (green, solid line) and {\\tt s}$r\\beta$-$T$ (orange, dash-dotted line). The vertical black dashed line marks the value $r_{\\rm sim}=0$.}\n    \\label{fig:synchrotron}\n\\end{figure}\n\nThis result shows that a full polarized foreground content can be treated at high frequencies when using a power law SED for the synchrotron coupled with the moment expansion of the MBB up to first order in both $\\beta$ and $T$ for the dust SED. A full study remains to be done in that direction using all the frequency bands of the \\textit{LiteBIRD}{} instrument. Eventually, Eq.~\\ref{eq:syncromoment} will also have to be expanded in moments with respect to its parameters. Doing so, one can expect to recover an unbiased value of $\\hat{r}$ associated with a decrease in $\\sigma_{\\hat{r}}$ down to a value compatible with the full success criterion of the mission.\n\n\\subsection{Limitations of this work and caveats}\n\n{As discussed in Sect.~\\ref{sec:formalismspectra}, we neglected polarization effects through this work by treating the $BB$ signal as an intensity signal. This is not problematic in the present work, because no variations along the lines of sight were present in the simulations. However, this point has to be addressed using complex simulations or real sky data.}\n\nThe choice of reference frequency $\\nu_0$ used for the normalization of the MBB in Eq.~\\ref{eq:MBB}, which is not discussed in this study, can potentially have a significant impact on the moment expansion and, in turn, on the measurement of $\\hat{r}$. Indeed, $\\nu_0$ is the pivot frequency of the moment expansion (moments are equal to zero at $\\nu_0$) and will determine the shape of the SED distortion around it. A poor choice for this reference frequency can have disastrous consequences for the moment fit: for example, if it is chosen far away from the observed bands, all the moments will become degenerated. In our case, the reference frequency (353\\,GHz) is within the observed frequency range (100 to 402\\,GHz), but we have not tried to optimize its position. In addition, the $\\nu_0$ pivot of our moment expansion coincides with the one used to extrapolate the dust template map in the {\\sc PySM} and we have not quantified how much this impacts our results. \n\nFinally, as pointed out several times in this work, the quantitative results depend strongly on the sky model of our simulations. Moreover, we lack dedicated sky models where we can control the complexity of the dust SED, either by directly including moments or by averaging the emission from the 3D structure of the Galaxy. However, both methods are beyond the scope of the present work.\n\n\n\\section{Conclusion}\n\nBeing able to precisely characterize the complexity of the Galactic thermal dust polarized SED has become critical for the measurement of the faint primordial $B$-mode signal of the CMB, especially at the sensitivity targeted by future CMB experiments such as the \\textit{LiteBIRD}{} satellite mission.\n\nIn this work, we applied the moment expansion formalism to the dust emission SED as a component-separation tool to recover the tensor-to-scalar ratio parameter $r$ in \\textit{LiteBIRD}-simulated data. This formalism, proposed by \\citet{Chluba} and implemented in harmonic space by \\citet{Mangilli}, allows us to deal with the spectral complexity of the Galactic dust signal by modeling its deviations from the canonical MBB model at the cross-angular power spectrum level. In the case of the data-driven realistic dust emission model ---we explore ({\\sc PySM} {\\tt d1}) \nhere---, suitably taking into account the dust SED distortions prevents the spurious detection of the primordial $B$-mode signal. \n\nWe show that the dust spectral index $\\beta$ and dust temperature $T$ spatial variations significantly distort the dust cross-power spectrum SED. The MBB is not a good model to describe the data in that case and the estimation of $r$ is dramatically affected. In the case where no primordial signal is included in the simulated data sets, not taking into account the dust SED complexity leads to a highly significant spurious detection of $r$ with \\textit{LiteBIRD}{} (from $\\hat{r}\\simeq5\\times10^{-3}$ to $1.25\\times10^{-2}$, with a 8.4 to 21.2\\,$\\sigma$ significance, from 50 to 70\\,\\% of the sky, respectively). \n\nTo overcome this obstacle, we applied the moment expansion formalism in order to model these SED distortions. We demonstrate that, at \\textit{LiteBIRD}{} sensitivity, the previously studied moment expansion with respect to the dust spectral index $\\beta$  \\citep{Mangilli,Azzoni} does not give satisfactory results. Indeed, expanding in $\\beta$ to first order (following the angular power spectrum definition of the order) leads to a significant bias on 70\\,\\% of the sky ($\\hat{r}=(3.29\\pm0.65)\\times10^{-3}$ when $r_{\\rm sim}=0$ and $\\hat{r}=(1.29\\pm0.08)\\times10^{-2}$, when $r_{\\rm sim}=10^{-2}$). At second order in $\\beta$, we observe a $\\sim$2\\,$\\sigma$ negative tension ($\\hat{r}=(-3.7\\pm1.9)\\times10^{-3}$ when $r_{\\rm sim}=0$ and $\\hat{r}=(6.25\\pm2.50)\\times10^{-3}$, when $r_{\\rm sim}=10^{-2}$). \n\nWe introduce for the first time in this work the expansion of the dust angular cross-power spectra with respect to both $\\beta$ and $T$. We show that by using this expansion up to first order, we correctly model the dust SED distortions due to spatial variations of both $\\beta$ and $T$ at the map level. This allows us to recover $r$ parameter without bias, with $\\hat{r}=(-3.3\\pm11.7)\\times10^{-4}$ if $r_{\\rm sim}=0$ and $\\hat{r}=(0.95\\pm0.15)\\times10^{-2}$ if $r_{\\rm sim}=10^{-2}$. Thus, despite the known degeneracy between the dust spectral index and its temperature in the Rayleigh-Jeans domain, it is important to correctly model the latter in order to accurately retrieve the tensor-to-scalar ratio $r$ at the unprecedented precision reached by experiments such as \\textit{LiteBIRD}{}. \n\nAdding parameters to tackle the dust SED complexity means an increase in the error budget. Given the \\textit{LiteBIRD}{} bands and sensitivities we consider in this work (frequency bands above 100\\,GHz), the ideal sensitivity on $r$ without delensing is $\\sigma_{\\hat{r}}=3.4\\times10^{-4}$. In the ideal case, where the dust $\\beta$ and $T$ are constant over the sky ({\\sc PySM} {\\tt d0}), separating the CMB from dust leads to $\\sigma_{\\hat{r}}=3.9\\times10^{-4}$ on 70\\,\\% of the sky. Adding the expansion to first order in $\\beta$ does not significantly increase the error ($\\sigma_{\\hat{r}}=4.5\\times10^{-4}$), but expanding to first order in both $\\beta$ and $T$ multiplies it by a factor of $\\sim2$ ($\\sigma_{\\hat{r}}=9.5\\times10^{-4}$) and to second order in $\\beta$ by a factor of $\\sim4$ ($\\sigma_{\\hat{r}}=16.4\\times10^{-3}$). We show that the surge of $\\sigma_{\\hat{r}}$ between the two latter cases, sharing the same number of free parameters, is due to strong correlations between the SED of the second-order moments in $\\beta$ and the CMB. This is an important point, as it could lead to some intrinsic limitation for component-separation algorithms based exclusively on the modeling of the SED. Furthermore, when dealing with real data, if the dust SED is complex enough to have significant second-order distortions with respect to $\\beta$, CMB experiments might reach a dilemma: either include the second order in the modeling at the cost of losing sensitivity on $r,$ or neglect it at the cost of a potential spurious detection. Coupling the SED-based separation with methods exploiting the diversity of spatial distribution between components \\citep[e.g.,][]{Wavelets} seems a natural way to overcome this issue.\n\nNevertheless, moment expansion at the cross-angular power spectrum level provides a powerful and agnostic tool, allowing us to analytically recover the actual dust complexity without making any further assumptions. We additionally show that this method is robust, in the sense that it can effectively distinguish the primordial tensor signal from dust when $r_{\\rm sim} \\neq 0$, as in the case of \\textit{LiteBIRD}{} simulations. The dust moments in $\\beta$ and $T$ at first order are needed in order to retrieve a reliable measure of $r$; they are significantly detected for $\\ell\\lesssim100$. We can therefore define a cut in $\\ell$ above which we do not fit for the whole complexity of the dust (we fit only the expansion up to first order in $\\beta$ and not in $\\beta$ and $T$). Doing so, we can reduce the error on $\\hat{r}$ while keeping the bias negligible ($\\hat{r}=(-0.9\\pm8.8)\\times10^{-4}$). We could imagine other ways to reduce the number of free parameters in our model \\cite[e.g., assuming a power-law of $\\ell$ behavior for the moments, as in][]{Azzoni} and hence reduce the error on $r$. However, this optimization really depends on the simulated sky complexity and has not been comprehensively explored in the present work.\n\nThe {\\sc PySM} {\\tt d1} sky simulations, being data-driven, are widely used by the CMB community as they contain some of the real sky complexity. Nevertheless, at high-Galactic latitudes, the dust spectral index and temperature templates from \\textit{Planck}{} are dominated by systematic errors (uncertainty on the assumed zero-level of the \\textit{Planck}{} intensity maps, residual cosmic infrared background (CIB), anisotropies, instrumental noise, etc.). Therefore, some of the complexity we observe far from the Galactic plane in this sky model is not real. On the other hand, the modeled SED of the dust is {exactly} a MBB in each pixel, and line-of-sight averages or more complex dust models are ignored. As a consequence, our method and CMB $B$-mode component-separation algorithms in general need to be confronted with more complex models in order to really assess their performances in a  quantitative manner. \n\nFinally, although we demonstrate that the synchrotron component can be tackled at frequencies above 100\\,GHz with a minimal model under our assumptions, a study over the full \\textit{LiteBIRD}{} frequency bands, including synchrotron and the potential moment expansion of its SED, will be considered as a natural next step for a further application.\n\n\n\\section*{Acknowledgments}\n\n\nThis work is supported in Japan by ISAS\/JAXA for Pre-Phase A2 studies, by the acceleration program of JAXA research and development directorate, by the World Premier International Research Center Initiative (WPI) of MEXT, by the JSPS Core-to-Core Program of A. Advanced Research Networks, and by JSPS KAKENHI Grant Numbers JP15H05891, JP17H01115, and JP17H01125. The Italian \\textit{LiteBIRD}{} phase A contribution is supported by the Italian Space Agency (ASI Grants No. 2020-9-HH.0 and 2016-24-H.1-2018), the National Institute for Nuclear Physics (INFN) and the National Institute for Astrophysics (INAF). The French \\textit{LiteBIRD}{} phase A contribution is supported by the Centre National d'Etudes Spatiale (CNES), by the Centre National de la Recherche Scientifique (CNRS), and by the Commissariat \u00e0 l'Energie Atomique (CEA). The Canadian contribution is supported by the Canadian Space Agency. The US contribution is supported by NASA grant no. 80NSSC18K0132. \nNorwegian participation in \\textit{LiteBIRD}{} is supported by the Research Council of Norway (Grant No. 263011). The Spanish \\textit{LiteBIRD}{} phase A contribution is supported by the Spanish Agencia Estatal de Investigaci\u00f3n (AEI), project refs. PID2019-110610RB-C21 and AYA2017-84185-P. Funds that support the Swedish contributions come from the Swedish National Space Agency (SNSA\/Rymdstyrelsen) and the Swedish Research Council (Reg. no. 2019-03959). The German participation in \\textit{LiteBIRD}{} is supported in part by the Excellence Cluster ORIGINS, which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy (Grant No. EXC-2094 - 390783311). This research used resources of the Central Computing System owned and operated by the Computing Research Center at KEK, as well as resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy.\n\nMR acknowledges funding support from the ERC Consolidator Grant CMBSPEC (No. 725456) under the European Union's Horizon 2020 research and innovation program.\n\nThe authors would like to thank David Alonso, Josquin Errard, Nicoletta Krachmalnicoff and Giuseppe Puglisi, for useful discussions as well as Jens Chluba for insightful comments on earlier version of this work. \n\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Appendix}\\noindent\nHere we state expressions for the power spectrum defined in \\Eqref{eq:Hankelkx} (for simplicity with vanishing short distance cutoff $\\lambda=0$).\nUsing the exact expression for $g^{(1)}(r)$ of \\eqref{eq:integral} in the definition of \\ensuremath{\\mathcal{T}(k)}\\xspace, and interchanging the integrals we obtain an integral representation \nfor \\ensuremath{\\mathcal{T}(k)}\\xspace given by\n\\begin{equation}\\label{eq:Tkintegral}\n\\ensuremath{\\mathcal{T}(k)}\\xspace~=~\\frac{g\\,\\Lambda}{\\pi\\,k}\\int\\displaylimits_0^\\infty \\frac{dp\\, p}{\\mathrm{e}^{(E-\\mu)\/T}+1} \\left\\{j_0\\left[\\Lambda\\left(p-k\\right)\\right]-j_0\\left[\\Lambda\\left(p+k\\right)\\right]\\right\\}\\;.\n\\end{equation}\nTo leading order in Sommerfeld expansion this evaluates to (this is consistent with first expanding $g^{(1)}(r)$ to $\\mathcal{O}(T^2\/\\ensuremath{k_{\\mathrm{F}}}\\xspace^2)$ as in \\eqref{eq:g_Sommerfeld} \nand then performing \\eqref{eq:Hankelkx})\n\\begin{equation}\\label{eq:TkExpl}\n \\begin{split}\n\\ensuremath{\\mathcal{T}(k)}\\xspace~=~\n& \\frac{g}{\\pi}\\left\\{\\mathrm{Si}\\left[\\left(\\ensuremath{k_{\\mathrm{F}}}\\xspace-k\\right)\\Lambda\\right]+\\mathrm{Si}\\left[\\left(\\ensuremath{k_{\\mathrm{F}}}\\xspace+k\\right)\\Lambda\\right] \n- \\frac{2\\,\\sin\\left(\\ensuremath{k_{\\mathrm{F}}}\\xspace\\Lambda\\right)\\sin\\left(k\\Lambda\\right)}{k\\Lambda} \\right\\}+ \\\\\n& \\frac{g\\,\\pi\\,T^2}{6\\,\\ensuremath{k_{\\mathrm{F}}}\\xspace k}\\left\\{ \\Lambda\\mu^2 \\left[\n  \\frac{\\cos\\left[\\left(\\ensuremath{k_{\\mathrm{F}}}\\xspace-k\\right)\\Lambda\\right]}{\\ensuremath{k_{\\mathrm{F}}}\\xspace-k}-\\frac{\\cos\\left[\\left(\\ensuremath{k_{\\mathrm{F}}}\\xspace+k\\right)\\Lambda\\right]}{\\ensuremath{k_{\\mathrm{F}}}\\xspace+k}\\right] \\right.+ \\\\\n& \\hspace{1.2cm}\\left. \\frac{\\left(\\ensuremath{k_{\\mathrm{F}}}\\xspace^2-\\ensuremath{k_{\\mathrm{F}}}\\xspace k-\\mu^2\\right)\\sin \\left[\\left(\\ensuremath{k_{\\mathrm{F}}}\\xspace-k\\right)\\Lambda\\right]}{\\left(\\ensuremath{k_{\\mathrm{F}}}\\xspace-k\\right)^2} - \\frac{\\left(\\ensuremath{k_{\\mathrm{F}}}\\xspace^2+\\ensuremath{k_{\\mathrm{F}}}\\xspace k-\\mu^2\\right)\\sin \\left[\\left(\\ensuremath{k_{\\mathrm{F}}}\\xspace+k\\right)\\Lambda\\right]}{\\left(\\ensuremath{k_{\\mathrm{F}}}\\xspace+k\\right)^2}\\right\\}\\;.\n\\end{split}\n\\end{equation}\nAnalytic expressions for the critical temperatures are given by \n\\begin{equation}\\label{eq:Tcrit0}\n T^2_{c,0}(\\nu)=\\frac{6\\,\\ensuremath{k_{\\mathrm{F}}}\\xspace^4}{\\pi^2}\\frac{\\left[\\sin(\\B{\\nu})-\\Si{\\B{\\nu}}\\right]}{\\left[ \\left(2\\mu^2-\\ensuremath{k_{\\mathrm{F}}}\\xspace^2\\right)\\B{\\nu}\\cos(\\B{\\nu}) + \\left(\\mu^2\\B{\\nu}^2+\\ensuremath{k_{\\mathrm{F}}}\\xspace^2-2\\mu^2\\right)\\sin(\\B{\\nu})\\right]}\\;,\n\\end{equation}\n\\begin{equation}\\label{eq:TcritFermi}\n\\begin{split}\n T^2_{c,\\mathrm{F}}(\\nu)=& \\frac{24\\, \\ensuremath{k_{\\mathrm{F}}}\\xspace^4}{\\pi^2} \\frac{\\left[1-\\cos(2\\B{\\nu})-\\B{\\nu}\\Si{2\\B{\\nu}}\\right]}{\n \\B{\\nu}^2\\left[4\\ensuremath{k_{\\mathrm{F}}}\\xspace^2-2\\mu^2\\cos(\\B{\\nu})\\right]+\\B{\\nu}\\left[  \\left(\\mu^2-2\\ensuremath{k_{\\mathrm{F}}}\\xspace^2\\right)\\sin(2\\B{\\nu})\\right]}\\;,\n\\end{split}\n\\end{equation}\n\\begin{equation}\\label{eq:TcritDFermi}\n\\begin{split}\n & T^2_{c,\\mathrm{F}'}(\\nu)= \\frac{36\\, \\ensuremath{k_{\\mathrm{F}}}\\xspace^4}{\\pi^2} \\left[2\\B{\\nu}^2 - 4 \\sin(\\B{\\nu})^2 +\\B{\\nu} \\sin(2\\B{\\nu})\\right] \\times \\\\\n & \\left[ 4\\mu^2 \\B{\\nu}^4 - 6 \\B{\\nu}^2 (2\\ensuremath{k_{\\mathrm{F}}}\\xspace^2+(\\ensuremath{k_{\\mathrm{F}}}\\xspace^2-2\\mu^2)\\cos(2\\B{\\nu})) + 3 (3\\ensuremath{k_{\\mathrm{F}}}\\xspace^2- 2\\mu^2) \\B{\\nu} \\sin(2\\B{\\nu}) + 6 \\mu^2 \\B{\\nu}^3 \\sin(2\\B{\\nu})\\right]^{-1}\\;.\n\\end{split}\n \\end{equation}\nHere $\\B{\\nu}$ are the zeros of the Bessel function $J_{3\/2}$ typically called $j_{3\/2,\\nu}$ and $\\mathrm{Si}$ is the integral sine.\nNote that $T^2_{c,0}(\\nu)>0$ only for odd $\\nu$, while $T^2_{c,\\mathrm{F}}(\\nu),T^2_{c,\\mathrm{F}'}(\\nu)<0$ for all $\\nu$.\n\n\\twocolumngrid\n\\bibliographystyle{utphys}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\@startsection {section}{1}{\\z@}%\n                                   {-3.5ex \\@plus -1ex \\@minus -.2ex\n                                   {2.3ex \\@plus.2ex}%\n                                   {\\normalfont\\large\\bfseries}}\n\\renewcommand\\subsection{\\@startsection{subsection}{2}{\\z@}%\n                                     {-3.25ex\\@plus -1ex \\@minus -.2ex}%\n                                     {1.5ex \\@plus .2ex}%\n                                     {\\normalfont\\bfseries}}\n\n\n \n\\def1.2{1.2}\n\\parskip 6 pt\n\n\\marginparwidth 0pt\n\\oddsidemargin  0pt\n\\evensidemargin  0pt\n\\marginparsep 0pt\n\\topmargin   -0.5in\n\\textwidth   6.5in\n\\textheight  9.0 in\n\\newcommand{\\begin{equation}}{\\begin{equation}}\n\\newcommand{\\end{equation}}{\\end{equation}}\n\\newcommand{\\begin{eqnarray}}{\\begin{eqnarray}}\n\\newcommand{\\end{eqnarray}}{\\end{eqnarray}}\n\\newcommand{\\\"o}{\\\"o}\n\\newcommand{{\\rm Tr}}{{\\rm Tr}}\n\\newcommand{\\gone}[1]{{}}\n\\newcommand{\\noindent $\\bullet\\ $}{\\noindent $\\bullet\\ $}\n\\newcommand{{\\rm Re}}{{\\rm Re}}\n\\newcommand{{\\rm Im}}{{\\rm Im}}\n\\newcommand{{\\mathcal{N}}}{{\\mathcal{N}}}\n\n\\newcommand{\\ensuremath{H}\\xspace}{\\ensuremath{H}\\xspace}\n\\newcommand{\\ensuremath{f}\\xspace}{\\ensuremath{f}\\xspace}\n\\newcommand{\\ensuremath{Q}\\xspace}{\\ensuremath{Q}\\xspace}\n\\newcommand{\\ensuremath{R}\\xspace}{\\ensuremath{R}\\xspace}\n\n\n\n\\begin{document}\n\\begin{titlepage}\n\\begin{flushright}\nMAD-TH-11-02\n\\end{flushright}\n\n\\vfil\n\n\\begin{center}\n\n{\\bf \\large \nQuantization of charges and fluxes in  warped Stenzel geometry\n}\n\n\n\\vfil\n\n\nAkikazu Hashimoto$^a$ and Peter Ouyang$^b$\n\n\\vfil\n\n$^a$ \nDepartment of Physics, University of Wisconsin, Madison, WI\n53706, USA\n\n$^b$ \nDepartment of Physics, Purdue University, West Lafayette, IN 47907, USA\n\n\n\\vfil\n\n\\end{center}\n\n\\begin{abstract}\n\\noindent We examine the quantization of fluxes for the warped Stiefel\ncone and Stenzel geometries and their orbifolds, and distinguish the\nroles of three related notions of charge: Page, Maxwell, and brane.\nThe orbifolds admit discrete torsion, and we describe the associated\nquantum numbers which are consistent with the geometry in its large\nradius and small radius limits from both the type IIA and the M-theory\nperspectives.  The discrete torsion, measured by a Page charge, is\nrelated to the number of fractional branes.  We relate the shifts in\nthe Page charges under large gauge transformations to the\nHanany-Witten brane creation effect.\n\n\\end{abstract}\n\\vspace{0.5in}\n\n\n\\end{titlepage}\n\\renewcommand{1.2}{1.05} \n\n\\section{Introduction}\n\nRecently, a holographic duality for superconformal Chern-Simons-Matter\ntheories in 2+1 dimensions with ${\\cal N}=6$ and ${\\cal N}=8$\nsupersymmetry was proposed \\cite{Aharony:2008ug,Aharony:2008gk}. These\nfield theories have $U(N)_k \\times U(N+l)_{-k}$ product gauge symmetry\n(where the subscripts refer to the Chern-Simons level associated with\neach gauge group) and bifundamental matter fields.  In the large $N$\nlimit, the field theory has a dual gravity description in terms of\nM-theory as $N$ M2-branes on the orbifold $C^4\/Z_k$ (where the\norbifold acts by rotating each of the complex planes by an angle $2\n\\pi\/k$ simultaneously) and $l$ fractional branes.  The supergravity\nsolution corresponding to this brane system is $AdS_4 \\times S^7\/Z_k$,\nand the quantum number $l$ is encoded in the discrete torsion of the\n$H_3(Z) = Z_k$ homology group of $S^7\/Z_k$.\n\nThe M-theory background can also be described in type IIA supergravity\nby dimensionally reducing along the Hopf-fiber of $S^7\/Z_k$. In this\ndescription, the geometry has the form $AdS_4 \\times CP^3$ and is the\neffective description when $1 \\ll N \\ll k^5$. Homologically, $CP^3$ is\nvery different from $S^7\/Z_k$, particularly in that $CP^3$ has no\ndiscrete torsion cycles, but it does possess integral homology. Even\nfor this simple example, the relationship of the spectrum of charges\nand fluxes in the M-theory and the type IIA descriptions is subtle.\n\nOne way to gain some perspective on the physical meaning of the\ndefining data of the gravity side of these correspondences is to\nrealize the superconformal field theory as either the UV or IR fixed\npoint of a holographic renormalization group flow.  For example, a\nsuperconformal Chern-Simons theory can arise as the IR fixed point of\nan RG flow from a Chern-Simons-Yang-Mills-Matter theory\n\\cite{Hashimoto:2008iv,Aharony:2009fc}. Several related realizations\nhave also been considered \\cite{Hashimoto:2010bq}.  These\nrenormalization group flows are dual to transverse geometries which\ndiffer from $R^8$, and many of these constructions have the\ninteresting property that the dual geometry admits a normalizable\n4-form.  In M-theory, this allows one to introduce a nontrivial 4-form\nflux.  The freedom of tuning the 4-form flux has a specific\ninterpretation in terms of tuning the parameters of the dual field\ntheory, and in some examples, one can explore dynamical features such\nas phase transitions in the low energy effective field theory from the\ngeometry of the supergravity dual\n\\cite{Aharony:2009fc,Hashimoto:2010bq}.\n\nIn this article, we investigate the duality of ${\\cal N}=2$\nChern-Simons quiver theories dual to $AdS_4 \\times V_{5,2}\/Z_k$ where\n$V_{5,2}$ is a homogeneous Sasaki-Einstein seven-manifold. This\nduality was originally considered by Martelli and Sparks in\n\\cite{Martelli:2009ga}. On the field theory side, it generalizes the\nmodel of ABJM by adding a chiral multiplet in the adjoint\nrepresentation to each factor of the $U(N)_k \\times U(N+l)_{-k}$ gauge\ngroup. The gravity dual description can be deformed in the IR, giving\nrise to a geometry known as the warped Stenzel metric. At the moment,\nlittle is known about the field theory interpretation of this IR\ndeformation. In order to facilitate in its interpretation, it is\nuseful to enumerate the the discrete and continuous parameters\nassociated with this system. This is related to the problem of\nquantizing charges and fluxes in the gravity dual.\n\nIn type IIA (and IIB) supergravity, there is a well-known subtlety in\nimposing charge quantization, which arises in the example studied in\nthis paper.  The $V_{5,2}\/Z_k$ geometry, reduced to IIA along the\n$U(1)$ isometry along which the $Z_k$ acts, is a space $M_2$ which has\nthe same homology structure as $CP^3$ \\cite{Martelli:2009ga}; in\nparticular there is a nontrivial 4-cycle.  Now, one might want to\nquantize the four-form flux through this cycle, but the natural\ngauge-invariant four-form\n\\begin{equation} \\tilde F_4 = d A_3 + H_3 \\wedge A_1 \\end{equation}\nis not closed, and therefore its integral through the 4-cycle is not\nconserved and cannot be quantized.  A similar issue arises in the flux\nof $* \\tilde F_4$ through $M_2$. These apparent difficulties have also\nappeared in earlier examples considered in\n\\cite{Aharony:2009fc,Hashimoto:2010bq} and their resolution is well\nunderstood.  The four-form flux satisfies a modified Bianchi identity,\n\\begin{equation} d \\tilde F_4 = -H_3 \\wedge F_2 \\end{equation}\nso to define a conserved charge we should not integrate $\\tilde{F}_4$\nbut a modified flux which is chosen to be closed:\n\\begin{equation} Q_4^{Page} = {1 \\over (2 \\pi l_s)^3 g_s } \\int (-\\tilde F_4 + B\n\\wedge F_2) \\label{q4page}\\ .\\end{equation}\nThis new charge, known as the Page charge, is one of the three subtle\nnotion of charges identified by Marolf \\cite{Marolf:2000cb}. The three\ncharges being referred here are the Maxwell charge, brane charge, and\nthe Page charge, and they can take distinct values in gauge theories\ninvolving Chern-Simons terms as is the case for type IIA\nsupergravity. Each of these charges respects some, but not all, of the\nproperties commonly associated with charges in simpler contexts: gauge\ninvariance, conservation, localization, and integer quantization. Page\ncharge turns out to respect conservation, localization, and integer\nquantization, but fails to be invariant with respect to large gauge\ntransformations which shift the period of $B_2$. This ambiguity is\nprecisely what is required to interpret the Hanany-Witten brane\ncreation effects in the brane construction of these models is\nintimately connected to the duality of the field theory.\n\nIn this article, we will analyze the quantization of fluxes in $AdS_4\n\\times V_{5,2}\/Z_k$ geometry and its Stenzel deformation from the type\nIIA perspective, and relate the gauge ambiguity to Hanany-Witten brane\ncreation effects along the lines of\n\\cite{Aharony:2009fc,Hashimoto:2010bq}. In \\cite{Martelli:2009ga}, it\nwas argued that the Stenzel deformation is incompatible with the\npresence of discrete torsion which gives rise to a non-vanishing value\nof $l$ in $U(N)_k \\times U(N+l)_{-k}$.  On the contrary, we find that\nsome values of $l$ are allowed, and explain the source of this\napparent discrepancy.  We will also examine the compatibility of the\nIIA and the M-theory perspectives.\n\n\\section{Stiefel, Stenzel, and the ${\\cal N}=2$ Chern-Simons-Quiver theory}\n\nIn this section, we briefly review the construction of ${\\cal N}=2$\nChern-Simons-Quiver theory, its gravity dual, and its Stenzel\ndeformation. We closely follow the presentation of\n\\cite{Martelli:2009ga}.\n\n\\subsection{Stiefel cone}\n\\label{sec2.1}\nThe starting point is a non-compact Calabi-Yau 4-fold \n\\begin{equation} z_0^n + z_1^2+z_2^2+z_3^2 + z_4^2 = 0 \\label{curve} \\end{equation}\nwhere we take $n=2$. This geometry is a cone whose base is a\nSasaki-Einstein seven manifold $V_{5,2}$, also known as the Stiefel\nmanifold. Had one taken $n=1$, the geometry of the Calabi-Yau 4-fold\nwould have been $R^8$ which is formally a cone over $S^7$.  For $n\n>2$, the geometry is not a cone over a Sasaki-Einstein manifold\n\\cite{Martelli:2009ga}.\n\nWhen M2 branes are placed at the tip of the cone, we obtain a warped\ngeometry $AdS_4 \\times V_{5,2}$.  The base $Y_2=V_{5,2}$ has a torsion\n3-cycle $H_3(Y_2,Z)=Z_2$. \n\n\nThe $Z_k$ orbifold is taken on the $U(1)_b$ isometry which rotates\n\\begin{equation} (z_0,z_1+z_2, z_3+z_4, z_1-z_2,z_3-z_4) \\end{equation}\nwith weights $(0,1,1,-1,-1)$.  On $Y_2\/Z_k$, this changes the torsion\ngroup from $Z_2$ to $H_3(Y_2\/Z_k,Z)=Z_{2k}$, so\n\\begin{equation} {1 \\over (2 \\pi l_p)^3} \\int_{\\Sigma_3} C_3 = {l -k \\over 2 k} \\end{equation}\nfor $\\Sigma_3$ which generates $H_3(Y_2\/Z_k)$. Here we have shifted\n$l$ by $k$ compared to what is written in (3.26) of\n\\cite{Martelli:2009ga}. Both $l$ and $k$ are integers so this shift is\na matter of convention in describing the supergravity background.\n\nWhen reduced to IIA along the $U(1)_b$ direction parametrized by\n$\\gamma$, the Sasaki-Einstein space $Y_2\/Z_k$ decomposes into\n\\begin{equation} ds^2(Y_2\/Z_k) = ds^2(M_2) + {w \\over k^2}(d \\gamma + kP)^2 \\end{equation}\nand the IIA string frame metric becomes\n\\begin{equation} ds^2 = \\sqrt{w} {R^3 \\over k} \\left({1 \\over 4} ds^2(\\mbox{AdS}_4)+ds^2(M_2)^2\\right) \\end{equation}\nwith\n\\begin{equation} F_2 = k g_s l_s \\Omega_2, \\qquad \\Omega_2 = d P \\  . \\end{equation}\n\nSince\n\\begin{equation} C_3 = A_3 + B_2 \\wedge d \\gamma \\end{equation}\nwith this dimensional reduction, $B_2$ turns out to have the period\n\\begin{equation} {1 \\over (2 \\pi l_s)^2} \\int B_2 = {l \\over 2k} - {1 \\over 2} \\ . \\end{equation}\n\n\\subsection{Brane construction and the Hanany-Witten effect}\n\nThe field theory dual is conjectured in \\cite{Martelli:2009ga} to\narise from the low energy limit of a network of D3-branes, an NS5-brane and\na $(1,k)$ 5-brane in type IIB on $R^{1,2} \\times S^1 \\times R^2\\times\nC^2$. The D3-branes wind along $R^{1,2} \\times S^1$. The NS5 is\nextended along $R^{1,2}$, one of the $R$ in $R^2$ and along the curve\n$w_1=- i w_0^2$ where $C^2$ is parametrized by $(w_0,w_1)$. The\n$(1,k)$ 5-brane is extended along $R^{1,2}$, a line at an angle\nrelative to the NS5-brane in $R^2$, and along $w_1=i w_0^2$ in\n$C^2$. There may also be fractional D3-branes stretching between the\nNS5 and the $(1,k)$ 5-brane at $(w_0,w_1)=(0,0)$.\n\nIn a brane configuration of this type, the Hanany-Witten brane\ncreation effect occurs when one of the 5-branes are moved around the\ncircle $S^1$ keeping the other 5-brane fixed. If there were $N$\ninteger and $l$ fractional branes to start with, moving the 5-brane\nonce around the circle will give rise to a shift\n\\begin{eqnarray} N & \\rightarrow & N + l  \\cr\nl & \\rightarrow & l + 2k \\ . \n\\end{eqnarray}\n\n\\subsection{Stenzel Deformation}\n\\label{sec2.3}\n\nIn this subsection, we will briefly review the IR deformation of the\nStiefel cone.  As an algebraic curve, it amounts to deforming\n(\\ref{curve}) to\n\\begin{equation} z_0^2 + z_1^2+z_2^2+z_3^2 + z_4^2 = \\epsilon^2 \\label{stenzeleq}\\ . \\end{equation}\nThe tip of the cone is blown up by an $S^4$ parametrized by \n\\begin{equation} \\sum_{i=0}^4 ({\\rm Re} z_i)^2 =\\epsilon^2, \\qquad {\\rm Im} z_i = 0 \\end{equation}\nand the full geometry can be viewed as the cotangent bundle over\n$S^4$. This geometry is also known as the Stenzel geometry\n\\cite{Stenzel} and admits an explicit metric \\cite{Cvetic:2000db}. In\nthe notation adopted in \\cite{Klebanov:2010qs}, the metric takes the\nform\n\n\\begin{equation} ds^2 = c^2( dr^2 +  \\nu^2) + a^2 \\sum_{i=1}^3 \\sigma_i^2 + b^2 \\sum_{i=1}^3 \\tilde \\sigma_i^2 \\end{equation}\nwhere\n\\begin{eqnarray} \na^2(r) &=& 3^{-1\/4} \\lambda^2 \\epsilon^{3\/2}  (2 + \\cosh 2r)^{1\/4} \\cosh r, \\cr\nb^2(r) &=& 3^{-1\/4} \\lambda^2 \\epsilon^{3\/2}  (2 +\\cosh 2r)^{1\/4} \\sinh r \\tanh r \\cr\nc^2(r) &=& 3^{3\/4} \\lambda^2 \\epsilon^{3\/2}(2+\\cosh 2r)^{-3\/4} \\cosh^3 r \\end{eqnarray}\nand $\\nu$, $\\sigma_i$, and $\\tilde \\sigma_i$ are left-invariant\none-forms of the coset $SO(5)\/SO(3)$ (for which a nice explicit basis\nappears in \\cite{Klebanov:2010qs}.)\n\n\nAt $r=0$, the geometry collapses to an $S^4$. At large $r$, the\ngeometry asymptotes to a cone over $V_{5,2}$.  Formally, this geometry\nis similar to the deformed $B_8$ space \\cite{Gibbons:1989er} which\ncollapses to an $S^4$ near the tip, and asymptotes to cone over a\nsquashed 7-sphere, but there are a few important differences. One is\nthe fact the $Z_k$ orbifold along the $U(1)_b$ of the Stenzel geometry\nhas fixed points at antipodal points of $S^4$ at $r=0$. We will\ncomment on other differences below.\n\nOne important feature of the Stenzel geometry is that it admits a\nself-dual 4-form which can be written, explicitly, as\n\\begin{eqnarray} G_4 &=& m \\left\\{ {3  \\over \\epsilon^3 \\coth^4{r \\over 2}} \\left[ a^3 c \\nu \\wedge \\sigma_1 \\wedge \\sigma_2 \\wedge \\sigma_3 + {1 \\over 2} b^3 c d r \\wedge \\tilde \\sigma_1 \\wedge \\tilde \\sigma_2 \\wedge \\tilde \\sigma_3 \\right] \\right. \\cr\n&& \\left. + {1 \\over 2 \\epsilon^3 \\coth^4{r \\over 2}} \\left[{1 \\over 2} a^2 b c \\epsilon^{ijk} d r \\wedge \\sigma_i \\wedge \\sigma_j \\wedge \\tilde \\sigma_k + a b^2 c \\epsilon^{ijk} \\nu \\wedge \\sigma_i \\wedge \\tilde \\sigma_j \\wedge \\tilde \\sigma_k \\right]\\right\\} \\ . \n\\label{g4}\n\\end{eqnarray}\n\n\nBecause the four-form is self-dual, and the background geometry is\nCalabi-Yau, one can turn on this flux in eleven-dimensional\nsupergravity without breaking supersymmetry \\cite{Becker:2001pm}.\nMoreover, it gives rise to a solution where the background geometry is\nunmodified except for the presence of a warp factor $H$, as in the\nstandard warped product ansatz\n\\begin{eqnarray} ds^2 &=& H^{-2\/3} (-dt^2+dx_1^2 + dx_2^2) + H^{1\/3} ds_8^2 \\cr\nF_4 & = & dt \\wedge dx_1 \\wedge dx_2 \\wedge d \\tilde H^{-1} + m G_{4} \\ . \\label{ansatz} \\end{eqnarray}\nThe warp factor itself can be determined by solving the four-form field equation,\n\\begin{equation} d * G = {1 \\over 2} G \\wedge G \\ , \\end{equation}\nwhere in general there can be additional source terms due to the\npresence of explicit M2-branes.\n\n\\section{Quantization of fluxes in Stiefel cones and Stenzel space}\n\nLet us now consider the quantization of fluxes in the warped Stiefel\ncones and Stenzel geometries in order to identify the discrete\nparameters characterizing the background. There are two guiding\nprinciples which we follow in carrying out the quantization. One is\nthat quantized fluxes should be invariant under deformation of\nGaussian surfaces unless the discrete unit of charge crosses the\nsurface. The other is for the quantization condition to be invariant\nunder string dualities.\n\n\\subsection{Review of Maxwell, brane, and Page charges}\n\nWe begin by considering the quantization of fluxes for the Stenzel\ngeometry in the IIA description. While the IIA description of the\nStenzel geometry is singular near the core, one still expects Gauss\nlaw considerations to lead to a consistent picture far away from the core\nregion, where the geometry looks essentially like the warped Stiefel\ncone.\n\nThe relevant fluxes to consider then are the flux of $\\tilde F_4$ through\nthe generator of $H_4(M_2,Z)$ and the flux of $*\\tilde F_4$ through\nthe six cycle $M_2$. As we mentioned earlier, however, these fluxes\ndepend on the radius $r$ at which we identify the base $M_2$ for the\nbackground in consideration.\n\nThe resolution to these apparent difficulties is the realization that\none is dealing with a situation where the Maxwell, brane, and Page\ncharge are distinct from one another, and that care is required in\napplying  quantization conditions on the appropriate charge.\n\nLet us recall the specific definition of three charges. In type IIA\nsupergravity, the four form $\\tilde F_4 =d A_3 + H_3 \\wedge A_1$ is\ngauge invariant and well defined but is not closed and does not\nrespect Gauss' law. One can nonetheless compute the period of $\\tilde\nF_4$ on the generator of $H_4(M_2,Z)$ in the $r \\rightarrow \\infty$\nlimit. This defines the Maxwell charge. In contrast, the period of\nPage flux $-(\\tilde F_4 +B_2 \\wedge F_2)$ on $H_4(M_2,Z)$ is\nindependent of $r$, although it is ambiguous with respect to large\ngauge transformation of $B_2$. This quantity defines the Page\ncharge. Finally, the amount of charge carried by a brane source\nthrough its Wess-Zumino couplings defines the brane charge. Brane\ncharge includes the contribution of lower-brane charges from the\npull-back of the NSNS 2-form in the Wess-Zumino coupling. Therefore,\nif the background contains a non-uniform NSNS 2-form $B_2$, the brane\ncharge is not conserved with respect to changes in the position of the\nbranes. Some of these subtleties appeared originally in\n\\cite{Bachas:2000ik}.\n\nThe triplet of charges exists for the other forms, e.g. the six form\n$F_6 = * \\tilde F_4$ and are summarized in appendix B of\n\\cite{Aharony:2009fc}. For the flux of $F_6 =*F_4$, is is also useful\nto introduce the notion of bulk charge $Q_{bulk}$ which is the total\ncharges carried by the bulk fields\n\\begin{equation} Q_2^{bulk} = \\int_{Y_2} {1 \\over 2} G_4 \\wedge G_4 \\ . \\end{equation}\nThen, the bulk charge can be understood as being related to the brane\nand Maxwell charges via\n\\begin{equation} Q_2^{Maxwell} = Q_2^{brane} + Q_2^{bulk} \\ . \\end{equation}\nTo correctly quantize the supergravity solution, one should impose the\ndiscreteness condition on the Page charges, and not on Maxwell, brane,\nor bulk charges.\n\n\\subsection{Quantization on the Stiefel cone}\n\nTo illustrate the integrality of Page charges and the non-integrality\nof the other charges, let us first carryout the quantization procedure\nfor the Stiefel cone.\n\nFirst, consider the flux of $\\tilde F_4$. The Stiefel geometry has\nvanishing fourth Betti number, so there is no $G_4$ to consider in\nM-theory, and after dimensional reduction, the IIA flux $\\tilde F_4$\nalso vanishes.  We are not done yet, however, because we still have to\nconsider the Page flux (\\ref{q4page}), which contains a term $B_2\n\\wedge F_2$, and $F_2$ is nonvanishing in the dimensional reduction of\nthe orbifolded Stiefel cone.  Requiring the Page flux to be integer\nquantized imposes the quantization condition\n\\begin{equation} \\int B_2 = -{l \\over 2k} + {1 \\over 2} \\end{equation}\nwhich we inferred independently from M-theory considerations\nearlier in section \\ref{sec2.1}.\n\nNext, we consider the quantization of flux of D2 charge through\n$M_2$. We are interested in determining the Maxwell charge when the\nPage charge is set to $N$. One finds\n\\begin{equation} Q_2^{Maxwell} = N - {l(l-2k) \\over 2k} \\label{max1}\\end{equation}\nwhich can essentially be viewed as the sum of a contribution from $N$ M2-branes\nand a contribution from the discrete torsion, along the lines of\n\\cite{Bergman:2009zh}. The Maxwell charge $Q_2^{Maxwell}$ has several\nnotable features. First, it is not necessarily integer valued. Second,\nit is invariant under\n\\begin{equation} N \\rightarrow N+l, \\qquad l \\rightarrow l+2k \\ . \\end{equation}\nThis is consistent with the property of Maxwell charge that it is\nconserved under continuous deformations corresponding to moving one of\nthe 5-branes around the $S^1$ in the type IIB brane\nconstruction. Finally, $Q_2^{Maxwell}$ can go to zero or negative for\nsome range of $(N, l, k)$. This suggests that the entropy of the\nsuperconformal field theory is going to zero or negative, signaling a\nphase transition. The condition that $Q_2^{Maxwell}$ is positive is\nalso related to the condition necessary for supersymmetry to be\nunbroken as was highlighted in related contexts in\n\\cite{Aharony:2009fc,Hashimoto:2010bq}.\n\n\\subsection{Quantization in the Stenzel geometry}\n\nLet us now extend our analysis of flux quantization to the case where\nthe Stiefel cone is deformed into the Stenzel geometry, as described\nin Section \\ref{sec2.3}.  To keep a general set of charges under\nconsideration, we will study the case where the Stenzel manifold has\nbeen quotiented by a $Z_k$ orbifold action.\n\nThe most important feature of the geometry in the deep IR is its\nsingularity structure after the orbifold has been taken.  At the tip\nof the deformed orbifolded cone, the geometry has the local structure\n$(R^4\\times S^4)\/Z_k$, and in particular it has two fixed points\nwhich we can think of as the north and south poles of the $S^4\/Z_k$.\nAt each of the fixed points, the local geometry is $R^8\/Z_k$\n\\cite{Martelli:2009ga}.  This geometric feature has a nice\nimplication.  The supersymmetry of the deformed Stenzel cone is\nconsistent with adding some mobile M2-branes, and we are free to move\nsome number of them to either of the orbifold fixed points.  Then the\ntheory on the M2-branes in the deep IR should simply be two copies of\nthe ABJM theory.\n\n\nAt any finite excitation energy the theory should feel the effects of\ncurvature and the self-dual four-form flux in the background which\nbreak the supersymmetry from $\\mathcal{N} = 6$ to $\\mathcal{N}=2$.\nHowever, for issues such as charge quantization, we should be able to\nwork in the extreme low energy limit and use our intuition from the\nABJM case.  In particular one might expect that it is possible to turn\non discrete torsion at each singularity, and we will see that this is\ncorrect, although the torsion will be subject to some global\nconstraints.\n \n\nFirst we will consider the type IIA reduction of this geometry. This\ngeometry develops a dilaton and curvature singularity near the\ntip. However, because the geometry asymptotes to the Stiefel cone away\nfrom the tip, and because quantization of Page fluxes in type IIA\ndescription appropriately respects Gauss law\/localization of charge\nsources, we are able to partially constrain the discrete parameters of\nthe supergravity ansatz. We will then continue to consider the\ngeometry and fluxes near the core region from the M-theory\nperspective, and identify additional constraints which further\nrestrict the parameters of the ansatz.\n\nThe Stenzel manifold admits the self-dual four form flux (\\ref{g4}) which\ncan be derived from a three-form potential $C_3$\n\\cite{Klebanov:2010qs}\n\\begin{equation} C_3 = m \\beta + \\alpha \\Omega_2 \\wedge d \\gamma \\label{3-form}\n\\end{equation}\n\\begin{equation} \\beta = {a c \\over \\epsilon^3 \\cosh^4{r \\over 2}} \\left[ (2 a^2 + b^2) \\tilde \\sigma_1 \\wedge \\tilde \\sigma_2 \\wedge \\tilde \\sigma_3 + {a^2 \\over 2} \\epsilon^{ijk} \\sigma_i \\wedge \\sigma_j \\wedge \\tilde \\sigma_k \\right] \\ , \\label{beta}\\end{equation}\nwhere $\\Omega_2$ and $\\gamma$ are as defined in\n\\cite{Martelli:2009ga}.\\footnote{For the interested reader, $\\gamma$\nis the angular coordinate which is quotiented by the orbifold action,\nand $\\Omega_2$ is proportional to the geometric flux associated with\n$\\gamma$.}  Here we have added an exact term proportional to $\\alpha$,\nwhich does not affect the gauge invariant four-form flux.  This exact\nterm is present in the $AdS_4 \\times V_{5,2}\/Z_k$ system with discrete\ntorsion \\cite{Martelli:2009ga} which is the UV limit of the Stenzel\nsolution.\n\nIn quantizing the flux of the type IIA Page flux through the four\ncycle of $M_2$, we impose the condition\n\\begin{equation} \\int_{S^4} G_4 + n k \\int_{\\tilde S^3\/Z_k} C_3 = (2 \\pi) n \\alpha=\n-(2 \\pi l_p)^3 (l-k), \\qquad n=2 \\label{aquant} \\end{equation}\nwhich constrains $\\alpha$.  Note that in the asymptotically conical limit,\nthe torsion is $Z_{2k}$-valued, and so $l$ takes integer values in the\nrange $0\\le l \\le2k-1$.\n\n\nIn addition to this, the flux of $G_4$ through $S^4\/Z_k$ is\nindependently quantized to be integral.  This implies\n\\begin{equation} \\int_{S^4\/Z_k} G_4 = {8 \\pi^2 \\over 3 \\sqrt{3}k} m = (2 \\pi l_p)^3\nq  \\end{equation}\nfor integer $q$.  This constraint has no counterpart in the Stiefel\ncone as neither the $S^4$ cycle nor the self-dual 4-form exist in that\ncase.\n\n\nNow let us consider the quantization conditions that arise from\nconsidering M-theory near the orbifold fixed points; we will show that\nthe expected charges at the singularities are compatible with the IIA\ncalculations.  At the north pole of $S^4\/Z_k$, the pull-back of $G_4$\non the $R^4\/Z_k$ fiber was computed in \\cite{Martelli:2009ga}:\n\\begin{equation} {1 \\over (2 \\pi l_p)^3} \\int_{R^4\/Z_k} G_4 = {q \\over 2} \\equiv {\\tilde M \\over 2} = M \\end{equation}\nwhere $M$ and $\\tilde M$ are the variables used in\n\\cite{Martelli:2009ga}.  This means that the integral of $C_3$\n(including both the nontrivial flux and the discrete torsion\ncontribution) on $S^3\/Z_k$ at the north pole is\n\\begin{equation} {1 \\over (2 \\pi l_p)^3} \\int_{\\tilde S^3\/Z_k} C_3 = -{l \\over 2k} + {1 \\over 2} -{q \\over 2} \\ \\label{c3north}.\\end{equation}\n\n\nSuppose that at the north pole we impose the condition that the system is described by charges as in the ABJ case with $l^N$ units of discrete torsion (including a shift by 1\/2 a unit as discussed in \\cite{Aharony:2009fc}.)  This is compatible with (\\ref{c3north}) provided that\n\\begin{equation} -{l^N \\over k} + {1 \\over 2}  = -{l \\over 2k} + {1 \\over 2} -{q \\over 2} \\end{equation}\nor equivalently\n\\begin{equation} l^N = {l + kq \\over 2}. \\label{lN}\\end{equation}\n\nAt the south pole, the computation is very similar, except that the\nflux quantum $q$ appears with a minus sign:\n\\begin{equation} l^S = {l - kq \\over 2} \\label{lS}\\end{equation}\nThe difference in the pull-back of $C_3$ between the north and the\nsouth pole is just the total flux $q$, while the discrete torsion\ncontribution must be the same at the north and south poles because the\ntorsion has no associated flux.\\footnote{In the coordinates of\n\\cite{Bergman:2001qi,Klebanov:2010qs}, the $U(1)_b$ quotient as\ndefined in \\cite{Martelli:2009ga} is imposed on the angular coordinate\n$\\phi_2$.  With this choice of $U(1)$ action, the poles of the\n$S^4\/Z_k$ are located at $(\\tau=0,\\alpha = \\pi\/2,\\psi = 0,\\pi)$.  In\nthe vicinity of the poles, one can check that the one-forms\n$\\tilde{\\sigma_i}$ differ by a sign,\n$\\tilde{\\sigma}_i(N)=-\\tilde{\\sigma}_i(S)$, so the three-form $\\beta$\nin (\\ref{beta}) also changes by a sign from the north pole to the\nsouth pole.}\n\nHow should we interpret the formulas (\\ref{lN}) and (\\ref{lS})?  The\nfirst thing to note is that $l^N$ and $l^S$ are equal mod $k$, so if\nthey had described decoupled systems we would have said that they were\nequivalent up to a large gauge transformation.  However, they are not\ndecoupled, and there is no large gauge transformation that sets them\nequal to each other.  Instead, the picture that has emerged is that\n$l^N$ and $l^S$ locally appear to describe the same torsion, but\nglobally there is a topologically nontrivial twist relating them, and\nthe winding number of the twist is just the number of units of $G_4$\nflux in $S^4\/Z_k$.\n\nThe second thing to note is that in the local ABJ models at the north\nand south poles, $l^N$ and $l^S$ should themselves be integers, or in\nother words $l-kq$ must be even.  This means that for a given $q$ and\n$k$, $l$ must take either only even or only odd values.  In the\nundeformed theory, $l$ described a $Z_{2k}$-valued discrete torsion,\nbut we see that our local considerations at the tip of the Stenzel\ngeometry remove half of the possible values of $l$, and the discrete\ntorsion in the deformed case is $Z_k$-valued.  This phenomenon is\nreminiscent of the deformed conifold; the ``singular'' conifold admits\na $Z_2$-valued discrete torsion which is not present in the deformed\nconifold \\cite{Vafa:1994rv}.\n\n\nWe can now examine the quantization of the six form flux though $M_2$\nin IIA or the 7-form flux through $V_{5,2}$ in M-theory which measures\nthe charge of D2\/M2 branes in this background.\n\nOne way to approach this issue is to first examine the brane charges\npresent in this setup. Before adding any explicit 2-branes, there are\n2-brane charges arising from the discrete torsion at the $Z_k$ fixed\npoints at north and south poles \\cite{Bergman:2009zh}. These should\nhave the same form as what was computed in \\cite{Aharony:2009fc}, so\nwe find\n\\begin{equation} Q_2^{torsion} = \\left(-{l^N \\over k} + {1 \\over 2}\\right)+\\left(-{l^S \\over k} + {1 \\over 2}\\right) = -{l(l-2k) \\over 4k}- {k q^2 \\over 4} \\ . \\end{equation}\nIf, in addition, we were to introduce $N$ 2-branes which can be at any\npoint in the Stenzel geometry, there will be an additional\ncontribution of $N$ to the brane charge\n\\begin{equation} Q_2^{brane} = N-{l(l-2k) \\over 4k}- {k q^2 \\over 4} \\ . \\end{equation}\nSince Maxwell charge is the sum of brane charge, and since the bulk\ncharge is given by\n\\begin{equation} Q_2^{bulk} ={1 \\over (2 \\pi l_p)^6} \\int_{{\\cal M}_8} {1 \\over 2} G\\wedge G =  {2^{11} m^2 \\mbox{vol}(V_{5,2}) \\over (2 \\pi l_p)^6 3^6} = {k q^2 \\over 4}\\end{equation}\nwe infer that\n\\begin{equation} Q_2^{Maxwell}=N-{l(l-2k) \\over 4k} \\ .  \\label{max2}\\end{equation}\nIt also follows that the Page charge $Q_2^{Page}=N$.\n\nThis result is gratifying for several reasons. First, this result\nreflects the accounting of all identifiable charge sources in an\notherwise consistent and smooth M-theory background aside from the\norbifold fixed point. The final answer is the same as what we inferred\nfor the undeformed Stiefel cone (\\ref{max1}). It then follows that\nthe gauge invariant Maxwell charge is invariant under the shifts\n\\begin{equation} N \\rightarrow N+l, \\qquad l \\rightarrow l + 2k\\end{equation}\nwhich arises naturally from several perspectives mentioned earlier.\n\nThe only additional constraint imposed by the Stenzel deformation is\nthe restriction on the parity of $l$ so that $l$ is congruent to $kq$\nmod 2. This is far milder than what was found in\n\\cite{Martelli:2009ga}. \n\n\\section{Discussion}\n\nIn this article, we reviewed the quantization of fluxes in warped\nStiefel cone and its Stenzel deformation which is conjectured to be\nthe holographic dual of ${\\cal N}=2$ Chern-Simons matter theory in 2+1\ndimensions.  We described the subtle difference between several\ndifferent yet related notions of charges, and recovered a structure\ncompatible with the pattern of Hanany-Witten brane creation effects\nand duality cascades.\n\nThere are a number of interesting features which one can infer from\nthe structure of the gravity solution. $Q_2^{brane}$ is a measure of\nthe number of degrees of freedom in the deep infrared of this\nsystem. When $Q_2^{brane}$ is zero or negative, we expect the system\nto break supersymmetry and flow to a different universality class of\nvacuum as was the case for many related system\n\\cite{Aharony:2009fc,Hashimoto:2010bq}. It would be very interesting\nto better understand the nature of the effective low energy physics\nwhen the system is in this new phase. This question can be addressed in\nthe simple context of $k=1$ where there are no $Z_k$ orbifold fixed\npoints, and by taking $q$ to be even, we can even set $l=0$ and\ndisregard the contribution from the discrete torsion.\n\nOne way to probe the fate of pushing the system which is slightly\nperturbed into this new phase is to start with a background with $q$\nlarge but $Q_2^{brane}=0$ (which can easily be arranged for $q$ even\nand $k=1$). Consider now adding $p\\ll q$ anti M2-brane as a\nprobe. This setup is very similar to adding anti D3-brane in warped\ndeformed conifolds \\cite{Kachru:2002gs} which has received a lot of\nattention (and controversy) as a possible prototype as a gravity dual\nof a metastable vacua\n\\cite{DeWolfe:2008zy,Bena:2009xk,Bena:2010gs}. For the Stenzel\nmanifold, the effective action of the brane probe undergoing a KPV-like\ntransition \\cite{Kachru:2002gs} works essentially in the same way as\nis illustrated in figure \\ref{figa}. However, from the point of view\nof the bound $Q_2^{brane}>0$, one expects the stable supersymmetric\nminima not to exist when $p$ anti M2-branes are introduced. \n\n\\begin{figure}\n\\centerline{\\begin{tabular}{ccc}\n\\includegraphics[width=1.8in]{kpvV1} & \n\\includegraphics[width=1.8in]{kpvV2} & \n\\includegraphics[width=1.8in]{kpvV3} \\\\\n(a) & (b) & (c)\n\\end{tabular}}\n\\caption{Potential $V(\\psi)$ for $p$ anti D3-brane blowing up to an\nNS5-brane wrapping an $S^2$ of fixed latitude in $\\psi$ in $S^3$ at\nthe tip of the Klebanov-Strassler solution. (a), (b), and (c)\ncorresponds to $p\/M=0.03$, $p\/M=0.08$, and $p\/M=-0.03$, respectively.\nThese figures originally appeared in figure 2 of \\cite{Kachru:2002gs}.\n\\label{figa}}\n\\end{figure}\n\nTentatively, we interpret these facts as follows.  The computation of\nthe potential $V(\\psi)$ neglected the backreaction of the anti-branes,\nand when the number of anti-branes is parametrically small ($p \\ll q$)\nthis probe approximation is valid.  In particular, the existence of\nthe non-BPS local minimum in \\ref{figa}.(a) is a robust prediction in\nthis limit.  However, when the state in the metastable false vacuum\nillustrated in figure \\ref{figa}.(a) tunnels to the putative ``true''\nvacuum, the amount of charge carried by the probe grows to $q-p$ which\nis not parametrically small compared to $q$.  The backreaction due to\nthis charge can be significant, and so the computation of the\ntunneling potential is (at least) not obviously self-consistent.  It\nis tempting to speculate that the supersymmetric vacuum might actually\nbe spurious and that the non-BPS local minimum is the global minimum\nwhich characterizes the dynamics in the $Q_2^{brane}<0$ phase of these\ntheories up to corrections suppressed by $p\/q$.\n\nSimilar considerations apply to the BPS domain wall one constructs for $p\n< 0$ for which the KPV potential has the form illustrated in figure\n\\ref{figa}.(c). This domain wall can also be viewed as arising from\nwrapping an M5-brane on $S^4$ at the tip of the Stenzel manifold. A\n5-brane wrapped on a 4-cycle is effectively a string, and in 2+1\ndimensions, a string forms a domain wall. It would be very interesting\nto understand the nature of vacua separated by these domain\nwalls. Since M5-brane wrapped on $S^4$ with $q$ units of flux must\nhave $q$ additional M2-branes ending on it to cancel the anomaly, some\nquantum numbers of the vacuum must shift to reflect this. Nonetheless,\none expects the Maxwell charges $Q_2^{Maxwell}$ and $Q_4^{Maxwell}$ to\nbe invariant as one crosses the domain wall, as these charges are\nconserved. Making complete sense of these expectations requires taking\nthe full back reaction of the M5-brane and the $q$ anomalous M2-branes\ninto account. Unfortunately, $q$ M2-branes can not be treated reliably\nas a probe, making systematic analysis of these issues a challenge.\n\nLet us also mention that similar issues of stable\/metastable non-BPS\nvacua, domain wall, and low energy effective field theories can be\ndiscussed in the closely related $B_8$ system building on the analysis\nof \\cite{Hashimoto:2010bq} and \\cite{Gukov:2001hf}. Quantization of\ncharges and the enumeration of brane, Maxwell, and Page charges for\nthis system was carried out in \\cite{Hashimoto:2010bq}. Here, however,\nwe encounter one additional puzzle. It was argued in\n\\cite{Gukov:2001hf} that the 4-form flux through $S^4$ at the tip of\nthe $B_8$ cone is half integral as a result of the shift originally\ndue to Witten \\cite{Witten:1996md}. This would appear to require half\ninteger units of M2-branes to end on the domain wall made by wrapping\nthe M5 on the $S^4$. Of course, the number of M2's ending on an M5 is\nconstrained to be an integer.  Perhaps this is indicating that odd\nnumber of M5-branes are forbidden from wrapping the\n$S^4$. Alternatively, this paradox is another manifestation of not\nsystematically taking the back reaction of the domain wall into\naccount.\n\n\nFinally, let us emphasize that for the time being, the concrete field\ntheory interpretation of the Stenzel deformation and the quantum\nnumber $q$ is not known.  The gravity dual suggests that the parameter\n$q$ is important for both the IR and the UV physics.  At large radius,\n$q$ is related to the total number of units of M2-brane charge\ngenerated by the cascade, which in turn affects the UV gauge symmetry.\nNear the tip of the Stenzel geometry, the $G_4$ flux is nonvanishing\nso $q$ should also appear in the data of the IR field theory.  Of\ncourse, $q$ can only be nonzero when the geometry is deformed.\nMartelli and Sparks conjectured that this deformation was related to\nturning on a particular mass term on the field theory side.  One can\nindeed see that the null geodesic can travel from boundary at infinity\nto the core in finite field theory time, and so the spectrum of\nglueball-like states will exhibit a discrete structre whose scale is\nset by the deformation.  If this conjecture is correct, it would\nsuggest that the field theory confines because of a mass deformation\n(reminiscent of the $\\mathcal{N}=1^*$ theory in\n$d=4$\\cite{Polchinski:2000uf} and the mass deformed ABJM theory\n\\cite{Lin:2004nb,Gomis:2008vc,Kim:2010mr,Cheon:2011gv}) rather than as\na dynamical effect, as is the case in the Klebanov-Strassler system\n\\cite{Klebanov:2000hb}. It should be very interesting to understand\nthis theory better.\n\n\n\n\n\n\\section*{Acknowledgements}\n\nWe would like to thank  Ofer Aharony and Shinji Hirano for collaboration on\nrelated issues and for discussions at the early stage of this work.\nWe also thank \nIgor Klebanov,\nDon Marolf,\nDario Martelli, and\nJames Sparks\nfor useful comments and discussions. The work of AH is\nsupported in part by the DOE grant DE-FG02-95ER40896 and PO is supported in part by DOE grant\nDE-FG02-91ER40681.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\subsection{General results}\n\\label{sec:4D-general}\n\nThis section will follow the same logic as Secs.~\\ref{sec:Z2-general} and~\\ref{sec:ON-general} devoted to 3d CFTs. Historically however the very first attempt to study crossing relations using numerical techniques focused on 4d CFTs. This analysis, pioneered in \\textcite{Rattazzi:2008pe} and then refined in \\textcite{Rychkov:2009ij}, was spurred by high energy physics motivations which will be reviewed in Sec.~\\ref{sec:4D-bsm}. But first let us discuss general conformal bootstrap results for 4d CFTs with various global symmetries.\n\nConsider first the simple case of a 4d CFT containing a scalar operator $\\phi$ with dimension $\\Delta_\\phi$. We further assume that it is charged under a global symmetry (e.g., a $\\mathbb{Z}_2$ symmetry) so that the OPE $\\phi\\times\\phi$ does not contain $\\phi$. Then it is interesting to ask how high can one push the dimension of the first scalar operator in this OPE. It is also interesting to ask how large the OPE coefficient of the stress tensor $\\lambda_{\\phi\\phi T}\\propto \\Delta_\\phi\/\\sqrt{C_T}$ is allowed to be~\\cite{Poland:2010wg,Rattazzi:2010gj}, which translates into a lower bound on the central charge $C_T$. The best bounds to date were computed in \\textcite{Poland:2011ey} and are shown in Fig.~\\ref{fig:4D-singlecorr}.\\footnote{The bound on $C_T$ can be somewhat strengthened by incorporating the assumption that $\\phi$ is the lowest dimension scalar, as in \\textcite{Rattazzi:2010gj}.} When $\\Delta_\\phi$ approaches the unitarity bound, both bounds approach the free theory value for $\\Delta_{\\phi^2}$ and $C_T$. This is consistent with the fact that a scalar with dimension $(d-2)\/2$ satisfies $\\del^2\\phi=0$ whenever inserted in a correlation function, and must therefore be a free scalar.\n\n\\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig46a-simplescalar.pdf}(a)\n\\includegraphics[width=\\figwidth]{fig46b-ctscalar.pdf}(b)\n    \\caption{\\label{fig:4D-singlecorr} (Color online) (a) Upper bound on the dimension of the first scalar in the $\\phi\\times \\phi$ OPE as a function of $\\Delta_\\phi$ in 4d unitary CFTs; (b) lower bound on the central charge $C_T$, computed by maximizing the OPE coefficient $\\lambda_{\\phi\\phi T}$~\\cite{Poland:2011ey}.\n}\n  \\end{figure}\n\nAnalogous bounds have been obtained for CFTs assuming various continuous global symmetries. \\textcite{Poland:2011ey} studied the 4pt functions of a scalar $\\phi$ transforming in the fundamental representation of $SO(N)$ or $SU(N)$, deriving an upper bound on the dimension of the lowest singlet scalar in the OPEs {$\\phi_i \\times \\phi_j$} (or  {$\\phi^{\\dagger i}\\times \\phi_j$} in the case of $SU(N)$), as well as a lower bound on the central charge, shown in Fig.~\\ref{fig:4D-SON}.\\footnote{Numerics indicate that $SU(N)$ and $SO(2N)$ singlet and central charge bounds coincide~\\cite{Poland:2011ey}. A priori, because $SU(N)\\subset SO(2N)$, and because only singlets give rise to singlets when representations are reduced, these $SO(2N)$ bounds must be at least as strong as for $SU(N)$, but the exact coincidence is unexpected and remains unexplained.} As expected, the bounds scale with $N$, the size of the fundamental representation, at least when the dimension of the external scalar approaches the free value. It should be possible to extend this analysis to obtain upper bounds on the dimensions of operators transforming in other representations. For scalars in the symmetric traceless representation of $SO(4)$ this was done in \\textcite{Poland:2011ey}.\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\figwidth]{fig47a-scalarSUN.pdf}(a)\n\\includegraphics[width=\\figwidth]{fig47b-CT-SON.pdf}(b)\n    \\caption{\\label{fig:4D-SON} (Color online) (a) Upper bounds on the singlet scalar dimension in $SO(N)$ and $SU(N)$ symmetric 4d CFTs, as a function of $\\Delta_\\phi$ in the fundamental; (b) lower bounds on $C_T$ in the same theories \\cite{Poland:2011ey}.}\n  \\end{figure}\n \nIt is also possible to place upper bounds on the OPE coefficients of conserved vectors {of dimension 3} in the OPE of $\\phi$ with its conjugate. This class of operators includes the conserved currents of the considered global symmetry $G=SO(N)$ or $SU(N)$, transforming in the adjoint representation of $G$. Upper bounds on their OPE coefficients translate into the lower bounds on the central charges $C_J$. These bounds are shown in Fig.~\\ref{fig:4D-CJ-SON}. Once again the $SU(N\/2)$ bounds coincide with $SO(N)$ ones~\\cite{Caracciolo:2014cxa}. For $\\Delta_\\phi$ close to the free value, these bounds smoothly approach the free $SO(N)$ value.\n\nIn addition, for $G=SU(N)$ the OPE {$\\phi^{\\dagger i}\\times\\phi_j$} may also contain conserved currents of some other global symmetry which may exist in the theory, which are singlets under $G$. The lower bound on their inverse-square OPE coefficient is given in Fig.~\\ref{fig:4D-CJ-singlet-SUN}. Close to the free theory dimension, these bounds approach the value corresponding to the theory of $N$ massless complex scalar fields, whose full symmetry $SO(2N)$ is indeed larger that $SU(N)$. \n\nAdditionally,~\\textcite{Caracciolo:2014cxa} derived lower bounds on $C_J$ in the presence of a gap in the scalar singlet sector, as well as for extended global symmetries $SO(N)\\times SO(M)$ and $SO(N)\\times SU(M)$.\n\n  \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig48-CJboundsSON.pdf}\n    \\caption{\\label{fig:4D-CJ-SON} (Color online) Lower bound on $C_J$ in $SO(N)$-symmetric unitary 4d CFTs as a function of the dimension of a scalar in the fundamental~\\cite{Poland:2011ey}. $SU(N\/2)$ adjoint currents satisfy the same bound~\\cite{Caracciolo:2014cxa}.}\n  \\end{figure}\n\n\\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig49-CJ-singlet-SUN.pdf}\n    \\caption{\\label{fig:4D-CJ-singlet-SUN}\n    (Color online) Lower bound on the inverse square OPE coefficient of a singlet current in $SU(N)$-symmetric unitary 4d CFTs as a function of dimension of a scalar in the fundamental~\\cite{Poland:2011ey}.}\n  \\end{figure}\n  \nUnlike in 3d, most of the 4d bounds computed so far do not display any prominent kink or other dramatic feature, suggesting that  existing 4d CFTs may lie inside the allowed regions and not on the boundary. Note however that some unexplained features are visible in the $C_J$ lower bounds in Fig.~\\ref{fig:4D-CJ-SON}, as well as in the bounds on supersymmetric CFTs discussed in Sec.~\\ref{sec:4Dsusy}. \n\nThe bounds discussed in this section have been obtained by studying a 4pt function {$\\langle \\phi_i\\phi_j\\phi_k\\phi_l\\rangle$ or $\\langle \\phi_i\\phi^{\\dagger j}\\phi_k\\phi^{\\dagger l}\\rangle$}, where {$\\phi_i$} is a single primary operator or a global symmetry multiplet of primary operators. As far as we are aware, a systematic study of numerical bootstrap constraints from mixed correlators in 4d CFTs has not yet been performed outside of the supersymmetric context~\\cite{Lemos:2015awa,Li:2017ddj}. It will be important to do so in the future, and to study the impact on such bounds of assuming only a limited set of relevant operators. \n  \n\\subsection{Applications to the hierarchy problem}\n\\label{sec:4D-bsm}\nNext we will review some bootstrap results which shed light on the attempts to alleviate the hierarchy problem of the Standard Model (SM) of particle physics, which historically was one of the motivations for the development of the numerical bootstrap in 4d.\n\nFor the purposes of our discussion, the hierarchy problem can be briefly summarized as follows. The SM is certainly not the complete description of fundamental interactions, as it doesn't account for dark matter, baryogenesis, neutrino masses, and gravity. Instead it can be regarded as an effective description, valid at least up to the electroweak scale, where it has been extensively tested, including at the ongoing Large Hadron Collider (LHC) experiments. According to the effective field theory paradigm, the leading effects in this description are captured by the relevant and marginal operators, while all higher-dimensional operators correspond to subleading effects and are suppressed by powers of the electroweak scale ($\\Lambda_{\\text{IR}}\\sim $ {100 GeV}) over the scale of new physics ($\\Lambda_{\\text{UV}}$). The incredible success of the SM in precisely describing all phenomena observed so far is elegantly explained by simply pushing the scale of new physics to high values. In particular,  electroweak precision tests and more importantly bounds from flavor physics (in particular from $K$-$\\bar K$ mixing) generically require $\\Lambda_{\\text{UV}}\\gtrsim 10^{5}\\text{ TeV}$. \n\nThis simple assumption creates however a tension (called the hierarchy problem) with the other energy scale in the theory, namely the scale associated with the only relevant operator present in the SM---the Higgs mass term $H^\\dagger H$. Indeed, whenever a relevant deformation exists, it is generically expected to be generated at the fundamental scale with order one strength, unless some symmetry prevents this from happening. The contrary is usually considered an unnatural tuning of the model, similarly to how, in condensed matter systems, one typically needs to adjust a control parameter to approach a critical point.\n\nThe quest for a solution to the hierarchy problem \nhas been and remains an important goal in theoretical high energy physics. Strategies for solving it can be broadly divided in two categories: the first makes use of an additional symmetry that prevents the Higgs mass term from appearing, and then slightly breaks it in order to generate a scale parametrically smaller than $\\Lambda_{\\text{UV}}$. The second strategy instead removes altogether the dangerous relevant deformation by increasing the scaling dimension of the Higgs mass term. An example of the first strategy is low-energy supersymmetry, while the second one is realized in technicolor, which replaces the Higgs field with a fermion bilinear operator, of scaling dimension close to three. \n\nWhile technicolor solves the hierarchy problem by making the Higgs mass term irrelevant, it also raises the SM Yukawa operator dimensions from 4 to 6. To generate heavy quark masses of needed size, these operators need to originate at an energy scale not much above $\\Lambda_{\\text{IR}}$. This leads to a tension with flavor observables, due to four-fermion operators expected to originate at about the same scale unless yet additional structure is added. To elegantly solve this problem, \\textcite{Luty:2004ye} proposed the ``conformal technicolor\" scenario, in which the Higgs field has a scaling dimension close to the free value, while the Higgs mass term is close to marginality or irrelevant. \n\nMore precisely, to realize this scenario one would need a unitary CFT which contains a scalar operator $H$ replacing the SM Higgs field. To preserve the SM custodial symmetry, the CFT must have an $SO(4)$ global symmetry, with $H$ transforming in the fundamental. The scaling dimension requirements are as follows: $\\Delta_H$ has to be close to 1, while $\\Delta_{S}\\gtrsim 4$, where $S$ is the first scalar $SO(4)$-singlet operator in the OPE $H^\\dagger \\times H$, playing the role of the Higgs mass term in this setup. Given the scaling dimension requirements, this hypothetical CFT must necessarily be strongly coupled, while its coupling to the rest of the SM (gauge fields and fermions) can be treated as a small perturbation. \n\nThe 4d numerical bootstrap grew out from the attempts to show that the most optimistic requirements $\\Delta_H\\to 1$, $\\Delta_S>4$ are impossible to realize. A proof of this theorem about unitary 4d CFTs is visible in the upper bound on $\\Delta_S$ provided by the $N=4$ curve in Fig.~\\ref{fig:4D-SON}, which approaches 2 for $\\Delta_H\\to 1$. \n\nIt is phenomenologically acceptable to have $\\Delta_H$ slightly deviate from 1 without violating flavor constraints, and to allow $\\Delta_S$ somewhat below 4 at the price of some moderate tuning~\\cite{Luty:2004ye,Rattazzi:2008pe,Rychkov-talk}. Although this freedom helps to alleviate bootstrap constraints, some tension remains. Fig.~\\ref{fig:4D-ConformalTechnicolor} from \\textcite{Poland:2011ey} shows the regions of $\\{\\Delta_H,\\Delta_S\\}$ allowed under different degrees of tuning and different assumptions about the structure of the flavor sector. The conclusion is that compatibility with the bootstrap bound can be achieved only under optimistic flavor assumptions and with a moderate tuning.\n\nAn additional phenomenological constraint on conformal technicolor comes from the existence of the Higgs boson particle. While a SM-like Higgs boson may appear in conformal technicolor as a resonance of the strong dynamics at the electroweak scale associated with breaking of conformal invariance~\\cite{Luty:2004ye}, it is expected to be somewhat heavier than the experimentally observed value 125 GeV, and to have some deviations in its coupling to the top quark, which were not seen so far. This further reduces the likelihood that the conformal technicolor scenario is realized in nature. Still, the above analysis, performed prior to the Higgs boson discovery, remains a beautiful example of how theoretical investigations can lead to first-principles constraints on strongly coupled scenarios for particle physics beyond the SM.\n\n\n  \\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\figwidth]{fig50-ConformalTechnicolor.pdf}\n    \\caption{\\label{fig:4D-ConformalTechnicolor}\n    (Color online) Viable regions in the $\\{\\Delta_H,\\Delta_S\\}$ plane for conformal technicolor models in the flavor-generic (red) and flavor-optimistic (cross-hatched green) cases, superimposed with the $SO(4)$ bound. Regions for no tuning, $10\\%$, and $1\\%$ tuning are shown in successively lighter shades of each color, with the largest region corresponding to  $1\\%$ tuning in each case. Flavor-generic models are ruled out~\\cite{Poland:2011ey}.}\n  \\end{figure}\n  \n\n\\subsection{Theories with 4d $\\mathcal{N}=1$ supersymmetry}\n\nBefore discussing the numerical bootstrap results, let us spend a few words on the structure of representations of the superconformal algebra. For concreteness we will give this discussion for SCFTs with 4d $\\mathcal{N}=1$ supersymmetry.\\footnote{For similar results in other dimensions or with extended supersymmetry see \\textcite{Minwalla:1997ka} and the summary of recent progress in Sec.~\\ref{sec:other}. Many results described here can also be treated in a uniform way across dimensions for algebras with the same number of supercharges, see \\textcite{Bobev:2015jxa}.} Superconformal primary operators (annihilated by the special superconformal generators $\\cal S, \\bar{\\cal S}$) are labelled by four numbers $(q,\\bar{q},\\ell,\\bar{\\ell})$, where $\\ell,\\bar{\\ell}$ are the usual Lorentz quantum numbers and $q,\\bar{q}$ are related to the scaling dimension $\\Delta$ and $R-$charge of the superconformal primary operator:\n\\begin{equation}\n\\Delta = q+\\bar{q} \\qquad R = \\frac23(q-\\bar{q})\\,.\n\\end{equation}\nUnitarity bounds on these operators were worked out by~\\textcite{Flato:1983te} and~\\textcite{Dobrev:1985qv}, taking the form\n\\begin{align}\n&q \\geqslant \\frac12 \\ell+1\\,,\\, \\bar{q}\\geqslant \\frac12 \\bar{\\ell}+1 &(\\ell\\bar{\\ell}\\neq0)\\,,\\nonumber\\\\\n&q \\geqslant \\frac12 \\ell+1  &(\\bar{q} = \\bar{\\ell} =0)  \\,,\\\\\n&\\bar{q} \\geqslant \\frac12 \\bar{\\ell}+1  &(q = \\ell =0)  \\,.\\nonumber\n\\end{align}\nThe second and third lines in the above expression identify chiral ($\\Phi_{\\alpha_1....\\alpha_\\ell}$) or antichiral ($\\bar{\\Phi}_{\\dot{\\alpha}_1....\\dot{\\alpha}_{\\bar{\\ell}}}$) operators, which are annihilated by the supercharge $\\bar{\\mathcal{Q}}$ or $\\mathcal{Q}$, respectively.\n\nFinally, we would like to mention a few theoretical results for superconformal blocks present in the literature, focusing on those relevant for the 4d $\\mathcal{N}=1$ bootstrap. Superconformal blocks for correlation function of scalar superconformal primaries can be expressed in terms of finite linear combinations of ordinary scalar conformal blocks with suitable dimensions and spin; however, computing these coefficients can be a challenging task. The work of \\textcite{Poland:2010wg,Vichi:2011ux} obtained the superconformal blocks for 4pt functions of a scalar chiral supermultiplet $\\Phi$. Shortly after, \\textcite{Fortin:2011nq} computed superconformal blocks for 4pt functions of the multiplet associated to global symmetry conserved currents, whose lowest component is again a scalar field.\\footnote{Some incorrect coefficients and missing blocks were later pointed out in \\textcite{Berkooz:2014yda,Khandker:2014mpa}.} A similar analysis applicable to 4pt functions of $R$-current multiplets (containing the stress tensor) was also recently carried out in \\textcite{Manenti:2018xns}. The general approach in these works was to classify the possible 3-point functions in superspace using the formalism of \\textcite{Osborn:1998qu} and then expand in the Grassmann variables $\\theta_i$ to compute relations between OPE coefficients of conformal primaries. In this approach one must also carefully compute the norm of each conformal primary in the multiplet.\\footnote{Such norms were worked out for general multiplets in \\textcite{Li:2014gpa}.} \n\nThe work of \\textcite{Fitzpatrick:2014oza} developed alternate techniques based on either solving the super-Casimir equation or writing the blocks as superconformal integrals using a super-embedding formalism. The latter approach was employed in \\textcite{Khandker:2014mpa} to find the blocks appearing in the more general correlation function $\\langle\\Phi_1\\bar{\\Phi}_2 \\Psi_1\\bar{\\Psi}_2\\rangle$, where $\\Phi_i$ and $\\Psi_i$ are scalar superconformal primary operators with arbitrary dimension and $R$-charge,\\footnote{The first and second pair have the same conformal weights $q,\\bar q$, hence the notation.} with the restriction that the exchanged operator is neutral under $R$-symmetry. This analysis was later extended in \\textcite{Li:2016chh} to the more general case of four distinct scalar superconformal primary operators with arbitrary scaling dimensions and $R$-charges, with no restriction on the exchanged operators besides those imposed by superconformal symmetry. However, this analysis was missing a particular class of superconformal blocks, associated to exchanged primaries in representations of the Lorentz group with $\\ell\\neq\\bar\\ell$. In this case the corresponding superconformal primary does not enter the OPE of the external operators, but some of its superconformal descendants do. This issue was fixed in \\textcite{Li:2017ddj}.\n\n\n\\subsubsection{Bounds without global symmetries}\n\nNow we will summarize numerical results for correlation functions involving scalar chiral superfields. The first numerical studies, starting with \\textcite{Poland:2010wg} and improved in \\textcite{Vichi:2011ux} and \\textcite{Poland:2011ey}, focused on 4pt functions containing a single scalar chiral supermultiplet $\\langle\\Phi\\bar{\\Phi}\\Phi\\bar{\\Phi}\\rangle$. Crossing symmetry for this correlation function involves two OPE channels. Because of the chirality conditions, the $\\Phi\\times\\bar{\\Phi}$ OPE only receives contributions from traceless symmetric tensor superconformal primaries together with their $\\mathcal Q\\bar{\\mathcal{Q}}$ and  $\\mathcal Q^2 \\bar{\\mathcal{Q}}^2$ superdescendants, giving rise to the superconformal blocks described above.  The $\\Phi\\times\\Phi$ OPE on the other hand is more subtle and can receive three different contributions: 1) the chiral superfield $\\Phi^2$; 2) $\\bar{\\mathcal{Q}}$ descendants of semi-short multiplets; 3) $\\bar{\\mathcal{Q}}^2$ descendants of generic (long) multiplets. As a result, this channel allows conformal blocks of even spin $\\ell = \\bar{\\ell}$ at either the protected dimensions $\\Delta=2\\Delta_\\Phi+\\ell$ or at unprotected dimensions satisfying the unitarity bound $\\Delta\\geqslant|2\\Delta_\\Phi-3|+\\ell+3$.\n\n\\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig53a-dim_phibphi.pdf} (a)\n\\includegraphics[width=\\figwidth]{fig53b-phi2_ope_coef.pdf} (b)\n    \\caption{\n    \\label{fig:4chiral_longscalar} (Color online) (a) Upper bound on the dimension of the operator $\\mathcal R$ as a function of $\\Delta_\\Phi$~\\cite{Poland:2011ey,Li:2017ddj}.  The shaded area is excluded. The dashed line at $\\Delta_{\\mathcal R}=2 \\Delta_\\Phi$ corresponds to generalized free theories. (b) Lower and upper bounds on the OPE coefficient of the chiral operator $\\Phi^2$ entering the $\\Phi\\times\\Phi$ OPE. The vertical dotted line is at $\\Delta_\\Phi = 1.407$ and the horizontal dashed line is at the free theory value $\\lambda_{\\Phi\\Phi\\Phi^2}\\equiv\\lambda_\\phi^2=\\sqrt{2}$~\\cite{Poland:2015mta}.\n}\n  \\end{figure}\n\n  \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig54a-cc_assum.pdf} (a)\n\\includegraphics[width=\\figwidth]{fig54b-cT_upper_lower.pdf} (b)\n    \\caption{\n    \\label{fig:4chiral_ct} \n    (Color online) (a) Lower bound on the central charge as a function of $\\Delta_\\Phi$ assuming that $\\Delta_{\\mathcal R}$ is consistent with the unitarity bound (thin line) or it saturates the upper bound in Fig.~\\ref{fig:4chiral_longscalar} (thick line). The shaded area is excluded~\\cite{Li:2017ddj}. (b) Lower and upper bounds on the central charge as a function of $\\Delta_\\Phi$, with the assumption that there is no $\\Phi^2$ operator. The upper bounds correspond to different gaps until the second spin-1 superconformal primary $\\Delta_{\\ell=1}\\geqslant  3.1, 3.3, 3.5, 3.7, 3.9, 4, 4.1$ (from left to right). The shaded area is excluded~\\cite{Poland:2015mta}.\n}\n  \\end{figure}\n\nIn Fig.~\\ref{fig:4chiral_longscalar} we show an upper bound on the dimension of the first real scalar supermultiplet $\\mathcal R$ entering the OPE $\\Phi\\times\\bar{\\Phi}$. A first important consequence of this result is that in any perturbative SCFT, the combination $2\\Delta_\\Phi -\\Delta_{\\mathcal R}$ must be positive (or very suppressed) to satisfy the bound. Secondly, one can observe  a minor kink-like feature on the boundary of the allowed region. The same feature appears in the lower bound on the central charge, Fig~\\ref{fig:4chiral_ct}, and it also coincides with the minimal value of $\\Delta_\\Phi$ consistent with the absence of the chiral operator $\\Phi^2$, as shown in the bottom panel of Fig~\\ref{fig:4chiral_longscalar}.  \n  \n  In light of this, it is tempting to conjecture the existence of a ``minimal SCFT\" that realizes the chiral ring relation $\\Phi^2=0$ and saturates these bounds. This conjecture has been seriously addressed by~\\textcite{Poland:2015mta} and \\textcite{Li:2017ddj}, who studied the properties of this hypothetical theory. Notice that the minimal value of $\\Delta_\\Phi$ consistent with the chiral ring assumption, let us call it $\\Delta_\\Phi^\\text{mSCFT}$, represents an extremal solution, and it is therefore uniquely determined. In addition, to coincide with the kinks it should agree with the solution obtained from maximizing the dimension of the first neutral unprotected operator and the solution obtained from minimizing the central charge, at the same value of $\\Delta_\\Phi$. \n  \nFig~\\ref{fig:4chiral_ct} (a) shows that the two extremization procedures generically lead to two different solutions, except at $\\Delta_\\Phi^\\text{mSCFT}$. This confirms our expectation of a unique solution coinciding with the kinks. Furthermore, by inputting a gap between the stress-tensor multiplet (whose lowest component is the spin-1 $U(1)_R$ current) and the next spin-1 supermultiplet, one is able to extract an upper bound on the central charge. While this bound is gap-dependent at generic $\\Delta_\\Phi$, it almost coincides with the lower bound at $\\Delta_\\Phi^\\text{mSCFT}$, as shown in Fig~\\ref{fig:4chiral_ct} (b). By extrapolating these results at large $\\Lambda$,~\\textcite{Poland:2015mta} obtained the prediction \\mbox{$\\Delta_\\Phi^\\text{mSCFT}  \\approx 1.428,\\, c \\approx 0.111$}, perhaps consistent with \\mbox{$\\Delta_\\Phi^\\text{mSCFT} = 10\/7,\\, c = 1\/9$}.\n\nRecently a few theories have been proposed as mSCFT candidates \\cite{Xie:2016hny,Buican:2016hnq}, which implement the chiral ring condition $\\Phi^2=0$; however, they don't quite match the bootstrap predictions presented here. In particular the central charge is much larger than $1\/9$.\n\nIt is also worth noticing that, in any solution saturating the dimension bound of Fig.~\\ref{fig:4chiral_longscalar}, the chiral operator $\\Phi$ is not charged under any global symmetry. If it was, in fact, the solution would contain a spin-$1$ conserved current, which in $\\mathcal N=1$ SUSY happens to be the superdescendant of a dimension-2 real scalar which would appear in the $\\Phi\\times\\bar{\\Phi}$ OPE. \n\nTo conclude this section, let us mention that the work of \\textcite{Li:2017ddj} also performed a bootstrap study of a system of mixed correlators involving a scalar chiral superfield $\\Phi$ and a long real scalar superfied $\\mathcal R$, identified with the first scalar operator appearing in the $\\Phi\\times\\bar{\\Phi}$ OPE. Unlike in 3d, this analysis didn't seem to allow one to easily isolate a closed region. A preliminary inspection of the extremal solution doesn't reveal any obvious low-lying operators decoupling from the spectrum, but rather it involves a rearrangement of higher dimensional operators~\\cite{Stergiou2017}. It will be interesting to study this rearrangement further and understand how to robustly isolate the conjectured mSCFT in future work. \n  \n\\subsubsection{Bounds with global symmetries}\n\nAs mentioned in the previous section, conserved currents of global symmetries $j_\\mu^a $ sit in real supermultiplets $\\mathcal J^a$ whose lowest component is a dimension-2 scalar $J^a$. In addition, the multiplet satisfies the conservation condition $D^2 \\mathcal J^a =\\bar{D}^2 \\mathcal J^a =0$. Bootstrapping correlation functions of the scalars $J^a$ allows one access to the space of local SCFTs with a given global symmetry. Hence, due to supersymmetry, one can apply the same machinery encountered so far, with no need to deal with spinning conformal blocks. \n\nBounds on OPE coefficients of $SU(N)$ currents were explored in \\textcite{Berkooz:2014yda} and dimension bounds (and coefficient bounds assuming gaps) from single 4pt functions $\\<JJJJ\\>$ were explored in \\textcite{Li:2017ddj}. The latter work also studied the case of mixed correlators involving $J$ and a chiral field $\\Phi$ charged under the global symmetry. Note that this charge necessarily differs from the $R$-symmetry, which instead is part of the conformal algebra: the conserved current associated with the latter is the lowest component of the Ferrara-Zumino multiplet which contains the stress-tensor and supercurrents.\n\nA key result, shown in Fig.~\\ref{fig:JJJJ_ct}, shows that any local SCFT with a continuous global symmetry must contain a real scalar multiplet $\\mathcal O$ with dimension $\\Delta_\\mathcal{O} \\leqslant 5.246$.  The same figure also shows upper bounds on the OPE coefficient associated to $J$ itself as well as the one associated to the stress tensor multiplet, denoted as $V$. Interestingly, both bounds on $c_J$ and $c_V$ show plateaus for small values of the gap in the scalar sector. These are perhaps consistent with the existence of SCFT solutions shaping the bounds. On the other hand, the values extracted from Fig.~\\ref{fig:JJJJ_ct} are much smaller then the limits one obtains by inspecting the correlation functions of chiral superfields. For instance, the relation between $c_V$ and Fig.~\\ref{fig:4chiral_ct} is $c_V^2=1\/(90c)$, making the bound on the central charge very weak.\\footnote{The OPE coefficient bounds obtained in \\textcite{Berkooz:2014yda} for $SU(N)$ current 4pt functions were also relatively weak.}\n\n  \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig55a-cJ_from_JJJJ.pdf} (a)\n\\includegraphics[width=\\figwidth]{fig55b-cT_from_JJJJ.pdf} (b)\n    \\caption{\n    \\label{fig:JJJJ_ct} \n    (Color online) Upper bounds on the OPE coefficients $c_{\\mathcal{O}} \\equiv 2^{-\\ell\/2} \\lambda_{JJ\\mathcal{O}}$ appearing in the $J\\times J$ OPE arising from (a) $J$ itself or (b) the stress-tensor supermultiplet $V$, as a function of the dimension of the first unprotected scalar $\\mathcal O$ in the $J\\times J$ OPE. The region to the right of the dotted vertical line at $\\Delta_\\mathcal{O} = 5.246$ is not allowed~\\cite{Li:2017ddj}.\n}\n  \\end{figure}\n\nAn alternative method to study SCFTs with global symmetries is to consider external scalar operators in nontrivial representations of the symmetry. An important target is to make contact with supersymmetric QCD theories, e.g.~supersymmetric gauge theories with gauge group $SU(N_c)$ and $N_f$ flavors of quarks \\mbox{$Q_i, \\overline{Q}^{\\bar{j}}$}, with $N_f$ in the conformal window \\mbox{$3N_c\/2 \\leqslant N_f \\leqslant 3N_c$}~\\cite{Seiberg:1994pq}. The simplest gauge-invariant operators are the mesons $M_i^{\\bar j}=Q_i \\overline{Q}^{\\bar{j}}$, which transform as bi-fundamentals of \\mbox{$SU(N_f)_L\\times SU(N_f)_R$} and have dimension \\mbox{$\\Delta_M = 3(1-N_c\/N_f)$}. Due to supersymmetry, both the central charge and current central charge can be exactly computed due to their relation to anomaly coefficients.\n\nA partial bootstrap analysis applicable to meson 4pt functions was performed in \\textcite{Poland:2011ey}, which considered chiral scalar multiplets transforming in the fundamental representation of $SU(N)$ and obtained bounds on the OPE coefficients associated to conserved currents transforming in both the singlet and the adjoint representations of $SU(N)$. As can be seen in Fig.~\\ref{fig:SQCD}, these bounds are still somewhat far from the exact results of supersymmetric QCD (SQCD) theories, most likely because this study didn't utilize the full symmetry. It will be very interesting in future work to extend these analyses of chiral 4pt functions to use the whole SQCD global symmetry, together with mixed correlators containing the $SU(N_f)_{L\/R}$ current multiplets and\/or the stress-tensor multiplet.\n\\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig56-SQCDcomparison.pdf}\n    \\caption{\n    \\label{fig:SQCD} \n    (Color online) Lower bounds on the effective 2pt function coefficient $\\kappa_{\\text{eff}} = 1\/\\lambda_{\\Phi\\Phi J}^2$ of $SU(N)$ singlet currents appearing in $\\Phi \\times \\bar{\\Phi}$, where $\\Phi$ is a chiral scalar of dimension $d$ in the fundamental of $SU(N)$, for $N = 2, \\ldots, 14$. The bounds are normalized to the value $\\kappa_{\\text{chiral}}$ corresponding to a free chiral superfield. Each dot connected to a bound corresponds to the exact value in an SQCD theory with the same symmetry~\\cite{Poland:2011ey}.\n}\n  \\end{figure}\n\n\\subsection{Theories with 3d $\\mathcal{N}=2$ supersymmetry}\n\\label{sec:Fermions-WZ}  \n\nAnother interesting set of targets for the conformal bootstrap are the zoo of 3d CFTs with $\\mathcal{N}=1$ or $\\mathcal{N}=2$ supersymmetry. We made initial contact with the former in Sec.~\\ref{sec:Fermions-GN}, where there were no constraints from supersymmetry used other than relations between scaling dimensions {(see however footnote~\\ref{footnote:3dN1})}. The superconformal representation theory of the latter has a similar structure to that of 4d $\\mathcal{N}=1$ SCFTs; for details see \\textcite{Minwalla:1997ka} and \\textcite{Bobev:2015jxa}. Perhaps the simplest such theory is the $\\mathcal{N}=2$ supersymmetric Wess-Zumino model described in Sec.~\\ref{sec:FermionModels}. This CFT can be thought of as the IR fixed point of a theory of a single chiral superfield $\\Phi = \\phi + \\psi \\theta + F \\theta^2$ and superpotential $W = \\lambda \\Phi^3$. The fixed point has a $U(1)_R$ symmetry under which $\\Phi$ has charge 2\/3, implying exact dimensions for the complex scalar $\\phi$ and the Dirac fermion $\\psi$: $\\Delta_{\\phi} = q_{\\phi} = 2\/3$, $\\Delta_{\\psi} = \\Delta_{\\phi} + 1\/2 = 7\/6$.\n\n     \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig57-PRL_epsbound_3d_nmax9}\n    \\caption{\\label{fig:Fermions-SUSY3kinks} (Color online) Bound on the dimension of the first unprotected scalar $\\Phi \\bar{\\Phi}$ in the $\\Phi\\times\\bar{\\Phi}$ OPE in 3d SCFTs with $\\mathcal{N}=2$ supersymmetry \\cite{Bobev:2015vsa}.\n  }\n  \\end{figure}\n\nApplying the numerical bootstrap to the 4pt function $\\<\\Phi \\bar{\\Phi} \\Phi \\bar{\\Phi}\\>$ and incorporating the unitarity bounds and superconformal blocks of $\\mathcal{N}=2$ superconformal symmetry,~\\textcite{Bobev:2015vsa,Bobev:2015jxa} and \\textcite{Li:2017kck} studied general bounds on the dimension of the leading unprotected scalar operator $\\Phi \\bar{\\Phi}$, with the basic result shown in Fig.~\\ref{fig:Fermions-SUSY3kinks}. Curiously, the resulting bound has three distinct features, the first of which occurs at a scaling dimension $\\Delta_{\\Phi} \\simeq 2\/3$.  This gives a sharp upper bound $\\Delta_{\\Phi\\bar{\\Phi}} < 1.91$ for the $\\mathcal{N}=2$ supersymmetric Wess-Zumino model and a plausible conjecture that the model saturates the optimal version of this bound. Further analysis of the extremal spectrum of this kink can be found in \\textcite{Bobev:2015vsa,Bobev:2015jxa}, while \\textcite{Li:2017kck} found that an isolated island around $\\{\\Delta_{\\Phi}, \\Delta_{\\Phi \\bar{\\Phi}}\\} = \\{0.6678(13), 1.903(10)\\}$ could be obtained by assuming a modest gap in the spectrum of spin-1 superconformal primaries $\\Delta_{J'} \\geqslant 3.5$.\n\nThe middle kink occurs near $\\Delta_{\\Phi} = 3\/4$, and coincides with a kinematic threshold beyond which superconformal descendants of anti-chiral operators $Q^2 \\bar{\\Psi}$ can no longer appear in the $\\Phi \\times \\Phi$ OPE. It is not yet clear if any CFT sits at this kink. The right-most kink, occurring near $\\Delta_{\\Phi} \\sim .86$, also still lacks a clear interpretation, but seems to interpolate to the kink in the 4d $\\mathcal{N}=1$ bounds discussed above. Notably, the extremal spectrum of this kink seems to satisfy the chiral ring relation \\mbox{$\\Phi^2 = 0$}~\\cite{Bobev:2015jxa} and an island around the point can also be isolated using a set of gap assumptions~\\cite{Li:2017kck}, making it a plausible candidate for a new CFT.  Finally let us mention that this analysis was also extended to 3d $\\mathcal{N}=2$ supersymmetric CFTs with $O(N)$ global symmetry by~\\textcite{Chester:2015qca,Chester:2015lej}, who found similar features at each value of $N$. A related 3d $\\mathcal{N}=2$ theory with multiple interacting chiral superfields and a conformal manifold was also recently studied using bootstrap methods in~\\textcite{Baggio:2017mas}.\n\n\n\n\\section{Introduction}\n\\input{intro\/intro.tex}\n\n\\section{Conformal field theory techniques in $d$ dimensions}\n\\input{theory\/theory.tex}\n\n\\subsection{Conformal transformations}\n\\input{theory\/conformaltransformations.tex}\n\n\\subsection{Operators: primaries and descendants}\n\\input{theory\/primaries.tex}\n\n\\subsection{Correlation functions}\n\\input{theory\/correlation.tex}\n\n\\subsection{Operator product expansion}\n\\input{theory\/OPE.tex}\n\n\\subsection{Constraints from unitarity}\n\\input{theory\/unitarity.tex}\n\n\\subsection{Conformal blocks}\n\\input{theory\/conformalblocks.tex}\n\n\\subsection{Global symmetry}\n\\input{theory\/global.tex}\n\n\\subsection{Conserved local currents}\n\\input{theory\/ward.tex}\n\n\\subsection{Crossing relations}\n\\input{theory\/crossing.tex}\n\n\\section{Numerical methods} \n\\input{numerical\/numerical.tex}\n\n\\section{Applications in $d=3$}\n\\input{3D\/3D.tex}\n\n\\subsection{Bounds on critical vs multicritical behavior}\n\\input{multicrit\/multicrit.tex}\n\n\\subsection{$\\bZ_2$ global symmetry} \n\\input{Z2\/Z2.tex}\n\n\\subsection{$O(N)$ global symmetry}\n\\input{ON\/ON.tex}\n\n\\subsubsection{$O(2)$ global symmetry}\n\\input{O2\/O2.tex}\n\n\\subsubsection{$O(3)$ global symmetry}\n\\input{O3\/O3.tex}\n\n\\subsection{CFTs with fermion operators}\n\\input{Fermions\/Fermions.tex}\n\n\\subsection{QED${}_3$}\n\\input{QED3\/QED3.tex}\n\n\\subsection{Current and stress-tensor bootstrap}\n\\input{JandT\/JandT.tex}\n\n\\subsection{Future targets}\n\\input{targets\/targets.tex}\n\n\\section{Applications in $d=4$}\n\\input{4D\/4D.tex}\n\n\\subsection{Constraints on the QCD${}_4$ conformal window}\n\\input{4Dwindow\/4Dwindow.tex}\n\n\\section{Applications to superconformal theories}\n\\input{4Dsusy\/4Dsusy.tex}\n\n\\section{Applications to nonunitary models}\n\\input{nonunitary\/nonunitary.tex}\n\n\\section{Other applications}\n\\input{brief\/brief.tex}\n\n\\begin{acknowledgments}\n\nWe would like to thank Connor Behan, John Chalker, Shai Chester, Subham Dutta Chowdhury, Luigi Del Debbio, Mykola Dedushenko, Yin-Chen He, Kuo-Wei Huang, Denis Karateev, Zohar Komargodski, Petr Kravchuk, Adam Nahum, Yu Nakayama, Miguel Paulos, Silviu Pufu, Leonardo Rastelli, Subir Sachdev, David Simmons-Duffin, Andreas Stergiou, Tin Sulejmanpasic, Senthil Todadri, Emilio Trevisani, Ettore Vicari, and William Witczak-Krempa for useful discussions and comments. \n\nDP is supported by NSF grant PHY-1350180 and Simons Foundation grant 488651 (Simons Collaboration on the Nonperturbative Bootstrap). \nSR is supported by the Simons Foundation grant 488655 (Simons Collaboration on the Nonperturbative Bootstrap), and by Mitsubishi Heavy Industries as an ENS-MHI Chair holder.\nAV is supported by the Swiss National Science Foundation under grant no. PP00P2-163670 and by the ERC-STG under grant no. 758903.\n\n\\end{acknowledgments}\n\n\n\n\n\\section{Outlook}\n\\label{sec:outlook}\n\nThe conformal bootstrap is still in its infancy and there remains much low-hanging fruit to pick along with many important open questions. For instance, can we use the bootstrap to fully classify the space of critical CFTs with a given symmetry, placing universality on a rigorous footing? Can the bootstrap solve the conformal windows of QED${}_3$ and QCD${}_4$? Can it be used as a discovery tool to find new, perhaps non-Lagrangian, CFTs? Is there an analytical understanding of the kinks in numerical bounds or why certain CFTs such as the 3d Ising model live in them? Which CFTs can be found using extremal spectrum or truncation methods? And is there a fruitful way to incorporate developments in the analytical bootstrap with rigorous numerical methods?\n\nFor newcomers to the numerical bootstrap who want to quickly get started, after learning CFT basics we recommend becoming familiar with the available software tools,\\footnote{Many of the currently available tools are collected at the webpage:~\\url{http:\/\/bootstrapcollaboration.com\/activities\/}.} particularly \\texttt{SDPB} {which is under active development,} along with one of the efficient methods to compute conformal blocks in the dimension of interest as described in Sec.~\\ref{sec:cb}. Then one can start reproducing numerical bounds and thinking about how they can be generalized to say something new about situations of physical interest. For this purpose it is also helpful to get used to restating the physical properties of critical systems using symmetries and the spectrum of scaling dimensions, so questions can be sharply rephrased in the language of the bootstrap. Good luck!\n\n\\subsubsection{Models}\n\n\\label{sec:FermionModels}\nThe preceding sections discussed constraints from crossing relations for 4pt functions of scalar operators. Many 3d or (2+1)d CFTs of theoretical and experimental interest also contain fermionic operators, and here we will discuss what the bootstrap has so far been able to say about them.\n\nPerhaps the simplest example is the family of CFTs described by the Gross-Neveu model at criticality~\\cite{Gross:1974jv}.\\footnote{This model and its variations are frequently invoked to describe quantum phase transitions in condensed matter systems with emergent Lorentz symmetry in (2+1)d. Some examples of its applications include models for phase transitions in graphene~\\cite{herbut2006interactions, herbut2009relativistic}, the Hubbard model on the honeycomb and $\\pi$-flux lattice~\\cite{toldin2015fermionic}, models of time-reversal symmetry breaking in d-wave superconductors~\\cite{vojta2000quantum, vojta2003quantum}, and models of 3-dimensional gapless semiconductors~\\cite{Moon:2012rx,Herbut:2014lfa}.} While the critical theory is often described as the UV fixed point in a theory of fermions with 4-fermi interactions $\\mathcal{L} \\sim (\\bar{\\psi} \\psi)^2$, a better nonperturbative definition is as an IR fixed point in a theory with a scalar field coupled to fermions via Yukawa interactions. The latter Gross-Neveu-Yukawa (GNY) model contains a scalar $\\phi$ and $N$ Majorana fermions $\\psi_i$:\n\\beq\n    {\\cal L}_{\\rm GNY} =\\frac 12 \\sum_{i = 1}^N \\bar \\psi_i (\\slashed{\\partial} + g \\phi ) \\psi_i + \\frac 12 \\partial^\\mu \\phi\\partial_\\mu \\phi  + \\frac 12 m^2 \\phi^2 + \\lambda \\phi^4 \\,.\n     \\label{GNYLag}\n \\end{equation}\n This model has an $O(N)$ symmetry rotating the fermions. A fixed point can be established perturbatively at large $N$ in a $1\/N$ expansion, see e.g.~\\textcite{Gracey:1992cp,Gracey:1993kc} and \\textcite{Derkachov:1993uw}. This model has also been studied extensively from the perspective of the $\\epsilon$-expansion, with recent results by~\\textcite{fei2016yukawa,Mihaila:2017ble,Zerf:2017zqi}.\n \nAn interesting special case is $N=1$ (a single Majorana fermion coupled to a real scalar). It is expected \\cite{fei2016yukawa} that this model may contain a fixed point with $\\mathcal{N}=1$ supersymmetry. This supersymmetric fixed point has been proposed to described a critical point on the boundary of topological superconductors~\\cite{Grover:2013rc}. \n\nThere are variations of this model containing multiple scalar order parameters. One notable example is the $\\mathcal{N}=2$ supersymmetric critical Wess-Zumino model, containing a complex scalar related to a 3d Dirac fermion by supersymmetry.\\footnote{We will describe some of the implications of supersymmetry and a bootstrap analysis connecting to this model later in Sec.~\\ref{sec:4Dsusy}.} This theory has been proposed to describe a critical point on the surface of topological insulators~\\cite{Ponte:2012ru,Grover:2013rc}, and a superconducting critical point in (2+1)d Dirac semimetals with an attractive Hubbard interaction \\cite{Li:2017dkj}. Another important example is the Gross-Neveu-Heisenberg model, described in Sec.~\\ref{sec:O3}.\n\n\\subsubsection{General results}\n\\label{sec:Fermions-general}\n\nWe first discuss general results following from the existence of fermionic operators. Specialized bounds where the critical GNY and other models are featured more prominently will be discussed below.\n\nA bootstrap analysis of 4pt functions of identical Majorana fermions $\\<\\psi\\psi\\psi\\psi\\>$ was performed in \\textcite{Iliesiu:2015qra} and extended to 4pt functions $\\< \\psi_i \\psi_j \\psi_k \\psi_l \\>$ containing fermions that are vectors under an $O(N)$ symmetry in \\textcite{Iliesiu:2017nrv}. These studies both assumed a general (2+1)d CFT with parity symmetry. Tensor structures and conformal blocks for 4pt functions were derived using a spinorial embedding-space formalism also developed in \\textcite{Iliesiu:2015qra}, similar in logic to the vectorial embedding space reviewed in Appendix~\\ref{sec:embedding}. \n\nIn Fig.~\\ref{fig:Fermions-oddScalar} we show general upper bounds on the leading parity-odd and parity-even scalars in the $\\psi \\times \\psi$ OPE, called $\\sigma$ and $\\epsilon$ respectively. The bound on $\\sigma$ is nearly saturated by the MFT line $\\Delta_{\\sigma} = 2\\Delta_{\\psi}$, at least at small values of $\\Delta_{\\sigma}$. As $\\Delta_{\\psi} \\rightarrow 1$ the bound approaches the free theory value $\\Delta_{\\sigma} = 2$, where we can identify $\\sigma = \\bar{\\psi} \\psi$. On the other hand, there is an abrupt discontinuity in the bound around $\\Delta_{\\psi} \\sim 1.27$ occurring when $\\Delta_{\\sigma}$ approaches 3. This jump also coincides with a kink in the bound on $\\Delta_{\\epsilon}$. The interpretation of these features is currently an open question -- it is tempting to speculate that a CFT may live at the top of the jump in the bound on $\\Delta_{\\sigma}$ and in the kink in the bound on $\\Delta_{\\epsilon}$ but no concrete candidate CFTs have yet been identified. If it exists, this CFT would appear to have an unusual property of not possessing any relevant scalar deformations.\\footnote{Hypothetical theories with this property were recently named ``dead-end\" CFTs by~\\textcite{Nakayama:2015bwa}. They should be distinguished from ``self-organized\" CFTs which do not have any relevant \\emph{singlet} scalars as defined in Sec.~\\ref{sec:multicrit}.}\n\n \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig28a-oddScalar}(a)\n\\includegraphics[width=\\figwidth]{fig28b-evenScalar}(b)\n    \\caption{\\label{fig:Fermions-oddScalar}\n(Color online) Upper bounds on the dimension of (a) the first parity-odd scalar $\\sigma$ and (b) the first parity-even scalar $\\epsilon$ in the OPE $\\psi\\times \\psi$, as a function of $\\Delta_\\psi$ \\cite{Iliesiu:2015qra}. Here $\\psi$ is a Majorana fermion primary operator in a 3d parity-invariant unitary CFT.} \n  \\end{figure}\n\nIn Fig.~\\ref{fig:Fermions-centralCharge} we also show the general lower bounds on the central charge $C_T$ (normalized to its value in the theory of a free Majorana fermion), obtained by bounding the coefficient of the stress-tensor conformal block. These lower bounds approach the free values as $\\Delta_{\\psi} \\rightarrow 1$ and disappear completely for $\\Delta_{\\psi} \\gtrsim 1.47$. In the case of $O(N)$ symmetry they can be seen to grow linearly with $N$ and are compatible with values computed in the $1\/N$ expansion of the GNY model. Generalizations to the current central charge $C_J$ for fermions charged under $O(N)$ symmetry were also computed in \\textcite{Iliesiu:2017nrv}.\n\n   \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig29a-centralCharge}(a)\n\\includegraphics[width=\\figwidth]{fig29b-cTBound}(b)\n    \\caption{\\label{fig:Fermions-centralCharge}\n  (Color online) Lower bounds on $C_T$ as a function of $\\Delta_\\psi$, where $\\psi$ is (a) Majorana fermion or (b) a multiplet of Majorana fermions in the fundamental representation of an $O(N)$ global symmetry group \\cite{Iliesiu:2015qra,Iliesiu:2017nrv}.}\n  \\end{figure}\n\n\n\\subsubsection{Gross-Neveu-Yukawa models}\n\\label{sec:Fermions-GN}\n\nIn the critical GNY model at large $N$, $\\psi_i$ has dimension $1 + 4\/(3\\pi^2 N) + \\ldots$, while the leading parity-odd scalars in the $\\psi_i \\times \\psi_j$ OPE are the $O(N)$ singlet $\\phi$ with dimension $1-32\/(3\\pi^2 N)+\\ldots$ and the $O(N)$ symmetric tensor $\\bar{\\psi}_i \\psi_j$ with dimension $2 + 32\/(3\\pi^2 N) + \\ldots$~\\cite[{Table 1}]{Iliesiu:2017nrv}. The accumulation point $(\\Delta_{\\psi}, \\Delta_{\\sigma}) \\rightarrow (1,1)$ sits well in the interior of Fig.~\\ref{fig:Fermions-oddScalar}(a), but by imposing a gap until the second parity-odd scalar, $\\Delta_{\\sigma'} \\geqslant 2 + \\delta$ for different positive values of $\\delta$, we have the possibility of obtaining an allowed region that rules out critical GNY models with $N$ sufficiently large.\n\nThis is realized in Fig.~\\ref{fig:Fermions-grossNeveuSmallSigPrime}, where the effect of gaps ranging from $\\Delta_{\\sigma'} \\geqslant 2.01$ to $\\Delta_{\\sigma'} \\geqslant 2.9$ are shown. At very small values of $\\delta$ the lower bounds of the allowed regions possess a kink whose location matches very well to the large-$N$ GNY model prediction. At larger values of $\\delta$, the precise map between $\\delta$ and $N$ is not known but it is plausible that the kinks continue to match to the GNY model even at small values of $N$. However, starting around $\\Delta_{\\sigma'} \\geqslant 2.3$, a second lower feature also appears in these curves, where they all intersect and have an additional kink at a point near $(1.08, .565)$. \n\nThis structure of an ``upper\" and ``lower\" kink can be seen clearly in Fig.~\\ref{fig:Fermions-oneRelevantOddScalar}, specialized to the case \\mbox{$\\Delta_{\\sigma'} \\geqslant 3$}. In fact, in this case the line \\mbox{$\\Delta_{\\psi} = \\Delta_{\\sigma} + 1\/2$} expected for theories with supersymmetry comes very close to (but just misses) the upper kink. Thus, it is tempting to conjecture that the $\\mathcal{N}=1$ supersymmetric Gross-Neveu-Yukawa model, see Sec.~\\ref{sec:FermionModels}, may sit in this feature and has $\\Delta_{\\sigma'}$ slightly smaller than 3. This picture seems consistent with estimate \\mbox{$\\Delta_{\\sigma} \\approx 0.59$} from a Pad\\'e-extrapolation of the $\\epsilon$-expansion \\cite{fei2016yukawa}, as well as with the rigorous lower bound \\mbox{$\\Delta_\\sigma\\geqslant 0.565$} \\cite{Bashkirov:2013vya}, which follows using another supersymmetric relation $\\Delta_\\epsilon=\\Delta_\\sigma+1$ together with the bootstrap bound in Fig.~\\ref{fig:Z2-epsbound}, applicable with parity playing the role of a $\\mathbb{Z}_2$ symmetry.\\footnote{\\label{footnote:3dN1}Further progress on this CFT was made very recently in~\\textcite{Rong:2018okz} and~\\textcite{Atanasov:2018kqw}, where it was understood how to obtain an island in the scalar mixed-correlator bootstrap around $\\Delta_{\\sigma} = 0.584444(30)$. In these studies in addition to relations between scaling dimensions it is important to incorporate nontrivial 3d $\\mathcal{N}=1$ superconformal blocks.}  An additional speculation is that the lower feature may coincide with a non-supersymmetric fixed point, called GNY${}^*$ in \\textcite{Iliesiu:2017nrv}, which is seen in the $\\epsilon$-expansion as a nonunitary fixed point at large $N$, but whose fate at small $N$ and $\\epsilon \\rightarrow 1$ is not known. \n\nAdditional evidence for this picture comes from the generalization of the bounds to $O(N)$ symmetry~\\cite{Iliesiu:2017nrv}, where one can place independent bounds on different $O(N)$ representations. In Fig.~\\ref{fig:Fermions-ONtwokinks} we show computed bounds on the leading singlet dimension $\\Delta_{\\sigma}$, assuming that the next singlet is irrelevant, $\\Delta_{\\sigma'} \\geqslant 3$. These bounds also show both an upper and lower kink, which appear not too far from the $\\epsilon$-expansion estimates for the GNY and GNY${}^*$ models. In Fig.~\\ref{fig:Fermions-SigmaT} we also highlight the bounds on the leading $O(N)$ symmetric tensor $\\sigma_T$, which display mysterious and unexplained jumps when $\\Delta_{\\sigma_T}$ reaches marginality and at smaller values of $\\Delta_{\\psi_i}$ show a series of kinks which match to the large-$N$ GNY models. Understanding the mechanism behind these jumps is an important open problem, which may be related to the spectrum rearrangement phenomena from Sec.~\\ref{sec:Z2-spectrum}.\n\n   \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig30a-grossNeveuSmallSigPrime}\n\\includegraphics[width=\\figwidth]{fig30b-grossNeveuSigPrime2To3}\n    \\caption{\\label{fig:Fermions-grossNeveuSmallSigPrime}\n   (Color online) Effect of imposing a gap until the second pseudoscalar $\\sigma'$ on the parameter space of 3d parity-invariant CFTs~\\cite{Iliesiu:2015qra}.}\n  \\end{figure}\n\n   \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig31a-oneRelevantOddScalar}\n\\includegraphics[width=\\figwidth]{fig31b-sigmaVsPsiWithSigPrimeGT3Nmax12}\n    \\caption{\\label{fig:Fermions-oneRelevantOddScalar}\n   (Color online) Effect of imposing that there is only one relevant pseudoscalar, $\\Delta_{\\sigma'}\\geqslant 3$, in 3d parity-invariant CFTs~\\cite{Iliesiu:2015qra}.}\n  \\end{figure}\n\n     \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig32-gapOnSigmaPrimeSigBound}\n    \\caption{\\label{fig:Fermions-ONtwokinks}\n   (Color online) Effect of imposing a gap $\\Delta_{\\sigma'}\\geqslant 3$ in the singlet pseudoscalar sector of $O(N)$-symmetric fermionic CFTs \\cite{Iliesiu:2017nrv}. The kinks at low $N$ may perhaps be identified with the GNY and GNY${}^*$ CFTs.}\n  \\end{figure}\n\n   \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig33a-universalBoundsOnSigmaT}(a)\n\\includegraphics[width=\\figwidth]{fig33b-universalBoundsOnSigmaTZoomedIn}(b)\n    \\caption{\\label{fig:Fermions-SigmaT} \n   (Color online) Upper bounds on the dimension of the symmetric traceless pseudoscalar $\\sigma_{T}$ in the OPE $\\psi_i\\times \\psi_j$ in $O(N)$-symmetric fermionic CFTs~\\cite{Iliesiu:2017nrv}. Notice the mysterious jumps in the wide view of the bounds (a) when they cross marginality. (b) gives a zoom on the small $\\Delta_\\psi$ region, where the bounds exhibit kinks, in agreement with the GNY dimensions at large $N$.}\n  \\end{figure}\n \n  \n  \n\n\n\\subsection{Outline}\n\nDue to the overwhelming number of results in various incarnations of the conformal bootstrap, our review will necessarily be limited in scope. Let us briefly outline the topics that we will cover. We begin in Sec.~\\ref{sec:informal} with an informal overview of the conformal bootstrap. Sec.~\\ref{sec:cft} provides a concise introduction to the conformal field theory techniques that are needed to set up the bootstrap in $d$ dimensions. We follow in Sec.~\\ref{sec:numerical} with an overview of the various numerical methods that have been employed in studies of the bootstrap. Secs.~\\ref{sec:appl} and~\\ref{sec:appl4d} review results obtained from applying these methods to 3d and 4d CFTs. Sec.~\\ref{sec:4Dsusy} reviews results obtained with the stronger assumptions of 4d $\\mathcal{N}=1$ or 3d $\\mathcal{N}=2$ superconformal symmetry. We comment on applications to nonunitary models in Sec.~\\ref{sec:nonunitary}. Notably absent from our main review are CFTs in other dimensions (e.g., $d=2$ or $d>4$), CFTs with extended supersymmetry, analytical progress in the bootstrap, logarithmic and nonrelativstic CFTs, and other related topics. We finish with a brief overview of progress in these related lines of research in Sec.~\\ref{sec:other} and give some concluding words in Sec.~\\ref{sec:outlook}.\n\n\n\\section{Conformal bootstrap: informal overview}\n\\label{sec:informal}\n\nIn this section we will give a brief outline of the conformal bootstrap approach to critical phenomena in $d$ dimensions.\nWe will be rather informal in this section, while in the subsequent sections the same material will be treated in more depth and at a higher level of rigor. For another short introduction to these matters, see \\textcite{Poland:2016chs}. For longer pedagogical introductions see \\textcite{Rychkov:2016iqz,Simmons-Duffin:2016gjk}.\n\nAs a simple physical setup where these methods would be applicable, we can consider a statistical physics system in $d$ spatial dimensions which is (a) in thermodynamic equilibrium and (b) at a temperature corresponding to a continuous phase transition (so that the correlation length is infinite). Suppose that we are interested in equal-time correlation functions of some local quantities characterizing this system:\n\\beq\n\\langle {\\cal O}_1(x_1)\\ldots {\\cal O}_n(x_n) \\rangle\\,,\n\\label{eq:corr}\n\\end{equation}\nwhere $x_i$ are positions in $\\mathbb{R}^d$. For example, one can think of the 3d Ising model at the critical point, with ${\\cal O}_i(x)$ the local magnetization, local energy density, etc. In general, the ${\\cal O}_i(x)$ are called local operators. \n\nWe are interested in the behavior of the correlators~\\reef{eq:corr} at distances large compared to any microscopic (such as lattice) scale $a$. According to Wilson's RG theory, continuous phase transitions are fixed points of RG flows, which means that the long-distance behavior of~\\reef{eq:corr} will have scale invariance (as well as rotation and translation invariance). Using scale invariance, we can formally extend the long-distance behavior of these correlators from distances $|x_i-x_j|\\gg a$ to arbitrary short distances. In what follows we work in the so-defined \\emph{continuous limit} theory, which is exactly scale invariant at all distances from $0$ to $\\infty$.\\footnote{However, in this paper we will not consider behavior of correlators at coincident points.} \n\nAs discussed in the introduction, we expect that the critical theory is also conformally invariant (i.e., a CFT). This means that for any \\emph{conformal transformation} of $d$-dimensional space $x\\to x'$ (see Sec.~\\ref{sec:conformaltransformations} for the definition), Eq.~\\reef{eq:corr} is related to the same correlation function evaluated at points $x_1',\\ldots,x_n'$. This invariance property (or covariance) of correlation functions is expressed as a transformation rule for local operators, in the next section appearing in Eq.~\\reef{eq:fieldrepresentation}. For scalar operators, we have\n\\beq\n{\\cal O}(x')=\\Omega(x)^{-\\Delta_{\\cal O}} {\\cal O}(x)\\,,\n\\label{eq:invsc}\n\\end{equation}\n where $\\Omega(x)=|\\partial x'\/\\partial x|^{1\/d}$ is the $x$-dependent scale factor of the conformal transformation, and $\\Delta_{\\cal O}$ is a fixed parameter characterizing the operator ${\\cal O}$, called its \\emph{scaling dimension}.\\footnote{To be precise, such transformation rules hold for \\emph{primary} local operators, a subtlety which will not play a role in this informal discussion.}\n\n\\textcite{Polyakov:1970xd} noticed that invariance under \\reef{eq:invsc} strongly restricts two-point (2pt) and three-point (3pt) correlation functions. The 2pt function is nonzero only for identical {operators}\nand can be normalized to one:\n\\beq\n\\langle {\\cal O}_i(x_1){\\cal O}_j(x_2)\\rangle =\\delta_{ij}{|x_{1}-{x_2}|^{-2\\Delta_i}}\\,,\n\\label{eq:2pt}\n\\end{equation}\nwhile the 3pt function is fixed up to a numerical coefficient:\n\\beq\n\\langle {\\cal O}_1(x_1){\\cal O}_2(x_2){\\cal O}_3(x_3)\\rangle =\\frac{\\lambda_{123}}{|x_{12}|^{h_{123}} |x_{13}|^{h_{132}} |x_{23}|^{h_{231}}}\\,,\n\\label{eq:3pt}\n\\end{equation}\nwhere $x_{ij} \\equiv x_i-x_j$ and $h_{ijk} \\equiv \\Delta_i+\\Delta_j-\\Delta_k$. Similar equations hold for operators with indices, see Sec.~\\ref{sec:correlations}. \n\nThe set of numerical parameters $\\Delta_i$ and $\\lambda_{ijk}$ appearing in \\reef{eq:2pt} and \\reef{eq:3pt} is called the \\emph{CFT data}. It turns out that the CFT data determine not only 2pt and 3pt functions, but are also sufficient to compute all local observables in CFTs in flat space, by which we mean all correlation functions of local operators, including four-point (4pt) and higher-order correlation functions.\\footnote{It should be mentioned that CFTs also possess nonlocal observables in addition to the local ones, which are not necessarily determined by the OPE data. For example, one can probe a CFT by extended operators, such as boundaries or defects, or put it in a space of nontrivial geometry or topology. In this review we will focus on the local observables, although the bootstrap philosophy can also be useful in the study of some nonlocal observables; see~Sec.~\\ref{sec:bdry} for boundaries and defects and Sec.~\\ref{sec:other} for the modular bootstrap.} \n \nTo see this, one uses the OPE, which says that we can replace the insertion of two nearby local operators inside a correlation function by a series of single local operators:\n\\beq\n{\\cal O}_i(x_1){\\cal O}_j(x_2)=\\sum_k f_{ijk} {\\cal O}_k(y)\\,.\n\\label{eq:OPE}\n\\end{equation}\nThe coefficients of the series $f_{ijk}$ may and will depend on the relative positions of the operators ${\\cal O}_i$, ${\\cal O}_j$, ${\\cal O}_k$, and on their quantum numbers. Crucially, however, these coefficients are not supposed to depend on which other operators appear in the correlation function, as long as they are sufficiently far away from $x_1,x_2,y$. The precise criterion in the CFT context will be given in Eq.~\\reef{eq:converge}.\n\nNotice the freedom in where we put operators appearing on the r.h.s.~of the OPE: we can choose $y={\\textstyle\\frac 12}(x_1+x_2)$, $y=x_1$, or any other point nearby. Different choices of $y$ can be related by Taylor-expanding ${\\cal O}_k$, and thus can be compensated by changing {the} coefficients of derivatives of ${\\cal O}_k$ in the OPE. In what follows we will group the operator ${\\cal O}_k$ together with all its derivatives, formally thinking of $f_{ijk}$ as an infinite power series in $\\del_y$. \n\nThere are two things that make OPE in conformal field theories more powerful than in a generic QFT. Firstly, compatibility of the OPE with conformal invariance determines the functions $f_{ijk}$ up to a numerical prefactor, coinciding with the 3pt function coefficient $\\lambda_{ijk}$ (for this reason it is also called an OPE coefficient):\n\\beq\nf_{ijk}(x_1,x_2,y,\\del_y) =  \\lambda_{ijk} \\hat{f}_{ijk}(x_1,x_2,y,\\del_y)\\,.\n\\end{equation}\nThe reduced functions $\\hat f_{ijk}$ depend only on {the} operator dimensions $\\Delta_i,\\Delta_j,\\Delta_k$, the spins of these operators (which are kept implicit in this informal discussion), and on the space dimension $d$.\\footnote{Note that here we are assuming the normalization in Eq.~(\\ref{eq:2pt}).}\n\nSecondly, the OPE in conformal theories has a finite radius of convergence, which is determined by the distance to the next operator insertions. For example, in the correlator of Eq.~\\reef{eq:npt} given below, the OPE will converge if \n\\beq\n|x_1-y|,\\,|x_2-y|<\\min_{i=3\\ldots n} |x_i-y|\\,,\n\\label{eq:converge}\n\\end{equation}\ni.e.~if there exists a sphere centered at $y$ and separating $x_1$, $x_2$ from any other operator insertion.\n\nBecause of these two reasons we can compute any correlation function recursively using the OPE, provided that we know the CFT data. For example, suppose we want to compute the $n$-point function\n\\beq\n\\langle {\\cal O}_1(x_1) {\\cal O}_2(x_2) {\\cal O}_3(x_3)\\ldots {\\cal O}_n(x_n)\\rangle\\,.\n\\label{eq:npt}\n\\end{equation}\nApplying the OPE to ${\\cal O}_1(x_1) {\\cal O}_2(x_2)$, we reduce this correlator to a sum of correlators containing $n-1$ operators\n\\beq\n\\langle {\\cal O}_k(y) {\\cal O}_3(x_3)\\ldots {\\cal O}_n(x_n)\\rangle\\,.\n\\end{equation}\nProceeding in this way, we will eventually get down to 2pt functions, which are determined by the CFT data. The only parameters which will enter this computation are the operator positions and quantum numbers, the CFT data, and the space dimension $d$.\\footnote{Notice that although the presented scheme solves the problem of computing $n$-point functions in principle, it is not trivial to do in practice. For 4pt functions, the necessary techniques will be presented in Sec.~\\ref{sec:cb}.}\n \nConsider now the case of a 4pt function (Eq.~\\reef{eq:npt} with $n=4$)\n and compute it in two different ways. The first way is to apply the OPE to the pairs of operators ${\\cal O}_1 {\\cal O}_2$ and ${\\cal O}_3{\\cal O}_4$. This reduces the 4pt function to an infinite sum of 2pt functions of operators which appear in these OPEs. A second way is to apply the OPE to the pairs ${\\cal O}_1{\\cal O}_4$ and ${\\cal O}_2{\\cal O}_3$. Since we are dealing with the same 4pt function, the two expansions must agree in their overlapping regions of convergence. This \\emph{crossing relation} represents a consistency condition on the CFT data and is illustrated in Fig.~\\ref{fig:bootstrap}.\n\nThe main idea of the conformal bootstrap is that by imposing the crossing relation, we should be able to significantly winnow down the set of all possible CFT data. In the subsequent sections of this review, we will see how the crossing relation can be written in a mathematically manageable form, and how numerical algorithms can be applied to extract from it concrete constraints. \n\nIdeally, if we impose crossing relations for \\emph{all} 4pt functions of the theory, we will be left with the CFT data corresponding to the actually existing critical theories. In practice, it has so far been possible to impose crossing relations on only a handful of 4pt functions at a time. However, we will see that even this limited procedure produces nontrivial constraints, which are in some cases surprisingly strong.\n\n\\begin{figure}[t!]\n    \\includegraphics[width=0.49\\textwidth]{fig01-bootstrapO.pdf}\n    \\caption{\\label{fig:bootstrap}\n   Crossing relation for the 4pt function $\\<{\\cal O}_1 {\\cal O}_2 {\\cal O}_3 {\\cal O}_4\\>$.}\n  \\end{figure}\n\n \\subsection{Universality and the role of microscopic input}\n\\label{sec:universality}\nA fundamental concept in the theory of critical phenomena is universality: all continuous phase transitions can be grouped into universality classes which share the same critical exponents. This is neatly explained in Wilson's RG theory: two phase transitions will fall into the same universality class if they are described by the same fixed point. On the other hand, the conformal bootstrap provides a different perspective on the same phenomenon: each universality class corresponds to a different CFT, with a different set of CFT data. \n\nThese two points of view are clearly complementary, and it is important to establish the correspondence between them. Consider for example the critical exponents. In RG theory they can be related to the eigenvalues $\\lambda^{y_i}$ of the RG transformation linearized around the fixed point, where $\\lambda>1$ is the RG rescaling factor. As is well known, these eigenvalues are simply related to the scaling dimensions of the local operators: $y_i=d-\\Delta_i$. Thus, information about the critical exponents can be easily extracted from CFT data, and agreement of their values between an RG fixed point and a CFT may give us confidence that the two describe the same critical universality class.\n\nThere are however three more fundamental structural characteristics which can be used to identify universality classes, even before considering the numerical values of critical exponents. These characteristics may not be sufficient to uniquely classify the different CFTs, but they will give us a convenient starting point.\n\n\\emph{1. The global (or internal) symmetry group.} It can be discrete, as for the $\\bZ_2$ symmetry of the Ising model, or continuous, as for the $O(N)$ models. In RG studies, the global symmetry group is specified by considering an RG flow in the space of microscopic theories described by an action possessing a given symmetry. The global symmetry group for a CFT is the same group $G$ as for the corresponding RG fixed point, although it is specified in a different way: by demanding that each local operator transform in an irreducible representation of $G$ and that OPE coefficients respect this symmetry structure.\n\nWe note in passing that unlike the global symmetry, the presence of a \\emph{gauge symmetry} in a microscopic description does not manifest itself in the conformal bootstrap, because physically observable local CFT operators are gauge invariant.\\footnote{Gauge symmetries can make themselves known more indirectly, through anomaly coefficients which show up in the correlation functions of local operators or the existence of higher-form symmetries.}\n\n\\emph{2. The number of relevant singlet scalars.} The number of scalar operators which are relevant (i.e., have dimension $\\Delta_i<d$) and transform as singlets under the global symmetry determines whether the universality class has critical as opposed to multicritical behavior. This will be discussed in more detail in Sec.~\\ref{sec:multicrit}. Here it suffices to note that this number is easy to identify from both the RG and CFT perspectives.\n\n\\emph{3. Unitarity.} Unitarity is of course a required property when quantum mechanics is involved, which is the case for theories of interest to high-energy physics and quantum condensed matter. Many universality classes of interest to statistical physics also happen to be unitary.\\footnote{In statistical physics, the role of unitarity is played by its Euclidean counterpart called reflection positivity.} Importantly, the existence of unitarity can be established at the microscopic scale, and is then inherited by the RG fixed point. In the CFT description, unitarity is imposed via lower bounds on the operator dimensions and reality constraints on the OPE coefficients, see Sec.~\\ref{sec:unitarity}.\n \nFinally, let us comment on the OPE coefficients $\\lambda_{ijk}$. From the CFT point of view, they are an integral part of the CFT data, on par with the scaling dimensions. In the conformal bootstrap approach, the crossing relation involves both $\\lambda_{ijk}$ and $\\Delta_i$. In the examples below, when we are able to determine the $\\Delta_i$'s to some accuracy (as for the 3d Ising and the $O(N)$ models), we can typically determine the $\\lambda_{ijk}$'s to a comparable accuracy. This can be contrasted with the RG approach, where the OPE coefficients do not appear to play such a fundamental role, and they have received relatively little attention.\n\n\n\n\n\\subsection{Convex optimization and linear programming}\n\\label{sec:LP}\n\nThe CFT crossing relations describe a continuously infinite number of constraints on the CFT data parametrized by the cross ratios. In order to study the crossing relations numerically one must discretize this set of constraints. Starting with~\\textcite{Rattazzi:2008pe}, the common approach adopted in numerical studies is to Taylor expand the crossing relations around a point in cross-ratio space, typically taken to be the symmetric configuration $u=v=1\/4$ or equivalently $z=\\bar{z}=1\/2$. If one takes derivatives only up to a certain order $\\Lambda$, then one obtains a finite set of constraints. \n\nBefore proceeding, let us highlight that a number of other choices could be made here, e.g.~evaluating the crossing equations at different values of the cross ratios, Taylor expanding around other points, integrating the crossing equations, etc.\\footnote{Some of these alternative ideas have been explored by~\\textcite[{section 4.2}]{Hogervorst:2013sma}, \\textcite{Echeverri:2016ztu}, \\textcite{Mazac:2016qev}, \\textcite{Li:2017ukc}, and \\textcite{Mazac:2018mdx}.} Here we focus on the approach of Taylor expanding around the symmetric configuration since it works well in practice and is the most common approach in the literature. One justification for this choice of the expansion point is that it makes both the direct and the crossed channel in the conformal block expansion to converge maximally fast~\\cite{Pappadopulo:2012jk}. However, it is by no means obvious that it is the most efficient way to discretize the crossing relations.\n\nIn the case of a single 4pt function of identical scalars $\\<\\sigma\\s\\sigma\\s\\>$, the resulting constraints take the form\n\\begin{eqnarray}\\label{eq:crossingvec}\n0 &=& \\sum_{{\\cal O}} \\lambda_{\\sigma\\s{\\cal O}}^2 \\vec{F}^{\\Delta_\\sigma}_{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}},\n\\end{eqnarray}\nwhere $\\vec{F}^{\\Delta_\\sigma}_{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}}$ can be thought of as a vector with components\n\\begin{eqnarray}\n\\label{eq:F}\n\\left(\\vec{F}^{\\Delta_\\sigma}_{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}}\\right)^{mn} &=& \\partial_{z}^m \\partial_{\\bar{z}}^n F^{\\Delta_\\sigma}_{\\Delta_{{\\cal O}}, \\ell_{{\\cal O}}}(z,\\bar{z})\\big|_{z=\\bar{z}=1\/2}\\,,\n\\end{eqnarray}\nwhere we take derivatives of the functions \\reef{eq:Feq} and we keep components up to a cutoff $m+n \\leqslant \\Lambda$.\\footnote{{Since the functions \\reef{eq:Feq} are odd under $z\\to 1-z$, $\\bar{z}\\to 1-z$, only components with {$m+n$ odd} lead to nontrivial equations.}} {This computation will thus involve derivatives of conformal blocks up to some finite order.}\n \n{Computing the vectors $\\vec{F}^{\\Delta_\\sigma}_{\\Delta,\\ell}$ constitutes a nontrivial preliminary step for analyzing Eq.~\\reef{eq:crossingvec}. This step is handled starting from one of the many exact or approximate expressions for conformal blocks discussed in Sec.~\\ref{sec:cb}. The state-of-the-art approach is to use the rational approximation, see Sec.~\\ref{sec:rational}, where available software packages are also described. This approach gives rise to approximate expressions which reproduce $\\vec{F}^{\\Delta_\\sigma}_{\\Delta,\\ell}$ with any desired precision. These expressions can be efficiently evaluated ``on the fly\", as needed in the {continuous} simplex algorithm from Sec.~\\ref{sec:modified-simplex}. They can also be used as an input to the semidefinite programming methods described in Sec.~\\ref{sec:SDP}.}\n\nWe now proceed to {describe} strategies on how to decide if Eq.~\\reef{eq:crossingvec} has solutions, i.e.~if there {exists} some choice of the exchanged CFT spectrum $\\{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}\\}$ and OPE coefficients $\\lambda_{\\sigma\\sigma{\\cal O}}$ which makes it satisfied. First, let us remark that Eq.~(\\ref{eq:crossingvec}) is a set of linear equations in $\\lambda_{\\sigma\\s{\\cal O}}^2$. This is at the heart of both the linear programming approaches described in this subsection as well as the extremal functional and truncation methods described below. In particular, if one has a candidate CFT spectrum for operators appearing in $\\sigma \\times \\sigma$ but does not know the OPE coefficients, one can straightforwardly solve a linear algebra problem to find the coefficients.\n\nIn unitary (or reflection positive) CFTs, Eq.~(\\ref{eq:crossingvec}) states that a sum of vectors must add to zero with {\\it positive} coefficients, due to $\\lambda_{\\sigma\\s{\\cal O}}$ necessarily being real. For some choices of the CFT spectrum $\\{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}\\}$ this is not possible, as illustrated in Fig.~\\ref{fig:separating-plane}. When it is not possible one can identify a separating plane $\\alpha$ through the origin such that all vectors point to one side of the plane.\\footnote{Some vectors may point in the plane but at least one must point outside of it.}\n\n\\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=0.49\\textwidth]{fig05-separating-plane}\n    \\caption{\\label{fig:separating-plane}\nLeft: A case where vectors can sum to zero with positive coefficients. Right: A case where vectors cannot sum to zero with positive coefficients and there exists a separating plane $\\alpha$ such that all vectors point on one side of the plane. Figure from \\textcite{Poland:2016chs}.}\n  \\end{figure}\n\nThis observation forms the basis for the first numerical strategy of analyzing the crossing relation \\cite{Rattazzi:2008pe}: input some assumption about the CFT spectrum (e.g., a gap in the scalar spectrum with all other operators satisfying unitarity bounds) and numerically search for a separating plane $\\alpha$. Equivalently we can say that we are applying a linear functional $\\sum_{mn} \\alpha_{mn} \\partial_{z}^m \\partial_{\\bar{z}}^n \\left[ \\cdot \\right]\\big|_{z=\\bar{z}=1\/2}$ to the crossing relations and checking if it is possible to derive a contradiction. Concretely, one can look for a vector $\\vec{\\alpha}$ such that the scalar product is strictly positive on at least one operator whose OPE coefficient is nonzero (this may be the identity, the stress tensor, or any other operator that we assume appears in the OPE):\n\\beq\n\\vec{\\alpha} \\cdot \\vec{F}^{\\Delta_\\sigma}_{\\Delta_{{\\cal O}^*},\\ell_{{\\cal O}^*}} > 0\\,,\n\\end{equation}\nwhile it is nonnegative for all other $\\{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}\\}$ allowed by our assumptions:\n\\beq\n\\vec{\\alpha} \\cdot \\vec{F}^{\\Delta_\\sigma}_{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}} \\geqslant 0\\,.\n\\end{equation}\nEach inequality $\\vec{\\alpha} \\cdot \\vec{F}^{\\Delta_\\sigma}_{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}} \\geqslant 0$ identifies a half-space and their intersection carves out a convex cone. \n\nThere is still one issue before the vector $\\vec{\\alpha}$ can be searched for numerically -- a priori there are an infinite number of allowed vectors labeled by all $\\{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}\\}$. \nThe first numerical bootstrap studies\\footnote{See~\\textcite{Rattazzi:2008pe}, \\textcite{Rychkov:2009ij}, \\textcite{Caracciolo:2009bx}, \\textcite{Poland:2010wg}, \\textcite{Rattazzi:2010gj,Rattazzi:2010yc}, \\textcite{Vichi:2011ux,Vichi:2011zza}, and \\textcite{ElShowk:2012ht}.} employed a discretization approach: namely, they discretized the set $\\{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}\\}$ using some small spacing between allowed dimensions so that there are a finite number of linear inequalities satisfied by a finite number of unknown coefficients $\\vec{\\alpha}$. Then the problem becomes a standard linear programming problem and can be solved using standard algorithms. These include simplex algorithms, where one moves from vertex to vertex on the edge of the feasible region, or interior point algorithms, where one instead traverses the interior of the feasible region. Software packages that have been used {in the past} for this purpose are \\texttt{Mathematica}, the \\texttt{GNU Linear Programming Kit (GLPK)}, and the \\texttt{IBM ILOG CPLEX Optimizer}. {This discretization approach is currently considered to be obsolete, although it retains pedagogical value. More efficient approaches avoiding discretization will be discussed below.}\n\nOne can slightly modify the problem in order to place bounds on OPE coefficients \\cite{Caracciolo:2009bx}. By isolating one particular contribution ${\\cal O}^*$ and again applying a functional $\\vec{\\alpha}$ one rewrites the equation as\n\\beq\n\\label{eq:OPEbound}\n\\lambda_{\\sigma\\s{\\cal O}^*}^2 \\vec{\\alpha} \\cdot \\vec{F}^{\\Delta_\\sigma}_{\\Delta_{{\\cal O}^*},\\ell_{{\\cal O}^*}} = -\\vec{\\alpha} \\cdot \\vec{F}^{\\Delta_\\sigma}_{0,0} - \\sum_{{\\cal O}} \\lambda_{\\sigma\\s{\\cal O}}^2 \\vec{\\alpha} \\cdot \\vec{F}^{\\Delta_\\sigma}_{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}}.\n\\end{equation}\nThen by imposing the normalization condition \\mbox{$\\vec{\\alpha} \\cdot \\vec{F}^{\\Delta_\\sigma}_{\\Delta_{{\\cal O}^*},\\ell_{{\\cal O}^*}} = 1$} and the positivity constraints \\mbox{$\\vec{\\alpha} \\cdot \\vec{F}^{\\Delta_\\sigma}_{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}} \\geqslant 0$} one obtains the upper bound \\mbox{$\\lambda_{\\sigma\\s{\\cal O}^*}^2 \\leqslant  -\\vec{\\alpha} \\cdot \\vec{F}^{\\Delta_\\sigma}_{0,0}$}. The strongest upper bound is obtained by {\\it minimizing} $ -\\vec{\\alpha} \\cdot \\vec{F}^{\\Delta_\\sigma}_{0,0}${, which yields an optimization problem that can be solved with linear programming algorithms, adopting the above-mentioned discretization approach or other methods discussed below}.  Alternatively, one can also seek lower bounds by instead imposing  $\\vec{\\alpha} \\cdot \\vec{F}^{\\Delta_\\sigma}_{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}} \\leqslant 0$ and maximizing $ -\\vec{\\alpha} \\cdot \\vec{F}^{\\Delta_\\sigma}_{0,0}$ \\cite{Poland:2011ey}. However, in general it is not possible to obtain lower bounds on OPE coefficients unless the operator ${\\cal O}^*$ is isolated in the allowed spectrum, since one could always imagine that ${\\cal O}^*$ has a zero OPE coefficient but operators infinitesimally close to it have nonzero coefficients. \n\n\\subsubsection{Continuous primal simplex algorithm}\n\\label{sec:modified-simplex}\n\nInstead of looking for a vector $\\vec{\\alpha}$ with the desired positivity properties, an alternate strategy is to search directly for a set of vectors $\\{\\vec{F}^{\\Delta_\\sigma}_{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}}\\}$ appearing in Eq.~(\\ref{eq:crossingvec}), subject to the positivity conditions $\\lambda_{\\sigma\\s{\\cal O}}^2 \\geqslant 0$. This search can be viewed as a ``primal'' formulation of the linear program, whereas the search for $\\vec{\\alpha}$ described above can be viewed as the related ``dual'' problem. Note that in this formulation there are a continuously infinite number of possible vectors $\\vec{F}^{\\Delta_\\sigma}_{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}}$ in the search space. \\textcite{El-Showk:2014dwa} {developed} a modification of Dantzig's simplex algorithm in order to handle such a continuous search space.\\footnote{Such linear programming problems are called `continuous' or `semi-infinite'  \\cite{reemtsen_numerical_1998}.} The essential idea is to use Newton's method at each step of the algorithm to identify a vector to add, which is optimal over some continuous interval of scaling dimensions $\\left[\\Delta_{\\text{min}}, \\Delta_{\\text{max}}\\right]$ and discrete set of spins $\\left[0, \\ell_{\\text{max}}\\right]$. {For reasons explained in \\textcite{El-Showk:2014dwa}, it is necessary to perform computations at a precision higher than machine precision. This continuous simplex algorithm is one of two state-of-the art methods for the conformal bootstrap, the other one being the semidefinite programming method described below.} Three implementations of this algorithm are available: \na \\texttt{C++} code \\texttt{SIPSolver} \\cite{DSDSIPsolver} and a \\texttt{Python\/Cython} code \\cite{ElShowkRychkov} which were used for the computations in \\cite{El-Showk:2014dwa}, as well as a \\texttt{Julia} package \\texttt{JuliBootS}~\\cite{Paulos:2014vya}.\n\n\\subsection{Semidefinite programming}\n\\label{sec:SDP}\n\nWhile the linear programming techniques described above are adequate for crossing relations of single 4pt functions (possibly charged under some global symmetry), they are more difficult to adapt for systems of crossing relations containing multiple operators. The reason is that the resulting crossing relations for mixed correlators are no longer linear in the positive squares of OPE coefficients.\\footnote{However, they can be made linear at the expense of introducing additional continuous parameters~\\cite{El-Showk:2016mxr}. {This observation has not yet been implemented and it is not known how it would perform in practice.}} The same issue arises when considering 4pt functions of spinning operators, where multiple 3pt function tensor structures exist. In these situations one can phrase the optimization problem needed to obtain bounds using the language of semidefinite programming rather than linear programming \\cite{Kos:2014bka}.\\footnote{For a related problem of multiple internal symmetry coupling structures this was observed in \\textcite{Rattazzi:2010yc}.}\n\nAnother use of semidefinite programming \\cite{Poland:2011ey} is to avoid needing to discretize and impose a cutoff on the exchanged operator dimensions appearing in the positivity constraints such as $\\vec{\\alpha} \\cdot \\vec{F}_{\\Delta_{{\\cal O}}, \\ell_{{\\cal O}}} \\geqslant 0$. We will describe both of these uses of semidefinite programming, as well as how they can be combined, below.\n\nIn most applications to the bootstrap, it has proven necessary for numerical stability to solve the semidefinite programs described below at a precision higher than machine precision. The first numerical studies made use of the software \\texttt{SDPA-GMP}~\\cite{SDPAGMP} (a variant of \\texttt{SDPA}~\\cite{SDPA}) for this purpose. \n{The state-of-the-art is} an efficient software package \\texttt{SDPB}, described in \\textcite{Simmons-Duffin:2015qma}, which improves on the \\texttt{SDPA}'s primal-dual interior point algorithm primarily by taking advantage of matrix block structure and parallelization.\\footnote{{Further development of \\texttt{SDPB} is being carried out within the Simons Collaboration on the Nonperturbative Bootstrap (\\url{http:\/\/bootstrapcollaboration.com\/}), and this package will likely remain at the forefront of the numerical bootstrap studies in the coming years.}} In order to set up the problems so that they can be solved by \\texttt{SDPB}, recent studies have typically used either \\texttt{Mathematica} {notebooks}, or the interfaces \\texttt{PyCFTBoot}~\\cite{Behan:2016dtz} or \\texttt{cboot}~\\cite{CBoot}. \n\n\\subsubsection{Mixed correlators}\n\n\\label{sec:mixed-correlators}\n\nWe will illustrate the use of semidefinite programming for mixed correlators with a  simple example. Consider a system of 4pt functions containing two operators $\\sigma$ and $\\epsilon$, where $\\sigma$ is odd under a $\\mathbb{Z}_2$ symmetry and $\\epsilon$ is even. The resulting system of crossing relations for $\\<\\sigma\\s\\sigma\\s\\>$, $\\<\\sigma\\s\\epsilon\\e\\>$, and $\\<\\epsilon\\e\\epsilon\\e\\>$ takes the form\n\\cite{Kos:2014bka}\n\\beq\n\\label{eq:crossingequationwithv}\n 0  =  \\sum_{{\\cal O}^+} \\begin{pmatrix}\\lambda_{\\sigma\\s{\\cal O}} & \\lambda_{\\epsilon\\e{\\cal O}}\\end{pmatrix} \\vec{V}_{+,\\Delta,\\ell}\\begin{pmatrix} \\lambda_{\\sigma\\s{\\cal O}} \\\\ \\lambda_{\\epsilon\\e{\\cal O}} \\end{pmatrix}+ \\sum_{{\\cal O}^-} \\lambda_{\\sigma\\epsilon{\\cal O}}^2 \\vec{V}_{-,\\Delta,\\ell}\\,,\n\\end{equation}\nwhere the components of the vectors $\\vec{V}_{\\pm,\\Delta,\\ell}$ run over 5 independent crossing relations,\\footnote{In this section we are using vector notation to describe the vector of crossing relations, rather than derivatives.} ${\\cal O}_{\\pm}$ denote operators even\/odd under $\\mathbb{Z}_2$ symmetry, and each $\\vec{V}_{+,\\Delta,\\ell}$ is a 5-vector of $2 \\times 2$ matrices:\n\n\\beq\n\\vec{V}_{-,\\Delta,\\ell} = \\begin{pmatrix} 0  \\\\ 0 \\\\ F_{-,\\Delta,\\ell}^{\\sigma\\epsilon,\\sigma\\epsilon}(z,\\bar{z}) \\\\ (-1)^{\\ell} F_{-,\\Delta,\\ell}^{\\epsilon\\sigma,\\sigma\\epsilon}(z,\\bar{z}) \\\\ - (-1)^{\\ell} F_{+,\\Delta,\\ell}^{\\epsilon\\sigma,\\sigma\\epsilon}(z,\\bar{z}) \\end{pmatrix},\\nonumber\n\\end{equation} \n\\beq\n\\vec{V}_{+,\\Delta,\\ell} = \\begin{pmatrix} \\begin{pmatrix}  F^{\\sigma\\s,\\sigma\\s}_{-,\\Delta,\\ell}(z,\\bar{z}) & 0 \\\\ 0 & 0  \\end{pmatrix} \\\\ \\begin{pmatrix}  0 & 0 \\\\ 0 & F^{\\epsilon\\e,\\epsilon\\e}_{-,\\Delta,\\ell}(z,\\bar{z})  \\end{pmatrix}\\\\ \\begin{pmatrix}  0 & 0 \\\\ 0 & 0  \\end{pmatrix}  \\\\ \\begin{pmatrix}  0 & \\frac12 F^{\\sigma\\s,\\epsilon\\e}_{-,\\Delta,\\ell}(z,\\bar{z}) \\\\ \\frac12 F^{\\sigma\\s,\\epsilon\\e}_{-,\\Delta,\\ell}(z,\\bar{z}) & 0 \\end{pmatrix} \\\\ \\begin{pmatrix} 0 & \\frac12 F^{\\sigma\\s,\\epsilon\\e}_{+,\\Delta,\\ell}(z,\\bar{z}) \\\\ \\frac12 F^{\\sigma\\s,\\epsilon\\e}_{+,\\Delta,\\ell}(z,\\bar{z}) & 0  \\end{pmatrix} \\end{pmatrix}.\n\\end{equation}\nThe appearing functions $F^{ij,kl}_{\\pm,\\Delta,\\ell}$ are given in \\reef{eq:Funeq}. One can then look for bounds by making some assumption about the spectrum and searching for a functional $\\vec{\\alpha} = \\sum_{mn} \\vec{\\alpha}_{mn} \\partial_{z}^m \\partial_{\\bar{z}}^n \\left[\\cdot\\right] \\big|_{z=\\bar{z}=1\/2}$ satisfying the properties\n\\begin{align}\n&\\begin{pmatrix} 1 & 1\\end{pmatrix} \\vec \\alpha \\cdot \\vec{V}_{+,0,0} \\begin{pmatrix} 1 \\\\ \n1 \\end{pmatrix}  > 0 \\,, \\nn\\\\\n&\\vec \\alpha \\cdot \\vec{V}_{+,\\Delta,\\ell} \\succeq 0 \\quad\\textrm{for all $\\mathbb{Z}_2$-even operators with $\\ell$ even,} \\nn\\\\\n&\\vec \\alpha \\cdot \\vec{V}_{-,\\Delta,\\ell}  \\geqslant 0 \\quad \\textrm{for all $\\mathbb{Z}_2$-odd operators.} \n\\label{eq:functionalinequalities1}\n\\end{align}\nThe novel feature is now that $\\vec \\alpha \\cdot \\vec{V}_{+,\\Delta,\\ell} \\succeq 0$ must be a {\\it positive semidefinite} $2 \\times 2$ matrix, which makes the search in Eq.~(\\ref{eq:functionalinequalities1}) a semidefinite programming problem. Similar structure appears for more general systems of mixed\/spinning correlators, where if an exchanged operator has $N$ OPE coefficients appearing in the system then the needed positivity condition will be phrased in terms of positive semidefinite $N \\times N$ matrices.\n\n\n\\subsubsection{Polynomial approximations}\n\\label{sec:poly}\nA different use of semidefinite programming, relevant for both single correlators or mixed correlators, is to avoid any discretization of the exchanged operator dimensions~\\cite{Poland:2011ey}. We will first explain the idea for single correlators, where one imposes inequalities of the form\n\\begin{eqnarray}\n\\sum_{mn} \\alpha_{mn} \\partial_z^m \\partial_{\\bar{z}}^n F^{\\Delta_\\sigma}_{\\Delta,\\ell}(z,\\bar{z}) \\big|_{z=\\bar{z}=1\/2} &\\geqslant& 0\\,.\n\\end{eqnarray} \nDue to the pole expansion of the conformal blocks described in Sec.~\\ref{sec:analyticrecursion}, if one keeps a finite number of poles{,} then by reorganizing $h_{\\Delta,\\ell}$ into a rational function of $\\Delta$, such derivatives can be rewritten in the form (see Sec.~\\ref{sec:rational})\n\\begin{eqnarray}\n\\partial_z^m \\partial_{\\bar{z}}^n F^{\\Delta_\\sigma}_{\\Delta,\\ell}(z,\\bar{z}) \\big|_{z=\\bar{z}=1\/2} \\approx \\chi_{\\ell}(\\Delta) P_{\\ell}^{mn}(\\Delta)\\,,\n\\end{eqnarray}\nwhere $P_{\\ell}^{mn}(\\Delta)$ is a {\\it polynomial} in $\\Delta$, and\n$\\chi_{\\ell}(\\Delta)$ is a positive function for all $\\Delta$ and $\\ell$ satisfying the unitarity bounds. The degree of the polynomial depends on the number of poles kept in the expansion of the conformal block. Then one simply needs to impose the polynomial inequalities\n\\begin{eqnarray}\n\\sum_{mn} \\alpha_{mn} P^{mn}_{\\ell}(\\Delta^{\\text{min}}_{\\ell}+x) \\geqslant 0\n\\end{eqnarray}\n for all $x \\geqslant 0$, where the minimum dimension at each spin $\\Delta^{\\text{min}}_{\\ell}$ depends on the assumptions being made. \n \nSuch inequalities for polynomials can be rewritten in terms of positive semidefinite matrices following a theorem of~\\textcite{Hilbert:1888}. The relevant theorem states that any polynomial $P(x)$ that is nonnegative on the interval $[0,\\infty)$ can be written in the form\n\\beq\nP(x) = a(x) + x b(x),\n\\end{equation}\nwhere $a(x)$ and $b(x)$ are sums-of-squares of polynomials. Such sums-of-squares can in turn always be expressed in the form\n\\beq\na(x) = \\text{Tr}(A Q_{d_1}(x)), \\quad b(x) = \\text{Tr}(B Q_{d_2}(x))\\,,\n\\end{equation}\n\\\\[2pt]\nwhere $Q_{d}(x)\\equiv [x]_d [x]_d^T$ is a matrix built out of the monomials $[x]_d = (1,x,\\ldots, x^d)^T$, $d_1 = \\left\\lfloor \\frac12 \\deg P \\right\\rfloor$, $d_2 = \\left\\lfloor \\frac12 (\\deg P-1) \\right\\rfloor$, and $A$ and $B$ are some positive semidefinite matrices.\n\nWith this rewriting, one needs to search for coefficients $\\alpha_{mn}$ and positive semidefinite matrices $A_{\\ell}, B_{\\ell} \\succeq 0$ such that\n\\begin{multline}\n\\sum_{mn} \\alpha_{mn} P^{mn}_{\\ell}(\\Delta^{\\text{min}}_{\\ell}+x) =\\\\\n \\text{Tr}(A_{\\ell} Q_{d_1}(x)) + x \\text{Tr}(B_{\\ell} Q_{d_2}(x)).\n\\end{multline}\nIn practice one must also impose a cutoff on the set of included spins $0 \\leqslant \\ell \\leqslant \\ell_{\\text{max}}$. This search, combined with a normalization condition such as $\\vec{\\alpha} \\cdot \\vec{F}_{0,0}^{\\Delta_\\sigma} = 1$, is now in the form of a semidefinite programming problem.  \n\nAs explained in detail in \\textcite{Kos:2014bka}, this idea can also be applied to systems of mixed or spinning correlators where exchanged operators have multiple OPE coefficients appearing in the system. In those cases, after truncating the conformal block pole expansions one {imposes} constraints of the form\n\\begin{gather}\n\\sum_{mn}\\vec{\\alpha}_{mn} \\cdot\n\\begin{pmatrix}\n\\vec{P}^{(11;mn)}_\\ell(\\Delta) &\\dots& \\vec{P}_\\ell^{(1N;mn)}(\\Delta)\\\\\n\\vdots & \\ddots &\\vdots\\\\\n\\vec{P}^{(N1;mn)}_\\ell(\\Delta) & \\dots & \\vec{P}_\\ell^{(NN;mn)}(\\Delta)\n\\end{pmatrix}\\succeq 0\n\\label{eq:matrixpolynomialconstraints1}\\\\\n\\textrm{for}\\quad \\Delta\\geqslant \\Delta^{\\text{min}}_{\\ell}.\\nn\n\\end{gather}\nAgain there is a theorem that such positive semidefinite matrix polynomials can always be written as sums-of-squares of matrix polynomials. A consequence, worked out in \\textcite{Kos:2014bka}, is that each entry can be written as \n\\begin{multline}\n\\sum_{mn} \\vec{\\alpha}_{mn} \\cdot \\vec{P}^{(ij;mn)}_\\ell(\\Delta^{\\text{min}}_{\\ell}+x) =\\\\ \\text{Tr}(A^{ij}_{\\ell} Q_{d_1}(x)) + x \\text{Tr}(B^{ij}_{\\ell} Q_{d_2}(x))\n\\end{multline}\nin terms of positive semidefinite matrices \\mbox{$A_{\\ell}^{ij}, B_{\\ell}^{ij} \\succeq 0$}, \nand the problem is again phrased as a semidefinite programming problem.\n\n\\subsection{Bounds and allowed regions}\n\n\\label{sec:allowed}\nThe algorithms described in the previous sections can be used to establish if a given point in the space of CFT data, parametrized by the dimensions of external operators and by assumptions on the exchanged spectrum, belongs to the region allowed by crossing and unitarity. Since the exchanged spectrum contains infinitely many operators, there are infinitely many assumptions one can test. The art of the numerical bootstrap is to choose an interesting assumption, and then to delineate as precisely as possible the allowed region corresponding to this assumption. \n\nAs we will see in the next sections, one of the most frequently asked questions is the following: given an OPE $\\mathcal O\\times \\mathcal O$, derive an upper bound $\\Delta_\\text{max}$ on the dimension of the first operator appearing in this OPE having specified transformation properties under $SO(d)$ (and eventually under a global symmetry $G$),\\footnote{In particular, the existence of a bound with $\\Delta_{\\max}<\\infty$ provides a proof that such an operator exists. See \\textcite{Rattazzi:2008pe} and \\cite[{section 10.5}]{Simmons-Duffin:2016gjk} for intuitive explanations involving some numerics of why such bounds should exist at all, and \\textcite{Hogervorst:2013sma} and \\cite[{section 4.3.3}]{Rychkov:2016iqz} for an approximate analytic argument. At present, while the existence of bounds can sometimes be understood via such simple means, their actual values can only be precisely computed using the powerful numerical techniques described in the previous sections. Only in a handful of cases, e.g.~\\textcite{Mazac:2016qev} and \\textcite{Mazac:2018mdx}, have the best possible bounds been proven analytically.} assuming e.g.~that other operator {dimensions are allowed to take any values} allowed by unitarity. One can answer this question, for instance, as a function of $\\Delta_{\\mathcal O}$. This defines an allowed region with a boundary $\\Delta_\\text{max}(\\Delta_\\mathcal O)$. Similarly, when one obtains an upper (or lower) bound on an OPE {coefficient} as discussed in Sec.~\\ref{sec:LP}, this represents the boundary of an allowed region. These boundaries give us a view into the intricate underlying geometry of the space of CFT data allowed by crossing and unitarity.\n\n\n\\subsection{Spectrum extraction}\n\\label{sec:spectrum-extraction}\n\nA point in the allowed region (see Sec.~\\ref{sec:allowed}) is specified by external operator dimensions and by a handful of other numbers characterizing the assumptions, such as gaps on the exchanged operator spectrum. Once we ascertained that a point belongs to the allowed region, in some cases it is important to be able to go one step further and to extract an explicit solution to crossing, i.e.~the whole spectrum of exchanged operator dimensions and their OPE coefficients. The precise way of doing this depends on which algorithm one uses. An important point is that we expect this solution to be {non-unique} inside the allowed region, but it should generically become unique on its boundary (see below).\n\nThe spectrum extraction is simplest in the primal simplex method,  Sec.~\\ref{sec:modified-simplex}. In this case the spectrum is encoded directly in the set of basic vectors and is available in each step of the algorithm.\n\nIn the dual formulation of the linear programming method, one does not have access to the spectrum strictly inside the allowed region. However, one can extract a solution to crossing symmetry from the limiting functional when one approaches a {\\it boundary} of this region, by extremizing either an operator dimension or OPE coefficient. This is called the {\\it extremal functional method}, introduced in \\textcite{Poland:2010wg} and \\textcite{ElShowk:2012hu}.\n\nNamely, when approaching the boundary from the disallowed region, the system is on the verge of no longer allowing a separating plane, and the vector on which we require strict positivity is degenerating into the plane. In the case of a single crossing relation where we have imposed strict positivity on the identity operator, as we approach a dimension boundary we can find a vector $\\vec{\\alpha}$ such that $\\vec{\\alpha} \\cdot \\vec{F}^{\\Delta_{\\sigma}}_{0,0} \\rightarrow 0$, together with the sum rule\n\\begin{eqnarray}\\label{eq:extremal}\n\t0 &=& \\sum_{{\\cal O}} \\lambda_{\\sigma\\s{\\cal O}}^2 \\vec{\\alpha} \\cdot \\vec{F}^{\\Delta_{\\sigma}}_{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}},\n\\end{eqnarray}\nwhere $\\vec{\\alpha} \\cdot \\vec{F}^{\\Delta_{\\sigma}}_{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}} \\geqslant 0$ for all other possible (non-identity) operators in the spectrum. One obtains a similar condition from the OPE coefficient bound in Eq.~(\\ref{eq:OPEbound}) if one sets the OPE coefficient to its extremal value $\\lambda_{\\sigma\\s{\\cal O}^*}^2 = -\\vec{\\alpha} \\cdot \\vec{F}_{0,0}^{\\Delta_{\\sigma}}$. \n\nIn fact, it is easy to see that in order for these sums to hold along the boundary of the allowed region, it is necessary for either $\\lambda_{\\sigma\\s{\\cal O}}^2$ to be zero or for $\\vec{\\alpha} \\cdot \\vec{F}^{\\Delta_{\\sigma}}_{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}}$ to be zero. Thus, the zeroes of $\\vec{\\alpha} \\cdot \\vec{F}^{\\Delta_{\\sigma}}_{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}}$ tell us the scaling dimensions and spins at which the OPE coefficients are allowed to be nonzero.  The resulting {\\it extremal spectrum} is generically unique~\\cite{ElShowk:2012hu}.\n\nIn the above-mentioned primal simplex method, the extremal spectrum is reached from within the allowed region, and is encoded in the set of basic vectors that remain after the algorithm terminates. That this should agree with the dual approach via extremal functionals is guaranteed by the strong duality of linear programs.\n\nThe extremal functional method for extracting the spectrum is also applicable when using semidefinite programming. In this case the extremal functional is constructed in the dual formulation.\\footnote{It is not understood at present how to formulate an algorithm to extract the extremal spectrum along a dimension bound directly from the allowed region in the context of semidefinite programming. The currently used procedure is to sit in the interior of the space allowed by scaling dimension bounds and extremize an OPE coefficient to find an extremal functional.} Once the extremal spectrum is known, it is straightforward to reconstruct the OPE coefficients of the exchanged operators by either directly solving the bootstrap equations after inputting the extremal spectrum or extracting them from the primal solution of the primal-dual algorithm. {\\textcite{Simmons-Duffin:2016wlq} gives a precise algorithm for doing this using functionals output by \\texttt{SDPB}, realized in a \\texttt{Python} code\n\t\\cite{DSD-spectrum-extraction}.}\n\nAn important {open question is to understand} which CFTs are described by spectra which are extremal with respect to some extremization condition. As we will see in subsequent sections, empirically this seems to be the case for a variety of interesting CFTs including the 3d Ising and $O(N)$ models. Although it is not currently understood why it should be so, some speculations are given in Sec.~\\ref{sec:Z2-spectrum}.\n\n \\subsubsection{Flow method}\n \n \\label{sec:flow}\nAn interesting idea was proposed in \\textcite{El-Showk:2016mxr}, where given one extremal solution one can efficiently ``flow'' along the boundary to reconstruct nearby extremal solutions. The idea is to perturb the extremal spectrum and then impose that the perturbed spectrum is also extremal using (\\ref{eq:extremal}) as well as the tangency conditions $\\vec{\\alpha} \\cdot \\left(\\partial_{\\Delta_{{\\cal O}}} \\vec{F}^{\\Delta_{\\sigma}}_{\\Delta_{{\\cal O}},\\ell_{{\\cal O}}}\\right)$.  By linearizing perturbations of these conditions, the search for a nearby extremal spectrum (or a more precise extremal spectrum) can be efficiently solved using Newton's method. This approach then avoids the use of convex optimization after the initial step of finding an initial extremal solution, and can also be used to flow to nonunitary extremal solutions. This idea was shown to work well in $d=1$ in \\textcite{El-Showk:2016mxr,Paulos:2016fap}.\\footnote{The code is implemented as a separate module of \\texttt{JuliBoots}~\\cite{Paulos:2014vya}, available on request from its author.} It appears very promising and it needs to be further explored and extended, especially into higher dimensions.\n\n\\subsection{Truncation method}\n\\label{sec:truncation}\n\nFinally we wish to turn to an idea introduced by \\textcite{Gliozzi:2013ysa} and explored in a variety of works,\\footnote{See \\textcite{Gliozzi:2014jsa}, \\textcite{Gliozzi:2015qsa}, \\textcite{Gliozzi:2016cmg}, \\textcite{Esterlis:2016psv}, \\textcite{Hikami:2017hwv,Hikami:2017sbg,Hikami:2018mrf}, \\textcite{Li:2017agi,Li:2017ukc}, and \\textcite{LeClair:2018edq}.} which we will call the truncation method. The basic idea is to truncate the bootstrap equations to a finite number of operators $\\{\\Delta_{\\sigma}, {\\cal O}_I \\}$ with $N$ unknown scaling dimensions.  After normalizing by the identity contribution $f^{\\Delta_{\\sigma}}_{\\Delta_{{\\cal O}_I},\\ell_{{\\cal O}_I}}(z,\\bar{z}) \\equiv F^{\\Delta_{\\sigma}}_{\\Delta_{{\\cal O}_I},\\ell_{{\\cal O}_I}}(z,\\bar{z})\/\\left(-F^{\\Delta_{\\sigma}}_{0,0}(z,\\bar{z})\\right)$, let us write the crossing equations as\n\\begin{eqnarray}\\label{eq:linearsystem}\n\\sum_{{\\cal O}_I} \\lambda_{\\sigma\\s{\\cal O}_I}^2 f^{\\Delta_{\\sigma}}_{\\Delta_{{\\cal O}_I},\\ell_{{\\cal O}_I}} &=& 1,\\nonumber\\\\\n\\sum_{{\\cal O}_I} \\lambda_{\\sigma\\s{\\cal O}_I}^2 \\vec{f}^{\\Delta_{\\sigma}}_{\\Delta_{{\\cal O}_I},\\ell_{{\\cal O}_I}} &=& 0.\n\\end{eqnarray}\nHere the first equation containing $f^{\\Delta_{\\sigma}}_{\\Delta_{{\\cal O}_I},\\ell_{{\\cal O}_I}} \\equiv f^{\\Delta_{\\sigma}}_{\\Delta_{{\\cal O}_I},\\ell_{{\\cal O}_I}}(1\/2,1\/2)$ is viewed as an ``inhomogeneous\" equation containing the identity contribution on the right-hand side, and the second ``homogeneous\" equation contains the vector of derivatives $\\left(\\vec{f}^{\\Delta_{\\sigma}}_{\\Delta_{{\\cal O}_I},\\ell_{{\\cal O}_I}}\\right)^{mn} = \\partial_{z}^m \\partial_{\\bar{z}}^n f^{\\Delta_\\sigma}_{\\Delta_{{\\cal O}_I}, \\ell_{{\\cal O}_I}}(z,\\bar{z})\\big|_{z=\\bar{z}=1\/2}$. If one keeps $M$ derivatives with $M > N$ then the system becomes over-constrained, and only has solutions if all of the minors of order $N$ of the linear system vanish,\n\\begin{eqnarray}\\label{eq:det}\n\\text{det}A_i=0,\\quad A_i\\subset A=\\left[ \\left(\\vec{f}^{\\Delta_{\\sigma}}_{\\Delta_{{\\cal O}_I},\\ell_{{\\cal O}_I}}\\right)^{mn} \\right]_{N\\times M}.\n\\end{eqnarray}\nHere the ``rows\" of $A$ would run over different choices of $N$ derivatives $mn$ and the ``columns\" run over the $N$ unknown scaling dimensions. Note that the set of unknown scaling dimensions will include the external dimension $\\Delta_{\\sigma}$ in addition to the ${\\cal O}_I$, but may exclude exchanged operators of known dimension, such as the stress tensor of known dimension $\\Delta_T = d$. The general strategy is to solve the determinant conditions Eq.~(\\ref{eq:det}) to obtain an approximate spectrum of scaling dimensions, and then use the system in Eq.~(\\ref{eq:linearsystem}), including the inhomogeneous equation, in order to fix the OPE coefficients.\n\nA big advantage of the truncation approach over the linear and semidefinite programming approaches of the previous sections is that it does not require unitarity, i.e.~it works equally well for any sign of the OPE coefficients. For example, the idea has been successfully applied to the nonunitary Lee-Yang model, as well as to bulk-boundary bootstrap problems where there is no positivity in the coefficients. Another advantage is that it is relatively simple to implement, and the idea can be explored e.g.~using fairly simple \\texttt{Mathematica} notebooks. \n\nOn the other hand, we also see several disadvantages with this approach in its current incarnation. One is that the resulting spectrum can have a strong sensitivity to the set of included operators (e.g., the choices of spins) and to the set of derivatives included. It is also very difficult to assign reliable errors to the spectrum output from the method.\\footnote{Comparison with the rigorous results obtained using the linear and semidefinite programming methods, when possible, shows that the published truncation method errors are often underestimated.} Thus, it would be desirable to find ways to make the approach more systematic with errors under control. Some steps in this direction were recently taken in \\textcite{Li:2017ukc}. Applications to the boundary bootstrap also seem to be less sensitive to these issues~\\cite{Gliozzi:2015qsa,Gliozzi:2016cmg}. \n\nAnother issue is that simple implementations of numerical studies of the nonlinear determinant conditions~(\\ref{eq:det}), such as using the iterative Newton method implemented in \\texttt{Mathematica}'s \\texttt{FindRoot} function, do not scale very well with increasing the number of operators and the method likely needs a more efficient numerical implementation in order to push beyond $\\sim 10$ operators.\\footnote{One can view the flow method described in Sec.~\\ref{sec:flow} as a kind of more efficient implementation where additional extremality conditions have been added.}  \n\nNotice that since we are truncating the spectrum, we cannot generally expect to find \\emph{exact} solutions of Eq.~\\reef{eq:det}. On the other hand, the set of determinant conditions is in fact redundant because of the Pl\\\"ucker relations satisfied by the minors of a matrix, see \\textcite{Hikami:2017hwv}. A cleaner numerical formulation can be obtained by replacing Eq.~\\reef{eq:det} with the problem of minimizing the smallest singular value of the matrix $A$~\\cite{Esterlis:2016psv,LeClair:2018edq}.\n\nFinally, similarly to the extremal spectra methods above, it is not clear which CFT spectra are ``truncable\" in the sense that they can be found with this approach.\n\n\n\n\n\n\\subsubsection{General results}\n\\label{sec:ON-general}\n\nAs discussed in Sec.~\\ref{sec:global}, correlation functions of CFT operators that are in irreducible representations of the global symmetry $G$ can be organized using group theory and decomposed into different $G$-invariant tensor structures. Sec.~\\ref{sec:crossing} explained how these structures enter the crossing relations. The first numerical analyses of the resulting equations occurred in the context of 4d CFTs,\\footnote{See \\textcite{Poland:2010wg}, \\textcite{Rattazzi:2010yc}, \\textcite{Vichi:2011ux}, and \\textcite{Poland:2011ey}.} but the group theoretic structure is $d$-independent. The bootstrap for $O(N)$ symmetry in 3d was investigated by \\textcite{Kos:2013tga, Kos:2015mba,Kos:2016ysd} and \\textcite{Nakayama:2014yia}.\n\nWe will start our analysis assuming that the CFT contains an operator $\\phi\\equiv(\\phi_a)_{a=1}^N$ \nin the fundamental representation of $O(N)$, of dimension $\\Delta_\\phi$. Mimicking the discussion in Sec.~\\ref{sec:Z2-general}, we would like to learn about the operators in the OPE $\\phi_{a} \\times \\phi_{b}$. By group theory, operators of even spin $\\ell$ in this OPE will transform as $O(N)$ singlets or symmetric traceless tensors of rank 2, while odd-spin operators will transform in the rank-2 antisymmetric representation. \n\nFrom the crossing relations for the 4pt function of $\\phi$ one can put upper bounds on the dimensions of various operators. For the lowest dimension scalars ($\\ell=0$) in the singlet ($s$) and symmetric traceless tensor ($t$) sector, these bounds are shown in Figs.~\\ref{fig:ONsinglet}, \\ref{fig:ONsymtrace} as a function of $\\Delta_\\phi$ for various values of $N$. The ``kinks\" in these bounds will be interpreted in the next section. \n\\begin{figure}\n\\includegraphics[width=\\figwidth]{fig20-singletbound.pdf}\n\\caption{\\label{fig:ONsinglet}(Color online) Upper bound on the dimension of $s$ \\cite{Kos:2013tga}.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\figwidth]{fig21-symtracelessbound.pdf}\n\\caption{\\label{fig:ONsymtrace}(Color online) Upper bound on the dimension of $t$ \\cite{Kos:2013tga}.}\n\\end{figure}\n\nThe $\\phi\\times\\phi$ OPE also contains two interesting operators of spin $\\ell\\geqslant 1$: the stress tensor $T$ and the conserved current $J$. Using the bootstrap one can put lower bounds on their two-point function coefficients $C_T$ and $C_J$ (defined in Sec.~\\ref{sec:ward}) given in Figs.~\\ref{fig:ONCT}, \\ref{fig:ONCJ}. This is done by bounding from above the OPE coefficients $\\lambda_{\\phi\\phi T}\\propto \\Delta_{\\phi}\/\\sqrt{C_T}$ and $\\lambda_{\\phi\\phi J}\\propto 1\/\\sqrt{C_J}$ {(see Eqs.~(\\ref{eq:lambdaT}, \\ref{eq:lambdaJ}))}.\n\nLet's discuss the monotonicity of these bounds with $N$. Since $O(N+1)\\supset O(N)$, the bounds on $C_T$, $C_J$, and on $\\Delta_t$ should get stronger with increasing $N$, and indeed they do (notice that $C_T$ is plotted divided by $N$). Although it may seem counterintuitive that the $\\Delta_s$ bound gets weaker with $N$, there is no contradiction. The point is that the symmetric traceless tensor of $O(N+1)$ contains a singlet $\\tilde s$ when decomposed with respect to $O(N)$. Therefore the only constraint is that the $O(N)$ singlet bound should be weaker than the $O(N+1)$ symmetric traceless bound, which is satisfied by inspection.\n\nNotice also that the scaling of the $C_T$, $C_J$ bounds with $N$ close to $\\Delta_\\phi=1\/2$ is consistent with the fact that in the theory of $N$ free scalars, $C_T$ grows linearly with $N$ while $C_J$ is constant.\n\n\\begin{figure}\n\\includegraphics[width=\\figwidth]{fig22-CT.pdf}\n\\caption{\\label{fig:ONCT}(Color online) Lower bound on $C_T$ computed under the assumption $\\Delta_s, \\Delta_t\\geqslant 1$ \\cite{Kos:2013tga}.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\figwidth]{fig23-CJ.pdf}\n\\caption{\\label{fig:ONCJ}(Color online) Lower bound on $C_J$ \\cite{Nakayama:2014yia}.}\n\\end{figure}\n\n\\subsubsection{Critical $O(N)$ model}\n\nThe most famous 3d CFT with $O(N)$ symmetry is the critical point of the $O(N)$ lattice model, which is the generalization of Eq.~(\\ref{eq:Ising}) to $N$-component spins satisfying the constraint $|\\vec{s}|=1$.\nThis CFT is also known as the Wilson-Fisher fixed point, being an IR fixed point of the $O(N)$ symmetric scalar field theory with quartic interaction~\\cite{Wilson:1971dc}. For any integer $N$ this 3d CFT is unitary, given that the microscopic realizations are unitary.\\footnote{\\label{note:ONnonunitary}Sometimes one discusses analytic continuation of $O(N)$ models to noninteger $N$. These analytic continuations are nonunitary~\\cite{Maldacena:2011jn}, and fall outside the range of validity of the linear\/semidefinite methods. Although such attempts were made~\\cite{Shimada:2015gda}, we would advise caution. Here we will only consider integer $N\\geqslant 2$. We will discuss nonunitary CFTs in Sec.~\\ref{sec:nonunitary}.}\n\nIt's natural to ask where the critical $O(N)$ models lie in the parameter space of $O(N)$ symmetric CFTs allowed by the general bounds from the previous section. In the Wilson-Fisher description, $\\phi_{a}$ is the fundamental scalar field appearing in the Lagrangian, $s=\\phi^2$, and $t$ is the traceless part of $\\phi_a \\phi_b$. Dimensions of these fields have been previously estimated using RG methods (in particular the $\\epsilon$-expansion and the large $N$ expansion), Monte Carlo studies, and experiments. Comparing the $s$ and $t$ bounds and these prior determinations, marked with crosses in Figs.~\\ref{fig:ONsinglet}, \\ref{fig:ONsymtrace}, one is led to conjecture that the critical $O(N)$ models correspond to the ``kinks\". Similar kink-like features are visible in the lower bounds on $C_J$ and $C_T$. In the latter case the kinks can be made sharper by imposing that the $S$ operator saturate the gap, see Fig.~5 in \\textcite{Kos:2013tga}. This conjecture can be used to extract values of the $\\phi$, $s$, $t$ dimensions and of $C_T$, given in Table 3 of \\textcite{Kos:2013tga}. \n\nWe will now discuss how to isolate the critical $O(N)$ models without relying on the kink conjecture. The idea is to exploit the crucial physical feature of these CFTs --- that they possess robust gaps in the operator spectrum. The singlet scalar $s$ corresponds to the temperature deformation of the critical point and is relevant. The next singlet scalar, $s'$, must necessarily be irrelevant (otherwise the critical point would be multicritical), implying the gap $\\Delta_{s'}\\geqslant 3$ in the singlet scalar sector. We also expect a gap in the fundamental representation scalar sector. The order parameter $\\phi_{a}$ belongs to this sector and is relevant, while most likely the next fundamental scalar is irrelevant: $\\Delta_{\\phi'}\\geqslant 3$. This can be also deduced using the Wilson-Fisher description, using a nonrigorous but suggestive equation of motion argument~\\cite{Kos:2015mba}.\n\n\\textcite{Kos:2015mba}~studied bootstrap constraints for the system of three correlators \n{$\\{\\langle \\phi_a\\phi_b\\phi_c\\phi_d\\rangle$, $\\langle \\phi_a\\phi_b ss\\rangle$, $\\langle ssss\\rangle\\}$. } \nImposing the assumptions $\\Delta_{s'}\\geqslant 3$, $\\Delta_{\\phi'}\\geqslant 3$, they found small allowed regions (``islands\") shown in Fig.~\\ref{fig:ONarchipelago}. Improved versions of these islands for $O(2)$ and $O(3)$, discussed in the next sections, were subsequently obtained in \\textcite{Kos:2016ysd}. It's important to stress that, like in Fig.~\\ref{fig:Z2-mixed-sigpgap} for the Ising model, there are disconnected allowed regions outside the shown part of the parameter space; see e.g.~Fig.~\\ref{fig:O2singlet} below for the $O(2)$ case. These regions are practically unexplored and they might contain other interesting CFTs.\n\\begin{figure}\n\\includegraphics[width=\\figwidth]{fig24-ONarchipelago.pdf}\n\\caption{\\label{fig:ONarchipelago}(Color online) The $O(N)$ archipelago \\cite{Kos:2015mba}.}\n\\end{figure}\n\n\n\\subsubsection{Monopole bootstrap for QED${}_3$}\n\\label{sec:QED3-monopole}\n\nAn alternate approach, pursued in \\textcite{Chester:2016wrc} and \\textcite{Chester:2017vdh}, is to focus on monopole operators. When dealing with a compact $U(1)$ gauge field, these operators create topologically nontrivial configurations of the gauge field having magnetic flux emerging from a spacetime point.\\footnote{Thus they could also be called instantons, but the common terminology refers to them as monopoles.} Such operators are charged under a topological $U(1)_T$ global symmetry with symmetry current \\mbox{$J_T^{\\mu} = \\frac{1}{8\\pi} \\epsilon^{\\mu \\nu \\rho}F_{\\nu\\rho}$}. Taking the monopole operators to have charge $q \\in \\mathbb{Z}\/2$, the scalar monopoles transform in representations of $SU(N_f)$ corresponding to fully rectangular Young diagrams with $N_f\/2$ rows and $2|q|$ columns~\\cite{Dyer:2013fja}. Thus, the lightest scalar monopoles $M^I_{\\pm1\/2}$ are expected to be in $SU(N_f)$ representations with $N_f\/2$ fully antisymmetric indices.\\footnote{Monopoles with spin transform in other nontrivial flavor representations, see \\textcite{Chester:2017vdh}.}\n\nThe bootstrap for 4pt functions of $M^I_{\\pm1\/2}$ was studied in \\textcite{Chester:2016wrc} for $N_f = 2, 4, 6$, where they focused on placing bounds on the dimension of the second monopole operator $\\Delta_{M_{1}}$, making various assumptions about gaps in the uncharged $(q=0)$ sector. These bounds are shown in Figs.~\\ref{fig:QED3-monopole26}, \\ref{fig:QED3-monopole4}, where for $N_f=4,6$ they can be compared with the large $N_f$ estimate (black cross). Intriguingly, there is a kink-like discontinuity in the bound which comes close to the large $N_f$ estimate for certain values of the gap in the uncharged sector for operators in the same $SU(N_f)$ representation. By increasing the gap above $M_1$, the allowed region could also be turned into a peninsula around the kink. Similar bounds for the lightest spinning monopoles in the case $N_f = 4$, along with a comparison to the large $N_f$ predictions, were presented in \\textcite{Chester:2017vdh}.\n\nWhile these results are not definitive, they seem promising and show that the bootstrap for QED${}_3$ has a reasonable chance to be successful, perhaps after a few more ingredients are added. Some possible directions would be to consider a multiple correlator bootstrap involving $M_{\\pm1\/2}$, $M_{\\pm1}$, and\/or $\\bar{\\psi}_i \\psi^j$. It may also be fruitful to combine these with constraints from 4pt functions containing the $U(1)_T$ current, the $SU(N_f)$ current, or the stress tensor. \n\n \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig34a-MN2plot}(a)\n\\includegraphics[width=\\figwidth]{fig34b-MN6plot}(b)\n    \\caption{\\label{fig:QED3-monopole26} (Color online) Bounds on $\\Delta_{M_1}$ in terms of $\\Delta_{M_{1\/2}}$ in $d=3$ for $N_f=2,6$ (a,b) with various assumptions on the gaps in the uncharged sector in the same $SU(N_f)$ representation as $M_1$ \\cite{Chester:2016wrc}.}\n  \\end{figure}\n  \n   \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig35a-MN4plot}(a)\n\\includegraphics[width=\\figwidth]{fig35b-MN4Daggerplot}(b)\n    \\caption{\\label{fig:QED3-monopole4} (Color online) (a) is the analogue of Fig.~\\ref{fig:QED3-monopole26} for $N_f=4$. (b) starts from the $\\Delta_2\\geq3$ case of (a), and shows that placing an additional gap $\\Delta_{M'_1}$ above $\\Delta_{M_1}$ turns the kink into a peninsula \\cite{Chester:2016wrc}.}\n  \\end{figure}\n  \n\\subsubsection{Bosonic QED${}_3$ and deconfined quantum critical points}\n\\label{bQED3}\n\nFinally we would like to review the rich physics of bosonic QED${}_3$, where some bootstrap insights have recently been obtained. Bosonic QED${}_3$ is obtained by coupling the $U(1)$ gauge field to $N$ complex scalars $\\phi_i$ with an $SU(N)$ invariant potential $m^2 |\\phi|^2 +\\lambda (|\\phi|^2)^2$. This is also known as the $N$-component abelian Higgs model, and is believed to flow to a CFT for large enough $N$. Unlike for fermions, the boson mass term preserves all the symmetries and has to be fine-tuned to reach the fixed point.\n\nThis model has been much discussed in the condensed matter literature as the ``non-compact CP$^{N-1}$ model'' (NCCP$^{N-1}$) in connection with the phenomenon of ``deconfined criticality\" \\cite{deconfined}. To briefly review this connection, the physical systems of interest are certain quantum antiferromagnets in $(2+1)$ dimensions, which have a quantum phase transition between N\\'eel and Valence-Bond-Solid (VBS) phases.\\footnote{The absence of a disordered phase in such transitions can be understood using `t Hooft anomalies, see \\textcite{Komargodski:2017dmc}. This perspective also gives insight into the rich physics of interfaces in these theories \\cite{Komargodski:2017smk}.} The transition can be described by the $O(3)$ nonlinear sigma model (NLSM) for the N\\'eel order parameter, modified by the inclusion of Berry phase effects which suppress topological defects (hedgehogs), which will play an important role below. \n\nThe $O(3)$ NLSM can be written as the CP$^1$ model, which has two complex vectors $\\mathbf{z}=(z_1,z_2)$ subject to the constraint $|z_1|^2+|z_2|^2=1$ and a $U(1)$ gauge invariance $\\mathbf{z}\\sim e^{i\\phi} \\mathbf{z}$. This explains the emergence of the gauge field. Replacing the constraint by a quartic potential, and adding a Maxwell kinetic term for the gauge field (expected to be generated by the RG flow), one obtains bosonic QED$_{3}$ with $N=2$. \n\nIn the language of QED${}_{3}$, the above-mentioned topological defects are the monopole operators of quantized charge, similar to the ones in Sec.~\\ref{sec:QED3-monopole}. Of course the dimensions of monopole operators differ in bosonic and fermionic QED. Also here we will normalize the topological charge to be integer $q\\in \\bZ$. \n\nIf a monopole of charge $q$ appears in the action, it breaks the global topological $U(1)_T$ symmetry to the $\\bZ_q$ subgroup. Microscopic descriptions of quantum antiferromagnets may realize a discrete subgroup of $U(1)_T$ at the lattice level. On cubic lattices, a $\\bZ_{q_0}$ with $q_0=4$ is preserved, while for the hexagonal and rectangular lattices we have $q_0=3$ and $q_0=2$. The $\\bZ_{q_0}$ symmetry is also visible in the VBS phase where it permutes the vacua. This microscopic symmetry means that only monopoles with charges multiple of $q_0$ appear. Monopoles with different charges have their fugacity killed by the above-mentioned Berry phases~\\cite{Read-Sachdev}.\n\nIn light of the above discussion, the analysis of the critical behavior of QED${}_3$ can be split into two parts. First, does bosonic QED${}_3$, with all monopoles suppressed, have a fixed point?\nIf the answer is yes, then one can ask: can this fixed point be reached provided that one allows monopoles with charges in multiples of $q_0$? For this to happen, the monopole of charge $q_0$ has to be irrelevant.\n \nOne can study these questions analytically at large $N$: one finds a fixed point and computes the critical exponents in the $1\/N$ expansion.\\footnote{See \\textcite{Murthy:1989ps}, \\textcite{Kaul:2008xw}, \\textcite{Metlitski:2008dw}, and \\textcite{Dyer:2015zha}} At small $N$ one resorts to Monte Carlo simulations. The bootstrap at present cannot by itself resolve the question of the fixed point existence. However, it can provide valuable consistency checks on the other studies. Suppose that a certain Monte Carlo simulation is done on a lattice preserving a $\\bZ_{q_0}$ subgroup, finds a second order phase transition, and measures the scaling dimensions $\\Delta_q$ of monopole operators $M_q$ for a subset of charges $q$ (we denote by $M_0$ the relevant singlet scalar driving the transition). We have the following OPE algebra in the scalar sector, omitting the OPE coefficients ($M_{-q}=M_q^\\dagger$):\n\\beq\nM_{q} \\times M_{q'} \\sim \\delta_{q+q'}\\mathds{1}+ M_{q+q'}+\\ldots\\,.\n\\end{equation}\nBy the above discussion, the operator $M_{q_0}$ has to be irrelevant, as well as the higher charge monopoles. We can use the bootstrap to study the consistency of this algebra given the measured operator dimensions. \n\n\\subsubsection{Aside: constraints on symmetry enhancement}\n\\label{sec:enhancement}\n\nWhat we just presented is an instance of a more general question: under which conditions can the global symmetry of the fixed point $G$ be larger than the microscopically realized symmetry $H$? The case of interest for the previous section is $G=U(1)$ and $H=\\bZ_{q_0}$. For the symmetry enhancement to happen, operators which break $G$ to $H$ must be irrelevant. The bootstrap is a powerful tool to study whether this irrelevance assumption is consistent with conformal symmetry and with other information which may be available about the fixed point. We will see further applications of this philosophy in Secs.~\\ref{sec:4D-bsm} and~\\ref{sec:4Dwindow}.\n\nWe will now describe bootstrap constraints on the symmetry enhancement from $\\bZ_{q_0}$ to $U(1)$ derived by~\\textcite{Nakayama:2016jhq}. Enhancement from $\\bZ_2$ requires that $M_2$ is irrelevant. Since $M_1\\times M_1\\sim M_2$, one can bound $\\Delta_2$ given $\\Delta_1$, by studying the 4pt function $\\langle M_1 M_1^\\dagger M_1 M_1^\\dagger\\rangle$. The resulting bound is given in Fig.~\\ref{fig:QED3-emergent}. Imposing $\\Delta_2>3$, one gets a necessary condition \\mbox{$\\Delta_1>1.08$} for enhancement from $\\bZ_2$ to $U(1)$. \n\n\\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig36-charge_2}\n    \\caption{\\label{fig:QED3-emergent} (Color online) The 3d upper bound on $\\Delta_2$ as a function of $\\Delta_1$~\\cite{Nakayama:2016jhq}. It may be possible to improve this bound if $\\Delta_0$ is known. The same bound applies to $M_2\\times M_2\\sim M_4$.}\n  \\end{figure}\n  \n The same plot in Fig.~\\ref{fig:QED3-emergent} can be used to derive rough necessary conditions on the enhancement from $\\bZ_4$ to $U(1)$. Indeed, the bound applies also to $M_2\\times M_2\\sim M_4$. If $M_4$ is irrelevant, then we must have $\\Delta_2>1.08$, which in turn implies $\\Delta_1>0.504$. \n  \nTo study enhancement from $\\bZ_3$, one analyzes simultaneously three 4pt functions $\\langle M_1 M_1^\\dagger M_1 M_1^\\dagger\\rangle$, $\\langle M_1 M_1^\\dagger M_2 M_2^\\dagger\\rangle$, $\\langle M_2 M_2^\\dagger M_2 M_2^\\dagger\\rangle$. It is reasonable to assume that $M_4$ is irrelevant (as would be the case if $M_3$ is irrelevant and $\\Delta_q$ is monotonic in $q$), and to impose $\\Delta_0>1.044$ (which follows from an assumption that the fixed point is critical and not multicritical, see Sec.~\\ref{sec:multicrit}). Under these assumptions, the upper bound on $\\Delta_3$ as a function of $\\{\\Delta_1,\\Delta_2\\}$ is shown in Fig.~\\ref{fig:QED3-emergent-ch3}. From this bound, irrelevance of $M_3$ requires $\\Delta_1>0.585$. \n  \n \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig37-charge3}\n    \\caption{\\label{fig:QED3-emergent-ch3} (Color online) An upper bound on $\\Delta_3$ as a function of $\\{\\Delta_1,\\Delta_2\\}$ under the assumptions that $\\Delta_0>1.044$, $\\Delta_4>3$ \\cite{Nakayama:2016jhq}. It follows from Fig.~\\ref{fig:QED3-emergent} that the range of $\\Delta_2$ is restricted by the latter assumption from below, and, for fixed $\\Delta_1$, from above.}\n  \\end{figure}\n\n\\subsubsection{Back to deconfined criticality: is the transition second order?}\n \n The necessary conditions described in the previous section have been compared with available Monte Carlo and large $N$ data on the N\\'eel-VBS transition which claim to see a second-order transition and measure some critical exponents. For square and hexagonal lattices, there is nice consistency, as for rectangular lattices for $N\\leqslant 4$ and $N\\geqslant 6$, while some $N=5$ simulations are inconsistent with the bootstrap. The conclusion is that there must either be an error in the $N=5$ Monte-Carlo measurement or in the assumption that the transition is second-order. See \\textcite{Nakayama:2016jhq} for this survey and for further details. \n\nIt should be emphasized that while the bootstrap results may point out an inconsistency in Monte Carlo simulations, they cannot, at present, validate them and prove that the phase transition is indeed second order. It is still possible that even in the above cases when there is a nice agreement between Monte Carlo results and the bootstrap necessary conditions, the transition is still very weakly first order and not second order. \n\nLet us focus on the case $N=2$ which presents a controversy. Large-scale Monte Carlo simulations for $N=2$ were performed in \\textcite{Nahum:2015jya}, using a loop model on a cubic lattice which is in the same universality class as NCCP${}^1$ and has monopole suppression up to $q_0=4$, and going up to very large lattices of linear size up to $L=640$.\\footnote{See also \\textcite{Harada2013, Sreejith:2018ypg} for simulations of other microscopic models in the same universality class.} While they have not seen signs of a finite correlation length or a conventional first order transition, and observed scaling behavior of correlation functions at distances $1\\ll r \\ll L$, they have seen scaling violation for observables at larger distances $r\\sim L$, inconsistent with a conventional second order transition. \n\nSo, is the transition second order or weakly first order? Assuming a second order transition, \\textcite{Nahum:2015jya} extracted the scaling dimension of the monopole operator $\\Delta_1=0.625(15)$, which is consistent with the bootstrap condition $\\Delta_1>0.504$ necessary for the enhancement from $\\bZ_4$ to $U(1)$. However there is an extra piece of information which allows one to set up an even more stringent bootstrap test: further symmetry enhancement at the transition from $SO(3)\\times U(1)$ to $SO(5)$. Here $SO(3)$ acts on the N\\'eel order parameter $N_a = \\mathbf{z}^\\dagger\\sigma_a \\mathbf{z}$. Empirically, the scaling dimension of $N$ is very close to $\\Delta_1$~\\cite{Nahum:2015jya} and, moreover, the joint probability distribution of $(N, M_1)$ is very close to the spherical one after a rescaling~\\cite{Nahum:2015vka}, which can be explained if $N$ and $M_1$ belong to a vector multiplet $\\Phi$ of $SO(5)$ of dimension $\\Delta_\\Phi=\\Delta_1$. \n\nIn this description, the relevant scalar which drives the transition is a component of the symmetric traceless tensor (roughly $\\Phi_A\\Phi_B-\\text{trace}$).\\footnote{\\textcite{Nahum:2015vka} measured its scaling dimension to be $\\sim 1.5$.} For the $SO(5)$ enhancement to happen, any other scalar which breaks $SO(5)$ back to $SO(3)\\times U(1)$ must be irrelevant. In addition, the $SO(5)$ singlet $S$ (roughly $\\Phi_A\\Phi^A$) must be irrelevant for the transition to be second order, since otherwise the fixed point will not be reached. See \\textcite{Wang:2017txt} for further discussion. Given the dimension $\\Delta_\\Phi=\\Delta_1$ as above, it is straightforward to compute an upper bound on the dimension of $S$ which occurs in the OPE $\\Phi\\times\\Phi$. This is the same bound as for $N=5$ in Fig.~\\ref{fig:ONsymtrace} except the plot has to be extended to larger $\\Delta_\\Phi$. \\textcite{Nakayama:2016jhq} and~\\textcite{DSD2016} performed this analysis and report that $\\Delta_S>3$ is excluded for $\\Delta_\\Phi$ as above. In fact $\\Delta_S>3$ requires $\\Delta_\\Phi>0.76$~\\cite{Nakayama2016}.\n\nTo summarize, the bootstrap excludes a second-order phase transition described by a unitary 3d CFT with symmetry enhanced to $SO(5)$ and the order parameter scaling dimension taking the above value suggested by the Monte Carlo simulations. \nIn our opinion, the most compelling interpretation of available data is a weakly first-order transition due to walking RG behavior which ensues when the RG flow has no fixed points for a real value of the coupling but two complex conjugate fixed points with small imaginary parts. This is the same mechanism as for the weakly first-order transition in the 2d Potts model with $Q\\gtrsim 4$. As discussed in \\textcite{Nahum:2015jya}, this scenario may resolve the observed scaling violations at distances $r\\sim L$. It can also accommodate the enhancement to $SO(5)$~\\cite{Wang:2017txt}. In this scenario, there is no unitary 3d CFT (the complex fixed points being nonunitary), and the bootstrap bounds do not apply, resolving the contradiction.\n\nFinally, let us note that a similar analysis can constrain another scenario outlined in \\textcite{Wang:2017txt}, in which a variant called the easy-plane NCCP${}^1$ model is conjectured to have a fixed point with enhanced $O(4)$ symmetry and be dual to $N_f=2$ fermionic QED${}_3$. In this scenario, the conjectured fixed point cannot contain any fully $O(4)$-invariant scalar perturbations. However, recent Monte Carlo simulations~\\cite{Qin:2017cqw} point to a dimension for the $O(4)$ vector order parameter $\\Delta_\\Phi = 0.565(15)$ which seems incompatible with the bootstrap bound assuming irrelevance of the $O(4)$ singlet (Fig.~\\ref{fig:ONsinglet} extended further to the right), which requires $\\Delta_\\Phi > 0.868$~\\cite{PolandUnpublished}. In the future it will be interesting to further study the fate of these models using both bootstrap and Monte Carlo data.\n\n\n\\subsubsection{Multifield Landau-Ginzburg models}\n\nThere exists rich phenomenology of fixed points arising from Lagrangians with multiple scalar fields transforming under product group symmetries, e.g.~$SO(n) \\times SO(m)$. One can consider Lagrangians involving two coupled scalar multiplets, one transforming in the fundamental of $SO(n)$ and another of $SO(m)$. {Alternatively,} one can consider a field transforming in the bifundamental of $SO(n) \\times SO(m)$. Such Lagrangians have been invoked to describe phase transitions in many physical systems{; see \\textcite{Vicari:2007ma} for further details.}\n\n{When studying these fixed points using the RG,} a recurrent feature is that many of {them} do not exist in the $4-\\epsilon$ expansion and have to be studied directly in 3d. Since such computations lack a manifestly small expansion parameter, there seems to be no consensus about the existence of these fixed points. So this appears to be a perfect target for a nonperturbative approach like the bootstrap. Some preliminary bootstrap studies of 3d CFTs with \\mbox{$SO(n) \\times SO(m)$} were carried out in \\textcite{Nakayama:2014lva,Nakayama:2014sba}, but in our opinion more work is needed before firm conclusions can be drawn.\n\n\\subsubsection{Projective space models}\n\nAn interesting 3d lattice model is the CP$^{n}$ model, where microscopic lattice variables belong to CP$^{n}$ and have ferromagnetic interactions preserving the symmetry (see below for the antiferromagnetic case). Recall that CP$^{n}$ can be realized by starting with $(n+1)$-dimensional complex vectors $\\mathbf{z}=(z_1,\\ldots,z_{n+1})$ and imposing the constraint $\\mathbf{z}^\\dagger \\cdot \\mathbf{z} =1$, preserved up to the equivalence $\\mathbf{z}\\sim e^{i\\phi}\\mathbf{z}$. A simple lattice Hamiltonian is \n\\beq\nH=-J\\sum_{\\langle i j\\rangle} |\\mathbf{z}^\\dagger_i \\cdot \\mathbf{z}_j|^2\\,,\n\\end{equation}\nwith $J>0$ in the considered ferromagnetic case. The physics of this model is influenced by defects (hedgehogs), which are possible because $\\pi_2(CP^{n})=\\mathbb{Z}$. Here we consider the CP$^{n}$ model with defects allowed. It should be distinguished from the ``non-compact CP$^{n}$ model\" which results when defects are suppressed, see Sec.~\\ref{bQED3}.\n\nThe CP${}^1$ model is equivalent to the $O(3)$ model, with the order parameter $N_a=\\mathbf{z}^\\dagger \\sigma_a \\mathbf{z}$, and it has a second-order phase transition described by the same CFT.\n\nThe CP${}^2$ model has an internal symmetry $SU(3)$ (modulo global issues), with traceless hermitian matrix $Q_{ab}=z_a \\bar{z}_b-\\delta_{ab}$ as an order parameter. The Landau-Ginzburg description contains a cubic invariant ${\\rm Tr}(Q^3)$ and would suggest a first-order transition, but Monte Carlo simulations \\cite{Nahum:2013qha} indicate that the phase transition is continuous. This is similar to what happens for the 3-state Potts model in 2d and can be explained as an effect of fluctuations. Monte Carlo results for the critical exponents are $\\eta=0.23(2)$ and $\\nu=0.536(13)$, translating into the dimensions of $Q$ and of the relevant singlet scalar. Can this model be isolated using the numerical bootstrap?\n\nOne can also consider antiferromagnetic projective space models, taking $J<0$ in the above Hamiltonian.\nAntiferromagnetic CP$^{n}$ models \\cite{Delfino:2015gba} don't give rise to new universality classes.\\footnote{The ACP$^{1}$ model on a cubic lattice is equivalent to the ferromagnetic model and, as the latter, has a phase transition in the $O(3)$ universality class. For higher $n$ there is no equivalence between the antiferromagnetic and ferromagnetic models. The ACP${}^2$ model has a second-order transition which belongs to the $O(8)$ universality class (and so is different from CP${}^2$). For still higher $n$ the transition is first order.} On the other hand, a new class is observed for the antiferromagnetic RP${}^4$ model \\cite{Pelissetto:2017pxb}, and it could constitute a target for the bootstrap.\\footnote{The RP${}^n$ models are versions of $O(n)$ models with a gauged $\\bZ_2$ symmetry. Their second-order phase transitions for $n=2,3$ belong to the $O(2)$ and $O(5)$ classes respectively, but the $n=4$ class is mysterious.}\n\n\\subsubsection{Nonabelian gauge and Chern-Simons matter theories}\n\nWhile we have focused our attention on QED${}_3$, there is a whole landscape of 3d gauge theories coupled to various types of matter. An interesting case is QCD${}_3$ with a simple gauge group $G$ coupled to $N_f$ fundamental fermions. Such theories may for example play a role in the physics of cuprate superconductors~\\cite{Chowdhury:2014efa,Chowdhury:2014jya}. A fixed point can be established and the properties studied at large $N_f$~\\cite{Appelquist:1989tc}. For example, in \\textcite{Dyer:2013fja} a systematic study of monopole operators in such theories was performed, allowing for estimates of the bottom of the conformal window for different choices of $G$ by imposing irrelevance of the monopole operators~\\cite[{Table 4}]{Dyer:2013fja}. QCD${}_3$ coupled to both fermions and scalars was also proposed to describe the critical point of the `orthogonal semi-metal' (OSM) confinement transition in \\textcite{2018arXiv180401095G}, with critical exponents extracted in quantum Monte Carlo simulations. It would be very interesting to understand how to isolate these theories using bootstrap techniques and test these estimates. \n\nAnother natural set of targets consists of Chern-Simons gauge fields coupled to matter. Such theories are known to have conformal fixed points and sit in an intricate web of dualities.\\footnote{See for example \\textcite{Aharony:2015mjs}, \\textcite{Aharony:2016jvv}, \\textcite{Seiberg:2016gmd}, \\textcite{Hsin:2016blu}, \\textcite{Benini:2017dus}, and \\textcite{Gomis:2017ixy}. Duality means that two different microscopic descriptions lead to the same IR CFT (perhaps after tuning some parameters). Why should dualities exist? One reason may be the paucity of CFTs. If so, some dualities may perhaps be explained by the bootstrap, providing evidence that there is a single CFT satisfying certain constraints (symmetry, the number of relevant operators, etc). Then any microscopic theory satisfying these constraints should flow to this CFT at criticality. In this sense, the results of Section \\ref{sec:O2} provide an explanation for the particle-vortex duality of the Abelian Higgs model, originally proposed by \\textcite{Peskin:1977kp} and \\textcite{Dasgupta:1981zz}.} Some possible experimental realizations of these theories as transitions between fractional quantum Hall states were proposed in \\textcite{Lee:2018udi}. A hallmark of these theories is the existence of parity-violation; it would be interesting to see if they can be found after introducing parity-violating couplings into the bootstrap. Monopole operators in these theories were also recently studied in~\\textcite{Chester:2017vdh} and would constitute natural targets for the bootstrap.\n\n\\subsubsection{Other models}\n\nAnother theory briefly mentioned in Sec.~\\ref{sec:O3} is the Gross-Neveu-Heisenberg (GNH) model, a variant of the GN models with a 3-component scalar order parameter. For a pedagogical review of the model, its applications, and its connection to the lattice Hubbard model, see \\textcite{Sachdev:2010uz}. This constitutes another interesting target for the bootstrap.\n\nA 3d CFT with $SU(4)$ global symmetry and an order parameter in the symmetric tensor representation was considered in \\textcite{Basile:2004wa,Basile:2005hw}. It was proposed to describe a continuous chiral phase transition in 4d $SU(N)$ gauge theory coupled to $N_f=2$ massless quarks in the adjoint representation at finite temperature. The existence of this CFT and some information about critical exponents was found using RG methods; it would be interesting to explore it using the bootstrap. \n\n\n\\subsubsection{The Casimir equation}\n\\label{sec:casimir}\n\n{Let us consider} the following alternative representation of CPWs. \nIn radial quantization, as mentioned in Sec.~\\ref{sec:OPE}, the above 4pt function is expressed as a scalar product of two states \n\\beq\n\\<\\phi_3(x_3)\\phi_4(x_4)|\\phi_1(x_1)\\phi_2(x_2)\\>\\,\n\\end{equation}\nliving on a sphere separating $x_1,x_2$ from $x_3,x_4$. The CPW then corresponds to inserting an orthogonal projector $\\calP_{\\Delta,\\ell}$ {onto} the conformal multiplet of ${\\cal O}_{\\Delta,\\ell}$:\n\\beq\n\\lambda_{12{\\cal O}}\\lambda_{34{\\cal O}}\\,{\\rm W}_{\\cal O} = \\<\\phi_3(x_3)\\phi_4(x_4)|\\calP_{\\Delta,\\ell}|\\phi_1(x_1)\\phi_2(x_2)\\>\\,.\n\\label{eq:CPWrad}\n\\end{equation}\nFor future reference, the projector can be written as\n\\beq \n\\label{eq:proj}\n\\calP_{\\Delta,\\ell}=\\sum_{\\alpha,\\beta={\\cal O},P{\\cal O},PP{\\cal O},\\ldots}|\\alpha\\>G^{\\alpha\\beta}\\<\\beta|\\,,\n\\end{equation}\nwhere $G_{\\alpha\\beta}=\\<\\alpha|\\beta\\>$ is the Gram matrix of the multiplet and $G^{\\alpha\\beta}$ is its inverse.\n\nFurthermore, consider the quadratic Casimir\\footnote{The quartic Casimir operator \n\t${\\mathcal C}_4 = \\frac12 {\\cal J}_{AB}{\\cal J}^{BC}{\\cal J}_{CD}{\\cal J}^{DA}$ has also proved useful in some conformal block studies \\cite{DO3, Hogervorst:2013kva}\\,.}\n\\begin{eqnarray}\n\\mathcal C_2 = \\frac12 {\\cal J}_{AB}{\\cal J}^{BA}\\,,\n\\end{eqnarray}\nwhere ${\\cal J}_{AB}$ are the $SO(d+1,1)$ generators, Eq.~(\\ref{eq:SODalgebra}). Insert this operator into Eq.~\\reef{eq:CPWrad} right after $\\calP_{\\Delta,\\ell}$. The resulting expression can be computed in two ways.\nWhen we act with $\\mathcal C_2$ on the left we have\n\\beq\n\\calP_{\\Delta,\\ell}\\,{\\mathcal C}_2=C_{\\Delta,\\ell} \\calP_{\\Delta,\\ell}\\,,\n\\end{equation}\nwhere $C_{\\Delta,\\ell}$ is the quadratic Casimir eigenvalue: \n\\begin{eqnarray}\nC_{\\Delta,\\ell} = \\Delta(\\Delta-d) + \\ell(\\ell+d-2)\\,.\n\\end{eqnarray}\nOn the other hand, the action of $\\mathcal C_2$ on the right can be computed using the representation of the conformal generators on primaries as first-order differential operators, mentioned in Sec.~\\ref{sec:primaries}.\nWe conclude that the CPW, and hence the conformal block, satisfies a second-order partial differential equation.\\footnote{We followed the presentation in \\cite[{section 9.3}]{Simmons-Duffin:2016gjk}. The same conclusion can be reached using the OPE \\cite{Costa:2011dw}.} The actual form of this ``Casimir equation\" is most conveniently found using the embedding formalism \\cite{DO2}. In the $z,\\bar{z}$ coordinates of Eq.~(\\ref{eq:zzb}) it takes the form\n\\begin{eqnarray}\\label{eq:cb_diffeq}\n&&  \\mathcal D\\, g^{\\Delta_{12},\\Delta_{34}}_{\\Delta,\\ell}(z,\\bar{z}) =  C_{\\Delta,\\ell}\\,\\, g^{\\Delta_{12},\\Delta_{34}}_{\\Delta,\\ell}(z, \\bar{z})\\,, \n\\end{eqnarray}\nwhere\n\\begin{gather}\n \\mathcal D = \\mathcal D_z + \\mathcal D_{\\bar{z}} + 2(d-2) \\frac{z \\bar{z}}{z-\\bar{z}} [(1-z)\\partial_z - (1-\\bar{z})\\partial_{\\bar{z}} ]\\,, \\nonumber\\\\\n \\mathcal D_z = \\textstyle 2 z^2(1-z)\\partial_z^2 -(2+\\Delta_{34}-\\Delta_{12}) z^2\\partial_z + \\frac{\\Delta_{12}\\Delta_{34}}{2}z\\,.\n\\end{gather}\n\n{Moreover}, the leading $z,\\bar{z}\\to 0$ behavior of the conformal block can be easily determined using the OPE, and this provides boundary conditions for Eq.~\\reef{eq:cb_diffeq}. Considering the $x_{12},x_{34}\\rightarrow 0$ limit in Eq.~\\reef{eq:CPW} and using Eqs. (\\ref{eq:2points}) and \\reef{eq:OPEnorm}, one obtains\\footnote{The limit is worked out carefully in e.g.~\\textcite{DO1} or \\textcite{Costa:2011dw}.}\n\\beq\n\\label{eq:cb_ope}\ng^{\\Delta_{12},\\Delta_{34}}_{\\Delta,\\ell}(z, \\bar{z})\n\\underset{z,\\bar{z}\\rightarrow0}{\\sim}\n\\calN_{d,\\ell} \\, (z\\bar{z})^{\\frac\\Delta2} {\\rm Geg}_\\ell\\left(\\frac{z+\\bar{z}}{2\\sqrt{z\\bar{z}}}\\right)\\,,\n\\end{equation}\nwhere ${\\rm Geg}_\\ell(x)$ is a Gegenbauer polynomial,\n\\beq\n\\label{eq:Geg}\n{\\rm Geg}_\\ell(x)=C^{(d\/2-1)}_\\ell(x)\\,,\n\\end{equation}\nand the normalization factor $\\calN_{d,\\ell}$ is given by\\footnote{Here $(a)_n$ stands for the Pochhammer symbol.}\n\\beq\n\\calN_{d,\\ell} = \\frac{\\ell!}{{(-2)^\\ell}(d\/2-1)_\\ell}\\,.\n\\end{equation}\nWe warn the reader that many different {normalization choices can be found} in the literature. Different conformal block normalizations correspond to different normalizations of OPE coefficients as compared with the one in Eq.~\\reef{eq:OPEnorm}. In this review we will use the above normalization unless {mentioned} otherwise. For the reader's convenience, we have collected some other frequently used normalizations in Table~\\ref{tab:cb_norm}.\n\n\\begingroup\n\\squeezetable\n\\begin{table}[htp]\n\\begin{center}\n\\renewcommand{\\arraystretch}{2}\n\\begin{tabular}{|c|c|}\n\\hline\n$\\calN_{d,\\ell}$ & Reference \\\\\n\\hline\n $\\frac{\\ell!}{(-2)^\\ell(d\/2-1)_\\ell}$ & \n \\begin{minipage}{0.37\\textwidth}\n \t\\textcite{DO1,DO2}, \\\\\n \t\\textcite{Rattazzi:2008pe}, \\\\\n \t\\textcite{Penedones:2015aga}, {\\bf this review} \n \\end{minipage}\\\\\n    \\hline\n $ \\frac{\\ell!}{(d-2)_\\ell}$ &  \n \\begin{minipage}{0.37\\textwidth} \n \t\\textcite{DO3},\\\\ \n \t\\textcite{Hogervorst:2013sma}, \\\\ \n \t\\textcite{ElShowk:2012ht,El-Showk:2014dwa}, \\\\ \n\t\\textcite{Costa:2016xah}, \\\\\n \t\\texttt{JuliBoots} \\cite{Paulos:2014vya}, \\texttt{cboot} \\cite{CBoot}\n \\end{minipage}\\\\\n  \\hline\n $ \\frac{(-1)^\\ell \\ell!}{4^{\\Delta}(d\/2-1)_\\ell}$ & \\begin{minipage}{0.37\\textwidth} \\textcite{Kos:2014bka,Kos:2015mba,Kos:2016ysd}, \\textcite{Li:2017ddj}\\\\\\texttt{PyCFTBoot} \\cite{Behan:2016dtz}\\end{minipage} \\\\\n \\hline\n  $\\frac{\\ell!}{(d\/2-1)_\\ell}$ & \\begin{minipage}{0.37\\textwidth} \\textcite{Poland:2011ey}, \\textcite{Poland:2015mta} \\end{minipage}\\\\\n \\hline\n $ \\frac{\\ell!}{4^{\\Delta}(d-2)_\\ell}$ &\\begin{minipage}{0.37\\textwidth}  \\textcite{Kos:2013tga}\\\\ {\\texttt{Mathematica} notebook \\cite{DSDnotebookLink}} \\end{minipage} \\\\\n \\hline\n $ \\frac{(-1)^\\ell\\ell!}{(d\/2-1)_\\ell}$ & \\textcite{Simmons-Duffin:2016wlq}\\\\\n \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Summary of various conformal block normalizations $\\calN_{d,\\ell}$, Eqs.~(\\ref{eq:cb_ope}, \\ref{eq:cb_ope_e1}), used in the literature. \\label{tab:cb_norm}}\n\\end{table}\n\\endgroup\n \nBy solving Eq.~(\\ref{eq:cb_diffeq}) one can find conformal blocks for even $d$ \\cite{DO2}. They are expressed in terms of the basic function\n\\begin{eqnarray}\nk_\\beta(x) = x^{\\beta\/2} {}_2F_1\\left(\\frac{\\beta-\\Delta_{12}}2,\\frac{\\beta+\\Delta_{34}}2,\\beta;x\\right)\\,,\n\\end{eqnarray}\nwhich satisfies\n\\begin{eqnarray}\n\\mathcal D_x k_\\beta(x) = \\frac12\\beta(\\beta-2) k_\\beta(x)\\,, \\quad k_\\beta(x) \\underset{x\\rightarrow0}{\\sim} x^{\\beta\/2}\\,.\n\\end{eqnarray}\nIn the simplest case of $d=2$, we have $\\calD=\\calD_z+\\calD_{\\bar z}$, so the conformal blocks factorize. They take the form\\footnote{A partial case of this result was first found in \\textcite{Ferrara:1974ny} by another method. See also \\textcite{Osborn:2012vt} for general conformal blocks in 2d. Notice that the 2d global conformal blocks discussed here should be distinguished from the Virasoro conformal blocks.}\n\\begin{multline} \\label{eq:cb_d2}\nd=2:\\quad g^{\\Delta_{12},\\Delta_{34}}_{\\Delta,\\ell}(z,\\bar{z}) = \\frac{1}{(-2)^\\ell(1+\\delta_{\\ell 0})}\\\\ \\times\\left(k_{\\Delta+\\ell}(z)k_{\\Delta-\\ell}(\\bar{z})+z\\leftrightarrow \\bar{z}\\right)\\,.\n\\end{multline}\nResults for higher even $d$ can then be found using recursion relations relating blocks in $d$ and $d+2$ dimensions \\cite{DO2}. The important case of $d=4$ reads\\footnote{This result was first found in \\textcite{DO1} by resumming the OPE expansion.}\n\\begin{multline} \\label{eq:cb_d4}\nd=4:\\quad  g^{\\Delta_{12},\\Delta_{34}}_{\\Delta,\\ell}(z,\\bar{z})  =\\frac{1}{(-2)^\\ell}\\\\\n\\times \\frac{z\\bar{z}}{z-\\bar{z}} \\left(k_{\\Delta+\\ell}(z)k_{\\Delta-\\ell-2}(\\bar{z})- z\\leftrightarrow \\bar{z}\\right)\\,.\n\\end{multline}\n\nIn odd $d$, general closed-form solutions of the Casimir equation are so far unavailable. \nSometimes, one can get closed-form solutions along the ``diagonal\" $z=\\bar{z}$, as e.g.~in $d=3$ for all equal external dimensions \\cite[{Eqs.~(3.7-3.10)}]{Rychkov:2015lca}. Other {expressions} along the diagonal{,} valid for any $d${,} can be found in~\\textcite{Hogervorst:2013kva}. Using these results as a starting point, one can compute derivatives of conformal blocks orthogonal to the diagonal using the Casimir equation, by the Cauchy-Kovalevskaya method, see Sec.~\\ref{sec:rational}. The knowledge of these derivatives is usually sufficient for numerical conformal bootstrap applications. Other techniques used to access the conformal blocks numerically will be discussed below.\n\nFinally, let us mention that conformal blocks have simple transformation properties under the interchange of external operators $1\\leftrightarrow 2$ and $3\\leftrightarrow 4$ \\cite{DO1,DO3}:\n\\begin{gather}\n g^{\\Delta_{12},\\Delta_{34}}_{\\Delta,\\ell}(u\/v,1\/v) = (-1)^\\ell v^{\\frac{\\Delta_{34}}2 } g^{-\\Delta_{12},\\Delta_{34}}_{\\Delta,\\ell}(u,v)\\nonumber\\\\\n\\hspace{3cm}=  (-1)^\\ell v^{-\\frac{\\Delta_{12}}2 } g^{\\Delta_{12},-\\Delta_{34}}_{\\Delta,\\ell}(u,v)\\,.\n \\end{gather}\nThis follows from the symmetry of the OPE under the same interchange. As a check, the explicit expressions in Eqs.~(\\ref{eq:cb_d2}-\\ref{eq:cb_d4}) satisfy these relations.\n\n\n\\subsubsection{Radial expansion for conformal blocks}\n\\label{sec:radial}\n\nWhile closed-form expressions for conformal blocks in general $d$ are unknown, there exist rapidly convergent power series expansions. Following \\textcite{Hogervorst:2013sma}, we will describe a particular conformal frame used to generate such expansions.\n\n\\begin{figure}[t]\n\\begin{centering}\n\\includegraphics[width=0.6\\columnwidth]{fig04-radial.pdf}\n\\caption{\\label{fig:radial}\n(Color online) Conformal frame defining the radial coordinate. Figure from \\cite{Hogervorst:2013sma}.\n}\n\\end{centering}\n\\end{figure}\n\nStarting from the conformal frame \\reef{eq:conformalframe1}, we apply an additional conformal transformation which keeps the four points in the same 2-plane but moves them into a configuration symmetric around the origin as in Fig.~\\ref{fig:radial}. So the points $x_1=-x_2$ are now on a circle of radius $r<1$, while $x_3=-x_4$ lie on the unit circle.\n\nLet us call ${\\bf n}$ and $\\bf{n}'$ the unit vectors pointing to $x_2$ and $x_3$, and introduce the complex \\emph{radial coordinate} \\cite{Pappadopulo:2012jk}\n\\beq\n\\label{eq:reta}\n\\rho = r e^{i \\theta}\\,, \\qquad {\\bf n}\\cdot {\\bf n}'=  \\cos \\theta=\\eta\\,,\n\\end{equation}\nwhich is related to the variable $z$ in Eq.~\\reef{eq:z} via\n\\begin{eqnarray}\n\\rho = \\frac{z}{(1-\\sqrt{1-z})^2} \\,,\n\\qquad \nz = \\frac{4\\rho}{(1+\\rho)^2}\\,.\\nonumber\n\\end{eqnarray}\nSee \\textcite{Hogervorst:2013sma} for why $\\rho$ is preferable to $z$ for constructing rapidly convergent power series expansions for conformal blocks. \n\nIn this configuration, the 4pt function is interpreted as a matrix element between two radial quantization states: $\\<\\phi_3(1,{\\bf n}')\\phi_4(1,-{\\bf n}')|$ and  $|\\phi_1(r,-{\\bf n})\\phi_2(r,{\\bf n})\\>= r^D|\\phi_1(1,-{\\bf n})\\phi_2(1,{\\bf n})\\>$. The factor $r^D$, with $D$ the dilatation generator, takes care of the radial dependence.\\footnote{$D$ plays the role of the Hamiltonian operator in radial quantization and $\\log r$ is time.}\n\nConsider then the conformal partial wave given in Eq.~\\reef{eq:CPWrad}.\nThe conformal multiplet of the operator ${\\cal O}_{\\Delta,\\ell}$ at level $m$ contains descendants $|\\Delta+m,j\\>$ of spin $j$ varying from $\\max(0,\\ell-m)$ to $\\ell+m$. We need to know the matrix elements between these descendants and the above in and out states. Leaving aside the overall normalization of these matrix elements, their dependence on the unit vector $\\mathbf{n}$\nmust be proportional to the traceless symmetric tensor $({\\bf n}_{\\mu_1} \\ldots {\\bf n}_{\\mu_j}-\\text{traces})$. Contracting two such tensors for $\\mathbf{n}$ and $\\mathbf{n}'$ {gives}, up to a constant factor, the Gegenbauer polynomial ${\\rm Geg}_j({\\bf n}\\cdot {\\bf n'})$ from Eq.~\\reef{eq:Geg}.\n\nWe conclude that the conformal block has a power series expansion of the form\n\\beq\n\\label{eq:radial_exp}\n g^{\\Delta_{12},\\Delta_{34}}_{\\Delta,\\ell}(u,v)= r^{\\Delta}\\sum_{m=0}^\\infty   r^{m}\n \\sum_{j} w(m,j) \\; {\\rm Geg}_j(\\eta)\\,,\n\\end{equation}\nwhere $w(m,j)\\ne 0$ only for $\\max(0,\\ell-m)\\leqslant j \\leqslant \\ell+m$. Using unitarity, one can also conclude that $w(m,j)\\geqslant 0$ if $\\Delta$ is above the unitarity bound and $\\Delta_{12}=-\\Delta_{34}$.\n\nSince $z\\sim 4\\rho$ at small $z$, the OPE limit \\reef{eq:cb_ope} becomes\n\\begin{eqnarray}\n\\label{eq:cb_ope_e1}\ng^{\\Delta_{12},\\Delta_{34}}_{\\Delta,\\ell}(r, \\eta)\\displaystyle \\underset{r\\rightarrow0}{\\sim} \n\\calN_{d,\\ell}(4r)^{\\Delta}{\\rm Geg}_\\ell\\left(\\eta \\right) \\,,\n\\end{eqnarray}\nwhich fixes $w(0,\\ell)=\\calN_{d,\\ell} 4^\\Delta$. To find higher $w(m,j)$, one must determine the normalization of the descendant matrix elements and not just their dependence on $\\mathbf{n},\\mathbf{n}'$. While in principle this can be done using conformal algebra, two more efficient techniques will be discussed below. \n\nThe expansion \\reef{eq:radial_exp} converges for $|\\rho|<1$, showing that conformal blocks are smooth and real-analytic functions in this region.\\footnote{An exception occurs at the origin because of the $r^\\Delta$ factor.} The conformal block decomposition \\reef{eq:CBdec} can be similarly argued to converge for $|\\rho|<1$.\\footnote{This can be shown rigorously in unitary CFTs \\cite{Pappadopulo:2012jk}. While there are no general results concerning the convergence of conformal block decomposition is nonunitary theories, it appears reasonable to assume that it remains convergent in the same region.} In terms of the $z$ coordinate, this covers the whole complex plane minus the cut $(1,+\\infty)$, improving the convergence result argued below Eq.~\\reef{eq:CBdec} using the $z$ frame.\n\n\n\\subsubsection{Recursion relation from the Casimir equation}\n\\label{sec:casimirrecursion}\n\nThe first method to find the coefficients $w(m,j)$ is to substitute the expansion \\reef{eq:radial_exp} into the Casimir equation.\nThis gives rise to recurrence relations, obtained in \\textcite{Hogervorst:2013sma} and \\textcite{Costa:2016xah}, which determine $w(m,j)$ for $m>0$ starting from $w(0,\\ell)$.\n\nNamely, defining the functions $ f_{m,j} \\equiv r^m {\\rm Geg}_j(\\eta)$, it is straightforward to show that any of the operators $\\{r,\\eta,\\partial_r,\\partial_\\eta\\}$ acting on these functions produces linear combinations of $f_{m\\pm1,j\\pm1}$. Similarly, the operator $\\calD$ in \\reef{eq:cb_diffeq}, when written in radial coordinates, maps $f_{m,j}$ into a linear combination of $f_{m+\\hat{m},j+\\hat{\\jmath}}$ functions with suitable shifts. Eq.~\\reef{eq:cb_diffeq} then gives rise to a relation which can be economically written in the form \\cite{Costa:2016xah}\n\\begin{eqnarray}\n\\label{recrelScalar}\n\\sum_{(\\hat{m},\\hat{\\jmath}) \\in \\mathcal{S}} c(\\hat{m},\\hat{\\jmath}) \\; w(m+\\hat{m},j+\\hat{\\jmath})=0 \\ ,\n\\end{eqnarray}\nwhere the set $\\mathcal{S}=\\{(0,0),(-1,1),(-1,-1),\\ldots\\}$ contains $30$ points, all of which but the first have $\\hat m<0$. The coefficients $c(\\hat{m},\\hat{\\jmath})$ are known functions of the variables $\\Delta_{12}$, $\\Delta_{34}$, $\\Delta$, $\\ell$, $d$, $m$, and $j$ \\cite[{attached \\texttt{Mathematica} notebook}]{Costa:2016xah}. Using Eq.~\\reef{recrelScalar}, the coefficient $w(m,j)$ can then be recursively expressed in terms of $w(m',j)$ with $m'<m$. \n\n\n\\subsubsection{Recursion relation from analytic structure}\n\\label{sec:analyticrecursion}\n\nThe second method exploits the analytic structure in $\\Delta$ to obtain a recursion relation for the conformal blocks. A similar approach was first applied by \\textcite{Zamolodchikov:1985ie, Zamolodchikov:1987} to the 2d Virasoro conformal blocks, by considering them as meromorphic functions of the central charge $c$ or of the scaling dimension $\\Delta$. For conformal blocks of external scalars in arbitrary $d$, this idea was introduced in \\textcite{Kos:2013tga,Kos:2014bka}. It was formalized and extended to conformal blocks for external operators with spin in \\textcite{Penedones:2015aga}. Here we will explain the external scalar case.\n\nEqs.~\\reef{eq:CPWrad}, \\reef{eq:proj} provide a convenient starting point for discussing the analytic structure of a conformal block as a function of the exchanged primary dimension $\\Delta$. When $\\Delta$ is above the unitarity bound, the Gram matrix $G_{\\alpha\\beta}(\\Delta)$ is positive-definite and invertible. However it turns out that for special values of $\\Delta=\\Delta_A^*$ at or below the unitarity bounds, the Gram matrix becomes degenerate, in the sense that some states are null (i.e.~have zero norm). The conformal block then develops a pole in $\\Delta-\\Delta^*_A$. Here we will assume that there are only simple poles, as is true for example in {odd $d$}, while for even $d$ simple poles coalesce into double ones, see below. \n\nThe crucial observation is that the residue of the pole is proportional to another conformal block{:}\n\\begin{equation}\n\\label{eq:resid}\ng^{\\Delta_{12},\\Delta_{34}}_{\\Delta,\\ell}(r,\\eta) \\sim  \\frac{R_A}{\\Delta-\\Delta^*_A}  g^{\\Delta_{12},\\Delta_{34}}_{\\Delta_A,\\ell_A}(r,\\eta)\\,.\n\\end{equation}\nNamely, we identify the first descendant state ${\\cal O}_A^{\\rm null}$ of ${\\cal O}$ which becomes null when $\\Delta\\to\\Delta_A^*$. Let $\\Delta_A=\\Delta_A^*+n_A$ be its dimension in this limit, and $\\ell_A$ its spin. \nIt can be shown that ${\\cal O}_A^{\\rm null}$ is annihilated by $K_\\mu$ when $\\Delta\\to\\Delta_A^*$ and so can be thought of as both a descendant and a primary. Consider then a fictitious primary ${\\cal O}_A$ which has quantum numbers $(\\Delta_A,\\ell_A)$ and which is unit-normalized. It is the conformal block of such a primary, with standard normalization \\reef{eq:cb_ope_e1}, that appears in the residue. \n\nTo be more precise, consider the rate with which ${\\cal O}_A^{\\rm null}$ becomes null as $\\Delta\\to\\Delta_A^*$:\n\\beq\n\\label{eq:QA}\n\\<{\\cal O}_A^{\\rm null}|{\\cal O}_A^{\\rm null}\\>\\sim Q_A (\\Delta-\\Delta_A^*)\\,,\n\\end{equation} \nwith $Q_A$ some constant. When ${\\cal O}_A^{\\rm null}$ becomes null, all of its descendants become null too, with the rate proportional to \\reef{eq:QA}. Moreover, it can be shown that the Gram matrix in the submultiplet consisting of these descendants is equal to \\reef{eq:QA} times the (non-singular) Gram matrix of the multiplet of ${\\cal O}_A$, up to corrections of higher order in $\\Delta-\\Delta_A^*$. This explains why the residue in \\reef{eq:resid} involves the whole conformal block of ${\\cal O}_A$.\\footnote{The Casimir equation gives another argument for why the residue is a conformal block. Near the pole the Casimir equation for the block reduces to the Casimir equation for the residue \\cite{Rychkov-OIST}. The Casimir eigenvalue of the null descendant is the same as for the original block (since it's a descendant): $C_{\\Delta_*,\\ell}=C_{\\Delta_A,\\ell_A}$. Finally, the boundary condition at $r\\to0$ is consistent with the residue being the conformal block.}\n\nThe coefficient $R_A$ in $\\reef{eq:resid}$ is a product of three factors:\n\\begin{equation}\n R_A=M_A^{(L)} Q^{-1}_A M_A^{(R)} \\ ,\n \\label{QMM}\n\\end{equation}\nwhere $Q_A$ is defined in \\reef{eq:QA}, while $M_A^{(L)}$ and $ M_A^{(R)}$ come from the 3pt functions \n $\\langle \\phi_3\\phi_4| {\\cal O}_A^{\\rm null} \\rangle $ and $\\langle  {\\cal O}_A^{\\rm null} | \\phi_1 \\phi_2  \\rangle$. \n\nUsing information about the poles, we can now write a complete formula for the conformal block. It is convenient to define the regularized conformal block $h_{\\Delta,\\ell} \\equiv h^{\\Delta_{12},\\Delta_{34}}_{\\Delta,\\ell}$ by removing a $(4r)^\\Delta$ prefactor:\n\\begin{equation}\ng^{\\Delta_{12},\\Delta_{34}}_{\\Delta, \\ell}(r,\\eta)= (4r)^{\\Delta} h_{\\Delta, \\ell}(r,\\eta) \\,.\n\\end{equation}\n\nThe {function} $h_{\\Delta, \\ell}$ has the same poles in $\\Delta$ as $g_{\\Delta,\\ell}$. Moreover it is a meromorphic function of $\\Delta$, and is therefore fully characterized by its poles and the value at infinity:\n\\begin{multline}\n\\label{eq:recscalar}\nh_{\\Delta,\\ell}(r,\\eta)=h_{\\infty,\\ell}(r,\\eta)\\\\\n+\\sum_{A} \\frac{R_A }{\\Delta-\\Delta^*_A} (4 r)^{n_A}\\,\nh_{\\Delta^*_A+n_A,\\ell_A}(r,\\eta)\\,.\n\\end{multline}\nDetailed analysis shows that the poles occurring in this equation organize into one finite and two infinite sequences:\n\\begin{equation}\\label{eq:polestable}\n\\begin{array}{lccc}\nA&\\Delta^*_A &n_A &\\ell_A\t \\\\ \n\\hline\n\\mbox{I${}_n$ }\\ \\  (n\\in\\hbox{$\\bb N$})\t\t& 1-\\ell-n \t& n \t  &\\ell + n \\\\ \n\\mbox{II${}_n$ }\\  (1\\leqslant n\\leqslant \\ell) \\;\\;  \t&   \\ell+d-1-n  \t & n &\\ell - n\\\\\n\\mbox{III${}_n$ } (n\\in\\hbox{$\\bb N$}) \\; \t& \\frac{d}2-n  & 2n  &\\ell       \\\\ \n\\end{array}\n\\end{equation}\nUsing this definition, it is easy to check that the residues of the poles themselves are nonsingular (except in even dimensions, see below).\n\nThe $h_{\\infty,\\ell}$ term and the constants $R_{A}$ are given by \\cite{Kos:2014bka, Penedones:2015aga} \n\t\\begin{align}\n\t\t\th_{\\infty, \\ell}(r,\\eta)&= \\textstyle \\frac{\\left(1-r^2\\right)^{1-\\frac{d}2}\n\t\t\t\\calN_{d,\\ell}{\\rm Geg}_\\ell(\\eta)}\n\t\t{\\left(r^2-2 \\eta  r+1\\right)^{\\frac{1-\\Delta_{12}+\\Delta_{34}}{2}} \\left(r^2+2 \\eta  r+1\\right)^{\\frac{1+\\Delta_{12}-\\Delta_{34}}{2}}}\\,, \\nn\\\\\n\t\tR_{\\text{I}_n}&=\\textstyle \\frac{-n (-2)^n }{  (n!)^2} \\left(\\frac{\\Delta_{12}+1-n}{2}\\right)_n \\left(\\frac{\\Delta_{34}+1-n}{2}\\right)_n \\  ,\\nn \\\\\n\t\tR_{\\text{II}_n}&= \\textstyle\\frac{-n \\,\\ell!  }{ (-2)^n (n!)^2 (\\ell-n)!   } \\frac{(d+\\ell-n-2)_n}{\\left(\\frac{d}2+\\ell-n\\right)_n\n\t\t\\left(\\frac{d}2+\\ell-n-1\\right)_n}  \t     \\label{eq:residues}\\\\\n\t\t&\\quad\\textstyle\\times \\left(\\frac{\\Delta_{12}+1-n}{2}\\right)_n  \\left(\\frac{\\Delta_{34}+1-n}{2}\\right)_n \\ ,\\nn \\\\\n\t\tR_{\\text{III}_n}&=\\textstyle\\frac{-n (-1)^{n} \\left(\\frac{d}2-n-1\\right)_{2 n}}{(n!)^2 \\left(\\frac{d}2+\\ell-n-1\\right)_{2 n} \\left(\\frac{d}2+\\ell-n\\right)_{2 n}}  \\nn\\\\\n\t\t&\\quad\\times\\textstyle\\left( \\frac{\\Delta _{12}-\\frac{d}2-\\ell-n+2}{2} \\right)_n \\left( \\frac{\\Delta _{12}+\\frac{d}2+\\ell-n}{2} \\right)_n \\nn\\\\\n\t\t&\\quad\\times\\textstyle \\left( \\frac{\\Delta _{34}-\\frac{d}2-\\ell-n+2}{2} \\right) _n \\left( \\frac{\\Delta _{34}+\\frac{d}2+\\ell-n}{2} \\right)_n  \\ . \\nonumber\n\t\\end{align}\n\nThe key property of Eq.~(\\ref{eq:recscalar}) is that each pole residue comes with a factor $r^{n_A}$. This means that it can be used as a recursion relation to generate the regularized conformal block as a power series in $r$. Indeed, suppose we want to compute $h_{\\Delta,l}(r,\\eta)$ up to $O(r^{N})$. We use Eq.~(\\ref{eq:recscalar}) keeping all poles with $n_A\\leqslant N$, of which there are finitely many. The residues of these poles themselves are needed up to smaller order $O(r^{N-n_A})$, so we get a recursion relation. This is one of the most elegant and efficient currently known methods to compute the conformal blocks {outside of even $d$.}\n\nThe described recursion relation is adequate for computing conformal blocks in odd dimensions and also in generic $d$. It cannot be {applied} directly in even $d$, since some simple poles coalesce into double poles. This is not a problem, since even $d$ conformal blocks are known in closed form. Alternatively, one can apply the recursion relation $\\epsilon$ away from an even $d$, and take the limit $\\epsilon\\to0$ after the coefficients of the $r$ expansion have been generated. This gives the correct result because the conformal blocks vary analytically with $d$.\n\n\n\\subsubsection{Rational approximation of conformal blocks and their derivatives}\n\\label{sec:rational}\n\nWe will now describe {how to construct} rational approximations to conformal blocks and their derivatives at a given point $(r_*,\\eta_*)$, {which permit an efficient numerical evaluation of these quantities as a function of $\\Delta$. This will play an important role in the numerical techniques described in Sec.~\\ref{sec:numerical}.} Our focus will be on rational approximations to scalar conformal blocks, but later in Sec.~\\ref{sec:spinning} we will also describe how they can be extended to blocks for external spinning operators.\n\nA rational approximation for conformal block derivatives at a given point can be obtained by combining the radial expansion (\\ref{eq:radial_exp}) and the recursion relation (\\ref{eq:recscalar}). It can be expressed in the form\n\\beq\n\\label{eq:rationalApprox}\n\\partial_r^m \\partial_\\eta^n g_{\\Delta,\\ell}(r_*,\\eta_*) =(4 r_*)^\\Delta \\left(\\frac{P_N^{mn}(\\Delta)}{Q_N(\\Delta)} + O(r_*^{N-m}) \\right)\\,.\n\\end{equation}\nHere $Q_N$ is a polynomial made by the product of poles given in Eq.~(\\ref{eq:polestable}) up to order $N$,\n\\begin{eqnarray}\nQ_N(\\Delta) \n= \n\\prod\n_{\n\tA=(\\text{I,II,III})\n\t{}_n\n\t,\\ n\\leqslant N\n} \n(\\Delta-\\Delta_A^*)\\,,\n\\end{eqnarray} \nand $P_N^{mn}(\\Delta)$ is a polynomial with ${\\rm deg}(P_N^{mn})\\leqslant{\\rm deg}(Q_N)+m$. The approximation can be made arbitrarily precise by increasing $N$, at the expense of increasing the order of the polynomials.\n\nIn numerical applications it is often desirable to keep the polynomial order relatively small while maintaining a precise approximation. This can be accomplished using a trick introduced in~\\textcite{Kos:2013tga}, where one discards some number of poles but compensates by modifying the residues of the kept poles. For example, if one keeps $n$ poles, one can choose their new residues by demanding that the first $n\/2$ $\\Delta$-derivatives match between the old and new functions at both {the unitarity bound} and $\\Delta = \\infty$. \n\nAn important property that will be exploited in Sec.~\\ref{sec:numerical} is that the denominator $Q_N(N)$ is always positive in unitary theories. This follows from the fact that all the poles are at values of $\\Delta$ below the unitarity bound.  \n\nThe techniques introduced in the previous sections allow one to compute conformal blocks either in closed form or as {a} power series in the variable $r$. Starting from these expressions one can take a direct approach of first analytically computing the $r$ expansion to order $N$, taking $r,\\eta$ derivatives of the resulting expression, and evaluating the result at the point $r_*,\\eta_*$. The result can then be recombined to the form in Eq.~(\\ref{eq:rationalApprox}). Since the crossing relations will be more simply written in $z,\\bar{z}$ coordinates, one then typically converts to $z,\\bar{z}$ derivatives at the corresponding point $z_*, \\bar{z}_*$ using a suitable transformation matrix. This approach, while somewhat inefficient at large $N$ due to the need to compute the analytical dependence on $\\eta$, has been successfully used in the literature, almost always at the crossing symmetric point $z_*=\\bar{z}_*=1\/2$ which corresponds to $\\eta_*=1,r_*=3-2\\sqrt{2}$.\n\n{A somewhat more efficient algorithm is the following: \\\\\n(i) Compute the $r$ expansion to order $N$ and take derivatives only along the radial direction $\\eta=1$ ($z=\\bar{z}$) using either the methods of Sections \\ref{sec:casimirrecursion} or \\ref{sec:analyticrecursion}.\\footnote{In even dimensions one can start from the closed form expression of Sec.~\\ref{sec:casimir} evaluated at $\\eta=1$, and expand in $r$.} \\\\\n(ii) Convert to $z,\\bar{z}$ derivatives along the diagonal $z=\\bar{z}$ using a suitable transformation matrix. \\\\\n(iii) Use the Casimir equation to recursively compute derivatives in the transverse direction. }\n\nLet us briefly discuss the last step, also called Cauchy-Kovalevskaya method. Consider the Casimir differential equation, Eq.~(\\ref{eq:cb_diffeq}), and express it in the variables $a=z+\\bar z,\\,\\sqrt b=(z-\\bar z)$. The radial direction corresponds to $b=0$. Moreover, since conformal blocks are symmetric in $z\\leftrightarrow \\bar z$, their power series expansion away from the $z=\\bar z$ line will contain only integer powers of $b$.\nLet us denote the $\\partial_a^m\\partial_b^n$ derivative of the conformal block evaluated at $z=\\bar z=1\/2$ by $h_{m,n}$. From step (i) we know $h_{m,0}$ for any $m$.  Then, we can translate the Casimir equation into a recursion relation for $h_{m,n}$ (with $n>0$) in terms of $h_{m,n}$ with lower values of $n$. This recursion relation was obtained in {\\cite[{Appendix C}]{ElShowk:2012ht} for $\\Delta_{12}=\\Delta_{34}=0$, and generalized in \\cite[{Eq. (2.17)}]{Behan:2016dtz}.} It has the general structure: \n\\begin{align}\\label{eq:Cauchy-Kovalevskaya}\n\th_{m,n}&=\\sum_{m'\\leqslant m-1} m (\\ldots) h_{m',n} \\\\\n\t&+\\sum_{m'\\leqslant m+2}\\left[ (\\ldots) h_{m',n-1} + (n-1) (\\ldots) h_{m',n-2} \\right]\\,.\\nonumber\n\\end{align}\nSince the Casimir equation is of second order, $m'$ can only take values up to $m+2$. Also the recursion relation for $h_{0,n}$ only involves $h_{m',n'}$ with {$n'<n$}. \nEq.~(\\ref{eq:Cauchy-Kovalevskaya}) is then all that we need to perform step (iii).\n\nWe conclude this section by mentioning a few software packages that implement the above efficient algorithm. Their functionality for solving convex optimization problems will be discussed in Sec.~\\ref{sec:numerical}, so here we focus on how they compute conformal blocks.\n\nA \\texttt{Mathematica} notebook by \\textcite{DSDnotebookLink} can be used for general scalar conformal blocks; at step (i) it implements the recursion relation from analytic structure discussed in Sec.~\\ref{sec:analyticrecursion}. It also implements the trick of shifting pole residues described above. \n\nAnother \\texttt{Mathematica} notebook \\textcite{JuliBoots} can also be used for general scalar conformal blocks. At step (i) it implements the recursion relation from the Casimir equation discussed in Sec.~\\ref{sec:casimirrecursion}. This notebook accompanies the \\texttt{Julia} package \\texttt{JuliBoots} for bootstrap computations using linear programming \\cite{Paulos:2014vya}.\n\nA \\texttt{Python} package \\texttt{PyCFTBoot} \\cite{Behan:2016dtz} and a \\texttt{Sage} package \\texttt{cboot} \\cite{CBoot} contain integrated functions that compute general scalar conformal blocks derivatives using the above procedure. These two packages are designed as frontends to the semidefinite program solver \\texttt{SDPB} \\cite{Simmons-Duffin:2015qma}.\n\n\\subsubsection{Shadow formalism}\n\\label{sec:shadow}\n\nNext we will briefly review the shadow formalism, which was historically the very first technique to access the conformal blocks \\cite{Ferrara:1973vz}, and it continues to play a role conceptually and also in explicit computations.\n\nSuppose we want to compute the conformal block $g_{\\Delta,\\ell}$ of a primary operator ${\\cal O}_{\\Delta,\\ell}$ in a scalar 4pt function $\\<\\phi_1\\phi_2\\phi_3\\phi_4\\>$. Consider a primary ``shadow operator\" $\\widetilde{{\\cal O}}_{d-\\Delta,\\ell}$ which has the same spin $\\ell$ as ${\\cal O}$ and dimension $d-\\Delta$. We stress that this operator is fictitious, it does not belong to the theory as a local operator, and in particular {the fact} that its dimension is below the unitarity bound is of no concern.\n\nThe starting point of the shadow formalism is the following integral:\n\\begin{align}\\label{eq:partial_wave}\n{\\rm U}_{\\Delta,\\ell}(x_1,x_2,x_3&,x_4) = \\int d^d x \\langle \\phi_1(x_1)\\phi_2(x_2)\\mathcal O_{\\Delta,\\ell}^{\\mu_1\\ldots\\mu_\\ell}(x)\\rangle \\nn\\\\\n &\\times\\langle \\widetilde{\\mathcal O}_{d-\\Delta,\\ell;\\mu_1\\ldots\\mu_\\ell}(x)  \\phi_3(x_3)\\phi_4(x_4)\\rangle\\,,\n\\end{align}\nwhere under the integral sign we have a product of the conformal scalar-scalar-(spin $\\ell$) 3pt functions in Eq.~\\reef{eq:3points}, with the spin-$\\ell$ operators having dimensions $\\Delta$ and $d-\\Delta$.\n\nThe function ${\\rm U}_{\\Delta,\\ell}$ has two special properties. First, it conformally transforms in the same way as the 4pt function $\\langle \\phi_1(x_1)\\phi_2(x_2)\\phi_3(x_3)\\phi_4(x_4)\\rangle$.\nThis is because the product (operator $\\times$ shadow) transforms as a dimension $d$ primary scalar, which compensates for the Jacobian in the transformation of $d^d x$. Consequently we can write\n\\beq\n\\label{eq:Uf}\n{\\rm U}_{\\Delta,\\ell} = f_{\\Delta,\\ell}(u,v)\\,{\\bf K}_4 \\,,\n\\end{equation}\nwhere ${\\bf K}_4$ is as in Eq.~(\\ref{eq:4pf_scalars}) and $f_{\\Delta,\\ell}(u,v)$ is some function of $u$ and $v$.\n\nSecond, it is straightforward to see that ${\\rm U}_{\\Delta,\\ell}$ is an eigenfunction of the Casimir operator acting at $x_1$, $x_2$, with eigenvalue $C_{\\Delta,\\ell}$. Since the latter property is also true for the CPW ${\\rm W}_{\\cal O}$, it is tempting to identify $f_{\\Delta,\\ell}(u,v)$ in \\reef{eq:Uf} with the conformal block (up to a proportionality factor). However, this is not quite true. The point is {that} the conformal blocks of the operator and of its shadow satisfy the same Casimir equation, since their Casimir eigenvalues coincide: $C_{\\Delta,\\ell}=C_{d-\\Delta,\\ell}$. For this reason $f_{\\Delta,\\ell}$ is a linear combination of the block $g_{\\Delta,\\ell}$ and of the shadow block $g_{d-\\Delta,\\ell}$; see \\cite[{Eq. (3.25)}]{DO3} for the precise relation. \n\nFrom the practical viewpoint, the main advantage of the shadow formalism is that the integrand in Eq.~\\reef{eq:partial_wave} is quite easy to write. The downside is that the resulting conformal integrals are not always easy to evaluate, and that it is necessary to disentangle the contribution of a proper conformal block from the shadow one. \n\nEfficient ways to deal with these problems were proposed by \\textcite{SimmonsDuffin:2012uy}. First of all, the integrals become much easier to evaluate when written using the embedding formalism. Second, to separate the block from the shadow one uses that they transform differently under a monodromy transformation\n\\begin{gather}\nz\\rightarrow e^{2\\pi i} z,\\quad \\bar z=\\text{fixed}\\,,\\\\\ng_{\\Delta,\\ell}(z,\\bar z) \\rightarrow e^{2\\pi \\Delta i} \\,g_{\\Delta,\\ell}(z,\\bar z)\\,.\n\\end{gather}\nThe wanted conformal block is isolated via a \\emph{monodromy projector}, implemented as a proper choice of the integration contour in Eq.~(\\ref{eq:partial_wave}).\nThis prescription allows one to extract integral expressions for generic conformal blocks in arbitrary $d$. In some cases the conformal integrals can be performed exactly, and the results match the known formulas from other techniques.\n\n\n\\subsubsection{Spinning conformal blocks}\n\\label{sec:spinning}\n\nAlthough in this review we will mostly deal with scalar 4pt functions, the bootstrap has also been successfully applied to 4pt functions of operators {with spin; e.g., see Secs.~\\ref{sec:Fermions} for $j=1\/2$ spinors and \\ref{sec:JandT} for $\\ell=1,2$ tensors in 3d.} Here we will review the theory of the associated conformal blocks, referred to as ``spinning\", which present additional difficulties compared to the blocks of external scalars. \n\nAs in the scalar case, spinning conformal blocks correspond to the contribution of an entire conformal multiplet to a 4pt function. They are defined by the equation\n\\begin{align}\n\\label{eq:spinning_cb}\n&\\langle {\\cal O}_3 {\\cal O}_4 | \\mathcal P_{\\Delta,r} |{\\cal O}_1{\\cal O}_2 \\rangle \n=\\\\  \n&{\\bf K}_4 \\sum_{a=1}^{n_3}\\sum_{b=1}^{n'_3} \\sum_{c=1}^{n_4} \\lambda_{12 \\mathcal O^\\dagger}^{(a)} \\lambda_{34 \\mathcal O}^{(b)}{\\bf T}_{4}^{(c)}(x_i,\\zeta_i) {\\bf G}_{c,\\Delta,r}^{a,b}(\\Delta_i,r_i,u,v).\\nn\n\\end{align}\n\nHere, the external operators ${\\cal O}_i={\\cal O}_{\\Delta_i,r_i}(x_i,\\zeta_i)$ are positioned {at} $x_i$ and have their indices contracted with auxiliary polarization vectors (or spinors) $\\zeta_i$. They transform in some general $SO(d)$ {(or $Spin(d)$)} representations $r_i$. On the other hand $\\mathcal O_{\\Delta,r}$ is the exchanged operator (and $\\mathcal O_{\\Delta,r^\\dagger}^\\dagger$ its conjugate, see the discussion in Sec.~\\ref{sec:2pt}), and $\\mathcal P_{\\Delta,r}$ is the projector onto its conformal multiplet similar to Eq.~(\\ref{eq:proj}).\n\nThe prefactor ${\\bf K}_4$ is as in Eq.~(\\ref{eq:4pf_scalars}); it captures the scaling properties of the 4pt function, leaving everything else dimensionless. Eq.~\\reef{eq:spinning_cb} also contains a sum over possible conformally invariant 4pt tensor structures ${\\bf T}_4^{(c)}$, and a double sum over possible 3pt function structures\n\\begin{eqnarray}\n\\label{eq:3pt-spinning}\n&&\\langle \\mathcal O_{\\Delta_1,\nr_1}(x_1,\\zeta_1) \\mathcal O_{\\Delta_2,\nr_2}(x_2,\\zeta_2) \\mathcal O^\\dagger_{\\Delta,\nr^\\dagger}(x_3,\\zeta_3)\\rangle =\\\\\n&&\\hspace{3em}\\displaystyle \\sum_{a=1}^{n_3} \\lambda_{12 \\mathcal O^{\\dagger}}^{(a)} {\\bf T}_3^{(a)}(x_i,\\zeta_i,\\{\\Delta_1,r_1\\},\\{\\Delta_2,r_2\\},\\{\\Delta,r^{\\dagger}\\}),\\nn\n\\end{eqnarray}\nand similarly for $n'_3$. Finally, the functions ${\\bf G}_{c,\\Delta,r}^{a,b}(\\Delta_i,r_i,u,v)$ are the spinning conformal blocks. \n\nAccording to the above definition, when $r$ is not a real representation, both ${\\bf G}_{c,\\Delta,r}^{a,b}$ and ${\\bf G}_{c,\\Delta,r^\\dagger}^{a,b}$ have to be considered and generally both these blocks are nonzero. We will see a 4d example for $r=(\\ell,\\ell+p)$ below.\n\nSpinning blocks can be computed by reducing them to ``seed\" blocks. Consider the simplest case when the exchanged primary is a traceless symmetric spin $\\ell$. To understand the reduction to seeds, the key observation is that the 3pt tensor structures \\reef{eq:3pt-spinning} can be produced by differentiating the more elementary scalar-scalar-(spin $\\ell$) 3pt functions \\reef{eq:3points}. Namely \\textcite{Costa:2011dw} showed that there exist ``spinning-up\" differential operators $D^{(a)}_{r_1,r_2}$, depending on $x_i$ and $\\zeta_i$, such that\n\\begin{eqnarray}\n&&{\\bf T}_3^{(a)}(x_i,\\zeta_i,\\{\\Delta_1,r_1\\},\\{\\Delta_2,r_2\\},\\{\\Delta,r\\}) =\\\\\n&&\\hspace{3em} D^{(a)}_{r_1,r_2} \\mathcal {\\bf T}_3(x_i,\\zeta_3,\\{\\Delta'_1,0\\},\\{\\Delta'_2,0\\},\\{\\Delta,r\\}),\\nn\n\\end{eqnarray}\nfor a suitable basis of 3pt structures and choice of $\\Delta_i'$.\\footnote{In a {generic} basis of 3pt structures, e.g.~one that would be naturally constructed using the embedding space or conformal frame formalisms, there would be a linear combination of terms like the r.h.s. with different shifts.} Notice that in the above expression the third point is not affected. Therefore, in the definition (\\ref{eq:spinning_cb}), the differential operators do not interfere with the sum over descendants in $\\mathcal P_{\\Delta,\\ell}$. One concludes that the spinning blocks can be obtained by differentiating the scalar blocks:\n\\begin{multline}\n\\label{eq:spinning_cb_p0}\n{\\bf K}_4 (\\Delta_i)\\sum_{c=1}^{n_4} {\\bf T}_{4}^{(c)}(x_i,\\zeta_i){\\bf G}_{c,\\Delta,\\ell}^{a,b}(\\Delta_i,r_i,u,v) \\\\\n= D^{(a)}_{r_1,r_2}  D^{(b)}_{r_3,r_4} {\\bf K}_4 (\\Delta_i') g_{\\Delta,\\ell}^{\\Delta_{12}',\\Delta_{34}'}(u,v)\\,,\n\\end{multline}\nwhere $g_{\\Delta,\\ell}^{\\Delta_{12}',\\Delta_{34}'}(u,v)$ is the scalar conformal block discussed at length in the previous sections, referred to as a seed block in this situation. \n\nIn 3d, traceless symmetric tensors exhaust all bosonic $SO(d)$ representations, and therefore all bosonic spinning blocks can be obtained from scalar seeds via \\reef{eq:spinning_cb_p0}.\\footnote{Some explicitly worked out cases in 3d are for external operator pairs being (current)-(current) \\cite{Costa:2011dw}, scalar-(current or stress tensor) \\cite{Li:2015itl}, and (stress tensor)-(stress tensor) \\cite{Dymarsky:2017yzx}.} The 3d spinning-up operators were also extended to external spinors and exchanged spin $\\ell$ by \\textcite{Iliesiu:2015qra}. \n\nIf a representation $r$ does not couple to two scalars, its conformal block cannot be reduced to the scalar seed using this method. One therefore needs more seed blocks for such representations. As an example, consider the half-integer spin representations in 3d. The simplest pair of external operators to which they couple are a scalar $\\phi$ and a Majorana fermion $\\psi$. The corresponding conformal block $ \\langle\\phi_3 \\psi_4 | \\mathcal P_{\\Delta,j}|\\phi_1 \\psi_2\\rangle$ for half-integer $j$ can be taken as a seed. It was computed by \\textcite{Iliesiu:2015akf}, using recursion relations as in Sec.~\\ref{sec:analyticrecursion}, making the list of 3d seeds complete. \n\nA similar discussion holds in 4d. In this case the complete set of seed blocks corresponds to the representations $r=(\\ell,\\ell+p)$ and $(\\ell+p,\\ell)$ appearing in the 4pt function of two scalars, one $(p,0)$ tensor, and one $(0,p)$ tensor:\n\\beq\n\\langle \\phi_3(x_3)\\mathcal O_{\\Delta_4,(0,p)}(x_4)| \\mathcal P_{\\Delta,r}|\n \\phi_1(x_1)\\mathcal O_{\\Delta_2,(p,0)}(x_2) \\rangle\\,.\n\\end{equation}\nAll of these seeds were computed in closed form by \\textcite{Echeverri:2016dun}, making use of the shadow formalism from Sec.~\\ref{sec:shadow}.\n\nOnce the seeds are known, a relation analogous to (\\ref{eq:spinning_cb_p0}) allows one to relate any conformal block to a combination of seed blocks thorough a suitable set of spinning-up operators $D^{(a)}_{r_i,r_j}$. The latter can be nicely written in the embedding formalism discussed in Appendix~\\ref{sec:embedding} or one of its generalizations. The precise expressions can be found in \\textcite{Costa:2011dw,Iliesiu:2015akf} in 3d or \\textcite{Echeverri:2015rwa} in 4d. In 4d there is also available a comprehensive {\\tt Mathematica} package {\\tt CFTs4D} \\cite{Cuomo:2017wme} designed to facilitate general spinning 4d conformal block computations. {Spinors and spinor-tensor correlators in aribtrary dimensions were instead studied in \\textcite{Isono:2017grm}}\n\nLet us mention briefly several other ideas which have proved useful when dealing with spinning blocks. \\textcite{Karateev:2017jgd} introduced a more general class of ``weight-shifting\" operators which act on correlation functions. In addition to reproducing the spinning-up operators as a special case, they have a further interesting consequence: when acting on a conformal block these operators can change the $SO(d)$ {(or $Spin(d)$)} representation of the exchanged state {by utilizing the $6j$ symbols of the conformal group. Through} repetitive use of these operators, it is possible to express any conformal block, including the seeds, in terms of the scalar ones.\\footnote{Explicit formulas expressing the seed blocks in 3d and 4d are provided in \\textcite{Karateev:2017jgd}.} {These methods also lead to efficient derivations of various recursion relations satisfied by the conformal blocks.}\n\nThe Casimir recursion approach from Sec.~\\ref{sec:casimirrecursion} was extended to arbitrary external bosonic operators by \\textcite{Costa:2016xah}. More recently, \\textcite{Kravchuk:2017dzd} considered similar expansions for arbitrary external operators, and related the recursion relation coefficients to the $6j$ symbols of $Spin(d-1)$, which are known in closed form for arbitrary representations in {$d=3,4$}, and for representations entering the seed blocks in arbitrary $d$. He also discusses how to convert from the $z$ to the $\\rho$ coordinate, as is needed for practical applications. \n\nThe pole expansion of Sec.~\\ref{sec:analyticrecursion} has also been generalized to spinning conformal blocks \\cite{Penedones:2015aga,Costa:2016xah}. Although no closed form expressions are known for the analogues of $h_\\infty$ and of the residues $R_A$ in Eq.~(\\ref{eq:recscalar}), these ingredients can sometimes be found by combining this approach with the spinning-up\/weight-shifting operators, as in {\\textcite{Iliesiu:2015akf}, \\textcite{Dymarsky:2017xzb}, and \\textcite{Karateev:2017jgd}.} Commuting these operators with the pole expansion sum, one obtains the expected pole expansion for spinning conformal blocks. By truncating the pole expansion, rational approximations similar to those considered in Sec.~\\ref{sec:rational} can then be constructed for each spinning block tensor structure.\n\nFinally, the shadow block technique discussed in Sec.~\\ref{sec:shadow} has been used to compute the conformal blocks appearing in 4pt function of two scalars and two identical conserved currents \\cite{Rejon-Barrera:2015bpa}.\n\n\n\n\\subsubsection{2pt functions}\n\\label{sec:2pt}\n\nIt follows from Eq.~(\\ref{eq:correlations_covariance}) that the 2pt function of two operators ${\\cal O}_{\\Delta_1,r_1}$ and ${\\cal O}_{\\Delta_2,r_2}$ vanishes unless $\\Delta_1=\\Delta_2$ and $r_1=r_2^\\dagger$.\\footnote{Here $\\dagger$ means complex conjugation in Lorentzian signature, or taking the dual reflected representation in Euclidean signature, where reflected means replacing generators $R_{1\\nu}$ by $-R_{1\\nu}$. In 3d all representations are real, so the requirement $r_1 = r_2^{\\dagger}$ reduces to $r_1=r_2$, while in 4d if $r_1=(\\ell,\\bar \\ell)$ then $r_2=(\\bar\\ell,\\ell)$.} As a consequence, for every physical operator ${\\cal O}_{\\Delta,r}$, one can identify an operator ${\\cal O}^\\dagger_{\\Delta,r^\\dagger}$ which transforms in the conjugate representation.\\footnote{The precise action of Hermitian conjugation on Hilbert space operators depends on the signature and choice of quantization surface. For a detailed discussion see \\textcite{Simmons-Duffin:2016gjk}.}\n\nFurther, one can almost always work in a basis of operators such that ${\\cal O}$ has a nonzero 2pt function only with ${\\cal O}^\\dagger$, which is usually stated as ``the 2pt function is diagonal\".\\footnote{Examples of nonunitary conformal theories in which the 2pt functions cannot be so diagonalized occur in logarithmic CFTs, see e.g.~\\textcite{Hogervorst:2016itc}. We will not consider them in this review.} \nFor example, this is always possible in unitary theories. For operators in real $SO(d)$ representations $r^{\\dagger} = r$, like traceless symmetric tensors, we can choose a real operator basis so that ${\\cal O}^\\dagger = {\\cal O}$.\n\nSpecializing to traceless symmetric tensors, the 2pt function takes the form\\footnote{For the purposes of this review, it is sufficient to consider correlation functions in Euclidean signature. Most equations can also be used in Lorentzian signature, provided that all points are spacelike separated. For timelike separation one needs modifications, such as an $i\\epsilon$ prescription, which we will not discuss.}\n\\begin{gather}\n\\langle  {\\cal O}_{\\Delta,\\ell}(x_1,\\zeta_1)  {\\cal O}_{\\Delta,\\ell}(x_2,\\zeta_2) \\rangle = \\frac{\\left( I_{\\mu\\nu}(x_{12})\\zeta_1^\\mu \\zeta_2^\\nu\\right)^\\ell-\\text{traces} }{(x_{12}^2)^{\\Delta}}\\,,\\nn\\\\\nI_{\\mu\\nu}(x) =\\eta_{\\mu\\nu} -2 {x_\\mu x_\\nu}\/{x^2}\\,, \\label{eq:2points}\n\\end{gather}\nwhere $x_{ij} \\equiv x_i-x_j$, and ``traces\" are terms proportional to $\\zeta_1^2$, $\\zeta_2^2$, which are uniquely fixed by the tracelessness of ${\\cal O}_{\\Delta,\\ell}$. This generalizes Eq.~\\reef{eq:2pt} for scalars. It is customary to normalize such 2pt functions to unity, with exceptions being conserved currents and the stress tensor, see Sec.~\\ref{sec:ward}. The nontrivial part of the correlator is its numerator, which {specifies the dependence on the operator indices}. We will refer to such numerators as ``tensor structures\".\n\nIf the CFT contains a global symmetry, operators are grouped into global symmetry multiplets $\\pi$. In this case Eq.~\\reef{eq:2points} still applies to the individual components of the multiplets, with obvious appropriate modifications.\\footnote{If $\\pi$ is a complex representation, then it is not convenient to use the real operator basis. The nonzero 2pt function will then be between ${\\cal O}$ and ${\\cal O}^\\dagger$ transforming in $\\bar\\pi$.} We will discuss the consequences of global {symmetries} further in Sec.~\\ref{sec:global}.\n\n\n\\subsubsection{3pt functions}\n\\label{sec:3pt}\n\nNext we turn to 3pt functions, focusing on the case where the first two operators are scalars. Then it turns out that the third operator can only be a traceless symmetric tensor. {Generalizing Eq.~\\reef{eq:3pt} for three scalars, the 3pt function takes the form \\cite{Mack:1976pa}}\n\\begin{multline}\n\\label{eq:3points}\n\\langle  \\mathcal O_{\\Delta_1}(x_1)  \\mathcal O_{\\Delta_2}(x_2)  \\mathcal O_{\\Delta_3,\\ell}(x_3, \\zeta) \\rangle= \\\\\n\\lambda_{123}\\, [\\left(Z^\\mu_{123} \\zeta_\\mu \\right)^\\ell -\\text{traces}]\\, {\\bf K}_3\\,,\n\\end{multline} \nwhere ${\\bf K}_3={\\bf K}_3(\\Delta_i,x_i)$ is given by\n\\beq\n{\\bf K}_3 =\\frac{1} {(x^2_{12})^{\\frac{h_{123}+\\ell}2} (x^2_{13})^{\\frac{h_{132}-\\ell}2}\\,\n\t(x^2_{23})^{\\frac{h_{231}-\\ell}2}}\\,,\n\\end{equation}\n$h_{ijk} \\equiv \\Delta_i+\\Delta_j-\\Delta_k$, and $Z_{123}^{\\mu}= \\frac{x^\\mu_{13}}{x_{13}^2}-\\frac{x^\\mu_{23}}{x_{23}^2}$. This 3pt function is unique up to the overall coefficient $\\lambda_{123}$. Notice that as defined,\n\\beq\n\\label{eq:flipsign}\n\\lambda_{123} = (-1)^\\ell \\lambda_{213}\\,,\n\\end{equation}\nwhile if $\\ell=0$ we can exchange any pair of fields and $\\lambda_{123}$ is fully symmetric.\nThe normalization of these coefficients is unambiguous, since the operators are assumed to be unit-normalized according to Eq.~(\\ref{eq:2points}). \nTogether with the {spectrum}, the $\\lambda$'s constitute the \\emph{CFT data}, which distinguish one CFT from another, as discussed in Sec.~\\ref{sec:informal}. \n\nIn unitary theories, {the CFT data must satisfy a set of} general well-understood constraints, see Sec.~\\ref{sec:unitarity}. {Significantly more} nontrivial constraints on the CFT data come from the crossing relations to be discussed in Sec.~\\ref{sec:crossing}.\n\nFor operators in three general $SO(d)$ representations{, the 3pt} functions take a form more complicated than \\reef{eq:3points}. They are also in general not unique, although for any three representations there is at most a finite-dimensional space of allowed tensor structures. The problem of their construction has been completely solved in the most physically important cases of $d=3$~\\cite{Costa:2011mg,Iliesiu:2015qra} and $d=4$~\\cite{Elkhidir:2014woa}. For general $d$ there are partial results, e.g.~\\textcite{Costa:2011mg} for 3pt functions of traceless symmetric tensors,~\\textcite{Costa:2016hju} for traceless mixed-symmetry tensors, and~\\textcite{Kravchuk:2016qvl} for a general approach to classifying the structures.\n\n\\subsubsection{4pt functions}\n\nFinally let us consider 4pt functions, which as mentioned in Sec.~\\ref{sec:informal} play {a} fundamental role in the conformal bootstrap. Focusing here on the case of scalars, the 4pt function must take the general form\n\\beq\n\\label{eq:4p}\n\\langle  \\mathcal O_{\\Delta_1}(x_1)  \\mathcal O_{\\Delta_2}(x_2)  \\mathcal O_{\\Delta_3}(x_3)  \\mathcal O_{\\Delta_4}(x_4)\\rangle =  g(u,v)\\, \\mathbf{K}_4\\,.\n\\end{equation}\nThe factor $\\mathbf{K}_4=\\mathbf{K}_4(\\Delta_i,x_i)$ is given by\n\\beq\n\\label{eq:K4}\n\\mathbf{K}_4 = \n\\frac{1}{(x_{12}^2)^{\\frac{\\Delta_1+\\Delta_2}2}(x_{34}^2)^{\\frac{\\Delta_3+\\Delta_4}2}} \\left(\\frac{x^2_{24}}{x^2_{14}}\\right)^{\\frac{\\Delta_{12}}2}\n\\left(\\frac{x^2_{14}}{x^2_{13}}\\right)^{\\frac{\\Delta_{34}}2}\n\\,, \n\\end{equation}\nwhere {$\\Delta_{ij} \\equiv \\Delta_i - \\Delta_j$}. This factor by itself transforms under conformal transformations as {prescribed by Eq.~(\\ref{eq:correlations_covariance}). }\nThe remaining part of the correlator, $g(u,v)$, must be a function of two \\emph{cross ratios} $u,v$:\n\\begin{eqnarray}\n\\label{eq:uv}\nu=\\frac{x_{12}^2 x_{34}^2}{x_{13}^2 x_{24}^2}\\,, \\qquad v=\\frac{x_{14}^2 x_{23}^2}{x_{13}^2 x_{24}^2}\\,,\n\\end{eqnarray}\nwhich are invariant under all conformal transformations.\n\nWhile no further information about $g(u,v)$ can be obtained from conformal invariance alone, it can in fact be computed in terms of the CFT data using additional tools such as the OPE and conformal blocks. This will be discussed in Secs.~\\ref{sec:OPE} and~\\ref{sec:cb}.\n\n\n\\subsubsection{Conformal frames} \n\\label{sec:confframe}\nHere we will give a more group theoretical intuition of the number of degrees of freedom contained in a given correlator, and in particular of why conformal invariance fixes 2pt and 3pt functions up to a few constants, but allows arbitrariness in 4pt functions. \n \nGiven a set of $n$ points, we can make use of conformal transformations to arrange them in convenient configurations. {For instance, given} 3 arbitrary points we can find a conformal transformation which maps them to $x_{1,2,3}=0,\\hat{e},\\infty$, where $\\hat{e}$ is a fixed unit vector.\n\n\\begin{figure}[t]\n\t\\begin{centering}\n\t\t\\includegraphics[width=0.6\\columnwidth]{fig02-z-frame.pdf}\n\t\t\\caption{\\label{fig:z-frame}\n\t\t\t(Color online) Conformal frame defining the $z$ coordinate. Figure from \\cite{Hogervorst:2013sma}.\n\t\t}\n\t\\end{centering}\n\\end{figure}\n\nFor 4 points, we can first find a conformal transformation fixing 3 of them as above, and then rotate around the axis to put the fourth point into a fixed plane (we assume that $d\\geqslant 2$). The resulting configuration can be parametrized in Euclidean signature as ($\\mathbf{0} \\equiv \\mathbf{0}_{d-2}$)\\footnote{We define ${\\cal O}(\\infty)$ {by taking} the limit of $|x_4|^{2\\Delta_{\\cal O}} {\\cal O}(x_4)$ {as} $x_4\\to\\infty$, {which yields a finite value for the correlation function.}}\n\\begin{align}\nx_1 &= (0,0,\\mathbf{0})\\,,\\quad x_2 = (\\sigma,\\tau,\\mathbf{0})\\,,\\nn\\\\\nx_3 &= (1,0,\\mathbf{0})\\,,\\quad  x_4 = (\\infty,0,\\mathbf{0})\\,.\n\t\\label{eq:conformalframe1}\n\\end{align}\nIt is customary to define (see Fig.~\\ref{fig:z-frame})\n\\beq\n\\label{eq:z}\nz=\\sigma+i\\tau,\\quad \\bar{z}=\\sigma-i\\tau\\,,\n\\end{equation} \nwhich are complex conjugate variables if we are working in the Euclidean.\\footnote{Notice that we can analytically continue to {the Lorentzian} via $\\tau\\to i t$, and then $z$ and $\\bar z$ become independent real variables, but this will not play a role in this review.} The conformal cross ratios can be expressed in terms of $z$, $\\bar{z}$ as\n\\begin{eqnarray}\n\\label{eq:zzb}\n\tu= z\\bar{z} \\,, \\qquad v= (1-z)(1-\\bar{z})\\,.\n\\end{eqnarray}\n\nA choice of points $x_i$, as in Eq.~(\\ref{eq:conformalframe1}), is called a \\emph{conformal frame}. It can be thought of as a gauge fixing of most or all of the conformal symmetry. By construction, any coordinate configuration can be reduced to the conformal frame form. Therefore, the knowledge of a correlation function in the conformal frame is sufficient to reconstruct it at any other point through its covariance properties {\\cite{Osborn:1993cr}}. The functional forms of 2pt and 3pt functions are fixed because their conformal frames do not contain any free parameters. The 4pt conformal frame\n\\reef{eq:conformalframe1} has 2 real parameters, explaining the functional freedom of the conformal 4pt function. See Sec.~\\ref{sec:radial} for another frequently used conformal frame.\n\nConformal frames provide a way to construct conformal correlators {which is sometimes more convenient than the embedding formalism described in App.~\\ref{sec:embedding}.} This method can also be used to classify the allowed tensor structures. {An important role is then played by the stabilizer group, defined as the set of conformal transformations leaving the conformal frame configuration invariant. It is $SO(d-1)$ for 3pt functions and $SO(d-2)$ for 4pt functions. One classifies tensor structures invariant under the stabilizer group, and each of them lifts to an independent conformally invariant tensor structure \\cite{Kravchuk:2016qvl}. This method is particularly useful when dealing with 4pt functions of tensor operators: it does not overcount tensor structures, which may happen in the embedding formalism unless special care is taken.}\n\n\n\n\\subsubsection{Explicit solutions to crossing}\n\\label{sec:explicit}\n\nMany nontrivial 2d CFTs have exact solutions (e.g.~the minimal models), and the conformal block decompositions of their 4pt functions provide explicit solutions to crossing relations. \nHere we will discuss a few explicit solutions to crossing known in $d>2$. Their existence is important, even though as we will see they come from theories which are not physically the most interesting ones. For example, it is common to check the numerical algorithms against the known explicit solutions to exclude coding errors, before proceeding to study more physically interesting solutions numerically. \n\nEssentially all explicit solutions in $d>2$ are provided by scale-invariant ``gaussian theories\", i.e.~theories coming from a quadratic action written in terms of a fundamental field and not having any massive parameter.\\footnote{The only exceptions known to us are the ``fishnet theories\" ---  nonunitary bi-scalar field theories integrable in the large-$N$ limit \\cite{Gurdogan:2015csr}. Recently some conformally-invariant 4pt functions and their conformal block decompositions were computed in such theories in 4d \\cite{Grabner:2017pgm}, and in their nonlocal generalizations to arbitrary $d$ \\cite{Kazakov:2018qbr}.}\nThe correlation functions of such theories are generated by Wick's theorem from the basic 2pt function of the fundamental field. The simplest examples are the massless free scalar and massless free fermion theory, which are conformally invariant in any $d$, and the free abelian gauge theory, conformally invariant in $d=4$. In 4d, explicit conformal block decompositions of 4pt scalar correlation functions $\\< {\\cal O} {\\cal O} {\\cal O} {\\cal O} \\>$ in these theories (for ${\\cal O}=\\phi$, $\\phi^2$, $\\bar\\psi\\psi$, $F_{\\mu\\nu}^2$) were obtained by \\textcite{DO1}. \n\nAnother class of gaussian theories are mean field theories (MFTs), also called generalized free fields~\\cite{Heemskerk:2009pn}, \\cite[{section 4}]{ElShowk:2011a}. Correlation functions in these theories have the same disconnected structure generated by Wick's theorem as in the above mentioned free theories. The only difference is that the scaling dimension of the fundamental field, fixed to a particular value in free theories, becomes a free parameter in MFT.\\footnote{This structure naturally emerges in large-$N$ CFTs as a consequence of large-$N$ factorization. This is particularly transparent in CFTs with holographic duals, since MFT correlation functions are generated by free massive fields in AdS${}_{d+1}$ and the arbitrary scaling dimension is determined by the mass.}  For example, we can consider the MFT of a scalar field $\\phi$ of arbitrary dimension $\\Delta_\\phi$. Such a MFT is unitary as long as $\\Delta_\\phi$ satisfies the unitarity bound, and reduces to the free massless scalar for $\\Delta_\\phi=(d-2)\/2$. Just like for the usual free theories, the full space of operators in MFTs can be classified by considering normal-ordered products of the fundamental field and its derivatives.\\footnote{The OPE $\\phi\\times\\phi$ contains only operators of the schematic form $\\phi (\\partial^{2})^n\\partial^\\ell\\phi$, which have spin $\\ell$ and dimension \\mbox{$2\\Delta_\\phi+2n+\\ell$}.} For example there is an operator $\\phi^2$ which has dimension $2\\Delta_\\phi$. \n\nAlthough relatively trivial and nonlocal, MFTs satisfy most CFT axioms (except for for the existence of a local stress tensor). As we will see below, they frequently fall inside regions allowed by the bootstrap bounds, so it helps to be familiar with them. Explicit conformal block decompositions of MFT 4pt functions containing scalars were obtained by \\textcite{Heemskerk:2009pn} for $d=2,4$ and by \\textcite{Fitzpatrick:2011dm} in general $d$. \n\n\n\n\n\n\n\n\\subsubsection{Unitarity bounds}\n\\label{sec:UB}\n\nWe already get a simple and powerful constraint by considering radial quantization states $|\\Psi\\>$ produced by a local operator ${\\cal O}$ acting at the origin. In this case the conjugate operator is inserted at infinity. For {a} primary ${\\cal O}$ we recover that its 2pt function must have positive normalization and hence can be normalized to one as in \\reef{eq:2points}. Additional constraints arise from considering descendants of ${\\cal O}$. The conformal algebra computes the norms of descendants as polynomials in the primary dimension $\\Delta$. Imposing that all descendants have a non-negative norm gives a lower bound on $\\Delta$. This \n``unitarity bound\" depends on the representation $r$ of $SO(d)$ (or its double cover {for spinor representations}) in which the primary transforms.\\footnote{Standard CFT references are \\textcite{Ferrara:1974pt}, \\textcite{Mack:1975je}, and \\textcite{Minwalla:1997ka}. An early physics reference is \\textcite{doi:10.1063\/1.1705183}. In the mathematics literature, these bounds were derived by \\textcite{jantzen_kontravariante_1977}, although the relevance of this work for physics was realized only recently \\cite{Penedones:2015aga,Yamazaki:2016vqi}. See also \\textcite{Rychkov:2016iqz,Simmons-Duffin:2016gjk} for a review. Unitarity bounds can be equivalently derived by studying the positivity of the Fourier transform of the 2pt function analytically continued to Lorentzian signature (the Wightman function), see \\textcite{Ferrara:1974pt}, \\textcite{Mack:1975je} (in the sufficiency part of the argument), as well as \\textcite{Grinstein:2008qk} for a recent exposition emphasizing physics.} \\footnote{In Lorentzian signature, operators satisfying the unitarity bounds correspond to the unitary representations of the universal covering group of the Lorentzian conformal group $SO(d,2)$ having positive energy. Notice that in Euclidean signature operators satisfying the unitarity bounds have no relation to the representation of the Euclidean conformal group $SO(d+1,1)$ which are unitary in the usual mathematical sense of the term. This is already clear from looking at the principal series unitary representations of $SO(d+1,1)$ which have complex scaling dimensions $d\/2+i\\bR$.}\n\nIn 3d, the representation $r$ is labeled by a half-integer $j$, with $j=\\ell$ for traceless symmetric spin-$\\ell$ tensors. The unitarity bounds are\n\\begin{align}\n\td=3:\\quad &\\Delta\\geqslant 1\/2 &&(\\text{scalar, } j=0)\\,,\\nn\\\\\n\t&\\Delta\\geqslant 1 &&(\\text{smallest spinor, } j=1\/2) \\,,\\\\\n\t&\\Delta\\geqslant j+1\\  &&(j>1\/2)\\,.\\nn\n\t\\end{align}\n\nIn 4d, we can label the representation $r$ by two integers $(\\ell,\\bar \\ell)$, with traceless symmetric spin-$\\ell$ tensors having $\\ell=\\bar\\ell$.\\footnote{It is also common in the literature to label by half-integers $j=\\ell\/2,\\, \\bar{j}=\\bar{\\ell}\/2$.} The unitarity bounds then read\n\\begin{align}\n\td=4:\\quad &\\Delta\\geqslant 1&&(\\text{scalar, } \\ell=\\bar\\ell=0)\\,,\\nn\\\\\n\t&\\Delta\\geqslant {\\textstyle\\frac 12} \\ell+1 &&(\\ell>0, \\bar\\ell=0) \\,,\\label{eq:4dunitarity}\\\\\n\t&\\Delta\\geqslant {\\textstyle\\frac 12}(\\ell+\\bar\\ell)+2 &&(\\ell \\bar\\ell\\ne0)\\,.\\nn\n\\end{align}\n\nFor the 5d and 6d unitarity bounds see \\textcite{Minwalla:1997ka}. For some representations occurring in all dimensions the unitarity bounds can be written in dimension-independent form as follows:\n\\begin{align}\n\t &\\Delta\\geqslant {\\textstyle\\frac 12}(d-2)&&(\\text{scalar})\\,,\\nn\\\\\n\t&\\Delta\\geqslant {\\textstyle\\frac 12}(d-1) &&(\\text{smallest spinor}) \\,,\\\\\n\t&\\Delta\\geqslant \\ell+d-2 &&(\\text{traceless symmetric, spin }\\ell\\geqslant 1)\\,.\\nn\n\\end{align}\n\nAs a final comment, in physics literature the unitarity bounds are often derived by imposing positivity of the descendant norm{s} on the first (and the second, for scalars) level. It is a nontrivial fact that no further constraints arise from higher levels. See \\textcite[{Tables 3 and 5}]{Bourget:2017kik} for a review of rigorous mathematical results for unitary bounds in any $d$.\n\n\\subsubsection{OPE coefficients}\n\nUnitarity also gives reality constraints on OPE coefficients of real operators. Consider the 3pt function \\reef{eq:3points} between two scalars and a traceless symmetric tensor, assuming all three operators are real. Then the 3pt function coefficient must be real:\n\\beq\n\\label{eq:reality}\n\\lambda_{123}\\in \\bR\\,.\n\\end{equation}\nTo argue this, we can consider a 6pt function $\\<{\\cal O}_1 {\\cal O}_2 (\\Theta {\\cal O}_3) {\\cal O}_3 {\\cal O}_2 {\\cal O}_1\\>$, with the operators arranged mirror-symmetrically against a plane into two compact groups positioned a large distance from each other (see Fig.~\\ref{fig:6pt}). Here $\\Theta$ is the reflection factor mentioned in footnote \\ref{note:Theta}. Reflection positivity implies that this 6pt function should be real and positive.\\footnote{For this argument we are thus using the standard Osterwalder-Schrader reflection positivity and not the ``inversion-positivity\".} On the other hand, by cluster decomposition this 6pt function is equal to the product of two distant 3pt functions, which is easily seen to be $\\lambda_{123}^2$ times a positive number. So \\reef{eq:reality} follows. We stress that this conclusion holds for both even and odd $\\ell$.\\footnote{In essence we argued that the complex conjugate of a 3pt function is equal to the 3pt function of conjugate fields at reflected positions. This (for general $n$-point functions) is sometimes taken as an additional axiom for unitary theories, encoded by the equation ${\\cal O}(\\tau,\\mathbf{x})^\\dagger = {\\cal O}^\\dagger(-\\tau,\\mathbf{x})$ valid in Euclidean quantization by planes. Upon analytic continuation to Lorentzian signature, this leads to commutativity of operators at spacelike separation, used to prove reality of OPE coefficients in \\textcite{Rattazzi:2008pe}. Our 6pt argument shows that this axiom is not independent but follows from reflection positivity and cluster decomposition.}\n\n\\begin{figure}[t]\n\t\\begin{centering}\n\t\t\\includegraphics[width=0.6\\columnwidth]{fig03-6pt.pdf}\n\t\t\\caption{\\label{fig:6pt}\n       Positivity of this 6pt function implies reality of the 3pt function coefficient $\\lambda_{123}$, see the text.\n\t\t}\n\t\\end{centering}\n\\end{figure}\n\nIt was important for the above argument that the tensor structure entering \\reef{eq:3points} was parity invariant (i.e., it did not involve the $\\epsilon$-tensor). This argument can be generalized to OPE coefficients for general 3pt tensor structures. The OPE coefficients of tensor structures must be purely imaginary or real depending on whether they involve the $\\epsilon$-tensor or not. One must similarly be careful with OPE coefficients involving spinors.\n\nConsider now the 4pt function $\\<{\\cal O}_2 {\\cal O}_1 {\\cal O}_1 {\\cal O}_2\\>$ where ${\\cal O}_1$ and ${\\cal O}_2$ are real scalars and the point configuration is reflection-symmetric or inversion-symmetric. This 4pt function should be non-negative as a basic consequence of unitarity, and Eq.~\\reef{eq:reality} implies that a more nuanced statement is true: the individual contribution of every primary ${\\cal O}$ to this 4pt function is non-negative, see Eq.~\\reef{eq:CBdec} below. This can be generalized to external operators in general $SO(d)$ (or $Spin(d)$) representations, including the case when there are multiple 3pt function tensor structures.\n\nTo summarize, the unitarity bounds say that the CFT Hilbert space has a positive-definite norm, and the OPE coefficient reality constraints say that {the} OPE preserves this positive-definite structure. If the CFT data satisfies both of these constraints, we are guaranteed that the CFT will be unitary. The bootstrap {obtains} further constraints on CFT data by combining unitarity with crossing relations.\n\n\n\\subsubsection{Averaged null energy condition}\n\\label{sec:ANEC}\n\nIn a QFT in Lorentzian signature, we can consider the integral of the stress tensor component $T_{++}$ along a light ray: the light-like direction $x^+$ {with} all other coordinates fixed to zero. The averaged null energy condition (ANEC) says that this light-ray operator has a non-negative expectation value in any state:\\footnote{Such conditions were first introduced in general relativity, with integration along a null geodesic, in connection with singularity theorems and wormholes. Here we focus on the ANEC in flat space, first discussed by \\textcite{Klinkhammer:1991ki}.}\n\\beq\n\\label{eq:ANEC}\n\\<\\Phi|\\int_{-\\infty}^\\infty dx^+\\,T_{++} |\\Phi\\>\\geqslant 0\\,.\n\\end{equation}\nThe ANEC should hold in any unitary QFT. Two general proofs of the ANEC were given recently, one via quantum information \\cite{Faulkner:2016mzt}, and one by causality \\cite{Hartman:2016lgu}.\\footnote{See also \\textcite{Kravchuk:2018htv} for a recent discussion of light-ray operators in Lorentzian CFTs and an alternative proof of the ANEC.} \nSpecializing to CFTs, the causality argument makes it clear that the ANEC is not an extra assumption but follows from other CFT axioms such as unitarity, the OPE, and crossing relations for correlation functions involving $T_{\\mu\\nu}$.\\footnote{This is also suggested by the fact that {bounds following from the ANEC can be reproduced in the numerical bootstrap, see Sec.~\\ref{sec:JandT}.}}  Notice however that any results following from the ANEC will require the existence of a local stress-tensor operator.\n\nChoosing $|\\Phi\\>$ in \\reef{eq:ANEC} to be generated by a local operator ${\\cal O}$ acting on the vacuum, the ANEC leads to positivity constraints on 3pt functions $\\<{\\cal O} T_{\\mu\\nu} {\\cal O}\\>$ called  ``conformal collider bounds\" \\cite{Hofman:2008ar}.\\footnote{Conformal collider bounds in general dimensions for states created by the stress tensor or global symmetry currents were obtained in \\textcite{Buchel:2009sk} and \\textcite{Chowdhury:2012km}. A proof of these bounds independent from the ANEC was given in \\textcite{Hofman:2016awc}; see also \\textcite{Hartman:2015lfa,Hartman:2016dxc}. Other generalizations of these bounds have been explored in~\\textcite{Li:2015itl}, \\textcite{Komargodski:2016gci}, \\textcite{Chowdhury:2017vel}, \\textcite{Cordova:2017zej}, \\textcite{Meltzer:2017rtf}, and \\textcite{Cordova:2017dhq}. Sum rules involving the same coefficients were also recently presented in~\\textcite{Witczak-Krempa:2015pia}, \\textcite{Chowdhury:2016hjy}, \\textcite{Chowdhury:2017zfu}, and \\textcite{Gillioz:2016jnn,Gillioz:2018kwh}.  } \n\nRecently, \\textcite{Cordova:2017dhq} used the ANEC to argue that primaries of high chirality (large $|\\ell-\\bar\\ell|$) in unitary 4d CFTs should satisfy unitarity bounds stronger than \\reef{eq:4dunitarity}. From partial checks for $\\bar\\ell=0,1$, they conjecture the general bound (assuming $\\ell\\geqslant \\bar\\ell$)\n\\beq\n\\label{eq:Cordova}\n\\Delta \\geqslant \\ell\\,.\n\\end{equation}\nIf $\\bar\\ell=0$ this becomes stronger than \\reef{eq:4dunitarity} for $\\ell>2$ and for $\\ell>\\bar\\ell+4$ otherwise. This can be viewed as a CFT strengthening of the theorem of \\textcite{Weinberg:1980kq}.\n\n\\subsubsection{Stress tensor}\n\nIn the axiomatic approach considered here, a local CFT is simply defined as a CFT having a local conserved stress tensor operator $T_{\\mu\\nu}$. In the operator classification, $T_{\\mu\\nu}$ is a traceless symmetric spin-2 primary of scaling dimension $d$.\\footnote{Conformal invariance allows one to consistently impose conservation of the stress tensor. In technical language, the divergence of the dimension $d$ traceless symmetric spin-2 primary is a null descendant and can be set to zero.}\n\nIn local CFTs, the conformal algebra generators \\reef{eq:algebra} are obtained by integrating the stress tensor against {a vector field $\\epsilon^{\\cal J}_\\nu(x)$, describing the corresponding infinitesimal conformal transformation, over a surface $\\Sigma$ surrounding the origin. Thus we have}\n\\begin{eqnarray}\n\\label{eq:TwardIdentites}\n{\\cal J} = -\\int_{\\Sigma} dS_\\mu  \\epsilon^{\\cal J}_\\nu(x) T^{\\mu\\nu}(x)\\,,\n\\end{eqnarray}\nwhich is independent of the shape of $\\Sigma$. See \\textcite{Simmons-Duffin:2016gjk} for a detailed review of this way of introducing the conformal algebra.\\footnote{However, it should be stressed that there are physically interesting theories which satisfy all CFT axioms except for the existence of the local stress tensor. Examples include defect and boundary CFTs (see Sec.~\\ref{sec:bdry} and footnote \\ref{note:defects}), and critical points of models with long-range interactions, see~e.g.~\\textcite{Paulos:2015jfa} and~\\textcite{Behan:2017dwr,Behan:2017emf}.} In particular, the dilatation generator $D$ is given by \\reef{eq:TwardIdentites} with $\\epsilon^D_\\nu=x_\\nu$.\n\nIt is conventional to normalize the stress tensor via Eq.~\\reef{eq:TwardIdentites}. Namely, inserting the above surface operator in any correlator should have the effect of replacing the operator at the origin by $[{\\cal J},{\\cal O}(0)]$, assuming the other operators are outside of the region enclosed by $\\Sigma$. This constraint is called an (integrated) Ward identity.\n\nA frequently occurring case is to consider the 3pt function $\\<{\\cal O}(0) T_{\\mu\\nu}(x){\\cal O}(y)\\>$ which by the Ward identity should reduce to $\\<[{\\cal J},{\\cal O}(0)] {\\cal O}(y)\\>$ after integration. Since $[{\\cal J},{\\cal O}(0)]$ is known, this provides constraints on the coefficients of various tensor structures in the 3pt function. \n\nThese constraints should be imposed in addition to constraints from conservation of $T_{\\mu\\nu}$. Vanishing of the divergence is automatic for 2pt functions, while in general it must be imposed on 3pt functions containing $T_{\\mu\\nu}$, placing constraints on the allowed tensor structures. Such constraints are not independent if the other operators are scalars, but become nontrivial if they have spin, see \\textcite{Osborn:1993cr} and \\textcite{Costa:2011mg}.\\footnote{Some important cases are when ${\\cal O}$ is a conserved spin-1 vector or the stress tensor itself. In both these cases there are several tensor structures allowed by conformal invariance and conservation, and only one independent Ward identity, see \\textcite{Osborn:1993cr} and \\textcite{Dymarsky:2017xzb,Dymarsky:2017yzx}. Ward identity constraints on 3pt functions $\\<\\psi T \\bar\\psi\\>$ with $\\psi$ a fermion were studied in 3d by \\textcite{Iliesiu:2015qra} and in 4d by \\textcite{Elkhidir:2017iov}. In these cases there are two independent tensor structures allowed by conservation, and their coefficients can both be fixed by considering the Ward identity for $D$ as well as for $P_{\\mu}$ or $M_{\\mu\\nu}$.}\n\nIn particular, when ${\\cal O} = \\phi$ is a scalar, there is just one tensor structure. Using the Ward identity e.g.~for ${\\cal J}=D$ one fixes the OPE coefficient completely. In the notation of \\reef{eq:3points} we have \\cite{Osborn:1993cr}\n\\begin{gather}\n\t\\langle\\phi(x_1) \\phi(x_2) T(x_3,\\zeta)\\rangle =\\lambda_{\\phi\\phi T} [(Z_{123} \\cdot \\zeta)^2-{\\textstyle\\frac 12} \\zeta^2] \\, \\mathbf{K}_3\\,,\\nonumber\\\\\n\t\\lambda_{\\phi \\phi T} = - \\frac{d \\Delta_\\phi }{(d-1)S_d},\\quad S_d = \\frac{2\\pi^{d\/2}}{\\Gamma(d\/2)} \\,.\n\t\t\\label{eq:TOO1}\n\\end{gather}\nIt can also be shown that the stress tensor does not couple to two scalars of unequal dimension, as the 3pt function structure \\reef{eq:3points} is then incompatible with conservation.\n\nSince we normalize via Eq.~\\reef{eq:TwardIdentites}, the stress tensor 2pt function will not be unit-normalized but will contain a constant $C_T$ called the central charge:\\footnote{This corresponds to one of the central charge definitions in $d=2$. Notice however that in $d>2$, there is no known analogue of the Virasoro algebra interpretation of the central charge.}\n\\begin{eqnarray}\n\\label{eq:CT}\n\\langle T(x_1,\\zeta_1)T(x_2,\\zeta_2)\\rangle = \\frac{C_T}{S_d^2}\\frac{(\\zeta_1 \\cdot I \\cdot \\zeta_2)^2-\\frac1d \\zeta_1^2\\zeta_2^2}{(x_{12}^2)^{d}}\\,.\n\\end{eqnarray}\nA similar convention will be set below for conserved spin-1 currents, while the rest of primaries are kept unit-normalized. \n\nIn the normalization Eqs.~\\reef{eq:TOO1} and~\\reef{eq:CT}, the contribution of the stress tensor to 4pt functions of scalars is given by:\\footnote{This is easy to find by rescaling $T_{\\mu\\nu}$ to match the normalization in Eq.~(\\ref{eq:2points}).} \n\\begin{eqnarray}\n&&\\langle\\phi(x_1)\\phi(x_2) \\phi'(x_3)\\phi'(x_4)\\rangle \\supset  p_{d,2}\\, g^{0,0}_{d,2}(u,v) \\, \\mathbf K_4\\,,\\nonumber\\\\\n&&p_{d,2} = \\lambda_{\\phi\\phi T}  \\lambda_{\\phi'\\phi' T}\\frac{S_d^2}{ C_T} = \\frac{d^2}{(d-1)^2}\\frac{\\Delta_\\phi \\Delta_{\\phi'}}{C_T}\\,.\n\\label{eq:lambdaT}\n\\end{eqnarray}\nAs usual, the conformal block is normalized according to Eq.~(\\ref{eq:cb_ope}). This constraint can play an important role in bootstrap analyses involving multiple 4pt functions, as it implies that the stress tensor contributes to different 4pt functions in a correlated way. \n\nWhile outside of 2d there is no analogue of the ``$c$-theorem\"~\\cite{Zamolodchikov:1986gt} for $C_T$,\\footnote{Instead, it is known that in 3d the sphere free energy satisfies an ``$F$-theorem\", see \\textcite{Jafferis:2011zi}, \\textcite{Klebanov:2011gs}, and \\textcite{Casini:2012ei}, while in 4d the $a$ anomaly coefficient satisfies an ``$a$-theorem\", see \\textcite{Cardy:1988cwa}, \\textcite{Osborn:1989td}, \\textcite{Jack:1990eb}, \\textcite{Komargodski:2011vj}, and \\textcite{Komargodski:2011xv}.}  the central charge typically scales with the number of degrees of freedom. This is illustrated by the values of the central charge of a free theory containing $n_\\phi$ scalars, $n_\\psi$ Dirac fermions, and $n_A$ gauge vectors (in 4d only), given by~\\cite{Osborn:1993cr}\n\\beq\nC_T= \\frac{d}{d-1} n_\\phi +2^{\\left\\lfloor{d\/2}\\right\\rfloor-1} d \\, n_\\psi +16 \\,\\delta_{d,4}\\, n_A \\,.\n\\end{equation}\n\n\n\\subsubsection{Global symmetry currents}\n\nThe case of a continuous global symmetry in a local CFT is analogous. In this case there are conserved spin-1 currents $J_\\mu^A$ which transform in the adjoint representation of $G$ {and have scaling dimension $d-1$.} Global symmetry generators are obtained by integrating $J_\\mu^A$ {over} a surface, which defines a normalization for the current and leads to Ward identities. \n\nFor concreteness, consider scalar operator $\\phi_i$ with generators $(T^A)_{i}^{j}$  transforming in some representations $r$ of $G$ as well as $\\phi^\\dagger{}^{j}$ transforming in $\\bar r$. We assume that the scalar 2pt function is unit-normalized{, $\\langle\\phi_i \\phi^{\\dagger}{}^{j} \\rangle \\propto \\delta_{i}^{j}$,} as discussed in Sec.~\\ref{sec:global}. The generators of the global symmetry transformations are {then} $Q^A=-i\\int_\\Sigma dS^\\mu J^A_\\mu $ and the associated Ward identity requires $[Q^A,\\phi_i] = -(T^A)_{i}^{j} \\phi_j$. The 3pt function with $J^A$ is then fixed to be \\cite{Osborn:1993cr,Poland:2010wg}\n\\beq\n\\label{eq:JOO}\n\\langle \\phi_i(x_1) \\phi^{\\dagger}{}^{j}(x_2)  J^A(x_3,\\zeta)\\rangle =-\\frac{i}{S_d} (T^A)_{i}^{j} [Z_{123} \\cdot \\zeta ] \\, \\mathbf{K}_3 \\,.\n\\end{equation}\nIn this normalization one can define a \\emph{current central charge} $C_J$ by\n\\begin{eqnarray}\\label{eq:CJ}\n&& \\langle J^A(x_1,\\zeta_1)J^B(x_2,\\zeta_2)\\rangle =\\tau^{AB}\\frac{C_J}{S_d^2} \\frac{\\zeta_1 \\cdot I \\cdot \\zeta_2}{(x_{12}^2)^{d-1}}\\,,\\hspace{2em} \n\\end{eqnarray}\nwhere $\\tau^{AB}= {\\rm Tr}\\left[T^{A}T^{B}\\right]$. \n\nIn the end, rescaling $J^A_\\mu$ to match the normalization of Eq.~(\\ref{eq:2points}), the contribution of a spin-1 conserved current to the scalar 4pt function is\n\\begin{eqnarray}\n&&\\langle \\phi_i(x_1) \\phi^{\\dagger}{}^{j}(x_2) \\phi_k(x_3) \\phi^{\\dagger}{}^{l}(x_4) \\rangle \\supset - \\frac{\\mathcal T_{ik}^{jl} }{C_J } \\, g_{d-1,1}(u,v) \\,\\mathbf K_4 \\,,\n\\nonumber\\\\\n&&\\mathcal T_{ik}^{j l}  =  (\\tau^{-1})_{AB}(T^A)_{i}^{j}(T^B)_{k}^{l}\\,.\n\\label{eq:lambdaJ}\n\\end{eqnarray}\nNotice that $\\tau^{AB}$ in \\reef{eq:CJ}, $(T^A)_{i}^{j}$ in \\reef{eq:JOO}, and $\\calT_{ik}^{jl}$ are examples of 2pt, 3pt, {and} 4pt $G$-invariant tensors as discussed in Sec.~\\ref{sec:global}.\n\nFor example, if $\\phi$ is a complex scalar charged under a $U(1)$ with charge 1, then $\\mathcal T_{ik}^{jl}=\\mathcal T=1$. In the case in which $\\phi_i$ is in the fundamental representation of $SU(N)$ or $SO(N)$ (where $\\bar{r} = r$) we have instead \n\\begin{align}\n\t\\mathcal T_{ik}^{jl} &=  \\delta_i^{l} \\delta_k^{j} - \\frac{1}N \\delta_i^{j} \\delta_k^{l} \\qquad  &(G =  SU(N))\\,,\\\\\n\t\\mathcal T_{ijkl} &= \\frac12\\left(\\delta_{il} \\delta_{kj} -  \\delta_{ik} \\delta_{jl}\\right) \\qquad &(G =  SO(N)) \\,.\n\\end{align}\n\nA note about normalization is in order: once the generators $T^A$ are chosen, the Ward identity fixes the normalization of $J^A$ and determines $C_J$ according to our definition. Clearly, if we use a different generator normalization, then the value of $C_J$ would be modified accordingly. Moreover, once Eq.~(\\ref{eq:JOO}) is established, the Ward identity fixes the normalization of any other generator in any other representation. \n\nFinally, it should be mentioned that while free theories contain higher-spin conserved currents, there exist no-go theorems showing that interacting CFTs in $d\\ge3$ dimensions do not have conserved currents of spin $\\ell\\geqslant 3$, see \\textcite{Maldacena:2011jn} and \\textcite{Alba:2013yda}. This can be thought of as a CFT analogue of the Coleman-Mandula theorem for S-matrices.\n\n\n\\subsubsection{General results}\n\\label{sec:Z2-general}\n\nWe are not aware of any unitary 3d CFTs which do not possess any global symmetry.\\footnote{The Lee-Yang model has no global symmetry but is nonunitary, see Sec.~\\ref{sec:nonunitary}. {In any CFT with a global symmetry $G$, the singlet sector is closed under OPE. From the bootstrap point of view the singlet sector can be studied in isolation, results of Sec.~\\ref{sec:multicrit} being an example, and would appear as a perfectly consistent CFT with no global symmetry. Dealing only with local operators, we do not consider this construction as defining a complete CFT, as the singlet sector can in principle be extended back by including the other sectors (although it is not known how to decide in practice whether such an extension is possible by looking at the correlators of the singlet sector).}} Actually, most 3d CFTs have \\emph{continuous} global symmetries. Here we will start by considering the effect of having a discrete $\\bZ_2$,\\footnote{The bounds described in this section will also hold if the $\\bZ_2$ is taken to be a parity or time-reversal symmetry.} which may be a full symmetry as for the 3d Ising model, or a subgroup of a larger group.\\footnote{Another physically important discrete symmetry is cubic symmetry, see Sec.~\\ref{sec:O3} and footnote~\\ref{note:B3}.} \n\nIn the CFT context, a $\\bZ_2$ symmetry imposes selection rules on the possible operators appearing in different OPE channels. Let us take a {$\\bZ_2$-odd scalar} operator $\\sigma$ and consider the $\\sigma \\times \\sigma$ OPE. It can only contain $\\bZ_2$-even operators: \n\\beq\n\\sigma \\times \\sigma \\sim \\unit + \\lambda_{\\sigma\\s\\epsilon} \\epsilon + \\lambda_{\\sigma\\s T} T^{\\mu\\nu} + \\ldots .\n\\end{equation}\nHere, $\\unit$ is the identity operator, $\\epsilon$ is the leading $\\bZ_2$-even scalar, $T^{\\mu\\nu}$ is the stress-energy tensor, and so on. In particular, unlike in \\reef{eq:O0}, $\\sigma$ does not appear in the OPE.\n\n \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig07-3dkinklarge}\n    \\caption{\\label{fig:Z2-epsbound}\n(Color online) Upper bound on $\\Delta_\\epsilon$ as a function of $\\Delta_\\sigma$ in 3d CFTs~\\cite{ElShowk:2012ht}.}\n  \\end{figure}\n\nIn this setup, we would like to ask what is the maximal allowed value of $\\Delta_\\epsilon$. A numerical bootstrap analysis of the 4pt function function $\\<\\sigma\\s\\sigma\\s\\>$ (via linear or semidefinite programming) produces an upper bound on $\\Delta_{\\epsilon}$ as a function of $\\Delta_{\\sigma}$, shown in Fig.~\\ref{fig:Z2-epsbound}.\\footnote{\\textcite{Nakayama:2016jhq} observed empirically that the bounds in Figs.~\\ref{fig:multicrit} and~\\ref{fig:Z2-epsbound} coincide. A priori one may have expected a stronger bound in Fig.~\\ref{fig:Z2-epsbound} due to the extra constraint of not allowing $\\sigma$ in the r.h.s.~of the OPE.} The point $\\{1\/2,1\\}$ corresponds to the theory of a free massless scalar while the point $\\sim\\{0.518, 1.413\\}$, sitting near a discontinuity in the boundary, corresponds to the critical 3d Ising model which we discuss further below. Other theories that live in the interior of this region are the critical $O(N)$ models (see Sec.~\\ref{sec:ON}), where we can identify $\\sigma$ with a component of the $O(N)$ fundamental $\\phi_i$ and $\\epsilon$ with a component of the $O(N)$ symmetric tensor $t_{ij}$, as well as the line of mean field theory CFTs with $\\Delta_{\\epsilon} = 2 \\Delta_{\\sigma}$ (see Sec.~\\ref{sec:explicit}).\n\n \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig08-deltae-eprime-above3}\n    \\caption{\\label{fig:Z2-epsgap3bound}\n(Color online) Allowed region in the $\\{\\Delta_\\sigma,\\Delta_\\epsilon\\}$ plane under the assumption that $\\epsilon$ is the only relevant scalar \\cite{ElShowk:2012ht}.}\n  \\end{figure}\n\nA particularly physically interesting class of $\\bZ_2$-symmetric CFTs are those with only one relevant $\\bZ_2$-even operator (i.e., they have $S=1$). If the microscopic realization of the theory preserves the $\\bZ_2$ symmetry, then this condition ensures that only one parameter must be tuned in order to reach the critical point. This allowed region in $\\{\\Delta_\\sigma,\\Delta_\\epsilon\\}$ space was also computed in \\textcite{ElShowk:2012ht} from the $\\<\\sigma\\s\\sigma\\s\\>$ correlator, assuming that all scalars aside from the contribution at $\\Delta_{\\epsilon}$ are irrelevant. This region is shown in Fig.~\\ref{fig:Z2-epsgap3bound}, with the assumption having the effect of carving into the allowed region from both the left and from the bottom.\n\n  \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig09-ctmin-old}\n    \\caption{\\label{fig:Z2-cbound}\n    (Color online) Lower bound on the central charge as a function of $\\Delta_\\sigma$ \\cite{ElShowk:2012ht}.}\n  \\end{figure}\n\nAnother general result from this 4pt function is a lower bound on the central charge $C_T$ shown in Fig.~\\ref{fig:Z2-cbound}, obtained by computing an upper bound on the coefficient $\\lambda_{\\sigma\\s T} \\propto \\frac{\\Delta_{\\sigma}}{\\sqrt{C_T}}$ (see Sec.~\\ref{sec:ward} and Eq.~\\reef{eq:lambdaT}). As $\\Delta_{\\sigma} \\rightarrow 1\/2$, the lower bound on $C_T$ approaches the free scalar value, while near the critical 3d Ising dimension $\\Delta_{\\sigma} \\sim 0.518$, the lower bound on $C_T$ is seen to have a minimum. This particular bound was computed with the mild assumption $\\Delta_{\\epsilon} \\geqslant 1$, so it is applicable to any theory living in the allowed region seen in Fig.~\\ref{fig:Z2-epsbound}. \n\n   \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig10-allowed-Tprime-vs-sigma-corrected1}\n    \\caption{\\label{fig:Z2-spin2bound}\n    (Color online) Upper bound on the dimension $\\Delta_{T'}$ of the first $\\bZ_2$-even spin-2 operator after the stress tensor, as a function of $\\Delta_\\sigma$ \\cite{ElShowk:2012ht}.}\n  \\end{figure}\n\n   \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig11-spin4-bound}\n    \\caption{\\label{fig:Z2-spin4-bound}\n   (Color online) Upper bound on the dimension of the leading $\\bZ_2$-even spin-4 operator \\cite{ElShowk:2012ht}.}\n  \\end{figure}\n\nBefore we move on to discussing what can be learned from systems of several 4pt functions, we would like to highlight that upper bounds on the leading unknown scaling dimension in other channels can also be computed and are often quite strong. For example, an upper bound on the leading unknown spin-2 dimension $\\Delta_{T'}$ (the first $\\bZ_2$-even spin-2 operator after the stress tensor) is shown in Fig.~\\ref{fig:Z2-spin2bound}, and an upper bound on the leading spin-4 dimension $\\Delta_C$ is shown in Fig.~\\ref{fig:Z2-spin4-bound}. The bound on $\\Delta_{T'}$ shows a sharp jump near the critical 3d Ising value, while no such transition is seen in the bound on $\\Delta_C$ (which is close to being saturated by MFT: $\\Delta_C = 2\\Delta_{\\sigma} + 4$). The jump in $\\Delta_{T'}$ shows that it is possible for the low-dimension spin-2 operator present in the spectrum for $\\Delta_{\\sigma} \\gtrsim 0.52$ to decouple at smaller values {of $\\Delta_{\\sigma}$}. We discuss operator decoupling phenomena further in Sec.~\\ref{sec:Z2-spectrum} .\n\n   \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig12-mixed-sigpgap}\n    \\caption{\\label{fig:Z2-mixed-sigpgap}\n    (Color online) Allowed region following from the analysis of three 4pt functions assuming $\\Delta_{\\sigma'}\\geqslant 3$ with no assumption on $\\Delta_{\\epsilon'}$ \\cite{Kos:2014bka}.}\n  \\end{figure}\n\nNext one can ask what is the effect of adding constraints from other 4pt functions. So far the main system that has been studied in the literature is $\\{\\<\\sigma\\s\\sigma\\s\\>, \\<\\sigma\\s\\epsilon\\e\\>, \\<\\epsilon\\e\\epsilon\\e\\>\\}$, though other systems may also prove interesting. An advantage of including the correlator $\\<\\sigma\\s\\epsilon\\e\\>$ is that it allows one to probe the $\\bZ_2$-odd operators appearing in the OPE: \n\\beq\n\\sigma \\times \\epsilon \\sim \\lambda_{\\sigma\\epsilon\\sigma} \\sigma + \\lambda_{\\sigma\\epsilon\\sigma'} \\sigma' + \\ldots .\n\\end{equation}\nIn \\textcite{Kos:2014bka} it was found that with no assumptions this system leads to an allowed region identical to Fig.~\\ref{fig:Z2-epsbound}, while by inputting the assumption of a single relevant $\\bZ_2$-odd operator (i.e., $\\Delta_{\\sigma'} \\geqslant 3$) it leads to the allowed region shown in Fig.~\\ref{fig:Z2-mixed-sigpgap}. In this plot one can see a detached ``island\" containing the critical Ising model as well as a ``bulk\" region further to the right. This ``bulk\" region has so far not been systematically explored in the literature: it would be very interesting to understand what other CFTs lie inside of it.\n\n     \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig13-mixed-differentgaps}\n    \\caption{\\label{fig:Z2-mixed-differentgaps}\n    (Color online) This plot assumes $\\Delta_{\\epsilon'} \\geqslant 3$ (light blue), $\\Delta_{\\sigma'} \\geqslant 3$ (medium blue), or both gaps simultaneously (dark blue). Figure from \\cite{Kos:2014bka}.}\n  \\end{figure}\n\nIn Fig.~\\ref{fig:Z2-mixed-differentgaps} we also show the difference between assuming $\\Delta_{\\epsilon'} \\geqslant 3$, $\\Delta_{\\sigma'} \\geqslant 3$, and both assumptions simultaneously. One can see that the assumption of a gap in the $\\bZ_2$-odd spectrum is primarily responsible for creating the detached region. In the next section we describe the connection to the critical Ising model in more detail, as well as the techniques and additional inputs that can be used to make this detached island as small as possible.\n\n\n\\subsubsection{Critical Ising model}\n\\label{sec:Z2-Ising}\n\nPerhaps the most well-known 3d CFT is the critical 3d Ising model. The study of this model has a long history~\\cite{DombGreenVol3}, in part because it describes critical behavior in uniaxial magnets, liquid-vapor transitions, binary fluid mixtures, the quark-gluon plasma, and more \\cite{Pelissetto:2000ek}. While these applications are predominantly for systems in three spatial dimensions at finite temperature, described by a 3d Euclidean CFT, the critical Ising model can also be realized as a Lorentzian (2+1)d quantum critical point~\\cite{Fradkin-Susskind,Henkel-Ising3d}. Here we work in the Euclidean signature; the Lorentzian version is obtainable by Wick rotation and has the same set of CFT data.\n\nIn its original formulation as a model of ferromagnetism, the 3d Ising model is described using a set of spins $s_i=\\pm 1$ on a cubic lattice in $\\bR^3$ with nearest neighbor interactions, with partition function\n\\beq\n\\label{eq:Ising}\nZ = \\sum_{\\{s_i\\}} \\exp\\Bigl(- J \\sum_{\\<ij\\>} s_i s_j \\Bigr).\n\\end{equation}\nAt a critical value of the coupling $J$, the model becomes a nontrivial CFT at long distances. Notice that the lattice model has a manifest $\\bZ_2$ symmetry under which $s_i \\rightarrow -s_i$. This symmetry is inherited by the CFT, which contains local operators that are either even or odd under its $\\bZ_2$ global symmetry.\n\nAnother microscopic realization is in terms of a continuous scalar field theory in 3 dimensions, with action\n\\beq\nS = \\int d^3 x \\left( \\frac12 (\\partial \\sigma)^2 + \\frac12 m^2 \\sigma^2 + \\frac{1}{4!} \\lambda \\sigma^4 \\right),\n\\end{equation}\nwhich also has a $\\bZ_2$ symmetry under which $\\sigma \\rightarrow -\\sigma$. Because both $m^2$ and $\\lambda$ describe relevant couplings, this theory is described by a free scalar at short distances but has nontrivial behavior at long distances. At a critical value of the dimensionless ratio $m^2\/\\lambda^2$ the long-distance behavior is described by a CFT, which is the same as for the above lattice model.\n\nFrom the conformal bootstrap perspective, the Ising CFT has a $\\bZ_2$ global symmetry, one relevant $\\bZ_2$-odd scalar operator $\\sigma$, and one relevant $\\bZ_2$-even scalar operator $\\epsilon$. This is evident from experimental realizations, where $\\bZ_2$-preserving microscopic realizations require one tuning (e.g., tuning the temperature in uniaxial magnets) and $\\bZ_2$-breaking microscopic realizations require two tunings (e.g., tuning both temperature and pressure in liquid-vapor transitions).\\footnote{The existence of $\\bZ_2$-breaking liquid-vapor experimental realizations, allowing one to get $\\bZ_2$ as an emergent symmetry and predict the total number of relevant scalars, is a {nice feature of} the Ising model which does not have analogues for the $O(N)$ models.}   Note that the assumption that the only relevant scaling dimensions are $\\Delta_{\\sigma}$ and $\\Delta_{\\epsilon}$ is the same assumption that went into producing the dark blue detached region of Fig.~\\ref{fig:Z2-mixed-differentgaps}.\n\n\\textcite{Kos:2016ysd} pursued a numerical analysis of the mixed-correlator bootstrap system containing $\\sigma$ and $\\epsilon$ to high derivative order. In addition, they studied the impact of scanning over different possible values of the ratio $\\lambda_{\\epsilon\\e\\epsilon}\/\\lambda_{\\sigma\\s\\epsilon}$. This scan effectively inputs the information that there is a single operator in the OPE occurring at the scaling dimension $\\Delta_{\\epsilon}$,  whereas the plot of Fig.~\\ref{fig:Z2-mixed-differentgaps} allowed for the possibility of multiple degenerate operator contributions at the dimension $\\Delta_{\\epsilon}$.\\footnote{More precisely, the scan inputs that the outer product of OPE coefficients $\\left(\\lambda_{\\sigma\\s\\epsilon} \\quad \\lambda_{\\epsilon\\e\\epsilon}\\right) \\otimes \\left(\\lambda_{\\sigma\\s\\epsilon} \\quad \\lambda_{\\epsilon\\e\\epsilon}\\right)$ appearing in Eq.~(\\ref{eq:crossingequationwithv}) at dimension $\\Delta_{\\epsilon}$ is a rank 1 matrix, rather than the more generic rank 2 possibility which occurs if there are degenerate contributions.} This led to the three-dimensional allowed region shown in Fig.~\\ref{fig:Z2-3dIsingIsland} and its projection to the $\\{\\Delta_{\\sigma},\\Delta_{\\epsilon}\\}$ plane shown in Fig.~\\ref{fig:Z2-IsingIsland}. In addition, for each point in this region the magnitude of the leading OPE coefficients were also bounded, with the result shown in Fig.~\\ref{fig:Z2-OPEBound}. These world-record numerical determinations are summarized below in Table.~\\ref{tab:lowestdim}.\n\nFinally let us mention that recent studies of the conformal bootstrap for stress-tensor 4pt functions have also made contact with the 3d Ising model. In particular, after inputting  known values of the leading parity-even spectrum, \\textcite{Dymarsky:2017yzx} gave a new bound on the leading parity-odd $\\mathbb{Z}_2$-even scalar, $\\Delta_{\\text{odd}} < 11.2$, and constrained the independent coefficient in the stress-tensor 3pt function (parametrized by the variable $\\theta$) to be in the range $0.01 < \\theta < 0.05$ if $\\Delta_{\\text{odd}} > 3$ and in a tighter range $0.01 < \\theta < 0.018-0.019$ if $\\Delta_{\\text{odd}}$ is close to saturating its bound. We will discuss these constraints in more detail in Sec.~\\ref{sec:JandT}.\n\n       \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig14-3dIsingIsland}\n    \\caption{\\label{fig:Z2-3dIsingIsland}\n    (Color online) Allowed region in the $\\{\\Delta_\\sigma,\\Delta_\\epsilon,\\lambda_{\\epsilon\\e\\epsilon}\/\\lambda_{\\sigma\\s\\epsilon}\\}$ space obtained in \\textcite{Kos:2016ysd}. \n  }\n  \\end{figure}\n\n     \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig15-IsingIsland}\n    \\caption{\\label{fig:Z2-IsingIsland}\n    (Color online) Projection of the 3d region in Fig.~\\ref{fig:Z2-3dIsingIsland} on the $\\{\\Delta_\\sigma,\\Delta_\\epsilon\\}$ plane and its comparison with a Monte Carlo prediction for the same quantities \\cite{Kos:2016ysd}.\n  }\n  \\end{figure}\n\n         \\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig16-IsingOPEBound}\n    \\caption{\\label{fig:Z2-OPEBound}\n    (Color online) Variation of $\\lambda_{\\epsilon\\e\\epsilon}$ and $\\lambda_{\\sigma\\s\\epsilon}$ within the allowed region in Fig.~\\ref{fig:Z2-3dIsingIsland} \\cite{Kos:2016ysd}.\n  }\n  \\end{figure}\n\n\\subsubsection{Spectrum extraction and rearrangement}\n\\label{sec:Z2-spectrum}\n\nWe have seen in the previous section the remarkable precision with which the leading scaling dimensions of the critical 3d Ising model can be determined. This raises the immediate question of how well we can extract other operator dimensions and OPE coefficients in the spectrum using bootstrap methods (specifically the strategies described in Sec.~\\ref{sec:spectrum-extraction})\n\nEven prior to the mixed-correlator studies mentioned above, \\textcite{El-Showk:2014dwa} extracted the spectrum using the primal simplex method strategy, from a solution to crossing for the $\\<\\sigma\\s\\sigma\\s\\>$ correlator which minimizes the central charge $C_T$. For example, Fig.~\\ref{fig:Z2-ct-spin0} shows the scalar operators in the extracted spectrum as a function of $\\Delta_\\sigma$ near the 3d Ising model. A fascinating feature of these plots is the bifurcation of operators that occurs at the Ising value of $\\Delta_\\sigma$, which can be interpreted as a decoupling of one of the operators in the spectrum. This ``spectrum rearrangement\" phenomenon has yet to be fully understood, but speculatively it could be connected to the nonperturbative equations of motion (i.e., the 3d analogue of the relation $\\sigma \\partial^2 \\sigma \\sim \\sigma^4$ at the Wilson-Fisher fixed point) or a higher-dimensional extension of the null state conditions in the 2d Ising CFT (see also Sec.~\\ref{sec:why}).\\footnote{The 2d analogue of Fig.~\\ref{fig:Z2-epsbound} also displays a sharp kink exactly at the location of the 2d Ising model \\cite{Rychkov:2009ij}, at which the corresponding extremal solution displays a decoupling of states expected from the null state conditions \\cite{El-Showk:2014dwa}. The upper bound to the right of the kink can be interpreted as a one-parameter family of unitary 4pt functions which for a discrete sequence of $\\Delta_\\sigma$'s reduce to the 4pt function of the $\\phi_{1,2}$ operator in the higher unitary minimal models, see \\textcite{Liendo:2012hy} and \\textcite{Behan:2017rca}. While these higher minimal models exhibit further null state conditions, they are not visible in this 4pt function, and hence do not lead to kinks in this bound. \n\t\n\t}\n\n\\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig17-ct-spin0}\n    \\caption{\\label{fig:Z2-ct-spin0}\n    (Color online) The spectrum of $\\bZ_2$-even scalar operators appearing in the solution to crossing minimizing $C_T$ near $\\Delta_\\sigma$ corresponding to the 3d Ising model \\cite{El-Showk:2014dwa}. Line 1 corresponds to the $\\epsilon$ operator and shows little variation on the scale of this plot. All other lines exhibit the spectrum rearrangement phenomenon.\n  }\n  \\end{figure}\n\nOn the other hand, spectrum extraction using the extremal functional method was applied to the critical 3d Ising model by \\textcite{Komargodski:2016auf,Simmons-Duffin:2016wlq}. In the latter work, for a set of 20 trial points distributed within the island of Fig.~\\ref{fig:Z2-3dIsingIsland}, $C_T$-minimization was performed and the zeros of the extremal functional were found. While some zeros jump significantly when moving from point to point, many of them are found to be present in all families with tiny variations. About a hundred such ``stable zeros\" were identified, and are believed to represent operators which truly exist in the 3d Ising CFT, providing a remarkable view of the spectrum of this theory. The subset of stable operators with dimensions $\\Delta \\leqslant 8$, and their OPE coefficients, are shown in Table~\\ref{tab:lowestdim}. \n\nThis approach, while not fully rigorous, is intuitively justified as a means to extend the reach of rigorous analysis which produced the island in Fig.~\\ref{fig:Z2-3dIsingIsland}. The errors on stable operator dimensions and OPE coefficients are assigned as standard deviations in the set of trial points. Although these errors are not rigorous, as opposed to rigorous errors implied by Figs.~\\ref{fig:Z2-3dIsingIsland}, \\ref{fig:Z2-IsingIsland}, and \\ref{fig:Z2-OPEBound}, we believe that they represent realistic estimates. In future studies the error estimates can be further checked by enlarging the set of trial points and by extremizing multiple quantities as opposed to just $C_T$.\n\nResults of this approach for the leading towers of low-twist operators (of increasing spin) have also been tested against the analytical bootstrap computations in the lightcone limit\\footnote{{See~\\textcite{Fitzpatrick:2012yx}, \\textcite{Komargodski:2012ek}, \\textcite{Alday:2015ewa}, and \\textcite{Simmons-Duffin:2016wlq}.}} which yield analytical expressions for the large-spin asymptotics. In Fig.~\\ref{fig:tauSigSig0}, the data points extracted from the extremal functional approach show the leading twist ($\\tau=\\Delta-\\ell$) trajectory in the $\\hbox{$\\bb Z$}_2$-even sector as a function of $\\bar{h} = \\ell+\\tau\/2$, while the curve shows the analytical computation, displaying excellent agreement with the data even down to small spins. Similar good agreement was also found with the extracted OPE coefficients and subleading trajectories, as well as in the $\\hbox{$\\bb Z$}_2$-odd sector. \n\nWe also report here the prediction for the central charge from the above $C_T$-minimization over the 20 points in the island \\cite{Simmons-Duffin-private1}:\n\\beq\n\\label{eq:central-charge}\nC^{\\rm Ising}_T\/C_T^{\\text{free boson}} =0.9465389(12)\\,,\n\\end{equation}\nimproving the previous $C_T$-minimization determination by \\textcite{El-Showk:2014dwa}.\\footnote{One can also extract $C_T$ from Table \\ref{tab:lowestdim} using $\\lambda_{\\sigma\\sigma T}\\propto {\\Delta_\\sigma}\/{\\sqrt{C_T}}$. While consistent with \\reef{eq:central-charge}, this would have a larger error, because the errors on $\\lambda_{\\sigma\\sigma T}$ and $\\Delta_\\sigma$ are correlated.}\n\n\\begin{table}\n\\begin{center}\n{\\small\n\\begin{tabular}{|c|c|l|l|l|l|l|}\n\\hline\n${\\cal O}$ & $\\hbox{$\\bb Z$}_2$ & $\\ell$ & $\\Delta$ &  $f_{\\sigma\\s{\\cal O}}$ & $f_{\\epsilon\\e{\\cal O}}$ \\\\\n\\hline\n$\\epsilon$ & $+$ & 0 & $1.412625{\\bf\\boldsymbol(10\\boldsymbol)}$ & $1.0518537{\\bf\\boldsymbol(41\\boldsymbol)}$ & $1.532435{\\bf\\boldsymbol(19\\boldsymbol)}$ \\\\\n$\\epsilon'$ & $+$ & 0 & $3.82968(23)$ &  $0.053012(55)$ & $1.5360(16)$ \\\\\n& $+$ & 0 & $6.8956(43)$ &  $0.0007338(31)$ & $0.1279(17)$ \\\\\n& $+$ & 0 & $7.2535(51)$ &  $0.000162(12)$ & $0.1874(31)$ \\\\\n$T_{\\mu\\nu}$ & $+$ & 2 & $3$ &  $0.32613776(45)$ & $0.8891471(40)$ \\\\\n$T'_{\\mu\\nu}$ & $+$ & 2 & $5.50915(44)$  & $0.0105745(42)$ & $0.69023(49)$ \\\\\n& $+$ & 2 & $7.0758(58)$ & $0.0004773(62)$ & $0.21882(73)$ \\\\\n$C_{\\mu\\nu\\rho\\sigma}$ & $+$ & 4 & $5.022665(28)$ & $0.069076(43)$ & $0.24792(20)$ \\\\\n& $+$ & 4 & $6.42065(64)$  & $0.0019552(12)$ & $-0.110247(54)$ \\\\\n& $+$ & 4 & $7.38568(28)$ & $0.00237745(44)$ & $0.22975(10)$ \\\\\n& $+$ & 6 & $7.028488(16)$  & $0.0157416(41)$ & $0.066136(36)$ \\\\\n\\hline\n\\hline\n${\\cal O}$ & $\\hbox{$\\bb Z$}_2$ & $\\ell$ & $\\Delta$  & $f_{\\sigma\\epsilon{\\cal O}}$ &-\\\\\n\\hline\n$\\sigma$ & $-$ & 0 & $0.5181489{\\bf\\boldsymbol(10\\boldsymbol)}$  & $1.0518537{\\bf\\boldsymbol(41\\boldsymbol)}$ &\\\\\n$\\sigma'$ & $-$ & 0 & $5.2906(11)$  & $0.057235(20)$  &\\\\\n& $-$ & 2 & $4.180305(18)$  & $0.38915941(81)$  &\\\\\n& $-$ & 2 & $6.9873(53)$  & $0.017413(73)$  &\\\\\n& $-$ & 3 & $4.63804(88)$ & $0.1385(34)$  &\\\\\n& $-$ & 4 & $6.112674(19)$ & $0.1077052(16)$  &\\\\\n& $-$ & 5 & $6.709778(27)$ & $0.04191549(88)$ &\\\\\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\caption{\nStable operators in the critical 3d Ising model with dimensions $\\Delta\\leqslant 8$ \\cite{Simmons-Duffin:2016wlq}. Conventional names are shown in the leftmost column when available. Errors in bold are rigorous. All other errors are non-rigorous but, in our opinion, realistic. See Eq.~\\reef{eq:central-charge} for the central charge prediction from the same study.\nBecause we have chosen a different conformal block normalization convention, the OPE coefficients are related to our convention by $\\lambda_{ij{\\cal O}} = 2^{\\ell\/2} f_{ij{\\cal O}}$ (see Table \\ref{tab:cb_norm}). \n}\n\\label{tab:lowestdim}\n\\end{table}\n\n\\begin{figure}[t!]\n    \\centering\n    \\scalebox{0.8}{\n\\includegraphics[width=\\figwidth]{fig18-tauSigSig0}\n}\n    \\caption{\\label{fig:tauSigSig0}\n    (Color online) Comparison of the extremal functional spectrum with the analytic bootstrap \\cite{Simmons-Duffin:2016wlq}; see the text.\n  }\n  \\end{figure}\n\n\n\\subsubsection{Why kink? Why island?}\n\\label{sec:why}\n\nOne may be wondering why the 3d Ising model happens to live at a kink in Fig.~\\ref{fig:Z2-epsbound}. Plausibly, this has to do with the minimality of the spectrum of exchanged operators required to satisfy the crossing relation. In the interior of the allowed region in Fig.~\\ref{fig:Z2-epsbound}, the solution to crossing is not unique. When working numerically, a typical solution contains as many operators as the number of derivatives at $z=\\bar z=1\/2$ one is keeping in \\reef{eq:crossingvec}, \\reef{eq:F}. On the other hand, when one approaches the boundary of the allowed region in Fig.~\\ref{fig:Z2-epsbound}, the nature of the solution changes in that the operators first organize into pairs with nearby dimensions, and the pairs then merge into single operators at the boundary \\cite{El-Showk:2014dwa}. Thus the extremal solutions to crossing are quite economical, containing {many} fewer operators than the interior solutions, roughly {by} a half.\\footnote{It is a bit more than half because doubling never occurs for operators which remain at the unitarity bound, such as the stress tensor {(if present in the extremal solution)}, and for operators which saturate the gaps that one is imposing. In general, whether doubling occurs in the bulk of the spectrum depends on how many second-order zeros the extremal functional has. If there are too many zeros, then for some of them, called ``singles\" in \\textcite{El-Showk:2016mxr}, doubling will not occur.  See also Sec.~\\ref{sec:flow} for the flow method which uses such considerations to move along the boundary of the allowed region.}\n\nFurther reduction of the spectrum occurs at the kink. When one approaches the kink moving along the boundary, squared OPE coefficients of certain operators tend to zero. Further analytic continuation of the solution beyond the kink would be inconsistent with unitarity. Thus two different solution branches meet at the kink \\cite{El-Showk:2014dwa}, and the spectrum exhibits rearrangement phenomena mentioned in Sec.~\\ref{sec:Z2-spectrum}.\n\nTo summarize, that the 3d Ising model lives at a kink suggests that it is a CFT with a particularly minimal spectrum of operators. If this idea can be made precise, perhaps it can pave the way to an exact solution.\n\nLeaving the kink aside, let us discuss the island. It is perhaps not surprising that considering crossing for several 4pt functions the allowed region shrinks compared to what was allowed when considering just one 4pt function. It is however altogether unexpected and remarkable that considering only three 4pt functions, plus a {physically motivated} and robust\\footnote{Islands can be also produced for the Ising and other CFTs using a single 4pt function and reasonable assumptions about gaps in the spin-1 and spin-2 operator spectrum \\cite{Li:2017kck}. The robustness of these results (i.e.~their independence of the numerical values of the assumed gaps in a certain range) needs further investigation.} assumption of only two relevant operators, allows one to produce the tiny island shown in Figs.~\\ref{fig:Z2-mixed-sigpgap} and \\ref{fig:Z2-IsingIsland}.\n\nIt is not currently understood why this happens. Would the island continue to shrink indefinitely with increasing the number of included derivatives? Or would it stabilize, requiring one to add further correlators to fully fix the CFT? More generally, is it sufficient to include only 4pt functions of relevant operators or are external irrelevant operators also needed to have a unique solution? These are fascinating questions for the future.\n\nWe will see many kinks and islands in the subsequent sections of this review, about which similar considerations can be made.\n\n\\subsubsection{Nongaussianity}\n\n\\label{sec:nongauss}\nSince the leading spectrum and OPE coefficients of the critical 3d Ising model are now known to a high degree of precision, it is possible to reconstruct the full 4pt function $\\<\\sigma\\s\\sigma\\s\\>$ over a wide range of cross ratios. One can then probe the question of how much this 4pt function deviates from the ``gaussian\", i.e.~fully disconnected, form $\\<\\sigma\\s\\sigma\\s\\> = \\<\\sigma\\s\\>\\<\\sigma\\s\\> + \\text{perms}$. This question is also motivated by the fact that the Ising model contains higher-spin operators with dimensions that deviate by a small amount from those of higher-spin currents, see Fig.~\\ref{fig:tauSigSig0}. The first two of these operators are the $\\bZ_2$-even spin-4 and spin-6 operators in Table \\ref{tab:lowestdim}, of dimension close to 5 and 7 respectively.\n\n\\textcite{Rychkov:2016mrc} probed this question quantitatively using bootstrap data to reconstruct the ratio $Q(z,\\bar{z}) = \\frac{g(z,\\bar{z})}{1+(z\\bar{z})^{\\Delta_\\sigma} + (z\\bar{z}\/(1-z)(1-\\bar{z}))^{\\Delta_\\sigma}}$ in the critical 3d Ising model, where the denominator corresponds to the ``gaussian\" expectation. A plot of this deviation over a fundamental domain in the complex $z$ plane is shown in Fig.~\\ref{fig:3d}. They found e.g. that $Q < 0.75$ over a wide range of cross-ratio space and that it attains a minimum value of $Q_{\\min} \\approx 0.683$. Thus, any attempt to explain the small anomalous dimensions of higher-spin operators must account for this significant nongaussianity.\n\n\\begin{figure}[t!]\n    \\centering\n\\includegraphics[width=\\figwidth]{fig19-nongaussianity.pdf}\n    \\caption{\\label{fig:3d}\n    (Color online) The nongaussianity ratio $Q$ in the critical 3d Ising model \\cite{Rychkov:2016mrc}.\n  }\n  \\end{figure}\n\n\\subsubsection{Boundary and defect bootstrap, nontrivial geometries, off-criticality}\n\n\\label{sec:bdry}\n\nIt is also interesting to study the physics of defects in the 3d Ising model. These include both co-dimension one defects (e.g., flat 2d boundaries or interfaces) and co-dimension two defects (i.e., 1d line defects).\\footnote{\\label{note:defects}See~\\textcite{Gadde:2016fbj}, \\textcite{Billo:2016cpy}, \\textcite{Lauria:2017wav}, \\textcite{Fukuda:2017cup}, \\textcite{Rastelli:2017ecj}, \\textcite{Herzog:2017xha}, \\textcite{Herzog:2017kkj}, and \\textcite{Lemos:2017vnx} for some recent general discussions of defects in CFT.} Here we would like to highlight for the reader some recent numerical bootstrap studies of such defects.\n\nBootstrap constraints in the 3d Ising model in the presence of a flat 2d boundary were first studied using linear programming techniques in \\textcite{Liendo:2012hy}, where a number of rigorous bounds were placed on the scaling dimensions and OPE coefficients of boundary operators for different choices of boundary conditions, corresponding to the ``special\" and ``extraordinary\" transitions, assuming positivity of the bulk channel expansion coefficients. Estimates of the leading boundary data using the truncation method were also computed by \\textcite{Gliozzi:2015qsa} and \\textcite{Gliozzi:2016cmg}, where precise estimates applicable to the boundary condition of the ``ordinary\" transition could also be made. \n\nStudies have also been performed of the $\\mathbb{Z}_2$ twist line defect in the 3d Ising model, constructed on the lattice by reversing the Ising coupling on a semi-infinite half-plane. The 1d boundary of this half-plane then yields the twist line defect, which can also be defined in terms of its simple monodromy properties. Local operators living on this defect were studied using both Monte Carlo techniques~\\cite{Billo:2013jda} and numerical bootstrap (linear programming) techniques~\\cite{Gaiotto:2013nva}, with excellent agreement.  \n   \nA related line of inquiry is to study CFTs such as the critical 3d Ising model on nontrivial geometries, {the nontrivial case being manifolds not globally conformally equivalent to infinite flat space.}\\footnote{CFT correlation functions on manifolds conformally equivalent to flat space, such as the sphere $S^d$ or the ``cylinder\" $S^{d-1}\\times \\bR$, can be obtained from the flat space correlators via a Weyl transformation.} This is motivated in part by the search for a higher-dimensional analogue of modular invariance. One concrete realization has been to study the 3d Ising model on real projective space~\\cite{Nakayama:2016cim}. In this case the unknown coefficients in one-point functions of scalar primary operators $\\<\\mathcal{O}\\> \\propto A_{\\mathcal{O}}$ enter into a variant of the bootstrap equations called the cross-cap bootstrap equations. Numerical truncation studies of the cross-cap bootstrap equations in this work have yielded new nontrivial predictions, e.g. $A_{\\epsilon} = 0.667(2)$ and $A_{\\epsilon'} = 0.896(5)$ in the critical 3d Ising model on real projective space. Another interesting geometry is $S^1\\times \\bR^{d-1}$, which corresponds to putting the CFT at finite temperature. This was studied for the 3d Ising model and other higher-dimensional CFTs in \\textcite{Iliesiu:2018fao}. \\textcite{Gobeil:2018fzy} also discussed a generalization of the conformal block concept relevant for this geometry.\n\nLet us finally mention an interesting study \\cite{Caselle:2016mww} which combined the knowledge of the 3d Ising model CFT data acquired by the bootstrap with conformal perturbation theory. They achieved a remarkable agreement with the experimental data describing the 2pt function $\\<\\sigma\\sigma\\>$ off criticality, i.e.~at temperatures slightly different from the critical temperature, which corresponds to perturbing the CFT by a $\\int d^3x\\, \\epsilon(x)$ perturbation.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\nIn the last years, the development and deployment of Artificial Intelligence (AI) has grown dramatically and the European Commission (EC) expects that by 2025 the economic impact will reach between 6.5 and 12 trillion annually \\cite{Factsheet_EU}.\nTo shape the future of the technological innovation produced by AI, the EC took an active role in 2018 by introducing its own strategy on AI in which they proposed to work with all Member States on a coordinated plan to foster synergies across the European Union (EU) and identify common priorities to address the societal challenges with AI solutions, while taking into account the ethical implications. The European initiative is based on three pillars \\cite{AIStrategy}:\n\n\\begin{enumerate}[nolistsep]\n    \\item Boost the EU's technological and industrial capacity and AI uptake across the economy by private and public sectors. This implies to strengthen research and development investments in AI in the EU.\n    \\item Prepare for socio-economic changes brought by the transformation of AI in the labour market. Member States will need to prepare the Society to develop basic digital skills; re-skill or up-skill workers affected by automation, robotics and AI; and train more AI specialists, aiming for academic excellence.\n    \\item Ensure an appropriate ethical and legal framework to promote trustworthy and accountable AI made in Europe.\n\\end{enumerate}\n\nThis approach aims to make most of the opportunities offered by AI to develop solutions for social good, i.e. technology that has a positive impact on the society and the environment, based on European values and respecting fundamental rights. As a result of this coordinated action, the EC invited all Member States to develop their national strategies, including the expected investments and the implementation measures \\cite{Coordinated_Plan}.\nThis effort builds upon an ambitious goal set out in the European AI Strategy: ``to become the world-leading region for developing and deploying cutting-edge, ethical and secure AI, promoting a human-centric approach in the global context'' \\cite{Coordinated_Plan}. In these regards, the EC has made important steps. First, it tasked a group of independent experts to set up an ethical AI framework (the so-called Trustworthy AI guidelines \\cite{Trustworthy_EU}) and, more recently, put forward a proposal to regulate high-risk AI applications \\cite{Whitepaper_EU}.\n\nThe aim of this paper is to explore how Member States are approaching AI through the lens of their National Strategies. In particular, we focus on their investment plans and how these commit to the human-centric approach proposed by the Coordinated Plan on AI. Our guiding questions are: What do Member States plan to do for a responsible development of AI? Do they translate the ethical and social concerns into actual prevention measures? What are their plans to make AI developments more democratic and open to society? In other words, is AI made in Europe truly fostering social good? Though the presented work is a limited investigation, which is part of a wider project and has no pretension of exhaustiveness, we took it as an excuse to open a discussion on the Social, Ethical, Legal, Economic and Cultural issues (ELSEC) of AI within Europe.\n\nIn \\S\\ref{sec:soa} we introduce similar studies and the main differences from our work. In \\S\\ref{sec:methods} we describe our method along with the data that has been used. We present the main findings of our qualitative analysis in \\S\\ref{sec:results} and discuss them in \\S\\ref{sec:disc}. We conclude with a brief summary in \\S\\ref{sec:concl}.   \n\n\\section{State of the Art}\\label{sec:soa}\n\nThis work connects to a large literature dealing with a variety of guidelines and frameworks, which have rapidly sprung up worldwide to promote a responsible and sustainable development of AI. Several studies carried out a comparative analysis to identify similarities and divergences among these initiates.\nExamples include studies mapping keywords of different guidelines \\cite{zeng2018linking} and broad scope reviews \\cite{Human_right_standford,jobin2019global,hagendorff2020ethics}.\nMost of these studies consider an heterogeneous set of documents released by a variety of entities, including private companies, non-profit organisations and public institutions. Also, their main purpose is to study common ethical topics and their coverage across principles and guidelines issued in the few last years.\n\nCompared to these works, our study presents some points of contact, but also important distinctions. On the one hand, it shares the attention towards the ethical development of AI. But, on the other hand, it focuses on a more homogeneous set of documents (i.e. National Strategies) which are all part of a challenging European strategy. So, rather than (dis)agreements on AI ethical principles, our focus is more on how these principles translate into plans and measures taken by the European countries.\n\nThe present analysis is part of a wider research comparing National Strategies on distinct topics, such as re-skilling and education plans, and aims at supporting the ongoing debate on ethical AI principles. In particular, we do agree with \\cite{fjeld2020principled} that principles are better understood in their cultural, linguistic, geographic, and organizational context and investigating Europe's AI strategy from the perspective of different Member States adds value to the study of the European AI landscape.\n\n\\section{Method} \\label{sec:methods}\n\\subsection{Document selection}\nWe conducted a qualitative analysis of the investment plans stated in the European AI National Strategies.\nIn order to generate our data-set, we have evaluated all the EU National Strategies currently available. According to AI Watch \\cite{van2020ai}, 23 nations out of 27 have presented their strategy so far.\nHowever, we have reduced the selection to 15 nations based on the following requirements: \n\n\\begin{enumerate}[nolistsep]\n    \\item The national strategy needs to be official. Neither draft nor action plans were considered.\n    \\item Only AI strategies from Member States of the EU were included to ensure a common commitment towards the objectives of the Commission. \n   \n    \\item The documents need to be available in English to avoid language misinterpretations. \n\\end{enumerate}\nWe have monitored the release of new strategies from May 2020 to August 2020 to keep the data-set updated. This process generated the following list of countries: Austria\\cite{Austria}, Belgium\\cite{Belgium}, Czech Republic\\cite{CzechRepublic}, Denmark\\cite{Denmark}, Finland\\cite{Finland}, France\\cite{France}, Germany\\cite{Germany}, Lithuania\\cite{Lithuania}, Luxembourg\\cite{Luxembourg}, Malta\\cite{Malta}, Portugal\\cite{Portugal}, Slovakia\\cite{Slovakia}, Spain\\cite{Spain}, Sweden\\cite{Sweden}, and the Netherlands\\cite{Netherlands}.\n\n\\subsection{Qualitative Analysis}\nIn order to evaluate the documents, two researchers conducted an independent analysis to minimize the risks of bias. In the first place, the researchers selected the portion of text in each strategy associated to investments. This purposeful selection was guided by the following criteria: \\textit{(i)} investments with clear estimations in money allocation; \\textit{(ii)} investments made by the nation under consideration; and \\textit{(iii)} investments planned for the year of the publication or in the future. \n \nAs a first step, the researchers analyzed the text by assigning a label to each portion of the text that referred to a similar theme, according to the open code method \\cite{strauss1998basics}.\nThen the labels for each analysis were inspected and confronted to create a final and unified version. The final version resulted in a total of 18 first-order codes labels. As a second step, we aggregated the labels according to common characteristics in 8 high-level second-order codes categories using the axial coding \\cite{strauss1998basics}. Table \\ref{tab:qualitative} represents the first and second-order codes, along with the nations they refer to.\nThe table includes the results from 11 Member States which report information investment based on our requirements. \n\n\\subsection{Limitations} \\label{subsec:limit}\nOur analysis is bounded by some limits that arise from the nature of the selected documents.\nEven if the European Commission provided some guidance in \\cite{AIStrategy} of the objectives that should be covered, each document is structured in its own way, with differences in the level of uniformity and details provided.\nTherefore it was not possible to obtain a full comparison of monetary investments to be considered a representative picture of the European landscape. Also, we do not discard the hypothesis that some nations could have decided to not present economic estimations, or broadly appoint to them in these strategies. However, we have decided to remove from our research any discussion that did not include an economic value to focus our data on concrete actions. Nevertheless, we suggest that research with broader analysis on the investment (e.g. including examples of investments from other countries or intention to invest in specific fields) could bring benefit to have a holistic vision of the topic. \n\n\\begin{table*}[t]\n\\centering\n\\caption{Qualitative analysis of the investment areas in the European AI National Strategies}\n\\small\n\\begin{tabular}{|p{4cm}|p{3.9cm}|p{1cm}|p{2.5cm}|p{1cm}|}\n\\hline\n\\multicolumn{5}{|c|}{\\textbf{Investment Codes}}\\\\\n\\hline \\textbf{Nations (\\# of occurrences)} & \\textbf{1st Order Codes} & \\textbf{N(\\#occ)} & \\textbf{2nd Order Codes} & \\textbf{N(\\#occ)}\\\\\n\\hline\n\\multicolumn{1}{|l|}{Netherlands} & {\\cellcolor[gray]{0.9} Social Impact} & {\\cellcolor[gray]{0.9} 1} & {\\cellcolor[gray]{0.8}Society} & {\\cellcolor[gray]{0.8} 2}\\\\ \\cline{1-3}\n\\multicolumn{1}{|l|}{Denmark} & {\\cellcolor[gray]{0.9} Digital Welfare Solution} & {\\cellcolor[gray]{0.9} 1} & {\\cellcolor[gray]{0.8}} & {\\cellcolor[gray]{0.8}} \\\\ \\cline{1-3}\n\\hline\n\\multicolumn{1}{|l|}{Netherlands} & {\\cellcolor[gray]{0.9}Public Collaboration} & {\\cellcolor[gray]{0.9} 1} & {\\cellcolor[gray]{0.8} Cooperation} & {\\cellcolor[gray]{0.8} 3}\\\\ \\cline{1-3}\n\\multicolumn{1}{|l|}{Denmark} & {\\cellcolor[gray]{0.9} Public-Private Collaboration} & {\\cellcolor[gray]{0.9} 1} & {\\cellcolor[gray]{0.8}} & {\\cellcolor[gray]{0.8}}\\\\ \\cline{1-3}\n\\multicolumn{1}{|l|}{Denmark} & {\\cellcolor[gray]{0.9}International Collaboration} & {\\cellcolor[gray]{0.9}1} & {\\cellcolor[gray]{0.8}} & {\\cellcolor[gray]{0.8}}\\\\ \\cline{1-3}\n\\hline\n\\multicolumn{1}{|l|}{Belgium, Denmark (2), Germany, Malta, Spain} & {\\cellcolor[gray]{0.9} Current Investment} & {\\cellcolor[gray]{0.9}6} & {\\cellcolor[gray]{0.8}National Fund} & {\\cellcolor[gray]{0.8}11}\\\\ \\cline{1-3}\n\\multicolumn{1}{|l|}{Belgium (2), Denmark, Finland, Netherlands } & {\\cellcolor[gray]{0.9}Future Investments} & {\\cellcolor[gray]{0.9}5} & {\\cellcolor[gray]{0.8}} & {\\cellcolor[gray]{0.8}}\\\\ \\cline{1-3}\n\\hline\n\\multicolumn{1}{|l|}{Belgium, Denmark (3)} & {\\cellcolor[gray]{0.9}Digital Technology} & {\\cellcolor[gray]{0.9}4} & {\\cellcolor[gray]{0.8}Innovation} & {\\cellcolor[gray]{0.8}7}\\\\ \\cline{1-3}\n\\multicolumn{1}{|l|}{Denmark} & {\\cellcolor[gray]{0.9}Cybersecurity} & {\\cellcolor[gray]{0.9}1} & {\\cellcolor[gray]{0.8}} & {\\cellcolor[gray]{0.8}}\\\\ \\cline{1-3}\n\\multicolumn{1}{|l|}{Denmark} & {\\cellcolor[gray]{0.9}Data Collection} & {\\cellcolor[gray]{0.9}1} & {\\cellcolor[gray]{0.8}} & {\\cellcolor[gray]{0.8}}\\\\ \\cline{1-3}\n\\multicolumn{1}{|l|}{Netherlands} & {\\cellcolor[gray]{0.9}Supercomputing} & {\\cellcolor[gray]{0.9}1} & {\\cellcolor[gray]{0.8}} & {\\cellcolor[gray]{0.8}}\\\\ \\cline{1-3}\n\\hline\n\\multicolumn{1}{|l|}{Malta} & {\\cellcolor[gray]{0.9}National Promotion} & {\\cellcolor[gray]{0.9}1} & {\\cellcolor[gray]{0.8}International Representation} & {\\cellcolor[gray]{0.8}1}\\\\ \n\\hline\n\\multicolumn{1}{|l|}{Malta, Netherlands (5)} & {\\cellcolor[gray]{0.9}Employee Training} & {\\cellcolor[gray]{0.9}6} & {\\cellcolor[gray]{0.8}Education} & {\\cellcolor[gray]{0.8}12}\\\\ \\cline{1-3}\n\\multicolumn{1}{|l|}{Denmark, France} & {\\cellcolor[gray]{0.9}AI Literacy for Citizens} & {\\cellcolor[gray]{0.9}2} & {\\cellcolor[gray]{0.8}} & {\\cellcolor[gray]{0.8}}\\\\ \\cline{1-3}\n\\multicolumn{1}{|l|}{Denmark, Finland, Netherlands (2)} & {\\cellcolor[gray]{0.9}Educational Fund} & {\\cellcolor[gray]{0.9}4} & {\\cellcolor[gray]{0.8}} & {\\cellcolor[gray]{0.8}}\\\\ \\cline{1-3}\n\\hline\n\\multicolumn{1}{|l|}{Denmark (3), Germany, Malta (2), Netherlands (2)} & {\\cellcolor[gray]{0.9}Companies Investment} & {\\cellcolor[gray]{0.9}8} &  {\\cellcolor[gray]{0.8}Private} & {\\cellcolor[gray]{0.8}8}\\\\ \\cline{1-3}\n\\hline\n\\multicolumn{1}{|l|}{Denmark} & {\\cellcolor[gray]{0.9}Investment for Local Administrations} & {\\cellcolor[gray]{0.9}1} & {\\cellcolor[gray]{0.8}Public} & {\\cellcolor[gray]{0.8}5}\\\\ \\cline{1-3}\n\\multicolumn{1}{|l|}{Austria, Lithuania, Denmark, Sweden} & {\\cellcolor[gray]{0.9}AI Research} & {\\cellcolor[gray]{0.9}4} & {\\cellcolor[gray]{0.8}} & {\\cellcolor[gray]{0.8}}\\\\ \\cline{1-3}\n\\hline\n\\end{tabular}\n\\label{tab:qualitative}  \n\n\\end{table*} \n\n\\section{Results} \\label{sec:results}\nAccording to the results, 11 National Strategies report investment plans that meet our requirements. These include Austria, Belgium, Denmark, Finland, France, Germany, Lithuania, Malta, Spain, Sweden and the Netherlands. In the following subsections we highlight the main findings distinguishing between general investment plans and investments with an explicit commitment to society (e.g. welfare solutions, education and social impact)\n\n\\subsection{General Trends}\nMost of the National Strategies (7 out of 11) report packages of investments in AI initiatives (\\textbf{National Fund}).\nThese investments vary depending on whether they refer to ongoing efforts (\\textbf{Current Investment}) or future plans (\\textbf{Future investment}). Their description is usually generic and reports total volumes which often cover different areas of application (e.g. Belgium plans to invest at least EUR 1 billion by 2030 focusing on specific areas such as healthcare\/life sciences). \n\nSome strategies provide figures which connect to the digital transformation (\\textbf{Innovation}). For example, the Netherlands is investing EUR 18 million in a new national supercomputer (\\textbf{Supercomputing}), while Denmark allocated DKK 1.5 billion on cyber and information security (\\textbf{Cybersecurity}) and DKK 250 million in data quality and cross-sectoral cooperation on health data (\\textbf{Data Collection}).\nAnother emerging trend regards to the investments in the private sector (\\textbf{Private}), putting especial attention to support start-ups and SMEs in the uptake of AI, as they make 99\\% of business in Europe \\cite{AIStrategy}. Thus, it is clear that the early adoption of new technologies will help boost innovation and competition in the AI landscape. \nIn some strategies there are figures which refer more specifically to the public sector (\\textbf{Public}). For instance, Denmark allocates resources for testing and deploying digital welfare solutions in municipalities and regions (\\textbf{Investment for Local Administration}), while Austria, Lithuania, Denmark, and Sweden report investments in academic research (\\textbf{AI Research}). Another interesting case regards Malta, which plans to spend EUR 1 million per annum to promote their international visibility and become an emerging hub for technologies in Europe (\\textbf{International Representation}).  \n \n\n\\subsection{Social Goods}\nWhile documents include details about general investments (like AI research and public collaboration), just a few of them report quantified investments with regard to the social good. For example, the Netherlands reports investments to study the impact of AI on work and employment (\\textbf{Social Impact}). Denmark specifies allocated resources for digital welfare solutions (\\textbf{Digital Welfare Solution}), which connect to a wider reform in the Public Sector aimed at contributing to better and more cohesive welfare services. Regarding education, seven strategies propose economic plans. For example, Denmark and France describe investments to support the population with the challenge of obtaining new digital competences and be prepared to the new work places that are expected to be created with the rise of AI technologies, and that will require a new generation of experts in different fields. (\\textbf{AI Literacy for Citizens)}.\nThe Netherlands describe multiple economic initiatives (5 times) for training workers and promoting a learning culture in SMEs (\\textbf{Employee Training}), and, along with Denmark and Finland, they propose concrete investments in higher education. For example, the Danish government set aside a pool of DKK 190 million (EUR 25 million) to cover all technical fields, including new technologies like AI (\\textbf{Educational Fund}). \nAnother interesting proposal regards the investments for cooperation. Indeed, the Netherlands reported an open call, worth EUR 2.3 million, on explainable, socially aware and responsible AI (\\textbf{Public collaboration}). \n\n\\section{Discussion} \\label{sec:disc}\nIn this section we consider the collected results in the light of European strategy \n(see the three pillars outlined in \\S\\ref{sec:intro}) and provide our vision for the way forward.\\\\\nIn order to boost the EU's technological and industrial capacity, Member States seem to embrace the direction suggested by the EC to encourage the progress of AI in the private and public sector. This is well reflected in our analysis. Indeed, 7 countries out of 11 provide specific estimates for their investments in the private and public sector and 3 of them (Denmark, Netherlands and Malta) report more than one measure (e.g. the Danish strategy includes 5 instances of the codes \\textbf{``Private''} and \\textbf{``Public''}, see Table \\ref{tab:qualitative}). Our results align also with a recent survey by the EC that found that 42\\% of the European companies are already using AI \\cite{European2020enterprise}.\nEducation is the key concept to prepare the society for the socio-economic changes, and align with the second pillar. Even though it is a recurrent objective included in AI strategies, we observed fewer nations reporting investments in numeric terms (4 nations out of 11). Those plans regard the re-training and upskilling of the population will play an important role to include the society in the transformation. Indeed, AI literacy and education can contribute to fill the gap created by the rapid growth in AI between the producers, who know the strengths and limits of this technology, and the consumers, who may lack knowledge about AI and be more exposed to harmful applications.\nThis will lead, on one side, to new opportunities for citizens to develop AI-based competences at work and contribute to the digital transformation that will shape our society. On the other side, culture will bring a faster acceptance of new technology and penetration in society, bringing to life the aim to improve society that Europe is wishing.\nDifferent nations point out the importance of including citizens in the process of defining the future applications of AI, especially those that will be deployed and used by public administrations \n(e.g Austria intends to support societal discussion to increase acceptance, or Czech Republic involving the employee in technological transformation). \n\nThe third pillar relies on the creation of ethical and legal framework, which is tackled with different initiatives in the National Strategies. For example, 5 of them (Belgium, Denmark, Luxembourg, Malta, Spain) state they want to create an ethical committee to supervise the use and development of AI systems. Malta puts forward the proposal of an national AI certification program based on its Ethical AI Framework. However all these propositions lack details about allocated resources. While some of these proposals build upon existing initiatives and investments schemes we expect to see further measures as the  ambitious goal of Trustworthy AI cannot be achieved without costs. The set up of an appropriate ethical and legal framework is in fact a demanding effort which implies a long-term view and the mobilization of huge resources (e.g. experts in different fields, new business processes, holistic assessment methodologies, audits, etc.)   \n\n\\subsection*{Our Vision for a better strategy}\nWe have identified four key areas that could improve the impact of the current investments to develop AI technology for social good.\n\n\\textbf{Global benefit:} We suggest to envision an accessible and inclusive approach that \\textit{(i)} includes the needs and opinions of different actors and stakeholders (e.g the Netherlands and Denmark report investments for citizens and for international cooperation); \\textit{(ii)} focus on diversified fields aligned with the Sustainable Goals for Development \\cite{SGDs} (health, agriculture, environment etc.), and \\textit{(iii)} considers direct and indirect consequences of the use and development of AI-based solutions. In our opinion, it is important to understand the knock-on effect that those investments have to predict possible opportunities and limitations in the long-term.\n\n\\textbf{Legal Frameworks\/Support:} We suggest to provide measurable efforts in the promotion of the legal and ethical aspects of AI, including \\textit{(i)} transparency and trustworthiness of the AI system; \\textit{(ii)} safeguard of the physical or psychological integrity and the dignity of the human being, and \\textit{(iii)} dissemination to the society. We recommended that each initiative and decision affected by an AI system, especially those coming form the public sector, should be easily accessible for the citizens. To progress towards a competitive AI landscape in Europe requires to build upon an empowered society able to interact with the technology, be aware of its technical and ethical limits and the legal processes that protect it.\n\n\\textbf{Social Implications:} The implications of the use of AI could be unpredictable, therefore we believe important to understand the challenges that our society will face. Initiatives, such as the one presented by the Netherlands to invest in research lines to study the social impact of AI, will be necessary to promote a responsible use of new technologies. Even if expert consultations can offer a general overview of the risks, real case applications could bring light to new issues. To obtain a more comprehensive view of these implications, we believe that it will be necessary to create multidisciplinary teams that can analyse different dimensions of this challenge.\n     \n\\textbf{Societal Participation:} Citizens are already playing a key role in the AI landscape as users, but also by generating and sharing data that is used for multiple purposes. Thus it would be fair to include them in the definition of the ethical, legal, socio-economic and cultural strategies that will shape the future of Europe. To promote a human-centric approach of AI, the society should be put in the loop of the life cycle of AI systems trough direct participation or open consultations. As part of the AI4EU Observatory we are planing a citizen consultation to further analyse this vision of the European strategy of AI.  \n\n\\section{Conclusions} \\label{sec:concl}\nIn this paper we present an ongoing study on how European countries are approaching the field of AI, with its promises and risks, through the lens of their national AI strategies. In particular, we aimed to investigate how European countries are investing in AI and to what extent the stated plans can contribute to the benefit of the whole society. To understand how Member States are investing in AI-based technologies for social good, we conducted a qualitative on 15 nations highlighting the distribution of economic investments. Although the sources we used were limited (see \\S\\ref{subsec:limit}), our findings show that National Strategies are aligned to the pillars of the European strategic vision of AI. However, these still lack of concrete actions that will define the path of a human-centric and trustworthy AI. There is a need for a stronger commitment to boost the collaboration among public and private sector to reach a network of excellence in Europe, able to attract talent and generate innovation, without leaving behind an ethical and legal framework able to protect and prioritise European citizen's rights and interests.\n\n\\section*{Acknowledgement}\nThe authors are partially supported by the project A European AI On Demand Platform and Ecosystem (AI4EU) H2020-ICT-26 \\#825619. The views expressed in this paper are not necessarily those of the consortium AI4EU.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzjpge b/data_all_eng_slimpj/shuffled/split2/finalzzjpge
new file mode 100644
index 0000000000000000000000000000000000000000..273587876be766794996b24aeb0b09cab7c0df4c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzjpge
@@ -0,0 +1,5 @@
+{"text":"\\section{#1}}\n\\newcommand{\\subsect}[1]{\\subsection{#1}}\n\\newcommand{\\subsubsect}[1]{\\subsubsection{#1}}\n\\renewcommand{\\theequation}\n  {\\arabic{section}.\\arabic{equation}}\n\\renewcommand{\\thefootnote}{\\fnsymbol{footnote}}\n\n\n\n\\def{\\it i.e.}{{\\it i.e.}}\n\\def{\\it e.g.}{{\\it e.g.}}\n\n\\def\\begin{equation}{\\begin{equation}}\n\\def\\end{equation}{\\end{equation}}\n\\def\\begin{array}{\\begin{array}}\n\\def\\end{array}{\\end{array}}\n\n\\def{\\cal U}{{\\cal U}}\n\\def\\hbox{Fun}{\\hbox{Fun}}\n\\def\\co{\\Delta}                      \n\n\\def\\alpha{\\alpha}\n\\def{\\beta}{{\\beta}}\n\n\\def\\bar{\\alpha}{\\bar{\\alpha}}\n\\def\\bar{\\beta}{\\bar{\\beta}}\n\n\\def\\gamma{\\gamma}\n\\def\\g{{\\cal G}} \n\\def\\varepsilon{\\varepsilon}\n\n\\def\\k{\\omega}                \n\n\\def\\J{{\\extr J}}            \n\\def\\P{{\\extr P}}            \n\n\\def\\R{{\\extr R}}  \n\\def\\T{{\\extr T}}  \n\\def\\X{{\\extr X}}  \n\\def\\W{{\\extr W}}  \n\n\\def\\C{\\mbox{\\ C}}      \n\\def\\S{\\mbox{\\ S}} \n\n\\def\\s{{{so}}}           \n\\def\\is{{{iso}}}           \n\n\\def\\k_1,\\k_2,\\dots,\\k_N{\\k_1,\\k_2,\\dots,\\k_N}\n\\def0,\\k_2,\\dots,\\k_N{0,\\k_2,\\dots,\\k_N}\n\\def\\k_2,\\dots,\\k_N{\\k_2,\\dots,\\k_N}\n\\def\\k_2,\\k_3,\\k_4{\\k_2,\\k_3,\\k_4}\n\\def\\k_2,\\k_3{\\k_2,\\k_3}\n\\def\\k_2{\\k_2}\n\n\\def\\lambda{\\lambda}\n\n\n\\def\\Gamma^{(m)}_\\pardeform{\\Gamma^{(m)}_\\lambda}\n\\def\\Gamma^{(m)}{\\Gamma^{(m)}}\n\n\n\\catcode`\\@=11\n\\font\\tenmsa=msam10\n\\font\\sevenmsa=msam7\n\\font\\fivemsa=msam5\n\\font\\tenmsb=msbm10\n\\font\\sevenmsb=msbm7\n\\font\\fivemsb=msbm5\n\\newfam\\msafam\n\\newfam\\msbfam\n\\textfont\\msafam=\\tenmsa  \\scriptfont\\msafam=\\sevenmsa\n  \\scriptscriptfont\\msafam=\\fivemsa\n\\textfont\\msbfam=\\tenmsb  \\scriptfont\\msbfam=\\sevenmsb\n  \\scriptscriptfont\\msbfam=\\fivemsb\n\n\\def\\hexnumber@#1{\\ifnum#1<10 \\number#1\\else\n \\ifnum#1=10 A\\else\\ifnum#1=11 B\\else\\ifnum#1=12 C\\else\n \\ifnum#1=13 D\\else\\ifnum#1=14 E\\else\\ifnum#1=15 F\\fi\\fi\\fi\\fi\\fi\\fi\\fi}\n\n\\def\\msa@{\\hexnumber@\\msafam}\n\\def\\msb@{\\hexnumber@\\msbfam}\n\\mathchardef\\blacktriangleright=\"3\\msa@49\n\\mathchardef\\blacktriangleleft=\"3\\msa@4A\n\\catcode`\\@=12\n\n\n\\def\\triangleright\\!\\!\\!\\blacktriangleleft{\\triangleright\\!\\!\\!\\blacktriangleleft}\n\\def\\triangleright\\!\\!\\!<{\\triangleright\\!\\!\\!<}\n\\def>\\!\\!\\blacktriangleleft{>\\!\\!\\blacktriangleleft}\n\\def\\triangleright{\\triangleright}\n\\def\\triangleright\\!\\!\\!\\blacktriangleleft{\\triangleright\\!\\!\\!\\blacktriangleleft}\n\\def\\blacktriangleright\\!\\!\\!\\triangleleft{\\blacktriangleright\\!\\!\\!\\triangleleft}\n\\def\\triangleleft{\\triangleleft}\n\\def\\bar\\triangleright{\\bar\\triangleright}\n\\def\\triangleright\\!\\!\\!<{\\triangleright\\!\\!\\!<}\n\\def>\\!\\!\\!\\blacktriangleleft{>\\!\\!\\!\\blacktriangleleft}\n\\def>\\!\\!\\!\\triangleleft{>\\!\\!\\!\\triangleleft}\n\\def\\blacktriangleright\\!\\!\\!<{\\blacktriangleright\\!\\!\\!<}\n\n\n\\defH{H}\n\\defA{A}\n\\defK{K}\n\\defh{h}\n\\defg{g}\n\\defa{a}\n\\defb{b}\n\\defc{c}\n\n\n\n\n\\begin{document}\n\n\\rightline{DAMTP 96--100}\n\\rightline{To appear in J. Phys. {\\bf A}}\n\\vspace{1.5cm}\n\n\\begin{center} \n{\\large{\\bf{GRADED CONTRACTIONS AND BICROSSPRODUCT STRUCTURE}}}\n{\\large{\\bf{OF DEFORMED INHOMOGENEOUS ALGEBRAS \n\\footnote{PACS numbers: 02.20, 02.20Sv, 02.40}\n}}}\n\\end{center}  \n\n\\bigskip\\bigskip\n\n\\begin{center} \nJ. A. de Azc\\'arraga$^{1}$ \n\\footnote{St. John's College Overseas Visiting Scholar.}\n\\footnote{On sabbatical (J.A.) leave and on leave of absence (J.C.P.B.)\nfrom Departamento de F\\'{\\i}sica Te\\'orica and IFIC, Centro Mixto Univ. \nde Valencia-CSIC, E--46100 Burjassot (Valencia), Spain. E-mails: \nj.azcarraga@damtp.cam.ac.uk (azcarrag@evalvx.ific.uv.es), \npbueno@lie.ific.uv.es.},\nM.A. del Olmo$^2$  \\footnotemark[4],  \nJ.C. P\\'erez Bueno$^{1}$ \\footnotemark[3], \nand M. Santander$^2$ \n\\footnote{e-mails: olmo@cpd.uva.es, santander@cpd.uva.es.}\n\\end{center}\n\n\\begin{center} {\\it $^1$ Department of Applied Mathematics and Theoretical \nPhysics, \\\\\nSilver St., Cambridge CB3 9EW, UK}\n\\end{center}\n\n\\begin{center}{\\it{$^2$ Departamento de F\\'{\\i}sica Te\\'orica,\nUniversidad de Valladolid}\\\\ E--47011, Valladolid, Spain} \n\\end{center}\n\n\n\n\\begin{abstract}\nA family of deformed Hopf algebras corresponding to the classical\nmaximal isometry algebras of zero-curvature $N$-dimensional spaces (the\ninhomogeneous algebras ${iso}(p,q), \\ p+q=N,$  as well as some of their\ncontractions) are shown to have a bicrossproduct structure. This is done for\nboth the algebra and, in a low-dimensional example, for the (dual) group\naspects of the deformation.\n\\end{abstract}\n\n\\vfill\\eject\n\\sect{Introduction}\n\nThe procedure to deform simple algebras and groups\nwas established by Drinfel'd\n\\cite{Dri}, Jimbo \\cite{Ji} and Faddeev, Reshetikhin and Takhtajan \\cite{FRT}. \nThe algorithm, which leads to the\nso-called  `quantum' algebras, does not cover, however, the case of\nnon-semisimple algebras. Since the contraction process leads to\ninhomogeneous algebras by starting from  simple ones, it is natural\nto use it as a way to deform inhomogeneous Lie  ({\\it i.e.}, `classical'\nor undeformed) algebras. This path of extending the classical idea\nof the Lie algebra contraction to the case of deformed  algebras\nwas proposed by Celeghini {\\em et al.} \\cite{CGST}.  The basic\nrequirement to define a deformed inhomogeneous algebra  is\nthe commutativity of the processes  of contraction and\ndeformation: when considering a simple algebra and one of their\ninhomogeneous contractions, both at classical and deformed\nlevels, the deformation of the contracted inhomogeneous Lie algebras \nshould coincide with the contraction of the deformed simple algebra.\nThis commutativity is not always guaranteed, and in general requires \n\\cite{CGST} a redefinition of the deformation parameter $q$ in \nterms of the contraction parameter and the  new deformation one, so that\n$q$  is not a passive element in the contraction. \nThis was used, for instance,  to obtain the $\\kappa$-Poincar\\'e algebra\n\\cite{LNRT}, for which the  deformation parameter\n$\\kappa$ has dimensions of inverse of length.  \n\nThe concept of contraction of Lie\nalgebras (or groups) was discussed in the early fifties by \\.In\\\"on\\\"u \nand Wigner \\cite{IW} (see also \\cite{Saletan}).   \nThe idea of group contraction itself arose in the group analysis\nof the non-relativistic limit, and its applications to\nmathematical physics problems have been very fruitful. \nThe study of details\nbehind this procedure unveils interesting mathematical structures,\nwhich in many important cases are linked to physical properties. \nIn particular, the contraction process may increase the group cohomology\n\\cite{AA} (see also \\cite{AHPS}), \nas it is the case in the standard non-relativistic limit. \nSeveral attempts have been made to systematise the study of\ncontractions \nand recently a new approach has\nbeen put forward by Moody, Montigny and Patera \\cite{dMPMP},\nunder the name of graded contractions. \nThe key idea there is to preserve\na given grading of the original Lie algebra. This condition may fit\nneatly with physical requirements and is automatically\nsatisfied in the simplest case of the \\.In\\\"on\\\"u--Wigner contractions,\nwhich correspond to the simplest ${\\extr Z}_2$-grading.\n\nA class of Lie algebras describing a whole family of\ncontractions is the so-called orthogonal Cayley-Klein (CK) algebras.  The\nname is due to historical reasons:  these are the Lie algebras of the\nmotion groups of real spaces with a \nprojective metric \\cite{SommerYRY} (see also \\cite{SHO}). \nThe same family appears as a natural subset of all ${\\extr Z}_2^{\\otimes\nN}$-graded contractions which can be obtained from $\\s(N+1)$ \\cite{HMOS}. \nAnd furthermore, among orthogonal CK\nalgebras we find not only all simple pseudo-orthogonal algebras,\nbut many non-semisimple algebras of physical importance, as the\nkinematical Poincar\\'e and Galilei algebras in $(N-1,1)$ dimensions, the\nEuclidean algebra in $N$ dimensions, etc. The CK scheme\ndoes not deal with a single Lie algebra, but with a whole\nfamily of them simultaneously,\neach of which is parametrised by a set of real\nnumbers with a well-defined geometrical and physical significance. \nThe main point to be stressed is the ability of this kind of approach to\ndescribe some properties of many Lie algebras in a single unified form.\nThis is possible as the Lie algebras in the CK family, though not\nsimple, are `very near' to the simple ones, and many structural\nproperties of the simple algebras, when suitably reformulated, still\nsurvive for the CK algebras.\n\nIt is possible to give deformations of algebras in the CK\nfamily; naturally enough these will be \nsaid to belong to the CK family of Hopf `quantum'  algebras. \nIn \\cite{BHOSab} deformations of the\nenveloping algebras of all algebras in the CK family of \n$\\s(p,q),\\ p+q=3, 4$ were given. For higher dimensions, {\\it i.e.}\\ for \nalgebras in the family of $\\s(p,q),\\ (p+q=N+1)$ with $N>3$, a\nquantum deformation of the general parent member of the CK family \nis still not known, yet there exists a scheme of quantum deformations\nencompassing all  motion algebras of flat affine spaces in $N$\ndimensions, which include the ordinary inhomogeneous \n$\\is(p,q),\\ (p+q=N)$ \n\\cite{BHOScd}. This scheme provides a Hopf algebra\ndeformation for each algebra in the family. Some of its members \nare physically relevant non-semisimple algebras, and include as\nparticular cases most of the deformations of these algebras found in the\nliterature.   \n\nAn important fact in quantum algebra\/group theory is the (co)existence\nof two closely linked algebraic structures: the algebra (as expressed by the \ncommutators or the commuting properties of the algebra \nof functions on the group) and the coalgebra (as given by the coproduct). \nMost of the complications found when doing quantum contractions can\nbe traced to the need of dealing simultaneously with these two aspects.\nFor instance, a naive contraction might lead to divergences either in\nthe coproduct or in the $R$--matrix \\cite{CGST,BGHOS}.\nOne of the main motivations behind the CK scheme was to\nbe able to describe at the same time a family of algebras, including\nsome simple and some contracted algebras, in such a way that the\npossible origin of divergences under contractions is clearly seen\nand controlled. \n\nIn this paper we address a specific problem where the advantages of a\nCK type scheme are  exhibited.\nIn the classical case, an \\.In\\\"on\\\"u--Wigner (IW) contraction\nof a simple algebra leads to a non-semisimple one which is the\nsemidirect sum of an abelian algebra and the preserved subalgebra of\nthe original algebra with respect which the contraction was made.\n{\\it All} IW  contractions of simple algebras have a semidirect\nstructure. \nIt is then natural to ask: is there a similar pattern for the contracted \ndeformations {\\it i.e.},\nfor the Hopf algebra deformations of contracted simple Lie algebras? \nThe analogue of the semidirect product is an example of the bicrossproduct \nof Hopf algebras, introduced by Majid \\cite{MajMajid} \n(see also \\cite{SingMoln,bicII}).\nThe aim of this paper is to show that all deformed algebras in\nthe affine\\footnote{We use the word `affine' in the sense of inhomogeneous. \nNot all deformed inhomogeneous groups have a bicrossproduct structure; this \nis, for instance, the case of ${\\cal U}_q({\\cal E}(2))$ \nas discussed in \\cite{APb}.} \nCK family $\\is_{\\k_2,\\dots,\\k_N}(N)$ have indeed a\nbicrossproduct structure, as is the case of the $\\kappa$-Poincar\\'e \n\\cite{KPoinBicros}.  \nThis result opens the possibility of\nrecovering more easily the deformed dual groups\n$\\hbox{Fun}_q(ISO_{\\k_2,\\dots,\\k_N}(N))$ by using the dual bicrossproduct\n`group-like' expressions (see \\cite{APb} for some group-like (rather\nthan algebra-like) examples  of this construction).\nClassically, the $\\is_{\\k_2,\\dots,\\k_N}(N)$ family includes all inhomogeneous\nLie algebras $\\is (p,q),\\ (p+q=N)$, so we will refer loosely to the aim of \nthe paper as showing the bicrossproduct structure of deformed inhomogeneous\ngroups. It should be kept in mind, however, that we are referring to a\nspecific deformation, and that there exist examples \n(see \\cite{APb}) where \na contraction of a deformed algebra has no bicrossproduct structure. \n\nThe paper is organised as follows. In Section II we  \nbriefly describe the classical Cayley--Klein algebras and  \npresent a discussion on  contractions and \ndimensional analysis since this is relevant for the assignment of physical \ndimensions to the deformation parameters. \nIn Sec. III we give the explicit expressions for their\n$q$--deformations. The bicrossproduct structure of these $q$--deformed\nCayley--Klein Hopf algebras is shown in Section IV.  Examples of this structure\nfor physically interesting algebras are presented in Section V. \nIn Section VI we show, as an example,  how to obtain the (dual) \ngroup deformation in the case of lowest dimension $N=2$. \nSome conclusions close the paper. \n\n\n\n\\sect{ Affine Cayley--Klein Lie algebras and dimensional analysis} \n\n\n\\subsect{The CK scheme of geometries and Lie algebras}\n\n\\bigskip\nThe complete family of the $so(N+1)$ CK algebras is a set of real Lie algebras \nof dimension $(N+1)N\/2$, characterised by\n$N$ real parameters $(\\k_1,\\k_2,\\dots,\\k_N)$ \\cite{SHO}.  \nThis family appears, {\\it e.g.}, as a\nnatural subfamily \\cite{HMOS} of all the graded\ncontractions from the Lie algebra $so(N+1)$ \\cite{HS} corresponding to a\n${\\extr Z}_2^{\\otimes N}$ grading of $so(N+1)$, \nand its elements will be denoted\n$\\s_{\\k_1,\\k_2,\\dots,\\k_N} (N+1)$; \nin particular, \n$\\s_{1, 1, \\dots, 1} (N+1) \\equiv \\s(N+1)$. \nIn terms of a basis of\n$\\s_{\\k_1,\\k_2,\\dots,\\k_N} (N+1)$ adapted to the grading, $\\{ \\J_{ab};\\ a<b,\\ a,b= 0,\n1, \\dots, N\\}$,  \nthis family of algebras is defined by\n\\begin{equation}\n[\\J_{ab},\\J_{ac}] =  \\k_{ab} \\J_{bc}, \\quad\n[\\J_{ab},\\J_{bc}] = -\\J_{ac} ,  \\quad\n[\\J_{ac},\\J_{bc}] = \\k_{bc}\\J_{ab},\\label{aa}\n\\end{equation}\nwhere now $a<b<c, \\ a,b,c =0,1,\\dots ,N$,\n$\\k_{ab}:=\\k_{a+1}\\k_{a+2}\\ldots\\k_b=\\prod_{l=a+1}^{b}\\k_{l}$\n(thus, $\\k_{ab}\\k_{bc}=\\k_{ac}$) and $[\\J_{ab}, \\J_{cd}]=0$ if the four indices \nare different. \nBy simple rescaling of the generators, all the numerical values of the\nconstants $\\k_i$ may be brought to one of the  values $1, 0, -1$, hence the\ncomplete CK family contains $3^N$ algebras which are  different as graded\ncontractions, even if \nsome of them may still be isomorphic.\n\nWhen all the $\\k_i$ are non-zero but some of them\nare negative, the algebra $\\s_{\\k_1,\\k_2,\\dots,\\k_N} (N+1)$ is isomorphic to a certain\npseudo-orthogonal algebra $\\s (p,q)$ $(p+q=N+1,\\, p\\ge q > 0)$. \nIf all the $\\k_i$ are non-zero  we can introduce also\n$\\J_{ba}, (a<b)$ by $\\J_{ba}:= -\\frac{1}{\\k_{ab}}\\J_{ab}$ and $\\J_{aa}\n:=0$, so that the commutation relations can be written in the familiar form: \n\\begin{equation}\n[\\J_{ij},\\J_{lm}]=\\delta_{im}\\J_{lj}-\\delta_{jl}\\J_{im}+\\delta_{jm}\\k_{lm} \\J_{il}\n+\\delta_{il}\\k_{ij} \\J_{jm}.\n\\label{etiqueta}\n\\end{equation}\nIf, however, some constant(s) $\\k_i=0$, the\nalgebras (\\ref{aa}) become inhomogeneous and correspond to algebras that\nare obtained from $\\s (p,q)$\nthrough a sequence of \\.In\\\"on\\\"u-Wigner (IW) contractions. \nTo describe them let us denote ${\\fra h}^{(m)}$ ($m=1, \\dots, N$) the \nsubalgebra generated by the $\\J_{ab}$ ($a<b$) for which\n$a,b$  satisfy either $b< m$ {\\it or} $a \\geq m$. \nA complement for ${\\fra h}^{(m)}$ is the vector\nsubspace ${\\fra p}^{(m)}$ (not always a subalgebra) spanned by the\nelements $\\J_{ab}$ with $a < m$ \n{\\it and} $b \\geq m$. The decomposition\n$\\s_{\\k_1,\\k_2,\\dots,\\k_N} (N+1) = {\\fra p}^{(m)} \\oplus {\\fra h}^{(m)}$ is in fact a\nCartan-like decomposition, and there exists an involutive automorphism\nof the Lie algebra $\\s_{\\k_1,\\k_2,\\dots,\\k_N} (N+1)$ with  ${\\fra p}^{(m)}$ and \n${\\fra h}^{(m)}$ as the \nanti-invariant and invariant subspaces. \nThe structure of the subalgebra ${\\fra h}^{(m)}$ and of the vector subspace \n${\\fra p}^{(m)}$ of the Lie\nalgebra $\\s_{\\k_1,\\k_2,\\dots,\\k_N} (N+1)$ can be graphically displayed by arranging the \ngenerators of $so(N+1)$ in the form of a  triangle\n\n\\bigskip\n\n\\noindent\\hskip 1.8truecm \n\\begin{equation}\\begin{tabular}{cccc|cccc}\n$J_{01} $&$ J_{02} $&$\\ldots$&\n$J_{0\\, (m-1)} $&\n $J_{0m}$&$J_{0\\, (m+1)}$&\n$\\ldots$&$J_{0N}$\\\\\n &$ J_{12} $&$\\ldots$& $J_{1\\, (m-1)} $& $J_{1m}$&$J_{1\\, (m+1)}$&\n$\\ldots$&$J_{1N}$\\\\\n &&$\\ddots $&$\\vdots$&  $\\vdots$&$\\vdots$&\n$ $&$\\vdots$ \\\\\n & &$ $&$J_{(m-2)\\,(m-1)}$&  $J_{(m-2)\\,m}$&$J_{(m-2)\\,(m+1)}$&\n$\\ldots$&$J_{(m-2)\\,N}$\\\\\n & &$ $& &  $J_{(m-1)\\,m}$&$J_{(m-1)\\,(m+1)}$&\n$\\ldots$&$J_{(m-1)\\,N}$ \\label{triangulo}\\\\\n\\cline{5-8}\n & &$ $&\\multicolumn{1}{c}{\\,}&    $ $&$J_{m\\,(m+1)}$&\n$\\ldots$&$J_{m  N}$\\\\\n & &$ $&\\multicolumn{1}{c}{\\,}& $ $ &$ $&\n$\\ddots$&$\\vdots$\\\\\n & &$ $&\\multicolumn{1}{c}{\\,}& $ $&$ $& \n$ $&$J_{N-1\\,N}$\\\\\n\\end{tabular}\n\\end{equation}\n\n\\bigskip\n\\noindent\nWe see that the generators which \nspan the subspace ${\\fra p}^{(m)}$ are the\n$m(N+1-m)$ generators in the rectangle \ndetermined by the corner $J_{(m-1)m}$. \nThe triangles at its left and below correspond to the subalgebras\n$\\s_{\\k_1\\ldots \\k_{m-1}}(m)$ and \n$\\s_{\\k_{m+1}\\ldots \\k_N}(N+1-m)$ respectively, \nthe direct sum of which is the subalgebra \n${\\fra h}^{(m)}$. The  subspace\n${\\fra p}^{(m)}$ \ncorresponding to the $\\k_m$-rectangle in the\ndiagram, can be identified \nwith the Lie algebra quotient space \n$so_{\\k_1,\\k_2,\\dots,\\k_N} (N+1) \/ \\s_{\\k_1\\ldots \\k_{m-1}}(m) \\oplus \\s_{\\k_{m+1}\\ldots\n\\k_N}(N+1-m)$.\n\nFor each decomposition $\\s_{\\k_1,\\k_2,\\dots,\\k_N} (N+1) = {\\fra p}^{(m)} \\oplus {\\fra h}^{(m)}, \n\\ m=1,\\dots,N$, \nthere is a possible IW contraction,  denoted $\\Gamma^{(m)}$, to be performed on the\nalgebra $\\s_{\\k_1,\\k_2,\\dots,\\k_N} (N+1)$. Specifically, if we denote the generators of \nthe standard $\\s_{\\k_1,\\k_2,\\dots,\\k_N} (N+1)$ algebra by $\\extr X$, the IW contraction\n$\\Gamma^{(m)}$ of $\\s_{\\k_1,\\k_2,\\dots,\\k_N} (N+1)$ is given by the $\\epsilon \\to 0$\nlimit of the replacements \n\\begin{equation} \n\\begin{array}{l}\n\\Gamma^{(m)} (\\extr X)\\equiv \\extr X' = \\left\\{\\begin{array}{lc}\n\\  \\extr X \\quad &\\mbox \\ \\mbox{if}\\ \\extr X\\in \\fra h^{(m)}\\\\\n\\epsilon   \\extr X \\quad &\\ \\mbox{if}\\ \\extr X \\in \\fra p^{(m)}\n\\end{array}\n\\right. ,\\qquad   m=1,\\dots,N, \n\\label{ae}\n\\end{array}\\end{equation}\nUnder the contraction $\\Gamma^{(m)}$, the algebra $so_{\\k_1,\\k_2,\\dots,\\k_N}(N+1)$ goes to another\nalgebra in the CK family with  the same values of the $\\k_i$ constants except\nfor $\\k_{m} =0$. \nThus, in the triangular arrangement of\ngenerators, the\n$N$ possible IW contractions correspond to the\n$N$ different rectangles that can be selected inside the large triangle.\nThese rectangles are completely abelianised by the contractions, while the\ncommutators with\none or two generators outside ${\\fra p}^{(m)}$ remain unchanged. \nAs an example, the contraction given by\n(\\ref{ae}) with $m=1$ and starting from a $so(p,q)$ algebra, where all $\\k_i$ are\ndifferent from zero, corresponds to the limit $\\epsilon \\to 0$ of \n$J_{0i}\\mapsto J_{0i}'=\\epsilon   J_{0i}$, $J_{ij}=J_{ij}'\\ (i\\ne 0)\\,,\\,\nJ_{0i},J_{ij}\\in so(p,q)$. This leads to\n$[J_{0i}',J_{0j}']=\\pm \\epsilon^2 J_{ij}'$ and hence\n$J_{0i}'$ \n$i=1,\\ldots,N$ determines the abelian $N$-dimensional ideal ${\\fra p}^{(1)}$.\n\nLet us now consider the homogeneous space ${\\cal S} \\equiv SO_{\\k_1,\\k_2,\\dots,\\k_N} (N+1) \/\nSO_{\\k_2,\\dots,\\k_N}(N)$, where $SO_{\\k_2,\\dots,\\k_N}(N)$ is the subgroup generated by\nthe subalgebra $\\fra h^{(m)}$ with $m=1$. This space has an invariant\ncanonical connection, and a hierarchy of metrics, coming after suitable rescalings\nfrom the Cartan-Killing form  in the algebra\n$so_{\\k_1,\\k_2,\\dots,\\k_N}(N+1)$.  When \nthe constants $\\k_{2},\\dots, \\k_{N}$  are\ndifferent from zero, then the `main' metric is non-degenerate, the invariant\ncanonical connection turns out to be the corresponding Levi-Civita metric\nconnection, and the space ${\\cal S}$ has a curvature which is constant and \nequal to $\\k_1$. \n\nIn the particular case $( \\k_1,\\k_2,\\dots,\\k_N ) = (\\k_1, 1, \\dots, 1 )$ the space\n${\\cal S}$ reduces to the Riemannian space (positive definite metric) of \nconstant curvature $\\k_1$ and dimension $N$. \nWhen $\\k_1=0$ the algebras $\\s_{0,\\k_2,\\dots,\\k_N} (N+1)$, can be realized as algebras  \nof groups of affine transformations on $\\R ^N$  \\cite{SHO}; in this case we \nshall rename the generators as \n$\\{\\P_{i}:=\n\\J_{0i},\\\n\\J_{ij};\\ i<j,\\ i,j= 1, \\dots, N\\}$, the new names stressing the role of\n$\\P_i$ as generating translations and of $\\J_{ij}$ as the generators of\nrotations around the origin of the space. \nEach subalgebra ${\\fra h}^{(m)},\\  m=1, \\dots ,N$, is spanned by the set of \ngenerators $\\{\\P_i,\\J_{ij},\\ i,j=1,\\dots,m-1;\\ \\J_{kl},  \\ k,l=m,\\dots,N \\}$, \nand hence the collection of subalgebras ${\\fra h}^{(m)}$ can be clearly \nidentified in the space\n${\\cal S}$ as the isotropy subalgebras of a point (for $m=1$), of a line (for\n$m=2$), \\dots , of a hyperplane (for $m=N$).  \n\nThe non-zero Lie brackets of $\\s_{0,\\k_2,\\dots,\\k_N} (N+1)$ are given by \n\\begin{equation}\n[\\J_{ij},\\P_i] = \\P_j,  \\quad\n[\\J_{ij},\\P_j] = \\!-\\!\\k_{ij}\\,\\P_i,  \t\\quad  \n[\\J_{ij},\\J_{ik}] =  \\k_{ij} \\J_{jk}, \\quad\n[\\J_{ij},\\J_{jk}] = -\\J_{ik} ,  \\quad\n[\\J_{ik},\\J_{jk}] = \\k_{jk}\\J_{ij},\n\\label{ab}\n\\end{equation}\nwhere the indices $i, j, k =1,\\ldots, N$ are always assumed to be ordered,\n$i<j<k$.  \nNote in particular that all translation generators commute\n(as witnessing the zero curvature). \nIt will be convenient to denote\nthis $\\s_{0,\\k_2,\\dots,\\k_N}(N+1)$ Lie algebra by $\\is_{\\k_2,\\dots,\\k_N}(N)$ , and by \n$ISO_{\\k_2,\\dots,\\k_N}(N)$ the\ncorresponding group. There are $3^{N-1}$  different\n$N$--dimensional affine CK geometries, and from relations (\\ref{ab})\nit is clear that the groups $ISO_{\\k_2,\\dots,\\k_N} (N)$ have a semidirect\nproduct structure \n\\begin{equation}\nISO_{\\k_2,\\dots,\\k_N} (N) = SO_{\\k_2,\\dots,\\k_N} (N)\\odot T_N\\quad,\\label{ac}\n\\end{equation}\nwhere $T_N$ is the abelian subgroup generated by $\\{ \\P_i\\ ; \\\ni=1,\\dots,N\n\\}$ (in the case $\\k_1=0$, this abelian subgroup can be identified with\nthe CK homogeneous space $\\cal S$ itself) and\n$SO_{\\k_2,\\dots,\\k_N} (N)$ is a general CK group with $N-1$ constants $\\k_i$,  \ngenerated by $\\{ \\J_{ij}\\ ; \\ \ni,j=1,\\dots,N\\} $. The `main' \nmetric which is kept invariant by\nthe action of this group is described by the quadratic form given\nby a matrix with diagonal entries \n$(1, \\ \\k_2, \\ \\k_2 \\k_3,\\ \\dots, \\ \\k_2 \\cdots \\k_N)$. Among these\ninhomogeneous groups we can recognise the Euclidean group in $N$\ndimensions for which \n$(\\k_1,\\k_2,\\dots,\\k_N)=(0,1, 1, \\dots, 1 )$, the Poincar\\'e group in $(N-1,1)$ \ndimensions (appearing several times in the CK affine scheme as\n{\\it e.g.}, for $(0,-(1\/c^2), 1,\\dots, 1 )$) \nor the Galilei group in $(N-1,1)$ dimensions\nwhich corresponds to the values \n$(\\k_1,\\k_2,\\dots,\\k_N)=(0, 0, 1, \\dots, 1 )$;\nwe recall that in all these examples $\\k_1=0$. \nThe geometrical meaning of the\ncontractions $\\Gamma^{(m)}$, $m=1, \\dots, N$, is to describe the behaviour of the space\n${\\cal S}\n\\equiv SO_{\\k_1,\\k_2,\\dots,\\k_N} (N+1) \/ SO_{\\k_2,\\dots,\\k_N}(N)$ around a point, a line,\n\\dots, an hyperplane. In particular, within the inhomogeneous CK family\n$\\k_1 = 0$, only those contractions $\\Gamma^{(m)}, m=2, \\dots, N$ may produce a \ndifferent algebra. In other words, these inhomogeneous algebras can be\nthought of as the result of a  `local' contraction (around\na point, $m=1$) \nwhich made to vanish the \nassociated curvature, although they   \ncan still be contracted to describe the behaviour of the\nspace around a line, \\dots, an hyperplane, and hence the remaining \ncontractions $\\Gamma^{(m)}, m=2, \\dots, N$ may be relevant. \nFor instance, the non-relativistic limit, where the behaviour of spacetime \ngeometry is approximated in the neighbourhood of a given (time-like) line\ncorresponds to the contraction where $\\k_2 \\to 0$. \n\n\\medskip\n\nA second order central element for the algebra\n$\\is_{\\k_2,\\dots,\\k_N} (N)$, coming after a specialisation to this case of a\nsuitable rescaling of the general CK Killing's form, reads \n\\begin{equation}\n{\\cal{\\extr C}}= \\sum_{i=1}^{N-1}\\k_{iN}\\P_i^2  + \\P_N^2\\quad;\n\\label{ad}\n\\end{equation}\nnotice that this Casimir only involves generators from the abelian translation\nsubalgebra. \n\n\\medskip\n\nSummarising, we see that the graded contraction language allows us to describe\ncontractions simply by setting some parameters equal  to zero.\nThese contractions may still be described by the standard IW framework, \nalthough the graded contraction scheme is more economical and permits a \nunified discussion of the different contractions.\n\n\\subsect{Contractions and dimensional analysis}\n\nThe minimal possible approach to study\nthe dimensional structure in CK algebras is by enforcing the dimensional\nhomogeneity of the commutation relations in all algebras in the CK family. In\nthis approach, all generators as well as the structure constants are\ndimensional, in such a way that these dimensions are the same in all CK\nalgebras. Consider the redefinition\n$\\J_{ab}=\\eta_{ab}\\J'_{ab}$ for any CK algebra. If we now want the second \ncommutator in (\\ref{aa}) to be preserved (we still choose the structure\nconstants equal to $1$ as dimensionless, for we are interested here in\nalgebras in the CK family, and not beyond), we need \n$\\eta_{ab}\\eta_{bc}=\\eta_{ac}$, so that we see that $\\eta_{ab}$ may be \nexpressed as  $\\eta_{ab}=\\eta_{a+1}\\eta_{a+2}\\ldots\\eta_{b}$. If we make now\nthis change in the first and third commutators, we get \n\\begin{equation}\n[\\J'_{ab},\\J'_{ac}]=\\frac{\\k_{ab}}{\\eta_{ab}^2}\\J'_{bc},\\qquad\n[\\J'_{ac},\\J'_{bc}]=\\frac{\\k_{bc}}{\\eta_{bc}^2}\\J'_{ab}. \\label{cambio}\n\\end{equation}\nIn the special case when all $\\k_i$ are different from\nzero (the case of simple algebras), the choice $\\eta_{ab}^2=|\\k_{ab}|$\nleads to the standard commutators for the $\\J'_{ab}$ of the real form \n$so(p,q)$ of the specific algebra considered, with all non-zero structure \nconstants equal to $\\pm 1$, and the $\\J'_{ab}$'s are dimensionless. \n\nIn the general case (for generic CK algebras) by virtue of the above\nredefinition, the generators in (\\ref{aa}) have as dimensions\n$[\\J_{ab}]=[\\k_{ab}]^{1\/2}= \\prod_{i=a+1}^{b} [\\k_{i}]^{1\/2}$. In this\napproach, each constant $\\k_a$ has dimensions, and if the dimension of the\ngenerator $\\J_{a-1 a}$ is written as $D_{a}^{-1}$, then it is clear that the\ndimensions of each $\\k_a$ are $[\\k_a]=D_{a}^{-2}$, irrespective of $\\k_a$ \nbeing zero or not. The dimension of each generator $\\J_{ab}$\nincludes a factor $[\\k_a]$ for each of the\n$\\k_a$-rectangles in (\\ref{triangulo}) to which $\\J_{ab}$ belongs.\n\n\\bigskip\n\nAnother possibility is\nto allocate dimensions to generators and\/or canonical parameters for each CK\nalgebra independently, in such a way as to make dimensionless all non-zero\nstructure constants in the algebra.\nThe idea of basing the dimensional analysis of a theory on the structure of its \nunderlying Lie group\/algebra has its \nroots in the well known examples of the \nPoincar\\'e and Galilei groups, which are obtained by contracting with respect \ntwo dimensionful parameters, the de Sitter radius and the velocity of light $c$, \nand has been discussed in \\cite{CSAnalDim}; see also \\cite{AldAzb}. \n\nIf for a simple Lie algebra in the CK\nfamily (\\ref{aa}) with non-zero\n$\\k_i$ constants we adopt this hypothesis, then as a consequence all the\ngenerators of the algebra, as well as their associated canonical parameters \nare also without dimensions (as the $\\J'_{ab}$'s in the first approach). \nIf the same requirement is applied to a non-simple CK Lie algebra, then we get \nthe result that some generators are also dimensionless, while others get a \ndimension.\nFor example, if this is done on a CK algebra with a {\\em a single} $\\k_{a}$ is\nequal to zero, it is clear from the commutation relations that those\ngenerators which acquire in this case a non-trivial dimension $D_a^{-1}$\n($D_a$ is then the dimension of corresponding canonical parameter) are exactly\nthose inside the $\\k_a$-rectangle corresponding to the\nconstant $\\k_a$ which vanished in the triangular arrangement of generators. \n\nIf there are two constants equal to zero, say\n$\\k_{a}=\\k_{b}=0$, there will be two non-trivial dimensions, and so on.\nRemark that now each $\\k_a$ has a dimension which is still $D_a^{-1}$ when \n$\\k_a=0$ but it is dimensionless when $\\k_a\\neq0$. In this alternative choice\nthe dimensions of the generators will still be given by  \n$[\\J_{ab}]=[\\k_{ab}]^{1\/2}= \\prod_{i=a+1}^{b} [\\k_{i}]^{1\/2}$ but those\nfactors where $\\k_i \\neq 0$ are $[\\k_{i}]=1$.  This situation is  exemplified\nin the transition from the Poincar\\'e to the  Galilei algebras.  These are\ngiven by the values\n$(\\k_1,\\k_2,\\k_3,\\k_4)=  (0,-1\/c^2,1,1)$ and\n$(0,0,1,1)$ respectively, and the arrangement of the Galilei and or\nPoincar\\'e generators written in the usual physical notation is given by: \n\n\\noindent \n\\begin{equation}\n\\begin{tabular}{cccc}\n$H $ & $ P_1 $ & $  P_2  $ & $  P_3  $  \\\\ \n$  $ & $ K_1 $ & $  K_2  $ & $  K_3  $  \\\\ \n$  $ & $     $ & $ J_{12}$ & $ J_{13}$  \\\\ \n$  $ & $     $ & $       $ & $ J_{23}$  \\\\ \n\\end{tabular}\n\\label{physNot}  \n\\quad.\n\\end{equation}\nThe assignment of dimensions made this way for Poincar\\'e and  for  Galilei\nalgebras is\n$$\n\\noindent\n\\begin{tabular}{cccc}\n$ D_1^{-1}$ & $ D_1^{-1}$ & $ D_1^{-1}$ &  $ D_1^{-1}$ \\\\ \n$  $ & $ 1 $ & $ 1   $ & $  1   $  \\\\ \n$  $ & $     $ & $   1  $ & $    1  $  \\\\ \n$  $ & $     $ & $       $ & $    1  $  \\\\ \n\\end{tabular}\n\\qquad\\qquad \n\\begin{tabular}{cccc}\n$ D_1^{-1}$ & $ D_1^{-1}D_2^{-1} $ & $ D_1^{-1}D_2^{-1} $ & \n$ D_1^{-1}D_2^{-1} $ \\\\ \n$  $ & $ \\phantom{D_1^{-1}}D_2^{-1}  $ & \n$ \\phantom{D_1^{-1}}D_2^{-1}   $ & $ \n\\phantom{D_1^{-1}}D_2^{-1}   $  \\\\ \n$  $ & $     $ & $   1  $ & $    1  $  \\\\ \n$  $ & $     $ & $       $ & $    1  $  \\\\ \n\\end{tabular}\n$$\nwhich gives the single `length' dimension in\nrelativistic physics and the customary\n$T, L$ dimensions of  non-relativistic physics ($D_1 \\equiv T$ and $D_2 \\equiv\nLT^{-1}$, so \n$D_1 D_2 = L$). \n\n\\bigskip\n\n\nThe relation between both perspectives to the dimensional analysis of\nCK algebras is as follows: All constants $\\k_a$ can be considered at the\nbeginning as dimensionful, and then all generators are also dimensionful.\nHowever, when a given $\\k_a \\neq 0$, the dimension $D_a$ can be\nremoved by taking $\\k_a$ as a pure number, which can be set equal to \n$\\pm 1$; this  is tantamount to fixing the scale of the  generators or, in\nother words, to measuring the associated group parameter in  terms of the\ncorresponding unit much in the same way as in a relativistic  theory we may\nadopt units in which $c=1$ ({\\it i.e.}, $\\k_2=-1$ above). In the former example,\nsetting $\\k_3=1=\\k_4$ may be understood as having hidden universal constants\nin the theory (cf. \\cite{LeviLeblond}).  However, once a dimensionful $\\k_a$\nhas been set equal to zero ({\\it i.e.}, a  contraction has been made), the\ngenerators in the corresponding box retain a  dimension $[\\k_a]^{1\/2}$ since\nthey cannot be re-scaled any longer. This is why some generators\nin the former Galilei example retain the non-removable dimensions $D_1, D_2$,\nwhile $D_2$ disappears in the Poincar\\'e case while $D_3, D_4$ have already\ndisappeared in both cases\\footnote{The above is not the only group-theoretical \nmechanism for the introduction of \ndimensions.\nWhere centrally extended groups are physically relevant, the dimensions of the \ntwo-cocycle realizing the extension play a role. \nFor instance, in the \n$(1+1)$-dimensional extended Galilei group we find two \nparameters characterising the two-dimensional cohomology space, which \ncorrespond to the mass and a (constant) force.}.\n\n\n\\sect{Deformed N--dimensional affine CK algebras}\n\nAll the family of affine $N$--dimensional ($N\\geq 2$) CK algebras\n$\\is_{\\k_2,\\dots,\\k_N} (N)$ can be endowed with a standard \ndeformed Hopf algebra structure which has been called `quantum'\ninhomogeneous CK structure and which has been given in\n\\cite{BHOSab,BHOScd}.  \nIn order to avoid repeating statements on\nthe index ranges, we will conform in Sections III and IV\nto the following convention: the range of a latin index $i, j, k$ will\nbe $1, \\dots, N-1$, and the index $N$ will be dealt with separately,\nunless otherwise stated explicitly. Also, when two indices $i,j$\nappear  in a generator, we will always assume that $i<j$.  \n\nLet ${\\cal A}$ be the algebra of the formal power series in the deformation \nparameter $\\lambda$ with coefficients in the enveloping algebra \n${\\cal U}(\\is_{\\k_2,\\dots,\\k_N} (N))$ of the Lie algebra \n$\\is_{\\k_2,\\dots,\\k_N} (N)$ of (\\ref{ab}). \nThen the\ncoproduct, counit, antipode and deformed commutation relations of the\nalgebra \n${\\cal U}_\\lambda(\\is_{\\k_2,\\dots,\\k_N}(N))$, which is a Hopf algebra, are given\nby\n\n\\medskip\n\n\\noindent\n1) {\\it Coproduct}: \\hfill\n\\begin{equation}\\begin{array}{l}\n\\co(\\P_N)=1\\otimes \\P_N+ \\P_N\\otimes 1,\n   \\qquad \\qquad \n\\co(\\P_i)=e^{-\\frac \\lambda 2  \\P_N}\\otimes \\P_i +\n            \\P_i\\otimes e^{\\frac \\lambda 2  \\P_N},\\\\[0.3cm]\n\\co(\\J_{ij})=1\\otimes \\J_{ij}+ \\J_{ij}\\otimes 1      \\\\[0.3cm]\n\\co(\\J_{iN})=e^{-\\frac \\lambda 2  \\P_N}\\otimes \\J_{iN} +\n    \\J_{iN} \\otimes e^{\\frac \\lambda 2  \\P_N} \\!-\n   \\frac \\lambda 2  \\sum_{s=1}^{i-1} \\J_{si}e^{-\\frac \\lambda 2  \\P_N}\n                                     \\otimes \\k_{iN} \\P_s +\n     \\frac \\lambda 2 \\sum_{s=1}^{i-1}\\k_{iN} \\P_s \n              \\otimes e^{\\frac \\lambda 2  \\P_N} \\J_{si}  \\\\[0.3cm]\n\\qquad\\qquad\\ \n  + \\frac \\lambda 2 \\sum_{s=i+1}^{N-1}\\J_{is}e^{-\\frac \\lambda 2  \\P_N}\n                          \\otimes \\k_{sN} \\P_s\n   -\\frac \\lambda 2 \\sum_{s=i+1}^{N-1}\\k_{sN} \\P_s \\otimes \n                e^{\\frac \\lambda 2  \\P_N} \\J_{is}\\quad;\n\\label{ba}\n\\end{array}\n\\end{equation}\n\n\\noindent\n2) {\\it Counit}: \n\\begin{equation}\n\\varepsilon(\\P_i)=\n\\varepsilon(\\P_N)=\n\\varepsilon(\\J_{ij})=\n\\varepsilon(\\J_{iN})=0 \\quad;\n\\label{bb}\n\\end{equation}\n\n\\noindent\n3) {\\it Antipode}:\n\\begin{equation}\\begin{array}{l}\n\\gamma(\\P_i) = -\\P_i, \\quad \n\\gamma(\\P_N) = -\\P_N, \\quad\n\\gamma(\\J_{ij}) = -\\J_{ij}, \\quad \n\\gamma(\\J_{iN}) = -\\J_{iN}-\\k_{iN}(N-1)\\frac {\\lambda}2\\P_i \\quad;\n\\label{bc}\n\\end{array}\\end{equation}\n(it may be written in a compact way as \n$\\gamma(\\X)=-e^{(N-1)\\frac {\\lambda}2 \\P_N} \\X \\, e^{-(N-1)\\frac {\\lambda}2 \\P_N}$),\n\n\\medskip\n\n\\noindent\n4) {\\it Deformed commutators}:\n\\begin{equation}\\begin{array}{l}\n[\\J_{iN},\\P_{j}]=\\delta_{ij}\\frac 1\\lambda\\sinh(\\lambda\\P_N), \\cr\n[\\J_{iN},\\J_{jN}]=\\k_{jN}\\biggl\\{\\J_{ij}\\cosh(\\lambda\\P_N)\n+\\frac{\\lambda^2}4\\biggl(\\sum_{s=1}^{i-1}\\k_{iN}\\P_s\\W_{sij}\\cr\n\\qquad\\qquad\\qquad \n-\\sum_{s=i+1}^{j-1}\\k_{sN}\\P_s\\W_{isj}+\\sum_{s=j+1}^{N-1}\n\\k_{sN}\\P_s\\W_{ijs}\\biggr)    \\biggr\\},\\quad i<j, \n\\label{bd}\n\\end{array}\\end{equation}\nwhere\n\\begin{equation}\n\\W_{ijk}=\\k_{ij}\\P_i \\J_{jk}-\\P_j\\J_{ik}+\\P_k\\J_{ij},\\quad i<j<k,\n\\qquad i,j,k=1,\\dots,N-1 \\ \\ \\  \\quad.\n\\label{lubanski}\n\\end{equation}\nThe remaining commutators are non-deformed and as given in (\\ref{ab}). \nIt may be checked that $(A,\\ \\co, \\ \\varepsilon,\\  \\gamma)$ satisfies the \nHopf algebra axioms \nand hence eqs. (\\ref{ba})-(\\ref{lubanski}) may be taken as the definition of \nthe deformation ${\\cal U}_\\lambda(\\is_{\\k_2,\\dots,\\k_N} (N))$ of \n${\\cal U}(\\is_{\\k_2,\\dots,\\k_N} (N))$.\nThe parameter $\\lambda$ has a dimension inverse to that of $P_N$ \nso that the product $\\lambda \\P_N$ is dimensionless, and may be\ninterpreted as the  parameter left after contracting \nthe deformed Hopf algebra ${\\cal U}_q({so}_{\\k_1\\ldots\\k_N}(N+1))$  \nby previously redefining $q$ in terms of $\\lambda$ and the contraction \nparameter.\nHowever, the expression of the deformation \n${\\cal U}_q({so}_{\\k_1\\ldots\\k_N}(N+1))$  \nin the `physical' basis is not known and this precludes us for the moment from \nderiving  (\\ref{ba})-(\\ref{lubanski}) by contracting its deformed simple \nparent algebra ${\\cal U}_q({so}_{\\k_1\\ldots\\k_N}(N+1))$.\nNevertheless, it may be seen that the deformed  Hopf algebra\n${\\cal U}_\\lambda(\\is_{\\k_2,\\dots,\\k_N} (N))$ is a  quantisation of the coboundary Lie\nbialgebra \n$({\\cal U}(\\is_{\\k_2,\\dots,\\k_N} (N)),r)$ generated by the (non-degenerate) classical \n$r$--matrix\n\\begin{equation}\nr=\\lambda\\sum_{s=1}^{N-1} \\J_{sN}\\wedge \\P_s\\quad.\\label{bf}\n\\end{equation}\nDue to the structure of $r$ and our convention about dimensions, it\nturns out that $r$ is dimensionless, regardless of the values of the constants\n$\\k_i$ \nsince the product of $J_{sN}$ and $P_s$ will always have the same dimensions \nof $P_N$.\n\nWe remark that the above deformation (\\ref{ba})-(\\ref{lubanski}) is not the \nonly one possible for the $\\is_{\\k_2,\\dots,\\k_N}(N)$ family.\nHowever, it is distinguished by the fact that all its members present\ndeformed algebra and coalgebra sectors.\n\nThe quantum analogue of the second order Casimir (\\ref{ad}) is expressed by\n\\begin{equation}\n{\\cal {\\extr C}}_\\lambda=\n\\sum_{i=1}^{N-1} \\k_{iN}\\P_i^2 + \n\\frac 4{\\lambda^2}\\left[\\sinh(\\frac \\lambda 2  \\P_N)\\right]^2\\quad.\n\\label{ch}\n\\end{equation}\n\nAs far as their action on the algebra generators is concerned, \nthe quantum versions\n$\\Gamma^{(m)}_\\pardeform$ of the classical  IW contractions $\\Gamma^{(m)}$ are defined to coincide\nwith the classical one  (\\ref{ae}). In particular, the generator\n$\\P_N$ is re-scaled by the corresponding\ncontraction parameter $\\epsilon$ in any of the contractions in the family\n$\\Gamma^{(m)}_\\pardeform$, $\\Gamma^{(m)}(\\P_N)\\equiv\\P_N'=\\epsilon \\P_N$ (for $\\epsilon\\to 0$).\nThis means that since one has to replace $\\P_N$ in (\\ref{ba}) by \n$\\P'_N\/\\epsilon$, the exponents there will diverge. \nIt is therefore natural to replace simultaneously $\\lambda$ by \n$\\epsilon\\lambda'$, {\\it i.e.}\\ to re-scale the deformation parameter \nby $\\Gamma^{(m)}(\\lambda)\\equiv\\lambda'=\\lambda\/\\epsilon$ \n(all primes are removed after taking the contraction limit),\nas the simplest possibility to preserve the coproduct\n(\\ref{ba}). \nSumming up, the quantum contraction $\\Gamma^{(M)}_\\lambda$ is defined as\nthe result of taking the limit $\\epsilon  \\to 0$ in (\\ref{ba})-(\\ref{bd}) once\nthe transformations\n\\begin{equation}\n\\Gamma^{(m)}_\\pardeform(\\epsilon , \\X)=\\Gamma^{(m)} (\\epsilon , \\X)\\quad,\\quad\n\\X=(\\P,\\J)\\quad,\\quad\n\\Gamma^{(m)}_\\pardeform(\\epsilon , \\lambda)= \\lambda\/\\epsilon \n\\label{bh}\n\\end{equation}\nare performed.\nWe conclude this section with three observations.  \nFirst, we have made constant reference to the IW procedure only because up to \nnow the graded contraction theory had not been extended to deformed algebras. \nSecond, as far as the generators $\\X$ are concerned, \n$\\Gamma^{(m)}_\\pardeform=\\Gamma^{(m)}$, so that only the action of \n$\\Gamma^{(m)}_\\pardeform$ on $\\lambda$ makes  $\\Gamma^{(m)}$ and $\\Gamma^{(m)}_\\pardeform$ different.\nThe third comment is that the re-scaling of $\\lambda$ and $\\P_N$\nimplied by  $\\Gamma^{(m)}_\\pardeform$ may change their dimensions (see Sec. 2) while consistently\nkeeping a dimensionless $\\lambda\\P_N$ exponent.\n\n  \n\n\\sect{Bicrossproduct structure of ${\\cal U}_\\lambda(\\is_{\\k_2,\\dots,\\k_N} (N))$}\n\nIt is not obvious to see \nwhether the Hopf algebra ${\\cal U}_\\lambda(\\is_{\\k_2,\\dots,\\k_N}(N))$ \nhas a bicrossproduct structure by a simple inspection of\n(\\ref{ba})--(\\ref{bd}). The clue in this direction is provided by the\nbicrossproduct structure \\cite{KPoinBicros} of the\n$\\kappa$-Poincar\\'e algebra \\cite{LNRT} \n(appearing in our scheme when $(\\k_1,\\k_2, \\k_3, \\k_4) = (0, 1, 1, -1)$), \nwhich is clearly displayed in terms of a new set of\ngenerators. \n\n\nThe aim of this section is to show that {\\it all} the deformed\nHopf algebras  \n${\\cal U}_\\lambda(\\is_{\\k_2,\\dots,\\k_N} (N))$ in the CK family have indeed a \nbicrossproduct structure. The basic bicrossproduct \nformulae used are recalled in the Appendix; for a\ndetailed exposition, see \\cite{MajMajid}.\nLet ${P}_i$, ${J}_{ij}$, $P_N$, $J_{iN}$\nbe the new set of generators defined in terms of the old ones ${\\P}_i$ and \n${\\J}_{ij}$ by \n\\begin{equation}\n\\begin{array}{l}\n{P}_{i}= e^{-(\\lambda\/2)\\P_N}\\P_{i}, \n\\qquad {P}_{N}=\\P_{N}, \\qquad  \n{J}_{ij}=\\J_{ij},\\\\[0.3cm]\n{J}_{iN}=\\frac{1}{2}\\{\\J_{iN},e^{-(\\lambda\/2)\\P_N}\\} + \n  \\frac{\\lambda}{4}\\sum_{s=1}^{i-1} \\k_{iN}\\{\\J_{si},\\P_s\\}e^{-(\\lambda\/2)\\P_N} - \n  \\frac{\\lambda}{4}\\sum_{s=i+1}^{N-1}\\k_{sN}\\{\\J_{is},\\P_s\\}e^{-(\\lambda\/2)\\P_N}\\quad, \n\\label{ca}\n\\end{array}\\end{equation}\nA straightforward but tedious computation leads to the following \nnew expressions \n(where $i,j,k=1,\\ldots,N-1$) for the coproduct (\\ref{ba}), counit (\\ref{bb}), \nantipode (\\ref{bc}) and algebra commutators ((\\ref{bd}) and\/or (\\ref{ab})): \n\n\\noindent\n1) {\\it Coproduct}: \\hfill\n\\begin{equation}\\begin{array}{l}\n\\co(P_i)=e^{-\\lambda P_N}\\otimes P_i + P_i\\otimes  1 \\quad, \\quad  \n\\co({P_N})=1\\otimes{P_N}+{P_N}\\otimes 1 \\quad,\n\\\\[0.3cm]    \n\\co({ J_{ij}})=1\\otimes{ J_{ij}}+{J_{ij}}\\otimes 1\\quad,\\\\[0.3cm]\n\\co(J_{iN})=e^{-\\lambda P_N}\\otimes J_{iN} +J_{iN}\\otimes 1 \n  +\\lambda\\sum_{s=1}^{i-1}\\k_{iN} P_s \\otimes J_{si}\n  -\\lambda\\sum_{s=i+1}^{N-1}\\k_{sN} P_s \\otimes J_{is},\n\\label{cb}\n\\end{array}\\end{equation}\n\n\\noindent\n2) {\\it Counit}: \n\\begin{equation}\n\\varepsilon(P_i)=\\varepsilon(P_N)=\\varepsilon(J_{ij})=\\varepsilon(J_{iN})=0,\n\\label{cc}\n\\end{equation}\n\n\\noindent\n3) {\\it Antipode}:\n\\begin{equation}\\begin{array}{l}\n\\gamma(P_i)=-\\exp(\\lambda P_N) P_i\\quad,\\quad \n\\gamma(P_N)=-P_N \\quad,\\quad \n\\gamma(J_{ij})=-J_{ij}\\quad,\n  \\\\[0.3cm]\n\\gamma(J_{iN}) = - e^{\\lambda P_N}J_{iN} + \n \\lambda e^{\\lambda P_N}\\sum_{s=1}^{i-1}\\k_{iN} P_s J_{si} -\n \\lambda e^{\\lambda P_N}\\sum_{s=i+1}^{N-1}\\k_{sN} P_s J_{is}, \n\\label{cd}\n\\end{array}\\end{equation}\n\n\\noindent\n4) {\\it Commutators}: \n\\begin{eqnarray}\n&& [P_i,P_j]=0 \\quad, \\qquad [P_i,P_N]=0\\quad, \\nonumber \\\\{} \n&& [J_{ij},J_{ik}] =  \\k_{ij} J_{jk} \\qquad,\n   [J_{ij},J_{jk}] = -J_{ik}   \\quad,\n   [J_{ik},J_{jk}] = \\k_{jk}J_{ij}, \\nonumber \\\\{} \n&& [J_{ij},J_{iN}] =  \\k_{ij} J_{jN}, \\qquad\n   [J_{ij},J_{jN}] = -J_{iN} ,  \\qquad\n   [J_{ik},J_{jN}] = \\k_{jN}J_{ij}, \\label{ce} \\\\{} \n&& [J_{ij},P_k]=\\delta_{ik}P_k - \\delta_{jk}\\k_{ij}P_i, \\qquad\n   [J_{ij},P_N]=0, \\nonumber \\\\{} \n&& [J_{iN},P_{j}]=\\delta_{ij}\\left(\\frac{1-e^{-2\\lambda P_N}}{2\\lambda}- \n     \\frac \\lambda 2  \\sum_{s=1}^{N-1}\\k_{sN}P_s^2 \\right) \n     + \\lambda \\k_{iN}P_iP_j\\quad, \\qquad  \n   [J_{iN},P_N]=-\\k_{iN}P_i \\quad . \\nonumber \n\\end{eqnarray}\nThus, all brackets for the new generators $ P_i, P_N, J_{ij}, J_{iN} $ \ncoincide with the non-deformed ones given in (\\ref{ab}) \n(substituting  everywhere the new $X$'s for their counterparts $\\extr X$) \nexcept for $[J_{iN},P_{j}]$,  \nwhich is now the only deformed commutation relation.\nThe effect of (\\ref{ca}) is to modify the second commutator in \n(\\ref{bd}), so that one recovers the undeformed $\\s_{\\k_1,\\k_2,\\dots,\\k_N}(N+1)$\nalgebra commutators, and to replace the commutators in the first line in\n(\\ref{bd}) by those in the last line in (\\ref{ce}).  As a result, terms with\nthe\n$\\W$ symbols are no longer present in the deformed commutators.\n\nIt may be checked that for  $\\lambda={1\\over\\kappa}$ and $N=4$ with \n$(\\k_1,\\k_2,\\k_3,\\k_4)=(0,1,1,-1)$ eqs. (\\ref{cb})-(\\ref{ce}) reproduce the \n$\\kappa$-Poincar\\'e algebra in the basis of \\cite{KPoinBicros} \nfor which $[\\kappa]=L^{-1}$, $[P_4]=L^{-1}$.\nIf we want $P_4$ to have dimensions of inverse of time we may take \n$(\\k_1,\\k_2,\\k_3,\\k_4)=(0,1,1,-c^2)$ instead\nsince $\\k_1$, before being set equal to zero, was \n$\\k_1={1\\over R^2}$; in this case $[\\kappa]=T^{-1}$. \nWe check that the metric after (\\ref{ac}) will diverge in a non-relativistic \nlimit with $\\k_4=-c^2$, which explains why a non-relativistic limit of the \n$\\kappa$-Poincar\\'e algebra \\cite{LNRT} requires a further redefinition of the \ndeformation parameter $\\kappa$ (see the end of Sec. 5).\n\nThe new expressions for the coproduct, counit, antipode and \ncommutation relations of ${\\cal U}_\\lambda(\\is_{\\k_2,\\dots,\\k_N} (N))$ now allow us to \nuncover its bicrossproduct structure.\nTo this aim, consider\nthe translation sector, generated by $\\{ P_1, \\dots, P_N \\}$.\nAccording to the expressions (\\ref{cb})--(\\ref{ce}),  it defines a commutative\nbut non co-commutative Hopf subalgebra of \n${\\cal U}_\\lambda(\\is_{\\k_2,\\dots,\\k_N} (N))$ which will be denoted as\n${\\cal U}_\\lambda(T_N)$.  Let now ${\\cal U}( {so}_{\\k_2,\\dots,\\k_N} (N))$ be the \nnon-commutative and co-commutative non-deformed  \nCK Hopf algebra spanned by the remaining\ngenerators \n$\\{J_{ij};\\ i<j,\\  i,j=1,\\dots, N\\}$, hence with commutation relations given by \n(\\ref{ab}) and primitive coproduct (when all\n$\\k$'s are non-zero, this is a pseudo-orthogonal algebra). Let us define a\nright action \n$\\alpha:{\\cal U}_\\lambda(T_N)\\otimes {\\cal U}({so}_{\\k_2,\\dots,\\k_N} (N)) \\to {\\cal U}_\\lambda(T_N)$ by \n\\begin{equation}\n\\alpha(P_i,J_{jk})\\equiv P_i \\lhd J_{jk} :=[P_i,J_{jk}] ; \\quad\nj<k,\\ \\ i,j,k= 1,2,\\dots, N,\\label{cf}\n\\end{equation}       \nwhere the commutators is given in (\\ref{ce}), \nand a left coaction ${\\beta}:{\\cal U}({so}_{\\k_2,\\dots,\\k_N} (N)) \\to {\\cal U}_\\lambda(T_N)\\otimes \n{\\cal U}({so}_{\\k_2,\\dots,\\k_N} (N))$ by\n\\begin{equation}\\begin{array}{l}\n{\\beta}(J_{ij}) :=1\\otimes J_{ij}\\quad,\\quad\\\\[0.3cm]\n{\\beta}(J_{iN}) :=e^{-\\lambda P_N}\\otimes J_{iN} +\n\\lambda\\sum_{s=1}^{i-1}\\k_{iN} P_s \\otimes J_{si}\n-\\lambda\\sum_{s=i+1}^{N-1}\\k_{sN} P_s \\otimes J_{is}\\quad.\n\\end{array} \\label{cg} \\end{equation}\nIt may be checked that ${\\cal U}_\\lambda(T_N)$ is a right \n${\\cal U}({so}_{\\k_2,\\dots,\\k_N} (N))$--module \nalgebra (${\\cal U}({so}_{\\k_2,\\dots,\\k_N} (N))\\triangleright\\!\\!\\!<{\\cal U}_\\lambda(T_N)$) and \nthat ${\\cal U}({so}_{\\k_2,\\dots,\\k_N}(N))$ is \na left ${\\cal U}_\\lambda(T_N)$-comodule coalgebra  (${\\cal U}({so}_{\\k_2,\\dots,\\k_N} \n(N))>\\!\\!\\blacktriangleleft{\\cal U}_\\lambda(T_N)$) under the action (\\ref{cf}) and coaction (\\ref{cg}),\nrespectively,\nand that the compatibility conditions \\cite{MajMajid} \n(\\ref{apix})--(\\ref{apxiii}) between $\\alpha$ y \n$\\beta$ needed for\n${\\cal U}({so}_{\\k_2,\\dots,\\k_N} (N))\\otimes {\\cal U}_\\lambda(T_N)$ to have a\nbicrossproduct  structure are fulfilled.\nFor instance, (\\ref{apxiii}) is automatically satisfied, since\n${\\cal U}(\\s_{\\k_2,\\dots,\\k_N}(N))$ is undeformed and hence co-commutative and ${\\cal U}_\\lambda(T_N)$ \nis abelian.\nThis case, specially relevant here, was discussed in \\cite{SingMoln}.\nThen, the bicrossproduct structure of ${\\cal U}_\\lambda(\\is_{\\k_2,\\dots,\\k_N} (N))$ may\nbe stated  in the form of the following \n\n\\bigskip\n\\noindent\n{\\bf Theorem}. \n{\\it\nThe deformed Hopf CK family of algebras ${\\cal U}_\\lambda(\\is_{\\k_2,\\dots,\\k_N} (N))$ has a\nbicrossproduct structure \n$$\n{\\cal U}_\\lambda(\\is_{\\k_2,\\dots,\\k_N} (N)) =  {\\cal U}({so}_{\\k_2,\\dots,\\k_N} (N))\n^\\beta\\triangleright\\!\\!\\!\\blacktriangleleft_\\alpha  \n{\\cal U}_\\lambda(T_N)\n$$\nrelative to the right action $\\alpha$ and left coaction $\\beta$ given by \n(\\ref{cf}) and (\\ref{cg}) respectively.}\n\n\\medskip\n\\noindent\n{\\sl Proof}:\nAs mentioned, the mappings $\\alpha$ and ${\\beta}$ satisfy the bicrossproduct \nconditions as may be checked by direct computation. \nThen the expressions\n(\\ref{apxiv})-(\\ref{apxvii})  give the associated coproduct, counit and \nantipode. \nIt is then verified that the resulting expressions \nare in agreement with (\\ref{cb})--(\\ref{ce}) {\\it q.e.d.}\n\nThe interesting consequence of the above discussion is that, as the direct \ninspection of expressions (\\ref{cf}) and\n(\\ref{cg}) shows, the action and the coaction mappings depend on\nthe parameters $\\k_i$ in such a way that the bicrossproduct structure\nis formally invariant under any contraction $\\k_i=0$. \nIn other words, for $\\lambda$-deformations in the affine CK family,\nthe bicrossproduct structure is preserved by {\\it all} \nthe successive contractions: \ncontracting and taking bicrossproduct of the appropriate Hopf algebras with the \nresulting actions and coactions are commuting processes. \nThis is well within the\nspirit of the CK scheme, the aim of which is to state properties which\nhold simultaneously for a large number of algebras.\n\nThe expression of the deformed Casimir (\\ref{ch}) in the new basis \nis\n\\begin{equation}\n{\\cal C}_\\lambda=\n\\sum_{i=1}^{N-1} \\k_{iN}e^{-\\lambda P_N}{P}_i^2 + \n\\frac 4{\\lambda^2}\\left[\\sinh(\\frac \\lambda 2  {P}_N)\\right]^2\n\\quad;\n\\end{equation}\nit only depends on the generators of the \ndeformed Hopf subalgebra ${\\cal U}_\\lambda(\\T_N)$.\n\nOn the other hand,  the expression for the $r$--matrix is similar to\nthe former (\\ref{bf}) but in terms of the new generators, \n\\begin{equation}\nr=\\lambda\\sum_{s=1}^{N-1} J_{sN}\\wedge P_s\\quad. \\label{bfnew}\n\\end{equation}\n\n\\sect{Applications}\nThe quantum algebras we are dealing with range from deformations of the\ninhomogeneous algebras \n${iso}(p,q), \\ p+q=N$ (when all constants $\\k_2, \\k_3, \\dots \\k_N$\nare different from zero) to the extreme case of a Hopf deformation of \nthe algebra, where all constants are equal to zero, which\ncan be called flag space algebra \n(in this case the group action preserves a complete flag). \n\nClassically, all these algebras are semidirect products, and indeed there\nis a semidirect structure in the CK algebras associated to the\nvanishing of each constant $\\k_i$. We have restricted here to the\nalgebras with $\\k_1=0$, all of which have the semidirect structure\ndisplayed in (\\ref{ab}). When {\\em all} remaining constants\n$\\k_i$ are different from zero, say $\\k_i=\\pm 1$, \nthe algebra $\\is_{\\k_2,\\dots,\\k_N}(N)$ is\nisomorphic to an inhomogeneous pseudo-orthogonal algebra\n${iso}(p,q), \\  p+q=N$, \nwith the semidirect structure given by  the natural\naction of\n${so}(p,q)$ on $\\R^N$. These algebras are physically very relevant\nand some of their deformations have been thoroughly studied. \nIn particular, the $\\lambda$-deformed structures given in\n(\\ref{ba})--(\\ref{bd}) include a deformed \n$N$-dimensional Euclidean algebra, a deformed $(N-1,1)$ Galilei algebra and\nseveral\ndeformed $(N-1,1)$ Poincar\\'e algebras, as well as their analogues for any\nsignature. \n\nThe action and coaction mappings associated with the bicrossproduct \nare given by (\\ref{cf}) and (\\ref{cg}).\nExplicitly, eq. (\\ref{ce}) gives\n\\begin{eqnarray}\n&& \\alpha(P_N,J_{ij})\\equiv P_N \\lhd J_{ij} := 0 \\quad,\\quad\n   \\alpha(P_N,J_{iN})\\equiv P_N \\lhd J_{iN} := \\k_{iN} P_i \\quad,\\nonumber \\\\\n&& \\alpha(P_k,J_{ij})\\equiv P_k \\lhd J_{ij} := \n            -\\delta_{ki} P_j + \\delta_{kj} \\k_{ij} P_i\\quad,  \\nonumber \\\\\n&& \\alpha(P_k,J_{iN})\\equiv P_k \\lhd J_{iN} :=  \n      -\\delta_{ki}\\left(\\frac{1-e^{-2\\lambda P_N}}{2\\lambda}- \\frac \\lambda 2  \n       \\sum_{s=1}^{N-1}\\k_{sN}P_s^2 \\right) - \\lambda \\k_{iN}P_iP_k\\quad.\n        \\label{includ}\n\\end{eqnarray}\n\nIf we consider the special case $N=4$, this set of algebras\nincludes {\\em four} deformed Poincar\\'e algebras \n${\\cal U}_\\lambda({\\fra p}^{(s)}(3,1)), \\  s=1,2,3,4$. These are\ndeformations of the four undeformed CK algebras, denoted ${\\fra p}^{(s)}(3,1),\\ \ns=1,2,3,4$, which are isomorphic to the \n(3,1) Poincar\\'e algebra, and correspond to\nidentifying one of the generators $P_i$ to the time translation generator, the other\nthree being space translations.  \nIf the time generator is taken successively to be our  \n$P_1$, $P_2$, $P_3$, $P_4$, these four algebras correspond to the four sets\nof values of  $(\\k_1,\\k_2,\\k_3,\\k_4)=$ \n$(0,-1\/c^2,1,1)$,\n$(0,-c^2,-1\/c^2,1)$, \n$(0,1,-c^2,-1\/c^2)$, and\n$(0,1,1,-c^2)$. The set of four deformed Poincar\\'e algebras \n${\\cal U}_\\lambda({\\fra p}^{(s)}(3,1)), \\  s=1,2,3,4$ \n\\cite{BHOScd} includes three `space-like' Poincar\\'e deformed algebras, \nthe last one being the $\\kappa$-Poincar\\'e algebra once  \n$\\lambda={1\/\\kappa}$ with\n$[P_N]=T^{-1}$). In each case, the rotation generators comprise the\nboost and space rotation generators and the identification is made\naccording to the choice of the time generator ({\\it e.g.}, in the \n$\\kappa$-Poincar\\'e the boosts are the $N_i=J_{i4}$).  \nThe $N$-dimensional $\\kappa$--Poincar\\'e  \\cite{MasLuk} \nis associated to the $\\k_i$ values  $(0,1,\\dots, 1,-c^2)$.\n\n\nThe Euclidean algebra $\\fra e(4)$ appears only once (up to rescalings) for \n$(\\k_1,\\k_2,\\k_3,\\k_4)=(0,1,1,1)$\nand the bicrossproduct structure of their Hopf CK quantum deformation  \n${\\cal U}_\\lambda(\\fra e(4))$, is  \n$$\n{\\cal U}_\\lambda(\\fra e(4))=  {\\cal U}({so}(4)) \\triangleright\\!\\!\\!\\blacktriangleleft {\\cal U}_\\lambda(\\R_4)\n$$\n(see \\cite{APb} in the lower dimensional case).\n\nThe remaining quantum Hopf algebras in the CK family are quantum\ndeformations of less known undeformed Lie algebras, yet their\nbicrossproduct structure is described in parallel to the former cases.\nSpecially relevant from the physical point of view \nis the Galilei algebra \n$\\fra g (3,1)$, appearing within the affine CK family for the $\\k$ values\n$(0,0,1,1)$ (and only for these). It is worth remarking that this\nGalilei algebra is obtained from the \n${\\cal U}_\\lambda({\\fra p}^{(1)} (3,1))$ \nassociated to the $\\k$ values $(0,-1\/c^2,1,1)$ by means of the\ncontraction $\\k_2\n\\to 0$ ({\\it i.e.}, $c \\to \\infty$), which gives a deformation different from that in \n\\cite{POLACOS,APa}. \nThe bicrossproduct structure of the resulting ${\\cal U}_\\lambda(\\fra g (3,1))$\nHopf algebra (obtained for $(0,0,1,1)$ is  \n$$\n{\\cal U}_\\lambda(\\fra g (3,1)) = \n{\\cal U}({so}_{(0,1,1)} (4)) \\triangleright\\!\\!\\!\\blacktriangleleft {\\cal U}_\\lambda(\\R_4)\n\\equiv \n{\\cal U}({iso}_{(1,1)}(3))\\triangleright\\!\\!\\!\\blacktriangleleft  {\\cal U}_\\lambda(\\R_4)\\quad.\n$$  \n\nIt is interesting to check how the action and coaction mappings for the\nPoincar\\'e Hopf algebra ${\\cal U}_\\lambda (\\fra p^{(1)} (3,1))$ \nreduce to the corresponding Galilean ones\nunder the contraction $\\k_2 \\to 0$: ${\\cal U}_\\lambda (\\fra p^{(1)} (3,1))$ to\n${\\cal U}_\\lambda (\\fra g (3,1))$. \nWe give explicitly the complete Hopf structure of both \n${\\cal U}_\\lambda({\\fra p}^{(1)}(3,1))$ and ${\\cal U}_\\lambda({\\fra g}(3,1))$, which\ncorrespond to the choices $(0,-1\/c^2,1,1)$ and $(0,0,1,1)$ for the $\\k_i$'s. We\nwill present the results in the usual physical basis, constituted by the \ngenerators of time translation,\n$H$, space translations $P_1, P_2, P_3$, boosts $K_1, K_2, K_3$, and\nspace rotations $J_1, J_2, J_3$, \nwhich are related to the CK original generators $\\{ P_i,\\ J_{ij};\\\ni,j=1,2,3,4\\}$ as follows: the three {\\em space} translations, now denoted\nas $P_1, P_2, P_3$ in order to conform with the standard physical notation, \ncorrespond to those formerly denoted as $P_2, P_3, P_4$ in (\\ref{cb})-(\\ref{ce}),\nwhile the time translation generator\n$H$  now corresponds to the former $P_1$\nand the rest in (\\ref{cb})-(\\ref{ce}) correspond to \n$J_{12}=K_1\\,,\\,J_{13}=K_2\\,,\\,J_{14}=K_3\\,,\\,\nJ_{34}=J_1\\,,\\,J_{24}=-J_2\\,,\\,J_{23}=J_3$, as given in the diagram\n(\\ref{physNot}).  \n \nThe Hopf structure of the Poincar\\'e algebra \n${\\cal U}_\\lambda(\\fra p^{(1)} (3,1))$ now follows \nfrom expressions (\\ref{cb})-(\\ref{ce}):\n\n\n\\medskip\n\\noindent\n1) {\\it Coproduct}: \\hfill\n\\begin{equation}\\begin{array}{l}\n\\co(X)=e^{-\\lambda P_3}\\otimes X + X\\otimes  1, \\qquad X\\in \\{H,P_1,P_2\\},\n\\\\[0.3cm]    \n\\co({P_3})=1\\otimes{P_3}+{P_3}\\otimes 1, \\\\[0.3cm]\n\\co({X})=1\\otimes{X}+{X}\\otimes 1, \\qquad X\\in \\{K_1,K_2,J_3\\},\n\\\\[0.3cm] \\co(K_{3})=e^{-\\lambda P_3}\\otimes K_{3} +K_{3}\\otimes 1 \n  -\\lambda  P_1 \\otimes K_{1}\n  -\\lambda P_2 \\otimes K_{2}, \\\\[0.3cm]\n\\co(J_{1})=e^{-\\lambda P_3}\\otimes J_{1} +J_{1}\\otimes 1 \n +\\lambda  H \\otimes K_{2}\n  +\\lambda P_1 \\otimes J_{3}, \\\\[0.3cm]\n\\co(J_{2})=e^{-\\lambda P_3}\\otimes J_{2} +J_{2}\\otimes 1 \n -\\lambda  H \\otimes K_{1}\n  +\\lambda P_2 \\otimes J_{3}. \n\\label{fb}\n\\end{array}\\end{equation}\n\n\\noindent\n2) {\\it Counit}: \n\\begin{equation}\n\\varepsilon(H)=\\varepsilon(P_i)=\\varepsilon(K_{i})=\\varepsilon(J_{i})=0,\\qquad i=1,2,3.\n\\label{fc}\n\\end{equation}\n\n\\noindent\n3) {\\it Antipode}:\n\\begin{equation}\\begin{array}{l}\n\\gamma(X)=-e^{\\lambda P_3} X,\\qquad X\\in \\{H,P_1,P_2\\}; \\qquad \n\\gamma(P_3)=-P_3, \\\\ [0.3cm]\n\\gamma(X)=-X,\\quad X\\in \\{K_1,K_2,J_3\\}, \\\\[0.3cm]\n\\gamma(K_{3}) = - e^{\\lambda P_3}K_{3}\n- \\lambda e^{\\lambda P_3} P_1 K_{1} \n- \\lambda e^{\\lambda P_3}P_2 K_{2}, \\\\[0.3cm] \n\\gamma(J_{1}) = - e^{\\lambda P_3}J_{1} \n+ \\lambda e^{\\lambda P_3} H K_{2} \n+ \\lambda e^{\\lambda P_3} P_1 J_{3}, \\\\[0.3cm] \n\\gamma(J_{2}) = - e^{\\lambda P_3}J_{2} \n- \\lambda e^{\\lambda P_3} H K_{1} \n+ \\lambda e^{\\lambda P_3}P_2 J_{3}.\\label{fd}\n\\end{array}\\end{equation}\n\n\\noindent\n4) {\\it Commutators}: \n\\begin{equation}\n\\begin{array}{l}\n[H,P_i]=0,\\quad [P_i,P_j]=0\\quad i,j=0,1,2,3 ,\\\\[0.3cm]\n[K_{i},H]=P_i,\\ \\ \\ i=1,2; \\qquad \n   [K_{3},H]=\\frac{1-e^{-2\\lambda P_3}}{2\\lambda}  \n   -\\frac{\\lambda}{2c^2}H^2- \\frac{\\lambda}{2}P_1^2 \n   -\\frac{\\lambda}{2}P_2^2, \\\\[0.3cm]\n[K_{i},P_j]=\\frac{1}{c^2} \\delta_{ij}H, \\quad \ni,j=1,2, \\qquad [K_{i},P_3]=0,\\\\[0.3cm] \n[K_{3},P_1]=-\\frac{\\lambda}{c^2}HP_1,\\qquad  \n  [K_{3},P_2]=-\\frac{\\lambda}{c^2}HP_2,\\qquad\n  [K_{3},P_3]=\\frac{1}{c^2}H, \\\\[0.3cm]\n[J_{1},H]={\\lambda}P_2H,\\qquad\n  [J_{2},H]=-{\\lambda}P_1H,\\qquad [J_{3},H]=0,\\\\[0.3cm]\n [J_{1},P_1]={\\lambda}P_2P_1, \\quad \n  [J_{1},P_2]=\\frac{1-e^{-2\\lambda P_3}}{2\\lambda}  \n  +\\frac{\\lambda}{2c^2}H^2- \\frac{\\lambda}{2}P_1^2 \n  +\\frac{\\lambda}{2}P_2^2,\\quad \n  [J_{1},P_3]=-P_2,  \\\\[0.3cm]\n [J_{2},P_1]=-\\frac{1-e^{-2\\lambda P_3}}{2\\lambda}  \n  -\\frac{\\lambda}{2c^2}H^2 - \\frac{\\lambda}{2}P_1^2 \n  +\\frac{\\lambda}{2}P_2^2, \\quad  [J_{2},P_2]=-{\\lambda}P_1P_2,\n  \\quad [J_{2},P_3]=P_1, \\\\[0.3cm]\n[J_{3},P_1]=P_2, \\quad [J_{3},P_2]=-P_1, \\quad[J_{3},P_3]=0, \n  \\\\[0.3cm]  \n[K_{i},K_j]=-\\frac{1}{c^2}\\varepsilon_{ijk}K_k, \\quad \n  [J_{i},K_j]=\\varepsilon_{ijk}K_k,\\quad\n  [J_{i},J_j]=\\varepsilon_{ijk}J_k,\\ \\ \\ i,j=1,2,3. \n\\end{array}\n\\label{fe}\n\\end{equation}\n\nThe explicit form of the action and the coaction for ${\\cal U}_\\lambda({\\fra\np}^{(1)}(3,1))$, given by (\\ref{cf}) and (\\ref{cg}) is obtained from  the\ncommutators (\\ref{fe}) and the coproduct (\\ref{fb}). The bicrossproduct structure\nof   ${\\cal U}_\\lambda({\\fra p}^{(1)}(3,1))$ is\n$$\n{\\cal U}_\\lambda({\\fra p}^{(1)}(3,1)) = \n{\\cal U}({so}_{(-1\/c^2,1,1)} (4)) \\triangleright\\!\\!\\!\\blacktriangleleft {\\cal U}_\\lambda(\\R_4).\n$$  \n\nIn the non-relativistic limit, $\\k_2=-1\/c^2 \\to 0$, this deformed algebra goes \nto a \nnew deformed Galilei algebra, ${\\cal U}_\\lambda({\\fra g}(3,1))$, whose coproduct,\ncounit and antipode are the same as \nin  (\\ref{fb})--(\\ref{fd})). With respect to the Lie commutators we write\nonly those that are different from the Poincar\\'e \n${\\cal U}_\\lambda({\\fra p}^{(1)}(3,1))$ ones in (\\ref{fe})\n\n\\noindent\n\\begin{eqnarray}\n&&  [K_{3},H]=\\frac{1-e^{-2\\lambda P_3}}{2\\lambda}  \n- \\frac{\\lambda}{2}P_1^2 \n-\\frac{\\lambda}{2}P_2^2,\\nonumber\\\\[0.1cm]\n&&[K_{i},P_j]=0, \\quad \ni, j=1,2,3,\\nonumber\\\\[0.1cm] \n&& [J_{1},P_2]=\\frac{1-e^{-2\\lambda P_3}}{2\\lambda}  \n- \\frac{\\lambda}{2}P_1^2 \n+\\frac{\\lambda}{2}P_2^2, \n\\label{ff}\n\\\\[0.1cm]\n&&[J_{2},P_1]=-\\frac{1-e^{-2\\lambda P_3}}{2\\lambda}  \n - \\frac{\\lambda}{2}P_1^2 \n+\\frac{\\lambda}{2}P_2^2,\\nonumber\\\\[0.1cm]\n&&[K_{i},K_j]=0,\\quad  i,j=1,2,3. \\nonumber\n\\end{eqnarray}\n{}From simple inspection we see that the \nGalilean action (as the commutators) has\nchanged from \nthat in ${\\cal U}_\\lambda({\\fra p}^{(1)}(3,1))$,\nyet the coaction is the same than in\nPoincar\\'e since the coproduct has remained the same.\n\nAll other Poincar\\'e quantum algebras (including the $\\kappa$-Poincar\\'e) do \nnot allow a direct `non-relativistic' contraction; this is clear when the \nconstant $c$ is explicitly written as above (see \\cite{APa} for a \ndiscussion).\n\n\\sect{The group deformation aspect: the case of \\hfil\\break\n      $\\hbox{Fun}_\\lambda (ISO_{\\k_2}(2))$}\n\nThe bicrossproduct structure of \n${\\cal U}_\\lambda (\\is_{\\k_2,\\dots,\\k_N}(N))$ opens the way \nto the possibility of recovering more easily the dual group \n$\\hbox{Fun}_\\lambda(ISO_{\\k_1,\\k_2,\\dots,\\k_N} (N))$ from the dual bicrossproduct `group-like'\nexpressions. We show here the explicit calculation in the lowest\ndimension $N=2$ to exhibit the procedure. \nAs shown in Sec. 4, the deformed Hopf algebra has the bicrossproduct\nstructure \n\\begin{equation} \n{\\cal U}_\\lambda (\\is_{\\k_2}(2)) =  {\\cal U}({so}_{\\k_2} (2)) \\triangleright\\!\\!\\!\\blacktriangleleft \n {\\cal U}_\\lambda (\\T_2)\\quad.\n\\label{da}\n\\end{equation}\nLet us now recover from it the quantum dual group, \n$\\hbox{Fun}_\\lambda(ISO_{\\k_2}(2))$. \n\nThe dual algebra $\\hbox{Fun}_\\lambda(\\T_2)$ of ${\\cal U}_\\lambda(\\T_2)$ is easily found from \n${\\cal U}_\\lambda(\\T_2)$ to be\n\\begin{equation}\n\\co(a_1)=1\\otimes  a_1 +  a_1\\otimes 1\\quad,\\quad\n\\co(a_2)=1\\otimes  a_2 +  a_2\\otimes 1\\quad; \\label{db}\n\\end{equation}\n\\begin{equation}\n\\varepsilon(a_i)=0\\quad;\\quad \\gamma(a_i)=-a_i\\quad,\\quad i=1,2\\quad;\\label{dc}\n\\end{equation}\nand \n\\begin{equation}\n[a_1,a_2]=\\lambda a_1\\quad,\n\\end{equation}\nwhere $a_1,a_2$ are a system of coordinates of $\\hbox{Fun}_\\lambda (\\T_2)$. \n\nOn the other hand, let \n$\\hbox{Fun}(SO_{\\k_2}(2))$ be the dual of\n${\\cal U}(SO_{\\k_2}(2))$ generated by $\\varphi$,  \nwith a non-deformed Hopf structure defined by\n\\begin{equation} \n\\co(\\varphi)= 1\\otimes \\varphi +  \\varphi\\otimes 1\\quad, \\quad \n\\varepsilon(\\varphi)=0\\quad,\\quad \\gamma(\\varphi)=-\\varphi\\quad. \\label{dd} \n\\end{equation}   \nNow, the problem is finding a pair of mappings $\\bar{{\\beta}}$ and\n$\\bar{\\alpha}$ (see Appendix),  \n\\begin{equation}\\begin{array}{l}\n\\bar{{\\beta}}\\ :\\ \\hbox{Fun}_\\lambda (\\T_2)\\to \\hbox{Fun}_\\lambda (\\T_2)\\otimes\n\\hbox{Fun}(SO_{\\k_2}  (2)),\\\\[0,3cm]\n\\bar{\\alpha} \\ :\\ \\hbox{Fun}_\\lambda (\\T_2)\\otimes \\hbox{Fun}(SO_{\\k_2}  (2))\\to \n\\hbox{Fun}(SO_{\\k_2} (2)).\\label{de}\n\\end{array}\\end{equation} \nduals, respectively, to $\\alpha$ and ${\\beta}$ as given in (\\ref{includ}), (\\ref{cg}) \nfor $N=2$. A calculation shows that they have the form \n\\begin{equation}\\begin{array}{l}\n\\bar{{\\beta}}(a_1)=a_1\\otimes \\C_{\\k_2}(\\varphi) +\na_2\\otimes {\\k_2} \\S_{\\k_2}(\\varphi)\\quad,\\\\[0.3cm]\n\\bar{{\\beta}}( a_2)=-a_1\\otimes \\S_{\\k_2}(\\varphi) +\na_2\\otimes  \\C_{\\k_2}(\\varphi)\\quad,\\label{df}\n\\end{array}\\end{equation}\nand \n\\begin{equation}\n\\bar{\\alpha}( a_1\\otimes \\varphi)=\\lambda(1- \\C_{\\k_2}(\\varphi))\n\\quad,\\quad\n\\bar{\\alpha}(a_2\\otimes \\varphi)=\\lambda\\S_{\\k_2}(\\varphi)\n\\quad,\n\\label{dh}\n\\end{equation}\nwhere the functions $\\C_{\\k}(\\varphi)$ and $\\S_{\\k}(\\varphi)$ reduce to\nthe trigonometric cosine and sine functions for $\\k=1$ and to the\nhyperbolic ones for\n$\\k=-1$ (see \\cite{BHOSe} for more details). Note that \n$$\n(\\bar{{\\beta}}( a_1),\\bar{{\\beta}}( a_2))=(a_1,a_2)\\dot\\otimes\n\\left( \\begin{array}{rr} \n\\C_{\\k_2}(\\varphi) & -\\S_{\\k_2}(\\varphi) \\\\\\k_2 \\S_{\\k_2}(\\varphi) & \n\\C_{\\k_2}(\\varphi) \\end{array}  \\right)\\quad,\\label{dj}\n$$\nwhere the $2\\times 2$ matrix is the transpose of the matrix\nrepresenting the generic element of the group $SO_{\\k_2} (2)$.   \n\nSince $\\bar\\beta$ modifies the originally co-commutative coproduct in \n$\\hbox{Fun}_\\lambda(\\T_2)$ and $\\bar\\alpha$ the commutation relations between the \ngenerators of the two algebras $\\hbox{Fun}_\\lambda(\\T_2)$ and that generated by \n$\\varphi$ (see eqs. (\\ref{abpxv}) and (\\ref{abpxiv})),\nwe may now determine through its bicrossproduct structure the deformed Hopf \nalgebra $\\hbox{Fun}_\\lambda (SO_{\\k_2} (2+1))$ which has the form \n\\begin{equation}\\begin{array}{l}\n\\co( a_1)=1\\otimes a_1 +\na_1\\otimes  \\C_{\\k_2}(\\varphi)+  a_2\\otimes\n{\\k_2}\\S_{\\k_2}(\\varphi),\\\\[0.3cm]\n\\co( a_2)=1\\otimes a_2 -\na_1\\otimes  \\S_{\\k_2}(\\varphi) + a_2\\otimes\n\\C_{\\k_2}(\\varphi),\\\\[0.3cm]\n\\co(\\varphi)=1\\otimes \\varphi + \\varphi \\otimes  1;\\label{dk}\n\\end{array}\\end{equation}\n\\begin{equation}\n\\varepsilon(a_i)=0\\quad,\\ \\ \\  i=1,2\\quad;\\quad \\varepsilon(\\varphi)=0\\quad;\\label{dl}\n\\end{equation}\n\\begin{equation}\\begin{array}{l}\n\\gamma(a_1) = \n    - \\C_{\\k_2}(\\varphi) a_1 -{\\k_2} \\S_{\\k_2}(\\varphi) a_2\\quad,  \\\\[0.3cm]\n\\gamma(a_2) = \n     \\S_{\\k_2}(\\varphi) a_1 -\\C_{\\k_2}(\\varphi) a_2\\quad;\n\\label{dm}\n\\end{array}\\end{equation}\n\\begin{equation}\n[a_1,\\varphi]=  \\,\\lambda \\,(1-\\C_{\\k_2}(\\varphi))\\quad,\\quad\n[ a_2,\\varphi]= \\,  \\lambda \\S_{\\k_2}(\\varphi)\\quad,\\quad\n[{a_1},{a_2}]=  \\,\\lambda  \\,  a_1\\quad.\\label{dn}\n\\end{equation}\nIn this way the results obtained in \\cite{BHOSe} (and \\cite{APb} for $\\k_2=1,\\ \na_1\\to -a_2,\\ a_2\\to a_1$) are recovered.\n\n\n\n\\sect{Conclusions}\n\nThe bicrossproduct structure holds\ntrue for all deformed Hopf algebras in the inhomogeneous CK family \n${\\cal U}_\\lambda (\\is_{\\k_2,\\dots,\\k_N}(N))$. \nThe theorem in Sec. 4 follows from the fact that \nthe action and the coaction mappings that characterise the\nbicrossproduct structure of ${\\cal U}_\\lambda(iso_{\\k_2\\dots\\k_N}(N))$ depend on \n$\\k_2,\\k_3,\\ldots \\k_N$ in such a way that the contractions are simply\ndescribed by setting (some of) them equal to zero. \nIn this sense we may say that\nthe bicrossproduct structure  (\\ref{cf})--(\\ref{cg}) is compatible with\nthe contractions  (\\ref{bh}).  As a result, any contracted\n$\\lambda$-algebra is also  a bicrossproduct for  the `contracted'\naction and coaction.  We can also state this result by saying that the\nbicrossproduct structure does not diverge under any of the \ncontractions within the CK affine family. \nThese results extend obviously to the dual \n$\\hbox{Fun}_\\lambda(ISO_{\\k_2,\\ldots,\\k_N}(N))$ family of deformed Hopf algebras.\n\nThe deformation parameter $\\lambda$ behaves in the same way under the \ncomplete family of CK contractions of the inhomogeneous algebras\n${\\cal U}_\\lambda(\\is_{\\k_2,\\dots,\\k_N} (N))$; if the (first, for instance) \ncontraction has been realized as a limit,\nthe deformation parameter $z$ in $q=e^z$ is redefined by \n$\\Gamma^{(m)}_\\pardeform(z)=z'={\\lambda\\over R}$ \n(cf. eq. (\\ref{bh})) and the limit corresponds to $[\\k_1]^{1\/2}=1\/ R\\to 0$.\nAs discussed, this can be understood as a mechanism assigning \ndimensions to the deformation parameter characterising the deformed \n${\\cal U}_\\lambda(iso_{\\k_2\\dots\\k_N}(N))$ CK family. \nIn any case, we wish to stress here that the dimensions of\n$\\lambda$ which\nresult from the {\\it first} contraction process (that involving $\\k_1$) \ndepend crucially on the assumed dependence of the dimensionless $q$ (or $z$) \non the new deformation parameter \n$\\lambda $ and of the contraction one.\nFor instance, if $q=\\exp(\\lambda \/R)$ as above, where $R$ is a radius \n(length), $[\\lambda ]=L^{1}$.\nAny other dimensions assigned to $\\lambda $ necessarily include hidden \nhypotheses on the dependence of $q$ on {\\it other} fundamental constants \n($\\hbar$ and $c$ are necessary, for instance, in order to have \n$[\\lambda ]$=\n(mass)$^{-1}$).\nThis is specially important because the appearance of Planck's $\\hbar$, \nfor instance, implies quantum considerations in the strict \n({\\it i.e.}, physical) sense of the word.\nThese go beyond the purely mathematical deformation process, and \nshould be accordingly discussed separately in any physical application of \na $\\lambda$-deformed algebra \n(see \\cite{AKRA} in connection with deformed Minkowski spaces).\n \nIt would be interesting to know whether the bicrossproduct structure is \npresent in other cases, {\\it i.e.}, for $\\k_1\\ne 0$ different from the  $\\k_1=0$ \ndeformations considered here.\nHowever, the general form of the deformation \n${\\cal U}_q(so_{\\k_1\\dots\\k_N}(N+1))$ of the\ngeneral CK algebra in the `physical' (rather than the Cartan-Weyl) basis \nis unknown, and this precludes us for the moment from discussing this point.\n\n\\section*{Acknowledgments}\n\nThis work has been partially supported by the Spanish \nCICYT, DGICYT and DGES (grants AEN96--1669, \nPB94--1115, PR95--439 and PB95--0719).\nJ.A. and J.C.P.B. wish to thank the kind hospitality extended to them at DAMTP.\nThe support of St. John's College (J.A.) and an FPI grant from the Spanish\nMinistry of Education and Science and the CSIC (J.C.P.B.) are also gratefully\nacknowledged.\n\n\\renewcommand{\\thesection}{Appendix:}\n\\setcounter{section}{0}\n\\section{Some bicrossproduct formulae}\n\\renewcommand{\\theequation}{A.\\arabic{equation}}\n\nWe reproduce here some of the\nformulae needed in the main text, and refer to \n\\cite{MajMajid} for details.\nIn these formulae ${\\cal A}\\equiv{\\cal U}_\\lambda (T_N)$ and \n${\\cal H}\\equiv{\\cal U}({so}_{\\k_2,\\dots,\\k_N} (N))$. The right action is \n$\\alpha:{\\cal A}\\otimes{\\cal H}\\to {\\cal A}\\,,\\, \\alpha(a\\otimes h)\\equiv \na\\triangleleft h$, and the left coaction $\\beta:{\\cal H}\\to {\\cal \nA}\\otimes{\\cal H}\\,,\\, \\beta(h)=h^{(1)}\\otimes h^{(2)}\\,,\\,\n(h^{(1)}\\in {\\cal A}\\,,\\,h^{(2)}\\in {\\cal H})$\nso that $({\\cal A},\\alpha)$ $[({\\cal H},\\beta)]$ is a right ${\\cal H}$-module\n[left ${\\cal A}$-comodule].\nThe compatibility conditions are\n\\begin{equation}\n\\varepsilon_{\\cal A}(a\\triangleleft h)=\\varepsilon_{\\cal A}(a)\\varepsilon_{\\cal H}(h)\n\\quad,\n\\label{apix}\n\\end{equation}\n\\begin{equation}\n\\Delta(a\\triangleleft h)\\equiv (a\\triangleleft h)_{(1)}\\otimes \n(a\\triangleleft h)_{(2)}\n=(a_{(1)}\\triangleleft  h_{(1)})h_{(2)}^{\\ (1)}\\otimes a_{(2)}\\triangleleft \nh_{(2)}^{\\ (2)}\\quad,\n\\label{apx}\n\\end{equation}\n\\begin{equation}\n\\beta(1_{\\cal H})\\equiv 1_{\\cal H}^{(1)}\\otimes 1_{\\cal H}^{(2)}=\n1_{\\cal A}\\otimes 1_{\\cal H}\\quad,\n\\label{apxi}\n\\end{equation}\n\\begin{equation}\n\\beta(hg)\\equiv (hg)^{(1)}\\otimes (hg)^{(2)}\n=(h^{(1)}\\triangleleft g_{(1)})g_{(2)}^{\\ (1)}\\otimes h^{(2)}g_{(2)}^{\\ (2)}\n\\quad,\\label{apxii}\n\\end{equation}\n\\begin{equation}\nh_{(1)}^{\\ (1)}(a\\triangleleft  h_{(2)})\\otimes h_{(1)}^{\\ (2)}=\n(a\\triangleleft  h_{(1)})\nh_{(2)}^{\\ (1)}\\otimes h_{(2)}^{\\ (2)}\n\\label{apxiii}\n\\end{equation}\n(subindices refer to coproduct as usual; {\\it superindices} refer to the \ncomponents of $\\beta(h)$).\nWhen they are satisfied the right-left bicrossproduct structure \n${\\cal H}^\\beta\\triangleright\\!\\!\\!\\blacktriangleleft_\\alpha {\\cal A}$\non ${\\cal K}\\equiv{\\cal H}\\otimes{\\cal A}$ is determined by\n\\begin{equation}\n(h\\otimes a)(g\\otimes b)=hg_{(1)}\\otimes (a\\triangleleft  g_{(2)})b\\quad,\n\\quad h,g\\in {\\cal H}\n\\; ; a,b\\in {\\cal A}\\quad,\n\\label{apxiv}\n\\end{equation}\n\\begin{equation}\n\\Delta_{\\cal K}(h\\otimes a)=h_{(1)}\\otimes h_{(2)}^{\\ (1)}a_{(1)}\n\\otimes h_{(2)}^{\\ (2)}\\otimes a_{(2)}\\quad,\n\\label{apxv}\n\\end{equation}\n\\begin{equation}\n\\varepsilon_{\\cal K}=\\varepsilon_{\\cal H}\\otimes\\varepsilon_{\\cal A}\\quad,\\quad\n 1_{\\cal K}=1_{\\cal H}\\otimes 1_A\\quad,\n\\label{apxvi}\n\\end{equation}\n\\begin{equation}\nS(h\\otimes a)=(1_{\\cal H}\\otimes S_{\\cal A}(h^{(1)}a))(S_H(h^{(2)})\\otimes \n1_{\\cal A})\n\\quad.\n\\label{apxvii}\n\\end{equation}\n\n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{A'.\\arabic{equation}}\n\nLet $A,H$ be the duals of ${\\cal A},{\\cal H}$.\nThe dual `group' aspect of the above formulae imply the existence of mappings \n$\\bar\\alpha:A\\otimes H\\to H\\,,\\,\\bar\\beta:A\\to A\\otimes H$, dual to \n$(\\beta,\\alpha)$ respectively, which satisfy the conditions\n\\begin{equation}\n\\varepsilon_{H}(a\\bar\\triangleright h)=\\varepsilon_{A}(a)\\varepsilon_{H}(h)\\quad,\n\\label{abpix}\n\\end{equation}\n\\begin{equation}\n\\Delta(a\\bar\\triangleright h)\\equiv (a\\bar\\triangleright h)_{(1)}\\otimes (a\\bar\\triangleright h)_{(2)}\n=(a_{(1)}^{\\ (1)}\\bar\\triangleright h_{(1)})\\otimes a_{(1)}^{\\ (2)}(a_{(2)}\n\\bar\\triangleright h_{(2)})\n\\quad,\n\\label{abpx}\n\\end{equation}\n\\begin{equation}\n\\bar{\\beta}(1_{A})\\equiv 1_{A}^{(1)}\\otimes 1_{A}^{(2)}=\n1_{A}\\otimes 1_{H}\\quad,\n\\label{abpxi}\n\\end{equation}\n\\begin{equation}\n\\bar{\\beta}(ab)\\equiv (ab)^{(1)}\\otimes (ab)^{(2)}\n=a_{(1)}^{\\ (1)}b^{(1)}\\otimes a_{(1)}^{\\ (2)}(a_{(2)}\\bar\\triangleright b^{(2)})\n\\quad,\n\\label{abpxii}\n\\end{equation}\n\\begin{equation}\na_{(2)}^{\\ (1)}\\otimes(a_{(1)}\\bar\\triangleright h)a_{(2)}^{\\ (2)}=\na_{(1)}^{\\ (1)}\\otimes a_{(1)}^{\\ (2)}(a_{(2)}\\bar\\triangleright h)\\quad.\n\\label{abpxiii}\n\\end{equation}\nThen, there is a left-right bicrossproduct structure \n$H_{\\bar\\alpha}\\blacktriangleright\\!\\!\\!\\triangleleft^{\\bar\\beta} A$ on $K=H\\otimes A$ defined by\n\\begin{equation}\n(h\\otimes a)(g\\otimes b)=h(a_{(1)}\\bar\\triangleright g)\\otimes \na_{(2)}b\\quad,\n\\quadh,g\\in {H}\\;;\\;a,b\\in {A}\\quad,\n\\label{abpxiv}\n\\end{equation}\n\\begin{equation}\n\\Delta_{K}(h\\otimes a)=h_{(1)}\\otimes a_{(1)}^{\\ (1)}\n\\otimes h_{(2)}\na_{(1)}^{\\ (2)}\\otimes a_{(2)}\\quad,\\label{abpxv}\n\\end{equation}\n\\begin{equation}\n\\varepsilon_{K}=\\varepsilon_{H}\\otimes\\varepsilon_{A}\\quad,\\quad\n 1_{K}=1_{H}\\otimes 1_{A}\\quad,\\label{abpxvi}\n\\end{equation}\n\\begin{equation}\nS(h\\otimes a)=(1_{H}\\otimes S_{A}(a^{(1)}))(S_{H}(ha^{(2)})\n\\otimes 1_{A})\n\\quad.\n\\label{abpxvii}\n\\end{equation}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nGlobal navigation satellite systems (GNSSs), particularly the global positioning system (GPS) of the United States of America, play an essential role in various aspects of our lives \\cite{Park2020800, Yoon20, Jeong2020958, Shamaei21, Kim2020796, Maaref20, Park2020824, Son20191828, Han2019, Rhee21:Enhanced, Schmidt20, Park2021919, Braasch19, Park2018387, Kim2019, Knoop17, Kang21:Indoor, Sun21:Markov, Lee22:Optimal}. Recently, as autonomous vehicles, such as unmanned aerial vehicles (UAVs), have emerged, the use of GNSS has increased further \\cite{Kim2020784, Sun2020889, Moon2019157, Causa21, Moon21, Lee20191187, Moon22:Fast, Lee2018:Simulation, Moon202013, Savas21, Moon2019258, Moon20181530, Park21:Indoor, Moon22:Speeding, Moon22:Sample}.\n\nHowever, in urban areas, it is difficult to achieve high positioning accuracy with the GNSS because the signals get reflected by buildings in many cases \\cite{Lee2020:Preliminary, Lee2020939, Jia21:Ground, Lee20202347, Kim22:First, Lee22:SFOL, Jeong21:Development, Kim21:GPS}. Using the reflected GNSS signals for positioning could result in a positioning error greater than 100 m \\cite{MacGougan02}. From the perspective of signal reception, the GNSS signal can be received in the following three cases: when only line-of-sight (LOS) signal is received (that is, \\textit{LOS-only condition}), only non-line-of-sight (NLOS) signal is received (that is, \\textit{NLOS-only condition}), and both LOS and NLOS signals are simultaneously received (that is, \\textit{LOS+NLOS condition}). \nAmong these conditions, the performance degrades only in the NLOS-only, and LOS+NLOS conditions. Therefore, various methods for detecting multipath signals (that is, NLOS-only or LOS+NLOS signals) and mitigating the corresponding GNSS errors are being actively studied \\cite{Massarweh20, Shen20, Kubo20}.\n\nThe traditional method of detecting multipath signals uses the carrier-to-noise-density ratio ($C\/N_0$) measurement of the GNSS signal \\cite{Groves10}. By defining a threshold for the $C\/N_0$ values, a signal with $C\/N_0$ greater than the threshold is classified as an LOS-only signal, and a signal with $C\/N_0$ lesser than the threshold is classified as an NLOS-only or LOS+NLOS signal. \nSimilarly, there are numerous software-based detection algorithms for GNSS measurements, such as the Doppler shift and code-minus-carrier values \\cite{Xu19, Lee20}. \nAdditionally, there are hardware-based detection methods that use a special antenna \\cite{Closas11}, a sky-pointing camera  \\cite{Bai20}, and a method that uses a three-dimensional (3D) model of a city and ray-tracing techniques \\cite{Kumar14}. Furthermore, GNSS multipath signal detection techniques using machine learning approaches have been emerging recently \\cite{Sun21, Suzuki20, Munin20}.\n\nVarious machine learning techniques have been used for classifying GNSS signal reception conditions, and their multipath detection performances were better than those of traditional methods \\cite{Guermah18, Sun19, Hsu17}. \nThe performance of a machine learning technique is greatly influenced by the features that are used, and the algorithms that are applied to it. Guermah \\textit{et al.} \\cite{Guermah18} proposed a method that used the $C\/N_0$ difference value obtained by a dual-polarized antenna, as a feature for the machine learning model. Sun \\textit{et al.} \\cite{Sun19} classified LOS-only, NLOS-only, and LOS+NLOS signals by applying the gradient boosting decision tree (GBDT) algorithm, and used $C\/N_0$ values, pseudorange residuals, and satellite elevation angles as features for machine learning. \n\nIn this study, we proposed a machine learning-based GPS multipath detection method that uses dual antennas.\nUsing dual antennas instead of a single antenna helps to obtain more diverse GPS measurements, and utilize them as features for machine learning.\nResultantly, it is expected that the performance of classifying signal reception conditions would be improved.\nUsing more than two antennas might be more effective, but it has disadvantage in terms of the size and cost.\n\nWe used four features (namely, $C\/N_0$, temporal difference of $C\/N_0$, difference between delta pseudorange and pseudorange rate, and satellite elevation angle) that could be obtained from a single antenna, and one feature (namely, double-difference pseudorange residual) that could be obtained from dual antennas.\nFour algorithms were applied as the machine learning algorithms in this study, and their performances were compared.\nThere has been a study on GNSS multipath detection using dual antennas \\cite{Chen21}. However, unlike our study, it was based on a statistical approach, and not a machine learning approach.\nCompared to statistical methods, machine learning methods can identify dependencies in data sets that cannot be modeled mathematically \\cite{Siemuri21}.\nWe classified signal reception conditions by a machine learning method that utilizes five features, whereas the existing method \\cite{Chen21} classified the conditions by a statistical method that uses a single metric.\n\n\\section{GPS Signal Collection and Labeling}\n\\subsection{GPS Signal Collection}\n Fig.~\\ref{fig:Hardware} shows the GPS signal acquisition hardware that was set up to collect the large amount of GPS data for machine learning. It consisted of two right-hand circularly polarized (RHCP) antennas produced by Antcom, two NovAtel PwrPak7 GNSS receivers, and a laptop for storing raw GPS data. The distance between the two antennas was maintained at 19 cm, which was equal to the wavelength of the GPS L1 signal. We collected GPS L1, L2, and L5 signals at intervals of 1 s, in a static environment.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.8\\linewidth]{figures\/Hardware.png}\n  \\caption{Hardware setup for GPS signal collection}\n  \\label{fig:Hardware}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=1.0\\linewidth]{figures\/Dataset.png}\n  \\caption{Training and test data set formulation based on the collected data at five locations}\n  \\label{fig:Dataset}\n\\end{figure}\n\nFig.~\\ref{fig:Dataset} illustrates the training and test data sets. The GPS signals were collected at five locations within Yonsei University, Incheon, Korea, which were labeled as A, B, C, D, and E, all having similar multipath environments.\nThe data collected at three of these locations (that is, A, B, and C) were combined into one data set so that the number of LOS-only, NLOS-only, and LOS+NLOS samples within the data set could be similar to each other.\nThen, the same number of LOS-only, NLOS-only, and LOS+NLOS samples were randomly selected from the combined data set to form the training data set, T0, and the test data set, T1.\nTable \\ref{table:LabelResult} lists the number of data samples included in each data set.\n\nThe data collected at the remaining two locations (that is, D and E) were used only for testing. T2 and T3 in Table \\ref{table:LabelResult} represent the data set collected at locations D and E, respectively.\n\n\\begin{table}\n\\centering\n\\caption{The number of data samples for each data set}\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabular}{l||cccc}\\hline\n GPS data set & T0 & T1 & T2 & T3\\\\\n \\hline\n Total samples & 7500 & 7500 & 6311 & 11575 \\\\\n NLOS-only samples & 2500 & 2500 & 2038 & 3608 \\\\\n LOS-only samples & 2500 & 2500 & 2195 & 4033 \\\\\n LOS+NLOS samples & 2500 & 2500 & 2078 & 3934 \\\\\n \\hline\n\\end{tabular}\n\\label{table:LabelResult}\n\\end{table}\n\n\\subsection{Data Labeling Based on Ray-Tracing Simulation}\n\nIt is essential to label the true signal reception conditions on the GPS data collected for machine learning. In this study, labeling was performed through ray-tracing simulation, using a commercial ray-tracing software called Wireless InSite \\cite{WirelessInsite}. \nThe ray-tracing simulator displays the propagation paths of the signal from the GPS satellite to the receiver, based on the 3D building information around the receiver.\nWe used a commercial 3D building map provided by 3dbuilings \\cite{3Dbuildings}, and calculated the locations of the GPS satellites based on almanac data.\n\n\n\\section{Feature Selection and Machine Learning Algorithms}\n\\subsection{Features to Classify the Signal Reception Conditions}\n\nThe features used in this study were as follows: $C\/N_0$, temporal difference of $C\/N_0$, difference between delta pseudorange and pseudorange rate, satellite elevation angle, and double-difference pseudorange residual.\n\n\\begin{itemize}\n\\item $C\/N_0$: $C\/N_0$ is defined as the carrier-to-noise density ratio. It is a parameter containing received signal strength information, and its unit is dB-Hz. Generally, since the propagation loss increases when a signal is reflected by an obstacle, the $C\/N_0$ value of the LOS-only signal is larger than that of the NLOS-only signal.\n\n\\item Temporal difference of $C\/N_0$: This parameter is denoted as $\\Delta C\/N_0$, which indicates the change in $C\/N_0$ between two consecutive epochs. In a static environment, $C\/N_0$ values of NLOS-only or LOS+NLOS signals can increase with respect to time, based on the characteristics of the receiver tracking loop \\cite{Hsu17}.\n\n\\item Difference between delta pseudorange and pseudorange rate: This parameter represents the consistency between the pseudorange measurement and the Doppler shift. It has been proposed as a feature in \\cite{Hsu17}, and can be calculated as follows:\n\\begin{equation}\n  \\Delta \\rho - \\dot \\rho \\cdot \\Delta t\n  \\label{eqn:PRRate}\n\\end{equation}\nwhere $\\Delta \\rho$ is the change in pseudorange between two epochs; $\\dot \\rho$ is the pseudorange rate, which is related to the Doppler shift; and $\\Delta t$ is the time difference between the two epochs.\n\n\\item Satellite elevation angle: Generally, the higher the elevation angle of the satellite, the lower is the probability that the signal is reflected or blocked by a building.\n\n\\item Double difference pseudorange residual: The above four features can be obtained from a single antenna, and they have been proposed in previous literature. However, dual antennas are necessary to obtain the double difference pseudorange residual, which was proposed as a feature for multipath detection for the first time in this study. \nThis feature can be calculated as follows: \n\\begin{equation}\n\\begin{split}\n  \\rho_{12,\\mathrm{residual}}^{kj}\n  & = \\rho_{12,\\mathrm{m}}^{kj} - \\rho_{12,\\mathrm{e}}^{kj} \\\\\n  & = \\{(\\rho_{1,\\mathrm{m}}^{k}-\\rho_{2,\\mathrm{m}}^{k}) - (\\rho_{1,\\mathrm{m}}^{j}-\\rho_{2,\\mathrm{m}}^{j})\\} \\\\\n  & \\, \\; \\quad - \\{(\\rho_{1,\\mathrm{e}}^{k}-\\rho_{2,\\mathrm{e}}^{k}) - (\\rho_{1,\\mathrm{e}}^{j}-\\rho_{2,\\mathrm{e}}^{j})\\}\n  \\label{eqn:DDPR}\n\\end{split}\n\\end{equation}\nwhere $\\rho_{12,\\mathrm{m}}^{kj}$ and $\\rho_{12,\\mathrm{e}}^{kj}$ are the measured and expected double difference pseudorange values between satellites $k$ and $j$, and receivers 1 and 2, respectively. The expected pseudorange is the distance between the satellite and the receiver, where the satellite and receiver positions are obtained based on the ephemeris data and the receiver position fix, respectively.\nSince this feature was calculated based on the signals from two satellites, the satellite with the highest elevation angle was selected as the reference satellite for the given epoch and the double difference pseudorange residual was calculated between each satellite and the reference satellite.\n\nIf the received signal was NLOS-only or LOS+NLOS, its propagation path from the satellite to each of the two antennas would be noticeably different, which caused the absolute value of the double difference pseudorange residual to increase. Therefore, this feature can be used for the classification of signal reception conditions. \n\\end{itemize}\n\n\\subsection{Machine Learning Algorithms}\nThe four machine learning algorithms used in this study were GBDT, random forest, decision tree, and K-nearest neighbor (KNN). We used the algorithms in a machine learning library called scikit-learn \\cite{Pedregosa11}, and performed hyperparameter tuning using the GridSearchCV function included in scikit-learn.\n\n\\section{Results}\nAs shown in Fig.~\\ref{fig:Dataset}, the machine learning model was trained using the data set T0, and the trained model was tested using the data sets T1, T2, and T3.\nFig.~\\ref{fig:Accuracy} shows the classification accuracy of each machine learning algorithm for the test data set T1, T2, and T3.\nWhen the trained model was tested using the data set T1, the classification accuracy was 82\\%\u201396\\%, which was significantly high.\nThis was because the training and test data sets, T0 and T1, respectively, consisted of data collected at the same locations (that is, locations A, B, and C).\nHowever, when the trained model was tested using the test data sets T2 and T3, which were collected at locations different from the training data locations (that is, locations D and E), classification accuracies of 46\\%\u201377\\%, and 44\\%\u201355\\% were obtained, respectively. The decision tree algorithm performed better than other algorithms in this case.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.8\\linewidth]{figures\/Accuracy.png}\n  \\caption{Classification accuracy of each algorithm for the three test data sets}\n  \\label{fig:Accuracy}\n\\end{figure}\n\nFigs.~\\ref{fig:AccT1}--\\ref{fig:AccT3} show the classification accuracy of each machine learning algorithm according to the signal reception conditions.\nFor the data set T1, as shown in Fig.~\\ref{fig:AccT1}, an accuracy of 70\\% or greater was achieved.\nHowever, for the data sets T2 and T3, as shown in Figs.~\\ref{fig:AccT2} and  \\ref{fig:AccT3}, respectively, the classification accuracy of the LOS+NLOS signals was only 18\\%\u201353\\%, which was significantly lower than that of the other cases.\nThis result implied that the training data set should include more LOS+NLOS data samples to achieve a better training result. This was understandable because the LOS+NLOS environment had greater diversity than the LOS-only or NLOS-only environments, and thus required more training.\n\n\\section{Conclusion}\n\nIn this study, we proposed a machine learning-based GPS multipath detection method that uses dual antennas.\nWe used five features for machine learning, comprising four features that could be obtained from a single antenna, and one feature that could be obtained from dual antennas.\nThe performances of four machine learning algorithms (that is, GBDT, random forest, decision tree, and KNN) using the proposed features were evaluated.\nThe classification accuracy for LOS+NLOS signals was found to be low. However, it is expected to improve if more LOS+NLOS data samples are used for the training, and this can be investigated in a future work.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.8\\linewidth]{figures\/AccT1.png}\n  \\caption{Classification accuracy of each algorithm according to the signal reception conditions in the test data set T1}\n  \\label{fig:AccT1}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.8\\linewidth]{figures\/AccT2.png}\n  \\caption{Classification accuracy of each algorithm according to the signal reception conditions in the test data set T2}\n  \\label{fig:AccT2}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.8\\linewidth]{figures\/AccT3.png}\n  \\caption{Classification accuracy of each algorithm according to the signal reception conditions in the test data set T3}\n  \\label{fig:AccT3}\n\\end{figure}\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOne of the present mysteries of modern cosmology is the recent accelerated expansion of the Universe predicted by the observations of supernovae Ia \\cite{riess} and confirmed by the observations of Cosmic Microwave Background Radiation \\cite{koma}, Large-scale Structure \\cite{aba}, Baryon Acoustic Oscillation \\cite{per} and Planck data \\cite{ade}. According to observations, the present epoch of evolution of the Universe is dominated by an exotic energy content dubbed as ``dark energy\" (DE), which has negative pressure required to explain the accelerated expansion. In the literature, a variety of DE candidates are available some of which are: cosmological constant \\cite{car}, quintessence \\cite{cald}, phantom \\cite{well}, chaplygin gas \\cite{kam} etc. The standard $\\Lambda$-cold dark matter ($\\Lambda$CDM) model containing the cosmological constant $\\Lambda$ is the most natural and successful model of DE. However, it has some shortcomings in the form of fine-tuning and cosmic coincidence problems. The other DE candidates are also not free from problems. For a review on DE and DE candidates, see \\cite{cope}.\\\\\n\\indent Recently, the holographic dark energy (HDE), which possesses some significant properties of the quantum theory, has been proposed as a candidate of DE to explain the recent phase transition of the Universe. The HDE is based on the holographic principle proposed by 't Hooft \\cite{hooft} and further discussed by Susskind \\cite{suss} in the context of string theory. The origin of the HDE contains the more scientific approach in comparison of other DE candidates  and presents a better way to deal with the accelerated expansion. Cohen et al. \\cite{cohen} have shown that the formation of black hole imposes an upper bound on the total energy of size L and should not exceed the mass of black hole of the same size. In the paper \\cite{li}, the author assumed the largest infra-red (IR) cut-off to saturate the inequality imposed by black hole formation and obtained the density of HDE $\\rho_{h}=3c^2 M_{p}^2 L^{-2}$, where $c^2$ is a dimensionless constant, $M_{p}$ stands for the reduced Planck mass and L denotes IR cut-off.\\\\\n\\indent The Hubble horizon is a natural candidate for IR cut-off which is also free from causality but Hsu \\cite{hsu} found that it gives wrong equation of state (EoS) of DE. Later on, it was shown \\cite{pavon} that if there is an interaction between two dark components of the Universe the identification of $L$ with Hubble horizon, $L=H^{-1}$, may give suitable EoS of DE. It also was shown that it necessarily implies a constant ratio of the energy densities of the two components regardless of the details of the interaction. Thus, the HDE models may also alleviate the cosmic coincidence problem which provides an advantage to HDE models over the other DE models.\\\\\n\\indent The Brans-Dicke (BD) theory proposed by Brans and Dicke in 1961 \\cite{brans} is a natural extension of general relativity (GR). In this theory, the gravitational constant $G$ is replaced with a scalar field $\\phi$ called BD scalar field which couples to gravity with coupling parameter $\\omega$. The inflationary epoch has been studied widely in this theory \\cite{math}. Recently, the BD theory has got interest to explain the accelerated expansion due to its association with the string theory and extra dimensional theories. This theory explains the recent accelerated expansion of the Universe and accommodates the observational data as well \\cite{berto}. The BD theory provides a dynamical framework which is more suitable to study the HDE models as HDE belongs to the family of dynamical DE candidates. Therefore, it is quite natural to study the HDE models in the framework of BD theory. The HDE models have been studied in the framework of BD theory to explain the recent accelerated expansion and to alleviate the problems associated with the DE models like cosmic coincidence problem \\cite{seta, shey, ban, lxu, sheykh, sheykhi}.\\\\\n\\indent In the literature, most of the models have been discussed in BD theory by assuming the power-law form of BD scalar field $\\phi \\propto a^n$, where $a$ is the scale factor and $n$ is a constant. It has been shown in the papers \\cite{berman} that the assumption $\\phi \\propto a^n$ naturally leads to a constant deceleration parameter (DP) in BD theory irrespective of the matter content of the Universe. In the recent papers \\cite{ban, lxu, sheykh, sheykhi}, authors have used the same power-law form of BD scalar field in HDE model and found a time-dependent DP. Thus, it has been observed that a constant as well as a time-dependent DP may be obtained with this power-law form of BD scalar field in HDE model. In our point of view there should not be two different values of DP with this form of BD scalar field. Therefore, it is natural to explore more options for the BD scalar field to over come from the shortcoming of this power-law form of BD scalar field. The very first purpose of this paper is to propose a suitable form of BD scalar field which gives only time-dependent DP to discuss the evolution of the Universe in the HDE models.\\\\\n\\indent  In this context, we propose a logarithmic form $\\phi \\propto ln(\\alpha+\\beta a)$ of BD scalar field, where $a$ as usual denotes the scale factor, and $\\alpha$ and $\\beta$ are positive constants such that $\\alpha>1$. Using this assumption, we explore the cosmological consequences of our model. This form of BD scalar field is free from the constant value of DP which naturally arises in the power-law form. We successfully obtain the equation of state (EoS) of DE and the time-dependent deceleration parameter which explain the recent phase transition of the Universe. Moreover, the unification of early time acceleration (inflation) and late time acceleration has been observed including matter dominated era. In the early and late time evolution we find that the density ratio of holographic dark energy to dark matter is a constant of order one ($r\\sim \\mathcal{O}(1)$) which successfully solve the cosmic coincidence problem. Therefore, we have successfully alleviated the problem associated with the power-law maintaining the good features of it. Further, the early time inflation has been explained in contrast to power-law form. The long lasting cosmic coincidence problem associated with DE models have been alleviated in a well manner in the present model in comparison to existing models.\\\\\n\\indent The outline of our work is as follows. Section II presents the field equations of HDE model in BD theory by assuming the interaction between DM and HDE. In section III, we propose a logarithmic form of BD scalar field and discuss its cosmological consequences. Section IV is devoted to the cosmic coincidence problem. The summary of the results is discussed in section V.\n \\section{Holographic dark energy in Brans-Dicke Theory}\n\\noindent The modified Einstein-Hilbert action for the BD theory is given by \\cite{sheykh}\n \\begin{equation}\n S=\\int d^{4}x\\sqrt{-g}\\left[\\frac{1}{2}(-\\phi R+\\frac{\\omega}{\\phi}g^{\\mu\\nu}\\partial_{\\mu}\\phi\\; \\partial_{\\nu}\\phi)+\\mathcal{L}_{m}\\right],\n\\end{equation}\n  where $R$ denotes the Ricci scalar curvature, $\\phi=(8\\pi G)^{-1}$ is a time-dependent scalar field called BD scalar field which couples with gravity, $\\omega$ is a coupling parameter between scalar field and gravity called BD parameter and $\\mathcal{L}_{m}$ represents the matter lagrangian density.\\\\\n\\indent We consider a homogeneous and isotropic flat Friedmann-Robertson-Walker (FRW) Universe given by the line element\n\\begin{equation}\nds^{2}=dt^{2}-a^{2}(t)(dx^{2}+dy^{2}+dz^{2}),\n\\end{equation}\nwhere $a$ denotes the cosmic scale factor of the Universe. We assume that the Universe is filled with perfect fluid containing pressureless dark matter (DM) excluding baryonic matter and HDE.\\\\\n\\indent The variation of the action (1) with respect to the metric tensor, $g_{\\mu\\nu}$ for the line element (2) with the energy-momentum tensor of dust and HDE yield the following field equations.\n\\begin{equation}\nH^2+H\\frac{\\dot\\phi}{\\phi}-\\frac{\\omega}{6}\\frac{\\dot\\phi\\;^2}{\\phi\\;^2}=\\frac{\\rho_{m}+\\rho_{h}}{3\\phi},\n\\end{equation}\n\\begin{equation}\n2\\frac{\\ddot a }{a}+H^2+2 H\\frac{\\dot\\phi}{\\phi}+\\frac{\\omega}{2}\\frac{\\dot\\phi\\;^2} {\\phi\\;^2}+\\frac{\\ddot\\phi}{\\phi}=-\\frac{p_{h}}{\\phi},\n\\end{equation}\nwhere $\\rho_{m}$ and $\\rho_{h}$ are, respectively the energy density of DM and HDE, and $p_{h}$ denotes the pressure of HDE. Here, the over dot denotes the derivative with respect to the cosmic time $t$. Many cosmologists \\cite{amen} have considered the interaction between DM and DE. The recent cosmological observations \\cite{oliv} also support this interaction. Considering the interaction factor $Q$ between DM and HDE, the conservation equations of DM and HDE are respectively given by \\cite{ban, sheykhi}:\n\\begin{equation}\n\\dot\\rho_{m}+3H\\rho_{m}=Q,\n\\end{equation}\n\\begin{equation}\n\\dot\\rho_{h}+3H(\\rho_{h}+p_{h})=-Q,\n\\end{equation}\nwhere $Q=\\Gamma \\rho_{h}$, $\\Gamma$ stands for interaction rate. The sign of interaction rate $\\Gamma$ is crucial and defines the direction of the energy transfer, i.e., for $\\Gamma>0$, there is an energy transfer from HDE to DM, and for $\\Gamma<0$, there is an energy transfer from DM to HDE. In the literature \\cite{shey,sheykhi}, $\\Gamma$ has been assumed to be proportional to the Hubble parameter, i.e., $\\Gamma \\propto H$ to maintain the interaction term $Q$ as a function of a quantity with units of inverse of time multiplied with the energy density. Therefore, let us consider $\\Gamma=3b^2 H$ so that the interaction term becomes $Q=3b^2 H \\rho_h$, where $b^2$ is a coupling constant. This assumption relies purely on dimension basis in the absence of a suitable theory.\\\\\n\\indent The dynamical equation for the BD scalar field $\\phi$ is given by\n\\begin{equation}\n\\ddot\\phi+3H\\dot\\phi=\\frac{\\rho_{m}+\\rho_{h}-3p_{h}}{2\\omega+3}.\n\\end{equation}\\\\\n\\indent Motivated by the holographic principle \\cite{hooft}, Cohen et al. \\cite{cohen} obtained an upper bound on the total energy of size $L$ imposed by black hole formation. Li \\cite{li} assumed the largest possible IR cut-off to saturate this inequality and obtained the density of HDE, $\\rho_{h}=3c^2 M_{p}^2L^{-2}$. In the framework of BD theory, the HDE density has the form $\\rho_{h} =3c^2 \\phi L^{-2}$, where $\\phi= M_{p}^2=(8\\pi G)^{-1}$ is a time dependent scalar field which couples to gravity with a coupling parameter $\\omega$. In the literature, there are various forms of HDE depending on the different choices of IR cut-off ($L$) like particle horizon, future event horizon, Hubble horizon, Granda-Oliveros cut-off etc. The Hubble horizon is the most natural and viable candidate of IR cut-off because it is free from causality problem. Here, in this paper we choose Hubble horizon as an IR cut-off which gives HDE density as:\n\\begin{equation}\n\\rho_{h}=3\\;c^2 \\phi H^2,\n\\end{equation}\nwhere $c^2$ is a dimensionless constant.\n\\section{Logarithmic form of BD scalar field and cosmological consequences}\n\\noindent  Many authors \\cite{berman} have assumed that the BD scalar field $\\phi$ evolves as a power-law of the scale factor $a$, i.e., $\\phi\\propto a^n$. They have observed that this assumption leads to a constant value of DP. The constant value can be obtained irrespective of matter content of the Universe. However, some authors \\cite{ban,lxu,sheykh,sheykhi} have studied HDE model in BD theory with the same form of BD scalar field and have obtained a time-dependent DP. Now, the question is that why does the same form of BD scalar field lead to two different values of DP, constant and time-dependent in a same model?\\\\\n \\indent In spite of several advantages of this form of BD scalar field, it seems from the above mentioned works that this may not be a suitable assumption to discuss the evolution of the Universe in HDE models. In other words, this form may not be suitable for those models where we want to study the phase transition of the Universe. Taking into consideration to this problem, we hereby propose that the BD scalar field evolves as a logarithmic function of the scale factor which is given by\n \\begin{equation}\n \\phi \\propto ln(\\alpha+\\beta a),\n \\end{equation}\n \\noindent where $\\alpha>1$ and $\\beta>0$ are constants.\\\\\n\\indent In principle, the BD scalar field should evolve slowly to observe a slow variation of $G$. This logarithmic form of $\\phi$ fulfills this requirement. In this process, the value of $\\beta$ also plays an important role. It is worth noting that GR can be recovered for $\\beta=0$. One can observe that this form does not give a constant value of deceleration parameter when we combine Eqs. (3) and (4) with (7). Thus, we have resolved the constant value problem of the power-law form. It provides an initial advantage to the logarithmic form over the power-law form of BD scalar field. Therefore, it will be interesting to investigate the role of this form in the evolution of the Universe in the framework of BD theory within the formalism of interacting HDE model.\\\\\nFrom (6), we have\n \\begin{equation}\n\\dot\\rho_{h}+3H(1+w_{h})\\rho_{h}=-3b^2 H \\rho_{h},\n\\end{equation}\n where $w_{h}=p_{h}\/\\rho_{h}$ is the EoS parameter of the HDE. Using (8) and (9) into (10), we get\n\\begin{equation}\n\\frac{\\beta a}{(\\alpha+\\beta a)ln(\\alpha+\\beta a)}+2\\frac{\\dot H}{H^2}+3(1+w_{h})=-3b^2.\n\\end{equation}\nFrom (3) and (4), we obtain\n\\begin{widetext}\n\\begin{eqnarray}\n\\frac{\\dot H}{H^2}&=&\\frac{1}{2+\\frac{\\beta a}{(\\alpha+\\beta a)\\;ln(\\alpha+\\beta a)}}\\Big[-3-\\frac{\\omega \\beta^2 a^2}{2(\\alpha+\\beta a)^2\\;[ln(\\alpha+\\beta a)]^2}-\\frac{3\\beta a}{(\\alpha+\\beta a)\\;ln(\\alpha+\\beta a)}\\\\ \\nonumber\n&+&\\frac{\\beta^2a^2}{(\\alpha+\\beta a)^2\\; ln(\\alpha+\\beta a)}-\\frac{3w_{h}}{1+r}\\Big\\{1+\\frac{\\beta a}{(\\alpha+\\beta a)\\;ln(\\alpha+\\beta a)}-\\frac{\\omega \\beta^2 a^2}{6(\\alpha+\\beta a)^2\\;[ln(\\alpha+\\beta a)]^2}\\Big\\}\\Big],\n\\end{eqnarray}\n\\end{widetext}\nwhere $r=\\rho_{m}\/\\rho_{h}$ represents the energy density ratio. Using (12) into (11), we get\n\\begin{widetext}\n\\begin{equation}\nw_{h}=\\frac{(r+1)\\left[-3b^2\\left(2+\\frac{\\beta a}{(\\alpha+\\beta a)\\;ln(\\alpha+\\beta a)}\\right)-\\frac{2\\beta^2a^2}{(\\alpha+\\beta a)^2\\; ln(\\alpha+\\beta a)}+\\frac{\\beta a}{(\\alpha+\\beta a)\\;ln(\\alpha+\\beta a)}+\\frac{(\\omega-1)\\beta^2a^2}{(\\alpha+\\beta a)^2\\; [ln(\\alpha+\\beta a)]^2}\\right]}{3\\big[2r+\\frac{(r-1)\\beta a}{(\\alpha+\\beta a)\\;ln(\\alpha+\\beta a)}+\\frac{\\omega \\beta^2a^2}{3(\\alpha+\\beta a)^2\\; [ln(\\alpha+\\beta a)]^2}\\big]}.\n\\end{equation}\n\\end{widetext}\nTo discuss the behavior of $w_h$, let us first discuss the terms present in (13) which have significant contribution in the behavior of $w_h$. We observe that the terms $\\frac{\\beta a}{(\\alpha+\\beta a)\\;ln(\\alpha+\\beta a)}$ and $\\frac{\\beta^2a^2}{(\\alpha+\\beta a)^2\\;ln(\\alpha+\\beta a)}$ are zero at $a=0$ and both the terms converse to zero as $a\\rightarrow \\infty$. We find that both the terms start from zero, achieve maximum value during the evolution and then converge to zero in the late time evolution. Further, we observe that both the terms attain the maximum value asymptotically for sufficiently small values of $\\alpha$ and sufficiently large values of $\\beta$. The maximum value of the term $\\frac{\\beta a}{(\\alpha+\\beta a)\\;ln(\\alpha+\\beta a)}$ lies in the interval ]0, 1[ whereas the maximum value of the term $\\frac{\\beta^2a^2}{(\\alpha+\\beta a)^2\\;ln(\\alpha+\\beta a)}$ lies in the interval ]0, 0.41[ depending on the value of $\\alpha$ only. It is to be noted that the maximum value does not depend on the value of $\\beta$ in either term. \\\\\n\\indent The values of $\\omega$ and $b^2$ also play an important role in the value of $w_{h}$. The value of $b^2$ is expected to be a small positive constant. The value of BD parameter $\\omega$ has been constrained by various astronomical and cosmological observations. The solar system experiment Cassini gave a very stringent high bound $\\omega>40000$ \\cite{bert} for spherically symmetric solution in the parameterized post Newtonian formalism. The solar system constraints on $\\omega$ may not be consistent at the cosmological scales, therefore, the cosmological constraints are required to study the large scale properties of the Universe. In this context, Acquaviva et al. \\cite{aqua} have found $\\omega>120$ at 95\\% confidence level, Wu and Chen \\cite{wu} have found $\\omega <-120$ or $\\omega>97.8$ at 95.5\\% confidence level whereas Li et al. \\cite{Li} have obtained $\\omega>181.65$ at 95\\% confidence level. However, all these constraints on $\\omega$ depend on the choice of a model. In the present paper, we will consider only the value $\\omega>0$.\\\\\n\\indent Now, let us discuss the behavior of EoS parameter $w_{h}$ of HDE as obtained in Eq. (13). The sign of $w_{h}$ depends on the sign of numerator as the denominator has only positive values. In the beginning of the evolution ($a=0$), all the terms of the numerator are zero except the first term which has negative sign. Therefore, we observe a negative value of $w_h$ as\n\\begin{equation}\nw_h= -b^2\\left(1+\\frac{1}{r}\\right).\n\\end{equation}\nThus, HDE has EoS of DE in the beginning which is required to explain the inflation supposed to happen in very early time of the evolution. As the Universe evolves, the terms $\\frac{\\beta a}{(\\alpha+\\beta a)\\;ln(\\alpha+\\beta a)}$ and $\\frac{\\beta^2a^2}{(\\alpha+\\beta a)^2\\;ln(\\alpha+\\beta a)}$ attain their maximum value, therefore, the last term of numerator containing $\\omega$ starts to dominate as a large value of $\\omega$ has been expected from the observations. Thus, we may observe a change in the sign of EoS parameter $w_h$ from negative to positive and HDE may start behaving like radiation ($w_{h}>0$). It is evident from (13) that the first term of the numerator has a constant part, and the terms $\\frac{\\beta a}{(\\alpha+\\beta a)\\;ln(\\alpha+\\beta a)}$ and $\\frac{\\beta^2a^2}{(\\alpha+\\beta a)^2\\;ln(\\alpha+\\beta a)}$ converse to zero in the late time evolution. Therefore, ultimately the sign of $w_h$ changes from positive to negative. Thus, HDE achieves the EoS of DE again during the evolution which is required to explain the late time acceleration of the Universe. It is also obvious that HDE achieves dust like behavior ($w_{h}=0$) whenever $w_h$ changes sign from positive to negative or viceversa. Zimdahl and Pav\\'{o}n \\cite{zim} for HDE, and Zlatev et al. \\cite{zla} for tracker quintessence scalar field have also observed radiation and dust like behavior in the early time evolution.\\\\\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=8.5cm, height=6cm]{PRD4.eps}\\\\\n  \\caption{ The evolution of $w_h$ verses $a$ for different values of $\\alpha$, $\\beta$ and $\\omega$ with $b=1.05$ and $c=0.77$.}\\label{1}\n\\end{figure}\n  \\begin{figure}\n  \\centering\n  \\includegraphics[width=8.5cm, height=6cm]{PRD6.eps}\n  \\caption{The evolution of $w_h$ verses $a$ for different values of $b$, $c$ and $\\omega$ with $\\alpha=1.15$ and $\\beta=5$.}\\label{2}\n\\end{figure}\n\\indent We also emphasise that for a small value of $\\omega$ or a large value of $b^2$ or a suitable combination of their values, the EoS parameter $w_h$ is negative throughout the evolution. Thus, the HDE behaves like DE throughout the evolution. As it is mentioned that the maximum value of the terms $\\frac{\\beta a}{(\\alpha+\\beta a)\\;ln(\\alpha+\\beta a)}$ and $\\frac{\\beta^2a^2}{(\\alpha+\\beta a)^2\\;ln(\\alpha+\\beta a)}$ depend on the value of $\\alpha$, the change in sign of $w_h$  also depends on the value of $\\alpha$. The possible behaviors of $w_h$ are shown in Figs. 1 and 2 for different values of parameters which show the similar behaviors as discussed above. We have used value of $r$ from (21) and have assumed the present value of scale factor $a_0=1$ here and thereafter for our convenience.\\\\\n \\indent It is worthy to note that if we take constant value of $\\Gamma$ as taken in \\cite{pavon}, the Eq. (13) in the late time evolution ($a\\rightarrow \\infty$) reduces to its respective expression in GR for HDE model, that is,\n\\begin{equation}\nw_{h}\\approx-\\left(1+\\frac{1}{r}\\right)\\frac{\\Gamma}{3H}.\n\\end{equation}\nNow, for time-dependent $\\Gamma=3b^2 H$, we get the expression of $w_h$ for late time evolution as\n\\begin{equation}\nw_{h}\\approx-b^2\\left(1+\\frac{1}{r}\\right).\n\\end{equation}\n  One can observe from (16) that the EoS parameter converses to a negative constant value in the late time evolution. It means that as the Universe enters in the late time accelerating phase, the HDE behave like DE forever. In the absence of interaction ($b^2=0$), we obtain a positive EoS parameter of HDE ($w_h>0$) from (13) through out the evolution. It means accelerated expansion is not possible in this case as observed in the papers \\cite{pavon,lxu}. It is also observed that $w_{h}$ may cross the phantom divide ($w_{h}=-1$) for a suitable value of $b^2$ in the late time evolution.\\\\\n\\indent In the present model, the dynamics of the Universe depends not only on DM and HDE, but also on the BD scalar field. It will be early to conclude about the evolution of the Universe only on the basis of EoS of HDE. Therefore, it is important to discuss the behavior of deceleration parameter, $q=-\\frac{a\\ddot{a}}{\\dot{a}^2}$ to make a precise conclusion. We obtain the value of $q$ after dividing Eq. (4) by $H^2$, and using (8) and (9) which is given as\n\\begin{widetext}\n\\begin{equation}\nq=\\frac{3c^2 w_{h}+1-\\frac{\\beta^2a^2}{(\\alpha+\\beta a)^2\\;ln(\\alpha+\\beta a)}+\\frac{2\\beta a}{(\\alpha+\\beta a)\\;ln(\\alpha+\\beta a)}+\\frac{\\omega \\beta^2 a^2}{2(\\alpha+\\beta a)^2\\;[ln(\\alpha+\\beta a)]^2}}{2+\\frac{\\beta a}{(\\alpha+\\beta a)\\;ln(\\alpha+\\beta a)}}\n\\end{equation}\n\\end{widetext}\nLet us examine the sign of deceleration parameter to discuss the early and late time evolution. In the beginning of evolution at $a=0$, we obtain $q=\\frac{3c^2 w_h+1}{2}$. Using the initial value of $w_h$ given by Eq. (14), we get $q=\\frac{1}{2}\\left[1-3b^2c^2\\left(1+\\frac{1}{r}\\right)\\right]$, which produces accelerated expansion for $3b^2c^2\\left(1+\\frac{1}{r}\\right)>1$. Thus, we observe inflationary era which has been expected to happen in the beginning of the evolution to resolve the problems of Big-Bang cosmology. As $w_{h}$ may show sign change from negative to positive during the evolution and the last term containing $\\omega$ in numerator of Eq. (17) starts to dominate due to large value of $\\omega$, a change in the sign of deceleration parameter from negative to positive may also be observed. If $w_h$ is negative throughout the evolution then also we observe sign change in $q$ as shown in Figs. 3 and 4. Thus, we observe a decelerated expansion of the Universe after inflation.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=8.5cm, height=6cm]{PRD1.eps}\\\\\n  \\caption{The evolution of $q$ verses $a$ for different values of $\\alpha$ and $\\beta$ with $b=1.05$ and $c=0.77$.}\\label{3}\n\\end{figure}\nSince the terms $\\frac{\\beta a}{(\\alpha+\\beta a)\\;ln(\\alpha+\\beta a)}$ and $\\frac{\\beta^2a^2}{(\\alpha+\\beta a)^2\\;ln(\\alpha+\\beta a)}$ converse to zero in the late time evolution, the late time value of $q$ is obtained as\n \\begin{equation}\n q\\approx\\frac{3c^2 w_h+1}{2},\n \\end{equation}\n Using the late time value of $w_h$ given by (16) into (18), we get\n \\begin{equation}\n q\\approx\\frac{1}{2}\\left[1-3b^2c^2\\left(1+\\frac{1}{r}\\right)\\right].\n \\end{equation}\n \\begin{figure}\n  \\centering\n  \\includegraphics[width=8.5cm, height=6cm]{PRD5.eps}\\\\\n  \\caption{The evolution of $q$ verses $a$ for different values of $b$ and $c$ with $\\alpha=1.15$ and $\\beta=5$.}\\label{4}\n\\end{figure}\n  Here, $q$ will be negative if $3b^2c^2\\left(1+\\frac{1}{r}\\right)>1$, the same as for early time. Therefore, a sign change of $q$ from positive to negative may be observed which successfully explains the late time phase transition (deceleration to acceleration) of the Universe. Indeed, HDE shows almost same behavior in the early and late time evolution of the Universe. If there is no interaction ($b^2=0$), we observe $q>0$ through out the evolution. Thus, using the value of $q$ we reconfirm that the accelerated expansion is not possible in the case of no interaction. It can be seen that it is not only for $b^2=0$, but also for a small value of $b^2<<1$, a decelerating Universe may be observed throughout the evolution. The other parameters of the model will not change this scenario. Further, we observe that a sufficiently large value of $b^2$ is able to accelerate the expansion through out the evolution. Therefore, the value of coupling parameter $b^2$ plays an important role to observe the evolution of the Universe which is consistent with the observations. It means that a suitable interaction between the dark components of the Universe, i.e., a suitable value of $b^2$ is required to be consistent with the observations.\\\\\n   \\indent After having a suitable interaction, the most important parameters are $\\alpha$ and $\\beta$, which are due to the logarithmic form of BD scalar field. The terms containing $\\alpha$ and $\\beta$ in Eq. (17) start from zero, evolve up to maximum value and converse to zero in the late time which lead to observe two phase transitions of the Universe, i.e., the early time inflation and the late time acceleration. It is worthy to mention here that for the power-law form of BD scalar field, only a late time acceleration is possible. The values of $\\alpha$ and $\\beta$ are important to decide the time when the phase transitions, inflation and late time acceleration, will happen which can be seen in Figs. 3 and 4. Thus, to accommodate the observed history of the Universe the value of $\\alpha$ and $\\beta$ are significant. It is also evident from (17) that for a large value of $\\omega$ the inflation ends early and the late time acceleration occurs late, and viceversa. So far we have observed that a number of combination of the parameters are possible to explain the whole history of the Universe but let observations to decide the best combination of the parameters.\\\\\n  \\indent We have shown that the present model of dark energy is able to explain the whole known history of the Universe starting from inflation to recent accelerated expansion of the Universe including the matter dominated era (radiation+dust). Thus, this model presents a unified model of interacting HDE in the framework of BD theory with logarithmic form of BD scalar field. We emphasise that the logarithmic form proposed in this paper has played an important role to explain the evolution of the Universe in more better way. In the next section, we will show that this form also plays a significant role to resolve the coincidence problem.\n  \\section{Cosmic Coincidence Problem}\n\\noindent Let us investigate the present model on the ground of cosmic coincidence problem \\cite{zla,stein}. Using the conservation Eqs. (5) and (6), the evolution of energy density ratio $r$ can be obtained as\n\\begin {equation}\n\\dot r=3Hr\\left[w_{h}+\\left(\\frac{1+r}{r}\\right)b^2\\right],\n\\end{equation}\nwhich is the same expression as obtained in \\cite{pavon,ban} except we have $b^2$ at the place of $\\Gamma\/3H$ because they have chosen $\\Gamma$ as a constant whereas we have taken $\\Gamma=3b^2 H$. The nature of evolution of $r$ in our model is different from the mentioned works as we have taken time-dependent $\\Gamma$ and it also depends on the value of $w_{h}$. Using the late time value of $w_{h}$ given by (16) into (20) we obtain $\\dot r=0$, i.e., in the late time evolution $r$ has a constant value which is a significant feature to solve the coincidence problem. Pav\\'{o}n and Zimdahl \\cite{pavon} have obtained a constant value of $r$ in GR, however, $r$ is expected to be a time-varying value. Therefore, the authors assumed a time-varying value of $c$ to obtain a time-varying $r$. It has also been suggested to replace the Hubble horizon by the future event horizon to achieve a time-varying value of $r$ \\cite{wang}. Although Banerjee and Pav\\'{o}n \\cite{ban} have obtained time-dependent value of $r$ in Brans-Dicke theory but they achieved a soft coincidence only. They argued that $r$ can vary more slowly in their model than in the conventional $\\Lambda$CDM model. It have been demonstrated by Zhang et al. \\cite{zhang} that the interacting chaplygin gas model has a better chance to solve the coincidence problem in comparison of interacting quintessence and interacting phanton models. In the literature, many proposals have been made to solve the coincidence problem \\cite{campo,velten} but the problem still exists.\\\\\n\\indent Let us find the value of $r$ for our model to analyse coincidence problem in more detail. From (3), the value of $r$ is obtained as\n\\begin{widetext}\n\\begin {equation}\nr=-1+\\frac{1}{c^2}+\\frac{\\beta a}{c^2(\\alpha+\\beta a)\\;ln(\\alpha+\\beta a)}-\\frac{\\omega\\beta^2 a^2}{6c^2(\\alpha+\\beta a)^2\\;[ln(\\alpha+\\beta a)]^2}.\n\\end{equation}\n\\end{widetext}\nHere, we obtain a time-dependent value of $r$ which has a constant and finite value $r=-1+\\frac{1}{c^2}$ at the beginning of the evolution as the last two terms are zero at $a=0$. In the late time evolution, we also obtain a constant and finite value $r\\approx-1+\\frac{1}{c^2}$ as the last two terms converge to zero. The same constant value of $r$ has been obtained through out the evolution in GR \\cite{pavon} but we have obtained a time-dependent value of $r$ which has constant values in the early and late time evolution. Here, it will be interesting to quote a paper of Campo et al. \\cite{campo} ``Obviously, a mechanism that makes $r$ tends to a constant today or decrease its rate to a lower value than the scale factor expansion rate ameliorates the coincidence problem significantly, but it does not solve it in full. To do so the said mechanism must also achieve $r_{0}\\sim \\mathcal{O}(1)$\". We have obtained $r_{0}\\sim -1+\\frac{1}{c^2}$ which excellently satisfies the requirement $r_{0}\\sim \\mathcal{O}(1)$ as most of the observational constraints on $c$ obtain $0.5<c<1$ \\cite{Xu,mli,lixin}. In the other words, the cosmic coincidence problem has been resolve completely. In fact, we observe $r\\sim \\mathcal{O}(1)$ in the early time as well as in the late time evolution in our model. Therefore, the cosmic coincidence problem does not seem a problem because it is not a coincidence that we are living in a time where $r\\sim \\mathcal{O}(1)$, it have been observed in the early time also. Thus, in conclusion we can say that the cosmic coincidence problem has been resolved in a significantly well manner in the present model.\\\\\n\\indent From (16) and (20), it is clear that whatever the value of $b^2$ be chosen, we obtain $\\dot r=0$ in the late time evolution. It means the coupling parameter $b^2$ between DM and HDE does not play a significant role in the alleviation of coincidence problem. This is also evident from the value of $r$ given by (21) which has no $b^2$ term. Also, one may observe that the parameter $c$ is not significant to resolve the coincidence problem. Although, it plays an important in the value of $r$ at any given time. Actually, the last two terms in Eq. (21) which come due to the assumption of logarithmic form of BD scalar field,  play an important role to decide the coincidence problem.  These two terms converse to zero in the late time and may vary sufficiently slow at present due to which $r$ converses to a constant value in the late time and vary sufficiently slow at present. The suitable values of the parameters $\\alpha$ and $\\beta$  ensure the slow variation of $r$ at present time. The variation of $r$ is shown in the Fig. (5) which clearly verify our claim.\\\\\n \\begin{figure}\n  \\centering\n  \\includegraphics[width=8.5cm, height=6cm]{PRD2.eps}\\\\\n  \\caption{ The graph of $r$ verses $a$ for different values of $\\alpha$, $\\beta$ and $c$ with $\\omega=12$.}\\label{3}\n\\end{figure}\n \\indent Let us find the value of $c$ to match with the observational values using the observed value of $r$. According to recent observations, the present value of $r$ is $\\approx$ $\\frac{3}{7}$, where matter has $\\approx$ $30\\%$ and DE has $\\approx$ $70\\%$ of total energy content of the Universe. We obtain the value $c\\approx 0.8367$ using the theoretical relation $r\\approx -1+\\frac{1}{c^2}$ from (21) for late time evolution. Xu at al. \\cite{Xu} have obtained similar value $c=0.807^{+0.165}_{-0.160}$ for the HDE model in the framework of Brans-Dicke theory. Using the observations from various probes like WMAP, BAO, Planck etc. Li et al. \\cite{mli} have obtained values of $c$ which also show similarity with the above values, however, all these models are different.\\\\\n\\section{Conclusion}\n\\indent In this paper, we have studied interacting HDE model with Hubble horizon as an IR cut-off in the framework of BD theory. We have pointed out a serious problem associated with the assumption of power-law form ($\\phi\\propto a^n$) of the BD scalar field. It has been observed that this form of BD scalar field gives constant and time-dependent deceleration parameter for same model. Therefore, taking into consideration of this problem, we have proposed a logarithmic form of BD scalar field which always gives a time-varying deceleration parameter. We have observed that this form of BD scalar field is consistent with the cosmic evolution of the Universe. We have obtained EoS parameter and deceleration parameter to discuss the early and late time evolution of the Universe. We have also discussed the cosmic coincident problem. A summary of the main findings is as follows.\\\\\n\\indent In the early time evolution, we observe $w_h<0$ and $q<0$ which explains the inflationary era of the cosmic evolution supposed to happened in the very early Universe to resolve the problems of Big-Bang cosmology. Further, the sign change of $q$ from negative to positive has been observed. Thus, we observe the matter dominated era of the evolution. In the late time evolution, we have observed another sign change of $q$ from positive to negative which explain the late time acceleration of the Universe. Thus, this model presents a unified model of interacting HDE in the framework of BD theory with logarithmic form of BD scalar field. If we consider $\\Gamma$ as a constant then the EoS parameter $w_h$ reduces to the corresponding form of GR in the late time. The EoS parameter converges to a constant value and for a suitable value of $b^2$ it may cross the phantom-divide line $w_h=-1$ in the late time evolution. If there is no interaction between HDE and DM, i.e., $b^2=0$, it has been observed that the EoS parameter $w_h$ and deceleration parameter $q$ are always positive, therefore, the accelerated expansion is not possible in this case.\\\\\n\\indent We have also discussed the cosmic coincidence problem, a long standing problem with DE models, which has been resolved effectively in this paper. We have observed a time-varying energy density ratio $r$ which has a constant value in early and late time evolution of the Universe. The present value $r_{0}$ satisfies the condition $r_{0}\\sim \\mathcal{O}(1)$ which is consistent with the observations. Using the present value of $r$ obtained by observations, we get theoretical value $c=0.8367$ which shows similarity with the observed values of $c$.\\\\\n\\indent In conclusion, the advantage of our model is that we have alleviated the problem associated with the power-law containing the good features of it. In addition, our model presents a unification of early time inflation and late time acceleration including the matter dominated era which have not been obtained through power-law form. In the present model, the cosmic coincidence problem has been alleviated more effectively in comparison of existing models.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nNeutron stars (NSs), one of the most compact objects in the Universe, provide a suitable environment to study matter at densities ranging from subsaturation to few times nuclear saturation densities \\citep{1996cost.book.....G, Sedrakian2007PrPNP, weber2017pulsars}.\nThe well-known strategy to comprehend dense matter behaviour above nuclear saturation density ($n_0$) is understanding compact astrophysical objects. Constraints from observation of compact stellar objects help the understanding of dense matter above $n_0$. One important constraint is the lower limit of maximum stellar mass of compact object family obtained from several recent observations. On the high density regime, the primary constraint on equation of state (EOS) comes from the observations of massive compact stars with masses near $2 M_{\\odot}$. Observations of massive compact objects constrain the EOS to be stiff at a higher density limit. In addition, the recent NICER (Neutron star Interior Composition ExploreR) observations provide the mass and radius measurements of two compact objects (PSR J$0030+0451$ and PSR J$0740+6620$), which further constrain the EOS.\n  \nMassive compact star observations eradicate the softer EOSs failing to support \nthe observational lower bound on maximum mass stars. \nOn the other hand, the recent detection of gravitational wave (GW) emissions from binary neutron star (BNS) inspiral of the event GW170817 \\citep{LIGO_Virgo2017c, LIGO_Virgo2017a, LIGO_Virgo2017b} and GW190425 \\citep{2020ApJ...892L...3A} by LIGO-Virgo Collaboration (LVC) imposes bounds (although weakly) on intermediate densities of dense matter EOS.\nIn the BNS merger scenario, the response of a compact star to a strong gravitational field exerted by its companion is related to the EOS via induced tidal deformations during the inspiral phase. Recent observations of GW from binary star coalescence directly provide the estimate of combined dimensionless tidal deformability ($\\tilde{\\Lambda}$) of participating stars indicating matter to be soft at a lower density regime. \n\nDue to insufficient knowledge and restrictions about matter properties at density regimes higher than $n_0$, it is tough and challenging to predict the behaviour of dense matter. Several studies have been done to comprehend the NS matter composition in phenomenological \\citep{1972PhRvC...5..626V, 2001A&A...380..151D,2014PhRvC..89d5807B} as well as microscopic approaches \\citep{1998PhRvC..58.1804A, 2015RvMP...87.1067C, 2019PhRvC.100d5803L}. Phenomenological models such as the relativistic mean-field (RMF) model \\citep{1974AnPhy..83..491W, 1977NuPhA.292..413B, 1997IJMPE...6..515S} based on effective interactions between baryons via meson exchange can conveniently describe various finite nuclei properties and can be extrapolated from subsaturation to higher density regimes \\citep{2016PhR...621..127L}. Within this framework, there are a good number of admissible phenomenological models, each with a good number of parameter value sets. With recent observational constraints related to compact stars it is possible to narrow down the models of dense matter EOS. \n\nObservations of massive NSs open the likelihood of exotic matter such as hyperons \\citep{PhysRevLett.67.2414, Bonanno2012A&A, Weissenborn2012a, Colucci_PRC_2013, Oertel2015,2016PhRvC..94c5804F, Tolos2017b,  2018MNRAS.475.4347R,2018EPJA...54..133L}, $\\Delta$-resonances \\citep{Drago_PRC_2014, Cai_PRC_2015, Zhu_PRC_2016, Sahoo_PRC_2018, Kolomeitsev_NPA_2017, Li_PLB_2018, Li2019ApJ, Ribes_2019, Li2020PhRvD, particles3040043} and meson ($\\pi,\\bar{K},\\rho$-mesons) condensations \\citep{particles2030025, 1982ApJ...258..306H, 1999PhRvC..60b5803G, 2001PhRvC..63c5802B, 1997PhR...280....1P, 2021EPJST.tmp...32M, PhysRevD.102.123007, PhysRevD.103.063004}) in the NS interior details of which are still not completely understood. Although hyperonization in NSs may seem to be inescapable, it leads to softening of the EOS. Following similar energetic reasoning employed for nucleation of strange baryons, another interesting aspect is the onset of non-strange $\\Delta$-baryons in dense NS matter. The role of $\\Delta$-resonances in NS matter is still a matter of debate \\citep{Li_PLB_2018,2020PhLB..80235266M}. Inclusion of $\\Delta$-baryons softens the EOS at lower matter density regimes leading to higher compactness at those densities, consequently satisfying the GW170817 event observables \\citep{Li2019ApJ}. Similar to other exotic degrees of freedom, the inclusion of meson condensates also softens the EOS extensively. For recent reviews regarding the composition of compact stars, the reader may refer to \\citet{2017PhR...681....1Y, 2016EPJA...52...29C, Sedrakian2017_EPJWeb, Baym_2018} and \\citet{2021arXiv210514050S}.\n\nIn this work, we employ the available constraints obtained so far from compact object observations to narrow down the high-density matter models.\nWe explore the possible parametrization models based on constraints from terrestrial experiments and astrophysical observations (viz. massive NSs, radii estimations of NSs, \nGW emitted during the inspiral phase of BNS coalescence) \non dense matter EOS. To construct the EOS, we consider non-linear Walecka (NLW) type and density-dependent (DD) meson-baryon couplings within covariant density functional (CDF) model with exotic degrees of freedom in addition to nucleons. \nThe phenomenological EOS models based on density functional theories and realistic nuclear potentials have been analysed considering matter composition to be only nucleonic \\citep{PhysRevC.98.035804, PhysRevC.99.052802, hnps2989}.\nTherefore, this work will explore the novel aspects of CDF model parametrizations that satisfy the recent astrophysical observable estimations considering hyperonization of dense matter to be a viable energetic argument.\n\nThe paper is organized as follows. In Section \\ref{sec:observations}, we describe a few estimated parameters of compact stellar objects from observational constraints relevant to this work. The CDF model formalism (NLW and DD), its extension to heavier baryons in $\\beta$-equilibrated nuclear matter, aspects of tidal deformability of NSs and coupling parameters incorporated in this work are described in Section \\ref{sec:formalism}. Section \\ref{sec:results} provides the results and our conclusions and future perspectives are summarized in Section \\ref{sec:summary}. \n\n\\textit{Conventions}: We implement the natural units $G=\\hbar=c=1$ throughout the work.\n\n\\section{Observational constraints}\\label{sec:observations}\n\nThe understanding of dense matter above $n_0$ can be reasonably improved from recent compact star observations as the constraints on lower limit of maximum mass, observed range of mass-radius of certain compact stars and most importantly the compactness from observed tidal deformibility. The soft EOS of highly dense matter can be ruled out from the observations of massive stars viz. PSR J$1614-2230$ ($M=1.908\\pm 0.016~M_\\odot$) \\citep{2010Natur.467.1081D, Arzoumanian_2018}, PSR J$0348+0432$ ($2.01\\pm 0.04~M_\\odot$) \\citep{2013Sci...340..448A}, millisecond pulsar J$0740+6620$ ($2.14^{+0.20}_{-0.18}~M_\\odot$ with 95$\\%$ credibility \\citep{2020NatAs...4...72C}, $2.08^{+0.07}_{-0.07}$ $M_\\odot$ with 68.3\\% credibility \\citep{2021arXiv210400880F}) and PSR J$1810+1744$ ($2.13 \\pm 0.04~M_\\odot$ with 68$\\%$ credibility) \\citep{2021ApJ...908L..46R}.\n\nGW observations also can constrain well the models of highly dense matter as already stated. For low-spin prior systems from GW170817 event, it is estimated that the combined dimensionless tidal deformability ($\\tilde{\\Lambda}$) parameter value has an upper and lower bounds of $900$ (TaylorF2 model) \\citep{LIGO_Virgo2017c} and $400$ (AT2017gfo event) \\citep{Radice2018} respectively. Reanalysis of the GW170817 data by LVC has set new limits as $110 \\leq \\tilde{\\Lambda} \\leq 720$ (PhenomPNRT model) \\citep{PhysRevX.9.011001}. Another estimation of $\\tilde{\\Lambda}$ based on the viability of chiral effective field theory results provides the limit to be in the range $80 \\leq \\tilde{\\Lambda} \\leq 570$ \\citep{Tews2018}. In addition, an ameliorated analysis of GW170817 event data implementing identical EOS for both the compact stars producing rational waveforms provides a limit on the dimensionless tidal deformability ($\\Lambda$) for a $1.4M_{\\odot}$ NS to be in the range $70 \\leq \\Lambda_{1.4} \\leq 580$ with 90$\\%$ credibility \\citep{LIGO_Virgo2018a}. \\citet{Raithel_2018} reported that the $\\tilde{\\Lambda} \\leq 800$ constraint implies the radius of primary compact object to be $< 13$ km.\n\nAnother GW observation (GW190814) by the LIGO Livingston detector (LVC) inferred to be from a coalescence of a black-hole (BH) and lighter compact object appendage with mass of the latter to be $2.59^{+0.08}_{-0.09}~M_{\\odot}$ \\citep{2020ApJ...896L..44A} which falls in the `mass-gap'. The nature of lighter companion is still not resolved \\citep{2020MNRAS.499L..82M, Tews_2021, 2020PhRvD.102d1301S, LI2020135812, 2021PhRvC.103b5808D, 2020arXiv201001509B, 2020PhRvC.102f5805F}.\n\nAlong with this, we get good information about mass-radius relation of compact stars from NICER observations. This space mission recently provided adequate information to estimate the mass-radius of PSR J$0030+0451$ to be in the range of $1.44^{+0.15}_{-0.14}$ $M_\\odot$, $13.02^{+1.24}_{-1.06}$ km \\citep{2019ApJ...887L..24M} and $1.34^{+0.15}_{-0.16}$ $M_\\odot$, $12.71^{+1.14}_{-1.19}$ km (with 68.3$\\%$ credibility) \\citep{2019ApJ...887L..21R} respectively. \nLatest estimate of mass-radius of PSR J$0740+6620$ by NICER is in the range of $2.072^{+0.067}_{-0.066}$ $M_\\odot$, $12.39^{+1.30}_{-0.98}$ km \\citep{2021arXiv210506980R} and $2.08\\pm0.07$ $M_\\odot$, $13.71^{+2.62}_{-1.50}$ km (with 68$\\%$ credibility) \\citep{2021arXiv210506979M} respectively.\nSeveral works \\citep{2021arXiv210511031Z, 2021arXiv210502886B, 2021arXiv210508688P, 2021arXiv210506981R, 2021arXiv210413612S, 2021arXiv210504629L} have been performed based on different analyses on astrophysical observations inclusive of the latest NICER measurements to extract new information regarding dense matter EOS.\n \nAn estimation for tidal deformability from NICER observations of PSR J$0030+0451$ jointly with GW170817 event provides $240 \\leq \\Lambda_{1.4} \\leq 730$ with radius range as $R_{1.4}=12.1^{+1.2}_{-0.8}$ km \\citep{Jiang_2020}. Another recent analysis of the same PSR J$0030+0451$ data (NICER) reveals $R_{1.4}=12.32^{+1.09}_{-1.47}$ km \\citep{2020PhRvD.101l3007L}. The radius constraint on the $1.4 M_{\\odot}$ NSs from the GW170817 event has been derived to be in the range $10.5 \\leq R_{1.4}\/\\text{km} \\leq 13.4$ \\citep{PhysRevX.9.011001} while \\citet{PhysRevC.98.035804} provide the radius limit to be $11.82 \\leq R_{1.4}\/\\text{km} \\leq 13.72$. Considering similar low-spin prior systems for the GW190425 event, an upper bound of $600$ \n(PhenomPv2NRT model) has been placed on $\\tilde{\\Lambda}$ and the radius upper limit is derived as $R<15$ km \\citep{2020ApJ...892L...3A}.  \n\nFrom the analysis of GW170817 event data, bounds on matter pressure ranges are derived at $2n_0$ and $6n_0$ to be $3.5^{+2.7}_{-1.7} \\times 10^{34}$ dyn-cm$^{-2}$ and $9.0^{+7.9}_{-2.6} \\times 10^{35}$ dyn-cm$^{-2}$ respectively \\citep{LIGO_Virgo2018a}. \nRecent results obtained from GW170817 event by Bayesian analysis suggest matter pressure at $2n_0$ to be $\\sim 3.81^{+1.18}_{-2.32} \\times 10^{34}$ dyn\/cm$^2$ and $133 \\leq \\Lambda_{1.4} \\leq 686$ \\citep{2021EPJA...57...31L}.\nThe analysis of GW190425 event data reveals core matter density of the primary component involved to be in the range $3\\leq n\/n_0 \\leq 6$ and matter pressure to be in $10^{35} \\leq P(3-6~n_0) \\leq 8 \\times 10^{35}$ dyn\/cm$^2$ range \\citep{2020ApJ...892L...3A}.\n\nWe take into consideration the source properties at $90\\%$ credible intervals with low-spin posterior distributions from PhenomPNRT \\citep{PhysRevX.9.011001} and PhenomPv2NRT \\citep{2020ApJ...892L...3A} waveform models for GW170817 and GW190425, respectively, to evaluate various GW observables.\nBased on recent observable estimations of GW events, in addition to setting radius bounds on the \nNSs involved in GW170817 and GW190425 events,\nwe also find limits on compactness parameter for a $1.4~M_{\\odot}$ NS.\n\n\\section{Formalism}\\label{sec:formalism}\n\n\\subsection{EOS model}\n\nIn this section, we briefly discuss the NLW and DD CDF models implemented to contrive the EOS in this work. The dense matter possible constituents considered here are nucleons ($N\\equiv n,p$), hyperons ($Y\\equiv \\Lambda^0, \\Sigma^{\\pm,0},\\Xi^{-,0}$) and $\\Delta$-resonances ($\\Delta \\equiv \\Delta^{++}, \\Delta^{+}, \\Delta^{-}, \\Delta^{0}$) alongside leptons ($l\\equiv e^-,\\mu^{-}$) to maintain $\\beta$-equilibrium. The interaction between non-strange baryons are described via the exchange of isoscalar-scalar $\\sigma$, isoscalar-vector $\\omega$, and isovector-vector $\\rho$ mesons. For the hyperonic sector interactions, an additional hidden strangeness isoscalar-vector $\\phi$ meson is taken into consideration. In general, the total Lagrangian density describing the baryon-meson interactions is given by \\citep{1999PhRvC..60b5803G,2001PhRvC..63c5802B,Li_PLB_2018}\n\\begin{equation}\\label{eqn.1}\n\\begin{aligned}\n\\mathcal{L} & = \\sum_{b\\equiv N,Y} \\bar{\\psi}_b(i\\gamma_{\\mu} D^{\\mu}_{(b)} - m^{*}_b) \\psi_b + \\sum_{l} \\bar{\\psi}_l (i\\gamma_{\\mu} \\partial^{\\mu} - m_l)\\psi_l \n\\\\\n& + \\sum_{\\Delta} \\bar{\\psi}_{\\Delta \\nu}(i\\gamma_{\\mu} D^{\\mu}_{(\\Delta)} - m^{*}_{\\Delta}) \\psi^{\\nu}_{\\Delta} + \\frac{1}{2}(\\partial_{\\mu}\\sigma\\partial^{\\mu}\\sigma - m_{\\sigma}^2 \\sigma^2) \\\\\n & -  \\frac{1}{4}\\omega_{\\mu\\nu}\\omega^{\\mu\\nu} + \\frac{1}{2}m_{\\omega}^2\\omega_{\\mu}\\omega^{\\mu} - \\frac{1}{4}\\boldsymbol{\\rho}_{\\mu\\nu} \\cdot \\boldsymbol{\\rho}^{\\mu\\nu} + \\frac{1}{2}m_{\\rho}^2\\boldsymbol{\\rho}_{\\mu} \\cdot \\boldsymbol{\\rho}^{\\mu}  \\\\\n & - \\frac{1}{4}\\phi_{\\mu\\nu}\\phi^{\\mu\\nu} + \\frac{1}{2}m_{\\phi}^2\\phi_{\\mu}\\phi^{\\mu} - \\text{U}(\\sigma)\n\\end{aligned}\n\\end{equation}\nwhere, the last term $\\text{U}(\\sigma)$ corresponds to the self-interactions of scalar mesons only for NLW CDF models. $\\psi_b,~\\psi_{l},~\\psi_{\\Delta}^{\\nu}$ represent Dirac-fields of the baryon-octet, leptons and Schwinger-Rarita fields of $\\Delta$-quartet respectively. $m_b,~m_l$, and $m_{\\Delta}$ denote the bare masses of baryon octet, leptons and $\\Delta$-quartet species respectively. The covariant derivative mentioned in eq.-\\eqref{eqn.1} is given by\n\\begin{equation}\\label{eqn.2}\nD_{\\mu (j)} = \\partial_\\mu + ig_{\\omega j} \\omega_\\mu + ig_{\\rho j} \\boldsymbol{\\tau}_{j3} \\cdot \\boldsymbol{\\rho}_{\\mu} + ig_{\\phi j} \\phi_\\mu\n\\end{equation}\nwith $j$ denoting the baryon octet ($b$) and $\\Delta$-quartet ($\\Delta$). The coupling constants are represented by $g_{pj}$ with index-$p$ being the considered mesons. $\\boldsymbol{\\tau}_{j3}$ is the iso-spin projection of third component of isovector-vector meson fields. The scalar self-interaction term for NLW CDF model is given by\n\\begin{equation}\\label{eqn.3}\n\\text{U}(\\sigma) = \\frac{1}{3} g_2 \\sigma^3 + \\frac{1}{4} g_3 \\sigma^4\n\\end{equation}\nwhere $g_2$ and $g_3$ are the coefficients of self-interactions. \n\nWith the monotonic increase in baryon chemical potentials interior to NSs, the nucleonic matter may well transform to heavier strange and non-strange baryons leading to new hadronic degrees of freedom. \nThe conditions which are necessary to maintain strong $\\beta$-equilibrium between different particle species without strangeness being conserved are \\citep{2001PhRvC..63c5802B, Drago_PRC_2014}\n\\begin{equation}\\label{eqn.7}\n\\begin{aligned}\n\\mu_e & = \\mu_n - \\mu_p = \\mu_{\\mu}, \\quad \\mu_{\\Sigma^+} = \\mu_{\\Delta^+} = \\mu_p, \\\\\n\\mu_{\\Sigma^-} & = \\mu_{\\Xi^-} = \\mu_{\\Delta^-} = \\mu_n + \\mu_e, \\quad \\mu_{\\Delta^{++}} = \\mu_p - \\mu_e, \\\\\n\\mu_{\\Sigma^0} & =\\mu_{\\Xi^0}=\\mu_{\\Lambda^0} = \\mu_{\\Delta^0}= \\mu_n\n\\end{aligned}\n\\end{equation}\nwith $\\mu_j$ denoting chemical potential of the $j$-th baryon and defined as\n\\begin{equation}\\label{eqn.8}\n\\begin{aligned}\n& \\mu_{j} = \\sqrt{p_{F_j}^2 + m_{j}^{*2}} + \\Sigma_{B}.\n\\end{aligned}\n\\end{equation}\nHere, $\\Sigma_B = \\Sigma^{0} + \\Sigma^{r}$ represents the vector self energies with\n$\\Sigma^{0} = g_{\\omega j}\\omega_{0} + g_{\\phi j}\\phi_{0} + g_{\\rho j} \\boldsymbol{\\tau}_{j3} \\rho_{03}$ and\n$\\Sigma^{r}$ denotes the rearrangement term (present in DD CDF models only) necessary to maintain thermodynamic consistency which is given by\n\\begin{equation}\n\\begin{aligned}\\label{eqn.rear}\n\t\\Sigma^{r} & = \\sum_{b} \\left[ \\frac{\\partial g_{\\omega b}}{\\partial n}\\omega_{0}n_{b} - \\frac{\\partial g_{\\sigma b}}{\\partial n} \\sigma n_{b}^s + \\frac{\\partial g_{\\rho b}}{\\partial n} \\rho_{03} \\boldsymbol{\\tau}_{b3} n_{b} \\right. \\\\\n\t& \\left. + \\frac{\\partial g_{\\phi b}}{\\partial n}\\phi_{0}n_{b} \\right] + \\sum_{\\Delta} (\\psi_b \\longrightarrow \\psi_{\\Delta}^{\\nu}),\n\\end{aligned}\n\\end{equation}\nwith $n^s= \\langle\\bar{\\psi} \\psi \\rangle$ and $n=\\langle\\bar{\\psi} \\gamma^0 \\psi \\rangle$ denoting the scalar and vector (number) densities respectively.\n\nTwo additional constraints, viz. charge neutrality and global baryon number conservation, are administered while evaluating the EOS self-consistently. The re-arrangement term considered in DD CDF model contributes explicitly to the pressure term. For details, the readers may refer to \\cite{2001PhRvC..64b5804H}.\n\n\\subsection{Tidal deformability}\n\nThe compact stars in a binary system experience tidal deformations due to the gravitational fields of their respective companions. These tidal effects can be quantified in terms of tidal deformability parameter ($\\lambda$) defined as the ratio of induced mass quadrupole moment $Q_{ij}$ to external perturbing tidal field $\\mathcal{E}_{ij}$ \\citep{Hinderer_2008, PhysRevD.77.021502, PhysRevD.81.123016}, \n\\begin{equation}\\label{eqn.9}\n\\lambda = - \\frac{Q_{ij}}{\\mathcal{E}_{ij}} = \\frac{2}{3}k_2 R^5,\n\\end{equation}\nwhere,\n\\begin{equation}\\label{eqn.10}\n\\begin{aligned}\nk_2 = & \\frac{8 C^5}{5}(1-2C^2)[2+2C(y-1)-y] \\cdot \\{2C [6-3y+3C(5y-8)] \\\\\n& + 4C^3 [13 - 11y + C(3y-2) + 2C^2(1+y)] + 3(1-2C^2) \\cdot \\\\\n& [2 - y + 2C(y-1)] \\log (1-2C)\\}^{-1}\n\\end{aligned}\n\\end{equation}\nwith $C=M\/R$ being the compactness parameter, $M$ and $R$ being mass and radius of the star respectively and $k_2$, the EOS dependent tidal Love number. $y=y(R)$ is the function obtained after solving the differential equation \\citep{PhysRevD.80.084018, PhysRevD.81.084016, PhysRevC.95.015801},\n\\begin{equation}\\label{eqn.11}\nr \\frac{dy(r)}{dr} + y(r)^2 + y(r)F(r) + r^2 Q(r) = 0,\n\\end{equation}\nwhere the functions are\n\\begin{equation}\\label{eqn.12}\nF(r) = \\frac{r - 4\\pi r^3 [\\varepsilon(r) - P(r)]}{r - 2M(r)},\n\\end{equation}\n\\begin{equation}\\label{eqn.13}\n\\begin{aligned}\nQ(r) = & \\frac{4\\pi r [5\\varepsilon(r) + 9P(r) + \\frac{\\varepsilon(r) + P(r)}{\\partial P(r)\/\\partial \\varepsilon(r)}]}{r - 2M(r)} - 4 \\left[ \\frac{M(r) + 4\\pi r^3 P(r)}{r^2 (1 - 2M(r)\/r)} \\right].\n\\end{aligned}\n\\end{equation}\n\\begin{table*}\n\\caption{The nuclear properties of the CDF models at respective $n_0$.}\n\\centering\n\\begin{tabular}{cccccccc}\n\\hline \\hline\n\\multicolumn{2}{c}{CDF Model} & $n_0$ & $-E_0$ & $K_0$ & $E_{\\text{sym}}$ & $L_{\\text{sym}}$ & $K_{\\text{sym}}$ \\\\\n& & (fm$^{-3}$) & (MeV) & (MeV) & (MeV) & (MeV) & (MeV) \\\\\n\\hline\n& GM1 & 0.153 & $16.30$ & 300.00 & 32.50 & 93.857 & 17.91 \\\\\n& GM2 & 0.153 & $16.30$ & 300.00 & 32.50 & 89.289 & $-$ 11.98 \\\\\n& GM3 & 0.153 & $16.30$ & 240.00 & 32.50 & 89.627 & $-$ 6.46 \\\\\n& NL3 & 0.148 & $16.29$ & 271.76 & 37.40 & 118.317 & 100.53 \\\\\nNLW & NL3-II & 0.149 & $16.28$ & 272.15 & 37.70 & 119.563 & 103.19 \\\\\n& NL-SH & 0.146 & $16.346$ & 355.36 & 36.10 & 113.654 & 79.81 \\\\\n& NL-RA1 & 0.1466 & $16.15$ & 285.00 & 36.10 & 115.305 & 95.57 \\\\\n& NL3* & 0.150 & $16.31$ & 258.27 & 38.68 & 122.71 & 105.73 \\\\\n& GMT & 0.145 & $16.30$ & 281.00 & 36.90 & 112.796 & 63.04 \\\\\n\\hline\n& DD1 & 0.1487 & $16.021$ & 240.00 & 31.60 & 55.949 & $-$ 95.24 \\\\\n& DD2 & 0.149065 & $16.02$ & 242.70 & 32.73 & 54.966 & $-$ 93.24 \\\\\n& DD-ME1 & 0.152 & $16.20$ & 244.50 & 33.10 & 55.370 & $-$ 101.07 \\\\\n& DD-ME2 & 0.152 & $16.14$ & 250.89 & 32.30 & 51.253 & $-$ 87.31 \\\\\nDD & PKDD & 0.149552 & $16.267$ & 262.181 & 36.79 & 90.139 & $-$ 80.56 \\\\\n& TW99 & 0.153 & $16.247$ & 240.00 & 33.39 & 55.309 & $-$ 124.68 \\\\\n& DDV & 0.151 & $16.097$ & 240.00 & 33.589 & 71.463 & $-$ 93.97 \\\\\n& DDF & 0.1469 & $16.024$ & 223.10 & 31.60 & 55.919 & $-$ 139.66 \\\\\n& DD-MEX & 0.152 & $16.14$ & 267.059 & 32.269 & 49.576 & $-$ 71.47 \\\\\n\\hline\n\\end{tabular}\n\\label{tab:1}\n\\end{table*} \nDue to paramount dependence on the stellar radius, $\\lambda$ imposes stringent constraints on dense matter EOS. $\\lambda$ or, equivalently $k_2$ provides the ease of induced deformation estimate of bulk matter. This parameter is evaluated self-consistently alongside the Tolman-Oppenheimer-Volkov (TOV) equations \\citep{1996cost.book.....G}. Another dimensionless quantity $\\Lambda$ is much more expedient as it relates $\\lambda$ with $C$ through the relation \n\\begin{equation}\\label{eqn.14}\n\\Lambda=\\frac{\\lambda}{M^5}=\\frac{2}{3} \\frac{k_2}{C^5}.  \n\\end{equation}\n\nGW signal encodes information regarding deformation of both compact objects in the binary system as the weighted tidal deformability ($\\tilde{\\lambda}$) and is given by\n\\citep{Hinderer_2008, PhysRevD.81.123016}\n\\begin{equation}\\label{eqn.15}\n\\tilde{\\lambda} = \\frac{1}{26} \\left[ \\frac{M_1 + 12 M_2}{M_1} \\lambda_1 + \\frac{M_2 + 12 M_1}{M_2} \\lambda_2 \\right]\n\\end{equation}\nwhere $\\lambda_1$ and $\\lambda_2$ are the tidal deformabilities corresponding to stars with masses $M_1$ and $M_2$ respectively. In order to relate to $\\Lambda$, we incorporate the combined dimensionless tidal deformability ($\\tilde{\\Lambda}$) defined as \\citep{PhysRevLett.112.101101}\n\\begin{equation}\\label{eqn.16}\n\\begin{aligned}\n\\tilde{\\Lambda} & = 32 \\frac{\\tilde{\\lambda}}{(M_1 + M_2)^5} \\\\\n& = \\frac{16}{13} \\frac{(M_1 + 12 M_2) M_{1}^4 \\Lambda_1 + (M_2 + 12 M_1) M_{2}^4 \\Lambda_2}{(M_1 + M_2)^5} .\n\\end{aligned}\n\\end{equation}\n\n\\subsection{Coupling parameters} \\label{cp}\n\nIn this work, we adopt the NLW model with GM1, GM2, GM3 \\citep{PhysRevLett.67.2414}, NL3, NL3-II \\citep{1997PhRvC..55..540L}, NL-SH \\citep{SHARMA1993377}, NL-RA1 \\citep{PhysRevC.63.044303}, NL3* \\citep{LALAZISSIS200936}, GMT \\citep{2000NuPhA.674..553P} parametrizations and the DD CDF model with DD1 \\citep{PhysRevC.71.064301}, DD2 \\citep{PhysRevC.81.015803}, DD-ME1 \\citep{PhysRevC.66.024306}, DD-ME2 \\citep{2005PhRvC..71b4312L}, PKDD \\citep{2004PhRvC..69c4319L}, TW99 \\citep{TYPEL1999331}, DDV \\citep{Typel:2020ozc}, DDF \\citep{PhysRevC.74.035802}, DD-MEX \\citep{TANINAH2020135065} parametrizations for meson-baryon couplings. \n\n\\begin{figure}\n  \\begin{center}\n\\includegraphics[width=8.5cm,keepaspectratio ]{Figures\/sym_energy.pdf}\n\\caption{Symmetry energy coefficient ($E_{\\text{sym}}$) and its corresponding slope parameter ($L_{\\text{sym}}$) for all the parametrization models considered in this work. The shaded region represents the current empirical range in $E_{\\text{sym}}-L_{\\text{sym}}$ plane following recent experimental and microscopic model calculations \\citep{2017RvMP...89a5007O}.}\n\\label{fig-012}\n\\end{center}\n\\end{figure} \n\nComparison of experimental data \\citep{2017RvMP...89a5007O} from finite nuclei and heavy-ion collisions with different microscopic model calculations have provided bounds on nuclear saturation properties of symmetric nuclear matter (SNM):\n\\begin{enumerate}\n\\item Incompressibility, $ 210$ MeV $\\leq K_0(n_0) \\leq 280$ MeV,\n\\item Symmetry energy coefficient, $28.5$ MeV $\\leq E_{\\text{sym}} (n_0) \\leq 34.9$ MeV,\n\\item Slope parameter of $E_{\\text{sym}}$, $30.6$ MeV $\\leq L_{\\text{sym}} (n_0) \\leq 86.8$ MeV.\n\\end{enumerate}\nTable-\\ref{tab:1} displays the nuclear saturation properties obtained for models with different CDF parametrizations considered in this work with $E_0$, the saturation energy.\nThe bounds on curvature of symmetry energy, $K_{\\text{sym}}(n_0)=-111.8 \\pm 71.3$ MeV \\citep{2017PhRvC..96b1302M}, $-85^{+82}_{-70}$ MeV \\citep{2019ApJ...887...48B} and $-102^{+71}_{-72}$ MeV \\citep{2020arXiv200203210Z} deduced from nuclear data and various NS observations provide additional constraint on dense matter EOS.\n\n\\begin{figure*}%\n    \\centering\n    \\subfloat{\\includegraphics[width=8.5cm,keepaspectratio ]{Figures\/MR_N.pdf}}%\n    \\qquad\n    \\subfloat{\\includegraphics[width=8.5cm,keepaspectratio ]{Figures\/MR_NYD.pdf}}%\n    \\caption{The family of solutions of TOV equations for matter composed of, left-hand panel: only nucleons and right-hand panel: $\\Delta$-admixed hypernuclear matter alongside leptons to maintain $\\beta$-equilibrium. The solid curves denote the $M$-$R$ curves for different density-dependent model parametrizations, while the dot-dashed curves denote the cases with non-linear model parametrizations. The astrophysical constraints from GW190814 \\citep{2020ApJ...896L..44A}, PSR J$1810+1744$ \\citep{2021ApJ...908L..46R}, PSR J$0030+0451$ \\citep{2019ApJ...887L..24M, 2019ApJ...887L..21R, Jiang_2020, 2020PhRvD.101l3007L}, PSR J$0740+6620$ \\citep{2021arXiv210506980R, 2021arXiv210506979M} are represented by the shaded regions.}%\n    \\label{fig-001}%\n\\end{figure*}\n\nThe density-dependent meson-nucleon couplings implemented in DD CDF model are defined as \\citep{TYPEL1999331, PhysRevC.66.024306},\n\\begin{equation}\\label{eqn.17}\ng_{i N}(n)= g_{i N}(n_{0}) f_i(n\/n_0) \\quad \\quad \\text{for }i=\\sigma,\\omega\n\\end{equation}\nwhere, $n$ is the baryon number density and\n\\begin{equation}\\label{eqn.18}\nf_i(n\/n_0)= a_i \\frac{1+b_i (n\/n_0+d_i)^2}{1+c_i (n\/n_0 +d_i)^2}.\n\\end{equation}\nIn the case of $\\rho$-meson couplings, the functional is defined as\n\\begin{equation}\\label{eqn.19}\ng_{\\rho N}(n)= g_{\\rho N}(n_{0}) e^{-a_{\\rho}(n\/n_0 -1)}.\n\\end{equation}\nFor the coefficient values in equations \\eqref{eqn.17}-\\eqref{eqn.19}, the readers may refer to \\citet{PhysRevC.71.064301, PhysRevC.81.015803, PhysRevC.66.024306, 2005PhRvC..71b4312L, 2004PhRvC..69c4319L, TYPEL1999331, Typel:2020ozc, PhysRevC.74.035802} and \\citet{TANINAH2020135065}. These coefficients are associated with different DD CDF model parametrizations and fitted to reproduce various nuclei properties. The hidden strangeness meson $\\phi$ does not couple with nucleons, so $g_{\\phi N}=0$.\n\nThe meson-hyperon and meson-$\\Delta$ couplings are considered similar to the density-dependence footing in the case of nucleons in DD CDF model. In the case of the meson-hyperon vector couplings, we implement the SU(6) symmetry and quark counting rule \\citep{SCHAFFNER199435}. \nFor the scalar meson-hyperon couplings, we consider the optical potentials of $\\Lambda,\\Sigma$ and $\\Xi$-hyperons in SNM to be $-28,+30$ and $-14$ MeV, respectively, at nuclear saturation \\citep{Feliciello_2015, RevModPhys.88.035004}. \nThe optical potential depths $U_{\\Xi}^{(N)} (n_0)=-18$ MeV and $U_{\\Lambda}^{(N)} (n_0) =-30$ MeV \\citep{2000PhRvC..62c4311S, 2015ApJ...808....8G} are also widely implemented in dense matter studies. \nRecently \\citet{2021arXiv210400421F} reported an attractive optical potential depth of $\\Xi$-hyperons in SNM to be $\\gtrsim -20$ MeV. \nFor a more recent review on the aspects of strangeness in dense matter, the reader may refer to \\citet{2020PrPNP.11203770T}.\n\nDue to scarce information regarding the $\\Delta$-nucleon interactions, we treat the meson-$\\Delta$ resonances couplings as parameters. \\citet{PhysRevC.81.035502, KOCH1985765, WEHRBERGER1989797} have reported the data to constrain meson-$\\Delta$ baryon couplings at $n_0$ based on pion-nucleus scattering, electron scattering on nuclei and excitation studies of $\\Delta$-quartet experiments. Recent reviews \\citep{Drago_PRC_2014, PhysRevC.74.035802, Kolomeitsev_NPA_2017} on this aspect have reported the $\\Delta$-potential ($V_{\\Delta}$) in nuclear medium to be in the range $-30~\\text{MeV} + V_N \\leq V_{\\Delta} \\leq V_N$ ($V_N$ being the nucleon potential), the values of factor $x_{\\sigma \\Delta} - x_{\\omega \\Delta}$ to be between $0$ and $0.2$ with $x_{\\sigma \\Delta}=g_{\\sigma \\Delta}\/ g_{\\sigma N}$ and $x_{\\omega \\Delta}=g_{\\omega \\Delta}\/ g_{\\omega N}$. Many works \\citep{Drago_PRC_2014, Kolomeitsev_NPA_2017, Li_PLB_2018, Ribes_2019, PhysRevC.75.035806, Cai_PRC_2015, 2021PhLB..81436070R} have considered the ranges for $x_{\\omega \\Delta} \\in [0.6-1.2]$ and $x_{\\rho \\Delta} \\in [0.5-3.0]$. \nFor recent development regarding $\\Delta$-potential in dense matter, the reader may refer to \\citet{2021arXiv210108679C}.\nIn the ensuing discussion, we will consider $x_{\\omega \\Delta}= 1.10$, $x_{\\rho \\Delta}=1.00$ for the vector meson-$\\Delta$ baryon couplings and $x_{\\sigma \\Delta}=1.20$ for the scalar coupling. $\\Delta$-quartet being non-strange baryons does not couple with $\\phi$-meson, so $g_{\\phi \\Delta}=0$.\n\n\\section{Constraints on dense matter models}\\label{sec:results}\n\nNext, we sort out the different EOSs with various possible compositions and parametrizations based on terrestrial experimental and stellar observational values. From table-\\ref{tab:1} it can be noticed that all the models reproduce $n_0$ and $E_0$ in the correct range of empirical values. However, among the NLW parametrizations considered in this work, GM1 and GM2 cannot reproduce the empirical range of $K_0$ at $n_0$ mentioned in Section \\ref{cp}, while all the DD CDF models considered in this work satisfy the empirical range of $K_0$ at $n_0$. On the other hand, the empirical range of symmetry energy coefficient is satisfied by GM1, GM2 and GM3 parameter sets among NLW models, while in the case of DD models, all except PKDD parametrization lies within the bounds of this particular saturation property. The current empirical bound on $L_{\\text{sym}}$ is satisfied by all DD parametrization models considered in this work except PKDD parametrization. \nIn addition, the current empirical bounds on $K_{\\text{sym}}$ are satisfied by GM2, GM3 (among NLW models) and all DD parameter sets considered in this work.\nFig.-\\ref{fig-012} displays the parameter sets which are compatible with the current bounds on $E_{\\text{sym}}$ and $L_{\\text{sym}}$. All the DD models (except PKDD) considered in this work satisfy the current bounds on nuclear saturation properties. \n\n\\begin{table}\n\\caption{Summary of TOV results evaluated from parameter sets considered in this work (pure $N$-matter, refer to left-hand panel of fig.-\\ref{fig-001}). Fulfillment of the mass-radius constraints from various astrophysical observations are marked by $+(-)$.}\n\\centering\n\\begin{threeparttable}[b]\n\\begin{tabular}{cccccc}\n\\hline \\hline\n\\multicolumn{2}{c}{CDF Model} & \\parbox[t]{1mm}{\\rotatebox[origin=c]{90}{ PSR J$0030+0451$ }} & \\parbox[t]{1mm}{\\rotatebox[origin=c]{90}{ PSR J$0740+6620$ }} & \\parbox[t]{1mm}{\\rotatebox[origin=c]{90}{ PSR J$1810+1744$ }} & \\thead{GW190814 \\\\ (secondary)\\tnote{$^{**}$} } \\\\\n\\hline\n& GM1 & $+$ & $+$ & $+$ & $-$ \\\\\n& GM2 & $+$ & $+$ & $-$ & $-$ \\\\\n& GM3 & $+$ & $-$ & $-$ & $-$ \\\\\n& NL3 & $-$ & $+$ & $+$ & $+$ \\\\\nNLW & NL3-II & $-$ & $+$ & $+$ & $+$ \\\\\n& NL-SH & $-$ & $+$ & $+$ & $+$ \\\\\n& NL-RA1 & $-$ & $+$ & $+$ & $+$ \\\\\n& NL3* & $-$ & $+$ & $+$ & $+$ \\\\\n& GMT & $-$ & $+$ & $+$ & $+$ \\\\\n\\hline\n& DD1 & $+$ & $+$ & $+$ & $-$ \\\\\n& DD2 & $+$ & $+$ & $+$ & $-$ \\\\\n& DD-ME1 & $+$ & $+$ & $+$ & $-$ \\\\\n& DD-ME2 & $+$ & $+$ & $+$ & $-$ \\\\\nDD & PKDD & $+$ & $+$ & $+$ & $-$ \\\\\n& TW99 & $+$ & $-$ & $-$ & $-$ \\\\\n& DDV & $+$ & $-$ & $-$ & $-$ \\\\\n& DDF & $+$ & $-$ & $-$ & $-$ \\\\\n& DD-MEX & $+$ & $+$ & $+$ & $+$ \\\\\n\\hline\n\\end{tabular}\n  \\begin{tablenotes}\n    \\item[$^{**}$] The nature of secondary component of GW190814 is still in tension\n  \\end{tablenotes}\n  \\end{threeparttable}\n\\label{tab:4}\n\\end{table} \nThe mass-radius ($M$-$R$) relations corresponding to the various parametrizations for only nucleonic matter composition, are shown in left panel of fig.-\\ref{fig-001} obtained by solving the TOV \nequations for spherically symmetric, non-rotating stars. \nFor the crustal region, we consider the \\citet{1971ApJ...170..299B} EOS. The transition from crust to the core is modelled in a way that is thermodynamically consistent following \\citet{2016PhRvC..94c5804F}.\nWe consider the recent massive NS observation (PSR J$1810+1744$) as the lower bound for maximum mass configurations. If we consider pure nucleonic matter, all the EOSs considered in this work except a few satisfy the lower bound constraint of maximum mass as evident from the left-hand panel of fig.-\\ref{fig-001}. \nGM2 ($M_{\\text{max}}\\sim2.08~M_{\\odot}$), GM3 ($2.02~M_{\\odot}$) among NLW type and TW99 ($2.07~M_{\\odot}$), DDV ($1.93~M_{\\odot}$), DDF ($1.96~M_{\\odot}$) among DD type models fail to satisfy this mass constraint. However, the constraints obtained from NICER observations (PSR J$0030+0451$) are seen to be not satisfied by the NLW models satisfying mass constraint, except the GM1 parametrization, while all the DD type models satisfying the mass constraint satisfy this constraint too. Moreover, these models satisfy \nthe recent NICER results of PSR J$0740+6620$ simultaneously. Here, PSR J$0740+6620$ observation suggests the dense matter at higher densities is repulsive enough to produce large radii of heavier NSs. Most of the NLW models not satisfying the NICER observations of PSR J$0030+0451$ are observed to satisfy this recent constraint with a wider radii range. However, \neven though they are observed to fulfil the constraint from \\citet{2021arXiv210506979M}, they fail to fulfil another NICER measurement of PSR J$0030+0451$ from \\citet{2019ApJ...887L..24M, 2019ApJ...887L..21R, Jiang_2020, 2020PhRvD.101l3007L}. \nStiffer EOSs ($N$-matter) obtained from NLW type parametrizations (except GM1, GM2, GM3) and DD-MEX (DD type) yield $M_{\\text{max}} \\geq 2.5~M_{\\odot}$ satisfying the GW190814 event secondary component's mass constraint. Although this constraint is not unequivocal as the nature of secondary compact object is not found to be NS explicitly \\citep{2020PhRvD.102d1301S, LI2020135812}.\nThe TOV results involving different parametrizations with pure $N$-matter are summarized in table-\\ref{tab:4}.\nAs a result of EOS softening due to the inclusion of hyperons and $\\Delta$-quartet, the only NLW model EOS with the GM1 parametrization, satisfying all mass-radius constraints, produce stars that fail to fulfil the lower bound constraint of maximum mass as shown in the right-hand panel of fig.-\\ref{fig-001}. Except PKDD ($M_{\\text{max}}\\sim1.93~M_{\\odot}$), DD1 ($2.05~M_{\\odot}$) and DD2 ($2.06~M_{\\odot}$) parametrizations all other parametrizations from DD CDF models which satisfy the joint constraints from PSR J$1810+1744$, PSR J$0740+6620$ and PSR J$0030+0451$ with pure $N$-matter satisfy the mass constraints even after softening due to hyperonization. The incorporation of $\\Delta$-quartet reduces the radii of NSs, enhancing their respective compactness parameter. \n\n\\begin{figure}\n  \\begin{center}\n\\includegraphics[width=8.5cm,keepaspectratio ]{Figures\/EOS_N.pdf}\n\\caption{The variation of matter pressure as a function of energy density (EOS) for $\\beta$-equilibrated matter composed of only nucleons and leptons. The solid curves denote the EOSs for different density-dependent model parametrizations, while the dot-dashed curves denote the cases with non-linear model parametrizations. The constraints on matter pressure evaluated from GW170817 event are denoted by the vertical lines and provided in sec.-\\ref{sec:intro}.}\n\\label{fig-003}\n\\end{center}\n\\end{figure}\n\n\\begin{figure*}\n  \\begin{center}\n\\includegraphics[width=12.2cm,keepaspectratio ]{Figures\/lam_mass_ratio_GW170817.pdf}\n\\caption{Combined dimensionless tidal deformability as a function of mass-ratio ($q$) considering a fixed chirp mass, $\\mathcal{M}=1.188~M_{\\odot}$ (GW170817 event). Left panel: nucleonic matter, right panel: $\\Delta$-resonances and baryon octet matter. The solid, dot-dashed lines depicts the DD and NLW type parametrizations respectively. The parametrizations yielding soft EOSs are not shown in the right panel. The horizontal dotted lines denote bounds on $\\tilde{\\Lambda}$ (with $90\\%$ credibility, low-spin priors). The vertical dotted lines represent the mass-ratio, $q=0.73$ boundary \\citep{PhysRevX.9.011001}.}\n\\label{fig-004}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure}\n  \\begin{center}\n\\includegraphics[width=8.5cm,keepaspectratio ]{Figures\/lam_mass_ratio_GW190425.pdf}\n\\caption{Similar to fig.-\\ref{fig-004} but considering a fixed chirp mass, $\\mathcal{M}=1.43~M_{\\odot}$ (GW190425 event) with pure nucleonic matter. The solid, dot-dashed lines depicts the DD and NLW type parametrizations respectively. $\\tilde{\\Lambda}$ constraints are similar to fig.-\\ref{fig-004}. The vertical dotted lines represent the mass-ratio, $q=0.80$ boundary \\citep{2020ApJ...892L...3A}.}\n\\label{fig-005}\n\\end{center}\n\\end{figure}\n\n\\begin{figure*}\n  \\begin{center}\n\\includegraphics[width=12.20cm,keepaspectratio ]{Figures\/lam_2_lam_1.pdf}\n\\caption{Tidal deformability parameters $\\Lambda_1$, $\\Lambda_2$ corresponding to the binary components, $M_1,~M_2$ for GW170817 event with matter composition as, left panel: pure nucleonic matter, right panel: $\\Delta$-baryon admixed hypernuclear and assuming a fixed chirp mass, $\\mathcal{M}=1.188~M_{\\odot}$. The solid curves denote the DD type parametrizations while the NLW type are denoted by dash-dotted curves. The dotted, short-dashed curves denote the $\\tilde{\\Lambda} \\sim 900$ (TaylorF2), $720$ (PhenomPNRT) upper bounds at $90\\%$ confidence level (low-spin priors) respectively \\citep{LIGO_Virgo2017c,PhysRevX.9.011001}. The diagonal long-dashed line marks $\\Lambda_1=\\Lambda_2$ case.}\n\\label{fig-006}\n\\end{center}\n\\end{figure*}\n\nWe now look into the pressure bounds in lower and higher matter density regimes derived from GW170817 event data. Fig.-\\ref{fig-003} shows that all the EOSs with pure nuclear matter satisfying mass-radius constraints also satisfy both the pressure bounds. It is to be noted that among NLW models, GM1, GM2 and GM3 satisfy only the lower bound at $n\\sim 2n_0$ while they fail to satisfy the bound at $n\\sim 6n_0$.\nIt has to be kept in mind that the estimated bound at $6n_0$ is more than the central pressures of compact stars involved in GW170817 event.\n\nWe have evaluated $\\tilde{\\Lambda}$ with the range of primary and secondary masses $1.36-1.60~M_{\\odot}$ and $1.17-1.36~M_{\\odot}$ respectively to provide the chirp mass, $\\mathcal{M}=(M_1 M_2)^{3\/5}(M_T)^{-1\/5}=1.188~M_{\\odot}$ where the total mass $M_T=M_1 + M_2$, is in the range $2.73-2.78~M_{\\odot}$ for GW170817 event. \nIn this work, we have considered the source properties to be circumscribing within $90\\%$ credible intervals.\nFig.-\\ref{fig-004} shows the $\\tilde{\\Lambda}$ variation with mass-ratio parameter ($q$). \n$\\tilde{\\Lambda}$ is found to be almost  independent of the mass asymmetry factor $q$ \n(refer to table-\\ref{tab:2} for numerical results). In the left panel, curves are for pure nucleonic matter and in the right panel, they are for $\\Delta$-admixed hypernuclear matter. For pure nucleonic matter, only GM2, GM3 models lie within the observational $\\tilde{\\Lambda}\\sim 900$ bound among the NLW CDF models. However, they do not satisfy the lower bound constraint of maximum mass. \nIn DD CDF models, all parametrizations considered in this work follow the upper bound of $900$. The stringent limit of $\\tilde{\\Lambda}\\sim 720$ is observed to be satisfied by only TW99, DDV, DDF models which do not satisfy lower bound constraint for maximum mass. \nWith $\\Delta$-baryons coming into the picture, the models fulfil the latter stringent upper bound on $\\tilde{\\Lambda}$ as shown from the right panel of the figure.\nSince the effect of hyperon inclusion on $\\tilde{\\Lambda}$ is similar to that of nucleons for the NSs with mass bounds obtained from GW170817 event, so they are not shown in fig.-\\ref{fig-004}.\n\\begin{figure}\n  \\begin{center}\n\\includegraphics[width=8.5cm,keepaspectratio ]{Figures\/lam_radius.pdf}\n\\caption{Effective tidal deformability variation with radius of the binary's primary component for GW170817 and GW190425 events. The solid, dotted, stripped squares represent the DD parametrizations with pure nucleonic, baryon octet, $\\Delta$-admixed hypernuclear matter respectively. The solid circles denote the NLW parametrizations with nucleonic matter case. The quadratic correlations for the GW170817, GW190425 cases are given by short-dashed and dash-dotted curves respectively. The horizontal dotted lines represent bounds on $\\tilde{\\Lambda}$ similar to figs.-\\ref{fig-004},\\ref{fig-005}. The vertical short-dashed lines mark the points where the quadratic fits intersect $\\tilde{\\Lambda}$ bounds. \n}\n\\label{fig-007}\n\\end{center}\n\\end{figure}\n\n\\begin{table*}\n\\caption{Observational properties of various CDF models with different matter compositions. Here, $q$ represents the mass ratio of the secondary component ($M_2$) to primary one ($M_1$) involved in GW event. $C_{1.4}$ denotes the compactness parameter for a $1.4M_{\\odot}$ NS.}\n\\centering\n\\begin{tabular}{cccccccccc}\n\\hline \\hline\nMatter & CDF Model & $M_{\\text{max}}$ & $R_{1.4}$ & $\\Lambda_{1.4}$ & $C_{1.4}$ & \\multicolumn{2}{c}{$\\tilde{\\Lambda}~(q=0.8)$} & \\multicolumn{2}{c}{$\\tilde{\\Lambda}~(q=1.0)$}  \\\\\ncomposition & & ($M_{\\odot}$) & (km) & & & GW170817 & GW190425 & GW170817 & GW190425 \\\\\n\\hline\n& GM1 & 2.387 & 13.939 & 966.34 & 0.148 & 1109.39 & 355.77 & 1121.63 & 358.93 \\\\\n& GM2 & 2.079 & 13.468 & 743.19 & 0.153 & 870.59 & 246.12 & 871.51 & 244.33 \\\\\n& GM3 & 2.019 & 13.146 & 607.88 & 0.157 & 724.64 & 189.05 & 728.79 & 184.81 \\\\\n& NL3 & 2.774 & 14.430 & 1222.72 & 0.143 & 1394.86 & 489.88 & 1410.87  & 495.58 \\\\\n& NL3-II & 2.773 & 14.408 & 1217.59 & 0.143 & 1381.14 & 483.69 & 1401.64 & 494.33 \\\\\n& NL-SH & 2.799 & 14.630 & 1368.94 & 0.141 & 1547.36 & 554.41 & 1563.07  & 565.93 \\\\\n& NL-RA1 & 2.785 & 14.490 & 1278.05 & 0.143 & 1440.78 & 508.58 & 1465.28 & 518.21 \\\\\nPure & NL3* & 2.762 & 14.355 & 1181.41 & 0.144 & 1341.70 & 465.63 & 1358.48 & 476.51 \\\\\nNucleonic & GMT & 2.662 & 14.355 & 1177.91 & 0.144 & 1342.24 & 456.92 & 1359.93 & 462.36 \\\\\nMatter & DD1 & 2.410 & 13.126 & 678.92 & 0.157 & 772.75 & 259.42 & 784.91 & 261.88 \\\\\n& DD2 & 2.418 & 13.133 & 683.97 & 0.157 & 778.13 & 261.79 & 787.99 & 265.77 \\\\\n& DD-ME1 & 2.443 & 13.086 & 672.37 & 0.158 & 765.14 & 261.09 & 776.04  & 266.31 \\\\\n& DD-ME2 & 2.483 & 13.146 & 706.08 & 0.157 & 798.93 & 276.48 & 811.16  & 281.47 \\\\\n& PKDD & 2.328 & 13.461 & 750.66 & 0.154 & 869.05 & 273.56 & 874.03 & 275.28 \\\\\n& TW99 & 2.076 & 12.245 & 402.46 & 0.169 & 474.17 & 134.40 & 474.57 & 133.73 \\\\\n& DDV & 1.929 & 12.360 & 398.67 & 0.167 & 485.52 & 117.40 & 479.76 & 112.95 \\\\\n& DDF & 1.956 & 11.871 & 311.618 & 0.174 & 372.22 & 94.59 & 375.27 & 91.84 \\\\\n& DD-MEX & 2.556 & 13.293 & 773.49 & 0.156 & 869.28 & 309.58 & 883.91  & 316.32 \\\\\n\\hline\n& DD1 & 2.039 & 13.124 & 680.44 & 0.158 & 777.19 & 262.33 & 783.48 & 264.47 \\\\\n& DD2 & 2.046 & 13.132 & 684.61 & 0.157 & 784.18 & 265.08 & 793.09 & 267.20 \\\\\nHypernuclear & DD-ME1 & 2.075 & 13.086 & 673.16 & 0.158 & 773.05 & 265.19 & 778.05 & 267.71 \\\\\nMatter & DD-ME2 & 2.115 & 13.146 & 707.53 & 0.157 & 808.16 & 277.39 & 813.83 & 282.33 \\\\\n& PKDD & 1.943 & 13.444 & 744.97 & 0.154 & 866.23 & 269.07 & 870.23 & 273.08 \\\\\n& DD-MEX & 2.186 & 13.293 & 773.49 & 0.156 & 873.62 & 312.38 & 884.72 & 319.36 \\\\\n\\hline\n& DD1 & 2.052 & 12.254 & 398.11 & 0.169 & 462.63 & 137.06 & 465.33 & 138.56 \\\\\n& DD2 & 2.059 & 12.260 & 402.03 & 0.169 & 466.22 & 139.86 & 469.16 & 140.69 \\\\\n$\\Delta$-admixed & DD-ME1 & 2.085 & 12.320 & 418.19 & 0.168 & 485.03 & 146.52 & 486.65 & 148.15 \\\\\nHypernuclear & DD-ME2 & 2.126 & 12.400 & 444.20 & 0.167 & 514.21 & 160.55 & 517.89 & 163.58 \\\\\nMatter & PKDD & 1.934 & 12.832 & 501.68 & 0.161 & 617.18 & 144.53 & 602.05 & 136.74 \\\\\n& DD-MEX & 2.198 & 12.588 & 503.30 & 0.164 & 577.19 & 189.24 & 586.07 & 192.52 \\\\\n\\hline\n\\end{tabular}\n\\label{tab:2}\n\\end{table*} \n\n\\begin{figure}\n  \\begin{center}\n\\includegraphics[width=8.5cm,keepaspectratio ]{Figures\/lam_radius_m2.pdf}\n\\caption{Similar to fig.-\\ref{fig-007} but for secondary components of GW170817 and GW190425 events. The solid, dotted, stripped squares represent the DD parametrizations with pure nucleonic, baryon octet, $\\Delta$-admixed hypernuclear matter respectively. The solid circles denote the NLW parametrizations with nucleonic matter case. The quadratic correlations for the GW170817, GW190425 cases are given by short-dashed and dash-dotted curves respectively. $\\tilde{\\Lambda}$ constraints are similar to fig.-\\ref{fig-007}.\n}\n\\label{fig-011}\n\\end{center}\n\\end{figure}\n\nSimilar to fig.-\\ref{fig-004}, the effective tidal deformability as a function of mass-asymmetry factor $q$ corresponding to the GW190425 event with a fixed chirp mass, $\\mathcal{M}=1.43~M_{\\odot}$ for all the parametrizations considered in this work is shown in fig.-\\ref{fig-005} with only nucleonic matter. In this case, masses of the two compact stars are varied in the ranges $1.60 \\leq M_1\/M_{\\odot} \\leq 1.87$ (primary) and $1.46 \\leq M_2\/M_{\\odot} \\leq 1.69$ (secondary) \\citep{2020ApJ...892L...3A}. Weak dependece of $\\tilde{\\Lambda}$ on $q$ can be inferred. It is observed that all the parametrizations satisfy the upper bound constraint on $\\tilde{\\Lambda}$ provided by GW190425 event data. Consequently this GW event does not provide enough information to put strict limits on constraining dense matter EOSs. Inclusion of $\\Delta$-quartet will decrease $\\tilde{\\Lambda}$ compared to pure nucleonic case.\n\nWe next evaluate the tidal deformabilities ($\\Lambda_1$, $\\Lambda_2$) of binary components involved in GW170817 event with different matter compositions. For the evaluation of $\\Lambda_1$ and $\\Lambda_2$, we consider \n$\\mathcal{M}=1.188~M_{\\odot}$ where $M_T=2.73-2.78~M_{\\odot}$. The masses of the two components are varied in $1.36 \\leq M_1\/M_{\\odot} \\leq 1.60$ (primary) and $1.17 \\leq M_2\/M_{\\odot} \\leq 1.36$ (secondary) ranges \\citep{LIGO_Virgo2017c}. \nFrom the left panel of fig.-\\ref{fig-006}, it is observed that for pure nucleonic matter the NLW CDF model parametrization GM1 do not lie within the $90\\%$ probability contours of $\\tilde{\\Lambda}\\sim 900$, although GM2 and GM3 do, while in case of DD CDF models, all the parametrizations lie inside these $\\tilde{\\Lambda}\\sim 900$ contours. \nIn the case of matter composition as $NY$, the tidal deformability is quite similar to the ones with pure nucleonic matter and hence not shown in fig.-\\ref{fig-006}.\nIn case of $\\Delta$-resonance admixed hypernuclear matter, for all the relevant EOSs $\\Lambda_1$, $\\Lambda_2$ falls well even within the  $\\tilde{\\Lambda}\\sim 720$ probability contour (obtained from recent reanalysis) as shown in the right panel of the figure.\nTable-\\ref{tab:2} provides the numerical estimates of various observational properties of different CDF models considering matter composition to be purely nucleonic, hypernuclear and $\\Delta$-resonance admixed hypernuclear matter.\n\\begin{figure}\n  \\begin{center}\n\\includegraphics[width=8.5cm,keepaspectratio ]{Figures\/Tidal_NYD.pdf}\n\\caption{Dimensionless tidal deformability ($\\Lambda$) \nas a function of the NS mass ($M$) for CDF model \nparametrizations considered in this work \n(with $N$, $NY\\Delta$-matter). DD, NLW type models with $N$-matter are represented similar to fig.-\\ref{fig-001},\\ref{fig-003}. While DD models with $NY\\Delta$-matter are represented by long-dashed curves. The horizontal dotted lines denote the upper bounds on $\\Lambda_{1.4}=800,580$ obtained from \\citet{LIGO_Virgo2017c} and recent reanalysis \\citet{LIGO_Virgo2018a} respectively (GW170817 event). The vertical range denotes joint constraints from NICER observations (PSR J$0030+0451$) $\\&$ GW170817 event \\citep{Jiang_2020} and another obtained implementing Bayesian analysis \\citep{2021EPJA...57...31L} for a $1.4~M_{\\odot}$ NS. The various astrophysical constraints are similar as in fig.-\\ref{fig-001}.}\n\\label{fig-008}\n\\end{center}\n\\end{figure}\n\nNext we attempt to restrict radius of compact stars by evaluating $\\tilde{\\Lambda}$ with particular star combinations for both the events considering different parametrization models. For the event GW170817, we have taken $M_1=1.50~M_{\\odot}$ and $M_2=1.24~M_{\\odot}$ and for the event GW190425 we have taken $M_1=1.70~M_{\\odot}$ and $M_2=1.59~M_{\\odot}$ where, $M_1$, $M_2$ correspond to primary and secondary components respectively. We plot the values of $\\tilde{\\Lambda}$ with radius of the primary star for respective EOSs in fig.-\\ref{fig-007}. Tight correlations between $\\tilde{\\Lambda}$ and $R(M_1)$ are given by the fits\n\\begin{equation*}\n\\begin{aligned}\n\\tilde{\\Lambda}^{\\text{(GW170817)}}_{\\text{fit}} & = 7571 - 1423R(M_1) + 68.93(R(M_1))^2, \\\\\n\\tilde{\\Lambda}^{\\text{(GW190425)}}_{\\text{fit}} & = 2638 - 501.2R(M_1) + 24.34(R(M_1))^2 ,\n\\end{aligned}\n\\end{equation*}\nwith maximum deviations, $(|\\tilde{\\Lambda}_{\\text{fit}}-\\tilde{\\Lambda}|\/\\tilde{\\Lambda})\\sim 2.65\\%,~7.45\\%$, \n$\\chi^2 = \\sum_{i}^{\\mathcal{N}}\\left[(\\tilde{\\Lambda}_{\\text{fit}}^i-\\tilde{\\Lambda}^i)^2\/\\tilde{\\Lambda}^i \\right] = 4.16,~7.16$ with $\\mathcal{N}=30$ EOS models and coefficients of determination, $\\mathcal{R}^2 = 1 - SS_{\\text{res}}\/SS_{\\text{total}}$\n$\\sim 0.999,~0.996$ for GW170817, GW190425 event respectively. Here, $SS_{\\text{res}}=\\sum_i (\\tilde{\\Lambda}_i-\\tilde{\\Lambda}_i^{\\text{fit}})^2$, $SS_{\\text{total}}=\\sum_i (\\tilde{\\Lambda}_i-\\bar{\\tilde{\\Lambda}}_i)^2$ are sum of squares of the residual errors and squared error of the mean line respectively.\nThe point where the curves cross $\\tilde{\\Lambda}$ bounds corresponds to limits on primary component's radius and hence the EOSs. \nThe figure shows that the upper bound on $\\tilde{\\Lambda} \\sim 900$ results in radius $\\leq 13.44$ km, which excludes certain NLW models except for GM2 and GM3 EOSs. On the other hand, if we consider the upper bound on $\\tilde{\\Lambda} \\sim 720$, the radius bound $\\leq 12.98$ km not only excludes all the NLW EOSs but also matter without $\\Delta$ resonances with DD parametrizations. From the lower bound of $\\tilde{\\Lambda} \\sim 400$, the lower bound on radius $\\geq 11.89$ km which excludes dense matter composed of only nucleons with DDF (DD type) parametrization. \nFrom the observation of GW190425, the upper bound on $\\tilde{\\Lambda}\\sim 600$ provides the upper bound on the primary component's radius $\\leq 15.00$ km, which gives no limit on the EOSs.\nSimilar to fig.-\\ref{fig-007}, effective tidal deformability as a function of secondary component's radius in GW170817 and GW190425 events is shown in fig.-\\ref{fig-011}. The correlation fits between $\\tilde{\\Lambda}$ and $R(M_2)$ are given by\n\\begin{equation*}\n\\begin{aligned}\n\\tilde{\\Lambda}^{\\text{(GW170817)}}_{\\text{fit}} & = 6249 - 1251R(M_2) + 63.62(R(M_2))^2, \\\\\n\\tilde{\\Lambda}^{\\text{(GW190425)}}_{\\text{fit}} & = 2783 - 527.5R(M_2) + 25.48(R(M_2))^2 ,\n\\end{aligned}\n\\end{equation*}\nwith maximum deviations of $\\sim 17.76\\%,~19.49\\%$, $\\chi^2 = 54.49,~17.44$ and $\\mathcal{R}^2 \\sim 0.983,~0.988$ for GW170817 and GW190425 events respectively. The upper bounds on radius for the secondary components in GW170817 case are estimated to be $13.37$ km ($\\tilde{\\Lambda} \\sim 900$) and $12.94$ km ($\\tilde{\\Lambda} \\sim 720$) while the lower bound is evaluated to be $11.99$ km ($\\tilde{\\Lambda} \\sim 400$). In case of GW190425 case, radius of the secondary component, $R_{1.59} \\leq 14.99$ km ($\\tilde{\\Lambda} \\sim 600$).\nFrom both the correlations it can be inferred that the radius bounds on NSs involved in GW events are approximately $12 \\leq R_{*}\/\\text{km} \\leq 13$ and $R_{*} \\leq 15$ km in GW170817 and GW190425 events respectively.\n\n\\begin{table}\n\\caption{Threshold densities denoted by $n_{u}^{Y}$ (in units of $n_0$) for onset of hyperons in hypernuclear dense matter for various DD CDF models. $n_{1.4}^{c(Y)}$ represents the central number density for a $1.4M_{\\odot}$ NS with hypernuclear matter composition. $\\Lambda_{1.4}^{N}$, $\\Lambda_{1.4}^{NY}$ are the dimensionless tidal deformability of a $1.4M_{\\odot}$ NS with nucleonic and hypernuclear matter respectively.}\n\\centering\n\\begin{tabular}{cccc}\n\\hline \\hline\nCDF Model & $n_{1.4}^{c(Y)}~(n_0)$ & $n_{u}^{Y}~(n_0)$ & $\\Lambda_{1.4}^{N}\/\\Lambda_{1.4}^{NY}$ \\\\\n\\hline\nDD1 & 2.42 & 2.26 & 0.998 \\\\\nDD2 & 2.40 & 2.25 & 0.999 \\\\\nDD-ME1 & 2.33 & 2.23 & 0.999 \\\\\nDD-ME2 & 2.25 & 2.21 & 0.998 \\\\\nPKDD & 2.48 & 2.15 & 1.01 \\\\\nDD-MEX & 2.13 & 2.15 & 1.00 \\\\\n\\hline\n\\end{tabular}\n\\label{tab:3}\n\\end{table} \n\nFig.-\\ref{fig-008} depicts the dimensionless tidal deformability parameter as a function of NS mass evaluated from the DD and NLW type models with matter composition to be nucleonic and $\\Delta$-commixed baryon octet. It is observed that among the NLW CDF models, GM2 and GM3 parametrizations fulfil the $\\Lambda_{1.4}=800$ upper bound (with $N$-matter), while they fail to satisfy the recent $\\Lambda_{1.4}=580$ bound. Other NLW parametrizations produce larger radii NSs and, as a result, are more inclined to be easily deformable since $\\lambda\\sim R^5$ (i.e. higher tidal deformability values). Hence no NLW model considered in this work satisfy the mass and tidal deformability constraints simultaneously. In the case of DD CDF models (with $N$-matter), all the models satisfy the upper bound constraint on $\\Lambda_{1.4}=800$. However, except TW99, DDV and DDF parametrizations none of them fulfil the upper bound ($\\Lambda_{1.4}=580$) obtained from reanalysis of GW170817 event data. \nNo coupling parameter set considered in this work is seen to satisfy the more strict constraints of $\\Lambda_{1.4}=580$ and maximum mass simultaneously with pure nucleonic matter.\nAnother joint constraint from NICER (PSR J$0030+0451$) and GW170817 data sets an upper bound on $\\Lambda_{1.4}=730$. \nRecent constraint on $\\Lambda_{1.4}$ obtained from Bayesian analysis provides an upper bound of $686$.\nDD CDF models (DD1, DD2, DD-ME1, DD-ME2) are observed to satisfy these criteria inclusive with the lower bound on $M_{\\text{max}}$\n(see table-\\ref{tab:2} for numerical results). It should be noted, as evident from table-\\ref{tab:3} appearance of heavier baryons are inevitable in a $1.4$ $M_\\odot$ star with all DD CDF parametrizations except DD-MEX.\n\n\\begin{figure}\n  \\begin{center}\n\\includegraphics[width=8.5cm,keepaspectratio ]{Figures\/lam_lam14.pdf}\n\\caption{Correlation between weighted average tidal deformability and tidal deformability for $1.4~M_{\\odot}$ NSs. The short-dashed line denotes the best fit for the EOS data sets. The bounds on ${\\tilde{\\Lambda}}\\sim 900,720$ (upper), $400$ (lower) corresponding to GW170817 event are represented by the horizontal dotted lines similar to fig.-\\ref{fig-007}. The vertical short-dashed lines mark the points where the linear fit intersects $\\tilde{\\Lambda}$ bounds. Parametrization models involved are similar as in fig.-\\ref{fig-007}.}\n\\label{fig-013}\n\\end{center}\n\\end{figure}\nFig.-\\ref{fig-013} displays the tight correlation between weighted average $\\tilde{\\Lambda}$ and $\\Lambda_{1.4}$ tidal deformability for GW170817 event data. In this case, we have considered $M_1=1.40~M_{\\odot}$, $M_2=1.33~M_{\\odot}$ with $\\mathcal{M}=1.1878~M_{\\odot}$. The tight linear correlation between $\\tilde{\\Lambda}$ and $\\Lambda_{1.4}$ is given by\n\\begin{equation*}\n\\begin{aligned}\n\\tilde{\\Lambda}^{\\text{(GW170817)}}_{\\text{fit}} & = 16.28 + 1.138\\Lambda_{1.4},\n\\end{aligned}\n\\end{equation*}\nwith maximum deviation of $\\sim 3.11\\%$, $\\chi^2=2.39$ and $\\mathcal{R}^2 \\sim 0.999$. The upper bounds on $\\Lambda_{1.4}$ are deduced to be $777,~619$ corresponding to $\\tilde{\\Lambda}\\sim 900,~720$ respectively. While the lower bound on $\\Lambda_{1.4}$ based on $\\tilde{\\Lambda}\\sim 400$ (AT2017gfo) is estimated to be $337$. The upper bound on $\\tilde{\\Lambda}$ favours the DD parametrizations as evident from fig.-\\ref{fig-013}.\n\n\\begin{figure}\n  \\begin{center}\n\\includegraphics[width=8.50cm,keepaspectratio ]{Figures\/tidal_compact.pdf}\n\\caption{Dimensionless tidal deformability as a function of compactness parameter ($C$) for an isolated NS (with $N$-matter) for the CDF parametrizations implemented in this work. The bounds on ${\\Lambda}\\sim 800,580$ corresponding to GW170817 event are represented by the horizontal dotted lines similar to fig.-\\ref{fig-008}.}\n\\label{fig-009}\n\\end{center}\n\\end{figure}\n\nWe have shown the dependence of $\\Lambda$ on compactness of the star for different parametrizations (with $N$-matter) in fig.-\\ref{fig-009}. From the figure, it is clear that the dependence of $\\Lambda$ on compactness is almost independent of EOSs. \nThis relates with the result from \\citet{2013PhRvD..88b3007M}.\nThen in fig.-\\ref{fig-010}, we plot the $\\Lambda$ for an isolated NS of mass $1.4 M_\\odot$ which shows a general trend with almost all EOSs. In order to find lower bound on the compactness parameter of a $1.4~M_{\\odot}$ NS ($C_{1.4}$), we compute the correlations between $\\Lambda_{1.4}$ and $C_{1.4}$ \nas $\\Lambda_{1.4}\/10^4 = 2.094 - 22.67 C_{1.4} +62.2 C^{2}_{1.4}$. \nThe maximum deviation estimated is $\\sim 11.94\\%$ with $\\chi^2=41.03$, $\\mathcal{R}^2$ corresponding to $0.989$. The stiffer NLW type parametrization models produce compact stars with larger mass and radius leading to higher tidal deformabilities.\nObviously, these stiffer EOSs do not fulfil the upper limit of $\\Lambda_{1.4} \\leq 800$. Softer parametrizations GM2 and GM3 satisfy the upper limit of $\\Lambda$.\nOn the other hand, DD type parametrizations considered in this work generate compact stars with $\\Lambda_{1.4} \\leq 800$. Inclusion of $\\Delta$-resonances reduces $R_{1.4}$ resulting in increase of $C_{1.4}$ keeping $\\Lambda$ less than its upper limit. \nFrom fig.-\\ref{fig-010}, it can be inferred that for a $1.4~M_\\odot$ NS, the lower bound in compactness is $0.153(0.154)$  \ncorresponding to $\\Lambda_{1.4}\\sim 800(777)$ and it is $0.160(0.159)$ following the stringent upper bound of $\\Lambda_{1.4}\\sim 580(619)$. \nIt also shows that the upper bound in compactness for a $1.4~M_\\odot$ NS is $0.173$ following the lower bound $\\Lambda_{1.4}\\sim 337$. \nThe points which lie away from the correlation fit are from GM2, GM3, PKDD coupling models. \nBased on the derived bounds of $C_{1.4}$, the range of $R_{1.4}$ is found to be $11.95-13.00$ km ($C_{1.4}\\sim 0.159-0.173$) and $11.95-13.42$ km ($0.154-0.173$).\nThe estimated bounds on $R_{1.4}$ satisfy the range $11.5 \\leq R_{1.4}\/\\text{km} \\leq 13.6$ as reported in \\citet{2006PhLB..642..436L} with the latter constrained from terrestrial experimental data.\n\n\\begin{figure}\n  \\begin{center}\n\\includegraphics[width=8.5cm,keepaspectratio ]{Figures\/tidal_compact_1.pdf}\n\\caption{Correlation between tidal deformability and compactness parameter for $1.4~M_{\\odot}$ NSs. The short-dashed line denotes the best fit for the EOS data sets. The bounds on ${\\Lambda_{1.4}}$ corresponding to GW170817 event are represented by the horizontal dotted lines similar to figs.-\\ref{fig-008},\\ref{fig-009} in addition to ones deduced from fig.-\\ref{fig-013}. The vertical short-dashed lines mark the points where the quadratic fit intersects $\\Lambda_{1.4}$ bounds. Parametrization models involved are similar as in fig.-\\ref{fig-007}.}\n\\label{fig-010}\n\\end{center}\n\\end{figure}\n\nIn addition to heavier baryons, meson such as (anti)kaon condensations may also come into the picture with rising energy density towards the interior of NSs. \\citet{PhysRevD.102.123007, PhysRevD.103.063004} reported that these meson condensations affect the lowering of maximum mass NS configurations and do not have any significant reduction in radii of $\\sim 1.4~M_{\\odot}$ NSs compared to pure hadronic matter cases.\n\n\\section{Conclusions and Outlook}\\label{sec:summary}\n\n\nThe BNS mergers provide intriguing information to constrain various theoretical formulations of dense nuclear matter EOSs. In this work, we have analysed two schemes of coupling parametrizations and tried to constrain purely nucleonic, hypernuclear, and $\\Delta$-baryon admixed hypernuclear matter EOSs within the CDF framework model via employing GW and other astrophysical observations.\n\nConfronting the coupling parametrizations considered in this work with the recent bounds on nuclear saturation properties infers that the NLW parameter sets fail to satisfy these empirical ranges while DD coupling models (except PKDD) are seen to fulfil these bounds. In addition, imposing the lower limit of maximum mass constraint rules out the relatively softer EOSs viz. GM2, GM3 among NLW models and DDF, DDV, TW99 among DD CDF models. On the other hand, very stiff NLW EOSs, except GM1 fail to satisfy the measured radius range from observation of PSR J$0030+0451$. In the core of NSs, with enhanced matter density, the appearance of heavier baryons are inevitable. Their appearance softens the matter, excluding some more parametrization viz. DD1, DD2, PKDD among DD CDF models and only admissible NLW model parametrization GM1. Even with pure nucleonic matter, the parametrization GM1 is ruled out because it is not as soft as to provide $\\tilde{\\Lambda}\\le 900$ from GW170817 event. Hence, none of the parameter sets within NLW model considered in this work can satisfy all the observational constraints simultaneously. The upper bound on $\\tilde{\\Lambda}\\le 900$ allow all the parametrizations of DD CDF models, although the strict upper limit $\\tilde{\\Lambda}\\le 720$ does not allow any parametrization with pure nuclear matter maintaining lower bound constraints of maximum mass. However, the appearance of heavier non-strange baryons allows all the DD CDF parametrizations to lie within the stringent upper limit.\n\nThe advent of heavier baryons (hyperons) leads to softening of EOSs, which in turn reduce the NS maximum masses values by $\\sim 0.38~M_{\\odot}$ than that of pure nucleonic cases. \nThe effect on dimensionless tidal deformability ($\\Lambda_{1.4}$) due to the incorporation of hyperons is very marginal.\nIt can be seen that the central number densities of $1.4M_{\\odot}$ NSs are quite close to the onset of hyperons. This attributes to the marginal changes in $\\Lambda_{1.4}$. \nHowever, the appearance of $\\Delta$-resonances in dense matter is worthy of mentioning. \nFrom fig.-\\ref{fig-008}, it can be seen that the value of $\\Lambda_{1.4}$ lies above the stringent $580$ upper bound for the stiffer DD CDF EOSs for pure nucleonic matter. With the advent of $\\Delta$-quartet, $\\Lambda_{1.4}$ reduces sufficiently and moves below $580$ for all stiff DD CDF parametrizations considered in this work. This relates with the results from \\citet{Li2019ApJ} and \\citet{2021PhLB..81436070R}. \n$\\Delta$-baryon admixed hypernuclear NSs are observed to have smaller radius values in comparison to the NSs with only baryon octet particle spectrum (refer to table-\\ref{tab:2}). This is due to the early onset of $\\Delta^-$ baryon, which relatively softens the EOSs at lower densities.\n\n\\cite{2021ApJ...915L...5A} recently reported the observation of GWs from two NS-BH coalescences (GW200105 and GW200115). The masses of the NSs involved are deduced to be $1.9^{+0.3}_{-0.2}~M_{\\odot}$, $1.5^{+0.7}_{-0.3}~M_{\\odot}$ for the GW200105 and GW200115 events respectively. However, no constraints on tidal deformation of the secondary components have been reported. The coupling parametrizations considered in this work satisfy the NS mass constraints set by these recent GW events. \n\nBased on the coupling data sets contemplated in this work and observed strict restrictions on $\\tilde{\\Lambda}$, a radius range of the NSs involved in GW170817 event is deduced to be around $12 \\leq R_{*}\/\\text{km} \\leq 13$ and that with GW190425 event's data is found to be $R_* \\leq 15.00$ km. The lower bound on $\\tilde{\\Lambda}$ considered in this work is evaluated from the electromagnetic counterpart (kilonova) signal in GW170817 event.\nThe strong correlation between weighted average $\\tilde{\\Lambda}$ and $\\Lambda_{1.4}$ sets bounds as $337 \\leq \\Lambda_{1.4} \\leq 619$ corresponding to $400 \\leq \\tilde{\\Lambda} \\leq 720$.\nFurthermore, similar analysis with strict bounds on $\\Lambda_{1.4}$ translate to compactness parameter value of a $1.4~M_{\\odot}$ NS to be in the range $0.159 \\leq C_{1.4} \\leq 0.173$. This yields $R_{1.4}$ in the range $11.95-13.00$ km.\n\nAnother inference from the GW observational constraints is the possible existence of quark matter in the interior of NSs. Many works \\citep{Nandi_2018, 2019ApJ...877..139G,2018PhRvD..97h4038P, 2019MNRAS.489.4261M, 2021EPJST.tmp....5N, PhysRevC.103.055814} reported hybrid star configurations consistent with the strict upper bounds on $\\tilde{\\Lambda}$, $\\Lambda_{1.4}$ and massive NS observations. Since the knowledge regarding high-density quark matter is still inadequate, we could explore and constrain the parameter space of the functionals involved in describing the quark matter behaviour at such density regimes based on GW events and recent NICER data. Further analysis on the aspect is beyond the scope of this work and will be discussed in future studies.\n\n\\section*{Acknowledgements}\nThe authors thank the anonymous referee for constructive comments that significantly contributed to enhancing the manuscript's quality.\nThe authors acknowledge financial support from Science and Engineering Research Board, Department of Science and Technology, Government of India through Project No. EMR\/2016\/006577 and Ministry of Education, Government of India. VBT and MS are also thankful to Sarmistha Banik for fruitful discussions.\n\n\\section*{Data Availability}\nData sharing not applicable to this article as no data sets were generated during this study.\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\par\nThe unconventional thermal transport properties of low dimensional \nquantum magnets have drawn the attention of the condensed matter \nsociety.\\cite{zotos-prelovsek, \nPhysRevLett.89.156603, \nPhysRevB.66.140406,PhysRevB.67.134426, PhysRevB.67.224410,  \nPhysRevB.67.064410,PhysRevB.71.184415, PhysRevB.68.104401, \nPhysRevLett.94.087201, PhysRevLett.96.067202, \nPhysRevB.79.024425, sologubenko-2007, hess-2007, nov-Otter2009796, \nJPSJ.71.2485} %\nIn particular, the transport properties of spin chain materials  \nare described by the one-dimensional (1D) \n\\mbox{spin-$\\frac{1}{2}$} Heisenberg model, where the \nlarge  exchange coupling along the \nchain\\cite{PhysRevLett.73.332, PhysRevB.53.5116, PhysRevLett.76.3212} %\naccounts for an extraordinary high and anisotropic thermal \nconductivity.\\cite{PhysRevB.64.054412}\nActually, the pure anisotropic Heisenberg model (AHM) was shown to \nexhibit ballistic  thermal  transport at any temperature $T$,  attributed\nto its  integrability.\\cite{PhysRevB.55.11029, 0305-4470-35-9-307, 0305-4470-36-46-006} %\nIn accord with the theoretical predictions, evidence of \nballistic thermal transport in real materials has been  consolidated lately using \nsamples of very high purity.\\cite{PhysRevB.81.020405}\nHowever, defects may have a dramatic effect on the transport properties\nof these systems due to the reduced dimensionality.\n\\par\nIn this work, we try to shed light on the  effect of a single, \\mbox{spin-$S$} \nmagnetic impurity, on the transport properties of a \\mbox{spin-$\\frac{1}{2}$} \nHeisenberg chain. We address the behavior of the system on various problem \nparameters at high and intermediate $T$,  while we support our numerical \nresults  with analytical arguments. \nIt is well known that magnetic impurities get \nscreened by the chain at low $T$ forming with their neighbors an effective \nimpurity of different \nspin.\\cite{PhysRevB.46.10866, PhysRevLett.86.516, PhysRevB.64.092410} %\nNotwithstanding, as we show in this work, conspicuous excitations, \nwhich cannot be described by the effective impurity picture,  \nsurvive at low $T$; this holds for a single impurity in a finite system or \nplausibly for a finite concentration of impurities in the thermodynamic limit. %\nFurthermore, the screening of the impurity triggers \nKondo-type, many body,  phenomena below a crossover \ntemperature, and the chain becomes perfectly (insulating)transmitting  at $T=0$\nfor (anti)ferromagnetic  easy axis anisotropy  \n$\\Delta$.\\cite{PhysRevB.58.5529, PhysRevB.81.205101} %\nThis is an analog of the behavior of a Luttinger liquid \nin the presence of a non-magnetic impurity  where the chain is (cut)healed for \n(repulsive)attractive interactions.\\cite{PhysRevLett.68.1220} %\nWe seek evidence of the cutting-healing behavior of the chain  \nusing the  thermal conductivity  as a probe, which is ideal for this purpose \nsince the  only energy current relaxation mechanism  is induced by the impurity. \n\\par\nFrom the above it is obvious that the role of the  thermal conductivity in these \nsystems is exceptional and twofold. Not only is the thermal conductivity \na very useful theoretical tool but it is of high experimental \nand technological interests as well. Thus, our results could serve \nas  qualitative guidelines for forthcoming thermal conductivity measurements \nin doped cuprates, e.g., $\\mathrm{Ni}$ doping in the renowned \n \\mbox{$\\mathrm{SrCuO}_{2}$}  and   \n \\mbox{$\\mathrm{Sr}_{2}\\mathrm{CuO}_{3} $} cuprates. \nFurthermore,  magnetic impurities in highly heat conducting \nmaterials could function as potential switching mechanisms  \nenabling tunable heat transport and the emergence of \nnumerous technological applications.  \nTherefore, it is vital  to understand  the fundamental \nproperties of  a prototype model in the presence of magnetic impurities \nbefore  trying to exploit the transport properties of these truly  unique systems. \n\\section{Model}\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[scale=0.24]{model}\n\\caption{Spin-$S$ impurity embedded in a spin-$\\frac{1}{2}$ chain. \n}\\label{fig-1} \n\\end{figure}\n\\par\nIn a previous work,\\cite{PhysRevB.81.205101} the thermal transport properties \nof a \\mbox{spin-$\\frac{1}{2}$} Heisenberg chain was studied in the presence \nof a spin-$S$ magnetic impurity located out of the chain. In addition,  \na uniform chain with two consecutive weak links was discussed, \nwhich can be considered as a special case of a spin-$\\frac{1}{2}$ impurity \nembedded in the chain. In the present \nwork, we deal with the generic case of a spin-$S$ magnetic impurity embedded \nin the chain, Fig.\\ \\ref{fig-1}. Many of the conclusions and the basic ideas reached \nin Ref.\\cite{PhysRevB.81.205101} can be applied in the present work, as well,  \ndue to the single impurity character of the problem. However, significant \ndifferences in the physical results arise due to the different geometries of the \nproblems. A brief discussion and comparison between the two models is given \nat the end of the manuscript. \n\\par\nThe Hamiltonian of an  AHM on a ring of $L$ sites, \nwhere  a \\mbox{spin-$\\frac{1}{2}$} operator, say the $\\mathbf{s}_{0}$, \nis substituted by another one of different spin, say $\\mathbf{S}$, \nis given by \n\\begin{equation}\\label{hamiltonian}\nH=\\sum_{l=1}^{L-2} h_{l}+J'~\\big(\\mathbf{s}_{1}+\\mathbf{s}_{L-1}\\big)\\cdot \\tilde{\\mathbf{S}}\\:, \\quad\n\\big(h_{l}=J \\mathbf{s}_{l}\\cdot\\tilde{\\mathbf{s}}_{l+1}\\big)\\:,\n\\end{equation}\nsee  Fig.\\ \\ref{fig-1} as well. In Eq.\\ (\\ref{hamiltonian}), \n$J$  is the antiferromagnetic exchange coupling---with energy units---between  \n\\mbox{spin-$\\frac{1}{2}$} operators,  $J'$ is the host-impurity coupling,  \nwhile the tilded operators $\\tilde{\\mathbf{s}}$, $\\tilde{\\mathbf{S}}$ denote  the vector operators \n$(s^{x},s^{y},\\Delta s^{z})$,  $(S^{x},S^{y},\\Delta S^{z})$, respectively.   \nIn addition, we work in a system of units where the lattice constant $a$,  \nand the Planck and Boltzmann  constants are  $a,\\hslash,k_{B}=1$; yet numerical \nresults are presented for $J=1$.\n\\par\nThe spin $\\jmath^{s}$, energy $\\jmath^{\\epsilon}$  current operators \nare determined by the continuity  equation \n$\\partial_{t}{\\hat{O}}_{l}+\\nabla\\mathcal{J}_{l}=0$. Taking \n$\\hat{O}_{l}$ to be  the local spin, energy operators and \n$\\mathcal{J}_{l}$ to be the local spin, energy current operators \nrespectively, we arrive at\n\\begin{equation}\\label{spcur}\n\\jmath^{s}= \\sum_{l=1}^{L-2} J\\big(\\mathbf{s}_{l}\\times\\mathbf{s}_{l+1}\\big) \\cdot \\hat{e}_{z}+\nJ'\\big[\\big(\\mathbf{s}_{L-1}-\\mathbf{s}_{1}\\big)\\times\\mathbf{S}\\big]\\cdot\\hat{e}_{z}\\:,\n\\end{equation}\nwhere $\\hat{e}_{z}$ is the unit vector along the $z$-axis, and \n\\begin{eqnarray}\n\\jmath^{\\epsilon}&=&\\sum_{l=2}^{L-2}J^{2}\\mathbf{s}_{l}\\cdot\\big(\\tilde{\\mathbf{s}}_{l+1}\\times\\tilde{\\mathbf{s}}_{l-1}\\big)+\nJJ'\\mathbf{s}_{1}\\cdot\\big(\\tilde{\\mathbf{s}}_{2}\\times\\tilde{\\mathbf{S}}\\big)\\nonumber\\\\ &+&\nJJ'\\mathbf{s}_{L-1}\\cdot\\big(\\tilde{\\mathbf{S}}\\times\\tilde{\\mathbf{s}}_{L-2}\\big)+\nJ'^{2}\\mathbf{S}\\cdot\\big(\\tilde{\\mathbf{s}}_{1}\\times\\tilde{\\mathbf{s}}_{L-1}\\big)\\:. \\label{encur}\n\\end{eqnarray}\n\\par \nWithin linear response theory, the real part of the spin $\\sigma'$, \nthermal $\\kappa'$ conductivities are given \nby\\cite{JPSJ.12.570,PhysRev.135.A1505,PhysRevB.73.085117}\n\\begin{equation}\n\\sigma'(\\omega)=2\\pi D_S \\delta(\\omega)+\\sigma(\\omega)\\:,~~\n\\kappa'(\\omega)=2\\pi D_E \\delta(\\omega)+\\kappa(\\omega)\\:, \\nonumber\n\\end{equation}\nwhere the corresponding spin $D_S$, energy $D_E$ stiffnesses denote the \npresence of ballistic transport in the system. The regular components \n$\\sigma$, $\\kappa$ of the corresponding spin, thermal conductivities  are \ngiven by\n\\begin{eqnarray}\n\\sigma(\\omega)\\hskip-0.1cm&=&\\hskip-0.1cm\\frac{\\pi}{L}\\frac{1-e^{-\\beta\\omega}}{\\omega}\n\\hskip-0.15cm\\sum_{\\substack{n,m \\\\ (\\epsilon_{n}\\neq \\epsilon_{m})}}\\hskip-0.15cm\np_n|\\langle n|\\jmath^{s}| m\\rangle|^{2}\n\\delta(\\omega_{mn}-\\omega),~~  \\label{RegularConductivitys}\\\\\n\\kappa(\\omega)\\hskip-0.1cm&=&\\hskip-0.1cm\\frac{\\pi\\beta}{L}\\frac{1-e^{-\\beta\\omega}}{\\omega}\n\\hskip-0.3cm\\sum_{\\substack{n,m \\\\ (\\epsilon_{n}\\neq \\epsilon_{m})\n}}\\hskip-0.2cm\np_{n} |\\langle n|\\jmath^{\\epsilon}| m\\rangle|^{2}\n\\delta(\\omega_{mn}-\\omega),~~\\label{RegularConductivityk}\n\\end{eqnarray}\nwhere $\\epsilon_n$ are the eigenvalues and \n$|n\\rangle$ are the eigenstates  of the Hamiltonian (\\ref{hamiltonian}), \n$p_n=\\exp(-\\beta\\epsilon_n)\/Tr \\exp(-\\beta H)$, \n$\\omega_{mn}=\\epsilon_m-\\epsilon_n$, and $\\beta=1\/T$. %\n\\par\nWhile in the pure AHM it is clear that $D_E$ is finite for any value \nof the easy axis anisotropy \n$\\Delta$,\\cite{PhysRevB.55.11029, 0305-4470-35-9-307, 0305-4470-36-46-006} \nthere is an ongoing debate on  \nthe behavior of the spin stiffness.\\cite{PhysRevLett.103.216602,\nPhysRevB.68.134436,JPSJS.74S.181, \nPhysRevE.82.040103, PhysRevLett.88.077203, PhysRevLett.65.243, PhysRevB.53.983,\nPhysRevB.59.7382, PhysRevB.58.R2921} \nNevertheless, a single impurity renders ballistic transport \nincoherent and both $D_S$, $D_E$ vanish.\\cite{PhysRevB.80.125118} %\nThus, the static component of the transport quantities is given \nby the resistive dc spin, thermal  conductivities obtained by \nthe zero frequency limit of the regular components \n(\\ref{RegularConductivitys}),  (\\ref{RegularConductivityk}), \nviz.,  $\\sigma_{dc}=\\sigma(\\omega\\rightarrow0)$, \n$\\kappa_{dc}=\\kappa(\\omega\\rightarrow0)$.\n\\par\nTo  numerically study transport quantities for  systems with a  Hilbert space \n dimension up to $\\mathcal{D}\\sim10^{4}$, at high temperatures,   \nwe employ the  exact  diagonalization (ED) technique. \nThe \\mbox{$\\delta$-peaks} at the excitation frequencies are binned    \nin windows $\\delta\\omega=0.01$ while we introduce an additional \nbroadening  $\\eta=0.03$ using the well-known Kramers-Kronig relations. \nFor $\\mathcal{D}\\gtrsim10^{4}$, \nwe use  the  microcanonical   Lanczos method\\cite{PhysRevB.68.235106} \n (MCLM)   at high temperatures and the finite \ntemperature Lanczos method\\cite{sic-ftlm} (FTLM) at  finite temperatures;   \nyet we   keep the same additional  broadening. Lastly,  all results are obtained \n in the  $S^{z}_{\\text{total}}=0$ subsector.   \n\\section{Numerical results}\n\\subsection{Spin-1 impurity}\n\\par\nFirst, we would like to address the $S=1$ impurity case  since this may be \nthe most appealing magnetic impurity doping  for experiments. \nIn Fig.\\ \\ref{ks-dc}, we present results for the dc spin $\\sigma_{dc}$, \nthermal $\\kappa_{dc}$  conductivities, while in Figs.~ \\ref{klor}, and \\ref{k-ws} we \npresent the dynamical  \nthermal conductivity of the  isotropic ($\\Delta=1$) Heisenberg chain.   \nA  wide range  of host-impurity \ncouplings, $J'\/J=0.4-2.0$, is shown at high temperatures,  $\\beta\\rightarrow0$,  \nfor various lattice sizes, $L=15-21$.   \nResults for $L=15$ are obtained \n via the ED technique, while results for $L>15$ are obtained by \n employing the MCLM.  In order to eliminate  unimportant \n for this  discussion prefactors  of the \ntransport quantities, we plot either the normalized thermal conductivity,   \n$\\bar{\\kappa}(\\omega)=\\kappa(\\omega)\/\\kappa_{0}$ where the normalization \n$\\kappa_{0}$ \nis given by $\\kappa_{0}=\\int \\kappa(\\omega) d\\omega$, %\nor the non-trivial at high temperatures \n$T\\sigma_{dc}$, $T\\sigma(\\omega)$,  $T^{2}\\kappa_{dc}$ and $T^{2}\\kappa(\\omega)$. \nFinally, for the discussion of the lattice size scaling we plot \nthe scaled thermal conductivity  \n$\\kappa(\\omega L)\/L$.\\cite{PhysRevB.81.205101, PhysRevB.80.125118} %\n\\begin{figure\n\\centering\n\\includegraphics[scale=0.30]{fig-ks-dc}\n\\caption{(color online).\\  \nThe dc value of the thermal $T^{2}\\kappa_{dc}$, spin  $T\\sigma_{dc}$ \nconductivity is shown as a function of the host-impurity coupling $J'\/J$,  obtained \nvia ED for $L=15$, $S=1$, $\\Delta=1$, and  $\\beta\\rightarrow0$. }\\label{ks-dc}\n\\end{figure} %\n\\par \nFor a very weak or very strong $J'$,  a severe reduction of  $\\kappa_{dc}$ \nfrom its maximum value, which occurs at $J'=0.8J$, is illustrated \n in Fig.\\ \\ref{ks-dc}. %\nOn the contrary, the behavior of $\\sigma_{dc}$ is qualitatively different   \nfrom the one of $\\kappa_{dc}$.  \nBesides the qualitative difference, it is rather impressive that $\\sigma_{dc}$ hardly \nchanges in the wide range of $J'$ shown in Fig.\\ \\ref{ks-dc}. \nAlthough it is reasonable that \n$\\sigma$ will be less sensitive to the effect of the  single impurity  due to the \nbulk scattering---$[H,\\jmath^{s}]\\neq0$ in the pure model---its rigidity is still surprising.  \n\\begin{figure\n\\centering\n\\includegraphics[scale=0.30]{fig-klor}\n\\caption{(color online).\\  \nThe frequency dependence of the normalized and scaled \nthermal conductivity $\\bar{\\kappa}(\\omega L)\/L$ is shown \nfor $L=15,19,21$ and $J'\/J=0.8$; \nin addition  a Lorentzian fit is depicted. \nResults are obtained for $S=1$, $\\Delta=1$, and  $\\beta\\rightarrow0$. \n$L=15$ results are obtained  via ED and $L>15$ via MCLM.}\\label{klor}\n\\end{figure} %\n\\par\nIn Fig.\\ \\ref{klor},   $\\bar{\\kappa}(\\omega L)\/L$ is depicted\nfor  $J'=0.8J$ and a  Lorentzian fit as well,  \nsignifying  the Lorentzian behavior of $\\kappa$, \n$\\kappa(\\omega)=\\kappa_{dc}\/[1+(\\omega\\tau^{\\epsilon})^{2}]$ with \n$\\tau^{\\epsilon}$ the scattering time. \nThe thermal conductivity retains  its Lorentzian shape, approximately, \nin a range of values of the host-impurity coupling,  $J'\/J\\simeq0.8\\pm0.2$. \nAs previously proposed,\\cite{PhysRevB.81.205101}  a Lorentzian \nform of $\\kappa(\\omega)$ is an indication of a weak perturbation, \nwhile on the contrary a non-monotonic form  implies that the system has flown \nto  the strong perturbation regime.\nThe Lorentzian behavior is retained, with an almost constant $\\tau^{\\epsilon}$,  \nfor temperatures as  low as the limiting FTLM temperature for finite size systems \n$T_{fs}$ below which FTLM results are not reliable;\\cite{sic-ftlm}\nwe estimate $T_{fs}\/J\\simeq0.3$.\nAlthough $T_{fs}$ is sufficient to reveal the cutting \nof the chain for strong perturbations,\\cite{PhysRevB.46.10866}\n for weak perturbations a lower $T_{fs}$ is  required.\\cite{PhysRevB.81.205101}\nTo obtain an impression of  $T_{fs}$, consider that  \nfor systems like the \\mbox{$\\mathrm{Sr}_{2}\\mathrm{CuO}_{3}$} compound   \nwith $J\/k_{B}\\approx2500$ K, $T_{fs}\\approx750$ K. However, \ndefects may reduce $J$ of  doped samples\\cite{PhysRevB.64.092410}  \nbringing  $T_{fs}$  closer to  room  temperature. \n\\begin{figure\n\\centering\n\\includegraphics[scale=0.30]{fig-k-ws}\n\\caption{(color online).\\  \na) The frequency dependence of the normalized and scaled \nthermal conductivity $\\bar{\\kappa}(\\omega L)\/L$  is shown \nfor $L=15,19,21$ and $J'\/J=1.5$. \nb) The frequency dependence of $T^{2}\\kappa(\\omega L)\/L$ \nis shown for $L=15$ and two \nhost-impurity  couplings $J'\/J=0.4,2.0$. %\nResults are obtained for $S=1$, $\\Delta=1$, and $\\beta\\rightarrow0$. \n$L=15$ results are obtained  via ED and $L>15$ via MCLM. \n}\\label{k-ws} \n\\end{figure}\n\\par\nFor extreme values of the coupling $J'$, either strong or weak, $\\kappa(\\omega)$\nexhibits a strongly non-monotonic behavior, \nFigs.\\ \\ref{k-ws}(a), and \\ref{k-ws}(b), indicating  \nthat the system couples strongly to the impurity in both cases.  \nWhereupon, the low frequency \nbehavior ($L\\omega\/J \\lesssim\\pi$)\ncorresponding to an open-like chain,\\cite{PhysRevB.77.161101} \nis similar  for a weak and a strong \n$J'$,  Fig.\\ \\ref{k-ws}(b). %\nOn the other hand, the high frequency behavior of $\\kappa(\\omega)$ for  \na weak and a strong $J'$ is  strikingly different due to the emergence   \nof a conspicuous secondary structure at high frequencies. \nThe frequency of this  structure shifts with $J'$,\nindicating that its origin is \nlocal excitations of the impurity. Moreover,  the larger the $J'$, the more \nweight is accumulated  at this structure,\nwhich  becomes the prevalent contribution to $\\kappa(\\omega)$ \nfor fairly strong couplings, despite being only a $1\/L$ effect. \n\\subsection{Spin-$S$ impurity}\n\\par\nThe fact that the contribution of a $1\/L$ effect to $\\kappa(\\omega)$\nsurpasses the bulk contribution may seem quite bizarre, however, we\ncan comprehend the origin of this effect from the analytical expression  \n of  the  sum rule of $\\kappa(\\omega)$.\nSince the energy stiffness $D_{E}$ vanishes in  the presence of impurities \nthe sum rule of the thermal conductivity $\\kappa'(\\omega)$ is equal to the \nzeroth moment ($\\kappa_{0}$) of the regular part $\\kappa(\\omega)$.\nStarting from Eq. (\\ref{RegularConductivityk}) and taking the high temperature  \nlimit  the sum rule of the thermal conductivity can be written  as \n the thermodynamic \naverage of the square of the energy current operator\\cite{PhysRevLett.92.067202} \n\\begin{equation}\\label{Sumrule1}\n\\int_{-\\infty}^{+\\infty}\\hskip-0.25cm \\kappa(\\omega)d\\omega=\\frac{\\pi\\beta^2}{L} \\langle \\jmath^{\\epsilon} \\jmath^{\\epsilon} \\rangle,~~\n\\text{with, }~~~\n\\langle \\hat{O} \\rangle = \\frac{Tr e^{-\\beta H} \\hat{O}}{Tr e^{-\\beta H}},   \n\\end{equation}\nand  $\\pi\\beta^2 \\langle \\jmath^{\\epsilon} \\jmath^{\\epsilon} \\rangle\/L=\\kappa_{0}$. \nTaking the infinite temperature limit ($\\beta\\rightarrow0$) in the \nevaluation of the thermodynamic average, Eq.\\ (\\ref{Sumrule1}),  one arrives at \n\\begin{equation}\\label{SumRule}\n\\kappa_{0}=\\pi\\frac{1+2\\Delta^{2}}{T^{2}}\\left(\n\\frac{J^{4}}{32}\\Big(1-\\frac{3}{L}\\Big)\n+\\frac{\\mathcal{B}^{2}}{8L}\\Big(2J^{2}\n+J'^{2}\\Big)\\right)\\:. \n\\end{equation}\n$\\mathcal{B}^{2}=\\frac{1}{3}J'^{2}S(S+1)$ is  the characteristic spin \nimpurity dependence. The bulk contribution, $\\sim J^{4}$, to $\\kappa_{0}$ \nis equal to the impurity contribution, $\\sim J'^{4}$   \n(we omit  the  $\\sim (JJ')^{2}$ terms), \nfor a coupling \n$J'\\simeq J^{*}=J\\sqrt[4]{\\frac{3}{4}\\frac{L-3}{S(S+1)}}$. For one thing, \nfor a finite system studied via ED of \n$L=13$, $S=2$ we have  $J^{*}\\simeq J$. Thus, $\\kappa(\\omega)$  will \nexhibit resonant modes for  strong perturbations, at frequencies \n$\\omega\\sim J'$---at least at high  temperatures.  %\n\\begin{figure\n\\centering\n\\includegraphics[scale=0.3]{fig-kvrs}\n\\caption{(color online).\\ The frequency dependence of the thermal conductivity \n$T^2\\kappa(\\omega)$ is shown for two impurities $S=3\/2,2$ with \n$J'\/J=0.6,0.5$ and $L=14, 13$, respectively.  \nInset: $T^2\\kappa_{dc}$ as a function of the coupling $J'\/J$ for $S=3\/2,2$. \nResults are obtained via ED for  $\\Delta=1$, $\\beta\\rightarrow0$.\n}\\label{kvrs} %\n\\end{figure} %\n\\begin{figure\n\\centering\n\\includegraphics[scale=0.3]{fig-svrs}\n\\caption{(color online).\\  \nThe frequency dependence of the spin conductivity $T\\sigma(\\omega)$  is shown \nfor various perturbations: $S=1,3\/2,2$ with  $J'\/J=1.0$ and $L=15, 14, 13$, respectively,  as well as the case  $S=1$, $L=15$ with $J'\/J=1.5$. %\nResults are obtained via ED for $\\Delta=1$, $\\beta\\rightarrow0$. \n}\\label{svrs} \n\\end{figure}\n\n\\par\nAnother important conclusion that arises from the evaluation of $J^{*}$ is \nthat the larger the $S$ the smaller the $J^{*}$, since $J^{*}\\sim 1\/\\sqrt{S}$. \nThus, the local excitations of the impurity dominate even more easily \nover the bulk contribution for larger $S$. As a matter of fact,  we did not \nobtain a Lorentzian  $\\kappa(\\omega)$ for any $S>1$ by adjusting \nthe host-impurity coupling, implying that any $S>1$ constitutes a strong \nperturbation for $\\kappa(\\omega)$ while $S=1$ seems to be quite unique.  \nAs far as $\\sigma(\\omega)$ is concerned, its high frequency regime  is \ndominated  by  the impurity local excitations, similarly to $\\kappa(\\omega)$,  \nbut its low frequency  $\\sigma(\\omega\\rightarrow0)$ regime remains \nvirtually unaffected for higher $S$. This is a striking  difference between \n$\\sigma_{dc}$ and $\\kappa_{dc}$\nwhere the former shows a surprising rigidity to the influence of the \nimpurity (large $S$, strong\/weak $J'$) while the latter is severely reduced \nin the strong perturbation regime. %\nThe above arguments are summarized in Figs.\\ \\ref{kvrs}, and \\ref{svrs} where  \nthe frequency dependence of the thermal, spin conductivity is shown respectively. \nIn  Fig.\\ \\ref{kvrs},  $T^2\\kappa(\\omega)$ is shown for $S=3\/2,2$\nand $J'\/J=0.6,0.5$, namely, the corresponding couplings for which $\\kappa_{dc}$ is \nmaximum, Fig.\\ \\ref{kvrs} inset. For the case of the spin transport we present \n$T\\sigma(\\omega)$ in Fig. \\ref{svrs} for various moderate-strong \nperturbations. %\n\\subsection{Lattice size scaling}\n\\par\nLet us now turn our attention to the lattice size scaling. A \nLorentzian $\\kappa(\\omega)$ trivially\nobeys  a universal $L$ scaling, with the size independent \nquantity to be $\\kappa(\\omega L)\/L$, since $\\kappa_{dc}, \\tau^{\\epsilon}\\sim L$, \nFig.\\ \\ref{klor}.\\cite{PhysRevB.81.205101} %\nOn the other hand, for strong $J'$, $\\kappa(\\omega)$ exhibits two $L$-scalings; \nthe prominent  impurity contribution at $\\omega\\sim J'$, which is $\\mathcal{O}(1)$, \ndoes not scale with $L$, Fig.\\ \\ref{k-ws}(a), while \n the low frequency  bulk contribution obeys the proposed  scaling for any $J'$.\nMoreover, in accord with the evaluation of $J^{*}$  \n(Eq.\\ (\\ref{SumRule})),  Fig.\\ \\ref{k-ws}(a) shows that the contribution of a \nsingle impurity dwindles  with respect to the bulk contribution with increasing \n$L$, becoming  negligible in the limit $L\\rightarrow\\infty$. However, considering the \nthermodynamic limit and a  finite but dilute impurity concentration \n$c_{I}$---which actually  would be a more pragmatic approach to a real \nsystem---one could plausibly assume that  the high frequency structure \nwill be present, and similar scaling   behaviors would hold with the substitution  \n$1\/L\\rightarrow c_{I}$. %\n\\section{Strong coupling limit}\n\\begin{figure\n\\centering\n\\includegraphics[scale=0.7]{fig-spectrum}\n\\caption{The energy spectrum of the isotropic \\mbox{3-spin} system, \nEq.\\ (\\ref{LocalHamiltonian}).}\\label{fig-3} \n\\end{figure}\n\\par\nInteresting conclusions can be reached in the strong host-impurity \nlimit, $J'\\rightarrow\\infty$, and are rather useful  to understand \nthe high frequency behavior of $\\kappa(\\omega)$, $\\sigma(\\omega)$. \nStarting from Eq.\\ (\\ref{hamiltonian}) for the isotropic point ($\\Delta=1$)  \nand taking $J=0$, we end up in the \nHamiltonian of a \\mbox{3-spin} system \n\\begin{equation}\\label{LocalHamiltonian}\n\\mathcal{H}=J'(\\mathbf{s}_{L-1}+\\mathbf{s}_{1})\\cdot\\mathbf{S}\\:.\n\\end{equation}\nExploiting the rotational symmetry and the limited degrees \nof freedom of $\\mathcal{H}$,  we  perform an analytical \ndiagonalization into different \\mbox{$S_{z}$-total} subsectors. \nThe lowest\/highest $|S_{z}|=S+1$ subsectors are  \n$\\mathcal{D}=1$ Hilbert spaces. The second lowest\/highest  $|S_{z}|=S$ \nsubsectors are $\\mathcal{D}=3$ Hilbert spaces. The rest $2\\mathcal{S}+1$ $S_{z}$  subsectors, \nwith $\\mathcal{S}=S-1$,  are  $\\mathcal{D}=4$  Hilbert  spaces. The energy spectrum \nof the \\mbox{3-spin} system is shown in  Fig.\\ \\ref{fig-3}.\n\\par \nConsidering the local energy current operator   \n$\\tilde{\\jmath^{\\epsilon}}=J'^{2}\\mathbf{S}\\cdot(\\tilde{\\mathbf{s}}_{1}\\times\\tilde{\\mathbf{s}}_{L-1})$, we can see that \nits matrix elements vanish \nbetween non-zero eigenvalues, \nand, consequently, only transitions  between zero and non-zero eigenvalues survive; \nthis holds apart from the isotropic point as well.  Particularly, for \nthe isotropic Heisenberg model, the only non-vanishing transitions \ncorrespond to an energy difference $\\delta\\epsilon=\\pm J'$. \nThus, for $\\Delta=1$, $\\kappa(\\omega)$  \nwill exhibit only one sharp peak at high frequencies located at \n$\\omega\\simeq J'$ and it will be independent of $S$. \nTransitions corresponding to $\\delta\\epsilon=\\pm J'$ are between elevated \neigenstates, Fig.\\ \\ref{fig-3}, thus the peak at $\\omega\\simeq J'$  \nis expected to diminish  with decreasing temperature  \nand finally to vanish  as the system flows towards its ground state. These \nselection rules do not hold for the spin transport where there are more \nallowed transitions. These transitions  involve the ground state $\\epsilon_{g}$ \nas well and consequently  resonant peaks  will be present at low $T$. \nLastly, for  $\\Delta\\neq1$ there are transitions of  $\\tilde{\\jmath^{\\epsilon}}$ \nwhich do not  necessarily correspond  to  $\\delta\\epsilon=\\pm J'$ \nand yield a more   complicated high frequency structure for $\\kappa(\\omega)$. \n\\begin{figure\n\\centering\n\\includegraphics[scale=0.3]{fig-sk-2t}\n\\caption{(color online).\\  a, b) The frequency dependence of \nthe thermal conductivity $T\\kappa(\\omega)$, spin conductivity \n$\\sigma(\\omega)$ respectively for an $S=2$ impurity  with $J'=2J$, $L=19$, \n$\\Delta=1$ and two temperatures $T\/J=0.7,2$. Results are obtained via FTLM.}\n\\label{sk-2t} \n\\end{figure}%\n\\par\nThe previous analytical arguments are verified in Fig.\\ \\ref{sk-2t} where \nresults are obtained via FTLM for $L=19$ and various perturbations. \nThe  frequency dependence of the thermal conductivity \n$T\\kappa(\\omega)$ Fig.\\ \\ref{sk-2t}(a) and the spin conductivity \n$\\sigma(\\omega)$ Fig.\\ \\ref{sk-2t}(b) is shown, for $S=2$,  $J'=2J$, \n$\\Delta=1$ at $T\/J=0.7,2$.  \n\\par\nStarting with Fig. \\ref{sk-2t}(a) we can see that \nat  high  temperatures the prevalent  contribution to $\\kappa(\\omega)$ \ncomes from the operator $\\tilde{\\jmath^{\\epsilon}}$ yielding  a prominent peak at $\\omega=J'$\nindependent of  $S$ (compare Figs.\\ \\ref{k-ws}(b) and  \\ref{sk-2t}(a)). %\nAs the temperature decreases  and the system  flows to its ground state \nthe peak at $\\omega=J'$ decreases  gradually and eventually vanishes. \nNote that the high frequency leap in Fig.\\ \\ref{sk-2t}(a), present at $T\/J=0.7$,  \nemerges from  $\\propto JJ'$ terms of  the $\\jmath^{\\epsilon}$ current, Eq. (\\ref{encur}). \nIn contrast to  the thermal transport,  the local excitations of the spin current in the  \nreduced  system, Eq. (\\ref{LocalHamiltonian}),  \ninvolve the ground state giving a sharp peak at $\\omega=J' (S+1)$ \nwhich does  not  vanish with decreasing temperature, Fig.\\ \\ref{sk-2t}(b). \n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.3]{fig-ch}\n\\caption{(color online).\\ \nThe frequency \ndependence of the normalized thermal conductivity $\\bar{\\kappa}(\\omega)$ \nfor $S=1$, $J'=2J$, $L=19$ and $\\Delta=\\pm0.5$ at $T\/J=0.7,50$. Results\n are obtained via FTLM. %\n}\\label{ch} \n\\end{figure}\n\\par\nIn Fig.\\ \\ref{ch} we present the \nnormalized thermal conductivity $\\bar{\\kappa}(\\omega)$ for $S=1$, $J'=2J$, \n$\\Delta=\\pm0.5$ at $T\/J=0.7,50$; results are obtained via FTLM for $L=19$.  \nFirst, in connection with the previous arguments, we can say that the complicated    \nhigh frequency structure of $\\kappa(\\omega)$  for $\\Delta\\neq1$ \nindicates that the single excitation at $\\omega=J'$  \nis  a  unique property of the isotropic Heisenberg model. %\nSecond, we focus on the low frequency part of \n$\\kappa(\\omega)$ and particularly on the  stark difference for \n$\\Delta\\lessgtr0$  at low temperatures. The chain exhibits \ncutting(healing) behavior for $\\Delta>0$($\\Delta<0$) for an $S=1$ impurity \nembedded in the chain which was previously  reported for an $S=\\frac{1}{2}$\n impurity out of the  chain.\\cite{PhysRevB.58.5529, PhysRevB.81.205101} %\nAt high temperatures the curves for $\\Delta=\\pm0.5$ are one on top of the other. \nAs the temperature decreases, $\\kappa(\\omega)$ for $\\Delta=-0.5$ tends to \nobtain a  more Lorentzian-like form, characteristic of the  weak perturbation \nregime. On the contrary, for $\\Delta=+0.5$, $\\kappa(\\omega)$ obtains a \nstrongly non-monotonic  behavior with decreasing temperature resembling \nthe thermal conductivity of an open chain and signifying  the flow to the strong \nperturbation regime.\n\nNote that  $\\sigma(\\omega)$ exhibits a similar low frequency behavior for \n$\\Delta=\\pm0.5$ which could be plausibly read as evidence of a finite \n$D_S$, yielding a $\\sim\\delta(\\omega)$ contribution to $\\sigma'(\\omega)$, \nin the pure AHM  for $|\\Delta|<1$. %\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.3]{fig-kdc-vst}\n\\caption{(color online).\\ \nThe temperature \ndependence of  $T\\kappa_{dc}(T)$ for $S=1$, $J'\/J=0.8, 2$ and \n$S=2, J'\/J=1, 2$; results are obtained for $\\Delta=1, L=19$ via FTLM. }\\label{kdc-vst}\n\\end{figure} %\n\\par\nIt is also interesting to present the temperature dependence of the dc \nvalue of the thermal conductivity $\\kappa_{dc}$ itself since this is a directly \nmeasurable quantity in experiments. In Fig. \\ref{kdc-vst}, $T\\kappa_{dc}(T)$ is \nshown for four  characteristic perturbations and $\\Delta=1$. \nThe main conclusion is that the maximum of $\\kappa_{dc}$ occurs at \ndifferent temperatures for different perturbations. Although for $S=1$ \nthe host-impurity coupling   quantitatively only  affects   the behavior of \n$\\kappa_{dc}$ this is not the case for other impurities like the $S=2$ impurity.\n Generally the maximum of $\\kappa_{dc}$ may occur at different \n temperatures even for the same impurity if this is coupled to the chain with \n different host-impurity couplings $J'$. \n\\par\nFinally, we would like to address the screening of the \nimpurity by the chain which has been discussed previously in the \nliterature.\\cite{PhysRevB.46.10866, PhysRevB.64.092410, PhysRevLett.86.516} %\nThe ground state of the \\mbox{3-spin}  system $\\epsilon_{g}$ \nis \\mbox{$(2\\mathcal{S}+1)$-fold} degenerate, \nFig.\\ \\ref{fig-3},    implying  that  the impurity with its neighbors \nform an effective spin $\\mathcal{S}$  at low  \nenergies.  \nThus, for a strong $J'$ one can assume that the degrees of freedom of the \nsystem at low energies will be described by states of the  form \n$|\\epsilon_{g}\\rangle\\otimes|\\psi_{L-3}\\rangle$, where the \npseudo spin $\\mathcal{S}$ couples with the rest of the chain \nwith an effective coupling $\\tilde{J}$. $\\tilde{J}$ can be evaluated considering \nthe matrix elements of the operator $h_{1}$ at low \nenergies.\\cite{PhysRevB.56.654}\nFor  an $S=2$ impurity we obtain the effective coupling to be \n{\\it ferromagnetic}, $\\tilde{J}=-0.25J$, while the larger the $S$ the weaker the \n$\\tilde{J}$.  Thus, for a single impurity in a \nfinite system, or a finite concentration of impurities in the thermodynamic limit, \nthe  picture of the effective spin clearly fails to describe the whole \nfrequency range since the weak $\\tilde{J}$, $|\\tilde{J}|\\ll |J'|$,    cannot reproduce \nthe high frequency, conspicuous, excitations yielded by a strong  $J'$  \nat $\\omega\\sim J'$.    \n\\section{Discussion and Conclusions} \n\\par\nIt is worthwhile to devote a few lines to contrast the \nbasic points of the present model, where the impurity \nis embedded in the chain (IEC),  \nwith those of the model studied previously with the impurity located \nout of the chain (IOC).\\cite{PhysRevB.81.205101} %\nStark differences arise in the transport properties of the two \nmodels due to the position of the impurity and the \nway it couples to the pure system. For instance, a weak \nhost-impurity coupling $J'$ is a strong perturbation for the \nIEC model, Fig.\\ \\ref{k-ws}(b), while taking $J'=0$ in the IOC model \nwe end up in the pure AHM. Note also that it would not be accurate to perceive \nthe difference $|J-J'|$ as a perturbative parameter, simply,  \nbecause the case $J'=J$ does not correspond to the minimal \nperturbation of the heat transport. In addition, \nfor different impurities the maximum $\\kappa_{dc}$ occurs \nat different $J'$ which does not necessarily correspond to \nthe maximum of $\\sigma_{dc}$, Figs.\\ \\ref{ks-dc}, \n\\ref{kvrs} inset. %\nFor spin-$S$ impurities with $S>1$ $\\kappa(\\omega)$ exhibits a non-monotonic \nform. The absence of a Lorentzian $\\kappa(\\omega)$  for $S>1$ and any \nhost-impurity coupling $J'$ is an indication that $S>1$ impurities constitute \na strong perturbation for the heat transport of the Heisenberg model. \nThis is in sharp contrast to the behavior of \n$\\kappa(\\omega)$ in the IOC model which obeys a universal  \nscaling with the $\\mathcal{B}^{2}$ parameter, at least \nfor weak-intermediate $J'$.  \n\\par\nAnother significant  difference between the two models arises from \nthe absence of a $\\propto J'^2$ term in the energy current \nof the IOC model. As we have shown in this work, the $\\tilde{\\jmath^{\\epsilon}}$ \nterm accounts for the prominent high frequency excitations \nwhich become the prevalent contribution to $\\kappa(\\omega)$ for \nstrong perturbations. Similarly, the high frequency behavior \nof $\\sigma(\\omega)$ for strong perturbations is different in \nthe IEC and the IOC models due to the absence of a perturbative \nspin current term $\\propto J'$ in the latter. From all the above \none could conclude that the  magnetic impurity in the IEC model \nis a much stronger perturbation for a Heisenberg chain than \nin the IOC model. \n\\par\nTo summarize, studying the thermal $\\kappa$ \nand spin $\\sigma$ conductivities of the 1D,  \\mbox{spin-$\\frac{1}{2}$}, AHM \nwith an embedded  \\mbox{spin-$S$} impurity at finite temperatures $T$, \nwe have reached the  following  conclusions: \n\\setcounter{conc}{1}(\\roman{conc}) %\nAn $S=1$ impurity  can be considered as a relatively weak perturbation \nfor some host-impurity couplings $J'$, in contrast to  $S>1$ impurities \nwhich have a more drastic effect on $\\kappa$. Furthermore, the difference \nbetween $\\sigma_{dc}$ and  $\\kappa_{dc}$ is remarkable \nsince  the former shows an impressive rigidity  under the influence of the \nimpurity. %\n\\addtocounter{conc}{1} (\\roman{conc}) %\n$\\kappa(\\omega)$ obeys a universal scaling $\\kappa(\\omega L)\/L$\nfor weak and intermediate  $J'$, \nwhile a strong  $J'$ triggers  the emergence of impurity local excitations \nruining  the  \\mbox{$L$-scaling} of $\\kappa(\\omega)$.\n\\addtocounter{conc}{1} (\\roman{conc}) %\nWe have demonstrated  the origin of  these \nresonant modes for a  large $J'$ and a finite system---their position,  \ntheir magnitude, and their  temperature behavior as well, Eq.\\ (\\ref{SumRule}), \nFig.\\ \\ref{sk-2t}.  \n\\addtocounter{conc}{1} (\\roman{conc}) %\nThe presence of these sharp excitations at low $T$ \nmakes the picture of the effective spin $\\mathcal{S}=S-1$ insufficient to describe \nthe finite frequency transport properties, Fig.\\ \\ref{sk-2t}---at least for a single \nimpurity in a  finite system or plausibly for a finite concentration of impurities in the \nthermodynamic limit.\n\\addtocounter{conc}{1} (\\roman{conc}) %\nFinally, we observe the cutting-healing behavior of the chain \naccording to the sign of $\\Delta$ as it was previously reported  for magnetic \nimpurities out of the chain, \nFig.\\ \\ref{ch}.\\cite{PhysRevB.58.5529, PhysRevB.81.205101} %\n\\paragraph*{Acknowledgements.}\nThe author thanks P. Prelov\\v{s}ek and X. Zotos for \ntheir overall contribution  to this paper.  %\nThis work was supported by  the FP6-032980-2  NOVMAG project. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzjval b/data_all_eng_slimpj/shuffled/split2/finalzzjval
new file mode 100644
index 0000000000000000000000000000000000000000..ff8c45b500602ef2a249cef45cdbb47fa62609bd
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzjval
@@ -0,0 +1,5 @@
+{"text":"\\section{\\label{intro}Introduction}\n\nCurrently favored fundamental theories suggest that we live on a four-dimensional slice of a higher-dimensional universe, and while most forces are confined to our slice, gravity may leak off. These so-called brane world models may provide natural explanations for inflation (here known as brane inflation), and topological defects may at the end of it. The type of defect network which is formed and its basic properties (from its energy scale to whether or not it is long-lived) will depend on the specific details of the model in question \\cite{KIBBLE}. A thorough overview of the subject may be found in the book by Vilenkin and Shellard \\cite{VSH} and in more recent review articles \\cite{REV1,REV2,REV3}.\n\nMost of the past work on defects concerns the simplest models of cosmic strings, on the grounds that they are cosmologically benign, and are a generic prediction of inflationary models based on Grand Unified Theories \\cite{Rachel1,Rachel2} or branes \\cite{Branes1,Branes2}, while domain walls and monopoles tend to be cosmologically dangerous and tightly constrained. However, it is clear, particularly in the context of models with extra dimensions, that networks containing more than one type of defect will often be produced. Two examples that have attracted considerable interest are semilocal strings \\cite{SEMILOCAL1,SEMILOCAL2,SEMILOCALSIM} and cosmic necklaces \\cite{HINDMARSH}. For the latter these claims have been made both in the context of brane inflation \\cite{Dasgupta1,Chen,Dasgupta2} and in string theory itself \\cite{LEBLOND,LAKE}.\n\nThis is the third report on an ongoing project which is addressing some of these issues. In the past we have developed \\cite{MONOPOLES} an analytic model for the evolution of networks of local and global monopoles \\cite{DIRAC,THOOFT,POLYAK}. The model is analogous to the velocity-dependent one-scale model for cosmic strings \\cite{MS0,MS1,MS2}, which has been extensively tested against field theory \\cite{ABELIAN,MS3} and Goto-Nambu simulations \\cite{MS3,MS4}. This was then extended \\cite{HYBRIDS} to the case of monopoles attached to one string (the so-called hybrid networks \\cite{VILENKIN}), as well to vortons \\cite{VORTONS,MSVORTONS}. Here we study defect networks where monopoles are attached to two or more strings. Networks of the first type are commonly called cosmic necklaces; we shall refer to networks of of monopoles attached to three or more strings as cosmic lattices.\n\nThe behavior of necklaces and lattices is qualitatively similar, and will be for the most part treated together in this paper, through we will point out the small existing differences. However, their evolution differs in several key aspects from both that of individual monopoles and that of monopoles attached to a single string (usually called hybrid networks) \\cite{MONOPOLES,HYBRIDS}. The main difference is that necklaces and lattices form stable, long-lived networks which usually reach a scaling solution.\n\n\\section{\\label{review}Cosmic Necklaces and Lattices}\n\nWe start with a brief overview of previous results on the evolution of necklaces and lattices. This is by no means exhaustive; the aim is to highlight the dynamical aspects we will need to model. A more detailed discussion can be found in \\cite{VSH} as well as in other earlier references that we will point out where appropriate.\n\nThe defect networks of interest form via the symmetry breaking pattern $G\\to K\\times U(1)\\to K\\times Z_N$. If $G$ is  a semi-simple group, the first phase transition produces monopoles while in the second each monopole becomes attached to $N$ strings. If $K$ is trivial all the (Abelian) magnetic flux of the monopoles is confined into the strings, and there are no unconfined fluxes. However, unconfined non-Abelian magnetic fluxes can exist in the generic case. The previously discussed hybrid case \\cite{HYBRIDS} corresponds to $N=1$. Here we will discuss the case $N\\ge2$; $N=2$ corresponds to cosmic necklaces and $N\\ge3$ to cosmic lattices.\n\nThe corresponding defect masses will be $m\\sim ({4\\pi}\/{e})\\eta_m$ and $\\mu\\sim 2\\pi\\eta^2_s$, while the characteristic monopole radius and string thickness are $\\delta_m\\sim (e\\eta_m)^{-1}$ and $\\delta_s\\sim (e\\eta_s)^{-1}$. There are also scenarios where the intermediate phase transition is absent, $G\\to K\\times Z_N$, in which case an analogous network still forms but the role of the monopoles is now played by solitons that are usually called 'beads'. In this case the two energy scales are obviously similar, that is $\\eta_s\\sim\\eta_m$.\n\nUp to the second transition (if it exists) the models for plain monopoles \\cite{MONOPOLES} apply, but once the strings form a separate treatment is needed. Two key differences are immediately apparent. The evolution of isolated monopoles can be divided into a 'free' (pre-capture) and a post-capture period, with captured monopoles effectively decoupling from the network and losing energy radiatively until they decay (this is analogous to the evolution of cosmic string loops). In the present context the monopoles are effectively captured by the strings, and one needs to account explicitly not only for radiative losses (for example gauge radiation if there are unconfined magnetic charges) but also for the force the strings exert on the monopoles---depending on the context, the forces due to the string(s) or the other monopoles may be the dominant ones.\n\nIf all the strings attached to each monopole have the same tension (which we will assume to be the case in the present paper) then all the strings pull it with equal forces, and therefore there is no tendency for a monopole to be captured by the nearest antimonopole, unless their separation is of order $\\delta_s$. If there are $N$ strings attached to each monopole, its proper acceleration is given by the vector sum of the tension forces exerted by the strings. At a back-of-the-envelope level, each force is of order $f\\sim\\mu$, and hence one expects that $a\\sim\\mu\/m$. Monopoles should therefore be accelerated to relativistic speeds provided that the characteristic length of string segments, $L_s$, is such that $\\mu L_s\\gg m$, that is ${L_s}\/{\\delta_s}\\gg{\\eta_m}\/{\\eta_s}$.\n\nAryal et al. \\cite{ARYAL} first studied the formation and statistical properties of these networks, for $N=2$ and $N=3$, showing that for $N\\ge 3$ a single network is formed. In all cases they find that the system is dominated by one infinite network comprising more than $90\\%$ of the string length. Some finite networks and closed loops do exist, in numbers rapidly decreasing with their size. Finally, most of the string segments have a length comparable to the typical distance between monopoles (much larger segments being exponentially suppressed). This justifies our assumption of an inter-monopole separation, $L_m$, comparable to $L_s$.\n\nThe cosmological evolution of these networks was first discussed by Vachaspati \\& Vilenkin \\cite{VACHVIL}, who argued that assuming that the radiation of gauge quanta is the dominant energy loss mechanism of the networks, they reach scaling with a characteristic lengthscale $L\\sim ({\\eta^2_s}\/{\\eta^2_m}) t$ and the monopoles become highly relativistic. These networks can also lose energy by producing closed loops of string and small nets. The effect of these is harder to estimate, but as we shall discuss it is fairly easy to model phenomenologically..\n\nSpecifically they divide the network's energy into string and monopole parts, with $\\rho_m=\\beta\\rho$, $\\rho_s=(1-\\beta)\\rho$ and an effective equation of state $3{p}\/{\\rho}=\\beta+(1-\\beta)(2v^2_s-1)$ with $V_s$ being the string velocity and assuming $v_m\\sim1$. The evolution equation is then\n\\be\\label{vvdynamics}\n{\\dot\\rho}=-3H(\\rho+p)-nw\n\\ee\nwhere $w\\sim{(ga)^2}\/{6\\pi}$ for gauge radiation losses and $L\\sim n^{-1\/3}$ is both the length of string segments and the average monopole distance. Assuming a self-similar evolution one can also set $\\rho_s\\sim\\mu n^{2\/3}$.\n\nFor the case without unconfined magnetic fluxes (that is, no radiation), they claim $L\\propto t^\\alpha$, expecting $\\alpha<1$: there is no scaling and the defects eventually dominate the energy density of the universe. But note that in saying this they are specifically thinking of the radiation epoch (it is clear that the behavior of their solution depends on several parameters including the expansion rate).\n\nThe specific case of cosmic necklaces has subsequently been studied by Brezinsky \\& Vilenkin \\cite{BEREVIL}. They assume no unconfined magnetic fluxes (hence no Coulomb forces between the monopoles) and characterize the networks by a dimensionless ratio $r=m\/(\\mu L)$ with the average mass per unit length of the necklaces being $(r+1)\\mu$. They also neglect the effect of annihilations (though the validity of this assumption has been challenged \\cite{SIEMENS,BLANCOPILLADO}), and find that the system tends to evolve towards large $r$. Again the necklaces are expected to evolve in a scaling regime, with a characteristic network lengthscale $\\xi$. The force per unit length of string is $f\\sim{\\mu}\/{\\xi}$ and the acceleration is $a^{-1}\\sim (r+1)\\xi$ so we expect that\n\\be\\label{berez3}\n\\xi\\sim\\frac{t}{\\sqrt{1+r}}\\,,\\quad v\\sim\\frac{1}{\\sqrt{1+r}}\\,.\n\\ee\nIn the limit $r<<1$ the monopoles are sub-dominant and the strings will behave approximately as ordinary ones. In the limit $r>>1$ the strings are very slow and their separation is small. This is a very simple toy model, as in fact $r$ is generically not a constant parameter, since $d\\sim\\xi$, so these solutions are only approximate. Nevertheless, this approach has the advantage of algebraic simplicity, and we shall show below that in appropriate circumnstances it can be related to more robust models. Analogous results have been found with a somewhat different toy model \\cite{LEBLOND}.\n\n\\section{\\label{models}Quantitative Evolution}\n\nIt's easy to start modeling these networks by using the evolution equation derived in our previous work \\cite{MONOPOLES,HYBRIDS}. Most authors at this point focus on the evolution of the strings, treating the monopoles (as it were) as a small correction. Our approach, justified in \\cite{HYBRIDS}, is precisely the opposite---we focus on the evolution of the separation between monopoles.\n\nFor this context the evolution equations for the characteristic separation $L$ and root-mean squared velocity $v$ of the monopoles are\n\\be\\label{modell}\n3\\frac{dL}{dt}=(3+v^2)HL+Q_\\star\n\\ee\n\\be\\label{modelv}\n\\frac{dv}{dt}=(1-v^2)\\left(k_s\\frac{\\eta^2_s}{\\eta_m}-Hv\\right)\\,.\n\\ee\n\nWe have neglected the term due to friction in both equations (it's easy to show this is subdominant at late times). The energy loss term $Q_\\star$ in the lengthscale equation is a renormalized quantity, accounting for the various losses present (including string intercommutings, radiation and annihilations), as discussed in our previous work. There may be a velocity-dependence of some of these contributions, but as we shall see monopoles will typically have ultra-relativistic velocities $v\\sim1$ and therefore this dependence can be neglected. The velocity equation includes the force due to the strings (with a phenomenological curvature parameter $k_s$ that is discussed in \\cite{MS2}) but we have neglected that due to monopoles, since if it exists (which is only the case for unconfined fluxes) it's always smaller than that due to the strings. Indeed, using the above definitions of mass scales and thicknesses one finds that the ratio of the two forces is\n\\be\n\\frac{f_m}{f_s}\\sim\\frac{k_m}{k_s}\\left(\\frac{\\delta_s}{L}\\right)^2\\ll1\\,;\n\\ee\nwe expect the $k_i$ to be (dimensionless) coefficients of order unity, though note that they should be different for necklaces and lattices.\n\nFrom the velocity equation we immediately confirm that the monopole velocities will be driven towards unity, $v\\to1$, as previously stated. As for the monopole lenght scale, assuming a generic expansion rate $a\\propto t^\\lambda$, we find two different regimes for slow and fast expansion rates\n\\be\nL=\\frac{Q_\\star}{3-4\\lambda}t\\,,\\quad \\lambda<3\/4\n\\ee\n\\be\nL\\propto a^{4\/3}\\propto t^{4\\lambda\/3}\\,,\\quad \\lambda\\ge 3\/4\\,.\n\\ee\nThe former explicitly requires a non-zero energy loss rate---we will return to this point later. We therefore have linear scaling both in the radiation and matter eras (as generically claimed by previous authors, based on simpler qualitative arguments). For monopoles $L\\propto t$ corresponds to the monopole density decreasing relative to that of the background. However, for fast expansion rates the growth is superluminal, and the network will eventually disappear. An analogous scaling solution was already discussed in our previous work on hybrid networks.\n\nIts easy to establish a link between this analysis and that of Vachaspati \\& Vilenkin \\cite{VACHVIL} (which is embodied in Eq. \\ref{vvdynamics}) and thus to carry out a more detailed analysis of of the possible scaling solutions. Let's apply Eq. \\ref{vvdynamics} to the monopoles. We use both the notation and definitions of our previous work and those of \\cite{VACHVIL}. Since $\\rho_m=mn={m}\/{L^3}$ and the monopole equation of state is $3p=v^2_m\\rho$, we get by substitution\n\\be\\label{modelmon}\n3\\frac{dL}{dt}=(3+v^2)HL+\\frac{L}{\\eta_m}w\\,.\n\\ee\nNow, Vachaspati \\& Vilenkin are assuming energy losses through gauge radiation; noting that\n\\be\nw\\sim\\frac{(ga)^2}{6\\pi}\\sim\\left(\\frac{\\mu}{\\eta_m}\\right)^2\\sim{\\dot\\epsilon}_{gauge}\n\\ee\n\\be\nQ_{gauge}\\sim L\\frac{{\\dot\\epsilon}_{gauge}}{{\\epsilon}_{gauge}}\\sim\\frac{L}{\\eta_m}w\n\\ee\nwe see that this evolution equation for $L$ is exactly the same as Eq. \\ref{modell}, matching the $Q$ terms (which in our case can phenomenologically account for further energy loss channels).\n\nWe can also apply Eq. \\ref{vvdynamics} to the strings. In this case $\\rho_s=\\mu n^{2\/3}={\\mu}\/{L^2}$ and the string equation of state is $3p=(2v^2_s-1)\\rho$; again we find\n\\be\\label{modelstr}\n2\\frac{dL}{dt}=2HL(1+v^2_s)+{w}{\\mu}\\,.\n\\ee\nIn this case we have\n\\be\n{w}{\\mu}\\sim \\left(\\frac{\\eta_s}{\\eta_m}\\right)^2\\sim Q\\,.\n\\ee\nNote the interesting fact that the dimensionless parameter $Q$ determines the energy loss term for both the strings and the monopoles. The above is the usual evolution equation for the cosmic string correlation length \\cite{MS0,MS1,MS2}, if one assumes a constant string velocity---otherwise the energy loss term should depend linearly on velocity.\n\nThe scaling solution for the monopoles has already been discussed. For the case of the strings the solution is also the expected linear scaling\n\\be\nL=\\frac{Q}{2-2\\lambda(1+v^2_s)}t\\,,\n\\ee\nfor constant velocities and provided $\\lambda(1+v^2_s)<1$. But this solution is an attractor, as in the case of normal strings: for very large lengthscales the string velocity would no longer be a constant (the string velocity evolution equation would drive it to smaller values), and a new equilibrium value with a smaller lengthscale would be reached.\n\nWe can also consider more generic scaling solutions of the above equations, ie allowing for the possibility of zero energy losses ($Q=0$). We will confirm that scaling ($L\\propto t$) generically requires $Q\\neq0$. Starting again with the monopoles, for $Q\\neq0$ we have the two branches of the solution discussed above. For $Q=0$ the solution is always\n\\be\nL\\propto a^{4\/3}\\propto t^{4\\lambda\/3};\n\\ee\nnote that for $\\lambda<3\/4$ the lengthscale grows subluminally while for $\\lambda>3\/4$ it grows superluminally. In the absence of radiative energy losses, only a fast enough expansion can dilute the network. We can also compare the evolution of the monopole and background densities\n\\be\n\\frac{\\rho_m}{\\rho_b}\\sim\\frac{\\eta_m}{m^2_{Pl}}\\frac{t^2}{L^3}\\,.\n\\ee\nFor the linear scaling solution $L\\propto t$ this has the form \n\\be\n\\frac{\\rho_m}{\\rho_b}\\sim\\frac{1}{Q^3_\\star}\\left(\\frac{\\eta_m}{m_{Pl}}\\right)^2\\left(\\frac{T}{m_{Pl}}\\right)^2\\propto\\frac{1}{t}\\,.\n\\ee\nFrom this we see that if gauge radiation is present then the energy density of the network is smaller than the background density. However, if the only radiative channel available is gravitational radiation (in which case, as discussed in \\cite{HYBRIDS}, $Q_\\star=Q_{grav}\\sim(\\eta_s\/m_{Pl})^2$) then the network energy density is in fact the dominant one. For the non-scaling branch the density is\n\\be\n\\frac{\\rho_m}{\\rho_b}\\propto t^{2-4\\lambda}\\,,\n\\ee\nand the behavior depends on the cosmological epoch. Notice that during the radiation era the density is a constant fraction of that of the background.\n\nThese results can now be related with the toy-model analysis of Berezinsky \\& Vilenkin. In the linear scaling regime we have $r=\\rho_m\/\\rho_s\\propto t^{-1}$ and therefore $r\\to0$ and $v\\to1$ in agreement with our quantitative analysis. In the non-scaling branch $r\\propto t^{2-4\\lambda}$, whose behavior again depends on the cosmological epoch. The radiation epoch corresponds to the interesting case $r=const.$, while faster expansion rates (say, the matter-dominated epoch) dilute the monopole density relative to that of strings. Slower expansion rates ($\\lambda<1\/2$), which sometimes occur in the very early universe in string cosmology models, would have a growing density ratio $r$.\n\n\\section{\\label{concl}Summary and outlook}\n\nWe have extended a recently developed analytic model for the evolution of monopole \\cite{MONOPOLES} and hybrid networks \\cite{HYBRIDS} to the case of monopoles attached to several strings. We discussed their possible scaling solutions, generically confirming the expectation that the network will reach linear scaling (with its characteristic lengthscale $L\\propto t$), but also showing that other scaling behaviors can occur, depending on the expansion rate of the universe and on the energy loss mechanisms available to the network.\n\nThis completes the basic structure of analytic tools needed to study defect networks containing both strings and monopoles. These models have the advantage of conceptual simplicity, in addition to that of allowing a quantitative description of the network's evolution. Nevertheless, a further generalization will be required in order to more accurately describe the possible effects of a hierarchy of string tensions \\cite{TYE,LEBLOND,TASOS}. A further interesting case, which requires additional dynamics  is that of semilocal strings \\cite{SEMILOCAL1,SEMILOCAL2}, which we will address in a subsequent publication.\n\n\\begin{acknowledgments}\nThe work of C.M. is funded by a Ci\\^encia2007 Research Contract, funded by FCT\/MCTES (Portugal) and POPH\/FSE (EC). I am grateful to Ana Ach\\'ucarro for valuable discussions and collaboration on related issues, and to CTC, Cambridge (where part of this work was done) for the hospitality.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{par:1}\n\\subsection{}\nConsider the Sturm--Liouville problem\n\\begin{gather}\\label{eq:eq1}\n\t-y''+(q-\\lambda)y=0,\\\\ \\label{eq:eq2}\n\t\\left\\{\\begin{aligned}\n\t\ty'(0)-k_0^2y(0)=0,\\\\ y'(1)+k_1^2y(1)=0,\n\t\\end{aligned}\\right.\n\\end{gather}\nwhere real coefficients \\(k_0\\geq 0\\) and \\(k_1\\geq k_0\\) are fixed, the solution \\(y\\)\nbelongs to space \\(W_1^2[0,1]\\), equality \\eqref{eq:eq1} is considered as holding\nalmost everywhere at \\([0,1]\\), and the potential \\(q\\in L_1[0,1]\\)\nis a constant-sign function such that one of the integral conditions holds:\n\\begin{equation}\\label{eq:int}\n\t\\int_0^1 q\\,dx=1 \\;\\text{ or } \\int_0^1 q\\,dx=-1.\n\\end{equation}\nThe aim of this paper is to get the infima and suprema of the first eigenvalue\nof problem \\eqref{eq:eq1}--\\eqref{eq:int}.\n\n\\subsection{}\\label{pt:vrai1.2}\nProblem \\eqref{eq:eq1}--\\eqref{eq:int} is a partial case of problem\n\\eqref{eq:eq1}, \\eqref{eq:eq2} with \\(q\\in A_{\\gamma}\\) or \\(-q\\in A_{\\gamma}\\),\nwhere \\(\\gamma\\in\\mathbb R\\setminus\\{0\\}\\) and\n\\begin{equation}\\label{eq:Agamma}\n\tA_{\\gamma}\\rightleftharpoons\\left\\{q\\in L_1[0,1]\\::\\: q(x)\\geq 0\n\t\t\\text{\\ a.\\,e.,\\ and\\ }\\int_0^1 q^{\\gamma}\\,dx=1\\right\\}.\n\\end{equation}\nDenote by \\(\\lambda_1(q)\\) the minimal eigenvalue of problem \\eqref{eq:eq1} or\n\\begin{equation}\\label{eq:EK}\n\t-y''-\\lambda qy=0\n\\end{equation}\nwith some self-adjoint boundary conditions. Consider for each \\(\\gamma\\in\n\\mathbb R\\setminus\\{0\\}\\) four values \\(m_{\\gamma}^{\\pm}\\rightleftharpoons\n\\inf_{q\\in A_{\\gamma}}\\lambda_1(\\pm q)\\) and \\(M_{\\gamma}^{\\pm}\n\\rightleftharpoons\\sup_{q\\in A_{\\gamma}}\\lambda_1(\\pm q)\\). The estimates of\n\\(m_{\\gamma}^+\\) and \\(M_{\\gamma}^+\\) for equation \\eqref{eq:EK} with\nDirichlet boundary conditions were obtained in \\cite{EK:1996}. The analogous\nresults about Dirichlet problem for equation \\eqref{eq:eq1} were obtained\nin \\cite{VS:2003}, \\cite{Ez:2005}. In \\cite{Mur:2005} problem \\eqref{eq:EK},\n\\eqref{eq:eq2} was studied.\n\nThe values \\(m_{\\gamma}^+\\) and \\(M_{\\gamma}^+\\) for problem \\eqref{eq:eq1},\n\\eqref{eq:eq2} with \\(q\\in A_\\gamma\\) were considered by one of the authors\nin \\cite{Kar:2011} for all \\(\\gamma\\neq 0\\). The most detailed and precise results\nwere obtained for the case \\(\\gamma\\ne 1\\).\n\nThe case \\(\\gamma=1\\) is in some kind special. In \\cite{Ez:2005} and\n\\cite{Kar:2011}, for \\eqref{eq:eq1} with various boundary conditions, the precise\nresults for \\(M_1^+\\) were obtained by the method quite different from used for\n\\(\\gamma\\ne 1\\). In \\cite{Kar:2011} for \\(m_1^+\\) only inequality \\(m_1^+\\geq\n1\/4\\) was obtained. In \\cite{Ez:2005} for \\(m_1^-\\) it was proved that this infimum\nis attained at the non-summable potential \\(q^*=-\\boldsymbol\\delta_{1\/2}\\).\n\nIn this paper we extend the class of considered potentials from \\(L_1[0,1]\\)\nto space \\(W_2^{-1}[0,1]\\) (see \\cite{SaSh:2003} and \\ref{pt:1.2} later). Space\n\\(W_2^{-1}[0,1]\\), in particular, contains a Dirac delta function\n\\(\\boldsymbol\\delta_{\\zeta}\\) with support located at an arbitrary point\n\\(\\zeta\\in [0,1]\\). This generalization of the problem lets us to get the precise\ndescription of \\(M_1^{-}\\) and \\(m_1^{\\pm}\\) and to prove that they are attained\nat the potentials from the extended class.\n\n\\subsection{}\nThe main results of the paper are following four theorems:\n\n\\subsubsection{}\\label{prop:M1+}\n{\\textbf{Theorem.~}\\itshape By definition, put\n\\begin{equation}\\label{eq:ab}\n\t\\alpha_{\\mu}\\rightleftharpoons\\dfrac{1}{\\sqrt{\\mu}}\\arctan\n\t\t\\dfrac{k_0^2}{\\sqrt{\\mu}},\n\t\\qquad\\beta_{\\mu}\\rightleftharpoons\\dfrac{1}{\\sqrt{\\mu}}\\arctan\n\t\t\\dfrac{k_1^2}{\\sqrt{\\mu}}.\n\\end{equation}\nThen \\(M^+_1\\) is a unique solution to the equation\n\\begin{equation}\\label{eq:M1}\n\t1-\\alpha_{\\mu}-\\beta_{\\mu}=\\mu^{-1}\n\\end{equation}\nand is attained at the potential \\(q^*\\in L_1[0,1]\\) such that\n\\[\n\t\\belowdisplayskip=-0.4cm\n\tq^*(x)=\\left\\{\\begin{aligned}&M^+_1&\n\t\t&\\text{for }x\\in[\\alpha_{M_1^+},1-\\beta_{M_1^+}],\\\\\n\t\t&0&&\\text{otherwise}.\n\t\\end{aligned}\\right.\n\\]\n}\n\n\\subsubsection{}\\label{prop:M1-}\n{\\textbf{Theorem.~}\\itshape If \\(k_0^2+k_1^2\\leq 1\\), then \\(M^{-}_1=\nk_0^2+k_1^2-1\\) and is attained at the potential\n\\[\n\tq^*\\rightleftharpoons -k_0^2\\boldsymbol{\\delta}_0-\n\t\tk_1^2\\boldsymbol{\\delta}_1-(1-k_0^2-k_1^2).\n\\]\n\nIf \\(k_0^2+k_1^2\\geq 1\\) and \\(k_1^2-k_0^2\\leq 1\\), then \\(M^{-}_1\\)\nis the minimal eigenvalue of the problem\n\\begin{gather}\\label{eq:propM1-1}\n\t-y''=\\lambda y,\\\\ \\label{eq:propM1-2}\n\t2y'(0)-(k_0^2+k_1^2-1)y(0)=2y'(1)+(k_0^2+k_1^2-1)y(1)=0\n\\end{gather}\nand is attained at the potential\n\\[\n\tq^*\\rightleftharpoons -(1+k_0^2-k_1^2)\\boldsymbol{\\delta}_0\/2-\n\t\t(1-k_0^2+k_1^2)\\boldsymbol{\\delta}_1\/2.\n\\]\n\nIf \\(k_1^2-k_0^2\\geq 1\\), then \\(M^{-}_1\\) is the minimal eigenvalue\nof problem \\eqref{eq:propM1-1} with\n\\begin{equation}\\label{eq:propM1-3}\n\ty'(0)-k_0^2y(0)=y'(1)+(k_1^2-1)y(1)=0\n\\end{equation}\nand is attained at the potential \\(q^*\\rightleftharpoons-\\boldsymbol{\\delta}_1\\).\n}\n\n\\subsubsection{}\\label{prop:m1+}\n{\\textbf{Theorem.~}\\itshape \\(m^+_1\\) is the minimal eigenvalue of problem\n\\eqref{eq:propM1-1} with\n\\begin{equation}\\label{eq:propm1+}\n\ty'(0)-k_0^2y(0)=y'(1)+(k_1^2+1)y(1)=0\n\\end{equation}\nand is attained at the potential \\(q^*\\rightleftharpoons\\boldsymbol{\\delta}_1\\).\n}\n\n\\subsubsection{}\\label{prop:m1-}\n{\\textbf{Theorem.~}\\itshape If for some \\(\\mu\\geq-k_0^4\\) and some \\(\\zeta\\in\n(0,1)\\) the problem\n\\begin{gather}\\label{eq:grmm1}\n\t-y''=\\mu y\\quad\\text{at }(0,\\zeta)\\cup(\\zeta,1),\\\\ \\label{eq:grmm2}\n\ty'(0)-k_0^2y(0)=2y'(\\zeta-0)-y(\\zeta)=2y'(\\zeta+0)+y(\\zeta)=\n\t\ty'(1)+k_1^2y(1)=0\n\\end{gather}\nhas a continuous positive solution, then \\(m^{-}_1=\\mu\\) and \\(m^{-}_1\\)\nis attained at the potential \\(q^*\\rightleftharpoons-\\boldsymbol{\\delta}_{\\zeta}\\).\nOtherwise \\(m^{-}_1\\) is the minimal eigenvalue of problem \\eqref{eq:propM1-1}\nwith\n\\[\n\ty'(0)-(k_0^2-1)y(0)=y'(1)+k_1^2y(1)=0\n\\]\nand is attained at the potential \\(q^*\\rightleftharpoons -\\boldsymbol{\\delta}_0\\).\n}\n\n\\medskip\nSome additional remarks on solvability of boundary problem \\eqref{eq:grmm1},\n\\eqref{eq:grmm2} will be given in subsection \\ref{pt:3.5}.\n\n\\subsection{}\nLet us give some examples that illustrate the theorems from previous subsection.\nIn the case \\(k_0=k_1=0\\) we get \\(m_1^+=\\lambda_1(\\boldsymbol\\delta_1)=\n0.740174(\\pm 10^{-6})\\). In the case \\(k_0^2=k_1^2>1\/2\\) we get \\(m_1^-=\n\\lambda_1(-\\boldsymbol\\delta_{1\/2})\\). In the case \\(k_0^2=k_1^2=1\/2\\) we have\n\\(m_1^-=\\lambda_1(-\\boldsymbol\\delta_{\\zeta})=-1\/4\\) for any \\(\\zeta\\in [0,1]\\).\nIn the case \\(k_0^2=k_1^2<1\/2\\) we have \\(m_1^-=\\lambda_1(-\\boldsymbol\\delta_0)\\).\n\n\n\\section{The set \\protect\\(\\Gamma_1\\protect\\) and related topics}\\label{par:2}\n\\subsection{}\\label{pt:1.2}\nWe suppose that all considered functional spaces are real.\n\nBy \\(W_2^{-1}[0,1]\\) denote the Hilbert space that is a completion of \\(L_2[0,1]\\)\nin the norm\n\\[\n\t\\|y\\|_{W_2^{-1}[0,1]}\\rightleftharpoons\\sup\\limits_{\\|z\\|_{W_2^1[0,1]}=1}\n\t\t\\int_0^1 yz\\,dx.\n\\]\nWhen \\(y\\in W_2^{-1}[0,1]\\), by \\(\\int_0^1 yz\\,dx\\) we sometimes denote the result\n\\[\n\t\\langle y,z\\rangle\\rightleftharpoons\\lim\\limits_{n\\to\\infty}\n\t\t\\int_0^1 y_nz\\,dx \\quad (\\text{where } y=\\lim_{n\\to\\infty}y_n,\n\t\t\\quad y_n\\in L_2[0,1])\n\\]\nof applying the linear functional \\(y\\) to the function \\(z\\in W_2^1[0,1]\\).\n\nFor any fixed \\(q\\in L_1[0,1]\\) and \\(\\lambda\\in\\mathbb R\\) the map taking each\n\\(y\\in W_1^2[0,1]\\) satisfying \\eqref{eq:eq2} to\n\\[\n\t-y''+(q-\\lambda)y\\in L_1[0,1]\n\\]\ncan be extended by continuity to the bounded operator \\(T_q(\\lambda):\nW_2^1[0,1]\\to W_2^{-1}[0,1]\\). Using integration by part, we get\n\\begin{equation}\\label{eq:kv}\n\t(\\forall y,z\\in W_2^1[0,1])\\qquad\\langle T_q(\\lambda)y,z\\rangle=\n\t\t\\int_0^1 \\left[y'z'+(q-\\lambda)\\,yz\n\t\t\\right]\\,dx+k_0^2\\,y(0)z(0)+k_1^2\\,y(1)z(1).\n\\end{equation}\nConsider the linear operator pencil\\footnote{A linear operator pencil \\(L\\)\nis an operator-valued function such that \\(L(\\lambda)=A+\\lambda B\\), where\n\\(\\lambda\\in\\mathbb R\\), \\(A\\) and \\(B\\) are some operators not depending\non \\(\\lambda\\).} \\(T_q:\\mathbb R\\to \\mathcal B(W_2^1[0,1],\nW_2^{-1}[0,1])\\) that takes any \\(\\lambda\\in\\mathbb R\\) to the operator\n\\(T_q(\\lambda)\\) described by \\eqref{eq:kv}. The spectral problem for \\(T_q\\)\nmay be considered as a reformulation (or as a generalization in case when\n\\(q\\in W_2^{-1}[0,1]\\) is not summable) of boundary value problem \\eqref{eq:eq1},\n\\eqref{eq:eq2}. We can do this due to the following two facts.\n\n\\subsubsection{}\\label{prop:1.1}\n{\\itshape For all \\(q\\in L_1[0,1]\\) and \\(\\lambda\\in\\mathbb R\\) the function\n\\(y\\in W_2^1[0,1]\\) belongs to the kernel of the operator \\(T_q(\\lambda)\\), iff\n\\(y\\in W_1^2[0,1]\\) and \\(y\\) is a solution of problem \\eqref{eq:eq1},\n\\eqref{eq:eq2}.\n}\n\n\\begin{proof}\nIt directly follows from the definition of the operator \\(T_q(\\lambda)\\) that\nfor any solution \\(y\\in W_1^2[0,1]\\) of problem \\eqref{eq:eq1}, \\eqref{eq:eq2}\nthe equality \\(T_q(\\lambda)y=0\\) holds.\n\nLet us prove the converse. Consider some \\(y\\in\\ker T_q(\\lambda)\\), and put\n\\begin{equation}\\label{eq:eq3}\n\tw(x)\\rightleftharpoons y'(x)-\\int_0^x (q-\\lambda)y\\,dt.\n\\end{equation}\nFor any \\(z\\in{\\mathpalette\\Wo@{}}W_2^1[0,1]\\), using \\eqref{eq:kv}, we have\n\\begin{equation}\\label{eq:eq3l}\n\t0=\\langle T_q(\\lambda)y,z\\rangle=\\int_0^1 wz'\\,dx.\n\\end{equation}\nSince the set of the derivatives of all functions \\(z\\in{\\mathpalette\\Wo@{}}W_2^1[0,1]\\)\nis an orthogonal complement in \\(L_2[0,1]\\) of the set of all constants, from\n\\eqref{eq:eq3l} it follows that the function \\(w\\in L_2[0,1]\\) is constant.\nCombining this with \\eqref{eq:eq3}, we get that the function \\(y'\\) is absolutely\ncontinuous and its generalized derivative equals \\((q-\\lambda)y\\). Now, using\n\\eqref{eq:kv}, we see that for any \\(z\\in W_2^1[0,1]\\) we get\n\\[\n\t0=\\langle T_q(\\lambda)y,z\\rangle=[-y'(0)+k_0^2y(0)]\\,z(0)+\n\t\t[y'(1)+k_1^2y(1)]\\,z(1),\n\\]\nso \\(y\\) satisfies conditions \\eqref{eq:eq2}.\n\\end{proof}\n\n\\subsubsection{}\\label{prop:1.2}\n{\\itshape For any \\(q\\in W_2^{-1}[0,1]\\) the spectrum of the linear operator\npencil \\(T_q\\) is purely discrete, simple and bounded from below.\n}\n\n\\begin{proof}\nNote that for any \\(y\\in W_2^1[0,1]\\) we have\n\\[\n\t\\|y^2\\|_{W_2^1[0,1]}\\leq\\sup_{x\\in [0,1]}|y(x)|\\cdot\n\t\t\\sqrt{\\int_0^1\\left[y^2+4(y')^2\\right]\\,dx}\\leq\n\t\t2\\|y\\|_{C[0,1]}\\cdot\\|y\\|_{W_2^1[0,1]},\n\\]\nthen, by the embedding theorem, we get\n\\begin{equation}\\label{eq:norms}\n\t\\|y^2\\|_{W_2^1[0,1]}\\leq C\\,\\|y\\|_{W_2^1[0,1]}^2,\n\\end{equation}\nwhere \\(C\\) is some constant.\n\nSince \\(C[0,1]\\) is densely embedded in \\(W_2^{-1}[0,1]\\), for any\n\\(\\varepsilon\\in (0,1)\\) there exists a function \\(\\tilde q\\in C[0,1]\\)\nsuch that \\(\\|\\tilde q-q\\|_{W_2^{-1}[0,1]}\\leq\\varepsilon\/C\\). Using this\nand inequality \\eqref{eq:norms}, for any \\(y\\in W_2^1[0,1]\\) we get\n\\begin{equation}\\label{eq:1_1_2_2}\n\t\\left|\\int_0^1 (\\tilde q-q)\\,y^2\\,dx\\right|\\leq\n\t\\|\\tilde q-q\\|_{W_2^{-1}[0,1]}\\cdot\\|y^2\\|_{W_2^1[0,1]}\n\t\\leq\\varepsilon\\,\\|y\\|^2_{W_2^1[0,1]}.\n\\end{equation}\nFurther, for any \\(\\kappa>\\|\\tilde q\\|_{C[0,1]}+1\\) we have\n\\(\\int_0^1\\tilde q y^2\\,dx\\geq (1-\\kappa)\\int_0^1 y^2\\,dx\\). Combining\nthis with \\eqref{eq:kv} and \\eqref{eq:1_1_2_2}, we obtain\n\\begin{flalign}\\label{eq:kappa}\n\t&& IT_q(-\\kappa)&\\geq 1-\\varepsilon,&&\n\\end{flalign}\nwhere by \\(I:W_2^{-1}[0,1]\\to W_2^1[0,1]\\) we denote an isometry that satisfies\n\\[\n\t(\\forall y\\in W_2^{-1}[0,1])\\,(\\forall z\\in W_2^1[0,1])\\qquad\n\t\t\\langle Iy,z\\rangle_{W_2^1[0,1]}=\\langle y,z\\rangle.\n\\]\nThe existence and uniqueness of this isometry follows from the Riesz theorem\nabout the representation of a functional in a Hilbert space\n\\cite[\\S\\S~30,\\,99]{RN:1979}.\n\nFrom estimate \\eqref{eq:kappa} it follows \\cite[\\S~104]{RN:1979} that the operator\n\\(S\\rightleftharpoons IT_q(-\\kappa)\\) is boundedly invertible.\nTaking into account \\eqref{eq:kv}, we have \\(IT_q(\\lambda)\\equiv S-(\\lambda+\n\\kappa)J^*J\\), where \\(J:W_2^1[0,1]\\to L_2[0,1]\\) is the embedding operator.\nSo for any \\(\\lambda\\in\\mathbb R\\) the existence of a bounded inverse of\nthe operator \\(T_q(\\lambda)\\) is equivalent to the existence of a bounded inverse\nof the operator \\(1-(\\lambda+\\kappa)S^{-1\/2}J^*JS^{-1\/2}\\). Since \\(J\\)\nis compact, it follows that the spectrum of \\(T_q\\) is purely discrete, semi-simple\nand bounded from below.\n\nThe spectrum of the pencil \\(T_q\\) is simple since (see \\cite{SaSh:2003},\n\\cite[Propositions~2, 10]{Vl:2009}) for any \\(\\lambda\\in\\mathbb R\\) the kernel\nof the operator \\(T_q(\\lambda)\\) is formed by the first components \\(Y_1\\)\nof the solutions to the boundary value problem\n\\begin{gather}\\label{eq:Matr1}\n\t\\begin{pmatrix}Y_1 \\\\ Y_2\\end{pmatrix}' = \\begin{pmatrix}u&1\\\\\n\t\t-u^2&-u\\end{pmatrix} \\begin{pmatrix}Y_1 \\\\ Y_2\\end{pmatrix},\\\\\n\t\t\\label{eq:Matr2}\n\t\tY_2(0)-k_0^2 Y_1(0)=Y_2(1)+[k_1^2+\\omega]\\,Y_1(1)=0.\n\\end{gather}\nHere \\(u\\in L_2[0,1]\\) and \\(\\omega\\in\\mathbb R\\) are taken from the representation\n\\begin{equation}\\label{eq:Matr3}\n\t(\\forall y\\in W_2^1[0,1])\\qquad\\int_0^1 (q-\\lambda)y\\,dx\n\t\t=-\\int_0^1 uy'\\,dx+\\omega\\,y(1)\n\\end{equation}\nof the potential \\(q\\in W_2^{-1}[0,1]\\).\n\\end{proof}\n\n\\subsection{}\nFor eigenvalues\n\\[\n\t\\lambda_1(q)<\\lambda_2(q)<\\ldots<\\lambda_n(q)<\\ldots\n\\]\nof the pencil \\(T_q\\) we have the following propositions.\n\n\\subsubsection{}\\label{prop:varpr}(see \\cite[Proposition~10]{Vl:2009})\n{\\itshape For any \\(n\\geq 1\\), \\(q\\in W_2^{-1}[0,1]\\) and \\(\\lambda\\in\n\\mathbb R\\) the inequality \\(\\lambda>\\lambda_n(q)\\) is equivalent to the existence\nof \\mbox{\\(n\\)-di}\\-men\\-ti\\-o\\-nal subspace \\(\\mathfrak N\\subset W_2^1[0,1]\\)\nthat satisfies\n\\[\n\t\\belowdisplayskip=-0.5cm\n\t(\\forall y\\in\\mathfrak N\\setminus\\{0\\})\\qquad\n\t\t\\langle T_q(\\lambda)y,y\\rangle<0.\n\\]\n}\n\n\\subsubsection{}\\label{prop:cont}\n{\\itshape For any \\(n\\geq 1\\) the function \\(\\lambda_n:W_2^{-1}[0,1]\\to\n\\mathbb R\\) is continuous.\n}\n\n\\begin{proof}\nConsider some \\(q\\in W_2^{-1}[0,1]\\) and \\(\\varepsilon\\in (0,1\/2)\\). For any\n\\(y\\in W_2^1[0,1]\\), \\(\\lambda\\in\\mathbb R\\) and \\(\\tilde q\\in W_2^{-1}[0,1]\\)\nsuch that \\(\\|\\tilde q-q\\|_{W_2^{-1}[0,1]}<\\varepsilon\/C\\), where \\(C\\) is\nthe same as in \\eqref{eq:norms}, we get\n\\begin{flalign*}\n\t&& \\langle T_{\\tilde q}(\\lambda)y,y\\rangle&\\geq\\langle T_q(\\lambda)y,\n\t\ty\\rangle-\\varepsilon\\,\\|y\\|_{W_2^1[0,1]}^2&&\\\\\n\t&& &\\geq\\langle T_q(\\lambda)y,y\\rangle-\\varepsilon\\,\n\t\t\\|y\\|_{W_2^1[0,1]}^2-\\varepsilon\\cdot\\langle T_{2q}(\\lambda_1(2q))\n\t\ty,y\\rangle-\\varepsilon k_0^2y^2(0)-\\varepsilon k_1^2y^2(1)&&\\\\\n\t&& &=(1-2\\varepsilon)\\cdot\\left\\langle T_q\\left(\\dfrac{\\lambda+\n\t\t\\varepsilon\\cdot [1-\\lambda_1(2q)]}{1-2\\varepsilon}\\right)y,y\n\t\t\\right\\rangle.\n\\end{flalign*}\nConsequently, from variational principle \\ref{prop:varpr} it follows that any\n\\(\\lambda>\\lambda_n(\\tilde q)\\) satisfies\n\\[\n\t\\dfrac{\\lambda+\\varepsilon\\cdot [1-\\lambda_1(2q)]}{1-2\\varepsilon}>\n\t\t\\lambda_n(q).\n\\]\nSince we can choose \\(\\lambda\\) arbitrarily close to \\(\\lambda_n(\\tilde q)\\),\nwe have \\(\\lambda_n(\\tilde q)\\geq (1-2\\varepsilon)\\,\\lambda_n(q)-\\varepsilon\n\\cdot[1-\\lambda_1(2q)]\\). By the same method we get \\(\\lambda_n(\\tilde q)\\leq\n(1+2\\varepsilon)\\,\\lambda_n(q)+\\varepsilon\\cdot[1-\\lambda_1(2q)]\\).\n\\end{proof}\n\n\\subsection{}\\label{pt:2.2}\nLet \\(\\Gamma_1\\) be the closure in \\(W_2^{-1}[0,1]\\) of the set \\(A_1\\)\ndefined by \\eqref{eq:Agamma}. Put by definition\n\\[\n\t\\Lambda(X)\\rightleftharpoons\\{\\lambda\\in\\mathbb R\\::\\:\n\t\t(\\exists q\\in X)\\quad\\lambda=\\lambda_1(q)\\},\n\\]\nwhere \\(X\\subseteq W_2^{-1}[0,1]\\) is some set of generalized functions.\nThe set \\(\\Lambda(X)\\) is formed by all the possible values of \\(\\lambda_1(q)\\)\nfor all \\(q\\in X\\). By \\(-X\\) we, as usually, denote the set\n\\[\n\t\\{q\\in W_2^{-1}[0,1]\\::\\:(\\exists r\\in X)\\quad q=-r\\}.\n\\]\n\n\\subsubsection{}\\label{prop:2.2}\n{\\itshape Suppose \\(X\\) is a dense subset of\\\/ \\(\\Gamma_1\\), then the closures\nof\\\/ \\(\\Lambda(\\pm X)\\) and\\\/ \\(\\Lambda(\\pm \\Gamma_1)\\) coincide.\n}\n\n\\subsubsection{}\\label{prop:2.3.2}\n{\\itshape The extrema \\(m^{\\pm}_1\\rightleftharpoons\\inf\\Lambda(\\pm A_1)\\) and\n\\(M^{\\pm}_1\\rightleftharpoons\\sup\\Lambda(\\pm A_1)\\), defined in \\ref{pt:vrai1.2},\nsatisfy the equalities \\(m^{\\pm}_1=\\inf\\Lambda(\\pm\\Gamma_1)\\) and \\(M^{\\pm}_1=\n\\sup\\Lambda(\\pm\\Gamma_1)\\).\n}\n\n\\medskip\nProposition \\ref{prop:2.2} immediately follows from \\ref{prop:cont}. Proposition\n\\ref{prop:2.3.2} immediately follows from \\ref{prop:2.2}.\n\n\\subsubsection{}\\label{prop:4.2}\n{\\itshape The set\\\/ \\(\\Gamma_1\\) consists of all\nnon-negative\\footnote{The generalized function \\(q\\in W_2^{-1}[0,1]\\) is called\nnon-negative if for any non-negative function \\(y\\in W_2^1[0,1]\\) the inequality\n\\(\\langle q,y\\rangle\\geq 0\\) holds.} distributions \\(q\\in W_2^{-1}[0,1]\\)\nsuch that \\(\\int_0^1 q\\,dx=1\\).\n}\n\n\\begin{proof}\nSince for any \\(q\\in\\Gamma_1\\) there exists a sequence of functions from \\(A_1\\)\nsuch that its limit equals \\(q\\), it follows that the generalized function \\(q\\)\nis non-negative and satisfies \\(\\int_0^1 q\\,dx=1\\).\n\nLet us prove the converse. Suppose \\(q\\in W_2^{-1}[0,1]\\) is a non-negative\ngeneralized function and satisfies \\(\\int_0^1 q\\,dx=1\\).\nThen (see \\cite{SaSh:2003}, \\cite[\\S\\,2.3]{Vl:2009}) there exists a function\n\\(u\\in L_2[0,1]\\) such that\n\\begin{equation}\\label{eq:L2}\n\t(\\forall y\\in W_2^1[0,1])\\qquad\\int_0^1 qy\\,dx= -\\int_0^1 uy'\\,dx+y(1).\n\\end{equation}\nPut by definition\n\\[\n\t\\Pi_{\\gamma,\\eta,\\theta}(x)\\rightleftharpoons\\left\\{\\begin{aligned}&\n\t\t\\tfrac{x-\\gamma}{\\eta-\\gamma}&&\\text{for }x\\in [\\gamma,\\eta],\\\\\n\t\t&\\tfrac{\\theta-x}{\\theta-\\eta}&&\\text{for }x\\in [\\eta,\\theta],\\\\\n\t\t&0&&\\text{otherwise}\\end{aligned}\\right.\n\\]\nfor any reals \\(\\gamma<\\eta<\\theta\\). Suppose \\(0<a<b<c<d<1\\). Substituting\nthe functions \\(\\Pi_{-1,0,a}+\\Pi_{0,a,b}\\), \\(\\Pi_{a,b,c}+\\Pi_{b,c,d}\\) and\n\\(\\Pi_{c,d,1}+\\Pi_{d,1,2}\\) for \\(y\\) in \\eqref{eq:L2}, we get\n\\[\n\t0\\leq\\dfrac{1}{b-a}\\int_a^b u\\,dx\\leq\n\t\t\\dfrac{1}{d-c}\\int_c^d u\\,dx\\leq 1.\n\\]\nFrom these inequalities it follows that the function \\(u\\in L_2[0,1]\\)\nis non-decreasing and satisfies \\(\\vrai{inf}_{x\\in [0,1]}u(x)\\geq 0\\)\nand \\(\\vrai{sup}_{x\\in [0,1]}u(x)\\leq 1\\).\n\nSince there exists a sequence \\(\\{u_n\\}_{n=0}^{\\infty}\\) of non-decreasing\npiecewise linear functions such that \\(u_n(0)=0\\), \\(u_n(1)=1\\)\nand \\(u=\\lim\\limits_{n\\to\\infty}{u_n}\\), it follows that \\(q=\\lim\\limits_{n\\to\n\\infty}{u'_n}\\), where \\(u'_n \\in A_1\\).\n\\end{proof}\n\n\\subsection{}\nConsider the function \\(F\\) implicitely defined by the equation\n\\begin{equation}\\label{eq:implF}\n\t\\lambda_1(F(\\mu,\\zeta)\\boldsymbol{\\delta}_{\\zeta})=\\mu,\n\\end{equation}\nwhere \\(\\mu\\in\\mathbb R\\) and \\(\\zeta\\in [0,1]\\). The following three propositions\ngive us some information about this function.\n\n\\subsubsection{}\\label{prop:F1}\n{\\itshape For any \\(\\zeta\\in [0,1]\\) the function \\(F(\\cdot,\\zeta)\\)\nis single-valued, strictly increasing, and its domain is the interval \\((-\\infty,\nf^+)\\) with some \\(f^+>0\\).\n}\n\n\\begin{proof}\nFor any \\(a\\in\\mathbb R\\) there exists \\cite[Proposition~11]{Vl:2009} a positive\neigenfunction \\(y\\in\\ker T_{a\\boldsymbol{\\delta}_{\\zeta}}(\\mu)\\) corresponding\nto the eigenvalue \\(\\mu\\rightleftharpoons\\lambda_1(a\\boldsymbol{\\delta}_{\\zeta})\\),\nso for any \\(b<a\\) we have\n\\[\n\t\\langle T_{b\\boldsymbol{\\delta}_{\\zeta}}(\\mu)y,y\\rangle =\n\t\t\\langle T_{a\\boldsymbol{\\delta}_{\\zeta}}(\\mu)y,y\\rangle\n\t\t+(b-a)\\cdot y^2(\\zeta)<0.\n\\]\nUsing \\ref{prop:varpr}, we now get \\(\\lambda_1(b\\boldsymbol{%\n\\delta}_{\\zeta})<\\mu\\). So the function \\(F(\\cdot,\\zeta)\\) is the inverse\nof the strictly increasing and, according to \\ref{prop:cont}, continuous\nmap \\(a\\mapsto \\lambda_1(a\\boldsymbol\\delta_{\\zeta})\\). Therefore, the function\n\\(F(\\cdot,\\zeta)\\) is single-valued and strictly increasing.\n\nFurther, for any \\(a\\in\\mathbb R\\) from the equality\n\\[\n\t\\langle T_{a\\boldsymbol{\\delta}_{\\zeta}}(a+k_0^2+k_1^2)1,\n\t\t1\\rangle=a-(a+k_0^2+k_1^2)+k_0^2+k_1^2 =0\n\\]\nand proposition \\ref{prop:varpr} it follows that \\(\\lambda_1(a\\boldsymbol{%\n\\delta}_{\\zeta})\\leq a+k_0^2+k_1^2\\). Therefore the domain of \\(F(\\cdot,\n\\zeta)\\) is unbounded from below. Also for any \\(a>0\\) we have \\(\\lambda_1(a\n\\boldsymbol\\delta_{\\zeta})>0\\), so the right bound of \\(\\mathop{\\operatorname{dom}} F(\\cdot,\\zeta)\\)\nis positive.\n\\end{proof}\n\n\\subsubsection{}\\label{prop:F2}\n{\\itshape The function \\(F\\) is continuous.\n}\n\n\\begin{proof}\nConsider an arbitrary point \\((\\mu_0,\\zeta_0)\\in\\mathop{\\operatorname{dom}} F\\) and suppose \\(a^{\\pm}\\)\nsatisfy \\(a^{-}<F(\\mu_0,\\zeta_0)<a^{+}\\). For any point \\((\\mu,\\zeta)\\in\\mathbb R\n\\times [0,1]\\) sufficiently close to \\((\\mu_0,\\zeta_0)\\) from \\ref{prop:F1} and\n\\ref{prop:cont} we obtain the inequalities \\(\\lambda_1(a^{-}\\boldsymbol{\\delta}_{%\n\\zeta})<\\mu<\\lambda_1(a^{+} \\boldsymbol{\\delta}_{\\zeta})\\). Hence there exists\n\\(a\\in (a^{-},a^+)\\) such that \\(\\mu=\\lambda_1(a\\boldsymbol{\\delta}_{\\zeta})\\),\nso for the point \\((\\mu,\\zeta)\\) equation \\eqref{eq:implF} has a solution\n\\(F(\\mu,\\zeta) = a\\).\n\\end{proof}\n\n\\subsubsection{}\\label{prop:F3}\n{\\itshape A point \\((\\mu,\\zeta)\\in (0,+\\infty)\\times [0,1]\\) belongs to domain\nof the function \\(F\\) iff the following conditions hold:\n\\begin{equation}\\label{eq:sign}\n\t\\sqrt{\\mu}\\cdot(\\zeta-\\alpha_{\\mu})\\in (-\\pi\/2,\\pi\/2),\\qquad\n\t\\sqrt{\\mu}\\cdot(1-\\beta_{\\mu}-\\zeta)\\in(-\\pi\/2,\\pi\/2),\n\\end{equation}\nwhere \\(\\alpha_{\\mu}\\) and \\(\\beta_{\\mu}\\) are defined by \\eqref{eq:ab}.\nIn this case the equality\n\\begin{equation}\\label{eq:F}\n\tF(\\mu,\\zeta)=\\sqrt{\\mu}\\cdot\\left\\{\\tan[\\sqrt{\\mu}\\cdot(\\zeta-\n\t\t\\alpha_{\\mu})]+\\tan[\\sqrt{\\mu}\\cdot(1-\\beta_{\\mu}-\\zeta)]\\right\\}\n\\end{equation}\nholds.\n\nFor any \\(\\zeta\\in [0,1]\\) the equality\n\\begin{equation}\\label{eq:F0}\n\tF(0,\\zeta)=-\\dfrac{k_0^2}{1+k_0^2\\zeta}-\\dfrac{k_1^2}{1+k_1^2\\,\n\t\t(1-\\zeta)}\n\\end{equation}\nholds.\n\nFor any \\(\\mu<0\\) and \\(\\zeta\\in [0,1]\\) the equality\n\\begin{equation}\\label{eq:Fmin}\n\tF(\\mu,\\zeta)=-\\sqrt{|\\mu|}\\cdot\\{G(\\sqrt{|\\mu|},k_0^2,\\zeta)\n\t\t+G(\\sqrt{|\\mu|},k_1^2,1-\\zeta)\\},\n\\end{equation}\nwhere\n\\[\n\tG(\\nu,\\kappa,x)\\rightleftharpoons\\left\\{\\begin{aligned}\n\t\t&\\tanh\\left(\\nu x+\\ln\\sqrt{\\dfrac{\\nu+\n\t\t\\kappa}{\\nu-\\kappa}}\\right)&&\\text{for }\\nu>\\kappa,\\\\\n\t\t&1&&\\text{for }\\nu=\\kappa,\\\\\n\t\t&\\coth\\left(\\nu x+\\ln\\sqrt{\\dfrac{\\kappa+\\nu}{\\kappa-\n\t\t\\nu}}\\right)&&\\text{for }\\nu<\\kappa,\\end{aligned}\\right.\n\\]\nholds.\n}\n\n\\begin{proof}\nConsider \\(\\mu\\in\\mathbb R\\) and \\(\\zeta\\in (0,1)\\) such that \\((\\mu,\\zeta)\\in\n\\mathop{\\operatorname{dom}} F\\). According to \\eqref{eq:Matr1}--\\eqref{eq:Matr3}, the equality\n\\(T_{q}(\\mu)y=0\\), where \\(q\\rightleftharpoons F(\\mu,\\zeta)\n\\boldsymbol\\delta_{\\zeta}\\), is equivalent to the boundary problem\n\\begin{gather}\\label{eq:delt1}\n\t-y''=\\mu y\\quad\\text{at }(0,\\zeta)\\cup (\\zeta,1),\\\\\n\t\\label{eq:delt2} y'(\\zeta+0)-y'(\\zeta-0)=F(\\mu,\\zeta)y(\\zeta),\\\\\n\t\\label{eq:delt3} y'(0)-k_0^2y(0)=y'(1)+k_1^2y(1)=0.\n\\end{gather}\nFrom \\cite[Proposition~11]{Vl:2009} and \\eqref{eq:implF} it follows that\nany non-trivial solution to problem \\eqref{eq:delt1}--\\eqref{eq:delt3}\nis constant-sign.\n\nIn the case \\(\\mu>0\\) any solution to problem \\eqref{eq:delt1}, \\eqref{eq:delt3}\nhas the form\n\\begin{equation}\\label{eq:2.19}\n\ty(x)=\\left\\{\\begin{aligned}&A\\cdot\\cos[\\sqrt{\\mu}\\cdot(1-\\beta_{\\mu}-\\zeta)]\n\t\t\\cdot\\cos[\\sqrt{\\mu}\\cdot(x-\\alpha_{\\mu})]&&\\text{for }x<\\zeta,\\\\\n\t\t&A\\cdot\\cos[\\sqrt{\\mu}\\cdot(1-\\beta_{\\mu}-x)]\\cdot\\cos[\\sqrt{\\mu}\n\t\t\\cdot(\\zeta-\\alpha_{\\mu})]&&\\text{for }x>\\zeta,\\end{aligned}\\right.\n\\end{equation}\nwhere \\(A\\) is some constant. This function is constant-sign iff conditions\n\\eqref{eq:sign} hold. Using \\eqref{eq:delt2}, we now get \\eqref{eq:F}. The values\n\\(\\zeta\\in\\{0,1\\}\\) are finally included in the consideration using propositions\n\\ref{prop:F2} and \\ref{prop:cont}.\n\nThe cases \\(\\mu=0\\) and \\(\\mu<0\\) are considered on the base of\n\\eqref{eq:delt1}--\\eqref{eq:delt3} by analogous way using the solution\n\\begin{equation}\\label{eq:2.20}\n\ty(x)=\\left\\{\\begin{aligned}&A\\cdot [1+k_1^2(1-\\zeta)]\\cdot[1+k_0^2x]\n\t\t&&\\text{for }x<\\zeta,\\\\ &A\\cdot [1+k_1^2(1-x)]\\cdot[1+k_0^2\\zeta]\n\t\t&&\\text{for }x>\\zeta\\end{aligned}\\right.\n\\end{equation}\nin the case \\(\\mu=0\\), and the solution\n\\begin{equation}\\label{eq:2.21}\n\ty(x)=\\left\\{\\begin{aligned}&A\\cdot g(\\sqrt{|\\mu|},k_1^2,1-\\zeta)\\cdot\n\t\tg(\\sqrt{|\\mu|},k_0^2,x)&&\\text{for }x<\\zeta,\\\\\n\t\t&A\\cdot g(\\sqrt{|\\mu|},k_1^2,1-x)\\cdot g(\\sqrt{|\\mu|},k_0^2,\n\t\t\\zeta)&&\\text{for }x>\\zeta,\\end{aligned}\\right.\n\\end{equation}\nwhere\n\\[\n\tg(\\nu,\\kappa,x)\\rightleftharpoons\\left\\{\\begin{aligned}\n\t\t&\\cosh\\left(\\nu x+\\ln\\sqrt{\\dfrac{\\nu+\n\t\t\\kappa}{\\nu-\\kappa}}\\right)&&\\text{for }\\nu>\\kappa,\\\\\n\t\t&e^{\\nu x}&&\\text{for }\\nu=\\kappa,\\\\\n\t\t&\\sinh\\left(\\nu x+\\ln\\sqrt{\\dfrac{\\kappa+\\nu}{\\kappa-\n\t\t\\nu}}\\right)&&\\text{for }\\nu<\\kappa,\\end{aligned}\\right.\n\\]\nin the case \\(\\mu<0\\).\n\\end{proof}\n\n\n\\section{Proofs of the main results}\\label{par:3}\n\\subsection{}\nIn this section we prove theorems \\ref{prop:M1+}--\\ref{prop:m1-}.\nWe use the notation\n\\begin{align*}\n\t\\Omega^+(y)&\\rightleftharpoons\\textstyle\\left\\{x\\in[0,1]\\::\\:\n\t\ty(x)=\\sup_{t\\in[0,1]}y(t)\\right\\},\\\\\n\t\\Omega^-(y)&\\rightleftharpoons\\textstyle\\left\\{x\\in[0,1]\\::\\:\n\t\t y(x)=\\inf_{t\\in[0,1]}y(t)\\right\\},\n\\end{align*}\nwhere \\(y\\in W_2^1[0,1]\\) is an arbitrary positive function. Also we take\ninto account proposition \\ref{prop:2.3.2}.\n\n\\subsection{Proof of theorem \\ref{prop:M1+}.}\nConsider some potential \\(q^*\\in\\Gamma_1\\), and some positive eigenfunction\n\\(y\\in\\ker T_{q^*}(\\lambda_1(q^*))\\). Suppose that the support of the generalized\nfunction \\(q^*\\) is a subset of \\(\\Omega^+(y)\\). Then for any \\(q\\in\\Gamma_1\\)\nwe, using \\ref{prop:4.2}, have\n\\begin{flalign*}\n\t&&0&=\\langle T_{q^*}(\\lambda_1(q^*))y,y\\rangle\\\\\n\t&& &=\\int_0^1\\left[(y')^2- \\lambda_1(q^*)\\,y^2\\right]\\,dx+\n\t\t\\sup\\limits_{x\\in [0,1]}y^2(x)+k_0^2\\,y^2(0)+k_1^2\\,y^2(1)\\\\\n\t&& &\\geq\\int_0^1\\left[(y')^2+(q-\\lambda_1(q^*))\\,y^2\\right]\\,dx+\n\t\tk_0^2\\,y^2(0)+k_1^2\\,y^2(1),&&\n\\end{flalign*}\nhence \\(\\langle T_q(\\lambda_1(q^*))y,y\\rangle\\leq 0\\). It follows that\n\\(\\lambda_1(q)\\leq\\lambda_1(q^*)\\), therefore \\(\\lambda_1(q^*)=M_1^+\\).\nThus we proved that \\(M_1^+\\) is attained at any potential \\(q^*\\) such that\n\\(\\mathop{\\operatorname{supp}} q^*\\subseteq\\Omega^+(y)\\).\n\nSuppose that \\(\\Omega^+(y) = [\\tau_0,\\tau_1]\\), where \\(\\tau_0 \\ne \\tau_1\\).\nAlso suppose that the potential \\(q^*\\) is summable and has the form\n\\[\n\tq^*(x)= \\left\\{\\begin{aligned}&\\mu& &\\text{for }x\\in[\\tau_0,\\,\\tau_1],\\\\\n\t\t&0& &\\text{otherwise},\\end{aligned}\\right.\n\\]\nwhere \\(\\mu\\) is some positive constant. Since \\(y''(x)=0\\) for all \\(x\\in\n(\\tau_0,\\tau_1)\\), it follows that \\(\\mu=\\lambda_1(q^*)\\).\nTherefore the eigenfunction \\(y\\) has the form\n\\[\n\ty(x)=\\left\\{\\begin{aligned}\n\t\t&A\\cdot\\cos[\\sqrt{\\mu}\\cdot(x-\\alpha_{\\mu})]&\n\t\t&\\text{for }x<\\tau_0,\\\\\n\t\t&B&&\\text{for }x\\in[\\tau_0,\\,\\tau_1],\\\\\n\t\t&C\\cdot\\cos[\\sqrt{\\mu}\\cdot(1-\\beta_{\\mu}-x)]&\n\t\t&\\text{for }x>\\tau_1,\\end{aligned}\\right.\n\\]\nwhere \\(A\\), \\(B\\) and \\(C\\) are some positive constants, and \\(\\alpha_{\\mu}\\),\n\\(\\beta_{\\mu}\\) are defined by \\eqref{eq:ab}. From the continuity of \\(y'\\)\nit follows that \\(\\tau_0=\\alpha_{\\mu}\\) and \\(\\tau_1=1-\\beta_{\\mu}\\), hence\n\\(A=B=C\\). Finally, from the condition \\(\\int_0^1 q^*\\,dx=1\\) we have\nthe equation \\eqref{eq:M1}.\n\nTo conclude the proof, it remains to note that equation \\eqref{eq:M1} has\na unique solution, because \\(\\alpha_{\\mu}\\) and \\(\\beta_{\\mu}\\), considered\nas functions of \\(\\mu>0\\), are non-negative, continuous, non-increasing\nand tend to zero as \\(\\mu\\to+\\infty\\).\n\n\\subsection{Proof of theorem \\ref{prop:M1-}.}\nConsider some potential \\(q^*\\in-\\Gamma_1\\), and some positive eigenfunction\n\\(y\\in\\ker T_{q^*}(\\lambda_1(q^*))\\). Suppose that \\(\\mathop{\\operatorname{supp}} q^*\\subseteq\n\\Omega^-(y)\\). Then for any \\(q\\in-\\Gamma_1\\) we, using \\ref{prop:4.2}, have\n\\begin{flalign*}\n\t&&0&=\\langle T_{q^*}(\\lambda_1(q^*))y,y\\rangle\\\\\n\t&& &=\\int_0^1\\left[(y')^2- \\lambda_1(q^*)\\,y^2\\right]\\,dx-\n\t\t\\inf\\limits_{x\\in [0,1]}y^2(x)+k_0^2\\,y^2(0)+k_1^2\\,y^2(1)\\\\\n\t&& &\\geq\\int_0^1\\left[(y')^2+(q-\\lambda_1(q^*))\\,y^2\\right]\\,dx+\n\t\tk_0^2\\,y^2(0)+k_1^2\\,y^2(1),&&\n\\end{flalign*}\nhence \\(\\langle T_q(\\lambda_1(q^*))y,y\\rangle\\leq 0\\). It follows that\n\\(\\lambda_1(q)\\leq\\lambda_1(q^*)\\), therefore \\(\\lambda_1(q^*)=M_1^-\\).\nThus we proved that \\(M_1^-\\) is attained at any potential \\(q^*\\) such that\n\\(\\mathop{\\operatorname{supp}} q^*\\subseteq\\Omega^-(y)\\).\n\nSuppose \\(k_0^2+k_1^2\\leq 1\\). Consider the generalized function\n\\(q^*\\rightleftharpoons -k_0^2\\boldsymbol{\\delta}_0-k_1^2\\boldsymbol{\\delta}_1-\n(1-k_0^2-k_1^2)\\), which in this case belongs to \\(-\\Gamma_1\\). Using\n\\eqref{eq:kv}, we get that the first eigenfunction of the pencil \\(T_{q^*}\\)\nis \\(y\\equiv\\mathrm{const}\\), so \\(\\mathop{\\operatorname{supp}}{q^*}\\subseteq\\Omega^-(y)\\). It follows\nthat \\(M_1^-\\) is attained at the potential \\(q^*\\) and is equal to\nthe corresponding first eigenvalue \\(\\lambda_1(q^*)=k_0^2+k_1^2-1\\).\n\nSuppose\n\\begin{gather}\\label{eq:first}\n\tk_0^2+k_1^2\\geq 1,\\\\ \\label{eq:second}\n\tk_1^2-k_0^2\\leq 1.\n\\end{gather}\nConsider the generalized function \\(q^*\\rightleftharpoons-(1+k_0^2-k_1^2)\n\\boldsymbol{\\delta}_0\/2-(1-k_0^2+k_1^2)\\boldsymbol{\\delta}_1\/2\\), which, due to\n\\eqref{eq:second}, belongs to \\(-\\Gamma_1\\). For such \\(q^*\\) the equation\n\\(T_{q^*}(\\lambda)y=0\\) is equivalent to problem \\eqref{eq:propM1-1},\n\\eqref{eq:propM1-2}. The first eigenvalue \\(\\lambda_1(q^*)\\), due to\n\\eqref{eq:first} and \\eqref{eq:propM1-2}, is non-negative and the corresponding\neigenfunction is\n\\begin{equation}\\label{eq:100500}\n\ty(x)\\equiv\\cos[\\sqrt{\\lambda_1(q^*)}\\cdot (x-\\zeta)],\n\\end{equation}\nwhere \\(\\zeta=1\/2\\). Hence \\(\\mathop{\\operatorname{supp}} q^*\\subseteq\\Omega^-(y)\\). It follows that\n\\(M_1^-\\) is attained at the potential \\(q^*\\) and is equal to the corresponding\nfirst eigenvalue \\(\\lambda_1(q^*)\\).\n\nSuppose \\(k_1^2-k_0^2\\geq 1\\). Consider the generalized function\n\\(q^*\\rightleftharpoons-\\boldsymbol{\\delta}_1\\in-\\Gamma_1\\). For such \\(q^*\\)\nthe equation \\(T_{q^*}(\\lambda)y=0\\) is equivalent to problem \\eqref{eq:propM1-1},\n\\eqref{eq:propM1-3}. The corresponding first eigenfunction is defined by\n\\eqref{eq:100500}, where \\(\\zeta\\in [0,1\/2]\\), since \\(k_1^2-1\\geq k_0^2\\).\nHence \\(\\mathop{\\operatorname{supp}} q^*\\subseteq\\Omega^{-}(y)\\). It follows that \\(M_1^-\\) is attained\nat the potential \\(q^*\\) and is equal to the corresponding first eigenvalue\n\\(\\lambda_1(q^*)\\).\n\n\\subsection{Proof of theorem \\ref{prop:m1+}.}\nConsider some potential \\(q\\in\\Gamma_1\\), and some positive eigenfunction \\(y\\in\n\\ker T_{q}(\\lambda_1(q))\\). Then for any \\(\\lambda>\\lambda_1(q)\\), according to\n\\ref{prop:4.2}, we have\n\\begin{flalign*}\n\t&& 0&>\\int_0^1 \\left[(y')^2+(q-\\lambda)\\,y^2\\right]\\,dx\n\t\t+k_0^2\\,y^2(0)+k_1^2\\,y^2(1)\\\\\n\t&& &\\geq\\int_0^1 \\left[(y')^2-\\lambda y^2\\,\\right]\\,dx\n\t\t+\\inf\\limits_{x\\in[0,1]} y^2(x)+k_0^2\\,y^2(0)\n\t\t+k_1^2\\,y^2(1).&\n\\end{flalign*}\nIt follows that there exists \\(\\zeta\\in [0,1]\\) such that\n\\[\n\t\\int_0^1 \\left[(y')^2+(\\boldsymbol{\\delta}_{\\zeta}-\\lambda)\\,\n\t\ty^2\\right]\\,dx+k_0^2\\,y^2(0)+k_1^2\\,y^2(1)<0.\n\\]\nSo for any \\(\\lambda>m^+_1\\) there exists \\(\\zeta\\in [0,1]\\) such that\n\\(\\lambda_1(\\boldsymbol{\\delta}_{\\zeta})<\\lambda\\). Hence, using \\ref{prop:4.2},\nwe get \\(m^+_1=\\inf_{x\\in [0,1]}\\lambda_1(\\boldsymbol{\\delta}_x)\\). This equality\nis equivalent, according to \\ref{prop:F1}, to the following fact: \\(F(m^+_1,x)\\)\nis defined for all \\(x\\in [0,1]\\) and satisfies \\(\\sup_{x\\in [0,1]}F(m^+_1,x)=1\\).\n\nSince \\(m_1^+>0\\), from \\ref{prop:F3} it follows that if \\(\\mu=m^+_1\\), then for\nany \\(\\zeta\\in [0,1]\\) conditions \\eqref{eq:sign} hold. According to \\eqref{eq:F},\n\\eqref{eq:sign} and\n\\begin{equation}\\label{eq:diff}\n\t\\dfrac{\\partial F(\\mu,\\zeta)}{\\partial\\zeta}\\equiv\\mu\\cdot\\dfrac{%\n\t\t\\cos^2[\\sqrt{\\mu}\\cdot(1-\\beta_{\\mu}-\\zeta)]-\\cos^2[\\sqrt{\\mu}\n\t\t\\cdot(\\zeta-\\alpha_{\\mu})]}{\\cos^2[\\sqrt{\\mu}\\cdot(\\zeta-\n\t\t\\alpha_{\\mu})]\\cdot\\cos^2[\\sqrt{\\mu}\\cdot(1-\\beta_{\\mu}-\\zeta)]},\n\\end{equation}\nit follows that the function \\(F(\\mu,\\cdot)\\) can have at some point \\(\\zeta\\in\n(0,1)\\) a local extremum satisfying \\(F(\\mu,\\zeta)>0\\) only if \\(\\zeta=(1-\n\\beta_{\\mu}+\\alpha_{\\mu})\/2\\), \\(\\zeta>\\alpha_{\\mu}\\) and \\(\\zeta<1-\\beta_{\\mu}\\).\nBut this conditions imply, according to \\eqref{eq:diff}, that such \\(\\zeta\\)\nmust be a point of strict local minimum of the function \\(F(\\mu,\\cdot)\\). Therefore\n\\(F(\\mu,\\cdot)\\) can't have a supremum in \\((0,1)\\), so we get \\(m^+_1=\\inf\\{%\n\\lambda_1(\\boldsymbol{\\delta}_0),\\lambda_1(\\boldsymbol{\\delta}_1)\\}\\). Note that\nfor the potential \\(q^*\\rightleftharpoons\\boldsymbol\\delta_i\\), where \\(i\\in\n\\{0,1\\}\\), the equation \\(T_{q^*}(\\lambda)y=0\\) is equivalent to the problem\n\\begin{gather*}\n\t-y''=\\lambda y,\\\\ y'(0)-[k_0^2+(1-i)]\\,y(0)=y'(1)+[k_1^2+i]\\,y(1)=0.\n\\end{gather*}\nTherefore we have\n\\[\n\t\\dfrac{\\lambda_1(\\boldsymbol{\\delta}_i)-k_0^2k_1^2-k_{1-i}^2}{%\n\t\tk_0^2+k_1^2+1}=\\sqrt{\\lambda_1(\\boldsymbol{\\delta}_i)}\n\t\t\\cot\\sqrt{\\lambda_1(\\boldsymbol{\\delta}_i)},\n\\]\nso \\(m^+_1=\\lambda_1(\\boldsymbol{\\delta}_1)\\).\n\n\\subsection{Proof of theorem \\ref{prop:m1-}.}\nConsider some potential \\(q\\in-\\Gamma_1\\), and some positive eigenfunction\n\\(y\\in\\ker T_{q}(\\lambda_1(q))\\). Then for any \\(\\lambda>\\lambda_1(q)\\), according\nto \\ref{prop:4.2}, we have\n\\begin{flalign*}\n\t&& 0&>\\int_0^1 \\left[(y')^2+(q-\\lambda)\\,y^2\\right]\\,dx\n\t\t+k_0^2\\,y^2(0)+k_1^2\\,y^2(1)\\\\\n\t&& &\\geq\\int_0^1 \\left[(y')^2-\\lambda y^2\\,\\right]\\,dx\n\t\t-\\sup\\limits_{x\\in[0,1]} y^2(x)+k_0^2\\,y^2(0)\n\t\t+k_1^2\\,y^2(1).&&\n\\end{flalign*}\nIt follows that there exists \\(\\zeta\\in [0,1]\\) such that\n\\[\n\t\\int_0^1 \\left[(y')^2+(-\\boldsymbol{\\delta}_{\\zeta}-\\lambda)\\,\n\t\ty^2\\right]\\,dx+k_0^2\\,y^2(0)+k_1^2\\,y^2(1)<0.\n\\]\nSo for any \\(\\lambda>m^-_1\\) there exists \\(\\zeta\\in [0,1]\\) such that\n\\(\\lambda_1(-\\boldsymbol{\\delta}_{\\zeta})<\\lambda\\). Hence, using \\ref{prop:4.2},\nwe get \\(m^-_1=\\inf_{x\\in [0,1]}\\lambda_1(-\\boldsymbol{\\delta}_x)\\). This equality\nis equivalent, according to \\ref{prop:F1}, to the following fact: \\(F(m^-_1,x)\\)\nis defined for all \\(x\\in [0,1]\\) and satisfies \\(\\sup_{x\\in [0,1]}F(m^-_1,x)=-1\\).\n\nFor any fixed value \\(\\mu\\in\\mathbb R\\) we consider the conditions\n\\begin{gather}\\label{eq:Fstat1}\n\tF(\\mu,\\zeta)<0,\\\\ \\label{eq:Fstat2}\n\t\\partial F(\\mu,\\zeta)\/\\partial\\zeta=0.\n\\end{gather}\nIt is clear that some point \\(\\zeta\\in (0,1)\\) can satisfy the equalities\n\\(F(\\mu,\\zeta)=\\sup_{x\\in [0,1]} F(\\mu,x)=-1\\) only if \\eqref{eq:Fstat1}\nand \\eqref{eq:Fstat2} hold.\n\nSuppose \\(\\mu>0\\). Then, according to \\eqref{eq:diff}, \\eqref{eq:F},\n\\eqref{eq:2.19} and \\eqref{eq:delt2}, for any point \\(\\zeta\\in (0,1)\\)\nsatisfying \\eqref{eq:Fstat1} condition \\eqref{eq:Fstat2} holds iff the problem\n\\begin{gather}\\label{eq:grmm3}\n\t-y''=\\mu y\\quad\\text{at } (0,\\zeta)\\cup(\\zeta,1),\\\\ \\label{eq:grmm4}\n\t\\hbox to 0.8\\textwidth{$y'(0)-k_0^2y(0)=2y'(\\zeta-0)+\n\t\tF(\\mu,\\zeta)y(\\zeta)=$\\hfill}\\\\ \\notag\n\t\\hbox to 0.8\\textwidth{\\hfill$=2y'(\\zeta+0)-F(\\mu,\\zeta)y(\\zeta)=\n\t\ty'(1)+k_1^2y(1)=0$}\n\\end{gather}\nhas a continuous positive solution. Besides, for any point \\(\\zeta\\in (0,1)\\)\nsatisfying \\eqref{eq:Fstat1} and \\eqref{eq:Fstat2} we have\n\\begin{equation}\\label{eq:39}\n\t\\alpha_{\\mu}>\\zeta>1-\\beta_{\\mu}.\n\\end{equation}\nTherefore, according to \\eqref{eq:diff} and \\eqref{eq:sign}, this stationary point\n\\(\\zeta\\) is a strict maximum of \\(F(\\mu,\\cdot)\\). Since for any \\(x\\in [0,1]\\),\nusing \\eqref{eq:39}, we get\n\\[\n\t-\\pi\/2<-\\sqrt{\\mu}\\alpha_{\\mu}\\leq\\sqrt{\\mu}\\cdot(x-\\alpha_{\\mu})\n\t<\\sqrt{\\mu}\\cdot(x-1+\\beta_{\\mu})\\leq\\sqrt{\\mu}\\beta_{\\mu}<\\pi\/2,\n\\]\nit follows from proposition \\ref{prop:F3} that the function \\(F(\\mu,\\cdot)\\)\nis defined everywhere on \\([0,1]\\).\n\nSuppose \\(\\mu=0\\). Let us use the same method as in previous case, changing\n\\eqref{eq:F} to \\eqref{eq:F0}, and \\eqref{eq:2.19} to \\eqref{eq:2.20}. Then we get\nthat for any point \\(\\zeta\\in (0,1)\\) satisfying \\eqref{eq:Fstat1} condition\n\\eqref{eq:Fstat2} holds iff problem \\eqref{eq:grmm3}, \\eqref{eq:grmm4} has\na continuous positive solution. Using \\eqref{eq:F0} we also get that the second\nderivative of \\(F(0,\\cdot)\\) is negative. Hence any stationary point \\(\\zeta\\in\n(0,1)\\) is a strict maximum of \\(F(0,\\cdot)\\).\n\nSuppose \\(\\mu\\in(-k_0^4,0)\\). Then, using \\eqref{eq:Fmin}, we get\n\\[\n\t\\dfrac{\\partial F(\\mu,\\zeta)}{\\partial\\zeta}\\equiv-\\mu\\left\\{\n\t\t\\sinh^{-2}\\left(\\sqrt{|\\mu|}\\zeta+\\alpha_{\\mu}\\right)\n\t\t-\\sinh^{-2}\\left(\\sqrt{|\\mu|}(1-\\zeta)+\\beta_{\\mu}\\right)\\right\\},\n\\]\nwhere\n\\[\n\t\\alpha_{\\mu}\\rightleftharpoons\\dfrac12\\,\\ln\\dfrac{k_0^2+\\sqrt{|\\mu|}}{%\n\t\tk_0^2-\\sqrt{|\\mu|}},\\qquad\n\t\\beta_{\\mu}\\rightleftharpoons\\dfrac12\\,\\ln\\dfrac{k_1^2+\\sqrt{|\\mu|}}{%\n\t\tk_1^2-\\sqrt{|\\mu|}}.\n\\]\nTherefore, according to \\eqref{eq:2.21}, for any point \\(\\zeta\\in (0,1)\\)\nsatisfying \\eqref{eq:Fstat1} condition \\eqref{eq:Fstat2} holds iff problem\n\\eqref{eq:grmm3}, \\eqref{eq:grmm4} has a continuous positive solution. Since\n\\(\\partial^2 F(\\mu,\\zeta)\/\\partial\\zeta^2<0\\), it follows that any stationary\npoint \\(\\zeta\\in (0,1)\\) is a strict maximum of \\(F(\\mu,\\cdot)\\).\n\nSuppose \\(0>\\mu=-k_0^4=-k_1^4\\). Then the function \\(F(\\mu,\\cdot)\\) is a negative\nconstant, and for any point \\(\\zeta\\in (0,1)\\) problem \\eqref{eq:grmm3},\n\\eqref{eq:grmm4} has a continuous positive solution.\n\nSuppose \\(\\mu\\in [-k_1^4,-k_0^4]\\), also \\(\\mu<0\\) and \\(k_1>k_0\\). Then from\n\\eqref{eq:Fmin} and \\eqref{eq:2.21} it follows that \\({\\partial F(\\mu,\\zeta)}\/\n{\\partial\\zeta}<0\\), and problem \\eqref{eq:grmm3}, \\eqref{eq:grmm4} has no positive\nsolutions for any \\(\\zeta\\in (0,1)\\).\n\nSuppose \\(\\mu<-k_1^4\\). Then, using \\eqref{eq:Fmin}, we get\n\\[\n\t\\dfrac{\\partial F(\\mu,\\zeta)}{\\partial\\zeta}\\equiv\\mu\\left\\{\n\t\t\\cosh^{-2}\\left(\\sqrt{|\\mu|}\\zeta+\\alpha_{\\mu}\\right)\n\t\t-\\cosh^{-2}\\left(\\sqrt{|\\mu|}(1-\\zeta)+\\beta_{\\mu}\\right)\\right\\},\n\\]\nwhere\n\\[\n\t\\alpha_{\\mu}\\rightleftharpoons\\dfrac12\\,\\ln\\dfrac{\\sqrt{|\\mu|}+k_0^2}{%\n\t\t\\sqrt{|\\mu|}-k_0^2},\\qquad\n\t\\beta_{\\mu}\\rightleftharpoons\\dfrac12\\,\\ln\\dfrac{\\sqrt{|\\mu|}+k_1^2}{%\n\t\t\\sqrt{|\\mu|}-k_1^2}.\n\\]\nTherefore, according to \\eqref{eq:2.21}, for any point \\(\\zeta\\in (0,1)\\)\nsatisfying \\eqref{eq:Fstat1} condition \\eqref{eq:Fstat2} holds iff problem\n\\eqref{eq:grmm3}, \\eqref{eq:grmm4} has a continuous positive solution. Since\n\\(\\partial^2 F(\\mu,\\zeta)\/\\partial\\zeta^2>0\\), it follows that any stationary\npoint \\(\\zeta\\in (0,1)\\) is a strict minimum of \\(F(\\mu,\\cdot)\\).\n\nFrom proposition \\ref{prop:F1} we also get that for any \\(\\mu\\leq 0\\) the function\n\\(F(\\mu,\\cdot)\\) is defined everywhere on \\([0,1]\\).\n\nCombining all this, we obtain the following: the existence of a continuous positive\nsolution to problem \\eqref{eq:grmm1}, \\eqref{eq:grmm2} for some \\(\\mu\\geq-k_0^4\\)\nand \\(\\zeta\\in (0,1)\\) implies that \\(F(\\mu,\\zeta)=-1\\), the function \\(F(\\mu,\n\\cdot)\\) is defined everywhere on \\([0,1]\\), and \\(\\sup_{x\\in [0,1]}F(\\mu,x)\\leq\n-1\\). Therefore \\(m_1^-=\\lambda_1(-\\boldsymbol\\delta_{\\zeta})\\). In converse,\nif for any \\(\\mu\\geq-k_0^4\\) and \\(\\zeta\\in (0,1)\\) the positive solution of\n\\eqref{eq:grmm1}, \\eqref{eq:grmm2} doesn't exist, we get \\(m^{-}_1=\n\\inf\\{\\lambda_1(-\\boldsymbol{\\delta}_0),\\lambda_1(-\\boldsymbol{\\delta}_1)\\}\\).\nFrom the equation\n\\[\n\t\\lambda_1(-\\boldsymbol{\\delta}_i)-k_0^2k_1^2+k_{1-i}^2\n\t\t=(k_0^2+k_1^2-1)\\cdot\\psi(\\lambda_1(-\\boldsymbol{\\delta}_i)),\n\\]\nwhere \\(i\\in\\{0,1\\}\\) and\n\\[\n\t\\psi(x)\\rightleftharpoons\\left\\{\\begin{aligned}\n\t\t&\\sqrt{x}\\cot\\sqrt{x}&&\\text{for }x>0,\\\\\n\t\t&1&&\\text{for }x=0,\\\\\n\t\t&\\sqrt{|x|}\\coth\\sqrt{|x|}&&\\text{for }x<0,\\end{aligned}\\right.\n\\]\nwe obtain that \\(\\inf\\{\\lambda_1(-\\boldsymbol{\\delta}_0),\n\\lambda_1(-\\boldsymbol{\\delta}_1)\\}=\\lambda_1(-\\boldsymbol{\\delta}_0)\\).\n\n\\subsection{}\\label{pt:3.5}\nNow we get some conditions for the existence of a continuous positive solution\nto problem \\eqref{eq:grmm1}, \\eqref{eq:grmm2} considered in theorem \\ref{prop:m1-}.\n\nSuppose \\(\\mu_0(\\zeta)\\), where \\(\\zeta\\in (0,1]\\), is the minimal eigenvalue\nof the problem\n\\begin{gather}\\label{eq:grmm5}\n\t-y''=\\lambda y,\\\\ \\label{eq:grmm6}\n\ty'(0)-k_0^2y(0)=2y'(\\zeta)-y(\\zeta)=0,\n\\end{gather}\nand suppose \\(\\mu_1(\\zeta)\\), where \\(\\zeta\\in [0,1)\\), is the minimal eigenvalue\nof problem \\eqref{eq:grmm5} and\n\\[\n\t2y'(\\zeta)+y(\\zeta)=y'(1)+k_1^2y(1)=0.\n\\]\nIt is clear that for some \\(\\mu\\in\\mathbb R\\) and \\(\\zeta\\in (0,1)\\) a continuous\npositive solution to \\eqref{eq:grmm1}, \\eqref{eq:grmm2} exists iff the equalities\n\\(\\mu_0(\\zeta)=\\mu_1(\\zeta)=\\mu\\) hold.\n\n\\subsubsection{}\\label{prop:3.2.1}\n{\\itshape If \\(k_0^2=1\/2\\), then \\(\\mu_0(\\zeta)\\equiv-1\/4\\).\n\nIf \\(k_0^2>1\/2\\), then the function \\(\\mu_0\\) strictly decreases and satisfies\n\\(\\lim\\limits_{\\zeta\\to 0}\\mu_0(\\zeta)=+\\infty\\) and \\(\\mu_0(1)>-1\/4\\).\n\nIf \\(k_0^2<1\/2\\), then for any \\(\\zeta\\in (0,1]\\) the inequality\n\\(\\mu_0(\\zeta)<-1\/4\\) holds.\n}\n\n\\begin{proof}\nSuppose \\(k_0^2=1\/2\\). Then for any \\(\\zeta\\in (0,1]\\) problem\n\\eqref{eq:grmm5}, \\eqref{eq:grmm6} has the positive eigenfunction \\(y(x)\\equiv\ne^{x\/2}\\) corresponding to the eigenvalue \\(-1\/4\\).\n\nSuppose \\(k_0^2>1\/2\\). Since the eigenvalues of problem \\eqref{eq:grmm5},\n\\eqref{eq:grmm6} increase by \\(k_0^2\\), it follows that \\(\\mu_0(\\zeta)>-1\/4\\).\nThen let \\(y_0\\in W_2^1[0,\\zeta]\\) be an eigenfunction of problem\n\\eqref{eq:grmm5}, \\eqref{eq:grmm6} corresponding to the eigenvalue\n\\(\\mu_0(\\zeta)\\). Continuing the function \\(y_0\\) for any \\(\\theta\\in (\\zeta,1]\\)\nto the interval \\((\\zeta,\\theta]\\) in the form \\(y(x)\\rightleftharpoons y_0(\\zeta)\ne^{(x-\\zeta)\/2}\\), for the obtained function \\(y\\in W_2^1[0,\\theta]\\) we get\n\\[\n\t\\int_0^{\\theta} \\left[(y')^2-\\mu_0(\\zeta)\\,y^2\\right]\\,dx+\n\t\tk_0^2\\,y^2(0)-\\dfrac{y^2(\\theta)}{2}=[-1\/4-\\mu_0(\\zeta)]\\cdot\n\t\t[e^{\\theta-\\zeta}-1]\\cdot y^2(\\zeta)<0,\n\\]\nhence \\(\\mu_0(\\theta)<\\mu_0(\\zeta)\\). Finally, for \\(\\zeta\\to 0\\) we have uniform\nby \\(y\\in W_2^1[0,\\zeta]\\) asymptotic estimate\n\\begin{flalign*}\n\t&& \\int_0^{\\zeta} (y')^2\\,dx+k_0^2\\,y^2(0)-\\dfrac{y^2(\\zeta)}{2}&=\n\t\\left[\\int_0^{\\zeta}\\dfrac{(y')^2}{2}\\,dx+(k_0^2-1\/2)\\,y^2(0)\\right]\n\t\t{+}\\left[\\int_0^{\\zeta}\\dfrac{(y')^2}{2}\\,dx+\\dfrac{y^2(0)-\n\t\ty^2(\\zeta)}{2}\\right]\\\\\n\t&& &\\geq\\dfrac{k_0^2-1\/2+o(1)}{\\zeta}\\int_0^{\\zeta} y^2\\,dx-\n\t\t\\dfrac{1}{2}\\int_0^{\\zeta} y^2\\,dx,\n\\end{flalign*}\ntherefore \\(\\mu_0(\\zeta)\\geq [k_0^2-1\/2+o(1)]\\cdot\\zeta^{-1}\\).\n\nThe inequality \\(\\mu_0(\\zeta)<-1\/4\\) for the case \\(k_0^2<1\/2\\) is proved\nlikewise  the inequality  \\(\\mu_0(\\zeta)>-1\/4\\) for the case \\(k_0^2>1\/2\\).\n\\end{proof}\n\n\\subsubsection{}\\label{prop:3.2.2}\n{\\itshape If \\(k_1^2=1\/2\\), then \\(\\mu_1(\\zeta)\\equiv-1\/4\\).\n\nIf \\(k_1^2>1\/2\\), then the function \\(\\mu_1\\) strictly increases and satisfies\n\\(\\lim\\limits_{\\zeta\\to 1}\\mu_1(\\zeta)=+\\infty\\) and \\(\\mu_1(0)>-1\/4\\).\n\nIf \\(k_1^2<1\/2\\), then for any \\(\\zeta\\in [0,1)\\) the inequality\n\\(\\mu_1(\\zeta)<-1\/4\\) holds.\n}\n\n\\medskip\nThe proposition \\ref{prop:3.2.2} is proved likewise \\ref{prop:3.2.1}.\n\nCombining \\ref{prop:3.2.1} and \\ref{prop:3.2.2}, we get the last proposition:\n\n\\subsubsection{}\\label{prop:3.2.3}\n{\\itshape Problem \\eqref{eq:grmm1}, \\eqref{eq:grmm2} has a continuous positive\nsolution for some \\(\\mu\\geq -k_0^4\\) and \\(\\zeta\\in (0,1)\\) iff\none of these two conditions holds: \\(k_0^2>1\/2\\) or \\(k_0^2=k_1^2=1\/2\\).\n}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIt is well known \\cite{AkhGl} that every self-adjoint or unitary\noperator with a simple spectrum acting on some separable Hilbert\nspace is unitarily equivalent to the operator of multiplication by\nthe independent variable on the Hilbert space $L^2(\\dR,d\\mu)$ or\n$L^2(\\dT,d\\mu)$, respectively, where $d\\mu$ is a probability measure\non the real line $\\dR$ or on the unit circle\n$\\dT=\\{\\zeta\\in\\dC:|\\zeta|=1\\}$. The matrix representation of\nself-adjoint operators with simple spectrum was established for the\nfirst time by Stone \\cite{Akh}. He proved that every self-adjoint\noperator with a simple spectrum is unitarily equivalent to a certain\nJacobi (tri-diagonal) matrix of the form\n\\begin{equation}\n\\label{01}\n  J=\\begin{pmatrix} b_1 & a_1 & 0 &0   & 0 &\n\\cdot &\n\\cdot  \\\\\na_1 & b_2 & a_2 & 0 &0& \\cdot &\n\\cdot  \\\\\n0    & a_2 & b_3 & a_3 &0& \\cdot &\n\\cdot   \\\\\n\\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot\n\\end{pmatrix},\n\\end{equation}\nwhere $a_k>0$, and $b_k$ are real numbers for all $k\\in\\dN$. The\nnon-self-adjoint version of the Stone theorem has been recently\nobtained in \\cite{ArlTsek} for dissipative non-self-adjoint\noperators with rank one imaginary part. It turned out that the\nmatrix representation of such operators is a non-self-adjoint Jacobi\nmatrix of the form (\\ref{01}) with only nonreal first entry $b_1$\nsatisfying ${\\rm Im\\,} b_1>0$.\n\nThe problem of the canonical matrix representation  of a unitary\noperator with a simple spectrum has been recently solved by M.\nCantero, L. Moral and L. Vel\\'azquez in \\cite{CMV1}. They introduced\nand studied five-diagonal unitary matrices of the form\n\\begin{equation}\\label{02}\n\\cC=\\cC(\\{\\alpha_n\\})=\\begin{pmatrix}\n\\bar{\\alpha}_0&\\bar{\\alpha}_1\\rho_0&\\rho_1\\rho_0&0&0&\\ldots\\cr\n\\rho_0&-\\bar{\\alpha}_1\\alpha_0&-\\rho_1\\alpha_0&0&0&\\ldots\\cr\n0&\\bar{\\alpha}_2\\rho_1&-\\bar{\\alpha}_2\\alpha_1&\\bar{\\alpha}_3\\rho_2&\\rho_3\n\\rho_2& \\ldots\\cr\n0&\\rho_2\\rho_1&-\\rho_2\\alpha_1&-\\bar{\\alpha}_3\\alpha_2&-\\rho_3\\alpha_2&\\ldots\n\\cr 0&0&0&\\bar{\\alpha}_4\\rho_3&-\\bar{\\alpha}_4\\alpha_3&\\ldots\\cr\n\\ldots&\\ldots&\\ldots&\\ldots&\\ldots&\\ldots\n\\end{pmatrix}.\n\\end{equation}\nSuch matrix appears as a matrix representation of the unitary\noperator $(Uf)(\\zeta)=\\zeta f(\\zeta)$ in $L_2(\\dT,d\\mu)$ with\nrespect to the orthonormal system $\\{\\chi_n\\}$ obtained by\northonormalization  of the sequence\n$\\{1,\\zeta,\\zeta^{-1},\\zeta^2,\\zeta^{-2},\\ldots\\}$. The so called\nSchur parameters or Verblunsky coefficients $\\{\\alpha_n\\}$,\n$|\\alpha_n|< 1$, arise in the Szeg\\H{o} recurrence formula\n$$\\zeta \\Phi_n(\\zeta)=\\Phi_{n+1}(\\zeta)+\n\\bar\\alpha_n\\zeta^n\\overline{\\Phi_n(1\/\\bar\\zeta)},\\qquad n=0,1, \\ldots\n$$\nfor monic orthogonal with respect to $d\\mu$ polynomials\n$\\{\\Phi_n\\}$, and $\\rho_n:=\\sqrt{1-|\\alpha_n|^2}$. The matrices\n$\\cC(\\{\\alpha_n\\})$ are called the {\\it CMV matrices}. The spectral\nanalysis of CMV matrices has recently attracted much attention, and\nwe refer on this matter to the papers \\cite{CMV1, CMV2, GZ, S1, S2,\nS3}.\n\nThe spectral theory of non-self-adjoint and nonunitary operators and\ntheir models is based on the concept of {\\it characteristic\nfunction} of the corresponding operator or the operator colligation\n\\cite{BrR2, Br, Br1, L1, L2, L, LYan, N1, NKh, Pavl, Pol, SF}.\n\nIn this paper we employ the Sz.-Nagy--Foias theory \\cite{SF} and the\nunitary colligations approach \\cite{Br1} to the spectral analysis of\ncontractions acting on Hilbert spaces. The corresponding\ncharacteristic function belongs to the Schur class of\noperator-valued functions holomorphic in the open unit disk ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$.\nBy Sz.-Nagy--Foias theorem \\cite[Proposition VI.2.1]{SF} each\ncompletely nonunitary contraction $T$ with rank one defect operators\n$D_T=(I-T^*T)^{1\/2}$ and $D_{T^*}=(I-TT^*)^{1\/2}$ (shortly, with\nrank one {\\it defects}) is unitarily equivalent to the operator\n(functional model) of the form\n\\[\n\\begin{split}\n&\\sH_\\Theta=\\left(H^2\\oplus{\\rm{clos}}\\,\\Delta\nL^2(\\dT))\\right)\\ominus\n\\left\\{\\Theta u\\oplus\\Delta u:u\\in H^2\\right\\}=\\\\\n&=\\left\\{\\begin{pmatrix}f\\cr g \\end{pmatrix}: f\\in H^2,\\;g\\in\n{\\rm{clos}}\\,\\Delta L^2(\\dT)), P_{H^2}(\\overline \\Theta f+\\Delta g)=0\\right\\},\\\\\n&\\sT_\\Theta \\begin{pmatrix}f\\cr g \\end{pmatrix}=P_{\\sH_\\Theta}\\,\n\\zeta\\begin{pmatrix}f\\cr g \\end{pmatrix},\\;\\sT^*_\\Theta\n\\begin{pmatrix}f\\cr g\n\\end{pmatrix}=\\begin{pmatrix}\\bar \\zeta (f-f(0))\\cr \\bar \\zeta g\n\\end{pmatrix} \\begin{pmatrix}f\\cr g\n\\end{pmatrix}\\in \\sH_\\Theta,\n\\end{split}\n\\]\nwhere $H^2$ is the Hardy space,\n$$\\Theta=\\Theta_T(z)=\\left(-T+zD_{T^*}(I-zT^*)^{-1}D_{T}\\right){\\upharpoonright\\,}{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T}$$\nis the characteristic function of $T$, $\\Delta^2=1-|\\Theta|^2$,\n$P_{H^2}$ is the orthogonal projection onto $H^2$ in $L^2(\\dT)$, and\n$P_{\\sH_\\Theta}$ is the orthogonal projection onto the model space\n$\\sH_\\Theta$.\n\n{\\it We obtain a new functional model that complements the above\nmentioned Sz.-Nagy--Foias functional model, and show that every\ncompletely nonunitary contraction $T$ with rank one defects is\nunitarily equivalent to the compression $f(\\zeta)\\to P_\\cK\n\\left(\\zeta\\, f(\\zeta)\\right)$ on the Hilbert space $L^2(\\dT,d\\mu)$\nwith a probability measure $\\mu$ onto subspace\n$\\cK=L^2(\\dT,d\\mu)\\ominus \\dC$.}\n\nWe study the so called {\\it truncated} CMV matrix $\\cT$ obtained\nfrom the ``full'' CMV matrix $\\cC=\\cC(\\{\\alpha_n\\})$ (\\ref{02}) by\ndeleting the first row and the first column:\n\\[\n\\cT=\\cT (\\{\\alpha_n\\})=\\begin{pmatrix}\n-\\bar{\\alpha}_1\\alpha_0&-\\rho_1\\alpha_0&0&0&\\ldots\\cr\n\\bar{\\alpha}_2\\rho_1&-\\bar{\\alpha}_2\\alpha_1&\\bar{\\alpha}_3\\rho_2&\\rho_3\\rho_2&\n\\ldots\\cr\n\\rho_2\\rho_1&-\\rho_2\\alpha_1&-\\bar{\\alpha}_3\\alpha_2&-\\rho_3\\alpha_2&\\ldots\\cr\n\\ldots&\\ldots&\\ldots&\\ldots&\\ldots\n\\end{pmatrix}.\\]\nIn the semi-infinite case $\\cT$ takes on the block-matrix form (see\nSection 4.3)\n\\[\n  \\cT=\\begin{pmatrix} \\cB_1 & \\cC_1 & 0 &0   & 0 &\n\\cdot &\n\\cdot  \\\\\n{\\mathcal A}}   \\def\\cB{{\\mathcal B}}   \\def\\cC{{\\mathcal C}_1 & \\cB_2 & \\cC_2 & 0 &0& \\cdot &\n\\cdot  \\\\\n0    & {\\mathcal A}}   \\def\\cB{{\\mathcal B}}   \\def\\cC{{\\mathcal C}_2 & \\cB_3 & \\cC_3 &0& \\cdot &\n\\cdot   \\\\\n\\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot\n\\end{pmatrix}.\n\\]\n{\\it It turned out that the truncated CMV matrix $\\cT(\\{\\alpha_n\\})$\nis a contraction with rank one defects, and the Sz.-Nagy--Foias\ncharacteristic function that agrees with the Schur function $f$\nassociated with the measure $\\mu$ \\cite{S2}. Moreover, we show that\nthe sub-matrix $\\cT^{(k)}(\\{\\alpha_n\\})$ obtained from\n$\\cT(\\{\\alpha_n\\})$ by deleting the first $k$ rows and columns is\nalso a contraction with rank one defects, and its characteristic\nfunction agrees with the well known $k^{th}$ Schur iterate}\n\\[\nf_{k}(z)=\\frac{f_{k-1}(z)-\\alpha_{k-1}}{z(1-\\bar\\alpha_{k-1}\nf_{k-1}(z))},\\qquad f_0(z)=f(z).\n\\]\nThis relation is an analog of the corresponding relation between the\n$m$-function of a Jacobi matrix and the m-function of its sub-matrix\n(cf. \\cite{GS}).\n\n{\\it Our main result states that each completely nonunitary\ncontraction $T$ with rank one defects is unitarily equivalent to the\none-parameter family $\\cT(\\{e^{it}\\alpha_n\\})$, where $\\{\\alpha_n\\}$\nare the Schur parameters of the Sz.-Nagy--Foias characteristic\nfunction of $T$.\n We develop direct and inverse spectral analysis for\nfinite and  semi-infinite truncated CMV matrices. It is shown that\ngiven an arbitrary set of $N$ not necessarily distinct numbers from\n${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$ there is a one-parameter family of unitarily equivalent\n$N\\times N$ truncated CMV matrices having those numbers as the\neigenvalues counting algebraic multiplicity. We prove the uniqueness\nof $N\\times N$ truncated CMV matrix $\\cT$ with given not necessarily\ndistinct eigenvalues $z_1,\\ldots,z_r$ and given  {\\it first} $N-r+1$\nSchur parameters $\\alpha_0(\\cT),\\ldots,\\alpha_{N-r}(\\cT)$. This\nresult on inverse spectral analysis of finite truncated CMV matrices\nis an analog of the Hochstadt \\cite{H3} and Gesztesy-Simon \\cite{GS}\nuniqueness theorems for finite self-adjoint Jacobi matrices as well\nas for established in \\cite{ArlTsek} uniqueness theorem for finite\nnon-self-adjoint Jacobi matrices with rank one imaginary part. We\nobtain the existence of $N\\times N$ truncated CMV matrix $\\cT$ when\nits eigenvalues $z_1,\\ldots,z_m$ and the {\\it last} Schur parameters\n$\\alpha_m(\\cT),\\ldots, \\alpha_N(\\cT)$ are known.}\n\nHere is a summary of the rest of the paper. In Sections 2 and 3 we\ndiscuss some basics from the Sz.-Nagy--Foias theory and the unitary\ncolligations with the focus upon the characteristic function and its\nproperties. Section 4 provides a brief overview of the theory of\northogonal polynomials on the unit circle and CMV matrices. The main\nresults concerning truncated CMV matrices and the models of\ncompletely nonunitary contractions with rank one defects are\npresented in Section 5 and 6. The final section 7 deals with the\ninverse spectral analysis for truncated CMV matrices.\n\n\n\\section{Contractions, unitary colligations, and their characteristic\nfunctions}\n\\subsection{Contractions and the Sz.- Nagy -- Foias\ncharacteristic functions}\n\nLet $H$ be a separable Hilbert space with the inner product\n$(\\cdot,\\cdot)$. A bounded linear operator $T$ in $H$ is called a\n\\textit{contraction} if $\\|T\\|\\le 1$ (for the basic properties of\ncontractions see \\cite [Chapter I]{SF}). If $T$ is a contraction\nthen the operators\n\\[\nD_T:=(I-T^*T)^{1\/2},\\qquad D_{T^*}:=(I-TT^*)^{1\/2}\n\\]\nare called the \\textit{defect operators} of $T$ or shortly {\\it\ndefects}, and the subspaces ${\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T={\\rm \\overline{ran}\\,} D_T,$ ${\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}={\\rm \\overline{ran}\\,}\nD_{T^*}$ the \\textit{defect subspaces} of $T$. The dimensions\n$\\dim{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T,$ $\\dim{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}$ are known as the \\textit{defect\nnumbers} of $T$. Given a pair of numbers $n,n^*=0,1,\\ldots,\\infty$\nit is easy to construct a contraction with $n=\\dim{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T$,\n$n^*=\\dim{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}$. Each contraction $T$ acting on a finite\ndimensional Hilbert space has equal defect numbers: $n=n^*$.\n\nThe defect operators satisfy the following intertwining relations\n\\begin{equation}\n\\label{defect} TD_{T}=D_{T^*}T,\\qquad T^*D_{T^*}=D_{T}T^*,\n\\end{equation}\nand the block-operators\n\\[\n\\begin{pmatrix}-T^*&D_T\\cr D_{T^*} &T \\end{pmatrix}:\n\\begin{pmatrix}{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}\\cr H\\end{pmatrix}\n\\to \\begin{pmatrix}{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T}\\cr\nH\\end{pmatrix},\\qquad\\begin{pmatrix}-T&D_{T^*}\\cr D_{T} &T^*\n\\end{pmatrix}:\\begin{pmatrix}{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T}\\cr H\\end{pmatrix} \\to\n\\begin{pmatrix}{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}\\cr H\\end{pmatrix}\n\\]\nare unitary operators in the corresponding orthogonal sums of the\nspaces. It follows from \\eqref{defect} that\n$T{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T\\subset{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}$, $T^*{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}\\subset{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T$, and $T(\\ker\nD_T)=\\ker D_{T^*},$ $T^*(\\ker D_{T^*})=\\ker D_{T}$. Moreover,\n$T{\\upharpoonright\\,}\\ker D_{T}$ and $T^*{\\upharpoonright\\,}\\ker D_{T^*}$ are isometric\noperators. It follows that $T$ is a \\textit{quasi-unitary extension}\n\\cite{L1} of the isometric operator $V=T{\\upharpoonright\\,}\\ker D_{T}$ (for the\ndefinition see Section 6.2).\n\nA contraction $T$ is called \\textit{completely nonunitary} if there\nis no nontrivial reducing subspace of $T$, on which $T$ generates a\nunitary operator. One of the fundamental results of the contractions\ntheory \\cite [Theorem I.3.2]{SF} reads that, given a contraction $T$\nin $H$, there is a canonical orthogonal decomposition\n\\[\nH=H_0\\oplus H_1, \\qquad T=T_0\\oplus T_1, \\quad T_j=T{\\upharpoonright\\,} H_j,\n\\quad j=0,1,\n\\]\nwhere $H_0$ and $H_1$ reduce $T$, $T_0$ is a completely nonunitary\ncontraction, and $T_1$ is a unitary operator. Moreover,\n\\[\nH_1= \\left(\\bigcap\\limits_{n\\ge 1}\\ker\nD_{T^n}\\right)\\bigcap\\left(\\bigcap\\limits_{n\\ge 1}\\ker\nD_{T^{*n}}\\right),\n\\]\nso,\n\\begin{equation}\n\\label{cu} T \\;\\mbox{is completely\nnonunitary}\\;\\iff\\left(\\bigcap\\limits_{n\\ge 1}\\ker\nD_{T^n}\\right)\\bigcap\\left(\\bigcap\\limits_{n\\ge 1}\\ker\nD_{T^{*n}}\\right)=\\{0\\}.\n\\end{equation}\nClearly,\n\\begin{equation}\n\\label{perp}\n\\begin{split}\n&\\bigcap\\limits_{n\\ge 1}\\ker D_{T^n}=H\\ominus{\\rm \\overline{span}\\,}\\left\\{T^{*n}D_T\nH,\\;\nn=0,1,\\ldots\\right\\},\\\\\n&\\bigcap\\limits_{n\\ge 1}\\ker\nD_{T^{*n}}=H\\ominus{\\rm \\overline{span}\\,}\\left\\{T^nD_{T^*} H,\\;\nn=0,1,\\ldots\\right\\}.\n\\end{split}\n\\end{equation}\nLet $V$ be an isometry in $H$. A subspace $\\Omega$ in $H$ is called\nwandering for $V$ if $V^p\\Omega\\perp V^q\\Omega$ for all $p,q\\in\n\\dZ_+$, $p\\ne q$. Since $V$ is an isometry, the latter is equivalent\nto $V^n\\Omega\\perp \\Omega$ for all $n\\in\\dN$. If\n$H=\\sum_{n=0}^\\infty\\oplus V^n\\Omega$, then $V$ is called a\n\\textit{unilateral shift} and $\\Omega$ is called the generating\nsubspace. The dimension of $\\Omega$ is called the multiplicity of\nthe unilateral shift $V$. It is well known \\cite[Theorem I.1.1]{SF}\nthat $V$ is a unilateral shift if and only if $\\bigcap_{n=0}^\\infty\nV^n H=\\{0\\}.$ Clearly, if an isometry $V$ is the unilateral shift in\n$H$, then $\\Omega=H\\ominus VH$ is the generating subspace for $V$.\n\nGiven a contraction $T$ in $H$ and a subspace $\\sH\\subset H$, the\nunilateral shift $V:\\sH\\to\\sH$ is said to be \\textit{contained in\n$T$}, if $\\sH$ is invariant for $T$, and $T{\\upharpoonright\\,}\\sH= V$ \\cite{D0}.\n The subspaces $\\bigcap\\limits_{n\\ge 1}\\ker D_{T^n}$ and\n$\\bigcap\\limits_{n\\ge 1}\\ker D_{T^{*n}}$ are invariant for $T$ and\n$T^*$, respectively, and the operators $V_T:=T{\\upharpoonright\\,}\n\\bigcap\\limits_{n\\ge 1}\\ker D_{T^n}$ and\n$V_{T^*}:={\\upharpoonright\\,}\\bigcap\\limits_{n\\ge 1}\\ker D_{T^{*n}}$ are\nunilateral shifts. Moreover, $V_T$ and $V_{T^*}$ are the maximal\nunilateral shifts contained in $T$ and $T^*$. The multiplicities of\nthe shifts $V_T$ and $V_{T^*}$ do not exceed the defect numbers\n$\\dim{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}$ and $\\dim{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T}$, respectively \\cite{DR}. If $T$ is\na completely nonunitary contraction with rank one defects, then (see\n\\cite{D0}, \\cite[Theorem 1.7]{DR})\n\\begin{equation}\n\\label{shift}\n\\begin{split}\n& T\\;\\mbox{does not contain the unilateral shift}\\iff\nT^*\\;\\mbox{does\nnot contain the unilateral shift}\\\\\n& \\iff\\bigcap\\limits_{n\\ge 1}\\ker D_{T^n}=\\{0\\}\\iff\n\\bigcap\\limits_{n\\ge 1}\\ker D_{T^{*n}}=\\{0\\}.\n\\end{split}\n\\end{equation}\n\n The function (see \\cite[Chapter VI]{SF})\n\\[\n\\Theta_T(z)=\\left(-T+zD_{T^*}(I-zT^*)^{-1}D_{T}\\right){\\upharpoonright\\,}{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T}\n\\]\nis known as the \\textit{characteristic function} of the Sz.-Nagy --\nFoias type of a contraction $T$. This function belongs to the\n\\textit{Schur class} ${\\bf S}({\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T,{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*})$ of\n$\\cL({\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T,{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*})$-valued holomorphic in the unit disk ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$\noperator-functions, i.e., $\\|\\Theta_T(z)\\|\\le 1$ for $z\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$.\nMoreover, the function $\\Theta_T$ satisfies the condition\n$\\|\\Theta_T(0)f\\|<\\|f\\|$ for all $f\\in{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T\\setminus\\{0\\}$. The\ncharacteristic functions of $T$ and $T^*$ are connected by the\nrelation\n\\[\n \\Theta_{T^*}(z)=\\Theta^*_T(\\bar z),\\quad z\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}.\n\\]\n\nTwo operator-valued functions $\\Theta_1\\in{\\bf S}({\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}_1,\\sN_1)$ and\n$\\Theta_2\\in{\\bf S}({\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}_2,\\sN_2)$ are said to \\textit{agree} if\nthere are two unitary operators $V:\\sN_1\\to\\sN_2$ and\n$W:{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}_2\\to{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}_1$ such that\n\\[\nV\\Theta_1(z)W=\\Theta_2(z),\\quad z\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}.\n\\]\nIt is well known \\cite[Theorem VI.3.4]{SF}, that two completely\nnonunitary contractions $T_1$ and $T_2$ are unitary equivalent if\nand only if their characteristic functions $\\Theta_{T_1}$ and\n$\\Theta_{T_2}$ agree.\n\nEvery operator-valued function $\\Theta$ from the Schur class ${\\bf\nS}({\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN)$ has almost everywhere nontangential strong limit values\n$\\Theta(\\zeta)$, $\\zeta \\in \\dT$. A function $\\Theta \\in {\\bf\nS}({\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN)$ is called \\textit{inner} if\n$\\Theta^*(\\zeta)\\Theta(\\zeta)=I_{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}$ for a.e. $\\zeta\\in\\dT$, and\n\\textit{co-inner} if $\\Theta(\\zeta)\\Theta^*(\\zeta)=I_\\sN$ for a.e.\n$\\zeta\\in\\dT$. A function $\\Theta\\in {\\bf S}({\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN)$ is called\n\\textit{bi-inner}, if it is both inner and co-inner. A contraction\n$T$ on a Hilbert space $\\sH$ belongs to the classes $C_{0\\,\\cdot}$\n($C_{\\cdot\\, 0}$), if\n\\[\ns-\\lim\\limits_{n\\to\\infty}T^n=0  \\qquad\n(s-\\lim\\limits_{n\\to\\infty}T^{*n}=0),\n\\]\nrespectively. By definition $C_{00}:=C_{0\\,\\cdot}\\cap C_{\\cdot\\,\n0}$. The completely nonunitary part of a contraction $T$ belongs to\nthe class $C_{\\cdot\\,0}$, $C_{0\\,\\cdot}$, or $C_{00}$  if and only\nif its characteristic function $\\Theta_T(z)$  is inner, co-inner, or\nbi-inner, respectively (cf. \\cite[Section VI.2]{SF}).\n\nIn the following statement \\cite[Theorem VI.4.1]{SF} the spectrum of\ncompletely nonunitary contractions is described.\n\\begin{theorem}\n\\label{Spectr} Let $T$ be a completely nonunitary contraction on\n$H$. Denote by $S_T$ the set of points $z \\in {\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$ for which the\noperator $\\Theta_T(z)$ is not boundedly invertible, together with\nthose $z\\in \\dT$ not lying on any of the open arcs of $\\dT$ on which\n$\\Theta_T$ is a unitary operator valued analytic function.\nFurthermore, denote by $S_T^0$ the set of points $z\\in {\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$ for\nwhich $\\Theta_T(z)$ is not invertible at all. Then the spectrum\n$\\sigma(T)$ of $T$ agrees with $S_T$, and the point spectrum\n$\\sigma_p(T)$ with $S_T^0$.\n\\end{theorem}\n\nIf $T$ is a completely nonunitary contraction with rank one defects,\nand if $z_0$ is an eigenvalue of $T$, then the geometric\nmultiplicity of $z_0$ is one, the algebraic multiplicity is finite,\nand the characteristic function $\\Theta_T$ admits the following\nfactorization\n\\[\n\\Theta_T(z)=c\\prod\\frac{\\bar z_k}{z_k}\\,\\frac{z_k-z}{1-\\bar z_k\nz}\\,\\exp\\left(-\\int\\limits_0^{2\\pi}\\frac{e^{it}+z}{e^{it}-z}d\\mu(t)\\right)\\,\\\n\\exp\\left(\\frac{1}{2\\pi}\n\\int\\limits_0^{2\\pi}\\frac{e^{it}+z}{e^{it}-z}\\ln k(t)dt\\right),\n\\]\nwhere $|c|=1$, $k(t)\\ge 0$, $\\ln k(t)\\in L_1[0,2\\pi]$, $\\mu$ is a\nfinite nonnegative measure singular with respect to the Lebesgue\nmeasure, and $\\{z_k\\}$ are the eigenvalues of $T$. In addition, if\n$\\dim H=N<\\infty$, and $T$ is a completely nonunitary contraction in\n$H$ with rank one defects, then its characteristic function is the\nfinite Blaschke product of order $N$ of the form\n\\[\nb(z)=e^{i\\varphi}\\prod\\limits_{k=1}^m\\left(\\frac{z-z_k}{1-\\bar z_k\nz}\\right)^{l_k},\n\\]\nwhere $z_1,\\ldots,z_m$ are distinct eigenvalues of $T$ with the\nalgebraic multiplicities $l_1,\\ldots, l_m$, respectively,\n$l_1+\\ldots+l_m=N$, and $\\varphi\\in [0,2\\pi)$. Hence, a\nfinite-dimensional completely nonunitary contraction $T$ with rank\none defects belongs to the class $C_{00}$, and\n$\\lim\\limits_{n\\to\\infty}||T^n||=0.$ It is easily seen from Theorem\n\\ref{Spectr} that the point spectrum of a contraction $T$ with rank\none defects agrees with ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$ if and only if $\\Theta_T\\equiv 0$.\n\n\n\\subsection{Unitary colligations and their\ncharacteristic functions} Every contraction $T$ acting on Hilbert\nspace $H$ can be included into the \\textit{unitary operator\ncolligation} \\cite{Br1} \\footnote{also known as the\n\\textit{conservative system \\cite{A}}}\n\\[\n\\Delta=\\left\\{\\begin{pmatrix} S&G\\cr F&T\\end{pmatrix};{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN,\nH\\right\\},\n\\]\nwhere ${\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}$ and $\\sN$ are separable Hilbert spaces, and\n\\[\nU=\\begin{pmatrix} S&G\\cr F&T\\end{pmatrix}:\\begin{pmatrix}{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}\\cr H\n\\end{pmatrix}\\to \\begin{pmatrix}\\sN\\cr H\n\\end{pmatrix}\n\\]\nis a unitary operator. $T$ is called the \\textit{basic operator} of\nthe unitary colligation $\\Delta$. The spaces ${\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}$ and $\\sN$ are\ncalled the \\textit{left outer space} and \\textit{right outer space},\nrespectively. The unitarity of $U$ means\n\\[\nU^*U=\\begin{pmatrix}I_{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}&0\\cr 0& I_H\\end{pmatrix},\\qquad\nUU^*=\\begin{pmatrix}I_\\sN&0\\cr 0& I_H\\end{pmatrix}.\n\\]\nor equivalently,\n\\begin{equation}\n\\label{conserv}\n\\begin{split}\n&T^*T+G^*G=I_H,\\quad F^*F+S^*S=I_{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\quad T^*F+G^*S=0,\\\\\n&TT^*+FF^*=I_H,\\quad GG^*+SS^*=I_\\sN,\\quad TG^*+FS^*=0.\n\\end{split}\n\\end{equation}\nThe colligation\n\\begin{equation}\n\\label{delta0} \\Delta_0=\\left\\{\\begin{pmatrix}-T^*&D_T\\cr D_{T^*} &T\n\\end{pmatrix};\\;{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T},{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}, H\\right\\}\n\\end{equation}\nprovides an example of the unitary colligation with given basic\noperator $T$.\n\nLet $\\Delta=\\left\\{\\begin{pmatrix} S&G\\cr F&T\\end{pmatrix};\n{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN,H\\right\\}$ be a unitary colligation. Define the following\nsubspaces in $H$\n\\begin{equation}\n\\label{co} H^{(c)}={\\rm \\overline{span}\\,}\\{T^nF{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\; n=0,1,\\ldots\\},\\qquad\nH^{(o)}={\\rm \\overline{span}\\,}\\{T^{*n}G^*\\sN,\\; n=0,1,\\ldots\\}.\n\\end{equation}\nThe subspaces $H^{(c)}$ and $H^{(o)}$ are called the\n\\textit{controllable} and the \\textit{observable} subspaces,\nrespectively. Let\n\\begin{equation}\n\\label{ORTCOMP}\n (H^{(c)})^\\perp:=H\\ominus\nH^{(c)},\\qquad (H^{(o)})^\\perp :=H\\ominus H^{(o)}.\n\\end{equation}\n A unitary colligation $\\Delta$ is called\n\\textit{prime} if $\\overline{H^{(c)}+H^{(o)}}=H$. Clearly, the\nlatter condition is equivalent to\n\\[\n(H^{(c)})^\\perp\\cap(H^{(o)})^\\perp=\\{0\\},\n\\]\n From \\eqref{conserv} and \\eqref{ORTCOMP} we get\n\\begin{equation}\n\\label{SP}\n\\begin{split}\n&(H^{(c)})^\\perp=\\bigcap\\limits_{n\\ge 0}\\ker\n(F^*T^{*n})=\\bigcap\\limits_{n\\ge\n0}\\ker(D_{T^*}T^{*n})=\\bigcap\\limits_{n\\ge 1}\\ker(D_{T^{*n}}),\\\\\n& (H^{(o)})^\\perp=\\bigcap\\limits_{n\\ge 0}\\ker\n(GT^n)=\\bigcap\\limits_{n\\ge 0}\\ker(D_{T}T^n)=\\bigcap\\limits_{n\\ge\n1}\\ker(D_{T^n}).\n\\end{split}\n\\end{equation}\nIt follows now from \\eqref{cu} that \\textit{the unitary colligation\n$\\Delta=\\left\\{\\begin{pmatrix} S&G\\cr F&T\\end{pmatrix};\n{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN,H\\right\\}$ is prime if and only if $T$ is a completely\nnonunitary operator}.\n\nGiven a unitary colligation $\\Delta=\\left\\{\\begin{pmatrix} S&G\\cr\nF&T\\end{pmatrix}; {\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN,H\\right\\}$, its characteristic function\n\\footnote{\\textit{the transfer function} of the system \\cite{A}}\n\\cite[Section 3]{Br1} is defined by\n\\[\n\\Theta_\\Delta(z)=S+zG(I_H-zT)^{-1}F,\\quad z\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}.\n\\]\nThis function belongs to the Schur class ${\\bf S}({\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN)$ of\n$\\cL({\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN)$-valued holomorphic in the unit disk ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$\noperator-functions. In particular, the characteristic function of\nthe unitary colligation $\\Delta_0$ \\eqref{delta0}\n\\[\n\\Theta_0(z)=\\left(-T^*+zD_{T}(I-zT)^{-1}D_{T^*}\\right){\\upharpoonright\\,}{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}\n\\]\nis in fact the Sz.-Nagy -- Foias characteristic function of the\noperator $T^*$.\n\nTwo prime unitary colligations\n\\[\n\\Delta_1=\\left\\{\\begin{pmatrix} S&G_1\\cr F_1&T_1\\end{pmatrix};\n{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN,H_1\\right\\}\\quad\\mbox{and}\\quad\n\\Delta_2=\\left\\{\\begin{pmatrix} S&G_2\\cr F_2&T_2\\end{pmatrix};\n{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN,H_2\\right\\}\n\\]\nwhich have equal characteristic functions are unitarily equivalent\nin the following sense \\cite[Theorem 3.2]{Br1}: there exists a\nunitary operator $V:H_1\\to H_2$ such that\n\\[\nVT_1=T_2V, \\;VF_1=F_2,\\; G_2V=G_1\\iff \\begin{pmatrix} I_\\sN&0\\cr 0\n&V\\end{pmatrix}\\begin{pmatrix} S&G_1\\cr\nF_1&T_1\\end{pmatrix}=\\begin{pmatrix} S&G_2\\cr\nF_2&T_2\\end{pmatrix}\\begin{pmatrix} I_{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}&0\\cr 0 &V\\end{pmatrix}\n\\]\nBesides, given $\\Theta\\in {\\bf S}({\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN)$, there exists a prime\nunitary colligation\n\\[\n\\Delta=\\left\\{\\begin{pmatrix} S&G\\cr F&T\\end{pmatrix};\n{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN,H\\right\\}\n\\]\nsuch that $\\Theta_\\Delta=\\Theta$ in ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$ \\cite[Theorem 5.1]{Br1}.\n\nLater on in Section 3 we will need the following result.\n\\begin{theorem}\n\\label{CH1} Let $T$ be a contraction acting on Hilbert space $H$\nwith finite defect numbers. Suppose that ${\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}$ and $\\sN$ are two\ngiven Hilbert spaces such that $\\dim \\sN=\\dim{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T$ and\n$\\dim{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}=\\dim{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}$. Then all unitary colligations with the\nbasic operator $T$ and outer subspaces ${\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}$ and $\\sN$ take the form\n\\begin{equation}\n\\label{UN}\\Delta=\\left\\{\\begin{pmatrix} -KT^*M&KD_T\\cr\nD_{T^*}M&T\\end{pmatrix}; {\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN,H\\right\\},\n\\end{equation}\nwhere $K:{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T\\to\\sN$ and $M:{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}\\to{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}$ are unitary operators.\nThe characteristic function of $\\Delta$ is\n\\[\n\\Theta_\\Delta(z)=K\\Theta_{T^*}(z)M,\\quad z\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F},\n\\]\ni.e., $\\Theta_\\Delta$ agrees with the characteristic function\n$\\Theta_{T^*}$ of \\ $T^*$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\Delta=\\left\\{\\begin{pmatrix} S&G\\cr F&T\\end{pmatrix};\n{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN,H\\right\\}$ be a unitary colligation. From the relation\n$G^*G+T^*T=I_H$ it follows that\n\\[\n||Gf||^2=||D_{T}f||^2,\\quad f\\in H.\n\\]\nHence, the operator $K:{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T\\to \\sN$ defined by\n\\[\nKD_Tf=Gf,\\quad f\\in H,\n\\]\nis isometric, and ${\\rm ran\\,} K=\\sN$. Similarly, the relation\n$FF^*+TT^*=I_H$ yields that the operator $N:{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}\\to{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}$ given\nby the relation\n\\[\nND_{T^*}f=F^*f,\\quad f\\in H\n\\]\nis isometric, and ${\\rm ran\\,} N={\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}$. So  $M=N^*:{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}\\to{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}$ is\nunitary, and $F=D_{T^*}M$.\n\nFrom the relation $T^*F+G^*S=0$ we get $T^*D_{T^*}M+D_{T}K^*S=0$.\nHence by \\eqref{defect} $T^*M+K^*S=0$. As ${\\rm ran\\,} M={\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}$, ${\\rm ran\\,}\nK^*={\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T$, and $T{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}\\subset{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T$, we have\n\\[\nS=-KT^*M.\n\\]\nObserve also that\n\\[\n\\begin{split}\n&TG^*+FS^*=TD_{T}K^*-D_{T^*}MM^*TK^*=0\\\\\n&SS^*+GG^*=KT^*MM^*TK^*+KD^2_{T}K^*=K(T^*T+I-T^*T)K^*=I_\\sN,\\\\\n&S^*S+F^*F=M^*TK^*KT^*M+M^*D_{T^*}M=M^*(TT^*+I-TT^*)M=I_{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}.\n\\end{split}\n\\]\nThus, all conditions \\eqref{conserv} are satisfied, i.e., the\ncolligation $\\Delta$ is of the form \\eqref{UN}.\n\nConversely, if $\\dim\\sN=\\dim{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T<\\infty$,\n$\\dim{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}=\\dim{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}<\\infty$, and $K:{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T\\to\\sN$ and\n$M:{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}\\to{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}$ are unitary operators, then one can easily see\nthat $U=\\begin{pmatrix} -KT^*M&KD_T\\cr\nD_{T^*}M&T\\end{pmatrix}:\\begin{pmatrix}{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}\\cr H\\end{pmatrix}\\to\n\\begin{pmatrix}\\sN\\cr H\\end{pmatrix}$ is a unitary operator, i.e.,\nthe relations \\eqref{conserv} are satisfied. It follows that\n$$\\Delta=\\left\\{\\begin{pmatrix} S&G\\cr\nF&T\\end{pmatrix}; {\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN,H\\right\\}$$ is a unitary colligation,\nwhere $G=KD_T,$ $F=D_{T^*}M$, $S=-KT^*M$.\n\nFor the characteristic function $\\Theta_\\Delta$ we obtain for all\n$z\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$\n\\[\n\\Theta_\\Delta(z)=S+zG(I-zT)^{-1}F=-KT^*M+zKD_{T}(I-zT)^{-1}D_{T^*}M=K\\Theta_\n{T^*}(z)M.\n\\]\n\\end{proof}\n\\begin{corollary}\n\\label{COL1} Let $T$ be a contraction with finite defect numbers,\n$\\dim \\sN=\\dim{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T$, $\\dim{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}=\\dim{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}$, and let\n$\\Delta=\\left\\{\\begin{pmatrix} S&G\\cr F&T\\end{pmatrix};\n{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN,H\\right\\}$ be a unitary colligation. Then all other unitary\ncolligations with the basic operator $T$ and outer subspaces ${\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}$\nand $\\sN$ take the form\n\\[\n\\widetilde \\Delta=\\left\\{\\begin{pmatrix} C_1SC_2&C_1G\\cr\nFC_2&T\\end{pmatrix}; {\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN,H\\right\\},\n\\]\nwhere $C_1$ and $C_2$ are unitary operators in $\\sN$ and ${\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}$,\nrespectively.\n\\end{corollary}\n\\begin{proof} By Theorem \\ref{CH1} we have\n\\[\nG=KD_T,\\;F=D_{T^*}M,\\; S=-KT^*M,\n\\]\nwhere $K:{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T\\to{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}$ and $M:\\sN\\to{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}$ are unitary operators.\nIf $\\widetilde\\Delta=\\left\\{\\begin{pmatrix} \\widetilde S&\\widetilde G\\cr \\widetilde\nF&T\\end{pmatrix}; {\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O},\\sN,H\\right\\}$ is some other unitary\ncolligation then $\\widetilde G=\\widetilde KD_T,\\;\\widetilde F=D_{T^*}\\widetilde M,\\; \\widetilde S=-\\widetilde\nKT^*\\widetilde M,$  where $\\widetilde K:{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T\\to{\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}$ and $\\widetilde M:\\sN\\to{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}$\nare unitary operators. Let $C_1:=\\widetilde KK^{-1},\\; C_2:=M^{-1}\\widetilde M$.\nThen $C_1$ and $C_2$ are unitary operators in $\\sN$ and ${\\mathfrak M}}   \\def\\sN{{\\mathfrak N}}   \\def\\sO{{\\mathfrak O}$,\nrespectively, and\n\\[\n\\widetilde G=C_1 G,\\quad \\widetilde F=FC_2,\\quad \\widetilde S=C_1SC_2,\\] as needed.\n\\end{proof}\n\n\\section{Completely nonunitary contractions with rank one defects\nand the corresponding unitary colligations}\n\n\\begin{theorem}\n\\label{cyclic} Each contraction $T$ with rank one defects on the\nHilbert space $H$ can be included into the unitary colligation\n\\[\n\\Delta=\\left\\{\\begin{pmatrix} S&G\\cr F&T\\end{pmatrix};\\ \\dC,\\dC,\nH\\right\\}.\n\\]\nLet $\\vec 1=\\begin{pmatrix}1\\cr 0\\end{pmatrix}\\in\\dC\\oplus H$, and\nlet the subspaces $(H^{(c)})^\\perp$ and $(H^{(o)})^\\perp$ in $H$ be\ndefined by \\eqref{ORTCOMP}.\n Then\n\\begin{equation}\n\\label{coperp} \\begin{split} & (H^{(c)})^\\perp=(\\dC\\oplus\nH)\\ominus{\\rm \\overline{span}\\,}\\{U^n\\vec 1;\\; n=0,1,\\ldots\\},\\\\\n& (H^{(o)})^\\perp=(\\dC\\oplus H)\\ominus{\\rm \\overline{span}\\,}\\{U^{*n}\\vec 1;\\;\nn=0,1,\\ldots\\},\n\\end{split}\n\\end{equation}\nand so the following conditions are equivalent:\n\\begin{enumerate}\n\\def\\rm (\\roman{enumi}){\\rm (\\roman{enumi})}\n\\item the unitary colligation $\\Delta=\\left\\{\\begin{pmatrix}\nS&G\\cr F&T\\end{pmatrix};\\ \\dC,\\dC, H\\right\\}$ is prime; \\item $T$ is\na completely nonunitary contraction; \\item $\\vec 1$ is the cyclic\nvector for $U$: ${\\rm \\overline{span}\\,}\\{U^n \\vec 1,\\; n\\in\\dZ\\}=\\dC\\oplus H$.\n\\end{enumerate}\nAll other unitary colligations with the basic operator $T$ and the\nouter spaces $\\dC$ are of the form\n\\begin{equation}\n\\label{OTHER} \\widetilde \\Delta=\\left\\{\\begin{pmatrix} c_1c_2S&c_1G\\cr\nc_2F&T\\end{pmatrix};\\ \\dC,\\dC,H\\right\\},\n\\end{equation}\nwhere $|c_1|=|c_2|=1$.\n\\end{theorem}\n\\begin{proof} Since $\\dim{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T=\\dim{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}=1$, by Theorem \\ref{CH1}\nwe can choose the unitary colligation $\\Delta=\\left\\{\\begin{pmatrix}\nS&G\\cr F&T\\end{pmatrix};\\dC,\\dC, H\\right\\}$ of the form \\eqref{UN},\ni.e., $S=-KT^*M ,\\;G=KD_T,\\; F=D_{T^*}M$, and $K:{\\rm ran\\,} D_T\\to \\dC$,\n$M:\\dC\\to {\\rm ran\\,} D_{T^*}$ are isometric operators. So\n$U=\\begin{pmatrix} S&G\\cr F&T\\end{pmatrix}:\\begin{pmatrix}\\dC\\cr H\n\\end{pmatrix}\\to \\begin{pmatrix}\\dC\\cr H\n\\end{pmatrix}$ is the unitary operator.\n\nTo prove \\eqref{coperp}, suppose that the vector $\\vec\nh=\\begin{pmatrix} z\\cr h\\end{pmatrix}\\in\\dC\\oplus H$ is orthogonal\nto the subspace ${\\rm \\overline{span}\\,}\\{U^n \\vec 1,\\; n=0,1,\\ldots\\}$. Then $\nU^{*n}\\vec h\\perp \\vec 1$, $n=0,1,\\ldots$ so $z=0$ and $\\vec\nh=\\begin{pmatrix}0 \\cr h\\end{pmatrix}$. By using\n$U^*=\\begin{pmatrix} S^*&F^*\\cr G^*&T^*\\end{pmatrix}$, we get\nconsequently\n\\[\nF^*h=0,\\quad F^*T^*h=0,\\quad F^*T^{*2}h=0,\\quad\\ldots, \\quad\nF^*T^{*k}h=0,\\ldots\n\\]\nIt follows from (\\ref{SP}) that $h\\in (H^{(c)})^\\perp.$ Conversely,\nif $h \\in (H^{(c)})^\\perp$ then $h\\perp {\\rm \\overline{span}\\,}\\{U^n \\vec 1,\\;\nn=0,1,\\ldots\\}$. Similarly, $(H^{(o)})^\\perp=(\\dC\\oplus\nH)\\ominus\\left({\\rm \\overline{span}\\,}\\{U^{*n}\\vec 1,\\; n=0,1,\\ldots\\}\\right),$  as\nneeded.\n\nWe arrive at the following conclusion:\n\\[\n\\begin{split}\n&\\vec 1\\:\\mbox{is a cyclic vector for}\\; U \\iff (H^{(c)})^\\perp\\cap\n(H^{(o)})^\\perp=\\{0\\}\\iff\\\\\n&\\mbox{the unitary colligation}\\; \\Delta=\\left\\{\\begin{pmatrix}\nS&G\\cr\nF&T\\end{pmatrix};\\dC,\\dC, H\\right\\}\\;\\mbox{is prime}\\iff \\\\\n&\\mbox{the operator}\\; T\\;\\mbox{is completely nonunitary}.\n\\end{split}\n\\]\nBy Corollary \\ref{COL1} all other unitary colligations with the\nbasic operator $T$ and the outer subspace $\\dC$ are given by\n\\eqref{OTHER} with $|c_1|=|c_2|=1.$\n\\end{proof}\n\\begin{remark}\n\\label{VD1} In terms of the Naimark dilations of a probability\noperator-valued measure on the unit circle, the main result of\nTheorem \\ref{cyclic} is proved in \\cite[Theorem 1.20]{D1}.\n\\end{remark}\nLet us give more precise expressions for the operators $F,G$, and\n$S$. Let $\\widehat\\varphi_1\\in{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_T$, $\\widehat\\varphi_2\\in {\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{T^*}$. Put\n\\[\n\\varphi_1=\\frac{\\widehat\\varphi_1}{\\|\\widehat\\varphi_1\\|},\\qquad\\varphi_2=\\frac{\\widehat\\varphi_2}{\\|\\widehat\\varphi_2\\|}.\n\\]\nThen\n\\[\n\\begin{split}\n&Kh=b_1(h,\\varphi_1),\\quad h\\in{\\rm ran\\,} D_T,\\\\\n&M^*g=b_2(g,\\varphi_2),\\quad g\\in{\\rm ran\\,} D_{T^*},\n\\end{split}\n\\]\nwhere $|b_1|=|b_2|=1$. Observe that $T\\varphi_1=-\\alpha_0 \\varphi_2$ and\n$T^*\\varphi_2=-\\bar{\\alpha}_0\\varphi_1$, where $\\alpha_0$ is a complex number\nfrom ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$. It follows that\n\\[\nD^2_T\\varphi_1=(1-|\\alpha_0|^2)\\varphi_1,\\qquad\nD^2_{T^*}\\varphi_2=(1-|\\alpha_0|^2)\\varphi_2.\n\\]\nLet $\\rho_0=\\sqrt{1-|\\alpha_0|^2}.$  Since $\\dim({\\rm ran\\,}\nD^2_T)=\\dim({\\rm ran\\,} D^2_{T^*})=1$, the number $\\rho_0$ is a unique\npositive eigenvalue of $D_T $($D_{T^*}$).\nNext,\n\\[\n\\begin{split}\n&Gh=b_1(D_T h,\\varphi_1)=b_1(h, D_T\\varphi_1)=b_1\\rho_0(h,\\varphi_1),\\\\\n&F^*h=b_2(D_{T^*}h,\\varphi_2)=b_2(h, D_{T^*}\\varphi_2)=b_2\\rho_0(h,\\varphi_2),\\quad\nh\\in H.\n\\end{split}\n\\]\nHence $F 1=\\rho_0\\bar b_2\\varphi_2$. Since $S=-KT^*M$, we get\n\\[\nS1=-b_1 \\bar b_2(T^*\\varphi_2,\\varphi_1)=b_1 \\bar b_2\\bar\\alpha_0.\n\\]\nIn the case $\\dim H=N<\\infty$ the operator $T$ can be given by the\n$N\\times N$ matrix with respect to some orthonormal basis and  we\ncan choose $\\widehat\\varphi_1$(respectively, $\\widehat\\varphi_2)$, as one of the nonzero\ncolumns of the matrix $I-T^*T\\;$  $(I-TT^*)$. In addition,\n\\[\nTrace(I-T^*T)=Trace(I-TT^*)=\\rho^2_0.\n\\]\nThus, if\n$\\varphi_2=\\begin{pmatrix}\\varphi^{(1)}_{2}\\cr\\varphi^{(2)}_{2}\\cr\\ldots\\cr\n\\varphi^{(N)}_{2}\n   \\end{pmatrix}$, then the column $F$ takes the form\n$F=\\bar b_2\\rho_0\\begin{pmatrix}\\varphi^{(1)}_{2}\n\\cr\\varphi^{(2)}_{2}\\cr\\ldots\\cr \\varphi^{(N)}_{2} \\end{pmatrix}$. \\newline If\n$\\varphi_1=\\begin{pmatrix}\\varphi^{(1)}_{1}\\cr\\varphi^{(2)}_{1}\\cr\\ldots\\cr\n\\varphi^{(N)}_{1}\n   \\end{pmatrix}$, then\n   the row $G$ takes the form  $\nG=b_1\\rho_0\n\\begin{pmatrix}\\bar\\varphi^{(1)}_{1}&\\bar\\varphi^{(2)}_{1}&\\ldots&\\bar\\varphi^{(N)}_{1}\n\\end{pmatrix}.\n$ Finally, the number $S$ is given by $-b_1 \\bar b_2(T^*\\varphi_2,\\varphi_1)$.\n\nIf $\\dim H=N$ and $T$ is a completely nonunitary contraction with\nrank one defects, then $\\Theta_\\Delta$ is a finite Blaschke product\n\\[\n\\Theta_\\Delta(z)=e^{i\\varphi}\\prod\\limits_{k=1}^N\\frac{z-\\bar z_k}{1-\nz_k\\, z},\n\\]\nwhere the numbers $ z_1,\\dots,  z_N$ are the eigenvalues of $T$.\nSince all other unitary colligations are of the form \\eqref{OTHER},\nfor the characteristic function $\\Theta_{\\widetilde\\Delta}(z)$ we get\n$\\Theta_{\\widetilde\\Delta}(z)=c_1c_2\\Theta_\\Delta(z)=e^{it}\\Theta_\\Delta(z),$\n$z\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$, and $t\\in[0,2\\pi).$\n\nLet $U$ be a unitary operator with a cyclic vector $e$, acting on\nthe Hilbert space $H$. The spectral measure $\\mu$ associated with\n$U$ and $e$ provides the relation\n\\[\n\\left(F(U)e,e\\right)=\\int_{\\dT} F(\\zeta)d\\mu(\\zeta),\n\\]\nwhich is the Spectral Theorem for unitaries. For instance,\n\\begin{equation}\\label{cfun} F(z)=\\left((U+zI)(U-zI)^{-1}e,e\\right)=\n\\int_{\\dT}\\frac{\\zeta+z}{\\zeta-z}\\,d\\mu(\\zeta)\\,, \\quad z\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F} \\end{equation}\nis the Carath\\'eodory function (\\ref{carat}), i.e., $F$ is\nholomorphic in the unit disc ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$, ${\\rm Re\\,} F>0$ in ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$, and\n$F(0)=1$.\n\\begin{theorem}\n\\label{char}  Let $T$ be a completely nonunitary contraction with\nrank one defects, $\\Delta=\\left\\{\\begin{pmatrix} S&G\\cr\nF&T\\end{pmatrix};\\ \\dC,\\dC,H\\right\\}$ be the prime unitary\ncolligation, and $\\Theta_\\Delta$ be its characteristic function. Put\n\\begin{equation}\n\\label{Carat}\n F(z)=\\left((U+zI)(U-zI)^{-1}\\vec 1,\\vec\n1\\right),\\quad z\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F},\n\\end{equation}\nwhere $U=\\begin{pmatrix} S&G\\cr\nF&T\\end{pmatrix}:\\begin{pmatrix}\\dC\\cr H\n\\end{pmatrix}\\to \\begin{pmatrix}\\dC\\cr H\n\\end{pmatrix}$.\nThen\n\\begin{equation}\n\\label {REL} \\overline{\\Theta_\\Delta(\\bar\nz)}=\\frac{1}{z}\\,\\frac{F(z)-1}{F(z)+1},\\qquad\nF(z)=\\frac{1+z\\overline{\\Theta_\\Delta(\\bar\nz)}}{1-z\\overline{\\Theta_\\Delta(\\bar z)}},\\quad z\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}.\n\\end{equation}\n\n\\end{theorem}\n\\begin{proof} We use the well known Schur--Frobenius formula for the inverse\nof block operators (see, e.g., \\cite[Section 0.2]{D}, \\cite[p.\n57]{Gant}). Let $\\sH_1$ and $\\sH_2$ be two Hilbert spaces, and\n$\\Phi$ an operator in $\\sH_1\\oplus\\sH_2$ given by the block operator\nmatrix\n\\[\n\\Phi=\\begin{pmatrix}A& B\\cr C&D \\end{pmatrix}: \\begin{pmatrix}\n\\sH_1\\cr\\sH_2\\end{pmatrix}\\to \\begin{pmatrix}\n\\sH_1\\cr\\sH_2\\end{pmatrix}.\n\\]\nSuppose that $D^{-1}\\in\\cL(\\sH_2)$ and\n$(A-BD^{-1}C)^{-1}\\in\\cL(\\sH_1)$. Then $\\Phi^{-1}\\in\n\\cL(\\sH_1\\oplus\\sH_2,\\sH_1\\oplus\\sH_2)$ and\n\\[\n\\Phi^{-1}=\\begin{pmatrix}K^{-1}& -K^{-1}BD^{-1}\\cr\n-D^{-1}CK^{-1}&D^{-1}+D^{-1}CK^{-1}BD^{-1}\n\\end{pmatrix},\n\\]\nwhere $K=A-BD^{-1}C$.\n\nApplying this formula for\n\\[\n\\Phi=I-zU=\\begin{pmatrix}1-zS&-zG\\cr-zF&I-zT\\end{pmatrix}:\\begin{pmatrix}\n\\dC\\cr H\\end{pmatrix}\\to \\begin{pmatrix} \\dC\\cr H\\end{pmatrix},\n\\quad z\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F},\n\\]\nwe get $K=1-zS-z^2G(I-zT)^{-1}F=1-z\\Theta_\\Delta(z)$. Therefore\n\\[\n\\left((I-zU)^{-1}\\vec 1,\\vec 1\\right)=\\frac{1}{1-z\\Theta_\\Delta(z)},\n\\quad z\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}.\n\\]\nLet\n\\[\n\\Psi(z)=\\left((I+z U)(I-zU)^{-1}\\vec 1,\\vec 1\\right),\\quad z\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}.\n\\]\nClearly, the equality  $F(z)=\\overline{\\Psi(\\bar z)}$ holds, which yields\n\\eqref{REL}.\n\\end{proof}\n\\begin{remark}\n\\label{VD2} Relations \\eqref{REL} is proved in \\cite[Theorem 1.20,\nComments 2.8]{D1}. Our proof is different.\n\\end{remark}\n\n\n\\section{OPUC and CMV matrices}\n\\subsection{Basics of OPUC}\nIt is well recognized now that the theory of orthogonal polynomials\non the real line plays an important role in the spectral theory of\nself-adjoint operators (and close to such operators) acting on\nHilbert spaces. Likewise, the theory of orthogonal polynomials on\nthe unit circle (OPUC) appears in the same fashion in the study of\nunitary operators and close to such operators. Here we recall some\nrudiments and advances of the OPUC theory.\n\nIf $\\mu$ is a nontrivial probability measure on $\\dT$ (that is, not\nsupported on a finite set), the monic orthogonal polynomials\n$\\Phi_n(z,\\mu)$ (or $\\Phi_n$ if $\\mu$ is understood) are uniquely\ndetermined by \\begin{equation}\\label{monic} \\Phi_n(z)=\\prod_{j=1}^n (z-z_{n,j}),\n\\qquad \\int_{\\dT} \\zeta^{-j}\\Phi_n(\\zeta)\\,d\\mu=0, \\quad\nj=0,1,\\ldots,n-1, \\end{equation} so on the Hilbert space $L^2(\\dT, d\\mu)$,\n$\\langle\\Phi_n, \\Phi_m\\rangle=0$, $n\\not=m$. We also consider the\northonormal polynomials $\\phi_n$ of the form\n$\\phi_n=\\Phi_n\/\\|\\Phi_n\\|$.\n\nIn case when $\\mu$ is supported on a finite set, that is,\n\\begin{equation}\\label{triv} \\mu=\\sum_{k=1}^N \\mu_k\\delta(\\zeta_k), \\quad\n\\zeta_k\\in\\dT, \\end{equation} a finite number of orthogonal polynomials\n$\\{\\Phi_k\\}_{k=0}^{N-1}$ can be defined in the same manner.\n\nClearly, (\\ref{monic}) and the fact that the space of polynomials of\ndegree at most $n$ has dimension $n+1$ imply \\begin{equation}\\label{monic1}\n\\deg(P)=n, \\quad P\\bot\\,\\zeta^j, \\quad j=0,1,\\ldots, n-1 \\Rightarrow\nP=c\\Phi_n. \\end{equation}\n\nOn $L^2(\\dT, d\\mu)$ the anti-unitary map\n$f^*(\\zeta):=\\zeta^n\\overline{f(\\zeta)}$ (which depends on $n$) is\nnaturally defined. The set of polynomials of degree at most $n$ is\nleft invariant: \\begin{equation}\\label{starpoly} P(z)=\\sum_{j=0}^n p_jz^j\n\\Rightarrow P^*(z)=\\sum_{j=0}^n \\bar p_{n-j}z^j. \\end{equation} (\\ref{monic1})\nnow implies \\begin{equation}\\label{starort} \\deg(P)\\leq n, \\quad P\\bot\\,\\zeta^j,\n\\quad j=1,\\ldots, n \\Rightarrow P=c\\Phi_n^*. \\end{equation}\n\nA key feature of the unit circle is that the multiplication $Uf=z f$\nin $L^2(\\dT, d\\mu)$ is a unitary operator. So the difference\n$\\Phi_{n+1}(z)-z\\Phi_n(z)$ is of degree $n$ and orthogonal to $z^j$\nfor $j=1,2,\\ldots,n$, and by (\\ref{starort}) \\begin{equation}\\label{szego}\n\\Phi_{n+1}(z)=z\\Phi_n(z)-\\bar\\alpha_n(\\mu)\\Phi_n^*(z) \\end{equation} with some\ncomplex numbers $\\alpha_n(\\mu)$, called the {\\it Verblunsky\ncoefficients} \\cite{S2}. (\\ref{szego}) is known as the {\\it Szeg\\H{o}\\\nrecurrences} after its first occurrence in the celebrated book\n\\cite{Sz} of G. Szeg\\H{o}. (\\ref{szego}) at $z=0$ imply\n\\begin{equation}\\label{vc} \\alpha_n(\\mu)=\\alpha_n=-\\overline{\\Phi_{n+1}(0)}. \\end{equation} It is\nknown that for nontrivial measures $|\\alpha_n|<1$ for all\n$n=0,1,2,\\ldots$, and for trivial measures (\\ref{triv}) one has a\nfinite set of Verblunsky coefficients $\\{\\alpha_n\\}_{n=0}^{N-1}$\nwith $|\\alpha_n|<1$, $n=0,1,\\ldots,N-2$, and $|\\alpha_{N-1}|=1$.\nSince it arises often, define \\begin{equation}\\label{rho}\n\\rho_j:=\\sqrt{1-|\\alpha_j|^2}, \\qquad 0\\leq\\rho_j\\leq 1, \\qquad\n|\\alpha_j|^2+\\rho_j^2=1. \\end{equation} The inverse Szeg\\H{o}\\ recurrences are also\nof interest (cf. \\cite [Theorem 1.5.4]{S2}): \\begin{equation}\\label{invszego}\nz\\Phi_n(z)=\\rho_n^{-2}\\left(\\Phi_{n+1}(z)+\\bar\\alpha_n\\Phi_{n+1}^*(z)\\right)\n. \\end{equation} The norm of the polynomials $\\Phi_n$ in $L^2(\\dT,d\\mu)$ can be\ncomputed by:\n\\[\n||\\Phi_n||=\\prod\\limits_{j=0}^{n-1}\\rho_j,\\qquad n=1,2,\\ldots.\n\\]\n\nLet ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}^\\infty$ be the set of complex sequences\n$\\{\\alpha_j\\}_{j=0}^\\infty$ with $|\\alpha_j|<1$. The map ${\\mathcal S}}   \\def\\cT{{\\mathcal T}}   \\def\\cU{{\\mathcal U}$, from\n$\\mu\\rightarrow\\{\\alpha_j(\\mu)\\}_{j=0}^\\infty$, is a well defined\nmap from the set ${\\mathcal P}}   \\def\\cQ{{\\mathcal Q}}   \\def\\cR{{\\mathcal R}$ of nontrivial probability measures on $\\dT$\nto ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}^\\infty$. It was S.~Verblunsky who proved that ${\\mathcal S}}   \\def\\cT{{\\mathcal T}}   \\def\\cU{{\\mathcal U}$ is a\nbijection. As a matter of fact, ${\\mathcal S}}   \\def\\cT{{\\mathcal T}}   \\def\\cU{{\\mathcal U}$ is a homeomorphism, provided\n${\\mathcal P}}   \\def\\cQ{{\\mathcal Q}}   \\def\\cR{{\\mathcal R}$ is equipped with the weak*-topology, and ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}^\\infty$ with the\ntopology of component convergence. Moreover, it follows directly\nfrom (\\ref{szego}) that for two measures $\\mu_1$ and $\\mu_2$\n\\[\n\\alpha_j(\\mu_1)=\\alpha_j(\\mu_2), \\quad j=0,1,\\ldots,n-1 \\\n\\Rightarrow \\ \\Phi_j(z,\\mu_1)=\\Phi_j(z,\\mu_2), \\quad j=0,1,\\ldots,n.\n\\]\nConversely, by (\\ref{invszego})\n\\[\n\\Phi_n(z,\\mu_1)=\\Phi_n(z,\\mu_2) \\ \\Rightarrow \\\n\\alpha_j(\\mu_1)=\\alpha_j(\\mu_2), \\quad j=0,1,\\ldots,n-1.\n\\]\n\nThe orthonormal set $\\{\\phi_n\\}_{n\\geq0}$ does not necessarily form\na basis in $L^2(\\dT, d\\mu)$ (e.g., if $d\\mu=dm$ is the normalized\nLebesgue measure on $\\dT$, then $\\phi_n=\\zeta^n$ and $\\zeta^{-1}$ is\northogonal to all $\\phi_n$). A celebrated result of Szeg\\H{o} --\nKolmogorov -- Krein reads that $\\{\\phi_n\\}$ is a basis in $L^2(\\dT,\nd\\mu)$ if and only if $\\log \\mu'\\not\\in L^1(\\dT)$, where $\\mu'$ is\nthe Radon -- Nikodym derivative of $\\mu$ with respect to $dm$. In\naddition, the following result holds true (cf. \\cite[Theorem\n1.5.7]{S2}).\n\\begin{theorem}\n\\label{complete} For any nontrivial probability measure $\\mu$ on the\nunit circle, the following are equivalent\n\\begin{enumerate}\n\\def\\rm (\\roman{enumi}){\\rm (\\roman{enumi})}\n\\item $\\lim\\limits_{n\\to\\infty}||\\Phi_n||=0;$ \\item\n$\\sum\\limits_{n=0}^\\infty |\\alpha_n|^2=\\infty;$ \\item the system\n$\\{\\phi_n\\}_{n=0}^\\infty$ is the orthonormal basis in $L^2(\\dT,\nd\\mu)$.\n\\end{enumerate}\n\\end{theorem}\nNote that if $\\sum\\limits_{n=0}^\\infty |\\alpha_n|^2<\\infty$  and $P$\nis the orthogonal projection in $L^2(\\dT, d\\mu)$ onto\n${\\rm \\overline{span}\\,}\\{\\zeta^n,\\; n=0,1,\\ldots\\}$, then (see \\cite{S1})\n\\begin{equation}\n\\label{PROJ}\n \\|(I-P)\\bar \\zeta\\|=\\prod\\limits_{n=0}^\\infty\\rho_n.\n\\end{equation}\n\nLet us now turn to the basic properties of zeros\n$\\{z_{n,j}\\}_{j=1}^n$ of OPUC. It is well known (cf., e.g., \\cite\n[Theorem 1.7.1]{S2}) that $|z_{n,j}|<1$ for all $n$ and $j$.\nMoreover, a result of Geronimus \\cite [Theorem 1.7.5]{S2} reads that\ngiven a monic polynomial $P_n$ of degree $n$ with all its zeros\ninside ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$, there is a (nontrivial) probability measure $\\mu$ on\n$\\dT$ such that $P_n=\\Phi_n(\\mu)$. Actually, there are infinitely\nmany such measures, all of them have the same Verblunsky\ncoefficients up to the order $n-1$, and the same moments up to the\norder $n$. Given a monic polynomial $P_n$ with all its zeros inside\nthe disk, let us call a monic polynomial $Q_{n+m}$ an {\\it\nextension} of $P_n$, if there is a measure $\\mu$ such that\n\\[\nP_n=\\Phi_n(\\mu), \\quad Q_{n+m}=\\Phi_{n+m}(\\mu).\n\\]\nTo obtain all such extensions one just has to extend a sequence of\nVerblunsky coefficients $\\alpha_0,\\ldots,\\alpha_{n-1}$, which are\ncompletely determined by $P_n$, by a sequence\n$\\beta_0,\\ldots,\\beta_{m-1}$ with arbitrary $\\beta_j\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$ and then\napply (\\ref{szego}).\n\nOne of the most recent advances in the study of zeros of OPUC is the\ntheorem of Simon and Totik \\cite [Theorem 1.7.15]{S2}, which claims\nthat given a polynomial $P_n$ as above, and an arbitrary set of\npoints $z_1,\\ldots,z_m$ in the unit disk, not necessarily distinct,\nthere is an extension $Q_{n+m}$ of $P_n$ such that $Q_{n+m}(z_j)=0$,\n$j=1,2,\\ldots,m$, counting the multiplicity. The latter as usual\nmeans that\n\\[\nz_k=z_{k+1}=\\ldots=z_{k+p}\\Rightarrow\nQ_{n+m}(z_k)=Q_{n+m}'(z_k)=\\ldots=Q_{n+m}^{(p)}(z_k)=0.\n\\]\nThe uniqueness of such extension is an open problem. A particular\ncase $m=1$ appeared earlier in \\cite{AV}. Now $\\beta_0=\\alpha_n$ is\ndefined uniquely from (\\ref{szego}) by\n\\[\n0=Q_{n+1}(z_1)=z_1P_n(z_1)-\\bar\\alpha_nP_n^*(z_1).\n\\]\nThis result will play a key role in the inverse problems with mixed\ndata in Section 7.\n\n\\subsection{Geronimus theory}\nThere is an important analytic aspect of the OPUC theory which was\ndeveloped by Geronimus \\cite{ger1, ger2} in 1940's.\n\nGiven a probability measure $\\mu$ on $\\dT$, define the {\\it\nCarath\\'eodory function} by \\begin{equation}\\label{carat} F(z)=\nF(z,\\mu):=\\int_{\\dT}\\frac{\\zeta+z}{\\zeta-z}\\,d\\mu(\\zeta)=\n1+2\\sum_{n=1}^\\infty \\beta_nz^n, \\quad \\beta_n=\\int_{\\dT}\n\\zeta^{-n}d\\mu \\end{equation} the moments of $\\mu$. $F$ is an analytic function\nin ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$ which obeys ${\\rm Re\\,} F>0$, $F(0)=1$. The {\\it Schur function}\nis then defined by \\begin{equation}\\label{schur} f(z)=\nf(z,\\mu):=\\frac{1}{z}\\,\\frac{F(z)-1}{F(z)+1}, \\qquad F(z)=\\frac{1+z\nf(z)}{1-z f(z)}\\,, \\end{equation} so it is an analytic function in ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$ with\n$\\sup_{{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}}|f(z)|\\leq1$. A one-to-one correspondence can be easily\nset up between the three classes (probability measures,\nCarath\\'eodory and Schur functions). Under this correspondence $\\mu$\nis trivial, that is, supported on a finite set, if and only if the\nassociate Schur function is a finite Blaschke product. Moreover,\nthis Blaschke product has the order $N-1$ for measures (\\ref{triv}).\n\nWe proceed with the Schur algorithm. Given a Schur function $f=f_0$,\nwhich is not a finite Blaschke product, define inductively\n  \\begin{equation}\\label{schuralg}\n  f_{n+1}(z)=\\frac{f_n(z)-\\gamma_n}{z(1-\\bar\\gamma_n f_n(z))}\\,, \\qquad\n  \\gamma_n=f_n(0). \\end{equation}\nIt is clear that the sequence $\\{f_n\\}$ is an {\\it infinite}\nsequence of Schur functions (called the $n^{th}$ Schur iterates) and\nneither of its terms is a finite Blaschke product. The numbers\n$\\{\\gamma_n\\}$ are called the {\\it Schur parameters:}\n\\[\n{\\mathcal S}}   \\def\\cT{{\\mathcal T}}   \\def\\cU{{\\mathcal U} f=\\{\\gamma_0,\\gamma_1,\\ldots,\\}.\n\\]\nIn case when\n\\[\nf(z)=e^{i\\varphi}\\prod_{k=1}^N \\frac{z-z_k}{1-\\bar z_k z}\n\\]\nis a finite Blaschke product of order $N$, the Schur algorithm\nterminates at the N-th step. The sequence of Schur parameters\n$\\{\\gamma_k\\}_{k=0}^N$ is finite, $|\\gamma_k|<1$ for\n$k=0,1,\\ldots,N-1$, and $|\\gamma_N|=1$.\n\nIf a Schur function $f$ is not a finite Blaschke product, the\nconnection between the non-tangential limit values $f(\\zeta)$\n and its Schur parameters $\\{\\gamma_n\\}$ is given by the\nformula\n\\begin{equation}\n\\label{L1} \\prod\\limits_{n=0}^\\infty\n(1-|\\gamma_n|^2)=\\exp\\left\\{\\int_\\dT\\ln(1-|f(\\zeta)|^2)dm\\right\\}\n\\end{equation}\n(see \\cite {Boy}). It follows that\n\\[\n\\sum\\limits_{n=0}^\\infty|\\gamma_n|^2=\\infty\\iff\n\\ln(1-|f(\\zeta)|^2)\\notin L^1(\\dT).\n\\]\nIn addition, if one of the conditions\n\\begin{enumerate}\n\\item $\\limsup_{n\\to\\infty}|\\gamma_n|=1,$ \\item\n $\\lim_{n\\to\\infty}\\gamma_n\\gamma_{n+m}=0$ for each\n$m=1,2,\\ldots$, but $\\limsup_{n\\to\\infty}|\\gamma_n|>0$,\n\\end{enumerate}\nis fulfilled, then $f$ is the inner function (see \\cite{Rakh,\nKhfund}).\n\nLater in Section 7 we will make use of the following fundamental\nresult of Schur \\cite{Schur}: the set of all Schur functions $f$\nwith prescribed first Schur parameters $\\gamma_0,\\ldots,\\gamma_n$ is\ngiven by the linear fractional transformation \\begin{equation} \\label{set}\nf(z)=\\frac{A(z)+zB^*(z)s(z)}{B(z)+zA^*(z)s(z)}\\,, \\end{equation} where $s$ is\nan arbitrary Schur function, and $A,B$ are polynomials of degree at\nmost $n$. Moreover,\n\\[\n{\\mathcal S}}   \\def\\cT{{\\mathcal T}}   \\def\\cU{{\\mathcal U} f=\\{\\gamma_0,\\ldots,\\gamma_n,\\gamma_0(s),\\gamma_1(s),\\ldots\\}\n\\]\nThe pair $(A,B)$, known as the Wall pair, is completely determined\nby $\\{\\gamma_j\\}_{j=0}^n$. Specifically,\n\\[\nW(z):=\\begin{pmatrix}zB^*(z)& A(z)\\cr zA^*(z)& B(z)\\end{pmatrix}=\nQ_{\\gamma_0}(z)\\,Q_{\\gamma_1}(z)\\cdots Q_{\\gamma_n}(z),\n\\]\nwhere\n\\[\nQ_\\omega(z)=\\frac{1}{\\sqrt{1-|\\omega|^2}}\\,\\begin{pmatrix}z&\\omega\\cr\nz\\bar\\omega&\\, 1\\end{pmatrix}, \\quad \\omega\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}.\n\\]\nBy computing determinants, we see that\n\\[\nB^*(z)B(z)-A^*(z)A(z)=z^n\\prod_{j=0}^n(1-|\\gamma_j|^2)^{1\/2},\n\\]\nso $A$ and $B$ have no common zeros in $\\dC\\setminus\\{0\\}$. In fact\nthey have no common zeros at all since $B(0)=1$. It is known also\nthat $B\\not=0$ in $\\overline{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$, and both $AB^{-1}$ and $A^*B^{-1}$ are\nSchur functions.\n\nA straightforward computation shows that $Q_\\omega$ (and hence $W$)\nare $j$-inner matrix functions:\n\\[\n\\begin{split}\n&W^*(z)jW(z)\\ge j\\quad\\mbox{for}\\quad z\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F},\\\\\n&W^*(z)jW(z)=j\\quad\\mbox{for}\\quad z\\in\\dT\n\\end{split}\n\\]\nwith the signature matrix\n\\[\nj=\\begin{pmatrix}-1&0\\cr 0& 1 \\end{pmatrix}.\n\\]\nFor further properties of the Wall pairs see \\cite[Section\n4]{Khfund}, \\cite[Chapter 1.3.8]{S2}.\n\nA curious situation when the Schur parameters for a finite Blaschke\nproduct can be computed explicitly was found by Khrushchev \\cite\n[formula (1.12)]{Kh}. Let $\\mu$ be a nontrivial probability measure\n(or measure of the form (\\ref{triv}) with big enough $N$) with\nVerblunsky coefficients $\\{\\alpha_k\\}$, and $\\Phi_n$ be its $n$th\nmonic orthogonal polynomial. Consider the following Blaschke product\nof order $n$\n\\[\nb_0(z):=\\frac{\\Phi_n(z)}{\\Phi_n^*(z)}=\\prod_{j=1}^n\\frac{z-z_{n,j}}{1-\\bar\nz_{n,j} z}, \\qquad b_0(0)=-\\bar\\alpha_{n-1}.\n\\]\n It is a matter of a simple computation based on (\\ref{invszego})\nto make sure that\n\\[\nb_{1}(z)=\\frac{b_0(z)-b_0(0)}{z(1-\\bar b_0(0)\nb_0(z))}=\\frac{\\Phi_{n-1}(z)}{\\Phi_{n-1}^*(z)}\\,.\n\\]\nHence the Schur parameters of $b_0$ are of the form \\begin{equation}\\label{kh}\n{\\mathcal S}}   \\def\\cT{{\\mathcal T}}   \\def\\cU{{\\mathcal U}\nb_0=\\{-\\bar\\alpha_{n-1},-\\bar\\alpha_{n-2},\\ldots,-\\bar\\alpha_0,1\\}.\n\\end{equation}\n\nThe fundamental paper of Schur \\cite{Schur} had appeared a few years\nbefore Szeg\\H{o} introduced the notion of orthogonal polynomials on\nthe unit circle. Amazingly, neither of them benefited from the ideas\nof the other. Only 20 years later Geronimus put them together and\ncame up with the following fundamental result (see \\cite [Theorem\nIX, p. 111]{ger1})\n\n\\begin{theorem}\n\\label{TT} Let $\\mu$ be a nontrivial probability measure on $\\dT$\nand $f$ its Schur function with the Schur parameters $\\gamma_n(f)$.\nThen $\\gamma_n(f)=\\alpha_n(\\mu)$. For measures (\\ref{triv}) the\nlatter equality holds for $n=0,1,\\ldots,N-1$.\n\\end{theorem}\nIt is clear now why a minus and conjugate is taken in (\\ref{szego}).\n\nWe complete with the result which will be used later on in Section\n7.\n\\begin{theorem}\n\\label{TTT} Given two sets $\\alpha_0,\\ldots,\\alpha_{n-1}$ and\n$z_1,\\ldots,z_m$ of complex numbers in ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$, and $\\gamma\\in\\dT$,\nthere exists a finite Blaschke product $b$ of order $n+m$ such that\n\\begin{enumerate}\n\\def\\rm (\\roman{enumi}){\\rm (\\roman{enumi})}\n    \\item ${\\mathcal S}}   \\def\\cT{{\\mathcal T}}   \\def\\cU{{\\mathcal U}\n    b=\\{\\omega_0,\\ldots,\\omega_{m-1},\\alpha_0,\\ldots,\\alpha_{n-1},\\gamma\\}$,\n    \\item $b(z_j)=0$, $j=1,\\ldots,m$, counting multiplicity.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nDenote $\\beta_k:=-\\bar\\gamma\\bar\\alpha_{n-k-1}$, $k=0,1,\\ldots,n-1$\nand construct a system of monic orthogonal polynomials\n$\\{\\Phi_k(z,\\beta)\\}_{k=0}^n$  by (\\ref{szego}). The theorem of\nSimon-Totik claims that there is a measure $\\mu$ with\n\\[\n\\Phi_n(z,\\mu)=\\Phi_n(z,\\beta), \\qquad \\Phi_{n+m}(z_j,\\mu)=0, \\quad\nj=1,\\ldots,m,\n\\]\ncounting the multiplicity. The first equality means that\n$\\alpha_k(\\mu)=\\beta_k$, $k=1,\\ldots,n-1$. Finally, put\n\\[\nb(z):=\\gamma\\,\\frac{\\Phi_{n+m}(z,\\mu)}{\\Phi_{n+m}^*(z,\\mu)}\\,.\n\\]\nThe result now follows from Khrushchev's formula (\\ref{kh}).\n\\end{proof}\n\nNote that for $m=1$ the Blaschke product is uniquely determined.\n\n\\subsection{CMV matrices}\n\nOne of the most interesting developments in the OPUC theory in\nrecent years is the discovery by Cantero, Moral, and Vel\\'azquez\n\\cite{CMV1, CMV2} of a matrix realization for the operator of\nmultiplication by $\\zeta$ on $L^2(\\dT,d\\mu)$ which is a unitary\nmatrix of finite band size (i.e.,\n$|\\langle\\zeta\\chi_m,\\chi_n\\rangle|=0$ if $|m-n|>k$ for some $k$);\nin this case, $k=2$ to be compared with $k=1$ for the Jacobi\nmatrices, which correspond to the real line case. The CMV basis\n(complete, orthonormal system) $\\{\\chi_n\\}$ is obtained by\northonormalizing the sequence\n$1,\\zeta,\\zeta^{-1},\\zeta^2,\\zeta^{-2},\\ldots$, and the matrix,\ncalled the {\\it CMV matrix},\n$$ \\cC={\\cC}(\\mu)=\\|c_{n,m}\\|_{m,n=0}^\\infty =\\|\\langle \\zeta\\chi_m,\\chi_n\n\\rangle\\|,\n    \\qquad m,n\\in\\dZ_+  $$\nis five-diagonal. Remarkably, the $\\chi$'s can be expressed in terms\nof $\\phi$'s and $\\phi^*$'s:\n$$ \\chi_{2n}(z)=z^{-n}\\phi_{2n}^*(z), \\qquad\n\\chi_{2n+1}(z)=z^{-n}\\phi_{2n+1}(z), \\quad n\\in\\dZ_+, $$ and the\nmatrix elements in terms of $\\alpha$'s and $\\rho$'s:\n\\begin{equation}\n\\label{cmvmatr} \\cC=\\cC(\\{\\alpha_n\\})=\\begin{pmatrix}\n\\bar{\\alpha}_0&\\bar{\\alpha}_1\\rho_0&\\rho_1\\rho_0&0&0&\\ldots\\cr\n\\rho_0&-\\bar{\\alpha}_1\\alpha_0&-\\rho_1\\alpha_0&0&0&\\ldots\\cr\n0&\\bar{\\alpha}_2\\rho_1&-\\bar{\\alpha}_2\\alpha_1&\\bar{\\alpha}_3\\rho_2&\\rho_3\n\\rho_2& \\ldots\\cr\n0&\\rho_2\\rho_1&-\\rho_2\\alpha_1&-\\bar{\\alpha}_3\\alpha_2&-\\rho_3\\alpha_2&\\ldots\n\\cr 0&0&0&\\bar{\\alpha}_4\\rho_3&-\\bar{\\alpha}_4\\alpha_3&\\ldots\\cr\n\\ldots&\\ldots&\\ldots&\\ldots&\\ldots&\\ldots\n\\end{pmatrix},\n\\end{equation}\n$\\alpha$'s are the Verblunsky coefficients and $\\rho$'s are given in\n(\\ref{rho}).\n\nIt is not hard to write down a general formula for the matrix\nentries $c_{ij}$ (see \\cite{Gol}). Let $2\\epsilon_m:=1-(-1)^m$,\\\n$m\\in\\mathbb{Z}_+$, and $\\epsilon_{-1}=1$, so $\\{\\epsilon_m\\}_{m\\geq\n0}=\\{0,1,0,1,\\ldots\\}$,\n$$ \\epsilon_m+\\epsilon_{m+1}=1, \\qquad \\epsilon_m\\epsilon_{m+1}=0,\n   \\qquad \\epsilon_m-\\epsilon_{m+1}=(-1)^{m+1}. $$\nThen\n\\begin{equation}\\label{cmventr1} \\begin{split}\nc_{mm} &= -\\overline\\alpha_m\\alpha_{m-1},\\\\\nc_{m+2,m} &= \\rho_m\\rho_{m+1}\\epsilon_m, \\qquad\nc_{m,m+2}=\\rho_m\\rho_{m+1}\\epsilon_{m+1},\n\\end{split}\\end{equation}\nand\n\\begin{equation}\\label{cmventr2} \\begin{split}\n  c_{m+1,m} &=\n\\overline\\alpha_{m+1}\\rho_m\\epsilon_m-\\alpha_{m-1}\\rho_m\\epsilon_{m+1},\\\\\n  c_{m,m+1} &=\n\\overline\\alpha_{m+1}\\rho_m\\epsilon_{m+1}-\\alpha_{m-1}\\rho_m\\epsilon_m.\n\\end{split}\\end{equation}\nIt is clear (cf. \\cite[Theorem 1]{BD}), that any semi-infinite CMV\nmatrix $\\cC$ (\\ref{cmvmatr}) can be written in the three-diagonal\nblock-matrix form\n\n\\begin{equation}\n\\label{THREE}\n  \\cC=\\begin{pmatrix} \\cB_0 & \\cC_0 & 0 &0   & 0 &\n\\cdot &\n\\cdot  \\\\\n{\\mathcal A}}   \\def\\cB{{\\mathcal B}}   \\def\\cC{{\\mathcal C}_0 & \\cB_1 & \\cC_1 & 0 &0& \\cdot &\n\\cdot  \\\\\n0    & {\\mathcal A}}   \\def\\cB{{\\mathcal B}}   \\def\\cC{{\\mathcal C}_1 & \\cB_2 & \\cC_2 &0& \\cdot &\n\\cdot   \\\\\n\\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot\n\\end{pmatrix}\n\\end{equation}\nwith\n\\begin{equation}\n\\label{BLOCKS} \\begin{split} \\cB_0\n&=\\begin{pmatrix}\\bar\\alpha_0\\end{pmatrix},\\quad\n\\cC_0=\\begin{pmatrix}\\bar\\alpha_1\\rho_0&\\rho_1\\rho_0\\end{pmatrix},\\quad\n{\\mathcal A}}   \\def\\cB{{\\mathcal B}}   \\def\\cC{{\\mathcal C}_0=\\begin{pmatrix}\\rho_0\\\\0\\end{pmatrix},\\\\\n{\\mathcal A}}   \\def\\cB{{\\mathcal B}}   \\def\\cC{{\\mathcal C}_n\n&=\\begin{pmatrix}\\rho_{2n}\\rho_{2n-1}&-\\rho_{2n}\\alpha_{2n-1}\\cr 0&0\n\\end{pmatrix},\\quad\n\\cB_n=\\begin{pmatrix}-\\bar\\alpha_{2n-1}\\alpha_{2n-2}&-\\rho_{2n-1}\\alpha_{2n-2}\\cr\n\\bar\\alpha_{2n}\\rho_{2n-1}&-\\bar\\alpha_{2n}\\alpha_{2n-1}\n\\end{pmatrix},\\\\\n\\cC_n\n&=\\begin{pmatrix}0&0\\cr\\bar\\alpha_{2n+1}\\rho_{2n}&\\rho_{2n+1}\\rho_{2n}\n\\end{pmatrix}, \\qquad n=1,2,\\ldots.\n\\end{split}\n\\end{equation}\nThere is a nice multiplicative structure of the CMV matrices. In the\nsemi-infinite case $\\cC$ is the product of two matrices:\n$\\cC=\\cL{\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O}$, where\n\\begin{equation}\n\\label{lm}\n\\begin{split}\n\\cL\n&=\\Psi(\\alpha_0)\\oplus\\Psi(\\alpha_2)\\oplus\\ldots\\Psi(\\alpha_{2m})\n\\oplus\\ldots,\\\\\n{\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O} &={\\bf 1}_{1\\times 1}\\oplus\\Psi(\\alpha_1)\\oplus\\Psi(\\alpha_3)\n\\oplus\\ldots\\oplus\\Psi(\\alpha_{2m+1})\\oplus\\ldots,\n\\end{split}\n\\end{equation}\nand\n$\\Psi(\\alpha)=\\begin{pmatrix}\\bar\\alpha&\\rho\\cr\\rho&-\\alpha\\end{pmatrix}.$\nThe finite $(N+1)\\times (N+1)$ CMV matrix $\\cC$ obeys\n$\\alpha_0,\\alpha_1,\\ldots,\\alpha_{N-1}\\in {\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$ and $ |\\alpha_{N}|=1$\nis also the product $\\cC=\\cL{\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O}$, where in this case\n$\\Psi(\\alpha_{N})=\\left(\\bar\\alpha_{N}\\right)$.\n\nIt is just natural to take the ordered set\n$1,\\zeta^{-1},\\zeta,\\zeta^{-2},\\zeta^2,\\ldots$ instead of\n$1,\\zeta,\\zeta^{-1},\\zeta^2,\\zeta^{-2},\\ldots$, that leads to the\nalternate CMV basis $\\{x_n\\}$ and the alternate CMV matrix\n\\begin{equation}\n\\label{altcmvmatr} \\widetilde\\cC=\\|\\langle \\zeta x_m,x_n\n\\rangle\\|=\\begin{pmatrix} \\bar{\\alpha}_0&\\rho_0&0&0&0&\\ldots\\cr\n\\bar{\\alpha_1}\\rho_0&-\\bar{\\alpha}_1\\alpha_0&\\bar{\\alpha}_2\\rho_1&\\rho_2\\rho_1&0&\\ldots\\cr\n\\rho_1\\rho_0&-\\rho_1\\alpha_0&-\\bar{\\alpha}_2\\alpha_1&-\\rho_2\\alpha_1&0&\\ldots\\cr\n0&0&\\bar{\\alpha}_3\\rho_2&-\\bar{\\alpha}_3\\alpha_2&\\bar{\\alpha_4}\\rho_3&\\ldots\n\\cr\n0&0&\\rho_3\\rho_2&-\\rho_3\\alpha_2&-\\bar{\\alpha_4}\\alpha_3&\\ldots\\cr\n\\ldots&\\ldots&\\ldots&\\ldots&\\ldots&\\ldots\n\\end{pmatrix},\n\\end{equation}\nwhich turns out to be the transpose of $\\cC$ (see \\cite[Corollary\n4.2.6]{S2}). Furthermore, $\\cL=\\cL^t$ and ${\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O}={\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O}^t$ imply\n$\\widetilde\\cC=\\cC^t={\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O}\\cL$.\n\nAn important relation between CMV matrices and monic orthogonal\npolynomials similar to the well-known property of orthogonal\npolynomials on the real line\n\\[\n\\Phi_n(z)=\\det(zI_n-{\\cC}^{(n)})\n\\]\nholds, where ${\\cC}^{(n)}$ is the principal $n\\times n$ block of\n${\\cC}$.\n\nOne of the most important results of Cantero, Moral, and Vel\\'azquez\n\\cite{CMV1} states that each unitary operator $U$ with the simple\nspectrum (i.e. having a cyclic vector $e_1$) acting on some\ninfinite-dimensional separable Hilbert space (respectively,\nfinite-dimensional Hilbert space) is unitarily equivalent to a\ncertain CMV matrix in $l_2(\\dZ_+)$ (respectively, in $\\dC^n$). The\ncorresponding $\\alpha$'s come up as the Verblunsky coefficients of\nthe spectral measure $d\\mu$ of $U$ associated with $e_1$. This is\nthe analog of Stone's self-adjoint cyclic model theorem. To be more\nprecise, let us, following \\cite{S3}, call a {\\it cyclic unitary\nmodel} a unitary operator $U$ acting on a separable Hilbert space\n$\\cH$ with the distinguished cyclic unit vector $v_0$. Two cyclic\nunitary models, $(\\cH,U,v_0)$ and $(\\tilde\\cH, \\tilde U, \\tilde\nv_0)$ are called equivalent if there is a unitary operator $W$ from\n$\\cH$ onto $\\tilde\\cH$ such that $Wv_0=\\tilde v_0$ and\n$WUW^{-1}=\\tilde U$. It is clear that $\\delta_0=(1,0,0,\\ldots)^t$ is\ncyclic for any CMV matrix $\\cC$. Moreover, every class of equivalent\nunitary models contains exactly one CMV model\n$(\\ell^2,\\cC,\\delta_0)$.\n\n\\section{A model in the space $L^2(\\dT, d\\mu)$ of a completely nonunitary contraction\nwith rank one defects} \\label{s5}\n\\begin{theorem}\n\\label{L2} Let $T$ be a completely nonunitary contraction with rank\none defects. Then there exists a probability measure $\\mu$ on $\\dT$\nsuch that $T$ is unitarily equivalent to the following operator\n\\begin{equation} \\label{UNMODL2}\n\\sT h(\\zeta)=P_\\sH\\left(\\zeta h(\\zeta)\\right),\\qquad\nh\\in\\sH:=L^2(\\dT,d\\mu)\\ominus\\dC,\n\\end{equation}\nwhere $P_\\sH$ is the orthogonal projection in $L^2(\\dT,d\\mu)$ onto\n$\\sH$. The Schur function associated with $\\mu$ is exactly the\ncharacteristic function of $T$.\n\\end{theorem}\n\\begin{proof}\nInclude $T$ into a prime unitary colligation\n$\\Delta=\\left\\{\\begin{pmatrix} S&G\\cr F&T\\end{pmatrix};\n\\dC,\\dC,H\\right\\}$. The characteristic function $\\Theta_\\Delta$\nagrees with the characteristic function of $T^*$. By Theorem\n\\ref{cyclic} the vector $\\vec 1=\\begin{pmatrix}1\\cr 0\\end{pmatrix}$\nis cyclic for the unitary operator $U=\\begin{pmatrix} S&G\\cr\nF&T\\end{pmatrix}$.\n\nLet $E_U(\\zeta)$ be the resolution of identity for $U$. Define\n$d\\mu(\\zeta):=(dE_U(\\zeta)\\vec 1,\\vec 1)$ and put\n\\[\n\\cU f(\\zeta)=\\zeta f(\\zeta)\n \\]\nthe unitary multiplication operator in $L^2(\\dT, d\\mu)$. By the\nspectral theorem for unitaries with cyclic vectors (cf.\n\\cite[Section 1.4.5]{S2}) there exists a unitary operator\n$W:\\dC\\oplus H\\to L^2(\\dT,d\\mu)$ such that\n\\[\nU=W^{-1}\\cU W \\;\\mbox{and}\\; W\\vec 1=1.\n\\]\nIt follows that $W$ takes the block-operator form\n\\[\nW=\\begin{pmatrix} 1&0\\cr 0&V \\end{pmatrix}:\\begin{pmatrix}\\dC\\cr H\n\\end{pmatrix}\\to \\begin{pmatrix}\\dC\\cr\\sH\n\\end{pmatrix},\n\\]\nwhere $\\sH=L^2(\\dT,d\\mu)\\ominus\\dC$, $V:H\\to\nL^2(\\dT,d\\mu)\\ominus\\dC$ is a unitary operator. If $\\sT$ is given by\n\\eqref{UNMODL2}, then\n\\[\n\\sT:=P_\\sH\\cU{\\upharpoonright\\,}\\sH=VTV^{-1},\n\\]\ni.e., $T$ is unitarily equivalent to $\\sT$. Clearly, $\\cU$ has the\nblock form\n\\[\n\\cU=\\begin{pmatrix}P_\\dC \\cU{\\upharpoonright\\,}\\dC &P_\\dC \\cU{\\upharpoonright\\,}\\sH\\cr\nP_\\sH\\cU{\\upharpoonright\\,}\\sH&\\sT\n\\end{pmatrix},\n\\]\nwhere $P_\\dC$ is the orthogonal projection in $L^2(\\dT,d\\mu)$ onto\nthe subspace $\\dC$ of the constant functions in $L^2(\\dT,d\\mu)$. The\nunitary colligation $\\Delta$ is unitarily equivalent to the unitary\ncolligation\n\\begin{equation}\n\\label{unmod} \\left\\{\\begin{pmatrix}P_\\dC \\cU{\\upharpoonright\\,}\\dC &P_\\dC\n\\cU{\\upharpoonright\\,}\\sH\\cr P_\\sH\\cU{\\upharpoonright\\,}\\sH&\\sT\n\\end{pmatrix},\\dC,\\dC,\\sH\\right\\}.\n\\end{equation}\nNote that\n\\[\nP_\\dC(\\cU\\, 1)=\\int_\\dT\\zeta \\,d\\mu,\\; P_\\sH(\\cU\\,\n1)=\\zeta-\\int_\\dT\\zeta \\,d\\mu,\\;\nP_\\dC(\\cU^*\\,1)=\\bar\\zeta-\\int_\\dT\\bar\\zeta \\,d\\mu.\n\\]\n Let\n$F(z)=\\left((U+zI)(U-zI)^{-1}\\vec 1,\\vec 1\\right)$. Then\n\\[\nF(z)=\\left((\\cU+zI)(\\cU-zI)^{-1}\n1,1\\right)=\\int_{\\dT}\\frac{\\zeta+z}{\\zeta-z}\\,d\\mu(\\zeta),\n\\]\n i.e., $F$ is the Carath\\'eodory function associated with $\\mu$.\n From Theorem \\ref{char} we conclude\n\\[\n\\overline{\\Theta_\\Delta(\\bar z)}=\\frac{1}{z}\\frac{F(z)-1}{F(z)+1},\n\\]\nand so by \\eqref{schur} $\\overline{\\Theta_\\Delta(\\bar z)}$ agrees with\nthe Schur function associated with $\\mu$.\n\\end{proof}\nLet $\\{\\Phi_n\\}$ be the system of monic polynomials orthogonal with\nrespect to $\\mu$, and let $\\{\\alpha_n\\}$ be the corresponding\nVerblunsky coefficients. By Geronimus' theorem $\\{\\alpha_n\\}$ are\nthe Schur parameters of $f$. Let $\\sH^{(c)}$ be the controllable\nsubspace of the unitary colligation \\eqref{unmod}. From\n\\eqref{coperp} it follows that\n\\[\n(\\sH^{(c)})^\\perp=L^2(\\dT, d\\mu)\\ominus{\\rm \\overline{span}\\,}\\{\\zeta^n,\\;\nn=0,1,\\ldots\\}.\n\\]\nIf $\\mu$ is a nontrivial measure, then in view of \\eqref{PROJ} we\nobtain\n\\[\n\\|P_{(\\sH^{(c)})^\\perp}\n\\bar\\zeta\\|=\\prod\\limits_{n=0}^\\infty(1-|\\alpha_n|^2)^{1\/2}.\n\\]\nThe latter is equivalent to\n\\[\n\\|P_{(\\sH^{(c)})^\\perp}P_\\dC(\\cU^*\n\\,1)\\|=\\prod\\limits_{n=0}^\\infty(1-|\\alpha_n|^2)^{1\/2}.\n\\]\nHence, from \\eqref{UN} and \\eqref{co} we have the equivalence\n\\begin {equation}\n\\label{poln}\n {\\rm \\overline{span}\\,} \\{\\sT^n{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{\\sT^*},\\;n=0,1,\\ldots\\}=\\sH\\iff\n\\sum\\limits_{n=0}^\\infty|\\alpha_n|^2=\\infty.\n\\end{equation}\n\n\n\\begin{remark}\n\\label{Anmodel} By the construction of Theorem \\ref{L2}, the Schur\nfunction $f$ associated with $\\mu$ is exactly\n$\\overline{\\Theta_\\Delta(\\bar z)}$. Another (unitary equivalent) models\nof $T$ are connected with the operators $U_\\lambda=\\begin{pmatrix}\n\\bar\\lambda S&G\\cr \\bar\\lambda F&T\\end{pmatrix}$, where\n$|\\lambda|=1$. The characteristic function of the unitary\ncolligation\n$$\n\\Delta_\\lambda=\\left\\{\\begin{pmatrix} \\bar\\lambda S&G\\cr \\bar\\lambda\nF&T\\end{pmatrix},\\dC,\\dC, H\\right\\}\n$$\nis $\\bar\\lambda\\Theta_\\Delta$. The model operator $\\sT_\\lambda$\ntakes the form\n\\[\n\\sH_\\lambda=L^2(\\dT,d\\mu_\\lambda)\\ominus\\dC,\\; \\sT_\\lambda\nh(\\zeta)=P_{\\sH_\\lambda}\\left(\\zeta h(\\zeta)\\right),\\quad\nh(\\zeta)\\in\\sH_\\lambda.\n\\]\nThe Schur function $f_\\lambda$ associated with $\\mu_\\lambda$ is\n$f_\\lambda=\\lambda\\, f$. The connection between the Carath\\'eodory\nfunctions $F_\\lambda(z)=\\left((U_\\lambda+zI)(U_\\lambda-zI)^{-1}\\vec\n1,\\vec 1\\right)$ and $F$ is given by\n\\[\nF_\\lambda(z)=\\frac{(1-\\lambda)+(1+\\lambda)\nF(z)}{(1+\\lambda)+(1-\\lambda)F(z)}.\n\\]\nThe measures $\\mu_\\lambda$ are known as the \\textit{Aleksandrov\nmeasures} associated with $\\mu$ \\cite[Section 1.3.9]{S2}.\n\\end{remark}\n\n\n\\section{Truncated CMV matrices}\n\\subsection{Truncated CMV matrix as a model for contractions with\nrank one defects} \\label{ss6}\n\nLet $\\cC=\\cC(\\{\\alpha_n\\})$ be the CMV matrix given by\n\\eqref{cmvmatr}. Recall that $\\cC(\\{\\alpha_n\\})$ is the matrix\nrepresentation of the unitary operator $\\cU$ of multiplication by\n$\\zeta$ in $L^2(\\dT, d\\mu)$, where $\\mu$ is the probability measure\nwith Verblunsky coefficients $\\{\\alpha_n\\}$. By the Geronimus\ntheorem the Schur parameters of the Schur function \\eqref{schur}\nassociated with $\\mu$ are $\\{\\alpha_n\\}$.\n\nThe matrix $\\cC$ determines the unitary operator in the space\n$l_2(\\dZ_+)$ (respectively in $\\dC^{N+1}$ in the case of\n$(N+1)\\times (N+1)$ matrix). The vector $\\delta_0=(1,0,0,\\ldots)^t$\nis cyclic for $\\cC$. Consider the matrix\n\\begin{equation}\n\\label{MODEL} \\cT=\\cT (\\{\\alpha_n\\})=\\begin{pmatrix}\n-\\bar{\\alpha}_1\\alpha_0&-\\rho_1\\alpha_0&0&0&\\ldots\\cr\n\\bar{\\alpha}_2\\rho_1&-\\bar{\\alpha}_2\\alpha_1&\\bar{\\alpha}_3\\rho_2&\\rho_3\\rho_2&\n\\ldots\\cr\n\\rho_2\\rho_1&-\\rho_2\\alpha_1&-\\bar{\\alpha}_3\\alpha_2&-\\rho_3\\alpha_2&\\ldots\\cr\n0&0&\\bar{\\alpha}_4\\rho_3&-\\bar{\\alpha}_4\\alpha_3&\\ldots\\cr\n\\ldots&\\ldots&\\ldots&\\ldots&\\ldots\n\\end{pmatrix}\n\\end{equation}\nobtained from $\\cC$ by deleting the first row and the first column.\nIt is clear from \\eqref{THREE} that a semi-infinite $\\cT$ takes on\nthe three-diagonal $2\\times 2$ block-matrix form\n\\[\n  \\cT=\\begin{pmatrix} \\cB_1 & \\cC_1 & 0 &0   & 0 &\n\\cdot &\n\\cdot  \\\\\n{\\mathcal A}}   \\def\\cB{{\\mathcal B}}   \\def\\cC{{\\mathcal C}_1 & \\cB_2 & \\cC_2 & 0 &0& \\cdot &\n\\cdot  \\\\\n0    & {\\mathcal A}}   \\def\\cB{{\\mathcal B}}   \\def\\cC{{\\mathcal C}_2 & \\cB_3 & \\cC_3 &0& \\cdot &\n\\cdot   \\\\\n\\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot & \\cdot\n\\end{pmatrix},\n\\]\nwhere ${\\mathcal A}}   \\def\\cB{{\\mathcal B}}   \\def\\cC{{\\mathcal C}_n$, $\\cB_n$, and $\\cC_n$ are defined in \\eqref{BLOCKS}.\nHenceforth $\\cT$ is called a \\textit{truncated} CMV matrix. $\\cT$ is\nthe matrix of the operator $\\sT=P_\\sH\\cU{\\upharpoonright\\,}\\sH$, where $P_\\sH$ is\nthe orthogonal projection in $L^2(\\dT, d\\mu)$ onto the subspace\n$\\sH=L^2(\\dT, d\\mu)\\ominus\\dC$.\n\nIt is easy to see that given $\\cT$ \\eqref{MODEL}, the values\n$\\alpha_n$ are uniquely determined. Indeed, from $(2,2)$ and $(3,2)$\nentries we have by (\\ref{rho}) $\n|\\alpha_1|^2=|\\bar\\alpha_2\\alpha_1|^2+\\rho_2^2|\\alpha_1|^2, $ so\n$|\\alpha_1|$ and $\\rho_1>0$ are known, and we find $\\alpha_0$,\n$\\alpha_2$ from $(1,2)$ and $(2,1)$ entries of (\\ref{MODEL}). From\n$(2,1)$ and $(2,2)$ entries we get $\\rho_2>0$, then $\\alpha_1$,\n$\\alpha_3$, etc. We call $\\alpha_n=\\alpha_n(\\cT)$ the\n\\textit{parameters} of $\\cT$ (\\ref{MODEL}).\n\nAs it was mentioned in Section 4.3, $\\cC=\\cL{\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O}$, $\\cL$ and ${\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O}$\nare defined in \\eqref{lm}. Given a matrix $A$, we denote by $A_r$\n($A_c$) the matrix obtained from $A$ by deleting the first row\n(column). Clearly, $A_{rc}={(A_r)}_c$. So we have\n$\\cT=\\cC_{rc}=\\cL_r{\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O}_c$. ${\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O}_c$ is isometric with $\\dim{\\rm ran\\,}(I-\n{\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O}_c{\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O}^*_c)=1$, whereas $\\cL_r$ is coisometric with\n$\\dim{\\rm ran\\,}(I-\\cL^*_r\\cL_r)=1$.\n\nLet $P_{{\\delta_0}^\\perp}$ be the orthogonal projection in\n$l_2(\\dZ_+)$ ($\\dC^{N+1})$ onto the subspace ${\\delta_0}^\\perp\\cong\nl_2(\\dN)$ $(\\dC^{N})$. Then the matrix $\\cT$ determines on the\nHilbert space $\\delta_0^\\perp$ the operator $\\cT=\nP_{\\delta_{0}^\\perp}\\cC{\\upharpoonright\\,}{\\delta_{0}}^\\perp$. Let the operators\n(matrices) ${\\mathcal S}}   \\def\\cT{{\\mathcal T}}   \\def\\cU{{\\mathcal U}:\\dC\\to\\dC$, $\\cF:\\dC\\to\\delta^\\perp_0$ and\n${\\mathcal G}}   \\def\\cH{{\\mathcal H}}   \\def\\cI{{\\mathcal I}:\\delta^\\perp_0\\to\\dC$ be given by\n\\[\n\\cS1=\\bar\\alpha,\\quad \\cF1=\\begin{pmatrix}\\rho_0\\cr 0\\cr \\ldots\\cr\n0\\end{pmatrix},\\quad {\\mathcal G}}   \\def\\cH{{\\mathcal H}}   \\def\\cI{{\\mathcal I} \\begin{pmatrix}h_1\\cr h_2\\cr \\ldots\\cr\nh_n\\cr\\ldots\\end{pmatrix}=\\bar\\alpha_1 \\rho_0 h_1+\\rho_1\\rho_0 h_2.\n\\]\nHence, the matrix $\\cC$ takes the block form\n\\[\n\\cC=\\begin{pmatrix} {\\mathcal S}}   \\def\\cT{{\\mathcal T}}   \\def\\cU{{\\mathcal U}&{\\mathcal G}}   \\def\\cH{{\\mathcal H}}   \\def\\cI{{\\mathcal I}\\cr \\cF&\\cT\\end{pmatrix}.\n\\]\nFrom \\eqref{UN} it follows that\n\\begin{equation}\n\\label{FORMU}\n\\begin{split}\n&\\left\\|{\\mathcal G}}   \\def\\cH{{\\mathcal H}}   \\def\\cI{{\\mathcal I}\\begin{pmatrix}h_1\\cr h_2\\cr \\ldots\\cr\nh_n\\cr\\ldots\\end{pmatrix}\\right\\|^2=\\left\\|D_{\\cT}\\begin{pmatrix}h_1\\cr\nh_2\\cr \\ldots\\cr h_n\\cr\\ldots\\end{pmatrix}\\right\\|=\\rho^2_0|\\bar\n\\alpha_1 h_1+\\rho_1 h_2|^2,\\quad\n{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{\\cT}=\\left\\{\\lambda(\\alpha_1\\delta_1+\\rho_1\\delta_2,\\;\\lambda\\in\\dC)\\right\\},\\\\\n& \\left\\|\\cF^*\\begin{pmatrix}h_1\\cr h_2\\cr \\ldots\\cr\nh_n\\cr\\ldots\\end{pmatrix}\\right\\|^2=\\left\\|D_{\\cT^*}\\begin{pmatrix}h_1\\cr\nh_2\\cr \\ldots\\cr\nh_n\\cr\\ldots\\end{pmatrix}\\right\\|=\\rho^2_0|h_1|^2,\\quad\n{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{\\cT^*}=\\left\\{\\lambda\\delta_1,\\;\\lambda\\in\\dC\\right\\},\\\\\n&D_{\\cT}\nh=\\rho_0(h,\\alpha_1\\delta_1+\\rho_1\\delta_2)(\\alpha_1\\delta_1+\\rho_1\\delta_2),\\quad\nD_{\\cT^*}h=\\rho_0(h,\\delta_1)\\delta_1,\\quad h\\in \\ell^2(\\dN)\\;(\\dC^N),\\\\\n &\\cT(\\alpha_1\\delta_1+\\rho_1\\delta_2)=-\\alpha_0\\delta_1.\n\\end{split}\n\\end{equation}\nSince $\\delta_0$ is the cyclic vector for $\\cC$, then by Theorem\n\\ref{cyclic} the unitary colligation\n\\begin{equation}\n\\label{DELTAC} \\Delta_\\cC=\\left\\{\\begin{pmatrix} {\\mathcal S}}   \\def\\cT{{\\mathcal T}}   \\def\\cU{{\\mathcal U}&{\\mathcal G}}   \\def\\cH{{\\mathcal H}}   \\def\\cI{{\\mathcal I}\\cr\n\\cF&\\cT\\end{pmatrix};\\dC,\\dC, \\delta^\\perp_0\\right\\}\n\\end{equation}\nis prime, and $\\cT$ is a completely nonunitary operator with rank\none defects on the Hilbert spaces $l_2(\\dN)$ or $\\dC^{N}$.\n\nLet \\begin{equation}\\label{C-Sh}\nF(z)=\\left((\\cC+zI)(\\cC-zI)^{-1}\\delta_0,\\delta_0\\right),\\quad\nf(z)=\\frac{1}{z}\\frac{F(z)-1}{F(z)+1} \\end{equation} be the Carath\\'eodory and\nthe Schur functions associated with $\\cC$. By Theorems \\ref{CH1} and\n\\ref{char} $f$ agrees with the characteristic function of $\\cT$.\n\n\\begin{proposition}\n\\label{PROP} \\begin{enumerate}\n\\item\n For a semi-infinite truncated CMV\nmatrix $\\cT=\\cT(\\{\\alpha_n\\})$ the following statements are\nequivalent:\n\\begin{enumerate}\n\\def\\rm (\\roman{enumi}){\\rm (\\roman{enumi})}\n\\item the matrix $\\cT$ does not contain a unilateral\nshift;\n\\item\n the matrix $\\cT^*$ does not contain a unilateral\nshift;\n\\item ${\\rm \\overline{span}\\,}\\{\\cT^n\\delta_1,\\; n=0,1,\\ldots\\}=\\ell^2(\\dN);$\n\\item ${\\rm \\overline{span}\\,}\\{\\cT^{*n}(\\alpha_1\\delta_1+\\rho_1\\delta_2),\\;\nn=0,1,\\ldots\\}=\\ell^2(\\dN)$;\n\\item\n$\\sum\\limits_{n=0}^\\infty|\\alpha_n|^2=\\infty;$\n\\item $\\ln(1-|f(e^{it})|^2)\\notin L^1[-\\pi,\\pi]$.\n\\end{enumerate}\n\\item If $\\cT$ is a semi-finite truncated CMV matrix, and one of\nthe conditions\n\\begin{enumerate}\n\\item $\\limsup_{n\\to\\infty}|\\alpha_n|=1,$ \\item\n $\\lim_{n\\to\\infty}\\alpha_n\\alpha_{n+m}=0$ for\n$m=1,2,\\ldots$, but $\\limsup_{n\\to\\infty}|\\alpha_n|>0$\n\\end{enumerate}\nis fulfilled, then\n\\[\ns-\\lim\\limits_{n\\to\\infty} \\cT^n=s-\\lim\\limits_{n\\to\\infty}\n\\cT^{*n}=0.\n\\]\n\\item\n If $\\cT$ is a finite truncated CMV matrix, then\n$\\lim\\limits_{n\\to\\infty}||\\cT^n||=0.$\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n(1) Since $\\{\\alpha_n\\}$ are the Schur parameters of the Schur\nfunction $f$ associated with the full CMV matrix\n$\\cC(\\{\\alpha_n\\})$, and $f$ agrees with the characteristic function\nof $\\cT(\\{\\alpha_n\\})$, the equivalence of the statements (a)--(f)\nfollows from \\eqref{perp}, \\eqref{shift}, \\eqref{co}, \\eqref{SP},\n\\eqref{L1}, \\eqref{FORMU}, \\eqref{poln}, and Theorems \\ref{cyclic}\nand \\ref{complete}.\n\n(2) Each condition (a) or (b) implies $f$ is inner (see subsection\n4.2). Hence $\\cT$ belongs to the class $C_{00}$, i.e.,\n$s-\\lim\\limits_{n\\to\\infty} \\cT^n=s-\\lim\\limits_{n\\to\\infty}\n\\cT^{*n}=0.$\n\n(3) The function $f$ is a finite Blaschke product and so inner.\nSince $\\cT$ is finite-dimensional, we get\n$\\lim\\limits_{n\\to\\infty}\\|\\cT^n\\|=0.$\n\\end{proof}\n\n\n\\begin{proposition}\n\\label{unieq1} Let $\\cT(\\{\\alpha_n\\})$, and $\\cT(\\{\\beta_n\\})$ be\ntruncated CMV matrices. Then $\\cT(\\{\\alpha_n\\})$ and\n$\\cT(\\{\\beta_n\\})$ are unitarily equivalent if and only if\n$\\beta_n=e^{it}\\alpha_n$ for all $n$ and $t\\in [0,2\\pi)$. Moreover,\nif ${\\mathcal V}}   \\def\\cW{{\\mathcal W}}   \\def\\cX{{\\mathcal X}$ is the diagonal unitary matrix of the form\n\\begin{equation}\n\\label{V} {\\mathcal V}}   \\def\\cW{{\\mathcal W}}   \\def\\cX{{\\mathcal X}={\\rm{diag}}(e^{it},1,e^{it},1,\\ldots),\n\\end{equation}\nthen\n\\begin{equation}\n\\label{VV} {\\mathcal V}}   \\def\\cW{{\\mathcal W}}   \\def\\cX{{\\mathcal X}\\cT(\\{\\alpha_n\\}){{\\mathcal V}}   \\def\\cW{{\\mathcal W}}   \\def\\cX{{\\mathcal X}}^{-1}=\\cT(\\{e^{it}\\alpha_n\\}).\n\\end{equation}\n\\end{proposition}\n\\begin{proof}  Consider two CMV matrices $\\cC(\\{\\alpha_n\\})$ and\n$\\cC(\\{\\beta_n\\})$, and associated with them Schur functions\n$f_\\alpha$ and $f_\\beta$. Since these functions agree with the\ncharacteristic functions of $\\cT(\\{\\alpha_n\\})$ and\n$\\cT(\\{\\beta_n\\})$, respectively, the operators $\\cT(\\{\\alpha_n\\})$\nand $\\cT(\\{\\beta_n\\})$ are unitarily equivalent if and only if\n$f_\\alpha$ and $f_\\beta$ differ by a scalar unimodular factor, which\nin turn yields $\\beta_n=e^{it}\\alpha_n$ for all $n$ and $t\\in\n[0,2\\pi)$.\n\nEquality \\eqref{VV} with ${\\mathcal V}}   \\def\\cW{{\\mathcal W}}   \\def\\cX{{\\mathcal X}$ \\eqref{V} can be verified by the\ndirect calculation based on \\eqref{cmventr1}, \\eqref{cmventr2}. So\n$\\cT(\\{\\alpha_n\\})$ and $\\cT(\\{e^{it}\\alpha_n\\})$ are unitarily\nequivalent.\n\\end{proof}\n\n\\begin{remark} The similar problem for ``full'' CMV matrices\ncan be considered as well. Let two CMV matrices $\\cC(\\{\\alpha_n\\})$\nand $\\cC(\\{\\beta_n\\})$ be unitarily equivalent by a unitary\npreserving $\\delta_0$. Then they are identical (see \\cite[Theorem\n2.3]{S3}). In general, two unitaries with simple spectra are\nunitarily equivalent if and only if their spectral measures are in\nthe same measure class. This is a standard issue in what is called\nmultiplicity theory.  So, two CMV matrices are unitarily equivalent\nif and only if their measures are mutually absolutely continuous.\nFor instance, a CMV matrix is unitarily equivalent to the free one\n($\\alpha_n\\equiv 0$) if and only if the associated measure $\\mu$ has\nthe property $\\mu'>0$ a.e. and does not have a singular part.\n\\end{remark}\n\nFrom \\eqref{VV} it follows that\n\\[\n\\cT(\\{e^{it}\\alpha_n\\})=e^{it{\\mathcal A}}   \\def\\cB{{\\mathcal B}}   \\def\\cC{{\\mathcal C}}\\cT(\\{\\alpha_n\\})e^{-it{\\mathcal A}}   \\def\\cB{{\\mathcal B}}   \\def\\cC{{\\mathcal C}},\\;\nt\\in\\dR,\n\\]\nwhere ${\\mathcal A}}   \\def\\cB{{\\mathcal B}}   \\def\\cC{{\\mathcal C}$ is a self-adjoint diagonal matrix\n${\\mathcal A}}   \\def\\cB{{\\mathcal B}}   \\def\\cC{{\\mathcal C}=\\rm{diag}(1,0,1,0\\ldots)$. Hence the matrix\n$\\cT(\\{e^{it}\\alpha_n\\})$ satisfies the differential equation\n\\[\n\\frac{d\\cT(t)}{dt}=i\\left({\\mathcal A}}   \\def\\cB{{\\mathcal B}}   \\def\\cC{{\\mathcal C}\\cT(t)-\\cT(t){\\mathcal A}}   \\def\\cB{{\\mathcal B}}   \\def\\cC{{\\mathcal C}\\right)\n\\]\nand $\\cT(0)=\\cT(\\{\\alpha_n\\}).$\n\nThe next theorem states that truncated CMV matrices are models of\ncompletely nonunitary contractions with rank one defects.\n\\begin{theorem}\n\\label{cmv1} Let $T$ be a completely nonunitary contraction with\nrank one defects acting on infinite-dimensional separable Hilbert\nspace $H$(respectively, finite-dimensional Hilbert space). Then $T$\nis unitarily equivalent to the operator acting on $l_2(\\dN)$\n(respectively, on $\\dC^N$ in the case $\\dim H=N$) determined by the\ntruncated CMV matrix $\\cT=\\cT(\\{\\alpha_n\\})$, where $\\{\\alpha_n\\}$\nare the Schur parameters of the characteristic function of $T$. In\nparticular, every completely nonunitary contraction with rank one\ndefects is a product of co-isometric and isometric operators with\nrank one defects.\n\\end{theorem}\n\\begin{proof}\nInclude $T$ into a prime unitary colligation\n$\\Delta=\\left\\{\\begin{pmatrix} S&G\\cr F&T\\end{pmatrix};\n\\dC,\\dC,H\\right\\}$. By Theorem \\ref{cyclic} the vector $\\vec\n1=\\begin{pmatrix}1\\cr 0\\end{pmatrix}$ is a cyclic for the unitary\noperator $U=\\begin{pmatrix} S&G\\cr F&T\\end{pmatrix}$. From the\nresults of \\cite{CMV1, CMV2} (see also \\cite{S1, S2}) there exists a\nunique CMV matrix $\\cC$ such that\n\\[\nU=W^{-1}\\cC W,\\quad \\delta_0=W\\vec 1,\n\\]\nwhere $W$ is a unitary  operator from $\\dC\\oplus H$ onto\n$l_2(\\dZ_+)$ $(\\dC^{N+1})$, and $\\delta_0=(1,0,0,\\ldots)^t$. It\nfollows that the operator $W$ takes the block-operator form\n\\[\nW=\\begin{pmatrix} 1&0\\cr 0&\\cX \\end{pmatrix}:\\begin{pmatrix}\\dC\\cr H\n\\end{pmatrix}\\to \\begin{pmatrix}\\dC\\cr\\delta^\\perp_0\n\\end{pmatrix},\n\\]\nwhere $\\cX:H\\to\\delta^\\perp_0$ is a unitary operator. Hence $\\cT=\\cX\nT\\cX^{-1}$, i.e., the operator $T$ is unitarily equivalent to the\noperator in $\\l_2(\\dN)$ $(\\dC^N)$ given by the truncated CMV matrix\n$\\cT=\\cT(\\{\\alpha_n\\})$. From representation \\eqref{carat} of\n$F(z)=\\left((U+zI)(U-zI)^{-1}\\vec 1,\\vec 1\\right)$ and Theorem\n\\ref{char} it follows that $\\{\\alpha_n\\}$ are the Schur parameters\nof the function $\\overline{\\Theta_\\Delta(\\bar z)}$ that agrees with the\ncharacteristic function of $T$.\n\nLet $\\cQ$ be an arbitrary unitary operator in $\\delta_0^\\perp$.\nSince $\\cT=\\cL_r{\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O}_c$, we get\n\\[\nT=\\cX^{-1}\\cT \\cX=\\cX^{-1}\\cL_r{\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O}_c\\cX =\\cX^{-1}\\cL_r\n\\cQ\\cQ^{-1}{\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O}_c \\cX=L\\,M,\n\\]\nwhere $M=\\cQ^{-1}{\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O}_c\\cX$ is an isometric operator with rank one\ndefect, and $L=\\cX^{-1}\\cL_r\\cQ$ is a co-isometric operator with\nrank one defect.\n\\end{proof}\nNote that the unitary colligation \\eqref{DELTAC} is unitary\nequivalent to the unitary colligation \\eqref{unmod}.\n\n\\subsection{The Liv$\\rm{\\check{s}}$ic theorem for quasi-unitary contractive\nextensions and the corresponding truncated CMV matrix}\n\n\nLet $V$ be an isometric operator acting on some Hilbert space $H$\nwith the domain ${\\rm dom\\,} V$ and the range ${\\rm ran\\,} V$. The numbers\n$\\dim(H\\ominus{\\rm dom\\,} V)$ and $\\dim (H\\ominus{\\rm ran\\,} V)$ are called the\ndefect indices of $V$. The isometric operator $V$ is called prime if\nthere is no nontrivial subspace on which $V$ is unitary. In\n\\cite{L1, L2} M.~Liv$\\rm{\\check{s}}$ic developed the spectral theory\nof isometric operators with equal defect indices, and their\nquasi-unitary extensions. A nonunitary operator $S$ on $H$ is called\na quasi-unitary extension of the isometric operator $V$ with the\ndefect indices $(n,n)$, if $S$ agrees with $V$ on ${\\rm dom\\,} V$ and maps\n$H\\ominus {\\rm dom\\,} V$ into $H\\ominus{\\rm ran\\,} V$.\n\nLet $\\vec U$ be the bilateral shift in $\\ell^2(\\dZ)$, i.e., $\\vec\nU\\delta_k=\\delta_{k-1}$, $k\\in\\dZ$, where $\\{\\delta_k,\\; k\\in\\dZ\\}$\nis the canonical orthonormal basis in $\\ell^2(\\dZ)$. Define $\\vec\nV_0$ by\n\\[\n{\\rm dom\\,}\\vec V_0=\\delta^\\perp_o, \\qquad \\vec V_0=\\vec U{\\upharpoonright\\,} {\\rm dom\\,}\\vec\nV_0,\n\\]\nThen ${\\rm ran\\,} \\vec V_0=\\delta^\\perp_{-1}$. Let the quasi-unitary\nextension $\\vec S_0$ of $\\vec V_0$ be given by $\\vec S_0\\delta_0=0$,\n$\\vec S_0{\\upharpoonright\\,}{\\rm dom\\,}\\vec V_0=\\vec V_0$. Then each point of ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$ is\nthe eigenvalue of $\\vec S_0$. So the spectrum of $\\vec S_0$ agrees\nwith $\\overline{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$. The following result is essentially due to M.\nLiv$\\rm{\\check{s}}$ic \\cite{L1}.\n \\begin{theorem}\nLet $S$ be a quasi-unitary contractive extension of a prime\nisometric operator $V$ with the defect indices $(1,1)$. If the whole\nopen disk ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$ consists of the point spectrum of $S$, then $V$ and\n$S$ are unitarily equivalent to $\\vec V_0$ and $\\vec S_{0}$,\nrespectively.\n \\end{theorem}\nClearly, the rank of the defect operators $(I-\\vec S^*_0\\vec\nS_0)^{1\/2}$ and $(I-\\vec S_0\\vec S^*_0)^{1\/2}$ is equal to one.\nSince the point spectrum of $\\vec S_0$ is ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$, the Sz.-Nagy--Foias\ncharacteristic function $\\Theta$ of $\\vec S_0$ is identically equal\nto zero. On the other hand, one can easily show (and it is well\nknown) that a completely nonunitary contraction with rank one\ndefects and zero characteristic function is unitarily equivalent to\nthe operator $S\\oplus S^*$, where $S$ is the unilateral shift in\n$\\ell^2(\\dN)$. So the operators $\\vec S_0$ and $S\\oplus S^*$ are\nunitarily equivalent. Since all Schur parameters of the function\n$\\Theta=0$ are zeros, the corresponding truncated CMV matrix\n$\\cT_0=\\|t_0(i,j)\\|$ takes the form\n\\[\n\\cT_0=\\begin{pmatrix} 0&0&0&0&0&0&\\ldots\\cr 0&0&0&1&0&0&\\ldots\\cr\n1&0&0&0&0&0&\\ldots\\cr 0&0&0&0&0&1&\\ldots\\cr 0&0&1&0&0&0&\\ldots \\cr\n\\ldots&\\ldots&\\ldots&\\ldots&\\ldots&\\ldots&\\ldots\n\\end{pmatrix},\n\\]\ni.e., $t_0(2k,2k+2)=t_0(2k+1, 2k-1)=1,$ $k\\ge1$, and the rest\n$t_0(i,j)=0$. The matrix $\\cT_0$ is a submatrix of the free CMV\nmatrix $\\cC_0$ corresponding to zero Schur parameters. Each point\n$z$ of ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$ is the eigenvalue of $\\cT_0$. The corresponding\neigensubspace is\n\\[\n\\sN_z=\\{\\lambda\\, (0,1,0,z,0,z^2,0,z^3,\\ldots)^t,\\quad \\lambda\n\\in\\dC\\}.\n\\]\n Hence, the spectrum\nof $\\cT_0$ is the closed unit disk $\\overline {\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$.\n\nLet ${\\mathcal V}}   \\def\\cW{{\\mathcal W}}   \\def\\cX{{\\mathcal X}_0$ be the operator in $\\ell^2(\\dN)$\n\\begin{equation}\n\\label{VISO}\n {\\rm dom\\,}{\\mathcal V}}   \\def\\cW{{\\mathcal W}}   \\def\\cX{{\\mathcal X}_0=\\ell^2(\\dN)\\ominus\\{c\\delta_2\\}=\\ker D_{\\cT_0},\\;\n{\\mathcal V}}   \\def\\cW{{\\mathcal W}}   \\def\\cX{{\\mathcal X}_0=\\cT_0{\\upharpoonright\\,}{\\rm dom\\,}{\\mathcal V}}   \\def\\cW{{\\mathcal W}}   \\def\\cX{{\\mathcal X}_0.\n\\end{equation}\nThen ${\\rm ran\\,}{\\mathcal V}}   \\def\\cW{{\\mathcal W}}   \\def\\cX{{\\mathcal X}_0=\\ell^2(\\dN)\\ominus\\{c\\delta_1\\}=\\ker D_{\\cT^*_0}$,\nand ${\\mathcal V}}   \\def\\cW{{\\mathcal W}}   \\def\\cX{{\\mathcal X}_0$ is isometric with the defect indices $(1,1)$. The\ncontraction $\\cT_0$ is the quasi-unitary extension of ${\\mathcal V}}   \\def\\cW{{\\mathcal W}}   \\def\\cX{{\\mathcal X}_0$ with\nthe zero characteristic function. Therefore, the truncated CMV\nmatrix $\\cT_0$ is unitarily equivalent to the operator $\\vec S_0$,\nand by Liv$\\rm{\\check{s}}$ic theorem \\cite{L1} the isometric\noperator ${\\mathcal V}}   \\def\\cW{{\\mathcal W}}   \\def\\cX{{\\mathcal X}_0$ is unitarily equivalent to $\\vec V_0$.\n\nAll other quasi-unitary contractive extensions of ${\\mathcal V}}   \\def\\cW{{\\mathcal W}}   \\def\\cX{{\\mathcal X}_0$ are given\nby the truncated CMV matrices $\\cT=\\|t(i,j)\\|$\n\\begin{equation}\n\\label{TTTO} \\cT=\\begin{pmatrix}0&-re^{i\\varphi}&0&0&0&0&\\ldots\\cr\n0&0&0&1&0&0&\\ldots\\cr 1&0&0&0&0&0&\\ldots\\cr 0&0&0&0&0&1&\\ldots\\cr\n0&0&1&0&0&0&\\ldots \\cr\n\\ldots&\\ldots&\\ldots&\\ldots&\\ldots&\\ldots&\\ldots\n\\end{pmatrix},\n\\end{equation}\ni.e., $t(2k,2k+2)=t(2k+1, 2k-1)=1,$ $k\\ge 1,$ $t(1,2)=-re^{i\\varphi}$,\n$r\\in (0,1)$, $\\varphi$ is an arbitrary number from the interval $[0,\n2\\pi)$, and the rest $t(i,j)=0$. The characteristic function of\n$\\cT$ is the constant function $\\Theta=r e^{i\\varphi}$. The spectrum of\neach such matrix is the unit circle $\\dT$. Because\n$|\\Theta^{-1}|=r^{-1}$, each of such matrix is similar to unitary\nmatrix \\cite [Theorem IX.1.2]{SF}.\n\nThe matrices $\\cT_0$ and $\\cT$ contain the shift\n\\[\n{\\rm dom\\,} \\cW={\\rm \\overline{span}\\,}\\{\\delta_1,\\delta_3,\\ldots,\\delta_{2n-1},\\ldots\\},\\;\n\\cW\\left(\\sum_{n=1}^\\infty h_n\\delta_{2n-1}\\right)=\\sum_{n=1}^\\infty\nh_n\\delta_{2n+1}.\n\\]\nThe matrices $\\cT^*_0$ and $\\cT^*$ contain the shift\n\\[\n{\\rm dom\\,} \\cW_*={\\rm \\overline{span}\\,}\\{\\delta_2,\\delta_4,\\ldots,\\delta_{2n},\\ldots\\},\\;\n\\cW_*\\left(\\sum_{n=1}^\\infty h_n\\delta_{2n}\\right)=\\sum_{n=1}^\\infty\nh_n\\delta_{2n+2}.\n\\]\n\nLet $T$ be a completely nonunitary contraction with rank one defects\nand the constant characteristic function $\\Theta$,\n$0<|\\Theta(z)|=r<1$. Then by Theorem \\ref{cmv1} $T$ is unitarily\nequivalent to the truncated CMV matrices \\eqref{TTTO}.\n\n\\subsection{Sub-matrices of truncated CMV matrices and iterates of their Schur\nfunctions} Along with truncated CMV matrices $\\cT(\\{\\alpha_n\\})$\n\\eqref{MODEL}, we consider here truncated CMV matrices\n$\\widetilde\\cT(\\{\\alpha_n\\})$ obtained from the alternate CMV matrix\n$\\widetilde\\cC(\\{\\alpha_n\\})$ \\eqref{altcmvmatr} by the same procedure. The\nmatrix $\\widetilde\\cT(\\{\\alpha_n\\}$ is the transpose of $\\cT(\\{\\alpha_n\\})$\n\\begin{equation}\n\\label{TMODEL} \\widetilde\\cT=\\begin{pmatrix}\n-\\bar{\\alpha}_1\\alpha_0&\\bar{\\alpha}_2\\rho_1&\\rho_2\\rho_1&0&\\ldots\\cr\n-\\rho_1\\alpha_0&-\\bar{\\alpha}_2\\alpha_1&-\\rho_2\\alpha_1&0&\\ldots\\cr\n0&\\bar{\\alpha}_3\\rho_2&-\\bar{\\alpha}_3\\alpha_2&\\bar{\\alpha_4}\\rho_3&\\ldots\\cr\n0&\\rho_3\\rho_2&-\\rho_3\\alpha_2&-\\bar{\\alpha}_4\\alpha_3&\\ldots\\cr\n\\ldots&\\ldots&\\ldots&\\ldots&\\ldots\n\\end{pmatrix},\n\\end{equation}\nand\n\\[\n\\widetilde\\cT(\\{\\alpha_n\\})=\\cT^t(\\{\\alpha_n\\})=({\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O}_c)^t(\\cL_r)^t={\\mathcal M}}   \\def\\cN{{\\mathcal N}}   \\def\\cO{{\\mathcal O}_r\\cL_c.\n\\]\nAs in Section 6.1, it is not hard to show that\n$\\widetilde\\cT(\\{\\alpha_n\\})$ is a completely nonunitary contraction with\nrank one defects, and its characteristic function $\\widetilde f$ agrees\nwith the Schur function associated with Verblunsky coefficients\n(Schur parameters) $\\{\\alpha_n\\}$. Indeed (cf. \\eqref{C-Sh})\n\\[\n(\\widetilde\\cC+zI)(\\widetilde\\cC-zI)^{-1}=(\\cC^t+zI)(\\cC^t-zI)^{-1}=\n\\left((\\cC+zI)(\\cC-zI)^{-1}\\right)^t, \\]\n and so\n $\\widetilde F(z):=\\left((\\widetilde\\cC+zI)(\\widetilde\\cC-zI)^{-1}\\delta_0,\\delta_0\\right)=F(z)$,\n$\\widetilde f=f$, as claimed. So, the matrices $\\cT(\\{\\alpha_n\\})$ and\n$\\widetilde\\cT(\\{\\alpha_n\\})$ are unitarily equivalent.\n\nDenote by $\\cT^{(k)}$ ($\\widetilde\\cT^{(k)}$) the matrix obtained from\n$\\cT$ ($\\widetilde\\cT)$ by deleting the first $k$ rows and columns. The\nfollowing result provides the characteristic function of\n$\\cT^{(k)}$.\n\\begin{theorem}\n\\label{DER} Let $\\mu$ be a probability measure on $\\dT$ with\nVerblunsky coefficients $\\{\\alpha_n\\}_{n=0}^N$, $N\\le\\infty$, and\nlet $f$, $\\cC(\\{\\alpha_n\\})$, $\\widetilde\\cC(\\{\\alpha_n\\})$,\n$\\cT(\\{\\alpha_n\\})$, $\\widetilde\\cT(\\{\\alpha_n\\})$ be the corresponding\nSchur function, CMV and truncated CMV matrices, respectively. Then\n$\\cT^{(k)}$, $\\widetilde\\cT^{(k)}$ are completely nonunitary contractions\nwith rank one defects, and the following relations hold:\n\\[\n\\cT^{(2m-1)}(\\{\\alpha_n\\}_{n=0}^N)=\\widetilde\\cT(\\{\\alpha_n\\}_{n=2m-1}^N),\n\\quad\n\\cT^{(2m)}(\\{\\alpha_n\\}_{n=0}^N)=\\cT(\\{\\alpha_n\\}_{n=2m}^N),\\quad\nm=1,2,\\ldots.\n\\]\nSo, the characteristic function of $\\cT^{(k)}$ agrees with the\n$k^{th}$ Schur iterate of $f$.\n\\end{theorem}\n\\begin{proof}\nThe relations\n\\[\n\\cT^{(1)}(\\{\\alpha_n\\}_{n=0}^N)=\\widetilde\\cT(\\{\\alpha_n\\}_{n=1}^N), \\qquad\n\\widetilde\\cT^{(1)}(\\{\\alpha_n\\}_{n=1}^N)=\\cT(\\{\\alpha_n\\}_{n=2}^N)\n\\]\nfollows directly from \\eqref{MODEL} and \\eqref{TMODEL}. The rest is\na matter of simple induction and the definition of the $k^{th}$\nSchur iterates.\n\\end{proof}\n\nThe relation between characteristic functions of the sub-matrices\n$\\cT^{(k)}(\\{\\alpha\\}_{n=0}^N)$ and the $k^{th}$ Schur iterates\nestablished in the above mentioned theorem is a complete analog of\nthe result concerning the connections between m-functions of a\nJacobi matrix and its sub-matrices \\cite{GS}.\n\nLet us now go back to the model of Section \\ref{s5}.\n\\begin{theorem}\nLet $\\mu$ be a probability measure on $\\dT$ with Verblunsky\ncoefficients $\\{\\alpha_n\\}_{n=0}^N$, $N\\le \\infty$. Consider three\nsubspaces in $L^2(\\dT,\\mu)$\n\\[\n\\cH_{2m}:={\\rm span\\,} \\{1,\\zeta,\\bar \\zeta,\\zeta^2,\\bar\\zeta\n^2,\\ldots,\\zeta^m,\\bar\\zeta^m\\},\n\\]\n\\[\n\\cH_{2m-1}:={\\rm span\\,} \\{1,\\zeta,\\bar \\zeta,\\zeta^2,\\bar\\zeta\n^2,\\ldots,\\bar\\zeta^{m-1},\\zeta^m\\}, \\quad \\widetilde\\cH_{2m-1}:={\\rm span\\,}\n\\{1,\\bar\\zeta,\n\\zeta,\\bar\\zeta^2,\\zeta^2,\\ldots,\\zeta^{m-1},\\bar\\zeta^m\\}.\n\\]\nDenote by $\\sH_{2m}$ $(\\sH_{2m-1},\\ \\widetilde\\sH_{2m-1})$ their orthogonal\ncomplements in $L^2(\\dT,\\mu)$, and by $P_{2m}$ $(P_{2m-1},\\ \\widetilde\nP_{2m-1})$ the orthogonal projections onto $\\sH_{2m}$ $(\\sH_{2m-1},\\\n\\widetilde\\sH_{2m-1})$, respectively. Then the operators\n\\begin{equation}\n\\label{TK}\n\\sT_k h(\\zeta)= P_k \\left(\\zeta h(\\zeta)\\right),\\;\nh(\\zeta)\\in\\sH_k,\\qquad \\widetilde\\sT_{2m-1} h(\\zeta)= \\widetilde P_m \\left(\\zeta\nh(\\zeta)\\right),\\; h(\\zeta)\\in\\widetilde \\sH_{2m-1},\n\\end{equation}\nare completely nonunitary contractions with rank one defects. The\ncharacteristic function of  $\\sT_k$ agrees with the $k^{th}$ Schur\niterate of the Schur function $f(\\mu)$, the characteristic function\nof $\\widetilde\\sT_{2m-1}$ with $(2m-1)^{th}$ Schur iterate of $f(\\mu)$. So,\nthe operator $\\sT_{k}$ is unitarily equivalent to the operator\n\\begin{equation}\n\\label{mmmod} h(\\zeta)\\to P_0^{(k)}\\left(\\zeta h(\\zeta)\\right),\n\\quad h(\\zeta)\\in\nL^2\\left(\\dT,d\\mu(\\{\\alpha_n\\}_{n=k}^N)\\right)\\ominus \\dC,\n\\end{equation}\nwhere $P_0^{(k)}$ is the orthogonal projection onto\n$L^2\\left(\\dT,d\\mu(\\{\\alpha_n\\}_{n=k}^N)\\right)\\ominus \\dC$. In\naddition, $\\sT_{2m-1}$ is unitarily equivalent to $\\widetilde\\sT_{2m-1}$.\n\\end{theorem}\n\\begin{proof} Recall that  CMV matrices\n$\\cC(\\{\\alpha_n\\}$ and $\\widetilde\\cC(\\{\\alpha_n\\})$ represent the unitary\noperator $Uh(\\zeta)=\\zeta h(\\zeta)$ in $L^2(\\dT,d\\mu(\\{\\alpha_n\\}))$\nwith respect to the complete orthonormal systems $\\{\\chi_n\\}$ and\n$\\{x_n\\}$, respectively. Moreover\n\\[\n\\begin{split}\n&\\cH_{2m}=\n{\\rm span\\,}\\{\\chi_0,\\chi_1,\\ldots,\\chi_{2m}\\}={\\rm span\\,}\\{x_0,x_1,\\ldots,\nx_{2m}\\},\\\\\n&\\cH_{2m-1}={\\rm span\\,}\\{\\chi_0,\\chi_1,\\ldots,\\chi_{2m-1}\\},\\\\\n&\\widetilde\\cH_{2m-1}={\\rm span\\,}\\{x_0,x_1,\\ldots, x_{2m-1}\\}.\n\\end{split}\n\\]\nSince $\\cT(\\{\\alpha_n\\}_{n=0}^N\\})$\n($\\widetilde\\cT(\\{\\alpha_n\\}_{n=0}^N\\})$ is the matrix of $\\sT$\n\\eqref{UNMODL2} with respect to the basis $\\{\\chi_n\\}_{n=1}^N$\n$(\\{x_n\\}_{n=1}^N$), the operators $\\sT_{2m}$, $\\sT_{2m-1}$, and\n$\\widetilde\\sT_{2m-1}$ have the matrices $\\cT^{(2m)}$, $\\cT^{(2m-1)}$, and\n$\\widetilde\\cT^{(2m-1)}$, respectively. From Theorem \\ref{DER} it follows\nthat $\\sT_k$ are completely nonunitary contractions with rank one\ndefects for all $k$, and their characteristic functions agree with\nthe $k^{th}$ Schur iterates of $f$.\n By Theorems \\ref{DER} and \\ref{L2} the operator $\\sT_k$ is unitarily\n equivalent to the operator\n given by \\eqref{mmmod}. We also have\n\\[\n\\widetilde\\cT^{(2m-1)}(\\{\\alpha_n\\}_{n=0}^N)=\\cT(\\{\\alpha_n\\}_{n=2m-1}^N).\n\\]\nTherefore, the characteristic function of\n$\\widetilde\\sT^{(2m-1)}(\\{\\alpha_n\\}_{n=0}^N)$ agrees with $(2m-1)^{th}$\niterate $f_{2m-1}$ of $f$, and hence the operators\n$\\widetilde\\sT^{(2m-1)}(\\{\\alpha_n\\}_{n=0}^N)$ and\n$\\sT^{(2m-1)}(\\{\\alpha_n\\}_{n=0}^N)$ are unitarily equivalent.\n\\end{proof}\n\nWe complete the section with the general result from the\ncontractions theory which is proved with the help of the truncated\nCMV model.\n\\begin{proposition}\nLet $T$ be a completely nonunitary contraction with rank one defects\nin a separable Hilbert space $H$, $\\dim H\\ge 2$, and let $P_{\\ker\nD_{T^*}}$, $P_{\\ker D_{T}}$ be the orthogonal projections onto $\\ker\nD_{T^*}$ and $\\ker D_{T}$ in $H$, respectively. Then the operators\n\\[\n T_1:=P_{\\ker D_{T^*}}\\,T{\\upharpoonright\\,} \\ker D_{T^*},\\; \\widetilde T_1:=P_{\\ker\n {D_T}}T{\\upharpoonright\\,} \\ker D_{T}\n\\]\nare  unitarily equivalent completely nonunitary contractions with\nrank one defects, and their characteristic functions agree with the\nfunction\n\\[\nh_1(z):=\\frac{1}{z}\\,\\frac{h(z)-h(0)}{1-\\overline{h(0)} h(z)},\n\\]\nwhere $h$ is the characteristic function of $T$.\n\\end{proposition}\n\\begin{proof} By Theorem \\ref{cmv1} the operator $T$ is unitarily\nequivalent to the truncated CMV matrices\n$\\cT=\\cT(\\{\\alpha_n\\}_{n=0}^N)$ and\n$\\widetilde\\cT=\\widetilde\\cT(\\{\\alpha_n\\}_{n=0}^N)$, where $\\{\\alpha_n\\}_{n=0}^N$\nare the Schur parameters of $h$, $N\\le\\infty$. So, there exists a\nunitary operators $V,\\,\\widetilde V:\\delta_0^\\perp\\to H$ such that\n \\[\nV\\cT V^{-1}=\\widetilde V\\widetilde\\cT \\widetilde V^{-1}=T.\n \\]\nIt follows that\n \\[\nVD_{\\cT^*}V^{-1}=D_{T^*},\\qquad \\widetilde VD_{\\widetilde\\cT}\\widetilde V^{-1}=D_{T},\n \\]\nand  hence $V\\ker D_{\\cT^*}=\\ker D_{T^*}$,  $\\widetilde V\\ker\nD_{\\widetilde\\cT}=\\ker D_{T}$. Due to \\eqref{FORMU} we have\n\\[\n{\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{\\cT^*}={\\mathfrak D}}   \\def\\sE{{\\mathfrak E}}   \\def\\sF{{\\mathfrak F}_{\\widetilde\\cT}={\\rm span\\,}\\{\\delta_1\\} \\]\n and\n\\[\n\\cT^{(1)}=P_{\\ker D_{\\cT^*}}\\cT{\\upharpoonright\\,}\\ker D_{\\cT^*},\\qquad \\widetilde\n\\cT^{(1)}=P_{\\ker D_{\\widetilde\\cT}}\\widetilde\\cT{\\upharpoonright\\,}\\ker D_{\\widetilde\\cT}.\n\\]\nHence\n\\[\nV\\cT^{(1)}V^{-1}=T_1,\\qquad \\widetilde V\\widetilde\\cT^{(1)}\\widetilde V^{-1}=\\widetilde T_1.\n\\]\nNow from Theorem \\ref{DER} it follows that $T_1$ and $\\widetilde T_1$ are\ncompletely nonunitary contractions with rank one defects, and their\ncharacteristic functions agree with the first Schur iterate $h_1$ of\n$h$. Hence $T_1$ and $\\widetilde T_1$ are unitarily equivalent.\n\n\\end{proof}\n\n\n\\section{Inverse spectral problems for finite and semi-infinite truncated CMV matrices}\nConsider a $N\\times N$ truncated CMV matrix\n\\begin{equation}\n\\label{NMODEL} \\cT=\\cT (\\{\\alpha_n\\})=\\begin{pmatrix}\n-\\bar{\\alpha}_1\\alpha_0&-\\rho_1\\alpha_0&0&\\ldots&0\\cr\n\\bar{\\alpha}_2\\rho_1&-\\bar{\\alpha}_2\\alpha_1&\\bar{\\alpha}_3\\rho_2&\\ldots&0\\cr\n\\rho_2\\rho_1&-\\rho_2\\alpha_1&-\\bar\\alpha_3\\alpha_2&\\ldots&0\\cr\n\\ldots&\\ldots&\\ldots&\\ldots&\\bar\\alpha_N\\rho_{N-1}\\cr\n\\ldots&\\ldots&\\ldots&-\\rho_{N-1}\\alpha_{N-2}&-\\bar\\alpha_N\\alpha_{N-1}\n\\end{pmatrix}\n\\end{equation}\n(for even $N$ it looks a bit different). The problem under\ninvestigation in the present section is the reconstruction of the\nmatrix $\\cT$ \\eqref{NMODEL} from either the complete set of its\neigenvalues or from the mixed spectral data: the part of the\nspectrum and the part of the parameters $\\alpha_n(\\cT)$.\n\n\\subsection{Existence of a finite truncated CMV matrix\n with the given spectrum}\n\\begin{theorem}\n\\label{InvSp} Let $z_1, z_2,\\ldots, z_N$ be not necessarily distinct\nnumbers from the open unit disk. Then there exists a truncated\n$N\\times N$ CMV matrix $\\cT$ \\eqref{NMODEL} which has eigenvalues\n$z_1, z_2,\\dots,z_N$, counting their algebraic multiplicities. Such\nmatrix is determined uniquely up to multiplication of its parameters\n$\\alpha_n(\\cT)$ by the same unimodular factor.\n\\end{theorem}\n\\begin{proof}\nLet\n \\begin{equation}\\label{blash} b(z)=e^{i\\varphi}\\prod\\limits_{k=1}^N\\frac{z-{z}_k}{1-\\bar z_k\\,\nz},\\quad z\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F},\\quad \\varphi\\in [0,2\\pi). \\end{equation} We want to show that $b$\nis the characteristic function of a truncated CMV matrix $\\cT$\n(\\ref{NMODEL}). Put\n\\[\nF(z)=\\frac{1+zb(z)}{1-zb(z)}\\,,\n\\]\nwhich is a rational function with $N+1$ distinct simple poles lying\non $\\dT$, ${\\rm Re\\,} F(z)>0$, $z\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$, and $ F(0)=1$. It follows that\nthere exists a probability measure $d\\mu$ on the unit circle\nsupported at those poles, so that\n\\[\nF(z)=\\int_{\\dT}\\frac{\\zeta+z}{\\zeta-z}\\,d\\mu(\\zeta).\n\\]\nLet $\\{\\alpha_0,\\ldots,\\alpha_{N-1},\\alpha_N\\}$ be the Schur\nparameters of $b$, that is the same as the Verblunsky coefficients\nof $\\mu$. Construct the $(N+1)\\times (N+1)$ unitary CMV matrix $\\cC\n$ of the form \\eqref{cmvmatr}. Then\n\\[\nF(z)=\\left((\\cC+zI)(\\cC-zI)^{-1}\\delta_0,\\delta_0\\right),\\;|z|<1,\n\\]\nwhere $\\delta_0=(1,0,\\ldots,0)^t\\in\\dC^{N+1}$. Let $\\cT$ be $N\\times\nN$ be truncated CMV matrix of the form \\eqref{NMODEL}. $\\cC$ has the\nblock form\n\\[\n\\cC=\\begin{pmatrix}{\\mathcal S}}   \\def\\cT{{\\mathcal T}}   \\def\\cU{{\\mathcal U}& {\\mathcal G}}   \\def\\cH{{\\mathcal H}}   \\def\\cI{{\\mathcal I}\\cr  \\cF&\\cT\\end{pmatrix},\n\\]\nwhere $ {\\mathcal S}}   \\def\\cT{{\\mathcal T}}   \\def\\cU{{\\mathcal U}=\\bar\\alpha_0$, $\n{\\mathcal G}}   \\def\\cH{{\\mathcal H}}   \\def\\cI{{\\mathcal I}=\\begin{pmatrix}\\bar\\alpha_1\\rho_0,&\\rho_1\\rho_0,&0,&\\ldots,&0\\end{pmatrix}$,\nand $\\cF=\\begin{pmatrix}\\rho_0\\cr 0\\cr \\ldots\\cr 0\\end{pmatrix}$.\nSince $\\delta_0$ is a cyclic vector for $\\cC$, the unitary\ncolligation $\\Delta=\\left\\{\\begin{pmatrix} {\\mathcal S}}   \\def\\cT{{\\mathcal T}}   \\def\\cU{{\\mathcal U}& {\\mathcal G}}   \\def\\cH{{\\mathcal H}}   \\def\\cI{{\\mathcal I}\\cr\n\\cF&\\cT\\end{pmatrix},\\;\\dC,\\dC,\\dC^{N}\\right\\}$ is prime. Hence\n$\\cT$ is a completely nonunitary contraction with rank one defect\noperators. Let $\\Theta_{\\Delta}(z)$ be the transfer function of\n$\\Delta$. By Theorem \\ref{char} we have\n\\[\n\\overline{\\Theta_{\\Delta}(\\bar z)}=\\frac{1}{z}\\frac{F(z)-1}{F(z)+1}\\,,\n\\qquad \\Theta_\\Delta(z)=\\overline{b(\\bar z)}.\n\\]\nSo $b(z)$ agrees with the characteristic function of $\\cT$.\nTherefore $\\cT$ has eigenvalues $z_1,\\ldots, z_N$, counting their\nalgebraic multiplicities \\cite{SF}.\n\nFinally, let $\\cT(\\{\\alpha_n\\})$ and $\\cT(\\{\\beta_n\\})$ be two such\nmatrices. Each of them is a completely nonunitary matrix with rank\none defects, and their characteristic functions agree with $b$\n\\eqref{blash}. Hence they are unitarily equivalent, and Proposition\n\\ref{unieq1} completes the proof.\n\\end{proof}\n\\begin{example} Let $T$ be a completely nonunitary contraction with rank\none defects on $N$-dimensional Hilbert space, and let $T$ have just\none eigenvalue $z=0$ of the algebraic multiplicity $N$. Then its\ncharacteristic function agrees with $f(z)=e^{i\\varphi}z^N$. The\ncorresponding Schur parameters are\n$\\{\\underbrace{0,\\dots,0}_{N},e^{i\\varphi}\\}.$ It follows that $\\rho_n=1$\nfor $n=0,\\ldots, N-1$. Hence $T$ is unitarily equivalent to the\n$N\\times $N truncated CMV matrix $\\cT_N$ (see the expressions for\n$\\cT_5$ and $\\cT_6$):\n\\[\n\\cT_5=\\begin{pmatrix} 0&0&0&0&0\\cr 0&0&0&1&0\\cr 1&0&0&0&0\\cr\n0&0&0&0&e^{i\\varphi}\\cr 0&0&1&0&0\\end{pmatrix},\\; \\cT_6=\\begin{pmatrix}\n0&0&0&0&0&0\\cr 0&0&0&1&0&0\\cr 1&0&0&0&0&0\\cr 0&0&0&0&0&1\\cr\n0&0&1&0&0&0\\cr 0&0&0&0&e^{i\\varphi}&0\\end{pmatrix}.\n\\]\n\\end{example}\n\\subsection{Uniqueness and reconstruction of a finite truncated CMV matrix from\nmixed spectral data}\n\nIt is easily seen from \\eqref{NMODEL} that a truncated $N\\times N$\nCMV matrix $\\cT$ is completely determined by $N+1$ independent\nparameters $\\alpha_j(\\cT)$, $j=0,1,\\ldots,N$. The problem we discuss\nhere is whether $\\cT$ can be restored from the part of its spectrum\n(the eigenvalues $z_1,\\ldots,z_m$, of the algebraic multiplicity\n$l_k$, $k=1,\\ldots,m$, with $l_1+\\ldots+l_m=r$), and the {\\it first}\n$N-r+1$ parameters $\\alpha_0(\\cT),\\ldots,\\alpha_{N-r}(\\cT)$. As we\nwill see later on, the solution of this problem is unique (if it\nexists).\n\nWe begin with a simple result from complex analysis. We don't know\nwhere exactly it appears in the literature, but by all means it is\nknown to experts.\n\n\\begin{lemma}\\label{np}\nLet $z_1,\\ldots,z_m$ be distinct points in ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$, $l_1,\\ldots,l_m$\npositive integers, and $r=l_1+\\ldots+l_m$. Suppose that the\nNevanlinna-Pick interpolation problem with multiple nodes\n\\begin{equation}\\label{np1} b^{(j)}(z_k)=w_k^{(j)}, \\qquad j=0,1,\\ldots,l_k-1,\n\\quad k=1,2,\\ldots,m \\end{equation} has two solutions $b_1$ and $b_2$, both the\nBlaschke products of order $\\leq r-1$. Then $b_1=b_2$.\n\\end{lemma}\n\\begin{proof}\nAssume first that $z_k\\not=0$, $w_k^{(0)}\\not=0$, $k=1,\\ldots,m$.\nGiven a Blaschke product $s$, we see by differentiating the equality\n$\\overline{s(1\/\\bar z)}=s^{-1}(z)$ that\n\\[\n\\overline{s^{(j)}\\left(\\frac1{\\bar\nz}\\right)}=\\frac{P_j(s(z),s'(z),\\ldots,s^{(j)}(z))}{s^{2^j}(z)}\\,,\n\\]\nwhere $P_j$ is a polynomial of its variables. Hence\n\\[\n\\overline{s^{(j)}\\left(\\frac1{\\bar\nz_k}\\right)}=\\frac{P_j(s(z_k),\\ldots,s^{(j)}(z_k))}{s^{2^j}(z_k)}\\,,\n\\qquad k=1,2,\\ldots,m\n\\]\nso we have\n\\[\nb_1^{(j)}(z_k)=b_2^{(j)}(z_k), \\quad b_1^{(j)}\\left(\\frac1{\\bar\nz_k}\\right)=b_2^{(j)}\\left(\\frac1{\\bar z_k}\\right), \\qquad\nj=0,1,\\ldots,l_k-1, \\quad k=1,2,\\ldots,m.\n\\]\nThen for the difference $u=b_1-b_2$ the relations \\begin{equation}\\label{np2}\nu^{(j)}(z_k)=u^{(j)}\\left(\\frac1{\\bar z_k}\\right)=0, \\qquad\nj=0,1,\\ldots,l_k-1, \\quad k=1,2,\\ldots,m. \\end{equation} hold. Let now\n\\[\nb_l(z)=\\frac{p_l(z)}{q_l(z)}\\,, \\quad l=1,2, \\qquad\nu(z)=\\frac{p_1(z)q_2(z)-p_2(z)q_1(z)}{q_1(z)q_2(z)}=\\frac{p(z)}{q(z)}\\,,\n\\]\nwhere $p,q$ are polynomials of degree $\\leq 2r-2$. The Leibniz\nformula\n\\[\nu^{(n)}(z)=\\sum_{k=0}^n\n\\frac{n!}{k!(n-k)!}\\,p^{(k)}(z)\\left(\\frac1{q}\\right)^{(n-k)}(z)\n\\]\nshows by induction that \\eqref{np2} imply \\begin{equation}\\label{np3}\np^{(j)}(z_k)=p^{(j)}\\left(\\frac1{\\bar z_k}\\right)=0, \\qquad\nj=0,1,\\ldots,l_k-1, \\quad k=1,2,\\ldots,m. \\end{equation} But $\\deg p\\leq 2r-2$,\nand there are $2r$ conditions in \\eqref{np3}, so $p\\equiv 0$, as\nneeded.\n\nAssume next that $z_k\\not=0$, $k=1,\\ldots,m$ and some of $w_k^{(0)}$\nare zero. Take $\\varepsilon\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$, $\\varepsilon\\not=w_k^{(0)}$ and\nput\n\\[\ns_0:=\\frac{z-\\varepsilon}{1-\\bar\\varepsilon z}\\,, \\qquad \\widehat\nb_l(z):=s_0(b_l(z)), \\qquad l=1,2.\n\\]\nThen both $\\widehat b_1$ and $\\widehat b_2$ are Blaschke products of order\n$\\leq r-1$ which solve the interpolation problem\n\\[\n\\widehat b_l^{(j)}(z_k)=\\widehat w_k^{(j)}, \\qquad j=0,1,\\ldots,l_k-1, \\quad\nk=1,2,\\ldots,m, \\quad l=1,2,\n\\]\nwhere $\\widehat w_k^{(0)}=s_0(w_k^{(0)})\\not=0$ and $\\widehat w_k^{(j)}=\\left(\ns_0(b_l(z))\\right)^{(j)}\\Bigr|_{z=z_k}$. The above argument applied\nto $\\widehat b_l$ gives $\\widehat b_1=\\widehat b_2 \\Rightarrow b_1=b_2$, as needed.\n\nFinally, assume that $z_1=0$. Let $\\varepsilon\\not=-z_k$ for all\n$k$, and put\n\\[\n\\widetilde b_l(z):=b_l(s_0(z)), \\qquad l=1,2.\n\\]\nThen the Blaschke products $\\widetilde b_1$, $\\widetilde b_2$ of order $\\leq r-1$\nsatisfy\n\\[\n\\widetilde b_l^{(j)}(\\widetilde z_k)=\\widetilde w_k^{(j)}, \\qquad j=0,1,\\ldots,l_k-1,\n\\quad k=1,2,\\ldots,m, \\quad l=1,2\n\\]\nand $\\widetilde z_k=(z_k+\\varepsilon)(1+\\bar\\varepsilon z_k)^{-1}\\not=0$.\nHence $\\widetilde b_1=\\widetilde b_2$, and so $b_1=b_2$. The proof is complete.\n\n\\end{proof}\n\n\\begin{theorem}\n\\label{mix1} Let $z_1,\\ldots,z_m$ be distinct nonzero points in\n${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$, $l_1,\\ldots,l_m$ be positive integers, and\n$r=l_1+\\ldots+l_m\\leq N$. Let $\\alpha_0,\\ldots,\\alpha_{N-r}\\in{\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$.\nIf there exists a $N\\times N$ truncated CMV matrix $\\cT$\n\\eqref{NMODEL} such that $z_1,\\ldots z_m$ are eigenvalues of $\\cT$\nwith the algebraic multiplicities $l_1,\\ldots, l_m$, and\n$\\alpha_j(\\cT)=\\alpha_j$, $j=0,\\ldots,N-r$, then this matrix is\nunique.\n\\end{theorem}\n\\begin{proof}\nIf the required $\\cT$ exists then its characteristic function\n$\\Theta_{\\cT}(z)$ is the Blaschke product of order $N$ and of the\nform\n\\begin{equation}\n\\label{one} b(z)=e^{it}\\prod\\limits_{k=1}^m\\left(\\frac{z-z_k}{1-\\bar\nz_k z}\\right)^{l_k}\\,\\prod\\limits_{j=1}^{N-r}\\frac{z-v_j}{1-\\bar v_j\nz},\n\\end{equation}\nwith the given first $N-r+1$ Schur parameters\n$\\alpha_0(b),\\ldots,\\alpha_{N-r}(b)$. Our goal is to prove the\nuniqueness of such function $b$.\n\nAccording to the result of Schur \\cite{Schur} (see Section 4.2) the\nset of all Schur functions $b$ with given first $N-r+1$ Schur\nparameters is parametrized by \\begin{equation}\\label{shu}\nb(z)=\\frac{A(z)+zB^*(z)s(z)}{B(z)+zA^*(z)s(z)}\\,, \\end{equation} where $s(z)$\nis an arbitrary Schur function, and $A,B$ are polynomials of degree\nat most $N-r$. Since $b$ is the Blaschke product of order $N$, it is\nclear that so is $s(z)$, $\\deg s(z)=r-1$, and\n\\[\n{\\mathcal S}}   \\def\\cT{{\\mathcal T}}   \\def\\cU{{\\mathcal U} b=\\{\\alpha_0,\\ldots,\\alpha_{N-r},\\alpha_0(s),\\ldots,\n\\alpha_{r-1}(s)\\}.\n\\]\nLet us solve \\eqref{shu} for $s$:\n\\[\ns(z)=\\frac{A(z)-B(z)b(z)}{-zB^*(z)+zA^*(z)b(z)}\\,,\n\\]\nso $s(z)$ satisfies the Nevanlinna-Pick interpolation problem\n\\eqref{np1}, where $w^{(j)}_k$ are completely determined from the\ngiven nonzero $z_k$'s and $\\alpha_j$'s. By Lemma \\ref{np} there is\nat most one such $s(z)$, and the uniqueness of $b$ is proved.\n\\end{proof}\n\n\\begin{remark}\nSuppose that $z_1,\\ldots,z_m$ are distinct nonzero points in ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$,\nand $l_1+\\ldots+l_m=N$, so the only $\\alpha_0$ is prescribed. It is\nclear that $\\alpha_0$ is completely determined by the choice of\n$z_j$ and their multiplicities $l_j$:\n\\[\nb(z)=e^{it}\\prod\\limits_{k=1}^m\\left(\\frac{z-z_k}{1-\\bar z_k\nz}\\right)^{l_k}, \\quad\n\\alpha_0=b(0)=e^{it}\\prod_{j=1}^m(-z_k^{l_k}).\n\\]\nSo for all other $\\alpha_0$ the inverse problem has no solution.\n\\end{remark}\n\nIn the case when one of the eigenvalues is zero, all three\npossibilities (no solution, unique solution, and infinitely many\nsolutions) may occur for the inverse problem in question. For\ninstance, there is no solution at all as long as $z_1=0$,\n$\\alpha_0\\not=0$. Assume next, that $r=l_1=1$, $z_1=0$, and the\npoints $\\alpha_0,\\alpha_1,\\ldots,\\alpha_{N-1}$ are taken in ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$,\nwith the only restriction $\\alpha_0=0$, $\\alpha_1\\not=0$. The\nBlaschke products $b_\\gamma$ with the Schur parameters\n$\\{\\alpha_0,\\alpha_1,\\ldots,\\alpha_{N-1};\\gamma\\}$ and arbitrary\n$\\gamma\\in\\dT$ are of the form\n\\[\nb_\\gamma(z)=e^{it}z\\prod_{j=1}^{N-1}\\frac{z-v_j}{1-\\bar v_j z},\n\\]\nand the corresponding $N\\times N$ truncated CMV matrices\n$\\cT_\\gamma$ solve the problem.\n\nFinally, assume that except for the zero eigenvalue of multiplicity\n$k$ ($z_1=z_2=\\ldots=z_k=0$), a few more nonzero (and not\nnecessarily distinct) eigenvalues $\\lambda_1,\\ldots,\\lambda_r$ are\ngiven, as well as the points $\\alpha_0=\\ldots=\\alpha_{k-1}=0$,\n$\\alpha_k\\not=0,\\ldots,\\alpha_{N-r}$ in ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$. If the solution of\nthe corresponding mixed inverse problem $\\cT$ exists, its\ncharacteristic function takes the form\n\\[\nb(z)=e^{it}z^k\\prod\\limits_{j=1}^r\\frac{z-\\lambda_j}{1-\\bar\n\\lambda_j z}\\,g(z),\n\\]\nwhere $g$ is the Blaschke product of order $N-k-r$, $g(0)\\not=0$,\nand the first $N-k-r+1$ Schur parameters of $h=z^{-k}b$ are given\nnumbers $\\alpha_k,\\ldots,\\alpha_{N-r}$\\ . Clearly, $h$ is exactly\nthe $k^{th}$ Schur iterate of $b$. If the required truncated CMV\nmatrix $\\cT$ exists, then by Theorem \\ref{DER} the characteristic\nfunction of $\\cT^{(k)}$ agrees with $h$. It follows now from Theorem\n\\ref{mix1} that $\\cT^{(k)}$ is unique, and since $\\alpha_j(\\cT)=0$,\n$j=0,\\ldots,k-1$, the matrix $\\cT$ is unique as well.\n\nThe situation changes dramatically if we assume that the {\\it last}\nparameters of $\\cT$ \\eqref{NMODEL} are known. In this case we can\nprove the existence, but not the uniqueness of the solution.\n\n\\begin{theorem}\n\\label{mix2} Let $z_1, \\ldots, z_m$ and\n$\\alpha_m,\\ldots\\alpha_{N-1}$ be two collections of arbitrary\ncomplex number from the open unit disk, and let $\\alpha_N\\in\\dT$.\nThen there exists a $N\\times N$ truncated CMV matrix $\\cT$ of the\nform \\eqref{NMODEL} such that\n\\begin{enumerate}\n\\def\\rm (\\roman{enumi}){\\rm (\\roman{enumi})}\n\\item $z_1,\\ldots,z_m$ are eigenvalues of $\\cT$, counting the\nalgebraic multiplicity, \\item $\\alpha_n(\\cT)=\\alpha_n$,\n$n=m,m+1,\\ldots,N$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof} By Theorem \\ref{TTT} there exists a Blaschke product\n$b(z)$ of order $N$ such that $b(z_k)=0$, $k=1,\\ldots,m$, with the\nSchur parameters\n\\[\n\\alpha_n(b)=\\alpha_n, \\qquad n=m,m+1,\\ldots,N.\n\\]\nTake now the matrix $\\cT$ \\eqref{NMODEL} with\n$\\alpha_n(\\cT)=\\alpha_n$, $n=0,1,\\ldots,N$. By Theorem \\ref{char}\nthe characteristic function of $\\cT$ agrees with $b(z)$, that\ncompletes the proof.\n\\end{proof}\nTheorem 7.6 thereby says that a $N\\times N$ truncated CMV matrix\n$\\cT$ can be reconstructed from its $m$ eigenvalues and the lower\nprincipal block of order $N-m$. The latter is either the truncated\nCMV matrix $\\cT(\\{\\alpha_n\\}_{n=m}^N)$ or its transpose $\\widetilde\\cT$.\n\n\n\\subsection{Inverse problem for semi-infinite truncated CMV matrix}\n In this subsection we consider the criterion when given\n  complex numbers $z_n$, $n=1,2,\\ldots$ from ${\\mathbb D}}   \\def\\dE{{\\mathbb E}}   \\def\\dF{{\\mathbb F}$ are the\n  eigenvalues counting algebraic multiplicity of some semi-infinite\n  truncated CMV matrix.\n  \\begin{proposition}\n   Given complex numbers $z_n$, $n=1,2,...$ are eigenvalues counting\n   algebraic multiplicity of some semi-infinite truncated CMV matrix\n   if and only if\n \\[\n \\sum\\limits_{n=1}^{\\infty}(1-|z_n|)< \\infty.\n \\]\n \\end{proposition}\n\\begin{proof}\n The convergence of the sum is equivalent to the convergence of the Blashke\n product\n$$ b(z)=\\prod\\limits_{k=1}^{\\infty}\\frac{\\bar z_k}{z_k}\\,\\frac{z_k-z}{1-\\bar z_k\nz}.\n$$\n Let $\\{\\alpha_n\\}$ be the Schur parameters of $b$. The\ncharacteristic function of the truncated CMV matrix\n$\\cT(\\{\\alpha_n\\})$ agrees with $b$. Hence the eigenvalues of\n$\\cT(\\{\\alpha_n\\})$ are precisely the complex numbers $\\{z_n\\}$.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn many real world applications, data often emerges in the form of high-dimensional tensors.\nIt is typically very expensive to generate or collect such data, and we assume that we might be given a rather small amount of test and training data.\nNevertheless, it remains crucial to be able to classify tensorial data points.\nA prototypical example of this type is fMRI brain images~\\citep{GLOVER2011133}, which consist of three-dimensional tensors of voxels, and may also be equipped with an additional temporal dimension,\nin contrast to traditional two-dimensional pixel images.\n\nOne of the most popular methods for classifying data points are Support Vector Machines (SVM)~\\citep{vapnik,vapnik98}.\nThese are based on margin maximization and the computation of the corresponding weights via an optimization framework, typically the SMO algorithm~\\citep{Platt98sequentialminimal}.\nThese methods often show outstanding performance, but the standard SVM model~\\citep{vapnik95} is designed for vector-valued rather than tensor-valued data.\nAlthough tensor objects can be reshaped into vectors, much of the information inherent in the tensorial data is lost.\nFor example, in an fMRI image, the values of adjacent voxels are often close to each other~\\citep{DuSK}.\nAs a result, it was proposed to replace the vector-valued SVM by a tensor-valued SVM.\nThis area was called Supervised Tensor Learning (STL)~\\citep{Tao07,Zhou13,Guo12}.\nIn~\\citet{wolf}, the authors proposed to minimize the rank of the weight parameter with the orthogonality constraints on the columns of the weight parameter instead of the classical maximum-margin criterion, and~\\citet{Pirisiavash} relaxed the orthogonality constraints to further improve the Wolf\\textquotesingle s method.\n~\\citet{Hao13} consider a rank-one tensor factorization of each input tensor, while~\\citet{kotsia} adopted the Tucker decomposition of the weight parameter instead of rank-one tensor decompositions to retain a more structural information.\n~\\citet{Zeng17} extended this by using a Genetic Algorithm (GA) prior to the Support Tucker Machine (STuM) for the contraction of the input feature tensor.\nAlong with these rank-one and Tucker representations, recently the weight tensor of  STL has been approximated using the Tensor Train (TT) decomposition~\\citep{chen2018support}.\nWe point out that these methods are mainly focusing on a linear representation of the data.\nIt is well known that a linear decision boundary is often not suitable for the separation of complicated real world data~\\citep{MLbook1}.\n\nNaturally, the goal is to design a nonlinear transformation of the data, and we refer to~\\citet{Signoretto11,Signoretto12,QZhao13a}, where kernel methods have been used for tensor data.\nAll these methods are based on the Multi-linear Singular Value Decomposition\/Higher Order Singular Value Decomposition, which rely on the flattening of the tensor data.\nTherefore, the resulting vector and matrix dimensions are so high that the methods are prone to over-fitting.\nMoreover, the intrinsic tensor structure is typically lost. Thus, other approaches are desired.\n\nThe approximation of tensors based on low-rank decompositions has received a lot of attention in scientific computing over  recent years~\\citep{cichocki2016tensor,Kolda09,Cichocki13,Liu15}.\n\\kk{A Dual Structure-preserving Kernel (DuSK) for STL, which is\nparticularly tailored to SVM and tensor data, was introduced in~\\citep{DuSK}.\nThis kernel is defined on the Canonical Polyadic (CP) tensor format, also known as Parallel Factor Analysis, or PARAFAC~\\citep{Hitchcock1927,Hitchcock1928}.}\n\\kk{Once the CP format is available, DuSK delivers an accurate and efficient classification, but the CP approximation of arbitrary data can be numerically unstable and difficult to compute~\\citep{desilva2008}. In general, any optimization method (Newton, Steepest Descent or Alternating Least Squares) might return only a locally optimal solution, and it is difficult to assess whether this is a local or global optimum.}\nLater on, kernelized tensor factorizations, specifically a Kernelized-CP (KCP) factorization, have been introduced in~\\citet{MMK}, and the entire technique has been called the Multi-way Multi-level Kernel (MMK) method.\nFurther elaboration and understanding of the KCP approach~\\citep{KSTM} is provided by a kernelized Tucker model, inspired by~\\citet{Signoretto2013}.\n\nRecently, kernel approximations in the TT format have been introduced in~\\citet{chen2020kernelized}.\nInitially, we had pursued a similar idea for fMRI data sets, but\nwe observed that the nonlinear SVM classification\nusing directly the TT factors leads to poor accuracy,\nsince different TT factors have different dimensions and scales,\nmaking the feature space more complicated.\nHence, we have come up with a better exploitation of the data structure, as we explain in this paper.\n\nTensor decompositions and kernel-based methods have become an indispensable tool in many learning tasks.\nFor example,~\\citet{exponential} uses the TT decomposition for both the input tensor and the corresponding weight parameter in generalized linear models in machine learning.\nA Kernel Principal Component Analysis (KPCA), a kernel-based nonlinear feature extraction technique, was proposed in~\\citet{Wu07}.\nThe authors of~\\citet{speedingupCNN} propose a way to speed up Convolutional Neural Networks (CNN) by applying a low-rank CP decomposition on the kernel projection tensor.\n\\subsection{Main Novelty}\nIn this paper, we \\kk{develop an efficient structure-preserving nonlinear kernel function for SVM classification of tensorial data.\nWe start with the TT approximation of the data points, which can be computed reliably by the TT-SVD algorithm.\nMoreover, we enforce uniqueness of the SVD factors,\nsuch that ``close'' tensors yield ``close'' TT factors.\nSecond, we perform an exact expansion of the TT decomposition into the CP format.\nThis unifies the dimensions of the data used in classification.\nFinally, we redistribute the norms of the CP factors to equilibrate the actual scales of the data elements.\nThis yields  a CP decomposition that is free from scaling indeterminacy, while being a reliable approximation of the original data.\nWe have observed that using this decomposition in DuSK significantly increases the classification accuracy and stability of the STL.}\n\n\nThe paper is structured as follows.\nIn~\\secref{preliminaries}, we set the stage introducing basic definitions and important tools.\nAn extension to the tensor format SVM is explained in~\\secref{framektt},\nwhere we also introduce the Kernelized Support Tensor Machine (KSTM) via the kernel trick (\\secref{fmapkertrick}).\nIn~\\secref{overalgo} we explain the entire proposed algorithm step by step.\nIn particular, we introduce the uniqueness enforcing TT-SVD algorithm (\\secref{sec: USVD}), the TT-CP expansion (\\secref{TT-CP}) and the norm equilibration (\\secref{sec:normequ}).\nIn~\\secref{numel} we benchmark the different steps of the proposed algorithm and compare it to a variety of competing methods using two data sets each from two different fields with a limited amount of training data, which are known to be challenging for classification.\n\\section{Preliminaries} \\label{preliminaries}\nThis section introduces terminology and definitions used throughout the paper.\n\\subsection{Tensor Algebra}\nA tensor  is a multidimensional array~\\citep{Kolda09} which is a higher order generalization of vectors and matrices. We denote an $M^{th}$-order tensor ($M \\geq 3$) by a calligraphic letter \\tensx $\\in$ \\dims R I M, its entries by \\tenem x, a matrix by a boldface upper case letter \\bX $\\in$ \\RIJ, and a vector by a boldface lower case letter $\\bx \\in \\RI$. Matrix and vector elements are denoted by $x_{ij} = \\bX(i,j)$ and $x_i = \\bx(i)$, respectively. The order of a tensor is the number of its \\emph{dimensions}, \\emph{ways} or \\emph{modes}. The \\emph{size} of a tensor stands for the maximum index value in each mode. For example, \\tensx \\  is of order $M$ and the size in each mode is $I_m$, where $  m \\in \\expect M:=\\left\\lbrace 1,2, \\ldots , M \\right\\rbrace $. For simplicity, we assume that all tensors are real valued.\n\\begin{definition}{An $m$-mode matricization}\n        \\nten \\tensx m $\\in$ \\mdims m M for $m \\in  \\expect M$\n\tis the unfolding (or flattening) of an $M^{th}$-order tensor into a matrix in the appropriate order of elements\\kk{, \\ie~ a tensor element $(i_1, i_2, \\ldots i_M)$ maps to an element $(i_m,j)$ of a matrix as follows~\\citep{Kolda09}:}\n\t\\kk{$$ j = 1+ \\sum_{k = 1, k \\neq m}^{M} (i_k - 1) J_k ~~ \\text{with} ~~ J_k= \\prod_{\\ell = 1, \\ell \\neq m}^{k-1} I_\\ell.$$}\n\t \n\\end{definition}\n\\begin{definition}{An $m$-mode product $\\tensx \\nmod m \\bA  \\  \\in\n\t\\dimnmod R m$}, given\n\t\\tensx \\ $\\in$ \\dims R I M and $\\bA \\in$ \\matdim R, is defined as a tensor-matrix product in $m^\\text{th}$ way:\n\t$$ \\nten \\tensy m  = \\nten {(\\tensx \\nmod m  \\bA)} m = \\bA \\nten \\tensx m.$$\n\\end{definition}\n\\begin{definition}{A mode-($M$,1) contracted product $\\tenz   = \\tensx \\times_{M}^{1} \\tensy = \\tensx \\times^{1} \\tensy \\in$ \\modnprod \\R},\n\tfor given tensors $\\tensx \\in \\dims R I M$ and  $\\tensy \\in \\dims R J M$, with $I_M = J_1$, yields a tensor $\\tenz$ with entries  $$z_{i_1, \\ldots, i_{M-1}, j_2, \\ldots, j_M} = \\sum_{i_M = 1}^{I_M} x_{i_1,\\ldots,i_M} y_{i_M, j_2, \\ldots, j_M}.$$\n\\end{definition}\n\\begin{definition}{The inner product}\n\tof given tensors \\tensx, \\tensy \\ $\\in$ \\dims R I M is defined as\n\t$$\\langle \\tensx , \\tensy \\rangle = \\teninner.$$\n\\end{definition}\n\\begin{definition}{The outer product}\n\tof given tensors $\\tensx \\in \\dims R I M  \\text{and} ~ \\tensy \\in \\dims R J N $ generates an $\\brac{M + N}^{th}-$ order tensor $\\tenz = \\tensx \\circ \\tensy$  with entries  $$z_{i_1, \\ldots, i_{M}, j_1, \\ldots, j_N} =  x_{i_1,\\ldots,i_M} y_{j_1, \\ldots, j_N}.$$\n\\end{definition}\n\\begin{definition}{The Kronecker Product}\n\tof matrices $\\bA \\in \\RIJ, \\bB \\in \\RKL$ is defined as usual by\n\t$$\n\t\\bA \\otimes \\bB =  \\begin{bmatrix}\n\ta_{1,1} \\bB & \\cdots & a_{1,J} \\bB\\\\\n\t\\vdots & \\ddots & \\vdots\\\\\n\ta_{I,1} \\bB & \\cdots & a_{I,J} \\bB\\\\\n\t\\end{bmatrix} \\in \\RIJKL.\n\t$$\n\tSimilarly, the Kronecker product of two tensors $\\tensx \\in \\dims R I M, \\tensy \\in \\dims R J M$ returns a tensor $\\tenz = \\tensx \\otimes \\tensy \\in \\krodimten R I J$.\n\\end{definition}\nMoreover, the Khatri-Rao product is a column-wise Kronecker product,\n$$\\bA \\odot \\bB = [\\ba_1  \\otimes \\bb_1, \\ba_2 \\otimes \\bb_2, \\cdots, \\ba_R \\otimes \\bb_R ] \\in  \\RIKR.$$\nThese notations are summarized in Table~\\ref{notation}.\n\\begin{table}[ht]\n\n\t\\caption{Tensor Notations.}\n\t\\label{notation}\n\t\\vskip 0.15in\n\t\\begin{center}\n\t\t\\begin{small}\n\t\n\t\t\t\t\\begin{tabular}{lcr}\n\t\t\t\t\t\\toprule\n\t\t\t\t\tSymbol & Description\\\\\n\t\t\t\t\t\\midrule\n\t\t\t\t\t$x$    & Lower case letter for scalar value\\\\\n\t\t\t\t\t\\bx  & Lower case bold letter for vector\\\\\n\t\t\t\t\t\\bX    & Upper case bold letter for matrix\\\\\n\t\t\t\t\t\\tensx    & Calligraphic bold letter for tensor\\\\\n\t\t\t\t\t\\nten \\tensx m    & Calligraphic bold letter with subscript m \\\\\n\t\t\t\t\t& for $m$-mode matricization\\\\\n\t\t\t\t\t$\\circ$   & Outer product\\\\             \n\t\t\t\t\t$\\otimes$      & Kronecker product\\\\\n\t\t\t\t\t$\\odot$      & Khatri-Rao product\\\\\n\t\t\t\t\t$\\times^{1}_{M}$ & Mode-$(M,1)$ contracted product\\\\\n\t\t\t\t\t$\\langle M \\rangle$ & Integer values from 1 to $M$\\\\\n\t\t\t\t\t$\\langle \\tensx, \\tensy \\rangle$      & Inner product for tensors \\tensx \\ and \\tensy \\\\\n\t\t\t\t\t\\bottomrule\n\t\t\t\t\\end{tabular}\n\t\n\t\t\\end{small}\n\t\\end{center}\n\t\\vskip -0.1in\n\\end{table}\n\\subsection{Tensor Decompositions}\nTensor decomposition methods have been significantly enhanced during the last two decades, and applied\nto solve problems of varying computational complexity.\nThe main goal is the linear (or at most polynomial) scaling of the computational complexity in the dimension (order) of a tensor.\nThe key ingredient is the separation of variables via approximate low-rank factorizations.\nIn this paper we consider two of these decompositions.\n\\subsubsection{Canonical Polyadic decomposition}\nThe Canonical Polyadic (CP) decomposition of an $M^{th}-$order tensor $\\tensx \\in \\dims R I M$ is a factorization into a sum of rank-one components~\\citep{Hitchcock1927}, which is given element-wise as\n\\begin{align}\\label{cpkrush}\n\\tenem x & \\cong \\sum_{r=1}^{R} \\ba^{(1)}_{i_1,r} \\ba^{(2)}_{i_2,r} \\cdots \\ba^{(M)}_{i_M,r},  \\nonumber\\\\\n \\mbox{or shortly,} ~~\\qquad \\quad \\quad \\tensx & \\cong \\llbracket \\cpd \\rrbracket,\n\\end{align}\nwhere  $\\mathbf{A}^{(m)} = \\left[ \\ba^{(m)}_{i_m,r} \\right] \\in \\R^{I_m \\times R}$, $m=1,\\ldots,M$, are called \\textit{factor matrices} of the CP decomposition, see~\\figref{figcp}, and $R$ is called the CP-rank. The notation $\\llbracket \\cpd \\rrbracket$ is also called the Kruskal representation of the CP factorization.\nDespite the simplicity of the CP format, the problem of the\nbest CP approximation is often ill-posed~\\citep{desilva2008}.\nA practical CP approximation can be computed via the Alternating Least Squares (ALS) method~\\citep{cp_als}, but the convergence may be slow.\nIt may also be difficult to choose the rank $R$.\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\includegraphics[scale = 1.12]{cp_tensor.pdf}\n\t\t\\caption{CP decomposition of a 3-way tensor.}\n\t\t\\label{figcp}\n\t\\end{center}\n\t\\vskip -0.2in\n\\end{figure}\n\\subsubsection{Tensor Train decomposition}\nTo alleviate the difficulties of the CP decomposition mentioned above, we build\nour proposed algorithm on\nthe Tensor Train (TT)~\\citep{oseledets2011tensor} decomposition.\nThe TT approximation of an $M^{th}-$order tensor  $\\tensx \\in \\dims R I M$ is defined element-wise as\n\\begin{align}\\label{eq:tt}\n\\tenem x & \\cong \\sum_{r_0,\\ldots,r_{M}}\\tendtt G 1_{r_0,i_1,r_1} \\tendtt G 2_{r_1,i_2,r_2} \\cdots \\tendtt G M_{r_{M-1},i_M,r_M}, \\nonumber\\\\\n\\tensx & \\cong\\llangle \\tendtt G 1, \\tendtt G 2, \\ldots, \\tendtt G M \\rrangle,\n\\end{align}\nwhere \\tendtt G m $\\in$ \\ttcdim m, $m=1,\\ldots,M,$ are 3rd-order tensors called \\emph{TT-cores} (see~\\figref{figtt}), and $R_0,\\ldots,R_M$ with $R_0 = R_M = 1$ are called \\emph{TT-ranks}.\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\includegraphics[scale = 0.05]{TT_dec.pdf}\n\t\t\\caption{TT decomposition of a 3-way tensor.}\n\t\t\\label{figtt}\n\t\\end{center}   \n\t\\vskip -0.2in\n\\end{figure}\nThe alluring capability of the TT format is its ability to perform algebraic operations directly on TT-cores avoiding full tensors.\nMoreover, we can compute a quasi-optimal TT approximation of any given tensor using the SVD.\nThis builds on the fact that the TT decomposition constitutes a recursive matrix factorization, where each TT-rank is the matrix rank of the appropriate unfolding of the tensor, and hence the TT approximation problem is well-posed~\\citep{oseledets2011tensor}.\n\\subsection{Support Vector Machine}\\label{svm}\nIn this section, we recall the SVM method.\nFor a given training data set $\\lbrace \\left( \\bx_i, y_i\\right) \\rbrace _ {i = 1}^N$, with \\emph{input data} $\\bx_i \\in \\R^m$ and \\emph{labels} $y_i \\in \\lbrace-1,1\\rbrace$,  the dual-optimization problem for the \\textit{nonlinear} binary classification can be defined as,\n\\begin{align}\\label{eq: dualPhiOpt}\n&\\underset{\\alpha_1,\\ldots,\\alpha_N}{\\text{max}}  \\quad \\sum_{i = 1}^{N} \\balpha_i - \\frac{1}{2} \\sum_{i = 1}^{N}\\sum_{j = 1}^{N} \\balpha_i \\balpha_j y_i y_j \\langle \\phi {(\\bx_i)},\\phi {(\\bx_j)} \\rangle \\nonumber \\\\\n&  \\text{subject to}   \\quad  0 \\leq \\balpha_i \\leq C, \\quad  \\sum_{i = 1}^{N} \\balpha_i y_i = 0,\n\\end{align}\nwhere a tuning function $\\phi$ defines the nonlinear decision boundary with $\\phi\\colon  \\bx_i \\rightarrow \\phi\\left( \\bx_i\\right)$.\nIn practice, we compute directly\n$\\langle \\phi \\brac{\\bx_i},\\phi \\brac{\\bx_j} \\rangle$ using the so-called \\emph{Kernel Trick}~\\citep{bernhard2001}.\n\\subsubsection{Feature Map and Kernel Trick}\\label{fmapkertrick}\nThe function $\\phi \\colon \\R^m \\rightarrow \\mathbb{F}$ is called\n\\textbf{feature map}, and the \\textit{feature space} $\\mathbb{F}$ is a Hilbert Space (HS).\nEvery feature map is defined via a kernel such that $\\nk_{i,j}=\\nk \\brac {\\bx_i, \\bx_j} = \\langle \\phi(\\bx_i), \\phi(\\bx_j) \\rangle_{\\mathbb{F}} $. Employing the properties of the inner product, we conclude that $[\\nk_{i,j}]$ is a symmetric positive semi-definite matrix.\nThe \\emph{kernel trick} lies in\ndefining and computing directly $\\nk\\brac{\\bx_i,\\bx_j}$ instead of $\\phi(\\bx)$.\nIt is used to get a linear learning algorithm to learn a  \\emph{nonlinear boundary}, without explicitly knowing the nonlinear function $\\phi$.\nThe only task needed for the SVM is thus to choose a legitimate kernel function. That is how we work with the input data in the high-dimensional space while  doing all the computation in the original low dimensional space.~\\figref{kertrick} illustrates the linear separation in a higher dimensional space.\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\centerline{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\node at (-4,0) {\\includegraphics[scale= 0.15]{.\/twodim.jpg}};\n\t\t\t\\node at (4,0) {\\includegraphics[scale=0.15]{.\/threedim.jpg}};\n\t\t\t\\node at (0, 0.2) {$\\nk(x,y)$};\n\t\t\t\\node at (-4,-2.2) {\\scalebox{0.5}{\\brac{a}}};\n\t\t\t\\node at (4,-2.2) {\\scalebox{0.5}{ \\brac{b}}};\n\t\t\t\\draw[->]  (-1,0) -- (1,0);\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\t\\caption{Nonlinear mapping using kernel trick: \\brac{a} Nonlinear classification of data in $\\R^2$, \\brac{b} Linear classification in higher dimension ($\\R^3$).}\n\t\t\\label{kertrick}\n\t\\end{center}\n\t\\vskip -0.2in\n\\end{figure}\n\\subsection{Kernelized Support Tensor Machine}\\label{framektt}\nIn our case, we have a data set $\\lbrace \\left( \\tensx_i, y_i\\right) \\rbrace _ {i = 1}^N$ with input data in the form of a tensor $\\tensx_i \\in \\dims R I M$.\n\\kk{ We take the maximum margin approach to get the separation hyperplane. Hence, the objective function for a nonlinear boundary in the tensor space can be written as follows~\\citep{Cai06supporttensor}:}\n\\kk{\\begin{align}\\label {eq: tenPrimOpt}\n\t&\\underset{w,b}{\\text{min}} \\quad \\frac{1}{2} \\norm{w}^2 + C \\sum_{i=1}^{N} \\xi_i \\\\  \\nonumber\n\t  \\text{subject to} \\quad & y_i (\\langle \\Psi(\\tensx_{i}), w \\rangle + b) \\geq 1-\\xi_{i} ~~ \\quad \\xi_{i}~\\geq 0 ~~~ \\forall i.\t\n\\end{align}}\n\\kk{The classification setup given in~\\eqref{eq: tenPrimOpt} is known as Support Tensor Machine (STM)~\\citep{Tao05}. The dual formulation of the corresponding primal problem can be given as follows:}\n\\begin{align}\\label {eq: tenDualOpt}\n&\\underset{\\alpha_1,\\ldots,\\alpha_{N}}{\\text{max}} \\quad  \\sum_{i = 1}^{N} \\balpha_i - \\frac{1}{2} \\sum_{i = 1}^{N}\\sum_{j = 1}^{N} \\balpha_i \\balpha_j y_i y_j \\langle \\Psi(\\tensx_i),\\Psi(\\tensx_j) \\rangle \\nonumber\\\\\n&  \\text{subject to} \\quad 0 \\leq \\balpha_i \\leq C,  \\quad \\sum_{i = 1}^{N} \\balpha_i y_i = 0 ~~ \\forall i. \n\\end{align}\nThe nonlinear feature mapping $\\Psi \\colon \\dims R I M \\rightarrow \\mathbb{F}$ takes tensorial input data to a higher dimensional space similarly to the vector case. Therefore, by using the kernel trick, explained in~\\secref{fmapkertrick}, STM can be defined as follows:\n\\begin{align} \\label{eq: kerSTM}\n&\\underset{\\alpha_1,\\ldots,\\alpha_{N}}{\\text{max}} \\quad  \\sum_{i = 1}^{N} \\balpha_i - \\frac{1}{2} \\sum_{i = 1}^{N}\\sum_{j = 1}^{N} \\balpha_i \\balpha_j y_i y_j K(\\tensx_i,\\tensx_j) \\nonumber \\\\\n&  \\text{subject to} \\quad 0 \\leq \\balpha_i \\leq C, \\quad  \\sum_{i = 1}^{N} \\balpha_i y_i = 0 ~~ \\forall i.\n\\end{align}\nWe call this setup the \\emph{Kernelized STM (KSTM)}. \\kk{Once we have the real-valued function (kernel) value for each pair of tensors, we can use state-of-the-art LIBSVM~\\citep{libsvm}, which relies on the \\emph{Sequential Minimal Optimization} algorithm to optimize the weights $\\alpha_i$.} Hence, the preeminent part is the kernel function $K(\\tensx_i,\\tensx_j)$.\n\\kk{However, the direct treatment of large tensors can be both numerically expensive and inaccurate due to overfitting.\nTherefore, we need to choose a kernel that exploits the tensor decomposition.}\nIn the next section we propose a particular choice of the kernel for tensor data.\n\\section{The Proposed Algorithm}\\label{overalgo}\n\nThe first essential step towards using tensors is to approximate them in a low-parametric representation.\nTo achieve a stable learning model, we start with computing the TT approximations of all data tensors.\nThe second most expensive part is the computation of $K\\brac{\\tensx_{i}, \\tensx_{j}}$ for each pair of tensors.\nTherefore, an approximation of the kernel is required.\nBesides, we would like the kernel to exploit the factorized tensor representation.\nThese issues are resolved in the rest of this section.\n\\subsection{Uniqueness Enforcing TT- SVD }\\label{sec: USVD}\nSince the TT decomposition is computed using the SVD~\\citep{oseledets2011tensor}, the particular factors $\\teng^{(1)},\\teng^{(2)},\\ldots, \\teng^{(M)}$ are defined only up to a sign indeterminacy.\nFor example, in the first step, we compute the SVD of the $1$-mode matricization,\n\\[\n\\tensx_{(1)}  = \\sigma_1 u_1 v_1^\\top + \\cdots + \\sigma_{I_1} u_{I_1} v_{I_1}^\\top,\n\\]\nfollowed by truncating the expansion at rank $R_1$ or according to the accuracy threshold $\\varepsilon$, choosing $R_1$ such that $\\sigma_{R_1+1}<\\varepsilon$.\nHowever, any pair of vectors $\\{u_{r_1}, v_{r_1}\\}$ can be replaced by $\\{-u_{r_1}, -v_{r_1}\\}$ without changing the whole expansion.\nWhile this is not an issue for data compression,\nclassification using TT factors can be affected significantly by this indeterminacy.\nFor example, tensors that are close to each other should likely produce the same label.\nIn contrast, even a small difference in the original data may lead to a different sign of the singular vectors.\n\\kk{Such a qualitative difference in the data which is actually used for classification\nmay significantly inflate the dimension of the feature space $\\mathbb{F}$ in~\\eqref{eq: tenDualOpt} required for the accurate separation of the classes.}\n\n\\kk{We fix the signs of the singular vectors as follows.\nFor each $r_1=1,\\ldots,R_1$, we find the position of the maximum in modulus element in the left singular vector, $i_{r_1}^* = \\arg\\max_{i=1,\\ldots,I_1} |u_{i,r_1}|$,\nand make this element positive,}\n\\[\n\\kk{\\bar u_{r_1}:=u_{r_1} \/ \\mathrm{sign}(u_{i^*_{r_1},r_1}), \\quad \\bar v_{r_1}:= v_{r_1} \\cdot \\mathrm{sign}(u_{i^*_{r_1},r_1}).}\n\\]\n\n\\kk{Finally, we collect $\\bar u_{r_1}$ into the first TT core, $\\teng^{(1)}_{r_0,i_1,r_1} = \\bar u_{i_1,r_1}$, and continue with the TT-SVD algorithm using $\\bar v_{r_1}$ as the right singular vectors.\nIn contrast to the sign, the whole dominant singular terms $u_{r_1}v_{r_1}^\\top$ depend continuously on the input data, and so do the maximum absolute elements. The procedure is summarized in Algorithm~\\ref{UETTsvd}.}\n\n\n\\begin{algorithm}[ht]\n\t\\caption{\\kk{Uniqueness Enforcing TT-SVD}}\n\t\\label{UETTsvd}\n\t\\begin{algorithmic}[1]\n\t\t\\small\t\t\n\t\t\\STATE {\\bfseries Input:} $M$-dimensional tensor $\\tensx \\in \\dims R I M$, relative error threshold $\\epsilon$.\n\t\t\\STATE {\\bfseries Ensure:} Cores $\\ttbra$ of the TT-approximation $\\tensx'$ to $\\tensx$ in the TT-format with\n\t\tTT- rounding ranks $r_m$ equal to the $\\delta$-ranks of the unfoldings $\\tensx_{(m)}$ of $\\tensx$, where $\\delta = \\sqrt{\\frac{\\epsilon}{M-1}} \\norm{\\bA}_F$.\n\t\t\\STATE Initialize $\\hat \\bZ_1 = \\tensx_{(1)}, R_0 = 1$.\n\t\t\\FOR { $m = 1$ to $M-1$}\n\t\t\\STATE $\\bZ_m : = \\text{reshape} \\left(\\hat \\bZ_m, [R_{m-1} I_m,~I_{m+1}\\cdots I_M]\\right)$\n\t\t\\STATE Compute $\\delta$-truncated SVD: $\\bZ_m = \\bU_m \\bS_m \\bV_m^T + \\mathbf{E}_m, ~\\norm{\\mathbf{E}_m}_F \\leq \\delta$, where \\\\\n\t\t$\\bU_m = [u_1^{(m)}, u_2^{(m)}, \\ldots, u_{R_m}^{(m)}],\n\t\t\\bS_m = \\text{diag}(\\sigma_1^{(m)}, \\sigma_2^{(m)}, \\ldots, \\sigma_{R_m}^{(m)}), \\bV_m = [v_1^{(m)}, v_2^{(m)}, \\ldots, v_{R_m}^{(m)}]$\n\t\t\\FOR {$r_m = 1$ to $R_m$}\n\t\t\\STATE $i_{r_m}^* = \\arg\\max_{i=1,\\ldots,R_{m-1}I_m} |u_{i,r_m}^{(m)}|$ (with ties broken to first element) \\label{abs}\n\t\t\\STATE $\\bar u_{r_m}^{(m)}:=u_{r_m}^{(m)} \/ \\mathrm{sign}(u_{i_{r_m}^*,r_m}^{(m)}), \\quad \\bar v_{r_m}^{(m)}:= v_{r_m}^{(m)} \\cdot \\mathrm{sign}(u_{i^*_{r_m},r_m}^{(m)})$\n\t\t\\STATE $\\teng^{(m)}_{r_{m-1},i_m,r_m} = \\bar u_{r_{m-1}+(I_m-1)R_{m-1},~r_m}^{(m)}$, \\quad $\\bar \\bV_m = [\\bar v_1^{(m)}, \\bar v_2^{(m)}, \\ldots, \\bar v_{R_m}^{(m)}]$ \\label{ttcore}\n\t\t\\ENDFOR\n\t\t\\STATE $\\hat \\bZ_{m+1} := \\bS_m \\bar \\bV_m^T$\n\t\t\\ENDFOR \n\t\t\\STATE $\\teng^{(M)} = \\hat \\bZ_M$\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\\begin{lemma}\\label{lem}\n\\kk{Assume that the singular values $\\sigma_1^{(m)}, \\ldots, \\sigma_{R_m}^{(m)}$ are simple for each $m=1,\\ldots,M-1$.\nThen Algorithm~\\ref{UETTsvd} produces the unique TT decomposition.}\n\\end{lemma}\n\\begin{proof}\nThe $m$-th TT core produced in TT-SVD is a reshape of the left singular vectors of the Gram matrix of the current unfolding,\n$\n\\bA_m := \\bZ_m \\bZ_m^\\top.\n$\nTo set up an induction, we notice that $\\bZ_1 = \\hat \\bZ_1 = \\tensx_{(1)}$ is unique,\nand assume that $\\bZ_m$ is unique too.\nConsider the eigenvalue decomposition\n$\\bA_m \\bU_m = \\bU_m \\Lambda_m$,\n$\\Lambda_m = \\text{diag}(\\lambda_1^{(m)}, \\ldots, \\lambda_{R_{m-1}I_m}^{(m)})$.\nSince the eigenvalues $\\lambda_{i}^{(m)} = (\\sigma_{i}^{(m)})^2$ are simple,\neach of them corresponds to an eigenspace of dimension $1$, spanned by the corresponding column of $\\bU_m$.\nThis means that each eigenvector is unique up to a scalar factor, and, if the eigenvector is real and has Euclidean norm, the scalar factor can only be $1$ or $-1$.\nThe latter is unique if we choose it as the sign of the largest in modulus element of the eigenvector (which is always nonzero), with ties broken to take the first of identical elements.\nIt remains to establish the uniqueness of $\\bZ_{m+1}$ to complete the induction.\nBy the orthogonality of $\\bar\\bU_m=[\\bar u_1^{(m)}, \\ldots, \\bar u_{R_m}^{(m)}]$,\nwe get $\\hat \\bZ_{m+1} = \\bar\\bU_{m}^T \\bZ_{m}$, and since the reshape is unique, so is $\\bZ_{m+1}$.\n\\end{proof}\n\n\n\\begin{remark}\n\\kk{Most of the data featuring in machine learning are noisy.\nTherefore, the singular values of the corresponding matricizations are simple almost surely, and hence the TT decomposition delivered by Algorithm~\\ref{UETTsvd} is unique almost surely.}\n\\end{remark}\n\n\t\n\n\n\n\n\\subsection{TT-CP Expansion}\\label{TT-CP}\nDespite the difficulties in \\emph{computing} a CP approximation, its simplicity makes the CP format a convenient and powerful tool for revealing hidden classification features in the input data.\nHowever, as long as the TT decomposition is available, it can be\nconverted into the CP format suitable for the kernelized classification.\n\n\n\\begin{proposition}\\label{lem2}\n\\kk{For a given TT decomposition~\\eqref{eq:tt}, we can write a CP decomposition}\n\\begin{equation}\\label{eq:TT-CP}\n\\sum_{r_0,\\ldots,r_{M}}\\tendtt G 1_{r_0,i_1,r_1} \\tendtt G 2_{r_1,i_2,r_2} \\cdots \\tendtt G M_{r_{M-1},i_M,r_M} =  \\sum_{r=1}^{R} \\hat H^{(1)}_{i_1,r} \\hat H^{(2)}_{i_2,r} \\cdots \\hat H^{(M)}_{i_M,r}\n\\end{equation}\n\\kk{by merging the ranks $r_1, r_2, \\ldots r_M$ into one index $r = r_1 + (r_2-1) R_1 + \\ldots + (r_M-1) \\prod_{\\ell=1}^{M-1} R_{\\ell}$, $r=1,\\ldots,R$, $R=R_1\\cdots R_M$, and introducing the CP factors}\n$$\n\\kk{\\hat H^{(m)}_{i_m,r}  = {\\teng^{(m)}_{r_{m-1},i_m,r_m}}, \\quad m = 1, \\ldots, M.}$$\n\\end{proposition}\n\n\\kk{This transformation is free from any new computations, and needs simply rearranging and replicating the original TT cores. Although this expansion is valid for arbitrary dimension, higher dimensions may increase the number of terms massively.\nHowever, many experimental datasets are usually three or four dimensional tensors, for which the TT-CP expansion is feasible.}\n\n\n\\kk{Note that the number of terms $R$ in the CP decomposition~\\eqref{eq:TT-CP} can be larger than the minimal CP rank of the exact CP decomposition of the given tensor.\nHowever, the nonlinear kernel function is more sensitive to the features of the data rather than the number of CP terms \\emph{per se}.\nIn the numerical tests, we observe that the expansion~\\eqref{eq:TT-CP} gives actually a better classification accuracy than an attempt to compute an optimal CP approximation using an ALS method.\n}\n\n\n\\subsection{Norm Equilibration}\\label{sec:normequ}\nIn our preliminary experiments, we tried using directly the TT-CP expansion as above with the CP kernel from~\\citep{MMK}.\nHowever, this did not lead to better classification results.\nThe DuSK kernel~\\citep{MMK}\nintroduces the same width parameter for all CP factors.\nThis requires all CP factors to have identical (or at least close) magnitudes.\nIn contrast, different TT cores have different norms in the plain TT-SVD algorithm~\\citep{oseledets2011tensor}.\nHere, we rescale the TT-CP expansion\nto ensure that the columns of the CP factors have equal norms,\nand hence produce the same kernel features with the same width parameter.\nWe have found this to be a key ingredient for the successful TT-SVM classification.\n\nGiven a rank-$r$ TT-CP decomposition $\\llbracket \\hat H^{(1)}, \\hat H^{(2)},\\cdots, \\hat H^{(M)}\\rrbracket$,\nwe compute the total norm of each of the rank-1 tensors\n\\begin{equation}\n\\kk{n_r  = \\norm{\\hat H_{r}^{(1)}} \\cdots \\norm{\\hat H_{r}^{(M)}},\\label{norm1}}\n\\end{equation}\nand distribute this norm equally among the factors,\n\\begin{equation}\n\\kk{H^{(m)}_{r} :=  \\frac{\\hat H_{r}^{(m)}}{\\norm{\\hat H_{r}^{(m)}}} \\cdot n_r^{1\/M}, \\qquad m=1,2, \\cdots, M.\\label{norm2}}\n\\end{equation}\n\n\n\\subsection{Noise-aware Threshold and Rank Selection}\nGenerally, data coming from real world applications are affected by measurement or preprocessing noise.\nThis can affect both computational and modeling aspects,\nincreasing the TT ranks (since a tensor of noise lacks any meaningful TT decomposition), and spoiling the classification if the noise is too large.\nHowever, the SVD can serve as a de-noising algorithm automatically:\nthe dominant singular vectors are often ``smooth'' and hence represent a useful signal, while the latter singular vectors\nare more oscillating and capture primarily the noise.\nTherefore, it is actually beneficial to compute the TT approximation with deliberately low TT ranks \/ large truncation threshold.\nOn the other hand, the TT rank must not be too low in order to approximate the features of the tensor with sufficient accuracy.\nCross-validation is a technique to evaluate the effectiveness of the model, which is done by re-sampling the data into training-testing data sets. Since the precise magnitude of the noise is unknown,\nwe carry out a k-fold cross-validation test (k = 5) to find the optimal TT rank. \n\n\n\\subsection{Nonlinear Mapping}\\label{kermapping}\nEquipped with the homogenized TT-CP decompositions of the input tensors,\nwe are ready to define a nonlinear kernel function.\nWe follow closely the rationale behind DuSK proposed in~\\citet{DuSK,MMK} \\kk{and generalize it for tensors of arbitrary dimension}.\nWe assume that the feature map function \\kk{from the space of tensors to a \\emph{tensor product Reproducing Kernel Hilbert Space}~\\citep{Signoretto2013} $\\Psi  \\colon \\mathbb{R}^{I_1} \\times \\cdots \\times \\mathbb{R}^{I_M} \\mapsto \\mathbb{F}$} consists of separate feature maps acting on different CP factors,\n\\kk{\n\\begin{align}\\label{eq:NM}\n\\Psi & \\colon\\hspace{-0.1cm}\\sum_{r = 1}^{R} H_r^{(1)}\\otimes H_r^{(2)}\\otimes \\cdots \\otimes H_r^{(M)}\\mapsto\\sum_{r = 1}^{R} \\phi(H_r^{(1)})\\otimes \\phi(H_r^{(2)})\\otimes \\cdots \\otimes  \\phi(H_r^{(M)}).\n\\end{align}}\nThis allows us to exploit the fact that the data is given in the CP format to aid the classification.\nHowever, the feature function $\\phi(\\ba)$ is to be defined implicitly through a kernel function.\nSimilarly to the standard SVM, applying the kernel trick to~\\eqref{eq:NM} gives us a practically computable kernel.\nGiven CP approximations of two tensors $\\tensx = [x_{i_1,\\ldots,i_M}]$ and $\\tensy = [x_{i_1,\\ldots,i_M}]$,\n\\begin{align*}\nx_{i_1,\\ldots,i_M} \\approx \\sum_{r=1}^{R} H^{(1)}_{i_1,r} H^{(2)}_{i_2,r} \\cdots  H^{(M)}_{i_M,r}, \\qquad\ny_{i_1,\\ldots,i_M} \\approx \\sum_{r=1}^{R} P^{(1)}_{i_1,r} P^{(2)}_{i_2,r} \\cdots  P^{(M)}_{i_M,r},\n\\end{align*}\nwe compute\n\\kk{\n\\begin{align}\\label{eq:kerAPRX}\n\\langle \\Psi(\\tensx), \\Psi(\\tensy) \\rangle & =  K(\\tensx,\\tensy)\\nonumber\\\\\n& = K \\left( \\sum_{r=1}^{R} H_r^{(1)}\\otimes H_r^{(2)}\\otimes \\cdots \\otimes H_r^{(M)}, \\sum_{r=1}^{R} P_r^{(1)}\\otimes P_r^{(2)}\\otimes \\cdots \\otimes P_r^{(M)}  \\right),\\nonumber\\\\\n& = \\langle{\\Psi(\\sum_{r=1}^{R} H_r^{(1)}\\otimes H_r^{(2)}\\otimes \\cdots \\otimes H_r^{(M)}) ,\\Psi(\\sum_{r=1}^{R} P_r^{(1)}\\otimes P_r^{(2)}\\otimes \\cdots \\otimes  P_r^{(M)})}\\rangle \\nonumber\\\\\n& = \\sum_{i,j=1}^{R}\\langle\\phi(H_{i}^{(1)}),\\phi(P_{j}^{(1)})\\rangle \\langle\\phi(H_{i}^{(2)}),\\phi(P_{j}^{(2)})\\rangle \\cdots  \\langle\\phi(H_{i}^{(M)}),\\phi(P_{j}^{(M)})\\rangle \\nonumber\\\\\n& = \\sum_{i,j = 1}^{R}\\nk(H_{i}^{(1)},P_{j}^{(1)}) \\nk(H_{i}^{(2)},P_{j}^{(2)}) \\cdots \\nk(H_{i}^{(M)},P_{j}^{(M)}),\n\\end{align}}\nwhere \\begin{center}\n $ \\quad \\nk(\\bh,\\bp)  =  \\exp\\left(-\\frac{\\norm{{\\bh - \\bp}}^2}{2 \\sigma^2}\\right).$\n\\end{center}\n\nThis kernel approximation is computed for each pair of the tensor input data, represented by its CP factors.\nThe width parameter $\\sigma>0$ needs to be chosen judiciously to ensure accurate learning.\n\nSince the entire calculation starts from the TT decomposition,\nwe call this proposed model the \\emph{Tensor Train Multi-way Multi-level Kernel (TT-MMK)}.\nIt fulfills the objectives of extracting optimal low-rank features, and of building a more accurate and efficient classification model.\nPlugging the kernel values~\\eqref{eq:kerAPRX} into the STM optimizer \\eqref{eq: kerSTM} completes the algorithm.\nThe overall idea is summarized in Algorithm~\\ref{alg:KTTCP}.\n\n\\begin{algorithm}[ht]\n\t\\caption{TT-CP approximation of the STM Kernel}\n\t\\label{alg:KTTCP}\n\t\\begin{algorithmic}\n\t\t\\STATE {\\bfseries Input:} data ${\\{\\tensx_n\\}}_{n = 1}^{N}  \\dims R I M$, TT-rank $R$. \n\t\t\\STATE {\\bfseries Output:} Kernel matrix approximation $\\left[K(\\tensx_u,\\tensx_v)\\right] \\in \\R^{N\\times N}$\n\t\t\\FOR{$n = 1$ {\\bfseries to} $N$}\n\t\t\\STATE Compute TT approximation $\\tensx_n  \\cong \\llangle \\teng^{(1,n)}, \\teng^{(2,n)}, \\cdots,  \\teng^{(M,n)} \\rrangle$ using Algorithm~\\ref{UETTsvd}.\n\t\t\\STATE Compute TT-CP expansion $\\llbracket H^{(1,n)}, H^{(2,n)}, \\cdots, H^{(M,n)}\\rrbracket = \\llangle \\teng^{(1,n)}, \\teng^{(2,n)},\\cdots,  \\teng^{(M,n)} \\rrangle$ with equilibrated norms.\n\t\t\\ENDFOR\n\t\t\\FOR{$u, v = 1$ {\\bfseries to} $N$}\n\t\t\\STATE $K\\left(\\tensx_u,\\tensx_v\\right) \\approx$ \n\t\t$\\sum_{i,j = 1}^{R}\\nk(H_{i}^{(1,u)},H_{j}^{(1,v)}) \\nk(H_{i}^{(2,u)},H_{j}^{(2,v)}) \\cdots \\nk(H_{i}^{(M,u)},H_{j}^{(M,v)}).$\n\t\t\\ENDFOR\t\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Numerical Tests}\\label{numel}\n\\begin{itemize}\\setlength\\itemsep{0.5em}\n\t\\item \\textbf{Experimental Settings}\\\\\n\tAll numerical experiments have been done in \\texttt{MATLAB 2016b}. In the first step, we compute the TT format of an input tensor using the \\texttt{TT-Toolbox\\footnote{\\url{https:\/\/github.com\/oseledets\/TT-Toolbox}}}, where we modified the function \\verb+@tt_tensor\/round.m+ to enforce the uniqueness enforcing TT-SVD (\\secref{sec: USVD}).\n\tMoreover, we have implemented the TT-CP conversion, together with the norm equilibration.\n\tFor the training of the TT-MMK model, we have used the \\texttt{svmtrain} function available in the \\texttt{LIBSVM\\footnote{\\url{https:\/\/www.csie.ntu.edu.tw\/~cjlin\/libsvm\/}}} library.\n\tWe have run all experiments on a machine equipped with Ubuntu release 16.04.6 LTS 64-bit, 7.7 GiB of memory, and an Intel Core i5-6600 CPU @ 3.30GHz$\\times$4 CPU. The codes are available publicly on \\texttt{GitHub\\footnote{\\url{https:\/\/github.com\/mpimd-csc\/Structure-preserving_STTM}}}.\n\n\t\\item \\textbf{Parameter Tuning}\\\\\n\tThe entire TT-SVM model depends on three parameters. First, to simplify the selection of TT ranks, we take all TT ranks equal to the same value $R \\in \\lbrace 1,2, \\ldots 10 \\rbrace$. Another parameter is the width of the Gaussian Kernel $\\sigma$.\n\tFinally, the third parameter is a trade-off constant $C$ for the KSTM optimization technique \\eqref{eq: kerSTM}. Both $\\sigma$ and $C$ are chosen from $\\lbrace 2^{-8},2^{-7}, \\ldots, 2^{7}, 2^{8} \\rbrace$. For tuning $R,\\sigma$ and $C$ to the best classification accuracy, we use the \\emph{$k$-fold cross validation} with $k=5$. Along with this, we repeat all computations  50 times and average the accuracy over these runs.\n\tThis ensures a confident and reproducible comparison of different techniques.\n\n\\end{itemize}\n\\subsection{Data Collection}\n\\begin{enumerate}\\setlength\\itemsep{0.5em}\n\t\\item \\kk{Resting-state fMRI Datasets}\n\t\\begin{itemize}\n\t\t\\item \\textbf{Alzheimer Disease (ADNI):} The ADNI\\footnote{\\url{http:\/\/adni.loni.usc.edu\/}} stands for Alzheimer Disease Neuroimaging Initiative. It contains the resting state fMRI images of 33 subjects. The data set was collected from the authors of the paper~\\citep{MMK}. The images belong to either Mild Cognitive Impairment (MCI) with Alzheimer Disease (AD), or normal controls. Each image is a tensor of size $61 \\times 73 \\times 61$, containing $271633$ elements in total. The AD+MCI images are labeled with $-1$, and the normal control images are labeled with $1$. Preprocessing of the data sets is explained in~\\citep{DuSK}.\n\t\t\n\t\t\\item \\textbf{Attention Deficit Hyperactivity Disorder (ADHD):} The ADHD data set is collected from the ADHD-200 global competition data set\\footnote{\\url{http:\/\/neurobureau.projects.nitrc.org\/ADHD200\/Data.html}}. It is a publicly available preprocessed fMRI data set from eight different institutes, collected at one place. The original data set is unbalanced, so we have chosen 200 subjects randomly, ensuring that 100 of them are ADHD patients (assigned the classification label $-1$) and the 100 other subjects are healthy (denoted with label $1$). Each of the 200 resting state fMRI samples contains $49  \\times 58 \\times 47  = 133574$ voxels.\n\t\\end{itemize}\n\t\\item \\kk{Hyperspectral Image (HSI) Datasets: \n\tWe have taken the} \\verb+mat+ \\kk{file for both the datasets and their corresponding labels\\footnote{\\url{http:\/\/www.ehu.eus\/ccwintco\/index.php\/Hyperspectral\\_Remote\\_Sensing\\_Scenes}}. The following datasets have three dimensional tensor structure of different sizes, where each tensor data point represents a pixel value. Therefore, for our experiment we have taken a patch of size $5 \\times 5$ for two different pixel values, in order to get a binary classification dataset. }\n\t\\begin{itemize}\n\t\t\\item \\kk{\\textbf{Indian Pines:} The HSI images were collected via the Aviris Sensor\\footnote{\\url{https:\/\/aviris.jpl.nasa.gov\/}} over the Indian Pines test site. The size of the dataset is $145 \\times 145$ pixels over 224 spectral values. Hence, the size of the tensor data is $145 \\times 145 \\times 224$. The mat file we have collected for our experiment has reduced band size 200. This excludes bands covering the region of water absorption: [104-108], [150-163]. The original dataset contains 16 different labels to identify different corps and living areas. We have taken only 50 datapoints for each of the two labels 11 (Soybean-mintill ) and 7 (Grass-pasture-mowed). }\n\t\t\n\t\t\\item \\kk{\\textbf{Salinas:} This HSI images data was collected by 224 band Aviris Sensor over Salinas valley, California. Similar to Indian Pines, in this case, we have also collected samples  for two GroundTruths, namely 9 (Soil-vinyard-develop) and 15 (Vinyard-untrained) each with 50 datapoints. The size of the dataset is $512 \\times 217$ pixels over 224 spectral values. Hence, the size of the tensor data is $512 \\times 217 \\times 224$. }\n\t\\end{itemize}\n\\end{enumerate}\n\n\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\centering\n\t\t\\subfloat[\\kk{GroundTruth of Indian Pines dataset} ]{\\label{fig:hsispatial}\\includegraphics[width=.5\\linewidth]{.\/indian_pines.png}}\n\t\t\\subfloat[\\kk{GroundTruth of Salinas dataset}]{\\label{fig:fmrispatial}\t\n\t\t\\includegraphics[scale = 0.45]{.\/salines.png}}\n\t\\end{center}   \n\t\\vskip -0.2in\n\\end{figure}\n\\subsection{Influence of Individual Algorithmic Steps}\\label{methodology}\nIn the first test we investigate the impact of each individual transformation of the TT decomposition, outlined in~\\secref{sec: USVD}--\\secref{sec:normequ}.\nIn~\\figref{fig:a}, we apply a counterpart of the DuSK kernel~\\eqref{eq:kerAPRX}\ndirectly to the initial TT approximation of the data tensors.\nGiven TT decompositions\n$$\nx_{i_1,i_2,i_3} = \\sum_{r_1,r_2=1}^{R_1,R_2} \\teng^{(1)}_{i_1,r_1} \\teng^{(2)}_{r_1,i_2,r_2} \\teng^{(3)}_{r_2,i_3} \\quad\\mbox{and}\\quad\ny_{i_1,i_2,i_3} = \\sum_{t_1,t_2=1}^{R_1,R_2} \\tens^{(1)}_{i_1,t_1} \\tens^{(2)}_{t_1,i_2,t_2} \\tens^{(3)}_{t_2,i_3},\n$$\nwe compute a separable kernel similarly to~\\eqref{eq:kerAPRX} via\n\\begin{equation}\\label{eq:ttdusk}\nk(\\tensx, \\tensy) = \\sum_{r_1,t_1=1}^{R_1} \\sum_{r_2,t_2=1}^{R_2} \\nk(\\teng^{(1)}_{r_1}, \\tens^{(1)}_{t_1}) \\nk(\\teng^{(2)}_{r_1,r_2}, \\tens^{(2)}_{t_1,t_2}) \\nk(\\teng^{(3)}_{r_2}, \\tens^{(3)}_{t_2}).\n\\end{equation}\n\\kk{A similar approach was also proposed recently in~\\citet{chen2020kernelized}.\nIn~\\figref{fig:a} we have compared the difference between applying the DuSK kernel with TT decomposition and uniqueness enforced TT-SVD decomposition. However, we observe a rather poor classification accuracy both with and without uniqueness enforcing TT-SVD  (\\secref{sec: USVD}). }\n\n\n\n\n\n\n\n\\kk{Applying TT-UoSVD DuSK was not sufficient to produce the state-of-the-art results. Therefore, in~\\figref{fig:b} we convert the TT decomposition into the CP format as per~\\secref{TT-CP}.\nThis makes a more significant difference, in particular it slightly increases the accuracy and its distribution is much more uniform across the TT ranks.\nIn~\\figref{fig:c}, both  TT and TTCP with uniqueness enforced constraint  indicates that although UoSVD helps to stabilize the decomposition, the 'naive' TT DuSK~\\eqref{eq:ttdusk} suffers from inhomogeneous contribution of the TT cores of different size because DuSK relies essentially on the number of univariate kernel terms being equal across different variables.}\n\n\\kk{To equilibrate the scales of different terms even further,\nin~\\figref{fig:d} we enable the norm distribution as shown in~\\secref{sec:normequ}. This gives the highest classification accuracy,\nwhich relies on the internal features of the images instead of the scale indeterminacy of their representations. The comparison in~\\figref{fig:d} makes the impact of UoSVD also dominant to achieve the state-of-the-art.\nWe compare our algorithm to other methods next. }\n\n\n \\begin{figure}[ht]\n \t\\centering\n \t\\subfloat[TT with and without enforced uniqueness ]{\\label{fig:a}\\includegraphics[width=.5\\linewidth]{ttvsttsvd-eps-converted-to.pdf}}\n \t\\subfloat[TTCP with and without enforced uniqueness]{\\label{fig:b}\t\\includegraphics[width=.5\\linewidth]{ttcp_vs_ttcpsvd-eps-converted-to.pdf}}\n \t\\newline\n \t\\subfloat[ TT vs TTCP with enforced uniqueness]{\\label{fig:c}\t\\includegraphics[width=.5\\linewidth]{ttsvd_vs_ttcpsvd-eps-converted-to.pdf}}\n \t\\subfloat[TTCP norm equilibrium with and without eforced uniqueness]{\\label{fig:d}\t\\includegraphics[width=.5\\linewidth]{ttcp-ne_vs_ttcp-svd-ne-eps-converted-to.pdf}}\n \t\\caption{\\kk{Effect of each step of~\\secref{overalgo} on the classification accuracy for ADNI data set.}}\n \t\\label{fig:sec3compare}\n \\end{figure}\n\\subsection{Comparison to Other Methods}\nNext, we compare the classification accuracy of the whole proposed TT-MMK method (as per~\\figref{fig:d})\nwith the accuracy of the following existing approaches.\n\\begin{itemize}\\setlength\\itemsep{0.1em} \n\t\\item[] \\textbf{SVM:} the standard SVM with  Gaussian Kernel. This is the most used optimization method for vector input based on the maximum margin technique.\n\t\\item[] \\textbf{STuM:} The Support Tucker Machine (STuM)~\\citep{kotsia} uses the Tucker decomposition. The weight parameters of the SVM are computed for optimization into Tucker factorization form.\n\t\\item[] \\textbf{DuSK:} The idea of DuSK~\\citep{DuSK} is based on defining the kernel approximation for the rank-one decomposition. This is one of the first methods in this direction.\n\t\\item[] \\textbf{MMK:} This method is an extension of DuSK to the KCP input. It uses the covariance\/random matrix projection over the CP factor matrices, to get a KCP for the given input tensor~\\citep{MMK}.\n\tWe used the original DuSK and MMK codes provided by the authors of the paper~\\citep{MMK}.\n\t\\item[] \\textbf{Improved MMK:} This is actually a simplified MMK, where the projection of the CP onto the KCP is omitted (the covariance\/random matrices are replaced by the identity matrices).\n\t\\item[] \\textbf{TT-MMK:} This is Algorithm~\\ref{alg:KTTCP} proposed in this paper.\n\\end{itemize}\n\nThe results are shown in~\\figref{kttcp} and Table~\\ref{notation-table}.\nOur key observations are as follows.\n\\begin{itemize}\\setlength\\itemsep{0.1em}\n\t\\item[]\\textbf{(In)sensitivity to the TT Rank Selection:}~\\figref{kttcp}  shows that the proposed method gives almost the same accuracy for different TT ranks.\n\tFor some samples, even the TT rank of $2$ gives a good classification.\n\tNote that this is not the case for MMK, which requires a careful selection of the CP rank.\n\t\\item[]\\textbf{Computational Robustness:} while the CP decomposition can be computed using only iterative methods in general, all steps of the kernel computation in TT-MMK are ``direct'' in a sense that they require a fixed number of linear algebra operations, such as the SVD and matrix products.\n\t\\item[]\\textbf{Computational Complexity:} approximating the full tensor in the TT format has the same $\\cO(n^{M+1})$ cost as the Tucker and CP decompositions.\n\tAll further operations with factors scale linearly in the dimension $M$ and mode sizes, and polynomially in the ranks.\n\t\\item[]\\textbf{Classification Accuracy:} the proposed method gives the best average classification accuracy\n        compared to five other state of the art techniques.\n\t\\item[]\\kk{\\textbf{Generalization:} Top accuracy in datasets from two different areas (fMRI and HSI) shows that the method is suitable for a wide range of binary tensor classification problems.}\n\\end{itemize}\n\\begin{table}[ht]\n\t\\caption{\\kk{Average classification accuracy in percentage for different methods and data sets}}\n\t\\label{notation-table}\n\t\\vskip 0.15in\n\t\\begin{center}\n\t\t\\begin{small}\n\t\t\t\\begin{sc}\n\t\t\t\t\\begin{tabular}{lccccr}\n\t\t\t\t\t\\toprule\n\t\t\t\t\tMethods & ADNI & ADHD & Indian Pines & Salinas \\\\\n\t\t\t\t\t\\midrule\n\t\t\t\t\tSVM &  49  &  52 & 46 & 47\\\\\n\t\t\t\t\tSTuM & 51  & 54 & 57&74\\\\\n\t\t\t\t\tDuSK & 55 & 58  & 60& 92\\\\\n\t\t\t\t\tMMK& 69  &  60 & 93& 98 \\\\\n\t\t\t\t\tImproved  MMK & 71  &61  & 94 &98\\\\\n\t\t\t\t\tTT-MMK &  \\textbf{73} &  \\textbf{64} & \\textbf{99} & \\textbf{99}\\\\\n\t\t\t\t\t\\bottomrule\n\t\t\t\t\\end{tabular}\n\t\t\t\\end{sc}\n\t\t\\end{small}\n\t\\end{center}\n\t\\vskip -0.1in\n\\end{table}\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\centerline{\\includegraphics[width=0.5\\textwidth, height = 0.4\\textwidth]{final_result_ADNI-eps-converted-to.pdf}}\n\t\t\\caption{Accuracy of TT-MMK (TTCP), MMK and Improved MMK methods for the ADNI data set using different TT\/CP ranks respectively. TT-MMK and Improved MMK methods use identity kernel filtering while the MMK uses random kernel filtering.}\n\t\t\\label{kttcp}\n\t\\end{center}\n\t\\vskip -0.2in\n\\end{figure}\n\\section{Conclusions}\n\t\n\tWe have proposed a new kernel model for SVM classification of tensor input data. Our kernel extends the DuSK approach~\\citep{MMK} to the TT decomposition of the input tensor\nwith enforced uniqueness and norm distribution. The TT decomposition can be computed more reliably than the CP decomposition used in the original DuSK kernel.\nUsing fMRI \\kk{and Hyperspectral Image} data sets, we have demonstrated that the new TT-MMK method provides higher classification accuracy for an unsophisticated choice of the\nTT ranks\nfor a wide range of classification problems.\nWe have found out that the each component of the proposed scheme (uniqueness \\kk{enforced TT, TT-CP expansion and norm equilibration}) is crucial for achieving this accuracy.\n\nFurther research will consider improving the computational\ncomplexity of the current scheme, as well as a joint optimization of the TT cores and SVM weights. Similarly to the neural network compression in the TT format~\\citep{novikov-TT-NN-2015}, such a targeted\niterative refinement of the TT decomposition may improve the prediction accuracy.\n\n\n\\acks{This work is supported by the International Max Planck Research School for Advanced Methods in Process and System Engineering-\\href{https:\/\/www.mpi-magdeburg.mpg.de\/imprs}{\\textbf{IMPRS ProEng}}, Magdeburg and is a part of Max Planck research network on Big Data Driven Material Science (\\href{https:\/\/www.bigmax.mpg.de\/}{\\textbf{BiGmax}}) project. The authors are grateful to Lifang He for providing codes for the purpose of comparison.}\n\n\n\\vskip 0.2in\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction} \\label{intro}\n\n\\section*{Introduction} \\label{sec:introduction}\nThe paradigm shift that is taking place in the energy sector as part of the energy transition due to the increasing penetration of \\glspl{der} poses new challenges for grid operators, especially at the distribution grid level~\\cite{1_ourahou2020review}.\nTo meet these challenges, a more active role of the distribution grid operator is required through increased expansion of sensors and actuators, which provide telecontrol connections via \\glspl{ict} to resources such as controllable \\glspl{der}~\\cite{2_smartrgird2020book}.\nThis transformation from traditional grid structures to intelligent networked energy information systems -- \\glspl{sg} -- using \\glspl{ict} not only opens up new opportunities and solutions to master the energy transition, but also new dangers that threaten resilience and cyber-security~\\cite{3_van2020methods}.\n\nReliable and secure grid operation increasingly depends on properly functioning communication technologies and processes due to the high penetration of \\glspl{ict}, making it more vulnerable to failures and cyber-attacks~\\cite{4_klaer_sgam_2020}.\nIn particular in the context of \\gls{ics}, which also includes process networks of power grids, a threat landscape against cyber-attacks becomes apparent, which is essentially characterized by a long lifetime of assets and the use of legacy components with limited security mechanisms~\\cite{5_eder2017cyber}:\nIn 2015, unauthorized third parties exploited these vulnerabilities to gain control of remotely controlled equipment, such as circuit breakers, to disrupt the power supply of more than 225,000 customers~\\cite{6_case2016analysis}.\n\nTo counter new threats such as cyber-attacks, and to protect basic security objectives, i.e., confidentiality, availability, and integrity, cyber-security countermeasures are required, which are divided into preventive and reactive or active and passive measures.\nVarious guidelines and standards, e.g., the IEC 62531 series of standards~\\cite{30_wg152016iec}, specify countermeasures such as the use of cryptography and authentication procedures in the telecontrol protocols.\nHowever, given the long-standing legacy devices with performance and resource constraints, countermeasures with high performance overhead involve high expenditures and costs to implement upgrades or workarounds~\\cite{8_tanveer2020secure}.\nMore passive and reactive security measures are network-based \\glspl{ids}, which passively record communication traffic and perform attack detection within the process networks at selected points~\\cite{wolsing2022ipal}.\n\nIntrusion detection methods can be broadly divided into blacklisting, in which observations are compared with known attack signatures, and whitelisting, in which observations are compared with the established understanding of the system's characteristics under normal conditions~\\cite{11_krause2021cybersecurity}.\nFor process networks with deterministic network structures and physically plausible verifiable payloads, whitelisting is a promising methodology to detect attacks or anomalies without prior knowledge of patterns and signatures~\\cite{9_eckhart2018specification}.\nFurthermore, the impact of missing or hard-to-access data on attacks against power grids on the effectiveness of detection methods can be reduced, as whitelisting approaches do not primarily require such data.\n\nA challenge in applying a whitelisting approach is the need for a holistic capture of the characteristics of the system and its formalism, which includes technical and operational specifications of the infrastructure, as well as the behavior of the devices under normally defined states.\nPossibilities for this capture can be machine-learning-based or domain-specific knowledge-based approaches~\\cite{11_krause2021cybersecurity}.\nThe first approach is essentially characterized by the automated generation of a model in defined learning periods, which is trained and generated from recorded communication data~\\cite{10_baraneetharan2020role}.\nIn particular, in combination with \\gls{dpi}, industrial protocols such as \\gls{iec104} are decomposed into relevant fields to collect training data for models to detect the derivation of a standard pattern and suspicious processes in the form of anomalies~\\cite{mochalski2020cybersicherheit}.\nHowever, this may imply a potential vulnerability to data manipulation during the learning phases and incompleteness of non-observable legitimate situations such as maintenance in the training datasets.\nIn the second approach, specifications are defined that are used as a set of rules to define the characteristics of the system under normal conditions to detect anomalies.\nE.g., the IEC 61850 standard, which is mainly used in substations, describes data models and communication parameters in a format known as \\gls{scl}, which can provide the \\gls{sb} for normal conditions~\\cite{hokama2020cybersecurity}.\nHowever, the thoroughly available data prescribed by the standard's data model is not applicable to all industrial protocols, such as \\gls{iec104}.\nThus, the \\gls{sids} approach requires proper domain knowledge deposited and validated, but potentially achieves higher precision rates compared to machine-learning-based \\gls{ids} approaches~\\cite{12_verma2020machine,kus2022false}.\n\nHowever, the holistic capture of infrastructure specification and behavior of components in normal states requires high manual efforts in bundles of cross-domain knowledge and their maintenance, resulting in technically complex implementations without suitable accessible formalization.\nWe identify the following challenges:\n\\begin{itemize}\n\t\\item Concentration of dispersed domain-specific knowledge in a holistic \\gls{sb} description of the infrastructure.\n\t\\item Automated extraction of detection rules from infrastructure knowledge to detect anomalies and suspicious events.\n\t\\item Provision of explanations for issued alerts through coherent rule matching of infrastructure knowledge and alert generation.\n\t\\item Maintain high flexibility in detection capabilities through modular enrichment of infrastructure knowledge.\n\\end{itemize}\n\nTo address these challenges, we propose an approach for a \\gls{sids}, which, supported by an infrastructure specification and \\glspl{am} for component behavior w.r.t.\\ communication flows, captures characteristics of the \\gls{sg} for cyber-attack detection.\nMore precisely, our contributions are:\n\\begin{enumerate}\n\t\\item We identify relevant domain knowledge for the \\gls{sids} of cyber-attacks and intrusions in \\glspl{sg} by extracting domain-specific data based on a \\gls{gim} approach (\\nameref{subsec:framework_specification}).\n \t\\item We present and describe a structured approach for detecting anomalies in communication behavior in \\glspl{sg} process networks that uses a holistic \\gls{gim} as a \\gls{sb} and \\glspl{am} for flow consistency checks (\\nameref{subsec:framework_deeppacketinspect}).\n\t\\item We demonstrate and discuss the performance of our proposed approach against different attack scenarios in a physical testbed by evaluating the detection quality and performance within the scenarios (\\nameref{sec:result}).\n\\end{enumerate}\n\n\\section*{Smart Grid \\& Process-awareness in Detection} \\label{sec:background}\nAs the basis of our work, we describe the infrastructure specification of process networks based on \\gls{scada}, discuss their security and related research work.\n\n\\subsection*{SCADA-based Process Networks} \\label{subsec:background_scada}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.95\\linewidth]{S31-1.pdf}\n\\caption{Illustration of the \\gls{sg} infrastructure based on the Purdue reference model, representing a future-oriented \\gls{scada} system that is connected to the primary equipment via dedicated and\/or public communication infrastructure\\anc{~\\cite{26_vandervelde2021towards}}. The right side represents the graph-based formalism of the infrastructure as a \\gls{sb}\\anc{~\\cite{4_klaer_sgam_2020}}.}\n\\label{fig:smartgrid_spec}\n\\end{figure}\nBased on the Purdue reference model~\\cite{13_williams1994purdue}, process networks based on \\gls{scada} in \\glspl{sg} can be described as hierarchical control structures consisting of primary technology, secondary technology, and \\gls{ict}~\\cite{2_smartrgird2020book} as presented in Figure~\\ref{fig:smartgrid_spec}.\nDivided into several levels, the primary technology, such as circuit breaker switches, is monitored and controlled utilizing secondary technology, such as sensors and actuators.\n\nAt the lowest level of the control hierarchy, the infrastructure is divided into the \\gls{ot} network (equipment and systems with telecontrol connection through \\gls{wan}).\nDirectly connected to the \\gls{ot} network is the \\gls{scada} network (control systems and communication stations with operator stations).\nConnected via firewalls, a security-controlled level, the \\gls{dmz}, is present (logically segmented area between corporate office and \\gls{scada} network).\nFinally, the corporate network is connected to \\gls{dmz}.\n\nThe primary facilities are connected to the \\gls{ot} network through \\glspl{ied} used for control and measurement tasks, aggregated within the control hierarchy by \\glspl{rtu}.\nData are then forwarded to the \\gls{scada} system through the \\gls{ot} network using appropriate \\gls{ot} protocols such as \\gls{iec104}~\\cite{22_IEC104_2006} or Modbus~\\cite{23_Modbus_2012}.\nWithin the \\gls{ot} network, the \\gls{mtu} counterpart to the \\gls{rtu} acts as a gateway for the \\gls{scada} system.\n\n\\subsection*{Cyber-Security in Process Networks} \\label{subsec:background_cybersec}\nIn the European energy sector, the \\gls{iec104} protocol is often used to monitor and control geographically widely distributed processes~\\cite{21_matouvsek2017description}.\n\\gls{iec104} as a legacy industrial protocol does not provide security features such as encryption or authentication~\\cite{30_wg152016iec}.\nTherefore, without an encryption or authentication mechanism, unauthorized third parties can intercept critical \\gls{iec104} traffic and potentially endanger the grid.\nE.g., the attacker can intercept existing communication channels by a \\gls{mitm} attack or establish new connections to manipulate the traffic.\nThus, the attacker would be able to read, modify, inject, or discard new or sent messages between the intercepted or newly connected endpoints~\\cite{31_yang2012man}.\n\nTo address the critical security issues within the process network, especially the legacy protocols, the IEC 62351 standard discusses new security principles and requirements.\nE.g., the IEC 62351 standard requires secure end-to-end communication using the \\gls{tls} protocol, which provides secure key exchange, encryption, and authentication~\\cite{32_iec62351_2018}.\nHowever, large-scale implementation and adaptation of the new standards in traditional process networks are hampered by the large number of resource-constrained devices, which may jeopardize service availability.\nThese approaches can overwhelm resource-limited field devices such as \\glspl{rtu} or \\glspl{ied} and cause higher communication latency, preventing \\gls{scada} applications from meeting real-time requirements~\\cite{8_tanveer2020secure}.\n\nDifferent studies investigated the performance issues caused by \\gls{tls} protocol integration, and negative impacts on the performance of industrial protocol communications (e.g., IEC 61850, \\gls{iec104}) have been observed~\\cite{34_todeschini2020securing}.\nPower grids often contain performance-limited assets with long depreciation periods that cannot be replaced or upgraded without high costs, which require legacy compliant solutions~\\cite{35_castellanos2017legacy}.\n\\gls{ids} can provide passive security via detective capabilities to identify possible attack indicators or anomalies that do not actively interfere with the process network~\\cite{36_fernandes2019comprehensive}.\nThere are several \\gls{ids} approaches to identify potentially suspicious events, either by comparing observation with knowledge that represents normal system behavior, or by directly comparing the signature with known classified attacks~\\cite{37_zuech2015intrusion}.\n\nHowever, the latter approach requires attack data for detection, which limits flexibility in detecting unknown attacks such as zero-days~\\cite{akshaya2019study}.\nMoreover, comparing observations with known normal conditions based on trained models using data-driven machine learning approaches also has the disadvantage of low accuracy and limitation due to the scenarios included in the training data~\\cite{38_khraisat2019survey}.\nTherefore, a specification-based approach that relies on verified expert knowledge has the potential to provide high accuracy in detection and reduce the flexibility constraint by relying on domain-specific knowledge.\nThe challenge with \\gls{sids} approaches is to provide a standardized \\gls{sb} for different SG use cases, based on which anomaly detection conditions can be automatically derived.\nTherefore, in this paper, we present a \\gls{sids} that uses a defined \\gls{gim} to automatically derive the set of rules.\n\n\\subsection*{Related Work} \\label{subsec:background_relatedwork}\nMany studies and research works have investigated detection mechanisms based on process-awareness of \\gls{cps} for their suitability as \\gls{ids}.\n\nOne of these research directions involves addressing process-aware \\glspl{ids} that evaluates the attractiveness and criticality of \\gls{ics} devices that underlie industrial processes that could be modified to achieve adversary goals~\\cite{15_cook2017industrial}.\nOn this basis, the necessary signatures or heuristics that an adversary will leave as traces in its compromise attempt are identified.\nAnother research approach uses the degradation and functionality features of control signals to extract the meaning of the process of commands and determine the nondegradation pattern of the control signal within the action chain~\\cite{16_escudero2018process}.\nThe goal is to detect the unlegitimacy of the control signal issued by \\gls{ied} to the action chains before it controls the equipment.\n\nToward a holistic coverage of \\glspl{cps}, there are approaches that replicate the program states from physical devices to their digital twins using passive data sources and system specifications~\\cite{9_eckhart2018specification}.\nUsing stimuli and replication in a virtual environment, detailed testing is enabled in the context of \\gls{ids}.\nMore advanced research approaches address cyber-attack classification prepared in laboratory experiments and performed in tests to design various \\gls{ids} rules~\\cite{18_mohan2020distributed}.\nThe approach is based on rule generation algorithms in a distributed architecture to accommodate \\gls{scada} traffic.\n\nAnother approach pursues process-aware \\gls{ids} by modeling \\gls{ics}\/\\gls{scada} communication using probabilistic automata~\\cite{19_matouvsek2021efficient}.\nThe model represents normal communication with a small number of states and edges whose semantics are extracted from the headers of the protocol and detect state-based anomalies.\nIn the context of state interpolations in the automaton, an approach is presented that uses a combination of fuzzy interpolation with fuzzy automata~\\cite{20_almseidin2019fuzzy}.\nUsing automata theory and the fuzzy system for reasoning as part of the detection mechanism, a state transition rule base method is implemented to detect attacks.\n\nRegarding the anomaly detection methods for \\gls{iec104}, some multivariate access control and outlier detection approaches have been proposed using extracted packet information and communication statistics through Scapy~\\cite{rohith2018scapy} and CICFlowMeter~\\cite{lashkari2017cicflowmeter} for anomaly detection~\\cite{grammatikis2020anomaly}.\nIn the area of statistically based anomaly detection on \\gls{iec104}, the work in~\\cite{burgetova2021anomaly} presents a 3-value detection method that independently compares the number of packets transmitted in three consecutive time windows against a statistical profile and reports anomalies when a deviation from the specified range is detected.\nTo address the problem of missing labeled data, the work of~\\cite{anwar2021comparison} explores the use of unsupervised machine learning on \\gls{iec104}, in particular, one-class support vector machines, isolation forest, histogram-based outlier detection, and k-nearest neighbor are investigated.\n\nWhen addressing security issues within the protocol \\gls{iec104}, research in~\\cite{scheben2017status} examined the detection qualities of a machine learning-based detection system compared to a misuse-based system such as Snort~\\cite{caswell2004snort}.\nThe result undermines the flexibility-accuracy dilemma described earlier, where the misuse-based system has high accuracy but low flexibility, whereas the machine learning-based system has higher flexibility but lower accuracy.\nFurthermore, the challenge with automated machine learning-based detection systems also lies in their explainability~\\cite{dang2021improving}, which challenges the plausibility check of the output~\\cite{holzinger2020measuring}.\n\nAlthough the proposed approaches provide different mechanisms to combine process knowledge with cyber-security, they still require significant additional analytical resources to provide the necessary information for their functionality.\nE.g., in addition to the infrastructure specification for which the operator can provide necessary knowledge, additional efforts must be made to develop an understanding of likely attack targets, details about stimuli, statistical data, or vendor-specific technical specifications such as equipment degradation that are often inaccessible.\nTherefore, our approach is entirely based on the utilization of domain-specific knowledge, which is accessible from standards and infrastructure knowledge from grid operators.\nWhile the implementation of \\glspl{am} enables the detection of inconsistencies within processes and flows, it does not take into account the semantics of the data points involved in the traffic.\nTherefore, the intrusion detection capabilities of our proposed \\gls{sids} rely not only on automata-based detection, but also on semantic verification of the data points.\nThe \\gls{gim} encapsulates the semantics of the data points, which is part of advanced detection.\nThrough the holistic formulation of a graph-based specification foundation that provides the required overall understanding of the process, semantics, and communication of \\gls{sg}, we design a process-aware \\gls{sids}.\n\n\n\\section*{Specification-based Intrusion Detection} \\label{sec:framework}\nIn this section, our approach of a \\gls{sids} for \\glspl{sg} is presented.\n\nIn general, our approach is based on \\gls{sb} derived from \\gls{gim} that encapsulates domain-specific knowledge.\nUsing \\gls{sb}, monitored network traffic is checked against \\gls{sb}, with a violation resulting in a specific and explainable alert.\nFurthermore, the consistency of communication behavior is checked against protocol-specific \\glspl{am} to ensure that the industrial packet flows comply with the state transitions.\nThus, intrusion detection is performed using a mixture of approaches leveraging the domain knowledge of a \\gls{gim}.\n\n\\subsection*{Framework Overview} \\label{subsec:framework_overview}\n\\begin{figure}\n    \\centerline{\\includegraphics[width=\\columnwidth]{S31-2.pdf}}\n    \\caption{This illustration presents the \\gls{sids} approach to observe network traffic based on specifications and indication of traffic, which is a hybrid approach of specification-based rule matching and behavior consistency checking via a Mealy automata.}\n    \\label{fig:framework_overview}\n\\end{figure}\nOur proposed \\gls{sids} (cf. Figure~\\ref{fig:framework_overview}) is a network-based \\gls{sids} that checks for the presence of malicious content in the various layers of the network protocol.\nThe \\gls{sids} detects anomalies based on the captured traffic by checking information from the packet headers and payloads within the industrial protocol stack (e.g., Ethernet\/IP\/TCP\/\\gls{iec104}) and discrepancies in the packet flow using Mealy automata~\\cite{grigorchuk2000automata}.\nThe Mealy automaton has no accepting states, and the sequence of outputs from the sequence of transitions leads to a reactive system with transitions as its core and is therefore a more appropriate model for protocols~\\cite{bieniasz2016towards}.\nBased on a variation of Angluin's L* algorithm that generates Mealy automata~\\cite{27_bollig2010libalf}, the state machine model is used to perceive the communication behavior.\n\nIn this paper, we focus on the \\gls{iec104} protocol, which is a widely used standard in the European energy sector for monitoring and controlling tasks in TCP\/IP-based networks~\\cite{21_matouvsek2017description}.\nHowever, the \\gls{sids} is not limited to the contents of the application layer of \\gls{iec104} traffic, but considers all layers that are included in typical TCP\/IP-based \\gls{iec104} packets.\nIn particular, packets that use IP at the network layer, TCP at the transport layer, and \\gls{iec104} at the application layer.\n\nTo distinguish between illegitimate and legitimate traffic, the \\gls{sids} uses a set of rules defined in a machine-readable input file derived from the \\gls{sb} (cf. Section~\\nameref{subsec:framework_specification}).\nHere, specifications are defined as sets of information that represent the known parameters and characteristics of the \\gls{sg} infrastructure to some extent.\nAnything specified in \\gls{sb} is considered valid;\nanything that does not conform to a specification is considered malicious traffic.\n\nWhen observing network traffic, the \\gls{sids} examines each packet using \\gls{dpi} (cf. Section~\\nameref{subsec:framework_deeppacketinspect}).\nIn this context, the conformance of the data packet to the \\gls{sb}, such as the protocols used, protocol fields, address validation, and payload consistency is checked.\nAfter the initial inspection of the packet, the next inspection step evaluates connection attributes and states.\nEach connection is defined in \\gls{sb}, specifying the properties of the connection, and two Mealy automata modeling the connection endpoints.\n\nRegarding \\gls{iec104} traffic within the \\gls{scada} network, the roles of endpoints are represented as \\gls{mtu} and \\gls{rtu}.\nFor correct semantic mapping of packets, the mapper component is responsible for translating the packet contents into an input symbol for the Mealy automata (cf. Section~\\nameref{subsec:framework_automata}).\nAfter the mapper receives the corresponding input symbol, the connection object passes it to the instantiated \\glspl{am} representing the connections.\nBecause of the use of Mealy automata, it provides immediate feedback by returning an output symbol.\nIf the output symbol indicates an error or suspicious behavior, the connection object triggers the alert generator with a specific alert reason to issue an alert (cf. Section~\\nameref{subsec:framework_alert}).\nAlert generation is triggered by various components for different reasons.\nThe cyber threat information database represents the collection of alerts combined with the specification of the infrastructure, which is part of a higher-level correlation~\\anc{ as presented in our previous work~\\cite{24_sen2021towards,sen2022contextual}}.\n\n\\subsection*{Specification Basis} \\label{subsec:framework_specification}\nBased on a formal \\gls{gim} of \\gls{sg}~\\cite{4_klaer_sgam_2020}, we extract the \\gls{sb} for the \\gls{sids} through the explicit data model definition.\nThus, whitelists can be created from the data model and anomaly detection through the whitelist configurations.\nThis includes communication (e.g., link quality, routing, packet flows), authentication (e.g., MAC\/IP addresses), and process data (e.g., control, measurement, state - plausibility).\nTable~\\ref{tab:framework_specification} describes the domains and information fields of the \\gls{sb}.\n\n\\begin{table}[h!]\n\\centering\\small\n\\caption{Domain-specific attribution of captured traffic.}\n\\begin{tabular}{||p{2.3cm} p{9cm}||}\n \\hline\n Domain & Field Attribution\\\\\n \\hline\\hline\n Communication                              & Address matching of packets (L2-L4, L7).\\\\\n                                            & Connection and established communication channel (client\/server, protocol, port).\\\\\n                                            & Packet flow according to protocol (L4, L7).\\\\\n Asset                                      & Data point matching.\\\\\n                                            & Integrity at data point level.\\\\\n                                            & Role-based verified operations.\\\\\n Operation                                  & Technical assets boundaries.\\\\\n                                            & Technical command execution capability.\\\\\n \\hline\n\\end{tabular}\n\\label{tab:framework_specification}\n\\end{table}\n\nThe \\gls{sb} provides information on the behavior of communication, assets, and operating limits, from which rules can be derived.\nIn the area of communication, e.g., the addresses of the relevant fields of the protocol layer (L2-L4, L7), such as the MAC address, the IP address, the Port number, and \\gls{ioa} of the \\gls{iec104} protocol, are specified.\nAdditionally, legitimate connection channels and routes are defined which specify allowed communication channels between the endpoints with protocol types and Port numbers.\nIn the dynamic scope, the standards assigned to the application layers (L7) are defined accordingly, which then sets the corresponding predefined \\glspl{am} for attack detection.\nE.g., for \\gls{iec104}, \\glspl{am} are used that represent the data transaction process during communication initialization and confirmation of control commands.\n\nThe use of protocols is also considered in the \\gls{sb}.\nE.g., the use of certain protocols such as SSH can either be whitelisted or even restricted to certain periods such as maintenance times on weekends.\nProtocol behavior is observed with \\glspl{am} that represent valid communication flows for specific protocols.\nCurrently, there is only one model for \\gls{iec104} traffic, but in general other state-full protocols can also be modeled through \\glspl{am}.\nThe \\gls{sb} can also be extended to include other criteria such as the maximum \\gls{rtt} for TCP packets.\n\nIn the context of resource behavior, data points are taken from \\gls{sb} and verified for legitimacy within industrial telecontrol protocols, that is, regarding known data points with correct addressing.\nConsequently, the integrity of the data points is defined according to \\gls{sb}, if the data characteristics within the data points (e.g. \\gls{ioa} in \\gls{iec104}) are correctly assigned to the asset in the right communication direction.\nThis provides the base for role-based verification of asset operations, in which the legitimacy of operation options of assets is also defined by the data points (e.g., sensors can send measurements but not commands).\n\nIn the scope of operational behavior, the technical operating boundaries of assets (e.g., maximum power rating for setpoints), and the execution plausibility of commands are also extracted from the domain-specific knowledge of the \\gls{sb} (e.g., nominal power-dependent plausibility range for the $\\cos\\phi$ setting of inverters~\\cite{scheben2017status}).\n\n\\subsection*{Deep Packet Inspection} \\label{subsec:framework_deeppacketinspect}\nThe functionality \\gls{dpi} is a key feature of \\gls{sids} and is anchored in the central organizer and forwarder of all internal intrusion detection processes (cf. Figure~\\ref{fig:framework_overview}).\nAfter a packet is received, it is categorized depending on the packet layers it contains.\nRelevant packets are those that correspond to one of the protocols described in the \\gls{sb}, e.g., \\gls{iec104} or SSH packets.\n\nThe categorization determines the checks that are performed on each packet.\nPackets that are classified as irrelevant are ignored, while the contents of packets containing industrial protocols such as \\gls{iec104} are checked more thoroughly.\nAlthough basic address detection and verification are performed at the first level of \\gls{dpi}, advanced and contextual checks are performed as part of connection-related checks.\nEach packet associated with a particular connection object is forwarded to the corresponding checks.\nConnection objects represent a connection between two endpoints.\nEach of the endpoints is assigned specific addresses for each network layer, including the application layer, e.g., the \\gls{iec104} protocol, for which an \\gls{am} is assigned.\nFor \\gls{iec104} traffic, this means that each connection contains two \\glspl{am}, an \\gls{mtu} model, and an \\gls{rtu} model.\n\nWhen a packet is assigned to a connection by the \\gls{dpi} component, all addresses are checked for consistency, both on the sender and the receiver side.\nIn addition, the flow control of the \\gls{iec104} layer is also checked.\nFor this task, connections store the current packet sequence control counters for each endpoint individually.\nWhen a connection receives a packet containing sequence control information, namely packets containing \\gls{apci} frames in I\\hyp{}frame and U\\hyp{}frame format, the connection objects are checked.\nThey are checked for both endpoints whether the sequence numbers match the current counters and transmission direction.\nIn addition, an I\\hyp{}framed \\gls{apci} indicating an \\gls{asdu} is checked for technical specification conformity.\n\nAfter all addresses, traffic sequence, and technical specifications are checked, the packet is passed to the packet mapper.\nThe mapper maps the packet to an input symbol of the automata alphabet as an automata input.\nIf any of the checks of \\gls{dpi} fails, e.g., the contained address information is unknown or the packet cannot be assigned to a connection, an alert is issued.\n\n\\begin{figure}\n    \\centerline{\\includegraphics[width=\\columnwidth]{S31-3.pdf}}\n    \\caption{Illustration of the rule generation process to automatically generate the \\gls{sb} based on the infrastructure model.}\n    \\label{fig:framework_rulegenerator}\n\\end{figure}\n\nThe automated process of generating the \\gls{sb} based on the infrastructure model is broken down into several components as shown in Figure~\\ref{fig:framework_rulegenerator}.\nThe input consists of an \\gls{gim}, which describes the infrastructure in the respective domains of power grid, IT, and \\gls{ot} devices.\nEach of these domains contains domain-specific information, such as asset data points, component networking, and their operational function in the process.\nIn addition, a configuration is required that specifies the rules that will be used later to detect attacks.\nThe rule configuration specifies the type of devices of interest for which rules are to be created that contain attributes and their environmental constraints.\n\nTo achieve the desired detection quality, the \\gls{sids} must be correctly configured by the given input.\nThe prerequisite for this is \\gls{sb}, which is to be generated by the rule generator.\nThe task of the rule generator is to convert a \\gls{gim} into a \\gls{sb} based on a given configuration.\nThis \\gls{sb} represents the set of rules that the \\gls{sids} uses to decide which communication and payload content is valid.\n\nThe rule generator consists of three modules, each serving a different purpose.\nThe importer is used to read the respective inputs - the \\gls{gim} and the configuration - and prepare them for further use.\nThe rule manager is the main module of the rule generator and is responsible for reading the relevant attributes from the \\gls{gim} and converting them into rules based on the specifications in the configuration.\nFinally, the exporter summarizes the generated rules in a \\gls{sb} that can be read by the \\gls{sids}.\nAfter generating the \\gls{sb}, the \\gls{sids} can apply the previously generated specifications to the packets of the captured network communication.\nAs soon as the given specifications are violated or the recorded communication deviates from the expected normal behavior, alarms are triggered.\n\n\\subsection*{Automata Model} \\label{subsec:framework_automata}\n\\begin{figure}\n    \\centerline{\\includegraphics[width=\\columnwidth]{S31-4.pdf}}\n    \\caption{Exemplary illustration of the mapper, that maps \\gls{iec104} data packets to input symbols for the automata, which here, in an example for the \\gls{mtu} endpoint connection, generates according to he protocol standard a Mealy \\gls{am} to stateful monitor e.g., the data transaction.}\n    \\label{fig:framework_automata}\n\\end{figure}\n\\glspl{am} are used to dynamically check multiple packets within industrial protocol traffic (cf. Figure~\\ref{fig:framework_automata}).\nThe goal of \\glspl{am} is to model flow-based processes within the communication process according to the selected protocol, such as \\gls{iec104}.\nThe states of the \\gls{am} represent, e.g., the start of a connection and data transmission, the tracking of pending acknowledgments of commands and measured values, and the stop of the data transmission.\nPackets identified as industrial protocol traffic are processed by the mapper component, which maps the packets to their corresponding input alphabet counterparts for the \\gls{am}.\n\nThe mapper translates the contents of the \\gls{iec104} layers into input symbols for the \\gls{mtu} and \\gls{rtu} automata used within the connection object.\nA data packet may contain multiple instances of \\gls{iec104} layers.\nThe mapper then returns an ordered list of symbols mapped from these packets.\nThe symbol is mapped based on the contents of the decoded \\gls{iec104} layer.\nWithin this process, the format of the observed \\gls{apci} frame is determined, where the \\gls{apci} frame can have three different formats: U\\hyp{}frame, I\\hyp{}frame, and S\\hyp{}frame.\nDepending on the determined format, the frames are checked for additional flags that indicate membership in specific groups of packets mapped to special input symbols.\nSubsequently, these input symbols represent the set of possible input symbols for the \\gls{iec104} automata.\nAdditionally, an error symbol is used to indicate that the packet does not match any of the criteria used to assign it to one of the known input symbols.\n\nAccording to the \\gls{iec104} protocol standard, two transition systems are modeled, one representing the \\gls{mtu} stations of \\gls{scada} networks and one modeling \\gls{rtu} station for each connection.\nThe use of this role-based modeling approach allows the states within the \\glspl{am} to be sufficiently differentiated, such as the states of connection establishment and valid data transmission.\nTo define the processes within the connection procedure of communication more accurately, the models must also be able to determine whether an input is sent or received.\nTo this end, each packet categorization is extended to include a prefix indicating whether the packet was sent or received.\nBoth automata use the same input alphabet and state sets, but differ slightly in some transitions.\n\nAfter receiving an input symbol and using a transition, the automata returns an output indicating whether the input results in a suspicious state or whether this packet type is invalid for the current protocol procedure.\nInternally, this is done by a status variable within the automaton object.\nWhen certain transitions are used, they trigger functions that change the internal state, which is always given as a return value after processing the input.\nE.g., the \\gls{am} requires the generation of 15 different input symbols for seven different packet types.\nThe packets are recognized by the mapper and then extended by a prefix indicating the direction of transmission and an error input.\nThese transitions do not change the internal state variable, so the output would be valid in the sense of Mealy automata.\n\nAll other transitions that trigger a change in the internal state variables of \\glspl{am} are undefined behavior, i.e., a violation of the protocol procedure.\nTherefore, the output for each of these transitions is invalid.\nThe error input indicates that the packet was not recognized as belonging to one of the defined packet types, therefore cannot be processed, and thus leads to an alert.\n\n\\begin{figure}\n    \\centerline{\\includegraphics[width=\\columnwidth]{S31-5.pdf}}\n    \\caption{Example illustration of packet flow conformance checking based on the \\gls{mtu} automaton model, showing a simple \\gls{mtu} and \\gls{rtu} communication scenario.}\n    \\label{fig:framework_automata_example}\n\\end{figure}\n\nFigure~\\ref{fig:framework_automata_example} illustrates an example of how \\glspl{am} works to check the consistency of normal traffic flow in the monitored communication channel.\nAs an example, a data transmission sequence is used that contains an interrogation operation, where the \\gls{mtu} initiates the data transmission.\nAfter starting the data transmission, both automata reach the ``STARTED'' state, which allows I-frames to be sent.\nThe \\gls{mtu} sends an interrogation command to \\gls{rtu}, which is acknowledged with the first I-frame back to \\gls{mtu}.\nAfter the first I-frame to the \\gls{mtu}, the \\gls{rtu} sends several I-frames containing the measurement data.\nThe \\gls{mtu} acknowledges the packet reception with an S-frame indicating that all previous frames have been sent correctly and that both automata should now be in the ``STARTED'' state, since both have no unacknowledged frames.\nSince the traffic flow conforms to the automaton model, the conformity of the packet flow in this example is therefore also classified as correct by \\gls{sids}.\n\n\\subsection*{Alert Generation} \\label{subsec:framework_alert}\nAlerts are the notifications of the \\gls{sids} that are triggered when certain packets violate the specification.\nThey are issued by the alert generator component of \\gls{sids} and recorded in a machine-readable log file.\n\nAll alerts are written to a log file that assigns a unique running ID to each new alert.\nEach alert begins with an ID tag, followed by the attributes specified in Table~\\ref{tab:framework_alert}.\n\n\\begin{table}[h!]\n\\centering\\small\n\\caption{Alert output from \\gls{sids}}\n\\begin{tabular}{||p{3.5cm} p{8cm}||}\n \\hline\n Alert Field & Description\\\\\n \\hline\\hline\n Alert Type                                 & The type of alert indicates what type of alert has occurred.\\\\\n Threat Level                               & Low, medium or high threat levels.\\\\\n Timestamp                                  & Each warning issued contains the timestamp when the warning was created.\\\\\n Alert Reason                               & A textual reason that triggered the creation of this warning message.\\\\\n Packet Content Information                 & Detailed information about which data packet content is related to the issued alert.\\\\\n \\hline\n\\end{tabular}\n\\label{tab:framework_alert}\n\\end{table}\n\nTo illustrate how alert messages are generated, we provide an example in Listing~\\ref{lst:alert_file}.\nWe use Metasploit's \\gls{iec104} Client Utility Module as the basis for this sample scenario~\\cite{29_metasploitiec104}.\nTherefore, the scenario underlying this example is that a new endpoint with unknown IP and MAC address acts as a \\gls{mtu} and attempts to establish a \\gls{iec104} connection to a \\gls{rtu}.\nIn doing so, the new \\gls{mtu} also sends a control command specifying a new setpoint, such as a new power injection for a \\gls{pv} inverter.\nIn this example, the generated alarms cause anomalies regarding the IP and MAC addresses of the new endpoint, as these are not specified in the \\gls{sb}.\nIn addition, the connection is also not valid because the communication channel between the new endpoint and the \\gls{rtu} is also not specified.\nThus, any commands sent from the new endpoint to the \\gls{rtu} are also considered invalid.\nFurthermore, the control command contained a setpoint that also violates the specified allowable range of valid setpoints.\nThus, all active interactions between the new endpoint and the \\gls{rtu} are classified as anomalies and output as alerts.\n\n\\lstset{language=XML, numbers=none, captionpos=b, xleftmargin=0.2em, caption=Example of alert messages generated by the alert generator. , label=lst:alert_file, frame=single , breaklines=true, basicstyle=\\scriptsize, multicols=2}\n\\begin{lstlisting}\n[ALERT_0]\nalert_type = IP_MISMATCH\nthreat_level = high\ntimestamp = 14.04.2022 10:47:09\nalert_reason = IP of this packet is unknown: 173.24.0.3\npacket_info = ETH \/ IP \/ TCP \/ IEC104-U\n\n[ALERT_1]\nalert_type = PORT_MISMATCH\nthreat_level = high\ntimestamp = 14.04.2022 10:47:09\nalert_reason = One of the Ports of this packet is unknown: 59478\npacket_info = ETH \/ IP \/ TCP \/ IEC104-U\n\n[ALERT_2]\nalert_type = NO_SUCH_CONNECTION\nthreat_level = high\ntimestamp = 14.04.2022 10:47:09\nalert_reason = Connection does not exist in whitelisting data!\npacket_info = ETH \/ IP \/ TCP \/ IEC104-U\n\n[ALERT_3]\nalert_type = INVALID_OPERATION\nthreat_level = high\ntimestamp = 14.04.2022 10:48:00\nalert_reason = Send packet contains invalid operation for the endpoint!\npacket_info = ETH \/ IP \/ TCP \/ IEC104-I\n\n[ALERT_4]\nalert_type = INVALID_SETPOINT\nthreat_level = high\ntimestamp = 14.04.2022 10:48:00\nalert_reason = Active control command contains invalid setpoint!\npacket_info = ETH \/ IP \/ TCP \/ IEC104-I\n\\end{lstlisting}\n\nOverall, our \\gls{sids}, which automatically derives its \\gls{sb} based on a formal \\gls{gim}, is designed to detect explainable anomalies from different domains.\nSpecifically, the domain-specific knowledge used for anomaly detection is extracted as appropriate rules from the \\gls{gim}, which represents the operator's existing knowledge of its infrastructure, without requiring the knowledge of cyber-security experts.\nMoreover, the dynamic nature of communication behavior is also validated by \\glspl{am} with respect to protocol conformance and flow consistency.\nSubsequently, both the dynamic packet flow and attributes, as well as the protocol field values within the payload, are validated and checked for potential inconsistencies or specification violations.\nThus, our \\gls{sids} detects critical violations of legitimate processes and infrastructure specifications at both the communication and operational levels, relying only on existing and available knowledge without requiring external expertise.\n\n\\section*{Evaluation \\& Discussion} \\label{sec:result}\nTo demonstrate and discuss the performance of our proposed \\gls{sids}, we evaluate its detection quality in a physical testbed for attack and non-attack scenarios.\n\n\\subsection*{Smart Grid Testbed Setup} \\label{subsec:result_testbed}\nWe evaluate \\gls{sids} in a cyber-physical testbed as shown in Figure~\\ref{fig:tesbed_setup}\\anc{ that is based on our previous work~\\cite{25_sen2021towards}}.\nThe testbed replicates an MV \/ LV grid consisting of physical components networked through a dedicated \\gls{ict} infrastructure.\nTherefore, neither simulations nor virtualized components are involved.\nSince our \\gls{sids} is designed to monitor specific communication channels within defined network segments and thus acts more like a sensor with more than one entity deployment, the limited complexity of the testbed does not limit the scope of the study.\n\nIn our test setup, we use electrical equipment such as a 640 kVA secondary substation, 22 kWh \\gls{bss}, 12 kVA and 36 kVA \\glspl{pv}, and several resistive\/inductive loads.\nThe power system topology consists of two strings to which the \\glspl{der} and loads are connected, and on which we can measure current and voltage via integrated three-phase measuring points.\nWe control the \\gls{der} through their Modbus interface via \\glspl{rtu}, which is provided by their respective inverter.\nFollowing a \\gls{scada} network, the testbed also includes a process network consisting of the \\gls{ict} infrastructure and the control room.\nThe control room represents an \\gls{mtu} that sends \\gls{iec104} control and query commands to \\glspl{rtu}.\n\nWe consider different attack scenarios based on attacks that have already gained access to the process network.\nThe external attacker represents a new entity within the system with unknown IP and MAC addresses.\nContrary, the \\gls{mitm} attack intercepts the communication between the control room and the selected \\glspl{rtu}.\n\nIn addition, our \\gls{sids} approach is deployed in the process network at the \\gls{scada} switch, where traffic is monitored via a mirrored SPAN port or dedicated network taps.\nActive inline network taps are used between the \\gls{scada} node and switches to capture \\gls{scada} traffic and perform timestamping with a high resolution of 8 ns~\\cite{profishark2018}.\nThe mirrored SPAN port allows all traffic passing through the target switch to be captured, but with lower timestamp resolution and possibly more jitter.\nPreferably, dedicated network taps are used for the main communication channel between the \\gls{mtu} and \\gls{rtu} to be monitored.\nThus, \\gls{scada}-related network traffic is continuously forwarded to \\gls{sids} for intrusion detection.\n\n\\begin{figure}\n    \\centerline{\\includegraphics[width=\\columnwidth]{S31-6.pdf}}\n    \\caption{Our setup of a \\gls{sg} testbed\n    replicates a distribution grid with \\glspl{der} and \\gls{ict} infrastructure for control and monitoring.}\n    \\label{fig:tesbed_setup}\n\\end{figure}\n\n\\subsection*{Methodology} \\label{subsec:result_proecdure}\nWe use the cyber-physical testbed and attack scenario described in~\\nameref{subsec:result_testbed}.\nIn the first scenario without attack induction, all stations and traffic are within the specification, where normal protocol behavior of a system operation under normal conditions is replicated (Scenario 1).\nWe use this scenario to examine the \\gls{sids} under normal operation conditions with an average traffic volume.\n\nIn the second scenario, we investigate the detection quality of \\gls{sids} under attack conditions.\nTo this end, this scenario is divided into replicating an attack from outside the testbed with limited knowledge of the internal network and data points (Scenario 2-a) and an attack with sophisticated knowledge (Scenario 2-b).\n\nScenario 2-a contains a communication attempt with unknown addresses in L2 and L3 (e.g., MAC and IP address) to \\gls{rtu}, sending an interrogation command to query measurement values.\nScenario 2-b is executed as a \\gls{mitm}-based \\gls{fdi} attack in which two different types of packet are injected.\nThe first type of injected packet contains an \\gls{ioa} that is not associated with the corresponding asset in the specification (Scenario 2-b-I).\nIn contrast, the second type contains an \\gls{ioa} included in the specification and also regularly used in the normal conversation of \\gls{mtu} and \\gls{rtu} with the correct mapping of devices (Scenario 2-b-II).\nHowever, the measurements transmitted with these packets contain measurements that are overlaid with small noise within the range of the technical specification.\n\nWe also measure the performance of \\gls{sids} in large traffic volumes per connection in a short time.\nAn important requirement for this investigation is that \\gls{sids} always observes the beginning of the connection to use \\glspl{am} for correct context and packet sequence tracking.\nOtherwise, it cannot find the correct initial state and therefore produces alerts for almost all data packets.\n\nTo evaluate the quality of detection, we rely on the confusion matrix and the performance metrics derived~\\cite{28_tharwat2020classification}.\nThus, we measure the following metrics:\n\\begin{itemize}\n \\item \\gls{tp}: event correctly classified as attack indicator\n \\item \\gls{fp}: event incorrectly classified as attack indicator\n \\item \\gls{tn}: event correctly not classified as attack indicator\n \\item \\gls{fn}: event incorrectly not classified as attack indicator\n\\end{itemize}\nTo evaluate the performance of our approach, we also use the following telemetry data of the captured network traffic in the scenarios.\n\\begin{itemize}\n \\item L2: MAC source and destination addresses\n \\item L3: IP source and destination addresses, checksum\n \\item L4: TCP sequence and acknowledge numbers, port number, checksum, \\gls{rtt}\n \\item L7: IEC 104 protocol fields of U- and I-frames\n\\end{itemize}\n\n\\subsection*{Evaluation of Performance \\& Classification Accuracy} \\label{subsec:result_res}\n\\begin{table}[h!]\n\\centering\\small\n\\caption{Confusion Table of Experiments.}\n\\begin{tabular}{||p{2.3cm} p{1cm} p{1cm} p{1cm} p{1cm}||}\n \\hline\n Scenarios & \\gls{tp} & \\gls{tn} & \\gls{fp} & \\gls{fn}\\\\\n \\hline\\hline\n 1                                          & 0 & 200 & 0 & 0\\\\\n 2-a                                        & 115 & 0 & 0 & 0\\\\\n 2-b-I                                      & 10 & 0 & 0 & 0\\\\\n 2-b-II                                     & 0 & 0 & 0 & 10\\\\\n \\hline\n\\end{tabular}\n\\label{tab:result}\n\\end{table}\nOur results are presented in Table~\\ref{tab:result}.\nWithin Scenario 1, our results indicate that \\gls{sids} does not generate alerts caused by addressing, automata errors, or sequence number violations.\nThe only parameter that may cause slight variations in the \\gls{fpr} is the maximum \\gls{rtt} parameter, which in our experiments was parameterized in the range of $150 ms$ and $200 ms$.\nNarrower ranges caused more \\glspl{fp} in our experiments due to varying \\gls{rtt} in the communication channels.\nWith this adjustment, using a sufficiently large value for the maximum \\gls{rtt} (e.g., upper $95\\%$ confidence interval of the \\gls{rtt} variance), no \\glspl{fp} were produced.\n\nIn Scenario 2-a, the attacker mimics the normal behavior of a \\gls{mtu} by starting a conversation and sending a query command for the measurement data.\nThe \\gls{rtu} responds with measurement data.\nAll $115$ malicious packets were correctly detected.\n\nWithin Scenario 2-b, we inject a total of $20$ packets (10 packets from each of the sub-scenarios).\n\\gls{sids} was able to correctly classify the $10$ packets from scenario 2-b-I (\\gls{tp}) due to incorrect addressing of \\gls{ioa}.\nScenario 2-b-II represents an edge case where the attacker performs perfect spoofing and adheres to the legitimate specification of the system.\nTherefore, the $10$ injected packets from Scenario 2-b-II were not correctly detected by \\gls{sids}, showing the limits of our approach (\\gls{fn}).\nHowever, limiting the range of attack actions so that the attacker can evade detection can shift the impact trajectory of the attack into a treatable scope.\nSubsequently, a larger scope of attack is required to cause more impact, imposing more actions on the attacker that can potentially reduce their stealthy movement.\n\nTo assess processing performance, we also evaluated the processing time of packets with and without specification compliance (Scenario 1 and Scenario 2).\nFor compliance with the specification (Scenario 1), each packet monitored by \\gls{sids} is processed on average at $0.3ms$ with an insignificant standard deviation.\nWith invalid traffic (Scenario 2), each packet is processed in an average of $1.5ms$ with also insignificant standard deviation.\nThe reason for this discrepancy is that when a packet violates the specification, several steps are triggered in the reporting mechanism to extract alert-relevant information, which is written to the alert log.\n\nIn addition, we have also performed a comparison with other intrusion detection approaches, which is shown in Table~\\ref{tab:evalation_comparision}.\nHowever, due to the lack of a standardized benchmark evaluation for countermeasures against cyber-attacks in \\gls{scada} systems, the comparison is qualitative.\n\\begin{table}[h!]\n\\centering\\small\n\\caption{Comparison with other intrusion detection approaches.}\n\\begin{tabular}{||p{1cm} p{1.5cm} p{1cm} p{1.5cm} p{1cm} p{0.75cm} p{0.75cm} p{0.75cm} p{0.75cm}||}\n \\hline\n Ref. & Tech. & Proto. & Environ. & Att. & Mat. & Exp. & Det. & Perf.\\\\\n \\hline\\hline\n ~\\cite{41_al2016oscids} & ontology & Modbus TCP & simulation & mixed & M & L & M & L \\\\\n ~\\cite{43_cruz2016cybersecurity} & machine-learning & Modbus TCP & testbed & protocol & H & H & M & H \\\\\n ~\\cite{46_udd2016exploiting} & specification & \\gls{iec104} & simulation & channel & H & L & M & H  \\\\\n ~\\cite{51_yang2016multidimensional} & specification & IEC 61850 & testbed & packet & H & M & L & H\\\\\n ~\\cite{52_adepu2018distributed} & automata & Eth \/ IP & testbed & location & H & H & L & N\/A\\\\\n ~\\cite{55_lin2016runtime} & semantic & DNP3 & simulation & control & H & L & M & L \\\\\n ~\\cite{57_wang2018intrusion} & time series & \\gls{iec104} & simulation & \\gls{scada} & L & L & M & N\/A\\\\\n our \\gls{sids} & specification & \\gls{iec104} & testbed & \\gls{fdi} & H & H & H & M \\\\\n  \\hline\n\\end{tabular}\n\\label{tab:evalation_comparision}\n\\end{table}\nThe comparison compares our \\gls{sids} qualitatively with other approaches based on the following metrics:\n\\begin{itemize}\n \\item Tech.: describes the detection basis of the methodology\n \\item Proto.: describes which protocol is the main target of protection\n \\item Environ.: describes whether a simulated or physical testbed was used\n \\item Att.: describes on which basis the attack scenarios were designed\n \\item Mat.: describes the degree of readiness of the approach in likert-scale\n \\item Exp.: describes the flexibility to be adapted to other protocols in likert-scale\n \\item Det.: describes the degree of detection quality of the approach in likert-scale\n \\item Perf.: describes the performance level of the approach in likert-scale\n\\end{itemize}\nAs the comparison shows, the conditions and environment under which the different approaches were evaluated are mostly different.\nThe attack scenarios also diverge in their scope, vectors used, and interaction with operational equipment.\nThe experiments conducted also differ within their respective environments where simulation was used with simplification and abstraction.\nMany of the approaches have a high degree of maturity and are capable of being deployed and operated in real grid environments.\nHowever, they lack the ability to be extended to other protocols.\nThe performance of the approaches shows the recognition capabilities of packets within the time span $0.1ms$ to $1s$, and the detection quality is also in the medium range, which is mainly due to the high \\gls{fn}.\nThus, the evaluation suggests that our \\gls{sids} enables reliable detection of cyber-attacks within a reasonable time.\n\n\\subsection*{Discussion} \\label{subsec:result_dis}\nThe results show that for normal operation, our \\gls{sids} has not triggered any (false) alert messages.\nDeviations were only caused by a too narrow \\gls{rtt} range and should be considered when carefully setting this parameter for detection quality.\n\nIn the attack-induced scenario (Scenario 2-b-II), where the attacker knows which addresses and \\gls{ioa} entries are valid, \\gls{sids} will have difficulty detecting them if the manipulated values are within the technical specification.\nHowever, any deviation from the addresses defined in the specifications will lead to high detection rates.\nIn general, the detection quality is very dependent on the provided \\gls{sb}.\nThe given structure of the \\gls{sb}, which defines the exact addressing for each allowed connection, are very strict rules that detect all connections that are not explicitly allowed.\nAttacks from outside with limited knowledge of the technical specifications of infrastructures can thus be reliably detected.\n\nTo create perfect spoofing conditions, the attacker must maintain complete consistency and compliance with the specification, which requires extensive knowledge.\nFurthermore, the attacker must perform prior steps, such as reconnaissance and lateral movement, to persist in the process network, potentially leaving traces in the communication layer.\nIn the context of situational awareness for intrusion detection, our \\gls{sids} can act as a low-level sensor that provides domain-specific indicators of multi-staged cyber-attacks.\nAlerts can be centrally processed with other indicators from other \\gls{ids} sensors through a correlation system based on \\gls{siem} to reconstruct the attack sequence~\\cite{24_sen2021towards,sen2022contextual}.\n\nWhile our evaluation focuses on \\gls{iec104}, the proposed \\gls{sids} can also be used for other \\gls{scada} protocols such as IEC-61850.\nThe semantics of \\gls{sb} is provided by \\gls{gim}, where the adaptation of a new protocol requires the mapping process of the data and the fields of the protocol.\nThus, an appropriate mapper must be developed to reference semantic data with protocol fields.\nIn addition, \\gls{am} can generally be adapted to stateful communication such as TCP-based protocols, where packet flows can be described with state transitions.\n\n\\section*{Conclusion} \\label{sec:conclusion}\nIn the context of power grids transitioning to \\glspl{sg}, countermeasures against sophisticated cyber-attacks based on reliable detection mechanisms are required.\nTo this end, we present a \\gls{sids} that uses a graph-based specification to holistically encapsulate the \\gls{sg} infrastructure to detect cyber-attacks.\nWe discuss the design and subsequent implementation of our \\gls{sids}, which consists of a \\gls{dpi} component and an \\gls{am}.\nUsing our implementation, we evaluated the detection quality within a physical testbed for different scenarios under attack and normal conditions.\n\nOur main findings are that our \\gls{sids} approach can reliably detect attackers injecting false data into intercepted \\gls{iec104} channels.\nThe performance and detection quality show the advantages of an approach \\gls{sids} and was validated in our study.\nMoreover, the disadvantage of high knowledge provisioning overhead is reduced by our novel approach of coupling infrastructure modeling with \\gls{sids}.\nFuture work includes investigating different methods for detecting \\gls{fdi} in a cooperative, neighborhood-oriented manner.\nIn addition, the generated alerts of the proposed \\gls{sids} will also be investigated in terms of providing a reliable basis for a higher-level correlation system for reconstructing complex attack campaigns.\n\n\n\\begin{backmatter}\n\n\\printglossary\n\n  \n\\section*{Funding} \n\\anon{This work has partly been funded by the German Federal Ministry for Economic Affairs and Climate Action (BMWK) under project funding reference 0350028.}\n\n\\section*{Availability of data and materials}\n  No data and materials are published.\n\n\\section*{Author's contributions}\n\\anon{\n \\begin{inparaenum}[i)]\n  \\item  Conceptualization: \u00d6mer Sen;\n  \\item  Methodology: \u00d6mer Sen, Maik L\u00fchmann, Florian Spr\u00fcnken;\n  \\item  Validation:  \u00d6mer Sen, Maik L\u00fchmann, Florian Spr\u00fcnken, Martin Henze;\n  \\item  Formal analysis: Maik L\u00fchmann, Florian Spr\u00fcnken, \u00d6mer Sen;\n  \\item  Investigation: Maik L\u00fchmann, Florian Spr\u00fcnken, \u00d6mer Sen, Martin Henze;\n  \\item  Resources: Maik L\u00fchmann, Florian Spr\u00fcnken, \u00d6mer Sen;\n  \\item  Data Curation: \u00d6mer Sen, Maik L\u00fchmann, Florian Spr\u00fcnken;\n  \\item  Writing - Original Draft: \u00d6mer Sen;\n  \\item  Writing - Review \\& Editing: \u00d6mer Sen, Maik L\u00fchmann, Florian Spr\u00fcnken, Martin Henze, Dennis van der Velde, Immanuel Hacker, Andreas Ulbig;\n  \\item  Visualization: \u00d6mer Sen, Immanuel Hacker;\n  \\item  Supervision: \u00d6mer Sen, Martin Henze, Andreas Ulbig;\n  \\item  Project administration: Michael Andres, Dennis van der Velde;\n  \\item  Funding acquisition: Michael Andres;\n \\end{inparaenum}\n}\n\n\\section*{Competing interests}\n  The authors declare that they have no competing interests.\n\n\\bibliographystyle{bmc-mathphys}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzkwsc b/data_all_eng_slimpj/shuffled/split2/finalzzkwsc
new file mode 100644
index 0000000000000000000000000000000000000000..9f911c5a34850005b8c02727965c3218798afb9f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzkwsc
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\n\\noindent This paper considers a sample-selection model \\citep[e.g.,][]{Heckman79} given by\n\\begin{eqnarray}\nY^* &=&\\theta_0+\\bm{X}^{\\top}\\bm{\\beta}_0+U,\\label{outcome}\\\\\nD &=& 1\\left\\{\\bm{Z}^{\\top}\\bm{\\gamma}_0\\ge V\\right\\},\\label{selection}\\\\\nY &=& DY^*,\\label{observation}\n\\end{eqnarray}\nwhere $[\\begin{array}{cccc} D & \\bm{X}^{\\top} & \\bm{Z}^{\\top} & Y\\end{array}]$ is an observed random vector, \nwhere \n$[\\begin{array}{cc} U & V \\end{array}]$ is an unobserved random vector such that $E\\left[U^2\\right]<\\infty$ and \n$E[U|\\bm{X}]=0$ and $E[U|\\bm{Z}]=E[U]$ almost surely.  Equations~(\\ref{outcome}) and \n(\\ref{selection}) are typically referred to as the {\\it outcome} and {\\it selection} equations, respectively.  \nVariants of the model given in (\\ref{outcome})--(\\ref{observation}) have been considered at least since the \ncontributions of \\citet{Gronau74,Heckman74} and \\citet{Lewis74}.  These authors were primarily concerned with the \nissue of selectivity bias in empirical analyses of individual labour-force participation decisions, particularly \nfor women.  This bias, which is present in least-squares estimates of the parameter \n$[\\begin{array}{cc} \\theta_0 & \\bm{\\beta}^{\\top}_0\\end{array}]$ appearing above in (\\ref{outcome}), arises from \nthe assumption that the observed wages of workers are affected by the self selection of \nthose workers into the workforce.  In particular, one can only observe a wage that exceeds the reservation wage \nof the individual in question.  In terms of the model given in (\\ref{outcome})--(\\ref{observation}), the wage \noffer is represented by the variable $Y^*$, while the observed wage is denoted by $Y$.  The employment status of \nthe individual in question is denoted by the binary variable $D$, which takes a value of one if the unobserved \ndifference $\\bm{Z}^{\\top}\\bm{\\gamma}_0-V$ between the wage offer and the reservation wage is positive; $D$ is \notherwise equal to zero.  Observed variables influencing individual participation decisions are collected in the \nvector $\\bm{Z}$, while observed determinants of individual wage offers are collected in $\\bm{X}$.  Variants of the \nsample-selection model appearing in (\\ref{outcome})--(\\ref{observation}) have been found useful for a wide variety of \napplied problems apart from the analysis of indidivual labour supply decisions \\citep[e.g.,][]{Vella98}.  Relatively \nrecent economic applications include those of \\citet{HelpmanMelitzRubinstein08}, \\citet{MulliganRubinstein08} and \n\\citet{JimenezOngenaPeydroSaurina14}.\n\nThis paper focuses on statistical inference regarding the intercept $\\theta_0$ appearing above in (\\ref{outcome}).  \nThe intercept in the outcome equation is often of inherent interest in various applications of the model given by \n(\\ref{outcome})--(\\ref{observation}).  For example, suppose that (\\ref{selection}) accurately describes the \nselection of individuals into some treatment group.  In this case, the difference between the intercepts in the \noutcome equations for treated and non-treated individuals may be interpreted as the causal effect of treatment \nwhen selection to treatment is mean independent of the unobservable $U$ in the outcome equation \\citep[e.g.,][p. 500]{AndrewsSchafgans98}.  The intercept in the outcome \nequation is similarly crucial in computing the average wage difference in problems where the sample-selection \nmodel given above is applied to the analysis of wage differences between workers in two different socioeconomic \ngroups, or between unionized and non-unionized workers \n\\citep[e.g.,][]{Oaxaca73,Lewis86,Heckman90}.  Finally, the intercept in the outcome equation permits the \nevaluation of the net benefits of a social program in terms of the differences between the observed \noutcomes of participants and their counterfactual expected outcomes had they chosen not to participate \n\\citep[e.g.,][]{HeckmanRobb85}.  \n\nEarly applied work generally proceeded from the assumption that the unobservables \n$[\\begin{array}{cc} U & V\\end{array}]$ appearing above in (\\ref{outcome})--(\\ref{selection}) are bivariate normal \nmean-zero with an unknown covariance matrix and independent of \n$[\\begin{array}{cc} \\bm{X}^{\\top} & \\bm{Z}^{\\top}\\end{array}]$.  This assumption in turn allowed for the \nestimation of the parameters appearing in (\\ref{outcome})--(\\ref{selection}) via the method of maximum likelihood \nor the related two-step procedure of \\citet{Heckman74,Heckman76}.  These estimates, however, are generally \ninconsistent under departures from the assumed bivariate normality of $[\\begin{array}{cc} U & V\\end{array}]$ \n\\citep[e.g.,][]{ArabmazarSchmidt82,Goldberger83,Schafgans04}.  The desirability of not imposing a parametric \nspecification on the joint distribution of the unobservables in the outcome and selection equations has led in turn \nto the development of distribution-free methods of estimating the parameters appearing above in \n(\\ref{outcome})--(\\ref{selection}).  Distribution-free methods of estimating the intercept $\\theta_0$ in \n(\\ref{outcome}) include the proposals of \\citet{GallantNychka87,Heckman90,AndrewsSchafgans98} and \\citet{Lewbel07}.  \n\nEstimators of the intercept of the outcome equation implemented by distribution-free procedures have \nlarge-sample behaviours that vary depending on the extent of endogeneity in the selection mechanism, \ni.e., on the nature and extent of any dependence between the random variables $U$ and $V$ appearing above in (\\ref{outcome}) \nand (\\ref{selection}), respectively.  Given that these features of the joint distribution of \n$[\\begin{array}{cc} U & V \\end{array}]$ are typically unknown \nin empirical practice, the dependence of the asymptotic behaviour of intercept estimators on these features potentially \ncomplicates statistical inference regarding $\\theta_0$.  This issue is easily and starkly illustrated in the case of \nordinary least squares (OLS).  In particular, suppose that selection in the model is based strictly on observables, which \nis equivalent to assuming that the unobservable $U$ appearing in (\\ref{outcome}) and the selection indicator $D$ are conditionally \nmean independent given $\\bm{X}$ and $\\bm{Z}$, i.e., $P\\left[E[U|D=1,\\bm{X},\\bm{Z}]=E[U|D=0,\\bm{X},\\bm{Z}]\\right]=1$.  In this case $\\theta_0$ can be \nconsistently estimated at a parametric rate with no additional assumptions \nimposed on the joint distribution of $[\\begin{array}{ccc} U & V & \\bm{Z}^{\\top}\\bm{\\gamma}_0\\end{array}]$ by applying OLS \nto the outcome equation using only those observations for which $D=1$.  On the other hand, the OLS estimate of $\\theta_0$ \nobtained in this way is inconsistent if the difference $1-P\\left[E[U|D=1,\\bm{X},\\bm{Z}]=E[U|D=0,\\bm{X},\\bm{Z}]\\right]$ \nis positive, even if arbitrarily small.  It follows that OLS generates inferences regarding $\\theta_0$ that vary drastically with respect \nto the degree to which $E[U|D=1,\\bm{X},\\bm{Z}]$ may differ from $E[U|D=0,\\bm{X},\\bm{Z}]$.\n\nThis paper develops a distribution-free estimator of the intercept $\\theta_0$ in the outcome equation that is \nconsistent and asymptotically normal with a rate of convergence that is \nthe same regardless of the joint distribution of $[\\begin{array}{ccc} U & V & \\bm{Z}^{\\top}\\bm{\\gamma}_0\\end{array}]$.  I show that there exists an \nimplementation of the proposed estimator of $\\theta_0$ that converges uniformly at the rate $n^{-p\/(2p+1)}$, where $n$ denotes the sample size and where $p\\ge 2$ is an integer \nthat indexes the strength of certain smoothness assumptions described below.  The uniformity of this convergence involves uniformity over a class of joint distributions of \n$[\\begin{array}{ccc} U & V & \\bm{Z}^{\\top}\\bm{\\gamma}_0\\end{array}]$ satisfying necessary conditions for the identification of $\\theta_0$.   In other words, the estimator developed below is {\\it adaptive} to the nature of \nselection in the model.  \n\nThis paper also shows that the uniform $n^{-p\/(2p+1)}$-rate attainable by the proposed estimator is in fact \nthe optimal rate of convergence of an estimator of $\\theta_0$ in terms of a minimax criterion.  It follows that \nthe proposed estimator may be implemented in such a way as to converge in probability to \n$\\theta_0$ at the fastest possible minimax rate.\n   \nThe estimator developed below differs from earlier proposals of \\citet{Heckman90} and \\citet{AndrewsSchafgans98} \nthat involve the {\\it rate-adaptive} estimation of the intercept in the outcome equation.  These proposals also \ninvolve estimators that are consistent and asymptotically normal regardless of the extent to which selection is \nendogenous, but converge to the limiting normal distribution at unknown rates; in particular, see \\citet[Theorems~1--2]{SchafgansZindeWalsh02} for the \nestimator of \\citet{Heckman90} and \\citet[Theorems~2, 3, 5 and 5*]{AndrewsSchafgans98}.  \n\nThe estimator developed below also differs from estimators of $\\theta_0$ that take the form of averages weighted by the \nreciprocal of an estimate of the density of the selection index $\\bm{Z}^{\\top}\\bm{\\gamma}_0$ appearing above in (\\ref{selection}).  Such \nestimators \\citep[e.g.,][]{Lewbel07}, in common with the estimators of \\citet{Heckman90} and \n\\citet{AndrewsSchafgans98}, are known to converge generically at unknown rates.  In addition, estimators \ntaking the form of inverse density-weighted averages may have sampling distributions that are not even \nasymptotically normal \\citep{KhanTamer10,KhanNekipelov13,ChaudhuriHill16}.  In general, the estimators of \n\\citet{Heckman90} and \\citet{AndrewsSchafgans98} and of estimators in the class of \ninverse density-weighted averages converge at rates that depend critically on conditions involving the relative tail \nthicknesses of the distributions of the selection index $\\bm{Z}^{\\top}\\bm{\\gamma}_0$ and of the latent selection \nvariable $V$ appearing in (\\ref{selection}) \\citep{KhanTamer10}.  These conditions may be difficult to verify in \napplications.  The estimator developed below converges by contrast at a known rate under conditions implied by the \nidentification of $\\theta_0$ to a normal distribution uniformly over the underlying parameter space regardless of the \nrelative tail behaviours of $\\bm{Z}^{\\top}\\bm{\\gamma}_0$ and $V$.  This facilitates statistical inference regarding $\\theta_0$.\n\nThe remainder of this paper proceeds as follows.  The following section discusses identification of the \nintercept parameter in the outcome equation and presents the new estimator along with its first-order asymptotic \nproperties.  Section~\\ref{opt} derives the minimax rate optimality of the new estimator.  Section~\\ref{mc} \npresents the results of simulation experiments that investigate the behaviour in finite samples of the proposed \nestimator in relation to other methods.  Section~\\ref{ee} considers an application of the new \nestimator to the analysis of gender wage gaps in Malaysia.  Section~\\ref{concl} concludes.  Proofs of \nall theoretical results are collected in the appendix.\n\n\\section{The New Estimator}\\label{est}\n\n\\noindent This section presents the new estimator of the intercept $\\theta_0$ appearing in (\\ref{outcome}) \nand describes its asymptotic behaviour to first order.  Let $\\hat{\\bm{\\beta}}_n$ and \n$\\hat{\\bm{\\gamma}}_n$ denote $\\sqrt{n}$-consistent estimators of the parameters $\\bm{\\beta}_0$ and \n$\\bm{\\gamma}_0$ appearing above in (\\ref{outcome}) and (\\ref{selection}), respectively, where $\\bm{\\gamma}_0$ is \nassumed to be identified up to a location and scale normalization.  The existence of such \nestimators has long been established; see e.g., the proposals of \\citet{Han87,Robinson88,PowellStockStoker89,Andrews91,IchimuraLee91,Ichimura93,KleinSpady93,Powell01} or \n\\citet{Newey09}.  In addition, suppose that\n$\\left\\{[\\begin{array}{cccc} D_i & \\bm{X}_i^{\\top} & Y_i & \\bm{Z}_i^{\\top} \\end{array}]:\\, i=1,\\ldots,n\\right\\}$ are iid \ncopies of the random vector $[\\begin{array}{cccc} D & \\bm{X}^{\\top} & Y & \\bm{Z}^{\\top} \\end{array}]$.  Let \n$\\bm{z}\\in\\mathbb{R}^{l}$ be an arbitrary vector, and define \n\\begin{equation}\\label{etahatn}\n\\hat{\\eta}_n(\\bm{z})\\equiv \\frac{1}{n}\\sum_{i=1}^n 1\\left\\{(\\bm{Z}_i-\\bm{z})^{\\top}\\hat{\\bm{\\gamma}}_n\\le 0\\right\\}\n\\end{equation}\n\nLet \n\\begin{equation}\\label{what}\n\\hat{W}_i\\equiv D_i\\left(Y_i-\\bm{X}^{\\top}_i\\hat{\\bm{\\beta}}_n\\right)\n\\end{equation}\nfor each $i\\in\\{1,\\ldots,n\\}$.  This paper proposes to estimate $\\theta_0$ via a locally linear smoother of the form\n\\begin{equation}\\label{mhat}\n\\hat{\\theta}_n\\equiv \\bm{e}_1^{\\top}\\left(\\sum_{i=1}^n \\bm{S}_i K_i\\bm{S}_i^{\\top}\\right)^{-1}\\sum_{i=1}^n\\bm{S}_i K_i\\hat{W}_i,\n\\end{equation}\nwhere $\\bm{e}_1=[\\begin{array}{cc} 1 & 0\\end{array}]^{\\top}$ and where \n\\begin{equation}\\label{si}\n\\bm{S}_i=[\\begin{array}{cc} 1 & \\hat{\\eta}_n(\\bm{Z}_i)-1\\end{array}]^{\\top}\n\\end{equation} \nand \n\\begin{equation}\\label{ki}\nK_i= K\\left(h_n^{-1}\\left(\\hat{\\eta}_n(\\bm{Z}_i)-1\\right)\\right)\n\\end{equation} \nfor $i=1,\\ldots,n$.  The quantity $h_n$ appearing in each \n$K_i$ denotes a bandwidth $h_n>0$ such that $h_n\\to 0$ with $nh_n^{3}\\to\\infty$ as $n\\to\\infty$, while for some \n$p\\ge 2$, $K(\\cdot)$ denotes a smoothing kernel of order $p$, i.e., one where \n$\\int_{-\\infty}^{\\infty} K(u) du=1$, \n$\\int_{-\\infty}^{\\infty} u^r K(u)=0$ for all $r\\in \\{1,\\ldots,p-1\\}$ and \n$\\int_{-\\infty}^{\\infty} u^{p} K(u)<\\infty$.\n\nAssume that the disturbance $U$ in the outcome equation (\\ref{outcome}) satisfies $E[|U|]<\\infty$ with \n$P\\left[E[U|\\bm{Z}]=0\\right]=1$.  In addition, let the selection index $\\bm{Z}^{\\top}\\bm{\\gamma}_0$ be \ndistributed with distribution $F_0$, assumed to be absolutely continuous.    The estimator of $\\theta_0$ given in (\\ref{mhat}) exploits \nthe fact that identification of $\\theta_0$ occurs ``at infinity'' \\citep{Chamberlain86}, or in any case depends crucially on the \nselection index $\\bm{Z}^{\\top}\\bm{\\gamma}_0$ being able to take values sufficiently large so that the corresponding \nconditional probabilities of selection take values close to one.  In particular, $\\theta_0$ is characterized by the equalities\n\\begin{eqnarray}\n & & E\\left[\\left.D\\left(Y-\\bm{X}^{\\top}\\bm{\\beta}_0\\right)\\right|F_0\\left(\\bm{Z}^{\\top}\\bm{\\gamma}_0\\right)=1\\right]\\nonumber\\\\\n &=& E\\left[\\left.1\\left\\{F_0(V)\\le 1\\right\\}\\left(\\theta_0+U\\right)\\right|F_0\\left(\\bm{Z}^{\\top}\\bm{\\gamma}_0\\right)=1\\right]\\nonumber\\\\\n &=&\\theta_0\\label{theta0_id}\n\\end{eqnarray} \nThe proposed estimator of $\\theta_0$ exploits the representation of the estimand in (\\ref{theta0_id}), which suggests the estimation of $\\theta_0$ by direct \nestimation of the quantity \n$E\\left[\\left.D\\left(Y-\\bm{X}^{\\top}\\bm{\\beta}_0\\right)\\right|F_0\\left(\\bm{Z}^{\\top}\\bm{\\gamma}_0\\right)=1\\right]$.\n\nOne can view the estimator given in (\\ref{mhat}) as an extension of the Yang--Stute symmetrized nearest-neighbours \n(SNN) estimator of a conditional mean \\citep{Yang81,Stute84} to the problem of estimating the intercept \nin the outcome equation (\\ref{outcome}).  SNN estimators are characterized by asymptotic behaviours that \nare asymptotically ``design adaptive'' in the sense that their asymptotic normality can generally be established without technical conditions on the \nprobability of the design variable taking values in regions of low density \\citep{Stute84}.  In the present context, it is shown under certain conditions \nthat the estimator $\\hat{\\theta}_n$ in (\\ref{mhat}) is asymptotically normal with a \nrate of convergence that depends neither on the extent to which \nthe unobservables $U$ and $V$ are dependent, nor on the relationship between the behaviours of the selection index\n$\\bm{Z}^{\\top}\\bm{\\gamma}_0$ and the unobservable $V$ in the right tails of their respective marginal distributions.  \nThis is essentially accomplished by transforming the estimated indices $\\bm{Z}^{\\top}_i\\hat{\\bm{\\gamma}}_n$ into random \nvariables $\\hat{\\eta}_n(\\bm{Z}_i)$ that are approximately uniformly distributed on $[0,1]$.  \n\nIt is worth noting that estimators of $\\theta_0$ that take the form of inverse density-weighted averages \n\\citep[e.g.,][]{Lewbel07} have rates of convergence that generally vary with the extent to which $U$ and $V$ are \ndependent, as well as with the relative right-tail behaviours of $\\bm{Z}^{\\top}\\bm{\\gamma}_0$ and $V$ \n\\citep{KhanTamer10}.  These estimators rely on an alternative representation of (\\ref{theta0_id}) and are consistent \nunder additional regularity conditions.  In particular, suppose \n$m_0\\left(\\bm{z}^{\\top}\\bm{\\gamma}_0\\right)\\equiv E\\left[\\left. D\\left(Y-\\bm{X}^{\\top}\\bm{\\beta}_0\\right)\\right|\\bm{Z}^{\\top}\\bm{\\gamma}_0=\\bm{z}^{\\top}\\bm{\\gamma}_0\\right]$ \nis everywhere differentiable in $\\bm{z}^{\\top}\\bm{\\gamma}_0$ with derivative given by \n$m_0^{(1)}\\left(\\bm{z}^{\\top}\\bm{\\gamma}_0\\right)\\equiv \\partial m_0\\left(\\bm{z}^{\\top}\\bm{\\gamma}_0\\right)\/\\partial\\bm{z}^{\\top}\\bm{\\gamma}_0$.  \nSuppose in addition that the distribution of $\\bm{Z}^{\\top}\\bm{\\gamma}_0$ is absolutely continuous with density \n$f_0(\\cdot)$ such that \n$E\\left[\\left|m_0^{(1)}\\left(\\bm{Z}^{\\top}\\bm{\\gamma}_0\\right)\/f_0\\left(\\bm{Z}^{\\top}\\bm{\\gamma}_0\\right)\\right|\\right]<\\infty$.  One \ncan then write\n\\begin{eqnarray}\n\\theta_0 &=& \\lim_{F_0(\\bm{z}^{\\top}\\bm{\\gamma}_0)\\to 1} E\\left[\\left. D\\left(Y-\\bm{X}^{\\top}\\bm{\\beta}_0\\right)\\right|\\bm{Z}=\\bm{z}\\right]\\nonumber\\\\\n\t&=&\\int_{-\\infty}^{\\infty} m^{(1)}_0\\left(\\bm{z}^{\\top}\\bm{\\gamma}_0\\right) d\\bm{z}^{\\top}\\bm{\\gamma}_0\\nonumber\\\\\n\t&=&\\int_{-\\infty}^{\\infty} \\frac{m_0^{(1)}\\left(\\bm{z}^{\\top}\\bm{\\gamma}_0\\right)}{f_0\\left(\\bm{z}^{\\top}\\bm{\\gamma}_0\\right)}\\cdot f_0\\left(\\bm{z}^{\\top}\\bm{\\gamma}_0\\right)d\\bm{z}^{\\top}\\bm{\\gamma}_0\\nonumber\\\\\n\t&=& E\\left[\\frac{m_0^{(1)}\\left(\\bm{Z}^{\\top}\\bm{\\gamma}_0\\right)}{f_0\\left(\\bm{Z}^{\\top}\\bm{\\gamma}_0\\right)}\\right],\\label{id5}\n\\end{eqnarray}\nwhich suggests estimating $\\theta_0$ via its approximate sample analogue in which $m_0^{(1)}(\\cdot)$ and \n$f_0(\\cdot)$ are replaced by suitable preliminary estimates, and in which the systematic trimming of observations \ncorresponding to small values of $f_0\\left(\\bm{Z}^{\\top}_i\\bm{\\gamma}_0\\right)$ may be required.  This approach \nto estimating $\\theta_0$ follows that proposed by \\citet{Lewbel97} for estimating a binary choice model arising \nfrom a latent linear model in which a mean restriction is imposed on the latent error term, and is the \napproach to estimating $\\theta_0$ considered in more recent work by \\citet{Lewbel07,KhanTamer10} and \n\\citet{KhanNekipelov13}.  Consistent estimators of $\\theta_0$ that exploit (\\ref{id5}) in this way naturally \ndepend critically on the assumed finiteness of \n$E\\left[\\left|m_0^{(1)}\\left(\\bm{Z}^{\\top}\\bm{\\gamma}_0\\right)\/f_0\\left(\\bm{Z}^{\\top}\\bm{\\gamma}_0\\right)\\right|\\right]$, \nan assumption that in turn leads the rates at which they converge to their limiting distributions to depend on the \nrelative tail behaviours of the variables in the selection equation or the extent to which selection is endogenous \n\\citep{KhanTamer10}.  The non-uniformity in the rate of convergence as one varies the relative tail behaviours of the \ndeterminants of selection or the dependence between the disturbance terms $U$ and $V$ is a feature that is also shared by \nestimators of $\\theta_0$ that involve locally constant or polynomial regressions of $\\hat{W}_i$ on the \nuntransformed estimated selection indices \n$\\bm{Z}^{\\top}_i\\hat{\\bm{\\gamma}}_n$ \\citep[e.g.,][]{Heckman90,AndrewsSchafgans98}.   Results of this \nnature significantly complicate the task of statistical inference regarding $\\theta_0$.  By way of contrast, the \ntransformations to $\\hat{\\eta}_n(\\bm{Z}_i)$ of the selection indices used in the locally linear \nSNN estimator given in (\\ref{mhat}) permit the locally linear SNN estimator to enjoy asymptotic normality with a rate \nof convergence that varies neither with the endogeneity of selection nor with the relative tail behaviours of the \nvariables appearing in the selection equation.\n\nIt should also be noted that the result given above in (\\ref{theta0_id}), in which the estimand is identified as\n$\\theta_0=E\\left[\\left.D\\left(Y-\\bm{X}^{\\top}\\bm{\\beta}_0\\right)\\right|F_0(\\bm{Z}^{\\top}\\bm{\\gamma}_0)=1\\right]$, \nmotivates the formulation of the estimator in (\\ref{mhat}) as the intercept in a locally linear regression.  \nOne could as easily in this context use (\\ref{theta0_id}) to motivate an estimator of $\\theta_0$ as the corresponding \nvariant of a Nadaraya--Watson (i.e., locally constant regression) estimator; see in particular the approach taken \nin \\citet{StuteZhu05}.  The focus on a locally linear regression estimator of $\\theta_0$ is purely to improve the \nrate at which the bias of the proposed estimator vanishes in large samples, given that locally linear regression estimators \nhave biases that converge at the same rate regardless of whether the conditioning variable is evaluated at an interior or at a limit point of \nits support \\citep[e.g.,][]{FanGijbels92}.  Nadaraya--Watson estimators, on the other hand, have biases that \nconverge relatively slowly when the conditioning variable is evaluated at a limit point of its support.\n\nAssumptions underlying the first-order asymptotic behaviour of the estimator given by $\\hat{\\theta}_n$ in \n(\\ref{mhat}) are given as follows.\n\n\\begin{assumption}\\label{a1}\n\\begin{enumerate}\n\\item\\label{a1a} \n\\begin{enumerate}\n\\item $\\bm{X}$ is $k$-variate, with support not contained in any proper linear subspace of \n$\\mathbb{R}^k$;\n\\item $E\\left[\\left\\|\\bm{X}\\right\\|\\right]<\\infty$.\n\\end{enumerate} \n\\item\\label{a1b}\n\\begin{enumerate} \n\\item $\\bm{Z}$ is $l$-variate;\n\\item the support of $\\bm{Z}$ is not contained in any proper linear subspace of $\\mathbb{R}^{l}$,;\n\\item the first component of $\\bm{\\gamma}_0$ is equal to one;\n\\item $\\bm{Z}$ does not contain a non-stochastic component;\n\\item\\label{a1b4} the distribution $F_0$ of the selection index $\\bm{Z}^{\\top}\\bm{\\gamma}_0$ is absolutely \ncontinuous, with a density function $f_0$ that is differentiable on the support of \n$\\bm{Z}^{\\top}\\bm{\\gamma}_0$.\n\\end{enumerate}\n\\item\\label{a1c} The set $\\left\\{(D_i,\\bm{X}^{\\top}_i, \\bm{Z}^{\\top}_i, Y_i):\\,i=1,\\ldots,n\\right\\}$ consists of \nindependent observations each with the same distribution as the random vector \n$(D, \\bm{X}^{\\top}, \\bm{Z}^{\\top}, Y)$, which is generated according to the model given above in \n(\\ref{outcome})--(\\ref{observation}), and where \n\\begin{eqnarray*}\nE\\left[U^2\\right] &<& \\infty;\\\\\nP\\left[E[U|\\bm{X}]=0\\right] &=& 1;\\\\\nP\\left[E[U|\\bm{Z}] = E[U]\\right] &=& 1.\n\\end{eqnarray*} \n\\item\\label{a1d} \n\\begin{enumerate}\n\\item The joint conditional distribution given \n$[\\begin{array}{cc} \\bm{X}^{\\top} & \\bm{Z}^{\\top}\\end{array}]=[\\begin{array}{cc} \\bm{x}^{\\top} & \\bm{z}^{\\top}\\end{array}]$ of the \ndisturbances $[\\begin{array}{cc} U & V\\end{array}]$ appearing in (\\ref{outcome}) and (\\ref{selection}) is absolutely continuous for all \n$[\\begin{array}{cc} \\bm{x}^{\\top} & \\bm{z}^{\\top}\\end{array}]$ in the support of $[\\begin{array}{cc} \\bm{X}^{\\top} & \\bm{Z}^{\\top}\\end{array}]$; \nmoreover the corresponding joint density $g_{U,V|\\bm{x},\\bm{z}}(\\cdot,\\cdot)$ is continuously differentiable in both arguments almost everywhere on $\\mathbb{R}^2$.\n\\item\\label{a1d2} The conditional distribution given $[\\begin{array}{cc} \\bm{X}^{\\top} & \\bm{Z}^{\\top}\\end{array}]=[\\begin{array}{cc} \\bm{x}^{\\top} & \\bm{z}^{\\top}\\end{array}]$ \nof $V$ is absolutely continuous for all points $[\\begin{array}{cc} \\bm{x}^{\\top} & \\bm{z}^{\\top}\\end{array}]$ in the support of \n$[\\begin{array}{cc} \\bm{X}^{\\top} & \\bm{Z}^{\\top}\\end{array}]$; the corresponding density $g_{V|\\bm{x},\\bm{z}}(\\cdot)$ is\ndifferentiable almost everywhere on $\\mathbb{R}$.\n\\end{enumerate}   \n\\end{enumerate}\n\\end{assumption} \n\n\\begin{assumption}\\label{a2}\n\\begin{enumerate}\n\\item\\label{a2a} There exist estimators $\\hat{\\bm{\\beta}}_n$ and $\\hat{\\bm{\\gamma}}_n$ such that \n$\\left\\|\\hat{\\bm{\\beta}}_n-\\bm{\\beta}_0\\right\\|=O_p\\left(n^{-1\/2}\\right)$ \nand $\\left\\|\\hat{\\bm{\\gamma}}_n-\\bm{\\gamma}_0\\right\\|=O_p\\left(n^{-1\/2}\\right)$.  \n\\item\\label{a2b} The smoothing kernel $K(\\cdot)$ is bounded and twice continuously differentiable with \n$K(u)>0$ on $[0,1]$, $K(u)=0$ for all $u\\not\\in [0,1]$, with $\\int K(u) du=1$ and where $\\int K^2(u) du<\\infty$.\n\\item\\label{a2c} The bandwidth sequence $\\left\\{h_n\\right\\}$ satisfies $h_n>0$ with $h_n\\to 0$ as $n\\to\\infty$, \nand $nh_n^3\\to\\infty$.\n\\item\\label{a2d}  \n\\begin{enumerate}\n\\item \\label{a2d1} There exists $p\\ge 2$ such that $\\int u^j K(u) du=0$ for all $j\\in\\{1,\\ldots,p-1\\}$ and $\\int u^p K(u) du <\\infty$.\n\\item\\label{a2d11} The following hold for $p^*$ equal to the smallest odd integer greater than or equal to the constant $p+1$ specified \nin part~\\ref{a2d1} of this \nassumption: \n\\begin{enumerate}\n\\item\\label{a2d2} The joint conditional density $g_{U,V|\\bm{x},\\bm{z}}(\\cdot,\\cdot)$ specified in Assumption~\\ref{a1}.\\ref{a1d} is $p^*$-times \ncontinuously differentiable in both arguments almost everywhere on $\\mathbb{R}$ for all \n$[\\begin{array}{cc} \\bm{x}^{\\top} & \\bm{z}^{\\top}\\end{array}]$ in the support of $[\\begin{array}{cc} \\bm{X}^{\\top} & \\bm{Z}^{\\top}\\end{array}]$.\n\\item\\label{a2d3} Similarly, the conditional density $g_{V|\\bm{x},\\bm{z}}(\\cdot)$ specified in Assumption~\\ref{a1}.\\ref{a1d} is $p^*$-times \ncontinuously differentiable almost everywhere on $\\mathbb{R}$ for all $[\\begin{array}{cc} \\bm{x}^{\\top} & \\bm{z}^{\\top}\\end{array}]$ in the support of \n$[\\begin{array}{cc} \\bm{X}^{\\top} & \\bm{Z}^{\\top}\\end{array}]$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{enumerate}\n\\end{assumption}\n\n\\noindent The conditions of Assumption~\\ref{a1} are largely standard and notably suffice for the selection parameter $\\bm{\\gamma}_0$ \nto be identified up to the particular location and scale normalization imposed by Assumption~\\ref{a1}.\\ref{a1b}.  \nAssumption~\\ref{a1} also does not restrict the components $V$ and $\\bm{Z}^{\\top}\\bm{\\gamma}_0$ of the selection equation to be \nindependent. \n\nAssumption~\\ref{a1} plays a crucial role in controlling the asymptotic bias of the proposed estimator $\\hat{\\theta}_n$.  In \nparticular, identification of $\\bm{\\gamma}_0$ subject to the conditions of Assumption~\\ref{a1}, along with the \ndifferentiability conditions of Assumption~\\ref{a2}.\\ref{a2d2}--\\ref{a2}.\\ref{a2d3}, imply a smoothness restriction on \nthe conditional mean function\n\\begin{equation}\\label{m0F}\nm_{F_0}(q)\\equiv E\\left[D\\left(Y-\\bm{X}^{\\top}\\bm{\\beta}_0\\right)\\left|F_0\\left(\\bm{Z}^{\\top}\\bm{\\gamma}_0\\right)=q\\right.\\right],\n\\end{equation}\nThis smoothness restriction takes the form of differentiability of $m_{F_0}(q)$ for $q\\in (0,1)$ up to order no less than $p$, where $p\\ge 2$ is the \nconstant specified in Assumption~\\ref{a2}.\\ref{a2d}, along with finiteness of the left-hand limit of \n$\\left(d^p\/dq^p\\right) E\\left[D\\left(Y-\\bm{X}^{\\top}\\bm{\\beta}_0\\right)\\left|F_0\\left(\\bm{Z}^{\\top}\\bm{\\gamma}_0\\right)=q\\right.\\right]$ \nat $q=1$.  This smoothness restriction, in other words, corresponds to a standard assumption in the literature on kernel estimation of \nconditional mean functions.  On the other hand, the differentiability to $p^*$-order in the second argument of \n$g_{U,V|\\bm{x},\\bm{z}}(\\cdot,\\cdot)$ is slightly stronger than the usual assumption of differentiability to \norder $p$.  This slight strengthening of the standard differentiability condition is used in the rate optimality arguments developed below in \nSection~\\ref{opt}.  Details are contained in the proof of Theorem~\\ref{ubthm} below. \n\nThe smoothness restriction on $m_{F_0}(q)$ given in (\\ref{m0F}) can also be seen to be implied by the identification of \n$\\bm{\\gamma}_0$ up to a location and scale normalization and by the smoothness conditions imposed in Assumption~\\ref{a2}.\\ref{a2d} on \nthe conditional densities $g_{U,V|\\bm{x},\\bm{z}}(\\cdot,\\cdot)$ and $g_{V|\\bm{x},\\bm{z}}(\\cdot,\\cdot)$ for any \n$[\\begin{array}{cc} \\bm{x}^{\\top} & \\bm{z}^{\\top}\\end{array}]$ in the support of $[\\begin{array}{cc} \\bm{X}^{\\top} & \\bm{Z}^{\\top}\\end{array}]$.  In \nparticular, one can show that under the conditions of Assumption~\\ref{a1} and \\ref{a2}.\\ref{a2d}, $U$ has a conditional distribution given \n$F_0(V)=q$ for any $q\\in [0,1]$ that is absolutely continuous with density\n\\begin{equation}\\label{condl_dens}\nr_{U|Q}(u|q)\\equiv\\frac{g_{UV}\\left(u,F_0^{-1}(q)\\right)}{g_V\\left(F^{-1}_0(q)\\right)},\n\\end{equation}\nwhere $g_{UV}(\\cdot,\\cdot)$ and $g_V(\\cdot)$ are respectively the joint density of $[\\begin{array}{cc} U & V\\end{array}]$ and the marginal density of \n$V$.  The conditional density $r_{U|Q}(u|q)$ is, given the absolute continuity of $F_0$ and the differentiability conditions on $g_{UV}$ and $g_V$ \nimplied by Assumption~\\ref{a2}.\\ref{a2d}, $(p+1)$-times differentiable in $q$ on $(0,1)$ for any $u\\in\\mathbb{R}$.  The $(p+1)$-times \ndifferentiability of $r_{U|Q}(u|q)$ in $q$ on $(0,1)$ in turn implies the finiteness of \n$\\left.\\left(\\partial^p\/\\partial q^p\\right) r_{U|Q}(u|q)\\right|_{q=1}$ for any $u\\in\\mathbb{R}$.  It is the finiteness of \n$\\left.\\left(\\partial^p\/\\partial q^p\\right) r_{U|Q}(u|q)\\right|_{q=1}$ that implies the smoothness restrictions on $m_{F_0}(q)$ mentioned above.  \nFurther details are supplied below in Appendix~\\ref{lemmabound}.\n\nLet $\\sigma^2_{U|F_0(\\bm{Z}^{\\top}\\bm{\\gamma}_0)}(q)\\equiv E\\left[U^2\\left|F_0\\left(\\bm{Z}^{\\top}\\bm{\\gamma}_0\\right)=q\\right.\\right]$, where $U$ is \nthe disturbance in the outcome equation (\\ref{outcome}).  The following result summarizes the large-sample behaviour to first order of the proposed \nestimator:\n\n\\begin{theorem}\\label{mainthm}\nUnder the conditions of Assumptions~\\ref{a1} and \\ref{a2}, the estimator $\\hat{\\theta}_n$ given above in \n(\\ref{mhat}) satisfies\n\\[\n\\sqrt{nh_n}\\left(\\hat{\\theta}_n-\\theta_0-\\frac{h_n^p}{p!}\\int u^p K(u) du\\cdot m^{(p)}_{F_0}(1)\\right)\\stackrel{d}{\\to} N\\left(0,\\sigma^2_{U|F_0(\\bm{Z}^{\\top}\\bm{\\gamma}_0)}(1)\\int K^2(u) du\\right)\n\\]\nas $n\\to\\infty$, where $m^{(p)}_{F_0}(1)=\\lim_{q\\uparrow 1}\\left.\\left(d^p\/(dq^{\\prime p}\\right) m_{F_0}(q^{\\prime})\\right|_{q^{\\prime}=q}$ for $m_{F_0}(\\cdot)$ as given above in (\\ref{m0F}).\n\\end{theorem}\n\nIt follows from Theorem~\\ref{mainthm} that the rate of convergence of $\\hat{\\theta}_n$ to its limiting normal distribution \nis unaffected by the dependence, if any, between the disturbance terms $U$ and $V$ in (\\ref{outcome}) and (\\ref{selection}), \nrespectively.  The rate of convergence of $\\hat{\\theta}_n$ is also unaffected \nby the relative upper tail thicknesses of the distributions of $V$ and of the selection index \n$\\bm{Z}^{\\top}\\bm{\\gamma}_0$.  \n\nWe also have from the statement of Theorem~\\ref{mainthm} that a necessary condition for the consistency of \n$\\hat{\\theta}_n$ is the finiteness of the derivative $m^{(p)}_{F_0}(1)$.  \nThe finiteness of $m^{(p)}_{F_0}(1)$, as discussed above, is implied by the identification of $\\bm{\\gamma}_0$ up to a location \nand scale normalization as well as by the differentiability conditions \nspecified in Assumptions~\\ref{a2}.\\ref{a2d2}--\\ref{a2}.\\ref{a2d3}.  From this it follows that the consistency of the \nproposed estimator is implied by natural restrictions on the joint distribution of \n$[\\begin{array}{ccc} U & V & \\bm{Z}^{\\top}\\bm{\\gamma}_0\\end{array}]$.  These distributional restrictions correspond \ncollectively to a standard smoothness restriction in the literature on kernel estimation of conditional mean functions.\n\nThe presence of $m^{(p)}_{F_0}(1)$ in the bias term appearing in Theorem~\\ref{mainthm}, however, \nindicates that the approximate large-sample bias of $\\hat{\\theta}_n$ does depend on the extent to which $U$ is mean \ndependent on $V$.  In particular, the conditional mean derivative $m^{(p)}_{F_0}(1)$ depends on the smoothness of the conditional \nmean $E\\left[U\\left|F_0(V)=q\\right.\\right]$ as a function of $q$ for values of $q$ near one; Appendix~\\ref{lemmabound} below\ncontains further discussion.  It is worth noting in this connection that $m^{(p)}_{F_0}(1)=0$ when the selection mechanism is exogenous to the extent \nthat $U$ is mean independent of $V$, i.e., when $P\\left[E[U|V]= E[U]\\right]=1$.  \n\nThe dependence of the approximate large-sample bias of $\\hat{\\theta}_n$ on the joint distribution of \n$[\\begin{array}{ccc} U & V & \\bm{Z}^{\\top}\\bm{\\gamma}_0\\end{array}]$ through the conditional mean derivative \n$m^{(p)}_{F_0}(1)$ can be ameliorated in practice by a judicious choice of variable\nbandwidth; see Corollary~\\ref{cormain} below and the corresponding discussion and simulation \nevidence presented in Section~\\ref{mc}.  \nTheorem~\\ref{mainthm} in any case indicates that the approximate large-sample bias, but not the variance, of the proposed estimator \ndepends on the joint distribution of $[\\begin{array}{ccc} U & V & \\bm{Z}^{\\top}\\bm{\\gamma}_0\\end{array}]$.  \nTheorem~\\ref{mainthm} as such distinguishes the asymptotic behaviour of $\\hat{\\theta}_n$ from those of existing estimators \nof $\\theta_0$ \\citep[e.g.,][]{Heckman90,Lewbel07} whose biases and variances both depend on the joint distribution \nof $[\\begin{array}{ccc} U & V & \\bm{Z}^{\\top}\\bm{\\gamma}_0\\end{array}]$.\n\nThe following corollary is immediate from Theorem~\\ref{mainthm}:\n\n\\begin{corollary}\\label{cormain}\nThe following hold under the conditions of Theorem~\\ref{mainthm}:\n\\begin{enumerate}\n\\item\\label{cor1} If the additional condition that $nh_n^{2p+1}\\to 0$ holds, we have\n\\[\n\\sqrt{nh_n}\\left(\\hat{\\theta}_n-\\theta_0\\right)\\stackrel{d}{\\to} N\\left(0,\\sigma^2_{U|F_0\\left(\\bm{Z}^{\\top}\\bm{\\gamma}_0\\right)}(1)\\int K^2(u) du\\right)\n\\]\nas $n\\to\\infty$.\n\\item\\label{cor2} The theoretical bandwidth $h^*_n$ minimizing the asymptotic mean-squared error of $\\hat{\\theta}_n$ is given by\n\\[\nh^*_n=\\left[\\frac{(p!)^2\\sigma^2_{U|F_0\\left(\\bm{Z}^{\\top}\\bm{\\gamma}_0\\right)}(1)\\int K^2(u) du}{2p\\left(\\int u^p K(u) du\\right)^2\\left(m^{(p)}_{F_0}(1)\\right)^2 \\cdot n}\\right]^{\\frac{1}{2p+1}}.\n\\]\n\\end{enumerate}\n\\end{corollary}\n\n\\section{Rate Optimality}\\label{opt}\n\n\\noindent Continue to let $p\\ge 2$ be as specified above in Assumption~\\ref{a2} in the \nprevious section, and $n$ the sample size.  This section shows that under the conditions of Assumptions~\\ref{a1} \nand \\ref{a2}, the rate $n^{-p\/(2p+1)}$ is the fastest achievable, or {\\it optimal}, rate of convergence of an \nestimator of the intercept $\\theta_0$ in (\\ref{outcome}).  The optimality in question is relative to the \nconvergence rates of all other estimators of $\\theta_0$ and excludes by definition those estimators that are asymptotically \nsuperefficient at particular points in the underlying parameter space.  The exclusion of \nsuperefficient estimators in this context notably rules out estimators that converge at the parametric \nrate of $n^{-1\/2}$ under conditions stronger than those given above in Assumptions~\\ref{a1} and \\ref{a2}.  For \nexample, the OLS estimator of $\\theta_0$ based on observations for which $D_i=1$ is superefficient for \nspecifications of (\\ref{outcome})--(\\ref{observation}) over submodels in which the disturbance $U$ and the selection indicator $D$ are conditionally \nmean independent given $\\bm{X}$ and $\\bm{Z}$, i.e., models where $P\\left[E[U|D=1,\\bm{X},\\bm{Z}]=E[U|D=0,\\bm{X},\\bm{Z}]=0\\right]=1$.  As noted in the \nIntroduction, the OLS estimator of $\\theta_0$ converges at the standard rate of $n^{-1\/2}$ when $U$ and $D$ are conditionally mean independent given \n$\\bm{X}$ and $\\bm{Z}$ but is otherwise inconsistent.  \n\nThe approach to optimality taken here follows that of \\citet{Horowitz93}, which was in turn based on the \napproach of \\citet{Stone80}.  In particular, let \n$\\left\\{\\Psi_n:\\,n=1,2,3,\\ldots\\right\\}$ denote a sequence of sets of the form \n\\begin{equation}\\label{Psin}\n\\Psi_n=\\left\\{\\psi:\\,\\psi=(\\bm{\\psi}_1,g)\\right\\},\n\\end{equation} \nwhere $\\bm{\\psi}_1=(\\theta,\\bm{\\beta}^{\\top},\\bm{\\gamma}^{\\top})^{\\top}\\in\\mathbb{R}^{1+k+l}$ and where \n$g$ denotes the joint conditional density given $\\bm{X}$ and $\\bm{Z}$ of the disturbances $U$ and $V$ appearing above in (\\ref{outcome}) \nand (\\ref{selection}), respectively.  The quantity $\\psi$ may depend generically on $n$. \n\nConsider the observable random variables $D$, $Y$, $\\bm{X}$ and $\\bm{Z}$ appearing above in \n(\\ref{outcome})--(\\ref{observation}).  Suppose that for each $n$, the joint conditional distribution of the \nvector $(D,Y)$ given $\\bm{X}$ and $\\bm{Z}$ is indexed by some $\\psi\\in\\Psi_n$.  Let \n$P_{\\psi}[\\cdot]\\equiv P_{(\\bm{\\psi}_1,g)}[\\cdot]$ denote the corresponding probability measure.  Following \n\\citet{Stone80}, one may in this context define a constant $\\rho>0$ to be an {\\it upper bound} on the rate of \nconvergence of estimators of the intercept parameter $\\theta_0$ if for every estimator sequence \n$\\left\\{\\theta_n\\right\\}$,\n\\begin{equation}\\label{ub1}\n\\liminf_{n\\to\\infty}\\sup_{\\psi\\in\\Psi_n} P_{\\psi}\\left[\\left|\\theta_n-\\theta\\right|> sn^{-\\rho}\\right]>0\n\\end{equation}\nfor all $s>0$, and if \n\\begin{equation}\\label{ub2}\n\\lim_{s\\to 0}\\liminf_{n\\to\\infty}\\sup_{\\psi\\in\\Psi_n} P_{\\psi}\\left[\\left|\\theta_n-\\theta\\right|>sn^{-\\rho}\\right]=1,\n\\end{equation}\nwhere $\\theta$ as it appears in (\\ref{ub1}) and (\\ref{ub2}) refers to the first component of the \nfinite-dimensional component $\\bm{\\psi}_1$ of $\\psi$.  \n\nIn addition, define $\\rho>0$ to be an {\\it achievable} rate of convergence for the intercept parameter if there \nexists an estimator sequence $\\left\\{\\theta_n\\right\\}$ such that\n\\begin{equation}\\label{ach}\n\\lim_{s\\to\\infty}\\limsup_{n\\to\\infty}\\sup_{\\psi\\in\\Psi_n} P_{\\psi}\\left[\\left|\\theta_n-\\theta\\right|> sn^{-\\rho}\\right]=0.\n\\end{equation}\nOne calls $\\rho>0$ the {\\it optimal} rate of convergence for estimation of the intercept parameter if it is both \nan upper bound on the rate of convergence and achievable.  In what follows, I first show that for large $n$, \n$p\/(2p+1)$ is an upper bound on the rate of convergence.  I then show that there exists an implementation of \nthe estimator given in (\\ref{mhat}) that attains the $n^{-p\/(2p+1)}$-rate of convergence uniformly \nover $\\Psi_n$ as $n\\to\\infty$. \n\nThe approach taken first involves the specification for each $n$ of a subset $\\Psi^*_n$ of the parameter set \n$\\Psi_n$ in which the finite-dimensional component $\\bm{\\psi}_1\\equiv\\bm{\\psi}_{1n}$ lies in a shrinking \nneighbourhood $\\Psi^*_{1n}$ of some point \n$[\\begin{array}{ccc} \\theta_0 & \\bm{\\beta}_0^{\\top} & \\bm{\\gamma}_0^{\\top}\\end{array}]^{\\top}\\in\\mathbb{R}^{1+k+l}$.  In addition, the \ninfinite-dimensional component $g$ is embedded in a curve (i.e., parametrization) indexed by a scalar $\\psi_{2n}$ on a shrinking \nneighbourhood $\\Psi^*_{2n}$ of a bivariate density function $g_0$ satisfying all relevant conditions of Assumptions~\\ref{a1} and \\ref{a2} for a \nconditional density of $U$ and $V$ given $\\bm{X}$ and $\\bm{Z}$.  \n\nIn particular, consider a parametrization of the conditional joint density $g$ of $(U,V)$ given $\\bm{X}$ and $\\bm{Z}$ given by $g_{\\psi_{2n}}$ for \n$\\psi_2\\in\\Psi^*_{2n}$, where for some $\\psi_{2n0}\\in\\Psi^*_{2n}$, we have $g_{\\psi_{2n0}}(u,v|\\bm{x},\\bm{z})=g_0(u,v|\\bm{x},\\bm{z})$ for each \n$[\\begin{array}{cccc} u & v & \\bm{x}^{\\top} & \\bm{z}^{\\top}\\end{array}]\\in\\mathbb{R}^{2+k+l}$; i.e., the curve on $\\Psi^*_{2n}$ given by \n$\\psi_{2n}\\to g_{\\psi_{2n}}$ passes through the true conditional joint density $g_0$ at some point $\\psi_{2n0}\\in\\Psi^*_{2n}$.\n\nNow let $\\Psi^*_n\\equiv \\Psi^*_{1n}\\times\\Psi^*_{2n}$.  Let $s>0$ be arbitrary, and let $\\left\\{\\theta_n\\right\\}$ denote an arbitrary sequence of estimators of \n$\\theta_0$.  Consider that if\n\\begin{equation}\\label{ub1b}\n\\liminf_{n\\to\\infty}\\sup_{\\psi_{n}\\in\\Psi^*_{n}} P_{\\psi_{n}}\\left[n^{\\frac{p}{2p+1}}\\left|\\theta_n-\\theta\\right|>s\\right]>0,\n\\end{equation}\nthen (\\ref{ub1}) holds with $\\rho=p\/(2p+1)$.  This is because the set $\\Psi_n$ in (\\ref{ub1}) contains the \nset over which the supremum is taken in (\\ref{ub1b}).  Similarly, if \n\\begin{equation}\\label{ub2b}\n\\lim_{s\\to 0}\\liminf_{n\\to\\infty}\\sup_{\\psi_{n}\\in\\Psi^*_{n}} P_{\\psi_{n}}\\left[n^{\\frac{p}{2p+1}}\\left|\\theta_n-\\theta\\right|>s\\right]=1\n\\end{equation}\nholds, then so does (\\ref{ub2}).\n\nIt follows that proving (\\ref{ub1b}) and (\\ref{ub2b}) suffices to show that $p\/(2p+1)$ is an upper bound on \nthe rate of convergence; the key step in the proof is the specification of a suitable parametrization \n$\\psi_{2n}\\to g_{\\psi_{2n}}$ for $\\psi_{2n}\\in\\Psi^*_{2n}$.  This is in fact the approach taken \nin Appendix~\\ref{ubthmpf}, which contains a proof of the following result:\n\n\\begin{theorem}\\label{ubthm}\n\nUnder the conditions of Assumptions~\\ref{a1} and \\ref{a2}, (\\ref{ub1b}) and (\\ref{ub2b}) hold.\n\\end{theorem}\nTheorem~\\ref{ubthm} implies that $p\/(2p+1)$ is an upper bound on the rate of convergence of an estimator \nsequence $\\left\\{\\theta_n\\right\\}$ in the minimax sense of (\\ref{ub1}) and (\\ref{ub2}) above.\n\nNext, it is shown that $p\/(2p+1)$ is an achievable rate of convergence in the sense of (\\ref{ach}) by \nexhibiting an estimator sequence $\\left\\{\\theta_n\\right\\}$ such that (\\ref{ach}) holds with $\\rho=p\/(2p+1)$.  \nIn this connection, let $\\hat{\\theta}_n^*$ denote the proposed estimator given above in (\\ref{mhat}) implemented \nwith a bandwidth $h_n^*=cn^{-1\/(2p+1)}$ for some constant $c>0$.  In this case, (\\ref{ach}) is satisfied with \n$\\theta_n=\\hat{\\theta}^*_n$ and $\\rho=p\/(2p+1)$:\n\n\\begin{theorem}\\label{achthm}\n\nSuppose Assumptions~\\ref{a1} and \\ref{a2} hold.  Then (\\ref{ach}) holds with $\\theta_n=\\hat{\\theta}_n^*$ and \n$\\rho=p\/(2p+1)$, where $\\hat{\\theta}_n^*$ denotes the estimator given above in (\\ref{mhat}) implemented with a bandwidth \n$h_n^*=c n^{-1\/(2p+1)}$ for some constant $c>0$. \n\\end{theorem}\n\nTheorems~\\ref{ubthm} and \\ref{achthm} jointly imply that $p\/(2p+1)$ is the optimal rate of convergence for \nestimation of $\\theta_0$.\n\n\\section{Numerical Evidence}\\label{mc}\n\n\\noindent This section reports the results of simulation experiments that compare the finite-sample \nbehaviour of the estimator in (\\ref{mhat}) to the behaviours of alternative estimators.  The simulations involved:\n\\begin{itemize}\n\\item variation in the correlation between the unobservable terms in the outcome and selection equations;\n\\item variation in the relative upper tail thicknesses of the selection index and the unobservable term in the \nselection equation, thus implying variation in the degree to which the parameter of interest is identified;\n\\item and the imposition of two different parametric families for the joint distribution of \n$[\\begin{array}{ccc} U & V & \\bm{Z}^{\\top}\\bm{\\gamma}_0\\end{array}]$, where $U$ is the unobservable term in the \noutcome equation, $V$ is the unobservable term in the selection equation and $\\bm{Z}^{\\top}\\bm{\\gamma}_0$ is the \nselection index.\n\\end{itemize}\nEach simulation experiment involved 1000 replicated samples of sizes $n\\in\\{100,400\\}$ from the model given above in \n(\\ref{outcome})--(\\ref{observation}), where the parameter of interest was fixed at $\\theta_0=1$, the variance \nof the unobservable term in the selection equation was fixed at $Var[V]=1$ and where for some constant \n$\\rho\\in [-1,1]$, the unobservable term in the outcome equation was specified as $U=\\rho V+E$ for a random variable \n$E$ independent of $V$, where $E\\sim N\\left(0,1-\\rho^2\\right)$.  The parameter $\\rho$ in this case is by construction \nthe correlation coefficient between $U$ and $V$.  The simulations considered the settings \n$\\rho\\in\\{0,.25,.50,.75,.95\\}$. \n\nIn addition, the vector $\\bm{Z}$ of observable predictors of selection was taken to be $l$-variate with \n$\\bm{Z}=[\\begin{array}{ccc} Z_1 & \\cdots & Z_l\\end{array}]^{\\top}$, \nwhile the vector $\\bm{X}$ of outcome predictors was specified to be $k$-variate with $k<l$ and \n$\\bm{X}=[\\begin{array}{ccc} Z_1 & \\cdots & Z_k\\end{array}]^{\\top}$.  The coefficient \nvector attached to $\\bm{X}$ was set to $\\bm{\\beta}_0=\\bm{\\iota}_k$, i.e., the $k$-dimensional \nunit vector.  The simulations imposed the settings $l=7$, $k=4$, which were intentionally set to equal the \ndimensions of the corresponding vectors appearing in the empirical example used below in Section~\\ref{ee}.\n\nThe selection index $\\bm{Z}^{\\top}\\bm{\\gamma}_0$ and the selection-equation disturbance term were simulated from two \ndata-generating processes (DGPs), considered in turn.  The distributions of $\\bm{Z}^{\\top}\\bm{\\gamma}_0$ under both \nDGPs, as required by Assumption~\\ref{a1}, are \nabsolutely continuous.  In addition, the parameter $\\alpha>0$ used in the specification of \nboth models is defined so as to index the degree to which the parameter of interest $\\theta_0$ is identified.  In particular, $\\alpha\\ge 1$ can be \nseen in this context to be a necessary condition for the identification of $\\theta_0$, with $\\alpha\\ge 1$ corresponding to the case where the \ntransformation $F_0(V)$ has an absolutely continuous distribution with density given by the ratio $r_Q(q)$ defined in (\\ref{marg}) below.  Values of \n$\\alpha<1$, on the other hand, correspond to the non-identifiability of $\\theta_0$, with $\\alpha\\in (0,1)$ in the context of \neither of the following two DGPs implying failure of the condition that $F_0(V)$ have an absolutely continuous \ndistribution supported on $[0,1]$.  In particular, $\\alpha\\in (0,1)$ in the following two DGPs implies that the quantity \n$r_Q(q)$ given in (\\ref{marg}) below has the property that $r_Q(1)=\\infty$:\n \n\\begin{itemize}\n\\item (DGP1) I take\n\\[\n\\left[\\begin{array}{cc}  \\bm{Z} & \\bm{0} \\\\ \\bm{0} & V \\end{array}\\right]\\sim N\\left(\\bm{0},\\bm{I}_{l+1}\\right).\n\\]\nIn addition, the selection parameter $\\bm{\\gamma}_0$ is set to $\\bm{\\gamma}_0=[\\begin{array}{ccc} \\sqrt{\\alpha\/l} & \\cdots & \\sqrt{\\alpha\/l}\\end{array}]^{\\top}$ for \na constant $\\alpha>0$.  In this way we have \n\\[\n\\left[\\begin{array}{c}\\bm{Z}^{\\top}\\bm{\\gamma}_0 \\\\ V\\end{array}\\right]\\sim N\\left(\\bm{0},\\left[\\begin{array}{cc} \\alpha & 0 \\\\ 0 & 1\\end{array}\\right]\\right).\n\\] \n\\item (DGP2)  $Z_1,\\ldots,Z_l$ are iid standard Cauchy and jointly mutually independent of $V$, while \n$V$ is absolutely continuous on $[1,\\infty)$ with Pareto type-I density given by \n\\[\ng_{0V}(v)=\\alpha v^{-\\alpha-1},\n\\] \nwhere $\\alpha>0$ is a constant.  In addition, the selection parameter $\\bm{\\gamma}_0$ is set to \n$\\bm{\\gamma}_0=[\\begin{array}{cc} \\bm{0}_{l-1}^{\\top} & 1\\end{array}]^{\\top}$.  In this way the selection index \n$\\bm{Z}^{\\top}\\bm{\\gamma}_0$ is standard Cauchy and independent of $V$.\n\\end{itemize}\nThe simulations under both DGPs considered the settings $\\alpha\\in\\{2.00,1.50,1.25,1.00\\}$.  The effect of variation in \nthe correlation coefficient $\\rho$ and the parameter $\\alpha$ on estimation of the intercept was a primary focus of \nthese simulations.  The outcome-equation nuisance parameter $\\bm{\\beta}_0$ and the selection parameter $\\bm{\\gamma}_0$ \nwere accordingly fixed at their true values in these simulations in order to provide a clearer picture of the effects of \nvariation in $\\rho$ and $\\alpha$ on the behaviour of the various intercept estimators considered.\n\nThe proposed intercept estimator given above in (\\ref{mhat}) was implemented with a standard \n(i.e., second-order) Epanechnikov kernel.  The bandwidth used to implement $\\hat{\\theta}_n$ was initially \nset to the sample analogue of the theoretical asymptotic MSE-optimal bandwidth $h^*_n$ specified above in \nCorollary~\\ref{cormain}.  In particular, the simulations involved the bandwidth $\\hat{h}^*_n$, where $\\hat{h}^*_n$ was taken to be \nthe sample analogue of $h^*_n$.  As such, $\\hat{h}^*_n$ was set \nto decay at the MSE-optimal rate of $n^{-1\/5}$ corresponding to the order of kernel employed (i.e., $p=2$), while its \nleading constant was specified as the sample analogue of the leading constant appearing in $h^*_n$.  The unknown \nparameters appearing in the leading constant of $h^*_n$ were estimated via auxiliary locally cubic regressions as \ndescribed in \\citet[\\S 4.3]{FanGijbels96}.  The sensitivity of the proposed estimator's sampling behaviour to the choice \nof bandwidth was also assessed by considering implementations using the bandwidth settings $h_n=(2\/3)\\hat{h}_{n}^*$ and \n$h_n=(3\/2)\\hat{h}_{n}^*$.  \n\nComparisons of the corresponding sampling behaviours in terms of squared bias, standard deviation and root mean-squared \nerror (RMSE) over 1000 Monte Carlo replications for values of \n$(\\rho,\\alpha)\\in\\{0,.25,.50,.75,.95\\}\\times\\{2.00,1.50,1.25,1.00\\}$  are presented below for samples of size $n=100$ in \nTables~\\ref{dgp1n100mhat1.p2} and \\ref{dgp2n100mhat1.p2} for DGP1 and DGP2, respectively.  The corresponding results for samples of size $n=400$ \nappear in Tables~\\ref{dgp1n400mhat1.p2} and \\ref{dgp2n400mhat1.p2}.  The RMSE figures displayed \nin these tables are multiplied by $\\sqrt{n}$ in order to provide a clearer indication of \nthe rate of convergence of the proposed estimator.  The increases in $\\sqrt{n}\\times$ RMSE as one moves from simulated \nsamples of size $n=100$ to those of $n=400$ indicate the slower-than-parametric rate of convergence of the proposed \nestimator regardless of the precise setting of $(\\rho,\\alpha)$.  It is also clear that the rate of convergence of \nthe proposed estimator is slower for settings of $(\\rho,\\alpha)$ with one or both of $\\rho$ and $\\alpha$ close to one.  \nIn addition, and as predicted by Theorem~\\ref{mainthm} above, one can see that the effect of variation in \n$(\\rho,\\alpha)$ on the squared bias of the proposed estimator is more pronounced than the corresponding effect \non the variance; indeed the standard deviation of the proposed estimator tends to be relatively stable  \nover the various settings of $(\\rho,\\alpha)$ used in the simulations, particularly for values of \n$(\\rho,\\alpha)\\in\\{.25,.50,.75\\}\\times\\{2.00,1.50,1.25\\}$.  Finally, the tabulated results indicate that the sampling performance of $\\hat{\\theta}_n$ \nis not sensitive to moderate variations in bandwidth.  \n\n\\FloatBarrier\n\\begin{table}[H]\n\\pagenumbering{gobble}\n{\\scriptsize\n\\centering\n\\caption{DGP1 (bivariate normal), $n=100$, 1000 replications. Proposed estimator with second-order Epanechnikov kernel ($p=2$).  (RMSE is multiplied by $\\sqrt{n}$.)}\\label{dgp1n100mhat1.p2}\n\\begin{tabular}{ccccccccccccc}\n  \\hline\n\\multirow{2}{*}{$\\rho$} &\\multicolumn{3}{c}{$\\alpha=2.00$} &\\multicolumn{3}{c}{$\\alpha=1.50$} &\\multicolumn{3}{c}{$\\alpha=1.25$} &\\multicolumn{3}{c}{$\\alpha=1.00$}\\\\  \n                        & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE \\\\ \n  \\hline\n\\multicolumn{13}{c}{(optimal bandwidth)} \\\\\n0.0000 & 0.0081 & 0.1859 & 2.0645 & 0.0018 & 0.1872 & 1.9196 & 0.0008 & 0.1951 & 1.9723 & 0.0006 & 0.1900 & 1.9154 \\\\ \n0.2500 & 0.0003 & 0.1890 & 1.8985 & 0.0004 & 0.1857 & 1.8674 & 0.0023 & 0.1911 & 1.9695 & 0.0064 & 0.1802 & 1.9724 \\\\ \n0.5000 & 0.0014 & 0.1812 & 1.8503 & 0.0071 & 0.1861 & 2.0432 & 0.0127 & 0.1802 & 2.1253 & 0.0227 & 0.1779 & 2.3324 \\\\ \n0.7500 & 0.0092 & 0.1768 & 2.0125 & 0.0216 & 0.1784 & 2.3125 & 0.0353 & 0.1761 & 2.5750 & 0.0459 & 0.1687 & 2.7272 \\\\ \n0.9500 & 0.0215 & 0.1821 & 2.3373 & 0.0404 & 0.1665 & 2.6105 & 0.0548 & 0.1730 & 2.9114 & 0.0812 & 0.1626 & 3.2811 \\\\ \n\\multicolumn{13}{c}{($2\/3\\times$ optimal bandwidth)} \\\\\n0.0000 & 0.0081 & 0.1860 & 2.0657 & 0.0018 & 0.1873 & 1.9206 & 0.0008 & 0.1952 & 1.9733 & 0.0006 & 0.1901 & 1.9164 \\\\ \n0.2500 & 0.0003 & 0.1891 & 1.8996 & 0.0004 & 0.1858 & 1.8682 & 0.0023 & 0.1912 & 1.9702 & 0.0064 & 0.1803 & 1.9727 \\\\ \n0.5000 & 0.0014 & 0.1813 & 1.8507 & 0.0071 & 0.1862 & 2.0430 & 0.0126 & 0.1803 & 2.1243 & 0.0227 & 0.1780 & 2.3314 \\\\ \n0.7500 & 0.0091 & 0.1770 & 2.0109 & 0.0215 & 0.1786 & 2.3100 & 0.0351 & 0.1762 & 2.5724 & 0.0457 & 0.1689 & 2.7242 \\\\ \n0.9500 & 0.0212 & 0.1822 & 2.3336 & 0.0401 & 0.1666 & 2.6058 & 0.0545 & 0.1732 & 2.9068 & 0.0809 & 0.1628 & 3.2764 \\\\ \n\\multicolumn{13}{c}{($3\/2\\times$ optimal bandwidth)} \\\\\n0.0000 & 0.0081 & 0.1858 & 2.0640 & 0.0018 & 0.1872 & 1.9192 & 0.0008 & 0.1951 & 1.9719 & 0.0006 & 0.1900 & 1.9150 \\\\ \n0.2500 & 0.0003 & 0.1890 & 1.8980 & 0.0004 & 0.1857 & 1.8670 & 0.0023 & 0.1910 & 1.9692 & 0.0064 & 0.1802 & 1.9723 \\\\ \n0.5000 & 0.0014 & 0.1811 & 1.8502 & 0.0071 & 0.1861 & 2.0433 & 0.0127 & 0.1802 & 2.1257 & 0.0228 & 0.1779 & 2.3328 \\\\ \n0.7500 & 0.0093 & 0.1768 & 2.0132 & 0.0217 & 0.1784 & 2.3136 & 0.0354 & 0.1760 & 2.5762 & 0.0460 & 0.1687 & 2.7285 \\\\ \n0.9500 & 0.0216 & 0.1820 & 2.3389 & 0.0406 & 0.1664 & 2.6126 & 0.0550 & 0.1729 & 2.9134 & 0.0814 & 0.1625 & 3.2831 \\\\    \n\\hline\n\\end{tabular}\n} \n\\end{table}\n\n\\begin{table}[H]\n\\pagenumbering{gobble}\n{\\scriptsize\n\\centering\n\\caption{DGP2 (non-normal), $n=100$, 1000 replications. Proposed estimator with second-order Epanechnikov kernel ($p=2$) and optimal bandwidth.  (RMSE is multiplied by $\\sqrt{n}$.)}\\label{dgp2n100mhat1.p2}\n\\begin{tabular}{ccccccccccccc}\n  \\hline\n\\multirow{2}{*}{$\\rho$} &\\multicolumn{3}{c}{$\\alpha=2.00$} &\\multicolumn{3}{c}{$\\alpha=1.50$} &\\multicolumn{3}{c}{$\\alpha=1.25$} &\\multicolumn{3}{c}{$\\alpha=1.00$}\\\\  \n                        & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE \\\\ \n  \\hline\n\\multicolumn{13}{c}{(optimal bandwidth)} \\\\\n0.0000 & 0.1541 & 0.1719 & 4.2854 & 0.1892 & 0.1805 & 4.7097 & 0.2268 & 0.1708 & 5.0593 & 0.2714 & 0.1612 & 5.4535 \\\\ \n0.2500 & 0.0202 & 0.2119 & 2.5523 & 0.0394 & 0.2070 & 2.8669 & 0.0445 & 0.2229 & 3.0684 & 0.0636 & 0.2280 & 3.3994 \\\\ \n0.5000 & 0.0079 & 0.2391 & 2.5503 & 0.0071 & 0.2741 & 2.8687 & 0.0006 & 0.2942 & 2.9521 & 0.0000 & 0.3010 & 3.0102 \\\\ \n0.7500 & 0.1113 & 0.2796 & 4.3536 & 0.0972 & 0.3386 & 4.6030 & 0.0735 & 0.3288 & 4.2615 & 0.0585 & 0.4767 & 5.3448 \\\\ \n0.9500 & 0.2747 & 0.3174 & 6.1273 & 0.2510 & 0.3605 & 6.1721 & 0.2317 & 0.4509 & 6.5956 & 0.1950 & 0.4976 & 6.6528 \\\\ \n\\multicolumn{13}{c}{($2\/3\\times$ optimal bandwidth)} \\\\\n0.0000 & 0.1522 & 0.1725 & 4.2652 & 0.1871 & 0.1813 & 4.6906 & 0.2247 & 0.1715 & 5.0405 & 0.2693 & 0.1619 & 5.4357 \\\\ \n0.2500 & 0.0190 & 0.2125 & 2.5336 & 0.0376 & 0.2078 & 2.8422 & 0.0426 & 0.2242 & 3.0466 & 0.0613 & 0.2293 & 3.3752 \\\\ \n0.5000 & 0.0091 & 0.2396 & 2.5796 & 0.0084 & 0.2754 & 2.9025 & 0.0010 & 0.2960 & 2.9769 & 0.0000 & 0.3032 & 3.0319 \\\\ \n0.7500 & 0.1178 & 0.2799 & 4.4284 & 0.1036 & 0.3404 & 4.6850 & 0.0789 & 0.3307 & 4.3392 & 0.0632 & 0.4805 & 5.4233 \\\\ \n0.9500 & 0.2874 & 0.3178 & 6.2324 & 0.2633 & 0.3623 & 6.2816 & 0.2435 & 0.4536 & 6.7023 & 0.2055 & 0.5015 & 6.7598 \\\\\n\\multicolumn{13}{c}{($3\/2\\times$ optimal bandwidth)} \\\\ \n0.0000 & 0.1550 & 0.1716 & 4.2942 & 0.1902 & 0.1801 & 4.7180 & 0.2277 & 0.1705 & 5.0675 & 0.2724 & 0.1609 & 5.4612 \\\\ \n0.2500 & 0.0208 & 0.2116 & 2.5607 & 0.0401 & 0.2066 & 2.8777 & 0.0453 & 0.2224 & 3.0780 & 0.0646 & 0.2274 & 3.4100 \\\\ \n0.5000 & 0.0073 & 0.2389 & 2.5382 & 0.0066 & 0.2736 & 2.8547 & 0.0005 & 0.2934 & 2.9420 & 0.0001 & 0.3000 & 3.0015 \\\\ \n0.7500 & 0.1086 & 0.2795 & 4.3217 & 0.0945 & 0.3379 & 4.5683 & 0.0712 & 0.3280 & 4.2285 & 0.0565 & 0.4750 & 5.3115 \\\\ \n0.9500 & 0.2693 & 0.3173 & 6.0825 & 0.2458 & 0.3598 & 6.1255 & 0.2268 & 0.4497 & 6.5502 & 0.1906 & 0.4960 & 6.6072 \\\\ \n   \\hline\n\\end{tabular}\n} \n\\end{table}\n\n\\FloatBarrier\n\\begin{table}[H]\n\\pagenumbering{gobble}\n{\\scriptsize\n\\centering\n\\caption{DGP1 (bivariate normal), $n=400$, 1000 replications. Proposed estimator with second-order Epanechnikov kernel ($p=2$).  (RMSE is multiplied by $\\sqrt{n}$.)}\\label{dgp1n400mhat1.p2}\n\\begin{tabular}{ccccccccccccc}\n  \\hline\n\\multirow{2}{*}{$\\rho$} &\\multicolumn{3}{c}{$\\alpha=2.00$} &\\multicolumn{3}{c}{$\\alpha=1.50$} &\\multicolumn{3}{c}{$\\alpha=1.25$} &\\multicolumn{3}{c}{$\\alpha=1.00$}\\\\  \n                        & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE \\\\ \n  \\hline\n\\multicolumn{13}{c}{(optimal bandwidth)} \\\\\n0.0000 & 0.0073 & 0.0972 & 2.5875 & 0.0021 & 0.0917 & 2.0524 & 0.0007 & 0.0942 & 1.9547 & 0.0000 & 0.0933 & 1.8683 \\\\ \n0.2500 & 0.0007 & 0.0924 & 1.9180 & 0.0001 & 0.0966 & 1.9371 & 0.0017 & 0.0916 & 2.0082 & 0.0052 & 0.0920 & 2.3367 \\\\ \n0.5000 & 0.0011 & 0.0962 & 2.0330 & 0.0057 & 0.0908 & 2.3585 & 0.0112 & 0.0959 & 2.8557 & 0.0199 & 0.0917 & 3.3640 \\\\ \n0.7500 & 0.0082 & 0.0919 & 2.5766 & 0.0194 & 0.0924 & 3.3416 & 0.0293 & 0.0894 & 3.8594 & 0.0460 & 0.0889 & 4.6434 \\\\ \n0.9500 & 0.0178 & 0.0922 & 3.2404 & 0.0345 & 0.0878 & 4.1101 & 0.0505 & 0.0840 & 4.7995 & 0.0731 & 0.0845 & 5.6663 \\\\ \n\\multicolumn{13}{c}{($2\/3\\times$ optimal bandwidth)} \\\\\n0.0000 & 0.0073 & 0.0972 & 2.5890 & 0.0021 & 0.0918 & 2.0535 & 0.0007 & 0.0942 & 1.9557 & 0.0000 & 0.0934 & 1.8694 \\\\ \n0.2500 & 0.0007 & 0.0924 & 1.9204 & 0.0001 & 0.0966 & 1.9382 & 0.0017 & 0.0916 & 2.0077 & 0.0052 & 0.0921 & 2.3354 \\\\ \n0.5000 & 0.0011 & 0.0963 & 2.0315 & 0.0056 & 0.0908 & 2.3546 & 0.0111 & 0.0960 & 2.8519 & 0.0198 & 0.0918 & 3.3594 \\\\ \n0.7500 & 0.0080 & 0.0920 & 2.5692 & 0.0192 & 0.0924 & 3.3330 & 0.0291 & 0.0894 & 3.8510 & 0.0458 & 0.0891 & 4.6352 \\\\ \n0.9500 & 0.0175 & 0.0923 & 3.2283 & 0.0342 & 0.0879 & 4.0973 & 0.0502 & 0.0841 & 4.7867 & 0.0727 & 0.0846 & 5.6532 \\\\ \n\\multicolumn{13}{c}{($3\/2\\times$ optimal bandwidth)} \\\\ \n0.0000 & 0.0073 & 0.0971 & 2.5869 & 0.0021 & 0.0917 & 2.0520 & 0.0007 & 0.0941 & 1.9542 & 0.0000 & 0.0933 & 1.8678 \\\\ \n0.2500 & 0.0007 & 0.0924 & 1.9170 & 0.0001 & 0.0965 & 1.9367 & 0.0017 & 0.0916 & 2.0084 & 0.0052 & 0.0920 & 2.3372 \\\\ \n0.5000 & 0.0011 & 0.0962 & 2.0337 & 0.0057 & 0.0908 & 2.3602 & 0.0112 & 0.0959 & 2.8574 & 0.0199 & 0.0917 & 3.3660 \\\\ \n0.7500 & 0.0082 & 0.0919 & 2.5799 & 0.0195 & 0.0923 & 3.3453 & 0.0293 & 0.0893 & 3.8632 & 0.0461 & 0.0889 & 4.6471 \\\\ \n0.9500 & 0.0179 & 0.0921 & 3.2458 & 0.0346 & 0.0878 & 4.1158 & 0.0507 & 0.0840 & 4.8052 & 0.0733 & 0.0844 & 5.6721 \\\\ \n\\hline\n\\end{tabular}\n} \n\\end{table}\n\n\\begin{table}[H]\n\\pagenumbering{gobble}\n{\\scriptsize\n\\centering\n\\caption{DGP2 (non-normal), $n=400$, 1000 replications. Proposed estimator with second-order Epanechnikov kernel ($p=2$).  (RMSE is multiplied by $\\sqrt{n}$.)}\\label{dgp2n400mhat1.p2}\n\\begin{tabular}{ccccccccccccc}\n  \\hline\n\\multirow{2}{*}{$\\rho$} &\\multicolumn{3}{c}{$\\alpha=2.00$} &\\multicolumn{3}{c}{$\\alpha=1.50$} &\\multicolumn{3}{c}{$\\alpha=1.25$} &\\multicolumn{3}{c}{$\\alpha=1.00$}\\\\  \n                        & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE \\\\ \n  \\hline\n\\multicolumn{13}{c}{(optimal bandwidth)} \\\\\n0.0000 & 0.1460 & 0.0908 & 7.8539 & 0.1864 & 0.0884 & 8.8133 & 0.2179 & 0.0881 & 9.5014 & 0.2658 & 0.0831 & 10.4445 \\\\ \n0.2500 & 0.0184 & 0.1045 & 3.4218 & 0.0317 & 0.1053 & 4.1369 & 0.0426 & 0.1153 & 4.7277 & 0.0651 & 0.1158 & 5.6037 \\\\ \n0.5000 & 0.0095 & 0.1255 & 3.1783 & 0.0048 & 0.1261 & 2.8790 & 0.0025 & 0.1498 & 3.1582 & 0.0004 & 0.1898 & 3.8157 \\\\ \n0.7500 & 0.1296 & 0.1433 & 7.7487 & 0.1058 & 0.1769 & 7.4044 & 0.0912 & 0.1956 & 7.1956 & 0.0686 & 0.2002 & 6.5943 \\\\ \n0.9500 & 0.2897 & 0.1551 & 11.2027 & 0.2707 & 0.1828 & 11.0287 & 0.2632 & 0.2245 & 11.1992 & 0.2507 & 0.2913 & 11.5847 \\\\ \n\\multicolumn{13}{c}{($2\/3\\times$ optimal bandwidth)} \\\\\n0.0000 & 0.1428 & 0.0913 & 7.7763 & 0.1829 & 0.0891 & 8.7369 & 0.2143 & 0.0888 & 9.4263 & 0.2621 & 0.0837 & 10.3754 \\\\ \n0.2500 & 0.0163 & 0.1050 & 3.3076 & 0.0289 & 0.1061 & 4.0066 & 0.0392 & 0.1164 & 4.5947 & 0.0611 & 0.1170 & 5.4683 \\\\ \n0.5000 & 0.0120 & 0.1259 & 3.3398 & 0.0068 & 0.1271 & 3.0284 & 0.0040 & 0.1515 & 3.2815 & 0.0010 & 0.1918 & 3.8896 \\\\ \n0.7500 & 0.1426 & 0.1434 & 8.0783 & 0.1180 & 0.1784 & 7.7409 & 0.1022 & 0.1968 & 7.5083 & 0.0780 & 0.2022 & 6.8948 \\\\ \n0.9500 & 0.3146 & 0.1552 & 11.6389 & 0.2946 & 0.1837 & 11.4600 & 0.2865 & 0.2260 & 11.6209 & 0.2723 & 0.2939 & 11.9783 \\\\ \n\\multicolumn{13}{c}{($3\/2\\times$ optimal bandwidth)} \\\\ \n0.0000 & 0.1473 & 0.0905 & 7.8874 & 0.1879 & 0.0882 & 8.8462 & 0.2195 & 0.0878 & 9.5336 & 0.2674 & 0.0828 & 10.4742 \\\\ \n0.2500 & 0.0193 & 0.1042 & 3.4714 & 0.0329 & 0.1049 & 4.1929 & 0.0440 & 0.1149 & 4.7847 & 0.0668 & 0.1153 & 5.6615 \\\\ \n0.5000 & 0.0085 & 0.1254 & 3.1136 & 0.0041 & 0.1256 & 2.8209 & 0.0020 & 0.1491 & 3.1118 & 0.0002 & 0.1890 & 3.7895 \\\\ \n0.7500 & 0.1243 & 0.1432 & 7.6116 & 0.1009 & 0.1763 & 7.2650 & 0.0868 & 0.1950 & 7.0661 & 0.0649 & 0.1993 & 6.4701 \\\\ \n0.9500 & 0.2797 & 0.1551 & 11.0222 & 0.2610 & 0.1824 & 10.8502 & 0.2537 & 0.2239 & 11.0246 & 0.2419 & 0.2902 & 11.4212 \\\\ \n   \\hline\n\\end{tabular}\n} \n\\end{table}\n\nI next consider the simulated performances over 1000 Monte Carlo replications across DGPs, sample sizes and settings \nof $(\\rho,\\alpha)$ of several alternative estimators of the intercept $\\theta_0$.  The results for samples of size $n=100$ are summarized below in \nTables~\\ref{dgp1n100balt} and \\ref{dgp2n100balt} for DGPs 1 and 2, respectively.  The corresponding results for samples of size $n=400$ \nappear in Tables~\\ref{dgp1n400balt} and \\ref{dgp2n400balt}.  The standard Heckman 2-step estimator is found under DGP1 to have a performance in terms of RMSE that is comparable to that of the \nproposed estimator in (\\ref{mhat}).  The proposed estimator under DGP2, on the other hand, is found to dominate in terms of RMSE the performance \nof the following alternative estimators under all combinations of $(\\rho,\\alpha)$ considered:  \n\n\\begin{itemize}\n\\item (OLS)  The ordinary least squares estimator of the intercept parameter using only those \nobservations for which $D=1$.  \n\nThese results are consistent with well established theory.  In particular, Tables~\\ref{dgp1n100balt}--\\ref{dgp2n100balt} indicate the good performance \nof OLS when $\\rho=0$ and the poor performance of OLS when $\\rho>0$.  In addition, the decrease in $\\sqrt{n}\\times$ RMSE for the OLS estimator when $\\rho=0$ as \none moves from $n=100$ to $n=400$ is suggestive of superefficiency, while at the same time the increases in $\\sqrt{n}\\times$RMSE \nwhen $\\rho>0$ is consistent with OLS being inconsistent under $\\rho>0$.\n\n\\item (2-step)  The estimator of the intercept based on the well known procedure of \n\\citet{Heckman76,Heckman79}, which is known to be $\\sqrt{n}$-consistent if $[\\begin{array}{cc} U & V\\end{array}]$ is \nbivariate normal (i.e., generated according to DGP1).  \n\nThe results for DGP1 given in Table~\\ref{dgp1n100balt} below are consistent with expectations; in particular, the \n2-step procedure exhibits an RMSE that is stable across the various configurations of $(\\rho,\\alpha)$ that were tried.  In addition, a comparison \nof the relevant sections of Table~\\ref{dgp1n100balt} and Table~\\ref{dgp1n400balt} highlights the stability \nof $\\sqrt{n}\\times$ RMSE as one moves from $n=100$ to $n=400$, which is consistent with the $\\sqrt{n}$-consistency of the procedure under DGP1.\n\nThe results for DGP2 appearing in Table~\\ref{dgp2n100balt}, on the other hand, show that the performance of \nthe 2-step procedure can vary dramatically with $(\\rho,\\alpha)$.  A comparison of the $\\sqrt{n}\\times$RMSE figures in \nTable~\\ref{dgp2n400balt} with those in Table~\\ref{dgp2n100balt} also suggests that the 2-step procedure under DGP2 is \nsuperefficient at $\\rho=0$ and converges at a slower-than-parametric rate for model specifications with $\\rho>0$.\n\n\\item (H90) The intercept estimator suggested by \\citet{Heckman90}, which in the context of the model specified in (\\ref{outcome})--(\\ref{observation}) \nhas the form\n\\begin{equation}\\label{h90}\n\\hat{\\theta}_{H90}\\equiv\\frac{\\sum_{i=1}^n D_i\\left(Y_i-\\bm{X}^{\\top}_i\\hat{\\bm{\\beta}}\\right)1\\left\\{\\bm{Z}_i\\hat{\\bm{\\gamma}}>b_n\\right\\}}{\\sum_{i=1}^n D_i 1\\left\\{\\bm{Z}_i\\hat{\\bm{\\gamma}}>b_n\\right\\}}\n\\end{equation}\nfor some sequence of positive constants $\\{b_n\\}$ with $b_n\\to\\infty$ as $n\\to\\infty$.  I present the results of \nsimulations in which the nuisance-parameter estimators $\\hat{\\bm{\\beta}}$ and $\\hat{\\bm{\\gamma}}$ are fixed at \nthe true values of the corresponding estimands.  These results appear below in Tables~\\ref{dgp1n100balt}, \\ref{dgp2n100balt}, \\ref{dgp1n400balt} and \n\\ref{dgp2n400balt} for $b_n$ equal to the sample .95-quantile of $\\bm{Z}^{\\top}_i\\bm{\\gamma}_0$.  \n\nTable~\\ref{dgp1n100balt} below indicates that the performance of $\\hat{\\theta}_{H90}$ is comparable to that of the 2-step procedure under DGP1 in that its \nRMSE is stable over changes in $(\\rho,\\alpha)$.  The stability in the $\\sqrt{n}\\times$RMSE figures in these tables as one moves from $n=100$ to \n$n=400$, which is evident from a comparison of Table~\\ref{dgp1n100balt} with Table~\\ref{dgp1n400balt}, also suggests that \n$\\hat{\\theta}_{H90}$ may be $\\sqrt{n}$-consistent under DGP1.\n\nTable~\\ref{dgp2n100balt}, on the other hand, shows that the performance of $\\hat{\\theta}_{H90}$ can deteriorate dramatically as $\\rho$ moves away from \nzero, although its performance under DGP2 appears to be unaffected by variation in $\\alpha$ for any given value of $\\rho$.  The $\\sqrt{n}\\times$RMSE \nfigures in Table~\\ref{dgp2n100balt} and Table~\\ref{dgp2n400balt} indicate that $\\hat{\\theta}_{H90}$ has a \nslower-than-parametric rate of convergence under DGP2 that is highly sensitive to variation in $\\rho$ but relatively insensitive to variation in \n$\\alpha$.\n\n\\item (AS98)  The intercept estimator developed by \\citet{AndrewsSchafgans98} as a generalization of the procedure of \n\\citet{Heckman90}.  The AS98 estimator in the context of the \nmodel given above in (\\ref{outcome})--(\\ref{observation}) has the form\n\\begin{equation}\\label{as98}\n\\hat{\\theta}_{AS98}\\equiv\\frac{\\sum_{i=1}^n D_i\\left(Y_i-\\bm{X}^{\\top}_i\\hat{\\bm{\\beta}}\\right) s\\left(\\bm{Z}_i^{\\top}\\hat{\\bm{\\gamma}}-b_n\\right)}{\\sum_{i=1}^n D_i s\\left(\\bm{Z}_i^{\\top}\\hat{\\bm{\\gamma}}_n-b_n\\right)},\n\\end{equation} \nwhere, following \\citet[eq. (4.1)]{AndrewsSchafgans98}, I set\n\\begin{equation}\\label{as98s}\ns(u)=\\left\\{\\begin{array}{ccc} 1-\\exp\\left(-\\frac{u}{\\tau-u}\\right) &,&\\,x\\in (0,\\tau)\\\\\n\t\t\t\t\t\t\t\t0 &,&\\,x\\le 0\\\\\n\t\t\t\t\t\t\t\t1 &,&\\,x\\ge \\tau\\end{array}.\\right.\n\\end{equation}\nNote that the setting $\\tau=0$ reduces $\\hat{\\theta}_{AS98}$ to $\\hat{\\theta}_{H90}$ as given earlier in (\\ref{h90}).  \nIn addition, the tuning parameter $b_n$ in (\\ref{as98}), as it does for $\\hat{\\theta}_{H90}$ in (\\ref{h90}) above, refers \nto a sequence of positive constants with $b_n\\to\\infty$ as $n\\to\\infty$.\n\nI present, in common with other simulations reported here, results for $\\hat{\\theta}_{AS98}$ in which the \nnuisance-parameter estimators $\\hat{\\bm{\\beta}}$ and $\\hat{\\bm{\\gamma}}$ are fixed at the true values of the corresponding \nestimands.  These simulations also involve setting the nuisance parameter $\\tau$ in (\\ref{as98s}) to the sample \nmedian of $\\bm{Z}^{\\top}_i\\bm{\\gamma}_0$ and the smoothing parameter $b_n$ in (\\ref{as98}) to the sample .95-quantile \nof $\\bm{Z}^{\\top}_i\\bm{\\gamma}_0$.  The corresponding results appear below in \nTables~\\ref{dgp1n100balt}--\\ref{dgp2n100balt} and also in Tables~\\ref{dgp1n400balt}--\\ref{dgp2n400balt}.\n\nIt is clear from Tables~\\ref{dgp1n100balt}--\\ref{dgp2n100balt} below that $\\hat{\\theta}_{AS}$ is numerically \nunstable under DGP1 but numerically stable under DGP2.  Table~\\ref{dgp2n100balt} also indicates the \nsensitivity of the performance of $\\hat{\\theta}_{AS}$ to variation in $(\\rho,\\alpha)$.  A comparison of Table~\\ref{dgp2n100balt} with \nTable~\\ref{dgp2n400balt} also underscores the slower-than-parametric rate of convergence of $\\hat{\\theta}_{AS98}$.\n\\end{itemize}\n\n\\FloatBarrier\n\\begin{table}[H]\n\\pagenumbering{gobble}\n{\\scriptsize\n\\centering\n\\caption{DGP1 (bivariate normal), $n=100$, 1000 replications.  Alternative estimators.  (RMSE is multiplied by $\\sqrt{n}$.)}\\label{dgp1n100balt}\n\\begin{tabular}{ccccccccccccc}\n  \\hline\n\\multirow{2}{*}{$\\rho$} &\\multicolumn{3}{c}{$\\alpha=2.00$} &\\multicolumn{3}{c}{$\\alpha=1.50$} &\\multicolumn{3}{c}{$\\alpha=1.25$} &\\multicolumn{3}{c}{$\\alpha=1.00$}\\\\  \n                        & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE \\\\ \n  \\hline\n\\multicolumn{13}{c}{(OLS)} \\\\\n0.0000 & 0.0000 & 0.1788 & 1.7882 & 0.0000 & 0.1693 & 1.6936 & 0.0000 & 0.1728 & 1.7280 & 0.0001 & 0.1746 & 1.7495 \\\\ \n0.2500 & 0.0219 & 0.1727 & 2.2745 & 0.0256 & 0.1698 & 2.3326 & 0.0265 & 0.1795 & 2.4239 & 0.0306 & 0.1651 & 2.4053 \\\\ \n0.5000 & 0.0936 & 0.1656 & 3.4796 & 0.1029 & 0.1593 & 3.5820 & 0.1152 & 0.1605 & 3.7550 & 0.1218 & 0.1565 & 3.8253 \\\\ \n0.7500 & 0.2162 & 0.1582 & 4.9117 & 0.2378 & 0.1609 & 5.1352 & 0.2559 & 0.1464 & 5.2658 & 0.2712 & 0.1431 & 5.4004 \\\\ \n0.9500 & 0.3443 & 0.1458 & 6.0461 & 0.3887 & 0.1362 & 6.3815 & 0.4124 & 0.1362 & 6.5651 & 0.4361 & 0.1325 & 6.7356 \\\\ \n\\multicolumn{13}{c}{(Heckman 2-step)} \\\\\n0.0000 & 0.0000 & 0.3007 & 3.0068 & 0.0001 & 0.3299 & 3.3007 & 0.0000 & 0.3503 & 3.5025 & 0.0003 & 0.3886 & 3.8898 \\\\ \n0.2500 & 0.0011 & 0.3143 & 3.1596 & 0.0023 & 0.3265 & 3.2994 & 0.0009 & 0.3564 & 3.5757 & 0.0016 & 0.3650 & 3.6722 \\\\ \n0.5000 & 0.0031 & 0.2963 & 3.0150 & 0.0046 & 0.3320 & 3.3892 & 0.0058 & 0.3338 & 3.4248 & 0.0064 & 0.3712 & 3.7973 \\\\ \n0.7500 & 0.0083 & 0.2969 & 3.1056 & 0.0106 & 0.3148 & 3.3116 & 0.0111 & 0.3270 & 3.4354 & 0.0111 & 0.3509 & 3.6635 \\\\ \n0.9500 & 0.0171 & 0.2678 & 2.9806 & 0.0169 & 0.2952 & 3.2264 & 0.0131 & 0.3221 & 3.4179 & 0.0207 & 0.3052 & 3.3747 \\\\ \n\\multicolumn{13}{c}{(\\citet{Heckman90} ($b_n=\\hat{F}^{-1}_{\\bm{Z}^{\\top}\\bm{\\gamma}_0}(.95)$))} \\\\\n0.0000 & 0.0003 & 0.4509 & 4.5124 & 0.0002 & 0.4524 & 4.5261 & 0.0004 & 0.4557 & 4.5611 & 0.0001 & 0.4689 & 4.6895 \\\\ \n0.2500 & 0.0004 & 0.4488 & 4.4929 & 0.0006 & 0.4507 & 4.5135 & 0.0003 & 0.4488 & 4.4918 & 0.0005 & 0.4513 & 4.5180 \\\\ \n0.5000 & 0.0000 & 0.4481 & 4.4815 & 0.0011 & 0.4454 & 4.4661 & 0.0015 & 0.4423 & 4.4403 & 0.0002 & 0.4342 & 4.3443 \\\\ \n0.7500 & 0.0011 & 0.4516 & 4.5283 & 0.0001 & 0.4372 & 4.3732 & 0.0025 & 0.4379 & 4.4076 & 0.0040 & 0.4315 & 4.3615 \\\\ \n0.9500 & 0.0030 & 0.4531 & 4.5641 & 0.0020 & 0.4363 & 4.3863 & 0.0030 & 0.4326 & 4.3604 & 0.0060 & 0.4327 & 4.3953 \\\\ \n\\multicolumn{13}{c}{(\\citet{AndrewsSchafgans98} ($b_n=\\hat{F}^{-1}_{\\bm{Z}^{\\top}\\bm{\\gamma}_0}(.95)$))} \\\\\n0.0000 & Inf    & Inf    & Inf    & Inf    & Inf    & Inf    & Inf    & Inf    & Inf    & Inf    & Inf    & Inf \\\\ \n0.2500 & Inf    & Inf    & Inf    & Inf    & Inf    & Inf    & Inf    & Inf    & Inf    & Inf    & Inf    & Inf \\\\ \n0.5000 & 0.0000 & 0.5650 & 5.6499 & Inf    & Inf    & Inf    & Inf    & Inf    & Inf    & 0.0000 & 0.5579 & 5.5796 \\\\ \n0.7500 & 0.0019 & 0.5757 & 5.7739 & Inf    & Inf    & Inf    & 0.0025 & 0.5465 & 5.4883 & Inf    & Inf    & Inf \\\\ \n0.9500 & Inf    & Inf    & Inf    & 0.0017 & 0.5772 & 5.7865 & 0.0007 & 0.5559 & 5.5656 & 0.0036 & 0.5365 & 5.3982 \\\\ \n   \\hline\n\\end{tabular}\n} \n\\end{table}\n\n\\begin{table}[H]\n\\pagenumbering{gobble}\n{\\scriptsize\n\\centering\n\\caption{DGP2 (non-normal), $n=100$, 1000 replications. Alternative estimators.  (RMSE is multiplied by $\\sqrt{n}$.)}\\label{dgp2n100balt}\n\\begin{tabular}{ccccccccccccc}\n  \\hline\n\\multirow{2}{*}{$\\rho$} &\\multicolumn{3}{c}{$\\alpha=2.00$} &\\multicolumn{3}{c}{$\\alpha=1.50$} &\\multicolumn{3}{c}{$\\alpha=1.25$} &\\multicolumn{3}{c}{$\\alpha=1.00$}\\\\  \n                        & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE \\\\ \n  \\hline\n\\multicolumn{13}{c}{(OLS)} \\\\\n0.0000 & 0.0001 & 0.4821 & 4.8221 & 0.0000 & 0.3909 & 3.9091 & 0.0014 & 1.0831 & 10.8368 & 0.0005 & 0.6892 & 6.8959 \\\\ \n0.2500 & 0.1510 & 0.3288 & 5.0909 & 0.1736 & 0.4121 & 5.8603 & 0.2394 & 0.7670 & 9.0977 & 0.2633 & 2.1317 & 21.9257 \\\\ \n0.5000 & 0.5682 & 0.3763 & 8.4250 & 0.7799 & 0.7149 & 11.3620 & 0.8475 & 0.4896 & 10.4266 & 1.1118 & 1.1903 & 15.9015 \\\\ \n0.7500 & 1.2924 & 0.2869 & 11.7248 & 1.7087 & 0.4380 & 13.7860 & 1.8913 & 1.8977 & 23.4365 & 2.3993 & 1.4258 & 21.0529 \\\\ \n0.9500 & 2.1300 & 0.3085 & 14.9169 & 2.6775 & 0.5903 & 17.3954 & 3.2066 & 1.2800 & 22.0111 & 3.6936 & 4.8277 & 51.9616 \\\\ \n\\multicolumn{13}{c}{(Heckman 2-step)} \\\\\n0.0000 & 0.0000 & 0.5162 & 5.1620 & 0.0002 & 0.5989 & 5.9913 & 0.0020 & 1.6850 & 16.8557 & 0.0055 & 1.1861 & 11.8844 \\\\ \n0.2500 & 0.1849 & 0.5086 & 6.6596 & 0.2433 & 0.6251 & 7.9627 & 0.3461 & 0.7657 & 9.6563 & 0.4278 & 1.4736 & 16.1227 \\\\ \n0.5000 & 0.7718 & 0.6221 & 10.7649 & 1.2359 & 1.9152 & 22.1445 & 1.3865 & 0.9240 & 14.9673 & 2.3721 & 2.9882 & 33.6172 \\\\ \n0.7500 & 1.6186 & 0.5040 & 13.6844 & 1.0515 & 16.7408 & 167.7214 & 3.2286 & 2.1692 & 28.1672 & 4.6267 & 2.7391 & 34.8276 \\\\ \n0.9500 & 2.7113 & 0.5127 & 17.2457 & 3.8929 & 0.9294 & 21.8101 & 5.4307 & 3.6028 & 42.9077 & 8.1794 & 4.7219 & 55.2045 \\\\\n\\multicolumn{13}{c}{(\\citet{Heckman90} ($b_n=\\hat{F}^{-1}_{\\bm{Z}^{\\top}\\bm{\\gamma}_0}(.95)$))} \\\\\n0.0000 & 0.0001 & 0.4388 & 4.3889 & 0.0001 & 0.4741 & 4.7418 & 0.0004 & 0.4621 & 4.6254 & 0.0000 & 0.4743 & 4.7428 \\\\ \n0.2500 & 0.2121 & 0.4677 & 6.5640 & 0.3145 & 0.5099 & 7.5793 & 0.4265 & 0.6050 & 8.9023 & 0.5743 & 0.6945 & 10.2790 \\\\ \n0.5000 & 0.8606 & 0.4819 & 10.4540 & 1.3574 & 0.8578 & 14.4677 & 1.5614 & 0.8493 & 15.1087 & 2.1035 & 0.9589 & 17.3869 \\\\ \n0.7500 & 1.8488 & 0.5264 & 14.5806 & 2.9232 & 1.0078 & 19.8465 & 3.4719 & 0.9811 & 21.0580 & 5.0749 & 1.8515 & 29.1596 \\\\ \n0.9500 & 3.0237 & 0.7279 & 18.8508 & 4.3979 & 1.0647 & 23.5192 & 5.6603 & 1.6073 & 28.7121 & 7.8382 & 1.9172 & 33.9323 \\\\\n\\multicolumn{13}{c}{(\\citet{AndrewsSchafgans98} ($b_n=\\hat{F}^{-1}_{\\bm{Z}^{\\top}\\bm{\\gamma}_0}(.95)$))} \\\\\n0.0000 & 0.0000 & 0.9234 & 9.2336 & 0.0015 & 1.0125 & 10.1326 & 0.0017 & 1.0037 & 10.0452 & 0.0001 & 0.9807 & 9.8079 \\\\ \n0.2500 & 0.2035 & 1.0138 & 11.0963 & 0.3754 & 1.1914 & 13.3967 & 0.6937 & 1.8209 & 20.0236 & 1.0490 & 2.1058 & 23.4170 \\\\ \n0.5000 & 0.8978 & 1.0810 & 14.3748 & 2.0336 & 2.9706 & 32.9515 & 2.3156 & 3.0879 & 34.4245 & 3.2494 & 2.6734 & 32.2432 \\\\ \n0.7500 & 1.8968 & 1.1576 & 17.9912 & Inf & Inf & Inf & 4.9692 & 2.9053 & 36.6200 & 10.1655 & 7.9701 & 85.8419 \\\\ \n0.9500 & 3.3558 & 2.1412 & 28.1792 & 5.0293 & 2.8442 & 36.2197 & 7.8162 & 3.3294 & 43.4752 & 13.5506 & 7.1477 & 80.3995 \\\\ \n   \\hline\n\\end{tabular}\n} \n\\end{table}\n\n\\begin{table}[H]\n\\pagenumbering{gobble}\n{\\scriptsize\n\\centering\n\\caption{DGP1 (bivariate normal), $n=400$, 1000 replications. Alternative estimators.  (RMSE is multiplied by $\\sqrt{n}$.)}\\label{dgp1n400balt}\n\\begin{tabular}{ccccccccccccc}\n  \\hline\n\\multirow{2}{*}{$\\rho$} &\\multicolumn{3}{c}{$\\alpha=2.00$} &\\multicolumn{3}{c}{$\\alpha=1.50$} &\\multicolumn{3}{c}{$\\alpha=1.25$} &\\multicolumn{3}{c}{$\\alpha=1.00$}\\\\  \n                        & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE \\\\ \n  \\hline\n\\multicolumn{13}{c}{(OLS)} \\\\\n0.0000 & 0.0000 & 0.0866 & 1.7318 & 0.0000 & 0.0809 & 1.6194 & 0.0000 & 0.0833 & 1.6676 & 0.0000 & 0.0756 & 1.5130 \\\\ \n0.2500 & 0.0232 & 0.0840 & 3.4805 & 0.0258 & 0.0823 & 3.6111 & 0.0289 & 0.0829 & 3.7818 & 0.0295 & 0.0779 & 3.7712 \\\\ \n0.5000 & 0.0935 & 0.0805 & 6.3227 & 0.1071 & 0.0772 & 6.7232 & 0.1149 & 0.0771 & 6.9534 & 0.1211 & 0.0750 & 7.1190 \\\\ \n0.7500 & 0.2163 & 0.0758 & 9.4248 & 0.2390 & 0.0712 & 9.8808 & 0.2530 & 0.0712 & 10.1604 & 0.2719 & 0.0684 & 10.5179 \\\\ \n0.9500 & 0.3417 & 0.0695 & 11.7732 & 0.3830 & 0.0664 & 12.4488 & 0.4087 & 0.0634 & 12.8487 & 0.4366 & 0.0624 & 13.2739 \\\\ \n\\multicolumn{13}{c}{(Heckman 2-step)} \\\\\n0.0000 & 0.0000 & 0.1552 & 3.1042 & 0.0000 & 0.1640 & 3.2810 & 0.0000 & 0.1784 & 3.5672 & 0.0000 & 0.1832 & 3.6645 \\\\ \n0.2500 & 0.0000 & 0.1522 & 3.0456 & 0.0001 & 0.1591 & 3.1876 & 0.0001 & 0.1720 & 3.4472 & 0.0001 & 0.1797 & 3.6009 \\\\ \n0.5000 & 0.0002 & 0.1525 & 3.0627 & 0.0005 & 0.1646 & 3.3198 & 0.0003 & 0.1709 & 3.4357 & 0.0002 & 0.1770 & 3.5502 \\\\ \n0.7500 & 0.0009 & 0.1435 & 2.9301 & 0.0006 & 0.1559 & 3.1585 & 0.0003 & 0.1641 & 3.2977 & 0.0011 & 0.1708 & 3.4785 \\\\ \n0.9500 & 0.0007 & 0.1368 & 2.7857 & 0.0008 & 0.1468 & 2.9915 & 0.0012 & 0.1558 & 3.1939 & 0.0010 & 0.1614 & 3.2917 \\\\ \n\\multicolumn{13}{c}{(\\citet{Heckman90} ($b_n=\\hat{F}^{-1}_{\\bm{Z}^{\\top}\\bm{\\gamma}_0}(.95)$))} \\\\\n0.0000 & 0.0000 & 0.1623 & 3.2457 & 0.0000 & 0.1627 & 3.2548 & 0.0000 & 0.1653 & 3.3058 & 0.0000 & 0.1608 & 3.2164 \\\\ \n0.2500 & 0.0001 & 0.1585 & 3.1769 & 0.0001 & 0.1594 & 3.1950 & 0.0002 & 0.1566 & 3.1470 & 0.0005 & 0.1557 & 3.1449 \\\\ \n0.5000 & 0.0005 & 0.1582 & 3.1931 & 0.0010 & 0.1556 & 3.1755 & 0.0016 & 0.1647 & 3.3916 & 0.0028 & 0.1696 & 3.5550 \\\\ \n0.7500 & 0.0007 & 0.1569 & 3.1826 & 0.0016 & 0.1607 & 3.3140 & 0.0032 & 0.1557 & 3.3125 & 0.0070 & 0.1502 & 3.4401 \\\\ \n0.9500 & 0.0009 & 0.1539 & 3.1347 & 0.0027 & 0.1458 & 3.0967 & 0.0052 & 0.1487 & 3.3033 & 0.0098 & 0.1515 & 3.6168 \\\\ \n\\multicolumn{13}{c}{(\\citet{AndrewsSchafgans98} ($b_n=\\hat{F}^{-1}_{\\bm{Z}^{\\top}\\bm{\\gamma}_0}(.95)$))} \\\\\n0.0000 & Inf & Inf & Inf & 0.0000 & 0.3118 & 6.2369 & Inf & Inf & Inf & 0.0000 & 0.2991 & 5.9817 \\\\ \n0.2500 & 0.0000 & 0.3043 & 6.0860 & 0.0000 & 0.3056 & 6.1114 & Inf & Inf & Inf & Inf & Inf & Inf \\\\ \n0.5000 & Inf & Inf & Inf & 0.0001 & 0.2973 & 5.9497 & Inf & Inf & Inf & 0.0004 & 0.3065 & 6.1438 \\\\ \n0.7500 & 0.0001 & 0.2888 & 5.7792 & 0.0000 & 0.3041 & 6.0817 & Inf & Inf & Inf & Inf & Inf & Inf \\\\ \n0.9500 & Inf & Inf & Inf & Inf & Inf & Inf & Inf & Inf & Inf & 0.0015 & 0.3021 & 6.0908 \\\\ \n   \\hline\n\\end{tabular}\n} \n\\end{table}\n\n\\begin{table}[H]\n\\pagenumbering{gobble}\n{\\scriptsize\n\\centering\n\\caption{DGP2 (non-normal), $n=400$, 1000 replications. Alternative estimators.  (RMSE is multiplied by $\\sqrt{n}$.)}\\label{dgp2n400balt}\n\\begin{tabular}{ccccccccccccc}\n  \\hline\n\\multirow{2}{*}{$\\rho$} &\\multicolumn{3}{c}{$\\alpha=2.00$} &\\multicolumn{3}{c}{$\\alpha=1.50$} &\\multicolumn{3}{c}{$\\alpha=1.25$} &\\multicolumn{3}{c}{$\\alpha=1.00$}\\\\  \n                        & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE            & sq bias   & sd     & RMSE \\\\ \n  \\hline\n\\multicolumn{13}{c}{(OLS)} \\\\\n0.0000 & 0.0000 & 0.1225 & 2.4496 & 0.0000 & 0.1295 & 2.5902 & 0.0000 & 0.1356 & 2.7151 & 0.0000 & 0.1423 & 2.8469 \\\\ \n0.2500 & 0.1464 & 0.1253 & 8.0533 & 0.1855 & 0.1377 & 9.0428 & 0.2246 & 0.1549 & 9.9707 & 0.2653 & 0.1651 & 10.8174 \\\\ \n0.5000 & 0.5746 & 0.1254 & 15.3660 & 0.7346 & 0.1398 & 17.3680 & 0.8895 & 0.2137 & 19.3406 & 1.0931 & 0.2617 & 21.5551 \\\\ \n0.7500 & 1.3338 & 0.1182 & 23.2184 & 1.6652 & 0.1752 & 26.0457 & 1.9547 & 0.2239 & 28.3186 & 2.3994 & 0.2910 & 31.5215 \\\\ \n0.9500 & 2.1328 & 0.1128 & 29.2955 & 2.6396 & 0.1762 & 32.6839 & 3.1089 & 0.2503 & 35.6178 & 4.0168 & 0.4762 & 41.1997 \\\\ \n\\multicolumn{13}{c}{(Heckman 2-step)} \\\\\n0.0000 & 0.0000 & 0.1900 & 3.8004 & 0.0000 & 0.2107 & 4.2133 & 0.0002 & 0.2368 & 4.7430 & 0.0000 & 0.2641 & 5.2828 \\\\ \n0.2500 & 0.1966 & 0.1881 & 9.6325 & 0.2873 & 0.2342 & 11.6982 & 0.4031 & 0.3386 & 14.3913 & 0.6158 & 0.4610 & 18.2023 \\\\ \n0.5000 & 0.7901 & 0.1930 & 18.1916 & 1.1622 & 0.2667 & 22.2114 & 1.6124 & 0.5121 & 27.3834 & 2.4130 & 0.6840 & 33.9460 \\\\ \n0.7500 & 1.8144 & 0.2124 & 27.2728 & 2.6411 & 0.4291 & 33.6168 & 3.4999 & 0.5283 & 38.8794 & 5.4115 & 0.8450 & 49.4998 \\\\ \n0.9500 & 2.8707 & 0.2172 & 34.1632 & 4.1789 & 0.4178 & 41.7300 & 5.6244 & 0.6344 & 49.0992 & 9.1834 & 1.2123 & 65.2785 \\\\ \n\\multicolumn{13}{c}{(\\citet{Heckman90} ($b_n=\\hat{F}^{-1}_{\\bm{Z}^{\\top}\\bm{\\gamma}_0}(.95)$))} \\\\\n0.0000 & 0.0000 & 0.1625 & 3.2499 & 0.0000 & 0.1668 & 3.3373 & 0.0001 & 0.1698 & 3.4017 & 0.0000 & 0.1703 & 3.4069 \\\\ \n0.2500 & 0.1881 & 0.1647 & 9.2789 & 0.2496 & 0.1838 & 10.6470 & 0.3165 & 0.2183 & 12.0686 & 0.4023 & 0.2231 & 13.4475 \\\\ \n0.5000 & 0.7417 & 0.1732 & 17.5695 & 0.9928 & 0.2013 & 20.3307 & 1.2631 & 0.3301 & 23.4275 & 1.6212 & 0.4082 & 26.7415 \\\\ \n0.7500 & 1.7112 & 0.1877 & 26.4303 & 2.2915 & 0.3216 & 30.9514 & 2.7843 & 0.3655 & 34.1634 & 3.5582 & 0.4415 & 38.7456 \\\\ \n0.9500 & 2.7126 & 0.1885 & 33.1547 & 3.6019 & 0.2978 & 38.4220 & 4.4380 & 0.4261 & 42.9862 & 6.0276 & 0.7106 & 51.1176 \\\\  \n\\multicolumn{13}{c}{(\\citet{AndrewsSchafgans98} ($b_n=\\hat{F}^{-1}_{\\bm{Z}^{\\top}\\bm{\\gamma}_0}(.95)$))} \\\\\n0.0000 & 0.0000 & 0.4867 & 9.7345 & 0.0001 & 0.5337 & 10.6766 & 0.0001 & 0.5335 & 10.6730 & 0.0004 & 0.5566 & 11.1378 \\\\ \n0.2500 & 0.2272 & 0.5277 & 14.2224 & 0.4256 & 0.7807 & 20.3484 & 0.7890 & 1.5216 & 35.2385 & 1.1932 & 1.4466 & 36.2534 \\\\ \n0.5000 & 0.9464 & 0.7039 & 24.0155 & 1.6530 & 1.0790 & 33.5688 & 2.9148 & 2.8347 & 66.1824 & 5.3105 & 3.1842 & 78.6128 \\\\ \n0.7500 & 2.2403 & 0.9842 & 35.8267 & 3.9629 & 2.3050 & 60.9133 & 6.0330 & 2.7449 & 73.6684 & 10.2806 & 3.3502 & 92.7458 \\\\ \n0.9500 & 3.5332 & 0.9977 & 42.5610 & 5.7880 & 1.8579 & 60.7943 & 9.2207 & 3.1294 & 87.2098 & 21.6148 & 5.9786 & 151.4702 \\\\  \n   \\hline\n\\end{tabular}\n} \n\\end{table}\n\nIn summary, the simulations presented here show the potential of the proposed estimator to exhibit good performance in \nterms of RMSE across two different parametric families of data-generating process, across variation in the degree to \nwhich the errors $U$ and $V$ are dependent and across variation in the extent to which the parameter of interest is identified.  This assessment is \nunaffected by moderate variation in the estimated MSE-optimal bandwidth used to \nimplement the proposed estimator.  Tables~\\ref{dgp1n100mhat1.p2} and \\ref{dgp2n100mhat1.p2} also support the conclusion of Theorem~\\ref{mainthm}  \nin indicating the sensitivity of the bias of the proposed estimator to variation in the parameter $(\\rho,\\alpha)$ \nunder both DGP1 and DGP2.  These results also show that the estimated MSE-optimal bandwidth used to implement the proposed \nestimator was effective in limiting the extent to which the RMSE of the proposed estimator was sensitive to variation in \n$(\\rho,\\alpha)$.  In particular, the RMSE of the proposed estimator was found under DGP2 to dominate those of the other \nestimators considered.\n\n\n\\section{Empirical Example}\\label{ee}\n\n\\noindent  This section reconsiders individual labour-market data from Malaysia that were originally \nanalyzed by \\citet{Schafgans00}.  The estimator developed above is applied to the problem of \nestimating the extent of plausible gender wage discrimination in Malaysia using data from \nthe Second Malaysian Family Life Survey (MFLS2) conducted between August 1988 and January 1989.  Inferences \navailable from the proposed estimator are compared with those obtained via \nthe same alternative estimators considered in Section~\\ref{mc}.  The proposed estimator was found to \ngenerate inferences that differ significantly from those obtainable via the alternative estimators used in \nthe simulation experiments described in Section~\\ref{mc}.  All estimators applied to the MFLS2 data considered \nin this section were implemented in precisely the same way in which they were implemented in the simulation \nexperiments presented earlier.\n\nI consider a decomposition of the female--male log-wage difference for ethnic Malay workers.  The \nconsideration of gender wage gaps for Malaysian workers of the same ethnicity is potentially important because of the differential treatment of \nMalays in the labour force after 1970 \\citep[see e.g.,][and references cited]{Schafgans00}.  I follow \\citet{Schafgans00} by analyzing gender wage gaps in the MFLS2 using the basic decomposition \ntechnique of \\citet{Oaxaca73}.  In particular, suppose that the generic model given above in \n(\\ref{outcome})--(\\ref{observation}) holds for both men and women, \ni.e., \n\\begin{eqnarray}\nY^*_j &=& \\theta_{j}+\\bm{X}^{\\top}_j\\bm{\\beta}_{j}+U_j,\\label{outcomej}\\\\\nD_j &=& 1\\left\\{\\bm{Z}^{\\top}_j\\bm{\\gamma}_{j}\\ge V_j\\right\\},\\label{selectionj}\\\\\nY_j &=& DY^*_j\\label{observationj}\n\\end{eqnarray}\nwhere $Y^*_j$ is the natural logarithm of the offered average hourly wage, and where the index $j\\in\\{0,1\\}$ denotes a given gender.  For $j\\in\\{0,1\\}$ let \n$\\bar{Y}_j\\equiv E\\left[\\left.Y_j\\right|D_j=1\\right]$, and let $\\bar{\\bm{X}}_j$ denote the average \n``endowments'' of wage-determining attributes for workers of gender $j$.  The observed log-wage gap \n$\\bar{Y}_1-\\bar{Y}_0$ between the two genders can then be decomposed as\n\\begin{eqnarray}\n\\bar{Y}_1-\\bar{Y}_0 &=&\\left[\\left(\\theta_1-\\theta_0\\right)+\\bar{\\bm{X}}^{\\top}_0\\left(\\bm{\\beta}_1-\\bm{\\beta}_0\\right)\\right]+(\\bar{\\bm{X}}_1-\\bar{\\bm{X}}_0)^{\\top}\\bm{\\beta}_1\\nonumber\\\\\n\t& &+\\left(E\\left[\\left.U_1\\right|D_1=1\\right]-E\\left[\\left.U_0\\right|D_0=1\\right]\\right)\\label{decomp1}\\\\\n\t&=&\\left[\\left(\\theta_1-\\theta_0\\right)+\\bar{\\bm{X}}^{\\top}_1\\left(\\bm{\\beta}_1-\\bm{\\beta}_0\\right)\\right]+(\\bar{\\bm{X}}_1-\\bar{\\bm{X}}_0)^{\\top}\\bm{\\beta}_0\\nonumber\\\\\n\t& &+\\left(E\\left[\\left.U_1\\right|D_1=1\\right]-E\\left[\\left.U_0\\right|D_0=1\\right]\\right)\\label{decomp2}\\\\\n\t&\\equiv & A+B+C,\\label{decomp}\n\\end{eqnarray}\nwhere $A$ is that part of the gap due to differences in wage structures between genders;\n$B$ is due to observable differences between men and women in wage-determining \ncharacteristics and $C$ is the contribution of differential self-selection into the labour force.  Following \n\\citet{Schafgans00} the quantity $\\bar{Y}_1-\\bar{Y}_0-C=A+B$ is referred to as the \n{\\it selection-corrected log-wage gap}.\n\nWage discrimination in favor of members of gender $j=1$ is empirically plausible if the overall log-wage gap \n$\\bar{Y}_1-\\bar{Y}_0$ cannot be entirely explained by differential self-selection into paid work, differences in \nobserved endowments or by differing returns to those endowments.  Moreover, given the definitions of the \nquantities $A$ and $B$ appearing above in (\\ref{decomp}), the extent of plausible wage discrimination favoring gender 1 may be equated with the difference in \nintercepts $\\theta_1-\\theta_0$.  \n\nThe analysis that follows considers a subset of the sample taken from the MFLS2 of 1988--89 that \nwas analyzed by \\citet{Schafgans00}.  This particular dataset is publicly available from the {\\it Journal of Applied Econometrics} Data Archive \nat \\url{http:\/\/qed.econ.queensu.ca\/jae\/1998-v13.5\/schafgans\/}.  Each observation in this sample corresponds to a member of the labour force.  I specifically consider ethnic \nMalays residing in non-urban settings who were observed to have some level of unearned household income in terms of dividends, interests or rents, \nand who were also observed to have passed the highest level of schooling (i.e., primary on the one hand, or secondary or above) corresponding to the \nnumber of years of schooling observed.  This subset of the MFLS2 consisted of 965 women and 878 men.\n\nI also use the same variable specifications used by \\citet{Schafgans00}.  In particular, the outcome variable $Y^*_j$ is LWAGE, the log hourly real \nwage in the local currency deflated using the 1985 consumer price index.  The selection variable $D_j$ is the indicator PAIDWORK for whether the \nindividual in question is in fact a wage worker.  The exogenous variables appearing in the selection equations for each gender include UNEARN, a \nmeasure of household unearned income in terms of dividends, interest and rents; HOUSEH, the value of household \nreal estate owned, computed as the product of an indicator variable for house ownership and the cost of the house \nowned; and AMTLAND, the extent of household landholding in hundreds of acres.  In addition, selection into wage \nwork is also assumed to be determined by AGE, in years; AGESQ, the square of AGE divided by 100; YPRIM, years of \nprimary schooling and YSEC, years of schooling at the secondary level or above.  The variables appearing on the \nright-hand side of the outcome equations for each gender or ethnic group are AGE, AGESQ, YPRIM and YSEC.  \n\\citet[Section~4]{Schafgans00} contains further details regarding variable definitions.\n\nFor each gender $j\\in\\{\\mbox{female, male}\\}$, the proposed estimator and that of \\citet{Heckman90} and \\citet{AndrewsSchafgans98} rely on \nthe preliminary procedure described in \\citet[p. 484--487]{Schafgans98} to estimate the nuisance parameters \n$\\bm{\\beta}_j$ and $\\bm{\\gamma}_j$ appearing in (\\ref{outcomej}) and (\\ref{selectionj}), respectively.  This \ninvolves estimating the selection equation for each group via the method of \\citet{KleinSpady93} and estimating \nthe slope parameters in each outcome equation using the method of \\citet{Robinson88}.  This is followed by \nestimation of the intercept parameter in each outcome equation via the proposed estimator.  Standard errors are \ncalculated by bootstrapping with replacement with $B=200$ replications.  \n\nEstimates of the outcome-equation \nparameters obtained via the proposed estimator are given in which the proposed estimator is implemented using the same \nkernel and estimated MSE-optimal bandwidth $\\hat{h}^*_n$ used in the simulations reported in Section~\\ref{mc}.  In \ncommon with the results given earlier in Section~\\ref{mc}, I also considered implementations of the proposed \nestimator in which the bandwidth was set to $h_n=(2\/3)\\hat{h}^*_n$ and $h_n=(3\/2)\\hat{h}^*_n$.\n\nThe decomposition of the observed gender log-wage gaps for ethnic Malay workers is presented in Table~\\ref{eedecomp1110}.  In keeping with the theory \ndeveloped above, the focus is on the extent of plausible gender wage discrimination, which is identified with the difference between the estimated \nintercepts.  A striking result is the evidence provided by the proposed estimator of positive wage discrimination in \nfavor of women.  In particular, Table~\\ref{eedecomp1110} indicates that all three implementations of the proposed estimator imply a large, positive \nand significant difference in intercepts, while the OLS and 2-step procedures generated estimated intercept differences \nthat were both insignificant.  The implementation of the H90 procedure with $b_n$ set to the .90-quantile of the estimated \nselection index generated a similarly insignificant estimate of the difference in intercepts.  The other implementation of the H90 procedure, \nalong with the AS98 procedure, proved to be numerically unstable.  Table~\\ref{eedecomp1110} indicates that there exists a clear difference in \ninferences regarding the extent of gender wage discrimination amongst ethnic Malay workers between estimates generated by the \nproposed estimator and those generated by established procedures.  \n\n\\FloatBarrier\n\\begin{landscape}\n\\begin{table}[H]\n\\pagenumbering{gobble}\n{\\scriptsize\n\\caption{Female--male log-wage decomposition, Malays.  Standard errors in parentheses}\\label{eedecomp1110} \n\\centering\n\\hspace*{-30pt}\\setlength\\tabcolsep{3pt}\n\\begin{tabular}{l|c|c|c|c|c|c|c|c}\n  \\hline\\hline\n\\rule{0pt}{4ex}\\multirow{2}{*}{Wage gap (overall)}  & \\multicolumn{8}{c}{-0.2882} \\\\ \n                                                    & \\multicolumn{8}{c}{(0.0425)} \\\\[4pt] \\hline \n\\rule{0pt}{4ex}\\multirow{2}{*}{Female (endowment)}  & \\multicolumn{8}{c}{-0.0638} \\\\ \n                                                    & \\multicolumn{8}{c}{(0.0337)} \\\\[4pt] \\hline \n\\rule{0pt}{4ex}\\multirow{2}{*}{Male (endowment)}    & \\multicolumn{8}{c}{-0.0369} \\\\ \n                                                    & \\multicolumn{8}{c}{(0.0377)}\\\\[4pt] \n\\hline\n\\rule{0pt}{4ex}\\multirow{2}{*}{ }             & \\multicolumn{3}{c|}{\\rule{0pt}{4ex} $\\hat{\\theta}_n$}  & \\multirow{2}{*}{OLS} & \\multirow{2}{*}{2-step} & \\multicolumn{2}{c|}{H90} & AS98 \\\\[2pt] \\cline{2-4}\\cline{7-9} \n                               &\\rule{0pt}{4ex} ($h_n=\\hat{h}^*_n$) & ($h_n=(2\/3)\\hat{h}^*_n$) & ($h_n=(3\/2)\\hat{h}^*_n$) & & & ($b_n=\\hat{F}^{-1}_{\\bm{Z}^{\\top}\\hat{\\bm{\\gamma}}_n}(.90)$) & ($b_n=\\hat{F}^{-1}_{\\bm{Z}^{\\top}\\hat{\\bm{\\gamma}}_n}(.95)$) &($b_n=\\hat{F}^{-1}_{\\bm{Z}^{\\top}\\hat{\\bm{\\gamma}}_n}(.95)$) \\\\[4pt]\n\\hline\\rule{0pt}{4ex}\\multirow{2}{*}{Wage gap (selection-corrected)} & 0.9283 & 0.9328 & 0.9263 & -1.337 & -10.1628 & -0.2709 & -0.0516 & -0.1015 \\\\ \n                                                               & (0.7516) & (0.7515) & (0.7517) & (0.0812) & (1.0385) & (1.0617) & (NaN) & (NaN)  \\\\[4pt]\\hline \n\\rule{0pt}{4ex}\\multirow{2}{*}{Female (coefficients)}          &\\multicolumn{3}{c|}{0.1098}   & -1.2553  & -8.0955  &\\multicolumn{2}{c|}{0.1098} & 0.1098 \\\\ \n                                                               &\\multicolumn{3}{c|}{(0.6483)} & (0.6977) & (2.2356) &\\multicolumn{2}{c|}{(0.6483)} & (0.6483) \\\\[4pt] \\hline \n\\rule{0pt}{4ex}\\multirow{2}{*}{Male (coefficients)}            &\\multicolumn{3}{c|}{0.0829}   & -1.2951 & -7.988    &\\multicolumn{2}{c|}{0.0829} & 0.0829 \\\\ \n                                                               &\\multicolumn{3}{c|}{(0.6457)} & (0.6989) & (2.2296) &\\multicolumn{2}{c|}{(0.6457)} & (0.6457) \\\\[4pt] \\hline \n\\rule{0pt}{4ex}\\multirow{2}{*}{Difference in intercepts}       & 0.8823   & 0.8867   & 0.8803   & -0.0364  & -2.1082  & -0.3169  & -0.0976 & -0.1475 \\\\ \n                                                               & (0.3787) & (0.3784) & (0.3789) & (0.7526) & (3.1712) & (0.8401) & (NaN)   & (NaN)  \\\\ \n   \\hline\n\\end{tabular}\n}\n\\end{table}\n\\end{landscape}\n\n\\section{Conclusion}\\label{concl}\n\n\\noindent This paper has developed a new estimator of the intercept of a sample-selection model in which the \njoint distribution of the unobservables and the selection index is unspecified.  It has been shown that the new \nestimator can be made under mild conditions to converge in probability at an $n^{-p\/(2p+1)}$-rate, where $p\\ge 2$ is an \ninteger that indexes the strength of certain smoothness assumptions as given above in Assumption~\\ref{a2}.\\ref{a2d}.  \nThis rate of convergence is shown to be the optimal rate of convergence for estimation of the intercept parameter in terms \nof a minimax criterion.  The new estimator is under mild conditions consistent and asymptotically normal with a rate of \nconvergence that is the same regardless of the joint distribution of the unobservables and the selection \nindex.  This differs from other proposals in the literature and is convenient in practice, as the extent to which \nselection is endogenous is typically unknown in applications.  In addition, the rate of convergence of the new estimator, \nunlike those of better known estimators, does not depend on assumptions regarding the relative tail behaviours of \nthe determinants of selection beyond those necessary for the identification of the estimand.  This similarly \nfacilitates statistical inference regarding the intercept.  Simulations presented above show the potential accuracy of \nthe proposed estimator relative to that of established procedures across different model specifications.  An \nempirical example using individual labour-market data from Malaysia shows the potential of the proposed estimator to \ngenerate inferences regarding the extent of plausible gender wage discrimination that differ from those available from  \nbetter known estimators. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nArtificial intelligence (AI) techniques require considerable training data to achieve satisfying performance, especially for industrial applications. Therefore, one of the major challenges in building AI models is collecting enough training data. Conventionally,  the model developers could aggregate data by purchasing from multiple data owners or directly collecting data from the service users. However, these ways raise the risk of data privacy leakage and are forbidden by regulations. For example, the General Data Protection Regulation (GDPR)~\\cite{voss2016european} enacted by the European Union (EU) rectifies the usage of personal data and prohibits the transfer of personal data collected from users. The data privacy protection regulations pose a new challenge for AI developers to utilize the scattered data to build AI models. \n\nTo tackle this challenge, researchers propose a new distributed machine learning paradigm, dubbed Federated Learning (FL), which allows multiple participants to jointly train a machine learning model without directly sharing their private data \\cite{yang2019federated,mcmahan2017communication}. With a rapid development, FL are gaining widespread applications, such as language modeling \\cite{mcmahan2017learning8}, topic modeling \\cite{jiang2019federated103,jiang2021industrial}, speech recognition \\cite{Jiang2021GDPR}, healthcare~\\cite{chen2020fedhealth}, etc. Apart from the development of FL in research areas, the high commercial value of FL also motivates companies and institutions to explore industrial-level FL platforms, such as Tensorflow Federated~\\cite{tff}, PaddleFL~\\cite{ma2019paddlepaddle}, and FATE~\\cite{liu2021fate}, which provide comprehensive infrastructures for businesses to set up their FL prototypes and services. However, the application of such platforms requires the AI developers to be knowledgeable and familiar with not only the federated learning algorithms but also the applied platforms \\cite{zhuang2022easyfl}. Therefore, it turns out to be a high barrier for many potential industrial businesses. On the other hand, industrial users usually have their existing AI models that have been sufficiently evaluated on real-world businesses. In contrast, the existing FL algorithms and platforms require the participants to train federated models from scratch, bringing extra overheads and risks to the running AI services. \n\nHence, these come up with two consequent requirements for applying FL to practical industrial businesses. First, the insertion of FL should be lightweight, without increasing significant efforts and expert knowledge. Second, the federalization should be pluggable, which means the user can seamlessly switch its models between the federated and the non-federated versions.\n\n\n\\begin{figure*}[!htp]\n    \\centering\n\t\\includegraphics[width=0.8\\linewidth]{framework.jpg}\n\t\\caption{The general framework of WrapperFL. The left part shows the existing local models that are deployed separately in different businesses. The right part depicts the federated system where the existing local models of different businesses are extended with WrapperFL.}\n\t\\label{fig:framework}\n\\end{figure*}\n\n\nMotivated by these, this paper proposes a novel, simple yet effective federated learning framework, dubbed WrapperFL, which can be easily plugged into an existing machine learning system with little extra effort. The framework is non-intrusive, simply attaching to and reusing the input and the output interfaces of an existing machine learning model. With the spirit of ensemble learning \\cite{sagi2018ensemble}, WrapperFL regards the existing machine learning models as the experts whose knowledge can be exchanged with the others in a federated manner. Different from existing federated learning algorithms, such as FedAvg\\cite{mcmahan2017communication},  which train a joint model from scratch, WrapperFL aims to exploit existing models, which further poses another challenge of dealing with the heterogeneity in data distributions and existing models \\cite{li2020federated-fedprox,karimireddy2020scaffold}. More specifically, we design two model-agnostic strategies of WrapperFL, i.e., StackingWrapper and BaggingWrapper. The former is suitable for centralized federated learning, where exists a central curator that coordinates all federated participants; the latter is suitable for decentralized federated learning without a central curator. The details are described in section~\\ref{sec:method}. \n\n\nAs the name states, the WrapperFL converts an existing non-federated model to the federated version by wrapping it with a highly integrated toolkit, which contains learning-related components, i.e., translator and aggregator, and communication-related components, i.e., scheduler and connection manager. As shown in Figure \\ref{fig:framework}, the WrapperFL wraps the existing model by attaching it with external components without changing the original data flow as well as the existing local model. This maintains the usability of the existing model in its production environment. Besides, WrapperFL preserves identical input and output interfaces as the original local model for the upstream and downstream services, avoiding extra costs of adjusting existing services. The scheduler and connection manager are responsible for coordinating the federated system as well as transferring model parameters, which are decoupled from the data flow and interface logic, and are agnostic to existing services.  \n\nTo demonstrate the efficacy and feasibility of our proposed WrapperFL, we conduct comprehensive experiments on diverse tasks with different settings, including heterogeneous data distributions and heterogeneous models. The experimental results demonstrate that WrapperFL can be successfully applied to a wide range of applications under practical settings and improves the local model with federated learning. Meanwhile, the cost of switching a non-federated local model to a federated version is low in terms of both developing efforts and training time. \n\nIn summary, the major contribution of this paper is that we propose a pluggable federated learning toolkit, WrapperFL, which can painlessly convert an existing local model to a federated version and achieves performance gain. \n\n\n\n\\section{Related Work}\n\\subsection{Federated Learning Platforms}\nWith the huge demand from academia and industry, developers have proposed several FL platforms. The FL platforms can be roughly categorized into research-oriented and industry-oriented. Tensorflow Federated (TFF)~\\cite{tff} and FederatedScope~\\cite{federatedscope}, for example, focus on the support of deep learning models and allow flexible FL mechanism designs, which are favorable for FL research. However, they lack flow management and support for vertical federated learning \\cite{yang2019federated}, which makes them not feasible for real-world industrial applications. Federated AI Technology Enabler (FATE)~\\cite{liu2021fate} is the first open-source industrial FL platform that provides comprehensive APIs and flow controls for constructing federated applications, leading the trend of FL applications and inspiring other platforms such as PaddleFL~\\cite{ma2019paddlepaddle}. However, they are not user-friendly due to their complicated system architecture and high barriers. Recently, Zhuang et al. propose EasyFL~\\cite{zhuang2022easyfl}, a low-code platform that simplifies the development of FL models and allows for rapid testing. \n\nCompared to FL platforms, WrapperFL focuses on providing an easy-to-use FL plugin for industrial users with existing models for services. WrapperFL is not a platform to implement or develop new FL algorithms. Therefore, if one wants to propose or invent new FL algorithms, WrapperFL might be inferior. Instead, WrapperFL is a plug-and-play toolkit that allows users to rapidly switch an existing model to the federated version and achieve performance gain. \n\n\\subsection{Ensemble Learning}\nThe spirit of ensemble learning is to achieve higher prediction performance by combining multiple trained models, and the ensemble result is better than any of the individual model~\\cite{sagi2018ensemble}. Ensemble learning provides a feasible solution to utilize existing models from other clients. First, the existing local models are similar to the weak models in ensemble learning, and thus the appropriate fusion of their results could lead to a better prediction. Second, typically, when there is a lot of variation among the models, ensembles get superior outcomes \\cite{sollich1995learning,kuncheva2003measures}, implying that heterogeneous local models might bring extra benefits. \n\nBased on the idea of ensemble learning, WrapperFL adopts two ensemble strategies, stacking \\cite{breiman2001random} and bagging \\cite{buhlmann2012bagging}. The stacking strategy introduces a shared model across clients that takes the raw data and the output of the local model as the input. The shared model performs like a virtual bridge that connects local models to ensemble their knowledge. The bagging strategy adopts a simple linear regressor to re-weight the outputs from all local models. Notably, WrapperFL is different from conventionally federated learning algorithms that jointly learn a model from scratch; in contrast, WrapperFL aims to utilize the knowledge in other existing models to enhance the prediction of the existing local model. \n\n\n\n\n\n\n\n\\section{Methodology}\n\\label{sec:method}\nIn this section, we propose the methodology of WrapperFL. Unlike FL platforms, such as Tensorflow Federated~\\cite{tff}, FATE~\\cite{liu2021fate}, PaddleFL~\\cite{ma2019paddlepaddle}, EasyFL~\\cite{zhuang2022easyfl}, etc., WrapperFL is not designed to provide a comprehensive and fundamental toolkit for developing federated learning algorithms. In contrast, WrapperFL aims to help businesses achieve performance gains by federating their existing AI models at a minimum cost. Therefore, the WrapperFL does not involve complicated machine learning operators (MLOps), nor does it provides customized flow control. As this paper focuses on the learning strategy used in WrapperFL, we leave the engineering details, such as life cycle and communication, in the extended version. \n\n\\subsection{Framework}\nAs shown in Figure~\\ref{fig:framework}, WrapperFL is attached to the local model by extending to the input and output interfaces. The existing system can regard the WrapperFL (i.e., the light green block in Figure~\\ref{fig:framework}) as the original model as WrapperFL does not change the data flow logic. Inside the WrapperFL, the local model is empowered by two specific ensemble learning-based federated strategies, Stacking Wrapper and Bagging Wrapper. In abstraction, WrapperFL contains four components:\n\\begin{itemize}\n    \\item \\textbf{Translator}: The translator is in charge of aligning heterogeneous models and producing the federated output.\n    \\item \\textbf{Aggregator}: The aggregator fuses the output of the original model and the federated model\\footnote{Notably, in this paper, we do not use the term \\textit{global model}. Generally, the global model refers to the horizontal federated learning with homogeneous architecture. While in our settings, the federated participants might have distinct local models. Therefore, we use the term federated model to represent the non-local model(s).}.\n    \\item \\textbf{Scheduler}: The scheduler controls the federated learning flow, including the status synchronization, life cycle management, federated update, etc.\n    \\item \\textbf{Connection Manager}: The connection manager contains the functions of network connection, data transmission, authorization, etc. \n\\end{itemize}\n\n\n\\subsection{Stacking Wrapper}\n\\label{sec:stacking}\nIn this section, we introduce details of the stacking strategy for WrapperFL, named Stacking Wrapper. Stacking Wrapper is inspired by the stacking method in ensemble learning that combines the predictions from multiple machine learning models. As Figure \\ref{fig:stacking} shows, the Stacking Wrapper essentially stacks two models, the original local model and the translator. The raw data and its feature provided by the existing model (e.g., the hidden vector of the last layer in a neural network) are concatenated as the extended input for the translator. Notably, although the local models could be heterogeneous, the translators are homogeneous across participants, and thus we can apply the Secure Aggregation  \\cite{bonawitz2017practical3} to the training of translators. This architecture implicitly conducts the feature space alignment \\cite{day2017survey} across different federated participants. \n\n\\begin{figure}[!htp]\n    \\centering\n\t\\includegraphics[width=\\linewidth]{stacking.png}\n\t\\caption{The data flow of Stacking Wrapper.}\n\t\\label{fig:stacking}\n\\end{figure}\n\nStacking Wrapper requires the local models to be parametric models that naturally generate feature vectors. It is not compliant with non-parametric models such as Decision Trees and Support Vector Machines (SVMs). \n\n\n\\subsection{Bagging Wrapper}\n\\label{sec:bagging}\nBagging Wrapper leverages the idea of Bagging in ensemble learning \\cite{buhlmann2012bagging}, where we treat the dataset in each client as a random sample from a global data distribution and ensemble the predictions from the local models. As shown in Figure \\ref{fig:bagging}, the translator essentially embeds local models received from other participants, infers predictions from all models locally, and fuses the outputs. Compared to Stacking Wrapper, Bagging Wrapper does not pose any assumption on the architecture of local models and thus supports both parametric and non-parametric models. However, a major issue is that transferred local models might raise potential privacy risks. Notably, the risk of privacy leakage from models could be eliminated by incorporating with Trusted Execution Environment (TEE) or Fully Homomorphic Encryption (FHE). \n\n\\begin{figure}[!htp]\n    \\centering\n\t\\includegraphics[width=0.9\\linewidth]{bagging.png}\n\t\\caption{The data flow of Bagging Wrapper.}\n\t\\label{fig:bagging}\n\\end{figure}\n\n\n\n\\begin{table*}[!htp]\n\\small\n\\centering\n\\caption{Results on bank telemarketing prediction of local models and WrapperFL models under imbalanced data settings.}\n\\label{tab:tab_imb}\n\\begin{tabular}{rrrrrrrrr}\n\\toprule\n\\multicolumn{1}{l}{\\multirow{2}{*}{\\#Clients}} & \\multicolumn{2}{c}{Accuracy}                              & \\multicolumn{2}{c}{Precision}                             & \\multicolumn{2}{c}{Recall}                                & \\multicolumn{2}{c}{F1}                                    \\\\ \\cline{2-9}\n\\multicolumn{1}{l}{}                           & \\multicolumn{1}{c}{Local} & \\multicolumn{1}{c}{WrapperFL} & \\multicolumn{1}{c}{Local} & \\multicolumn{1}{c}{WrapperFL} & \\multicolumn{1}{c}{Local} & \\multicolumn{1}{c}{WrapperFL} & \\multicolumn{1}{c}{Local} & \\multicolumn{1}{c}{WrapperFL} \\\\ \\midrule\n5  & 0.8888$\\pm$0.01 & 0.9032$\\pm$0.01 & 0.4489$\\pm$0.07 & 0.5998$\\pm$0.01 & 0.1530$\\pm$0.01 & 0.2846$\\pm$0.01 & 0.1980$\\pm$0.02 & 0.3831$\\pm$0.01\\\\\n10 & 0.8588$\\pm$0.01 & 0.9120$\\pm$0.01 & 0.1350$\\pm$0.06 & 0.6132$\\pm$0.05 & 0.1321$\\pm$0.04 & 0.3172$\\pm$0.02 & 0.1049$\\pm$0.02 & 0.4110$\\pm$0.03 \\\\\n20 & 0.7991$\\pm$0.02 & 0.8983$\\pm$0.01 & 0.1048$\\pm$0.04 & 0.5145$\\pm$0.12 & 0.1864$\\pm$0.08 & 0.2556$\\pm$0.03 & 0.1096$\\pm$0.02 & 0.3215$\\pm$0.05\\\\\\bottomrule\n\\end{tabular}\n\\end{table*}\n\n\n\n\\begin{table*}[]\n    \\small\n    \\centering\n    \\caption{Results on bank telemarking prediction of local models and WrapperFL models under non-IID data settings.}\n    \\label{tab:tab_non}\n    \\begin{tabular}{rrrrrrrrr}\n    \\toprule\n    \\multicolumn{1}{l}{\\multirow{2}{*}{\\#Clients}} & \\multicolumn{2}{c}{Accuracy}                              & \\multicolumn{2}{c}{Precision}                             & \\multicolumn{2}{c}{Recall}                                & \\multicolumn{2}{c}{F1}                                    \\\\ \\cline{2-9} \n    \\multicolumn{1}{l}{}                           & \\multicolumn{1}{c}{Local} & \\multicolumn{1}{c}{WrapperFL} & \\multicolumn{1}{c}{Local} & \\multicolumn{1}{c}{WrapperFL} & \\multicolumn{1}{c}{Local} & \\multicolumn{1}{c}{WrapperFL} & \\multicolumn{1}{c}{Local} & \\multicolumn{1}{c}{WrapperFL} \\\\ \\midrule\n    5  & 0.8968$\\pm$0.01 & 0.9064$\\pm$0.01 & 0.4753$\\pm$0.06 & 0.7173$\\pm$0.02 & 0.1523$\\pm$0.01 & 0.2532$\\pm$0.01 & 0.2097$\\pm$0.02 & 0.3642$\\pm$0.01\\\\\n    10 & 0.8637$\\pm$0.01 & 0.9040$\\pm$0.01 & 0.2079$\\pm$0.05 & 0.6573$\\pm$0.03 & 0.1768$\\pm$0.03 & 0.3105$\\pm$0.02 & 0.1752$\\pm$0.03 & 0.3862$\\pm$0.02\\\\\n    20 & 0.7967$\\pm$0.02 & 0.9056$\\pm$0.01 & 0.1286$\\pm$0.05 & 0.6592$\\pm$0.04 & 0.1769$\\pm$0.05 & 0.3504$\\pm$0.01 & 0.1052$\\pm$0.01 & 0.4375$\\pm$0.01\\\\\\bottomrule\n    \\end{tabular}\n\\end{table*}\n\n\n\\subsection{Examples of Using WrapperFL}\nThe WrapperFL is easy to use. To set up a federated client, the user only needs to configure very few parameters, e.g., the existing local model, the client id, client's and server's IP addresses and ports for connection, the type of translator, etc. The WrapperFL will automatically complete the registration, initialization, and building connections with clients. Once instantiating the WrapperFL class with a simple configuration, the user can begin training the federated model with one line of code. Hence, the WrapperFL model can be used the same way as the local model. \n\nThe following code block shows four examples of using WrapperFL, which remarkably demonstrate the ease of use of WrapperFL. Lines 6-12 and 25-30 exemplify Stacking Wrapper and Bagging Wrapper configurations. The user can register to a federated network by filling their identifier in \\texttt{client\\_id}, local address in \\texttt{client\\_addr}, server address in \\texttt{server\\_addr}, and other federated clients' ids in \\texttt{clients}. The fields of \\texttt{local\\_model} and \\texttt{train\\_dataset} are related to the local model and local training dataset. These are used to train the translator specified in the field \\texttt{translator}. \n\nApart from the configuration, the user can federalize an existing local model with only three lines of code (i.e., lines 14, 16, and 21), which provides superior convenience for developers compared to other FL toolkits. Notably, in the inference phase, the WrapperFL model performs identically to the original local model (shown in Example 3 and Example 4), significantly reducing the efforts to adjust the existing interfaces. \n\n{\\scriptsize\n\\begin{minted}[xleftmargin=20pt,linenos]{python}\nfrom wrapperfl import BaggingWrapper, StackingWrapper\n\n## Initialization\n\n# --- Example 1: Federalization with Stacking Wrapper  ---\nconfigs = {\"local model\": local_model, \n           \"train_dataset\":dataset, \n           \"translator\":\"LR\", # translator structure\n           'client_id':'0', \n           'clients':['1', '2', ...], \n           'client_addr':{'ip':'x.x.x.x', 'port':'x'}, \n           'server_addr':{'ip':'x.x.x.x', 'port':'x'}}\n# registration and initialization\nwrapper = StackingWrapper(configs) \n#train the translator\nwrapper.start(status='train', \n              train_config={'local_epochs':10, \n                            'rounds':10, \n                            ...}) \n#swtich to the inference mode\nwrapper.start(status='infer', \n              infer_config={'threshold':0.5, ...}) \n\n# --- Example 2: Federalization with Bagging Wrapper\nconfigs = {\"local model\": local_model, \n           \"train_dataset\":dataset,\n           'clients':['1', '2', ...], \n           'client_id':'0', \n           'client_addr':{'ip':'x.x.x.x', 'port':'x'}, \n           'server_addr':{'ip':'x.x.x.x','port':'x'}}\nwrapper = BaggingWrapper(configs) \nwrapper.start(status = 'train', \n              train_config={'local_epochs':10, ...})) \nwrapper.start(status = 'infer', \n              infer_config={'threshold':0.5, ...}) \n\n## Usage of WrapperFL model \n\n# --- Example 3: Sklearn-style local model\n# local model inference\ny = local_model.predict(X) \n# WrapperFL model inference\ny = wrapper.model.predict(X) \n\n# --- Example 4: PyTroch-style local model\n# local model inference\ny = local_model(X) \n# WrapperFL model inference\ny = wrapper.model(X) \n\\end{minted}\n}\n\n\n\n\n\\section{Experiment}\nTo evaluate the effectiveness of our proposed WrapperFL, we conduct comprehensive experiments on diverse tasks with different settings, including heterogeneous data distributions and heterogeneous models. \n\n\\subsection{Datasets}\nWe verify our proposed WrapperFL with three different forms of data, including the bank telemarketing prediction \\cite{moro2014bankdata} (tabular data), news classification~\\cite{tianchidata} (textual data), image classification with CIFAR-10~\\cite{cifar10} (image data). These datasets are large-scale and thus allow us to manipulate their data distributions to form heterogeneous data distributions including \\textbf{imbalanced} and \\textbf{non-IID} data distributions. Notably, although non-IID data contains imbalanced data in some literature~\\cite{zhu2021federated-noniid-survey}, in this paper, we refer \\textbf{imbalanced} data to the setting that class distributions across clients are (almost) identical, while \\textbf{non-IID} data to the setting that class distributions are different. \n\n\n\n\nTo simulate a different number of clients, we partition the tabular data and text data into 5, 10, and 20 clients, and we split the image data into 5 and 10 clients; as in the case of 20 clients, the heterogeneity in data distribution across clients are not apparent. \n\nTo synthesize imbalanced datasets, we first assign the number of instances for each client following the Dirichlet Process~\\cite{Teh2010}, and uniformly sample the instances for each client from the global data distribution. For non-IID datasets, we directly assign the number of samples of each class in each client following the Dirichlet Process. A special case is the bank telemarketing prediction data, which has already been class-imbalanced. For this case, we only sample the number of positive samples and then pad the local datasets with negative samples to the identical size. Figure~\\ref{fig:tab_dist}-\\ref{fig:image_dist} illustrate the overview of class-aware distributions of each client of three datasets, which are shown in appendix \\ref{apdx:figure}\n\n\\subsection{Evaluation Metrics}\nWe compute the accuracy, precision, recall, and F1-score for all tasks for each client's local and WrapperFL models under a balanced global testing set. We report the means and standard deviations of the metrics under each setting. \n\n\\subsection{Experiments on Tabular Dataset}\nTabular datasets are common in real-world industries. We evaluate the effectiveness of WrapperFL on the bank telemarketing prediction data~\\cite{moro2014bankdata} with heterogeneous model structures. \n\nMore specifically, we set 40\\% of the clients to use Logistic Regression (LR) as the local modes, and 60\\% of the clients to use the three-layer MLP with various hidden sizes (16, 18, and 24 each for 20\\% of the clients). Table~\\ref{tab:tab_imb} shows the performance comparison of the local models and our WrapperFL models under imbalanced data settings. In this case, the WrapperFL adopts a three-layer MLP with the hidden size of 16 as the translator, which is trained with ten communication rounds using FedAvg~\\cite{mcmahan2017communication}. \n\nNotably, when the number of clients increases, the precision of local models drops dramatically due to the highly imbalanced local data. In contrast, extended with WrapperFL, the participants achieve significantly better performance. For example, when there are ten clients, the mean F1 score of WrapperFL models is 2.92x better than that of local models. We can make similar conclusions in the case of non-IID data settings, which are shown in Table~\\ref{tab:tab_non}. In this case, the WrapperFL model shows strong robustness against the local models. For example, the mean F1-score of WrapperFL is 3.16x higher than that of local models in the setting of 20 clients. \n\n\n\n\\begin{table*}[]\n    \\small\n    \\centering\n    \\caption{Results on news classification of local models and WrapperFL models under imbalanced data settings.}\n    \\label{tab:text_imb}\n    \\begin{tabular}{rrrrrrrrr}\n    \\toprule\n    \\multicolumn{1}{l}{\\multirow{2}{*}{\\#Clients}} & \\multicolumn{2}{c}{Accuracy}                              & \\multicolumn{2}{c}{Precision}                             & \\multicolumn{2}{c}{Recall}                                & \\multicolumn{2}{c}{F1}                                    \\\\ \\cline{2-9} \n    \\multicolumn{1}{l}{}                           & \\multicolumn{1}{c}{Local} & \\multicolumn{1}{c}{WrapperFL} & \\multicolumn{1}{c}{Local} & \\multicolumn{1}{c}{WrapperFL} & \\multicolumn{1}{c}{Local} & \\multicolumn{1}{c}{WrapperFL} & \\multicolumn{1}{c}{Local} & \\multicolumn{1}{c}{WrapperFL} \\\\ \\midrule\n    5  & 0.2129$\\pm$0.08 & 0.2836$\\pm$0.05 & 0.1647$\\pm$0.10 & 0.1743$\\pm$0.08 & 0.1968$\\pm$0.06 & 0.2085$\\pm$0.05 & 0.1507$\\pm$0.07 & 0.1655$\\pm$0.06\\\\\n    10 & 0.2003$\\pm$0.08 & 0.2311$\\pm$0.07 & 0.1438$\\pm$0.07 & 0.1305$\\pm$0.06 & 0.1726$\\pm$0.04 & 0.1753$\\pm$0.04 & 0.1273$\\pm$0.05 & 0.1264$\\pm$0.05\\\\\n    20 & 0.2140$\\pm$0.03 & 0.3977$\\pm$0.01 & 0.0720$\\pm$0.03 & 0.4032$\\pm$0.01 & 0.1160$\\pm$0.01 & 0.2911$\\pm$0.01 & 0.0703$\\pm$0.02 & 0.2514$\\pm$0.01\n    \\\\\\bottomrule\n    \\end{tabular}\n\\end{table*}\n\n\n\\begin{table*}[]\n    \\small\n    \\centering\n    \\caption{Results on news classification of local models and WrapperFL models under non-IID data settings.}\n    \\label{tab:text_non}\n    \\begin{tabular}{rrrrrrrrr}\n    \\toprule\n    \\multicolumn{1}{l}{\\multirow{2}{*}{\\#Clients}} & \\multicolumn{2}{c}{Accuracy}                              & \\multicolumn{2}{c}{Precision}                             & \\multicolumn{2}{c}{Recall}                                & \\multicolumn{2}{c}{F1}                                    \\\\ \\cline{2-9} \n    \\multicolumn{1}{l}{}                           & \\multicolumn{1}{c}{Local} & \\multicolumn{1}{c}{WrapperFL} & \\multicolumn{1}{c}{Local} & \\multicolumn{1}{c}{WrapperFL} & \\multicolumn{1}{c}{Local} & \\multicolumn{1}{c}{WrapperFL} & \\multicolumn{1}{c}{Local} & \\multicolumn{1}{c}{WrapperFL} \\\\ \\midrule\n    5  & 0.6259$\\pm$0.01 & 0.6977$\\pm$0.01 & 0.4355$\\pm$0.01 & 0.4892$\\pm$0.01 & 0.3833$\\pm$0.01 & 0.4035$\\pm$0.01 & 0.3580$\\pm$0.01 & 0.3962$\\pm$0.01\\\\\n    10 & 0.5533$\\pm$0.01 & 0.6561$\\pm$0.01 & 0.3056$\\pm$0.01 & 0.3643$\\pm$0.01 & 0.3119$\\pm$0.01 & 0.3404$\\pm$0.01 & 0.2815$\\pm$0.01 & 0.3221$\\pm$0.01\\\\\n    20 & 0.5152$\\pm$0.01 & 0.6288$\\pm$0.01 & 0.2157$\\pm$0.01 & 0.3457$\\pm$0.01 & 0.2597$\\pm$0.01 & 0.3069$\\pm$0.01 & 0.2128$\\pm$0.01 & 0.2912$\\pm$0.01\n    \\\\\\bottomrule\n    \\end{tabular}\n\\end{table*}\n\n\n\\subsection{Experiments on Text Dataset}\n\n\n\nText data is also commonly used in the industry~\\cite{devlin2018bert55}. We select the news classification dataset~\\cite{tianchidata} to represent the NLP tasks as text classification is one of the most important tasks in NLP. Unlike the tabular data, the news classification dataset has 14 evenly distributed classes. Moreover, the feature distribution in news data (i.e., the word distributions) is more consistent across different classes. Therefore, it requires the classifier to capture the deep semantics of a sentence. \n\n\\begin{table}[!htp]\n\\small\n    \\caption{Results on image classification of local models and WrapperFL models under imbalanced data settings.}\n    \\label{tab:img_imb}\n    \\begin{tabular}{ccrr}\n        \\toprule\n        \\multicolumn{2}{l}{\\#Clients}          & \\multicolumn{1}{c}{5} & \\multicolumn{1}{c}{10} \\\\ \\midrule\n        \\multirow{2}{*}{Accuracy}  & Local     & 0.7444$\\pm$0.0049 & 0.6301$\\pm$0.0182  \\\\ \\cline{2-2}\n                                   & WrapperFL & 0.7798$\\pm$0.0018 & 0.8254$\\pm$0.0004  \\\\\\midrule\n        \\multirow{2}{*}{Precision} & Local     & 0.7412$\\pm$0.0055 & 0.6340$\\pm$0.0153  \\\\ \\cline{2-2}\n                                   & WrapperFL & 0.7823$\\pm$0.0016 & 0.8284$\\pm$0.004   \\\\ \\midrule\n        \\multirow{2}{*}{Recall}    & Local     & 0.7443$\\pm$0.0049 & 0.6304$\\pm$0.0177  \\\\ \\cline{2-2}\n                                   & WrapperFL & 0.7793$\\pm$0.0018 & 0.8258$\\pm$0.0004  \\\\ \\midrule\n        \\multirow{2}{*}{F1}        & Local     & 0.7416$\\pm$0.0054 & 0.6233$\\pm$ 0.0218 \\\\ \\cline{2-2}\n                                   & WrapperFL & 0.7781$\\pm$0.0019 & 0.8241$\\pm$0.0005\\\\ \\bottomrule\n        \\end{tabular}\n\\end{table}\n\n\n\n\n\nIn this experiment, we evaluate the compatibility of WrapperFL with different deep learning models. We randomly assign 20\\% of the clients with TextCNN~\\cite{jacovi2018understanding}, 40\\% of the clients with Transformer \\cite{vaswani2017attention} (with three stacking encoders), and 40\\% of the clients with Transformer (with two stacking encoders) as their local models. WrapperFL adopts a single-encoder Transformer as the translator, which is trained with ten communication rounds using FedAvg~\\cite{mcmahan2017communication}. \n\n\nTable \\ref{tab:text_imb} and Table \\ref{tab:text_non} display the performance comparisons between local models and WrapperFL models, from which we can confirm the high utility of WrapperFL on complex deep learning models. For example, in the setting of 20 clients, the WrapperFL models perform much better than local models no matter of accuracy, precision, recall, or F1 score. \n\n\\subsection{Experiments on Image Dataset}\n\\label{sec:exp:img}\nProcessing image data is also a huge requirement in the industry, such as face recognition. We adopt one of the most widely used image datasets, CIFAR-10~\\cite{cifar10}, to verify our WrapperFL on CNN models with different scales. We randomly assign 20\\% of clients with ResNet18~\\cite{he2016deep}, 40\\% of clients with VGG13~\\cite{simonyan2014very}, and 40\\% of clients with VGG16. WrapperFL adopts a much smaller CNN model, VGG11, as the translator, which is trained with ten communication rounds using FedAvg~\\cite{mcmahan2017communication}. \n\nTable~\\ref{tab:img_imb} and Table~\\ref{tab:img_non} display the performances of local models and WrapperFL models. Consistently, we can observe significant performance gains comparing the WrapperFL models to the local models. Moreover, the experimental results also demonstrate that using a much smaller homogeneous translator can efficiently and effectively distill knowledge from well-trained large-scale local models. \n\n\n\n\\begin{table}[]\n\\small\n    \\caption{Results on image classification of local models and WrapperFL models under non-IID data settings.}\n    \\label{tab:img_non}\n    \\begin{tabular}{ccrr}\n        \\toprule\n        \\multicolumn{2}{l}{\\#Clients}          & \\multicolumn{1}{c}{5} & \\multicolumn{1}{c}{10} \\\\ \\midrule\n        \\multirow{2}{*}{Accuracy}  & Local     & 0.6902$\\pm$0.0020 & 0.5747$\\pm$0.0031 \\\\ \\cline{2-2} \n                                   & WrapperFL & 0.7405$\\pm$0.0026 & 0.7853$\\pm$0.0015 \\\\ \\hline\n        \\multirow{2}{*}{Precision} & Local     & 0.6617$\\pm$0.0066 & 0.5563$\\pm$0.0055 \\\\ \\cline{2-2} \n                                   & WrapperFL & 0.7511$\\pm$0.0016 & 0.8017$\\pm$0.0005 \\\\ \\hline\n        \\multirow{2}{*}{Recall}    & Local     & 0.6877$\\pm$0.0020 & 0.5737$\\pm$0.0027 \\\\ \\cline{2-2} \n                                   & WrapperFL & 0.7404$\\pm$0.0025 & 0.7862$\\pm$0.0012 \\\\ \\hline\n        \\multirow{2}{*}{F1}        & Local     & 0.6628$\\pm$0.0042 & 0.5377$\\pm$0.0048 \\\\ \\cline{2-2} \n                                   & WrapperFL & 0.7249$\\pm$0.0053 & 0.7792$\\pm$0.0020 \\\\ \\bottomrule\n        \\end{tabular}\n\\end{table}\n\n\n\n\\subsection{Quantitative Comparison of Training Cost}\nWhile it is hard to quantify the reduction in the required manpower and resources of WrapperFL compared to existing FL platforms, we can empirically validate its feasibility. Most importantly, WrapperFL utilizes the existing local model, significantly reducing the training time and tuning the local model, which is well-known for both time and labor-consuming. Besides, with a simple plug-and-play package, ML developers without background knowledge of federated learning can quickly adapt their services to the federated version, which eliminates the barrier of manpower of FL experts.  \n\nTo further show the training cost (including the time cost and the potential performance loss), we conduct a quantitative comparison between WrapperFL and the FedAvg on the non-IID dataset mentioned in Section \\ref{sec:exp:img}. In Figure~\\ref{fig:time_cost}, we plot the accuracy points at training time intervals of WrapperFL and FedAvg, respectively, from which we can observe that WrapperFL achieves over 60\\% accuracy much faster than FedAvg. Moreover, we can also observe that the performance of WrapperFL is comparable to FedAvg when converged, which indicates that performance loss of WrapperFL is negligible. Notably, the dataset used in this experiment is much smaller than industrial-scale data and the training time reported here ignores the time cost of deployment and communication. Therefore, in practice, the time acceleration of WrapperFL could be much more significant. \n\n\\begin{figure}[!htp]\n    \\centering\n\t\\includegraphics[width=0.9\\linewidth]{time_cost.pdf}\n\t\\caption{The Accuracy Curves of WrapperFL and FedAvg on Image Dataset}\n\t\\label{fig:time_cost}\n\\end{figure}\n\n\n\n\n\\section{Conclusion}\nThis paper focus on the problem of applying federated learning to the industry, where the businesses have already had a mature and existing model and have no experts on federated learning algorithms and platforms. The problem is challenging from two aspects. First, the federated learning process should be completely decoupled from the existing system, and the federalization should be lightweight, low-intrusive, and require as less development workload as possible. Second, the federated learning toolkit should be model-agnostic. Therefore, it can be compatible with various model architectures, and the users can seamlessly change the underlying machine learning models. To tackle these challenges, we propose a simple yet practical federated learning plug-in inspired by ensemble learning, dubbed WrapperFL, allowing participants to build\/join a federated system with existing models at minimal costs. The experimental results of tabular, text, and image data demonstrate the efficacy of our proposed WrapperFL on both model heterogeneous and data heterogeneous settings. \n\n\n\\input{ijcai22.bbl}\n\\onecolumn\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\newtheorem{definicao}{Definition}[section]\n\\newtheorem{teorema}{Theorem}[section]\n\\newtheorem{lema}{Lemma}[section]\n\\newtheorem{corolario}{Corolary}[section]\n\\newtheorem{proposicao}{Proposition}[section]\n\\newtheorem{axioma}{Axiom}[section]\n\\newtheorem{observacao}{Remark}[section]\n\n\tThe gauge field copy question is one of those unexpected phenomena that creep up in mathematical physics when we go from linear objects to their nonlinear extensions. Linear gauge fields, say, the electromagnetic field, admit a single potential over a nice neighborhood modulo gauge tranformations. However when we go from the linear to the nonlinear domain, that nice relation between fields and potentials breaks down. (We emphasize the adjective: the relation between potentials and fields in linear gauge fields is a `nice' one because it reflects the very deep $\\partial^{2} = 0$ relation in homological algebra and in algebraic topology; for a simple application of that relation to mathematical physics see \\cite{Doria-78}.)\n\n\tIn the nonlinear, nonabelian case, some gauge fields admit two or more potentials which cannot be made equivalent (even locally) modulo gauge transformations. Such an ambiguity is known as the gauge field copy problem and was discovered in 1975 by T.T. Wu and C.N. Yang \\cite{Wu}. Gauge copies fall into two cases:\n\\begin{itemize}\n\\item{{\\em True copies}: Here the gauge field can be derived from at least two different potentials that aren't even locally related by a gauge transformation;}\n\\item{{\\em False copies}: In this case the field can be derived from potentials that are always locally related by a gauge transformation.}\n\\end{itemize}\nFor a review of the geometric phenomena behind the copy problem see \\cite{Costa-Amaral}.\n\n\tLet $P(M,G)$ be a principal fiber bundle, where $M$ is a finite-dimensional smooth real manifold and $G$ is a\nfinite-dimensional semi-simple Lie group. We denote by $(P,\\alpha)$ the principal fiber bundle $P(M,G)$ endowed with the connection-form $\\alpha$, and by $L$ the field that corresponds to the potential $A$ associated to $\\alpha$. We mean by that that $A$ is a family $\\{A_{\\lambda}\\}_{\\lambda\\in \\Lambda}$ of $l(G)$-valued one-forms, where $l(G)$ is the Lie algebra associated to $G$, and for each $\\lambda\\in\\Lambda$, $A_{\\lambda}$ is defined on an open subset $U_{\\lambda}$ of $M$ with $D_{A_{\\lambda}} = dA_{\\lambda}+\\frac{1}{2}[A_{\\lambda},A_{\\lambda}] = L$ on $U_{\\lambda}$ and with $M = \\bigcup_{\\lambda\\in\\Lambda}U_{\\lambda}$. We consider also that $A_{\\lambda}$ and $A_{\\lambda'}$ are gauge equivalent for $\\lambda\\neq\\lambda'$. We recall that an automorphism of a principal fiber bundle $P(M,G)$ with projection $\\pi$ is a diffeomorphism $f:P\\rightarrow P$ such that $f(pg) = f(p)g$ for all $g\\in G$, $p\\in P$. A gauge transformation is an automorphism $f:P\\rightarrow P$ such that $\\overline{f} = 1_{M}$, where $\\overline{f}:M\\rightarrow M$ is the diffeomorphism induced by $f$ given by $\\overline{f}(\\pi(p)) = \\pi(f(p))$. Given a principal fibre bundle $P(M,G)$ and a Lie subgroup $G'$ of $G$, we say that $P(M,G)$ is reducible to the bundle $P'(M,G')$ if and only if there is a monomorphism $f':G'\\rightarrow G$ and an imbedding $f'':P'\\rightarrow P$ such that $f''(u'a') = f''(u')f'(a')$, for all $u'\\in P'$ and $a'\\in G'$.\n\n\\begin{definicao}\n\\begin{enumerate}\n\\item{The field $L$ or the potential $A$ are said to be {\\bf reducible} if the corresponding bundle $(P,\\alpha)$ is reducible.}\n\\item{If $U\\subset M$ is a nonvoid open set then $L$ or $A$ are said to be {\\bf locally reducible over} $U$ whenever $(P,\\alpha)\\mid _{U}$ is reducible.}\n\\item{$L$ or $A$ are said to be {\\bf fully irreducible} if they are not locally reducible.}$\\Box$\n\\end{enumerate}\\label{def-redutibilidade}\n\\end{definicao}\n\n\tOur main results are essentially based on a theorem that gives a topological condition for the existence of false copies and on the Atiyah-Singer index theorem. But before we state the topological condition for the existence of false gauge field copies, we find interesting to recall the Ambrose-Singer theorem, since it will be used to derive such a result:\n\n\\begin{proposicao}\n(Ambrose-Singer) Let $P(M,G)$ be a principal fiber bundle, where $M$ is connected and paracompact. Let $\\Gamma$ be a connection in $P$, $L$ the curvature form, $\\Phi(u)$ the holonomy group with reference point $u\\in P$ and $P(u)$ the holonomy bundle through $u$ of $\\Gamma$. Then the Lie algebra of $\\Phi(u)$ is equal to the subspace of $l(G)$ spanned by all elements of the form $L_{v}(X,Y)$, where $v\\in P(u)$ and $X$ and $Y$ are arbitrary horizontal vectors at $v$.\\label{Ambrose-Singer}\n\\end{proposicao}\n\n\tThe proof of this theorem (also known as the holonomy theorem), can be found in \\cite{Kobayashi}. It is considered that we may assume $P(u) = P$, which means that $\\Phi(u) = G$.\n\n\tNow we establish the topological condition for the existence of false gauge field copies, based on a result due to Doria \\cite{Doria-81}:\n\n\\begin{proposicao}\nLet $P(M,G)$ be as above together with the fact that $G$ be semi-simple and $M$ is connected and paracompact. $L$ is falsely copied, that is, $L$ has different potentials that are locally related by a gauge transformation if and only if $L$ is reducible.\\label{Teo-Doria}\n\\end{proposicao}\n\n\t{\\em Proof}: If $L$ is reducible, then $P(M,G)$ is reducible (definition \\ref{def-redutibilidade}). This means that $P(M,G)$ can be reduced to a nontrivial $P'(M,G')$, where $G'$ is the Ambrose-Singer holonomy group (it corresponds to the group $\\Phi(u)$ in proposition \\ref{Ambrose-Singer}). If we assume\n\\[A_{\\mu} = B_{\\mu} + \\partial_{\\mu}h',\\] \nwhere $\\partial_{\\mu}=_{def}\\partial\/\\partial x^{\\mu}$, $x^{\\mu}$ is a coordinate of a coordinate system at $U\\subset M$ (such that the bundle is trivial over $U$), $B_{\\mu}$ takes values in $l(G)$ and $h'$ takes values on $l(G')$, then:\n\\[F_{\\mu\\nu}(A) = \\partial_{\\mu}(B_{\\nu}+\\partial_{\\nu}h') - \\partial_{\\nu}(B_{\\mu} + \\partial_{\\mu}h') + (B_{\\mu} + \\partial_{\\mu}h')(B_{\\nu} + \\partial_{\\nu}h') - (B_{\\nu} + \\partial_{\\nu}h')(B_{\\mu} + \\partial_{\\mu}h'),\\]\nwhere $F_{\\mu\\nu}(A)$ denotes the components of the curvature form $F$ associated to the connection form $A$.\n\n\tHence:\n\n\\[F_{\\mu\\nu}(A) = F_{\\mu\\nu}(B) + 0_{field}.\\]\n\n\tThe sufficient condition is proved as follows: if the holonomy group is semi-simple as indicated, and if $A_{\\mu} = B_{\\mu} + \\partial_{\\mu}h'$ as indicated, then there is a reducibility $G\\oplus G'\\rightarrow G'$.$\\Box$\n\n\tWe now suppose that $X$ is a compact smooth manifold and that $G$ is a compact Lie group acting smoothly on $M$. The Atiyah-Singer index theorem can be stated as \\cite{Shanahan}:\n\n\\begin{proposicao}\nLet $\\chi$ and $\\vartheta$ be complex vector bundles defined over $X$. If $D:C^{\\infty}(X;\\chi)\\rightarrow C^{\\infty}(X;\\vartheta)$ is a $G$-invariant elliptic partial differential operator on $X$, which sends cross-sections of $\\chi$ to cross-sections of $\\vartheta$, then {\\rm index}$_{G}D = t_{{\\rm ind}_{G}^{X}}(\\sigma(D))$, where $t_{{\\rm ind}}$ is the topological index defined on $K_{G}(TX)$ and $\\sigma(D)$ is the symbol of $D$.$\\Box$\\label{Atiyah-Singer-D}\n\\end{proposicao}\n\n\t(For the proof and notational features see \\cite{Shanahan}.) Another version of the index theorem \\cite{Atiyah-Index-I} asserts:\n\\begin{proposicao}\nThe analytical index $a_{{\\rm ind}_{G}}$ and the topological index $t_{{\\rm ind}_{G}^{X}}$ coincide as homomorphisms $K_{G}(TX)\\rightarrow R(G)$.$\\Box$\n\\end{proposicao}\n(Proof in the reference.)\n\n\\section{A necessary condition for false copies}\n\n\tGauge fields and gauge potentials can be seen and defined as cross-sections of vector bundles associated to the principal fiber bundle $P(M,G)$. More specifically, potential space (or connection space) coincides with the space of all $C^{k}$ cross-sections of the vector bundle $E$ of $l(G)$-valued 1-forms on $M$, where $l(G)$ is the group's Lie algebra, while field space (or curvature space) coincides with the space of all $C^{k}$ cross-sections of the vector bundle ${\\bf E}$ of $l(G)$-valued 2-forms on $M$.\n\n\tLet $F$ and ${\\bf F}$ be manifolds on which $G$ acts on the left and such that $E = P\\times_{G}F$ and ${\\bf E} = P\\times_{G}{\\bf F}$, where $P$ is the total space of $P(M,G)$. In other words, $E$ is the quotient space of $P\\times F$ by the group action. Similarly, ${\\bf E}$ is the quotient space of $P\\times {\\bf F}$ by the action of the group $G$.\n\n\tTo prove the following proposition we use the Atiyah-Singer index theorem. So, we are still assuming that $M$ and $G$ are compact.\n\n\tTherefore:\n\\begin{proposicao}\nIf a gauge field (a cross-section of ${\\bf E}$) is associated to copied potentials that are locally gauge-equivalent, then there is: a non-trivial sub-group of $G$, denoted by $G'$; a $G'$-manifold $P'$; two $G'$-vector spaces $F'$ and $\\bf F'$; and two elliptic partial\ndifferential operators,\n\\begin{equation}\n{\\cal D}_{G}:C^{\\infty}(P;P\\times F)\\rightarrow C^{\\infty}(P;P\\times\n{\\bf F})\n\\end{equation}\nand\n\\begin{equation}\n{\\cal D}_{G'}:C^{\\infty}(P';P'\\times F')\\rightarrow\nC^{\\infty}(P';P'\\times {\\bf F'})\n\\end{equation}\nrespectively $G$-invariant and $G'$-invariant, such that the\n${\\rm index}\\; {\\cal D}_{G'}$ can be defined as a nontrivial function of the ${\\rm index}\\; {\\cal D}_{G}$.\\label{main}\n\\end{proposicao}\n\n\t{\\em Proof}: If a gauge field is associated to copied potentials that are locally gauge-equivalent, then such a field is reducible (Proposition \\ref{Teo-Doria}). Therefore the bundle $P(M,G)$ is reducible (Definition \\ref{def-redutibilidade}). So, there is a non-trivial sub-group $G'$ of $G$ and a monomorphism $\\varphi:G'\\rightarrow G$ such that one can define a reduced principal fiber bundle $P'(M,G')$ and a reduction $f:P'(M,G')\\rightarrow P(M,G)$. Similarly we define $G'$-vector spaces $F'$ and $\\bf F'$ and maps ${\\bf f}:{\\bf F'}\\rightarrow {\\bf F}$ and ${\\rm f}: F'\\rightarrow F$ such that:\n\\begin{equation}\n{\\rm f}(g'\\xi') = \\varphi(g'){\\rm f}(\\xi')\n\\end{equation}\nand\n\\begin{equation}\n{\\bf f}(g'\\zeta') = \\varphi(g'){\\bf f}(\\zeta')\n\\end{equation}\nfor all $g'\\in G'$, $\\xi'\\in F'$ and $\\zeta'\\in {\\bf F'}$.\n\n\tNow consider $P\\times F$ as the total space of the trivial vector bundle $P\\times F\\rightarrow P$, with a canonical projection. That vector bundle is noted $\\chi$. Similarly the trivial vector bundles $P\\times {\\bf\nF}\\rightarrow P$, $P'\\times F'\\rightarrow P'$ and $P'\\times {\\bf F'}\\rightarrow P'$, are respectively noted $\\vartheta$, $\\chi'$ and $\\vartheta'$.\n\n\tTherefore, the diagram below commutes:\n\\[K_{G}(TP)\\stackrel{\\varphi^{*}}{\\longrightarrow}K_{G'}(TP')\\]\n\\[t_{{\\rm ind}_{G}^{P}}\\downarrow\\;\\;\\;\\;\\;\\;\\downarrow t_{{\\rm ind}_{G'}^{P'}}\\]\n\\[R(G))\\stackrel{\\varphi^{*}}{\\longrightarrow}R(G')\\]\n(Here $\\varphi^{*}$ is induced by $\\varphi$.)\n\n\t$\\sigma({\\cal D}_{G}) = \\vartheta-\\chi$ and $\\sigma({\\cal D}_{G'}) = \\vartheta'-\\chi'$. So, the homomorphisms $f$, $\\bf f$ and f and the diagram given above induce the relation:\n\\begin{equation}\n\\sigma({\\cal D}_{G'}) = \\varphi^{*}(\\sigma({\\cal D}_{G}))\n\\end{equation}\n\n\tThus, according to the diagram, $t_{{\\rm ind}_{G'}^{P'}}(\\sigma({\\cal D}_{G'})) = \\varphi^{*}(t_{{\\rm ind}_{G}^{P}}(\\sigma({\\cal D}_{G})))$. If we use the Atiyah-Singer index theorem (\\ref{Atiyah-Singer-D}), it can be noticed that\\\\\n\\begin{equation}\n{\\rm index}\\;{\\cal D}_{G'} = \\varphi^{*}({\\rm index}\\;{\\cal D}_{G}).\\Box\n\\end{equation}\n\n\\begin{observacao}\n{\\rm The condition that ${\\cal D}_{G}$ and ${\\cal D}_{G'}$ are elliptic partial differential operators is necessary in our proof of proposition 3.1 in order to apply the Atiyah-Singer index theorem given by proposition 1.3.}\n\\end{observacao}\n\n\tThe topological condition given in \\cite{Doria-81}, in order to check whether there are false gauge field copies does not impose that the manifold $M$ should be compact, or that the Lie group must be compact (a very common situation in gauge theories). But when we apply the Atiyah-Singer index theorem to obtain the analytical condition for false copies, we must consider that both $M$ and $G$ are compact.$\\Box$\n\n\\section{Conclusions}\n\n\tThere are several points of contact between classical physics and $K$-theory: a method to prove the index theorem based on the asymptotics of the heat equation \\cite{Atiyah-Calor} \\cite{Gilkey}; and the physical interpretation of nonvanishing characteristic classes in terms of magnetic monopoles, solitons and instantons \\cite{Mayer}. We present here a new $K$-theoretical result with consequence to physics: an analytical condition to the existence of `false' gauge field copies obtained from a topological condition.\n\n\tOur result has some limitations: it refers only to false copies; it is imposed that $M$ and $G$ are compact; and it is imposed that $G$ is semi-simple. We believe that it is possible to extend our results while eliminating those restrictions. (One possibility should be to modify the geometry of an irreducible principal fiber bundle in such a way as to handle true copies as false copies (see \\cite{Costa-Amaral} on this). That is a possible way to define an analytical condition for generic copies through the use of the Atiyah-Singer index theorem. On the structure of $M$, we just recall that $K$-theory has a technique for dealing with locally compact spaces. Moreover, a possible way of generalizing our results to any Lie group (compact or non-compact) amounts (we hope) to the use of some kind of compactification \\cite{Dikranjan}.\n\n\tAnother side-result of the present work has to do with the categorical approach that underlies $K$-theory. When we formalize gauge theory and the gauge copy problem with tools from category theory we notice that our categorical formalization ends up as being too complicated when compared with the traditional approach based on set theory \\cite{da Costa-92}.\n\n\tMoreover, we think it is possible to obtain the same result of Proposition \\ref{main} without $K$-theoretical concepts. Firstly, these results make no reference to $K$-theoretical elements. The index theorem only appears in the proof of those propositions. Secondly, there is an analytical (non-topological) proof (non-topological) of the index theorem by the use of Zeta functions $\\zeta(s) = \\sum\\lambda^{-s}$ ($\\lambda$ denotes eigenvalues of an operator) \\cite{Atiyah-Bott}. Therefore, we conjecture it may be possible to prove the analytical condition that we established for the existence of false gauge field copies in a similar way.\n\n\tStill another possibility has to do with proving our results with the help of the asymptotics of the heat equation \\cite{Gilkey}, since that technique was also used to prove the index theorem. But those are questions to be answered in future papers.\n\n\tAll those questions can be summed up in the following: the gauge field copy phenomenon is a remarkable feature of nonlinear gauge fields without a clear-cut physical interpretation. Copied fields belong to a nowhere dense, bifurcation-like domain in the space of all gauge fields in the usual topology \\cite{Doria-84}. Copied fields are actually involved in a (possible) symmetry-breaking mechanism that remains strictly within the bounds of classical field theory \\cite{Amaral}. Yet their contribution to path-integral computations remain unknown -- for instance, if Feynmann integrals are understood as Kurtzweil-Henstock (KH) integrals \\cite{Muldowney} \\cite{Peng-Yee}, the rather unusual topology of the space of all KH-integrable functions may lead to a new interpretation of the Feynman integrals over field and potential spaces in gauge theory, when given the adequate KH-structure.\n\n\tSo, when we try to establish a connection between Atiyah-Singer theory and gauge copies, we are trying to expand the perspectives from where we can observe and try to understand the field copy phenomenon.\n\n\\section{Acknowledgments}\n\n\tThe present work was partially written while A.S.S. was visiting Stanford University as a post-doctoral fellow. A.S.S. wishes to thank Professor Patrick Suppes for his hospitality. The authors acknowledge also support from CNPq, CAPES and Fapesp (Brazil). This paper was computer-formatted with the help of {\\em Project Griffo} at Rio's Federal University.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\bf{Introduction}}\nThe idea of paracompactness given by $\\text{Dieudonn}e^{'}$ in the year 1944 came out as a generalization of the notion of compactness. It has many implication in field of differential geometry and it plays important roll in metrization theory. \nThe concept of the Alexandroff space \\cite{ASF} $($i.e., a $\\sigma$-space or simply a space$)$ was introduced by A. D. Alexandroff in the year 1940 as a generalization of a topological space where the union of open sets were taken to be open for only countable collection of open sets instead of arbitrary collection. \nAnother kind of generalization of a topological space  is the idea of a bitopological space introduced by J.C. Kelly in \\cite{JCK}. Using these ideas Lahiri and Das \\cite{BN} introduced the idea of a bispace as a generalization of a $\\sigma$-space. More works on topological properties were carried out by many authors $( $\\cite{WJP}, \\cite{ILR}, \\cite{JW} etc.$)$ in the setting of a bitopological space and in the setting of a bispace $($\\cite{AKB2}, \\cite{AKB3}, \\cite{AKB4}, \\cite{RM}, \\cite{RM3} etc.$)$. Datta \\cite{MCD} studied the idea of paracompactness in a bitopological space and  tried to get analogous results of topological properties given by Michael \\cite{Michael} in respect of paracompactness. In 1986 Raghavan and Reilly \\cite{TGR} gave the idea of paracompactness in a bitopological space in another way. Later in 2008 M. K. Bose et al \\cite{MKBOSE} studied the same in a bitopological space as a generalization of pairwise compactness. Here we study the idea of pairwise paracompactness using the thoughts given by Bose et al \\cite{MKBOSE} and investigate some results of it in the setting of more general structure a bispace.\n\\section{\\bf{Preliminaries}}\\label{preli}\n\\begin{defi}\\cite{ASF} \nA set $X$ is called an Alexandroff space or $\\sigma $- space or simply space if in it is chosen a system $\\mathcal {F}$ of subsets of $X$, satisfying the following axioms\\\\\n$\\left(i\\right)$The intersection of countable number sets in $\\mathcal {F}$ is a set in $\\mathcal {F}$.\\\\\n$\\left(ii\\right)$The union of finite number of sets from $\\mathcal {F}$ is a set in $\\mathcal {F}$.\\\\\n$\\left(iii\\right)$The void set and $X$ are in $\\mathcal {F}$.\\end{defi}\nSets of $\\mathcal F$ are called closed sets. There complementary sets are called open.It is clear that instead of closed sets in the definition of a space, one may put open sets with  subject to the conditions of  countable summability, finite intersectability and the condition that $X$ and the void set should be open.\\\\ \nThe collection of such open will sometimes be denoted by $\\mathcal{P}$ and the space by $(X, \\mathcal{P})$.\nIt is noted that $\\mathcal{P}$ is not a topology in general as can be seen by taking $X=\\mathbb{R}$, the set of real numbers and $\\tau$ as the collection of all $F_\\sigma$ sets in $\\mathbb{R}$.\\\\\n\\begin{defi}\\cite{ASF} \nTo every set $M$ we correlate its closure $\\overline M$ as the intersection of all closed sets containing M.\\end{defi}  \nGenerally the closure of a set in a $\\sigma$-space is not a closed set. We denote the closure of a set $M$ in a space $(X, \\mathcal{P})$ by $\\mathcal{P}$-cl$(M)$ or cl$(M)$ or simply $\\overline M$ when there is no confusion about $\\mathcal{P}$.\nThe idea of limit points, derived set, interior of a set etc. in a space are similar as in the case of a topological space which have been thoroughly discussed in \\cite{PD}.\n\\begin{defi} \\cite{AKB}\nLet $(X, \\mathcal{P})$ be a space. A family of open sets $B$ is said to form a base $($open$)$ for $\\mathcal{P}$ if and only if every open set  can be expressed as countable union of members of $B$.\\end{defi}\n\\begin{thm} \\cite{AKB}\n A collection of subsets $B$ of a set $X$ forms an open base of a suitable space structure $\\mathcal{P}$ of $X$ if and only if \\\\\n\\noindent 1) the null set $\\phi \\in B$\\\\\n\\noindent 2) $X$ is the countable union of some sets belonging to $B$.\\\\\n\\noindent 3) intersection of any two sets belonging to $B$ is expressible as countable union of some sets belonging to $B$.\\\\\n\\end{thm}\n\\begin{defi} \\cite{BN}\nLet $X$ be a non-empty set. If $\\mathcal{P}$ and $\\mathcal{Q}$ be two collection of subsets of $X$ such that $(X,\\mathcal{P})$ and $(X,\\mathcal{Q})$ are two spaces, then $X$ is called a bispace. \n\\end{defi}\n\\begin{defi} \\cite{BN} \nA bispace $(X,\\mathcal{P},\\mathcal{Q})$ is called pairwise $T_{1}$ if for any two distinct points $x$, $y$ of $X$, there exist $U\\in \\mathcal{P}$ and $V\\in\\mathcal{Q}$ such that $x\\in U$, $y\\notin U$ and $y\\in V$, $x\\notin V$.\n\\end{defi}\n\\begin{defi} \\cite{BN}\nA bispace $(X,\\mathcal{P},\\mathcal{Q})$ is called pairwise Hausdorff if for any two distinct points $x$, $y$ of $X$, there exist $U\\in\\mathcal{P}$ and $V\\in\\mathcal{Q}$ such that $x\\in U$, $y\\in V$, $U\\cap V=\\phi$. \n\\end{defi}\n\\begin{defi} \\cite{BN} \nIn a bispace $(X,\\mathcal{P},\\mathcal{Q})$, $\\mathcal{P}$ is said to be regular with respect to $\\mathcal{Q}$ if for any $x\\in X$ and a $\\mathcal{P}$-closed set $F$ not containing $x$, there exist $U\\in \\mathcal{P}$, $V\\in\\mathcal{Q}$ such that $x\\in U$, $F\\subset V$, $U\\cap V=\\phi$. $(X,\\mathcal{P},\\mathcal{Q})$ is said to be pairwise regular if $\\mathcal{P}$ and $\\mathcal{Q}$ are regular with respect to each other. \n\\end{defi}\n\\begin{defi} \\cite{BN} \nA bispace $(X,\\mathcal{P},\\mathcal{Q})$ is said to be pairwise normal if for any $\\mathcal{P}$-closed set $F_{1}$ and $\\mathcal{Q}$-closed set $F_{2}$ satisfying $F_{1}\\cap F_{2}=\\phi$, there exist  $G_{1}\\in \\mathcal{P}$, $G_{2}\\in \\mathcal{Q}$ such that $F_{1}\\subset G_{2}$, $F_{2}\\subset G_{1}$, $G_{1}\\cap G_{2}=\\phi$.\n\\end{defi}\n\\section{\\bf{Pairwise paracompactness}}\n\\noindent We called a space $($ or a set $)$ is bicompact \\cite{BN} if every open cover of it has a finite subcover. Also similarly as \\cite{BN} a cover B of $(X,\\mathcal{P},\\mathcal{Q})$ is said to be pairwise open if $B \\subset \\mathcal{P} \\cup \\mathcal{Q}$ and B contains at least one nonempty member from each of $\\mathcal{P}$ and $\\mathcal{Q}$. Bourbaki and many authors defined the term paracompactness in a topological space including the requirement that the space is Hausdorff. Also in a bitopological space some authors follow this idea. But in our discussion we shall follow the convention as adopted in Munkresh\\cite{JRM} to define the following terminologies as in the case of a topological space.\n\\begin{defi} (cf.\\cite{JRM})\nIn a space $X$ a collection of subsets $\\mathcal{A}$ is said to be locally finite in $X$ if every point has a neighborhood that intersects only a finitely many elements of $\\mathcal{A}$.\n\\end{defi} \nSimilarly a collection of subsets $\\mathcal{B}$ in a space $X$ is said to be countably locally finite in $X$ if $\\mathcal{B}$ can be expressed as a countable union of locally finite collection. \n\\begin{defi} (cf.\\cite{JRM})\nLet $\\mathcal{A}$ and $\\mathcal{B}$ be two covers of a space $X$, $\\mathcal{B}$ is said to be a refinement of $\\mathcal{A}$ if for $B \\in \\mathcal{B}$ there exists a $A \\in \\mathcal{A}$ containing $B$.\n\\end{defi} \nWe call $\\mathcal{B}$ an open refinement of $\\mathcal{A}$ if the elements of $\\mathcal{B}$ are open and similarly if the elements of $\\mathcal{B}$ are closed  $\\mathcal{B}$ is said to be a closed refinement.\n\\begin{defi} (cf.\\cite{JRM})\nA space $X$ is said to be paracompact if every open covering $\\mathcal{A}$ of $X$ has a locally finite open refinement $\\mathcal{B}$ that covers $X$.\n\\end{defi}\nAs in the case of a topological space \\cite{MCD, MKBOSE} we define the following terminologies. Let $\\mathcal{A}$ and $\\mathcal{B}$ be two pairwise open covers of a bispace $(X,\\mathcal{P},\\mathcal{Q})$. Then $\\mathcal{B}$ is said to be a parallel refinement \\cite{MCD} of $\\mathcal{A}$ if for any $\\mathcal{P}$-open set$(\\text{respectively } \\mathcal{Q} \\text{-open set})$ $B$ in $\\mathcal{B}$ there exists a $\\mathcal{P}$-open set$(\\text{respectively } \\mathcal{Q} \\text{-open set})$ $A$ in $\\mathcal{A}$ containing $B$. Let $\\mathcal{U}$ be a pairwise open cover in a bispace $(X, \\mathcal{P}_{1}, \\mathcal{P}_{2})$. If $x$ belongs to $X$ and $M$ be a subset of $X$, then by ``$M$ is $\\mathcal{P}_{\\mathcal{U} x}$-open\" we mean $M$ is $\\mathcal{P}_{1}$-open$(\\text{respectively } \\mathcal{P}_{2} \\text{-open set})$ if $x$ belongs to a $\\mathcal{P}_{1}$-open set$(\\text{respectively } \\mathcal{P}_{2} \\text{-open set})$ in $\\mathcal{U}$.\n\\begin{defi} (cf. \\cite{MKBOSE})\nLet $\\mathcal{A}$ and $\\mathcal{B}$ be two pairwise open covers of a bispace $(X, \\mathcal{P}_{1}, \\mathcal{P}_{2})$. Then $\\mathcal{B}$ is said to be a locally finite refinement of $\\mathcal{A}$ if for each $x$ belonging to $X$, there exists a $\\mathcal{P}_{\\mathcal{A} x}$-open open neighborhood of $x$ intersecting only a finite number of sets of $\\mathcal{B}$.\n\\end{defi}\n\\begin{defi} (cf. \\cite{MKBOSE})\nA bispace $(X, \\mathcal{P}_{1}, \\mathcal{P}_{2})$ is said to be pairwise paracompact if every pairwise open cover of $X$ has a locally finite parallel refinement.\n\\end{defi}\n\\noindent To study the notion of paracompactness in a bispace the idea of pairwise regular and strongly pairwise regular spaces plays significant roll as discussed below.\\\\\n\\indent As in the case of a bitopological space a bispace $(X, \\mathcal{P}_{1}, \\mathcal{P}_{2})$ is said to be strongly pairwise regular\\cite{MKBOSE} if $(X, \\mathcal{P}_{1}, \\mathcal{P}_{2})$ is pairwise regular and both the spaces $(X, \\mathcal{P}_{1})$ and $(X, \\mathcal{P}_{2})$ are regular.\\\\\n\\indent Now we present two examples, the first one is of a strongly regular bispace and the second one is of a pairwise regular bispace without being a strongly pairwise regular bispace.\\\\ \n\\textbf{Example 3.1.}\nLet $X=\\mathbb{R}$ and $(x,y)$ be an open interval in $X$. We consider the collection $\\tau_{1}$ with sets $A$ in $\\mathbb{R}$ such that either $(x,y) \\subset \\mathbb{R} \\setminus A$ or $A \\cap (x,y)$ can be expressed as some union of open subintervals of $(x,y)$ and $\\tau_{2}$ be the collection of all countable subsets in $(x,y)$. Also if $\\tau$ be the collection of all countable union of members of $\\tau_{1} \\cup \\tau_{2}$ then clearly $(X,\\tau)$ is a $\\sigma$-space but not a topological space. Also consider the bispace $(X, \\tau, \\sigma)$, where $\\sigma$ is the usual topology on $X$. \\\\\n\\indent  We first show that $(X,\\tau)$ is regular. Let $p \\in X$ and $P$ be any $\\tau$-closed set not containing $p$. Then $A =\\{p \\}$ is a $\\tau$-open set containing $p$. Also $A =\\{p \\}$ is closed in $(X,\\tau)$ because if $p \\notin (x, y)$ then  $A^{c} \\cap (x,y)= (x,y)$ and if $p \\in (x, y)$ then $A^{c} \\cap (x,y)= (x,p) \\cup (p,y)$ and hence $A^{c}$ is a $\\tau$-open set containing $P$. \\\\\n\n\\indent Now we show that the bispace $(X, \\tau , \\sigma)$ is pairwise regular. Let $p\\in X$ and $M$ be a $\\tau$-closed set not containing $p$. Then $A =\\{p \\}$ is a $\\tau$-open set containing $p$ and also as every singleton set is closed in $(X,\\sigma)$, $A^{c}$ is a $\\sigma$-open set containing $M$.\\\\ \n \\indent Now let $p \\in X$ and $P$ be a $\\sigma$-closed set not containing $p$. Now consider the case when $P \\cap (x,y)=\\phi$ then $P$ is a $\\tau$-open set containing $P$ and $P^{c}$ is a $\\sigma$-open set containing $p$.\\\\\n\\indent Now we consider the case when $P \\cap (x, y) \\neq \\phi$. Since $p \\notin P$, $P^{c}$ is a $\\sigma$-open set containing $p$ and hence there exists an open interval $I$ containing $p$ be such that $p \\in I \\subset P^{c}$ and $p \\in \\overline{I} \\subset P^{c}$, where $\\overline{I}$ is the closer of $I$ with respect to $\\sigma$. If $I$ intersects $(x,y)$ then let $I_{1} = (x,y) \\setminus \\overline{I}$. Clearly $I_{1}$ is non empty because $ P \\cap (x,y) \\neq \\phi$. Also $\\overline{I} \\subset P^{c}$ and hence $(x,y) \\setminus P^{c} \\subset (x,y) \\setminus \\overline{I}$ and its follows that $P \\cap (x, y) \\subset I_{1}$. So clearly $P \\cup I_{1}$ is a $\\tau$-open set containing $P$ and $I$ is a $\\sigma$-open set containing $p$ and which are disjoint. Again if $I$ does not intersect $(x,y)$ then $P \\cup (x,y)$ is a $\\tau$-open set containing $P$ and $I$ itself a $\\sigma$-open set containing $p$ and which are disjoint. Therefore the bispace $(X, \\tau , \\sigma)$ is strongly pairwise regular.\\\\ \n\\noindent \\textbf{Example 3.2.}\nLet $X= \\mathbb{R}$ and $(X, \\tau_{1}, \\tau_{2})$ be a bispace where $(X, \\tau_{1})$ is cocountable topological space and $\\tau_{2}= \\{ X, \\phi \\} \\cup \\{ \\text{countable subsets of real numbers} \\}$. Clearly $\\tau_{2}$ is not a topology and hence $(X, \\tau_{1}, \\tau_{2})$ is not a bitopological space. We show that $(X, \\tau_{1}, \\tau_{2})$ is a pairwise regular bispace but not a strongly pairwise regular bispace. Let $p \\in X$ and $A$ be a $\\tau_{1}$-closed set not containing $p$. Then clearly $A$ itself a $\\tau_{2}$-open set containing $A$ and $A^{c}$ is a $\\tau_{1}$-open set containing $p$ and clearly they are disjoint.\\\\\n\\indent Similarly if $B$ is a $\\tau_{2}$-closed set such that $p \\notin B$, then $B$ being a complement of a countable set is $\\tau_{1}$-open set containing $B$. Also $B^{c}$ being countable is $\\tau_{2}$-open set containing $p$.\\\\\n\\indent Now let $p \\in X$ and $P$ be a closed set in $(X, \\tau_{2})$ such that $p \\notin P$. Then $P$ must be a complement of a countable set in $\\mathbb{R}$ and hence it must be a uncountable set. So clearly the only open set containing $P$ is $\\mathbb{R}$ itself. Therefore $(X, \\tau_{2})$ is not regular and hence $(X, \\tau_{1}, \\tau_{2})$ can not be strongly pairwise regular.\n\\begin{Note}\nIn a bitopological space, pairwise Hausdorffness and pairwise paracompactness together imply pairwise normality but similar result holds in a bispace if an additional condition C$(1)$ holds. \n\\end{Note}\n\\begin{thm}\nLet $(X,\\mathcal{P},\\mathcal{Q})$ be a bispace, which is pairwise Hausdorff and pairwise paracompact and satisfies the condition C$(1)$ as stated below then it is pairwise normal.\\end{thm}\n\\noindent C$(1):$ If $A\\subset X$ is expressible as an arbitrary union of $\\mathcal{P}$-open sets and $A \\subset B$, $B$ is an arbitrary intersection of $\\mathcal{Q}$-closed sets, then there exists a $\\mathcal{P}$-open set $K$, such that $A \\subset K \\subset B$, the role of $\\mathcal{P}$ and $\\mathcal{Q}$ can be interchangeable.\n\\begin{proof}\nWe first show that $X$ is pairwise regular. So let us suppose $F$ be a $\\mathcal{P}$-closed set not containing $x \\in X$. Since $X$ is pairwise Hausdorff for $\\xi \\in F$, there exists a $U_{\\xi}\\in \\mathcal{P}$ and $V_{\\xi}\\in \\mathcal{Q}$, such that $x \\in U_{\\xi}$ and $\\xi \\in V_{\\xi}$ and $U_{\\xi} \\cap V_{\\xi} = \\phi$. Then the collection $\\{V_{\\xi}   : \\xi \\in F\\} \\cup (X \\setminus F)$ forms a pairwise open cover of $X$. Therefore it has a locally finite parallel refinement $\\mathcal{W}$. Let $H= \\cup \\{W \\in \\mathcal{W} : W\\cap F\\neq \\phi \\}$. Now $x\\in X \\setminus F$ and $X \\setminus F$ is $\\mathcal{P}$-open set and hence there exists a $\\mathcal{P}$-open neighborhood $D$ of $x$ intersecting only a finite number of members $W_{1}, W_{2} \\dots W_{n}$ of $\\mathcal{W}$. Now if $W_{i} \\cap F = \\phi$ for all $n=1, 2 \\dots n$, then $H \\cap D = \\phi$. Therefore by C$(1)$ we must have a $\\mathcal{Q}$-open set $K$ such that $F \\subset H \\subset K \\subset D^{c}$. Hence we have a $\\mathcal{Q}$-open set $K$ containing $F$ and $\\mathcal{P}$-open set $D$ containing $x$ with $D \\cap K= \\phi$. If there exists a finite number of elements $W_{p_{1}}, W_{p_{2}} \\dots W_{p_{k}}$ from the collection $\\{W_{1}, W_{2} \\dots W_{n} \\}$ such that $W_{p_{i}} \\cap F \\neq \\phi$, then we consider $V_{\\xi _{p_{i}}}$ such that $W_{p_{i}} \\subset V_{\\xi _{p_{i}}}$, $\\xi _{p_{i}} \\in F$ and $i=1, 2 \\dots k$, since $\\mathcal{W}$ is a locally finite parallel refinement of $\\{V_{\\xi}   : \\xi \\in F\\} \\cup (X \\setminus F)$. Now if $U_{\\xi_{p_{i}}}$ are the corresponding member of $V_{\\xi_{p_{i}}}$, then $x \\in D \\cap (\\bigcap^{n}_{i=1}U_{\\xi_{p_{i}}})= G(say) \\in \\mathcal{P}$. Since $\\mathcal{W}$ is a cover of $X$ it covers also $D$ and since $D$ intersects only finite number of members $W_{1}, W_{2} \\dots W_{n}$, these $n$ sets covers $D$. Now since the members $W_{p_{1}}, W_{p_{2}} \\dots W_{p_{k}}$ be such that $W_{p_{i}} \\cap F \\neq \\phi$, we have $D \\cap F \\subset \\bigcup_{i=1}^{k}W_{p_{i}}$. Now let $W_{p_{i}} \\subset V_{\\xi_{p_{i}}}$ for some $\\xi_{p_{i}} \\in F$ and consider $U_{\\xi_{p_{i}}}$ corresponding to $V_{\\xi_{p_{i}}}$ be such that $U_{\\xi_{p_{i}}} \\cap V_{\\xi_{p_{i}}}= \\phi$. Now we claim that $G \\cap F = \\phi$. If not let $y \\in G \\cap F = [D \\cap (\\bigcap^{n}_{i=1}U_{\\xi_{p_{i}}})] \\cap F= [D \\cap F]\\cap (\\bigcap^{n}_{i=1}U_{\\xi_{p_{i}}})$. Then $y \\in D \\cap F$ and hence there exists $W_{p_{i}}$ for some $i=1,2, \\dots k$ such that $y \\in W_{p_{i}} \\subset V_{\\xi_{p_{i}}}$. Also $y \\in (\\bigcap^{n}_{i=1}U_{\\xi_{p_{i}}}) \\subset U_{\\xi_{p_{i}}}$ and hence $y \\in  U_{\\xi_{p_{i}}} \\cap V_{\\xi_{p_{i}}}$, which is a contradiction. So $G \\cap F = \\phi$. Now we have a $\\mathcal{P}$-open neighborhood $G$ of $x$ intersecting only a finite number of members $W_{r_{1}}, W_{r_{2}} \\dots W_{r_{k}}$ of $\\mathcal{W}$ where $W_{r_{i}} \\cap F= \\phi$. So by similar argument there exists a $\\mathcal{Q}$-open set $K$ such that $F \\subset H \\subset K \\subset G^{c}$. Thus we have a $\\mathcal{Q}$-open set $K$ containing $F$ and a $\\mathcal{P}$-open set $G$ containing $x$ such that $G \\cap K=\\phi$. \\\\\n\\indent Next let $A$ be a $\\mathcal{Q}$-closed set and $B$ be a $\\mathcal{P}$-closed set and $A\\cap B =\\phi$. Then for every $x \\in B$ and $\\mathcal{Q}$-closed set $A$ there exists $\\mathcal{P}$-open set $U_{x}$ containing $A$ and $\\mathcal{Q}$-open set $V_{x}$ containing $x$ with $U_{x} \\cap V_{x}= \\phi$. Now the collection $\\mathcal{U}= (X \\setminus B) \\cup \\{V_{x} : x \\in B \\}$ forms a pairwise open cover of $X$. Hence there exists a locally finite parallel refinement $\\mathcal{M}$ of $\\mathcal{U}$. Clearly $B \\subset Q$ where $Q= \\cup \\{ M \\in \\mathcal{M} : M\\cap B \\neq \\phi \\}$. Now for $x \\in X \\setminus B$, a $\\mathcal{P}$-open set there exists a $\\mathcal{P}$-open neighborhood of $x$ intersecting only a finite number of elements of $\\mathcal{M}$. Since $A \\subset X \\setminus B$, so for $x \\in A$ there exists a $\\mathcal{P}$-open neighborhood $D_{x}$ of $x$ intersecting only a finite number of elements $M_{x_{1}}, M_{x_{2}} \\dots M_{x_{n}}$ of $\\mathcal{M}$ with $M_{x_{i}} \\cap B\\neq \\phi$ for some $i=1,2,\\dots n$. Suppose if $M_{x_{i}} \\subset V_{x_{i}}$, $i=1, 2 \\dots n$ and let $P_{x}= D_{x} \\cap (\\bigcap^{n}_{i=1}U_{x_{i}})$ where $U_{x_{i}} \\cap V_{x_{i}}= \\phi$. If $M_{x_{i}} \\cap B= \\phi$ for all $i=1,2,\\dots n$, then we consider $P_{x}= D_{x}$. Now if $P= \\bigcup \\{P_{x} : x\\in A\\}$ then $A\\subset P$ and $P \\subset Q^{c}$.\\\\\n\\indent\tNow by the given condition C$(1)$ there exists a $\\mathcal{P}$-open set $R$ be such that $A \\subset P \\subset R \\subset Q^{c}$. Again by the same argument there exists a $\\mathcal{Q}$-open set $S$ be such that $B \\subset Q \\subset S \\subset R^{c}$. Hence there exists a $\\mathcal{P}$-open set $R$ containing $A$ and $\\mathcal{Q}$-open set $S$ containing $B$ with $R \\cap S= \\phi$.\n\\end{proof}\n\\begin{thm}\nIf the bispace $(X, \\mathcal{P}_{1}, \\mathcal{P}_{2})$ is strongly pairwise regular and satisfies the condition C$(2)$ given below, then the following statements are equivalent:\\\\\n$(i)$ X is pairwise paracompact.\\\\\n$(ii)$ Each pairwise open cover $\\mathcal{C}$ of $X$ has a countably locally finite parallel refinement.\\\\\n$(iii)$ Each pairwise open cover $\\mathcal{C}$ of $X$ has a locally finite refinement.\\\\\n$(iv)$  Each pairwise open cover $\\mathcal{C}$ of $X$ has a locally finite refinement $\\mathcal{B}$ such that if $B \\subset C$ where $B \\in \\mathcal{B}$ and $C \\in \\mathcal{C}$, then $\\mathcal{P}_{1}$-cl$(B) \\cup \\mathcal{P}_{2}$-cl$(B) \\subset C$.\\\\\n\\end{thm}\n\\noindent C$(2) :$ If $M \\subset X$ and $\\mathcal{B}$ is a subfamily of $\\mathcal{P}_{1} \\cup \\mathcal{P}_{2}$ such that $\\mathcal{P}_{i}$-$cl(B) \\cap M=\\phi$, for all $B \\in \\mathcal{B}$, then  there exists a $\\mathcal{P}_{i}$- open set $S$ such that $M \\subset S \\subset[\\bigcup_{B\\in \\mathcal{B}}\\mathcal{P}_{i}$-$cl(B)]^{c}$.\n\\begin{proof}\n$(i) \\Rightarrow (ii)$\\\\\nLet $\\mathcal{C}$ be a pairwise open cover of $X$. Let $\\mathcal{U}$ be a locally finite parallel refinement of $\\mathcal{C}$. Then the collection  $\\mathcal{V}= \\bigcup _{n=1} ^{\\infty} \\mathcal{V}_{n}$, where $\\mathcal{V}_{n} = \\mathcal{U}$ for all $n\\in \\mathbb{N}$, becomes the countably locally finite parallel refinement of $\\mathcal{C}$.\\\\\n$(ii) \\Rightarrow (iii)$\\\\\nWe consider a pairwise open cover $\\mathcal{C}$ of $X$. Let $\\mathcal{V}$ be a parallel refinement of $\\mathcal{C}$, such that $\\mathcal{V}= \\bigcup _{n=1} ^{\\infty} \\mathcal{V}_{n}$, where for each $n$ and for each $x$ there exists a $\\mathcal{P}_{\\mathcal{C}x}$-open neighborhood of $x$ intersecting only a finite number of members of $\\mathcal{V}_{n}$. For each $n \\in \\mathbb{N}$, let us agree to write $\\mathcal{V}_{n}$ as $\\mathcal{V}_{n}= \\{\\mathcal{V}_{n \\alpha} : \\alpha \\in \\wedge_{n} \\}$ and we consider $M_{n} = \\bigcup _{\\alpha \\in \\wedge_{n}} \\mathcal{V}_{n \\alpha}$, $n \\in \\mathbb{N}$. Clearly the collection $\\{ M_{n} \\}_{n \\in \\mathbb{N}}$ is a cover of $X$. Let $N_{n}= M_{n} - \\bigcup _{k<n} M_{k}$. Clearly for $x \\in X$ if $x \\in M_{n}$, where $n$ is the least positive integer then $x \\in N_{n}$ and hence $\\{ N_{n} : n \\in \\mathbb{N}\\}$ covers $X$. Also $N_{n} \\subset M_{n}$ for every $n$, so $\\{ N_{n} : n \\in \\mathbb{N}\\}$ is a refinement of  $\\{ M_{n} : n \\in \\mathbb{N}\\}$. The family  $\\{ N_{n} : n \\in \\mathbb{N}\\}$ is locally finite because for $x \\in X$ there exists a $\\mathcal{V}_{n \\alpha} \\in \\mathcal{V}$ which can intersects only some or all of $N_{1}, N_{2} \\dots N_{n}$. Now the collection $\\{ \\mathcal{V}_{n \\alpha} \\cap N_{n} : \\alpha \\in \\wedge_{n}, n \\in \\mathbb{N} \\}$ covers $X$ as if $x \\in \\mathcal{V}_{p \\alpha}$ for the least positive integer $p$ then $x \\in N_{p}$ and hence $x \\in \\mathcal{V}_{p \\alpha} \\cap N_{p}$. So clearly $\\{ \\mathcal{V}_{n \\alpha} \\cap N_{n} : \\alpha \\in \\wedge_{n}, n \\in \\mathbb{N} \\}$ is a refinement of $\\mathcal{V}$ and hence of $\\mathcal{C}$. Also for $x \\in X$ there exists a $\\mathcal{P}_{\\mathcal{C}x}$-open neighborhood $\\mathcal{V}_{ k \\alpha}$ intersecting only a finite number of members of $\\{ N_{n} : n \\in \\mathbb{N}\\}$ and hence it intersects only a finite number of members of $\\{ \\mathcal{V}_{n \\alpha} \\cap N_{n} : \\alpha \\in \\wedge_{n}, n \\in \\mathbb{N} \\}$.\\\\\n$(iii) \\Rightarrow (iv)$\\\\\nLet $\\mathcal{C}$ be a pairwise open cover of $X$. Let $x \\in X$ and suppose that $x \\in C_{x}$ for some $C_{x} \\in \\mathcal{C}$. Without any loss of generality let $C_{x} \\in \\mathcal{P}_{1}$. Then $x \\notin C_{x}^{c}$ and hence by using the condition of strongly pairwise regularity of $X$ there exists a $\\mathcal{P}_{1}$-open set $D_{1}$ containing $x$ and a $\\mathcal{P}_{1}$-open set $D_{1}^{'}$ containing $C_{x}^{c}$ with $D_{1} \\cap D_{1}^{'}= \\phi$. Now $(D_{1}^{'})^{c} \\subset C_{x}$ and hence $(D_{1}^{'})^{c}$ is a $\\mathcal{P}_{1}$-closed set such that $x \\in (D_{1}^{'})^{c} \\subset C_{x}$. Therefore $\\mathcal{P}_{1}$-cl$(D_{1}) \\subset C_{x}$ as $D_{1} \\subset (D_{1}^{'})^{c} \\subset C_{x}$. Again $x \\notin D_{1}^{c}$, a $\\mathcal{P}_{1}$-closed set and hence by pairwise regularity of $X$ there exists a $\\mathcal{P}_{1}$-open set $D_{2}$ containing $x$ and a $\\mathcal{P}_{2}$-open set $D_{2}^{'}$ containing $D_{1}^{c}$ with $D_{2} \\cap D_{2}^{'}= \\phi$. Now $D_{2} \\subset (D_{2}^{'})^{c}$ and $D_{2} \\subset (D_{2}^{'})^{c} \\subset D_{1} \\subset C_{x}$. Hence $\\mathcal{P}_{2}$-cl$(D_{2}) \\subset C_{x}$ and also $D_{2} \\subset D_{1}$. Therefore $\\mathcal{P}_{1}$-cl$(D_{2}) \\subset \\mathcal{P}_{1}$-cl$(D_{1})$ and hence $\\mathcal{P}_{1}$-cl$(D_{2}) \\cup \\mathcal{P}_{2}$-cl$(D_{2}) \\subset C_{x}$. Similarly if $C_{x} \\in \\mathcal{P}_{2}$ then there exists a $\\mathcal{P}_{2}$-open set $D_{2}$ containing $x$ such that $\\mathcal{P}_{1}$-cl$(D_{2}) \\cup \\mathcal{P}_{2}$-cl$(D_{2}) \\subset C_{x}$. Let us denote $D_{2}$ by a general notation $G_{x}$ and then we can write $\\mathcal{P}_{1}$-cl$(G_{x}) \\cup \\mathcal{P}_{2}$-cl$(G_{x}) \\subset C_{x}$. Then, since $\\mathcal{C}$ be a pairwise open cover $\\{ G_{x} : x \\in X , C_{x} \\in  \\mathcal{C}  \\}$ is a pairwise open cover of $X$ which refines of $\\mathcal{C}$. Therefore by $(iii)$ there exists a locally finite refinement $\\mathcal{B}$ of $\\{ G_{x} : x \\in X \\}$ and hence of $\\mathcal{C}$. If $B \\in \\mathcal{B}$ then for some $G_{x}$ we have $B \\subset G_{x} \\subset C_{x}$ and so $\\mathcal{P}_{1}$-cl$(B) \\cup \\mathcal{P}_{2}$-cl$(B) \\subset \\mathcal{P}_{1}$-cl$(G_{x}) \\cup \\mathcal{P}_{2}$-cl$(G_{x}) \\subset C_{x}$.\\\\\n$(iv) \\Rightarrow (i)$\\\\\nLet $\\mathcal{C}$ be a pairwise open cover of $X$ and without any loss of generality we assume that there does not exists any element of $\\mathcal{C}$ which is both $\\mathcal{P}_{1}$-open and $\\mathcal{P}_{2}$-open. So there exists a locally finite refinement $\\mathcal{A}$ of $\\mathcal{C}$. For $x \\in X$ we must have a $C \\in \\mathcal{C}$ containing $x$. Let us suppose $C$ is $\\mathcal{P}_{i}$-open. Let $W_{x}$ be a $\\mathcal{P}_{i}$-open neighborhood of $x$ intersecting only a finite number of elements of $\\mathcal{A}$. So the collection $\\mathcal{W}=\\{W_{x} : x\\in X\\}$ is a pairwise open cover of $X$ and let $E= \\{E_{\\lambda} : \\lambda \\in \\wedge\\}$ be a locally finite refinement of $\\mathcal{W}$ such that if $E_{\\lambda} \\subset W_{x}$ then $\\mathcal{P}_{1}$-cl$(E_{\\lambda}) \\cup \\mathcal{P}_{2}$-cl$(E_{\\lambda}) \\subset W_{x}$. Now for $A \\in \\mathcal{A}$ we consider $C_{A} \\in \\mathcal{C}$ such that $A \\subset C_{A}$. Then if $C_{A}$ is $\\mathcal{P}_{i}$-open, then we consider the set $F_{A} =\\cup \\{ \\mathcal{P}_{i}$-$cl(E_{\\lambda}) : E_{\\lambda} \\in E, \\mathcal{P}_{i}$-$cl(E_{\\lambda}) \\cap A=\\phi \\}$. Let $G_{A}= X \\setminus F_{A}$, then by the given condition C$(2)$ there exists a $\\mathcal{P}_{i}$-open set $S_{A}$ such that $A \\subset S_{A} \\subset G_{A}$. We write $H_{A}=S_{A} \\cap C_{A}$ and since $A \\subset H_{A}$ , the collection $\\{H_{A} : A \\in \\mathcal{A} \\}$ covers $X$. Also $H_{A} \\subset C_{A}$ and $H_{A}$ is $\\mathcal{P}_{i}$-open. Thus $\\{H_{A} : A \\in \\mathcal{A} \\}$ is a parallel refinement of $\\mathcal{C}$. Now we show that $\\{H_{A} : A \\in \\mathcal{A} \\}$ is a locally finite refinement of $\\mathcal{C}$. \\\\ \n\\indent We show that if $M$ is a $\\mathcal{P}_{\\mathcal{W}x}$-open set containing $x$ then it is also a $\\mathcal{P}_{\\mathcal{C}x}$-open set containing $x$. Let $M$ is a $\\mathcal{P}_{\\mathcal{W}x}$-open set containing $x$ and $M$ is $\\mathcal{P}_{i}$-open set then $x$ must be contained in a $\\mathcal{P}_{i}$-open set $W_{x}$ in $\\mathcal{W}$. So there exists a $\\mathcal{P}_{i}$-open set $C$ in $\\mathcal{C}$ containing $x$. This shows that $M$ is also a $\\mathcal{P}_{\\mathcal{C}x}$-open set containing $x$.\\\\ \n\\indent Now let $x \\in X$ and $J_{x}$ be a $\\mathcal{P}_{\\mathcal{W}x}$-open neighborhood of $x$ intersecting only a finite numbers of members $ E_{{\\lambda}_{1}}, E_{{\\lambda}_{2}} \\dots E_{{\\lambda}_{n}} $ of $E$. Hence $J_{x}$ is also a $\\mathcal{P}_{\\mathcal{C}x}$-open neighborhood of $x$ intersecting only a finite numbers of members $ E_{{\\lambda}_{1}}, E_{{\\lambda}_{2}} \\dots E_{{\\lambda}_{n}} $ of $E$.\nClearly $J_{x}$ can be covered by these members of $E$. Now each $ E_{{\\lambda}_{i}}$ is contained in some $W_{x}$ with $\\mathcal{P}_{1}$-cl$(E_{{\\lambda}_{i}}) \\cup \\mathcal{P}_{2}$-cl$(E_{{\\lambda}_{i}}) \\subset W_{x}$. Also $W_{x}$ can intersects only a finite number of members of $\\mathcal{A}$. Hence each $\\mathcal{P}_{1}$-cl$(E_{{\\lambda}_{i}})$ or $\\mathcal{P}_{2}$-cl$(E_{{\\lambda}_{i}})$ can intersect only a finite number of sets in $\\mathcal{A}$. So each $\\mathcal{P}_{1}$-cl$(E_{{\\lambda}_{i}})$ or $\\mathcal{P}_{2}$-cl$(E_{{\\lambda}_{i}})$ can intersect only a finite number of sets in $\\{G_{A} : A \\in \\mathcal{A} \\}$. Therefore $J_{x}$ can intersect only a finite number of sets of $\\{ G_{A} : A \\in \\mathcal{A}\\}$. Now $\\{H_{A} : A \\in \\mathcal{A}\\}$ covers $X$ and $H_{A} \\subset G_{A}$, hence $J_{x}$ can intersect only a finite number of sets in $\\{H_{A} : A \\in \\mathcal{A}\\}$. Also $H_{A} \\subset C_{A}$ and hence clearly $\\{H_{A} : A \\in \\mathcal{A}\\}$ refines $\\mathcal{C}$. Therefore $\\{H_{A} : A \\in \\mathcal{A}\\}$ is a locally finite parallel refinement of $\\mathcal{C}$.\n\\end{proof}\n\\begin{thm}\nLet $\\mathcal{A}$ be a locally finite collection in a $\\sigma$-space $X$. Then the collection $\\mathcal{B} =\\{ \\overline{A} \\}_{A \\in \\mathcal{A}}$ is also locally finite.\n\\end{thm}\n\\begin{proof}\nLet $x \\in X$ and $U$ be a neighborhood of $x$ intersecting only a finite number of members of $\\mathcal{A}$. Now if for $A \\in \\mathcal{A}$, $A \\cap U = \\phi$ then $A \\subset U^{c}$ and hence $A \\subset \\overline{A} \\subset U^{c}$. Therefore $\\overline{A} \\subset U^{c}$ so $\\overline{A} \\cap U= \\phi$. Therefore $U$ can intersect only a finite number of members of $\\mathcal{B}$. \n\\end{proof}\n\\begin{thm}\nIn a space any sub collection of a locally finite collection of sets is locally finite. \n\\end{thm}\n\\begin{proof}\nLet $\\mathcal{A}$ be a locally finite collection of sets in a space $X$ and $\\mathcal{B} = \\{ B_{\\alpha}  : \\alpha \\in \\Lambda, \\text{an indexing set}\\}$ be a sub collection of $\\mathcal{A}$. If $x \\in X$ then there exists a neighborhood $U$ of $X$ intersecting only a finite number of sets in $\\mathcal{A}$. Hence $U$ can not intersect infinite number of sets in $\\mathcal{B}$. If $U$ does not intersect any member of $\\mathcal{B}$, then consider $B_{p} \\in \\mathcal{B}$ such that $M=B_{p} \\setminus \\bigcup _{\\alpha \\in \\Lambda} ^{\\alpha \\neq p} B_{\\alpha}\\neq \\phi$. Then $M \\cup U$ is a neighborhood of $x$ intersecting only $B_{p}$ of $\\mathcal{B}$. Hence $\\mathcal{B}$ is locally finite.\n\\end{proof}\nIt has been discussed in \\cite{DJ66} that in a regular topological space $X$ the following four conditions are equivalent:\\\\\n$(i)$ The space $X$ is paracompact.\\\\\n$(ii)$ If $\\mathcal{U}$ is a open cover of $X$ then it has an open refinement $\\mathcal{V}=\\bigcup_{n=1} ^{\\infty} V_{n}$, where $V_{n}$ is a locally finite collection in $X$ for each $n$.\\\\ \n$(iii)$For every open cover of the space $X$ there exists a locally finite refinement of it.\\\\\n$(iv)$For every open cover of the space $X$ there exists a closed locally finite refinement of it.\\\\\nIn a $\\sigma$-space it is not true because closure of a set may not be closed. But a similar kind of result has been discussed below.\n\\begin{thm}\nIn a regular space $X$ for the following four conditions we have $(i)\\Rightarrow (ii) \\Rightarrow (iii) \\Rightarrow (iv)$:\\\\\n$(i)$ The space $X$ is paracompact.\\\\\n$(ii)$ If $\\mathcal{U}$ is a open cover of $X$ then it has an open refinement $\\mathcal{V}=\\bigcup_{n=1} ^{\\infty} V_{n}$, where $V_{n}$ is a locally finite collection in $X$ for each $n$.\\\\ \n$(iii)$For every open cover of the space $X$ there exists a locally finite refinement of it.\\\\\n$(iv)$For every open cover $\\mathcal{A}$ of the space $X$ there exists a locally finite refinement $S= \\{ S_{\\alpha} : \\alpha \\in \\Lambda \\}$ of it such that $\\{ \\overline{S_{\\alpha}} : S_{\\alpha} \\in S\\}$ is also a locally finite refinement of it, $\\Lambda$ being an indexing set.\\\\\n\\end{thm}\n\\begin{proof}\n$(i)\\Rightarrow (ii)$\\\\\nThe proof is straightforward.\\\\\n$(ii) \\Rightarrow (iii)$\\\\\nLet $\\mathcal{A}$ be an open cover of $X$. Then by $(ii)$ there exists an open refinement $\\mathcal{B}=\\bigcup_{n=1} ^{\\infty} B_{n}$ where $B_{n}$ is a locally finite collection in $X$ for each $n$. Let $B_{n}=\\{B_{n \\alpha} : \\alpha \\in \\Lambda_{n} \\}$ and $C_{n}= \\bigcup _{\\alpha \\in \\Lambda_{n}} B_{n \\alpha}$, $\\Lambda_{n}$ being an indexing set. Now clearly the collection $\\{ C_{n} \\}$ covers $X$. Let us consider $D_{n} = C_{n} \\setminus \\bigcup _{k <n} C_{k}$. For $x \\in X$, suppose that $k$ be the least natural number for which $x \\in B_{k \\alpha}$, then $B_{k \\alpha}$ can intersect at most $k$ members $D_{1}, D_{2} \\dots D_{k}$ of $\\{ D_{n}: n\\in \\mathbb{N} \\}$. Hence $\\{ D_{n}: n\\in \\mathbb{N} \\}$ is a locally finite refinement of $\\{ C_{n}: n\\in \\mathbb{N} \\}$. Now we show that $M= \\{ D_{n} \\cap B_{n \\alpha} : n \\in \\mathbb{N}, \\alpha \\in \\Lambda_{n} \\}$ is a locally finite refinement of $\\mathcal{B}$. For $n \\in \\mathbb{N}$ we have $\\bigcup _{\\alpha \\in \\Lambda_{n}}(D_{n} \\cap B_{n \\alpha}) =D_{n} \\cap (\\bigcup _{\\alpha \\in \\Lambda_{n}}B_{n \\alpha}) =D_{n} \\cap C_{n} =D_{n}$ as $D_{n} \\subset C_{n}$. Also $D_{n}$ covers $X$ and hence $\\bigcup_{n \\in \\mathbb{N}}\\bigcup _{\\alpha \\in \\Lambda_{n}}(D_{n} \\cap B_{n \\alpha} ) =X$. Let $x \\in X$ then there exists an  neighborhood $U$ of $x$ intersecting only a finite number members $D_{i_{1}}, D_{i_{2}} \\dots D_{i_{n}}(say)$ of $\\{ D_{n}: n\\in \\mathbb{N} \\}$. Also there exists an open set $U_{i_{n}}$ intersecting only a finite number of members of $B_{i_{n}}$. Now $U \\cap (\\bigcap_{k=1}^{n}U_{i_{k}})$ is an neighborhood of $x$ intersecting only a finite numbers of $M$ as $M$ covers $X$. Also $D_{n} \\cap B_{n \\alpha} \\subset B_{n \\alpha}$ and hence $M= \\{ D_{n} \\cap B_{n \\alpha} : n \\in \\mathbb{N}, \\alpha \\in \\Lambda_{n} \\}$ is a locally finite refinement of $\\mathcal{B}$. And also since $D_{n} \\cap B_{n \\alpha} \\subset B_{n \\alpha} \\subset A$ for some $A \\in \\mathcal{A}$, $M= \\{ D_{n} \\cap B_{n \\alpha} : n \\in \\mathbb{N}, \\alpha \\in \\Lambda_{n} \\}$ is a locally finite refinement of $\\mathcal{A}$.\\\\\n$(iii) \\Rightarrow (iv)$\\\\\nLet $\\mathcal{U}$ be an open cover of $X$. Now for $x \\in X$ we have a $U_{x} \\in \\mathcal{U}$ such that $x \\in U_{x}$. So $x \\notin (U_{x})^{c}$ and hence by regularity of $X$, there exist disjoint open sets $P_{x}$ and $Q_{x}$ containing $x$ and $(U_{x})^{c}$ respectively.  Hence $x \\in P_{x} \\subset (Q_{x})^{c} \\subset U_{x}$ and clearly $x \\in \\overline{P_{x}} \\subset U_{x}$. Now $P=\\{ P_{x}: x\\in X \\}$ is an open cover of $X$ and by $(iii)$ it has a locally finite refinement $S= \\{ S_{\\alpha} : \\alpha \\in \\Lambda, \\text{an indexing set} \\} \\text{(say)}$. Also the collection $\\{\\overline{S_{\\alpha}} : S_{\\alpha} \\in S\\}$ is locally finite by previous lemma. Now for $\\alpha \\in \\Lambda$, $S_{\\alpha}\\subset P_{x} \\subset U_{x}$ for some $P_{x} \\in P$ and hence $\\overline{S_{\\alpha}} \\subset \\overline{P_{x}} \\subset U_{x}$ for some $U_{x} \\in \\mathcal{U}$. Therefore $S$ is a locally finite refinement of $\\mathcal{U}$ such that $\\{\\overline{S_{\\alpha}} : S_{\\alpha} \\in S \\}$ is also a locally finite refinement of $\\mathcal{U}$.\\\\\n\\end{proof}\nWe have discussed some results associated with paracompactness in a $\\sigma$-space because our motivation was to establish the statement ``If $(X, \\mathcal{P}_{1}, \\mathcal{P}_{2})$ is a pairwise paracompact bispace with $(X, \\mathcal{P}_{2})$ regular, then every $\\mathcal{P}_{1}$-$F_{\\sigma}$ proper subset is $\\mathcal{P}_{2}$ paracompact''. This has been discussed in a bitopological space \\cite{MKBOSE}. But we have failed due to the fact that arbitrary union of open sets in a $\\sigma$-space may not be open.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nIn 1964, physicist Richard Feynman famously remarked ``\\textit{that all things are made of atoms and that everything that living things do can be understood in terms of the jigglings and wigglings of atoms}''.~\\cite{feynman1963feynman} As such, an atomically resolved picture can provide invaluable insights on biological (and other) processes. The first molecular dynamics (MD) study of a protein in 1977 by \\citet{mccammon1977dynamics} did not consider explicit solvent molecules and was limited to less than 10~ps of simulation.\nStill, it challenged the (at that time) common belief that proteins are essentially rigid structures\\cite{phillips1981biomolecular} and, instead, suggested that the interior of proteins behaves more fluid-like. \nSince then, systems consisting of more than a million atoms have been studied\\cite{schulz2009scaling}, simulation times extended to the millisecond regime\\cite{shaw2008anton}, and the study of entire viruses in atomic detail made possible\\cite{freddolino2006molecular,zhao2013mature}.\n\nTo perform MD simulations, typically, the Newtonian equations of motion are integrated numerically, which requires knowledge of the forces acting on individual atoms at each time step of the simulation\\cite{karplus2002molecular}. In principle, the most accurate way to obtain these forces is by solving the Schr\\\"odinger equation (SE), which describes the physical laws underlying most chemical phenomena and processes\\cite{dirac1929quantum}. Unfortunately, a solution in closed form is only possible for very small systems. For larger chemical structures, the SE can only be solved approximately. However, even with approximations, an accurate numerical solution is a computationally demanding task. For example, the CCSD(T) method, which is widely regarded as the ``gold standard'' of chemistry (as its predictions compare well with experimental results)\\cite{gordon2005theory}, scales as $\\propto N^7$ with system size $N$. Because of this, it is unfeasible to calculate the forces for many different configurations of large chemical systems, which is required for running MD simulations, with accurate methods. Instead, simple empirical functions are commonly used to model the relevant interactions. From these so-called force fields (FFs), atomic forces can be readily derived analytically. \n\nMost conventional FFs model chemical interactions as a sum over bonded and non-bonded terms\\cite{gonzalez2011force,unke2020high}. The former can be described with simple functions of the distances between directly bonded atoms, or angles and dihedrals between atoms sharing some of their bonding partners. The non-bonded terms consider pair-wise combinations of atoms, typically by modeling electrostatics with Coulomb's law (assuming a point charge at each atom's position) and dispersion with a Lennard-Jones potential\\cite{jones1924determination}. Due to the computational efficiency of these terms, such classical FFs allow to study systems consisting of many thousands of atoms. However, while offering a qualitatively reasonable description of chemical interactions, the quality of MD simulations, and hence the insights that can be obtained from them, are ultimately limited by the accuracy of the underlying FF.\\cite{vitalini2015dynamic} This is particularly problematic when polarization, or many-body interactions in general, are of major importance, as these effects are not adequately modeled by the terms described above. While it is possible to improve an FF's accuracy (in exchange for computational efficiency) by explicitly accounting for such effects\\cite{rasmussen2007force,shi2013polarizable,lopes2013polarizable,unke2017minimal}, it is not always clear \\textit{a priori} when this is necessary. Beyond that, conventional FFs require a preconceived notion of bonding patterns and thus cannot describe bond breaking or bond formation. While there exist reactive FFs offering an approximated description of reactions\\cite{warshel1980empirical,van2001reaxff,nagy2014multisurface}, they are often not sufficiently accurate for quantitative studies or restricted to specific types of reactions. Mixed quantum mechanics\/molecular mechanics (QM\/MM) treatments\\cite{senn2009qm} pose an alternative solution: Here, the SE is only solved for a small part of the system where high accuracy is required or reactions may take place, e.g.\\ the active site of an enzyme. Meanwhile, all remaining atoms are treated at the FF level of accuracy. Although this is more efficient than a pure quantum-mechanical approach, it is often necessary to model a large number of atoms at the QM level for converged results\\cite{kulik2016large}, which is still highly computationally demanding.\\\\\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{overview.png}\n\t\\caption{Accurate \\textit{ab initio} methods are computationally demanding and can only be used to study small systems in gas phase or regular periodic materials. Larger molecules in solution, such as proteins, are typically modeled by force fields, empirical functions which trade accuracy for computational efficiency. Machine learning methods are closing this gap and allow to study increasingly large chemical systems at \\textit{ab initio} accuracy with force field efficiency.}\n\t\\label{fig:overview}\n\\end{figure*}\n\nIn a ``dream scenario'' for computational chemists and biologists, it would be possible to treat even large systems with the CCSD(T) method, which would require prohibitively large computational resources in the real world. Machine learning (ML) methods could help to achieve this dream in a principled manner, thus closing the gap between the accuracy of \\textit{ab initio} methods and the efficiency of classical FFs (Fig.~\\ref{fig:overview}). ML methods aim to learn the functional relationship between inputs (chemical descriptors) and outputs (properties) from patterns or structure in the data. Ideally, a trained learning machine would then reflect the underlying effective ``rules'' of quantum mechanics.\\cite{schutt2020machine} Practically, ML models can take a shortcut by not having to solve any equations that follow from the physical laws governing the structure-property relation. Because of this unique ability, ML methods have been enjoying growing popularity in the chemical sciences in recent years. They allow to explore chemical space and predict the properties of compounds with unprecedented efficiency and simultaneously with high accuracy.\\cite{rupp2012fast,montavon2013machine,hansen2013assessment,hansen2015machine,von2020exploring,schutt2020machine} ML has also been used to augment and accelerate traditional methods used in molecular simulations, e.g.\\ for sampling equilibrium states\\cite{noe2019boltzmann,kohler2020equivariant} and rare events\\cite{zhang2019targeted}, computing reaction rates\\cite{koner2019exhaustive}, exploring protein folding dynamics\\cite{noe2020machine} and other biophysical processes\\cite{sonderby2015convolutional,almagro2017deeploc,botlani2018machine,boninsegna2018sparse,senior2020improved}, Markov state modeling\\cite{scherer2015pyemma,mardt2018vampnets,wehmeyer2018time,wu2018deep,chen2019nonlinear,klus2019kernel,sidky2019high}, and coarse-graining\\cite{chen2018learning,wang2019machine,nuske2019coarse,wang2020ensemble} (for a recent review on applications of ML in molecular simulatons, see Ref.~\\citenum{noe2020machinemolsim}). Recent advances made it even possible to predict molecular wave functions, which can act as an interface between ML and quantum chemistry\\cite{schutt2019unifying,gastegger2020deep}, as knowledge of the wave function allows to deduce many different quantum mechanical observables at once. ML can also be combined with more traditional semi-empirical methods, for example by predicting accurate repulsive potentials for density functional tight-binding approaches.\\cite{stohr2020accurate} Instead of circumventing equations, ML methods can also help when solving them: They have been used to find novel density functionals\\cite{snyder2012finding,brockherde2017bypassing,bogojeski2019density} and solutions of the Schr\\\"odinger equation\\cite{hermann2019deep,carleo2017solving}. Other promising applications include the generation of molecular structures to tackle inverse design problems,\\cite{gebauer2018generating,gebauer2019symmetry,hoffmann2019generating,winter2019efficient,simm2020reinforcement} or planning chemical syntheses\\cite{strieth2020machine}.\n\nFor constructing ML-FFs, suitable reference data to learn the relevant structure-property relation include energy, forces, or a combination of both, obtained from \\textit{ab initio} calculations. Contrary to conventional FFs, no preconceived notion of bonding patterns needs to be assumed. Instead, all chemical behavior is learned from the reference data. This allows to reconstruct the important interactions purely from atomic positions without imposing a restricted analytical form on the interatomic potential and enables a natural description of chemical reactions. For example, it is now possible to construct ML-FFs for small molecules from CCSD(T) reference data close to spectroscopic accuracy and with computational efficiency similar to conventional FFs\\cite{chmiela2018, sauceda2019molecular}. While this review focuses on FFs, i.e.\\ the prediction of energies and forces, other properties of chemical systems can be predicted as well. For example, dipole moments, which are a measure for how polar molecules are, can be used to calculate infrared spectra from MD simulations\\cite{gastegger2017machine, christensen2019operators,kaser2020reactive} and allow a comparison to experimental measurements. Other prediction targets could be used to screen large compound databases for molecules with desirable properties several orders of magnitude faster than with \\textit{ab initio} methods. The HOMO\/LUMO gap, which is important for materials used in electronic devices such as solar cells\\cite{agnihotri2014computational}, is only one prominent example of many potentially interesting prediction targets.\n\nThis review will focus on the construction of ML-FFs for the usage in MD simulations and other applications (for details on how to set up such simulations or how to extract physical insights from them, refer to Refs.~\\citenum{payne1986molecular,adcock2006molecular,paquet2015molecular}). The text is structured as follows: Section~\\ref{sec:applications_of_machine_learned_force_fields} lists several applications of ML-FFs described in the literature and highlights physical and chemical insights made possible through the use of ML, which could not have been obtained by conventional methods. Next, Section~\\ref{sec:mathematical_and_coceptual_framework} reviews fundamental concepts of chemistry (\\ref{subsec:chemistry_foundations}) and machine learning (\\ref{subsec:machine_learning_foundations}) relevant to the construction of ML-FFs and discusses special considerations when the two are combined (\\ref{subsec:combining_ml_and_chemistry}). It provides readers new to these concepts with the necessary background to understand the remainder of the review. The next part (Section~\\ref{sec:best_practices_and_pitfalls}) serves as a step-by-step guide and reference for readers that want to apply ML-FFs in their own research. Here, the best practices for constructing ML-FFs are outlined, possible problems that may be encountered along the way (and how to avoid them) are discussed and an overview of several software packages, which may be used to accelerate the construction of ML-FFs, is provided. The review is concluded in Section~\\ref{sec:challenges} by a discussion of obstacles that still need to be overcome to extend the applicability of ML-FFs to an even broader context.\n\n\\section{Physical and Chemical Insights from Machine Learned Force Fields}\n\\label{sec:applications_of_machine_learned_force_fields}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{fig_applications.pdf}\n\\caption{ML-FFs combine the accuracy of \\textit{ab initio} methods and the efficiency of classical FFs. They provide easy access to a system's potential energy surface (PES), which can in turn be used to derive a plethora of other quantities.\nBy using them to run MD simulations on a single PES, ML-FFs allow chemical insights inaccessible to other methods (see gray box). For example, they accurately model electronic effects and their influence on thermodynamic observables and allow a natural description of chemical reactions, which is difficult or even impossible with conventional FFs. \nTheir efficiency also allows them to be applied in situations where the Born--Oppenheimer approximation begins to break down and a single PES no longer provides an adequate description. An example is the study of nuclear quantum effects and electronically excited states (upper right).\nFinally, ML-FFs can be further enhanced by modeling additional properties. This provides direct access to a wide range of molecular spectra, building a bridge between theory and experiment (lower right).\nIn general, such studies would be prohibitively expensive with \\textit{ab initio} methods.}\n\\label{fig:achieved}\n\\end{figure*}\n\nIn nature, the atoms in chemical systems are in constant motion, giving rise to various configurations and reactive events. A large number of experimental observations are not based on a single molecule or atom, but instead on ensembles of various species subject to external conditions, such as temperature or pressure. Consequently, properties associated with individual structures are not sufficient to characterize macroscopic systems. One way to compute ensemble averages are molecular dynamics (MD) simulations, where the time evolution of a system is governed by the atomic forces derived from its associated potential energy surface (PES). From the ergodic hypothesis\\cite{boltzmann2017vorlesungen} it is known that the expected value of an observable $A$ can also be obtained from the time average $\\langle A\\rangle_{t}=N_t^{-1}\\sum_{t}^{N_t}{A_t}$, where $A_t$ is the value of $A$ corresponding to the structure at time step $t$ of the dynamics trajectory and $N_t$ is their total number. Of course, this relation is valid only when the dynamics is long enough to visit all configurations of the system accessible under the simulation conditions. \n\t\nIn order to obtain meaningful statistics with MD simulations, many thousands (or millions) of successive PES evaluations are necessary. Due to their high computational cost, accurate electronic structure PESs quickly become intractable for such simulations, which is why highly efficient classical force fields (FFs) are usually employed for running MD simulations. However, this efficiency comes at a cost: Conventional FFs completely neglect or misrepresent some potentially relevant contributions to the potential energy, such as polarization, charge transfer, or electronic effects, which limits their usefulness in modeling complex chemical phenomena. Machine-learned FFs (ML-FFs) offer a unique combination of computational efficiency and high accuracy, opening up tantalizing new possibilities in the simulation of the dynamics of molecules, surfaces, materials and condensed phases. They are able to model \\emph{all} chemical interactions -- including those that are typically neglected by conventional FFs. The high accuracy of ML-FFs allows to obtain qualitatively different and novel insights, which would otherwise only be accessible from computationally infeasible \\emph{ab initio} MD (AIMD) simulations. \nIn the following, some chemical insights made possible by ML-FFs, which could not have been obtained with conventional FFs, are highlighted in greater detail. A brief overview is given in Table~\\ref{tab:overview_applications}.\n\n\\begin{table*}\n\\resizebox{0.9\\textwidth}{!}{%\n\\begin{tabular}{lllrl}\n\\toprule\nCategory                         & Reference                                                      & ML-FF  & Max. $N_\\mathrm{atoms}$ & Reference theory \\\\\n\n\\midrule\n\\multirow{3}{*}{Electronic Effects}      & Sauceda \\emph{et. al.}\\cite{sauceda2019molecular}      & sGDML       & 21      & CCSD(T), CCSD      \\\\\t\n\t\t                                 & Sauceda \\emph{et. al.}\\cite{sauceda2020molecular}      & sGDML       & 21      & CCSD(T), CCSD      \\\\\t\t\n\t\t                                 & Sauceda \\emph{et. al.}\\cite{chmiela2018}               & sGDML       & 21      & PBE, CCSD(T), CCSD \\\\\n\\midrule\n\\multirow{8}{*}{Thermodynamics}          & Morawietz \\emph{et. al.}\\cite{morawietz2016van}        & BP-NNP      & 6\\,912  & RPBE, BLYP \\\\\n                                         & Andrade \\emph{et. al.}\\cite{andrade2020free}           & DeepMD      & 426     & SCAN \\\\                              \n                                         & Deringer \\emph{et. al.}\\cite{deringer2017machine}      & GAP         & 1\\,000  & LDA \\\\\n                                         & Behler \\emph{et. al.}\\cite{behler2008pressure}         & BP-NNP      & 64      & LDA \\\\\n                                         & Bartok \\emph{et. al.}\\cite{bartok2018machine}          & GAP         & 23\\,496 & PW91 \\\\\n                                         & Deringer \\emph{et. al.}\\cite{deringer2018realistic}    & GAP         & 4\\,096  & PW91 \\\\                                \n                                         & Bonati \\emph{et. al.}\\cite{bonati2018silicon}          & DeepMD      & 680     & SCAN \\\\\n                                         & Brickel \\emph{et. al.}\\cite{brickel2019reactive}       & PhysNet     & 6       & MP2 \\\\\n\\midrule\n\\multirow{7}{*}{Reactions}               & Unke \\emph{et. al.}\\cite{unke2016collision}            & RKHS        & 3       & UCCSD(T) \\\\\n\t\t\t\t\t\t\t\t\t     & Denis \\emph{et. al.}\\cite{denis2017quantum}            & RKHS        & 3       & UCCSD(T)-F12a \\\\\n\t\t\t\t\t\t\t\t\t     & Lu \\emph{et. al.}\\cite{lu2020comprehensive}            & PIP-NN      & 7       & UCCSD(T)-F12a \\\\\n\t\t\t\t\t\t\t\t\t     & Sweeny \\emph{et. al.}\\cite{sweeny2020thermal}          & PhysNet     & 7       & MP2 \\\\\n\t\t\t\t\t\t\t\t\t     & K{\\\"a}ser \\emph{et. al.}\\cite{kaser2020isomerization}  & PhysNet     & 7       & MP2 \\\\\n\t\t\t\t\t\t\t\t\t     & Rivero \\emph{et. al.}\\cite{rivero2019reactive}         & PhysNet     & 19      & M06-2X \\\\\n\t\t\t\t\t\t\t\t\t     & Liu \\emph{et. al.}\\cite{liu2018constructing}           & BP-NNP      & 38      & RPBE \\\\\n\\midrule\n\\multirow{5}{*}{Nuclear Quantum Effects} & Chmiela \\emph{et. al.}\\cite{chmiela2017}               & GDML        & 21      & PBE \\\\\n                                         & Chmiela \\emph{et. al.}\\cite{chmiela2018}               & sGDML       & 21      & CCSD, CCSD(T) \\\\\n                                         & Sch{\\\"u}tt \\emph{et. al.}\\cite{schutt2018}             & SchNet      & 20      & PBE \\\\\n                                         & Sauceda \\emph{et. al.}\\cite{sauceda2020}               & sGDML       & 21      & CCSD, CCSD(T) \\\\\n                                         & Hellstr{\\\"o}m \\emph{et. al.}\\cite{hellstrom2018nuclear}& BP-NNP      & 1\\,700  & RPBE \\\\\n\\midrule\n\\multirow{3}{*}{Excited States}          & Chen \\emph{et. al.}\\cite{chen2018deep}                 & HDNN        & 5       & CASSCF \\\\\n                                         & Westermayr \\emph{et. al.}\\cite{westermayr2019machine}  & NN          & 6       & MR-CISD \\\\\n                                         & Westermayr \\emph{et. al.}\\cite{westermayr2020combining}& SchNet      & 6       & MR-CISD, CASSCF \\\\\n\\midrule   \n\\multirow{4}{*}{Spectroscopy}            & Gastegger \\emph{et. al.}\\cite{gastegger2017machine}    & BP-NNP      & 209     & BLYP, BP86, B2PLYP \\\\\n                                         & Yao \\emph{et. al.}\\cite{yao2018tensormol}              & BP-NNP      & 60      & $\\omega$B97X-D \\\\\n                                         & Raimbault \\emph{et. al.}\\cite{raimbault2019using}      & SOAP        & 80      & PBE \\\\\n                                         & Sommers \\emph{et. al.}\\cite{sommers2020raman}          & DeepMD      & 512     & SCAN \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{Overview of different topics and applications of ML-FFs discussed in this section. In all cases, the type of employed ML-FF is given along with the number of atoms of the largest system used to study the respective phenomenon. The basic level of reference theory (neglecting basis sets and dispersion corrections for clarity) are also reported.}\n\\label{tab:overview_applications}\n\\end{table*}\n\t\n\\paragraph{Electronic Effects}\nA good example for the power of ML-FFs is a recent study of the dynamics of small molecules (malondialdehyde, ethanol, salicylic acid, paracetamol, aspirin) with atomic forces at CCSD(T) quality.\\cite{sauceda2019molecular} AIMD simulations were run at 500~K at the DFT\/PBE+TS level of theory\\cite{PBE1996,tkatchenko2009accurate} and the collected configurations randomly subsampled to calculate energies and forces at the CCSD(T) level of theory (reference data for aspirin was calculated at CCSD accuracy). For each molecule, an ML-FF was constructed from 1000 data points with the sGDML\\cite{chmiela2019} method (see Section~\\ref{subsubsec:symmetric_gradient_domain_machine_learning}) and used to run MD simulations at 300~K. Running simulations of this quality with \\textit{ab initio} methods is impossible, as they would require up to a billion times more computation time. Conventional FFs were shown to be no viable alternative to ML-FFs, as they do not adequately describe, or even completely neglect, effects which strongly influence the dynamics -- and hence the properties -- of the studied molecules (Fig.~\\ref{fig:notmodeledbyffs}).\n\t\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\columnwidth]{notmodeledbyff.png}\n\\caption{Visualization of electronic effects, which are accurately modeled by ML-FFs, but neglected by conventional FFs. Electron lone pairs, hybridization changes and orbital donation effects all influence the dynamics of molecules and hence the properties that are computed from MD simulations. When predicting, e.g.\\ free energy surfaces (FESs) or molecular spectra, neglecting them will lead to qualitatively different results.}\n\\label{fig:notmodeledbyffs}\n\\end{figure}\n\t\nFor example, in ethanol, the lone pairs of the oxygen atom interact with the partially positively charged hydrogen atoms of the methyl group. Due to this attraction, the configuration where both lone pairs are adjacent to a hydrogen atom is visited most frequently during a dynamics simulation. Any derived property, e.g.\\ the free energy surface (FES) or the infrared spectrum, is only accurate when this effect is properly described. Conventional FFs do not account for lone pairs and are thus unable to predict the molecular properties correctly. \n\t\nA similar effect can be observed in malondialdehyde. Here, the lone pairs of the two oxygen atoms strongly repel each other, which drives the dynamics away from configurations where they are close. While conventional FFs can crudely model electrostatic repulsion between the oxygen atoms with negative partial charges, the steric contributions from the overlap of the electron clouds is not described, causing a qualitatively different dynamics.\n\t\nParacetamol is another molecule where lone pairs influence the stability of specific configurations: The partially positively charged phenyl-hydrogen adjacent to the oxygen atom of the acetamide group interacts with its lone pairs and favors a specific dihedral angle. Additionally, the nitrogen atom of the acetamide group is $sp^2$ hybridized, which allows conjugation to the electrons in the phenyl system and leads to the planar geometry of paracetamol. When the nitrogen hybridization state is changed to $sp^3$, the energetically favorable interaction is broken and corresponding configurations are thus rarely visited during room temperature dynamics. However, at higher temperatures, the hybridization state may switch frequently -- conventional FFs are unable to describe this.\n\t\nAnother important electronic effect can be observed in aspirin. Here, an occupied (lone pair) $n$ orbital of the carbonyl group overlaps with an unoccupied antibonding $\\pi^*$ orbital in the ester group. This $n\\rightarrow\\pi^*$ interaction dictates the relative arrangement of these functional groups in the global minimum structure.\\cite{sauceda2020molecular} The effect is even amplified during dynamics, since thermal fluctuations enhance the overlap.\\cite{chmiela2018} \n\nThese and many other electronic effects, e.g.\\  $n\\rightarrow\\sigma^*$ interactions, hyperconjugation, and Jahn-Teller distortions are captured automatically by ML-FFs. In contrast, including them in conventional FFs would require additional terms, whose functional form (and even which effects need to be modeled) are typically unknown \\textit{a priori}.\n\t\n\\paragraph{Thermodynamics}\nA typical application of classical FFs is the study of thermodynamic properties of bulk systems, such as enthalpies, entropies, and phase diagrams. However, their limited accuracy is a major obstacle for quantitative predictions, as small inaccuracies in the interaction of a few particles will inevitably lead to big discrepancies when studying many particles. A good example for this are van der Waals (vdW) interactions. They are weak contributions to the total potential energy for small molecules in gas phase, but they add up in large condensed systems and bulk materials and can strongly influence their properties and dynamics.\\cite{hermann2017first} While conventional FFs account for vdW interactions, they typically do so with a relatively crude model based on the Lennard-Jones potential\\cite{jones1924determination}, which is insufficient for quantitative predictions in many cases. A prime example is water: It is the most studied liquid in literature and many different conventional FFs for water (some with additional special-purpose terms) have been proposed in the last decades, yet none of them is able to reproduce all experimentally measured properties of water in MD simulations\\cite{guillot2002reappraisal}.\n\t\nHere, ML-FFs offer a promising alternative. \\citet{morawietz2016van} trained a descriptor-based NNP (see Section~\\ref{subsubsec:high_dimensional_neural_network_potentials}) on periodic configurations of liquid and crystalline water, for which reference data was calculated with different DFT functionals.\nMD simulations with the ML-FF revealed that the thermodynamic anomalies of water, such as its density maximum and negative volume of melting, are due to a delicate balance of weak vdW forces. The study was able to accurately predict experimentally measured radial distribution functions, as well as temperature dependent shear viscosities and diffusion coefficients. As ML-FFs are naturally able to describe bond breaking and formation, the study could even investigate proton transfer between different water molecules. \n\t\nThe ability to analyze thermodynamic properties of reactive events is a major advantage of ML-FFs over conventional methods. For example, a recent study investigated the free energy of proton transfer in liquid water at a titanium oxide surface.\\cite{andrade2020free} A descriptor-based NNP was trained using reference data collected through an adaptive sampling approach (see Section~\\ref{subsec:data_collection}) and used to run MD simulations. The study revealed that a significant fraction of water molecules forms short-lived hydroxyl groups on the titanium oxide surface, which strongly influence its surface chemistry. Such insights are key to understanding phenomena such as surface functionalization and photocatalytic processes.\n\t\nAnother application where the flexibility of ML-FFs is a major advantage is the modeling of bulk materials. For example, Gaussian approximation potentials (GAPs, see Section~\\ref{subsubsec:gaussian_approximation_potentials}) and NNPs have been constructed for elemental carbon\\cite{deringer2017machine} and silicon\\cite{behler2008pressure,bartok2018machine,deringer2018realistic,bonati2018silicon}. They allow to investigate a wide range of phenomena of liquid, crystalline, and amorphous solid phases, including defects and crack propagation. Modeling these effects accurately is only possible with ML-FFs or prohibitively expensive AIMD simulations. It is even possible to predict accurate phase diagrams of such systems with ML-FFs\\cite{behler2008pressure,bartok2018machine}. Since this requires a correct model of bond formation and breaking, as well as changes of bonding patterns, such insights could not be obtained from conventional FFs. \n\t\n\\paragraph{Reactions}\nOne of the most significant advantages of ML-FFs over conventional FFs is their natural ability to model chemical reactions. Even in cases where it is possible to construct special purpose classical FFs that are able to describe reactions, they are typically much less accurate than their ML-FF counterparts. For example, a recent study compared an ML-FF constructed with a message-passing NNP (see Section~\\ref{subsubsec:high_dimensional_neural_network_potentials}) with two classical methods to obtain a reactive FF for the Cl--CH$_3$--Br transformation.\\cite{brickel2019reactive} Here, the ML-FF achieved up to three orders of magnitude lower errors and yielded qualitatively and quantitatively different predictions for the free energy surface along the reaction path.  \nIt is therefore no surprise that one of the first fields where ML-FFs were employed with great success are reaction dynamics. Here, the chemical transformations associated with molecular collisions over short time and length scales are studied. These simulations offer detailed atomistic insights into the reaction mechanism, providing access to rate constants and scattering cross sections, as well as insights on how the molecular energy is distributed between different modes, all of which can be directly related to experiments. In order to yield quantitative predictions, sufficient statistics and highly accurate PESs are required, making them an excellent application for ML-FFs. Studies typically involve small molecular systems, which are treated at high levels of accuracy, such as the collision of N$_2^+$ and Ar\\cite{unke2016collision,denis2017quantum} or the Cl+CH$_3$OH~$\\rightarrow$~HCl+CH$_3$O\/CH$_2$OH reaction\\cite{lu2020comprehensive}. Typical conventional FFs require fixed bonding patterns and are thus intrinsically unsuited for studying chemical reactions. While there also exist reactive variants of classical FFs, they do not reach the accuracy of ML-FFs. For example, a recent study investigated the thermal activation of methane by MgO$^+$ with a message-passing NNP (see Section~\\ref{subsubsec:high_dimensional_neural_network_potentials}) and a reactive classical FF.\\cite{sweeny2020thermal} Here, the ML-FF achieved prediction errors up to two orders of magnitude lower than the classical variant compared to \\textit{ab initio} data. In addition, the disagreement between experimental rate constants and those predicted from MD simulations was lower by a factor of two with the ML-FF compared to the values obtained from the classical FF. The remaining discrepancy between prediction and experiment was further investigated and it was determined that the deviation was not due to inaccuracies of the ML-FF \\textit{per se}, but instead could be traced back to the multireference character of the transition state, i.e.\\ problems with the \\textit{ab initio} reference data itself.\n\t\nEven though it is possible to construct classical reactive FFs for specific reactions, there are cases where this is exceedingly difficult. A good example is a recent study where the photo-tautomerization reaction of acetaldehyde was investigated, which is speculated to be a major pathway for formic acid formation in the atmosphere\\cite{kaser2020isomerization}. After being photo-excited, acetaldehyde contains enough energy that it may not only tautomerize to ethenol, but also dissociate into carbon monoxide and methane, or into hydrogen and ethenone. An accurate description of all three possible reaction pathways with the same FF is extremely difficult to achieve with conventional methods. The NNP used for the study on the other hand was trained on MP2\/aug-cc-pVTZ\\cite{moller1934note,dunning1989gaussian} reference data and allowed an unbiased description of all relevant processes at \\textit{ab initio} quality. Analyzing a total of 12000 individual trajectories, the study concluded that the formation of ethenol from photo-tautomerization of acetaldehyde is unlikely under atomospheric conditions. This insight could not have been obtained by running AIMD simulations in a reasonable time frame: The combined simulation time of 1~\\textmu s would amount to ten billion single point calculations (a time step of $\\Delta t=0.1$~fs was used due to the large excitation energies). In contrast, less than 500k structures were used for training the ML-FF, i.e.\\ the time spent for running \\textit{ab initio} calculations was reduced by more than five orders of magnitude by employing an ML-FF.\n\t\nDue to the efficiency of ML-FFs, scattering simulations can now even be extended to involve larger organic molecules. For example, a study of the minimum dynamic path\\cite{unke2019sampling} of Diels-Alder reactions of 1,3-dibromo-1,3-butadiene and maleic anhydride with an end-to-end NNP has revealed that molecular rotations are a major driving force for the formation of products\\cite{rivero2019reactive}, an effect which had not been described previously in the literature for this type of reaction. ML-FFs can even be applied to reactions between molecules and surfaces. For example, a study by \\citet{liu2018constructing} investigated (reactive) HCl scattering on a gold surface using a descriptor-based NNPs.\n\n\\paragraph{Nuclear Quantum Effects}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=\\textwidth]{Fig_PIMD.pdf}\n\\caption{\n\\textbf{A)} Schematic description of the path integral (ring polymer) molecular dynamics (PIMD) method, where quantum particles are approximated by a classical ring polymer with $P$ beads. There is an exact isomorphism between these two systems when $P\\to\\infty$, i.e.\\ their statistical properties become equivalent. \\textbf{B)} PIMD simulations using DFT or coupled cluster calculations. 1: Coupled cluster PIMD simulations of the Zundel model to compute $^1H$ magnetic shielding tensor (adapted from Ref.~\\citenum{Spura_CC-PIMD_PCCP2015}). 2: Example of hydrogen-bond networks and their NQE implications on biological functions and enzyme catalysis (adapted from Ref.~\\citenum{Wang_Ketosteroid-PIMD_JPCB2017}). 3: IR spectrum of the porphycen molecule computed from PIMD simulations (adapted from Ref.~\\citenum{Litman_Porphycen-PIMD_JACS2019}). \\textbf{C)} PIMD simulations using ML-FFs trained on DFT or coupled cluster data. 1: Ultra-low temperature dynamics of the Zundel model obtained from PIMD simulations (adapted from Ref.~\\citenum{Schran_Zundel-PIMD_JCTC2018}). 2: Comparison of the statistical sampling of different conformers of ethanol between experiment and simulations (adapted from Ref.~\\citenum{chmiela2018}). 3: Schematic description of the enhancement in intra- and inter-molecular interactions due to NQEs (adapted from Ref.~\\citenum{sauceda2020}).\n}\n\\label{fig:pimd}\n\\end{figure*}\n\nPredictive simulations of molecular systems and materials require not only highly accurate representations of the potential energy surface (PES), but also appropriate statistical sampling of the PES. While classical MD simulations are sufficient for this in some cases, the quantum nature of nuclei plays an important role in many systems. Nuclear quantum fluctuations are a fundamental phenomenon in nature resulting from Heisenberg's uncertainty principle\\cite{heisenberg1985anschaulichen}, hence physical and chemical properties of molecular or biological systems, as well as nano- and bulk-materials, may be affected by them up to certain extent. In particular, light elements, such as protons and atoms in the first row of the periodic table, are prone to display nuclear quantum effects (NQEs) even at room temperature. Furthermore, materials or molecules formed by heavier atoms, but having strong bonds or being at low temperatures, exhibit significant NQEs\\cite{Merchant1973,Kirchner2000,NanotubeCTE2004,ThermalExpSi2018,ThermalExpZnSb2015,Poltavsky-graphene,NQEreview2018,Freitas_disloc-PIMD_npjCM2018}.\n\nConsequently, in order to generate predictive simulations of many physical properties, NQEs must be incorporated. A widely used methodology to perform quantum dynamics is path integral molecular dynamics (PIMD). This method is based on the isomorphism between a quantum particle and a classical harmonic ring polymer of $P$ beads (i.e.\\ $P$ harmonically-coupled copies of the particle), where the equality holds for $P\\to\\infty$ (see Fig.~\\ref{fig:pimd}A)\\cite{chandler1981exploiting}. In practice, convergence of thermodynamical properties can be achieved using only a small number of beads. For light atoms at room temperature for example, $P\\sim$16--32 is often sufficient to converge mechanical properties\\cite{chmiela2019,Spura_CC-PIMD_PCCP2015,Litman_Porphycen-PIMD_JACS2019,Poltavsky-graphene}. This number can be reduced even further by using more sophisticated thermodynamic estimators\\cite{pPI_2018}.\n\nGiven that PIMD simulations require energies and forces for $P$ copies of the system of interest, it is infeasible to use \\textit{ab initio} methods to derive them in most cases. There are some exceptions: For example, PIMD simulations to study the IR spectrum of the porphycen molecule have been performed using DFT with the B3LYP functional and it was shown that the correct free energy and vibrational spectrum can only be recovered by considering NQEs (see Fig.~\\ref{fig:pimd}B:3)\\cite{Litman_Porphycen-PIMD_JACS2019}. Another example are PIMD simulations of the Zundel model at the CCSD level of theory to study the impact of NQEs on its structure and the $^1$H magnetic shielding tensor\\cite{Spura_CC-PIMD_PCCP2015}. However, \nboth of these studies required supercomputers to make the calculations possible in a reasonable time frame. On the other hand, ML-FFs can replicate the same results at a fraction of the computational cost, i.e.\\ speed-ups by a factor of 10$^5$--10$^7$ (depending on the reference level of theory) can be achieved\\cite{schutt2017schnet,chmiela2018}.\nThis gain in computational efficiency makes it possible to run PIMD simulations for a wide range of systems and offers the chance to reveal new chemical and physical insights.\n\nFor example, \\citet{chmiela2017} performed room temperature PIMD simulations of aspirin using a GDML model (see Section~\\ref{subsubsec:sgdml_package}) trained on PBE+TS\\cite{PBE1996,tkatchenko2009accurate} reference data to investigate the paths followed between different minima on its PES. In a followup study, \\citet{chmiela2018} compared free energies and vibrational density of states of a variety of medium-sized molecules obtained from PIMD simulations with a model trained on CCSD or CCSD(T) reference data to the same quantities obtained from a model trained on DFT\/PBE+TS data. The authors found that even though the PESs at the two different levels of theory are very similar, tiny differences may still lead to largely different free energies. Additionally, it was shown that the experimentally determined populations for different conformations of ethanol can only be recovered from simulations when including NQEs (see Fig.~\\ref{fig:pimd}C:2).\n\nIn another study, \\citet{schutt2018} investigated the dynamics of C$_{20}$ fullerene using a NNP trained on DFT\/PBE+TS reference data. Here, including NQEs broadens the radial distribution function significantly, which also increases the molecular polarizability\\cite{sauceda2020}. A change in the distribution of interatomic distances also influences electronic effects: A follow-up study of Ref.~\\citenum{sauceda2019molecular} (mentioned earlier in the paragraph on electronic effects) investigated NQEs in small organic molecules\\cite{sauceda2020}. The study revealed that NQEs can dynamically strengthen molecular interactions by enhancing $n\\rightarrow\\pi^*$ donation through increasing orbital overlap, or by strengthening electrostatic interactions between neighboring charge densities (see Fig.~\\ref{fig:pimd}C:3). \nAnother interesting observed effect is a temporary change of bond orders, which can lead to emerging localized transient states of methyl rotors. The study also showed that vdW interactions are strengthened by NQEs: Since interatomic distances expand on average due to thermal and quantum dilations, the molecular polarizability is also increased (see Fig.~\\ref{fig:pimd}C:3). Other observed implications of NQEs include ``bonding'' between hydroxyl groups and hindered rotor dynamics, which leads to molecular stiffening and smoother free energy surfaces. \n\nML-FFs also make it possible to go far beyond the system size accessible with standard electronic structure methods. In Ref.~\\citenum{hellstrom2018nuclear}, a descriptor-based NNP was used to study the influence of NQEs on aqueous NaOH solutions of different concentrations ($\\sim$1000 atoms).  It could be shown that NQEs exert a subtle influence on the solvation structure in the Na$^+$ environment and significantly increase the proton transfer rates and hence diffusion coefficients of the different species. The accuracy of the ML-FF also made it possible to identify error cancellation effects in the reference method, leading to artificially good agreement with experiment in the absence of NQEs.\n\n\\paragraph{Excited States}\nThe Born--Oppenheimer approximation breaks down when modeling the dynamics of molecular excited states, which are essential for understanding photochemical processes. An extension to classical MD, which allows for the simulation of such phenomena, is quantum-classical surface hopping MD.\nIn this approach, the excited state dynamics of a molecule are simulated by letting it evolve on a set of PESs associated with the different electronic states.\nTo describe the distribution of the molecule between the different states, the effective PES governing the time evolution changes according to stochastic criteria, e.g.\\ based on coupling terms between the relevant states.\nThe correct quantum statistics are then recovered from multiple independent simulations.\nThese simulations are computationally intensive, as they do not only require the computation of multiple PES, but also different coupling terms.\nThis is further amplified by the need for a large number of trajectories in order to obtain reliable statistics. As such, quantum-classical surface hopping simulations can profit greatly from the efficiency and versatility of ML-FFs.\n\t\nIn Ref.~\\citenum{chen2018deep}, for example, the authors used descriptor-based NNPs to study the excited state dynamics of the methylene imine molecule, as well as regions close to the conical intersection between the singlet ground and excited states. It could be shown that the NNPs are able to recover the effective PES with high accuracy and allow for efficient simulations to estimate the state populations of the system. Here, the coupling between the different surfaces was computed based on the Zhu\u2013Nakamura approximation\\cite{zhu1993two}, which relies on the energy differences between states. \nMore accurate quantum mechanical descriptions of the inter-state couplings rely on so-called non-adiabatic coupling vectors (NACs), which introduce several additional challenges from an ML perspective. First, NACs exhibit the same rotational equivariance as molecular forces. Second, they grow rapidly for states lying close in energy. And finally, as a quantity computed between different states, they are determined only up to an arbitrary phase. The latter property in particular complicates the construction of ML models, as the random nature of the phase factor needs to be compensated during training. Early works relied on a costly preprocessing of the reference data\\cite{westermayr2019machine}. Ref.~\\citenum{westermayr2020combining}, however, demonstrated that the phase problem can be overcome by introducing phase-less loss functions during the training procedure. Using a modified end-to-end NNP to describe the excited state dynamics of the methylenimmonium cation, it could be shown that using such loss terms completely eliminates the need for a preprocessing step. In addition, the work modeled the NACs as derivatives of a proxy potential, thus accounting for their transformation under rotations of the molecule. The combination of these approaches not only made it possible to obtain accurate population statistics for the studied system, but could also greatly extend the time scales accessible by the simulation beyond the limits of conventional electronic structure approaches.\n\t\n\\paragraph{Spectroscopy}\nAs stated  at the beginning of this section, MD simulations are an excellent tool to model the temporal autocorrelation functions of various quantities, which can in turn be used to predict experimental observables, such as diffusion coefficients. These quantities need not be restricted to properties derived from the PES, but encompass other electronic properties such as dipole moments or polarizabilities. Access to the corresponding time autocorrelation functions enables the simulation of a wide range of molecular spectra, which can be directly related to experiment. The most prominent examples are infrared and Raman spectra derived from the autocorrelation functions of dipole moments and polarizabilities, respectively. Both types of vibrational spectra are of great practical interest, since they can be measured accurately via experiment and provide insights into the atomic structure of molecules and materials. However, these spectra can be subject to a series of complex quantum mechanical effects, such as vibrational anharmonicities. \nHence, high level electronic structure treatments are required in order to obtain quantitatively accurate predictions of experimental results. Unfortunately, computing the required autocorrelation functions based purely on electronic structure calculations quickly grows prohibitively expensive, as simulations covering sufficient time scales are required in order to yield reliable spectra. In addition, if the influence of temperature or other phenomena should be studied in detail, a large number of such simulations is required. \nRecently, ML-FFs  have emerged as invaluable tools for obtaining reliable molecular spectra. A growing number of ML-FFs now provide access to quantities beyond the PES, e.g.\\ dipole moments or polarizabilities. As such, they offer the possibility to perform these simulations in only a fraction of the time required by an \\textit{ab initio} approach or even make them possible at all.\n\t\nRef.~\\citenum{gastegger2017machine} demonstrates the potential inherent to ML-FFs based on the prediction of infrared spectra for organic molecules including the protonated alanine tripeptide. By combining a descriptor-based NNP model of the PES with a dipole moment model based on latent NN-predicted atomic charges, highly accurate infrared spectra could be obtained for all studied systems. The efficiency of such an approach was demonstrated based on an alkane containing more than 200 atoms, where it was possible to reduce a projected computation time of 9000 years with the original \\textit{ab initio} method to only a few days (including the reference calculations needed for training the ML models). Moreover, the high accuracy of the predictions made it possible to identify shortcomings of the original reference methods and study how they influence the infrared spectrum of the tripepdtide. A similar latent charge based approach was employed in Ref.~\\citenum{yao2018tensormol} to model infrared spectra of various amino acids. This study could not only obtain accurate spectra, but also demonstrated that the latent charges predicted by the dipole model constitute a valid ML driven scheme for deriving atomic partial charges, which can be used to model long-range electrostatic interactions explicitly. This scheme has since been employed in many physically augmented models (e.g.\\ TensorMol\\cite{yao2018tensormol} or PhysNet\\cite{unke2019}).\n\t\nIn a similar manner, ML models capable of predicting polarizability tensors offer access to Raman spectra. Ref.~\\citenum{raimbault2019using} introduces a symmetry adapted approach for modeling polarizability tensors using Gaussian process regression (GPR) (see Section~\\ref{subsubsec:kernel_based_methods}) based on the SOAP\\cite{bartok2013representing} kernel. The authors use this model to study the Raman spectra of paracetamol in gas phase and various molecular crystals and achieve excellent agreement with electronic structure methods in both cases. \nNot only is the proposed approach highly data efficient, requiring only a small number ($<$1000) of reference data, but it could also be shown that the resulting model is transferable between different polymorphic forms of the crystal.\nRef.~\\citenum{sommers2020raman} models Raman spectra of liquid water using descriptor-based NNPs to predict molecular polarizabilities. The computational efficiency of the approach made it possible to obtain Raman spectra for a system containing 416 water molecules based on two nanosecond trajectories at DFT level accuracy, a feat which would be infeasible with the original reference method. \nAs a consequence, the influence of temperature effects on the Raman spectra of water and heavy water could be studied in detail. The atomic resolution of the employed ML approach made it possible to decompose the simulated spectra into intramolecular and intermolecular contributions, offering insights into the mechanisms governing the temperature dependence of the different spectral features.\n\n\\section{Mathematical and conceptual framework}\n\\label{sec:mathematical_and_coceptual_framework}\nSection~\\ref{subsec:chemistry_foundations} reviews important chemical concepts such as the potential energy surface and invariance properties of physical systems, which are essential for constructing physically meaningful models. It is meant as a short summary of the most important physical principles and fundamental chemical knowledge for readers with a primarily ML-focused background who are interested in constructing ML-FFs. On the other hand, to offer readers with a chemical background a first orientation, an overview of two important methodologies in Machine Learning, namely kernel-based learning approaches and artificial neural networks, is given in Section~\\ref{subsec:machine_learning_foundations}. Finally, Section~\\ref{subsec:combining_ml_and_chemistry} lists constraints related to the physical invariances mentioned earlier and gives examples of models for constructing ML-FFs and how they implement these constraints in practice.\n\n\\subsection{Chemistry foundations}\n\\label{subsec:chemistry_foundations}\nThe Schr\\\"odinger equation (SE)\\cite{schrodinger1926undulatory}, which describes the interaction of atomic nuclei and electrons, is sufficient for understanding most chemical phenomena and processes\\cite{dirac1929quantum}. Unfortunately, it can only be solved analytically for very simple systems, such as the hydrogen atom. For more complex systems like molecules, exact numerical solutions are often impractical due to a steep increase of computational costs as a function of system size. For this reason, numerous approximation schemes have been devised to enable insights into more complicated chemical systems. Virtually all of these are based on the Born-Oppenheimer (BO) approximation\\cite{born1927quantentheorie}, which decouples electronic and nuclear motion, neglecting the latter completely. It is assumed that electrons adjust instantaneously to changes in the nuclear positions, which is motivated by the observation that atomic nuclei are heavier than electrons by several orders of magnitude, thus moving on a vastly different timescale. Hence, the nuclear positions appear almost stationary to the electrons and therefore enter the resulting ``electronic SE'' only parametrically: The energy of the electrons depends on the external potential caused by the nuclei, which in turn is fully determined by their positions and nuclear charges. By summing the electronic energy and the Coulomb repulsion between the nuclei, the total potential energy of the system is obtained. The potential energy is one of the most important properties of molecular systems. Alongside entropic contributions, it determines the relative stability of different compounds, whether reactions are exothermic or endothermic, and can even serve as proxy for more complex properties. For example, the potency of some drugs can be estimated from their binding energy to biomolecules\\cite{meng2011molecular}.\n\n\\subsubsection{The potential energy surface}\n\\label{subsubsec:the_potential_energy_surface}\nBy introducing a parametric dependency between energy and nuclei, the BO approximation implies the existence of a functional relation $f:\\{Z_i,\\mathbf{r}_i\\}_{i=1}^{N}\\mapsto E$, which maps the nuclear charges $Z_i$ and positions $\\mathbf{r}_i$ of $N$ atoms directly to their potential energy $E$. This function, called the potential energy surface (PES), governs the dynamics of a chemical system, similar to a ball rolling on a hilly landscape. Minima (``valleys'') on the PES correspond to stable molecules and significant structural changes (even chemical reactions) occur when a system crosses over a transition state (``ridge'') from one minimum into another (Fig.~\\ref{fig:various_PES}).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Fig_ChemRev_PESs.pdf}\n\\caption{Top: Two-dimensional projections of the PESs of different molecules, highlighting rich topological differences and various possible shapes. Bottom: Cut through the PES of keto-malondialdehyde for rotations of the two aldehyde groups. Note that the shape repeats periodically for full rotations. Regions with low potential energy are drawn in blue and high energy regions in yellow. Structure (a) leads to\na steep increase in energy due to the proximity of the two oxygen\natoms carrying negative partial charges. Local minima of the PES are shown in (b) and (c), whereas (d) displays structural fluctuations around the global minimum. By running molecular dynamics simulations, the most common transition paths (F1, F2, and F3) between the different minima could be revealed. Figure adapted from Ref.~\\citenum{sauceda2020construction}.\n}\n\\label{fig:various_PES} \n\\end{figure}\n\nKnowledge of the PES therefore also allows to predict how a system evolves over time. For example, by studying a thermal ensemble of molecules starting from the same minimum on the PES, it is possible to determine which fraction of them will reach different minima and in what time frame, allowing to assess their reactivity and which products are formed. It is also possible to deduce the macroscopic thermodynamic properties of a system by studying how it behaves at an atomic level. In such molecular dynamics (MD) simulations, a classical treatment of nuclear dynamics is sometimes sufficiently accurate. In case of significant nuclear delocalization, which may occur in systems with light atoms, strong bonds, or for shallow potential energy landscapes\\cite{tuckerman1993efficient}, nuclear quantum effects (NQEs) must be included as well. Even then, methods like path-integral MD establish a one-to-one correspondence between the properties of a quantum object and a classical system with an extended phase space, eliminating the need to solve the nuclear SE\\cite{chandler1981exploiting, habershon2013ring, poltavsky2016modeling}.\n\nAt each time step of the simulation, the forces $\\mathbf{F}_i$ acting on each atom~$i$ must be known so that the equations of motion can be integrated numerically (e.g.\\ using the Verlet algorithm\\cite{verlet1967computer}). They can be derived from the PES by using the relation $\\mathbf{F}_i=-\\nabla_{\\mathbf{r}_i} E$, i.e.\\ the forces are the negative gradient of the potential energy $E$ with respect to the atom positions $\\mathbf{r}_i$ (see Section~\\ref{subsubsec:invariances_of_physical_systems}). Forces can also be used to perform geometry optimizations, e.g.\\ to find special configurations of atoms which correspond to critical points on the PES. For example, the height of a reaction barrier can be computed from the energy difference between the saddle point (transition state) and either of the two minima (equilibrium structures) which are connected by it.\n\nAlthough the BO approximation simplifies the SE, even approximate solutions can be computationally demanding, in particular for large systems containing many degrees of freedom. Thus, it is often unfeasible to derive \\textit{ab initio} energies and forces for each time step of an MD simulation. For this reason, analytical functions, so-called force fields (FFs) are typically used to represent the PES, circumventing the problem of solving an equation altogether. The difficulty is then shifted to finding an appropriate functional form and parametrization of the FF. Doing so manually requires chemical intuition and knowledge and is time- and labor-intensive. An alternative is to regard the task as a regression problem and use machine learning (ML) methods (see Section~\\ref{subsec:machine_learning_foundations}) to learn a suitable functional form for the PES from appropriate reference data automatically. These machine-learned force fields (ML-FFs) are the main focus of this review.\n\n\\subsubsection{Invariances of physical systems}\n\\label{subsubsec:invariances_of_physical_systems}\nClosed physical systems are governed by various conservation laws that describe invariant properties. They are fundamental principles of nature that characterize symmetries that must not be violated. As such, conservation laws provide strong constraints that can be used as guiding principles in search of physically plausible theories. The basic invariances of molecular systems are directly derived from Noether's theorem\\cite{noether1918invarianten}, which states that each conserved quantity is associated with a differentiable symmetry of the action of a physical system. The action is represented as the integral of the Lagrangian $L$ over time\n\\begin{equation}\n\\mathcal{S}=\\int_{t_{1}}^{t_{2}} L[q(t), \\dot{q}(t), t] \\mathrm{d}t\\,,\n\\label{eq:lagrangian_integral_over_time}\n\\end{equation}\nwhere $\\dot{q}=\\partial q \/ \\partial t$ is the time derivative of the system state $q$. Within that framework, the behavior of any dynamical system is described by the trajectories through phase space for which the action is stationary. A symmetry is defined as any coordinate $q_{k}$ that does not appear in the Lagrangian, with the result that $\\partial L \/ \\partial q_{k}=0$. Then, it follows for the Euler-Lagrange equation of motion\n\\begin{equation}\n\\frac{\\partial}{\\partial t}\\left(\\frac{\\partial L}{\\partial \\dot{q}_{k}}\\right)=\\frac{\\partial L}{\\partial q_{k}}=0 \\quad \\rightarrow \\quad \\frac{\\partial L}{\\partial \\dot{q}_{k}}=C\\,,\n\\label{eq:euler_lagrange}\n\\end{equation}\nwhere $C$ is a constant and $\\partial L \/ \\partial \\dot{q}_{k}$ is a conserved quantity, i.e.\\ independent of time. For example, in Cartesian coordinates, the conserved quantity $\\partial L \/ \\partial \\dot{x}=m \\dot{x}$ is the linear momentum. Conserved quantities in Lagrangian systems include the total energy (following from temporal invariance), as well as angular and linear momentum (roto-translational invariance). \n\nEnergy conservation imposes a particular structure on vector fields in order for them to be valid force fields with corresponding potentials. Namely, forces must be the negative gradient of the potential energy with respect to atomic positions. This relation ensures that when atoms move, they always acquire the same amount of kinetic energy as they lose in potential energy (and \\textit{vice versa}), i.e.\\ the total energy is constant (the work done along closed paths is zero). The conservation of linear and angular momentum implies that the potential energy of a molecule only depends on the relative position of its atoms to each other, i.e.\\ it does not change with rigid rotations and\/or translations.\nAnother invariance (not derived from Noether's theorem) follows from the fact that, from the perspective of the electrons, atoms with the same nuclear charge appear identical to each other. They can thus be exchanged without affecting the energy and forces, which makes the PES symmetric with respect to permutations of some of its arguments. To ensure physically meaningful predictions, ML-FFs must be made invariant under the same transformations as the true PES by introducing appropriate constraints.\n\n\\subsection{Machine learning foundations}\n\\label{subsec:machine_learning_foundations}\nA question that frequently arises for researchers new to the field of ML concerns the difference of ML modeling to plain interpolation in the noise free regression case. After all, the Shannon sampling theorem gives bounds for the number of ``training samples'' needed to reconstruct a band-limited signal exactly\\cite{shannon1949communication}. Since the regression tasks considered in this review use \\textit{ab initio} data as reference, they can be considered practically noise-free. Furthermore, PESs are usually smooth, i.e.\\ there is a well-defined frequency cutoff in the spectrum of this ``signal''. Thus, both requirements for Shannon interpolation are satisfied and it should in principle be possible to reconstruct FFs via interpolation of the training samples without error, provided there are enough of them.\n\nThis is where ML diverges from signal interpolation theory. In practice, there is often not enough data available to fully capture all the necessary information for a perfect reconstruction. In that case, the goal of ML methods is not to recover the training data, but rather to estimate the true process with its underlying regularities that also describes all new and unseen data -- this is often denoted as generalization.\nThe key to generalization is selecting a model based on the well known principle of Occam's razor, i.e.\\ the notion that simpler hypotheses are more likely to be correct\\cite{schaffer2015not}. The capacity of the model can be controlled using the bias--variance trade-off\\cite{geman1992neural} (a compromise between expressiveness and complexity) and is practically done by exercising model selection techniques (see Section~\\ref{subsubsec:model_selection_how_to_choose_hyperparameters}) such as cross-validation that leave out part of the  data  from the ML training process and use it later to obtain a valid estimate of the generalization error.\\cite{muller2001introduction,hansen2013assessment}\n\nML methods typically rely on the fact that nonlinear problems, such as predicting energy from nuclear positions, can be ``linearized'' by mapping the input\nto a (often higher-dimensional) ``feature space'' (see\nFig.~\\ref{fig:linear_separability_in_feature_space}).\\cite{scholkopf1997kernel,scholkopf1998nonlinear,scholkopf1999input,muller2001introduction} \n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{linear_separability.pdf}\n\t\\caption{\\textbf{A}: The blue and red points with coordinates\n\t\t($x_1$, $x_2$) are linearly\n\t\tinseparable. \\textbf{B}: By defining a suitable mapping from the\n\t\tinput space ($x_1$, $x_2$) to a higher-dimensional feature\n\t\tspace ($x_1$, $x_2$, $x_3$), blue and red points become\n\t\tlinearly separable (gray plane at $x_3=0.5$).}\n\t\\label{fig:linear_separability_in_feature_space}\n\\end{figure}\n\nNote that such feature spaces are explicitly constructed for kernel-based learning methods (see Section~\\ref{subsubsec:kernel_based_methods}) or learned respectively for deep learning models \\cite{bishop1995neural} (see Section~\\ref{subsubsec:artificial_neural_networks}):\nKernel-based methods achieve this by taking advantage of the so-called \\textit{kernel trick}\\cite{muller1997predicting,boser1992training,scholkopf1998nonlinear,theodoridis2008pattern,theodoridis2020machine} which allows implicitly operating in a high-dimensional feature space without explicitly performing any computation there. In contrast, artificial neural networks (NNs) decompose a complex non-linear function into a composition of linear transformations with learnable parameters connected by nonlinear activation functions. With increasingly many of such nonlinear transformations organized in ``layers'' (\\emph{deep} NNs), it is possible to efficiently  learn highly  complex feature spaces.\n\nWhile NNs tend to require more training data to reach the same accuracy as kernel methods (see Fig.~\\ref{fig:md17_scatter}), they typically scale better to larger data sets. In general, neither method is strictly superior over the other\\cite{wolpert1996lack} and both have advantages and disadvantages that must be weighed against each other for a specific application. Recently, it has even been discovered that in the limit of infinitely wide layers, deep NNs are equivalent to kernel methods, which shifts the main differentiating factor between both methodologies to how they are constructed and trained\\cite{lee2017deep, matthews2018gaussian} and makes deep NNs accessible to kernel-based analysis methods \\cite{braun2008relevant,montavon2011kernel}.\n\nIn the following, kernel methods and neural networks are described in more detail to highlight the most important properties that differentiate both methodologies.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{md17_scatter3b.png}\n\t\\caption{Mean absolute force prediction errors (MAEs) of different ML models trained on molecules in the MD17 dataset\\cite{chmiela2017}, colored by model type. Overall, kernel methods (GDML\\cite{chmiela2017}, sGDML\\cite{chmiela2018}, FCHL18\/19\\cite{faber2018alchemical,christensen2020fchl}) are slightly more data efficient, i.e.\\ they produce more accurate predictions with smaller training datasets, but neural network architectures (PhysNet\\cite{unke2019}, SchNet\\cite{schutt2017schnet}, DimeNet\\cite{klicpera_dimenet_2020}, EANN\\cite{zhang2019embedded}, DeePMD\\cite{zhang2018deep}, DeepPot-SE\\cite{zhang2018end}, ACSF\\cite{behler2007generalized}, HIP-NN\\cite{lubbers2018hierarchical}) catch up quickly with increasing training set size and continue to improve when more data for training is available.}\n\t\\label{fig:md17_scatter}\n\\end{figure}\n\n\n\n\\subsubsection{Kernel-based methods}\n\\label{subsubsec:kernel_based_methods}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{kernel.pdf}\n    \\caption{Overview of the mathematical concepts that form the basis of kernel methods.\n    \\textbf{A}: Gaussian process regression of a one-dimensional function $f(x)$ (red line) from $N=5$ data samples $(x_i,y_i)$. The black line $\\hat{f}(x)$ depicts the mean (Eq.~\\ref{eq:gaussian_process_mean}) of the conditional probability $p(y_*|\\mathbf{y})$ (see Eq.~\\ref{eq:gaussian_process_distribution}), whereas the grey area depicts the 95\\% confidence interval (see Eq.~\\ref{eq:gaussian_process_variance}). Note that predictions are most confident in regions where training data is present. \\textbf{B}: The function $\\hat{f}(x)$ can be expressed as a linear combination of $N$ kernel functions $K(x,x_i)$ weighted with regression coefficients $\\alpha_i$ (see Eq.~\\ref{eq:kernel_regression}). In this example, the Gaussian kernel (Eq.~\\ref{eq:gaussian_kernel}) is used (the hyperparameter $\\gamma$ controls its width). \\textbf{C}: Influence of noise on prediction performance. Here, the function $f(x)$ (thick gray line) is learned from $N=25$ samples, however, each data point $(x_i,y_i)$ contains observational noise (see Eq.~\\ref{eq:gaussian_process_noise_model}). When the coefficients $\\alpha_i$ are determined without regularization, i.e.\\ no noise is assumed to be present, the model function reproduces the training samples faithfully, but undulates wildly between data points (orange line, $\\lambda=0$). The regularized solution (blue line, $\\lambda=0.1$, see Eq.~\\ref{eq:krr_coefficient_relation_regularized}) is much smoother and stays closer to the true function $f(x)$, but individual data points are not reproduced exactly. When the regularization is too strong (green line, $\\lambda=1.0$), the model function becomes unable to fit the data. Note how regularization shrinks the magnitude of the coefficient vectors  $\\lVert\\boldsymbol{\\alpha}\\rVert$. \\textbf{D}: For constructing force fields, it is necessary to encode molecular structure with a representation $\\mathbf{x}$. The choice of this structural descriptor may strongly influence model performance. Here, the potential energy $E$ of a diatomic molecule (thick gray line) is learned from $N=5$ data points by two kernel machines using different structural representations (both models use a Gaussian kernel). When the interatomic distance $r$ is used as descriptor (orange line, $x=r$), the predicted potential energy oscillates between data points, leading to spurious minima and qualitatively wrong behavior for large $r$. A model using the descriptor $x=e^{-r}$ (blue line) predicts a physically meaningful potential energy curve that is qualitatively correct even when the model extrapolates.}\n    \\label{fig:kernel_basics}\n\\end{figure*}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{regression_operators.png}\n    \\caption{Kernel ridge regression can be understood as a linear integral operator $T_k$ that is applied to the (only partially known) target function of interest $f(\\mathbf{x})$. Such operators are defined as convolutions with a continuous kernel function $K$, whose response is the regression result. Because the training data is typically not sampled on a grid, this convolution task transforms to a linear system that yields the regression coefficients $\\mathbf{\\alpha}$. Because only $T_k f(\\mathbf{x})$ and not the true $f(\\mathbf{x})$ is recovered, the challenge is to find a kernel that defines an operator that leaves the relevant parts of its original function invariant. This is why the Gaussian kernel (Eq.~\\ref{eq:gaussian_kernel}) is a popular choice: Depending on the chosen length scale $\\gamma$, it attenuates high frequency components, while passing through the low frequency components of the input, therefore making only minimal assumptions about the target function. However, stronger assumptions (e.g.\\ by combining kernels with physically-motivated descriptors) increase the sample efficiency of the regressor.}\n    \\label{fig:regression_operators}\n\\end{figure}\n\nGiven a data set $\\{(y_i; \\mathbf{x}_i)\\}_{i=1}^{N}$ of $N$ reference values $y_i \\in \\mathbb{R}$ for inputs $\\mathbf{x}_i \\in\n\\mathbb{R}^D$, kernel regression aims to estimate $y_*$ for unknown inputs $\\mathbf{x}_*$. For example, for PES construction, $y$ is the potential energy and $\\mathbf{x}$ encodes structural information about the atoms, i.e.\\ their nuclear charges and relative positions in space. Popular choices for such ``descriptors'' are vectors of internal coordinates, Coulomb matrices\\cite{rupp2012fast}, representations of atomic environments (e.g.\\ symmetry functions\\cite{behler2011atom},\nSOAP\\cite{bartok2013representing} or\nFCHL\\cite{faber2018alchemical,christensen2020fchl}), or an encoding of crystal structure\\cite{schutt2014represent,faber2015crystal,faber2016machine}. \n\nThe representer theorem states that the functional relation \n\\begin{equation}\n    y = f(\\mathbf{x}) + \\epsilon\\,,\n    \\label{regressionwithnoise}\n\\end{equation}\nwhere $\\epsilon$ denotes measurement noise,  can be optimally approximated as a linear combination\n\\begin{equation}\nf(\\mathbf{x}) \\approx \\hat{f}(\\mathbf{x}) = \\sum_{i = 1}^{N}\n\\alpha_i K(\\mathbf{x},\\mathbf{x}_i)\\,,\n\\label{eq:kernel_regression}\n\\end{equation}\nwhere $\\alpha_i$ are coefficients and $K(\\mathbf{x},\\mathbf{x'})$ is a (typically nonlinear) symmetric and positive semi-definite function\\cite{wahba1990spline,scholkopf2001generalized,argyriou2009there} that measures the similarity of two compound descriptors $\\mathbf{x}$ and $\\mathbf{x'}$ (see Fig.~\\ref{fig:regression_operators}).\\footnote{The function $K(\\mathbf{x},\\mathbf{x'})$ computes the inner product of two points $\\phi(\\mathbf{x})$ and $\\phi(\\mathbf{x'})$ in some Hilbert space $\\mathcal{H}$ (the feature space) without the need to evaluate (or even know) the mapping $\\phi: \\mathbb{R}^D \\mapsto \\mathcal{H}$ explicitly, i.e.\\ $K(\\mathbf{x},\\mathbf{x'})$ is a reproducing kernel of $\\mathcal{H}$.\\cite{berlinet2011reproducing,muller2001introduction}}\nExamples for such functions $K$ are the polynomial kernel \\begin{equation}\nK(\\mathbf{x},\\mathbf{x'}) = (\\langle\\mathbf{x},\\mathbf{x'}\\rangle+ c)^d\\,,\n\\label{eq:polynomial_kernel}\n\\end{equation}\nwhere hyperparameter $d$ is the degree of the polynomial and $\\langle \\cdot, \\cdot \\rangle$ is the dot product, or the Gaussian kernel given by\n\\begin{equation}\nK(\\mathbf{x},\\mathbf{x'}) = e^{-\\gamma\\lVert\n\t\\mathbf{x}-\\mathbf{x'}\\rVert^2}\n\\label{eq:gaussian_kernel}\n\\end{equation}\nwith hyperparameter $\\gamma$ controlling its width\/scale and $\\lVert \\cdot \\rVert$ denoting the $L^2$-norm (see Refs.~\\citenum{muller2001introduction,hansen2013assessment,scholkopf2002learning,theodoridis2008pattern} for more examples).\n\nThe structure and number of dimensions of the associated Hilbert space $\\mathcal{H}$ depends on the choice of $K(\\mathbf{x},\\mathbf{x'})$ and dimension of the inputs $\\mathbf{x}$ and $\\mathbf{x'}$. As an example, consider the polynomial kernel (Eq.~\\ref{eq:polynomial_kernel}) with degree $d=2$ and two-dimensional inputs. The corresponding homogeneous $(c=0)$ polynomial mapping is given by $\\phi:(x_1,x_2)\\mapsto(x_1^2, \\sqrt{2}x_1x_2, x_2^2)$, so the associated $\\mathcal{H}$ is three-dimensional. While in this case, it is still possible to compute $\\phi$ and evaluate the inner product of two points $\\phi(\\mathbf{x})$ and $\\phi(\\mathbf{x'})$ explicitly, the advantage of using kernels becomes apparent when the Gaussian kernel (Eq.~\\ref{eq:gaussian_kernel}) is considered. Rewriting Eq.~\\ref{eq:gaussian_kernel} as \n\\begin{equation}\nK(\\mathbf{x},\\mathbf{x'}) = e^{-\\gamma\\lVert \\mathbf{x}\\rVert^2}e^{-\\gamma\\lVert \\mathbf{x'}\\rVert^2}e^{2\\gamma \\langle\\mathbf{x},\\mathbf{x'}\\rangle}\n\\label{eq:gaussian_kernel_rewritten}\n\\end{equation}\nand expanding the third factor in a Taylor series\n$e^{2\\gamma\\langle\\mathbf{x},\\mathbf{x'}\\rangle}=\\sum_{d=0}^\\infty\\frac{1}{d!}\\left(\n2\\gamma\\langle\\mathbf{x},\\mathbf{x'}\\rangle\\right)^d$ reveals that the Gaussian kernel is equivalent to an infinite sum over (scaled) polynomial kernels (see Eq.~\\ref{eq:polynomial_kernel}) and the associated $\\mathcal{H}$ is $\\infty$-dimensional. Fortunately, by using the kernel function $K(\\mathbf{x},\\mathbf{x'})$, it is possible to operate in\n$\\mathcal{H}$ implicitly and evaluate $\\hat{f}(\\mathbf{x})$ (Eq.~\\ref{eq:kernel_regression}) without computing the mapping $\\phi$. This is often referred to as the\n\\textit{kernel trick}\\cite{muller1997predicting,boser1992training,scholkopf1998nonlinear,muller2001introduction,scholkopf2002learning,theodoridis2008pattern}. \n\nIt remains the question how the coefficients $\\alpha_i$ in Eq.~\\ref{eq:kernel_regression} are determined. One way to do so is by adopting a Bayesian, or probabilistic, point of view.\\cite{rasmussen2003gaussian,murphy2012machine} Here, it is assumed that the reference data $\\{(y_i; \\mathbf{x}_i)\\}_{i=1}^{N}$ are generated by a Gaussian process (GP), i.e.\\ drawn from a multivariate Gaussian distribution. For simplicity, it can be assumed that this distribution has a mean of zero, as other values can be generated by simply adding a constant term. Further, the possibility that the reference data might be contaminated by noise (for example due to uncertainties in measuring $y_i$) is accounted for explicitly. Typically, Gaussian noise is assumed, i.e.\\\n\\begin{equation}\ny_i =  f(\\mathbf{x}_i) + \\mathcal{N}(0,\\lambda)\\,,\n\\label{eq:gaussian_process_noise_model}\n\\end{equation}\nwhere $\\lambda$ is the variance of the normally distributed noise $\\mathcal{N}$. In the GP picture, the choice of $K(\\mathbf{x},\\mathbf{x'})$ expresses an assumption about the underlying function class. For example, choosing the Gaussian kernel implies that $f(\\mathbf{x})$ does not change drastically over a length scale controlled by $\\gamma$ (see Eq.~\\ref{eq:gaussian_kernel}). As such, a particular kernel function $K$ corresponds to an implicit regularization, i.e.\\ an assumption about the underlying smoothness properties of the function to be estimated.\\cite{smola1998connection} The challenge lies in finding a kernel that represents the structure in the data that is being modeled as good as possible.\\cite{smola1998connection,braun2008relevant} Many kernels are able to approximate continuous functions on a compact subset arbitrarily well,\\cite{smola1998connection,micchelli2006universal} but a strong prior has the advantage of restricting the hypothesis space, which drastically improves the convergence of the learning task with respect to the available training data.\\cite{huang2016communication} \n\nUnder these conditions, it is now possible to rigorously answer the question\n``given the data $\\mathbf{y}=\\left[y_1 \\cdots y_N\\right]^{\\top}$,\nhow likely is it to observe the value $y_*$ for input\n$\\mathbf{x}_*$?'' As $y_*$ is generated by the same GP as the reference data, the conditional probability $p(y_*|\\mathbf{y})$ can be expressed as\n\\begin{equation}\n\\begin{bmatrix}\n\\mathbf{y}\\textbf{}\\\\\ny^*\n\\end{bmatrix}\n\\sim\n\\mathcal{N}\\left(0,\n\\begin{bmatrix}\n\\mathbf{K} + \\lambda\\mathbf{I}_N & \\mathbf{K}_*^{\\top}\\\\\n\\mathbf{K}_* &  K(\\mathbf{x}_*,\\mathbf{x}_*)\n\\end{bmatrix}\n\\right)\\,,\n\\label{eq:gaussian_process_distribution}\n\\end{equation}\nwhere $\\mathbf{I}_N$ is the identity matrix of size $N$, $\\mathbf{K}$ is the $N\\times N$ kernel matrix\\cite{hofmann2008kernel,muller2001introduction} with entries $K_{ij}=K(\\mathbf{x}_i,\\mathbf{x}_j)$ and $\\mathbf{K}_*=\\left[K(\\mathbf{x}_*,\\mathbf{x}_1) \\cdots\nK(\\mathbf{x}_*,\\mathbf{x}_N)\\right]$. In other words, Eq.~\\ref{eq:gaussian_process_distribution} expresses a probability distribution over possible predictions, where the mean value\n\\begin{equation}\n\\bar{y}_* = \\mathbf{K}_* \\left(\\mathbf{K} + \\lambda\\mathbf{I}_N\\right)^{-1}\\mathbf{y}\n\\label{eq:gaussian_process_mean}\n\\end{equation}\nis the most likely estimate for $y_*$ and the variance \n\\begin{equation}\n\\mathrm{var}(y_*) = K(\\mathbf{x}_*,\\mathbf{x}_*) - \\mathbf{K}_* \\left(\\mathbf{K} + \\lambda\\mathbf{I}_N\\right)^{-1} \\mathbf{K}_*^{\\top}\n\\label{eq:gaussian_process_variance}\n\\end{equation}\nprovides information about how certain the prediction is. The optimal coefficients $\\boldsymbol{\\alpha} = \\left[\\alpha_1 \\cdots\n\\alpha_N\\right]^{\\top}$ in Eq.~\\ref{eq:kernel_regression} are thus given by \n\\begin{equation}\n\\boldsymbol{\\alpha} = \\left(\\mathbf{K}+\\lambda\\mathbf{I}_N\\right)^{-1}\\mathbf{y}\n\\label{eq:krr_coefficient_relation_regularized}\n\\end{equation}\nor simply $\\boldsymbol{\\alpha} = \\mathbf{K}^{-1}\\mathbf{y}$ in the noise-free case ($\\lambda=0$). However, even in the absence of noise, it can be beneficial to choose a non-zero $\\lambda$ to obtain a regularized solution. The addition of $\\lambda > 0$ to the diagonal of $\\mathbf{K}$ increases numerical stability and has the effect of damping the magnitude of the coefficients and thereby increasing the smoothness of $\\hat{f}(\\mathbf{x})$, with the downside that the known reference values $y_i$ are only approximately reproduced. This, however, also decreases the chance of overfitting and can lead to better generalization, i.e.\\ increased accuracy when predicting unknown values. \n\nMatrix factorization methods like Cholesky decomposition\\cite{golub2012matrix} are typically used to efficiently solve the linear problem in Eq.~\\ref{eq:krr_coefficient_relation_regularized} in closed form. However, this type of approach scales as $\\mathcal{O}(N^3)$ with the number of reference data and may become problematic for extremely large data sets. Iterative, e.g.\\ gradient-based, solvers can reduce the complexity to $\\mathcal{O}(N^2)$.\\cite{raykar2007fast} Once the coefficients have been determined, the value $y_*$ for an arbitrary input $\\mathbf{x}_*$ can be estimated according to\nEq.~\\ref{eq:kernel_regression} with $\\mathcal{O}(N)$ complexity (a sum over all $N$ reference data points is required). \n\nAlternatively, a variety of approximation techniques exploit that kernel matrices usually have a small \\emph{numerical} rank, i.e.\\ a rapidly decaying eigenvalue spectrum. This enables approximate factorizations $\\mathbf{R}\\mathbf{R}^T \\approx \\mathbf{K}$, where $\\mathbf{R}$ is either a rectangular matrix $\\in \\mathbb{R}^{N \\times M}$ with $M < N$ or sparse. As a result Eq.~\\ref{eq:krr_coefficient_relation_regularized} becomes easier so solve, albeit the result will not be exact\\cite{williams2001using, quinonero2005unifying, snelson2006sparse, rahimi2008random, rudi2017falkon}. \n\nA straightforward approach to approximate a linear system is to pick a representative or random subset of $M$ points $\\tilde{\\mathbf{x}}$ from the dataset (in principle, even arbitrary $\\tilde{\\mathbf{x}}$ could be chosen) and construct a rectangular kernel matrix $\\mathbf{K}\n_{MN}\\in\\mathbb{R}\n^{M\\times N}$ with entries $K_{MN,ij} = K(\\tilde{\\mathbf{x}}_i, \\mathbf{x}_j)$. Then, the corresponding coefficients can be obtained via the Moore-Penrose pseudoinverse\\cite{moore1920reciprocal,penrose1955generalized}\n\\begin{equation}\n\\tilde{\\boldsymbol{\\alpha}} = (1+\\lambda)^{-1}\\left(\\mathbf{K}_{MN}\\mathbf{K}_{NM}\\right)^{-1}\\mathbf{K}_{MN}\\mathbf{y}\\,,\n\\label{eq:krr_coefficient_relation_approximated}\n\\end{equation}\nwhere $\\mathbf{K}_{NM}=\\mathbf{K}_{MN}^{\\top}$. Solving Eq.~\\ref{eq:krr_coefficient_relation_approximated} scales as $\\mathcal{O}(NM^2)$ and is much less computationally demanding than inverting the original matrix in Eq.~\\ref{eq:krr_coefficient_relation_regularized}. Once the $M$ coefficients $\\tilde{\\boldsymbol{\\alpha}}$ are obtained, the model can be evaluated with $\\hat{f}(\\mathbf{x})=\\sum_M \\tilde{\\alpha}_iK(\\mathbf{x},\\tilde{\\mathbf{x}}_i)$, i.e.\\ an additional benefit is that evaluation now scales as $\\mathcal{O}(M)$ instead of $\\mathcal{O}(N)$ (see Eq.~\\ref{eq:kernel_regression}).\n\nHowever, the approximation above gives rise to an over-determined system with fewer parameters than training points and therefore reduced model capacity. Strictly speaking, the involved matrix does not satisfy the properties of a kernel matrix anymore, as it is neither symmetric nor positive semi-definite. To obtain a kernel matrix that still maintains these properties, the Nystr\\\"om~\\cite{williams2001using} approximation\n\\begin{equation}\n\\mathbf{K} \\approx \\tilde{\\mathbf{K}}=\\mathbf{K}_{N M} \\mathbf{K}_{M M}^{-1} \\mathbf{K}_{N M}^{\\top}\n\\label{eq:nystroem_approx}\n\\end{equation}\ncan be used instead. Here, the sub-matrix $\\mathbf{K}_{M M}$ is a true kernel matrix between all inducing points $\\tilde{\\mathbf{x}}_i$. Using the Woodbury matrix identity\\cite{cutajar2016preconditioning}, the regularized inverse is given by\n\\begin{equation}\n\\begin{aligned}\n&(\\tilde{\\mathbf{K}}+\\lambda \\mathbf{I}_N)^{-1} = \\lambda^{-1}[\\mathbf{I}_N-\\mathbf{K}_{N M} \\\\ \n& \\: \\left(\\lambda \\mathbf{K}_{M M}+\\mathbf{K}_{N M}^{\\top} \\mathbf{K}_{N M}\\right)^{-1} \\mathbf{K}_{N M}^{\\top}]\n\\end{aligned}\n\\label{eq:nystroem_regularized_inverse}\n\\end{equation}\nand $\\tilde{\\boldsymbol{\\alpha}} = (\\tilde{\\mathbf{K}}+\\lambda \\mathbf{I})^{-1} \\mathbf{y}$. The computational complexity of solving the Nystr\\\"om approximation is $\\mathcal{O}(M^3 + NM^2)$.\n\nIt should be mentioned that kernel regression methods are known under different names in the literature of different communities. Due to their relation to GPs, some authors prefer the name Gaussian process regression (GPR). Others favor the term kernel ridge regression (KRR), since determining the coefficients with Eq.~\\ref{eq:krr_coefficient_relation_regularized} corresponds to solving a least-squares objective with $L^2$-regularization in the kernel feature space $\\phi$ and is similar to ordinary ridge regression\\cite{tikhonov1977solutions}. Sometimes, the method is also referred to as reproducing kernel Hilbert space (RKHS) interpolation, since Eq.~\\ref{eq:kernel_regression} ``interpolates'' between known reference values (when coefficients are determined with $\\lambda=0$, all known reference values are reproduced exactly). All these methods are formally equivalent and essentially differ only in the philosophical manner the relevant equations are derived. The most important concepts discussed in this section are summarized visually in Fig~\\ref{fig:kernel_basics}.\n\n\\subsubsection{Artificial neural networks}\n\\label{subsubsec:artificial_neural_networks}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{neuralnets.pdf}\n    \\caption{Schematic representation of the mathematical concepts underlying artificial (feed-forward) neural networks.\n    \\textbf{A}: A single artificial neuron can have an arbitrary number of inputs and outputs. Here, a neuron that is connected to two inputs $i_1$ and $i_2$ with ``synaptic weights'' $w_1$ and $w_2$ is depicted. The bias term $b$ can be thought of as the weight of an additional input with a value of 1. Artificial neurons compute the weighted sum of their inputs and pass this value through an activation function $\\sigma$ to other neurons in the neural network (here, the neuron has three outputs with connection weights $w'_1$, $w'_2$, and $w'_3$). \\textbf{B}: A possible activation function $\\sigma(x)$. The bias term $b$ effectively shifts the activation function along the $x$-axis. Many non-linear functions are valid choices, but the most popular are sigmoid transformations such as $\\tanh(x)$ or (smooth) ramp functions, e.g.\\ $\\mathrm{max}(0,x)$ or $\\ln(1+e^x)$. \\textbf{C}: Artificial neural network with a single hidden layer of three neurons (gray) that maps two inputs $x_1$ and $x_2$ (blue) to two outputs $y_1$ and $y_2$ (yellow), see Eq.~\\ref{eq:single_layer}. For regression tasks, the output neurons typically use no activation function. Computing the weighted sums for the neurons of each layer can be efficiently implemented as a matrix vector product. Some entries of the weight matrices ($\\mathbf{W}$ and  $\\mathbf{W'}$) and bias vectors ($\\mathbf{b}$ and  $\\mathbf{b'}$) are highlighted in color with the corresponding connection in the diagram. \\textbf{D}: Schematic depiction of a \\emph{deep} neural network with $L$ hidden layers (Eq.~\\ref{eq:deep_feedforward_nn}). Compared to using a single hidden layer with many neurons, it is usually more parameter-efficient to connect multiple hidden layers with fewer neurons sequentially.}\n    \\label{fig:nn_basics}\n\\end{figure*}\nOriginally, artificial neural networks (NNs) were, as suggested by their name, intended to model the intricate networks formed by biological neurons\\cite{mcculloch1943logical}. Since then, they have become a standard ML algorithm\\cite{mcculloch1943logical,kohonen1988introduction,abdi1994neural,bishop1995neural,clark1999neural,ripley2007pattern,haykin2009neural,lecun2012efficient,theodoridis2020machine} only remotely related to their original biological inspiration. In the simplest case, the fundamental building blocks of NNs are \ndense (or ``fully-connected'') layers -- linear transformations from input vectors\n$\\mathbf{x}\\in \\mathbb{R}^{n_{\\rm in}}$ to output vectors\n$\\mathbf{y}\\in \\mathbb{R}^{n_{\\rm out}}$ according to\n\\begin{equation}\n\\mathbf{y} = \\mathbf{W}\\mathbf{x} + \\mathbf{b}\\,,\n\\label{eq:dense_layer}\n\\end{equation}\nwhere both weights $\\mathbf{W}\\in \\mathbb{R}^{n_{\\rm out} \\times n_{\\rm in}}$ and biases $\\mathbf{b}\\in \\mathbb{R}^{n_{\\rm out}}$ are parameters, and $n_{\\rm in}$ and $n_{\\rm out}$ denote the\nnumber of dimensions of $\\mathbf{x}$ and $\\mathbf{y}$, respectively. Evidently, a single dense layer can only express linear functions. Non-linear relations between inputs and outputs can only be modeled when at least two dense layers are stacked and combined with a non-linear \\emph{activation function} $\\sigma$:\n\\begin{equation}\n\\label{eq:single_layer}\n\\begin{aligned}\n\\mathbf{h} &= \\sigma\\left(\\mathbf{W}\\mathbf{x} + \\mathbf{b}\\right)\\,,\\\\\n\\mathbf{y} &= \\mathbf{W'}\\mathbf{h} + \\mathbf{b'}\\,.\n\\end{aligned}\n\\end{equation}\nProvided that the number of dimensions of the ``hidden layer'' $\\mathbf{h}$ is large enough, this arrangement can approximate any mapping between inputs $\\mathbf{x}$ and outputs $\\mathbf{y}$ to arbitrary precision, i.e.\\ it is a general function approximator\\cite{gybenko1989approximation,hornik1991approximation}.\n\nIn theory, \\emph{shallow} NNs as shown above are sufficient to approximate any functional relationship. In practice, however, \\emph{deep} NNs with multiple hidden layers are often superior and were shown to be exponentially more\nparameter-efficient.\\cite{eldan2016power} To construct a deep NN, $L$ hidden layers are combined sequentially\n\\begin{equation}\n\\label{eq:deep_feedforward_nn}\n\\begin{aligned}\n\\mathbf{h}_1 &= \\sigma\\left(\\mathbf{W}_1\\mathbf{x} + \\mathbf{b}_1\\right)\\,,\\\\\n\\mathbf{h}_2 &= \\sigma\\left(\\mathbf{W}_2\\mathbf{h}_1 + \\mathbf{b}_2\\right)\\,,\\\\\n&\\vdots\\\\\n\\mathbf{h}_L &= \\sigma\\left(\\mathbf{W}_L\\mathbf{h}_{L-1} + \\mathbf{b}_L\\right)\\,,\\\\\n\\mathbf{y} &= \\mathbf{W}_{L+1}\\mathbf{h}_L + \\mathbf{b}_{L+1}\\,,\n\\end{aligned}\n\\end{equation}\nmapping the input $\\mathbf{x}$ to several intermediate feature representations $\\mathbf{h}_l$, until the output $\\mathbf{y}$ is obtained by a linear regression on the features $\\mathbf{h}_L$ in the final layer. For PES construction, typically, the NN maps a representation of chemical structure $\\mathbf{x}$ to a one-dimensional output representing the energy. Contrary to the coefficients $\\boldsymbol{\\alpha}$ in kernel methods (see Eq.~\\ref{eq:krr_coefficient_relation_regularized}), the parameters $\\{\\mathbf{W}_l, \\mathbf{b}_l\\}_{l=1}^{L+1}$ of an NN cannot be fitted in closed form. Instead, they are initialized randomly and optimized (usually using a variant of stochastic gradient descent) to minimize a loss function that measures the discrepancy between the output of the NN and the reference data.\\cite{montavon2012neural} A common choice is the mean squared error (MSE), which is also used in kernel methods. During training, the loss and its gradient are estimated from randomly drawn batches of training data, making each step independent of the number of training data $N$. On the other hand, finding the coefficients for kernel methods scales as $\\mathcal{O}(N^3)$ due to the need of inverting the $N\\times N$ kernel matrix. Evaluating an NN according to\nEq.~\\ref{eq:deep_feedforward_nn} for a single input $\\mathbf{x}$ scales linearly with respect to the number of model parameters. The same is true for kernel methods, but here the number of model parameters is tied to the number of reference data $N$ used for training the model (see Eq.~\\ref{eq:kernel_regression}), which means that evaluating kernel methods scales $\\mathcal{O}(N)$. As the evaluation cost of NNs is independent of $N$ and only depends on the chosen architecture, they are typically the method of choice for learning large datasets. A schematic overview of the mathematical concepts behind NNs is given in Fig.~\\ref{fig:nn_basics}.\n\n\\subsubsection{Model selection: How to choose hyperparameters}\n\\label{subsubsec:model_selection_how_to_choose_hyperparameters}\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{cross_validation.png}\n\t\\caption{Overview of model selection process.}\n\t\\label{fig:cross_validation} \n\\end{figure*}\n\nIn addition to the parameters that are determined when learning an ML model for a given dataset, e.g.\\ the weights $\\mathbf{W}$ and biases $\\mathbf{b}$ in NNs or the regression coefficients $\\boldsymbol{\\alpha}$ in kernel methods, many models contain hyperparameters that need to be chosen before training. They allow to tune a given model to the prior beliefs about the dataset\/underlying physics and thus play a significant role in how a model generalizes to different data patterns. Two types of hyperparameters can be distinguished: The first kind influences the composition of the model itself, such as the type of kernel or the NN architecture, whereas the second kind affects the training procedure and thus the final parameters of the trained model. Examples for hyperparameters are the width (number of neurons per layer) and depth (number of hidden layers) of an NN, the kernel width $\\gamma$ (see Eq.~\\ref{eq:gaussian_kernel}), or the strength of regularization terms (e.g.\\ $\\lambda$ in Eq.~\\ref{eq:krr_coefficient_relation_regularized}). \n\nThe range of valid values is strongly dependent on the hyperparameter in question. For example, certain hyperparameters might need to be selected from the positive real numbers (e.g.\\ $\\gamma$ and $\\lambda$, see above), while others are restricted to positive integers or have interdependencies (such as depth and width of an NN). This is why hyperparameters are often optimized with primitive exhaustive search schemes like grid search or random search in combination with educated guesses for suitable search ranges. Common gradient-based optimization methods can typically not be applied effectively. Fortunately, model performance is fairly robust to small changes for many hyperparameters and good default values can be determined which work across many different datasets.\n\nBefore any hyperparameters may be optimized, a so-called test set must be split off from the available reference data and kept strictly separate. The remainder of the data is further divided into a training and a validation set. This is done because the performance of ML models is  not judged by how well they predict the data they were trained on, as it is often possible to achieve arbitrarily small errors in this setting. Instead, the generalization error, i.e.\\ how well the model is able to predict unseen data, is taken as indicator for the quality of a model. For this reason, for every trial combination of hyperparameters, a model is trained on the training data and its performance measured on the validation set to estimate the generalization error. Finally, the best performing model is selected. To get better statistics for estimates of the generalization error, instead of splitting the remaining data (reference data excluding test set) into just two parts, it is also possible to divide it into $k$ parts (or folds). Then, a total of $k$ models is trained, each using $k-1$ folds as the training set and the last fold as validation set. This method is known as $k$-fold cross validation\\cite{hastie2009elements,hansen2013assessment}.\n\nAs the validation data influences model selection (even though it is not used directly in the training process), the validation error may give too optimistic estimates and is no reliable way to judge the true generalization error of the final model. A more realistic value can be obtained by evaluating the model on the held-out test set, which has neither direct nor indirect influence on model selection. To not invalidate this estimate, it is crucial not to further tweak any parameters or hyperparameters in response to test set performance. More details on how to construct ML models (including the selection of hyperparameters and the importance of keeping an independent test set) can be found in Section~\\ref{sec:best_practices_and_pitfalls}. The model selection process is summarized in Fig.~\\ref{fig:cross_validation}.\n\n\\subsection{Combining machine learning and chemistry}\n\\label{subsec:combining_ml_and_chemistry}\nThe need for ML methods often arises from the lack of theory to describe a desired mapping between input and output. A classical example for this is image classification: It is not clear how to distinguish between pictures of different objects, as it is unfeasible to formulate millions of rules by hand to solve this task. Instead, the best results are currently achieved by learning statistical image characteristics from hundreds of thousands of examples that were extracted from a large dataset representing a particular object class. From that, the classifier learns to estimate the distribution inherent to the data in terms of feature extractors with learned parameters like convolution filters that reflect different scales of the image statistics.\\cite{lee2017deep,theodoridis2020machine,bishop1995neural} This working principle represents the best approach known to date to tackle this particular challenge in the field of computer vision.\n\nOn the other hand, the benchmark for solving molecular problems is set by rigorous physical theory that provides essentially exact descriptions of the relationships of interest. While the introduction of approximations to exact theories is common practice and essential to reduce their complexity to a workable level, those simplifications are always physical or mathematical in nature. This way, the generality of the theory is only minimally compromised, albeit with the inevitable consequence of a reduction in predictive power. In contrast, statistical methods can be essentially exact, but only in a potentially very narrow regime of applicability. Thus, a main role of ML algorithms in the chemical sciences has been to shortcut some of the computational complexity of exact methods by means of empirical inference, as opposed to providing some mapping between input and output at all (as is the case for image classification). Notably, recent developments could show that machine learning can provide novel insight beyond providing efficient shortcuts of complex physical computations.\\cite{schutt2017quantum,carleo2017solving,brockherde2017bypassing,chmiela2017,schutt2019unifying,noe2019boltzmann,sauceda2019molecular,sauceda2020} \n\nForce field construction poses unique challenges that are absent from traditional ML application domains, as much more stringent demands on accuracy are placed on ML approaches that attempt to offer practical alternatives to established methods. Additionally, considerable computational cost is associated with the generation of high-level \\textit{ab initio} training data, with the consequence that practically obtainable datasets with sufficiently high quality are typically not very large. This is in stark contrast with the abundance of data in traditional ML application domains, such as computer vision, natural language processing etc. The challenge in chemistry, however, is to retain the generality, generalization ability and versatility of ML methods, while making them accurate, data-efficient, transferable, and scalable.\n\n\\subsubsection{Physical constraints}\n\\label{subsubsec:physical_constraints}\nTo increase data efficiency and accuracy, ML-FFs can (and should) exploit the invariances of physical systems (see Section~\\ref{subsubsec:invariances_of_physical_systems}), which provide additional information in ways that are not directly available for other ML problems. Those invariances can be used \nto reduce the function space from which the model is selected, in this manner effectively reducing the degrees of freedom for learning,\\cite{chmiela2018,anselmi2016invariance} i.e. making the learning problem easier and thus also solvable with a fraction of data. As ML algorithms are universal approximators with virtually no inherent flexibility restrictions, it is important that physically meaningful solutions are obtained. In the following, important physical constraints of such solutions and possible ways of their realization are discussed in detail. Furthermore, existing kernel-based methods and neural network architectures tailored for the construction of FFs and how they implement these physical constraints in practice are described.\n\n\\paragraph{Energy conservation} A necessary requirement for ML-FFs is that, in the absence of external forces, the total energy of a chemical system is conserved (see Section~\\ref{subsubsec:invariances_of_physical_systems}). When the potential energy is predicted by any differentiable method and forces derived from its gradient, they will be conservative by construction. However, when forces are predicted directly, this is generally not true. This makes deriving energies from force samples slightly more complicated, but this \napproach also carries some advantages. The main challenge to overcome is that not every vector field is necessarily a valid gradient field. Therefore, the learning problem cannot simply be cast in terms of a standard multiple output regression task, where the output variables are modeled without enforcing explicit correlations. \n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{commutative_diagramm.pdf}\n\t\\caption{Differentiation of an energy estimator (blue) versus direct force reconstruction (red). The law of energy conservation is trivially obeyed in the first case, but requires explicit \\emph{a priori} constraints in the latter scenario. The challenge in estimating forces directly lies in the complexity arising from their high $3N$-dimensionality (three force components for each of the $N$ atoms) in contrast to predicting a single scalar for the energy.}\n\t\\label{fig:e_vs_f_learning}\n\\end{figure*}\nA big advantage of predicting forces directly is that they are true quantum-mechanical observables within the BO approximation by virtue of the Hellmann-Feynman theorem\\cite{hellman1937einfuhrung,feynman1939forces}, i.e.\\ they can be calculated \\emph{analytically} and therefore at a relatively low additional cost when generating \\textit{ab initio} reference data. As a rough guideline, the computational overhead for analytic forces scales with a factor of only around $\\sim$1--7 on top of the energy calculation.\\cite{chmiela2019}. In contrast, at least $3N + 1$ energy evaluations would be necessary for a numerical approximation of the forces by using finite differences. For example, at the PBE0\/DFT level of theory\\cite{adamo1999toward}, calculating energy and analytical forces for an ethanol molecule takes only $\\sim$1.5 times as long as calculating just the energy (the exact value is implementation-dependent), whereas for numerical gradients, a factor of at least $\\sim$10 would be expected. \n\nAs forces provide additional information about how the energy changes when an atom is moved, they offer an efficient way to sample the PES, which is why it is desirable to formulate ML models that can make \\emph{direct} use of them in the training process. Another benefit of a direct reconstruction of the forces is that it avoids the amplification of estimation errors due to the derivative operator that would otherwise be applied to the PES reconstruction (see Fig.~\\ref{fig:e_vs_f_learning}).\\cite{chmiela2017,sauceda2019molecular,snyder2012finding}\n\n\\paragraph{Roto-translational invariance}\nA crucial requirement for ML-FFs is the rotational and translational invariance of the potential energy, i.e.\\ $E\\left(\\mathbf{R}\\right) = E\\left(\\mathcal{R}\\mathbf{R} +\\mathcal{T}\\right)$, where $\\mathcal{R}$ and $\\mathcal{T}$ are rigid rotations and translations and $\\mathbf{R}$ are the Cartesian coordinates of the atoms. As long as the representation $\\mathbf{x}(\\mathbf{R})$ of chemical structure chosen as input for the ML model itself is roto-translationally invariant, ML-FFs inherit its desired properties and even the gradients will automatically behave in the correct equivariant way due to the outer derivative $\\partial (\\mathbf{x}(\\mathcal{R}\\mathbf{R}+\\mathcal{T}) \/ \\partial \\mathbf{R} = \\mathcal{R}\\partial\\mathbf{x}(\\mathbf{R})\/\\partial\\mathbf{R}$. One example of appropriate features to construct a representation $\\mathbf{x}$ with the desired properties are pairwise distances. For a system with $N$ atoms, there are $\\binom{N}{2}$ different pairwise distances, which results in reasonably sized feature sets for systems with a few dozen atoms. Apart from very few pathological cases, this representation is complete, in the sense that any possible configuration of the system can be described exactly and uniquely\\cite{bartok2013representing}. However, while pairwise distances serve as an efficient parametrization of some geometry distortions like bond stretching, they are relatively inefficient in describing others, e.g.\\ rotations of functional groups. In the latter case, many distances are affected even for slight angular changes, which can pose a challenge when trying to learn the geometry-energy mapping. Complex transition paths or reaction coordinates are often better described in terms of bond and torsion angles in addition to pairwise distances. The problem is that the number of these features grows rather quickly, with $\\binom{N}{3}$ and $\\binom{N}{4}$, respectively. At that rate, the size of the feature set quickly becomes a bottleneck, resulting in models that are slow to train and evaluate. While an expert choice of relevant angles would circumvent this issue, it reduces some of the \n``data-driven'' flexibility that ML models are typically appreciated for. Note that models without roto-translational invariance are practically unusable, as they will start to generate spurious linear and\/or angular momentum during dynamics simulations.\n\n\\paragraph{Indistinguishability of identical atoms}\nIn the BO approximation, the potential energy of a chemical system only depends on the charges and positions of the nuclei. As a consequence, the PES is symmetric under permutation of atoms with the same nuclear charge. However, symmetric regions are not necessarily sampled in an unbiased way during MD simulations (see Fig.~\\ref{fig:FigBiasedSampl}). Consequently, ML-FFs that are not constrained to treat all symmetries equivalently may (due to the uneven sampling) predict different results when permuting atoms.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{Fig_BiasedSampl.pdf}\n\t\\caption{Regions of the PESs for ethanol, keto-malondialdehyde and aspirin visited during a 200~ps \\textit{ab initio} MD simulation at 500~K using the DFT\/PBE+TS level of theory~\\cite{PBE1996,tkatchenko2009accurate}. The black dashed lines indicate the symmetries of the PES. Note that regions related by symmetry are not necessarily visited equally often.}\n\t\\label{fig:FigBiasedSampl}\n\\end{figure}\nWhile it is in principle possible to arrive at a ML-FF that is symmetric with respect to permutations of same-species atoms indirectly via data augmentation\\cite{montavon2012learning,montavon2013machine} or by simply using datasets that naturally include all relevant symmetric configurations in an unbiased way, there are obvious scaling issues with this approach. It is much more efficient to impose the right constraints onto the functional form of the ML-FF, such that all relevant symmetric variants of a certain atomic configuration appear equivalent automatically. Such symmetric functions can be constructed in various ways, each of which has advantages and disadvantages.\n\nAssignment-based approaches do not symmetrize the ML-FF \\textit{per se}, but instead aim to normalize its input, such that all symmetric variants of a configuration are mapped to the same internal representation. In its most basic realization, this assignment is done heuristically, i.e.\\ by using inexact, but computationally cheap criteria. Examples for this approach are the Coulomb matrix\\cite{rupp2012fast} or the Bag-of-Bonds\\cite{hansen2015machine} descriptors, that use simple sorting schemes for that purpose. Histograms\\cite{huo2017unified, christensen2020fchl} and some density-based\\cite{bartok2013representing,hirn2017wavelet,eickenberg2017solid} approaches follow that same principle, although not explicitly. All of those schemes have in common that they compare the features in aggregate as opposed to individually. A disadvantage is that dissimilar features are likely to be compared to each other or treated as the same, which limits the accuracy of the prediction. Such weak assignments are better suited for datasets with diverse conformations rather than gathered from MD trajectories that contain many similar geometries. In the latter case, the assignment of features might change as the geometry evolves, which would lead to discontinuities in the prediction and would effectively be treated by the ML model as noise (see $\\epsilon$ in Eq.~\\ref{regressionwithnoise}).\n\nAn alternative path is to recover the true correspondence of molecular features using a graph matching approach\\cite{kriege2016valid,Vert2008}. Each input $\\mathbf{x}$ is matched to a canonical permutation of atoms $\\tilde{\\mathbf{x}} = \\mathbf{P}\\mathbf{x}$ before generating the prediction. This procedure effectively compresses the PES to one of its symmetric subdomains (see dashed black lines in Fig.(\\ref{fig:FigBiasedSampl})), but in an exact way. Note that graph matching is in all generality an NP-complete problem which can only be solved approximately. In practice, however, several algorithms exist to ensure at least consistency in the matching process if exactness can not be guaranteed\\cite{pachauri2013solving}. A downside of this strategy is that any input must pass through a matching process, which is relatively costly, despite being approximate. Another issue is that the boundaries of the symmetric subdomains of the PES will necessarily lie in the extrapolation regime of the reconstruction in which prediction accuracy tends to degrade. As the molecule undergoes symmetry transformations, these boundaries are frequently crossed, to the detriment of prediction performance.\n\nArguably the most universal way of imposing symmetry, especially if the functional form of the model is already given, is via invariant integration over the relevant symmetry group $f_\\text{sym}(\\mathbf{x}) = \\int_{\\pi \\in \\mathcal{S}} f(\\mathbf{P}_\\pi \\mathbf{x})$. Typically, $\\mathcal{S}$ would be the permutation group and $\\mathbf{P}_\\pi$ the corresponding permutation matrix that transforms each vector of atom positions $\\mathbf{x}$. Some approaches\n\\cite{bartok2013representing, bartok2015g, de2016comparing} avoid this implicit ordering of atoms in $\\mathbf{x}$ by adopting a three-dimensional density representation of the molecular geometry defined by the atom positions, albeit at the cost of losing rotational invariance, which then must be recovered by integration. \nInvariant integration gives rise to functional forms that are truly symmetric and do not require any pre- or post-processing of the in- and output data. A significant disadvantage is however, that the cardinality of even basic symmetry groups is exceedingly high, which affects both training and prediction times. \n\nThis combinatorial challenge can be solved by limiting the invariant integral to the physical point group and fluxional symmetries that actually occur in the training dataset. Such a sub-group of meaningful symmetries can be automatically recovered and is often rather small\\cite{chmiela2019}. For example, each of the molecules benzene, toluene and azobenzene have only 12 physically relevant symmetries, whereas their full symmetric groups have orders $6!6!$, $7!8!$ and $12!10!2!$ symmetries respectively.\n\n\\subsubsection{(Symmetric) Gradient Domain Machine Learning ((s)GDML)}\n\\label{subsubsec:symmetric_gradient_domain_machine_learning}\nGradient domain machine learning (GDML) is a kernel-based method introduced as a data efficient way to obtain accurate reconstructions of flexible molecular force fields from small reference datasets of high-level \\textit{ab initio} calculations\\cite{chmiela2017}. Contrary to most other ML-FFs, instead of predicting the energy and obtaining forces by derivation with respect to nuclear coordinates, GDML predicts the forces directly. As mentioned in Section~\\ref{subsubsec:physical_constraints}, forces obtained in this way may violate energy conservation. To ensure conservative forces, the key idea is to use a kernel $\\mathbf{K}\\left(\\mathbf{x},\\mathbf{x}^{\\prime}\\right) = \\nabla_{\\mathbf{x}}K_{E}\\left(\\mathbf{x},\\mathbf{x}^{\\prime}\\right)\\nabla_{\\mathbf{x}^{\\prime}}^{\\top}$ that models the forces $\\mathbf{F}$ as a transformation of an unknown potential energy surface $E$ such that\n\\begin{equation}\n\\begin{aligned}\n\\boldsymbol{\\mathbf{F}}&=-\\nabla E\\\\&\\sim\\mathcal{G}\\mathcal{P}\\left[-\\nabla\\mu_{E},(\\mathbf{x}),\\nabla_{\\mathbf{x}}K_{E}\\left(\\mathbf{x},\\mathbf{x}^{\\prime}\\right)\\nabla_{\\mathbf{x}^{\\prime}}^{\\top}\\right]\\,.\n\\end{aligned}\n\\label{eq:gdml_key_idea}\n\\end{equation}\nHere, $\\mu_{E}: \\mathbb{R}^{d} \\rightarrow \\mathbb{R}$ and $K_{E}: \\mathbb{R}^{d} \\times \\mathbb{R}^{d} \\rightarrow \\mathbb{R}$ are the prior mean and covariance functions of the latent energy-based Gaussian process $\\mathcal{G}\\mathcal{P}$, respectively. The descriptor of chemical structure $\\mathbf{x}\\in \\mathbb{R}^{d}$ consists of the inverse of all $d$ pair-wise distances, which guarantees roto-translational invariance of the energy. Training on forces is motivated by the fact that they are available analytically from electronic structure calculations, with only moderate computational overhead atop energy evaluations. The big advantage is that for a training set of size $M$, only $M$ reference energies are available, whereas there are three force components for each of the $N$ atoms, i.e.\\ a total of $3NM$ force values. This means that a kernel-based model trained on forces contains more coefficients (see Eq.~\\ref{eq:kernel_regression}) and is thus also more flexible than an energy-based variant. Additionally, the amplification of noise due to the derivative operator is avoided.\n\nA limitation of the GDML method is that the structural descriptor $\\mathbf{x}$ is not permutationally invariant, because the values of its entries (inverse pairwise distances) change when atoms are re-ordered. An extension of the original approach, sGDML\\cite{chmiela2018,chmiela2019}, additionally incorporates all relevant rigid space group symmetries, as well as dynamic non-rigid symmetries of the system at hand into the kernel, to further improve its efficiency and ensure permutational invariance. Usually, the identification of symmetries requires chemical and physical intuition about the system at hand, which is impractical in an ML setting. Here, however, a data-driven multi-partite matching approach is employed to automatically recover permutations of atoms that appear within the training set\\cite{chmiela2019}. A matching process finds permutation matrices $\\mathbf{P}$ that realize the assignment between adjacency matrices $(\\mathbf{A})_{ij} = \\|\\vec{r}_i - \\vec{r}_j\\|$ of molecular graph pairs $G$ and $H$ in different energy states\n\\begin{equation}\n\\operatorname*{arg\\,min}_{\\tau} \\mathcal{L}(\\tau) = \\|\\mathbf{P}(\\tau)\\mathbf{A}_G\\mathbf{P}(\\tau)\\tran - \\mathbf{A}_H\\|^2\n\\label{eq:matching_objective}\n\\end{equation}\nand thus between symmetric transformations\\cite{Umeyama1988}. The resulting approximate local pairwise assignments are subsequently globally synchronized using transitivity as the consistency criterion\\cite{pachauri2013solving} to eliminate impossible assignments. By limiting this search to the training set, combinatorially feasible, but physically irrelevant permutations $\\tau$ are automatically excluded (ones that are inaccessible without crossing impassable energy barriers). Such hard symmetry constraints (derived from the training set) greatly reduce the intrinsic complexity of the learning problem without biasing the estimator, since no additional approximations are introduced.\n\n\\subsubsection{Gaussian approximation potentials (GAPs)}\n\\label{subsubsec:gaussian_approximation_potentials}\nGaussian approximation potentials (GAPs)\\cite{bartok2010gaussian} were originally developed for materials such as bulk crystals, but were later also applied to molecules\\cite{bartok2017machine}.\nThey scale linearly with the number of atoms of a systems and can accommodate for periodic boundary conditions.\nSimilar to high-dimensional neural network potentials\\cite{behler2007generalized} (see Section~\\ref{subsubsec:high_dimensional_neural_network_potentials}), GAPs decompose each system into atom-centered environments $i$ such that its energy can be written as the sum of atomic contributions\n\\begin{equation}\nE=\\sum_{i=1}^N E_i(\\{\\mathbf{r}_{ij}\\}_{j \\in [1, N]})\\,,\n\\label{eq:gap_local_contributions}\n\\end{equation}\nwith $\\mathbf{r}_{ij} = \\mathbf{r}_{j} - \\mathbf{r}_{i}$ and $\\mathbf{r}_{i}$ being the position of atom $i$. A smooth cutoff function is applied to the pairwise distances $\\lVert \\mathbf{r}_{ij} \\rVert$ to ensure that the contributions $E_i$ are local and no discontinuities are introduced when atoms enter or leave the cutoff radius. Even though such a decomposition is inherently non-unique and no labels for atom-wise energies are available in the reference data, they can still be approximated \nby a Gaussian process: The sum over atomic environments can be moved into the kernel function, yielding a kernel for systems $\\mathbf{x}$ and $\\mathbf{x}'$ with $N$ and $N'$ atoms, respectively:\n\\begin{equation}\nK(\\mathbf{x}, \\mathbf{x}') = \\sum_{i=1}^N \\sum_{j=1}^{N'} K_\\text{local}(x_i, x'_j)\\,.\n\\label{eq:gap_kernel}\n\\end{equation}\nThus, reference energies for the whole system are sufficient for the model to learn a suitable energy decomposition into atomic environments.\n\nSeveral descriptors and kernels for GAPs have been developed based on a local ``atomic density'' $\\rho(\\mathbf{r}) = \\sum_j \\delta(\\mathbf{r} - \\mathbf{r_j})$.\nInitially, \\citet{bartok2010gaussian} proposed to employ local atomic coordinates projected onto a 4D hyper sphere. \nSince this projection can represent the volume of a 3D sphere, the introduction of an additional radial basis can be avoided.\nTo achieve rotational invariance, the bispectrum of 4D spherical harmonics of these coordinates was used as a descriptor. Alternatively, the SOAP (\\textit{smooth overlap of atomic positions}) kernel\\cite{bartok2013representing} is defined as the integral over rotations $\\mathcal{R}$ of atomic densities\n\\begin{equation}\nK(\\rho, \\rho') = \\int d\\mathcal{R} \\left| \\int \\rho(\\mathbf{r}) \\rho'(\\mathcal{R}\\mathbf{r}) \\mathrm{d}\\mathbf{r} \\right|^n\\,.\n\\label{eq:soap_kernel}\n\\end{equation}\nGiven smoothed local densities $\\rho(\\mathbf{r})=\\sum_j\\exp(-\\gamma\\|\\mathbf{r}-\\mathbf{r_j}\\|^2)$, it has been shown that the SOAP kernel is equivalent to the linear kernel over the SO(3) power spectrum and bispectrum for $n=2$ and $n=3$, respectively\\cite{bartok2013representing}.\nBoth approaches are invariant to permutation of neighboring atoms as well as the rotation of the local environment.\nFurther representations include best matches of the atomic densities over rotations\\cite{de2016comparing} and kernels for symmetry-adapted prediction of tensorial properties\\cite{grisafi2018symmetry,csanyi2020machine}.\n\n\\subsubsection{Neural Network Potentials}\n\\label{subsubsec:high_dimensional_neural_network_potentials}\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.99\\textwidth]{nn_architectures.png}\n    \\caption{Overview of descriptor-based (top) and end-to-end (bottom) NNPs. Both types of architecture take as input a set of $N$ nuclear charges $Z_i$ and Cartesian coordinates $\\mathbf{r}_i$ and output atomic energy contributions $E_i$, which are summed to the total energy prediction $E$ (here $N=9$, an ethanol molecule is used as example). In the descriptor-based variant, pairwise distances $r_{ij}$ and angles $\\alpha_{ijk}$ between triplets of atoms are calculated from the Cartesian coordinates and used to compute hand-crafted two-body ($G^2$) and three-body ($G^3$) ACSFs (see Eqs.~\\ref{eq:two_body_symmetry_function}~and~\\ref{eq:three_body_symmetry_function}). For each atom $i$, the values of $M$ different $G^2$ and $K$ different $G^3$ ACSFs are collected in a vector $\\mathbf{x}_i$, which serves as a fingerprint of the atomic environment and is used as input to an NN predicting $E_i$. Information about the nuclear charges is encoded by having separate NNs and sets of ACSFs for all (combinations of) elements. In end-to-end NNPs, $Z_i$ is used to initialize the vector representation $\\mathbf{x}_i^0$ of each atom to an element-dependent (learnable) embedding (atoms with the same $Z_i$ start from the same representation). Geometric information is encoded by iteratively passing these descriptors (along with pairwise distances $r_{ij}$ expanded in radial basis functions $\\mathbf{g}(r_{ij})$) in $T$ steps through NNs representing interaction functions $\\mathcal{F}^t$ and atom-wise refinements $\\mathcal{G}^t$ (see Eq.~\\ref{eq:general_message_passing}). The final descriptors $\\mathbf{x}^T_i$ are used as input for an additional NN predicting the atomic energy contributions (typically, a single NN is shared among all elements).}\n    \\label{fig:nnp_overview}\n\\end{figure*}\n\nThe first Neural Network Potentials (NNPs) used a set of internal coordinates, e.g.\\ distances and angles, as structural representation to model the PES\\cite{blank1995neural,brown1996combining,tafeit1996neural,no1997description,prudente1998fitting}. While being roto-translationally invariant, internal coordinates impose an arbitrary order on the atoms and are thus not reflecting the equivalence of permuted inputs. As a result, the NNP might assign different energies to symmetrically equivalent structures.\nBeyond that, the number of atoms influences the dimensionality of the input $\\mathbf{x}$, limiting the applicability of the PES to chemical systems of the same size.\nDecomposing the energy prediction in the spirit of a many-body expansion circumvents these issues\\cite{manzhos2006random,manzhos2007using,malshe2009development}, however, it scales imperfectly with system size and number of chemical species, because each term in the many-body expansion has to be modeled by a separate NN. \n\n\\citet{behler2007generalized} where the first to propose so-called High-dimensional Neural Networks Potentials (HDNNPs), where the total energy of a chemical system is expressed as a sum over atomic contributions $E=\\sum_{i}E_i$, predicted by the same NN (or one for each element). The underlying assumption is that the energetic contribution $E_i$ of each atom depends mainly on its local chemical environment. As all atoms of the same type are treated identically and summation is commutative, the output does not change when the input is permuted.\nDue to the decomposition into atomic contributions, systems with varying numbers of atoms can be predicted by the same NNP. In principle, this framework also enables transferability between system sizes, e.g.\\ a model can be trained on small systems, but applied to predict energies and forces for larger systems. \nHowever, this requires sufficient sampling of the local environments to remove spurious correlations caused by the training data distribution, as well as corrections for long-range effects.\n\nThe introduction of HDNNPs inspired many NN architectures that can be broadly categorized into two types. \\emph{Descriptor-based} NNPs\\cite{behler2011atom,khorshidi2016amp,artrith2017efficient,unke2018reactive} rely on fixed rules to encode the environment of an atom in a vector $\\mathbf{x}$, which is then used as input for an ordinary feed-forward NN (see Eq.~\\ref{eq:deep_feedforward_nn}). These architectures include many variants of the original Behler-Parrinello network, such as ANI\\cite{smith2017ani} and TensorMol\\cite{yao2018tensormol}. \nOn the other hand, \\textit{end-to-end} NNPs\\cite{duvenaud2015convolutional,kearnes2016molecular,schutt2017quantum,gilmer2017neural}, take nuclear charges and Cartesian coordinates as input and learn a suitable representation from the data. \n\nMany end-to-end NNPs have been inspired by the graph neural network model by \\citet{scarselli2008graph} and were later collectively cast as message-passing neural networks (MPNNs)\\cite{gilmer2017neural}. A prominent example is the Deep Tensor Neural Network (DTNN)\\cite{schutt2017quantum}. Since its introduction, this approach has been refined to create new architectures, such as\nSchNet\\cite{schutt2017schnet, schutt2018quantum},\nHIP-NN\\cite{lubbers2018hierarchical} or\nPhysNet\\cite{unke2019}.\nEnd-to-end NNPs that do not directly fall into the category of MPNNs are covariant compositional networks that are able to employ features of higher angular momentum\\cite{hy2018predicting,anderson2019cormorant,weiler20183d} as well as models using a pseudo-density as input\\cite{eickenberg2017solid}.\n\nBecause no fixed rule is used to construct descriptors, end-to-end NNPs are able to automatically adapt the environment representations $\\mathbf{x}$ to the reference data (in contrast to the descriptor-based variant). However, as long as $\\mathbf{x}$ is invariant with respect to translation, rotation, and permutation of symmetry equivalent atoms, both types of NNPs adhere to all physical constraints outlined in Section~\\ref{subsubsec:physical_constraints}. NNPs are commonly used to predict energies, while conservative forces are obtained by derivation. Despite being energy-based, it is still possible to incorporate information from \\textit{ab initio} forces by including them in the loss term optimized during training. \nAt this point, it should be noted that the requirement for differentiable models excludes the use of certain activation functions, for example the popular ReLU activation\\cite{nair2010rectified}, when constructing ML-FFs based on neural networks. To avoid discontinuities in the forces, activation functions used for NNPs must always be smooth.\n\n\\paragraph{Descriptor-based NNPs}\nThe first descriptor-based NNP introduced by \\citet{behler2007generalized} uses atom-centered symmetry functions (ACSFs)\\cite{behler2011atom} consisting of two-body terms\n\\begin{equation}\nG^2_i = \\sum_{j \\neq i}^{} e^{-\\eta(r_{ij}-r_s)^2}f_{\\rm cut}(r_{ij})\n\\label{eq:two_body_symmetry_function}\n\\end{equation}\nand three-body terms\n\\begin{equation}\n\\begin{split}\nG^3_i = 2^{1-\\zeta}\\sum_{j,k \\neq i}^{} \\left(1+\\lambda\\cos\\theta_{ijk}\\right)^\\zeta e^{-\\eta\\left(r_{ij}^2+r_{ik}^2+r_{jk}^2\\right)} \\\\ \n\\times f_{\\rm cut}(r_{ij})f_{\\rm cut}(r_{ik})f_{\\rm cut}(r_{jk})\n\\end{split}\n\\label{eq:three_body_symmetry_function}\n\\end{equation}\n to encode information about the chemical environment of each atom $i$. Here, $r_{ij}$ is the distance between atoms $i$ and $j$, $\\theta_{ijk}$ the angle spanned by atoms $i$, $j$ and $k$ centered around $i$, and the summations run over all atoms within a cutoff distance $r_{\\rm cut}$. As the atom order is irrelevant for the values of $G^2_i$ and $G^3_i$ and only internal coordinates are used to calculate them, all physical invariants are satisfied. A cutoff function such as\n\\begin{equation}\nf_{\\rm cut}(r) = \\begin{cases} \n\\dfrac{1}{2}\\left[\\cos\\left(\\dfrac{\\pi r}{r_{\\rm cut}}\\right)+1\\right]\\,, & r\\leq r_{\\rm cut} \\\\\n0\\,, & r > r_{\\rm cut}\n\\end{cases}\n\\label{eq:behler_parrinello_cutoff}\n\\end{equation}\nensures that $G^2_i$ and $G^3_i$ vary smoothly when atoms enter or leave the cutoff sphere and the parameters $\\eta$, $r_s$, $\\zeta$,  and $\\lambda(=\\pm1)$ determine to which distances, or combinations of angles and distances, the ACSFs are most sensitive. When sufficiently many $G^2_i$ and $G^3_i$ with different parameters are combined and stored in a vector $\\mathbf{x}_i$, they form a ``fingerprint'' of the local environment of atom $i$. This environment descriptor is then used as input for a neural network for predicting the energy contributions $E_i$ of atoms~$i$ and the total energy $E=\\sum_{i}E_i$ is obtained by summation.\n\nSince the ACSFs only use geometric information, they work best for systems containing only atoms of one element, for example crystalline silicon\\cite{behler2007generalized}. To describe multi-component systems, typically, the symmetry functions are duplicated for each combination of elements and separate NNs are used to predict the energy contributions for atoms of the same type\\cite{behler2015constructing}. Since the combinatorial explosion can lead to a large number of ACSFs for systems containing many different elements, an alternative is to modify the ACSFs with element-dependent weighting functions\\cite{gastegger2018wacsf}. Most descriptor-based NNPs, such as ANI\\cite{smith2017ani} and TensorMol\\cite{yao2018tensormol}, use variations of Eqs.~\\ref{eq:two_body_symmetry_function}~and~\\ref{eq:three_body_symmetry_function} (sometimes allowing parameters of ACSFs to be optimized during training) to construct the environment descriptors $\\mathbf{x}_i$. Different ways to encode the structural information are possible, for example using three-dimensional Zernike functions\\cite{khorshidi2016amp}, or the coefficients of a spherical harmonics expansion\\cite{unke2018reactive}, but the general principle remains the same. Also, while most descriptor-based NNPs use separate parametrizations for different elements, it is also possible to use a single NN to predict all atomic energy contributions\\cite{unke2018reactive}. The common feature for all variations of this approach is that the functional form of the environment descriptor is predetermined and manually designed.\n\n\\paragraph{End-to-end NNs}\nA potential drawback of the previously introduced ACSFs is their restriction to collective symmetry functions, i.e.\\ the choice of descriptor may limit the expressive power of the neural network and designing a good descriptor often requires expert knowledge. Additionally, a growing number of input dimensions can quickly become computationally expensive, both for calculating the descriptors and for evaluating the NN. This is especially the case when modeling multi-component systems, where commonly orthogonality is assumed between different elements (which increases the number of input dimensions) or the descriptors are simply weighted by an element-dependent factor (which may limit the structural resolution of the descriptor).\n\nIn contrast, end-to-end NNPs directly take atomic types and positions as inputs to learn suitable representations from the reference data. Similar to descriptor-based NNPs, many end-to-end NNPs obtain the total energy $E$ as a sum of atomic contributions $E_i$. However, those are predicted from learned features $\\mathbf{x}_i$ encoding information about the local chemical environment of each atom $i$. This allows them to adapt the features based on the size and distribution of the training set as well as the chemical property of interest during the training process. The idea is to learn a mapping to a high-dimensional feature space, so that structurally (and energetically) similar atomic environments lie close together and dissimilar ones far apart.\n\nWithin the deep tensor neural network framework\\cite{schutt2017quantum}, this is achieved by iteratively refining the atomic features $\\mathbf{x}_i$ based on neighboring atoms. The features are initialized to $\\mathbf{x}_i^0 = \\mathbf{e}_{Z_i}$, where $\\mathbf{e}_{Z}$ are learnable element-dependent representations that are updated for $T \\in [3, 6]$ steps. This procedure is inspired by diffusion graph kernels\\cite{kondor2002diffusion} as well as the graph neural network model by \\citet{scarselli2008graph}.\nMany end-to-end networks have adapted this approach which can be written in general as\n\\begin{equation}\n\\mathbf{x}_{i}^{t+1} = \\mathcal{G}^t\\left(\\mathbf{x}_{i}^{t} + \\sum_{j \\neq i} \\mathcal{F}^{t}\\left(\\mathbf{x}_{i}^{t}, \\mathbf{x}_{j}^{t},\\mathbf{g}(r_{ij})\\right)f_{\\rm cut}(r_{ij})\\right)\\,,\n\\label{eq:general_message_passing}\n\\end{equation}\nwhere the summation runs over all atoms within a distance $r_{\\rm cut}$ and a cutoff function $f_{\\rm cut}$ ensures smooth behavior when atoms cross the cutoff.\nHere, the ``atom-wise'' function $\\mathcal{G}^{t}$ is used to refine the atomic features after they have been updated with information from neighboring atoms through the interaction-function $\\mathcal{F}^{t}$. Usually, the interatomic distance $r_{ij}$ is not used directly as input to $\\mathcal{F}^{t}$, but expanded in a set of uniformly spaced radial basis functions\\cite{schutt2017quantum,schutt2017schnet,unke2019} to form a vectorial input $\\mathbf{g}(r_{ij})$.\nBoth $\\mathcal{F}^{t}$ and $\\mathcal{G}^{t}$ functions are NNs with the specific implementations varying between different end-to-end NNP architectures. As only pair-wise distances are used and the order of atoms is irrelevant due to the commutative property of summation, the features $\\mathbf{x}_i$ obtained by Eq.~\\ref{eq:general_message_passing} are automatically roto-translationally and permutationally invariant (and thus also the energy predictions).\n\n\\citet{gilmer2017neural} have cast graph networks of this structure as message-passing neural networks and proposed a variant that uses a set2set decoder instead of a sum over energy contributions to achieve permutational invariance of the energy.\nSchNet\\cite{schutt2017schnet} takes an alternative view of the problem: the atoms of the molecule sparsely populate the continuous space while their interactions are modeled with convolutions. The convolution filters need to be continuous (to have smooth predictions) but are evaluated at finite points, i.e.\\ the positions of neighboring atoms. To ensure rotational invariance, only radial filters are used, leading again to an interaction function that is a special case of Eq.~\\ref{eq:general_message_passing}.\n\nWhile the previously introduced approaches aim to learn as much as possible from the reference data guaranteeing only basic quantum-chemical regularities, several models have been proposed to better exploit chemical domain knowledge.\nThe hierarchical interacting particle neural network (HIP-NN)\\cite{lubbers2018hierarchical} obtains the prediction as a sum over atom-wise contributions $E_i^t$ that are predicted after every update step $t$. A regularizer penalizes larger energy contributions in deeper layers, i.e.\\ enforcing a declining, hierarchical prediction of the energy.\nPhysNet\\cite{unke2019} modified the energy function to include explicit terms for electrostatic and dispersion interactions,\n\\begin{equation}\nE = \\sum_{i=1}^N E_i + k_e \\sum_{i=1}^N \\sum_{j>i}^N \\tilde{q}_i \\tilde{q}_j \\chi(r_{ij}) + E_\\text{D3}\\,,\n\\end{equation}\nwhere $E_\\text{D3}$ is Grimme's D3 dispersion correction\\cite{grimme2010consistent}, $k_e$ is Coulomb's constant and $\\tilde{q}_i$ are corrected partial charges predicted by the network that are guaranteed to sum to the total charge of the molecule. In an ablation study on a dataset of S$_\\text{N}$2 reactions, \\citet{unke2019} showed that the inclusion of long-range terms improves prediction accuracy for energies and forces while models without these terms show qualitatively wrong asymptotic behavior.\n\n\\section{Best practices and Pitfalls}\n\\label{sec:best_practices_and_pitfalls}\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{best_practices.png}\n    \\caption{Overview of the most important steps when constructing and using ML-FFs.}\n    \\label{fig:best_practices}\n\\end{figure*}\n\nA number of careful modeling steps are necessary to construct an ML-FF for a particular problem of interest (Fig.~\\ref{fig:best_practices}). Even before starting this process, some forethought is appropriate due to certain limitations of \\textit{ab initio} methods themselves. This section gives an overview about all steps necessary to construct an ML-FF from scratch and highlights possible ``pitfalls'', i.e.\\ issues that may occur along the way, in particular when the recommended practices are not followed. First, some preliminary considerations, which should be taken before starting with the construction of an ML-FF, are discussed (\\ref{subsec:preliminary_considerations}). Next, basic principles for choosing an appropriate ML method for a specific task are given (\\ref{subsec:choosing_an_appropriate_ml_method}). Then, the importance of high quality reference data, different strategies to collect it (\\ref{subsec:data_collection}), and how the data has to be prepared (\\ref{subsec:data_preparation}), are outlined. This is followed by an overview of how to train an ML model on the collected data (\\ref{subsec:training_the_model}) and guidelines for using the trained ML-FF in a production setting, e.g.\\ for running MD simulations (\\ref{subsec:using_the_model_in_production}). Finally, popular software packages for constructing ML-FFs are briefly described and code examples are given (\\ref{subsec:example_code_and_software_packages}).\n\n\\subsection{Preliminary considerations}\n\\label{subsec:preliminary_considerations}\nBefore running any \\textit{ab initio} calculations to collect data for training learning models, it is advisable to think about the limitations of the chosen level of theory itself. The issues discussed here are problem-specific and often not unique to ML-FFs, but PES reconstruction in general. As such, a comprehensive list is not possible, but a few examples are given below.\n\n\\paragraph{Practicability}\nOn the spectrum of quantum chemistry methods, ML-FFs fit into the niche between highly efficient conventional FFs\\cite{monticelli2013force} and accurate, but computationally expensive \\textit{ab initio} methods.\\cite{friesner2005ab} Efficiency-wise, they are still inferior to classical FFs, because their functional forms are considerably more complex and thus more expensive to evaluate. Even the fastest ML-FFs are still one to three orders of magnitude slower\\cite{chmiela2019,brickel2019reactive,sauceda2020molecular}. On the other end, ML-FFs are lower bounded by the accuracy of the reference data used for training, which means that the underlying \\textit{ab initio} method will always be at least equally accurate. In practical terms, this means that in order to be useful, ML-FFs need to offer time savings over directly running \\textit{ab initio} calculations and an improved accuracy compared to conventional FFs. For this purpose, the full procedure of data generation, training and inference must be taken into account, as opposed to just regarding inference speed, which will almost certainly be much quicker than \\textit{ab initio} methods. While this consideration sounds trivial at first, it is still advisable to think about whether constructing an ML-FF really is economical. For example, if the goal is to run just a single short MD trajectory, the question is how much data is necessary for the model to reach the required accuracy. Some models may require several thousands of training points to produce accurate enough predictions, even for fairly small molecules. Then, when factoring in the overall time required for going through the process of creating the ML model, testing it, and running the MD simulation, it might be more efficient to simply run an \\textit{ab initio} MD simulation in the first place.\n\n\\paragraph{Multireference effects}\nMany \\textit{ab initio} methods use a single Slater determinant to express the wave function of a system. The problem with this approach is that different determinants may be dominant in different regions of the PES, leading to a poor description of the wave function if the wrong determinant is chosen. Especially when many calculations are performed for various strongly distorted geometries, for example when a reaction is studied and bonds need to be broken, it may happen that the solution ``jumps'' discontinuously from one electronic state to another, leading to inconsistent reference data. When an ML model is trained on such a data set, it will try to find a compromise between the inconsistencies and its performance typically be unsatisfying. It is therefore advisable to check for possible multireference effects prior to generating data and, if necessary, switch to a multireference method (for a comprehensive review on multireference methods, see Ref.~\\citenum{szalay2012multiconfiguration}).\n\n\\paragraph{Strong delocalization}\nThe models discussed in this review all assume that energy contributions are local to some degree. This assumption is either introduced explicitly by a cutoff radius, or it enters the model through the use of a specific structural descriptor. For example, by using inverse distances to encode chemical structures for kernel methods (as is done e.g.\\ in GDML, see Section~\\ref{subsubsec:symmetric_gradient_domain_machine_learning}), relative changes between close atoms are weighted more strongly when comparing two conformations. While assuming locality is valid in many practical applications, there exist many cases where this assumption breaks down. An example are extensive conjugated $\\pi$-systems, where a rotation around certain bonds might break the favorable interaction between electrons, leading to a ``non-local'' energy contribution. If such effects exist, an appropriate model should be chosen, for example the cutoff radius may need to be larger than usual or a different structural descriptor must be picked. \n\n\\subsection{Choosing an appropriate ML method}\n\\label{subsec:choosing_an_appropriate_ml_method}\nSeveral different variants of ML-FFs have been discussed in Section~\\ref{subsec:combining_ml_and_chemistry} and many more are described in the literature. Although all these methods can be applied to construct ML-FFs for any chemical system, some methods might be more promising than others depending on the very situation. For researchers who want to apply ML methods to a specific problem for the first time, the abundance of different models to choose from may be overwhelming and it might be difficult to find an appropriate choice. \n\nIn the following, possible applications of ML-FFs are broadly categorized based on simple questions about the task at hand. For each case, advantages and disadvantages of individual models are discussed to provide help and guidance to the reader for identifying an appropriate model for their use case.\n\n\\paragraph{How much reference data can be used for training?} \\textit{\n\tWhen in doubt which method to use, a rule of thumb could be to prefer kernel methods when there are less than $\\sim10^3$--$10^4$ training points and NN-based approaches otherwise (but this may also be a matter of preference).\n}\n\nDepending on the desired accuracy, the amount of \\emph{ab initio} reference data which can be collected within a reasonable time frame may be vastly different. For example, if reference calculations are performed at the DFT level of theory, it is often feasible to collect several thousands of data points, even for relatively large molecules. On the other hand, if CCSD(T) accuracy and a large basis set is required, already a few hundred reference calculations for small molecules can require a considerable amount of computing time. Although it is of course always desirable to perform as few reference calculations as possible, for some tasks, collecting a large data set is unavoidable. For example, if a model should be able to predict a variety of different molecules containing many different elements, the relevant chemical space must be sampled sufficiently.\n\nIn general, kernel-based models tend to achieve good prediction accuracies even with few training points, whereas NNs often need more data to reach their full potential (although there may be exceptions for both model variants, see also Fig.~\\ref{fig:md17_scatter}). Further, the optimal model parameters for kernel models can be determined analytically (see Eq.~\\ref{eq:krr_coefficient_relation_regularized}) which, for small datasets, is typically faster than training a NN via (a variant of) stochastic gradient descent. However, when the data set size $N$ is very large, solving Eq.~\\ref{eq:krr_coefficient_relation_regularized} can become prohibitively expensive as it scales $\\mathcal{O}(N^3)$ and training an NN may become more efficient. Further, evaluating kernel models scales with $\\mathcal{O}(N)$ (see Eq.~\\ref{eq:kernel_regression}), whereas the cost of evaluating NN-based methods has (as long as the number of parameters does not have to be increased for larger datasets) constant complexity. For this reason, NNs tend to be more suitable for large datasets. Note that there are approximations which improve the scaling of kernel methods (so they can be applied even to very large datasets), this however at the cost of accuracy (see Eqs.~\\ref{eq:krr_coefficient_relation_approximated}-\\ref{eq:nystroem_regularized_inverse}).\n\n\n\n\\paragraph{Should the model be able to predict a single type of chemical system or multiple different ones?}\n\\textit{\n\tTo be applicable to multiple systems, a model must decompose its prediction into atomic contributions. For global models, either a fixed size descriptor must be used or several separate models need to be trained.\n}\n\n\nSome ML-FFs only need to be able to predict systems with a fixed composition and number of atoms, for example to study the dynamics of a single molecule, whereas other applications require the ability to predict different systems with varying size, e.g.\\ when clusters consisting of a different number and kind of molecules are studied with the same model.\n\nWhile all ML-FFs can be applied in the first case, the latter requires either that the length of chemical descriptors is independent of the number of atoms, or that model predictions are decomposed into local contributions based on fixed-size fingerprints of atomic environments (which naturally makes them extensive). Most NNPs (see Section~\\ref{subsubsec:artificial_neural_networks}) and many kernel methods, e.g.\\ GAPs (see Section~\\ref{subsubsec:gaussian_approximation_potentials}) or FCHL\\cite{faber2018alchemical,christensen2020fchl}, use such an approach and can be applied to differently sized chemical systems without issues. An exception are e.g.\\ (s)GDML models (see Section~\\ref{subsubsec:symmetric_gradient_domain_machine_learning}), which encode chemical structures as vectors of inverse distances between atomic pairs. Consequently, the length of the descriptor changes with the number of atoms and the model can only be applied to a single type of system. In some special cases, it may be possible to choose a maximum descriptor length and pad descriptors of smaller molecules with zeros, but this may introduce other problems and\/or reduce the accuracy. \n\n\n\\paragraph{Will the model be applied to single or multi-component systems?}\n\\textit{\n\tIf only a handful of elements is relevant, all models are equally suitable. When a large number of elements needs to be considered, the model should be able to encode and use information about atom types efficiently.\n}\n\nAs long as an ML-FF is only applied to single-component systems (consisting of a single element), for example elemental carbon or silicon, all relevant information is contained in the relative arrangement of atoms and nuclear charges need not be encoded explicitly. However, as soon as there are multiple atom types (as is common for most applications of ML-FFs), the model must have some way to distinguish between them. A notable exception are some global models such as (s)GDML. Here, although only inverse pairwise distances are used as a descriptor, information about atom types is implicitly contained, because specific entries always correspond to the same combination of atom types. \n\nMany local descriptors of atomic environments only use geometric information in the form of distances and angles between pairs and triplets of atoms (see Eqs.~\\ref{eq:two_body_symmetry_function}~and~\\ref{eq:three_body_symmetry_function}). To include information about atomic types, descriptor entries are duplicated for every possible combination of elements (the same approach is used for kernel machines based e.g.\\ on SOAP\\cite{bartok2013representing} or FCHL\\cite{faber2018alchemical,christensen2020fchl}). Many descriptor-based NNPs further use separate NNs to predict atomic contributions of different elements (see Fig.~\\ref{fig:nnp_overview}). A disadvantage of these approaches is that the number of terms in the descriptor increases combinatorially with the number of elements covered by the model (in particular if three-body or even four-body terms are used), which impacts the computational cost of training and evaluating the model (in particular larger amounts of training data become necessary). As long as only a few elements need to be considered, this is not an issue, but if a model for a significant fraction of the periodic table is required, a more efficient representation is desirable. Most end-to-end NNPs employ so-called element embeddings (see Fig.~\\ref{fig:nnp_overview}), which do not become more complex when the number of elements is increased. This has the additional benefit of  potentially increasing the data efficiency of the model by utilizing alchemical information. Another alternative is to introduce element-dependent weighting functions (instead of duplicating terms in the descriptor)\\cite{gastegger2018wacsf}. \n\n\\paragraph{Will the model be applied to small or large systems?} \\textit{Models for very large target systems should be able to exploit chemical locality, so that reference calculations for fragments can be used as training data. Additionally, this allows trivial parallelization of predictions over multiple machines.}\n\nOften, ML-FFs are used to study small or medium-sized molecules. In such cases, all models are equally applicable. For very large systems containing many atoms however, some methods have particularly advantageous properties. For example, it might be infeasible to run \\textit{ab initio} calculations for the full target system. In this case, being able to fragment the system into smaller parts, for which reference calculations are affordable, would be very useful. \n\nTo be trainable on such fragments, ML-FFs must introduce an explicit assumption about chemical locality by introducing a cutoff radius. Every method that decomposes predictions into a sum of local atomic contributions can thus be trained in this way. Global ML-FFs on the other hand need reference data for the full system (see above). Another advantage of local models is that their predictions are \\emph{embarrassingly parallel}: The contributions of individual atoms can be calculated on separate machines (storing a copy of the model), each requiring only information about neighboring atoms within the cutoff radius. Apart from possible efficiency benefits, this may even become necessary if the computations to handle all atoms do not fit into the memory of a single machine (for example when the system of interest consists of millions of atoms\\cite{jia2020pushing}). Note that while not all ML methods to construct FFs are embarrassingly parallel, most models contain mostly linear operations, which are amenable to other parallelization methods, e.g.\\ by utilizing GPUs.\n\nAt this point, a subtle difference between cutoffs used in NNPs of the message-passing type (see Section~\\ref{subsubsec:high_dimensional_neural_network_potentials}) and descriptor-based NNPs (as well as kernel machines based on local atomic environments) should be pointed out. In message-passing schemes, information between all atoms within the cutoff radius is exchanged over $T$ iterations, thus the \\emph{effective} cutoff radius increases by a factor of $T$. This means that in order to distribute the computation over multiple machines, it is either necessary to communicate updates to other machines after each iteration, or a sufficiently large subdomain needs to be stored on all machines. \n\n\n\\paragraph{Are long-range interactions expected to be important for the system of interest?} \\textit{If strong long-range contributions to the energy are present, it is advisable to either use a global model, or augment a local ML method by explicitly including physical interaction terms.}\n\nAs described earlier, many ML-FFs introduce cutoffs to exploit chemical locality. An obvious downside of this approach is that all interactions beyond the cutoff cannot be represented. For uncharged molecules without strong dipole moments, relevant interactions are usually sufficiently short-ranged that this is not problematic. However, when strong long-ranged (e.g.\\ charge-dipole) interactions are important, cutoffs may introduce significant errors. Models such as (s)GDML, which consider the global chemical structure, do not suffer from this issue. \n\nWhile it is possible to simply increase the cutoff distance until more long-ranged contributions can be neglected, this decreases the computational efficiency and data efficiency of models. A better alternative could be to include the relevant physical interaction terms explicitly in the model. For example, TensorMol\\cite{yao2018tensormol} and PhysNet\\cite{unke2019} include such correction terms by default, but other models can be augmented in a similar fashion. Although not strictly necessary, even global models may profit from such terms by an increased data efficiency.\n\n\n\\subsection{Data collection}\n\\label{subsec:data_collection}\nA fundamental component of any ML model is the reference data. While its architecture and other technical details are responsible for the potential accuracy of a model, the choice of reference data and its quality defines the reliability and range of applicability of the final model. Any deficiencies that are present in the data will inevitably also lead to artifacts in models trained on it, a principle often colloquially stated as ``garbage in, garbage out''\\cite{sanders2017garbage}. As such, the reference data is one of the most important components of an ML-FF. The generation of datasets in computational chemistry and physics are challenges on their own. First of all, each reference point is a result of computationally expensive and often nontrivial calculations (see Section~\\ref{subsec:chemistry_foundations}), which limits the amount of data that can be collected. Furthermore, the dimensionality of the configurational space of molecules, solids, or liquids is so vast that -- except for trivial cases -- it is not apparent how to identify the representative geometries in the ocean of possibilities. The optimal choice of reference data might even need to be adapted to the individual properties of the respective ML model that consumes it and\/or its intended application. In the following, several strategies for sampling the PES and generating reference datasets are outlined (multiple of these approaches can be combined). Afterwards, problems that may occur due to insufficient sampling are highlighted and general remarks about the importance of a consistent reference dataset are given.\n\n\\paragraph{AIMD sampling}\nA good starting point to assemble the reference dataset is by sampling the PES using \\textit{ab initio} molecular dynamics (AIMD) simulations. Albeit expensive in terms of the amount of necessary reference calculations, this technique constitutes a straightforward way to explore configurational space. Here, the temperature of the simulation determines which regions of the PES and what energy ranges (according to the Boltzmann distribution) are explored (see Fig.~\\ref{fig:FigEthSampl}). For example, if the aim is to construct an ML-FF for calculating the vibrational spectrum of ethanol at 300~K, generating the database at 500~K is a safe option since the subspace of configurations relevant at 300~K is contained in the resulting database (see Fig.~\\ref{fig:FigEthSampl}A). Sampling at higher temperatures ensures that the model does not enter the extrapolation regime during production runs, which is practically guaranteed to happen when a lower temperature is used for sampling. In general, the resulting dataset will be biased towards lower energy regions of the PES, where the system spends most of the simulation time. For this reason, pure AIMD sampling is only advisable when the intended application of the final ML model involves MD simulations for equilibrium or close to equilibrium properties, where rare events do not play a major role. Examples of this are the study of vibrational spectra, minima population, or thermodynamic properties.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{Fig_Sampling_Ethanol.pdf}\n\t\\caption{\\textbf{A}: Two-dimensional projection of the sampled regions of the PES of ethanol at 100~K, 300~K and 500~K from running AIMD simulations with FHI-aims\\cite{FHIaims2009} at the DFT\/PBE+TS level of theory\\cite{PBE1996,tkatchenko2009accurate}. The length of the simulation was 500 ps. \\textbf{B}: Distribution of sampled potential energies for the three different temperatures.}\n\t\\label{fig:FigEthSampl}\n\\end{figure}\n\n\\paragraph{Sampling by proxy}\nConstructing reliable reference datasets from AIMD simulations can be computationally expensive. While system size plays a major role, other phenomena, such as the presence of intramolecular interactions and fluxional groups, can also influence how quickly the PES is explored. Because of this, long simulation times may be required to visit all relevant regions. For example, generating $2\\times10^5$ conformations from AIMD using a relatively affordable level of theory (e.g.\\ PBE+TS with a small basis set) can take between a few days to several weeks (depending on the size of the molecule). With higher levels of theory, the required computation time may increase to months, or, when  highly accurate methods such as CCSD(T) are required, even become prohibitively long (several years).\n\nTo resolve this issue, a possible strategy is to sample the PES at a lower level of theory to generate a long trajectory that covers many regions on the PES. The collected dataset is then subsampled to generate a small, but representative set of geometries, which serve as input for performing single-point calculations at a higher level of theory (see Fig.~\\ref{fig:FigSubSampl}).\nThis strategy works best when the PES has a similar topology at both levels of theory, so it can be expected that configurations generated at the lower level are representative of configurations that would be visited in an AIMD simulation at the higher level (see the two-dimensional projections of the PES in Fig.~\\ref{fig:FigSubSampl}). When the two PESs are topologically very different, e.g.\\ when a semi-empirical method or even a conventional FF is used to generate the initial trajectory, it may happen that the relevant regions of the PES at the higher level of theory are not covered sufficiently. Then, when an ML-FF is trained on the collected dataset and used for running an MD simulation, the trajectory may enter the extrapolation regime and the model might give unphysical predictions. Thus, extra care should be taken when two very different levels of theory are used for sampling by proxy.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{Fig_SubSampling.pdf}\n\t\\caption{Procedure followed to generate a database at the CCSD(T) level of theory for keto-malondialdehyde using sampling by proxy. An AIMD simulation at 500~K computed at the DFT\/PBE+TS level of theory is used to sample the molecular PES. Afterwards, the trajectory is sub-sampled (black dots) to generate a subset of representative geometries, for which single-point calculations at the CCSD(T) level of theory are performed (red dots). This highly accurate reference data is then used to train an ML-FF.}\n\t\\label{fig:FigSubSampl}\n\\end{figure}\n\n\\paragraph{Adaptive sampling}\nAnother method to minimize the amount of expensive \\textit{ab initio} calculations is called adaptive sampling or \\textit{on-the-fly} ML\\cite{csanyi2004learn}. Here, a preliminary ML-FF is trained on only a small initial set of reference data and then used to run an MD simulation. During the dynamics, additional conformations are collected whenever the model predictions become unreliable according to an uncertainty criterion. Then, new reference calculations are performed for the collected structures and the training of the ML model is continued or started from scratch on the augmented dataset. The process is repeated until no further unreliable regions can be discovered during MD simulations.\n\nWhen following this strategy, the quality of the uncertainty estimate is crucial for an efficient sampling of the PES: If the estimate is overconfident, deviations from the reference PES might be missed. If the estimate is overly cautious, many redundant \\textit{ab initio} calculations have to be performed. There exist several ways to estimate the uncertainty of an ML-FF. For example, Bayesian methods learn a probability distribution over models, which enables straightforward uncertainty estimates (see the predictive variance of a Gaussian process, Eq.~\\ref{eq:gaussian_process_variance}). For models where an explicit uncertainty estimate is not available, e.g.\\ neural networks, a viable alternative is \\textit{query-by-committee}\\cite{seung1992query,behler2015constructing}. Here, an ensemble of models is trained, for example on different subsets of the reference data and each starting from a different parameter initialization. Then, the discrepancy between their predictions can be used as uncertainty estimate. Query-by-committee has been successfully employed to sample PESs using neural networks for water dimers\\cite{morawietz2012neural}, organic molecules\\cite{gastegger2017machine,unke2019} as well as across chemical compound space\\cite{smith2018less}. Other alternatives, for example using dropout\\cite{srivastava2014dropout} as a Bayesian approximation\\cite{gal2016dropout}, could also be used.\n\nCollecting data ``on-the-fly'' is even possible without uncertainty estimates. Instead, additional reference calculations are performed at fixed intervals during the MD simulation\\cite{csanyi2004learn,li2015molecular}. This relies on the assumption that the probability of reaching the extrapolation regime of an ML model rises with increasing length of the MD trajectory.\nWhile performing \\textit{ab initio} calculations in regular intervals will discover all deviations of the model eventually, this variant of on-the-fly ML does not exploit any information about the already collected reference set and may thus lead to many redundant data points. More detailed reviews on uncertainty estimation and active sampling of PESs can be found in Refs.~\\citenum{gastegger2020molecular}\n~and~\\citenum{shapeev2020active}.\n\n\\paragraph{Normal mode sampling}\nIt is also possible to sample the PES without running any kind of MD simulation. In normal mode sampling\\cite{smith2017ani}, the idea is to start from a minimum on the PES and generate distorted structures by randomly displacing atoms along the normal modes. They are the eigenvectors of the mass-weighted Hessian matrix obtained at the minimum position, i.e.\\ a harmonic approximation of the molecular vibrations. From the associated force constants (related to the eigenvalues), the increase in potential energy for displacements along individual normal modes can be estimated. Since they are orthogonal to each other, it is straightforward to combine multiple random displacements along different normal modes such that the resulting structures are sampled from a Boltzmann distribution at a certain temperature. In other words, structures generated like this are drawn from the same distribution as if an ``approximated PES'' was sampled with a (sufficiently long) MD simulation. This approximated PES is equivalent to a Taylor expansion of the original PES around the minimum position, truncated after the quadratic term (the contribution of the linear term vanishes at extrema). \n\nStructures generated from random normal mode sampling are not correlated, in contrast to those obtained from adjacent time steps in MD simulations, which makes this approach an efficient way to explore the PES.  However, the disadvantage is that only regions close to minima can be sampled. Additionally, the harmonic approximation is only valid for small distortions, i.e.\\ the larger the temperature, the more the sampled distribution diverges from the Boltzmann distribution on the true PES. Because of these limitations, it is best to combine normal mode sampling with other sampling methods, for example to generate an initial reference dataset, which is later expanded by adaptive sampling.\n\n\\paragraph{Problems due to insufficient sampling} Because their extrapolation capabilities are limited, ML methods only give reliable predictions in regions where training data is present.\\cite{sugiyama2007covariate} When generating reference data, it is therefore important that all regions of the PES that may be relevant for a later study are sampled sufficiently. For example, when studying a reaction, the data should not only cover configurations corresponding to educt and product structures, but also the region around the transition state and along the transition pathway. When the reaction coordinate defining the transition process is already known, a straightforward way to generate the reference data would be running metadynamics\\cite{barducci2011metadynamics}. However, even when an ML model can reproduce the entire reference dataset with the required accuracy, it is still possible to run into issues when the model is used to study the reaction. If the rare transition process was not sampled sufficiently, it is not guaranteed that MD simulations with the ML-FF reproduce it correctly. The reference data may be restricted to a specific subset of molecular configurations along the transition pathway. Hence, the model can enter the extrapolation regime somewhere between the boundary states and the transition pathways generated by an MD simulation might be unreasonable. Another potential issue is that after passing the transition state region, typically, a large amount of potential energy is converted to internal motions such as bond vibrations. As a result, the effective temperature defined by the kinetic energy exceeds the ambient conditions by orders of magnitude. Even when using a thermostat in the simulation, thermal energy increases so rapidly that it may not be able to handle the increase in temperature immediately. As a consequence, the trajectory visits high-energy configurations, which may not be included in the reference data, and the model again has to extrapolate.\n\nWhen ML-FFs enter the extrapolation regime, i.e.\\ they are used to predict values outside the sampled regions of the PES, unphysical effects may be observed. Consider for example the dissociation of the O\\nobreakdash--H bond in the hydroxyl group of ethanol (Fig.~\\ref{fig:etoh_dissociation_curve}). \nHere, different models were trained on data gathered from an MD simulation of ethanol at 500~K and used to predict how the energy changes when the O\\nobreakdash--H distance of the hydroxyl group is shortened or elongated to extreme values well outside the range sampled during the dynamics. In this example, while the sGDML model is able to accurately extrapolate to much shorter distances than are present in the training data, it still fails to predict the bond dissociation. The NNP models (PhysNet and SchNet) exhibit qualitatively wrong short-range behavior and spurious minima on the PES, which may trap trajectories during MD simulations. Because of these limited extrapolation capabilities, it is advisable to sample larger regions of the PES than are expected to be visited during MD simulations, so that there is a ``buffer'' and models never enter the unreliable extrapolation regime during production runs. For example, when an ML-FF is to be used for a study at a temperature of 300~K, the PES should be sampled around 500~K or higher.\n\\begin{figure}[t]\n\t\\includegraphics[width=\\columnwidth]{ethanol_OH_dissociation_curve.png}\n\t\\caption{One-dimensional cut through the PES of ethanol along the O--H bond distance for different ML-FFs (solid blue, yellow and orange lines) compared to \\textit{ab initio} reference data (dashed black line). Close to the region sampled by the training data (range highlighted in gray), all model predictions are virtually identical to the reference method (see zoomed view). When extrapolating far from the sampled region, the different models have increasingly large prediction errors and behave unphysically.}\n\t\\label{fig:etoh_dissociation_curve}\n\\end{figure}\n\n\n\\paragraph{Importance of data consistency} Although it may appear trivial, it is crucial that all data used for training a model is internally consistent: A single level of theory (method and basis set) should be used to calculate the reference data. When multiple quantum chemical codes (or even different versions of the same code) are used for data generation, it should be checked that their output is numerically identical when given the same input geometry (if they do not then this will effectively manifest itself like noisy outputs, severely deteriorating the precision of the ML model). Further, many \\textit{ab initio} codes automatically re-orient the input geometry such that the principal moments of inertia are aligned with the $x$-, $y$- and $z$\\nobreakdash-axes, so extra care should be taken when forces or other orientation-dependent quantities (i.e.\\ electric moments) are extracted to verify they are consistent with the input geometry. When some calculation settings need to be adapted for a subset of the data, e.g.\\ for cases with difficult convergence, it is important to check that values computed with the modified settings are consistent with the rest of the data. Additionally, for training some ML models, it may be essential that atoms are ordered in a particular way throughout the data set. For example, the permutational symmetry of (s)GDML models is limited to the transformations recovered from the training set, whereas the NN models discussed in this review are fully agnostic with respect to atom indexing.\n\n\\subsection{Data preparation}\n\\label{subsec:data_preparation}\nAfter the reference data is collected, it has to be prepared for the training procedure. This includes splitting the data into different subsets, which are reserved for separate purposes. Some models may also require that the data is preprocessed in some way before the training can start. In the following, important aspects of these preparation steps are highlighted.\n\n\\paragraph{Splitting the data}\nPrior to training any ML model, it is necessary to split the reference data into disjoint subsets for training\/validation and testing (see Section~\\ref{subsubsec:model_selection_how_to_choose_hyperparameters}). While the training\/validation set is used for fitting the model, the test set is only ever used after a model is trained to estimate its generalization error, i.e.\\ to judge how well the model performs on unseen data.\\cite{hansen2013assessment,hastie2009elements} It is very important to keep the two splits separate, as it is easily possible to achieve training errors that are several orders of magnitude lower than the true generalization error when the model is not properly regularized. Many models also feature hyperparameters, such as kernel widths, regularization terms or learning rates, that must be tuned by comparing several trained model variants on a third dataset used purely for validation (split from the training\/validation set). Note that information from the validation set will still enter the model indirectly, i.e.\\ it also participates in the training process. This is why a strict separation of the training\/validation set from the test set is crucial. Undetected duplicates in the dataset can complicate splitting, as the contamination of the test set with training data (``data leakage'') might go unnoticed. In this case, the model is effectively trained on part of the test set and estimates of the generalization error might be too optimistic and unreliable. Such a scenario can occur even when no obvious mistakes were made, e.g.\\ when the structures for a dataset are sampled by running a long MD simulation where snapshots are written very frequently. Structures collected from adjacent time steps may be highly correlated in this case and when splitting the data randomly into training and test sets, a large portion of both sets will be almost identical. In such a case, instead of using a random split, a better approach would be to use a time-split of the dataset,\\cite{lemm2011introduction} e.g.\\ using the first 80\\% of the MD trajectory as the training\/validation set and reserving the last 20\\% for testing. \n\n\\paragraph{Data preprocessing}\nPrior to training a model, the raw data is often processed in some way to improve the numerical stability of the ML algorithm. For example, a common practice is normalization, where inputs (or prediction targets) are scaled and shifted to lie in the range $-1\\dots1$ or to have a mean of zero and unit variance. The constants required for such transformations must never be extracted from the complete dataset. Instead, only the training set may be used to obtain this information.\\cite{muller2001introduction,lemm2011introduction,hansen2013assessment} Otherwise, estimates of the generalization error on the test set may be overconfident (this is another form of data leakage). While normalization may be less common for the purpose of constructing ML-FFs, any ``data-dependent`` transformation must be done carefully. For example, it may be desirable to subtract the mean energy of structures from the energy labels in order to obtain numbers with smaller absolute values (for numerical reasons). This mean energy should be calculated only from the structures in the training set.\n\nIf a model is trained using a hybrid loss that incorporates multiple interdependent properties, such as energy and forces, it is important to consider the effects of the normalization procedure on the functional relationship of those values. For example, multiplying the energy labels by a factor requires that the forces are treated in the same way, because the factor carries over to the derivative. Also, while subtracting the mean value from energy labels is valid, it is not correct to add any constant to the force labels, because that would translate into a linear term in the energy domain (the energy is related to the forces through integration). Consequently, the consistency between both label types would be broken and an energy conserving model would be incapable of learning. Even when doing simple unit transformations, care should be taken not to introduce any inconsistencies. For example, when energy labels are converted from $E_{\\rm h}$ to kcal~mol$^{-1}$ and atom coordinates from $a_0$ to \\AA, force labels have to also be converted to kcal~mol$^{-1}$~\\AA$^{-1}$ so that all data is consistent. Depending on which code was used to obtain the reference data, it is even possible that units for some labels \\emph{must} be converted, because they may be given in different unit systems in the raw data (\\textit{ab initio} codes often report energy and forces in atomic units, whereas for coordinates, angstroms are popular).\n\n\\subsection{Training the model}\n\\label{subsec:training_the_model}\nAfter the data has been collected and prepared, the next step is training the ML-FF. During the training process, the parameters of the model are tuned to minimize a loss function, which measures the discrepancy between the training data and the model predictions. In some cases, the optimal solution can be found analytically. When this is not possible, the parameters are typically optimized by gradient descent or a similar algorithm (see Ref.~\\citenum{ruder2016overview} for an overview). The hyperparameters of a model (e.g.\\ the number of layers or their width in the case of NNs) can also be selected in this step, albeit by checking the model performance on the validation set after training (instead of optimizing them directly). This section details the training process and highlights important points to consider, e.g.\\ the choice of loss function or how to prevent overfitting of the model to the training data.\n\n\\paragraph{Choosing the loss function}\nFor regression tasks, a standard choice for the loss function is the mean squared error (MSE) given by $\\mathcal{L}=\\frac{1}{N}\\sum_{i=1}^{N} (y_i-\\hat{y}_i)^2$, because it punishes outliers disproportionately. Here, the index $i$ runs over all $N$ samples of the training data, $y_i$ is the reference value for data point $i$ and $\\hat{y}_i$ is the corresponding model prediction. When the MSE is used as loss function, it is implicitly assumed that any noise present in the reference data is distributed normally, which without additional information, is a sensible guess for most data. Further, the MSE loss allows finding the optimal parameters analytically (due to convexity) for linear ML algorithms, such as kernel ridge regression (see Eq.~\\ref{eq:krr_coefficient_relation_regularized} in Section~\\ref{subsubsec:kernel_based_methods}). However, the MSE is not necessarily the best choice for all cases. For example, to make the model less sensitive to outliers, a common alternative is to use a mean absolute error (MAE) loss given by $\\mathcal{L}=\\frac{1}{N}\\sum_{i=1}^{N} \\lvert y_i-\\hat{y}_i \\rvert$. Other functional forms, such as Huber loss\\cite{huber1992robust} or even an adaptive loss\\cite{barron2019general}, are also possible, provided they are a meaningful measure of model performance.\n\nAfter deciding on the general form of the loss function, the question remains which labels $y$ to use as a reference. \nWhile the potential energy is an obvious choice, in classical MD, the PES is explored via integration of Newton's second law of motion, which exclusively involves atomic forces. Since an important objective of ML-FFs is to reproduce the dynamical behavior of molecules in MD simulations as well as possible, it could even be argued that accurate force predictions should take priority over energy predictions in MD applications. However, since energy labels are usually available as a byproduct of force calculations, it seems reasonable to include both label types in the hope that this will help improve the overall prediction performance for both quantities. This gives rise to models based on hybrid loss functions that simultaneously penalize force $\\mathbf{F}$ and energy $E$ training errors. Assuming an MSE loss, it generally takes the form\n\\begin{equation}\n\\mathcal{L} = \\frac{1}{N}\\sum_{i=1}^{N} \\underbrace{\\lVert \\hat{\\mathbf{F}}_i-\\mathbf{F}_{i}\\rVert^{2}}_{\\mathcal{L}_{\\mathbf{F}_{i}}}+\\eta\\underbrace{(\\hat{E}_i-E_{i})^{2}}_{\\mathcal{L}_{E_{i}}}\\,,\n\\label{eq:multi_objective_loss}\n\\end{equation}\nwhere the hyperparameter $\\eta$ determines the relative weighting between both loss terms to account for differences in units, information content, and noise level of the label types. A bilateral reduction of both loss terms is only possible if the objectives are non-competing, i.e.\\ when the optimal parameter set is equally effective across both tasks. For this to be true, it must hold that \n\\begin{equation}\n\\begin{aligned}\n\\lVert\\hat{\\mathbf{F}}_i - \\mathbf{F}_i\\rVert^2 =& \\eta(\\hat{E}_i - E_i)^2\\,,\\\\\n\\Rightarrow \\lVert-\\nabla_{\\mathbf{R}}(\\hat{E}_i - E_i)\\rVert^2 =&  \\eta(\\hat{E}_i - E_i)^2\n\\end{aligned}\n\\label{eq:loss_relation}\n\\end{equation}\n at every training point $i$ (here, the relation $\\mathbf{F}=-\\nabla_{\\mathbf{R}}E$ was substituted). Otherwise, the objectives $\\mathcal{L}_{\\mathbf{F}_{i}}$ and $\\mathcal{L}_{E_{i}}$ are necessarily minimized by a different set of model parameters. Eq.~\\ref{eq:loss_relation} is only true in general when $\\mathcal{L}_{E_{i}}=\\mathcal{L}_{\\mathbf{F}_{i}}=0$ for all~$i$, which is not fulfilled in practice, because both labels may contain noise and they can usually not be fitted perfectly. A model trained using a hybrid loss (Eq.~\\ref{eq:multi_objective_loss}) will thus have to compromise between fulfilling both objectives on the training data, as opposed to joining energy and force labels for a performance gain on both. For this reason, the use of hybrid loss functions (or how to weight different contributions) warrants careful consideration depending on the intended application of the final model. Some models, e.g.\\ (s)GDML (see \\ref{subsubsec:symmetric_gradient_domain_machine_learning}), do not even include energy constraints in their loss function at all and are trained on forces only. The energy can still be recovered via integration, but it does not participate in the training procedure except for determining the integration constant. In the end, the ultimate measure of a model's quality should not be how well it minimizes a particular loss function, but instead how well it is able to reproduce the experimental observables of interest. Also, it is important to keep in mind that the loss function measured on the training data is only a proxy for the true objective of any model, which is to generalize to unseen data. Compromising between the energy and force labels of the training data can even improve prediction accuracy for \\emph{both} label types on unseen data. For a more thorough discussion on the role of gradient reference data and how it can improve prediction performance, see Ref.~\\cite{chmiela2019towards,christensen2020on,meyer2020machine}.\n\n\\paragraph{Tuning hyperparameters}\nHyperparameters, such as kernel widths or the depth and width of a neural network, are typically optimized independently of the parameters that determine the model fit to the data: A hyperparameter configuration is chosen, the model is trained, and its performance is measured on the validation set. This process is repeated for as many trials as are affordable or until the desired accuracy is reached. Here it is crucial that no test data is used to measure model performance when tuning hyperparameters, so the ability to estimate the generalization error on the test set is not compromised. Choosing good values for the hyperparameter regimes requires some experience and intuition of the problem at hand. Fortunately, many models are quite robust and good default hyperparameters exist, which do not require any further tuning to arrive at good results. In other cases, hyperparameter tuning can be automated (for example via grid or random search\\cite{lecun2012efficient,muller2001introduction,bergstra2012random,hansen2013assessment,chmiela2019}) and does not need to be performed manually. See also Section~\\ref{subsubsec:model_selection_how_to_choose_hyperparameters} for a more detailed discussion on tuning hyperparameters.\n\n\\paragraph{Regularization}\nBecause ML models contain many parameters (sometimes even more than the number of data points used for training), it is possible or even likely that they ``overfit'' to the training data. An overfitted model achieves low prediction errors on the training set, but performs significantly worse on unseen data (Fig.~\\ref{fig:overfitting}A). The aim of regularization methods is to prevent this unwanted effect by limiting or decreasing the complexity of a model. \n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{regularization.pdf}\n\t\\caption{\\textbf{A}: One-dimensional cut through a PES predicted by different ML models. The overfitted model (red line) reproduces the training data (black dots) faithfully, but oscillates wildly in between reference points, leading to ``holes'' (spurious minima) on the PES. During an MD simulation, trajectories may become trapped in these regions and produce unphysical structures (inset). The properly regularized model (green line) may not reproduce all training points exactly, but fits the true PES (gray line) well, even in regions where no training data is present. However, too much regularization may lead to underfitting (blue line), i.e.\\ the model becomes unable to reproduce the training data at all. \\textbf{B}: Typical progress of the loss measured on the training set (blue) and on the validation set (orange) during the training of a neural network. While the training loss decreases throughout the training process, the validation loss saturates and eventually increases again, which indicates that the model starts to overfit. To prevent overfitting, the training can be stopped early once the minimum of the validation loss is reached (dotted vertical line).}\n\t\\label{fig:overfitting} \n\\end{figure}\n\nWhen the loss function is minimized iteratively by gradient descent or similar algorithms, as is common practice for training NNs, one of the most simple methods to prevent overfitting is \\emph{early stopping}(see e.g. \\cite{prechelt1998early}): In the beginning of the training process, prediction errors typically decrease on both training and validation data. At some point however, because the validation set is not used to directly optimize parameters, the performance on the training data will continue to improve, whereas the loss measured on the validation set will stagnate at a constant value or even begin to increase again. This indicates that the model starts overfitting. Early stopping simply halts the training process as soon as the validation error converges (instead of waiting for convergence of the training error), see Fig.~\\ref{fig:overfitting}B. Early stopping also limits the size of the neural network weights and thus implicitly limits the complexity of the underlying function class. Similar to tuning hyperparameters, only the validation set, but never the test set, must be used for determining the stopping point.\n\nAnother method of regularization is the introduction of penalty terms to the loss function.  Since overfitted models often are characterized by high variance in the prediction (see Fig.~\\ref{fig:overfitting}A), the idea is to penalize large model parameters. For example, $L^2$ regularization (adding the squared magnitude of parameters to the loss) shrinks the $L^2$-norm of the parameter vectors towards zero and prevents very large parameter values. On the other hand, $L^1$ regularization (adding the absolute values of parameters to the loss) shrinks their $L^1$-norm, i.e.\\ it favors sparse parameter combinations. Typically, the regularization term is weighted by an additional hyperparameter $\\lambda$ that determines its strength (like all hyperparameters, $\\lambda$ has to be tuned on the validation set). Note that solving Eq.~\\ref{eq:krr_coefficient_relation_regularized} to determine the parameters of a kernel method will result in an $L^2$-regularized model trained on the MSE loss function.\n\n\\subsection{Using ML-FFs in production}\n\\label{subsec:using_the_model_in_production}\nThe main motivation for training an ML-FF is to use it for some production task, such as running an MD simulation. Before doing so however, it is advisable to verify that it fulfills the accuracy requirements for its intended application. At this point in time, the test set becomes important: Since it was neither used directly nor indirectly during the training process, the data in the test set allows to estimate the performance of a model on truly unseen data, i.e.\\ how well it generalizes. For this, it is common practice to compute summary errors on the test set, for example the mean absolute error (MAE) or root mean squared error (RMSE), as a measure of the global accuracy of a model. In general, such a way of quantifying accuracy gives an overview of the ML model's performance on the given dataset and provides a simple way to benchmark. \n\nHowever, summary errors are biased towards the densely sampled regions of the PES, whereas much larger errors can be expected for less populated regions. Therefore, while summary errors measured on the test set are typically a good indicator for the quality of a model, they are not necessarily the best way to judge how well an ML-FF performs at its primal objective, namely capturing the relevant quantum interactions present in the original molecular system. In other words, performance measures evaluated on the test set should not be trusted blindly. They are only reliable when the test set is representative of the new data encountered during production tasks, i.e.\\ when they are drawn from the same distribution. When a model has to extrapolate, it might give unreliable predictions, even when its performance on the test set is satisfactory. When in doubt, especially when an ML-FF is used for a different task than it was originally constructed for, it is better to collect a few new reference data points to verify that a model is still valid for its use case. Because of the generally limited extrapolation capabilities of ML models, results obtained from studies with ML-FFs should always be scrutinized more carefully than e.g.\\ results obtained with conventional FFs. For example, it is advisable to randomly select a few trajectories and verify that the sampled structures look ``physically sensible'', e.g.\\ no extremely short or long bonds are present and atoms have no unusual valencies. Since the PES is a high-dimensional object, rare events, where a trajectory visits a part of configurational space that is not sampled in the reference data, are always possible, even when the PES was carefully sampled. If any questionable model predictions are found, it is advisable to double-check their accuracy with additional reference calculations. \n\n\\subsection{Example code and software packages}\n\\label{subsec:example_code_and_software_packages}\nWhile many modern ML-FFs are conceptually simple, their implementation is often not straightforward, involving many intricate details that can not be exhaustively covered in  publications. Instead, those details are best conveyed by a reference implementation of the respective model. Publicly available well maintained codes allow to replicate numerical experiments and to build on top of existing models with minimal effort. \n\nIn this section, example code snippets for training and evaluating kernel- and NN-based ML-FFs with the \\texttt{sGDML}\\cite{chmiela2019} (\\ref{subsubsec:sgdml_package}) and \\texttt{SchNetPack}\\cite{schutt2018schnetpack} (\\ref{subsubsec:schnetpack_package}) software packages are given. This is followed by a short description of other popular software packages for the construction of ML-FFs (\\ref{subsubsec:other_software_packages}). Note that the list is not comprehensive and a number of other similar packages exist.\n\n\\subsubsection{The \\texttt{sGDML} package}\n\\label{subsubsec:sgdml_package}\nA reference implementation of the (s)GDML model is available as Python software package at \\url{http:\/\/www.sgdml.org}~\\cite{chmiela2019}. It includes a command-line interface that guides the user through the complete process of model creation and testing, in an effort to make this ML approach accessible to broad practitioners.  Interfaces to the Atomic Simulation Environment (ASE)\\cite{ase17} or i-PI\\cite{ipiv2} make it straightforward to perform MD simulations, vibrational analyses, structure optimizations, nudged elastic band computations, and more. \n\nTo get started, only user-provided reference data is needed, specifically a set of Cartesian geometries with corresponding total energy and atomic-force labels. Force labels are necessary, because sGDML implements energy conservation as an explicit linear operator constraint by modeling the FF reconstruction as the transformation of an underlying energy model (see Section~\\ref{subsubsec:symmetric_gradient_domain_machine_learning}). The trained model will give predictions at the accuracy of the reference data and can  be queried like any other FF.\n\n\\paragraph{Dataset preparation} The \\texttt{sGDML} package uses a proprietary format for its datasets, but scripts to import and export to all file types supported by the ASE package\\cite{ase17}, which covers most popular standards, are included. To convert a \\texttt{<dataset>}, simply call\n\\begin{verbatim}\n\tsgdml_dataset_via_ase.py <dataset>\n\\end{verbatim}\nand follow the instructions.\n\n\\paragraph{Training} The most convenient way to reconstruct a FF is via the command line interface:\n\\begin{verbatim}\nsgdml all <dataset> <ntrain> <nvalid>\n\\end{verbatim}\nThis command will automatically generate a fully trained and cross-validated model and save it to a file, i.e.\\ model selection and hyperparameter tuning (see Section~\\ref{subsubsec:model_selection_how_to_choose_hyperparameters}) are performed automatically. The parameters \\texttt{<ntrain>} and \\texttt{<nvalid>} specify the sample sizes for the training and validation subsets, respectively. All remaining points are reserved for testing. Each subset is sampled from the provided reference \\texttt{<dataset>} without overlap.\n\n\\paragraph{Using the model} To use the trained model, the sGDML predictor is instantiated from the \\texttt{<model>} file  generated above and energy and forces are queried for a given geometry (for example stored in an XYZ file \\texttt{<xyz>}):\n\\begin{verbatim}\nimport numpy\nfrom sgdml.predict import GDMLPredict\nfrom sgdml.utils import io\n\n# Load model from file\nparameters = numpy.load(\"<model>\")\ngdml = GDMLPredict(parameters)\n\n# Load structure from xyz file\nr,_ = io.read_xyz(\"<xyz>\")\n\n# Evaluate model \n# (energies e and forces f)\ne,f = gdml.predict(r)\n\\end{verbatim}\n\nIt is also possible to run MD simulations using ASE and the \\texttt{Calculator} interface included with the \\texttt{sGDML} package:\n\n\\begin{verbatim}\nimport ase\nfrom sgdml.intf.ase_calc import \\\n    SGDMLCalculator\nfrom ase.md.velocitydistribution import \\\n    MaxwellBoltzmannDistribution\n\n# Load sGDML model as ASE calculator\ncalc = SGDMLCalculator(\"<model>\")\n\n# Load structure and attach calculator\natoms = ase.io.read(\"<xyz>\")\natoms.set_calculator(calc)\n\n# Initialize momenta at 300 K\nMaxwellBoltzmannDistribution(\n    atoms, 300*units.kB)\n\n# Setup MD using the velocity Verlet \n# integrator and a time step of 0.2 fs \ndyn = ase.md.verlet.VelocityVerlet(\n    atoms, 0.2*ase.units.fs,\n    trajectory=\"<trajectory>\")\n\n# Simulate for 1000 steps\ndyn.run(1000)\n\\end{verbatim}\n\nTo run this script, a trained model (\\texttt{<model>}) and an initial geometry (\\texttt{<xyz>}) are needed. The resulting MD trajectory is stored in a file \\texttt{<trajectory>}. For more details and applications examples, please visit the documentation at \\url{www.sgdml.org\/doc\/}.\n\t\n\\subsubsection{The \\texttt{SchNetPack} package}\n\\label{subsubsec:schnetpack_package}\n\\texttt{SchNetPack}\\cite{schutt2018schnetpack} is a toolbox for developing and applying deep neural networks to the atomistic modeling of molecules and materials available from \\url{https:\/\/schnetpack.readthedocs.io\/}. It offers access to models based on (weighted) atom-centered symmetry functions and the deep tensor neural network SchNet, which can be coupled to a wide range of output modules to predict potential energy surfaces and forces, as well as a growing number of other quantum-chemical properties. \\texttt{SchNetPack} is designed to be readily extensible to other neural network potentials such as the DTNN~\\cite{schutt2017quantum} or PhysNet~\\cite{unke2019}. It provides extensive functionality for training and deploying these models, including access to common benchmark datasets. It also provides an Atomic Simulation Environment (ASE)\\cite{ase17} \\texttt{Calculator} interface, which can be used for performing a wide variety of tasks implemented in ASE. Moreover, \\texttt{SchNetPack} includes a fully functional MD suite, which can be used to perform efficient MD and PIMD simulations in different ensembles.\n\nAs it is based on the PyTorch deep learning framework\\cite{paszke2019pytorch}, \\texttt{SchNetPack} models are highly efficient and can be applied to large datasets and across multiple GPUs. Combined with the modular design paradigm of the code package, these features also allow for a straightforward implementation and evaluation of new models. Similar to the \\texttt{sGDML} package, the central commodity for training models in \\texttt{SchNetPack} is a dataset containing the Cartesian geometries (including unit cells and periodic boundary conditions, if applicable) and atom types, as well as the target properties to be modeled (e.g.\\ energies, forces, dipole moments, etc.). More information can be found in Ref.~\\citenum{schutt2018schnetpack}.\n\n\\paragraph{Dataset preparation} \\texttt{SchNetPack} uses an adapted version of the ASE database format to handle reference data. The package provides several routines for preparing custom datasets, as well as a range of pre-constructed dataset classes for popular benchmarks (e.g.\\ QM9\\cite{ramakrishnan2014quantum} and MD17\\cite{chmiela2017}), which will automatically download and format the data. For example, molecular data from the MD17 dataset can be loaded via\n\\begin{verbatim}\nspk_load.py md17 <molecule> <path>\n\\end{verbatim}\nwhere \\texttt{<molecule>} indicates the molecule for which data should be loaded (e.g.\\ \\texttt{ethanol}), while the second argument specifies where the data is stored locally.\n\n\\texttt{SchNetPack} also provides a utility script for converting data files in the extended XYZ format, which is able to handle a wide variety of properties, to the database format used internally. Conversion can be invoked with the command\n\\begin{verbatim}\nspk_parse.py <input> <target>\n\\end{verbatim}\nwhere the arguments specify the file paths to the \\texttt{<input>} data file and \\texttt{<target>} database in \\texttt{SchNetPack} format, respectively.\n\n\\paragraph{Training} \nAs for the \\texttt{sGDML} package, training and evaluating ML models in \\texttt{SchNetPack} can be performed via a command line interface. For example, a basic model can be trained with the script:\n\\begin{verbatim}\nspk_run.py train [model_type]\n    [dataset_type] <dataset> <model>\n    --split <ntrain> <nvalid>\n\\end{verbatim}\nHere, \\texttt{[model\\_type]} specifies which kind of NNP to use (\\texttt{wacsf} for a descriptor-based NNP using wACSFs\\cite{gastegger2018wacsf}, or \\texttt{schnet} for the SchNet\\cite{schutt2017schnet} end-to-end NNP architecture) and\n \\texttt{[dataset\\_type]} specifies either a preexisting dataset (e.g.\\ \\texttt{qm9} or \\texttt{md17}), or a \\texttt{custom} dataset provided by the user. The next two arguments are the paths to the reference \\texttt{<dataset>} and the file the trained \\texttt{<model>} will be written to. The arguments\n\\texttt{<ntrain>} and \\texttt{<nvalid>} specify the sample sizes for the training and validation subsets, while the remaining points are reserved for testing. \\texttt{SchNetPack} offers a wide range of additional settings to modify the training process (e.g.\\ model composition, use of GPU, how different properties should be treated etc.), see \\url{https:\/\/schnetpack.readthedocs.io\/}.\n\n\\paragraph{Using the model} \nOnce a model has been trained, it can be evaluated in several different ways. The most basic method is to perform predictions via:\n\\begin{verbatim}\nimport torch\nimport ase\nfrom schnetpack.data.atoms \\\n\timport AtomsConverter\n\n# Load model from file\nspk_model = torch.load(\"<model>\")\n\n# Set up converter for ASE atoms\nconverter = AtomsConverter()\n\n# Load structure from xyz file\natoms = ase.io.read(\"<xyz>\")\ninputs = converter(atoms)\n\n# Evaluate model and collect\n# predictions in a dictionary\nresults = spk_model(inputs)\n\\end{verbatim}\n\nIt is also possible to use the \\texttt{SchNetPack} MD suite to perform various simulations with the trained model. \nContinuing the above example, a basic MD run can be carried out as:\n\\begin{verbatim}\nimport ase\nimport schnetpack.md as md\n\n# Set up the system using 1 replica\n# (use n_replicas > 1 for PIMD)\nmd_system = md.System(n_replicas=1)\n\n# Load structure from xyz file\natoms = ase.io.read(\"<xyz>\")\nmd_system.load_molecules(atoms)\n\n# Initialize momenta at 300 K\ninit = md.MaxwellBoltzmannInit(300)\ninit.initialize_system(md_system)\n\n# Set up integrator \ntstep = 0.2 # fs\nintegrator = md.VelocityVerlet(tstep)\n\n# Prepare model for MD (specify \n# units, required properties, etc.)\ncalculator = md.SchnetPackCalculator(\n    spk_model,\n    required_properties=[\"forces\"],\n    force_handle=\"forces\",\n    position_conversion='A',\n    force_conversion='kcal\/mol\/A')\n\n# Combine everything\nsimulator = md.Simulator(\n    md_system, integrator, calculator)\n\n# Simulate for 1000 steps\nsimulator.simulate(1000)\n\\end{verbatim}\n\nSimulations can be further modified via hooks, which introduce temperature and pressure control, as well as various sampling schemes.\nFurther documentation of the code package and usage tutorials can be found at \\url{https:\/\/schnetpack.readthedocs.io\/}.\n\n\\subsubsection{Other software packages}\n\\label{subsubsec:other_software_packages}\n\\paragraph{AMP: Atomistic Machine-learning Package}\nAMP is a Python package designed to integrate closely with the Atomistic Simulation Environment\\cite{ase17} (ASE) and aims to be as intuitive as possible. Its modular architecture allows many different combinations of structural descriptors and model types. The main idea of AMP is to construct ML-FFs \\emph{on-demand}, i.e.\\ simulations are first started with an \\textit{ab initio} method and later switched to the ML-FF once the model is sufficiently accurate. The package is described in greater detail in Ref.~\\citenum{khorshidi2016amp} and on its official website \\url{https:\/\/amp.readthedocs.io\/}.\n\n\\paragraph{DeePMD-kit}\nThe DeePMD-kit is a package written in Python\/C++ aiming to minimize the effort required to build deep NNPs with different structural descriptors. It is based on the TensorFlow deep learning framework\\cite{abadi2016tensorflow} and offers interfaces to the high-performance classical and path-integral MD packages LAMMPS\\cite{plimpton1993fast} and i-PI\\cite{ipiv2}. More details on the DeePMD-kit can be found in Ref.~\\citenum{wang2018deepmd} or on \\url{https:\/\/github.com\/deepmodeling\/deepmd-kit\/}.\n\n\\paragraph{Dscribe} Dscribe is a Python package for transforming atomic structures into fixed-size numerical fingerprints.\\cite{dscribe} These descriptors can then be used as input for neural networks or kernel machines to construct ML-FFs. Supported representations include the standard Coulomb matrix\\cite{rupp2012fast} and variants for the description of periodic systems\\cite{faber2015crystal}, ACSFs\\cite{behler2011atom}, SOAP\\cite{bartok2013representing}, and MBTR\\cite{huo2017unified}. More details can be found on the official website \\url{https:\/\/singroup.github.io\/dscribe\/} or in Ref.~\\citenum{dscribe}.\n\n\\paragraph{RKHS toolkit}\nThe RKHS toolkit is mainly intended for constructing highly accurate and efficient PESs for studying scattering reactions of small molecules. As described in section~\\ref{subsubsec:kernel_based_methods}, the evaluation of kernel-based methods scales linearly with the number of training points $N$ (see Eq.~\\ref{eq:kernel_regression}). By using special kernel functions and precomputed lookup tables, the RKHS toolkit allows to bring this cost down to $\\mathcal{O}(\\log N)$. However, it requires that the training data has grid structure, which limits its applicability to small systems, where it is meaningful to sample the PES by scanning a list of values for each internal coordinate. The implemented kernel functions also allow to encode physical knowledge about the long-range decay behavior of certain coordinates, which enables accurate extrapolation well beyond the range covered in the training data. A Fortran90 implementation of the toolkit can be downloaded from \\url{https:\/\/github.com\/MMunibas\/RKHS\/} and the algorithmic details are described in Ref.~\\citenum{unke2017toolkit}.\n\n\\paragraph{QML}\nQML is a toolkit for learning properties of molecules and solids written in Python.\\cite{christensen2017qml} It supplies building blocks to construct efficient and accurate kernel-based ML models, such as different kernel functions and premade implementations of many different structural representations, e.g.\\ Coulomb matrix\\cite{rupp2012fast}, SLATM\\cite{huang2016communication}, and FCHL\\cite{christensen2020fchl}. The package is primarily intended for the general prediction of chemical properties, but can also be used for the construction of ML-FFs. For further details, refer to the official website \\url{https:\/\/www.qmlcode.org} or the github repository \\url{https:\/\/github.com\/qmlcode\/qml\/}.\n\n\\section{Challenges}\n\\label{sec:challenges}\nFollowing the best practices outlined in the previous section, the current generation of ML-FFs is applicable to a wide range of problems in  chemistry that involve small- to medium-sized systems. While this space of chemical compounds is already significant in size, the ``dream scenario'' of chemists and biologists referenced in the introduction can only be realized with access to larger system sizes. Not only does the number of stable structures increase exponentially with added atomistic degrees of freedom,\\cite{blum2009970,ruddigkeit2012enumeration} many interesting phenomena play out at nanoscale resolution, which is inaccessible to ML methods as of yet. This is because some steps involved in the construction of ML-FFs,  like sampling the reference data, which are solvable at small scale, become seemingly insurmountable obstacles at larger scales due to unfeasible computing times. The complexity of interactions, e.g.\\ the non-classical behavior of nuclei, as well as significant contributions from large fluctuations, increase the space of conformations that need to be learned. To further complicate things, the cost of accurate \\textit{ab initio} calculations increases steeply with expanding system size, limiting the amount of reference data that can be collected within a reasonable time frame. This also means that a growing number of atom correlations need to be represented by a model in order to capture the full scope of interactions present in the real system. Below, some considerations in reconciling the somewhat contradicting demands of scalability, transferability, data efficiency and accuracy in large-scale ML-FFs are outlined.\n\n\\subsection{Locality and smoothness assumptions}\n\\label{subsec:locality_and_smoothness_assumptions}\nA fundamental challenge that must be faced by \\textit{ab initio} methods, conventional FFs, and ML models alike is the many-body problem. Most properties of a physical system are determined by the interaction of many particles, whether those are electrons or, on a higher abstraction level, atoms. In fact, the reason that \\textit{ab initio} calculations are expensive to obtain is due to the challenging computational scaling properties of high-dimensional many-body problems. As a result, the hierarchy of different levels of theory is directly defined by the level of correlation treatment in the respective wave function parametrization. Because the electronic degrees of freedom of a system are much higher than the number of atoms, the computational limitations of \\textit{ab initio} methods become evident very quickly, even for small systems. Atomistic approximations scale more favorably, because they need to correlate less particles, but they are subject to the same scaling laws. The only escape is to neglect some correlations in favor of a reduced problem size. Unfortunately, it is to date impossible to reliably determine which interactions can be removed with minimal impact, without compromising the full many-body solution. Thus, the ideal of a local model is in conflict with the very nature of many-body systems. While not fully justified from a physics perspective, assuming locality is still a useful inductive bias, which can help generalization and computational efficiency. It also helps when collecting reference data, as it implies that larger systems can be predicted using the information learned from smaller systems. Another assumption, which all ML-FFs discussed in this review make, is that the PES is smooth. This is a necessary requirement for most practical applications, since a non-smooth PES implies force discontinuities, which would lead to instabilities during MD simulations. Smoothness is also a requirement from the ML perspective, as only regular signals can be reconstructed from limited observations. \n\nFor most commonly used NNPs and many kernel-based ML-FFs, locality is built into the design explicitly through the introduction of a cutoff radius. The global interactions between atoms are modeled by accumulating individual local atomic contributions. In this ``mean-field approximation'', the interaction of a particle with its surroundings is reduced to an effective one-body problem, i.e.\\ an interaction of that particle with the average effect of its neighbors. As similar neighborhoods can be identified in different compounds across chemical space, these assumptions allow to build models from reference calculations of small molecules, which are transferable to much larger structures.\\cite{huang2017dna,huang2020quantum} However, the lack of explicit higher-order terms comes at the cost of potentially loosing some important interaction effects, similar to the Hartree-Fock method and Kohn-Sham DFT in \\textit{ab initio} calculations. \n\nOn the other hand, some models (e.g.\\ (s)GDML) capture global correlations in the sense that a single prediction is obtained for the whole structure. Of course, this relies on reference calculations that are accurate enough to contain the relevant information. Global interactions of large systems can not be accurately inferred from a training set of small molecules or molecular fragments, which is why reference calculations for the exact target structure are necessary. It can therefore become difficult to collect enough reference data for large structures. In addition, global models might still implicitly assume that interactions are local to some degree due to their chemical descriptor. For example, in (s)GDML models, systems are encoded as a vector of inverse pair-wise distances. Therefore, structural changes between distant atoms contribute less strongly to changes in the overall descriptor than proximal atoms.\n \nWhile locality and smoothness are valid assumptions for the majority of chemical systems, there are pathological cases where they break down and ML models that rely on them perform poorly. As an example, consider cumulenes -- hydrocarbons of the form C$_{2+n}$H$_4$ ($n \\ge 0$) with $n+1$ cumulative double bonds. These molecules have a rigid linear geometry with the two terminal methylene groups forming an equilibrium dihedral angle of 0$^{\\circ}$ (when $n$ is even) or 90$^{\\circ}$ (when $n$ is odd). Rotating the dihedral angle out of its equilibrium position results in a sharp increase in potential energy even though the methylene groups may be separated by several angstroms when $n$ is large. This is due to the energetically favorable overlap of $\\pi$-orbitals along the carbon chain (a highly non-local interaction), which is broken when the methylene groups are rotated against each other. Additionally, the potential energy exhibits a sharp ``cusp'' at the maximum energy (i.e.\\ it is not smooth), because the ground state electronic configuration switches abruptly from one state to another (strictly speaking, multi-reference calculations would be necessary here). One-dimensional projections of the PESs predicted by ML-FFs along the rotation of the dihedral angle reveal several problems (Fig.~\\ref{fig:cumulene_scans}). For example, all models predict smooth approximations by design, which is beneficial for running MD simulations, but results in large prediction errors around the cusp. Further, when the number of double bonds ($n+1$), i.e.\\ the ``non-locality'' of relevant interactions, is increased, the quality of predictions decreases dramatically, until all models are unable to reproduce the energy profile. \n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{cumulene_scans.pdf}\n\\caption{Energy profiles of different ML-based PESs for a rotation of the dihedral angle between the terminal methylene groups of cumulenes (C$_{2+n}$H$_4$) of different sizes ($0\\le n \\le 7$). All reference calculations were performed with the semi-empirical MNDO method\\cite{dewar1977ground} and models were trained on 4500 structures (with an additional 450 structures used for validation) collected from MD simulations at 1000~K. Because rotations of the dihedral angle are not sufficiently sampled at this temperature, the dihedral was rotated randomly before performing the reference calculations. Instead of a sharp cusp at the maximum of the rotation barrier, all models predict a smooth curve. Predictions become worse for increasing cumulene sizes with the cusp region being over-smoothed more strongly. For $n=7$, all models fail to predict the angular energy dependence. Note that NNP models (such as PhysNet and SchNet) may already fail for smaller cumulenes when the cutoff distance is chosen too small ($r_{\\rm cut} = 6$~\\AA), as they are unable to encode information about the dihedral angle in the environment-descriptor. However, it is possible to increase the cutoff ($r_{\\rm cut} = 12$~\\AA) to counter this effect.}\n\\label{fig:cumulene_scans} \n\\end{figure}\n\nNote that by design, NNPs relying on message-passing are unable to resolve information about the dihedral angle if information between hydrogen atoms on opposite ends of the molecule cannot be exchanged directly (i.e.\\ $r_{\\rm cut}$ is too small) and predict constant energies in this case. The same is true for descriptor-based NNPs, as fingerprints of chemical environments also only consider atoms up to a cutoff (see Eqs.~\\ref{eq:two_body_symmetry_function}~and~\\ref{eq:three_body_symmetry_function}). Any kernel method taking as input local structural descriptors relying on cutoff radii (e.g.\\  SOAP\\cite{bartok2013representing} or  FCHL19\\cite{christensen2020fchl}) will suffer from the same problems. Even when a ``global'' descriptor such as inverse pair-wise distances is chosen (e.g.\\ Coulomb matrix\\cite{rupp2012fast}), changes in the dihedral angle between distant groups of atoms are not resolved sufficiently for accurate predictions (see sGDML model in Fig.~\\ref{fig:cumulene_scans}). The only way to fix this problem in general is to drop the locality assumption completely, for example by including all $\\binom{N}{4}$ possible dihedral angles in the structural descriptor (without introducing additional factors that decrease the weight of these features with increasing distance between atoms). However, due to the combinatorial explosion of the number of possible dihedral angles, this would lead to extremely large descriptors whenever the number of atoms $N$ is not very small. The resulting models would be slow to evaluate and require a lot of reference data to give robust predictions (to prevent them from entering the extrapolation regime). An expert choice, i.e.\\ including only a single relevant dihedral angle in the descriptor, is a possible way around this issue, but requires prior knowledge of the problem at hand and goes somewhat against ML philosophy. \n\nAs a final remark, it should be mentioned that conventional FFs only include terms for dihedral angles between directly bonded atoms, so they are equally unable to predict the energy profiles of the larger cumulenes shown in Fig.~\\ref{fig:cumulene_scans}. As such, relying on chemical locality is an assumption made by virtually all methods for approximating PESs and is not specific to just ML methods.\n\n\\subsection{Transferability, scalability and long-range interactions}\n\\label{subsec:transferability_scalability_and_long_range_interactions}\nThe concept of chemical locality discussed above also plays a central role in the transferabilty and scalability of ML models for atomistic systems. Transferability indicates how well models can adapt to compounds varying in their chemical composition, while scalability indicates how efficiently these models scale with respect to the size of systems modeled. Both concepts are closely related and inherently rooted in chemical locality. The assumption that interactions between atoms are local implies that similar structural motifs will give rise to comparable interactions and hence similar contributions to the global properties of a molecule or material. In an ML context, chemical locality allows a model to reuse the information learned for different parts of a molecule for similar features in different systems. In this manner, a large atomistic system could in principle be assembled from smaller components like a jigsaw puzzle.\\cite{huang2017dna} The former aspect is crucial in order to make models transferable, while the latter allows for the development of architectures whose evaluation cost scales linearly with system size.\n\nML-FFs exploiting chemical locality offer several advantages compared to global models. If trained properly, they can be applied to systems of different size and composition. The training procedure benefits in a similar manner, as local models can be trained on structures containing different numbers of atoms. Moreover, it is also possible to use only fragments of the original system during construction of a model. This property is very attractive in situations where accurate reference computations for the whole system are infeasible due to system size and\/or scaling of the computational method. Local chemical environments are also less diverse than global atomistic structures, potentially reducing the need for extensive sampling and decreasing the chances that models enter the extrapolation regime in a production setting.\nIn addition, local models scale linearly with system size, as interactions are limited to the cutoff radius and can be evaluated efficiently. In contrast, global models are typically more limited in their practical applicability for extended systems. They always require reference computations to be performed for the whole system and, once trained, can only be reused for this particular molecule or material.\n\nDespite these advantages, local ML models suffer from several inherent problems. In order to construct models which exploit locality, a chemical system needs to be partitioned in one way or another. This can for example be achieved by limiting interactions to terms involving only a certain number of atoms (similar to conventional FFs) or by restricting them to local atom-centered environments.\nThese approximations place strong limitations on which kind of interactions can be described. As a result, local ML models have difficulty when dealing with the situations where non-local effects are important, such as strongly conjugated systems and excited states (see Section~\\ref{subsec:locality_and_smoothness_assumptions}). For standard simulations, the presence of long-range interactions, such as electrostatic and dispersion effects, are much more common phenomena. These are particularly important for modeling extended systems, where ML models are typically believed to offer a significant advantage over more conventional FFs. Since the structure and dynamical behavior of such systems is influenced greatly by long-range interactions, ML models need to be able to account for them in a satisfying manner.\n\nRecovering long-range effects necessitates a balancing act between physical accuracy and computational efficiency, as the scalability of local models hinges on there being a limited number of interactions which need to be evaluated. This feat is further complicated by the typical energy scales of these interactions, which are small compared to local contributions such as bond energies. \nFor these reasons, it is not advisable to account for long-range interactions by simply increasing the size of local environments. While local models with sufficiently large cutoffs are able to learn the relevant effects in principle, it may require a disproportionately large amount of data to reach an acceptable level of accuracy for an interaction with a comparably simple functional form. The reason is that average gradients and curvature in different regions of the PES may differ by several orders of magnitude, which makes it difficult to achieve uniformly low prediction errors across all regions. Hence, an optimal description would require to employ different characteristic scales. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\columnwidth]{LJ3.pdf}\n\\caption{Lennard-Jones potential (thick gray line) predicted by KRR with a Gaussian kernel. In (a), the potential energy is decomposed into short- (red), middle- (magenta) and long-range (blue) parts, which are learned by separate models (symbols show the training data and solid lines the model predictions). The mean squared prediction errors (in arbitrary units) for the respective regions are shown in the corresponding colors. In (b), the entire potential is learned by a single model using the same training points (green). All models in (a) and (b) use $r$ as the structural descriptor. Panel (c) shows a single model learning the potential, but using $r^{-1}$ as structural descriptor (yellow). The mean squared errors (a.u.) for different parts of the potential in (b) and (c) are reported independently to allow direct comparison with the values reported in (a).}\n\\label{fig:LJ}\n\\end{figure}\n\nFor illustration, consider the following toy examples: In the first variant, a Lennard-Jones (LJ) potential\\cite{jones1924determination} is separated into a region around its minimum, a repulsive short-range, and an attractive long-range part. The task is to learn each of the three regions with a separate model (see Fig.~\\ref{fig:LJ}a). In the second variant, a single model is trained on all regions at once (see Fig.~\\ref{fig:LJ}b). Here, all models are kernel-based and use a Gaussian kernel (Eq.~\\ref{eq:gaussian_kernel}). The kernel hyper-parameter $\\gamma$ is optimized by a grid-search and cross-validation. Compared to the models trained on the first task, the prediction errors of the global model increase by around an order of magnitude. Further, the global model shows spurious oscillations between training points in the long-range region. When the optimal values of $\\gamma$ for the different models are compared, the reason for the failure of the global model becomes apparent: The optimal values of $\\gamma$ are $198.88$, $75.47$, $0.08$ for the short-, middle-, and long-range models, respectively, which highlights the multi-scale nature of the PES. On the other hand, the global model necessarily has to compromise between the different regions, which leads to an optimal value of $\\gamma=22.12$. In this toy example, the multi-scale problem can be solved by switching from using $r$ as a structural descriptor to the more appropriate inverse distance $r^{-1}$ (Fig.~\\ref{fig:LJ}c). Unfortunately, for realistic (high-dimensional) PESs with multiple minima, it can be difficult to find an appropriate descriptor to address the multi-scale nature of the PES, which leads to data-inefficient models. As a result, more training data is needed to reach an acceptable accuracy, which is problematic considering the computational cost of high-quality reference calculations.\n\nOne possibility of overcoming these limitations is by instead partitioning the energy into contributions modeled entirely via ML (short-range) and contributions described via explicit physical relations based on local quantities predicted via ML (long-range). A prime example for such an approach is the treatment of electrostatics, as was first introduced in Ref.~\\citenum{morawietz2012neural}. Here, an ML model is used to predict partial charges for each atom based on their local environment. These charges can then be used in standard Coulomb and Ewald summation to compute the long-range electrostatic energy of a system. While such schemes initially relied on point charge reference data obtained from (arbitrary) partitioning methods of the \\textit{ab initio} electron density (e.g.\\ Hirshfeld charges\\cite{hirshfeld1977bonded}), they have since been extended to operate on charges derived from an ML model for dipole moments (a true quantum mechanical observable).\\cite{gastegger2017machine,yao2018tensormol,unke2019} Here, scalar partial charges $q_i$ are predicted for each atom $i$ and the molecular dipole moment is constructed as $\\boldsymbol{\\mu}=\\sum_i q_i \\mathbf{r}_i$, where $\\mathbf{r}_i$ are the atomic positions (the predicted $q_i$ can be corrected to guarantee charge conservation\\cite{unke2019}). The discrepancy between reference and predicted dipole moments is included in the loss function used for training the model (see Section~\\ref{subsec:training_the_model}) and the partial charges consequently derived in a purely data-driven manner.\n\nContrary to electrostatics, accounting for dispersion interactions is not as straightforward. First, the exact physical form of dispersion interactions is still debated and a variety of approximate schemes have been proposed.\\cite{hermann2017first} Second, dispersion corrections typically depend on coefficients computed from atomic polarizabilities as local properties. Unfortunately, these are not as amenable to ML driven partitioning, as the corresponding quantum mechanical observable is the molecular polarizability. In contrast to charges derived from dipole moments (a vector quantity), molecular polarizabilities are scalars and offer insufficient constraints to a purely data driven partitioning. Because of these reasons, ML approaches often rely on the same empirical pair-wise potentials employed in correcting density functional theory computations.\\cite{morawietz2013density,uteva2017interpolation,unke2019}\n\nTo summarize, local ML architectures are a promising approach towards transferable and scalable models, but they have a number of drawbacks which will still need to be addressed in the future. Promising alternative approaches to achieve transferability are ML models based directly on electronic structure methods, i.e.\\ ``semi-empirical ML''\\cite{li2018density,zubatyuk2019machine,stohr2020accurate} and models for electron density and Hamiltonians\\cite{schutt2019unifying}.\nThese approaches express fundamental quantum chemical quantities in a local representation, e.g.\\ Hamiltonian matrix elements in an atomic orbital basis. Non-locality can then be introduced via the ``correct'' mathematical mechanism, e.g.\\ matrix diagonalization in the case of Hamiltonians. This physically motivated structure allows such models to recover a wide range of interactions while still being transferable. They are also better suited to predict intensive properties of molecules (whose magnitude is independent of system size), where assuming additive atomic contributions is not valid. A downside of such models compared to conventional ML-FFs is the increased computational cost due to the additional matrix operations. \n\nWith respect to scalability, hybrid approaches similar to QM\/MM\\cite{senn2009qm} might constitute valid alternatives to pure ML models. Although several orders of magnitude more efficient than electronic structure theory, even local ML models encounter problems when faced with systems containing tens of thousands of atoms. Compared to conventional FFs, the more complex functional form underlying ML-FFs leads to an increased computational cost. In such cases, partitioning the system into regions treated at different levels of approximation can lead to a significant speedup. ML models can for example be embedded into regions modeled by classical force fields (yielding ML\/MM like simulation protocols) or even coarse-grained environments generated via another ML model\\cite{wang2020ensemble}. Restricting elaborate ML approaches to only a subset of a chemical systems would make it possible to employ more accurate approximations in a manner analogous to conventional QM\/MM.\n\n\\section{Concluding remarks}\n\\label{sec:concluding_remarks}\nThe last decades have witnessed significant advances in statistical learning that allowed ML techniques to enter our daily lives, industrial practice and scientific research. \n\nClassically, automation in industry and scientific fields relied on hand-crafted rules that represented human knowledge.\\cite{nilsson1982principles} Not only is the creation of rule-based systems laborious and may require to cover an excessive number of cases, it often leads to rigid structures that are unable to adapt well to new situations. Even worse, some concepts are difficult or impossible to formalize, such as human perception for image classification. \n\nModern statistical ML algorithms\\cite{murphy2012machine,theodoridis2020machine} such as deep learning\\cite{bishop1995neural,lecun2015deep, schmidhuber2015deep,goodfellow2016deep} or kernel-based learning\\cite{cortes1995support,vapnik1995nature,scholkopf1998nonlinear,muller2001introduction,scholkopf2002learning} enable models that freely adapt to knowledge that is implicitly contained in datasets (in an abstract form) and thus offer a more robust way of solving problems than rule-based reasoning. \nFor the field of molecular simulations, the potential of ML methods may help to bridge the accuracy-efficiency gap between first-principles electronic structure methods and conventional (rule-based) FFs. Bringing both fields together has raised many questions and still poses some fundamental challenges for new generations of ML-FFs. At this point in time, ML-FFs have already become a successful and practical tool in computational chemistry. \n\nStarting from a broad perspective, this review has focused on the role of ML for constructing force fields and assessed what can be achieved with these new techniques at the current stage of development. This has been contrasted with problems that are (so far) beyond the reach of present methods. Illustrative examples of the relevant chemistry and ML concepts have been discussed to demonstrate the practical usefulness that modern ML techniques can bring to chemistry and physics. This includes an overview of the most important considerations behind the construction of modern ML-FFs, such as the incorporation of physical invariances, choice of ML algorithms, and loss functions. Special attention has been given to the topic of validating ML-FFs, which requires particular care in scientific applications.\\cite{lapuschkin2019unmasking} Furthermore, a comprehensive list of best practices, pitfalls, and challenges has been provided, which will serve as a useful guideline for practitioners standing on either side of this growing interdisciplinary field. These ``tricks of the trade''\\cite{montavon2012neural} are often assumed to be obvious and thus omitted from publications -- here they have been deliberately spelled out to avoid unnecessary barriers to enter the field. Additionally, a small catalog of software tools that can enable and accelerate the implementation of ML-FFs has been provided as a pointer for readers wishing to adopt  ML methods in their own research.\n\nWhile routinely performing computational studies of condensed phase systems (e.g.\\ proteins in solution) at the highest levels of theory is still beyond reach, ML methods have already made other ``smaller dreams'' a reality: Just a decade ago, it would have been unthinkable to study the dynamics of molecules like aspirin at coupled cluster accuracy. Today, a couple hundred \\textit{ab initio} reference calculations are enough to construct ML-FFs that reach this accuracy within a few tens of wavenumbers.\\cite{sauceda2020construction}  In the past, even if suitable reference data was available, constructing accurate force fields was labor-intensive and required human effort and expertise. Nowadays, by virtue of automatic ML methods, the same task is as effortless as the push of a button. Thanks to the speed-ups offered by ML methods over conventional approaches, studies that previously required supercomputers to be feasible in a realistic time frame\\cite{Spura_CC-PIMD_PCCP2015,Litman_Porphycen-PIMD_JACS2019} can now be performed on a laptop computer\\cite{schutt2017schnet,chmiela2018}. \n\nIn addition to enabling studies that were prohibitively expensive in the past, ML methods have also led to new chemical insights on systems that were thought to be already well understood. Even relatively small molecules were shown to display non-trivial electronic effects, influencing their dynamics and allowing a better understanding of experimental observations.\\cite{sauceda2020} Many other unknown chemical effects potentially wait to be discovered by studies now possible with ML-FFs. At the speed at which improvements to existing ML-FFs are published, it is not unreasonable to expect significant advances that will make similar studies possible for larger systems and help realize many more ``dreams'' in the near future. \n\nConcluding, ML-FFs are a highly active line of research with many unexplored avenues and attractive applications in chemistry, with possibilities to contribute to a better understanding of fundamental quantum chemical properties and ample opportunity for novel theoretical, algorithmic and practical improvement. Given the success of this relatively young interdisciplinary field, it is to be expected that ML-FFs will become a fundamental part of modern computational chemistry. \n\n\\begin{acknowledgement}\nOTU acknowledges funding from the Swiss National Science\nFoundation (Grant No. P2BSP2\\_188147).\nKRM was supported in part  by the Institute of Information \\& Communications Technology Planning \\& Evaluation (IITP) grant funded by the Korea Government (No. 2019-0-00079,  Artificial Intelligence Graduate School Program, Korea University), and was partly supported by the German Ministry for Education and Research (BMBF) under Grants 01IS14013A-E, 01GQ1115, 01GQ0850, 01IS18025A, 031L0207D and 01IS18037A; the German Research Foundation (DFG) under Grant Math+, EXC 2046\/1, Project ID 390685689. We would like to thank Stefan Ganscha for his valuable input to the manuscript. Correspondence to AT and KRM. \n\\end{acknowledgement}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzlygr b/data_all_eng_slimpj/shuffled/split2/finalzzlygr
new file mode 100644
index 0000000000000000000000000000000000000000..ee2ffa35e892eeaa6874407c21d52efbd457ad00
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzlygr
@@ -0,0 +1,5 @@
+{"text":"\n\\subsection{Related Works}\n\\label{sec:related}\n\\paragraph{Relation to Tensor Networks. }\nTensor networks are widely used in quantum physics~\\cite{orus2014practical},  numerical analysis~\\cite{grasedyck2013literature} and recently machine learning~\\cite{cichocki2016low, cichocki2017tensor}. Different from existing tensor networks, (1) our {TNN\\xspace}s have nonlinearity on the edges between the tensors; and (2) {TNN\\xspace}s are constructed as deep compositions of interleaving {\\em generalized tensor operations} and nonlinear transformations, similar to feedforward neural networks. Furthermore, while tensor networks are still multi-linear and thus can be learned by algorithms such as power iteration~\\cite{wang2017tensor} or alternative least squares~\\cite{comon2009tensor}, {TNN\\xspace}s require decomposition of nonlinear (no longer multi-linear) hierarchical tensor networks.\n\n\\paragraph{Compression of Neural Networks. }\nA recent survey~\\cite{cheng2017survey} reviews state-of-the-art techniques for compressing neural networks. These techniques can be grouped into two categories: (1) compressing an existing model and (2) novel compact designs. The first category includes \\textit{low-rank approximations}~\\cite{jaderberg2014speeding, denton2014exploiting, lebedev2014speeding, kim2015compression}, \\textit{knowledge distillation}~\\cite{romero2014fitnets, ba2014deep, hinton2015distilling, furlanello2018born} and \\textit{quantization}~\\cite{han2015deep, courbariaux2015binaryconnect, zhu2016trained, rastegari2016xnor, hubara2017quantized}; while the second category includes \\textit{compact designs of filters}~\\cite{cheng2015exploration, yang2015deep, sindhwani2015structured} and \\textit{compact designs of architectures}~\\cite{szegedy2015going, he2016deep, huang2017densely, chollet2016xception, szegedy2017inception}. When our {TNN\\xspace}s are used for compression, we project an existing neural network to the class of {TNN\\xspace}s with fewer parameters, and in Section~\\ref{sec:interpretation}, we demonstrate this projection naturally corresponds novel compact architecture.  \n\n\n\\section{Introduction}\n\\label{sec:introduction}\n\nModern neural networks~\\cite{krizhevsky2012imagenet, simonyan2014very, szegedy2015going, he2016deep, szegedy2017inception, huang2017densely} achieve unprecedented accuracy over many difficult learning problems at the cost of deeper and wider architectures with overwhelming number of model parameters. \nThe large number of model parameters causes repeated high-cost in predictions, which becomes a practical bottleneck when these neural networks are deployed on constrained devices, such as smartphones and Internet of Things (IoT) devices.\n\nOne fundamental problem in deep learning research is to design neural network models with compact architectures, but still maintain comparable expressive power as large models.\nTo achieve this goal, two complementary approaches are adopted in the community: one approach is to compress well-trained neural networks while preserving their predictive performance as much as possible~\\cite{cheng2017survey}.\nAnother approach is to find better architectures for neural network designs such as grouping the filters into inception modules~\\cite{szegedy2015going, szegedy2017inception} or bottleneck layers~\\cite{lin2013network, he2016identity}. \n\nTo address this aforementioned fundamental problem, we propose \\textit{tensorial neural networks\\xspace} ({TNN\\xspace}s), which allows not only compression of well-trained networks, but also exploration of better designs in network architectures.\nTNN\\xspace is a generalization of existing neural networks\\xspace ({NN\\xspace}s) where matrix-vector multiplication (in fully connected layer and recurrent layers) and convolution (in convolutional layer) are extended to \\textit{generalized tensor operations}. To achieve this, we introduce new tensor algebra to extend existing operations with low order operands to those with high order operands (see Section~\\ref{sec:preliminary} and Appendix~\\ref{app:operations} for details).\n\n\\begin{figure}[!htbp]\n\t\\centering\n\t\\psfrag{c1}[][][0.8]{$\\bm{\\mathcal{G}}^{p}$}\n\t\\psfrag{c2}[][][0.8]{$\\bm{\\mathcal{H}}^{p}$}\n\t\\psfrag{c3}[][][0.8]{$\\bm{\\mathcal{G}}^{q}$}\n\t\\psfrag{c4}[][][0.8]{$\\bm{\\mathcal{H}}^{q}$}\n\t\\psfrag{f}[][][0.8]{$f$}\n\t\\psfrag{g1}[][][0.8]{$g^{q}$}\n\t\\psfrag{g2}[][][0.8]{$g^p$}\n\t\\psfrag{h1}[][][0.8]{$h^{q}$}\n\t\\psfrag{h2}[][][0.8]{$h^p$}\n\t\\psfrag{c1:compressedNN}[][][0.8]{$\\bm{\\mathcal{G}}^{p}$:compressed NN}\n\t\\psfrag{c2:compressedTNN}[][][0.8]{$\\bm{\\mathcal{H}}^{p}$:compressed TNN}\n\t\\psfrag{c3:NN}[][][0.8]{$\\bm{\\mathcal{G}}^{q}$: NN}\n\t\\psfrag{c4:TNN}[][][0.8]{$\\bm{\\mathcal{H}}^{q}$: TNN}\n\t\\includegraphics[width=0.4\\textwidth]{.\/\/circle.eps}\n\\caption{Relationship between {NN\\xspace}s and {TNN\\xspace}s. In the figure, $f$ is the target concept. (1) \\textbf{Learning} of a NN\\xspace with $q$ parameters results in $g^{q}$ that is closest to $f$ in $\\mathcal{G}^{q}$, while learning of a TNN\\xspace with $q$ number of parameters results in $h^{q}$ that is closest to $f$ in $\\mathcal{H}^{q}$. Apparently, $h^{q}$ is closer to $f$ than $g^{q}$.  (2) \\textbf{Compression} of a pre-trained {NN\\xspace} $g^{q}$ to {NN\\xspace}s with $p$ parameters ($p \\le q$) results in $g^{p}$ that is closest to $g^{q}$ in $\\mathcal{G}^p$, while compressing of $g^{q}$ to {TNN\\xspace}s with $p$ parameters results in $h^p$ that is closest to $g^{q}$ in $\\mathcal{H}^p$.  Apparently, the compressed {TNN\\xspace} $h^{p}$ is closer to the pre-trained {} $g^{q}$ than the compressed NN $g^{p}$. }\n\\label{fig:framework}\n\\end{figure}\n\nFigure~\\ref{fig:framework} illustrates the relationship between existing neural networks\\xspace and our proposed tensorial neural networks\\xspace. \nLet $\\mathcal{G}^{q}$ and $\\mathcal{H}^{q}$ denote the sets of functions that can be represented by existing {NN\\xspace}s and our {TNN\\xspace}s, both with at most $q$ parameters.\nSince existing {NN\\xspace}s are special cases of {TNN\\xspace}s, we have the following properties: \n(1) for any $q > 0$, $\\mathcal{G}^{q} \\subseteq \\mathcal{H}^{q}$ and \n(2) there exists $p \\leq q$ such that $ \\mathcal{H}^{p} \\subseteq \\mathcal{G}^{q}$. \nThe first property indicates that {TNN\\xspace}s are generalization of {NN\\xspace}s while the second property guarantees that {TNN\\xspace}s can be used for compression of {NN\\xspace}s.\n\nThe input to a {TNN\\xspace} is a tensor of any order $m$, and the TNN\\xspace reduces to a normal {NN\\xspace} if the input is a vector ($m = 1$) or a matrix ($m = 2$).\n\n\\textbf{Prediction} in {TNN\\xspace}s is similar to traditional {NN\\xspace}s: the input is passed through the layers of a TNN\\xspace in a feedforward manner, where each layer is a generalized tensor (multilinear) operation between the {\\em high-order} input and the {\\em high-order} weight kernels followed by a nonlinear activation function such as $\\mathsf{ReLU}$.\n\\textbf{Learning} parameters in a {TNN\\xspace} is equivalent to hierarchical nonlinear tensor decomposition, which is hard for arbitrary TNN\\xspace architectures. We introduce a suite of generalized tensor algebra, which allows easy derivation of backpropagation rules and a class of TNN architectures (detailed in Appendices~\\ref{app:derivatives}, \\ref{app:dense-tensorized} and \\ref{app:convolutional-tensorized}). With these backpropagation rules, {TNN\\xspace}s can be efficiently learned by standard {\\em stochastic gradient descent}. \n\n{TNN\\xspace}s can be used for compression of traditional neural networks\\xspace, since our proposed {TNN\\xspace}s naturally identify \\emph{invariant structures} in neural networks\\xspace (justified in section~\\ref{sec:invariant}).\nGiven a pre-trained {NN\\xspace} $g^{q} \\in \\mathcal{G}^{q}$, compressing it to a {TNN\\xspace} with $p$ parameters results in $h^{p}$ that is closest to $g^{q}$ in $\\mathcal{H}^p$ as depicted in Figure~\\ref{fig:framework}. It proceeds in two steps: (1) \\textbf{data tensorization}: the input is reshaped into an $m$-order tensor; and (2) \\textbf{knowledge distillation}: mapping from a NN\\xspace to a TNN\\xspace using layer-wise data reconstruction. \n\nWe demonstrate the effectiveness of compression using tensorial neural networks\\xspace by conducting a set of experiments on several benchmark computer vision datasets.\n We compress ResNet\\xspace-32 on the CIFAR10 dataset by 10x with a degrading of only 1.92\\% (achieving an accuracy of 91.28\\%). \nExperiments on LeNet-5 (MNIST), ResNet-32 (CIFAR10) and ResNet-50 (ImageNet) demonstrate that our TNN\\xspace compression outperforms (5\\% test accuracy improvement universally on CIFAR10) the state-of-the-art low-rank approximation techniques under same compression rate, besides achieving orders of magnitude faster convergence rates due to the efficiency of {TNN\\xspace}s. \n\n\\paragraph{Contributions of this paper}\n\\begin{compactenum}\n\\item We propose a new framework of {tensorial neural networks\\xspace} that extends traditional {neural networks\\xspace}, which naturally preserve multi-dimensional structures of the input data (such as videos).\n\\item We introduce a system of {\\em generalized tensor algebra} for efficient learning and prediction in {TNN\\xspace}s. In particular, we are the first to derive and analyze backpropagation rules for generalized tensor operations.  \n\\item We apply {tensorial neural networks\\xspace} to effectively compress existing neural networks by exploiting additional invariant structures in both data and parameter spaces, therefore reduce the complexity of the model (the number of parameters).\n\\item We provide interpretations of famous neural network architectures in computer vision using our proposed {TNN\\xspace}s. Understanding of why some existing neural network architectures are successful in practice could lead to more insightful neural network designs.\n\\end{compactenum}\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Generalized Tensor Algebra}\n\\label{sec:preliminary}\n\n\\begin{figure}[!htbp]\n\\begin{minipage}{0.45\\textwidth}\n\n\t\\begin{subfigure}[b]{0.2\\textwidth}\n\t\\centering\n\t\\psfrag{n1}{$a$}\n\t\\includegraphics[width=0.4\\textwidth]{.\/\/diagram-1-scalar.eps}\n\t\\caption{Scalar}\n\t\\label{fig:diagram-scalar}\n\t\\end{subfigure}\n\t\\hfill\n\n\t\\begin{subfigure}[b]{0.24\\textwidth}\n\t\\centering\n\t\\psfrag{n1}{$\\myvector{v}$}\n\t\\psfrag{S1}{$I$}\n\t\\includegraphics[width=0.66\\textwidth]{.\/\/diagram-2-vector.eps}\n\t\\caption{Vector}\n\t\\label{fig:diagram-vector}\n\t\\end{subfigure}\n\t\\hfill\n\n\n\t\\begin{subfigure}[b]{0.24\\textwidth}\n\t\\centering\n\t\\psfrag{n1}{$\\mymatrix{M}$}\n\t\\psfrag{S1}{$I$}\n\t\\psfrag{S2}{$J$}\n\t\\includegraphics[width=\\textwidth]{.\/\/diagram-3-matrix.eps}\n\t\\caption{Matrix}\n\t\\label{fig:diagram-matrix}\n\t\\end{subfigure}\n\t\\hfill\n\n\t\\begin{subfigure}[b]{0.24\\textwidth}\n\t\\centering\n\t\\psfrag{n1}{$\\tensor{T}$}\n\t\\psfrag{S1}{$I$}\n\t\\psfrag{S2}{$J$}\n\t\\psfrag{S3}{$K$}\n\t\\includegraphics[width=\\textwidth]{.\/\/diagram-4-tensor.eps}\n\t\\caption{Tensor}\n\t\\label{fig:diagram-tensor}\n\t\\end{subfigure}\n\\caption{\\small{\\textbf{Tensor Diagrams} of \na scalar $a\\! \\in\\! \\R$,\na vector $\\myvector{v}\\! \\in\\! \\R^{I}$,\na matrix $\\mymatrix{M} \\in \\R^{I \\times J}$, \nand a $3$-order tensor $\\tensor{T} \\in \\R^{I \\times J \\times K}$.}}\n\\label{fig:diagrams}\n\\end{minipage}\n\\end{figure}\n\n\\paragraph{Notations.} \nAn $m$-dimensional array $\\tensor{T}$ is defined as an $m$-{\\em order} tensor\n$\\tensor{T} \\in \\R^{I_0 \\times \\cdots \\times I_{m-1}}$.\nIts $(i_0, \\cdots, i_{n-1}, i_{n+1}, \\cdots, i_{m-1})^{{\\mbox{\\tiny th}}}$ {\\em mode}-$n$ {\\em fiber}, a vector along the $n^{{\\mbox{\\tiny th}}}$ axis, is denoted as $\\tensorSub{T}{i_0,\\cdots, i_{n-1}, \\mathbf{:}, i_{n+1},\\cdots, i_{m-1}}$.\n\n\\paragraph{Tensor Diagrams.}\nFollowing the convention in quantum physics~\\cite{orus2014practical, grasedyck2013literature}, Figure~\\ref{fig:diagrams} introduces {\\em tensor diagrams}, graphical representations for multi-dimensional objects. In {tensor diagrams}, an array (scalar\/vector\/matrix\/tensor) is represented as a {\\em node} in the graph, and its {\\em order} is denoted by the number of {\\em edges} extending from the node, where each edge corresponds to one {\\em mode} (whose {\\em dimension} is denoted by the number associated to the edge). \n\n\\input{.\/\/figure_diagrams_operations}\n\n\\paragraph{Generalized Tensor Operations.}\nWe introduce {\\em generalized tensor operations} on {\\em high-order tensor operands} that consist of four primitive operations in Figure~\\ref{fig:operations}, extending the usual operations with vector or matrix operands.\nIn tensor diagrams, an operation is represented by linking edges from the input tensors, where the type of operation is denoted by the shape of line that connects the nodes: solid line stands for {\\em tensor contraction} or {\\em tensor multiplication}, dashed line represents {\\em tensor convolution}, and curved line is for {\\em tensor partial outer product}. \nWe illustrate our generalized tensor operations via simple examples where third-order operands $\\tensor{X}$ and $\\tensor{Y}$ are considered in Figure~\\ref{fig:operations}. However, our generalized tensor operations can extend to higher-order tensors as rigorously defined in Appendix~\\ref{app:operations}.\n\\section{Tensorial Neural Networks} \n\\label{sec:tnn}\n\nA traditional convolutional neural network using tensor diagram is depicted in Figure~\\ref{fig:nn}. We propose a richer class of functions, tensorial neural networks\\xspace ({TNN\\xspace}s), whose the input are tensors of any order (denoted $m$) and operations are generalized tensor operations.\n{TNN\\xspace} are generalization of traditional {neural networks\\xspace}: if the input is a vector and operation is matrix-vector multiplication, TNN\\xspace reduces to {\\em multi-layer perceptrons (MLP)}; and if the input is a feature map and operation is convolution, TNN\\xspace reduces to {\\em convolutional neural networks\\xspace (CNN\\xspace)}. \nThe order of the input tensor and the type of the generalized tensor operation are hyperparameters that define the {\\em architecture} of a TNN\\xspace. \nIn this paper, we design a number of successful architectures of {tensorial neural networks\\xspace} (see Appendices~\\ref{app:dense-tensorized} and \\ref{app:convolutional-tensorized}),\nand an example of \\textit{Tensor-train\\xspace TNN\\xspace} is illustrated in Figure~\\ref{fig:tnn}.\n\n\\input{.\/\/figure_tnn}\n\n\\subsection{One-layer of TNN\\xspace vs One-layer of CNN\\xspace}\nIn CNN\\xspace, each linear layer is a special convolution operation between a {\\em low-order} input and another {\\em low-order} weights kernel.\nIn contrast, each layer in TNN\\xspace is characterized by a generalized tensor operation between a {\\em high-order} input and another {\\em high-order} weights kernel, allowing for  higher expressive power. \n\n\\paragraph{One-layer of CNN}\nA convolutional layer in traditional neural networks\\xspace is parameterized by a $4$-order kernel $\\tensor{K} \\in \\R^{H \\times W \\times S \\times T}$, \nwhere $H, W$ are height\/width of the filters, and $S, T$ are the numbers of input\/output channels. \nThe layer maps a $3$-order input tensor $\\tensor{U} \\in \\R^{X \\times Y \\times S}$ to another $3$-order output tensor $\\tensor{V} \\in \\R^{X^{\\prime} \\times Y^{\\prime} \\times T}$, where $X, Y$ and $X^{\\prime}, Y^{\\prime}$ are heights\/widths of the input and output feature maps. \n\\begin{equation}\n\\tensor{V} = \\tensor{U} ~ (\\ast^{0}_{0} \\circ \\ast^{1}_{1} \\circ \\times^{2}_{2}) ~ \\tensor{K}\n\\label{def:convolutional-main}\n\\end{equation}\\noindent\nWe define such an operation as \\textit{compound operation}, where multiple edges are linked simultaneously (see Figure~\\ref{fig:process-original} for the tensor diagram of a convolutional layer).\nGeneral compound operations are discussed systematically in Appendices~\\ref{app:operations} and~\\ref{app:derivatives}, and in general {TNN\\xspace}s allow arbitrary compound operation to be used at each linear layer. \n\n\\paragraph{One-layer of TNN\\xspace} \nWe illustrate one layer of the {Tensor-train\\xspace TNN\\xspace} in Figure~\\ref{fig:process-rtt} (with input tensor of order $m = 5$), where the parameters are characterized by $(m-1)$ kernels $\\{ \\tensorSub{K}{i} \\}_{i = 0}^{m - 2}$.\nThe multi-dimensional structure lying in the higher-order input tensor $\\tensor{U}^{\\prime}$ is effectively preserved:\neach mode $i$ of the input tensor $\\tensor{U}^{\\prime}$ contracts with the corresponding kernel $\\tensorSub{K}{i}$. {Tensor-train\\xspace TNN\\xspace} allows interactions between the modes by the contraction edges between adjacent kernels $\\tensorSub{K}{i}$ and $\\tensorSub{K}{i+1}$. These edges are crucial for modeling general multi-dimensional transformations while preserving the structures of the input data. \nEffectively, each layer of {Tensor-train\\xspace TNN\\xspace} implements a multi-dimensional propagation of the high-order input.\n\n\\paragraph{Relationship between {TNN\\xspace}s and {CNN\\xspace}s}\n(1) \\textit{{TNN\\xspace} generalizes {NN\\xspace}}. Formally, for any $q > 0$, $\\mathcal{G}^{q} \\subseteq \\mathcal{H}^{q}$ holds. \nConsider a special case of Figure~\\ref{fig:process-rtt} where $\\tensor{K}_0$ and $\\tensorSub{K}{1}$ are identical mappings and contraction of $\\tensorSub{K}{2}$ and $\\tensorSub{K}{m-2}$ equals to a kernel $\\tensor{K}$ as in Figure~\\ref{fig:process-original}, the TNN\\xspace reduces to a CNN\\xspace.  \n(2) \\textit{NN\\xspace can be mapped to TNN\\xspace with fewer parameters}. Formally, there exists $p \\leq q$ such that $\\mathcal{H}^{p} \\subseteq \\mathcal{G}^{q}$.\nGiven an $h^{p} \\in \\mathcal{H}^{p}$, we could factorize $\\tensor{K}$ into $\\{\\tensorSub{K}{i} \\}_{i = 0}^{m - 2}$ using generalized tensor decomposition (as detailed in Section~\\ref{sec:invariant} and Appendix~\\ref{app:decompositions}), therefore find $q \\leq p$ and $g^{q} \\in \\mathcal{G}^q$ ($\\{\\tensorSub{K}{i} \\}_{i = 0}^{m - 2}$ altogether have fewer parameters than $\\tensor{K}$).\n\n\\paragraph{Why are {TNN\\xspace}s more flexible than {NN\\xspace}s?}\n(1) {TNN\\xspace}s are more flexible than {NN\\xspace}s as the input can be tensors of arbitrary order, which naturally deals with data represented by multi-dimensional array without breaking their internal structures. \n(2) Even in the scenario that the input data is a vector or a matrix, one can reshape the input into higher-order tensor in order to exploit the invariant structures within the data discussed in Section~\\ref{sec:invariant}.\n\n\\subsection{Prediction in TNN\\xspace}\nPrediction with a TNN\\xspace proceeds similarly as a normal neural network: the input is passed through the layers in TNN\\xspace in a feedforward manner, where each layer is a generalized tensor operation (multilinear operation) between the input and the model parameters followed by a nonlinear activation function such as $\\mathsf{ReLU}$.  \nWhen the number of operands in a generalized tensor operation is greater than two (i.e. the layer is characterized by more than one kernel), it is generally NP-hard to find the best order to evaluate the operation. We identify efficient strategies for all TNN\\xspace architectures in this paper. For example, each layer in {Tensor-train\\xspace TNN\\xspace} can be efficiently evaluated as:\n\\begin{subequations}\n\\begin{gather}\n\\tensorSub{U}{i+1} = \\tensorSup{U}{i} ~ \\left( \\times^{2}_{1} \\circ \\times^{-1}_{0} \\right) ~ \\tensorSub{K}{i} \\\\\n\\tensor{V}^{\\prime} = \\tensorSub{U}{m-1} ~ \\left( \\ast^{0}_{1} \\circ \\ast^{1}_{2} \\circ \\times^{-1}_{0} \\right) ~ \\tensorSub{K}{m}\n\\end{gather}\n\\end{subequations}\nwhere $\\tensorSub{U}{0} = \\tensor{U}^{\\prime}$ and $\\tensor{V}^{\\prime}$ are the input\/output of the layer, $\\tensorSub{U}{i}$ is the intermediate result after interacting with $\\tensorSub{K}{l}$. The forward pass for other architectures are derived in Appendix~\\ref{app:dense-tensorized} and~\\ref{app:convolutional-tensorized}, with their complexities summarized in Table~\\ref{table:convolutional-main}. \n\n\\subsection{Learning in TNN\\xspace}\nLearning in a TNN\\xspace corresponds to {\\em hierarchical nonlinear general tensor decomposition}, which recovers all tensors in the TNN\\xspace. \nSolving the decomposition problem in closed form is difficult, therefore we derive the backpropagation rules for general tensor operations (in Appendix~\\ref{app:derivatives}) such that the tensors can be recovered by {\\em stochastic gradient descent} approximately.\nFor each layer in TNN\\xspace, backpropagation requires computations of the derivatives of some loss function $\\mathcal{L}$ w.r.t. the input ${\\partial \\mathcal{L}}\/{\\partial \\tensor{U}^{\\prime}}$ and kernel factors $\\{ {\\partial \\mathcal{L}}\/{\\partial \\tensor{K}_{i}}\\}_{i = 0}^{m-2}$ given ${\\partial \\mathcal{L}}\/ \\partial \\tensor{V}^{\\prime}$.\nAs in forward pass, identifying the best strategy to compute these backpropagation equations simultaneously is NP-hard, and we need to develop efficient algorithm for each architecture. For the case of {Tensor-train\\xspace TNN\\xspace}, the algorithm proceeds as follows:\n{\\small\n\\begin{subequations}\n\\begin{gather}\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSub{U}{m-2}} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}^{\\prime}} \\left( (\\ast^{0}_{1})^{\\top} \\circ (\\ast^{2}_{1})^{\\top} \\right) \\tensorSub{K}{m-2} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSub{K}{m-2}} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}^\\prime} \\left( (\\ast^{0}_{0})^{\\top} \\circ (\\ast^{1}_{1})^{\\top} \\circ \\times^{2}_{2} \\cdots \\times^{m-1}_{m-1} \\right) \\tensorSub{U}{m-2} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSub{U}{i}} = \\mathsf{swapaxes} \\big( \\tensorSub{K}{i} \\left( \\times^{2}_{-2} \\circ \\times^{3}_{-1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSub{U}{i+1}} \\big) \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSub{K}{i}} = \\mathsf{swapaxes} \\big( \\tensorSub{U}{i} \\left( \\times^{0}_{0} \\circ \\times^{1}_{1} \\circ \\times^{3}_{2} \\cdots \\times^{m-1}_{m-2} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSub{U}{i+1}} \\big) \n\\end{gather}\n\\end{subequations}\n}\nwhere $\\mathsf{swapaxes}(\\cdot)$ permutes the order as needed (explained in Appendix~\\ref{app:notations}). The algorithms for other architectures are derived in appendices~\\ref{app:dense-tensorized} and~\\ref{app:convolutional-tensorized}, whose time complexities are summarized in Table~\\ref{table:convolutional-main}. \n\n\\input{.\/\/table_convolutional_main}\n \n\n\n\\section{Compression of Neural Networks via TNN\\xspace}\n\\label{sec:invariant}\nSuppose we are given a pre-trained neural network $g^{q} \\in \\mathcal{G}^{q}$ with $q$ parameters (as described in Figure~\\ref{fig:framework}) and we want to compress it to $p$ parameters such that $p \\ll q$. \nIf we are looking for a compressed model in the class of traditional neural networks\\xspace $\\mathcal{G}^{p}$, any compression algorithm can at best find $g^p$, while if we consider a boarder class of compressed models in our proposed {TNN\\xspace}s $\\mathcal{H}^{p}$, a good enough compression algorithm finds $h^p \\in \\mathcal{H}^{p} \\setminus \\mathcal{G}^{p}$.\nAs is demonstrated in Figure~\\ref{fig:framework}, $h^p$ is closer to the uncompressed NN\\xspace $g^{q}$ than $g^{p}$, therefore outperforms $g^{p}$ in its predictive accuracy.\n\nWe introduce a compression algorithm that projects a pre-trained NN\\xspace $g \\in \\mathcal{G}^{q}$ to a TNN\\xspace $h^{\\star} \\in \\mathcal{H}^{p}$. Superscripts on $g$ and $h$ are omitted for notational simplicity. Suppose the input to $g$ is $\\tensor{U}$, our goal is to find a  $h^{\\star}$ such that\n\\begin{equation}\nh^{\\star} = \\arg \\min_{h \\in \\mathcal{H}^p} \\mathsf{dist}(h(\\tensor{U}^{\\prime}), g(\\tensor{U}))\n\\label{opt:}\n\\end{equation}\\noindent\nwhere $m$-order $\\tensor{U}^\\prime$ is the reshaped version of $\\tensor{U}$ (to fit the input requirement of $h$), and $ \\mathsf{dist}(\\cdot, \\cdot)$ denotes any distance, such as $\\ell_2$ distance, between the outputs of $h$ and $g$.\n\n\\subsection{Tensorization: Exploiting Invariant Structures}\nWe first reshape the input data $\\tensor{U}$ to $m$-order tensor $\\tensor{U}^{\\prime}$. \nConsider this toy example of a vector with periodic structure $[1,2,3,1,2,3,1,2,3]$ or modulated structure $[1,1,1,2,2,2,3,3,3]$ in Figure~\\ref{fig:demo-invariant}. The number of parameters needed to represent this vector, naively, is 9. However if we map or \\emph{reshape} the vector into a higher order object, for instance, a matrix ($m = 2$)$[1,1,1;2,2,2;3,3,3]$ where the columns of the matrix are repeated, then apparently this reshaped matrix can be decomposed into rank one without losing information. Therefore only 6 parameters are needed to represent the original length-9 vector.\n\\begin{figure}[!htbp]\n\\centering\n\t\\includegraphics[width=0.46\\textwidth]{.\/\/demo-invariant.eps}\n\t\\caption{{A toy example of invariant structures. The periodic and modulated structures are picked out by exploiting the low rank structure in the reshaped matrix.}}\n\\label{fig:demo-invariant}\n\\end{figure}\n\n\\noindent Now we are ready to learn the mapping from NN\\xspace to TNN\\xspace.\n\n\\subsection{Mapping NN to TNN}\n\n\\paragraph{End-to-End (E2E\\xspace) Learning}\nLet the $l$-th layer of TNN\\xspace $h$ be parameterized by $(m-1)$ kernels $\\{\\tensorInd{K}{i}{l}\\}_{i=0}^{m-2}$. if $\\ell_2$ distance is used, the compression reduces to an optimization problem : \n\\begin{equation}\n\\{ {\\tensorInd{K}{i}{l}}^{\\star} \\}_{i, l} = \\arg \\min_{\\{ \\tensorInd{K}{i}{l} \\}_{i, l}} \\lVert h(\\tensor{U}^\\prime) - g(\\tensor{U})\\rVert_2^2\n\\label{eq:end-to-end}\n\\end{equation}\\noindent\nSolving the problem by backpropagation is very expensive as it requires end-to-end gradients flow in TNN\\xspace, therefore we relax it into a sequence of problems as follows.\n\n \\vspace{-0.5em}\n \n\\paragraph{Sequential\\xspace (Seq\\xspace) Learning} Let $\\tensorSup{V}{l}$ and $\\tensorSup{V^\\prime}{l}$ be the outputs of the $l$-th layers of $g$ and $h$. We propose to project each layer of $g$ to a layer of $h$ bottom-up sequentially, i.e. for $l = 0, \\cdots, L$:\n\\begin{equation}\n\\{ {\\tensorInd{K}{i}{l}}^{\\star} \\}_{i} =\\arg \\min_{\\{\\tensorInd{K}{i}{l}\\}_{i}} \\lVert \\tensorSup{V}{l} - \\tensorSup{V^\\prime}{l}\\rVert_2^2\n\\label{eq:sequential}\n\\end{equation}\\noindent\nwhere the input to these layers are fixed to $\\tensorSup{V}{l-1}$ and $\\tensorSup{V^{\\prime}}{l-1}$ respectively. \n \n \\vspace{-0.5em}\n \n\\paragraph{Generalized Tensor Decomposition}\nSince each layer of TNN\\xspace should approximate the pre-trained NN\\xspace, our goal is to find ${\\tensorInd{K}{i}{l}}^{\\star}$ such that their composition is closed to the uncompressed kernel $\\tensorSup{K}{l}$, and in {Tensor-train\\xspace TNN\\xspace}:\n\\begin{equation}\n\\tensorSup{K}{l} \\approx {\\tensorInd{K}{0}{l}}^{\\star} \\times^{-1}_{0} {\\tensorInd{K}{1}{l}}^{\\star} \\times^{-1}_{0} \\cdots \\times^{-1}_{0} {\\tensorInd{K}{m-2}{l}}^{\\star}\n\\label{eq:convolutional-rtt-main}\n\\end{equation}\\noindent\nThe equation above is known as {\\em generalized tensor decomposition}, because it reverses the mapping of the general tensor operations proposed earlier: given a set of operations and a tensor, generalized tensor decomposition aims to recover the factors\/components such that the operations on these factors result in a tensor approximately equal to the original one.  (See Appendix~\\ref{general_decomposition} for details).  \n\n\\section{Interpretation of Neural Networks Designs}\n\\label{sec:interpretation}\nRecent advances in novel architecture design of neural networks include {\\em Inception}~\\cite{szegedy2017inception}, {\\em Xception}~\\cite{chollet2016xception} and {\\em Bottleneck structures}~\\cite{lin2013network, he2016identity}. We will show, in this section, that all aforementioned architectures can be naturally derived as special cases of our TNN\\xspace framework (with small modifications). \n\n\\paragraph{Inception Network.} \nInception network~\\cite{szegedy2017inception} is a special case of {Tensor-train\\xspace TNN\\xspace} as shown in Figure~\\ref{fig:inception_all}. For instance, an Inception network in Figure~\\ref{fig:inception} can be represented using tensor diagram as in Figure~\\ref{fig:inception-tensordiagram}, a simplified {Tensor-train\\xspace TNN\\xspace} with input tensor of size $3 \\times 3$ and the dimension of the connecting edge is $1$. \n\n\\input{.\/\/figure_inception}\n\n\\paragraph{Xception Network.} Xception~\\cite{chollet2016xception} is an architecture that further simplifies Inception, where local connection at the \\textit{depthwise layers} are further limited to one neuron (or one feature map).\nSimilar to Inception, the Xception network in Figure~\\ref{fig:xception} can be represented using tensor diagram as in Figure~\\ref{fig:xception-tensordiagram}, a simple Tensor-train\\xspace TNN\\xspace with input tensor of size $2 \\times 2$ and the dimension of the connecting edge is $2$. \n\n\\input{.\/\/figure_xception}\n\n\\paragraph{Bottleneck Network.} Finally, bottleneck architecture in~\\cite{lin2013network, he2016identity} can be interpreted as decomposition of a wider architecture into a tensor network with $3$ kernels.\n\n\\input{.\/\/figure_bottleneck}\n\n\\paragraph{Insights for NN designs.} Our TNN\\xspace provides some insights on how to design a compact NN architecture in practice: \n(1) First, we can start our design with a traditional neural network (with wide architecture); (2) Second, we factorize the kernel into a tensor network by general tensor decomposition, which converts the original neural network into a TNN\\xspace that naturally bear nice compact structures as we see in Figures~\\ref{fig:inception} and~\\ref{fig:xception}. (3) Since each tensor network corresponds to a compact design, therefore exploration of different designs can be done by designing different tensor networks. \n\\section{Experiments}\n\\label{sec:experiments} \n\n\nIn this section, we evaluate the effectiveness of our proposed compression algorithm (TNN based compression\\xspace) proposed in section~\\ref{sec:invariant} on several benchmark deep neural networks and datasets. We evaluate fully connected layer compression on MNIST (discussed in Appendix~\\ref{app:dense-tensorized}); we evaluate convolutional layer compression on ResNet\\xspace-32~\\cite{he2016identity} for CIFAR-10; and we evaluate the scalability of our compression algorithm on ResNet\\xspace-50~\\cite{he2016identity} for ImageNet (2012) dataset. \n\n\nWe conduct experiments on various compression rates and compare the performance of our TNN based compression\\xspace (TNN-C\\xspace) methods with that of the state-of-the-art low-rank approximation based compression\\xspace methods (NN-C\\xspace~\\cite{jaderberg2014speeding, lebedev2014speeding, kim2015compression}) under the same compression rate. More details of the state-of-the-art low-rank approximation based compression\\xspace are provided in appendix~\\ref{app:convolutional}. For all experiments, we use the Adam optimizer~\\cite{kingma2014adam} with 1e-3 learning rate and decay the learning rate by 10x every 50 epochs. Overall, we show that our TNN based compression\\xspace maintain high accuracy even when the networks are highly compressed. \n\nWe denote \\nnPrefix\\CP, \\nnPrefix\\TK, and \\nnPrefix\\TT as the compressed neural network obtained by baseline low-rank approximation based compression\\xspace, which uses classical types of tensor decompositions (CP\\xspace, TK\\xspace, and TT\\xspace) as the layer-wise projecting methods (mentioned in section~\\ref{sec:invariant}). While \\tnnPrefix\\rCP, \\tnnPrefix\\rTK and \\tnnPrefix\\rTT are compressed tensorial neural networks\\xspace obtained by TNN based compression\\xspace, which uses our proposed modified tensor decomposition (\\reshapePrefix\\CP, \\reshapePrefix\\TK, and \\reshapePrefix\\TT). \n\nAs mentioned in section~\\ref{sec:invariant}, we refer to traditional back propogation-based compression of the network as end-to-end\\xspace (E2E\\xspace) compression, and refer to our strategy of data reconstruction-based sequential compression as Sequential\\xspace (Seq\\xspace) compression.\n\\smallskip \n\n\\noindent \\textbf{Our algorithm achieves 5\\% higher accuracy than baseline on CIFAR10 using ResNet-32.}\nThe results from table~\\ref{baseline} demonstrate that our TNN-C\\xspace maintains high accuracy even after the networks are highly compressed on CIFAR-10. Given a well-trained 32-layer ResNet\\xspace network and a goal of reducing the number of parameters to 10\\% of the original size, the \\nnPrefix\\CP obtained by NN-C\\xspace using end-to-end\\xspace compression reduces the original accuracy from 93.2\\% to 86.93\\%; while the \\tnnPrefix\\rCP obtained by TNN-C\\xspace paired with Seq\\xspace compression increases the accuracy to 91.28\\% with the same compression rate --- \\textbf{a performance loss of 2\\% with only 10\\% of the number of parameters}. Furthermore, TNN-C\\xspace achieves further aggressive compression --- \\textbf{a performance loss of 6\\% with only 2\\% of the number of parameters}.\nWe observe similar trends (higher compression and higher accuracy) are observed for \\tnnPrefix\\rTT as well.   \nThe structure of the Tucker\\xspace decomposition (see section~\\ref{app:convolutional-tensorized}) makes \\tnnPrefix\\rTK less effective with very high compression, since the {``internal structure''} of the network reduces to very low rank, which may lose necessary information. Increasing the network size to 20\\% of the original provides reasonable performance on CIFAR-10 for \\tnnPrefix\\rTK as well.\n\n\n\\begin{table*}[!htbp]\n\\centering\n\\begin{tabular}{l}\n\\begin{tabular}{c |c  c  c  c || c | c  c  c  c}\n\\multirow{2}{3.6em}{Architect.} & \\multicolumn{4}{c||}{Compression rate} & \\multirow{2}{3.6em}{Architect.} & \\multicolumn{4}{c}{Compression rate} \\\\ \n& \\multicolumn{1}{c}{5\\%} & \\multicolumn{1}{c}{10\\%} & \\multicolumn{1}{c}{20\\%} & \\multicolumn{1}{c||}{40\\%} & \\multicolumn{1}{c|}{} & \\multicolumn{1}{c}{2\\%} & \\multicolumn{1}{c}{5\\%} & \\multicolumn{1}{c}{10\\%} & \\multicolumn{1}{c}{20\\%} \\\\\n\\hline\n\\nSVD~\\cite{jaderberg2014speeding} & 83.09 & 87.27 & 89.58 & 90.85 & TNN-TR{$^\\dag$}~\\cite{wang2018wide} & - & 80.80{$^\\dag$} & - & 90.60 \\\\\n\\nnPrefix\\CP~\\cite{lebedev2014speeding}   & 84.02 & 86.93 & 88.75 & 88.75 & \\tnnPrefix\\rCP & 85.7 & 89.86 & \\textbf{91.28} & - \\\\\n\\nnPrefix\\TK~\\cite{kim2015compression}   & 83.57 & 86.00 & 88.03 & 89.35 & \\tnnPrefix\\rTK & 61.06 & 71.34 & 81.59 & 87.11 \\\\\n\\nnPrefix\\TT   & 77.44 & 82.92 & 84.13 & 86.64 & \\tnnPrefix\\rTT & 78.95 & 84.26 & 87.89 & - \\\\\n\\end{tabular} \\\\\n\\rule{0in}{1.2em}$^\\dag$\\scriptsize The tensor ring (TR) results are cited from~\\cite{wang2018wide}, and the accuracy of 80.8\\% is achieved by 6.67\\% compression rate.\n\\end{tabular}\n\\vspace{-0.8em}\n\\caption{Percentage test accuracy of baseline \\textbf{NN-C\\xspace} with\nE2E\\xspace compression vs. our \\textbf{TNN-C\\xspace} with  Seq\\xspace compression on CIFAR10. \nThe uncompressed ResNet\\xspace-32 achieves {93.2\\%} accuracy with 0.46M parameters. \n\\label{baseline}}\n\\end{table*}\n\n\n\\begin{table*}[!htbp]\n\\centering\n\\begin{tabular}{c | c  c | c  c | c c | c c}\n\\multirow{3}{3.6em}{Architect.} & \\multicolumn{8}{c}{Compression rate} \\\\ \n& \\multicolumn{2}{c|}{5\\%} & \\multicolumn{2}{c|}{10\\%} & \\multicolumn{2}{c|}{20\\%} & \\multicolumn{2}{c}{40\\%} \\\\\n& Seq & E2E & Seq & E2E & Seq & E2E & Seq & E2E \\\\\n\\hline\n\\nSVD & 74.04 & \\textbf{83.09} & 85.28 & \\textbf{87.27} & \\textbf{89.74} & 89.58  & \\textbf{91.83} & 90.85 \\\\\n\\nnPrefix\\CP   & 83.19 & \\textbf{84.02} & \\textbf{88.50} & 86.93 & \\textbf{90.72} & 88.75 & \\textbf{89.75} & 88.75 \\\\\n\\nnPrefix\\TK   & 80.11 & \\textbf{83.57} & \\textbf{86.75} & 86.00 & \\textbf{89.55} & 88.03 & \\textbf{91.3} & 89.35 \\\\\n\\nnPrefix\\TT   & \\textbf{80.77} & 77.44 & \\textbf{87.08} & 82.92 & \\textbf{89.14} & 84.13 & \\textbf{91.21} & 86.64\\\\  \n\\end{tabular}\n\\caption{Percentage accuracy of \\textbf{our Seq} vs. \\textbf{baseline E2E} tuning using NN-C\\xspace on CIFAR10. \n\\label{seq-e2e-noreshape}}\n\\end{table*}\n\n\t\n\\begin{table*}[!htbp]\n\\centering\n\\begin{tabular}{c | c c || c | c c}\n\\multirow{2}{3.6em}{Architect.} & \\multicolumn{2}{c||}{Compression rate} & \\multirow{2}{3.6em}{Architect.} & \\multicolumn{2}{c}{Compression rate}\\\\  \n& \\multicolumn{1}{c}{5\\%} & \\multicolumn{1}{c||}{10\\%} & \\multicolumn{1}{c|}{} & \\multicolumn{1}{c}{5\\%} & \\multicolumn{1}{c}{10\\%} \\\\ \\hline\n\\nnPrefix\\CP & 83.19 & 88.50 & \\tnnPrefix\\rCP & \\textbf{89.86} & \\textbf{91.28} \\\\\n\\nnPrefix\\TK & 80.11 & 86.73 & \\tnnPrefix\\rTK & 71.34 & 81.59 \\\\\n\\nnPrefix\\TT & 80.77 & 87.08 & \\tnnPrefix\\rTT & \\textbf{84.26} & \\textbf{87.89} \\\\\n\\end{tabular}\n\\caption{Percentage accuracy of \\textbf{our TNN-C\\xspace} vs.  \\textbf{baseline NN-C\\xspace} using Seq tuning on CIFAR10.}\n\\label{reshape-vs-non}\n\\end{table*}\n\t\n\n\\begin{table*}[!htbp]\n\\begin{center}\n\\begin{tabular}{c|c||c||c||c}\n&  &  {Uncompressed}  &{\\nnPrefix\\TT} (E2E\\xspace) &{\\tnnPrefix\\rTT} (Seq\\xspace)  \\\\\n\\# epochs  & \\# samples & \\# params. = 25M & \\# params. = 2.5M & \\# params. = 2.5M \\\\\n\\hline\n0.2 & 0.24M & 4.22& 2.78 & 44.35  \\\\\n0.3 & 0.36M & 6.23& 3.99 & 46.98   \\\\\n0.5 & 0.60M & 9.01& 7.48 & 49.92  \\\\\n1.0 & 1.20M & 17.3& 12.80 & 52.59   \\\\\n2.0 & 2.40M & 30.8& 18.17 & \\textbf{54.00}\\\\\n\\end{tabular}\t\t\n\\caption{Convergence of percentage accuracies of \\textbf{uncompressed} vs. \\textbf{NN-C\\xspace} (TT\\xspace decomposition) vs. \\textbf{TNN-C\\xspace} (\\tnnPrefix\\rTT decomposition) achieving 10\\% compression rate for ResNet\\xspace-50 ImageNet.\n\\label{imagenet}}\n\\end{center}\n\\end{table*}\n\n\n\n\n\\begin{figure}[!htbp]\n\\begin{minipage}[]{\\linewidth}\n\t\t\\centering\n\t\t\\psfrag{y-axis}{\\scriptsize{Test Error}}\n\t\t\\psfrag{x-axis}[Bl]{\\scriptsize{\\# Gradient Updates ($\\times 10^{11}$)}}\n\t\t\\psfrag{cp-seq}[Bl]{\\scriptsize{Seq-CP\\xspace}}\n\t\t\\psfrag{cp-e2e}[Bl]{\\scriptsize{E2E-CP\\xspace}}\n\t\t\\psfrag{tt-seq}[Bl]{\\scriptsize{Seq-TT\\xspace}}\n\t\t\\psfrag{tt-e2e}[Bl]{\\scriptsize{E2E-TT\\xspace}}\n\t\t\\psfrag{tk-e2e}[Bl]{\\scriptsize{E2E-TK\\xspace}}\n\t\t\\psfrag{tk-seq}[Bl]{\\scriptsize{Seq-TK\\xspace}}\n\t\t\\includegraphics[width=0.8\\textwidth]{.\/\/faster_convergence_new.eps}\n\t\t\\captionof{figure}{\\label{convergence}Convergence rate for Seq vs. E2E\\xspace compression on CIFAR10.}\n\t\\end{minipage}\n\n\\end{figure}\n\n\nTable~\\ref{baseline} shows that \\emph{TNN-C\\xspace with Seq\\xspace compression} outperforms \\emph{NN-C\\xspace with end-to-end\\xspace compression}.  Now we address the following question: is one factor (Seq\\xspace compression or TNN-C\\xspace) primarily responsible for increased performance, or is the benefit due to synergy between the two?\n\n\\smallskip \n\n\\noindent \\textbf{Seq\\xspace compression, TNN based compression\\xspace, or both?} (1) \nWe present the effect of different compression methods on accuracy in Table~\\ref{seq-e2e-noreshape}.  \nOther than at very high compression rate (5\\% column in\nTable~\\ref{seq-e2e-noreshape}), Seq\\xspace compression (Seq) consistently\noutperforms end-to-end\\xspace (E2E) compression. In addition, Seq\\xspace compression is also\n{\\em much\\\/} faster and leads to more stable convergence compared to end-to-end\\xspace compression.\nFigure~\\ref{convergence} plots the compression error over the number of gradient\nupdates for various compression methods.\n(2) We present the effect of different compression methods on accuracy in Table~\\ref{reshape-vs-non}.\nInterestingly, as what is demonstrated in Table~\\ref{reshape-vs-non}, if TNN based compression\\xspace method is used, the test accuracy is restored for even very high compression ratios~\\footnote{Note that \\tnnPrefix\\rTK remains an exception for aggressive compression due to the low rank internal structure that we previously discussed.}.  \nThese results confirm that TNN\\xspace is more flexible than NN as TNN\\xspace allows exploitation of additional invariant structures in the parameter space of deep neural networks, and such invariant structures are picked up by our proposed TNN based compression\\xspace (our TNN-C\\xspace), but not by low-rank approximation based compression\\xspace (NN-C\\xspace). \nTherefore, our results show that TNN-C\\xspace and Seq\\xspace compression are\nsymbiotic, and {\\em both\\\/} are necessary to simultaneously obtain a\nhigh accuracy and a high compression rate.\n\n\n\\paragraph{Scalability}\nFinally, we show that our methods scale to state-of-the-art large\nnetworks, by evaluating performance on the ImageNet 2012 dataset on a\n50-layer ResNet\\xspace (uncompressed with 76.05\\%  accuracy).\nTable~\\ref{imagenet} shows the accuracy of \\tnnPrefix\\rTT obtained by TNN-C\\xspace\nwith Seq\\xspace tuning compared to that of \\nnPrefix\\TT obtained by low-rank approximation based compression\\xspace with E2E\\xspace tuning, and the accuracy of the uncompressed network (ResNet\\xspace-50) with 10\\% compression rate.  \nTable~\\ref{imagenet} shows that TNN-C\\xspace paired with Seq tuning is much faster than the alternative. This is an important\nresult because it empirically validates our hypotheses that (1)\nour TNN based compression\\xspace captures the invariant structure of the ResNet\\xspace (with few redundancies) better and faster than the baseline NN-C\\xspace compression, \n(2) data reconstruction Seq\\xspace tuning is effective even on the largest networks and datasets, and \n(3) our proposed efficient TNN-C\\xspace can scale to the state-of-the-art neural networks. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion and Perspectives}\n\\label{sec:conclusion}\nWe define a new generalized tensor algebra extending existing tensor operations.  We extend vector\/matrix operations to their\nhigher order tensor counterparts, providing systematic notations and libraries for\ntensorization of neural networks and higher order tensor\ndecompositions. \nUsing these generalized tensor operations, we introduce tensorial neural networks\\xspace (TNN\\xspace) which extends existing neural networks. \nOur TNN is more flexible than NN and more compact than neural networks allowing same amount expressive power with fewer number of parameters. \nTherefore mapping NN to its closest TNN is a compression of NN as the resulting TNN will carry fewer number of parameters. \nOther compression techniques as mentioned in the related work can naturally be used on the compressed TNN to further compress the TNN.  \n As a future step, we will explore optimizing the order of (parallel) implementations of the tensor algebra.\n\n\n\\section{Notations}\n\\label{app:notations}\n\n\\paragraph{Symbols:}\nLower case letters (e.g. $\\myvector{v}$) are used to denote column vectors, while upper case letters (e.g. $\\mymatrix{M}$) are used for matrices, and curled letters (e.g. $\\tensor{T}$) for multi-dimensional arrays (tensors). For a tensor $\\tensor{T} \\in \\R^{I_0 \\times \\cdots \\times I_{m-1}}$, we will refer to the number of indices as \\textit{order}, each individual index as \\textit{mode} and the length at one mode as \\textit{dimension}. Therefore, we will say that $\\tensor{T} \\in \\R^{I_0 \\times \\cdots \\times I_{m-1}}$ is an $m$-order tensor which has dimension $I_k$ at mode-$k$. {\\em Tensor operations} are extensively used in this paper: The {\\em tensor (partial) outer product} is denoted as $\\otimes$, {\\em tensor convolution} as $\\ast$, and finally $\\times$ denotes either {\\em tensor contraction} or {\\em tensor multiplication}. Each of these operators will be equipped with subscript and superscript when used in practice, for example $\\times^{m}_{n}$ denotes mode-$(m, n)$ tensor contraction (defined in Appendix~\\ref{app:operations}). Furthermore, the symbol $\\circ$ is used to construct {\\em compound operations}. For example, $(\\ast \\circ \\otimes)$ is a compound operator simultaneously performing tensor convolution and tensor partial outer product between two tensors.     \n\n\\paragraph{Indexing:} In this paragraph, we explain the usages of subscripts\/superscripts for both multi-dimensional arrays and operators, and further introduce several functions that are used to alter the layout of multi-dimensional arrays. \n\n\\begin{itemize}[leftmargin=*]\n\n\\item \n(1) Nature indices start from 0, but reversed indices are used occasionally, which start from $-1$. Therefore the first entry of a vector $\\myvector{v}$ is ${v}_{0}$, while the last one is ${v}_{-1}$. (2) For multi-dimensional arrays, the {\\em subscript} is used to denote an entry or a subarray within an object, while {\\em superscript} is to index among a sequence of arrays. For example, $\\mymatrix{M}_{i, j}$ denotes the entry at $i^{{\\mbox{\\tiny th}}}$ row and $j^{{\\mbox{\\tiny th}}}$ column of a matrix $\\mymatrix{M}$, and $\\matrixSup{M}{k}$ is the $k^{{\\mbox{\\tiny th}}}$ matrix in a set of $N$ matrices $\\{ \\matrixSup{M}{0}, \\matrixSup{M}{1}, \\cdots \\matrixSup{M}{N-1} \\}$. For operators, as we have seen, both subscript and superscript are used to denote the modes involved in the operation. (3) {\\em The symbol colon '$:$'} is used to slice a multi-dimensional array. For example, $\\tensorSub{M}{:, k}$ denotes the $k^{{\\mbox{\\tiny th}}}$ column of $\\mymatrix{M}$, and $\\tensorSub{T}{:, :, k}$ denotes the $k^{{\\mbox{\\tiny th}}}$ {\\em frontal slice} of a 3-order tensor $\\tensor{T}$. (4) {\\em Big-endian notation} is adopted in conversion between multi-dimensional array and vectors. Specifically, the function $\\vectorize(\\cdot)$ flattens (a.k.a. {\\em vectorize}) a tensor $\\tensor{T} \\in \\R^{I_0 \\times \\cdots \\times I_{m-1}}$ into a vector $\\myvector{v} \\in \\R^{\\prod_{l=0}^{m-1} I_l}$ such that $\\tensorSub{T}{i_0, \\cdots, i_{m-1}} = v_{i_{m-1} + i_{m-2} I_{m-1} + \\cdots + i_0 I_1 \\cdots I_{m-1}}$.\n \n\n\\item \n(1) The function $\\mathsf{swapaxes}(\\cdot)$ is used to permute ordering of the modes of a tensor as needed. For example, given two tensors $\\tensor{U} \\in \\R^{I \\times J \\times K}$ and $\\tensor{V} \\in \\R^{K \\times J \\times I}$, the operation $\\tensor{V} = \\mathsf{swapaxes}(\\tensor{U})$ convert the tensor $\\tensor{U}$ into $\\tensor{V}$ such that $\\tensorSub{V}{k, j, i} = \\tensorSub{U}{i, j, k}$.\n(2) The function $\\mathsf{flipaxis}(\\cdot, \\cdot)$ flips a tensor along a given mode. For example, given a tensor $\\tensor{U} \\in \\R^{I \\times J \\times K}$ and $\\tensor{V} = \\mathsf{flipaxis}(\\tensor{U}, 0)$, the entries in $\\tensor{V}$ is defined as $\\tensorSub{V}{i, j, k} = \\tensorSub{U}{I - 1 - i (mod ~ I), j, k}$.\n\n\\end{itemize}\n\\section{Tensor operations}\n\\label{app:operations}\n\n\\input{.\/\/table_operations}\n\nTo begin with, we describe several {\\em basic tensor operations} that are natural generalization to their vector\/matrix counterparts. These basic operations can be further combined to construct {\\em compound operations} that serve as building blocks of {\\em tensorial neural networks}. \n\n\\paragraph{Tensor contraction} \nGiven an $m$-order tensor $\\tensorSup{T}{0} \\in \\R^{I_0 \\times \\cdots \\times I_{m-1}}$ and another $n$-order tensor $\\tensorSup{T}{1} \\in \\R^{J_0 \\times \\cdots \\times J_{n-1}}$, which share the same dimension at mode-$k$ of $\\tensorSup{T}{0}$ and mode-$l$ of $\\tensorSup{T}{1}$( i.e. $I_k = J_l$), the mode-$(k, l)$ contraction of $\\tensorSup{T}{0}$ and $\\tensorSup{T}{1}$, denoted as $\\tensor{T} \\triangleq \\tensorSup{T}{0} \\times^{k}_{l} \\tensorSup{T}{1}$, returns a $(m + n - 2)$-order tensor $\\tensor{T}\\ \\in R^{I_0 \\times \\cdots \\times I_{k-1} \\times I_{k+1} \\times \\cdots \\times I_{m-1} \\times J_0 \\times \\cdots \\times J_{l-1} \\times J_{l+1} \\times \\cdots \\times J_{n-1}}$, whose entries are computed as\n\\begin{subequations}\n\\begin{align}\n& \\tensorSub{T}{i_0, \\cdots, i_{k-1}, i_{k+1}, \\cdots, i_{m-1}, j_0, \\cdots, j_{l-1}, j_{l+1}, \\cdots, j_{n-1}} \\nonumber \\\\\n= ~ & \\sum_{r=0}^{I_k - 1} \\tensorInd{T}{i_0, \\cdots, i_{k-1}, r, i_{k+1}, \\cdots, i_{m-1}}{0} ~ \\tensorInd{T}{j_0, \\cdots, j_{l-1}, r, j_{l+1}, \\cdots, j_{n-1}}{1} \\label{def:tensor-contraction-1} \\\\\n= ~ & \\langle  \\tensorInd{T}{i_0, \\cdots, i_{k-1}, :, i_{k+1}, \\cdots, i_{m-1}}{0}, \\tensorInd{T}{j_0, \\cdots, j_{l-1}, :, j_{l+1}, \\cdots, j_{n-1}}{1} \\rangle \\label{def:tensor-contraction-2}\n\\end{align}\n\\end{subequations}\nNotice that tensor contraction is a direct generalization of matrix multiplication to higher-order tensor, and it reduces to matrix multiplication if both tensors are $2$-order (and therefore matrices). As each entry in $\\tensor{T}$ can be computed as inner product of two vectors, which requires $I_k = J_l$ multiplications, the total number of operations to evaluate a tensor contraction is therefore $O( ( \\prod_{u = 0}^{m - 1} I_u ) ( \\prod_{v = 0, v \\neq l}^{n - 1} J_v ) )$, taking additions into account. \n\n\\paragraph{Tensor multiplication (Tensor product)} \nTensor multiplication (a.k.a. tensor product) is a special case of tensor contraction where the second operant is a matrix. Given a $m$-order tensor $\\tensor{U} \\in \\R^{I_0 \\times \\cdots \\times I_{m-1}}$ and a matrix $\\mymatrix{M} \\in \\R^{I_k \\times J}$, where the dimension of $\\tensor{U}$ at mode-$k$ agrees with the number of the rows in $\\mymatrix{M}$, the mode-$k$ tensor multiplication of $\\tensor{U}$ and $\\myvector{M}$, denoted as $\\tensor{V} \\triangleq \\tensor{U} \\times_k \\mymatrix{M}$, yields another $m$-order tensor $\\tensor{V} \\in \\R^{I_0 \\times \\cdots \\times I_{k-1} \\times J \\times I_{k+1} \\times \\cdots I_{m-1}}$, whose entries are computed as\n\\begin{subequations}\n\\begin{align}\n& \\tensorSub{V}{i_0, \\cdots, i_{k-1}, j, i_{k+1}, \\cdots, i_{m-1}} \\nonumber \\\\\n= ~ & \\sum_{r=0}^{I_k - 1} \\tensorSub{U}{i_0, \\cdots, i_{k-1}, r, i_{k+1}, \\cdots, i_{m-1}} ~ \\mymatrix{M}_{r, j} \\label{def:tensor-multiplication-1} \\\\\n= ~ & \\langle \\tensorSub{U}{i_0, \\cdots, i_{k-1}, :, i_{k+1}, \\cdots, i_{m-1}}, ~ \\mymatrix{M}_{:, j} \\rangle \\label{def:tensor-multiplication-2}\n\\end{align}\n\\end{subequations}\nFollowing the convention of multi-linear algebra, the mode for $J$ now substitutes the location originally for $I_k$ (which is different from the definition of tensor contraction). Regardlessly, the number of operations for tensor multiplication follows tensor contraction exactly, that is $O ( ( \\prod_{u = 0}^{m - 1} I_u ) J )$.\n\n\\paragraph{Tensor convolution}\nGiven a $m$-order tensor $\\tensorSup{T}{0} \\in \\R^{I_0 \\times I_1 \\times \\cdots \\times I_{m-1}}$ and another $n$-order tensor $\\tensorSup{T}{1} \\in \\R^{J_0 \\times J_1 \\times \\cdots \\times J_{n-1}}$. The mode-$(k, l)$ convolution of $\\tensorSup{T}{0}$ and $\\tensorSup{T}{1}$, denoted as $\\tensor{T} \\triangleq \\tensorSup{T}{0} \\ast^{k}_{l} \\tensorSup{T}{1}$, returns a $(m + n - 1)$-order tensor $\\tensor{T} \\in \\R^{I_0 \\times \\cdots \\times I^{\\prime}_k \\times \\cdots \\times I_{m-1} \\times J_0 \\times \\cdots \\times J_{l-1} \\times J_{l+1} \\times \\cdots \\times J_{n-1}}$. The entries of $\\tensor{T}$ can be computed using any convolution operation $\\ast$ that is defined for two vectors.\n\\begin{subequations}\n\\begin{align}\n& \\tensorSub{T}{i_0, \\cdots, i_{k-1}, :, i_{k+1}, \\cdots, i_{m-1}, j_0, \\cdots, j_{l-1}, j_{l+1}, \\cdots, j_{n-1}} \\nonumber \\\\\n= ~ & \\tensorInd{T}{i_0, \\cdots, i_{k-1}, :, i_{k+1}, \\cdots, i_{m-1}}{0} \\ast \\tensorInd{T}{j_0, \\cdots, j_{l-1}, :, j_{l+1}, \\cdots, j_{n-1}}{1} \\label{def:tensor-convolution} \\\\\n= ~ & \\tensorInd{T}{j_0, \\cdots, j_{l-1}, :, j_{l+1}, \\cdots, j_{n-1}}{1} ~\\overline{\\ast}~ \\tensorInd{T}{i_0, \\cdots, i_{k-1}, :, i_{k+1}, \\cdots, i_{m-1}}{0} \\label{def:tensor-convolution-reversed}\n\\end{align}\n\\end{subequations}\nHere we deliberately omit the exact definition of vector convolution $\\ast$ (and its conjugate $\\overline{\\ast}$), because it can be defined differently depending on the user case (Interestingly, the \"convolution\" in convolutional layer indeed computes {\\em correlation} instead of convolution).\nCorrespondingly, the resulted dimension $I^{\\prime}_k$ at mode-$k$ is determined by the chosen type of convolution. \n For example, the \"convolution\" in convolutional layer typically yields $I^{\\prime}_k = I_k$ (with same padding) or $I^{\\prime}_k = I_k - J_l + 1$ (with valid padding). \nWithout {\\em Fast Fourier Transform (FFT)}, the number of operations is $O ( ( \\prod_{u = 0}^{m - 1} I_u ) ( \\prod_{v = 0}^{n - 1} J_v ) )$.\n\n\\paragraph{Tensor outer product}\nGiven a $m$-order tensor $\\tensorSup{T}{0} \\in \\R^{I_0 \\times I_1 \\times \\cdots \\times I_{m-1}}$ and another $n$-order tensor $\\tensorSup{T}{1} \\in \\R^{J_0 \\times J_1 \\times \\cdots \\times J_{n-1}}$, the outer product of $\\tensorSup{T}{0}$ and $\\tensorSup{T}{1}$, denoted $\\tensor{T} \\triangleq \\tensorSup{T}{0} \\otimes \\tensorSup{T}{1}$, concatenates all the indices of $\\tensorSup{T}{0}$ and $\\tensorSup{T}{1}$, and returns a $(m + n)$-order tensor $\\tensor{T} \\in \\R^{I_0 \\times \\cdots \\times I_{m-1} \\times J_0 \\times \\cdots \\times J_{n-1}}$ whose entries are computed as \n\\begin{equation}\n\\label{def:tensor-outer-product}\n\\tensorSub{T}{i_0, \\cdots, i_{m-1}, j_0, \\cdots,  j_{n-1}} = \\tensorInd{T}{i_0, \\cdots, i_{m-1}}{0} ~ \\tensorInd{T}{j_0, \\cdots, j_{n-1}}{1} \n\\end{equation}\\noindent\nIt is not difficult to see that tensor outer product is a direct generalization for outer product for two vectors $\\mymatrix{M} = \\myvector{u} \\otimes \\myvector{v} = \\myvector{u} ~ \\myvector{v}^\\top$. Obviously, the number of operations to compute a tensor outer product explicitly is $O ( ( \\prod_{u = 0}^{m - 1} I_u ) ( \\prod_{v = 0}^{n - 1} J_v ) )$. \n\n\\paragraph{Tensor partial outer product}\nTensor partial outer product is a variant of tensor outer product defined above, which is widely used in conjunction with other operations. Given a $m$-order tensor $\\tensorSup{T}{0} \\in \\R^{I_0 \\times I_1 \\times \\cdots \\times I_{m-1}}$ and another $n$-order tensor $\\tensorSup{T}{1} \\in \\R^{J_0 \\times J_1 \\times \\cdots \\times J_{n-1}}$, which share the same dimension at mode-$k$ of $\\tensorSup{T}{0}$ and mode-$l$ of $\\tensorSup{T}{1}$ (i.e. $I_k = J_l$), the mode-$(k, l)$ partial outer product of $\\tensorSup{T}{0}$ and $\\tensorSup{T}{1}$, denoted as $\\tensor{T} \\triangleq \\tensorSup{T}{0} \\otimes^{k}_{l} \\tensorSup{T}{1}$, returns a $(m + n - 1)$-order tensor $\\tensor{T} \\in \\R^{I_0 \\times \\cdots \\times I_{m-1} \\times J_0 \\times \\cdots \\times J_{l-1} \\times J_{l+1} \\times \\cdots \\times J_{n-1}}$, whose entries are computed as\n\\begin{subequations}\n\\begin{align}\n& \\tensorSub{T}{i_0, \\cdots, i_{k-1}, r, i_{k+1}, \\cdots, i_{m-1}, j_0, \\cdots, j_{l-1}, j_{l+1}, \\cdots, j_{n-1}} \\nonumber \\\\\n= ~ & \\tensorInd{T}{i_0, \\cdots, i_{k-1}, r, i_{k+1}, \\cdots, i_{m-1}}{0} ~ \\tensorInd{T}{j_0, \\cdots, j_{l-1}, r, j_{l+1}, \\cdots, j_{n-1}}{1} \n\\label{def:tensor-partial-outer-product} \\\\\n= ~ & \\tensorInd{T}{\\cdots, r, \\cdots}{0} \\otimes \\tensorInd{T}{\\cdots, r, \\cdots}{1}\n\\end{align}\n\\end{subequations}\nThe operation bears the name \"partial outer product\" because it reduces to outer product once we fix the indices at mode-$k$ of $\\tensorSup{T}{0}$ and mode-$l$ of $\\tensorSup{T}{1}$.\nReferring to the computational complexity of tensor outer product, the number of operations for each fixed index is $O ( ( \\prod_{u = 0, u \\neq k}^{m - 1} I_u ) ( \\prod_{v = 0, v \\neq l}^{n - 1} J_v ) )$, therefore the total time complexity for the tensor partial outer product is $O ( ( \\prod_{u = 0}^{m - 1} I_u ) ( \\prod_{v = 0, v \\neq l}^{n - 1} J_v ) )$.\n\n\n\\paragraph{Compound operations:}\n As building blocks, the basic tensor operations defined above can further combined to construct compound operations that perform multiple operations on multiple tensors simultaneously. We illustrate their usage using two representative examples in this section, and we will see more examples when we derive backpropagation rules for tensor operations in Appendix~\\ref{app:derivatives}.\n \n\\begin{itemize}[leftmargin=*]\n\n\\item \\textbf{Simultaneous multi-operations between two tensors. } For example, given two $3$-order tensors $\\tensorSup{T}{0} \\in \\R^{R \\times X \\times S}$ and $\\tensorSup{T}{1} \\in \\R^{R \\times H \\times S}$, we can define a compound operation $\\left( \\otimes^0_0 \\circ \\ast^1_1 \\circ \\times^2_2 \\right)$ between $\\tensorSup{T}{0}$ and $\\tensorSup{T}{1}$, where mode-$(0,0)$ partial outer product, mode-$(1,1)$ convolution and mode-$(2,2)$ contraction are performed simultaneously, which results in a $2$-order tensor $\\tensor{T}$ of $\\R^{R \\times X^{\\prime}}$ (it is indeed a matrix, though denoted as a tensor). The entries of $\\tensor{T} \\triangleq \\tensorSup{T}{0} \\left( \\otimes^0_0 \\circ \\ast^1_1 \\circ \\times^2_2 \\right) \\tensorSup{T}{1}$ are computed as\n\\begin{equation}\n\\tensorSub{T}{r, :} = \\sum_{s = 0}^{S - 1} \\tensorInd{T}{r, :, s}{0} \\ast \\tensorInd{T}{r, :, s}{1} \n\\label{def:compound-1}\n\\end{equation}\\noindent\nFor commonly used vector convolution, it is not difficult to show that number of operations required to compute the result $\\tensor{T}$ is $O\\left( R \\max(X, H) \\log(\\max(X, H)) S \\right)$ with FFT and $O(RXHS)$ without FFT, as each of the $R$ vectors in $\\tensor{T}$ is computed with a sum of $S$ vector convolutions. \n\n\\item \\textbf{Simultaneous operations between a tensor and a set of multiple tensors}. For example, given a $3$-order tensor $\\tensor{U} \\in \\R^{R \\times X \\times S}$ and a set of three tensors $\\tensorSup{T}{0} \\in \\R^{R \\times P}$, $\\tensorSup{T}{1} \\in \\R^{K \\times Q}$ and $\\tensorSup{T}{2} \\in \\R^{S \\times T}$, we can define a compound operation on $\\tensor{U}$ as $\\tensor{V} \\triangleq \\tensor{U} ( \\otimes^{0}_{0} \\tensorSup{T}{0} \\ast^{1}_{0} \\tensorSup{T}{1} \\times^{2}_{0} \\tensorSup{T}{2} )$, which performs mode-$(0,0)$ partial outer product with $\\tensorSup{T}{0}$, mode-$(1, 0)$ convolution with $\\tensorSup{T}{1}$ and mode-$(2, 0)$ contraction with $\\tensorSup{T}{2}$ simultaneously. In this case, a $5$-order tensor $\\tensor{V} \\in \\R^{R \\times X^{\\prime} \\times P \\times Q \\times T}$ is returned, with entries calculated as \n\\begin{equation}\n\\tensorSub{V}{r, :, p, q, t} = \\sum_{s=0}^{S-1} \\tensorInd{T}{r, p}{0} \\left( \\tensorSub{U}{r, :, s} \\ast \\tensorInd{T}{:, q}{1} \\right) \\tensorInd{T}{s, t}{2}\n\\label{def:compound-2}\n\\end{equation}\\noindent\nIdentifying the best order to evaluate a compound operation with multiple tensors is in general an NP-hard problem, but for this example we can find it using exhaustive search: \n\\begin{equation}\n\\tensorSub{V}{r, :, p, q, t} = \\left( \\left( \\sum_{s=0}^{S-1}  \\tensorSub{U}{r, :, s} \\tensorInd{T}{s, t}{2} \\right) \\ast \\tensorInd{T}{:, q}{1} \\right) \\tensorInd{T}{r, p}{0}\n\\end{equation}\\noindent\nIf we follows the supplied brackets and break the evaluation into three steps, these steps take $O(RXST)$, $O(RXHPT)$ and $O(RX^{\\prime}HPT)$ operations respectively, therefore result in a total time complexity of $O( RXST + RXHPT  + RX^{\\prime} PQT)$ for the compound operation.\n\n\\end{itemize}\n\nGenerally, compound operations over multiple tensors are difficult to flatten into mathematical equations, and usually described by {\\em tensor diagrams} as in Section~\\ref{sec:preliminary}, which are usually called {\\em tensor networks}~\\cite{cichocki2016low}in the physics literature. \n\n\n\\section{Backpropagation of tensor operations}\n\\label{app:derivatives}\n\nIn order to use the operations in Appendix~\\ref{app:operations} in tensorial neural networks, we derive in this section their backpropagation rules. For brevity, we use tensor contraction as a tutorial example, and omit the derivations for other operations.   \n\n\n\n\n\n\\paragraph{Tensor contraction}\nRecall the definition of tensor contraction in Equations~\\eqref{def:tensor-contraction-1} and~\\eqref{def:tensor-contraction-2} in tensor algebra:\n\\begin{equation}\n\\tensor{T} = \\tensorSup{T}{1} \\times \\tensorSup{T}{2}\n\\label{def:tensor-contraction}\n\\end{equation}\nThe partial derivatives of the result $\\tensor{T}$ w.r.t. its operants $\\tensorSup{T}{0}$, $\\tensorSup{T}{1}$ can be computed at the entries level:\n\\begin{subequations}\n\\begin{align}\n\\frac{\\partial \\tensorSub{T}{i_0, \\cdots, i_{k-1}, i_{k+1}, \\cdots, i_{m-1}, j_0, \\cdots, j_{l-1}, j_{l+1}, \\cdots, j_{n-1}}}{\\partial \\tensorInd{T}{i_0, \\cdots, i_{k-1}, r, i_{k+1}, \\cdots, i_{m-1}}{0}} & = \\tensorInd{T}{j_0, \\cdots, j_{l-1}, r, j_{l+1}, \\cdots, j_{n-1}}{1}\n \\label{eq:derivative-tensor-contraction-1} \\\\\n\\frac{\\partial \\tensorSub{T}{i_0, \\cdots, i_{k-1}, i_{k+1}, \\cdots, i_{m-1}, j_0, \\cdots, j_{l-1}, j_{l+1}, \\cdots, j_{n-1}}}{\\partial \\tensorInd{T}{j_0, \\cdots, j_{l-1}, r, j_{l+1}, \\cdots, j_{n-1}}{1}} & = \\tensorInd{T}{i_0, \\cdots, i_{k-1}, r,  i_{k+1}, \\cdots, i_{m-1}}{0} \n\\label{eq:derivative-tensor-contraction-2}\n\\end{align}\n\\end{subequations}\nWith chain rule, the derivatives of $\\mathcal{L}$ w.r.t. $\\tensorSup{T}{0}$ and $\\tensorSup{T}{1}$ can be obtained through ${\\partial \\mathcal{L}}\/{\\partial \\tensor{T}}$.\n\\begin{subequations}\n\\begin{gather}\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorInd{T}{i_0, \\cdots, i_{k-1}, r, i_{k+1}, \\cdots, i_{m-1}}{0}} = \\sum_{j_0 = 0}^{J_0 - 1} \\cdots \\sum_{j_{l-1} = 0}^{J_{l-1} - 1} \\sum_{j_{l+1} = 0}^{J_{l+1} - 1} \\cdots \\sum_{j_{n-1} = 0}^{J_{n-1} - 1} \\nonumber \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSub{T}{i_0, \\cdots, i_{k-1}, i_{k+1}, \\cdots, i_{m-1}, j_0, \\cdots, j_{l-1}, j_{l+1}, \\cdots, j_{n-1}}} ~ \\tensorInd{T}{j_0, \\cdots, j_{l-1}, r, j_{l+1}, \\cdots, j_{n-1}}{1} \\label{eq:backprop-tensor-contraction-1-1} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorInd{T}{j_0, \\cdots, j_{l-1}, r, j_{l+1}, \\cdots, j_{n-1}}{1}} = \\sum_{i_0 = 0}^{I_0 - 1} \\cdots \\sum_{i_{k-1} = 0}^{I_{k-1} - 1} \\sum_{i_{k+1} = 0}^{I_{k+1} - 1} \\cdots \\sum_{i_{m-1} = 0}^{I_{m-1} - 1} \\nonumber \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSub{T}{i_0, \\cdots, i_{k-1}, i_{k+1}, \\cdots, i_{m-1}, j_0, \\cdots, j_{l-1}, j_{l+1}, \\cdots, j_{n-1}}} ~ \\tensorInd{T}{i_0, \\cdots, i_{k-1}, r, i_{k+1}, \\cdots, i_{m-1}}{0} \\label{eq:backprop-tensor-contraction-2-1} \n\\end{gather}\n\\end{subequations}\nThese tedious equations can be greatly simplified with tensor notations introduced before.\n\\begin{subequations}\n\\begin{align}\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{T}{0}} & = \\mathsf{swapaxes} \\left( \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{T}} \\left( \\times^{m-1}_{0} \\circ \\cdots \\circ \\times^{m+l-2}_{l-1} \\circ \\times^{m+l-1}_{l+1} \\circ \\cdots \\circ \\times^{m+n-3}_{n-1} \\right) \\tensorSup{T}{1} \\right) \n\\label{eq:backprop-tensor-contraction-1-2} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{T}{1}} & = \\mathsf{swapaxes} \\left( \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{T}} \\left( \\times^{0}_{0} \\circ \\cdots \\circ \\times^{k-1}_{k-1} \\circ \\times^{k}_{k+1} \\circ \\cdots \\circ \\times^{m-2}_{m-1} \\right) \\tensorSup{T}{0} \\right) \n\\label{eq:backprop-tensor-contraction-2-2}\n\\end{align}\n\\end{subequations}\nwhere $\\mathsf{swapaxes}(\\cdot)$ aligns the modes of outputs. Notice that the backpropagation equations are compound operations, even if the original operation is a basic one. The number of operations required for both backpropagation equations are $O ( ( \\prod_{u = 0}^{m - 1} I_u ) \\allowbreak ( \\prod_{v = 0, v \\neq l}^{n - 1} J_v ) )$, which are exactly the same as in the forward computation in Equation~\\ref{def:tensor-contraction}. \n\n\n\\paragraph{Tensor multiplication} \nRecall the definition of tensor multiplication in Equations~\\eqref{def:tensor-multiplication-1} and~\\eqref{def:tensor-multiplication-2} in tensor algebra:\n\\begin{equation}\n\\tensor{V} = \\tensor{U} \\times_{k} \\mymatrix{M}\n\\label{def:tensor-multiplication}\n\\end{equation}\nBackpropagation equations:\n\\begin{subequations} \n\\begin{gather}\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{U}} \\times_{k} \\mymatrix{M}^{\\top} \\label{eq:backprop-tensor-multiplication-1-2} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\mymatrix{M}} = \\tensor{U} \\left( \\times^{0}_{0} \\circ \\cdots \\circ \\times^{k-1}_{k-1} \\circ \\times^{k+1}_{k+1} \\circ \\cdots \\circ \\times^{m-1}_{m-1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}} \\label{eq:backprop-tensor-multiplication-2-2}\n\\end{gather}\n\\end{subequations}\nThe time complexities for both equations are $O ( ( \\prod_{u = 0}^{m - 1} I_u ) J )$, which is identical to the forward computation in Equation~\\eqref{def:tensor-multiplication}.\n\n\\paragraph{Tensor convolution} \nRecall the definition of tensor convolution in Equation~\\eqref{def:tensor-convolution} in tensor algebra:\n\\begin{equation}\n\\tensor{T} = \\tensorSup{T}{1} \\ast^{k}_{l} \\tensorSup{T}{2} \n\\label{def:tensor-convolution-2}\n\\end{equation}\nBackpropagation equations:\n\\begin{subequations} \n\\begin{align}\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{T}{0}} \n& = \\mathsf{swapaxes} \\left( \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{T}} \\big( \\times^{m}_{0} \\cdots \\times^{m+l-1}_{l-1} \\circ \\ast^{k}_{l} \\circ \\times^{m+l-2}_{l+1} \\circ \\cdots \\circ \\times^{m+n-2}_{n-1} \\big) \\mathsf{flipaxis}(\\tensorSup{T}{1}, l) \\right) \n\\nonumber \\\\\n& = \\mathsf{swapaxes} \\left( \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{T}} \\left( \\times^{m}_{0} \\cdots \\times^{m+l-1}_{l-1} \\circ (\\ast^{k}_{l})^{\\top} \\circ \\times^{m+l-2}_{l+1} \\cdots \\times^{m+n-2}_{n-1} \\right) \\tensorSup{T}{1} \\right) \n\\label{eq:backprop-tensor-convolution-1-2} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{T}{1}} \n& = \\mathsf{swapaxes} \\left( \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{T}} \\big( \\times^{0}_{0} \\circ \\cdots \\circ \\times^{k-1}_{k-1} \\circ \\ast^{k}_{k} \\circ \\times^{k+1}_{k+1} \\circ \\cdots \\circ \\times^{m-1}_{m-1} \\big) \\mathsf{flipaxis}(\\tensorSup{T}{0}, k) \\right) \n\\nonumber \\\\\n & = \\mathsf{swapaxes} \\left( \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{T}} \\left( \\times^{0}_{0} \\circ \\cdots \\circ \\times^{k-1}_{k-1} \\circ (\\ast^{k}_{k})^{\\top} \\circ \\times^{k+1}_{k+1} \\circ \\cdots \\circ \\times^{m-1}_{m-1} \\right) \\tensorSup{T}{0} \\right) \n\\label{eq:backprop-tensor-convolution-2-2}\n\\end{align}\n\\end{subequations}\nwhere $(\\ast^{k}_{l})^{\\top}$ is the {\\em adjoint operator} of $\\ast^{k}_{l}$. Without FFT, \nthe time complexities for the equations are $O ( ( \\prod_{u = 0, u \\neq k}^{m - 1} I_u ) \\allowbreak ( \\prod_{v = 0, v \\neq l}^{n - 1} J_v ) I_k^{\\prime} J_l  )$ and $O ( ( \\prod_{u = 0, u \\neq k}^{m - 1} I_u ) \\allowbreak ( \\prod_{v = 0, v \\neq l}^{n - 1} J_v ) I_k^{\\prime} I_k )$ respectively. \n\n\n\n\\paragraph{Tensor outer product} \nRecall the definition of tensor outer product in Equation~\\eqref{def:tensor-outer-product} in tensor alegebra:\n\\begin{equation}\n\\tensor{T} = \\tensorSup{T}{0} \\otimes \\tensorSup{T}{1}\n\\label{def:tensor-outer-product-2}\n\\end{equation}\nBackpropagation equations:\n\\begin{subequations}\n\\begin{align}\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{T}{0}} & = \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{T}} \\left( \\times^{m}_{0} \\circ \\cdots \\circ \\times^{m+n-1}_{n-1} \\right) \\tensorSup{T}{1} \n\\label{eq:backprop-tensor-outer-product-1-2} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{T}{1}} & = \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{T}} \\left( \\times^{0}_{0} \\circ \\cdots \\circ \\times^{m-1}_{m-1} \\right) \\tensorSup{T}{0} \n\\label{eq:backprop-tensor-outer-product-2-2}\n\\end{align}\n\\end{subequations}\nThe number of operations required for both equations are $O ( ( \\prod_{u = 0, u \\neq k}^{m - 1} I_u ) \\allowbreak ( \\prod_{v = 0}^{n - 1} J_v ) )$.\n\n\\paragraph{Tensor partial outer product}\nRecall the definition of tensor partial outer product in Equation~\\eqref{def:tensor-partial-outer-product}:\n\\begin{equation}\n\\tensor{T} = \\tensorSup{T}{0} \\otimes^{k}_{l} \\tensorSup{T}{1}\n\\label{def:tensor-partial-outer-product}\n\\end{equation}\nBackpropagation equations:\n\\begin{subequations}  \n\\begin{align}\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{T}{0}} & = \\mathsf{swapaxes} \\left( \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{T}} \\left( \\times^{m}_{0} \\circ \\cdots \\circ \\times^{m+l-1}_{l-1} \\circ \\otimes^{k}_{l} \\circ \\times^{m+l-2}_{l+1} \\circ \\cdots \\circ \\times^{m+n-2}_{n-1} \\right) \\tensorSup{T}{1} \\right) \\label{eq:backprop-tensor-partial-outer-product-1-2} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{T}{1}} & = \\mathsf{swapaxes} \\left( \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{T}} \\left( \\times^{0}_{0} \\circ \\cdots \\circ \\times^{k-1}_{k-1} \\circ \\otimes^{k}_{k} \\circ \\times^{k+1}_{k+1} \\circ \\cdots \\circ \\times^{m-1}_{m-1} \\right) \\tensorSup{T}{0} \\right) \\label{eq:backprop-tensor-partial-outer-product-2-2}\n\\end{align}\n\\end{subequations}\nIt is not difficult to show the time complexity for both equations above are $O( ( \\prod_{u = 0}^{m - 1} I_u ) \\allowbreak ( \\prod_{v = 0, v \\neq l}^{n - 1} J_v ) )$.\n\n\n\\section{Tensor decompositions}\n\\label{app:decompositions}\n\nTensor decompositions are natural extensions of matrix factorizations for multi-dimensional arrays. In this section, we will review three commonly used tensor decompositions, namely {\\em CANDECOMP\/PARAFAC\\xspace (CP\\xspace) decomposition}~\\cite{kolda2009tensor}, {\\em Tucker\\xspace (TK\\xspace) decomposition}~\\cite{kolda2009tensor} and {\\em Tensor-train\\xspace (TT\\xspace) decomposition}~\\cite{oseledets2011tensor}. \n\n\n\\paragraph{CP\\xspace decomposition} \nCP\\xspace decomposition is a direct generalization of singular value decomposition (SVD\\xspace) which decomposes a tensor into additions of rank-1 tensors (outer product of multiple vectors). Specifically, given an $m$-order tensor $\\tensor{T} \\in \\R^{I_0 \\times I_1 \\times \\cdots \\times I_{m-1}}$, CP\\xspace decomposition factorizes it into $m$ factor matrices $\\{ \\mymatrix{M}^{(0)} \\}_{l = 0}^{m - 1}$, where $\\mymatrix{M}^{(l)} \\in \\R^{R \\times I_l}, \\forall l \\in [m]$, where $R$ is called the \\textit{canonical rank} of the CP\\xspace decomposition, which is allowed to be larger than the $I_l$'s.\n\\begin{subequations}\n\\begin{gather}\n\\tensorSub{T}{i_0, \\cdots, i_{m-1}} \\triangleq \\sum_{r = 0}^{R - 1} \\mymatrix{M}^{(0)}_{r, i_0} \\cdots \\mymatrix{M}^{(m-1)}_{r, i_{m-1}} \n\\label{def:cp-decomposition-1} \\\\\n\\tensor{T} \\triangleq \\sum_{r = 0}^{R - 1} \\mymatrix{M}^ {(0)}_{r, :} \\otimes \\cdots \\otimes \\mymatrix{M}^{(m-1)}_{r, :} = \\myvector{1} \\times^{0}_{0} \\left( \\mymatrix{M}^{(0)} \\otimes^{0}_{0} \\cdots \\otimes^{0}_{0} \\mymatrix{M}^{(m-1)} \\right)  \\label{def:cp-decomposition-2}\n\\end{gather}\n\\end{subequations}\nwhere $\\myvector{1} \\in \\R^{R}$ is an all-ones vector of length $R$. With CP\\xspace decomposition, $\\tensor{T}$ can be represented with only $( \\sum_{l = 0}^{m - 1} I_l ) R$ entries instead of $( \\prod_{l = 0}^{m - 1} I_l )$ as in the original tensor. \n\n\n\\paragraph{Tucker\\xspace decomposition} Tucker\\xspace decomposition provides more general factorization than CP\\xspace decomposition. Given an $m$-order tensor $\\tensor{T} \\in \\R^{I_0 \\times I_1 \\times \\cdots \\times I_{m-1}}$, Tucker\\xspace decomposition factors it into $m$ factor matrices $\\{ \\matrixSup{M}{l} \\}_{l = 0}^{m - 1}$, where $\\matrixSup{M}{l} \\in \\R^{R_l \\times I_l}, \\forall l \\in [m]$ and an additional $m$-order core tensor $\\tensor{C} \\in \\R^{R_0 \\times R_1 \\times \\dots \\times R_{m-1}}$, where the {\\em Tucker\\xspace ranks} $R_l$'s are required to be smaller or equal than the dimensions at their corresponding modes, i.e. $R_{l} \\leq I_l, \\forall l \\in [m]$. \n\\begin{subequations}\n\\begin{gather}\n\\tensorSub{T}{i_0, \\cdots, i_{m-1}} \\triangleq \\sum_{r_0 = 0}^{R_0 - 1} \\dots \\sum_{r_{m-1} = 0}^{R_{m-1}  - 1} \\tensorSub{C}{r_0, \\dots, r_{m-1}} ~ \\mymatrix{M}^{(0)}_{r_0, i_0} \\cdots \\mymatrix{M}^{(m-1)}_{r_{m-1}, i_{m-1}} \n\\label{def:tucker-decomposition-1} \\\\\n\\tensor{T} \\triangleq \\tensor{C} \\left( \\times_{0} \\mymatrix{M}^{(0)} \\times_{1} \\mymatrix{M}^{(1)} \\cdots \\times_{m-1} \\mymatrix{M}^{(m-1)} \\right)\n\\label{def:tucker-decomposition-2}\n\\end{gather}\n\\end{subequations}\nNotice that when $R_0 = \\cdots = R_{m-1} = R$ and $\\tensor{C}$ is a super-diagonal tensor with all super-diagonal entries to be ones (a.k.a. identity tensor), Tucker\\xspace decomposition reduces to CP\\xspace decomposition, and therefore CP\\xspace decomposition is a special case of Tucker\\xspace decomposition. With Tucker\\xspace decomposition, a tensor is approximately by $(\\prod_{l=0}^{m-1} R_l + \\sum_{l=0}^{m-1} I_l R_l)$ entries. \n\n\n\\paragraph{Tensor-train\\xspace decomposition} \nTensor-train\\xspace decomposition factorizes a $m$-order tensor into $m$ interconnected low-order tensors $\\{ \\tensorSup{T}{l}  \\}_{l = 0}^{m - 1}$, where $\\tensorSup{T}{l} \\in \\R^{R_l \\times I_l \\times R_{l+1}}, l = 1, \\cdots, m-2$ with $\\tensorSup{T}{0} \\in \\R^{I_0 \\times R_0}$,  and $\\tensorSup{T}{m-1} \\in \\R^{R_{m-1}\\times I_{m-1}}$ such that\n\\begin{subequations}\n\\begin{gather}\n\\tensorSub{T}{i_0, \\dots, i_{m-1}} \\triangleq \\sum_{r_0 = 1}^{R_0 - 1} \\dots \\sum_{r_{m-2} = 1}^{R_{m-2} - 1} \\tensorInd{T}{i_0, r_0}{0} ~ \\tensorInd{T}{r_0, i_1, r_1}{1} \\cdots \\tensorInd{T}{r_{m-2}, i_{m-1}}{m-1} \n\\label{def:Tensor-train-decomposition-1} \\\\\n\\tensor{T} \\triangleq \\tensorSup{T}{0} \\times^{-1}_{0} \\tensorSup{T}{1} \\times^{-1}_{0} \\cdots \\times^{-1}_{0} \\tensorSup{T}{m-1} \n\\label{def:Tensor-train-decomposition-2}\n\\end{gather}\n\\end{subequations}\nwhere the $R_l$'s are known as \\textit{Tensor-train\\xspace ranks}, which controls the tradeoff between complexity and accuracy of the representation. With Tensor-train\\xspace decomposition, a tensor is represented by $(R_0 I_0 + \\sum_{l=1}^{m-2} R_l I_l R_{l+1} + R_{m-1} I_{m-1})$ entries. \n\n\n\n\\paragraph{General tensor decompositions}\n\\label{general_decomposition}\nIn this paper, we use the term tensor decomposition in a more general way, i.e. we do not stick to the standard formats defined in the previous paragraphs. Indeed, we consider tensor decomposition as a reverse mapping of tensor operations: consider a general tensor operation $f$ on $m$ input tensors $\\{ \\tensorSup{T}{l} \\}_{l = 0}^{m - 1}$ such that $\\hat{\\tensor{T}} = f(\\tensorSup{T}{0}, \\cdots, \\tensorSup{T}{m-1})$ (i.e. $\\hat{\\tensor{T}}$ is linear in each operant $\\tensorSup{T}{l}$), the corresponding general tensor decomposition aims to recover the input factors $\\{ \\tensorSup{T}{l} \\}_{l = 0}^{m - 1}$ from a given tensor $\\tensor{T}$ such that $\\tensor{T} \\approx \\hat{\\tensor{T}} = f(\\tensorSup{T}{0}, \\cdots, \\tensorSup{T}{m-1})$. In Figure~\\ref{fig:decompositions}, we demonstrate a few examples of general tensor decomposition using tensor diagrams. \n\n\n\\input{.\/\/figure_decompositions}\n\n\\input{.\/\/table_decompositions}\n\\section{Low-rank approximations of convolutional layer}\n\\label{app:convolutional}\n\nIn our paper, compact architecture of neural networks can be derived in two steps: (1) proposing an arbitrary architecture; and (2) decomposing the weights kernel at each layer into lower-order factors. When the proposed architecture is a standard convolutional neural network and the kernel is decomposed by traditional tensor decomposition, the derived architectures are known as low-rank approximations, in which each layer in the derived architecture is still a traditional operation (i.e. not general tensor operation).    \n\n\\paragraph{Standard convolutional layer} \nWe review the standard convolutional neural network (CNN) for reference. In CNN, a convolutional layer is parameterized by a 4-order kernel $\\tensor{K} \\in \\R^{H \\times W \\times S \\times T}$, where $H$ and $W$ are height and width of the filters (which are typically equal), $S$ and $T$ are the number of input and output channels respectively. A convolutional layer maps a 3-order tensor $\\tensor{U} \\in \\R^{X \\times Y \\times S}$ to another 3-order tensor $\\tensor{V} \\in \\R^{X^{\\prime} \\times Y^{\\prime} \\times T}$, where $X$ and $Y$ are the height and width for the input feature map, while $X^{\\prime}$ and $Y^\\prime$ are the ones for the output feature map, with the following equation:\n\\begin{equation}\n\\tensor{V} = \\tensor{U} \\left( \\ast_{0}^{0} \\circ \\ast_{1}^{1} \\circ \\times_{2}^{2} \\right) \\tensor{K} \n\\label{def:convolutional-2}\n\\end{equation}\\noindent\nNotice that the number of parameters in a standard convolutional layer is $HWST$ and the number of operations needed to evaluate the output $\\tensor{V}$ is $O(HWSTXY)$.\n\n\n\n\n \n\\paragraph{SVD\\xspace-convolutional layer} \nIn SVD\\xspace-convolutional layer~\\cite{jaderberg2014speeding}, the weights kernel $\\tensor{K}$ is factorized by SVD.\n\\begin{equation}\n\\tensor{K} = \\mathsf{swapaxes} \\left( \\tensorSup{K}{0} \\times^{2}_{1} \\tensorSup{K}{1} \\right) \\label{def:convolutional-svd-2}\n\\end{equation}\nwhere $\\tensorSup{K}{0} \\in \\R^{H \\times S \\times R}$ and $\\tensorSup{K}{1} \\in \\R^{W \\times R \\times T}$ are the two factor tensors. With two factors, the forward pass of SVD\\xspace-convolutional layer is computed in two steps:\n\\begin{subequations}\n\\begin{align}\n\\tensorSup{U}{0} & = \\tensor{U} \\left( \\ast_{0}^{0} \\circ \\times^{2}_{1} \\right) \\tensorSup{K}{0} \\\\\n\\tensor{V} & = \\tensorSup{U}{0} \\left( \\ast_{1}^{1} \\circ \\times^{2}_{1} \\right) \\tensorSup{K}{1}\n\\end{align}\n\\end{subequations}\nwhere $\\tensorSup{U}{0}$ is an intermediate tensor after the first step.\nThe backpropagation equations for these two steps are:\n\\begin{subequations}\n\\begin{align}\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}} \\left( (\\ast^{1}_{0})^{\\top} \\circ \\times^{2}_{2} \\right) ~ \\tensorSup{K}{1}, ~ & \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{1}} = \\tensorSup{U}{0} \\left( \\times^{0}_{0} \\circ (\\overline{\\overline{\\ast}^{1}_{1})^{\\top}} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensor{U}} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}}\\left( (\\ast^{0}_{0})^{\\top} \\circ \\times^{2}_{2} \\right) ~ \\tensorSup{K}{0}, ~ & \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{0}} = \\tensor{U} \\left( \\overline{(\\overline{\\ast}^{0}_{0})^{\\top}} \\circ \\times^{1}_{1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}}\n\\end{align}\n\\end{subequations}\n\n\\paragraph{CP\\xspace-convolutional layer} \nIn CP\\xspace-convolutional layer~\\cite{lebedev2014speeding, denton2014exploiting}, the weights kernel $\\tensor{K}$ is factorized by CP decomposition.\n\\begin{equation}\n\\tensor{K} = \\myvector{1} \\times^{0}_{2} \\left( \\tensorSup{K}{1} \\otimes^{2}_{1} \\tensorSup{K}{0} \\otimes^{2}_{0} \\tensorSup{K}{2} \\right) \\label{def:convolutional-cp-2}\n\\end{equation}\nwhere $\\tensorSup{K}{0} \\in \\R^{S \\times R}$, $\\tensorSup{K}{1} \\in \\R^{H \\times W \\times R}$ and $\\tensorSup{K}{2} \\in \\R^{R \\times T}$ are three factor tensors.\nWith three factors, the forward pass of CP\\xspace-convolutional consists of three steps:\n\\begin{subequations}\n\\begin{align}\n\\tensorSup{U}{0} & = \\tensor{U} \\times_2 \\tensorSup{K}{0} \\\\\n\\tensorSup{U}{1} & = \\tensorSup{U}{0} \\left( \\ast_{0}^{0} \\circ \\ast_{1}^{1} \\circ \\otimes_{2}^{2} \\right) \\tensorSup{K}{1} \\\\\n\\tensor{V} & = \\tensorSup{U}{1} \\times_2 \\tensorSup{K}{2}\n\\end{align}\n\\end{subequations}\nwhere $\\tensorSup{U}{0} \\in \\R^{X \\times Y \\times R}$ and $\\tensorSup{U}{1} \\in \\R^{X^{\\prime} \\times Y^{\\prime} \\times R}$ are two intermediate tensors, and the corresponding backpropagation equations can then be computed as:\n\\begin{subequations}\n\\begin{gather}\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{1}}  =  \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}} \\times_2 (\\tensorSup{K}{2})^{\\top}, ~ \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{2}} = \\tensorSup{U}{1} \\left( \\times^{0}_{0} \\circ \\times^{1}_{1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{1}} \\left( (\\ast^{0}_{0})^{\\top} \\circ ({\\ast}^{1}_{1})^{\\top} \\circ \\otimes^{2}_{2} \\right) \\tensorSup{K}{1}, \n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{1}} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{1}} \\left( \\overline{(\\overline{\\ast}^{0}_{0})^{\\top}} \\circ \\overline{(\\overline{\\ast}^{1}_{1})^{\\top}} \\circ \\otimes^{2}_{2} \\right) \\tensorSup{U}{1} \\nonumber \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensor{U}} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}} \\times_2 (\\tensorSup{K}{0})^{\\top}, ~ \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{0}} = \\tensor{U} \\left( \\times^{0}_{0} \\circ \\times^{1}_{1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}}\n\\end{gather}\n\\end{subequations}\n\n\\paragraph{TK\\xspace-convolutional layer} \nIn TK\\xspace-convolutional layer~\\cite{kim2015compression}, the kernel $\\tensor{K}$ is factorized by a partial Tucker decomposition,\n\\begin{equation}\n\\tensor{K} = \\tensorSup{K}{1} \\left( \\times_{2} (\\tensorSup{K}{0})^\\top \\times_{3} \\tensorSup{K}{2} \\right) \n\\label{def:convolutional-tk-2}\n\\end{equation}\nwhere $\\tensorSup{K}{0} \\in \\R^{S \\times R_s}$, $\\tensorSup{K}{1} \\in \\R^{H \\times W \\times R_s \\times R_t}$ and $\\tensorSup{K}{2} \\in \\R^{R_t \\times T}$ are three factor tensors.\nAgain, the forward pass of Tucker\\xspace-convolutional layer has three steps:\n\\begin{subequations}\n\\begin{align}\n\\tensorSup{U}{0} & = \\tensor{U} \\times_{2}  \\tensorSup{K}{0} \\\\\n\\tensorSup{U}{1} & = \\tensorSup{U}{0} \\left( \\ast_{0}^{0} \\circ \\ast_{1}^{1} \\circ \\times_{2}^{2} \\right) \\tensorSup{K}{1} \\\\\n\\tensor{V} & = \\tensorSup{U}{1} \\times_{2} \\tensorSup{K}{2}\n\\end{align}\n\\end{subequations}\nwhere $\\tensorSup{U}{0} \\in \\R^{X \\times Y \\times R_s}$ and $\\tensorSup{U}{1} \\in \\R^{X^{\\prime} \\times Y^{\\prime} \\times R_t}$ are two intermediate tensors. Their backpropagation equations are summarized as follows:\n\\begin{subequations}\n\\begin{align}\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{1}}  =  \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}} \\times_2 (\\tensorSup{K}{2})^{\\top}, ~ & \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{2}} = \\tensorSup{U}{1} \\left( \\times^{0}_{0} \\circ \\times^{1}_{1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{1}} \\left( (\\ast^{0}_{0})^{\\top} \\circ (\\ast^{1}_{1})^{\\top} \\circ \\times^{2}_{3} \\right) \\tensorSup{K}{1}, ~ & \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{1}} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{1}} \\left( \\overline{(\\overline{\\ast}^{0}_{0})^{\\top}} \\circ \\overline{(\\overline{\\ast}^{1}_{1})^{\\top}} \\right) \\tensorSup{U}{1} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensor{U}} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}} \\times_2 (\\tensorSup{K}{0})^{\\top}, ~ & \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{0}} = \\tensor{U} \\left( \\times^{0}_{0} \\circ \\times^{1}_{1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}}\n\\end{align}\n\\end{subequations}\n\n\\paragraph{TT\\xspace-convolutional layer}\nTT\\xspace-convolutional layer is derived by factorizing $\\tensor{K}$ using Tensor-train\\xspace decomposition.\n\\begin{equation}\n\\tensor{K} = \\mathsf{swapaxes} \\left( \\tensorSup{K}{0} \\times^{-1}_{0} \\tensorSup{K}{1} \\times^{-1}_{0} \\tensorSup{K}{2} \\times^{-1}_0 \\tensorSup{K}{3} \\right)\n\\label{def:convolutional-tt-2}\n\\end{equation}\nwhere $\\tensorSup{K}{0} \\in \\R^{S \\times R_s}$, $\\tensorSup{K}{1} \\in \\R^{R_s \\times H \\times R}$, $\\tensorSup{K}{2} \\in \\R^{R \\times W \\times R_t}$ and $\\tensorSup{K}{3} \\in \\R^{R_t \\times T}$ are four factor tensors.\nThe forward pass now contains four steps:\n\\begin{subequations}\n\\begin{align}\n\\tensorSup{U}{0} & = \\tensor{U} \\times_2 \\tensorSup{K}{0} \\\\\n\\tensorSup{U}{1} & = \\tensorSup{U}{0} ~ (\\ast^{0}_{1} \\circ \\times^{2}_{0}) ~ \\tensorSup{K}{1} \\\\\n\\tensorSup{U}{2} & = \\tensorSup{U}{1} ~ (\\ast^{1}_{1} \\circ \\times^{2}_{0}) ~ \\tensorSup{K}{2} \\\\\n\\tensor{V} & = \\tensorSup{U}{2} \\times_2 \\tensorSup{K}{3}\n\\end{align}\n\\end{subequations}\nwhere $\\tensorSup{U}{0} \\in \\R^{X \\times Y \\times R_s}$, $\\tensorSup{U}{1} \\in \\R^{X^{\\prime} \\times Y \\times R}$ and $\\tensorSup{U}{2} \\in \\R^{X^{\\prime} \\times Y^{\\prime} \\times R_t}$ are three intermediate results.\nCorresponding backpropagation equations for these computations are:\n\\begin{subequations}\n\\begin{align}\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{2}}  =  \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}} \\times_2 (\\tensorSup{K}{3})^{\\top}, ~ & \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{3}} = \\tensorSup{U}{2} \\left( \\times^{0}_{0} \\circ \\times^{1}_{1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{1}} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{2}} \\left( (\\ast^{1}_{0})^{\\top} \\circ \\times^{2}_{2} \\right) \\tensorSup{K}{2}, ~ & \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{2}} = \\tensorSup{U}{1} \\left( \\times^{0}_{0} \\circ \\overline{(\\overline{\\ast}^{1}_{1})^{\\top}} \\right)\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{2}} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{1}}\\left( (\\ast^{0}_{0})^{\\top} \\circ \\times^{2}_{2} \\right) \\tensorSup{K}{1}, ~ & \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{2}} = \\tensorSup{U}{0} \\left(\\overline{(\\overline{\\ast}^{0}_{0})^{\\top}} \\circ \\times^{1}_{1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{1}} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensor{U}} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}} \\times_2 (\\tensorSup{K}{0})^{\\top}, ~ & \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{0}} = \\tensor{U} \\left( \\times^{0}_{0} \\circ \\times^{1}_{1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}}\n\\end{align}\n\\end{subequations}\n\n\\input{.\/\/table_convolutional}\n\\section{TNN based compression on dense layer}\n\\label{app:dense-tensorized}\n\n\n\nWhen the same technique in Appendix~\\ref{app:convolutional} is applied to compress the dense layer (a.k.a fully connected layer), we encounter one difficulty: the kernel in dense layer is a matrix and can not be further factorized by any tensor decomposition. In order to address this difficult, we define an equivalent {\\em tensorized dense layer} that maps an $m$-order input tensor $\\tensor{U} \\in \\R^{S_0 \\times \\cdots \\times S_{m-1}}$ to another $m$-order tensor $\\tensor{V} \\in \\R^{T_0 \\times \\cdots \\times T_{m-1}}$ with a $2m$-order kernel $\\tensor{K} \\in \\R^{S_0 \\times \\cdots S_{m-1} \\times T_0 \\times \\cdots \\times T_{m-1}}$. \n\\begin{equation}\n\\tensor{V} = \\tensor{U} \\left( \\times^{0}_{0} \\circ \\cdots \\circ \\times^{m-1}_{m-1} \\right) \\tensor{K} \n\\label{def:dense-tensorized-2}\n\\end{equation}\nThe layer is equivalent to a standard dense layer if input $\\tensor{U}$, kernel $\\tensor{K}$ and output $\\tensor{V}$ are reshaped versions of their counterparts. Now we are ready to derive compact architectures for compression using general tensor decompositions. \n\n\\paragraph{\\reshapePrefix\\CP-dense layer}\nThe layer is derived if the kernel $\\tensor{K}$ is factorized by a modified CP\\xspace decomposition.\n\\begin{equation}\n\\tensor{K} = \\mathsf{swapaxes} \\left( \\myvector{1} \\times^{0}_{0} (\\tensorSup{K}{0} \\otimes^{0}_{0} \\cdots \\otimes^{0}_{0} \\tensorSup{K}{m-1} \\right)\n\\label{def:dense-cp-2}\n\\end{equation}\nwhere $\\tensorSup{K}{l} \\in \\R^{R \\times S_l \\times T_l}, \\forall l \\in [m]$ are $m$ factors. It results in a multi-steps procedure for forward pass: \n\\begin{subequations}\n\\begin{align}\n\\tensorSup{U}{0} & = \\myvector{1} \\otimes \\tensor{U} \\\\\n\\tensorSup{U}{l+1} & = \\tensorSup{U}{l} \\left( \\otimes_{0}^{0} \\circ \\times^{1}_{1} \\right) \\tensorSup{K}{l} \\\\\n\\tensor{V} & = \\myvector{1} \\times^{0}_{0} \\tensorSup{U}{m}\n\\end{align}\n\\end{subequations}\nwhere the $m$ intermediate results $\\tensorSup{U}{l} \\in \\R^{R \\times S_{l} \\times \\cdots S_{m-1} \\times T_0 \\times \\cdots \\times T_{l-1}}, \\forall l \\in [m]$ are all $(m+1)$-order tensors. Correspondingly, their backpropagation equations are computed as\n\\begin{subequations}\n\\begin{align}\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{l}} & = \\tensorSup{K}{l} \\left( \\otimes^{0}_{0} \\circ \\times^{2}_{-1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{l+1}} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{l}} & = \\tensorSup{U}{l} \\left( \\otimes^{0}_{0} \\circ \\times^{2}_{1} \\circ \\cdots \\circ \\times^{m}_{m-1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{l+1}} \\\\ \n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{m}} & = \\myvector{1} \\otimes \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}}, ~ \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{U}} = \\myvector{1} \\times^{0}_{0} \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}} \\nonumber\n\\end{align}\n\\end{subequations}\n\n\\paragraph{\\reshapePrefix\\TK-dense layer} The layer is obtained when a modified Tucker\\xspace decomposition is used.\n\\begin{equation}\n\\tensor{K} = \\tensor{C} \\left( \\times_0 (\\matrixSup{P}{0})^{\\top} \\cdots \\times_{m-1} (\\matrixSup{P}{m-1})^{\\top} \\times_{m} \\matrixSup{Q}{0} \\cdots \\times_{2m - 1} \\matrixSup{Q}{m-1} \\right) \n\\label{def:dense-tensorized-tk-2}\n\\end{equation}\nwhere $\\matrixSup{P}{l} \\in \\R^{S_l \\times R^s_l}, \\forall l \\in [m]$ are named as input factors, $\\tensor{C} \\in \\R^{R^s_0 \\times \\cdots \\times R^s_{m-1} \\times R^t_0 \\times \\cdots \\times R^t_{m-1}}$ as core factor, and lastly $\\matrixSup{Q}{l} \\in \\R^{R^t_l \\times T_l}, \\forall l \\in [m]$ as output factors.\nThe forward pass of \\reshapePrefix\\TK-dense layer can then be evaluated in three steps\n\\begin{subequations}\n\\begin{align}\n\\tensorSup{U}{0} & = \\tensor{U} \\left( \\times_0 \\matrixSup{P}{0} \\cdots \\times_{m-1} \\matrixSup{P}{m-1} \\right) \\\\\n\\tensorSup{U}{1} & = \\tensorSup{U}{0} \\left( \\times_{0}^{0} \\circ \\cdots \\circ \\times_{m-1}^{m-1} \\right) \\tensor{C} \\\\\n\\tensor{V} & = \\tensorSup{U}{1} \\left( \\times_0 \\matrixSup{Q}{0} \\cdots \\times_{m-1} \\matrixSup{Q}{m-1} \\right) \n\\end{align}\n\\end{subequations}\nwhere $\\tensorSup{U}{0}$ and $\\tensorSup{U}{1}$ are two intermediate results. \nThe backpropagation equations can then be derived accordingly:\n\\begin{subequations}\n\\begin{gather}\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{1}} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}} \\left( \\times_{0} (\\matrixSup{Q}{0})^{\\top} \\cdots \\times_{m-1} (\\matrixSup{Q}{m-1})^{\\top} \\right) \\\\\n\\left( \\frac{\\partial \\mathcal{L}}{\\partial \\matrixSup{Q}{l}} \\right)^{\\top} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}} \\left( \\times^{0}_{0} \\circ \\cdots \\circ \\times^{l-1}_{l-1} \\circ \\times^{l+1}_{l+1} \\circ \\cdots \\circ \\times^{m-1}_{m-1} \\right) \\nonumber \\\\\n\\left( \\tensorSup{U}{1} \\left( \\times_{0} \\matrixSup{Q}{0} \\cdots \\times_{l-1} \\matrixSup{Q}{l-1} \\times_{l+1} \\matrixSup{Q}{l+1} \\cdots \\times_{m-1} \\matrixSup{Q}{m-1} \\right) \\right) \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensor{U}} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}} \\left( \\times_{0} (\\matrixSup{P}{0})^{\\top} \\cdots \\times_{m-1} (\\matrixSup{P}{m-1})^{\\top} \\right) \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}} = \\tensor{C} \\left( \\times^{m}_{0} \\circ \\cdots \\circ \\times^{2m - 1}_{m - 1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{1}} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensor{C}} = \\tensorSup{U}{0} \\otimes \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{1}} \\\\\n\\left( \\frac{\\partial \\mathcal{L}}{\\partial \\matrixSup{P}{l}} \\right)^{\\top} = \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}} \\left( \\times^{0}_{0} \\circ \\cdots \\circ \\times^{l-1}_{l-1} \\circ \\times^{l+1}_{l+1} \\circ \\cdots \\circ \\times^{m-1}_{m-1} \\right) \\nonumber \\\\\n\\left( \\tensor{U} \\left( \\times_{0} \\matrixSup{P}{0} \\cdots \\times_{l-1} \\matrixSup{P}{l-1} \\times_{l+1} \\matrixSup{P}{l+1} \\cdots \\times_{m-1} \\matrixSup{P}{m-1} \\right) \\right)\n\\end{gather}\n\\end{subequations}\n\n\\paragraph{\\reshapePrefix\\TT-dense layer}\nThe layer~\\cite{novikov2015tensorizing} is derived by factorizing $\\tensor{K}$ using a modified Tensor-train\\xspace decomposition. \n\\begin{equation}\n\\tensor{K} = \\mathsf{swapaxes} \\left( \\tensorSup{K}{0} \\times^{-1}_{0} \\tensorSup{K}{1} \\times^{-1}_{0} \\cdots \\times^{-1}_{0} \\tensorSup{K}{m-1} \\right) \\label{def:dense-tensorized-tt-2}\n\\end{equation}\nwhere the factor tensors are $\\tensorSup{K}{l} \\in \\R^{R_{l-1} \\times S_l \\times T_l \\times R_l},\\forall l = 1, \\cdots, m-2$, with two corner cases $\\tensorSup{K}{0} \\in \\R^{S_0 \\times T_0 \\times R_0}$ and $\\tensorSup{K}{m-1} \\in \\R^{R_{m-2} \\times S_{m-1} \\times T_{m-1}}$.\n\\begin{equation}\n\\tensorSup{U}{l+1} = \\tensorSup{U}{l} \\left( \\times^{0}_{1} \\circ \\times^{-1}_{0} \\right) \\tensorSup{K}{l}\n\\end{equation}\\noindent\nwhere $\\tensorSup{U}{0} = \\tensor{U}$,  $\\tensorSup{U}{m} = \\tensor{V}$, and the other $m$ tensors $\\tensorSup{U}{l}$'s are intermediate results. Following Appendix~\\ref{app:derivatives}, its backpropagation equations can be derived as:\n\\begin{subequations}\n\\begin{align}\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{l}} & = \\tensorSup{K}{l} \\left( \\times^{1}_{-2} \\circ \\times^{2}_{-1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{l+1}} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{l}} & = \\mathsf{swapaxes} \\left( \\tensorSup{U}{l} \\left( \\times^{1}_{0} \\circ \\cdots \\circ \\times^{m-1}_{m-2} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{l+1}} \\right)\n\\end{align}\n\\end{subequations}\n\n\n\n\\input{.\/\/table_tensorized_dense}\n\n\\section{TNN compression on convolutional layer}\n\\label{app:convolutional-tensorized}\n\n\n\n\nInspired by the reshaping trick in Appendix~\\ref{app:dense-tensorized}, we develop three novel architectures that compress convolution layer by first proposing its higher-order counterpart: a {\\em tensorized convolutional layer} maps $(m+2)$-order tensor $\\tensor{U} \\in \\R^{X \\times Y \\times S_0 \\times \\cdots \\times S_{m-1}}$ to another $(m+2)$-order tensor $\\tensor{V} \\in \\R^{X^{\\prime} \\times Y^{\\prime} \\times T_0 \\times \\cdots \\times T_{m-1}}$ with a $(2m + 2)$-order kernel $\\tensor{K} \\in \\R^{H \\times W \\times S_0 \\times \\cdots \\times S_{m-1} \\times T_0 \\times  \\cdots \\times T_{m-1}}$.\n\\begin{equation}\n\\tensor{V} = \\tensor{U} \\left( \\ast_{0}^{0} \\circ \\ast_{1}^{1} \\circ \\times_{2}^{2} \\circ \\cdots \\circ \\times^{m+1}_{m+1} \\right) \\tensor{K} \n\\label{def:convolutional-tensorized-2}\n\\end{equation}\nThe tensorized convolutional layer is equivalent to standard convolutional layer if input $\\tensor{U}$, kernel $\\tensor{K}$ and output $\\tensor{V}$ are reshaped from their counterparts.. \n\n\n\\paragraph{\\reshapePrefix\\CP-convolutional layer}\nThe layer is derived if the kernel $\\tensor{K}$ is factorized by a modified CP decomposition as in Figure~\\ref{fig:decomposition-rcp}. \n\\begin{equation}\n\\tensor{K} = \\mathsf{swapaxes} \\left( \\myvector{1} \\times^{0}_{0} \\left( \\tensorSup{K}{0} \\otimes^{0}_{0} \\cdots \\otimes^{0}_{0} \\tensorSup{K}{m} \\right) \\right) \\label{def:convolutional-tensorized-cp-2}\n\\end{equation}\nwhere $\\tensorSup{K}{l}  \\in \\R^{R \\times S_l \\times T_l}, \\forall l \\in [m]$ and $\\tensorSup{K}{m} \\in \\R^{R \\times H \\times W}$ are $(m + 1)$ factors.\nAccordingly, the multi-steps procedure to evaluate the output $\\tensor{V}$ now has $(m + 2)$-steps:\n\\begin{subequations}\n\\begin{align}\n\\tensorSup{U}{0} & = \\myvector{1} \\otimes \\tensor{U} \\\\\n\\tensorSup{U}{l+1} & = \\tensorSup{U}{l} \\left( \\otimes_{0}^{0} \\circ \\times^{3}_{1} \\right) \\tensorSup{K}{l} \\\\\n\\tensor{V} & = \\tensorSup{U}{m} \\left( \\times_{0}^{0} \\circ \\ast^{1}_{1} \\circ \\ast^{2}_{2} \\right) \\tensorSup{K}{m}\n\\end{align}\n\\end{subequations}\nwhere $\\tensorSup{U}{l} \\in \\R^{R \\times S_{l} \\times \\cdots \\times S_{m+1} \\times T_{0} \\times \\cdots \\times T_{l-1}}, \\forall l \\in [m]$ are $m$ intermediate tensors. \nThe backpropagation equations of these steps above are\n\\begin{subequations}\n\\begin{align}\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{m}} & = \\tensorSup{K}{m} \\left( (\\ast^{1}_{0})^{\\top} \\circ (\\ast^{2}_{1})^{\\top} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}} \\\\ \n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{m}} & = \\tensorSup{U}{m} \\left( (\\ast^{1}_{0})^{\\top} \\circ (\\ast^{2}_{1})^{\\top} \\times^{3}_{2} \\circ \\cdots \\circ \\times^{m+2}_{m+1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{l}} & = \\mathsf{swapaxes} \\left( \\tensorSup{K}{l} \\left( \\otimes^{0}_{0} \\circ \\times^{2}_{-1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{l+1}} \\right) \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{l}} & = \\tensorSup{U}{l} \\left( \\otimes^{0}_{0} \\circ \\otimes^{1}_{1} \\circ \\otimes^{2}_{2} \\circ \\times^{4}_{3} \\circ \\cdots \\circ \\times^{m+3}_{m+2} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{l+1}} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensor{U}} & = \\myvector{1} \\times^{0}_{0} \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}}\n\\end{align}\n\\end{subequations}\n\n\\paragraph{\\reshapePrefix\\TK-convolutional layer} \nThe layer is derived if a modified TK decomposition as in Figure~\\ref{fig:decomposition-rtk} on kernel $\\tensor{K}$. \n\\begin{equation}\n\\tensor{K} = \\tensor{C} \\left( \\times_2 (\\matrixSup{P}{0})^\\top \\cdots \\times_{m+1} (\\matrixSup{P}{m-1})^\\top \\times_{m+2} \\matrixSup{Q}{0} \\cdots \\times_{2m + 1} \\matrixSup{Q}{m-1} \\right) \n\\label{def:convolutional-tensorized-tk-2}\n\\end{equation}\nwhere $\\matrixSup{P}{l} \\in \\R^{S_l \\times R^s_l}, \\forall l \\in [m]$, $\\tensor{C} \\in \\R^{H \\times W \\times R^s_0 \\times \\cdots \\times R^s_{m-1} \\times R^t_{0} \\times \\cdots \\times R^t_{m-1}}$ and $\\matrixSup{Q}{l} \\in \\R^{R^t_l \\times T_l}, \\forall l \\in [m]$ are named as input factors, core factor and output factors respectively. The forward pass of \\reshapePrefix\\TK-convolutional layer takes three steps:\n\\begin{subequations}\n\\begin{align}\n\\tensorSup{U}{0} & = \\tensor{U} \\left( \\times_2 \\matrixSup{P}{0} \\cdots \\times_{m+1} \\matrixSup{P}{m-1} \\right) \\\\\n\\tensorSup{U}{1} & = \\tensorSup{U}{0} \\left(\\ast^{0}_{0} \\circ \\ast^{1}_{1} \\times_{2}^{2} \\circ \\cdots \\circ \\times_{m-1}^{m-1} \\right) \\tensor{C} \\\\\n\\tensor{V} & = \\tensorSup{U}{1} \\left(\\times_2 \\matrixSup{Q}{0} \\cdots \\times_{m+1} \\matrixSup{Q}{m-1} \\right) \n\\end{align}\n\\end{subequations}\nwhere $\\tensorSup{U}{0} \\in \\R^{X \\times Y \\times R^s_0 \\times \\cdots \\times  R^s_{m-1}}$ and $\\tensorSup{U}{1} \\in \\R^{X^{\\prime} \\times Y^{\\prime} \\times R^t_0 \\times \\cdots \\times R^t_{m-1}}$ are two intermediate tensors. The backpropagation equations can be derived similarly:\n\\begin{subequations}\n\\begin{gather}\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{0}} = \\tensor{C} \\left( (\\ast^{0}_{0})^{\\top} \\circ (\\ast^{1}_{1})^{\\top} \\circ \\times^{m+2}_{2} \\circ \\cdots \\circ \\times^{2m + 1}_{m + 1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{1}} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensor{C}} = \\tensorSup{U}{0} \\left( (\\ast^{0}_{0})^{\\top} \\circ (\\ast^{1}_{1})^{\\top} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{1}}\n\\end{gather}\n\\end{subequations}\n\n\\paragraph{\\reshapePrefix\\TT-convolutional layer} \nThe layer is derived by factoring $\\tensor{K}$ according to a modified TT\\xspace decomposition as in Figure~\\ref{fig:decomposition-rtt}.\n\\begin{equation}\n\\tensor{K} = \\mathsf{swapaxes} \\left( \\tensorSup{K}{0} \\times^{-1}_{0} \\tensorSup{K}{1} \\times^{-1}_{0} \\cdots \\times^{-1}_{0} \\tensorSup{K}{m} \\right) \n\\label{def:convolutional-tensorized-tt-2}\n\\end{equation}\nwhere $\\tensorSup{K}{0} \\in \\R^{S_0 \\times T_0 \\times R_0}$, $\\tensorSup{K}{l} \\in \\R^{R_{l-1} \\times S_l \\times T_l \\times R_l}$ and $\\tensorSup{K}{m} \\in \\R^{R_{m-1} \\times H \\times W}$ are $(m + 1)$ factor tensors. The multi-stages forward pass to evaluate $\\tensor{V}$ now has $(m+1)$ steps:\n\\begin{subequations}\n\\begin{align}\n\\tensorSup{U}{l} & = \\tensorSup{U}{l} \\left( \\times^{2}_{1} \\circ \\times^{-1}_{0} \\right) \\tensorSup{K}{l} \\\\\n\\tensor{V} & = \\tensorSup{U}{m} \\left( \\ast^{0}_{1} \\circ \\ast^{1}_{2} \\circ \\times^{-1}_{0} \\right) \\tensorSup{K}{m}\n\\end{align}\n\\end{subequations}\nwhere $\\tensorSup{U}{l} \\in \\R^{X \\times Y \\times S_{l} \\times \\cdots \\times S_{m-1} \\times T_{0} \\times \\cdots \\times T_{l-1} \\times R_{l-1}}, \\forall l \\in [m]$ are the intermediate results. We summarize the corresponding backpropagation equations as follows:\n\\begin{subequations}\n\\begin{align}\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{m}} & = \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}} \\left( (\\ast^{0}_{1})^{\\top} \\circ (\\ast^{2}_{1})^{\\top} \\right) \\tensorSup{K}{m} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{m}} & = \\frac{\\partial \\mathcal{L}}{\\partial \\tensor{V}} \\left( (\\ast^{0}_{0})^{\\top} \\circ (\\ast^{1}_{1})^{\\top} \\circ \\times^{2}_{2} \\cdots \\times^{m+1}_{m+1} \\right) \\tensorSup{U}{m} \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{l}} & = \\mathsf{swapaxes} \\left( \\tensorSup{K}{l} \\left( \\times^{2}_{-2} \\circ \\times^{3}_{-1} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{l+1}} \\right) \\\\\n\\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{K}{l}} & = \\mathsf{swapaxes} \\left( \\tensorSup{U}{l} \\left( \\times^{0}_{0} \\circ \\times^{1}_{1} \\circ \\times^{3}_{2} \\cdots \\times^{m+1}_{m} \\right) \\frac{\\partial \\mathcal{L}}{\\partial \\tensorSup{U}{l+1}} \\right) \n\\end{align}\n\\end{subequations}\n\n\\input{.\/\/table_tensorized_convolutional}\n\\section{Supplementary experiments}\n\\label{app:supp_experiments}\n\n\\paragraph{Convergence Rate}\nCompared to end-to-end\\xspace, an ancillary benefit of sequential\\xspace tuning is\n{\\em much} faster and leads to more stable convergence.\nFigure~\\ref{convergence} plots compression error over number of gradient\nupdates for various methods.  (This experiment is for NN-C\\xspace with 10\\% compression rate.)\nThere are three salient points: first, sequential\\xspace tuning has very\nhigh error in the beginning while the ''early'' blocks of the\nnetwork are being tuned (and the rest of the network is left\nunchanged to {tensor decomposition values}).\nHowever, as the final block is tuned\n(around $2\\times 10^{11}$ gradient updates) in the figure, the errors drop\nto nearly minimum immediately. In comparison, end-to-end\\xspace tuning\nrequires 50--100\\% more gradient updates to achieve stable\nperformance. \nFinally, the result also shows that for each block,\nsequential\\xspace tuning achieves convergence very quickly (and nearly\nmonotonically), which results in the stair-step pattern since extra\ntuning of a block does not improve (or appreciably reduce)\nperformance.  \n\n\\paragraph{Performance on Fully-Connected Layers} \nAn extra advantage of TNN based compression\\xspace is that it can apply flexibly to\nfully-connected as well as convolutional layers of a neural network.\nTable~\\ref{table:exp-dense} shows the results of applying TNN based compression\\xspace to\nvarious tensor decompositions on a variant of LeNet-5\nnetwork~\\cite{lecun1998gradient}.  The convolutional layers of the\nLeNet-5 network were {\\em not} compressed, trained or updated in\nthese experiments.  The uncompressed network achieves 99.31\\% accuracy.\nTable~\\ref{table:exp-dense} shows \\textbf{the fully-connected layers can be compressed\nto 0.2\\% losing only about 2\\% accuracy}.  In fact, compressing the\ndense layers to 1\\% of their original size reduce accuracy by less\nthan 1\\%, demonstrating the extreme efficacy of TNN based compression\\xspace when\napplied to fully-connected neural network layers.\n\n\\begin{table}[!htbp]\n\\centering\n\\begin{tabular}{ c | c | c | c }\n& \\multicolumn{3}{c}{Compression rate} \\\\ \nArchitect.  & 0.2\\% & 0.5\\% & 1\\% \\\\ \n\\hline\n\\tnnPrefix\\rCP & 97.21 & 97.92 & 98.65 \\\\\n\\tnnPrefix\\rTK &  97.71 & 98.56 & 98.52 \\\\\n\\tnnPrefix\\rTT & 97.69 & 98.43 & 98.63 \\\\\n\\end{tabular}\n\\caption{TNN compression of fully-connected layer in LeNet-5. \nThe uncompressed network achieves 99.31\\% accuracy.}\n\\label{table:exp-dense}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nUnderstanding how the brain of mammals, including humans, represents, processes and stores information is one of the main challenges of contemporary science. In addition to the obvious direct interest in such an ambitious goal, any progress made towards elucidating the brain working principles would also help developing a new generation of artificial intelligence devises. Reversely, advances in computer science help shedding light on the analogies and differences between our present operational  knowledge on \"artificial intelligence\u00b7 and  \"natural intelligence\". This two-sided dialogue is hoped to guide exciting breakthroughs in the next coming years in both fields. \n\nA popular idea, coming from the world of artificial neural networks \\citep{Langton,Mitchell} and then exported to the realm of biological systems (see \\citep{RMP} and refs. therein), is that information-processing complex systems, composed of many individual interacting units, are best suited to encode, respond, process, and store information  if they operate in a dynamical regime nearby the critical point of a phase transition, i.e. at the edge between \"order\" and \"disorder\"\\citep{Mora-Bialek,Plenz-Functional,RMP,Chialvo2010,KC,Shriki,Breakspear-review,Shew2015b,Serena-LG,Martinello}.  In a nutshell, one can say that \"ordered phases\" encode information in  a robust or stable way, but they are not flexible enough as to accommodate for or respond to input changes; on the other hand, \"disordered phases\" are dominated by noise, thus hindering information storage and making retrieval exceedingly difficult.  Therefore, there needs to be some kind of  trade-off between order and disorder that can be formulated in a number of different ways, e.g., between \"stability and responsiveness\" or between \"robustness and flexibility\". \nThe criticality hypothesis poses that such a trade-off is best resolved near criticality or \"at the edge of chaos\", where combined advantages from the two alternative phases can be obtained \\citep{RMP}. Furthermore, at  critical points there is a concomitant scale invariance  ---with its characteristic power-law distributions and scaling---  entailing the existence of broadly different time and length scales, which seem much convenient for the representation of multiscale complex inputs.  Let us remark, that the terms \"criticality\" and \"edge of chaos\" are sometimes used indistinctly, though the last one applies to deterministic systems, in which a transition occurs between ordered and chaotic states, but as recently emphasized, they can describe two  sides of the same coin (we refer to \\citep{Moritz} for an illuminating recent discussion).  \n\nEmpirical evidence that actual neural networks might operate close to criticality has kept accumulating in recent years \\citep{BP,Petermann,Taglia,Plenz-synchro1,Shew2015b}. Most of this evidence (though not all) relies on the concept of neuronal avalanches \\citep{BP} which are empirically observed to be scale invariant across species, brain regions, resolution levels and experimental techniques \\citep{Schuster,Breakspear-review,RMP}. However, as of today, smoking-gun evidence is still needed to validate or dismiss this fascinating conjecture, and it remains controversial \\citep{Touboul}; thus, novel theoretical and data-oriented analyses are much needed \\citep{RMP}. \n\n\n\n\nIn a seemimgly-unrelated remarkable work, Stringer {\\emph et al.} have recently made a step forward in understanding how neuronal networks actually represent complex inputs. In particular, these authors proved mathematically that the statistics of spiking neurons representing external sensory inputs (such as natural images represented in the mouse visual cortex) need to obey certain constraints for the input representation (or \"neural code\") to be \"continuous and differentiable\" \\citep{Stringer}. These abstract mathematical properties are the formal counterpart of a much-desired property of neural networks: i.e., robustness of the representation against small perturbations of the inputs. Such \na robustness is well-known to be often violated in artificial neural networks (ANNs); in particular, so-called, \\emph{adversarial attacks}, consisting in  tiny variations in the input or their statistics, can fool the network, leading to wrong predictions and mis-classifications \\citep{adversarial}.\nWe refer to Stringer \\emph{et al.} \\citep{Stringer} for an in-depth explanation and justification of these important ideas,  as well as to \\citep{nassar_1n_2020} for an application of them onto multi-layer ANNs. In any case, the conclusion of Stringer \\emph{et al.} is that, in order to achieve robust input representations, the covariance matrix of neuronal  activities measured across time when the network is exposed  to a sequential series of inputs, must obey the following spectral property: its rank-ordered eigenvalues should decay as a power law of their rank, with an exponent $\\alpha$ strictly larger than $1+2\/d$, where $d$ is the embedding dimension of the input. Thus,  $\\alpha=1$ sets a lower bound for the possible values of the eigenspectrum decay-exponent for complex, high-dimensional inputs.\n\n\n\nRather remarkably, these theoretical predictions are verified to be fulfilled in experimental  recordings of more than $10000$ individual neurons in the mouse visual cortex exposed to a very large sequence  of natural images. This confirms that information encoding occurs in as mathematically predicted, i.e. in a continuous and differentiable manifold.\n \nThe main question we pose here is: are the internal representations of ANNs trained to classify images  similar to those of the mouse visual cortex? More specifically, is the spectrum of eigenvalues of the associated covariance matrix a power law of the rank? Is the exponent in all cases larger than (and close to) $1$? If so, do the exponent values change with the images dimensionality in the way predicted by Stringer et al.? \n\nHere, in order to tackle all these questions within the simplest possible scenario, we analyze the neural encoding of inputs with different dimensions in a paradigmatic example of ANN: the \\emph{Echo state network} (ESN)  \\citep{jaeger__2001}. This type of networks, together with \\emph{liquid state machines} \\citep{Maass,maass_real-time_2002}, constitute the prototype of \\emph{reservoir computing} (RC) approaches \\citep{lukosevicius_reservoir_2009}, a paradigm of computation that seems particularly well suited for exploiting the putative advantages of operating at the \"edge of chaos\" \\citep{RMP}.\n\n \n\\section{Materials and Methods}\n The Echo State Network, in its original formulation, was devised by Jaeger as a flexible and easy-trainable recurrent neural network (RNN) for time-series prediction tasks \\citep{jaeger__2001, jaeger_harnessing_2004}. More specifically, the architecture of ESNs is described as consisting of: \n\n\\begin{itemize}\n\n\\item An input layer, which scales a number $L_{1}$ of inputs at each time step before they arrive in the reservoir, according to some random weights  $W^{in}\\in\\mathbb{R}^{N\\times L_{1}}$.\n\n\\item A reservoir consisting of $N$ internal units connected with random weights  $W^{res}\\in\\mathbb{R}^{N\\times N}$, whose states usually evolve according to a non-linear, time-discrete dynamical equation under the influence of a time-dependent input. In this way, the reservoir maps the external input into a high-dimensional space.  \n\n\\item An output layer, that converts the information contained in the high-dimensional states of the neurons (which serve as an internal representation of the inputs) to generate the final output.\n\\end{itemize}\nThus, unlike in other ANNs, the internal weights or \"synaptic connections\" in ESNs do not need to be updated during the learning process, and training is achieved by just modifying the layer of output weights that readout the network internal states. \n\nIn order to adapt this architecture ---usually employed in time-series analyses--- for image classification tasks, we used black and white images with $L_{1}\\times L_{2}$ pixels (each of them characterized by a value in the $[0,1]$ interval, representing a normalized gray-scale) and  converted them into multivariate time series by considering their vertical dimension as a vector of  $L_{1}$ elements or features, that \"evolve\" along $T=L_{2}$ discrete \"time\" steps. One can then define a standard training protocol in which, as illustrated in Fig.\\ref{Figure_0}, at each time $t \\in[0,T]$, vectors $\\mathbf{u}(t) \\in[0,1]^{L_{1}}$ corresponding to columns of the image are fed as inputs to the ESN. In this way, the network dynamics for the reservoir states is  given by the following non-linear activation function: \n\\begin{equation}\n\\mathbf{x}(t)=\\tanh(\\varepsilon W^{in}\\mathbf{u}(t)+W^{res}\\mathbf{x}(t-1))\\label{eq:States_Update}\n\\end{equation}\nwhere $\\varepsilon$ is an input scaling factor. \n\\begin{figure*}\n\\begin{centering}\n\\includegraphics[scale=0.5]{Images\/Figure_0}\n\\par\\end{centering}\n\\caption{{\\bf Sketch of the Echo State Network and the image classification task.} Left: Images are converted to multivariate time series and then fed into the reservoir. Right: for each processed image a set of parameters $\\theta_{x}$ is generated, which characterizes the high-dimensional state of the reservoir, i.e. the ``\\emph{reservoir model space}''. These are then fed into the readout module, that linearly transforms the information in the reservoir model space into an output label. Finally, output weights $\\tilde{W}_{out}$ are generated by minimizing the error between the predicted and target labels. Red arrows indicate steps in which a Ridge regression is performed.}\\label{Figure_0}\n\\end{figure*}\nUsing a supervised learning scheme, the goal of the ESN is to generate an output label $\\mathbf{y}\\in\\mathbb{\\mathbb{N}}^{F}$ that correctly classifies each image in the test set as belonging to one of the $F$ existing categories or classes. This label consists of a vector in which every element is zero, except for a value of one at the position corresponding to the assigned class (i.e., \"one-hot-encoded\" in the machine learning jargon). Several readout methods have been proposed in the literature to transform the information contained in the reservoir dynamics into the expected target output $\\mathbf{y}^{target}\\in\\mathbb{\\mathbb{N}}^{F}$, ranging from linear regressions methods over the reservoir states \\citep{reinhart_constrained_2010,reinhart_reservoir_2011}, to the use of \"support vector machines\" or \"multilayer perceptrons as decoders \\citep{babinec_merging_2006}. Here, we use a simple Ridge regression (see Appendix I for a detailed explanation of the algorithm) over the \\emph{\"reservoir model space\"}, a method that has been recently proposed for the classification of multivariate-time series \\citep{bianchi_reservoir_2021}. \n\nThe \"reservoir model space\" is a set of parameters $\\theta_{x}$ that encodes the reservoir \\emph{dynamical} state for a given input (image). Such parameters are obtained from a Ridge regression to predict the next reservoir state from the past one at discrete time steps,\n\\begin{equation}\n\\mathbf{x}(t+1)=W_{x}\\mathbf{x}(t)+\\mathbf{w}_{x},\n\\end{equation}\nin such a way that $\\theta_{x}=\\left[\\mathrm{vec}(W_{x});\\mathbf{w}_{x}\\right]\\in\\mathbb{R}^{N\\left(N+1\\right)}$  provides a characterization of  the internal reservoir dynamical state during the presentation of a given input, where $\\mathrm{vec}(\\cdot)$ denotes reshaping to a one-column vector and $\";\"$  vertical concatenation. Then, for each image a readout module or decoder can transform this internal representation into an output label:\n\\begin{equation}\n\\mathbf{y}=W_{out}\\theta_{x}+\\mathbf{w}_{out}.\\label{Eq_Readout}\n\\end{equation}\nThe parameters $\\theta_{out}=\\left[\\mathrm{vec}\\left(W_{out}\\right);\\mathbf{w}_{out}\\right]$ ---where $W_{out}\\in\\mathbb{\\mathbb{R}}^{F\\times N(N+1)}$ and $\\mathbf{w}_{out}\\in\\mathbb{\\mathbb{R}}^{F}$ are defined as output weights and biases, respectively--- are determined again through Ridge regression, minimizing the error between the produced and target label for all the presented images in the training set. \n\nLet us remark that the presented framework can be naturally extended to include, for instance, leakage and noise terms in Eq.\\ref{eq:States_Update}, feedback connections from the output to the reservoir, or plastic rules that modify the reservoir weights according to the inputs  \\citep{lukosevicius_practical_2012,morales_unveiling_2021}, among other possible extensions. However and since our aim here is not to reach state-of-the-art classification accuracy ---but rather highlight the link between optimal input representation and the internal dynamical state--- for the sake of parsimony, we refrain from adding further features to our model for the time being.\n\n\\section{Results}\n\n\\subsection{Non-trivial scaling at the edge of chaos.}\nAlthough relatively simple, our proposed ESN model has several hyperparameters than can be tuned, affecting its performance. More specifically, the \\emph{spectral radius} $\\rho$ of the reservoir internal weight matrix and the \\emph{scaling factor} $\\varepsilon$ of the input weights are two variables that usually determine the dynamical regime within the reservoir \\citep{lukosevicius_reservoir_2009}. The spectral radius ---or largest eigenvalue of the reservoir weight matrix--- controls the dynamical stability inside the reservoir when no input is fed into the network. Thus, a spectral radius exceeding unity has been often regarded as a source of instability in ESNs due to the loss of the so-called \\textquotedblleft \\emph{echo state property}\\textquotedblright, a mathematical condition ensuring that the effect of initial conditions on the reservoir states fades away asymptotically in time \\citep{jaeger__2001,jaeger_short_2001,yildiz_re-visiting_2012}. Nevertheless, later studies have shown that the echo state property can be actually maintained over a unitary spectral radius, and different sufficient conditions have been proposed  \\citep{buehner_tighter_2006,yildiz_re-visiting_2012,gallicchio_chasing_2018} (see in particular \\citep{manjunath_echo_2013}, where the authors analyze the problem from the lens of non-autonomous dynamical systems, deriving a sufficient condition for the echo state property with regard to a given input). On the other hand, increasing the value of  $\\varepsilon$  can convert an initially expanding mapping into a contracting dynamics, as stronger inputs tend to push the activities of the reservoir units towards the tails of the non-linearity.\n\nIn what follows,  we analyze the input representation that the reservoir codifies in terms of the trade-off between $\\rho$ and $\\varepsilon$, which \\emph{together} determine the dynamical operating regime of the ESN and the presence or absence of the echo state property. For the rest of parameters, the number of units in the reservoir is kept fixed to $N=2000$ and the density of the reservoir-weight-matrix elements (i.e., the percentage of non-zero connections) to $10\\%$, while both reservoir and input weights are extracted at random from a uniform distribution in the interval  $[-1,1]$. \n\\begin{figure}\n\\centering{}\\includegraphics[scale=0.5]{Images\/Figure_5_Map_Full}\\caption{Exponent for the power-law decay of the spectrum of the activity covariance matrix  as a function of the spectral radius ($\\rho$) and input scaling factor ($\\varepsilon$) of the reservoir, plotted together with the maximum Lyapunov exponent (MLE) color-coded within the surface. The insets correspond to the activity covariance matrix eigenspectrum measured in three different points of the parameter space, where the variance in the n-th dimension (n-th eigenvalue) scales as a power-law  $n^{-\\alpha}$  of the rank.  For ease of visualization, the plane separating the region $\\alpha < 1$  in which the representation is no longer continuous nor differentiable was plotted in purple.}\\label{Figure_1}\n\\end{figure}\n\nFollowing the same methodology as in Stringer {\\emph et al.}\\citep{Stringer}, the ESN was first presented with a large set of high-dimensional, natural images, and the activity of the internal units in the reservoir  was stored for each step of the training. Then, principal component analysis (PCA) was performed directly over the full set of neuron activities $X\\in\\mathbb{R}^{N\\times( T\\times M)}$,  where $T=90$ is the number of pixels in the horizontal dimension of the images and\n$M=2800$  is the total number of images.  In this way, we obtained the variance along each principal component or eigenvector of the covariance matrix, that serves as a basis for the activity inside the reservoir (see \\citep{shlens_tutorial_2014} for a very gentle but rigorous introduction to PCA).\n\nNotably, we found that the spectrum of eigenvalues as a function of their rank (i.e., the variance associated to the n-th principal component, when ordered from the largest to the smallest) can be well fitted to a power-law  $n^{-\\alpha} $ (see insets in Fig.\\ref{Figure_1}), whose  associated exponent $\\alpha$ decays (flatter spectrum) with the spectral radius $\\rho$; \nwhile, the exponent increases (faster decay) with the input scaling factor $\\varepsilon$  for most of the parameter space (see Fig. 2). In \\citep{Stringer} it was found that the exponent of this power-law relation is close to $1$  when natural, high-dimensional images were shown to the mouse as an input. As discussed above, the authors proved mathematically that $\\alpha>1+2\/d$ is a necessary condition for the neural manifold that emerges from the representation of a $d$-dimensional input to be continuous and differentiable. For natural images, $d$  is very large and one can approximate the critical exponent by $\\alpha_{c}\\approx1$ (this condition is marked by the purple plane in Fig.\\ref{Figure_1}). \n\nAs a following step, one might  naturally wonder if there is some\n aspect of our model that characterizes such a regime of robust representations in the parameter space $(\\rho,\\varepsilon)$, for which an exponent $\\alpha$ close to unity is found. In other words: is the dynamics of the system inherently different in the regions for which the input representation manifold is found to be non-analytical (i.e. below the purple plane in Fig.\\ref{Figure_1})?\n\nAs it turns out, rather remarkably, the exponent $\\alpha$ characterizing the decay of the eigenspectrum approaches unity for choices of $\\rho$ and $\\varepsilon$ that drive the network dynamics towards the so-called \"edge-\n of instability\" or \"edge of chaos\", this is, near a transition point between an ordered and a chaotic regime. Traditionally, chaotic regimes are  characterized by their average sensitivity to perturbations in the initial conditions; to quantify this effect, one usually measures the rate of divergence of two trajectories with a very small difference in their initial conditions:\n\\begin{equation}\n\\lambda=\\lim_{k\\to\\infty}\\dfrac{1}{k}log\\left(\\dfrac{\\gamma_{k}}{\\gamma_{0}}\\right)\n\\end{equation}where $\\lambda$ is termed the maximum Lyapunov exponent (MLE), $\\gamma_{0}$ is the initial distance between the perturbed and unperturbed trajectories, and $\\gamma_{k}$ is the distance between the trajectories at step $k$  (we refer the reader to \\citep{boedecker_information_2011} and \\citep{sprott_chaos_2003} for a detailed explanation of the algorithm used to compute the MLE). Thus, chaotic dynamics is typically associated with a positive MLE, while the system is said to be stable to local perturbations provided $\\lambda<0$. It can be clearly seen from Fig.\\ref{Figure_1} that the region in which one finds non-analytical representations of the input (below the purple plane) matches almost perfectly with the region (colored in green) in which a positive MLE is found. \n\nThe transition order-to-chaos can be also visualized looking directly at the activities inside the reservoir (see Fig.\\ref{Figure_5}). Observe that, when the network is in an \"ordered\" state, with $\\lambda<0$, the responses of the neurons are quite heterogeneous when compared among them, but they are highly localized within each neuron, i.e., individual neurons experience a limited response to stimuli. On the other hand, dynamical states characterized by $\\lambda>0$ have neurons whose response extends across the full range of the non-linearity (with higher probability along the tails, reflecting a saturated behavior), but it is this same \"phase space expansion\" that makes units almost indistinguishable from each other. It is only around the critical point or f-chaos, that we find a trade-off between dynamical richness in individual units and variability across units. \n\nComing back to the results in Stringer \\emph{et al.}, one may also wonder whether the continuity and differentiability condition $\\alpha>1+2\/d$ holds also for low-dimensional inputs, for which the expected bound $\\alpha_{c}=1+2\/d$  deviates considerably from unity.  To this purpose, Fig.\\ref{Figure_2} shows the measured eigenspectrum of the reservoir activity covariance matrix (i.e. eigenvalues as a function of their rank) when images of different dimensionality (the same ones used by Stringer \\emph{et al.} in their experiments) are presented as inputs, and the reservoir is tuned to operate at the onset of a chaotic regime, i.e., for values in the parameter space $\\left(\\rho,\\varepsilon\\right)$ for which $\\lambda$ was near zero but still negative. Remarkably, we find in all cases that \\emph{the exponents observed in the mouse visual-cortex activity are best reproduced when the reservoir dynamics is tuned close to the \"edge of chaos\"}. \n\nThis finding suggests that one can set the network parameters in such a way that the neural activity manifold in which the input is represented is almost as high-dimensional as possible without loosing its \"smoothness\", and that such optimal solution is found at the edge of chaos.\n\\begin{figure}\n\\centering{}\\includegraphics[scale=0.8]{Images\/Figure_2_Full}\\caption{From left to right: (A) sample from the $M=2800$  images in the training set; (B) eigenspectrum of the images pixel intensities; (C) eigenspectrum for the activities of an ESN (blue line) and real, V1 mouse neurons (yellow line, plotted after  \\citep{Stringer} applying cvPCA) when subject to images of dimensionality $d$ ; (D) same analysis, but now zero-centered white noise of amplitude $\\epsilon = 0.4$  is added to the neuron dynamics (blue line), and no cvPCA is performed over the experimental values (yellow line); (E) same analysis as in (D), but now noise has been substracted using cvPCA. From top to bottom: results for natural, high-dimensional images; the same images projected onto 8 dimensions; the same images projected onto 4 dimensions. To obtain the ESNs eigenspectra, parameters were chosen so that the networks operated near the edge of chaos, with $\\lambda \\sim -5 \\cdot 10^{-3}$. }\\label{Figure_2}\n\\end{figure}\n\nAt this point, it is pertinent and timely to dig a bit deeper on the similarities and differences between the results presented in \\citep{Stringer} for real, V1-cortex neurons in the mouse, and the power-law exponents obtained through our reservoir computing model. \n\n(i) First of all, as in the case of real neurons, the observed correlations between the internal units are  not just a byproduct emerging from scale-free features of natural images (see second column in Fig.\\ref{Figure_2}). In particular,  one can see that the power-law decay of the eigenspectrum persists even in response to low-dimensional inputs whose embedding vector space can be spanned with just a few principal components (i.e. without a power-law decaying intrinsic spectrum).\n\n(ii) In our model, images are processed sequentially in time along their horizontal dimension, so that for each image one can measure the activity of the $N$ internal units over $T=L_{2}$ time steps. In contrast, activity of V1 neurons in \\citep{Stringer} is scanned at a relatively low rate, so that for each image the neural representation in characterized by just one amplitude value in each neuron. \n\n(iii) To avoid confusion, let us remark that the variance observed by Stringer et al. is not directly measured over the raw activity of the neurons. Instead, the authors first project out the network spontaneous activity from the data, and then perform a cross-validated PCA (cvPCA) that allows them to filter out the trial-to-trial variability or \"noise\". The cvPCA method is thus able to estimate the stimulus-related variance confined in an n-dimensional manifold by first computing the eigenvectors spanning this manifold from a first repeat of the full training set, and then measuring the amount of a second repeat's variance that is confined to this plane (we refer to \\citep{Stringer} for a detailed explanation and derivation of the cvPCA method). However, since our model is completely deterministic for a given initialization of an ESN, the stimulus-related variance computed through cvPCA  trivially matches that of a standard PCA. \n\n\nA natural question then arises from this last point: what happens when a noise term is included in Eq.\\ref{eq:States_Update}, so that the dynamics becomes stochastic? Are the  power-law exponents robust to the introduction of noise? To answer these questions, we considered stochastic versions of the ESNs  ---including an independent small additive noise term in their inputs--- and presented them with two repeats of the same input training set. The internal states of the noisy  reservoirs were collected at each time step. We then performed the same type of cvPCA analyses proposed in \\citep{Stringer} to estimate the signal variance in our reservoirs (see fourth and fifth columns in Fig.\\ref{Figure_2}). Just as in the case  of real V1 neurons, the exponents measured over the raw, noisy activity are lower and below the critical threshold for continuity and differentiability of the neural manifold. Nevertheless, a cvPCA over the internal states retrieves the expected exponents after noise has been filtered out.\nWe will further comment on the possible implications of this finding in the Discussion section, but for now, let us wrap up our findings tackling what we believe is a fundamental question from the perspective of machine learning: does working at the edge of chaos (or, equivalently, having maximal, continuous and differentiable neural manifolds) provide any functional advantage?\n\n\\subsection{Solving a bench-mark classification task.}\n\nThe advantages of working at the so-called edge of chaos were first pointed in general dynamical systems and cellular automata \\citep{crutchfield_computation_1988,Langton}, and only later analyzed in reservoir computer models with binary \\citep{bertschinger_real-time_2004, legenstein_edge_2007, busing_connectivity_2010} and analog \\citep{busing_connectivity_2010, boedecker_information_2011} internal units. In particular, in \\citep{boedecker_information_2011} the authors showed that ESNs presented maximal information storage and transfer, as well as enhanced memory capacity right at the edge of chaos. However, while ESNs and other RC approaches have been previously applied to classification tasks with very good results \\citep{schaetti_echo_2016, skowronski_automatic_2007, aswolinskiy_time_2016, ma_functional_2016, yusoff_modeling_2016,bianchi_reservoir_2021}, to the best of our knowledge an analysis of the influence of the dynamical regime on the performance of RC architectures for classification tasks is still missing. To this end, we measure the performance of ESNs in a classification task over the canonical MNIST dataset, which includes $60,000$ handwritten instances of  the first $10$ digits as training (although only a third of them was used) and $10,000$  for testing, following the training and testing procedure as described in Materials and methods. \n\nThe results, showed in Fig.\\ref{Figure_4}, highlight the fact that optimal performance ($\\sim 2.2 \\%$ error rate) is found just below the onset of chaos, when $\\lambda \\lesssim 0$. Most notably, the plot also evinces that the decay in performance is not only preceded by a positive MLE, but coincides too with exponents $\\alpha$  for the fit of the covariance-matrix eigenpectrum that are below the  limiting value $\\alpha_{c}\\approx1$, indicating the loss of continuity and differentiability of the neural representation manifold for high-dimensional images. Let us remark that, the slower the decay (i.e. the larger the exponent) the more weight is given to fine details of the input, but if the decay is too slow (smaller than the lower bound above), an excessive importance is given to such fine details at the cost of hampering the existence of a \"smooth manifold\" representation. Thus, operating near the edge of instability could provide the network with an optimal trade-off between representing as much details as possible and constructing operative, smooth representations.\n\n\nWe finally remark that the results shown here were obtained with a reservoir consisting only of $500$ internal units and using only one-third of the training set, with no pre-processing of the images. In contrast, the current best performance in MNIST digit recognition ($0.81\\%$ error rate) using reservoir computing networks has been achieved with a two-layer architecture, each consisting of $16,000$ units, which amounts to a total $880,000$ trainable parameters \\citep{jalalvand_design_2015}. In this sense, a simple ESN with readouts over the reservoir model space, when tuned near the edge of chaos, is able to outperform ESNs with a greater number of units and much more complicated dynamics (including feedback connections and leakage terms), trained over the full MNIST dataset \\citep{schaetti_echo_2016}. \n\n\\begin{figure}\n\\centering{}\\includegraphics[scale=1]{Images\/Figure_3_New}\\caption{Curves for the accuracy in MNIST testset (blue dots), maximum Lyapunov\nexponent (orange line) and best fit exponent for power-law spectrum\nof activity covariance matrix (purple line). Training was performed\nover 20,000 randomly chosen images of the MNIST training dataset, while classification error was assessed\nover the full test set (10,000 images). Errors in each case were estimated as the standard deviation from the mean over ten different initializations of the ESN.}\\label{Figure_4}\n\\end{figure}\n\n\\section{Discussion}\n Stringer {\\emph{et al.}} observed in  \\citep{Stringer} that neural coding of different inputs in the mouse V1-cortex is close to optimal for each type of input, constrained by requirements of continuity and differentiability of the neural response manifold. In this paper, we open the door to the possibility that optimal, continuous and differentiable response manifolds emerge for neuron dynamics laying close to an edge of chaos type of critical point. Indeed, we have shown that a simple non-linear model of  randomly-connected neurons, when subject to an external input, is able to reproduce power-law exponents similar to those found in mouse V1-cortex for the decay of the covariance matrix eigenspectrum.\n\nWe find nevertheless important to clarify that the term edge of chaos ---and the concept of chaos itself--- should be taken with caution as it is not devoid of criticism in this context.  As pointed out in \\citep{manjunath_echo_2013}, ESNs are an example of nonautonomous dynamical systems, for which typical concepts based in the theory of autonomous systems (e.g., \"sensitivity to initial conditions\", \"attractor\" and \"deterministic chaos\") do not directly apply \\citep{clemson_discerning_2014,gandhi_theory_2012}. In fact, the authors  of \\citep{manjunath_echo_2013} claim that local perturbation experiments cannot represent an ultimate evidence of chaotic dynamics  in non-autonomous systems, since it might well be the case that the input drives the system towards and expanding dynamics for a certain time span, while the system shows on average a contracting, non-chaotic dynamics. Despite these caveats, at the light of the presented results it appears like there is indeed an actual dynamical phase transition occurring as the maximum Lyapunov exponent crosses zero. Thus, in any case, it seems a sensible choice to use such a quantity as a control parameter when analyzing the underlying neural representation of external inputs. \n\nOn the other hand, adding stochasticity in the form of small-amplitude white noise naturally leads to flatter eigenspectra, much like those found when PCA is performed over raw experimental data. Nevertheless, one can use the same cvPCA technique introduced also in \\citep{Stringer} to extract the input-related variance of the activity, thus obtaining similar exponents to the fully deterministic case. This remarkable result therefore suggests that the role of spontaneous activity and trial-to-trial variability on the representation of external inputs can be easily accounted for in our simple echo-state-network model.\n\n\nFinally, results obtained on a benchmark classification task suggest that input-representation manifolds that are critically high-dimensional (from the point of view of their analytic properties) may serve a bigger purpose than just being a mathematical curiosity, as ESNs show a better performance when poised near such a critical point, while the accuracy falls rapidly as soon as the representation manifold becomes fractal.\n\nTherefore, the presented results open the path for very exciting research avenues at the boundary of biology and machine learning, calling for theoretical formulations that can shed light into the fascinating properties of these input-representation neural manifolds and their relation with the criticality hypothesis.\n\n\n\\section*{Appendix A: Ridge regression}\n\nReadouts in ESNs are typically linear, as it can be seen from Eq.\\ref{Eq_Readout}. To simplify the forthcoming derivations, let us rewrite Eq.\\ref{Eq_Readout} as:\n\\begin{equation}\n\\mathbf{y}=\\tilde{W}_{out}\\left[\\theta_{x};1\\right]\n\\end{equation}\nwhere $\\tilde{W}_{out}\\in\\mathbb{\\mathbb{R}}^{F\\times\\left(N(N+1)+1\\right)}$ and \" ;  \" indicates vertical vector concatenation. Thus, for a given input image a reservoir representation of the states $\\theta_{x}$ is constructed, and one can use the above equation to generate the corresponding output (which in our case will be a one-hot-encoded label $\\mathbf{y}\\in\\mathbb{\\mathbb{N}}^{F}$ classifying the image). We can trivially generalize the above equation to apply to the full set of $M$ images in the training set $Y\\in\\mathbb{\\mathbb{N}}^{F\\times M}$:\n\\begin{equation}\nY=\\tilde{W}_{out}\\Theta_{x}\n\\end{equation}\nwhere $\\Theta_{x}\\in\\mathbb{R}^{\\left(N\\left(N+1\\right)+1\\right)\\times M}$ contains as columns the vectors $\\left[\\theta_{x};1\\right]$ generated for each of the $M$ input images. Finding the optimal weights\n$\\tilde{W}_{out}$ that minimize the squared error between the produced and target labels, $\\mathbf{y}$ and $\\mathbf{y}^{target}$, is then reduced to a standard linear regression problem, which is the greatest strength of the reservoir-computing approach:\n\\begin{equation}\nY^{target}=\\tilde{W}_{out}\\Theta_{x}.\n\\end{equation}\nOwing to the fact that large output weights are commonly associated to over-fitting of the training data \\citep{lukosevicius_practical_2012}, it is a common practice to add a regularization term to the error in the target reconstruction, usually defined in terms of the root-mean-squared error (RMSE). Although several methods have been proposed to achieve this regularization \\citep{lukosevicius_practical_2012, reinhart_constrained_2010,reinhart_reservoir_2011}, one of the most efficient and stable algorithms is Ridge regression, which aims to solve:\n\\begin{equation}\n\\tilde{W}_{out}=\\underset{\\left\\{ \\tilde{W}_{out}\\right\\} }{\\arg\\min}\\dfrac{1}{M}\\sum_{n=1}^{M}\\sum_{i=1}^{F}\\left(y_{i}[n]-y_{i}^{target}[n]\\right)^{2}+\\beta\\left\\Vert \\tilde{w}_{i}^{out}\\right\\Vert ^{2}=Y^{target}\\Theta_{x}^{T}\\left(\\Theta_{x}\\Theta_{x}^{T}+\\beta I\\right)^{-1},\\label{eq:MSE_Error}\n\\end{equation}\nwhere $\\left\\Vert \\text{\\ensuremath{\\cdot}}\\right\\Vert $ stands for the Euclidian norm, $I$ is the identity matrix and $\\beta$ is the regularization coefficient. Notice that choosing $\\beta=0$ removes the regularization, turning the Ridge regression into a standard generalized linear regression problem (we used $\\beta = 1$ across all simulations in the paper).\n\nWe finally remark that, in order to obtain the reservoir model space parameters $\\Theta_{x}$, the same Ridge regression is also implemented to solve Eq. 2 for each of the input images.\n\n\\section*{Appendix B: Phase space of reservoir units}\nIn Fig.5, we reproduce the phase space of $4$ different internal units of a reservoir operating (from top to bottom lines) in a sub-critical, critical and super-critical regimes, respectively.  Each plot represents the activity of the corresponding neuron at each time step against the total input (sum of external input plus reverberating activity in Eq.\\ref{eq:States_Update}) arriving to the neuron at the previous time. Since each image is first transformed into a multivariate time series of $T=L_{2}=90$ time steps ---and we plot the activity along the first three images only--- each panel in Fig.\\ref{Figure_5} contains 270 points.\n\\begin{figure}\n\\centering{}\\includegraphics[scale=1]{Images\/Figure_6_Neurons_Activity.png}\\caption{Activity of four different neurons operating in the sub-critical, critical and super-critical regimes (from top to bottom) when presented with three different high-dimensional, natural images. Each point in the panels represents the activity $x_{i}(t)$  of neuron $i$  at time step $t$  as a function of the total input $f_{i}(t)=\\varepsilon \\sum_{l=1}^{L_{1}} w_{il}^{in} u_{l}(t) + \\sum_{j=1}^{N} w_{ij}^{res} x_{j}(t-1)$ arriving to it.} \\label{Figure_5}\n\\end{figure}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nA system of two mutually coupled semiconductor lasers is a simple example of interacting nonlinear oscillators. This system has been experimentally realised using for example edge emitting \\cite{HEI01}, quantum dot \\cite{HEG07}, VCSEL \\cite{FUJ03} and DBR \\cite{VAU09} lasers and the observed dynamical phenomena include leader-laggard dynamics \\cite{HEI01}, frequency locking and intensity pulsations \\cite{WUE05} and bubbling \\cite{FLU09,TIA12}.  For recent reviews on the rich dynamical features of coupled semiconductor lasers we refer to \\cite{LUE12,SOR13}.  \n\nIn the current paper we study this problem theoretically on the basis of a well established rate equation model.  In general, there are two time scales which govern the character of the dynamics in coupled semiconductor lasers, namely the period of the relaxation oscillation $T_R=1\/\\nu_R$  and the delay time $\\tau$ due to the separation between the two lasers.  In the long delay case $\\tau\\gg T_R$, it is well known that the synchronous state of two coupled lasers is in general not stable \\cite{WHI02}, which gives rise to  leader-laggard dynamics and noise induced switching between the  asymmetric states \\cite{MUL04}.  In view of applications in small-scale and high-speed devices, we are however interested in the opposite limit, where the delay time is comparable to or even much smaller than the relaxation period.  In this limit the rate equation approach is justified if the distance between the lasers is significantly larger than the optical wavelength itself.  At even shorter distances, composite cavity \nmodels have recently been used to describe effects due to evanescent lateral coupling \\cite{ERZ08,BLA12}.  Using typical values for the relaxation oscillation in the GHz regime and an optical wavelength of around 1~$\\mu$m, we therefore now focus on the case of two coupled lasers at distances between 10~$\\mu$m and 500~$\\mu$m. \n\nWe observe that for small spatial separation and strong optical coupling, the two lasers can mutually lock into one of two stable one-colour states. These states spontaneously break the original symmetry of exchanging the two lasers, and both lasers emit light at precisely the same frequency but at different intensities. In this bi-stable regime the current state of the system can be conveniently observed optically from the amplitude of the light output of the lasers.  There also exists a region of hysteresis, where the symmetric one-colour state is jointly stable with the two symmetry broken one-colour states, thus giving rise to a parameter region of tri-stability. For more moderate coupling strength, the symmetry broken one-colour states become unstable, and instead symmetry broken two-colour states appear, which are similar to the ones observed numerically in \\cite{ROG03}.  In this case, both lasers emit light at the same two optical frequencies, however the intensities of the respective colours are \nnot identical in both lasers.  \n\nIn order to study these and similar transitions in the dynamics  systematically, we introduce a reduced five dimensional model, which allows for a bifurcation analysis using the continuation software AUTO \\cite{DOE06}.  The transition from a one-colour to a two-colour symmetry broken state then corresponds to a Hopf bifurcation from a symmetry-broken fixed-point state to a symmetry-broken limit cycle state in the language of bifurcation theory.  We are in particular interested in the fundamental bifurcations, which bound the domain of symmetry broken two-colour states. We identify the relevant codimension two points, which organise the bifurcation scenario in this region, and explain the mechanism which gives rise to the large region of bi-stability between symmetry-broken two-colour states. \n\nFrom the technological side there exists currently a strong interest in the development of small-scale devices which are capable of all-optical signal processing.  One particular challenge, which has attracted a significant amount of recent research  activities,  is the design of all-optical memory elements  \\cite{HIL04,OSB09,LIU10,CHE11,HEI11,NOZ12,PER12}.  The goal is to design fast and efficient memory elements which can be switched between at least two different stable states via an external optical signal.  At the same time it is also desired that the  state of the memory element is accessible optically, and the optical output from one memory element should be able to trigger the switching in further elements, with little or no intermediate processing.  Based on these criteria, the question arises, if all-optical memory elements can be realised using two mutually interacting identical single-mode lasers.  After having identified a number of interesting regions of multi-stabilities involving one-colour \nand two-colour states, we numerically demonstrate that these states can indeed be exploited for the design of all-optical memory units. By injecting suitable pulses of light into the coupled lasers we are able to switch between different multi-stable states.  In order to assess the technologically relevant speed of the switching events, we define the write time as the minimal injected pulse duration required to trigger the switching event, and the read time as the minimum time after which the state of the memory can be obtained optically.  We find read and write times of less than 100ps, which suggests the possibility of fast and simple all-optical memory elements on the basis of identical closely coupled single-mode lasers. \n\n\n\\section{Rate Equation Model}\n\n\\begin{figure}\n\\includegraphics[width=1.0\\columnwidth]{fig01}\n\\caption{\\label{fg:scheme} Schematic diagram of two coupled lasers of wavelength $\\lambda_0$ separated by a distance $D$, where  $C_p \\in [0,2\\pi)$ and $j \\in \\field{N}$.}\n\\end{figure}\n\nTwo identical single-mode semiconductor lasers with free-running wavelength $\\lambda_0$ are placed in a face-to-face alignment with a separation $D$ as sketched in Fig.~\\ref{fg:scheme}. They are mutually coupled via a certain amount of the light of each entering the other cavity after a delay $\\tau = D\/c$, where $c$ denotes the speed of light. This scenario has been successfully studied in the literature on the basis of rate equation models \\cite{HOH97,MUL02,ROG03,YAN04,ERZ06} and we will use the following system of delay differential equations:\n\n\\begin{subequations}\\label{eq:mLK}\n\\begin{eqnarray}\n\\dot{E_1}(t) =& (1 + i \\, \\alpha) \\, N_1(t) \\, E_1(t) + \\kappa e^{-i \\, {C_p}} E_2 (t -\\tau) \\label{eq:E1dot}\\\\\n\\dot{E_2}(t) =& (1 + i \\, \\alpha) \\, N_2(t) \\, E_2(t) + \\kappa e^{-i \\, {C_p}} E_1 (t -\\tau) \\label{eq:E2dot} \\\\\n\\dot{N_1}(t) =& \\frac{1}{T}\\left[P - N_1(t) - \\left( 1 + 2N_1(t) \\right) \\, |E_1(t)|^2\\right]  \\label{eq:N1dot}\\\\\n\\dot{N_2}(t) =& \\frac{1}{T}\\left[P - N_2(t) - \\left( 1 + 2N_2(t) \\right) \\, |E_2(t)|^2\\right] \\label{eq:N2dot}\n\\end{eqnarray}\n\\end{subequations}\n\nThe parameters include $P=0.23$, the pumping of electron-hole pairs into each laser; $\\alpha=2.6$, the line-width enhancement factor; $T=392$, the ratio of the photonic and carrier lifetimes; $\\tau$ the dimensionless delay time between the lasers. The coupling strength $\\kappa$ and the coupling phase $C_p\\,=\\,2\\pi ( D\\,mod\\,\\lambda_0)$ are the two main bifurcation parameters in subsequent sections. The model~\\eqref{eq:mLK} is dimensionless with time measured in units of the photon lifetime $\\tau_p = 1.0204$ ps.  The dynamical variables $N_1$ and $N_2$ denote the population inversions, and $E_1$ and $E_2$ are the slowly varying complex optical fields in laser 1 and laser 2.  The rapidly oscillating physical fields can be recovered via $\\tilde{E}_{1,2} = E_{1,2}(t) e^{i \\omega_0 t}$, where the optical angular frequency is given by $ \\omega_0= 2\\pi c \/ \\lambda_0$. The phase factor $e^{-i C_p}$  in system~\\eqref{eq:mLK} is therefore a consequence of expressing the time delayed physical fields using slowly \nvarying fields via $\\tilde{E}_{1,2}(t-\\tau) e^{-i \\omega_0 t} = e^{-i C_p} E_{1,2}(t-\\tau)$. \n\n\nAny solution of equations~\\eqref{eq:mLK} can be multiplied by a common phase factor in both optical fields leading to a $S^1$ symmetry. In addition, as the two lasers are identical, a $\\field{Z}_2$ symmetry exists due to the ability to swap the lasers. Mathematically  these two phase space symmetries can be formulated as \\cite{ERZ06},\n\\begin{equation}\n\\label{eq:symm}\n\\begin{aligned}\n & \\left(E_1,E_2\\right) \\rightarrow \\left(e^{i\\,b}E_1,e^{i\\,b}E_2\\right), \\: b \\in [0,2\\pi) \\:  & S^1 \\text{ symmetry} \\\\\n & \\left(E_1,E_2,N_1,N_2\\right) \\rightarrow \\left(E_2,E_1,N_2,N_1\\right) \\: & \\field{Z}_2 \\text{ symmetry}\n\\end{aligned}\n\\end{equation}\nBoth the $S^1$ and $\\field{Z}_2$ symmetries are frequently used and referred to in subsequent sections. \n\n\\section{One-Colour States}\\label{sec:CLM}\n\n\\begin{figure}\n\\includegraphics[width=1.0\\linewidth,type=pdf,ext=.pdf,read=.pdf]{fig02}\n\\caption{\\label{fg:CLMs} Time-traces (top panels) and frequency spectra (bottom panels) showing symmetric (left column) and symmetry-broken (middle and right columns) CLMs for $\\tau=0.1$, $\\kappa=0.3$ and ${C_p}=0.33\\,\\pi$.  These parameters are consistent with a point in region 7 of Fig.~\\ref{fg:map}.}\n\\end{figure}\n\nOne-colour states, which are also known as compound laser modes (CLMs)~\\cite{ERZ06} are constant amplitude and single frequency solutions to~\\eqref{eq:mLK} whereby the two lasers are frequency locked. They are characterised by the following ansatz,  \n\\begin{equation}\n\\label{eq:1C}\n\\begin{aligned}\n& E_1(t) = A_1 e^{i\\,\\omega_A\\,t}& \\qquad &N_1(t)= N_1^c\\\\\n& E_2(t) = A_2 e^{i\\,\\omega_A\\,t} e^{i\\,\\delta_A}& &N_2(t)= N_2^c\n\\end{aligned}\n\\end{equation}\nwith real constants $A_1$, $A_2$, $N_1^c$, $N_2^c$, $\\delta_A$ and $\\omega_A$. The frequency $\\omega_A$ is the common locked  frequency of the slowly varying optical fields of the two lasers and $\\delta_A$ allows for a constant phase difference between them. This ansatz can be split into two complementary classes; (i) symmetric CLMs ($A_1=A_2$) which are invariant under the $\\mathbb{Z}_2$ symmetry and (ii) symmetry-broken CLMs where both lasers lase at different intensities ($A_1\\neq A_2$). Further details on this classification are given in Appendix~\\ref{ap:ssbCLM}. \n\nSymmetric CLMs can be further sub-divided into ``in-phase'' ($\\delta_A=0$) and ``anti-phase'' ($\\delta_A=\\pi$) solutions and their stability is extensively studied in \\cite{ERZ06,YAN04}.  In particular, it was found that symmetric CLMs can lose their stability via Hopf, saddle node or pitchfork bifurcations, and the stability boundaries were obtained via numerical continuation techniques \\cite{ERZ06} or in the instantaneous limit $\\tau=0$ analytically by using the characteristic equation of the system \\cite{YAN04}.  In the literature, symmetry-broken CLMs play a role in the stability analysis of symmetric CLMs, but are themselves not stable. \n\nA principal result of this paper is that \\emph{symmetry-broken} CLMs are shown to be stable for small delay and relatively high coupling between the lasers. This is demonstrated numerically in the middle and right columns of Fig.~\\ref{fg:CLMs} where optical field intensities and frequency spectra of two stable symmetry-broken states are plotted.  Due to the $\\mathbb{Z}_2$ symmetry of exchanging the two lasers, symmetry-broken CLMs always exist in pairs.  For purposes of display, a parameter set was chosen where a symmetric CLM is also stable as shown in the left column of Fig.~\\ref{fg:CLMs}, giving rise to a tri-stability in CLMs.  \n\nThe frequencies of the CLMs in the bottom panels of Fig.~\\ref{fg:CLMs}, can be analytically determined by plugging the ansatz~\\eqref{eq:1C} into the system of ODEs~\\eqref{eq:mLK} \\cite{ERZ06,YAN04}. The equations for $\\dot{E}_1$ and  $\\dot{E}_2$ yield  the following relation between the frequency $\\omega_A$ of the CLM and the phase difference $\\delta_A$ between the lasers \n\\begin{equation}\n\\label{eq:freqs}\n{\\frac{\\omega^{2}_A}{ \\kappa^{ 2} \\left( 1+ \\alpha^2 \\right)} } = \\sin^{2}\\left( {C_p} + \\omega_A\\tau + \\arctan\\alpha\\right) - \\sin^{ 2}\\delta_A\n\\end{equation}\nFor in-phase $\\omega_{in}=\\omega_A(\\delta_A=0)$ and anti-phase $\\omega_{an}=\\omega_A(\\delta_A=\\pi)$ CLMs, this immediately gives an implicit function for the frequency.  For the  symmetry-broken one-colour states one additionally needs to consider the fixed point solutions $\\dot{N}_{1,2}=0$ for the inversions. An implicit solution is calculated in \nAppendix~\\ref{ap:omegaA} for the system with delay time $\\tau$. A secant method is then used to obtain the numerical values appearing as dot-dashed vertical lines in Fig.~\\ref{fg:CLMs}.\n\n\\section{Two-Colour States}\\label{sec:2C}\n\n\\begin{figure}\n\\includegraphics[width=1.0\\linewidth,type=pdf,ext=.pdf,read=.pdf]{fig03}\n\\caption{\\label{fg:2CSB} Numerical solution of system~\\eqref{eq:mLK} for $\\tau=0.3, \\kappa=0.15, C_p=0.33 \\pi$ which is consistent with region 4 of Fig.~\\ref{fg:map}. The top left panel contains the magnitude of the electric fields, and the top right their inversions. The bottom panel shows the optical spectrum relative to the frequency of the free running lasers. The dot-dashed vertical lines are the frequencies calculated from ansatz~\\eqref{eq:2C}.}\n\\end{figure}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{fig04}\n\\caption{\\label{fg:BS} Transition between in-phase (left) and anti-phase (right) CLMs for $\\tau=0.2$, $\\kappa=0.4$.\nThe in-phase ($\\omega_{in} = -74.19~GHz$) and anti-phase ($\\omega_{an} = +49.46~GHz$) one-colour frequencies are obtained from Eq.~\\eqref{eq:freqs}.}\n\\end{figure*}\n\nIn this section we introduce two-colour states which are stable for small delay and moderate to low coupling strength between the lasers.  We stress that the two-colour states are induced by the coupling between the lasers alone, the uncoupled lasers are single-mode only.  Like the one-colour states, the two-colour states can either be symmetric or symmetry-broken with respect to the $\\mathbb{Z}_2$ symmetry of exchanging the two lasers. \n\nAn example of a stable symmetry-broken two-colour state is shown in Fig.~\\ref{fg:2CSB}.  Due to the $\\field{Z}_2$ symmetry of being able to exchange the two lasers, a twin two-colour state is also stable.  In the optical spectrum in the bottom panel of Fig.~\\ref{fg:2CSB} we see that there are indeed only two dominating frequencies $\\omega_A$ and $\\omega_B$ and both lasers lase at these two frequencies, but with unequal intensities.  In the time traces of the field amplitudes this gives rise to beating oscillations as shown in the upper left panel of Fig.~\\ref{fg:2CSB}.   The corresponding inversions shown in the upper right panel of Fig.~\\ref{fg:2CSB} also reflects the symmetry-broken nature of this state and in addition shows small oscillations at the beating frequency.\n\nAn example for symmetric two-colour states and their connection with symmetric CLMs is shown in Fig.~\\ref{fg:BS}.  Starting from the in-phase CLM in the left hand panel of Fig.~\\ref{fg:BS} and increasing the coupling phase $C_p$, we observe a torus bifurcation where the frequency of the anti-phase CLM is turned on. Increasing $C_p$ further smoothly transfers power from the in-phase mode to the anti-phase mode until a second torus bifurcation kills the in-phase CLM's frequency and only the anti-phase CLM remains.  Frequency spectrum snapshots as the parameter $C_p$ is changed are shown from left to right in Fig.~\\ref{fg:BS}.  We note that symmetric two-colour states exist only over very small ranges of the coupling phase $C_p$, however we will show in Sect.~\\ref{sec:map} that they are crucial for the overall understanding of the bifurcation structure in closely coupled single mode lasers. The symmetric two-colour states can be interpreted as beating between symmetric CLMs. This is similar to the beating \nbetween delay-created external cavity modes for a single laser with mirror \\cite{ERN00}.\n\nA useful ansatz~\\cite{ROG03} which approximates the dynamics of symmetric and symmetry-broken two-colour states is given by\n\\begin{equation}\n\\label{eq:2C}\n\\begin{aligned}\nE_1(t)=A_1 e^{i\\,\\omega_A\\,t}& + B_1 e^{i\\,\\omega_B\\,t},  &N_1(t)= N_1^c,\\\\\nE_2(t)=A_2 e^{i\\,\\omega_A\\,t}& e^{i\\,\\delta_A} + B_2 e^{i\\,\\omega_B\\,t} e^{i\\,\\delta_B}, &N_2(t)= N_2^c,\n\\end{aligned}\n\\end{equation}\nwith real constants $A_{1,2}$, $B_{1,2}$, $N_{1,2}^{c}$, $\\omega_{A,B}$ and $\\delta_{A,B}$.  While this ansatz is a straightforward generalisation of the CLM ansatz \\eqref{eq:1C}, with a second frequency $\\omega_B$, we stress that in contrast to the CLM ansatz, equations~\\eqref{eq:2C} fulfil the original system~\\eqref{eq:mLK} only approximately.   The presence of two frequencies give rise to oscillations in the intensities of the electric fields of the form \n\\begin{equation}\\label{eq:l2norm}\n\\left\\vert E_{1}\\left(t\\right) \\right\\vert^2 = A_{1}^2 + B_{1}^2 + 2 A_{1} B_{1} \\cos\\left(\\left(\\omega_A -\\omega_B\\right) t\\right)\n\\end{equation}\nand similarly for $\\left|E_2\\right|^2$.   According to \\eqref{eq:mLK} this then also leads to oscillations in the population inversions, as we have seen in the upper left panel of Fig.~\\ref{fg:2CSB}, and thereby contradicts the assumptions of constant $N_{1,2}(t)= N_{1,2}^c$.  However, as the parameter $T$ is large, ansatz  \\eqref{eq:2C} is often well justified in practice, in particular if the beating frequency $\\omega_A - \\omega_B$ is also large.  In the same way as for CLMs, the ansatz~\\eqref{eq:2C} can be split into symmetric ($A_1=A_2$, $B_1=B_2$) and symmetry-broken solutions. \n\nIn the case of symmetric two-colour states, the frequencies  $\\omega_A$ and $\\omega_B$ correspond to the frequencies of in-phase and anti-phase CLMs and are obtained from \\eqref{eq:freqs}.  For symmetry-broken states the analytical calculation of  $\\omega_A$ and $\\omega_B$ is shown in Appendix~\\ref{ap:omegaA_omegaB}.\n\n\\section{Reduced Coordinate System}\\label{sec:coord}\n\nOur aim is to understand the bifurcation structure associated with the various one-colour states and two-colour states presented in the previous sections.  As we are mostly interested in the closely coupled limit, we now formally set $\\tau=0$.  System \\eqref{eq:mLK} then becomes a six dimensional system of ordinary differential equations.  \n\nThe system still possesses the  $S^1$ symmetry~\\eqref{eq:symm} which can be exploited in order to reduce the number of dimensions of the system.  One popular way of achieving this is by rewriting the system \\eqref{eq:mLK} with the dynamical variables $\\left(\\left| E_{1} \\right|,\\left| E_{2} \\right|,\\phi_D, \\phi_A, N_1, N_2 \\right)$ using $E_{1,2}=\\left| E_{1,2} \\right| e^{\\phi_A\\mp \\phi_D\/_2}$.  Then the dynamical variable for the absolute phase $\\phi_A$ decouples from the rest of the system and we are left with a five dimensional system. \n\n\\begin{figure}\n\\includegraphics[width=0.6\\linewidth]{fig05}\n\\caption{\\label{fg:singularity} Demonstration of the discontinuity of $\\phi_D$ at the origin. A small change in the electric field $E_1$ of the first laser can lead to a large change in the polar coordinate $\\phi_D$ to $\\phi_D+\\pi$.}\n\\end{figure}\n\nAlthough widely used, this approach is problematic, because the dynamical variable for the phase difference between the electric fields $\\phi_D$ is not well defined if either $E_1$ or $E_2$ vanishes. As a consequence, $\\phi_D$  can jump discontinuously as one of the electric fields goes through the origin.  This is schematically demonstrated in Fig.~\\ref{fg:singularity}.  To see this mathematically  consider the differential equation for $\\phi_{ D}$ which is given by\n\\begin{equation}\n\\begin{aligned}\n\\dot{\\phi}_{ D} = &\\alpha \\left(N_2 - N_1\\right)  \\\\\n& + \\kappa \\left[ \\frac{\\left|E_2\\right|}{ \\left|E_1\\right| } \\sin({ C_p} { -}  \\phi_{ D} ) - \\frac{\\left|E_1\\right| }{ \\left|E_2\\right| } \\sin({ C_p} { +}  \\phi_{ D} ) \\right] \\label{eq:phidot} \\nonumber     \n  \\end{aligned}\n\\end{equation}\nThe discontinuity in $\\phi_D$ manifests itself in the form of singularities at $\\left|E_{1\/2}\\right|=0$, which make it difficult to use numeric continuation software to explore the dynamical features of the system.\n\nIn order to avoid these singularities, we introduce a five dimensional coordinate system $(q_x,q_y,q_z,N_1,N_2)$, where the variables are defined via \n\\begin{subequations}\\label{eq:q}\n\\begin{eqnarray}\nq_x + i q_y &=& 2 E_1^* E_2 \\label{eq:qxy}\\\\\nq_z &=& \\vert E_1 \\vert^2 - \\vert E_2 \\vert^2 \n\\end{eqnarray}\n\\end{subequations}\nThe multiplication by 2 in \\eqref{eq:qxy} ensures that the Euclidean length of the q-vector equals the total intensity output of both lasers, ${R}=\\left( q_x^2 + q_y^2 + q_z^2 \\right)^\\frac{1}{2} = \\vert E_1 \\vert^2 + \\vert E_2 \\vert^2$.  The coordinates $q_x, q_y, q_y$ are analogous to the usual Poincar{\\'e} sphere representation of polarised light, and per definition do not depend on the absolute phase $\\phi_A$.   Therefore the new variables are invariant under the $S^1$ symmetry of the original system \\eqref{eq:symm}.  The $\\field{Z}_2$ symmetry  now operates as follows\n\\begin{equation}\n\\label{eq:symm_new}\n \\left(q_x,q_y,q_z,N_1,N_2\\right) \\rightarrow \\left(q_x,-q_y,-q_z,N_2,N_1\\right) \n\\end{equation}\n\nRewriting the system \\eqref{eq:mLK} for $\\tau=0$  in terms of the new coordinates~\\eqref{eq:q} yields the five dimensional system\n\\begin{equation}\\label{eq:qdot}\n\\begin{aligned}\n\\dot{q_x} &= q_x \\left( N_1 + N_2 \\right) + \\alpha q_y \\left( N_1 - N_2 \\right) + 2 {\\kappa} {R} \\cos({C_p}) \\\\\n\\dot{q_y} &= q_y \\left( N_1 + N_2 \\right) - \\alpha q_x \\left( N_1 - N_2 \\right) - 2 {\\kappa} q_z \\sin({C_p}) \\\\\n\\dot{q_z} &= q_z \\left( N_1 + N_2 \\right) + {R} \\left( N_1 - N_2 \\right) + 2 {\\kappa} q_y \\sin(C_p) \\\\\nT \\dot{N_1} &= P - N_1 - \\left( 1 + 2N_1 \\right) \\, \\left({R + q_z}\\right)\/2  \\\\\nT \\dot{N_2} &= P - N_2 - \\left( 1 + 2N_2 \\right) \\, \\left({R - q_z}\\right)\/2.\n\\end{aligned}\n\\end{equation}\n\nAs intended, there are now no singularities in the dynamical variables, and the dynamical equations are invariant under the $\\field{Z}_2$ symmetry operation \\eqref{eq:symm_new}.  It is this reduced system that will be used for bifurcation analysis in the next section.\n\n\\section{Bifurcation Diagram}\\label{sec:map}\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{fig06}\n\n\\caption{\\label{fg:map} Bifurcation diagram of system \\eqref{eq:qdot} in the $(C_p,\\kappa)$ parameter plane. The bifurcation lines separate parameter space into eight distinct dynamical regions.  The codimension one bifurcation lines are organised by a set of codimension two points.  The labelling used for the dynamical regions and for the codimension two points are explained in the left and right legends below the main graph.}\n\\end{figure}\n\n\nIn this section the bifurcation structure at $\\tau =0$ is explored using the 5 dimensional model \\eqref{eq:qdot}. In the reduced coordinate system \\eqref{eq:q}, one-colour states become fixed point solutions and the two-colour states become limit cycles.  This reduction in complexity allows us use continuation software AUTO \\cite{DOE06} to obtain a comprehensive overview of the involved bifurcations. \n\nIn Fig.~\\ref{fg:map}, the bifurcation diagram with the two bifurcation parameters of coupling phase $C_p$ and coupling strength $\\kappa$ is presented. Due to the parameter symmetry\n\\begin{equation}\\label{eq:psymm}\n\\left( q_x, q_y, {C_p} \\right) \\rightarrow \\left( -q_x, -q_y, {C_p}+\\pi \\right) \n\\end{equation}\nit is sufficient to consider the parameter range of $C_p$ in the interval $[0,1\\pi)$ only.  The (codimension one) bifurcation lines in the diagram separate the parameter space into eight distinct regions.  Only bifurcations which affect stable dynamical states are plotted.  In region 1 and 2 in-phase one-colour states and anti-phase one-colour states are stable respectively. For the new coordinates \\eqref{eq:q}, symmetric one colour states are confined to  the $q_x$ axis ($q_y=0; q_z=0$), with $q_x>0$ in the in-phase case. \n\nAt very high coupling strengths ($\\kappa>PH1$), the in-phase region 1 and the anti-phase region 2 are separated by two supercritical Hopf bifurcations in close proximity as shown in the lower right panel of Fig.~\\ref{fg:cuts}. Between these two Hopf bifurcations a stable limit cycle of low amplitude and high frequency exists which corresponds to the symmetric two-colour dynamics seen in Fig.~\\ref{fg:BS} of Sect.~\\ref{sec:2C}, where the torus bifurcations of the original system \\eqref{eq:mLK} have become the Hopf bifurcations in the reduced system. In Fig.~\\ref{fg:map} these two vertical Hopf bifurcation lines appear as a single line at the scale of the diagram with the tiny region 3 of symmetric two-colour states nestled between them. In addition symmetry-broken one-colour states ($q_y\\neq 0; q_z \\neq 0$) are stable in region 6 which is accessible from the symmetric one-colour states via a supercritical bifurcation at the top of the blue pitchfork line between the SNP and PH1 points. \n\n\\begin{figure}\n\\includegraphics[width=1.0\\linewidth]{fig07}\n\\caption{\\label{fg:cuts} Several cuts of constant $\\kappa$ as indicated across Fig.~\\ref{fg:map} are shown. The top panels and lower-left panel show bifurcation diagrams of maximum $N_{1,2}$ versus $C_p$. Solid lines indicate stable states and dashed lines unstable states. Symmetric in-phase and anti-phase one-colour states are red and blue respectively. Symmetry-broken one-colour states appear as purple lines whilst symmetry-broken limit cycles are orange. Green lines indicate symmetric limit cycles. Hopf, pitchfork, pitchfork of limit cycles, saddle-node, saddle-node of limit cycles are denoted by H, P, PLC, SN, SNLC. Point B contains several bifurcations, see main text. The lower-left panel shows a 3-dimensional graph in q-vector space~\\eqref{eq:q} showing symmetric limit cycles (green) between in-phase (red) and anti-phase (blue) one-colour states. The orange ring marks the pitchfork of limit cycles. The $C_p$ range corresponds to the same as right-most inset of Fig.~\\ref{fg:cut}. \nSymmetry-broken \nlimit cycles \nnot drawn.}\n\\end{figure}\n\n\n\n\n\\begin{figure}\n\\includegraphics[width=0.9\\linewidth]{fig08}\n\\caption{\\label{fg:blowup} Sketch of bifurcation structure in vicinity of the pitchfork-Hopf codimension two point, labelled PH1 in Fig.~\\ref{fg:map}. Subscripts ${in}$, ${an}$ and ${sb}$ refer to bifurcations acting on in-phase, anti-phase and symmetry-broken states respectively.  The solid bifurcation lines affect stable states, while the dashed lines only affect unstable states.  }\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=1.0\\linewidth]{fig09}\n\\caption{\\label{fg:cut} An inversion cut at $\\kappa=0.1$ across the entire $1\\pi$ range of $C_p$. Colours and labels are as in Fig.~\\ref{fg:cuts}. T represents a torus bifurcation. Insets are blow-ups of the regions indicated.}\n\\end{figure}\n\n\nA sketch of the situation in the vicinity of the PH1 point is provided in the left part of Fig.~\\ref{fg:blowup}, where one of these supercritical Hopf bifurcations and the supercritical pitchfork bifurcation intersect. The second supercritical Hopf bifurcation although nearby does not play a role. The PH1 point is a pitchfork Hopf codimension two bifurcation and analytical expressions are obtained in Appendix~\\ref{ap:location-ph1-point}. In order to classify this bifurcation, we note that the pitchfork Hopf and the Hopf-Hopf bifurcation share the same reduced normal form. PH1 corresponds to a Hopf-Hopf ``simple'' case III of~\\cite{KUZ04} or equivalently case II\/III of~\\cite{GUC83}. Figure~\\ref{fg:blowup} illustrates that this pitchfork Hopf point is responsible for the symmetry-broken two colour dynamics which originates between a pitchfork of limit cycles (PLC) and a supercritical Hopf bifurcation ($2H_{sb}$). Symmetry-broken two-colour states are stable in region 4 of Fig.~\\ref{fg:map}. \n\nThe point labelled SNP is a saddle-node pitchfork codimension two bifurcation. It shares the same reduced normal form as a generalised (Bautin) Hopf bifurcation without rotation. At this point the supercritical pitchfork bifurcation becomes subcritical, thus spawning the tri-stable region 7, where both symmetry-broken and symmetric one-colour states are stable. An integration with noise of the original system~\\eqref{eq:mLK} showing the three stable one-colour states was presented in Fig.~\\ref{fg:CLMs} of section~\\ref{sec:CLM}. \n\nThe third corner of symmetry-broken one-colour states is closed by a saddle-node Hopf (also known as fold Hopf) codimension two bifurcation labelled SNH in  Fig.~\\ref{fg:map}. At this point a saddle-node and a Hopf bifurcation meet tangentially and also a torus bifurcation line emerges from this point.  The unfolding of saddle-node Hopf bifurcations have been studied extensively in the literature, and we identify the SNH point with the third case in~\\cite{KUZ04}\\footnote{\\cite{KUZ04} does not enumerate the cases. Here we refer to the case of Figure 8.16 in the 3rd version of the book.}. These three codimension two bifurcations organise the multi-stabilities.\n\nNext we provide a series of cuts of constant coupling strength $\\kappa$ across the diagram (Fig.\\ref{fg:map}) to further explain the dynamics in each of the regions. In Fig.~\\ref{fg:cut}, a plot at $\\kappa=0.1$ of the inversions $N_{1,2}$ across the entire $1\\pi$ span of the coupling phase $C_p$ is presented.  This plot illustrates that symmetry-broken two-colour states are stable over a large range of $C_p$. The bottom inset highlights region 5 which is relatively small at $\\kappa=0.1$. In this region a symmetric one-colour is stable in addition to the symmetry-broken two-colour states. In the right-most inset a blow-up of region 3 is provided. The limit cycle born after a supercritical Hopf bifurcation is still invariant under $\\field{Z}_2$ symmetry  \\eqref{eq:symm_new} and therefore obeys \n\\begin{equation}\n\\label{eq:symm_lc}\n \\left(q_x,q_y,q_z,N_1,N_2\\right)\\left(t\\right) = \\left(q_x,-q_y,-q_z,N_2,N_1\\right)\\left(t+\\frac{\\tau_l}{2}\\right),\n\\end{equation}\nwhere $\\tau_l$ is the period of the limit cycle.  In a projection to the $(q_x,q_y,q_z)$ coordinates, the limit cycles therefore form rings around the $q_x$ axis, as shown in the lower left panel of Fig.~\\ref{fg:cuts}. These limit cycles corresponds to the symmetric two-colour states. They lose stability at a pitchfork of limit cycles bifurcation and create the large regions 4 and 5 of symmetry-broken two-colour states. In the bifurcation diagram (Fig.~\\ref{fg:map}) the GH point is a codimension two generalised Hopf (Bautin) bifurcation. As shown in top right panel of Fig.~\\ref{fg:cuts},  the scenario remains very similar for $\\kappa$ below this point. However here the supercritical Hopf bifurcation becomes subcritical and a saddle-node of limit cycles bifurcation occurs.  Finally in the top left of Fig.~\\ref{fg:cuts}, we look at an inversion cut of much higher $\\kappa$ just beneath the SNP \npoint. Here the supercritical Hopf and supercritical pitchfork of limit cycles bifurcations occur in close proximity at the point B. In contrast to Fig.~\\ref{fg:cut}, symmetry-broken one-colour states are stable limited on the left by a saddle-node bifurcation and on the right by a supercritical Hopf bifurcation. Region 7 of Fig.~\\ref{fg:map} corresponds the small area of hysteresis between these symmetry-broken one-colour and symmetric one-colour states. \n\nThe remaining two codimension two points plotted in Fig.~\\ref{fg:map} are a period-doubling torus (PDT) and a second pitchfork Hopf (PH2) bifurcation. PH2 gives rise to a tiny triangular region of bi-stability between symmetric one-colour states which occurs below the green horizontal Hopf as shown in~\\cite{YAN04}. We identify this point as being equivalent to the Hopf-Hopf ``difficult'' case VI of~\\cite{KUZ04} which is the same as case VIa of~\\cite{GUC83}. This completes the bifurcation picture for moderate to high coupling strength. We note that different dynamics occurs for lower coupling strengths, in particular region 8 of Fig.~\\ref{fg:map} which is outside the scope of the current paper.\n\n\\section{All-Optical Switching}\\label{sec:opticalswitch}\n\nThe realisation of stable, fast and scaleable all optical memory elements would significantly increase the scope of photonic devices and has therefore attracted considerable interest in the laser community. Examples of optical memory designs include hetero-structure photonic crystal lasers~\\cite{CHE11}, coupled micro-ring lasers~\\cite{HIL04} and dual mode semiconductor lasers with delayed feedback~\\cite{BRA12}. In this section, we propose a conceptually {\\em simple} all optical memory element on the basis of two closely coupled single-mode lasers.\n\n\\begin{figure}\n\\includegraphics[width=1.0\\linewidth,type=pdf,ext=.pdf,read=.pdf]{fig10}\n\\caption{\\label{fg:scheme2} Sketch showing the two laser coupling schemes considered as optical switches.}\n\\end{figure}\n\nMemory units require at least two stable states and a mechanism to switch between them. In the previous sections, a number of different parameter regions displaying multi-stabilities were discovered for closely coupled lasers. In particular, in section~\\ref{sec:2C} corresponding to region 4 of Fig.~\\ref{fg:map} we established the presence of symmetry-broken two-colour states. Switching between them is achieved via optical injection from two master lasers as shown schematically in Fig.~\\ref{fg:scheme2}(a). Initially the lasers are in a symmetry-broken state with an optical spectrum displayed in the top left panel of Fig.~\\ref{fg:switch_2c}. The oscillations in the magnitude of the electric fields, shown in the graph beneath, are due to the beatings between the two optical frequencies (Sec.~\\ref{sec:2C}). Between 5~ns and 6~ns an optical pulse from a master laser is injected into laser 1 which causes laser 1 to lock to the external frequency and for its carrier density to significantly reduce. This is shown in \nthe top panel of Fig~\\ref{fg:os1speed_N}, where we denote the point ``A'' at approximately $50$~ps  when the carrier density of laser 1 is pushed beneath that of laser 2.  After the pulse is switched off at 6~ns, a transient of about 3~ns is visible in the bottom panel of Fig.~\\ref{fg:switch_2c} for the coupled lasers to completely settle to the twin symmetry-broken two-colour state which corresponds to the two lasers having been exchanged (top centre panel of Fig.~\\ref{fg:switch_2c}).  The contrast ratio between the two stable states is relatively large.  For example, the intensity of the colour with higher frequency changes by a factor of more than four. \n\n\\begin{figure}\n\\includegraphics[width=1.0\\linewidth,type=pdf,ext=.pdf,read=.pdf]{fig11}\n\\caption{\\label{fg:switch_2c} Optical switching between symmetry-broken two-colour states for a two laser configuration as outlined in Fig.~\\ref{fg:scheme2}(a) for parameters $\\tau=0.2$, $\\kappa=0.1$, $C_p=0.35\\pi$. The large bottom panel shows the magnitude of the electric field in both lasers. Laser 1 in red and laser 2 mirrored underneath in blue. The central black line dividing the two shows the injection strength and duration of the master lasers. At 5~ns a pulse of 1~ns is injected from a master laser into laser 1. At 13~ns a pulse of 50~ps from a master laser is injected into laser 2. The top three panels show the corresponding frequency states. The black arrow indicates the frequency of injected light which is 19.5~GHz larger than the lasers' free running frequency.}\n\\end{figure}\n\nOne important aspect from an application point of view is the ability to switch between two states with a very short external optical pulse and we therefore define the \\emph{write time} as the minimum pulse duration to ensure switching.  In order to demonstrate that a shorter pulse is sufficient to trigger the switch, we inject at 13~ns a pulse of 50ps duration into laser 2.  In the lower panel Fig~\\ref{fg:os1speed_N}, we see that $N_2$ is indeed pushed below $N_1$ during the pulse. After the pulse is switched off, both  $N_1$ and $N_2$ oscillate strongly but $N_2$ remains consistently below $N_1$. Therefore the write time is determined by the minimum time needed to reduce the carrier density of one laser below that of the other laser which for our setup is approximately 50~ps. We have thus demonstrated a basic mechanism for an all-optical memory element with a large contrast ratio, a short write time and a low coupling strength between the lasers. \n\n\\begin{figure}\n\\includegraphics[width=1.0\\linewidth,type=pdf,ext=.pdf,read=.pdf]{fig12}\n\\caption{\\label{fg:os1speed_N} Diagrams showing the dynamics of the carrier densities for both lasers during the first switching event (top panel) and second switching event (bottom panel) of Fig.~\\ref{fg:switch_2c}. Black vertical lines mark the time when the external injection is turned on and then off. Point labelled ``A'' indicates where the carrier density of laser~1 becomes less than laser~2.}\n\\end{figure}\n\nFor certain applications it may be desirable to inject into only one of the two lasers as in Fig.~\\ref{fg:scheme2}(b). To achieve this, we choose parameters consistent with region 5 of Fig.~\\ref{fg:map}. In this region, in addition to the two-colour symmetry-broken states of the previous paragraph, a one-colour symmetric state is also stable. In Fig.~\\ref{fg:switch_1c2c}, these states form the basis for the optical memory unit. Initially the two lasers start in a degenerate state with the same amplitude and the same single frequency in both lasers. Using a pulse with a positive detuning relative to the central frequency ($+83$~GHz in the case of Fig.~\\ref{fg:switch_1c2c}) the symmetric state of the two lasers symmetry breaks to a two-colour state. As the number, position and intensities of frequencies change, the two states can be easily distinguished which is desirable from an application point of view. \n\n\\begin{figure}\n\\includegraphics[width=1.0\\linewidth,type=pdf,ext=.pdf,read=.pdf]{fig13}\n\\caption{\\label{fg:switch_1c2c} Optical switching between a symmetric one-colour state and a symmetry-broken two-colour state as sketched in Fig.~\\ref{fg:scheme2}(b) for parameters $\\tau=0.2$, $\\kappa=0.2$, $C_p=0.25\\pi$. Panel layout as in Fig.~\\ref{fg:switch_2c}. At 5~ns a pulse of 1~ns with frequency offset of +83~GHz is injected from the master laser into laser 1. At 20~ns a pulse of 1~ns with frequency offset of -32~GHz is injected from the master laser into laser 1.}\n\\end{figure}\n\nTo discuss how quickly the two states can be ascertained, we introduce the \\emph{read time} which is the minimum duration needed to differentiate optically between the two states after the injection has turned off. In Fig.~\\ref{fg:os1speed_N} for the first optical switch after the external injection was removed at 6~ns and 13.05~ns, large amplitude oscillations in the carrier densities at a frequency consistent with the relaxation oscillations for the coupled system (5.7~GHz) ensue. These oscillations are directly related to the length of transient observed in the optical fields for the system to completely settle to the twin symmetry-broken two-colour state. We stress that one does not need to wait for all relaxation oscillations in the system to die out before reading. Small amplitude high frequency oscillations (40~GHz) which are due to the beatings between the two optical colours (see Sec.~\\ref{sec:2C}) are observable before each switching event and are discernible in Fig.~\\ref{fg:os1speed_N} as small \ndeformities in the larger amplitude oscillations within $1$~ns of the external injection being turned off. In Fig.~\\ref{fg:os2_freq}, two Fourier modes consistent with the peak frequencies of $-32$~GHz and $-83$~GHz are traced out for each switching event for the second optical switch. All other frequencies are filtered out. During the first injection episode from $5$~ns to $6$~ns, the frequency centred at $-83$~GHz is turned off within $400$~ps. We also observe that after injection, the frequency at $-32$~GHz has reached a stable intensity within 100~ps. To switch back to the symmetric one-colour state, a pulse with negative detuning (-38~GHz in the case of Fig.~\\ref{fg:switch_1c2c}) is injected into laser 1. In the lower panel of Fig.~\\ref{fg:os2_freq} the frequency component at $-83$~GHz relaxes to its equilibrium value within 1.5~ns. The frequency at $-34$~GHz is completely off within $100$~ps. This constitutes a robust memory element where we inject into one laser only. An external injection pulse with \npositive detuning causes the coupled lasers to enter a symmetry-broken state whilst negative detuning causes the coupled lasers to enter a symmetric state. Compared to the switching scenario in the previous paragraph (Fig.~\\ref{fg:switch_2c}) the two lasers are more strongly coupled but the contrast ratio between the two states is greater.\n\n\\begin{figure}\n\\includegraphics[width=1.0\\linewidth,type=pdf,ext=.pdf,read=.pdf]{fig14}\n\\caption{\\label{fg:os2_freq} Intensity plot tracing frequencies $-32$~GHz and $-83$~GHz for the uninjected laser 2 during the first switching event (top panel) and second switching event (bottom panel) of Fig.~\\ref{fg:switch_1c2c}. A Fourier transform using a Hann window is executed every $100$~ps.}\n\\end{figure}\n\nIn this section we have shown that two closely coupled identical lasers can operate as an optical memory element and have provided two specific examples. The high degree of multi-stability discovered in the previous sections enables many other designs. As the multi-stabilities persist in the limit of $\\tau\\rightarrow0$, the distance between the lasers can be reduced as far as is technologically possible. Memory units of this kind are therefore open to miniaturisation and allow integrability. For the optical switch of Fig.~\\ref{fg:switch_2c}, a fast write time of $50$~ps was demonstrated in the second switching event and was discussed with reference to the carrier density in Fig.~\\ref{fg:os1speed_N}. A significant speed increase on this number may be possible by increasing the injection strength of the external pulse or a fine tuning of the coupled lasers' parameters. For the optical switch of Fig.~\\ref{fg:switch_1c2c}, we showed via Fig.~\\ref{fg:os2_freq} that it is possible to distinguish optically between \nthe states  within a read time of $100$~ps after the external injection was turned off. Again significant improvements on this number may be possible. Indeed the larger the frequency separation between states the shorter the time needed to differentiate between them. Therefore choosing parameters consistent with larger frequency separations which normally occur at higher coupling strength may substantially decrease the read time. We conclude that closely coupled lasers offer a promising approach for the realisation of scaleable and fast all optical memory elements.\n\n\n\n\n\n\\section{Conclusion}\n\nIn this paper, a comprehensive overview of the bifurcation scenarios in a system of two closely coupled single-mode lasers is provided.  For moderate to high coupling strength the four characteristic stable states are symmetric one-colour, symmetry-broken one-colour, symmetric two-colour and symmetry-broken two-colour states.  We introduce a new coordinate representation which accounts for the $S^1$ symmetry in the system without creating unnecessary singularities. This allows us to study the bifurcation structure of this system using conventional numerical continuation techniques.  \n\nOur results show that the bifurcations between the various stable states are organised by a number of codimension two bifurcation points which are identified with reference to the literature.  In particular it is found that the interplay between a pitchfork Hopf, a saddle-node Hopf and a saddle-node pitchfork codimension two points give rise to regions of multi-stabilities.  Detailed knowledge of the bifurcations and their boundaries of closely coupled lasers may open several technological applications. We propose two all-optical switch candidates. \n\n\\begin{acknowledgments}\nThis work was supported by Science Foundation Ireland under grant number 09\/SIRG\/I1615. We thank P. Harnedy and E. P. O'Reilly for supporting discussions. \n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn a previous study \\cite{dashti2018implied}, we introduced the ratio of realized variance to implied variance, represented by squared volatility indices VIX or VXO, as a measure of their correlations. We pointed out that the realized variance is calculated for trading dates while the implied variance covers every day, so one of them needs rescaled for a proper comparison. We argued that studying the distribution of the ratios produces a deeper insight into these correlations than a simple regression analysis \\cite{christensen1998relation}, which arrives at an obvious conclusion that VIX\/VXO are a slightly better predictor of the future realized volatility (RV) than the past RV since it builds on the latter with the benefit of additional information. \n\nIn \\cite{dashti2018implied} we concluded that the ratio of the actual realized variance $RV^2$ to $VIX^2$ and $VXO^2$, that is to its predicted values, was best described by the fat-tailed inverse Gamma (IGa) distribution and its inverse by the Gamma (Ga) distribution.We speculated that this is due to unanticipated spikes of realized volatility. In this paper we show that a Beta Prime (BP) distribution provides a better fit both for the ratio and its inverse. For the former, the exponent of the  power dependence for small values of the ratio is very large, which mimics exponential behavior of IGa. For the latter, the exponent of the power-law tails is very large, which mimics exponential decay of Ga.\n\nWe also concluded in \\cite{dashti2018implied} that the ratio of $RV^2$ of the preceding month to $VIX^2$ and $VXO^2$ was best described by the lognormal (LN) distribution. Its inverse was also best described by the LN with similar parameters. We argued that while the spikes in the past realized volatility lead to spikes of implied volatility, there was enough uncertainty for the ratio to have heavy tails. In this paper we show that BP, with similar parameters for the ratio and its inverse, provides a better fit for both. The exponents of the power law for both small values and the fat tails are large, so that BP mimics the LN behavior. Additionally, for both BP and LN the distribution of the inverse variable is BP and LN respectively as well. \n\nAs a reminder to the reader, PDF of BP distribution is given by\n\\begin{equation}\nBP(p, q, \\beta; x) = \\frac{(1+\\frac{x}{\\beta})^{-p-q}(\\frac{x}{\\beta})^{p-1}}{\\beta \\space B[p,q]}\n\\label{BetaPrimePDF}\n\\end{equation}\nwhere $\\beta$ is a scale parameter, $p$ and $q$ are shape parameters and $B[p,q]$ is the Beta function; $BP \\propto x^{p-1}$ for $x \\ll \\beta$ and $BP \\propto x^{-q-1}$ for $x \\gg \\beta$.\n\nThis paper is organized as follows. In Section \\ref{EO} we summarize empirical observations regarding distributions of $RV^2$ and volatility indices $VIX^2$ and $VXO^2$ and their ratios. In Section \\ref{RV} we show the results of statistical fits of the ratios. In Section \\ref{Correlations} we summarize correlations between the quantities and their ratios.\n\n\\section{Empirical Observations \\label{EO}}\n\nIn \\cite{dashti2018implied} we presented empirical distributions (PDF) of $RV^2$ vis-a-vis $VIX^2$ and $VXO^2$, as well as the ratios, and in \\cite{dashti2018realized} we will have described their fits. We observe the following features, in agreement with \\cite{russon2017nonlinear} (see also \\cite{vodenska2013understanding,kownatzki2016howgood}):\n\n\\begin{itemize}\n  \\item $VIX^2$ and $VXO^2$ have lower high-volatility probabilities relative to $RV^2$, including shorter fat tails, indicating that volatility indices do not predict accurately large values of RV, including the largest volatility spikes. In other word, volatility indices underestimate future large RV. \n   \\item $VIX^2$ and $VXO^2$ have higher mid-volatility probabilities relative to $RV^2$, indicating that volatility indices overestimate future mid-level RV.\n  \\item $VIX^2$ and $VXO^2$ have lower low-volatility probabilities relative to $RV^2$, indicating that volatility indices underestimate future low RV.\n\\end{itemize}\n\nFor the distributions of the ratios $RV^2\/VIX^2$, $RV^2\/VXO^2$, it is important to notice that, since realized and implied volatilities are correlated, we cannot construct them simply as the quotient distributions of two independent variable. We observe the following:\n\n\\subsection{Predicted Month}\n\nFor the month predicted by the volatility indices\n\n\\begin{itemize}\n  \\item The distributions have fat tails, indicating again that VIX and VXO underestimate future values of RV, in particular volatility spikes.\n  \\item Very small ratios are suppressed, as manifested by a very large power exponent, indicating that it is rare that RV is considerably smaller than the one predicted by the volatility indices. \n  \\item The tail exponents of the ratio distributions is larger than that of either $RV^2$ or $VIX^2$ and $VXO^2$, pointing to that for the $RV^2$ values taken from the tails, the values of $VIX^2$, $VXO^2$ are also more likely to come from the tails. \n\\end{itemize}\n\n\\subsection{Preeceding Month}\n\nFor RV of the preceding month:\n\n\\begin{itemize}\n  \\item The tails of the distributions are much shorter than those for the predicted month, reflecting the fact that volatility indices account for past RV.\n  \\item The tail exponents of the distributions are almost identical to those of their inverse, $VIX^2\/RV^2$ and $VXO^2\/RV^2$ distributions, indicating, as above, strong correlations. \n\\end{itemize}\n\nFor the ratio distribution of $RV^2$ of the predicted (next) month to $RV^2$ of the preceding month (see below), we observe that \n\n\\begin{itemize}\n  \\item The exponent of the fat tail is smaller than those of the $RV^2\/VIX^2$ and $RV^2\/VXO^2$ distributions, that is the tails are longer. \n  \\item The power-law exponent at very small ratios is much smaller for this distribution than for $RV^2\/VIX^2$ and $RV^2\/VXO^2$, that is those ratios are far less suppressed. \n\\end{itemize}\n\nBy both measures, VIX and VXO are better predictors of the future RV than the past RV. \n\n\\section{Statistical Fits of Ratio Distributions \\label{RV}} \n\nBelow, figures show the plots of the ratios and their distribution fits and tables contain parameters of the distribution and KS statistics. The two novel elements here, relative to \\cite{dashti2018implied}, is the inclusion of BP in the fits of $RV^2\/VIX^2$ and $RV^2\/VXO^2$ distributions and the fits of the ratio distribution of $RV^2$ of the predicted (next) month to $RV^2$ of the preceding month (and its inverse).\n\n\\clearpage\n\\subsection{Predicted Month}\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width = 0.49 \\textwidth]{.\/RVOverVIXList19902016}\n\\includegraphics[width = 0.49 \\textwidth]{.\/histogramRVOverVIX19902016}\n\\end{tabular}\n\\caption{$\\mathrm{RV}^2 \/ \\mathrm{VIX}^2$, from Jan 2nd, 1990 to Dec 30th, 2016.}\n\\label{RVOverVIXListSRV2OverVIX21990}\n\\end{figure}\n\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width = 0.49 \\textwidth]{.\/VIXOverRVList19902016}\n\\includegraphics[width = 0.49 \\textwidth]{.\/histogramVIXOverRV19902016}\n\\end{tabular}\n\\caption{$ \\mathrm{VIX}^2 \/ \\mathrm{RV}^2$, from Jan 2nd, 1990 to Dec 30th, 2016.}\n\\label{VIXOverRVListSVIX2OverRV21990nn}\n\\end{figure}\n\n\\begin{table}[!htb]\n\\caption{MLE results for ``$\\mathrm{RV}^2 \/ \\mathrm{VIX}^2$\" and ``$\\mathrm{VIX}^2 \/ \\mathrm{RV}^2$\"}\n\\label{MLESRV2OverVIX21990nn}\n\\begin{minipage}{0.5\\textwidth}\n\\begin{center}\n\\begin{tabular}{ c c c} \n\\multicolumn{2}{c}{} \\\\\n\\hline\n             &       parameters &          KS test  \\\\\n\\hline\nNormal & N(          1.0000,           0.9067) &           0.1940 \\\\\n\\hline\nLogNormal & LN(         -0.2027,           0.5867) &           0.0446 \\\\\n\\hline\nIGa & IGa(          3.3595,           2.3466) &           0.0246 \\\\\n\\hline\nGamma & Gamma(          2.6219,           0.3814) &           0.0978 \\\\\n\\hline\nWeibull & Weibul(          1.1124,           1.4009) &           0.1224 \\\\\n\\hline\nIG & IG(          1.0000,           2.3168) &           0.0607 \\\\\n\\hline\nBP & BP(         27.2279,           3.8055,           0.1014) &           0.0198 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\\begin{minipage}{.5\\textwidth}\n\\begin{center}\n\\begin{tabular}{ c c c} \n\\multicolumn{2}{c}{} \\\\\n\\hline\n             &       parameters &          KS test  \\\\\n\\hline\nNormal & N(          1.0000,           0.5626) &           0.0972 \\\\\n\\hline\nLogNormal & LN(         -0.1562,           0.5867) &           0.0446 \\\\\n\\hline\nIGa & IGa(          2.6219,           1.8314) &           0.0978 \\\\\n\\hline\nGamma & Gamma(          3.3595,           0.2977) &           0.0246 \\\\\n\\hline\nWeibull & Weibul(          1.1306,           1.8882) &           0.0500 \\\\\n\\hline\nIG & IG(          1.0000,           2.3168) &           0.0734 \\\\\n\\hline\nBP & BP(          3.8055,          27.2279,           6.8913) &           0.0198 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n  \\end{minipage}\n\n\\end{table}\n\n\n\n\\clearpage\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width = 0.49 \\textwidth]{.\/RVOverVXOList19902016}\n\\includegraphics[width = 0.49 \\textwidth]{.\/histogramRVOverVXO19902016}\n\\end{tabular}\n\\caption{$\\mathrm{RV}^2 \/ \\mathrm{VXO}^2$, Jan 2nd, 1990 to Dec 30th, 2016.}\n\\label{RVOverVXOListSRV2OverVXO22016}\n\\end{figure}\n\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width = 0.49 \\textwidth]{.\/VXOOverRVList19902016}\n\\includegraphics[width = 0.49 \\textwidth]{.\/histogramVXOOverRV19902016}\n\\end{tabular}\n\\caption{$ \\mathrm{VXO}^2 \/ \\mathrm{RV}^2$, from Jan 2nd, 1990 to Dec 30th, 2016}\n\\label{VXOOverRVListSVXO2OverRV22016n}\n\\end{figure}\n\n\\begin{table}[!htb]\n\\caption{MLE results for ``$\\mathrm{RV}^2 \/ \\mathrm{VXO}^2$\" and ``$\\mathrm{VXO}^2 \/ \\mathrm{RV}^2$\"}\n\\label{MLESRV2OverVXO22016n}\n\\begin{minipage}{0.5\\textwidth}\n\\begin{center}\n\\begin{tabular}{ c c c} \n\\multicolumn{2}{c}{} \\\\\n\\hline\n             &       parameters &          KS test  \\\\\n\\hline\nNormal & N(          1.0000,           0.8747) &           0.1910 \\\\\n\\hline\nLogNormal & LN(         -0.1973,           0.5795) &           0.0449 \\\\\n\\hline\nIGa & IGa(          3.4629,           2.4438) &           0.0224 \\\\\n\\hline\nGamma & Gamma(          2.6897,           0.3718) &           0.0971 \\\\\n\\hline\nWeibull & Weibul(          1.1150,           1.4256) &           0.1230 \\\\\n\\hline\nIG & IG(          1.0000,           2.3981) &           0.0611 \\\\\n\\hline\nBP & BP(         47.6001,           3.7157,           0.0563) &              0.0177 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\\begin{minipage}{.5\\textwidth}\n\\begin{center}\n\\begin{tabular}{ c c c} \n\\multicolumn{2}{c}{} \\\\\n\\hline\n             &       parameters &          KS test  \\\\\n\\hline\nNormal & N(          1.0000,           0.5467) &           0.0925 \\\\\n\\hline\nLogNormal & LN(         -0.1513,           0.5795) &           0.0449 \\\\\n\\hline\nIGa & IGa(          2.6897,           1.8982) &           0.0971 \\\\\n\\hline\nGamma & Gamma(          3.4629,           0.2888) &           0.0224 \\\\\n\\hline\nWeibull & Weibul(          1.1308,           1.9374) &           0.0499 \\\\\n\\hline\nIG & IG(          1.0000,           2.3981) &           0.0729 \\\\\n\\hline\nBP & BP(          3.7157,          47.6002,          12.5409) &           0.0177 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n  \\end{minipage}\n\n\\end{table}\n\n\\clearpage\n\\subsection{Preceding Month}\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width = 0.49 \\textwidth]{.\/RVOverVIXList19902016}\n\\includegraphics[width = 0.49 \\textwidth]{.\/histogramRVOverVIX19902016}\n\\end{tabular}\n\\caption{$\\mathrm{RV}^2 \/ \\mathrm{VIX}^2$, from Jan 2nd, 1990 to Dec 30th, 2016.}\n\\label{RVOverVIXListSRV2OverVIX21990}\n\\end{figure}\n\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width = 0.49 \\textwidth]{.\/VIXOverRVList19902016}\n\\includegraphics[width = 0.49 \\textwidth]{.\/histogramVIXOverRV19902016}\n\\end{tabular}\n\\caption{$ \\mathrm{VIX}^2 \/ \\mathrm{RV}^2$, from Jan 2nd, 1990 to Dec 30th, 2016.}\n\\label{VIXOverRVListSVIX2OverRV21990nn}\n\\end{figure}\n\n\\begin{table}[!htb]\n\\caption{MLE results for ``$\\mathrm{RV}^2 \/ \\mathrm{VIX}^2$\" and ``$\\mathrm{VIX}^2 \/ \\mathrm{RV}^2$\"}\n\\label{MLESRV2OverVIX21990nn}\n\\begin{minipage}{0.5\\textwidth}\n\\begin{center}\n\\begin{tabular}{ c c c} \n\\multicolumn{2}{c}{} \\\\\n\\hline\n             &       parameters &          KS test  \\\\\n\\hline\nNormal & N(          1.0000,           0.4974) &           0.0992 \\\\\n\\hline\nLogNormal & LN(         -0.1099,           0.4689) &           0.0147 \\\\\n\\hline\nIGa & IGa(          4.6889,           3.7619) &           0.0431 \\\\\n\\hline\nGamma & Gamma(          4.7110,           0.2123) &           0.0381 \\\\\n\\hline\nWeibull & Weibul(          1.1325,           2.1250) &           0.0672 \\\\\n\\hline\nIG & IG(          1.0000,           4.0580) &           0.0215 \\\\\n\\hline\nBP & BP(          9.2230,           9.9855,           0.9742) &           0.0117 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\\begin{minipage}{.5\\textwidth}\n\\begin{center}\n\\begin{tabular}{ c c c} \n\\multicolumn{2}{c}{} \\\\\n\\hline\n             &       parameters &          KS test  \\\\\n\\hline\nNormal & N(          1.0000,           0.4999) &           0.1059 \\\\\n\\hline\nLogNormal & LN(         -0.1104,           0.4689) &           0.0147 \\\\\n\\hline\nIGa & IGa(          4.7110,           3.7796) &           0.0381 \\\\\n\\hline\nGamma & Gamma(          4.6889,           0.2133) &           0.0431 \\\\\n\\hline\nWeibull & Weibul(          1.1329,           2.1186) &           0.0751 \\\\\n\\hline\nIG & IG(          1.0000,           4.0580) &           0.0163 \\\\\n\\hline\nBP & BP(          9.9855,           9.2230,           0.8236) &           0.0117 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n  \\end{minipage}\n\n\\end{table}\n\n\n\n\\clearpage\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width = 0.49 \\textwidth]{.\/RVOverVXOList19902016}\n\\includegraphics[width = 0.49 \\textwidth]{.\/histogramRVOverVXO19902016}\n\\end{tabular}\n\\caption{$\\mathrm{RV}^2 \/ \\mathrm{VXO}^2$, Jan 2nd, 1990 to Dec 30th, 2016.}\n\\label{RVOverVXOListSRV2OverVXO22016}\n\\end{figure}\n\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width = 0.49 \\textwidth]{.\/VXOOverRVList19902016}\n\\includegraphics[width = 0.49 \\textwidth]{.\/histogramVXOOverRV19902016}\n\\end{tabular}\n\\caption{$ \\mathrm{VXO}^2 \/ \\mathrm{RV}^2$, from Jan 2nd, 1990 to Dec 30th, 2016}\n\\label{VXOOverRVListSVXO2OverRV22016n}\n\\end{figure}\n\n\\begin{table}[!htb]\n\\caption{MLE results for ``$\\mathrm{RV}^2 \/ \\mathrm{VXO}^2$\" and ``$\\mathrm{VXO}^2 \/ \\mathrm{RV}^2$\"}\n\\label{MLESRV2OverVXO22016n}\n\\begin{minipage}{0.5\\textwidth}\n\\begin{center}\n\\begin{tabular}{ c c c} \n\\multicolumn{2}{c}{} \\\\\n\\hline\n             &       parameters &          KS test  \\\\\n\\hline\nNormal & N(          1.0000,           0.4915) &           0.1064 \\\\\n\\hline\nLogNormal & LN(         -0.1041,           0.4539) &           0.0150 \\\\\n\\hline\nIGa & IGa(          5.0351,           4.0948) &           0.0331 \\\\\n\\hline\nGamma & Gamma(          4.9618,           0.2015) &           0.0454 \\\\\n\\hline\nWeibull & Weibul(          1.1316,           2.1383) &           0.0730 \\\\\n\\hline\nIG & IG(          1.0000,           4.3548) &           0.0203 \\\\\n\\hline\nBP & BP(         11.1694,           9.4027,           0.7520) &           0.0133 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\\begin{minipage}{.5\\textwidth}\n\\begin{center}\n\\begin{tabular}{ c c c} \n\\multicolumn{2}{c}{} \\\\\n\\hline\n             &       parameters &          KS test  \\\\\n\\hline\nNormal & N(          1.0000,           0.4768) &           0.0933 \\\\\n\\hline\nLogNormal & LN(         -0.1026,           0.4539) &           0.0150 \\\\\n\\hline\nIGa & IGa(          4.9618,           4.0352) &           0.0454 \\\\\n\\hline\nGamma & Gamma(          5.0351,           0.1986) &           0.0331 \\\\\n\\hline\nWeibull & Weibul(          1.1319,           2.2099) &           0.0689 \\\\\n\\hline\nIG & IG(          1.0000,           4.3548) &           0.0212 \\\\\n\\hline\nBP & BP(          9.4027,          11.1694,           1.0814) &           0.0133 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n  \\end{minipage}\n\n\\end{table}\n\n\\clearpage\n\\subsection{Ratio of Realized Variances of Two Adjacent Months}\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width = 0.49 \\textwidth]{.\/RVOverRVList19902016next}\n\\includegraphics[width = 0.49 \\textwidth]{.\/histogramRVOverRV19902016next}\n\\end{tabular}\n\\caption{Ratio of next-month realized variance to that of the preceding month, from Jan 2nd, 1990 to Dec 30th, 2016.}\n\\label{RVOverVIXListSRV2OverRV21990}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width = 0.49 \\textwidth]{.\/RVOverRVList19902016}\n\\includegraphics[width = 0.49 \\textwidth]{.\/histogramRVOverRV19902016}\n\\end{tabular}\n\\caption{Ratio of preceding-month realized variance to that of the following month, from Jan 2nd, 1990 to Dec 30th, 2016.}\n\\label{RVOverVIXListSRV2OverRV21990}\n\\end{figure}\n\n\\begin{table}[!htb]\n\\caption{MLE results for ``$\\mathrm{nRV}^2 \/ \\mathrm{RV}^2$\" and ``$\\mathrm{RV}^2 \/ \\mathrm{nRV}^2$\"}\n\\label{MLESRV2OverRV21990nn}\n\\begin{minipage}{0.5\\textwidth}\n\\begin{center}\n\\begin{tabular}{ c c c} \n\\multicolumn{2}{c}{} \\\\\n\\hline\n             &       parameters &          KS test  \\\\\n\\hline\nNormal & N(          1.3175,           1.2580) &           0.1809 \\\\\n\\hline\nLogNormal & LN(         -0.0037,           0.7211) &           0.0244 \\\\\n\\hline\nIGa & IGa(          2.1291,           1.6472) &           0.0472 \\\\\n\\hline\nGamma & Gamma(          1.9390,           0.6795) &           0.0801 \\\\\n\\hline\nWeibull & Weibul(          1.4403,           1.2869) &           0.0922 \\\\\n\\hline\nIG & IG(          1.3175,           1.8743) &           0.0340 \\\\\n\\hline\nBP & BP(          5.8771,           3.4893,           0.5556) &           0.0123 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\\begin{minipage}{.5\\textwidth}\n\\begin{center}\n\\begin{tabular}{ c c c} \n\\multicolumn{2}{c}{} \\\\\n\\hline\n             &       parameters &          KS test  \\\\\n\\hline\nNormal & N(          1.2925,           1.0777) &           0.1422 \\\\\n\\hline\nLogNormal & LN(          0.0037,           0.7211) &           0.0244 \\\\\n\\hline\nIGa & IGa(          1.9390,           1.4717) &           0.0801 \\\\\n\\hline\nGamma & Gamma(          2.1291,           0.6071) &           0.0472 \\\\\n\\hline\nWeibull & Weibul(          1.4300,           1.3951) &           0.0608 \\\\\n\\hline\nIG & IG(          1.2925,           1.8387) &           0.0513 \\\\\n\\hline\nBP & BP(          3.4893,           5.8771,           1.7999) &           0.0123 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n  \\end{minipage}\n\\end{table}\n\n\\section{Correlation Analysis \\label{Correlations}}\nTables \\ref{PCCVIX} and \\ref{PCCVXO} list Pearson correlation coefficients (PCC). Here \"n\" labels the \"next\" month, that is the month for which VIX and VXO were predicting the implied RV; \"r\" a \"random\" month and unlabeled RV is the one of the preceding month. All $RV^2$ are scaled, as explained earlier and in \\cite{dashti2018implied}. Tables \\ref{KSVIX} and \\ref{KSVXO} list Kolmogorov-Smirnov (KS) statistic for comparison of the two plots.\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{PCC VIX}\n\\label{PCCVIX}\n\\begin{tabular}{ccccc} \n& \\multicolumn{4}{c}{}\\\\\n\\hline\n             &       $RV^2$ & $nRV^2$ & $VIX^2$ & $rRV^2$  \\\\\n\\hline\n$RV^2$ &1 & 0.70 & 0.88 & 0.0055 \\\\\n\\hline\n$nRV^2$ &0.70 & 1 & 0.71 & 0.0025 \\\\\n\\hline\n$VIX^2$ &0.88  & 0.71 & 1 & 0.003 \\\\\n\\hline\n$rRV^2$ &0.0055  & 0.0025 & 0.003 & 1 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{PCC VXO}\n\\label{PCCVXO}\n\\begin{tabular}{ccccc} \n& \\multicolumn{4}{c}{}\\\\\n\\hline\n             &       $RV^2$ & $nRV^2$ & $VXO^2$ & $rRV^2$  \\\\\n\\hline\n$RV^2$ &1 & 0.70 & 0.87 & 0.0015 \\\\\n\\hline\n$nRV^2$ &0.70  & 1 & 0.72 & 0.004 \\\\\n\\hline\n$VXO^2$ &0.87  & 0.72 & 1 & 0.002 \\\\\n\\hline\n$rRV^2$ &0.0015  & 0.004 & 0.002 & 1 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{KS VIX}\n\\label{KSVIX}\n\\begin{tabular}{ccccccc} \n& \\multicolumn{3}{c}{}\\\\\n\\hline\n             &       $\\frac{RV^2}{VIX^2}$ & $\\frac{nRV^2}{VIX^2}$ & $\\frac{RV^2}{nRV^2}$ & $\\frac{rRV^2}{rVIX^2}$ & $\\frac{rRV^2}{rRV^2}$& $\\frac{nRV^2}{RV^2}$  \\\\ \n\\hline\n$\\frac{RV^2}{VIX^2}$&0 & 0.056& 0.13& 0.20&0.26& -\\\\\n\\hline\n$\\frac{nRV^2}{VIX^2}$&0.056 & 0&- &0.18 & 0.23&0.13\\\\\n\\hline\n$\\frac{RV^2}{nRV^2}$  &0.13 &- & 0&  0.17&0.15&-\\\\          \n\\hline\n$\\frac{rRV^2}{rVIX^2}$ &0.20  & 0.18 &   0.17& 0 &0.063& 0.17\\\\\n\\hline\n$\\frac{rRV^2}{rRV^2}$ & 0.26 & 0.23 & 0.15 & 0.063& 0 &0.16 \\\\\n\\hline\n$\\frac{nRV^2}{RV^2}$  &- &0.13 & -& 0.17 &0.16&0 \\\\  \n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{KS VXO}\n\\label{KSVXO}\n\\begin{tabular}{ccccccc} \n& \\multicolumn{3}{c}{}\\\\\n\\hline\n             &       $\\frac{RV^2}{VXO^2}$ & $\\frac{nRV^2}{VXO^2}$ & $\\frac{RV^2}{nRV^2}$ & $\\frac{rRV^2}{rVXO^2}$ & $\\frac{rRV^2}{rRV^2}$ & $\\frac{nRV^2}{RV^2}$ \\\\ \n\\hline\n$\\frac{RV^2}{VXO^2}$&0 & 0.063& 0.13& 0.22&0.26&- \\\\\n\\hline\n$\\frac{nRV^2}{VXO^2}$&0.063 & 0&- &0.19 & 0.23&0.16\\\\\n\\hline\n$\\frac{RV^2}{nRV^2}$  &0.13 &- & 0&  0.17&0.16&-\\\\          \n\\hline\n$\\frac{rRV^2}{rVXO^2}$ &0.22  & 0.19 &   0.17& 0 &0.057& 0.18\\\\\n\\hline\n$\\frac{rRV^2}{rRV^2}$ & 0.26 & 0.23 & 0.16  & 0.057& 0& 0.17 \\\\\n\\hline\n$\\frac{nRV^2}{RV^2}$  &- &0.16 & -&  0.18&0.17&0\\\\    \n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\clearpage\n\n\\section{Conclusions \\label{Conclusions}}\nBeta Prime distribution provides the best fit to the distributions of ratios of realized variance (squared realized volatility) to squared implied volatility indices VIX and VXO, as well as of ratio of realized variances of two consecutive months. \n\nFor realized variance of the month for which volatility indices calculate implied realized variance, distributions have very slowly decaying fat tails. This indicates that volatility indices tends to underestimate future volatility, especially its large spikes. Conversely, probability of having very small ratios is suppressed due to a large power-law exponent. Comparing this to the ratio of realized variance of the preceding month to the following month, the latter has even longer tails, while the small ratios are significantly more populated. By both measures, VIX and VXO are better predictors of future realized volatility. \n\nFor realized variance of the preceding month, the power law exponents for small ratios and for those in the tails are nearly identical, which reflects the fact that distributions of the ratio and its inverse are nearly identical and that the distribution of the inverse variable of Beta Prime is also Beta Prime.\n\nCorrelation and Kolmogorov-Smirnov statistics are in excellent agreement with empirical analysis of Section \\ref{EO} and fitting in Section \\ref{RV}. In a future work we will more closely identify the months whose ratios are responsible for the tails and the low-ratio regions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this paper we establish the \\emph{universality} of certain probability distributions on Hilbert spaces known as \\emph{GAP measures} \\cite{Gold1,Rei08}. This makes precise some statements and mathematical considerations outlined in our earlier paper \\cite{Gold1} on the thermal equilibrium distribution of the wave function of an open quantum system. \n\nBy saying that GAP measures are universal we mean that the distributions $\\mu_1$ \\x{(described below)} are typically close to a GAP measure,\n\\begin{equation}\n\\mu_1 \\approx GAP\n\\end{equation}\nwhen the system's environment is sufficiently large. To illustrate the terminology of universality, one can say that the central limit theorem conveys a sense in which the Gaussian probability distribution on the real line is universal: many physically relevant probability distributions are approximately Gaussian. \\x{Instead} of universality, one also often speaks of typicality; we use these two terms more or less interchangeably.\n\nThe family of GAP measures is a family of probability measures on\nHilbert spaces. There is one GAP measure for every density matrix $\\rho$ in a Hilbert space $\\mathscr{H}$, denoted $GAP(\\rho)$; it is concentrated on the unit sphere in $\\mathscr{H}$,\n\\begin{equation}\n  \\mathbb{S}(\\mathscr{H}) = \\{\\psi \\in \\mathscr{H}: \\|\\psi\\|=1\\}\\,.\n\\end{equation}\nThe density matrix of $GAP(\\rho)$ is $\\rho$, in the usual sense that, for any probability measure $\\mu$ on $\\mathbb{S}(\\mathscr{H})$, its density matrix is\n\\begin{equation}\\label{covmatrix}\n\\rho_\\mu=\\int_{\\mathbb{S}(\\mathscr{H})} \\mu(d\\psi) \\ket{\\psi}\\bra{\\psi}\\,,\n\\end{equation}\nwhich is also the covariance matrix of $\\mu$ provided $\\mu$ has mean zero. The GAP measures relevant to thermal equilibrium are those associated with canonical density matrices\n\\begin{equation}\\label{rhobeta}\n  \\rho_\\beta = \\frac{1}{Z} e^{-\\beta H}\\,,\n\\end{equation}\nwhere $Z= \\tr e^{-\\beta H}$ is the normalization constant,\n$\\beta$ the inverse temperature and $H$ the Hamiltonian. Detailed\ndiscussions of GAP measures and their physical applications can be \nfound in \\cite{Gold1,Rei08}. See\n\\cite{TZ05} for a study about the support of GAP measures, that is,\nabout what $GAP(\\rho_\\beta)$-distributed wave functions typically look like.\n\nThe main application of GAP measures\nis the characterization of the wave functions of systems we encounter in\nnature. In most cases we do not know a system's wave function,\nfor example because it is a photon coming from the sun (or another star,\nor the cosmic microwave background, or a lamp), or because it is an electron\nthat has escaped from a piece of metal. But in many cases the system is more or less in thermal equilibrium, and then, according to the considerations presented in \\cite{Gold1} and here,\nits wave function should be GAP distributed.\n\n\n\n\n\\subsection{Conditional Wave Function}\n\\label{sec:condwf}\n\nConsider a composite quantum system consisting of two subsystems, system 1 and system 2, with associated Hilbert spaces $\\mathscr{H}_1$ and $\\mathscr{H}_2$. Suppose that the system is in a pure state $\\psi \\in \\mathscr{H}_{\\mathrm{total}} = \\mathscr{H}_1 \\otimes \\mathscr{H}_2$. We ask what might be meant by the wave\nfunction of system 1. An answer is provided by the notion of \\emph{conditional wave function}, defined as follows \\cite{Gold1}:\\footnote{This definition is inspired by Bohmian mechanics, a formulation of quantum mechanics with particle trajectories, where the (non-normalized) conditional wave function $\\psi_1$ of system 1 is defined as \\cite{DGZ}\n\\begin{equation}\n  \\psi_1(x) = \\psi(x,Y)\n\\end{equation}\nfor $x$ in the configuration space of system 1, with $Y$ the actual configuration of system 2.} \nLet $b=\\{b_j\\}$ be an orthonormal basis of $\\mathscr{H}_2$. For\neach choice of $j$, the partial inner product $\\scp{b_j}\n{\\psi}$, taken in $\\mathscr{H}_2$, is a vector belonging to\n$\\mathscr{H}_1$. Regarding $j$ as random (and therefore writing $J$), we are led to\nconsider the random vector $\\psi_1\\in \\mathscr{H}_1$ given by\n\\begin{equation}\\label{Psi1def}\n  \\psi_1 = \\frac{\\scp{b_J} {\\psi}}{\\bigl\\|\\scp{b_J} {\\psi}\\bigr\\|}\n\\end{equation}\nwhere \n$b_J$ is a random element of the basis $\\{b_j\\}$, chosen with the quantum distribution\n\\begin{equation}\\label{marg}\n  \\mathbb{P}^{\\psi,b}(J = j) = \\bigl\\| \\scp{b_j} {\\psi} \\bigr\\|^2.\n\\end{equation}\nWe refer to $\\psi_1$ as the conditional wave function of system 1.\\footnote{The conditional wave function can be regarded as a precise version of the ``collapsed'' wave function in the standard quantum formalism: Suppose that system 1 has interacted with system 2, and their joint wave function, as produced by the appropriate Schr\\\"odinger evolution, is now \n\\begin{equation}\\label{superposition}\n\\sum_j c_j \\psi_j^{(1)} \\otimes \\psi_j^{(2)}\\,,\n\\end{equation}\nwhere the $c_j$ are complex coefficients and all $\\psi$s are normalized. If system 2 is a macroscopic system and the $\\psi_j^{(2)}$s are macroscopically different states then in the standard formalism one regards $j$ as random with distribution $|c_j|^2$, and says accordingly that system 1 can be attributed the ``collapsed'' wave function $\\psi_j^{(1)}$ with probability $|c_j|^2$. The conditional wave function of system 1, according to the above definition in the case that the $\\psi_j^{(2)}$s are among the $\\{b_j\\}$, is indeed $\\psi_j^{(1)}$ with probability $|c_j|^2$.}\n\nThe distribution of $\\psi_1$ corresponding to \\eqref{Psi1def} and\n\\eqref{marg} is given by \\x{the following} probability measure on $\\mathbb{S}\n(\\mathscr{H}_1)$: The probability that $\\psi_1\\in A\\subseteq \\mathbb{S}(\\mathscr{H}_1)$ is\n\\begin{align}\\label{mu1}\n  \\mu_1(A)=\\mu_1^{\\psi,b}(A)= \\mathbb{P}(\\psi_1 \\in A) \n  &= \\sum_{j} \\bigl\\|\n  \\scp{b_j} {\\psi} \\bigr\\|^2 \\, \\delta_{\\scp{b_j}{\\psi}\/\\|\\scp{b_j}{\\psi}\\|}(A) \\\\\n  &= \\sum_{j} \\bigl\\|\n  \\scp{b_j} {\\psi} \\bigr\\|^2 \\, 1_A\\biggl(\\frac{\\scp{b_j}{\\psi}}{\\|\\scp{b_j}{\\psi}\\|}\\biggr) \\,,\n\\end{align}\nwhere $\\delta_\\phi$ denotes the Dirac ``delta'' measure (a point\nmass) concentrated at $\\phi$ \\x{and $1_A$ denotes the characteristic function of the set $A$}. While the density matrix $\\rho_{\\mu_1}$ associated with ${\\mu_1}$ always equals the reduced\ndensity matrix $\\rho_1^\\psi$ of system 1, given by\n\\begin{equation}\\label{rhoreddef}\n  \\rho_1^\\psi = \\tr_2 |\\psi \\rangle \\langle \\psi | = \\sum_{j}\n  \\scp{b_j}{\\psi} \\scp{\\psi}{b_j} \\,,\n\\end{equation}\nthe measure $\\mu_1$ itself usually depends on the choice of the basis $b$, so $\\mu_1=\\mu_1^{\\psi,b}$. \n\n\n\\subsection{Summary of Results}\n\nIn this paper, we prove several \\emph{universality theorems} about GAP measures, Theorems~\\ref{thm1}--\\ref{thm4}, formulated in Section~\\ref{sec:results}. These are statements to the effect that for \\emph{most} wave functions $\\psi$ from relevant subsets of $\\mathscr{H}_1\\otimes\\mathscr{H}_2$ and\/or \\emph{most} orthonormal bases $b$ of $\\mathscr{H}_2$, $\\psi_1$ is approximately GAP-distributed. Here, ``most'' means that the set of exceptions is small with respect to the appropriate natural uniform measure. \n\nThe basic universality property is expressed in Theorem~\\ref{thm1}, which asserts that \\y{for sufficiently large $\\dim \\mathscr{H}_2$, for any orthonormal basis $b$ of $\\mathscr{H}_2$, and for any density matrix $\\rho_1$ on $\\mathscr{H}_1$, most $\\psi$ in $\\mathbb{S}(\\mathscr{H}_1\\otimes\\mathscr{H}_2)$ with the reduced density matrix $\\tr_2\\ket{\\psi}\\bra{\\psi} = \\rho_1$ are such that} the distribution $\\mu_1^{\\psi,b}$ of $\\psi_1$ is arbitrarily close to $GAP(\\rho_1)$,\n\\begin{equation}\\label{mu1GAPrho1}\n\\mu_1^{\\psi,b} \\approx GAP(\\rho_1)\\,.\n\\end{equation}\nThis fact was derived \\x{(but not rigorously proven) in Section 5.1.3 of \\cite{Gold1}.}\nThe rigorous proof of Theorem~\\ref{thm1} is based on the fact,\nfound independently by several authors \\cite{Y72,W78,DEL92,Colthesis,PR04} \n(see also \\cite{Ol90,ZS00,Jia06,MT07}),\nthat for a random $n\\times n$ unitary matrix with\ndistribution given by the Haar measure on the unitary group $U(n)$,\nthe upper left (or any other) $k\\times k$ submatrix, multiplied by a normalization factor $\\sqrt{n}$, converges as $n\\to\\infty$ to a\nmatrix of independent complex Gaussian random variables with mean 0\nand variance 1. (To understand the factor $\\sqrt{n}$, note that a column of a unitary $n \\times n$ matrix is a unit vector, and thus a single entry should be of order $1\/\\sqrt{n}$.) \n\nTheorem~\\ref{corbasis} asserts that the conclusion of Theorem~\\ref{thm1}---that \\eqref{mu1GAPrho1} holds with arbitrary accuracy for sufficiently large $\\dim\\mathscr{H}_2$---is also true for \\emph{every} $\\psi$ with $\\tr_2 \\ket{\\psi}\\bra{\\psi}=\\rho_1$ for \\emph{most} $b$ (instead of for \\emph{every} $b$ for \\emph{most} $\\psi$).\n\nTheorems~\\ref{thm3} and \\ref{thm4} justify the physical conclusion that, if a system (system 1) is weakly coupled to a very large (but finite) second system (the ``heat bath,'' system 2) then, for most wave functions of the composite system with energy in a given narrow energy range $[E,E+\\delta E]$, the conditional wave function of the system is approximately GAP-distributed for most orthonormal bases of the heat bath. In more detail, let the interaction between the two systems be negligible so that the Hamiltonian can be taken to be\n\\begin{equation}\nH=H_1\\otimes I_2 + I_1\\otimes H_2\n\\end{equation}\n(with $I_{1\/2}$ the identity operator on $\\mathscr{H}_{1\/2}$), and let $\\mathscr{H}_R\\subset \\mathscr{H}_1\\otimes \\mathscr{H}_2$ be a micro-canonical energy shell of the composite system, i.e., the subspace spanned by the eigenstates of the total energy with eigenvalues in $[E,E+\\delta E]$. Assume that the eigenvalues of $H_2$ are sufficiently dense and that the dimensions of $\\mathscr{H}_2$ and $\\mathscr{H}_R$ are sufficiently large. Then, for most $\\psi\\in\\mathbb{S}(\\mathscr{H}_R)$,\n\\begin{equation}\n\\mu_1^{\\psi,b} \\approx GAP(\\rho_\\beta)\n\\end{equation}\nfor most bases $b$ of $\\mathscr{H}_2$; here, $\\rho_\\beta$ is the canonical density matrix \\eqref{rhobeta} and $\\beta=\\beta(E)$ . \n\nIn Theorems~\\ref{thm3} and \\ref{thm4} we relax the condition that $\\psi$ have a prescribed reduced density matrix, and exploit instead \\emph{canonical typicality}. This is the fact, found independently by several groups \\cite{GMM04,Gold2,PSW05,PSW06}\nand anticipated long before by Schr\\\"odinger \\cite{schrbook},\nthat for most $\\psi\\in\\mathbb{S}(\\mathscr{H}_R)$, the reduced density matrix $\\tr_2\\ket{\\psi}\\bra{\\psi}$ is approximately of the canonical form \\eqref{rhobeta}.\nMore generally, in Theorems~\\ref{thm3} and \\ref{thm4} we may regard $\\mathscr{H}_R$ as \\emph{any} subspace of $\\mathscr{H}_1\\otimes\\mathscr{H}_2$ of sufficiently high dimension. Canonical typicality then refers to the fact that for most $\\psi\\in\\mathbb{S}(\\mathscr{H}_R)$,\n$\\tr_2\\ket{\\psi}\\bra{\\psi}$ is close to $\\tr_2\\rho_R$, where $\\rho_R$ denotes $1\/\\dim\\mathscr{H}_R$ times the projection to $\\mathscr{H}_R$; the precise version of canonical typicality that we use in the\nproof of our Theorems~\\ref{thm3} and \\ref{thm4} is due to Popescu, Short,\nand Winter \\cite{PSW05,PSW06}. Theorem~\\ref{thm3} asserts that for most $\\psi\\in\\mathbb{S}(\\mathscr{H}_R)$,\n\\begin{equation}\\label{mu1GAPtr2rhoR}\n\\mu_1^{\\psi,b} \\approx GAP(\\rho^{(1)}_R)\\,,\n\\end{equation}\nwith $\\rho^{(1)}_R=\\tr_2 \\rho_R$, for most bases $b$ of $\\mathscr{H}_2$.\n\nTheorem~\\ref{thm4} is a very similar statement but differs in the detailed meaning of ``$\\approx$'' and refers to a fixed density matrix, such as $\\rho_\\beta$, in place of $\\rho^{(1)}_R$ in \\eqref{mu1GAPtr2rhoR}.\n\n\\subsection{Remarks}\n\n\\begin{itemize}\n\\item \\textit{Time evolution.} It may be interesting to consider how $\\mu_1^{\\psi,b}$ evolves with time if the wave function $\\psi=\\psi_t$ of systems 1 and 2 together evolves according to the Schr\\\"odinger equation\n\\begin{equation}\ni\\hbar\\frac{\\partial \\psi_t}{\\partial t} = H \\psi_t\\,.\n\\end{equation}\nIn a situation in which most $\\psi\\in\\mathbb{S}(\\mathscr{H}_R)$ have $\\mu_1^{\\psi,b}\\approx GAP(\\rho^{(1)}_R)$, we may expect that even for $\\psi_0\\in\\mathbb{S}(\\mathscr{H}_R)$ with $\\mu_1^{\\psi_0,b}$ far from any GAP measure, $\\mu_1(t)=\\mu_1^{\\psi_t,b}$ will approach $GAP(\\rho^{(1)}_R)$ and stay near $GAP(\\rho^{(1)}_R)$ most of the time (though not forever, as follows from the recurrence property \\y{(almost-periodicity)} of the Schr\\\"odinger evolution in a finite-dimensional Hilbert space). We leave this problem open but briefly remark that one can already conclude by interchanging the time average and the average over $\\psi_0$ that whenever it is true for most $\\psi\\in\\mathbb{S}(\\mathscr{H}_R)$ that $\\mu_1^{\\psi,b}\\approx GAP(\\rho^{(1)}_R)$, then for most $\\psi_0\\in\\mathbb{S}(\\mathscr{H}_R)$, $\\mu_1^{\\psi_t,b} \\approx GAP(\\rho^{(1)}_R)$ for most times $t$; the open problem is to prove a statement that concerns \\emph{all}, rather than \\emph{most}, $\\psi_0$.\n\n\n\\item \\textit{The role of interaction.}\nAnother remark concerns the role of interaction (between the system and the heat bath) for obtaining the distribution $GAP(\\rho_\\beta)$. The nature of the interaction is relevant to our discussion in two places---although our theorems do not depend on it, as they do not mention the Hamiltonian at all. First, interaction is relevant for creating typical wave functions, as it helps evolve atypical wave functions into typical ones. This is closely related to the fact that a system coupled to a heat bath (i.e., a big second system) will typically go from non-equilibrium to thermal equilibrium only in the presence of interaction; see Section~4 of \\cite{GLMTZ09b} for further discussion and examples. Second, it depends on the interaction which subspace of $\\mathscr{H}_1\\otimes\\mathscr{H}_2$ is the micro-canonical energy shell that we want $\\mathscr{H}_R$ to be, and thus also which density matrix $\\tr_2 \\rho_R$ is. In the limit of negligible interaction, $\\tr_2 \\rho_R$ has the canonical form $\\rho_\\beta = (1\/Z) e^{-\\beta H}$, while interaction makes it deviate from \\x{this form}. As a consequence of these two roles, when we want to obtain from non-equilibrium a wave function $\\psi\\in\\mathscr{H}_1\\otimes\\mathscr{H}_2$ such that the distribution of the conditional wave function $\\psi_1$ is close to $GAP(\\rho_\\beta)$, we may want that the interaction be not too large (or else there will be deviations from $\\rho_\\beta$) and that the interaction be not too small (or else it may take too long, say longer than the present age of the universe, to reach thermal equilibrium). \n\\end{itemize}\n\n\n\n\\subsection{Definition of the GAP Measure}\n\\label{sec:GAPdef}\n\nFor any density matrix $\\rho$ on $\\mathscr{H}$, the measure $GAP(\\rho)$ on (the Borel $\\sigma$-algebra of) $\\mathbb{S}(\\mathscr{H})$ is built \\x{by} starting from the measure $G(\\rho)$,\nwhich is the Gaussian measure on $\\mathscr{H}$ with mean 0 and covariance matrix\n$\\rho$. In this paper, we are interested only in the case $\\dim\\mathscr{H}< \\infty$.\nThen $G(\\rho)$ can be explicitly defined as follows: Let $S$ be the subspace of \n$\\mathscr{H}$ on which $\\rho$ is supported, i.e., its positive spectral subspace, or\nequivalently the orthogonal complement of its kernel, or\nequivalently its range; let $d'=\\dim S$ and $\\rho_+$ the restriction\nof $\\rho$ to $S$; then $G(\\rho)$ is the measure on $\\mathscr{H}$\nsupported on $S$ with the following density relative to the Lebesgue\nmeasure $\\lambda$ on $S$:\n\\begin{equation}\n  \\frac{dG(\\rho)}{d\\lambda}(\\psi) = \\frac{1}{\\pi^{d'} \\,\n  \\det \\rho_+} \\exp(-\\langle \\psi |\\rho^{-1}_+| \\psi \\rangle)\\,.\n\\end{equation}\n\\y{Equivalently, a $G(\\rho)$-distributed random vector $\\psi$ is one whose coefficients $\\scp{\\chi_i}{\\psi}$ relative to an eigenbasis $\\{\\chi_i\\}$ of $\\rho$ (i.e., $\\rho\\chi_i = p_i \\chi_i$ with $0\\leq p_i\\leq 1$) are independent complex Gaussian random variables with mean 0 and variances $\\mathbb{E}|\\scp{\\chi_i}{\\psi}|^2=p_i$; by a \\emph{complex Gaussian} random variable we mean one whose real and imaginary parts are independent real Gaussian random variables with equal variances.}\n\nNoting that \n\\begin{equation}\\label{Gpsi1}\n\\int_\\mathscr{H} G(\\rho)(d\\psi) \\, \\|\\psi\\|^2 = \\tr \\rho =1\\,,\n\\end{equation}\nwe now define the adjusted Gaussian measure $GA(\\rho)$ on $\\mathscr{H}$ as:\n\\begin{equation}\n  GA(\\rho)(d\\psi) = \\|\\psi\\|^2 G(\\rho)(d\\psi)\\,.\n\\end{equation}\nIf $\\psi^{GA}$ is a $GA(\\rho)$-distributed vector, then $GAP(\\rho)$\nis the distribution of this vector projected on the unit sphere; that is, $GAP(\\rho)$ is the distribution of\n\\begin{equation}\n  \\psi^{GAP} = \\frac{\\psi^{GA}}{\\|\\psi^{GA}\\|}\\,.\n\\end{equation}\nLike $\\G{\\rho}$ and unlike $\\GA{\\rho}$, $\\GAP{\\rho}$ has covariance matrix $\\rho$.\n\nMore generally, one can define for any measure $\\mu$ on $\\mathscr{H}$\nthe ``adjust-and-project'' procedure. We denote by $A\\mu$ the\nadjusted measure\n\\begin{equation}\\label{Adef}\nA\\mu(d\\psi) = \\|\\psi\\|^2 \\, \\mu(d\\psi)\\,.\n\\end{equation}\nThe projection on the unit sphere is defined as:\n\\begin{equation}\\label{Pdef}\nP:\\mathscr{H} \\setminus \\{0\\} \\to \\mathbb{S}(\\mathscr{H})\\, , \\quad\nP(\\psi)= \\frac{\\psi}{\\|\\psi\\|}\\,.\n\\end{equation}\nThen the adjusted-and-projected measure is $P_* ( A\\mu) = A\\mu \\circ\nP^{-1}$, where $P_*$ denotes the action of $P$ on measures, thus\ndefining a mapping $P_* \\circ A$ from the\nmeasures on $\\mathscr{H}$ with $\\int \\mu(d\\psi) \\, \\|\\psi\\|^2 = 1$\nto the probability measures on $\\mathbb{S}(\\mathscr{H})$.\n\n\n\n\n\n\n\n\n\\section{Results}\n\\label{sec:results}\n\n\n\\subsection{GAP Measure From a Typical Wave Function of a Large System, Given the Reduced Density Matrix}\n\\label{sec:typical}\n\n\nLet $\\mathscr{H}_\\mathrm{total}=\\mathscr{H}_1\\otimes\\mathscr{H}_2$, where\n$\\mathscr{H}_1$ and $\\mathscr{H}_2$ have respective dimension $d_1$\nand $d_2$, with $d_1< d_2<\\infty$. For any given density matrix $\\rho_1$ on\n$\\mathscr{H}_1$, let\n\\begin{equation}\n\\mathscr{R}(\\rho_1)=\\bigl\\{\\psi\\in\\mathbb{S}(\\mathscr{H}_\\mathrm{total}):\n\\rho_1^\\psi=\\rho_1\\bigr\\}\n\\end{equation}\nbe the set of all normalized wave functions in $\\mathscr{H}_{\\mathrm{total}}$ with reduced density matrix $\\rho_1^\\psi=\\rho_1$. We will see that $\\mathscr{R}(\\rho_1)$ is always non-empty.\n\nTheorem~\\ref{thm1} below concerns typical wave functions in\n$\\mathscr{R}(\\rho_1)$, i.e., typical wave functions with fixed\nreduced density matrix. The concept of ``typical'' refers to the \nuniform distribution $u_{\\rho_1}$ on $\\mathscr{R}(\\rho_1)$;\nan explicit definition of this distribution will be given in Section~\\ref{sec:urho1def}.\n\n\\x{Before we formulate Theorem~\\ref{thm1}, we introduce some notation. First, for} any Hilbert space $\\mathscr{H}$, let $\\mathscr{D}(\\mathscr{H})$ denote the set of all density operators on $\\mathscr{H}$,\ni.e., of all positive operators on $\\mathscr{H}$ with trace 1. \\x{Second, when $\\mu$ is a measure on $\\mathscr{H}$ or $\\mathbb{S}(\\mathscr{H})$ and $f(\\psi)$ is} a measurable function on $\\mathscr{H}$ or $\\mathbb{S}(\\mathscr{H})$ then we use the notation\n\\begin{equation}\\label{muf}\n\\mu(f):=\\int \\mu(d\\psi) f(\\psi)\\,.\n\\end{equation}\n\\x{Third, let} $\\|f\\|_\\infty=\\sup_{x} |f(x)|$.\n\n\\begin{thm}\\label{thm1}\nFor every $0<\\varepsilon<1$, $0<\\delta<1$, and $d_1\\in\\mathbb{N}$, there is $D_2=D_2(\\varepsilon,\\delta,d_1)>0$ such that for all $d_2\\in\\mathbb{N}$ with $d_2>D_2$, for every $\\mathscr{H}_1$ and $\\mathscr{H}_2$ with $\\dim\\mathscr{H}_{1\/2}=d_{1\/2}$, for every orthonormal basis $b=\\{b_1,\\ldots,b_{d_2}\\}$ of $\\mathscr{H}_2$, for every $\\rho_1\\in \\mathscr{D}(\\mathscr{H}_1)$, and for every \\y{bounded measurable} function $f:\\mathbb{S}(\\mathscr{H}_1)\\to\\mathbb{R}$,\n\\begin{equation}\\label{ineqthm1}\nu_{\\rho_1} \\Bigl\\{ \\psi\\in \\mathscr{R}(\\rho_1): \\bigl|\\mu_1^{\\psi,b}(f) - \\GAP{\\rho_1}(f)\\bigr|< \\varepsilon \\, \\|f\\|_\\infty \\Bigr\\} \\geq 1-\\delta\\,.\n\\end{equation}\n\\end{thm}\n\nWe give the proof, as well as those of Theorems~\\ref{corbasis}--\\ref{thm4}, in Section~\\ref{sec:proofs}. \n\nIt follows from Theorem~\\ref{thm1} that, for every sequence $(\\mathscr{H}_{2,n})_{n\\in\\mathbb{N}}$ of Hilbert spaces with $d_{2,n}=\\dim \\mathscr{H}_{2,n} \\to \\infty$ as $n\\to\\infty$ and every sequence $(b_n)_{n\\in\\mathbb{N}}$ of orthonormal bases $b_n=\\{b_{1,n},\\ldots,b_{d_{2,n},n}\\}$ of $\\mathscr{H}_{2,n}$, for every $\\rho_1\\in\\mathscr{D}(\\mathscr{H}_1)$, and for every \\y{bounded measurable} function $f:\\mathbb{S}(\\mathscr{H}_1)\\to\\mathbb{R}$, the sequence of random variables $\\mu_1^{\\Psi_n,b_n}(f)$, where $\\Psi_n$ has distribution $u_{\\rho_1}$ on $\\mathbb{S}(\\mathscr{H}_1\\otimes\\mathscr{H}_{2,n})$, converges in distribution, as $n\\to\\infty$, to the constant $\\GAP{\\rho_1}(f)$, in fact uniformly in $\\rho_1$, $b_n$ and those $f$ with $\\|f\\|_\\infty\\leq 1$. Because of the convergence for every $f$, we can say that the sequence of random measures $\\mu_1^{\\Psi_n,b_n}$ converges ``weakly in distribution'' to the fixed measure $GAP(\\rho_1)$. \n\nA few comments about notation. In \\cite{Gold1}, $d_1$ was called $k$,\n$d_2$ was called $m$, and the notation for the\nbasis $\\{b_1, \\ldots, b_{d_2}\\}$ was $\\{\\ket{1}, \\ldots, \\ket{m}\\}$. For\nenumerating the basis, we will use the letter $j$, and thus write\n$b_j$; in \\cite{Gold1}, the notation was $q_2$ for $j$ (subscript 2\nbecause it refers to $\\mathscr{H}_2$). For a random choice of $j$, we\nwrite $J$; the corresponding notation in \\cite{Gold1} was $Q_2$.\n\n\n\n\\subsection{GAP Measure From a Typical Basis of a Large System}\n\nAs already explained in \\cite{Gold1}, instead of considering a\ntypical wave function and a fixed basis one can consider a fixed\nwave function and a typical basis. Let $ONB(\\mathscr{H}_2)$ be the set\nof all orthonormal bases of $\\mathscr{H}_2$, and recall the notation $\\rho_1^\\psi=\\tr_2 \\ket{\\psi}\\bra{\\psi}$.\n\n\\begin{thm}\\label{corbasis}\nFor every $0<\\varepsilon<1$, $0<\\delta<1$, $d_1\\in\\mathbb{N}$, \n$d_2\\in\\mathbb{N}$ with $d_2>D_2(\\varepsilon,\\delta,d_1)$ as in Theorem~\\ref{thm1}, every $\\mathscr{H}_1$ and $\\mathscr{H}_2$ with $\\dim\\mathscr{H}_{1\/2}=d_{1\/2}$, every $\\psi\\in \\mathbb{S}(\\mathscr{H}_1\\otimes\\mathscr{H}_2)$, and every \\y{bounded measurable} function $f:\\mathbb{S}(\\mathscr{H}_1)\\to\\mathbb{R}$,\n\\begin{equation}\\label{ineqcorbasis}\nu_{ONB} \\Bigl\\{ (b_1,\\ldots,b_{d_2}) \\in ONB(\\mathscr{H}_2): \\bigl|\\mu_1^{\\psi,b}(f) - \\GAP{\\rho_1^\\psi}(f)\\bigr|< \\varepsilon \\, \\|f\\|_\\infty \\Bigr\\} \\geq 1-\\delta\\,,\n\\end{equation}\nwhere $u_{ONB}$ is the uniform probability measure on $ONB(\\mathscr{H}_2)$,\ncorresponding to the Haar measure on the unitary group $U(\\mathscr{H}_2)$.\n\\end{thm}\n\n\n\n\n\\subsection{GAP Measure From a Typical Basis and a Typical Wave Function in a Large Subspace}\n\nIn our main physical application, the reduced density matrix $\\rho_1^\\psi$ is not fixed, although---by a \\x{fact} known as canonical typicality---most of the relevant $\\psi$s have a reduced density matrix $\\rho_1^\\psi$ that is close to a certain fixed density matrix, for example to the canonical density matrix $\\rho_\\beta= (1\/Z)e^{-\\beta H}$. In this section, we present two further universality theorems that are appropriate for such situations, in which the relevant set of $\\psi$s is a subspace of $\\mathscr{H}_1\\otimes\\mathscr{H}_2$ that will be denoted $\\mathscr{H}_R$.\n \nThe physical setting to have in mind is this. A system with Hilbert space $\\mathscr{H}_1$ is entangled with a large system whose Hilbert space is $\\mathscr{H}_2$. The Hamiltonian $H$ is thus defined on $\\mathscr{H}_{\\mathrm{total}}=\\mathscr{H}_1\\otimes\\mathscr{H}_2$; suppose the total system is confined to a finite volume, so that $H$ has pure point spectrum. Let $[E,E+\\delta E]$ be a narrow energy window, located at a suitable energy $E$ such as one corresponding to a more or less fixed energy per particle or per volume. Then the \\emph{micro-canonical energy shell} is the spectral subspace of $H$ associated with this interval, i.e., the subspace spanned by the eigenvectors with eigenvalues between $E$ and $E+\\delta E$, and this is our subspace $\\mathscr{H}_R$. The \\emph{micro-canonical density matrix} $\\rho_R$ is the density matrix associated with $\\mathscr{H}_R$, i.e., $1\/\\dim\\mathscr{H}_R$ times the projection to $\\mathscr{H}_R$. Canonical typicality then asserts that for most wave functions in $\\mathbb{S}(\\mathscr{H}_R)$, the reduced density matrix is approximately $\\rho_\\beta$ for \\x{an} appropriate value of $\\beta$.\n\nFor more general $\\mathscr{H}_R$, canonical typicality means that for most $\\psi\\in\\mathbb{S}(\\mathscr{H}_R)$, the reduced density matrix $\\rho_1^\\psi$ is close to $\\tr_2\\rho_R$. The precise statement that we make use of is Theorem~1 of \\cite{PSW05} or the ``main theorem'' of \\cite{PSW06}, which asserts, in a somewhat specialized and simplified form that suffices for our purposes:\n\n\\begin{lemma}\\label{lem:can}\nConsider a Hilbert space $\\mathscr{H}_1$ of dimension $d_1\\in\\mathbb{N}$, another Hilbert space $\\mathscr{H}_2$ of dimension $d_2\\in\\mathbb{N}$ and a subspace $\\mathscr{H}_R\\subseteq \\mathscr{H}_1 \\otimes \\mathscr{H}_2$ of dimension $d_R$. Let $\\rho_R$ be $1\/d_R$ times the projection to $\\mathscr{H}_R$, and $u_R$ the uniform distribution on $\\mathbb{S}(\\mathscr{H}_R)$. Then for every $\\eta>0$,\n\\begin{equation}\n  u_R \\left\\{ \\psi \\in \\mathbb{S}(\\mathscr{H}_R): \\Bigl\\|\\rho_1^\\psi -\n  \\tr_2 \\rho_R  \\Bigr\\|_{\\tr} \\geq \\eta + \\frac{d_1}{\\sqrt{d_R}}\n  \\right\\} \\leq 4 \\exp\\Bigl(-\\frac{d_R\\eta^2}{18\\pi^3}\\Bigr)\\,.\n\\end{equation}\n\\end{lemma}\n\nHere, the \\emph{trace norm} is defined by\n\\begin{equation}\n  \\|M\\|_{\\tr} = \\tr |M| = \\tr \\sqrt{M^* M}\\,.\n\\end{equation}\nBy the \\emph{uniform distribution} $u_R$ we mean the $(2d_R-1)$-dimensional surface area measure on $\\mathbb{S}(\\mathscr{H}_R)$, normalized so that $u_R(\\mathbb{S}(\\mathscr{H}_R))=1$.\n\n\n\n\\begin{thm}\\label{thm3}\nFor every $0<\\varepsilon<1$, $0<\\delta<1$, $d_1\\in\\mathbb{N}$, every Hilbert space $\\mathscr{H}_1$ with $\\dim \\mathscr{H}_1=d_1$, and every continuous function $f:\\mathbb{S}(\\mathscr{H}_1)\\to\\mathbb{R}$,  there is a number $D_R=D_R(\\varepsilon,\\delta,d_1,f)>0$ such that for all $d_R,d_2\\in\\mathbb{N}$ with $d_R>D_R$ and $d_2>D_2(\\frac{\\varepsilon}{2\\|f\\|_\\infty},\\delta\/2,d_1)$ as in Theorem~\\ref{thm1}, and for every $\\mathscr{H}_2$ and $\\mathscr{H}_R\\subseteq \\mathscr{H}_1\\otimes\\mathscr{H}_2$ with $\\dim\\mathscr{H}_{2\/R} = d_{2\/R}$, \n\\begin{multline}\\label{eq:thm3}\n  u_R \\times u_{ONB} \\Bigl\\{ \\bigl( \\psi, b \\bigr) \\in\n  \\mathbb{S}(\\mathscr{H}_R) \\times ONB (\\mathscr{H}_2) : \\\\\n  \\bigl|\\mu_1^{\\psi,b}(f) - \\GAP{\\tr_2 \\rho_R}(f) \\bigr| < \\varepsilon \\Bigr\\}\n  \\geq 1-\\delta\\,.\n\\end{multline}\n\\end{thm}\n\nIt follows that, for every sequence $(\\mathscr{H}_{2,n})_{n\\in\\mathbb{N}}$ of Hilbert spaces with $d_{2,n}=\\dim \n\\mathscr{H}_{2,n} \\to \\infty$ as $n\\to\\infty$, every sequence $(\\mathscr{H}_{R,n})_{n\\in\\mathbb{N}}$ of subspaces of $\\mathscr{H}_1\\otimes\\mathscr{H}_{2,n}$ with $d_{R,n}=\\dim\\mathscr{H}_{R,n} \\to\\infty$ as $n\\to\\infty$, and every continuous function $f:\\mathbb{S}(\\mathscr{H}_1)\\to\\mathbb{R}$, the sequence of random variables\n\\begin{equation}\n\\mu_1^{\\Psi_n,B_n}(f)-\\GAP{\\tr_2 \\rho_{R,n}}(f)\\,,\n\\end{equation}\nwhere $(\\Psi_n,B_n)$ has distribution $u_{R,n}\\times u_{ONB,n}$ on $\\mathbb{S}(\\mathscr{H}_{R,n})\\times ONB(\\mathscr{H}_{2,n})$, converges to zero in distribution as $n\\to\\infty$. We say that the sequence of random signed measures $\\mu_1^{\\Psi_n,B_n}-\\GAP{\\tr_2 \\rho_{R,n}}$ converges ``weakly in distribution'' to zero.\n\n\\bigskip\n\nFor $0<\\gamma<1\/\\dim\\mathscr{H}$ let $\\mathscr{D}_{\\geq \\gamma}(\\mathscr{H})$ denote the set of density matrices $\\rho\\in\\mathscr{D}(\\mathscr{H})$ whose eigenvalues are all greater than or equal to $\\gamma$ (so that, in particular, zero is not an eigenvalue of $\\rho$).\n\n\\begin{thm}\\label{thm4}\nFor every $0<\\varepsilon<1$, $0<\\delta<1$, $d_1\\in\\mathbb{N}$,  \\x{and $0<\\gamma<1\/d_1$, there} are numbers $D_R'=D_R'(\\varepsilon,\\delta,d_1,\\gamma)>0$ and $r'=r'(\\varepsilon,d_1,\\gamma)>0$ such that for all $d_R,d_2\\in\\mathbb{N}$ with $d_R>D_R'$ and $d_2>D_2(\\varepsilon\/2,\\delta\/2,d_1)$ as in Theorem~\\ref{thm1}, \\x{for every Hilbert space $\\mathscr{H}_1$ with $\\dim \\mathscr{H}_1=d_1$, for every} $\\Omega \\in \\mathscr{D}_{\\geq\\gamma}(\\mathscr{H}_1)$, for every $\\mathscr{H}_2$ and $\\mathscr{H}_R\\subseteq \\mathscr{H}_1\\otimes\\mathscr{H}_2$ with $\\dim\\mathscr{H}_{2\/R} = d_{2\/R}$ satisfying\n\\begin{equation}\\label{ROmega}\n\\bigl\\|\\tr_2 (\\rho_R)-\\Omega\\bigr\\|_{\\tr}<r'\\,,\n\\end{equation} \nand for every bounded measurable function $f:\\mathbb{S}(\\mathscr{H}_1)\\to\\mathbb{R}$, \n\\begin{multline}\\label{eq:thm4}\n  u_R \\times u_{ONB} \\Bigl\\{ \\bigl( \\psi, b \\bigr) \\in\n  \\mathbb{S}(\\mathscr{H}_R) \\times ONB (\\mathscr{H}_2) : \\\\\n  \\bigl|\\mu_1^{\\psi,b}(f) - \\GAP{\\Omega}(f) \\bigr| < \\varepsilon \\,\\|f\\|_\\infty \\Bigr\\}\n  \\geq 1-\\delta\\,.\n\\end{multline}\n\\end{thm}\n\nIf we want to consider just one particular density matrix $\\Omega$ (of which zero is not an eigenvalue) then we can set $\\gamma$ equal to the smallest eigenvalue of $\\Omega$. It then follows that, for every sequence $(\\mathscr{H}_{2,n})_{n\\in\\mathbb{N}}$ of Hilbert spaces with $d_{2,n}=\\dim \\mathscr{H}_{2,n} \\to \\infty$ as $n\\to\\infty$, and every sequence $(\\mathscr{H}_{R,n})_{n\\in\\mathbb{N}}$ of subspaces of $\\mathscr{H}_1\\otimes\\mathscr{H}_{2,n}$ with $d_{R,n}=\\dim\\mathscr{H}_{R,n} \\to\\infty$ and $\\tr_2 \\rho_{R,n} \\to \\Omega$ as $n\\to\\infty$, the sequence of random measures $\\mu_1^{\\Psi_n,B_n}$ converges weakly in distribution to the fixed measure $GAP(\\Omega)$. In short,\n\\begin{equation}\n\\mu_1^{\\psi,b}\\stackrel{u_R\\times u_{ONB}}{\\Longrightarrow} GAP(\\Omega)\\,.\n\\end{equation}\n\nOf the two theorems above, Theorem~\\ref{thm3} is the simpler and perhaps more natural mathematical statement: it does not even mention any other density matrix than $\\tr_2\\rho_R$; its structure is to ask first that $\\varepsilon$, $\\delta$, and $f$ be specified, which define the accuracy of the desired approximations;\\footnote{How $f$ defines a sense of accuracy becomes manifest if we consider finitely many test functions $f_1,\\ldots,f_\\ell$, assume $d_R>\\max(D_R(f_1),\\ldots,D_R(f_\\ell))$, and then apply Theorem~\\ref{thm3} to obtain that $\\mu_1^{\\psi,b}$ and $GAP(\\tr_2\\rho_R)$ agree approximately on all linear combinations of $f_1,\\ldots,f_\\ell$.} and it applies to all subspaces $\\mathscr{H}_R$ of sufficient dimension. For the physical application, though, we often want to compare $\\mu_1^\\psi$ to $GAP(\\Omega)$ rather than $GAP(\\tr_2\\rho_R)$, for example because $\\Omega$ is the thermal density matrix $\\rho_\\beta=(1\/Z)e^{-\\beta H}$ while $\\tr_2\\rho_R$ is something complicated; we usually do not need that the estimate applies uniformly to all spaces $\\mathscr{H}_R$ of sufficient dimension, but instead consider only one fixed $\\mathscr{H}_R$; and in that situation we can, in fact, obtain an estimate, the one provided by Theorem~\\ref{thm4}, that is uniform in $f$.\n\n\n\n\n\\subsection{GAP Measure as the Thermal Equilibrium Distribution}\n\\label{sec:thermo}\n\nTheorem~\\ref{thm4} justifies regarding $GAP(\\rho_\\beta)$ as the thermal\nequilibrium distribution of the wave functions of the system 1 in the following way.\nLet $\\mathscr{H}_R$ be the microcanonical subspace, i.e., the spectral\nsubspace of $H$ associated with the interval $[E,E +\\delta E]$. It\nis a standard fact (e.g., \\cite{G95,ML79}) that when the interaction energy between system 1 and system 2 is sufficiently small, i.e., when we may set\n\\begin{equation}\nH = H_1 \\otimes I_2+ I_1\\otimes H_2\n\\end{equation}\non $\\mathscr{H}_\\mathrm{total}=\\mathscr{H}_1\\otimes\\mathscr{H}_2$,  and when system 2 is a large heat bath, i.e., when the eigenvalues of $H_2$ are sufficiently dense, then $\\tr_2 \\rho_R$ is approximately of the exponential form $Z^{-1}\\exp(-\\beta\nH_1)$ with $Z = \\tr \\exp(-\\beta H_1)$ for suitable $\\beta>0$, i.e., is approximately the\ncanonical density matrix $\\rho_\\beta$. Then by Theorem~\\ref{thm4} in\nthis special case of negligible interaction we have that for most wave functions $\\psi\\in\\mathbb{S}(\\mathscr{H}_R)$,\n\\begin{equation}\n\\mu_1^{\\psi,b}\\approx GAP(\\rho_\\beta)\n\\end{equation}\nfor most orthonormal bases $b$ of $\\mathscr{H}_2$.\n\n\n\n\\section{Proofs}\n\\label{sec:proofs}\n\n\\subsection{Definition of $u_{\\rho_1}$}\n\\label{sec:urho1def}\n\nAccording to the Schmidt decomposition \\cite{Schmidt}, every $\\psi\\in\\mathscr{H}_\\mathrm{total}$ can\nbe written in the form\n\\begin{equation}\\label{Schmidt}\n\\psi = \\sum_{i=1}^{d_1} c_i \\, \\tilde{\\chi}_i\\otimes\\tilde{\\phi}_i\n\\end{equation}\nwhere $\\{\\tilde{\\chi}_i\\}$ is an orthonormal basis in $\\mathscr{H}_1$,\n$\\{\\tilde{\\phi}_i\\}$ is an orthonormal system in $\\mathscr{H}_2$ (i.e., a set of\northonormal vectors that is not necessarily complete), and the $c_i$ are\ncoefficients which can be chosen to be real and non-negative. If $\\|\\psi\\|=1$, the reduced\ndensity matrix of the system 1 is then\n\\begin{equation}\n\\rho_1^\\psi=\\sum_{i=1}^{d_1} c_i^2 \\ket{\\tilde{\\chi}_i}\\bra{\\tilde{\\chi}_i}\\,.\n\\end{equation}\nThus, $\\{\\tilde{\\chi}_i\\}$ is an eigenbasis of $\\rho_1^\\psi$, and $c_i^2$ are the corresponding eigenvalues. \n\nNow let a density matrix $\\rho_1$ be given, let $\\{\\chi_i\\}$ be an eigenbasis for $\\rho_1$, and let $0\\leq p_i\\leq 1$ be the corresponding eigenvalues. Then every $\\psi\\in\\mathscr{R}(\\rho_1)$ possesses a Schmidt decomposition of the form\n\\begin{equation}\\label{Schmidt2}\n\\psi=\\sum_{i=1}^{d_1} \\sqrt{p_i} \\, \\chi_i \\otimes \\phi_i\n\\end{equation}\nwith some orthonormal system $\\{\\phi_i\\}$ in $\\mathscr{H}_2$. Indeed, we know it has a Schmidt decomposition \\eqref{Schmidt} in which $\\{\\tilde{\\chi}_i\\}$ is an eigenbasis of $\\rho_1$, and $c_i^2$ are the eigenvalues. Reordering the terms in \\eqref{Schmidt}, we can make sure that $c_i=\\sqrt{p_i}$. Any two eigenbases $\\{\\chi_i\\}$ and $\\{\\tilde{\\chi}_i\\}$ of $\\rho_1$ are related by a block unitary; more precisely, for every eigenvalue $p$ of $\\rho_1$, $\\{\\chi_i:i\\in\\mathscr{I}(p)\\}$ and $\\{\\tilde{\\chi}_i:i\\in\\mathscr{I}(p)\\}$ (using the index set $\\mathscr{I}(p)=\\{i:c_i^2=p\\}=\\{i:p_i=p\\}$) are two orthonormal bases of the eigenspace of $p$, and thus related by a unitary matrix $(U_{ij}^{(p)})_{i,j\\in\\mathscr{I}(p)}$:\n\\begin{equation}\\label{chitildechi}\n\\tilde{\\chi}_i = \\sum_{j\\in\\mathscr{I}(p)} U^{(p)}_{ij} \\, \\chi_j\\,.\n\\end{equation}\nSetting\n\\begin{equation}\\label{phitildephi}\n\\phi_i = \\sum_{j\\in\\mathscr{I}(p)} U^{(p)}_{ji} \\tilde{\\phi}_j\\,,\n\\end{equation}\nwe obtain \\eqref{Schmidt2}, and that $\\{\\phi_i\\}$ is an orthonormal system.\n\nConversely, every orthonormal system $\\{\\phi_i\\}$ in $\\mathscr{H}_2$ defines, by \\eqref{Schmidt2}, a $\\psi\\in\\mathscr{R}(\\rho_1)$. Thus, \\eqref{Schmidt2} defines a bijection $F_{\\rho_1,\\{\\chi_i\\}}:ONS(\\mathscr{H}_2,d_1)\\to \\mathscr{R}(\\rho_1)$. The Haar measure on the unitary group of $\\mathscr{H}_2$ defines the uniform distribution on the set of orthonormal bases of $\\mathscr{H}_2$, of which the uniform distribution on $ONS(\\mathscr{H}_2,d_1)$ is a marginal; let $u_{\\rho_1,\\{\\chi_i\\}}$ be its image under $F_{\\rho_1,\\{\\chi_i\\}}$. \n\nWe note that $u_{\\rho_1,\\{\\chi_i\\}}$ actually does not depend on the choice of the eigenbasis $\\{\\chi_i\\}$. Indeed, if $\\{\\tilde{\\chi}_i\\}$ is any other eigenbasis of $\\rho_1$ (without loss of generality numbered in such a way that the eigenvalue of $\\tilde{\\chi}_i$ is $p_i$) then, as explained above, it is related to $\\{\\chi_i\\}$ by a block unitary $d_1\\times d_1$ matrix $U$ consisting of the blocks $(U_{ij}^{(p)})$. Let $\\overline{U}$ be the matrix whose entries are the complex conjugates of the entries of $U$, and let $\\hat{\\overline{U}}$ denote the action of $\\overline{U}$ on $ONS(\\mathscr{H}_2,d_1)$ given by\n\\begin{equation}\n\\hat{\\overline{U}}\\Bigl(\\bigl\\{\\phi_i:i=1,\\ldots,d_1\\bigr\\}\\Bigr) = \n\\biggl\\{\\sum_{j=1}^{d_1}\\overline{U}_{ij} \\phi_j:i=1,\\ldots,d_1\\biggr\\}\\,.\n\\end{equation}\nThen\n\\begin{equation}\nF_{\\rho_1,\\{\\chi_i\\}}=F_{\\rho_1,\\{\\tilde{\\chi}_i\\}}\\circ \\hat{\\overline{U}}\\,.\n\\end{equation}\nSince the Haar measure is invariant under left multiplication, its marginal on $ONS(\\mathscr{H}_2,d_1)$ is invariant under $\\hat{\\overline{U}}$. We thus define $u_{\\rho_1}$ to be $u_{\\rho_1,\\{\\chi_i\\}}$ for any eigenbasis $\\{\\chi_i\\}$.\n\n\n\\subsection{Proof of Theorem~\\ref{thm1}}\n\nWe will obtain Theorem~\\ref{thm1} from the corresponding\nstatement about the Gaussian measures $G(\\rho_1)$:\n\n\\begin{lemma}\\label{lemma1}\nFor every $0<\\tilde{\\varepsilon}<1$, $0<\\tilde{\\delta}<1$, and $d_1\\in\\mathbb{N}$, there is $\\tilde{D}_2=\\tilde{D}_2(\\tilde{\\varepsilon},\\tilde{\\delta},d_1)>0$ such that for all $d_2\\in\\mathbb{N}$ with $d_2>\\tilde{D}_2$, for every $\\mathscr{H}_1$ and $\\mathscr{H}_2$ with $\\dim \\mathscr{H}_{1\/2}=d_{1\/2}$, for every orthonormal basis $b=\\{b_1,\\ldots,b_{d_2}\\}$ of $\\mathscr{H}_2$, for every $\\rho_1\\in \\mathscr{D}(\\mathscr{H}_1)$, and for every bounded \\y{measurable} function $\\tilde{f}:\\mathscr{H}_1 \\to \\mathbb{R}$,\n\\begin{equation}\\label{ineqlemma1}\nu_{\\rho_1} \\Bigl\\{ \\psi\\in\\mathscr{R}(\\rho_1): \\bigl| \\tilde{\\mu}^{\\psi,b}_1(\\tilde{f}) -\\G{\\rho_1}(\\tilde{f}) \\bigr| < \\tilde{\\varepsilon} \\,\\|\\tilde{f}\\|_\\infty \\Bigr\\} \\geq 1-\\tilde{\\delta}\\,,\n\\end{equation}\nwhere $\\tilde{\\mu}^\\psi_1$ is the distribution of $\\sqrt{d_2}\\langle\nb_{J}|\\psi\\rangle \\in \\mathscr{H}_1$ (not normalized) with respect to\nthe uniform distribution of $J\\in\\{1,\\ldots,d_2\\}$.\n\\end{lemma}\n\n\nWe now collect the tools needed for the proof of Lemma~\\ref{lemma1}. \nWe note that $\\tilde{\\mu}^\\psi_1$ is the sum of $d_2$ delta measures\nwith equal weights,\n\\begin{equation}\\label{mubar}\n\\tilde{\\mu}^\\psi_1 = \\frac{1}{d_2} \\sum_{j=1}^{d_2} \\delta_{\\psi_1(j)}\\, ,\n\\end{equation}\nlocated at the points\n\\begin{equation}\\label{psi1}\n\\psi_1(j) = \\sqrt{d_2}\\scp{b_j}{\\psi}\\,.\n\\end{equation}\nLet $\\rho_1\\in\\mathscr{D}(\\mathscr{H}_1)$, let $\\{\\chi_i\\}$ be an eigenbasis of $\\rho_1$ with eigenvalues $p_i$, and let $\\psi\\in\\mathscr{R}(\\rho_1)$. According to \\eqref{Schmidt2},\n\\begin{equation}\n\\psi_1(j)=\\sum_{i=1}^{d_1} c_i \\sqrt{d_2}\n\\langle b_j | \\phi_i \\rangle \\chi_i\n\\end{equation}\nwith $c_i=\\sqrt{p_i}$. Now we regard $\\psi$ as random with\ndistribution $u_{\\rho_1}$; then the $\\psi_1(j)$ are $d_2$ random\nvectors, and $\\tilde{\\mu}^\\psi_1$ is their empirical distribution.\nAs noted above, $\\{\\phi_i\\}$ is a random orthonormal system\ndistributed according to a marginal of the Haar measure; in other words, the\nexpansion coefficients $\\langle b_j | \\phi_i \\rangle$ of\n\\begin{equation}\n\\phi_i = \\sum_{j=1}^{d_2} \\langle b_j | \\phi_i \\rangle b_j\n\\end{equation}\nform a $d_1\\times d_2$ matrix that arises as the first $d_1$ rows of a\nHaar-distributed unitary $d_2\\times d_2$ matrix.\n\nIt is therefore interesting to us how a submatrix of a Haar-distributed unitary matrix is distributed. We cite the relevant result from Olshanskij \\cite[Lemma 5.3]{Ol90} and Collins \\cite[Chap. 4]{Colthesis} (the corresponding fact for orthogonal matrices was established independently by Diaconis, Eaton, and Lauritzen \\cite{DEL92}). To formulate this result, we use the following terminology. Recall that a \\emph{complex Gaussian} random variable is one whose real and imaginary parts are independent real Gaussian random variables with equal variances. The \\emph{variation distance} of two measures $\\mu,\\nu$ on the same $\\sigma$-algebra $\\mathscr{A}$ is defined to be\n\\begin{equation}\\label{defvariationdistance}\n\\|\\mu-\\nu\\| \n= \\sup_{A\\in\\mathscr{A}} (\\mu(A)-\\nu(A))\n+\\sup_{A\\in\\mathscr{A}} (\\nu(A)-\\mu(A))\\,.\n\\end{equation}\nIn case $\\mu$ and $\\nu$ possess densities $f$ and $g$ relative to some measure $\\lambda$ on $\\mathscr{A}$, this coincides with the $L^1$ norm of $f-g$,\n\\begin{equation}\n\\|\\mu-\\nu\\| = \\int \\lambda(dx) \\, |f(x)-g(x)|\\,.\n\\end{equation}\n\n\\begin{lemma}\\label{lem:asymp}\nFor $k,n\\in\\mathbb{N}$ with $k \\leq n$, let the random matrix $(U_{ij})$ be $\\mathrm{Haar}(U(n))$ distributed, and let $X$ be the upper left $k\\times k$ submatrix multiplied by the normalization factor $\\sqrt{n}$, $X_{ij}=\\sqrt{n}U_{ij}$ for $1\\leq i,j \\leq k$. Let $G$ a random $k\\times k$ matrix whose entries $G_{ij}$ are independent complex Gaussian random variables with mean 0 and variance $\\mathbb{E} |G_{ij}|^2 =1$. Let $\\mu_{k,n}$ denote the distribution of $X$ and $\\mu_k$ that of $G$. Then $\\mu_{k,n}$ converges, as $n\\to\\infty$, to $\\mu_k$ in the variation distance. In fact, as soon as $n\\geq 2k$, $\\mu_{k,n}$ and $\\mu_k$ possess densities $f_{k,n}$ and $f_k$ relative to the Lebesgue measure in $\\mathbb{C}^{k\\times k}$ given by\n\\begin{equation}\\label{det}\nf_{k,n}(X) = \\mathscr{N}_{k,n}\\,1_{\\|X\\|_\\infty<\\sqrt{n}} \\biggl(\\det\\Bigl(I-\\frac{XX^*}{n}\\Bigr)\\biggr)^{n-2k} \n\\end{equation}\n(where $\\mathscr{N}_{k,n}$ is the appropriate normalization factor, $\\displaystyle\\|X\\|_\\infty=\\sup_{v\\in\\mathbb{C}^k\\setminus\\{0\\}}|Xv|\/|v|$, and $I$ denotes the $k\\times k$ unit matrix) and\n\\begin{equation}\\label{Gaussian}\nf_k(X) = \\frac{1}{\\pi^k} e^{-\\tr XX^*}\\,,\n\\end{equation}\nand\n\\begin{equation}\n\\|f_{k,n}-f_k\\|_{L^1(\\mathbb{C}^{k\\times k})}\\to 0\\text{ as }n\\to \\infty.\n\\end{equation}\n\\end{lemma}\n\nA random matrix such as $G$ is $\\sqrt{k}$ times what is sometimes called a ``standard non-selfadjoint Gaussian matrix.'' \nWe will use only the following consequence of Lemma~\\ref{lem:asymp}, concerning the convergence of the upper left $k\\times 2$ entries of $(U_{ij})$ to a matrix of independent complex Gaussian random variables:\n\n\\begin{cor}\\label{cor:AB}\nFor every $0<\\varepsilon<1$ and $k\\in\\mathbb{N}$ there is $n_0=n_0(\\varepsilon,k)>0$ such that for every $n\\in\\mathbb{N}$ with $n>n_0$ and every bounded measurable function $g:\\mathbb{C}^{k}\\to\\mathbb{R}$,\n\\begin{equation}\\label{LemmaA}\n  \\Bigl|\\mathbb{E} g(\\sqrt{n}U_{11},\\ldots, \\sqrt{n} U_{k1}) -\n  \\mathbb{E} g(G_{11},\\ldots,G_{k1}) \\Bigr|< \\varepsilon \\, \\|g\\|_\\infty\n\\end{equation}\nand\n\\begin{multline}\\label{LemmaB}\n  \\Bigl|\\mathbb{E} [g(\\sqrt{n}U_{11},\\ldots, \\sqrt{n} U_{k1}) \n  g(\\sqrt{n}U_{12},\\ldots, \\sqrt{n} U_{k2})] -\\\\\n  \\mathbb{E} [g(G_{11},\\ldots,G_{k1})g(G_{12},\\ldots,G_{k2})] \\Bigr|< \\varepsilon \\, \\|g\\|_\\infty^2\\,,\n\\end{multline}\nwhere $(U_{ij})$ is Haar distributed in $U(n)$, and $G_{ij}$ ($i=1,\\ldots,k$; $j=1,2$) are independent complex Gaussian random variables with mean 0 and variance $\\mathbb{E} |G_{ij}|^2=1$.\n\\end{cor}\n\n\\proof\nChoose $n_0=n_0(\\varepsilon,k)\\geq 2k$ so that, for all $n>n_0$, $\\|f_{k,n}-f_k\\|_{L^1(\\mathbb{C}^{k\\times k})}<\\varepsilon$. This is possible by Lemma~\\ref{lem:asymp}. Then \\eqref{LemmaA} and \\eqref{LemmaB} follow.\n\\endproof\n\n\n\\proof[Proof of Lemma~\\ref{lemma1}]\nSet\n\\begin{equation}\\label{tildeDdef}\n\\tilde{D}_2(\\tilde{\\varepsilon},\\tilde{\\delta},d_1) \n= \\max \\biggl(n_0\\Bigl(\\frac{\\tilde{\\varepsilon}}{2},d_1\\Bigr),\nn_0\\Bigl(\\frac{\\tilde{\\delta}\\tilde{\\varepsilon}^2}{16},d_1\\Bigr),\n\\frac{32}{\\tilde{\\delta}\\tilde{\\varepsilon}^2}\\biggr)\\,.\n\\end{equation}\n\nLet us first introduce some abbreviations and notation. We write $n$ for $d_2$ and $k$ for $d_1$. Let $G_{11},\\ldots,G_{k1},G_{12},\\ldots,G_{k2}$ again be independent complex Gaussian random variables with mean 0 and variance $\\mathbb{E}|G_{ij}|^2=1$. We use the basis $\\{\\chi_i\\}$ to identify $\\mathscr{H}_1$ with\n$\\mathbb{C}^{k}$. Let $\\tilde{f}:\\mathbb{C}^{k} \\to \\mathbb{R}$ be a bounded \\y{measurable} function,\nand let $c_i\\geq 0$ be as before (the square roots of the\neigenvalues of $\\rho_1$). Set\n\\begin{equation}\ng(z_1,\\ldots,z_{k}) = \\tilde{f}(c_1z_1, \\ldots, c_{k}z_{k})\\,.\n\\end{equation}\nThen $g$, too, is \\y{measurable} and bounded with bound $\\|g\\|_\\infty = \\|\\tilde{f}\\|_\\infty$. As we said above, the\ndistribution of the $\\scp{b_j}{\\phi_i}$ is the same as that of\n$U_{ij}$ with $j \\in \\{1,\\ldots,n\\}$, that is, the first $k$\nrows of a Haar distributed unitary $n\\times n$ matrix. We thus\nwrite $U_{ij}$ instead of $\\scp{b_j}{\\phi_i}$, and obtain\n\\begin{equation}\\label{mumeans}\n\\tilde{\\mu}^\\psi_1 (\\tilde{f}) =: \\tilde{\\mu}(\\tilde{f}) = \\frac{1}{n} \\sum_{j=1}^{n}\ng(\\sqrt{n} U_{1j},\\ldots, \\sqrt{n} U_{kj}) \\,.\n\\end{equation}\nWe write $X$ for $\\sqrt{n}$ times the upper left $k\\times n$ submatrix of $(U_{ij})$, $\\vec{X}_j$ for the $j$-th column of $X$, i.e.,\n\\begin{equation}\n\\vec{X}_j = (\\sqrt{n}U_{1j},\\ldots,\\sqrt{n}U_{kj})\\,,\n\\end{equation}\nand $\\vec{G}_i$ for $(G_{1i},\\ldots,G_{ki})$ ($i=1,2$). In this notation,\n\\begin{equation}\\label{mumeans2}\n\\tilde{\\mu}(\\tilde{f}) = \\frac{1}{n} \\sum_{j=1}^n g(\\vec{X}_j)\n\\end{equation}\nand\n\\begin{equation}\\label{Gmeans}\nG(\\rho_1)(\\tilde{f}) = \\mathbb{E} \\tilde{f}(c_1 G_{11},\\ldots, c_{k} G_{k1}) = \\mathbb{E}\ng(\\vec{G}_1)\\,.\n\\end{equation}\n\nThe expression to be estimated is\n\\begin{equation}\n\\Bigl|\\tilde{\\mu} (\\tilde{f})-G(\\rho_1)(\\tilde{f})\\Bigr|\\,,\n\\end{equation}\nfor which we have by the triangle inequality that\n\\begin{equation}\\label{triangle}\n\\Bigl|\\tilde{\\mu} (\\tilde{f})-G(\\rho_1)(\\tilde{f})\\Bigr|\\leq \\Bigl|\\tilde{\\mu} (\\tilde{f})\n-\\mathbb{E}\\tilde{\\mu}(\\tilde{f}) \\Bigr| + \\Bigl|\\mathbb{E}\\tilde{\\mu}(\\tilde{f}) -\nG(\\rho_1)(\\tilde{f})\\Bigr|\\,.\n\\end{equation}\nNote that the second contribution in \\eqref{triangle} is nonrandom.\n\nWe first show that\n\\begin{equation}\\label{nonrandomsmall}\n\\Bigl|\\mathbb{E}\\tilde{\\mu}(\\tilde{f}) - G(\\rho_1)(\\tilde{f})\\Bigr| \n<\\frac{\\tilde{\\varepsilon}}{2} \\|\\tilde{f}\\|_\\infty\n\\end{equation}\nif $n$ is sufficiently large: By \\eqref{mumeans2},\n\\begin{equation}\\label{Emu}\n\\mathbb{E} \\tilde{\\mu}(\\tilde{f}) = \\frac{1}{n} \\sum_{j=1}^{n} \\mathbb{E} g(\\vec{X}_j) =  \\mathbb{E} g(\\vec{X}_1)\n\\end{equation}\nbecause each $\\vec{X}_j$ has the same distribution---because the columns of $(U_{ij})$ are exchangeable due to the invariance of the Haar\nmeasure. By \\eqref{LemmaA} in Corollary~\\ref{cor:AB}, if $n>n_0(\\tilde{\\varepsilon}\/2,k)$ then the absolute difference between \\eqref{Gmeans} and \\eqref{Emu} is less than $\\tfrac{\\tilde{\\varepsilon}}{2}\\|g\\|_\\infty$, i.e., \\eqref{nonrandomsmall} holds.\n\nConcerning the first contribution in \\eqref{triangle}, Chebyshev's\ninequality (see, e.g., \\cite[p.~65]{Bill}) asserts that\n\\begin{equation}\\label{Chebyshev}\n\\mathbb{P}\\Bigl( \\bigl| \\tilde{\\mu}(\\tilde{f}) - \\mathbb{E}\\tilde{\\mu}(\\tilde{f}) \\bigr| >\n\\frac{\\tilde{\\varepsilon}}{2}\\|\\tilde{f}\\|_\\infty \\Bigr) \n< \\frac{4}{\\tilde{\\varepsilon}^2\\|\\tilde{f}\\|_\\infty^2} \\mathrm{var}(\\tilde{\\mu}(f)) \\,,\n\\end{equation}\nwhere $\\mathbb{P}$ is the Haar measure for $U_{ij}$ and var($Y$) is the\nvariance of the random variable $Y$. Now Lemma~\\ref{lemma1} follows if we can show that\n\\begin{equation}\\label{varto0}\n \\mathrm{var}(\\tilde{\\mu}(\\tilde{f}))  \n \\leq \\frac{\\tilde{\\delta}\\tilde{\\varepsilon}^2}{4}\\|\\tilde{f}\\|_\\infty^2 \n\\end{equation}\nfor sufficiently large $n$. We find that\n\\begin{equation}\n\\mathrm{var}(\\tilde{\\mu}(\\tilde{f})) \n= \\mathbb{E} \\bigl[\\tilde{\\mu}(\\tilde{f})^2\\bigr] -\n(\\mathbb{E}\\tilde{\\mu}(\\tilde{f}))^2 = \\frac{1}{n^2} \\sum_{j,j'=1}^{n} \\mathbb{E}\n[g(\\vec{X}_j) g(\\vec{X}_{j'})] - [\\mathbb{E} g(\\vec{X}_1)]^2\\,.\n\\end{equation}\nSince the $\\vec{X}_j$ are exchangeable, the joint\ndistribution of $\\vec{X}_j$ and $\\vec{X}_{j'}$ for $j\\neq j'$ is the same as the joint\ndistribution of $\\vec{X}_1$ and $\\vec{X}_2$, so that all summands with\n$j\\neq j'$ are equal (and all summands with $j=j'$ are equal), and\nwe can write\n\\begin{align}\n\\mathrm{var}(\\tilde{\\mu}(\\tilde{f})) \n&= \\frac{n^2 - n}{n^2} \\mathbb{E} \\bigl[g(\\vec{X}_1) \\, g(\\vec{X}_2)\\bigr]  \n+ \\frac{n}{n^2} \\underbrace{\\mathbb{E} \\bigl[g(\\vec{X}_1)^2\\bigr]}_{\\leq \\|g\\|_\\infty^2} \n- [\\mathbb{E} g(\\vec{X}_1)]^2\\\\\n&\\leq \\mathbb{E} \\bigl[g(\\vec{X}_1) \\, g(\\vec{X}_2)\\bigr] \n-\\frac{1}{n} \\underbrace{\\mathbb{E} \\bigl[g(\\vec{X}_1) \\, g(\\vec{X}_2)\\bigr]}_{\\geq -\\|g\\|_\\infty^2} \n+ \\frac{1}{n} \\|g\\|_\\infty^2 - [\\mathbb{E} g(\\vec{X}_1)]^2\\\\\n&\\leq \\mathbb{E} \\bigl[g(\\vec{X}_1) \\, g(\\vec{X}_2)\\bigr] \n+\\frac{2}{n} \\|g\\|_\\infty^2 - [\\mathbb{E} g(\\vec{X}_1)]^2\\\\\n&= \\mathbb{E} \\bigl[g(\\vec{X}_1) \\, g(\\vec{X}_2)\\bigr] \n\\underbrace{- \\mathbb{E} \\bigl[ g(\\vec{G}_1) \\, g(\\vec{G}_2) \\bigr]\n + \\bigl[\\mathbb{E}  g(\\vec{G}_1) \\bigr]^2}_{=0} \\: +\\nonumber\\\\\n&\\quad +\\: \\frac{2}{n} \\|g\\|_\\infty^2 \n- [\\mathbb{E} g(\\vec{X}_1)]^2\\\\\n&\\leq \n\\Bigl|\\mathbb{E}  \\bigl[g(\\vec{X}_1) \\, g(\\vec{X}_2)\\bigr]  - \\mathbb{E}  \\bigl[g(\\vec{G}_1) \\, g(\\vec{G}_2)\\bigr]  \\Bigr| \n+\\frac{2}{n} \\|g\\|_\\infty^2 \\: + \\nonumber\\\\ \n&\\quad +\\:  \\bigl[\\mathbb{E}  g(\\vec{G}_1) \\bigr]^2\n- [\\mathbb{E} g(\\vec{X}_1)]^2\\\\\n&\\leq \n\\Bigl|\\mathbb{E}  \\bigl[g(\\vec{X}_1) \\, g(\\vec{X}_2)\\bigr]  - \\mathbb{E}  \\bigl[g(\\vec{G}_1) \\, g(\\vec{G}_2)\\bigr]  \\Bigr| \n+\\frac{2}{n} \\|g\\|_\\infty^2 \\: + \\nonumber\\\\\n&\\quad +\\: \\bigl|\\mathbb{E}  g(\\vec{G}_1) - \\mathbb{E} g(\\vec{X}_1)\\bigr| \\, \n\\underbrace{\\bigl|\\mathbb{E}  g(\\vec{G}_1) + \\mathbb{E} g(\\vec{X}_1)\\bigr|}_{\\leq 2 \\|g\\|_\\infty} \\\\\n&\\leq \\frac{\\tilde{\\delta}\\tilde{\\varepsilon}^2}{16} \\|g\\|_\\infty^2 \n+\\frac{2}{n} \\|g\\|_\\infty^2 \n+ \\frac{\\tilde{\\delta}\\tilde{\\varepsilon}^2}{16} \\|g\\|_\\infty \\, 2\\|g\\|_\\infty \n\\end{align}\nfor $n>n_0(\\tilde{\\delta}\\tilde{\\varepsilon}^2\/16,k)$ by Corollary~\\ref{cor:AB}. If in addition $n>32\/\\tilde{\\delta}\\tilde{\\varepsilon}^2$, we thus have that\n\\begin{equation}\n\\mathrm{var}(\\tilde{\\mu}(\\tilde{f})) \n\\leq \\tilde{\\delta}\\tilde{\\varepsilon}^2 \\|g\\|_\\infty^2 \\Bigl(\\frac{1}{16} + \\frac{1}{16} + \\frac{1}{8}\\Bigr)\n= \\frac{\\tilde{\\delta}\\tilde{\\varepsilon}^2}{4} \\|g\\|_\\infty^2\n= \\frac{\\tilde{\\delta}\\tilde{\\varepsilon}^2}{4} \\|\\tilde{f}\\|_\\infty^2\\,.\n\\end{equation}\nThus, for $d_2=n>\\tilde{D}_2$ in \\eqref{tildeDdef}, both \\eqref{nonrandomsmall} and \\eqref{varto0} hold; Lemma~\\ref{lemma1} follows using \\eqref{triangle} and \\eqref{Chebyshev}.\n\\endproof\n\n\nWe now collect the tools needed for the proof of Theorem~\\ref{thm1}.\n\n\\begin{lemma}\\label{lem:n} \n$\\tilde{\\mu}_1^\\psi\\bigl(\\|\\cdot\\|^2\\bigr)=1.$\n\\end{lemma}\n\n\\proof\nRecall that $\\tilde{\\mu}_1^\\psi$ is the distribution of $\\sqrt{d_2}\\scp{b_J}{\\psi}$ with $\\mathbb{P}(J=j)=1\/d_2$. Thus,\n\\begin{equation}\\label{mu1psisquare}\n\\int_{\\mathscr{H}_1} \\tilde{\\mu}_1^\\psi (d\\psi_1) \\|\\psi_1\\|^2 =\n\\mathbb{E}\\left(d_2\\|\\scp{b_J}{\\psi}\\|^2\\right) =\n\\sum_{j=1}^{d_2}\\|\\scp{b_j}{\\psi}\\|^2 = \\|\\psi\\|^2 = 1\\,.\n\\end{equation}\n\\endproof\n\n\\begin{lemma}\\label{lem:tildemu1}\n$\\mu_1^\\psi = P_* A \\tilde{\\mu}_1^\\psi$.\n\\end{lemma}\n\n\\proof\nBy definition, $A\\tilde{\\mu}_1^\\psi$ is the distribution of $\\sqrt{d_2}\\scp{b_J}{\\psi}$ with $\\mathbb{P}(J=j)=\\|\\scp{b_j}{\\psi}\\|^2$, and $P_* A \\tilde{\\mu}_1^\\psi$ is the distribution of $\\scp{b_J}{\\psi}\/\\|\\scp{b_J}{\\psi}\\|$ with $\\mathbb{P}(J=j)=\\|\\scp{b_j}{\\psi}\\|^2$. The latter is the definition of $\\mu_1^\\psi$. \n\\endproof\n\nRecall from \\eqref{Gpsi1} that, for every $\\rho\\in\\mathscr{D}(\\mathscr{H})$,\n\\begin{equation}\n\\int_\\mathscr{H} G(\\rho)(d\\psi) \\, \\|\\psi\\|^2 =1.\n\\end{equation}\n\n\\begin{lemma}\\label{lem:R}\nFor every $0<\\varepsilon<1$ and $d\\in\\mathbb{N}$, there is $R=R(\\varepsilon,d)>0$ such that, for every $\\mathscr{H}$ with $\\dim\\mathscr{H}=d$ and every $\\rho\\in\\mathscr{D}(\\mathscr{H})$,\n\\begin{equation}\n\\int_{\\{\\psi\\in\\mathscr{H}:\\|\\psi\\|<R\\}} \\G{\\rho}(d\\psi) \\, \\|\\psi\\|^2 > 1-\\varepsilon\\,.\n\\end{equation}\n\\end{lemma}\n\n\\proo\nLet $X_1,\\ldots, X_d$ be independent complex Gaussian random variables with mean 0 and variance 1. Then $X=(X_1,\\ldots,X_d)$ has Gaussian distribution $\\G{I}$ with covariance matrix $I$, the identity matrix; note that\n\\begin{equation}\n\\int_\\mathscr{H} \\G{I}(d\\psi) \\, \\|\\psi\\|^2 = \\mathbb{E} \\sum_{i=1}^d |X_i|^2=d.\n\\end{equation}\nThus, there is $R>0$ with\n\\begin{equation}\n\\int_{\\{\\psi\\in\\mathscr{H}:\\|\\psi\\|<R\\}} \\G{I}(d\\psi) \\, \\|\\psi\\|^2\n > d-\\varepsilon\\,.\n\\end{equation}\n\nFor any $\\mathscr{H}$ with $\\dim\\mathscr{H}=d$ and $\\rho\\in\\mathscr{D}(\\mathscr{H})$, choose an eigenbasis of $\\rho$ to identify $\\mathscr{H}$ with $\\mathbb{C}^d$, so $\\rho=\\mathrm{diag}(p_1,\\ldots,p_d)$. Set $Z_i=\\sqrt{p_i} X_i$, so $Z=(Z_1,\\ldots,Z_d)$ has distribution $\\G{\\rho}$. Then\n\\begin{align}\n\\int_{\\{\\psi\\in\\mathscr{H}:\\|\\psi\\|\\geq R\\}} \\G{\\rho}(d\\psi) \\, \\|\\psi\\|^2\n&=\\mathbb{E}\\Bigl( 1_{\\sum |Z_j|^2\\geq R^2} \\sum_{i} |Z_i|^2\\Bigr)\\\\\n&=\\mathbb{E}\\Bigl( 1_{\\sum p_j |X_j|^2\\geq R^2} \\sum_i p_i |X_i|^2\\Bigr)\\\\\n&\\leq \\mathbb{E}\\Bigl( 1_{\\sum |X_j|^2\\geq R^2} \\sum_i |X_i|^2\\Bigr)\\\\\n&= \\int_{\\{\\psi\\in\\mathscr{H}:\\|\\psi\\|\\geq R\\}} \\G{I}(d\\psi) \\, \\|\\psi\\|^2\n <\\varepsilon\\,.\n\\end{align}\n\\endproof\n\n\nAs an abbreviation, we write $M(\\mathscr{H}_1,\\mathscr{H}_2,\\rho_1,b,f,\\varepsilon)$ or shorter $M(f,\\varepsilon)$ for the set considered in \\eqref{ineqthm1}, and $\\tilde{M}(\\mathscr{H}_1,\\mathscr{H}_2,\\rho_1,b,\\tilde{f},\\tilde{\\varepsilon})$ or shorter $\\tilde{M}(\\tilde{f},\\tilde{\\varepsilon})$ for the set considered in \\eqref{ineqlemma1}.\n\n\\begin{lemma}\\label{lem:M}\nFix $0<\\varepsilon<1$, $\\mathscr{H}_1$, $\\mathscr{H}_2$, $b$, and $\\rho_1$. Let $R=R(\\varepsilon\/4,d_1)$ with $R(\\cdot,\\cdot)$ provided by Lemma~\\ref{lem:R}. Then, for every \\y{bounded measurable} function $f:\\mathbb{S}(\\mathscr{H}_1)\\to\\mathbb{R}$,\n\\begin{equation}\\label{M}\nM(f,\\varepsilon) \\supseteq \n\\tilde{M}\\Bigl((f\\circ P) 1_{N<R} N^2,\\frac{\\varepsilon}{4R^2}\\Bigr) \\cap \n\\tilde{M}\\Bigl(1_{N<R} N^2,\\frac{\\varepsilon}{4R^2}\\Bigr)\\,,\n\\end{equation}\nwhere $P$ denotes the projection to the unit sphere as defined in \\eqref{Pdef} and\n$N(\\psi_1)=\\|\\psi_1\\|$ on $\\mathscr{H}_1$.\n\\end{lemma}\n\nNote that while $f\\circ P$ is not defined at $0\\in\\mathscr{H}_1$, $(f\\circ P) N^2$ is---as 0.\n\n\n\\proof[Proof of Lemma~\\ref{lem:M}]\nIf we want to show of a particular $\\psi\\in\\mathscr{R}(\\rho_1)$ that $\\psi \\in M(f,\\varepsilon)$ then, by Lemma~\\ref{lem:tildemu1}, we need to show that\n\\begin{equation}\\label{toshow}\n\\bigl| P_* A \\tilde{\\mu}^{\\psi,b}_1(f) - GAP(\\rho_1)(f)\\bigr| < \\varepsilon \\, \\|f\\|_\\infty\\,.\n\\end{equation}\nWe note that, for any probability measure $\\tilde{\\mu}$ on $\\mathscr{H}_1$ with $\\tilde{\\mu}(N^2)=1$ (in particular, by Lemma~\\ref{lem:n}, for $\\tilde{\\mu}=\\tilde{\\mu}_1^{\\psi,b}$), writing $g=f\\circ P$, we obtain that\n\\begin{align}\n&\\bigl| P_* A \\tilde{\\mu}(f) - \\GAP{\\rho_1}(f)\\bigr| = \n\\bigl|\\tilde{\\mu}(g N^2)-\\G{\\rho_1}(g N^2) \\bigr|\\leq\\\\\n&\\leq \\Bigl| \\tilde{\\mu}\\bigl(g 1_{N<R} N^2\\bigr) - \\G{\\rho_1}\\bigl(g 1_{N<R} N^2 \\bigr)\\Bigr|+\\label{term1}\\\\\n&\\quad +\\Bigl|\\tilde{\\mu}\\bigl( g 1_{N\\geq R} N^2 \\bigr) \\Bigr| +\\label{term2}\\\\\n&\\quad + \\Bigl| \\G{\\rho_1}\\bigl( g 1_{N\\geq R} N^2 \\bigr) \\Bigr|\\,.\\label{term3}\n\\end{align}\n\nNow suppose $\\psi\\in \\tilde{M}\\Bigl(g 1_{N<R} N^2,\\varepsilon\/4R^2\\Bigr)$, which means that $\\psi\\in\\mathscr{R}(\\rho_1)$ and\n\\begin{equation}\\label{cond1}\n\\Bigl| \\tilde{\\mu}^{\\psi,b}_1\\bigl(g 1_{N<R} N^2\\bigr) \n-\\G{\\rho_1}\\bigl(g 1_{N<R} N^2\\bigr) \\Bigr| \n< \\frac{\\varepsilon}{4R^2} \\,\\|g 1_{N<R} N^2\\|_\\infty\\,.\n\\end{equation}\nSince $g 1_{N<R} N^2$ is a bounded function with\n\\begin{equation}\n\\bigl\\|g 1_{N<R} N^2\\bigr\\|_\\infty \\leq R^2\\, \\|f\\|_\\infty\\,,\n\\end{equation}\nit follows from \\eqref{cond1} that for $\\tilde{\\mu}=\\tilde{\\mu}^{\\psi,b}_1$ the term \\eqref{term1} is less than $\\|f\\|_\\infty\\, \\varepsilon\/4$.\n\nThe term \\eqref{term2} is less than or equal to\n\\begin{align}\n&\\|f\\|_\\infty \\, \\tilde{\\mu}(1_{N\\geq R}N^2)\\\\ \n&= \\|f\\|_\\infty \\, \\bigl(\\tilde{\\mu}(N^2) - \\tilde{\\mu}(1_{N<R} N^2) \\bigr)\\\\\n&= \\|f\\|_\\infty \\, \\bigl(1 - \\tilde{\\mu}(1_{N<R} N^2) \\bigr)\\\\\n&\\leq \\|f\\|_\\infty \\, \\Bigl( 1- \\G{\\rho_1}(1_{N<R} N^2) + \\bigl| \\tilde{\\mu}(1_{N<R} N^2) - \\G{\\rho_1} (1_{N<R} N^2)\\bigr| \\Bigr)\\,,\n\\end{align}\nwhere we have used Lemma~\\ref{lem:n}. By Lemma~\\ref{lem:R},\n\\begin{equation}\\label{Grho1chiN2}\n\\G{\\rho_1}(1_{N<R}N^2)\n> 1-\\varepsilon\/4\\,,\n\\end{equation}\nso the term \\eqref{term2} is less than or equal to\n\\begin{equation}\\label{bound9}\n\\|f\\|_\\infty \\, \\Bigl( \\varepsilon\/4 + \\bigl| \\tilde{\\mu}(1_{N<R} N^2) - \\G{\\rho_1} (1_{N<R} N^2)\\bigr| \\Bigr)\\,.\n\\end{equation}\nNow suppose \\y{that also} $\\psi\\in \\tilde{M}\\Bigl(1_{N<R} N^2,\\varepsilon\/4R^2\\Bigr)$, which means that $\\psi\\in\\mathscr{R}(\\rho_1)$ and\n\\begin{equation}\\label{cond2}\n\\bigl| \\tilde{\\mu}^{\\psi,b}_1(1_{N<R} N^2) -\\G{\\rho_1}(1_{N<R} N^2) \\bigr| < \\frac{\\varepsilon}{4R^2} \\,\\|1_{N<R} N^2\\|_\\infty = \\frac{\\varepsilon}{4}\\,.\n\\end{equation}\nIt follows from \\eqref{bound9} and \\eqref{cond2} that for $\\tilde{\\mu}=\\tilde{\\mu}^{\\psi,b}_1$, the term \\eqref{term2} is less than $\\|f\\|_\\infty\\, \\varepsilon\/2$.\n\nThe term \\eqref{term3} is less than or equal to\n\\begin{align}\n&\\|f\\|_\\infty \\, \\G{\\rho_1}\\bigl(1_{N\\geq R}N^2\\bigr)\\\\ \n&= \\|f\\|_\\infty \\, \\bigl( 1-\\G{\\rho_1}(1_{N<R} N^2) \\bigr)\\\\\n&< \\|f\\|_\\infty\\, \\frac{\\varepsilon}{4}\\,,\n\\end{align}\nwhere we have used $\\G{\\rho_1}(N^2)=1$ \\y{and \\eqref{Grho1chiN2}.}\n\nPutting together the above bounds for the terms \\eqref{term1}, \\eqref{term2}, and \\eqref{term3}, we obtain that their sum is bounded by $\\varepsilon\\|f\\|_\\infty(\\tfrac{1}{4}+\\tfrac{1}{2}+\\tfrac{1}{4})= \\varepsilon \\|f\\|_\\infty$, so \\eqref{toshow} is satisfied and $\\psi\\in M(f,\\varepsilon)$.\n\\endproof\n\n\n\\proof[Proof of Theorem~\\ref{thm1}]\nSuppose we are given $0<\\varepsilon<1$, $0<\\delta<1$, and $d_1\\in\\mathbb{N}$. Set\n\\begin{equation}\nD_2=D_2(\\varepsilon,\\delta,d_1) = \n \\tilde{D}_2\\Bigl(\\frac{\\varepsilon}{4R^2},\\frac{\\delta}{2},d_1\\Bigr) \n\\end{equation}\nwith $\\tilde{D}_2$ as provided by Lemma~\\ref{lemma1} and $R$ as in Lemma~\\ref{lem:M}. \nNow consider any $d_2\\in\\mathbb{N}$ with $d_2>D_2$, any $\\mathscr{H}_1$ and $\\mathscr{H}_2$ with $\\dim\\mathscr{H}_{1\/2}=d_{1\/2}$, any orthonormal basis $b$ of $\\mathscr{H}_2$, any $\\rho_1\\in\\mathscr{D}(\\mathscr{H}_1)$, and any \\y{bounded measurable} function $f:\\mathbb{S}(\\mathscr{H}_1)\\to\\mathbb{R}$. Then, by two applications of Lemma~\\ref{lemma1},\n\\begin{align}\nu_{\\rho_1}\\Bigl( \\tilde{M}\\Bigl(g 1_{N<R} N^2,\\frac{\\varepsilon}{4R^2}\\Bigr) \\Bigr) &\\geq 1-\\delta\/2\\,,\\\\\nu_{\\rho_1}\\Bigl(\n\\tilde{M}\\Bigl(1_{N<R} N^2,\\frac{\\varepsilon}{4R^2}\\Bigr)\\Bigr) &\\geq 1-\\delta\/2\\,.\n\\end{align}\nTherefore, $u_{\\rho_1}$ of the intersection of these sets is greater than or equal to $1-\\delta$. By Lemma~\\ref{lem:M}, this intersection is contained in $M(f,\\varepsilon)$. Thus, $u_{\\rho_1}(M(f,\\varepsilon))\\geq 1-\\delta$, which is what we wanted to show.\n\\endproof\n\n\n\n\\subsection{Proof of Theorem~\\ref{corbasis}}\n\n\n\\proof \nNote that for any unitary $U$ on $\\mathscr{H}_2$\n\\begin{equation}\\label{unitaryequiv}\n\\scp{U^{-1}b_j}{\\psi} = \\scp{b_j}{\\,I_1\\otimes U\\,\\psi}\\, .\n\\end{equation}\n\\y{From this fact and the fact that the Haar measure is invariant under $U\\mapsto\nU^{-1}$ it follows that the distribution of $\\mu_1^{\\psi,b}$, when $\\psi\\in\\mathscr{R}(\\rho_1)$\nis $u_{\\rho_1}$-distributed and $b$ is fixed, is the same as when $b$ is $u_{ONB}$-distributed and $\\psi\\in\\mathscr{R}(\\rho_1)$ is fixed. Thus, Theorem~\\ref{corbasis} is equivalent to Theorem~\\ref{thm1}. (It also follows that the distribution of $\\mu_1^{\\psi,b}$, when $\\psi\\in\\mathscr{R}(\\rho_1)$ is $u_{\\rho_1}$-distributed and $b$ is fixed, does not depend on $b$.)}\n\\endproof\n\n\n\n\n\\subsection{Continuity of GAP}\n\nFor the proofs of Theorems~\\ref{thm3} and \\ref{thm4}, we will exploit canonical typicality, i.e., the fact that for most $\\psi\\in\\mathbb{S}(\\mathscr{H}_R)$, the reduced density matrix $\\rho_1^\\psi$ is close to $\\tr_2\\rho_R$. Theorems~\\ref{thm3} and \\ref{thm4} then follow from Theorem~\\ref{corbasis} via suitable continuity of the mapping $\\rho\\mapsto GAP(\\rho)$. The following two lemmas provide somewhat different statements about continuity:  \\y{Recall that $\\mathscr{D}_{\\geq \\gamma}(\\mathscr{H})$ is the set of density matrices with all eigenvalues greater than or equal to $\\gamma$. Lemma~\\ref{lem:contLinfty} asserts that $GAP(\\rho)(f)$ depends in a \\emph{uniformly} continuous way on both $\\rho$ and $f$ when we restrict $\\rho$ to $\\mathscr{D}_\\gamma(\\mathscr{H})$ for arbitrarily small $\\gamma>0$; continuity is not uniform without this restriction. However, Lemma~\\ref{lem:cont} asserts that for any \\emph{fixed} and \\emph{continuous} test function $f$, continuity is uniform in $\\rho$ without restrictions.}\n\n\n\\begin{lemma}\\label{lem:cont}\nFor every $0<\\varepsilon<1$, every $d\\in\\mathbb{N}$, every Hilbert space $\\mathscr{H}$ with $\\dim \\mathscr{H}=d$, and every continuous function $f:\\mathbb{S}(\\mathscr{H})\\to\\mathbb{R}$ there is $r=r(\\varepsilon,d,f)>0$ such that for all $\\rho,\\Omega\\in\\mathscr{D}(\\mathscr{H})$,\n\\begin{equation}\n\\text{if }\\|\\rho-\\Omega\\|_{\\tr} < r\n\\text{ then }\n\\bigl| GAP(\\rho)(f)-GAP(\\Omega)(f) \\bigr| < \\varepsilon\\,.\n\\end{equation}\n\\end{lemma}\n\nWhile all norms on $\\mathscr{D}(\\mathscr{H})$ are equivalent for $\\dim \\mathscr{H}< \\infty$, we use the trace norm $\\|\\cdot\\|_{\\tr}$ here because in this norm the continuity extends to $\\dim \\mathscr{H}=\\infty$ and because it is used in Lemma~\\ref{lem:can}.\n\nTo formulate the other continuity statement, let $u_{\\mathbb{S}(\\mathscr{H})}$ denote the normalized uniform measure on the unit sphere in $\\mathscr{H}$. For any density matrix $\\rho\\in\\mathscr{D}(\\mathscr{H})$ of which zero is not an eigenvalue, $GAP(\\rho)$ possesses a density relative to $u_{\\mathbb{S}(\\mathscr{H})}$ \\cite{Gold1}.\n\n\\begin{lemma}\\label{lem:contLinfty}\nFor every $0<\\varepsilon<1$, every $d\\in\\mathbb{N}$, every Hilbert space $\\mathscr{H}$ with $\\dim \\mathscr{H}=d$, and every $0<\\gamma<1\/d$, there is $r=r(\\varepsilon,d,\\gamma)>0$ such that for all $\\rho,\\Omega\\in\\mathscr{D}_{\\geq\\gamma}(\\mathscr{H})$,  \n\\begin{equation}\\label{eq:contLinfty}\n\\text{if }\\|\\rho-\\Omega\\|_{\\tr}<r\n\\text{ then }\n\\biggl\\| \\frac{dGAP(\\rho)}{du_{\\mathbb{S}(\\mathscr{H})}}\n-\\frac{dGAP(\\Omega)}{du_{\\mathbb{S}(\\mathscr{H})}} \n\\biggr\\|_{\\infty} < \\varepsilon\\,.\n\\end{equation}\nAs a consequence, for such $\\rho$ and $\\Omega$,\n\\begin{equation}\\label{eq:contL1}\n\\bigl| GAP(\\rho)(f)-GAP(\\Omega)(f) \\bigr|<\\varepsilon \\, \\|f\\|_1\n\\end{equation}\nfor every $f\\in L^1\\bigl(\\mathbb{S}(\\mathscr{H}),u_{\\mathbb{S}(\\mathscr{H})}\\bigr)$.\n\\end{lemma}\n\nIt follows in particular that for any \\emph{fixed} density matrix $\\Omega$ of which zero is not an eigenvalue and any sequence $(\\rho_n)$ of density matrices with $\\rho_n\\to\\Omega$, the density of $GAP(\\rho_n)$ converges to that of $GAP(\\Omega)$ in the $\\|\\cdot\\|_\\infty$ norm: Take $\\gamma>0$ to be less than the smallest eigenvalue of $\\Omega$ and note that only finitely many $\\rho_n$ can lie outside $\\mathscr{D}_{\\geq \\gamma}(\\mathscr{H})$.\n\nTo see that in Lemma~\\ref{lem:contLinfty} $\\mathscr{D}_{\\geq\\gamma}(\\mathscr{H})$ cannot be replaced by $\\mathscr{D}(\\mathscr{H})$ (i.e., that continuity is not uniform without restrictions), note that, when 0 is an eigenvalue of $\\Omega$, $\\GAP{\\Omega}$ does not have a density with respect to $u_{\\mathbb{S}(\\mathscr{H})}$, so that at such an $\\Omega$, $\\rho\\mapsto\\GAP{\\rho}$ is certainly not continuous in $L^\\infty\\bigl(\\mathbb{S}(\\mathscr{H}),u_{\\mathbb{S}(\\mathscr{H})}\\bigr)$ or in the variation distance \\eqref{defvariationdistance}.\n\nTo see that in Lemma~\\ref{lem:cont} one cannot drop the assumption that $f$ is continuous, consider an $\\Omega$ that has zero as an eigenvalue and a $\\rho$ that does not. Then $GAP(\\Omega)$ is concentrated on a subspace of dimension less than $d$ while $GAP(\\rho)$ has a density on the sphere and lies \\emph{near} (rather than \\emph{in}) that subspace. Thus, for a test function $f$ that is bounded measurable but not continuous, $GAP(\\rho)(f)$ does not have to be close to $GAP(\\Omega)(f)$.\n\n\n\\label{sec:contproof}\n\nAs part of the proof of Lemma~\\ref{lem:cont}, we will need the continuity property of Gaussian measures expressed in the next lemma. When $\\mu_n,\\mu$ are measures on a topological space $X$, we write $\\mu_n \\Rightarrow \\mu$ to denote that the sequence of measures $\\mu_n$ converges weakly to $\\mu$. This means that $\\mu_n(f) \\to \\mu(f)$ for every bounded continuous function $f:X\\to \\mathbb{R}$ and implies that the same thing is true for every bounded measurable\nfunction $f:X\\to\\mathbb{R}$ such that $\\mu(D(f))=0$, where $D(f)$ is the set of\ndiscontinuities of $f$.\n\n\\begin{lemma}\\label{contGk}\nThe mapping $\\rho\\mapsto\\G{\\rho}$ is continuous in the weak topology on measures: If $\\rho_n \\in \\mathscr{D}(\\mathbb{C}^d)$ for every $n\\in\\mathbb{N}$ and $\\rho_n \\to \\rho$ then $G(\\rho_n) \\Rightarrow G(\\rho)$.\n\\end{lemma}\n\n\\proof \nWe use characteristic functions; as usual, the characteristic\nfunction $\\hat{\\mu}:\\mathbb{R}^{2d} \\to \\mathbb{C}$ of a probability measure\n$\\mu$ on $\\mathbb{R}^{2d}$ is defined by\n\\begin{equation}\n  \\hat{\\mu}(k_1,\\ldots, k_{2d}) = \\int \\mu(dx_1 \\cdots dx_{2d}) \\,\n  \\exp\\Bigl(i\\sum_{j=1}^{2d} k_j x_j \\Bigr)\\,,\n\\end{equation}\nor, in our notation on $\\mathscr{H} = \\mathbb{C}^{d}$,\n\\begin{equation}\n  \\hat{\\mu}(\\phi) = \\int \\mu(d\\psi) \\,\n  \\exp\\Bigl(i\\Re \\scp{\\phi}{\\psi} \\Bigr)\\,,\n\\end{equation}\nwhere $\\Re$ denotes the real part. We write $\\mu_n = G(\\rho_n)$ and\n$\\mu=G(\\rho)$; their characteristic functions are:\n\\begin{equation}\\label{hatGauss}\n  \\hat{\\mu}_n(\\psi) = \\exp\\bigl(-\\bra{\\psi}\\rho_n\\ket{\\psi}\\bigr)\\,, \\quad\n  \\hat{\\mu}(\\psi) = \\exp\\bigl(-\\bra{\\psi}\\rho\\ket{\\psi}\\bigr)\\,.\n\\end{equation}\nIf $\\rho_n \\to \\rho$ then $\\bra{\\psi}\\rho_n\\ket{\\psi} \\to\n\\bra{\\psi}\\rho\\ket{\\psi}$ for every $\\psi$ and thus $\\hat{\\mu}_n \\to \\hat{\\mu}$ pointwise. Since (e.g., \\cite{Bill}) pointwise convergence of\nthe characteristic functions is equivalent (in finite dimension) to\nweak convergence of the associated measures, it follows that\n$G(\\rho_n) \\Rightarrow G(\\rho)$, which is what we wanted to show.\n\\endproof\n\n\n\\proof[Proof of Lemma~\\ref{lem:cont}]\nSince $\\mathscr{D}(\\mathscr{H})$ is compact, uniform continuity follows from continuity. That is, it suffices to show that, assuming $\\rho_n \\in \\mathscr{D}(\\mathscr{H})$ for every $n\\in\\mathbb{N}$,\n\\begin{equation}\\label{GAPcont}\n\\text{if }\\rho_n \\to \\rho\n\\text{ then }GAP(\\rho_n) \\Rightarrow GAP(\\rho)\\,.\n\\end{equation}\nThis follows from Lemma~\\ref{contGk}, the continuity of the adjustment mapping $A$ defined in \\eqref{Adef} in Section~\\ref{sec:GAPdef}, and the continuity of the projection $P:\\mathscr{H}\\setminus\\{0\\}\\to\\mathbb{S}(\\mathscr{H})$. Our first step is to establish the continuity of $A$ on the set of probability measures $\\mu$ on $\\mathscr{H}$ such that $\\int \\mu(d\\psi) \\, \\|\\psi\\|^2 = 1$: If, for every $n\\in\\mathbb{N}$, $\\mu_n$ is a probability measure on the Borel $\\sigma$-algebra of $\\mathscr{H}$ such that $\\int \\mu_n(d\\psi) \\, \\|\\psi\\|^2 = 1$, then\n\\begin{equation}\\label{Acont}\n\\text{if }\\mu_n \\Rightarrow \\mu\n\\text{ and }\\int \\mu(d\\psi) \\,\\|\\psi\\|^2 = 1\n\\text{ then }A\\mu_n\\Rightarrow A\\mu\\,.\n\\end{equation}\n\nFix $\\varepsilon>0$ and an arbitrary non-zero, bounded, continuous\nfunction $f: \\mathscr{H} \\to \\mathbb{R}$. As before, we use the notation $N(\\psi) = \\|\\psi\\|$. Since,\nby hypothesis, $\\mu(N^2) = 1$, there\nexists $R>0$ so large that\n\\begin{equation}\n  \\int_{\\{\\psi\\in \\mathscr{H}: \\|\\psi\\|<R\\}} \\mu(d\\psi) \\, \\|\\psi\\|^2 > 1- \\frac{\\varepsilon}{6\\|f\\|_\\infty}\\,.\n\\end{equation}\nLet the ``cut-off function''\n$\\chi_0: [0,\\infty) \\to [0,1]$ be any continuous function such that\n$\\chi_0(x)=1$ for $x\\leq R$ and $\\chi_0(x)=0$ for $x\\geq 2R$; set $\\chi(\\psi)=\\chi_0(\\|\\psi\\|)$. Because $\\chi  N^2$ and $f\\chi N^2$ are bounded continuous functions, and because\n$\\mu_n \\Rightarrow \\mu$, we have that $\\mu_n(\\chi  N^2) \\to \\mu(\\chi \nN^2)$ and $\\mu_n(f\\chi N^2) \\to \\mu(f\\chi N^2)$; that is, there is an $n_1\\in \\mathbb{N}$ such that, for all $n>n_1$,\n\\begin{equation}\n  \\left| \\mu_n(\\chi  N^2) - \\mu(\\chi  N^2) \\right| < \\frac{\\varepsilon}{3\\|f\\|_\\infty} \n\\end{equation}\nand\n\\begin{equation}\n  \\left| \\mu_n(f\\chi  N^2) - \\mu(f\\chi  N^2) \\right| < \\frac{\\varepsilon}{3}\\,.\n\\end{equation}\nThus, for all $n>n_1$, we have that\n\\begin{align}\n  &\\left| A\\mu_n(f) - A\\mu(f) \\right| \n  = \\left| \\mu_n(f N^2) - \\mu(f N^2) \\right| \\\\\n  &\\leq \\left| \\mu_n(f\\chi  N^2) - \\mu(f\\chi  N^2) \\right| +\n  \\left| \\mu_n\\bigl( f(1-\\chi) N^2 \\bigr) \\right| +\n  \\left| \\mu\\bigl( f(1-\\chi)N^2 \\bigr) \\right| \\\\\n  &< \\frac{\\varepsilon}{3} + \\|f\\|_\\infty \\mu_n\\bigl( (1-\\chi)N^2 \\bigr) \n  + \\|f\\|_\\infty \\mu\\bigl( (1-\\chi)N^2\\bigr)\\\\\n  &= \\frac{\\varepsilon}{3} + \\|f\\|_\\infty \\bigl(1-\\mu_n(\\chi N^2)\\bigr)\n  + \\|f\\|_\\infty \\bigl(1-\\mu(\\chi N^2)\\bigr)\\\\\n  &\\leq \\frac{\\varepsilon}{3} + 2\\|f\\|_\\infty \\bigl(1-\\mu(\\chi N^2)\\bigr) \n  + \\|f\\|_\\infty \\bigl|\\mu_n(\\chi N^2)-\\mu(\\chi N^2)\\bigr|\\\\\n  &\\leq \\frac{\\varepsilon}{3}+\\frac{\\varepsilon}{3}+\\frac{\\varepsilon}{3}\n  = \\varepsilon\\,.\n\\end{align}\nThis proves \\eqref{Acont}.\\footnote{We remark that the \nhypothesis $\\int \\mu(d\\psi) \\|\\psi\\|^2=1$ cannot\nbe dropped, that is, does not follow from $\\int \\mu_n(d\\psi)\n\\|\\psi\\|^2=1$. An example is $\\mu_n = (1-1\/n) \\delta_0 + (1\/n)\n\\delta_{\\psi_n}$, where $\\delta_\\phi$ means the Dirac delta measure\nat $\\phi$ and $\\psi_n$ is any vector with $\\|\\psi_n\\|^2 = n$; then\n$\\mu_n$ is a probability measure with $\\int \\mu_n(d\\psi)\n\\|\\psi\\|^2=1$ but $\\mu_n \\Rightarrow \\delta_0$, which has $\\int\n\\delta_0(d\\psi) \\|\\psi\\|^2=0$.}\n\n\nWe are now ready to establish \\eqref{GAPcont}. Suppose $\\rho_n\\to\\rho$. We\nhave that $\\GAP{\\rho_n}=P_*A(\\G{\\rho_n})$ and that $\\left(AG(\\rho)\\right)(0)=0$. Since\n$\\psi\\mapsto P\\psi$ is continuous for $\\psi\\neq 0$, \\eqref{GAPcont}\nfollows from \\eqref{Acont} and Lemma~\\ref{contGk}. This completes the proof of Lemma~\\ref{lem:cont}.\n\\endproof\n\n\\bigskip\n\n\\proof[Proof of Lemma~\\ref{lem:contLinfty}]\nWe first note that, for any self-adjoint $d\\times d$ matrix $A$ and $\\psi\\in\\mathbb{S}(\\mathbb{C}^d)$,\n\\begin{equation}\\label{trnormineq}\n\\Bigl| \\scp{\\psi}{A|\\psi} \\Bigr| \\leq \\|A\\| \\leq \\|A\\|_{\\tr}\\,.\n\\end{equation}\n\nFor any density matrix $\\rho\\in\\mathscr{D}(\\mathscr{H})$ of which zero is not an eigenvalue, the density of $GAP(\\rho)$ relative to $u_{\\mathbb{S}(\\mathscr{H})}$ is given by \\cite{Gold1}\n\\begin{align}\n  \\frac{dGAP(\\rho)}{du_{\\mathbb{S}(\\mathscr{H})}}(\\psi) &= \\frac{1}{\\pi^d \\, \\det \\rho}\n  \\int\\limits_0^\\infty dr \\, r^{2d-1} \\, r^2 \\exp(-r^2 \\langle \\psi |\n  \\rho^{-1} | \\psi \\rangle) = \\\\ &= \\frac{d!}{2\\pi^{d} \\, \\det\n  \\rho} \\, \\langle \\psi | \\rho^{-1} | \\psi \\rangle^{-d-1}\n  \\,.\\label{mupowerlaw}\n\\end{align}\n\nUsing the last expression, we will now show that \\eqref{eq:contLinfty} holds when $\\rho$ is sufficiently close to $\\Omega$. \nThis follows from the facts (i)~that, on $\\mathscr{D}_{\\geq \\gamma}(\\mathscr{H})$, the functions $\\rho\\mapsto1\/\\det\\rho$ and $\\rho\\mapsto\\rho^{-1}$ are uniformly continuous, (ii)~that\n\\begin{equation}\n\\Bigl| \\scp{\\psi}{\\rho^{-1}|\\psi}-\\scp{\\psi}{\\Omega^{-1}|\\psi}\\Bigr| \\leq \\|\\rho^{-1}-\\Omega^{-1}\\|_{\\tr}\n\\end{equation}\nfor all $\\psi\\in\\mathbb{S}(\\mathscr{H})$, (iii)~that the function $x\\mapsto x^{-d-1}$ is uniformly continuous on the interval $[1,\\infty)$, and (iv)~that $\\scp{\\psi}{\\rho^{-1}|\\psi}\\geq 1$, $\\scp{\\psi}{\\Omega^{-1}|\\psi}\\geq 1$. \nThis establishes the existence of $r(\\varepsilon,d,\\gamma)>0$ as described in Lemma~\\ref{lem:contLinfty}. \n\nNow \\eqref{eq:contL1} follows from \\eqref{eq:contLinfty} according to\n\\begin{align}\n&\\bigl| GAP(\\rho)(f)-GAP(\\Omega)(f) \\bigr|\\nonumber\\\\\n&=\\Biggl|\\,\\,\\int\\limits_{\\mathbb{S}(\\mathscr{H})}du_{\\mathbb{S}(\\mathscr{H})} \\biggl( \\frac{dGAP(\\rho)}{du_{\\mathbb{S}(\\mathscr{H})}}(\\psi)\n-\\frac{dGAP(\\Omega)}{du_{\\mathbb{S}(\\mathscr{H})}}(\\psi)\\biggr) f(\\psi) \\Biggr|\\\\\n&\\leq\\int\\limits_{\\mathbb{S}(\\mathscr{H})}du_{\\mathbb{S}(\\mathscr{H})} \\biggl| \\frac{dGAP(\\rho)}{du_{\\mathbb{S}(\\mathscr{H})}}(\\psi)\n-\\frac{dGAP(\\Omega)}{du_{\\mathbb{S}(\\mathscr{H})}}(\\psi)\\biggr| \\, |f(\\psi)|\n<\\varepsilon \\, \\|f\\|_1\\,.\n\\end{align}\n\\endproof\n\n\n\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm3}}\n\n\\proof[Proof of Theorem~\\ref{thm3}.] \nSuppose we are given $0<\\varepsilon<1$, $0<\\delta<1$, $d_1\\in\\mathbb{N}$, a Hilbert space $\\mathscr{H}_1$ of dimension $d_1$, and a continuous function $f:\\mathbb{S}(\\mathscr{H}_1)\\to\\mathbb{R}$. Set\n\\begin{equation}\nD_R=D_R(\\varepsilon,\\delta,d_1,f) = \\frac{4}{r(\\varepsilon\/2,d_1,f)^2}\\max\\Bigl(d_1^2, 18\\pi^3 \\log (8\/\\delta) \\Bigr)\\,,\n\\end{equation}\nwith $r(\\varepsilon,d,f)$ as provided by Lemma~\\ref{lem:cont}. Now consider any $d_R,d_2\\in\\mathbb{N}$ with $d_R>D_R$ and $d_2>D_2(\\varepsilon\/2\\|f\\|_\\infty,\\delta\/2,d_1)$, any $\\mathscr{H}_2$ and $\\mathscr{H}_R\\subseteq \\mathscr{H}_1\\otimes\\mathscr{H}_2$ with $\\dim\\mathscr{H}_{2\/R}=d_{2\/R}$. Let $M(f,\\varepsilon)$ be the set mentioned in \\eqref{eq:thm3}, \n\\begin{equation}\nM(f,\\varepsilon)=\n\\left\\{ \\bigl( \\psi, b \\bigr) \\in\n  \\mathbb{S}(\\mathscr{H}_R) \\times ONB (\\mathscr{H}_2) : \\bigl|\n  \\mu_1^{\\psi,b}(f) - \\GAP{\\tr_2 \\rho_R}(f) \\bigr| < \\varepsilon \\right\\},\n\\end{equation}\nlet\n\\begin{equation}\nM'(f,\\varepsilon) =\n\\left\\{\\bigl( \\psi, b \\bigr) \\in\n  \\mathbb{S}(\\mathscr{H}_R) \\times ONB (\\mathscr{H}_2) : \\bigl|\n  \\mu_1^{\\psi,b}(f) - \\GAP{\\rho_1^\\psi}(f) \\bigr| < \\varepsilon \\right\\}\n\\end{equation}\nand\n\\begin{equation}\nM''(\\varepsilon) =\n\\Bigl\\{\\psi\\in\\mathbb{S}(\\mathscr{H}_R): \\|\\rho_1^\\psi -\n\\tr_2 \\rho_R\\|_{\\tr}<\\varepsilon \\Bigr\\}.\n\\end{equation}\nThen, by Lemma~\\ref{lem:cont},\n\\begin{equation}\\label{MM'M''}\nM(f,\\varepsilon) \\supseteq \nM'\\Bigl(f,\\frac{\\varepsilon}{2}\\Bigr) \\cap \n\\Bigl[M''\\Bigl(r\\bigl(\\frac{\\varepsilon}{2},d_1,f\\bigr)\\Bigr)\\times ONB(\\mathscr{H}_2)\\Bigr]\\,.\n\\end{equation}\n\nTheorem~\\ref{corbasis} yields, using our assumption $d_2>D_2(\\varepsilon\/2\\|f\\|_\\infty,\\delta\/2,d_1)$, that for every $\\psi\\in\\mathbb{S}(\\mathscr{H}_R)$,\n\\begin{equation}\nu_{ONB} \\Bigl\\{ b \\in ONB(\\mathscr{H}_2): \\bigl|\\mu_1^{\\psi,b}(f) - \\GAP{\\rho_1^\\psi}(f)\\bigr|< \\frac{\\varepsilon}{2} \\Bigr\\} \\geq 1-\\delta\/2\\,.\n\\end{equation}\nThus, averaging over $\\psi\\in\\mathbb{S}(\\mathscr{H}_R)$ according to $u_R$,\n\\begin{equation}\\label{M'}\nu_R\\times u_{ONB} \\Bigl( M'(f,\\varepsilon\/2) \\Bigr) \\geq 1 -\\delta\/2\\,.\n\\end{equation}\n\nLemma~\\ref{lem:can} with $\\eta = r\/2$ for $r=r(\\varepsilon\/2,d_1,f)$ yields, using our assumption $d_R > 4d_1^2\/r^2$, which implies that $d_1\/\\sqrt{d_R}\\leq r\/2$, that\n\\begin{equation}\nu_R(M''(r)) \\geq 1 - 4\\exp\\Bigl(-\\frac{d_R r^2}{18\\pi^34}\\Bigr)\\,.\n\\end{equation}\nUsing our assumption $d_R > 18\\pi^34\\log(8\/\\delta)\/r^2$, the right hand side is greater than or equal to $1-\\delta\/2$, and thus\n\\begin{equation}\\label{M''}\nu_R\\times u_{ONB} \\Bigl[M''(r)\\times ONB(\\mathscr{H}_2)\\Bigr] \\geq 1- \\delta\/2\\,.\n\\end{equation}\n\nFrom \\eqref{M'}, \\eqref{M''}, and \\eqref{MM'M''} together we have that\n\\begin{equation}\nu_R\\times u_{ONB} \\Bigl[ M(f,\\varepsilon) \\Bigr] \\geq 1 -\\delta\\,,\n\\end{equation}\nwhich is what we wanted to show.\n\\endproof\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm4}}\n\n\\proof[Proof of Theorem~\\ref{thm4}]\nSuppose we are given $0<\\varepsilon<1$, $0<\\delta<1$, $d_1\\in\\mathbb{N}$, $0<\\gamma<1\/d_1$, and a Hilbert space $\\mathscr{H}_1$ of dimension $d_1$. Set\n\\begin{align}\nD'_R&=D'_R(\\varepsilon,\\delta,d_1,\\gamma) = \\frac{4}{(r')^2}\\max\\Bigl(d_1^2, 18\\pi^3 \\log (8\/\\delta) \\Bigr)\\,,\\\\\nr'&=r'(\\varepsilon,d_1,\\gamma)=\\frac{1}{2}r(\\varepsilon\/2,d_1,\\gamma)\\,,\n\\end{align}\nwith $r(\\varepsilon,d,\\gamma)$ as provided by Lemma~\\ref{lem:contLinfty}. Now consider any $d_R,d_2\\in\\mathbb{N}$ with $d_R>D'_R$ and $d_2>D_2(\\varepsilon\/2,\\delta\/2,d_1)$, any $\\Omega\\in\\mathscr{D}_{\\geq\\gamma}(\\mathscr{H}_1)$, any $\\mathscr{H}_2$ and $\\mathscr{H}_R\\subseteq \\mathscr{H}_1\\otimes\\mathscr{H}_2$ with $\\dim\\mathscr{H}_{2\/R}=d_{2\/R}$, and any bounded measurable function $f:\\mathbb{S}(\\mathscr{H}_1)\\to\\mathbb{R}$. Let $M_0(f,\\varepsilon)$ be the set mentioned in \\eqref{eq:thm4}, \n\\begin{equation}\nM_0(f,\\varepsilon)=\n\\left\\{ \\bigl( \\psi, b \\bigr) \\in\n  \\mathbb{S}(\\mathscr{H}_R) \\times ONB (\\mathscr{H}_2) : \\bigl|\n  \\mu_1^{\\psi,b}(f) - GAP(\\Omega)(f) \\bigr| < \\varepsilon \\, \\|f\\|_\\infty \\right\\},\n\\end{equation}\nlet, as in the proof of Theorem~\\ref{thm3},\n\\begin{equation}\nM'(f,\\varepsilon) =\n\\left\\{\\bigl( \\psi, b \\bigr) \\in\n  \\mathbb{S}(\\mathscr{H}_R) \\times ONB (\\mathscr{H}_2) : \\bigl|\n  \\mu_1^{\\psi,b}(f) - GAP(\\rho_1^\\psi)(f) \\bigr| < \\varepsilon \\right\\},\n\\end{equation}\nlet\n\\begin{equation}\nM_0''(\\varepsilon) =\n\\Bigl\\{\\psi\\in\\mathbb{S}(\\mathscr{H}_R): \\|\\rho_1^\\psi -\n\\Omega\\|_{\\tr}<\\varepsilon \\Bigr\\},\n\\end{equation}\nand let, as in the proof of Theorem~\\ref{thm3},\n\\begin{equation}\nM''(\\varepsilon) =\n\\Bigl\\{\\psi\\in\\mathbb{S}(\\mathscr{H}_R): \\|\\rho_1^\\psi -\n\\tr_2 \\rho_R\\|_{\\tr}<\\varepsilon \\Bigr\\}.\n\\end{equation}\nNow assume $\\bigl\\|\\tr_2 \\rho_R-\\Omega\\bigr\\|_{\\tr}<r'$. Then\n\\begin{equation}\\label{M0''M''}\nM_0''(2r') \\supseteq M''(r')\n\\end{equation}\nand, by Lemma~\\ref{lem:contLinfty} and $\\|f\\|_1\\leq \\|f\\|_\\infty$,\n\\begin{equation}\\label{M0M'M0''}\nM_0(f,\\varepsilon) \\supseteq \nM'\\biggl(f,\\frac{\\varepsilon\\|f\\|_\\infty}{2}\\biggr) \\cap \n\\Bigl[M_0''(2r')\\times ONB(\\mathscr{H}_2)\\Bigr]\\,.\n\\end{equation}\n\nAs in the proof of Theorem~\\ref{thm3}, Theorem~\\ref{corbasis} yields \\eqref{M'} \\y{with $\\varepsilon$ replaced by $\\varepsilon\\|f\\|_\\infty$} using our assumption $d_2>D_2(\\varepsilon\/2,\\delta\/2,d_1)$, and Lemma~\\ref{lem:can} yields \\eqref{M''} \\y{with $r$ replaced by $r'$}, using our assumption $d_R>D'_R$. From \\eqref{M'}, \\eqref{M''}, \\eqref{M0''M''}, and \\eqref{M0M'M0''} together we have that\n\\begin{equation}\nu_R\\times u_{ONB} \\Bigl[ M_0(f,\\varepsilon) \\Bigr] \\geq 1 -\\delta\\,,\n\\end{equation}\nwhich is what we wanted to show.\n\\endproof\n\n\n\n\n\n\n\\bigskip\n\n\\noindent\\textit{Acknowledgments.} \nWe are grateful to Benoit Collins for helpful discussions.\nS.~Goldstein was supported in part by the National Science Foundation [grant DMS-0504504].\nJ.~L.~Lebowitz and C.~Mastrodonato were supported in part by the National Science Foundation [grant  DMR 08-02120] and the Air Force Office of Scientific Research [grant AF-FA 49620-01-0154].\nN.~Zangh\\`\\i\\ was supported in part by Istituto Nazionale di Fisica Nucleare. \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzznmjv b/data_all_eng_slimpj/shuffled/split2/finalzznmjv
new file mode 100644
index 0000000000000000000000000000000000000000..9c74d3536e52f30eecf5364a2bcdcea404cef734
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzznmjv
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\nWe shall consider a boundary value problem\n\\begin{equation}\n \\begin{cases}\n   u_{rr}+ \\dfrac{n-1}{r}u_r -\\omega u+u^p-u^{2p-1}=0 & \\text{for $r>0$}, \\\\\n   u_r(0)=0, \\\\\n   \\displaystyle \\lim_{r \\to \\infty} u(r) =0, \\label{b}\n \\end{cases} \n\\end{equation} \nwhere $n \\in \\mathbb{N}$, $p>1$ and $\\omega >0$.\nThe above problem arises in the study of \n\\begin{equation}\n \\begin{cases}\n   \\triangle u- \\omega u+u^p-u^{2p-1}=0  & \\text{in  $\\mathbb{R}^n$},\\\\\n   \\displaystyle \\lim_{\\lvert x \\rvert \\to \\infty} u(x)  =0. \\label{a}\n \\end{cases}  \n\\end{equation}\nIndeed, the classical work of Gidas, Ni and Nirenberg~\\cite{G1,G2} tells us that any positive solution to \\eqref{a} \nis radially symmetric. \nOn the other hand, for a solution $u(r)$ of \\eqref{b}, $v(x):=u(\\lvert x \\rvert)$ is a solution to \\eqref{a}.  \n\nThe condition to assure the existence of positive solutions to \\eqref{a} (and so \\eqref{b}) was given by \nBerestycki and Lions~\\cite{B1} and Berestycki, Lions and Peletier~\\cite{B2}:\nA solution to \\eqref{b} with fixed $p>1$ exists if and only if $\\omega \\in (0, \\omega_p)$, where \n\\begin{equation*}\n\\omega_p=\\dfrac{p}{(p+1)^2}.\n\\end{equation*}\nWe shall review what this $\\omega_p$ is for in Section 2.\nThroughout this paper, \\textit{a solution} means a classical solution.\n\nUniqueness of positive solutions to \\eqref{b} had long remained unknown.\nFinally in 1998 Ouyang and Shi~\\cite{OS} proved uniqueness for \\eqref{b} with all $\\omega \\in (0, \\omega_p)$, $p>1$.\nSee also Kwong and Zhang~\\cite{KZ}.\n\nIn this present paper, we prove that for $\\omega$ close to $\\omega_p$, the uniqueness result is obtained directly \nfrom the classial result given by Peletier and Serrin~\\cite{PS} in 1983.\nFor another attempt to obtain the uniqueness when $\\omega$ is close to $\\omega_p$, see Mizumachi~\\cite{M}.\nOur result of the present paper is the following:\n\\begin{Thm}\nLet $n \\in \\mathbb{N}$, $p>1$ and $\\omega\\in[a_p,\\omega_p)$, where $a_p:=\\dfrac{p(7p-5)}{4(p+1)(2p-1)^2}$.\nThen \\eqref{b} has exactly one positive solution.  \\label{thm}\n\\end{Thm}\n\n\\begin{Rem}\nNote that \n\\begin{equation*}\n0<a_p<\\omega_p=\\dfrac{p}{(p+1)^2}, \\qquad p>1.\n\\end{equation*}\n\\end{Rem}\n\nIn the next section we clarify the definitions of $\\omega_p$ and $a_p$ from the point of view from~\\cite{PS}.\n\n\\section{Study of the nonlinearity as a function}\n\nIn this section, we study the properties of the function $f(u):=-\\omega u+u^p-u^{2p-1}$ in $(0, \\infty)$,\nwhere $\\omega>0$ and $p>1$ are given constants.\n\nFirst we define $F(u):=\\displaystyle \\int_0^u f_{\\omega,p}(s)ds$, and by a direct calculation we have\n\\begin{align}\n   F(u) &= -\\frac{\\omega}{2}u^2+\\frac{u^{p+1}}{p+1}-\\frac{u^{2p}}{2p} \\notag   \\\\\n        &= \\frac{u^2}{2p(p+1)}\\left[ -\\omega p(p+1)+2pu^{p-1}-(p+1)u^{2(p-1)}\\right].  \\label{c}\n\\end{align}\nThere are two cases of concern:\n\\begin{itemize}\n\\item[(a)]~~~~~ $\\omega < \\omega_p$ $\\Longleftrightarrow$ $F$ has two zeros in $(0,\\infty)$.\n\\item[(b)]~~~~~ $\\omega \\ge \\omega_p$ $\\Longleftrightarrow$ $F$ has at most one zero in $(0,\\infty)$.\n\\end{itemize}\n\nThe condition to assure the existence of positive solutions of \\eqref{b} given in~\\cite{B1,B2} is the following;\n\\begin{Lem}\nThe problem \\eqref{b} has a positive solution if and only if both of the following hypotheses are fulfilled:\n\\begin{itemize}\n\\item[(H1)] $\\displaystyle \\lim_{u \\to +0}\\frac{f(u)}{u}$ exists and is negative,\n\\item[(H2)] $F(\\delta)>0$ for some positive constant $\\delta$.\n\\end{itemize} \\label{key}\n\\end{Lem}\n\\begin{Lem}\nThe problem \\eqref{b} has a positive solution if and only if \n\\begin{equation*}\n\\omega \\in (0,\\omega_p)   \n\\end{equation*}    \nfor $p>1$.\n\\label{condexlem}\n\\end{Lem}\n\\begin{proof}\n(H1) is equivalent to the condition $\\omega > 0$.\n(H2) is equivalent to the condition (a) above.\n\\end{proof}\n\nThis is the origin of $\\omega_p$. Next we turn to the exponent $a_p$.\n\nAs a preparation, we calculate the derivatives of $f(u)=-\\omega u+u^p-u^{2p-1}$:\n\\begin{align*}\n  f'(u)    &= -\\omega+pu^{p-1}-(2p-1)u^{2(p-1)},  \\\\\n  f''(u)   &= 2(p-1)(2p-1)u^{p-2}\\left[ \\dfrac{p}{2(2p-1)} -u^{p-1} \\right].\n\\end{align*}\n\nWe shall introduce four positive constants $\\alpha$, $b$, $c$ and $\\beta$. \n\\begin{itemize}\n\\item Let $\\alpha$ denote the unique zero of $f''$ in $(0,\\infty)$: \n$\\displaystyle \\alpha=\\left[ \\frac{p}{2(2p-1)}\\right]^{\\frac{1}{p-1}}$.\n\\item Let $b$ denote the first zero of $f$ in $(0,\\infty)$:\n$\\displaystyle b=\\left[ \\frac{1-\\sqrt{1-4\\omega}}{2}\\right]^{\\frac{1}{p-1}}$.\n\\item Let $c$ denote the last zero of $f$ in $(0,\\infty)$:\n$\\displaystyle c=\\left[ \\frac{1+\\sqrt{1-4\\omega}}{2}\\right]^{\\frac{1}{p-1}}$.\n\\item Let $\\beta$ denote the first zero of $F$ in $(0,\\infty)$:\n$\\displaystyle \\beta=\\left[ \\frac{p}{p+1} \\left(1-\\sqrt{1-\\frac{(p+1)^2}{p}\\omega} \\right)\\right]^{\\frac{1}{p-1}}$.\n\\end{itemize}\nIt is easy to check that \n\\begin{equation}\n  \\beta \\in (b, c) \\label{ineq}  \n\\end{equation}\neither by observing the graphs or by a straightforward calculation.\nFrom \\eqref{ineq} we deduce\n\\begin{equation}\nf(\\beta)>0,\\label{z}\n\\end{equation} \nwhich will be used later.\n\nWe are not able to give a clear explanation on the relation between $\\alpha$ and $\\beta$.\n\\begin{Lem}\nThe condition $\\alpha \\le \\beta$ is equivalent to \n$\\omega \\ge a_p=\\displaystyle \\frac{p(7p-5)}{4(p+1)(2p-1)^2}$. \\label{eta}\n\\end{Lem}\n\\begin{proof}\nA simple calculation.\n\\end{proof}\nThis is where our $a_p$ comes into play. In the next section, we see what this condition stands for.\n\n\\section{Proof of Theorem~\\ref{thm}.}\n\nFirst we state the result by Peletier and Serrin~\\cite{PS}, which assures the uniqueness of solutions of \\eqref{b}.\n\\begin{Lem}\nLet $f$ satisfy (H1-3),\nwhere (H1), (H2) are in Lemma~\\ref{key}., and (H3) is the following:  \n\\begin{itemize}\n\\item[(H3)] $G(u):=\\displaystyle \\frac{f(u)}{u-\\beta}$ is nonincreasing in \n$(\\beta, c)$.\n\\end{itemize}\nThen \\eqref{b} has exactly one positive solution. \\label{lem}\n\\end{Lem}\nNow we are in a position to prove Theorem~\\ref{thm}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm}]\nWe shall see that for $\\omega\\in[a_p,\\omega_p)$, (H1-3) are satisfied.\nIt is enough to show that if $\\omega \\ge a_p$, then \n\\begin{equation}\nk(u):=f'(u)(u-\\beta )-f(u)\\le 0 \\qquad \\text{in}  ~~(\\beta, c).\\label{reallast}\n\\end{equation}\nTo prove \\eqref{reallast} we calculate the derivative of $k(u)$\n\\begin{equation*}\nk'(u)=f''(u)(u-\\beta),\n\\end{equation*}\nand note that\n\\begin{alignat*}{2}\n  f''(u) &> 0 & \\qquad & \\text{in} ~(0,\\alpha); \\\\\n  f''(u) &< 0 & \\qquad & \\text{in} ~(\\alpha,\\infty).\n\\end{alignat*}\nSo if $\\alpha\\le \\beta$ (i.e. $\\omega \\ge a_p$, see Lemma~\\ref{eta}), \nthen $k'(u)<0$ in $(\\beta, c)$, i.e. $k$ is decreasing in the interval. \nTherefore\n\\begin{equation*}  \n k(u)<k(\\beta)=-f(\\beta)<0 \\qquad \\text{in~~  $(\\beta, c)$},\n\\end{equation*}\nwhere the last inequality follows by \\eqref{z}.\n\nThis proves \\eqref{reallast} and completes the proof.\n\\end{proof}\nIf $\\alpha>\\beta$, we need to check that $k(\\alpha)\\le 0$, i.e.\n\\begin{equation}\n\\alpha -\\frac{f(\\alpha)}{f'(\\alpha)}\\le \\beta.   \\label{last}\n\\end{equation}\nThis condition provides an implicit relation between $\\omega$ and $p$.\nBesides, \n\\begin{Rem}\nThe condition~\\eqref{last} does not cover all $\\omega \\in (0,\\omega_p)$.\nThat is for $\\omega$ close to zero, $\\alpha -\\dfrac{f(\\alpha)}{f'(\\alpha)}> \\beta$.  \n\\end{Rem} \n\\begin{proof}\nThe left hand side of \\eqref{last} is estimated from below as\n\\begin{align*}\n \\alpha-\\frac{f(\\alpha)}{f'(\\alpha)} &= \\frac{(p-1){\\alpha}^p(1-2{\\alpha}^{p-1})}{-\\omega +p{\\alpha}^{p-1}-(2p-1){\\alpha}^{2(p-1)}} \\\\\n                                     &> \\frac{(p-1){\\alpha}^p(1-2{\\alpha}^{p-1})}{p{\\alpha}^{p-1}-(2p-1){\\alpha}^{2(p-1)}}>0,\n\\end{align*}\nfor all $\\omega\\in(0,\\omega_p)$, \nwhereas the right hand side $\\beta$ decreases to zero as $\\omega$ decreases to zero.\n\\end{proof}\nWhen $\\omega$ is close to zero, a very delicate observation is needed. \nSee Ouyang and Shi~\\cite{OS} for details. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIt is believed that most of our Universe is made of cold dark matter\nparticles. Big bang cosmology implies that these particles have interactions\nof order the weak scale, i.e.\\ they are WIMPS.\\cite{Seckel} When our galaxy\nwas formed the cold dark matter inevitably clustered with the luminous matter\nto form a sizeable fraction of the\n\\begin{equation}\n        \\rho_{\\chi}=0.4\\rm~GeV\/cm^3  \\label{density}\n\\end{equation}\ngalactic matter density implied by observed rotation curves. Unlike the\nbaryons, the dissipationless WIMPS fill the galactic halo which is believed to\nbe an isothermal sphere of WIMPS with average velocity\n\\begin{equation}\n         v_{\\chi}=300\\rm\\ km\/sec \\,. \\label{velocity}\n\\end{equation}\nParticle physics provides us with rather compelling candidates for WIMPS. The\nStandard Model is not a model. A most elegant and economical way to revamp it\ninto a consistent and calculable framework is to make the model\nsupersymmetric. If supersymmetry is indeed Nature's extension of the Standard\nModel it must produce new phenomena at or below the TeV scale. A very\nattractive feature of supersymmetry is that it provides cosmology with a\nnatural dark matter candidate in form of a stable lightest supersymmetric\nparticle.\\cite{Seckel} There are a priori six candidates: the (s)neutrino,\naxi(o)n(o), gravitino and neutralino. They are, in fact, the only candidates\nbecause supersymmetry completes the Standard Model all the way to the GUT\nscale where its forces apparently unify. Therefore because supersymmetry\nlogically completes the Standard Model with no other new physics threshold up\nto the GUT-scale, it must supply the dark matter. So, if supersymmetry, dark\nmatter and accelerator detectors are on a level playing field. Here we will\nfocus on the neutralino which, along with the axion, is for various reasons\nthe most palatable candidate.\\cite{Berezinsky} The supersymmetric partners of\nthe photon, neutral weak boson and the two Higgs particles form four neutral\nstates, the lightest of which is the stable neutralino\n\\begin{equation}\n\\chi = z_{11} \\tilde W_3 + z_{12} \\tilde B + z_{13} \\tilde H_1 + z_{14} \\tilde\nH_2 \\,.\n\\end{equation}\n\nIn the minimal supersymmetric model (MSSM)\\cite{Haber} down- and up-quarks\nacquire mass by coupling to different Higgs particles, usually denoted by\n$H_1$ and $H_2$, the lightest of which is required to have a mass of order the\n$Z$-mass. Although the MSSM provides us with a definite calculational\nframework, its parameters are many. For the present discussion we only have to\nfocus on the following terms in the MSSM lagrangian\n\\begin{equation}\nL = \\cdots \\mu\\tilde H_1 \\tilde H_2 -{1\\over2} M_1 \\tilde B \\tilde B\n-{1\\over2}M_2 \\tilde W_3 \\tilde W_3 - {1\\over\\sqrt2} g v_1 \\tilde H_1 \\tilde\nW_3 - {1\\over\\sqrt2} g v_2 \\tilde H_2 \\tilde W_3 + \\cdots \\,,\n\\end{equation}\nwhich introduce the (unphysical) masses $M_1,\\ M_2$ and $\\mu$ associated with\nthe neutral gauge bosons and Higgs particles, respectively. $M_1$ and $M_2$ are\nrelated by the Weinberg angle. The lagrangian introduces two Higgs vacuum\nexpectation values $v_{1,2}$; the coupling $g$ is the known Standard Model\nSU(2) coupling. Although the parameter space of the MSSM is more complex, a\nfirst discussion of dark matter uses just 3 parameters\n\\begin{equation}\n                \\mu,\\ M_2,\\ {\\rm and}\\ \\tan\\beta=v_2\/v_1 \\,.\n\\end{equation}\nFurther parameters which can also be varied include the masses of top, Higgs,\nsquarks, etc.\n\nNeutralino masses less than a few tens of GeV have been excluded by\nunsuccessful collider searches. For supersymmetry to resolve the\nfine-tuning problems of the Standard Model the masses of supersymmetric\nparticles must be of order the weak scale and therefore, in practice, at the\nTeV scale or below. Also, if neutralinos have masses of order a few TeV and\nabove, they overclose the Universe. Despite its rich parameter space\nsupersymmetry has therefore been framed inside a well defined GeV--TeV mass\nwindow with\n\\begin{equation}\n                \\mbox{tens of GeV} < m_{\\chi} < \\rm TeV \\,.\n\\end{equation}\nParticles produced by the annihilation of WIMPS represent the\nexperimental signature for the presence of halo dark matter. For the present\ndiscussion it is sufficient to focus on the dominant annihilation\nchannels\\cite{Drees}\n\\begin{equation}\n\\chi + \\bar\\chi \\to b + \\bar b \\label{chi to b}\n\\end{equation}\nor, if the WIMPS are sufficiently massive,\n\\begin{equation}\n\\chi + \\bar \\chi \\to W^+ + W^-\n\\end{equation}\n\n\\section{Balloon-borne experiments}\n\nNASA runs a vigorous program of measurements of cosmic antiproton and\npositron fluxes using balloon-borne particle detectors operating near the top\nof the Earth's atmosphere.\\cite{Gaisser} The instruments typically consist of\na magnet, tracking and calorimetry which can separate electrons and protons.\nSuch experiments provide us with an excellent opportunity to discover halo\ndark matter. Two possibilities are illustrated in Fig.~1. The $b$-quarks from\nhalo dark matter annihilation hadronize into jets in ways that have been\nstudied in much detail in accelerator experiments. Among the $b$-quark's\nhadronization products are low energy antiprotons which can be detected by\nballoon-borne antimatter detectors.\\cite{Kamionkowski} Presence of dark matter\nis signaled by an excess $\\bar p$-flux in the sub-GeV region where the\nbackground is conveniently small. The origin of the background is the\nproduction of antiprotons in the interactions of cosmic rays with matter in\nthe galaxy, e.g.\\ the hydrogen in the disk. (The additional background of\n$\\bar p$'s, produced in interactions of cosmic rays in the residual atmosphere\nabove the balloon, can be calculated and subtracted.)\n\n\\begin{figure}[h]\n\\centering\n\\epsfxsize=5in\\hspace{0in}\\epsffile{fig1.eps}\n\nFig.~1\n\\end{figure}\n\n\nAn excess of positrons over and above the positron flux produced by the\npropagation of cosmic rays in the galaxy may also be a signature of dark\nmatter. Semi-leptonic decay of heavy quarks is the dominant\ncontribution to the signal unless the neutralino mass is above threshold for\nannihilation into $W$'s. In this case the purely leptonic decay of the $W \\to\ne^+ + \\nu_e$ produces a ``positron line\" associated with the 2-body\ndecay.\\cite{Turner} In reality this ``line\" is broadened by the motion of the\nweak bosons into a high energy feature as sketched in Fig.~1.\n\nThe antiproton and positron flux of WIMP origin is proportional to the square\nof their halo density (\\ref{density}) and to their confinement time in the\nhalo. The latter is unfortunately uncertain by roughly four orders of\nmagnitude. Due to such astrophysical uncertainties the detection of WIMPS in\nthese experiments can be by no means guaranteed. No exclusion limits can be\nderived from negative experiments. As we will see further on, this is in\ncontrast with searches using the sun and Earth rather than the halo as the\nsource of WIMPS. The low energy antiproton and high energy positron signatures\nare nevertheless virtually a ``smoking gun\" for particle dark matter in the\nhalo and thus very much worthy of note.\n\n\n\\section{Neutrino Signature of WIMP-Annihilation in the Sun}\n\nWIMPS, scattering off protons in the sun, loose energy. They may fall below\nescape velocity and be gravitationally trapped. Trapped dark matter particles\neventually come to equilibrium temperature, and therefore to rest at the\ncenter of the sun. While the neutralino density builds up, their\nannihilation rate increases until equilibrium is achieved where the\nannihilation rate equals half of the capture rate. The sun has thus become a\nreservoir of neutralinos which annihilate into any open fermion, gauge boson\nor Higgs channels. The leptonic decays from annihilation channels such as\n$b\\bar b$ and $W^+W^-$ turn the sun into a source of high energy neutrinos.\nTheir energies are in the GeV\nto TeV range, rather than in the familiar KeV to MeV range\nfrom its nuclear burning.\nThese neutrinos can\nbe detected in deep underground experiments. Figure~2 shows a cartoon of the\nchain of events leading from dark matter in the halo to a muon track, pointing\nback to the sun, in the Earth-based Cherenkov detector.\n\nWe will illustrate the power of neutrino telescopes as dark matter detectors\nusing as an example the search for a 500~GeV neutralino, a mass outside the\nreach of present accelerator and future LHC experiments. It is a reasonable\nchoice given that there are arguments, mostly related to fine-tuning of unruly\nradiative corrections in the Higgs sector of the Standard Model, that the mass\nof the neutralino should be well below 1~TeV.\n\nNo long and complex code is necessary to qualitatively evaluate the potential\nof high energy neutrino telescopes as dark matter detectors. For\nquantitative calculations such code exists\nand can be distributed to anyone interested.\\cite{Halzen}\n\n\n\\begin{figure}[t]\n\\centering\n\\epsfxsize=3.5in\\hspace{0in}\\epsffile{fig2.eps}\n\n\\small Fig.~2\n\\end{figure}\n\nA calculation of the rate of high energy muons of neutralino origin triggering\na detector is calculated in 5 easy steps.\n\n\\break\n\\noindent\n{\\bf {\\it Step 1:} the halo neutralino flux \\boldmath{$\\phi_{\\chi}$} }\n\nIt is given by their number density and average velocity of (\\ref{density}),\n(\\ref{velocity}). From (\\ref{density})\n\\begin{equation}\nn_\\chi = 8\\times 10^{-4} \\left[ 500{\\rm\\ GeV}\\over m_\\chi \\right]\\rm\\\ncm^{-3}\n\\end{equation}\nand therefore\n\\begin{equation}\n\\phi_\\chi = n_\\chi v_\\chi = 2\\times 10^{4} \\left[ 500{\\rm\\ GeV}\\over m_\\chi\n\\right] \\rm\\ cm^{-2}\\, s^{-1} \\,. \\label{flux}\n\\end{equation}\n\n\\smallskip\n\\noindent\n{\\bf {\\it Step 2:} cross section \\boldmath{$\\sigma_{\\rm sun}$} for the capture\nof neutralinos by the\nsun }\n\nThe probability that a neutralino is captured is proportional to the number of\ntarget hydrogen nuclei in the sun (i.e.\\ the solar mass divided by the nucleon\nmass) and the neutralino-nucleon scattering cross section. $\\sigma(\\chi N)$\nreceives contributions from 2 classes of diagrams: the exchange of Higgses and\nweak bosons, and the exchange of squarks; see Fig.~3. The result is often\ndominated by the large coherent cross section associated with the exchange of\nthe lightest Higgs particle $H_2$ and is of the form\n\\begin{equation}\n\\sigma = \\alpha_H (G_F m_N^2)^2 {m_\\chi^2\\over(m_N + m_\\chi)^2} \\, {m_Z^2\\over\nm_H^4}\n\\end{equation}\nor, for large $m_{\\chi}$\n\\begin{equation}\n\\sigma = \\alpha_H \\left(G_F m_N^2\\right)^2 {m_Z^2\\over m_H^4} \\,. \\label{large}\n\\end{equation}\nThe proportionality parameter $\\alpha_H$ is of order unity, but can become\nas small as $10^{-2}$ in some regions of the MSSM parameter space. This is\nillustrated in Fig.~4 where the MSSM parameter space is parametrized in terms\nof the unphysical masses $M$($\\mu$) of the unmixed wino(Higgsino). (The ratio\nof the vacuum expectation values associated with the two Higgs particles\n$v_2\/v_1=2$ is here fixed to some arbitrary value.) The relation of these\nparameters to the neutralino mass is shown in the figure. The full lines show\nfixed values of the neutralino mass $m_{\\chi}$. The lines labelled by squares\ntrace fixed values of the ``coupling'' $\\alpha_H$. The dashed area\nindicates $M$, $\\mu$ values which are excluded by cosmological considerations.\nIn standard big bang cosmology neutralinos with the corresponding parameters\nwill overclose the Universe. Note that for a given $\\chi$ mass there are two\npossible states with the same $\\alpha_H$ value. One of them will\npreferentially annihilate into weak bosons, the other into fermions.\nTherefore, their neutrino signature is provided by $W,Z$ decay and\nsemi-leptonic heavy\nquark decays, respectively. Figure~4 illustrates that for heavy neutralinos,\nwhich can only be searched for by the indirect methods discussed here and are\ntherefore of prime interest, any detector which can study dark matter with\n$\\alpha_H$ as small as 0.1 can exclude the bulk of the phase space currently\navailable to MSSM dark matter candidates.\n\n\\begin{figure}\n\\centering\n\\epsfxsize=3.75in\\hspace{0in}\\epsffile{fig3.eps}\n\n\\small Fig.~3\n\\end{figure}\n\n\\begin{figure}\n\\hfil\\epsfxsize=3.75in\\epsffile{fig4.eps}\\hfill\n\n{\\small Fig.~4. Contours in the $M,\\mu$ plane of constant $\\alpha_{H_2}=1.0,\\\n0.1,\\ 0.01$ (boxes) and constant neutralino mass $M_\\chi=30, 100, 500$ and\n1000~GeV (solid). The shaded region is excluded by cosmological\nconsiderations.}\n\n\\end{figure}\n\n\n\nIn the end we can just estimate the neutralino-nucleon cross section by\ndimensional analysis which gives $\\left(G_F m_N^2\\right)^2\/m_Z^2$. This also\nfollows from (\\ref{large}) as $m_H$ is of order $m_Z$ in the MSSM. We obtain\nfor the solar capture cross section\n\\begin{equation}\n\\Sigma_{\\rm sun} = n\\sigma = {M_{\\rm sun}\\over m_N} \\sigma(\\chi N)\n= \\left[1.2\\times 10^{57}\\right] \\left[10^{-41}\\,\\rm cm^2\\right] \\,.\n\\label{capture}\n\\end{equation}\n\n\\smallskip\n\\noindent\n{\\bf {\\it Step 3:} Capture rate \\boldmath{$N_{\\rm cap}$} of neutralinos by the\nsun }\n\n$N_{\\rm cap}$ is determined by the neutralino flux (\\ref{flux}) and the\nsun's capture\ncross section (\\ref{capture}) obtained in the first 2 steps\n\\begin{equation}\nN_{\\rm cap} = \\phi_\\chi \\Sigma_{\\rm sun} = 3\\times 10^{20}\\,\\rm s^{-1} \\,.\n\\end{equation}\n\n\\smallskip\n\\noindent\n{\\bf {\\it Step 4:} Number of solar neutrinos of neutralino origin }\n\nOne can check that the sun comes to a steady state where capture and\nannihilation of neutralinos are in equilibrium. For a 500~GeV neutralino the\ndominant annihilation rate is into weak bosons; each produces muon-neutrinos\nwith a leptonic branching ratio which is roughly 10\\%:\n\\begin{equation}\n\\chi\\bar\\chi \\to WW \\to \\mu \\nu_\\mu \\,. \\label{branch}\n\\end{equation}\nTherefore, as we get 2 $W$'s for each capture, the number of neutrinos\ngenerated in the sun is\n\\begin{equation}\nN_\\nu = {1\\over 5} N_{\\rm cap}\n\\end{equation}\nand the corresponding neutrino flux at Earth is given by\n\\begin{equation}\n \\phi_\\nu = {N_\\nu\\over 4\\pi d^2} = 2\\times 10^{-8}\\,\\rm cm^{-2} s^{-1} \\,,\n\\label{nu flux}\n\\end{equation}\nwhere the distance $d$ is 1 astronomical unit.\n\n\n\\smallskip\n\\noindent\n{\\bf {\\it Step 5:} Event rate in a high energy neutrino telescope }\n\nIt is evident that this flux is small enough to require the use of large\nvolumes of natural water or ice as a $\\nu_\\mu$ interaction volume. The\nsecondary muons produced in the interaction provide the experimental signature\nof the neutrinos in the underground detector; see Fig.~2. Optical modules view\nthe water or ice target and detect the muon by the Cherenkov light emitted as\nit traverses the detector. By mapping the Cherenkov cone the solar origin of\nthe neutrino can be established, at least for the higher energies where muon\nand neutrino directions are approximately aligned. The number of muons\nobserved in a detector of a given area is\n\\begin{equation}\nN = \\hbox{area}\\int dE\\,{dN_\\nu\\over dE}\\,P_{\\nu\\to\\mu} \\,.\n\\end{equation}\nThe quantity $P_{\\nu\\to\\mu}$ represents the combined probabilities that a\n``solar'' neutrino interacts in the volume viewed by the optical modules and\nthat the muon, produced in the interaction, has a sufficient range to reach\nthe detector. It depends on the particle density of the target, the neutrino\ninteraction cross section and the range of the muon, therefore\n\\begin{equation}\nP_{\\nu\\to\\mu} = \\rho_{\\rm H_2O}\\,\\sigma_{\\nu\\to\\mu}(E)\\,R_\\mu \\,.\n\\label{P}\n\\end{equation}\nEvaluating Eq.~(\\ref{P}) only involves standard particle physics.\n\nFor (\\ref{branch}) the $W$-energy is approximately $m_{\\chi}$ and the\nneutrino energy half that by 2-body kinematics. The energy of the detected\nmuon is given by\n\\begin{equation}\n E_\\mu \\simeq {1\\over2} E_\\nu \\simeq {1\\over4}m_\\chi \\,.\n\\end{equation}\nwhere we used the fact that, in this energy range, roughly half of the neutrino\nenergy is transferred to the muon. Simple estimates of the neutrino\ninteraction cross section and the muon range can be obtained as follows\n\\begin{eqnarray}\n  \\sigma_{\\nu\\to\\mu} &=& 10^{-38}\\,{\\rm cm^2}\\,{E_\\nu\\over \\rm GeV} =\n2.5\\times10^{-36}\\,\\rm cm^2  \\label{sigma numu} \\\\\n\\noalign{\\hbox{and}}\nR_\\mu &=& 5\\ {\\rm m}\\,{E_\\mu\\over\\rm GeV} = 625\\ \\rm m \\,, \\label{R mu}\n\\end{eqnarray}\nwhich is the distance covered by a muon given that it loses 2~MeV for each\ngram of matter traversed. We have now collected all the information to compute\nthe number of events in a detector of area $10^4\\,$m$^2$, typical for those\npresently under construction. For the neutrino flux given by (\\ref{nu flux})\nwe obtain\n\\begin{equation}\n {\\rm \\#\\ events\/year = area} \\times \\phi_\\nu \\times \\rho_{\\rm H_2O} \\times\n\\sigma_{\\nu\\to\\mu} \\times R_\\mu \\simeq 10 \\,.\n\\end{equation}\nTen 125~GeV muons from the direction of the sun! It is a pretty safe bet that\nsuch a signal will not be drowned by whatever real world experimental problems\ndilute this naive estimate.\n\nThe above exercise is just meant to illustrate that present high energy\nneutrino\ntelescopes compete with present and future accelerator experiments in the\nsearch for supersymmetry. Especially for heavier neutralinos the technique is\npowerful because underground high energy neutrino detectors have been\noptimized to be sensitive in the energy region where the neutrino interaction\ncross section and the range of the muon are large; notice the\n$E$-factors in (\\ref{sigma numu}), (\\ref{R mu}). Also, for high energy\nneutrinos the muon and neutrino are nicely aligned along a direction pointing\nback to the sun with good angular resolution. Direct searches will have to\ndeliver detectors reaching better than 0.05~events\/kg\\,day sensitivity to\ncompete.\\cite{Bottino}\n\nIn many places we made assumptions that lead to an underestimate of the\nsignal. We neglected, for instance, squark exchange contributions in (13) and\nneglected other decay channels contributing neutrinos in (16). We did not\ninclude the signal from annihilation of neutralinos trapped in the center of\nthe Earth.\\cite{Gould} We did, on the other hand, assume that $\\alpha_H$ is\nunity while in some corners of SUSY-space the value is 2 orders of magnitude\nsmaller; see Fig~4. Therefore the natural scale of the complete SUSY detector\nis two orders of magnitude larger than the $10^4\\,$m$^2$ area of present\ndetectors, i.e.\\ $1\\,$km$^2$. This is the size next-generation neutrino\ntelescopes already under discussion.\\cite{Learned} Such an instrument can be\nbuilt with the number of phototubes of the SNO experiment in Canada and the\nbudget of the Superkamiokande experiment in Japan. We will return to this\nfurther on.\n\n\n\n\n\n\\section{Now for those who do not trust back-of-the-envelope . . . .}\n\n\nAbove estimates can, of course, be done rigorously. This is tedious and\nstraightforward. Our main conclusions are corroborated by the results of such\na calculation shown in Fig.~5 which exhibits, as a function of the\nneutralino mass, the detector area required to observe one event per year.\nThe two branches in this and the following figures correspond to the two\nsolutions for a fixed neutralino mass; see Fig.~4. Various annihilation\nthresholds are clearly visible, most noticeable is the threshold associated\nwith the $W,Z$ mass near 100~GeV.  The graphs confirm that, realistically, a\ndetector of km-scale is required to study the full neutralino mass range. It\nis clear from Fig.~5, however, that even detectors of more modest size can\nradically improve on accelerator results.\\cite{Mori}$^,$\\cite{LoSecco}\nNeutralinos\nof 1~TeV mass are observable in a detector of area a few times\n$10^3\\,\\rm m^2$. The\nenergy of the produced neutrinos is typically ``a fraction'' of the neutralino\nmass, e.g.\\ 1\/2 for neutralino annihilation into a $W$ followed by a leptonic\n$e \\nu$ decay. For lower masses the event rates are small because the detection\nefficiency for low energy neutrinos is reduced. This mass range has, however,\nalready been excluded by accelerator experiments. For very high masses the\nnumber density of neutralinos, and therefore the event rate, becomes small.\nThis is not a problem as problematically large masses are excluded by\ntheoretical arguments as previously discussed.\n\n\n\\begin{figure}[t]\\hfil%\n\\epsfxsize=4in\\hspace{0in}\\epsffile{fig5.eps}\\hfill\n\n\\small Fig.~5. As a function of the neutrino mass we show the telescope size\nrequired to be sensitive at the one event per year level. We fix $\\tan\n\\beta=2,\\ \\alpha_{H_2}=0.1$. The two branches correspond to the two solutions\nfor fixed $\\alpha_{H_2}$.\n\\end{figure}\n\nThe same results are shown in Fig.~6~a as contours in the $M, \\mu$ plane which\ndenote the neutrino detection area required for observation of 1~event per\nyear. Clearly the $10^5\\,\\rm m^2$ contour covers the parameter space. The\nproblematic large $\\mu, M_2$-region does not represent a problem as its\nparameters lead to values of the matter density $\\Omega$ exceeding unity as\nshown in the accompanying Fig.~6~b.\n\n\\begin{figure}[t]\\hfil%\n\\epsfxsize=3.175in\\hspace{0in}\\epsffile{fig6.eps}\\hfill\n\n\\small Fig.~6. In the $M_2,\\mu$ plane for $M_{\\tilde q}=\\infty$, (a)~contours\nof constant detection rate  (events m$^{-2}$ yr$^{-1}$) and (b)~regions of\n$\\Omega_\\chi h^2 > 1$ and $\\Omega_\\chi h^2 < 0.02$ which are ruled out by\ncosmological considerations.\n\\end{figure}\n\nClearly a realistic evaluation of the reach of an underground detector\nrequires more than counting events per year. Realistic simulations of\nstatistics and systematics must be done. Also a more complete mapping of the\nMSSM parameter space is required. For those interested we refer to\nreference~\\ref{Halzen}.\n\n\n\\section{High Energy Neutrino Telescopes}\n\nHigh-energy neutrino telescopes are multi-purpose instruments which can make\ncontributions to astronomy, astrophysics and particle physics. It is\nintriguing that all of these missions, including the search for dark matter\ndiscussed here, point to the necessity of building 1~km$^3$\ndetectors.\\cite{Learned} We close with a discussion of the possibility of\nbuilding a 1~km scale neutrino detector based on the experience gained in\ndesigning the instruments now under construction, specifically AMANDA, Baikal,\nDUMAND and NESTOR which we will briefly review.\\cite{Learnedprime} One can\nconfidently predict that such a telescope can be constructed at a reasonable\ncost, e.g.\\ a cost similar to Superkamiokande,\\cite{Suzuki} to which it is\ncomplimentary in the sense that its volume is over four orders of magnitude\nlarger while its threshold is in the GeV, rather than the MeV range. The\nthreshold is in the 2--10~GeV energy range for AMANDA and is about 10~GeV for\nDUMAND. Relative to a 1~km scale detector, the experiments under construction\nare only ``few\" percent prototypes. Yet, using natural water or ice as a\ndetection medium, these neutrino detectors can be deployed at roughly 1\\% of\nthe cost of conventional accelerator-based neutrino detectors which use\nshielding and some variety of tracking chambers.  It is thus not hard to\nbelieve that the Cherenkov detectors can be extended to a larger scale at\nreasonable cost.\n\nDetectors presently under construction have a nominal effective area of\n$10^4$~m$^2$. Baikal has been operating 18 optical modules for almost\none year and the South Pole AMANDA experiment started operating 4 strings with\n20 optical modules each in January 94. The first generation telescopes will\nconsist of roughly 200 optical modules (OM) sensing the Cherenkov light of\ncosmic muons. The experimental advantages and challenges are different for\neach experiment and, in this sense, they nicely complement one another.\nBriefly,\n\n\\begin{itemize}\n\\item  DUMAND will be positioned under 4.5~km of ocean water, below most\nbiological\nactivity and well shielded from cosmic ray muon backgrounds. One nuisance of\nthe ocean is the background light resulting from radioactive decays, mostly\nK$^{40}$, plus some bioluminescence, yielding an OM noise rate of\n50--100~kHz. Deep ocean water is, on the other hand, fantastically clear, with\nan attenuation length of order 40~m in the blue. The deep ocean is a\ndifficult location for access and service, not at all like a laboratory\nexperiment.  Detection equipment must be built to high reliability standards,\nand the data must be transmitted to the shore station for processing.  It has\nrequired years to develop the necessary technology and learn to work in an\nenvironment foreign to high-energy physics experimentation, but hopefully this\nwill be accomplished satisfactorily.\n\n\\item AMANDA is operating in deep clear ice. The ice provides a\nconvenient mechanical support for the detector. The immediate advantage is\nthat all electronics can be positioned at the surface. Only the optical\nmodules are deployed into the deep ice. Polar ice is a sterile medium with a\nconcentration of radioactive elements reduced by more than $10^{-4}$ compared\nto sea or lake water. The low background results in an improved sensitivity\nwhich allows for the detection of high energy muons with very simple trigger\nschemes which are implemented by off-the-shelf electronics. Being positioned\nunder only 1~km of ice it is operating in a cosmic ray muon background which\nis over 100 times larger than DUMAND. The challenge is to reject the\ndown-going muon background relative to the up-coming neutrino-induced muons by\na factor larger than $10^6$. The group claims to have met this challenge with\nan up\/down rejection which is at present superior to that of the deep\ndetectors. The task is, of course, facilitated by the low\nbackground noise. The polar environment is difficult as well, with\nrestricted access and one shot deployment of photomultiplier strings. The\ntechnology has, however, been satisfactorily demonstrated with the deployment\nof the first 4 strings. It is now clear that the hot water drilling technique\ncan be used to deploy OM's larger than the 8~inch photomultiplier tubes now\nused to any depth in the 3~km deep ice cover.\n\n\\item NESTOR is similar to DUMAND, being placed in the deep ocean (the\nMediterranean), except for two critical differences.  Half of its optical\nmodules point up, half down. The angular response of the detector is being\ntuned to be much more isotropic than either AMANDA or DUMAND, which will give\nit advantages in, for instance, the study of neutrino oscillations. Secondly,\nNESTOR will have a higher density of photocathode (in some substantial volume)\nthan the other detectors, and will be able to make local coincidences on lower\nenergy events, even perhaps down to the supernova energy range (tens of\nMeV).\n\n\\item BAIKAL shares the shallow depth with AMANDA, and has half its optical\nmodules pointing up like NESTOR. It is in a lake with 1.4~km\nbottom, so it cannot expand downwards and will have to grow horizontally.\nOptical backgrounds similar in magnitude to ocean water have been discovered\nin Lake Baikal. The Baikal group has been operating for one year an array with\n18 Quasar photomultiplier (a Russian-made 15~inch tube) units in April 1993,\nand may well count the first neutrinos in a natural water Cherenkov detector.\n\n\\item Other detectors have been proposed for near surface lakes or ponds\n(e.g.\\break GRANDE, LENA, NET, PAN and the Blue Lake Project), but at this time\nnone are in construction.\\cite{Learnedprime}  These detectors all would have\nthe\ngreat advantage of accessibility and ability for dual use as extensive air\nshower detectors, but suffer from the $10^{10}$--10$^{11}$ down-to-up ratio of\nmuons, and face great civil engineering costs (for water systems and\nlight-tight containers). Even if any of these are built it would seem that the\ncosts may be too large to contemplate a full-km scale detector.\n\\end{itemize}\n\nIn summary, there are four major experiments proceeding with construction,\neach of which have different strengths and face different challenges. For the\nconstruction of a 1~km scale detector one can imagine any of\nthe above detectors being the basic building block for the ultimate 1~km$^3$\ntelescope. The present AMANDA design, for example, consists of 9 strings on a\n30~meter radius circle with a string at the center (referred to as a $1+9$\nconfiguration).  Each string contains 20 OMs separated by 12~m.  Imagine\nAMANDA ``supermodules'' which are obtained by extending the basic string\nlength (and module count per string) by a factor of 4.5.  Supermodules would\nthen consist of $1+9$ strings with, on each string, 90 OMs separated by\n12~meters for a length of 1080~meters. A 1~km scale detector then might\nconsist of a $1+7$ configuration of supermodules, with the 7 supermodules\ndistributed on a circle of radius 540~meters, and have a total of about 7200\nphototubes. Such a detector can be operated in a dual mode:\n\n\n\\begin{itemize}\n\\item it obviously consists of $4.5\\times8$ of the presently planned\nAMANDA array modules, leading to an effective area\nof $\\sim0.75$~km$^2$.  Importantly, the characteristics of the detector,\nincluding threshold, are the same as those of the original AMANDA array module.\n\n\\item the $1+7$ Supermodule configuration, looked at as a whole, instruments\nwith\nOMs a 1~km$^3$ cylinder with a diameter and height of 1080~m. High-energy\nmuons will be superbly reconstructed as they can produce triggers in 2 or more\nof the modules spaced by large distance.  Reaching more than one supermodule\nrequires 100~GeV energy to cross 500~m.  For a 1~km deep detector the\nthreshold for downgoing muons is thus raised from 200 to 300~GeV.  We note\nthat this is the\nenergy for which a neutrino telescope has optimal sensitivity to a typical\n$E^{-2}$ source (background falls with threshold energy, and until about\n1~TeV little signal is lost).\n \\end{itemize}\n\n\\noindent\nAlternate methods to reach the 1~km scale have been discussed by Learned and\nRoberts.\\cite{Roberts}\n\n\n\nHow realistic are the construction costs for such a detector?  AMANDA's\nstrings cost \\$150,000 including deployment. By naive scaling the final cost of\nthe postulated $1+7$ array of supermodules is of order \\$50 million and still\nbelow that of Superkamiokande (with $11{,}200 \\times 20$~inch photomultiplier\ntubes in a 40~m diameter by 40~m high stainless steel tank in a deep mine). It\nis clear that the naive estimate makes several approximations over- and\nunderestimating the actual cost.\n\n\\begin{figure}[h]\n\\centering\n\\epsfxsize=4in\\hspace{0in}\\epsffile{modules.eps}\n\n\\small Fig.~7\n\\end{figure}\n\nAt the 1~km$^2$ size it seems inescapable that supersymmetry will be found or,\nalternatively, will be forced to ``escape'' in rather special regions of its\nvast parameter space. The appeal of this beautiful theoretical idea would be\ndiminished.\n\n\n\n\n\\section*{Acknowledgements}\n\nWe would like to thank Steve Barwick, Manuel Drees, Concha Gonzalez-Garcia and\nMarc Kamionkowski for useful conversations. This research was supported in\npart by the U.S.~Department of Energy under Contracts No.~DE-FG02-91ER40626\nand No.~DE-AC02-76ER00881, in part by the Texas Research National Commission\nunder Grant No.~RGFY9173, and in part by the University of Wisconsin Research\nCommittee with funds granted by the Wisconsin Alumni Research Foundation.\n\n\\section*{References}\n\\frenchspacing\n\\begin{enumerate}\n\\addtolength{\\itemsep}{-.05in}\n\n\\item\\label{Seckel}\nJ.~R.~Primack, B.~Sadoulet, and D.~Seckel,\n Ann. Rev. Nucl. Part. Sci. {\\bf B38} (1988) 751.\n\n\\item\\label{Berezinsky}\nV.~S.~Berezinsky,\n {\\it Proc.\\ of the Fourth\nInternational Symposium on Neutrino Telescopes}, Venice (1992),\ned.\\ by M.~Baldo-Ceolin.\n\n\\item\\label{Haber}\nH.~E.~Haber and G.~L.~Kane,\n Phys. Rep. {\\bf 117}, 75 (1985).\n\n\\item\\label{Drees}\nM.~Drees and M.~M.~Nojiri,\n Phys.\\ Rev.\\ {\\bf D47}, 376 (1993).\n\n\\item\\label{Gaisser}\n For a recent review, see T.~K.~Gaisser {\\it et al.}, {\\it The Astrophysics and\nParticle Physics of High Energy Cosmic Radiation}, National Research Council\nResearch Briefing, Washington (1994).\n\n\\item\\label{Kamionkowski}\n For updated calculations and references, see G.~Jungman and M.~Kamionkowski,\nPrinceton Preprint IASSNS-HEP-93\/54, 1993.\n\n\\item\\label{Turner}\n M. Kamionkowski and M. S. Turner, Phys.\\ Rev.\\ {\\bf D43}, 1774 (1991) and\nreferences therein.\n\n\\item\\label{Halzen}\n F.~Halzen, M.~Kamionkowski, and T.~Stelzer, Phys.\\ Rev.\\ {\\bf\nD45}, 4439 (1992).\n\n\\item\\label{Bottino}\n A.~Bottino {\\it et al.}\nMod. Phys. Lett. A {\\bf 7}, 733 (1992).\n\n\n\\item\\label{Gould}\n A. Gould,\n Astrophys. J. {\\bf 321}, 571 (1987);\nAstrophys.~J. {\\bf 368}, 610 (1991); Astrophys.~J. in\npress (1991).\n\n\\item \\label{Learned}\n F.~Halzen and J.G.~Learned, {\\it Proc.\\ of the Fifth\nInternational Symposium on Neutrino Telescopes}, Venice (1993),\ned.\\ by  M.~Baldo-Ceolin.\n\n\\item\\label{Mori}\n M. Mori {\\em et al.}, KEK Preprint 91-62; N. Sato {\\em et al.},\nPhys. Rev. {\\bf D44}, 2220 (1991).\n\n\\item\\label{LoSecco}\n IMB Collaboration: J. M. LoSecco {\\em et al.}, Phys. Lett.  {\\bf\nB188}, 388 (1987); R. Svoboda {\\em et al.}, Astrophys.~J. {\\bf 315}, 420\n(1987).\n\n\\item\\label{Learnedprime}\n See recent reports on these experiments in the following reference, and a\ncritical summary of the various projects in J.~G.~Learned, {\\it Proceedings of\nthe European Cosmic Ray Symposium}, Geneva, Switzerland, July 1992, ed. by\nP.~Grieder and B.~Pattison,  Nucl.\\ Phys.\\ B (1993); see also presentations in\n{\\it Proceedings of the High Energy Neutrino Astrophysics Workshop}, ed. by\nV.~J.~Stenger, J.~G.~Learned, S.~Pakvasa, and X.~Tata, World Scientific,\nSingapore (1992).\n\n\\item\\label{Suzuki}\n Y.~Suzuki, {\\it Proceedings of the 3$^{rd}$ International\nWorkshop on Neutrino Telescopes}, Venice, March 1992, ed.\\ by M.~Baldo-Ceolin,\nVenice (1992).\n\n\\item\\label{Roberts} J.~G.~Learned and A.~Roberts, {\\it Proceedings of the\n23$^{rd}$ International Cosmic Ray Conference}, Calgary, Canada (1993).\n\n\n\n\\end{enumerate}\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{Introduction}\n\nWhat's common among the great batsman Sachin Tendulkar, the great athlete Usion Bolt, the great tennis player Roger Fedrer and the great footballer Maradonna?\n\n{\\bf Answer : They spend hours after hours, days after days training!} \n\nOver the years, the author has taught material presented here to the graduate and undergraduate students at IISER Pune and in various summer and winter schools meant for graduate students and\/or teachers. The author would like to thank the participants of various ATM schools who enthusiastically responded to the lectures and tutorials.  \n\\begin{enumerate}\n\\item {\\bf Why this?} The idea behind these notes is to introduce matrix groups. The groups we mostly encounter are matrix groups: most of the finite simple groups, algebraic groups, Lie groups etc. Yet, in our undergraduate course on group theory we do not introduce matrix groups. This is an attempt to correct that mistake.\n\\item {\\bf For whom?} This book grew out of an attempt to convince audience that they can make use of Linear Algebra they know, to study some of the matrix groups. Any PhD student, undergraduate student or teacher will benefit from this. The topics presented here form bread and butter for group theorists, number theorists, physicists and engineers.  \n\\item {\\bf How to read this?}\nThis note is meant to be a supplement for those who have studied first course in Linear Algebra and Group Theory. That is why its in the form of exercises. The author confesses that some of the exercises throw big, unknown words but they are not meant to scare you. It's hoped that this will enthuse you to look further.\n\\end{enumerate}\nWe present some motivation behind this attempt. Author hopes that these stories will enthuse the audience towards this material.\n\n\\section{CFSG}\nOne of the biggest achievements of the last century is ``The Classification of Finite Simple Groups''. It was born out of the following: \n\n{\\bf Can we classify all finite groups?}\n\nLet $G$ be a finite group. If $G$ has a proper normal subgroup $N$, it can be broken into two pieces: $N, G\/N$. However, usually it is not easy to construct the group $G$ given the smaller groups. They fit in an exact sequence:\n$$1\\rightarrow N \\rightarrow G \\rightarrow G\/N \\rightarrow 1.$$\nThe problem of constructing $G$ out of given other two is called \"Extension Problem\" and is a difficult problem. For example, to determine central extension one uses some cohomology theory. \n\nHowever, using the above idea we reduce the problem of classifying all finite groups to the classification of finite simple groups. This has been achieved now. It has taken efforts of more than a hundred year and several hundred mathematicians. Thus it is usually not attributed to any one of them.\n\\begin{theorem}[CFSG]\nA finite simple group is one of the following:\n\\begin{enumerate}\n\\item Cyclic groups $\\mathbb Z\/p\\mathbb Z$, where $p$ is a prime.\n\\item Alternating groups $A_n$ for $n\\geq 5$.\n\\item Finite groups of Lie type\n\\begin{enumerate}\n\\item classical type \n\\item exceptional type\n\\end{enumerate} \n\\item The sporadic groups $26$ of them.\n\\end{enumerate}\n\\end{theorem}\nIn our undergraduate group theory course we learn the first two family. The idea here is to attempt to convince the audience that they can get familiar with the third family as well using their knowledge of Linear Algebra, at least for sure the group of classical type. For the history and more information please read the wikipedia article. Here are the classical groups:\n\\begin{enumerate}\n\\item Special Linear Groups\n\\item Orthogonal Groups of odd dimension\n\\item Symplectic Groups\n\\item Orthogonal Groups of even dimension\n\\item Unitary Groups\n\\end{enumerate}\n\n\\section{Representation theory}\nLet $G$ be a group. While working with some examples of groups we quickly realise that there is not much we know about them. One way is possibly to compare them with some known groups. In this case we take the known groups as linear group, $GL(V)$, and the comparison is made via all group homomorphisms from $G$ to linear groups. More precisely, we define a representation.\n\nFix a field $k$. A representation of $G$ is a group homomorphism $\\rho \\colon G \\rightarrow GL(V)$ where $V$ is a vector space over the field $k$. A subspace $W\\subset V$ is said to be invariant if $\\rho(g)(W)\\subset W$, for all $g\\in G$. Further, a representation is said to be irreducible if it has no proper invariant subspace.\n\nFor a finite group $G$, it turns out that over algebraically closed field $k$ when $char(k)\\nmid |G|$ the representations of the group can be understood via character theory. In this case every representation can be written as a direct sum of irreducible representations. Thus, the character theory is built on knowing basic group theory and linear algebra well.\n\n\\section{Computational group theory}\nTo understand a subject it is often important to know many examples. Even better, if one could compute with these examples and verify before making any conjecture or trying to solve a problem, would be divine.\nDue to advancement in computational power the idea is to implement groups in the computer so that one is able to compute with those examples. There are several computational softwares SAGE, GAP and MAGMA. To make the idea clear: like any software there are two group of people, developers and users. It goes without saying that knowing these packages adds an extra dimension to one's own research. It's interesting to learn some of the fundamental problems in CGT. \n\n\n\n\\section{Robotics, Computer graphics etc}\nIn robotics basic mathematics suc as rotation and translation in 3 dimension is used. One often works with the groups $O_2(\\mathbb R)$ and $O_3(\\mathbb R)$, and slightly more general, isometry groups. In computer graphics Hamilton's quaternions are used to achieve the desired results.\n\n\n\\chapter{The Group $SL_2$}\nThe group $SL_2$ is ubiquitous in mathematics. It appears \n\\begin{enumerate}\n\\item in algebraic groups as a smallest example of simple algebraic groups $SL_2$,\n\\item in Lie groups as the group $SL_2(\\mathbb R)$,\n\\item in complex analysis as mobius transformations $SL_2(\\mathbb Z)$,\n\\item in topology as an example of free product $PSL_2(\\mathbb Z)\\cong \\mathbb Z\/2\\mathbb Z * \\mathbb Z\/3\\mathbb Z$,\n\\item in finite group theory as a part of the finite simple groups $PSL_2(q)$,\n\\item in number theory as the action of $PSL_2(\\mathbb Z)$ on the upper half plane, \n\\item in K-theory as the group $SL_2(R)$ over a ring $R$ with identity etc.\n\\end{enumerate}\nThe exercises below are devoted to the ubiquity of this group.\n\n\\section{Generation and Automorphisms of the group $SL_2(k)$}\nLet $k$ be a field. The group $SL_2(k)=\\left\\{X\\in M_2(k)\\mid \\det(X)=1 \\right\\}$. Now define $x_{12}(t):= \\begin{pmatrix} 1&t\\\\ &1\\end{pmatrix}$ and $x_{21}(s):= \\begin{pmatrix} 1&\\\\s &1\\end{pmatrix}$ for $t, s\\in k$. The matrices $x_{12}(t)$ and $x_{21}(s)$ are called {\\bf elementary matrices}.\n\n\\begin{exercise} With the notation as above, \n\\begin{enumerate}\n\\item The maps $\\mathfrak{X}_{12}\\colon k \\rightarrow SL_2(k)$ given by $t\\mapsto x_{12}(t)$ and $\\mathfrak{X}_{21}\\colon k \\rightarrow SL_2(k)$ given by $t\\mapsto x_{21}(t)$ are group homomorphisms where $k$ is additive group of the field.\n\\item Compute $w(t)=x_{12}(t)x_{21}(-t^{-1})x_{12}(t)$. What is $w(-1)$?\n\\item Compute $h(t)=w(t)w(-1)$.\n\\item For $c\\neq 0$, verify the following for an element in $SL_2(k)$, \n$$\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}= \\begin{pmatrix}1&(a-1)c^{-1}\\\\&1\\end{pmatrix} \\begin{pmatrix}1&\\\\c&1\\end{pmatrix} \\begin{pmatrix}1&(d-1)c^{-1}\\\\ &1\\end{pmatrix}.$$\n\\item Show that the group $SL_2(k)$ is generated by the set of all elementary matrices $\\{x_{12}(t), x_{21}(s) \\mid t,s\\in k\\}$. \n\\end{enumerate}\n\\end{exercise}\n\\noindent This exercise shows that the elementary matrices generate the group. These generators are also called {\\bf Chevalley generator}.\n\n\\begin{exercise}[Bruhat decomposition]\nLet $B=\\left\\{\\begin{pmatrix}a&b\\\\&a^{-1}\\end{pmatrix}\\right\\}$ be the set of upper triangular matrices in $SL_2(k)$. Show the following double coset decomposition of $SL_2(k)$,\n$$B\\backslash SL_2(k)\/B = B\\bigcup B \\begin{pmatrix}&-1\\\\1&\\end{pmatrix} B.$$\n\\end{exercise}\n\\noindent To solve this exercise take a matrix which is not upper triangular. Now, by multiplying appropriate $x_{12}(t)$ from, first left and then right, reduce it to of the form $w(s)$. Among other things, this exercise shows that the group is generated by mostly upper triangular matrices provided we throw in the permutation matrix. However, as $t$ is varying over $k$ it seems we need a large number of generating elements. The following exercise says that that is not necessarily the case.\n\n\\begin{exercise}\nProve that the group $SL_2(\\mathbb F_p)$ is generated by the two of its elements $\\begin{pmatrix} 1& 1 \\\\ &1\\end{pmatrix}$ and $\\begin{pmatrix} 1&  \\\\1 &1\\end{pmatrix}$.\n\\end{exercise}\n\\noindent \n\n\\begin{exercise}\nProve that the commutator subgroup $[SL_2(k), SL_2(k)]=SL_2(k)$ if $|k|\\geq 5$.\n\\end{exercise}\n\\noindent In the view of earlier exercises it is enough to show that the elementary matrices are commutator.\n\n\\begin{exercise} Let $k$ be a field with $|k| \\geq 5$.\nShow that any group homomorphism $\\rho \\colon SL_2(k) \\rightarrow k^*$ is trivial.\n\\end{exercise}\n\n\\begin{exercise}[Automorphism]\n\\begin{enumerate}\n\\item Let $g\\in GL_2(k)$. The map $\\iota_g \\colon SL_2(k) \\rightarrow SL_2(k)$ defined by $x\\mapsto gxg^{-1}$ is a group automorphism.\n\\item Show that this gives a group homomorphism $GL_2(k) \\rightarrow \\textup{Aut}(SL_2(k))$ defined by $g\\mapsto \\iota_g$. What is the kernel of this map?\n\\item Show that the map $d\\colon A\\mapsto \\ensuremath{{}^t\\!} A^{-1}$ is an automorphism of $SL_2(k)$.\n\\item Let $\\sigma\\in \\textup{Aut}(k)$ be a field automorphism of $k$. Define a map $\\sigma \\colon SL_2(k) \\rightarrow SL_2(k)$ by $\\begin{pmatrix} a&b\\\\c&d\\end{pmatrix} \\mapsto \\begin{pmatrix} \\sigma(a)&\\sigma(b)\\\\\\sigma(c)&\\sigma(d)\\end{pmatrix}$. Show that it is an automorphism.\n\\end{enumerate} \n\\end{exercise}\n\\noindent \nThis exercise describes automorphisms of the group $SL_2(k)$. The first kind of automorphisms, $\\iota_g$, are called inner automorphism if $g\\in SL_2(k)$ and diagonal automorphism otherwise. The automorphism $d$ given by transpose-inverse is called the graph automorphism. This comes from the Dynkin-diagram automorphism. The maps induced by field automorphisms are called field automorphism. It is a theorem of Steinberg and further extended by Humphreys that any abstract automorphism of the group $SL_2(k)$ is of the form $\\iota_g d \\sigma$, that is, compositions of the above maps. Further, if we consider only algebraic group automorphisms the field automorphisms are not there, as they are not morphism of varieties. \n \n\\section{The group $SL_2(\\mathbb Z)$} \nThe group $SL_2(\\mathbb Z)=\\left\\{X\\in M_2(\\mathbb Z)\\mid \\det(X)=1 \\right\\}$.\nLet us recall some transformations from complex analysis. We denote by $\\mathbb C^+$, the extended complex plane.\n\\begin{exercise}\nConsider the upper half plane $U=\\{z\\in \\mathbb C \\mid Im(z)>0\\}\\cup \\{\\infty\\}$. Denote $U_1=\\{z\\in U\\mid |z|<1\\}$.\n\\begin{enumerate}\n\\item[(a)] Show that a M\\\"{o}bius Transformation $T\\colon \\mathbb C^+ \\rightarrow \\mathbb C^+$ defined by $T(z)=\\frac{az+b}{cz+d}$ where $a,b,c,d\\in \\mathbb R$ with the property $ad-bc=1$ maps $U$ to $U$. Thus, the group of M\\\"{o}bius Transformations, $SL_2(\\mathbb Z)$ as well as $SL_2(\\mathbb R)$, act on $U$.\n\\item[(b)] Consider $s\\colon U\\rightarrow U$ given by $s(z)=-\\frac{1}{z}$ and $\\tau\\colon U\\rightarrow U$ given by $\\tau(z)=z+1$. Find the image of an element $z\\in U_1$ under $s,\\tau,s\\tau, \\tau s, s\\tau s$ and $\\tau s\\tau$. Draw pictures in each case as what these elements do to $U_1$ and $\\mathbb C\\backslash U_1$.\n\\item[(c)] Show that $-1$ acts trivially.\n\\item[(d)] Show that $s^2=1, (s\\tau)^3=-1$ and for any integer \n$n>0$,  $(s\\tau s)^n\\neq 1, (\\tau s\\tau)^n\\neq 1$.\n\\end{enumerate}\n\\end{exercise}\n\n\\begin{exercise} Using the exercise above solve the following.\n\\begin{enumerate}\n\\item Show that the group $SL_2(\\mathbb Z)$ is generated by the two matrices $S:=\\begin{pmatrix} & -1 \\\\1&\\end{pmatrix}$ and $T:=\\begin{pmatrix} & -1\\\\ 1& 1\\end{pmatrix}$. Show that the subgroup generated by $<S,T> = <S,U>$ where $U=\\begin{pmatrix} 1& 1\\\\ & 1\\end{pmatrix}$.\n\\item Use this to prove that the group $PSL_2(\\mathbb Z)$ is a free product of $\\mathbb Z\/2\\mathbb Z$ and $\\mathbb Z\/3\\mathbb Z$, i.e., $PSL_2(\\mathbb Z)\\cong \\mathbb Z\/2\\mathbb Z*\\mathbb Z\/3\\mathbb Z$.\n\\item Use this to prove that the group $SL_2(\\mathbb Z)$ is an amalgamated free product of $\\mathbb Z\/4\\mathbb Z$ and $\\mathbb Z\/6\\mathbb Z$ over $\\mathbb Z\/2\\mathbb Z$, i.e., $SL_2(\\mathbb Z)\\cong \\mathbb Z\/4\\mathbb Z*_{\\mathbb Z\/2\\mathbb Z}\\mathbb Z\/3\\mathbb Z$.\n\\end{enumerate}  \n\\end{exercise}\n\\noindent Here is a brief idea how to go about it. The elements, $\\pm I$, are obtained by computing power of $S$. Then show that the elements of the form $\\begin{pmatrix} 1& 1\\\\ & 1\\end{pmatrix}$ can be obtained by computing $S^{-1}T$. Use this to prove that $\\begin{pmatrix} 1& t\\\\ & 1\\end{pmatrix}$ matrices can be generated by taking power (including inverse) of the previous one. By multiplying with $-I$ show that any upper triangular matrix can be produced. Now to get a general matrix $\\begin{pmatrix} a& b\\\\ c&d\\end{pmatrix}$ we may assume $|a|\\geq |c|$. Use division algorithm and multiplication by the upper triangular matrices (from right) to reduce the size of $a$. \n\n\\begin{exercise}\nFinite groups are algebraic groups, thus $<S>$ and $<T>$ are algebraic subgroups of $SL_2(\\mathbb C)$. However, the subgroup $<S,T>=SL_2(\\mathbb Z)$ is not an algebraic subgroup. \n\\end{exercise}\nNote that this exercise also points out that subgroup generated by two finite order elements could be infinite.\n\n\\section{Topological group $SL_2(\\mathbb R)$}\nWe consider $M_2(\\mathbb R)$ as Euclidean space $\\mathbb R^4$ with usual metric topology. \n\\begin{exercise}\nShow that the map $\\det \\colon M_2(\\mathbb R) \\rightarrow \\mathbb R$ given by $A\\mapsto \\det(A)$ is a continuous map.  \n\\end{exercise}\n\n\\begin{exercise}\nThe set $GL_2(\\mathbb R)$ is an open set and $SL_2(\\mathbb R)$ is a closed set.\n\\end{exercise}\n\n\n\\begin{exercise}\n\\begin{enumerate}\n\\item Show that $SL_2(\\mathbb R)$ is connected. (You can use the idea of Chevalley generators from previous section which gives one-parameter subgroups.)\n\\item Show that $SL_2(\\mathbb R)$ is not compact.\n\\end{enumerate}\n\\end{exercise}\n\n\\begin{exercise}\nShow that $SO_2(\\mathbb R)$ is a maximal compact connected subgroup of $SL_2(\\mathbb R)$.\n\\end{exercise}\n\n\\begin{exercise}[Iwasawa Decomposition]\nLet $A=\\begin{pmatrix} a& b \\\\ c& d\\end{pmatrix} \\in SL_2(\\mathbb R)$. Use Gram-orthonormalisation process on the row vectors of the matrix $A$ to convert it to orthonormal vectors. This idea will prove the following results:\n\\begin{enumerate}\n\\item Write $A=PS$ where $P=\\begin{pmatrix} \\alpha & \\beta \\\\ & \\delta \\end{pmatrix}$ with $\\alpha, \\delta > 0$ and $S\\in O_2(\\mathbb R)$. \n\\item This shows that as a topological space $SL_2(\\mathbb R) \\cong \\mathbb R^{+}\\times \\mathbb R^{+} \\times \\mathbb R \\times O_2(\\mathbb R) $.\n\\item Using this prove that $SL_2(\\mathbb R)$ is connected.\n\\item Prove that the fundamental group $\\pi_1(SL_2(\\mathbb R))\\cong \\mathbb Z$.\n\\end{enumerate}\n\\end{exercise}\n\n\n\\section{Conjugacy classes and representations}\n\n\\begin{exercise}\nDetermine the conjugacy classes in $GL_2(\\mathbb C), GL_2(\\mathbb R), SL_2(\\mathbb C), SL_2(\\mathbb R)$.\n\\end{exercise}\n\\begin{exercise}\nShow that the unipotents $\\begin{pmatrix} 1&a\\\\&1\\end{pmatrix}$ form one conjugacy class in $GL_2(k)$. However, they form $k^*\/{k^*}^2$ many conjugacy classes in $SL_2(k)$. Thus when $k=\\mathbb Q$, they form infinitely many conjugacy classes of unipotents.\n\\end{exercise}\n\n\\begin{exercise}\nConsider the group $SL_2(\\mathbb R)$. This group has a natural representation on $2$-dimensional real vector space $V_2$. We write this, say with basis $\\{x,y\\}$, as follows: for $g\\in SL_2(\\mathbb R)$ write $g=\\begin{pmatrix} a& b \\\\c&d\\end{pmatrix}$ then, $$g.x=ax+cy \\ \\ {\\rm and}\\ \\ g.y=bx+dy.$$ \nNow consider a real vector space $V_{n+1}$ for $n\\geq 1$ with basis \n$$\\{x^n, x^{n-1}y, x^{n-2}y^2,\\ldots, xy^{n-1}, y^n\\}.$$\nWe define a representation $\\rho_{n+1}\\colon SL_2(\\mathbb R) \\rightarrow GL(V_{n+1})$ as follows: \n$$\\rho_{n+1}(g)(x^iy^j):=(g.x)^i(g.y)^j$$\nwhere right hand side is expanded as polynomials. Show that $\\rho_{n+1}$ is an irreducible representation.\n\\end{exercise}\n\\noindent In this exercise one may consider the vector space $V_{n+1}$ as the space of all homogeneous polynomials of degree $n$. Another way to think about this space is that $V_{n+1}\\cong Sym^n(V_2)$, the $n^{th}$ symmetric power. To prove irreducibility one makes use of the two nice subgroups, the diagonals and the compact subgroup $SO_2(\\mathbb R)$. \n\n\\begin{exercise}\nCompute all conjugacy classes of the group $GL_2(\\mathbb F_q)$ and $SL_2(\\mathbb F_q)$ and number of elements in each conjugacy class. Make a table for the same.\n\\end{exercise}\n\n\nNow we introduce more general version of this group. Let $R$ be a commutative ring with identity. Then $SL_2(R)$ is a group. Further the elementary matrices $x_{12}(t)$ and $x_{21}(t)$ are in $SL_2(R)$ for $t\\in R$. The subgroup generated by all of the elementary matrices is called the elementary subgroup $E_2(R)$. In general, $E_2(R)$ is a subgroup of $SL_2(R)$. One of the questions is to determine for what $R$ they are equal. Certainly this is the case when $R$ is a field. \n\n\\section{$SL_2(\\mathbb Q)$}\nLet $K=\\mathbb Q(\\sqrt d)$ be a degree two field extension of $\\mathbb Q$ where $d$ is a squarefree integer. Denote by $K^1=\\{x\\in K \\mid N(x)=1\\}$ where $N(a+b\\sqrt d)=a^2-b^2d$.\n\\begin{exercise}\nShow that the map $\\phi\\colon K^* \\rightarrow GL_2(\\mathbb Q)$ defined by $a+b \\sqrt d \\mapsto \\begin{pmatrix} a & bd \\\\b& a  \\end{pmatrix}$ is an injective group homomorphism.\n\\end{exercise}\n\n\\begin{exercise}\nShow that the map $\\phi \\colon K^1 \\rightarrow SL_2(\\mathbb Q)$ defined by $a+b \\sqrt d \\mapsto \\begin{pmatrix} a & bd \\\\b& a  \\end{pmatrix}$ is an injective group homomorphism.\n\\end{exercise}\n\n\\begin{exercise}\n Show that all elements of $K^*$ (identified with the image via $\\phi$) are diagonalisable in $GL_2(\\mathbb C)$.\n\\end{exercise}\n\n\\begin{exercise}\nShow that $\\phi(K^*)$ is a centraliser of an element. \n\\end{exercise}\n\n\\begin{exercise}\n Let $K$ and $L$ be two non-isomorphic field extensions of $\\mathbb Q$. Show that the image subgroups of $K^*$ and $L^*$ are not conjugate in $GL_2(\\mathbb Q)$. \n\\end{exercise}\n\n\n\\chapter{$GL_n(k)$ and $SL_n(k)$}\n\nLet $k$ be a field. Let $V$ be a vector space of finite dimension over $k$, say dimension of $V$ is $n$. By fixing a basis $\\mathcal B=\\{e_1,\\ldots, e_n\\}$ we can identify $V$ with column vectors $k^n$. We denote the set $GL(V)$ (general linear group) by the set of all invertible linear transformation. We know that using a fixed basis we can represent any element of $GL(V)$ be an invertible matrix, thus we can identify $GL(V)$ with $GL_n(k)$. We wish to study $GL_n(k)$ as a group. The group $SL(V)$ (special linear group) is the linear transformations of determinant $1$ and are written in matrix form as $SL_n(k)$.\nThe identity matrix is usually written as $I$ and $e_{i,j}$ represents the matrix with $1$ at $ij^{th}$ place and $0$ elsewhere.\n\n\\section{Some actions} \n\\begin{exercise}\nThe group $GL(V)$ acts on $V$ by evaluation. Show that there are exactly two orbits $\\{0\\}$ and $V\\backslash \\{0\\}$. \n\\end{exercise}\nThe set of all one dimensional subspaces of $V$ is written as $\\mathbb P(V)$, called projective space. This can be defined in another way as follows. Define an equivalence relation on $V\\backslash \\{0\\}$ by declaring $v$ is related to $w$ if there exists an scalar $\\lambda \\in k^*$ such that $v=\\lambda w$. Then the set $\\mathbb P(V)$ is all equivalence classes. \n\\begin{exercise}\nShow that $GL(V)$ acts on $\\mathbb P(V)$ transitively. This action has kernel which is the center of the groups. \n\\end{exercise}\n\\begin{exercise}\nShow that a matrix $A\\in M_n(k)$ is invertible if and only if its row vectors form a basis of $k^n$ if and only if its column vectors form a basis of $k^n$. \n\\end{exercise}\n\\begin{exercise}\nLet $k=\\mathbb F_q$ be the finite field with $q$ elements. \n\\begin{enumerate}\n\\item Show that $|GL_n(\\mathbb F_q)| = (q^n-1)(q^n-q)(q^n-q^2)\\cdots (q^n-q^{n-1})$.\n\\item Compute the size of $|\\mathbb P(\\mathbb F_q^{n})|$?\n\\end{enumerate}\n\\end{exercise}\n\\noindent Now we introduce some more groups: projective general linear group \n$$PGL_n(k) = \\frac{GL_n(k)}{\\mathcal Z(GL_n(k))} $$\nand projective special linear group \n$$PSL_n(k) = \\frac{SL_n(k)}{\\mathcal Z(SL_n(k))}.$$\n\\begin{exercise}\nShow that the center $\\mathcal Z(GL_n(k)) =\\{\\lambda.I_n \\mid \\lambda \\in k^*\\} \\cong k^*$ and  $\\mathcal Z(SL_n(k)) =\\{\\lambda.I_n \\mid \\lambda \\in k^*, \\lambda^n=1\\}$.\n\\end{exercise}\n\\begin{exercise}\n\\begin{enumerate}\n\\item Show that the group $PGL_n(k)$ acts transitively on the set of all $(n+1)$-tuples of points from $\\mathbb P(k^n)$.   \n\\item Show that $PGL_n(k)$ is $2$-transitive on the points of $\\mathbb P(k^n)$.\n\\item Show that $PGL_2(k)$ is $3$-transitive on $\\mathbb P(k^1)$.\n\\end{enumerate}\n\\end{exercise}\n\\begin{exercise}\nLet $G=GL_n(k)$ and $H=SL_n(k)$. Show that $H$ is a normal subgroup of $G$. Consider the subgroup $K=\\{\\textup{diag}(\\lambda, 1\\ldots,1)\\}$. Show that $G$ is a semi-direct product of $H$ and $K$. Thus, the following short exact sequence is split:\n$$1\\rightarrow SL_n(k) \\rightarrow GL_n(k) \\stackrel{\\det}{\\rightarrow} k^* \\rightarrow 1.$$\n\\end{exercise}\n\n\\begin{exercise}\nLet $G=GL_n(\\mathbb F_p)$ where $p$ is a prime.\n\\begin{enumerate}\n\\item What is the $|G|$?\n\\item Show that the set of upper triangular matrices with all diagonal entries $1$ form a Sylow $p$-subgroup of the group $GL_n(\\mathbb F_p)$.\n\\item Show that $G$ has $(1+p)(1+p+p^2)(1+p+p^2+p^3)\\cdots (1+p+\\cdots+p^{n-1})$ Sylow $p$-subgroups.\n\\end{enumerate}\n\\end{exercise}\nFor the above exercise, consider $U$ as a Sylow $p$-subgroup consisting of  upper triangular matrices with all diagonals $1$. Consider its normalizer $N_G(U)$. Show that $N_G(U)=B$, the set of all upper triangular matrices. Then the number $n_p=|G|\/|N_G(U)|=|G|\/|B|$.\n\n\\begin{exercise}[Grassmanian]\nLet $V$ be a vector space over a field $k$ of dimension $n$. Consider the set of all $r$-dimensional subspaces denoted as $G^r(V)$. \n\\begin{enumerate}\n\\item Show that $GL(V)$ acts on $G^r(V)$ given by $g.W=g(W)$ transitively. \n\\item Determine the stabiliser. Can we restrict this action to $SL(V)$?\n\\item Show that the number of elements in $G^r(\\mathbb F_q^n)$ is \n$$\\frac{(q^n-1)(q^n-q)\\cdots (q^n-q^{r-1})}{(q^r-1)(q^r-q)\\cdots (q^{r}-q^{r-1})}.$$ \n\\item Show that $|G^r(\\mathbb F_q^n)| = |G^{n-r}(\\mathbb F_q^n)|$. \n\\end{enumerate}\n\\end{exercise}\n\\noindent The space $G(V)=\\bigoplus G^r(V)$ is called Grassmanian. It is a projective variety.\n\n\\begin{exercise}\nLet $V$ be a vector space and $V^*$ its dual space. For a subspace $W$ of $V$ define $W^*=\\{f\\in V^*\\mid f(w)=0\\forall w\\in W\\}$. Show that $W\\mapsto W^*$ is a inclusion reversing one-one map between subspaces of $V$ and $V^*$.\n\\end{exercise}\n\n\\begin{exercise}[Cayley-Hamilton theorem]\n\\begin{enumerate}\n\\item[(a)] Let $A=(a_{ij})$ be a $n\\times n$ upper triangular matrix, i.e., $a_{ij}=0$ for all $i>j$. Find its characteristic polynomial $\\chi_A(X)$.\n\\item[(b)] Prove the Cayley-Hamilton theorem for an upper triangular matrix $A$ (by direct calculation), that is, $\\chi_A(A)\\equiv 0$.\n\\item[(c)] Use the part (b) to prove Cayley-Hamilton theorem for any complex matrix $S$, i.e., show that $\\chi_S(S)\\equiv 0$.\n\\item[(d)] Let $R$ be a commutative ring with $1$. Let $S\\in M_n(R)$. Let $\\chi_S(x)=\\det(x.I-S)$ be the characteristic polynomial over $R$. Show that $\\chi_S(S)\\equiv 0$. \n\\end{enumerate}\n\\end{exercise}\n\n\n\\section{Bruhat Decomposition}\nDenote the subgroup of all upper triangular matrices in $GL_n(k)$ by $B$. It is also called a Borel subgroup. The set of permutation matrices obtained by permuting the rows of identity matrix is a copy of the symmetric group in $GL_n(k)$. Then,\n\\begin{theorem}\nThe double coset decomposition\n$$B\\backslash GL_n(k) \/ B = \\bigsqcup_{w\\in S_n} BwB $$\nwhere $S_n$ denotes the set of all permutation matrices. \n\\end{theorem}\n\\noindent Denote by $T$ the set of all diagonal matrices in $GL_n(k)$. It is called a maximal torus.\n\\begin{exercise}\n\\begin{enumerate}\n\\item Prove that the normaliser of $T$ is the set of all monomial matrices. Show that $\\mathcal Z_{GL_n(k)}(T) = T$. \n\\item Show that $N_{GL_n(k)}(T)\/T \\cong S_n$.\n\\end{enumerate}\n\\end{exercise}\n\nLet us recall the traditional row-column operations. There are three row operations and three column operations each. The row operation $R1$ is to multiply an $i^{th}$ row by some $t\\in k$ and add to $j^{th}$ row where $i\\neq j$. The row operation $R2$ interchanges an $i^{th}$ row with $j^{th}$ for $i\\neq j$. The row operation $R3$ multiplies a row by a scalar $\\lambda\\in k^*$. Similarly, there are three column operations $C1, C2$ and $C3$ analogous to the row operations.\nLet $k$ be a field and for $t\\in k$, define $x_{i,j}(t)=I+te_{i,j}\\in GL_n(k)$ for $i\\neq j$ where $e_{i,j}$ represents the matrix with $1$ at $ij^{th}$ place and $0$ elsewhere. \n\\begin{exercise}\n\\begin{enumerate}\n\\item Show that the multiplication by $x_{i,j}(t)$ from left (right) to a matrix $A$ is the row (column) operation $R1$ ($C1$).\n\\item Show that the multiplication by a permutation matrix from left (right) is the row operation $R2$ ($C2$).\n\\item Show that the multiplication by a diagonal matrix $\\textup{diag}(1,\\ldots, 1, \\lambda, 1, \\ldots, 1)$ where $\\lambda$ is at $i^{th}$ place, from left (right) is the row (column) operation $R3$ ($C3$). \n\\item Let $A$ be an invertible matrix. Show that by using the row-column operations we can reduce it to identity. \n\\item Show that inverse of row-column operations are themselves. This algorithm is used to compute inverse of a matrix.\n\\end{enumerate}\n\\end{exercise}\n\\noindent However, now we allow only the first row and column operations $R1$ and $C1$. \n\\begin{exercise}\nBy computing $n_{ij}(1)=x_{ij}(1)x_{ji}(-1)x_{ij}(1)$ show that we can still do the $R2$ and $C2$ (well ! almost).\n\\end{exercise}\n\\noindent Thus we introduce our new row-column operations $NR2$ and $NC2$ where we multiply by the above matrices $n_{ij}$ replacing $R2$ and $C2$. \n\\begin{exercise}\nBy computing $h_{ij}=n_{ij}(t)n_{ij}(-1)$ where $n_{ij}(t)=x_{ij}(t)x_{ji}(-t^{-1})x_{ij}(t)$ show that $h_{ij}(t)$ is a diagonal matrix with $t$ at $i^{th}$ place and $t^{-1}$ at $j^{th}$ place. Use this to show we can still do $R3$ and $C3$, again almost.\n\\end{exercise}\n\\noindent Now, we introduce new row-column operations $NR3$ and $NC3$ where we multiply by the above matrices $h_{ij}(t)$ replacing $R3$ and $C3$. \n\\begin{exercise}[Gaussian Elimination]\nNow we have row-column operations as $R1, NR2, NR3$ and $C1, NC2, NC3$. \n\\begin{enumerate}\n\\item Show that, using row-column operations, any invertible matrix $A$ can be reduced to a diagonal matrix $\\textup{diag}(1,\\ldots,1,\\det(A))$.\n\\item Show that if we use row-column operations $x_{i,j}(t)$ for $i<j$, that is only half of the $R1$ and $C1$, then any invertible matrix $A$ can be written as $U_1WU_2$ where $U_1$ and $U_2$ are upper triangular matrices and $W$ is a monomial (permutation of a diagonal) matrix.\n\\item Show that any matrix $A$ (not necessarily invertible) can be reduced to a diagonal matrix of kind $\\textup{diag}(1,\\ldots,1,\\det(A))$ or $\\textup{diag}(1,\\ldots,1,0,\\ldots,0)$ by row-column operations. Determine the number of non-zeros on the diagonal.\n\\item Determine when an invertible matrix can be written as product of an upper triangular matrix and a lower triangular matrix?\n\\item Prove that $SL_n(k)$ is generated by the elementary matrices $x_{ij}(t)$ where $i\\neq j$ and $t\\in k$.\n\\end{enumerate}\n\\end{exercise}\n\n\\begin{exercise}\nLet $N\\in M_n(k)$ be a nilpotent matrix. Show that $\\exp(N):=\\sum_{r\\geq 0} \\frac{N^r}{r!}$ has only finitely many terms and is an invertible matrix. \n\\end{exercise}\n\\begin{exercise}\nConsider the set $sl_n(k)=\\{A\\in M_n(k) \\mid trace(A)=0\\}$. Compute the exponentials of $e_{ij}$ where $i\\neq j$. \n\\end{exercise}\n\n\n\\section{Parabolic subgroups}\nNow we introduce the parabolic subgroups.\nLet $V$ be a vector space over field $k$ of dimension $n$. A flag $\\mathcal F =(V_0, V_1, V_2, \\cdots, V_r)$ is a strictly increasing sequence of subspaces satisfying \n$$0=V_0 \\subsetneq V_1 \\subsetneq V_2\\subsetneq \\cdots \\subsetneq V_i\\subsetneq V_{i+1} \\subsetneq \\cdots \\subsetneq V_r=V.$$\nWe say that this flag is of length $r$ (this counts the number of proper inclusions). To a flag $\\mathcal F =(V_0, V_1, \\cdots, V_r)$ we associate a sequence of positive integers $(n_1, n_2, \\ldots, n_r)$ such that $\\dim(V_i)=n_1+n_2\\cdots +n_i$. Note that this sequence is a partition of $n$. The flag $(0, V)$ is called the trivial flag. It has length $1$ and corresponds to partition $n=n$. A flag is said to be complete if it has length $n$ and corresponds to the partition $n=1+1+\\cdots +1$. We count unordered partitions in the sense that the partitions $1+2$ and $2+1$ of $3$ are distinct. Now, we fix a basis of $V$, say, $\\{v_1,v_2,\\ldots, v_n\\}$. Then to each partition of $n=n_1+n_2+\\cdots +n_r$ we associate a standard flag as follows: We define the subspace $V_i=<v_1,\\ldots, v_{n_1+\\cdots+n_i}>$. Conversely, every flag can be thought of this way with respect to some basis. Let us denote by $\\mathfrak F$, the set of all flags.\n\\begin{exercise}\n\\begin{enumerate}\n\\item Show that $GL(V)$ acts on the set of all flags $\\mathfrak F$ given by $g(V_1, V_2, \\cdots, V_r)= (gV_1, gV_2, \\cdots, gV_r)$.\n\\item Show that each orbit contains a standard flag. Hence the orbits are in one-one correspondence with partitions. How many orbits are there?\n\\item Determine the stabilisers in each case. Note that it is enough to determine this for the standard flag. We get the stair-case subgroups which in block matrix form looks like this\n$$P_{(n_1,\\ldots, n_r)} = \\begin{pmatrix} GL_{n_1} & * & *& \\cdots &* \\\\ 0 &GL_{n_2} & *&\\cdots & * \\\\ \\vdots & 0&\\ddots & & *\\\\ \\vdots & \\vdots&&\\ddots  & *\\\\0 & 0  &\\cdots& 0& GL_{n_r} \\end{pmatrix}.$$\nThis subgroup is called standard parabolic subgroups of $GL_n$.\n\\end{enumerate}  \n\\end{exercise}\n\\noindent The stabiliser of a flag is called a parabolic subgroup. The parabolic subgroup obtained by stabiliser of a standard flag is called standard parabolic. All other parabolics are conjugate of standard parabolics. The parabolic corresponding to a complete flag is called a Borel subgroup. Note that the standard Borel subgroup is the set of all upper triangular matrices. Further, every parabolic contains a Borel subgroup. \n\n\\begin{exercise}[Levi decomposition]\nShow that the standard parabolic subgroups have a following decomposition:\n\\begin{eqnarray*}\nP_{(n_1,\\ldots, n_r)}&=& \\begin{pmatrix} GL_{n_1} & * & *& \\cdots &* \\\\ 0 &GL_{n_2} & *&\\cdots & * \\\\ \\vdots & 0&\\ddots & & *\\\\ \\vdots & \\vdots&&\\ddots  & *\\\\0 & 0  &\\cdots& 0& GL_{n_r} \\end{pmatrix} \\\\\n&=& \\begin{pmatrix} GL_{n_1} & 0 & 0& \\cdots &0 \\\\ 0 &GL_{n_2} & 0&\\cdots & 0 \\\\ \\vdots & 0&\\ddots & & 0\\\\ \\vdots & \\vdots&&\\ddots  & 0\\\\0 & 0  &\\cdots& 0& GL_{n_r} \\end{pmatrix}  \\ltimes          \\begin{pmatrix} I_{n_1} & * & *& \\cdots &* \\\\ 0 & I_{n_2} & *&\\cdots & * \\\\ \\vdots & 0&\\ddots & & *\\\\ \\vdots & \\vdots&&\\ddots  & *\\\\0 & 0  &\\cdots& 0& I_{n_r} \\end{pmatrix}.\n\\end{eqnarray*}\nThis is obtained by defining a group homomorphism $P_{(n_1,\\ldots,n_r)} \\rightarrow \\prod_{i=1}^r GL_{n_i}$. \n\\end{exercise}\n\\noindent This kind of decomposition is not unique. We have simply exhibited one such. For example, in the case of Borel this amounts to fixing a maximal torus.\n\n\n\\section{Embedding field extensions in $GL_n(k)$}\nLet $K$ be a field extension of $k$ of degree $n$. That is to say, considered $K$ over $k$ as a vector space is of dimension $n$. Let $\\alpha\\in K$. We define left multiplication map $l_{\\alpha} \\colon K \\rightarrow K$ defined by $x\\mapsto \\alpha x$.\n\\begin{exercise}\n\\begin{enumerate}\n\\item What is the relation between minimal polynomial of the element $\\alpha$ and the minimal polynomial of linear map $l_{\\alpha}$?\n\\item What is the relation between the traces of $\\alpha$ and $l_{\\alpha}$?\n\\item What is the relation between the norm of $\\alpha$ and determinant $l_{\\alpha}$?\n\\end{enumerate}\n\\end{exercise}\n\\begin{exercise}\nProve that the map $\\mathbb C \\rightarrow M_2(\\mathbb R)$ defined by $a+ib \\mapsto \\begin{pmatrix} a & -b \\\\ b&a\\end{pmatrix}$ is an injective ring homomorphism.\n\\end{exercise}\n\n\\begin{exercise}[Trace Bilinear form]\nDefine the bilinear form on the $k$-vector space $K$ as follows: $B\\colon K\\times K \\rightarrow k$ by $B(x,y)=trace(l_xl_y)$.\n\\begin{enumerate}\n\\item Show that $B$ is a symmetric bilinear form.\n\\item Show that $B$ is non-degenerate if and only if $K$ is a separable extension over $k$. \n\\end{enumerate}\n\\end{exercise}\n\\noindent Since all field extensions of characteristics $0$ are separable the trace bilinear form is always non-degenerate. It's the case for finite extensions of finite fields. \n\\begin{exercise}\nTake the example of field $\\mathbb F_p(t)$ and consider the extension given by splitting the polynomial $X^p-t$. Show that the trace bilinear form is degenerate.\n\\end{exercise} \n\n\\section{Metric Topology}\n\nConsider the set $M_n(\\mathbb R)$ with Euclidean topology (identified with $\\mathbb R^{n^2})$. \n\\begin{exercise}\nDecide if $GL_n(\\mathbb R)$ and $SL_n(\\mathbb R)$ are compact and\/or connected. \n\\end{exercise}\n\n\\begin{exercise}[Iwasawa decomposition]\nUse Gram-orthogonalisation to prove that every matrix $A$ in $SL_n(\\mathbb R)$ can be written as $A=PS$ where $P$ is an upper triangular matrix with all diagonals $>0$ and $S\\in O_n(\\mathbb R)$. \n\\end{exercise}\n\n\n\\begin{exercise}\nConsider $M_n(\\mathbb C)$ (as $\\mathbb C^{n^2}$) with usual metric topology. \n\\begin{enumerate}\n\\item[(a)] Show that $GL_n(\\mathbb C)$ is a dense open subset of $M_n(\\mathbb C)$.\n\\item[(b)] Show that the set of all diagonalisable matrices in $GL_n(\\mathbb C)$ is a dense subset.\n\\end{enumerate}\n\\end{exercise}\n\n\n\n\n\n\n\\section{Conjugacy classes in $GL_n(k)$}\n\nWhen $k$ is an algebraically closed field (for example $k=\\mathbb C$ or $\\bar{\\mathbb F_q}$) the conjugacy classes are determined by Jordan canonical forms. Let us recall this from linear algebra. We define the following: The linear transformations $T$ and $S$ on a finite dimensional vector space $V$ are said to be similar if there exists an invertible linear transformation $P$ such that $PTP^{-1}=S$. We note that if $T$ and $S$ are in $GL(V)$ they are similar if and only if they are conjugate. The theory of Jordan canonical forms uniquely (almost) associates a representative to each conjugacy class. \n\\begin{theorem}[Jordan canonical forms]\nRecall?\n\\end{theorem}\n\n\\begin{exercise}\nLet $J_r(\\lambda)$ be the $r\\times r$ matrix with all diagonals $\\lambda$ and $1$ at supper diagonals (above diagonal). \n\\begin{enumerate}\n\\item Find minimal and characteristic polynomial of $J_r(\\lambda)$?\n\\item Find minimal and characteristic polynomial of $J_{\\bar r}(\\lambda):=J_{r_1}(\\lambda)\\oplus J_{r_2}(\\lambda)\\oplus \\cdots\\oplus J_{r_k}(\\lambda)$ where notation $\\bar r=(r_1,\\ldots,r_k)$ written in decreasing order is a partition.\n\\item Find minimal and characteristic polynomial of $J:=J_{n_1}(\\lambda_1)\\oplus J_{n_2}(\\lambda_2)\\oplus \\cdots\\oplus J_{n_m}(\\lambda_m)$.\n\\item Find minimal and characteristic polynomial of $J:=J_{\\bar n_1}(\\lambda_1)\\oplus J_{\\bar n_2}(\\lambda_2)\\oplus \\cdots\\oplus J_{\\bar n_m}(\\lambda_m)$.\n\\end{enumerate}\n\\end{exercise}\n\nThe above definitions are extended to matrices. Two matrices $A$ and $B$ are said to be similar if there exists $P\\in GL_n(k)$ such that $PAP^{-1}=B$. We note that the similarity relation on $GL_n(k)$ is same as conjugacy relation. However, the next question is to determine the conjugacy classes over any arbitrary field, specially when it is not algebraically closed. Since the eigen values do not exist we can not use the Jordan canonical form theory. Thus, one makes use of Rational canonical forms. \n\\begin{exercise}\nRecall rational canonical forms?\n\\end{exercise}\n\n\n\\begin{exercise}\nLet $V$ be a vector space over field $k$ of dimension $\\geq 2$. \n \\begin{enumerate}\n\\item If $k=\\mathbb C$ every linear transformation has an eigen-vector. Show by an example that this may not be true over $\\mathbb R$.\n\\item Not every linear transformation is diagonalisable. \n\\item Over $\\mathbb C$, every linear transformation is triangulable. Show by an example that this may not be true over $\\mathbb R$. \n\\item A linear transformation $T$ is called nilpotent if $T^r=0$ for some $r$. Every nilpotent linear transformation is triangulable. \n\\item A linear transformation $T$ is called unipotent if $T-I$ is nilpotent. Every unipotent linear transformation is triangulable. \n\\end{enumerate}\n\\end{exercise}\n\n\n\\begin{exercise}\nLet $k$ be an algebraically closed field and $G\\subset M_n(k)$ consisting of commuting set of matrices.\n\\begin{enumerate}\n\\item If all elements of $G$ are diagonalisable then prove that $G$ is simultaneously diagonalisable.\n\\item Give an example that this need not be true if $G$ is not a commuting set.\n\\item In general, show that $G$ is triangulable in such a way that all diagonalisable elements of $G$ become diagonal at the same time.\n\\end{enumerate}\n\\end{exercise}\n\n \n \n\\section{Jordan-Chevalley decomposition}\nLet $V$ be a finite dimensional vector space over a field $k$. \n\\begin{definition}\nAn element $X\\in \\textup{End}(V)$ is said to be {\\bf semisimple} if it is diagonalisable over $\\bar k$. An element $X\\in \\textup{End}(V)$ is said to be nilpotent if $X^r=0$ for some $r$. An element $X\\in \\textup{End}(V)$ is said to be unipotent if $X-1$ is nilpotent.\n\\end{definition}\n\\begin{exercise}\nLet $X\\in \\textup{End}(V)$. Prove the following,\n\\begin{enumerate}\n\\item $X$ is nilpotent if and only if all eigen-values of $X$ are $0$. Prove that any such transformation can be always upper-tiangulaised.\n\\item $X$ is unipotent if and only if all eigen-values of $X$ are $1$. Prove that any such transformation can be always upper-tiangulaised and it is always invertible. \n\\end{enumerate}\n\\end{exercise}\n\\begin{exercise}\nLet $X\\in \\textup{End}(V)$. Suppose $W$ is an $X$-invariant subspace, i.e., $X(W)\\subset W$. Prove the following:\n\\begin{enumerate}\n\\item If $X$ is semisimple then $X|_W$ is also semisimple.\n\\item If $X$ is nilpotent then $X|_W$ is also nilpotent.\n\\item If $X$ is unipotent then $X|_W$ is also unipotent.\n\\end{enumerate}\n\\end{exercise}\n\\begin{exercise}\nLet $X\\in \\textup{End}(V)$. Suppose $W_1$ and $W_2$ are $X$-invariant subspaces such that $V=W_1\\bigoplus W_2$. Prove the following:\n\\begin{enumerate}\n\\item If $X|_{W_1}$ and $X|_{W_2}$ both are semisimple then $X$ is also semisimple.\n\\item If  $X|_{W_1}$ and $X|_{W_2}$ both are nilpotent then $X$ is also nilpotent.\n\\item If $X|_{W_1}$ and $X|_{W_2}$ both are unipotent then $X$ is also unipotent.\n\\end{enumerate}\n\\end{exercise}\n\\noindent We can easily find examples and show that not every linear transformation is of one of the above form. However, it is very interesting to note that over algebraically closed field $k=\\bar k$ every transformation is made of using only these transformations. We prove this below. This idea is useful in algebraic groups as when we requre to prove something for a group elementwise we first prove them for semisimple and unipotent elements and then try to guess if we can prove for a general element.\n\\begin{proposition}[Additive Jordan decomposition]\nLet $k=\\bar k$. Let $T\\in \\textup{End}(V)$. Then,\n\\begin{enumerate}\n\\item There exists $S,N\\in \\textup{End}(V)$ such that $T=S+N$ where $S$ is semisimple and $N$ is nilpotent satisfying the property that they commute, i.e., $SN=NS$.\n\\item The transformations $S$ and $N$ are unique with the above property.\n\\item There exists polynomials $p(X)$ and $q(X)$ without constants such that $S=p(T)$ and $N=q(T)$.\n\\end{enumerate}\n\\end{proposition}\n\\noindent We sketch the proof of this in the following exercises.\n\\begin{exercise}\nLet $m_T(X)$ be the minimal polynomial of $T$ and its factorisation over $k=\\bar k$ be\n$$m_T(X) = \\prod_{\\alpha\\in k} (X-\\alpha)^{n_{\\alpha}}.$$\nConsider $V_{\\alpha}=\\{v\\in V \\mid (T-\\alpha)^m(v)=0,\\ for\\ some\\ m\\geq 1\\}$.\n\\begin{enumerate}\n\\item Show that $V=\\bigoplus_{\\alpha\\in k} V_{\\alpha}$ is a direct sum decomposition. \n\\item Show that $V_{\\alpha}=ker(T-\\alpha)^{n_\\alpha}$ and $T(V_{\\alpha})\\subset V_{\\alpha}$.\n\\item Show that the minimal polynomial of $T_{\\alpha}=T|_{V_{\\alpha}}$ is $(X-\\alpha)^{n_{\\alpha}}$.\n\\item Let us construct $S$ now. Define a transformation $S$ on $V$ by defining it on each $V_{\\alpha}$ to be the scalar transformation $\\alpha.I_{V_{\\alpha}}$, where $I_{V_{\\alpha}}$ is identity on $V_{\\alpha}$.\n\\begin{enumerate}\n\\item Prove that $S$ is semisimple.\n\\item $S$ and $T$ commute. Check this on each $V_{\\alpha}$.\n\\item Show that $T-S$ is nilpotent on each $V_{\\alpha}$.\n\\end{enumerate}\n\\item Define $N=T-S$ and prove that it is a nilpotent linear transformation.\n\\item Thus we have $T=S+N$. Now, show that $N$ and $S$ commute. This prove part 1.\n\\item Proof of part 2 is easy. Use the fact that only linear transformation which is semisimple and nilpotent both is the $0$ transformation.\n\\end{enumerate}\n\\end{exercise}\nTo complete the proof we still need to prove part 3. Which we do in the following,\n\\begin{exercise}\n\\begin{enumerate}\n\\item Prove that the polynomials $\\{(X-\\alpha)^{n_{\\alpha}}\\}$ are coprime. Then use Chinese Remainder Theorem to show that there exists a polynomial $p(X)$ such that $p(X) \\equiv \\alpha \\imod{ (X-\\alpha)^{n_{\\alpha}}}$ for all $\\alpha$ and $p(X) \\equiv 0 \\imod{X}$.  \n\\item Show that $S=p(T)$. \n\\item Now define $q(X)=X-p(X)$ and show that $q(T)=N$.\n\\end{enumerate}\n\\end{exercise}\n\\begin{exercise}\nShow that if $T$ is invertible then $S$ is also invertible. Further show that $S^{-1}$ is a polynomial in $T$. For this use that $p(X)$ and $m_T(X)$ are coprime. \n\\end{exercise}\n\\noindent Using the above ideas conclude the following:\n\\begin{proposition}[Jordan decomposition]\nLet $k=\\bar k$. Let $T\\in GL(V)$. Then,\n\\begin{enumerate}\n\\item There exists $S, U\\in GL(V)$ such that $T=SU$ where $S$ is semisimple and $U$ is unipotent satisfying the property that they commute, i.e., $SU=US$.\n\\item The transformations $S$ and $U$ are unique with the above property.\n\\item There exists polynomials $p(X)$ and $q(X)$ without constants such that $S=p(T)$ and $U=q(T)$.\n\\end{enumerate}\n\\end{proposition}\nThe importance of this theorem in algebraic groups is this, Chevalley proved that Jordan decomposition  continues to hold for algebraic subgroups of $GL(V)$. Which is to say, that if we have an algebraic subgroup $G$ of $GL(V)$ and $T\\in G$ then the components $T$ and $U$ are again in $G$. Further he proved that this is a functorial property, that is, semisimple and unipotent elements gets mapped to semisimple and unipotent elements respectively under algebraic group homomorphisms. For example the classical groups, such as, orthogonal groups and symplectic groups (over $k=\\bar k$) are algebaric subgroups. Hence the Chevalley's theorem would imply that the Jordan decomposition of elements in these groups can be done with in those groups. \n\n\\begin{exercise}\n\\begin{enumerate}\n\\item Let $char(k)=0$. Show that every finite order element in $GL_n(k)$ is semisimple.\n\\item Show that the matrices $\\begin{pmatrix} \\cos(\\theta) & \\sin(\\theta) \\\\ -\\sin(\\theta) & \\cos(\\theta)\\end{pmatrix}$ are semisimple in $SL_2(\\mathbb C)$.\n\\end{enumerate}\n\\end{exercise}\n\\begin{exercise}\nConsider the subgroup $H$ generated by the matrix $\\begin{pmatrix} \\pi &1 \\\\ \\pi^{-1}\\end{pmatrix}$ of $SL_2$. Does $H$ contain semisimple and unipotent parts of each of its elements?\n\\end{exercise}\n\n\n\\chapter{Root Datum}\nIn the theory of Lie algebras, we associate a combinatorial data to a semisimple Lie algebra, called {\\bf root system}. The root systems are classified using their Dynkin diagram and in turn we classify (semi)simple Lie algebras. In the theory of Linear algebraic groups (over $k=\\bar k$) we associate {\\bf root datum} to a reductive group. In general root datum is more general than root system.\n\n\\section{Root Datum vs Root System}\n\\begin{definition}[Root Datum]\nA root datum $\\Psi=(X, R, Y, R^{\\vee})$ is a quadrpule where\n\\begin{enumerate}\n\\item[(a)] $X$ and $Y$ are free Abelian groups of finite rank with a perfect pairing $<,>\\colon X\\times Y \\rightarrow \\mathbb Z$, and,\n\\item[(b)] $R$ and $R^{\\vee}$ are finite subsets of $X$ and $Y$ respectively, along-with a bijection $R\\rightarrow R^{\\vee}$ denoted by $\\alpha\\mapsto \\alpha^{\\vee}$.\n\\end{enumerate}\nThis quadrpule satisfies the following properties:\n\\begin{enumerate}\n\\item $\\forall \\alpha, <\\alpha, \\alpha^{\\vee}>=2$.\n\\item $s_{\\alpha}(R)=R$ and $s_{\\alpha}^{\\vee}(R^{\\vee})=R^{\\vee}$ for all $\\alpha \\in R$, where $s_{\\alpha}\\colon X\\rightarrow X$ is in $\\textup{Aut}(X)$ given by $s_{\\alpha}(x)=x-<x,\\alpha^{\\vee}>\\alpha$ and $s_{\\alpha}^{\\vee}\\in \\textup{Aut}(Y)$ given by  $s_{\\alpha}^{\\vee}(y)=y-<\\alpha,y>\\alpha^{\\vee}$.\n\\end{enumerate}\n\\end{definition}\nRecall that a perfect pairing here is a $\\mathbb Z$-bilinear map which induces isomorphism $X$ to $Y^*$ when we fix first variable, and $Y$ to $X^*$, when we fix the second variable.\nThe Weyl group of a root datum is defined as a subgroup $W(\\Psi)=<s_{\\alpha} \\mid \\alpha\\in R>\\subset \\textup{Aut}(X)$. \n\\begin{exercise}\n$0\\notin R$ and $W(\\Psi)$ is finite.\n\\end{exercise}\n\n\n\\section{Root datum of rank $1$ semisimple algebraic groups}\nLet $k$ be an algebraically closed field. Let $G$ be a semisimple algebraic group of rank $1$. There are two such groups: $SL_2(k)$ and $PGL_2(k)$. We try to understand their root datum and how they differ a bit from each other. \n\n\\subsection{$SL_2(k)$} Let us first begin working with the group $SL_2(k)$. We fix the maximal torus $T=\\left\\{ \\begin{pmatrix} t & \\\\ & t^{-1} \\end{pmatrix} \\mid t\\in k^*\\right\\}$. The Lie algebra is $sl_2(k)=\\{X\\in M_2(k) \\mid tr(X)=0\\}$ and the $Ad$ action is given by $Ad \\colon SL_2(k) \\rightarrow GL(sl_2(k))$ where $Ad_g(X)=gXg^{-1}$. We restrict this action to $T$ and decompose the Lie algebra $sl_2(k)$ as simultaneous eigen-spaces. \n\\begin{exercise}\nCompute the character group $X(T)$ of the torus and show that it is isomorphic to $\\mathbb Z$ generated by $\\chi(\\textup{diag}(t,t^{-1}))=t$.\n\\end{exercise}\n\n\\begin{exercise}\nShow the decomposition $sl_2(k)=<h> \\bigoplus <e_{12}> \\bigoplus <e_{21}>$ with respect to the $Ad$ action of $T$. Here $h=\\textup{diag}(1,-1)$. The following computation will be useful to conclude this:\n\\begin{enumerate}\n\\item $Ad(\\textup{diag}(t,t^{-1}))(h)=h$,\n\\item $Ad(\\textup{diag}(t,t^{-1}))(e_{12})=t^2e_{12} = 2\\chi(t) e_{12}$,\n\\item $Ad(\\textup{diag}(t,t^{-1}))(e_{21})=t^{-2}e_{21} = -2\\chi(t) e_{21}$.\n\\end{enumerate}\n\\end{exercise}\n\\noindent Hence, $\\Phi=\\{2\\chi, -2\\chi\\}\\subset X(T)=<\\chi>\\cong \\mathbb Z$. Verify that $<\\Phi>\\cong 2\\mathbb Z \\subset \\mathbb Z$. Now, the question is to determine the co-roots. \n\\begin{exercise}\nShow that the co-character group $Y(T)$ of the torus is $<\\psi>\\cong \\mathbb Z$ where $\\psi\\colon \\mathbb G_m \\rightarrow T$ given by $t\\mapsto \\textup{diag}(t,t^{-1})$. \n\\end{exercise}\n\\begin{exercise}\nFor the root $\\alpha=2\\chi$ determine the co-root $\\alpha^{\\vee}\\in Y(T)$ using the condition that it must satisfy $<\\alpha, \\alpha^{\\vee}>=2$. \n\\end{exercise}\n\\noindent Recall that the $<,> \\colon X(T)\\times Y(T) \\rightarrow \\mathbb Z$ is defined by $<m\\chi, n\\psi> = mn$, since $(m\\chi)(n\\psi)(t) = t^{mn}$. Hence we get $\\alpha^{\\vee}=\\psi$ and $\\Phi^{\\vee}= <\\psi>$. Thus, the root datum for the group $SL_2$ is \n$$(X(T), \\Phi, Y(T), \\Phi^{\\vee}) = (<\\chi>, \\{2\\chi\\}, <\\psi>, \\{\\psi\\}) = (\\mathbb Z, 2\\mathbb Z, \\mathbb Z, \\mathbb Z).$$\nNow we determine the root datum for $PGL_2$.\n\n\\subsection{$PGL_2(k)$}\nThe group \n$$PGL_2(k) = \\frac{GL_2(k)}{ \\mathcal Z(GL_2(k))} = \\frac{GL_2(k)}{\\{\\textup{diag}(\\lambda, \\lambda)\\mid \\lambda\\in k^*\\}}.$$ \nWe represent the maximal torus by $T=\\{\\textup{diag}(\\lambda , 1)\\mid \\lambda\\in k^*\\}$ thought of as the image of diagonal torus in $GL_2(k)$ identified as follows: $\\textup{diag}(\\lambda_1, \\lambda_2) \\mapsto \\textup{diag}(\\lambda_1 \\lambda_2^{-1}, 1)$.\n\\begin{exercise}\nShow that the character group $X(T)=<\\chi> \\cong \\mathbb Z$ where the character $\\chi$ is given by $\\chi(\\textup{diag}(\\lambda_1 \\lambda_2^{-1}, 1)) = \\frac{\\lambda_1}{\\lambda_2}$. \n\\end{exercise}\nThe Lie algebra is $pgl_2= M_2(k)\/\\{\\textup{diag}(\\lambda, \\lambda)\\mid \\lambda\\in k\\}$ and hence we can take $e_{11}, e_{12}, e_{21}$ as a basis of this Lie algebra.\nUse the restriction of $Ad$ map to the maximal torus and do the simultaneous diagonalisation to prove the following.\n\\begin{exercise}\nThe root-space decomposition of the Lie algebra is\n$$pgl_2 = <h> \\bigoplus <e_{12}> \\bigoplus <e_{21}>.$$\nTo prove this verify the following,\n$Ad(\\textup{diag}(\\lambda_1, \\lambda_2))(e_{11})=e_{11}$ and $Ad(\\textup{diag}(\\lambda_1, \\lambda_2))(e_{12})=\\frac{\\lambda_1}{\\lambda_2}e_{12}$.\n\\end{exercise}\nHence, $\\Phi=\\{\\chi, -\\chi\\} \\subset X(T)$ is the corresponding root system. We also note that $<\\Phi> =<\\chi> = X(T)\\cong \\mathbb Z$. Now the question is to find the co-roots.\n\\begin{exercise}\nShow that the co-character group $Y(T)=<\\psi>\\cong \\mathbb Z$ where $\\psi\\colon \\mathbb G_m \\rightarrow T$ given by $\\psi(\\lambda) \\mapsto \\textup{diag}(\\lambda, 1)$. \n\\end{exercise}\n\\noindent Now we need to compute the co-root $\\chi^{\\vee}$. This can be done by the equation $<\\chi, a\\psi>=2$ gives the value $a=2$. Hence $\\chi^{\\vee}=2\\psi$. Hence the co-roots are $\\Phi^{\\vee}=\\{2\\psi, -2\\psi\\}$. Hence the root datum for the group $PGL_2(k)$ is \n$$(X, \\Phi, Y, \\Phi^{\\vee}) = (<\\chi>, \\{\\chi, -\\chi\\}, <\\psi>, \\{2\\psi, -2\\psi\\}) \\cong (\\mathbb Z, \\mathbb Z, \\mathbb Z, 2\\mathbb Z).$$\nThis example shows that the root datum of the two groups $SL_2(k)$ and $PGL_2(k)$ are different and they can be distinguished by knowing the root datum.\n\n\n\\section{Root datum of the reductive algebraic group $GL_n$}\nIn this section we have $k=\\bar k$. Some of the exercises might be repetition from previous section but they have been included here for the sake of completeness.\n\n\\begin{exercise}\nShow that $GL_n(k)$ is a Zariski-closed subset of $k^{n^2+1}$. Thus its an affine algebraic group of dimension $n^2$.\n\\end{exercise}\n\\begin{exercise}\n\\begin{enumerate}\n\\item Show that the set $D_n(k)$ of all diagonals in $GL_n(k)$ is a maximal torus.\n\\item Any two maximal tori in $GL_n(k)$ are conjugate.\n\\end{enumerate}\n\\end{exercise}\nHint: Since the centraliser of $D_n(k)$ is itself, the first one follows.\n\\begin{exercise}\nThe set of all semisimple elements \n$$G_s=\\bigcup_{g\\in GL_n}g D_n(k) g^{-1} = \\bigcup_{S\\ maximal\\ tori} S. $$\n\\end{exercise}\n\\begin{exercise}\nThe set $N_n(k)$ of all upper triangular matrices with $1$ on diagonal is a maximal connected unipotent subgroup of $GL_n(k)$. In fact, any unipotent subgroup of $GL_n(K)$ can be conjugated to a subgroup of $N_n(k)$. Hence, the set of all unipotents $$G_u=\\bigcup_{g\\in GL_n} gN_n(k)g^{-1} .$$\n\\end{exercise}\n\\begin{exercise}\n$$N_{GL_n(k)}(N_n(k)) = T_n(k)$$\nwhere $T_n(k)$ is the set of all upper triangular matrices.\n\\end{exercise}\n\\begin{exercise}\nShow that the group $T_n(k)\\cong D_n(k) \\ltimes N_n(k)$.\n\\end{exercise}\n\\begin{exercise}[Weyl group]\nConsider $T=D_n(k)$, a maximal torus. Then,\n\\begin{enumerate}\n\\item $\\mathcal Z_{GL_n(k)}(T)=T$.\n\\item $N_{GL_n(k)}(T)$ is the set of all monomial matrices.\n\\item The Weyl group \n$$W(GL_n, T)=\\frac{N_{GL_n}(T)}{\\mathcal Z_{GL_n}(T)} \\cong S_n.$$\n\\item The Weyl group is independent of choice of a maximal torus.\n\\end{enumerate}\n\\end{exercise}\n\\noindent Recall that a matrix is said to be monomial if its each row and each column has exactly one non-zero entry. It can be also thought of as permutations of a diagonal matrix.\n\nWe recall the notation of unordered partition. A partition of $n$ is a sequence $(n_1,\\ldots, n_r)$ such that $n=n_1+n_2+\\cdots +n_r$. For us the partition $(1,2)$ and $(2,1)$ of $3$ are different.\n\n\\begin{exercise}\nFor a partition $n=n_1+n_2+\\cdots + n_r$ consider the set \n$$T_{(n_1,\\ldots, n_r)}=\\{\\textup{diag}(\\lambda_1,\\ldots, \\lambda_1, \\ldots, \\lambda_r,\\ldots, \\lambda_r)\\mid \\lambda_1,\\ldots, \\lambda_r\\in k^*\\}$$ of diagonal matrices where $\\lambda_i$ repeats $n_i$ many times. \n\\begin{enumerate}\n\\item Show that $T_{(n_1,\\ldots, n_r)}$ is a torus.\n\\item Any torus is conjugate to a subtorus of this form. \n\\end{enumerate}\n\\end{exercise}\n\n\\begin{exercise}\nShow that\n$$\\mathcal Z_{GL_n}(T_{(n_1,\\ldots, n_r)}) \\cong GL_{n_1}\\times \\cdots \\times GL_{n_r}.$$\n\\end{exercise}\n\\begin{exercise}\n\\begin{enumerate}\n\\item Compute the normaliser $N_{GL_n(k)}(T_{(n_1,\\ldots, n_r)})$.\n\\item Compute the quotient group \n$$W(GL_n, T_{(n_1,\\ldots, n_r)})=\\frac{N_{GL_n}(T_{(n_1,\\ldots, n_r)})}{\\mathcal Z_{GL_n}(T_{(n_1,\\ldots, n_r)})}.$$\n\\end{enumerate}\n\\end{exercise}\n\nFor $i\\neq j$ define the $ij^{th}$-elementary matrices $x_{ij}(t)=I + te_{ij}$.\n\\begin{exercise}[Root subgroups]\nShow that the map $\\psi_{ij} \\colon \\mathbb G_a \\rightarrow GL_n(k)$ given by $t\\mapsto x_{ij}(t)$ is a morphism of algebraic groups. \n\\end{exercise}\n\\begin{exercise}\nThe group $SL_n(k)$ is generated by the set of all elementary matrices $\\{x_{ij}(t) \\mid i\\neq j, t\\in k\\}$. In fact, the Gaussian elimination algorithm using row-column operations provides a proof of this.\n\\end{exercise}\n\\begin{exercise}\nThe commutator subgroup $[GL_n(k), GL_n(k)]=SL_n(k)$.\n\\end{exercise}\n\\begin{exercise}[Lie Algebra]\nThe set of all matrices $gl_n(k)$ and the set of all trace $0$ matrices $sl_n(k)$ are the Lie algebra of the group $GL_n(k)$ and $SL_n(k)$ respectively.\n\\end{exercise}\n\\begin{exercise}\nFix $i\\neq j$. Show that the map $\\phi_{ij}\\colon SL_2(k)\\rightarrow GL_n(k)$ defined by extending \n$$\\begin{pmatrix} 1& t \\\\ &1\\end{pmatrix} \\mapsto x_{ij}(t),\\hskip10mm  \\begin{pmatrix} 1&  \\\\ t &1\\end{pmatrix} \\mapsto x_{ji}(t)$$\nis a morphism of algebraic groups.\n\\end{exercise}\n\\begin{exercise}\nWith the notation as above,\n\\begin{enumerate}\n\\item Compute $n_{ij}(t)=x_{ij}(t)x_{ji}(-t^{-1})x_{ij}(t)$ and show that $n_{ij}(t)\\in N_{GL_n}(D_n)$.\n\\item Compute $h_{ij}(t)=n_{ij}(t)n_{ij}(-1)$ and show that it belongs to $D_n(k)$.\n\\item The elements $n_{ij}:=n_{ij}(1)$ gives all possible row permutations and hence generates the Weyl group $W=\\frac{N_{GL_n}(D_n)}{D_n}$.\n\\end{enumerate}\n\\end{exercise}\n\\begin{exercise}[Bruhat decomposition]\nLet $G=GL_n$ and $B=T_n(k)$ be a Borel subgroup. Show that the double coset decomposition of $G$ by $B$ is \n$$G=\\bigcup_{w\\in S_n} BwB.$$\n\\end{exercise}\n\\noindent Notice that, in this case the double cosets have a group structure induced from the corresponding Weyl group. In general this is not true for an arbitrary finite group.\n\nA character of an algebraic group $G$ is a morphism of algebraic group $\\chi\\colon G \\rightarrow \\mathbb G_m$. \n\\begin{exercise}\n\\begin{enumerate}\n\\item Show that the determinant map is a character on $GL_n$.\n\\item Show that the character group $X(GL_n)\\cong \\mathbb Z$.\n\\end{enumerate}\n\\end{exercise} \n\n\\begin{exercise}\nShow that the set of all flags $\\mathfrak F$ is a projective variety and hence complete. Thus the quotient $GL_n\/P_{(n_1,\\ldots,n_r)}$ is a complete variety. Hence the parabolic subgroups $P_{(n_1,\\ldots, n_r)}$ are parabolic in algebraic group sense.\n\\end{exercise}\n\n\\begin{exercise}[Levi decomposition]\nThe parabolic subgroups $P_{(n_1,\\ldots, n_r)}$ have decomposition $L_P\\ltimes U_P$ where $L_P$ is called a Levi component which is a reductive group and $U_P$ is the unipotent radical of $P_{(n_1,\\ldots,n_r)}$.\n\\end{exercise}\n\\begin{exercise}[Perfect Pairing]\nConsider the maximal torus $T=D_n(k)$. \n\\begin{enumerate}\n\\item The character group $X(T)=<\\chi_1,\\ldots, \\chi_n> \\cong \\mathbb Z^n$ where $\\chi_i\\colon T \\rightarrow \\mathbb G_m$ is given by $\\chi(\\sum a_ie_{ii})=a_i$.\n\\item The co-character group $Y(T)=<\\psi_1, \\ldots, \\psi_n> \\cong \\mathbb Z^n$ where $\\psi_j\\colon \\mathbb G_m \\rightarrow T$ given by $\\psi_j(t)=te_{jj}$.\n\\item The pairing of $X(T)$ and $Y(T)$ is a map induced from the map \n$$< , >\\colon X(T) \\times Y(T) \\rightarrow \\textup{Aut}(\\mathbb G_m) \\text{\\ given by\\ } (\\chi,\\psi)\\mapsto \\chi\\psi .$$ Thus, computing with the above notation this gives the map $<,>\\colon \\mathbb Z^n \\times \\mathbb Z^n \\rightarrow \\mathbb Z$ by $<(a_1,\\ldots,a_n), (b_1,\\ldots b_n)> = \\sum a_ib_i$.\n\\end{enumerate}\n\\end{exercise}\n\n\\begin{exercise}\\label{weylgroup}\nThe map $w\\colon W(GL_n, T) \\rightarrow \\textup{Aut}(X(T))$ given by $n\\mapsto w_n$ defined by \n$$(w_n(\\chi))(t)=\\chi(n^{-1}tn)$$\nis an injective map.\n\\end{exercise}\n\n\\begin{exercise}\nWe consider $Ad$ action of $GL_n(k)$ on $M_n(k)$ as follows: $Ad_g(X)=gXg^{-1}$. The maps $Ad_g$ are vector space isomorphisms and hence we have $Ad\\colon GL_n(k) \\rightarrow GL(M_n(k))$. In fact, they are algebra isomorphism of $M_n(k)$.\n\\end{exercise}\n\\noindent We remark that the Lie algebra of the algebraic group $GL_n(k)$ is the set of all matrices $M_n(k)$ with Lie bracket defined by $[A,B]=AB-BA$.\n\\begin{exercise}\nLet us consider the restriction of $Ad$ action to the maximal torus $T=D_n(k)$. Thus we have $Ad \\colon T \\rightarrow GL(M_n(k))$. Since $T$ consists of commuting set of semisimple elements, its image is simultaneously diagonalisable. Obtain the following simultaneous eigenspace decomposition  with respect to $T$,\n$$M_n(k) = d_n(k) \\bigoplus \\bigoplus_{i\\neq j} <e_{ij}>.$$\nTo get this prove the following:\n\\begin{enumerate}\n\\item For the trivial character $\\chi=0$ (in additive notation which maps everything to the constant $1$) we get $V_{\\chi}=d_n(k)$. This is because, $\\{X\\in M_n(k) \\mid Ad_t(X)=X \\forall t\\in T\\} = \\mathcal Z_{M_n(k)}(T) = d_n(k)$.\n\\item For $i\\neq j$, $V_{\\chi_i-\\chi_j}= <e_{ij}>$ where $\\chi_i-\\chi_j \\in X(T)$ given by \n$$(\\chi_i-\\chi_j)(\\textup{diag}(\\lambda_1,\\ldots, \\lambda_n))=\\lambda_i\\lambda_j^{-1}.$$ \nFor this we need to verify \n$$Ad_t(e_{ij}) = te_{ij}t^{-1} = (\\chi_i-\\chi_j)(t)e_{ij}.$$\n\\end{enumerate}\n\\end{exercise}\n\\noindent Now, we denote the set of non-zero characters appearing in the above decomposition by $\\Phi=\\{\\chi_i-\\chi_j \\mid i\\neq j\\}\\subset X(T)$.\n\\begin{exercise}\nThe set $\\Phi$ has rank $n-1$, i.e., it spans $n-1$ dimensional Abeilan subgroup in $\\mathbb Z^n$. Further, this can be obtained as kernel of the map $X(T) \\rightarrow \\mathbb Z$ given by $\\sum a_i\\chi_i \\mapsto \\sum a_i$.\n\\end{exercise}\n\n\\begin{exercise}\nTake $\\chi\\in \\Phi$, say $\\chi=\\chi_i-\\chi_j$. \n\\begin{enumerate}\n\\item Then show that $S= ker(\\chi) = \\{\\textup{diag}(\\lambda_1,\\ldots ,\\lambda_n) \\in T \\mid \\lambda_i=\\lambda_j\\}$ is a torus of rank $n-1$.\n\\item Compute $G_{\\chi}=\\mathcal Z_{GL_n}(S)$ which for $i=1, j=2$ looks like \n$$= \\left\\{ \\begin{pmatrix} GL_2 & & & & \\\\ &* &&& \\\\ &&\\ddots&&  \\\\&&&&* \\end{pmatrix} \\right\\}$$\nand show that it is isomorphic to $GL_2 \\times \\mathbb G_m^{n-2}$.\n\\item Note that $S\\subset T \\subset G_{\\chi}$. In fact, $S \\subset \\mathcal Z(G_{\\chi})$.\n\\end{enumerate}\n\\end{exercise}\n\n\\begin{exercise}\nObserve that $T\\subset G_{\\chi}$ is a maximal torus. Now define $W_{\\chi}= \\frac{N_{G_{\\chi}}(T)}{\\mathcal Z_{G_{\\chi}}(T)}$, the Weyl group of $G_{\\chi}$ with respect to $T$. Being isomorphic to the Weyl group of $G_{\\chi}\/S$ with respect to $T\/S$, it is generated by $W_{\\chi}=<n_{\\chi}>$ where $n_{\\chi}=x_{ij}(t)x_{ji}(-t^{-1})x_{ij}(t)$.\n\\end{exercise}\n\n\\begin{exercise}\nFor each $\\chi\\in \\Phi$, we have $n_{\\chi}$ obtained as above. Clearly, $n_{\\chi} \\in W(GL_n, T)$. Compute $w_{n_{\\chi}}$ (as in Exercise~\\ref{weylgroup}) and show that for $\\chi=\\chi_i-\\chi_j$, it maps $\\chi_i\\leftrightarrow \\chi_j$ and fixes all others. Thus, in $\\textup{Aut}(X)$ it is a permutation of basis elements.\n\\end{exercise}\n\n\\begin{exercise}\nLet us fix a Borel $B=T_n(k)$ containing our fixed diagonal maximal torus $T$. Then for $\\chi=\\chi_i-\\chi_j$,\n\\begin{enumerate}\n\\item $G_{\\chi} \\bigcap N_n(k) = X_{ij}$, the root subgroup.\n\\item $G_{\\chi} \\bigcap N_n(k)^{-} = X_{ji}$. \n\\end{enumerate}\n\\end{exercise}\n\nRecall that the co-root $\\alpha^{\\vee}$ for $\\alpha$ is defined by $<\\alpha, \\alpha^{\\vee}>=2$ and $w_{n_{\\alpha}}(x)=x-<x, \\alpha^{\\vee}>\\alpha$ where $w_{n_{\\alpha}}\\in \\textup{Aut}(X)$ is already determined.\n\\begin{exercise}[Co-roots]\nThus, for  $\\chi=\\chi_i-\\chi_j$ we get $\\chi^{\\vee}=\\psi_i-\\psi_j$.\n\\end{exercise}\nHint: Let $\\chi^{\\vee}=\\sum_{r} m_r\\psi_r$. Now, we know $w_{n_{\\chi}}(\\chi_r)=\\chi_{r}$ for $r\\neq i, j$ which gives, $\\chi_r= w_{n_{\\chi}}(\\chi_r)=\\chi_r-<\\chi_r, \\sum m_s\\psi_s>(\\chi_i-\\chi_j) = \\chi_r - m_r(\\chi_i-\\chi_j)$ gives $m_r=0$. Now $\\chi_j= w_{n_{\\chi}}(\\chi_i)=\\chi_i-<\\chi_i, \\sum m_s\\psi_s>(\\chi_i-\\chi_j) = \\chi_i - m_i(\\chi_i-\\chi_j)$, gives $m_i=1$. Similarly, we get $m_j=-1$.\n\nThe quadruple $(X(T), \\Phi, Y(T), \\Phi^{\\vee})$ obtained here is the root datum of $GL_n$ and determines it completely. \n\n\n\\chapter{(Real compact) Orthogonal group and Symmetry}\n\nWe begin with the set $\\mathbb R^n$. To denote the elements of $\\mathbb R^n$ we use coordinates. This coordinate system provides a unique representation of each element with respect to the coordinate system $\\{e_1,\\ldots,e_n\\}$. This coordinate system is called the standard basis. The set $\\mathbb R^n$ has many structures. \n\nThe {\\bf Euclidean space} $(\\mathbb R^n, d)$ is a metric space with the distance function \n$$d(x,y):=\\sqrt{(x_1-y_1)^2+\\cdots +(x_n-y_n)^2}$$\nwhere $x=(x_1,\\ldots,x_n)$ and $y=(y_1,\\ldots,y_n)$.\nThis induces a topology on $\\mathbb R^n$. One can study functions with various properties such as continuity, differentiability etc. Usually we take $n\\geq 2$. An {\\bf isometry} of $\\mathbb R^n$ is a map $f\\colon \\mathbb R^n\\rightarrow \\mathbb R^n$ such that $d(f(x),f(y))=d(x,y)$ for all $x,y\\in\\mathbb R^n$. Here we want to study isometries. We prove that isometries are bijection and form a group. They are also called affine linear maps.\n\nWe recall that $\\mathbb R^n$ is an $n$-dimensional vector space over the field $\\mathbb R$. Thus on $\\mathbb R^n$ we already have ``nice'' linear maps. The coordinate system provides a standard basis. These linear maps can be represented as matrices with respect to the fixed standard basis. First, we would like to determine which linear maps are isometry. For this purpose we introduce norm on the vector space. The {\\bf norm} on the vector space $\\mathbb R^n$ is given by $||x||:=\\sqrt{x_1^2+\\cdots+x_n^2}$. We also have corresponding symmetric bilinear\nform given by $B(x,y):=\\sum_{i=1}^n x_iy_i$. The norm and symmetric bilinear form are related and can be obtained from each other.\n\n\\section{$O_2(\\mathbb R)$ and $\\mathbb C$}\nThe orthogonal group $O_2(\\mathbb R)=\\{A\\in M_2(\\mathbb R) \\mid \\ensuremath{{}^t\\!} AA=I\\}$. \n\\begin{exercise}\nFor $a\\in \\mathbb R^2$ define the translation map $\\tau_a\\colon \\mathbb R^2\\rightarrow \\mathbb R^2$ by $\\tau_a(x)=x+a$. Show that $\\tau_a$ is an isometry but not a linear map. Further show\nthat $\\tau_a\\tau_b=\\tau_{a+b}$ and $\\tau_{a}^{-1}=\\tau_{-a}$.\n\\end{exercise}\n\n\\begin{exercise}\nConsider the rotation map $\\rho_{\\theta}\\colon \\mathbb R^2\\rightarrow \\mathbb R^2$ (around origin) given\nby $$\\rho_{\\theta}(x_1,x_2)=(\\cos(\\theta)x_1-\\sin(\\theta)x_2, \\sin(\\theta)x_1+\\cos(\\theta)x_2).$$\n\\begin{enumerate}\n\\item Show that $\\rho_{\\theta}$ is an isometry. Also $\\rho_{\\theta}=\\rho_{\\theta+2\\pi}$.\n\\item Show that $\\rho_{\\theta}$ is a linear, one-one and onto map. Write down\nits matrix.\n\\item $\\rho_{\\theta_1}\\rho_{\\theta_2}=\\rho_{\\theta_3}$ where $\\theta_3=\\theta_1+\\theta_2$ modulo $2\\pi$. \n\\end{enumerate}\n\\end{exercise}\n\n\\begin{exercise}\nGiven a non-zero vector $v\\in\\mathbb R^2$, consider the reflection map $r_v\\colon \\mathbb R^2\\rightarrow \\mathbb R^2$ defined by\n$r_v(x)=x-2\\frac{B(x,v)}{||v||^2}v$. This is a reflection in the line passing through origin perpendicular to the vector $v$.\n\\begin{enumerate}\n\\item Show that $r_v$ is an isometry and $r_v=r_{\\alpha v}$ for any $\\alpha\\neq 0$.\n\\item Show that $r_v$ is a linear, one-one and onto map.\n\\item Show that $r_v$ is characterised (among linear maps) by $r_v(v)=-v$ and $r_v(x)=0$ for $x\\perp v$ (i.e., $B(x,v)=0$).\n\\item Show that $r_v^2=1$. What do you get if you compose two reflections.\n\\end{enumerate}\n\\end{exercise}\n\\begin{exercise}\n\\begin{enumerate}\n\\item Use the definition and show that \n$$SO_2(\\mathbb R)=\\left \\{\\begin{pmatrix} \\cos(\\theta) & -\\sin(\\theta) \\\\ \\sin(\\theta) & \\cos(\\theta)\\end{pmatrix} \\mid 0\\leq \\theta < 2\\pi \\right\\}.$$\nThese elements are called rotations. \n\\item Use the map $\\det \\colon O_2(\\mathbb R) \\rightarrow \\{\\pm 1\\}$ to show that $|\\frac{O_2(\\mathbb R)}{SO_2(\\mathbb R)}|=2$.\n\\item By fixing an element $s\\in O_2(\\mathbb R)\\backslash SO_2(\\mathbb R)$ prove that $O_2(\\mathbb R) = SO_2(\\mathbb R) \\bigcup s.SO_2(\\mathbb R)$. Prove that we can take $s=\\begin{pmatrix} 1 &  \\\\ & -1 \\end{pmatrix}$.\n\\item Hence, \n$$O_2(\\mathbb R)=\\left \\{\\begin{pmatrix} \\cos(\\theta) & -\\sin(\\theta) \\\\ \\sin(\\theta) & \\cos(\\theta)\\end{pmatrix}, \\begin{pmatrix} \\cos(\\phi) & -\\sin(\\phi) \\\\ -\\sin(\\phi) & -\\cos(\\phi)\\end{pmatrix} \\mid 0\\leq \\theta, \\phi < 2\\pi \\right\\}.$$\n\\item Prove that every element of $O_2(\\mathbb R)\\backslash SO_2(\\mathbb R)$ has a fixed line. These are reflections.\n\\item Show that $SO_2(\\mathbb R)$ is an Abelian group but $O_2(\\mathbb R)$ is not.\n\\item Compute $s\\rho_{\\theta}s^{-1}$ for $\\rho_{\\theta}\\in SO_2(\\mathbb R)$?\n\\item Prove that every element of $SO_2(\\mathbb R)$ is a product of two  reflections. This is simply writing $\\rho_{\\theta}=s.s\\rho_{\\theta}$. \n\\end{enumerate}\n\\end{exercise}\n\nWe explore its relation with complex numbers $\\mathbb C$. It is a coincidence that the space $\\mathbb R^2$ has a multiplication which makes it a field. The important fact is that the norm $N(x)=||x||^2$ is multiplicative with respect to the multiplication defined by the complex.\n\\begin{exercise}\nShow that with complex multiplication $N(xy)=N(x)N(y)$. This will not be satisfied if we took co-ordinate wise multiplication on $\\mathbb R^2$. \n\\end{exercise}\n\\noindent Hurwitz proved that the only space which has multiplicative property for norm is $n=1,2, 4$ and $8$. Later in the next chapter we introduce Hamilton's quaternion on $\\mathbb R^4$ and provide the next example. The multiplication on $\\mathbb R^8$ which makes norm multiplicative makes it Octonion.\n\\begin{exercise}\nConsider $\\mathbb C$ as one dimensional vector over itself and $2$ dimensional vector space over $\\mathbb R$ with basis $\\{1,i\\}$.\n\\begin{enumerate}\n\\item Let $0\\neq \\alpha \\in \\mathbb C$. Write down the real matrix of left multiplication by $\\alpha$.\n\\item Show that $SO_2(\\mathbb R)$ can be identified with the group $SO_2(\\mathbb R)$.\n\\item Consider the conjugation map. What is its real matrix. \n\\end{enumerate}\n\\end{exercise}\n\n\\begin{exercise}\n\\begin{enumerate}\n\\item Prove that any finite subgroup of $SO_2(\\mathbb R)\\cong S^1$ is cyclic.\n\\item Prove that any finite subgroup of $O_2(\\mathbb R)$ which is not contained in $SO_2(\\mathbb R)$ is a dihedral group.\n\\item Fix a prime $p$. Take the set $\\{z\\in S^1 \\mid z^{p^r}=1 \\ for \\ some \\ r\\}$. Show that its a group.\n\\end{enumerate}\n\\end{exercise}\n\n\\begin{exercise}\nProve that $O_2(\\mathbb R)$ is not connected whereas $SO_2(\\mathbb R)$ is connected.  \n\\end{exercise}\n\\begin{exercise}\nProve that $O_2(\\mathbb R)$ and $SO_2(\\mathbb R)$ both are compact groups. \n\\end{exercise}\n\n\\section{$O_n(\\mathbb R)$}\n\nA linear transformation $S\\colon \\mathbb R^n \\rightarrow \\mathbb R^n$ is called {\\bf orthogonal}\nif $||S(x)||=||x||$ for all $x\\in \\mathbb R^n$.\n\\begin{exercise}\nShow that the following are equivalent for a linear transformation $S\\colon \n\\mathbb R^n \\rightarrow \\mathbb R^n$:\n\\begin{enumerate}\n\\item $S$ is orthogonal.\n\\item $S$ is an isometry.\n\\item $B(S(x),S(y))=B(x,y)$ for all $x,y\\in \\mathbb R^n$.\n\\item $S$ maps an orthonormal basis to an orthonormal basis.\n\\item The matrix $A$ of $S$ with respect to the standard basis satisfies $\\ensuremath{{}^t\\!} AA=I$.\n\\end{enumerate}\n\\end{exercise}\n\nWe define the {\\bf orthogonal group} $O(n)$ as the set of all orthogonal linear transformations.\nIn the matrix form (with respect to the fixed standard basis) \n$$O(n):=\\{A\\in M_n(\\mathbb R)\\mid \\ensuremath{{}^t\\!} AA=I\\}.$$\n\\begin{exercise}\nLet $A, B\\in O(n)$. Show that:\n\\begin{enumerate}\n\\item $\\det(A)=\\pm 1$.\n\\item $AB\\in O(n)$ and $A^{-1}\\in O(n)$.\n\\end{enumerate}\n\\end{exercise}\n\nWe define the set $SO(n):=\\{A\\in O(n)\\mid \\det(A)=1\\}$.\n\n\\begin{exercise}\nUse the reflection defined in the next exercise to show that $SO(n)\\subset O(n)$ is proper.\n\\end{exercise}\n\\section{Isometries}\n\n\\begin{exercise}[Examples of isometry]\n\\begin{enumerate}\n\\item For $a\\in \\mathbb R^n$ we define translation $\\tau_a\\colon \\mathbb R^n\\rightarrow\n\\mathbb R^n$ by $\\tau_a(x)=x+a$. Show that it's an isometry but not a linear map.\n\\item For a non-zero vector $v\\in \\mathbb R^n$ define reflection \n$r_v\\colon \\mathbb R^n \\rightarrow \\mathbb R^n$ by \n$$r_v(x)=x-2\\frac{B(x,v)}{||v||^2}v.$$\nShow that $r_v(v)=-v$ and $r_v$ fixes a $(n-1)$ dimensional hyperplane (which one!) pointwise. Further it\nis an isometry and a linear map.\n\\item For $\\theta\\in [0,2\\pi)$ and fixed $1\\leq l\\leq n-1$ we define \n$\\rho_{l,\\theta}\\colon \\mathbb R^n\\rightarrow \\mathbb R^n$\nby $\\rho_{l,\\theta}(e_l)= \\cos(\\theta)e_l-\\sin(\\theta)e_{l+1}$, \n$\\rho_{l,\\theta}(e_{l+1})= \\sin(\\theta)e_l+\\cos(\\theta)e_{l+1}$ and\n$\\rho_{l,\\theta}(e_i)=e_i$ for $i\\neq l,l+1$. Show that $\\rho_{l,\\theta}$\nis an isometry and is a linear map.\n\\end{enumerate}\n\\end{exercise}\n\\noindent However the distance function and the norm $||.||$, both are related on $\\mathbb R^n$.\n\\begin{exercise}\nShow the following:\n\\begin{enumerate}\n\\item $d(x,y)=||x-y||$. Further $d$ satisfies the $3$ properties of being a distance function.\n\\item $B(x,x)=||x||^2$.\n\\item $B(x,y)=\\frac{||x+y||^2-||x||^2-||y||^2}{2}$. Further $B$ is a symmetric bilinear map.\n\\end{enumerate}\n\\end{exercise}\n\\begin{exercise}\nShow that if $f\\colon \\mathbb R^n\\rightarrow \\mathbb R^n$ is a linear map which is also an isometry, then $f\\in O_n(\\mathbb R)$. \n\\end{exercise}\n\n\n\\noindent Now we prove that any isometry is composition of a translation and an orthogonal map. We remark that it is not clear from the definition of an isometry that it is a bijection.\n\\begin{exercise}\nLet $T\\colon \\mathbb R^n\\rightarrow \\mathbb R^n$ be an isometry. Suppose $T$ fixes origin\nand also every element of the standard basis, i.e., $T(0)=0$ and $T(e_i)=e_i\\forall i$.\nThen show that $T$ is identity.\n\\end{exercise}\n\\begin{proof}\nLet $x=(x_1,\\ldots, x_n)\\in \\mathbb R^n$ and $T(x)=y=(y_1,\\ldots, y_n)$. We want to prove $y=x$. We are given that $T$ is an isometry. Hence we have following equations, $d(0,x)=d(T(0),T(x))=d(0,y)$ implies $||x||=||y||$. Similarly,  $d(x,e_i)=d(T(x), T(e_i))=d(y,e_i)$ implies $||x-e_i||=||y-e_i||$. This gives the equation \n$$x_1^2+\\cdots+x_{i-1}^2+(x_i-1)^2+x_{i+1}^2+ \\cdots +x_n^2 = y_1^2+ \\cdots + y_{i-1}^2+ (y_i-1)^2+ y_{i+1}^2 + \\cdots +y_n^2$$\nwhich after expanding and combining with $||x||=||y||$ gives $x_i=y_i$. This proves the required result. \n\\end{proof}\n\\begin{exercise}\nLet $T\\colon \\mathbb R^n\\rightarrow \\mathbb R^n$ be an isometry. Suppose $T$ fixes origin.\nThen show that $T$ is an orthogonal linear transformation.\n\\end{exercise}\n\\begin{proof}\nConsider a matrix $S$ of which columns are $y_i=T(e_i)$. Then $S$ defines a linear map $S\\colon \\mathbb R^n \\rightarrow \\mathbb R^n$ with the property $S(e_i)=y_i= T(e_i)$ for all $i$. We claim that $S$ is an orthogonal transformation. For this verify that $\\{y_1,\\ldots, y_n\\}$ is an orthonormal basis, thus $S$ maps an orthonormal basis to an orthonormal basis.\n\nNow let us consider the map $S^{-1}T\\colon \\mathbb R^n \\rightarrow \\mathbb R^n$. This is an isometry. Further this map fixes origin and all of the $e_i$. Hence from previous exercise it must be the identity map. This implies that $T=S$.\n\\end{proof}\n\\begin{exercise}\nLet $f\\colon \\mathbb R^n\\rightarrow \\mathbb R^n$ be an isometry. \nShow that $f=\\tau_a T$ where $a=f(0)$ and $T\\in O_n(\\mathbb R)$. \n\\end{exercise}\n\\begin{proof}\nConsider the map $\\tau_{-a}f$ which is an isometry. Clearly $(\\tau_{-a}f)(0)=\\tau_{-a}(f(0))=f(0)-a=0$. Thus from previous exercise  $\\tau_{-a}f$ is an isometry, say $T\\in O_n(\\mathbb R)$. Hence, $f=\\tau_aT$.\n\\end{proof}\n\\noindent Thus we have the following,\n\\begin{exercise}\nLet $f\\colon \\mathbb R^n\\rightarrow \\mathbb R^n$ be an isometry. \n\\begin{enumerate}\n\\item Show that there exists $a\\in \\mathbb R^n$ and $A\\in O_n(\\mathbb R)$ such that $f(x)=Ax+a$. \n\\item Let $f, f'$ be isometry given by $f(x)=Ax+a$ and $f'(x)=A'x+a'$. Show that \n$$(ff')(x) = AA'x+(Aa'+a).$$ \n\\item Show that an isometry is one-one and onto map.\n\\item Show that the set of isometries $Iso(\\mathbb R^n)$ is closed under composition. \n\\item Show that the inverse of an isometry is again an isometry. In particular, $f^{-1}(x) = A^{-1}x - A^{-1}a$.\n\\item Show that $Iso(\\mathbb R^n)$ is a group.\n\\item Show that $(f\\tau_b f^{-1})(x) = \\tau_{Ab}(x)$ where $f(x)=Ax+a$. Use this to show that the set of translations is a normal subgroup of $Iso(\\mathbb R^n)$.\n\\item Prove that $Iso(\\mathbb R^n)\\cong \\mathbb R^n \\rtimes O_n(\\mathbb R)$.\n\\end{enumerate}\n\\end{exercise}\n\n\n\n\n\\section{Symmetries and Platonic Solids}\n\n\\begin{exercise}\nThis exercise leads to the {\\bf Classification of Platonic solids}.\n\\begin{definition}\nA {\\bf convex polyhedron} in 3d is said to be {\\bf regular} if:\n\\begin{itemize}\n\\item Its faces are convex regular $n$-gons (for some $n \\in \\mathbb N, n \\geq 3$) and any two faces are congruent.\n\\item Only possible intersections of faces are at edges.\n\\item At each vertex the same number (say $m$) of faces meet.\n\\end{itemize}\n\\end{definition}\nIt is clear that $n$ and $m$ are at least $3$. Regular convex polyhedra are also called Platonic solids. Cube is an example of a Platonic solid with $n = 4$ and $m = 3$.\nFor a Platonic solid the pair $(n,m)$ is called its Schl{\\\"a}fli symbol.  \n\nLet $P$ be a Platonic solid with Schl{\\\"a}fli symbol $(n,m)$. Let $v$ be the number of vertices of $P$, $e$ be the number of edges of $P$ and $f$ be the number of faces of $P$.\nSolve the following:\n\\begin{enumerate}\n\\item[(5a)] Show that $nf=2e=mv$.\n\\item[(5b)] Show that $\\frac{1}{m}+\\frac{1}{n}>\\frac{1}{2}$.\n\\item[(5c)] Find all solutions $(n,m)$.\n\\end{enumerate}\n\\end{exercise}\n\n\\begin{exercise}\nCompute the symmetry group of $n$-gons and Platonic solids. \n\\end{exercise}\n\n\\section{Finite subgroups of $O_2(\\mathbb R)$ and $O_3(\\mathbb R)$}\nThese are some well known theorems worth attempting to give a prove.\n\n\\begin{theorem}\nAny finite subgroup of $O_2(\\mathbb R)$ is isomorphic to either the cyclic group $\\mathbb Z\/m\\mathbb Z$ or the dihedral group $D_m$. \n\\end{theorem}\n\n\\begin{theorem}\nAny finite subgroup of  $O_3(\\mathbb R)$ is isomorphic to one of the following: the cyclic group $\\mathbb Z\/m\\mathbb Z$, the dihedral group $D_m$, $A_4, S_4$ or $A_5$. \n\\end{theorem}\n\nA more general theory at this point is trying to understand finite Coxeter groups. The finite Coxeter groups are defined as a subgroup generated by a bunch of finitely many reflections in an Orthogonal group. This is very welll understood and leads to classification via root systems, similar to simple Lie algebras. Interested reader can refer to the book by Humphreys on this subject.\n\n\\iffalse\n\\chapter{The alternating group $A_5$}\n\n\\section{Alternating group as $PGL_2$}\n\\begin{exercise}\nProve the following:\n$$A_5 \\cong SL_2(\\mathbb F_4) \\cong PGL_2(\\mathbb F_4) \\cong PSL_2(\\mathbb F_5).$$\n\\end{exercise}\n\n\\begin{exercise}\nProve the following:\n\\begin{enumerate}\n\\item $S_3\\cong GL_2(\\mathbb F_2)$.\n\\item $A_4\\cong PSL_2(\\mathbb F_3)$.\n\\item $S_4\\cong PGL_2(\\mathbb F_3)\\cong AGL_2(\\mathbb F_2)$.\n\\item $S_6\\cong Sp_4(\\mathbb F_2)$.\n\\end{enumerate}\n\\end{exercise}\n\\section{Alternating group as symmetries of Dodecahedron}\n\\fi\n\n\n\\chapter{Hamilton's Quaternion}\nThe set $\\mathbb H= \\{ a= a_0+a_1 i + a_2 j +a_3 k \\mid a_0, a_1, a_2, a_3 \\in \\mathbb R \\}$, with addition and multiplication defined below, is called the set of real quaternions or Hamilton's quaternion. These operations make it an associative algebra (non-commutative ring as well as a vector space over $\\mathbb R$). There is a much more general definition of quaternions over any field (not just over $\\mathbb R$) which we describe in Chapter~\\ref{quaternion}. The algebra operations on $\\mathbb H$ are defined as follows: for $a=a_0+a_1 i + a_2 j +a_3 k$ and $b = b_0+ b_1 i + b_2 j +b_3 k$, we have,\n\\begin{eqnarray*}\na + b &=& (a_0+b_0) + (a_1+b_1) i + (a_2+b_2) j + (a_3+b_3) k \\\\\na.b &=& (a_0b_0 - a_1b_1 -a_2b_2 -a_3b_3) + (a_0b_1 + a_1b_0 + a_2b_3 -a_3b_2) i   \\\\ && + (a_0b_2 -a_1b_3 + a_2b_0 + a_3b_1) j + (a_0b_3 + a_1b_2 -a_2b_1 + a_3b_0)k.\n\\end{eqnarray*}\nThe multiplication can be simply obtained by using the following relations: $i^2=-1=j^2=k^2$ and $ij=-ji=k$. \n\n\\begin{exercise}\nShow that $\\mathbb H$ is a division algebra. That is, it has all properties of field except the multiplication is not commutative. \n\\end{exercise}\n\nIt has several properties similar to complex numbers $\\mathbb C$. We define conjugation by \n$$\\bar a = a_0-a_1 i - a_2 j- a_3 k.$$ \n\\begin{exercise}\nCompute and show that $a \\bar a=a_0^2+a_1^2 + a_2^2 + a_3 ^2=\\bar a a$ is a real number. \n\\end{exercise}\nThis leads to the definition of norm, $N(a)=a\\bar a$, and trace $tr(a)=a+\\bar a = 2a_0$.\n\n\\begin{exercise}\n\\begin{enumerate}\n\\item Show that the map $N\\colon \\mathbb H^* \\rightarrow \\mathbb R^*$ defined by $a\\mapsto N(a)$ is a multiplicative group homomorphism. \n\\item Show that the map $Tr \\colon \\mathbb H \\rightarrow \\mathbb R$ defined by $a \\mapsto Tr(a)$ is an $\\mathbb R$-linear map.\n\\end{enumerate}\n\\end{exercise}\n\n\\begin{exercise}\n\\begin{enumerate}\n\\item How many solutions does the equation $X^2 + 1$ have over $\\mathbb H$? Compare your answer with solving equations over a field.\n\\item For an $a\\in \\mathbb H$ show that it always satisfies the quadratic equation $X^2 - (tr(a)) X + N(a) =0$.\n\\end{enumerate}\n\\end{exercise}\n\n\\section{Realization of quaternions}\nThere are several ways to think about quaternions. In various situation they can be helpful in computation.\n\n\\subsection{As double complex numbers}\nFor $a\\in \\mathbb H$ we write \n$$a=a_0+a_1 i + a_2 j +a_3 k = (a_0+a_1 i) + (a_2+a_3 i)j = z_1 + z_2 j$$\nwhere $z_1, z_2$ can be thought of as elements in $\\mathbb C$. Thus we may think of $\\mathbb H = \\mathbb C + \\mathbb C j$ and note that the new multiplication formula would be, for $a=z_1 + z_2 j$ and $b = w_1 + w_2 j$,\n$$a.b = (z_1 + z_2 j).(w_1 + w_2 j) = (z_1w_1-\\bar w_2 z_2) + (w_2z_1 + z_2\\bar w_1)j.$$\nWe remark that this formula can be further generalised to define an octonion algebra. \n\n\\begin{exercise}\nVerified that the new formula is same as usual multiplication on $\\mathbb H$. In this notation, $N(a)= N(z_1 + z_2 j) = N(z_1)+N(z_2)$ and $\\bar a = \\overline{z_1+z_2 j} = \\bar z_1 - z_2 j$ where the norm and conjugation on complex numbers are defined in usual way. \n\\end{exercise}\n\n\n\\subsection{As $2\\times 2$ complex matrices}\n\nFor $a\\in \\mathbb H$ we write $a= z_1 + z_2 j$ where $z_1, z_2 \\in \\mathbb C$ as above. Now we identify  $a$ with the matrix $a=\\begin{pmatrix} z_1 & -z_2 \\\\ \\bar z_2 & \\bar z_1 \\end{pmatrix}$ in $M_2(\\mathbb C)$. Thus we can think of $\\mathbb H$ to be the set of $2\\times 2$ complex matrices of this kind. The multiplication in this case is simply given by the matrix multiplication.\n\\begin{exercise}\nVerify that the quaternion multiplication is same as matrix multiplication represented in this form, i.e, \n$$a.b = \\begin{pmatrix} z_1 & -z_2 \\\\ \\bar z_2 & \\bar z_1 \\end{pmatrix} \\begin{pmatrix} w_1 & -w_2 \\\\ \\bar w_2 & \\bar w_1 \\end{pmatrix} = \\begin{pmatrix} z_1w_1 -z_2\\bar w_2 & -z_1w_2-z_2\\bar w_1 \\\\ \\bar z_2w_1+ \\bar z_1\\bar w_2 & -\\bar z_2 w_2 + \\bar z_1\\bar w_1 \\end{pmatrix}.$$\n\\end{exercise}\n\\begin{exercise}\nThe norm $N(a)=\\det\\begin{pmatrix} z_1 & -z_2 \\\\ \\bar z_2 & \\bar z_1 \\end{pmatrix} = z_1\\bar z_1 + z_2\\bar z_2$ and conjugation is $\\bar a = \\begin{pmatrix} \\bar z_1 & z_2 \\\\ -\\bar z_2 & z_1 \\end{pmatrix} = \\ensuremath{{}^t\\!}(\\bar a)$. Also, trace is simply trace of the matrix.\n\\end{exercise}\nRemember the following association in the matrix notation:\n$$1=\\begin{pmatrix} 1 & \\\\ & 1 \\end{pmatrix}, i=\\begin{pmatrix} i & \\\\ & -i \\end{pmatrix}, j=\\begin{pmatrix}  & -1 \\\\ 1 &  \\end{pmatrix}, j = \\begin{pmatrix}  & -i \\\\ -i & 0 \\end{pmatrix}.$$\n\n\\begin{exercise}[Four Square Identity]\n\\begin{enumerate}\n\\item Show that $\\overline{ab}=\\bar b \\bar a$, and hence, $N(ab) = ab. \\overline{ab} = ab \\bar b \\bar a = a N(b) \\bar a = N(a)N(b)$.\n\\item Since the norm map is multiplicative, ie., $N(ab)=N(a)N(b)$, write this in expanded form to get the Four Square Identity.\n\\begin{eqnarray*}\n&&(a_0^2+a_1^2 + a_2^2 +a_3^2). (b_0^2+b_1^2 + b_2^2 +b_3^2)  \\\\\n&=&   (a_0b_0 - a_1b_1 -a_2b_2 -a_3b_3)^2 + (a_0b_1 + a_1b_0 + a_2b_3 -a_3b_2)^2   \\\\ && + (a_0b_2 -a_1b_3 + a_2b_0 + a_3b_1)^2  + (a_0b_3 + a_1b_2 -a_2b_1 + a_3b_0)^2.\n\\end{eqnarray*}\n\\item Show that, over $\\mathbb Z$, the product of a sum of $4$-squares is again a sum of $4$-squares.\n\\end{enumerate}\n\\end{exercise}\n\n\\section{Application to Number Theory}\n\nOne of the famous problem in number theory is Waring problem. It asks the following question. Given a natural number $r$, does there exists a natural number $f(r)$, such that the polynomial function $x_1^r+x_2^r +\\cdots + x_{f(r)}^r$ takes value all positive integers if we substitute integers. If yes, find the smallest value $f(r)$. For $r=2$, the answer is $f(2)=4$ which is the four square theorem. There are several proofs of this result and here we present one. \n\n\\begin{theorem}[Four Square theorem]\nEvery natural number is a sum of four integer squares. \n\\end{theorem}\n\\begin{lemma}\nLet $p$ be an odd prime, say $p=2n+1$. Then there exists $l,m \\in \\mathbb Z$ such that $p | (1+l^2+m^2)$. \n\\end{lemma}\n\\begin{proof}\nWe know that $\\mathbb Z\/p\\mathbb Z$ is a field. Consider the set $[n]=\\{0,1,\\ldots, n\\}$. Let $x\\neq y \\in [n]$. Then $x^2\\not\\equiv y^2 \\imod p$. For if $x^2\\equiv y^2 \\imod p$ then $(x^2-y^2)\\equiv 0\\imod p$ and hence either $x-y=0$ or $x+y=0$ in $\\mathbb Z\/p\\mathbb Z$. Since $0< x+y < n+n < p$ we can't have $x+y=0$.\n\nHence $T=\\{0^2, 1^2, \\ldots, n^2\\}$ are all distinct mod $p$. Similarly the set $-1 - T=\\{-1-0^2, -1-1^2, -1-2^2, \\ldots, -1-n^2\\}$ has all elements distinct. Both sets $T$ and $-1-T$ have $n+1$ elements and hence they have a common element. That is there exists $l^2\\in T$ and $-1-m^2 \\ in -1-T$ such that $l^2=-1-m^2$ and we are done. \n\\end{proof}\n\n\\begin{proof}[Proof of the theorem]\nLet $n$ be a natural number. One can easily verify this theorem for $n=0, 1, 2, 3, 4$. Thus we may assume $n\\geq 5$.\n\nStep 1. We can reduce this theorem to prove only for primes. For if $n$ is not a prime, write $n=n_1 n_2$ and use the four square identity. Since $n_1$ and $n_2$ both are sum of four squares, so is the product $n$.\n\nStep 2. If $2m$ is a sum of two squares then so is $m$. Let $2m=x^2+y^2$ then either both $x,y$ are even or both are odd. Then write $m=(\\frac{x-y}{2})^2 + (\\frac{x+y}{2})^2$.\n\nStep 3. Let $p$ be an odd prime. Then from the lemma a multiple of $p$ is a sum of four squares. That is, there exists $l,m$ such that $p | (1+l^2+m^2)$ and hence $kp = 1 +l^2 +m^2$ for some $k$. Further we assume $0<k<p$ (else we may go mod $p$). Thus if we already have $k=1$ we are done. Otherwise we do the following.\n\nStep 4. We apply decent argument to arrive at a contradiction. Suppose there exists $m , a, b ,c, d$ integers such that $mp=a^2 + b^2 + c^2 + d^2$ and $1< m < p$. We show that we can find $n$ such that $1\\leq n < m$ and $np=a'^2 + b'^2 + c'^2 + d'^2$ for some $a', b', c',d'$. This would provide contradiction if we begin with minimal $m$ with the required property. \n\nWithout loss of generality $m>1$. If $m$ is even, say $m=2n$, then $mp$ is even which is a sum of $4$ squares and hence step 2 implies that $np$ is a sum of four squares and we are done. Now we assume that $m$ is odd $>2$. Now pick $w, x, y, z$ such that $w\\equiv a\\imod m$, $x\\equiv b\\imod m$, $y\\equiv c\\imod m$ and $z\\equiv d\\imod m$ and $-\\frac{m}{2} < w,x,y,z < \\frac{m}{2}$. Thus, $w^2+x^2+y^2+z^2 < 4.\\frac{m^2}{4} = m^2$ and $w^2+x^2+y^2+z^2 \\equiv a^2+b^2+c^2+d^2 \\equiv 0 \\imod m$. Hence $w^2+x^2+y^2+z^2=nm$ for some $n$. Since $w^2+x^2+y^2+z^2< m^2$ we have $n<m$. Check that $n\\neq 0$. \n\nThus, $(a^2+b^2+c^2+d^2)(w^2+x^2+y^2+z^2)=mp.nm = nm^2p$ is a sum of $4$ squares. Expanding out the left side (and replacing $x,y,z$ by their negative) we get $ (aw + bx +cy +dz)^2 + (-ax + bw - cz +dy)^2  + (-ay +bz + cw - dx)^2  + (-az - by + cx + dw)^2 = m^2.np$. Claim: $m$ divides each term on left, $(aw + bx +cy +dz), (-ax + bw - cz +dy), (-ay +bz + cw - dx), (-az - by +cx + dw)$. For this, first we write each of this term mod $m$. We get, \n$(aw + bx +cy +dz) = a^2+b^2+c^2+d^2 \\imod m =0\\imod m, (-ax + bw - cz +dy) = 0 \\imod m, (-ay +bz + cw - dx)=0 \\imod m, (-az - by +cx + dw)= 0 \\imod m$. Hence, by dividing on the both side in the equation by $m^2$ we get that $np$ is a sum of $4$-squares. This proves the required result.\n\\end{proof} \n\n\\begin{exercise}\nCan you write every natural number as a sum of $2$ or $3$ squares? What about sum of $2$ or $3$ cubes?\n\\end{exercise}\n\n\n\\section{Application to Geometry}\n\nLet $\\mathbb H =\\{a=a_0+a_1i+a_2j+a_3k \\mid a_r\\in \\mathbb R, i^2=-1=j^2, ij=-ji=k\\}$ be the real quaternions defined above. Recall that the conjugation is defined by $\\bar a = a_0-a_1i-a_2j-a_3k$. We also denote $tr(a)=a+\\bar a$ and $N(a)=a\\bar a$. Now, we can think of $\\mathbb H$ as a $4$-dimensional vector space over $\\mathbb R$. Further we have a non-degenerate symmetric bilinear form $B$ on $\\mathbb H$ induced from $N$ given as follows: \n\\begin{eqnarray*}\nB(x,y)&=& \\frac{N(x+y)-N(x)-N(y)}{2}\\\\\n&=& \\frac{1}{2}[ (x_0+y_0)^2 + (x_1+y_1)^2+ (x_2+y_2)^2 + (x_3+y_3)^2 -   (x_0^2 +x_1^2+x_2^2+x_3^2)\\\\&& - (y_0^2 +y_1^2+y_2^2+y_3^2)] \\\\\n&=& x_0y_0+x_1y_1+ x_2y_2 + x_3y_3.\n\\end{eqnarray*}\nThe three dimensional subspace spanned by $i,j,k$ is called imaginary subspace. \n\\begin{exercise}\nLet $V:= Im(\\mathbb H)=<i,j,k>$ be the vector subspace of $\\mathbb H$.\n\\begin{enumerate}\n\\item[(a)] For $a=\\cos(\\theta)j+\\sin(\\theta)k$, compute the matrix of the map $\\phi_a\\colon V\\rightarrow V$ defined by $\\phi_a(v)=ava^{-1}$. Show that $\\phi_a = \\begin{pmatrix} -1 && \\\\ &\\cos(2\\theta) & \\sin(2\\theta) \\\\ &\\sin(2\\theta) & -\\cos(2\\theta) \\end{pmatrix}$.\n\\item[(b)] Compute the matrix of the map $\\psi\\colon V\\rightarrow V$ given by $\\psi(v)=ivi$. \nShow that $\\psi$ is a reflection. The matrix of $\\psi$ is $\\textup{diag}(-1, 1, 1)$.\n\\end{enumerate}\n\\end{exercise}\n\n\\begin{exercise}\nUse the group homomorphism given by norm $N\\colon \\mathbb H^*\\rightarrow \\mathbb R^*$ defined by $a\\mapsto N(a)=a\\bar a$ to show that $\\mathbb H^1:= ker(N)$ is a group. Hence, show that the sphere $\\mathbb S^3$ is a group.\n\\end{exercise}\n\n\\noindent It is an amazing fact that the sphere $\\mathbb S^n \\subset \\mathbb R^{n+1}$ is a group if and only if $n=1, 3$ or $7$. This phenomena is related to being able to put composition algebra structure on $\\mathbb R^{m}$ which is possible if and only if $m=1, 2, 4$ or $8$. The composition algebras of dimension $1$ is the field $\\mathbb R$, of dimension $2$ is either $\\mathbb R\\times \\mathbb R$ or $\\mathbb C$, of dimension $4$ is either $M_2(\\mathbb R)$ or $\\mathbb H$ and of dimension $8$, two possibilities, called octonions. \n\\begin{exercise}\nFor a non-zero $a\\in \\mathbb H$ define a map $\\phi_a\\colon \\mathbb H\\rightarrow \\mathbb H$\nby $\\phi_a(x)=axa^{-1}$. Prove the following:\n\\begin{enumerate}\n\\item Show that $\\phi_a$ preserves norm, hence is in $O_4(\\mathbb H, N)$.\n\\item Show that $\\phi_a(1)=1$.\n\\item Show that if $x\\perp 1$ then $\\phi_a(x) \\perp 1$. Thus $\\phi_a$ restricts to $1^{\\perp}=V$.\n\\end{enumerate}\n\\end{exercise}\n\\noindent Thus, we define a map $a\\mapsto \\phi_a$ from $\\mathbb H^1$ to $SO_3(\\mathbb R)$ and we get the following exact sequence:\n$$1\\rightarrow \\{\\pm 1\\} \\rightarrow \\mathbb H^1 \\rightarrow SO_3(\\mathbb R) \\rightarrow 1.$$\n\\begin{exercise}\nConsider $V=Im(\\mathbb H)$. Compute the conjugation map $\\phi_a\\colon V\\rightarrow V$ for\n$a=a_0+a_1i$ and $a=a_2j+a_3k$.\n\\end{exercise}\n\\begin{exercise}\nShow that the map $\\phi \\colon \\mathbb H^1 \\rightarrow SO_3(\\mathbb R)$ defined by $a\\mapsto \\phi_a$ is a surjective group homomorphism with kernel $\\{\\pm 1\\}$. Thus $\\mathbb H^1$ is a double cover of the group $SO_3(\\mathbb R)$.  \n\\end{exercise}\n\\noindent In the above exercise, if needed, you may use connectedness.\n\\begin{exercise}\nShow that an element of $SO(3)$ always fixes a non-zero vector, i.e., $1$ is an eigenvalue. Thus up to conjugacy it is of the following form \n$$\\begin{pmatrix}1&&\\\\&\\cos(\\theta) &-\\sin(\\theta)\\\\ & \\sin(\\theta)&\\cos(\\theta) \\end{pmatrix}.$$\n\\end{exercise}\n\n\\begin{exercise}\nFor $a\\in \\mathbb H^1$ define the map $l_a\\colon \\mathbb H \\rightarrow \\mathbb H$ by $x\\mapsto ax$. Show that its an isometry. This defines a group homomorphism $l\\colon \\mathbb H^1 \\rightarrow O_4(\\mathbb R)$.\n\\end{exercise}\n\n\\section{The group $SU_2(\\mathbb C)$} \nThe group $SU_2(\\mathbb C) = \\{X\\in M_2(\\mathbb C) \\mid \\ensuremath{{}^t\\!}{\\bar X} X=I, \\det(X)=1\\}$. \n\\begin{exercise}\nCompute and show that $X\\in SU_2(\\mathbb C)$ is of the form $\\begin{pmatrix} w & -z \\\\ \\bar z & \\bar w\\end{pmatrix}$ for some $w, z\\in \\mathbb C$ satisfying $w\\bar w + z\\bar z =1$. \n\\end{exercise}\n\n\\begin{exercise}\n\\begin{enumerate}\n\\item Recall the realization of $\\mathbb H$ as $2\\times 2$ complex matrices and note that $\\mathbb H^1$ is identified with $SU_2(\\mathbb C)$.\n\\item Show that $\\mathbb H^1\\cong S^3$ is group isomorphic to $SU_2(\\mathbb C)$.\n\\item Use this to get the double cover map $SU_2(\\mathbb C)\\rightarrow SO_3(\\mathbb R)$.\n\\begin{center}\n\\begin{tikzcd}\n\\mathbb H^1 \\arrow[rr] \\arrow[rd]&  & \\arrow[dl]SU_2(\\mathbb C) \\\\ \n&SO_3(\\mathbb R)&\n\\end{tikzcd}\n\\end{center}\n\\end{enumerate}\n\\end{exercise}\n\n\\begin{exercise}\nShow that $SU_2(\\mathbb C)$ is simply connected and $SO_3(\\mathbb R)\\cong \\mathbb RP^3$.  \n\\end{exercise}\n\n\nHere we describe the topology of $SO_3(\\mathbb R)$. Let us first recall how we get projective spaces. We begin with $\\mathbb RP^2$. This is suppose to be obtained by antipodal identification of $S^2$. We can further think of this as the upper hemisphere with boundary circle being identified, which after projecting down in the plane becomes solid circle (disc) in the plane with points on the boundary being identified. We can carry forward this idea to any higher dimension.\n\nNow, any element $A\\in SO_3(\\mathbb R)$ has a fixed axis and it is rotation in the perpendicular plane. So, $A$ can be associated to $(v,\\theta)\\in S^2\\times [-\\pi, \\pi]$ where the first coordinate $v$ represents a direction and the second coordinate represents the rotation amount. Notice that the rotation by $\\pi$ and $-\\pi$, in this representation, is same and hence in this representation we have to further identify $\\pi$ with $-\\pi$. Thus the object, $S^2\\times [-\\pi, \\pi]$, is a solid sphere (disc) of radius $\\pi$ together with the property that points on the surface are identified. This object is nothing but $\\mathbb RP^3$.  \n\n\n\n\n\n\n\\chapter{Symplectic groups}\n\nLet $V$ be a vector space of dimension $2l$ over a field $k$. We also assume $char(k)\\neq 2$. There is a unique non-degenerate skew-symmetric linear form $\\beta$ up to equivalence on $V$. We fix a basis of $V$ indexed as follows: $\\{e_1, \\ldots, e_l, e_{-1}, \\ldots, e_{-l}\\}$ which gives the following skew-symmetric matrix $J$ corresponding to the form $\\beta$:\n$$J=\\begin{pmatrix} &I_l \\\\-I_l & \\end{pmatrix}.$$\nThe symplectic group is the set of isometries of the form $\\beta$, which in matrix form is the set of symplectic matrices with respect to this form:\n$$Sp_{2l}=\\{X\\in GL_{2l} \\mid \\ensuremath{{}^t\\!} XJX = J\\}.$$\nIn the algebraic groups notation, this is a group of type $C_l$. We also define, the projective symplectic group, \n$$PSp_{2l}=\\frac{Sp_{2l}}{\\mathcal Z(Sp_{2l})}.$$\n\n\\begin{exercise}\nWhen $l=1$, show that $Sp_2=SL_2$.\n\\end{exercise}\nThus, we always assume $l>1$. \n\\begin{exercise}\nFor $g\\in Sp_{2l}$, we have $\\det(g)=1$. Hence, $Sp_{2l}$ is a subgroup of $SL_{2l}$.\n\\end{exercise}\nHint: Use Pfaffian to prove this. The Pfaffian $Pf(A)$ is a polynomial associated to a skew-symmetric matrix $A$ which is non-zero when the matrix is invertible of even size. It also has the propery that $Pf(\\ensuremath{{}^t\\!} X A X) = \\det(X) Pf(A)$. \n\n\\begin{exercise}\nShow that the map $GL_l \\rightarrow Sp_{2l}$ given by $A\\mapsto \\begin{pmatrix} A & \\\\ & \\ensuremath{{}^t\\!} A^{-1}\\end{pmatrix}$ is an injective group homomorphism.\n\\end{exercise}\n\n\\begin{exercise}\nCompute the center $\\mathcal Z(Sp_{2l})=\\{\\pm I\\}$.\n\\end{exercise}\n\n\\section{Symplectic Lie algebra}\nNotice that we are going to use $\\{1, \\ldots, l, -1, \\ldots, -l\\}$ as indices instead of using $1$ upto $2l$. The Lie algebra of the symplectic group $Sp_{2l}$ is \n$$\\mathfrak{sp}_{2l} = \\{X\\in \\mathfrak{gl}_{2l} \\mid \\ensuremath{{}^t} XJ = -JX\\}$$\nwhere $\\mathfrak{gl}_{2l}$ is the set of $2l$ size matrices.\n\\begin{exercise}\nFor $X\\in \\mathfrak{sp}_{2l}$ show that $tr(X)=0$.\n\\end{exercise}\n\\begin{exercise}\nLet $\\begin{pmatrix} W&X \\\\Y & Z \\end{pmatrix}$ be in $\\mathfrak{sp}_{2l}$ written in block form of size $l$. Show that $Z=-\\ensuremath{{}^t} W$, $X=\\ensuremath{{}^t} X$ and $Y=\\ensuremath{{}^t} Y$. \n\\end{exercise}\nHint: We can do block multiplication here.\n\\begin{exercise}\\label{basis-sp}\nShow that the following set of elements for $1\\leq i, j \\leq l$, \n$$\\{(e_{i,j}-e_{-j,-i}) \\text{\\ for}\\; i\\neq j, \\} \\cup \\{(e_{i,-j}+e_{j,-i}) \\text{\\ for}\\; i<j,\\} \\cup  \n\\{(e_{-i,j}+e_{-j,i}) \\text{\\ for}\\; i<j\\}, \\cup \\{e_{i,-i}, e_{-i,i}\\}\n$$ \nform a basis of the Lie algebra together with the diagonals $\\{e_{i,i} - e_{-i,-i}\\}$.\n\\end{exercise}\n\n\\begin{exercise}\nCompute the dimension of the Lie algebra $\\mathfrak{sp}_{2l}$.\n\\end{exercise}\n\n\\begin{exercise}\nShow that each of the matrices in the basis above (except the diagonal ones) is nilpotent.\n\\end{exercise}\n\n\\begin{exercise}[Chevalley generators]\\label{ch-gen-sp}\nThe following elements belong to the symplectic group for any $t\\in k$ and for any $1\\leq i,j \\leq l$:\n\\begin{eqnarray*}\nx_{i,j}(t)&=& I +t(e_{i,j}-e_{-j,-i}) \\text{\\ for}\\; i\\neq j, \\\\ \nx_{i,-j}(t)&= &I +t(e_{i,-j}+e_{j,-i}) \\text{\\ for}\\; i<j,  \\\\ \nx_{-i,j}(t)&=& I +t(e_{-i,j}+e_{-j,i}) \\text{\\ for}\\; i<j, \\\\ \nx_{i,-i}(t) &=& I +t e_{i,-i}, \\\\ \nx_{-i,i}(t)&= & I+te_{-i,i}.\n\\end{eqnarray*}\nIn fact, these matrices generate the symplectic group. This is a non-trivial exercise but not very difficult. See https:\/\/arxiv.org\/pdf\/1504.03794.pdf for an algorithmic proof similar to row-column operations in the $SL_n$ case.\n\\end{exercise}\n\n\\begin{exercise}\nAssume that the Chevallley generators in the above exercise~\\ref{ch-gen-sp} generate the symplectic group. Use this to prove the following:\n\\begin{enumerate}\n\\item For $X\\in Sp_{2l}$, we have $\\det(X)=1$.\n\\item For $X\\in Sp_{2l}$, we have $\\ensuremath{{}^t\\!} X\\in Sp_{2l}$.\n\\end{enumerate} \n\\end{exercise}\n\n\\section{Root Datum}\n\\begin{exercise}\nThe set $T=\\{\\textup{diag}(\\lambda_1, \\ldots, \\lambda_l, \\lambda_1^{-1}, \\ldots, \\lambda_l^{-1})\\mid \\lambda_i\\in k^*\\}$ is a maximal torus in $Sp_{2l}$. Thus, rank of $Sp_{2l}$ is $l$.\n\\end{exercise}\n\n\\begin{exercise}\nFix characters $\\chi_i \\colon T \\rightarrow \\mathbb G_m$ by $\\chi_i(\\textup{diag}(\\lambda_1, \\ldots, \\lambda_l, \\lambda_1^{-1}, \\ldots, \\lambda_l^{-1}))= \\lambda_i$. Show that the character group $X^*(T)\\cong \\mathbb Z^l$.\n\\end{exercise}\n\n\\begin{exercise}\nFix co-characters $\\mu_i \\colon \\mathbb G_m \\rightarrow T$ by $\\chi_i(t) = \\textup{diag}(1, \\ldots, t\\ldots ,1 \\ldots, \\ldots , 1)$ where $t$ is at $i^{th}$ place. Show that the co-character group $X_*(T)\\cong \\mathbb Z^l$.\n\\end{exercise}\n\n\\begin{exercise}\nConsider the $Ad \\colon Sp_{2l} \\rightarrow GL(\\mathfrak{sp}_{2l})$ given by $Ad(g)(X)=gXg^{-1}$. We restrict this map to the maximal torus $T$ and obtain the simultaneous decomposition of $\\mathfrak{sp}_{2l}$,\n$$\\mathfrak{sp}_{2l}=\\mathfrak t \\bigoplus \\left ( \\bigoplus_{\\chi\\in X^*(T)} \\mathfrak g_{\\chi} \\right ).$$\nShow that the diagonals $\\mathfrak t$ form a $T$-invariant subspace with respect to the $0$ character. Show that each of the basis vectors in the exercise~\\ref{basis-sp}, other than the diagonals, form a $1$ dimensional $T$-invariant subspace $\\mathfrak g_{\\chi}$.\n\\end{exercise}\nHint: Use the action to compute the eigen-values which are the charcters in the exercise below. Since these turn out to be distinct they are, each one of them, all possible eigen-spaces.\n\\begin{exercise}\nCompute the set of all non-trivial characters appearing in the above decomposition $P\\subset X^*(T)$. Show that \n$$P=\\{\\chi_i-\\chi_j \\mid 1\\leq i\\neq j \\leq l\\} \\bigcup \\{\\pm (\\chi_i+\\chi_j) \\mid 1\\leq i < j \\leq l\\} \\bigcup \\{\\pm 2\\chi_i \\mid 1\\leq i \\leq l\\}. $$\n\\end{exercise}\n\\begin{exercise}[Co-characters]\nCompute the co-charactrs $Q=P^{\\vee} \\subset X_*(T)$ and show that $P^{\\vee}=\\{\\mu_i-\\mu_j \\mid 1\\leq i\\neq j \\leq l\\} \\bigcup \\{\\pm (\\mu_i+\\mu_j) \\mid 1\\leq i < j \\leq l\\} \\bigcup \\{\\pm \\mu_i \\mid 1\\leq i \\leq l\\}$.\n\\end{exercise}\n\n\\begin{exercise}\nConsider the isotropic flag \n$${0}\\subset V_1 \\subset \\ldots \\subset V_i \\subset \\ldots \\subset V_l $$\nwhere $V_i=<e_1, \\ldots, e_i >$.\nThen $B$, the stabiliser of this flag, is a Borel subgroup. Let us explicitely compte this subgroup. For $g\\in B$,\n$g(e_i)\\in V_i= <e_1, \\ldots, e_i >$ for $1\\leq i \\leq l$. Now, we know $G(W^{\\perp})\\subset W^{\\perp}$. Thus, \n$$V_l^{\\perp} \\subset V_{l-1}^{\\perp} \\subset \\ldots \\subset V_i^{\\perp} \\subset \\ldots \\subset V_1^{\\perp} \\subset {0}^{\\perp}$$\nand is naturally fixed by $g$ as well. That is,\n$$<e_{1}, \\ldots, e_{l}> \\subset <e_{1}, \\ldots, e_{l}, e_{-l}> \\subset \\ldots \\subset <e_1, \\ldots, e_l, e_{-2}, \\ldots, e_{-l}> \\subset V.$$\nThus we get $g(e_{-1})\\in <e_1, \\ldots, e_l, e_{-1}, \\ldots, e_{-l}> $, $g(e_{-2}) \\in <e_1, \\ldots, e_l, e_{-2}, \\ldots, e_{-l}> $, and, in general $g(e_{-i})\\in <e_1, \\ldots, e_l, e_{-i}, \\ldots, e_{-l}> $. Hence, the elements of $B$ are of the form $\\begin{pmatrix} W&X \\\\ & \\ensuremath{{}^t} W^{-1} \\end{pmatrix}$ where $W$ is upper triangular.\nThus, a Borel in $Sp_{2l}$ containing the fixed diagonal maximal torus is\n$$\\left \\{\\begin{pmatrix} W&X \\\\ & \\ensuremath{{}^t} W^{-1} \\end{pmatrix} \\mid W \\right\\}\\bigcap Sp_{2l}.$$\n\\end{exercise}\n\\begin{exercise}\nThus a system of positive roots is\n$$R^{+}=\\{\\chi_i \\pm \\chi_j \\mid 1\\leq i < j \\leq l\\} \\bigcup \\{2\\chi_i \\mid 1\\leq i \\leq l\\} $$\nand a set of simple roots is \n$$\\Delta=\\{\\chi_1-\\chi_2, \\ldots, \\chi_{l-1}-\\chi_{l}, 2\\chi_l \\} $$\n\\end{exercise}\n\n\\begin{exercise}[Weyl group]\nShow that the Weyl group is $S_l \\ltimes (\\mathbb Z\/2\\mathbb Z)^l$.\n\\end{exercise}\n\n\\section{$GSp_{2l}$}\nThe similitude symplectic group is\n $$GSp_{2l}=\\{X\\in GL_{2l} \\mid \\ensuremath{{}^t\\!} XJX = \\lambda J, \\text{\\ for some\\ } \\lambda\\in k^*\\}.$$\nThe scalar $\\lambda=\\lambda_X$ varies with $X$ and is called similitude factor of the element $X$. \n\\begin{exercise}\nThe map $\\Psi \\colon GSp_{2l} \\rightarrow k^*$ given by $X\\mapsto \\lambda_X$ is a group homomorphism with kernel $Sp_{2l}$. \n\\end{exercise}\n \\begin{exercise}\n The set of scalars $\\{\\alpha. I \\in GL_{2l} \\mid \\alpha\\in k^*\\}$ is the center of $GSp_{2l}$.\n\\end{exercise}\n\\begin{exercise}\nThe elements of $GSp_{2l}$ induce automorphisms of $Sp_{2l}$ acting via conjugation.\n\\end{exercise}\n \n \n\n\\chapter{Orthogonal groups}\n\nLet $k$ be an algebraically closed field of $char\\neq 2$. \nLet $V$ be a vector space of dimension $n$. We write $n=2l$ or $2l+1$ depending on when $n$ is even or odd. We fix a basis $\\{e_1, \\ldots, e_l, e_{-1}, \\ldots, e_{-l}\\}$ when $n$ is even and $\\{e_0, e_1, \\ldots, e_l, e_{-1}, \\ldots, e_{-l}\\}$ when $n$ is odd. Recall that the orthogonal group is $$O_{n} = \\{g\\in GL_{n} \\mid \\ensuremath{{}^t} g J g =J\\}$$ where $J=\\begin{pmatrix} & I_l \\\\ I_l &\\end{pmatrix}$ when $n=2l$ and $J=\\begin{pmatrix} 1&&\\\\& & I_l \\\\& I_l &\\end{pmatrix}$ when $n=2l+1$. Its connected component is the special orthogonal group $SO_n=O_n \\cap SL_n$. Its Lie algebra is $$\\mathfrak{o}_{n} = \\{X\\in \\mathfrak{gl}_{n} \\mid \\ensuremath{{}^t} XJ = -JX\\}.$$\n\n\\section{Root datum of $D_l$ type}\nLet us first deal with even dimensional one which is also called $D_l$ type.\n\\begin{exercise}\\label{basis-o2l}\nLet $\\begin{pmatrix} W&X \\\\Y & Z \\end{pmatrix}$ be in $\\mathfrak{o}_{2l}$. Show that $Z=-\\ensuremath{{}^t} W$, $X= -\\ensuremath{{}^t} X$ and $Y= -\\ensuremath{{}^t} Y$. Hence, the following set of elements for $1\\leq i, j \\leq l$,\n\\begin{eqnarray*}\n(e_{i,j}-e_{-j,-i}) &\\text{for}\\; i\\neq j,\\\\ \n(e_{i,-j}-e_{j,-i})&\\text{for}\\; i<j,\\\\ \n(e_{-i,j}-e_{-j,i})& \\text{for}\\; i<j,\n\\end{eqnarray*}\nform a basis of the Lie algebra together with the diagonals $e_{i,i}-e_{-i,-i}$.\n\\end{exercise}\n\n\\begin{exercise}\nThe set $T=\\{\\textup{diag}(\\lambda_1, \\ldots, \\lambda_l, \\lambda_1^{-1}, \\ldots, \\lambda_l^{-1})\\mid \\lambda_i\\in k^*\\}$ is a maximal torus in $O_{2l}$. Thus, the rank of $O_{2l}$ is $l$.\n\\end{exercise}\n\n\\begin{exercise}\nFix characters $\\chi_i \\colon T \\rightarrow \\mathbb G_m$ by $\\chi_i(\\textup{diag}(\\lambda_1, \\ldots, \\lambda_l, \\lambda_1^{-1}, \\ldots, \\lambda_l^{-1}))= \\lambda_i$. Show that $X^*(T)\\cong \\mathbb Z^l$.\n\\end{exercise}\n\n\\begin{exercise}\nFix co-characters $\\mu_i \\colon \\mathbb G_m \\rightarrow T$ by $\\chi_i(t) = \\textup{diag}(1, \\ldots, t\\ldots ,1 \\ldots,  , 1)$ where $t$ is at $i^{th}$ place. Show that $X_*(T)\\cong \\mathbb Z^l$.\n\\end{exercise}\n\n\\begin{exercise}\nConsider the $Ad \\colon O_{2l} \\rightarrow GL(\\mathfrak{o}_{2l})$ given by $Ad(g)(X)=gXg^{-1}$. We restrict this map to the maximal torus $T$ and obtain the simultaneous decomposition of $\\mathfrak{o}_{2l}$. Show that the diagonals form a $T$-invariant subspace with respect to $0$ character. Show that each of the basis vector in the above exercise ~\\ref{basis-o2l}, other than diagonals, forms a $1$ dimensional $T$-invariant subspace.\n\\end{exercise}\n\n\\begin{exercise}\nCompute $P\\subset X^*(T)$. Show that \n$$P=\\{\\chi_i-\\chi_j \\mid 1\\leq i\\neq j \\leq l\\} \\bigcup \\{\\pm (\\chi_i+\\chi_j) \\mid 1\\leq i < j \\leq l\\}. $$\n\\end{exercise}\n\\begin{exercise}\nCompute $Q=P^{\\vee} \\subset X_*(T)$. This gives us the root datum $(X^*(T), P, X_*(T), P^{\\vee})$.\n\\end{exercise}\n\n\\begin{exercise}[Borel subgroup]\nConsider the isotropic flag \n$${0}\\subset V_1 \\subset \\ldots \\subset V_i \\subset \\ldots \\subset V_l $$\nwhere $V_i=<e_1, \\ldots, e_i >$.\nThen $B$, the stabiliser of this flag, is a Borel subgroup. Explicitely compute this and get the elements of $B$  which are of the form $\\begin{pmatrix} W&X \\\\ & \\ensuremath{{}^t} W^{-1} \\end{pmatrix}$ where $W$ is upper triangular.\n\\end{exercise}\n\n\\begin{exercise}\nThe Weyl group is isomorphic to $S_l\\ltimes (\\mathbb Z\/2\\mathbb Z)^{l-1}$.\n\\end{exercise}\n\\section{Root datum of type $B_l$}\nNow to deal with $n=2l+1$. \n\\begin{exercise}\nThe Lie algebra elements are  \n$$\\mathfrak{o}_{2l+1}=\\left\\{\\begin{pmatrix} &\\ensuremath{{}^t} c& -\\ensuremath{{}^t} b \\\\ b& W &X \\\\-c& Y & - \\ensuremath{{}^t} W \\end{pmatrix}\\mid  X= -\\ensuremath{{}^t} X, Y= -\\ensuremath{{}^t} Y \\right\\}.$$\nThus along with diagonals, the following form a basis, $1\\leq i, j \\leq l$,\n\\begin{eqnarray*}\ne_{i,0} -e_{0,-i} \\\\\ne_{0,i} - e_{-i,0}\\\\\n(e_{i,j}-e_{-j,-i}) &\\text{for}\\; i\\neq j,\\\\ \n(e_{i,-j}-e_{j,-i})&\\text{for}\\; i<j,\\\\ \n(e_{-i,j}-e_{-j,i})& \\text{for}\\; i<j,\n\\end{eqnarray*}\n\\end{exercise}\n\n\\begin{exercise}\nThe set $T=\\{\\textup{diag}(1, \\lambda_1, \\ldots, \\lambda_l, \\lambda_1^{-1}, \\ldots, \\lambda_l^{-1})\\mid \\lambda_i\\in k^*\\}$ is a maximal torus in $O_{2l+1}$. Thus, the rank of $O_{2l+1}$ is $l$.\n\\end{exercise}\n\n\n\\begin{exercise}\nCompute $P\\subset X^*(T)$ by the action of maximal torus on the Lie algebra. Show that \n$$P=\\{\\pm \\chi_i \\mid 1\\leq i\\leq l\\} \\bigcup \\{\\chi_i-\\chi_j \\mid 1\\leq i\\neq j \\leq l\\} \\bigcup \\{\\pm (\\chi_i+\\chi_j) \\mid 1\\leq i < j \\leq l\\}. $$\n\\end{exercise}\nThis root datum is dual of $C_l$ type.\n\\begin{exercise}\nCompute the co-characters $P^{\\vee}$.\n\\end{exercise}\n\\begin{exercise}\nCompute the centraliser of the element $\\textup{diag}(1, -1, -1)$ in $SO(3)$.  \n\\end{exercise}\n\n\n\\chapter{Quaternion Algebra}\\label{quaternion}\n\n\nQuaternion algebras are central object in mathematics. The real quaternions were discovered by Hamilton in 1843. However one can define quaternions over any field $k$. There are several ways to construct a quaternion algebra. We collect  some of them here for fields of characteristic $\\neq 2$.\n\n\\section{Introduction and Definition}\n\nLet $k$ be a field. In this note we consider algebras over $k$ which are associative (unless mentioned otherwise) with identity and need not be commutative. A $k$-algebra which has no proper two-sided ideal with center $k$ is called a central simple algebra. Artin-Wedderburn theorem gives structure of such a finite dimensional algebra. It says that any central simple algebra is isomorphic to $M_n(D)$ where $D$ is a central division algebra over $k$. A quaternion algebra (sometimes called generalised quaternion algebra)  over $k$ is a central simple algebra over $k$ of dimension four. The Artin-Wedderburn theorem implies that any quaternion algebra is either isomorphic to $M_2(k)$ (called split quaternion algebra) or is isomorphic to a division algebra over $k$ of dimension $4$. In this note we will describe several ways of constructing quaternions over a field of characteristic $\\neq 2$.\n\nApart from being a central object in mathematics, specially in number theory and algebraic groups, quaternions play fundamental role in physics and computer graphics. We will briefly indicate these in the section on applications. \n\nLet $k$ be a field of characteristic $\\neq 2$. \nLet $a,b\\in k^*$. Then we can define a multiplication on the four dimensional $k$-vector space $Q=k.1\\oplus ki\\oplus kj \\oplus kij$  by $i^2=a$, $j^2=b$, $ij=-ji$ and $1$ is multiplicative identity. This makes $Q$ a central simple algebra and hence is a quaternion algebra. We denote this algebra $Q=\\left(\\frac{a,b}{k}\\right)$. In fact, any quaternion algebra is of this form. \n\nWe have conjugation operation on $Q$ defined by $\\bar x:=x_0-x_1i-x_2j-x_3ij$ where $x=x_0+x_1i+x_2j+x_3ij$. This satisfies the following relation $\\overline{xy}=\\bar y\\bar x$. We can define trace and norm of an element by $tr(x):=x+\\bar x =2x_0$ and $N(x):=x\\bar x=x_0^2-x_1^2a-x_2^2b+x_3^2ab$ and check that every element $x\\in Q$ satisfies a quadratic polynomial $X^2-tr(x)X+N(x)=0$. The norm on $Q$ is a quadratic form $<1,-a,-b, ab>$ which, in fact, is a Pfister form $<1,-a><1,-b>$. Further the norm is multiplicative, i.e., $N(xy)=N(x)N(y)$. We denote the space of pure quaternions by $Q_0=\\{x\\in Q\\mid tr(x)=0\\}=\\{x\\in Q\\mid x_0=0\\}$. The space $Q_0$ is characterised by $x\\in Q$ such that $x^2\\in k$ and has quadratic form $N_0=<-a, -b, ab>$. Observe that $N_0$ is restriction of $N$ to $Q_0$. We abuse notation and denote the corresponding bilinear form by $N_0$ as well.\n\nIn the constructions below we would need to understand quadratic \\'{e}tale algebras. Any such algebra is of the form $K\\cong k[X]\/<X^2-a>$ for some $a\\in k^*$. Thus $K\\cong k\\times k$ (also called split) or is a quadratic field extension. The algebra $K$ is split if and only if $a\\in (k^*)^2$. By taking norm we get norm subgroup $D_a=N_{K\/k}(K^*)=\\{x-ay^2\\}\\subset k^*$. To get a quaternion algebra we need $b\\in k^*$ as well. If we take $b\\in k^*-D_a$ then we get quaternion division algebras.\n\n\n\\section{Graded tensor product}\n\nWe refer to~\\cite{gr} chapter 8, page 71 for this section. \nLet $A$ and $B$ be two finite dimensional associative (not necessarily commutative) $k$-algebras with identity. Then the algebra $A\\otimes B$ is an algebra where multiplication is defined by $(a\\otimes b). (a'\\otimes b') = aa'\\otimes bb'$. Further if $A$ and $B$ both are commutative then $A\\otimes B$ is commutative.\n \nNow suppose $A=A_0\\oplus A_1$ and $B=B_0\\oplus B_1$ are $\\mathbb Z\/2\\mathbb Z$ graded where the subscript denotes the degree; $0$ degree term and degree $1$ term. Then we have graded tensor product $A\\hat\\otimes B$ with algebra product defined as follows:\n$$(a\\otimes b). (a'\\otimes b') = (-1)^{\\deg(b)\\deg(a')}aa'\\otimes bb'$$ where degree $0$ term is $(A_0\\otimes B_0) \\oplus (A_1\\otimes B_1)$ and degree $1$ term is $(A_0\\otimes B_1) \\oplus (A_1\\otimes B_0)$. Notice that even if $A$ and $B$ are commutative $A\\hat\\otimes B$ need not be. \n\nWe begin with two quadratic \\'{e}tale algebras $K$ and $L$ over $k$, say $K=k[\\alpha]=k\\oplus k\\alpha$ and $L=k[\\beta]=k\\oplus k\\beta$ with $\\alpha^2=a$ and $\\beta^2=b$. We have natural grading on $K$ and $L$. We consider $K\\hat\\otimes L$ with basis $1=1\\otimes 1$, $i=\\alpha\\otimes 1$, $j=1\\otimes \\beta$  and $ij=\\alpha\\otimes \\beta$. We claim that this gives us a quaternion algebra $Q\\cong\\left(\\frac{a,b}{k}\\right)$. We note that,\n\\begin{eqnarray*}\ni^2&=&(\\alpha\\otimes 1).(\\alpha\\otimes 1)=(-1)^{0.1}\\alpha^2\\otimes 1^2=a.1\\\\\nj^2&=&(1\\otimes\\beta).(1\\otimes\\beta)=(-1)^{1.0}1^2\\otimes\\beta^2=b.1\\\\\nij&=& (\\alpha\\otimes 1).(1\\otimes\\beta)=(-1)^0\\alpha\\otimes\\beta=\\alpha\\otimes\\beta\\\\\nji&=& (1\\otimes\\beta).(\\alpha\\otimes 1)=(-1)^{1.1}\\alpha\\otimes\\beta=-ij.\n\\end{eqnarray*}\n\\begin{proposition}\nLet $K$ and $L$ be two quadratic \\'{e}tale extensions of $k$. Then $K\\hat\\otimes L$ is a quaternion algebra. Conversely, if $Q$ is a quaternion algebra over $k$ then there exist quadratic \\'{e}tale extensions $K$ and $L$ of $k$ such that $Q\\cong K\\hat\\otimes L$.\n\\end{proposition}\n\\begin{proof}\nForm the discussion above it follows that given $K$ and $L$, the graded tensor product $K\\hat\\otimes L$ is a quaternion algebra. \nFor converse, let $Q=\\left(\\frac{a,b}{k}\\right)$. Then consider $K=k[i]$ and $L=k[j]$. This will give us $Q\\cong K\\hat\\otimes L$.\n\\end{proof}\nFurther if either $K$ or $L$ is split then $Q$ is definitely split. In case both $K$ and $L$ are field and either $b\\not\\in D_a$ or $a\\not\\in D_b$ then $Q$ is division. \n\n\\section{Composition Algebras and Doubling Method}\n\nA composition algebra $C$ over a field $k$ is a not necessirly associative algebra over $k$ with identity together with a non-degenerate quadratic form $N$ on $C$ which is multiplicative, i.e., $N(xy) = N(x)N(y)$ for all $x,y \\in C$.\nIt turns out that the possible dimensions of a composition algebra over $k$ are $1, 2, 4, 8$. \nComposition algebras of dimension $1$ and $2$ are commutative and associative, those of dimension $4$ are associative but not commutative and are quaternion algebras, and those of dimension $8$ are neither commutative nor associative called octonion algebra or  Cayley algebra. For more details we refer to the chapter 1 of \\cite{sv}.\n\nLet $K$ be a quadratic \\'{e}tale algebra over $k$. Denote the non-trivial $k$-automorphism of $K$ by $x\\mapsto \\overline x$. Let $\\lambda\\in k^*$ and $Q=K\\oplus K$. Define multiplication on $Q$ as follows:\n$$(x_1,y_1).(x_2,y_2)=(x_1x_2+\\lambda y_1\\overline y_2, x_1y_2+\\overline x_2y_1).$$\nThis makes $Q$ an algebra and is called doubling method. \nSuppose $K=k[\\alpha]$ with $\\alpha^2=a\\in k^*$. The automorphism of $K$ is $x=x_1+\\alpha y_1\\mapsto \\overline x=x_1-\\alpha y_1$, i.e., $\\overline \\alpha=-\\alpha$. Clearly $(1,0)$ is identity of $Q$. Denote $i=(\\alpha,0)$ and $j=(0,1)$. Then\n\\begin{eqnarray*}\ni^2&=& (\\alpha,0)(\\alpha,0)= (\\alpha^2, 0)= a(1, 0)\\\\\nj^2 &=& (0,1)(0,1)=(\\lambda .1, 0)=\\lambda (1, 0) \\\\\nij&=& (\\alpha, 0)(0,1) = (0, \\alpha)=\\alpha(0,1)\\\\\nji&=& (0,1)(\\alpha, 0) = (0,\\overline\\alpha)=-\\alpha(0,1)=-ij\n\\end{eqnarray*}\nThus $Q\\cong\\left(\\frac{a,b}{k}\\right)$ where $a=\\alpha^2$ and $b=\\lambda$.\nFurther any quaternion algebra can be constructed this way. Clearly if $K$ is split or $\\lambda\\in D_a$ we get split quaternion algebra.\n\n\n\\section{Clifford Algebra}\n\nLet $V$ be a vector space of dimension $n$ over $k$. Suppose $V$ has a quadratic form $q$ on it. Then we have Clifford algebra $C(V,q)$ of dimension $2^n$ defined by:\n$$C(V,q)=\\frac{T(V)}{\\langle\\{v\\otimes v-q(v).1\\mid v\\in V\\}\\rangle}$$\nwhere $T(V)$ is tensor algebra.\nThis algebra is $\\mathbb Z\/2\\mathbb Z$ graded where grading is induced from that of $T(V)$. We refer to~\\cite{gr}, chapter 8 for more details.\n\nLet $V$ be a two dimensional vector space with a non-degenerate quadratic form $q$ on it. The Clifford algebra construction above gives us an algebra of dimension $4$. We claim that this algebra is a quaternion algebra. Suppose $\\{x,y\\}$ is an orthogonal basis of $V$ with $q(x)=a$ and $q(y)=b$ both $a,b\\in k^*$ as $q$ is non-degenerate. Then $T(V)\\cong k[x,y]$, non-commutative polynomial ring in two variables, and $C(V,q)\\cong k[x,y]\/\\langle x^2-a, y^2-b, xy+yx\\rangle$. By denoting $i=x$ and $j=y$ we see that $C(V,q)\\cong \\left(\\frac{a,b}{k}\\right)$. Further any quaternion algebra can be obtained this way by taking $V$ as space generated by orthogonal basis $i$ and $j$.\n\nWhat condition on $(V,q)$ gives a division quaternion algebra?\n\n\\section{Cyclic Algebra}\n\nLet $K\/k$ be a cyclic field extension of degree $n$, i.e., $K\/k$ is Galois with Galois group $Gal(K\/k)\\cong \\mathbb Z\/n\\mathbb Z=<\\sigma>$. Take $b\\in k^*$. We define multiplication on $n$-dimensional $K$-vector space $A=K.1\\oplus Ke\\oplus Ke^2\\oplus\\cdots\\oplus Ke^{n-1}$ to make it an algebra as follows:\n$$e^n=b, \\ e.\\beta = \\sigma(\\beta) e.$$ \nThe algebra $A=(K\/k, \\sigma, b)$ is called a cyclic algebra. For further details we refer to~\\cite{pi} chapter 15, in particular, section 15.1.\n\nWe begin with a quadratic \\'{e}tale algebra $K=k[\\alpha]\\cong k[X]\/<X^2-a>$ over $k$ with $\\sigma$ the non-trivial automorphism of $K$. We fix $b\\in k^*$. Consider a two-dimensional $K$-vector space $Q=K.1\\oplus K.e$ and define multiplication by $e^2=b$ and $e.\\beta =\\sigma(\\beta)e$. We claim that the algebra obtained this way is a quaternion algebra. Define $i=\\alpha 1$ and $j=1e$. Then,\n$i^2=\\alpha^2=a$, $j^2=1e.1e= e^2=b$, $ij=\\alpha 1.1e=\\alpha e$, $ji=1e.\\alpha 1 = \\sigma(\\alpha)e=-\\alpha e=-ij$.\nFurther any quaternion algebra can be obtained this way. Note that if $K$ is a field and $b\\not\\in D_a$ we get a division algebra.\n \n\n\n\\section{3-dimensional quadratic space with trivial discriminant}\n\nTo motivate the construction of this section we begin with analysing the multiplication on quaternions. We have $Q= k\\oplus Q_0$. Thus multiplication on $Q$ \n\\begin{eqnarray*}\nxy&=&(x_0+x_1i+x_2j+x_3ij)(y_0+y_1i+y_2j+y_3ij)\\\\\n&=&(x_0y_0+x_1y_1a+x_2y_2b-x_3y_3ab ) + (x_0y_1+x_1y_0-x_2y_3b+x_3y_2b)i \\\\\n&& (x_0y_2+x_1y_3a+x_2y_0-x_3y_1a)j + (x_0y_3+x_1y_2-x_2y_1 + x_3y_0)ij\n\\end{eqnarray*}\ncan be rewritten as: \n$$xy=(x_0y_0-N_0(x,y), x_0w+y_0v+v\\times w)$$\nwhere $x=(x_0,v)$ and $y=(y_0,w)$ for $v,w\\in Q_0$ and $\\times\\colon Q_0\\times Q_0\\rightarrow Q_0$ is a cross product defined by $v\\times w=-(x_2y_3-x_3y_2)bi+(x_1y_3-x_3y_1)aj+(x_1y_2-x_2y_1)ij$. We need to check that this is a unique well defined cross product (i.e., $(Q_0,\\times)$ satisfies properties of being a Lie algebra) on $Q_0$ with respect to $N_0$. Other properties are easy to verify using the above formula except the Jacobi identity: \n$u\\times(v\\times w)+w\\times (u\\times v)+v\\times (w\\times u)=0.$\nWe verify the following formal formula to compute the cross product: \n\\begin{eqnarray*} v\\times w &= & \\det\\begin{pmatrix}-bi&-aj&ij\\\\ x_1& x_2& x_3\\\\ y_1&y_2 &y_3\\end{pmatrix}\\\\\nu\\times(v\\times w) &= & \\det\\begin{pmatrix}-bi&-aj&ij\\\\ u_1& u_2& u_3\\\\ -b(x_2y_3-x_3y_2) & a(x_1y_3-x_3y_1) &x_1y_2-x_2y_1\\end{pmatrix} \\\\ \n&=& vN_0(u,w)-wN_0(u,v).\n\\end{eqnarray*}\nHence,\n\\begin{eqnarray*}\n && u\\times(v\\times w)+w\\times (u\\times v)+v\\times (w\\times u)\\\\\n&=& vN_0(u,w)-wN_0(u,v) + uN_0(w,v) - vN_0(w,u) + wN_0(v,u) - uN_0(v,w)\\\\\n&=&0\n\\end{eqnarray*} \n\nWe see that $(Q_0,N_0)$ has trivial discriminant which determines the cross product. Check that for any $u=u_1i+u_2j+u_3ij$ we get $N_0(u,v\\times w)=u_1(x_2y_3-x_3y_2)ba-u_2(x_1y_3-x_3y_1)ab +u_3(x_1y_2-x_2y_1)ab=ab\\det\\begin{pmatrix}u_1 &u_2&u_3\\\\mathfrak X_1& x_2& x_3\\\\y_1&y_2 &y_3\\end{pmatrix}$. Note that $\\bigwedge^3V=<i\\wedge j\\wedge ij>\\cong k$ and $\\bigwedge^3N_0(i\\wedge j\\wedge ij)=(-a)(-b)ab=(ab)^2$. Hence the map $\\psi\\colon \\bigwedge^3(Q_0,N_0)\\rightarrow (k,<1>)$ given by $i\\wedge j\\wedge ij\\mapsto ab.1$ (i.e., multiplication by $ab$) is an isomorphism of one-dimensional quadratic spaces where $<1>$ denotes the bilinear form $(x,y)\\mapsto xy$ on $k$. Thus $v\\times w$ is uniquely determined by $N_0(u,v\\times w)=\\psi(u\\wedge v\\wedge w)$.\n\nWe give a construction for quaternion algebra starting from a rank $3$ quadratic space $V$ over $k$, with trivial discriminant. This construction is adaptation of the construction for Cayley algebras in \\cite{th}.  \nLet $B\\colon V\\times V\\rightarrow k$ be a non-degenerate bilinear form. Suppose that the discriminant of $(V,B)$ is trivial. That is, there exists $\\psi\\colon \\bigwedge^3(V,B)\\rightarrow (k,<1>)$ which is an isomorphism of quadratic spaces. We use this to define a vector product $\\times\\colon V\\times V\\longrightarrow V$ by the formula, $B(u,v\\times w)=\\psi(u\\wedge v\\wedge w)$, for $u,v,w\\in V$. \nLet $Q= k\\oplus V$ be a $k$-vector space of dimension $4$. \nWe define a multiplication on $Q$ by,\n$$\n(a,v)(b,w)=(ab-B(v,w), aw+bv+v\\times w)\n$$\nwhere $a,b\\in k$ and $v,w\\in V$. \nWith this multiplication, $Q$ is a quaternion algebra over $k$, with norm $N(a,v)=a^2+B(v,v)$. \nThe isomorphism class of $Q$ is independent of $\\psi$ chosen. \nOne can show that all quaternion algebras arise this way. \n\n\\subsection{$3$-dimensional vector space and cross product}\n\nLet $V$ be a $3$ dimensional vector space with a non-degenerate quadratic form $N=\\langle a,b,c\\rangle$ with respect to basis $\\{e_1,e_2,e_3\\}$. We wish to define a cross product on $V$ uniquely determined by $N$.\n\nDenote $\\tilde N=\\langle \\frac{c}{b}, \\frac{c}{a},ab\\rangle $. Now define the cross product $\\times \\colon V\\times V \\rightarrow V$ by the following formal formula: \n$$v\\times w = \\det \\begin{pmatrix} b e_1 & ae_2 & \\frac{c}{ab}e_3 \\\\ x_1 &x_2 & x_3 \\\\ y_1 & y_2 & y_3 \\end{pmatrix}$$\nwhere $v=(x_1,x_2,x_3)$ and $w=(y_1,y_2,y_3)$. \nIt is easy to verify that this is bilinear and $v\\times v=0$. To verify the Jacobi identity we first check the following formula $u\\times(v\\times w) = v\\tilde N(u,w) - w\\tilde N(u,v)$ for all $u,v,w\\in V$. \n\\begin{eqnarray*}\nu\\times (v\\times w) &=& \\det\\begin{pmatrix} b e_1 & ae_2 & \\frac{c}{ab}e_3 \\\\ u_1 & u_2 & u_3 \\\\ b(x_2y_3-x_3y_2) & -a(x_1y_3 - x_3y_1) & \\frac{c}{ab}(x_1y_2-x_2y_1) \\end{pmatrix} \\\\\n&=& x_1e_1(\\frac{c}{a}u_2y_2 + ab u_3y_3) - y_1e_1 (\\frac{c}{a}u_2x_2 + ab u_3x_3) + x_2e_2(\\frac{c}{b}u_1y_1 + ab u_3y_3) \\\\&& - y_2e_2 (\\frac{c}{b}u_1x_1 + ab u_3x_3) +\nx_3e_3(\\frac{c}{b}u_1y_1 + \\frac{c}{a} u_2y_2) - y_3e_3 (\\frac{c}{b}u_1x_1 + \\frac{c}{a} u_2x_2)   \\\\\n&=& v\\tilde N(u,w) - w\\tilde N(u,v)\n\\end{eqnarray*}\n\nFurther we want the cross product to be uniquely determined by $N$, i.e., $N(u, v\\times w)=0$ if $u=v$ or $u=w$. We verify the following formal formula:\n$$N(u,v\\times w)=\\det \\begin{pmatrix}abu_1 & ab u_2 & \\frac{c^2}{ab}u_3 \\\\ x_1 & x_2 & x_3 \\\\ y_1 & y_2 & y_3\\end{pmatrix}$$\nand the equation $N(v,v\\times w)=0$ gives $c^2=(ab)^2$, i.e., $Q$ should have trivial discriminant. \n\n\\section{Geometric Method} \nTo a quaternion algebra $Q=\\left(\\frac{a,b}{k}\\right)$ one associates a conic $C(a,b)$ given by the equation $ax^2+by^2=z^2$ in the projective space $P^2$.  The attached conic does not depend on choice of a basis and isomorphism of $Q$ is equivalent to $k$-isomorphism of associated conics. Further one can prove that the quaternion algebra $Q$ is split if and only if the attached conic $C$ has a $k$-rational point. We refer the section 1.3 in \\cite{gz} for this construction and proofs.\n\n\\section{Embedding inside Matrix algebra}\n\nLet $Q=\\left(\\frac{a,b}{k}\\right)$ be a quaternion algebra. Denote $K=k[X]\/<X^2-a>=k[\\alpha]$. The following map gives an embedding of $Q$ inside $M_2(K)$: \n$$(x_0+x_1i)+(x_2+x_3i)j=z_1+z_2j\\mapsto \\begin{pmatrix}z_1 &bz_2\\\\ \\overline{z_2} & \\overline{z_1}\\end{pmatrix}.$$\nOne can also check that norm and trace is determinant and trace of the corresponding matrix.\n\n\n\n\\section{Some Facts and Applications}\n\n\\subsection{Classification over certain fields}\nClassifying quaternion algebras over a field up to isomorphism is not an easy task. Here we mention some for example.\n\\begin{theorem}\n\\begin{enumerate}\n\\item Over an algebraically closed field the only quaternion is matrix algebra. \n\\item Over a finite field the only quaternion algebra is matrix algebra.\n\\item Over reals any quaternion algebra is either isomorphic to $M_2(\\mathbb R)$ or $\\mathbb H=\\left(\\frac{-1,-1}{\\mathbb R}\\right)$.\n\\end{enumerate}\n\\end{theorem}\n\nFor the airthmetic aspect of quaternions the standard source of reference is~\\cite{vi}.\n\n\\subsection{Groups of type $A_1$}\n\nLet $G$ be an algebraic group defined over $k$. It is one of the important problem in the theory of algebraic groups to classify them. Given a quaternion algebra $Q$ over $k$ one can form the group $SL_1(Q)$ which is a group defined over $k$ of type $A_1$. In fact any group of type $A_1$ can be obtained this way.\n\n\\subsection{Merkurjev-Suslin Theorem}\n\\begin{theorem}\nAny $2$-torsion element in Brauer group of $k$ is equivalent to $Q_1\\otimes Q_2\\otimes\\cdots \\otimes Q_n$ where $Q_i$ are quaternion algebra over $k$.\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nOver the past several years, multiple $B$-physics experiments, including BaBar, LHCb, and Belle, have reported anomalies in decays associated with the $b \\to c \\ell \\nu$ and $b \\to s \\ell \\ell$ transitions. Violations of lepton flavor universality (LFU), known to be theoretically clean probes of New Physics (NP), are of particular interest. In the Standard Model (SM) LFU is only broken by the lepton masses. Hints for additional sources of LFU violation have been observed in the ratios of branching ratios of flavor-changing charged current and neutral current decays of $B$ mesons, $R_D$, $R_{D^*}$, $R_{K}$, and $R_{K^*}$,\n\\begin{equation}\n    R_{D^{(*)}} = \\frac{\\text{BR}(B \\to D^{(*)} \\tau \\nu)}{\\text{BR}(B \\to D^{(*)} \\ell \\nu)}\\,, \\qquad R_{K^{(*)}} = \\frac{\\text{BR}(B \\to K^{(*)} \\mu^+\\mu^-)}{\\text{BR}(B \\to K^{(*)} e^+e^-)} \\,.\n\\end{equation}\nThe experimental world averages of $R_D$ and $R_{D^*}$ from the heavy flavor averaging group (HFLAV) are based on measurements from BaBar~\\cite{Lees:2012xj}, Belle~\\cite{Huschle:2015rga,Hirose:2016wfn,Belle:2019rba}, and LHCb~\\cite{Aaij:2015yra,Aaij:2017uff}, and read~\\cite{Amhis:2019ckw}\n\\begin{equation}\n    R_D = 0.340 \\pm 0.027 \\pm 0.013 \\,, \\qquad R_{D^*} = 0.295 \\pm 0.011 \\pm 0.008 \\,,\n\\end{equation}\nwith an error correlation of $\\rho = -38\\%$.\nThe corresponding SM predictions are known with high precision~\\cite{Bernlochner:2017jka,Bigi:2017jbd,Jaiswal:2017rve}. The values adopted by HFLAV are~\\cite{Amhis:2019ckw}\n\\begin{equation}\n    R_D^\\text{SM} = 0.299 \\pm 0.003  \\,, \\qquad R_{D^*}^\\text{SM} = 0.258  \\pm 0.005 \\,.\n\\end{equation}\nThe combined discrepancy between the SM prediction and experimental world averages of $R_D$ and $R_{D^*}$ is at the $3.1\\sigma$ level.\n\nThe most precise measurement to date of the LFU ratio $R_K$ has been performed by LHCb~\\cite{Aaij:2019wad}\n\\begin{equation}\n R_K = 0.846^{+0.060}_{-0.054}{}^{+0.016}_{-0.014} \\,, \\qquad \\text{for}~~~ 1.1\\,\\text{GeV}^2 < q^2 < 6\\,\\text{GeV}^2 \\,,\n\\end{equation}\nwith $q^2$ being the dilepton invariant mass squared.\nThe SM predicts $R_K^\\text{SM} \\simeq 1$ with theoretical uncertainties well below the current experimental ones~\\cite{Bordone:2016gaq}. \nThe above experimental value is closer to the SM prediction than the Run-1 result~\\cite{Aaij:2014ora}. However, the reduced experimental uncertainties still imply a tension between theory and experiment of $2.5\\sigma$.\n\nThe most precise measurement of $R_{K^*}$ is from a Run-1 LHCb analysis~\\cite{Aaij:2017vbb} that finds\n\\begin{equation}\\label{eq:RKstarEXP}\n R_{K^*} = \\begin{cases} 0.66^{+0.11}_{-0.07}\\pm0.03 \\,, \\qquad \\text{for}~ 0.045\\,\\text{GeV}^2 < q^2 < 1.1\\,\\text{GeV}^2 \\,, \\\\ 0.69^{+0.11}_{-0.07}\\pm0.05 \\,, \\qquad \\text{for}~ 1.1\\,\\text{GeV}^2 < q^2 < 6\\,\\text{GeV}^2 \\,. \\end{cases}\n\\end{equation}\nThe result for both $q^2$ bins are in tension with the SM prediction~\\cite{Bordone:2016gaq}, $R_{K^*}^\\text{SM} \\simeq 1$, by $\\sim 2.5\\sigma$ each. Recent measurements of $R_{K^*}$ and $R_K$ by Belle~\\cite{Abdesselam:2019wac,Abdesselam:2019lab}\\footnote{Here we quote the isospin average of $B^0 \\to K^{(*)0} \\ell^+ \\ell^-$ and $B^\\pm \\to K^{(*)\\pm} \\ell^+ \\ell^-$ decays.}\n\\begin{equation}\n R_{K^*} = \\begin{cases} 0.90^{+0.27}_{-0.21}\\pm0.10 \\,, \\qquad \\text{for}~ 0.1\\,\\text{GeV}^2 < q^2 < 8\\,\\text{GeV}^2 \\,, \\\\ 1.18^{+0.52}_{-0.32}\\pm0.10 \\,, \\qquad \\text{for}~ 15\\,\\text{GeV}^2 < q^2 < 19\\,\\text{GeV}^2 \\,, \\end{cases}\n\\end{equation}\n\\begin{equation}\n R_{K} = \\begin{cases} 0.98^{+0.27}_{-0.23}\\pm0.06 \\,, \\qquad \\text{for}~ 1\\,\\text{GeV}^2 < q^2 < 6\\,\\text{GeV}^2 \\,, \\\\ 1.11^{+0.29}_{-0.26}\\pm0.07 \\,, \\qquad \\text{for}~ 14.18\\,\\text{GeV}^2 < q^2 \\,, \\end{cases}\n\\end{equation}\nare compatible with both the SM prediction and the LHCb results. Several papers have re-analyzed the status of the $B$ anomalies in light of the latest experimental updates, and found preference for new physics with high significance~\\cite{Alguero:2019ptt,Alok:2019ufo,Ciuchini:2019usw,Datta:2019zca,Aebischer:2019mlg,Kowalska:2019ley,Arbey:2019duh}.\n\nWhile the anomalies detailed upon above persist, the question of the origin of the observed baryon asymmetry \\cite{Ade:2015xua} also remains a long standing problem in cosmology.  Any dynamical explanation requires sizable C- and CP-violating interactions in the early universe \\cite{Sakharov:1967dj}.  In light of upcoming low-energy experiments with much greater sensitivity to electric and magnetic dipole moments of elementary particles, it is interesting to ask whether solutions to the flavor anomalies may also be associated with sizable CP violating complex phases that may be probed by these experiments.  \n\nThe only known viable, single-mediator explanation of all flavor anomalies is a $U_1$ vector leptoquark~\\cite{Alonso:2015sja, Bhattacharya:2016mcc, Buttazzo:2017ixm,Calibbi:2017qbu,Crivellin:2018yvo,Angelescu:2018tyl, Cornella:2019hct}.  This leptoquark generically introduces new sources of CP violation in the Lagrangian in the form of complex parameters \\cite{Panico:2018hal}.  The scope of the present study is to explore, for the first time, the prospects of observing electric dipole moments (EDMs) induced by a $U_1$ vector leptoquark that could explain the flavor anomalies reviewed above.  We additionally explore collider constraints, as well as constraints from measurements of the magnetic moments, and other flavor observables. Implications for EDMs and other CPV observables in scalar leptoquark scenarios have recently been discussed in~\\cite{Bobeth:2017ecx,Fuyuto:2018scm,Dekens:2018bci,Crivellin:2019kps}.\n\nThis paper is organized as follows: In Sec.~\\ref{sec:model}, we  introduce the CP violating $U_1$ model and discuss its effects on the B-physics anomalies. In Sec.~\\ref{sec:dipoles}, we give an overview of the effects of the CP violating leptoquark on EDMs of quarks, leptons, and neutrons. We also include a discussion of the present status of the experimental searches and the prospects for future measurements. In Sec.~\\ref{Sec:Pheno}, we report the main results of our paper, showing the leptoquark parameter space that can be probed by B-physics and EDM measurements. In Sec.~\\ref{Sec:LHC}, we discuss the LHC bounds on our leptoquark model. Finally, we reserve Sec.~\\ref{Sec:Conclusions} for our conclusions.\n\n\\section{The CP violating \\boldmath $U_1$ Vector Leptoquark Model} \\label{sec:model}\n\nWe consider the vector leptoquark $U_1 = ({\\mathbf 3}, {\\mathbf 1})_{2\/3}$ (triplet under $SU(3)_c$, singlet under $SU(2)_L$, and with hypercharge $+2\/3$).  This model may be viewed as the low energy limit of Pati-Salam models described in Ref.~\\cite{Bordone:2017bld, Bordone:2018nbg} (see also~\\cite{Assad:2017iib,DiLuzio:2017vat,Barbieri:2017tuq,Blanke:2018sro,Greljo:2018tuh,Heeck:2018ntp,Balaji:2018zna,Fornal:2018dqn}).\nThe most general dimension-4 Lagrangian describing the vector leptoquark of mass $M_{U_1}$ is (see e.g.~\\cite{Dorsner:2016wpm} for a recent review)\n\\begin{eqnarray} \\label{eq:U1}\n \\mathcal L_{U_1} &=& -\\frac{1}{2} U_{\\mu\\nu}^\\dagger U^{\\mu\\nu} + M_{U_1}^2 U_\\mu^\\dagger U^{\\mu} \\nonumber \\\\\n && + i g_s U_\\mu^\\dagger T_a U_\\nu \\left( \\kappa_s G^{\\mu\\nu}_a + \\tilde \\kappa_s \\tilde G^{\\mu\\nu}_a \\right)+ i g' \\frac{2}{3} U_\\mu^\\dagger U_\\nu \\left( \\kappa_Y B^{\\mu\\nu} + \\tilde \\kappa_Y  \\tilde B^{\\mu\\nu}\\right) \\,, \\nonumber \\\\ \n && + \\sum_{i,j} \\left( \\lambda_{ij}^q (\\bar Q_i \\gamma_\\mu P_L L_j) U^\\mu + \\lambda_{ij}^d (\\bar D_i \\gamma_\\mu P_R E_j) U^\\mu \\right) + \\text{h.c.}\\,,\n\\end{eqnarray}\nwhere $U^{\\mu\\nu} = D^\\mu U^\\nu - D^\\nu U^\\mu$ is the leptoquark field strength tensor in terms of its vector potential $U^\\mu$ and gauge covariant derivative $D^\\mu = \\partial^\\mu + i g_s T_a G_a^\\mu + i g' \\frac{2}{3} B^\\mu$.   $G_a^\\mu$ and $B^\\mu$, and $G_a^{\\mu\\nu}$ and $B^{\\mu\\nu}$ are the gluon and hypercharge vector potentials and field strengths, respectively.  The dual field strength tensors are $\\tilde G_a^{\\mu\\nu} = \\frac{1}{2} \\epsilon^{\\mu\\nu\\rho\\sigma} G_{a\\,\\rho\\sigma}$ and $\\tilde B^{\\mu\\nu}= \\frac{1}{2} \\epsilon^{\\mu\\nu\\rho\\sigma} B_{\\rho\\sigma}$.\n\nThe third line in Eq.~\\eqref{eq:U1} contains couplings of $U_1$ with the SM quarks and leptons. Specifically, $Q_i$ and $L_i$ are the left-handed quark and lepton doublets, while $D_j$ and $E_j$ are the right-handed down quark and charged lepton singlets.\nWe assume that the model does not contain light right-handed neutrinos. (If right-handed neutrinos are introduced, additional couplings of $U_1$ with right-handed neutrinos and right-handed up quarks are possible~\\cite{Robinson:2018gza}.)\nThe couplings $\\lambda_{ij}^q$ and $\\lambda_{ij}^d$ are in general complex and are therefore a potential source of CP violation of the model.   We work in the fermion mass eigenstate basis and define the leptoquark couplings $\\lambda_{ij}^q$ and $\\lambda_{ij}^d$ in a way such that\n\\begin{equation}\\label{eq:yuk}\n \\mathcal L_{U_1} \\supset \\sum_{ijk} (V_{ik} \\lambda_{kj}^q) (\\bar u_i \\gamma_\\mu P_L \\nu_j) U^\\mu +  \\sum_{ij} \\lambda_{ij}^q (\\bar d_i \\gamma_\\mu P_L \\ell_j) U^\\mu +  \\sum_{ij} \\lambda_{ij}^d (\\bar d_i \\gamma_\\mu P_R \\ell_j) U^\\mu + \\text{h.c.}\\,,\n\\end{equation}\nwhere $V$ is the CKM matrix.\n\nThe second line in Eq.~\\eqref{eq:U1} encodes the chromo- and hypercharge- magnetic and electric dipole moments of the $U_1$ leptoquark.\n\nIf the leptoquark arises from the spontaneous breakdown of a gauge symmetry, gauge invariance requires these couplings to be fixed to $\\kappa_s = \\kappa_Y = 1$, $\\tilde \\kappa_s = \\tilde \\kappa_Y = 0$.  In more generic scenarios where $U_1$ is composite, the values of $\\kappa_s$, $\\tilde \\kappa_s$, $\\kappa_Y$, $\\tilde \\kappa_Y$ are free parameters.\nNon-zero values for $\\tilde \\kappa_s$ and $\\tilde \\kappa_Y$ are an additional potential source of CP violation.  However, since they do not directly influence flavor physics, we will focus our attention to CP-violation contained in $\\lambda_{ij}^q$ and $\\lambda_{ij}^d$ (even though in Sec.~\\ref{sec:dipoles} we will present fully generic expressions for the EDMs, including their dependence on $\\tilde \\kappa_s$ and $\\tilde \\kappa_Y$).\n\n\n \\subsection{Leptoquark Effects in B-meson Decays} \\label{sec:LQBdecays}\n \n \nThe $U_1$ leptoquark can simultaneously address the hints for LFU violation in charged current decays $R_{D^{(*)}}$ and in neutral current decays $R_{K^{(*)}}$\\footnote{Note that the small anomaly in the low $q^2$ bin of $R_K^*$ in (\\ref{eq:RKstarEXP}) cannot be fully addressed by the $U_1$ leptoquark, but it requires the presence of light NP \\cite{Sala:2017ihs,Ghosh:2017ber,Datta:2017ezo,Altmannshofer:2017bsz}.}. Here we will use the results of a recent study~\\cite{Aebischer:2019mlg} that identified a benchmark point in the leptoquark parameter space that gives a remarkably consistent new physics explanation of these hints. We will explore the parameter space around this benchmark point (supplemented by a few more points), focusing on the implications for dipole moments. As we discuss below, not all leptoquark couplings in (\\ref{eq:U1}) are required to address the anomalies.\n\nExplaining the observed values of $R_{D^{(*)}}$ by non-standard effects in the $b \\to c \\tau \\nu$ transition is possible if the leptoquark has sizable couplings to the left-handed tau. Avoiding strong constraints from leptonic tau decays $\\tau \\to \\nu_\\tau \\ell \\bar \\nu_\\ell$ and the $B \\to X_s \\gamma$ decay is possible in a well defined parameter space around the benchmark point with $\\lambda^q_{33} \\simeq 0.7$, $\\lambda^q_{23} \\simeq 0.6$ with a leptoquark mass of $M_{U_1} = 2$\\,TeV~\\cite{Aebischer:2019mlg}.\nThis corresponds to the following non-standard value for $R_{D^{(*)}}$\n\\begin{equation}\n \\frac{R_{D^{(*)}}}{R_{D^{(*)}}^\\text{SM}} = \\left| 1 + \\frac{v^2}{2 M_{U_1}^2} \\frac{\\lambda^{q*}_{33}\\lambda^q_{23}}{V_{cb}}  \\right|^2 \\simeq 1.2 \\,,\n\\end{equation}\nwhich is in good agreement with observations (in this equation we normalize $v=246$ GeV).\n\nThe results for $R_{K^{(*)}}$ can be accommodated by a non-standard effect in the $b \\to s \\mu\\mu$ transition if the couplings to the left-handed muon obey Re$(\\lambda^q_{22} \\times \\lambda^q_{32}) \\simeq -2.5 \\times 10^{-3}$ for $M_{U_1} = 2$\\,TeV~\\cite{Aebischer:2019mlg}.\nThe leptoquark effects for this choice of couplings are described by a shift in the Wilson coefficients of the effective Hamiltonian relevant for $b \\to s \\ell \\ell$ transitions (see e.g.~\\cite{Aebischer:2019mlg} for the precise definition)\n\\begin{equation}\nC_9^{bs\\mu\\mu} = - C_{10}^{bs\\mu\\mu} = - \\frac{4\\pi^2}{e^2} \\frac{v^2}{M_{U_1}^2} \\frac{\\lambda_{32}^q \\lambda_{22}^{q*}}{V_{ts}^* V_{tb}} \\simeq -0.4 \\,.\n\\end{equation}\nThis agrees well with the best fit value for the Wilson coefficients found in~\\cite{Aebischer:2019mlg}.\n\nThe muonic couplings $\\lambda^q_{22}, \\lambda^q_{32}$ (that can explain the $R_{K^{(*)}}$ anomalies) in combination with the tauonic couplings $\\lambda^q_{23}, \\lambda^q_{33}$ (that are required to explain the $R_{D^{(*)}}$ anomalies) lead to lepton flavor violating decays. The strongest constraints arise from the decays $\\tau \\to \\phi \\mu$ and $B \\to K \\tau \\mu$. For the $\\lambda_{33}^q,~\\lambda_{23}^q$ benchmark mentioned above, existing limits on those decay modes result in the bounds on the leptoquark couplings $|\\lambda^q_{22}| \\lesssim 0.16$ and $ |\\lambda^q_{32}| \\lesssim 0.40$ for $M_{U_1} = 2$\\,TeV~\\cite{Aebischer:2019mlg}.\n\nThe experimental values of $R_{K^{(*)}}$ may also be explained by new physics in the $b \\to s ee$ transition as opposed to modifying the $b \\to s \\mu\\mu$ transition. Focusing on left-handed couplings, the required shifts in the relevant Wilson coefficients is~\\cite{Aebischer:2019mlg}\n\\begin{equation} \\label{eq:C9e}\nC_9^{bsee} = - C_{10}^{bsee} = -\\frac{4\\pi^2}{e^2} \\frac{v^2}{M_{U_1}^2} \\frac{\\lambda_{31}^q \\lambda_{21}^{q*}}{V_{ts}^* V_{tb}} \\simeq +0.4 \\,,\n\\end{equation}\ncorresponding to the couplings Re$(\\lambda^q_{21} \\times \\lambda^q_{31}) \\simeq +2.5 \\times 10^{-3}$ for $M_{U_1} = 2$\\,TeV.\nThe experimental bounds on the lepton flavor violating processes $\\tau \\to \\phi e$ and $B \\to K \\tau e$ are comparable to those of $\\tau \\to \\phi \\mu$ and $B \\to K \\tau \\mu$~\\cite{Aubert:2009ap,Miyazaki:2011xe,Lees:2012zz}. We therefore expect that the constraints on the left-handed electron couplings $|\\lambda^q_{21}|$ and $|\\lambda^q_{31}|$ are similar to the muon couplings mentioned above, i.e. $|\\lambda^q_{21}| \\lesssim 0.16$ and $ |\\lambda^q_{31}| \\lesssim 0.40$ for $M_{U_1} = 2$\\,TeV.\n\nMotivated by this discussion, in the next sections we will explore the leptoquark parameter space in the neighborhood of four benchmark scenarios:\n\\begin{subequations}\\label{eq:BM}\n\\begin{align}\n\\text{BM1:}&~~\n\\lambda_{33}^q =0.7\\,,~\\lambda_{23}^q=0.6\\,, \\quad\n\\lambda_{32}^q = -0.25\\,,~ \\lambda_{22}^q = 0.01\\,, \\quad\n\\lambda_{31}^q = \\lambda_{21}^q = 0\\,,\\label{eq:BM1} \\\\ \n\\text{BM2:}&~~\n\\lambda_{33}^q =0.7\\,,~\\lambda_{23}^q=0.6\\,, \\quad \n\\lambda_{32}^q = \\lambda_{22}^q = 0\\,, \\quad \n\\lambda_{31}^q = 0.05\\,, \\lambda_{21}^q = 0.05\\,,\\label{eq:BM2}\\\\ \n\\text{BM3:}&~~\n\\lambda_{33}^q = \\lambda_{23}^q=0\\,, \\quad\n\\lambda_{32}^q = -1.4\\,,~ \\lambda_{22}^q = 10^{-3}\\,, \\quad\n\\lambda_{31}^q = \\lambda_{21}^q = 0\\,, \\label{eq:BM3}\\\\ \n\\text{BM4:}&~~\n\\lambda_{33}^q = \\lambda_{23}^q=0\\,, \\quad\n\\lambda_{32}^q = \\lambda_{22}^q = 0 \\,, \\quad\n\\lambda_{31}^q = 0.5\\,,~ \\lambda_{21}^q = 5.0\\times 10^{-3}\\,, \\label{eq:BM4}\\\\[6pt] \n& ~~M_{U_1} =2~{\\rm{TeV}}, \\enspace \\kappa_{Y,s}=1, \\enspace \\tilde\\kappa_{Y,s}=0 ~\\text{for all benchmarks } \\nonumber\n\\end{align}\n\\end{subequations}\nwith all the other fermionic couplings of the leptoquark in Eq.~(\\ref{eq:U1}) set to zero.\nIn BM1 and BM2 both the $R_{D^{(*)}}$ and $R_{K^{(*)}}$ anomalies are addressed. The $R_{K^{(*)}}$ explanations involve new physics in the $b \\to s \\mu\\mu$ transition (BM1) or in the $b \\to s ee$ transition (BM2). For benchmark points BM3 and BM4 we forgo an explanation of $R_{D^{(*)}}$. This allows us to increase the couplings to muons\/electrons while avoiding the strong constraints from lepton flavor violating tau decays. Note that in benchmark BM3, the $R_{K^{(*)}}$ anomalies are only partially addressed. For BM3 we have  $R_K \\simeq R_K^{*} \\simeq 0.88$ which is in good agreement with the latest $R_K$ measurement, but $\\sim 2\\sigma$ away from the measured $R_{K^*}$ value. As we discuss below in Sec.~\\ref{sec:muon} benchmark BM3 is motivated because it can accommodate the longstanding discrepancy in the anomalous magnetic moment of the muon.\n\nFor all benchmark scenarios we explicitly checked compatibility with the measurements of the di-lepton~\\cite{Greljo:2017vvb} and di-tau \\cite{Faroughy:2016osc} invariant mass distributions at the LHC and searches for electron-quark contact interactions at LEP~\\cite{Schael:2013ita}. In the case of the di-lepton invariant mass distributions at the LHC, the value of $\\lambda^q_{32}$ in BM3 is close to the exclusion bound.\n\nStarting with these benchmark points, in the following sections we turn on couplings to right-handed taus $\\lambda^d_{33}$, muons $\\lambda^d_{32}$, and electrons $\\lambda^d_{31}$ and determine the expected size of electric and magnetic dipole moments of the leptons as function of the real and imaginary part of the new couplings.  In principle, also the couplings $\\lambda^d_{23}$, $\\lambda_{22}^d$ and $\\lambda_{21}^d$ influence the dipole moments; we comment on $\\lambda_{22}^d$ and $\\lambda_{21}^d$ in Secs. \\ref{sec:muon} and \\ref{sec:electron}, but we do not consider $\\lambda^d_{23}$ since it does not play any role in explaining the flavor anomalies.\n \n\\section{Dipole Moments of Quarks and Leptons} \\label{sec:dipoles}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.50\\textwidth]{fermion-EDM.pdf}\n\\caption{Feynman diagrams contributing to the dipole moments of quarks and leptons from leptoquark exchange.}\n\\label{fig:diagrams}\n\\end{figure}\n\nIn this section, we calculate and present new and original formulae for shifts in the electric and magnetic dipole moments of leptons and quarks induced by the leptoquark.  We then estimate the size of the neutron electric dipole.  Finally, we review experimental limits on the dipole moments.\n\nThe leptoquark radiatively induces dipole moments starting at one loop order as shown in Fig.~\\ref{fig:diagrams}.  After integrating out the leptoquark, effective interactions encoding the dipole moments are given by the effective Lagrangian\n\\begin{equation}\n    \\mathcal L_\\text{eff} = \\sum_f \\left( a_f \\frac{ e Q_f }{4 m_f} (\\bar f \\sigma^{\\mu\\nu} f) F_{\\mu\\nu} -\\frac{i d_f}{2}  (\\bar f \\sigma^{\\mu\\nu} \\gamma_5 f) F_{\\mu\\nu} \\right)\\,,\n\\end{equation}\n where $a_f$ is the anomalous magnetic dipole moment, and $d_f$ is the electric dipole moment of SM fermion $f$.  In the absence of right-handed neutrinos, the $U_1$ leptoquark does not generate dipole moments for neutrinos. \n\nThrough its coupling with the gluons, the leptoquark induces chromomagnetic, $\\hat a_q$, and chromoelectric, $\\hat d_q$, dipole moments of quarks \n\\begin{equation}\n\\mathcal L_\\text{eff} = \\sum_q \\left(  \\frac{\\hat a_q}{4 m_q} (\\bar q \\sigma^{\\mu\\nu} T^a q) G^a_{\\mu\\nu} - \\frac{i \\hat d_q}{2} (\\bar q \\sigma^{\\mu\\nu} T^a \\gamma_5 q) G^a_{\\mu\\nu} \\right)\\,.\n\\end{equation}\n\n\\subsection{Leptoquark Contribution to Dipole Moments of SM Leptons and Quarks}\n\n\nIn the large $M_{U_1}$ limit, the leptoquark contribution to the anomalous magnetic moment of the muon is\n\\begin{multline} \\label{eq:a_lepton}\na_\\mu = \\frac{N_C}{16\\pi^2}\\sum_i\\Big[2\\text{Re}( \\lambda_{i2}^q  \\lambda_{i2}^{d*})\\frac{m_{d_i} m_\\mu}{M_{U_1}^2}\\Big(2 Q_d + Q_{U}\\big((1-\\kappa_Y)\\ln(\\frac{\\Lambda_\\text{UV}^2}{M_{U_1}^2})+\\frac{1-5\\kappa_Y}{2}\\big)\\Big) \\\\\n+2Q_{U}{\\tilde\\kappa}_Y\\text{Im}( \\lambda_{i2}^q  \\lambda_{i2}^{d*})\\frac{m_{d_i} m_\\mu}{M_{U_1}^2}\\Big(\\ln(\\frac{\\Lambda_\\text{UV}^2}{M_{U_1}^2})+\\frac{5}{2}\\Big) \\\\\n-(| \\lambda_{i2}^q|^2 + | \\lambda_{i2}^d|^2)\\frac{m_\\mu^2}{M_{U_1}^2}\\Big(\\frac{4}{3} Q_d + Q_{U}\\big((1-\\kappa_Y)\\ln(\\frac{\\Lambda_\\text{UV}^2}{M_{U_1}^2})-\\frac{1+9\\kappa_Y}{6}\\big)\\Big)\\Big]\\,,\n\\end{multline}\nwhere $Q_d = -1\/3$, $Q_U = +2\/3$ is the leptoquark electric charge, and $N_C = 3$.  Note that if $\\kappa_Y \\neq 1$ or $\\tilde \\kappa_Y \\neq 0$, relevant for scenarios in which the leptoquark is not a gauge boson, the dipole moment exhibits logarithmic dependence on the cut-off scale $\\Lambda_{UV}$ not far above the leptoquark mass.  Our formula is in agreement with \\cite{Queiroz:2014zfa,Kowalska:2018ulj} when specialized to the vector leptoquark model with $\\kappa_Y=1$ and $\\tilde\\kappa_Y=0$.  Similarly, the muon electric dipole moment is\n\\begin{multline} \\label{eq:d_lepton}\nd_\\mu = \\frac{e N_C}{16\\pi^2}\\sum_i\\Big[ \\text{Im}( \\lambda_{i2}^q  \\lambda_{i2}^{d*})\\frac{m_{d_i}}{M_{U_1}^2}\\Big(2Q_d + Q_{U}\\big((1-\\kappa_Y)\\ln(\\frac{\\Lambda_\\text{UV}^2}{M_{U_1}^2})+\\frac{1-5\\kappa_Y}{2}\\big)\\Big) \\\\\n + Q_U {\\tilde\\kappa}_Y \\text{Re}( \\lambda_{i2}^q  \\lambda_{i2}^{d*})\\frac{m_{d_i}}{M_{U_1}^2}\\big(\\ln(\\frac{\\Lambda_\\text{UV}^2}{M_{U_1}^2})+\\frac{5}{2}\\big) \\\\\n + Q_U {\\tilde\\kappa}_Y (| \\lambda_{i2}^q|^2 + | \\lambda_{i2}^d|^2)\\frac{m_\\mu}{M_{U_1}^2}\\big(\\frac{1}{2}\\ln(\\frac{\\Lambda_\\text{UV}^2}{M_{U_1}^2})+\\frac{3}{4}\\big)\\Big]\\,.\n\\end{multline}\nCP violation is provided either by the imaginary part of the fermion coupling combination $ \\lambda_{i2}^q  \\lambda_{i2}^{d*}$, or by the CP violating hypercharge coupling $\\tilde \\kappa_Y$.  Dipole moments of other charged leptons are obtained by the appropriate replacement of the muon mass, $m_\\mu$, and leptoquark couplings to muons, $\\lambda_{i2}$. \n\nThe bottom quark electric dipole moment induced by the leptoquark is\n\\begin{multline}\\label{eq:strangequark-edm}\nd_b = \\frac{e}{16\\pi^2}\\sum_i\\Big[\\text{Im}( \\lambda_{3i}^q  \\lambda_{3i}^{d*})\\frac{m_{\\ell_i}}{M_{U_1}^2}\\Big(2Q_\\ell + Q_{U}\\big((1-\\kappa_Y)\\ln(\\frac{\\Lambda_\\text{UV}^2}{M_{U_1}^2})+\\frac{1-5\\kappa_Y}{2}\\big)\\Big) \\\\\n + Q_U {\\tilde\\kappa}_Y \\text{Re}( \\lambda_{3i}^q  \\lambda_{3i}^{d*})\\frac{m_{\\ell_i}}{M_{U_1}^2}\\big(\\ln(\\frac{\\Lambda_\\text{UV}^2}{M_{U_1}^2})+\\frac{5}{2}\\big) \\\\\n - Q_U {\\tilde\\kappa}_Y (| \\lambda_{3i}^q|^2 + | \\lambda_{3i}^d|^2)\\frac{m_b}{M_{U_1}^2}\\big(\\frac{1}{2}\\ln(\\frac{\\Lambda_\\text{UV}^2}{M_{U_1}^2})+\\frac{3}{4}\\big)\\Big]\\,,\n\\end{multline}\nand the chromoelectric dipole moment (cEDM) is\n\\begin{multline}\\label{eq:strangequark-cedm}\n\\hat{d}_b = \\frac{g_s}{16\\pi^2}\\sum_i \\Big[ \\text{Im}( \\lambda_{3i}^q  \\lambda_{3i}^{d*})\\frac{m_{\\ell_i}}{M_{U_1}^2}\\Big((1-\\kappa_Y)\\ln(\\frac{\\Lambda_\\text{UV}^2}{M_{U_1}^2})+\\frac{1-5\\kappa_Y}{2}\\Big) \\\\\n + {\\tilde\\kappa}_Y \\text{Re}( \\lambda_{3i}^q  \\lambda_{3i}^{d*})\\frac{m_{\\ell_i}}{M_{U_1}^2}\\big(\\ln(\\frac{\\Lambda_\\text{UV}^2}{M_{U_1}^2})+\\frac{5}{2}\\big) \\\\\n - {\\tilde\\kappa}_Y (| \\lambda_{3i}^q|^2 + | \\lambda_{3i}^d|^2)\\frac{m_b}{M_{U_1}^2}\\big(\\frac{1}{2}\\ln(\\frac{\\Lambda_\\text{UV}^2}{M_{U_1}^2})+\\frac{3}{4}\\big)\\Big].\n\\end{multline}\nThe other down-type quark (chromo-)electric dipole moments can be obtained by appropriate replacements of flavor indices.\n\nAnalogously, up-type quark (chromo-)electric dipole moments are obtained from the bottom quark result by the replacement $ \\lambda_{ij}^q \\rightarrow V_{ik}  \\lambda_{kj}^q$, $ \\lambda_{ij}^d \\rightarrow 0$, $m_\\ell \\rightarrow m_\\nu = 0$, $m_b \\rightarrow m_u$ yielding\n\\begin{equation}\nd_u = - \\frac{e}{16\\pi^2} Q_U {\\tilde\\kappa}_Y \\sum_i | (V\\lambda^q)_{1i}|^2 \\frac{m_u}{M_{U_1}^2}\\Big(\\frac{1}{2}\\ln(\\frac{\\Lambda_\\text{UV}^2}{M_{U_1}^2})+\\frac{3}{4}\\Big)\\,,\n\\end{equation}\n\nand\n\\begin{equation}\n\\hat {d}_u = - \\frac{g_s}{16\\pi^2} {\\tilde\\kappa}_Y \\sum_i |(V\\lambda^q)_{1i}|^2 \\frac{m_u}{M_{U_1}^2}\\Big(\\frac{1}{2}\\ln(\\frac{\\Lambda_\\text{UV}^2}{M_{U_1}^2})+\\frac{3}{4}\\Big)\\,.\n\\end{equation}\nWe do not consider anomalous (chromo-)magnetic moments of the quarks as they are experimentally not constrained.\n\n\n\\subsection{Connecting Quark Dipole Moments to the Neutron EDM}\nIn the following, we determine the neutron electric dipole moment due to quark-level dipole moments.  We neglect the running of quark dipole moments from the leptoquark scale to the hadronic scale, since the neglected logarithm of order $\\alpha_s \\ln(M_{U_1}^2\/M_n^2) \\approx 1.6$ leads to corrections which are small compared to the relevant hadronic uncertainties discussed below.  \n\nThe dominant contributions to the neutron EDM are from the short range QCD interactions involving quark EDMs, $d_i$, and cEDMs, $\\hat{d}_i$,  given by\n\\begin{equation}\\label{eq:shortRangeEDM}\nd_n \\sim -\\frac{v}{\\sqrt{2}}\\Big[\\beta^{uG}_n \\hat{d}_u + \\beta^{dG}_n \\hat{d}_d + \\beta^{sG}_n \\hat{d}_s + \\beta_n^{u\\gamma} d_u + \\beta_n^{d\\gamma} d_d + \\beta_n^{s\\gamma} d_s\\Big]\\,,\n\\end{equation}\nwhere the $\\beta_i^{(k)}$ are the hadronic matrix elements.  Estimates from quark cEDM are given by $\\beta^{uG}_n \\approx 4^{+6}_{-3}\\times10^{-4}\\,e\\text{ fm}$ and $\\beta^{dG}_n \\approx 8^{+10}_{-6}\\times10^{-4}\\,e\\text{ fm}$ \\cite{Engel:2013lsa}.  The most recent lattice evaluations of the matrix elements involving the electromagnetic EDMs are \\cite{Bhattacharya:2015esa, Bhattacharya:2016zcn} $-\\frac{v}{\\sqrt{2}}\\beta^{u\\gamma}_n \\approx -0.233(28)$, $-\\frac{v}{\\sqrt{2}}\\beta^{d\\gamma}_n \\approx 0.776(66)$ and $-\\frac{v}{\\sqrt{2}}\\beta^{s\\gamma}_n \\approx 0.008(9)$.\n\nContributions from heavy quark cEDM are estimated by integrating out the heavy quark, $Q=c,b$, to generate the three gluon Weinberg (gluon cEDM) operator,\n\\begin{equation}\n\\mathcal{L} = \\frac{c_{\\tilde{\\text{G}}}}{m_Q^2} \\frac{g_sf^{abc}}{3}\\tilde{G}^a_{\\mu\\nu}G^b_{\\nu\\rho}G_{\\rho}^{c\\,\\mu}\\,,\n\\end{equation}\nwhere the Wilson coefficient is given by \\cite{Chang:1990jv,Boyd:1990bx,Dine:1990pf}\n\\begin{equation}\nc_{\\tilde{\\text{G}}} = \\frac{g_s^2}{32\\pi^2} m_Q \\hat{d}_Q\\,.\n\\end{equation}\nContributions to $c_{\\tilde{\\text{G}}}$ from CP-violating leptoquark gluon interactions proportional to $\\tilde{\\kappa}_s$ are also present, but we do not consider them since they are unrelated to flavor anomalies.\nIn terms of $c_{\\tilde{\\text{G}}}$, the neutron EDM is given by \\cite{Engel:2013lsa}\n\\begin{equation}\nd_n = \\frac{v^2}{m_Q^2} \\beta^{\\tilde{G}}_n c_{\\tilde{\\text{G}}}\n\\end{equation}\nwhere $\\beta^{\\tilde{G}}_n\\approx [2,\\, 40]\\times10^{-20} \\text{ e cm}$ is the nucleon matrix element estimated using QCD sum rules and chiral perturbation theory \\cite{Demir:2002gg}.\n\nTo compare the relative sizes of contributions from light and heavy quark to the neutron EDM, we take the strange and bottom quark contributions, and assume for simplicity that $\\kappa_Y = 1$, $\\tilde\\kappa_Y = 0$.  We also assume $M_{U_1} \\sim  2\\text{ TeV}$ for the leptoquark scale.\n\nPutting together Eqs. (\\ref{eq:strangequark-edm}) and (\\ref{eq:strangequark-cedm}) with Eq. (\\ref{eq:shortRangeEDM}), we find that the strange quark EDM contribution to the neutron EDM is \n\n\\begin{eqnarray}\\nonumber\nd_n^{\\rm{strange}} &\\approx& -\\frac{5}{24\\pi^2 M_{U_1}^2 \\text{ cm}}\\left[m_\\tau \\text{Im}( \\lambda_{23}^q  \\lambda_{23}^{d*}) +m_\\mu \\text{Im}( \\lambda_{22}^q  \\lambda_{22}^{d*})\\right]\\times 0.008  \\text{ $e$ cm}\\\\\n\\label{eq:s-neturon-edm}&\\sim& -\\left(\\text{Im}( \\lambda_{23}^q  \\lambda_{23}^{d*})+0.06~\\text{Im}( \\lambda_{22}^q  \\lambda_{22}^{d*})\\right)\\times 1.5\\times 10^{-24}\\text{ $e$ cm}\\,.\n\\end{eqnarray}\nThe bottom quark cEDM contribution to the neutron EDM is instead given by\n\n\\begin{eqnarray}\\nonumber\nd_n^{\\rm{bottom}} &\\approx& -\\frac{g_s^3  v^2}{(16\\pi^2)^2 m_b M_{U_1}^2}\\left[m_\\tau \\text{Im}( \\lambda_{33}^q  \\lambda_{33}^{d*})+m_\\mu \\text{Im}( \\lambda_{32}^q  \\lambda_{32}^{d*})\\right] \\times[2, 40]\\times 10^{-20}\\\\\n\\label{eq:b-neturon-edm}&\\sim& -\\left(\\text{Im}( \\lambda_{33}^q  \\lambda_{33}^{d*})+0.06~\\text{Im}( \\lambda_{32}^q  \\lambda_{32}^{d*})\\right)\\times [2,40]\\times5\\times 10^{-27}\\text{ $e$ cm}\\,.\n\\end{eqnarray}\nFor generic $\\mathcal{O}(1)$ sized leptoquark couplings $\\lambda^q_{ik}$ and $\\lambda^d_{ik}$ the strange quark contribution (\\ref{eq:s-neturon-edm}) to the neutron EDM is much larger than the bottom quark contribution (\\ref{eq:b-neturon-edm}).  However, in the region of parameter space we are exploring, the bottom quark contribution is typically bigger than the strange quark contribution. \n\n\\subsection{Experimental Status and Prospects}\nWe review here the current experimental status of dipole moments of Standard Model fermions. \nThe anomalous magnetic moments of the electron, $a_e$, and the muon, $a_\\mu$, are measured extremely precisely~\\cite{Hanneke:2008tm,Bennett:2006fi}, and are predicted to similarly high precision within the SM, with new physics contributions constrained to lie within the range~\\cite{Keshavarzi:2019abf,Davoudiasl:2018fbb} (see also~\\cite{Davier:2019can,Crivellin:2018qmi,Jegerlehner:2017lbd})\n\\begin{equation}\n \\Delta a_\\mu = (28.0 \\pm 6.3_{\\rm{exp}}\\pm 3.8_{\\rm{th}}) \\times 10^{-10} \\,, \\qquad \\Delta a_e  = (-8.9 \\pm 3.6_{\\rm{exp}}\\pm 2.3_{\\rm{th}} ) \\times 10^{-13} \\,,\n\\end{equation}\n In addition to the long standing discrepancy in the muon magnetic moment with a significance of more than $3\\sigma$, a discrepancy in the electron magnetic moment arose after a recent precision measurement of the fine structure constant~\\cite{Parker:2018vye} with a significance of $\\sim 2.4\\sigma$. Combining the expected sensitivity from the running $g-2$ experiment at Fermilab~\\cite{Grange:2015fou} with expected progress on the SM prediction (see~\\cite{Blum:2016lnc,Blum:2018mom,Davies:2019efs,Shintani:2019wai,Gerardin:2019rua,Blum:2019ugy} for recent lattice efforts and \\cite{Colangelo:2015ama,Colangelo:2017qdm,Hoferichter:2018dmo,Colangelo:2018mtw,Hoferichter:2019gzf} for recent efforts using the framework of dispersion relations) the uncertainty on $\\Delta a_\\mu$ will be reduced by a factor of a few in the coming years. Similarly, for $\\Delta a_e$ we expect an order of magnitude improvement in the sensitivity~\\cite{Gabrielse:2019cgf}.\n\nThe anomalous magnetic moment of the tau, $a_\\tau$, is currently only very weakly constrained. The strongest constraint comes from LEP and reads at 95\\% C.L.~\\cite{Abdallah:2003xd}\n\\begin{equation}\n -0.055 < a_\\tau < 0.013 \\,.\n\\end{equation}\nImprovements in sensitivity by an order of magnitude or more might be achieved at Belle II or future electron positron colliders (see~\\cite{Pich:2013lsa} for a review).\n\nStrong experimental constraints exist for the EDM of the electron. The strongest bound is inferred from the bound on the EDM of ThO obtained by the ACME collaboration which gives at 90\\% C.L.~\\cite{Andreev:2018ayy}\n\\begin{equation}\n |d_e| < 1.1 \\times 10^{-29} \\, e\\,\\text{cm}\\,.\n\\end{equation}\nSignificant improvements by an order of magnitude or more can be expected from ACME in the future~\\cite{Andreev:2018ayy}.\n\nOnly weak constraints exist for the EDMs of the muon and the tau, $d_\\mu$ and $d_\\tau$. Analyses by the Muon g-2 collaboration~\\cite{Bennett:2008dy} and the Belle collaboration~\\cite{Inami:2002ah} give the following bounds at 95\\% C.L.\n\\begin{equation}\n |d_\\mu| < 1.9 \\times 10^{-19} \\, e\\,\\text{cm}\\,, \\qquad -2.2 \\times 10^{-17} \\, e\\,\\text{cm} < d_\\tau < 4.5 \\times 10^{-17}  \\,e\\,\\text{cm}\\,.\n\\end{equation}\nThe proposed muon EDM experiment at PSI aims at improving the sensitivity to the muon EDM by 4 orders of magnitude, $d_\\mu \\lesssim 5 \\times 10^{-23}  e$\\,cm~\\cite{Adelmann:2010zz}. Improving the sensitivity to the tau EDM by roughly two orders of magnitude ($d_\\tau<2\\times 10^{-19}~e$\\,cm) might be possible at Belle II or at future $e^+e^-$ colliders~\\cite{Kou:2018nap}.\n\n\nTurning to quarks, we note that the magnetic and chromo-magnetic dipole moments of quarks, $a_q$ and $\\hat a_q$, are very weakly constrained and we therefore do not consider them in this work.\nAs discussed in the previous section, the EDMs and cEDMs of quarks, $d_q$ and $\\hat d_q$, lead to EDMs of hadronic systems like the neutron and are therefore strongly constrained. In the following we will focus on the neutron EDM which is bounded at 95\\% C.L. by~\\cite{Afach:2015sja}\n\\begin{equation}\n |d_n| < 3.6 \\times 10^{-26} \\, e\\,\\text{cm}\\,.\n\\end{equation}\nExperimental sensitivities should improve by two orders of magnitude to a few $10^{-28}e\\,\\text{cm}$ in the next decade \\cite{Strategy:2019vxc}.\n\nWe collect the SM predictions, the current experimental results, and expected future experimental sensitivities to the dipole moments in Table~\\ref{tab:expstatus}.\n\\begin{table}[tb]\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nobservable & SM theory & current exp. & projected sens. \\\\\n\\hline\n$a_e-a_e^{\\text{SM}}$ & $\\pm 2.3\\times{10^{-13}}$ \\cite{Keshavarzi:2019abf,Parker:2018vye}& $(-8.9\\pm3.6)\\times 10^{-13}$ \\cite{Hanneke:2008tm} & $\\sim10^{-14}$ \\cite{Gabrielse:2019cgf}\n\\\\\n$a_\\mu - a_\\mu^{\\text{SM}}$ & $\\pm3.8\\times10^{-10}$  \\cite{Keshavarzi:2019abf} & $(28.0\\pm 6.3)\\times10^{-10}$ \\cite{Bennett:2006fi}&$ 1.6\\times10^{-10}$ \\cite{Grange:2015fou}\\\\\n$a_\\tau - a_\\tau^\\text{SM}$ &$\\pm3.9 \\times 10^{-8}$  \\cite{Keshavarzi:2019abf} & $(-2.1\\pm1.7)\\times10^{-2}$ \\cite{Abdallah:2003xd}& \\\\\n\\hline\n$d_e$ & \n $< 10^{-44}~e$ cm \\cite{Pospelov:2013sca,Smith:2017dtz} & $< 1.1 \\times 10^{-29} \\,e\\,\\text{cm}$ \\cite{Andreev:2018ayy} & $ \\sim 10^{-30} \\,e\\,\\text{cm}$ \\cite{Andreev:2018ayy}\\\\\n$d_\\mu$ &$< 10^{-42}~e$ cm \\cite{Smith:2017dtz} & $ < 1.9 \\times 10^{-19} \\,e\\,\\text{cm}$ \\cite{Bennett:2008dy} & $ \\sim 10^{-23} \\,e\\,\\text{cm}$ \\cite{Adelmann:2010zz} \\\\\n$d_\\tau$ &$< 10^{-41}~e$ cm \\cite{Smith:2017dtz} & $ (1.15 \\pm 1.70) \\times 10^{-17} \\,e\\,\\text{cm}$ \\cite{Inami:2002ah} & $ \\sim 10^{-19} \\,e\\,\\text{cm}$ \\cite{Kou:2018nap} \\\\\n\\hline\n$d_n$ & $\\sim 10^{-32}~e$ cm \\cite{Dar:2000tn}& $< 3.6 \\times 10^{-26} \\, e\\,\\text{cm}$~\\cite{Afach:2015sja} & few$\\times 10^{-28}e\\,\\text{cm}$ \\cite{Strategy:2019vxc}\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\caption{Summary of Standard Model theory errors\/bounds (first column), current experimental measurements\/limits (second column) and projected precision of next-generation experiments (third column) of magnetic moment anomalies and electric dipole moments of the charged leptons and the neutron.  For clarity, for the anomalous magnetic moments, the Standard Model central values have been subtracted. We are not aware of any experimental analysis for the projected sensitivity of the tau magnetic moment.}\n\\label{tab:expstatus}\n\\end{table}\n\n\n\\section{Flavor Anomalies and Electric Dipole Moments}\\label{Sec:Pheno}\nIn this section, we study the impact of leptoquarks on (c)EDMs and $B$-physics measurements at the benchmark points presented in Sec. \\ref{sec:LQBdecays}.\n\n\\subsection{Probing the Parameter Space Using Tau Measurements}\\label{sec:flavor-edm-pheno}\n\nGiven the BM1 and BM2 benchmarks for the leptoquark couplings to left-handed taus, $\\lambda^q_{33} \\simeq 0.7$, $\\lambda^q_{23} \\simeq 0.6$, we begin by turning on the coupling to right-handed taus $\\lambda^d_{33}$ while setting the right-handed couplings to muons and electrons ($\\lambda^d_{32}$  and $\\lambda^d_{31}$, respectively) to zero. The coupling $\\lambda^d_{33}$ will induce the dipole moments of the tau as in Eqs.~\\eqref{eq:a_lepton} and~\\eqref{eq:d_lepton}, as well as transition dipole moments leading to the lepton flavor violating decay modes $\\tau \\to \\mu \\gamma$ and $\\tau \\to e \\gamma$. In the limit $m_e, m_\\mu \\ll m_\\tau \\ll m_b$, the partial width for the $U_1$ contribution to $\\tau \\to \\mu \\gamma$ is given by\n\\begin{multline} \\label{eq:taumugamma}\n\\Gamma_{\\tau\\rightarrow\\mu\\gamma} = \\frac{\\alpha m_\\tau^3 N_C^2}{256\\pi^4 M_{U_1}^4} m_b^2 |\\lambda_{32}^q \\lambda_{33}^{d*}|^2 \\Bigg[ \\Big(2 Q_b - Q_{U}\\big((1-\\kappa_Y)\\ln(\\frac{\\Lambda_\\text{UV}^2}{M_{U_1}^2})+\\frac{1-5\\kappa_Y}{2}\\big)\\Big)^2 \\\\ \n+ Q_{U}^2 \\tilde{\\kappa}_Y^2 \\big(\\ln(\\frac{\\Lambda_\\text{UV}^2}{M_{U_1}^2})+\\frac{5}{2}\\big)^2\\Bigg].\n\\end{multline}\nThis expression is in agreement with \\cite{Cornella:2019hct}, when specialized to the vector leptoquark model with $\\kappa_Y=1$ and $\\tilde\\kappa_Y = 0$. \nThe expression for the decay mode $\\tau \\to e \\gamma$ is obtained by an appropriate replacement of the lepton flavor index. \nThe experimental upper limits on the branching ratios of the $\\tau \\to \\mu \\gamma$ and $\\tau \\to e \\gamma$ decays are $5.0 \\times 10^{-8}$ and $5.4 \\times 10^{-8}$, respectively~\\cite{Amhis:2019ckw}.\n\nIn addition to inducing lepton flavor violating tau decays, the $\\lambda^d_{33}$ coupling will modify the new physics contributions to charged current decays based on the $b \\to c \\tau \\nu$ and $b \\to u \\tau \\nu$ transitions and neutral current decays based on $b \\to s \\tau \\tau$. The decay modes that are particularly sensitive to right-handed currents are the helicity suppressed two body decays $B_c \\to \\tau \\nu$~\\cite{Li:2016vvp,Alonso:2016oyd}, $B^\\pm \\to \\tau \\nu$, and $B_s \\to \\tau^+ \\tau^-$. We find\n\\begin{eqnarray} \\label{eq:Bctaunu}\n\\frac{\\text{BR}(B_c \\to \\tau \\nu)}{\\text{BR}(B_c \\to \\tau \\nu)_\\text{SM}} &=& \\left| 1 - \\frac{\\sum_{j}V_{cj}\\lambda^q_{j3}}{V_{cb}} \\frac{v^2}{M_{U_1}^2} \\left(\\frac{\\lambda^{q\\ast}_{33}}{2}+ \\frac{\\lambda^{d\\ast}_{33} m_{B_c}^2}{m_\\tau(m_b+m_c)} \\right) \\right|^2 \\,, \\\\ \\label{eq:Btaunu}\n\\frac{\\text{BR}(B^\\pm \\to \\tau \\nu)}{\\text{BR}(B^\\pm \\to \\tau \\nu)_\\text{SM}} &=& \\left| 1 - \\frac{\\sum_j V_{uj} \\lambda^q_{j3} }{V_{ub}} \\frac{v^2}{M_{U_1}^2} \\left(\\frac{\\lambda^{q*}_{33}}{2}+ \\frac{\\lambda^{d*}_{33} m_{B^\\pm}^2}{m_\\tau m_b} \\right) \\right|^2 \\,,\n\\end{eqnarray}\nUsing the expression for the branching ratio in terms of the Wilson coefficients from~\\cite{Altmannshofer:2017wqy}, we find\n\\begin{eqnarray} \\label{eq:Bstautau}\n\\frac{\\text{BR}(B_s \\to \\tau^+ \\tau^-)}{\\text{BR}(B_s \\to \\tau^+ \\tau^-)_\\text{SM}} &=& \\left| 1 + \\frac{4\\pi^2}{e^2 C_{10}^\\text{SM}} \\frac{v^2}{M_{U_1}^2} \\left(\\frac{\\lambda_{33}^{q*} \\lambda_{23}^{q} + \\lambda_{33}^{d*} \\lambda_{23}^d}{V_{ts}^* V_{tb}} - \\frac{m_{B_s}^2}{m_\\tau m_b} \\frac{\\lambda_{33}^{q*} \\lambda_{23}^d + \\lambda_{33}^{d*} \\lambda_{23}^q}{V_{ts}^* V_{tb}}\\right) \\right|^2  \\nonumber \\\\\n&&  + \\frac{16\\pi^4}{e^4 (C_{10}^\\text{SM})^2} \\frac{v^4}{M_{U_1}^4} \\frac{m_{B_s}^4}{m_\\tau^2 m_b^2} \\left|\\frac{\\lambda_{33}^{q*} \\lambda_{23}^d - \\lambda_{33}^{d*} \\lambda_{23}^q}{V_{ts}^* V_{tb}} \\right|^2 \\left( 1 - \\frac{4 m_\\tau^2}{m_{B_s}^2} \\right) \\,,\n\\end{eqnarray}\nwhere we neglected the finite life time difference in the $B_s$ system. We use a normalization such that the SM value for the Wilson coefficient is $C_{10}^\\text{SM} \\simeq -4.1$~\\cite{Altmannshofer:2008dz}. Renormalization group running from the leptoquark scale down to the $b$-scale can be incorporated by evaluating the quark masses in Eqs.~\\eqref{eq:Bctaunu}-\\eqref{eq:Bstautau} at the scale $\\mu \\simeq 2$\\,TeV. Note that the terms containing both left-handed and right-handed couplings enjoy a mild chiral enhancement by factors $m_{B_c}^2\/(m_\\tau (m_b+m_c))$, $m_{B^\\pm}^2\/(m_\\tau m_b)$, and $m_{B_s}^2\/(m_\\tau m_b)$, respectively. \n\nThe measured BR$(B^\\pm \\to \\tau \\nu) = (1.09 \\pm 0.24)\\times 10^{-4}$~\\cite{Tanabashi:2018oca} agrees well with the SM prediction BR$(B^\\pm \\to \\tau \\nu)_\\text{SM} = (0.838^{+0.039} _{-0.029})\\times 10^{-4}$~\\cite{Charles:2004jd}, yielding\n\\begin{equation}\n\\frac{\\text{BR}(B^\\pm \\to \\tau \\nu)}{\\text{BR}(B^\\pm \\to \\tau \\nu)_\\text{SM}} = 1.30 \\pm 0.29 \\,.\n\\end{equation}\n\nSo far no direct measurement of the $B_c \\to \\tau \\nu$ branching ratio has been performed. We impose the bound BR$(B_c \\to \\tau \\nu) < 30\\%$~\\cite{Alonso:2016oyd}.\nThe SM branching ratio is\n\\begin{equation}\n\\text{BR}(B_c \\to \\tau \\nu)_\\text{SM} = \\tau_{B_c} m_{B_c}  \\frac{f_{B_c}^2 G_F^2}{8\\pi} |V_{cb}|^2 m_\\tau^2 \\left(1 - \\frac{m_\\tau^2}{m_{B_c}^2} \\right)^2= (2.21 \\pm 0.09) \\times 10^{-2}\\,,\n\\end{equation}\nwith the lifetime of the $B_c$ meson $\\tau_{B_c}=(0.507 \\pm 0.009)\\times 10^{-12}$\\,s~\\cite{Tanabashi:2018oca}, the $B_c$ decay constant $f_{B_c} = (0.427\\pm0.006)$\\,GeV~\\cite{McNeile:2012qf} and we used $|V_{cb}| = (41.6 \\pm 0.56 )\\times 10^{-3}$~\\cite{Charles:2004jd}.\n\nSimilarly, the $B_s \\to \\tau^+ \\tau^-$ decay has not been observed so far. The first direct limit on the branching ratio was placed by LHCb~\\cite{Aaij:2017xqt} and is $\\text{BR}(B_s \\to \\tau^+ \\tau^-) < 6.8 \\times 10^{-3}$, while the SM branching ratio is  $\\text{BR}(B_s \\to \\tau^+ \\tau^-)_\\text{SM} = (7.73\\pm 0.49)\\times 10^{-7}$ \\cite{Bobeth:2013uxa}. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{tauLQ1.pdf}\n\\caption{Constraints on the $U_1$ leptoquark parameter space in the plane of the complex coupling $\\lambda^d_{33}$ divided by the leptoquark mass, $M_{U_1}$, and all other parameters fixed as in BM1 (\\ref{eq:BM1}) or BM2 (\\ref{eq:BM2}). The gray region represents parameter space that is excluded by  $B_s \\to \\tau^+ \\tau^-$. The red hatched region is excluded by the bound on the tau lepton anomalous magnetic moment. The dashed blue line is the projected sensitivity of future experiments to the tau EDM. The region above the solid purple line is excluded by bounds on the neutron EDM, and the dashed purple line is the projected sensitivity of future neutron EDM experiments. The surrounding purple bands reflect the theoretical uncertainty in the nucleon matrix element $\\beta^{\\tilde{G}}_n$. Note that the observables shown in the figure are independent of $\\lambda^q_{32},\\lambda^q_{22}$ and $\\lambda^q_{31},\\lambda^q_{21}$, and the change from benchmark BM1 to BM2 has no effect on the exclusion curves.}\n\\label{fig:tau}\n\\end{figure}\n\nIn Fig.~\\ref{fig:tau}, we show current and projected constraints on the $U_1$ leptoquark in the plane of the complex $\\lambda^d_{33}$ coupling divided by the leptoquark mass for BM1 and BM2 benchmark points. The figure represents both BM1 and BM2, since the shown constraints are independent of the muon couplings $\\lambda^q_{32},\\lambda^q_{22} $ and electron couplings $\\lambda^q_{31},\\lambda^q_{21}$ and changing from BM1 to BM2 does not affect our results.\nThe most stringent constraint comes from $B_s \\to \\tau^+ \\tau^-$ and is shown in gray in the figure. Constraints from $B^\\pm \\to \\tau \\nu$, $B_c \\to \\tau \\nu$, and lepton flavor violating tau decays ($\\tau \\to \\mu \\gamma$ for benchmark BM1 and $\\tau \\to e \\gamma$ for BM2) are slightly weaker and exclude values of $\\lambda^d_{33}$ that are a factor of a few larger than those excluded by $B_s \\to \\tau^+ \\tau^-$. (In Fig.~\\ref{fig:tau} we show only the strongest constraint coming from $B_s \\to \\tau^+ \\tau^-$.) Once the bounds are imposed, the allowed values of the right-handed coupling $\\lambda^d_{33}$ are sufficiently small such that they do not affect $R_{D^{(*)}},~R_{K^{(*)}}$ in a significant way. Therefore, in all the allowed region in Fig.~\\ref{fig:tau}, the anomalies are satisfied.\n\nFrom the figure, we observe that the current experimental bounds on $d_\\tau$ and $a_\\tau$ do not constraint the parameter space in a relevant way. The constraint from $a_\\tau$ is depicted by the red hatched region in Fig.~\\ref{fig:tau}, while the experimental bound on $d_\\tau$ constrains values of $\\Im(\\lambda^d_{33})\/M_{U_1}$ that are $\\mathcal{O}(10^5)$~TeV$^{-1}$, and, therefore, beyond the range of the plot. Projected sensitivities of next-generation experiments to the tau EDM \\cite{Kou:2018nap} (shown by the dashed blue line) are still far from being able to probe the viable new physics parameter space. \n\nIn addition to the tau electric and anomalous magnetic dipole moments, the $U_1$ leptoquark coupling, $\\lambda^d_{33}$, will contribute to the neutron EDM, $d_n$. The constraint from the current bound on the neutron EDM is shown by the solid purple line in Fig.~\\ref{fig:tau}, where the region above this line is excluded due to the leptoquark generating a contribution to the neutron EDM that is too large. The surrounding purple bands reflect the theoretical uncertainty in the nucleon matrix element $\\beta^{\\tilde{G}}_n$. We observe that the currend bound on the neutron EDM leads to a constraint that is weaker than $B_s \\to \\tau^+ \\tau^-$ and is not yet probing the allowed parameter space. On the other hand, the projected sensitivity of future neutron EDM experiments \\cite{Strategy:2019vxc} (shown by the dashed purple line) will begin probing the new physics parameter space and can lead to stronger constraints on the amount of CP violation present in the right-handed couplings of $U_1$ to tau leptons. \n\n\n\\subsection{Probing the Parameter Space Using Muon Measurements} \\label{sec:muon}\n\nNext we focus on the BM1 and BM3 benchmarks, and investigate the impact of the leptoquark couplings to right-handed muons, $\\lambda^d_{32}$, while setting the right-handed tau and electron couplings ($\\lambda^d_{33}$ and $\\lambda^d_{31}$, respectively) to zero.\nThe coupling $\\lambda^d_{32}$ will lead to a shift in the anomalous magnetic moment of the muon, $\\Delta a_\\mu$, in the muon EDM, $d_\\mu$, and in the EDM of the bottom quark given in Eqs.~\\eqref{eq:a_lepton},~\\eqref{eq:d_lepton}, and~\\eqref{eq:strangequark-edm}, as well as the lepton flavor violating decay mode $\\tau\\to\\mu\\gamma$ given in  Eq.~\\eqref{eq:taumugamma} with $|\\lambda_{32}^q \\lambda_{33}^{d*}|^2\\rightarrow|\\lambda_{32}^d \\lambda_{33}^{q*}|^2$.  In the presence of the coupling  $\\lambda^d_{32}$, the muon dipole moment enjoys a sizable chiral enhancement by $m_b\/m_\\mu$. \n\nIn addition, the coupling $\\lambda^d_{32}$ can also give sizable non-standard effects in the $B_s \\to \\mu^+\\mu^-$ decay. The corresponding expression is analogous to the one for the $B_s \\to \\tau^+\\tau^-$ decay given in Eq.~(\\ref{eq:Bstautau})\n\\begin{eqnarray} \\label{eq:Bsmumu}\n\\frac{\\text{BR}(B_s \\to \\mu^+ \\mu^-)}{\\text{BR}(B_s \\to \\mu^+ \\mu^-)_\\text{SM}} &=& \\left| 1 + \\frac{4\\pi^2}{e^2 C_{10}^\\text{SM}} \\frac{v^2}{M_{U_1}^2} \\left(\\frac{\\lambda_{32}^{q*} \\lambda_{22}^{q} + \\lambda_{32}^{d*} \\lambda_{22}^d}{V_{ts}^* V_{tb}} - \\frac{m_{B_s}^2}{m_\\mu m_b} \\frac{\\lambda_{32}^{q*} \\lambda_{22}^d + \\lambda_{32}^{d*} \\lambda_{22}^q}{V_{ts}^* V_{tb}}\\right) \\right|^2  \\nonumber \\\\\n&&  + \\frac{16\\pi^4}{e^4 (C_{10}^\\text{SM})^2} \\frac{v^4}{M_{U_1}^4} \\frac{m_{B_s}^4}{m_\\mu^2 m_b^2} \\left|\\frac{\\lambda_{32}^{q*} \\lambda_{22}^d - \\lambda_{32}^{d*} \\lambda_{22}^q}{V_{ts}^* V_{tb}} \\right|^2 \\,.\n\\end{eqnarray}\nThe terms that contain both left-handed and right-handed couplings are chirally enchanced by a factor $m_{B_s}^2\/(m_\\mu m_b)$. \n\nThe branching ratio BR$(B_s \\to \\mu^+ \\mu^-)$ has been measured at LHCb, CMS and ATLAS~\\cite{Chatrchyan:2013bka,CMS:2014xfa,Aaij:2017vad,Aaboud:2018mst}. We use the average of these results from~\\cite{Aebischer:2019mlg}, that, combined with the SM prediction~\\cite{Bobeth:2013uxa,Beneke:2017vpq}, reads\n\\begin{equation}\n\\frac{\\text{BR}(B_s \\to \\mu^+ \\mu^-)}{\\text{BR}(B_s \\to \\mu^+ \\mu^-)_\\text{SM}} = 0.73 ^{+0.13} _{-0.10}\\,,\n\\end{equation}\nwhich is in slight tension ($\\sim 2\\sigma$) with the SM prediction. Interestingly enough, in the region of parameter space where the couplings to left-handed muons $\\lambda^q_{22}$, $\\lambda^q_{32}$ provide an explanation of $R_{K^{(*)}}$, the tension in $B_s \\to \\mu^+ \\mu^-$ is largely lifted.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{muonLQ1.pdf} ~~~~\n\\includegraphics[width=0.45\\textwidth]{muonLQ2.pdf}\n\\caption{Constraints on the $U_1$ leptoquark parameter space in the plane of the complex $\\lambda^d_{32}$ coupling divided by the leptoquark mass for the benchmark points BM1 (left panel) and BM3 (right panel). The gray region is excluded by $B_s \\to \\mu^+ \\mu^-$ at the 95\\% C.L.. The dashed blue line is the projected sensitivity of future experiments to the muon EDM. The red shaded region corresponds to the parameter space the can address the anomaly in the anomalous magnetic moment of the muon. The solid (dashed) purple lines represent the current constraint (projected sensitivity) from the neutron EDM, with the purple bands reflecting the uncertainty in the nucleon matrix element $\\beta^{\\tilde{G}}_n$.}\n\\label{fig:muon}\n\\end{figure}\n\nIn Fig.~\\ref{fig:muon} we show the current and projected constraints on the $U_1$ leptoquark for BM1 (left) and BM3 (right) in the plane of the complex coupling $\\lambda^d_{32}$ divided by the leptoquark mass. For both benchmarks, the most stringent constraint arises from $B_s \\to \\mu^+\\mu^-$. The region that is excluded at the $95\\%$ C.L. is shaded in gray. Once the constraints from $B_s \\to \\mu^+ \\mu^-$ are imposed, the allowed values of $\\lambda^d_{32}$ are sufficiently small that they do not affect $R_{K^{(\\ast)}}$ in a significant way. The region that is shaded in red is the region of parameter space that is able to address the anomaly in the anomalous magnetic moment of the muon, while the blue dashed lines are the projected sensitivities of future experiments to the muon EDM. Similar to Fig.~\\ref{fig:tau}, the solid (dashed) purple line is the current constraint (projected sensitivity) of the neutron EDM. The current bound on the muon EDM, $d_\\mu$, is very weak and constrains values of $\\Im (\\lambda^d_{32})\/M_{U_1}$ outside from the range of the plot ($\\Im (\\lambda^d_{32})\/M_{U_1}\\sim \\mathcal{O}(10^3)$~TeV$^{-1}$ for BM1 and $\\Im(\\lambda^d_{32})\/M_{U_1}\\sim \\mathcal{O}(10^2)$~TeV$^{-1}$ for BM3). \n\nIn the left plot of Fig.~\\ref{fig:muon} we observe that, once the constraints from $B_s \\to \\mu^+ \\mu^-$ is imposed, the BM1 benchmark cannot address the $a_\\mu$ anomaly. We conclude that the $U_1$ leptoquark can not explain the $B$ anomalies and the $(g-2)_\\mu$ anomaly simultaneously with the parameters fixed to those of BM1. This is mainly due to limits on lepton flavor violating decays $\\tau \\to \\phi \\mu$ and $B \\to K \\tau \\mu$ that impose stringent constraints on the size of the left-handed muonic couplings $\\lambda^q_{32}$ and $\\lambda^q_{22}$ (see discussion in Sec.~\\ref{sec:LQBdecays}). \n\nIn order to avoid these constraints, we can instead set the $U_1$ couplings to left-handed tau leptons, $\\lambda^q_{33}$ and $\\lambda^q_{23}$, to zero as in BM3 in (\\ref{eq:BM3}). The decay rates $\\tau \\to \\phi \\mu$, $B \\to K \\tau \\mu$, and $\\tau \\to \\mu \\gamma$ mediated by $U_1$ then go to zero, allowing the muonic couplings $\\lambda^q_{32}$ and $\\lambda^q_{22}$ to have larger values.\nHowever, by switching off $\\lambda^q_{33}$ and $\\lambda^q_{23}$ we forgo an explanation of $R_{D^{(\\ast)}}$. \n\nIn the right plot of Fig.~\\ref{fig:muon} we show that, for BM3, the region of parameter space that can address the $a_\\mu$ anomaly (the red shaded region) overlaps with the region of parameter space that is allowed by $B_s \\to \\mu^+ \\mu^-$, and the $U_1$ leptoquark can therefore address both the $(g-2)_\\mu$ anomaly and (at least partially, cf. discussion in Sec.~\\ref{sec:LQBdecays}) the $R_{K^{(\\ast)}}$ anomalies.\nFinally, we notice that, for this benchmark, projected sensitivities to the neutron EDM might start to probe the viable parameter space.\n\nWe also explored the region of parameter space with nonzero $\\lambda_{22}^d$ instead of $\\lambda_{32}^d$.  In this case, for BM1 and BM3, the neutron EDM is dominated by the strange quark contribution (\\ref{eq:s-neturon-edm}), so its projected sensitivity covers larger region of parameter space.  However in this case, we did not find any viable region of parameter space explaining the anomaly in $a_\\mu$.\n\n\\subsection{Probing the parameter space using electron measurements} \\label{sec:electron}\n\nInstead of muon specific couplings that address the discrepancies in the LFU ratios $R_{K^{(*)}}$ by new physics that suppresses the $b\\to s\\mu\\mu$ transitions, one can also entertain the possibility that new physics addresses the anomaly by enhancing the $b\\to see$ transitions. \nThis can be achieved with the leptoquark couplings $\\lambda^d_{31}$, $\\lambda^d_{21}$ as given in Eq.~\\eqref{eq:C9e} and by our benchmark points BM2 and BM4. \n\nThese couplings will also lead to shifts in the anomalous magnetic moment of the electron, $\\Delta a_e$, and, in the presence of CP violation, induce an electron EDM, $d_e$, (see  Eqs.~\\eqref{eq:a_lepton} and ~\\eqref{eq:d_lepton}, respectively), and the lepton flavor violating mode $\\tau\\to e\\gamma$ (see Eq.~\\eqref{eq:taumugamma} with $|\\lambda_{32}^q \\lambda_{33}^{d*}|^2\\rightarrow|\\lambda_{31}^d \\lambda_{33}^{q*}|^2$).  Note that the chiral enhancement of the dipole moments $m_b\/m_e$ can be particularly pronounced in the case of the electron. \n\nIn this scenario, potentially important constraints arise from the $B_s \\to e^+ e^-$ decay.\nThe effect of the leptoquark is given by an expression analogous to Eq.~\\eqref{eq:Bsmumu} with $m_\\mu \\to m_e$ and $\\lambda^f_{32}, \\lambda^f_{22} \\to \\lambda^f_{31}, \\lambda^f_{21}$, with the SM prediction given by BR$(B_s \\to e^+e^-) = (8.54 \\pm 0.55) \\times 10^{-14}$~\\cite{Bobeth:2013uxa}. \nExperimentally, the $B_s \\to e^+e^-$ branching ratio is bounded at the 90\\% C.L. by~\\cite{Aaltonen:2009vr}\n\\begin{equation} \\label{eq:Bsee_exp}\n    \\text{BR}(B_s \\to e^+e^-) < 2.8 \\times 10^{-7} ~.\n\\end{equation}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{electronLQ1.pdf} ~~~~\n\\includegraphics[width=0.45\\textwidth]{electronLQ2.pdf}\n\\caption{Constraints on the $U_1$ leptoquark parameter space in the plane of the complex coupling $\\lambda^d_{31}$ divided by the leptoquark mass for the benchmark points BM2 and BM4, left and right panel, respectively). The gray region is excluded by $B_s \\to e^+e^-$ at the 95\\% C.L.. The red shaded region corresponds to the parameter space the can address the anomaly in the anomalous magnetic moment of the electron. The solid (dashed) blue lines represent the current constraint (projected sensitivity) from the electron EDM. In the right panel, the dashed purple line represents the projected sensitivity from the neutron EDM, with the purple band reflecting the uncertainty in the nucleon matrix element $\\beta^{\\tilde{G}}_n$.}\n\\label{fig:electron}\n\\end{figure*}\n\nThe plots in Fig.~\\ref{fig:electron} show the current and projected constraints on the $U_1$ leptoquark in the plane of the complex coupling $\\lambda^d_{31}$ divided by the leptoquark mass for BM2 (left) and BM4 (right).  In both panels the gray region is excluded by the bound from $B_s \\to e^+ e^-$, while the red shaded region is the region of parameter space that can address the $2.4 \\sigma$ anomaly in the electron magnetic moment, $a_e$. The blue solid (dashed) lines are the current constraint (projected sensitivity) of the electron electric dipole moment, $d_e$. In the right panel, the dashed purple line and the surrounding purple band is the projected sensitivity of the neutron EDM, $d_n$.\n\nFor BM2 (left plot of Fig.~\\ref{fig:electron}) we observe that the region of parameter space that is able to address the anomaly in $a_e$ is excluded by constraints from $B_s \\to e^+ e^- $ and a simultaneous explanation of all the $B$ anomalies and $a_e$ is not possible. This is due to stringent constraints on the size of $\\lambda^q_{31}$ from the lepton flavor violating decays $\\tau \\to \\phi e$ and $B \\to K \\tau e$ (see discussion in Sec.~\\ref{sec:LQBdecays}).  Constrains from the $\\tau\\to e\\gamma$ are slightly weaker.\n\nTo avoid the stringent constraints from lepton flavor violating decays, we can set all the $U_1$ couplings to tau leptons to zero. Then, the $\\tau \\to \\phi e$ and $B \\to K \\tau e$ rates as well as the $\\tau \\to e \\gamma$ rate go to zero, and the left-handed couplings to electrons can be larger. However, by setting $\\lambda^q_{33}$ and $\\lambda^q_{23}$ to zero, we forgo an explanation of $R_{D^{(\\ast)}}$. This scenario is given by BM4, and the resulting constraints are shown in the right plot of Fig.~\\ref{fig:electron}. We observe that the smaller value of $\\lambda^q_{21} = 0.005$ in BM4 leads to weaker constraints on $\\lambda^d_{31}$ from $B_s \\to e^+ e^-$. In addition, the larger value of $\\lambda^q_{31} = 0.5$ generates a larger contribution to the electron magnetic moment necessary to explain the slight tension in $a_e$. In moving from BM2 to BM4 the bound from $B_s \\to e^+e^-$ opens up a wide region in parameter space favorable for the electron magnetic moment, $a_e$. We conclude that BM4 can address the anomalies in both $R_{K^{(\\ast)}}$ and $a_e$.\n\nWe also investigated the region of parameter space with nonzero $\\lambda_{21}^d$ instead of $\\lambda_{31}^d$.  We find in BM2 and BM4 that sensitivity to $d_e$ is reduced because it is chirally enhanced by $m_s$ rather than $m_b$ in Eq. (\\ref{eq:d_lepton}).  We also find no region of parameter space where the $U_1$ leptoquark explains the tension of the measured $a_e$ with theory.\n\n\\section{LHC Bounds on the Leptoquark}\\label{Sec:LHC}\n\n{\\renewcommand{\\arraystretch}{1.1} \n\\begin{table*}[tbp] \n\\begin{ruledtabular}\n\\begin{tabular}{c c c }\n\\toprule\n    \\multicolumn{3}{c}{\\textbf{LHC Bounds on Scalar Leptoquarks}} \\\\ \\hline\n   Channel & Experiment & Limit \\\\\\hline\n       \\multicolumn{3}{c}{\\textit{First Generation Leptoquarks}} \\\\ \\hline\n   \\multirow{2}{*}{$eejj~(\\beta =1 ) $} & ATLAS \\cite{Aaboud:2019jcc} &  1400 GeV\\\\[-2.mm]\n   & CMS \\cite{Sirunyan:2018btu} & 1435 GeV \\\\   \n  \\multirow{2}{*}{$e\\nu jj~(\\beta =0.5)$}    &  ATLAS \\cite{Aaboud:2019jcc} &  1290 GeV\\\\[-2.mm]\n  &   CMS \\cite{Sirunyan:2018btu} & 1270 GeV   \\\\\n   \\hline\n      \\multicolumn{3}{c}{\\textit{Second Generation Leptoquarks}} \\\\ \\hline\n   \\multirow{2}{*}{$\\mu\\mu jj ~(\\beta = 1)$}&  ATLAS \\cite{Aaboud:2019jcc}&1560 GeV\\\\[-2.0mm]\n   & CMS \\cite{Sirunyan:2018ryt} & 1530 GeV\\\\\n  \\multirow{2}{*}{$\\mu\\nu jj~(\\beta = 0.5)$}  & ATLAS \\cite{Aaboud:2019jcc}&1230 GeV \\\\[-2.0mm]\n  &  CMS \\cite{Sirunyan:2018ryt} & 1285 GeV \\\\\n  \\hline\n   \\multicolumn{3}{c}{\\textit{Third Generation Leptoquarks}} \\\\ \\hline\n     \\multirow{2}{*}{$b\\tau b\\tau$}  & ATLAS \\cite{Aaboud:2019bye}& 1030 GeV \\\\\n     &  CMS \\cite{Sirunyan:2018vhk}& 1020 GeV\\\\  \\hline\n       \\multicolumn{3}{c}{\\textit{Reinterpreted SUSY searches}} \\\\ \\hline\n    $q\\nu q\\nu$&  CMS \\cite{Sirunyan:2018kzh} &  980 GeV \\\\\n     \\multirow{2}{*}{$t\\nu t\\nu$} &  ATLAS \\cite{Aaboud:2019bye} & 1000 GeV \\\\[-1.5mm]\n     & CMS \\cite{Sirunyan:2018kzh} &  1020 GeV  \\\\\n     \\hline \\hline\n     \\multicolumn{3}{c}{\\textbf{LHC Bounds on Vector Leptoquarks}} \\\\ \\hline\nChannel & Experiment & Limit \\\\\\hline\n\\multicolumn{3}{c}{\\textit{Reinterpreted SUSY searches}} \\\\ \\hline\n\\multirow{2}{*}{$q\\nu q\\nu$}&  \\multirow{2}{*}{CMS \\cite{Sirunyan:2018kzh}} & 1410 GeV~($\\kappa_s = 0$)\\\\[-2.0mm]\n    && 1790 GeV~($\\kappa_s = 1$)\\\\[2.5mm]\n     \\multirow{2}{*}{$t\\nu t\\nu$} \n     &  \\multirow{2}{*}{CMS \\cite{Sirunyan:2018kzh}} & 1460 GeV~($\\kappa_s = 0$)\\\\[-2.0mm]\n    && 1780 GeV~($\\kappa_s = 1$)\n\\end{tabular}\n\\end{ruledtabular}\n\\caption{LHC bounds on pair-production of scalar and vector leptoquarks. For scalar leptoquarks, the first three sections correspond to bounds from dedicated leptoquark searches, while the last section corresponds to bounds derived from the reinterpretation of squark pair production searches. For vector leptoquarks, only reintepreted SUSY searches exist. The parameter $\\beta$ denotes the branching ratio of the leptoquark to a quark and a charged lepton. We do not report the bounds on the decays of the LQ to down-type quarks and a neutrino since these decays do not exist in our model.}\n\\label{tab:LHCbounds}\n \\end{table*}}\n \nLow-energy flavor observables like those discussed in the previous sections provide an indirect probe of the $U_1$ leptoquark. A complementary approach to probe the existence of $U_1$ is direct production at high energy colliders and looking for signatures of their decay products. The goal of this section is to compute the lower bound on the leptoquark mass in the allowed regions of parameter space in Figs. \\ref{fig:tau} -\\ref{fig:electron}.\n\nThe two main production mechanisms are single production in association with a lepton ($gq \\rightarrow \\ell~U_1$), and pair production ($ g g, q \\bar q \\rightarrow U_1~\\overline{U_1}$). For a recent review see \\cite{Diaz:2017lit}. \nOnce produced, the leptoquark will decay into a pair of SM fermions. The interactions of the $U_1$ leptoquark with SM quarks and leptons in Eq.~(\\ref{eq:yuk}) generate the decays of $U_1$ into an up-type quark and a neutrino, or a down-type quark and a charged lepton. In the limit where $M_{U_1}$ is much larger than the masses of the decay products, the partial widths of $U_1$ are given by\n\\begin{subequations}\\label{eq:decays}\n\\begin{align}\n\\Gamma(U_1 \\rightarrow u_i \\nu_j) &= \\frac{M_{U_1}}{24 \\pi} \\bigg| \\sum_{k = 1,2,3} V_{ik} \\lambda^q_{kj} \\bigg|^2\\,, \\\\\n\\Gamma(U_1 \\rightarrow d_i \\ell_j) &= \\frac{M_{U_1}}{24 \\pi} \\Big(  \\big|\\lambda^q_{ij}\\big|^2 +  \\big|\\lambda^d_{ij}\\big|^2 \\Big)\\,,\n\\end{align}\n\\end{subequations}\nwhere $i,j = 1,2,3$ label the three generations.\n\nSeveral dedicated searches for singly and pair produced scalar leptoquarks have been performed by the LHC, and are classified according to whether the leptoquark decays to first, second, or third generation fermions. The strongest bounds on leptoquark pair-production from ATLAS and CMS have been compiled in Tab. \\ref{tab:LHCbounds}, where the searches are organized according to whether the branching ratio into a quark and a charged lepton (denoted by $\\beta$) is 100\\% or 50\\%, with the remaining 50\\% to a quark and a neutrino. In addition, in the table we also report the CMS reinterpretation of the squark pair production searches to place constraints on pair produced vector leptoquarks decaying to a quark and a neutrino, $t\\nu$, or $q\\nu~(q  = u,c,d,s)$  \\cite{Sirunyan:2018kzh}. Similarly, ATLAS have presented reinterpretations of squark searches  \\cite{Aaboud:2019bye}, although they only consider the decay of a leptoquark into 3rd generation quarks. We note that the ATLAS and CMS searches also consider leptoquark decays into down-type quarks and a neutrino (e.g $b \\nu b \\nu$ final states), but the corresponding couplings do not exist in our model and, therefore, we do not consider them here.\n\nSingly produced scalar leptoquarks have been searched in $ej$, $\\mu j$, and $b \\tau$ final states.  The bounds on the leptoquark mass from single production depends on the coupling of the leptoquark to quarks and leptons. For unit couplings, 8 TeV searches for single production of first and second generation scalar leptoquarks constrain the leptoquark mass to be above $\\sim 1700$ GeV and $\\sim 700$ GeV, respectively \\cite{Khachatryan:2015qda}, while the 13 TeV search for third generation scalar leptoquarks constrains the mass to be above 740 GeV \\cite{Sirunyan:2018jdk}. In our benchmark models, the leptoquarks are mainly coupled to bottom or strange quarks. For this reason, the searches for singly produced\nleptoquarks are less sensitive to our benchmark models than the searches for pair produced leptoquarks. In the following, we will discuss in some details the bounds from searches of pair produced leptoquarks in all benchmarks.\n\nFor BM1 and BM2, the dominant non-zero couplings of $U_1$ are couplings involving tau leptons ($\\lambda_{33}^q,\\lambda_{23}^q$) and the dominant decay modes are $U_1 \\to b \\tau, s \\tau, t \\nu_\\tau, c \\nu_\\tau$. At small values of $\\lambda^d_{33}$ (see Fig. \\ref{fig:tau}), the branching ratios of the $b\\tau$ and $\\tau \\nu_\\tau$ decay modes are similar in value ($\\sim 0.25$) and dominate over the $s\\tau$ and $c \\nu_\\tau$ decays modes, which themselves have similar branching ratios ($\\sim 0.18$). For values of $\\lambda^d_{33}$ near the border of the region allowed by $B_s \\to \\tau^+ \\tau^-$ (see Fig. \\ref{fig:tau}), the decay into $b\\tau$ becomes the dominant decay mode with BR$(U_1\\to b\\tau)\\sim 0.4$.\n\nThe reinterpreted SUSY search for pair production of vector leptoquarks decaying to $t \\nu$ \\cite{Sirunyan:2018kzh} and the CMS search for leptoquarks decaying to $b \\tau$ \\cite{Sirunyan:2018vhk} are the most sensitive searches. We find that these searches yield a similar lower bound on the mass of $U_1$ at around 1.2 TeV in the region of parameter space with small $\\lambda^d_{33}$. The exact bound varies by at most $\\sim$100 GeV in the region allowed by $B_s\\to\\tau^+ \\tau^-$.\n\nIn BM3, $U_1$ couples dominantly to 2nd generation leptons and the main decay modes are $U_1 \\to b\\mu, s\\mu, t \\nu_\\mu, c \\nu_\\mu$, with the $b \\mu$ and $t \\nu_\\mu$ decays modes being the dominant ones since $\\lambda^q_{32} \\gg \\lambda^q_{22}$, BR$(U_1\\to t\\nu_\\mu)\\sim{\\rm{BR}}(U_1\\to b\\mu)\\sim 0.5$. The most stringent LHC constraint on this benchmark comes from the search for pair produced leptoquarks in final states with two muons and two jets in  \\cite{Sirunyan:2018ryt}\\footnote{The search does not require any anti-$b$ tagging, and, therefore, we can simply apply it to our benchmark.}. This search leads to the bound $m_{U_1}\\gtrsim 1.9$ TeV. This bound is valid in the entire parameter space shown in the right panel Fig. \\ref{fig:muon}, since $\\lambda^d_{32}$ is constrained to be very small, and therefore does not affect the leptoquark branching ratios.\n\nFinally, in BM4, $U_1$ couples dominantly to 1st generation leptons and the main decay modes are $U_1 \\to b e$ and $U_1 \\to t \\nu_e$.  In particular, at small values of $\\lambda^d_{31}$ (see Fig. \\ref{fig:electron}), the branching ratios of these decay modes are very similar in value ($\\sim 0.5$). At larger values of $\\lambda^d_{31}$, the branching ratio into $be$ becomes the dominant one, with BR$(U_1\\to be)\\sim 0.7$ at the border of the allowed region for $\\lambda^d_{31}$, as shown in the right plot of Fig.~\\ref{fig:electron}. The search for pair produced leptoquarks decaying in an electron and a jet in \\cite{Sirunyan:2018btu} provides the strongest constraint on the mass of $U_1$ and gives a lower bound of $\\sim 1.8$ TeV at small values of $\\lambda^d_{31}$. The exact bound varies by at most $\\sim$100 GeV in the region allowed by $B_s\\to e^+ e^-$. \n\n\n\n\n\n\n\n\\section{Conclusions}\\label{Sec:Conclusions}\nIn this study, we focused on the possible, and quite likely, existence of new sources of CP violation if the flavor anomalies in $b \\to c$ and $b \\to s$ decays are due to new physics,  specifically in the case where the new physics consists of a $U_1$ vector leptoquark. The underpinning of our study is that the $U_1$ vector leptoquark is one of the only (if not {\\em the} only) new physics scenarios known to us that can provide a simultaneous explanation of the anomalies observed in lepton flavor universality ratios in $b \\to c \\ell \\nu$ and $b \\to s \\ell \\ell$ decays, $R_{D^{(*)}}$ and $R_{K^{(*)}}$.\nSince the couplings of the $U_1$ to quarks and leptons are generically CP violating, they are expected just as generically to produce potentially observable electric dipole moments (EDMs) in leptonic and hadronic systems. Here, we have first provided new, original, and complete formulae for the calculation of the relevant EDMs, and carried out a phenomenological study of a few benchmark cases of how EDMs can constrain the $U_1$ leptoquark interpretation of the anomalies.\n\nWe note that the expressions we provided are the most general expressions for dipole moments induced by vector leptoquarks at one loop level, accounting for the most generic set of leptoquark couplings, which can accomodate scenarios for which the leptoquark may be composite.  \n\nWe explored the parameter space of the $U_1$ leptoquark in the vicinity of 4 benchmark points that explain the $R_{D^{(*)}}$ and $R_{K^{(*)}}$ anomalies (or a subset of them). \nWe identified viable regions of parameter space where the existing discrepancies in the anomalous magnetic dipole moments of the electron $a_e$ and the muon $a_\\mu$ can be explained in addition to $R_K^{(*)}$.\nHowever, we concluded that a simultaneous explanation of all three classes of discrepancies ($R_{D^{(*)}}$, $R_{K^{(*)}}$, $a_{e,\\mu}$) is not possible.\n\nWe found that, in the presence of non-zero CP-violating phases in the leptoquark couplings, EDMs play an important role in probing the parameter space of the model.  Existing bounds on the electron EDM already exclude large parts of parameter space with CP violating leptoquark couplings to electrons.  The expected sensitivities to the neutron EDM can probe into motivated parameter space and probe imaginary parts of leptoquark couplings to taus and muons.\n\n\n\\section*{Acknowledgements}\nThe research of WA is supported by the National Science Foundation under Grant No. PHY-1912719. WA acknowledges support by the Munich Institute for Astro- and Particle Physics (MIAPP) which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC-2094 - 390783311.\nThe research of SG and DT is supported in part by the NSF CAREER grant PHY-1915852. SG would like to thank the Aspen Center for Physics under NSF grant PHY-1607611, where part of this work was performed.\nHHP and SP are partly supported by the U.S.\\ Department of Energy grant number de-sc0010107.\nThe work of DT is supported in part by the U.S. Department of Energy, Office of Science, Office of Workforce Development for Teachers and Scientists, Office of Science Graduate Student Research (SCGSR) program. The SCGSR program is administered by the Oak Ridge Institute for Science and Education for the DOE under contract number de-sc0014664. \n\n\\bibliographystyle{atlasBibStyleWithTitle.bst}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe parallelogram law states that $\\|x+y\\|^2+\\|x-y\\|^2=2(\\|x\\|^2 +\n\\|y\\|^2)$ holds for all vectors $x$ and $y$ in a Hilbert space.\nThis law implies that the so-called parallelogram inequality\n$\\|x+y\\|^2\\leq 2(\\|x\\|^2 + \\|y\\|^2)$ trivially holds. S. Saitoh\n\\cite{SAI} noted the inequality $\\|x+y\\|^2\\leq\n2(\\|x\\|^2+\\|y\\|^2)$ may be more suitable than the usual triangle\ninequality. He used this inequality to the setting of a natural\nsum Hilbert space for two arbitrary Hilbert spaces.\n\nObviously the classical triangle inequality in an arbitrary\nnormed space implies the above inequality. This motivates us to\nintroduce an apparently extension of the triangle inequality.\nMore precisely, we introduce the notion of a $q$-norm, by\nreplacing, in the definition of a norm, the triangle inequality\nby $\\| x+y\\| ^{q}\\leq 2^{q-1}\\left( \\| x\\| ^{q}+\\| y\\|\n^{q}\\right)$, where $q \\geq 1$. We establish that every q-norm is\na norm in the usual sense, and that the converse is true as well.\nThe reader is referred to \\cite{J-L} for undefined terms and\nnotations.\n\n\\begin{definition} Let ${\\mathcal X}$ be a real or complex linear space and $q \\in [1, \\infty)$. A mapping $%\n\\| \\cdot \\| :{\\mathcal X}\\rightarrow \\left[ 0,\\infty \\right) $ is\ncalled a $q$-norm on ${\\mathcal X}$\\ if it satisfies the following\nconditions:\n\n\\begin{enumerate}\n\\item $\\| x\\| =0\\Leftrightarrow x=0,$\n\n\\item $\\| \\lambda x\\| =\\| \\lambda \\|\n\\| x\\| \\ \\ $for all $x\\in {\\mathcal X}$ and all scalar $\\lambda ,$\n\n\\item $\\| x+y\\| ^{q}\\leq 2^{q-1}\\left( \\|\nx\\| ^{q}+\\| y\\| ^{q}\\right) \\ $for all $x,y\\in {\\mathcal X}.$\n\\end{enumerate}\n\\end{definition}\n\nWe first prove a rather trivial result.\n\n\\begin{proposition}\nEvery norm in the usual sense is a $q$-norm.\n\\end{proposition}\n\\begin{proof} One can easily verify that the\nfunction $f(t) = \\frac{1 + t^q}{2} - (\\frac{1 + t}{2})^q$ has a\nnonnegative derivative and so it is monotonically increasing on\n$[0, \\infty)$. It follows that $(\\frac{1 +\n\\frac{\\|y\\|}{\\|x\\|}}{2})^q \\leq \\frac{1 +\n(\\frac{\\|y\\|}{\\|x\\|})^q}{2}$ whenever $\\|x\\| \\leq \\|y\\|$.\nTherefore $\\|\\frac{x + y}{2}\\|^q \\leq (\\frac{\\|x\\| + \\|y\\|}{2})^q\n\\leq \\frac{\\|x\\|^q + \\|y\\|^q}{2}$ for all $x, y \\in {\\mathcal X}$.\nIt follows that $\\|.\\|$ is a $q$-norm.\n\\end{proof}\n\nNow we state the following lemma which is interesting on its own\nright.\n\n\\begin{lemma} Let ${\\mathcal X}$ be a real or complex linear space. Let $%\n\\| \\cdot \\| :{\\mathcal X}\\rightarrow \\left[ 0,\\infty \\right) $ be\na mapping satisfying (1) and (2) in the definition of a $q$-norm.\nThen $\\| \\cdot \\| $ is a norm if and only if the set $B=\\left\\{\nx\\mid \\| x\\| \\leq 1\\right\\}$ is convex.\n\\end{lemma}\n\n\\begin{proof} If $\\| \\cdot \\| $ is a norm, then $B$ is clearly a\nconvex set. Conversely, let $B$ be convex and $x,y\\in {\\mathcal\nX}.$ We can assume that\n$x \\neq 0, y\\neq 0$. Putting $x^{\\prime }=\\frac{x}{\\| x\\| }$ and $%\ny^{\\prime }=\\frac{y}{\\| y\\| }$ we have $x^{\\prime },y^{\\prime\n}\\in B.$\n\nNow $\\lambda x^{\\prime }+\\left( 1-\\lambda \\right) y^{\\prime }\\in\nB$ for all $0 \\leq \\lambda \\leq 1.$ In particular, for $\\lambda\n=\\frac{\\|x\\|}{\\|x\\| +\\| y\\| }$ we obtain%\n\\[\n\\| \\frac{x}{\\| x\\| +\\| y\\| }+\\frac{%\ny}{\\| x\\| +\\| y\\| }\\| =\\| \\lambda x^{\\prime }+\\left( 1-\\lambda\n\\right) y^{\\prime }\\| \\leq 1.\n\\]\n\nSo that $\\| x+y\\| \\leq \\| x\\| +\\| y\\|.$\n\\end{proof}\n\nWe are just ready to prove our main result.\n\n\\begin{theorem}\nEvery $q$-norm is a norm in the usual sense.\n\\end{theorem}\n\\begin{proof} We shall show that $B = \\left\\{ x : \\|\nx\\|\n\\leq 1\\right\\} $ is convex. Let $x,y\\in B.$ Then we have%\n\\[\n\\| x+y\\| ^{q}\\leq 2^{q-1}\\left( \\| x\\| ^{q}+\\| y\\| ^{q}\\right)\n\\leq 2^{q-1}\\left( 1+1\\right) =2^{q}.\n\\]%\nwhence $\\| \\frac{x+y}{2}\\| ^{2}\\leq 1,$ so $\\frac{1}{2}%\nx+\\left( 1-\\frac{1}{2}\\right) y\\in B.$ Thus if $A=\\left\\{ \\frac{k}{2^{n}}%\n\\mid n=1,2,\\ldots ;k=0,1,\\ldots ,n\\right\\}$, then for each\n$\\lambda \\in A$ we have $\\lambda x+\\left( 1-\\lambda \\right) y\\in\nB.$\n\nLet $0\\leq \\lambda \\leq 1$ and $z=\\lambda x+\\left( 1-\\lambda\n\\right) y.$ Since $A$\\ is dense in $\\left[ 0,1\\right]$, there\nexists a decreasing\nsequence $\\left\\{ r_{n}\\right\\} $ in $A$ such that $\\lim\\limits_{n}r_{n}=%\n\\lambda .$ Put $\\beta _{n}=\\frac{1-r_{n}}{1-\\lambda }.$ Obviously $0\\leq\n\\beta _{n}\\leq 1,$ $\\lim\\limits_{n}\\beta _{n}=1$ and $\\frac{r_{n}+\\beta\n_{n}-1}{r_{n}} \\leq 1.$ Since $\\frac{r_{n}+\\beta _{n}-1}{r_{n}}x\\in B$ and $%\nr_{n}\\in A$ we conclude that%\n\\[\n\\beta _{n}z = \\lambda \\beta _{n}x+\\left( 1-\\lambda \\right) \\beta _{n}y=r_{n}%\n\\frac{r_{n}+\\beta _{n}-1}{r_{n}}x+\\left( 1-r_{n}\\right) y\\in B.\n\\]%\nThus $\\beta _{n} \\| z\\| =\\| \\beta _{n}z\\| \\leq 1$ for all $n.$\nTending $n$ to infinity we get $\\| z\\| \\leq 1,$ i.e. $z\\in B.$\n\\end{proof}\n\n{\\bf Acknowledgment.} We would like to sincerely thank Professor\nSaburou Saitoh for his encouragement.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzokfv b/data_all_eng_slimpj/shuffled/split2/finalzzokfv
new file mode 100644
index 0000000000000000000000000000000000000000..49a1c4bbcf0c9d2b5340da1b590c18202b0b738d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzokfv
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\nVarious problems in science and engineering lead to\nnonlocal operators and corresponding operator equations.\nExamples arise from physical problems like field calculations\nand Riesz energy problems, from machine learning, and also \nfrom stochastic simulations and uncertainty quantification.\n\nTraditional discretizations of nonlocal operators result in densely \npopulated system matrices. This feature renders the computation very costly in \nboth respects, the computation time and computer memory requirements. \nTherefore, over recent decades, different ideas for the data sparse approximation \nof nonlocal operators have been developed. Most prominent examples of \nsuch methods are the fast multipole method \\cite{GR}, the panel clustering \n\\cite{HN}, the wavelet matrix compression \\cite{BCR,DHS1}, and the hierarchical \nmatrix format \\cite{HA2}. These techniques are able to represent nonlocal \noperators in linear or almost linear cost with respect to the \nnumber of degrees of freedom used for their discretization.\n\nThe present article relies on a compression of the system matrix by\nwavelets. Especially, the matrix representation of the nonlocal operator \nin wavelet coordinates is quasi-sparse, i.e.\\ most matrix entries are \nnegligible and can be treated as zero without compromising the overall\naccuracy. Discarding the non-relevant matrix entries is called matrix \ncompression. Roughly speaking, nonlocal operators become local operators \nin wavelet coordinates. A fully discrete version of the wavelet \nmatrix compression has been developed in \\cite{HS2}. It computes the \ncompressed operator within discretization accuracy with linear cost.\n\nBased on the sparsity pattern of the system matrix which is \nsolely determined by the order of the underlying operator, we\nemploy a fill-in reducing reordering of the matrix entries by\nmeans of nested dissection, see \\cite{Geo73,LRT79}. This \nreordering in turn allows for the rapid inversion of the system \nmatrix by the Cholesky decomposition or more generally by \nthe LU decomposition. In particular, besides the rigorously\ncontrollable error for the matrix compression in the wavelet \nformat and the roundoff errors in the computation of the matrix\nfactorization, no additional approximation errors are introduced.\nThis is a major difference to other approaches for the discretization \nand the arithmetics of nonlocal operators, e.g.\\ by means of hierarchical \nmatrices. As the hierarchical matrix format is not closed under \narithmetic operations, a recompression step after each arithmetic \n(block) operation has to be performed, which results into \naccumulating and hardly controllable consistency errors \nfor matrix factorizations, see \\cite{HA1,HA2}. \n\nIn order to demonstrate the efficiency of the suggested approach, \nwe consider applications from different fields. Namely, we consider \n(i) a boundary integral equation arising from the polarizable\ncontinuum model in quantum chemistry as a classical example \nfor a nonlocal operator equation, (ii) a parabolic problem for the \nfractional Laplacian, and finally (iii) the fast numerical simulation of \nGaussian random fields as an important example from computational \nuncertainty quantification.\n\nOne of the most widespread methods to include solvent effects in \nquantum chemistry is by making use of a continuum dielectric: the \nsolvent is represented by a continuum which surrounds the molecule.\nSolute-solvent interactions are then described through appropriate \nfunctions supported on the molecule's surface. For an overview of \ncontinuum solvation models, we refer the reader to \\cite{TMC}, and \nin particular for the polarizable continuum model to \\cite{CM,CMT,MST}.\nWavelet matrix compression for the polarizable continuum model has been\nconsidered in \\cite{WaveletPCM2,WaveletPCM1}. Especially, in \\cite{ILU}, \nthe use of an incomplete Cholesky decomposition based preconditioner has \nbeen suggested, which is however inferior to the approach presented here. \n\nThe fractional Laplacian is an operator which generalizes the notion of \nspatial derivatives to fractional orderss. It appears in image asnalysis, \nkinetic equations, phase transitions and nonlocal heat conduction, \njust to mention some applications. We refer to the review article\n\\cite{DGZ} and the references therein for further details. In particular,\nwe will focus here on the definition of the fractional Laplacian in its\nintegral form, as it can be found in \\cite{EG} and also in \\cite{DGZ}. \nTo the best of our knowledge, the numerical treatment of the parabolic \nproblem for the fractional Laplacian by means of wavelets has not \nbeen addressed in literature yet. \n\nThe rapid simulation of (Gaussian) random fields with a prescribed \ncovariance structure is of paramount importance in computational \nuncertainty quantification. The fast methods, which have been \nsuggested so far are based on the computation of matrix square \nroots employing low-rank factorizations, block-wise low-rank \nfactorizations, such as obtained by hierarchical matrices, or the \ndiscretization of the action of the matrix square root on a given \nvector by Krylov subspace methods. Other approaches compute the \nKarhunen-Lo\\`eve expansion by circulant embeddings and fast Fourier \ntechniques or employ the contour integral method. For more details on \nthese methods, we refer to \\cite{FKS18,GKNS+18,HPS15,HKS20,RH}.\nIn contrast to the previously mentioned approaches, we consider here \nthe direct simulation of the random field by the Cholesky decomposition \nof the covariance matrix, which is very sparse in wavelet coordinates.\n\nThis article is organized as follows. Wavelet bases and \ntheir properties are specified in Section~\\ref{sec:wavelets}.\nSection~\\ref{sec:WEM} briefly repeats the main features of the \nfully discrete wavelet matrix compression scheme from \\cite{HS2}. \nThen, in Section~\\ref{sect:ND}, for the sake of completeness,\nthe idea of nested dissection is briefly outlined.\nSection~\\ref{sct:applications} presents the three different \napplications considered in this article, while Section~\\ref{sec:numerix} \nis devoted to related numerical experiments. Finally, Section~\\ref{sec:conclusion} \ncontains some concluding remarks.\n\nIn the following, in order to avoid the repeated use of generic but\nunspecified constants, we write $C \\lesssim D$ to indicate that $C$ \ncan be bounded by a multiple of $D$, independently of parameters\nwhich $C$ and $D$ may depend on. Then, $C \\gtrsim D$\nis defined as $D \\lesssim C$, while we write $C \\sim D$, iff\n$C \\lesssim D$ and $C \\gtrsim D$.\n\n\\section{Wavelets and multiresolution Analysis}\n\t\\label{sec:wavelets}\nLet $D$ denote a domain in $\\mathbb{R}^n$ or \na manifold in $\\mathbb{R}^{n+1}$. A \n\\emph{multiresolution analysis} consists of a nested \nfamily of finite dimensional approximation spaces\n\\begin{equation}\t\\label{eq:hierarchy}\n\\{0\\} = V_{-1} \\subset V_0 \\subset V_1 \\subset \\cdots \n\\subset V_j\\subset \\cdots \\subset L^2(D),\n\\end{equation}\nsuch that\n\\[ \\overline{\\bigcup_{j\\ge0} V_j} = L^2(D)\\quad\\text{and}\\quad\n\\dim V_j\\sim 2^{jn}.\n\\] We will\nrefer to \\(j\\) as the \\emph{level} of \\(V_j\\) in the\nmultiresolution analysis.\nEach space $V_j$ is endowed with a \\emph{single-scale basis}\n\\[\n\\Phi_j = \\{\\varphi_{j,{\\bs k}}:{\\bs k}\\in\\Delta_j\\},\n\\]\ni.e.\\ $V_j=\\operatorname{span}\\Phi_j$, where $\\Delta_j$ denotes a suitable \nindex set with cardinality $|\\Delta_j| \\sim 2^{jn}$. For convenience, \nwe shall in the sequel write bases on the form of row vectors, such \nthat, for ${\\bs v} = [v_k]_{k\\in\\Delta_j}\\in\\ell^2(\\Delta_j)$, the \ncorresponding function can simply be written as a dot product according\nto\n\\[\nv_j = \\Phi_j{\\bs v}=\\sum_{k\\in\\Delta_j} v_k\\varphi_{j,k}. \n\\]\nIn addition, we shall assume that the single-scale \nbases $\\Phi_j$ are \\emph{uniformly stable}, this means that \n\\[\n\\|{\\bs v}\\|_{\\ell^2(\\Delta_j)}\n\\sim \\|\\Phi_j{\\bs v}\\|_{L^2(D)}\\quad\\text{for all }\n{\\bs v}\\in\\ell^2(\\Delta_j)\n\\]\nuniformly in $j$, and that they satisfy the locality condition \n\\[\n\\operatorname{diam}(\\operatorname{supp}\\varphi_{j,{\\bs k}})\\sim 2^{-j}.\n\\]\n\nAdditional properties of the spaces $V_j$ are required\nfor using them as trial spaces in a Galerkin scheme.\nThe approximation spaces shall have the \\emph{regularity} \n\\[\n\\gamma\\mathrel{\\mathrel{\\mathop:}=}\\sup\\{s\\in\\mathbb{R}: V_j\\subset H^s(D)\\}\n\\]\nand the \\emph{approximation order} $d\\in\\mathbb{N}$, \nthat is\n\\[\n       d = \\sup\\Big\\{s\\in\\mathbb{R}: \\inf_{v_j\\in V_j}\\|v-v_j\\|_{L^2(D)} \n       \t\t\\lesssim 2^{-js}\\|v\\|_{H^s(D)}\\Big\\}.\n\\]\n\nRather than using the multiresolution analysis corresponding to the \nhierarchy in \\eqref{eq:hierarchy}, the pivotal idea \nof wavelets is to keep track of the increment of information \nbetween two consecutive levels $j-1$ and $j$. Since we have\n$V_{j-1}\\subset V_j$, we may decompose \n\\[\nV_j = V_{j-1}\\oplus W_j,\\ \\text{i.e.}\\ V_{j-1}\\cup W_j = V_{j}\\ \\text{and}\\  V_{j-1}\\cap W_j = \\{0\\},\n\\]\nwith an appropriate \\emph{detail space} $W_j$. Of practical interest \nis the particular choice of the basis of the detail space  $W_j$ in $V_j$. This basis \nwill be denoted by\n\\[\n  \\Psi_j = \\{\\psi_{j,{\\bs k}}:\n  \t{\\bs k} \\in \\nabla_j\\mathrel{\\mathrel{\\mathop:}=} \\Delta_j\\setminus \\Delta_{j-1}\\}.\n\\]\nIn particular, we shall assume that the collections $\\Phi_{j-1}\\cup\\Psi_j$\nform uniformly stable bases of $V_j$, as well. If $\\Psi = \n\\bigcup_{j\\ge 0}\\Psi_j$, where $\\Psi_0\\mathrel{\\mathrel{\\mathop:}=} \\Phi_0$, is even a \nRiesz-basis of $L_2(D)$, then it is called a \\emph{wavelet \nbasis}. We require the functions $\\psi_{j,{\\bs k}}$ to be \nlocalized with respect to the corresponding level $j$, i.e.\\ \n\\[\n\\operatorname{diam}(\\operatorname{supp}\\psi_{j,{\\bs k}}) \\sim 2^{-j}, \n\\]\nand we normalize them such that \n\\[\n\\|\\psi_{j,{\\bs k}}\\|_{L_2(D)}\\sim 1.\n\\]\n\nAt first glance it would be very convenient to deal with a single \northonormal system of wavelets. However, it has been shown\nin \\cite{DHS1,DPS4,S} that orthogonal wavelets are not optimal \nfor the efficient approximation nonlocal operator equations. For this \nreason, we rather use \\emph{biorthogonal wavelet bases}.\nIn this case, we also have a dual, multiresolution\nanalysis, i.e.\\ dual single-scale bases and wavelets\n\\[\n\\widetilde{\\Phi}_j = \\{\\widetilde{\\varphi}_{j,{\\bs k}}:{\\bs k}\\in\\Delta_j\\},\\quad\n\\widetilde{\\Psi}_j = \\{\\widetilde{\\psi}_{j,{\\bs k}}:{\\bs k}\\in\\nabla_j\\},\n\\] \nwhich are coupled to the primal ones by the orthogonality \ncondition \n\\[\n(\\Phi_j,\\widetilde{\\Phi}_j)_{L^2(D)} = {\\bs I},\\quad\n(\\Psi_j,\\widetilde{\\Psi}_j)_{L^2(D)} = {\\bs I}. \n\\]\nThe corresponding spaces $\\widetilde{V}_j\\mathrel{\\mathrel{\\mathop:}=} \\operatorname{span}\\widetilde{\\Phi}_j$\nand $\\widetilde{W}_j\\mathrel{\\mathrel{\\mathop:}=}\\operatorname{span}\\widetilde{\\Psi}_j$ satisfy\n\\begin{equation}\t\\label{eq:space-coupling}\n  V_{j-1}\\perp \\widetilde{W}_j, \\quad \\widetilde{V}_{j-1}\\perp W_j.\n\\end{equation}\nMoreover, the dual spaces are supposed to exhibit some approximation order\n$\\widetilde{d}\\in\\mathbb{N}$ and regularity $\\widetilde{\\gamma}>0$.\n\nDenoting in complete analogy to the primal basis $\\widetilde{\\Psi} = \n\\bigcup_{j\\ge 0}\\widetilde{\\Psi}_j$, where $\\widetilde{\\Psi}_0 \n\\mathrel{\\mathrel{\\mathop:}=}\\widetilde{\\Phi}_0$, then every $v \\in L^2(D)$ has\nunique representations \n\\[\nv = \\widetilde{\\Psi} (v,\\Psi)_{L^2(D)} = \\Psi (v,\\widetilde{\\Psi})_{L^2(D)}\n\\]\nsuch that\n\\[\n \\|v\\|_{L^2(D)}^2 \\sim \\sum_{j\\ge 0}\\sum_{k\\in\\nabla_j}\n\t\\big\\|(v,\\widetilde\\psi_{j,{\\bs k}})_{L^2(D)}\\big\\|_{\\ell^2(\\nabla_j)}^2\n \\sim \\sum_{j\\ge 0}\\sum_{k\\in\\nabla_j}\n\t\\big\\|(v,\\psi_{j,{\\bs k}})_{L^2(D)}\\big\\|_{\\ell^2(\\nabla_j)}^2.\n\\]\nIn particular, relation \\eqref{eq:space-coupling} implies that the \nwavelets exhibit {\\em vanishing moments} of order $\\widetilde{d}$, i.e.\\\n\\begin{equation}\t\\label{eq:cancellation}\n  \\big|(v,\\psi_{j,{\\bs k}})_{L^2(D)}\\big| \\lesssim 2^{-j(1+\\widetilde{d})}\n\t|v|_{W^{\\widetilde{d},\\infty}(\\operatorname{supp}\\psi_{j,{\\bs k}})}.\n\\end{equation}\nHerein, the quantity $|v|_{W^{\\widetilde{d},\\infty}(D)}\n\\mathrel{\\mathrel{\\mathop:}=} \\sup_{|\\boldsymbol\\alpha|=\n\\widetilde{d}}\\|\\partial^{\\boldsymbol\\alpha}v\\|_{L^\\infty(D)}$\nis the semi-norm in $W^{\\widetilde{d},\\infty}(D)$.\nWe refer to \\cite{DA1} for further details.\n\nPiecewise constant and bilinear wavelets which \nprovide the above properties have been constructed in \n\\cite{HS1,HS3}. In what follows, we will refer to the\nwavelet basis of $V_J$ by $\\Psi_J = \\{\\psi_\\lambda:\n\\lambda\\in\\nabla_J\\}$, where the multi-index \n$\\lambda = (j,{\\bs k})$ incorporates the scale \n$j=|\\lambda|$ and the spatial location \n${\\bs k} = {\\bs k}(\\lambda)$.\n\n\\section{Wavelet Matrix Compression}\t\t\\label{sec:WEM}\nFor a given domain or manifold $D$ and $q\\in\\mathbb{R}$, let\n\\[\n\\mathcal{A}:H^q(D)\\to H^{-q}(D)\n\\]\ndenote a given (continuous and bijective) nonlocal operator of\norder $2q$. According to the Schwartz kernel theorem, it can \nbe represented in accordance with\n\\begin{equation}\t \\label{eq:integral equation}\n (\\mathcal{A}u)({\\bs x}) = \\int_D k({\\bs x},{\\bs y})\n \tu({\\bs y})\\d{\\bs y},\n \\quad {\\bs x}\\in D,\n\\end{equation}\nfor a suitable kernel function $k\\colon D\\times D\\to\\mathbb{R}$.\nThe kernel functions under consideration are supposed \nto be smooth as functions in the variables ${\\bs x}$ and \n${\\bs y}$, apart from the diagonal $\\{({\\bs x},{\\bs y})\\in\nD\\times D: {\\bs x}={\\bs y}\\}$ and may exhibit a singularity \non the diagonal. Such kernel functions arise, for instance, from \napplying a boundary integral formulation to a second order elliptic \nproblem \\cite{SS,ST}. Typically, they decay like a negative power \nof the distance of the arguments which depends on the order \n$2q$ of the operator. More precisely, there holds\n\\begin{equation}\t\\label{eq:decay}\n  \\big|\\partial_{\\bs x}^{\\boldsymbol\\alpha}\\partial_{\\bs y}^{\\boldsymbol\\beta} \n  \tk({\\bs x},{\\bs y})\\big| \\le c_{\\boldsymbol\\alpha,\\boldsymbol\\beta}\n\t\t\\|{\\bs x}-{\\bs y}\\|^{-n-2q-|\\boldsymbol\\alpha|-|\\boldsymbol\\beta|}.\n\\end{equation}\nWe emphasize that this estimate remains valid for the kernels of\narbitrary pseudodifferential operators, see \\cite{DHS2} for the details.\n\nCorresponding to the nonlocal operator from \\eqref{eq:integral equation},\nwe may consider the operator equation \\[\\mathcal{A}u=f\\] which gives rise\nto the Galerkin approach:\n\\begin{align*}\n&\\text{find $u_J\\in V_J$ such that}\\\\\n&\\quad (\\mathcal{A}u_J,v_J)_{L^2(D)} = (f,v_J)_{L^2(D)}\n\\quad\\text{for all $v_J \\in V_J$}.\n\\end{align*}\nTraditionally, this equation is discretized \nemploying the single-scale basis of $V_J$ which results in densely\npopulated system matrices. If $N_J\\sim 2^{Jn}$ denotes the \nnumber of basis functions in the space $V_J$, then the system \nmatrix contains $\\mathcal{O}(N_J^2)$ nonzero matrix entries. In \ncontrast, by utilizing a wavelet basis in the Galerkin discretization,\nwe end up with a matrix that is quasi-sparse, i.e.\\ it is compressible \nto $\\mathcal{O}(N_J)$ nonzero matrix entries without compromising \nthe overall accuracy. More precisely, by combining \\eqref{eq:cancellation}\nand \\eqref{eq:decay}, we arrive at the decay estimate\n\\begin{equation}\t\\label{eq:decay estimate}\n  (\\mathcal{A}\\psi_{\\lambda'},\\psi_\\lambda)_{L^2(D)}\n  \t\\lesssim \\frac{2^{-(|\\lambda|+|\\lambda'|)(\\widetilde{d}+n\/2)}}\n\t{\\operatorname{dist}(D_\\lambda,D_{\\lambda'})^{n+2q+2\\widetilde{d}}}\n\\end{equation}\nwhich is the foundation of the compression estimates in \\cite{DHS1}. \nHerein, $D_\\lambda\\mathrel{\\mathrel{\\mathop:}=}\\operatorname{supp}\\psi_\\lambda$ and \n$D_{\\lambda'}\\mathrel{\\mathrel{\\mathop:}=}\\operatorname{supp}\\psi_\\lambda$ denote the \nconvex hulls of the supports of the wavelets $\\psi_\\lambda$ and\n$\\psi_{\\lambda'}$.\n\nBased on \\eqref{eq:decay estimate}, we shall neglect\nall matrix entries for which the distance of \nthe supports between the associated ansatz and test \nwavelets is larger than a level dependent cut-off \nparameter $\\mathcal{B}_{j,j'}$. An additional compression, \nreflected by a cut-off parameter $\\mathcal{B}_{j,j'}^s$, \nis achieved by neglecting several of those matrix entries, \nfor which the corresponding trial and test functions \nhave overlapping supports. \n\nTo formulate this result, we introduce the abbreviation\n$D_\\lambda^s\\mathrel{\\mathrel{\\mathop:}=} \\operatorname{sing}\\operatorname{supp}\\psi_\\lambda$\nwhich denotes the {\\em singular support} of the wavelet \n$\\psi_\\lambda$, i.e.\\ that subset of $D$ where \nthe wavelet is non-smooth.\n\n\\begin{theorem}[A-priori compression \\cite{DHS1}]\\label{thm:a-priori}\nLet $D_\\lambda$ and $D_\\lambda^s$ be given as \nabove and define the compressed system matrix ${\\bs A}_J$, \ncorresponding to the boundary integral operator $\\mathcal{A}$, by\n\\begin{equation}\t\t\\label{eq:a-priori}\n  [{\\bs A}_J]_{\\lambda,\\lambda'}\\mathrel{\\mathrel{\\mathop:}=} \\begin{cases}\n  \\qquad \\quad 0,& \\operatorname{dist}(D_\\lambda,D_{\\lambda'}) > \\mathcal{B}_{|\\lambda|,|\\lambda'|}\\ \\text{and $|\\lambda|,|\\lambda'|> 0$}, \\\\\n  \\qquad \\quad 0,& \\operatorname{dist}(D_\\lambda,D_{\\lambda'}) \\le 2^{-\\min\\{|\\lambda|,|\\lambda'|\\}}\\ \\text{and}\\\\\n                 & \\operatorname{dist}(D_\\lambda^s,D_{\\lambda'}) >\n\t\t \\mathcal{B}_{|\\lambda|,|\\lambda'|}^s\\ \\text{if $|\\lambda'| > |\\lambda|\\geq 0$}, \\\\\n\t\t & \\operatorname{dist}(D_\\lambda,D_{\\lambda'}^s) >\n\t\t \\mathcal{B}_{|\\lambda|,|\\lambda'|}^s\\ \\text{if $|\\lambda| > |\\lambda'|\\geq 0$}, \\\\\n  (\\mathcal{A}\\psi_{\\lambda'},\\psi_\\lambda)_{L^2(D)}, &\\text{otherwise}. \\end{cases}\n\\end{equation}\nFixing\n\\begin{equation}\t\\label{eq:parameters}\n  a > 1, \\qquad d < \\delta < \\widetilde{d} + 2q,\n\\end{equation}\nthe cut-off parameters $\\mathcal{B}_{j,j'}$ and\n$\\mathcal{B}_{j,j'}^s$ are set according to\n\\begin{equation}\t\\label{eq:cut-off parameters}\n  \\begin{aligned}\n  \\mathcal{B}_{j,j'} \\mathrel{\\mathrel{\\mathop:}=} a \\phantom{'}\\max\\left\\{ 2^{-\\min\\{j,j'\\}},\n\t2^{\\frac{2J(\\delta-q)-(j+j')(\\delta+\\widetilde{d})}{2(\\widetilde{d}+q)}}\\right\\}, \\vspace*{2mm} \\\\\n  \\mathcal{B}_{j,j'}^s \\mathrel{\\mathrel{\\mathop:}=} a \\max\\left\\{ 2^{-\\max\\{j,j'\\}},\n\t2^{\\frac{2J(\\delta-q)-(j+j')\\delta-\\max\\{j,j'\\}\n\t\\widetilde{d}}{\\widetilde{d}+2q}}\\right\\}.\n  \\end{aligned}\n\\end{equation}\nThen, the system matrix ${\\bs A}_J$ only has $\\mathcal{O}(N_J)$\nnonzero entries. In addition, the error estimate\n\\begin{equation}\t\\label{eq:errest}\n    \\|u-u_J\\|_{H^{2q-d}(D)} \\lesssim 2^{-2J(d-q)}\\|u\\|_{H^d(D)}\n\\end{equation}\nholds for the solution $u_J$ of the compressed Galerkin system\nprovided that $u$ and $D$ are sufficiently regular.\n\\end{theorem}\n\nThe compressed system matrix can be assembled with linear \ncost if the exponentially convergent \n$hp$--quadrature method proposed in \\cite{HS2} is employed for\nthe computation of matrix entries. Moreover, \nfor performing faster matrix-vector multiplications, an\nadditional a-posteriori compression might be applied\nwhich reduces again the number of nonzero entries \nby a factor 2--5, see \\cite{DHS1}. The pattern of the compressed \nsystem matrix shows the typical {\\em finger structure}, see \nthe left hand side of Figure~\\ref{fig:compression}.\n\n\\begin{figure}\n\\begin{center}\n\\begin{minipage}{0.49\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.55,clip,trim=0 0 0 13,frame]{SWpattern_PCM}\n$\\nnz({\\bs V}_J) = 15\\,363\\,141$\n\\end{center}\n\\end{minipage}\n\\begin{minipage}{0.49\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.55,clip,trim=0 0 0 13,frame]{Spattern_PCM}\n$\\nnz({\\bs V}_{J,\\text{ND}}) = 15\\,363\\,141$\n\\end{center}\n\\end{minipage}\n\\caption{\\label{fig:compression}Sparsity patterns of \\({\\bs V}\\) (left) and its nested\ndissection reordering\n\\({\\bs V}_{J,\\text{ND}}\\) (right) for the single layer operator on the benzene \ngeometry and \\(N_J=93184\\). Each dot corresponds to a submatrix of size \n\\(64\\times 64\\). Lighter blocks have less entries than darker blocks.}\n\\end{center}\n\\end{figure}\n\n\\section{Nested dissection}\\label{sect:ND}\nThe representation of the system matrix corresponding to\na nonlocal operator with respect to an appropriate wavelet\nbasis leads to a quasi-sparse matrix, i.e.\\ a matrix with many\nsmall entries which can be neglected without compromising\naccuracy. Performing a thresholding procedure as discussed \nin the previous section then yields a sparse system matrix\nwhose symmetric sparsity pattern is solely determined by the \norder of the underlying operator, see the left hand side of \nFigure~\\ref{fig:compression}.\n\nThe factorization of the system matrix represented \nin the canoncial levelwise ordering leads to a massive \nfill-in. This means that a huge amount of nonzero \nentries is generated by a Cholesky decomposition or an LU decomposition, \ntypically resulting in dense matrix factors. In order to obtain \nmuch sparser factorizations, we employ a nested dissection \nordering, cf.\\ \\cite{Geo73,LRT79}, see the right hand side of \nFigure~\\ref{fig:compression}.\n\nNested dissection is a divide and conquer algorithm whose foundation\nis a graph theoretical observation. To each matrix\n\\({\\bs A}\\in\\mathbb{R}^{N\\times N}\\) with a symmetric sparsity\npattern, we may assign an undirected graph \\(G=(V,E)\\) with\nvertices \\(V=\\{1,2,\\ldots,N\\}\\) and edges\n\\(E=\\big\\{\\{i,j\\}:a_{i,j}\\neq 0\\big\\}\\). Then, a symmetric permutation\n\\({\\bs P}{\\bs A}{\\bs P}^\\intercal\\) of the rows and columns of \\({\\bs A}\\)\namounts to a permutation \\(\\pi(V)\\) of the nodes in \\(V\\).\nIn particular, we have the following important result from \\cite{RTL76},\nsee also \\cite{Par61}, which we formulate here only for the Cholesky decomposition\n\\({\\bs P}{\\bs A}{\\bs P}^\\intercal={\\bs L}{\\bs L}^\\intercal\\).\n\n\\begin{lemma}[\\cite{RTL76}]\nAssuming that no cancellation of nonzero entries\nin the Cholesky decomposition\nof \\({\\bs P}{\\bs A}{\\bs P}^\\intercal\\) takes place, then \n\\(\\ell_{i,j}\\neq 0\\) for \\(i>j\\), iff there is a path\n\\(i=v_1,v_2,\\ldots, v_{k+1}=j\\), \\(k\\geq 0\\), in \\(G\\)\nsuch that \\(\\pi(v_t)<\\min\\{\\pi(i),\\pi(j)\\}\\) for \\(2\\leq t\\leq k\\).\n\\end{lemma}\n\nThe lemma states that the Cholesky decomposition connects all\nnodes \\(i\\) and \\(j\\), resulting in a nonzero entry \\(\\ell_{i,j}\\),\nfor which there exists a path of nodes that have been eliminated\nbefore \\(i\\) and \\(j\\).\n\nFinding an optimal ordering is a hard problem in general. Therefore, we \nresort to the following strategy, which is known as nested dissection ordering:\nWe split \\(V=V_1\\cup V_2\\cup S\\)\nsuch that \\(E\\cap\\big\\{\\{v_1,v_2\\}: v_1\\in V_1,v_2\\in V_2\\big\\}=\\emptyset\\),\ni.e.\\ the removal of the vertices of the \\emph{separator} \\(S\\) and \nits adjacent edges results into two disjoint subgraphs.\nHence, employing an ordering which puts first the nodes into \\(V_1\\) and \\(V_2\\)\nand afterwards the nodes in \\(S\\), leads to a matrix structure of the form\n\\[\n{\\bs P}{\\bs A}{\\bs P}^\\intercal=\n\\begin{bmatrix}{\\bs A}_{V_1,V_1} & & {\\bs A}_{V_1,S}\\\\\n & {\\bs A}_{V_2,V_2}& {\\bs A}_{V_2,S}\\\\\n{\\bs A}_{S,V_1} & {\\bs A}_{S,V_2}& {\\bs A}_{S,S}\\\\\\end{bmatrix}.\n\\]\nRecursively applying this procudeure then yields a structure similar to\nthe one on the right hand side of Figure~\\ref{fig:compression}.\nFor obvious reasons, it is desirable to have a minimal separator\n\\(S\\), which evenly splits the \\(V\\) into two subsets, we refer to \n\\cite{LRT79} and the references therein for a comprehensive \ndiscussion of this topic. In order to obtain suitable separators \nfor the computations in this article, we will adopt the strategy \nfrom \\cite{KK98}, which performs very well in terms of reducing \nthe fill-in.\n\n\\section{Applications}\\label{sct:applications}\n\\subsection{Polarizable continuum model}\nContinuum solvation models are widely used to model\nquantum effects of molecules in liquid solutions, compare\n\\cite{TMC} for an overview. In the {\\em polarizable continuum \nmodel} (PCM), introduced in \\cite{MST}, the molecule under \nstudy (the solute) is located inside a cavity $D$, surrounded \nby a homogeneous dielectric (the solvent) with dielectric \nconstant $\\epsilon\\ge 1$. The solute-solvent interactions \nbetween the charge distributions which compose the solute \nand the dielectric are reduced to those of electrostatic origin.\n\nFor a given charge $\\rho\\in H^{-1}(D)$, located inside\nthe cavity, the solute-solvent interaction is expressed by the\n{\\em apparent surface charge} $\\sigma\\in H^{-1\/2}(\\partial D)$.\nIt is given by the integral equation\n\\begin{equation}\t\\label{eq:PCM}\n \\mathcal{V}\\sigma = \\bigg(\\frac{1+\\epsilon}{2}+(1-\\epsilon)\\mathcal{K}\\bigg)^{-1}\n\t\t\\mathcal{N}_\\rho-\\mathcal{N}_\\rho\\quad\\text{on $\\partial D$},\n\\end{equation}\nwhere $\\mathcal{V}$ is the \\emph{single layer operator}\n\\[\n  (\\mathcal{V}u)({\\bs x}) = \\int_{\\partial D}\n  \t\\frac{u({\\bs y})}{4\\pi\\|{\\bs x}-{\\bs y}\\|^3}\\d\\sigma_{\\bs y},\n\\]\n$\\mathcal{K}$ is the \\emph{double layer operator}\n\\[\n  (\\mathcal{K}u)({\\bs x}) = \\int_{\\partial D} u({\\bs y})\n  \t\\frac{\\langle{\\bs n}({\\bs y}),{\\bs x}-{\\bs y}\\rangle}{4\\pi\\|{\\bs x}-{\\bs y}\\|^3}\\d\\sigma_{\\bs y},\n\\]\nand $\\mathcal{N}_\\rho$ denotes the \\emph{Newton potential} of the \ngiven charge\n\\[\n  \\mathcal{N}_\\rho({\\bs x})\\mathrel{\\mathrel{\\mathop:}=} \\int_{\\partial D}\\frac{\\rho({\\bs y})}{4\\pi\\|{\\bs x}-{\\bs y}\\|}\n  \t\\d{\\bs y}.\n\\]\n\nThe discretization of the boundary integral equation\n\\eqref{eq:PCM} by means of a Galerkin scheme is as \nfollows, compare \\cite{HM1,HM2}: We make the ansatz \n\\[\n  \\sigma_J = \\sum_\\lambda\\sigma_\\lambda\\psi_\\lambda\n\\]\nand introduce the mass matrix\n\\[\n{\\bs G}_J = [(\\psi_{\\lambda'},\\psi_\\lambda)_{L^2(D)}]_{\\lambda,\\lambda'}\n\\]\nand the system matrices\n\\[\n {\\bs V}_J = [(\\mathcal{V}\\psi_{\\lambda'},\\psi_\\lambda)_{L^2(D)}]_{\\lambda,\\lambda'},\\quad\n {\\bs K}_J = [(\\mathcal{K}\\psi_{\\lambda'},\\psi_\\lambda)_{L^2(D)}]_{\\lambda,\\lambda'}.\n\\]\nThen, for a given data vector ${\\bs f}_J = [(\\mathcal{N}_\\rho,\n\\psi_\\lambda)_{L^2(\\partial D)}]_\\lambda$, we need to solve the \nlinear system of equations\n\\begin{equation}\\label{eq:PCM2}\n  {\\bs V}_J\\boldsymbol\\sigma_J = {\\bs G}_J\n  \t\\bigg(\\frac{1+\\epsilon}{2}{\\bs G}_J+(1-\\epsilon) {\\bs K}_J\\bigg)^{-1}\n  \t{\\bs f}_J-{\\bs f}_J\n\\end{equation}\nin order to determine the sought apparent surface charge.\n\nIn quantum chemical simulations, for example when \nsolving the Hartree-Fock equations in a self consistent \nfield approximation, one has to compute the interaction \nenergies between the different particles. This amounts to \nthe determination of different apparent surface charges. \nTherefore, the fast solution of \\eqref{eq:PCM2} for multiple \nright hand sides is indispensable for fast simulations \nin quantum chemistry.\n\n\\subsection{Parabolic diffusion problem for the fractional Laplacian}\nFor a given domain $D\\subset\\mathbb{R}^n$ and $0<s<1\/2$,\nthe fractional Laplacian $\\mathcal{L}_s\\colon H^s(D)\/\\mathbb{R}\n\\to H^{-s}(D)\/\\mathbb{R}$ is given by\n\\[\n(\\mathcal{L}_su)({\\bs x})\\mathrel{\\mathrel{\\mathop:}=} 2 \\int_{D} \\frac{u({\\bs y}) - u({\\bs x})}{\\|{\\bs x}-{\\bs y}\\|^{n+2s}}\\d{\\bs y},\n\t\\quad {\\bs x} \\in D,\n\\]\ncompare \\cite{EG,DGZ}. We intent to solve the following \nparabolic diffusion problem\n\\[\n \\partial_t u - \\mathcal{L}_s u = f\\quad\\text{in $D$}\n\\] \nfor the fractional Laplacian.\nTo this end, we employ the $\\theta$-scheme in time and a \nGalerkin discretization of the problem in space. This leads to the\nlinear system of equations\n\\begin{equation}\\label{eq:theta}\n {\\bs G}_J\\frac{{\\bs u}_J(t_{i+1})-{\\bs u}_J(t_i)}{t_{i+1}-t_i} \n - {\\bs L}_J \\big\\{(1-\\theta) {\\bs u}_J(t_i) + \\theta{\\bs u}_J(t_{i+1})\\big\\}\n \t= (1-\\theta){\\bs f}_J(t_i) + \\theta{\\bs f}_J(t_{i+1}).\n\\end{equation}\nHere, \n\\[\n {\\bs L}_J = \\big[(\\mathcal{L}_s\\psi_{\\lambda'},\n \t\\psi_\\lambda)_{L^2(D)}\\big]_{\\lambda,\\lambda'},\\quad\n {\\bs G}_J = \\big[(\\psi_{\\lambda'},\\psi_\\lambda)_{L^2(D)}\\big]_{\\lambda,\\lambda'},\n\\]\nare the system matrix of the fractional Laplacian and the\nmass matrix, respectively. In each time step, we have hence to invert the\nmatrix ${\\bs A}_J={\\bs G}_J - \\theta(t_{i+1}-t_i) {\\bs L}_J$ for\ncomputing the new solution ${\\bs u}_J(t_{i+1})$ from the\nsolution  ${\\bs u}_J(t_i)$ of the previous time step $t_i$.\nA factorization of the system matrix \\({\\bs A}_J\\)\nis favourable in this situation since \nthe system matrix does not change with time.\n\n\\subsection{Gaussian random fields}\nLet \\((\\Omega,\\mathcal{F},\\mathbb{P})\\) denote a complete\nand separable probability space. \nWe consider a Gaussian random field\n\\[\na\\colon D\\times\\Omega\\to\\mathbb{R}\n\\]\nwith \\emph{expectation} \n\\[\n\\Ebb[a]({\\bs x})\\mathrel{\\mathrel{\\mathop:}=}\\int_\\Omega a({\\bs x},\\omega)\\d\\Pbb(\\omega)\n\\] and \\emph{covariance}\n\\[\n\\operatorname{Cov}[a]({\\bs x},{\\bs x}')\\mathrel{\\mathrel{\\mathop:}=}\\int_\\Omega\n\\big(a({\\bs x},\\omega)-\\Ebb[a]({\\bs x})\\big)\n\\big(a({\\bs x}',\\omega)-\\Ebb[a]({\\bs x}')\\big)\\d\\Pbb(\\omega).\n\\]\nIf the expectation and the covariance are known, we may represent\n \\(a\\) by its \\emph{Karhunen-Lo\\`eve expansion}\n\\[\na({\\bs x},\\omega)=\\Ebb[a]({\\bs x})+\\sum_{k=1}^\\infty\\sqrt{\\mu_k} a_k({\\bs x})Y_k(\\omega)\n\\]\nHerein,  \\((\\mu_k,a_k)\\), \\(k=1,2,\\ldots\\),\ndenote the eigen pairs of the Hilbert-Schmidt operator\n\\[\n(\\mathcal{C}v)({\\bs x})\\mathrel{\\mathrel{\\mathop:}=}\\int_D\\operatorname{Cov}[a]({\\bs x},{\\bs x}')v({\\bs x}')\\d{\\bs x}',\n\\]\nwhile \\(Y_1,Y_2,\\ldots\\) are independent and standard normally distributed random variables.\n\nIn order to discretize the Karhunen-Loeve expansion, we proceed in complete\nanalogy to \\cite{HPS15} and compute the orthogonal projection of \\(\\mathcal{C}\\) onto \\(V_J\\).\nLet \\({\\bs{C}}_J\\in\\mathbb{R}^{N_J\\times N_J}\\) denote the corresponding coefficient matrix. Obviously, a suitable\nbasis for the orthogonal projection is given by the orthogonalized wavelet basis\n\\(\\Psi_J{\\bs G}^{-1\/2}_J\\), where\n\\[\n{\\bs G}_J = \\big[(\\psi_{\\lambda'},\\psi_\\lambda)_{L^2(D)}\\big]_{\\lambda,\\lambda'}\n\\]\ndenotes the mass matrix and \\({\\bs G}^{-1\/2}_J\\) is an inverse matrix root.\nThus, in accordance with \\cite{HPS15}, we arrive at the discretized random field\n\\begin{equation}\\label{eq:KLexp}\na_J({\\bs x},\\omega)=\\Psi_J({\\bs x}){\\bs G}^{-1\/2}_J\\big(\\overline{\\bs a}_J + {\\bs V}{\\bs\\Sigma}{\\bs Y}(\\omega)\\big),\n\\end{equation}\nwhere \\({\\bs Y}\\mathrel{\\mathrel{\\mathop:}=}[Y_1,Y_2,\\ldots, Y_{N_J}]^\\intercal\\) is a standard normally\ndistributed Gaussian vector, \\(\\overline{\\bs a}_J\\) is the orthogonal projection of\n\\(\\mathbb{E}[a]\\) onto \\(V_J\\) and \\({\\bs V}{\\bs\\Sigma}{\\bs\\Sigma}{\\bs V}^\\intercal={\\bs C}_J\\)\nis the spectral decomposition of \\({\\bs C}_J\\). \n\nHence, for the Galerkin projection, we find\nthe identities\n\\[\n\\begin{aligned}\n\\overline{\\bs a}^{G}_J&\\mathrel{\\mathrel{\\mathop:}=} \\big[(\\mathbb{E}[a],\\psi_{\\lambda})_{L^2(D)}\\big]_\\lambda={\\bs G}^{1\/2}_J\n\\overline{\\bs a}_J,\\\\\n{\\bs C}^G_J&\\mathrel{\\mathrel{\\mathop:}=}\\big[(\\Ccal\\psi_{\\lambda'},\\psi_{\\lambda})_{L^2(D)}\\big]_{\\lambda,\\lambda'}\n={\\bs G}^{1\/2}_J{\\bs C}_J{\\bs G}^{1\/2}_J.\n\\end{aligned}\\] \nHence, we obtain for the expectation\n\\[\n\\overline{\\bs a}_J={\\bs G}^{-1\/2}_J\\overline{\\bs a}^{G}_J,\n\\]\nwhile the covariance satisfies\n\\[\n{\\bs C}_J =({\\bs V}{\\bs\\Sigma})({\\bs V}{\\bs\\Sigma})^\\intercal= {\\bs G}^{-1\/2}_J{\\bs C}^G_J{\\bs G}_J^{-1\/2}=({\\bs G}^{-1\/2}_J{\\bs L})({\\bs G}^{-1\/2}_J{\\bs L})^\\intercal,\n\\]\nwhere \\({\\bs L}{\\bs L}^\\intercal={\\bs C}^G_J\\) is the Cholesky decomposition of \\({\\bs C}^G_J\\).\n\nIt is\nwell known that any two matrix square roots\n\\({\\bs R}{\\bs R}^\\intercal=\\widehat{\\bs R}\\widehat{\\bs R}^\\intercal={\\bs C}_J\\)\nonly differ by an isometry. Consequently,\nthere exists an orthogonal matrix \\({\\bs Q}\\)\nsuch that \\({\\bs V}{\\bs\\Sigma}={\\bs G}^{-1\/2}_J{\\bs L}{\\bs Q}\\).\nIntroducing the transformed random vector \n\\(\n{\\bs X}(\\omega)\\mathrel{\\mathrel{\\mathop:}=}{\\bs Q}{\\bs Y}(\\omega)\n\\), we infer\n\\[\n\\int_\\Omega{\\bs X}(\\omega){\\bs X}^\\intercal({\\omega})\\d\\mathbb{P}(\\omega)\n={\\bs Q}\\int_\\Omega{\\bs X}(\\omega){\\bs X}^\\intercal({\\omega})\\d\\mathbb{P}(\\omega){\\bs Q}^\\intercal={\\bs Q}{\\bs I}{\\bs Q}^\\intercal={\\bs I}.\n\\]\nTherefore, \\({\\bs X}(\\omega)\\) is a standard normally\ndistributed Gaussian vector as well and we end up with the representation\n\\begin{equation}\\label{eq:CholFastField}\na_J({\\bs x},\\omega)=\\Psi_J({\\bs x}){\\bs G}^{-1}_J\n\\big(\\overline{\\bs a}^{G}_J + {\\bs L}{\\bs X}(\\omega)\\big).\n\\end{equation}\n\nIf the singular values in the Karhunen-Loeve expansion \\eqref{eq:KLexp} \ndecay only slowly, as it is typically the case for non-smooth\ncovariance functions, the numerical solution of the associated\neigenvalue problem becomes prohibitive. In such cases, the direct \nsimulation of the random field by means of a sparse Cholesky \ndecomposition in \\eqref{eq:CholFastField} is computationally \nsuperior. Therefore, the proposed method is in particular useful for\nrough random fields, which issue from non-smooth covariance \nfunctions. \n\nWe remark that in case of a smooth covariance function, the \nmatrix \\({\\bs C}_J^G\\) becomes numerically positive semidefinite.\nIn such cases, the pivoted Cholesky decomposition, see \\cite{HPS}, \nimmediately yields a low-rank decomposition of \\({\\bs C}_J^G\\) and \nis computationally superior to other data sparse representations,\nwe refer to \\cite{HPS15} for a comprehensive discussion of this \ncomplementary approach.\n\n\\section{Numerical Results}\\label{sec:numerix}\nIn order to obtain consistent computation times,\nall computations reported in this section have been\ncarried out on a single core of a compute server with \nIntel Xeon E5-2650 v3 @2.30GHz CPUs and 512GB DDR4 \n@2133MHz main memory. The implementation of the wavelet matrix\ncompression has been performed in ANSI C, while nested dissection\norderings, Cholesky decompositions and LU decompositions\nhave been computed using \\textsc{Matlab} 2020a, compare \n\\cite{MLB20}.\n\n\\subsection{Polarizable continuum model}\nFor PCM, we consider the assembly and the factorization of the\nmatrices\n\\[\n  {\\bs V}_J\\quad\\text{and}\\quad{\\bs A}_J\\mathrel{\\mathrel{\\mathop:}=}\n  \t\\frac{1+\\epsilon}{2}{\\bs G}_J+(1-\\epsilon) {\\bs K}_J,\n\\]\nwhich are required to set up the linear system\nof equations \\eqref{eq:PCM2}. The dielectric constant is chosen as \\(\\epsilon=78.39\\), \ncorresponding to the solvent water. The molecule under study is \nbenzene, whose solvent excluding surface is depicted in Figure\n\\ref{fig:benzene}. For the wavelet matrix compression, we use \npiecewise constant wavelets with three vanishing moments, \nas developed in \\cite{HS1}. \n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=0.16,clip,trim=290 30 280 110]{benzene}\n\\caption{\\label{fig:benzene}Solvent excluding surface of benzene.}\n\\end{center}\n\\end{figure}\n\n\\begin{table}[htb]\n\\begin{center}\n\\begin{tabular}{|r|r|r|r|r|r|r|}\\hline\n$N_J$ & $t_\\text{WEM}$ &  $t_\\text{ND}$ &   $t_\\text{Chol}({\\bs V}_J)$ &\n$t_\\text{LU}({\\bs A}_J)$ & $\\anz({\\bs V}_J)$  & $\\anz({\\bs L}_J)$ \\\\\\hline\n1456  &   0.96  &  0.032    &  0.19  &  0.22  & 134  &  209\\\\\n5824  &   7.10  &  0.15    &   0.19  &  1.59  & 163  &  328\\\\\n23296  &  52.32  &  0.74  &  1.05  & 14.03  & 159  &  419 \\\\\n93184  &  309.35 &   3.91   &  7.66  & 189.70  & 165  &  639\\\\\n372736  &  2265.91  &  19.38   & 58.28  & 2211.57  & 173  &  890\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:PCM}Computation times and \nnumbers of nonzero entries in case of PCM.}\n\\end{center}\n\\end{table}\n\nTable~\\ref{tab:PCM} shows the numerical results. The first \ncolumn labelled \\(N_J\\) corresponds to the number of surface\nelements. The second column labelled \\(t_\\text{WEM}\\) contains\nthe combined computation times in seconds for the assembly of \\({\\bs V}_J\\)\nand \\({\\bs A}_J\\) by the wavelet matrix compression. The third column, \nlabelled by \\(t_\\text{ND}\\), provides the computation times in \nseconds for the nested dissection ordering, as \\({\\bs V}_J\\)\nand \\({\\bs A}_J\\) have identical sparsity patterns, the same\nreordering can be applied to both of them. The fourth\ncolumn labelled by \\(t_\\text{Chol}({\\bs V}_J)\\) denotes\nthe computation times in seconds for the Cholesky factorization \nof \\({\\bs V}_J\\), while the fifth column shows the computation \ntimes in seconds of the LU decomposition of \\({\\bs A}_J\\). For \nthe purpose of measuring the fill-in relative to the matrix size, we\nintroduce for a sparse matrix \\({\\bs A}\\in\\mathbb{R}^{N\\times N}\\)\nthe \\emph{average number of nonzeros per row}\n\\[\n\\anz({\\bs A})\\mathrel{\\mathrel{\\mathop:}=} \\frac{\\nnz({\\bs A})}{N}.\n\\]\nThe values for \\(\\anz({\\bs V}_J)\\) and \\(\\anz({\\bs A}_J)\\) are \nidentical and can be found in the sixth row of Table~\\ref{tab:PCM},\nwhile the values \\(\\anz({\\bs L}_J)\\) for the Cholesky factor of \n\\(\\anz({\\bs V}_J)\\) are given in the last column of the table. The \nsparsity pattern of the\nL-factor of the LU decomposition of \\({\\bs A}_J\\) coincides with that\nof ${\\bs L}_J$, while the U-factor has somewhat less coefficients compared \nto the L-factor. Hence, the average number of nonzeros per row for \nthe LU factorization of \\({\\bs A}_J\\) matches that of \\(\\anz({\\bs L}_J)\\). \n\n\\begin{figure}[htb]\n\\begin{center}\n\\setlength{\\fboxrule}{0.2pt}\n\\begin{minipage}{0.31\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.4,clip,trim=0 0 0 13,frame]{Spattern_PCM}\n$\\nnz({\\bs V}_J) = 15\\,363\\,141$\n\\end{center}\n\\end{minipage}\n\\begin{minipage}{0.31\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.4,clip,trim=0 0 0 13,frame]{Lpattern_PCM}\n$\\nnz({\\bs L}_J) = 59\\,484\\,864$\n\\end{center}\n\\end{minipage}\n\\begin{minipage}{0.31\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.4,clip,trim=0 0 0 13,frame]{UKpattern_PCM}\n$\\nnz({\\bs U}_J) = 59\\,295\\,122$\n\\end{center}\n\\end{minipage}\n\\caption{\\label{fig:PCM}Sparsity patterns of \\({\\bs V}_J\\) as well as \\({\\bs A}_J\\) (left), \nthe Cholesky factor \\({\\bs L}_J\\) (middle), and the upper triangular matrix \\({\\bs U}_J\\)\n(right) for PCM and \\(N_J=93\\,184\\). Each dot corresponds to a submatrix of size \n\\(64\\times 64\\). Lighter blocks have less entries than darker blocks. }\n\\end{center}\n\\end{figure}\n\nThe sparsity patterns for \\({\\bs V}_J\\), \\({\\bs L}_J\\) and for the U-factor \nare shown in Figure~\\ref{fig:PCM} for \\(N_J=93\\,184\\). In order \nto obtain a neat representation in this figure, we have merged\nmatrix blocks of size \\(64\\times 64\\). Darker blocks have a higher\ndensity of entries, while lower blocks have a lower density of entries. \n\nAs can clearly be inferred from Table~\\ref{tab:PCM}, the \ntimes for the computing the nested dissection ordering and the \nsubsequent Cholesky decomposition are negligible with respect \nto the wavelet matrix compression. The average number of nonzero \nentries per row stays rather low and only increases for increasing \nnumbers of unknowns up to a factor of approximately 5 for \\(N_J=372\\,736\\).\n\nIt can be seen from the fifth column of Table~\\ref{tab:PCM} that\nthe LU decomposition is significantly slower than the Cholesky\ndecomposition. This issues from the fact that all matrices are \nstored in a sparse column major format, resulting in a large \noverhead for the access of matrix rows. We remark that, in \nprinciple, the LU decomposition could be accelerated by an \nappropriate data structure, which enables direct access \nto the rows of the matrix as well.\n \n\\subsection{Parabolic diffusion problem for the fractional Laplacian}\nWe consider the parabolic problem for the fractional Laplacian with \n\\(s=\\nicefrac{3}{8}\\). The right hand side is a Gaussian heat\nspot moving on a circular trajectory, given by\n\\[\nf({\\bs x},t)=100\\exp\\Big(-40\\big(x_1-\\cos(2\\pi t)\\big)^2-40\\big(x_2-\\sin(2\\pi t)\\big)^2\\Big),\n\\]\nwhile the initial condition is set to 0.\nFor the solution of the ordinary differential equation in time, \nwe employ the \\(\\theta\\)-scheme with \\(\\theta=\\nicefrac{1}{2}\\), which \nyields the Crank-Nicolson method \\cite{CN}. The time interval \nis given by \\([0,T]=[0,3]\\) and we discretize this time interval\nby 150 equidistant time steps. \n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=0.14,clip,trim=510 160 510 160]{disk}\n\\caption{\\label{fig:disc}Computational geometry for the fractional Laplacian.}\n\\end{center}\n\\end{figure}\n\nIn the \\(\\theta\\)-scheme \\eqref{eq:theta}, we need to invert the matrix\n\\[\n{\\bs A}_J={\\bs G}_J + \\theta(t_{i+1}-t_i) {\\bs L}_J,\n\\]\nwhich is assembled with the help of wavelet matrix compression\nusing Haar wavelets. The computational geometry is the unit disc \ndepicted in Figure~\\ref{fig:disc}. The solution on the space-time \ncylinder is visualized for \\(N_J= 20\\,480\\) in Figure~\\ref{fig:solFL}.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{tikzpicture}\n\\draw(0,0) node{\\includegraphics[scale=0.1,clip,trim=1050 120 100 120]{fractionalNew.png}};\n\\fill[color=white] (1.3,-2) rectangle (3,2);\n\\draw(1.45,1.2) node {$14$};\n\\draw(1.45,0.6) node {$10$};\n\\draw(1.45,0.25) node {$\\phantom{0}8$};\n\\draw(1.45,-0.1) node {$\\phantom{0}6$};\n\\draw(1.45,-0.45) node {$\\phantom{0}4$};\n\\draw(1.45,-0.8) node {$\\phantom{0}2$};\n\\draw(1.45,-1.15) node {$\\phantom{0}0$};\n\\draw(2.4,0) node[rotate=90] {Temperature};\n\n\\end{tikzpicture}\n\\caption{\\label{fig:solFL}Solution of the fractional Laplacian on the space-time\ncylinder.}\n\\end{center}\n\\end{figure}\n\n\\begin{table}[htb]\n\\begin{center}\n\\begin{tabular}{|r|r|r|r|r|r|r|}\\hline\n$N_J$ & $t_\\text{WEM}$ &  $t_\\text{ND}$ & $t_\\text{Chol}({\\bs A}_J)$ & \n\\(t_{\\theta}\\)  & $\\anz({\\bs A}_J)$  & $\\anz({\\bs L}_J)$ \\\\\\hline\n   1280 & 0.63       &  0.02   &   0.01  & 0.19     &      88   &  140\\\\\n   5120 &  3.35      &  0.11   &   0.10  & 0.96     &     106   &  222\\\\\n  20480 &  22.20     &  0.53   &   0.59  & 5.34     &     117   &  315\\\\\n  81920 &  161.83    &  2.71   &   3.89  & 27.94    &     129   &  474\\\\\n 327680 &   724.60   & 12.87   &  23.49  & 131.30   &     135   &  576\\\\\n1310720 &  3875.30   & 62.62   & 161.65  & 633.50   &     142   &  743\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:FLap}Computation times and numbers of \nnonzero entries in case of the fractional Laplacian.}\n\\end{center}\n\\end{table}\n\nWe have tabulated the similar characteristics from the previous \nexample for the matrix \\({\\bs A}_J\\) in Table~\\ref{tab:FLap}, while\nthe sparsity patterns of \\({\\bs A}_J\\) and \\({\\bs L}_J\\) for \\(N_J=81\\,920\\)\nafter reordering are shown in Figure~\\ref{fig:FractionalS}. We observe \na similar behaviour as for PCM, the computation times for nested \ndissection and the Cholesky decomposition are negligible compared \nto the wavelet matrix compression. In view of the fill-in, there is \nan increase of approximately a factor of 5 for the average number \nof nonzero entries per row for \\(N_J=1\\,310\\,720\\). The computation \ntimes for the \\(\\theta\\)-scheme in seconds for the 150 time steps are shown\nin the column labelled by \\(t_{\\theta}\\). As can be seen, we obtain a\nsolution time of roughly 11 minutes for \\(N_J=1\\,310\\,720\\) unknowns\nin the spatial discretization.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\setlength{\\fboxrule}{0.2pt}\n\\begin{minipage}{0.49\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.55,clip,trim=0 0 0 13,frame]{Spattern_Frac}\n$\\nnz({\\bs A}_J) = 10\\,518\\,832$\n\\end{center}\n\\end{minipage}\n\\begin{minipage}{0.49\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.55,clip,trim=0 0 0 13,frame]{Lpattern_Frac}\n$\\nnz({\\bs L}_J) = 38\\,819\\,059$\n\\end{center}\n\\end{minipage}\n\\caption{\\label{fig:FractionalS}Sparsity patterns of \\({\\bs A}_J\\) (left) and the Cholesky factor\n\\({\\bs L}_J\\) (right) for the fractional Laplacian and \\(N_J=81\\,920\\). Each dot corresponds to \na submatrix of size \\(64\\times 64\\). Lighter blocks have less entries than darker blocks.}\n\\end{center}\n\\end{figure}\n\n\\subsection{Gaussian random field}\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=0.14,clip,trim= 360 75 360 75]{plate}\n\\caption{\\label{fig:randomFields}Computational geometry for the Gaussian random field.}\n\\end{center}\n\\end{figure}\n\nFor the simulation of a Gaussian random field, we consider an\nL-shaped domain with three holes, compare Figure~\\ref{fig:randomFields}.\nThe domain has a side length of 4, while the holes have a diameter \nof 0.8. For the wavelet matrix compression, we use piecewise bilinear \nbiorthogonal wavelets with four vanishing moments, see \\cite{HS1}.\nThe expectation is set to \\(\\mathbb{E}[a]({\\bs x})\\equiv 0\\), while the\ncovariance is given by the exponential kernel\n\\[\n\\operatorname{Cov}[a]({\\bs x},{\\bs x}')=\\exp(-\\|{\\bs x}-{\\bs x}'\\|_2).\n\\]\nFour different realizations of the corresponding field in case of \n\\(N_J=792\\,688\\) nodes are depicted in Figure~\\ref{fig:FieldReals}.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{tikzpicture}\n\\draw(0,0) node {\\includegraphics[scale=0.11,clip,trim= 380 120 610 60]{field3}};\n\\draw(5,0) node {\\includegraphics[scale=0.11,clip,trim= 380 120 610 60]{field4}};\n\\draw(5,5) node {\\includegraphics[scale=0.11,clip,trim= 380 120 610 60]{field2}};\n\\draw(0,5) node {\\includegraphics[scale=0.11,clip,trim= 380 120 610 60]{field1}};\n\\draw(9.2,2.25) node {\\includegraphics[scale=0.194,clip,trim= 1600 400 50 420]{field1}};\n\\fill[color=white] (8.5,0.5) rectangle (10.8,4);\n\\draw(9,3.8) node {$\\phantom{-}3.8$};\n\\draw(9,2.98) node {$\\phantom{-}2.0$};\n\\draw(9,2.54) node {$\\phantom{-}1.0$};\n\\draw(9,2.1) node {$\\phantom{-}0\\phantom{.0}$};\n\\draw(9,1.65) node {$-1.0$};\n\\draw(9,1.22) node {$-2.0$};\n\\draw(9,0.75) node {$-3.0$};\n\\draw(10.1,2.25) node[rotate=90] {Value};\n\\end{tikzpicture}\n\\caption{\\label{fig:FieldReals}Four different realizations of the Gaussian random field.}\n\\end{center}\n\\end{figure}\n\nThe pattern of the covariance matrix \\({\\bs C}_J^G\\) is shown in \nFigure~\\ref{fig:RandomFieldC}, while the patterns of the reordered \nmatrix and its Cholesky factor are provided in Figure~\\ref{fig:CholRF}.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\setlength{\\fboxrule}{0.2pt}\n\\begin{minipage}{0.49\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.55,clip,trim=0 0 0 13,frame]{SWpattern_RF}\n$\\nnz({\\bs C}_J^G) = 122\\,684\\,812$\n\\end{center}\n\\end{minipage}\n\\caption{\\label{fig:RandomFieldC}Sparsity patterns of \\({\\bs C}_J^G\\) \nof the random field for \\(N_J=199\\,692\\). Each dot corresponds to a submatrix\nof size \\(108\\times 108\\). Lighter blocks have less entries than darker blocks.}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\setlength{\\fboxrule}{0.2pt}\n\\begin{minipage}{0.49\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.55,clip,trim=0 0 0 13,frame]{Spattern_RF}\n$\\nnz({\\bs C}_J^G) = 122\\,684\\,812$\n\\end{center}\n\\end{minipage}\n\\begin{minipage}{0.49\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.55,clip,trim=0 0 0 13,frame]{Lpattern_RF}\n$\\nnz({\\bs L}_J) = 722\\,864\\,637$\n\\end{center}\n\\end{minipage}\n\\caption{\\label{fig:CholRF}Sparsity patterns of \\({\\bs C}_J^G\\) (left) and \nthe Cholesky factor \\({\\bs L}_J\\) (right) for the random field and \\(N_J=199\\,692\\). \nEach dot corresponds to a submatrix of size \\(108\\times 108\\). Lighter blocks \nhave less entries than darker blocks.}\n\\end{center}\n\\end{figure}\n\nTable~\\ref{tab:RF} shows the computation times and the average \nnumbers of nonzero entries as in the previous examples. In addition,\nwe have the column \\(t_{\\text{Sample}}\\), which contains the times for \ncomputing a single realization of the random field in seconds. These \ntimes have been computed by averaging the computation times over 1000 samples.\nIn order to compute the inverse of the mass matrix, we could in\nprinciple reuse the nested dissection ordering which has been computed for\nthe system matrix, as the pattern of the mass matrix is a subset of the \npattern of the system matrix. However, this will result in a fill-in similar \nto the system matrix. Hence, it is favourable to use a different ordering \nfor the mass matrix, resulting in much less fill-in.\n\n\\begin{table}[htb]\n\\begin{center}\n\\scalebox{0.93}{\n\\begin{tabular}{|r|r|r|r|r|r|r|r|r|}\\hline\n$N_J$ & $t_\\text{WEM}$ &  $t_\\text{ND}$ &   $t_\\text{Chol}({\\bs C}_J^G)$ &\n$t_\\text{Sample}$ & $\\anz({\\bs C}_J^G)$  & $\\anz({\\bs L}_J)$ &$\\anz({\\bs G}_J)$  &\n$\\anz(\\widehat{\\bs L}_J)$\\\\\\hline\n    972 &  1.50 & 0.04 &  0.23 &$6.25\\!\\cdot\\!10^{-4}$  &      249  &     251\n&  53 & 31\\\\ \n   3468 & 14.49 & 0.16 &  0.26 &$3.97\\!\\cdot\\!10^{-3}$  &      338  &     498\n&  91 & 72\\\\\n  13068 &  119.44 & 0.83 &  2.22 &$3.13\\!\\cdot\\!10^{-2}$  &      429  &    1096\n& 126 &124\\\\\n  50700 & 777.69 & 4.45 & 21.45 &$2.18\\!\\cdot\\!10^{-1} $  &     517   &   1998\n&156  &196\\\\\n 199692 &  4368.40 &24.34 & 222.72 &  $1.51\\!\\cdot\\!10^{0\\phantom{-}} $  &     615   &   3620\n&184  &283\\\\\n792588 &   29376.29 &   130.23 & 4290.29 & $1.61\\!\\cdot\\!10^{1\\phantom{-}}$&   714 &   8336 &   211 &   390\\\\\n\\hline\n\\end{tabular}}\n\\caption{\\label{tab:RF}Computation times and numbers of \nnonzero entries in case of the Gaussian random field.}\n\\end{center}\n\\end{table}\n\nAs can be seen from Table~\\ref{tab:RF}, the times for sampling \nthe random field only increase moderately. We remark that, due to the \nlarger supports of the bilinear wavelets and the higher precision of the \ndiscretization, the system matrix contains more entries per row on average.\nThis also leads to higher computation times for the matrix assembly.\nHowever, as before the increase of nonzero entries in the Cholesky factor \nremains very moderate. In addition, we have provided the average number \nof nonzeros per row for the mass matrix in the column labelled by \n$\\anz({\\bs G}_J)$. The number of nonzeros for the corresponding \nCholesky factor of the mass matrix is given in the column labelled \nby $\\anz(\\widehat{\\bs L}_J)$.\n\n\\section{Conclusion}\\label{sec:conclusion}\nIn this article, we have proposed a very efficient direct\nsolver for nonlocal operators. The pivotal idea is to\ncombine the wavelet matrix compression with the nested dissection\nordering. Thereby, the fill-in resulting from a Cholesky\ndecomposition or an LU decomposition is drastically reduced. \nThis has been numerically investigated into depth for three \nrelevant applications, namely the polarizable continuum model, \na parabolic problem for the fractional Laplacian in integral \nform, and the fast simulation of Gaussian random fields.\nIn all three cases, wavelet matrix compression yields sparse \nsystem matrices while the fill-in in the matrix factorization \nremains very low thanks to the nested dissection ordering. \nThis behaviour could be observed for more then \\(10^6\\) \nunknowns in the discretization. \n\nA formidable application of the presented approach is the\nfast simulation of rough (Gaussian) random fields. In such \ncases, the numerical solution of the eigenvalue problem for \nthe covariance is computationally prohibitive and, hence, the \ncomputation of a Karhunen-Lo\\`eve expansion is not feasible. \nIn turn, the use of a wavelet basis yields a sparse representation \nof the covariance operator and a matrix root is rapidly computable \nby employing the nested dissection ordering and the Cholesky \ndecomposition.\n\n\\subsection*{Acknowledgement}\nThis research has been funded in parts by the Swiss National Science \nFoundation (SNF) under project number $407540\\_167186$ NFP 75 Big Data.\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{sec_intro}\n\nDirect imaging of exoplanets is a field of research in rapid development. The past year has seen a number of interesting planet candidates imaged directly around stars. Most notable is arguably HR 8799, showing three planetary companions (see Marois et al. 2008) so far. These planets have been shown to exhibit Keplerian motion around the star, and have estimated masses in the range of 7-10 $M_{\\rm jup}$ from theoretical models (Baraffe et a. 2003). The system is known to also host a debris disk (e.g. Moor et al. 2006; Rhee et al. 2007). Along with the fact that there are three almost equal-mass companions, all orbiting the star in what appears to be a co-planar manner, this implies that the objects most likely formed in a circumstellar disk, by a process distinct from the star or brown dwarf formation process. This can be contrasted with the case of the 2M1207 system (Chauvin et al. 2005), where the low primary-to-secondary mass ratio is more reminiscent of a brown dwarf binary system than a star-planet system. Two other intriguing planet candidates in systems with debris disks were reported around the same time as HR 8799 b\/c\/d: Fomalhaut b (Kalas et al. 2008) and Beta Pic b (Lagrange et al. 2009), though additional follow-up observations would be desirable to provide more information on these systems. \n\n$\\epsilon$ Ind A is a K4V-type southern sky star located at a distance of 3.6 pc, with a very high proper motion of 4.7$''$ yr$^{-1}$ (Perryman et al. 1997), see Table \\ref{indtab0}. Comoving on the sky at the same rate, at a separation of 6.7$'$, is $\\epsilon$ Ind B, which was first detected by Scholz et al. (2003), and shortly thereafter resolved into the brown dwarf binary $\\epsilon$ Ind Ba\/Bb (McCaughrean et al. 2004) as it is known today. Being the most nearby binary brown dwarf, and with a physical separation small enough to determine its dynamical mass within a reasonable timeframe, $\\epsilon$ Ind Ba\/Bb will be a benchmark object for the physical understanding of brown dwarfs. However, in addition to the Ba\/Bb components, the $\\epsilon$ Ind system may provide an additional possibility to study an even lower-mass and cooler object. $\\epsilon$ Ind A displays a linear radial velocity trend (Endl et al. 2002; Zechmeister et al., in prep.) which, unless due to some exotic stellar remnant, is indicative of a giant planet or very low-mass brown dwarf companion. If this object could be directly imaged, it would constitute yet another important benchmark object, given its probable low mass, low temperature, and the possibility to estimate both its luminosity as well as its dynamical mass within a reasonable timeframe. It would also be the closest planet or very low-mass brown dwarf companion directly detected outside of our Solar System.\n\n\\begin{table}\n   \\centering\n\\caption[]{Summary of $\\epsilon$ Ind A properties.}\n         \\label{indtab0}\n\\begin{tabular}{ll}\n            \\noalign{\\smallskip}\n            \\hline\n            \\noalign{\\smallskip}\n                 Property & Value \\\\\n            \\noalign{\\smallskip}\n            \\hline\n            \\noalign{\\smallskip}\nRight Ascension  & 22 03 21.66 \\\\\nDeclination   & -56 47 09.5 \\\\\nSpectral Type  & K4.5V \\\\\nDistance & $3.626 \\pm 0.009$ pc \\\\\nProper Motion  & 4705 mas yr$^{-1}$  \\\\\nMass & 0.7 $M_{\\rm sun}$  \\\\\nAge & 1-5 Gyr \\\\\n            \\noalign{\\smallskip}\n            \\hline\n\\end{tabular}\n\\end{table}\n\nHere, we will present our deep imaging campaign of $\\epsilon$ Ind, using narrow-band 4$\\mu$m imaging with two different high-contrast techniques. One technique is pure active angular differential imaging (aADI) as already implemented and demonstrated as a powerful technique for detecting close companions to bright nearby stars (Janson et al. 2008). In aADI (also known as roll subtraction), images are taken at two different instrument rotator angles, and one is subtracted from the other, thereby removing the bulk of the stellar PSF including static instrumental speckles. The other technique is a combination of PSF reference subtraction and aADI (PSFR+aADI), using the LOCI (locally optimized combination of images) algorithm developed by Lafreniere et al. (2007). We also compare the performance of aADI at 4$\\mu$m with the same technique in the L'-band. The concept of using aADI with L' was proven by Kasper et al. (2007; the usefulness of L' for high-contrast imaging purposes was also independently demonstrated by Hinz et al. 2006). The concept of using NB4.05 to enhance the physical contrast was introduced in Janson et al. (2008).\n\nThe outline is as follows: In Sect. \\ref{sec_obs}, we summarize the observational parameters and ambient conditions of the observing runs. The two paths of data reduction employed are described in Sect. \\ref{sec_datared}. This is followed by a presentation of the results and the associated analysis in Sect. \\ref{sec_results}, including a comparison between filters (Sect. \\ref{sec_filters}), a comparison between techniques (Sect. \\ref{sec_methods}), a discussion of what we can learn from the dynamical input in Sect. \\ref{sec_dynamics}, and the final detection limits and their interpretation in Sect. \\ref{sec_detlimits}. Finally, we conclude in Sect. \\ref{sec_conclusions}.\n\n\\section{Observations}\n\n\\label{sec_obs}\n\nThe data presented here are based on two different sets of VLT\/NACO observations of $\\epsilon$ Ind. One set of observations consisted of deep imaging with aADI in the NB4.05 filter, executed in service mode and split into two equal observing blocks (henceforth observations A1 and A2), on 31 Oct 2008 and 2 Nov 2008. The other set of observations were taken in visitor mode on 3 Jul 2008, as part of a larger survey searching for planets around a volume-limited sample of nearby stars (Apai et al., in prep.). Those observations were less deep, but consisted of aADI imaging in both the NB4.05 and broad-band L' filters (henceforth observations B1 and B2, respectively). For observations A1 and A2, the same strategy was used as in previous observations (see Janson et al. 2008): Jittering was applied to enable a good subtraction of the thermal background, and the aADI was performed at two different instrument rotator angles, using a differential angle of 33 degrees. For B1 and B2, a four-point large-throw dithering scheme was applied for the background subtraction purposes, and the differential angle used for aADI was 20 degrees. All observations were taken with the L27 objective, providing a pixel scale of 27 mas\/pixel, and a field of view of 28$''$ by 28$\"$. The weather conditions and observational parameters of each run are listed in Table \\ref{indtab1}.\n\n\\begin{table}\n   \\centering\n\\caption[]{Observational conditions for the four runs.}\n         \\label{indtab1}\n\\begin{tabular}{lllll}\n            \\noalign{\\smallskip}\n            \\hline\n            \\noalign{\\smallskip}\n                 ~ & A1   & A2   & B1 & B2\\\\\n            \\noalign{\\smallskip}\n            \\hline\n            \\noalign{\\smallskip}\nDate (2008)         & 31 Oct & 2 Nov & 3 Jul & 3 Jul \\\\\nFilter          & NB4.05 & NB4.05 & NB4.05 & L'  \\\\\nSeeing$^a$    & 0.9$''$    & 0.7$''$    & 1.1$''$ & 1.0$''$  \\\\\nStrehl$^b$    & 83\\%      & 79\\%      & 85\\% & 83\\% \\\\\nHumidity      & 8\\%         & 13\\%        & 3\\% & 3\\%  \\\\\nCoh. time$^a$ & 2.6 ms      & 3.4 ms      & 2.3 ms & 2.4 ms \\\\\nFrames             & 19          & 19$^c$          & 30 & 33 \\\\\n(per angle)        &      ~      &     ~       &       ~  & ~  \\\\\nDIT                & 1.0 s       & 1.0 s       & 0.2 s & 0.2 s \\\\\nNDIT               & 61          & 61          & 150  & 150  \\\\\nTot. time          & 1159 s      & 1159 s$^c$      & 900 s & 990 s  \\\\\n(per angle)        &      ~      &     ~       &       ~  & ~  \\\\\n\n            \\noalign{\\smallskip}\n            \\hline\n\\end{tabular}\n\\begin{list}{}{}\n\\item[$^{\\mathrm{a}}$] Values given by the atmospheric seeing monitor at a wavelength of 500 nm.\n\\item[$^{\\mathrm{b}}$] Strehl ratio given by the AO system, rescaled to the observing wavelengths.\n\\item[$^{\\mathrm{c}}$] Two frames were de-selected for 0$^{\\rm o}$, hence the effective time is 1037 s for that case.\n\\end{list}\n\\end{table}\n\n\\section{Data reduction}\n\n\\label{sec_datared}\n\nThe data reduction for A1 and A2 was done differently with respect to the two different techniques applied, hence separate descriptions are provided below. The reduction of B1 and B2 was only done with aADI.\n\n\\subsection{Reduction for the purpose of aADI}\n\n\\label{sec_aadired}\n\nSince observations A1 and A2 were taken in the same way as the observations of Janson et al. (2008), largely the same data reduction could be applied as in the case for aADI purposes: The most basic reduction steps (e.g. flat fielding, bad pixel removal, background subtraction) were provided by the ESO automatic pipeline. The subsequent steps were performed with our dedicated IDL pipeline: The images were shifted using bilinear interpolation to a common center determined through cross-correlation, and to an absolute center using center of gravity. Low-frequency filtering was applied by subtracting a smoothed counterpart of each image produced by convolution with a Gaussian kernel with 0.5$''$ FWHM (the FWHM of the stellar PSF was about 120 mas). All the images corresponding to different rotation angles were subtracted from each other, and the results from the two nights were coadded. As an alternative analysis, all images were also de-rotated back to a common angle and coadded. This procedure yields a single co-added signature of any companion, which provides a useful alternative way to look at the data in the background-limited regime, with respect to the aADI-subtracted data which instead produces two independent signatures of half the amplitude each.\n\nFor observations B1 and B2, the data reduction was performed with the IDL routines as described above, with the exception that the centering was determined based on a sub-frame of 200x200 pixels around the star instead of the whole frame. This was done in order to avoid influence from residual features from the sky subtraction and the different dithering scheme.\n\nThe residuals as function of separation from the star were calculated from the standard deviation of all pixels in an annulus corresponding to each separation step. The physical brightness contrast was derived from the 3$\\sigma$ residuals by division of the peak value of the primary. Since the primary was saturated in the science images (the saturation radius was about 5-6 pixels), this had to be calibrated. For runs A1 and A2, this was done by introducing a neutral density filter into the optical path during acquisition, thus getting non-saturated images of the primary, allowing to determine a renormalized peak value. For run B1, the same non-saturated images as for A1 and A2 were used, which could be done since the Strehl ratio was stable and almost equal between the epochs. For B2, an image was taken of a fainter photometric standard star during the night.\n\n\\subsection{Reduction for the purpose of PSFR+aADI}\n\n\\label{sec_psfrred}\n\nIn applications involving LOCI, it is preferable to maximize the number of PSF representations of a target or reference star, hence for this case, reduction was done on individual frames, with the combination of frames only performed at the very end. The PSF star used was $\\epsilon$ Eri, which is practically ideal for the purpose given the similar spectral type, brightness, and the fact that observations exist taken under almost identical circumstances. All individual target and reference star frames were manually subjected to flat fielding, dark subtraction and bad pixel correction using calibration frames provided by ESO. Low-frequency filtering was applied as described above. A master sky frame was then produced by taking the median of the frames, where the stellar image is randomly placed in each frame, thereby removing the star altogether. The individual frames revealed ring-like structures in the background that could be reproduced in the master sky frame. By subtracting the master sky frame from each individual frame, the pattern could be removed. The pattern was found to be constant during the extent of an observation, but variable between observations (e.g., the frames corresponding to $\\epsilon$ Ind and $\\epsilon$ Eri were different from each other), and is probably related to instrumental dust emitting at 4$\\mu$m. The full background subtraction obtained in this way was found to be equally good as that delivered by the ESO 'jitter' routine. It is interesting to note, that given the fact that the pattern is constant during an observation, it should be possible to calibrate it out of a generic observation by making a master sky observation directly before or after the target observation. Hence, for any observation dedicated to the detection of point sources, it should be possible to achieve the same degree of background subtraction with and without jittering. This is an important realization with respect to high-contrast imaging at these wavelengths using techniques such as passive ADI (pADI) or coronagraphy, where it is desirable to maintain the stellar primary at a fixed position on the detector, and simultaneously achieve the best possible background subtraction. In summary, there appears to be no conflict between these two requirements, as long as the master sky calibration step is performed during observations.\n\nPSFR and aADI were performed separately, in sequence. For PSFR, every target and reference image was de-rotated such that the spider patterns were aligned. For each target frame, an optimized PSF reference frame was then produced from the full set of reference frames using the LOCI (Lafreniere et al. 2007) algorithm and subtracted from the target frame. The optimization was performed in 10 regions, five for the image range contaminated by the four spiders, covering different radial sections of the PSF, and five for the image range not contaminated by spiders, also covering different radial sections. The spider optimization areas were rectangular with a fixed width of 25 pixels, inner radii of 10, 20, 40, 70, and 120 pixels, and outer radii of 60, 70, 90, 150, and 200 pixels. The remaining areas were annuli excluding the spider regions, between inner radii of 20, 30 40, 50, and 60 pixels and outer radii of 50, 60, 70, 80, and 100 pixels. The subtractions were performed sequentially outwards with the subtraction zone defined from the inner radius of the optimization zone and outwards. Following this procedure, each of the target frames were re-rotated to their true parallactic angle. The aADI step was then performed through a second LOCI PSF construction, using all 33$^{\\rm o}$ frames as PSF library for each 0$^{\\rm o}$ frame, and vice versa. For this case, the optimization regions were simply five annuli between inner radii of 10, 20, 30, 50, and 70 pixels and outer radii of 40, 50, 60, 70 and 100 pixels. The optimization regions were chosen to provide a good balance between the two main criteria of LOCI: to maximize the efficiency of stellar PSF structure subtraction, and minimize subtraction of actual companions. The latter was tested by generating a series of runs where false companions had been introduced in the target frame -- in total 3600 companions distributed between 10 and 100 pixels separation from the center of the star, and over all azimuthal angles. The partial subtractions in each case were used to construct a radial profile of conserved companion flux fractions. As expected, a significant flux loss occurs at 10 pixels, but decreases rapidly outwards. At 100 pixels, the fraction of restored companion flux approaches unity, as indeed expected, given that the LOCI optimization is not applied beyond 100 pixels for the vast majority of the image space. \n\nFinally, all frames corresponding to each rotator angle were combined using 3$\\sigma$-clipping. The radial profile of residuals was created in the same way as for aADI, but with the additional step that it was normalized by the radial profile of conserved companion flux fraction to provide an accurate measure of the actual achieved contrast.\n\n\\section{Results and discussion}\n\n\\label{sec_results}\n\nThe output images from runs A1+A2 from each of the two reduction paths are shown in Fig. \\ref{ei_imgs_mrg}. No companion candidates were detected in the images. In the following, we discuss the implications of this result, and compare the methods used.\n\n   \\begin{figure*\n   \\centering\n  \n   \\includegraphics[width=15cm]{mj_fig1.eps}\n\\caption{Final output images for aADI (top) and PSFR+aADI (bottom) to the same flux scale, for the images A1+A2. The most obvious difference between the methods is in the treatment of the spiders.}\n\\label{ei_imgs_mrg}\n    \\end{figure*}\n\n\\subsection{Comparison between filters}\n\n\\label{sec_filters}\n\nAlthough the B1 and B2 images are less deep than A1+A2, the fact that they were obtained for the same target at about the same time, and with an almost identical observational setup, makes them ideal for comparing L' and NB4.05 imaging for planet detection purposes around bright stars. A comparison was already made in Janson et al. (2008) between L' aADI, NB4.05 aADI, and SDI+aADI (from Janson et al. 2007). While a fully relevant comparison could be made between SDI+aADI and NB4.05 aADI, where NB4.05 aADI was found to perform better under all circumstances, the comparison with L' was preliminary, since no comparable data was available. Instead, the comparison between L' and NB4.05 was based entirely on physical contrast given by the theoretical models, and the instrumental contrast was assumed to be the same. While this is relevant for a large part of the parameter space, there will in reality be differences in instrumental contrast due to differences in Strehl ratio, PSF diffraction, and thermal background between the filters. Using the B1 and B2 observations, we can now provide a comparison that takes all these issues into account.\n\nThe comparison was done by translating the brightness contrasts into mass detection limits using the spectral and photometric evolutionary models of Baraffe et al. (2003) and Burrows et al. (2003) for various ages. The method is described in detail in Janson et al. (2008). Note that the comparison is done for almost identical observing time, and with virtually no difference in overheads, i.e. the telescope time investment is also the same in both cases. As expected, the instrumental contrast is almost identical in the contrast-linited range, confirming the assumptions of the previous analysis, and thus the difference in the inner range is almost entirely set by the expected flux distribution of the companion. We show an example that demonstrates the favourable spectral range of NB4.05 in Fig. \\ref{xmpl_spec_sm}, for 10$M_{\\rm jup}$ and 15$M_{\\rm jup}$ objects, at an age of 1 Gyr. The flux density is higher in NB4.05 than in both L'-band and M-band. For cooler objects, the bulk of the flux moves redward, hence M-band becomes better in terms of flux density, but the thermal background is also much worse in M-band. The improvement of NB4.05 over L'  increases further for cooler objects.\n\n   \\begin{figure\n   \\centering\n  \n   \\includegraphics[width=8.0cm]{mj_fig2.eps}\n\\caption{Example of two model spectra from Burrows et al. (2003), and the corresponding flux densities in filters L', NB4.05, and M. Upper lines: A 15$M_{\\rm jup}$ object. Lower lines: A 10$M_{\\rm jup}$ object. The age is 1 Gyr in both cases.}\n\\label{xmpl_spec_sm}\n    \\end{figure}\n\nWe show the results of the detection limit comparison for 1 Gyr, 3 Gyr, and 5 Gyr in Fig. \\ref{brg_vs_lband}. It is seen that for all these ages, NB4.05 performs better in the contrast-limited inner part, and L' performs better in the outer background-limited part, as expected. The crossover point for $\\epsilon$ Ind A in our dataset is at about 4$''$. The position of the crossover point will vary as a function of stellar brightness and integration time. The brighter the star and the longer the integration time, the larger the parameter range where NB4.05 will be favourable,  and vice versa. We conclude that NB4.05 is likely to be an excellent choice for very deep planet search imaging close to bright stars, although it should be noted that this depends on the validity of the theoretical models. A first test of the models could be provided by the HR 8799 system.\n\n   \\begin{figure\n   \\centering\n  \n   \\includegraphics[width=8.0cm]{mj_fig3.eps}\n\\caption{Comparison between L' (dashed line) and NB4.05 (solid line) imaging for high-contrast purposes, for 1 Gyr (top panel), 3 Gyr, (middle panel), and 5 Gyr (bottom panel). As expected, NB4.05 provides a better performance than L' in the inner image range, and the opposite is true in the outer range. The comparison is based on sets B1 and B2, note that the A1+A2 detection limits are better.}\n\\label{brg_vs_lband}\n    \\end{figure}\n\n\\subsection{Comparison of aADI and PSFR+aADI}\n\n\\label{sec_methods}\n\nAs can be seen in the images (Fig. \\ref{ei_imgs_mrg}), the main difference between aADI and PSFR+aADI is that spiders are more efficiently removed in the latter case. However, the impact of this is largely cosmetic, as a comparable amount of flux is lost from the companion in the spider regions. This can be clearly seen in a comparison of the respective contrast curves for the two methods (see Fig. \\ref{comp_aadi_loci}), which have been normalized with respect to flux losses. PSFR+aADI slightly improves the performance at large separations, but provides no improvement at all for small separations. It should be noted that the results are based on a single PSF reference star (though with multiple representations) -- it would be preferable to use multiple reference stars, and doing so might substantially improve the performance. In any case, we do not reach as promising results as those achieved with PSFR using LOCI on space-based HST data (see Lafreniere et al. 2009), where a significant improvement over aADI is readily seen. As we have demonstrated, 4 $\\mu$m imaging provides a very high Strehl ratio, so if this was the limiting PSF stability factor at this level of contrast, we should have expected an improvement in the inner image range. Hence, the results imply that other PSF effects become dominant once the Strehl ratio is high enough, such as low-order aberrations arising in the telescope, and differences in PSF representation resulting from dithering. This in turn implies that a stable telescope configuration is the best way forward for improving the contrast in 4 $\\mu$m imaging even further. There is an obvious and well-tested technique for achieving this, called passive ADI, in which the pupil is stabilized during observations, while the field is allowed to rotate (see Marois et al. 2006). Indeed, the LOCI algorithm was originally designed for this purpose (Lafreniere et al. 2007). In fact, we have a passive ADI sequence at 4 $\\mu$m showing exquisite performance at small separations, but those data are taken with a different telescope and of a different target, so a rigorous comparison can not be made. The passive ADI data will be part of a separate publication.\n\n   \\begin{figure\n   \\centering\n  \n   \\includegraphics[width=8.0cm]{mj_fig4.eps}\n\\caption{Comparison of contrast for aADI (solid line) and PSRF+aADI (dashed line). The performance is generally very similar.}\n\\label{comp_aadi_loci}\n    \\end{figure}\n\n\\subsection{Input from dynamics}\n\n\\label{sec_dynamics}\n\nThere exist extensive radial velocity measurements of $\\epsilon$ Ind A, as well as some limited astrometric information, which can be used to constrain the properties of any sufficiently massive companion, as discussed in the following.\n\n\\subsubsection{The radial velocity linear trend}\n\n\\label{sec_rvtrend}\n\nThe linear radial velocity trend of $\\epsilon$ Ind A was first reported by Endl et al. (2002). The original dataset covered an observational baseline of about 5.2 years, taken with the ESO CES instrument in the period 1992-1998. Since then, HARPS data have been taken from 2003 to 2008 (Zechmeister et al., in prep.). The linear trend of the HARPS data is consistent with that of the aforementioned CES data, with a slope of 4.4 m s$^{-1}$ yr$^{-1}$. Hence, we adopt this slope over the 16 year total baseline. One contributor to the linear trend is secular acceleration. This is the apparent acceleration that an observer measures over time in projected motion (in this case along the line of site) of an object with constant velocity, due to the actual motion in 3D space. Using all the measured spatial coordinates and velocity components of $\\epsilon$ Ind, the secular acceleration can be calculated to 1.8 m s$^{-1}$ yr$^{-1}$. This is quite large, due to the fast motion of $\\epsilon$ Ind in the plane of the sky, but still leaves a 2.6 m s$^{-1}$ yr$^{-1}$ trend that must be due to actual acceleration.\n\nSince $\\epsilon$ Ind Ba\/Bb is known to be physically bound to $\\epsilon$ Ind A, it needs to be tested whether it could be responsible for the observed trend. We do this with the following order-of-magnitude estimate: The projected separation between A and Ba\/Bb is about 1500 AU, hence for masses of 0.7$M_{\\rm sun}$, 0.047$M_{\\rm sun}$, and 0.028$M_{\\rm sun}$ respectively (see McCaughrean et al. 2004), the orbital period of the A\/B system is at least 66000 years for a circular Keplerian orbit. Such an orbit would lead to a radial velocity semi-amplitude for $\\epsilon$ Ind A of 62 m s$^{-1}$, which in turn gives an average peak-to-peak acceleration of $4*10^{-3}$ m s$^{-1}$ yr$^{-1}$. Thus, the gravitational influence of $\\epsilon$ Ind Ba\/Bb is several orders of magnitude too small to make any significant contribution to the observed trend.\n\nWith $\\epsilon$ Ind Ba\/Bb out of the picture, we are left with closer, as of yet unseen companions. A previous H- and K-band imaging campaign (Gei{\\ss}ler et al. 2007) has excluded the presence of stellar and massive brown dwarf companions, down to 53 $M_{\\rm jup}$ outside of a projected separation of 1.5 AU and 21 $M_{\\rm jup}$ outside of 4.7 AU. This also excludes white dwarfs, since at ages up to several Gyrs, they are much brighter in H-and K-band than a 50$M_{\\rm jup}$ object (see e.g. Holberg \\& Bergeron 2006 and Baraffe et al. 2003). Stellar objects outside of the field of view can be excluded, as they would be detectable with wide-field or all-sky surveys such as 2MASS (Skrutskie et al. 2006). While more exotic forms of stellar remnants (e.g. neutron stars) can perhaps not be categorically excluded, for the remainder of this paper we will assume that the observed acceleration is due to a low-mass brown dwarf or giant planet. The combined constraints from the imaging and the radial velocity trend are given in Sect. \\ref{sec_detlimits}.\n\n\\subsubsection{Astrometry}\n\n\\label{sec_astrometry}\n\nAs will be seen in the following, the companion is expected to have a mass in the range of $\\sim$5-20$M_{\\rm jup}$, and an orbital semi-major axis in the range of $\\sim$10-20 AU. At the distance of the  $\\epsilon$ Ind system, this corresponds to a strong astrometric amplitude signature imposed on the primary of about 15-60 mas. However, with an orbital period of a few decades, it would not be possible to detect orbital motion with, e.g., \\textit{Hipparcos} data alone. On the other hand, one might expect a systematic difference between the proper motion as measured by \\textit{Hipparcos} versus that measured in long-term ground based monitoring, such as from the Fifth Fundamental Catalog (FK5). This type of signature is referred to as $\\Delta \\mu$ binarity, see Wielen et al. (2001). For $\\epsilon$ Ind, an approximate conversion between the FK5 and HIP systems implies that there is a difference between the \\textit{Hipparcos} and FK5 proper motions of $\\Delta \\mu _{\\alpha} = -0.23 \\pm 1.68$ mas yr$^{-1}$, and $\\Delta \\mu _{\\delta} = -2.5 \\pm 0.98$ mas yr$^{-1}$. This corresponds in total to a significance level of $F=2.54$, where the $F$ value is roughly to the same level of confidence as the equivalent $\\sigma$-number for Gaussian statistics (Wielen et al. 2001). In other words, there is an indication of a companion in the data, but not at a very high level of significance. We can make an order-of-magnitude estimation of whether these numbers are consistent with the RV companion by assuming that the orbital motion is completely averaged out in the FK5 data, that the orbit is circular, and that a sufficiently small fraction of the orbit is covered by \\textit{Hipparcos} such that local curvature in the motion during that period is negligible. The limiting cases quoted above then yield astrometric motions of $\\pi*15$ mas in 32 years and $\\pi*60$ mas in 89 years respectively, i.e. 1.5 mas yr$^{-1}$ and 2.0 mas yr$^{-1}$, both of which are consistent with the given $\\Delta \\mu$ within the errors. Hence, the astrometry is indeed consistent with the RV trend, though we reiterate that the significance is rather limited for the astrometry.\n\n\\subsection{Detection limits}\n\n\\label{sec_detlimits}\n\nSince A1+A2 are the deepest images, they provide the strongest detection limits, and therefore we concentrate on them in this section. The instrumental contrast for A1+A2 is determined at each separation as the maximum performance out of the aADI and PSFR+aADI contrasts at that separation. The corresponding mass limits for A1+A2, calculated in the same way as for B1 and B2 in section \\ref{sec_filters}, are shown in Fig. \\ref{mlim_rv} for ages of 1, 3, and 5 Gyr. Several age determinations exist pointing to an age in the range of 1 Gyr for $\\epsilon$ Ind (e.g. Lachaume et al. 1999; Barnes 2007). However, preliminary analysis of the astrometric masses of $\\epsilon$ Ind Ba\/Bb (Cardoso et al. 2008) implies that the components are probably under-luminous with respect to model predictions (Baraffe et al. 2003) at 1 Gyr, such that the $\\epsilon$ Ind system has to be significantly older, perhaps up to 5 Gyr, if the models are accurate (which may not be the case, see e.g. Dupuy et al. 2009). On the other hand, such an old age would be incompatible with the observed spectra of $\\epsilon$ Ind Ba and Bb according to the analysis of Kasper et al. (2009). It is with these uncertainties in mind that we consider the full range of 1 to 5 Gyr in our analysis.\n\n   \\begin{figure\n   \\centering\n  \n   \\includegraphics[width=8.0cm]{mj_fig5.eps}\n\\caption{Detection limits at 1, 3, and 5 Gyr for $\\epsilon$ Ind A. The solid line that increases outwards is the mass as function of semi-major axis corresponding to the 2.6 m s$^{-1}$ yr$^{-1}$ slope of the observed RV trend, and the dashed line is the reference for minimum projected separation at a typical inclination of 60$^{\\rm o}$, both under the assumption of a circular orbit. The dotted vertical line is the minimum semi-major axis from the RV baseline at $q=2$.}\n\\label{mlim_rv}\n    \\end{figure}\n\nAlso plotted is the mass as function of semi-major axis derived from the slope of the linear trend, under the assumption that the inclination is $60^{\\rm o}$ (the mean inclination of randomly oriented orbits). The minimum possible semi-major axis is set by the minimum possible period, which in turn is some multiple $q$ of the observational baseline. The exact value of $q$ depends on the amount of curvature present in the trend, the determination of which would be an over-interpretation of the data at hand. As discussed in Janson et al. (2008), $q=1$ would be the most conservative limit possible to set, but it is unrealistic, since it would require a discrete change in velocity state. Here, we set $q=2$, which is still conservative, and more realistic.\n\nThe mass limits and RV trend shown in Fig. \\ref{mlim_rv} provide a good illustration of the detectability of the dynamical companion under the assumption of a circular orbit. However, given the large eccentricity spread of the exoplanet population outside of 0.1 AU, it is necessary to perform more detailed simulations in order to constrain the possible physical and orbital parameters of the companion. The method for doing so is described in detail in Janson et al. (2008), and we follow it here for $q=2$. In brief, based on the empirical distribution of eccentricities for known exoplanets outside of 0.1 AU, we simulate all possible orbits and orbital phases and test whether they are consistent with the observed linear trend. The fraction of such orbits as function of semi-major axis is named $\\phi$. Out of these allowed orbits, we test what fraction would lead to a detectable companion. This fraction as a function of semi-major axis is termed $\\chi$. One addition has been made to this procedure with respect to what was presented in Janson et al. (2008): In the case of $\\epsilon$ Eri, the plane of the disk, the rotational plane of the star, and the orbital plane of the planet candidate $\\epsilon$ Eri b all gave a consistent orbital inclination of about 30$^{\\rm o}$, hence this number was fixed in the simulations. In the case of $\\epsilon$ Ind, we have no prior information of the inclination, hence it is treated as a free parameter in the simulations. This is done by performing the simulations over several different inclination angles and averaging the results. The input inclination angles are set to correspond to the actual probability of a given inclination occurring (i.e., accurately taking into account that the inclination is more likely to be edge-on than face-on). \n\n   \\begin{figure\n   \\centering\n  \n   \\includegraphics[width=8.0cm]{mj_fig6.eps}\n\\caption{Detection probability in our images as function of semi-major axis for 1, 3, and 5 Gyr. Also plotted is $\\phi$, the fraction of orbits at a given semi-major axis that are consistent with the linear RV trend, denoted with the subscript \"All\" to signify that it is independent of age, in contrast to $\\chi$.}\n\\label{probfig_q2}\n    \\end{figure}\n\nThe results of the simulation are shown in Fig. \\ref{probfig_q2}. It can be seen that if the age is 1 Gyr, the probability of detecting the companion is always about 90\\% or higher for any semi-major axis, hence since no companion is detected, it is quite unlikely that the system is that young. On the other hand, if the age is 3 Gyr, or even 5 Gyr as discussed above, there is still a substantial parameter range in which the companion could hide. Given these results, in approximate numbers we can constrain the planet or brown dwarf mass to about 5-20 $M_{\\rm jup}$ and its semi-major axis to about 10-20 AU. Also, the inclination must be larger than at least 20$^{\\rm o}$, otherwise the projection effects could never bring the companion close enough to the star to hide it, and the actual mass would be sufficiently larger than the projected mass to make it brighter than the background at any reasonable age.\n\n\\section{Conclusions}\n\n\\label{sec_conclusions}\n\nWe have attempted to image the indirectly discovered companion to $\\epsilon$ Ind A, using imaging in the 4$\\mu$m filter as well as the L'-band. As expected, 4$\\mu$m imaging was found to be a preferable choice over L'-band in the inner, contrast-limited regime, whereas the opposite was found to be true in the outer, background-limited range. This conclusion is based on theoretical models that ultimately need to be confirmed through observations of known planets. The overlap occurs at a radius of 4$''$ in our images, a number that will depend on target brightness and integration time. Two PSF subtraction techniques were employed: regular active ADI as used previously, and a new combination of techniques, using PSF reference subtraction and aADI with the LOCI algorithm. While PSFR+aADI performs slightly better at large separations, the techniques are virtually indistinguishable for most of the contrast-limited regime. Using more than one PSF reference star may change this picture. In addition, the method of combining 4$\\mu$m imaging and LOCI is also well suited for passive ADI, which has the potential to substantially enhance the performance even further.\n\nIn spite of the high sensitivities achieved in our images, we did not detect any potential companion candidate. Unless the known radial velocity companion to $\\epsilon$ Ind A is a neutron star or even more exotic stellar remnant, the non-detection in all images implies that the system is probably older than 1 Gyr, possibly consistent with preliminary results presented by Cardoso et al. (2008). Furthermore, we can constrain the planet or brown dwarf mass to within approximately 5-20 $M_{\\rm jup}$, the semi-major axis to $\\sim$10-20 AU, and the inclination to $>$20$^{\\rm o}$. An analysis based on astrometry from FK5 and \\textit{Hipparcos} is consistent with such a companion. Given the high significance of the RV trend, the fact that we can exclude all stellar, white dwarf and high-mass brown dwarf companions, and the fact that exotic stellar remnants are rare, it seems very plausible that $\\epsilon$ Ind A is one of the nearest stars to host a massive giant planet or very low-mass object. Furthermore, it is likely that this companion would be detectable through further imaging with either the presently available facilities, or facilities that come online in the relatively near future. Hence, $\\epsilon$ Ind is a high-profile target for the study of substellar objects, even aside from the fact that it hosts the nearest binary brown dwarf.\n\nFinally, we note that no sophisticated coronagraph adapted for observations beyond 3$\\mu$m presently exists on any of the 8m-class or larger AO-assisted telescopes (although simple coronagraphs do exist, e.g. a Lyot coronagraph for NACO). The potential coronagraphic performance is intimately connected to the adaptive optics performance, which leads to an interest in coronagraphs in the context of 'extreme AO' facilities currently in development (e.g. Petit et al. 2008). However, given the fact that a demonstrated Strehl ratio in the range of 85\\% can be reached even with NACO at 4$\\mu$m, an 'extreme AO'-type performance in this particular wavelength range is available already today. The development of a coronagraph for this wavelength range could therefore be another promising avenue to further increase the near-future capacity of detecting extrasolar planets through direct imaging.\n\n\\section*{Acknowledgments}\n\nThe authors wish to thank Marten van Kerkwijk and Yanqin Wu for useful discussion. The study made use of the CDS and SAO\/NASA ADS online services. M.J. is supported through the Reinhardt postdoctoral fellowship from the University of Toronto.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMany phenomena may be described through networks, including investment interactions between bidders and firms in venture capital (VC) markets \\cite{liang2016} and professional relationships among firms \\cite{bonaventura2020}.\nRisk capital is an essential resource for the formation and growth of entrepreneurial venture and venture capital firms are often linked together in a network by their joint investments in portfolio companies \\cite{bygrave1988}. Through connections in such a network, they exchange resources and investment opportunities with one another. Many studies show the impact of network dynamics on investments, raising efficiency \\cite{wetzel1987informal} and providing precious information when there is a great level of information asymmetry \\cite{fiet1995reliance}. Also, differentiating connection types and avoiding tight cliques appear to help the success of an investor by providing more diverse information and reducing confirmation bias \\cite{bygrave1988}.\n\nCB Insights \\cite{cbins} provides records of all transactions in venture capital markets from 1948. Since data until 2000 are partial and discontinuous, we focus on the period 2000-2020, in order to minimize the impact of missing data on our analysis. Additionally, since different sectors may be characterized by different investment dynamics \\cite{dushnitsky2006does}, we focus on the healthcare sector, which is of great importance and has shown to be less sensitive to market oscillations \\cite{pisano2006}. This stability is also shared by returns of life science VC, where investments have a lower failure rate but are at the same time less likely to generate \"black-swan\" returns \\cite{booth2011defense}, offering more consistency but a lower likelihood of achieving billion-dollars evaluations. \n\nWhile the number of exits through an IPO or through a trade sale can be seen as a proxy for the success of an investor \\cite{hege2003determinants}, there are instead different definitions of \"success\" for startups, but a common factor seems to be the growth rate of the company \\cite{santi2017lit}. Our work aims to understand whether network features may affect \"success\" of investments in healthcare firms. In order to investigate this, we introduce progressively more nuanced definitions of \"success\", and analyze them with increasingly sophisticated statistical tools. \n\nThe paper is organized as follows. Section \\ref{sec:char} introduces and characterizes a network of investors and firms, describing its structure and salient properties, including the communities emerging from its topology. Then, Section \\ref{sec:success} focuses on the definition and analysis of \"successful\" firms. We first characterize \"success\" by looking at the funding trajectories of each firm, clustering these trajectories into two broad groups capturing a high and a low funding regime. The binary cluster membership labels provide a first, rough definition of \"success\". We run a logistic regression in order to explain \"success\" defined in this fashion with statistics computed on the network itself. We then move to more complex characterizations of \"success\": the total amount of money raised (a scalar) and the funding trajectory itself (a functional outcome). We run regressions also on these outcomes, to validate and refine our previous results. Finally, we discuss main findings and provide some concluding remarks in Section \\ref{sec:disc}.\n\n\\section{Network characterization}\n\\label{sec:char}\n\nThe 83258 agents in the healthcare sector are divided into two broad categories: 32796 bidders, or investors, and 50462 firms. Companies open investment calls in order to collect funds; investors answer such calls and finance firms. Each deal, i.e. each transaction from an investor to a company, is recorded in the CB Insights' database. This market dynamics can be described by a \\textit{bipartite network}, which indeed is built on the notion of dichotomous heterogeneity among its nodes. In our case, each node may be a firm or an investor, respectively. An undirected link exists between two nodes of different kinds when a bidder has invested into a firm. Of course, given the possibility for an investor to finance the same firm twice, the bipartite network is also a \\textit{multi-graph}. By knowing the date in which investments are made, we can produce yearly snapshots of the bipartite network. A company (investor) is included in a snapshot of a certain year only when it receives (makes) an investment that year. By projecting the bipartite network onto investors and firms, we produce the two projected graphs which are used to compute all the node statistics described in Table \\ref{tab:variables}. As the bipartite network is a multi-graph, defining projections on a subset of nodes requires an additional assumption. Specifically, we project the bipartite graph onto firms by linking them in a cumulative fashion: we iteratively add to each yearly projected snapshot a link between two companies in which a bidder has invested during that year. Concerning the projection of the bipartite network onto investors, we link two bidders whenever they invest in the same company in the same financing round. \n\n\\begin{table}[t]\n    \\centering\n    \\caption{Statistics computed on the projected graphs of investors and firms. Before running regressions in Section~\\ref{sec:success}, left-skewed variables are normalized through log-transformation.}\n    \\label{tab:variables}\n    \\begin{tabularx}{\\textwidth}{XX}\n    \\toprule\n    \\textbf{Variable} & \\textbf{Network meaning}\\\\\n    \\midrule\n    Degree centrality & Influence \\\\\n    Betweenness centrality \\cite{hannan1977population} & Role within flow of information \\\\\n    Eigenvector centrality \\cite{bonacich1987power} & Influence \\\\\n    VoteRank  \\cite{zhang2016identifying} & Best spreading ability \\\\\n    PageRank \\cite{page1999pagerank}& Influence \\\\\n    Closeness centrality \\cite{freeman1978centrality} & Spreading power (short average distance from all other nodes) \\\\\n    Subgraph centrality \\cite{estrada2005subgraph} & Participation in subgraphs across the network \\\\\n    Average neighbor degree \\cite{barrat2004architecture} & Affinity between neighbor nodes \\\\\n    Current flow betweenness centrality \\cite{newman2005measure} & Role within flow of information\\\\\n    \\bottomrule\n    \\end{tabularx}\n\\end{table}\n\nRoughly 75\\% of the companies in the network projected onto firms are North American and European (around 55\\% belong to the US market), while the remaining 25\\% is mostly composed of Asian companies. Around 60\\% of the companies operate within the sub-sectors of medical devices, medical facilities and biotechnology -- the pharmaceutical sub-sector alone accounts for 20\\% of the network. As of August, 2021, roughly of 80\\% the companies in the network are either active or acquired, with the remaining portion being inactive or having completed an IPO. We witness turnover of the active companies through the years, but this is expected: a company's status is evaluated as of 2021, and it is more likely to observe a dead company among those that received investments in 1999 than in 2018. Indeed, both death and IPO represent the final stage of the evolution of a company, so those that received funding in earlier years are more likely to have already reached their final stage. Finally, we do not observe marked changes in terms of graph sub-sectoral composition: the relative share of each sub-sector is rather stable through the years, with the exception of an increase in the shares of the internet software and mobile software sub-sectors (from 1\\% in 1999 to 8\\% in 2019 and from 0\\% in 1999 to 5\\% in 2019, respectively). \n\n\\subsection{Communities}\n\nBy employing the Louvain method \\cite{blondel2008fast}, we identify meso-scale structures for each yearly snapshot of the network projected onto firms. For each year, we rank communities by their size, from the largest to singletons. We then compare the largest communities across years, by looking at their relative sub-sectors, status and geographical composition.\n\nWhile the specific nodes in the biggest communities may vary throughout the years, we notice a relative stability in their features. The largest communities (which contain between 13\\% and 20\\% of the nodes) reflect the status composition of the general network, downplaying unsuccessful companies and giving higher relative weight to IPO ones, showing just a variation between acquired and active companies across years (i.e.~active companies are relatively over-represented in more recent largest communities than in older ones). Considering geographical information, the largest communities comprise mainly US companies, with an under-representation of other continents. This trait is quite consistent through the years, with the exception of\ntwo years (2013-2014). With respect to sub-sectors, the largest communities mainly contain medical device and biotechnology companies, and they are quite consistent through the years in terms of sub-sectoral composition. \n\nThe second largest communities (containing between 10\\% and 14\\% of nodes in the network) have a less consistent sub-sectoral composition through the years, although it is worth highlighting that they comprise companies operating within software and technology. Geographically, we are still witnessing communities of mostly US-based companies, although 5 years out of 20 show a remarkable (roughly 80\\%) presence of European companies. Finally, status composition is balanced between active and acquired until the later years, when active companies predominate within the second largest communities. IPOs are not present, while there are, in a small percentage (between 5\\% and 20\\%), dead startups.\n\nFinally, the third largest communities (containing between 7\\% and 12\\% of the nodes) present a clear change within the period considered: in the first ten years, they mostly comprise failed or acquired European companies within the fields of biotechnology and drug development, while, in the second decade, they comprise active US companies within the fields of medical devices and medical facilities. \n\n\\section{Success analysis}\n\\label{sec:success}\nGiven the bipartite network and its projections, we now turn to the analysis of success and of its main drivers. Because of the elusiveness of the definition of \"success\", we proceed in stages -- considering progressively more refined outcomes and comparing our findings. Moreover, since many of the records available in the CB Insights' data set are incomplete, and our aim is to capture the temporal dynamics leading a firm to succeed, we further restrict attention to those companies for which full information is available on birth year, healthcare market sub-sector and investment history for the first 10 years from founding. Although this filtering may introduce some biases, it still leaves us with a sizeable set of 3663 firms belonging to 22 different sub-sectors.\n\nNotably, we restrict our focus also in terms of potential predictors, due to the fact that our collection of network features exhibits strong multicollinearities. By building a feature dendrogram (Pearson correlation distance, complete linkage) and by evaluating the correlation matrix, we reduce the initial set to four representatives. In particular, we select two features related to the investors' projection (the maximum among the degree centralities of the investors in a company and the maximum among their current flow betweenness centralities, both computed in the company's birth year) and two features computed on the firms' projection (a company's eigenvector and closeness centralities, computed in the year in which the company received its first funding). \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{aligned_example_pharma.eps}\n    \\caption{Money raised cumulatively as a function of time, shown for 319 firms in the pharmaceuticals and drugs sub-sector. Funding trajectories are constructed over a period of $10$ years since birth, and aligned using birth years as registration landmarks.}\n    \\label{fig:aligned_pharma}\n\\end{figure}\n\nEach firm has its own funding history: after its birth, it collects funds over the years, building a \\textit{trajectory} of the amount of money it is able to attract. We treat these trajectories as a specific kind of structured data, by exploiting tools from a field of statistics called \\textit{Functional Data Analysis} (FDA) \\cite{ramsey2005functional}, which studies observations that come in the form of functions taking shape over a continuous domain. In particular, we focus on the \\textit{cumulative} function of the money raised over time by each company. As an example, Figure \\ref{fig:aligned_pharma} shows 319 such cumulative functions, for the firms belonging to the pharmaceuticals and drugs sub-sector. Trajectories are \\textit{aligned}, so that their domain (\"time\") starts at each company's birth (regardless of the calendar year it corresponds to). By construction, these functions exhibit two characterizing properties: first, they are monotonically non-decreasing; second, they are step functions, with jumps indicating investment events. \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{clustering_pharma.eps}\n    \\caption{$k$-means clustering ($k=2$) of the funding trajectories of firms belonging to the pharmaceuticals and drugs sub-sector. The green and red dashed lines represent firms in the high (\"successful\") and low regimes, respectively. Bold curves represent cluster centroids. To aid their visualization, centroids are shown again in the right panel with individual trajectories in gray.}\n    \\label{fig:clustered_pharma}\n\\end{figure}\n\nOur first definition of success is based on separating these trajectories into two regimes characterized by high (successful) vs. low investment patterns: the first runs at high levels, indicating successful patterns, and the second at low levels. Because of heterogeneity among healthcare sub-sectors, we accomplish this by running a \\textit{functional k-means clustering} algorithm \\cite{jacques2014func, hartigan1979} with $k=2$, separately on firms belonging to each sub-sector.\nAs an example, companies belonging to the sub-sector of pharmaceuticals and drugs are clustered in Figure \\ref{fig:clustered_pharma}. Throughout all sub-sectors, the algorithm clusters $89$ firms in the high-regime group and $3574$ in the low-regime one. \n\nThis binary definition of \"success\" turns out to be rather conservative; very few firms are labeled as belonging to the high investment regime. Consider the logistic regression\n\\begin{equation}\n\\label{eq:log}\n    \\log\\left(\\frac{P(y_i=1)}{1-P(y_i=1)}\\right)=\\beta_0+\\sum_{j=1}^{p} \\beta_j x_{ij} \\quad i=1,\\dots n\n\\end{equation}\nwhere $n$ is the number of observations, $y_i$, $i=1,\\dots n$, are the binary responses indicating membership to the high ($y_i=1$) or low ($y_i=0$) regime clusters; $\\beta_0$ is an intercept and $x_{ij}$, $i=1,\\dots n$ and $j=1,\\dots,p$ ($p=4$), are the previously selected scalar covariates.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{significance.eps}\n    \\caption{Scatter plots of logistic regression coefficient estimates (horizontal) and significance (vertical; $-log(p$-$value)$). Each point represents one of $1000$ fits run on data balanced by subsampling the most abundant class. Orange solid line mark averages across the fits, and orange dashed lines $\\pm1$ standard deviations about them. Green solid lines mark $0$ on horizontal axes. Blue line mark significance values associated to a p-value of $0.1$.}\n    \\label{fig:est_coef}\n\\end{figure}\n\nIf we fit this regression on our unbalanced data, results are bound to be unsatisfactory and driven by the most abundant class. Running such a fit, one obtains an explained deviance of only $0.10$. To mitigate the effects of unbalanced data \\cite{haibo2009}, we randomly subsample the most abundant class (the low-regime firms) as to enforce balance between the two classes, and then run the logistic regression in Equation \\ref{eq:log}. We repeat this procedure 1000 times, recording estimated coefficients, associated p-values and explained deviances. The average of the latter across the 1000 replications is substantially higher than on the unbalanced fit, reaching $0.18$\n(some fits produce deviance explained as high as $0.45$). Moreover, we can investigate significance and stability of the coefficient estimates through their distribution across the repetitions. Figure \\ref{fig:est_coef} shows scatter-plots of these quantities, suggesting that the two variables related to the firms' centrality have a modest yet stable, positive impact on the probability of belonging to the high-regime cluster. This is not the case for the variables related to the investors' centrality.\n\nThis first evidence of a positive relationship between the success of a firm and its centrality, or importance (in a network sense) is promising. However, the binary definition of \"success\" we employed is very rough -- and the unbalance in the data forced us to run the analysis relying on reduced sample sizes ($89+89=178$ observations in each repeated run). Thus, we next consider a scalar proxy for \"success\", which may provide a different and potentially richer perspective. Specifically, we consider the cumulative end point of a firm's funding trajectory, i.e. the total value of the investment received through its temporal domain.\n\n\nFor this scalar response, we run a \\textit{best subset selection} \\cite{friedman2005elements} considering all the network features in our initial set -- not just the $4$ selected to mitigate multicollinearity prior to the logistic regression exercise. Notably, despite the substantial change in the definition of \"success\", results are in line with those from the logistic regression. Indeed, the first selected variable, when the predictor subset is forced to contain only one feature, is the eigenvector centrality of firms. When the predictor subset size is allowed to reach $4$, the features selected are the closeness and the VoteRank of the firm, and the maximum current flow betweenness centrality among its investors (computed on the firm's birth year). Thus, the only difference compared to our previous choice is the selection of the firms' VoteRank centrality instead of the maximum among the investors' degree centrality. We compare the two alternative selections of four features as predictors of the scalar \"success\" response fitting two linear models of the form:\n\\begin{equation}\n    y_i =\\beta_0+\\sum_{j=1}^{p} \\beta_j x_{ij} +\\epsilon_i \\quad i=1,\\dots n\n\\end{equation}\nwhere $n$ is the number of observations, $y_i$, $i=1,\\dots n$, are the scalar responses (aggregate amount of money raised); $\\beta_0$ is an intercept; $x_{ij}$, $i=1,\\dots n$ and $j=1,\\dots,p$ ($p=4$), are the scalar covariates belonging to one or the other subset and $\\epsilon_i$, $i=1,\\dots n$, are i.i.d. Gaussian model errors. As shown in Table \\ref{tab:linreg}, the maximum degree centrality among a firm's investors is not statistically significant.\nSurprisingly, the maximum among investors' current flow betweenness centralities is significantly negative,  but its magnitude is close to 0. In contrast, the firms' closeness and eigenvector centralities are positive, statistically significant and sizeable. This is in line with what we expected, since it is reasonable to think that knowledge may indirectly flow from other startups through common investors, increasing the expected aggregate money raised. Finally, the firms' VoteRank centrality appears to have a negative, statistically significant impact on the aggregate money raised. This should not be surprising, given that the higher the VoteRank centrality is, the less influential the node will be. The variance explained by the two models is similar and still relatively low ($R^2 \\approx 0.13)$, which may be simply due to the fact that network characteristics are only one among the many factors involved in a firm's success \\cite{dosilimmancabile1994}. Nevertheless, the results obtained here through the scalar \"success\" outcome are consistent with those obtained through the binary one and logistic regression.\n\n\\begin{table}[t]\n\\centering \n\\caption{Linear regressions of aggregate money raised on \ntwo sets of predictors. All variables are scaled and some are log-transformed (as indicated parenthetically).} \n\\label{tab:linreg} \n\\begin{tabular}{@{\\extracolsep{5pt}}lcc} \n\\toprule\n & \\multicolumn{2}{c}{\\textit{Dependent variable:}} \\\\ \n\\cline{2-3} \n\\\\[-1.8ex] & \\multicolumn{2}{c}{Aggregate money raised (log)} \\\\ \n\\\\[-1.8ex] & (1) & (2)\\\\ \n\\midrule\n newman\\_max & $-$0.065$^{**}$ & $-$0.072$^{*}$ \\\\ \n  & (0.030) & (0.041) \\\\ \n  & & \\\\ \n voterank (log) & $-$0.140$^{***}$ &  \\\\ \n  & (0.033) &  \\\\ \n  & & \\\\ \n degcen\\_max (log) &  & 0.050 \\\\ \n  &  & (0.040) \\\\ \n  & & \\\\ \n closeness & 0.126$^{***}$ & 0.130$^{***}$ \\\\ \n  & (0.037) & (0.030) \\\\ \n  & & \\\\ \n eigenvector (log) & 0.214$^{***}$ & 0.255$^{***}$ \\\\ \n  & (0.034) & (0.028) \\\\ \n  & & \\\\ \n Constant & 0.113$^{***}$ & 0.062$^{**}$ \\\\ \n  & (0.030) & (0.025) \\\\ \n  & & \\\\ \n\\midrule \\\\[-1.8ex] \nObservations & 1,118 & 1,364 \\\\ \nR$^{2}$ & 0.136 & 0.127 \\\\ \nAdjusted R$^{2}$ & 0.133 & 0.125 \\\\ \nResidual Std. Error & 0.992 (df = 1113) & 0.923 (df = 1359) \\\\ \nF Statistic & 43.951$^{***}$ (df = 4; 1113) & 49.458$^{***}$ (df = 4; 1359) \\\\ \n\\bottomrule\n\\textit{Note:}  & \\multicolumn{2}{r}{$^{*}$p$<$0.1; $^{**}$p$<$0.05; $^{***}$p$<$0.01} \\\\ \n\\end{tabular} \n\\end{table} \n\nOur scalar outcome (aggregate money raised) has its own drawbacks. In particular, it implicitly assumes that the right time to evaluate success and investigate its dependence on network features is, cumulatively, at the end of the period considered (10 years). Note that this translates into a 10-year gap between the measurement of network features and financial success. \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{fos_plot.eps}\n    \\caption{Function-on-scalar regression, coefficient curve estimates. (a) intercept function (this can be interpreted as the sheer effect of time on the response); (b) maximum degree centrality among investors (company's birth year); (c) maximum across investors' current flow betweenness centrality (company's birth year); (d) company's eigenvector centrality; (e) company's closeness centrality. Dotted lines represent confidence bands. All the covariates are standardized.}\n    \\label{fig:fos_coef}\n\\end{figure}\n\nAlthough this issue could be approached relying on additional economic assumptions, we tackle it refining the target outcome and considering the full funding trajectories -- instead of just their end point. This requires the use of a more sophisticated regression framework from FDA; that is, function-on-scalar regression \\cite{kokoszka2017introduction}. In particular, we regress the funding trajectories on the same two sets of covariates considered in the scalar case above. The equation used for function-on-scalar regression is:\n\n\\begin{equation}\n    \\label{eq:fun_reg}\n        Y_i(t) = \\beta_0(t) + \\sum_{j=1}^{p} \\beta_j(t) x_{ij} + \\epsilon_i(t) \\quad i=1,\\dots n\n    \\end{equation}\nwhere $n$ is the number of observations; $Y_i(t)$, $i=1,\\dots n$, are the aligned funding trajectories; $\\beta_0(t)$ is a functional intercept; $x_{ij}$, $i=1,\\dots n$ and $j=1,\\dots,p$ ($p=4$), are the scalar covariates belonging to the one or the other set, and $\\epsilon_i(t)$, $i=1,\\dots n$, are i.i.d. Gaussian model errors.\n\nThe regression coefficient of a scalar covariate in this model, $\\beta_j(t)$, is itself a curve describing the time-varying relationship between the covariate and the functional response along its domain. Together with the functional coefficients, we also estimate their standard errors, which we use to build confidence bands around the estimated functional coefficients \\cite{refund2016}. Coefficient curve estimates for the covariate set including the maximum investors' degree centrality are shown in Figure~\\ref{fig:fos_coef} (results are very similar with the other set of covariates). The impacts of an increase in the maximum among the degree centralities and in the maximum among the current flow betweenness centralities of the investors in a firm are not statistically significant. Conversely, eigenvector and closeness centralities of firms have positive and significant impacts. The impact of the eigenvector centrality seems to be increasing during the first five years, reaching a \"plateau\" in the second half of the domain. These findings reinforce those obtained with the binary and scalar outcomes previously considered, confirming a role for firms' centrality in shaping their success.\n\n\\section{Discussion}\n\\label{sec:disc}\nThis paper exploits techniques from the fields of network and functional data analysis. We build a network of investors and firms in the healthcare sector and characterize its largest communities. Next, we progressively shape the concept of a firm's \"success\" using various definitions, and associate it to different network features. Our findings show a persistent positive relationship between the importance of a firm (measured by its centrality in the network) and various (binary, scalar and functional) definitions of \"success\". In particular, we cluster funding trajectories into a high (\"successful\") and a low regime, and find significant associations between the cluster memberships and firms' centrality measures. Then, we switch from this binary outcome to a scalar and then a functional one, which allow us to confirm and enrich the previous findings. Among centralities computed on the two network projections, our results suggest a preeminent role for those computed in the companies' projection. In particular, both a firm high closeness centrality, indicating a small shortest distances to other firms, and its eigenvector centrality, which may account for a firm's reputation, seem to be related to the propensity to concentrate capital.\n\nOur analysis can be expanded in several ways. First, we limit our study to the healthcare sector, while it may be interesting to investigate other fields, or more healthcare firms based on the availability of more complete records. It would also be interesting to account for external data (e.g. country, sub-sector, etc.) in two ways. One the one hand, these information would be useful as to compute more informative statistics on the network topology. On the other hand, they may be used in our regression, to control for these factors.\nMoreover, meso-scale communities may be analyzed in terms of their longitudinal evolution, as to characterize \"successful\" clusters of firms from a topological point of view. \n\n\\section*{Acknowledgments}\nF.C., C.E., G.F., A.M. and L.T. acknowledge support from the Sant'Anna School of Advanced Studies. \nF.C. acknowledges support from Penn State University. \nG.R. acknowledges support from the scheme \"INFRAIA-01-2018-2019: Research and Innovation action\", Grant Agreement n. 871042 \"SoBigData++: European Integrated Infrastructure for Social Mining and Big Data Analytics\".\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{}}\n\n\nThe $b\\to c$ transitions have been very important in the extraction of the $V_{cb}$ CKM matrix element. \nIn experiments aimed to determine $V_{cb}$, actually the product\n$|V_{cb} {\\cal F } (1) |$ is extracted, where  ${\\cal F } (1)$ denotes\nthe $B \\to D$ or $B \\to D^*$ hadronic form factors at zero recoil. A\nlack of precise information about the shapes of these form factors\naway from zero recoil is one of the main sources of uncertainties.\nFurthermore recent spectroscopic discoveries of low lying orbitally excited charmed meson states at the charm and B factories have prompted reevaluation of their contributions to B meson decays. \nThe leading order (LO) $SU(3)$ chiral corrections at leading and next to leading order (NLO) in $1\/m_Q$ to the $B_q\\to D_q^{(*)}$ semileptonic form factors have previously been computed~\\cite{Boyd:1995pq}. Also virtual effects of positive parity heavy meson states in these transitions have been considered before~\\cite{Falk:1993iu}.\nHowever, phenomenological discussion at the time was limited by the lack of experimental and lattice  QCD (LQCD) information available on the relevant phenomenological parameters. We are now able to provide the first more reliable estimates of these contributions, but we also extend the program to B meson transitions to positive parity charmed mesons~\\cite{EFK1}. In particular we address the issue of extrapolation of LQCD results on these quantities to the chiral regime.\nIn the $B_q\\to D_q \\tau\\nu$ decay mode due to the large tau mass one can discuss helicity suppressed contributions as such coming from the exchange of charged scalars~\\cite{Tanaka:1994ay,Kiers:1997zt,Nierste:2008qe,Kamenik:2008tj}. \nThis calls for high precision theoretical estimates for both the dominating vector\nas well as the subleading relative scalar form factor contribution.\nWe have estimated the leading chiral symmetry breaking corrections, governing the differences between the vector and scalar form factors in $B_q\\to D_q$ transitions~\\cite{EFK2}. These corrections can on the one hand be used to guide LQCD studies in their chiral extrapolations. On the other hand we use them to estimate qualitatively the relative scalar form factor values in $B_s\\to D_s$ transitions.\n\nFirst we present the most important results of our calculation~\\cite{EFK1} of\nleading chiral loop corrections to the form factors governing transitions of B mesons to positive and negative parity charmed mesons. In our heavy meson chiral perturbation theory (HM$\\chi$PT) description, heavy-light mesons appear in velocity ($v$) dependent spin-parity doublets due to heavy quark spin symmetry, while chiral $SU(N_f)$ symmetry determines their interactions with the multiplet of light pseudo-Goldstone bosons. Such a framework provides a systematic expansion terms of light and inverse heavy quark masses (for details c.f. ref.~\\cite{Fajfer:2006hi}).  \nThe relevant form factor contributions in the effective theory are obtained using operator bosonization procedure introducing the universal Isgur-Wise (IW) functions $\\xi(w)$, $\\tilde \\xi(w)$ and $\\tau_{1\/2}(w)$ (where $w=v\\cdot v'$ is the product of velocities of initial and final heavy-light states) governing transitions within the negative, positive parity doublets, and between states of opposite parity respectively. We calculate loop corrections to these effective weak vertices coming from the one loop diagram topologies shown in Fig. 1.\n\\psfrag{pi}[bl]{\\footnotesize $\\Red{ {\\pi^i(q)}}$}\n\\psfrag{Ha}[cc]{\\footnotesize $\\Red{H_a(v)}$}\n\\psfrag{Hb}[cc]{\\footnotesize $\\Red{H_b(v')}$}\n\\psfrag{Hc}[cc]{\\footnotesize $~\\Red{H_c(v)}$}\n\\psfrag{Hd}[cc]{\\footnotesize $~~\\Red{H_c(v')}$}\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=145mm]{xi_loop.eps}\n\\caption{Weak vertex correction diagrams. Crossed boxes represent effective weak vertices while filled circles represent LO strong vertices. Only diagrams of the utmost left topology contribute to the amplitude at the leading chiral log order.} \\label{JACpic2-f2}\n\\end{figure*}\nIn addition, wave function renormalization of the heavy-light fields needs to be taken into account. This has been done e.g. in ref.~\\cite{Fajfer:2006hi} and non-zero contributions come from the self energy diagrams with LO effective strong couplings in the loop.\nPresence of two opposite parity heavy-light meson multiplets in the theory introduces a new scale -- the mass gap between the ground and excited heavy meson states $\\Delta_{SH}$.\nIn order to tame the chiral behavior of the\namplitudes containing the mass gap we use the $1\/\\Delta_{SH}$\nexpansion of the chiral loop integrals~\\cite{Fajfer:2006hi}.\nWe have argued~\\cite{Becirevic:2006me}\nthat the expansion works well in a $SU(2)$ theory where kaons and etas,\nwhose masses would compete with $\\Delta_{SH}$, do not\npropagate in the loops. Therefore we can write down explicit expressions\nfor the chiral loop corrected IW functions specifically for\nthe strangeless states in the $SU(2)$ theory\\footnote{Following~\\cite{EFK1} we absorb the infinite and scale dependent pieces from one loop amplitudes into the appropriate counterterms at order $\\mathcal O (m_q)$.}:\n\\begin{eqnarray}\n[\\xi(w)]^{\\mathrm{Loop}}  &=&   \\xi(w) \\Bigg\\{ 1 + \\frac{3}{32\\pi^2 f^2} m^2_{\\pi} \\log \\frac{m^2_{\\pi}}{\\mu^2}  \\Bigg[ g^2 2 (r(w)-1)\n - h^2  \\frac{m^2_{\\pi}}{4\\Delta_{SH}^2} \\left(1-w\\frac{\\tilde \\xi(w)}{\\xi(w)}\\right) - h g \\frac{m^2_{\\pi}}{\\Delta_{SH}^2} w(w-1)\\frac{\\tau_{1\/2}(w)}{\\xi(w)}\\Bigg] \\Bigg\\},\\nonumber\\\\\n\\label{eq:5}\n\\end{eqnarray}\nwhere\n$r(x) = {\\log(x+\\sqrt{x^2-1})}\/{\\sqrt{x^2-1}}$ and all IW functions on the rhs should be considered tree-level, similarly\n\\begin{eqnarray}\n[\\tau_{1\/2 }(w)]^{\\mathrm{Loop}} &=&  \\tau_{1\/2}(w) \\Bigg\\{ 1 + \\frac{3}{32\\pi^2 f^2} m^2_{\\pi} \\log \\frac{m^2_{\\pi}}{\\mu^2}  \\Bigg[ - g\\tilde g(2r(w)-1) - \\frac{3}{2} (g^2+\\tilde g^2)\\nonumber\\\\\n&&\n\\hskip -0cm + h^2  \\frac{m^2_{\\pi}}{4\\Delta_{SH}^2} \\left(w-1\\right) - h g \\frac{m^2_{\\pi}}{2\\Delta_{SH}^2} \\frac{\\xi(w)}{\\tau_{1\/2}(w)} w(1+w) + h\\tilde g \\frac{m^2_{\\pi}}{2\\Delta_{SH}^2} \\frac{\\tilde \\xi(w)}{\\tau_{1\/2}(w)} w(1+w) \\Bigg] \\Bigg\\}.\n\\label{eq:6}\n\\end{eqnarray}\nThe first parts of Eqs.~(\\ref{eq:5}) and~(\\ref{eq:6}) contain the leading contributions while the calculated  $1\/\\Delta_{SH}$ corrections are contained in the second parts. Definitions and values for the LO strong couplings $g$, $\\tilde g$ and $h$ can be found in ref.~\\cite{EFK1}.\nWe present\nthe chiral behavior of the IW functions\nin the chiral limit below the $\\Delta_{SH}$ scale in Fig. 2.\n\\psfrag{xk1}[bc]{{$r\\sim m_{u,d}\/m_s$}}\n\\psfrag{xm1}[tc][tc][1][90]{{${\\xi'(1)}^{\\mathrm{Loop}}\/\\xi'(1)^{\\mathrm{Tree}}$}}\n\\psfrag{s1}[cl]{\\footnotesize{$(1\/2)^-$ contributions}}\n\\psfrag{s3}[cl]{\\footnotesize{$\\xi'(1)-\\tilde \\xi'(1)=1$}}\n\\psfrag{s4}[cl]{\\footnotesize{$\\xi'(1)-\\tilde \\xi'(1)=-1$}}\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=85mm]{plot1.eps}\n\\caption{Chiral extrapolation of the slope of the IW\nfunction at $w=1$ ($\\xi'(1)$). Negative parity heavy states'\ncontributions (black line) and a range of possible positive parity\nheavy states' contribution effects when the difference of slopes of\n$\\xi(1)$ and $\\tilde \\xi(1)$ is varied between $1$ (red dashed line)\nand $-1$ (blue dash-dotted line)} \\label{JACpic2-f3}\n\\end{figure*}\n\n\n\n\nNext we focus particularly on the matrix element of the vector $b\\to c$ quark current between a B and a D meson of velocity $v$ and $v'$ respectively, which can be parametrized in terms of two form factors:\n\\begin{eqnarray}\n&& \\bra{D(v')}\\bar c \\gamma_{\\mu} b \\ket{B(v)} = \\sqrt{m_B m_D} [ h_+(w) (v+v')_{\\mu} \n\n+ h_-(w) (v-v')_{\\mu} ]\\,.\\label{eq:h+expansion}\n\\end{eqnarray}\nIt is important to know that in heavy quark expansion \n$h_+(w)$ is the leading order term in $1\/m_Q$ (proportional to the IW function $\\xi(w)$) while $h_-(w) \\sim 1\/m_c - 1\/m_b$. \nAfter the bosonization of the relevant new operators which appear at $1\/m_Q$~\\cite{EFK2}, we determine leading logarithmic chiral corrections relevant for LQCD extraction of the $h_+(w)$ and $h_-(w)$ form factors in the $SU(2)$ theory:\n\\begin{eqnarray}\n\\label{eq:hpchi}\n\th_+ &=& h_+^{\\mathrm{Tree}} \\left[ 1 + 3 g^2 \\frac{r(w)-1}{(4\\pi f_{\\pi})^2}  m_{\\pi}^2\\log \\frac{m_{\\pi}^2}{\\mu^2} + m_{\\pi}^2 c_{+}(\\mu,w) \\right]\\,, \\\\\n\\label{eq:hmchi}\n\th_- &=&  h_-^{\\mathrm{Tree}} \\left\\{ 1 - 3 g^2 \\frac{2 - Y^{*}_+ [r(w)+1]}{(4\\pi f_{\\pi})^2}  m_{\\pi}^2\\log \\frac{m_{\\pi}^2}{\\mu^2} \\right\n+ m_{\\pi}^2 c_{-}(\\mu,w) \\Big\\}\\,,\n\\end{eqnarray}\nwhere $Y^{*}_+= h_-^{*\\mathrm{Tree}}\/h_-^{\\mathrm{Tree}}$ is the ratio of vector current matrix elements proportional to $v-v'$ between vector and pseudoscalar states respectively (see ref.~\\cite{EFK1} for details and other possible choices of parameterizing this quantity), and the $w$ dependence is implicit. Here and in the rest of the text $c_i(\\mu,w)$ denote the sums of local analytic counter-terms, which cancel the $\\mu$ dependence of the chiral log pieces. As stressed by the notation, they will in general have a non-trivial $w$ dependence, originating both from the finite analytic residuals of chiral loops as well as from local NLO chiral current operators, needed to cancel the UV divergences of chiral loops. \n\nThe ratio $h_-\/h_+$ is particularly important since it enters in the multiplicative contribution to the differential decay rate distribution distinguishing $B\\to D \\tau\\nu$ from $B \\to D e\\nu$ decays. The chiral corrections in this case read:\n\\begin{eqnarray}\n\t\\frac{h_-}{h_+} &=&  \\left(\\frac{h_-}{h_+}\\right)^{\\mathrm{Tree}} \\left[ 1 - 3 g^2 \\frac{r(w)+1}{(4\\pi f_{\\pi})^2}  m_{\\pi}^2\\log \\frac{m_{\\pi}^2}{\\mu^2}  \\right\n+ \\left(\\frac{h^*_-}{h_+}\\right)^{\\mathrm{Tree}} 3 g^2 \\frac{r(w)+1}{(4\\pi f_{\\pi})^2}  m_{\\pi}^2\\log \\frac{m_{\\pi}^2}{\\mu^2} + m_{\\pi}^2 c_{{\\pm}}(\\mu,w)\\,,\n\\end{eqnarray}\nas written in ~\\cite{EFK2} and where relevant quantities are defined. \n\n\nWe can summarized our results:\nwithin a HM$\\chi$PT  framework, which includes even and odd parity\nheavy meson interactions with light pseudoscalars as\npseudo-Goldstone bosons, we have calculated chiral loop corrections\nto the functions $\\xi$ and $\\tau_{1\/2}$. As in previous cases~\\cite{Fajfer:2006hi,Becirevic:2004uv}  we have  shown that\nthe leading pionic chiral logarithms are not changed by the\ninclusion of even parity heavy meson states we consider chiral\nextrapolation of IW functions. Our results are particularly\nimportant for the LQCD extraction of the form factors. The\npresent errors on the $V_{cb}$ parameter in the exclusive channels\nare of the order few percent. This calls for careful control over\ntheoretical uncertainties in its extraction. \nWe have found also~\\cite{EFK2} how calculating the ratio of current matrix elements in $B\\to D$ and $B^*\\to D^*$ transitions on the lattice can improve the extraction of the scalar form factor. \n\n\n\n\n\\begin{acknowledgments}\nThis work is supported in part by the European Commission RTN network, \nContract No. MRTN-CT-2006-035482 (FLAVIAnet).\nThe work of S.F. and J.K. is supported in part by the Slovenian\nResearch Agency. J.O.E. is supported in part by the Research Council\nof Norway. \n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction and Summary}\nThermalization in unitary quantum field theories has been a topic of great significance. Using AdS\/CFT correspondence, it has also been linked to black hole formation \\cite{Bhattacharyya:2009uu, Chesler:2008hg}.\nOne of the current views of thermalization is that of the thermalization of a finite subsystem, in which the conjugate subsystem is considered the heat bath.\nIn other words, it is the thermalization observed by an observer who has access to only a subsystem of the full system. It can also be considered as if the `fine-grained' observables{\\footnote{Observables which show the non-thermal behaviour of the pure state, in contrast to `coarse-grained' observables which cannot distinguish between the pure state and the thermal ensemble.}} are spatially widely separated bilocal or higher point observables. Starting from a pure state, in the high energy (high effective temperature) limit, the final thermal entropy observed by such an observer is actually the entanglement entropy of the subsystem with its conjugate. Obviously, the pure state has to be a time-dependent state. Closely related to thermalization(equilibration in general), the study of time-dependent states after a quantum quench has also been of great interest\\cite{Polkovnikov:2010yn, gogolin2015equilibration}. Quantum quench is the process in which the parameters of the Hamiltonian of a system in a certain state are changed with time. After the quantum quench, in the long time limit, if the subsystem of our interest looks like a thermal ensemble, in the sense that the expectation values of observables in the finite subsystem have the same expectation values as in a thermal ensemble, then we say that the system has thermalized. In this paper, we will be mainly considering quantum quench as the preparation of the time-dependent states of our interest.\n\nWe will also restrict ourself to critical quantum quenches, in which the final Hamiltonian is a critical Hamiltonian, i.e., the corresponding theory is a conformal field theory (CFT). More specifically, we will be considering free fermions in which starting from a certain state in the massive theory, the mass is set to zero gradually or suddenly. In more general theories, starting from the ground state of a gapped theory, it has been proposed \\cite{Calabrese:2005in, Calabrese:2006rx} that the state obtained after the critical quench is a Calabrese-Cardy(CC) state which has the form $e^{-\\kappa_2 H}|B\\rangle$, where $\\kappa_2$ is a scale given by the initial gap and the other scales of the quench process, $H$ is the Hamiltonian of the CFT and $|B\\rangle$ is a conformally invariant boundary state. It has been shown that such a state thermalizes to a thermal ensemble with temperature $T=1\/\\beta=1\/(4\\kappa_2)$. This result has also been generalised to the case in which the final theory has other conserved charges of local currents \\cite{Mandal:2015jla}. The corresponding ansatz for the state after quench from ground state is a generalized Calabrese-Cardy(gCC) states which have the form $e^{-\\kappa_2 H-\\kappa_4 W_4-\\kappa_6 W_6-...}|D\\rangle$ where again the parameters $\\kappa_2, \\kappa_4, \\kappa_6,\\cdots$ are given by the initial gap and other scales in the quench process, e.g. $\\delta t=1\/\\rho$ the time taken to set the mass to zero, and $W_4, W_6, \\cdots$ are the conserved charges of local currents. In this case also, it has been shown that the state thermalizes into a generalized Gibb's Ensemble(GGE) with the density matrix $e^{-\\beta H -\\mu_4 W_4 - \\mu_6 W_6-...}$ where the corresponding temperature and chemical potentials are $T=1\/\\beta=1\/(4\\kappa_2)$, $\\mu_4=4\\kappa_4$, $\\mu_6=4\\kappa_6, \\cdots$. \n\nThe gCC state ansatz has been shown to be true for mass quenches in free scalar and free fermion theories in a recent paper(MPS) \\cite{Mandal:2015kxi}. Starting from the ground state of the massive theories, the quenched states obtained are of the gCC form with infinite number of charges $W_{2n}$ with $n\\in\\mathbb{N}$ ($W_2=H$). For the scalar theory, it was also found that naively taking the sudden limit when the mass profile is taken to be a step function, the final state is non-normalizable. For massless free scalar theory, $W_{2n}=\\sum |k|^{2n-1}d^\\dagger_kd_k$, where $d^\\dagger_k$ and $d_k$ are the bosonic annihilation and creation operators.\\footnote{The normalization of the charges differ from the normalization in \\cite{Bakas:1990ry, Pope:1991ig}.} It was also shown that starting from specially prepared squeezed states of the massive scalar theory, CC state and gCC state with finite number of charges can also be created. By calculating correlators, thermalization of these states were explicitly shown.\n\nIn this paper, we find similar results for the fermionic mass quench. In the sudden limit, starting from the ground state, we observe that the final state has divergent energy density, $W_4, W_6, \\cdots$. Again, as in the case of scalar fields in MPS, starting from specially prepared squeezed states using the sudden quench limit, we can prepare CC state and gCC state with a finite number of charges of our choice. For the CC state and the gCC state with finite number of charges, we calculate correlators and explicitly show thermalization to thermal ensemble and GGE respectively.\n\nAmong the other calculable quantities, entanglement entropy(EE) is the most interesting one. The EE growth has been calculated(mostly numerically) in many dynamical systems, see for e.g. \\cite{Calabrese:2005in, Calabrese:2007-local, PhysRevA.78.010306, PhysRevX.3.031015, Nezhadhaghighi:2014pwa, Nahum:2016muy, Cotler:2016acd}. It has also been extensively examined in holographic systems \\cite{AbajoArrastia:2010yt, Hartman:2013qma, Caputa:2013eka, Liu:2013qca, Kundu:2016cgh}. Recently, non-monotonic EE growth consisting of an initial dip around the quench time has also been observed in a holographic set-up in \\cite{Bai:2014tla}.\n\nSince our final theory consists of only massless Dirac fermions, so using bosonization, we could calculate EE in some of our time-dependent states. We are interested in EE of a single interval only. For CC states, we find that EE grows monotonically. The asymptotic time limit is given by the well-known expression from CFT in a thermal ensemble, $S_A=\\frac{c}{3}\\log(\\sinh(\\frac{\\pi r}{\\beta}))$, where for Dirac fermions $c=1$ and the effective temperature $1\/\\beta =1\/4\\kappa$. In case of gCC states, we are not able to calculate EE with the charges of the usual fermionic bilinear $\\mathcal{W}_{1+\\infty}$ currents. But we are able to calculate the EE with the fermionic charge corresponding to the bosonic charges $W_{2n}=\\sum |k|^{2n-1}d^\\dagger_kd_k$. These are the charges of bosonic bilinear $\\mathcal{W}_{2n}$ currents for $n>1$. For such gCC states with the $W_4$ charge, we found a dynamical phase transition in which EE grows non-monotonically when the effective chemical potential $\\mu_4$ is greater than a critical value. Below this critical value, the EE growth is strictly monotonic.\n\n\\gap1\n\\noindent In summary, the key results of the present work are:\n\\begin{enumerate}\n \\item For ground state quench, similar to the scalar quench, a naive sudden quench limit gives divergent conserved charges. Calculation of the correlators show equilibration explicitly. But the long distance and time and ultimately the stationary limit is significantly different from thermalization to a thermal ensemble. This is the same manifestation of the UV\/IR mixing found in MPS.\n \\item Starting from appropriately prepared squeezed states of the massive theory, we can prepare CC and gCC states with specific $W_{2n}$ charges using quench. Calculation of correlators in CC state and gCC states explicitly show thermalization to thermal ensemble and GGE respectively. Here again, for gCC state, the long time and long distance limit of the correlators have significant dependence on the chemical potentials. This is again another avatar of the UV\/IR mixing.\n \\item For CC state, we are able to calculate the growth of entanglement entropy of a single interval explicitly in analytic form. The EE growth is strictly monotonically increasing for CC state. The stationary limit is, as expected, the entanglement entropy of a single interval in thermal ensemble.\n \\item We also calculate the EE growth of a single interval in gCC state with $W_4$ charge of the $\\mathcal{W}_{2n}$ representation of fermion corresponding to the $\\mathcal{W}_{2n}$ bilinear bosonic representation. We find dynamical phase transition in which the EE growth is monotonically increasing upto a critical value of $\\kappa_4$. Beyond the critical value, the EE growth is non-monotonic.\n\\end{enumerate}\n\n\n\\gap1\n\\noindent The outline of the paper is as follows:\n\n\\noindent In section \\ref{theo}, we solve the Dirac equation with time-dependent mass and from explicit solutions for a specific mass profile, we calculate the Bogoliubov coefficients for the transformation between the massive and massless modes. In section \\ref{qstates}, we find the final state after the quench starting from the ground state and a few squeezed states of our interest. In sections \\ref{ed} and \\ref{crf}, we calculate energy density and some correlators in the different quenched states that we obtained. The EE growth of a single subsystem in CC state is explicitly calculated in section \\ref{secEE_CC}. In section \\ref{secEE_gCC}, we show the dynamical phase transition in the EE growth of a subsystem in a particular gCC state. Section \\ref{cond} contains some discussions. The appendix contains details that we have omitted in the main sections.\n\n\\section{\\label{theo}Free Dirac fermions with time-dependent mass}\nThe action for Dirac fermions with time-dependent mass is\n\\begin{eqnarray}\nS=-\\int dx^2\\left[i\\bar{\\Psi}\\gamma^\\mu\\partial_\\mu\\Psi-m(t)\\bar{\\Psi}\\Psi\\right]\\\n\\label{action}\n\\end{eqnarray}\nThe equation of motion (EOM) is\n\\begin{eqnarray}\n \\left[ i\\gamma^0\\partial_t-i\\gamma^1\\partial_x-m(t)\\right]\\Psi(x,t)=0\\\n \\label{eom}\n\\end{eqnarray}\nand we are interested in the solvable mass profile\\cite{Duncan:1977fc, Das:2014hqa}\n\\begin{eqnarray}\nm(t)=m[1-\\tanh(\\rho t)]\/2\\\n \\label{mass}\n\\end{eqnarray}\n$m$ is the initial mass and $\\rho$ is the only scale of the quench process. $\\rho\\to\\infty$ is the sudden limit in which the mass is set to zero suddenly - much faster than any other length scale in the theory.\nIt is easier to solve (\\ref{eom}) in the Dirac basis in which $\\gamma_0$ is diagonal. Since the system is translation invariant in the  spatial $x$-direction, the solution ansatz is\n\\begin{eqnarray}\n \\Psi(x,t)=\\left[\\gamma^0\\partial_t-\\gamma^1\\partial_x-im(t)\\right]e^{\\pm i kx}\\Phi(t)\\\n \\label{spinor_antz}\n\\end{eqnarray}\nSubstitution in the EOM gives,\n\\begin{eqnarray}\n \\left[\\partial^2_t+k^2+m(t)^2-i\\gamma^0\\dot{m}(t)\\right]e^{\\pm i kx}\\Phi(t)=0\\nonumber\\\n\\end{eqnarray}\nwhere $\\dot{m}(t)=\\partial_tm(t)$.\n\n$\\Phi(t)$ is solved in the eigenbasis of $\\gamma^0$. For the two eigenvalues of $\\gamma^0$ (1 and -1), the two solutions $\\phi_+(t)$ and $\\phi_-(t)$ are given by,\n\\begin{eqnarray}\n &&\\left[\\partial^2_t+k^2+m(t)^2-i\\dot{m}(t)\\right]\\phi_+(t)=0\\nonumber\\\\\n &&\\left[\\partial^2_t+k^2+m(t)^2+i\\dot{m}(t)\\right]\\phi_-(t)=0\\\n \\label{pppm}\n\\end{eqnarray}\nwhere $\\Phi(t)=\\begin{bmatrix} \\phi_+(t) & \\phi_-(t)  \\end{bmatrix}^\\text{T}$. The eigenstates of $\\gamma^0$ are $u_0=\\begin{bmatrix} 1 \\\\ 0  \\end{bmatrix}$ and $v_0=\\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}$, they are the spinors in the rest frame.\n\nFor the mass profile (\\ref{mass}), there are two important bases of solutions in which we are interested in. The first one is the `in' basis in which the two independent solutions of the second order linear differential equations become different single frequency modes in the $t\\to -\\infty$ limit. In other words, one solution becomes the negative energy mode and the other solution becomes the positive energy mode. Similarly, there is also an `out' basis of solutions in which one solution becomes the negative energy mode and the other becomes the positive energy mode in the $t\\to\\infty$ limit. Accordingly, we will also have different `in' and `out' creation and annihilation operators.\nConsider the solutions of (\\ref{pppm}) in the two bases to be\n\\begin{eqnarray}\n&&\\phi_{\\pm}(t,k)=\\phi_{in,\\pm p}(t,k)+\\phi_{in,\\pm m}(t,k)\\\\\n&&\\phi_{\\pm}(t,k)=\\phi_{out,\\pm p}(t,k)+\\phi_{out,\\pm m}(t,k)\\\n\\end{eqnarray}\nwhere the limits are\n\\begin{eqnarray}\n&&\\lim_{t\\to-\\infty} \\phi_{in,\\pm p}(t,k)=e^{-i\\omega_{in} t}, \\quad \\lim_{t\\to-\\infty}\\phi_{in,\\pm m}(t,k)=e^{i\\omega_{in} t}\\nonumber\\\\\n&&\\lim_{t\\to\\infty} \\phi_{out,\\pm p}(t,k)=e^{-i\\omega_{out} t}, \\quad \\lim_{t\\to\\infty}\\phi_{out,\\pm m}(t,k)=e^{i\\omega_{out} t}\\nonumber\\\n\\end{eqnarray}\nwhere `p' means {\\it positive energy} and `m' means {\\it negative energy}. The above solutions are the same but written in two different bases for simplicity in the appropriate time limits, they are related by Bogoliubov transformations.\\\\\nBut from (\\ref{pppm}), we see that the equations of $\\phi_+$ and $\\phi_-$ are the complex conjugates of each other, so\n\\begin{eqnarray}\n &&\\phi_{in,+p}(t,k)=\\phi^*_{in,-m}(t,k), \\quad \\phi_{in,+m}(t,k)=\\phi^*_{in,-p}(t,k)\\\\\n \\label{conj1}\n &&\\phi_{out,+p}(t,k)=\\phi^*_{out,-m}(t,k), \\quad \\phi_{out,+m}(t,k)=\\phi^*_{out,-p}(t,k)\\\n \\label{conj2}\n\\end{eqnarray}\nThe Bogoliubov transformations are\n\\begin{eqnarray}\n \\phi_{in,+p}(t,k)&=&\\alpha'_+(k)\\phi_{out,+p}(t,k)+\\beta'_+(k)\\phi_{out,+m}(t,k)\\nonumber\\\\\n \\label{bop}&=&\\alpha'_+(k)\\phi_{out,+p}(t,k)+\\beta'_+(k)\\phi^*_{out,-p}(t,k)\\\\\n \\phi_{in,-p}(t,k)&=&\\alpha'_-(k)\\phi_{out,-p}(t,k)+\\beta'_-(k)\\phi_{out,-m}(t,k)\\nonumber\\\\\n &=&\\alpha'_-(k)\\phi_{out,-p}(t,k)+\\beta'_-(k)\\phi^*_{out,+p}(t,k)\\ \\label{bom}\n\\end{eqnarray}\nwhere $\\alpha'_{\\pm}(k)$ an $\\beta'_{\\pm}(k)$ are actually functions of $|k|$, since the equations of motion have only $k^2$ terms.\n\nNow, suppressing the basis labels `in' and `out' since they apply to both bases, we write the $u_0$ part of $\\Psi(x,t)$ as (upto normalization)\n\\begin{eqnarray}\n \\tilde{U}(x,t;k)=\\left[ \\gamma^0\\partial_t+\\gamma^1\\partial_x -im(t)\\right]e^{ikx}\\phi_{+p}(t)\\begin{bmatrix} 1 \\\\ 0  \\end{bmatrix}\\\n \\label{Uxt}\n\\end{eqnarray}\nAnd the $v_0$ part of $\\Psi(x,t)$ as\n\\begin{eqnarray}\n \\tilde{V}(x,t;k)&=&\\left[ \\gamma^0\\partial_t+\\gamma^1\\partial_x -im(t)\\right]e^{-ikx}\\phi_{-m}(t)\\begin{bmatrix} 0 \\\\ 1  \\end{bmatrix}\\nonumber\\\\\n &=&\\left[ \\gamma^0\\partial_t+\\gamma^1\\partial_x -im(t)\\right]e^{-ikx}\\phi^*_{+p}(t)\\begin{bmatrix} 0 \\\\ 1  \\end{bmatrix}\\\n \\label{Vxt}\n\\end{eqnarray}\nWe can define the spinors as (upto normalization)\n\\begin{eqnarray}\n \\tilde{u}(t,k)&=&\\frac{1}{e^{ikx}\\phi_{+p}(t)}\\tilde{U}(x,t;k)\\nonumber\\\\\n \\tilde{v}(t,k)&=&\\frac{1}{e^{-ikx}\\phi_{-m}(t)}\\tilde{V}(x,t;k)\\nonumber\\\n\\end{eqnarray}\nWith proper normalization, the final Dirac fermion mode expansion is\n\\begin{eqnarray}\n \\Psi(x,t)&=&\\int \\frac{dk}{2\\pi}\\frac{1}{\\sqrt{2\\omega}}\\left[a_k U(x,t;k)+b^\\dagger_k V(x,t;k)\\right]\\nonumber\\\\\n &=&\\int \\frac{dk}{2\\pi}\\frac{1}{\\sqrt{2\\omega}}\\left[a_k u(t;k)e^{ikx}\\phi_{+p}(t)+b^\\dagger_k v(t;k)e^{-ikx}\\phi_{-m}(t)\\right]\\\n \\label{modes}\n\\end{eqnarray}\n\n\\subsection{Bogoliubov transformation of oscillators}\nThe initial mass is taken to be $\\lim_{t\\to -\\infty}m(t)=m$. It is convenient to take the final mass $\\lim_{t\\to\\infty}m(t)$ to be some $m_{out}$, because of the spinor convention (in P\\&S), although we are interested in $m_{out}=0$.\n\nWith time-dependent mass, as mentioned above, the spinors are functionals of $m(t)$, but their normalizations are constants or else they will not solve the Dirac equations. So, we have to differentiate between `in' spinors and `out' spinors. Taking this into account, the mode expansion of $\\Psi(x,t)$ starting from `in' basis to `out' basis is\n\\begin{eqnarray}\n \\Psi(x,t)&=&\\int \\frac{dk}{2\\pi}\\frac{1}{\\sqrt{2\\omega_{in}}}\\left[a_{in,k} u_{in}(k,m) \\phi_{in,+p}(t,k) e^{ikx}+b^\\dagger_{in,k} v_{in}(k,m)\\phi^*_{in,+p}(t,k)e^{-ikx}\\right]\\nonumber\\\\\n &=&\\int \\frac{dk}{2\\pi}\\frac{1}{\\sqrt{2\\omega_{in}}}\\Huge{[} \\{\\alpha'_+(k)a_{in,k} u_{in}(k,m)\\phi_{out,+p}(t,k)+b^\\dagger_{in,-k} v_{in}(-k,m)\\beta'^*_+(k)\\phi_{out,-p}(t,k)\\} e^{ikx}\\nonumber\\\\\n &&\\quad+\\{\\alpha'^*_+(k)b^\\dagger_{in,k} v_{in}(k,m)\\phi^*_{out,+p}(t,k)+a_{in,-k} u_{in}(-k,m)\\beta'_+(k)\\phi^*_{out,-p}(t,k)\\}e^{-ikx}\\huge{]}\\nonumber\\\\\n\\end{eqnarray}\nwhere we have used the facts that the $k$ integral is from $-\\infty$ to $\\infty$ and $\\alpha'_{\\pm}$, $\\beta'_{\\pm}$ and $\\phi_{\\pm p}$ are functions of $|k|$. In $t\\to\\infty$ limit, $m(t)\\to m_{out}$, so,\n\\begin{eqnarray}\n \\lim_{t\\to\\infty}\\Psi(x,t)&=&\\int \\frac{dk}{2\\pi}\\frac{1}{\\sqrt{2\\omega_{out}}}\\sqrt{\\frac{\\omega_{out}}{\\omega_{in}}}\\huge{[} \\{\\alpha'_+(k)a_{in,k} u_{in}(k,m_{out})+b^\\dagger_{in,-k} v_{in}(-k,m_{out})\\beta'^*_+(k)\\} e^{-ik\\cdot x}\\nonumber\\\\\n &&\\quad+\\{\\alpha'^*_+(k)b^\\dagger_{in,k} v_{in}(k,m_{out})+a_{in,-k} u_{in}(-k,m_{out})\\beta'_+(k)\\}e^{ik\\cdot x}\\huge{]}\\nonumber\\\n\\end{eqnarray}\nComparing with the mode expansion in the `out' solution basis in the same limit $t\\to\\infty$,\n\\begin{eqnarray}\n \\lim_{t\\to\\infty}\\Psi(x,t)&=&\\int \\frac{dk}{2\\pi}\\frac{1}{\\sqrt{2\\omega_{out}}}\\left[ a_{out,k} u_{out}(k,m_{out}) \\phi_{out,+p}(t,k) e^{ikx}+b^\\dagger_{out,k} v_{out}(k,m_{out})\\phi^*_{out,+p}(t,k)e^{-ikx}\\right]\\nonumber\\\n\\end{eqnarray}\nwe get the Bogoliubov transformations of the creation and annihilation operators.\n\\begin{eqnarray}\n\\label{aout} a_{out,k}&=&\\alpha_+(k)a_{in,k}+b^\\dagger_{in,-k}\\, \\chi(k,m_{out})\\beta^*_+(k)\\\\\n b^\\dagger_{out,k}&=&\\alpha^*_+(k)b^\\dagger_{in,k} +a_{in,-k}\\, \\tilde{\\chi}(k,m_{out})\\beta_+(k)\\\n \\label{bdout}\n\\end{eqnarray}\nwhere $\\alpha_+(k)=\\sqrt{\\frac{\\omega_{out}(\\omega_{out}+m_{out})}{\\omega_{in}(\\omega_{in}+m_{in})}}\\,\\alpha'_+$ and $\\beta_+(k)=\\sqrt{\\frac{\\omega_{out}(\\omega_{out}+m_{out})}{\\omega_{in}(\\omega_{in}+m_{in})}}\\,\\beta'_+(k)$. Using (\\ref{norspinor})\n\\begin{eqnarray}\n \\chi(k,m_{out}) &=& \\frac{1}{2m_{out}}\\sqrt{\\frac{\\omega_{in}+m_{in}}{\\omega_{out}+m_{out}}}\\,\\bar{u}_{out}(k,m_{out},\\omega_{out})v_{in}(-k,m_{out},-\\omega_{out})\\nonumber\\\\\n &=&\\text{sgn}(k)\\qquad\\qquad \\text{when}\\quad m_{out}\\to0\\\n \\label{chi}\n\\end{eqnarray}\nwhere we have to be careful that $v_{in}(k,m_{out})$ is a functional of the accompanying mode, which is $\\sim e^{-i\\omega_{out}t}$ in the above case. Similarly, with $m_{out}\\to0$,\n\\begin{eqnarray}\n\\tilde{\\chi}(k)&=&-\\frac{1}{2m_{out}}\\sqrt{\\frac{\\omega_{in}+m_{in}}{\\omega_{out}+m_{out}}}\\,\\bar{v}(k,m_{out},\\omega_{out})u(-k,m_{out},\\omega_{out})=\\text{sgn}(k) \\\n\\label{chit}\n\\end{eqnarray}\ntaking into account the normalization of $\\bar{v}_{out}v_{out}=-2m_{out}$. Inverting (\\ref{aout}) and (\\ref{bdout}), we get\n\\begin{eqnarray}\n\\label{ain} a_{in,k}&=&\\alpha^*_+(k)a_{out,k}-\\text{sgn}(k)\\beta^*_+(k)b^\\dagger_{out,-k}\\\\\nb^\\dagger_{in,-k}&=&\\alpha_+(k)b^\\dagger_{out,-k}+\\text{sgn}(k)\\beta_+(k)a_{out,k}\\\n\\label{bdin}\n\\end{eqnarray}\nFrom here on, we will suppress the subscript `out' on creation and annihilation operators, so $a_{out,k}=a_k$, similarly for $b_{out,k}$ and their Hermitian conjugates. Also, since $\\chi(k)$ and $\\tilde{\\chi}(k)$ are simple sign functions, with a slight abuse of the nomenclature, we will call $\\alpha_+(k)$ and $\\beta_+(k)$ as the Bogoluibov coefficients.\nMoreover, $\\chi(k)^2$ and $\\tilde{\\chi}(k)^2$ are identically equal to 1. So, the fermionic anti-commutation relations of the `in' and `out' operators constraint the Bogoluibov coefficients as\n\\begin{eqnarray}\n\\label{ab1}|\\alpha_+(k)|^2+|\\beta_+(k)|^2=1\\ \n\\end{eqnarray}\n\n\\subsection{Explicit solutions}\nIn the `in' basis, for our choice of mass profile, the solutions are\n\\begin{eqnarray}\n\\phi_{in,+p} &=& e^{-i t \\left(\\omega+m\\right)} \\left(e^{-2 \\rho  t}+1\\right)^{-\\frac{i m}{2 \\rho }} \\ _2F_1\\left(\\frac{i\\left(|k|-m-\\omega\\right)}{2 \\rho },-\\frac{i \\left(|k|+m+\\omega\\right)}{2 \\rho };1-\\frac{i \\omega}{\\rho };-e^{2 t\\rho }\\right)\\nonumber\\\\\n\\phi_{in,-m} &=& e^{i t \\left(\\omega-m\\right)} \\left(e^{-2 \\rho  t}+1\\right)^{-\\frac{i m}{2 \\rho }}\\ _2F_1\\left(\\frac{i\\left(-|k|-m+\\omega\\right)}{2 \\rho },\\frac{i \\left(|k|-m+\\omega\\right)}{2 \\rho };\\frac{i \\omega}{\\rho }+1; -e^{2 t\\rho}\\right)\\nonumber\\\n\\end{eqnarray}\nwhere $\\omega = \\sqrt{k^2+m^2}$. While in the `out' basis, the solutions are\n\\begin{eqnarray}\n\\phi_{out,+p} &=& e^{-i |k| t} \\left(e^{-2\\rho t}+1\\right)^{-\\frac{i m}{2 \\rho }} \\ _2F_1\\left(\\frac{i |k|-i m+i \\omega}{2 \\rho },\\frac{i |k|-i m-i \\omega}{2 \\rho };1+\\frac{i |k|}{\\rho };-e^{-2\\rho t}\\right)\\nonumber\\\\\n\\phi_{out,-m} &=& e^{i |k| t} \\left(e^{-2\\rho t}+1\\right)^{-\\frac{i m}{2 \\rho }} \\ _2F_1\\left(\\frac{-i |k|-i m+i \\omega}{2 \\rho },\\frac{-i|k|-i m-i \\omega}{2 \\rho};1-\\frac{i |k|}{\\rho };-e^{-2\\rho t}\\right)\\nonumber\\\n\\end{eqnarray}\n\nUsing the properties of confluent hypergeometric functions $_2F_1$ given in \\cite{Abramowitz}, the Bogoliubov coefficients of the frequency modes as defined in (\\ref{bop}) are\n\\begin{eqnarray}\n\\label{alphap}\n\\alpha'_+ &=& \\frac{\\Gamma \\left(-\\frac{i |k|}{\\rho }\\right) \\Gamma \\left(1-\\frac{i \\omega}{\\rho }\\right)}{\\Gamma \\left(-\\frac{i\\left(|k|+m+\\omega\\right)}{2 \\rho }\\right) \\Gamma \\left(1+\\frac{-i |k|+i m -i \\omega}{2 \\rho }\\right)}\\\\\n\\label{betap}\n\\beta'_+ &=& \\frac{\\Gamma \\left(\\frac{i |k|}{\\rho }\\right) \\Gamma \\left(1-\\frac{i \\omega}{\\rho }\\right)}{\\Gamma \\left(1-\\frac{i\\left(-|k|-m+\\omega\\right)}{2 \\rho }\\right) \\Gamma \\left(-\\frac{i \\left(-|k|+m+\\omega\\right)}{2 \\rho }\\right)}\n\\end{eqnarray}\nIn the sudden limit($\\rho\\to\\infty$). the Bogoliubov coefficients of the frequency modes are\n\\begin{eqnarray}\n\\alpha'_+(k) &=& \\frac{|k|+m_{in}+\\sqrt{k^2+m^2}}{2|k|}\\\\\n\\beta'_+(k) &=& \\frac{|k|-m_{in}-\\sqrt{k^2+m^2}}{2|k|}\\\n\\end{eqnarray}\nAs mentioned above, for a quench starting from the ground state of the massive theory, the naive sudden limit gives a non-normalizable state in the massless theory \\cite{Mandal:2015kxi}. The problem arises only for a quench starting from the ground state. In case the quench is starting from squeezed states of our interest, the naive sudden limit given above works well.\nAs defined in (\\ref{aout}) and (\\ref{bdout}), the Bogoluibov coefficients of the oscillator modes differ from $\\alpha'_+(k)$ and $\\beta'_+(k)$ by an overall factor.\n\\begin{eqnarray}\n\\label{alpha}\n\\alpha_+ &=& \\sqrt{1-\\frac{m}{\\sqrt{k^2+m^2}}}\\frac{\\Gamma \\left(-\\frac{i |k|}{\\rho }\\right) \\Gamma \\left(1-\\frac{i \\omega}{\\rho }\\right)}{\\Gamma \\left(-\\frac{i\\left(|k|+m+\\omega\\right)}{2 \\rho }\\right) \\Gamma \\left(1+\\frac{-i |k|+i m -i \\omega}{2 \\rho }\\right)}\\\\\n\\label{beta}\n\\beta_+ &=& \\sqrt{1-\\frac{m}{\\sqrt{k^2+m^2}}}\\frac{\\Gamma \\left(\\frac{i |k|}{\\rho }\\right) \\Gamma \\left(1-\\frac{i \\omega}{\\rho }\\right)}{\\Gamma \\left(1-\\frac{i\\left(-|k|-m+\\omega\\right)}{2 \\rho }\\right) \\Gamma \\left(-\\frac{i \\left(-|k|+m+\\omega\\right)}{2 \\rho }\\right)}\n\\end{eqnarray}\nIn the sudden limit, they are\n\\begin{eqnarray}\n\\label{alphas} \\alpha_+(k) &=& \\sqrt{1-\\frac{m}{\\sqrt{k^2+m^2}}}\\, \\frac{|k|+m+\\sqrt{k^2+m^2}}{2|k|}\\\\\n\\label{betas} \\beta_+(k) &=& \\sqrt{1-\\frac{m}{\\sqrt{k^2+m^2}}}\\, \\frac{|k|-m-\\sqrt{k^2+m^2}}{2|k|}\\\n\\end{eqnarray}\n\nFor completeness, the expressions of $\\alpha'_-$ and $\\beta'_-$ in (\\ref{bom}) for our particular quench protocol are\n\\begin{eqnarray}\n\\alpha'_- &=& \\frac{\\Gamma \\left(-\\frac{i \\left| k\\right| }{\\rho }\\right) \\Gamma\\left(1-\\frac{i \\omega}{\\rho }\\right)}{\\Gamma \\left(-\\frac{i\\left(\\left| k\\right| -m+\\omega\\right)}{2 \\rho }\\right) \\Gamma \\left(1-\\frac{i \\left(\\left| k\\right| +m+\\omega \\right)}{2 \\rho }\\right)}\\\\\n\\beta'_-&=& \\frac{\\Gamma \\left(\\frac{i \\left| k\\right| }{\\rho }\\right) \\Gamma \\left(1-\\frac{i \\omega}{\\rho }\\right)}{\\Gamma \\left(\\frac{i\\left(\\left| k\\right| +m-\\omega\\right)}{2 \\rho }\\right) \\Gamma \\left(1-\\frac{i \\left(-\\left| k\\right|+m+\\omega\\right)}{2 \\rho }\\right)}\n\\end{eqnarray}\n\n\\section{\\label{qstates}Quenched states}\n\n\\subsection{From ground state}\nStarting from the ground state of the massive theory $|\\Psi\\rangle=|0,in\\rangle$, using Eq (\\ref{ain}), the state in terms of `out' operators is given by\n\\begin{align}\na_{in,k}|\\Psi\\rangle=0 \\quad&\\Rightarrow \\left[\\alpha^*_+(k)a_{k}-\\text{sgn}(k)\\beta^*_+(k)b^\\dagger_{-k}\\right]|\\Psi\\rangle=0\\nonumber\\\\\n\\label{psig}&\\Rightarrow |\\Psi\\rangle = e^{\\sum_k \\text{sgn(k)}\\gamma(k) a^\\dagger_{k}b^\\dagger_{-k}}|0\\rangle\\\\\n\\text{where} &\\quad\\gamma(k)=\\frac{\\alpha^*_+(k)}{\\beta^*_+(k)}\\\n\\label{gg}\n\\end{align}\nwhere we have taken $|0\\rangle$ to be the ground state of `out' oscillators. Using the Baker-Campbell-Hausdorff(BCH) formula derived in appendix (\\ref{bch}), the above state can be written in gCC form. For the particular mass profile (\\ref{mass}), $\\alpha_+(k)$ and $\\beta_+(k)$ are given in (\\ref{alpha}) and (\\ref{beta}). The gCC form which was first obtained in MPS is\n\\begin{eqnarray}\n\\label{psigg}|\\Psi\\rangle =e^{-\\kappa_2 H-\\kappa_4W_4-\\kappa_6W_6-...}|D\\rangle\\\n\\end{eqnarray}\nwhere\n\\begin{gather}\n\\kappa_2=\\frac{1}{2 m}+\\frac{\\pi ^2 m}{12 \\rho ^2}+\\frac{1}{m}\\mathcal{O}\\left(\\frac{m}{\\rho}\\right)^3, \\quad \\kappa_4=-\\frac{1}{12 m^3}+\\frac{\\pi ^2}{24 m \\rho ^2}+\\frac{1}{m^3}\\mathcal{O}\\left(\\frac{m}{\\rho}\\right)^3,\\nonumber\\\\\n\\kappa_6=\\frac{3}{80 m^5}-\\frac{\\pi ^2}{96 m^3 \\rho ^2}+\\frac{1}{m^5}\\mathcal{O}\\left(\\frac{m}{\\rho}\\right)^3,...\\\n\\label{kgs}\n\\end{gather}\nand $|D\\rangle$ is the Dirichelet state and the explicit expression is in Appendix \\ref{bstate}. It should be noted that since the mass does not go to zero at any finite time, the above state should is only valid in sufficiently long time limit and the correction due to the non-vanishing mass is $\\mathcal{O}(e^{-\\rho t})$.\n\n\\subsection{From squeezed states: CC state and gCC states}\nWe could start with specially prepared squeezed states so that after the quench, the states become CC states or gCC states. Here, we will consider only the simple case of sudden quench ($\\rho\\to\\infty$). For our aim of creating a CC state or a gCC state, finite `$\\rho$' quenches are an unnecessary complication.\\\\\n\nWe start with a squeezed state of `in' modes\n\\begin{eqnarray}\n |S\\rangle = \\exp\\left(\\sum_{k=-\\infty}^{\\infty}f(k)a^\\dagger_{in,k}b^\\dagger_{in,-k}\\right)|0,in\\rangle\\\n \\label{sqg}\n\\end{eqnarray}\nwhere unlike $\\gamma(k)$, $f(k)$ need not be an even function of $k$, but $|f(k)|^2$ is an even function of $k$.\n\nIt is easier to work with $|S\\rangle$ as an operator relation. $|S\\rangle$ can also be defined as\n\\begin{eqnarray}\n \\label{sstate} \\tilde{a}_{k}|S\\rangle=\\tilde{b}_{k}|S\\rangle=0\\\n \\text{and} && \\left\\{\\tilde{a}_k,\\tilde{a}^\\dagger_{k'}\\right\\}=\\left\\{\\tilde{b}_{-k},\\tilde{b}^\\dagger_{-k'}\\right\\}=\\delta(k-k')\\\n \\label{tacom}\n\\end{eqnarray}\nwhere the new operators in terms of the out modes using (\\ref{ain}) and (\\ref{bdin}) are\n\\begin{align}\n\\tilde{a}_{k}&=\\frac{1}{\\sqrt{(1+|f(k)|^2)}}a_{in,k}-\\frac{f(k)}{\\sqrt{(1+|f(k)|^2)}}b^\\dagger_{in,-k}\\nonumber\\\\\n&=A^*(k)a_{out,k}-\\text{sgn}(k)B^*(k)b_{out,-k}\\nonumber\\\\\n\\tilde{b}_{-k}&=\\frac{1}{\\sqrt{(1+|f(k)|^2)}}b_{in,-k}+\\frac{f(k)}{\\sqrt{(1+|f(k)|^2)}}a^\\dagger_{in,k}\\nonumber\\\\\n&=A^*(k)b_{out,-k}+\\text{sgn}(k)B^*(k)a^\\dagger_{out,k}\\\n\\label{tab}\n\\end{align}\nwhere $A(k)$ and $B(k)$ are the Bogoliubov coefficients for the transformation from `{\\it{tilde}}' operators to `out' operators and are given by\n\\begin{gather}\n\\label{AnB}A(k)=\\frac{\\alpha_+(k)-\\text{sgn}(k)\\beta_+^*(k)f^*(k)}{\\sqrt{(1+|f(k)|^2)}},\\quad B(k)=\\frac{\\beta_+(k)+\\text{sgn}(k)\\alpha_+^*(k)f^*(k)}{\\sqrt{(1+|f(k)|^2)}}\\\\\n|A(k)|^2+|B(k)|^2=1\\\n\\label{AB1}\n\\end{gather}\nNow using the BCH formula (\\ref{BCHeqn}) from appendix (\\ref{bch}),\n\\begin{gather}\n\\label{sqgo}|S\\rangle=\\exp\\left\\{-\\sum_k \\tilde{\\kappa}(k)\\left(a^\\dagger_{out,k}a_{out,k}+b^\\dagger_{out,k}b_{out,k}\\right)\\right\\}|D\\rangle\\\\\n\\text{where} \\quad \\tilde{\\gamma}(k)=\\frac{B^*(k)}{A^*(k)}, \\quad\\text{and}\\quad \\tilde{\\kappa}(k)=-\\frac{1}{2}\\log(\\tilde{\\gamma}(k))\\nonumber\\\n\\end{gather}\nFor a CC state, i.e., so that $|S\\rangle$ in eqn (\\ref{sqgo}) is $e^{-\\kappa_2 H}|D\\rangle$, $f(k)$ should be tuned as\n\\begin{eqnarray}\n f(k)=\\frac{\\left(\\sqrt{k^2+m^2}+m\\right) \\cosh (\\kappa_2 k )-k \\sinh (\\kappa_2 k)}{\\left(\\sqrt{k^2+m^2}+m\\right) \\sinh (\\kappa_2 k)+k\\cosh (\\kappa_2 k )}\\\n \\label{fcc}\n\\end{eqnarray}\nStarting with\n\\begin{eqnarray}\n f(k)=\\frac{k-k\\, e^{2 \\left| k\\right|  \\left(\\kappa_2+\\kappa_4 k^2\\right)}+\\text{sgn}(k)\\left(\\sqrt{k^2+m^2}+m\\right)\\left(e^{2 \\left| k\\right|  \\left(\\kappa_2+\\kappa_4k^2\\right)}+1\\right)}{\\left| k\\right|  \\left(e^{2 \\left| k\\right|  \\left(\\kappa_2 +\\kappa_4 k^2 \\right)}+1\\right)+\\left(\\sqrt{k^2+m^2}+m\\right) \\left(e^{2 \\left| k\\right|  \\left(\\kappa_2+\\kappa_4 k^2\\right)}-1\\right)}\\\n \\label{fgcc}\n\\end{eqnarray}\nwe get a gCC state of the form $e^{-\\kappa_2 H-\\kappa_4 W_4}|D\\rangle$, where as mentioned earlier, $W_4$ is the conserved charge of the $\\mathcal{W}_4$ current of free Dirac fermions{\\footnote{For the action (\\ref{action}), $H=\\sum_k |k|(a^\\dagger_k a_k+b^\\dagger_k b_k)$ or $H=\\int\\frac{dk}{2\\pi} |k|(a^\\dagger_k a_k+b^\\dagger_k b_k)$. $\\mathcal{W}_4$ has been normalized so that $W_4=\\sum_k |k|^3(a^\\dagger_k a_k+b^\\dagger_k b_k)$ or $W_4=\\int \\frac{dk}{(2\\pi)^3}|k|^3(a^\\dagger_k a_k+b^\\dagger_k b_k)$ in the continuum limit.}. Note that $f(k)$ are odd functions of $k$.\nFor future reference, we can invert Eq (\\ref{tab}) and we write down the `in' and `out' operators in terms of the `{\\it{tilde}}' operators.\n\\begin{gather}\n \\label{aintota}a_{in,k}=\\frac{1}{\\sqrt{(1+|f(k)|^2)}}\\tilde{a}_k+\\frac{f(k)}{\\sqrt{(1+|f(k)|^2)}}\\tilde{b}^\\dagger_{-k}\\\\\n \\label{bintotb} b^\\dagger_{in,-k}=\\frac{1}{\\sqrt{(1+|f(k)|^2)}}\\tilde{b}^\\dagger_{-k}-\\frac{f^*(k)}{\\sqrt{(1+|f(k)|^2)}}\\tilde{a}_k\\\\\n \\label{aouttota} a_{out,k}=A(k)\\tilde{a}_k+\\text{sgn}(k)B^*(k)\\tilde{b}^\\dagger_{-k}\\\\\n \\label{bdouttotbd}b^\\dagger_{out,-k}=A^*(k)\\tilde{b}^\\dagger_{-k}-\\text{sgn}(k)B(k)\\,\\tilde{a}_k\\\n\\end{gather}\n \n\\section{\\label{ed}Energy density}\nIn the post-quench theory, the occupation number is given by\n\\begin{eqnarray}\n\\hat{N}_k=a_k^\\dagger a_k +b_k^\\dagger b_k\\ \n\\end{eqnarray}\nusing the Bogoliubov transformations (\\ref{aouttota}) and (\\ref{bdouttotbd}) and definition of $|\\tilde{0}\\rangle$ in (\\ref{sstate}), the expectation value of the occupation number is given by\n\\begin{eqnarray}\n N_k&=&\\lim_{t\\to\\infty}\\langle\\tilde{0}|a_k^\\dagger a_k +b_k^\\dagger b_k|\\tilde{0}\\rangle\\nonumber\\\\\n &=&B^*(k)B(k)\\langle\\tilde{0}|\\tilde{b}_{-k}^\\dagger\\tilde{b}_{-k}|\\tilde{0}\\rangle+B^*(-k)B(-k)\\langle\\tilde{0}|\\tilde{a}_{-k}^\\dagger\\tilde{a}_{-k}|\\tilde{0}\\rangle+...\\nonumber\\\\\n &=& B^*(k)B(k)+B^*(-k)B(-k)\\\n\\end{eqnarray}\nThe expression of $B(k)$ is given in (\\ref{AnB}). For ground state, we have to use $f(k)=0$ in the expression of $B(k)$. So, energy density of the post-quench state is given by\n\\begin{eqnarray}\n E=\\int_{-\\infty}^\\infty\\frac{dk}{2\\pi}|k|\\left[B^*(k)B(k)+B^*(-k)B(-k)\\right]\\\n\\end{eqnarray}\n\n\\subsection*{Ground state quench}\nFor ground state quench, the occupation number is given by\n\\begin{eqnarray}\n N_k&=&\\lim_{t\\to\\infty}\\langle 0,in|\\hat{N}_k|0,in\\rangle=|\\beta_+(k)|^2+|\\beta_+(-k)|^2\\nonumber\\\n\\end{eqnarray}\nSince $\\alpha_+(k)$ and $\\beta_+(k)$ are even functions of $k$. Using (\\ref{ab1}), (\\ref{gg}) and (\\ref{forbch}), we have\n\\begin{eqnarray}\n|\\beta_+(k)|^2=\\frac{|\\gamma(k)|^2}{1+|\\gamma(k)|^2},\\qquad |\\gamma(k)|^2=e^{-4\\kappa(k)}\\\n\\end{eqnarray}\nHence, the occupation number in the ground state in the asymptotically long time limit is given by\n\\begin{eqnarray}\n N_k=\\frac{2}{e^{4\\kappa(k)}+1}\\\n\\end{eqnarray}\nThis is the occupation number in a GGE defined as\n\\begin{eqnarray}\n\\text{Tr}\\,e^{-\\sum_k 4\\kappa(k)\\hat{N}_k}=\\text{Tr} e^{- 4\\kappa_2 H-4\\kappa_4 W_4-\\kappa_6 W_6-\\cdots}\\\n\\end{eqnarray}\nwhere the $\\kappa$'s are given in (\\ref{kgs}).\nUsing the expressions of $\\beta_+(k)$ from (\\ref{beta}), the explicit expression of the occupation number is \n\\begin{eqnarray}\n N_k&=&\\text{csch}\\left(\\frac{\\pi  k}{\\rho }\\right) \\left(\\cosh \\left(\\frac{\\pi  m}{\\rho }\\right)-\\cosh \\left(\\frac{\\pi  \\left(k-\\sqrt{k^2+m^2}\\right)}{\\rho }\\right)\\right) \\text{csch}\\left(\\frac{\\pi  \\sqrt{k^2+m^2}}{\\rho }\\right)\\nonumber\\\\\n &\\xrightarrow{\\rho\\to\\infty}&1-\\frac{k}{\\sqrt{k^2+m^2}}\\nonumber\\\n\\end{eqnarray}\nIt is interesting that in $m\\to\\infty$ limit, $N_k\\to1$, not 2. This is because $\\lim_{\\rho\\to\\infty}|\\alpha_+(k)|^2=1\/2$ and we have the constraint $|\\alpha_+(k)|^2+|\\beta_+(k)|^2=1$.\\\\\nFor arbitrary $\\rho$, the energy density cannot be calculated in closed form. In the sudden limit $\\rho\\to\\infty$, the energy density diverges as $\\log(\\Lambda)$ where $\\Lambda$ is the UV cutoff. Hence, all other $W$ charges also diverge in the sudden limit. Hence, naively taking $\\rho\\to\\infty$ produce a non-renormalizable state. So, the sudden limit has to be taken as in MPS where $m\/\\Lambda\\to0$ while $m\/\\rho\\to\\epsilon^+$. Simply put, the quench rate parameter $\\rho$ should be much small than the UV cut-off.\n\n\\subsection*{Squeezed state quench: CC and gCC states}\nFor CC state given by (\\ref{fcc}), the expectation value of occupation number is given by\n\\begin{eqnarray}\n N_k=\\frac{2}{1+e^{4 \\kappa_2\\left|k\\right|}}\\\n\\end{eqnarray}\nThis is the occupation number of fermions in a thermal ensemble of temperature $1\/\\beta=1\/4\\kappa_2$. The enengy density is\n\\begin{eqnarray}\n E=\\int_{-\\infty}^{\\infty}\\frac{dk}{2\\pi}N_k=\\frac{\\pi }{96 \\kappa_2 ^2}\\\n\\end{eqnarray}\nSimilarly, for gCC state given by (\\ref{fgcc}), the expectation value of occupation number is given by\n\\begin{eqnarray}\n N_k=\\langle gCC|\\hat{N}_k|gCC\\rangle=\\frac{2}{1+e^{4 \\kappa_2\\left|k\\right|+4\\kappa_4\\left|k\\right|^3}}\\\n\\end{eqnarray}\nThis is same as the occupation number of fermions in a generalised Gibbs ensemble of temperature $1\/\\beta=4\\kappa_2$ and chemical potential $\\mu_4=4\\kappa_4$ of $W_4$ charge. The enengy density cannot be calculated in closed form.\n\n\\section{\\label{crf}Correlation functions}\nSince our theory is a free theory, all the observables can be explicitly calculated. In the following subsections we calculate $\\langle\\psi^\\dagger(r,t)\\psi(0,t)\\rangle$ correlation functions for the three different states obtained above. The quench process cannot differentiate between holomorphic dof(`left-movers') and anti-holomorphic dof(`right-movers'), so $\\langle\\bar{\\psi}^\\dagger(0,t)\\bar{\\psi}(r,t)\\rangle$ is equal to $\\langle\\psi^\\dagger(r,t)\\psi(0,t)\\rangle$ and they are time independent quantities.{\\footnote{A simple reason why these quantities are time independent is the fact that they are holomorphic-holomorphic and antiholomorphic-antiholomorphic quantities and they cannot `see' the presence of the boundary state $|D\\rangle$. They are already thermalized\/equilibrated.}} We also calculated $\\langle\\bar{\\psi}^\\dagger(r,t)\\psi(0,t)\\rangle$ which has non-trivial time-dependence. Also as expected, $-\\langle\\psi^\\dagger(0,t)\\bar{\\psi}(r,t)\\rangle$ is the complex conjugate of $\\langle\\bar{\\psi}^\\dagger(r,t)\\psi(0,t)\\rangle$. Since, we are calculating equal-time correllation functions, so for example for $\\langle\\psi^\\dagger(r,t)\\psi(0,t)\\rangle$, we would rather be calculating $\\frac{1}{2}\\langle\\psi^\\dagger(r,t)\\psi(0,t)-\\psi(0,t)\\psi^\\dagger(r,t)\\rangle$.\\\\\nUsing the Bogoluibov transformations (\\ref{aouttota}) and (\\ref{bdouttotbd}) in the chiral mode expansions (\\ref{crep}) and (\\ref{crepb})  we get \n\\begin{align}\n\\label{psioutt}\\psi(w)&=\\int_0^\\infty \\frac{dk}{2\\pi} \\Big[A(k) \\tilde{a}_{k}e^{-ikw}+\\text{sgn}(k)B^*(k)\\tilde{b}^\\dagger_{-k}e^{-ikw}+A^*(-k)\\tilde{b}^\\dagger_{k}e^{ikw}+\\text{sgn}(k)B(-k)\\tilde{a}_{-k}e^{ikw}\\Big]\\\\\n\\bar{\\psi}(\\bar{w})&=\\int^{\\infty}_0 \\frac{dk}{2\\pi} \\Big[A(-k) \\tilde{a}_{-k}e^{-ik\\bar{w}}-\\text{sgn}(k)B^*(-k)\\tilde{b}^\\dagger_{k}e^{-ik\\bar{w}}-A^*(k)\\tilde{b}^\\dagger_{-k}e^{ik\\bar{w}}+\\text{sgn}(k)B(k)\\tilde{a}_{k}e^{ik\\bar{w}}\\Big]\\\n\\label{bpsioutt}\n\\end{align}\nwhere $w=t-x$ and $\\bar{w}=t+x$. For the ground state quench, $f(k)=0,\\,\\tilde{a}_{k}=a_{in,k},\\,\\tilde{b}=b_{in,k}$ and $|\\tilde{0}\\rangle=|0,in\\rangle$.\\\\\nFor a general $f(k)$ corresponding to some $|\\tilde{0}\\rangle$, the correlation functions are\n \\begin{eqnarray}\n \\label{psidpsiS}\\langle\\tilde{0}|\\psi^\\dagger(0,t)\\psi(r,t)|\\tilde{0}\\rangle &=& \\frac{1}{2}\\int_{0}^{\\infty}\\frac{dk}{2\\pi}\\left[(2|B(k)|^2-1)e^{ikr}-(2|B(-k)|^2-1)e^{-ikr}\\right]\\\\\n \\label{bpsidpsiS}\\langle\\tilde{0}|\\bar{\\psi}^\\dagger(0,t)\\psi(r,t)|\\tilde{0}\\rangle&=& -\\int_0^{\\infty}\\frac{dk}{2\\pi}\\left[\\text{sgn}(k)A^*(-k)B(-k)e^{ik(2t-r)}+\\text{sgn}(k)A(k)B^*(k)e^{-ik(2t-r)}\\right]\\nonumber\\\\\n \\end{eqnarray}\nwhere we have used (\\ref{AB1}) to write $A(k)$ in terms of $B(k)$ in the first equation.\\\\\n\\subsection*{Ground state quench:}\nTaking careful limit, for ground state quench, we have\n\\begin{eqnarray}\n \\langle 0,in|\\psi^\\dagger(0,t)\\psi(r,t)|0,in\\rangle &&= -\\frac{1}{2}\\int_{-\\infty}^{\\infty}\\frac{dk}{2\\pi}\\frac{\\left| k\\right| }{\\sqrt{k^2+m^2}}\\nonumber\\\\\n &&=\\frac{1}{4} m \\left[\\pmb{L}_{-1}(m r)-I_1(m r)\\right]\\nonumber\\\\\n &&\\xrightarrow{m\\to\\infty}\\frac{1}{2 \\pi  m r^2}+\\frac{3}{2 \\pi  m^3 r^4}+O\\left(\\frac{1}{m^4}\\right)\\\\\n \\langle\\tilde{0}|\\bar{\\psi}^\\dagger(0,t)\\psi(r,t)|\\tilde{0}\\rangle&&=\\int_0^{\\infty}\\frac{dk}{2\\pi}\\,\\frac{i \\,\\text{sgn}(k) \\,m\\sin (k (2 t-r))}{\\sqrt{k^2+m^2}}\\nonumber\\\\\n &&=-\\frac{i m}{4} \\, \\left[\\text{sgn}(r-2 t) I_0(m (r-2 t))-\\pmb{L}_0(m (r-2 t))\\right]\\nonumber\\\\\n &&\\xrightarrow[t>r\/2]{m\\to\\infty}\\frac{i}{2 \\pi  (2 t-r)}+\\frac{i}{2 \\pi  m^2 (2 t-r)^3}+O\\left(\\frac{1}{m^4}\\right)\\\n\\end{eqnarray}\nwhere $I_\\nu(x)$ is Modified Bessel Function of the First Kind and $\\pmb{L}_\\nu(x)$ is Modified Struve Function.\n\\subsection*{Quenched squeezed state - CC state:}\nFor CC state, all the calculations are done in $|S_{CC}\\rangle$ defined as the state (\\ref{sqg}) with the expression of $f(k)$ given in (\\ref{fcc}).\n\\begin{eqnarray}\n \\langle CC|\\psi^\\dagger(0,t)\\psi(r,t)|CC\\rangle&=&-i\\int_{0}^{\\infty}\\frac{dk}{2\\pi}\\,\\tanh (2\\kappa_2 |k|)\\sin(kr)\\\\\n \\label{thermform}&=&-i\\int_{0}^{\\infty}\\frac{dk}{2\\pi}\\,\\sin(kr)\\left[\\frac{1}{e^{4\\kappa_2 |k|}+1}-\\frac{1}{2}\\right]\\\\\n &=& -\\frac{i\\, \\text{csch}\\left(\\frac{\\pi  r}{4 \\kappa _2}\\right)}{8 \\kappa _2}\\\\\n\\label{ccr=0} \\langle CC|\\bar{\\psi}^\\dagger(0,t)\\psi(r,t)|CC\\rangle&=&-i\\int_0^{\\infty}\\frac{dk}{2\\pi}\\,\\text{sech}(2 k \\kappa_2 ) \\cos (k(2 t-r))\\\\\n &=&-\\frac{i\\, \\text{sech}\\left(\\frac{\\pi  (2t-r)}{4 \\kappa_2 }\\right)}{8 \\kappa_2 }\\\n\\end{eqnarray}\nThese are exactly what have been calculated using BCFT techniques \\cite{Calabrese:2006quench}. It is evident from (\\ref{thermform}) that $\\psi^\\dagger\\psi$ expectation value is already the thermal expectation value at temperature $T=1\/\\beta =1\/(4\\kappa_2)$, i.e., it is already thermalized.\n\\subsection*{Quenched squeezed state - gCC state with $W_4$:}\nSimilarly, for gCC state, all the calculations are done in $|S_{fCC}\\rangle$ defined as the state (\\ref{sqg}) with the expression of $f(k)$ given in (\\ref{fgcc}).\n\\begin{eqnarray}\n \\langle\\psi^\\dagger(0,t)\\psi(r,t)\\rangle_{gCC}&=&-i\\int_{0}^{\\infty}\\frac{dk}{2\\pi}\\,\\tanh\\left(2\\kappa_2 |k| +2\\kappa_4 |k|^3\\right)\\,\\sin(kr)\\\\\n\\label{thermform2} &=&-i\\int_{0}^{\\infty}\\frac{dk}{2\\pi}\\,\\sin(kr)\\left[\\frac{1}{e^{4\\kappa_2 |k|+4\\kappa_4 |k|^3}+1}-\\frac{1}{2}\\right]\\\\\n \\langle\\bar{\\psi}^\\dagger(r,t)\\psi(0,t)\\rangle_{gCC}&=&-i\\int_0^{\\infty}\\frac{dk}{2\\pi}\\,\\text{sech}(2 \\kappa_2 k+2\\kappa_4 k^3 ) \\cos (k(2 t-r))\\\n \\end{eqnarray}\nAgain, it is evident from (\\ref{thermform2}) that $\\psi^\\dagger\\psi$ expectation value is already thermalized into the expectation value in a GGE with $T=1\/\\beta =1\/4\\kappa_2$ and $\\mu=4\\kappa_4$. A possible way of evaluating these integrals (which yield no closed form answer) is via the residue theorem. The integrands in both cases, have poles at the solutions of $2\\kappa_2 k +2\\kappa_4 k^3= \\frac{2n+1}{2} i\\pi$, where $n\\in\\mathbb{Z}$. These poles and their residues have been treated in detail in \\cite{Mandal:2015kxi}. The sum of residues is still an infinite sum which cannot be performed. In the perturbative regime ($\\kappa_4\/\\kappa_2^3<<1$), we see that our correlators match the general form presented in \\cite{Mandal:2015jla} with $h=1\/2$ as expected.\n\n\\gap3\n\nAs expected form MPS, here in the fermionic theory also we see the UV\/IR mixing. For the ground state quench, all the charges affect the long distance and long time limit of the correlators. This is explicit seen in the case of gCC state with $W_4$ charge only. The long time and large distance limit or the correlators are very much dependent upon $k_4$, although a naive Wilsonian RG argument would show that $k_4$ is an irrelevant coupling. \n\n\\section{\\label{secEE_CC}Exact Growth of Entanglement in CC state}\nWe will consider only a finite single interval or subsystem A, with its endpoints at $(w_1,\\bar{w}_1)$ and $(w_2,\\bar{w}_2)$ in light-cone coordinates, or $(0,t)$ and $(r,t)$ in space and time coordinates. Using the replica trick (\\cite{Holzhey:1994we}, \\cite{Calabrese:2004eu}), the $n\\textsuperscript{th}$ R\\'enyi entropy $S_n(A)$ of the interval is given by the logarithm of the expectation value of twist and antitwist operators inserted at the end-points.\n\\begin{eqnarray}\n\\label{sna} S_n(A)=\\frac{1}{1-n}\\log\\langle\\Psi(t)|\\mathcal{T}_{n}(w_1,\\bar{w}_1)\\tilde{\\mathcal{T}}_{n}(w_2,\\bar{w}_2)|\\Psi(t)\\rangle\\\n\\end{eqnarray}\nThe entanglement entropy(EE) $S_{A}$ is given by $\\lim_{n\\to1}S_n(A)$.\nWe can diagonalize the twist operators and write them as products of twist fields. Hence,\n\\begin{eqnarray}\n\\label{tntkn} \\mathcal{T}_{n}(w,\\bar{w})=\\prod_{k=-(n-1)\/2}^{k=(n-1)\/2}\\mathcal{T}_{k,n}(w,\\bar{w}), \\quad  \\tilde{\\mathcal{T}}_{n}(w,\\bar{w})=\\prod_{k=-(n-1)\/2}^{k=(n-1)\/2}\\tilde{\\mathcal{T}}_{k,n}(w,\\bar{w})\\\n\\end{eqnarray}\nIn CC state, in Heisenberg picture, the quantity of our interest is\n\\begin{eqnarray}\n\\label{tttf} Z_{k}=\\langle D_{f}|e^{-\\kappa_2 H_{f}}\\mathcal{T}_{k,n}(0,t)\\tilde{\\mathcal{T}}_{k,n}(r,t)e^{-\\kappa_2 H_{f}}|D_{f}\\rangle\\\n\\end{eqnarray}\nThe subscript `f' means we are working in the fermionic theory and the subscript `b' would mean we are working in the bosonic theory. To find the exact expression of the entanglement entropy of a spatial region in our free fermionic CFT, we will use the method using bosonization described in \\cite{Casini:2005rm}. Moreover, as shown in Appendix(\\ref{bbs}), Dirichlet state $|D_{f}\\rangle$ in fermionic theory corresponds to a Dirichlet state in the bosonic theory $|D_{b}\\rangle$ and $H_{f}$ corresponds to $H_{b}$. So, we get\n\\begin{eqnarray}\n\\label{tttf} Z_{k}&=&\\langle D_{b}|e^{-\\kappa_2 H_{b}}e^{i\\sqrt{4\\pi}\\frac{k}{n}\\left(\\phi(0,t)-\\phi(r,t)\\right)}e^{-\\kappa_2 H_{b}}|D_{b}\\rangle\\\n\\end{eqnarray}\nThis is a free scalar theory in a strip geometry with Dirichlet boundary conditions and operator insertions at $(0,t)$ and $(r,t)$. It can be calculated explicitly \n\\begin{eqnarray}\n \\log\\left[Z_{k}\\right]&=& -4\\pi\\,\\frac{2k^2}{n^2}\\left(\\langle\\phi(0,t)\\phi(0,t)\\rangle-\\langle\\phi(0,t)\\phi(r,t)\\rangle\\right)\\\n\\end{eqnarray}\n\\noindent The $n\\textsuperscript{th}$ R\\'enyi entropy of interval A is given by\n\\begin{eqnarray}\n  S_{n}(A)&=&-4\\pi\\frac{1}{1-n}\\sum_{k=-(n-1)\/2}^{k=(n-1)\/2}\\frac{2k^2}{n^2}\\left(\\langle\\phi(0,t)\\phi(0,t)\\rangle-\\langle\\phi(0,t)\\phi(r,t)\\rangle\\right)\\nonumber\\\\\n \\label{RE} &=& 4\\pi\\,\\frac{n+1}{6 n}\\,\\left(\\langle\\phi(0,t)\\phi(0,t)\\rangle-\\langle\\phi(0,t)\\phi(r,t)\\rangle\\right)\\\n\\end{eqnarray}\nTaking $n\\to1$ limit, we get the entanglement entropy,\n\\begin{eqnarray}\n\\label{EE} S_{A}=4\\pi\\,\\frac{1}{3}\\,\\left(\\langle\\phi(0,t)\\phi(0,t)\\rangle-\\langle\\phi(0,t)\\phi(r,t)\\rangle\\right)\\\n\\end{eqnarray}\n\n\\gap3\n\\noindent{\\small{\\bf{Remark on winding number:}} While the free boson considered in MPS \\cite{Mandal:2015kxi} is the uncompactified free boson, the boson in (\\ref{tttf}) is a compactified free boson. So, Hamiltonian of the compactifed boson has zero mode terms but the winding number is not important for our analysis.\nIn the large system size limit($L\\to\\infty$), the zero modes vanished.\nEven if we are taking the limiting case of a finite size system, the zero momentum modes do not play any role in our calculation. Using the mode expansion of the boson $\\phi(w,\\bar{w})=\\varphi(w)+\\bar{\\varphi}(\\bar{w})$ in \\cite{Senechal:1999us},\n\\begin{eqnarray}\n\\varphi(w)&=&Q+\\frac{P}{2L}\\,w +\\sum_{n>0}\\frac{1}{\\sqrt{4\\pi n}}\\left(d_n e^{-inw}+d^{\\dagger}_ne^{inw}\\right)\\\\\n\\bar{\\varphi}(\\bar{w})&=&\\bar{Q}+\\frac{\\bar{P}}{2L}\\,\\bar{w}+\\sum_{n>0}\\frac{1}{\\sqrt{4\\pi n}}\\left(d_{-n} e^{-in\\bar{w}}+d^{\\dagger}_{-n}e^{in\\bar{w}}\\right)\\\n\\end{eqnarray}\nFirst, $Q$ and $\\bar{Q}$ are cancelled identically in (\\ref{tttf}).\nMoreover, by bosonization formulae \\cite{Senechal:1999us, vonDelft:1998pk},\n\\begin{align}\n P&=\\sqrt{4\\pi}N_f & \\bar{P}&=\\sqrt{4\\pi}\\bar{N}_f\\\\\n N_f&=J_0=-\\sum_{k=0}^{\\infty}\\left[a^\\dagger_ka_k-b^\\dagger_kb_k\\right] & \\bar{N}_f&=\\bar{J}_0=-\\sum_{k=0}^{\\infty}\\left[a^\\dagger_{-k}a_{-k}-b^\\dagger_{-k}b_{-k}\\right]\\\n\\end{align}\n But for our particular CC state, from (\\ref{JbJ0}), $N_f|CC_f\\rangle = 0$ and $\\bar{N}_f|CC_f\\rangle = 0$. Now, $P$ and $\\bar{P}$ commute with all the other bosonic creation and annihilation operators of non-zero momentum, hence they don't play any role in the calculation of (\\ref{tttf}).\n If we still keep the system size finite, the winding number would be important to interpret the stationary limit as a thermal ensemble. But we must take the $L\\to\\infty$ limit, if we want to examine the stationary limit. In other words, $L$ is the largest length scale in our theory and time $t<<L$.}\n\\gap3\n\nThe bosonic propagator in CC state has been calculated in \\cite{Mandal:2015kxi}. It is given by\n\\begin{eqnarray}\n\\label{phiprop}\\langle CC|\\phi(0,t)\\phi(r,t)|CC\\rangle=-\\frac{1}{8\\pi}\\log \\left(\\frac{2\\sinh ^2\\left(\\frac{\\pi  r}{4 \\kappa_2 }\\right)}{\\cosh \\left(\\frac{\\pi  r}{2 \\kappa_2 }\\right)+\\cosh \\left(\\frac{\\pi  t}{\\kappa_2 }\\right)}\\right)\\\n\\end{eqnarray}\n$r\\to 0$ gives the UV divergence of scalar field theory in 2D spacetime. The $n\\textsuperscript{th}$ R\\'enyi entropy and entanglement entropy of interval A in CC state is given by\n\\begin{eqnarray}\n \\label{CC_RE} S_{n}(A)&=&\\frac{n+1}{12 n}\\,\\left[\\log \\left(\\frac{\\sinh ^2\\left(\\frac{\\pi  r}{4 \\kappa_2 }\\right)\\left(1+\\cosh \\left(\\frac{\\pi  t}{\\kappa_2 }\\right)\\right)}{\\cosh \\left(\\frac{\\pi  r}{2 \\kappa_2 }\\right)+\\cosh \\left(\\frac{\\pi  t}{\\kappa_2 }\\right)}\\right) -\\lim_{\\epsilon\\to 0+}2 \\log (\\epsilon )- \\log \\left(\\frac{\\pi^2}{16 \\kappa_2^2}\\right)\\right]\\nonumber\\\\\n \\\\\n\\label{CC_EE} S_{A}&=&\\frac{1}{3}\\,\\left[\\frac{1}{2}\\log \\left(\\frac{\\sinh ^2\\left(\\frac{\\pi  r}{4 \\kappa_2 }\\right)\\left(1+\\cosh \\left(\\frac{\\pi  t}{\\kappa_2 }\\right)\\right)}{\\cosh \\left(\\frac{\\pi  r}{2 \\kappa_2 }\\right)+\\cosh \\left(\\frac{\\pi  t}{\\kappa_2 }\\right)}\\right) -\\lim_{\\epsilon\\to 0+}\\log (\\epsilon )-\\frac{1}{2} \\log \\left(\\frac{\\pi^2}{16 \\kappa_2 ^2}\\right)\\right]\\\n\\end{eqnarray}\nTaking the stationary limit $t\\to\\infty$ gives the entanglement entropy of A in a thermal ensemble at temperature $T=1\/\\beta=1\/(4\\kappa_2)$.\n\\begin{eqnarray}\n S_{A}&=&\\frac{1}{3}\\,\\left[\\log \\left(\\sinh \\left(\\frac{\\pi  r}{4 \\kappa_2 }\\right)\\right)-\\lim_{\\epsilon\\to 0+}\\log (\\epsilon )-\\frac{1}{2} \\log \\left(\\frac{\\pi^2}{16 \\kappa_2^2}\\right)\\right]\\\n\\end{eqnarray}\nThis exactly matches the thermal value which has been calculated using CFT techniques \\cite{Calabrese:2004eu}. It is fixed only by the temperature, the central charge of the CFT($c=1$ for dirac fermion), and `$r$' the length of the interval.\n\\begin{figure}%\n    \\centering\n    \\subfloat[Low effective temperature, $\\kappa_2=1$]{{\\includegraphics[width=7cm]{CC_EEkap1.eps} }}%\n    \\qquad\n    \\subfloat[High effective temperature, $\\kappa_2=0.2$]{{\\includegraphics[width=7cm]{CC_EEkapdot2.eps} }}%\n    \\caption{Entanglement entropy growth of an interval(r=5) in CC state.}%\n    \\label{CC_EE}%\n\\end{figure}\nTaking the high temperature limit $\\kappa_2\\to 0$, we get the extensive thermal entropy formula $S_{therm}=\\frac{1}{3}\\frac{\\pi r}{\\beta}$.\n\nBesides the thermalization, the most interesting aspect of figure (\\ref{CC_EE}) is that the entanglement entropy grows monotonically. The first derivative of $S_A$ w.r.t. time is\n\\begin{eqnarray}\n \\left<\\frac{\\partial S_A}{\\partial t}\\right>_{CC} &=& \\frac{\\pi  \\sinh ^2\\left(\\frac{\\pi  r}{4 \\kappa_2 }\\right) \\tanh \\left(\\frac{\\pi  t}{2 \\kappa_2 }\\right)}{3 \\kappa_2 \\left[ \\cosh \\left(\\frac{\\pi  r}{2 \\kappa_2 }\\right)+  \\cosh \\left(\\frac{\\pi  t}{\\kappa_2 }\\right)\\right]}\\\\\n \\label{scctan} &=&\\frac{\\pi}{12 \\kappa_2 }  \\left[2 \\tanh \\left(\\frac{\\pi  t}{2 \\kappa_2 }\\right)-\\tanh \\left(\\frac{\\pi  (r+2 t)}{4 \\kappa_2 }\\right)+\\tanh \\left(\\frac{\\pi  (r-2 t)}{4 \\kappa_2 }\\right)\\right]\\\n\\end{eqnarray}\nFrom the first expression, as a function of time $t>0$, it is clear that there are no finite zero. Hence, the EE growth of CC state is always monotonically increasing. Also note that in the high effective temperature limit $\\kappa_2\\to 0$, the approach to thermal value is sharper. In the limiting case, from the second expression, it is clear that the thermalization time is \n\\begin{equation}\n\\label{thermtime}t=\\frac{r}{2}\\ \n\\end{equation}\nwhich has also been calculated using BCFT techniques in \\cite{Calabrese:2005in}.\n\nIt would be interesting to check the monotonicity of EE growth in gCC states. Unfortunately, even for the free fermions with explicit twist operators, the entanglement entropy in gCC state with $W_4$ charge cannot be explicitly calculated. The bilinear fermionic $\\mathcal{W}_4(w)$ current when bosonized gives $\\phi^4$ terms\\cite{Pope:1991ig}, so the bosonized theory is an intereacting theory.\n\n\\section{\\label{secEE_gCC}Non-Monotonic EE Growth and Dynamical Phase Transition}\nAlthough we could not calculate EE in gCC state with $W_4$ charge of the fermionic bilinear $\\mathcal{W}_4$ current, we can still calculate entanglement entropy explicitly with the fermionic charge corresponding to the  bosonic charge $W_4(w)=\\sum_k |k|^3 d^\\dagger_kd_k$, where $d^\\dagger_k$ and $d_k$ are the bosonic annihilation and creation operators. As mentioned above, the zero modes do not play any role. Refermionization of the bosonic bilinear $\\mathcal{W}_4$ is done in Appendix \\ref{refW4}.\\footnote{We would like to thank Justin David for informing us that this refermionization could be done in principle using U(1) currents and it has not been done anywhere.} So, the fermionic state that we are considering is\n\\begin{eqnarray}\n |\\Psi\\rangle=\\text{e}^{-\\kappa_2 H_f-\\kappa_4 \\tilde{W}_4}|D_f\\rangle\\\n\\end{eqnarray}\nwhere the expression for $\\tilde{W}_4$ is given in (\\ref{newW4ch}).\n\nAgain, the R\\'enyi and entanglement entropies are given by the expression (\\ref{RE}) and (\\ref{EE}). The scalar propagator with the bosonic $W_4$ charge has also been calculated in MPS.\n\\begin{eqnarray}\n \\langle\\phi(0,t)\\phi(r,t)\\rangle=\\int_{-\\infty}^{\\infty}\\frac{dk}{8\\pi}\\frac{e^{ikr}}{k} \\left[\\coth \\left(2 k \\left(\\kappa_2 +\\kappa_4  k^2\\right)\\right)-\\cos (2 k t) {\\rm cosech}\\left(2 k \\left(\\kappa_2 +\\kappa_4  k^2\\right)\\right)\\right]\\\n\\end{eqnarray}\nThe momentum integral cannot be done explicitly. But we still can plot the entanglement entropy numerically. Figure (\\ref{gCC_EE}) are the plots of EE growth with `small' and `large' values of $\\kappa_4$. As expected, the entanglement entropy reaches an equilibrium quickly.\n\\begin{figure}%\n    \\centering\n    \\subfloat[Monotonic behavior, $\\kappa_4=0.01$]{{\\includegraphics[width=7cm]{bp} }}%\n    \\qquad\n    \\subfloat[Non-monotonic behaviour, $\\kappa_4=0.30$]{{\\includegraphics[width=7cm]{ap} }}%\n    \\caption{Entanglement entropy growth of an interval(r=5) for different choice of $\\kappa_4$ and $\\kappa_2=1$.}%\n    \\label{gCC_EE}%\n\\end{figure}\n\nThe most interesting aspect of Figure (\\ref{gCC_EE}) is the non-monotonic growth of EE in the gCC state with `large' $\\kappa_4$. As in case of CC state, to study the monotonic or non-monotonic behaviour of $S_A$, the more appropriate quantity is not $S_A$ but rather $\\frac{\\partial S_A}{\\partial t}$, the expression also simplifies tremendously.\n\\begin{eqnarray}\n\\left<\\frac{\\partial S_A}{\\partial t}\\right>_{gCC}&=&\\frac{1}{3}\\int_{-\\infty}^{\\infty}\\,dk\\,(1-e^{ikr})\\text{cosech}(2\\kappa_2 k + 2\\kappa_4 k^3)\\sin(2kt)\\nonumber\\\\\n&=&\\frac{1}{3}\\int_{-\\infty}^{\\infty}\\,dk\\,(1-\\cos(kr))\\text{cosech}(2\\kappa_2 k + 2\\kappa_4 k^3)\\sin(2kt)\\\n\\label{dS_gCC}\n\\end{eqnarray}\nUnfortunately, the above integral still cannot be done in closed form. The objective is to find finite positive real zeroes of the above expression as a function of time $t$. But, calculating zeroes of Fourier transforms, unless it can be done in closed form, is notoriously hard, the most famous example being the Riemann hypothesis.\n\nThe most interesting question that can be asked in Figure (\\ref{gCC_EE}) is whether even a small infinitesimal $\\kappa_4$, although not visible in the numerical plot, gives rise to the non-monotonic EE growth or whether the non-monotonic behaviour starts from a sharp finite value of $\\kappa_4$. If it is the second case, then it is a dynamical phase transition. In other words, the question is whether (\\ref{dS_gCC}) has finite zeroes as a function of time even for an infinitesimal $\\kappa_4$ or do the finite zeroes appear for $\\kappa_4$ greater than a critical value.\n\n\\gap1\n\n{\\it{We found that the non-monotonic behaviour starts abruptly at a critical value of $\\kappa_4=16 \\kappa_2^3\/27 \\pi ^2$, i.e., it is a dynamical phase transition. In terms of the effective temperature and chemical potential in the stationary limit, $\\beta = 4\\kappa_2$ and $\\mu_4=4\\kappa_4$, the critical value is $\\mu_4=\\beta^3\/27 \\pi ^2$.}}\n\n\\gap1\n\nAlthought the integral (\\ref{dS_gCC}) cannot be done in closed form, we can take advantage of the fact that for our question we do not need to know the precise zeroes. Using contour integration, the integral is given by the sum of residues of the poles given by $2\\kappa_2 k+2\\kappa_4 k^3 =in\\pi$ where $n\\in\\mathbb{Z}-\\{0\\}$. $n=0$ is not a pole of (\\ref{dS_gCC}). The expressions of the poles(from MPS)\\footnote{The numerical values of the poles may get interchanged for specific values of the parameters but the result will always be the same set of roots. This arises from the particular method used for solving the cubic equation.} are\n\\begin{eqnarray}\nk_1&=&\\frac{-2\\ 6^{2\/3} \\kappa_2+\\sqrt[3]{6} \\left(\\sqrt{48 \\kappa_2^3-81 \\pi ^2 \\kappa_4 n^2}+9 i \\pi\\sqrt{\\kappa_4} n\\right)^{2\/3}}{6 \\sqrt[3]{\\sqrt{3} \\sqrt{\\kappa_4^3 \\left(16 \\kappa_2^3-27 \\pi ^2\\kappa_4 n^2\\right)}+9 i \\pi \\kappa_4^2 n}}\\\\\nk_2&=&\\frac{4 \\sqrt[3]{-6} \\kappa_2+i \\left(\\sqrt{3}+i\\right) \\left(\\sqrt{48 \\kappa_2^3-81 \\pi ^2 \\kappa_4 n^2}+9 i\\pi\\sqrt{\\kappa_4} n\\right)^{2\/3}}{2\\ 6^{2\/3} \\sqrt[3]{\\sqrt{3} \\sqrt{\\kappa_4^3 \\left(16 \\kappa_2^3-27\\pi^2 \\kappa_4 n^2\\right)}+9 i \\pi\\kappa_4^2 n}}\\\\\nk_3&=&-\\frac{\\sqrt[3]{-1} \\left(2 \\sqrt[3]{-6} \\kappa_2+\\left(\\sqrt{48 \\kappa_2^3-81 \\pi ^2 \\kappa_4 n^2}+9 i \\pi \\sqrt{\\kappa_4} n\\right)^{2\/3}\\right)}{6^{2\/3} \\sqrt{\\kappa_4} \\sqrt[3]{\\sqrt{48 \\kappa_2^3-81 \\pi ^2\\kappa_4 n^2}+9 i \\pi  \\sqrt{\\kappa_4} n}}\\\n\\label{poles}\n\\end{eqnarray}\nOut of the three poles, only one is perturbative. In $\\kappa_4\\to 0$ series expansion, the other two start with $\\mathcal{O}(\\frac{1}{\\sqrt{\\kappa_4}})$. One of the three poles is always imaginary for arbitrary $n$ and arbitrary positive $\\kappa_4$.\n\nThere are three important ingredients for the proof of the dynamical phase transition:\n\\begin{enumerate}\n\\item All three $n^\\text{th}$ poles become purely imaginary when $16 \\kappa_2^3-27 \\pi ^2\\kappa_4 n^2$ is negative, or $\\kappa_4$ is greater than $16 \\kappa_2^3\/27 \\pi ^2 n^2$, we will call this the $n^{\\text{th}}$ critical value $\\kappa_{4c,n}$,\n\\begin{align}\n\\label{kappa4cn} \\kappa_{4c,n}&= \\frac{16 \\kappa_2^3}{27 \\pi ^2 n^2}\\\n\\end{align}\nBelow this value, the residues of the $n^{\\text{th}}$ poles are exponential decaying functions of time $t$, with no oscillatory factor. Obviously, ($n=\\pm1$) critical\\footnote{We will call this value just `critical value' without the `$n^{\\text{th}}$' specification because, as shown below, this is the critical value of $\\kappa_4$ where the dynamical phase transition happens.} value $\\kappa_{4c}$ is larger than $\\kappa_{4c,n}$ for $|n|>1$. With $\\kappa$ scaled to $1$, $\\kappa_{4c}$ is $16 \\kappa_2^3\/27 \\pi ^2\\sim 0.0600422$.\n\\item With $\\kappa_4$ less than $(n=\\pm1)$ critical value, the sum of the residues of ($n=\\pm 1$) poles is larger than the sum of the residues of all the other ($|n|>1$) poles. Hence, the behaviour of the first poles of $n=\\pm1$ dictate the behaviour of the integral (\\ref{dS_gCC}) when $\\kappa_4<16 \\kappa_2^3\/27 \\pi ^2$.\n\\item Above this critical value, for each $n$, two of the poles have real parts while one of them, say $k_1$, is imaginary. The poles are\n\\begin{gather}\n\\label{poleR} k_1= -2i\\,\\text{sgn}(n)\\,b, \\qquad k_2=a+i\\,\\text{sgn}(n)\\,b, \\qquad k_3=-a+i\\,\\text{sgn}(n)\\,b\\\\\na=\\frac{B^{2\/3}-2 \\sqrt[3]{6} \\kappa _2}{2\\ 2^{2\/3} \\sqrt[6]{3} \\sqrt[3]{B} \\sqrt{\\kappa _4}}, \\qquad b= \\frac{B^{2\/3}+2 \\sqrt[3]{6} \\kappa_2}{2\\ 6^{2\/3} \\sqrt[3]{B} \\sqrt{\\kappa_4}}\\nonumber\\\\\nB=\\sqrt{81 \\pi ^2 \\kappa_4 n^2-48 \\kappa_2^3}+9 \\pi |n|\\sqrt{\\kappa_4}\\nonumber\\\n\\end{gather}\nwhere we have to take the real roots of the radicals. $k_1$'s have the largest imaginary parts and the exponential decay of their residues as a function of time are faster while the other poles $k_2$ and $k_3$ have comparatively large magnitudes and ocsillations.\\footnote{This competition between poles of each $n$ might be important, if we have turned on $W_6$ chemical potential instead of $W_4$, in which case there will be five poles, or $W_8$ in which case there will be seven poles and so on.} In the total integral, the contributions of the imaginary poles $k_1$'s cannot compete with the contributions of the oscillating poles.  Lastly, it would be a very special arrangment if all ocsillating terms conspire to give a non-oscillatory sum. Hence, the total integral is oscillatory as a function of time and the EE growth is non-monotonic.\n\\end{enumerate}\nFor future reference, we also note that the expansion of the real part `$a$' in (\\ref{poleR}) around the $n^\\text{th}$ critical value $\\kappa_{4c,n}$ is\n\\begin{multline}\n a=\\frac{\\sqrt[3]{\\pi } \\sqrt[3]{|n|} \\sqrt{\\kappa _4-\\kappa _{4 c,n}}}{2^{2\/3} \\sqrt{3} \\kappa _{4 c,n}^{5\/6}}-\\frac{35 \\left(\\sqrt[3]{\\pi } \\sqrt[3]{|n|}\\right) (\\kappa_4-\\kappa _{4 c,n})^{3\/2}}{54 \\left(2^{2\/3} \\sqrt{3} \\kappa _{4 c,n}^{11\/6}\\right)}\\\\\n +\\frac{1001 \\sqrt[3]{\\pi } \\sqrt[3]{|n|} (\\kappa_4-\\kappa _{4 c,n})^{5\/2}}{1944\\ 2^{2\/3} \\sqrt{3} \\kappa _{4 c,n}^{17\/6}}+\\mathcal{O}(\\kappa_4-\\kappa_{4c,n})^{7\/2}\n \\label{aexp}\n\\end{multline}\n\nFor all our calculations below, we have scaled $\\kappa_2$ to be 1. The first point is clear from figure (\\ref{RePol}). The real parts of ($n=\\pm1$) poles vanish at $\\kappa_4\\sim 0.060$, which is the critical value found above. The critical value of $(n=\\pm2$) poles is $\\kappa_4\\sim 0.015$.\n\n\\begin{figure}[H]\n\\centerline{\\includegraphics[scale=.5]{RePole}}\n\\caption{Real parts of poles of $n\\in\\{\\pm1, \\pm2\\}$ as a function of $\\kappa_4$ with $\\kappa_2$ scaled to $1$.}\n\\label{RePol}\n\\end{figure}\n\nBelow the critical value, we will show that the total contributions from $n=\\pm 1$ poles is larger than the sum of all residues of $|n|>1$ poles. We will concentrate on the late time period, $t>r\/2$. For $e^{i2kt}$ of $\\sin(2kt)$ factor in (\\ref{dS_gCC}), the contour is closed upward encircling the upper half plane, and for $e^{-i2kt}$, the contour is closed downward encircling the lower half plane. From the expansion of ${\\rm cosech}\\left(2 \\kappa _4 \\left(k-k_1\\right)\n\\left(k-k_2\\right) \\left(k-k_3\\right)+i \\pi n\\right)$ around $k_1$, the contribution from $k_1$ poles for arbitrary $n$ are the real parts of \n\\begin{align}\nP_n(k_1)&=2\\pi iR_1(k_1)=\\frac{(-1)^n}{6\\kappa_4(k_1-k_2)(k_1-k_3)}\\,\\left(e^{i2k_1t}-\\frac{e^{ik_1(r+2t)}+e^{ik_1(-r+2t)}}{2}\\right) \\; &\\text{if} \\;\\text{Im}[k_1]>0\\nonumber\\\\\n\\label{I1}\\\\\nQ_n(k_1)&=-2\\pi iR_2(k_1)=\\frac{(-1)^n}{6\\kappa_4(k_1-k_2)(k_1-k_3)}\\,\\left(e^{-i2k_1t}-\\frac{e^{ik_1(r-2t)}+e^{-ik_1(r+2t)}}{2}\\right)\\; &\\text{if} \\;\\text{Im}[k_1]<0\\nonumber\\\\\n\\label{I2}\\\n\\end{align}\nwhere $R_1$ and $R_2$ denote the residues. Similarly, cyclic replacements of $k_1$ with $k_2$ and $k_3$ give the contributions of $k_2$ and $k_3$ poles. For the poles in the lower half of the complex plane, since the contour is anticlockwise, $Q_n$ have an extra minus sign in the residue.\nWe will call the contributions to the integral form $n=\\pm1$ poles as $I_0(t)$ and the contributions of the $|n|>1$ poles as $I_1(t)$. The other parameters ($\\kappa_4$, $r$ and $\\kappa$ which is already scaled to 1) are suppressed.\n\nAs a first visual evidence, Figure (\\ref{ccritNnR}) is the comparison of numerical integration of (\\ref{dS_gCC}) and $I_0(t)$. It is evident that the residues of ($n=\\pm1$) poles dominate the contour integration. We have chosen $\\kappa_4=0.0600420$ which is very close to the critical value. As mentioned above, with this choice, all the poles except the $n=\\pm 1$ poles give ocsillating residues as a function of time. Although it is not very conspicuous, it is also evident from the graph that $I_1(t)$ is oscillating around $I_0(t)$, the value of the numerical integration is above the $I_0(t)$ curve in some regions and below in other regions of time $t$.\n\\begin{figure}[H]\n\\centerline{\\includegraphics[scale=.5]{NintRes}}\n\\caption{Comparison of numerical integration of $\\left<\\partial{S_A}\/\\partial t\\right>_{gCC}$ (blue curve) and $I_0(t)$ (purple curve) as a function of time $t$. The parameters are $\\kappa_4=0.0600420$, $r=5$.}\n\\label{ccritNnR}\n\\end{figure}\nThe numerical integration is unreliable in the long time limit. So, to complete our argument, we will calculate an upper bound of $I_1(t)$ and compare it with $I_0(t)$ for a specific time $t$. The choice of the parameters are\n\\begin{eqnarray}\n\\kappa=1, \\; \\kappa_4=0.0600420, \\; r=5,\\; t=4r=20,\\\n\\label{para1}\n\\end{eqnarray}\nWith these parameters, the $n=1$ and $n=-1$ poles are\n\\begin{align}\n k_1&= 2.3538234 i & k_2&=2.3585719 i& k_3&=-4.7123954 i &;\\; n&=1\\\\\n k_1&= 4.7123954 i& k_2&=-2.3538234 i& k_3&=-2.3585719 i &;\\; n&=-1\\\n\\end{align}\nand $I_0(t)$ is given by\n\\begin{eqnarray}\n I_0(t)|_{t=20}&=&P(k_1)|_{n=1}+P(k_2)|_{n=1}+Q(k_3)|_{n=1}+P(k_1)|_{n=-1}+Q(k_2)|_{n=-1}+Q(k_3)|_{n=-1}\\nonumber\\\\\n &=&6.646589\\times10^{-35}\\\n \\label{Inpm1}\n\\end{eqnarray}\nWe can show that $I_1(t)|_{t=20}$ is less than $I_0(t)|_{t=20}$. The first few poles are\n\\begin{align*}\nk_1&=5.5495551 i& k_2&=2.5383386 - 2.7747775 i & k_3&=-2.5383386 - 2.7747775 i  &;\\; n&=-3\\\\\nk_1&=5.1737935 i, & k_2&=1.8496206 - 2.5868967i & k_3&= -1.8496206 - 2.5868967 i &;\\; n&=-2\\\\\nk_1&=-5.1737935 i & k_2&= -1.8496206 + 2.5868967 i & k_3&=1.8496206 + 2.5868967 i &;\\; n&=2\\\\\nk_1&= -5.54955505 i & k_2&= -2.5383386 + 2.7747775 i & k_3&=2.5383386 +2.7747775 i &;\\; n&=3\\\n\\end{align*}\nThe residues of these ($|n|>1$) poles cannot be summed up into a closed form, as that would amount to doing the integral in closed form. We are interested in an upper bound. The residues of two of the three poles of every ($|n|>1$) have an oscillation factor. As we saw, even each residue has a separate 3-6 real oscillating terms as a function of time. So, we can represent the sum of the modulus (absolute value of the amplitude) of the oscillating terms of the three residues for each $n$, by a bigger function which has the analytic sum from $|n|>1$ to infinity. And if the sum is less $I_0(t)$, then $I_0(t)$ dominates the contribution from all the other poles.\\footnote{A simplified example of our strategy is the comparison between say $X$ and $a\\sin(x)+b\\cos(y)$ where $\\{X,a,b,x,y\\}\\in\\mathcal{R}$, while $A>|a|$ and $B>|b|$ and $\\{A,B\\}\\in\\mathcal{R}^+$, then $A+B>|a|+|b|>a\\sin(x)+b\\cos(y)$ and if $X>A+B$ then $X>a\\sin(x)+b\\cos(y)$.}\n\\begin{figure}[H]\n\\centerline{\\includegraphics[scale=.6]{upperbound}}\n\\caption{Comparison of sum of modulus of residues of ($|n|>1$) poles with the approximating function $f(n)=10^{-39}\/n^2$. The dots are the discrete $n$ values of the corresponding functions.}\n\\label{ResComp}\n\\end{figure}\nFigure (\\ref{ResComp}) are the plots of the sum of the moduli separately for the oscillating terms of the three residues as a function of $n$ and the approximating function $f(n)=10^{-39}\/n^2$. Now, we have\n\\begin{eqnarray}\n \\sum_{-\\infty}^{n=-2}\\frac{10^{-39}}{n^2}+ \\sum_{n=2}^{\\infty}\\frac{10^{-39}}{n^2} =1.289868\\times10^{-39}\\\n \\label{rescompsum}\n\\end{eqnarray}\nThis is much less than $I_0(t)|_{t=20}$ in (\\ref{Inpm1}) and is of the order of $10^{-5}$ of $I_0(t)|_{t=20}$. So, the non-oscillating $I_0(t)$ dominates $I_1(t)$, the contribution from the other poles. Hence, below $\\kappa_4=16 \\kappa_2^3\/27 \\pi ^2$, the EE growth is monotonic.\n\nVisually from figure (\\ref{ccritNnR}), $t=3.7$ is a time-slice where the difference between $I_0(t)$ and the numerical integration has a local maxima. At this time slice, repeating the above exercise, $I_0(t)|_{t=3.7}=0.109727$ and repeating the same exercise of estimating the upper bound of $I_1(t)|_{t=3.7}$ with the same parameters as (\\ref{para1}) except the change in $t$, we get a good upper bound to be $0.0064493$ which is less than $I_0(t)|_{t=3.7}$ and is of the order of $60\\%$ of $I_0(t)|_{t=3.7}$. So, the approximation of the full integral by $I_0(t)$ gets better with increasing time. In the long time limit, we can effectively take the only time-dependence to be the time-dependence of $I_0(t)$. {\\it{It is worth mentioning here that even $(n=\\pm1)$ pole calculations take into account $\\kappa_4$ non-perturbatively, since two of the poles of each $n$ are non-perturbative in $\\kappa_4$.}}\n \nAs listed above as one of the main points, above the critcal value, each $n$ has an imaginary pole but the other two poles have real parts and also have larger magnitudes so the total residue of the three poles of each $n$ is oscillatory. It would also be a very special arrangement if all the oscillatory contributions of each $n$ conspire to give a non-oscillatory $\\partial S_A\/\\partial t$. Hence, we conclude that the EE growth is non-monotonic above the critical value.\n\nNear the critical point $(\\kappa_4-\\kappa_{4c})\\to 0^+$, we can try to estimate an upper bound of the time upto which the EE growth is monotonic. The upper bound is half of the longest time period. Using the leading term in expansion of `$a$' from (\\ref{aexp}) and the expressions of the residues (\\ref{I1}) and (\\ref{I2}), the lowest frequency($|n|=1$) gives the upper bound as\n\\begin{align}\n\\frac{\\sqrt[3]{\\pi } \\sqrt{\\kappa _4-\\kappa _{4 c}}}{2^{2\/3} \\sqrt{3} \\kappa _{4 c}^{5\/6}}(2t-r)=\\pi\\quad\\Rightarrow\\quad t= \\frac{(2\\pi)^{2\/3} \\sqrt{3} \\kappa _{4 c}^{5\/6}}{2\\sqrt{\\kappa _4-\\kappa _{4 c}}}+\\frac{r}{2}\\;\\sim\\; \\frac{2.95\\,\\kappa _{4 c}^{5\/6}}{\\sqrt{\\kappa _4-\\kappa _{4 c}}}\\     \n\\end{align}\nwhere finite `$r$' can be neglected in the limit $(\\kappa_4-\\kappa_{4c})\\to 0^+$.\n\nThe critical value in terms the effective temperature $\\beta=4\\kappa_2$ and chemical potential $\\mu_4=4\\kappa^4$ in the stationary limit is\n\\begin{eqnarray}\n\\mu_4=\\frac{\\beta^3}{27 \\pi ^2}\\\n\\label{mucrit}\n\\end{eqnarray}\n\\begin{figure}[H]\n\\centerline{\\includegraphics[scale=.4]{pd_f}}\n\\caption{The critical curve $\\mu_4=\\beta^3\/27 \\pi ^2$ in terms of the effective temperature and chemical potential in the stationary limit and the phase diagram.}\n\\label{pd}\n\\end{figure}\nFor the early times $t<r\/2$, in the residue calculations (\\ref{I1}) and (\\ref{I2}), we have to replace the sign of the exponents with $r-2t$ so the magnitudes of the exponentials decreases as time increases. Upto the critical value of $\\kappa_4$, the EE growth is always monotonic for this time period.\n\n\\subsection{Turning on other charges}\nWe could also calculate the EE growth of gCC states with other charges of the fermionic theory corresponding to bosonic bilinear $W_{2n}=\\sum |k|^{2n-1}d^\\dagger_kd_k$ where $n=3,4,5,...$. Repeating the exercise of quenching tuned squeezed states of scalar field theory in MPS, the propagator with these charges are simply given by\n\\begin{multline}\n\\langle\\phi(0,t)\\phi(r,t)\\rangle=\\int\\frac{dk}{4\\pi}\\frac{e^{ikr}}{k} \\left(\\coth \\left(2 \\kappa_2 k +\\sum_{n=2}^{\\infty}\\kappa_{2n}  k^{2n-1}\\right)\\right.\\\\\n\\left.-\\cos (2 k t) {\\rm cosech}\\left(2 \\kappa_2 k +\\sum_{n=2}^{\\infty}\\kappa_{2n}  k^{2n-1}\\right)-1\\right)\n\\end{multline}\nSubstituting this propagator in the general formula (\\ref{RE}) and (\\ref{EE}) give the R\\'enyi entropy  and entanglement entropy. The first derivative of EE w.r.t. time is\n\\begin{eqnarray}\n\\left<\\frac{\\partial S_A}{\\partial t}\\right>_{gCC}=\\frac{1}{3}\\int_{-\\infty}^{\\infty}\\,dk\\,(1-\\cos(kr))\\text{cosech}\\left(2\\kappa k + \\sum_{n=2}^{\\infty}\\kappa_{2n}  k^{2n-1}\\right)\\sin(2kt)\\\n\\label{dS_gCCn} \n\\end{eqnarray}\nWe believe the dynamics will be much richer with these other charges, with much more complex phase diagrams which can be in a $n-1$ dimensional space.\nBut the general poles analysis cannot be done in these cases because the poles will be given by quintic and higher order equations.\nConsidering gCC states with $W_4$ and $W_6$ charges, the numerical plots of EE growth looks the same as (\\ref{gCC_EE}) where by trial and error method, some parameter subspace gives monotonic growth and some subspaces do not give monotonic growth. Considering $n=\\pm1$, the poles are given by $2\\kappa_2 k+2\\kappa_4 k^3 + 2\\kappa_6 k^5=i\\pi$. For $\\kappa_2=1$ and $\\kappa_4=0.06$, numerically we find two interesting parameter subspaces of $\\kappa_6$. The first one is when all the poles become imaginary when $\\kappa_4$ is decreased.\n\\begin{align}\n &k_1 =2.0887597\\,\\text{sgn}(n)\\,i, \\quad  k_2=2.9527785\\,\\text{sgn}(n)\\,i, \\quad k_3=-6.5425830 \\,\\text{sgn}(n)\\,i,\\nonumber\\\\\n &k_4= -6.6158300 \\,\\text{sgn}(n)\\,i, \\quad k_5=8.1168748 \\,\\text{sgn}(n)\\,i      \\qquad\\text{for}\\qquad\\kappa_6=0.0007249\\\\\n \\nonumber\\\\\n &k_1=-0.0076887-6.5788763 \\,\\text{sgn}(n)\\,i, \\quad k_2=2.0887456 \\,\\text{sgn}(n)\\,i, \\quad k_3=2.9528549 \\,\\text{sgn}(n)\\,i,\\nonumber\\\\\n &k_4=8.1161520 \\,\\text{sgn}(n)\\,i, \\quad k_5=0.0076887-6.5788763 \\,\\text{sgn}(n)\\,i     \\qquad\\text{for}\\qquad\\kappa_6=0.0007250\\\n\\end{align}\nThis looks like the same transition if $n=\\pm1$ dominates, but the poles with real parts have large imaginary part also, so they would be highly damped.\nThe  other case is\n\\begin{align}\n &k_1 =-0.8215058 + 1.9681831\\,\\text{sgn}(n)\\,i, \\quad  k_2=-5.2389645\\,\\text{sgn}(n)\\,i, \\quad k_3=-5.2472000 \\,\\text{sgn}(n)\\,i,\\nonumber\\\\\n &k_4= 6.5497983\\,\\text{sgn}(n)\\,i, \\quad k_5=0.8215058 + 1.9681831 \\,\\text{sgn}(n)\\,i    \\qquad\\text{for}\\qquad\\kappa_6=0.0019179\\\\\n &k_1=-0.8215060 + 1.9681836 \\,\\text{sgn}(n)\\,i, \\quad k_2=-0.0040372 - 5.2430724\\,\\text{sgn}(n)\\,i, \\nonumber\\\\\n &k_3=6.5497775 \\,\\text{sgn}(n)\\,i, \\quad k_4=0.0040372 - 5.2430724\\,\\text{sgn}(n)\\,i, \\nonumber\\\\\n &k_5=0.8215060 + 1.9681836\\,\\text{sgn}(n)\\,i    \\qquad\\text{for}\\qquad\\kappa_6=0.0019180\\\n\\end{align}\nfor the smaller $\\kappa_4$, although two of the poles have real parts, they have to compete with the three imaginary poles. So, this could also be phase transition.\n\n\\section{\\label{cond}Discussion}\nIn this paper, we have examined free fermionic mass quench. We find that the ground state quench equilibrates but not to a thermal ensemble. Starting from specially prepared squeezed states, we get CC and gCC states with fermionic bilinear $W_{2n}$ charges. Calculation of correlators in CC and gCC states explicitly shows thermalization to thermal emsemble and GGE respectively.\n\nFor CC state, we calculate EE growth exactly. The EE growth is strictly monotonically increasing. For gCC state with a particular charge, we find dynamical phase transition in which the EE growth is monotonic upto a critical value of the effective chemical potential. In the pure state, the effective chemical potential is the coupling constant of the current corresponding to the charge. Above the critical value, the EE growth is non-monotonic. It would be interesting to reproduce our result in large $c$ holographic CFTs and examine what it would mean for Black hole physics.\n\n\\section*{Acknowledgement}\nWe are especially grateful to Prof. Gautam Mandal for many discussions on this work and for proofreading the draft. This work is possible because of scholarship grants from the Government of India. This work was also partly supported by Infosys Endowment for the study of the Quantum Structure of Space Time.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzpejw b/data_all_eng_slimpj/shuffled/split2/finalzzpejw
new file mode 100644
index 0000000000000000000000000000000000000000..62b29848c16fa8cc861a282e7e4922508e743946
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzpejw
@@ -0,0 +1,5 @@
+{"text":"\\section{#1}\\setcounter{equation}{0}}\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\\def{\\hbox{ 1\\kern-.8mm l}}{{\\hbox{ 1\\kern-.8mm l}}}\n\\def{\\hbox{ 0\\kern-1.5mm 0}}{{\\hbox{ 0\\kern-1.5mm 0}}}\n \n\\begin{document}\n{}~\n{}~\n\\hfill\\vbox{\\hbox{hep-th\/0502126}\n}\\break\n \n\\vskip .6cm\n\\begin{center}\n{\\Large \\bf \nBlack Holes, Elementary Strings and Holomorphic Anomaly\n}\n\n\\end{center}\n\n\\vskip .6cm\n\\medskip\n\n\\vspace*{4.0ex}\n \n\\centerline{\\large \\rm\nAshoke Sen}\n \n\\vspace*{4.0ex}\n \n \n\\centerline{\\large \\it Harish-Chandra Research Institute}\n\n \n\\centerline{\\large \\it  Chhatnag Road, Jhusi,\nAllahabad 211019, INDIA}\n \n\\centerline{E-mail: ashoke.sen@cern.ch,\nsen@mri.ernet.in}\n \n\\vspace*{5.0ex}\n \n\\centerline{\\bf Abstract} \\bigskip\n\nIn a previous paper we had proposed a specific route to relating\nthe entropy\nof two charge black holes to the degeneracy of elementary string states\nin N=4 supersymmetric heterotic\nstring theory in four dimensions. For toroidal compactification\nthis proposal works\ncorrectly to all orders in a power series expansion in inverse\ncharges provided we take\ninto account the \ncorrections to the black hole entropy formula due to holomorphic anomaly.\nIn this paper we demonstrate that similar agreement holds also for other\nN=4 supersymmetric heterotic string compactifications.\n\n\\vfill \\eject\n \n\\baselineskip=18pt\n\n\\tableofcontents\n\n\\sectiono{Introduction} \\label{sintro}\n\nThe attempt to relate the Bekenstein-Hawking entropy of a black hole\nto the number of states of the black hole in some microscopic description\nof the theory is quite old. In string theory this takes a new direction\nas the theory already has a large number of massive states in the spectrum\nof elementary string, and hence one is tempted to speculate that for large\nmass we should be able to relate the degeneracy of these states to the\nentropy of a black hole with the same charge quantum \nnumbers\\cite{thooft,9309145,9401070,9405117,9612146}. However this\nproblem is complicated due to large renormalization effects which make\nit difficult to relate the quantum numbers of the black hole to those\nof the elementary string states. This problem can be avoided by \nconsidering a tower of BPS states\\cite{rdabh0,rdabh1,rdabh2}\nwhere such renormalization effects\nare absent. The entropy of the corresponding\nblack hole solution vanishes in the supergravity approximation;\nhowever one finds\nthat the curvature associated with the solution becomes large near\nthe horizon, indicating  breakdown of the supergravity approximation.\nAlthough the complete analysis of the problem has not been possible to this\ndate, a general argument based on symmetries of the theory shows that\nthe entropy of the black hole, modified by $\\alpha'$ corrections, has the\nright dependence on all the parameters (charges as well as the asymptotic\nvacuum expectation values of various fields) so as to agree with the\nlogarithm of the degeneracy of elementary string \nstates\\cite{9504147,9506200,9712150}. However the overall \nnormalization constant is not determined\nby the symmetry argument, and its\ncomputation requires inclusion of all order\n$\\alpha'$ corrections to the tree\nlevel heterotic string action.\n\nIt was later realized that instead of elementary strings, \nD-branes\\cite{9602052} provide\na much richer arena for the study of black holes. In particular, by\nconsidering a sufficiently complicated configuration of D-branes one can\nensure that the corresponding BPS black hole solution carrying the same \ncharge\nquantum numbers as the D-brane system has a finite area event\nhorizon where $\\alpha'$ and string loop corrections are small.\nComparison of the entropy of the black hole\nto the logarithm of the degeneracy of states of the D-brane configuration\n(which we shall call the statistical entropy)\nshows exact agreement\\cite{9601029} for large charges. This\nagreement has been verified for a variety of \nblack holes in different string\ntheories.\n\nInitial comparison between the black hole entropy and the statistical\nentropy was carried out in the limit of large charges. For a class of black\nholes in $N=2$\nsupersymmetric string compactification\\cite{9508072,9602111,9602136}\nref.\\cite{9711053} attempted to go beyond the large charge limit, and computed\ncorrections to the statistical entropy which are suppressed by the inverse\npower of the charges. The corresponding corrections to the black hole entropy\ncome from higher derivative terms in the effective\naction. By taking\ninto account a special class of higher derivative \nterms\\cite{9602060,9603191} which come from\nsupersymmetrization of the curvature squared terms in the\naction\\cite{rzwiebach,9610237}, \nrefs.\\cite{9812082,9904005,9906094,9910179,0007195,0009234,0012232}\ncomputed corrections\nto the black hole entropy and found precise agreement. One non-trivial\naspect of this calculation is that in order to reach this agreement\nwe need to also modify the Bekenstein-Hawking formula for the black\nhole entropy due to the presence of higher derivative\nterms\\cite{9307038,9312023,9403028,9502009}.\n\nRecently there has been renewed interest in the black hole solution\nrepresenting elementary string states. This followed\nthe observation by Dabholkar\\cite{0409148}\nthat if we take into account the special class of higher derivative\nterms which were used in the analysis of \n\\cite{9812082,9904005,9906094,9910179,0007195,0009234,0012232}\nand calculate their\neffect on the black hole solutions representing elementary string states,\nwe get a solution with a finite area event horizon.\nThe entropy of this black hole, calculated using the formul\\ae\\\ngiven in \\cite{9307038,9312023,9403028,9502009}, reproduces\nprecisely the leading term in the expression for the\nstatistical entropy obtained by taking the logarithm of the degeneracy\nof elementary string states. This analysis was developed further in\n\\cite{0410076,0411255,0411272,0501014}. An alternative viewpoint for these \nblack \nholes \ncan be found in \\cite{0412133,0502050}.\n\nOne of the advantages of using elementary string states for comparison\nwith black hole entropy is that for this system the degeneracy of states\nand hence the statistical entropy is known very precisely. Hence one\ncan try to push this comparison beyond the large charge approximation.\nHowever one problem that one encounters in this process is that even if\nwe know the degeneracies exactly, the\ndefinition of the statistical entropy is somewhat ambiguous since it depends\non the particular ensemble we use to define entropy. As in the case of\nan ordinary thermodynamic\nsystem, all definitions of entropy agree when the charge (analog of volume)\nis large, but finite `size' corrections to the entropy differ between the\nentropies defined through different ensemble. This is due to the fact that\nthe agreement between different ensembles ({\\it e.g.} microcanonical,\ncanonical and grand canonical ensembles)\nis proved using a saddle point\napproximation which is valid only in the `large volume' limit. Thus the\nquestion that we need to address is: which definition of \nstatistical entropy should\nwe use for comparison with the black hole entropy? There is no {\\it\na priori} answer to this question, and one has to proceed by trial and\nerror to discover if there is some natural definition of\nstatistical entropy which agrees with the\nblack hole entropy beyond leading order. For a class of black holes\nin $N=2$ supersymmetric string compactification,\nRefs.\\cite{0405146,0412139} proposed\nsuch a definition based on a mixed ensemble where we sum over\nhalf of the charges (the `electric' charges) by introducing\nchemical potentials for these charges and keep the other half of\nthe charges fixed. By applying the same prescription to the black\nholes representing elementary string states in $N=4$ supersymmetric\ntheories, \\cite{0409148} was able to reproduce the black hole entropy\nto all orders in the inverse charges up to an additive\nlogarithmic piece which\nappears as a multiplicative factor in the partition function involving\npowers of the winding number charge\\cite{ism04}. One\ndisadvantage of this prescription is that it destroys manifest\nsymmetry between the momentum and winding charges of the string\nsince in defining the ensemble\nwe sum over all momentum states but keep fixed the winding\ncharge. As a result\nT-duality invariance of the statistical entropy defined\nthis way is not guaranteed.\n\nA related but somewhat different proposal for relating the degeneracy\nof elementary string states to black hole entropy,\nwhich maintains manifest T-duality invariance,\nwas proposed in \\cite{0411255}. This also requires summing over\ncharges but in a manner that preserves manifest T-duality. In particular\nthe chemical potential couples to a T-duality invariant combination\nof the charges.\\footnote{We also sum over all angular momentum states\nwhich is equivalent to choosing an ensemble with a chemical potential\ncoupled to the angular momentum, and then extremizing \nthe corresponding\nfree energy with respect to this chemical potential. This\nsets the chemical potential\nto zero. This argument is due to B.~Pioline, and\nI wish to thank A.~Dabholkar for discussion on this\npoint.}\nIt was shown in \\cite{0411255} that up to terms which\nare non-perturbative in the inverse charges, this definition\nof the statistical entropy agrees with the black hole entropy including\nlogarithmic terms, provided we take into account the effect of\nholomorphic anomaly\\cite{9302103,9309140} in the effective action.\nA related duality invariant\npresecription for dealing with 1\/4 BPS black holes in N=4\nsupersymmetric heterotic string compactification was later given\nin \\cite{0412287}. \n\nIn order to put this proposal on a firm footing it is important that \nwe test it for other N=4 heterotic string compactifications. This is\nwhat we attempt to do in this paper. In particular we focus on a class\nof four dimensional CHL models with $N=4$ \nsupersymmetry\\cite{CHL,CP,9507027,9507050,9508144,9508154} and compare the\nstatistical entropy computed using the prescription of \\cite{0411255}\nwith the black hole entropy. We again find that after taking into account\ncorrections due to holomorphic anomaly, the black hole entropy and the\nstatistical entropy agree up to non-perturbative terms.\\footnote{In\nthe analysis of this paper as well as in the analysis of \\cite{0411255}\nan overall charge independent additive constant in the expression\nfor the entropy could not be fixed\ndue to our lack of precise knowledge of the effect of the\nholomorphic anomaly terms on black hole entropy.\nThus we could not compare this overall additive\nconstant between the black hole\nand the statistical entropy.} \n\nThe rest of the paper is organised as follows. In section \\ref{srev} we\nreview the proposal of \\cite{0411255} for relating the black hole\nentropy to the degeneracy of elementary string states. In section\n\\ref{sstat} we review CHL string compactifications, count the degeneracy\nof elementary string states in these models, and compute the statistical\nentropy using these results. In section \\ref{sbh} we calculate the entropy\nof the black holes of the CHL model carrying the same charge quantum numbers\nas the elementary string states, and show that the result agrees with the\nstatistical entropy found in section \\ref{sstat}. During this computation\nwe also determine the coefficient of the holomorphic anomaly term in\nthese CHL models. Section \\ref{sdiss} contains a discussion of\nthe results and possible extension to more general class of models\nand\/or states.\nThe two appendices are devoted to \nthe analysis of the errors\ninvolved in the various approximations used in this paper, and to\ndemonstrate that these corrections are all non-perturbative in the\ninverse charges. Of the two appendices, appendix \\ref{sa} analyses\nthe possible errors in the computation of the statistical entropy and\nappendix \\ref{sb} examines possible errors in the computation of\nthe black hole entropy. In appendix \\ref{sb} we also determine the\nS-duality invariant form of the curvature squared terms in the\nCHL models.\n\nI have been informed by A.~Dabholkar that for a general class of models,\nref.\\cite{private}\nhas successfully carried out the comparison between the black\nhole entropy and the entropy defined through the ensemble introduced\nin \\cite{0405146}. After completing this paper we also learned of\nref.\\cite{9708062}\nwhere some of the computations of section \\ref{sbh} and\nappendix \\ref{sb}, required for determining the form of the curvature\nsquared terms in the effective action, have been carried out.\n\n\\sectiono{Proposal for Relating Black Hole Entropy to the Degeneracy of \nElementary String States} \\label{srev}\n\nWe shall be considering $N=4$ supersymmetric\nheterotic string theory in four\ndimension, with a compactification manifold of the form $K_5\\times S^1$.\nIn this theory we consider a fundamental\nstring wound $w$ times along the circle $S^1$ and carrying $n$ units\nof momentum along the same circle.\nLet $d(n,w)$ denote the degeneracy\nof elementary string states satisfying the following properties:\n\\begin{itemize}\n\\item The state is invariant under half of the space-time supersymmetry\ntransformations.\n\\item The state carries gauge charges appropriate to an elementary\nheterotic string carrying $w$\nunits of winding and $n$ units of momentum along $S^1$. This means\nthat if $x^4$ denotes the coordinate along $S^1$ and $x^\\mu$ ($0\\le\\mu\\le \n3$) denote the coordinates of the non-compact part of the space-time, then \nthe state carries\ngauge charges proportional to $n$ and $w$ associated with the gauge fields\n$G^{(10)}_{4\\mu}$ and $B^{(10)}_{4\\mu}$ respectively, but does \nnot carry any other \ngauge charge. Here $G^{(10)}_{MN}$ and $B^{(10)}_{MN}$ denote the\nten dimensional string metric and the anti-symmetric tensor field\nrespectively.\n\\end{itemize}\nWe shall see in section \\ref{sstat} that the\ndegeneracy of such states\nis a function of the combination $N\\equiv nw$. \nDenoting this\nby $d_{N}$, we define the partition function associated with these\nstates as:\n\\begin{equation} \\label{e2a}\ne^{{\\cal F}(\\mu)} = \\sum_{N} d_N \\, e^{-\\mu N}\\, .\n\\end{equation}\nGiven ${\\cal F}(\\mu)$, we define the statistical entropy $\\widetilde S_{stat}$ as the\nLegendre transform of ${\\cal F}(\\mu)$:\n\\begin{equation} \\label{e11}\n\\widetilde S_{stat}(N) = {\\cal F}(\\mu) + \\mu \\, N\\, ,\n\\end{equation}\nwhere $\\mu$ is given by the solution of the equation\n\\begin{equation} \\label{e12}\n{\\partial{\\cal F}\\over \\partial\\mu} + N =0\\, .\n\\end{equation}\nThe proposal of ref.\\cite{0411255} is to identify \n$\\widetilde S_{stat}(nw)$ with\nthe entropy of the black hole solution carrying same charge quantum numbers\n$(n,w)$:\n\\begin{equation} \\label{eprop}\n\\widetilde S_{stat}(nw) = S_{BH}(n,w)\\, .\n\\end{equation}\nThis is the relation we shall try to verify in this paper for different\nheterotic string compactifications.\n\nThe definition of statistical entropy given above is appropriate for a kind\nof grand canonical ensemble where we introduce a chemical potential conjugate\nto $nw$.\nA \nmore direct definition of the statistical entropy would be the one based on\nthe microcanonical ensemble:\n\\begin{equation} \\label{e12a}\nS_{stat}(N) = \\ln d_{N}\\, .\n\\end{equation}\nThe two definitions agree in the limit of large $N$ \nwhere we can evaluate\nthe sum in \\refb{e2a} by a saddle point method. In this case the leading\ncontribution to $e^{{\\cal F}(\\mu)}$ is given by $d_{N_0} e^{-\\mu N_0}$ where\n$N_0$ is the value of $N$ that maximizes the summand:\n\\begin{equation} \\label{esaddle}\n{\\cal F}(\\mu) \\simeq \\ln d_{N_0} - \\mu N_0, \\qquad {\\partial\\over \\partial N_0}\n\\ln d_{N_0}-\\mu = 0\\, .\n\\end{equation}\nThus in this approximation ${\\cal F}(\\mu)$ is the Legendre transform of\n$\\ln d_{N_0}=S_{stat}(N_0)$. Hence $\\widetilde S_{stat}(N)$, \ndefined as the Legendre\ntransform of ${\\cal F}(\\mu)$, will be equal to $S_{stat}(N)$.\nHowever the complete ${\\cal F}(\\mu)$ defined through \\refb{e2a} has\nadditional contribution besides that given in \\refb{esaddle},\nand as a result\n$S_{stat}$ and\n$\\widetilde S_{stat}$ differ in non-leading terms. In particular\nthe coefficient of\nthe $\\ln N$ terms in $S_{stat}$ and \n$\\widetilde S_{stat}$ differ. It is not\n{\\it a priori} clear which definition of \nstatistical entropy we should be comparing\nwith the entropy of the black hole solution \ncarrying the same quantum numbers.\nIt was shown in \\cite{0411255} that for \nheterotic string theory compactified on a\ntorus, $\\widetilde S_{stat}$ agrees with the black hole entropy\nup to exponentially suppressed contributions. \nWe shall see in section \\ref{sbh}\nthat such agreement between $\\widetilde S_{stat}$ and \n$S_{BH}$ continues to hold also\nfor CHL compactification\\cite{CHL,CP,9508144,9508154} \nof the heterotic string theory.\n\nNote that given $S_{stat}=\\ln {d_N}$ we can calculate $\\widetilde S_{stat}$\nusing eqs.\\refb{e2a}-\\refb{e12}. Conversely, given $\\widetilde S_{stat}$ we\ncan compute ${\\cal F}(\\mu)$ by taking its Legendre transform and then\ncompute $d_N$ by solving \\refb{e2a}.\nThis gives $S_{stat}$. Thus $S_{stat}$\nand $\\widetilde S_{stat}$ contain complete information about each other and the \ndegeneracies\n$d_N$. This allows us to restate the proposal \\refb{eprop} in a slightly\ndifferent but equivalent form. Given $S_{BH}(n,w)$ (which turns out to be\na function of the combination $N=nw$) we define ${\\cal F}_{BH}(\\mu)$ by taking\nthe Legendre transform of $S_{BH}$ with respect to the variable $N$, and\nthen define $d^{BH}_N$ through an analog of eq.\\refb{e2a} with ${\\cal F}(\\mu)$ \nand \n$d_N$ replaced by ${\\cal F}_{BH}(\\mu)$ and $d_N^{BH}$ respectively. The\nproposal \\refb{eprop} then translates to the relation:\n\\begin{equation} \\label{enewp}\n{\\cal F}_{BH}(\\mu) = {\\cal F}(\\mu)\\, , \\qquad d^{BH}_N = d_N\\, .\n\\end{equation}\nAlthough we shall work with \\refb{eprop} for convenience, we should\nkeep in mind that verifying \\refb{eprop} amounts to verifying\n\\refb{enewp}.\n\n\\sectiono{Counting Degeneracy of BPS String States in CHL Models} \n\\label{sstat}\n\nIn this section we shall compute $d_N$ and hence ${\\cal F}(\\mu)$ for a\nclass of $N=4$ supersymmetric heterotic string compactification.\nFirst we shall illustrate the counting procedure in the context of a\nspecific CHL model\\cite{CP} \nand then generalize this to other models. The \nconstruction of the model\nbegins with\n$E_8\\times E_8$ heterotic string theory compactified on a six torus\n$T^4\\times \\widetilde\nS^1\\times S^1$. \nIn this theory the gauge fields in the Cartan subalgebra consist of\n22 gauge fields arising out of the left-moving U(1) currents of the\nworld-sheet theory, and six gauge fields arising out of the right-moving\nU(1) currents of the world-sheet theory. All the gauge fields associated\nwith the $E_8\\times E_8$ group arise out of the left-moving world-sheet\ncurrents.\nWe now mod out this theory by a $Z_2$ transformation\nthat involves a half shift along $\\widetilde S^1$ together with an exchange of\nthe two $E_8$ lattices\\cite{CP}.  The resulting theory still has $N=4$\nsupersymmetry. In particular the 6 U(1) \ngauge fields associated with the right-moving world-sheet currents\nare untouched by the $Z_2$ projection, and continue to provide\nus with the graviphoton fields of $N=4$ supergravity.\nOn the other hand only the diagonal sum of the two $E_8$\ngauge groups survive the projection. As a result the $E_8\\times E_8$\ncomponent of the gauge group is reduced to $E_8$, and the rank of the\nunbroken gauge group from the left-moving sector of the world-sheet is\nreduced to $14$ from its original value 22. 8 of these U(1) gauge fields\ncome from the\nsurviving $E_8$ gauge group\nand the other 6 come from appropriate linear combination\nof the metric and antisymmetric tensor field, with one index lying\nalong one of the six directions of\nthe internal torus and the other index lying along one of the\nnon-compact directions.\n\nWe now consider an elementary string state wound $w$ times along $S^1$ and \ncarrying $n$ units of momentum along the same $S^1$. The BPS excitations \nof this string state come from restricting the right-moving \noscillators \nto have total level 1\/2 (in the Neveu-Schwarz sector) or 0 (in the Ramond \nsector) but allowing arbitrary oscillator and momentum excitations in the \nleft-moving sector. We would like to count BPS states with a given set of \ngauge charges, notably those carried by an elementary string state with \n$w$ units of winding and $n$ units of momentum along $S^1$. First let us \ndo this calculation for heterotic \nstring theory on a torus\\cite{rdabh0}. In this case\nthe only possible \nexcitations are those created by left-moving oscillators, since any\nadditional momentum and \/ or winding will generate additional gauge\ncharges carried by the state.\nIf $N_L$ denotes \nthe total level of the left-moving oscillators then the level matching \ncondition gives $N_L=nw+1$, and hence the degeneracy of elementary string \nstates carrying quantum numbers $(n,w)$ is the number \nof ways we can get total oscillator level $N_L$ from the 24 left-moving \noscillators, multiplied by a factor of 16 that counts the degeneracy of\nthe ground state of the right-moving sector. \nWe shall call this number $d^{(0)}_{N_L}$.\nIt is \ngiven by the generating function\\cite{rdabh0}\n\\begin{equation} \\label{e4}\n\\sum_{N_L} d^{(0)}_{N_L} e^{-\\mu (N_L-1)} = \n16 \\left(\\eta(e^{-\\mu})\\right)^{-24}, \\qquad\n\\eta(q) = q^{1\/24} \n\\, \\prod_{n=1}^\\infty ( 1 - \nq^n)\\, .\n\\end{equation}\nFor the CHL string theory under consideration, the counting is a \nlittle more complicated. Since only the diagonal $E_8$ gauge group \nsurvives, we can \nsatisfy the condition for vanishing $E_8$ charge if we choose equal and \nopposite momentum vector $\\vec p$ and $-\\vec p$\nfrom the two $E_8$ lattices. We choose\nthe overall normalization of $\\vec p$ such that the $E_8$\nlattice is self-dual \nunder the inner product $(\\vec p,\\vec q)=\\vec p\\cdot \\vec q$. In this \nnormalization there is one lattice point per unit volume\nin the $\\vec p$ space, and the\ncontribution to the $\\bar L_0$ eigenvalue from the vector \n$\\vec s =(\\vec p, -\\vec p)$ is given by \n$\\vec p^2\/2+\\vec p^2\/2=\\vec p^2$.\nThe level matching condition now gives:\n\\begin{equation} \\label{e1}\nN_L + {\\vec p^2} = nw + 1\\, .\n\\end{equation}\nThus the degeneracy $d_{nw}$ for given $(n,w)$ is equal to the number of \nways we can satisfy \\refb{e1}, subject to the condition that the\nresulting state is even under the orbifold \ngroup:\n\\begin{equation} \\label{ednw}\nd_{nw} \\simeq {1\\over 2} \\sum_{N_L} \\sum_{\\vec p\\in \\Lambda_{E_8}} \nd^{(0)}_{N_L} \\, \\delta_{N_L+\\vec p^2 -nw, 1}\\, .\n\\end{equation}\nSince we only include\nstates which are even under the $Z_2$ transformation, we must symmetrize\nthe state under the exchange of the oscillators and momenta associated with\nthe two $E_8$ factors. As shown in appendix \\ref{sa}, up to exponentially \nsmall contribution\nthis introduces a multiplicative factor of 1\/2 in the counting of states\nwhich we have included in the right hand side of \\refb{ednw}.\nNote that the twisted sector states do not play any\nrole in this counting, since they carry half-integral winding number along\nthe circle $\\widetilde S^1$ and hence belong to a different charge sector. \nUsing \\refb{ednw} and \\refb{e4} the partition function $e^{{\\cal F}(\\mu)}$ \ndefined in \\refb{e2a} is now\ngiven by\n\\begin{equation} \\label{e3}\ne^{{\\cal F}(\\mu)} \\simeq {1\\over 2}\n\\sum_{N_L} d^{(0)}_{N_L} e^{-\\mu (N_L-1)} \\, \n\\sum_{\\vec p\\in \\Lambda_{E_8}} \ne^{-\\mu \\vec \np^2} = 8\\,\n\\left(\\eta(e^{-\\mu})\\right)^{-24} \\, \\sum_{\\vec p\\in \\Lambda_{E_8}} \ne^{-\\mu \\vec \np^2}\\, .\n\\end{equation}\n\nWe shall be interested in the behaviour of ${\\cal F}(\\mu)$ at small $\\mu$. In this\nlimit,\n\\begin{equation} \\label{e6}\n\\left(\\eta(e^{-\\mu})\\right)^{-24} \\simeq e^{{4\\pi^2\/\\mu}} \\, \\left(\n{\\mu\\over 2\\pi}\\right)^{12}\\, ,\n\\end{equation}\nand, using Poisson resummation formula,\n\\begin{equation} \\label{e7}\n\\sum_{\\vec p\\in \\Lambda_{E_8}} \ne^{-\\mu \\vec \np^2} = \\left({\\pi\\over \\mu}\\right)^4\\, \n\\sum_{\\vec q\\in \\Lambda_{E_8}} e^{-\\pi^2\\vec q^2\/\\mu}\n\\simeq \\left({\\pi\\over \\mu}\\right)^{4}\\, .\n\\end{equation}\nHere we have used the fact that the $E_8$ lattice is self-dual.\nThus we get, for small $\\mu$\n\\begin{equation} \\label{e8}\ne^{{\\cal F}(\\mu)} \\simeq {1\\over 2}\\,  \n\\left({\\mu\\over 2\\pi}\\right)^{8} \n\\, e^{{4\\pi^2\/\\mu}} \\, ,\n\\end{equation}\nand hence\n\\begin{equation} \\label{e9}\n{\\cal F}(\\mu) \\simeq {4\\pi^2\\over \\mu} + 8 \\ln {\\mu\\over 2\\pi}\n+\\ln{1\\over 2}\\, .\n\\end{equation}\n\nBefore we turn to the more general case, let us try to estimate the error\nin \\refb{e9}. The first source of error appears in \\refb{e3} where we have \nrepresented the symmetry requirement of the states under the $Z_2$\norbifold group by a factor of ${1\\over 2}$ in\n$e^{\\cal F}$. A more careful analysis described in appendix \\ref{sa} shows that \nthe error in ${\\cal F}$ due to this\napproximation involves powers of $e^{-\\pi^2\/\\mu}$. The second source of\nerror is in the small $\\mu$ approximation of $\\eta(e^{-\\mu})$ used\nin \\refb{e6}. The fractional error in \nthis formula is of order $e^{-4\\pi^2\/\\mu}$. \nFinally the approximation used in \\refb{e7} \nalso introduces a fractional error\ninvolving powers of $e^{-\\pi^2\/\\mu}$. \nThus we conclude that the net error in\neq.\\refb{e9} for ${\\cal F}(\\mu)$ is non-perturbative in the \nsmall $\\mu$ approximation.\n\n\nThe above analysis can be easily generalized to a class of\nother CHL compactifications.\nWe begin with heterotic string theory compactified on $T^4\\times \\widetilde S^1\n\\times S^1$, and tune the moduli associated with $T^4$ compactification such\nthat the twentyfour dimensional Narain lattice\\cite{narain,nsw}\n$\\Lambda_{20,4}$\nassociated with heterotic compactification on $T^4$ has a $Z_m$\nsymmetry. We now\nmod out the theory by a $Z_m$ symmetry group generated by\na shift $h$ of order $m$ along $\\widetilde S^1$,\naccompanied by the generator $g$ of the $Z_m$ automorphism group\nof $\\Lambda_{20,4}$. In order that the\nfinal theory has $N=4$ world-sheet supersymmetry, the $Z_m$\nautomorphism should act trivially on the right-moving U(1) currents\nof the world-sheet. However it could have non-trivial action on the\nleft-moving world-sheet currents, and as a result modding out by this\nsymmetry projects out certain number (say $k$) of the U(1) gauge fields\nbelonging to the Cartan subalgebra of the gauge group.\nIn the resulting quotient theory the rank of the gauge group associated\nwith the left-moving sector of the world-sheet theory is \nreduced to $(22-k)$.\nIf we now consider an elementary string wound $w$ times along $S^1$ and \ncarrying\n$n$ units of momentum along $S^1$, then the computation of \nthe degeneracy $d_N$ ($N=nw$) and the partition function\n$e^{{\\cal F}(\\mu)}$ associated with these states involves a sum over the oscillator\nlevels $N_L$ as well as a sum over the $k$-dimensional momentum lattice \n$\\Lambda$ whose vectors do not\ncouple to any massless gauge field of the resulting theory. \nThis gives\n\\begin{equation} \\label{ednnew}\nd_N \\simeq {1\\over m} \\, \\sum_{N_L} \\, \\sum_{\\vec s\\in \\Lambda}\nd_{N_L}^{(0)} \\, \\delta_{N_L+\\vec s^2\/2 -N, 1}\\, ,\n\\end{equation}\nand\n\\begin{equation} \\label{efnew}\ne^{{\\cal F}(\\mu)} \\simeq {1\\over m} \\, \\sum_{N_L} d_{N_L}^{(0)} \ne^{-\\mu (N_L-1)} \\, \\sum_{\\vec s\\in \\Lambda} \\, e^{-\\mu\\vec s^2\/2}\\, .\n\\end{equation}\nAs discussed in appendix \\ref{sa}, the factor\nof $1\/m$ approximately accounts for the fact that we need to count only \nthose states\nwhich are invariant under the orbifold group, and the error involved in \nthis approximation involves powers of $e^{-\\pi^2\/\\mu}$. The sum over $N_L$ \ncan \nbe performed using \\refb{e4}, whereas the sum over $\\vec s$ can be done\nusing Poisson resummation formula:\n\\begin{equation} \\label{epois}\n\\sum_{\\vec s\\in \\Lambda} e^{-\\mu \\vec s^2\/2} = {1\\over V}\n\\, \\left(\n{2\\pi\\over \\mu}\\right)^{k\/2} \n\\sum_{\\vec r \\in \\widetilde\\Lambda} \\, e^{-2\\pi^2 \\vec r^2\/\\mu} \\simeq\n{1\\over V}\\, \\left(\n{2\\pi\\over \\mu}\\right)^{k\/2}\\, ,\n\\end{equation}\nwhere $V$ denotes the volume of the unit cell in the lattice $\\Lambda$\nand $\\widetilde\\Lambda$ is the lattice dual to $\\Lambda$.\nThus\nthe final expression for\n${\\cal F}(\\mu)$ is given by\n\\begin{equation} \\label{e10}\n{\\cal F}(\\mu) \\simeq {4\\pi^2\\over \\mu} + {1\\over 2} \\, \n(24-k) \\ln {\\mu\\over 2\\pi} + \\ln {16\\over Vm}\\, .\n\\end{equation}\nThe errors in this equation come from errors in eqs.\\refb{e6}, \n\\refb{efnew}\nand \\refb{epois}. Each of these errors involves\npowers of $e^{-\\pi^2\/\\mu}$. Thus\nas in the first example, for small $\\mu$ the corrections to \\refb{e10}\ninvolve powers of $e^{-\\pi^2\/\\mu}$.\n \nGiven ${\\cal F}(\\mu)$, we define the statistical entropy $\\widetilde S_{stat}$ \nthrough \\refb{e11}, \\refb{e12}. This gives:\n\\begin{equation} \\label{e10a}\n\\widetilde S_{stat}(N) \\simeq \\mu \\, N +\n{4\\pi^2\\over \\mu} + {1\\over 2} \\, \n(24-k) \\ln {\\mu\\over 2\\pi} + \\ln {16\\over Vm}\\, ,\n\\end{equation}\nwhere $\\mu$ is the solution of the equation\n\\begin{equation} \\label{e10b}\n-{4\\pi^2\\over \\mu^2} +{24-k\\over 2\\mu} +N \\simeq 0 \\, ,\n\\end{equation}\nand $N=nw$.\nIn the limit of large $N$, the $\\mu$ obtained by solving \\refb{e10b} \nis\ngiven by\n\\begin{equation} \\label{e13}\n\\mu \\simeq {2\\pi \\over \\sqrt{N}} \n\\left(1+{\\cal O}\\left({1\\over \\sqrt N}\\right)\\right) \\, .\n\\end{equation}\nThus for large $N$, $\\mu$ is small. This justifies the small $\\mu$\napproximation used in arriving at \\refb{e10}.\nSince the error in ${\\cal F}(\\mu)$ involves powers of $e^{-\\pi^2\/\\mu}$, the\nerror in $\\widetilde S_{stat}$ computed from \\refb{e10a}, \\refb{e10b} will\ninvolve powers of $e^{-\\pi\\sqrt{N}}$.\n\nWe conclude this section by noting that $\\widetilde S_{stat}$ computed from\n\\refb{e10a}, \\refb{e10b} is of the form\n\\begin{equation} \\label{e14}\n\\widetilde S_{stat} = 4\\pi\\sqrt{N} - {24-k\\over 2} \\, \\ln \\sqrt{N} + {\\cal O}(1) \\, .\n\\end{equation}\nAlthough eq.\\refb{e14} gives more explicit expression for\n$\\widetilde S_{stat}$, this\nequation has corrections involving inverse powers of $\\sqrt N$. Thus\nthe comparison with the black hole entropy will be made with the formul\\ae\\\n\\refb{e10a}, \\refb{e10b} for $\\widetilde S_{stat}$ which\nare correct up to error terms involving\npowers of $e^{-\\pi\\sqrt N}$.\n\n\\sectiono{Analysis of Black Hole Entropy and Comparison with the \nStatistical Entropy} \\label{sbh}\n\nWe shall now turn to the analysis of the entropy $S_{BH}$\nof the BPS black hole\ncarrying the\nsame charge and mass as an elementary string state described above.\nThe entropy of such a black hole vanishes in the supergravity \napproximation\\cite{9504147}. However the curvature associated \nwith the string\nmetric becomes large near the horizon, showing that we must\ntake the higher derivative terms into account for computing\nthe entropy of such a black hole.\nIn contrast the string coupling near the horizon is small\nfor large\n$n$ and $w$ and hence to leading order we can ignore the string loop\ncorrections\\cite{9504147}. There is a general symmetry argument that shows \nthat at the tree level in heterotic string theory the modified entropy \nmust have the form $a\\sqrt{nw}$ for some numerical constant \n$a$\\cite{9504147,9712150,0411255}. However the value of the constant $a$ \nis not determined by this argument ($a$ could be zero for example). If \n$a=4\\pi$, the black hole entropy would agree with the leading term in the \nexpression \\refb{e14} for $\\widetilde S_{stat}$. Following \nthe formalism developed in\nrefs.\\cite{9812082,9904005,9906094,9910179,0007195,0009234}, \nref.\\cite{0409148} analyzed the effect of\na special class of higher\nderivative terms in the tree level effective action of heterotic \nstring theory which come from supersymmetric completion of the\nterm\\cite{9602060,9603191}\n\\begin{equation} \\label{ebh0}\n{1\\over 16\\pi} \n\\int d^4 x \\, \\sqrt{\\det g}\\, \\left(S\\, R^-_{\\mu\\nu\\rho\\sigma} \n\\, R^{-\\mu\\nu\\rho\\sigma} + \\bar S \\, R^+_{\\mu\\nu\\rho\\sigma} \n\\, R^{+\\mu\\nu\\rho\\sigma} \\right)\\, ,\n\\end{equation}\nwhere $g_{\\mu\\nu}$, $R^\\pm_{\\mu\\nu\\rho\\sigma}$ and $S$ \ndenote respectively \nthe canonical\nmetric, the self-dual and anti-self-dual components of the\nRiemann tensor and the complex scalar field whose real and\nimaginary parts are given by the exponential of the dilaton field \nand the axion field respectively. After taking into account the\nmodification of the equations of motion and supersymmetry transformation\nlaws due to these additional terms, the modified\nblack hole entropy is given by\nthe expression\\cite{0409148,0410076,0411255,0411272}:\n\\begin{equation} \\label{eb1}\nS_{BH} = {\\pi N \\over S_0} + 4\\pi \\, S_0\\, , \\qquad \nN \\equiv nw\\, ,\n\\end{equation}\nwhere $S_0$, defined as the value of the field $S$ at the horizon,\nis given by the solution of the equation\\footnote{Note that\nthe left hand side of \\refb{eb2} is equal to the derivative of the \nright hand side of \\refb{eb1} with respect to $S_0$. This feature\nsurvives even after including the correction due to \nholomorphic anomaly\\cite{0411255,0412287}.}\n\\begin{equation} \\label{eb2}\n-{\\pi\\, N\\over S_0^2} + 4 \\pi =0\\, .\n\\end{equation}\nThis gives $S_{BH}=4\\pi\\sqrt{N}$. This agrees with the leading term\nin the expression \\refb{e14} for $\\widetilde S_{stat}$\\cite{0409148}.\n\nRef.\\cite{0409148} checked this agreement for heterotic string \ncompactification on a torus. However once this has been checked\nfor  torus compactification, similar agreement for other heterotic\nstring compactifications is\nautomatic due to an argument in \\cite{9712150} where it was shown that at tree\nlevel in heterotic string theory the part of the effective action relevant\nfor computing the entropy of these states is identical in all heterotic string\ncompactification with $N=4$ or $N=2$ supersymmetry. Thus the leading \ncontribution\nto $S_{BH}$ will be given by $4\\pi\\sqrt{nw}$ for all heterotic string\ncompactifications. This clearly agrees with the leading term in \nthe expression \\refb{e14} for $\\widetilde S_{stat}$.\n\nWe now turn to the non-leading corrections to the entropy. \nFor this we need to go beyond the tree level effective action of\nthe heterotic string theory.\nA special class of\nsuch corrections come from a term in the action of the form:\n\\begin{equation} \\label{eb3}\n-{K\\over 128\\pi^2} \\, \\int d^4 x \\, \\sqrt{\\det g}\\, \\ln (S+\\bar S) \n\\, R_{\\mu\\nu\\rho\\sigma} \\, R^{\\mu\\nu\\rho\\sigma}\\, ,\n\\end{equation}\nthat arises from the so called holomorphic anomaly\\cite{9302103,9309140}.\nHere $K$ is a \nconstant that will be determined later. (For toroidal\ncompactification $K=24$\\cite{9610237}.)\nIn order to carry out a systematic analysis of the effect of this term on the\nexpression for the black hole entropy, we need to \n\\begin{itemize}\n\\item supersymmetrize this term,\n\\item study how these additional terms in the action\nmodify the expression for\nthe black hole entropy in terms of various fields near the horizon,\n\\item study how the various field configurations near the horizon are\nmodified by these extra terms in the equation of motion, \n\\item and finally evaluate the modified expression for \nthe black hole entropy for the\nmodified near horizon field configuration.\n\\end{itemize}\nThis however has not so far been carried out explicitly. \nIn order to appreciate the reason for this difficulty, one needs to\nknow the difference in the origin of the terms \\refb{ebh0} and \\refb{eb3}.\nIn fact both terms originate from a term of the form:\\footnote{In the\nconvention of ref.\\cite{9906094} this corresponds to choosing\n$ F^{(1)} = -{i\\pi\\over 4} \\, \\left( g(S) - {K\\over 64\\pi^2} \\ln(S+\\bar S)\n\\right) \\, \\Upsilon. $\nFor toroidal compactification $g(S) = -{3\\over 4\\pi^2} \\ln\\eta\n(e^{-2\\pi S})$ \nand $K=24$.}\n\\begin{equation} \\label{eb6}\n\\int d^4 x \\, \\sqrt{\\det g} \\left[\n\\phi(S, \\bar S) \\, R^-_{\\mu\\nu\\rho\\sigma} \nR^{-\\mu\\nu\\rho\\sigma} + c.c.\\right] \\, ,\n\\end{equation}\nwhere \n\\begin{equation} \\label{eb7a}\n\\phi(S,\\bar S)=g(S) -{K\\over 128\\pi^2} \\, \\ln(S+\\bar S)\n\\end{equation} \nis the sum of a piece $g(S)$ that is holomorphic\nin $S$ and a piece proportional to $\\ln(S+\\bar S)$\nthat is a function of both $S$ \nand $\\bar S$.\nFor large $S$,\n\\begin{equation} \\label{e7b}\ng(S) \\simeq {S\\over 16\\pi}\\, ,\n\\end{equation}\nso as to reproduce \\refb{ebh0}. Hence \n\\begin{equation} \\label{eb7}\n\\phi(S, \\bar S)\n\\simeq {1\\over 16\\pi} \\, \\left( S -{K\\over 8\\pi} \\ln(S+\\bar S)\n\\right)\\, .\n\\end{equation}\nA detailed analysis of the function $g(S)$ can be found in\nappendix \\ref{sb} where it has been shown that corrections to \\refb{e7b} \nare of order $e^{-2\\pi S}$. The contribution\n\\refb{ebh0} comes from the $g(S)\\simeq S\/16\\pi$ term in $\\phi(S,\\bar S)$.\nBeing holomorphic in $S$,\nthis part is easy to supersymmetrize, and was used in arriving at expressions\n\\refb{eb1}, \\refb{eb2} for $S_{BH}$. On the other hand \\refb{eb3} arises\nfrom the part of $\\phi(S,\\bar S)$ proportional to $\\ln(S+\\bar S)$ which cannot\nbe regarded as a holomorphic function. Supersymmetrization\nof this term has not been carried out completely.\nNevertheless,\nusing various consistency requirements, \\cite{9906094} guessed that \nsupersymmetric completion of the term \\refb{eb6} modifies\nequations \\refb{eb1} and \\refb{eb2} to\\footnote{The analysis of\n\\cite{9906094} was done in the context of toroidal compactification\nof heterotic string theory. We are using a generalization of\nthis result.}\n\\begin{equation} \\label{eb4aa}\nS_{BH} = {\\pi N \\over S_0} + \n64\\pi^2 \\, g(S_0) - {K\\over 2} \\, \\ln (2S_0)\\, ,\n\\end{equation}\nand\n\\begin{equation} \\label{eb5aa}\n-{\\pi\\, N\\over S_0^2} + 64 \\pi^2 g'(S_0) -{K\\over 2S_0} \\simeq 0\\, .\n\\end{equation}\nFor large $N$, $S_0$ computed from \\refb{eb5aa} is of order $\\sqrt{N}$\nand hence we can use the large $S_0$ approximation \\refb{e7b} for $g(S_0)$.\nThis gives\n\\begin{equation} \\label{eb4}\nS_{BH} \\simeq {\\pi N \\over S_0} + \n4\\pi \\, S_0 - {K\\over 2} \\, \\ln (2S_0)\\, ,\n\\end{equation}\nand\n\\begin{equation} \\label{eb5}\n-{\\pi\\, N\\over S_0^2} + 4 \\pi -{K\\over 2S_0} \\simeq 0\\, .\n\\end{equation}\n\n{}In order to complete the computation of $S_{BH}$ we need to calculate the\nconstant $K$.\\footnote{This calculation has been carried out earlier\nin \\cite{9708062} using direct analysis of one loop amplitudes in type II\nstring theory.} \nFortunately there is a simple expression for $K$ by virtue of\nthe fact that it appears as the coefficient of the non-holomorphic piece of\n$\\phi(S,\\bar S)$ and hence is directly related to the holomorphic anomaly\n$\\partial_S\\partial_{\\bar S} \\phi(S,\\bar S)$\\cite{9302103,9309140}. \nThis computation is carried out by mapping the heterotic string theory to\nthe dual type IIA description. As is well known, heterotic string theory\non $T^4$ is dual to type IIA string theory on \nK3\\cite{9410167,9501030,9503124,9504027,9504047}.\nUnder this duality the Narain lattice $\\Lambda_{20,4}$ of the heterotic \nstring theory\ngets mapped to the lattice of integer homology cycles of $K3$, and\nthe components of the\ngauge fields take value in the real cohomology group\nof $K3$\\cite{9503124}. Also the generator $g$ of the $Z_m$ symmetry\nof $\\Lambda_{20,4}$, that was used in section \\ref{sstat}\nfor the construction of the CHL model, gets mapped to an order $m$ symmetry\ngenerator $\\widetilde g$ of the conformal field theory describing type IIA string\ntheory on $K3$\\cite{9508154} with specific action \non the elements of the homology and the cohomology group of $K3$.\nSince $g$ preserves $(24-k)$ of the 24 directions of the Narain lattice\nassociated with heterotic string compactification on $T^4$,\n$\\widetilde g$ will preserve $(24-k)$ of the 24 basis vectors of the real\ncohomology group of $K3$. \nNow\ncompactifying both sides on $\\widetilde S^1\\times S^1$ we get a duality between\nheterotic string theory on \n$T^4\\times \\widetilde S^1\\times S^1$ and type IIA on $K3\\times \n\\widetilde S^1\\times S^1$. \nLet us denote by $h$ the generator of the\norder $m$ shift along $\\widetilde S^1$. Then the CHL model, obtained by\nmodding out heterotic string theory on $T^4\\times \\widetilde S^1\\times S^1$\nby the $Z_m$ symmetry group generated by \n$h\\cdot g$, is dual to\ntype IIA string theory on $K3\\times\n\\widetilde S^1\\times S^1$, modded out by the $Z_m$ group generated by\n$h\\cdot \\widetilde g$\\cite{9507027,9507050,9508144,9508154}.\nWe shall denote\nby ${\\cal C}$ the conformal field theory associated with the six compact directions\nof the type IIA string theory after taking this quotient. \n\nIt is well\nknown that the dilaton-axion field $S$ of the heterotic string theory\ngets mapped to the complexified Kahler modulus of the two torus \n$\\widetilde S^1\\times S^1$ on the type IIA side\\cite{9503124}. \nThus computation of\n$\\partial_S\\partial_{\\bar S}\\phi(S,\\bar S)$ requires computing the derivative \nof $\\phi$ with\nrespect to the Kahler modulus of $\\widetilde S^1\\times S^1$ and its complex conjugate\nin the type IIA description.\nThe detailed procedure for this computation can be found in \n\\cite{9302103,9309140}; \nhere we just summarize\nthe relevant result of these papers which lead to the value of $K$.\nLet\nus denote by $\\psi^4$, $\\psi^5$ the right-handed world-sheet fermions,\nand by $\\bar\\psi^4$, $\\bar \\psi^5$ the left-handed world-sheet fermions \nassociated with the directions along the circles $S^1$ and $\\widetilde S^1$ in the\ntype IIA theory. We define:\n\\begin{equation} \\label{ec1}\n\\psi^\\pm = {1\\over \\sqrt 2} (\\psi^4 \\pm i \\psi^5), \\qquad\n\\bar\\psi^\\pm = {1\\over \\sqrt 2} (\\bar\\psi^4 \\pm i \\bar\\psi^5)\\, .\n\\end{equation}\nIn the Ramond-Ramond (RR) sector\n$\\psi^\\pm$ as well as $\\bar\\psi^\\pm$ have zero modes. We denote them by\n$\\psi_0^\\pm$ and $\\bar\\psi_0^\\pm$ respectively. \nThey satisfy the usual anti-commutation relations\n\\begin{equation} \\label{ec1a}\n\\{\\psi_0^+, \\psi_0^-\\} = 1, \\qquad \\{\\bar\\psi_0^+, \\bar\\psi_0^-\\} = 1\\, ,\n\\end{equation}\nwith all other anti-commutators being zero.\nIf we now define\n\\begin{equation} \\label{ec2}\nC = \\psi_0^+ \\bar \\psi_0^-, \\qquad \\bar C = \\psi_0^- \\bar \\psi_0^+\\, ,\n\\end{equation}\nthen in the subspace\nof RR sector ground states they represent\nthe action of the\noperators $\\psi^+\\bar \\psi^-$ and $\\psi^-\\bar\\psi^+$ which appear in the\nvertex operator of the Kahler class of $\\widetilde S^1\\times S^1$ and its complex \nconjugate, -- the fields $S$, $\\bar S$\nwith respect to which we want to take derivatives of\n$\\phi(S,\\bar S)$. \nIn terms of these operators the coefficient $K$ is \ngiven by\\cite{9302103}\n\\begin{equation} \\label{ec3}\nK = -Tr_{RR} \\left[ (-1)^{F_L+F_R} \\, C \\, \\bar C \\right]\\, ,\n\\end{equation}\nwhere the trace is to be\ntaken over the RR sector ground states (with $L_0=\\bar \nL_0=0$) of the conformal field theory ${\\cal C}$, and $F_L$ and $F_R$ denote\nthe world-sheet fermion numbers in the left and the right-moving sector of\nthis conformal field theory. In arriving at \\refb{ec3} we have used\nthe fact that $Tr\\left( (-1)^{F_L+F_R}\\right)$ vanishes in the conformal\nfield theory ${\\cal C}$, since the action of the fermion zero modes\n$\\psi_0^\\pm$ pairs states with equal and opposite $(-1)^{F_L+F_R}$ \neigenvalues.\n\nThe states of the conformal field theory ${\\cal C}$ include both untwisted sector\nstates and twisted sector states. Of these the twisted sector states \nnecessarily carry fractional \nunit of winding along $\\widetilde S^1$. Hence the twisted RR states\nalways have strictly positive $L_0$, $\\bar L_0$ \neigenvalues and cannot contribute\nto \\refb{ec3}. \nThe untwisted sector\nstates are states associated with the original CFT with target\nspace $K3\\times \n\\widetilde S^1\\times S^1$ which are invariant under the $Z_m$ \nsymmetry generated by $h\\cdot \\tilde g$. \nThese can be divided into two classes: those which are invariant\nseparately under $h$ and $\\widetilde g$, and those which pick up \nequal and opposite non-trivial\nphases under the action of $h$ and $\\widetilde g$.\nThe latter class, being not invariant under $h$,\ncarries non-zero momentum along $\\widetilde S^1$, and hence\nthe RR sector states in this class have strictly positive \n$L_0$ and $\\bar L_0$ eigenvalues.\nThus they cannot contribute to the trace in \\refb{ec3}, and we are left with\nstates which are invariant separately under $h$ and $\\widetilde g$. \nSince the \noperators $C$ and $\\bar C$ appearing in \\refb{ec3} act on the Hilbert space\nof the CFT with target space $\\widetilde S^1\\times S^1$, the contribution to $K$\nfrom these states may be factorized as\n\\begin{equation} \\label{ec4}\nK = -Tr_{RR}^{K3;inv}\\left[(-1)^{F_L+F_R}\\right] \\, \nTr_{RR}^{\\widetilde S^1\\times S^1} \n\\left[(-1)^{F_L+F_R} C \\bar C\\right] \\, ,\n\\end{equation}\nwhere in $Tr_{RR}^{K3;inv}$\nthe trace is now taken over the $\\widetilde g$ invariant RR sector \nground states of the\nconformal field theory with target space $K3$, and in\n$Tr_{RR}^{\\widetilde S^1\\times S^1}$ the trace is to be taken over the\nRR sector ground states of the CFT with target space $\\widetilde S^1\\times S^1$.\nFor the later CFT the requirement of vanishing $L_0$ and $\\bar L_0$\nforces the states to carry vanishing momenta along $S^1$ and $\\widetilde S^1$ and\nhence they are automatically invariant under $h$.\n\nThere is a one to one map between the vector space of\nRR sector ground states in the CFT\nassociated with K3 and the\nreal cohomology group of $K3$. Under this map $(F_L+F_R)$ get mapped to\nthe degree of the cohomology element. Since K3 has non-trivial\ncohomology of even degree only,\n$Tr\\left[(-1)^{F_L+F_R}\\right]$ for K3 is equal to the dimension of the \ncohomology group of K3 which is 24. Here however we are interested in the\nRR sector ground states which are even under $\\widetilde g$, and hence we should\ncount the dimension of the cohomology group of K3 which is invariant under \n$\\widetilde g$. Since this number is equal to $(24-k)$, we have\n\\begin{equation} \\label{ec5}\nTr_{RR}^{K3;even}\\left[(-1)^{F_L+F_R}\\right] = 24-k\\, .\n\\end{equation}\nIn order to calculate the second factor appearing in \\refb{ec4}, we note that\nthe RR sector ground states \nassociated with $\\widetilde S^1\\times S^1$ consist of four\nstates. Defining the vacuum state $|0\\rangle$ to\nbe annihilated by $\\psi_0^-$ and\n$\\bar\\psi_0^-$, and have $F_L=F_R=-{1\\over 2}$, the states are\n\\begin{equation} \\label{ec6}\n|0\\rangle, \\quad \\psi_0^+|0\\rangle, \\quad \\bar\\psi_0^+|0\\rangle, \\quad \\psi_0^+\\bar\n\\psi_0^+|0\\rangle \\, , \n\\end{equation}\nwith $(F_L,F_R)$ values $\\left(-{1\\over 2}, -{1\\over 2}\\right)$, \n$\\left(-{1\\over 2}, {1\\over 2}\\right)$, $\\left({1\\over 2}, -{1\\over 2}\\right)$, \n$\\left({1\\over 2}, {1\\over 2}\\right)$ respectively. {}From the structure\nof $C$ and $\\bar C$ defined in \\refb{ec2} it is clear that only the\nstate $\\psi_0^+|0\\rangle$ will contribute to the trace appearing in\nthe second factor of \\refb{ec4}. Since $C\\bar C\\psi_0^+|0\\rangle=-\\psi_0^+|0\\rangle$,\nwe get\n\\begin{equation} \\label{ec7}\nTr_{RR}^{\\widetilde S^1\\times S^1} \n\\left[(-1)^{F_L+F_R} C \\bar C\\right] =-1\\, .\n\\end{equation}\nSubstituting \\refb{ec5} and \\refb{ec7} into \\refb{ec4}  we get\n\\begin{equation} \\label{ec8}\nK = (24-k)\\, .\n\\end{equation}\n\nEqs.\\refb{eb4}, \\refb{eb5} now give\n\\begin{equation} \\label{eb4a}\nS_{BH} \\simeq {\\pi N \\over S_0} + 4\\pi \\, S_0 - {1\\over 2}\n\\, (24-k) \\, \n\\ln (2S_0)\\, ,\n\\end{equation}\nand\n\\begin{equation} \\label{eb5a}\n-{\\pi\\, N\\over S_0^2} + 4 \\pi -{24-k\\over 2S_0} \\simeq 0\\, .\n\\end{equation}\nThese agree with eqs.\\refb{e10a}, \\refb{e10b} under the identification\n\\begin{equation} \\label{ec9}\n\\widetilde S_{stat} = S_{BH}, \\qquad \\mu = {\\pi\\over S_0}\\, .\n\\end{equation}\nThus we see that in this approximation the entropy $S_{BH}$\nof the black hole\nagrees with the statistical entropy $\\widetilde S_{stat}$ calculated following\nthe procedure given in section \\ref{srev} up to an overall constant.\n\nEarlier we had estimated the error in $\\widetilde S_{stat}$ calculated from\n\\refb{e10a}, \\refb{e10b} to be\nnonperturbative in $1\/\\sqrt{N}$. We shall now try to carry out a\nsimilar estimate of the error involved in eqs.\\refb{eb4a}, \\refb{eb5a} so\nthat we can determine up to what level the agreement between $S_{BH}$ and\n$\\widetilde S_{stat}$ holds.\nFirst of all we should remember that there is an\nuncertainty involved in the\noriginal formulae \\refb{eb4aa}, \\refb{eb5aa} since they have not been derived\nfrom first principles. \nIn particular \\cite{9906094} used an argument based on duality symmetry to\nderive the effect of the holomorphic anomaly terms, and this does not\nfix the additive constant on the right hand side\nof \\refb{eb4aa}. Thus there is an ambiguity in the overall additive\nconstant in the\nexpression for $S_{BH}$, and hence we cannot hope to compare the additive\nconstants in $\\widetilde S_{stat}$ and $S_{BH}$.\nAssuming that the formul\\ae\\ \\refb{eb4aa}, \\refb{eb5aa} are correct up to this\nadditive constant, we see\nthat the error in the determination of $S_{BH}$ lies essentially in the\nerror in the determination of the function $g(S)$. As reviewed in \nappendix \\ref{sb}, \nwe can determine the form of the corrections to $g(S)$ by the requirement \nof S-duality invariance of the\ntheory,\\footnote{As will be discussed in appendix \\ref{sb},\nthe S-duality group for these models is usually\na subgroup of SL(2,Z), and \ndepends on the specific\nmodel we are analysing\\cite{9507050}.}\nand typically corrections\nto \\refb{e7b} involve powers of $e^{-2\\pi S}$. Since $S_0$ obtained by \nsolving eq.\\refb{eb5a} is of order\n$\\sqrt{N}$, we see that the corrections to $S_{BH}$ will involve powers of\n$e^{-\\pi\\sqrt{N}}$. Thus for large $N$\nthe agreement between $S_{BH}$ and $\\widetilde S_{stat}$ \nholds up to an undetermined additive\nconstant, and terms which are non-perturbative in $1\/\\sqrt{N}$.\nIn particular if we express $\\widetilde S_{stat}$ and $S_{BH}$ in power series\nexpansion in $1\/\\sqrt{N}$ by solving eqs.\\refb{e10a}, \\refb{e10b} \nand \\refb{eb4a}, \\refb{eb5a}\nrespectively, then the results agree to all orders in $1\/N$ including\nterms proportional to $\\ln N$. This in turn implies similar agreement\nbetween $\\ln d_N$ and $\\ln d_N^{BH}$ defined in section \\ref{srev}.\n\n\\sectiono{Discussion} \\label{sdiss}\n\nGiven the agreement between the statistical entropy $\\widetilde S_{stat}$ and \nthe black hole entropy $S_{BH}$ up to non-perturbative terms, one might \nwonder if this correspondence also holds after we include non-perturbative \nterms. Unfortunately however even for toroidally compactified heterotic \nstring theory $\\widetilde S_{stat}$ and $S_{BH}$ differ once we include\nnon-perturbative corrections\\cite{0411255,0412287}, and hence we\nexpect that such disagreement will also be present\nfor CHL compactifications. One could contemplate several reasons for\nthis discrepancy:\n\\begin{enumerate}\n\\item First of all we should remember that the formul\\ae\\\n\\refb{eb4aa},\n\\refb{eb5aa}\nfor the black\nhole entropy in the presence of non-holomorphic terms, as given in\n\\cite{9906094}, have not been derived from first principles. Thus\nthere could be further corrections to these formul\\ae\\ which could\nmodify the expression for $S_{BH}$.\n\\item It could also be that the proposal \\refb{e2a} - \\refb{eprop}\nfor relating the black hole\nentropy to the degeneracy of elementary string states\nis not \ncomplete; and that the formul\\ae\\ needs to be\nmodified once non-perturbative effects are taken into account.\n\\item Besides supersymmetric\ncompletion of the curvature squared terms, the effective field\ntheory contains infinite number of other higher derivative terms\nwhich are in principle equally important, and\nat present there is no understanding as to why these terms do not\naffect the expression for the entropy. It could happen that while\nthese terms do not affect the perturbative corrections, their\ncontribution becomes significant at the non-perturbative level.\n\\item Finally there is always a possibility that the relation between\nthe black hole entropy and statistical entropy exists only as a\npower series expansion in inverse powers of various charges. In this\ncase we do not expect any relation between the non-perturbative\nterms in the expressions for $S_{BH}$ and $\\widetilde S_{stat}$. \n\\end{enumerate}\n\nAt present we do not know which (if any) of these possibilities\nis correct. This issue clearly needs further investigation.\nWe note however that if the fourth proposal is correct, namely the \nagreement between the black hole entropy and the statistical entropy holds \nonly as a power series expansion in inverse powers of the charges, then \nthe proposal \\refb{eprop} relating the black hole and the statistical \nentropy can be extended to more general models and more general states. \nThe essential point is that in our analysis we have been restricted to \ncompactifications of the form $K_5\\times S^1$ and to states carrying \nmomentum and winding along $S^1$ in order to ensure that the degeneracy of \nthe states depends only on the \ncombination $N=nw$. If\nwe consider more general $N=4$ supersymmetric compactification \n({\\it e.g.} where the orbifold group is $Z_m\\times Z_{m'}$\nand acts on both circles instead of just \none circle\\cite{9508154}) and\/or more general \nstates \ncarrying arbitrary gauge charges $(\\vec P_L, \\vec P_R)$ associated\nwith gauge fields arising out of the left and the right sectors of the\nworld-sheet, then the role of the T-duality invariant combination $nw$ is \nplayed by\n$N\\equiv {1\\over 2} (\\vec P_R^2 - \n\\vec P_L^2)$. However in this case the degeneracy $d(\\vec P_L, \\vec P_R)$\nof such states could depend on $\\vec P_L$ and $\\vec P_R$ separately\ninstead of being a function of the combination $N$ only. To see how\nsuch dependence can arise, we can\nconsider the class of models described in section \\ref{sstat} and consider\na state that carries $\\tilde n$ units of momentum along the circle $\\widetilde S^1$ \nbesides the charge\nquantum numbers $n,w$ associated with the circle $S^1$. \nFor such states we still have $N=nw$. However\nin this case the\npart of the wave-function associated with $\\widetilde S^1$\npicks up a phase $e^{2\\pi i \\tilde n\/m}$ \nunder the $Z_m$ shift along $\\widetilde S^1$,\nand we must compensate for it by introducing a factor of \n$e^{2\\pi i \\tilde n l\/m}$\nmultiplying $g^l$ in the projection operator \\refb{epr1} in order to \nensure that the complete state is invariant under the $Z_m$ \ntransformation. This introduces\na specific dependence of the partition function and hence of\nthe degeneracy of states on $\\tilde n$. If in addition the state carries\nsome gauge charges associated with the lattice $\\Lambda_{20,4}$, then\nthe sum over\nmomentum $\\vec s$ in \\refb{ednnew}, \\refb{efnew} \nmight run over a shifted lattice,\nwhich is equivalent to replacing the $\\exp(-\\mu\\vec s^2\/2)$ factor\nin \\refb{efnew} by $\\exp[-\\mu (\\vec s + \\vec K)^2\/2]$ for some fixed vector\n$\\vec K$ that depends on the component of\n$(\\vec P_L, \\vec P_R)$ along $\\Lambda_{20,4}$. However by following the\nanalysis given in section \\ref{sstat} and\nappendix \\ref{sa} one can see that the\ndependence on $(\\vec P_L, \\vec P_R)$ introduced by either of these effects\nis exponentially suppressed, and\nhence if we are interested in $\\ln d(\\vec P_L, \\vec P_R)$ as a power series\nexpansion in inverse powers of charges, the result depends only on the\ncombination $N=(\\vec P_R^2-\\vec P_L^2)\/2$. Similar analysis can also\nbe carried out for the $Z_m\\times Z_{m'}$ orbifold models.\nThis allows us to define\n$\\widetilde S_{stat}(N)$ through eqs.\\refb{e2a}-\\refb{e12} within this approximation.\nOn the other hand from the results of \\cite{9906094} it \nalso follows that the black hole entropy will also be a function of the \ncombination $N={1\\over 2} (\\vec P_R^2 -\n\\vec P_L^2)$. An analysis similar to the one described in this paper\ncan then be used to show that\nthe correspondence \\refb{eprop} between the statistical \nentropy and \nthe black hole entropy continues to hold for these more general class\nof states and\/or models.\n\n\n\n\n\n{\\bf Acknowledgement}: I would like to thank P.~Aspinwall, A.~Dabholkar,\nR.~Gopakumar, D.~Jatkar\nand D.~Surya Ramana for valuable discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{submission}\n\nWork on generating images with Generative Adversarial Networks (GANs) has shown promising results \\cite{GANS}.\nA generative net G and a discriminator D are trained simultaneously with conflicting objectives.\nG takes in a noise vector and outputs an image, while\nD takes in an image and outputs a prediction about whether the image is a sample from G.\nG is trained to maximize the probability that D makes a mistake, and D is trained to minimize that probability.\nBuilding on these ideas, one can generate good output samples using a cascade \\cite{LAPGAN} of convolutional neural networks.\nMore recently \\cite{DCGAN}, even better samples were created from a single generator network.\nHere, we consider the situation where we try to solve a semi-supervised classification task and learn a generative model simultaneously.\nFor instance, we may learn a generative model for MNIST images while we train an image classifier, which we'll call C.\nUsing generative models on semi-supervised learning tasks is not a new idea - Kingma et al. \\yrcite{SSVAE} expand work on variational generative techniques \n\\cite{VAES, VAES2} to do just that.\nHere, we attempt to do something similar with GANs.\nWe are not the first to use GANs for semi-supervised learning.\nThe CatGAN \\cite{CATGAN} modifies the objective function to\ntake into account mutual information between observed examples and their predicted\nclass distribution.\nIn Radford et al. \\yrcite{DCGAN}, the features learned by D are reused in classifiers.\n\n\nThe latter demonstrates the utility of the learned representations, but it has several undesirable properties.\nFirst, the fact that representations learned by D help improve C is not surprising - it seems\nreasonable that this should work. However, it also seems reasonable that learning a good C would help to improve the performance of D.\nFor instance, maybe images where the output of C has high entropy are more likely to come from G.\nIf we simply use the learned representations of D\nafter the fact to augment C, we don't take advantage of this.\nSecond, using the learned representations of D after the fact doesn't allow for training C and G simultaneously.\nWe'd like to be able to do this for efficiency reasons, but there is a more important motivation.\nIf improving D improves C, and improving C improves D (which we know improves G) then we may be able to\ntake advantage of a sort of feedback loop, in which all 3 components (G,C and D) iteratively make each other\nbetter.\n\nIn this paper, inspired by the above reasoning, we make the following contributions:\n\n\\begin{itemize}\n  \\item First, we describe a novel extension to GANs that allows them to learn a generative model and a classifier simultaneously.\nWe call this extension the Semi-Supervised GAN, or SGAN.\n\\item Second, we show that SGAN improves classification performance on restricted data sets over a baseline classifier with no generative component.\n\\item Finally, we demonstrate that SGAN can significantly improve the quality of the generated samples and reduce\n  training times for the generator.\n\\end{itemize}\n\n\\section{The SGAN Model}\n\nThe discriminator network D in a normal GAN outputs an estimated probability that the input image is\ndrawn from the data generating distribution. Traditionally this is implemented with a feed-forward network\nending in a single sigmoid unit, but it can also be implemented with a softmax output layer with one unit\nfor each of the classes [REAL, FAKE]. Once this modification is made, it's simple to\nsee that D could have N+1 output units corresponding to [CLASS-1, CLASS-2, \\dots CLASS-N, FAKE].\nIn this case, D can also act as C. We call this network D\/C.\n\nTraining an SGAN is similar to training a GAN.\nWe simply use higher granularity labels for the half of the minibatch that has been drawn from the data generating distribution.\nD\/C is trained to minimize the negative log likelihood with respect to the given labels and G is trained to maximize it, as shown in Algorithm \\ref{alg:example}.\nWe did not use the modified objective trick described in Section 3 of Goodfellow et al. \\yrcite{GANS}.\n\nNote: in concurrent work, \\cite{IMPROVEDTECHNIQUES} propose the same method for augmenting\nthe discriminator and perform a much more thorough experimental evaluation of the technique.\n\n\\begin{algorithm}[tb]\n   \\caption{SGAN Training Algorithm}\n   \\label{alg:example}\n\\begin{algorithmic}\n   \\STATE {\\bfseries Input:} $I$: number of total iterations\n   \\FOR{$i=1$ {\\bfseries to} $I$}\n   \\STATE Draw $m$ noise samples $\\{z^{(1)}, \\dots, z^{(m)}\\}$ from noise prior $p_g(z)$.\n   \\STATE Draw $m$ examples $\\{ (x^{(1)}, y^{(1)}) , \\dots, (x^{(m)}, y^{(m)}) \\}$ from data generating distribution $p_d(x)$.\n   \\STATE Perform gradient descent on the parameters of D w.r.t. the NLL of D\/C's outputs on the combined minibatch of size $2m$.\n   \\STATE Draw $m$ noise samples $\\{z^{(1)}, \\dots, z^{(m)}\\}$ from noise prior $p_g(z)$.\n   \\STATE Perform gradient descent on the parameters of G w.r.t. the NLL of D\/C's outputs on the minibatch of size $m$.\n   \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Results}\n\nThe experiments in this paper were conducted with \\href{https:\/\/github.com\/DoctorTeeth\/supergan}{https:\/\/github.com\/DoctorTeeth\/supergan}, which borrows heavily from \\href{https:\/\/github.com\/carpedm20\/DCGAN-tensorflow}{https:\/\/github.com\/carpedm20\/DCGAN-tensorflow} and which contains more details about the experimental setup.\n\n\\subsection{Generative Results}\n\nWe ran experiments on the MNIST dataset \\cite{MNIST} to determine whether an SGAN would result in better generative samples\nthan a regular GAN.\nUsing an architecture similar to that in Radford et al. \\yrcite{DCGAN}, we\ntrained an SGAN both using the actual MNIST labels and with only the labels REAL and FAKE.\nNote that the second configuration is semantically identical to a normal GAN.\nFigure \\ref{icml-historical} contains examples of generative outputs from both GAN and SGAN.\nThe SGAN outputs are significantly more clear than the GAN outputs.\nThis seemed to hold true across different initializations and network architectures,\nbut it is hard to do a systematic evaluation of sample quality for varying hyperparameters.\n\n\\begin{figure}[ht]\n\\vskip 0.2in\n\\begin{minipage}[b]{0.5\\linewidth}\n\\centerline{\\includegraphics[width=\\columnwidth]{figures\/labels_1_840.png}}\n\\end{minipage\n\\begin{minipage}[b]{0.5\\linewidth}\n\\centerline{\\includegraphics[width=\\columnwidth]{figures\/no-labels_1_840.png}}\n\\end{minipage} \n\\caption{Output samples from SGAN and GAN after 2 MNIST epochs.\nSGAN is on the left and GAN is on the right.}\n\\label{icml-historical}\n\\vskip -0.2in\n\\end{figure} \n\n\\subsection{Classifier Results} \n \nWe also conducted experiments on MNIST to see whether the classifier component of the SGAN would\nperform better than an isolated classifier on restricted training sets.\nTo train the baseline, we train SGAN without ever updating G.\nSGAN outperforms the baseline in proportion to how much we shrink the training set,\nsuggesting that forcing D and C to share weights improves data-efficiency.\nTable \\ref{sample-table} includes detailed performance numbers.\nTo compute accuracy, we took the maximum of the outputs not corresponding to the FAKE label.\nFor each model, we did a random search on the learning rate and reported the best result.\n\n\\begin{table}[t]\n\\caption{Classifier Accuracy}\n\\label{sample-table}\n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\begin{tabular}{lcccr}\n\\hline\n\\abovespace\\belowspace\nExamples & CNN & SGAN\\\\\n\\hline\n\\abovespace\n1000  &  0.965 & 0.964\\\\\n100   &  0.895 & 0.928\\\\\n50    &  0.859 & 0.883\\\\\n25    &  0.750 & 0.802\\\\\n\\belowspace\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\n\\section{Conclusion and Future Work} \nWe are excited to explore the following related ideas:\n\n\\begin{itemize}\n\\item Share some (but not all) of the weights between D and C, as in the dual autoencoder \\cite{TPUL}. This could allow some weights to be specialized to discrimination and some to classification.\n\\item Make GAN generate examples with class labels \\cite{CONDITIONAL}. Then ask D\/C to assign one of $2N$ labels [REAL-ZERO, FAKE-ZERO, \\dots ,REAL-NINE, FAKE-NINE].\n\\item Introduce a ladder network \\cite{LADDER} L in place of D\/C, then use samples from G as unlabeled data to train L with.\n\\end{itemize}\n\n\\section*{Acknowledgements} \n \nWe thank the authors of Tensorflow \\cite{TENSORFLOW}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA game models a scenario where multiple strategic controllers (or players) optimize their objective functionals, which depend not only on the self actions but also on the actions of other controllers. In stochastic static games, players observe the realization of some random state of nature, possibly through separate noisy channels, and use such observations to independently determine their actions so that the expected values of their individual cost (or utility) functions are optimized. In a stochastic dynamic game, on the other hand, the players act at multiple time steps, based on observation or measurement of some dynamic process which itself is driven by past actions as well as random quantities, which could again be called random states of nature. What information each player acquires at each stage of the game determines what is called the {\\it information structure} of the underlying game. If all the players acquire the same information at each time step, then the dynamic game is said to be a game of \nsymmetric information. However, in many real scenarios, the players do not have access to the same information about the underlying state processes and other players' observations and past actions. Such games are known as games with asymmetric information. For example, several problems in economic interactions \\cite{harsanyi1967,cole2001,myerson1997}, attacks on cyber-physical systems \\cite{gupta2012}, auctions, cryptography, etc. can be modeled as games of asymmetric information among strategic players.\n\nGames with symmetric and\/or perfect information have been well studied in the literature; see, for example, \\cite{Shapley:1953,Sobel1971, fudenberg1991,Basarbook,Filarbook}. In these games, the players have the same beliefs on the states of the game, future observations and future expected costs or payoffs. However, in games with asymmetric information, the players need not have the same beliefs on the current state and future evolution of the game. General frameworks to compute or refine Nash equilibria in stochastic games of symmetric or perfect information have received attention from several researchers, see for example, \\cite{maskin2001, myerson1997, fudenberg1991} among many others. However, by comparison, such general frameworks for games of asymmetric information are scant (for exceptions, see \\cite{Basar1975,Basaronestep,Basarmulti, basar1985}). This paper, in addition to its earlier finite-game version \\cite{nayyar2012a}, provides such a \nframework.\n\n\nIn our recent work \\cite{nayyar2012a}, we considered a general finite non-zero sum dynamic stochastic game of asymmetric information with stagewise additive cost functions. Under certain assumptions on the information structures of the players, we obtained a characterization of a particular class of Nash equilibria using a dynamic programming like approach. The key idea there was to use common information among the players to transform the original game of asymmetric information to a game of symmetric and perfect state information with an expanded state and action spaces of the players so that a more easily computable Markov perfect equilibrium of the latter can be used to obtain a Nash equilibrium for the former. The advantage of this technique is that instead of searching for equilibrium in the (large) space of strategies (which grows with the number of stages), we only need to compute Nash equilibrium in a succession of static games of complete information. This reduces the computational effort in \ncomputing a Nash equilibrium of the game. We call the Nash equilibria obtained with this approach as {\\it common information based Markov perfect equilibrium}.\n\nIn this work, we extend the framework and results of \\cite{nayyar2012a} to infinite games, particularly those with linear state and observation equations and Gaussian random variables. For quadratic cost functions of the players satisfying certain assumptions, we show that a unique common information based Markov perfect equilibrium exists. The general framework developed in the paper can be applied to obtain Nash equilibria of broader classes of stochastic dynamic games with asymmetric information, satisfying two general assumptions delineated in the paper.  \n\n\\subsection{Previous Work}\nIn the past, specific models of various classes of games have been studied, where different players acquire different information. Harsanyi, in his seminal paper \\cite{harsanyi1967}, studied one subclass of static games with finite state and action spaces of the players, and showed that under some technical conditions, Nash equilibrium exists in such games. Various authors \\cite{Behn68,Rhodes69,Willman69} have studied two-player zero-sum differential games with linear state dynamics and quadratic payoffs, where the players do not make the same measurements about the state. A zero sum differential game where one player's observation is nested in the other player's observation was considered in \\cite{Hojota}. A zero-sum differential game where one player makes a noisy observation of the state while the other one does not make any measurement was considered in \\cite{MintzBasar}.\n\nDiscrete-time non-zero sum LQG games with one step delayed sharing of observations were studied in \\cite{Basaronestep} and \\cite{Basarmulti}. A game with one-step delayed observation and action sharing among the players was considered in \\cite{Altman:2009}. A two-player finite game in which the players do not have access to each other's observations and  control actions was considered in \\cite{hespanha2001}, where a necessary and sufficient condition for existence of a Nash equilibrium in terms of two coupled dynamic programs was obtained.\n\nObtaining equilibrium solutions for  stochastic  games  when players make independent noisy observations of the state and do not  share all of their information (or even when they have access to the same noisy observation as in \\cite{Bjota}) has remained a challenge for general classes of games. Identifying classes of games which would lead to tractable solutions or feasible solution methods is therefore an important goal in this area.\n\n\\subsection{Contributions of this Paper}\nThis paper is a sequel to our earlier finite game work in \\cite{nayyar2012a}, where we make similar assumptions on the information structures of the players. We study games in which the state, the players' actions and primitive random variables take values in finite-dimensional Euclidean spaces. The state evolution and observation equations are taken to be linear in their arguments and all primitive random variables are assumed to be mutually independent zero-mean Gaussian random variables. We assume that the players have a stagewise additive total cost function. \n\nWe assume that the information structures of the players satisfy two sufficient conditions. For any dynamic game satisfying these assumptions, we show that we can decompose it into several static games using a backward induction algorithm. If there exists a Nash equilibrium for each of those static games, then there exists a common information based Markov perfect equilibrium for the original dynamic game. Furthermore, we present an algorithm that computes the common information based Markov perfect equilibrium in such games, provided that it exists. For games in which the cost functions of the players are quadratic satisfying certain assumption, we show that the static game at each time step admits a unique Nash equilibrium in the class of all Borel measurable strategies of the players at that time step, thereby proving the existence of a unique common information based Markov perfect equilibrium. We also show, by example, that there may be other Nash equilibria of such games that cannot be computed using \nthe conceptual method developed in this paper.\n\nTo sum up, common information based Markov perfect equilibria constitute a subclass of Nash equilibria of such games, and it can be thought of as a {\\it refinement of Nash equilibrium} for games with asymmetric information. However, we do not look into implementation issues of common information based Markov perfect equilibrium in this paper, and we leave this as a topic of further investigation.\n\n\n\\subsection{Notation}\nRandom variables are denoted by upper case letters and their realizations by the corresponding lower case letters. Random vectors are denoted by upper case bold letters and their realizations by lower case bold letters. Unless otherwise stated, the state, action and observations are assumed to be vector valued. \n\nLet $\\ALP X$ be a set. For a subset $\\SF X\\subset\\ALP X$, we let $\\SF X^\\complement$ denote the complement of the set $\\SF X$. We use $\\text{id}_{\\ALP X}$ to denote the identity map on the set $\\ALP X$. The transpose of a matrix $A$ is denoted by $A^\\transpose$. \n\n\n\nSubscripts are used as time indices and superscripts are used as player\/controller indices. Consider $a,b\\in\\mathbb{N}$. Let $\\VEC X_t$ be an element of a finite dimensional Euclidean space $\\ALP X_t$ for $a\\leq t\\leq b$. If $a\\leq b$, then we let $\\VEC X_{a:b}$ denote the set of vectors $\\{\\VEC X_a, \\VEC X_{a+1}, \\dots, \\VEC X_b\\}$. If $a>b$, then $\\VEC X_{a:b}$ is empty. On the other hand, we use $\\ALP X_{a:b}$ to denote the product space $\\prod_{t=a}^b\\ALP X_t$, which is a finite dimensional Euclidean space, with the understanding that $\\ALP X_{a:b}=\\emptyset$ if $a>b$. We use a similar convention for superscripts.\n\nWe use $\\mathds{P}\\{\\cdot\\}$ to denote the probability of an event and $\\mathds{E}[\\cdot]$ to denote the expectation of a random variable. For a collection of functions $\\boldsymbol{g}$, the notations $\\mathds{P}^{\\boldsymbol{g}}\\{\\cdot\\}$ and $\\mathds{E}^{\\boldsymbol{g}}[\\cdot]$ indicate that the probability\/expectation depends on the choice of functions in $\\boldsymbol{g}$. Similarly, for a probability measure $\\pi$, the notation $\\mathds{E}^{\\pi}[\\cdot]$ indicates that the expectation is with respect to the measure $\\pi$. The notation $\\ind{\\VEC x}$ denotes a Dirac measure at the point $\\VEC x$. For a set $\\ALP X$ and its subset $\\SF X$, $1_{\\SF X}:\\ALP X\\rightarrow\\{0,1\\}$ denotes the indicator function on the set $\\SF X$.\n\nLet $\\VEC X, \\VEC Y$ and $\\VEC Z$ be three random variables taking values, respectively, in the spaces $\\ALP X,\\:\\ALP Y$ and $\\ALP Z$. Then, $\\mathds{P}\\{\\SF X|\\VEC y,\\VEC z\\}$ denotes the probability of the event $\\SF X\\subset\\ALP X$ given the realizations $\\VEC y$ and $\\VEC z$ of the random variables $\\VEC Y$ and $\\VEC Z$. Similarly, $\\mathds{E}[\\cdot|\\VEC y]$ denotes the expected value of a real-valued function $(\\cdot)$ given the realization $\\VEC y$. We use $\\mathds{P}\\{d\\VEC x,d\\VEC y|\\VEC z\\}$ to denote the conditional probability measure over the space $\\ALP X\\times\\ALP Y$ given a realization $\\VEC z$ of another random variable $\\VEC Z$.\n\n\n\\subsection{Outline of the Paper}\nThe paper is organized as follows. In Section \\ref{sec:problem}, we formulate the two-player non-zero sum game problem with linear dynamics, linear observation equations, and asymmetric information among the players. We make two assumptions on the information structures of the controllers and an assumption on the admissible strategies of the agents. We also discuss consequences of the assumptions we make on the information structures. In Section \\ref{sec:main}, we state the main result of the paper and develop a backward induction algorithm that computes the common information based Markov perfect equilibrium of the game formulated in Section \\ref{sec:problem}, provided that it exists. In Section \\ref{sec:lqggames}, we specialize the result of Section \\ref{sec:main} to LQG games, and show that under further assumptions on cost functions, a unique common information based Markov perfect equilibrium exists in the class of measurable strategies of the players.\nIn Section \\ref{sec:information}, we show through an example that there may be other Nash equilibria of a game with asymmetric information, and that using our algorithm, we compute only a subclass of all Nash equilibria. We discuss some implications of our assumptions in Section \\ref{sec:discussion}. Finally, we conclude our discussion in Section \\ref{sec:conclude} and identify several directions for future research. Proofs of most of the results in the paper are given in appendices.\n\n\\section{Problem Formulation}\\label{sec:problem}\nLet $\\VEC X_t$ be the state of a linear system which is controlled by two controllers (players)\\footnote{In the paper, we use the term ``controller'' instead of ``player'', because we introduce another set of players in the symmetric information game introduced in the next section.}. At each time step $t$, Controller $i$, $i=1,2$, observes the state through a noisy sensor; this observation is denoted by $\\VEC Y^i_t$.  Controller $i$'s action at time step $t$ is denoted by $\\VEC U^i_t$. For each Controller $i\\in\\{1,2\\}$ at time $t\\in\\{1,\\ldots,T-1\\}$, the state, action and observation spaces are denoted by $\\ALP X_t$, $\\ALP U^i_t$ and $\\ALP Y^i_t$, respectively, and they are assumed to be finite dimensional Euclidean spaces. The dynamics and observation equations are given as\n\\beq{\\label{eq:lqgdynamics} \\VEC X_{t+1} &=& A_t\\VEC X_{t}+B^1_t\\VEC U^1_t+B^2_t\\VEC U^2_t+\\VEC W_t^0,\\\\\n\\label{eq:lqgobserv} \\VEC Y^i_t &=& H^i_t\\VEC X_t + \\VEC W^i_t, \\mbox{~~}i=1,2,}\nwhere $\\VEC W^i_t$ is a random variable taking values in a finite dimensional Euclidean space denoted by $\\ALP W^i_t$ for all $i\\in\\{0,1,2\\}$ and $t\\in\\{1,\\ldots,T-1\\}$, and $A_t,B^i_t,H^i_t,\\: i\\in\\{1,2\\},\\: t\\in\\{1,\\ldots,T-1\\}$ are matrices of appropriate dimensions. $\\VEC X_1,\\VEC  W^{0:2}_{1:T-1}$ are primitive random variables, and they are assumed to be mutually independent and zero-mean Gaussian random vectors.\n\n\\subsection{Information Structures of the Controllers}\nThe information available to each controller at time step $t\\in\\{1,\\ldots,T-1\\}$ is a subset of all information generated in the past, that is, $\\{\\VEC Y^{1:2}_{1:t},\\VEC U^{1:2}_{1:t-1}\\}$. Let $\\ALP E^i_t$ and $\\ALP F^i_t$, respectively, be defined as\n\\beqq{\\ALP E^i_t:=\\{(j,s)\\in\\{1,2\\}\\times\\{1,\\ldots,T\\}:\\text{ Controller } i \\text{ at time } t \\text{ knows } \\VEC Y^j_s\\},\\\\\n\\ALP F^i_t:=\\{(j,s)\\in\\{1,2\\}\\times\\{1,\\ldots,T\\}:\\text{ Controller } i \\text{ at time } t \\text{ knows } \\VEC U^j_s\\}.}\nDefine $\\ALP I^i_t$, $\\ALP C_t$ and $\\ALP P^i_t$ for $i=1,2$ and $t\\in \\{1,\\ldots,T-1\\}$ as\n\\beqq{\\ALP I^i_t  &=& \\prod_{(j,s)\\in\\ALP E^i_t} \\ALP Y^j_s\\times \\prod_{(j,s)\\in\\ALP F^i_t} \\ALP U^j_s,\\\\\n\\ALP C_t &=& \\prod_{(j,s)\\in\\ALP E^1_t\\cap\\ALP E^2_t} \\ALP Y^j_s\\times \\prod_{(j,s)\\in\\ALP F^1_t\\cap \\ALP F^2_t} \\ALP U^j_s,\\\\\n\\ALP P^i_t & = & \\prod_{(j,s)\\in\\ALP E^i_t\\setminus(\\ALP E^1_t\\cap\\ALP E^2_t)} \\ALP Y^j_s\\times \\prod_{(j,s)\\in\\ALP F^i_t\\setminus (\\ALP F^1_t\\cap \\ALP F^2_t)} \\ALP U^j_s.}\nNote that $\\ALP I^i_t$, $\\ALP C_t$ and $\\ALP P^i_t$ for $i=1,2$ and $t\\in \\{1,\\ldots,T-1\\}$ are finite dimensional Euclidean spaces. \n\nWe let $\\VEC I^i_t\\in\\ALP I^i_t$ denote the information available to Controller $i$ at time step $t\\in\\{1,\\ldots,T-1\\}$, which is a vector comprised of measurements and control actions that are observed by the controller. The common information of the controllers at a time step is defined as the vector of all random variables that are observed by both controllers at that time step. The private information of a controller at a time step is the vector of random variables that are not observed by the other controller. The common information is denoted by $\\VEC C_t\\in\\ALP C_t$, and the private information of Controller $i$ is denoted by $\\VEC P^i_t\\in\\ALP P^i_t$ at time step $t\\in\\{1,\\ldots,T\\}$.\n\nA dynamic game is said to be one of symmetric information if $\\VEC C_t=\\VEC I^1_t = \\VEC I^2_t$ at all time steps. We are interested in games where the controllers may have asymmetry in information, that is, $\\VEC I^1_t \\neq \\VEC I^2_t$. An extreme example of a game of asymmetric information is the case when $\\ALP P^1_t\\neq \\emptyset$ while $\\ALP P^2_t=\\emptyset$ for all time steps $t$. Another example of a game of asymmetric information is when the controllers recall their past information and share their observations after a delay of one time step, that is, $\\ALP C_t = \\ALP Y^{1:2}_{1:t-1}$ and $\\ALP P^i_t =\\ALP Y^i_t$ for all time steps $t$.\n\n\\subsection{Admissible Strategies of Controllers}\n\nAt every time step, Controller $i$ uses a {\\it control law} $g^i_t: \\ALP P^i_t\\times \\ALP C_t\\rightarrow \\ALP U^i_t$ to map its information to its action. We assume that the control law $g^i_t$ is a Borel measurable function, and denote the space of all such control laws by $\\ALP G^i_t$.\n\nA strategy of Controller $i$, which we define as the collection of its control laws over time, is denoted by $\\mathbf g^i = (g^i_1,\\ldots,g^i_{T-1})$ and the space of strategies of Controller $i$ is denoted by $\\ALP G^i_{1:T-1}$. The pair of strategies of both controllers, $(\\mathbf g^1,\\mathbf g^2)\\in\\ALP G^1_{1:T-1}\\times \\ALP G^2_{1:T-1}$, is called the strategy profile of the controllers.\n\nThe total cost to Controller $i$, as a function of the strategy profile of the controllers, is\n\\beqq{J^i(\\mathbf g^1,  \\mathbf g^2) := \\mathds{E}\\Bigg[c^i_T(\\VEC x_T) + \\sum_{t=1}^{T-1}c^i_t(\\VEC x_t,\\VEC u^1_t,\\VEC u^2_t)\\Bigg],}\nwhere $c^i_t$ is a non-negative continuous function of its arguments for $i\\in\\{1,2\\}$ and $t\\in\\{1,\\ldots,T-1\\}$, and the expectation is taken with respect to the probability measure on the state and action processes induced by the choice of strategy profile $(\\mathbf g^1,  \\mathbf g^2)$.\n\n\nA strategy profile $(\\mathbf g^1,  \\mathbf g^2)$ is said to be a Nash equilibrium of the game if it satisfies the following two inequalities\n\\beqq{J^1(\\mathbf g^1,  \\mathbf g^2) \\leq J^1(\\tilde{\\mathbf g}^1,  \\mathbf g^2),\\quad\\text{ and }\\quad J^2(\\mathbf g^1,  \\mathbf g^2) \\leq J^2(\\mathbf g^1,  \\tilde{\\mathbf g}^2),} \nfor all admissible strategies $\\tilde{\\mathbf g}^1\\in\\ALP G^1_{1:T-1}$ and $\\tilde{\\mathbf g}^2\\in\\ALP G^2_{1:T-1}$. \n\nWe assume that the state evolution equations, observation equations, the noise statistics, cost functions of the controllers and the information structures of the controllers are part of common knowledge. The game thus defined is referred to as game {\\bf G1}.\n\n\n\\subsection{Assumption on Evolution of Information}\nAs noted above, each controller's information consists of common information and private information. We place the following condition on the evolution of common and private information of the controllers in game {\\bf G1}.  \n \\begin{assumption}\\label{assm:lqginfoevolution}\nThe common and private information evolve over time as follows:\n\\begin{enumerate}\n\\item The common information increases with time, that is, $(\\ALP E^1_t\\cap \\ALP E^2_t) \\subset (\\ALP E^1_{t+1}\\cap \\ALP E^2_{t+1})$ and $(\\ALP F^1_t\\cap \\ALP F^2_t) \\subset (\\ALP F^1_{t+1}\\cap \\ALP F^2_{t+1})$ for all $t\\in\\{1,\\ldots,T-1\\}$. Let $\\ALP D_{1,t+1}:= (\\ALP E^1_{t+1}\\cap \\ALP E^2_{t+1})\\setminus(\\ALP E^1_{t}\\cap \\ALP E^2_{t})$ and $\\ALP D_{2,t+1}:= (\\ALP F^1_{t+1}\\cap \\ALP F^2_{t+1})\\setminus(\\ALP F^1_{t}\\cap \\ALP F^2_{t})$. Define\n\\beq{\\label{eqn:commoninfo}\\ALP Z_{t+1} = \\prod_{(j,s)\\in\\ALP D_{1,t+1}} \\ALP Y^j_s\\times \\prod_{(j,s)\\in\\ALP D_{2,t+1}} \\ALP U^j_s.}\nThen, $\\VEC Z_{t+1}\\in\\ALP Z_{t+1} $ denotes the increment in common information from time $t$ to $t+1$ and we have \n\\begin{equation}\n\\VEC Z_{t+1} = \\zeta_{t+1}([\\VEC P^{1\\transpose}_t, \\VEC P^{2\\transpose}_t, \\VEC U^{1\\transpose}_t, \\VEC U^{2\\transpose}_t, \\VEC Y^{1\\transpose}_{t+1}, \\VEC Y^{2\\transpose}_{t+1}]^\\transpose),\\label{eq:lqgcommoninfo}\n\\end{equation}\nwhere $\\zeta_{t+1}$ is an appropriate projection function.\n\\item The private information evolves according to the equation\n\\begin{equation}\n\\VEC P^i_{t+1} = \\xi^i_{t+1}([\\VEC P^{i\\transpose}_{t}, \\VEC U^{i\\transpose}_t,  \\VEC Y^{i\\transpose}_{t+1}]^\\transpose). \\label{eq:lqgprivateinfo}\n\\end{equation}\nwhere $\\xi^i_{t+1}$ is an appropriate projection function.\n\\end{enumerate}\n\\end{assumption}\n\n\nWe now introduce a few notations in order to prove an important result. Let $\\ALP S_t := \\ALP X_t\\times\\ALP P^1_t\\times\\ALP P^2_t$ for $t\\in\\{1,\\ldots,T-1\\}$. Fix the strategy profile of the controllers as $(\\VEC g^1,\\VEC g^2)$. Let $\\Pi_t$ be the conditional measure on the space of state and private informations, $\\ALP S_t$,  given the common information $\\VEC C_t$ at time step $t$. Thus,\n\\beqq{\\Pi_t(d\\VEC s_t) = \\mathds{P}^{g^{1:2}_{1:t-1}}\\left\\{d\\VEC s_t\\Big|\\VEC C_t\\right\\},}\nwhere the superscript denotes the fact that the probability measure depends on the choice of control laws. The conditional probability measure $\\Pi_t$ is a $\\VEC C_t$-measurable random variable, whose realization, denoted by $\\pi_t$, depends on the realization  $\\VEC c_t$ of the common information. We now have the following result, which is a consequence of Assumption \\ref{assm:lqginfoevolution}.\n\nLet $\\Gamma^i_t$ be a random measurable function from $\\ALP P^i_t$ to $\\ALP U^i_t$ defined as $\\Gamma^i_t(\\cdot):=g^i_t(\\cdot,\\VEC C_t)$ for $i=1,2$ and $t\\in\\{1,\\ldots,T-1\\}$, where the realization of $\\Gamma^i_t$ is denoted by $\\gamma^i_t$ and it depends on the realization of the random variable $\\VEC C_t$. Thus, $g^i_t(\\VEC P^i_t,\\VEC c_t) = \\gamma^i_t(\\VEC P^i_t)$. We now have the following result about the evolution of conditional measure $\\Pi_t$.\n\\begin{lemma}\\label{lem:piupdate}\nFix the strategy profile $(\\VEC g^1,\\VEC g^2)\\in\\ALP G^1_{1:T-1}\\times\\ALP G^2_{1:T-1}$. If Assumption \\ref{assm:lqginfoevolution} holds, then \n\\beqq{\\Pi_{t+1} = F_t(\\Pi_t,\\Gamma^1_t,\\Gamma^2_t,\\VEC Z_{t+1}),}\nwhere $F_t$ is a fixed transformation which does not depend on the choice of control laws.\n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{app:piupdate}.\n\\end{proof}\n\n\\subsection{Strategy Independence of Beliefs}\nA crucial assumption, which forms the basis of the analysis in this paper, is the following.\n\\begin{assumption}[Strategy Independence of Beliefs] \\label{assm:lqgseparation}\nAt any time $t$ and for any realization of common information $\\VEC c_t$, the conditional probability measure $\\pi_t$ on the state $\\VEC X_t$ and the private information $(\\VEC P^1_t, \\VEC P^2_t)$ given the common information does not depend on the choice of control laws. In particular, if $\\VEC z_{t+1}$ is a realization of the increment in the common information at time step $t+1$, then $\\pi_{t+1}$ evolves according to the equation\n\\beq{\\label{eqn:pit1}\\pi_{t+1} = F_t(\\pi_t, \\VEC z_{t+1}),}\nwhere $F_t$ is a fixed transformation that does not depend on the control laws.\n\\end{assumption}\n\nAssumption \\ref{assm:lqgseparation} allows us to define the conditional belief $\\pi_t$ without specifying the control laws used. Another important consequence of Assumption \\ref{assm:lqgseparation} is that these conditional beliefs on the state and private information admit Gaussian density functions. We make this precise in the following lemma.\n\n\\begin{lemma}\\label{lem:pitgaussian}\nFor any time step $t\\in \\{1,\\ldots,T\\}$ and any realization of common information $\\VEC c_t$, the common information based conditional measure $\\pi_t$ admits a Gaussian density function.\n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{app:pitgaussian}.\n\\end{proof}\n\nWe henceforth call $\\pi_t$ as {\\it common information based conditional belief}. In the next subsection, we prove a result on the evolution of the mean and variance of the common information based conditional beliefs.\n\n\\subsection{Evolution of Conditional Beliefs}\nSince the common information based conditional belief $\\pi_t$ admits a Gaussian density at any time step $t$, $\\pi_t$ is completely characterized by its mean ${\\VEC m}_t$ and the covariance matrix $\\Sigma_t$. The conditional covariance of a collection of jointly Gaussian random variables is data independent. \nAssumption \\ref{assm:lqgseparation} allows us to derive the following result for game {\\bf G1}.\n\\begin{lemma}\\label{lemma:lqgmean}\nThe evolution of the conditional mean \\[\\VEC M_t := (\\VEC M^0_t, \\VEC M^1_t, \\VEC M^2_t) =(\\mathds{E}[\\VEC X_t| \\VEC C_t],\\mathds{E}[\\VEC P^1_t| \\VEC C_t],\\mathds{E}[\\VEC P^2_t| \\VEC C_t])\\]  of the density function of common information based conditional belief is given as\n\\begin{equation}\n{\\VEC M}_{t+1} = F^1_t({\\VEC M}_t,\\VEC Z_{t+1}), \\label{eq:lqgevolution1}\n\\end{equation}\nwhere $F^1_t$ is a fixed affine transformation that does not depend on the strategies of the controllers. The evolution of conditional covariance matrix $\\Sigma_t$ is given as\n\\begin{equation}\n\\Sigma_{t+1} = F^2_t(\\Sigma_t), \\label{eq:lqgevolution2}\n\\end{equation}\nwhere $F^2_t$ is a fixed  transformation that does not depend on the strategies of the controllers.\n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{app:lqgmean}.\n\\end{proof}\n\n\n\nExamples of several classes of games that satisfy Assumptions \\ref{assm:lqginfoevolution} and \\ref{assm:lqgseparation} are given in \\cite{nayyar2012a}. For example, if each controller acquires the realizations of the observations and the actions of the other controller with zero or one-step delay, then the corresponding game satisfies Assumptions \\ref{assm:lqginfoevolution} and \\ref{assm:lqgseparation}.\n\n\n\\section{Main Results}\\label{sec:main}\nFollowing the approach introduced in \\cite{nayyar2012a}, we now construct a new game \\textbf{G2} with two virtual players, where at every time step $t\\in\\{1,\\ldots,T-1\\}$, each virtual player observes the common information $\\VEC C_t$, but not the private information of the controllers. Since the common information is nested (by Assumption \\ref{assm:lqginfoevolution}), game {\\bf G2} is a game of perfect recall. This game is intricately related to game {\\bf G1}, and we exploit the symmetric information structure of game {\\bf G2} to devise a computational scheme to compute a Nash equilibrium of game {\\bf G1}. The steps taken to devise the scheme are as follows:\n\\begin{enumerate}\n\\item We formulate game {\\bf G2} in the next three subsections. Further, we show that the common information based conditional mean $\\VEC M_t$ is a Markov state of game {\\bf G2} at time $t$.\n\\item In Subsection \\ref{sub:relation}, we show that any Nash equilibrium of game {\\bf G2} can be used to obtain a Nash equilibrium of game {\\bf G1}, and vice-versa.\n\\item We focus on Markov perfect equilibria of game {\\bf G2} and provide a backward induction characterization of such equilibria. An equilibrium of game {\\bf G1} obtained from a Markov perfect equilibrium of game {\\bf G2} is called {\\it common information based Markov perfect equilibrium} of game {\\bf G1}.\n\\item We interpret the backward induction characterization of common information based Markov perfect equilibrium in terms of a sequence of one-stage Bayesian games.\n\\end{enumerate}\n\nWe now turn our attention to formulating game {\\bf G2}. At time step $t$ and for each realization $\\VEC c_t$ of the common information, virtual player $i$ selects a measurable function $\\gamma^i_t:\\ALP P^i_t\\rightarrow\\ALP U^i_t$. The action space of virtual player $i$ at time $t$ is denoted by $\\ALP A^i_t$, and it is defined as\n\\beq{\\label{eqn:alpAit}\\ALP A^i_t :=\\{\\gamma^i_t:\\ALP P^i_t\\rightarrow\\ALP U^i_t \\text{ such that }\\gamma^i_t \\text{ is a Borel measurable map}\\}.}\n\nWe call the actions taken by virtual players as ``prescriptions'' due to the following reason: After observing the common information $\\VEC c_t$ at time step $t$, virtual player $i\\in\\{1,2\\}$ computes the equilibrium prescription $\\gamma^i_t\\in\\ALP A^i_t$, and prescribes it to Controller $i$. The controllers evaluate the prescriptions based on the realizations of their private informations, to compute their actions at that time step.\n\n\n\n\\subsection{Admissible Strategies of Virtual Players in Game {\\bf G2}}\nA map $\\chi^i_t:\\ALP C_t\\rightarrow \\ALP A^i_t$ denotes the control law of virtual player $i\\in\\{1,2\\}$ at time step $t\\in\\{1,\\ldots,T-1\\}$. The control law $\\chi^i_t$ maps common information at time $t$ to a prescription, which itself maps private information of Controller $i$ at time $t$ to the control action of Controller $i$. Thus, a choice of $\\chi^i_t$ induces a map from $\\ALP C_t\\times\\ALP P^i_t$ to $\\ALP U^i_t$, which we denote by $\\chi^i_t(\\cdot)(\\cdot)$. We say $\\chi^i_t$ is admissible if\n\\beqq{\\chi^i_t(\\cdot)(\\cdot) \\quad  \\text{ is a Borel measurable function from } \\ALP C_t\\times \\ALP P^i_t \\text{ to } \\ALP U^i_t. }\nThe set of all such admissible control laws is denoted by $\\ALP H^i_t$, $i\\in\\{1,2\\}$ and $t\\in\\{1,\\ldots,T-1\\}$. The collection of control laws at all time steps of virtual player $i$ is called the strategy of that virtual player, and it is denoted by $\\chi^i :=\\{\\chi^i_1,\\ldots,\\chi^i_{T-1}\\}$. The space of all strategies of the virtual player $i$, denoted by $\\ALP H^i_{1:T-1}$, is called the strategy space of that virtual player. A strategy tuple $(\\chi^1,\\chi^2)$ is called the strategy profile of virtual players. \n\n\\begin{definition}\\label{def:var}\nFor $i\\in\\{1,2\\}$ and $t\\in\\{1,\\ldots,T-1\\}$, let $\\varrho^i_t:\\ALP H^i_t\\rightarrow\\ALP G^i_t$ be an operator that takes a function $\\chi^i_t:\\ALP C_t\\rightarrow \\ALP A^i_t$ as its input and returns a measurable function $g^i_t:\\ALP P^i_t\\times \\ALP C_t\\rightarrow\\ALP U^i_t$ as its output, that is $ g^i_t = \\varrho^i_t(\\chi^i_t)$, such that $g^i_t(\\VEC p^i_t,\\VEC c_t) := \\chi^i_t(\\VEC c_t)(\\VEC p^i_t)$ for all $\\VEC c_t\\in\\ALP C_t$ and $\\VEC p^i_t\\in\\ALP P^i_t$. For a collection of functions $\\chi^i :=\\{\\chi^i_1,\\ldots,\\chi^i_{T-1}\\}$, let $\\varrho^i(\\chi^i)$ be defined as the set $\\{\\varrho^i_1(\\chi^i_1),\\ldots,\\varrho^i_{T-1}(\\chi^i_{T-1})\\}$.\n\nSimilarly, we let $\\varsigma^i_t:\\ALP G^i_t\\rightarrow\\ALP H^i_t$ be the operator such that $\\varsigma^i_t\\circ \\varrho^i_t = \\text{id}_{\\ALP H^i_t}$ and $\\varrho^i_t\\circ \\varsigma^i_t = \\text{id}_{\\ALP G^i_t}$. Thus, for $g^i_t\\in\\ALP G^i_t$, if $\\chi^i_t=\\varsigma^i_t(g^i_t)$, then $\\chi^i_t(\\VEC c_t)(\\VEC p^i_t) := g^i_t(\\VEC p^i_t,\\VEC c_t)$ for all $\\VEC c_t\\in\\ALP C_t$ and $\\VEC p^i_t\\in\\ALP P^i_t$. Similar to the expression above, for a collection of functions $\\VEC g^i :=\\{g^i_1,\\ldots,g^i_{T-1}\\}$, let $\\varsigma^i(\\VEC g^i)$ be defined as the set $\\{\\varsigma^i_1(g^i_1),\\ldots,\\varsigma^i_{T-1}(g^i_{T-1})\\}$.{\\hfill $\\Box$}\n\\end{definition}\n\n\n\\subsection{Cost Functions for Virtual Players}\nThe cost functions of the virtual players are defined as follows: Fix a time step $t\\in\\{1,\\ldots,T-1\\}$ and a virtual player $i$. Let $\\pi$ denote a normal distribution on the space $\\ALP S_t = \\ALP X_t\\times\\ALP P^1_t\\times\\ALP P^2_t$ with mean $\\VEC m$ and variance $\\Sigma_t$, where $\\Sigma_t$ is given by the result in Lemma \\ref{lemma:lqgmean}, and let $(\\gamma^1,\\gamma^2)$ be a prescription pair chosen by the virtual players. Define the cost function $\\tilde{c}^i_t:\\ALP S_t\\times\\ALP A^1_t\\times\\ALP A^2_t\\rightarrow\\Re_+$ of virtual player $i$ at that time step $t\\in\\{1,\\ldots,T-1\\}$ to be \n\\beqq{\\tilde{c}^i_t(\\VEC m,\\gamma^1,\\gamma^2) = \\int_{\\ALP S_t} c^i_t(\\VEC x_t,\\gamma^1(\\VEC p^1_t),\\gamma^2(\\VEC p^2_t))\\pi(d\\VEC s_t),}\nwhere one can view $\\VEC x_t,\\VEC p^1_t$ and $\\VEC p^2_t$ as appropriate projections of the variable $\\VEC s_t$. We now define the cost function of the virtual players at the final time step. Let $\\pi$ be a Gaussian distribution with mean $\\VEC m$ and variance $\\Sigma_T$. The cost functions of the virtual players at the final time step is $\\tilde{c}^i_T(\\VEC m) = \\int_{\\ALP X_T} c^i_T(\\VEC x_T)\\pi(d\\VEC x_T)$. The total cost for virtual player $i$ is given by\n\\beqq{\\tilde J^i(\\chi^1,\\chi^2) = \\ex{\\tilde{c}^i_T(\\VEC M_T)+\\sum_{t=1}^{T-1}\\tilde{c}^i_t(\\VEC M_t,\\Gamma^1_t,\\Gamma^2_t)},} \nwhere the expectation is taken with respect to the probability measure induced on the mean of common information based conditional belief by the choice of strategies $(\\chi^1,\\chi^2)$. We have the following claim about the expected cost of game {\\bf G2} given a pair of strategy profiles of controllers in game {\\bf G1} and {\\it vice versa}.\n\n\\begin{lemma}\\label{lem:vpcost}\nLet $(\\VEC g^1,\\VEC g^2)\\in\\ALP G^1_{1:T-1}\\times\\ALP G^2_{1:T-1}$, and let $(\\chi^1,\\chi^2)$ be defined as $\\chi^i:=\\varsigma^i(\\VEC g^i), i=1,2$. Then, $J^i(\\VEC g^1,\\VEC g^2) = \\tilde J^i(\\chi^1,\\chi^2)$ for $i=1,2$.\n\nConversely, let $(\\chi^1,\\chi^2)\\in\\ALP H^1_{1:T-1}\\times\\ALP H^2_{1:T-1}$, and let $(\\VEC g^1,\\VEC g^2)$ be defined as $\\VEC g^i:=\\varrho^i(\\chi^i), i=1,2$. Then, $\\tilde J^i(\\chi^1,\\chi^2)=J^i(\\VEC g^1,\\VEC g^2)$ for $i=1,2$.\n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{app:vpcost}.\n\\end{proof}\n\n\n\\subsection{A Markov State of Game {\\bf G2}}\nRecall from Lemma \\ref{lemma:lqgmean} that given a realization $\\VEC c_t$ of common information, the common information based conditional belief $\\mathds{P}\\{d\\VEC s_t|\\VEC c_t\\}$ admits a Gaussian density function with mean $\\VEC m_t$ and variance $\\Sigma_t$. Our next result is that the mean $\\VEC M_t$ is a controlled Markov chain, and it is controlled by the actions taken (that is, the prescriptions chosen) by the virtual players.\n\n\\begin{lemma}\\label{lemma:lqgmarkov}\nThe process $\\{\\VEC M_t\\}_{t\\in\\{1,\\ldots,T\\}}$ is a controlled Markov process with the virtual players' prescriptions as the controlling actions. In particular, conditioned on the realization  $\\VEC m_t$ of  $\\VEC M_t$ and the prescriptions $(\\gamma^1_t, \\gamma^2_t)$ of the virtual players, the conditional mean $\\VEC M_{t+1}$ at the next time step is independent of the current common information, the past conditional means and past prescriptions. Equivalently, this fact is expressed as\n\\beqq{\\mathds{P}\\{\\VEC M_{t+1}\\in \\SF M_{t+1}|\\VEC c_t, \\VEC m_{1:t},\\gamma^{1:2}_{1:t}\\} = \\mathds{P}\\{\\VEC M_{t+1}\\in \\SF M_{t+1}|\\VEC m_{t},\\gamma^{1:2}_t\\}}\nfor all Borel sets $\\SF M_{t+1}\\subset \\ALP S_{t+1}$.\n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{app:lqgmarkov}.\n\\end{proof}\n\nIt should be noted that the update equation of the Markov process $\\VEC M_t$ is induced by the state dynamics, observation equations, and information structure of the controllers of game {\\bf G1}. The main differences between the structures of the two games {\\bf G1} and {\\bf G2} are summarized in the table below.\\\\\n\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabular}{|l|c|c|}\n\\hline\nAt time step $t\\in\\{1,\\ldots,T-1\\}$ & Game {\\bf G1} & Game {\\bf G2}\\\\\\hline\nState of the game& $\\VEC X_t\\in\\ALP X_t$ & $\\VEC M_t\\in\\ALP S_t$\\\\\\hline\nAction of Player $i$ & $\\VEC U^i_t\\in\\ALP U^i_t$ & $\\gamma^i_t\\in\\ALP A^i_t$\\\\\\hline\nInformation of Player $i$ & $\\VEC I^i_t\\in\\ALP I^i_t$ & $\\VEC C_t\\in\\ALP C_t$\\\\\\hline\nCost function of Player $i$ & $c^i_t$ & $\\tilde c^i_t$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{10pt}\n\nSince both virtual players observe the same information (the common information between the controllers) and the common information always increases by Assumption \\ref{assm:lqginfoevolution}, game {\\bf G2} between virtual players is a game of symmetric information with perfect recall. \n\n\\subsection{Relation between Games {\\bf G1} and {\\bf G2}}\\label{sub:relation}\nIn the next theorem, we show that any Nash equilibrium of game {\\bf G1} can be used to compute a Nash equilibrium for game {\\bf G2} and vice versa.\n\\begin{theorem}\\label{thm:lqgequiv}\nLet $(\\chi^{1\\star}, \\chi^{2\\star})$ be a Nash equilibrium strategy profile of game \\textbf{G2}. Then, the strategy profile $(\\mathbf g^{1\\star}, \\mathbf g^{2\\star})$ for game \\textbf{G1}, defined as $\\VEC g^{i\\star} =\\varrho^i(\\chi^{i\\star}), i=1,2$, forms a Nash equilibrium strategy profile of game \\textbf{G1}. Conversely, if $(\\mathbf g^{1\\star}, \\mathbf g^{2\\star})$ is a Nash equilibrium strategy profile of game {\\bf G1}, then the strategy profile $(\\chi^{1\\star}, \\chi^{2\\star})$, defined by $\\chi^{i\\star}:=\\varsigma^i(\\VEC g^{i\\star}), i=1,2,$ is a Nash equilibrium strategy profile for game {\\bf G2}.\n\\end{theorem}\n\\begin{proof}\nSee Appendix \\ref{app:lqgequiv}.\n\\end{proof}\n\nIn light of the theorem above, we want to compute a Nash equilibrium of game {\\bf G2}, and then project the solution back to the original game using the operators $\\varrho^1$ and $\\varrho^2$ as introduced in Definition \\ref{def:var}.\n\nSince at any time step $t$, both virtual players observe the common information $\\VEC C_t$, the virtual players can compute the mean $\\VEC M_t$ of the common information based conditional belief. Since the mean $\\VEC M_t$ is a Markov state of game {\\bf G2} and both virtual players know its realization, game {\\bf G2} is a game of perfect state information. Also note that in game {\\bf G2}, the cost functions of the virtual players are stagewise-additive. A natural solution concept to compute the Nash equilibrium of a game of perfect information with stagewise-additive cost function is Markov perfect equilibrium \\cite{fudenberg1991}. We define the Markov perfect equilibrium of {\\bf G2} in the next subsection and prove the main result of the section.\n\n\n\\subsection{Markov Perfect Equilibrium of Game {\\bf G2}}\nFix virtual player $i$'s control laws $\\chi^i_{1:T-1}$ such that the prescription at time step $t$ is only a function of $\\VEC M_t$, say $\\chi^i_t(\\VEC C_t) = \\psi^i_t(\\VEC M_t)$ for some function $\\psi^i_t:\\ALP S_t\\rightarrow\\ALP A^i_t$, such that $\\psi^i_t(\\cdot)(\\cdot)$ is a measurable function from $\\ALP S_t\\times\\ALP P^i_t$ to $\\ALP U^i_t$ at all time step $t\\in\\{1,\\ldots,T-1\\}$. Let us use $\\bar{\\ALP H}^i_t$ to denote the set of all such maps $\\psi^i_t$, and note that $\\bar{\\ALP H}^i_t\\subset\\ALP H^i_t$. Then, virtual player $j$'s, $j\\neq i$, (one-person) optimization problem is to minimize its stagewise additive cost functional that depends on $\\VEC M_t$ and on virtual player $i$'s fixed strategy. Thus, virtual player $j$ needs to solve a finite horizon Markov decision problem with state space $\\ALP S_t$ and action space $\\ALP A^{j}_t$ at time step $t$. This is made precise in our next result.\n\n\n\\begin{lemma}\\label{lemma:lqginfostatelemma}\nConsider game {\\bf G2} among virtual players. Assume that virtual player $i$ is using the strategy $\\{\\psi^i_1,\\ldots,\\psi^i_{T-1}\\}\\in\\bar{\\ALP H}^i_{1:T-1}$, that is, virtual player $i$ selects the prescriptions at time step $t$ only as a function of the mean $\\VEC M_t$ of the common information based conditional belief $\\Pi_t$: \n\\[ \\Gamma^i_t = \\psi^i_t(\\VEC M_t),\\qquad t\\in\\{1,\\ldots, T-1\\}.\\]\nThen, for the fixed strategy of virtual player $i$, virtual player $j$'s ($j\\neq i, j\\in\\{1,2\\}$) one-sided optimization problem is a finite horizon Markov decision problem with state $\\VEC M_t$, control action $\\gamma^j_t$, and cost as $\\tilde{c}^j_t(\\VEC M_t,\\gamma^j_t,\\psi^i_t(\\VEC M_t))$ at time step $t\\in\\{1,\\ldots,T-1\\}$ and terminal cost $\\tilde{c}^j_T(\\VEC M_T)$.\n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{app:lqginfostatelemma}.\n\\end{proof}\n\nNote that game {\\bf G2} is a dynamic game of perfect information and perfect recall. A Markov strategy of a virtual player is defined as a collection of control laws of that virtual player at all time steps such that the control law at time step $t$ is a measurable map of common information based conditional mean (state of game {\\bf G2}) to its action space at that time step. Lemma \\ref{lemma:lqginfostatelemma} states that if one virtual player sticks to Markov strategy, then the other virtual player's one-sided optimization problem is a finite horizon Markov decision problem. Under certain assumptions on the cost functions of the virtual players\\footnote{See, for example, \\cite[Section 3.3]{lerma1996} for a set of such assumptions.}, there exists a Markov strategy of the other virtual player that achieves the minimum in its Markov decision problem. Thus, there is no incentive for the other virtual player to search for optimal strategies outside the class of Markov strategies. This is an important \nobservation for game {\\bf G2}, because one can define a refinement concept for Nash equilibrium, called Markov perfect equilibrium \\cite{fudenberg1991}, for game {\\bf G2}.\n\n\n\n\\begin{definition}\nA strategy profile $(\\psi^{1\\star}_{1:T-1},\\psi^{2\\star}_{1:T-1})\\in \\ALP H^1_{1:T-1}\\times\\ALP H^2_{1:T-1}$ is said to be a Markov perfect equilibrium \\cite{fudenberg1991} of game \\textbf{G2} if (i) at each time $t$, the control laws of the virtual players at time step $t$ are functions of the mean of the common information based conditional belief $\\VEC M_t$, that is, $\\psi^i_t\\in\\bar{\\ALP H}^i_t$, and (ii) for all time steps $t\\in\\{1,\\ldots,T-1\\}$, the strategy profiles $(\\psi^{1\\star}_{t:T-1},\\psi^{2\\star}_{t:T-1})$ form a Nash equilibrium for the sub-game starting at time step $t$ of game {\\bf G2}.\n\\end{definition}\n\nIt should be noted that Markov perfect equilibrium is a refinement concept for Nash equilibria of games in which players make perfect state observations. In game {\\bf G2} among virtual players, a strategy profile that is {\\it not} a Markov perfect equilibrium either depends on the common information (and not just on the mean $\\VEC M_t$), or is not a Nash equilibrium of every sub-game in game {\\bf G2}, or both. We reemphasize this point later in Section \\ref{sec:information}.  \n\nGiven a Markov perfect equilibrium of \\textbf{G2}, we can construct a corresponding Nash equilibrium of game \\textbf{G1} using Theorem~\\ref{thm:lqgequiv}. We refer to the class of Nash equilibria of \\textbf{G1} that can be constructed from the Markov perfect equilibria of \\textbf{G2} as the \\emph{common information based Markov perfect equilibria} of game \\textbf{G1}.\n\\begin{definition}\nIf $(\\psi^{1\\star}_{1:T-1},\\psi^{2\\star}_{1:T-1})$ is a Markov perfect equilibrium of game \\textbf{G2}, then the strategy profile $(\\VEC g^{1\\star}, \\VEC g^{2\\star})$ of the form $\\VEC g^{i\\star} = \\varrho^i(\\psi^{i\\star}_{1:T-1}), i=1,2,$  is called \\emph{common information based Markov perfect equilibrium} of game \\textbf{G1}.{\\hfill$\\Box$}\n\\end{definition}\n\nA similar concept was introduced for finite games with asymmetric information in our earlier work \\cite{nayyar2012a}.\n\n\\subsection{Computation of Markov Perfect Equilibrium of Game {\\bf G2}}\\label{sub:MPEG2}\nIn this subsection, we characterize Markov perfect equilibrium of game \\textbf{G2} using value functions that depend on the mean of the common information based conditional belief. \n\n\n\\begin{theorem}\\label{thm:mpevirtual}\nConsider a strategy pair $(\\psi^{1\\star}_{1:T-1},\\psi^{2\\star}_{1:T-1})\\in\\bar{\\ALP H}^1_{1:T-1}\\times\\bar{\\ALP H}^2_{1:T-1}$. Define functions $V^i_t:\\ALP X_t\\times\\ALP P^1_{t}\\times\\ALP P^2_{t}\\rightarrow\\Re$, called expected value functions of Controller $i$ at time $t$, as follows:\n\\begin{enumerate}\n\\item For each possible realization $\\VEC m = (\\VEC m^0,\\VEC m^1,\\VEC m^2)$ of $\\VEC M_{T}$, define the value functions:\n\\beq{\\label{eqn:viT} V^i_{T}(\\VEC m) := \\tilde c^i_T(\\VEC m) = \\mathds{E}[c^i_T(\\VEC X_T)|\\VEC M_T = \\VEC m]\\qquad i\\in\\{1,2\\}.}\n\\item For $t=T-1,\\ldots,1$, and for each possible realization $\\VEC m$ of $\\VEC M_{t}$, define  the value functions:\n\\beq{V^i_{t}(\\VEC m) &:=& \\min_{\\tilde\\gamma^i_t\\in\\ALP A^i_t} \\mathds{E}\\Big[ \\tilde c^i_t(\\VEC M_t,\\gamma^1_t,\\gamma^2_t) +V^i_{t+1}(F^1_t(\\VEC M_t, \\VEC Z_{t+1})) \\nonumber\\\\\n& & \\qquad\\qquad\\Big|\\VEC M_{t}=\\VEC m, \\gamma^i_t =\\tilde\\gamma^i_t , \\gamma^{-i}_t = \\psi^{-i\\star}_{t}(\\VEC m)\\Big]\\qquad i\\in\\{1,2\\},\\qquad\\label{eqn:vit}}\nassuming that the minimum exists in the equation above. Then, a necessary and sufficient condition for $(\\psi^{1\\star}_{1:T-1},\\psi^{2\\star}_{1:T-1})$ to be a Markov perfect equilibrium of \\textbf{G2} is that for every time step $t\\in\\{1,\\ldots,T-1\\}$, $i\\in\\{1,2\\}$ and for every realization $\\VEC m$ of $\\VEC M_t$,\n\\beq{\\label{eqn:psiitm}\\psi^{i\\star}_t(\\VEC m) \\in \\underset{\\tilde\\gamma^i_t\\in \\ALP A^i_t}{\\arg\\min} \\;\\;\\mathds{E}\\Big[ \\tilde c^i_t(\\VEC m,\\tilde\\gamma^i_t, \\psi^{-i\\star}_{t}(\\VEC m)) +V^i_{t+1}(F^1_t(\\VEC m, \\VEC Z_{t+1}))\\Big].}\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nSee Appendix \\ref{app:mpevirtual}.\n\\end{proof}\n\nTo show that the sub-game admits a Nash equilibrium at time step $t$ requires a fixed-point argument, which means that the reaction curves of the virtual players intersect in the product of their strategy spaces $\\bar{\\ALP H}^1_t\\times\\bar{\\ALP H}^2_t$ \\cite{Basarbook}. In the next section, we show that if the cost functions of the players are quadratic functions of their arguments, then under certain conditions on the cost functions, the reaction curves of the virtual players intersect at a unique point. Thus, under those assumptions, a unique common information based Markov perfect equilibrium exists in LQG games with asymmetric information.\n\n\\begin{remark}\\label{rem:multiequilibria}\nAs stated earlier, Markov perfect equilibrium is only a subclass of Nash equilibria of game {\\bf G2}. Game {\\bf G2} (and equivalently, the corresponding game {\\bf G1}) may have several other Nash equilibria besides Markov perfect equilibrium. An example of a two-player two-stage game of asymmetric information in which there is a continuum of Nash equilibria is presented in Section \\ref{sec:information}.{\\hfill $\\Box$}\n\\end{remark}\n\n\\subsection{One-stage Bayesian Games}\\label{sub:stagebayesian}\nAt any time step $t$, let $\\VEC m_t$ and $\\VEC x_t$, respectively, be realizations of the common information based conditional mean and the state. Let us rewrite the expressions of the expected cost-to-go functions in \\eqref{eqn:vit} as\n\\beq{\\label{eqn:hatcit}\\bar{c}^i_t(\\VEC m_t;\\VEC x_t,\\VEC u^1_t,\\VEC u^2_t):=c^i_t(\\VEC x_t,\\VEC u^1_t,\\VEC u^2_t) +\\ex{V^i_{t+1}(F^1_t(\\VEC m_t, \\VEC Z_{t+1}))\\Big| \\VEC x_t,\\VEC u^1_t,\\VEC u^2_t}.}\nThese are the cost-to-go functions for the two controllers in game {\\bf G1} if both controllers stick to the common information based Markov perfect equilibrium for all time step $s>t$. For a realization $\\VEC m_t$, Controller $i$ chooses a map $\\gamma^i_t:\\ALP P^i_t\\rightarrow\\ALP U^i_t$. We assume that the probability measure on $\\ALP X_t\\times\\ALP P^1_t\\times\\ALP P^2_t$ admits a Gaussian density function with mean $\\VEC m_t$ and variance $\\Sigma_t$. One can notice that this is precisely the setup of a Bayesian game. Therefore, we say that the game between Controllers $1$ and $2$, with cost functions $\\bar{c}^1_t(\\VEC m_t;\\cdot)$ and $\\bar{c}^2_t(\\VEC m_t;\\cdot)$, respectively, is the {\\it one-stage Bayesian game at time step $t$ with mean $\\VEC m_t$}. \n\n\n\\textbf{An algorithm to compute common information based Markov perfect equilibrium:}\n\nWe can now describe a backward induction process to find a common information based Markov perfect equilibrium of game \\textbf{G1} using a sequence of one-stage Bayesian games. We proceed as follows: \\\\\n\\underline{\\textbf{Algorithm 1:}}\n\\begin{enumerate}\n\\item At the terminal time $T-1$, for each realization $\\VEC m$ of the common information based conditional mean at time $T-1$, we define a one-stage Bayesian game  $SG_{T-1}(\\VEC m)$ where \n\\begin{enumerate}\n\\item The probability distribution on $(\\VEC X_{T-1}, \\VEC P^1_{T-1}, \\VEC P^2_{T-1})$, denoted by $\\pi$, is a Gaussian distribution with mean $\\VEC m$ and covariance $\\Sigma_{T-1}$.\n\\item  Agent\\footnote{Agent $i$ can be thought to be the same as Controller $i$. We use a different name here in order to maintain the distinction between games \\textbf{G1} and $SG_{T-1}(\\VEC m)$.} $i$  observes $\\VEC P^i_{T-1}$ and chooses action $\\VEC U^i_{T-1}$, $i=1,2$. \n\\item  Agent $i$'s cost is $\\bar{c}^i_{T-1}(\\VEC m;\\VEC X_{T-1},\\VEC U^1_{T-1},\\VEC U^2_{T-1})$, $i=1,2$.\n\\end{enumerate}\nA Bayesian Nash equilibrium\\footnote{See \\cite{fudenberg1991, Osborne, Myerson_gametheory} for a definition of Bayesian Nash equilibrium.} of this game is a pair of strategies $(\\gamma^{1*},\\gamma^{2*})$, where $\\gamma^{i*}:\\ALP P^i_{T-1}\\rightarrow\\ALP U^i_{T-1}$ is a measurable function such that for any realization $\\VEC p^i\\in\\ALP P^i_{T-1}$, $\\gamma^{i*}(\\VEC p^i)$ is a solution of the minimization problem \n\\[ \\min_{\\VEC u^i} \\mathds{E}^{\\pi}[\\bar{c}^i_{T-1}(\\VEC m;\\VEC X_{T-1},\\VEC u^i, \\gamma^{j*}(\\VEC P^{j}_{T-1}))|\\VEC P^i_{T-1} = \\VEC p^i ], \\]\nwhere $j \\neq i$ and the superscript $\\pi$ denotes that the expectation is with respect to the distribution $\\pi$. If a Bayesian Nash equilibrium $(\\gamma^{1*},\\gamma^{2*})$ of $SG_{T-1}(\\VEC m)$ exists, denote the corresponding expected equilibrium costs as $V^i_{T-1}(\\VEC m), i=1,2$, and define $\\psi^i_T(\\VEC m) := \\gamma^{i*}$, $i=1,2$.\n\n\\item At time $t < T-1$, for each realization $\\VEC m$ of the common information based conditional mean at time $t$, we define the one-stage Bayesian   game $SG_t(\\VEC m)$ where\n\\begin{enumerate}\n\\item The probability distribution on $(\\VEC X_t, \\VEC P^1_t, \\VEC P^2_t)$, denoted by $\\pi$, admits a Gaussian density function with mean $\\VEC m$ and covariance $\\Sigma_{t}$.\n\\item  Agent $i$ observes $\\VEC P^i_t$ and chooses action $\\VEC U^i_t$, $i=1,2$. \n\\item  Agent $i$'s cost is $\\bar{c}^i_t(\\VEC m;\\VEC X_t,\\VEC U^1_t,\\VEC U^2_t)$, $i=1,2$. \n\\end{enumerate}\nA Bayesian Nash equilibrium of this game is a pair of strategies $(\\gamma^{1*},\\gamma^{2*})$, where $\\gamma^{i*}:\\ALP P^i_{T-1}\\rightarrow\\ALP U^i_{T-1}$ is a measurable function such that for any realization $\\VEC p^i\\in\\ALP P^i_t$, $\\gamma^{i*}(\\VEC p^i)$ is a solution of the minimization problem \n\\[ \\min_{\\VEC u^i} \\mathds{E}^{\\pi}[\\bar{c}^i_t(\\VEC m;\\VEC X_t,\\VEC u^i, \\gamma^{j}(\\VEC P^{j}_t)) |\\VEC P^i_t = \\VEC p^i ], \\]\nwhere $j \\neq i, i,j=1,2,$ when control actions $\\VEC U^i_t =\\VEC u^i$ and $\\VEC U^j_t = \\gamma^{j}(\\VEC P^{j}_t)$ are used. The expectation is taken with respect to the Gaussian distribution with mean $\\VEC m$ and covariance $\\Sigma_t$. If a Bayesian Nash equilibrium $(\\gamma^{1*},\\gamma^{2*})$ of $SG_t(\\VEC m)$ exists, denote the corresponding expected equilibrium costs as $V^i_t(\\VEC m), i=1,2$ and define $\\psi^i_t(\\VEC m) := \\gamma^{i*}$, $i=1,2$.\n\\end{enumerate}\n\n\n\\begin{theorem}\\label{thm:backward_ind}\nThe strategies $\\boldsymbol \\psi^i = (\\psi^i_1,\\psi^i_2,\\ldots,\\psi^i_{T-1})$, $i=1,2,$ defined by the backward induction process described in Algorithm 1 form a Markov perfect equilibrium of game \\textbf{G2}. Consequently, strategies $\\VEC g^1$ and $ \\VEC g^2$ defined as\n\\[ g^i_t(\\cdot, \\VEC c_t) := \\psi^i_t(\\VEC m_t), \\quad i\\in\\{1,2\\},\\:t\\in\\{1,\\ldots,T-1\\} \\]\nform a common information based Markov perfect equilibrium of game \\textbf{G1}.\n\\end{theorem}\n\\begin{proof}\nTo prove the result, we just need to observe that the strategies defined by the backward induction procedure of Algorithm 1  satisfy the  conditions of Theorem \\ref{thm:mpevirtual} and hence form a Markov perfect equilibrium of game \\textbf{G2}.\n\\end{proof}\n\n\nIn the next section, we consider LQG games, in which the cost functions of the controllers at any time step are quadratic in the state and actions of the controllers. Under certain sufficient conditions on the cost functions of the controllers, we prove that the one-stage Bayesian game at any time $t\\in\\{1,\\ldots,T-1\\}$ with any mean $\\VEC m_t\\in\\ALP S_t$ admits a unique Bayesian Nash equilibrium. We follow the steps of the algorithm above to prove that every LQG game with cost functions satisfying certain conditions admits a unique common information based Markov perfect equilibrium.\n\n\\section{Game with Quadratic Cost Functions}\\label{sec:lqggames}\nLet us now consider the special class of games where the stagewise cost functions $c^i_T$ and $c^i_t$ are quadratic functions of their arguments:\n\\beqq{c^i_T(\\VEC X_{T}) = \\VEC X_{T}^\\transpose R^i_{11} \\VEC X_{T}, & & c^i_t(\\VEC X_t,\\VEC U^1_t,\\VEC U^2_t) = \\bmat{\\VEC X_t^\\transpose , \\VEC U^{1\\transpose}_t, \\VEC U^{2\\transpose}_t} R^i \\bmat{\\VEC X_t \\\\ \\VEC U^1_t\\\\ \\VEC U^2_t}  \\\\\n\\quad\\text{ where } R^i&:=&\\bmat{R^i_{11} & R^i_{12} & R^i_{13}\\\\R_{12}^{i\\transpose} & R^i_{22} & R^i_{23}\\\\R_{13}^{i\\transpose} & R_{23}^{i\\transpose} & R^i_{33}},}\n$R^i_{11}\\geq 0$ and $R^i_{ii}>0$ for $i\\in\\{1,2\\}$. We refer to Gaussian games in which the cost functions are of the form above as dynamic LQG games.\n\nBefore we analyze the dynamic LQG game, we first formulate and compute the Nash equilibrium of a static auxiliary (Bayesian) game in the next subsection. We use the result of this auxiliary game to compute the Nash equilibrium strategies of the dynamic game. One of the main results of this section is that any LQG game that satisfies a certain assumption on the cost functions in addition to Assumptions \\ref{assm:lqginfoevolution} and \\ref{assm:lqgseparation} admits a unique common information based Markov perfect equilibrium in the class of all Borel measurable strategy profiles of the controllers. We prove this in two steps:\n\\begin{enumerate}\n\\item The first step consists of computing a Bayesian Nash equilibrium of a particular two-player static game with asymmetric information. This is done in Subsection \\ref{sec:basargame}.\n\\item We then exploit the uniqueness of Nash equilibrium, the structure of the Nash equilibrium strategies of the controllers, and the expected equilibrium costs to the controllers to obtain the main result for LQG games in Subsection \\ref{sub:generallqg}. \n\\end{enumerate}\n\n\n\\subsection{An Auxiliary Game, {\\bf AG1}}\\label{sec:basargame}\n\nThe static Bayesian game is described as follows: $\\VSF X, \\VSF Y^1, \\VSF Y^2$ are jointly Gaussian random vectors such that $\\VSF Y^i = H^i \\VSF X$ for some matrix $H^i$ of appropriate dimensions. The mean and covariance of the three-tuple $\\VSF X, \\VSF Y^1, \\VSF Y^2$ are given by\n\\[ \\VSF m = \\bmat{ \\VSF m_{x}  \\\\ \\VSF m_{y^1} \\\\\\VSF m_{y^2}} \\quad \\Sigma =\\bmat{\\Sigma_{xx} & \\Sigma_{xy^1} & \\Sigma_{xy^2} \\\\\\Sigma_{y^1x} & \\Sigma_{y^1y^1} & \\Sigma_{y^1y^2} \\\\\\Sigma_{y^2x} & \\Sigma_{y^2y^1} & \\Sigma_{y^2y^2} },  \\]\n\\beq{\\label{eqn:sigmaij}\\text{where}\\qquad \\Sigma_{y^iy^j} = \\Sigma_{y^iy^i}^{\\frac{1}{2}}\\Sigma_{y^jy^j}^{\\frac{1}{2}\\transpose},\\qquad \\Sigma_{y^ix} = \\Sigma_{y^iy^i}^{\\frac{1}{2}}\\Sigma_{xx}^{\\frac{1}{2}\\transpose} \\quad\\text{for } i,j=1,2.}\nSince $\\VSF X, \\VSF Y^1, \\VSF Y^2$ are jointly Gaussian random variables, the conditional expectations $\\mathds{E}[\\VSF X|\\VSF Y^i]$ and $\\mathds{E}[\\VSF Y^{-i}|\\VSF Y^i] $ are  affine functions of $\\VSF Y^i$, given by\n\\beq{\\mathds{E}[\\VSF X|\\VSF Y^i] &=& \\VSF m_x + \\Sigma_{xy^i}\\Sigma_{y^iy^i}^{-1}(\\VSF Y^i-\\VSF m_{y^i}),\\\\\n\\mathds{E}[\\VSF Y^{-i}|\\VSF Y^i] &=& \\VSF m_{y^{-i}} + \\Sigma_{y^{-i}y^i}\\Sigma_{y^iy^i}^{-1}(\\VSF Y^i-\\VSF m_{y^i}),}\nwhere $\\Sigma_{y^iy^i}^{-1}$ is the generalized inverse (pseudo inverse) of $\\Sigma_{y^iy^i}$ \\cite{catlin1989} for $i=1,2$.\n\nThe cost functions are\n\\beq{c^1(\\VSF X,\\VSF U^1,\\VSF U^2) &=& \\bmat{\\VSF X^\\transpose,\\VSF U^{1\\transpose},\\VSF U^{2\\transpose}}C\\bmat{\\VSF X \\\\\\VSF U^1\\\\\\VSF U^2} + 2\\bmat{d_1, d_2, d_3}\\bmat{\\VSF X \\\\\\VSF U^1\\\\\\VSF U^2}+ r^1 \\label{eqn:j1aux},\\\\\n c^2(\\VSF X,\\VSF U^1,\\VSF U^2) &=& \\bmat{\\VSF X^\\transpose,\\VSF U^{1\\transpose},\\VSF U^{2\\transpose}}E\\bmat{\\VSF X \\\\\\VSF U^1\\\\\\VSF U^2} + 2\\bmat{f_1, f_2, f_3}\\bmat{\\VSF X \\\\\\VSF U^1\\\\\\VSF U^2} + r^2,\\label{eqn:j2aux}\\\\\n\\text{where}\\quad C &=& \\bmat{C_{11} &C_{12} &C_{13}\\\\C_{12}^{\\transpose} &C_{22} &C_{23}\\\\C_{13}^{\\transpose} &C_{23}^{\\transpose} & C_{33}} \\quad\\text{ and }\\quad E = \\bmat{E_{11} &E_{12} &E_{13}\\\\E_{12}^{\\transpose} &E_{22} &E_{23}\\\\E_{13}^{\\transpose} &E_{23}^{\\transpose} & E_{33}}.\\nonumber}\nwith $C\\geq 0, E\\geq 0$, $C_{22}>0, E_{33}>0$, $C_{ij},E_{ij}$ are matrices, $d_i,f_i$ are row vectors of appropriate dimensions and $r_1,r_2$ are scalar constants. \n\nController $i$ observes $\\VSF Y^i, i=1,2$ and selects $\\VSF U^i$ according to a decision rule $g^i$, that is, $\\VSF U^i = g^i(\\VSF Y^i)$, where $g^i$ is a measurable function of $Y^i$ satisfying $\\mathds{E}[g^{i\\transpose}(\\VSF Y^{i})g^{i}(\\VSF Y^{i})] < \\infty$. Let the space of all such measurable functions $g^i$ be denoted by $\\ALP A\\ALP G^i$. This game is referred to as game {\\bf AG1}$(\\VSF m,\\Sigma,c^1,c^2)$. We make the following assumption on the cost functions of the controllers in {\\bf AG1}$(\\VSF m,\\Sigma,c^1,c^2)$.\n\n\\begin{assumption}\\label{assm:eigcost}\nFor any square matrix $A$, let $\\bar{\\lambda}(A)$ denotes the positive square root of the maximum eigenvalue of $A^{\\transpose}A$. For the matrix tuple $(C,E)$, define $K_1 = C_{22}^{-1}C_{23}E_{33}^{-1}E_{23}^{\\transpose}$ and $K_2 = E_{33}^{-1}E_{23}^{\\transpose}C_{22}^{-1}C_{23}$. Let $\\ALP K_i$ be the space of all matrices that are similar to $K_i,\\: i=1,2$, that is, $\\tilde K\\in\\ALP K_i$ implies there exists a square invertible matrix $L$ of appropriate dimensions such that $\\tilde K = LK_iL^{-1}$. There exists an $i_0\\in\\{1,2\\}$ and a matrix $K\\in \\ALP K_{i_0}$ such that $\\bar{\\lambda}(K)<1$.{\\hfill$\\Box$}\n\\end{assumption}\n\nIn the next lemma, which builds on and follows from the earlier results in \\cite{Basar1975,Basarmulti}, we show that the Nash equilibrium of the auxiliary game {\\bf AG1}$(\\VSF m,\\Sigma,c^1,c^2)$ that satisfies Assumption \\ref{assm:eigcost} exists in the space $\\ALP A\\ALP G^1\\times\\ALP A\\ALP G^2$, is unique, and is affine in the information of the controllers.\n\n \n\\begin{lemma}\\label{lem:basargame}\nThe following statements hold:\n\\begin{enumerate} \n\\item For the 1-stage game \\textbf{AG1}, a pair of decision rules $g^{1\\star}, g^{2\\star}$ is a Nash equilibrium if and only if they simultaneously satisfy the following two equations,\n\\beqq{g^{1\\star}(\\VSF Y^1) &=& -C_{22}^{-1}d_2^{\\transpose} - C_{22}^{-1}C_{12}^\\transpose\\mathds{E}[\\VSF X|\\VSF Y^1] - C_{22}^{-1}C_{23}\\mathds{E}[g^{2\\star}(\\VSF Y^2)|\\VSF Y^1], \\nonumber \\\\\ng^{2\\star}(\\VSF Y^2) &=& -E_{33}^{-1}f_3^{\\transpose} - E_{33}^{-1}E_{13}^\\transpose\\mathds{E}[\\VSF X|\\VSF Y^2] - E_{33}^{-1}E_{23}^{\\transpose}\\mathds{E}[g^{1\\star}(\\VSF Y^1)|\\VSF Y^2].}\n\n\\item If the matrices $(C,E)$ in the cost functions of game {\\bf AG1}$(\\VSF m,\\Sigma,c^1,c^2)$ satisfy Assumption \\ref{assm:eigcost}, then the game has a unique Nash equilibrium in the class of all Borel measurable strategies $\\ALP A\\ALP G^1\\times\\ALP A\\ALP G^2$, given as\n\\beq{\\label{eqn:gistaraux} g^{i\\star}(\\VSF Y^i) =  T^i(\\VSF Y^i-\\VSF m_{y^i}) + b^i,}\nwhere $b^1,b^2$ are \nsolutions of the following pair of equations\n\\beqq{b^1 &=& -C_{22}^{-1}[d_2^{\\transpose}+C_{12}\\VSF m_x+C_{23}b^2], \\\\\nb^2 &=& -E_{33}^{-1}[f_3^{\\transpose}+E_{13}\\VSF m_x +E_{23}^{\\transpose}b^1 ],}\nand are of the form\n\\beq{\\label{eqn:b1b2}b^1 &=& l^1+L^1 \\VSF m_x,\\qquad  b^2 = l^2+L^2\\VSF m_x,}\nand $T^1, T^2$ are solutions of  the following pair of equations\n\\beq{\\label{eqn:t1}T^1 &=& -C_{22}^{-1}[C_{12}^\\transpose\\Sigma_{xy^1}\\Sigma_{y^1y^1}^{-1}+ C_{23}T^2\\Sigma_{y^2y^1}\\Sigma_{y^1y^1}^{-1}], \\\\\n\\label{eqn:t2}T^2 &=& -E_{33}^{-1}[E_{13}^\\transpose\\Sigma_{xy^2}\\Sigma_{y^2y^2}^{-1} +E_{23}^{\\transpose}T^1\\Sigma_{y^1y^2} \\Sigma_{y^2y^2}^{-1}].}\nHere, $l^i $ and $L^i$ are independent of $\\VSF m$ for both $i=1,2$. \n\n\\item The expected costs to the controllers when they play according to Nash equilibrium $(g^{1\\star},g^{2\\star})$ are\n\\beq{\\label{eqn:auxeqcost}\\mathds{E}[c^i(\\VSF X,g^{1\\star}(\\VSF Y^1),g^{2\\star}(\\VSF Y^2))] = \\VSF m^{\\transpose} \\Phi^i {\\VSF m} + \\Xi^{i}\\VSF m + \\Upsilon^i,}\nwhere the matrices $\\Phi^i$, $\\Xi^i$ and $\\Upsilon^i$ for $i=1,2$ are defined by\n\\beqq{&\\tilde{L} := \\bmat{0 &0 & 0\\\\ L^1 & -T^1 & 0\\\\ L^2 & 0 & -T^2}, \\quad \\tilde{T} := \\bmat{I &0 & 0\\\\ 0 & T^1 & 0\\\\0& 0 & T^2}, \\quad\n\\tilde{l} := \\bmat{0 \\\\ l^1 \\\\l^2},\\\\\n&\\Phi^1 = (\\tilde{T}+\\tilde{L})^{\\transpose}C(\\tilde{T}+\\tilde{L}), \\qquad \\Xi^1 = 2\\tilde{l}^{\\transpose}C(\\tilde{T}+\\tilde{L})+2[d_1,d_2, d_3](\\tilde{T}+\\tilde{L}),\\\\\n&\\Phi^2 = (\\tilde{T}+\\tilde{L})^{\\transpose}E(\\tilde{T}+\\tilde{L}), \\qquad \\Xi^2 = 2\\tilde{l}^{\\transpose}E(\\tilde{T}+\\tilde{L})+2[f_1,f_2, f_3](\\tilde{T}+\\tilde{L}),\\\\\n&\\Upsilon^1 = r^1 + 2[d_1,d_2, d_3]\\tilde{l} + \\textrm{trace}\\left(\\tilde{T}^{\\transpose}C\\tilde{T}\\Sigma\\right)+2\\tilde{l}^{\\transpose}C\\tilde{l}, \\\\\n& \\Upsilon^1 = r^2 + 2[f_1,f_2, f_3]\\tilde{l} + \\textrm{trace}\\left(\\tilde{T}^{\\transpose}E\\tilde{T}\\Sigma\\right)+2\\tilde{l}^{\\transpose}E\\tilde{l}.}\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nPart 1 of the lemma is proved by differentiating $\\ex{c^i(\\VSF X,\\VSF U^i, g^{-i}(\\VSF Y^{-i}))|\\VSF Y^i}$ with respect to $\\VSF U^i$ and setting it equal to zero. For the proof of Part 2 of the lemma, see Appendix \\ref{sec:auxiliary}. For proving Part 3 of the lemma, notice that if $\\VSF U^i = g^{i\\star}(\\VSF Y^i)$ for $i=1,2$, then\n\\beqq{\\bmat{\\VSF X \\\\\\VSF U^1\\\\\\VSF U^2} = \\tilde{T}\\bmat{\\VSF X \\\\\\VSF Y^1\\\\\\VSF Y^2}+\\tilde{L}\\VSF m+\\tilde{l}.}\nNow, substituting this in the expressions for $c^1$ and $c^2$ and taking the expectations, we get the expected costs of the controllers. This completes the proof of the lemma.\n\\end{proof}\n\n\\begin{remark}\\label{rem:taui}\nThe Nash equilibrium strategy of Player $i$ given in \\eqref{eqn:gistaraux} of the auxiliary game {\\bf AG1} can be rewritten as\n\\beqq{g^{i\\star}(\\VSF Y^i) = \\matrix{ccc}{l^i+L^i\\VSF m_x-T^i\\VSF m_{y^i} &|& T^i } \\matrix{c}{1 \\\\ \\VSF Y^i}.}\nIt should also be noted that the unique Nash equilibrium in the auxiliary game {\\bf AG1} exists in the class of all Borel measurable strategies of the controllers. {\\hfill$\\Box$}\n\\end{remark}\n\nIn the next subsection, we consider a class of dynamic LQG games satisfying certain assumptions, and show that each LQG game in that class admits a unique common information based Markov perfect equilibrium.\n\n\\subsection{Generalization to Dynamic LQG Games}\\label{sub:generallqg}\nIn this subsection, we consider LQG games that satisfy Assumptions \\ref{assm:lqginfoevolution} and \\ref{assm:lqgseparation}. In order to prove the main result of the section, we need the following lemma.\n\n\\begin{lemma}\\label{lem:bayesiangamet}\nConsider an LQG game {\\bf G1} that satisfies Assumptions \\ref{assm:lqginfoevolution} and \\ref{assm:lqgseparation}. Fix a time step $t\\in\\{1,\\ldots,T-2\\}$. If the expected value functions $V^i_t, i=1,2$ of the controllers at time $t+1$ are affine-quadratic functions of $\\VEC m_{t+1}$, then the one-stage Bayesian game at time step $t$ with any mean $\\VEC m_t\\in\\ALP S_t$ is an instance of auxiliary game {\\bf AG1}.\n\\end{lemma}\n\\begin{proof}\nConsider a time step $t\\in\\{1,\\ldots,T-1\\}$ and a realization $\\VEC c_t$ of the common information at time step $t$. Define\n\\beqq{\\VSF X = \\VEC S_t = \\bmat{\\VEC X_t \\\\ \\VEC P^1_t\\\\ \\VEC P^2_t},\\quad \\VSF Y^i = \\VEC P^i_t,\\quad \\VSF U^i = \\VEC U^i_t, \\qquad i=1,2.}\nThe probability measure on the state $\\VSF X$ is taken to be equal to the common information based conditional measure $\\pi_t(d\\VEC s_t) = \\pr{d\\VEC s_t|\\VEC c_t}$, that admits a Gaussian density function with mean $\\VEC m_t$ (dependent on $\\VEC c_t$) and variance $\\Sigma_t$, which are defined in Lemma \\ref{lemma:lqgmean}. The observation $\\VSF Y^i$ of auxiliary Controller $i$ is the private information $\\VEC P^i_t$.\n\nWe now prove that the one-stage Bayesian game defined above is an instance of the Auxiliary game {\\bf AG1}. First, note that $\\VSF Y^i = H^i\\VSF X$ for some appropriate matrix $H^i$, $i\\in\\{1,2\\}$, which implies that the assumption on the covariance matrix of the auxiliary game given in \\eqref{eqn:sigmaij} is satisfied by the auxiliary game defined above. We just need to verify that the cost functions of the controllers of the one-stage Bayesian game are of the same form as \\eqref{eqn:j1aux} and \\eqref{eqn:j2aux}. \n\nAt any time step $t\\leq T-1$, let us assume that the expected value function of the Controller $i$ is $V^i_{t+1}(\\VEC m_{t+1}) = \\VEC m^{\\transpose}_{t+1}\\Phi^i_{t+1}\\VEC m_{t+1}+\\Xi^i_{t+1}\\VEC m_{t+1}+\\Upsilon^i_{t+1}$ for some appropriate positive definite matrix $\\Phi^i_{t+1}$, matrix $\\Xi^i_{t+1}$ and a non-negative real number $\\Upsilon^i_{t+1}$. Recall from Lemma \\ref{lemma:lqgmean} that $\\VEC M_{t+1}:= F^1_t(\\VEC m_t,\\VEC Z_{t+1})$, where $F^1_t$ is an affine function of $\\VEC m_t$ and $\\VEC Z_{t+1}$. Thus, the cost-to-go for the Controller $i\\in\\{1,2\\}$ in the one-stage Bayesian game at time step $t< T$ with mean $\\VEC m_t$ is of the form\n\\beq{\\label{eqn:citlqg}\\check{\\check{c}}^i_{t}(\\VEC m_t;\\VEC S_t,\\VEC U^1_t,\\VEC U^2_t,\\VEC Z_{t+1}) &:=& c^i_t(\\VEC X_t,\\VEC U^1_t,\\VEC U^2_t)+(F^{1}_t(\\VEC m_t,\\VEC Z_{t+1}))^\\transpose \\Phi^i_{t+1} F^1_t(\\VEC m_t,\\VEC Z_{t+1})\\nonumber\\\\ & & +\\Xi^i_{t+1} F^{1}_t(\\VEC m_t,\\VEC Z_{t+1})+ \\Upsilon^i_{t+1}.\\qquad}\nNow, recall the definition of $\\VEC Z_{t+1}$ in Assumption \\ref{assm:lqginfoevolution}, and substitute for $\\VEC Y^1_{t+1}$ and $\\VEC Y^2_{t+1}$ in the expression for $\\VEC Z_{t+1}$ in terms of $\\VEC X_t$, $\\VEC U^1_t$, $\\VEC U^2_t$ and noises using \\eqref{eq:lqgdynamics} and \\eqref{eq:lqgobserv}. Thus, $\\VEC Z_{t+1}$ is an affine map of $\\VEC S_t$, $\\VEC U^1_t$, $\\VEC U^2_t$ and noises $\\VEC W^0_t,\\VEC W^1_{t+1}$ and $\\VEC W^2_{t+1}$. Also recall from Lemma \\ref{lemma:lqgmean} that $F^1_t$ is an affine map of its arguments. Define $\\bar{c}^i_t, i=1,2$ as\n\\beqq{\\bar{c}^i_{t}(\\VEC m_t;\\VEC S_t,\\VEC U^1_t,\\VEC U^2_t) = \\ex{\\check{\\check{c}}^i_{t}(\\VEC m_t;\\VEC S_t,\\VEC U^1_t,\\VEC U^2_t,\\VEC Z_{t+1})|\\VEC S_t,\\VEC U^1_t,\\VEC U^2_t}.}\nThus, the expression for cost function $\\bar{c}^i_t$, $i=1,2$ are precisely of the forms\n\\beq{\\label{eqn:barc1t}\\bar{c}^1_{t}(\\VEC m_t;\\VEC S_t,\\VEC U^1_t,\\VEC U^2_t) &=& \\bmat{\\VEC S_t^\\transpose,\\VEC U_t^{1\\transpose},\\VEC U_t^{2\\transpose}}C_t\\bmat{\\VEC S_t \\\\\\VEC U_t^1\\\\\\VEC U_t^2} + 2\\VEC m_t^\\transpose D_t \\bmat{\\VEC S_t \\\\\\VEC U_t^1\\\\\\VEC U_t^2}+ r^1_t \\VEC m_t+\\tilde\\Upsilon^1_{t+1},\\qquad\\\\\n\\label{eqn:barc2t}\\bar{c}^2_{t}(\\VEC m_t;\\VEC S_t,\\VEC U^1_t,\\VEC U^2_t) &=& \\bmat{\\VEC S_t^\\transpose,\\VEC U_t^{1\\transpose},\\VEC U_t^{2\\transpose}}E_t\\bmat{\\VEC S_t \\\\\\VEC U_t^1\\\\\\VEC U_t^2} + 2\\VEC m_t^\\transpose F_t \\bmat{\\VEC S_t \\\\\\VEC U_t^1\\\\\\VEC U_t^2} + r^2_t \\VEC m_t +\\tilde\\Upsilon^2_{t+1},\\qquad}\nwhere $C_t,D_t,E_t,F_t, r^1_t,r^2_t,\\tilde{\\Upsilon}^1_{t+1}, \\tilde{\\Upsilon}^2_{t+1}$ are dependent on matrices $R^i$, $\\Phi^i_{t+1}$, $\\Xi^i_{t+1}$ for $i=1,2$, the linear map $F^1_t$, the variances of noises $\\VEC W^0_t,\\VEC W^1_{t+1}$ and $\\VEC W^2_{t+1}$, and the projection function $\\zeta_{t+1}$, where $\\zeta_{t+1}$ is defined in \\eqref{eq:lqgcommoninfo}. The cost functions of the controllers given above are of the same form as considered in \\eqref{eqn:j1aux} and \\eqref{eqn:j2aux}. Thus, the one-stage Bayesian game at time $t$ with mean $\\VEC m_t$ is the same as auxiliary game {\\bf AG1}$(\\VEC m_t,\\Sigma_t,\\bar{c}^1_t(\\VEC m_t;\\cdot),\\bar{c}^2_t(\\VEC m_t;\\cdot))$. This completes the proof of the lemma.\n\\end{proof}\n\n\\begin{definition}\\label{def:agtg1}\nFor any LQG game {\\bf G1}, the corresponding one-stage Bayesian game at time step $t$ with mean $\\VEC m_t$ and cost functions of the controllers given by \\eqref{eqn:barc1t} and \\eqref{eqn:barc2t} is referred to as ${\\bf AG}_t({\\bf G1};\\VEC m_t)$. {\\hfill$\\Box$}\n\\end{definition}\n\nLemma \\ref{lem:bayesiangamet} implies that the one-stage Bayesian game of the LQG game {\\bf G1} at time $t$ with mean $\\VEC m_t$ is an instance of an auxiliary game {\\bf AG1}$(\\VEC m_t,\\Sigma_t,\\bar{c}^1_{t}(\\VEC m_t;\\cdot),\\bar{c}^2_{t}(\\VEC m_t;\\cdot))$, where $\\bar{c}^i_{t}$ is defined in \\eqref{eqn:citlqg}. If the matrix tuple $(C_t,E_t)$, as defined in \\eqref{eqn:barc1t} and \\eqref{eqn:barc2t}, satisfies Assumption \\ref{assm:eigcost}, then for any realization $\\VEC m_t$, Lemma \\ref{lem:basargame} implies that there exists a unique Nash equilibrium of the one-stage Bayesian game at time step $t$, which is affine in the private information of the controllers. Furthermore, the expected equilibrium costs are affine-quadratic in the mean $\\VEC m_t$. This crucial observation about LQG games leads us to the next theorem, which is also the main result of this section. First, we need the following assumption on the cost functions of the one-stage Bayesian games of game {\\bf G1}. \n\n\\begin{assumption}\\label{assm:lqgcost} \nAt every time step $t\\in\\{1,\\ldots,T-1\\}$ of game {\\bf G1}, the matrix tuple $(C_t,E_t)$, as defined in \\eqref{eqn:barc1t} and \\eqref{eqn:barc2t} and obtained using the procedure outlined in Algorithm 1 in Subsection \\ref{sub:stagebayesian}, satisfies Assumption \\ref{assm:eigcost}. {\\hfill$\\Box$}\n\\end{assumption}\n\nThe main result of this section is now captured by the following theorem.\n\n\\begin{theorem}\\label{thm:lqgg1}\nConsider an LQG game {\\bf G1} that satisfies Assumptions \\ref{assm:lqginfoevolution} and \\ref{assm:lqgseparation}. If Assumption \\ref{assm:lqgcost} also holds for game {\\bf G1}, then it admits a unique common information based Markov perfect equilibrium in the class of {\\it all Borel measurable strategies} of the controllers. Furthermore, the equilibrium strategy of Controller $i\\in\\{1,2\\}$ is affine in its information at all time steps.\n\\end{theorem}\n\\begin{proof}\nWe follow the procedure outlined in Algorithm 1 in Subsection \\ref{sub:stagebayesian}. At time step $T-1$, let $\\VEC m_{T-1}$ be a realization of the common information based conditional mean. Consider the one-stage Bayesian game ${\\bf AG}_{T-1}({\\bf G1};\\VEC m_{T-1})$ at time $T-1$. Since Assumption \\ref{assm:lqgcost} holds, we use the result of Lemma \\ref{lem:basargame} to conclude that a unique Nash equilibrium policies of the controllers exist. Furthermore, the Nash equilibrium policies of the one-stage Bayesian game ${\\bf AG}_{T-1}({\\bf G1};\\VEC m_{T-1})$ are affine in the conditional mean and the private information of the controllers at that time step. Since the conditional mean $\\VEC m_{T-1}$ is affine in the common information of the controllers, we conclude that the Nash equilibrium policies of the controllers in the one-stage Bayesian game ${\\bf AG}_{T-1}({\\bf G1};\\VEC m_{T-1})$ is affine in the information of the controllers. \n\nWe continue this process for all possible means $\\VEC m_t\\in\\ALP S_t$ at all time steps $t\\in\\{T-1,T-2,\\ldots,1\\}$ to conclude that there is a unique common information based Markov perfect equilibrium for game {\\bf G1}.\n\\end{proof}\n\n\\begin{remark}\nIt should be noted that one can compute the Bayesian Nash equilibrium of game {\\bf AG1} simply by solving a set of linear equations. In the dynamic LQG game {\\bf G1} satisfying the sufficient conditions of Theorem \\ref{thm:lqgg1}, the controllers can just solve a set of linear equations at successive time steps to obtain the unique common information based Markov perfect equilibrium; thus, computing the equilibrium is inexpensive in the class of LQG games.{\\hfill $\\Box$}\n\\end{remark}\n\n\\subsection{LQG Games not satisfying Assumption \\ref{assm:lqgcost}}\\label{sub:not}\nIn this subsection, we show that even if an LQG game does not satisfy Assumption \\ref{assm:lqgcost}, it may still admit a common information based Markov perfect equilibrium under some mild conditions. However, we cannot claim uniqueness of that equilibrium like we did in Theorem \\ref{thm:lqgg1} in the previous subsection.\n\nConsider the auxiliary game {\\bf AG1} discussed above. We needed Assumption \\ref{assm:eigcost} in two places in the result of Lemma \\ref{lem:basargame} above - (i) to conclude the uniqueness of Nash equilibrium provided that it exists, and (ii) to show the existence of matrices $l^1,l^2, L^1, L^2,T^1$ and $T^2$ as defined in \\eqref{eqn:b1b2}, \\eqref{eqn:t1} and \\eqref{eqn:t2} above. However, if we drop this assumption and instead make a milder assumption, then we can obtain a result that is weaker than what we got above. First, we state the assumption we make to obtain the weaker result.  \n\n\\begin{assumption}\\label{assm:eigcostb}\nIn game {\\bf AG1}, the either $(I - C_{22}^{-1}C_{23}E_{33}^{-1}E_{23}^{\\transpose})$ is invertible or $(I-E_{33}^{-1}E_{23}^{\\transpose}C_{22}^{-1}C_{23})$ is invertible, and there exists a unique solution to the coupled pair of equations \\eqref{eqn:t1} and \\eqref{eqn:t2}. {\\hfill$\\Box$}\n\\end{assumption}\n\nIf we make this assumption on the auxiliary game {\\bf AG1}, then we can conclude the following about the Nash equilibrium of the game.\n  \n\\begin{lemma}\\label{lem:basargameb}\nIf the auxiliary game {\\bf AG1}$(\\VSF m,\\Sigma,c^1,c^2)$ satisfies Assumption \\ref{assm:eigcostb}, then the game admits a Nash equilibrium. The expressions of Nash equilibrium control laws and expected costs are the same as in Lemma \\ref{lem:basargame}. \n\\end{lemma}\n\\begin{proof}\nThe proof is analogous to the proof of Lemma \\ref{lem:basargame}.\n\\end{proof}\n\nThis brings us to the following result for LQG games that may not satisfy Assumption \\ref{assm:lqgcost}.\n\\begin{theorem}\\label{thm:lqgg2}\nConsider an LQG game {\\bf G1} that satisfies Assumptions \\ref{assm:lqginfoevolution} and \\ref{assm:lqgseparation}. For all time steps $t\\in\\{1,\\ldots,T-1\\}$ and realizations of the mean $\\VEC m_t\\in\\ALP S_t$, obtain the one-stage Bayesian game ${\\bf AG}_t({\\bf G1};\\VEC m_t)$ by following the steps of Algorithm 1 in Subsection \\ref{sub:stagebayesian}. If ${\\bf AG}_t({\\bf G1};\\VEC m_t)$ satisfies Assumption \\ref{assm:eigcostb} for all $t\\in\\{1,\\ldots,T-1\\}$ and $\\VEC m_t\\in\\ALP S_t$, then game {\\bf G1} admits a common information based Markov perfect equilibrium.\n\\end{theorem}\n\\begin{proof}\nThe proof follows from the same arguments as in the proof of Theorem \\ref{thm:lqgg1}. It should be noticed that for a fixed affine strategy of Controller 1, the one-person dynamic optimization problem for Controller 2 can be solved using a dynamic programming (existence follows from Assumption \\ref{assm:eigcostb}) to obtain optimal strategies that are linear in its information. \n\\end{proof}\n\nNotice that we do not claim uniqueness of the common information based Markov perfect equilibrium for LQG games that do not satisfy Assumption \\ref{assm:lqgcost}.\n\\subsection{An Illustrative Example}\\label{sub:global}\nIn this section, we consider an example of a two-player non-zero sum game considered above. There are three states in the game, out of which one is a global state that is observed by both controllers, and two states are local states of the controllers. The state evolution is given by\n\\beqq{\\VEC X^0_{t+1} &=& A \\VEC X^0_t+B^1\\VEC U^1_t+B^2\\VEC U^2_t+\\VEC W^{0}_t,\\\\\n\\VEC X^i_{t+1} &=& A^i \\VEC X^0_t+D^{i}_1\\VEC U^1_t+D^i_2 \\VEC U^2_t+\\VEC W^i_t,\\quad i=1,2,}\nwhere $\\VEC X^0_t$ is the global state and $\\VEC X^i_t,i=1,2$ are the local states for $t=1,\\ldots,T-1$. The noise processes $\\VEC W^i_t$ are assumed to be mutually independent mean-zero Gaussian random variables with variances $\\Lambda^i_t$ for $i=0,1,2$ and $t=1,\\ldots,T-1$. Player $i$'s information $\\VEC I^i_t$ lies in a Euclidean space, that is $\\ALP I^i_t = \\ALP X^0_{1:t}\\times\\ALP X^i_{1:t}\\times\\ALP U^{1:2}_{1:t-1}$. The cost functions of the players are given by\n\\beqq{c^i_T(\\VEC x^0_T,\\VEC x^i_T) &=& \\VEC x^{0\\transpose}_T Q^i \\VEC x^0_T+\\VEC x^{i\\transpose}_T Q^i \\VEC x^i_T,\\\\\nc^i_t(\\VEC x^0_t,\\VEC x^i_t,\\VEC u^{1:2}_t) &=& \\VEC x^{0\\transpose}_t Q^i \\VEC x^0_t+\\VEC x^{i\\transpose}_t Q^i \\VEC x^i_t+\\VEC u^{1\\transpose}_{t}R^i\\VEC u^1_t+\\VEC u^{2\\transpose}_{t}S^i\\VEC u^2_t, \\quad i=1,2.}\nWe first verify in the following lemma that both assumptions on the information structure is satisfied by this game. Toward this end, recall that the space of common information of the players is $\\ALP C_t:=\\ALP X^0_{1:t}\\times\\ALP U^{1:2}_{1:t-1}$ and the space of private information of Player $i$ is $\\ALP P^i_t:=\\ALP X^i_{1:t}$.\n\\begin{lemma}\n\\label{lem:global}\nThe information structure of the controllers in the game defined above satisfies Assumptions \\ref{assm:lqginfoevolution} and \\ref{assm:lqgseparation}.\n\\end{lemma}\n\\begin{proof}\nSince the information structure is nested, Assumption \\ref{assm:lqginfoevolution} is automatically satisfied. Given the common information $\\VEC c_t$, it is easy to verify that the joint distribution of private informations of the players are Gaussian and independent of past strategies of the players since $\\VEC W^i_t$ are mutually independent Gaussian random variables for all time steps, and $i=1,2$. This implies that Assumption \\ref{assm:lqgseparation} is also satisfied.\n\\end{proof}\n\nWe are interested in computing common information based Markov perfect equilibrium of this game. At a time step $t$, since the private state of both controllers are affected by the global state at time step $t-1$ but not the private states at the previous time steps, the past realizations of private states of Controller $i$, $\\VEC x^i_1,\\ldots,\\VEC x^i_{t-1}$, do not affect the common information based Markov perfect equilibrium. Therefore, for ease of exposition, we make minor changes in the notations from what we have used in previous sections. Let us denote the mean of the random variable $\\VEC X^i_{t}$ given the common information $\\VEC C_t$ as $\\VEC M^i_{t}, i=1,2$. Note that the mean is given by $\\VEC M^i_{t} =  A^i \\VEC X^0_{t-1}+D^{i}_1\\VEC U^1_{t-1}+D^i_2 \\VEC U^2_{t-1}$. The variance of $\\VEC X^i_t$ given the common information $\\VEC C_t$ is $\\Lambda^i_t$. Define $\\VEC M_t := [\\VEC X^{0\\transpose}_t,\\VEC M^{1\\transpose}_t,\\VEC M^{2\\transpose}_t]^\\transpose$ to be the conditional mean of the states \n$[\\VEC X^{0\\transpose}_t,\\VEC X^{1\\transpose}_t,\\VEC X^{2\\transpose}_t]^\\transpose$ given the common information $\\VEC C_t$. Also note that $\\VEC Z_{t+1}=[\\VEC U^{1\\transpose}_t, \\VEC U^{2\\transpose}_t,\\VEC X^{0\\transpose}_{t+1}]^\\transpose$. The evolution of $\\VEC M_t$ is given by\n\\beq{\\VEC M_{t+1} &=& \\matrix{c}{\\VEC X^0_{t+1}\\\\A^1 \\VEC X^0_t+D^{1}_1\\VEC U^1_t+D^1_2 \\VEC U^2_t\\\\ A^2 \\VEC X^0_t+D^{2}_1\\VEC U^1_t+D^2_2 \\VEC U^2_t}\\nonumber\\\\\n\\label{eqn:f1tglobal}& =: & F^1_t(\\VEC X^0_t,\\VEC U^1_t,\\VEC U^2_t,\\VEC X^0_{t+1}).}\nThe conditional covariance matrix is $\\Sigma_{t+1} := \\text{diag}\\{0,\\Lambda^1_{t+1},\\Lambda^2_{t+1}\\}$. Let us assume that this game satisfies Assumption \\ref{assm:lqgcost}. By Theorem \\ref{thm:lqgg1}, we conclude that this game admits a unique common information based Markov perfect equilibrium. Now, we compute the unique common information based Markov perfect equilibrium of this game.\n\n\\begin{enumerate}\n\\item At the terminal time $T$, for each realization $\\VEC m:=[\\VEC x^{0\\transpose}_T, \\VEC m^{1\\transpose},\\VEC m^{2\\transpose}]^\\transpose$ of $\\VEC M_{T}$, the one-stage Bayesian game  $SG_{T}(\\VEC m)$ is defined as follows \n\\begin{enumerate}\n\\item The conditional probability distribution on $\\ALP X^0_{T}\\times \\ALP X^1_{T}\\times\\ALP X^2_{T}$ given the common information $\\VEC c_T$ is a Gaussian density with mean $\\VEC m$ and covariance $\\Sigma_{T}$.\n\\item Player $i$ observes $\\VEC X^0_{T}, \\VEC X^i_{T}$, $i=1,2$. No action is chosen.\n\\item Player $i$'s cost is $c^i_T(\\VEC x^0_T,\\VEC x^i_T)$. \n\\end{enumerate}\nThe expected costs as functions of beliefs are of the form $V^i_{T}(\\VEC m) =\\VEC x^{0\\transpose}_T Q^i \\VEC x^0_T +\\VEC m^{i\\transpose} Q^i\\VEC m^i +\\textrm{trace}(Q^i\\Lambda^i_T)$.\n\n\\item At time $t<T$, for each realization $\\VEC m$ of $\\VEC M_{t}$, we define a one-stage Bayesian game $SG_{t}(\\VEC m)$ where \n\\begin{enumerate}\n\\item The probability distribution on $\\ALP X^0_{t}\\times \\ALP X^1_{t}\\times\\ALP X^2_{t}$ is a Gaussian density with mean $\\VEC m$ and covariance $\\Sigma_{t}$.\n\\item  Player $i$ observes $\\VEC X^0_{t}, \\VEC X^i_{t}$ and chooses action $\\VEC U^i_{t}$, $i=1,2$. \n\\item  The cost of the subgame $SG_t(\\VEC m)$ accrued by Player $i$ is \n\\beqq{c^i_t(\\VEC x^0_t,\\VEC x^i_t,\\VEC u^{1:2}_t) + \\ex{V^i_{t+1}(F^1_{t}(\\VEC x^0_t,\\VEC u^1_{t},\\VEC u^2_{t},\\VEC X^0_{t+1}))},}\nwhere $F^1_{t}$ is defined in \\eqref{eqn:f1tglobal}. If we expand this cost function and write it in terms of $\\VEC x^0_t,\\VEC u^1_t$, $\\VEC u^2_t$ and noise variables, then we observe that the resulting cost function is of the same form as in the auxiliary game {\\bf AG1} considered in Subsection \\ref{sec:basargame} above. The Nash equilibrium of the Bayesian game $SG_t(\\VEC m)$ is computed using the result in Lemma \\ref{lem:basargame}. The value functions of the virtual players are of the form $V^i_{t}(\\VEC m) =\\VEC x^{0\\transpose}_t \\Phi^i_t \\VEC x^0_t +\\VEC m^{i\\transpose} \\Xi^i_t \\VEC m^i + \\Upsilon^i_t$, where $\\Phi^i_t$ and $\\Xi^i_t$ are non-negative definite matrices and $\\Upsilon^i_t$ is a non-negative real number.\n\\end{enumerate}\n Thus, we computed the unique common information based Markov perfect equilibrium of the game.\n\\end{enumerate}\nIn the next section, we show that there could be several other Nash equilibria of game {\\bf G1}.\n\n\\section{A Game with Multiple Nash Equilibria}\n\\label{sec:information}\nIn previous sections, we outlined an algorithm that can be used to compute Nash equilibria of games that satisfy Assumptions \\ref{assm:lqginfoevolution} and \\ref{assm:lqgseparation}. In this section, we exhibit an example of a game of asymmetric information that has several Nash equilibria and a unique common information based Markov perfect equilibrium. This reinforces Remark \\ref{rem:multiequilibria} stated after Theorem \\ref{thm:mpevirtual}, which points out that our algorithm computes only a subclass of those Nash equilibria that can be obtained using the Markov perfect equilibrium of the corresponding symmetric information game between the virtual players.\n\nTo illustrate the existence of multiple Nash equilibria, we follow the lines in Example 1 of \\cite[p. 241]{Basarmulti}, and consider the following two-stage game in which all variables are scalar:\n\\beqq{x_2 &=& x_1+u^1_1+u^2_1+w^0_1, \\quad y^1_1 = x_1+w^1_1,\\quad y^2_1 = x_1+w^2_1,\\\\\nx_3 &=& x_2+u^2_2+w^0_2,\\qquad\\quad y^2_2 = x_2+w^2_2.} \nThe primitive random variables $\\{X_1,W^0_1,W^0_2,W^1_1,W^2_1,W^2_2\\}$ are all mutually independent, mean zero Gaussian random variables with unit variance. The information structure of Controller $1$ is $\\VEC I^1_1 = Y^1_1$ and $\\VEC I^1_2 = [U^1_1,U^2_1,Y^1_1,Y^2_1]^\\transpose$, and the information structure of Controller $2$ is $\\VEC I^2_1 = Y^2_1$ and $\\VEC I^2_2 = [Y^2_2,U^1_1,U^2_1,Y^1_1,Y^2_1]^\\transpose$. Since the information structure is of the one-step delayed sharing pattern type, this game satisfies Assumptions \\ref{assm:lqginfoevolution} and \\ref{assm:lqgseparation}. The cost functions of the controllers are\n\\beqq{J^1(g^1_1,g^2_1,g^2_2) &=& \\ex{(X_3)^2+(U^1_1)^2},\\\\\nJ^2(g^1_1,g^2_1,g^2_2)  &=& \\ex{(X_3)^2+(U^2_1)^2+(U^2_2)^2}.}\nThe Kalman filter equations for evolution of common information based belief are given by\n\\beqq{m^0_2 &=& \\ex{X_2|\\VEC c_2} = \\frac{y^1_1+y^2_1}{3}+u^1_1+u^2_1,\\quad \\Sigma^0_2 = \\ex{(X_2-m^0_2)^2} = \\frac{4}{3}.}\nNow, we compute the common information based Markov perfect equilibrium of this game.\n\\subsection{Common Information based Markov Perfect Equilibrium}\nRecall that since this is an LQG game, there exists a unique common information based Markov perfect equilibrium. We now apply our algorithm to compute the common information based Markov perfect equilibrium of this game. In the sub-game starting at time step $t=2$, Controller 2 is the only player acting and the cost-to-go function for the sub-game starting at $t=2$ is strictly convex in the control action of Controller 2. The unique optimal control law for Controller 2 for the sub-game at $t=2$, given its information, is given by   \n\\beq{\\label{eqn:g2}g^{2\\star}_2(\\VEC i^2_2) = -\\frac{1}{2}\\ex{X_2|\\VEC i^2_2} = -\\frac{1}{2}\\ex{X_2|m^0_2,y^2_2}  = -\\frac{1}{2}\\left(m^0_2+\\frac{4}{7}(y^2_2-m^0_2)\\right).}\nThe expected value functions of the controllers at time step $2$ are\n\\beqq{V^1_2(m^0_2) = \\frac{1}{4}(m^0_2)^2+\\frac{37}{21},\\qquad V^2_2(m^0_2) = \\frac{1}{2}(m^0_2)^2+\\frac{41}{21}.}\nNow, using the result of Lemma \\ref{lem:basargame}, we obtain the unique Nash equilibrium of the sub-game starting at the first time step to be\n\\beq{\\label{eqn:g1g2}g^{1\\star}_1(y^1_1) = -\\frac{5}{59} y^1_1,\\qquad g^{2\\star}_1(y^2_1) = -\\frac{9}{59} y^2_1.}\nThus, this game has a unique common information based Markov perfect equilibrium $(g^{1\\star}_1,(g^{2\\star}_1,g^{2\\star}_2))$. In the next subsection, we show that there exists a continuum of Nash equilibria in this game, and those equilibria cannot be obtained using our approach.\n\n\\subsection{Other Nash Equilibria}\nIn this subsection, we show that if Controller 2 uses the common information (instead of only mean $m^0_2$) to construct its control law at time step $t=2$, then we have a continuum of Nash equilibrium in this game.\n\nWe now define a tuple of strategies $(g^{1\\dagger}_1,(g^{2\\dagger}_1,g^{2\\dagger}_2))\\in \\ALP G^1_1\\times\\ALP G^2_{1:2}$ of both controllers, parametrized by a real number $\\lambda\\neq -59\/22$:\n\\beq{g^{1\\dagger}_1(y^1_1) &=& -\\frac{10\\lambda +5}{22\\lambda +59} y^1_1,\\quad\\qquad g^{2\\dagger}_1(y^2_1) = -\\frac{2\\lambda+9}{22\\lambda+59} y^2_1,\\nonumber\\\\\n\\label{eqn:g2dagger} g^{2\\dagger}_2(\\VEC i^2_2) &=& -\\frac{1}{2}\\left(m^0_2+\\frac{4}{7}(y^2_2-m^0_2)\\right) + \\lambda\\left(u^1_1 -g^{1\\dagger}_1(y^1_1)\\right).}\n\n\nWe now have the following result:\n\\begin{lemma}\nThe strategy profile $(g^{1\\dagger}_1,(g^{2\\dagger}_1,g^{2\\dagger}_2))$ is a Nash equilibrium of the game formulated above for any value of $\\lambda\\neq -59\/22$. \n\\end{lemma}\n\\begin{proof}\nWe prove that for a fixed strategy $g^{1\\dagger}_1$, $(g^{2\\dagger}_1,g^{2\\dagger}_2)$ is the best response strategy of Controller 2 and vice versa. \n\nFirst, fix $u^1_1 = g^{1\\dagger}_1(y^1_1)$. Then, $g^{2\\dagger}_2$ minimizes the cost $\\ex{(X_3)^2\\big|u^1_1 = g^{1\\dagger}_1(y^1_1), \\VEC i^2_2}$ (note that $\\VEC i^2_2$ contains $y^1_1$). One can then verify that $g^{2\\dagger}_1$ minimizes the cost functional $J^2(g^{1\\dagger}_1,(g^2_1,g^{2\\dagger}_2))$. Therefore, we conclude that $J^2(g^{1\\dagger}_1,(g^{2\\dagger}_1,g^{2\\dagger}_2))\\leq J^2(g^{1\\dagger}_1,(g^{2}_1,g^{2}_2))$ for any $\\VEC g^2:=(g^{2}_1,g^{2}_2)\\in\\ALP G^2_{1:2}$.\n\nNext, fix $u^2_1 =g^{2\\dagger}_1(y^2_1), u^2_2 = g^{2\\dagger}_2(\\VEC i^2_2)$. Then, the cost-to-go function for Controller 1 is at time step $1$ for the fixed strategy of Controller 2 is\n\\beqq{\\frac{1}{4}(m^0_2)^2+\\frac{37}{21}+\\lambda m^0_2(u^1_1-g^{1\\dagger}_1(y^1_1))+\\lambda^2\\left(u^1_1-g^{1\\dagger}_1(y^1_1)\\right)^2.}\nThe control law $g^{1\\dagger}_1$ minimizes the cost functional of Controller 1 at time step $t=1$. Thus, we conclude $J^1(g^{1\\dagger}_1,(g^{2\\dagger}_1,g^{2\\dagger}_2))\\leq J^1(g^{1}_1,(g^{2\\dagger}_1,g^{2\\dagger}_2))$ for any $g^1_1\\in\\ALP G^1_1$. \n\\end{proof}\n\nThus, we have proved that, in fact, there are several Nash equilibria of this game, and the Nash equilibrium obtained using our algorithm is just one among them (notice that the common information based Markov perfect equilibrium corresponds to the choice of $\\lambda =0$).\n\n\\begin{remark}\nIt has been shown in \\cite{Basaronestep} that if the control actions are not shared among the controllers, then there exists a unique Nash equilibrium in the game formulated above. The unique Nash equilibrium is the same set of control laws given in \\eqref{eqn:g2} and \\eqref{eqn:g1g2} with $u^i_1$ substituted with $g^{i\\star}_1(y^i_1)$ in the expression of $m^0_2$. Thus, the existence of multiple Nash equilibria in this game is due to the information available to Controller 2 at time step $2$ about the action taken by the Controller 1 at time step 1. This example illustrates that more information to controllers may be harmful in a game as it gives rise to several other Nash equilibria! {\\hfill$\\Box$}\n\\end{remark}\n\n\n\\begin{remark}\nIt should also be noted that in case control actions are not shared, Assumption \\ref{assm:lqgseparation} does not hold. However, \\cite{Basaronestep} proved that an LQG game with one-step delayed observation sharing pattern admits a unique Nash equilibrium. Thus, Nash equilibrium in a dynamic game of asymmetric information may exist even in the absence of Assumption \\ref{assm:lqgseparation} on that game. {\\hfill$\\Box$}\n\\end{remark}\n\n\\subsection{Effects on Expected Costs}\nWe now compare the expected costs to the controllers, if the Controller 2 plays according to a Nash equilibrium given by \\eqref{eqn:g2dagger}. First, note that $\\ex{(Y^i_1)^2} = 2, i=1,2$ and $\\ex{Y^1_1Y^2_1} = 1$. This implies\n\\beqq{(m^0_2)^2 = \\frac{2}{9(22\\lambda+59)^2}\\left((-8\\lambda+44)^2+(16\\lambda+32)^2+(-8\\lambda+44)(16\\lambda+32)\\right).}\nThe expected costs to the controllers at Nash equilibrium $(g^{1\\dagger}_1,(g^{2\\dagger}_1,g^{2\\dagger}_2))$ are given by\n\\beqq{J^{1\\dagger}(\\lambda):=J^1(g^{1\\dagger}_1,(g^{2\\dagger}_1,g^{2\\dagger}_2)) &=& \\frac{1}{4}(m^0_2)^2+\\frac{2(10\\lambda +5)^2}{(22\\lambda +59)^2}+\\frac{37}{21},\\\\\nJ^{2\\dagger}(\\lambda):=J^2(g^{1\\dagger}_1,(g^{2\\dagger}_1,g^{2\\dagger}_2)) & = & \\frac{1}{2}(m^0_2)^2+\\frac{2(2\\lambda+9)^2}{(22\\lambda+59)^2}+\\frac{41}{21}.}\nIf we take the limit $\\lambda\\rightarrow\\infty$, we get\n\\beqq{\\lim_{\\lambda\\rightarrow\\infty}J^{1\\dagger}(\\lambda) = \\frac{200+\\frac{32}{3}}{484}+\\frac{37}{21} \\approx 2.197,\\quad\\lim_{\\lambda\\rightarrow\\infty}J^{2\\dagger}(\\lambda) = \\frac{8+\\frac{64}{3}}{484}+\\frac{41}{21} \\approx 2.013.} \nOn the other hand, $J^{1\\dagger}(0) \\approx 1.832$ and $J^{2\\dagger}(0) \\approx 2.092$, which corresponds to the expected costs to the controllers if they play according to the common information based Markov perfect equilibrium. Clearly, Controller 2, by choosing an appropriate (very large) value of $\\lambda$, can reduce its expected cost, while increasing the expected cost to Controller 1; this observation has connections to incentive designs where Controller 2 can be viewed as the designer (leader) in the game.\n\n\\section{Discussion}\\label{sec:discussion}\nOne of the crucial assumptions we made in the game formulation is the strategy independence of beliefs in Assumption \\ref{assm:lqgseparation}. As discussed in our companion paper \\cite{nayyar2012a}, this assumption plays a crucial role in computing the Markov perfect Nash equilibrium of game {\\bf G2} using a backward induction dynamic programming. In this section, we briefly describe the reason why this assumption is important. For a more detailed and technical discussion, we refer the reader to \\cite{nayyar2012a}.\n\nRecall that as a consequence of Assumption \\ref{assm:lqginfoevolution}, we have\n\\beqq{\\Pi_{t+1} = F_t(\\Pi_t,\\Gamma^1_t,\\Gamma^2_t,\\VEC Z_{t+1}).}\nSuppose Assumption \\ref{assm:lqgseparation} does not hold for some game {\\bf G1}, and the common information based conditional belief is dependent on the strategy of Controller 1 at some time step $t_0$. At any time step $t\\geq t_0$, in order for Controller 2 to know the belief $\\Pi_t$ exactly, the virtual player 1 needs to share its prescriptions in game {\\bf G2}. If virtual player 1 does not share its prescription, then virtual player 2 needs to know the precise strategy of virtual player 1. Now, if the virtual players neither share their prescriptions nor their strategies, then each controller has an incentive to deviate from the Nash equilibrium to reap the benefit of asymmetry in the beliefs caused by changing its strategy. In other words, each controller has an incentive to ``deceive'' the other controller if Assumption \\ref{assm:lqgseparation} does not hold. Assumption \\ref{assm:lqgseparation} assumes that the common information is ``rich enough'' so that controllers cannot deceive each other.\n\nIn case the cost functions of the controllers are aligned (that is, they are same at all time steps), then the game problem is just a team problem, and Assumption \\ref{assm:lqgseparation} is not required. In team problems, the agents can agree, prior to the start of the play, on what strategies they will use during the course of the play. Moreover, since the cost functions of all the agents are aligned, no agent has an incentive to deviate from the pre-agreed strategies. Consequently, we need not make Assumption \\ref{assm:lqgseparation}. \n\nWe now consider the case when the cost functions have opposite signs in game {\\bf G1}, that is $c^1_t = -c^2_t$ for all time steps $t\\in\\{1,\\ldots, T\\}$. This makes game {\\bf G1} as a zero-sum game with asymmetric information among the controllers. In this case, if a common information based Markov perfect equilibrium exists, then due to the ordered-interchangeability property of multiple saddle-point equilibria of zero-sum games \\cite{Basarbook}, the expected costs to the controllers remain the same for {\\it all} other saddle-point equilibrium strategies of the game. Thus, our backward induction algorithm in Theorem \\ref{thm:mpevirtual} provides a constructive method to compute saddle-point equilibrium in a zero-sum game of asymmetric information among the controllers, provided that such an equilibrium exists. \n\n\\section{Conclusion and Future Work}\\label{sec:conclude}\nWe studied dynamic two-player linear-Gaussian non-zero sum games in this paper, where the controllers have asymmetric information and their information structures satisfy two assumptions. We showed that under certain structural assumptions on the admissible strategies and stagewise additive cost functions of the players, the existence of a common information based Markov perfect equilibrium can be established by proving the existence of Nash equilibrium in a sequence of static games of symmetric information. For LQG games with cost functions satisfying certain assumptions, we showed that there exists a unique common information based Markov perfect equilibrium. We also gave analytical expressions for the common information based Markov perfect equilibrium for a class of LQG games.\n\nThe main idea consisted of defining a new game of symmetric information and perfect observations among virtual players, computing the Markov perfect equilibrium of that game, and then using the Markov perfect equilibrium strategies to obtain a Nash equilibrium of the original game of asymmetric information. We also developed a backward induction algorithm that computes the common information based Markov perfect equilibrium of the game, as long as a Nash equilibrium exists in the static symmetric information game between the virtual players at every time step in the backward induction algorithm. This conceptual approach can be extended to non-zero sum dynamic stochastic games with a finite number of players, which satisfy the two assumptions on the information structures and the assumptions on the cost functions and admissible strategies.\n\nWe showed that there may be other Nash equilibria in games, which however cannot be computed using the common information based approach. This is due to the fact that Markov perfect equilibria of the corresponding symmetric and perfect information game among virtual players is a small subclass of Nash equilibria of that game. The conceptual framework we developed in this paper can be used to compute the Nash equilibrium of several classes of games of asymmetric information with infinite and uncountable state and action spaces.\n\nThe case of multi-player LQG games can also be solved along similar lines using the result of \\cite{Basarmulti}. What still remains to be investigated is the value of information in games with asymmetric information. This is a challenging problem, because as we showed in Section \\ref{sec:information}, extra information given to one player generates several other Nash equilibria in the game. It will be interesting to identify a refinement concept for several Nash equilibria arising out of asymmetry in information or extra information to one player.\n\n\\section*{Acknowledgments}\nThis work was supported in part by the AFOSR MURI Grant FA9550-10-1-0573.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Detailed proofs}\\label{sec:proofs}\n\\subsection{Proof of Lemma~\\ref{lem:kappa-bound}}\n\\label{sec:proof-lem-1}\n\nAssuming that the actual MDP $M$ is in the extended MDP $M^+$, i.e., $\\bar{r}(s, a) \\in B_\\delta(s, a)$ and $p(\\cdot|s, a) \\in C_\\delta(s, a)$ for all $s \\in \\mathcal{S}, a \\in \\mathcal{A}$, we have\n$$ \\max_s u_i(s) - \\min_{s'} u_i(s') \\leq \\kappa(M) $$\nwhere $u_i(s)$ is the $i$-step optimal undiscounted value of state $s$.\n\n\\begin{proof}\n  By assumption, the actual mean rewards $\\bar{r}$ and transitions $p$ are contained in the extended MDP $M^+$, i.e., for any $s \\in \\mathcal{S}$ and $a \\in \\mathcal{A}$, $\\bar{r}(s, a) \\in B_\\delta(s, a)$ and $p(\\cdot|s, a) \\in C_\\delta(s, a)$.\n  Thus for any policy $\\pi : \\mathcal{S} \\rightarrow \\mathcal{A}$ in the actual MDP $M$, we can construct a corresponding policy $\\pi^+ : \\mathcal{S} \\rightarrow \\mathcal{A}^+$ in the extended MDP $M^+$\n  %\n  $$ \\pi^+(s) \\coloneqq \\Big( \\pi(s), p(\\cdot| s, \\pi(s)), \\bar{r}(s, \\pi(s)) \\Big). $$\n  %\n  Following $\\pi^+$ in $M^+$ induces the same stochastic process $(s_t, a_t, r_t)_{t \\geq 0}$ as following $\\pi$ in $M$. In particular they have the same expected hitting times and expected rewards.\n  By definition $u_i(s)$ is the value of following an \\emph{optimal} $i$-step non-stationary policy starting at $s$ in the extended MDP $\\mathcal{M}^+$. For any $s'$, by optimality, $u_i(s)$ must be no worse than first following $\\pi^+$ from $s$ to $s'$ and then following the optimal $i$-step non-stationary policy from $s'$ onward. Along the path from $s$ to $s'$, we receive rewards according to $\\sigma_3(\\pi^+) = \\bar{r}$ and after arriving at $s'$, we have missed at most $r_\\text{max} h_{s \\rightarrow s'}(M^+, \\pi^+)$-many rewards of $u_i(s')$ so in expectation\n  %\n  \\begin{align*}\n    u_i(s) &\\geq \\Exp\\left[ \\sum_{t=0}^{h_{s \\rightarrow s'}(M^+, \\pi^+) - 1} r_t \\right] + u_i(s') - \\Exp[r_\\text{max} h_{s \\rightarrow s'}(M^+, \\pi^+)] \\\\\n    &= \\Exp\\left[ \\sum_{t=0}^{h_{s \\rightarrow s'}(M^+, \\pi^+) - 1} r_t - r_\\text{max} \\right] + u_i(s') \\\\\n    &\\text{By definition of $\\pi^+$, hitting time $h_{s \\rightarrow s'}(M, \\pi) = h_{s \\rightarrow s'}(M^+, \\pi^+)$ } \\\\\n    &= \\Exp\\left[ \\sum_{t=0}^{h_{s \\rightarrow s'}(M, \\pi) - 1} r_t - r_\\text{max} \\right] + u_i(s').\n  \\end{align*}\n  %\n  Moving the terms around and we get\n  $$ u_i(s') - u_i(s) \\leq \\Exp\\left[ \\sum_{t=0}^{h_{s \\rightarrow s'}(M, \\pi) - 1} r_\\text{max} - r_t \\right]. $$\n\n  Since this holds for any $\\pi$ by optimality, we can choose one with the smallest expected hitting cost\n  $$ u_i(s') - u_i(s) \\leq \\min_{\\pi : \\mathcal{S} \\rightarrow \\mathcal{A}} \\Exp\\left[ \\sum_{t=0}^{h_{s \\rightarrow s'}(M, \\pi) - 1} r_\\text{max} - r_t \\right]. $$\n\n  Since $s, s'$ are arbitrary, we can maximize over pairs of states on both sides and get\n  $$ \\max_{s'} u_i(s') - \\min_s u_i(s) \\leq \\max_{s, s'} \\min_{\\pi : \\mathcal{S} \\rightarrow \\mathcal{A}} \\Exp\\left[ \\sum_{t=0}^{h_{s \\rightarrow s'}(M, \\pi) - 1} r_\\text{max} - r_t \\right] = \\kappa(M). $$\n\n  It should be noted that even in some cases where the hitting time is infinity---in a non-communicating MDPs for example---$\\kappa$ can still be finite and this inequality is still true! In these cases, $r_t = r_\\text{max}$ except for finitely many terms implying $\\rho^*(M, s) = r_\\text{max}$.\n\\end{proof}\n\n\\subsection{Proof of Theorem~\\ref{thm:factor-two}}\n\\label{sec:proof-thm-two}\n\nGiven an MDP $M$ with finite maximum expected hitting cost $\\kappa(M) < \\infty$ and an unsaturated optimal average reward $\\rho^*(M) < r_\\text{max}$, the maximum expected hitting cost of any PBRS-parametrized MDP $M^\\varphi$ is bounded by a multiplicative factor of two\n$$ \\frac{1}{2} \\kappa(M) \\leq \\kappa(M^\\varphi) \\leq 2 \\kappa(M). $$\n\n\\begin{proof}\n  We denote the expected hitting cost between two states $s, s'$ as\n  $$ c(s, s') \\coloneqq \\min_{\\pi : \\mathcal{S} \\rightarrow \\mathcal{A}} \\Exp \\left[ \\sum_{t=0}^{h_{s \\rightarrow s'}(M, \\pi) - 1} r_\\text{max} - r_t \\right]. $$\n\n  Suppose that the pair of states $(s, s')$ maximizes the expected hitting cost in $M$ which is assumed to be finite\n  $$ \\kappa(M) = c(s, s') < \\infty. $$\n  %\n  Furthermore, the condition that $\\rho^*(M) < r_\\text{max}$ implies that the hitting times are finite for the minimizing policies. This ensures that the destination state is actually hit in the stochastic process.\n\n  Considering the expected hitting cost of the reverse pair, $(s', s)$,\n  \\begin{equation}\\label{eq:max-pair}\n    \\kappa(M) = \\max \\{ c(s, s'), c(s', s) \\} \\leq c(s, s') + c(s', s)\n  \\end{equation}\n  since hitting costs are nonnegative.\n\n  With $\\varphi$-shaping,\n  \\begin{align*}\n    c^\\varphi(s, s') &= \\min_{\\pi : \\mathcal{S} \\rightarrow \\mathcal{A}} \\Exp \\left[ \\sum_{t=0}^{h_{s \\rightarrow s'}(M, \\pi) - 1} r_\\text{max} - r^\\varphi_t \\right] \\\\\n    &= \\min_{\\pi : \\mathcal{S} \\rightarrow \\mathcal{A}} \\Exp \\left[ \\sum_{t=0}^{h_{s \\rightarrow s'}(M, \\pi) - 1} r_\\text{max} - (r_t -\\varphi(s_t) + \\varphi(s_{t+1})) \\right] \\\\\n    %\n    &\\text{By telescoping sums} \\\\\n    %\n    &= \\min_{\\pi : \\mathcal{S} \\rightarrow \\mathcal{A}} \\Exp \\left[ \\varphi(s_0) - \\varphi(s_{h_{s \\rightarrow s'}(M, \\pi)}) + \\sum_{t=0}^{h_{s \\rightarrow s'}(M, \\pi) - 1} r_\\text{max} - r_t \\right] \\\\\n    %\n    &\\text{By definition of a finite hitting time, $s_{h_{s \\rightarrow s'}(M, \\pi)} = s'$} \\\\\n    %\n    &= \\varphi(s) - \\varphi(s') + \\min_{\\pi : \\mathcal{S} \\rightarrow \\mathcal{A}} \\Exp \\left[ \\sum_{t=0}^{h_{s \\rightarrow s'}(M, \\pi) - 1} r_\\text{max} - r_t \\right] \\\\\n    &= \\varphi(s) - \\varphi(s') + c(s, s') \\numberthis \\label{eq:shaped-cost}\n  \\end{align*}\n  and that the minimizing policy for a state pair will not change. Therefore,\n  %\n  \\begin{align*}\n    \\kappa(M^\\varphi) &\\\\\n    %\n    &\\text{By definition of MEHC} \\\\\n    %\n    &\\geq \\max \\{ c^\\varphi(s, s'), c^\\varphi(s', s) \\} \\\\\n    %\n    &\\text{By (\\ref{eq:shaped-cost})} \\\\\n    %\n    &= \\max \\{ c(s, s') + \\varphi(s) - \\varphi(s'), c(s', s) + \\varphi(s') - \\varphi(s) \\} \\\\\n    %\n    &\\text{The maximum is no smaller than half of the sum} \\\\\n    %\n    &\\geq \\frac{1}{2} [c(s, s') + c(s', s)] \\\\\n    %\n    &\\text{By (\\ref{eq:max-pair})} \\\\\n    %\n    &\\geq \\frac{1}{2} \\kappa(M).\n  \\end{align*}\n  We obtain the other half of the inequality by observing $M = (M^\\varphi)^{-\\varphi}$.\n\\end{proof}\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nIf we view RL as an engineering tool that ``compiles'' an arbitrary reward function into a behavior (as represented by a policy) in an environment, then a programmer's primary responsibility would be to craft a reward function that faithfully expresses the intended goal. However, this problem of reward design is complicated by practical concerns for the difficulty of learning. As recognized by \\citet[section 3.4]{kober2013reinforcement},\n\\begin{quote}\n    ``[t]here is also a trade-off between the complexity of the reward function and the complexity of the learning problem.''\n\\end{quote}\nAccurate rewards are often easy to specify in a sparse manner (reaching a position, capturing the king, etc), thus hard to learn, whereas dense rewards, providing more feedback, are harder to specify accurately, leading to incorrect trained behaviors. The recent rise of deep RL also exposes ``bugs'' in some of these designed rewards \\cite{clark2016faulty}. Our results show that the informativeness of rewards, an aspect of ``the complexity of the learning problem'' can be controlled to some extent by a well specified potential without inadvertently changing the intended behaviors of the original reward. Therefore, we propose to separate the definitional concern from the training concern. Rewards should be first defined to faithfully express the intended task, and then any extra knowledge can be incorporated via a shaping potential to reduce the sample complexity of training to obtain the same desired behaviors. That is not to say that it is generally easy to find a helpful potential making the rewards more informative.\n\nThough Theorem~\\ref{thm:factor-two} might be a disappointing result for PBRS, we wish to emphasize that this result most directly concerns algorithms whose regrets scale with MEHC, such as \\texttt{UCRL2}. It is conceivable that in a different setting such as discounted total rewards, or for a different RL algorithm, such as SARSA with epsilon-greedy exploration \\citep[footnote 4]{ng1999policy}, PBRS might have a greater impact on the learning efficiency.\n\n\\section{Introduction}\n\\label{sec:introduction}\nIn the average reward setting of reinforcement learning (RL) \\cite{puterman1994markov,sutton1998reinforcement}, an algorithm learns to maximize its average rewards by interacting with an \\emph{unknown} Markov decision process (MDP). Similar to analysis in multi-armed bandits and other online machine learning problems, (cumulative) regret provides a natural model to evaluate the efficiency of a learning algorithm. With the \\texttt{UCRL2} algorithm, \\citet{jaksch2010near} show a problem-dependent bound of $\\tilde{O}(DS\\sqrt{AT})$ on regret and an associated logarithmic bound on the expected regret, where $D$ is the diameter of the actual MDP (Definition~\\ref{def:diameter}), $S$ the size of the state space, and $A$ the size of the action space. Many subsequent algorithms \\cite{fruit2019exploitation} enjoy similar diameter-dependent bounds.\nThis establishes diameter as an important measure of complexity for an MDP.\nHowever, strikingly, this measure is independent of rewards and is a function of only the transitions.\nThis is obviously peculiar as two MDPs differing only in their rewards would have the same regret bounds even if one gives the maximum reward for all transitions.\nWe review the related key observation by \\citet{jaksch2010near}, and refine it with a new lemma (Lemma~\\ref{lem:kappa-bound}), establishing a reward-\\emph{sensitive} complexity measure that we refer to as the \\emph{maximum expected hitting cost} (MEHC, Definition~\\ref{def:mehc}), which tightens the regret bounds of \\texttt{UCRL2} and similar algorithms by replacing diameter (Theorem~\\ref{thm:regret-bound}).\n\nNext, with respect to this new complexity measure, we describe a notion of reward informativeness (Section~\\ref{sec:informative}). Intuitively speaking, in an environment, the \\emph{same} desired policies can be motivated by different (immediate) rewards.\nThese differing definitions of rewards can be more or less \\emph{informative} of useful actions, i.e., yielding high long-term rewards.\nTo formalize this intuition, we study a way to reparametrize rewards via potential-based reward shaping (PBRS)~\\cite{ng1999policy} that can produce different rewards with the same near-optimal policies (Section~\\ref{sec:pbrs}). We show that the MEHC changes under reparametrization by PBRS and, in turn, so do regret and sample complexity, substantiating this notion of informativeness. Lastly, we study the extent of its impact. In particular, we show that there is a factor-of-two limit on its impact on MEHC in a large class of MDPs (Theorem~\\ref{thm:factor-two}). This result and the concept of reward informativeness may be useful for a task designer crafting a reward function (Section~\\ref{sec:discussion}). The detailed proofs are deffered to Appendix~\\ref{sec:proofs}.\n\nThe main contributions of this work are two-fold:\n\\begin{itemize}\n  \\item We propose a new MDP structural parameter, maximum expected hitting cost (MEHC), that accounts for both transitions and rewards. This parameter replaces diameter in the regret bounds of several model-based RL algorithms.\n  \\item We show that potential-based reward shaping can change the maximum expected hitting cost of an MDP and thus the regret bound. This results in a set of equivalent MDPs with different learning difficulties as measured by regret. Moreover, we show that their MEHCs differ by a factor of at most two in a large class of MDPs.\n\\end{itemize}\n\n\\subsection{Related work}\n\\label{sec:related}\nThis work is closely related to the study of diameter as an MDP complexity measure~\\cite{jaksch2010near}, which is prevalent in the regret bounds of RL algorithms in the average reward setting~\\cite{fruit2019exploitation}. As noted by \\citet{jaksch2010near}, unlike some previous measures of MDP complexity such as the return mixing time~\\cite{kearns2002near,brafman2002r}, diameter depends only on the transitions, but not the rewards.\nThe core reason for the presence of diameter in the regret analysis is that it upper bounds the optimal value span of the extended MDP that summarizes the observations (Section~\\ref{sec:ofu} and Equation~\\ref{eq:diameter-bound}). We review and update this observation with a reward-dependent parameter we called maximum expected hitting cost (Lemma~\\ref{lem:kappa-bound}). Interestingly, the gap between diameter and MEHC can be arbitrarily large $\\kappa(M) \\leq r_\\text{max} D(M)$; there are MDPs with finite MEHC and infinite diameter. These MDPs are non-communicating but have saturated optimal average rewards $\\rho^*(M) = r_\\text{max}$. Intuitively, there is a state $s$ in these MDPs from which the learner cannot visit some other state $s'$, but can nonetheless achieve the maximum possible average reward, thus allowing for good regret guarantees; the unreachable states will not seem better than the reachable ones under the principle of optimism in the face of uncertainty (OFU).\nWe will use \\texttt{UCRL2} \\cite{jaksch2010near} as an example algorithm throughout the rest of the article, however the main results do not depend on it. In particular, with MEHC, its regret bounds are updated (Theorem~\\ref{thm:regret-bound}).\n\nAnother important comparison is with optimal bias span~\\cite{puterman1994markov,bartlett2009regal,fruit2018efficient}, a reward-dependent parameter of MDPs.\nHere, we again find that the gap can be arbitrarily large $sp(M) \\leq \\kappa(M)$.\\footnote{This inequality can be derived as a consequence of Lemma~\\ref{lem:kappa-bound} as $N(s,a) \\rightarrow \\infty$, $M^+$ has very tight confidence intervals around the actual transition and mean rewards of $M$. Observe that the span of $u_i$ is equal to $sp(M)$ at the limit of $i \\rightarrow \\infty$ \\cite[remark 8]{jaksch2010near}.} These non-communicating MDPs would have unsaturated optimal average reward $\\rho^*(M) < r_\\text{max}$.\nBut as shown elsewhere~\\cite{fruit2018near,fruit2018efficient}, extra knowledge of (some upper bound on) the optimal bias span is necessary for an algorithm to enjoy a regret that scales with this smaller parameter. In contrast, \\texttt{UCRL2}, which scales with MEHC, does not need to know the diameter or MEHC of the actual MDP.\n\nPotential-based reward shaping \\cite{ng1999policy} was originally proposed as a \\emph{solution technique} for a programmer to influence the sample complexity of their reinforcement learning algorithm, without changing the near-optimal policies in episodic and discounted settings.\nPrior theoretical analysis involving PBRS \\cite{ng1999policy,wiewiora2003potential,wiewiora2003principled,asmuth2008potential,grzes2017reward} mostly focuses on the consistency of RL against the shaped rewards, i.e., the resulting learned behavior is also (near-)optimal in the original MDP, while suggesting empirically that the sample complexity can be changed by a well specified potential.\nIn this work, we use PBRS to construct $\\Pi$-equivalent reward functions in the average reward setting (Section~\\ref{sec:informative}) and show that two reward functions related by a shaping potential can have different MEHCs, and thus different regrets and sample complexities (Section~\\ref{sec:pbrs}).\nHowever, a subtle but important technical requirement of $[0, r_\\text{max}]$-boundedness of MDPs makes it difficult to immediately apply our results (Section~\\ref{sec:pbrs} and Theorem~\\ref{thm:factor-two}) to the treatment of PBRS as a solution technique because an arbitrary potential function picked without knowledge of the original MDP may not preserve the $[0, r_\\text{max}]$-boundedness. Nevertheless, we think our work may bring some new perspectives to this topic.\n\n\\subsubsection*{Acknowledgments}\nThis work was supported in part by the National Science Foundation under Grant No.\\ 1830660. We thank Avrim Blum for many insightful comments. In particular, his challenge to finding a better example has led to Theorem~\\ref{thm:factor-two}. We also thank Ronan Fruit for a discussion on a concept similar to the proposed maximum expected hitting cost that he independently developed in his thesis draft.\n\n\n{\\small\n\\printbibliography\n}\n\n\\clearpage\n\n\n\\section{Results}\n\\label{sec:results}\n\n\\subsection{Markov decision process}\\label{sec:setting}\nA \\textit{Markov decision process} is defined by the tuple $M = (\\mathcal{S}, \\mathcal{A}, p, r)$, where $\\mathcal{S}$ is the state space, $\\mathcal{A}$ is the action space, $p : \\mathcal{S}\\times \\mathcal{A} \\rightarrow \\mathcal{P}(\\mathcal{S})$ is the transition function, and $r : \\mathcal{S} \\times \\mathcal{A} \\rightarrow \\mathcal{P}([0, r_\\text{max}])$ is the reward function %\nwith mean $\\bar{r}(s, a) \\coloneqq \\Exp[r(s, a)]$.\nWe assume that the state and action spaces are finite, with sizes $S \\coloneqq \\lvert \\mathcal{S}\\rvert$ and $A \\coloneqq \\lvert\\mathcal{A}\\rvert$, respectively.\nAt each time step $t = 0, 1, 2, \\ldots$, an algorithm $\\mathfrak{L}$ chooses an action $a_t \\in \\mathcal{A}$ based on the observations up to that point.\nThe state transitions to $s_{t+1}$ with probability $p(s_{t+1} \\vert s_t, a_t)$ and a reward $r_t \\in [0, r_\\text{max}]$ is drawn according to the distribution $r(s_t, a_t)$.\\footnote{It is important to assume that the support of rewards lies in a \\emph{known} bounded interval, often $[0, 1]$ by convention. This is sometimes referred to as a \\emph{bounded} MDP in the literature. Analogous to bandits, the details of the reward distribution often is unimportant, and it suffices to specify an MDP with the mean rewards $\\bar{r}$.} The transition probabilities and reward function of the MDP are unknown to the learner. The sequence of random variables $(s_t, a_t, r_t)_{t \\ge 0}$ forms a stochastic process. Note that a stationary deterministic policy $\\pi : \\mathcal{S} \\rightarrow \\mathcal{A}$ is a restrictive type of algorithm whose action $a_t$ depends only on $s_t$. We refer to stationary deterministic policies as policies in the rest of the paper.\n\nRecall that in a Markov chain, the \\emph{hitting time} of state $s'$ starting at state $s$ is a random variable $h_{s \\rightarrow s'} \\coloneqq \\inf \\{ t \\in \\mathbb{N}_{\\ge 0} \\vert s_t = s' \\text{ and } s_0 = s\\}$\\footnote{$0$-indexing ensures that $h_{s \\rightarrow s} = 0$. Note also that by convention, $\\inf \\varnothing = \\infty$.} \\cite{levin2008markov}.\n\n\\begin{defn}[Diameter, \\cite{jaksch2010near}] \\label{def:diameter}\nSuppose in the stochastic process induced by following a policy $\\pi$ in MDP $M$, the time to hit state $s'$ starting at state $s$ is $h_{s \\rightarrow s'}(M, \\pi)$. We define the \\emph{diameter} of $M$ to be\n$$ D(M) \\coloneqq \\max_{s, s' \\in \\mathcal{S}} \\min_{\\pi : \\mathcal{S} \\rightarrow \\mathcal{A}} \\Exp \\left[ h_{s \\rightarrow s'}(M, \\pi) \\right]. $$\n\\end{defn}\n\nWe incorporate rewards into diameter, and introduce a novel MDP parameter.\n\\begin{defn}[Maximum expected hitting cost] \\label{def:mehc}\nWe define the \\emph{maximum expected hitting cost} of a Markov decision process $M$ to be\n$$ \\kappa(M) \\coloneqq \\max_{s, s' \\in \\mathcal{S}} \\min_{\\pi : \\mathcal{S} \\rightarrow \\mathcal{A}} \\Exp \\left[ \\sum_{t=0}^{h_{s \\rightarrow s'}(M, \\pi) - 1} r_\\text{max} - r_t \\right]. $$\n\\end{defn}\nObserve that MEHC is a smaller parameter, that is, $\\kappa(M) \\leq r_\\text{max} D(M)$, since for any $s, s', \\pi$, we have $r_\\text{max} - r_t \\leq r_\\text{max}$.\n\n\\subsection{Average reward criterion, and regret} \\label{sec:regret}\nThe \\textit{accumulated reward} of an algorithm $\\mathfrak{L}$ after $T$ time steps in MDP $M$ starting at state $s$ is a random variable\n$$ R(M, \\mathfrak{L}, s, T) \\coloneqq \\sum_{t=0}^{T-1} r_t. $$\n\nWe define the \\textit{average reward} or \\textit{gain} \\cite{puterman1994markov} as\n\\begin{equation}\\label{eq:criterion}\n    \\rho(M, \\mathfrak{L}, s) \\coloneqq \\lim_{T \\rightarrow \\infty} \\frac{1}{T} \\Exp \\left[ R(M, \\mathfrak{L}, s, T) \\right].\n\\end{equation}\nWe will evaluate policies by their average reward. This can be maximized by a stationary deterministic policy and we define the \\textit{optimal average reward} of $M$ starting at state $s$ as\n\\begin{equation}\\label{eq:opt-avg-reward}\n   \\rho^*(M, s) \\coloneqq \\max_{\\pi : \\mathcal{S} \\rightarrow \\mathcal{A}} \\rho(M, \\pi, s).\n\\end{equation}\n\nFurthermore, we assume that the optimal average reward starting at any state to be the same, i.e., $\\rho^*(M, s) = \\max_{s'} \\rho^*(M, s')$ for any state $s$. This is a natural requirement of an MDP in the online setting to allow for any hope for a vanishing regret. Otherwise, the learner may take actions leading to states with a lower average optimal reward due to ignorance and incur linear regret when compared with the optimal policy starting at the initial state. In particular, this condition is true for communicating MDPs \\cite{puterman1994markov} by virtue of their transitions, but this is also possible for non-communicating MDPs with appropriate rewards. We will write $\\rho^*(M) \\coloneqq \\max_{s'} \\rho^*(M, s')$.\n\nWe will compete with the expected cumulative reward of an optimal policy \\emph{on its trajectory}, and define the \\emph{regret} of a learning algorithm $\\mathfrak{L}$ starting at state $s$ after $T$ time steps as\n\\begin{equation} \\label{eq:regret}\n   \\Delta(M, \\mathfrak{L}, s, T) \\coloneqq T \\rho^*(M) - R(M, \\mathfrak{L}, s, T).\n\\end{equation}\n\n\\subsection{Optimism in the face of uncertainty, extended MDP, and \\texttt{UCRL2}}\n\\label{sec:ofu}\nThe principle of optimism in the face of uncertainty (OFU) \\cite{sutton1998reinforcement} states that for uncertain state-action pairs, i.e., those that we have not visited enough up to this point, we should be optimistic about their outcome. The intuition for doing so is that taking reward-maximizing actions with respect to this optimistic model (in terms of both transitions and immediate rewards for these uncertain state-action pairs), we will have no regret if the optimism is well placed and will otherwise quickly learn more about these suboptimal state-action pairs to avoid them in the future. This fruitful idea has been the basis for many model-based RL algorithms \\cite{fruit2019exploitation} and in particular, \\texttt{UCRL2} \\cite{jaksch2010near}, which keeps track of the statistical uncertainty via upper confidence bounds.\n\nSuppose we have visited a particular state-action pair $(s, a)$ $N(s, a)$-many times. With confidence at least $1 - \\delta$, we can establish that a confidence interval for both its mean reward $\\bar{r}(s, a)$ and its transition $p(\\cdot|s, a)$ from the Chernoff-Hoeffding inequality (or Bernstein, \\cite{fruit2018near}). Let $b(\\delta, n) \\in \\mathbb{R}$ be the $\\delta$-confidence bound after observing $n$ i.i.d. samples of a $[0, 1]$-bounded random variable, $\\hat{r}(s, a)$ the empirical mean of $r(s, a)$, $\\hat{p}(\\cdot|s, a)$ the empirical transition of $p(\\cdot|s, a)$. The statistically plausible mean rewards are\n$$ B_\\delta(s, a) \\coloneqq \\bigl\\{r' \\in \\mathbb{R} : |r' - \\hat{r}(s, a)| \\leq r_\\text{max} \\, b(\\delta, N(s, a)) \\bigr\\} \\cap [0, r_\\text{max}] $$\nand the statistically plausible transitions are\n$$ C_\\delta(s, a) \\coloneqq \\bigl\\{p' \\in \\mathcal{P}(\\mathcal{S}) : ||p'(\\cdot) - \\hat{p}(\\cdot|s, a)||_1 \\leq b(\\delta, N(s, a)) \\bigr\\}. $$\n\nWe define an \\emph{extended MDP} $M^+ \\coloneqq (\\mathcal{S}, \\mathcal{A}^+, p^+, r^+)$ to summarize these statistics \\cite{givan2000bounded,strehl2005theoretical,tewari2007bounded,jaksch2010near}, where $\\mathcal{S}$ is the same state space as in $M$, the action space $\\mathcal{A}^+$ is a union over state-specific actions\n\\begin{equation}\\label{eq:a-plus}\n  \\mathcal{A}^+_s \\coloneqq \\bigl\\{ (a, p', r') : a \\in \\mathcal{A}, p' \\in C_\\delta(s, a), r' \\in B_\\delta(s, a) \\bigr\\},\n\\end{equation}\nwhere $\\mathcal{A}$ is the same action space in $M$, $p^+$ the transitions according to the selected distribution $p'$\n\\begin{equation}\\label{eq:p-plus}\n  p^+\\big(\\cdot|s, (a, p', r') \\big) \\coloneqq p'(\\cdot),\n\\end{equation}\nand $r^+$ is the rewards according to the selected mean reward $r'$\n\\begin{equation}\\label{eq:r-plus}\n  r^+\\big(s, (a, p', r')\\big) \\coloneqq r'.\n\\end{equation}\nIt is not hard to see that $M^+$ is indeed an MDP with an infinite but compact action space.\n\nBy OFU, we want to find an optimal policy for an optimistic MDP within the set of statistically plausible MDPs.\nAs observed in \\cite{jaksch2010near}, this is equivalent to finding an optimal policy $\\pi^+ : \\mathcal{S} \\rightarrow \\mathcal{A}^+$ in the extended MDP $M^+$, which specifies a policy in $M$ via $\\pi(s) \\coloneqq \\sigma_1(\\pi^+(s))$, where $\\sigma_i$ is the projection map onto the $i$-th coordinate (and an optimistic MDP $\\widetilde{M} = (\\mathcal{S}, \\mathcal{A}, \\widetilde{p}, \\widetilde{r})$ via transitions $\\widetilde{p}(\\cdot|s,\\pi(s)) \\coloneqq \\sigma_2(\\pi^+(s))$ and mean rewards $\\widetilde{r}(s, \\pi(s)) \\coloneqq \\sigma_3(\\pi^+(s))$ over actions selected by $\\pi$\\footnote{We can set transitions and mean rewards over actions $a \\neq \\pi(s)$ to $\\hat{p}$ and $\\hat{r}$, respectively.}).\n\nBy construction of the extended MDP $M^+$, $M$ is in $M^+$ with high confidence, i.e., $\\bar{r}(s, a) \\in B_\\delta(s, a)$ and $p(\\cdot|s, a) \\in C_\\delta(s, a)$ for all $s \\in \\mathcal{S}, a \\in \\mathcal{A}$.\nAt the heart of \\texttt{UCRL2}-type regret analysis, there is a key observation \\citep[equation (11)]{jaksch2010near} that we can bound the span of optimal values in the \\emph{extended} MDP $M^+$ by the diameter of the actual MDP $M$ under the condition that $M$ is in $M^+$.\nThis observation is needed to characterize how good following the ``optimistic'' policy $\\sigma_1(\\pi^+)$ in the actual MDP $M$ is.\nFor $i \\geq 0$, the \\emph{$i$-step optimal values} $u_i(s)$ of $M^+$ is the expected total reward by following an optimal non-stationary $i$-step policy starting at state $s \\in \\mathcal{S}$. We can also define them recursively (via dynamic programming\\footnote{In fact, the exact maximization of Equation~\\ref{eq:u-i} can be found via extended value iteration \\citep[section 3.1]{jaksch2010near}})\n$$ u_0(s) \\coloneqq 0 $$\n\\begin{align*}\n  u_{i+1}(s) &\\coloneqq \\max_{(a, p', r') \\in \\mathcal{A}^+_s} \\left[ r^+\\big(s, (a, p', r')\\big) + \\sum_{s'} p^+\\big(s'|s, (a, p', r')\\big) \\, u_i(s') \\right] \\\\\n  %\n  &\\text{By \\eqref{eq:p-plus} and \\eqref{eq:r-plus}} \\\\\n  %\n  &= \\max_{(a, p', r') \\in \\mathcal{A}^+_s} \\left[ r' + \\sum_{s'} p'(s') \\, u_i(s') \\right] \\\\\n  %\n  &\\text{By \\eqref{eq:a-plus}} \\\\\n  %\n  &= \\max_{a \\in \\mathcal{A}} \\left[ \\max_{r' \\in B_\\delta(s, a)} r' + \\max_{p' \\in C_\\delta(s, a)} \\sum_{s'} p'(s') \\, u_i(s') \\right] \\label{eq:u-i}\\numberthis\n\\end{align*}\n\nWe are now ready to restate the observation. If $M$ is in $M^+$, which happens with high probability, \\citet{jaksch2010near} observe that\n\\begin{equation}\\label{eq:diameter-bound}\n  \\max_s u_i(s) - \\min_{s'} u_i(s') \\leq r_\\text{max} D(M).\n\\end{equation}\n\nHowever, this bound is too conservative because it fails to account for the rewards collected. By patching this, we tighten the upper bound with MEHC.\n\n\\begin{lem}[MEHC upper bounds the span of values]\n\\label{lem:kappa-bound}\nAssuming that the actual MDP $M$ is in the extended MDP $M^+$, i.e., $\\bar{r}(s, a) \\in B_\\delta(s, a)$ and $p(\\cdot|s, a) \\in C_\\delta(s, a)$ for all $s \\in \\mathcal{S}, a \\in \\mathcal{A}$, we have\n$$ \\max_s u_i(s) - \\min_{s'} u_i(s') \\leq \\kappa(M) $$\nwhere $u_i(s)$ is the $i$-step optimal undiscounted value of state $s$.\n\\end{lem}\n\nThis refined upper bound immediately plugs into the main theorems of \\citep[equations 19 and 22, theorem 2]{jaksch2010near}.\n\n\\begin{thm}[Reward-sensitive regret bound of \\texttt{UCRL2}]\n\\label{thm:regret-bound}\nWith probability of at least $1 - \\delta$, for any initial state $s$ and any $T > 1$, and $\\kappa \\coloneqq \\kappa(M)$, the regret of \\texttt{UCRL2} is bounded by\n\\begin{align*}\n    &\\Delta(M, \\texttt{UCRL2}, s, T) \\\\\n    &\\quad \\leq \\sqrt{\\frac{5}{8} T \\log\\left(\\frac{8T}{\\delta}\\right)} + \\sqrt{T} + \\kappa \\sqrt{\\frac{5}{2} T \\log\\left(\\frac{8T}{\\delta}\\right)} + \\kappa SA \\log_2\\left( \\frac{8T}{SA} \\right) \\\\\n    &\\quad\\quad+ \\Bigg(\\kappa\\sqrt{14 S \\log\\left(\\frac{2AT}{\\delta}\\right)} + \\sqrt{14 \\log\\left(\\frac{2SAT}{\\delta}\\right)} + 2 \\Bigg) (\\sqrt{2} + 1) \\sqrt{SAT} \\\\\n    &\\quad \\leq 34 \\max\\{1, \\kappa\\} S \\sqrt{AT \\log\\left(\\frac{T}{\\delta}\\right)}.\n\\end{align*}\n\\end{thm}\n\nAs a corollary, Theorem~\\ref{thm:regret-bound} implies that \\texttt{UCRL2} offers $O\\left( \\frac{\\kappa^2 S^2 A}{\\varepsilon^2} \\log \\frac{\\kappa S A}{\\delta \\varepsilon} \\right)$ sample complexity~\\cite{kakade2003sample}, by inverting the regret bound by demanding that the per-step regret is at most $\\varepsilon$ with probability of at least $1-\\delta$ \\citep[corollary 3]{jaksch2010near}.\nSimilarly, we have an updated logarithmic bound on the expected regret \\citep[theorem 4]{jaksch2010near}, $ \\Exp [ \\Delta(M, \\texttt{UCRL2}, s, T) ] = O(\\frac{\\kappa^2 S^2 A \\log T}{g}) $ where $g$ is the gap in average reward between the best policy and the second best policy.\n\n\\subsection{Informativeness of rewards}\n\\label{sec:informative}\nInformally, it is not hard to appreciate the challenge imposed by delayed feedback inherent in MDPs, as actions with high immediate rewards do not necessarily lead to a high \\emph{optimal} value. Are there different but ``equivalent'' reward functions that differ in their \\emph{informativeness} with the more informative ones being easier to reinforcement learn? Suppose we have two MDPs differing only in their rewards, $M_1 = (\\mathcal{S}, \\mathcal{A}, p, r_1)$ and $M_2 = (\\mathcal{S}, \\mathcal{A}, p, r_2)$, then they will have the same diameters $D(M_1) = D(M_2)$ and thus the same diameter-dependent regret bounds from previous works. With MEHC, however, we may get a more meaningful answer.\n\nFirstly, let us make precise a notion of equivalence. We say that $r_1$ and $r_2$ are \\emph{$\\Pi$-equivalent} if for any policy $\\pi : \\mathcal{S} \\rightarrow \\mathcal{A}$, its average rewards are the same under the two reward functions $\\rho(M_1, \\pi, s) = \\rho(M_2, \\pi, s)$. Formally, we will study the MEHC of a class of $\\Pi$-equivalent reward functions related via a potential.\n\n\\subsection{Potential-based reward shaping}\n\\label{sec:pbrs}\nOriginally introduced by \\citet{ng1999policy}, potential-based reward shaping (PBRS) takes a potential $\\varphi : \\mathcal{S} \\rightarrow \\mathbb{R}$ and defines shaped rewards\n\\begin{equation}\\label{eq:pbrs}\n  r^\\varphi_t \\coloneqq r_t -\\varphi(s_{t}) + \\varphi(s_{t+1}).\n\\end{equation}\n\nWe can think of the stochastic process $(s_t, a_t, r^\\varphi_t)_{t\\geq 0}$ being generated from an MDP $M^\\varphi = (\\mathcal{S}, \\mathcal{A}, p, r^\\varphi)$ with reward function $r^\\varphi : \\mathcal{S} \\times \\mathcal{A} \\rightarrow \\mathcal{P}([0, r_\\text{max}])$\\footnote{One needs to ensure that $\\varphi$ respects the $[0, r_\\text{max}]$-boundedness of $M$.} whose mean rewards are\n$$ \\bar{r^\\varphi}(s, a) = \\bar{r}(s, a) -\\varphi(s) + \\Exp_{s' \\sim p(\\cdot|s,a)}\\left[ \\varphi(s') \\right]. $$\n\nIt is easy to check that $r^\\varphi$ and $r$ are indeed $\\Pi$-equivalent. For any policy $\\pi$,\n\\begin{align*}\n  \\rho(M^\\varphi, \\pi, s) &= \\lim_{T \\rightarrow \\infty} \\frac{1}{T} \\Exp \\left[ R(M^\\varphi, \\pi, s, T) \\right] \\\\\n    &= \\lim_{T \\rightarrow \\infty} \\frac{1}{T} \\Exp \\left[ \\sum_{t=0}^{T-1} r^\\varphi_t \\right] \\\\\n    &= \\lim_{T \\rightarrow \\infty} \\frac{1}{T} \\Exp \\left[ \\sum_{t=0}^{T-1} r_t - \\varphi(s_t) + \\varphi(s_{t+1}) \\right] \\\\\n    %\n    &\\text{By telescoping sums of potential terms over consecutive $t$} \\\\\n    %\n    &= \\lim_{T \\rightarrow \\infty} \\frac{1}{T} \\Exp \\left[ - \\varphi(s_0) + \\varphi(s_T) + \\sum_{t=0}^{T-1} r_t \\right] \\\\\n    &= \\lim_{T \\rightarrow \\infty} \\frac{1}{T} \\Big(  - \\varphi(s) + \\Exp[\\varphi(s_T)] + \\Exp\\left[ R(M, \\pi, s, T) \\right] \\Big) \\\\\n    %\n    &\\text{The first two terms vanish in the limit} \\\\\n    %\n    &= \\lim_{T \\rightarrow \\infty} \\frac{1}{T} \\Exp\\left[ R(M, \\pi, s, T) \\right] \\\\\n    &= \\rho(M, \\pi, s). \\numberthis \\label{eq:shaped-avg-reward}\n\\end{align*}\n\nTo get some intuition, it is instructive to consider a toy example (Figure~\\ref{fig:example}).\nSuppose $0 < \\beta < \\alpha$ and $\\epsilon \\in (0, 1)$, then the optimal average reward in this MDP is $1 - \\beta$, and the optimal stationary deterministic policy is $\\pi^*(s_1) \\coloneqq a_2$ and $\\pi^*(s_2) \\coloneqq a_1$, as staying in state $s_2$ yields the highest average reward.\nAs the expected number of steps needed to transition from state $s_1$ to $s_2$ and vice versa are both $\\nicefrac{1}{\\epsilon}$ via action $a_2$, we conclude that $\\kappa(M) = \\max\\{ \\alpha, \\nicefrac{\\alpha}{\\epsilon}, \\nicefrac{\\beta}{\\epsilon}, \\beta \\} = \\nicefrac{\\alpha}{\\epsilon}$.\nFurthermore, notice that taking action $a_2$ in either state transitions to the other state with probability of $\\epsilon$, however the immediate rewards are the same as taking the alternative action $a_1$ to stay in the current state---the immediate rewards are not \\emph{informative}.\nWe can differentiate the actions better by shaping with a potential of $\\varphi(s_1) \\coloneqq 0$ and $\\varphi(s_2) \\coloneqq \\nicefrac{(\\alpha - \\beta)}{2 \\epsilon}$.\nThe shaped mean rewards become, at $s_1$,\n$$ \\bar{r^\\varphi}(s_1, a_2) = 1 - \\alpha - \\varphi(s_1) + \\epsilon \\varphi(s_2) + (1 - \\epsilon) \\varphi(s_1) = 1 - \\nicefrac{(\\alpha + \\beta)}{2} > 1 - \\alpha = \\bar{r^\\varphi}(s_1, a_1) $$\nand at $s_2$,\n$$ \\bar{r^\\varphi}(s_2, a_2) = 1 - \\beta - \\varphi(s_2) + \\epsilon \\varphi(s_1) + (1 - \\epsilon) \\varphi(s_2) = 1 - \\nicefrac{(\\alpha + \\beta)}{2} < 1 - \\beta = \\bar{r^\\varphi}(s_2, a_1). $$\nThis encourages taking actions $a_2$ at state $s_1$ and discourages taking actions $a_1$ at state $s_2$ simultaneously. The maximum expected hitting cost becomes smaller\n\\begin{align*}\n    \\kappa(M^\\varphi) &= \\max \\left\\{ \\alpha, \\beta, \\varphi(s_1) - \\varphi(s_2) + \\frac{\\alpha}{\\epsilon}, \\, \\varphi(s_2) - \\varphi(s_1) + \\frac{\\beta}{\\epsilon} \\right\\} \\\\\n    &= \\max \\left\\{ \\alpha, \\beta, \\frac{\\alpha + \\beta}{2 \\epsilon}, \\, \\frac{\\alpha + \\beta}{2 \\epsilon} \\right\\} \\\\\n    &= \\frac{\\alpha + \\beta}{2 \\epsilon} \\\\\n    &< \\frac{\\alpha}{\\epsilon} = \\kappa(M).\n\\end{align*}\n\n\\begin{figure}[!tb]\n  \\centering\n  \\begin{tikzpicture}[->, >=stealth', shorten >=1pt, auto, semithick]\n    \\tikzstyle{action} = [draw=black,fill=none]\n       \n        \\node[state] at (-2, 0) (s1) {$s_1$};\n        \\node[state] at (2, 0) (s2) {$s_2$};\n\n       \n        \\node[action, left=of s1] (s1a1) {$a_1$};\n        \\node[action, above right=of s1] (s1a2) {$a_2$};\n\n        \\node[action, right=of s2] (s2a1) {$a_1$};\n        \\node[action, below left=of s2] (s2a2) {$a_2$};\n\n       \n        \\path (s1a2) edge [bend left] node {$\\epsilon$} (s2)\n                     edge [bend left] node {$1-\\epsilon$} (s1);\n        \\path (s2a2) edge [bend left] node {$\\epsilon$} (s1)\n                     edge [bend left] node {$1-\\epsilon$} (s2);\n        \\path (s1a1) edge [bend left] node {$1$} (s1);\n        \\path (s2a1) edge [bend left] node {$1$} (s2);\n\n       \n        \\path (s1) edge [-, dashed] node [text=red] {$1-\\alpha$} (s1a1);\n        \\path (s1) edge [-, dashed] node [text=red] {$1-\\alpha$} (s1a2);\n        \\path (s2) edge [-, dashed] node [text=red] {$1-\\beta$} (s2a1);\n        \\path (s2) edge [-, dashed] node [text=red] {$1-\\beta$} (s2a2);\n    \\end{tikzpicture}\n  \\caption{Circular nodes represent states and square nodes represent actions. The solid edges are labeled by the transition probabilities and the dashed edges are labeled by the mean rewards. Furthermore, $r_\\text{max} = 1$. For concreteness, one can consider setting $\\alpha = 0.11, \\beta = 0.1, \\epsilon = 0.05$.}\n  \\label{fig:example}\n\\end{figure}\n\nIn this example, MEHC is halved at best when $\\beta$ is made arbitrarily close to zero. Noting that the original MDP $M$ is equivalent to $M^\\varphi$ shaped with potential $-\\varphi$, i.e. $M = (M^\\varphi)^{-\\varphi}$ from (\\ref{eq:pbrs}), we see that MEHC can be almost doubled. It turns out that halving or doubling the MEHC is the most PBRS can do in a large class of MDPs.\n\n\\begin{thm}[MEHC under PBRS]\n\\label{thm:factor-two}\nGiven an MDP $M$ with finite maximum expected hitting cost $\\kappa(M) < \\infty$ and an unsaturated optimal average reward $\\rho^*(M) < r_\\text{max}$, the maximum expected hitting cost of any PBRS-parameterized MDP $M^\\varphi$ is bounded by a multiplicative factor of two\n$$ \\frac{1}{2} \\kappa(M) \\leq \\kappa(M^\\varphi) \\leq 2 \\kappa(M). $$\n\\end{thm}\n\nThe key observation is that the expected total rewards along a loop remains unchanged by shaping, which originally motivated PBRS \\cite{ng1999policy}. To see this, consider a loop as a concatenation of two paths, one from $s$ to $s'$ and the other from $s'$ to $s$. Under the shaping of a potential $\\varphi$, the expected total rewards of the former is increased by $\\varphi(s') - \\varphi(s)$ and the latter is decreased by the same amount. For more details, see Appendix~\\ref{sec:proof-thm-two}.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Review of van Kampen's microscopic theory}\\label{App1}\n\tWe here review the microscopic derivation of the Gaussian Langevin equation. \n\tLet us consider a system driven by a single stochastic environment as \n\t\\begin{equation}\n\t\t\\frac{d\\hv}{dt} = \\hFA,\n\t\\end{equation}\n\twhere $\\hFA$ is a stochastic force from the single environment. \n\tWe here make a critical assumption that $\\hFA$ is scaled by a positive small number $\\eps$ as \n\t\\begin{equation}\n\t\t\\hFA = \\eps \\hat{\\eta}_A(t;\\hv), \\label{spp:assmp:scale_SDE}\n\t\\end{equation}\n\twhere $\\hat{\\eta}_A$ is a Markovian jump force independent of $\\eps$. \n\tLet us write the jump rate of $\\hat{\\eta}_A(t;\\hv)$ as $\\overline{W}(v;\\mc{Y})$, \n\twhere $\\overline{W}(v;\\mc{Y})$ is the $\\eps$-independent transition probability per unit time on the condition $\\hat v(t)=v$ with the amplitude of the Poisson noise $\\mc{Y}$. \n\tWe assume that the master equation for the velocity $\\hat v$ is given by\n\t\\begin{equation}\n\t\t\\frac{\\partial P(v,t)}{\\partial t} = \\int_{-\\infty}^\\infty dy[P(v-y,t)W_\\eps(v-y;y)-P(v,t)W_\\eps(v,y)],\\label{spp:eq:master_eq}\n\t\\end{equation}\n\twhere $P(v,t)\\equiv P(\\hv(t)=v)$ is the probability distribution function (PDF) for the system's velocity and $W_\\eps(v;y)$ is the transition rate for $\\hv$ on the condition $\\hv(t)=v$ with velocity jump $y$. \n\tConsidering the relation~{(\\ref{spp:assmp:scale_SDE})}, $y$ and $\\mc{Y}$ are connected by $y=\\eps \\mc{Y}$. \n\tThen, the Jacobian relation holds as \n\t\\begin{equation}\n\t\tW_\\eps(v;y)dy = \\overline{W}(v;\\mc{Y})d\\mc{Y} \\Longleftrightarrow W_\\eps(v;y)=\\frac{1}{\\eps}\\overline{W}\\left(v;\\frac{y}{\\eps}\\right). \\label{spp:eq:scaling_kernel}\n\t\\end{equation}\n\tThe Kramers-Moyal expansion for the master equation is given by\n\t\\begin{equation}\n\t\t\\frac{\\partial P(v,t)}{\\partial t} = \\sum_{n=1}^\\infty \\frac{(-\\eps)^n}{n!}\\frac{\\partial^n}{\\partial v^n}[\\alpha_n(v)P(v,t)],\\label{spp:eq:Kramers_Moyal}\n\t\\end{equation}\n\twhere we have introduced the Kramers-Moyal coefficient\n\t\\begin{equation}\n\t\t\\alpha_n(v) \\equiv \\int_{-\\infty}^\\infty d\\mc{Y}\\overline{W}(v;\\mc{Y})\\mc{Y}^n. \n\t\\end{equation}\n\tWe here assume the stability condition for the noise around $\\hv=0$: \n\t\\begin{equation}\n\t\t\\alpha_1^{(0)} = 0, \\>\\>\\> \\alpha^{(1)}_1 = -\\bar{\\gamma}_A < 0, \\>\\>\\> \\alpha_2^{(0)}=2\\bar{\\gamma}_A\\bar{T}_A > 0,\n\t\\end{equation}\n\twhere $\\alpha_1(v)$ has a single zero point and we introduce the expansion $\\alpha_n(v)=\\sum_{k=0}^\\infty v^k\\alpha_n^{(k)}\/k!$. \n\tBy introducing the following scaled variables as $\\tau = \\eps t$, $V = v\/\\sqrt{\\eps}$, Eq.~{(\\ref{spp:eq:Kramers_Moyal})} is reduced to\n\t\\begin{align}\n\t\t&\\frac{\\partial P(V,\\tau)}{\\partial \\tau} \n\t\t\t\t\t\t\t\t\t\t\t\t= \\bigg[\\frac{\\partial }{\\partial V}\\bar{\\gamma}_AV - \\sum_{k=2}^\\infty \\frac{\\eps^{(k-1)\/2}}{k!}\\frac{\\partial }{\\partial V}\\alpha_1^{(k)}V^k\\notag\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t&+\\sum_{n=2}^\\infty\\sum_{k=0}^\\infty \\frac{(-1)^n\\eps^{(n+k)\/2-1}}{n!k!}\\frac{\\partial^n}{\\partial V^n}V^k\\alpha_n^{(k)}\\bigg]P(V,\\tau)\\notag\\\\\n\t\t\t\t\t\t\t\t\t\t\t\t &= \\bar{\\gamma}_A\\left[\\frac{\\partial }{\\partial V}V + \\bar{T}_A\\frac{\\partial^2 }{\\partial V^2}\\right]P(V,\\tau) + O(\\eps^{1\/2}),\\label{spp:eq:derived_FP}\n\t\\end{align}\n\twhich is equivalent to the Gaussian Langevin equation~{(5)} in the main text.\n\t\n\tWe here note that the scaling~{(\\ref{spp:eq:scaling_kernel})} is essentially equivalent to that introduced by van Kampen~\\cite{vanKampen,vanKampenB,Gardiner}, \n\twhere $\\eps$ corresponds to the inverse of the system size $\\Omega_{\\rm sys}$ as $\\Omega_{\\rm sys}\\equiv 1\/\\eps$. \n\tTo see this point, let us transform velocity variables\n\t\\begin{equation}\n\t\ta' \\equiv \\frac{v}{\\eps}=\\Omega_{\\rm sys} v, \\>\\>\\> \\Delta a \\equiv \\frac{y}{\\eps} = \\Omega_{\\rm sys} y.\n\t\\end{equation}\n\tThen, the scaling assumption~{(\\ref{spp:eq:scaling_kernel})} is equivalent to \n\t\\begin{equation}\n\t\tW(a';\\Delta a) = \\Omega_{\\rm sys} \\overline{W}\\left(\\frac{a'}{\\Omega_{\\rm sys}}, \\Delta a\\right),\\label{spp:eq:scaling_kernel_Omega}\n\t\\end{equation}\n\twhere we have introduced the transition rate for the transformed variable $a'$ as $W(a';\\Delta a)\\equiv W_\\eps(v;y)$. \n\tThe scaling~{(\\ref{spp:eq:scaling_kernel_Omega})} is exactly equal to that introduced in Ref.~\\cite{Gardiner} on page 277. \n\t\n\t\\subsection*{Note on the invalidity of the Langevin description for rare trajectories}\n\tWe here note that the Langevin equation is an effective description only for typical trajectories, where one does not observe rare trajectories. \n\tIn the above discussion, we have implicitly assumed that the scaled velocity $V$ is not much larger than the typical velocity $V^*\\equiv \\sqrt{\\overline{T}_A}$ (i.e., $|V|\\lesssim V^*$). \n\tIn fact, when $V$ is much larger than $V^*$ as $V\/V^* = O(1\/\\eps)$, all of the terms on the right hand side of Eq.~{(\\ref{spp:eq:derived_FP})} are relevant, which implies the invalidity of the system size expansion for $|V|\\gg V^*$ \n\t(i.e., the distribution of the tail cannot be well described by the Langevin description in general). \n\tThis is because the system size expansion is not a uniform asymptotic expansion in terms of the velocity. \n\tFortunately, however, the probability of such rare trajectories is estimated to be extremely small and is irrelevant to averages of ordinary physical quantities. \n\tThe Langevin description is proved to be effectively valid in this sense. \n\t\n\t\\subsection*{Note on the validity of the van Kampen's theory for systems without microscopic reversibility}\n\tWe note that the system size expansion is valid for genuine nonequilibrium systems without time-reversal symmetry because it is based on the central limit theorem. \n\tIndeed, the effectiveness of the Gaussian Langevin description is discussed for some granular systems in Refs.~\\cite{Brey,Brilliantov}, \n\twhere the starting points of the models are given by master equations without time-reversal symmetry. \n\tWe also note that these results do not contradict the time-reversal symmetry of the Langevin model, because the Langevin model is just an effective description for typical trajectories. \n\tEven if the system is well-described by the Langevin equation for typical trajectories, the time-irreversal symmetry can be observed in general for rare trajectories. \n\t\n\t\\subsection*{Note on the state-dependence of noise}\n\tWe here note that the fluctuation described by the master equation~{(\\ref{spp:eq:master_eq})} is state-dependent noise, which is not simple white noise. \n\tIn fact, the transition rate $W_\\eps(v;y)$ for the velocity jump $y$ depends on $v$, \n\twhich implies the strong correlation between the system and the environment. \n\tWe also note that the state-dependent noise cannot be generally written as a single multiplicative noise, \n\tbecause the time-series of the Poisson flights is independent of the state of the system for the single multiplicative noise. \n\t\n\t\t\n\\section{Relation to the independent kick model}\\label{App2}\n\tThe non-Gaussian Langevin equation reproduces the independent kick model in the strong friction limit. \n\tThe independent kick was originally introduced to explain the behavior of the granular motor in the presence of the dry friction~{\\cite{Talbot,Gnoli1,Gnoli2,Gnoli3}}. \n\tAccording to the main text, the steady distribution of the non-Gaussian Langevin equation is represented in the Fourier space as  \n\t\\begin{equation}\n\t\t\\tl{P}_{\\rm SS}(s) = \\exp{\\left[\\int_0^s \\frac{ds'\\Phi(s')}{\\gamma s'}\\right]}, \\label{spp:g_Fdist}\n\t\\end{equation}\n\twhere the cumulant function is given by\n\t\\begin{equation}\n\t\t\\Phi(s)=-\\gamma \\mc{T}s^2+\\int_{-\\infty}^\\infty d\\mc{Y}\\overline{W}(0;\\mc{Y})(e^{i\\mc{Y}s}-1)\n\t\\end{equation}\n\twith the Poisson transition rate $\\overline{W}(0;\\mc{Y})$.\n\tLet us consider the case without the Gaussian part ($\\mc{T}=0$). \n\tIn the strong friction limit $\\gamma \\rightarrow \\infty$, Eq.~{(\\ref{spp:g_Fdist})} is reduced to\n\t\\begin{align}\n\t\t\\tl{P}_{\\rm SS}(s) \t&= 1 + \\int_0^s \\frac{ds'\\Phi(s')}{\\gamma s'} + O(\\gamma^{-2})\\notag\\\\\n\t\t\t\t\t\t\t\t\t\t\t&= 1 + \\int_{-\\infty}^\\infty d\\mc{Y}\\int_0^s ds' \\frac{\\mc{W}(\\mc{Y})}{\\gamma s'}(e^{is'\\mc{Y}}-1) + O(\\gamma^{-2}).\\label{spp:eq:independent}\n\t\\end{align}\n\t\n\t\n\tOn the other hand, the system is kicked by rare collisions and instantly relaxes to the rest state in the independent kick model~{\\cite{Talbot,Gnoli1,Gnoli2,Gnoli3}}. \n\tThis implies the following scenario. The system is typically in the rest state $\\mc{V}=0$. \n\tHowever, an occasional collision at time $t=0$ with a Poisson flight distance $\\mc{Y}$ changes the state from $\\mc{V}(-0)=0$ to $\\mc{V}(+0)=\\mc{Y}$, \n\tand the system freely relaxes as $\\mc{V}(t)=\\mc{Y}e^{-\\gamma t}$. \n\tThis scenario leads to \n\t\\begin{equation}\n\t\t\\la h(\\hat{\\mc{V}})\\ra \\simeq \\int_{-\\infty}^\\infty d\\mc{Y}\\overline{W}(0;\\mc{Y})\\int_0^\\infty dth(\\mc{V}(t)),\\label{spp:eq:IKM_general}\n\t\\end{equation}\n\twhere $h(v)$ is an arbitrary function.\n\tSubstituting $h(\\mc{V})=e^{is\\mc{V}}-1$ into Eq.~{(\\ref{spp:eq:IKM_general})}, we obtain \n\t\\begin{align}\n\t\t\\tl{P}_{\\rm SS}(s) - 1 \t&\\simeq \\int_{-\\infty}^\\infty d\\mc{Y}\\overline{W}(0;\\mc{Y})\\int_0^\\infty dt\\left\\{\\exp{\\left[is\\mc{Y}e^{-\\gamma t}\\right]}-1\\right\\}\\notag\\\\\n\t\t\t\t\t\t\t\t&= \\int_{-\\infty}^\\infty d\\mc{Y}\\overline{W}(0;\\mc{Y})\\int_0^\\infty dt\\sum_{n=1}^\\infty \\frac{1}{n!}(is\\mc{Y})^ne^{-\\gamma nt}\\notag\\\\\n\t\t\t\t\t\t\t\t&= \\int_{-\\infty}^\\infty d\\mc{Y}\\overline{W}(0;\\mc{Y})\\sum_{n=1}^\\infty \\frac{(is\\mc{Y})^n}{n\\gamma n!}\\notag\\\\\n\t\t\t\t\t\t\t\t&= \\int_{-\\infty}^\\infty d\\mc{Y}\\int_0^s ds'\\frac{\\overline{W}(0;\\mc{Y})}{\\gamma s'}(e^{is'\\mc{Y}}-1),\n\t\\end{align}\n\twhich is equivalent to Eq.~{(\\ref{spp:eq:independent})}. \n\tThus, our theory is equivalent to the independent kick model in the strong friction limit.\n\n\\section{Granular motor under the viscous and dry frictions}\\label{App3}\n\t\\subsection{Setup}\n\t\tWe consider a granular motor under the viscous friction. \n\t\tThe motor is a cuboid of height $h$, width $w$, and length $l$ in the granular gas as in Fig~{\\ref{fig:gmotor}(a)}. \n\t\tThe cuboid rotates around the $z$-axis, and the rotational angle $\\hat \\theta$ fluctuates because of collisional impacts by surrounding granular particles. \n\t\tWe assume that there exists the Coulombic friction during the rotation around the axis. \n\t\tLet us first consider its collision rules (see Fig.~{\\ref{fig:gmotor}(b)}). \n\t\tWe assume that the motor and a particle collide at the position $\\vec r$.\n\t\tWe denote the motor's angular velocity and particle's velocity by $\\omega$ and $\\vec v$, respectively. \n\t\tThe moment of inertia along the $z$-axis and the radius of inertia are respectively given by $I$ and $R_I\\equiv \\sqrt{I\/M}$. \n\t\tThe conservation of the angular momentum and the definition of the restitution coefficient $e$ are respectively given by \n\t\t\\begin{equation}\n\t\t\tI\\omega \\vec e_z + m \\vec r\\times \\vec v = I\\omega' \\vec e_z + m \\vec r\\times \\vec v',\\>\\>\\>\\>\n\t\t\t-\\frac{(\\vec V' -\\vec v')\\cdot \\vec n}{(\\vec V-\\vec v)\\cdot \\vec n} = e,\\label{am_dr}\n\t\t\\end{equation}\n\t\twhere $\\omega'$, $\\vec V'$ and $\\vec v'$ are the angular velocity of the motor, the velocity of the motor and the velocity of the particle after the collision, respectively,\n\t\tand $\\vec n$ is the normal unit vector on the surface, and $\\vec e_z \\equiv (0,0,1)$.\n\t\tWe assume the non-slip condition for the collision: the velocity change of the particle is perpendicular to the surface as\n\t\t\\begin{equation}\n\t\t\t\\vec v' = \\vec v +\\beta \\vec n\\label{nsc}\n\t\t\\end{equation}\n\t\twith an appropriate coefficient $\\beta$. \n\t\tWe note the following relations: \n\t\t\\begin{equation}\n\t\t\t\\vec V = \\omega \\vec e_z \\times \\vec r,\\>\\>\\> \\vec V' = \\omega' \\vec e_z \\times \\vec r.\\label{r_v_av}\n\t\t\\end{equation}\n\t\tSolving Eqs.~{(\\ref{am_dr})}, {(\\ref{nsc})}, and~{(\\ref{r_v_av})}, we obtain \n\t\t\\begin{equation}\n\t\t\t\\Delta \\omega\\equiv  (1+e)\\frac{\\Delta \\vec V\\cdot \\vec n}{R_I}\\frac{\\epsilon (\\vec r\\cdot \\vec t\/R_I)}{1+\\epsilon (\\vec r\\cdot \\vec t\/R_I)^2},\\>\\>\\>\n\t\t\t\\beta = \\frac{(1+e)\\Delta \\vec V\\cdot \\vec n}{1+\\epsilon (\\vec r\\cdot \\vec t\/R_I)^2},\\label{col_rule}\n\t\t\\end{equation}\n\t\twhere we introduced $\\vec t\\equiv \\vec e_z \\times \\vec e$, $\\Delta \\vec V\\equiv \\vec V-\\vec v$, and $\\epsilon\\equiv m\/M$. \n\t\tBased on the collision rule~{(\\ref{col_rule})}, we model this setup as the Boltzmann-Lorentz equation~{\\cite{Talbot,Gnoli1,Gnoli2,Gnoli3,Brilliantov}}: \n\t\t\\begin{align}\n\t\t\t&\\frac{\\partial }{\\partial t}P(\\omega,t) = \\gamma\\left[\\frac{\\partial }{\\partial \\omega} \\omega +\\frac{T}{I}\\frac{\\partial^2 }{\\partial \\omega^2}\\right]P(\\omega,t) \\notag\\\\\n\t\t\t&\t+\\int \\!\\!dy\\left[P(\\omega\\!-\\!y,t)W_\\eps(\\omega\\!-\\!y;y)\\!-\\!P(\\omega,t)W_\\eps(\\omega;y)\\right],\\label{eq:Suppl:BLeq}\\\\\n\t\t\t&W_\\eps(\\omega ;y) \\!=\\!\\rho h\\!\\int\\! ds\\! \\int\\! d\\vec v f(\\vec v)\\Theta(\\Delta \\vec V\\cdot \\vec n)|\\Delta \\vec V\\cdot\\vec n|\\delta(y-\\Delta \\omega),\n\t\t\\end{align}\n\t\twhere $s$ is the coordinate along the cuboid, $f(\\vec v)$ is the granular distribution function, $\\gamma$ is the coefficient of the viscous friction, \n\t\t$\\vec n(s)$ is the normal unit vector to the surface at $s$, and we have introduced \n\t\t\\begin{align}\n\t\t\t&\\vec V(s) \\equiv \\omega\\vec e_z \\times \\vec r(s), \\>\\> g(s)\\equiv \\frac{\\vec r(s)\\cdot \\vec t(s)}{R_I}, \n\t\t\t\\>\\> \\vec t(s)\\equiv \\vec e_z\\times \\vec n(s), \\notag\\\\\n\t\t\t&\\Delta \\vec V(s) \\equiv \\vec V(s)-\\vec v,\n\t\t\t\\>\\> \\Delta \\omega \\equiv \\frac{\\Delta \\vec V\\cdot \\vec n}{R_I}\\frac{(1+e)\\epsilon g(s)}{1+\\epsilon g^2(s)}.\n\t\t\\end{align}\n\t\tAccording to the Kramers-Moyal expansion, we obtain the differential form of the master equation as\n\t\t\\begin{align}\n\t\t\t&\\frac{\\partial P(\\omega,t)}{\\partial t}= \\gamma\\left[\\frac{\\partial }{\\partial \\omega}\\omega + \\frac{T}{I}\\frac{\\partial^2 }{\\partial \\omega^2}\\right]P(\\omega,t)\\notag\\\\\n\t\t\t\t&+\\sum_{n=1}^\\infty \\frac{(-1)^n}{n!}\\frac{\\partial^n }{\\partial \\omega^n}K_n(\\omega)P(\\omega,t),\n\t\t\\end{align}\n\t\twhere we have introduced the Kramers-Moyal coefficients\n\t\t\\begin{align}\n\t\t\t&K_n(\\omega)\t\t= \\int ds \\int d\\vec v (\\Delta\\omega)^n \\rho h f(\\vec v)\\Theta(\\Delta \\vec V\\cdot \\vec n)|\\Delta \\vec V\\cdot \\vec n| \\notag\\\\\n\t\t\t\t\t\t\t&= \\rho h\\!\\!\\int ds\\!  \\left[\\frac{\\epsilon (1+e)g(s)}{R_I(1+\\epsilon g^2(s))}\\right]^n\n\t\t\t\t\t\t\t\t\\!\\!\\int \\!\\!d\\vec v f(\\vec v)\\Theta(\\Delta \\vec V\\cdot \\vec n)(\\Delta \\vec V\\cdot \\vec n)^{n+1}.\n\t\t\\end{align}\n\t\n\t\t\n\t\\subsection{Small noise expansion}\n\t\tWe consider the following four assumptions: \n\t\t(i) {\\it $\\epsilon$ is a small positive parameter,}\n\t\t(ii) {\\it $\\gamma$ is a small positive number independent of $\\epsilon$,}  \n\t\t(iii) {\\it $T$ is scaled as $T=\\epsilon^2\\mc{T}$, where $\\mc{T}$ is independent of $\\epsilon$, } and\n\t\t(iv) {\\it $f(\\vec v)$ is isotropic as $f(\\vec v)=\\phi(|\\vec v|)$.}\n\t\tIntroducing a scaled variable $\\hat \\omega=\\epsilon\\hat \\Omega$, we obtain the scaled master equation as \n\t\t\\begin{align}\n\t\t\t\\frac{\\partial \\mc{P}(\\Omega,t)}{\\partial t}&= \\gamma \\left[\\frac{\\partial }{\\partial \\Omega}\\Omega+\\frac{\\mc{T}}{I}\\frac{\\partial^2 }{\\partial \\Omega^2}\\right]\\mc{P}(\\Omega,t) \\notag\\\\\n\t\t\t\t&+\\sum_{n=1}^\\infty \\frac{(-1)^n}{n!}\\frac{\\partial^n }{\\partial \\Omega^n}\\mathcal{K}_n(\\Omega)\\mc{P}(\\Omega,t), \\label{eq:Suppl:scaled_ME}\n\t\t\\end{align}\n\t\twith the scaled Kramers-Moyal coefficients \n\t\t\\begin{align}\n\t\t\t&\\mathcal{K}_n(\\Omega) = \\rho h\\int ds  \\frac{(1+e)^ng^n(s)}{R_I^n(1+\\epsilon g^2(s))^n}\\times \\notag\\\\\n\t\t\t\\int d\\vec v &\\phi(|\\vec v|)\\Theta((\\epsilon \\vec {\\mathcal{V}}(s)-\\vec v)\\cdot \\vec n)[(\\epsilon\\vec {\\mathcal{V}}(s)-\\vec v)\\cdot \\vec n]^{n+1}, \n\t\t\\end{align}\n\t\twhere $\\mathcal{\\mathcal{V}}=\\Omega\\vec e_z \\times \\vec r(s)$. \n\t\tIn the limit $\\epsilon\\rightarrow+0$, Eq.~{(\\ref{eq:Suppl:scaled_ME})} is reduced to \n\t\t\\begin{align}\n\t\t\t\\frac{\\partial \\mc{P}(\\Omega,t)}{\\partial t}&= \\gamma \\left[\\frac{\\partial }{\\partial \\Omega} \\Omega+\\frac{\\mc{T}}{I}\\frac{\\partial^2 }{\\partial \\Omega^2}\\right]\\mc{P}(\\Omega,t) \\notag\\\\\n\t\t\t\t&+\\sum_{n=1}^\\infty \\frac{(-1)^n}{n!}\\frac{\\partial^n }{\\partial \\Omega^n}\\mathcal{K}_n\\mc{P}(\\Omega,t), \n\t\t\\end{align}\n\t\t\\begin{equation}\n\t\t\t\\mathcal{K}_n = \\frac{\\rho h(1+e)^n}{R_I^n}\\int ds g^n(s)\n\t\t\t\t\t\t\t\t\\int d\\vec v \\phi(|\\vec v|)\\Theta(-\\vec v\\cdot \\vec n)(-\\vec v\\cdot \\vec n)^{n+1}. \n\t\t\\end{equation}\n\t\tHere we can calculate the integral with the aid of\n\t\t\\begin{align}\n\t\t\t&\\int d\\vec v \\phi(|\\vec v|)\\Theta(-\\vec v\\cdot \\vec n)(-\\vec v\\cdot \\vec n)^{n+1}\\notag\\\\\n\t\t\t=& \\int_0^\\infty dvd\\theta d\\psi v^2\\sin{\\psi}   \\phi(v)\\Theta(-v\\cos{\\psi})(-v\\cos{\\psi})^{n+1}\\notag\\\\\n\t\t\t=& 2\\pi\\int_0^\\infty dvv^{n+3}\\phi(v)\\int_{\\pi\/2}^{\\pi} (-\\cos{\\psi})^{n+1}\\sin{\\psi}d\\psi\\notag\\\\\n\t\t\t=& \\frac{2\\pi}{n+2}\\int_0^\\infty dvv^{n+3}\\phi(v),\n\t\t\\end{align}\n\t\tand\n\t\t\\begin{align}\n\t\t\t&\\int ds g^n(s) \n\t\t\t= \\frac{2}{R_I^{n}}\\int_{-l\/2}^{l\/2} ds' s'^n + \\frac{2}{R_I^{n}}\\int_{-w\/2}^{w\/2} ds' s'^n \\notag\\\\\n\t\t\t&= \t\\begin{cases}\n\t\t\t\t\t\\frac{4}{R_I^n(n+1)}\\left[\\left(\\frac{l}{2}\\right)^{n+1} \\!+\\! \\left(\\frac{w}{2}\\right)^{n+1}\\right] & ({\\rm for\\>\\>even\\>\\>}n) \\cr\n\t\t\t\t\t0 & ({\\rm for\\>\\>odd\\>\\>}n)\n\t\t\t\t\\end{cases}.\n\t\t\\end{align}\n\t\twhere we have used the isotropic distribution $\\phi(\\vec v)=\\phi(v)$, and \n\t\tdecomposed the position vector $\\vec r$ as $\\vec r=x\\vec e+s'\\vec t+z\\vec e_z$ with $x=\\pm l\/2$ or $x=\\pm w\/2$. \n\t\tWe thus have the relation $g=s'\/R_I$. \n\t\tThen, the cumulant $\\mathcal{K}_n$ is simplified as \n\t\t\\begin{equation}\n\t\t\t\\mc{K}_n \\!=\\!\t\\begin{cases}\n\t\t\t\t\t\t\t\\frac{4\\pi \\rho h(1+e)^n (l^{n+1} + w^{n+1})}{2^n R_I^{2n}(n+1)(n+2)}\\!\\!\\int_0^\\infty \\!\\!dvv^{n+3}\\phi(v) & \\!\\!({\\rm for\\>\\>even\\>\\>}n) \\cr\n\t\t\t\t\t\t\t0 & \\!\\!({\\rm for\\>\\>odd\\>\\>}n)\n\t\t\t\t\t\t\\end{cases},\\label{BL_cum}\n\t\t\\end{equation}\n\t\twhich implies that the cumulant function is given as \n\t\t\\begin{align}\n\t\t\t&\\Phi(s) +\\mc{T}\\!s^2 =   \\sum_{n=1}^\\infty \\frac{(is)^n}{n!}K_n\\notag\\\\\n\t\t\t\t\t&= -\\!\\frac{16\\pi \\rho hR_I^4}{ls^2(1\\!+\\!e)^2}\\int_0^\\infty \\!\\!\\!\\!dv v\\phi(v) \\sum_{n=2}^\\infty \\frac{(-1)^n}{(2n)!}\\left[\\frac{s(1+e)lv}{2R_I^2}\\right]^{2n}\\notag\\\\\n\t\t\t\t\t   &-\\frac{16\\pi \\rho hR_I^4}{ws^2(1\\!+\\!e)^2}\\int_0^\\infty\\!\\!\\!\\! dv v\\phi(v) \\sum_{n=2}^\\infty \\frac{(-1)^n}{(2n)!}\\left[\\frac{s(1+e)wv}{2R_I^2}\\right]^{2n}\\notag\\\\\n\t\t\t\t\t&= -\\!\\frac{8\\pi \\rho hR_I^4}{ls^2(1\\!+\\!e)^2}\\int_{-\\infty}^\\infty \\!\\!\\!\\!\\!\\!\\!dv |v|\\phi(v) \\!\\!\\left[e^{\\frac{is(1+e)lv}{2R^2_I}}\\!\\!\\!\\!\\!-1+\\frac{s^2(1\\!+\\!e)^2l^2v^2}{8R^4_I}\\right]\\notag\\\\\n\t\t\t\t\t   &\\!-\\!\\frac{8\\pi \\rho hR_I^4}{ws^2(1\\!+\\!e)^2}\\int_{-\\infty}^\\infty \\!\\!\\!\\!\\!\\!\\!dv |v|\\phi(v) \\!\\!\\left[e^{\\frac{is(1+e)wv}{2R^2_I}}\\!\\!\\!\\!\\!-1+\\frac{s^2(1\\!+\\!e)^2w^2v^2}{8R^4_I}\\right].\\label{eq:Suppl:cumulant_general}\n\t\t\\end{align} \n\t\t\n\t\\subsection{In the case with the exponential velocity distribution}\n\t\tLet us consider the case of $\\mc{T}=l=0$, and assume that the velocity distribution of the granular gas is given by the exponential form as\n\t\t\\begin{equation}\n\t\t\tf(\\vec v) = \\frac{1}{8\\pi v_0^3}e^{-|v|\/v_0},\n\t\t\\end{equation}\n\t\twith the characteristic velocity $v_0$. \n\t\tThe non-Gaussian Langevin equation for the scaled angular velocity $\\hat \\Omega\\equiv\\omega\/\\epsilon$ is \n\t\t\\begin{equation}\n\t\t\t\\frac{d\\hat{\\Omega}}{dt} = -\\gamma \\hat{\\Omega} + \\hat {\\eta}_g.\n\t\t\\end{equation}\n\t\tHere, the cumulants function is given by \n\t\t\\begin{equation}\n\t\t\t\\Phi(s) = -\\frac{\\rho hwv_0\\Omega_g^2s^2 (5+3\\Omega_g^2s^2)}{2(1+\\Omega^2_gs^2)^2},\n\t\t\\end{equation}\n\t\twhere we have introduced $\\Omega_g\\equiv wv_0(1+e)\/2R_I^2$.  \n\t\tThen, we obtain the velocity distribution (11) in the main text for the scaled angular velocity $\\tl{\\Omega}\\equiv \\Omega\/\\Omega_g$ as \n\t\t\\begin{equation}\n\t\t\t\\mc{P}_{\\rm SS}(\\tl{\\Omega}) = \\int_{-\\infty}^\\infty \\frac{ds}{2\\pi} \\frac{1}{(1+s^2)^{3v_0\/2\\tl{v}}}\\exp{\\left[-is\\tl{\\Omega} - \\frac{v_0s^2}{\\tl{v}(1+s^2)}\\right]},\n\t\t\\end{equation}\n\t\twhere $\\mc{P}_{\\rm SS}(\\tl{\\Omega})\\equiv \\mc{P}_{\\rm SS}(\\Omega)\\Omega_g$, and $\\tl{v}\\equiv 2\\gamma\/\\rho hw$\n\t\t\\begin{figure*}\n\t\t\t\\includegraphics[width=175mm]{dist.eps}\n\t\t\t\\caption{\n\t\t\t\t\t\tNumerical data of the Monte Carlo simulation of Eq.~{(\\ref{eq:Suppl:BLeq})}. \n\t\t\t\t\t\t(a)~The steady distribution function of rotor's angular velocity. \n\t\t\t\t\t\t(b)~The numerical Fourier transform of $P_{\\rm SS}(\\Omega)$. \n\t\t\t\t\t\t(c)~The numerical data of $g(s)$. If we ignore the numerical fluctuation, $g(s)$ tends to converge to zero in the limit of $s\\rightarrow+\\infty$. \n\t\t\t\t\t\t(d)~The estimated data of the granular velocity distribution using Eq.~{(\\ref{spp:eq:estdist_num})}. \n\t\t\t\t\t\t    Because of the singularity at $v=0$ in Eq.~{(\\ref{spp:eq:estdist_num})}, the accuracy of the data near $v=0.5$ is not good.\n\t\t\t\t\t}\n\t\t\t\\label{fig:Suppl:dist}\n\t\t\\end{figure*}\n\t\\subsection{Inverse estimation formula for the spherical distribution}\n\t\tWe derive the inverse formula of the granular velocity distribution for the case of $\\mc{T}=l=0$ and an arbitrary $\\phi(v)$. \n\t\tThe non-Gaussian Langevin equation is given by \n\t\t\\begin{equation}\n\t\t\t\\frac{d\\hat{\\Omega}}{dt} = -\\gamma\\hat \\Omega +  \\hat \\eta_g, \n\t\t\\end{equation}\n\t\twhere the cumulant function of $\\hat{\\eta}_{g}$ is given by\n\t\t\\begin{equation}\n\t\t\t\\Phi(s) = -\\frac{2\\pi \\rho hw}{s^2F_g^2}\\int_{-\\infty}^\\infty dv |v|\\phi(v) \\left[e^{iF_gsv}-1+\\frac{F_g^2s^2v^2}{2}\\right],\n\t\t\\end{equation}\n\t\twith the typical collisional impact $F_g\\equiv w(1+e)\/2R_I^2$. \n\t\tFrom Eq.~{(\\ref{spp:g_Fdist})}, we obtain  the following relation between the granular velocity distribution and the Fourier representation of rotor's angular velocity distribution as\n\t\t\\begin{align}\n\t\t\t&-\\frac{2\\pi \\rho hw}{s^2F_g^2}\\int_{-\\infty}^\\infty dv |v|\\phi(v) \\left[e^{iF_gsv}-1+\\frac{F_g^2s^2v^2}{2}\\right] \\notag\\\\\n\t\t\t&=  \\gamma s\\frac{d}{ds}\\log{\\tl{P}_{SS}(s)}.\\label{eq:Suppl:est1}\n\t\t\\end{align}\n\t\tThis formula can be transformed into the following form:\n\t\t\\begin{equation}\n\t\t\t\\phi(v) \\!=\\! \\frac{1}{\\pi |v|} \\!\\int_{0}^\\infty \\!\\!ds \\!\\left[a\\!-\\!\\frac{bs^2}{2}\\!-\\!cs^3\\frac{d}{ds}\\log{\\tl{P}_{\\rm SS}(s\/F_g)}\\right]\\cos{(sv)},\\label{eq:Suppl:est2}\n\t\t\\end{equation}\n\t\twhere we introduced $a\\equiv\\int_{-\\infty}^{\\infty}dv|v|\\phi(v)$, $b\\equiv\\int_{-\\infty}^{\\infty}dv|v|^3\\phi(v)$, and $c\\equiv \\gamma\/2\\pi\\rho hw$. \n\n\t\tLet us explain how to determine the coefficients $a$ and $b$. \n\t\tAccording to the Riemann-Lebesgue lemma~{\\cite{RLlemma}}, the following relation holds if $|v|\\phi(v)$ is an $L^1$-function: \n\t\t\\begin{equation}\n\t\t\t\\lim_{s\\rightarrow+\\infty}\\int_{-\\infty}^{\\infty} ds |v|\\phi(v)e^{isv} = 0,\n\t\t\\end{equation}\n\t\tor equivalently, \n\t\t\\begin{equation}\n\t\t\t\\lim_{s\\rightarrow\\infty}\\left[a-\\frac{bs^2}{2}-cs^3\\frac{d}{ds}\\log \\tl{P}_{\\rm SS}(s\/F_g)\\right] = 0. \\label{eq:Suppl:RLlemma}\n\t\t\\end{equation}\n\t\tEquation~{(\\ref{eq:Suppl:RLlemma})} is practically useful to determine the coefficients $a$ and $b$ from the experimental data of $\\tl{P}_{\\rm SS}(s)$.\n\t\t\n\t\\subsection{The numerical technique for the inverse estimation formula}\n\t\tHere we explain our numerical procedure for the inverse formula~{(\\ref{eq:Suppl:est2})}. \n\t\tWe obtain the steady distribution function of rotor's angular velocity using the Monte Carlo simulation of Eq.~{(\\ref{eq:Suppl:BLeq})} on the following setup: $\\phi(v)=e^{-|v|}\/8\\pi$, $l=T=0$, $w=\\sqrt{12}$, $M=\\rho=h=e=I=1$, $m=0.01$, and $\\gamma=2$. \n\t\tThe numerical data is plotted in Fig.~{\\ref{fig:Suppl:dist}(a)}. \n\t\t\n\t\tTo obtain the Fourier transformation $\\tl{P}_{\\rm SS}(s)$, we have used the numerical distribution $\\mc{P}_{SS}(\\Omega)$ for $0\\leq \\Omega\\leq 30$. \n\t\t$\\tl{P}_{\\rm SS}(s)$ is numerically plotted in Fig.~{\\ref{fig:Suppl:dist}(b)}. \n\t\tWe numerically estimate the coefficients $a$ and $b$ as $a=0.080121$ and $b=0.464$, respectively, \n\t\tand obtain the following function as shown in Fig.~{\\ref{fig:Suppl:dist}(c)}: \n\t\t\\begin{equation}\n\t\t\tg(s)\t=a - \\frac{bs^2}{2} - cs^3\\frac{d}{ds}\\log{\\tl{P}_{\\rm SS}(s\/F_g)}. \n\t\t\\end{equation}\n\t\tFigure~{\\ref{fig:Suppl:dist}(c)} implies the asymptotic form $g(s) \\simeq 0$ for $s\\rightarrow\\infty$. \n\t\tOn the basis of the numerical data of $g(s)$, \n\t\twe estimate the granular velocity distribution function $\\phi(v)$ as\n\t\t\\begin{equation}\n\t\t\t\\phi(v) = \\frac{1}{\\pi |v|}\\int_0^\\infty dsg(s)\\cos{sv}.\\label{spp:eq:estdist_num}\n\t\t\\end{equation}\n\t\tWe plot the granular velocity distribution estimated from Eq.~{(\\ref{spp:eq:estdist_num})} in Fig.~{\\ref{fig:Suppl:dist}(d)}. \n\t\tWe note that Eq.~{(\\ref{spp:eq:estdist_num})} has a singularity at $v=0$, which explains that the numerical accuracy is not good around $v=0$. \n\t\n\t\n\t\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzqxcc b/data_all_eng_slimpj/shuffled/split2/finalzzqxcc
new file mode 100644
index 0000000000000000000000000000000000000000..0a544b165d79be74740e47e7ef64ec3a7867c807
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzqxcc
@@ -0,0 +1,5 @@
+{"text":"\n\nDedication\n\nTo my precious daughter, Zayla. To our precious daughters.\nEpigraph\n\nSeawater begs the pearl\n\nTo break its shell\n\n\u2014FROM THE ECSTATIC POEM\n\n\"SOME KISS WE WANT,\"\n\nBY JALAL AD-DIN MOHAMMAD RUMI,\n\nTHIRTEENTH-CENTURY PERSIAN POET\nContents\n\nDedication\n\nEpigraph\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\nChapter 32\n\nChapter 33\n\nChapter 34\n\nChapter 35\n\nChapter 36\n\nChapter 37\n\nChapter 38\n\nChapter 39\n\nChapter 40\n\nChapter 41\n\nChapter 42\n\nChapter 43\n\nChapter 44\n\nChapter 45\n\nChapter 46\n\nChapter 47\n\nChapter 48\n\nChapter 49\n\nChapter 50\n\nChapter 51\n\nChapter 52\n\nChapter 53\n\nChapter 54\n\nChapter 55\n\nChapter 56\n\nChapter 57\n\nChapter 58\n\nChapter 59\n\nChapter 60\n\nChapter 61\n\nChapter 62\n\nChapter 63\n\nChapter 64\n\nChapter 65\n\nChapter 66\n\nChapter 67\n\nChapter 68\n\nChapter 69\n\nAcknowledgments\n\nAbout the Author\n\nCredits\n\nCopyright\n\nAbout the Publisher\nCHAPTER 1\n\nShahla stood by our front door, the bright green metal rusting on the edges. She craned her neck. Parwin and I rounded the corner and saw the relief in her eyes. We couldn't be late again.\n\nParwin shot me a look and we picked up our hurried pace. We did the best we could without running. Rubber soles slapped against the road and raised puffs of dusty smoke. The hems of our skirts flapped against our ankles. My head scarf clung to beads of sweat on my forehead. I guessed Parwin's was doing the same, since it hadn't yet blown away.\n\nDamn them. It was their fault! Those boys with their shameless grins and tattered pants! This wasn't the first time they'd made us late.\n\nWe ran past the doors, blue, purple, burgundy. Spots of color on a clay canvas.\n\nShahla waved us toward her.\n\n\"Hurry!\" she hissed frantically.\n\nPanting, we followed her through the front door. Metal clanged against the door frame.\n\n\"Parwin! What did you do that for?\"\n\n\"Sorry, sorry! I didn't think it would be that loud.\"\n\nShahla rolled her eyes, as did I. Parwin always let the door slam.\n\n\"What took you so long? Didn't you take the street behind the bakery?\"\n\n\"We couldn't, Shahla! That's where he was standing!\"\n\nWe had gone the long way around the marketplace, avoiding the bakery where the boys loitered, their shoulders hunched and their eyes scouting the khaki jungle that was our village.\n\nBesides pickup games of street soccer, this was the main sport for school-age boys\u2014watching girls. They hung around waiting for us to come out of our classrooms. Once off school grounds, a boy might dart between cars and pedestrians to tail the girl who'd caught his eye. Following her helped him stake his claim. This is my girl, it told the others, and there's only room for one shadow here. Today, my twelve-year-old sister, Shahla, was the magnet for unwanted attention.\n\nThe boys meant it to be flattering. But it frightened the girl since people would have loved to assume that she'd sought out the attention. There just weren't many ways for the boys to entertain themselves.\n\n\"Shahla, where is Rohila?\" I whispered. My heart was pounding as we tiptoed around to the back of the house.\n\n\"She's taken some food to the neighbor's house. Madar-jan cooked some eggplant for them. I think someone died.\"\n\nDied? My stomach tightened and I turned my attention back to following Shahla's footsteps.\n\n\"Where's Madar-jan?\" Parwin said, her voice a nervous hush.\n\n\"She's putting the baby to sleep,\" Shahla said, turning toward us. \"So you better not make too much noise or she'll know you're just coming home now.\"\n\nParwin and I froze. Shahla's face fell as she looked at our widened eyes. She whipped around to see Madar-jan standing behind her. She had come out of the back door and was standing in the small paved courtyard behind the house.\n\n\"Your mother is very much aware of exactly when you girls have gotten home and she is also very much aware of what kind of example your older sister is setting for you.\" Her arms were folded tightly across her chest.\n\nShahla's head hung in shame. Parwin and I tried to avoid Madar-jan's glare.\n\n\"Where have you been?\"\n\nHow badly I wanted to tell her the truth!\n\nA boy, lucky enough to have a bicycle, had followed Shahla, riding past us and then circling back and forth. Shahla paid no attention to him. When I whispered that he was looking at her, she hushed me, as if speaking it would make it true. On his third pass, he got too close.\n\nHe looped ahead of us and came back in our direction. He raced down the dirt street, slowing down as he neared us. Shahla kept her eyes averted and tried to look angry.\n\n\"Parwin, watch out!\"\n\nBefore I could push her out of the way, the cycling stalker's front wheel rolled over a metal can in the street; he veered left and right, then swerved to avoid a stray dog. The bicycle came straight at us. The boy's eyebrows were raised, his mouth open as he struggled to regain balance. He swiped Parwin before toppling over on the front steps of a dried-goods shop.\n\n\"Oh my God,\" Parwin exclaimed, her voice loud and giddy. \"Look at him! Knocked off his feet!\"\n\n\"Do you think he's hurt?\" Shahla said. She had her hand over her mouth, as if she had never seen a sight so tragic.\n\n\"Parwin, your skirt!\" My eyes had moved from Shahla's concerned face to the torn hem of Parwin's skirt. The jagged wires holding the spokes of the bicycle together had snagged Parwin's dress.\n\nIt was her new school uniform and instantly Parwin began to weep. We knew if Madar-jan told our father, he would keep us home instead of sending us to school. It had happened before.\n\n\"Why are you all silent only when I ask you something? Do you have nothing to say for yourselves? You come home late and look like you were chasing dogs in the street!\"\n\nShahla had spoken on our behalf plenty of times and looked exasperated. Parwin was a basket of nerves, always, and could do nothing but fidget. I heard my voice before I knew what I was saying.\n\n\"Madar-jan, it wasn't our fault! There was this boy on a bicycle and we ignored him but he kept coming back and I even yelled at him. I told him he was an idiot if he didn't know his way home.\"\n\nParwin let out an inadvertent giggle. Madar-jan shot her a look.\n\n\"Did he come near you?\" she asked, turning to Shahla.\n\n\"No, Madar-jan. I mean, he was a few meters behind us. He didn't say anything.\"\n\nMadar-jan sighed and brought her hands to her temples.\n\n\"Fine. Get inside and start your homework assignments. Let's see what your father says about this.\"\n\n\"You're going to tell him?\" I cried out.\n\n\"Of course I am going to tell him,\" she answered, and spanked my backside as I walked past her into the house. \"We are not in the habit of keeping things from your father!\"\n\nWe whispered about what Padar-jan would say when he came home while we dug our pencils into our notebooks. Parwin had some ideas.\n\n\"I think we should tell Padar that our teachers know about those boys and that they have already gotten in trouble so they won't be bothering us anymore,\" Parwin suggested eagerly.\n\n\"Parwin, that's not going to work. What are you going to say when Madar asks Khanum Behduri about it?\" Shahla, the voice of reason.\n\n\"Well, then we could tell him that the boy said he was sorry and promised not to bother us again. Or that we are going to find another way to get to school.\"\n\n\"Fine, Parwin. You tell him. I'm tired of talking for all of you anyway.\"\n\n\"Parwin's not going to say anything. She only talks when no one's listening,\" I said.\n\n\"Very funny, Rahima. You're so brave, aren't you? Let's see how brave you are when Padar-jan comes home,\" Parwin said, pouting.\n\nGranted, I wasn't a very brave nine-year-old when it came time to face Padar-jan. I kept my thoughts bottled behind my pursed lips. In the end, Padar-jan decided to pull us out of school again.\n\nWe begged and pleaded with Padar-jan to let us return to school. One of Parwin's teachers, a childhood friend of Madar-jan, even showed up at the house and tried to reason with our parents. Padar-jan had relented in the past but this time was different. He wanted us to go to school but struggled with how to make that happen safely. How would it look for his daughters to be chased by local boys for all to see? Awful.\n\n\"If I had a son this would not be happening! Goddamn it! Why do we have a house full of girls! Not one, not two\u2014but five of them!\" he would yell. Madar-jan would busy herself with housework, feeling the weight of disappointment on her shoulders.\n\nHis temper was worse these days. Madar-jan would tell us to hush and be respectful. She told us too many bad things had happened to Padar-jan and it had made him an angry man. She said if we all behaved then he would go back to being his normal self soon. But it was getting harder and harder to remember a time when Padar-jan wasn't angry and loud.\n\nNow that we were home, I was given the extra chore of bringing the groceries from the store. My older sisters were quarantined since they were older and noticeable. I was, thus far, invisible to boys and not a risk.\n\nEvery two days I stuffed a few bills from Madar-jan in the pouch that she had sewn into my dress pocket so I would have no excuse for losing them. I would wind my way through the narrow streets and walk thirty minutes to reach the market I loved. The stores were bustling with activity. Women looked different now than they had a few years ago. Some wore long blue burqas and others wore long skirts and modest head scarves. The men all dressed like my father, long tunics with billowing pantaloons\u2014colors as drab as our landscape. Little boys wore ornate caps with small round mirrors and gold scrolling. By the time I got there, my shoes were again dusty and I would resort to using my head scarf as a filter for the clouds of dirt the hundreds of cars left in their wake. It was as if the khaki-colored landscape were dissolving into the air of our village.\n\nTwo weeks into our expulsion from school, the shop owners had gotten to know me. There were not many nine-year-old girls who would walk determinedly from shop to shop. And having watched my parents haggle prices down, I thought I could do the same. I would argue with the baker who tried to charge me double what I had seen him charge my mother. I bickered with the grocer trying to tell me that the flour I wanted was imported and, thus, subject to a surcharge. I pointed out that I could just as easily buy the same fancy flour from Agha Mirwais down the block and scoffed at the price he quoted. He gritted his teeth and put the flour in the bag along with the other groceries, muttering words under his breath that no child should hear.\n\nMadar-jan was pleased to have my help with the market. She was busy enough with Sitara, who was just taking her first steps. Madar-jan had Parwin look after Sitara while she and Shahla took care of the household chores of dusting, sweeping and preparing the night's meal. In the afternoons, Madar-jan made us all sit down with our books and notebooks and complete the homework she assigned us.\n\nFor Shahla, the days were isolating and difficult. She longed to see her friends and talk with her teachers. Shahla's strengths were her intuition and her intelligence. She wasn't at the very top of her class, but she usually charmed her teachers just enough to push her onto the short list of star pupils. She was average looking but put extra care into her appearance. She would spend at least five minutes brushing her hair every night, since someone told her it would make her locks grow longer. Shahla's face was what people would call pleasant, not beautiful or memorable. But her personality made her glow. People looked at her and couldn't help but smile. Polite and proper, she was a favorite in school. She had a way of looking at you and making you feel important. In front of family and friends, Shahla made Madar-jan proud as she would speak maturely and inquire after each member of the family.\n\n\"How is Farzana-jan doing? It's been so long since I've seen her! Please do tell her that I was asking about her,\" she would say. Grandmothers would nod in approval, praising Madar-jan for raising such a respectable girl.\n\nParwin was another story. She was striking. Her eyes were not the mud-brown color the rest of us had. Instead, hers were a hazel-gray blend that made you forget what it was that you were going to say. Her hair hung around her face in wavy locks with a natural luster. She was undeniably the best-looking girl in our whole extended family.\n\nBut she was completely lacking in social skills. If Madar-jan's friends stopped by, Parwin would shrink into a corner, busying herself with folding and refolding a tablecloth. If she could manage to escape before company made it into the room, even better. Nothing was more of a relief to her than avoiding the traditional three-kiss greeting. She kept her answers brief and all the while kept her eyes on the nearest escape route.\n\n\"Parwin, please! Khala Lailoma is asking you a question. Can you please turn around? Those plants do not need to be watered at this very moment!\"\n\nWhat Parwin lacked in social skill, she more than made up for in artistic ability. She was masterful with pencil and paper. Graphite turned into visual energy in her hands. Wrinkled faces, an injured dog, a house too damaged to repair. She had a gift, an ability to show you what you did not see, even though your eyes graced the same sights as hers. She could sketch a masterpiece in minutes but washing the dishes could take hours.\n\n\"Parwin is from another world,\" Madar-jan would say. \"She is a different kind of girl.\"\n\n\"What good is that going to do her? She's going to have to survive and make her way through this world,\" Padar-jan would retort, but he loved her drawings and kept a pile of them at his bedside to flip through from time to time.\n\nThe other problem with Parwin was that she'd been born with a bad hip. Someone had told Madar-jan she must have been lying on her side too much when she was pregnant. From the time Parwin started to crawl, it was obvious something was off. It took her much longer to learn to walk and to this day she hasn't lost her limp. Padar-jan had taken her to a doctor when she was five or six but they said it was too late.\n\nThen there was me. I didn't mind the expulsion as much as my sisters. I suppose this was because it gave me opportunity to venture out on my own, without two older sisters to chastise me or insist I hold their hands as we crossed the street. Finally, I had freedom\u2014even more than my sisters!\n\nMadar-jan needed help with the errands and lately it was impossible to depend on Padar-jan for anything. She would ask him to pick up some things from the market on his way home and inevitably he would forget, then curse her for having an empty pantry. But if she went to the bazaar by herself, he went into an even worse rage. From time to time, Madar-jan asked the neighbors to pick up an item or two for her but she tried not to do that too often, knowing they already whispered about the peculiar way Padar-jan had of walking up and down our small street, his hands gesturing wildly as he explained something to the birds. My sisters and I wondered about his behavior too, but Madar-jan told us our father needed to take a special medicine and that was why he sometimes acted strangely.\n\nAt home, I could not help but talk about my adventures in the outside world. It bothered Shahla more than Parwin, who was content with her pencils and paper.\n\n\"I think tomorrow I'll pick up some roasted chickpeas from the market. I have a few coins. If you like, I could bring you some, Shahla.\"\n\nShahla sighed and shifted Sitara from one hip to another. She looked like a young exasperated mother.\n\n\"Forget it. I don't want any. Just go and finish the chores, Rahima. I'm sure you're just dawdling out there. In no rush to come home, I bet.\"\n\n\"I'm not dawdling. I go and do the errands that Madar-jan tells me to do. But never mind. I'll see you later.\"\n\nIt wasn't so much that I wanted my sisters to be envious. It was more that I wanted to celebrate my new privileges to come and go, to wander through the shops without my sister's supervision. If I had a little more tact, I would have found another way to express myself. But my loud mouth caught Khala Shaima's attention. Maybe there was a higher purpose to my insensitivity.\n\nKhala Shaima was my mother's sister\u2014her older sister. Madar-jan was closer to her than anyone else in her family and we saw her often. Had we not grown up around her, we probably would have been frightened by her appearance. Khala Shaima was born with a crooked spine that wiggled through her back like a snake. Although our grandparents had hoped to find a suitor before her shape became too obvious, she was passed over time and again. Families would come to ask about my mother or Khala Zeba, the youngest of the sisters, but no one wanted Khala Shaima with her hunched back and one raised shoulder.\n\nShe understood early in life that she would not catch anyone's eye and decided not to bother fussing with appearances at all. She let her eyebrows grow in, left those few stray chin hairs and dressed in the same drab clothing day in and day out.\n\nInstead, she focused her energies on her nieces and nephews and taking care of my grandparents as they aged. Khala Shaima supervised everything\u2014making sure we were doing satisfactorily in school, that we had proper clothing for the winter and that lice hadn't nested in our hair. She was a safety net for anything our parents might not have been able to do for us and she was one of the few people who could stand being around Padar-jan.\n\nBut you had to know Khala Shaima to get her. I mean to really get her. If you didn't know that she had the best intentions at heart, you could be put off by the lack of pleasantries in her conversation, by her sharp criticisms or by the doubtful squint in her eyes while she listened to you talk. But if you knew how she'd been spoken to her whole life, by strangers and family, you wouldn't be surprised.\n\nShe was good to us girls and always came with candy-laden pockets. Padar-jan would comment snidely that her pockets were the only sweet thing about Khala Shaima. My sisters and I would feign patience while we waited for the rustle of chocolate wrappers. When she arrived, I had just returned from the market, and in plenty of time to get my share of the sweets.\n\n\"Shaima, honest to God, you're spoiling these girls! Where are you getting chocolates like these from these days! They can't be cheap!\"\n\n\"Don't stop a donkey that's not yours,\" she fired back. That was another thing about Khala Shaima. Everyone used those old Afghan proverbs, but Khala Shaima could hardly speak without them. It made conversations with her as circuitous as her spine. \"Stay out of it and let's let the girls get back to their homework.\"\n\n\"We're done with our homework, Khala Shaima-jan,\" Shahla said. \"We've been working on it all morning.\"\n\n\"All morning? Didn't you go to school today?\" Shaima's eyebrows furrowed.\n\n\"No, Khala Shaima. We don't go to school anymore,\" Shahla said, averting her eyes since she knew she was throwing Madar-jan into the fire.\n\n\"What does that mean? Raisa! Why aren't the girls in school?\"\n\nMadar-jan lifted her head from the teapot reluctantly.\n\n\"We had to take them out again.\"\n\n\"In God's name, what ridiculous excuse did you come up with this time to keep them from their studies? Did a dog bark at them in the street?\"\n\n\"No, Shaima. Don't you think I would much rather have them going to school? It's just that they're running into foolishness in the streets. You know how boys can be. And, well, their father is just not happy to send them out so they can be toyed with by the neighborhood boys. I don't blame him, really. You know, it's only been a year that the girls are even able to walk in the street. Maybe it's just too soon.\"\n\n\"Too soon? How about too late! They should have been going to school all this time but they haven't. Imagine how far behind they are and now that they can catch up, you're going to keep them at home to scrub the floors? There will always be idiots in the street saying all kinds of things and giving all kinds of looks. You can believe that. If you hold these girls back for that, you're no better than the Taliban who closed their schools.\"\n\nShahla and Parwin shot each other looks.\n\n\"Then what am I supposed to do? Arif's cousin Haseeb told him that\u2014\"\n\n\"Haseeb? That moron who's dumber than a Russian tank? You're making decisions for your children based on something Haseeb said? Sister, I thought more of you.\"\n\nMadar-jan huffed in frustration and rubbed her temples. \"Then you stay here till Arif gets home and you tell him yourself what you think we should do!\"\n\n\"Did I say I was leaving?\" Khala Shaima said coolly. She propped a pillow behind her uneven back and leaned against the wall. We braced ourselves. Padar-jan hated dealing with Khala Shaima's intrusions and he was just as blunt as she was about it.\n\n\"You're a fool to think these girls are better off rotting in this home instead of learning something in school.\"\n\n\"You never went to school and see how well you turned out,\" Padar-jan said facetiously.\n\n\"I've got a lot more sense than you, engineer-sahib.\" A low blow. Padar-jan had wanted to major in engineering when he finished high school but his marks didn't make the cut. Instead, he took some general classes for one semester and then dropped out to start working. He had a shop now where he fixed old electronics, and though he was pretty good at what he did, he was still bitter about not making it as an engineer, a highly regarded title for Afghans.\n\n\"Damn you, Shaima! Get out of my house! They're my daughters and I don't need to listen to a cripple tell me what I should do with them!\"\n\n\"Well, this cripple has an idea that may solve your problem\u2014let you keep your precious pride while the girls can get back into school.\"\n\n\"Forget it. Just get out so I don't have to look at your face anymore. Raisa! Where the hell is my food?\"\n\n\"What is your idea, Shaima?\" Madar-jan jumped in, eager to hear what she had to say. She did respect her sister, ultimately. More often than not, Shaima was right. She hurriedly fixed a plate of food and brought it over to Padar-jan, who was now staring out the window blankly.\n\n\"Raisa, don't you remember the story our grandmother told to us? Remember Bibi Shekiba?\"\n\n\"Oh, her! Yes, but how does that help the girls?\"\n\n\"She became what her family needed. She became what the king needed.\"\n\n\"The king.\" Padar-jan scoffed. \"Your stories get crazier every time you open your ugly mouth.\"\n\nKhala Shaima ignored his comment. She had heard much worse.\n\n\"Do you really think that would work for us too?\"\n\n\"The girls need a brother.\"\n\nMadar-jan looked away and sighed with disappointment. Her failure to bear a son had been a sore spot since Shahla's birth. She had not anticipated that it would be brought to everyone's attention again tonight. She avoided Padar-jan's eyes.\n\n\"That's what you've come here to tell me! That we need a son? Don't you think I know that? If your sister were a better wife, then maybe I would have one!\"\n\n\"Quit jabbering and let me finish.\"\n\nBut she didn't finish. She only started. That night Khala Shaima started a story of my great-great-grandmother Shekiba, a story that my sisters and I had never before heard. A story that transformed me.\nCHAPTER 2\n\nShekiba.\n\nYour name means \"gift,\" my daughter. You are a gift from Allah.\n\nWho could have known that Shekiba would become the name she was given, a gift passed from one hand to another? Shekiba was born at the turn of the twentieth century, in an Afghanistan eyed lasciviously by Russia and Britain. Each would take turns promising to protect the borders they had just invaded, like a pedophile who professes to love his victim.\n\nThe borders between Afghanistan and India were drawn and redrawn from time to time, as if only penciled in. People belonged to one country and then the other, nationalities changing as often as the direction of the wind. For Great Britain and the Soviet Union, Afghanistan was the playing field for their \"Great Game,\" the power struggle to control Central Asia. But the game was slowly coming to an end, the Afghan people ferociously resisting outside control. Chests expanded with pride when Afghans talked about their resilience.\n\nBut parts of Afghanistan were taken\u2014little by little until its borders shrank in like a wool sweater left in the rain. Areas to the north like Samarkand and Bukhara had been lost to the Russian Empire. Chunks of the south were chipped away and the western front was pushed in over the years.\n\nIn that way, Shekiba was Afghanistan. Beginning in her childhood, tragedy and malice chipped away at her until she was just a fragment of the person she should have been. If only Shekiba had been prettier, something at least pleasing for the eye to gaze upon. Maybe then, her father could have hoped to arrange a proper marriage for her when her time came. Maybe people would have looked at her with an ounce of kindness.\n\nBut Shekiba's village was unforgiving. To get to Kabul, one had to ride one week, crossing a river and three mountains. Most people spent their entire lives in the village, in the green fields surrounded by mountains, walking the dirt roads that connected one compound to another. Their village was in a valley, dark soil nurtured by the nearby river and tall peaks giving a sense of enclosure, privacy. There were a few dozen clans, extended families who had known each other over generations. Most people were related to each other, somehow, and gossip was one way to keep busy.\n\nShekiba's parents were second cousins, their marriage arranged by Shekiba's paternal grandmother. Their family, like many others, lived off the land. Each generation splintered the family's land so that people would have a place to build a home, if they decided to leave the clan's main house. Shekiba's father, Ismail Bardari, was the youngest in his home. His older brothers had married before him and filled the compound with their wives and children.\n\nSeeing there was no room for him and his new bride, Shafiqa, Ismail picked up his chisel and set to work. He was lucky though, in that his father bequeathed him a lot with such fertile soil that his share of crops would be guaranteed. He was the hardest working of his brothers and his father wanted to ensure that the land's potential would be realized. There were many hungry mouths to feed and a good yield could bring in extra income from the village. His brothers lacked Ismail's instincts. He had a gift. He knew just the right temperature at which to plant, how often to till the soil and the perfect amount of water to make crops grow. Ismail's brothers resented him for being their father's favorite. They pretended to prefer living in the main home. In the end, he surrounded the house with a wall of mud and stones to give it privacy, as a proper Afghan home needed.\n\nIsmail brought his nervous bride to their new home, surrounded by a small plot of land that bordered his brother's. Standing outside, she could see her in-laws coming and going from the house, their burqas blue spots on a khaki landscape. When the women headed in her direction she would hurry inside and cover herself, embarrassed that her belly was swollen with child. But Shafiqa's in-laws found her dull and timid, and over time they took less interest in her and her children. The women sighed heavily when they spoke with her and whispered to her husband when she wasn't near. Had Shekiba's father been like most other men, he might have heeded those whispers and taken a second wife. But Ismail Bardari was unlike some other men and stayed with the one wife he had, however his mother and sisters felt about her.\n\nShekiba's brothers, Tariq and Munis, were the only real link to the clan. Shafiqa watched over Shekiba and her little sister Aqela, nicknamed \"Bulbul\" because her light, melodic voice reminded Ismail of the local songbird. Tariq and Munis would come and go between their father's and their grandfather's homes, acting as couriers of clothing, vegetables and news. The boys were well liked by their grandparents and valued as male heirs. Ismail's mother, Bobo Shahgul, often said the two boys were the only good thing to come from Shafiqa. The boys overheard many hateful comments but they knew better than to share everything they heard. Shekiba and Aqela didn't realize how little their father's family cared about them since they spent their days close at their mother's side. Sometimes, too close.\n\nA clumsy two-year-old Shekiba changed her life in the blink of an eye. She woke from a midmorning nap and set off to find her mother. Shekiba heard the familiar sounds of peeling in the kitchen and stumbled into the cooking niche. Her small foot caught on the hem of her dress and her arm flailed into the air, knocking a pot of hot oil from a burner top before her mother could reach her. The oil flew out and melted the left half of Shekiba's cherub face into blistered and ragged flesh.\n\nShafiqa screamed and doused her daughter's face with cool water but it was too late. It took months to heal, as Shafiqa diligently kept Shekiba's face clean, using a compound the local alchemist had mixed for them. The pain got worse as her skin fought to recover. The itching drove Shekiba mad and her mother was forced to wrap her hands in cloth, especially while she picked away at the dead, blackened skin. Fevers came, so high they made the toddler's body tremble and writhe, and Shafiqa had nothing to offer, nothing she could do but pray at her daughter's side, her body rocking back and forth, and beseech Allah for mercy.\n\nBobo Shahgul came to see Shekiba when she heard about the incident. Shafiqa anxiously waited to hear any helpful advice her mother-in-law might offer but Bobo Shahgul had none. Before she left, she suggested Shafiqa pay closer attention to her children and muttered thanks that it hadn't been one of the boys.\n\nShekiba's survival was nothing short of a miracle, another gift from Allah. Though her face healed, she was not the same. From then on, Shekiba was halved. When she laughed, only half her face laughed. When she cried, only half her face cried. But the worst part was the change in people's expressions. People who saw her profile from the right would begin to smile, but as their view turned the corner, beyond her nose, their own faces would change. Every reaction reminded Shekiba that she was ugly, a horror. Some people would step back and cover a gaping mouth with a hand. Others would dare to lean in, eyes squinted, to get a better look. From across the road, people would stop in their tracks and point.\n\nThere. Did you see her? There goes the girl with half a face. Didn't I tell you she was horrid looking? God only knows what they did to deserve that.\n\nEven her aunts and uncles would shake their heads and cluck their tongues every time they saw her, as if every time they were freshly disappointed and shocked to see what she looked like. Her cousins came up with twisted names for her. \"Shola face,\" as her skin resembled the lumpy soft rice. \"Babaloo,\" or monster. That one she hated more than the others, since she too was afraid of the babaloo, the creature that frightened every Afghan child in the night.\n\nShafiqa tried to keep her sheltered from the comments, the jeers, the stares, but it was too late to save Shekiba's self-esteem, a commodity people didn't value much anyway. She covered Shekiba with a burqa when she saw people approaching their home or on the rare occasion when the family ventured into the village.\n\nRemember, \"Shekiba\" means \"a gift.\" You are our gift, my daughter. No need to let others gawk at you.\n\nShekiba knew she was horribly disfigured and that she was lucky to even be accepted by her immediate family. In the summers, the burqa was hot and stifling but she felt safer within it, protected. She was not exactly happy but was satisfied to stay in the house and out of sight. Her days passed with fewer insults that way. Her parents withdrew even more from the clan, and the resentment toward Shafiqa's aloofness grew.\n\nTariq and Munis were both energetic, and being just a year apart in age, they could pass for twins. When they were eight and nine, they were helping their father with the fieldwork and running errands in the village. They usually ignored the comments they heard about their \"cursed sister\" but Tariq had been known to throw back insults from time to time. On one occasion, Munis came home with scattered bruises and a foul temper. He'd had more than he could take of the local boys pestering him about his half-faced sister. Padar-jan had gone to the boy's home to make amends with his parents but he never reprimanded Tariq or Munis for defending their Shekiba.\n\nAqela, always smiling, would sing nursery rhymes in her sweet bulbul voice and kept her mother and Shekiba's spirits lifted as they did the chores. They were happy keeping to themselves. They didn't have much, but they had everything they needed and never felt lonely.\n\nIn 1903, a wave of cholera decimated Afghanistan. Children shriveled up within hours and succumbed in their mothers' weak arms. Shekiba's family had no choice but to use the poisoned water that coursed through their village. First Munis, then the others. The illness came quickly and it came strong. The smell was unbearable. Shekiba was stunned. She saw her siblings' faces grow pale and thin in days. Aqela was quiet, her songs reduced to a soft moan. Shafiqa was frantic; Ismail quietly shook his head. Word came from the compound that two children had died, one from each of Shekiba's uncles.\n\nShekiba and her parents waited for their own bellies to begin cramping. They nervously cared for the others, watching each other and waiting to see who else would become ill. Shekiba saw her father put his arms around his wife's shoulders as she rocked and prayed. Aqela's skin was graying, Tariq's eyes were sunken. Munis was quiet and still.\n\nShe was thirteen when she helped her parents wash and wrap Tariq, Munis and Aqela, the songbird, in white cloth, the traditional garb for the deceased. Shekiba sniffled quietly, knowing she would be haunted by the memory of helping her moaning father to dig the graves for her teenage brothers and delicate Aqela, who had just turned ten. Shekiba and her parents were among the survivors.\n\nIt was the first time in years that the clan made an appearance. Shekiba watched her uncles and their wives come in and out of the house, paying their obligatory respects before moving on to the next home grieving their dead. It went without saying that they pitied Shekiba's parents, not so much for the loss of their three children, but for the disappointment that Allah could not have spared one of the sons instead of the defective girl. Luckily, Shekiba was numb by then.\n\nThousands died that year. Her family's losses were notches on the epidemic's belt.\n\nOne week after her three children were buried, Shafiqa began to whisper to herself when no one was looking. She asked Tariq to help her with the water pails. She warned Munis to eat all his food so that he would grow up to be as tall as his brother. Her fingers moved through the yarn of the blanket as if she were braiding Aqela's hair.\n\nThen Shafiqa started sitting idly, plucking individual hairs from her head, one by one, until her scalp was bare; then her eyebrows and lashes disappeared. With nothing left to pluck, she resorted to picking at the skin of her arms and legs. She ate her food but gagged on pieces that she had forgotten to chew. Her whispers became louder and Shekiba and her father pretended not to notice. Sometimes she would listen and then giggle with a lightheartedness alien to their household. Shekiba slowly became her mother's mother, making sure she bathed and reminding her to go to sleep at night.\n\nA year later, in the same dismal month of Qows, Shekiba's languishing mother decided not to wake up from sleep. It came as no surprise.\n\nIsmail held his wife's hands and thought how tired they must be from all the wringing they had endured. Shekiba brought her cheek to her mother's and saw that her eyes had lost their desperate glassiness. Madar-jan must have died looking at the face of God, Shekiba thought. Nothing else could have brought the look of peace so quickly.\n\nThe house sighed in relief. Shekiba bathed her mother one last time, taking care to wash her bald head and realizing that her mother had even plucked the hairs from her womanly parts. The weight of sadness lifted. Her corpse was shockingly light.\n\nBy the following day, Shekiba and her father were back in the field to open the earth once more. They did not bother to tell the rest of the family. Her father read a prayer over the mound of dirt and they looked at each other, quietly wondering which of them would join the others first.\n\nShekiba was left with her father. A cousin stopped by to tell them of an upcoming wedding and took back news of the new widower to the rest of the clan. The hawks descended on the house within days, extending their condolences, but only after they advised Shekiba's father that he now had the opportunity to begin again with a new wife. They named a few families with eligible daughters in the village, most of them only a few years older than Shekiba, but her father was so heartbroken and fatigued that his family could not manage to arrange a new wife for him.\n\nShekiba came of age with only her father to turn to, his sparse words, his lonely eyes. She worked beside him day and night. The more she did, the easier it was for him to forget that she was a girl. He began to think of her as a son, sometimes even slipping and calling her by her brothers' names. The village chattered about them. How could a father and daughter live alone? Sympathy gave way to criticism and Ismail and Shekiba grew even more distant from the outside world. The clan did not want to be associated with them and the village had no interest in a scarred old man and his even more scarred daughter-son.\n\nOver the years, Ismail lulled himself into believing that he had always lived without a wife and that he had always had only one child. He managed by ignoring everything. He was the only person who did not see Shekiba's marred face and did not notice that, as a young woman, she might need some direction from a female. When she bled every month, he pretended not to smell the soiled rags that she would keep soaking and hidden behind a stack of logs in their two-room home. And when he heard her shed tears, he shrugged her sniffles off as a touch of flu.\n\nShekiba's father took his daughter-son to the fields to help him manage their small plot of land. She hoed, she slaughtered and she chopped as any strong-backed son would do for his father. She made it possible for Ismail to go on believing that life had always been father and son. Shekiba proved to be able-bodied, affirming her father's confidence in her ability to manage the farm. Her arms and shoulders knotted with muscle.\n\nYears passed. Shekiba's features grew coarser; her palms and soles were thick and callused. Every day, Ismail's back hunched more, his eyes saw less and his needs grew. There were days Shekiba was left to run the entire farm and house on her own.\n\nHad Shekiba been any other girl, she probably would have felt lonesome in this solitary life, but her circumstances were different. The children nearby would always point and tease, as would their parents. Her appearance was shocking everywhere, except at home.\n\nPeople who are beset by tragedy once and twice are sure to grieve again. Fate finds it easier to retrace its treads. Shekiba's father became weaker, his voice raspier, his breaths shallower. One day, as Shekiba watched from the wall of stone and mud, he grabbed his chest, took two steps and crumpled to the ground with a sickle in his grip.\n\nShekiba was eighteen years old but she knew what to do. She dragged her father's body back to the house on a large cloth, stopping every few steps to adjust her grasp and to wipe away the tears that trickled down the right side of her face. The left side of her face remained stoic.\n\nShe laid his body in the living room and sat at his side, repeating the four or five Qur'anic verses that her parents had taught her until the sun came up. In the morning, she began the ceremony she had performed too often in her short life. She undressed her father, careful to keep his private areas hidden beneath a rag. The ritual washing should have been done by a man but Shekiba had no one to call on. She would rather have invited Allah's wrath into her home than turn to those vile people.\n\nShe bathed him, turning away as she poured water onto his man parts and blindly wrapping his stiff body in a cloth, as she and her mother had done with her sister. She dragged him back outside and opened the earth one final time to complete her family's interment. Shekiba chewed her lip and debated digging one more spot for herself, thinking there would be no one left to do so when her turn came. Too tired to do anything more, Shekiba said a few prayers and watched her father disappear under clods of earth\u2014disappear like her sister, her brothers and her mother.\n\nShe walked back to the hollow house and sat silently\u2014afraid, angry and calm.\n\nShekiba was alone.\nCHAPTER 3\n\n\"We wouldn't be the first. It's been done before.\"\n\n\"You're listening to that lunatic Shaima and that story about your precious grandmother.\"\n\n\"It wasn't my grandmother. It was\u2014\"\n\n\"I don't care. All I know is that woman makes my head ache.\"\n\n\"Arif-jan, I think it would be wise for us to consider this. For everyone's sake.\"\n\n\"And what good will come of it? You see everyone else who has done it? They all have to change back in a few years. It doesn't help anything.\"\n\n\"But, Arif-jan, she could do things. She could go to the store. She could walk her sisters to school.\"\n\n\"Do what you want. I'm going out.\"\n\nI listened carefully from the hallway, just a few feet from the bedroom we all shared. Our kitchen was behind the sitting room, a few pots and a gas burner. Our home was spacious, built in a time when my grandfather's family had more. Now these walls were bare and cracking and looked more like those of our neighbors.\n\nWhen I heard Padar-jan strain to get up, I quickly tiptoed off, my bare toes silent on the carpet. When I was sure he was gone, I came back to the living room to find my mother lost in thought.\n\n\"Madar-jan?\"\n\n\"Eh? Oh. Yes, bachem. What is it?\"\n\n\"What were you and Padar-jan talking about?\"\n\nShe looked at me and bit her lip.\n\n\"Sit down,\" she said. I sat cross-legged in front of her, careful that the hem of my skirt reached over my knees and covered my calves. \"You remember the story your khala Shaima told the other night?\"\n\n\"The one about our great-great-great-great . . .\"\n\n\"You're worse than your father, sometimes. Yes, that one. I think it is time we change something for you. I think it would be best if we let you be a son to your father.\"\n\n\"A son?\"\n\n\"It's simple and it's done all the time, Rahima-jan. Just think how happy that would make him! And you could do so many things that your sisters wouldn't be able to do.\"\n\nShe knew how to pique my interest. I cocked my head to the side and waited for her to go on.\n\n\"We could change your clothes and we'll give you a new name. You'll be able to run to the store any time we need anything. You could go to school without worrying about the boys bothering you. You could play games. How does that sound?\"\n\nIt sounded like a dream to me! I thought of the neighbors' sons. Jameel. Faheem. Bashir. My eyes widened at the thought of being able to kick a ball around in the street as they did.\n\nMadar-jan wasn't thinking of the boys in the street. She was thinking of our empty cupboard. She was thinking of Padar-jan and how much he had changed. We were lucky when he brought home some money from an odd job here or there. Every once in a while, his mind focused enough that he was able to tinker with an old engine and breathe life back into it. His small earnings were spent, unevenly, on his medicine and keeping us clothed and fed. The more Madar-jan thought about it, the more she realized how desperate our situation was becoming.\n\n\"Come with me. There's no reason to delay anything. Your father is taking more and more . . . medicine these days. Your khala Shaima is right. We need to do something or we're going to be in real trouble.\"\n\nWe girls were nervous about getting sick. We worried that if we did, we would have to take the same medicine that Padar-jan took. It made him do funny things, behave in funny ways. Mostly he just wanted to lie about the house and sleep. Sometimes he said things that didn't make sense. And he never remembered anything we said. It was worse when he didn't take his medicine.\n\nHe had broken nearly everything in the house that could be broken. The dishes and glasses survived only because he lacked the energy to pull them from the cabinet. Anything within reach had already been thrown against a wall and smashed to pieces. A ceramic urn. A glass plate that Madar-jan had received as a gift. They were casualties of the war inside Padar-jan's head.\n\nPadar-jan had fought with the mujahideen for years, shooting at the Russian troops that bombarded our town with rockets. When the Soviets finally slinked back to their collapsing country, Padar-jan came home and prayed that life would return to normal, though few people could recall such a time. That was 1989.\n\nIn that year, he returned home to his parents, who barely recognized him as the seventeen-year-old boy who had left home with a gun slung over his shoulder in the name of God and his country. His mother and father hurriedly arranged a marriage for him. At twenty-four years old, he was long overdue and they thought a wife and children would bring him back to normal, but Padar-jan, just like the rest of the country, had forgotten what normal was.\n\nMadar-jan was barely eighteen when they were wed. I imagine she must have been as terrified on her wedding night as I was on mine. Sometimes I wonder why she did not warn me, but I suppose those are not things women should speak of.\n\nAs the country planned for new beginnings, so did my parents. My sister Shahla came first, followed by Parwin and me. Then came Rohila and Sitara. We were all a year apart and close enough in age that only our mother could tell us apart once we were walking. But with one daughter after another, Madar-jan did not become the wife that Padar-jan expected. Even more sorely disappointed was my grandmother, who had respectably borne five sons and only one daughter.\n\nThings fell apart at home, just as they did across the country when Russia left. While the Afghan warriors turned their guns and rockets on each other, Padar-jan tried to settle into life at home. He tried to work alongside his father as a carpenter but a man who had been taught only to destroy found it hard to create. Loud sounds jarred him. He grew frustrated and drifted back to the warlord, Abdul Khaliq, he had fought under.\n\nWarlords were Afghanistan's new aristocracy. Allegiance to a man with local clout meant a better life. It meant an income when there otherwise would be none. It wasn't long before Padar-jan had oiled his machine gun, slung it over his shoulder and gone off to fight again, this time in Abdul Khaliq's name. He returned home every so often. When he returned the first time and found that Madar-jan had given birth to yet another girl, me, he walked out again and returned to the killing fields with fresh anger.\n\nMadar-jan was left behind with a houseful of girls and only her bitter in-laws to turn to. We lived in a small two-room house, part of the family's compound. War pushed families together. Two of my uncles were killed in the fighting. My uncle's wife died giving birth to her sixth child. Until he could remarry two months later, his children were cared for by my mother and my other aunts. We should have felt like one big family. We should have been kind to each other. But there was resentment. There was anger. There was jealousy. There was, as there would be in the rest of the country, civil war.\n\nMadar-jan's family lived a few kilometers away, but they might as well have been on the other side of the Hindu Kush mountains. They had given their daughter to Padar-jan and did not want to interfere in her relationship with her new family. Madar jan's deformed sister, Shaima, was the exception.\n\nDeformities were not easily forgiven, so Khala Shaima steeled herself to resist the name-calling, the ridiculing, the gawking. Older than Madar-jan by nearly ten years, our aunt would tell us things that no one else would say. She would tell us about the war, how the warlords controlled everything and conquered without mercy, even attacking women in the most shameful way of all. Usually Madar-jan hushed her older sister with a pleading look. We were young, after all, and it wasn't Khala Shaima who would have to quiet our night terrors. Sometimes Khala Shaima forgot we were children and told us so much that we sat wide-eyed, frightened of our own father.\n\nWhen Padar-jan came home, we cowered. His moods ranged from jubilant to foul but there was no predicting where on the spectrum he would be or when he would make an appearance. Madar-jan was lonely and welcomed her sister's visits, even if her mother-in-law griped about them. My grandmother made sure to report to her son just how many times Khala Shaima had come to visit while he was away, clucking her tongue in disapproval and inciting his wrath. It was her way of showing Madar-jan that she was in control of our home, even if it sat fifty feet away from the main house.\n\nEveryone wanted control but it was hard to get. The only one who seemed to have any was Abdul Khaliq Khan, the warlord. He and his militia were able to gain control of our town and the neighboring towns, having pushed back their rivals. We were north of Kabul and hadn't seen any fighting in about four years but from what we heard, Kabul was besieged. People in our town shook their heads in dismay at the news but our homes were already pockmarked and turned to rubble. It was time for the privileged in Kabul to taste what we had survived.\n\nThose were ugly times. I can only imagine what my father must have seen from the time he was just a teenage boy. Like so many others, he numbed himself to the ugliness with the \"medicine\" that Madar-jan referred to. He clouded his mind with the opium that Abdul Khaliq kept around, as crucial to his men's ability to wage war as the ammunition strapped to their backs.\n\nMadar-jan grew weary of our father but all she could do was look after us girls. Khala Shaima brought her some concoction that she took so she wouldn't have any more children after me. I don't know what the medicine was, but it worked for six years. When Madar-jan felt her belly stretch again, she prayed and prayed and did all the things that Khala Shaima told her to do. Nothing worked. Disappointed and fearful, she named our youngest sister Sitara and dreaded the day that Padar-jan would come home to find out she had brought yet another daughter into his home.\n\nThen came the Taliban. They were just another faction in the civil war but they gained in strength and their regime crept across the country. It didn't affect us much until we were pulled out of school, windows were blackened and music was banned. Madar-jan sighed but carried on, her daily routine largely unaffected by the new codes.\n\nWhen word got out that our town had fallen to the Taliban, Abdul Khaliq brought his men home to fight back\u2014and to defend his honor as a warlord. There were weeks of explosions, crying, burying, and then the men came home, victorious. Our town was again our own.\n\nPadar-jan stayed home for a few months. He spent time with his brothers, tried to help his father recover some business and even helped some of the neighbors to rebuild their homes. Things were going well until the day that a young boy came knocking on our door with a message for Padar-jan. The next morning, Padar-jan oiled his machine gun, donned his pakol hat and headed back out to rejoin the war.\n\nHe came back here and there but his mood swings were worse with each visit. We saw him only two or three days at a time and we were children, too young to understand the rage he brought home. He was not the same person at all. Even Bibi-jan, my grandmother, would cry after his visits, saying she had lost another son to the war.\n\nIt was my cousin Siddiq who told us about the news. He had heard from our grandfather.\n\n\"Amrika. That's who. They came and they're bombing the Taliban. They have the biggest guns, the biggest rockets! And their soldiers are so strong!\"\n\n\"Why didn't Amrika come before?\" Shahla had asked. She was nearly twelve years old then. Wise enough to come up with questions that made us look at her with admiration.\n\nSiddiq was ten but had the confidence of a boy twice his age. His father had been killed years ago and he grew up under our grandfather's wing. He was the man of his house.\n\n\"Because the Taliban bombed Amrika. Now they're angry and they're bombing them back.\"\n\nOur grandfather entered the courtyard and overheard our conversation.\n\n\"Siddiq-jan, what are you telling your cousins?\"\n\n\"I was just telling them about Amrika, Boba-jan. That they're firing rockets at the Taliban!\"\n\n\"Padar-jan,\" Shahla asked timidly, \"did the Taliban destroy many homes in Amrika?\"\n\n\"No, bachem. Someone attacked a building in Amrika. Now they are angry and they've come after him and his people.\"\n\n\"Just one building?\"\n\n\"Yes.\"\n\nWe were silent. It sounded like good news. A big, powerful country had come to our rescue! Our people had an ally in the war against the Taliban!\n\nBut Boba-jan could see in Shahla's eyes that there was something that puzzled her and he knew just what it was. Why would Amrika be so upset after just one building was attacked? Half our country had crumbled under the Taliban. We were all thinking the same thing.\n\nIf only Amrika would have been upset about that too.\nCHAPTER 4\n\nShekiba continued to toil in the fields as if her father were at her side. She fed the chicken and the donkey and fixed the plow when the axle snapped on a stone in the field. The house was quiet, somber. Sometimes the silence grated on her nerves and she would try to break it with the sounds of chores, or by talking to the birds perched on the wall. Some days she felt content, almost happy, to be self-sufficient. She hoped her mother liked the small flowers she had planted while she listened to the bulbul sing over Aqela's grave.\n\nSome things were difficult. Without her father around, Shekiba had no connection with the village or its resources. She used the cooking oil sparingly and was careful with how much she harvested from their field so that she would not go hungry. She dug a small trench between the house and the wall and buried some potatoes so that she would have a stock for the coming winter months. She picked the beans and ate a few, leaving the rest to dry for later.\n\nHer father's death seemed to usher winter in sooner than usual, by Shekiba's warped sense of time. Shekiba had little reason to care about the month or year. The sun would rise and fall and she continued to do her chores, occasionally bothering to wonder what would come of her. How long would this existence last? More than once she thought of ending her life. Once, she'd pinched her nose and shut her mouth. She felt her chest tighten and tighten until she finally took a breath and continued to live, cursing her weakness.\n\nShe again contemplated digging her own plot, beside her father, and lying down in it. Maybe the dark angel Gabriel would see her and reunite her with her family. Shekiba wondered if she would see her mother again. If she did, she prayed it would be the mother who sang while she cooked their meals, not the bald, glassy-eyed woman Shekiba had buried.\n\nWinter came and Shekiba floundered along, subsisting on what she had managed to keep through the fall. Each time she bothered to undress and bathe, she noticed her ribs protruding more. She used her siblings' clothing to cushion her hip bones from the hard floor. She grew weak, her hair brittle and frayed. Her gums bled when she chewed but she barely noticed the taste of blood in her mouth.\n\nSpring came and Shekiba looked forward to the warmth of the sun and the tasks that came with it. But along with spring came a visitor, and the first hint that Shekiba would not be allowed to live like this for long.\n\nShe was feeding the chicken when she saw a young boy in the distance, coming toward her home from her grandfather's house. She could not tell who it was but went inside and donned her burqa. She paced back and forth, peeking through the door from time to time to confirm that the boy was still coming toward her. Indeed he was, and as he neared, Shekiba could see that he was no more than seven or eight years old. She marveled at how healthy he looked and wondered what her cousins were eating at the main house. Once more, Shekiba was thankful for the ability to hide behind the blue cloak.\n\n\"Salaaaaaam!\" he called out when he was near enough. \"I am Hameed! Dear uncle, I want to speak to you!\"\n\nHameed? Who was Hameed? It did not surprise Shekiba that she didn't recognize him. Likely many cousins had been born since she lost contact with the clan. Shekiba wondered how to reply. Should she answer or should she keep quiet? What would invite less inquiry?\n\n\"Salaaaaaaam! I am Hameed! Dear\u2014\"\n\nShekiba cut him off.\n\n\"Your uncle is not home. He cannot speak to you now.\"\n\nThere was no answer for a time. She wondered if Hameed had been warned about her. She could imagine the conversation.\n\nBut be careful. Your uncle has a daughter, a monster, really. She is terrible to look at, so don't be too frightened. She's insane and may say crazy things.\n\nShekiba put her ear to the wall, trying to hear if Hameed was still there or if he was walking away.\n\n\"Who are you?\"\n\nShekiba did not know how to answer.\n\n\"I said who are you?\"\n\n\"I am . . . I am . . .\"\n\n\"Are you my uncle's daughter? Are you Shekiba?\"\n\n\"Yes.\"\n\n\"Where is my uncle? I was told to bring him a message.\"\n\n\"He is not here.\"\n\n\"Where is he then?\"\n\nAt the edge of the field. Did you see the tree? The one that should be growing apples but grows nothing at all? That's where he is. You walked right past him, along with my mother, my sister and my two brothers. If you have anything to tell him, you can tell him as you make your way back to the house with all the food.\n\nBut Shekiba did not say what she was thinking. She had that much sense left in her.\n\n\"I said, where is he?\"\n\n\"He has gone out.\"\n\n\"When will he be back?\"\n\n\"I do not know.\"\n\n\"Well, tell him that Bobo Shahgul wants to see him. She wants him to come to the house.\"\n\nBobo Shahgul was Shekiba's paternal grandmother. Shekiba hadn't seen her since before the cholera took her family. Bobo Shahgul had come over to tell her son about a girl in the village, the daughter of a friend. She had wanted her son to take her on as a second wife, maybe even to have him move back into the family compound with the second wife and keep the first wife at this house. Shekiba remembered watching her mother listen to the conversation with her head bowed, saying nothing.\n\n\"Tell Bobo Shahgul that . . . that he is not here now.\"\n\nShe was skirting the truth.\n\n\"You will tell my uncle what I have said?\"\n\n\"I will.\"\n\nShe could hear his footsteps grow distant but waited a full hour before emerging from the wall, just in case. She wasn't the brightest girl, but even Shekiba knew it was just a matter of time before her grandmother sent another message.\n\nThree months passed.\n\nShekiba was attaching the harness to the donkey to begin tilling the soil when she saw two men walking toward the house. She darted inside and grabbed her burqa in a panic. Her heart fluttered as she waited for them to near. She kept her ear against the inner wall, listening for footsteps.\n\n\"Ismail! Come out and speak to us! Your brothers are here!\"\n\nHer father's brothers? Bobo Shahgul meant business. Shekiba frantically tried to think of something reasonable to say.\n\n\"My father is not at home!\"\n\n\"Enough with the nonsense, Ismail! We know you're here! You're too much of a coward to leave your home! Come on out or we'll barge in there and shake some sense into you!\"\n\n\"Please, my father is not home!\" She could hear her voice cracking. Would they force their way in? It wouldn't take much effort. The door would fold in at their slightest touch.\n\n\"Goddamn you, Ismail! What are you doing hiding behind your daughter! Move aside, girl, we are coming in!\"\nCHAPTER 5\n\nMadar-jan took me behind the house with Padar-jan's scissors and razor. I sat nervously while my sisters watched. She pulled my long hair into a ponytail behind my head, whispered a prayer and slowly began to shear away. Shahla looked astonished. Rohila looked entertained and Parwin watched only for a moment before running back into the house for her pencils and paper. She sketched furiously with her back turned to me.\n\nMadar-jan cut and trimmed, bending my ear forward to trim around it. She cut my bangs short and straight across my forehead. I looked at the ground around me and saw hair everywhere. She brushed the loose strands from my shoulders, blew at my neck and dusted off my back. My neck felt bare, exposed. I giggled with nervous excitement. Only Shahla noticed the single tear that trickled down Madar-jan's cheek.\n\nThe next step was my clothing. Madar-jan asked my uncle's wife for a shirt and pair of pants. My cousin had outgrown them, as had his older brother and my other cousin before him. She sent me inside to get dressed while she and my sisters swept my girl hair from the courtyard.\n\nI slipped one leg in and then the other. They were slimmer and heavier than the usual balloon pants I wore under my dresses. I cinched the strings at the waist and made a knot. I pulled the tunic over my head and realized there was no ponytail to pull through after it. I let my hand run against the back of my head, feeling the short ends.\n\nI looked down and saw my knobby knees through the pantaloons. I folded my arms across my chest and cocked my head, as I'd seen my cousin Siddiq do so many times. I kicked my foot, pretending there was a ball in front of me. Was that it? Was I a boy already?\n\nI thought of Khala Shaima. I wondered what she would say if she were to see me like this. Would she smile? Had she really meant it when she suggested I should be turned into a boy? She told us our great-great-grandmother had worked on the farm like a boy, that she'd been a son to her father. I had waited for her to go on, to get to the part where our great-great-grandmother turned into a boy. Khala Shaima said she would come back and tell us more of the story another day. I hated having to wait.\n\nI smoothed my shirt down and went back out to see what my mother thought.\n\n\"Well! Aren't you a handsome young boy!\" Madar-jan said. Even I could detect the hint of nervous uncertainty in her voice.\n\n\"Are you sure, Madar-jan? Don't I look odd?\"\n\nShahla covered her mouth with her hand at the sight of me.\n\n\"Oh my goodness! You look just like a boy! Madar-jan, you can hardly tell it's her!\"\n\nMadar-jan nodded.\n\n\"You won't have to get your knots taken out anymore,\" Rohila said enviously. Getting the knots brushed out of our hair was a painful morning routine. Her hair coiled into a mess of tiny birds' nests that Madar-jan struggled to brush out while Rohila winced and squirmed.\n\n\"Bachem, from now on we're going to call you Rahim instead of Rahima,\" Madar-jan said tenderly. Her eyes looked heavier than they should have at the age of thirty.\n\n\"Rahim! We have to call her Rahim?\"\n\n\"Yes, she is now your brother, Rahim. You will forget about your sister Rahima and welcome your brother. Can you do that, girls? It's very important that you speak only of your brother, Rahim, and never mention that you have another sister.\"\n\n\"Just in case we forget what she looked like, Parwin drew this picture of Rahima.\" Rohila handed Madar-jan the sketch Parwin had done while she was cutting my hair. It was an incredible likeness of me, the old me with long hair and na\u00efve eyes. Madar-jan looked at the drawing and whispered something we didn't understand. She folded the paper and placed it on the tabletop.\n\n\"Is that it? Just like that? She's a boy?\" Shahla looked skeptical.\n\n\"Just like that,\" Madar-jan said quietly. \"This is how things are done. People will understand. You'll see.\" She knew my sisters would be the hardest to convince. Everyone else\u2014teachers, aunts, uncles, neighbors\u2014they would accept my mother's new son without reservation. I wasn't the first bacha posh. This was a common tradition for families in want of a son. What Madar-jan was already dreading was the day they would have to change me back. But that would only be when I began to change into a young woman. That was still a few years away.\n\n\"Oh, wow.\" Parwin had returned to the courtyard to see what happened.\n\n\"So just like that. She's a boy.\"\n\n\"Nope, not yet,\" Parwin said calmly. \"She's not a boy yet.\"\n\n\"What do you mean?\" Rohila asked.\n\n\"She's got to walk under a rainbow.\"\n\n\"A rainbow?\"\n\n\"What are you talking about?\"\n\n\"My God, Parwin,\" Madar-jan said, smiling faintly. \"I don't remember telling you about that poem. How do you even know about it?\"\n\nParwin shrugged her shoulders. We weren't surprised. Parwin couldn't tell you if she had eaten breakfast but she often knew things that no one expected her to know.\n\n\"What is she talking about, Madar-jan?\" I asked, curious to find out if Parwin was right or if her imagination had gotten the best of her today.\n\n\"She's talking about an old poem. I don't know if I can even remember how the story goes but it's about what happens if you pass under a rainbow.\"\n\n\"What happens if you pass under a rainbow?\" Rohila asked.\n\n\"There's a legend that walking under a rainbow changes girls into boys and boys into girls.\"\n\n\"What? Is that true? Could that really happen?\"\n\nThis perplexed me. I hadn't walked under a rainbow. I'd never even seen one, for that matter. How was this change supposed to work?\n\n\"Tell us the poem, Madar-jan. I know you remember it. We drank in spirits . . .\" Parwin started her off.\n\nMadar-jan sighed and went into the living room. We followed. She sat with her back against the wall and looked to the ceiling, trying to recall the details. Her chador fell across her shoulders. We sat around her and waited expectantly.\n\n\"Afsaanah, see-saanah . . . ,\" she began. One story, thirty stories. And then she sang the poem.\n\nWe drank in spirits and played in fields\n\nEnamored of\n\nIndigos, saffrons and teals\n\nThere was fog in the space\n\nBetween them and I\n\nColors reach to touch God in the sky\n\nI envy the arc, stretched strong and wide\n\nAs one brilliance blends into another\n\nColors bow deeply to welcome a brother\n\nWe humble servants, meekly pass under\n\nRostam's bow changes girl to boy, makes one the other\n\nUntil the air grows dry and tires of the game\n\nAnd the mist opens its arms, colors reclaimed\nCHAPTER 6\n\nShekiba sat with her back against the cool wall. It was night and the house was quiet. Snoring came from every direction, some louder than others. By the soft glow of the moon, she could see the kettles and pots she had washed and stacked in the corner to make room for her blanket. Like most nights, her eyes were wide open while everyone else's were closed. This was the hour of night when she would wonder what she could have done differently.\n\nHer uncles had barged into the home that day, refusing to be turned away. Now that she had been reunited with her grandmother, she could hardly blame them for their persistence. No one wanted to disappoint Bobo Shahgul. She was horrid enough when she was satisfied.\n\nIt hadn't taken long for Shekiba's uncles to realize that something had happened to her father. The house smelled of rot and loneliness. Shekiba had stopped sweeping the floor and had let the potato peels collect in a corner, too disinterested to take them outside. After a time, she didn't notice the smell. But it wasn't just the house. Shekiba had become apathetic. She hadn't bothered to wash her dress, and for most of the winter, she had curled up in a ball under a blanket, letting her own stench fester. Daylight and warmth had inspired her to wash herself but it would take more than a few baths to undo what had become of her. Her hair was a tangled nest of lice and unbrushable for months.\n\nShekiba was pale and gaunt. For a moment, her uncles believed they may have been looking at a djinn, a spirit. How could living flesh look like that?\n\nThey asked for her father, their eyes scouting the room and realizing instantly that he was not there. Shekiba trembled and turned to the side, wanting to hide from them but making sure they were not approaching her. They couldn't see her, but they could smell fear, sweat and blood. They asked again, louder, angrier.\n\nThat was when Shekiba left. She heard a scream and a blue ghost ran into the wall that had sheltered her from the view of others\u2014the wall her father had built to guard his family. Another scream, and as she fell to the earth, hands grabbed the ghost, shocked at how easily their fingers circled bones. The ghost wanted to fight back, to run away and escape, but the men had meat on their bones. They gripped her and she let go, allowing them to roll her onto her blanket and carry her back to the family compound in much the same way that she had carried her father to his grave.\n\nAs she passed by the tree where her family lay buried, Shekiba moaned and called out to them. She tried to lift her head to see the rounded mounds of earth.\n\nMadar. Padar. Tariq. Munis. Bulbul.\n\nShe did not see her uncles look at each other, sharing a realization that the entire family was dead, even their brother Ismail. Shekiba didn't see them bite their tongues, hold back their tears and mutter that they should have been there to wash their brother's body and throw dirt on his grave. Shekiba was the last survivor\u2014the one who should not have survived. They wondered how long this girl had been living alone and shook their heads with the shame of the situation. A girl, by herself! What dishonor this could bring to their family if anyone in the village were to find out!\n\nThey laid her in the courtyard of the home while they went to notify Bobo Shahgul. Within minutes, the spry old lady stood over Shekiba, peering down through her cataract-clouded eyes to get a better look at the grandchild she could do without.\n\n\"Tell your wives to get her washed up. Warn them that her face will turn their stomachs. And tell them to feed her. We must deal with this creature now if we are to save our good name within the village. May God punish her for keeping her father from us, my son! Not even telling us when he left this world! She will pay for this.\"\n\nBobo Shahgul proved to be a woman of her word. Since her husband had died two years ago, she had happily taken on the role of the family matriarch. She presided over her sons' brides with her walking stick, though there was nothing at all wrong with her legs. She had earned the right to walk with her head high since she had given her husband six sons and two daughters. Now it was her turn to oversee the roost with the same iron fist she had survived.\n\nShekiba let herself be undressed and bathed. She found it much easier than resisting. The youngest wives were assigned the formidable task of deconstructing the beast Shekiba had become. Pails of water were brought in. Her hair was sheared, too far gone to salvage. They cursed her for the rank smells of every body recess, their nostrils seared. They put food in her mouth; someone moved her jaw, reminding her to chew.\n\nIn a few days, Shekiba's mind returned to her body. She began to hear what people were saying; she began to notice that her belly did not ache with hunger. Her fingers reached up and felt a head scarf covering the ragged edges of her chopped hair.\n\nI must look like one of my cousins, she thought.\n\nHer skin was raw and reddened from the brutal baths she had been given. Her aunts had scrubbed a layer of filth from her with a washcloth too rough for her frail skin. She had some scabs, while other areas stayed red and chafed, her body too malnourished to repair minor damage. At night, she slept on a blanket in the narrow kitchen, her feet often knocking against a pot and waking her up. In the morning, she was moved into one of many rooms where she would be out of the way while the wives prepared breakfast.\n\nI'm tired of lifting her. Get Farrah to help you. My back is aching.\n\nYou say the same thing every day! Your back, your back. Surely, it's not from doing anything around here. What has your husband been doing to you! Tell him to go easy.\n\nGiggling.\n\nShut your mouth and pick up her arms. Ugh. I am queasy enough today. I can't stand to look at her face.\n\nFine, but we'll put her in your room. My room still has her smell from yesterday and I cannot stand it.\n\nShekiba let herself be moved around and insulted. At least she was not being asked to participate in this existence. But that would not last. Bobo Shahgul had other plans for her.\n\nThe family home had a small kitchen where the wives all helped cook. There was one main family room where everyone sat around during the day, the children played and meals were shared. Surrounding those two main rooms were four or five other rooms, each assigned to one of Bobo Shahgul's sons. Families slept together in one room. Only Bobo Shahgul had a room of her own.\n\nShekiba was on her side in her uncle's room when she vaguely felt Bobo Shahgul's walking stick jab into her thigh.\n\n\"Get up, you insolent girl! Enough of your nonsense. You have been asleep for over a week. You're not going to get away with this kind of behavior in this house. God only knows what craziness your mother allowed.\"\n\nShekiba winced. A downside to her recovery was that her body now had the energy to sense pain. Again, the stick poked into her leg. Shekiba rolled onto her side and tried to push herself back, away from her grandmother. Her head was heavy with too much sleep.\n\n\"Insolent and lazy! Just like your mother!\"\n\nThere was no escape from this woman. Shekiba eased herself to a sitting position and managed to focus her eyes on her grandmother.\n\n\"Well? Have you nothing to say for yourself? Disrespectful and ungrateful. We have bathed and fed you and you can do nothing more than sit there and stare like an idiot?\"\n\n\"Salaam . . . ,\" Shekiba said meekly.\n\n\"Sit up straight and watch your legs. Although you may not know it, you are a girl and you should sit like one.\" Bobo Shahgul snapped her stick against her granddaughter's arm. Shekiba flinched and straightened her back as best she could. Bobo Shahgul leaned in close. Shekiba could see her deep-set wrinkles, the yellow of her eyes.\n\n\"I want you to tell me what happened to my son.\" Each syllable was punctuated by a fine spray of saliva.\n\nYour son? Your son? Shekiba thought, her mind suddenly clear and focused. Your son was my father. When was the last time you saw him? When was the last time you bothered to send him any food, any oil? You could see him in the field. You could see the pain in his movements. Did you bother to send him anything then? All you cared about was giving him another wife, saving the family name.\n\n\"He was my father.\" Shekiba left the rest unsaid.\n\n\"Your father? And a lot of good that did him! He could have had a decent life. He could have had a wife to look after him, to bear him sons who would grow our clan and work on our land. But you did your very best to keep him secluded, trapped with such a wild creature as yourself that no one would want to come near you or him! First your mother, then you! You killed my son!\"\n\nHer stick jabbed Shekiba's breastbone.\n\n\"Where is he? What did you do with him?\"\n\n\"He is with my mother. He is with my brothers and my sister. They are all there together, waiting for me.\"\n\nBobo Shahgul fumed at Shekiba's detachment. As she suspected, her son had been buried without her knowledge. Her eyes swelled with rage.\n\n\"Waiting for you, eh? Maybe God will see fit that your time come soon,\" she hissed.\n\nIf only, Shekiba thought.\n\n\"Zarmina! Come and get this girl! She is to help you with the chores around the house. It is time for her to start earning her stay here. She has caused this family enough grief and she needs to start making up for it.\"\n\nZarmina was married to Shekiba's oldest uncle. She had the strength of a mule and the face of one too. Shekiba guessed she was the one who had scrubbed her skin raw. Zarmina walked into the room, wiping her hands on a rag.\n\n\"Ahhh, so finally we can stop waiting on this girl hand and foot! About time. God has no use for the lazy. Get up and get into the kitchen. You can start peeling the potatoes. There is much to be done.\"\n\nThis was the beginning of a new phase in Shekiba's life. She was no stranger to hard work, to lifting and peeling, to scrubbing and hauling. She was assigned the least desirable chores in the house and accepted them without argument. Bobo Shahgul wanted her to pay for her father's death. She made this clear every day, sometimes calling out his name and clucking her tongue.\n\nShe would even wail and lament the tragedy of his death.\n\n\"He was taken too young. How could he have left his mother to grieve him? How could such a thing happen to our family? Have we not prayed enough? Have we not followed God's word? Oh, my dear son! How could this have happened to you?\"\n\nHer daughters-in-law would sit at her side, plead with her to be strong and tell her that Allah would care for him since his own family had not. They would fan her and warn her that she would make herself ill with all this grief. But Bobo Shahgul's sobbing came without tears and turned off just as easily as it turned on. Shekiba continued with the task of brushing the rug. She did not bother to look up.\n\nWhat happened to you? We heard that they call you shola-face. Did you put shola on your face?\n\nHer cousins asked the same question over and over again. Shekiba ignored them for the most part. Sometimes people answered for her.\n\nShe did not listen to her mother and that's what happened to her. Did you understand what I said? So you had better pay attention to what I say or your face will turn just as hideous as hers!\n\nShekiba became a very useful instrument for discipline in the house.\n\nLook at what you've done! Clean this up or you will be sleeping with Shekiba tonight!\n\nThere was no end.\n\nGod has punished Shekiba. That is why she has no mother or father. Now go wash for prayers or else God will do the same to you.\nCHAPTER 7\n\nMadar-jan kept me at home for a couple of weeks, wanting me to get used to the idea of being a boy before she let me test the waters outside of our home. She corrected my sisters when they called me Rahima and did the same with my younger cousins who had never before seen a bacha posh. They ran into their houses to report the news to their mothers, who smirked. Each had given her husband at least two sons to carry on the family name. They didn't need to make any of their daughters a bacha posh.\n\nBut Madar-jan ignored their looks and went about her chores. Bibi-jan hated that anyone in her family was forced to resort to the bacha posh tradition.\n\n\"We needed a son in the house, Khala-jan.\"\n\n\"Hmmph. Would be better if you could just have one as the others did.\"\n\nMadar-jan bit her tongue for the thousandth time.\n\nPadar-jan barely seemed to notice the change. He had been gone for a couple of days and came home exhausted. He sat in the living room and opened an envelope of small pellets. He squeezed them between his fingers and sprinkled the mix into a cigarette casing. He lit one end and sucked on it deeply. Thick, sweet smoke twisted around his face and wrapped around his head. My sisters and I came in from outside to find him sitting there. We stopped short and said hello, our heads bowed.\n\nHe looked at us and inhaled deeply. He squinted through the smoke as he noticed that something was different about his three daughters.\n\n\"So she's done it then.\" And that was all he said about the matter.\n\nKhala Shaima was the reassuring voice that Madar-jan needed to hear.\n\n\"Raisa, what else were you going to do? Your husband is delirious half the time and of no use to you. You can't send the girls to school or even to the market because you're afraid of what will happen. Your in-laws are all too busy talking about each other to help you out. This is your only option. Besides, it'll be better for her, you'll see. What can a girl do in this world, anyway? Rahim will appreciate what you've done for him.\"\n\n\"But my in-laws, I\u2014\"\n\n\"Forget them! The person who doesn't appreciate the apple doesn't appreciate the orchard. You'll never please them. The sooner you figure that out the better off you'll be.\"\n\nMy first errand as a boy was an exciting one. I was to go to the market for oil and flour. Madar-jan nervously handed me a few bills and watched me walk down the street. My sisters poked their faces around either side of her skirt trying to get a look as well. I kept glancing over my shoulder and waved at Madar-jan cheerfully, trying to inspire a little confidence in both of us that I could pull this off.\n\nThe streets were lined with shops. Copper pots. Baby clothes. Sacks of rice and dried beans. Colorful flags hung from front doors. The shops were two levels, with balconies on the second floor where men sat back and watched the comings and goings of their neighbors. None of the men walked with any urgency. The women, on the other hand, moved purposefully and carefully.\n\nI stepped into the first shop I recognized, a large sign overhead announcing the arrival of a new cooking oil.\n\n\"Agha-sahib, how much for a kilo of flour?\" I asked, remembering to keep my shoulders straight. I couldn't quite bring myself to look the man in the eye so I kept shifting my gaze to the tin cans he had stocked on the shelf behind him.\n\n\"Fifteen thousand afghanis,\" he said, barely looking up. Not too long ago, a kilo of flour had cost forty afghanis. But money was worthless now that everyone had bags of it.\n\nI bit my lip. This was double what I had seen him charge my mother, which she complained was already too much. I wasn't surprised. I had come to this same man twice before when my mother had reluctantly sent me out to the market and I had been able to bargain him down to half of what he originally demanded.\n\n\"That's too much, agha-sahib. Not even a king could pay that much. How about six thousand afghanis?\"\n\n\"You take me for a fool, little boy?\"\n\n\"No, sir.\" My chest puffed to hear him call me a boy. \"But I know that Agha Kareem has flour for sale too and he charges much less. I didn't want to walk all the way down to his shop, but . . .\"\n\n\"Ten thousand afghanis. That's it.\"\n\n\"Agha-sahib, it's only one kilo I'm asking for. Eight thousand afghanis is all I'll pay.\"\n\n\"Boy! You're wasting my time,\" he barked, but I knew he had nothing else to do. He'd been picking dirt from under his fingernails when I entered his shop.\n\n\"Then I'll pay you twelve thousand afghanis but I'll need a kilo of flour and a kilo of oil to go with it.\"\n\n\"And a kilo of oil? Have you\u2014\"\n\n\"I'm no fool, agha-sahib,\" I said, and forced myself to look him in the eye, as a boy should. He stopped short and his mouth tightened. His eyes narrowed as he took a good look at me. I felt myself shrink under his stare. Maybe I'd gone too far.\n\nSuddenly he let out a guffaw.\n\n\"You're a little smart-ass, aren't you?\" he said with a smirk. \"Whose son are you anyway?\"\n\nMy shoulders relaxed. He saw the bacha posh but it was just as Madar-jan had promised\u2014people understood.\n\n\"I'm Arif's son. From the other side of the field, past the stream.\"\n\n\"Well done, my boy. Here, take your oil and flour and run off before I come to my senses.\"\n\nI quickly counted out the bills, took my spoils and hurried back home to show Madar-jan. My walk turned to a jog as I realized I didn't have to be demure and proper. I tested an old man walking by. I looked directly at him, meeting his squinted eyes and seeing that he didn't react to my forwardness. Thrilled, I started to run faster. No one gave me a second glance. My legs felt liberated as I ran through the streets without my knees slapping against my skirt and without worrying about chastising eyes. I was a young man and it was in my nature to run through the streets.\n\nMadar-jan smiled to see me panting and grinning. I laid the goods before her and proudly showed her how much money I'd returned home with.\n\n\"Well, well. Looks like my son bargains better than his mother!\" she said.\n\nI started to understand why Madar-jan needed a son in the home. Certain chores she had left for my father had not been done in months. Now she could ask me.\n\nWhen my sister's shoes came undone, the rubber sole flopping like an open mouth, I took them to the old man down the street. With only three fingers on his right hand, he could fix any shoe in any condition. I brought bread from the baker and chased the stray dog down the street. My father would come home, his eyes red and small, and laugh when he saw me.\n\n\"Bachem, ask your sister to bring me a cup of tea. And tell her to fix me something to eat too,\" he said, ruffling my hair as he walked lazily to the corner of the living room and stretched out on the floor, his head thumping against the pillow cushion.\n\nI was confused for a moment. Why hadn't he asked me to bring the tea and food? But realization swept over me as I walked into the kitchen. I saw Rohila first.\n\n\"Hey, Rohila. Padar-jan wants some tea and something to eat. He's in the living room.\"\n\n\"So? Why didn't you put a plate together? You know there's some korma-katchaloo in the pot.\"\n\n\"He didn't ask me. He said for me to tell my sister. That's you. Anyway, I'm going out. Don't take all day. He looks like he's hungry,\" I said cheerfully. Rohila's hazel eyes gave me a look even as she turned to heat up a bowl of potato stew for our father. She was angry and part of me knew I was being a brat, but everything I was experiencing was new and I wanted to enjoy it. I ignored the shadow of guilt and headed out to see if the stray dog had returned for another game of chase.\n\nA month later, school was back in session and my nerves were again rattled. Madar-jan trimmed my hair and spoke to me cautiously.\n\n\"You'll be in the boys' classroom this year. Pay attention to your teacher and mind your studies,\" she warned me, trying to make this little talk sound routine. \"Remember that your cousin Muneer will be in your class as well. No one, the teacher, the students, no one will ask you about . . . about anything. Just remember that your father has decided to send you to school this year. You are one of the boys and . . . and . . . mind what the teacher tells you.\"\n\nIt would be different, I understood. Khala Shaima's plan had worked well within the confines of our family compound and even in my trips to the bazaar. School would put this charade to the test though, and I could sense my mother's trepidation. My sisters were furious. Padar-jan had decided they were to stay home even though I could have accompanied them to school.\n\nMuneer and I walked to school together. He wasn't the brightest of my cousins and I rarely saw him since his mother kept her children away from the rest of us. That probably worked in my favor. He needed to be told only once that I was his cousin Rahim and always had been, and in his mind there never had been a Rahima. I breathed a sigh of relief that I didn't have to worry about his giving me away.\n\n\"Salaam, Moallim-sahib,\" I said when we arrived.\n\nThe teacher grunted a reply in return, nodding as each student walked in. I wiped my moist palms on my pants.\n\nI felt the teacher's curious eyes follow the back of my head but it could have been my imagination. I scanned the room and stayed close behind Muneer, noting that none of the boys seemed fazed by me. I kept my head bowed and we made our way to the back of the classroom, where Muneer and I shared a long bench with three other boys. One boy was especially eager to show how much he knew about the teacher.\n\n\"Moallim-sahib is very strict. Last year he gave four boys bad marks because their fingernails weren't clean.\"\n\n\"Oh yeah?\" his friend whispered. \"Then you better keep your finger out of your nose!\"\n\n\"Boys! Sit up straight and pay attention,\" the teacher said. He was a rotund man, his shiny bald head rimmed with salt-and-pepper hair. His neatly groomed mustache matched his sparse hairs. \"You'll begin by writing your names. Then we'll see what, if anything, you learned in your last class.\"\n\nI quickly realized the male teachers were just as strict as the women. Class wasn't much different except that there was more whispering and shooting each other looks than I'd ever seen in a girls' classroom. I wrote my name carefully and watched Muneer struggle from the corner of my eye. His letters were awkwardly connected and an extra dot had changed \"Muneer\" to \"Muteer.\" I debated correcting him but the teacher looked in my direction before I could even begin to whisper. He walked around the room and looked at everyone's names, shaking his head at some and grunting at others. Very few seemed to meet his standards.\n\nHe looked over my shoulder and I could hear the air whistle through his nostrils, his belly casting a shadow over my paper. My name got no reaction, which I could take only to mean it had not severely disappointed him. Muneer's notebook, however, made him groan.\n\n\"What is your name?\" he demanded.\n\n\"M-M-Muneer.\" He stole a glance upward at the teacher but quickly looked down again.\n\n\"Muneer,\" he said dramatically. \"If you come back to this class tomorrow and make a single mistake in your name, I'll send you back to repeat last year's work. Understood?\"\n\n\"Yes, Moallim-sahib,\" Muneer whispered. I could feel the heat from his face.\n\nSo the boys weren't learning much more than the girls, I realized.\n\nAfter class, the boys were more interested in racing outside and kicking a ball around than questioning who I was or where I'd come from. Muneer and I walked home with two boys named Ashraf and Abdullah. They were neighbors who lived a half kilometer from our family's house. This was the first time I'd met them, though they knew Muneer and my other boy cousins.\n\n\"What's your name again?\" Ashraf asked. He was the shorter of the two and had light brown hair and round eyes. He was pretty enough to make me wonder if he was like me, a girl underneath those pants.\n\n\"My name is Rahim.\"\n\n\"Yeah, his name is Rahim. He's my cousin,\" Muneer added. The teacher's warnings had shaken him up but now that we were outside, he was breathing easier.\n\n\"Abdullah, have you ever seen Rahim before?\"\n\nAbdullah shook his head. He was dark haired, slim and calmer than his neighbor.\n\n\"No. Are you any good at soccer, Rahim?\"\n\nI stole a sidelong glance and shrugged my shoulders.\n\n\"Oh, he's really good at soccer,\" Muneer said emphatically. His reply caught me off guard. \"I bet he could beat you.\"\n\nI looked at Muneer, wondering if he was trying to set me up.\n\n\"Oh, yeah?\" Abdullah grinned. \"Well, he doesn't have to beat me but it would help if he could beat Said Jawad and his friends. They're probably over in the street playing if you want to join them.\"\n\n\"Yeah, let's do it!\" Muneer picked up his pace and headed down the side street that led to the makeshift field and away from our house. The field was actually an unused side street, too narrow for a car. The boys were accustomed to meeting there for pickup games.\n\n\"Muneer, don't you think we should\u2014\"\n\n\"C'mon, Rahim. Just for a little while! It'll be fun,\" Abdullah said, giving my shoulder a light shove.\n\nI suppose I could have been worse. The only thing I knew how to do was to run. Luckily, I did that well enough that the boys didn't notice that my foot never made contact with the ball or that I never shouted for the ball to be passed to me. I ran up and down the street, my shoulders scraping the clay wall of the alley. I kept expecting my mother or father to appear and drag me back home angrily.\n\nI liked feeling the breeze on my face. I liked feeling my legs stretch, trying to catch the others, trying to race ahead of them. My arms swung by my sides, free.\n\n\"Over here! Pass it over here!\"\n\n\"Don't let him get by! Catch him!\"\n\nI neared the ball. There were six feet kicking at it, trying to knock it back in their direction. I stuck my foot into the melee. I felt the leather against my sole. I kicked at it, sending it flying in Abdullah's direction. He stopped the ball with his heel and nudged it toward the opposite goal. He was running.\n\nI felt a thrill as I chased after him. I liked being part of the team. I liked the dust kicking up under my feet.\n\nI liked being a boy.\nCHAPTER 8\n\nQuickly, most of the household work was turned over to Shekiba. Her uncles' wives found that, once she'd recovered, she was quite capable and could manage even the chores that required the combined strength of two women. She could balance three pails of water, instead of just two. She could lift the wood into the stove. They whispered happily to each other when Bobo Shahgul was not listening, not wanting to appear lazy to the matriarch.\n\nShe has the strength of a man, but she does the chores of a woman. Could there be any better help for the house? Now we know what it must feel like to live like Bobo Shahgul!\n\nShekiba heard their comments but it was in her nature to work. She found that sunset came faster if she busied herself, no matter how laborious the task. Her back ached at the end of the day, but she did not let her face show it. She did not want to give them the satisfaction of exhausting her. Nor did she want to risk a beating for not being able to keep up with her work. In this home, there were many ready sticks to teach her that indolence would not be tolerated.\n\nKhala Zarmina, Kaka Freidun's wife, was the worst. Her thick hands came down with a surprising strength even though she claimed to be too old and tired to do any of the more cumbersome tasks in the house. Her temper was short and she seemed to be poised to take Bobo Shahgul's place when Allah finally decided to reclaim the bitter old woman. Bobo Shahgul realized as much and could see through her false flattery but she tolerated it, keeping Zarmina in line with an occasional berating in front of the others.\n\nKhala Samina was by far the mildest. She was wife to Bobo Shahgul's youngest living son, Kaka Zelmai. It took about a week for Shekiba to realize that Samina scolded or hit her only in the presence of the other daughters-in-law. When she raised her hand, Shekiba braced herself. Unnecessarily, she realized. Samina put no more weight into her blows than she would to swat a fly.\n\nShe doesn't want to look weak, Shekiba thought. But now I know she is.\n\nShekiba kept to herself, did the work assigned to her and tried to avoid eye contact. She did nothing to invite conversation, although she did provide a good topic for discussions in the house. Summer was a few weeks away when Bobo Shahgul interrupted her scrubbing the floor. Kaka Freidun stood beside her, arms crossed.\n\nShekiba instinctively pulled her head scarf across her face and turned her shoulders to face the wall.\n\n\"Shekiba, when you have finished with cleaning this floor, you are to go into the field and help your uncles with the harvest. I'm sure you will appreciate a chance to get fresh air outside and it seems you are experienced with this kind of work.\"\n\n\"But I still have to prepare the\u2014\"\n\n\"Then prepare it quickly and get outside. It is about time you helped to grow the food that has fattened your face.\"\n\nKaka Freidun smirked in agreement. This was all his idea. He had watched Ismail's land reap a harvest that most others would have thought impossible given last season's pitiful rainfall. It occurred to him that his brother's daughter-son may have inherited his instincts with the earth. Why not make use of her? After all, there were plenty of women to do the housework. Bobo Shahgul had agreed readily. The clan was in need of a good harvest. There were many mouths to feed and for the first time in years, their debts were growing.\n\nShekiba nodded, knowing that the new assignment would not mean a relief from her current ones. Her days would be longer. Khala Zarmina was especially angry about the new arrangement but she dared not contest Bobo Shahgul.\n\n\"There is more to be done here in the house! Bobo Shahgul has forgotten what it means to take care of the cooking and cleaning. I've left a pile of clothes in need of hemming and darning for Shekiba-e-shola but I suppose that will all have to wait if she is going to be out in the field during the day. She had better wake up earlier if she's going to get lunch ready too.\"\n\nThe family had quickly embraced her nickname. In Afghanistan, disabilities defined people. There were many others in the village who had such names. Mariam-e-lang, who had walked with a limp since childhood. Saboor-e-yek dista was born with one hand. And if you don't listen to your father, your hand will fall off just like his, mothers used to warn their sons. Jowshan-e-siyaa, or the black, for his dark complexion. Bashir-e-koor, the blind, had lost most of his sight in his thirties and despised the children who laughed at his stumbling gait. He knew, too, that their parents joined in the snickers.\n\nShekiba dried the floor hastily and tightened her head scarf under her chin. She went outside and saw that her uncles were taking a break, leaning against the outside wall and drinking tea that her cousin Hameed had brought out to them. Shekiba turned to assess the progress they had made.\n\nFrom this side of the house she could see her home. It looked small in comparison to the clan's house.\n\nThis is how it felt to watch us.\n\nShe noticed that there were new pieces of equipment in their field and that her father's tools had been carted over to this side of the land. The house had been emptied. A pile of their belongings lay outside the wall her father had built.\n\nThey're taking my home. They wanted our land.\n\nSuddenly, Shekiba realized why it was that Bobo Shahgul had summoned her youngest son after so much time. Her father was tilling the most fertile land the family had and they wanted it. They wanted more than the share of crops he sent over from time to time. They wanted it all. Now there was no one in their way. They were taking her home.\n\nShekiba thought she would feel nothing but inside, she seethed. No one had thought of her when the house's contents were thrown outside for trash. The few remaining items that had belonged to her mother, her father, her siblings all tossed aside to make way for something new. Was someone going to move into her home? Shekiba realized part of her was still hoping to return to that home, to live there independently as she had before. But, of course, that would never happen.\n\nShekiba found a container and walked into the field. There was much to be harvested. The onion plants had long yellow leaves and had probably dried up about three weeks ago, given their appearance.\n\nWhy haven't they pulled these onions out? Shekiba thought, and leaned over to get a closer look.\n\n\"Hey, Freidun! Look what she's doing! Tell her not to touch the onions! They aren't ready yet! This imbecile is going to ruin our lot!\" It was Kaka Sheeragha, the skinniest and laziest of the group.\n\nThe leaves were brittle in her fingertips. She reached at the base and began to pull the bulbs from the earth.\n\nAlmost too late. They're about to rot. No wonder our food tastes the way it does. God knows what they're doing with the rest of the crops.\n\nKaka Freidun walked over and looked at the three onions she had already unearthed. Shekiba did not turn to look at him. He grunted something and then walked away.\n\n\"You didn't say anything to her?\" Sheeragha yelled out.\n\n\"Enough,\" Freidun answered. \"They're ready.\"\n\nSheeragha looked at his elder brother and bit his tongue. The men returned to the fields and grunted instructions at each other. They kept a distance from Shekiba but watched her from the corners of their eyes. She moved nimbly through the rows, her callused fingers weaving between the stems and yanking with just the amount of force needed to bring the bulb to the surface. She stopped only to readjust her head scarf.\n\nBut when she had finished one square area, the sun was beginning to set and it was time to prepare dinner. Shekiba resumed her post in the kitchen and was dismayed, but not surprised, to see that nothing had been done for the evening dinner. She quickly started a flame and set some water to boil. Khala Zarmina walked past her and peered into the dim room.\n\n\"Oh, there you are! I was just about to boil some rice for dinner but I see that you're here now. I'll leave it up to you, then. I just hope you'll clean your hands well\u2014they're filthy.\"\n\nShekiba waited till Zarmina had walked away to let out a heavy sigh. How she wished she would have died on the cold floor of her own home, before her uncles had found her.\n\nJumaa prayers had just ended. Her uncles were returning home from the small masjid in town.\n\n\"Children, outside. We are speaking with your grandmother,\" Kaka Freidun snapped. Shekiba watched her cousins scamper out of the main living room. Kaka Sheeragha looked at her and seemed to be considering something. He followed his brothers into the living room.\n\nShekiba pretended to walk back into the kitchen with the clothes she had gathered from the clothesline. Before she reached the kitchen, she stopped and sat on the floor to fold the clothes. From there, she could hear some of what her uncles were saying.\n\n\"We need to settle this debt. Azizullah is losing patience with us. He says he's waited long enough.\"\n\n\"Hmm. What exactly were his demands?\"\n\n\"I spoke with him in the village two weeks ago and he told me that he is in need of a wife for his son. He wants one of the girls from this family.\"\n\n\"Is that what he said?\"\n\n\"Well, he said that there is a debt to settle. And that he was thinking of it more these days because he wants to secure a wife for his son.\"\n\n\"I see.\" Bobo Shahgul's voice was sharp, matter-of-fact. \"How old is his son?\"\n\n\"His son is ten.\"\n\n\"He still has time.\"\n\n\"Yes, but he wants to arrange the matter now.\"\n\nI could hear Bobo Shahgul tapping her walking stick on the floor in thought.\n\n\"Then we need to arrange a deal with him.\"\n\n\"Zalmai, your girls are the right age. Maybe one of them. The older one. She's eight, isn't she?\" Kaka Freidun's voice was unmistakable.\n\n\"Sheeragha's daughter is the same age. And your daughter is the same age as Azizullah's son. She would be a good match as well and would settle our debts sufficiently.\"\n\n\"Freidun's got more girls than anyone. It makes sense to give one of\u2014\"\n\n\"I don't think it is necessary to send one of the girls.\"\n\nThere was a pause as Bobo Shahgul's sons waited for her to explain.\n\n\"We will offer Shekiba.\"\n\nI am not one of the girls.\n\n\"Shekiba-e-shola? Are you joking? He'll take one look at her and come after us demanding twice what we owe! To offer Shekiba will offend him, for sure!\"\n\nShekiba closed her eyes and pressed the back of her head against the wall.\n\nYour name means \"gift,\" my daughter. You are a gift from Allah.\n\n\"Zalmai, I want you to speak to Azizullah and tell him that his son is still young. God willing, he and his father have long lives ahead of them with plenty of time to arrange for a suitable marriage. Tell him it would be more useful for them to have someone who can help them at home now. Tell him a happy wife bears more sons. Then you can offer Shekiba.\"\n\n\"But what if he says no?\"\n\n\"He won't. Just be sure to tell him that she is very capable. That she has the back of a young man and can manage a household. She is a reasonable cook and she keeps quiet, now that she's been tamed. Tell him that it is an honorable thing to take in an orphan and that Allah will reward him for bringing her into his home. She will be like a second wife without the price.\"\n\n\"And what about the work she's doing here? Who will do that?\"\n\n\"The same lazy women who were doing it before Shekiba came here!\" Bobo Shahgul snapped. \"Your wives have been spoiled. They have taken to lying about, drinking tea and making my ears ache with their chatter. It will be good for them to get back on their feet. This is a home, not the royal palace.\"\n\nThe brothers grunted. Would Azizullah really take the offer? they wondered. Better to try than to argue on whose daughter would be given as a bride otherwise.\n\n\"Say nothing to your wives now. No need to go stirring the henhouse yet. First let us discuss matters with Azizullah.\"\n\nShekiba picked herself up from the floor and hurried into the kitchen before her uncles emerged. She couldn't help but be thankful her parents were not alive to hear this conversation. She felt a tear well in her right eye.\n\nThat is the problem with gifts, Madar-jan. They are always given away.\nCHAPTER 9\n\nAzizullah took the deal.\n\nShekiba-e-shola packed her two dresses.\n\n\"Do not do anything that will bring shame to this family.\" Her grandmother's farewell to her was unceremonious.\n\nShekiba did something she never thought she would do. She lifted her burqa from her face and spat at her grandmother's wrinkled feet. A wad of saliva landed on her walking stick.\n\n\"My father was right to run from you.\"\n\nBobo Shahgul's mouth gaped as Shekiba turned and began walking toward her uncle, who was to escort her to Azizullah's home.\n\nShe knew it was coming but she did not care.\n\nShe also knew Khala Zarmina was watching. And smiling.\n\nThe walking stick came down on her shoulders twice before her Kaka Zalmai raised a hand to block his mother's revenge.\n\n\"Enough, Madar-jan, I cannot take the beast to Azizullah crippled. Her face is bad enough. If he sees her hobbling surely he will turn us down. Let Allah punish her for her insolence.\"\n\nShekiba kept her shoulders up and did not falter. She did not know what lay ahead for her but she knew she could not return to this home. She had closed this door for sure.\n\n\"You wretched creature! Allah in all His wisdom has marked your face as a warning to all! There is a monster within! Ungrateful, just like your despicable mother! Do you ever wonder why your entire family is gone, buried under the ground? It is you! You are cursed!\"\n\nShekiba felt something rise within her. She turned slowly and lifted her burqa again.\n\n\"Yes, I am!\" Shekiba smirked and pointed a finger at her grandmother. \"And with Allah as my witness, I curse you, Grandmother! May demons haunt your dreams, may your bones shatter as you walk and may your last breaths be painful and bloody!\"\n\nBobo Shahgul gasped. Shekiba could see the fear in her eyes. She stared at her granddaughter's portentous face and took a nervous step back.\n\nKaka Zalmai slapped her face with a mighty backhand. Even the deadened nerves on the left side of her face stung with his blow.\n\nClever, she thought as she tried to catch her balance. Won't leave a mark there.\n\nHe tightened his fingers around her arm and dragged her away from the house.\n\n\"We are leaving. Madar-jan, I'll be back when I have gotten rid of this monster. Samina, help my mother back into the house!\"\n\nShekiba had no trouble keeping up with her uncle's pace. She kept two steps behind and played the scene over and over again in her mind. Had she really done that? Had she really said those things?\n\nHer burqa hid a lopsided smile.\n\nThey walked the four kilometers to Azizullah's home in silence. Kaka Zalmai occasionally looked back and muttered something that Shekiba could not make out. They passed through the village Shekiba had not seen since early childhood. The shops looked more or less the same and there were a handful of people walking about, blue burqas following men dressed in loose flowing pants and long shirts.\n\nAs they moved further from her family's land, Shekiba wondered if she had done the right thing. What if she found herself alone again? What would she do? But she knew. She would do what she had intended to do months ago.\n\nI will find a way back to our land and bury myself with my family, Shekiba resolved.\n\nAzizullah's home was large in comparison to Bobo Shahgul's. And when she discovered that only Azizullah, his wife and four children lived in it, she was astonished. Azizullah had been given the home by his father, who had been a relatively wealthy man by village standards. Today, Azizullah made his living as a man of commerce. He bought and sold anything that was of any value to anyone. He made trades and loaned money as needed. He knew everyone in the village, but more important, everyone knew him. His family was well connected, with two brothers in the military service.\n\nIt was Azizullah himself who answered the outer gate.\n\nThe men shook hands and exchanged pleasantries. Shekiba stood just behind her uncle, feeling invisible.\n\nAzizullah was a burly man who looked to be in his thirties. He wore a brown lambskin hat of rippled fur that sat snugly on his head. His eyes were dark and he had a thick but neatly trimmed beard. His clothes and hands looked clean.\n\nHe does not look like a working man, thought Shekiba.\n\n\"Please come in, Zalmai-jan. Join me for a cup of tea.\"\n\nKaka Zalmai accepted the invitation and followed Azizullah into his courtyard. Shekiba stood behind, not sure what she should do, until she saw her uncle shoot her a look. She took a step into her new home. The men went into the living room but Shekiba thought it best if she remained outside. She stood with her back to the wall, her shoulders now starting to ache where Bobo Shahgul's walking stick had come down on her earlier. Again, a smile beneath her burqa. Nearly twenty minutes passed before she was summoned into the living room by her uncle.\n\n\"This is Shekiba, Azizullah-jan. You will see that, as we told you, she is a very hard worker and is sure to prove useful in your home. I trust your wife will be pleased with her.\"\n\n\"Zalmai-jan, we have lived in this village for many years and Shekiba-e-shola is no secret. I had heard of her scars before your brother spoke of it. Now I want to see exactly what it is that I am bringing into my home. Have your niece show her face.\"\n\nKaka Zalmai looked in Shekiba's direction and gave her a nod. His eyes warned her against disobeying. Shekiba took a deep breath, lifted her burqa and braced herself.\n\nHis reaction came slowly. At first, he saw only the right side of her face. Her high cheekbone. Skin with the delicacy and color of an eggshell. Her dark iris and naturally arched brow caught Azizullah by surprise. The infamous monster was half-beautiful.\n\nBut as Shekiba turned her face, her left side came into view. She moved slowly, deliberately\u2014anticipating a response. It suddenly occurred to her that Azizullah could be so repulsed as to send her back to her grandmother's house. She held her breath, unsure what to wish for.\n\nAzizullah's brows wove together.\n\n\"Impressive. Well, no matter. For our purposes, her face is insignificant.\"\n\nInsignificant?\n\n\"She has no other illnesses? Does she speak?\"\n\n\"No, Azizullah-jan. Aside from her face, she is healthy. She speaks but not enough to pester you. She should be an unobtrusive addition to your household.\"\n\nAzizullah stroked his beard. He took a moment to contemplate and then made his final decision.\n\n\"She will do.\"\n\n\"I am so happy that you see things this way, Azizullah-jan. You truly are a very open-minded person, may God grant you a long life.\"\n\n\"And you, my friend.\"\n\n\"I should be on my way then. I trust this will satisfy my family's debt to you. And please know that my mother sends her warmest regards to your wife as well.\"\n\nKaka Zalmai spoke so graciously, Shekiba could hardly recognize him as a member of her family.\n\n\"Our debts are settled, as long as this girl works as you've said she will.\"\n\nAnd she did. Mostly out of fear that she would be sent back to Bobo Shahgul's house. Soon Shekiba realized that she was much better off here in Azizullah's home anyway. Azizullah called his wife, Marjan, into the living room after Zalmai took his leave.\n\n\"This is Shekiba. You should acquaint her with the chores of the house so that she can get to work. Her family speaks highly of her abilities to keep a clean house and manage even heavy tasks. Let us see how she proves herself.\"\n\nMarjan eyed her carefully, wincing as her eyes fell upon Shekiba's face. She was a good-hearted woman and immediately took pity on Shekiba.\n\n\"Allah, dear girl! How terrible!\" she exclaimed, wiping her powdery hands on her skirt. She recovered quickly, though. \"Well, let me show you around. I was just kneading the dough but it's all done now. Follow me.\"\n\nMarjan was probably in her late twenties. Shekiba calculated that she must have had her first child at Shekiba's age.\n\n\"This is our bedroom. And this is the kitchen area,\" she said, pointing to a doorway on the left. Shekiba stepped in and looked around. \"Oh, for God's sake, look at your hips! How will you squeeze a baby through them?\"\n\nMarjan's girth was generous, probably having increased by inches with each new addition to their family.\n\nBut Marjan's statement surprised Shekiba. No one had ever mentioned the possibility of her bearing children\u2014not even in jest. She felt a heat rise into the right side of her face and lowered her head.\n\n\"Oh, you're embarrassed! That's sweet! Well, let's move on. There are many things to be done while we stand here chatting.\"\n\nMarjan listed the chores to be done around the house, but she spoke without the bitter condescension of Shekiba's own family. Despite the fact that she'd been brought here as a servant, Shekiba realized Azizullah's home would be a reprieve for her. She caught herself before she broke out into a full smile.\n\nAzizullah and Marjan had four children. Shekiba met the youngest first\u2014Maneeja, a two-year-old girl with soft dark curls that framed her rosy cheeks. Her eyes were thickly lined with kohl, which made the whites glow. Maneeja clung to her mother, her tiny fingers hanging on to her mother's skirt as she eyed the new face warily. Shekiba saw herself and Aqela doing the same with Madar-jan. Marjan and Shekiba sat down to finish rolling the dough into thin, long ovals. They would be taken to the baker later to be made into fresh-baked bread.\n\nThe eldest child, Fareed, was ten years old. He darted into the kitchen and grabbed a piece of bread before Marjan could chastise him. And before he could take stock of Shekiba's face. Shekiba tried to imagine which of her female cousins would possibly have been arranged as his future bride had her services not been offered instead. It was hard to guess.\n\nNext came eight-year-old Haris and seven-year-old Jawad. They were in a hurry to keep up with their older brother and barely noticed that there was a new person toiling away with their mother in the kitchen. They were energetic boys who froze in their father's presence. But when Azizullah was not around, they quibbled and tackled each other, teaming up against their stronger older brother.\n\nThe children seemed to have inherited their parents' attitude toward disfigurement. After their initial surprise and a few bold questions, they no longer seemed to notice.\n\nWithin two weeks, Shekiba felt quite at home with Azizullah's family. The boys reminded her of her own brothers, Tariq and Munis. Maneeja had Aqela's dark curly hair. But the resemblance brought Shekiba more pleasure than pain. It was almost as if she was living with her reincarnated siblings.\n\nYou did me a favor, Grandmother. The only decent thing you've ever done for me.\n\nJust as she had at Bobo Shahgul's house, Shekiba soon came to manage most of the household on her own. She busied herself with washing the clothes, scrubbing the floors, bringing the water from the well, cooking the meals\u2014just as she had done in the past. Things were considerably easier here, though, since there were only six people to look after. She could tell that Marjan was more pleased with her work than she wanted to show. Azizullah paid her no attention, as long as his wife had no complaints with their new servant.\n\nBut when the family took to their beds and the house settled into its night rhythm, Shekiba lay awake as the outsider she would always be. Shekiba had experienced upheaval and change before and each time, she adjusted. She was by now used to the idea that she was not truly part of any home, not truly part of any family. She would be sheltered by these walls only as long as she scrubbed them until her hands bled.\n\nBecause she was Shekiba, the gift that could be given away as easily as it had been accepted.\nCHAPTER 10\n\nKhala Shaima told us how Bibi Shekiba adjusted to the changes in her life. Now I had to adjust to the changes in mine. I had to learn how to interact with boys. It was one thing to play soccer with them, running alongside them and bumping elbows or shoulders. It was a whole other to be talking with them as we walked home from school. Abdullah and Ashraf would pat me on the back, sometimes even sling an arm around my neck as a friendly gesture. I would smile meekly and try not to look as uncomfortable as I felt. My instincts were to jerk back, to run away and never look them in the eye again.\n\nMy mother would raise an eyebrow if I came home before Muneer.\n\n\"Why are you home so early?\" she would say, wiping her wet hands on a rag.\n\n\"Because,\" I said vaguely, and tore off a piece of bread.\n\n\"Rahim!\"\n\n\"Sorry, I'm hungry!\"\n\nMadar-jan bit her tongue and resumed slicing potatoes into round chips with a hint of a smile on her face.\n\n\"Listen, Rahim-jan. You should be out with the boys, playing. That's what boys do\u2014do you understand what I'm saying?\"\n\nMadar-jan still spoke in circles when it came to talking about my shift from girl to boy. I think she was afraid she would stop believing the charade herself if she spoke of it too directly.\n\n\"Yes, Madar-jan, but sometimes I just don't want to. They . . . they push each other a lot.\"\n\n\"Then push back.\"\n\nI was surprised by her advice but the look on her face told me she was serious. Here sat my mother telling me the exact opposite of what she'd always said. I would have to toughen up.\n\nPadar-jan had been home for three days and everyone was on edge. Every sound, every smell jarred him, inciting a string of profanities and a few slaps when he mustered the effort. For most of the day, he sat in the living room and smoked his cigarettes. Our heads grew dizzy from the smell and Madar-jan had us spend more time in the courtyard. She swaddled Sitara in a blanket and turned her over to Shahla while she did the cooking on her own. Sometimes my uncles would sit with him, smoking and talking about the war, about the neighbors and the Taliban, but none of them smoked as much as Padar-jan.\n\n\"What do you think it would be like if Kaka Jamaal was our father?\" Rohila asked one day. She and Shahla were collecting the laundry from the clothesline. Shahla stopped in her tracks.\n\n\"Rohila!\"\n\n\"What?\"\n\n\"How could you say such a thing?\"\n\nI listened but kept my attention on the marbles in front of me. I flicked my finger and watched one send another off too far to the left. I let out a frustrated huff. Ashraf's aim was much better than mine.\n\nJust pay attention to where you want it to go, Abdullah had said. You're only looking at the marble in front you. You have to look at the target.\n\nI froze when he took my hand and showed me how to position my fingers, tucking my pinky under so it wouldn't get in the way. I still wondered what my mother would say if she were to see us. Was this okay too?\n\nAbdullah was right. Once I started looking in the direction I wanted the marble to roll, my shots were better. Marbles tapped against each other and rolled out of the circle. I would have won against Abdullah today. Well, maybe not Abdullah but definitely against Ashraf. My aim was improving.\n\n\"It's just a question, Shahla. You don't have to get so upset about it!\"\n\nShahla shot Rohila a chastising look.\n\n\"It's not just a question. If it were just a question, I'd like to see you go and ask it in front of Padar-jan. Anyway, Kaka Jamaal always looks like he's mad. Even when he's laughing. Have you noticed the way his eyebrows move?\" She cocked her head to the side and turned both her eyebrows inward, leaning toward Rohila, who burst into laughter.\n\n\"You can't ask for another father,\" Parwin interjected. Rohila's chuckles quieted as she turned to hear what Parwin was thinking. \"It would throw everything off.\"\n\nI sat up. My left side had gotten stiff from leaning in one position.\n\n\"What are you talking about, Parwin?\" I asked.\n\n\"You can't just have Kaka Jamaal as your father without making a lot of other changes. That means Khala Rohgul would be your mother and then Saboor and Muneer would be your brothers.\"\n\nParwin was Padar-jan's favorite\u2014if he had to pick one, that is. Maybe he'd already suffered enough disappointment by the time she was born that her being a girl hadn't stung him as the other two's had. But more than that, there was something about her temperament and drawings that calmed him. Maybe that's why she was more forgiving of him. Or it could have been the other way around.\n\n\"Anyway, you'd better stop before someone hears you,\" Shahla warned Rohila. Sitara had started to whine and wriggle in her blanket. Shahla bounced her over her shoulder expertly. She was about to enter adolescence, her body no longer an androgynous shape. Rohila, strangely enough, seemed to be two steps ahead of her. Madar-jan had started her wearing a bra a year and a half ago when her breasts began to poke through her dresses impertinently.\n\nI had tried her bra on once. Just out of curiosity. Rohila had left it behind in the washroom by accident again. Madar-jan had slapped her once for being so indecent. Still, she had forgotten. I laid it out in front of me and tried to make sense of the straps. I stuck my arms through the loops and tried to fasten it in the back, my arms reaching awkwardly, blindly for the clasp. After a few minutes I gave up and looked down at the lumps of cloth hanging loosely over my square chest.\n\nI stuck my chest out, trying to see if I could fill the miniature cups and realizing I didn't want to. Instead, I sat on the ground, cross-legged and comfortable, while my sisters became women.\n\nLater that night, I answered a knock at the door. Padar-jan lay in the living room, his loud snores rumbling through his chest. Sometimes he snorted so loudly that Rohila giggled and Shahla's hand instinctively clamped over her sister's mouth to stifle the sound. Parwin would shake her head, disappointed in her sister's behavior. Madar-jan shot both girls a warning look; Shahla's eyes widened in a declaration of innocence.\n\nThere was a man at the front gate. I recognized him as one of my father's friends. He was gruff and had skin the texture of our plaster walls.\n\n\"Salaam, Kaka-jan.\"\n\n\"Go and call your father,\" he said simply.\n\nI nodded and ran back into the house, taking a deep breath before I nudged Padar-jan's shoulder. I called out to him, louder and louder, before his snoring rhythm broke and he fumbled to rub his bloodshot eyes.\n\n\"What the hell is wrong with you?\"\n\n\"Excuse me, Padar-jan. Kaka-jan is at the gate.\"\n\nHis eyes began to focus. He sat up and scratched his nose.\n\n\"Fine, bachem. Go and bring me my sandals.\" I was his son and allowed to wake him up for important matters. I saw Shahla's eyebrows draw upward. She noted the difference too.\n\nI went to the courtyard to listen in on their conversation. I sat away from the gate where they were talking, out of the man's view.\n\n\"Abdul Khaliq has summoned everyone. We'll meet in the morning and then head out. They're bombarding an area north of here and it looks like they'll gain some ground if we don't fend them off. There's a lot of talk about that area. Seems the Americans are going to be sending us some weapons or something.\"\n\n\"The Americans? How do you know that?\" Padar-jan asked, his back against the gate. His guest had declined his invitation to come in.\n\n\"Abdul Khaliq met with one of their men last week. They want those people out of there. They're still looking for that Arab. Whatever the reason, at least they'll be helping out.\"\n\n\"When are we leaving?\"\n\n\"Sunrise. By the boulder on the road going east.\"\n\nPadar-jan was gone for two months that time but it felt different to me. I felt proud to know my father was fighting alongside a giant like America. My grandfather wasn't so sure it was a good idea. He seemed more suspicious of these Americans but I didn't see why.\n\nKhala Shaima was sitting in our living room when I came home that afternoon. Since my transformation, I had only seen her once, and that was before school had started.\n\n\"There you are! I've been aging waiting for you, Rahim-jan,\" she said, emphasizing the new twist on my name.\n\n\"Salaam, Khala Shaima!\" I was happy to see her but nervous to hear what she would say about my progress.\n\n\"Come sit next to me and tell me exactly what you've been doing. Your mother has obviously failed in getting your sisters to school, despite the fact that we came up with a plan to make everyone, even your intoxicated father, satisfied.\" She shot Madar-jan a look from the corner of her eye. Madar-jan sighed and moved Sitara to her left breast to nurse. She looked as if she'd already tired of this conversation.\n\n\"I've been going to class and Moallim-sahib is giving me good marks, right, Madar-jan?\" I wanted Khala Shaima to approve, especially since it had been her who had won me these new freedoms.\n\n\"Yes, he's been doing well.\" A small smile. Shahla and Parwin were sitting in the living room, their fingers nimbly sifting through lentils and removing stones. Shahla had done twice as much as Parwin, who had arranged her lentils into piles of different shapes. Rohila had come down with a cold and was sleeping in the next room.\n\n\"Well, I'm sorry I wasn't here sooner to check up on you all. My health hasn't been very good. I hate that it keeps me from doing what I want.\"\n\n\"Are you feeling better now, Khala-jan?\" Shahla asked politely.\n\n\"Yes, bachem, but for how long? My bones are tired and achy and the dust was so bad last month that each breath threw me into a hacking fit. Sometimes I coughed so hard I thought my intestines would fall right out of my body!\"\n\nThat was Khala Shaima's way of explaining things.\n\n\"But anyway, enough talk about old people. You know your sisters aren't as lucky as you, Rahim.\"\n\n\"Shaima! I told you, once things have settled down, I'll be able to send the girls back to school.\"\n\n\"Settled down? Settled down where? In this house or do you mean the whole country? And when do you think that will be, because as far as I can remember these children have been living under rocket fire for their entire lives! For God's sake, I can't even remember a day when this country wasn't at war.\"\n\n\"I know that, Shaima-jan, but I don't think you understand my situation. If their father forbids them from\u2014\"\n\n\"Their father can eat shit.\"\n\n\"Shaima!\"\n\nShahla and Parwin both froze. That was more than we would have expected, even from Khala Shaima.\n\n\"You're so defensive about him! Open your eyes, Raisa! Can't you see what he is?\"\n\n\"What he is, is my husband!\" Madar-jan yelled, louder than we'd ever heard her before. \"And you have to understand that! Please! Don't you think I know better than anyone what he is or isn't? What can I do?\"\n\n\"Your husband is an idiot. That's why I worry about these girls being around him. Sit with us and you'll be one of us. Sit with a pot and you'll be black.\"\n\n\"Shaima, please!\"\n\nKhala Shaima sighed and relented. \"Fine. All right then, Raisa. But that's why I keep coming here and harping after these girls. Somebody needs to oppose him.\"\n\n\"And who better than . . .\"\n\n\"That's right,\" Khala Shaima said with satisfaction. She turned her attention back to me. Shahla and Parwin resumed their work but at a slower pace, unnerved by Madar-jan's yelling. \"So, tell me then. Have you been adjusting well? No troubles with the boys?\"\n\n\"No, no trouble, Khala-jan. I've been playing soccer and I'm better than my cousin Muneer, I think.\"\n\n\"And no one's said anything to you?\"\n\n\"No, Khala-jan.\"\n\n\"Good. And what kinds of things are you doing to help your mother?\"\n\n\"Rahim's been going to the market for me. The store owners give him better prices than they do me.\"\n\n\"Don't forget, Madar-jan. I've been working with Agha Barakzai and he's been giving me a little money!\"\n\n\"I was getting to that, Rahim. You know Agha Barakzai has that little shop in the village. Well, he's been in need of help with errands and I asked Rahim to stop by there and see if he could pick up a bit of work. Agha Barakzai can hardly see anymore with his terrible eyes.\"\n\n\"You're a working boy! Now, that's news!\" Khala Shaima clapped her hands together.\n\n\"Yup, I go all around town and no one bothers me. I can do anything! I even saw Padar's friend Abdul Khaliq yesterday.\"\n\nMadar-jan stiffened and looked at me.\n\n\"Who did you see?\"\n\n\"Abdul Khaliq,\" I repeated, quieter this time. Khala Shaima looked as displeased as my mother. I wondered if I'd done something wrong.\n\n\"Did he say something to you?\"\n\n\"Not much. He bought me a snack and told me I was coming along nicely.\"\n\nMadar-jan shot another look at Khala Shaima, who shook her head.\n\n\"Raisa, that is not a man to have your children tagging along after. Not even Rahim!\"\n\n\"You'll stay away from that man, Rahim,\" Madar-jan said, warning me, her eyes wide and serious. \"Do you understand me?\"\n\nI nodded. My sisters fidgeted in the silence that followed.\n\n\"Khala Shaima, could you tell us more about Bibi Shekiba?\" Parwin asked.\n\n\"Bibi Shekiba? Ah, you want to know more? Well, let me see if I can remember where I left off . . .\"\n\nJust as Khala Shaima leaned back and closed her eyes to tell us more of the story, we heard the door open. My grandmother rarely came to visit us but Padar-jan had been gone two months and she felt compelled to check up on things, especially when she saw Khala Shaima hobble through the front gate. Khala Shaima treated my grandmother with respect, but it was measured and anything but warm. My grandmother, on the other hand, felt no obligation to put on airs with my aunt.\n\n\"Salaam,\" she called as she entered. My mother jumped to her feet, startling Sitara, who had nearly fallen asleep. She adjusted the top of her dress and walked to the door to greet her mother-in-law.\n\nKhala Shaima took her time but pushed herself up to greet her sister's mother-in-law.\n\n\"Salaam, Khala-jan. How is your health? Well, I hope.\" She almost sounded sincere. My sisters and I kissed her hands. She sat down across from my mother and Shahla brought a cup of tea from the kitchen.\n\n\"Oh, you're here, Shaima-jan! How nice of you to drop by again so soon.\"\n\nI could hear it in my grandmother's voice: You come too often. Khala Shaima said nothing.\n\n\"You've heard nothing from Arif-jan? Any word on when they'll return?\"\n\nMadar-jan shook her head. \"No, Khala-jan. Nothing at all. I pray they will return soon.\"\n\n\"In the meantime, I've spoken with Mursal-jan and her family has agreed to give their daughter's hand in marriage for Obaid.\" Obaid was my father's brother. This was surprising news.\n\n\"Obaid-jan? Oh, I didn't realize . . .\"\n\n\"Yes. So we'll be preparing for her arrival. We will have their nikkah in two months' time, inshallah. This will be a blessing for our family. A second wife will bring him more children and grow our family.\"\n\n\"They have five children, nam-e-khoda,\" Madar-jan said softly.\n\n\"Yes, but only two boys. Boys are blessings and Obaid wants more sons. Better to have more children than to try to change the ones you have. Anyway, I've made you aware. Fatima may call on you for help preparing a place for his new wife. This is happy news and we'll all take part in it.\"\n\n\"Of course, Khala-jan. It's wonderful news.\" Madar-jan's voice was soft. Khala Shaima watched the interaction with narrowed eyes.\n\n\"Hopefully, there will be more of it in the future,\" she said, nodding her head.\n\nMy grandmother got back up and walked to the door.\n\n\"Anyway, that's all for now. Shaima-jan, send my regards to the family, will you please? I guess you'll be leaving soon, as it is getting late.\"\n\n\"You're too kind, Khala-jan. You make me feel so welcomed here, it's difficult to leave.\"\n\nI saw my grandmother's shoulders stiffen before she left and the way Madar-jan and Khala Shaima looked at each other. Khala Shaima shook her head. This meant bad news for our household.\n\n\"Come, girls, let me tell you more about Bibi Shekiba. I'll tell you how easily women pass from one place to another, from one home to another. What happens once, happens twice and then a third time . . .\"\nCHAPTER 11\n\nAzizullah sat in the living room with his brother, Hafizullah. There were two other men with them as well but Shekiba did not know their names and had never seen them before. They had white turbans on their heads and pale blue tunics and pantaloons. Hafizullah wore a brown vest over his tunic, his prayer beads hanging from the pocket.\n\n\"Shekiba, Padar-jan wants the food to be ready in twenty minutes,\" announced Haris. \"He says they're going to leave soon so it better not take too long.\"\n\nShekiba nodded nervously, knowing the rice would have to be a touch undercooked. She added more oil to the pot, hoping that the extra grease would soften the grains.\n\nHaris leaned over her shoulder and tried to snatch a piece of meat from the bowl next to Shekiba. Her right arm went up instinctively and snagged him by the wrist.\n\n\"You know better, Haris. Not until after they've eaten.\" Her tone was gentle but firm. Haris was by far her favorite of the children. He would sometimes sit with her when he had tired of his siblings. She didn't mind his company. On the contrary, she enjoyed his chatter and the stories he would tell about his teacher.\n\n\"Just one piece!\" he pleaded.\n\n\"If you have a piece, then your brothers will want some too when they see you licking the sauce off your fingers.\"\n\n\"No, I promise! I won't tell them I had some! I'll lick my fingers clean here before I go back out!\" Haris was already an expert negotiator.\n\n\"Fine then. But just one\u2014\"\n\nHe had snatched the largest chunk before Shekiba could finish her sentence.\n\n\"Haris!\"\n\nHe grinned, his cheeks lumpy with lamb. How lucky she was to live in a house that could afford to eat lamb! Shekiba sighed and pretended to be annoyed.\n\n\"What are they talking about in there, anyway?\" she asked.\n\n\"Don't you know? The king is coming!\"\n\n\"The king?\" Shekiba asked. \"Which king?\"\n\n\"Which king? King Habibullah, of course!\"\n\n\"Oh.\" Shekiba had no idea who King Habibullah was. It had been years since her father had shown an interest in anything beyond their walls. \"Is he coming here?\"\n\n\"Here? Are you crazy, Shekiba? He is going to Kaka Hafizullah's house.\"\n\nAzizullah's brother had managed to secure himself a position as a friend of the monarchy. He served as a regional overseer and reported to the authorities in Kabul, the capital. For years, he had served as a loyal delegate and traveled frequently to the palace to meet with the king's advisers. He was vying for royal attention in the hopes of becoming hakim to their province. With such a title came an attractive amount of power, and so Hafizullah often shared hearty meals and lavished compliments on anyone with any influence.\n\nAzizullah had no patience for such highbrow relations but he did enjoy the secondary benefits that came with having a strategically placed brother. People in the village showed Azizullah deference, hoping they could curry favor with Hafizullah. This was how influence trickled down from the monarchy into the most insignificant of homes in the countryside.\n\nAnd while Shekiba had no knowledge of such diplomatic matters, she too became enchanted by the prospect of the king paying a local visit. She imagined horses and regal clothing, guards at his side.\n\nShe adjusted her head scarf and poured fresh cups of tea, hoping to distract their appetites for a few more minutes. She carried a tray into the living room and kept her head bowed, wanting to be as discreet as possible.\n\n\"It is a huge honor. This is the opportunity I have been waiting for. Thanks be to Allah, I have called in many favors and secured the makings of a fine feast for the night. We will make qurbani; a goat will be slaughtered in the king's good name. I am sparing no expense.\"\n\n\"How are you to pay for this? How many people will be with him? Surely, there will be at least a dozen pretentious mouths to feed!\"\n\n\"There is a price to pay for everything but it is a chance I could not let escape. Sharifullah has been hakim of this province for long enough. It is pure good fortune that he has traveled across the country now to attend the funeral of his cousin.\"\n\n\"Good fortune for you!\" Azizullah laughed. \"But not for his cousin!\"\n\n\"Forget about his cousin, dear brother. The point is that this is a chance for our family to reach the next level. That is what our father would have wanted to see, may Allah forgive him and keep him in peace. If I am made hakim, we will control the entire province! Imagine the life we would have.\"\n\n\"You would be an excellent hakim, certainly. And from what I have heard, many of the villages are displeased with Sharifullah's rulings.\"\n\n\"The man is spineless. The kingdom would all but forget our province were it not for the crops our land produces every season. Sharifullah has done nothing for us! When Agha Sobrani and Agha Hamidi disputed that land by the river, it was his idiotic idea that they should each take half.\"\n\nShekiba listened as she gathered the empty teacups and brought the dish of nuts closer to the men.\n\n\"Now, neither Sobrani nor Hamidi has any respect for him. They are equally dissatisfied with him. He should have given the land to Hamidi. His claim was reasonable and his family carries more clout than Sobrani's. Better to have Hamidi's full support and anger only Sobrani!\"\n\nIrrefutable logic. Shekiba quietly crept out of the room. She had grown accustomed to Hafizullah's animated speeches and found him entertaining in some way. At the same time, she was thankful that Allah hadn't placed her in his custody, as she was certain he was a brute in his home.\n\nAs soon as she left the room, she heard Hafizullah's tone change. She stopped and tilted her ear toward the living room.\n\n\"And how are things going with your new help? Shekiba-e-shola is fulfilling her duties around the house?\"\n\n\"Well enough,\" Azizullah answered. \"Marjan has not had many grievances about her.\"\n\n\"Hmmph. That family must be so relieved to have unloaded her. From what I have heard, Bobo Shahgul was heartbroken at her son's passing. Could not bear to have his child in her home because she was a constant reminder of her dead son.\"\n\n\"You would have heard more than me. The girl does not speak of her family. Actually, she hardly speaks at all. She has that much sense.\"\n\n\"At least your wife doesn't have to worry about your taking her as a second wife!\" Hafizullah joked, slapping his hand on his thigh loudly.\n\n\"No, she is not for marriage. She is able-bodied and does the work of a man. Sometimes it escapes us that she is, in fact, a girl. Her strength makes me marvel. I saw her just a few days ago carrying three pails of water and walking straight, as if it were no effort whatsoever. Her uncles told me she had been keeping up her father's farm along with him.\"\n\n\"More useful than a mule. Good,\" Hafizullah said. \"Whatever happened to her father? I remember running into him just after his children were taken in the cholera wave. He looked terrible. Too sensitive, that man was.\"\n\n\"His brother told me that he had not been feeling well in the last few months. Agha Freidun told me they had a conversation and he knew his time was coming. He made arrangements for his daughter to live with Bobo Shahgul and distributed his land, his tools and his animals among his brothers.\"\n\nShekiba's eyes widened.\n\nA lie! My father had no such conversation!\n\nHe had not seen his brothers after her mother died. She wondered if this story was Kaka Freidun's idea or Bobo Shahgul's. Her family was swooping in to pick up any scraps her father had left behind.\n\nThat land should be mine. My grandfather gave it to my father. My father wanted nothing to do with his family. I should be the owner of that land.\n\nShekiba wondered where the deed was. The deed was a simple document signed by her grandfather, her father, a few distant relatives and a village elder to confirm the transaction. Surely her uncles must have been looking for it when they dumped the contents of the house outside.\n\n\"Shekiba? What are you doing here?\"\n\nTeacups rattled in Shekiba's startled hands. Marjan had come up behind a very distracted Shekiba. She looked puzzled to see her frozen a few feet away from the living room.\n\n\"I just . . . chai . . . ,\" she mumbled, and headed directly for the kitchen, her head bowed to conceal her hurt eyes.\n\nThe scent of cumin and garlic filled the room. Azizullah and his brother shared their meal, tearing off chunks of flatbread and picking up morsels of rice and meat. Shekiba wondered if any would be left for the rest of the family. Meat was hard to come by, even in this household, and it seemed that the men were going to finish the week's stock in one sitting.\n\nHer mind began to wander as she dried the pots. What would happen if she were to try to claim that land? The thought almost made her laugh. Imagine that. A young woman trying to claim her father's land, snatching it from her uncles' greedy claws. She tried to imagine taking the deed to the local judge. What would he say? Most likely he would kick her out. Call her insane. Maybe even send her back to her family.\n\nBut what if he didn't? What if he listened to her? Agreed with her? Maybe he would think it was her right to have her father's land.\n\nMarjan was in the kitchen with her. She was sifting through the rice for any small stones.\n\n\"Khanum Marjan?\" Shekiba said meekly.\n\n\"Yes?\" Marjan paused and looked up. Shekiba spoke so rarely, one had to take notice.\n\n\"What happens to a daughter when her father . . . if her father has some land . . . if he is not . . .\"\n\nMarjan pursed her lips and cocked her head. She could sense the question buried in Shekiba's ramblings.\n\n\"Shekiba-jan, you are asking a ridiculous question. Your father's land will go to his family, since your brothers are dead, may Allah grant them peace.\" Marjan's response was blunt but it was reality\u2014regardless of what the laws might say. Her candor gave Shekiba confidence to speak openly.\n\n\"But what about me? Am I not rightfully an heir to the land? I am his child too!\"\n\n\"You are his daughter. You are not his son. Yes, the law says that daughters may inherit a portion of what the son would inherit but the truth is that women do not claim land. Your uncles, your father's brothers, have no doubt taken the property.\"\n\nShekiba let out a frustrated sigh.\n\n\"My dear girl, you are being quite ridiculous. What do you think you would do with a piece of land? First of all, you are living here now. This is your place. Secondly, you are unmarried and no woman could possibly live on a piece of land alone! That is simply absurd.\"\n\nI lived alone on that land for months. It didn't feel absurd. It felt like home.\n\nBut Marjan could not know about her time alone. Shekiba did not dare share the details, knowing it was unspeakable for her to have done so. No reason to give the village more fodder for gossip.\n\n\"But if I were a son?\" she asked, unwilling to let the matter go completely.\n\n\"If you were a son, you would inherit the land. But you are not a son and you cannot be a son and your life is now here as part of this home. You are asking questions that will invite nothing but anger. Enough!\" Marjan needed to put a stop to the discussion. If her husband heard them, he would surely be displeased. If these were the kinds of thoughts that ran through her head, Marjan was thankful Shekiba did not speak more often.\n\nBut I have always been my father's daughter-son. My father hardly knew I was a girl. I have always done the work a son would do. I am not to be considered for a wife, so what is the difference? What of me is a girl?\n\nShekiba gritted her teeth.\n\nI have lived alone. I have no need for anyone.\n\nAzizullah's family had been relatively kind to her but Shekiba was restless. She felt freshly resentful of her family.\n\nI cannot go on like this forever. I must find a way to make a life for myself.\nCHAPTER 12\n\nToo often, I missed the opportunity to learn from Bibi Shekiba's story. She was determined to make a life for herself and I seemed determined to unravel the one I had.\n\nI wonder how long I would have gone on as a boy had Madar-jan not seen us on that day. Most children who were made bacha posh were changed back into girls when their monthly bleeding started but Madar-jan had let me go on, bleeding but looking like a boy. My grandmother warned her it was wrong. Next month, my mother would promise. But I was too useful to her, to my sisters, to the whole family. She couldn't bear to give up having someone who could do for her what my father wouldn't. And I was happy to continue playing soccer and practicing tae kwon do with Abdullah and the boys.\n\nWe didn't have any hot pepper at home and Padar-jan liked his food spicy. Those peppers changed everything for me.\n\nAbdullah, Ashraf, Muneer and I were coming down our small street. The boys walked with us and then continued on to go to their own homes, smaller than ours but in as poor condition. People in our neighborhood weren't starving but we all thought twice before throwing a scrap to a stray dog. This was how it had been for years. Some days we walked lazily. Other days we were boisterous and raced each other to the tin can, to the old lady, to the house with the blue door.\n\nAbdullah and I stayed close together. In our circle of friends, we had something different. Something a little more. His arm across my shoulder, he would lean past me and tease Ashraf. I was a bacha posh but it had gone on too long, like a guest who had grown too comfortable to leave.\n\nIt was Ashraf who had started it. He had kicked his leg up into the air, though not as high as he thought it went. We tried to tell him he could barely reach our waists but he was certain he saw his foot swoop past our faces. Muneer shook his head. He was tired of Ashraf practicing on him.\n\nWe were fans of martial arts. We'd seen some magazines with fighters in different poses, their feet higher than their heads, their arms fired forward. We wanted to be like them and flipped through the pages copying their stances.\n\nWe had fought this way before. All of us. Playfully and without giving it much thought. I had started wrapping a tight cloth around my breast buds. I didn't want the boys to notice them or comment on them. It was awkward enough that my voice had not begun to change as theirs had. Sometimes I came away with bruises. Once, my ankle twisted in under me as I ducked a kick from Ashraf. For one week, I limped from home to school and back. I told Madar-jan I'd tripped on a rock, knowing I couldn't tell her how it had really happened.\n\nBut it was worth it. Worth it for that moment when, inevitably, Abdullah would have me cornered, or would twist my arm behind me and I could feel his breath on my neck. Somewhere inside I tingled to be that close to him. I didn't want him to let go, even if I could feel my arm pulling from its socket. I reached out and grabbed at his other arm, feeling his adolescent muscles flex under my fingers. When I was close enough to smell him, to smell the sweat on his neck, I felt dangerous and alive. That's why it was often me who started the sparring. I loved where it put me.\n\nThat was what we were doing when Madar-jan came out of the neighbor's house, a fistful of red peppers in her right hand and the corner of her chador in her left hand. It couldn't have been worse. She spotted us just as he'd tripped my foot. I lost my balance and fell to the ground. I looked up and saw Abdullah's handsome grin as he, victorious yet again, straddled me and laughed.\n\n\"Rahim!\"\n\nI heard my mother's voice, sharp and horrified. I saw her faded burgundy dress out of the corner of my eye. I felt my stomach drop.\n\nAbdullah must have seen the look on my face. He jumped to his feet and looked over at my mother. Her face confirmed that something had gone wrong. He reached his hand out to me so I could get up.\n\n\"That's all right,\" I mumbled, and got to my feet, dusting off my pants and trying to avoid my mother's accusing eyes.\n\n\"Salaam, Khala-jan,\" Abdullah called out. Ashraf and Muneer were reminded of their manners and echoed the same. She turned abruptly and went through our front gate.\n\n\"What happened? Your mother seems upset.\"\n\n\"Ah, it's nothing. She's always telling me that I come home with my clothes filthy. More to wash, you know.\"\n\nAbdullah looked skeptical. He knew a mother's angry face and could tell there was something more behind this.\n\nI didn't want to go home. I knew Madar-jan was upset but if I delayed facing her, things would be worse.\n\nI couldn't look at Abdullah, already feeling my face flush. My mother had seen something different than everyone else. She had seen her daughter pinned under a boy in the middle of the street. Few sights could have been more shameful.\n\nI felt a crunch and saw red peppers, crushed by my sandal, at our front gate. Where Madar-jan had dropped them. I collected what I could from the ground and went inside.\n\n\"Madar-jan, I'm going to wash up for dinner,\" I called out. I could see her in the kitchen and wanted to test the waters, without actually meeting her eyes.\n\nShe didn't answer me, which I could only take as a bad sign.\n\nI felt my hands start to shake. Sure, I knew better. Even dressed as a boy, I shouldn't have let things go so far. My aunts or uncles could have seen me. And it was possible they had. I would hardly have noticed with Abdullah up against me.\n\nI wondered if she would tell Padar-jan. That would be the end of me. Every possibility sent my brain spinning and drove me into a wild panic. I left the broken peppers on the family room table and went to wash up as I'd said I would. I tried to come up with a plan to talk my way out of this mess. I went to the kitchen, my face still wet.\n\n\"Madar-jan?\"\n\n\"Hmm.\"\n\n\"Madar-jan, what are you doing?\" My voice was meek and unsteady.\n\n\"Dinner. Go and finish your work now that you're done embarrassing yourself in the streets.\"\n\nThere it was. I felt a tiny bit relieved to hear her say it. Now I could start to defend myself.\n\n\"Madar-jan, we were just playing.\"\n\nMadar-jan looked up from the pot she was stirring. Her eyes were narrow and her lips tight.\n\n\"Rahim, you know better. Or at least I thought you did. This has gone on too long.\"\n\n\"Madar-jan, I\u2014\"\n\n\"I don't want to hear another word out of you. I will talk to you later. Right now, I've got to get your father's dinner ready or I'll have a second disaster on my hands.\"\n\nI retreated to the other room and worked on my homework assignments for a while before I decided to see if Agha Barakzai needed any help for the afternoon. I didn't want to be around while Madar-jan's anger festered. He kept me busy until the evening and I came home to find that Madar-jan had not saved me any food.\n\nShe saw me looking into the empty pots.\n\n\"There's a little soup left. You can have it with some bread.\"\n\n\"But, Madar-jan, there's nothing but onions and water in this soup. Wasn't there any meat left?\"\n\n\"We finished it all. Maybe next time there will be some for you.\"\n\nMy stomach growling painfully, I suddenly became very angry.\n\n\"You could have left me something! That's how you treat me? You want me to just go hungry?\"\n\n\"I'm not sure what it is you're hungry for!\" she whispered pointedly.\n\nPadar-jan walked in just then. He rubbed his eyes.\n\n\"What's all the yelling about?\" he asked. \"What's going on, bachem?\"\n\nI shot my mother a look and spoke without thinking.\n\n\"She didn't save me a single piece of meat. She wants me to have onion broth and bread! I was working at Agha Barakzai's shop and there's no dinner for me when I come home!\"\n\nI threw my wages on the table for good measure. The bills fluttered in the air and spread out dramatically.\n\n\"Raisa! Is this true? Is there nothing for my son to eat?\"\n\n\"Your son . . . your son . . .\" Madar-jan fumbled, trying to find a reasonable explanation for why she was punishing me. But Madar-jan wasn't quick enough or sly enough to come up with an alternative story on the spot. And as angry as she was, my mother couldn't bring herself to throw me into the fire.\n\nI saw it coming and instantly wished I could take back what I'd said. I saw his face redden with anger. I saw his head tilt and his shoulders rise. His arms began to wave with anger.\n\n\"My son is hungry! Look at the money he's brought home! And even with this you can't find a morsel of food for him? What kind of mother are you?\"\n\nA clap as the back of his hand swung across her face. She reeled from the blow. My stomach dropped.\n\n\"Padar!\"\n\n\"Find him something to eat or you'll be going hungry for a month!\" he barked. He struck again. A drop of blood trickled from my mother's lip. She covered her face with her hands and turned away from him. I trembled when he looked at me. From the corner of my eye, I saw Shahla and Rohila peeking from across the hall.\n\n\"Go, bachem. Go to your grandmother and ask her to fix you something to eat. Make sure you tell her what your mother has done. Not that she'll be surprised to hear it.\"\n\nI nodded and stole a glance at my mother, thankful she didn't meet my gaze.\n\nThat night I thought of Bibi Shekiba. I liked to compare myself to her, to feel like I was as bold and strong and honorable as her, but in my most honest moments I knew I wasn't.\nCHAPTER 13\n\nThe idea brewed for some time before Shekiba considered actually going ahead with it. The conversation with Marjan should have discouraged her but it hadn't. All she had gleaned from it was that, officially, she had a right to claim at least a portion of her father's land.\n\nShe lay awake every night thinking of the deed. A mere piece of paper with a handful of signatures, and yet it carried so much weight. Where would her father have kept it? Shekiba closed her eyes and imagined herself at home. She heard the clapping of the gate against the latch, the metal rusted over. She pictured her father's corner, his blankets laid out and ready for those chilly nights. She saw her mother's kitchen stool and her brother's sweaters, folded and stacked on a shelf.\n\nIt must be in his books, Shekiba thought. Since she'd been the only one to tend to it, she knew every inch of the house. She thought of the shelf and how she'd given up on dusting it after her mother died. Padar had collected three or four books over the years and that was where he kept them.\n\nWhen Shekiba made the realization, she nearly hit herself for how obvious it was.\n\nBut how do we know, Padar-jan?\n\nAll the answers are in the Qur'an, bachem.\n\nHer father taught them all to read, first with the Qur'an and next with the books he kept. She would follow along as his callused finger traced the words. Her brothers occasionally brought home a newspaper from their adventures into the village and the children would take turns poring over the pages and practicing making sense of the words and phrases. It was difficult but Padar-jan patiently let them make mistakes, peering over their shoulders when they faltered and filling in the pieces.\n\nIt's in the Qur'an, she realized. What were the chances her uncles had not yet found it? Unlikely\u2014but maybe there was a possibility those bullheaded men had not bothered looking for it. Surely they had no inkling that Shekiba would even think to assert any claim over the land.\n\nWhich meant Shekiba was thinking of returning to her home\u2014not a small undertaking.\n\nAnd if she were to find the deed, what would she do with it? She couldn't expect to show it to her uncles and have a rational discussion with them. No, she needed to bring the deed to an official, the local judge, so that she could argue her case.\n\nIt was just like Azizullah and his brother had discussed. A disagreement like this needed to be settled by an official, which meant Shekiba's plan became even more complex. How would she find this person?\n\nAnd how would she get to all these places? She needed to be out of the house for a day. Shekiba wondered if Marjan would let her venture out on her own. After their conversation, it was hard to imagine Marjan would be supportive of her idea. Shekiba would have to come up with something.\n\nTwo days later, Shekiba approached Marjan as she was knitting a sweater for Haris. She rehearsed her question in her mind before clearing her throat.\n\n\"Salaam, Khanum Marjan,\" she said, trying to keep her voice steady.\n\n\"Salaam, Shekiba,\" Marjan said, barely lifting her eyes from the needles as they crossed, uncrossed and crossed again in her hands.\n\n\"Khanum Marjan, I wanted to ask you something.\"\n\n\"What is it, Shekiba?\"\n\n\"I was wondering if I could take a day to visit my family. I have not seen my family in several months and I was hoping to visit them. Next week is Eid and I know it will be a busy time here, so perhaps this week?\" She folded her hands behind her to stop from wringing them.\n\nMarjan stopped her knitting and set the needles on her lap. She looked puzzled.\n\n\"Your family? Dear girl, since coming here you have never once mentioned your family. I was beginning to think you were so cold as to not have any affection for them! How is it that you now want to pay them a visit?\"\n\n\"Oh, I've missed them dearly,\" she said, trying her best to make her voice sound genuine. \"But in my first days here, I did not think it was proper to make such a request.\"\n\n\"And now?\"\n\n\"Well, now I have been here for some months and with the holiday coming . . . I wanted to pay a visit to my grandmother, out of respect.\" Shekiba wondered if she was giving omniscient Allah a good laugh or if she'd be damned for her lies.\n\n\"Your grandmother.\" Marjan sighed heavily and pressed her fingers to her temples.\n\nShekiba braced herself.\n\n\"We have much to do to prepare for the holiday. We need to bake some cookies, there will be many meals to prepare, the house needs to be spotless . . . ,\" she said, listing the tasks ahead. \"But I suppose it is only proper that you should pay a visit to Bobo Shahgul. She is your grandmother, after all. I will speak to Azizullah and present your request.\"\n\nShekiba tried not to smile. She bowed her head in gratitude.\n\n\"Thank you, Khanum Marjan,\" she said. \"I would really appreciate that.\"\n\nEvery once in a while, Shekiba became aware of how painfully na\u00efve she was. The following day was one such occasion.\n\nMarjan walked into the kitchen area as Shekiba sat on the floor, with a heap of potatoes before her. She stopped peeling when she heard her name being called.\n\n\"Shekiba, Azizullah agrees . . . hey, girl! What is wrong with you?\" Marjan took one look at Shekiba and froze. Her hands flew to her hips and her eyes narrowed.\n\n\"Huh? What is it, Khanum Marjan?\" Shekiba looked down at the pile before her, wondering what had offended the mistress of the house so.\n\n\"Is that how a girl sits?\" she said, waving an arm at Shekiba's sprawled legs.\n\nShekiba turned to look at herself. She was leaning against the wall and had her knees bent, the pile of potatoes in the valley her skirt formed between her legs.\n\n\"In the name of God, have some decency! Fix yourself before the children see you! Were you never taught how to sit?\"\n\nShekiba got up and fixed her skirt, tucking her legs under her, and looked up at Khanum Marjan for approval.\n\n\"That's better. I heard you had become your father's son but I did not think it had gone this far.\"\n\n\"Yes, Khanum Marjan.\" Shekiba felt half her face flush.\n\n\"Now, what was I saying? Oh, yes. Azizullah agrees that you should be allowed to pay respects to your grandmother for the holidays. You are to accompany him this Friday when he goes into the village for Jumaa prayers.\"\n\nAzizullah would take her there?\n\n\"Khanum Marjan, a world of thanks, but I do not wish to trouble your husband. I can find my own way and I will not bring him out of his way.\"\n\nMarjan looked at her incredulously. Shekiba never ceased to amaze her. The girl was quite handy and efficient in the house but when it came to common sense, she was seriously lacking.\n\n\"You expect to go wandering around the village by yourself? Have you lost your mind?\"\n\nShekiba remained silent. Her mind raced.\n\n\"He will take you, as you requested, and join you to pay a visit to your family, although your uncles usually come by on the holidays. Azizullah will accompany you back home. You cannot expect to be wandering around the village like a street dog!\"\n\nShekiba had done too much on her own while she lived with her father and before her uncles had claimed her. It had not occurred to her that she would have to be accompanied by someone. Her chest tightened with panic. She had not anticipated this stipulation.\n\n\"I . . . I had not meant to trouble . . .\"\n\n\"Well, if you do not want to trouble him then you should not have raised the question.\" Marjan walked out in exasperation. Shekiba's bizarre questions were getting on her nerves.\n\nShekiba was left to wonder. She could tell Marjan she no longer wanted to go. It would seem strange but it could work. Or maybe once she was there she could ask permission to collect some belongings from her father's home. But what about taking the deed to a hakim, the local official?\n\nMaybe on another day. But even if she were granted another day, she would still need to be accompanied. And she had no idea where to find the hakim.\n\nShekiba would have to ponder that one. One bridge at a time, she thought.\n\nJumaa came and Shekiba steeled herself. It would take all her resolve to face her family again, especially her grandmother. But this was her only hope at getting her hands on the deed.\n\nMarjan had instructed her to be ready in the morning, as Azizullah would not wait on her. He nodded in acknowledgment when he saw her waiting by the outside door, her burqa donned and her head bowed.\n\n\"Salaam,\" she said quietly.\n\n\"Let's go,\" he said, then opened the door and led the way.\n\nThey did not speak on the way to the masjid. Shekiba walked a few steps behind but paid close attention to the road. She tried to memorize everything on the way there. The road was wide and dusty but lined with tall trees. There were a handful of homes scattered on either side, about two acres apart. The homes were uniformly surrounded by six-foot-high clay privacy walls. Shekiba could see rows of plantings in their yards and could spot the potatoes, carrots and onion plants even from this distance. The weather was dry and crops were suffering, which meant the families were probably suffering too.\n\nA masjid, three shops and a bread baker constituted the village center. The storefronts were modest, with dull glass windows and handwritten signs. The bread baker didn't really have a store. He sat against a wall of another shop and pulled hot, golden round breads from his tandoor, buried in the ground. The smell of fresh bread coming from the open circle in the ground made Shekiba's mouth water. Two women stood waiting for their naan to bake. Shekiba recalled walking through the area when her uncle had taken her to Azizullah's as a means of repaying his debt.\n\nShekiba, the gift, she thought miserably.\n\nAzizullah took her past the masjid to a small home about a quarter of a kilometer away. He knocked at the front door.\n\n\"Salaam, Faizullah-jan,\" he said with his hand on his chest.\n\n\"Agha Azizullah, how nice to see you! Are you on your way to Jumaa prayers?\"\n\n\"Most certainly. But I had a favor to ask of you. This is my servant. I am taking her to visit her family after prayers have finished but I hoped I could trouble your wife to watch over her until then. I cannot leave her out in the street.\"\n\n\"Oh, of course! I heard you had taken in Bobo Shahgul's grandchild, the one with the half face. Have her stay in the courtyard. Not a good idea to leave an idle girl in the marketplace.\"\n\nShekiba was directed to a stool with a view of the outhouse. She rested her head against the wall. The smell from the outhouse was overwhelming but she dared not move her seat, afraid to anger her unseen hostess.\n\nShe never met the man's wife or children but she could hear them inside. Crying. Laughing. Running.\n\nThe sounds of a family.\n\nI could leave now, Shekiba thought. What if I just opened the door and left? I can find my home from here. I could look for the deed and maybe even make it back for the end of prayers.\n\nBut Azizullah would probably come back and find her gone. Or the lady of the house would notice that the burqa had disappeared from the courtyard and tell him. And then what? Shekiba feared angering Azizullah mostly because she feared being sent back to Bobo Shahgul's house. Nothing would be worse. At least, nothing she could think of.\n\nAzizullah returned and thanked his friend for allowing Shekiba to stay. He gave her a nod and again they were on the dirt road, this time headed toward Bobo Shahgul's house. When they arrived, Hameed answered the gate.\n\n\"Salaam!\" Hameed called out.\n\n\"Salaam, bachem. Where is your father? Your uncles? I did not see them at Jumaa prayers. Did they not go?\"\n\n\"No, sahib. No, and if you only knew what Bobo-jan told them for being so lazy.\" Hameed never could keep anything to himself.\n\nAzizullah chuckled. \"Well, may God forgive their sins even if your bobo-jan will not. Tell them your kaka Azizullah and your cousin are paying a visit.\"\n\nHameed led them into the courtyard and ran inside announcing their arrival at a volume that rivaled the mullah's azaan, call to prayer.\n\n\"Bobo-jaaaaaaaan! Bobaaaaaaaaaa! Kaka Azizullah brought Shekiba baaaack!\"\n\nShekiba panicked even more and turned to look at Azizullah's face. Had he really brought her for a visit or was he returning her to this house? Maybe Marjan had complained about her? About the way she sat? About her odd questions? Her palms grew sweaty. The burqa was suffocating.\n\nAzizullah's attention had turned to a flowering bush. He was examining the petals and did not seem to notice Hameed's announcement.\n\nKaka Freidun appeared in the doorway. He looked unsettled.\n\n\"Agha Azizullah! Welcome! How wonderful to see you.\" Kaka Freidun extended his arms in greeting. The men hugged and exchanged customary pecks on the cheek. \"How are you? How is your family?\"\n\n\"Everyone is well, thank you. And you? Bobo Shahgul is in good health, I pray?\"\n\n\"Ah, the usual aches and pains of age and unruly children,\" he joked, shooting me a glaring look. He thinks I have done something wrong. Already, he would love to punish me.\n\n\"Your family is blessed to have her at this age. I still grieve my mother, God rest her soul, and it has already been two years since she passed.\"\n\n\"May Allah forgive her and may heaven be her place of rest,\" Freidun said. \"Please come in. Join us for a cup of tea.\"\n\nThey walked toward the house and Shekiba stood a few meters back. She felt out of place and shifted on her feet. She was within her family's courtyard but she kept her burqa on. She preferred its cover for the time being.\n\n\"Azizullah-jan, we have not seen each other in some time. I hope that things are well at home.\" Freidun's statement was more of a question. He was trying to gauge the reason behind the visit.\n\n\"Yes, yes, things are well. And you? How is the family doing? How is the farm? Are your crops doing well this year?\"\n\n\"As well as can be expected, with the lack of rain. The dry skies do not help but we are hoping to make at least enough to get by.\"\n\n\"I have heard similar complaints from others around town. And where is Bobo Shahgul? Is she resting?\"\n\n\"She went to lie down after she finished her prayers,\" Freidun said. \"Did you want to speak to her?\" Again, he looked anxious.\n\nKaka Zalmai and Kaka Sheeragha entered the courtyard, their expressions mirroring their brother's. Azizullah stood and the men hugged and exchanged brief pleasantries.\n\nHer uncles pretended not to notice her in the background. Shekiba knew she should go through the back door and find the women but she had little interest in doing so.\n\n\"Shekiba wanted to pay a visit to the family, since Eid is coming next week. She missed everyone a great deal and wanted to say hello, especially to Bobo Shahgul.\"\n\nHer uncles could not conceal their surprise. After a moment, Kaka Freidun nodded smugly.\n\n\"Ah, I see. I am not surprised. Bobo Shahgul is much loved by all her grandchildren.\"\n\nHe thinks I regret how I left. He's even dumber than his wife.\n\n\"Her grandmother is probably about to wake up from her rest and will surely be surprised to see her,\" Freidun said.\n\nShekiba's lips tightened with frustration.\n\n\"Well, you have come all this way. Let us go inside and share a cup of tea with you, dear friend. Surely Bobo Shahgul will be happy for the time with her dear granddaughter!\" Freidun said glibly.\n\nZalmai and Sheeragha shared a smirk.\n\nShekiba felt like a puppet; her arms and legs were being directed by her uncle. What else could she do? Her every move was propelled by her desire to stay out of this house. If Azizullah saw her as an insolent girl, she risked being returned to her family.\n\nHer legs obeyed and she walked slowly through the back door of the house. She passed by Khala Samina's son, Ashraf, who was carrying a tray of steaming teacups and bowls of raisins and nuts. The cups rattled with his unbalanced nerves.\n\nShekiba walked into the hallway and paused. Should she really go to her grandmother? Would they check on her? She lifted her burqa and let it drape from her head.\n\nKhala Samina appeared in the hallway. She was thin framed, more petite than her sisters-in-law.\n\n\"Salaam, Shekiba,\" she said quietly. \"She knows you're here. She's waiting for you.\"\n\n\"Salaam,\" Shekiba answered.\n\n\"Shekiba . . .\"\n\nShe turned around to look at her aunt, who was scratching her forehead. She took a few steps toward Shekiba and lowered her voice.\n\n\"She is an ornery old lady. Don't give her any reasons. She knows no other means of entertaining herself.\"\n\nShekiba nodded, suddenly feeling her throat tighten. Samina's voice was gentle, a tone rarely used toward Shekiba. She suddenly felt a gaping hole where her mother should have been.\n\n\"Thank you, Khala Samina.\"\n\nSamina closed her eyes briefly and nodded her head in acknowledgment before she resumed her work in the kitchen.\n\nShekiba walked a few more meters to Bobo Shahgul's room. She could see through the gauzy curtain that her grandmother sat in a chair with her walking stick in her hand. Her bony fingers were wrapped tightly around the stick.\n\nShe knows I am here. I have no choice now.\n\nShekiba pulled the curtain aside and met her grandmother's icy stare.\n\n\"Well, well. Look who has decided to disrupt our peace yet again.\"\n\n\"Salaam.\" Shekiba decided she would take Samina's advice and try not to antagonize the old woman.\n\n\"Salaaaaam,\" Bobo Shahgul said mockingly. \"You stupid girl. How dare you come here? How dare you step foot in this house?\"\n\nShekiba steeled herself. She had taken worse. All she had to do was resist the temptation to fire back.\n\nYou need to get to your house and get the deed. Do not forget why you came here. Do not let the old lady distract you.\n\n\"Eid mubarak, Bobo-jan.\"\n\n\"As if I needed to see that face,\" she replied, turning away in repulsion. \"There is no Eid for a disrespectful creature like you\u2014you dare to disrespect the grandmother who took you in even after you robbed her of her son.\" She rose on her hobbled feet, fueled by rage.\n\n\"My father was a wise man who decided for himself.\"\n\nShekiba saw it coming but hardly flinched.\n\nBobo Shahgul's walking stick came crashing down on her shoulder.\n\nShe is weaker than a few months ago, Shekiba realized.\n\n\"Bobo-jan, how is your health? You're looking a bit frail, God forbid.\"\n\nA second blow. She was trying harder.\n\n\"You beast! Get out of my house!\"\n\n\"As you please,\" Shekiba said, turned and walked out with her chin held high. She had said nothing. And nothing could have made Bobo Shahgul more irate.\n\nShekiba stopped by the kitchen. She wondered if Khala Samina had heard the conversation.\n\n\"Dear girl, there is something about you that makes that old lady crazy.\"\n\nShe had heard.\n\n\"Khala Samina, I want to get a few things from my father's house. I will not take long.\" Shekiba looked in the direction of the living room. She could hear the men laughing.\n\nSamina shook her head. \"Do as you must\u2014you are not a child. But understand that there are many people willing to make your life more difficult. It is up to you to find a way to make things easier for yourself.\"\n\nShekiba nodded, wondering which one of them was more na\u00efve.\n\n\"I won't be long,\" she said, lowered her burqa and headed out the back door.\n\nShe crossed the fields quickly, peering over her shoulder every thirty seconds or so to see if anyone was coming after her. After about twenty meters, she broke into a jog, hoping she didn't attract attention. Her father's home looked smaller than she remembered it. She felt her heart quicken as she neared the rusted gate.\n\nFor a second, she saw her father standing outside, his face to the sky as he wiped the sweat from his brow with the back of his hand. She heard her mother call out her brothers' names. She saw Aqela's songbird face in the front window, watching their father toil in the fields.\n\nThere should have been a word for what she felt, the way her stomach jumped with anticipation to be somewhere she missed so much, to be around people who missed her as much as she missed them. It was a feeling that started sweet and finished bitter, when she realized that she stood in the ashes of those perfect times, as short as they'd been.\n\nNo one had claimed the home yet but it looked as if someone were trying to fix it up. Cracks in the walls had been filled with clay. The splintered table outside had a new plank nailed to it. Inside, the two solid chairs were gone, as were the few blankets she'd left strewn about to make believe her parents and siblings still slept in the house with her.\n\nShekiba wondered which vulture had his eyes on the house but pushed the thought aside for now.\n\nShe needed to find the Qur'an. Her father's books had not been touched. They still sat on the crooked shelf above where Padar-jan once slept. She looked out the window, half expecting to hear her uncles' angry voices.\n\nShe blinked back tears and used a step stool to reach the top shelf. Her fingers reached over the ledge and sought blindly.\n\nThat's it.\n\nShe pulled at a corner of fabric and the book slid toward her. She grabbed it with both hands and came down from the stool. The Qur'an was wrapped in a thin, emerald-green cloth embroidered with silver thread. This had been her mother's dismol, or wedding cloth. Shekiba brushed the dust away and kissed the holy book, then touched it to her left and right eyes as her parents had taught her.\n\nWhy do we keep the Qur'an all the way up there, Madar-jan? It is so hard to reach it there!\n\nBecause nothing is above the Qur'an. This is how we show our respect for the word of Allah.\n\nShekiba unfolded the cloth and opened the first page.\n\nTariq. Munis. Shekiba. Aqela.\n\nBeside each name, Padar-jan had penciled in the month and year of their birth.\n\nShekiba flipped through the pages, the corners frayed. The book opened to the second sura. She recognized the line that her father often quoted. She traced the calligraphy with her finger and heard his voice.\n\nIt means that we treasure many things in this world, but there is even more awaiting us in paradise.\n\nThe paper fell into her hands. Yellowed parchment with two columns of ornate signatures. She recognized her grandfather's name. This was the deed!\n\nShekiba's senses heightened now that she had what she'd come looking for. She took a quick look around and tucked the deed back into the pages of the Qur'an. It was time to get back to the house before her escapade incited too much anger. She covered the Qur'an again with her mother's dismol and tucked it irreverently under her shirt.\n\nGod, forgive me, she thought.\n\nAs she exited her rusted front door, she could see Kaka Sheeragha across the field.\n\nLazy, she thought, looking at her uncle. The others would have come after me.\n\nSheeragha met her at the door.\n\n\"What were you doing in that house?\" he demanded.\n\n\"Praying.\" Shekiba slipped past him and returned to the living room, hoping Azizullah was ready to leave.\n\n\"Where have you been? Bobo Shahgul said she had a pleasant but short visit with you.\" Azizullah took one last sip from his teacup. \"We should be going. We have taken up enough of your time.\"\n\n\"Time with you is time well spent,\" Zalmai said graciously while he eyed Shekiba with suspicion. Sheeragha nodded in tacit agreement. He was not blessed with the social graces of his brothers.\n\n\"You are very kind. Please pass my regards along to the rest of the family. I am sure I will see you in the masjid for Eid prayers next week.\"\n\n\"Yes, of course you will.\"\n\n\"Absolutely.\"\n\nShekiba followed Azizullah through the courtyard and into the street. Her uncles watched them leave, mumbling to each other.\n\nThey put on a good show, she thought, knowing they were wondering what spurred her return to the family home.\nCHAPTER 14\n\n\"Of course he hit her again! Why did you have to say something like that to him? You know how he is!\" Shahla was folding the laundry in the courtyard, her eyes moving back and forth between the clothes and Sitara, who was drawing circles in the dirt with a rock.\n\n\"I didn't mean for that to happen. I was just . . . I only meant to . . .\"\n\n\"Well, you should think before you say something. She couldn't even lift her arm this morning. God knows what he did to her.\"\n\nI bit my lip. I had gone to my grandmother's as my father instructed. I was hoping he would have left Madar-jan alone, but he hadn't. His toxic anger never went away, not without his medicine. I wanted Shahla to stop telling me how awful he had been to our mother. But I needed to hear. I needed to know what had happened.\n\n\"You've ruined everything for all of us. You don't think. You're so busy being a boy that you've forgotten what can happen to a girl. Now we all have to pay for your selfish mistakes.\"\n\n\"It has nothing to do with you. He was angry at Madar-jan so stop worrying about yourself.\"\n\nShahla was fighting back tears. \"You think it was all about Madar-jan? You think everything stops there? Well, it doesn't. What you do affects all of us.\"\n\n\"What are you talking about?\"\n\n\"You know what we all are? We're all dokhtar-ha-jawan. We're all young women. Me, Parwin. Even you, Rahim. Even you.\"\n\nShe was angry. I'd never seen Shahla so upset. Sitara looked up, sensing the tension.\n\n\"He hit her again. Parwin and I, we were scared to look but we could hear it. He went on yelling and screaming about how it wasn't bad enough that she had failed him as a wife. Now she was failing as a mother.\"\n\nI remembered how she'd looked, cowering under him. His face had been red with anger, his eyes bulging.\n\n\"She must have fallen to the floor. Her shoulder's hurt badly. I don't know. She tried to get him to calm down but he was . . . well, you know how he can get. And then she said something to him that made him stop.\"\n\n\"What did she say?\" I asked quietly.\n\n\"She said she was taking care of all of us. She said it was a house full of dokhtar-ha-jawan and it wasn't easy. All of a sudden, he got quiet. Then he started pacing the floor, saying his house was full of young women and that it wasn't right.\"\n\n\"What's not right?\"\n\n\"Don't you know what people say? They say it's not right to keep a dokhtar-e-jawan in your home.\"\n\n\"What are you supposed to do with them?\" I sensed the ugly turn this was taking.\n\n\"What do you think you're supposed to do? You're supposed to marry them off. That's what's in his head now. And it's all because you don't know what to do with yourself. You think just because you're wearing pants and you strap your breasts down every morning that no one will care what you do. But you're not a kid anymore. People won't pretend anymore. You're no different than me and Parwin.\"\n\n\"You think he wants us to get married?\"\n\n\"I don't know what he's thinking. He left the house after that and he hasn't come home yet. God knows where he is.\"\n\nParwin came out of the house with the second load and started hanging sheets on the clothesline. She reached the twine with difficulty. Most of the sheets she tossed over and then pulled the corners from below. Shahla looked as if she were about to help her, then paused, deciding against it. When Parwin finished, she looked up at the sky, blocking the sun with her hand, and mumbled something under her breath.\n\nI thought of a conversation I'd once overheard. Khala Shaima and my mother thought no one was awake but I was having a hard time sleeping.\n\n\"That's why it's important for these girls to go to school, Raisa. They'll have nothing otherwise. Be wise about it. Look at me and think of what might happen to Parwin.\"\n\n\"I know, I know. I worry about her more than the others.\"\n\n\"As you should. I was passed over despite everything Madar-jan did. All the friends she talked to, all the special prayers. And look at me, wrinkled and alone. No children of my own. Sometimes I think it's worked out best for me that your husband is away so much, that ass. At least it gives me more chances to come and spend time with your daughters.\"\n\n\"They love having you around, Shaima. They hunger for your stories. You're the best family they have.\"\n\n\"They're good girls. But be realistic. Before you know it, you'll have to seriously consider the suitors. Except for Parwin. You'll be lucky if anyone comes for her.\"\n\n\"She's a beautiful girl.\"\n\n\"Bah. The porcupine feels velvet when she rubs her baby's back. You're her mother. Parwin-e-lang. That's what she is. Allah as my witness, I love her as much as you do, but that's what people call her and you have to be honest with yourself and realize it. Just like I'm Shaima-e-koop. I've always been Shaima the hunchback. As long as she goes to school, that at least gives her something. At least she'll be able to pick up a book and read it. At least she'll have a chance to know something other than these four walls and the smell of her father's opium.\"\n\n\"She would make a good wife. And mother. She's a special girl. The way she draws, it's as if God guides her hands. Sometimes I think she still talks to angels, the way she used to when she was a baby.\"\n\n\"Men have little need for special girls. You should know that.\"\n\nI couldn't imagine Parwin married any more than I could imagine the rest of us married. I drifted off to sleep after that. I dreamed of girls in green veils, hundreds of them, climbing up the mountain to the north of our town. A stream of emerald on the trail to the summit, where, one by one, they fell off the other side, their arms outstretched like wings that should have known how to fly.\n\nIn a three-room house, I couldn't expect to avoid my mother for long. I saw her puffy lip, her long face, and hoped she saw the remorse in mine.\n\n\"Madar-jan . . . I . . . I'm sorry, Madar-jan.\"\n\n\"It's all right, bachem. It's as much my fault as it is yours. Look at what I've done to you. I should have put a stop to this long ago.\"\n\n\"But I don't want you to\u2014\"\n\n\"Things will be changing soon, I'm sure. I'm afraid everything is out of my hands now. We will see what naseeb, what destiny, God has in store for us. Your father acts rashly and it doesn't help to have your grandmother whispering things into his ear.\"\n\n\"What do you think he'll do?\" I asked nervously. I was relieved my mother wasn't angry with me. She lay on her side, my baby sister next to her. I resisted the urge to curl up with them.\n\n\"Men are unpredictable creatures,\" she said, her voice tired and defeated. \"God knows what he'll do.\"\nCHAPTER 15\n\nShekiba faced a new dilemma. She wanted to take the deed to the local hakim but she didn't know if Azizullah would allow such an act. Maybe he would. Men were, after all, unpredictable creatures.\n\nShe decided against asking Azizullah for permission but that meant she needed to get herself to the town's hakim. She had overheard his name in conversations between Azizullah and his brother, Hafizullah, but she had no inkling where she would find this man. Then there was the issue of getting to him. What possible excuse could she make this time?\n\n\"How was your visit with your family?\" Marjan asked.\n\n\"It was pleasant,\" Shekiba answered. She was elbow-deep in hot, sudsy water, washing the children's clothes.\n\n\"And how was Bobo Shahgul? Is she in good health?\"\n\n\"Yes,\" said Shekiba. Unfortunately, she thought.\n\n\"And the rest of the family? Did you see everyone? All your uncles?\"\n\n\"I saw Kaka Zalmai, Sheeragha and Freidun. My other two uncles are still away in the army.\"\n\nMarjan stood over her, a finger on her lip as she pondered something. Shekiba purposely avoided her gaze.\n\n\"You know, I ran into Zarmina-jan, your uncle's wife, at the hammam last week. She told me that she was surprised that you wanted to visit your family for Eid.\"\n\nShekiba's neck muscles tightened.\n\n\"She said that you did not adjust well to Bobo Shahgul's house after your father's death.\"\n\nKhala Zarmina. What are you up to?\n\n\"Were you angry to be sent here?\"\n\nShekiba shook her head.\n\n\"Well, I hope not. This was an arrangement that everybody agreed to so I hope that you are not intending to carry out the same kind of behaviors here in this home.\"\n\nShekiba felt a fire burn in her belly. \"This is a different place,\" she said in a bitter voice.\n\n\"Good. Just be warned that we do not tolerate disrespectful behavior. I will not have my children learning . . . such things!\"\n\nShekiba nodded.\n\nBut Marjan was uncomfortable with her. Maybe Khala Zarmina had said something more.\n\nShe prepared the family's dinner and ate quietly in the kitchen. She liked to listen to the children bickering with each other. Amid the din, she heard Marjan tell Azizullah that she had something she needed to discuss with him later.\n\nShekiba knew it would be about her.\n\nIn the night she heard Marjan's soft yelps and knew that Azizullah was taking his wife. This was something Shekiba had learned in her grandmother's house. From where she slept in the kitchen, she could hear the same grunts and pants through the wall and would see Kaka Zalmai emerge from their room refreshed while Samina avoided Shekiba's gaze and busied herself with her children. The women often joked about it when the children were out of earshot but they did not mind Shekiba hearing them.\n\n\"You've been working on that sweater for over a week, Zarmina! When are you going to finish?\"\n\n\"Sounds like what I hear you saying to your husband in the middle of the night, Nargis!\"\n\nLaughter and a hand clapped against someone's back. Shekiba listened closely, intrigued by the rare moments of camaraderie amongst the women.\n\nNargis giggled and shot back without hesitation.\n\n\"Mahtub-gul can hardly see beyond her huge breasts to know what is happening below.\"\n\nLaughter again. Samina looked in Shekiba's direction and seemed uncomfortable to have her in the room. Zarmina noticed and raised her teacup.\n\n\"I wouldn't worry about her, Samina dear. Remember, she was her father's son so it's in her best interests to learn the way things are from women. Imagine if you had no idea what your wedding night had in store for you! Let her be aware.\"\n\nSamina clucked her tongue. \"Knowing would only make it worse.\"\n\nShekiba had thought of her statement often. What was worse about it? Whatever it was, her aunts made it sound awful but tolerable. They were laughing about it, after all.\n\nHearing Marjan's soft sighs and gasps came as no surprise. It was the thing that transpired between a husband and wife and it was how women became heavy with child. This much Shekiba had pieced together.\n\nAfter a few moments, the grunts ceased and Shekiba could hear the sounds of a conversation. She pressed her ear to the wall.\n\n\"And Zarmina told you she did that?\"\n\n\"Yes, that's what she said. And now I know why Bobo Shahgul was so eager to make this arrangement. She didn't want to have this girl in her house.\"\n\n\"I've never trusted those boys. Especially Freidun. They think the world of themselves but not one of those sons is a quarter of the man their father was. Their mother is right to keep a close eye on them.\"\n\n\"But what are we to do with Shekiba-e-shola? True, she does her work around the house well enough but I am afraid that she will turn on us as she did with her own grandmother. What if she threatens to put a curse on our family as well?\"\n\nPut a curse on the family?\n\n\"Hmm. Interesting.\"\n\n\"And Zarmina said that even though she had been doing the chores as a son, that the girl has the spirit of a wild woman. The last thing this home needs to invite is scandal and rumor.\"\n\n\"And what is it that you think we should do?\"\n\n\"I think you should send her back.\"\n\n\"Send her back?\"\n\n\"Yes! For the sake of everyone in this house. Take her back and tell her uncles that they will have to settle their debt in another way. We cannot have her.\"\n\n\"I see.\" Marjan was wise to bring up the matter now, with Azizullah feeling spent and relaxed.\n\n\"But we mustn't tell them why we want to send her back. Zarmina specifically asked me to keep all this to myself.\"\n\n\"I bet she did.\"\n\nThere was silence. Shekiba felt betrayed and then wondered why she was surprised by her aunt's accusations.\n\nWhat does she want? Does Zarmina want me back in the house? Why?\n\n\"It's going to be a shame to lose her help but I have a bad feeling about this girl. I cannot shake Zarmina's words from my mind.\" Shekiba thought of Marjan's nervous behavior the last couple days and almost laughed.\n\nFor a while, she relished the idea that she could be so formidable a threat.\n\n\"If I take her back, it will create a rift between our families and that is not in our best interests. By the looks of their land, I anticipate that the family will be again knocking on our door to borrow money. Not a single one of them knows how to grow a crop. But I have another idea,\" Azizullah said.\n\n\"What is it?\"\n\n\"You worry about the children and look after the house. Did I not say I would take care of it?\" Marjan's window of opportunity was quickly closing. Azizullah's impatience was returning. \"Let me talk to Hafizullah about it but there may be a way to get rid of this girl if she is so bothersome to you. And at the same time, we may be able to secure our position in this community. There are changes coming and Hafizullah has high aspirations.\"\n\nShekiba kept her eyes and ears open in the next few days, looking for any sign of what Azizullah's plan might be. He was out of the house most of the time, undoubtedly meeting with Hafizullah about his mystery plan. Shekiba grew more and more frightened.\n\nWomen who brought scandal or trouble to a home were not tolerated. Even a na\u00efve girl like Shekiba knew as much. Shekiba began to fear for her life.\n\nShe tried to gauge her situation through Marjan.\n\n\"Khanum Marjan,\" Shekiba said quietly. Marjan was darning socks. She jumped at Shekiba's voice.\n\n\"I . . . excuse me! I did not mean to startle you! I was going to prepare dinner.\"\n\n\"Oh, Shekiba!\" Her hand covered her chest. Marjan shook her head. \"Why do you sneak around like that? Go ahead and begin dinner. Azizullah will be hungry when he returns from outside.\"\n\nShekiba fidgeted for a moment before daring to ask.\n\n\"Khanum Marjan? May I ask a question?\"\n\nMarjan looked up expectantly.\n\n\"When you . . . when you spoke to Khala Zarmina . . . what did she tell you? I mean, about me.\"\n\nMarjan turned back to her socks and looked up at Shekiba from the corner of her eye.\n\n\"What does it matter?\"\n\n\"I would like to know.\"\n\n\"She said that you argue.\"\n\n\"Argue? With who?\"\n\n\"You don't know?\"\n\n\"I did not argue with anyone there. I did everything they asked of me.\"\n\n\"Well, seems like you're arguing right now, aren't you?\"\n\n\"No,\" she replied adamantly. She was desperate to defend herself. \"I am not arguing! But whatever she said about me is not true!\"\n\n\"Shekiba! Lower your voice! Forget what they said. Just busy yourself with the chores.\"\n\nShekiba felt helpless. She retreated into the kitchen to begin dinner, angry and frustrated and forced to hide it.\n\nTwo days later Azizullah came home with his brother. They sat in the living room and shared a lunch of rice and eggplant. Shekiba frantically searched for excuses to loiter around the living room door, eager to hear their conversation.\n\n\"They will be traveling with around thirty people. I have asked that the house be readied. We are sparing no expense.\"\n\n\"Your home will suit them fine, my brother. Better than our simple home would. Have you enough food for the night?\"\n\n\"Yes, I've called in all my favors in town and we're going to have a meal that even the king himself will talk about! It is costing me more than I had anticipated but I think this will be a great opportunity. For the both of us, do not forget.\" Hafizullah was slick with confidence.\n\n\"I will be there for sure and if there's anything we can do, we will do it,\" Azizullah said. \"But there is something I would like to offer to the king.\"\n\n\"Oh? And what is that?\" Hafizullah said, a half-chewed morsel still in his mouth.\n\n\"I would like to offer King Habibullah a gift of a servant.\"\n\nShekiba's heart began to pound.\n\n\"A servant? Which servant?\"\n\n\"I do not have that many from which to choose,\" Azizullah said, chuckling.\n\n\"You mean Shekiba-e-shola?\"\n\n\"Yes, that's the one.\"\n\n\"Oh, I don't know about this. Brother, do you really think it is wise to make such a halfhearted offering to the king? You may anger him, you know.\"\n\n\"She is a good worker and will serve the palace well. Is there not a way to make an honorable gesture of her?\"\n\nShekiba, the gesture. Shekiba, the gift.\n\nShe felt insignificant and disposable to hear herself described that way. Again.\n\n\"Well, let me think on it. It is possible, I suppose. I mean, it's not as if he needs to see her face . . . but you know there may be a good use for this girl in the palace after all. Now that I think of it\u2014I just had a conversation with a general. You know General Homayoon, don't you?\"\n\n\"Yes, that no-good money-hungry fool. What were you doing with him?\"\n\n\"He is a money-hungry fool but he's likely going to be promoted, so watch what you say about him. Better to have this fool as your friend than your enemy. He told me that he has been placed in charge of recruiting soldiers to help guard King Habibullah's harem. The king doesn't trust men to watch over his women and he has collected a group of women who are kept as men. This way he need not worry that his guards will take advantage of his ladies.\"\n\n\"Ah, what a brilliant solution! I am telling you, my brother, this girl is well suited for such a role. She walks and breathes like a man, my wife tells me.\"\n\n\"Then we will arrange it,\" Hafizullah declared. \"I will speak to the general so that we can make the entourage aware of the gift before you present her to King Habibullah. This is a historic visit to our town and you will be making a mark. You can expect this to bring you many returns, I believe.\"\n\nShekiba had heard enough. She walked back to the kitchen, her legs wobbly beneath her. Her head was spinning.\n\nThe king? The palace?\n\nWords that were foreign to her.\n\nShekiba, the half face. The girl-boy who walks like a man.\n\nShekiba was not a whole anything, she realized.\nCHAPTER 16\n\nKhala Shaima liked to keep us hanging. I wondered what would happen to Bibi Shekiba almost as much as I wondered what would happen to us. It seemed that we were both about to leave our homes.\n\nPadar-jan spent more time away from home in the following weeks. When he did return, he scowled and barked orders more. Even Parwin's soft singing, which he usually secretly enjoyed, provoked him. Madar-jan tried to keep him placated with ready meals and a quiet home but he inevitably found another reason to explode.\n\nI spent more time at Agha Barakzai's shop. It was my way of avoiding the guys without explaining what was happening. I worried that my mother was going to change me back into a girl and I wondered how Abdullah and Ashraf would react. I hated to be away from them, mostly Abdullah, but I was scared to be with them, too.\n\nI lay awake at night, thinking about Abdullah and remembering the day Madar-jan had caught us play-fighting. Until the moment she called my name, it had been thrilling. I tingled to think of Abdullah's face over mine, his long legs trapping my hips under him, his hands pinning my wrists. And his grin. I blushed in the dark.\n\nI tried to make up to Madar-jan for what I had done. I tried to keep Padar-jan distracted from her, even if it meant him yelling at me. Even though I'd been relieved of housework when I became a bacha posh, I tried to help when I saw her washing clothes or beating the dust from the carpets.\n\nShahla didn't say more than a few words to me every day. She was still upset and could sense from Madar-jan's mood that trouble was brewing. She was quiet around Padar-jan, bringing him tea or food and leaving the room before he could realize she was one of those young women he had kept home for too long.\n\nMy grandmother stopped by more often. She was intrigued by the new wave of unrest in our home and wanted to see it for herself. Madar-jan tried to be as polite as she could.\n\n\"Tell my son that I want to talk to him. When he gets home, make sure he comes to see me.\"\n\n\"Of course. What is it that you want to talk to him about?\"\n\n\"Is it any business of yours? Just tell him what I've asked.\"\n\nMadar-jan knew what the topic was. Maybe this time her husband would be more interested in bringing another wife home.\n\nI listened in when Padar-jan went to see his mother. I pretended to be playing with a ball in the courtyard and slowly kicked it further and further until I was right outside my grandmother's living room. I heard her shrill voice loud and clear. My father, mumbling at times, was more difficult to make out.\n\n\"Bachem, it's high time. You've given her plenty of opportunity to give you a son and she's failed. Now, let's bring a second wife for you so that you can finally expand this family.\"\n\n\"And where am I going to put her? We have one room for all the girls as it is. There's no money to build another space behind our home or to buy something else in town. I can always find a new wife. It's the space and money that are harder to come by.\"\n\n\"What about Abdul Khaliq? Hasn't he promised to help you when you need?\"\n\nPadar-jan shook his head.\n\n\"The men are short on weapons, on supplies. There isn't money to spare.\"\n\n\"Psht. The hell there isn't money. I've heard what he does. I've heard from the people in town about his horses, his wives, all his children. He's got plenty!\"\n\n\"Madar! Be careful what you say! He's a powerful man and don't be part of any loose talk about him. Do you understand me?\"\n\n\"I'm not the one starting this talk. There are lots of tongues flapping about him. That's all I'm trying to tell you,\" she said, annoyed to be silenced by her son.\n\n\"Anyway, I'll be making some changes at home soon and things will be easier on my pockets. It's time I relieved myself of some of these girls.\"\n\n\"And how do you expect to do that?\"\n\n\"Just watch what Raisa does while I'm gone and I'll find a way to take care of the rest.\"\n\nShahla and Madar-jan were right. Padar-jan was about to shake up our home.\n\nEleven days later, Abdul Khaliq showed up at our home with seven other men. They pulled up in two black SUVs, their tires leaving clouds of dust in the street. Abdullah saw the car and knew immediately who they belonged to. Most people in our town traveled on foot.\n\nIt was my cousin Muneer who opened the front gate and pointed out our home. Not even my father was expecting him. Muneer watched openmouthed as Abdul Khaliq and his entourage walked by. Two men had black guns slung over their shoulders. Abdul Khaliq was a burly man in his late forties, judging by the lines around his eyes and the gray in his beard. He wore a white turban and a beige tunic over loose pants. An antenna stuck out from the pocket of his gray vest, another sign that this man was something other than common folk. He was the first person in our town to own a mobile phone. Few had access to any phone at all.\n\nWe usually sent one of the men to the front gate to greet a visitor. People didn't just barge in, since the women of the home could be wandering about the courtyard without their head coverings. But it was either Muneer's stupidity or Abdul Khaliq's presence that changed things from the usual routine. He and his men were in our courtyard, their eyes assessing the situation. I caught sight of them and recognized Abdul Khaliq from the bazaar. I darted inside to warn my mother and send my father out to meet his friend.\n\n\"Padar-jan, Abdul Khaliq is here\u2014with a lot of people.\"\n\nMy father sat up straight and pushed his newspaper aside. \"What are you talking about? Where?\"\n\n\"Out there. In the courtyard. He's got seven men with him. And guns.\"\n\nMy father's brow furrowed. He got to his feet faster than usual.\n\n\"Tell your mother to prepare something for our guests,\" he said, and went outside to meet the warlord.\n\nMadar-jan heard us and stood in the kitchen looking disturbed. She shot a look at the doorway to our bedroom, where Shahla and Rohila were putting Sitara to sleep. Parwin was peeling onions at Madar-jan's feet. She was the only one whose eyes didn't sting and tear when the layers came off.\n\n\"He's going to want more than tea,\" Parwin predicted without looking up.\n\nMadar-jan looked at Parwin almost as if she heard some prophesy in her daughter's words. She bit her lip and took out some cups.\n\n\"Bring these to them, Rahim-jan,\" she said nervously.\n\nI took the tray and willed my hands not to shake. I could feel their eyes boring into me when I entered the room, their conversation suddenly pausing. The men had spread out, Abdul Khaliq sitting on the cushion across from my father, his fingers nimbly working a string of prayer beads as he leaned back. On either side of him sat older men, more gray in their beards than black. The armed men were closest to the door. I didn't look at their faces and tried to keep my gaze off their weapons as well. Kneeling, I put a cup in front of each person and backed out of the room as quickly as I could to listen from the hallway. Madar-jan was doing the same.\n\n\"Arif-jan, I've come here today to discuss an important and honorable matter with you. For that reason, I have brought my elders with me, as well as a few members of my family whom you have met before. I'm sure you recognize my uncle's sons, my father and my uncle. You have fought with me for years and I respect you for that. From one man to another, we both know that there are traditions in our culture.\"\n\n\"You honor me with your visit, sahib, and I have been proud to fight under your leadership. We've done great things for our people thanks to you.\" I'd never heard Padar-jan speak in such a way with anyone. Abdul Khaliq unnerved him. \"And I am honored to have your family in my humble home. Dearest uncles, I appreciate you traveling this far to be our guests.\"\n\nThe men nodded, acknowledging my father's platitudes. Abdul Khaliq's father cleared his throat and began to speak. His voice was raspy and he had a light lisp.\n\n\"My son speaks highly of you, and of course, your family is well respected in this town. I've known your father for many years, Arif-jan. He is a good man. That is why I'm sure we will see eye to eye on this matter as well. As you know, my son is a man who takes pride in meeting his duties as a Muslim. And one of the duties that Allah has outlined for us is to build families and to provide for women and children.\"\n\nI could feel my heart pounding. Madar-jan stood behind me, one hand on my shoulder and the other covering her mouth, as if she thought she might let out a scream otherwise.\n\n\"Of course, dear uncle . . .\" Padar's voice trailed off; he was unsure what to say. Abdul Khaliq began to speak.\n\n\"And you came to me recently talking of your concerns. That you have young women at home and not enough money with which to provide for them. I have been thinking about your situation and am here to offer a solution.\"\n\nAbdul Khaliq's father gave him a look. Let me do the talking, his eyes said.\n\n\"We must often think of what is in everyone's best interests. In this case, you have a young woman whom my son would like to honor as his wife. Our family is large and well respected, as you know. Your daughter would do well to join our family and a union between us would be cause for celebration. Of course, as a result, you would be better able to provide for your family as well.\"\n\n\"My daughter?\"\n\n\"Yes. If you give it some thought, I'm sure you'll see it's the wisest choice.\"\n\n\"But my eldest is\u2014\"\n\n\"We are not here for your eldest daughter, Arif-jan. I'm speaking of your middle daughter. The bacha posh. My son has expressed an interest in her.\"\n\n\"The bacha posh . . .\"\n\n\"Yes. And do not be surprised. You have kept her as bacha posh beyond what anyone should accept. You are breaking tradition.\"\n\nI turned around and looked at my mother, my face drained of color. Padar was silent. I knew he was wondering how Abdul Khaliq knew about me but word had way of traveling. I remembered the day in the bazaar, the way Abdul Khaliq had looked at me and the way he had smirked and nodded when the man next to him leaned in and whispered something in his ear.\n\nMy mother's fingers tightened as she wrapped her arms around me. She was shaking her head, willing her husband to refuse this man and praying he could do so in a way that wouldn't offend him or his guns.\n\n\"With all due respect, sir . . . it's just that . . . well, she is a bacha posh . . . but I have two other daughters older than her. And as you said, we are people of tradition and usually the younger daughters are not given until the eldest . . . I just don't think . . .\"\n\nThere was a long pause before Abdul Khaliq's father began to speak again, slowly and deliberately.\n\n\"You are right. It would be improper to give your middle daughter's hand without the other two being wed as well.\"\n\nFor a second I could breathe. But it was only a second.\n\n\"But this can be easily arranged. My cousins are here, Abdul Sharif and his brother Abdul Haidar. They are looking for wives as well. We can arrange for each of them to take one of your daughters. They are strong men, able-bodied, and will provide well for your girls, who are now young women and should not be kept idle at home. Let these men bring honor to your home and ease your troubles.\"\n\n\"Abdul Khaliq, dear uncles, you know I hold you in the highest regard, but . . . but this is a matter . . . well, tradition dictates that I should consult my family, as you have done. I cannot make such decisions without the presence of my father and our gray-haired family members.\"\n\nAbdul Khaliq's father nodded in understanding.\n\n\"Reasonable. This is not a problem. We shall return in one week's time. Kindly arrange to have your father and your elders here so that we may meet with them.\"\n\nIt may have sounded like a request but Padar-jan knew it was more of a command. They would not take no for an answer.\n\nAs soon as the last man was out the door, Madar-jan ran up to my father.\n\n\"Arif, what are you going to do? The girls are so young!\"\n\n\"It's none of your business what I'm going to do! They're my daughters and I'll do what's right for them. It's not as if you're capable of doing anything.\"\n\n\"Arif, please, Rahim's only thirteen!\"\n\n\"And he's right! She shouldn't be a bacha posh any longer! She's a young woman and it's shameful to have her out on the streets and working with Agha Barakzai at this age. You've given no thought to her decency, have you? Do you know how this looks for my family's name?\"\n\nMadar-jan bit her tongue. If only my father knew . . .\n\n\"You think you can come up with a better plan for this family? There is no money, Raisa! You're thinking of nothing but yourself. And you've seen what happens to girls who stay in their fathers' homes for too long. There is talk about them. There is scandal. Or worse! What will you do if some bandits come and take your daughters by force? This man, this family, they can provide for your daughters! They can give your daughters a respectable life!\"\n\nMadar-jan searched for a way to argue back. But a lot of what her husband said was true. She was barely able to feed us with what he provided. Padar-jan's brothers were in no better a situation, not to mention the two widows and their children.\n\n\"Maybe I can ask my sister, Shaima, to be here when they return. She could reason with them.\"\n\n\"Khanum, if your insolent sister dares step foot in this house on that day, I swear to you I'll cut her tongue out and send her hunched back rolling down the street!\"\n\nMadar-jan shuddered to hear him talk about Khala Shaima in that way.\n\n\"Abdul Khaliq is a powerful man and he's in a position to improve our family's lot. This is a matter I'll discuss with my father. You should concern yourself only with fixing what you've done. It's time to undo Rahim.\"\n\nThere was nothing more my mother could say to him. He'd been intimidated by Abdul Khaliq, and from what we'd overheard, it sounded like my father had planted the idea in Abdul Khaliq's mind. I thought back to what Shahla had told me about their fight.\n\nHe wants this, I realized. My father wants to marry us off.\n\nThe thought sent a chill down my spine. I realized what my mother knew as well. Men could do what they wanted with women. There would be no stopping what Padar had set in motion.\nCHAPTER 17\n\nKing Habibullah had taken the throne in 1901, just as Shekiba turned eleven years old. This was two years before the cholera epidemic that claimed her family and half her village. That was all she knew about the man. She was a girl from a small village and knew nothing of the palace or life in the capital of Kabul.\n\nHaving overheard Hafizullah's brilliant plan for her, Shekiba became terrified. She had no reason to believe that life in the palace would be any better for her. The more powerful people were, the more harm they could do her. Shekiba sat in the night and chewed her lip, her fingers confirming the presence of the deed under her blanket.\n\nI have to get to the hakim. That's my only chance.\n\nShekiba did not know when the king would visit, but it would be soon. She had nothing to lose. She had a plan.\n\nShekiba tucked the deed into her dress and crept out of her room at first light. The azaan sounded, calling the town to prayer. She remembered the way from Azizullah's house to the village center. There were a few shops there and surely someone would be able to direct her to the house of the hakim.\n\nShe heard Azizullah's snores and crept past his and Marjan's room. Fortunately, he rarely woke for morning prayers, claiming he would make them up later in the afternoon. The children were still asleep.\n\nShe slipped her burqa over her head and slowly pushed open the heavy gate. She was outside the courtyard. She paused for a moment, waiting to hear the sound of footsteps behind her. When she heard nothing, she took a deep breath, said a quick prayer and headed down the small dirt road. Shekiba walked quickly, trying not to look back at the house since that might draw more suspicion. But no one was out yet and the two donkeys outside didn't even bray at the sight of her.\n\nAgha Sharifullah, the hakim. Shekiba hoped someone in town would be able, and willing, to direct her to him. She rehearsed her appeal in her mind for the thousandth time. She wondered what her mother would have thought of her plan.\n\nThe sky was bright by the time she entered the village center and she passed by a family of five, the mother and children following behind their father, probably on their way to visit relatives. They looked at her oddly from across the road but said nothing. Shekiba exhaled when they were finally out of view.\n\nA few moments later, two men exited a house and began walking ahead of her. They looked back at her and commented to one another. Shekiba bowed her head and slowed her gait, wanting to put more distance between them. The younger man pointed at her and shook his head. The older man nodded and fingered the beads on his tasbeh.\n\n\"Khanum, who are you?\" he called out.\n\nShekiba kept her gaze lowered and slowed her step even more.\n\n\"Khanum, where are you going by yourself? Who are you?\"\n\nShekiba debated asking these men if they knew Hakim-sahib. She stopped, afraid to get any closer to them.\n\n\"Khanum, this is very wrong! Whoever you are, you should not be wandering around alone,\" he scolded. \"What family are you from?\"\n\nShekiba felt her tongue loosen.\n\n\"I am from Agha Azizullah's home,\" she said shakily.\n\n\"Agha Azizullah? But you are not Khanum Marjan. Who are you?\" called out the older man.\n\n\"Khanum Marjan is not well,\" she lied. \"I have been sent to bring her medicine.\"\n\n\"Sent out for medicine? Well, this is just absurd.\" The younger man turned to his counterpart. \"He is a dear friend of mine but I can't imagine what Agha Azizullah was thinking.\"\n\n\"This is truly bizarre,\" he said, shaking his head. And then he made a decision. \"Follow us into town. I'll speak with Azizullah later.\"\n\nShekiba nodded and walked about five meters behind them, now doubly panicked. Surely, by now, Marjan had discovered her absence and she had probably shared the news with Azizullah. Would they come looking for her? Although it seemed this man believed her story, he would surely report back to Azizullah. Although Azizullah already had plans to get rid of her, he could do much worse if he were angered and shamed by Shekiba.\n\nThey led her to the village's dry-goods store owner, who doubled as the local apothecary. She entered behind the older man.\n\n\"Salaam, Faizullah-jan.\"\n\n\"Wa-alaikum as-salaam, Muneer-jan. How are you?\"\n\nSo it is Muneer who will report back to Azizullah.\n\nThey exchanged pleasantries before Azizullah addressed Shekiba's presence.\n\n\"Azizullah has sent this girl to bring medicine for his wife. I found her walking about in the streets alone. Can you imagine? I think the man has lost his mind.\"\n\nFaizullah shook his head.\n\n\"No doubt he is distracted by King Habibullah's visit. It is just two days from now and I'm sure his brother has him running in circles.\"\n\nTwo days from now?\n\n\"What illness does she have?\"\n\nShekiba nodded yes or no arbitrarily as he rattled off a few symptoms. She left with a small bottle of blended herbs and Faizullah made a note of the purchase in his records.\n\nAzizullah is going to kill me, Shekiba suddenly realized. She had gone too far.\n\n\"Excuse me, sahib,\" she said outside. There was no reason to stop now. \"I must take a paper to Hakim-sahib.\"\n\n\"What? What sort of paper?\"\n\n\"I was instructed to discuss this only with Hakim-sahib.\"\n\nThe younger man looked indignant.\n\n\"Padar, this is ridiculous!\" he said.\n\n\"It is indeed!\" said his father. Shekiba waited nervously.\n\nBut they pointed her toward Hakim-sahib's house, which, as Shekiba had prayed, was within the village's central area. They were fed up with her and decided to let her find her own way. Azizullah could clean up his own mess.\n\nA young boy answered the gate and Shekiba asked to speak to Hakim-sahib. The boy gave her a curious look before running back into the courtyard. A moment later, a puzzled man with a grayed beard appeared at the door. He peered out from behind the half-open door.\n\n\"Please, esteemed Hakim-sahib, I have come to you with a most serious request.\"\n\n\"You? Who are you and what are you doing here? Is there no one with you?\"\n\n\"No, sahib. But I have a paper that I need to show you.\"\n\n\"Who are you? Who is your husband?\"\n\n\"I have no husband.\"\n\n\"Who is your father?\" He still had not fully opened the door, uninterested in inviting this strange lone girl into his courtyard.\n\n\"Sahib, this paper is from my father. His name was Ismail Bardari.\"\n\n\"Ismail? Ismail Bardari?\"\n\n\"Yes, sir.\"\n\n\"You are his daughter? You are the one who . . .\"\n\n\"Yes, I am. Please, sahib, I have the deed to my father's land.\"\n\nIt all came in one breath. And then she heard her name.\n\n\"Shekiba!\"\n\nShekiba almost did not recognize Azizullah. She whipped around to see him walking quickly toward Hakim-sahib's house. Hakim-sahib pushed the door wide open. Shekiba turned to him and spoke quickly. Azizullah was a hundred meters away. Her words came fast and furious.\n\n\"Please, sahib, I have the deed to my father's land and I am his only surviving child. I want to claim my inheritance. That land should belong to me and my uncles are taking it without right.\"\n\nHakim's eyes widened. \"You want what? Azizullah-jan, may Allah grant you a long life,\" he called out.\n\nShekiba could not take much hope from his exasperated tone. She pulled the paper from under her burqa.\n\n\"It is my land and it is my right. Please, sahib, just look at the deed and you will see\u2014\"\n\nHakim-sahib took the paper from her hand and glanced at it. His eyes returned to fast-approaching Azizullah.\n\n\"Please, Hakim-sahib, I have nothing else. I have no one else. This land is my only\u2014\"\n\nA blow to the side of her head. Shekiba reeled.\n\n\"Goddamn you, girl!\" A second blow knocked Shekiba off her feet.\n\nShe lay on her side, curled. Her hands instinctively rose to cover her head beneath the burqa. She looked at Hakim-sahib. He was shaking his head.\n\n\"Azizullah-jan, what is going on with this girl?\"\n\n\"Hakim-sahib, those damned Bardari brothers gave this as repayment for their debt and never have I been so swindled in my life!\" he screamed, pointing at Shekiba. \"We have fed her and housed her and look at how she treats us!\" A kick to her flank. Shekiba yelped. \"What are you doing? What kind of girl sneaks out of a house? Have you no shame?\"\n\n\"What is this talk of a deed?\" the hakim said.\n\n\"What deed?\"\n\n\"This girl is here to claim her father's land,\" Hakim explained.\n\n\"To claim what? Is there no end to this girl's stupidity?\" He turned to Shekiba and landed another kick into her side.\n\nThe pain threw her into a rage.\n\n\"I am only here to claim what is rightfully mine! I am my father's daughter and that land should belong to me! My father would never have chosen his brothers over me! He never did!\"\n\n\"A family of fools!\" Azizullah shouted. He threw his arms into the air in exasperation.\n\nThe hakim sighed heavily and clucked his tongue.\n\n\"Girl, you know nothing of tradition,\" he said, and tore the deed into pieces.\nCHAPTER 18\n\nTradition hadn't lost importance between Bibi Shekiba's time and now.\n\nOur home was tense all week. Madar-jan's hands trembled. She dropped forks and food while her mind wandered and worried. I caught her watching me and my sisters. Shahla shook her head and Parwin made comments that made Madar-jan burst into tears.\n\n\"The pigeons look sad today. As if their friends all flew away and now they have no one to talk to.\" Parwin looked up from her paper. She'd sketched five birds, each flying off in a different direction.\n\nMy mother took one look at the drawing, covered her mouth with her hand and went to talk to Padar-jan. We heard yelling and the sound of glass breaking. She returned to us, her lip quivering and a dustpan full of glass shards in her hands.\n\nMy father spoke with our grandfather and summoned my uncles to join us at the house. Kaka Haseeb, Jamaal and Fareed showed up along with Boba-jan. They looked solemn. I wondered what Padar-jan had told them.\n\nAs promised, Abdul Khaliq's family returned in the afternoon. My sisters and I had Sitara look out the window and tell us what she saw.\n\n\"Lots of people,\" she said.\n\nMadar-jan came back into the room with us, leaving the discussion to the patriarchs of our compound. She had tried several times to talk to my father but to no avail. He was not interested in hearing her. She stood in our doorway and craned her neck to hear down the hall. In our small home, we could hear every word of the conversation.\n\n\"Thank you, agha-sahib, for coming today and joining your sons for this important discussion. Our family takes these matters very seriously and we come to you with the best of intentions. This is an issue of honor and family. We have known each other for many years. Our fathers were born and buried in the same soil. We are nearly kin,\" Abdul Khaliq's father said.\n\n\"I have a great deal of respect for your family and always have,\" Boba-jan said simply. It was up to the suitors to do the talking.\n\n\"And it is for this reason that we have come to this home. We believe that your granddaughter would make an excellent match for my son Abdul Khaliq, whom this village has come to respect and appreciate for defending our people and our homes for years.\"\n\n\"Our people owe him a debt of gratitude. He has shown great bravery.\"\n\n\"Then you will agree that he would be an honorable husband for your granddaughter.\"\n\n\"Well,\" Boba-jan said slowly. I could picture my father's eyes on my grandfather, hoping he would stick to what they had rehearsed. \"With the highest respect, Agha Khaliq . . . we have concerns, which I believe my son Arif expressed to you last week. I understand you are speaking of Rahim. We agree that he . . . she has been kept as a bacha posh for too long and should be returned to what Allah created. But, still, there are two sisters before her, and as you know tradition dictates that\u2014\"\n\n\"This is understood and we have already discussed your other two granddaughters. We have here again my nephews Abdul Sharif and Abdul Haidar. Each of them will be honored to take a daughter as a wife. Even better to further strengthen the ties between our families.\"\n\n\"Hmm,\" Boba-jan said, considering the proposal. My father cleared his throat.\n\n\"My second daughter\u2014you probably do not know this, but she was born with a lame leg. She limps . . .\"\n\n\"No matter. She will not be a first wife anyway. I've seen lame-legged women bear children. You should be happy then, anyway. Unlikely you would otherwise marry her off.\"\n\n\"Yes, unlikely . . .\"\n\nThree daughters married off at once would be a huge burden lifted from my father's inept shoulders. While his mind toyed with the idea, my uncle Fareed spoke.\n\n\"Abdul Khaliq Khan, sahib, you honor us with your proposals but . . . but my family also has traditions. I don't mean to insult you but there is something that has been passed down through generations . . .\"\n\n\"I can respect tradition. What is it?\" I could hear annoyance in his voice. He was losing patience with our family, having had to make a second trip. He'd acquired his last wife with much less fuss.\n\n\"Well, my family traditionally asks for a large bride price for our daughters and I am embarrassed to bring up matters of money with a man such as yourself, but it is something that I cannot brush under the carpet. This goes back generations and to break from what our ancestors . . .\"\n\nMy father must have been nervous. The bride price was the critical part he and his brothers had discussed.\n\nI could tell by my mother's face that my uncle was lying. She was trying to read through the wall if Abdul Khaliq was buying his story.\n\n\"What is it?\"\n\n\"Excuse me?\"\n\n\"How much is the bride price?\"\n\n\"It's\u2014as I've said, I'm embarrassed to be discussing this but it's quite hefty. It's . . . it's one million afghanis,\" he said finally. My mother and I nearly choked at the amount. We'd never heard of such a large figure!\n\n\"One million afghanis? I see,\" he said, and turned to one of the men with a gun slung over his shoulder. \"Bahram,\" he said simply. We heard the door open and close. The room was silent until Bahram returned. Abdul Khaliq was tired of cajoling.\n\nWe heard a soft thump. Abdul Khaliq began speaking again. \"That should cover it,\" he said simply. \"You'll have plenty there to cover the bride price of each of your three daughters. Of course, as family, we will share with you some of the products of the land to the north. Perhaps that would be of interest to you.\" I knew my father's eyes were bulging at the promise of opium. My mother shook her head.\n\n\"Now we need only arrange the nikkah date for these three unions. Wouldn't you agree?\"\n\n\"I . . . I suppose . . . Abdul Khaliq, sahib, what about a wedding? A celebration?\" Usually there was something. Guests, food, music.\n\n\"I don't think that's really necessary. My cousins and I, we've all had weddings. The most important thing is to have the marriage done properly with a mullah. For that, I'll bring my friend Haji-sahib.\" He waved his hand in the direction of the bag. \"Now that this matter has been settled, I'm sure you agree that the nikkah is the most important part.\"\n\nMy father, my grandfather and my uncles were silent. My mother and I felt our stomachs drop, knowing they could not resist what Abdul Khaliq was offering\u2014more money than our family had ever seen and the promise of a steady opium supply. I covered my face with my hands and pressed my head against the wall.\n\nI slipped out of Madar-jan's clutching fingers and left her standing there, stunned. Three daughters. Turning me into a boy hadn't protected me at all. In fact, it had put me right in front of this warlord who now demanded my hand in marriage. Barely a teenager, I was to be wed to this gray-haired fighter with bags of money and armed men to do his bidding.\n\nMy sisters looked at me, already crying. Shahla was trembling.\n\n\"It's terrible, Shahla!\" I sobbed. \"I'm so sorry, I'm so very sorry! It's so awful!\"\n\n\"They're really agreeing to it?\"\n\n\"It's . . . it's just like you said . . . there's too many . . . they're giving Padar so much money . . .\"\n\nI couldn't bring myself to form the words. Shahla understood though. I saw her eyes well up and her lip stiffen before she turned her back to me. She was angry.\n\n\"God help us,\" she said.\n\nI wanted to be outside with Abdullah. I wished I could be chasing stray dogs with him or kicking a ball down the street. I wondered what he would say if he knew I was to be married.\n\nThat night, I dreamed of Abdul Khaliq. He had come for me. He towered over me with a stick in his hand, laughing. He was pulling me by the arm. He was strong and I couldn't get away. The streets were empty but as I walked past the houses, gates opened one by one. My mother. Khala Shaima. Shahla. Bibi Shekiba. Abdullah. Each one stood in a doorway and watched me walk by; they all shook their heads.\n\nI looked at their faces. They were sad.\n\n\"Why aren't you helping me?\" I cried. \"Don't you see what's happening? Please, can't you do something? Madar-jan! Khala Shaima! Bibi-jan! I'm sorry! Shahla, I'm sorry!\"\n\n\"Allah has chosen this as your naseeb,\" they each called out in turn. \"This is your naseeb, Rahima.\"\nCHAPTER 19\n\nAbdul Khaliq Khan was a clever man. A clever man with many guns. He knew all the right buttons to push. My father had never seen so much money and would choose opium over food even if he hadn't eaten for days. What good were his daughters anyway?\n\nWe were young but not that young. Shahla was fifteen years old, Parwin was fourteen and I was thirteen. We were flower buds that had just started to open. It was time for us to be taken to our new homes, just like Bibi Shekiba.\n\nMy father had come into our room and ordered my mother to make a shirnee, something sweet he could put before the guests to show our family agreed to the arrangement. We didn't have much so Madar-jan gave him a small bowl of sugar, wet with tears, which he took and laid before Abdul Khaliq's father. The men embraced each other in congratulations. We girls huddled around my mother, looking to each other for comfort.\n\nThe arrangements moved quickly. Abdul Sharif was a rugged-looking man in his thirties and his brother Abdul Haidar was probably a few years older. Abdul Sharif had one other wife at home but was content to take on a second, especially since the bride price had been covered by his cousin. Abdul Haidar already had two wives at home. Parwin would be his third.\n\nCome back in two weeks for the nikkah, Padar-jan had said, his eyes darting back and forth from the guests to the black bag on the floor.\n\nShahla was so angry that she did not speak to me for four days.\n\nI tried to talk to her but she wouldn't look at me.\n\n\"Why did you have to make Padar so angry? I don't want to go with that man! Parwin doesn't want this either! We were fine! Leave me alone. Go and be with Abdullah now!\"\n\nI was stunned. My sister was right, though. I had pushed the situation without thinking about anyone else. I wanted to be allowed to wrestle with Abdullah, to walk to school with him and feel his arm around my shoulder. This was my doing.\n\n\"I'm sorry, Shahla. I'm really sorry! I didn't mean for any of this to happen! Please believe me!\"\n\nShahla wiped her cheeks and blew her nose.\n\nParwin watched us, her mouth in a tight pout.\n\n\"One by one, the birds flew off . . . ,\" she said quietly. I looked at her, her left leg tucked under her and her right stretched before her. I wondered how her husband would treat a wife with a lame leg. I could see in Shahla's eyes, she was thinking the same thing.\n\nShahla blamed me. If I hadn't pushed Padar-jan that day, then he and Madar-jan would not have had that argument. And we would not have been betrothed to Abdul Khaliq's family.\n\nI wondered if it would have made a difference. I wondered if one small difference in the sequence of events would have altered the paths we ended up on. If I hadn't let Abdullah, sweet, strong Abdullah, pin me down in the street for my mother to see, we wouldn't have argued. I would have eaten dinner with the family. My father would have gone on smoking his own paltry opium supply and he would not have thought to complain to Abdul Khaliq that he needed to marry his daughters off.\n\nMaybe I could have stayed a boy, running alongside Abdullah, making faces behind Moallim-sahib's back and having my father ruffle my hair when I walked by. As if he wanted me around.\n\nBut that wasn't my naseeb.\n\n\"It's all in Allah's hands, my girls. God has a plan for you. Whatever is in your naseeb will happen,\" my mother had sobbed.\n\nI wondered if Allah hadn't meant for us to choose our naseeb.\n\nWith my father standing over her shoulder, my mother reluctantly made three baskets of shirnee. She covered a cone-shaped block of sugar and loose candies from Agha Barakzai's shop with a layer of tulle she'd purchased with some of the bride price. She cut swatches from her nicest dress and edged the sides with some lace she'd been given as a gift. Three large squares, one for each basket. These were our dismols, as important as the sweets. My father nodded in approval. My mother avoided his eyes. I looked at them and wondered if that was how it would be for each of us with our husbands. Or if they would be more like Kaka Jameel, who never seemed to raise his voice and whose wife smiled more than any woman in our family.\n\nI wondered why they were different.\n\nPadar hardly noticed what was happening at home. He didn't even notice that Madar-jan slept in our room with us, instead of at his side. He was busy counting bills and smoking opium at least twice a day. Abdul Khaliq had made good on his promise and my father was enjoying his end of the bargain.\n\n\"I've brought home a chicken, Raisa! Make sure you send some to my mother, and not just the bones, mind you! And if the meat is dry and tough like last time, you'll have no more tomorrows.\"\n\nMy mother hadn't eaten more than a couple of bites since the suitors had left and her eyes looked heavy. She was civil with my father, afraid to rile his anger and risk losing her youngest daughters too.\n\nIn the meantime, Madar-jan had to undo what she had done to me. She gave me one of Parwin's dresses and a chador to hide my boyish hair. She gave my pants and tunics to my uncle's wife for her boys.\n\n\"You are Rahima. You are a girl and you need to remember to carry yourself like one. Watch how you walk and how you sit. Don't look people, men, in the eye and keep your voice low.\" She looked like she wanted to say more but stopped short, her voice breaking.\n\nMy father looked at me as if he saw a new person. No longer his son, I was someone he preferred to ignore. After all, I wouldn't be his for much longer.\n\nI lingered around Shahla, brought her food and helped with her share of the chores. I regretted the way things had happened and wanted her to know how sorry I was that I'd pushed her into Abdul Sharif's home. These things I told her while she stared off. But Shahla was too kind to stay angry long. And we didn't have long.\n\n\"Maybe we'll be able to see each other. I mean, they're all part of the same family. Maybe it will be like here and we can see each other every day\u2014you, me and Parwin.\"\n\n\"I hope so, Shahla.\"\n\nMy sister's round eyes looked pensive. I suddenly realized how much she resembled our mother and felt the urge to sidle up next to her. I felt better with her shoulder touching mine.\n\n\"Shahla?\"\n\n\"Hm?\"\n\n\"Do you think . . . do you think it will be terrible?\" I asked, my voice hushed so Madar-jan and Parwin wouldn't hear.\n\nShahla looked at me, then at the ground. She didn't answer.\n\nKhala Shaima came over. She'd heard rumblings through the town that Abdul Khaliq and his clan had paid our family two visits. She figured my father was up to something. Her knuckles whitened when Madar-jan told her, sobbing, that her three eldest daughters were to be wed next week.\n\n\"He's really done it. The ass made himself quite a deal, I'm sure.\"\n\n\"What was I to do, Shaima, with a room full of gray-haired men? And he is their father. How could I have stopped anything?\"\n\n\"Every man is king of his own beard,\" she said, shaking her head. \"Did you try to talk to him?\"\n\nMadar-jan just looked at her sister. Khala Shaima nodded in understanding.\n\n\"A council of asses. That's what you had gathered here. Just look at these girls!\"\n\n\"Shaima! What am I supposed to do? Clearly, this is what Allah has chosen as their naseeb\u2014\"\n\n\"Oh, the hell with naseeb! Naseeb is what people blame for everything they can't fix.\"\n\nI wondered if Khala Shaima was right.\n\n\"Since you know so much, tell me what you would have done!\" Madar-jan cried in exasperation.\n\n\"I would have insisted that I be present. And I would have told Abdul Khaliq's family that the girls were not yet of age for marriage!\"\n\n\"A lot of good that would have done. You know who we're dealing with. It's not some peasant from the streets. It's Abdul Khaliq Khan, the warlord. His bodyguards sat in our living room with machine guns. And Arif agrees with the plan. Do you honestly think they would have listened to anything I had to say?\"\n\n\"You are their mother.\"\n\n\"And that's all I am,\" Madar-jan said sadly. Her voice grew quiet. I'm sure she didn't think any of us could hear them. \"There is only one thing I could think of doing.\"\n\n\"What is that?\"\n\nMadar-jan looked down, her voice lowered.\n\n\"A death in the family would mean there could be no wedding for at least a year.\"\n\n\"A death? Raisa, what in the hell are you talking about?\"\n\n\"It happens all the time, Shaima. You and I have both heard stories. Remember Manizha from the other side of the village?\"\n\n\"Raisa, you've lost your mind! Just think about what you're saying! You think setting yourself on fire is going to solve any problems? You think orphaned girls are better off than married ones? And what about the little ones? What do you think they'll do without their mother? For God's sake, look at your in-laws! You've got two widows in this compound and your brothers-in-law are eyeing them already.\"\n\nMy heart pounded so loudly I was certain they could hear it.\n\n\"I just don't know what else to do, Shaima!\"\n\n\"You have to find a way to turn them down. To make Arif turn them down.\"\n\n\"Easier said than done, Shaima! Why don't you come for the nikkah? Bring your big mouth and I'll see what you do then.\"\n\n\"I will be here, Raisa. Don't think I won't.\"\n\nMadar-jan looked exhausted. She leaned her head against the wall and closed her eyes; the shadows under them had darkened since yesterday.\n\nWe gathered around Khala Shaima.\n\n\"My girls, let me tell you a little more about Bibi Shekiba. As much as I hate to think it, her story is your story.\" She sighed and shook her head. \"I suppose we all carry the story of our ancestors in us. Where did we leave off?\"\nCHAPTER 20\n\nTwo days passed before Shekiba could stand. Her lip was swollen and scabbed, her legs and back bore multiple bruises and each breath yanked her ribs in different directions.\n\nIt wasn't her naseeb to claim her father's land. Instead, Azizullah had dragged her back to the house and beaten her for an hour. Every time his strikes slowed, he would yell and huff about the humiliation she had caused him. His momentum would pick up again and he'd toss her left and right with each blow.\n\nMarjan had watched from the doorway, shaking her head. She had one hand over her eyes and when she could watch no more turned her back and left. Shekiba did not notice. She had let her mind drift long ago.\n\nMarjan came to her three times a day and brought her tea and bread. She would prop Shekiba up and dribble tea into her mouth with small lumps of wet bread. She rubbed an ointment on Shekiba's back and on her cut lip.\n\n\"Stupid girl. I warned you not to bring up such matters. Now look what you've done to yourself,\" she muttered over and over again.\n\nShekiba wished Azizullah would have killed her. She wondered why he hadn't.\n\nShe did not see him, but she could hear his voice. His mood was sour and the children avoided him. Marjan could not.\n\n\"Make sure she's up and ready today. No excuses.\"\n\n\"She is weak but I will see what she can\u2014\"\n\n\"Weak? If she's so weak, what was she doing walking through town, following Muneer and his son around? Why did I find her at Hakim's front door? She's a liar and the sooner we rid ourselves of her, the better. No excuses. She will be up and ready today!\"\n\nShekiba heard the words and the situation began to register. Today was the day King Habibullah would pay a visit to Hafizullah. Today was the day she would be gifted again.\n\nAzizullah left early in the morning and Marjan huffed for an hour before coming to Shekiba.\n\n\"Come on. Time to get washed up.\" Shekiba was lifted to her feet by a woman half her height but twice her width. Marjan guided her to the washroom and let her slide onto the floor. \"You stupid girl. You've made more work for me! God knows you won't last at the palace if you pull tricks like this.\"\n\n\"I only wanted what should be mine. You would have done the same,\" Shekiba said flatly.\n\n\"No, I would not have! You think you're the only girl who should have inherited land? My brothers divided our land and not one square inch of it was deemed mine. That's how things are! You accept it or you die. It's that simple.\"\n\n\"Then I should die.\"\n\n\"Maybe so, but not today. Now get undressed so you can take a decent bath.\"\n\nAzizullah returned in the evening, his mood much improved.\n\n\"What a day it's been! Hafizullah outdid himself! Never have I seen so much food. I even met with some of the king's advisers. Good people with a great deal of influence. I think this visit will bring good fortune to our family and our town. We have put ourselves under King Habibullah's nose and he will surely remember how hospitably he has been treated here.\"\n\n\"Did you speak to the king too?\"\n\n\"Of course I did! What kind of question is that? He's a wise man\u2014this I could see right away. But they'll be leaving at first light and I think the girl should be presented tonight, over dinner, so that everyone can see what a gift we have made to the king! We will make our mark while Hafizullah makes his. Bring the girl! I do not want to sit here and chatter with you now. I want to get back before dinner.\"\n\n\"The girl is ready,\" Marjan said, and went to bring her. She found Shekiba sitting against the cold wall, her legs tucked under her. \"Get up, Shekiba. It's time.\"\n\nShe looked at Marjan blankly. After a moment, she rose, ignoring the pain shooting through her ribs. Marjan led her by the elbow to the living room. She stopped short in the hallway.\n\n\"Shekiba, listen to me. You are a girl without mother or father, without brothers or uncles to look after you. Obey the word of God and let Him look after you. Bring your head out of the sky and understand your place in this world.\"\n\n\"I have no place in this world, Khanum Marjan.\"\n\nMarjan felt a chill run through her spine. Shekiba's words were cold, resolute. She wondered if this half-crazed girl had finally gone completely mad. Zarmina's warnings echoed in her mind and she decided to keep her mouth shut. If Shekiba was going into a frenzy, she didn't want to invite her wrath.\n\nAzizullah was standing at the door to the courtyard, putting a green and blue vest on over his tunic. His face and voice were stern.\n\n\"If this girl has any sense in her at all, she will give me no trouble tonight. And if she dares to walk with even the slightest limp, I'll take both her legs off.\"\n\nThe warning was issued. Marjan bit her lip and handed Shekiba her burqa. Shekiba slipped it over her head and followed behind her master with a resigned step.\n\nEvery footstep jolted her bruises and welts. Shekiba kept pace, though, too hurt to risk more punishment. Within twenty minutes, they approached a home with horses and armed soldiers outside. The horses were tall and muscular; their tails flicked side to side casually. But what caught Shekiba's eye was what stood behind them. For the first time in her life, Shekiba saw a carriage. Four large wheels, a cushioned seat and handsome carvings on the sides.\n\nThe king, she realized.\n\nThey entered the front gate and walked into a courtyard nearly twice the size of Azizullah's. Shekiba could not help but look around. There were benches and several bushes with striking purple flowers. From the living room came the sound of men laughing loudly.\n\nShe walked around to the back of the house to enter into the kitchen area.\n\n\"Stay outside, in the back. Behave yourself or I'll let the soldiers straighten you out.\"\n\nAzizullah went in through the living room door and rejoined the gathering. Shekiba closed her eyes and tried to eavesdrop on their conversation. The sky grew dim before she heard something that actually pertained to her.\n\n\"We will be leaving in the morning to head back to Kabul. The road ahead of us is long but we hope to reach home by nightfall.\"\n\n\"Amir-sahib, you and your esteemed generals have honored us with your visit to our humble village. We wish for many more visits in the future.\"\n\n\"With the roads project, travel will become easier. We anticipate that your village will be more involved in the agriculture projects that have begun. Amir-sahib has a new team of engineers that are looking at our current situation.\"\n\n\"Anything that we can do here to assist you, we are at your service. I was born and raised in this village, as was my dear brother, Azizullah. Our roots here are respected by the village and we can serve as your delegates for anything you may need.\"\n\n\"You have made that clear, Hafizullah-sahib. Your sentiments are appreciated.\" The voice was gruff and Shekiba detected a slight exasperation in it.\n\n\"I hope so, General-sahib. And I hope that you will accept my brother's gift to the amir-sahib. It is a small token.\"\n\n\"Yes, he mentioned this earlier. The servant will ride with our entourage in the morning to be taken to the palace.\"\n\n\"Wonderful. Please, General-sahib, your journey tomorrow is long and you will need your strength. Have some more sweets . . .\"\n\nHafizullah's wife came to the courtyard and found Shekiba slumped across a bench. She was a petite woman, her face lined with worry and fatigue. By the looks of her, she had done most of the preparation for the king's visit. She clucked her tongue in dismay.\n\n\"Merciful Allah. Follow me, girl. I will show you where you can sleep until you leave in the morning.\"\n\nShekiba slid to the floor in the corner of a dark room. She could see two small figures curled up and breathing softly. These were Hafizullah's daughters, but Shekiba never did meet them. In the early hours of morning, the mistress of the house came to wake her. Shekiba bolted upright when she felt a hand on her shoulder.\n\n\"Wake up. The men are leaving.\"\n\nShekiba focused. She heard the sounds of horses, men chattering outside the house.\n\nShe rose, made sure her Qur'an was tucked into her dress and walked outside to be taken to her new home.\nCHAPTER 21\n\nThere was barely enough room in our small home for Abdul Khaliq's family. They wanted to hold all three nikkahs at the same time and brought with them Abdul Khaliq's mother, a gray-haired woman with downturned lips and narrowed eyes. She needed a walking stick but refused to use one, preferring to lean on her daughter-in-law's forearm instead. They also brought Haji-sahib, a mullah. Khala Shaima scoffed at the mention of his name.\n\n\"Haji-sahib? If he's Haji, then I'm a pari!\" said Khala Shaima, whom no one would describe as an angel from heaven. The title haji was given to anyone who had made the religious pilgrimage to Mecca, God's house. Haji-sahib, Khala Shaima reported, had dubbed himself with the title after paying a visit to a shrine north of our town. But as a dear friend of Abdul Khaliq, no one contested his credentials. The two men chatted amicably outside.\n\nShahla kept her head down and pleaded with my crying mother not to give her away. Madar-jan's body shook, her voice trapped in her clenched throat. Shahla was more than a daughter to her. She was Madar-jan's best friend. They shared the housework, the child care and their every thought.\n\nParwin was her special girl. Part of Madar-jan had held on to Khala Shaima's prediction that no one would want Parwin as a wife. Sometimes it comforted her that she would have her singing, drawing daughter with her always.\n\nAnd me. I was Madar-jan's helper. Her spunky, troublemaking bacha posh. I know she wondered if she had made the right decision. If I were a little wiser, I would have told her it had been the best thing for me. I would have told her that I wished I could have stayed a bacha posh forever.\n\nThe family was here to claim their three sister brides. We listened to hear what Khala Shaima would say.\n\nHaji-sahib started with a prayer. Even Madar-jan cupped her hands and bowed her head to join in. I was pretty sure everyone was praying for different things. I wondered how Allah would sort it all out.\n\n\"Let us begin with a dua, a prayer. Bismillah al-rahman al-raheem . . .\"\n\nThe room echoed behind him. Haji-sahib, the mullah, went on to recite a sura from the Qur'an.\n\n\"Yaa Musabbibal Asaabi.\"\n\nAfter a moment, we heard Khala Shaima interrupt.\n\n\"Yaa Musabbibal Asbaabi.\"\n\nThere was a pause. The room had gone silent.\n\n\"Khanum, did you have reason to interrupt Haji-sahib?\"\n\n\"Yes, I did. Mullah-sahib is reading the sura incorrectly. Oh causer of the causes, the verse is meant to read. Not causer of the fingers. I'm sure he would want to know he was making such an egregious error, wouldn't you, Haji-sahib?\"\n\nThe mullah cleared his throat and tried to pick up where he had left off. He thought hard but recited the verse the exact same way, error and all.\n\n\"Yaa Musabbibal Asaabi.\"\n\nKhala Shaima corrected him again.\n\n\"Asbaabi, Mullah-sahib.\" Her tone was that of an annoyed schoolteacher. It didn't go unnoticed.\n\nI feared Padar-jan would make good on his threat to cut out Khala Shaima's tongue. I was nervous for her.\n\n\"Shaima-jan, please have a little respect for our esteemed mullah here,\" Boba-jan said.\n\n\"I have the utmost respect for him,\" she said facetiously. \"And I have the utmost respect for our Qur'an, as I'm sure you all do. What a disservice it would be for us to recite the verse incorrectly.\"\n\nOnce more, the mullah sighed and cleared his throat.\n\n\"Yaa Musabbibal Asbaabi Yaa Mufattihal Abwaabi.\"\n\n\"That's better,\" Khala Shaima interrupted loudly. I could hear the satisfaction in her voice.\n\nWe could hear the men beginning the nikkah in the next room. Padar-jan was giving his full name, his father's name and his grandfather's name to be written on the marriage contract.\n\nParwin tried to put on a strong front, seeing Madar-jan's condition. Khala Shaima, our only advocate in the nikkah, had strategically positioned herself between my grandfather and Abdul Khaliq's mother. No one knew what to make of her presence. Padar-jan huffed in frustration but thought it best not to make a scene in front of his guests.\n\nMadar-jan spoke softly. We had formed a tight circle in the next room.\n\n\"My daughters, I prayed this day would not come so soon for you but it is here and I'm afraid there's nothing I or Khala Shaima can do to stop this. I suppose this is God's will for you. Now, I haven't had much time to prepare you, but you are young women,\" she said, hardly believing her own words. \"Your husbands will expect things of you. As a wife, you have an obligation to your husband. It won't be easy at first but . . . but with time you'll learn how to . . . how to tolerate these things that Allah has created for us.\"\n\nWhen Madar-jan began to cry, we cried as well. I didn't want to know what it was Madar-jan was talking about. It sounded like it was something terrible.\n\n\"Please don't cry, my girls. These things are a part of life\u2014girls are married and then become part of another family. This is the way of the world. Just as I came to your father's home.\"\n\n\"Can I come back sometimes, Madar-jan?\" Parwin asked.\n\nMadar-jan exhaled slowly, her throat thick and tight.\n\n\"Your husband will want you at home but I hope that he is a man of heart and will bring you here from time to time to see your mother and your sisters.\"\n\nThis was as much as she could promise. Parwin and I sat on either side of our mother, her hands stroking our hair. I had my hands on her knee. Shahla kneeled in front of us, her head resting on Madar-jan's lap. Rohila and Sitara watched on nervously, Rohila understanding that something was about to happen.\n\n\"Now, my girls, there's one more thing. There will be other wives to deal with. Treat them well and I pray they will show kindness to you. Older women are spiteful toward younger girls, so be careful how much you trust them. Make sure you take care of yourselves. Eat, bathe, say your prayers and cooperate with your husbands. And your mothers-in-law. These are the people whom you will need to keep satisfied.\"\n\nA voice bellowed from the next room.\n\n\"Bring the eldest girl! Her husband, Abdul Sharif, is waiting. May their steps together as husband and wife be blessed. Congratulations to both your families.\"\n\n\"Shahla!\" my father called out unceremoniously.\n\nShahla wiped the tears from her face and bravely pulled her chador over her head. She kissed my mother's face and hands before she turned to us, her sisters. I squeezed my sister and felt her breath in my ear.\n\n\"Shahla . . . ,\" was all I could get out.\n\nIt was Parwin's turn next. They started over again, a new contract. For the sake of tradition, they repeated all the same questions, wrote down all the same names.\n\n\"Agha-sahib,\" Khala Shaima interrupted again. \"Allah has given my niece a lame leg and I can tell you better than anyone else that it is not easy to manage with such a disability. It would be in this girl's best interests for her to have some time to go to school, to learn to manage physically, before she is made into a wife.\"\n\nAbdul Khaliq's father was taken aback by the sudden objection, as were the others in the room.\n\n\"This has been discussed and I think my nephew has been more than generous in agreeing to give this girl a chance to be the wife of a respected man. School will not fix her lame leg, as it has not fixed your hunched back. Let's continue.\"\n\nThe nikkah resumed.\n\n\"Bring the girl! May Allah bless this nikkah and Abdul Khaliq, who has made this possible. May God give you many years, Abdul Haidar, for agreeing to take on a wife in the tradition of our beloved Prophet, peace be upon him. And a disabled wife at that; truly you are a great man, Abdul Haidar. What a relief this must be for your family, Arif-jan.\"\n\nMadar-jan kissed Parwin's forehead and stood up slowly, as if the ground was pulling her back. Parwin stood up and straightened her left leg as best she could. Madar-jan whispered to Parwin things she hated to say.\n\n\"Parwin-jan, my sweet girl, remember to do your chores in your new home. There may not be time for drawing, and sing softly and only to yourself. They'll say things to you, just as the others always have, about your leg, but pay no attention, my daughter.\"\n\n\"Agha-sahib, you are keeping this man waiting. Please bring him his new bride,\" the mullah ordered.\n\n\"Bring her out!\" My father's voice was cold and loud as he tried to assert control. Madar-jan's delaying made him look small in front of the mullah and Abdul Khaliq's family, as if Khala Shaima's behavior hadn't been enough.\n\n\"Please, my sweet daughter. Remember these things that I've told you. May Allah watch over you now,\" she whimpered, brushing away Parwin's tears and then her own. She fixed Parwin's chador and had her hold it close under her chin before she turned her around and led her down the hallway and into the living room, where she became the wife of a man as old as my father.\n\nI sat in the room with Rohila and Sitara. I listened to Parwin try to mask her limp, lifting her left leg so it wouldn't drag along the floor as it usually did. Our cousins always teased her, as did the children in the neighborhood. Even for those few months when she attended school, her classmates had mocked her gait and the teacher had doubted she would learn anything, as if walking and reading were related. They wouldn't treat her well, we knew. Our hearts broke for her.\n\n\"Rahim, where is Parwin going?\" Sitara asked.\n\nI looked at my youngest sister. She still called me by my bacha posh name.\n\n\"It's Rahima,\" Rohila reminded her. Her vacant eyes stayed glued to the door, willing Parwin to come back.\n\n\"Rahima, where did Parwin go?\" Sitara asked again.\n\n\"She's . . . she's gone to live with a new family.\" I couldn't say words like \"marriage\" or \"husband\" in the same sentence with my sister's name. It sounded awkward. Like a little girl wearing her mother's shoes.\n\nI knew my mother was watching Parwin from behind the doorway. Their voices faded as they walked out the door. I went to the window to see my sister one last time. Because of her limp, she was shorter than any other fourteen-year-old girl and looked to be half the size of her new husband. I shuddered to think how she would feel to be alone with him.\n\n\"When will she come back?\"\n\nI looked at my sisters blankly. Madar-jan returned, drained. I was next. Khala Shaima had not succeeded in saving my sisters from Abdul Khaliq's family. I knew I shouldn't hope for any better, but I did.\n\nI wish I could say that I put on as strong a front as Shahla or even Parwin, at least for my mother's sake. I wish I could have done something. After all, I'd been a boy for years. Boys were supposed to defend themselves and their families. I was more than just a girl, I thought. I was a bacha posh! I had been practicing martial arts with my friends in the streets. I didn't have to crumple as my sisters had.\n\nMy father had to drag me from my mother's arms while I cried, the chador falling from my head and revealing my absurdly short hair. Abdul Khaliq's family watched in consternation. This didn't bode well. My father dug his fingers into my arm. I only know because I saw the bruises later.\n\nI tried to pull my arms away, kick my legs, twist my body away. It wasn't the same as play-fighting with the boys. My father was stronger than Abdullah.\n\nAll we managed to do was embarrass my father. My mother sobbed, her hands in powerless fists. Khala Shaima shook her head and shouted that this, all of this, was wrong, a sin. She didn't stop until my father slapped her across the face. She reeled backward. Our guests looked on, feeling it was well deserved. My father had redeemed himself in their eyes.\n\nMy struggle changed nothing. I just made it harder on my mother. And Khala Shaima.\n\nMy father handed me over to my new husband. My mother-in-law stared with a critical eye. She would have a lot of work to do to set me straight.\n\nAnd Abdul Khaliq, my new husband, smirked to see me squirm under my father's grip. As if he liked what he saw.\n\nThat was my wedding.\nCHAPTER 22\n\n\"First things first. You need a proper bath.\"\n\nShekiba stood before a heavyset woman with cropped dark hair. She looked to be in her twenties. She wore ballooned pants and boots with a button-down shirt. If it weren't for her voice, Shekiba would have believed her to be a man. As it was, Shekiba was baffled and had been since Kabul came into view.\n\nNever could she have imagined such a place. All the homes and shops of her village could have fit in Kabul's belly. There were streets lined with stores, striped awnings and men walking through the maze of roads. There were houses with colorful doors at the front gate. People turned and raised their hands, a respectful acknowledgment of the king's entourage passing through. Kabul was a spectacle!\n\nWhen the royal compound came into view, Shekiba's mouth gaped. The gated entrance was flanked by stone pillars, layer after layer before the palace itself came into view. Through the main entrance, a wide path encircled an imposing tower. Shekiba craned her neck to get a good view.\n\nThat tower just about reaches the heavens!\n\nThe palace's fa\u00e7ade was embellished with carvings and arches, polished and bright. Bushes and greenery lined the path, including the portico that cut through the tower. The palace was an impressive structure with more windows than she had ever seen and incomparable in size to any home Shekiba had ever beheld.\n\nSoldiers guarded every corner. It was only when they came to the entrance of the palace that Shekiba actually saw King Habibullah. On the ride to Kabul, he had been at the head of the caravan, riding in the magnificent carriage that had been stationed outside Hafizullah's house. When they disembarked, Shekiba was sent in a different direction but she caught sight of him entering a main door.\n\nThat's the king, Shekiba thought.\n\nHe was a stocky man with a thick beard. He wore a military uniform with a row of medals pinned across his left chest and tassels at his shoulders. A broad yellow sash crossed from his right shoulder to his left hip and covered some of the stars on his jacket. A striped belt and medallion clasp sat snugly across the middle of his belly and a tall hat of sheep's wool added five inches to his stature. The soldiers stood at attention for King Habibullah's return.\n\nShekiba wondered if she would ever cross paths with him in this enormous place.\n\n\"Follow me.\"\n\nA soldier took her around the corner, behind the palace, where the path opened into a verdant and majestic courtyard. Shekiba's eyes widened. The courtyard had small ponds, flowering bushes and fruit trees. They followed a footpath that led to a smaller stone house, still much larger than even Agha Azizullah's home. The soldier knocked on the door and a guard answered.\n\n\"Take her. She is to be a guard with you. Fix her up.\" The guard nodded and waited for the soldier to turn before the door opened wide.\n\n\"Come in.\"\n\nA woman! Shekiba stood motionless.\n\n\"I said come in! What are you doing standing there?\"\n\nShekiba's feet unfroze and she followed the woman-man into the room. There were three women sitting on cushions around the floor, each older than Shekiba but younger than any of her uncle's wives. They had stopped their conversation when she entered. Shekiba noticed four other guards in the room. Were they women too?\n\n\"Well, let's take a look at you.\" She lifted Shekiba's burqa and took a step back. \"Well, well. That's quite a face. I suppose that's why you were sent here. Ladies, this is our newest guard.\"\n\nShekiba's surprise grew when she learned all of the guards in this house were actually women dressed in men's clothing. Ghafoor seemed to be in charge of the five guards. It was evening and she could see the exhaustion in Shekiba's face. Ghafoor had her rest for the night and told her work would begin in the morning. For the first time in a long time, Shekiba slept soundly, surrounded by women pretending to be men.\n\nHer transformation started at daybreak. Ghafoor led Shekiba to the wash area and cut her thick, knotted hair. She was instructed to bathe and given a set of clothing identical to what Ghafoor wore. Shekiba stared in wonder at the pants and could scarcely believe she should walk about in them. She slipped one leg in and then the other, fastening the buttons at the waist. She was given a corseted undergarment that pushed her modest bosom flat against her chest. She slipped her arms into the shirt and buttoned it closed. The boots felt heavy. Shekiba stood and stared down. Then she reached up and ran her fingers through her short hair.\n\nShe took two steps and turned. Her legs felt loose and she blushed when she looked down and saw the crotch of her pants. Her hands ran over her backside and she shuddered to think the shape of her limbs would be so visible in these ballooned pants. She had only ever seen women in skirts, draped enough to disguise the curves and crevices that hid underneath.\n\nAnd yet there was something liberating about her new clothes. She lifted her right leg and then her left. She thought of her brothers and how they would run about the fields in their flowing pants.\n\nGhafoor understood.\n\n\"It is awkward at first, but you'll adjust quickly. The uniform is comfortable enough with time.\"\n\n\"What are we guarding?\"\n\nGhafoor laughed. \"They've told you nothing? We are guards for King Habibullah's women.\"\n\n\"His wives?\"\n\n\"Not exactly. His women. These are women he spends time with, women he takes when he is struck by the mood.\" Shekiba must have looked confused. \"Men can take more than just their wives, dear girl. Sometimes wives are not enough.\"\n\nShekiba was certain she did not understand but kept her mouth shut for the time being.\n\nGhafoor looked at her thoughtfully.\n\n\"What happened to your face?\" she asked.\n\nShekiba looked down instinctively. \"I was burned as a child.\"\n\n\"Hmm. And where is your family?\"\n\n\"My village is one day's travel from here. My mother and father are dead. My brothers and sister are dead.\"\n\nGhafoor's brow furrowed. \"You have no other family?\"\n\n\"They gave me away to repay a debt. And that man gave me away to the king.\"\n\n\"And now you are one of us. Welcome, Shekiba. But here you will be Shekib, understand? Now let me introduce you to the others.\"\n\nFour women-men guarded the king's harem. Shekiba found herself staring at their faces as so many others had stared at hers. But with good reason. Ghafoor was actually Guljaan. She was the leader of the group, not only because she was tallest and loudest but also because she had been in the palace longer than the others. She was the most content with her role and seemed to take pride in doing a good job. Her face was smooth, but a fine, downy rim on her upper lip and untamed brows gave her the appearance of a young man, fresh with enthusiasm for his important post.\n\nGhafoor came from a modest family in a nearby village and had been given to the palace in exchange for a cow. It was midafternoon and her mother had been busy with her younger siblings. Ghafoor's father had interrupted her needlework. We are going to visit your grandmother, he had said. Ghafoor wondered why the others were not coming but shrugged her shoulders and followed her father two kilometers down the road, where she was delivered to a man dressed in a gray tunic and pants. Her father warned her sternly to follow the man's directions and turned to walk the two kilometers back to their family. She cried and screamed when she realized she would not see her mother or siblings again.\n\nGhafoor was brought to the palace and watched as a guard brought out a cow for the man in gray. It was a decent cow, not too sickly looking and plenty to satisfy her family's needs. She realized immediately what her father had done and wondered if her mother had been privy to the plan. She cursed him for his deceit and feared what would become of her, an adolescent girl, in the hands of strangers.\n\nIt did not take long, however, for Ghafoor to appreciate her father's barter. She missed her mother and siblings terribly but life behind the palace walls, even for a servant, was easier than life at home. The beatings were fewer, the food more plentiful, and she had taken on some authority.\n\nThe king needed guards to watch over his harem, but he believed no man to be above temptation. For months he paced and debated, the dilemma as perplexing as the tribal disputes in Kurram Valley. When an adviser came up with a plan to dress women as male guards, the king rewarded him for his stroke of genius and had him fill the positions as quickly as possible.\n\nGhafoor enjoyed the comfort of palace life. All she had to do was give up being a woman, an easy trade. Two other girls were recruited along with her, but they lasted only two or three months. One had argued with a woman of the harem and Benafsha, the other, had been so beautiful that the king took an immediate liking to her and decided she should be guarded as well. She was made to grow her hair long again and reassigned to her new position as a concubine.\n\nThen came two sisters, Karima, who would become Karim, and Khatol, who would be renamed Qasim. This time the king's representatives chose more wisely, recruiting girls who were tall enough to pass for men but homely enough that they would not tantalize the king. Karim and Qasim came from a family of four girls. Their mother cried violently as she told the girls they could not afford to feed all four and that their father had arranged for them to be taken to the king's palace, where they would have a much better life. The obedient girls had tearfully accepted their parents' decision and left home hand in hand.\n\nKarim was two years older and looked after her sister. She quickly overcame her timidity and became second in command, arguing with Ghafoor so that she would not dominate them completely. Qasim was quieter and missed the family. She was taller than her sister by an inch but hunched her shoulders, prompting Ghafoor to poke her repeatedly in the back until she learned to stand as a guard should.\n\nTariq, the newest addition, was different from the others. She carried out her duties well enough but fantasized that she would be noticed by the king and recruited to his court of women. She was the shortest of the group and plumper in the face, with chestnut hair that she had been told no man could resist. She would not say where the compliment had come from but she refused to let the defeminizing uniform spoil her chances. She made sure her hips swayed when she walked and batted her eyes when the king neared. Of all the women in the harem, she guarded Benafsha most, feeling kinship with the former guard who had enticed the king.\n\nGhafoor and Karim rolled their eyes at her often but tolerated her occasional fantasies. Every guard had her own way of coping.\n\nGhafoor introduced Shekiba to a few of the king's concubines, the women who kept the king satisfied. Benafsha was the youngest of the group. She knew why Tariq favored her over the others but refused to indulge any details of the king. Whenever Tariq asked her about the monarch, she would shake her head and adjust her skirt. She was lightest in complexion and her eyes were light brown with speckled irises. Tariq could see why she had attracted the king's attention. She was the most beautiful, now that Halima's face had begun to show her years.\n\nHalima, the eldest of the group, had borne the king two daughters over the years. The girls were two and four years old and bore a striking resemblance to their mother. Halima stroked their hair and sighed wistfully, realizing the king beckoned to her less often and wondering what that would mean for her and her daughters. Halima was kind and motherly and tempered the bickering of the others.\n\nBenazir, the darkest, had ebony eyes that teared easily these days. She was with child and terrified. Her belly had just started to swell but she had been ill for weeks, unable to keep down more than a few mouthfuls of rice at a time. She would stare at the walls and started when Halima put a hand on her shoulder.\n\nSakina and Fatima were feistier girls, but less beautiful than the others. Fatima had borne a son, which gave her an edge over the others. They were friendly enough, but unlike good-natured Halima, they were usually the instigators of any turbulence in the harem. Sakina in particular despised Benafsha, knowing that her ranking in the harem had dropped notches with the temptress's arrival. And Benafsha knew how to throw that fact in Sakina's face when she needed. Shekiba knew to keep her distance from these two, her instincts telling her they would be unforgiving in their comments about her face.\n\nThere were others, she was told. She would see more tomorrow.\n\nHarem life was relatively simple. Shekiba listened in amazement to hear what the women did. And, more important, what they did not do. They did not cook, nor did they carry buckets of water from a well. They did not tend to animals or spend hours peeling vegetables.\n\n\"Who does all the housework then?\" Shekiba asked Ghafoor as they watched Sakina and Benazir rouge their cheeks and stain their lips with crushed cherries.\n\n\"The people for the housework. Everyone has a purpose here in the palace. The guards, the servants, the women, and us. We all do our part in Arg.\" Ghafoor sat with her right ankle crossed over her left knee. She was comfortable as a man.\n\n\"Arg?\"\n\n\"Arg-e-Shahi. You do not know what Arg is?\" Ghafoor laughed with the self-satisfaction of someone who had once been as ignorant. \"This is Arg-e-Shahi, the palace! Arg is your new home, Shekib-jan!\"\nCHAPTER 23\n\n\"Take off your chador.\"\n\nI kept my face to the wall and pulled my legs in under me. The room was small enough that I could hear each raspy breath.\n\nAbdul Khaliq stood in the doorway, his hands on his hips. From this angle, he looked larger than life. He took two steps in and shut the door behind him.\n\n\"I said, take off your chador.\"\n\nI lowered my head and told myself to breathe. I prayed he would be frustrated and walk away, as he had yesterday.\n\n\"I will not tolerate insolence. Yesterday, I let you be. That was my gift to you, to show you I can be kind. Today, things are different. You are in your husband's home, my home. You will behave as a wife should.\"\n\nI was sharing a house with Abdul Khaliq's third wife. I was his fourth. The other wives lived in separate homes within the same compound, all interconnected. It had been nearly dark when we got to the compound and I hadn't seen much. Bibi Gulalai, his mother, had insisted on using me as a cane to get to the car. She was old and I was not rude enough to refuse, though I only answered her questions with one-or-two-word responses. She was sizing me up.\n\nBibi Gulalai led me to a small room at the end of a hallway. This was to be my room, she said. There was a bathroom just outside my door, the likes of which I had never before seen. It was modern, with running water and a toilet.\n\nWife number three was Shahnaz. I saw her for just a moment before I was ushered into my room. She turned her back to me and walked away, uninterested in introductions.\n\n\"That's Shahnaz. You'll meet her in the morning when she shows you around.\"\n\nMy room had a cushion in the corner, a pillow and a small table.\n\n\"We'll send you a plate of food for tonight. Tomorrow you become part of your new home,\" Bibi Gulalai said smugly.\n\nI doubted it.\n\nI had nearly screamed yesterday when Abdul Khaliq entered the room. I was crouched in the corner. He wiped the grease from his mouth with the back of his hand. He had just finished eating. My plate was untouched.\n\n\"You haven't eaten? My wife is not hungry, eh?\" He chuckled.\n\nI said nothing.\n\nHe squatted next to me and lifted my chin with two fingers. His touch was rough. I kept my gaze averted. He pulled my chador off my head and felt the back of my head.\n\n\"Tomorrow,\" he promised, and walked back out of the room. I shook with fright.\n\nNight came and went and I didn't sleep. I tossed and turned on the mattress, listening for the sound of footsteps, a hand on the doorknob, a knock. I thought of my mother, my sisters. I wondered if Shahla and Parwin were close by. I prayed we were all in the same compound and I would see them in the morning, every morning. I wondered what Rohila was telling Sitara, who every day had been asking more questions that we couldn't answer. I wished I could be laid out at Khala Shaima's feet, listening to her tell another chapter of Bibi Shekiba's story.\n\nI wished more than anything that I could be back in class, Moallim-sahib's back turned toward us, Abdullah and I shooting each other bored looks, kicking each other under the table and tilting our notebooks so the other could see the right answer.\n\nI wished I could be anywhere but here.\n\nWhen my bladder could wait no longer, I opened the door a crack. I looked into the hallway, saw that it was empty and crept out slowly to go to the washroom. Shahnaz caught me on the way out.\n\n\"Good morning,\" she said plainly. She looked a few years older than Shahla, with features that matched the dullness in her voice. She was thin and stood a couple inches taller than me. She balanced a baby on her hip, no more than six months old.\n\n\"Salaam,\" I replied cautiously. I knew who she was and I remembered my mother's warnings.\n\n\"Your name is Rahima?\"\n\nI nodded.\n\n\"All right, Rahima. Bibi Gulalai has asked me to show you around. So, let's get started. You've hid in your room long enough.\"\n\nShahnaz looked disinterested in me but she'd been given a task, and as Madar-jan had advised, she was doing what her mother-in-law\u2014our mother-in-law\u2014had asked of her.\n\n\"This has been my home for three years. I was told I wouldn't be sharing it with anyone else. This room is for my children and me. Here is the kitchen. That is our living room. That hallway leads to the rest of the houses, the better houses. I expect that you'll do your share of the cooking and cleaning. As you can see, I've already got my hands full.\"\n\nShe paused and looked at me carefully.\n\n\"Your hair. Why is it cut so short?\"\n\n\"I'm a bacha posh. I mean, I was a bacha posh.\"\n\n\"I've never seen a bacha posh before. Why were you made into a boy?\"\n\n\"My mother had only daughters and my father wanted a son.\"\n\n\"So they dressed you as a boy? And did you go out of the house like that?\"\n\nI could hear more curiosity than dislike in her voice. It gave me confidence to continue the conversation. Something about her reminded me of Shahla and I could already tell I would be desperate for an ally here.\n\n\"Sure. I went to school. I ran errands for my mother. I even worked and brought money home. I was learning how to fix electronics,\" I boasted. That was more than I had done for Agha Barakzai but Shahnaz wouldn't know the difference.\n\n\"Well, don't expect to be treated like the special son here.\"\n\nAs soon as she said it, I realized that's what I had secretly been hoping.\n\n\"Who else lives here in the compound?\" I asked, hoping my face didn't show my disappointment. The baby started to whine, her small hands batting at her mother's face.\n\nShahnaz led me into the living room, where she began to nurse the baby.\n\n\"Our home is one of three. Each wife has her own home. Or at least we did, until you came along. His first wife is Badriya. She has the biggest home, with the bedroom on the second level. His second is Jameela. She lives in the biggest part of the house too but on the lower level. Abdul Khaliq's room is in that main house. I thought you would have seen it last night but I'm sure you'll see it soon enough.\"\n\nI ignored her last comment, scared to think of what that meant. The memory of his touch made me shiver.\n\n\"Where does . . . where does Bibi Gulalai live?\"\n\n\"In the compound next door but she's here often, keeping an eye on her eldest son's affairs. Especially since he's gone so often. Be careful with her. She rules with a heavy hand.\"\n\n\"And what about the rest?\"\n\n\"What rest?\"\n\n\"I mean his cousins, Abdul Sharif and Abdul Haidar?\" I was nervous to ask. I prayed she would tell me they were next door as well.\n\n\"Oh, I heard what happened. So, it's true then? Sometimes Safiya gets the story all wrong. She told me two other sisters were married at the same time. And one of them has a limp, right? Hard to imagine how they arranged that deal. Well, Abdul Sharif lives on the other side of the hill, about four kilometers away. Abdul Haidar lives on the other side of that wall. He's here often since he's Abdul Khaliq's right-hand man.\"\n\nParwin was close by! She was on the other side of the wall. I wondered what she was doing and if she knew that I was meters away from her. Shahla. Shahla had been taken the farthest.\n\n\"Does Abdul Sharif come here sometimes?\"\n\n\"He does, but not as often as his brother. If you think you'll see your sisters, though, don't get your hopes up. Neither one of them brings their wives when they come by. The women of this family don't travel much. Get used to these walls. They're going to be all you see.\"\n\nShahnaz tired of me and went to put the baby to sleep. She had two children, a two-year-old son and the five-month-old girl I'd seen her holding.\n\nI found out weeks later that Abdul Khaliq had taken her from a village in the south. He and his men had gone there and successfully pushed back the Taliban forces. The village had been saved so Abdul Khaliq and his men felt they earned the right to take what they wanted. They looted houses, harassed women. The village had no one to defend it. Most of the men had perished in the war. The men took whatever caught their eye. In Abdul Khaliq's case, it was Shahnaz. She hadn't seen her family since the day of her nikkah.\n\nIt could have been worse, she said. At least he took her as his wife. She had heard of many women who had been raped and left with their families. There was nothing worse than that.\n\nI thought about Shahnaz's village often, knowing my father must have been party to that mission. I wondered if he pillaged as the others had. I wanted to believe he hadn't.\n\nI could start with cleaning, Shahnaz said. She needed to bathe her son. I found the broom and began to sweep the floors as I'd seen my sisters do. The broom felt awkward in my hands and I waited for someone to relieve me from the duty. When Shahnaz didn't come back out, I put the broom away and went back to my room to pout. I missed my old life.\n\nBefore long, it was evening again. Bibi Gulalai came to eat with us, around the cloth laid out on the living room floor. Shahnaz had prepared a meal of stew and rice. I reminded myself to fold my legs under me and sit like a lady. I could feel my mother-in-law watching me. I helped Shahnaz clear the dishes and wash up before I went back to my room. Bibi Gulalai sat in the living room with her cup of tea, watching her grandson play with a wooden spoon.\n\nI listened for the sound of her leaving but she didn't. My door opened.\n\n\"Your husband has asked for you. You should go and see to him as his bride. Shahnaz will take you there.\"\n\nWhen I didn't get up, she came after me, pulling me to stand by my ear.\n\n\"Did you not hear what I said? Do you want me to repeat myself?\"\n\nMy twisted ear stung under her gnarled fingers. I yelped and stumbled to my feet. Shahnaz was in the hallway. She looked mildly entertained.\n\nWe went down the hallway and into the main house. Had I been less nervous, I probably would have noticed more of my surroundings. I remember thinking that the hallways were wide, the ceilings tall. We passed by many doors. I'd never imagined a house so large!\n\nShahnaz pointed to a door and told me to go ahead and knock. Before I could ask a question she turned and headed back down the stairs. I ran after her and grabbed her arm.\n\n\"Shahnaz, please, let me go back with you!\"\n\nShe shook her arm free and looked at me with annoyance.\n\n\"Let go of me!\" she hissed. \"Your husband has asked for his new bride. You'd be making a big mistake to keep him waiting. That's my best advice to you.\"\n\n\"Please, Shahnaz-jan! I'm scared!\" I panicked. I didn't want to be alone here. I wanted to go back to my dark room and my small mattress. I felt out of place and I hated wearing a dress. It felt unnatural, awkward. I was a bacha posh! Just like Bibi Shekiba, the palace guard!\n\n\"Are you stupid? Get in there or you'll regret it. You'll be punished worse than you could imagine.\"\n\nShe walked away and left me in the hallway, scrambling for options that didn't exist.\n\nHe must have heard me. I gasped and jumped backward when the door opened. My reaction made him smile. He beckoned me in. I hesitated, but fearing that Shahnaz was right, I followed.\n\nIn subsequent visits, I would realize that Abdul Khaliq's bedroom looked like what I might have imagined of a palace. His mattress sat on a wooden platform a few feet off the ground. A plush armchair sat in the corner and a beautifully woven burgundy carpet covered the floor. Two windows overlooked the courtyard, where three armed men were on guard.\n\nI walked in, too terrified to see anything but Abdul Khaliq. He had already made himself comfortable on his bed. He was sitting up, propped up against pillows.\n\n\"Take your chador off,\" he ordered.\n\nI looked at the ground and stood motionless. I had wanted to rip the chador off my head when Madar-jan put it on me but now, with Abdul Khaliq eyeing me in this way, I couldn't let it go. I watched him from the corner of my eye and saw his intrigued but exasperated face.\n\n\"Listen,\" he said, leaning forward. With his turban off, I could see that his hair matched his salt-and-pepper beard. He wore a beige cotton tunic and pants. His legs were outstretched. The room was lit by a lamp on his bedside table. \"Maybe you haven't received any instruction on what it is to be a wife. From what I've seen of the women in your family, I wouldn't be surprised. Let me explain to you how things are here. I am your husband and this is your home. When I ask for something, you make it happen. In return, you will be given shelter and have the privilege of being wife to Abdul Khaliq.\"\n\nAgain he beckoned me closer. I fought the wave of nausea and took two steps toward him. I was within his reach. My muscles stiffened.\n\nHe turned my face toward him. He was so close I could see the lines on his face. I could make out each hair of his eyebrow. I tried to keep my eyes lowered.\n\n\"Do you understand what I'm saying?\"\n\nI nodded. My mind flashed back to his bodyguards and their guns. I was terrified.\n\n\"Good. Now, do as I say and take off your chador.\"\n\nHe could have done it. I thought about it later and realized he could have done all the things he made me do, but that wouldn't have served his purpose. One by one, he made me take off everything I'd been wearing. First the chador, then my socks, my pants, my dress. With every piece, I trembled more. When my pants came down, I began to cry, which didn't faze him in the least. I was humiliated. I stood before him, weak and vulnerable, my arms doing their best to cover as much as they could.\n\nHe nodded in approval, his lips wet with excitement.\n\n\"You're not a bacha posh any longer. Tonight I'll show you that you're a woman, not a boy.\"\nCHAPTER 24\n\nThe thought of him made me queasy. I hated the feeling of it. I hated his breath, his whiskers, his callused feet. But there would be no escape. He called for me when he pleased and made me do what he wanted. Thankfully, it rarely lasted more than a few minutes. I wished Madar-jan had told me exactly what to expect, but then I think if she had, I never would have made it to the nikkah.\n\nShahnaz seemed to pity me the following day. She must have known. My face reddened when my eyes met hers.\n\nMy insides hurt. Raw and angry. I nearly cried when I urinated into the fancy western toilet.\n\nShahnaz asked me to prepare lunch for the family. She had the children to tend to. I went into the kitchen and looked through the vegetables on the counter, almost thankful to have a task that would keep my mind off what I had endured. There were canisters of flour and sugar as well. I thought of my mother and sighed. Ever since I'd been converted into a bacha posh, I'd been relieved of all cooking duties as well. If my father had seen his \"son\" working in the kitchen, his temper would have turned our home upside down. I had no idea how to make even a simple meal.\n\nI tried to think of the foods my mother and Shahla cooked. Even Parwin could prepare a decent meal, although she spent more time sculpting shapes out of the potatoes than she did actually cooking them.\n\nI set out to make some potato stew. I put the rice in water, as I'd seen my mother do. I tried to focus but my eyes kept drifting to the kitchen window, with a view into the courtyard. Several boys, two of them looking to be almost my age, were kicking a ball around. They shouted and teased each other. I felt my heart beat faster, wanting to be with them instead of bent over a metal pot with potato peels stuck to my fingers.\n\nI wondered who the boys were. I could see they wouldn't have been much of a challenge on the field. They kicked clumsily, barely making contact with the ball.\n\n\"Rahima! Why are you sitting like that? For God's sake, aren't you embarrassed?\"\n\nShahnaz's voice jolted me. I looked down and snapped my legs together, bending my knees. I'd been sitting like a boy basking in the summer sun. A bolt of pain shot between my thighs.\n\n\"Oh, sorry, I was just\u2014\"\n\n\"Have a little decency!\"\n\nI hung my head, my face flushed again. I cursed myself. Thank goodness my mother hadn't witnessed this. She had warned me over and over again to carry myself as a proper girl in my new home but I'd been living as a boy for years. There was a lot of unlearning to do.\n\nOur mother-in-law joined us for lunch. She hobbled in, her fingers on the shoulder of a young boy, probably a grandson. I kissed her hands and mumbled a greeting, following Shahnaz's lead. Her visit was a surprise to me, but not to Shahnaz. I looked to her for guidance. She didn't offer much.\n\n\"She did the same thing to me,\" Shahnaz whispered. \"She wants to see if you're being a good wife. Go ahead and lay out the food, the plates. Sit with her.\" She went into the living room and spoke sweetly to Bibi Gulalai. \"Khala-jan, with your permission, I'm going to feed the baby. I'm sorry I can't sit with you but your new bride has prepared lunch for you.\"\n\nI took out the food as Shahnaz suggested, thinking to myself that she'd just fed the baby before our mother-in-law walked in. But I quickly forgot about it as I began to put the potatoes into a serving dish. Nothing looked like the food my mother prepared. My hands shook as I laid it out on the cloth. Bibi Gulalai fingered her prayer beads while she eyed my every move. Once I had spread out the potato stew and the rice, she spoke.\n\n\"A cup of tea would have been a nice start. Looks like you're rushing us to lunch.\"\n\n\"I . . . I'm sorry. I can bring a cup of\u2014\"\n\n\"Yes. Bring a cup of tea first. That's how you treat a guest.\"\n\nI got to my feet and went back to the kitchen to boil some water. I sprinkled tea leaves into the teapot and searched everywhere until I finally found a teacup.\n\n\"Did you add cardamom?\"\n\nI sighed.\n\n\"No, Khala-jan. I'm sorry, I forgot the cardamom . . .\"\n\n\"Tea without cardamom?\" She shook her head in disappointment and leaned back. \"Maybe that's how your family drank tea, but the rest of us\u2014\"\n\n\"No, my mother always puts cardamom.\"\n\nHer eyes narrowed.\n\n\"I was saying\"\u2014she was not happy with my interruption\u2014\"that the rest of us prefer our tea with cardamom. So pay attention next time.\"\n\nI nodded silently while she slowly sipped her unflavored tea, the disappointment showing in her eyes. I watched the steam waft from the rice.\n\n\"All right. Why don't we try this food that you've made now.\"\n\nI reached over and spooned some rice onto her plate. Large clumps stuck together. The potatoes looked more reasonable. I prayed her eyes were old enough that they couldn't clearly see what I'd made of the rice. She took two bites and shook her head in frustration.\n\n\"This is cold. Food doesn't taste good cold. And we're supposed to eat grains of rice, not balls of it. How long did you cook it for?\"\n\n\"I don't . . . I don't know . . .\"\n\n\"Too long. Too long. And the potatoes are still hard!\" She sighed heavily. \"Shahnaz! Shahnaz, come out here!\"\n\nShahnaz came into the living room, her eyebrows raised in curiosity.\n\n\"Yes, Khala-jan?\"\n\n\"This girl cannot cook! Did you try this food? It's terrible!\"\n\n\"No, Khala-jan, I didn't. She insisted on making lunch so I let her. Otherwise, I would have been happy to prepare something for you.\"\n\nI looked at Shahnaz and began to realize that she was not as benign as I had thought. She avoided looking at me. I had the urge to throw a punch at her but kept my cool.\n\n\"That's not true! She told me I should make lunch. And she just fed the baby! You did this on purpose!\"\n\n\"Rahima, this kind of behavior is exactly what I was worried about. You're a wild child and not a suitable wife for my son but he's taken you and now we have to undo what you are. Listen to me carefully. You are to behave like a proper bride and learn to keep house. That tantrum you threw in your father's home will not be tolerated here. I'm leaving now but know that I'll be keeping an eye on you.\" She got to her feet and wobbled to the door. She said nothing more and let the door slam behind her.\n\nShahnaz tossed her hair back and walked to her room, a smug look on her face. She had set me up.\n\nMadar-jan, you were right. And this is probably just the beginning.\n\nI confronted Shahnaz later that day.\n\n\"Why did you do that?\"\n\n\"Do what?\"\n\n\"You could have warned me. And you lied to her. You wanted me to look bad.\"\n\n\"What are you talking about? I didn't lie!\"\n\nI remembered one of Khala Shaima's favorite sayings: A liar is forgetful.\n\n\"Don't be upset, Rahima. You'll learn soon enough. God knows I did.\"\n\nShahnaz was a ball of contradictions. She was angry to have to share her home with me. It was bad enough that she had gotten the smallest of the houses. The other wives had more children and their marriages had been arranged by Bibi Gulalai. Only Shahnaz and I had been chosen by Abdul Khaliq himself and his mother clearly did not approve. Shahnaz was bitter one day but would sit and chat with me as if we were old girlfriends the next. She was lonely, I could see, and missed her sisters as much as I did mine.\n\n\"You know why she doesn't like you?\" she asked me one day.\n\n\"Because I'm a bad wife?\"\n\n\"No.\" Shahnaz chuckled. \"Although that's not helping matters any. She hates you because she wanted Abdul Khaliq to take her brother's daughter as his fourth wife. Instead, he took you.\"\n\n\"Why didn't he take his cousin?\"\n\n\"He was going to. That's what I'd heard from the others, at least. But something changed a few weeks ago and he made some excuse to his uncle's family. Next we heard he'd arranged for a nikkah with someone else\u2014you. And Bibi Gulalai's brother was more than a little disappointed, since they'd already courted his daughter.\"\n\nI knew I couldn't trust Shahnaz or anything she said but I was lonely too. She was the only person around me most of the time. Her son, Maroof, took to me quickly and I passed my time showing him how to kick a ball. Shahnaz would watch me suspiciously, as if waiting for me to do something wrong.\n\nAnd somehow it seemed I did everything wrong. I sat wrong, I cooked wrong, I cleaned wrong. All I wanted to do was get back to school and back to my family, my friends. I felt clumsy in a skirt, my breasts pointy in the brassiere my mother had purchased for me before my nikkah. I wanted to tie my chest down again. A lot of days, that's exactly what I did. I would wrap a long strip of fabric around my chest and pin it tight, trying to prevent full womanhood from setting in.\n\nMy mother-in-law came back often. When the house wasn't cleaned to her standards, she would pull me by my ear and make me scrub the floors while she watched. Shahnaz blamed everything on me and Bibi Gulalai was more than happy to believe anything she said.\n\nAbdul Khaliq returned, as determined as his mother to make me into a proper wife. I hated to feel his breath on my face, my neck. His teeth were yellowed and his beard rough on my face. I sometimes tried to pull away, to squirm from him like the fighters in the magazines. But the more I struggled, the more forceful he became. And worse than that was the smirk on his face. As if he enjoyed when I put up a fight. I shouldn't have been surprised. He was a man of war, after all.\n\nEach time, I felt dirty and weak. I hated that I was powerless under him. I was supposed to be this man's wife and that changed everything. I wasn't supposed to fight back. And the look on his face told me that fighting back would only make matters worse.\n\nSo many nights I lay curled on my side, crying quietly and waiting for morning to come so the man snoring beside me would stretch his arms and leave.\nCHAPTER 25\n\n\"Now taste this. See? It has no flavor at all. You've got to add some salt. Everything tastes better with a little salt. Mmm.\"\n\nShahnaz stirred the pot once more, the tomatoes melting into the simmering oil. She was teaching me a few basic dishes. It wasn't easy but I realized she took well to flattery. It was much better than antagonizing her.\n\n\"You see the difference? Now just touch the edge of the potato. It should be soft. See? It's cooked. My God, it really amazes me that you don't even know this much. You must have been so spoiled at home. I hope your sisters aren't such oafs in the kitchen!\"\n\nI wasn't worried about that at all. Shahla and Parwin could cook nearly as well as Madar-jan. But the mention of them made my heart ache. It had been two weeks since we were taken away from our home. I wondered what my mother was doing. I could picture my father, asleep in our living room with a satisfied smile on his face, clouds of heady smoke around him and his stomach heavy with food.\n\n\"Shahnaz, how can I see my sisters? I miss them so much! Parwin is so close by. Can I go to visit her?\"\n\n\"Not a question to ask me. Ask your husband. Or your mother-in-law,\" she said. I wasn't sure if that was really a good idea or if she was setting me up again.\n\nI saw my mother-in-law most afternoons. My third day at the compound, I was summoned back to the main house but this time by my husband's first wife, Badriya. There was laundry to be done. Badriya was also Bibi Gulalai's second cousin and, therefore, her favorite bride. Abdul Khaliq treated Badriya well, since she had been a good wife to him and because there was a family relation to respect. But as he added newer, younger wives to his compound, she spent fewer and fewer nights in his bed. This was a point of contention, though I couldn't understand why.\n\nBadriya was nothing near pretty. Her cheeks hung low and she had two moles above her mouth, a constellation that looked to me like the letter tay. Her face was as thick as her hips, but she didn't need looks. Now in her thirties, she was heavyset, her girth widened by the five sons and two daughters she had proudly borne her husband. Bibi Gulalai loved the grandchildren Badriya had given her and boasted about them to the other wives. This fed the tensions among Abdul Khaliq's wives and kept life interesting for Bibi Gulalai.\n\n\"Make sure she does a good job, Badriya. This girl has a lot to learn. She was a bacha posh, don't forget. Can you believe that? A bacha posh at this age! No wonder she has no clue how to carry herself as a woman. Look at the way she walks, her hair, her fingernails! Her mother should be ashamed of herself.\"\n\nBadriya was resentful of Abdul Khaliq taking me on as a fourth wife, but he was a warlord and this was common practice for anyone, so she bit her tongue as any good wife did. Badriya had nothing to complain about anyway. She had the best house in the compound, the one with an actual bed and sofas in the living room. She had a cook and a housekeeper to tend to all the chores in her house. She was the most esteemed wife, the one Abdul Khaliq would discuss matters with, and she made sure the others knew as much.\n\nThere was a rhythm and routine to life in the compound. The wives tended to their children while Abdul Khaliq tended to his affairs, whatever that meant. There hadn't been any armed fighting lately, but nearly every day he and his bodyguards sped off in his three black SUVs, clouds of dust in their wake. His entourage circled him as he walked, nodding when he gave out orders and keeping away from any of the women in the compound. The men ate meals together, served by the housekeepers that Abdul Khaliq had brought on. They ate in Abdul Khaliq's entertaining room, a carpeted room with a perimeter of cushions and pillows on which the men reclined, licking their fingers and sipping their tea as they discussed the day's affairs. When they were finished, the women and children ate what was left. The servants were the third round, hoping enough had slipped through the many greedy fingers before them.\n\nThe women never left the compound. The children played together and fought together as brothers and sisters but subdivided. Half brothers got along most of the time but a casual game of soccer could quickly disintegrate into a scuffle where the sons of the first wife teamed up against the sons of the second. The same held true for the girls, who could become catty in the blink of an eye.\n\nBadriya had no problem putting me to work. Nor did anyone else. Though they had plenty of help at the compound, the women seemed to derive a special pleasure from making me take on the most menial of tasks, especially since I fumbled with them. I swept the floors, washed the diapers and cleaned the western toilets as best I could. My hands burned at the end of the day and all I wanted to do was lay my head down. Most nights, that wasn't possible. Abdul Khaliq called for me to join him in his bedroom to repeat what he had done the night before. And the night before that.\n\nMy insides burned and I walked as if a shard of glass was stuck in my underwear. Sometimes I would wake up in the night remembering. It made it impossible to go back to sleep. I would pull my thighs together tight and curl up, praying he would tire of me. I wished my monthly bleeding would come more often but it had only started six months ago and came infrequently. My only escape was training my mind to wander when I was with him. I would close my eyes or stare at a stain on the wall, like seeing shapes in the clouds.\n\nDuring the day I watched the compound's walls, hoping for a glimpse of my sister. I prayed Parwin would hobble into our courtyard unannounced and surprise me with a visit, a drawing, a smile. I couldn't bear to think of what her days were like. I hoped she didn't have to do all the things I had to do. Parwin's legs moved slowly, clumsily. People didn't like that. If the people around her were anything like the people around me, she was sure to be punished. I'd been smacked around more than once for a job not done well enough.\n\nI couldn't bear knowing my sister was just over the wall. I wanted to see her. I wanted to look at a face that knew me, that loved me. I couldn't bear it anymore and worked up the nerve to ask Bibi Gulalai when I saw her walking through the courtyard.\n\n\"Khala-jan! Khala-jan!\" I panted, running up behind her.\n\nMy mother-in-law turned around, already displeased. When I reached her she wasted no time and slapped my face.\n\n\"What are you doing yelling and running like that? My God! You have absolutely no idea how to behave yourself! Have you learned nothing here yet?\"\n\nMy face stung and my mouth gaped as I searched for an apology that wouldn't make her angrier.\n\n\"Forgive me, Khala-jan, but I wanted to speak with you before you left. Good morning. How are you feeling?\" I asked, not really caring but trying to show her that I did have some manners.\n\n\"You came running across the yard like a rabid dog to ask me that?\"\n\nThere was no winning with her.\n\n\"Khala-jan, I wanted to ask you something. I really miss my sisters. It's been weeks since I've seen either one of them or anyone from my family. Would it be possible that I could see my sister Parwin, at least? She's just next door and I\u2014\"\n\n\"You were not brought here to go playing with your sister and taking her away from her responsibilities as well. It's bad enough that you can't manage what's asked of you here! This is your family now. Stop thinking about anything else and go finish your chores. Your sister is hardly a help over there with her limp leg. Forget about making things even worse.\"\n\n\"But, please, Khala-jan. Just to see her for a few moments. I promise I'll have all my work done. I've already washed the floors and beaten the dust from the carpets this morning. I could even go there and help her with whatever she needs to do\u2014\"\n\nAnother slap across my face. I took a step back and felt my eyes blur with tears. I was always surprised by the amount of force her wrinkled fingers brought.\n\n\"You had better learn to hear what I say the first time I say it.\"\n\nShe turned her back to me and walked out of the courtyard, shaking her head.\n\nI shouldn't have been surprised but I was. My sister was yards away but she might as well have been across the country. Bibi Gulalai made me wonder even more how she was doing, with her \"limp leg.\" I prayed the other wives had some sympathy for her, that there was at least one kind face.\n\nIn Abdul Khaliq's compound, there was only one person who was genuinely nice to me, Abdul Khaliq's second wife, Jameela. While Badriya and Shahnaz appeared friendly enough, it took a half day with each to see their true colors. Badriya, with her larger, second-story home, looked down on everyone but even more so on me, the young latecomer.\n\n\"Badriya was the same way with me,\" Shahnaz said when I came back to the house crying one day. \"It's not easy being the oldest wife.\"\n\n\"Why not? She's got everything! The best cook, the best maids, the best rooms!\"\n\n\"It's not about any of those things. Abdul Khaliq doesn't want her. He doesn't call for her, now that he's busy with you. He used to be the same way with me and she hated it. Hated me for it.\"\n\n\"But . . . but I don't want to be called to him. I would be happy if he ignored me. What does she do that he doesn't call for her?\"\n\nShahnaz laughed, her eyes lit up with amusement. \"Simple, just get old. You see how Abdul Khaliq doesn't like to eat food cooked yesterday? Men want something fresh, hot off the stove.\" She cocked her head to the side and gave a sly smile.\n\nThat night I prayed for Allah to make me old, as old as Badriya, who looked older than my own mother.\n\nBut Shahnaz was just as bitter toward me as Badriya was. She, too, hated being called by Abdul Khaliq, but it wasn't much better when she saw me going toward his quarters. She would bang the pots around, huff if I asked her anything and slam her door. The following day, more chores were piled on me than usual, even if I was also called to clean Badriya's house.\n\nJameela was the only one who was different. She was Abdul Khaliq's second wife and, being such, had the second-best accommodations of the compound. She lived downstairs and down the hall from Badriya. She had been given to Abdul Khaliq by her family as a token of gratitude. No one was sure exactly what they were grateful for\u2014it was always spoken of in very vague terms\u2014but she seemed content enough with the arrangement. She had borne him three sons and two daughters, making him satisfied enough that she was holding up her end of their arrangement.\n\nAt thirty, Jameela was much more beautiful than Badriya and even Shahnaz, who was at least ten years younger than her. Her eyes sparkled with kindness and good humor when she spoke. My mother's warnings had been sage advice when it came to the other wives of the compound, but when I met Jameela, I knew I could trust her.\n\nI had met Jameela last. She'd run into me coming out of Badriya's home.\n\n\"You must be Rahima! Ay, you're even younger than Badriya predicted.\"\n\n\"I'm not that young!\" I'd shot back. I was tired and sweaty and didn't need anyone else making comments about me. \"Who are you anyway?\"\n\n\"Looks like you're off to a good start.\" She'd smiled gently. Her reaction had embarrassed me. \"I'm Jameela. I live in the part of the house here with my children. My son Kaihan is probably your age. My daughter Laila, too. Have you met them?\"\n\nI shook my head. I hadn't seen anyone my age yet. I wondered if Laila was as nice as her mother.\n\n\"Laila!\" she called out. \"Laila-jan, what are you doing?\"\n\n\"Zarlasht dirtied her clothes, Madar-jan! I'm changing her!\"\n\n\"Come here for a second, janem, and bring Zarlasht with you. There's someone you should meet.\"\n\nI heard footsteps. Laila was indeed close to my age, probably a couple years younger than me, but the baby on her hip hid the difference. She looked like her mother\u2014her eyes and hair the color of night, dark and dramatic against her gauzy emerald head scarf. She looked at me with curiosity. Zarlasht was about a year old. Seeing them made me think of Shahla and Sitara. As a baby, Sitara spent just as much time in my sister's arms as she did in my mother's.\n\n\"This is Rahima-jan,\" Jameela said, taking Zarlasht from her daughter's arms. \"Remember the nikkah we heard about last week? This is your father's bride.\"\n\nLaila raised an eyebrow. \"You are?\"\n\nI stood still, unable to bring myself to admit to a title that seemed too heavy for my shoulders.\n\n\"She is, so you'll be seeing her around more.\"\n\n\"Why is your hair so short? Like a boy?\"\n\nI felt my face flush and turned away. I wasn't sure how much to share. Maybe it wasn't a good idea to tell everyone I'd been a bacha posh.\n\n\"That's . . . that's how I wore it when I was going to school!\" I blurted, hoping that was explanation enough but mostly wanting Laila to know that I'd been to school.\n\n\"School?\" she exclaimed. \"You were going to school like that? Madar-jan, she looks like Kaihan, doesn't she?\"\n\n\"You were a bacha posh, weren't you?\" Jameela asked. \"That's what I'd heard. Bibi Gulalai mentioned it before the nikkah. My children have never seen a bacha posh but I remember my neighbor's cousin had been one. Up until she was ten years old, that is. Then she changed back to a girl.\"\n\n\"What's a bacha posh?\"\n\n\"Laila-jan, I'll explain more later. I just wanted you to meet Rahima-jan for now. And this is Zarlasht, my youngest.\"\n\nMore footsteps came down the hallway as I tried not to stare too much at Laila, who reminded me how much I missed my sisters.\n\n\"Kaihan! Hashmat! Stop running inside! You boys are shaking the walls!\" Jameela turned to me and explained. \"Hashmat is about the same age as my son. He's Badriya's boy.\"\n\nI took one look at Hashmat and a knot formed in my stomach. He looked from Jameela to me and grinned.\n\n\"Who are you supposed to be?\" he said bluntly, his tongue slipping through his teeth and giving his words a wet lisp. It occurred to me that I'd seen him before, that I'd heard him before. We'd played soccer on more than one occasion in the streets a few blocks from our school. My voice escaped me. I wondered if he'd recognize me as well.\n\n\"This is Rahima, your father's bride,\" Jameela said. I turned my face and looked down, avoiding his gaze. Jameela was surprised by my modesty given how I'd spoken to her just a few moments ago.\n\n\"Oh. Yeah, I heard about you. You're . . . hey, aren't you . . . you're Abdullah's friend, aren't you?\"\n\nI didn't know how to respond. I fidgeted and looked to Jameela. I knew this looked strange to everyone. No girl my age should have been referred to as \"Abdullah's friend.\" Jameela looked at Laila, who seemed more confused now than before.\n\n\"Never mind that, Hashmat,\" she said intuitively. \"She's your father's bride and you'll be respectful of that. No one wants to hear anything else from your mouth.\"\n\nI stared at the ground, knowing now why he looked familiar. I remembered him pushing and shoving his way to the ball, his mouth open and his dirty fingernails clawing at anyone in his way. He had friends only because boys were afraid not to be friends with Abdul Khaliq's son, a lesson they'd learned from their parents. We had made a point to avoid him and his group entirely. It had been a year since I'd seen him.\n\n\"You're a girl?\" he exclaimed. \"What kind of girl are you? That's you, isn't it? That's why you're not answering!\"\n\n\"Hashmat! Do you want me to tell your mother\u2014\"\n\n\"Look at that! You've even got short hair and everything! What kind of bride are you? You've been running through the streets with Abdullah and his gang. No wonder you guys couldn't score a single goal!\" Saliva sprayed out when he spoke with excitement. I covered my face with my veil, wanting to hide from his wet assault.\n\n\"Hashmat! That's enough I said!\"\n\n\"Maybe Abdullah's a girl too! Maybe you all are!\" he laughed.\n\nI would think of lots of clever things to say later, when Hashmat was not around.\n\nInstead of saying any of those things now, I ran. I ran with the washrags still in my hand, my eyes blurring with tears. I wanted to get away from Hashmat, from this boy who knew me as I wished I still were\u2014a boy just as free as him. I hated that he lived here. I knew he would always bring it up. He would always look at me and laugh at the girl who used to be a boy.\n\nBy the time I got to my room and slammed the door behind me, I wondered if he would see Abdullah again. I imagined what he might say and felt my heart drop. I didn't want Abdullah to see me as a girl, as Abdul Khaliq's wife, as Hashmat's stepmother.\n\nI dropped my head into my hands and cried.\nCHAPTER 26\n\nIt drove me mad to think of Parwin. Months had passed and there was no hint that I would be allowed to see her. I knew where the adjacent compound was and tried to eavesdrop at the wall between the two homes to hear her voice or even someone speaking of her. I couldn't spend much time out there or Bibi Gulalai would come chasing after me to tend to something that no one else wanted to do. She had taken to using a walking stick these days, a change driven as much by her intensified desire to discipline me as her unsteady step.\n\nI waited a month before making another move. I needed to work up the nerve to try again and to figure out how I could manage to get out of our compound. I set off in the early morning, when I usually went to tend to the laundry. I took the pail and walked across the courtyard as casually as I could. My throat was dry as I scouted the area. A few servants here and there but no one seemed to be paying any attention to me. My husband had left earlier in the morning and wouldn't be back for hours.\n\nI moved closer and closer to the front gate, my palms sweaty.\n\nDon't hesitate, I told myself, and opened the gate to walk out. I waited but heard nothing. No one had even noticed.\n\nThe compound sat on an open dirt road, one that I hadn't seen since the day of my wedding. I looked to the right and saw the adjacent compound where Parwin lived. I pulled a burqa from the pail and donned it. I walked quickly and tried their gate but it was locked.\n\nI knocked lightly. This was the time of day that only the servants were in the courtyard and that was what I was counting on. If I could just get one of the servants to open the gate, I could find my way to my sister. I waited a moment but no one answered. I tried again, a little louder this time.\n\nOn my third try, beads of sweat trickling down the back of my neck, I heard footsteps and grumbling. I stepped back as I saw the gate open.\n\n\"As-salaam-alaikum,\" answered an older woman cautiously. From her worn clothing, I guessed she was one of the servants. I tried to see past her and into the compound. She squinted and narrowed the opening behind her. \"Forgive me, I haven't recognized you. Are you here to see someone?\"\n\nI cleared my throat and willed my voice not to betray me.\n\n\"Wa-alaikum as-salaam. Yes, I am. I am the sister of Khanum Parwin. I've come to pay her a visit.\"\n\n\"Ah, Khanum Parwin! Her sister? Welcome, welcome, but . . . have you come alone?\" she said curiously. She looked behind me, expecting to see a chaperone.\n\n\"My mother-in-law, Bibi Gulalai, was supposed to come with me and would be here if it weren't for her aching back. She had to rest. But she told me to go on without her,\" I said, trying to keep my voice steady. \"Is my sister around? I just want to see her for a few minutes.\"\n\nThe woman looked confused. Indeed it was strange for one of Abdul Khaliq's wives to show up at the front gate unaccompanied, but then who would imagine that a young girl would lie about such a thing? She opted not to give me, Abdul Khaliq's wife, a hard time and pulled the gate open to let me through.\n\n\"I think she's still in her room. I'll show you the way,\" she said.\n\nThe compound was much smaller than Abdul Khaliq's but set up in a similar way. My eyes looked for Parwin. I couldn't believe I'd made it this far! We walked past a few children, no more than six or seven years old. They glanced at me, too preoccupied with their own games to bother with the stranger in the burqa.\n\n\"Who is this with you, Rabia-jan?\" I stopped, as did my guide, whose name was apparently Rabia.\n\n\"Good morning, Khanum Lailuma. This is Khanum Parwin's sister. She's come from next door to pay a visit.\"\n\n\"By herself? You are Abdul Khaliq's bride?\" Lailuma said, her brows furrowed together with displeasure.\n\n\"Yes,\" I said. I reminded myself to look confident.\n\n\"Does anyone know you're here?\"\n\n\"Of course!\" I said. \"As I told Rabia-jan, Bibi Gulalai was going to accompany me but she was having backaches. I just wanted to pay my sister a short visit. It's been so long since I've seen her.\"\n\n\"Well, that's . . . I just don't think . . .\"\n\n\"I'm so glad to meet you! I've heard much about the family next door to our compound but I haven't had the chance to meet anyone. Were those your children I saw just now in the courtyard? So adorable, God bless them!\"\n\nLailuma was disarmed by my flattery, which to me sounded much more like something Shahla would say than anything I would have come up with.\n\n\"They are, yes, thank you. It's a shame we haven't met. Well, then go on but don't be long because your sister has responsibilities to tend to.\"\n\n\"Of course! I don't want to keep her,\" I said as sweetly as I could.\n\nRabia sighed and hurried me along, not wanting to be pulled away from her other duties any longer. We went down a short hallway and as soon as we turned the corner I saw her.\n\nParwin's back was to us but I could see her limping, a pail of water in one hand. The water sloshed with her wide gait, a trail of splashed water behind her.\n\n\"Parwin!\" I called out, running to her. My sister turned around, her face puzzled. She dropped the pail on the floor and I could see the servant shaking her head at Parwin's clumsiness.\n\n\"Rahima? Rahima! What are you doing here?\" she said, her eyes tearing up as I threw my arms around her thin frame.\n\n\"I came to visit you! I missed you so much, Parwin!\" I turned around and saw Rabia was already shuffling down the hallway. \"Let's go somewhere! I want to talk to you before I have to go back.\"\n\nParwin nodded and led me to her room, a small rectangular space without windows. It was even smaller than where I slept. We closed the door behind us and Parwin fell back onto her mattress with a sigh. She looked exhausted.\n\n\"Parwin, I've wanted to see you for so long but they wouldn't let me come! All they want me to do over there is work and work and I'm so tired of it! I scrub the floors and do the laundry and . . .\" My voice tapered as I realized my sister's life was probably no different from mine. I was being selfish to complain to her.\n\n\"I know, Rahima. It's terrible here too,\" she whispered. \"I pray every day that something will happen and I'll be able to go back home. I miss Madar-jan, Shahla and the girls! I even miss Padar-jan!\"\n\nI wanted to disagree with her but oddly, I missed our father too, even though I blamed him for putting us all through this.\n\n\"What is it like for you there, Rahima? They let you come here today?\"\n\n\"I snuck away, Parwin. I've asked so many times but Bibi Gulalai won't allow it. So today, I just walked over here. I told the servant that I'd gotten permission.\"\n\n\"Oh no! Won't they notice that you're not there? What are they going to do to you?\"\n\nI had given that some thought and only hoped that my reasoning worked.\n\n\"I've gotten in trouble a couple of times. The last time, Bibi Gulalai threatened to send me back to my parents. I'm hoping that if she finds out about this, that's what she'll do. I want to go back home. I hate it over there!\"\n\n\"Do you really think they'll send you back?\" Parwin seemed doubtful. My sister looked different, I realized. Her face looked thinner and her eyes lacked their sparkle. Her cheeks were marked with dark spots.\n\n\"I don't know but I really wanted to see you. And I thought it was worth a try,\" I added with a smile.\n\n\"I wish they would send me back too,\" she said wistfully.\n\n\"Are you . . . are you doing all right here? Are they nice to you?\"\n\n\"I would rather be home. Remember those birds that used to fly over our yard? Remember how mad Shahla got when their droppings got on the laundry\u2014twice in one day! That was so funny!\" She was looking past me. Seeing something that no longer existed.\n\n\"Parwin, are you still doing your drawings? Have you sketched anything new? I miss looking at your work.\"\n\nShe shook her head. \"There's too much to do and I don't want to disappoint anyone here. I have to keep up with my chores. Anyway, I don't really feel much like sketching.\"\n\nThis was completely unlike Parwin. I held her hands in mine and wondered what to say. There were questions I wanted to ask but the answers would only hurt us both. I stared at her while she smiled awkwardly. She talked about Rohila and Sitara, told me stories about them as if she'd seen them just days ago. I wondered what her husband was like. I wondered if she had to tolerate the same things I did.\n\n\"Khala Shaima said that Rohila is probably going to go to school now. Isn't that wonderful? She's going to love it.\"\n\n\"Khala Shaima? Did you see her? Did you talk to her?\" It sounded like Parwin had completely lost her mind.\n\n\"Yes, she came here. About two weeks ago. I just saw her by the front gate for a few moments and then she left again. She asked about you too but I told her I hadn't seen you.\"\n\n\"She came here? Why didn't she come see me too?\"\n\n\"She tried.\"\n\nOf course, they'd kept her away from me. They probably didn't want me telling Khala Shaima how they treated me.\n\n\"What else did she say?\"\n\n\"She said Padar-jan is the same, but happier now that he can get a lot more medicine. And Madar-jan and the girls are doing all right. We didn't really talk for very long. I wished she could have stayed and told more of her stories. I liked hearing about Bibi Shekiba, didn't you? I think about her a lot now.\"\n\nI thought about her more than anyone could know. I often wondered what she would have done in my place. Or what I would have done in hers. Or if there was much difference anyway.\n\n\"Parwin, maybe we should just run away!\" I whispered, interrupting her chatter. \"Just like I snuck out this morning. We could just take off!\"\n\nIf only I'd known then what the future held, I would have done just that. I would have snuck away with her in the night. At least that would have given her a chance.\n\n\"Rahima, you're always making trouble. I'm all right here. It's a lot of work but it's okay. Madar-jan said we should do what's asked of us and I am. You're going to get yourself into big trouble if you try anything.\"\n\nI felt my throat tighten to hear her talk this way. She wasn't herself but I realized there was no running away for us, especially her. Parwin wouldn't make it more than a few feet from the compound with her limp.\n\nVoices in the hallway grew louder.\n\n\"Where is she? Who let her in here?\"\n\n\"She came alone? Does Bibi Gulalai know about this?\"\n\nI heard the footsteps and knew my time was up, quicker than I thought. I didn't bother turning around to see who had come after me. I kissed my sister's face and squeezed her hands as the door flung open.\n\n\"I'm sorry, Parwin. I'm sorry about all of this,\" I said. \"I'm not far from you, Parwin, remember that, okay? I'm not far from you!\"\n\nI kept my eyes on her as I was yanked to my feet. Parwin looked oddly peaceful amidst the shouting.\n\n\"Birds fly away, one by one . . . ,\" she said meekly, watching as I was pulled away from her once again.\nCHAPTER 27\n\nBibi Gulalai seethed with anger.\n\nSomeone had seen me leaving the compound. Word got back to Badriya, who, probably happily, reported the news to Bibi Gulalai. It didn't matter much. Just made me hate them more. Badriya was a more spiteful person than I'd first thought. I prayed I'd one day find a way to get even with her. No wonder Hashmat was such a jerk.\n\nBut I'd invited this round of punishment. I'd asked for it. With every blow, every curse, I held out hope that my mother-in-law would blurt out that she'd had enough, that she was sending me back to my mother. I covered my head with my arms and waited to hear what she'd said the last time. When she didn't say it, I spoke up.\n\n\"If I'm so terrible then why don't you send me back?\"\n\nShe paused. At that moment, I realized I hadn't done myself any favors. She knew that was exactly what I wanted and refused to give it to me, even if it would shame my family and me before our entire community. No, at that moment she decided she would straighten out this pesky bride herself. My plan had backfired but at least I'd seen Parwin. Or what was left of her. My sister, so different and delicate in her disposition, had been changed by her new life. I knew it was partly my doing. This had all come about because of me, the bacha posh, and because of the argument I had with my mother. The rest of the blame sat on my father's addicted shoulders.\n\nI thought of Shahla. I wondered if she still blamed me too. She had forgiven me on the day of our nikkahs, but I wondered if things were different now. Maybe things were better for her than Parwin or me. Shahla had a way of pleasing people, making people smile. I found it hard to believe anyone could treat her badly.\n\nNow the relationship between Bibi Gulalai and me was forever soured. She focused her energies on making life miserable for me. My husband took from me what he wanted, did to me what he wanted and left the rest of my existence in his mother's hands. He was too busy to care about the details, now that he had even more lucrative business with some foreigners. His power and influence in our area were growing, and with it, so was his aggression and domination at home. We four wives shared a fear of his ready fist.\n\nThere was something else that worried me these days. For two weeks, I had been waking up in the mornings, my stomach reeling with nausea. The feeling frightened me and I finally confided in Jameela, who looked at me and sighed.\n\n\"Let me see your face,\" she said, cupping my cheeks in her hands. She turned my head from side to side, looking at my skin and my eyes. I let out a yelp when she felt my full breasts. \"Yes, it looks to be true. You're going to become a mother, Rahima-jan.\"\n\nHer gentle words stunned me. For some reason, this possibility had not occurred to me.\n\n\"What? How can you tell?\"\n\n\"Rahima-jan, how long has it been since your monthly illness?\"\n\nCome to think of it, I couldn't remember when I had last bled. It happened so irregularly that I never could keep track. I shrugged my shoulders.\n\n\"Well, it seems that you're pregnant now. The sickness will pass, you'll see, but other things will change for you.\"\n\nI felt light-headed. Jameela took me by the arm and sat me down on a stool in the courtyard.\n\n\"It's all right, dokhtar-jan,\" she said. \"Every woman goes through the same thing. All of us. This will help you, you'll see. Your husband and mother-in-law will be pleased. Bearing children is a wife's duty.\"\n\nOne that Parwin had not fulfilled. Maybe that was why they had made her life so miserable. I wondered if Bibi Gulalai would treat me any better knowing this.\n\n\"I don't want anyone to know!\" I whispered. I didn't want anyone to look at me differently. I felt ashamed.\n\n\"Do not say anything to anyone. It's not proper, anyway. We don't speak of these things. Keep quiet, do your work and let Allah handle the rest. In a few months you will see your child, if Allah wills it. May God keep you in good health,\" she whispered at the end.\n\nI had no idea what was ahead of me. Jameela looked worried, even as she tried to comfort me. In her wisdom, she kept from me the troubling things she'd seen before she was married. Her uncle had married two girls around my age. When the first delivered her child, she bled for three days until her veins went dry and she could bleed no more. Her baby, with no one to nurse him, followed her ten days later. His second bride survived childbirth but the baby had ripped through her immature body, leaving a hole in its wake. Her husband, repulsed by the constant trickle of urine down her leg, said she was \"unclean\" and sent her back to her family to hide from the world in shame. Young mothers did not fare well but Jameela did not want to frighten me.\n\nI took Jameela's advice but before long Shahnaz recognized the way my nose turned at the smell of food.\n\n\"You're pregnant!\" she laughed haughtily. \"Now you'll see how tough life can really be!\"\n\nSome days I hated her more than Bibi Gulalai. She shared the news with Badriya, knowing it would make her even more spiteful toward me. If I brought a son into Abdul Khaliq's compound, her husband and mother-in-law might not treat me as the lowly servant in the house. I doubted much would change. Bibi Gulalai looked at me as one would look at a flea-infested dog yapping at one's feet.\n\nBut surprises were always around the corner and a month later, I was permitted visitors. I wasn't sure if it was because my mother-in-law had learned I was with child. I was shocked to see Khala Shaima standing in our courtyard, looking around with a suspicious eye. Behind her stood Parwin, clutching her chador at her chin and keeping her eyes downcast, a model of modesty. I dropped the heap of laundry I was carrying and ran over to them. It felt so good to see their faces, though I prayed they wouldn't be able to see the change in mine. I didn't want to share the news with them.\n\nI held Parwin's hand tightly. Khala Shaima balked when I tried to kiss her hand. She grabbed my shoulders and looked me over, assessing the changes of the past few months.\n\nKhala Shaima shook her head and sighed when she saw my full face and rounded belly. My baby was three months away. She didn't look surprised in the least.\n\n\"Are you feeling all right?\"\n\nI nodded. We didn't speak any more on the matter. I was thankful for that.\n\nSatisfied that I was at least whole and fed, she pulled me aside so we three could talk with some privacy. I had so many questions for her. She was my link to my past life.\n\nOur first meeting was bittersweet. Or sweetbitter, which better represented the order of things. I was thrilled to have them here but I knew how painful it would be when they left. Parwin and I couldn't get close enough to Khala Shaima.\n\n\"How's Madar-jan? Why didn't she come with you?\"\n\n\"Your mother is fine. You know how she is. She manages things inside the house but she's been kept under your father's thumb for so long that sometimes she forgets to stand on her own two feet.\"\n\n\"What about Rohila and Sitara?\" Parwin asked. \"Do they ask about us?\"\n\n\"Of course they ask about you! They are your sisters. That hasn't changed just because you're living somewhere else now! Don't listen to the garbage that some people say about girls belonging to other people. Bah! Girls belong to their families and always do. You have a mother and sisters and nothing changes that\u2014I don't care who you've married.\"\n\nWe nodded but I looked around quickly to make sure no one was within earshot. I knew enough about Khala Shaima to know that her fiery comments invited trouble.\n\n\"But why didn't Madar-jan come then? Is she all right? Doesn't she miss us?\"\n\n\"Of course she misses you! She's . . . well, you might as well know. She's been very upset since you girls left. She's been so upset that she started taking some of your father's medicine.\"\n\n\"She what?\"\n\n\"That's how things go sometimes. Listen, girls, when things are rough, people look for an escape. A way out. Sometimes it's hard to find the right way. Your father's escape has been that damned medicine and now your mother too. It was just a matter of time. It's in front of her face every day.\"\n\nI was angry. Madar-jan was going to be just like our father. I pictured her glassy eyed and snoring on the couch, Rohila looking after the baby.\n\n\"What about all the money? What are they doing with it?\" I asked bitterly.\n\n\"They divided it up. Of course, your father took most of it but he gave some to his brothers and your grandfather. They feasted on greasy meals, showed off around the village thinking it's going to change the way people look at them. God knows what else he's spending it on. I know your mother hasn't had a finger on any of it.\"\n\n\"What about Shahla? Have you heard anything about her?\"\n\n\"No, I asked your father about her since he's more in touch with that family than anyone else but he just says she's doing all right. He hasn't seen her. So far away, that poor thing. At least you girls have each other.\"\n\n\"But, Khala Shaima, I never get to see Parwin! She's so close but it's like she's on the other side of the world.\"\n\n\"Hmph. Still? Well, I'll just have to stop by more often so we can all see each other then. How are they treating you girls otherwise? Parwin?\"\n\n\"I'm all right, Khala-jan. They're treating me just fine,\" she said so sweetly that no one would have believed it.\n\nKhala Shaima's eyes narrowed. \"Your mother-in-law? Does she beat you? Do you get enough to eat?\"\n\n\"She's kind to me, Khala Shaima. She shows me how to do things and I eat plenty. Most of the time I'm not hungry anyway.\"\n\nKhala Shaima turned to me, unsure what to make of Parwin's answers.\n\n\"I'm all right, Khala-jan. My mother-in-law, Bibi Gulalai, she's hit me a few times but I've figured out how to keep her happy. And she can't hit very hard anyway, that old witch.\" I lowered my voice instinctively. Bibi Gulalai always seemed to pop up when I least wanted her to.\n\n\"Witch is right,\" Khala Shaima hissed. \"Damn these people, taking such young girls.\"\n\n\"Khala Shaima, can you promise to come often? I miss seeing you so much!\" I blurted. Parwin nodded her head in agreement.\n\n\"Of course. I'll come as often as I can with this damned back of mine. Somebody's got to keep an eye on you girls. Abdul Khaliq may be the biggest man in this village but you girls have a family too. I want to make sure these people know that.\"\n\nHer words, her presence, were such a relief, although it did nothing to change our daily life.\n\n\"And maybe you could tell us more about Bibi Shekiba?\" I asked.\n\n\"Ah! Now, there's something we should pick up again. No one likes an unfinished story . . .\"\n\nPeriodically, she would pick up Parwin and bring her to Abdul Khaliq's compound, where the three of us could sit and talk. She was persistent and managed to get her way. I thanked God for that. Those were the rare occasions when I was able to see my sister. Each time broke my heart and almost made me wish I hadn't seen her at all. The weak smile she gave me and Khala Shaima looked ridiculous on her fragile frame, her sallow skin. I hated her husband's other wives for what they were doing to her.\n\nParwin never complained to us. She never told us just how things were.\n\nIn some ways, I think she was the bravest of all. She, my meek and timid sister, was the one who acted in the end. She was the one who showed those around her that she'd had enough of their abuse. As Khala Shaima said, everyone needed a way to escape.\nCHAPTER 28\n\nOver the next few weeks, and with Ghafoor's help, Shekib became familiar with her new home. Arg was a majestic building and Ghafoor had explored every niche. The palace had been built by Amir Abdul Rahman, while Shekiba was just a toddler. A trench of water surrounded the heavy walls and a tower stood watch at each of the four corners of the estate. At the top of each tower, Shekib could make out a canon aimed into the distance. There were ramparts all around the fortress and soldiers posted everywhere.\n\n\"That building over there, on the eastern side, that's Salaam Khana. That's where the king receives his visitors. There are a few smaller buildings behind it where he spends time with his family or his closest advisers. There is where the soldiers sleep and that building is all for weapons.\"\n\nThey walked onward; the soldiers kept their gaze averted but watched their movements with great interest. They crossed the vibrant gardens and walked to the other, western side of the palace grounds.\n\n\"What's that one over there?\" Shekib pointed to a larger structure, tall enough that you could see it looking over the palace walls. It was a beautiful piece of architecture, stately appearing and just a short walk from Arg.\n\n\"Ah, that one! That's Dilkhosha Palace.\"\n\n\"It looks amazing!\"\n\n\"It is. The inside of it is so beautiful that it can make your heart melt! There are paintings, carvings and gold vases. You could never imagine anything so beautiful!\"\n\n\"You've been inside?\"\n\n\"Well, I haven't exactly been inside . . . but that's what I've heard.\" Ghafoor's voice was full of conviction.\n\n\"Where does the king live?\"\n\n\"Oh, well, he travels a lot but when he's here, he stays in that building over there with his wife.\"\n\n\"His wife? Do the women ever go in there?\"\n\n\"For God's sake, no! What kind of crazy idea is that? The women of the harem stay in the harem. That's their place. They can wander into their courtyard and they have their own bathhouse that they can use whenever they please but they are not to be confused with the king's wife!\"\n\n\"The harem?\" Shekib took a deep breath. If she wanted to survive this new place, there was a lot she needed to learn.\n\n\"Yes, like haram. It means that it's forbidden for other men to enter. Except for the king of course. That's part of the reason why we're guarding it instead of his soldiers. But mostly it's because he knows men would be men and they can't be trusted around women, not even women who belong to the king.\"\n\nShekib had left the harem with Ghafoor early in the morning. The women were still sleeping and the other guards were just dressing to begin their duties.\n\n\"How many women are in the harem?\" Ghafoor had only pointed out five or six women yesterday evening but their quarters were large with many rooms. Shekib thought there could be more.\n\n\"How many? Hmm . . . by last count there are twenty-nine.\"\n\n\"Twenty-nine?!\"\n\n\"Sure. Twenty-nine. That is, if you still consider Benazir one of them right now!\" she laughed. \"She won't really get his attention now that her belly has begun to swell. He won't bother with her until after it's done.\"\n\n\"Until what's done?\"\n\n\"Until it is done. Until the baby is born,\" she said.\n\n\"Oh. And their children, they will live with their mothers in that house?\"\n\n\"Of course. Did you not see Halima's children there with her?\"\n\n\"Where did he find all the women? For the harem, I mean.\"\n\n\"Same way he found me. And you. Lots of families can do without their girls. Lots of families need things. Anyway, he is the king. He takes what he wants.\"\n\n\"And what of the children? Does he have anything to do with them?\"\n\n\"Surely. You know\"\u2014Ghafoor brought her voice to a whisper\u2014\"the king himself was born to a slave mother. He knows firsthand that any child can rise to greatness, not only those who are born to the first wife.\"\n\nA steady breeze began to blow and Shekib reminded herself that her backside was not exposed. It would take some time to get used to wearing pants, she decided. Ghafoor, on the other hand, looked entirely comfortable in her garb.\n\n\"Does it hurt?\" Ghafoor asked casually.\n\nShekiba knew to what she was referring but feigned ignorance. \"What?\"\n\n\"Your face. Does it hurt?\"\n\n\"No.\" Shekib kept her gaze straight ahead. It was no accident that Ghafoor was walking on her right side, her good side. With the head scarf gone, she had no cover for her deformity. She wanted Ghafoor to see her face as it should have been.\n\n\"That's good.\"\n\nShekib was glad the conversation ended there.\n\nThey returned to the harem, now bustling with chatter as the women had woken up. With so many new faces around, Shekiba's hand rose up instinctively to bring her head scarf over her cheek but there was nothing to pull.\n\nPast the foyer, there were women everywhere, sitting in groups of four or five. Two or three were feeding young children; one nursed in the corner. Some were in their thirties and some looked to be around Shekib's age. Some were slender and others were plump. Few bothered to look up. Ghafoor put a hand on her elbow and led her into a large room with stone floors. In the center was a large pool. Three women sat with their bosoms half submerged in the water. Their voices echoed against the walls.\n\n\"This is the pool room,\" Ghafoor announced, watching for the astonished reaction she knew the view would draw. Shekib's mouth opened slightly and Ghafoor chuckled. Shekib ignored her amusement. The stone walls rose tall and grand. A balcony on the second level overlooked the pool.\n\nThere were plants in the room, lush green leaves that drank in the room's moisture. The women looked over at Shekib and Ghafoor briefly, but seeing only Shekib's good half, their attention quickly returned to their conversation. The guards walked onward.\n\n\"These rooms are for the concubines. Some have to share, but the ones with children each get their own. In about a half hour, the palace will send over lunch. The palace has female servants who come to these quarters but sometimes we help them gather the plates when the meals are done.\"\n\n\"What else are we supposed to do?\" Shekib's eyes were busy looking at the maze of doors.\n\n\"Just keep an eye on things. Most important is to control the ins and outs. No one is to come in without our knowledge and approval, just as no one is to go out. Every once in a while, especially for someone who is new here, they may want to wander around. It is our responsibility to guard against things like that. And sometimes the women call on us for help with something. Nothing else, really. Like I said before, everybody has a role in the palace. This is ours.\"\n\nThe voices in the large room grew louder in unsynchronized excitement. Ghafoor's ears perked.\n\n\"Let's go see what has the women feisty this morning. That kind of chatter means something's going on.\"\n\nGhafoor was not wrong. Amanullah, the king's son, had returned to the palace.\nCHAPTER 29\n\n\"Why all this excitement for the king's son?\"\n\n\"Why? Do you not know about his son Amanullah? You poor thing. You have so much to learn still!\"\n\nShekib decided Ghafoor was a snob but tolerable.\n\n\"Tell me then. Why all the fuss for him?\"\n\n\"He's the one. All bets are that he will succeed the king. He's the governor of Kabul and he's in charge of the army and the treasury.\"\n\n\"What's the treasury?\" Shekib had never before heard the word.\n\n\"You know! It's the group that works with the army. They give out food and uniforms. And . . . and sometimes they take care of the horses too.\" Something about the way Ghafoor fidgeted told Shekib her answer could not be trusted. \"But the most important thing about him is that he is not yet married. Amanullah is of age and his father is in search of the right bride for him. What a lucky girl she will be!\"\n\n\"When will he marry?\"\n\n\"The king has not yet decided. But Amanullah is well loved among the women of the harem. He is kind and handsome, more so than his father. The women servants of the palace are always on their best behavior around him, wishing they could be his concubine instead of his father's.\"\n\n\"Does he have his own harem?\"\n\n\"No. He hasn't yet married. He probably will once he marries.\"\n\nAmanullah had been gone for two months. Traveling to the disputed British-Afghan borders had exhausted him and he did not care for the usual palace pomp. Shekib would not get a look at him today but two days later, she did see his father.\n\nAmanullah must have brought good news from the front line.\n\nShekib stood in the corner of the pool room shifting her weight and wondering how long she would be living in this palace. Life was comfortable enough. The rice and vegetables were plentiful, the cakes sweet. She had a blanket to keep her warm at night and the company of women-men who meant her no harm. But Shekib was still restless. She wondered what her mother and father would think to see her living in the king's palace. She wondered if they could see her from heaven, dressed as a man. Her father would likely not notice any difference. He had never seen her as girl or boy while he walked the earth. She still grew angry when she thought of her father's land. Her land. Seeing the torn deed scatter in the hakim's courtyard like fallen leaves had hurt more than Azizullah's beating.\n\nBring your head out of the sky and understand your place in this world, Khanum Marjan had said.\n\nEveryone has a purpose here in the palace, Ghafoor had told her.\n\nShekib wondered what her place in this world was. Something told her it was not her place to be a house servant. And it was not her place to be the unwanted granddaughter. Surely, being a harem guard could not be her fate either, as comfortable as it had seemed in the last couple of days. Shekib knew in her heart that she would need to act if she were to find her true purpose.\n\nIf she hadn't been so preoccupied with finding a way out of her current situation, she might have noticed the king sooner. As it was, she had no idea how long he had been on the balcony. She hadn't even noticed that the women in the pool had quieted their loud laughs and started behaving more demurely.\n\n\"Guard!\"\n\nShekib jolted at the sound of a man's voice. She looked up and recognized the man from the carriage. Her heart pounded.\n\nHad he seen her daydreaming? Her defenses went up instinctively.\n\n\"Guard! Come here!\"\n\nShekib straightened her back, bowed her head and walked up the narrow stairs that led to the balcony. The king had entered from a back stairway, unnoticed. He had his uniform on but no hat. He was leaning over the railing, eyeing the women in the pool with casual interest.\n\nShekib said nothing and kept her head lowered.\n\nA lifetime passed before he spoke.\n\n\"Bring me Sakina.\"\n\n\"Bring her here?\"\n\nThe king turned around sharply. He was not accustomed to hearing the guards speak. His squinted eyes bored into her face. She instinctively turned to the side.\n\n\"You are new?\" he said finally.\n\n\"Yes, sir.\"\n\n\"Hmph. Tell Sakina I have called for her. She will show you the way.\"\n\nShekib nodded and headed back down the stairs. The women had heard the king's voice and awaited Shekib's return. They were familiar with his perch and his habits. Shekib still had much to learn about the palace. The women looked at each other but dared not look up. They spoke in a hush.\n\nShekib stood at the poolside and looked at Sakina, her thick, dark hair pulled back in a braid and her pale shoulders beaded with moisture.\n\n\"He calls for you,\" Shekib said softly.\n\nSakina smiled slyly, her lips pulled to one corner. \"Me, again? Dear God, I thought he had had enough of me by now.\" She spoke loudly enough for the ladies in the pool to hear.\n\nShekib saw some eyes roll, some mouths tighten. Benafsha's green eyes fixated on Sakina's behind.\n\n\"Some days men crave qaimaq. Other days they make do with spoiled milk.\" Benafsha's voice was cool and even. The others tried to hide their giggles. Benafsha dipped her head back, her long dark locks fanning into the water. For now, she was the qaimaq, the cream of the harem.\n\nSakina turned and shot her a hateful look. She stepped out of the water and reached for her towel. She wrapped it around her naked body and patted herself dry before standing at Shekib's side. Some of the women noticed Shekib's face for the first time.\n\n\"Allah, have mercy! Look at that! I guess after Benafsha, they have made an extra effort to pick guards who won't tempt the king!\"\n\n\"Mercy, Allah, please! I can't even imagine . . .\"\n\nSakina looked at her expectantly, ignoring the jabbering behind them.\n\n\"He said . . . he said you would show me the way,\" Shekib said finally.\n\nSakina raised an eyebrow. \"Yes, I know the way.\"\n\nShekib heard the conversation continue as she turned around.\n\n\"It looks like haleem, doesn't it?\"\n\nShekib sighed. This was not the first time her face had been compared to the dish of slow-cooked meat with grains, mushy enough that it could be spoon-fed to infants.\n\n\"Her face?\"\n\n\"Oh, you're right! How awful!\"\n\n\"Damn you, you know how much I love haleem! You've just gone and ruined my breakfast for me!\"\n\nThere were quiet chuckles as Shekiba's new nickname was passed around and adopted by the group.\n\nSakina took the lead and Shekib followed her into a back corridor and up a separate flight of steps. At the top of the steps was a heavy wooden door. Sakina stopped and turned back to face Shekib. \"Now, knock on that door and when you hear an answer, you'll open the door, then turn around and return to the others. This is as far as you go.\"\n\nShekib nodded and did as Sakina instructed. From within, she heard the king's voice bellow something incomprehensible. She opened the door just enough for Sakina to enter, holding her towel around her and her head lowered. Shekib closed the door and waited for a moment. She could hear their voices talking quietly. A laugh. A squeal. Shekib felt her face blush remembering that Sakina had nothing but a towel to cover her. She turned back down the stairs, suddenly afraid that she would be discovered lingering outside the room.\n\nHer introduction to the harem was complete. She understood now that the king visited who he wanted, when he wanted. He came often but usually didn't stay long. There were some consorts he favored more than others, some he ignored for the most part but kept in the harem. The nine women who had borne him sons were treated better than the others. These nine women were given dresses with the finest embroidery and the choicest fruits, and they walked taller than the rest. Their places were more secure than the others', thanks to their gifted wombs. Neelab, whose three boys had not lived more than a month, was the exception. She had disappointed the king more than those women who had borne him daughters and she would receive no special treatment until she could give him a son who would live long enough to at least take a few steps.\n\nShekib watched and learned over the next few months. She paid attention to the way the palace functioned, the way the women interacted with each other and the habits of the king. She was stronger than the other guards and began to take on duties that the others struggled with. It was easy for her to lug the heavy pails of water into the harem. She had no trouble carrying the children when they fell asleep in the courtyard. She was not a threat to anyone, thanks to her disfigurement.\n\nBut Shekib did not stop thinking about her own plight. She watched the women of the harem. At least they belonged to someone. At least they had someone to care for them, to look after them. Daughters looked at their mothers' faces, nestled against their bosoms. How that must feel!\n\nBut what would become of the guards?\n\nShekib needed a plan. In the meantime, she made sure to fulfill her obligations and keep Ghafoor and the palace satisfied. She did not want to invite any punishment, thinking back to her grandmother and Azizullah. In more powerful households, the food might have been better but the penalties were that much harsher.\n\nShe was in the courtyard of the harem when she saw him. He walked casually with another man, a man with a wool hat and a short beard. Shekib had seen the man in the wool hat before. He was Amanullah's friend, she had been told. His name was Agha Baraan. Shekib wondered what they were talking about. This was the fifth time she had seen the prince and she now understood why his arrival had created such a stir.\n\nAmanullah, the king's son, was striking. He was solidly built, a few inches taller than Shekib. When he walked, his broad shoulders spoke of confidence, even though he seemed to be close to Shekib's age. He exuded a natural boldness, tempered with kind, rational eyes.\n\nShekib melted into Shekiba.\n\nShe had instinctively tried to cover her left face and lower her gaze the first time she saw him. After the third sighting, however, she changed her approach as she realized she could take advantage of her \"manhood.\" She stared at the prince, who did not see her gawking anyway.\n\nHe gave her something to think about instead of her father's land. Or her dead family.\n\nThey were headed into the palace gardens. Shekib's hand touched her face and hair, wondering what she looked like to him. She knew half her face was actually beautiful. She could tell by the reaction of those who only saw as much.\n\nShe had worried that if she were ever to have children, they would turn away from her, repulsed by the demi-mask she wore. But the children of the harem reached out to her, trusted her, laughed when she tickled them. Maybe her own children would do the same. Maybe her own children would see her as her mother had, as unflawed and worthy of love.\n\nAnd then Shekib realized how she could change her fate. How she could stop being gifted from one stranger to another. But to do so she needed to belong to someone, to a man. And if she had sons, she would seal her fate. A mother of sons would not be passed from hand to hand like livestock.\n\nAmanullah had paused. His companion was pointing to some bushes that had flowered in the last week. He bent over and touched the leaves with an attentiveness Shekib would not have expected from the commander of the army. And the treasury, whatever that really was.\n\nShe stood tall, the right side of her face turned in his direction. She willed him to turn and look at her, to see her. She walked a few steps forward, hoping movement would attract his attention. He stood up and, almost as if pulled by her thoughts, turned in her direction.\n\nShekib's heart leapt into her throat. She froze, watching him from the corner of her eye and wondering what she should do. She gave a half smile and bowed her head just slightly, without diverting her view.\n\nHe began to speak and turned back to the friend, without changing his expression. Was he saying something about her? What could it be? Could he tell her apart from the other guards at this distance? Maybe the king had told him about her, the newest of the women-men.\n\nShekib realized she was smiling and turned back to face the house. She did not want anyone to see her staring at Amanullah and his friend as they walked thoughtfully through the maze of bushes and flowers. She bit her lower lip and pulled her shoulders back. An idea was beginning to take shape in her mind but it would require some work.\nCHAPTER 30\n\nSeasons changed, two years passed and I feared I was forgetting what my mother looked like. I doubted I would recognize my younger sisters if I were ever to run into them. I got updates from Khala Shaima but it usually wasn't good news. She tempered what she told us but she felt we had a right to know. Madar-jan had become as much of an addict as my father. Rohila and Sitara were mostly left to fend for themselves, though my grandmother sometimes stepped in to pick up the slack. In return, Madar-jan was doing more work around the compound and the already strained relationship between her and her in-laws had deteriorated. Padar-jan, when clear-headed, made her life miserable. After all, as his mother pointed out, she wasn't being much of a wife or mother these days.\n\nPart of me was thankful that I wasn't around to see what had become of my mother. Part of me wondered if things would have been different had I been sent back. Once I started that line of thinking, I could go on for days with what-if scenarios. I always ended up in the same place\u2014wondering how things would have worked out had I never been made a bacha posh. I think that's where my family started to crumble. Inevitably, I would wonder if Shahla and Parwin had the same thoughts. And if they still resented me.\n\nI also wondered what Bibi Shekiba was planning. The walls around me were so stifling I couldn't imagine what had given her a spark of hope.\n\nIn the meantime, I learned the rhythm of the compound and found my niche within it. The crescent moon rounded and thinned many times over as I found ways to make my life easier, though nothing changed who I was to Bibi Gulalai.\n\nMy son, Jahangir, was ten months old at the time, a miracle in his own right. Carrying him for nine months and pushing him out of my body had nearly ripped me apart. I had never seen so much blood. Jameela delivered him, as she had Shahnaz's children. Abdul Khaliq did not like for his wives to go to hospitals and there were no midwives in our area. My husband's wife cut the umbilical cord while I lay exhausted and stunned. I'd never felt so weak. Jameela rubbed my belly and brought thick broths of flour, oil, sugar and walnuts to my lips, urging me to drink. I faintly remember her praying over me, mumbling something about my not having the same fate as her uncle's wives. I wonder if it was her prayers that protected me.\n\nJameela and Shahnaz cared for my little boy for the first week while I recovered. Even Bibi Gulalai left me alone for a while. At least I had borne a son, she said. Finally, I had done something right.\n\nJahangir was named after a character Abdullah, Ashraf and I had created, a figure born of our collective imaginations. Jahangir was a strong and mighty man who feared no one. He was the ultimate athlete, the strongest fighter and the cleverest person in the whole country. He was the conqueror of the world, as his name implied. We all wanted to be Jahangir. He could do anything.\n\nIt became a running joke between us. When Abdullah huffed that he couldn't learn the newest karate move we'd seen, we told him Jahangir wouldn't have given up so easily. When I couldn't get the soccer ball anywhere near the goalpost, I focused my thoughts on Jahangir and how he would kick the ball. Ashraf channeled Jahangir's persona when he tried to haggle his way through the market, gloating when he felt he'd gotten a real bargain out of a vendor.\n\nWhile I was pregnant, I hadn't given much thought to a name, as if I believed babies were born with names, just as they were born with two arms or two legs. I was so frightened by the prospect of having a baby that I didn't care much about its name. But Jameela got me thinking.\n\n\"You must have a name and it has to mean something,\" she said.\n\nBy the time she had finished washing the blood from my thighs, my son was named.\n\nIt took me a couple of weeks to adjust to him. I would always be grateful to Jameela for her help. Even Shahnaz, at nineteen, was an experienced mother and couldn't resist teaching me how to feed, bathe and hold this tiny person.\n\nI fell in love with him. Jahangir was my salvation\u2014his face became my escape. He gave me reason to rise in the morning and to hope for tomorrow.\n\nKhala Shaima hadn't been by for months, which was unlike her. I worried that she might be ill but I had no way of getting in touch with her or finding out. I could only wait for her to show up again. I hadn't even seen Parwin in about a month. I wanted them both to see Jahangir. He was starting to clap his hands and would grab on to tables to stand up. I wanted his aunt to see the things he could do now.\n\nI had made up my mind to arrange a visit with Parwin. I had been given a little more freedom these days, now that I'd borne the family a son. Abdul Khaliq was bringing a foreigner to the house to talk business and there would be a lot of preparations to attend to. I knew I would be summoned to help the cook and servants. I decided to put off my visit until the following day.\n\nJust after midday prayers, I began kneading the dough for dumplings when Bibi Gulalai came into the kitchen. I waited for her to point out what I was doing wrong. She looked perplexed, as if there was something she wanted to say.\n\n\"What are you doing now?\"\n\n\"I'm going to roll out the dough for the aushak, Khala-jan. I finished cleaning the living room. It's ready for tonight.\"\n\n\"Yes, well, maybe . . . I guess that's fine. Keep on doing what you're doing.\"\n\nI was puzzled by her behavior. \"Is everything all right?\"\n\n\"Yes, everything's fine. Why? Why do you ask that?\"\n\n\"No reason, just that I . . . well, I was just asking,\" I said, and turned my attention back to the dough. It was getting tough. It was time to cut it into ovals and stuff it with leeks and scallions.\n\n\"Fine then,\" Bibi Gulalai said, and went back out the door.\n\nThat was my first clue that something was wrong. I think my mother-in-law, as cold as she was, was working up the nerve to tell me the news. She returned two hours later. This time Jameela was with her. Jahangir was crawling around the kitchen. I had blockaded off the stove, remembering how Bibi Shekiba had been burned as a child. I didn't want my son to carry such a scar. Life was difficult for the disfigured, I'd learned.\n\nJahangir was pulling on my skirt hem, whining. He was hungry but I wanted to finish the aushak before the guests arrived. I kept an eye on him but the expression on Jameela's face put my nerves on edge.\n\n\"Rahima, my grandchild is looking for food. I'll have Shahnaz feed him something,\" Bibi Gulalai said. She looked almost as uneasy as I felt.\n\n\"I'm done now, Khala-jan. I'll make something for him,\" I said nervously. \"Jameela, what's going on? What is it?\"\n\n\"Oh, Rahima-jan, something terrible has happened! I don't know how to share this sad news with you . . .\"\n\nMadar-jan. My mind flashed to her.\n\n\"What's happened, Jameela? Tell me!\"\n\n\"Your sister! Your sister Parwin has been taken to the hospital! She's been very badly injured!\"\n\nParwin?\n\n\"What hospital? How was she hurt?\" I was on my feet, my son in my arms.\n\n\"I only know what I've heard from Bibi Gulalai.\" Jameela turned to our mother-in-law, who scowled and looked away.\n\n\"Go on, tell her already!\"\n\n\"They say she set herself on fire this morning . . .\"\n\nNothing Jameela said registered after that. I put Jahangir on the ground as my head closed in on itself. Parwin had tried to kill herself. All I could picture was her unconvincing smile, her feeble reassurance that she was doing all right, that people were treating her well enough. Why hadn't I gone to visit her this morning?\n\nI pieced things together much later. Jameela took me to her house to lie down. She brought Jahangir along and one of the older girls in the house watched him while she sat with me. I asked her over and over again what happened and she explained it as best she could. Parwin had doused herself with cooking oil in the morning, while most of the women and children were eating breakfast. Her husband, Abdul Haidar, had already left the house.\n\nAbdul Haidar's second wife, Tuba, came to help tell me what had happened. Some things she made clear. Others she twisted in vagueness but I understood that my sister had been seen that morning with a fresh bruise on her face.\n\nTuba claimed they had no idea Parwin would do such a thing to herself. There were no warning signs, no red flags. She hadn't said anything, and as a matter of fact, Tuba said Parwin had smiled at her last night. I wanted to call her a liar. I knew the empty smile Tuba was talking about. I wanted to call them blind and stupid but my tongue was tied with guilt. If I, her own sister, had ignored her behavior, what could I expect of her co-wives? What could I expect of her husband?\n\nThey had heard the screams. She'd lit the match in the courtyard and that's where they found her, tried to cover her with a blanket to put out the flames. She'd fallen to the ground. There was a lot of confusion, screaming, trying to help. She'd passed out. They had taken her back to the house and tried to undress her, clean her burns. But it had been too much. They talked about it and talked about it and finally someone had decided that Parwin needed to go to the hospital.\n\nThe nearest hospital was not near at all. Her husband was not happy about being called back to the house to deal with the situation.\n\nSomehow, they'd sent word to my parents.\n\nMadar-jan must have been crazed with worry. Even Padar-jan, who had given us away for a bag of money, had been partial to his artistic daughter. The news must have shaken him. Khala Shaima had been at the house when they sent the message. She was on her way over to see me. I wanted to be with her but feared her reaction.\n\nPlease don't make this worse, Khala Shaima.\n\nBut Khala Shaima was our voice. She said what others dared not say. I needed her. She arrived in the evening, out of breath and teary eyed.\n\n\"Oh, my dear girl. I heard what happened! This is just awful. I can't believe it. That poor thing!\"\n\nShe hugged me tight. I could feel her clavicles press into my face. I'd never realized just how little flesh she had on her frame.\n\n\"Why did she do this, Khala Shaima! I was going to go see her today but I didn't. How could she do such a thing?\"\n\nI shuddered thinking of how painful it must have been, how horrifyingly painful.\n\n\"Sometimes women are pushed too far, kicked too hard, and there's no escape for them. Maybe she thought this was her only way. Oh, my poor niece!\"\n\nWe all need an escape. Khala Shaima was right.\n\n\"What did my niece say?\" Khala Shaima demanded. \"Tell me, was she talking when she was taken to the hospital?\"\n\nTuba shook her head. It had been an ugly scene. The smell of burning flesh, agonizing moans, hysteria. She couldn't bring herself to describe the horror to us.\n\n\"She wasn't talking at all? Was she conscious?\"\n\n\"She was . . . she was just lying very still but she was awake. I was talking to her,\" Tuba explained. \"She was listening to me but she just wasn't saying anything.\"\n\n\"She must have been in so much pain! Allah save her, that poor thing!\"\n\n\"I'm sure they will give her medicines in the hospital, Shaima-jan. Allah is great and I'm sure he's watching over her.\"\n\nI resisted the urge to spit at her. Pretending. She was pretending that things hadn't been that bad. Parwin hadn't been in that much pain. The hospital, which was a day's journey away and itself in woeful condition, would patch her up in no time. Allah, who had let this happen in the first place, would fix everything. It was all a game of pretend, just as Parwin had pretended every time we'd seen her. There was no honesty in our lives.\n\nKhala Shaima began to lament. I wished she would stop. The sound of her wails made my head spin.\n\n\"You people destroyed her,\" she cried. \"If she dies, her blood is on the hands of this family. Do you understand? This young girl's blood will be on your hands!\"\n\nThe women were silent. Tuba bit her lip and fought back tears. I wondered if she could be honest with me.\n\nI asked Tuba, with one mournful whisper, if my sister would live.\n\nThrough tears, she told me that God was great and that the whole family was praying for Parwin and that she was on her way to the hospital, so they really were very hopeful.\n\nI wanted to believe her. I wanted to believe that my sister would be okay.\n\nTuba's eyes told me it wasn't in her naseeb.\nCHAPTER 31\n\nParwin had stopped pretending.\n\nAfter ten agonizing days, her peace finally came.\n\nHer body was brought back and buried in the local cemetery. My father attended the burial, as did a few of my uncles and my grandfather.\n\nAt the fateha, I saw my mother again. The first time since my wedding day. Had I had a life more ordinary, I would not have been able to believe what she'd become.\n\n\"Rahima! Rahima, my daughter, oh God! Can you believe this? Allah has taken my daughter, my precious Parwin! So young! Oh, Rahima-jan, thank God she at least had you nearby!\"\n\nMy mother's hair was thin and stringy. Her words came out wet and lisped. She was missing a few teeth. Her skin sagged and she looked much older.\n\n\"Madar-jan!\" I hugged her tightly, surprised at how much like Khala Shaima she felt. \"Madar-jan, I've missed you so much!\"\n\n\"I've missed you too, my daughter! I've missed all of you! This is your son? God bless my grandchild!\"\n\n\"His name is Jahangir, Madar-jan. I wish . . . I wish you could have come to see him. He's a sweet child.\"\n\nMy son smiled, showing off his two bottom teeth. I waited for my mother to reach out to hold him. She didn't. She touched his cheek with her trembling hand and then looked away. Jahangir looked as disappointed as I felt in her lack of interest.\n\n\"Oh, I've wanted to come and see you, Rahima-jan. Especially when I heard about my grandchild, but it's not easy for me to get away from home, you know that. And your husband's home is not very close. With two kids at home, it just hasn't been possible.\"\n\nI bit my tongue, wondering why the distance wasn't too much for Khala Shaima and knowing that she could have brought my sisters or left them with one of my uncles' wives if she'd wanted to. My mother was weaker than I'd ever realized.\n\nWe women in mourning sat in a row, a wall of misery and tears. Women from our village came to pay their respects, whispering the same words of condolence to each of us one by one. Some even cried. I wondered why. So many of them had laughed to see my sister try to keep up with the other children, had called her Parwin-e-lang and had thanked God out loud that their own children weren't similarly deformed. They had made her feel small and wrong. Today they pretended to share our pain. I despised the insincerity.\n\nWe prayed. The women sat in rows before us, rocking to the rhythm of the prayer, the gray haired in the group blowing their noses into handkerchiefs and shaking their heads. They cried for us, their hearts softened with age and they themselves one step closer to the grave than most others. In the last ten days, my eyes had dried up. I sat still, blankly watching the faces in front of me. Madar-jan reached over and held my hand.\n\nRohila and Sitara sat on my right. I shook my head. How wrong I was to think I wouldn't have recognized my sisters! They had grown taller, more mature, but their faces were unchanged. They spoke sweetly and I hurt to think what home was like for them. Rohila grabbed me and wouldn't let go.\n\n\"Rahima, is it true? Is Parwin really dead? That's what Madar-jan told us but I can't believe it!\"\n\n\"I wish it weren't true.\" Nothing good came of pretending, I'd decided. \"How are you, Rohila? How are things at home?\"\n\n\"Can't you come back home sometimes? It's been so lonely since you all left!\"\n\nI believed her. I'd felt the same loneliness. I bet we all had, each in her corner of the world, separated by so many walls.\n\n\"Are you taking care of Sitara?\"\n\n\"Yes.\" Rohila nodded. It occurred to me that she was now the same age I'd been when I was married off. I looked at her and wondered if I'd looked just as young. I could see that her breasts were just starting to bud. Her shoulders were hunched forward, her chest pulled in. I recognized her posture. She was uncomfortable with her changing body. I wondered if Madar-jan had given her a bra yet.\n\nSitara was now almost nine years old and clung to Rohila more than she did to Madar-jan. She looked unsure around me, as if she didn't trust anyone but Rohila.\n\n\"How's Madar-jan been, Rohila?\" I whispered. I knew I would draw looks for talking, even in a hushed voice, during the fateha but this was my only opportunity to see my sisters. What I saw worried me.\n\nRohila shrugged her shoulders and glanced over at Madar-jan. \"She just lies around most of the time, just like Padar-jan. She cries a lot, especially when Khala Shaima comes over. That just makes Khala Shaima angrier.\"\n\nAt the mention of her name, Khala Shaima looked in our direction. I expected her to give us a chastising look but she didn't. She didn't give a damn about decorum.\n\n\"Are you going to school?\"\n\n\"Sometimes. Depends on what Padar-jan says. Sometimes, when she's taken Padar-jan's medicine, I have to stay home to clean up and get Madar-jan up and dressed. If Bibi-jan sees her the way she is, there's always a big fight.\"\n\nSitara stared at the ground but I could tell she was listening to our hushed conversation. She looked so timid, so different from the inquisitive little girl I'd left behind. I looked back to see Madar-jan wiping tears away, muttering angrily and fidgeting in her chair. I stared at her cheekbones, the lost look in her eyes. She was every emotion and blank at once. She was as badly addicted as my father.\n\nMadar-jan, what's become of you?\n\nMy stomach sank when I thought of what might happen to my sisters. I prayed for God to keep Khala Shaima alive and present in their lives. I pushed away the thought that they would be addicts soon too.\n\nThings were worse than I'd let myself believe, even with Khala Shaima's dismal updates.\n\n\"Rahima, why isn't Shahla here?\"\n\nShahla hadn't been allowed to come. She had just delivered her second child and it wasn't proper for her to be out of the home in her condition. I wondered how she had taken the news, alone and so far from the rest of us.\n\nRespects had been paid. The prayers were complete. The women repeated the procession, again wishing for Allah to ease our suffering, praying for Parwin's place among the angels in heaven and to themselves thinking it was in her best interests that she put herself out of her handicapped and childless misery. I wanted them all to disappear so I could spend this precious time with my mother and sisters.\n\nThe fateha passed quickly. I was back at the compound, but even more miserable. Madar-jan was in bad shape. Rohila had taken over as matriarch. How had this happened to us? I was the only one of my sisters who'd had a chance to live any kind of childhood at all, and that was only because I'd been a bacha posh. I looked at my son and thanked God for making him a boy. His lips turned up in a cheerful smile, his eyelashes so long they looked like they could get tangled. At least he had a chance.\n\nI wanted to be alone but there was little possibility of that at the compound. With the fateha over, so was my period of mourning. I was expected to resume my duties. Bibi Gulalai treated me just as she always had, if not worse. I think she had convinced herself that Parwin's suicide had been a purposeful attack on her family. With Parwin gone, I took the fall for the tragedy she'd brought to her extended family.\n\nI ignored everything and everyone. I carried out my duties, often with Jahangir a few feet away, playing or napping. I watched him wistfully, vowing to be better to him than my mother was being to my sisters. Thankfully, Abdul Khaliq had no trouble clothing and feeding his family. Jahangir was his son, as much as the other boys in the house. He would go to school and enjoy the privileges that came with being a warlord's son.\n\nAnd his father loved him in a way that surprised and relieved me. Abdul Khaliq kept his daughters at arm's length but his sons stayed at his side. The older boys even joined their father in some of his meetings. The younger ones nervously scattered when Abdul Khaliq came home, afraid of getting yelled at for spending too much time playing. He didn't have much patience for crying babies but he would watch them while they slept. Except for my son. Often, I caught him gently stroking Jahangir's cheek or whispering something into his ear. He held him with the same adoration I did. He chuckled when Jahangir spilled things and his chest swelled with pride to hear him say \"baba,\" as if he were hearing the word for the first time. The rhythmic breathing of his sleeping son calmed even his foulest of moods. I was happy Jahangir was a favorite, knowing I never would be. At least my son was safe.\n\nThe older boys, my son's brothers, both feared and adored their father. They vied for his attention and looked for ways to make him happy\u2014or at least not angry. The older boys stood tall when they recited suras from the Qur'an and the younger ones would bring him his sandals when he asked. He was proud to have boys. He smiled for them, and for little else.\n\nMy husband was spending more and more time with foreigners and the men he kept around as close advisers. Plans were brewing. The wives were on edge, though only Badriya knew why. If things were not going well for Abdul Khaliq, then things would not be going well for us. When we asked Badriya, she brushed us off dismissively.\n\n\"Don't bother yourselves worrying about it. He's worked up because he's renegotiating the arrangement he has with some of these people. It's too complicated to explain to you,\" she would say, not wanting to divulge the knowledge that set her apart from the rest of us. As his first wife, he discussed these matters with her. It was really the only interaction he had with her since he rarely called her to his bed. Everyone had a role in the house. That was hers.\n\nBut walls were thin and I spent most of my time at the main part of the house. I started to hear things when Abdul Khaliq and his men sat in the living room.\n\n\"They've got five more open seats for the province. The seat from our region needs to be filled. There are a few other powerful men who will be looking to step in and challenge you, Abdul Khaliq, but a woman candidate would be a sure thing. She would take the seat without question because of these stupid rules they've created.\"\n\n\"I don't like this idea. Why should we put a woman in a man's place? And even worse, you're asking me to put my wife in my place? Since when do we have a woman do a man's job?\"\n\n\"I understand that, sahib, truly. And, believe me, I don't like it any more than you do, but these are the rules. I'm simply suggesting we find a way to work around the system so that we don't lose all control over this area. The elections are coming up soon. We must plan for this.\"\n\n\"Damn whoever decided on these shameful rules! Telling us we have to have women representatives? They have no business there! Who do they think is going to look after the children then?\"\n\nHis advisers were silent. I could hear my husband pacing, grunting. I was surprised at what I was hearing. It sounded like they were suggesting that one of Abdul Khaliq's wives run in the upcoming parliamentary election! Would he really even consider such a move? We wives rarely left the compound. How could he possibly send us out to interact with strangers?\n\nI looked at the clock on the wall. Jahangir had been sleeping for forty minutes. He would be waking up soon. And Khala Shaima had promised to come over today. Tomorrow would mark forty days since Parwin's death.\n\n\"I'm simply presenting an option, sahib. I know it's not an attractive one but it may be our only one. I just don't want you to lose the opportunity to have some influence in the government. You're already in good position with the contracts you've secured.\"\n\nSmoke wafted from under the doorway, the acrid, thick smell of opium. My mind drifted home, to my father asleep in the living room and my mother sewing our clothes.\n\n\"It's true,\" another voice chimed in. \"There's no one else who can guarantee the same security\u2014especially over the bridge. Those foreigners, they certainly don't want to send their own soldiers to guard it. They depend on us. This pipeline is not a small project. They've been talking about it for years and this time it looks like it's actually going to happen.\"\n\n\"It's true. There's a lot of money in that pipe. And this area belongs to you, sahib. It would be a shame to lose even part of that control.\" The voice was measured and cautious.\n\n\"I know that!\" Abdul Khaliq thundered. \"Don't you think I know that? I don't need you to tell me things I already know!\"\n\nI didn't want to be around for what I knew was coming next. I picked up my son and walked back toward my own quarters to wait for Khala Shaima. I wanted her to take my mind off things. To tell me about Bibi Shekiba's mysterious plan.\nCHAPTER 32\n\nShekib waited for the right time. Mahbuba was rarely alone but she was the right person, Shekib had decided. She had borne the king four sons.\n\nThe first stage of Shekib's plan was to find out what Mahbuba had done right. How was it that she came to have four sons while other women continued to bring girls? There must have been something she had done differently to not have a single girl in her brood.\n\nHer boys ranged in age from one to seven years old. When Shekib came upon them, Mahbuba was bathing her youngest son. Her eyes searched for a towel, while the older boys ran off to play.\n\n\"Thank you! I thought I had something here with me,\" she said as Shekib handed her a cloth from a nearby shelf. Mahbuba held Saboor's hand as she dried him off.\n\n\"Certainly,\" Shekib mumbled. She had made a point of being quietly helpful with the king's consorts. It was unlike her to start a conversation but she forced herself to speak the lines she had rehearsed.\n\n\"You have lovely sons.\"\n\n\"Thank Allah, they are blessings,\" she said, sighing. The boy was wiggling to escape his mother's grip. His eyes chased after his brothers.\n\n\"The others have daughters. Mostly. You are lucky.\"\n\n\"Yes, well, some of us are blessed with sons and others have to bear daughters.\"\n\n\"You have made the king very happy.\"\n\nIt dawned on Mahbuba that this conversation was peculiar. She turned around to see who she was talking to.\n\n\"Oh, you! What is your name?\"\n\n\"Shekib.\" She looked at Mahbuba straight on. The women of the harem had made her quite comfortable in the last few weeks. They were too busy picking each other apart to pay attention to the woman-man guard with the melted face. Shekib no longer missed being able to pull her head scarf over her cheek. She found it liberating to walk about, her hands in her pockets and the sun on her face.\n\n\"Right. Shekib. Let me ask you something. What's your real name, my dear? Your girl name?\"\n\nShekib fidgeted. Mahbuba had surprised her.\n\n\"My name is Shekiba.\"\n\n\"Clever. Bet that was Ghafoor's idea. Do you and the others get along well enough with her? She can be such a nuisance.\"\n\n\"Sure,\" Shekib said vaguely.\n\n\"It's so ridiculous that they have you wearing those uniforms. As if anyone would forget that you are not men. As if we need guards anyway. What we need are more servants to help us with the children. But that would offend the king's sense of security.\"\n\n\"Some people forget.\"\n\n\"Forget that you're women? Do you really think so?\" Mahbuba was struggling to dress her son. He scratched at his mother's face in angry protest. She turned him around and locked him between her knees. He looked at Shekib with a defeated pout.\n\n\"How did you make it so . . . how did you manage to have boys?\"\n\n\"What?\"\n\n\"I want to know how you managed to have all boys. What did you do?\"\n\nMahbuba laughed naughtily. \"Do you want me to start with the basics? You are dressed as a man but know nothing about their parts, eh?\"\n\nShekib blushed. \"I mean . . . no, that's not what I meant. I was asking how . . . the other women have girls. How did you manage to have boys instead of girls?\" she stammered.\n\n\"Do you think you are the first to ask me such a question? Most of the women in this house have come to me looking for that very answer. I have borne the king more sons than any other woman!\" Mahbuba needed a minute to sing her own praises. Shekib waited. \"I have given him son after son and nothing but sons! That is why he looks at me with fire in his eyes, with respect in his heart. You are a wise girl-boy. You are looking for the key to a satisfied man.\"\n\nThe humidity of the bathhouse made Shekib's breaths heavy. She wondered if Mahbuba would ever reveal her secret. Maybe this had been a mistake.\n\n\"But tell me. Why are you asking such a question? You are a man now, are you not? Are you to be turned back into a woman? Are you to be married?\"\n\nShekib shook her head.\n\n\"I did not think so. Then why bother to ask for answers you have no business using? Did someone send you to ask me this? Who was it? Was it Shokria? I've seen the way she looks at my boys. Five girls, she has. Can you imagine? That witch. I'll fix her if she casts a jealous eye on my sons!\"\n\n\"No, no one sent me!\" Shekib panicked. She did not want to cause any fuss among the women. If it led back to her, it would not help her situation.\n\n\"Farida? She's another one with her devil eyes . . . can't be trusted. You shouldn't go about the harem doing their dirty work for them!\" Her son had managed to finally twist himself free. Mahbuba sighed. He was missing one sock.\n\n\"Forgive me, I have not been sent by anyone. I was . . . I was just asking out of my own curiosity.\"\n\n\"Are you wanted by a man?\"\n\n\"Am I . . . no, I just\u2014\" Shekib decided she should close this conversation.\n\n\"I am teasing you. I will tell you a few tricks if you promise\u2014\" Mahbuba paused and looked from left to right dramatically. Her voice turned into a hush. \"If you promise that you will not share these secrets with anyone else. You can use them if you find yourself under a man one day and in the mood to give him a son.\"\n\nShekib squatted next to Mahbuba, her ears hot.\n\nSome of what she was told, she never would have anticipated. And would never have been able to repeat.\n\nBut she committed the tips to memory, hoping that they might prove useful. The shape of the moon, the seeds of the yellow flowered plant, the juice of an apple with no brown spots. These were simpler. But the other things, the things with the man, these made Shekib wonder if Mahbuba was not looking to make a fool out of her. But there had been no glimmer of trickery in her eye. She spoke casually, as if the things she talked about were commonplace and ordinary. To Mahbuba they were. To Shekib, they were not.\n\nDid the women really allow the king to do such things? She thought of Halima and could not imagine it. Then she thought of Sakina, the way she had walked, half-naked, to the king's chamber and knocked on the door with feigned timidity. It could be true.\n\nShe could not stop her mind from drifting to Amanullah, the governor of Kabul. She thought of the way he walked, the confidence of his step, his fingers grazing the petals with delicate respect. She wondered what it would be like to be near him, to feel his breath on her face, moist and warm like the air of the harem's bath room. She thought of her fingers tracing the borders of his neatly trimmed beard and the medals of his uniform pressing against her unfettered bosom.\n\nShekib shook her head and hoped her face did not betray her thoughts.\n\nAt night, the guards slept in a room just outside the concubines' quarters. They took turns standing guard. Tonight was Shekib's turn. Kabul's air was brisk but she did not mind it. She wrapped her coat tighter around her and rubbed her hands together. She thought back to her first night on duty, a night she spent standing at attention, terrified that someone would find her asleep or sneak up on her. By morning she had drawn her weapon, a heavy baton, a half dozen times, only to frighten the frog who had wandered too far from the pond. She nearly collapsed when Ghafoor came to ask her how her night had gone.\n\n\"Why are there so many noises at night? There are frogs and lizards and soldiers coughing and pacing! You said I should just stand in the quiet night until morning. It wasn't a quiet night at all!\"\n\nGhafoor had laughed uncontrollably. Two soldiers had turned, their brows furrowed in disapproval to hear a woman laugh so loudly, even if it was a woman-man.\n\n\"Did the frogs shake you up? Well, little girl from the village, I didn't think a few little night critters would make you so nervous!\"\n\nShekib had felt a little embarrassed. \"It wasn't the frogs, it was mostly the soldiers . . . they are loud but I couldn't see them. I just thought . . .\"\n\n\"Don't worry about it. The next night will be easier on you. You'll grow accustomed to the sounds of the palace at night. You might even enjoy it more than the days.\"\n\nGhafoor was right, although Shekib kept that revelation to herself. Over the next few months, she grew content to sit in the dark, the dim light from the king's main residence and a few oil lanterns casting enough of a glow to make a game of shadows. Shekib smiled when some resembled animals, laughed when one took the shape of her grandmother.\n\nTariq joined her on those nights when she could not sleep. She had been in the king's presence more than a few times and he hardly glanced at her. She was losing hope for being that rose that is plucked from the garden, as she had put it. She fretted, bit her nails and creased her forehead but Shekib did not mind her company.\n\n\"Ghafoor is snoring again.\"\n\nShekib nodded.\n\n\"It's like sleeping next to a congested horse. I can't take it. I don't know how the others ignore it.\"\n\n\"She'll deny it in the morning.\"\n\nTariq smiled. \"Anything happening in the palace?\"\n\n\"No, not so far.\" It was quiet in the gardens, but the palace was unpredictable. People came and went at odd hours sometimes. And from time to time, King Habibullah hungered for a concubine in the darkest hour.\n\nThe guards were silent. Tariq sighed. Something was on her mind.\n\n\"Are you happy here?\" she asked.\n\n\"Happy? What do you mean?\"\n\n\"I mean, are you happy? Are you satisfied with this?\"\n\n\"I've seen worse.\"\n\n\"You don't miss your family?\"\n\n\"I miss them as much as they miss me.\"\n\nTariq did not know how to interpret Shekib's response. She understood from her tone that she would not elaborate. She pulled at her bangs, tried to make them reach her eyebrows.\n\n\"But how much longer do you think we will be here?\"\n\n\"I don't know.\"\n\n\"I wonder sometimes.\"\n\n\"About what?\"\n\n\"I wonder what the palace will do with us. How long will they keep us here? I want to be married. I want to have children and a home. I want to live somewhere else, don't you?\"\n\nTariq, dressed as a man, was a woman after all. Her voice was nearly cracking. Shekib understood better than she let on. She had to protect her own plan.\n\n\"I don't know. We have a comfortable life here.\"\n\nTariq sighed heavily. \"It's comfortable, but this can't be it. I'm not like Ghafoor. Or even Karim. I don't want to wear pants for the rest of my life. I was happy as a girl.\"\n\nTariq's laments were interrupted by the sound of a door slamming shut. The guards froze and looked for the source of the noise. They focused their eyes in the dark, trying to locate the footsteps.\n\n\"Where was\u2014\"\n\n\"Shhh!\" Shekib hissed.\n\nA shadow scurried away from the harem's side door. The figure was running back into the palace.\n\n\"Do you think it's the king?\"\n\nShekib did not. King Habibullah never left from the side door. And he had no need to sneak past the guards either.\n\n\"Who goes there?\" she called out. She wrapped her fingers around her baton.\n\nThe figure scurried faster, passing under the yellow glow of a lantern. From the breadth of the shoulders and the shape of the pants, they could see it was a man. A man in the harem?\n\n\"This is bizarre. Stay here. I'm going to check on things inside,\" Shekib said.\n\nBut the harem was peaceful. Shekib could hear the light snores. The man had come from somewhere though. She waited, cocked her ear for any movements. She tiptoed through the hallway slowly. Carefully.\n\nWhen she had crossed through the bath and checked the hallway on the opposite side, she retraced her steps. Something stirred by the foyer. She focused her eyes in the dark as the figure turned toward her.\n\n\"Did you find anything?\"\n\nIt was Tariq.\n\nShekib sighed and shook her head. They stepped back into the night air and looked into the courtyard, across the gardens, to the palace. Nothing moved. Shekib wondered who it could have been. Someone had paid a visit to one of the king's concubines. Who could be so bold as to trespass here? And which woman had allowed him into her chambers?\n\nShekib and Tariq sat in silence, chewing over the same thought. If the palace were to find out, the guards would be held responsible.\nCHAPTER 33\n\nShekib and Tariq entered their sleeping quarters when daylight broke. They had neither seen nor heard anything else throughout the night. The soldiers were walking about now and the servants looked hurried. The king was likely expecting a visitor.\n\nGhafoor was awake, her arms stretched over her head as she yawned. The others rubbed their eyes.\n\n\"Tariq? You're up already? Did you not sleep last night?\" Ghafoor asked, puzzled.\n\n\"Something happened last night,\" Shekib said softly. \"Something you all need to know about.\"\n\nHer words, rare as they were, got everyone's attention.\n\n\"We saw someone leaving the harem through the side door, which should have been locked. It looked like a man. He ran off toward the palace but in the dark we couldn't make out his face.\"\n\n\"It must have been the king. You know his urges come at odd hours.\"\n\nTariq shook her head. \"It wasn't the king, trust me. I know his shape. This man was leaner, taller. And the king doesn't sneak in and out of the side door. He comes and goes as he pleases, even when the hour is late. This was someone else.\"\n\nGhafoor and Karim leaned forward; they were just now making the realization that Shekib and Tariq had made last night. Qasim looked at her sister's concerned face.\n\n\"Did you hear anything inside? Was anyone awake?\" Karim asked.\n\n\"Nothing. I walked through the hallways and heard nothing at all, saw no one. Whoever it was that let him in was not making a sound,\" Shekib said, her tone flat and serious.\n\n\"Of course not,\" Ghafoor said. \"But if this has happened once, then it has probably happened twice and three times and more. We have a serious problem on our hands, guards. If the king learns that someone has been sneaking past us and paying secret visits to his private harem, we can start saying our final prayers.\"\n\n\"Should we tell someone in the palace?\" Qasim asked nervously. \"No, absolutely not!\" Ghafoor cried. \"We have to find out what we can on our own and stop this from exploding on us.\"\n\nKarim and Tariq nodded in agreement. Shekib stood in silence. Ghafoor was taking charge now.\n\n\"First of all, we need to speak with the concubines, privately, one at a time, and see if anyone can give us any information.\"\n\n\"You think whoever brought him in is going to tell us?\" Qasim asked.\n\n\"No, she won't tell us anything, I'm sure. But if this has been happening, someone must have heard something and I'm sure that someone else will be willing to talk about it. You know how these women are with each other. They can't wait for a chance to rip the others to shreds.\"\n\n\"I can't believe we haven't already heard about this,\" Tariq said.\n\n\"This was bound to happen. It was just a matter of time. There are just too many women in one house. One of them was going to invite trouble.\" Ghafoor spoke confidently, as if she had predicted this months ago.\n\nShekib and Tariq lay down to get some rest. The others assumed their posts, rotating to cover Tariq's position as well so that she could close her reddened eyes for a few hours. The situation had given Ghafoor new energy. Her face was serious and her tone urgent. She gave orders as if she were a palace general commanding her soldiers.\n\nKarim and Qasim shot each other looks but let her be.\n\nShekib could not sleep. From the moment she had seen that shadowy figure, a feeling had taken root in her stomach. Something would come from this. She lay on her side, looking at the cracks and crevices of the stone wall. She was not in her village now. She was not even in Azizullah's house. She was in the king's palace. Bigger people meant bigger problems.\n\nSleep claimed her finally but briefly. In the afternoon, she rose and dressed. She found Karim in the bathhouse. Five women soaked in the pool. Shekib looked up and saw the balcony was empty.\n\n\"Have you heard anything?\"\n\nKarim shook her head. \"Ghafoor says she has her suspicions but no one is talking yet. I've asked two of the women, Parisa and Benazir, if they heard anything odd last night but they said they hadn't. We have only asked those women who have children since it's unlikely young ones would sleep through a visitor in the dark.\"\n\nShekib nodded. The reasoning made sense.\n\n\"But it is best not to create too much of a stir since one of the women might actually tell King Habibullah what we've been asking.\" Karim sighed heavily. \"There are just so many ways for this to turn on us.\"\n\n\"That's how things are. There always has to be someone to blame,\" Shekib said. She could still see Bobo Shahgul's crooked finger pointing at her, her beady eyes filled with hatred.\n\nThe next week brought no revelations, no clues as to who had come to visit the harem. The only proof that Shekib and Tariq had not imagined the whole thing was that the visitor had returned. Just five days after the first sighting, he was seen again leaving the house. This time it had been Qasim's turn for night duty.\n\nQasim's description seconded what Shekib and Tariq had described.\n\n\"Did you go after him? Did you see his face?\" Ghafoor had demanded.\n\n\"No . . . I only saw\u2014\"\n\n\"You just stood there? We're trying to find out who this is and you just stand there? Great job guarding the harem!\" She threw her arms up in exasperation.\n\n\"He was walking so fast. I didn't think I should . . .\"\n\n\"Forget it. It's fine. There's no point chasing this man down. He probably already knows that we've seen him and obviously he doesn't care. He's only concerned about getting caught by the palace. He knows we cannot do anything,\" Karim said with annoyance.\n\n\"What are you talking about? If Qasim would have had half a nerve, she could have\u2014\"\n\n\"Then you can take her night duty and you can chase him down yourself!\" Karim shouted. She was tired of hearing Ghafoor's complaints. Ghafoor pursed her lips but was silenced.\n\nThe bickering had seeped through the harem and into the guards' quarters. Their small troupe now felt the pressure and it was straining the thin friendships that had formed among them. Shekib watched as the cracks grew, week by week.\n\nThe man visited the harem about once a week and though we posted a guard at the side door, we were somehow unable to confront or identify him. By Ghafoor's account, he never appeared on her overnight shifts but the others doubted this. More likely, she was turning a blind eye since she too did not want to be the one to chase him down in the middle of the night. Better to find out from the woman and put a stop to it there.\n\nIn the meantime, Shekib decided to continue laying the groundwork for her own plans. She approached a few women with two purposes in mind. She asked if they had heard anything, any strange noises in the night. And she found ways to make mention of her own family. She awkwardly and clumsily told a story about them, about the string of boys her mother had borne, her aunts had borne, her grandmother had borne.\n\nWomen in our family have many sons. I was the only daughter.\n\nCurious looks. The women were not sure why the disfigured guard was sharing such information but they nodded politely and moved on. Or they shooed her off and crossed their brows. But Shekib persisted.\n\nSomething told her she did not have much time.\nCHAPTER 34\n\n\"She looked terrible, Khala-jan,\" I said. \"I just never thought I would see my mother looking like that. Shahla would have been in tears to see her!\"\n\nIt occurred to me, though, that Shahla too had likely changed. None of us were what we had been three years ago. Shahla now had two children. I thought of her when I looked at Shahnaz. I wondered how her new family was treating her. I prayed she was faring better than Parwin had.\n\n\"Rohila is a smart girl. I just wish they would send them to school.\" She sighed. \"That's all I wanted for each of you. A bit of education that you could carry with you through life.\"\n\n\"What good did it do me?\" I asked in frustration. \"I went to school for a few years and it did absolutely nothing to change where I am now.\"\n\n\"You'll see later in life. Every bit does some good. Look at me. I'm lucky I know how to read. It's a candle in a dark room. What I don't know, I can find out for myself. It's easier to fool someone who can't figure things out on his own.\"\n\nI bit my tongue. I still didn't see what use it was. Khala Shaima had been the only one of her sisters to make it to the eighth grade, since no suitors had come for her. Other than her being able to read a newspaper or a book here and there, I didn't see how her life was any better. She hadn't been able to stop anything from happening to my sisters or me.\n\n\"Your mother will be all right,\" Khala Shaima said, misreading the doubtful expression on my face. \"The human spirit, you know what they say about the human spirit? It is harder than a rock and more delicate than a flower petal.\"\n\n\"Sure.\"\n\n\"Your mother is protecting herself. She's protecting her spirit, making the delicate petal as hard as rock with the medicine your father brings home because it's the only way she knows to survive. You should do the same, in a different way, of course. Don't forget that you are part flower petal and part rock too.\"\n\nShe sighed.\n\n\"That damn medicine. Now that Abdul Khaliq is your father's damaat, he can get as much as he wants. There was just too much of it for your mother to resist.\"\n\n\"They made out well in this whole arrangement,\" I said with more cynicism than I'd intended. Sometimes I saw my mother as a victim. Other times I thought of her as my father's coconspirator. Either way, my sisters were the ones who suffered. I looked at Jahangir and swore never to do the same to him.\n\n\"You can blame your mother but it won't do any good. You don't know what it was like to be in her position. In an ant colony, dew is a flood.\"\n\n\"But you said it too! You were the one telling her that she shouldn't give us away. I remember you arguing with her!\"\n\nKhala Shaima sighed and looked away, frustrated. \"Of course I told her all those things! And she tried. She tried to talk to your father but he's\u2014\"\n\n\"I know what he is.\"\n\nKhala Shaima quieted. She bit her lip. It was time to change the subject. \"How has Abdul Khaliq been with you lately?\"\n\n\"He's so busy with his own affairs that he's hardly around the house at all.\"\n\n\"Good. Busy with what?\"\n\nI shrugged my shoulders. \"I'm not sure exactly but I heard him talking with his advisers and guards the other day. Something about his soldiers doing what the foreign soldiers can't do.\"\n\n\"Or don't want to do. He's got a good racket. These other countries come in here and throw a few bombs around. Friends today with yesterday's enemies. They just change their hats and all of a sudden, they're allies to these western countries. No one cares what Abdul Khaliq was doing for the past few years.\"\n\n\"What was he doing?\"\n\nKhala Shaima's lips pursed together. \"He's your husband, Rahima, so I would have thought you'd have a better idea by now. How do you think he got to be so rich and powerful? Off the blood of our own people, that's how. By ransoming, stealing, killing and then washing up and looking pretty for the westerners who either don't know any better or pretend not to. Your husband is not the only one and he's probably not even one of the worst. You were too young to really know how things worked and no one in your house would talk about it since your father was fighting under him.\" Khala Shaima's voice was a cautious whisper.\n\nI remembered how Shahnaz had come to be Abdul Khaliq's wife\u2014pillaged from her home as if she were a piece of jewelry or silver serving tray.\n\n\"You should know these things, Rahima, since you're living here in this house. As his wife, no less. But don't speak of them, ever. Not even with his other wives. Understand me?\"\n\nI nodded. Her warning was unnecessary. I already knew how loose the lips in this house were.\n\n\"His advisers were telling him he should have one of his wives run for parliament,\" I said, thinking of the conversation I'd overheard. \"It sounds like such a crazy idea.\"\n\n\"Run for parliament? Those conniving bastards!\"\n\n\"They really want him to. That would be a big change for him, Khala Shaima, wouldn't it? Imagine, one of his wives in the parliament.\"\n\n\"To hell it's a big change! It's a charade. There's a rule that a certain number of seats have to be filled by women. They made this rule part of the constitution because otherwise no one would give any woman the time of day. But he'll put one of his wives in and tell her exactly what to say, how to vote, who to talk to. It's no different than Abdul Khaliq taking the seat himself!\" Her words were bitter, underscored by the way she spat some letters out.\n\nI hadn't thought of the situation that way but Khala Shaima's reasoning made sense. And it explained why Abdul Khaliq was even considering the option. It was as his adviser had said\u2014this might be the only way to keep control over the region.\n\n\"Did he say which wife he wanted to have run?\"\n\n\"No, they didn't.\" I had wondered the same thing myself.\n\n\"Probably Badriya.\"\n\n\"Why Badriya?\"\n\n\"Because Jameela is too pretty. He won't want men's eyes on her. And you and Shahnaz are too young.\"\n\nShe was right.\n\nOver the next few weeks Badriya was groomed for the election. Abdul Khaliq spent more time with her behind closed doors. We didn't know what they were talking about and Badriya was tight-lipped, or at least put on the appearance of being so.\n\n\"It's going to be a difficult election,\" she said, tapping her finger against her lips. It was obvious she was feeling very special to have been chosen for the task. \"We've been discussing getting the word out, getting my name out.\"\n\n\"What kinds of things will you have to do if you're part of the parliament?\" Shahnaz asked. It was a warm afternoon and the children were all in the courtyard. Abdul Khaliq had gone on an overnight trip and Bibi Gulalai was in bed, recovering from a cold that she said had nearly killed her three times over. The compound could breathe now that Bibi Gulalai swore she couldn't.\n\n\"Silly thing! Don't you know what the parliament does? Good thing it's me and not you that's running!\"\n\nI saw Jameela swallow a smile. We both knew Badriya was trying to come up with an answer.\n\n\"It's a lot of work once you hold a jirga seat. There are things to vote on, decisions to make . . .\" She waved her hand about as if it was just too much trouble to bother explaining.\n\nShahnaz raised her eyebrows. \"But you're going to be covered, right?\"\n\n\"Of course! I'll be wearing my burqa.\"\n\n\"And if you make it into the parliament, then what? It's mostly men, isn't it? You'll have to go and meet with them?\"\n\n\"Yes, that'll be my responsibility as an elected official. We'll have to talk about the voting, the issues.\"\n\n\"When are the elections?\"\n\n\"In two months. There's a lot to be done.\" Badriya sighed as if she had just realized how much work awaited her.\n\nBadriya, the first wife, had been accustomed to a status within the compound but she had started to resent all the attention the other wives were getting. This development was just the boost she needed to reclaim her distinction. But not all attention was good attention.\n\nAbout a week after our conversation, I woke up in the morning, tied my hair behind my head and slipped on my work dress. I was to clean out the chicken coop. The smell always turned my stomach so I brought a square of cloth to tie around my nose and mouth.\n\nI walked outside and went to the far edge of the compound. The chickens were up early and clucked with excitement at my arrival. Feathers flew into the air, making me cough. I adjusted my mask and took a deep breath.\n\nBefore I could pick up my broom, the clucking heightened, and the chickens started to pace the area as if they'd been upset by something. I turned back toward the compound and saw Badriya walking behind the house. She had her left arm tucked in under her side and walked with a slight limp that made me think of Parwin.\n\nI watched her and realized she hadn't seen me. She stopped at the clothesline and reached up to pull off a chador and a dress. It took her three tries before she was able to get the dress down; each time she reached upward, she would stop short and withdraw her arm sharply, shaking her head. I wondered what had happened and was happy for an excuse to delay my task anyway.\n\n\"Badriya-jan! Sobh bakhair!\"\n\nBadriya whipped around, her surprised expression interrupted by a wince. \"Oh, Rahima! Yes, sobh bakhair. Good morning to you too. What are you doing back here?\"\n\nHer arm was still tucked in.\n\n\"I have to clean out the henhouse,\" I explained. \"It looks like you're having a hard time with your arm. What happened?\"\n\nBadriya frowned. \"It's nothing,\" she said unconvincingly. As she went to turn back to the clothesline, I caught a glimpse of her neckline and saw bruising around her collarbone. I started to say something about it but caught myself. She tried to move around as usual but her face betrayed her.\n\n\"Just get on with whatever you were doing, Rahima. I'm too busy to chat,\" she said dismissively. I walked back toward the chickens, looking over my shoulder to confirm that she was still limping. Hashmat met her at the door of the house and helped her in. He noticed me watching and shook his head. I kept my distance from him these days. By now I'd figured out that I shouldn't be around boys my own age or older, no matter what their relation to me. And I didn't want to invite any talk about Abdullah, who now seemed like a character I'd created in my imagination.\n\nIn the afternoon, I returned to Jameela's house. Most of the time, my son joined me while I did my chores, but cleaning the henhouse was impossible with him around. Jahangir had taken to spending time with Jameela while I was attending to some of the more taxing work. She enjoyed having him around now that her own children were grown and I trusted her more than anyone else. Even though I lived with Shahnaz, it was Jameela I turned to with every question about feeding and bathing Jahangir. She even knitted a sweater and cap for him to keep him warm through the winter.\n\n\"He hasn't been too much trouble, has he?\" I asked, knowing what her answer would be.\n\n\"Oh, he's getting sweeter every day, Rahima. Tomorrow we should espand him, to keep the evil eye away. Before you know it, he's going to be talking his mouth off. You should see him trying.\"\n\n\"Have you seen Badriya today?\" I asked, wanting to talk to someone about what I'd observed.\n\n\"No, are you looking for her?\" She was feeding morsels of tea-soaked bread into Jahangir's open mouth.\n\n\"I saw her this morning, outside behind the house. She looks like her arm's pretty badly hurt. And she's not walking right.\"\n\n\"Hm. Did you ask her about it?\" Jameela said, shaking her head.\n\n\"Yes, but she brushed me off.\"\n\n\"She expected too much.\" Jameela sighed. \"A man has to feel that he's in charge of his home, at the end of the day. Especially a man like Abdul Khaliq Khan.\"\n\n\"What are you talking about?\"\n\n\"You know it's not easy for him to agree to have her run for election. Her name has to be publicized in the area for people to vote for her; she will be talked about. And it will be big news that the powerful Abdul Khaliq's wife is out of the house, running for parliament. It's not what he wants.\"\n\nI felt stupid for not figuring this out on my own.\n\n\"Last night, I heard him.\"\n\n\"What happened?\"\n\n\"He warned her not to turn into one of those women, the kind who make a lot of noise, talk with lots of people. He wanted her to know that it was his decision to put her into the election and that it had nothing to do with her. I think he's heard her talking about it. That's not what he wants of his wives. I don't know what exactly she said but he was rough on her last night.\" Jameela shook her head and clucked her tongue. \"It sounded like he was at his worst.\"\n\nAs much as her smugness had irritated me, I pitied Badriya. We all knew Abdul Khaliq's heavy hand. I wondered if Badriya regretted being chosen to run for the jirga seat.\n\n\"Is he still going to go through with it? I mean, to have her run for the parliament?\"\n\n\"I think so. He wants the power. Through her, he would have his finger in a lot of different projects. He's not going to give that up, as much as he might hate to have his wife's name written on ballots and know that she's going to have to be away from the house sometimes to fulfill her duties. I'm sure he's trying to think of a way around all that anyway.\"\n\nAbdul Khaliq had indeed taught Badriya a lesson. She didn't talk about the upcoming election after that. He met with her from time to time as well as his advisers. I caught bits and pieces of conversations. Things weren't going well. His advisers weren't sure Badriya would win the seat in the jirga but Abdul Khaliq had a way of convincing them.\n\nMy husband was used to getting his way. If he wanted Badriya elected, she would be.\nCHAPTER 35\n\nAbdul Khaliq and Badriya traveled to Kabul frequently. He hated it. She claimed to enjoy it but we could see that she didn't. Abdul Khaliq was always tense before they left and even worse by the time they returned.\n\nBadriya had won the election, mostly thanks to the women's votes, according to local news. To me and my husband's two other wives, it seemed unreal that something as important sounding as the parliament would let women vote. Khala Shaima had come by again. I asked her about my family and Bibi Shekiba. She asked me about Badriya and Abdul Khaliq. By this time, my na\u00efvet\u00e9 had been washed away. I knew just what kind of man I was married to and I knew he had done terrible things to people. Jahangir, my son, was starting to look like his father, which frightened me. Sometimes I worried I might grow to dislike him if he did. I cringed when he became angry or frustrated, his shrieks taking on a familiar hostility. But his moods were nothing in comparison to his father's and he was otherwise very loving and affectionate, pulling my face to his and patting my head as if I were the child and he the parent.\n\nKhala Shaima's breathing was more labored today. It could have been the dust in the air, her waning health or my own paranoia. She was the only family that I had left and I often worried about what I would do without her visits. I prayed for her health selfishly.\n\n\"He's telling her exactly how to vote. She's got no choice but to follow his orders.\"\n\nI nodded. \"You should see how exhausted she looks every time they return from Kabul. She looks completely drained.\"\n\n\"But there must be some way, some way for her to vote on her own. He doesn't go into the parliament, you know. Once she's in a session, he's not there to sway her.\"\n\n\"I'm sure he's got ways of knowing or watching every little thing that happens behind those doors.\" I pried Jahangir's small hand open and took away the stone he had found. He had watched his older half brothers playing and now wanted to imitate them. His round eyes lit up when he saw them, his mouth broke into a wide grin and he would pull my face and point for me to look at what he was seeing.\n\n\"Yes, bachem, I see them. You're going to grow up to be just as big and strong. Just wait.\" Sometimes I tried to imagine what he would look like in ten years but my mind couldn't envision him as anything but the sweet toddler he was. When I tried to picture myself in ten years it was frightening. My hands were already rough and knobby. My back ached at night, partly from carrying Jahangir for nine months and partly from being bent over to wash clothes and scrub floors most days. This home, this life, had aged me. Maybe that was what Parwin had seen, life in ten years. Maybe it was a sight too ugly to bear.\n\nEveryone needs an escape.\n\n\"Maybe you can go to Kabul with her,\" Khala Shaima suggested. She started to cough, a rough cough that rattled her whole body. I put my hand over hers and pushed a glass of water closer to her. \"Thank you, dokhtar-jan. Bah! The dust is irritating me more than usual today.\"\n\nI hoped that was all it was.\n\n\"Anyway, what was I saying? Yes, why don't you see if you can go to Kabul with her?\"\n\n\"What am I going to do in Kabul, Khala-jan?\"\n\n\"Who knows,\" she said vaguely. \"But in Kabul you'll see different things. It's an education of sorts. See how people live there, see the buildings and see what the parliament is doing. It's an opportunity for you.\"\n\nThe idea was tempting. I wouldn't have minded seeing what the big city of Kabul looked like. I'd only heard about it through the story of Bibi Shekiba, which I hoped Khala Shaima would continue today. It was as if she read my mind.\n\n\"I know you enjoy hearing about your bibi Shekiba. She lived in Kabul, you know. It's a different life there.\"\n\n\"But you've never seen it, have you?\"\n\n\"Look at me, Rahima! I'm thankful my ragged bones bring me this far. When I was younger, though . . .\" Her voice softened. \"I did dream of going to Kabul. I wished a carriage would come down the road, pick me up and take me to see the presidential palace and the shops and the streets and the airport. I wanted to see all the places I had read about.\"\n\nThat was her escape, I realized. Where her body couldn't take her, her mind went.\n\n\"Maybe you could go now?\" I suggested. The yearning in her voice made me wish she could go.\n\n\"My time has passed. But think about it. Badriya is going back and forth between here and the city. It shouldn't be a big deal for her to take you along. Offer to help her.\"\n\n\"Help? The only help she needs of me is right here, washing, scrubbing, ironing, rubbing her back even . . .\" The list went on and on.\n\n\"I know Badriya's type. I doubt she can read. I wonder how she's managing that with her role in the parliament. Let her know you can read and write. That would be a much better way for you to be useful to her.\"\n\nThat was true. Badriya had never learned how to read. I'd once seen Hashmat reading her a letter from her family. She listened eagerly as he deciphered the scribble. She wasn't alone. Most women in our village didn't know how to read. My sisters and I had only learned thanks to Khala Shaima's insistence. Rohila and Sitara may not have been getting the same opportunity, I thought, now that Madar-jan had retreated into herself and Khala Shaima's health was not what it used to be.\n\n\"She can't read. Neither can Shahnaz. Jameela can read a little bit, I think.\"\n\n\"Well, there you go,\" she said. She leaned forward and exhaled slowly, her lips pursed. \"Talk to her, nicely. I think it would be good for you to see the places your bibi Shekiba saw.\"\n\nThe idea excited me even more once she brought up Bibi Shekiba. I had already experienced her double life, living as a boy. I wanted to see the places she'd seen. But I wanted more than she had too. I didn't want to be a pawn the way she had been, passed from one set of hands to another. I wanted to be bolder. I wanted to make my naseeb, not have it handed to me. But from what my mother had always said, I didn't know if that was possible.\n\n\"Khala Shaima, do you think you can change your naseeb?\"\n\nShe raised an eyebrow. \"Tell me this, how do you know what your naseeb is?\"\n\nI didn't have an answer for her. \"I don't know. Madar-jan said it was my naseeb to be married to Abdul Khaliq. And for Shahla to be married to Abdul Sharif and Parwin to be married to Abdul Haidar.\"\n\n\"And what about this morning? What did you eat for breakfast?\"\n\n\"I ate a piece of bread with tea.\"\n\n\"Did someone bring you the bread?\"\n\n\"No.\" I nearly laughed at the thought of someone bringing something to me. \"Of course not! I got it myself.\"\n\n\"So maybe this morning it seemed it was your naseeb that you shouldn't have any breakfast at all. And then what happened?\"\n\n\"I changed it?\"\n\n\"Maybe. Or maybe it was your naseeb all along that you should have the bread and tea. Maybe your naseeb is there but waiting for you to make it happen.\"\n\n\"But wouldn't people say that is blasphemous? To change the naseeb that Allah has for us?\"\n\n\"Rahima, you know how deeply I love Allah. You know I bow before God five times a day with all my heart. But you tell me which of those people who say such a thing have spoken with Allah to know what the true naseeb is.\"\n\nThat night I lay awake thinking of what Khala Shaima had said. Jahangir breathed softly, tucked in next to me, his small hand on my neck.\n\nWas it Parwin's naseeb to die that way, her skin a mess of melted flesh? Or had she missed an opportunity to change things? To realize her actual naseeb? Was it Madar-jan's naseeb to lie dazed with opium while Rohila and Sitara fended for themselves? Dodged my father's angry rages on their own?\n\nIt baffled me. I sighed and pulled the blanket over my son's shoulders. I traced his pink lips with my finger. His face twitched in his sleep and the corners of his mouth turned up in a dreamy grin. I smiled.\n\nI didn't know what my naseeb was, much less that of my son. But I decided that night I would do whatever I could to make it the best naseeb possible. For both of us. I was not going to miss any opportunities.\n\nFrom what Khala Shaima had told me about Bibi Shekiba, she looked for chances to make her own naseeb. I, her great-great-granddaughter, could do the same.\nCHAPTER 36\n\nShekib's heart pounded; her mouth was dry. Amanullah was again walking through the gardens. Shekib was standing at her post, just a border of shoulder-high shrubs between them. He was walking with the older man again, his friend. Shekib recognized him by his wool hat. They took a seat on a bench and made Shekib's palms sweat.\n\nIt was naseeb that they should walk through here now, while I am on guard.\n\n\"There are many forces at play here. Your father will have to tread carefully. We are mice in a field of elephants but if we are smart about our moves, we can save ourselves from their heavy feet.\"\n\n\"The problem is that we have unrest within our borders and unrest at the borders. Our attention cannot lag or we will be weakened.\" Shekib could hear the respect in Amanullah's voice. He trusted this man.\n\n\"This is true. But the two are linked. A country secure in itself will stand strong against those who eye it hungrily. And those who eye us know that troubles at home make for easy prey.\"\n\n\"Our army is weak compared to theirs.\"\n\n\"But our will is strong,\" he said firmly.\n\nAmanullah sighed thoughtfully.\n\nShekib stiffened at the sound of his breath. She took a step to the right and then two to the left, stirring to make her presence known.\n\n\"Our people know so little of what goes on outside these borders. They are barely aware what happens one province, one village away from their own.\"\n\nShekib held her breath. She wondered if Amanullah realized it was her. Her back was facing the two men but she kept her head turned just slightly, her right profile to them\u2014if they had bothered to look. They stood and walked back toward the palace. Shekib could not resist the opportunity to look at Amanullah when she was close enough to see the color of his eyes. She twisted at the waist and looked from the corner of her eye.\n\nHe looked back. A nod.\n\nHe looked! He nodded! He saw me!\n\nShekib felt her breath quicken. Nearly an hour passed before she realized that Agha Baraan, too, had nodded in her direction, a subtle acknowledgment. She rubbed her moist palms on her uniform pants. She had made contact with Amanullah. He had noticed her and nodded. She had not detected any repulsion in his expression, not an ounce of disgust. Was it possible? Could Amanullah have looked past her disfigurement?\n\nThe afternoon reenergized her. She needed more contact with the palace, with anyone outside the harem. But the guards were insulated, were they not? Shekib considered the situation. She had more freedom than the concubines. She could travel the palace grounds without restriction. She could interact with the servants who came to deliver meals to the harem.\n\nKarim came to relieve her of her post.\n\n\"You can get some dinner. I think they were going to bring the carts over soon.\"\n\n\"I am not that hungry yet, actually. I may just go for a stroll.\"\n\n\"Whatever you want. Just keep your eyes open. It's been weeks and we know nothing.\"\n\nThe women were tightlipped. Each guard had her own suspicions but the questions they asked had gotten a spectrum of useless and curious answers.\n\nShekib traversed the gardens, passed the statues, the pond, two soldiers talking quietly to each other, eyeing her from afar. She looked out at Dilkhosha Palace, impressive and forbidden. She wanted to see inside but she had no business there. She let her imagination tell her what might be within.\n\nMaybe there were doves inside, graceful white winged birds that fed on warm palace bread and chirped blessings for the monarch. Or perhaps there were mountains of food, delicacies baked by cooks to tickle the king and queen's palates.\n\nThings were so different here in Kabul, in the palace. So many things Shekib had never before heard of, things she had never heard her parents speak of. She wondered if the palace thought of the villages as much as they thought of these other things. Why were they so preoccupied by these Russians, whoever they were, when villages were struggling without water?\n\nShe was so lost in thought that she hadn't noticed Agha Baraan sitting on a bench, sheets of paper in his hand.\n\n\"As-salaam-alaikum,\" he said gently.\n\nShekib turned sharply. When she realized who it was that had startled her, she turned her shoulders and head so her right faced him.\n\n\"Wa . . . wa-alaikum as-salaam,\" she whispered.\n\nHe turned back to his papers, reading thoughtfully.\n\nShekib took a step to leave but realized she had walked into a rare opportunity. Here was a link to the palace, a man very close to Amanullah. There were no walls between them, no interferences. She could speak to him, if she could make her voice follow her command.\n\n\"I . . . I guard the harem,\" she said simply.\n\nBaraan looked up, his brown eyes surprised. \"Yes, I remember. We saw you earlier today by the courtyard. You have an important position here in the palace.\"\n\nEverybody has a role in the palace.\n\n\"Yes. And it seems you do as well.\"\n\nHe chuckled. \"That will depend on who you speak with.\"\n\n\"What is it that you do?\"\n\n\"What do I do? Well, I am an adviser, you could say. I work with one of the viziers. An assistant to the assistant, so to speak.\"\n\nDo palace people always speak in riddles? Shekib wondered, thinking of his earlier conversation with Amanullah. \"Are you in the army?\" she asked. Her voice no longer trembled. His demeanor, his voice, his words told her he was not a threat.\n\n\"I am not. I work with them but I am not a soldier myself.\"\n\n\"I don't know anything about Kabul.\"\n\n\"You are from a village. That is not surprising.\"\n\nThere was condescension in his voice but Shekib chose to ignore it.\n\n\"What is your name?\"\n\nShe paused before she answered. \"Shekib.\"\n\n\"Shekib, I see. And the name your parents gave you?\"\n\n\"Shekiba.\"\n\n\"Shekiba-jan. My name is Agha Baraan. I am pleased to make your acquaintance. Is your family nearby?\"\n\n\"I have no family.\" The words rolled off her tongue before she could reconsider. But it was the truth. Bobo Shahgul and her uncles had made that abundantly clear.\n\n\"I am sorry to hear that.\"\n\nShekib suddenly remembered her plan. If she wanted to change her naseeb, she could not waste an opportunity like this. She tried to recover from her misstep.\n\n\"I mean, I had a family but now I live here. I no longer see my family. But I had many brothers. I am the only daughter in a long line of sons. My aunts all had boys. My grandmother too.\"\n\nAgha Baraan's lips tightened slightly. He looked away for a moment before returning to Shekib. \"Their husbands must be happy.\"\n\n\"They were.\" She fidgeted; her tongue felt thick with lies. He watched her. She wondered if he had sensed the dishonesty in her voice.\n\n\"Are you content here in the palace?\"\n\n\"Yes . . . mostly.\" Shekib hesitated. She was not sure how much to say. \"The palace is beautiful.\"\n\n\"It is. You are in Kabul, in the king's palace, the heart of Afghanistan. It is here within these walls that history is made.\"\n\nSuch grandiose talk, she thought, but she let her expression reveal nothing. \"The king's son.\" She could not bring herself to utter his name. \"He is an important man?\"\n\n\"He is and he is not.\"\n\n\"That's not possible.\"\n\nBaraan raised an eyebrow. \"Why is that?\"\n\n\"Because he either is or he isn't. He cannot be both,\" she said bluntly.\n\nHe chuckled again. \"You disapprove of contradictions. Well, you are ill prepared for life in the palace then. These walls are home to all that is and is not.\"\n\nTwo soldiers walked by and looked at them curiously. Shekib saw one whisper something to the other. She turned away from Agha Baraan abruptly and straightened her back.\n\n\"I need to get back to the harem.\"\n\nShe was clumsy and unrefined, Baraan thought, but interesting in an odd way. He wondered how she had gotten her scar and how much of what she had said was true.\nCHAPTER 37\n\nBadriya looked surprised.\n\n\"It's just that you look like it's bothering you. You've been holding your back all day long. I think you'd feel better if you let me rub it.\"\n\n\"That's just what I need. You're right. I have some oil here. Let me lie down.\" She wasted no time leading the way back to her bed, where she stretched out on her side, her back to me. She wiggled her dress up to her neck, looking over to make sure the door was closed.\n\nI dipped my fingers into the tin of animal fat and started to knead into her back. Rolls of skin hung loosely around her waist.\n\n\"Wooeee wooooeee,\" she moaned. I rolled my eyes. She tended to complain about her back only when there was something to be done around the house. Other times, she loved to point out that she was more active than Jameela and even Shahnaz, who were both much younger than her. Just another one of her contradictions.\n\nShe was putting on a good show now, although it wasn't necessary.\n\n\"Akkkh, you're young. You have no idea what aches and pains are. Have a couple more children and you'll see. My back, my knees, even my neck! Every part of me hurts from morning to night. And the road from here to Kabul is long and bumpy. My muscles get so stiff that by the time we get to the city, I can barely straighten my legs.\"\n\nI kneaded harder, knowing she loved the attention. She had brought up Kabul, though, and I searched for a way to broach Khala Shaima's idea.\n\n\"Are you going back to Kabul soon?\"\n\n\"In about three weeks. The parliament is meeting again. We have to vote on a couple of laws and there are some subjects up for discussion. Things you wouldn't understand.\"\n\nMy massage must have relaxed her. She was falling into her old habits and boasting about her position. This was what had gotten her black and blue under Abdul Khaliq's fist before she'd even taken the seat in the jirga.\n\n\"It must be a lot of work for you to do while you're there.\"\n\n\"Oh, it is. It's a huge responsibility. And going back and forth from here to Kabul is exhausting. It's not easy.\"\n\n\"You must be so tired.\" My tongue felt heavy and awkward saying things I didn't mean. Badriya hardly lifted a finger around the house and her children were mostly grown. They helped her with what little she had to do. And if she was so happy to have been elected to the seat in the jirga, then she should have been happy to travel to Kabul.\n\n\"I am\u2014I am so tired. Push harder here,\" she said, pointing somewhere in her lower back.\n\nI told myself not to huff. My fingers started to cramp but I dug my palms in where she had pointed. I needed her cooperation for the plan that was starting to take shape in my mind. Khala Shaima had planted a seed.\n\n\"You know, I was thinking, maybe I could help you in Kabul.\"\n\n\"You? Help me?\" Badriya balked. I gritted my teeth. \"You're young, just a girl! You know nothing about the jirga or what goes on there. It's government business, not child's play.\"\n\nIt had been a long time since I'd had time for any kind of child's play. And, as Khala Shaima had explained, Badriya had no experience or knowledge to qualify her to participate in the parliament. She was there only because Abdul Khaliq wanted her to be.\n\n\"I just thought I could help you with some of the smaller things, like filling out any paperwork or reading through the Kabul newspapers . . .\"\n\nBadriya's breathing paused. I could feel her hips tense under my hands. \"You . . . you wouldn't mind that kind of thing? You can read?\"\n\n\"Sure.\"\n\n\"And you can write too?\"\n\n\"Yes.\"\n\n\"And you can do it well? Not just a couple letters here and there?\"\n\n\"Yes. I got high marks in school on writing and reading. Better than my classmates,\" I said before reminding myself not to reminisce too long on that time of my life.\n\n\"Hm. I'll think about it. It's a demanding job and I could use some help with it . . . but I wonder what Abdul Khaliq would say. You know he doesn't like for us to be away from home. He made an exception for me,\" she said with an arrogance that could not be contained.\n\n\"He is different with you. I think it would be best if you would explain to him that I would be there for you, to make things easier on you. Because obviously he likes you best.\"\n\nShe looked satisfied with my reasoning. For a moment she forgot how often Abdul Khaliq called me to spend the night with him. As if the night was not bad enough, I always had Badriya's bitterness to look forward to the following morning. Once she beat me with her sandal for breaking a plate when she had seen her son knock it from my hand. Everything was reported back to our mother-in-law, who took exceptional pride in reinforcing my punishments.\n\n\"What about your son? Jahangir is still young. You would leave him behind? Bibi Gulalai is not going to like that idea.\"\n\nShe was seriously considering my proposition. I hadn't thought my idea through so I spoke slowly, making it up as I went along.\n\n\"I think I could bring him along. He's not a difficult child so I don't think he would disturb you much. I could look after him in Kabul and still help you out.\" I stopped myself before I said anything about Bibi Gulalai. She hated everything I did anyway.\n\n\"I don't know if Abdul Khaliq wants his son traveling to Kabul.\" She seemed skeptical but I felt an opening. I pushed it.\n\n\"Just bring it up with him. Please. I think I could be useful to you.\"\n\n\"But why? Why do you want to do this?\" She turned around to see my face. Her eyes narrowed to slits. I shifted and moved my hands to her shoulders, trying to divert her attention.\n\n\"Because . . . because you have so much to do and I thought . . . well, I've always wanted to see Kabul. I thought this would be a good chance. As you said, Abdul Khaliq makes exceptions for you, so maybe if you discussed it with him, and told him that I could help you . . . maybe he would agree?\"\n\nShe closed her eyes and let out a sigh as I worked my way over to her shoulders. She liked the idea. Now we just had to work on our husband.\n\nI hoped she would be as convincing as I was.\n\nEvery time I asked her about it, she shrugged me off. She either hadn't had a chance to ask him or she had forgotten, or he wasn't in the right mood to bring it up. Her next trip to Kabul was coming up soon. Two weeks away. One week away. I became discouraged. She didn't have the nerve, even though I knew she liked the idea. A few days after I approached her, she'd asked me to read a few things around the house. I think she was testing me. Not that she could tell the difference but she seemed reassured that I actually could make sense of letters.\n\nWhen just two days remained Badriya finally approached Abdul Khaliq. The way Badriya told it, he wasn't excited by the idea, but after much cajoling, she managed to get him to agree. I asked her again when I brought her the dresses she'd asked me to press.\n\n\"Make no mistake, he wasn't for the idea at all. And for all the reasons I had predicted. I really didn't think that even I could get him to agree, but he did. So there you go. You got your wish. We'll leave Sunday to be there in time for Monday's session. You'd better make yourself very useful to me there or I'll regret going to such trouble for you.\"\n\n\"You won't regret it\u2014you'll see! Thank you so much! I'd better pack some things for me and Jahangir!\"\n\n\"Just you,\" she said, and turned her back toward me. She put the clothes directly into a duffel bag. \"You won't need to pack anything for Jahangir.\"\n\n\"Why not?\" I asked, confused.\n\n\"He's not going. Abdul Khaliq says he's too young to travel. He said Jameela can take care of him while we're away.\"\n\nI became tense. I'd never been apart from Jahangir. The mention of leaving him made my heart fall. Should I insist? Should I stay behind?\n\n\"Oh, I didn't think . . . he said that? For sure?\"\n\n\"For sure? Do you think there's any mistaking what Abdul Khaliq says? It's always for sure, Rahima. Just pack a few clothes together. Jahangir will be fine with Jameela. She's got a soft spot for little children.\"\n\nI squirmed still. \"How long would we be gone for?\"\n\n\"Rahima, enough with these idiotic questions. Parliament is in session for four months. I've been going back and forth to get things prepared and we get breaks.\"\n\n\"Breaks for what?\"\n\n\"Breaks are for us to come back to the areas we represent. To meet with people and get an idea of the issues at home.\"\n\n\"But you've never met with anyone.\"\n\n\"Do you think Abdul Khaliq would have me running around town talking with this and that one? Honestly, it doesn't matter. No one is checking up on us and I don't think any of the others actually go back to talk to constituents. Who needs to? I'm sure everyone in this region has the same issues we do.\"\n\n\"And what issues are those?\"\n\nBadriya looked frustrated. \"Maybe you don't have enough to do around the house! Have you been sitting and thinking of nonsense to ask me? You won't be talking with people in Kabul but you might be seen, so bring your nicer clothes. Not that ratty blue housedress you always wear.\"\n\nThe ratty blue housedress. I'd worn it so much I could almost see through the material, as Shahnaz had snickered and pointed out one day. I'd been embarrassed but it was hard retiring it. The navy blue reminded me of a pair of blue jeans I'd happily worn for a few months. Denim. In denim, I had been free to run down the block, to walk with my best friend's arm around my shoulder, to kick a soccer ball between the goalkeeper's legs. That ratty blue housedress was my freedom flag, but no one else knew it.\n\n\"How long would we be gone?\" I was calculating. I knew Badriya had made several trips back and forth in the last session but I'd never paid attention to how often she came back for a break.\n\n\"Two weeks, I think. Then we come back for a short break before heading back to Kabul . . . that's the way it goes.\"\n\n\"Two weeks? Oh, wow. Two weeks . . . I suppose I could . . .\"\n\n\"You suppose? You brought this all up, so don't be such a child about it.\"\n\nShe wants me to go with her, I realized, and nearly smiled. She needs me. It almost felt like I had a card to play.\n\nI learned later how things really worked. Badriya, like all the other parliamentarians, was given a stipend to hire one assistant, one driver and two personal security guards. So far, Abdul Khaliq had been garnering her stipend and salary since he'd already sent her with his own driver and guards. Unable to do any of her own paperwork, Badriya had been going to the director general's office more often than any other member. They were tired of seeing her and had insisted that she find an assistant as soon as possible or they would take part of the stipend away.\n\nIt was an empty threat but an assistant would make things easier for everyone.\n\nBut I didn't know how things worked at the time. I surely didn't know that Abdul Khaliq and Badriya were doing what so many other parliamentarians were doing too. It seemed that no one in Kabul followed up on money. Or promises.\n\nAll I could think was that I could do this. I trusted Jameela would take good care of my son. Maybe this would be good for Jahangir and me in the end. Anything had to be better than waiting on every person in this house.\n\n\"All right,\" I said in agreement, thinking this might be my crossroads, my naseeb.\nCHAPTER 38\n\nWhen Shekib had first arrived at the palace, she could barely make eye contact with anyone she came across, even the women. She had been veiled for so long and had worked in homes where people wanted neither to hear nor see her. The first time she'd crossed paths with a soldier, her heart had nearly pounded out of her chest because he'd muttered some unintelligible greeting to her. The second time, it was a gardener. It took an hour for her hands to stop trembling and for her to get over the awkward eye contact they had made.\n\nIt was hard for Shekib to believe that she could look directly at a stranger and speak. Instinctively, she wanted to run away. But as days passed and her legs grew more confident in their pants, she slowly became more accustomed to small interactions. She forced herself to speak with the other guards and listened when they spoke with each other.\n\nSome days, Shekib came across people who worked in the palace, not just outside it. Each time, it became the slightest bit easier for her to strike up a conversation. And inevitably, she would find a way to interject something about the long lineage of sons in her family. She wasn't very deft about it, but that was of no concern to her.\n\nIt had been a year since she had first arrived at the king's palace. She walked the grounds with confidence. She knew more about each concubine than she had thought possible. She had watched their children, the king's children, take their first steps, write their first words. Habibullah seemed to be a good king, according to the palace workers. He had expanded the network of roads across the country. He had founded a military academy and other schools.\n\nKing Habibullah was gone for weeks at a time and occasionally returned with a new concubine, girls, doe-eyed and nervous. Shekib watched as the new consorts floundered until they settled into the harem.\n\nEverybody has a role in the palace.\n\nNew concubines made older concubines purse their lips and reconsider their position. Sakina grew feistier, gave the newcomers facetious advice and stayed silent for days when King Habibullah passed her over for a fresh face. Benazir had given birth to a little girl. She had named her Mezhgan and lined her eyes with kohl, as Halima advised.\n\nFatima had grown pale in the last few weeks. Her son had just turned a year old but spent a good deal of time with Halima, since she rarely had enough energy to keep up with him. Her illness unnamed, she was visited often by the harem's physician, a British woman named Mrs. Brown. Kabul had only male doctors, which would not suit the king's insecurities. Mrs. Brown had been brought in from abroad, a kind but firm woman who satisfied the monarch in both her competence as well as her demeanor. She stayed at the palace and traveled back to England infrequently. Mrs. Brown (\"Khanum Behrowen,\" as the women called her) placed her stethoscope on Fatima's chest and back, her hands pressed into her belly. She would sigh and tap her forefinger against her lips, thoughtful.\n\nDespite its tensions, the harem was a family. The older women were mothers to the younger concubines, while the younger consorts had rivalries with each other like siblings with only one toy. King Habibullah visited when he chose, appearing sometimes in daylight and other times well into the night. He came with minimal fanfare but he made no secret of his visits. Unlike the other man.\n\nThe other visitor, whoever he was, came rarely. The guards would almost believe he had tired of his mistress when he would make another appearance, always under the cloak of darkness. He must have known the guards had seen him and probably surmised that they felt powerless to stop him. Whoever he was, he boldly betrayed the king with the most sinful trespass and then returned to slumber in his palace.\n\nShekib wondered who could be so brazen. And why.\n\nAmanullah stayed closer to the palace while his father ventured out into the country to check on the roads he had commissioned. He came to the harem's courtyard from time to time, leaning over to pat his younger half siblings on the back, ruffle their hair and kick a stray ball back in their direction. Shekib watched him, her heart beating in odd tempos, wistful and hopeful. He would acknowledge her and give her a light smile, a formal nod. Like a secret handshake between them, Shekib thought.\n\nI probably look a bit older than him but I haven't outgrown the possibility of marriage. I am young yet, able-bodied and strong. I hope the others have told him about me, how I help the gardeners replant shrubs, how I carry the children when they grow sleepy, how I bring trays of food into the ladies' quarters. My back is as strong as that of any soldier in the palace, my arms solid and my mind rational. Think of me, Amanullah-jan, and I am certain I would not disappoint a man like you.\n\nShekib was not the only one thinking of Amanullah's naseeb.\n\nKing Habibullah also believed it was time for his son to be given a bride. In his mind, there were a handful of contenders\u2014daughters of the viziers or his closest advisers. In his own words, words Shekib overheard one day as she stood outside his suite in the harem, \"I cannot force his hand. He will choose for himself, my boy. Amanullah is different than his brothers. He is more like me than the others. And so unlike me in other ways. I sometimes wonder how I would feel about him were he not my own son.\"\n\nShekib felt a clock ticking. Amanullah would choose a bride soon. She charged ahead with her humble efforts. She found a reason to speak to nearly anyone who crossed her path and made certain to mention that women in her family rarely bore anything but boys.\n\nShe saw him again with Agha Baraan. They crossed the palace's grounds, returning from a meeting at Dilkhosha Palace. Shekib dug her hands into her pockets and looked around. She floated in and out of genders easily now, aware of her flattened bosom and hidden curves only in Amanullah's presence. She tingled for him. She hoped he knew.\n\nThe men stopped at the bench. Agha Baraan plucked a red rose, breathed in its perfume and stuck it in his blazer pocket. Shekib was a good distance away but slowly and casually made her way toward them, pretending to inspect the shrubbery as she wandered over. Once seated, their view was blocked by the greens and they were unaware of the woman-man guard at their side, eavesdropping and flirting.\n\n\"So you have decided?\"\n\n\"I am ready, Agha Baraan. I think it is time for me to take a wife. I want to have a legacy of my own and I must start a family to do so. I want to have at my side a woman who is thoughtful and who will be as dedicated to Kabul as I am. I am confident in my decision. She is strong-willed and has undergone hardship; people have turned against her and yet she walks with her head held high. When I see her face, I see that she brings with her a gentle understanding because of what she has experienced.\"\n\nShekib froze. Her face? Could he be talking about my face? Yes, people have turned against me! Nearly everyone has turned against me! But I would work so hard for Kabul! I would do anything he needs! She did not move, terrified that she would give her presence away.\n\nMaybe Agha Baraan had told him about her? Maybe he had shared those morsels she had laid out for him and maybe they knew she was listening at this very moment.\n\n\"And what will your father say? I mean, given where she comes from . . .\"\n\n\"I know that, but it was my father and this palace that introduced me to her.\"\n\nShekib's eyes widened. Indeed, it was King Habibullah who had brought her to the palace and into his son's life. She straightened her shoulders, wanting to comport herself as a palace woman would.\n\n\"I will speak with him again tonight. I have brought this up before but he did not believe I was serious.\"\n\nBaraan took a deep breath.\n\nShekib said nothing to the other guards but for two days they shot each other looks, noticing a change in her. Ghafoor had to repeat herself three times before Shekib would notice she was talking. Karim and Qasim watched her meals go barely touched and shared her leftovers when she walked away. Tariq tried to approach her, to talk about her dreams of motherhood. Shekib blankly nodded and shook her head in a way that told Tariq she might as well have been talking to the pigeons.\n\nTwo days passed as such. In the nights, Shekib stared at the wall, pictured Amanullah's face and imagined how someone from the palace might approach her with his proposal. Where would she live? She would grow her hair. She would wear makeup, as some of the women in the harem did from time to time. A British woman visiting the palace had brought rouge and powder, showing the women how to lighten their complexions and bring an alluring tint of color to their cheeks. Shekib wondered if the powder could conceal her disfigurement, her half mask.\n\nOn the third night, Shekib was on duty. She stood outside the harem, watching the palace and wishing her mother was alive. She took longer than she should have to react to the footsteps and talking inside the harem. Halima was at the front entrance just when Shekib was starting to realize something was going on.\n\n\"It's Fatima! She's not well. We need to send for the doctor!\"\n\nFatima had taken a dramatic turn for the worse and with it, Shekib's naseeb changed course.\nCHAPTER 39\n\nThe road was bumpy. My sides ached with every jolt. Badriya watched me from the corner of her eye. The experienced first wife wasn't surprised. Last night Abdul Khaliq had asked me to visit him. I entered his room quietly. Although I was no longer a new bride, the nights with my husband still repulsed me. I had to take my mind elsewhere, think of the chores I still needed to do or school days when the moallim would teach us to sing our multiplication tables to memory.\n\nWhenever my wifely obligations were fulfilled, I would wait to hear my husband's snores, a signal that I could put myself back together and retreat to my room. Last night was different.\n\nBadriya and I were set to leave in the morning for my first trip to Kabul. I was excited but anxious about leaving Jahangir behind. Abdul Khaliq's even breaths told me he was relaxed but not yet asleep. I took a chance.\n\n\"I wanted to ask something . . . ,\" I said hesitantly. I looked for the combination of words that wouldn't anger him right away. He looked surprised to hear me speak. With a raised eyebrow, he told me to explain.\n\n\"Tomorrow . . . because I will be helping Badriya-jan . . . I was hoping I could take Jahangir with me so that\u2014\"\n\n\"Jameela will watch him.\"\n\n\"But I didn't want to trouble her. She's already got her own to look after.\"\n\n\"He'll be fine.\"\n\n\"And I want to be sure that Jahangir eats well. Sometimes he can be so picky . . .\"\n\nI had said too much.\n\n\"Then don't go!\" he thundered. \"It was an idiotic idea to start with! Now I have to listen to you nag! You appreciate nothing!\"\n\nHe was up now, the sheets pulled behind him, leaving my legs uncovered.\n\n\"I'm sorry\u2014\" I started to say, hoping to stymie the reaction I could see coming.\n\nIt was too late. Abdul Khaliq spent the next thirty minutes making me regret I'd spoken.\n\nI realized then my husband understood people. He knew just how to get to people to do what he wanted, to make them angry or sad or fearful. I realized that was probably how he had been successful at whatever it was that he did.\n\nMorning came and I kissed my sleeping son before laying him on a cushion in Jameela's bedroom. I touched his cheek and watched as his lips turned slightly in a dreamy smile.\n\nJameela bit her lip when she saw my face. My cheek was starting to turn a deeper red, a bruise in the outline of a hand taking shape.\n\n\"He'll be fine, Rahima-jan,\" she said warmly. \"I'll have Jahangir sleep right next to me with your blanket. We'll talk about you until you come back. This will be good for you, you'll see.\"\n\nI was grateful and knew Jahangir loved being with her and her children. Still, I hated to leave my son.\n\nTwo weeks, I thought. We'll be back in two weeks for our first break. It's not that long, right?\n\nI ran my fingers through his dark locks once more and leaned over to kiss his head. He turned onto his side, his perfect lips parting just wide enough for me to see his petite teeth.\n\n\"It's okay, Rahima-jan. He'll be fine, you'll be fine. You'll see,\" Jameela said. She hugged me delicately, knowing one bruise heralded the presence of others.\n\nI carried my duffel bag out to the car. Bibi Gulalai and Badriya were outside, as was Hashmat. He looked over and smiled snidely.\n\n\"Good morning!\" he called out.\n\n\"Good morning,\" I mumbled, my mind still on Jahangir's soft face. I was in no mood for Hashmat's facetiousness today. \"Salaam, Khala-jan.\"\n\nShe ignored my greeting. \"You're ready for your trip to Kabul I see. I don't know how you could leave a young boy to go off doing things you've no business doing. My son is being kind in allowing this, so you better make yourself very useful to Badriya.\"\n\n\"That's right,\" Badriya echoed.\n\n\"I doubt she'll be worth the trouble she'll cause,\" Bibi Gulalai muttered.\n\nHashmat laughed. \"Isn't that nice that you'll be joining Madar-jan in Kabul! I bet all your classmates would be jealous if they knew you were going to see the city,\" he said.\n\nI shot him a sharp look that went unnoticed by Bibi Gulalai and Badriya. Hashmat made a point of talking about my bacha posh days and my male classmates as often as he could. He used to do it in front of his father but it sometimes resulted in such an explosion of anger that he would be caught in the overflow. Something about me as a bacha posh had piqued Abdul Khaliq's interest, but now he could not tolerate hearing about me even sitting next to boys in school.\n\nAbdul Khaliq's guards put our bags in the back of the car. We donned our burqas and climbed into the backseat.\n\nDon't speak to the guards. They'll watch out for you but if you do anything . . . let me assure you . . . you'll regret it. And in Kabul, I have people. I will hear about everything you do. If you do anything to embarrass me, I promise you that you'll wish you never stepped foot in that city.\n\nHe was clear. I was thankful Jahangir was too young to cause much trouble. Abdul Khaliq's temper came hard and fast and often without warning. I had asked Jameela to make sure Jahangir did not get in his father's way. I wouldn't be there to shield him.\n\nThese thoughts played over and over until the rough road finally lulled me to sleep in the backseat. Badriya was in no mood to talk. She leaned her head on the window and started to snore lightly.\n\nI don't know how many hours passed before structures came into view again. There were buildings, houses, horses and cars. I sat up straight. We were in a jeep with tinted windows so I dared to look out and see what the people of Kabul looked like. My mind jumped to Bibi Shekiba and her first impressions of the capital, as Khala Shaima told it.\n\nI was the same, wide-eyed and amazed, but in a different way. I had never seen so many cars and people in one place! It looked as if everyone who lived in Kabul owned a car. And store after store, the streets were lined with exotic wares and different foods. Bakeries, tailors, even a beauty salon! This was so different from home. I wished Shahla could be here to see it all with me. Or the boys. There were so many places we could have explored if we'd grown up here!\n\n\"Kabul is . . . Kabul is amazing!\" I exclaimed.\n\nBadriya seemed entertained by my reaction. \"Of course it is! There's a lot going on here. We won't have time for me to point everything out to you.\" I saw Maroof and Hassan in the front seat look at each other. It was unlikely Badriya had actually seen any of Kabul. She had complained to Jameela that the guards took her from her hotel to the parliament building and back. \"We're almost there. We're going to be staying at a guesthouse run by some Europeans.\"\n\nDown a tree-lined street, a building came into view.\n\nIt had a gated entrance flanked by porticos with stone pillars, Through the main entrance, a wide path led to and encircled an imposing tower with a flag flapping from its summit. I craned my neck to get a good view.\n\nThat tower reaches the sky! I thought.\n\nThe palace's fa\u00e7ade was embellished with carvings and arches, dull and chipped, but it surely once looked very majestic. A woman walked past the front gate, her green-yellow head scarf pulled across her face, hiding everything below her nose and cascading down her shoulders. As we drove past, she turned slightly and looked directly at my tinted window, her eyes meeting mine as if she could see through it. This first glimpse of a Kabuli woman was exciting for me, a girl from a village.\n\n\"What's that building?\" I asked, already knowing the answer.\n\n\"That's Arg-e-Shahi, the presidential palace.\"\n\n\"Bibi Shekiba . . . ,\" I whispered. I got a chill thinking of how my great-great-grandmother must have felt when she first saw those gates. And to think of what she had seen on the other side. As usual, Khala Shaima had left her story unfinished. The turn of events in her life was unpredictable. I wanted to know what became of her almost as much as I wanted to know what would become of me.\n\n\"God have mercy, what the hell are you mumbling about?\"\n\nBadriya's question went unanswered. I stared at the palace, where my legacy began.\n\nWhat happened to you here? I wondered.\n\nMaroof turned left, then right and left again, weaving through the crowded streets and cursing every car in his path. There were tanks and soldiers in fatigues and helmets. They didn't look Afghan. These were the foreign soldiers Badriya had told us about. Just like my husband's guards, they had large guns hanging at their sides. Little boys stood in front of them, looking curious. The soldiers laughed and chatted casually.\n\n\"Are they American?\" I asked Badriya.\n\n\"They're from everywhere. Some are American, some European or whatever they are.\" She pointed to a building coming up on our left. \"We're here,\" she announced.\n\n\"Is this where you always stay?\"\n\n\"Yes, it's a nice place. You'll see.\"\n\nBadriya was right. We pulled up to a metal gate on a small street, tucked away from the busy market.\n\nOur driver rolled down his window when we pulled up to the blue-uniformed guard at the gate. He mentioned Abdul Khaliq's name. I thought the men were shaking hands but I realized Hassan's fingers held a folded stack of bills that the man slipped into his pocket.\n\nI looked over at Badriya but she either hadn't noticed the exchange or didn't care.\n\nHassan opened the gate and our driver, Maroof, pulled into a circular drive that looped in front of the largest building I'd ever seen. It was three stories tall with rows of windows lined up like a hundred eyes. Two columns framed the glass, double-door entrance.\n\n\"And this is where the meetings are?\"\n\n\"No, you fool. The parliament meets in the parliament building.\" I was too excited to be annoyed with her condescending tone.\n\nWe were led into an elegant lobby with a reception desk. A man wearing a crisp dress shirt and slim pants was talking on the phone, but he nodded when he saw our driver and the other guard. He cradled the receiver and looked up at our guards. I stood behind Badriya, not wanting to make an inappropriate move. Three women walked in from outside dressed in fitted tunic tops and denim pants. Their head scarves were demurely tied under their chins but wisps of hair framed their faces and their delicately arched brows. Their shoes got my attention most. Black leather pumps broke the silence in the room.\n\nLooking at their clothes, I was thankful the burqas hid our faded, baggy dresses. I felt suddenly unsophisticated and awkward. I tried to hide my feet behind Badriya. The women were busy talking and hardly noticed us.\n\nThe conversation between Abdul Khaliq's bodyguards and the man at the reception desk went back and forth until finally there was another handshake. Another wad of bills slipped into the receptionist's palm and from there was quickly tucked into his jacket pocket, while he made a quick glance around the room to make sure no one else was watching, not that anyone would have cared.\n\nWe were led to a room on the third floor with two single beds and a bathroom with a western toilet. The window looked out on the courtyard behind the hotel, a small stone area surrounded by flowering plants and shrubs. I saw a pigeon waddling in the shade of a tree.\n\nLike the palace gardens where Bibi Shekiba used to stand guard, I thought.\n\n\"I can't believe this is where you stay in Kabul! No wonder you like coming here so much!\"\n\n\"Don't get used to this,\" she said, opening her duffel bag and pulling out a sweater.\n\n\"Why not?\"\n\n\"Because we'll be in an apartment soon. Abdul Khaliq is only using this place temporarily. He's been looking to find a place in Kabul where we can stay with more privacy, only his guards outside.\"\n\n\"Has he found a place yet?\" I asked.\n\n\"How the hell should I know?\" she replied. She sat on the bed and took her sandals off. Her heels were cracked and yellow. She rubbed one of her soles and sighed. \"Look, Rahima, I know why you're doing this. Don't think I'm stupid.\"\n\nI looked at her but said nothing. I thought it best I let her explain.\n\n\"But as long as you help me with what I need to read and write for these meetings, then I don't care much. Just don't expect to see much of Kabul.\"\n\nBadriya was right. Our personal guards kept to themselves but were never more than twenty feet away. Most of the time they stayed in the small seating area on the third floor, just two doors away from our room. I hated knowing that Abdul Khaliq was keeping tabs on us at all times, but Jameela had told me about the threats against parliament members, especially the women, so there was something comforting about knowing Abdul Khaliq's trusted bodyguards were watching over us in this new, busy town. I felt safer because of them.\n\nWork started the following day. Our guards drove us to the parliament building in the morning. We wore our burqas until we got there. Badriya slipped hers off and instructed me to do the same. I looked over at the guards to see their reaction. They had turned away but watched peripherally while we entered a long and stately building with a row of columns before it.\n\nPeople walked in and out, men and women who looked to be from all different regions. Some of the men were dressed in the flowing caftans and pants common to our village, their heads wrapped in turbans, one end cascading over a shoulder. But it was the women who made my jaw drop. Some were dressed as we were, in simple flowing calf-length dresses with loose pants underneath. But others wore button-down shirts and long flowing skirts. Some even wore jackets and slacks. They wore their colorful head scarves smartly. As we got nearer, I could see that a few women wore lipstick or rouge, while others had outlined their eyes with kohl. I wondered what their husbands thought of them walking uncovered, with painted faces.\n\nWe came to a security station. Four uniformed guards stood at the entrance, two men and two women. The mass of people slowly melded into three lines. Badriya took me by the elbow and led me past the others. She paused briefly when she came to the security guard, dressed in the same khaki color as her male counterparts but in a long skirt.\n\nA woman guard. Just like Bibi Shekiba, I thought. I couldn't help but stare at her face, wondering if she looked anything like the woman I'd heard so much about.\n\nBadriya muttered a quick greeting and waved at her. The guard nodded and turned her attention back to the woman in front of her. She pulled her behind a partition.\n\n\"What are they doing?\"\n\n\"They're here for security. They're checking people for weapons. That room back there is where the female guards check the women. We're not supposed to bring anything into this building. And we're not supposed to take anything out of it either.\"\n\n\"We don't have to go through the checkpoint?\"\n\n\"Well, we're supposed to but I don't. The guards know me. And no one else from the parliament goes through either. We are the parliamentarians, after all! How ridiculous if we were to be patted down every time we walked in! I wouldn't stand for it!\"\n\nI bit my tongue, knowing she would stand for it if she were ordered to.\n\nBadriya smiled politely to a few people she knew. Two women, wearing dresses and longer head scarves, approached with bright and cheerful faces.\n\n\"Badriya-jan! Good to see you again! How are you? How's the family doing?\" They were of similar height and build and even face structures. But their ages differed by about ten years, the older woman's face with more lines, her hair with more wisps of gray.\n\nCheeks pressed to one another, kisses in the air, an arm around a shoulder. The women greeted each other.\n\n\"Sufia-jan, qandem, salaam!\" My eyes widened to hear Badriya greet her with such syrupy sweetness. \"Thanks be to God, everyone is well. How are you and your family doing? And you, Hamida-jan? How are you?\"\n\n\"Fine, thank you. Ready for another busy session?\" Hamida replied. Her face was plain, unpainted and serious.\n\n\"Yes, I am. When do you think it will start?\"\n\n\"They said we should be starting in half an hour,\" Sufia said, scanning the entryway. She was the older of the two. There was gentleness in her eyes that put me at ease. \"But my guess is that we don't have enough people here. We'll probably begin in about an hour. Maybe two. You know how it is.\"\n\nBadriya nodded politely and was silent.\n\nShe doesn't know what else to say to them, I thought.\n\n\"And who do you have here with you? Is this your daughter?\"\n\nHamida and Sufia were looking at me expectantly and smiling. I looked at Badriya and felt the urge to step away. I didn't like the idea of her being mistaken for my mother. She didn't like it either, but for different reasons.\n\n\"Her? Oh, no, she's not my daughter. She's my husband's wife.\"\n\n\"Your husband's wife? Oh!\" Hamida's smile tightened. She disapproved.\n\n\"Have you brought her to see how the parliament runs?\" Sufia asked, trying to distract us from Hamida's reaction.\n\n\"Yes, er . . . she wanted to see for herself what it is that I do. That we do. So I've decided to hire her as my assistant.\"\n\n\"Oh, she's going to be your assistant! What's your name?\"\n\n\"Rahima,\" I said. \"I am pleased to meet you.\"\n\n\"And we're pleased to meet you as well,\" Sufia said, looking impressed with my manners. \"I think it's a great idea for you to come see what the parliament does. Maybe you would want to join your . . . er . . . Badriya-jan and take a seat in the jirga. We need women to get involved in our government.\"\n\nBadriya nodded but looked uncomfortable.\n\n\"Why don't you both come to the resource center tonight? After the session is over.\"\n\nBadriya shook her head. \"No, we can't make it. Some other time.\"\n\n\"Why not, Badriya-jan? They have some instructors there who have helped us very much. Tonight we're going to work on the computers. It's not easy. You really have to spend some time to figure those machines out. It would be good to get familiar with it.\"\n\n\"I know that. I've seen computers. It's not that hard,\" she said, her eyes shifting nervously.\n\nThe look on my face confirmed for Hamida and Sufia that Badriya was not in the least familiar with computers. Hamida decided to ignore the obvious lie.\n\n\"What else do they teach there?\" I asked. I had been away from school for so long. The idea of instructors and lessons excited a part of me that Abdul Khaliq's compound had buried.\n\n\"They teach lots of things,\" Sufia said, happy to hear my curiosity. \"How to speak English, how to do research, how the parliament is supposed to function . . .\"\n\n\"It's a school? Can anyone go?\"\n\nHamida nodded. \"You could come, as her assistant. It's only for women parliamentarians. It's run by a foreign organization and it's open after the sessions end for the day. Maybe you can convince Badriya-jan to come along. There are too many people doing nothing in this building. We all need to do something more.\"\n\n\"Excuse us, ladies. I want to show Rahima-jan around the building and then we're going to get to our seats,\" Badriya said, her fingers wrapped around my elbow firmly. She wanted out of this conversation.\n\nI followed her lead but my heart lightened at the talk of classes. I was starting to taste the possibility of change here.\nCHAPTER 40\n\nShekib stood frozen.\n\n\"Don't just stand there! She needs the doctor. Go and get Khanum Behrowen!\" Halima threw her hands up in frustration. Shekib nodded and turned around but stopped short, realizing she had no way of summoning the doctor without walking right into the palace in the middle of the night. She turned back to the guards' quarters.\n\n\"Ghafoor! Ghafoor, wake up. We need to get the doctor for Fatima. She's ill and needs help.\"\n\nGhafoor, the consummate guard, bolted upright and answered the call to take charge.\n\n\"She's ill? Worse than before?\"\n\n\"I suppose so. I haven't seen her.\"\n\n\"What? You haven't even gone in to check on her? What were you . . . never mind! Karim, get up. Go and see how Fatima is doing. Take Qasim with you. I'll go to the palace and ask for the doctor.\"\n\n\"What should I do?\" Shekib asked.\n\n\"Nothing. You can do that much, can't you?\" Ghafoor said with annoyance. She brushed past her and went to pull on her uniform quickly. She fastened her belt brusquely before shooting Shekib one last glare.\n\nPeople from the palace will be awakened. I should resume my duties, she thought, and returned to her post outside the harem. Karim and Qasim soon walked past her and entered the harem. Tariq, hating to be alone, followed after them, her arms folded against her chest in the cool night air. She half smiled through tight lips as she passed Shekib.\n\nShekib tapped her foot. She could see the way they looked at her, the distance. The same look Khanum Marjan had given her\u2014some pity, but no friendship.\n\nI am on my own, Shekib thought. Nothing has changed. She began to pace in front of the harem, walking around to the side entrance and making sure she looked like she was actively guarding the building.\n\nGhafoor and Dr. Behrowen emerged from the darkness. Ghafoor carried a lantern and Khanum Behrowen a black bag, their pace hurried. Two men followed behind them, sent to observe and bring word back to the palace. Shekib turned to walk back to the front entrance when she heard a door open. Before she could turn, she had been pushed aside, just roughly enough to make her stumble. She braced her fall with her hands and knees and looked up to see the man's back as he darted off.\n\nShe started to call out after him and then caught herself. She looked over at Ghafoor and the approaching team from the palace. They hadn't seen the man knock her down, nor did they see him disappear behind the shrubs. She kept her mouth shut and scrambled to get back to her feet. She wanted to meet them by the front entrance.\n\n\"The others are inside with Khanum Fatima,\" Shekib announced as they approached. \"I am standing guard here.\" She made sure to be loud enough for the men to hear. They stood back, rubbing their hands together and speaking quietly as they watched the women enter the harem. \"Should I come in with you?\"\n\nGhafoor did not pause. \"Do whatever you wish,\" she called out from the foyer.\n\nShekib followed them in. The hallways were lit with multiple lanterns. They followed the sounds to Fatima's chamber. It was a small room toward the back of the house. Nabila and a few others stood in the narrow hallway, shaking their heads and muttering to each other. Inside the crowded room, Shekib could see a circle of women. Sakina sat behind Fatima, holding her head up on her lap. Fatima's face looked wan, even in the amber glow of the lanterns.\n\nDr. Behrowen knelt at Fatima's side and opened her bag. She laid a hand on her forehead and called out for some wet cloths in rudimentary Dari. Halima rushed past Ghafoor to bring them. The doctor picked up Fatima's wrist and placed two fingers on it, her forehead tightened and her lips pursed. She took out her stethoscope and bent over Fatima with her head turned to the side, listening carefully. The chatter in the room had grown louder with the doctor's arrival. She finally lifted her head up and pointed an angry finger toward the door.\n\n\"Hush! Step outside if you wish to chat!\" Though they did not understand English, the room was immediately silent.\n\nBeads of sweat lined Fatima's brow, like soldiers preparing for battle. She moaned softly and turned her head to the side. Her son began to whimper and pulled at her sleeve. Benazir picked him up and whispered something into his ear that quieted him, his bottom lip still curled out.\n\n\"She is feverish. I want to bring her to a cool bath. Ladies! Help her to the bath area!\"\n\nThe women looked at Dr. Behrowen, puzzled by her instruction. Dr. Behrowen had learned a few words of Dari over time but most of her communication with the harem was through gesture. Sighing with frustration, she motioned for Sakina and Nabila to lift Fatima and then pointed at the doorway. They nodded and Karim and Qasim jumped in to help. Grabbing limp arms and legs, they carried Fatima into the hallway. Dr. Behrowen pointed toward the bath area.\n\n\"Aab, aab!\" she called out.\n\n\"She wants us to go to the baths!\" Qasim called out. They hurried down the hallway. Khanum Behrowen pointed at a shallow bath and instructed the women to place Fatima in the water.\n\n\"We have got to bring her temperature down,\" the doctor muttered to herself. \"She is burning up.\"\n\nFatima reacted to the water, but Qasim held on, her hands under Fatima's armpits to keep her head above water. She looked more awake, more alert, turning her face to Khanum Behrowen.\n\n\"I feel so weak, doctor,\" she said.\n\nDr. Brown nodded. That much was already clear to her.\n\n\"What's going on in there?\" It was a man's voice echoing from the front entrance. The women jolted at the sound. Ghafoor looked up at Shekib.\n\n\"Go and tell them she has a fever and that Dr. Behrowen is bringing it down. Go!\"\n\nShekib nodded and hurried to the front door. The two men were pacing just outside the entrance. They were growing impatient.\n\n\"She has a fever. She is in the bath now to bring it down. She is weak.\"\n\n\"Is she going to be all right?\"\n\n\"I don't know much more than this. Dr. Behrowen will have to tell you.\"\n\nThey huffed, dissatisfied with the answer but helpless to find out more for themselves.\n\nShekib returned to the baths. They had pulled Fatima out of the water.\n\n\"Let's have her lie down!\" Dr. Behrowen pointed at the nearest door, just a few feet down the hallway. It was Benafsha's room. The door was closed.\n\n\"Khanum Benafsha, open the door please!\" Ghafoor called out. She knocked a second time, louder, when there was no answer. \"Khanum Benafsha!\"\n\n\"Please, I am sleeping!\" she called back. The women looked at one another in surprise.\n\n\"Khanum Benafsha, please, this is an emergency. Khanum Fatima is\u2014\"\n\n\"Oh, may God have mercy. Just open the door!\" Sakina said angrily, and pushed the door open into Benafsha's chambers. Benafsha's mouth opened in surprise to see them lay a pale-faced Fatima on her floor. Benafsha's face was flushed and she had pulled a robe around her nightgown. Someone had thought enough to bring dry cloths and a dry gown for Fatima. They began to strip her wet clothes when Sakina looked over at Benafsha.\n\n\"What is the matter with you? You can hear us, can't you? She's not well!\"\n\nBenafsha bit her bottom lip and rubbed her eye. \"I was sleeping. I didn't hear anything.\"\n\n\"You must sleep well if you\u2014\" Sakina paused. \"What is that?\"\n\nA dozen eyes followed her finger.\n\nOn the floor of the room, behind the door, was a gray lamb's-wool hat. A man's hat.\n\nBenafsha's mouth gaped. Her face grew as pale as Fatima's.\n\n\"This is a man's hat!\"\n\nShe was speechless. The women looked at each other, slowly realizing the implications. Benafsha tried to recover.\n\n\"It is his, it belongs to our dear Habibullah . . . come on, Sakina, what are you trying to . . .\"\n\n\"It's you, isn't it? The guards have all been asking about strange noises, any odd events! It's you they've been asking about! Where's Ghafoor? Where's Karim? Here!\" Sakina stormed to the door, picked up the hat and waved it wildly in the air. \"Is this what you've been looking for? Benafsha dares to have a lover!\"\n\n\"Sakina, you tramp! Watch what you say or you'll regret it! I don't have to answer to you! You of all people with your . . . your . . .\" Her eyes searched the room for an ally. Unfortunately, in her time in the harem, Benafsha's haughty attitude had made her no real friends. She looked at Tariq, her eyes pleading. Tariq looked away, her face conflicted.\n\nBenafsha's attempts to retaliate failed. Her eyes welled with tears and her tongue floundered as she looked at a room of hostile stares. Only Dr. Behrowen's attentions stayed on Fatima, who had been stirred both by the cool bath and now by the fresh scandal. She had propped herself up on her elbows, her bleary eyes looking around with the others.\n\nSakina could hear the panic in Benafsha's voice. She pounced.\n\n\"Well, if it is Habibullah's hat, then we can just take it to him and ask him to confirm it. That's easy enough,\" she said sweetly. She pushed the hat into Benafsha's face and then tossed it to Ghafoor. Ghafoor looked at the gray hat with almost as much trepidation as Benafsha. Her mind scrambled, knowing nothing good came from bringing bad news to the palace.\n\nBenafsha was crazed.\n\n\"Sakina, sisters,\" she cried, looking around the room. \"You can't possibly think . . . please, don't say such things about me to Habibullah! He will think things . . . he will . . . please! I have never been unkind to any of you! Please just stop and think before you act on such wild ideas!\"\n\n\"Wild? Look who talks of wild!\"\n\n\"Khanum-ha, please! Hush!\" Dr. Behrowen grew frustrated with the storm of tears and screaming. Her patient still needed attention. \"I don't know what you women are arguing about but surely it can wait,\" she muttered.\n\n\"Sakina, let's consider this a moment,\" Halima said, her voice feigning calm. \"Let's just stop for now and focus on Fatima-jan. We will address this later. Let's see what Khanum Behrowen needs for the moment.\"\n\nShekib watched but her ears tuned out the talking. She saw nervous glances, hot whispers, tongues clucking. There was pacing, head shaking and hot tea. Children walked in and were sent back out. Benafsha's green eyes blurred behind tears. She pitied herself. She hated Sakina.\n\nShekib noticed something the others had not. A single red rose petal on the floor, trampled under the many slippers of the king's concubines.\n\nShekib knew exactly whom Benafsha had welcomed into her bed.\nCHAPTER 41\n\nFatima's condition improved. Benafsha's worsened.\n\nThe harem was tense. Periodic updates were sent to the men outside. Nothing had been said yet about Benafsha, but it was just a matter of time. A matter of hours. Some of the more sensitive women had regressed into their own chambers, knowing the palace, the king, would not take Benafsha's transgressions lightly. She had made a fatal mistake and they could do nothing for her.\n\nNo one wanted to break the news to the palace, fearing they would strike broadly at anyone remotely involved.\n\n\"Have mercy, please. Have mercy,\" Benafsha whimpered in the corner. She was on her knees, her head touching the ground in supplication.\n\nThe guards and a few of the concubines had gathered outside her room. Fatima had been returned to her own chambers, Dr. Behrowen still at her side.\n\n\"It should be one of the guards,\" Sakina decided. \"You are the ones responsible for the happenings of the harem. It is your duty to report back to the palace what happens here.\"\n\n\"What if we say nothing?\" Nabila suggested meekly. \"I am sure she will put an end to this sinful behavior after tonight. She looks as if she's suffered enough now.\"\n\n\"You would dare hide this from the king? And what if he comes to find out some other way? We will all be blamed!\" Sakina said. \"I can't take that chance with my life.\"\n\nA few others nodded, agreeing with Sakina's reasoning. What if the men outside had actually heard everything? What if they were going to report everything to the king? The harem had to be forthcoming if they were to save their own hides.\n\n\"Khanum Sakina, maybe it would go easier on the king's ears if he were to hear it from someone he fancies. And since you were the one who made the discovery, I am sure he will reward you for sharing this with him and putting an end to such a shame,\" Ghafoor said.\n\nShe is impressive, thought Shekib. She could have Bobo Shahgul's blood in her veins.\n\n\"You are talking as if it is your first day in the palace. You know very well that you are the ones who report to the king's people. We, women of the harem, are not to be involved in this discussion. I will not hide anything from my dear Habibullah but it is not my place to march up to his chambers and make such an announcement.\"\n\nGhafoor chewed her lip and looked at Karim. She shook her head, having nothing to add to the argument. Ghafoor grew more nervous, knowing that, as leader of the guards, she was responsible for direct communications with the palace. The onus sat heavily on her shoulders. She could be rewarded for her service or she could be struck down for bearing such devastating news. She motioned with a subtle tilt of her head for the other guards to follow her into the foyer. A quick peek outside confirmed that the two men were idling by the far end of the courtyard, their backs to the harem.\n\n\"Karim, why don't you and Qasim go and ask those men for a chance to speak with the king directly. It may be worse for this message to go through too many ears before reaching his.\"\n\n\"With all due respect, Ghafoor-jan,\" Karim said facetiously, \"since you have always been in charge of our troupe, this does not seem to be something you can delegate out like a night shift. Neither one of us would dare infringe on your responsibilities.\"\n\n\"Nor us,\" Tariq said, glancing at Shekib. She, too, felt the need to pair up with someone.\n\nGhafoor huffed. \"Fine. Fine! Cowards. I'll go and speak to them myself.\" Her eyes betrayed her confidence. She paced the foyer for ten minutes before putting her hand on the doorknob.\n\nKarim had her ear to the door, trying to listen in, but the voices in the courtyard were hushed. The guards looked at each other, paced and sighed frequently. Eyes were bloodshot with fatigue and conflict. When Karim cracked the door open ten minutes later, the courtyard was empty. They had taken Ghafoor to the palace.\n\nAn hour passed, painfully, before Ghafoor reappeared. Qasim and Karim had fallen asleep leaning against the foyer wall. Tariq sat near the door as if ready to make a quick escape. She tapped her foot nervously. Her eyelids were heavy and dark. Shekib sat against the wall opposite the sisters, her stomach uneasy. A house under stress had never boded well for her. She had no reason to believe she would emerge from this unscathed.\n\nGhafoor looked about nervously and took stock of the situation.\n\n\"How is Fatima?\" she asked quietly, her eyes shifting around the room, hesitant to land on anyone in particular.\n\n\"She's a little better. She's had some tea with sugar and was talking for a bit. Now she's fallen asleep. Dr. Behrowen left a few minutes ago. You probably passed her on the way here,\" Tariq said, her voice as exhausted as her eyes.\n\n\"Good.\"\n\n\"Aren't you going to tell us what happened?\" Karim asked impatiently.\n\n\"I spoke with the men outside and they took me back to Agha Ferooz, our king's most trusted adviser. They did not want to disturb the king himself. I explained the situation and they are, of course, very upset. They notified the king.\"\n\n\"And? What will happen now?\" Qasim asked.\n\n\"He is angry. He wishes to speak with Shekib.\"\n\nShekib was not in the least surprised.\n\n\"What is that they want to speak to me about?\" Her tone was measured, even. It made Ghafoor nervous. She looked at the others while Shekib saw through her act.\n\nShe's done something.\n\n\"How should I know?\" she said defiantly. \"They asked me who was on night duty tonight and I gave them an answer. I had better check on Khanum Fatima. There's a soldier waiting outside, Shekib. He'll escort you to the palace. I'm sure it's not a big deal.\"\n\nShekib was sure that it was.\n\nBut she said nothing, staring at the back of Ghafoor's bobbing head as she scampered down the hall, putting distance between them as quickly as she could.\n\nThe others watched her leave and then turned to Shekib. She said nothing but rose and walked to the door. As Ghafoor had promised, outside stood a soldier. A baby's face in a man's uniform. He looked nervous in the brisk dawn air. He motioned for her to follow, turning back once to steal a glance at her face.\n\nHe walked her to the palace's heavy front doors, intricately carved and oddly inviting even at this moment. He opened the door and led her in, down one long hallway with ornate patterns on the walls, gilded pedestal tables and richly embroidered chairs. Shekib noticed her surroundings with vague interest.\n\n\"In this room,\" he announced, and cracked the door open enough for her to enter. He stayed back and looked thankful that his duties ended there.\n\nShekib entered, remembering to keep her back straight and her eyes focused. Weariness was blurring her judgment as well as her vision.\n\nIn the room, King Habibullah paced behind a handsome wood-carved desk, his fingers pulling at the fringes of his beard. Two men sat anxiously in armchairs to his left, opposites of each other. One was heavyset and short, the other tall and lanky. Had Shekib been less nervous, she might have noticed how ridiculous they looked as a pair. They looked up at Shekib, their lips tightening.\n\n\"You!\" King Habibullah called out. He had stopped pacing abruptly, his blue chappan flapping as he whirled to a stop.\n\n\"As-salaam-alaikum, Your Highness,\" she said in a hush, keeping her head bowed and her eyes downcast.\n\n\"As-salaam-alaikum, eh? As if nothing has happened? Do you know the meaning of the words, you idiot?\"\n\n\"I apologize, esteemed sir. I meant no disrespect\u2014\"\n\n\"Don't patronize me, guard! You are here to answer questions, to speak up for your actions\u2014or inaction, as it appears! It was you who was on guard tonight, when a man somehow managed to evade your attention and enter my harem!\"\n\nA conversation began to take shape in Shekib's mind. She could imagine Ghafoor standing in this very room, not too long ago, painting a picture of an idle guard, passively allowing a man to violate the king's sanctuary, to indulge in his private stock of women.\n\n\"Dear king, I was on guard tonight but I saw no one enter.\"\n\n\"You saw no one enter? But someone did enter, didn't he!\" His face was the color of the carpets on the floor. A blue vein pulsed across his forehead like a lightning bolt. He fell into his chair and looked at his two counselors expectantly.\n\n\"Guard, did you see someone leave the harem tonight?\" The thinner man rose to his feet and spoke up.\n\nShekib did not have much time to consider her answer. \"No, sir.\"\n\n\"And you saw no one enter?\"\n\n\"No, sir.\"\n\n\"Are these the kinds of guards we have for my harem!\" The king exploded, his fist rattling the table with a thunderous clap. \"We might as well have brought donkeys!\"\n\n\"Guard, explain to our dear king what happened tonight. Was there a man in the harem?\" the lanky man demanded.\n\nShekib searched for the right answer, her hands trembling at her sides. She was afraid to move. They took turns shouting questions at her.\n\n\"Answer!\"\n\n\"I . . . I did not see\u2014\"\n\n\"Don't tell us what you did not see! Tell us what happened!\"\n\n\"Tonight we found a hat in one of the chambers.\" Shekib was not sure how to phrase such a finding. It was a sensitive matter and the wrong words could be dangerous. They were waiting for her to continue. \"There was no one there but the hat . . . the hat suggested that someone . . . a person had been there. We asked but\u2014\"\n\n\"Whose chambers were you in?\" the king asked, his eyes slits. He spoke slowly and precisely.\n\n\"We were in Khanum . . . Khanum Benafsha's chambers,\" she answered, her eyes cemented to the marble floor. Benafsha had shamed the palace with her iniquity but Shekib still felt reluctant to expose her. She pictured Benafsha back at the harem, prostrated, her face wet with misery.\n\nWhy did you do this? Why did you bring this upon us?\n\n\"Benafsha.\" Habibullah turned his back and faced the window. Heavy burgundy drapes framed his silhouette. \"That vixen.\"\n\n\"Have you seen anyone before? Coming in and out of the harem?\"\n\nWhat did you tell them, Ghafoor?\n\n\"I . . . I have not.\"\n\n\"This was the first time you learned of this?\"\n\n\"Yes, sir.\"\n\nThree men brooded. Shekib could hear their measured breaths.\n\n\"You. You believe this happened once?\"\n\n\"I . . . I . . . believe so.\"\n\n\"And who was it guarding the harem tonight?\"\n\n\"I was, sir.\"\n\n\"You are a liar. We have heard differently. Ghafoor has already told us that you saw this man before! And you kept it from everyone until tonight!\" the short man shouted.\n\n\"With respect, agha-sahib, I had not seen\u2014\"\n\n\"Liar!\"\n\nGhafoor, you scoundrel! You fed me to the lions!\n\nIt was clear now. Her word against Ghafoor's, and they were taking Ghafoor's. Shekib was not a bystander. She was a guilty party.\n\n\"Did you know of Benafsha's activities? Did she ask you to cover up for her?\"\n\n\"No, sir! I had no\u2014\"\n\n\"What about the man? Who is he? Did he bribe you?\"\n\n\"Please, dearest king, I had nothing to do\u2014\"\n\nHe barely heard anything she said. He was more interested in how this made him look.\n\n\"Know this, guard! An offense this grave does not go unpunished. My name has been besmirched. One look at your face and it is obvious you are damned! Have her locked up! And Benafsha too! We'll make swift examples of them both.\"\nCHAPTER 42\n\n\"Why did you have to do such a thing?\"\n\n\"You wouldn't understand.\"\n\nThe room was dark and smelled of rotted meat. The stench reminded Shekib of cholera, of mourning and loneliness.\n\nBenafsha's face had changed. Shekib was struck by the difference. Just eight hours ago, she had been the most striking woman in the harem. How quickly her face had grayed! Her hair was stringy and her green eyes looked defeated and bloodshot.\n\nOne of the king's most prized concubines. A life of luxury by any standards. The choicest foods, clothes. What had driven her to take all these for granted?\n\nAn hour passed in silence. Shekib wanted to ask her about Agha Baraan. She was sure it was him. The hat. The rose petal. But why? He was Amanullah's friend. Why would a man like him commit such an act against his friend's family, especially when his father was the most powerful man in Afghanistan?\n\n\"I am sorry you are here.\"\n\nShekib looked up. \"So am I.\"\n\nShe thought of Amanullah. What would he think when he heard of the night's events? How disappointed he would be in her! She wasn't much of a guard, according to the palace. What made her think she could be much of a wife? Benafsha had ruined everything. She looked at the girl with disgust and pity. Then there was Ghafoor, that split-tongued viper. She had set Shekib up, saving herself. No wonder she had run off. Coward.\n\nThe dank room was unfamiliar but the rest of the experience was not. Angry fingers had often pointed at Shekiba.\n\nOn the king's orders, Shekib had been led away\u2014through the hallways, through the kitchen and into the small room where the cooks once kept cured meats and vegetables. The room smelled of flesh and earth. Shekib closed her eyes and imagined her father's house. Her mind floated to those bare walls, her brother's shirt thrown across a chair as if he would run through the door looking for it. Her sister's amulet on the table. Her father, sitting in the corner clicking the beads of his tasbeh while he stared through the window onto fallow fields, a fallow home.\n\nShekib stood up and began to pace. The walls were tight but light crept in, framing the door with a yellow glow. The palace had electricity courtesy of a foreign company commissioned by the king. All of Afghanistan twinkled by lanterns but the palace shone, a beacon for the rest of the country.\n\nThe king must have his way. How much it must burn him that another man has had his way with his precious Benafsha. She's pretty, I suppose. If she doesn't show her teeth when she smiles. All pushed together, her teeth look like chickens climbing over one another in a crowded coop.\n\nBenafsha had her head between her knees. Shekib couldn't tell if she was awake or asleep.\n\n\"What do you think they're going to do with us?\" Shekib asked quietly.\n\nBenafsha shoulders lifted and fell with a deep breath.\n\n\"How long do you think we'll have to be in here?\"\n\nBenafsha looked up. Her eyes were flat with resignation. \"You really don't know?\"\n\nShekib shook her head.\n\n\"When the crime is adultery, the punishment is sangsaar. I will be stoned.\"\nCHAPTER 43\n\nThe large auditorium, a room larger than any I'd ever seen, held hundreds of parliamentarians. Their chairs were arranged in rows that went from one side of the room to the other, leather chairs behind a row of desks. Each member had a microphone and a bottle of water.\n\nBadriya's and mine sat in the center of the room, sharing our row with Hamida and Sufia. In the front of the room sat a man with a neatly trimmed mustache and salt-and-pepper hair. He listened, nodding his head from time to time.\n\nThe men intimidated me. Some of them were my husband's age, gray haired with beards that nearly touched their chests. Others were younger, their faces shaved and their clothes different from what the men in my village wore. Pants, button-down shirts, jackets.\n\nAs we broke for lunch during the first week, Hamida had asked me what I thought so far. I was nervous to tell her, afraid I would sound stupid. And I worried that if they saw me reading and writing, they would realize how basic my knowledge was.\n\n\"They come from where?\" I asked, astounded by the accents I was hearing.\n\n\"What do you mean?\" She looked to see where I was pointing.\n\n\"I mean, I've never seen men dressed like . . . dressed like that.\" I pointed with my head to a man wearing brown pants and a military-style vest over a white shirt.\n\n\"That's what you'll see in Kabul, Rahima-jan,\" Hamida said, proudly. \"This parliament is where every corner of Afghanistan comes together.\"\n\n\"Comes together?\" Sufia scoffed. \"More like this is where Afghanistan comes apart!\"\n\nHamida laughed. A man one row away turned around and shot her a look. He shook his head and leaned over to mutter something to the man seated next to him, sharing his disapproval.\n\nThe session was called to order. Rahima tried to look around without anyone noticing. Badriya picked up a pen and held it to the blank paper before her as she watched the speaker. She was playing the part.\n\n\"Ladies and gentlemen, the matter of the president's cabinet members will now be introduced. Seven people have been nominated by the president. It is up to this parliament to approve or reject the nominations.\"\n\n\"Badriya, are we going to see the president?\" I whispered. It was hard to believe I might come face-to-face with our nation's most powerful man.\n\n\"No, you fool! This is the parliament. He does his work and we do ours! Why should he come here?\"\n\n\"We'll talk about the candidates one by one. I'll call on you to ask whatever questions you may have. We need to decide if these individuals are suited for the job. And if they'll help take our country in the right direction. First up is Ashrafullah Fawzali, nominated for position of minister of justice.\"\n\nThe speaker went on to talk about Fawzali's background, his home province and his role in training the police force.\n\nA woman parliamentarian sat in the seat beside me. I heard her huff, frustrated. I watched her from the corner of my eye, slouched back in her chair and shaking her head. As the candidates' virtues and experience were extolled by a man who had taken the floor, she became more and more displeased, fidgeting in her seat and tapping her pen.\n\nThe next candidate was introduced, someone equally distressing to her. She raised her hand to speak but the director looked past her. She waved her hand more dramatically.\n\n\"Excuse me, but I would like to say something about this candidate,\" she said, leaning forward and speaking into her microphone. \"Excuse me!\"\n\n\"Khanum, the time for the discussion of this candidate is up. We're getting close to ending today's session. Thank you all, please return for tomorrow's voting. The parliament is dismissed.\"\n\n\"Of course it is! God forbid we actually talk about these candidates!\" the woman hissed.\n\n\"Who is she?\" I asked Badriya.\n\n\"The one next to you? Oh, that's Zamarud Barakati. She's trouble. Make sure you stay away from her,\" Badriya leaned forward to tell me. \"She's one of those you don't want to get mixed up with.\"\n\n\"Why? What's wrong with her?\"\n\n\"She's a troublemaker. You see what she did today? Always interrupting things. That woman's lucky they haven't condemned her to sangsaar.\"\n\nStoning. I shuddered and thought of Bibi Shekiba.\n\nAs far as I had seen, Zamarud hadn't done anything that several other parliamentarians hadn't done. Just like the men, she had raised her hand and asked to speak. But I could see many people didn't appreciate hearing from her. Several men had rolled their eyes or waved their hand in annoyance to hear her ask for the floor's attention.\n\n\"She pushes her ideas too much. People don't want to listen to her all the time.\" We were walking out the security checkpoint by this time. Our driver saw us coming and went to turn the car on. Our guard was already with him. Zamarud walked angrily past us, her own security guards struggling to keep up.\n\nShe reminded me of Khala Shaima, the only woman I'd known who would speak up to men outside her own family. I wondered what Khala Shaima would have thought of Zamarud. Picturing the two of them in the same room made me smile. They could have the entire parliament up in arms.\n\nBut what I saw in that first day was just the beginning. The parliament was a fiery mix of personalities and politics. There were so many women there, but only a few of them spoke during the sessions. And there was only one Zamarud.\n\nAs the discussion of the cabinet nominations went on, Zamarud became more and more agitated. She was given opportunity to speak and took the floor like a storm, questioning the intentions and honesty of the candidates. She implied that the candidates had been chosen for reasons other than their qualifications, since one was the president's brother-in-law while another was the president's childhood friend. And there was no diversity, she said critically. They were all from one sect of the Afghan population. Afghanistan needed to represent all of its many colors, Zamarud insisted, or it would fall apart. Again.\n\nOn the fifth day of sessions, we took our seats. I missed my son more today and saw his round cheeks and almond eyes when I closed my eyes. I wondered if he was walking at this moment, one hand tightly gripping Jameela's. I wanted to hear his voice, the tiny sound of \"maada\"; he was still unable to roll his tongue to produce the proper \"madar.\"\n\nZamarud's voice brought me back.\n\n\"It's imperative that we think of the future of this country. We Afghans have become complacent, letting almost anyone take on these positions of power and influence. Let's think about it carefully and then decide.\"\n\n\"Khanum, I believe it would be wise for you to consider before you speak. There are many people here and you're not thinking\u2014\"\n\n\"I'm not thinking? I'm thinking about it a great deal! It's you and the rest of you that need to start thinking. I'm going to speak my mind right now.\"\n\nBadriya looked over at me. Waves of anger were rippling through the room. The men were leaning over and complaining to their neighbors. Hamida and Sufia looked over at Zamarud nervously.\n\n\"From what I have seen, the nominations that have been presented thus far have been of men who worked alongside the most sinister characters in our country's recent history. The money in their pockets comes from drugs, from alliances with warlords and mercenaries. They have the blood of their fellow Afghans on their hands.\n\n\"And there are candidates who are family members, getting special treatment from those in the highest position.\"\n\nIt was obvious she was talking about the president's brother-in-law, who traveled between Kabul and other cities like Dubai, Paris, London and Islamabad, importing and exporting goods. He had built a successful trade business and a life of luxury for his family. But everyone knew his business didn't account for all of his income.\n\n\"We must watch who we place in these official positions. They should be there for the right reasons, for the development and protection of our beloved Afghanistan. We have suffered enough in the hands of others in the last decades. Our people deserve to have right-minded individuals in power. I wonder, as do so many others, how it is that some of our nominees have been able to amass a fortune when our people go hungry. How is it that they are able to live lavishly when they are engaged in simple businesses? We all know the answer. We know that there are sources of money that are not talked of, that are not openly discussed. Bribes. Nepotism. Drugs. These practices will bring our country down.\"\n\nThe room began to talk. Zamarud continued, louder.\n\n\"I will not stand for this. I will not approve the election of such people, brothers and cousins taking under the table what rightfully belongs to our country. Are we to sit here quietly and let them suck the blood of the Afghan people? Getting fat off of government money?\"\n\n\"That's enough!\" one man called out. Others echoed after him.\n\n\"Shut her up.\"\n\nZamarud went on, unfazed by their comments. She raised her voice over the protests.\n\n\"Every person in this room, every man and every woman, who would dare to approve these nominations will share the responsibility for keeping those lips greasy with the money that should go to the Afghan people, to the Afghan country. And for what? For a chance of fattening your own pockets! You know who you are. You come here and pretend to represent your provinces when really you represent nothing but your own pockets!\"\n\n\"Who does this woman think she is?\"\n\n\"I will not listen to this harlot babble on!\"\n\nThe yelling became angrier. Hamida and Sufia, not far from Zamarud, had gone over and pulled her back to her seat. Sufia was talking to her, saying something in her ear, while Hamida put a hand over the microphone. We were close enough that we could still hear her.\n\n\"I will not be silenced! I have had enough of their nonsense! Which of you will speak up if I do not? Call me what you like but you know I speak the truth and it is you all that are damned for what you're doing! It's a sin! It's a sin!\"\n\nTwo men went to confront her directly. Fingers were pointed, just inches from her face. I felt my body tense with their aggression. I wanted to pull Zamarud back but I sat frozen, my eyes wide. I prayed for her to stop talking.\n\nThe room was on its feet. Arms were waving. A group of men had gathered in a corner of the auditorium, pointing in Zamarud's direction and shaking their heads. Two other women had joined Hamida and Sufia in trying to restrain a belligerent Zamarud. Others were on their feet, watching the fray with interest or enjoyment.\n\nI was nervous for her, as was every other woman in the room. I'd never seen a woman speak so boldly, so directly, and in a room full of men! Everything I'd ever seen in my life told me Zamarud wouldn't make it out the door.\n\n\"This is bad,\" Badriya muttered, keeping her head low. We had not stirred from our seats. \"We can have no part in this, understand me? Just stay where you are. We're going to leave just as soon as things calm down.\"\n\nI nodded. The last thing we wanted was for Abdul Khaliq to get word we'd been involved in a shouting match between the parliament's most outspoken woman member and the group gathered by the door. They were men like my husband, older and with fearful constituencies back home. They were warlords.\n\nHamida walked over to us when things calmed.\n\n\"Unbelievable,\" she said. \"These people are wild!\"\n\nBadriya nodded politely, not wanting to weigh in with an opinion.\n\n\"I mean, she's a bit bold, I'll give them that. Actually, she's a bulldozer. But she's right. Especially about Qayoumi. He has friends in the Ministry of Defense and they fed him every contract that came through their office. As if he needs any more money. Have you seen his car? His house?\"\n\n\"No, I haven't,\" I said, intrigued. Badriya was so silent around these women that I almost forgot she was there. It was completely unlike her but she tensed, fearful that Abdul Khaliq would hear about any idle chatter.\n\n\"Let me tell you, his house is one of the nicest houses in Kabul. He tore down an old, run-down home in Shahr-e-Naw and then built himself a two-story mansion! And you know how expensive that area is! No Afghan can buy anything there. All those properties go for at least half a million U.S. dollars. At least!\"\n\nHalf a million U.S. dollars? My mind reeled at the staggering amount.\n\n\"Half a million . . . ?\"\n\n\"Yes, that's right! He'll do anything to get what he wants. Anything. He was a Taliban ally not too long ago and they pillaged one town, robbed the people of everything they had. Setting fires, lining up the men and killing them. By the time they finished with that town, whoever they left alive had only the clothes on their back. Sinful!\"\n\n\"And they want to vote him in?\" If this was common knowledge, why weren't people more upset about him?\n\n\"Yes, they do. That's how it is. For God's sake, warlords make up at least a third of the parliament right now. Those people who led the rocket attacks, the bloodshed\u2014they're all sitting in this assembly room. Now they want to fix what they broke. It's almost comical,\" she said, shaking her head. \"If I thought of it too much, I'd go crazy. Like Zamarud!\"\n\nHad I been anyone else, I might have been more surprised. But I was a wife of Abdul Khaliq, a man who inspired fear in every corner of our province. And I was sure I didn't know a quarter of what he had done in the years of war. Actually, I still didn't know what he did when he set out with his guards and his automatic weapons. Someone could nominate him for a post as well.\n\n\"What can you do? Our politics are full of people like that. But I can tell you, I won't be approving the nomination of that corrupt butcher. Sufia's talked to the other women. They're going to be rejecting him as well.\"\n\n\"If so many people are going to vote against his nomination, he won't stand a chance, right?\" I watched Badriya, her lips pulled down in a frown. I was asking too many questions.\n\n\"He stands a very good chance, actually. Warlords make agreements, alliances, to serve their own purposes.\"\n\nI wondered if Hamida knew who Abdul Khaliq was. I wasn't sure how far his name had reached. Where we came from, he held a lot of power and he was trying to grow that. Badriya's involvement in the parliament was a step in that direction.\n\n\"Hamida-jan, we're going to get a cup of tea from the cafeteria, if you don't mind,\" Badriya said. The conversation had touched a nerve. Her voice was stiff. \"Can I bring you anything?\"\n\n\"No, I'm fine, thank you. Let me go see what Sufia is up to. The session will probably resume in another thirty minutes.\"\n\nIn our hotel room that night, I asked Badriya about Zamarud's allegations.\n\n\"Is it true? Are there that many people in politics who are that corrupt?\"\n\n\"Don't bother yourself with things like that. It's none of your business.\"\n\nThat angered me. I was fairly sure Sufia would not have agreed with her. \"But it's yours, isn't it? You're going to be voting on those nominations tomorrow. Are you going to approve them?\"\n\n\"Of course I am.\"\n\n\"Why?\"\n\n\"Why? Because that's who I will vote for! Have you finished filling out that form yet? The director's office has been asking about it all week.\"\n\n\"It's almost done.\" I sighed. I wondered how Badriya had coped in her last stay in Kabul. She could barely scratch out her own signature. \"But how do you decide how you will vote?\"\n\n\"I decide, all right? I know what the issues are and then I choose.\"\n\nI thought back to today's heated session, Zamarud's determined look. \"Does she have a husband?\"\n\n\"Who? Zamarud?\" she snickered. \"They say she does but I can only imagine what a mouse of a man he must be! Can you believe the way she behaves?\"\n\n\"She's not afraid of them.\"\n\n\"She should be. Zamarud's gotten more threats than any other woman in that assembly. Not surprising, the way she carries on. Shameless,\" she said, clucking her tongue.\n\n\"You haven't gotten any threats, have you? Hamida said most of the women have. Her family begged her not to run for parliament again but she wanted to.\"\n\n\"She's another mule of a woman. I haven't gotten any threats because I know what I'm doing. I mind my own business and do only what needs to be done. I'm not here to embarrass myself or my husband.\"\n\nI shuddered to think how Abdul Khaliq would put Zamarud in her place. But I didn't think Badriya had any special business in the parliament. My instincts told me it had something to do with our husband.\n\n\"This form asks if you want to join the group traveling other countries with parliaments. As a learning experience, it says. Europe. It says, 'the director highly recommends that all parliamentarians go to learn how other assemblies function.' \"\n\nNow that I was in Kabul, I was hearing of places even grander and more unimaginable, like Europe. I wondered what a place like that could look like. We'd come all the way to Kabul. Maybe we could go to Europe too? Badriya lifted her head, as intrigued as I was by the exotic name.\n\n\"Go to Europe? Really?\" Once she'd said it, Badriya realized how ridiculous it sounded. \"Forget it. Not interested. Put that damn thing away. I'm tired. You can finish it in the morning. I'm going to bed.\"\nCHAPTER 44\n\n\"We will now take a vote on the candidate Ashrafullah Fawzali. Please raise your paddles with your vote on his nomination.\"\n\nThe parliamentarians each had two paddles, one red and one green, which they raised to vote aye or nay in the assembly. This was the first vote to be taken and Badriya looked nervous.\n\n\"Are you going to vote for him?\" I whispered.\n\n\"Shhh!\" she hissed at me, her eyes scouting the room. Paddles were going up, many at a time. Badriya reached for the green paddle and raised it halfway, still unsure.\n\nI followed her eyes to a man sitting toward the front of the room. From our position, we could see his profile. He was a burly man with a heavy beard and thick features. His gray turban sat coiled on his head like a serpent. He held a green paddle.\n\nI saw him look in our direction, giving Badriya a subtle nod. Her green paddle went up and she kept her eyes fixed on the front of the room. I was puzzled. I didn't recognize this man but it looked like Badriya did.\n\n\"Badriya, what are you doing? Who is that?\"\n\n\"Shut up! Just take notes or whatever it is that you're doing.\"\n\n\"But he's looking over here!\"\n\n\"Shut up I said!\"\n\nI crossed my arms, shut my mouth and watched. That's how things went for the rest of the session. Each time the director asked the parliament to vote on a candidate, Badriya waited until this man raised his paddle. And each time she would pick the paddle that matched his. Green, green, red, green, red, red. And each time he looked over, his face was smug with approval to see her vote his way.\n\nThe ladies looked over at Badriya, seemingly confused. Sufia whispered something to Hamida, who shrugged her shoulders.\n\nQayoumi. It was time to vote on his nomination. I looked over at Hamida and Sufia. They were shaking their heads as the director prepared to take a vote. A small murmur wove through the assembly as the parliamentarians prepared to decide on one of the most controversial figures in Kabul. Tongues clucked with disapproval even before the paddles went up.\n\n\"Ladies and gentlemen, please raise your votes. Raise them high so we can see them!\"\n\nThe man voted green.\n\nI looked at Badriya. I was sure she could feel my eyes on her but she avoided my gaze.\n\nShe watched as the ladies both raised their paddles red. The representatives around them raised their red paddles as well. There were pockets of green here and there, almost all men.\n\nThe mumbling got louder as green paddles went up.\n\nBadriya kept her head down and picked up her paddle. I opened my mouth to say something.\n\nGreen.\n\n\"Badriya! What are you doing? Didn't you hear what they said about him? Why are you voting for him?\"\n\n\"Please, Rahima, shut up!\"\n\nHamida and Sufia looked over, eyebrows raised. They looked away and leaned toward each other. I thought of our conversation with Hamida. I couldn't ignore everything she'd told us.\n\n\"But Hamida said\u2014\"\n\n\"If you can't shut your mouth, leave then! Just get out!\" she snarled. \"I don't need you.\"\n\nI stared at her. There was nowhere for me to go. I sat beside her, fuming, even though I had no right to. Maybe I would have done the same if I were her. Maybe I would have aped the votes of the man in the corner.\n\nAbdul Khaliq. He set her up for this. That man must have something to do with that security contract he wanted to land. Just like Hamida talked about.\n\nI was surprised only that my husband's influence was this far-reaching, into the parliament building of Kabul. And wherever that man was from.\n\nHamida looked over, her lips pursed.\n\nMaybe I wouldn't have been like Badriya if I were in parliament. Maybe I could have been more like Hamida. Or Sufia. Or even Zamarud. Maybe I would have sat in that assembly seat and made up my own mind.\n\nBut I probably would not have. It wouldn't be easy to go home to Abdul Khaliq after going against his instructions. Especially in a matter this big.\n\nThe session closed for the day. Badriya rose quickly from her seat and gathered her bag. She made her way down the row and out the main aisle without turning around to see if I was following.\n\nWe ran into the ladies near the security check. Not even a polite smile. It was obvious they were disappointed in Badriya's voting. They could tell her reds and greens were decided by outside forces. She was part of the problem.\n\n\"I'm glad the day is finally over,\" Sufia said neutrally.\n\n\"Yes, so am I,\" Badriya said, agreeing demurely.\n\n\"Interesting day,\" Hamida murmured, adjusting her head scarf.\n\nI watched the exchange, wanting to shout out that I wasn't part of this. I wanted to say that I wouldn't have voted for Qayoumi. Even though I was almost certain I would have. I was learning that cosmopolitan Kabul was, at least in that way, no different from my obscure village. Many of our decisions were not decisions at all. We were herded into one choice or another, to put it gently. I wondered if the other women representatives truly felt free to make their own judgments.\n\nI sat in the car and leaned back, wishing I was home with Jahangir. He was probably taking a nap now, his mouth half open and his eyelids fluttering with innocent dreams. Thank God Jameela was there to look after him.\n\nBadriya got in from the other side, slid across the seat, turned and slapped my face so hard I fell against the car door.\n\n\"Rahima, you question me again and I swear I will go straight to Abdul Khaliq and tell him you're opening your idiot mouth in the assembly. We'll see if you're so eager to wag your tongue then! Learn to control yourself, you bitch.\"\n\nMaroof looked into the rearview mirror. An expression of surprise twisted into a smirk. He was entertained. My face stung but I said nothing. I had the rest of our stay to get through and I refused to become a spectacle for our bodyguards.\n\nThe following morning, we wove through clusters of foreign soldiers and returned to the parliament building. Late, because of Badriya. But there was no voting today, only discussions. Nothing of importance to her, though she was obligated to make an appearance.\n\nI wasn't speaking to her, just answering her questions and keeping out of the way. I was beginning to reconsider if being in Kabul was worth putting up with her attitude. As bad as she was at the compound, she was worse here. There was only me to take all her attention and the pressure of following our husband's plan was getting to her.\n\nI took notes for her and filled out a survey distributed by some international organization looking to improve the parliament, and then we broke for lunch. I gravitated toward Hamida and Sufia. Badriya reluctantly followed with her tray.\n\n\"How are you two doing?\" Hamida said. They looked at us differently now. Yesterday had changed things.\n\n\"Fine, thanks. You?\" Badriya was curt. It wasn't helping the situation.\n\n\"Still surprised from yesterday. We were hoping to block more of those nominations. But I guess it was their naseeb to get approved.\"\n\nNaseeb. Did Sufia really believe that? If she did, why bother voting?\n\n\"Maybe so,\" Badriya said in agreement.\n\nI searched for something to say that would tell the ladies I was on their side but without riling Badriya's nerves.\n\n\"Sometimes people surprise you, don't they?\" I said. \"Maybe something good will come of it.\"\n\n\"An optimist\u2014there's something we don't see often.\"\n\nI had no reason to think Qayoumi was anything but the bastard they said he was. I had almost no reason to believe anyone would do anything good, really. My \"optimism\" was just words, strung together in hopes of making me look neutral. I wanted to be friendly with these women. They were independent and happy, something I'd tasted only as a young boy.\n\n\"Sufia and I are going to the resource center this evening. Maybe you would want to join us?\"\n\n\"Thank you but I can't,\" said Badriya. \"I'm going to my cousin's home tonight. I haven't seen her in over two years.\"\n\nI looked at her, surprised. Was she telling the truth? She spoke up, seeing the look on my face. \"My mother's cousin lives here in Kabul. I haven't seen them in so long and my aunt is getting older. They've insisted that I come by and visit them. They live on the other side of the river, by the women's hospital.\"\n\n\"Well, if you ladies are going there tonight then maybe another\u2014\"\n\nBadriya looked startled. \"Us? Oh, no. I'm going alone. Since it's my cousin, you know,\" she said, fumbling her words as she tried to undo my accompanying her. \"And Rahima-jan said she didn't want to go anyway.\"\n\nEyes on me for confirmation.\n\n\"Well, you kept saying they were such nice people. Maybe I should go after all, huh?\"\n\nBadriya's eyes widened. \"Really? You want to go? Are you sure that's what you want to do?\" she said. Her glare told me the answer she expected.\n\n\"No,\" I said. \"You know what? I think I've changed my mind. You should go and see your aunt and cousins. Maybe I'll go to the resource center instead. It would be great to see what they offer. I wouldn't mind taking some lessons while we're here.\"\n\nHamida's eyes lit up. It was as if she saw me in a new light.\n\n\"That's a great idea! That's what we'll do. While Badriya visits her aunt, we'll go to the resource center. We can go meet directly after the session closes today and then head over to their office. You'll be ready to go then, right?\"\n\nI agreed, satisfied that I'd gotten my way, even if Badriya had gotten hers as well. We parted ways when the session closed and I followed Hamida and Sufia. Badriya had taken Maroof and the guard. I was left with no one, which made me feel more free than alone. We picked up some dinner from the cafeteria and carried the plastic bags with us.\n\n\"Do they have these classes all the time? Is it like a school?\" I asked. I was getting more and more excited at the thought of returning to a classroom. Even if nothing came from the lessons.\n\n\"They have different instructors. Haven't you heard Sufia speak English? Where do you think she learned to say so nicely, 'Hello, how are you?' \" Hamida mimicked cheerfully.\n\nI had no idea what she'd said but I was impressed that they were learning English. Even more than that, I wanted to learn how to use the computers I'd seen in the parliament's library. The library was a small room in the basement level with three bookcases, two of which were empty. The book collection was sparse but the woman in charge was determined to amass a collection with works on politics, law and history. I thumbed through the books and realized how much there was to learn about government. It was not as simple as raising paddles.\n\nThe computers caught my eye. There were three of them but more were coming, we were told. The three were all being used by men whom I recognized from the assembly. I tried not to stare over their shoulders but I wanted to know what they were looking at on those screens. I watched from the corner of my eye as they punched slowly and carefully at the keyboard, piecing letters together in a way I'd never before seen.\n\nThe women took me to a small, newly constructed building with small windows and a sign out front in both English and Dari.\n\nWomen's Training Center, it read.\n\n\"This is really just for women?\" I asked. \"The men can't come here?\"\n\n\"Absolutely not, just like the hammam.\" Sufia chuckled. \"Thank God, someone finally took our involvement seriously. You know, Rahima-jan, international organizations send teachers and computers. All of it is available. We just have to use it.\"\n\n\"Do many of the women from the parliament come here?\"\n\n\"Hardly!\" Hamida said. \"So many of those women have no idea what they're doing. I had no idea what I was doing either but now it's my second term and I am just starting to realize how much we still have to learn before this assembly is really functional. We're like babies, just learning to crawl.\"\n\nAn image of Jahangir, his knees rough and dark from crawling about, his palms slapping against the floor with excitement. I missed my son.\n\nSufia must have read my face.\n\n\"You have children?\"\n\nI nodded. \"I have a son.\"\n\n\"How long have you been married?\"\n\n\"Almost three years.\"\n\n\"Hm. You were how old when you married?\"\n\n\"Thirteen,\" I answered quietly, my mind still on my little boy's face. I wondered what he was doing.\n\n\"Your husband must be much older, judging by Badriya's age,\" Hamida said, pausing before she opened the door to the training center.\n\nI nodded. I realized they both were trying not to look as curious as they were.\n\n\"Your husband . . . what does he do?\"\n\nI drew a blank. I wasn't quite certain what he did and I was even less certain how to avoid explaining it.\n\n\"I don't know,\" I said. I blushed when I saw the way they looked at me.\n\n\"You don't know? How can you not know?\"\n\n\"I never asked him.\"\n\n\"Never asked him? But you live there! You must have some idea what he does.\"\n\nThis outing was not as innocent as it appeared. They were interested\u2014probably after seeing Badriya's bizarre voting trend. But talking too much would come back to haunt me.\n\n\"He has some land. And he provides security for some foreigners, some people who are trying to build something in our province. I don't really know the details. I keep out of his business.\"\n\n\"I see,\" Sufia said in a way that made me feel like I had just given everything away.\n\nI needed to stop talking.\n\n\"Did Badriya talk to you about the candidates? The people she voted for?\" Hamida tried to sound casual.\n\n\"No,\" I said, reaching for the door. This conversation had to end. \"We don't really discuss the parliament issues. I'm just here to help her with paperwork and reading the documents.\"\n\n\"Can she not read?\"\n\nFrom the first day, I'd liked these women. I really had. But they were making me very uncomfortable right now, hitting every nerve. I was certain I was going to pay for this later.\n\n\"Let's go in, please. I can't wait to see what they have inside.\"\n\nThey relented. I followed them into the center, where an American woman was sitting at a computer, her fingers flying across the keyboard. She looked up and smiled brightly to see us, the first visitors she'd had all week.\n\nShe came over and hugged us, greeted us warmly. Sufia, her confidence growing, practiced her English and asked her how she was doing, how her family was getting along.\n\n\"Why isn't anyone else here?\" I whispered.\n\n\"Not interested. They just show up to the sessions and then go home. No one cares to learn anything new. They think they know what they're doing, even though they've never done it before. Born experts!\" Hamida laughed.\n\nThe ladies introduced me to Ms. Franklin and explained to her that I was an assistant to another parliamentarian. She seemed thrilled to have me there. I stared at her light brown hair, soft bangs peeking out from under her head scarf. She looked to be in her thirties, with a brightness in her eyes that made me think she'd never experienced sadness.\n\nIf that's true, lucky her, I thought.\n\n\"Salaam-alaikum, Rahima-jan,\" she said, her accent so thick it made me giggle. \"Chotoor asteen?\"\n\n\"I'm fine, thank you,\" I answered, and looked at Hamida. I'd never before seen an American. I was amazed to hear her speak our language. My reaction looked familiar to Hamida.\n\n\"Her Dari is good, isn't it?\" she laughed. \"Now, dear teacher, what can you show us today?\"\n\nWe spent almost two hours there, Ms. Franklin patiently guiding us through the basics of using a computer, guiding the mouse across a table to move a pointer on the screen. I was thrilled, feeling an excitement I hadn't felt since my days as a bacha posh.\n\nImagine if I learned to use this machine. Imagine if I could work like this woman, Ms. Franklin. To know so much that I could teach it to others!\n\nI felt privileged. A new feeling! I doubted even Hashmat had ever seen a computer, much less received personalized instruction on how to use it. I would have loved to see the look on his face if he ever learned what I was doing in Kabul.\n\nBut it was going to get dark soon and it was time to leave. The women had promised Badriya that one of them would escort me back to the hotel with her guard and driver. I hugged Ms. Franklin before we left, making her laugh out loud, her blue eyes twinkling with kindness.\n\n\"I want to come back here, please! I like it very much!\"\n\nIf only our day had ended with that sentiment.\n\nSufia had a hand on the door when a large explosion startled us all. We dropped to the ground, out of the way of windows. Nervous stares.\n\n\"What was that?\"\n\n\"Something. Couldn't have been too far. But it didn't sound like a rocket.\"\n\nWe were a people of war; explosions were familiar to our senses. But not for Ms. Franklin. Her face drained of color and she was shaking. Hamida put an arm around her young teacher, trying to reassure her. Sufia squeezed my hand. No other sounds came. Sufia got up cautiously and went to the door. People in the street were yelling, pointing. Her driver and guard jogged over to the door. They looked frustrated. They were panting.\n\n\"What is it? What's happened?\" she asked.\n\n\"Some kind of bomb. Looks like it was right by the parliament building. Stay here. We're going to find out what happened.\"\n\nWe were huddled by the window, trying to read the faces of the pedestrians. Hamida called out into the street.\n\n\"What's going on? Was that a bomb?\"\n\nThe street was chaotic. Either no one heard her or no one bothered to answer.\n\nWe inched out the door, our curiosity overwhelming. I was nervous. Although my father and my husband had been in the throes of battle, the war had always been at least one village away from me. I wondered if Badriya was anywhere near here.\n\nSufia's driver came back, shaking his head and muttering something under his breath. Hamida's guard stopped him, wanting to know what he'd found out.\n\n\"It was two blocks from the parliament building. A bomb in a car. Looks like they were trying to get Zamarud.\"\n\nMy stomach lurched. I pictured her storming out of the building, remembered the hateful leers she'd drawn from some of the men. Even some of the women had shaken their heads as she walked by. People thought she was out of line, and the punishment for being out of line was severe in our world. It always had been.\n\n\"Zamarud! Not surprised, with the finger-pointing she's been doing. This isn't good. Is she all right?\"\n\n\"I don't know. Someone said she was killed. They took her away. I didn't see her there, or her guards. We'd better get out of here.\"\nCHAPTER 45\n\nWhen Abdul Khaliq got wind of what happened, he told his driver and guard to bring us back to the compound. The bombing had scared me badly. Badriya and I stayed in our hotel room, afraid that other women parliamentarians would be targeted, and heard a hundred versions of yesterday's events from our guard and the hotel staff.\n\nShe was dead. She was alive but had lost a leg. She was unscathed but three children walking by had been killed. It was the Taliban. It was a warlord. It was the Americans.\n\nI didn't know what to believe. Badriya believed each and every story wholly until the next one came along. My head spun. I prayed for Zamarud, thinking there was something inspiring about the way she had riled the entire parliament with her irreverent behavior. The guards came back; our driver was smoking a cigarette, his eyes red from the lingering smoke of the bomb. He'd been too curious to walk away. When he nodded that the car was ready, I sighed with relief. I wanted to hold Jahangir.\n\nI imagined how my little boy might squeal and laugh when he saw me, how he might run to my arms. I couldn't wait to hold him, praying he didn't hate me for leaving. Though I regretted thinking it, I hoped I wasn't like my mother. I didn't want to abandon him and leave him to raise himself. I opened my small bag and checked for the ballpoint pen and a few sheets of paper I'd taken from the parliament building. I smiled, thinking how happy it would make Jahangir to scribble.\n\nMy son was the bright spot in my return to the compound. I went directly to Jameela's room and called his name. He froze at the sound of my voice and toddled to greet me at the door, an innocent grin on his face and a twinkle in his eye.\n\n\"Maa-da! Maa-da!\"\n\nMy heart melted to hear him call for me.\n\n\"Play ball outside! Maa-da!\"\n\nHe wasted no time trying to recruit me to play with him. I smiled, wishing I could join him in the courtyard, where we could kick his brother's soccer ball back and forth. I was close enough to childhood that playing still appealed to me. But they had just killed a chicken for Abdul Khaliq's dinner guests and there was little time to pluck and clean the bird for tonight's dinner.\n\n\"Forgive me, bachem, but maybe later I can come outside with you. Right now I have to get some work done. Maybe your brother can go outside with you.\"\n\nI think I secretly hoped that my work in Kabul would change how I was treated at the compound, but that idea was quickly dispelled. Bibi Gulalai stopped by the next day to make sure my time in the city hadn't undone all the work she'd put into me.\n\n\"That was Kabul, this is here. In this house, remember who I am. There are no meetings or papers for you to look after here. Now go wash your face. You look filthy. How embarrassing.\"\n\nI sighed, nodded and walked away before she talked herself into a worse mood on my account.\n\nI kept to my room. My last night here with my husband had been exceptionally unpleasant, exceptionally violent, and I didn't want to put myself in his way again. I wondered if he'd let us go back to Kabul after this. I still didn't know if Zamarud was alive or dead.\n\nAnd something was happening at home. I didn't know what it was but Jameela looked anxious and distracted. She was polite with Bibi Gulalai but she would quickly excuse herself. Bibi Gulalai seemed to be surveying our home with a discriminating eye. I asked Jameela but she smiled and changed the subject. Shahnaz, bitter because I had been allowed to travel to Kabul, was curt and snide with me. There was no use approaching her.\n\nThrough Jameela's younger son, I sent word to Khala Shaima that I'd returned from Kabul. I wished very much to see her. Where before I was a listener, we could now exchange stories. I wanted to tell her about Zamarud and the bombing. And about Hamida and Sufia and the resource center. But a week passed and Khala Shaima had not come. I asked Jameela if she'd heard anything about my aunt but she hadn't. A second week went by and still nothing.\n\nI was worried and frustrated but there was nothing I could do. Already I was feeling the differences between home and Kabul. That taste of independence, even the possibility of it, made me yearn to go back.\n\nThree weeks passed. Badriya and I were waiting on Abdul Khaliq's decision. He was most likely going to allow us to return to Kabul and complete the remaining three months of sessions. He hadn't said anything to Badriya and she was my source of information. Abdul Khaliq did not discuss these matters directly with me. In that respect, he treated me more like a daughter than a wife. It didn't matter to me. The less interaction I had with him, the better.\n\nBadriya finally approached Bibi Gulalai and asked her what was going on. Leaning against the living room wall, her shawl across her lap, Bibi Gulalai began to speak to her in a hush. When I paused by the doorway, the two of them looked up, annoyed.\n\n\"Go on, get to the carpets! And do it right this time. I don't want them looking dingy,\" Bibi Gulalai said.\n\nI moved away from the door but lingered in the hallway.\n\n\"When did this come up?\" Badriya said when I was out of view.\n\n\"Right when you all left. He knows her brother. I wish he never would have taken this stray. I don't know what he wanted with Rahima. Such a worthless family.\"\n\n\"I agree. Why he wanted a bacha posh for his wife, I'll never understand. But, Khala-jan, why do you think he would want to get rid of her? She is the youngest here and he wanted her for something . . .\"\n\n\"He will. I think he knows now that she was a mistake. And he wants to make up for it with this one. He's going to marry her.\"\n\n\"But why not just keep her and marry this girl?\"\n\n\"Because he's living by the hadith! He is a respected man in this village, in this province! He leads by example, so he is doing as the Prophet said. And the Prophet, peace be upon him, said that a man should take no more than four wives at a time. This wouldn't have been a problem if he wouldn't have taken that bacha posh.\"\n\nMy throat went dry. What was my husband planning? A fifth wife?\n\n\"Well, God bless him. It's admirable that he wants to be such an upright, devout Muslim.\"\n\nBibi Gulalai gave a quick hum, agreeing with Badriya's praise of her son.\n\n\"Just don't say anything to Rahima about this. She's wild enough as it is. We don't need her or her insane aunt Shaima making a fuss about this. It's none of their business anyway.\"\n\n\"I won't say a word but she'll find out soon enough . . .\"\n\nThe kids were coming down the hall. I slinked away from the door and melded into their footsteps.\n\nI needed to talk to Jameela. Would Abdul Khaliq really try to get rid of me? How?\n\n\"Why? What did you hear?\" Jameela said, her eyes narrowed.\n\nI recounted the conversation for her. She listened intently.\n\n\"I don't know anything more than that. Bibi Gulalai will only talk to Badriya, of course, her angel. The rest of us will only hear when something happens. But God help us all. If he really does this, it's going to be a disaster.\"\n\n\"But do you think he'll take a fifth wife? He wants to get rid of me, Jameela-jan. Can he do that?\"\n\n\"He can do\u2014\" Jameela started to say, but changed her response after a brief pause. \"I don't know, Rahima-jan. I really don't know.\"\n\nWe left it unsaid. If he wanted to take another wife without going over the limit, that would mean getting rid of one of us, and Bibi Gulalai had already made it clear that I was the expendable one. I'd once prayed my husband would send me back to my parents. Now that would mean leaving my son behind. Jameela had told me of one girl who had been sent back to her father's house, her husband dissatisfied with her as a wife. The girl's family, unable to tolerate the shame, refused to take her in. No one knew what happened to her.\n\nFour weeks since our return. Jahangir came into our bedroom, where I was mending a tear in my dress, the blue housedress Badriya had warned me against wearing in Kabul. And after seeing how most of the women parliamentarians dressed, I could see why. But it was in fair shape and there was little chance of new fabric coming my way.\n\nJahangir called out to me. I looked up, surprised to see Khala Shaima hobbling a few steps behind him. She had never come into this part of the house.\n\n\"Khala-jan! Salaam, Khala Shaima-jan, you came! I was so worried about you!\" I scrambled to greet her.\n\nKhala Shaima put a hand on the door frame, leaning forward and steadying her breathing.\n\n\"Salaam . . . ah . . . salaam, dokhtar-jan. Damn Abdul Khaliq for building his compound so far from town,\" she panted as I kissed her hands. I could hear the air whistle in her lungs. I quickly glanced in the hallway to be sure no one heard her curse my husband.\n\n\"I'm so sorry, Khala Shaima-jan. I wish I could come to you.\"\n\n\"Eh, forget it. I'll walk as long as my feet allow. Now, let me sit and get myself together. You must have something to tell me from your trip. And what the hell are you doing back here for so long?\"\n\nI told her about everything, the hotel, the guards, the buildings and the foreign soldiers. Then I told her about the bombing and the reason we came back.\n\n\"I heard about that on the radio. Bastards. Can't handle a woman with a voice.\"\n\n\"Who do you think was responsible?\"\n\n\"Does it matter? They may not know who brought the bomb there but we all know why. She's a woman. They don't want to hear from her. The last thing this country needed is one more cripple. And that's what we've got now.\"\n\n\"She's not dead? What happened to her?\"\n\n\"You don't know?\"\n\n\"We heard so many things before we left. And here no one cares to find out. I'm sure Abdul Khaliq knows, but . . .\"\n\n\"But you're not going to ask him.\"\n\nI shook my head.\n\n\"It seems the bomb went off just next to her car. Exploded and killed one of her guards. But she survived the attack. I think they said her leg was burned but nothing more.\"\n\n\"Is she going to come back to the parliament?\"\n\n\"She wants to.\"\n\nI didn't doubt that. Zamarud was not one to be scared off easily. I wished I could be more like her\u2014so determined and brave.\n\nAnd I should be, I thought. I'd been so self-assured when I was a bacha posh. Walking around with the boys, I feared nothing. If they had dared me to wrestle a grown man to the ground, I would have done it. I thought I could do anything.\n\nAnd now I trembled before my husband, before my mother-in-law. I had changed. I had lost my confidence. The dress I wore felt like a costume, something that disguised the confident, headstrong boy I was supposed to be. I felt ridiculous, like someone pretending to be something he was not. I despised what I was.\n\nKhala Shaima had read my mind.\n\n\"She's taking risks and she just might be a total lunatic, but she's doing what she wants. And I bet she doesn't regret it. I bet she'll keep doing it. That's what people have to do sometimes to get what they want. Or to be what they want.\"\n\nKhala Shaima was like no one else. Everyone else thought Zamarud was a fool to say the things she did and an even bigger idiot for willingly offending men.\n\nCarefully, quietly, I told Khala Shaima about Abdul Khaliq wanting to take another wife and what Badriya and Bibi Gulalai had said about me.\n\nShe said nothing but I could tell the news unsettled her. She looked anxious.\n\n\"Did they say how soon?\"\n\nI shook my head.\n\n\"Dear God, Rahima. This is not good.\"\n\nHer words made me more nervous.\n\n\"We have to figure something out. But keep this to yourself for now. Remember, the walls have mice and the mice have ears.\"\n\nI nodded, blinking back tears. I had hoped Khala Shaima would say something else. That the rumor was absurd. That I was safe here as Abdul Khaliq's wife.\n\n\"Things don't always work out the way you think they will. I bet you've been wondering what became of Bibi Shekiba. Shall I pick up where I left off?\"\n\nI half listened to my great-great-grandmother's story. My mind was preoccupied.\n\nI did have to figure something out. And I should be able to, shouldn't I? Why did it matter if I wore a dress now? Why did it matter that I no longer bound my breasts flat? I wanted to be the same person I had been. Zamarud let nothing get in her way. She wore a dress and she had married and she campaigned to get a seat in the jirga. A seat she occupied as a real parliamentarian.\n\nThe dress didn't hold her back as it did me. I felt restless. I thought how much more comfortable I would be if I could just button my shirt and walk into the street. If I could just slip into my old clothes . . . how much more capable I would be. Zamarud might have disagreed but the clothes meant something different to me because I'd lived in them.\n\nThe dress, the husband, the mother-in-law. I wished I could toss them all aside.\nCHAPTER 46\n\nWhen Shekib had been a girl, she'd heard about a woman in a nearby village condemned to stoning. It was the talk of their town as well as the neighboring towns.\n\nThe woman had been buried shoulder-deep in the earth and encircled by a crowd of onlookers. When it was time, her father had thrown the first stone, striking her squarely in her temple. The line continued until she slumped over in atonement.\n\nShekiba had listened to the story being recounted by her uncle's wife. Her mouth gaped at the horror of such a punishment and the grains of rice she was sifting fell through her drifting fingers and missed the bowl. An anthill of rice collected on the floor.\n\n\"What had she done?\"\n\nHer uncles' wives turned around and paused their conversation, surprised. They often forgot she was there.\n\nBobo Shahgul's eyes narrowed seeing the wasted rice on the floor.\n\n\"She ruined her father's life and gave her entire family nothing but grief!\" she said brusquely. \"Watch what you're doing, you absentminded fool!\"\n\nShekiba looked down to see the mess she had made. Her mouth closed sharply and she turned her attention back to the rice. Bobo Shahgul tapped her walking stick in warning.\n\nSangsaar? A chill coursed through Shekib's veins as she looked at Benafsha and pictured her half-buried. Stones hurled at her head.\n\nShe asked no more questions of Benafsha. The room was silent but for the grumbling of two empty stomachs.\n\nTwo days passed without food or water. The door did not open once, though Shekib could see people walking behind it, stopping and listening before walking off. From the slit beneath the door, Shekib could make out the soles of army boots and knew soldiers were guarding them.\n\nOn the third day, the door opened. An army officer looked down at the two women, curled up on the floor. Shekib pulled herself to stand. Benafsha barely stirred.\n\n\"Guard. Khanum Benafsha.\"\n\nShekib dusted her pants off and straightened her back.\n\n\"Your offenses against our dear king are grave and reprehensible. You are both to be stoned tomorrow afternoon.\"\n\nShekib gasped. Her eyes widened in disbelief. \"But, sir, I\u2014\"\n\n\"I did not ask you to speak. You have shamed yourself enough, have you not?\"\n\nHe turned around abruptly and slammed the door shut behind him. Shekib heard him order a soldier to lock the door. A chain clanged and a key turned, leaving the two women with their fate.\n\nBenafsha let out a soft moan once the door closed. She had known.\n\n\"They're going to stone us both!\" Shekib whispered, her voice tight and unbelieving. \"Even me? I did nothing!\"\n\nBenafsha had her elbow tucked under her head. Her eyes gazed at the wall in front of her. She had known exactly what they would do to her. Why had she brought this upon herself?\n\n\"This is your fault! They're going to stone me because of you!\" She knelt at Benafsha's side and grabbed her shoulders roughly. \"Because of you!\"\n\nBenafsha rattled limply in her hands. \"With Allah as my witness, I am sorry that you are here,\" she said softly, her voice tearful and resigned.\n\nShekib pulled back and stared at Benafsha. \"Why? You knew what they would do to you. Why did you do this? How could you do such a thing in the king's own palace?\"\n\n\"You would not understand,\" she said for the second time.\n\n\"No, I do not understand how you could do something so stupid!\"\n\n\"It is impossible to understand if you do not know love,\" Benafsha whispered. Her eyes closed and she started to recite lyrics I'd never heard before. Phrases that I memorized because they echoed in my mind after she'd stopped talking and meant different things to me at different moments.\n\nThere is some kiss we want with our whole lives,\n\nThe touch of Spirit on the body.\n\nSeawater begs the pearl to break its shell.\n\nAnd the lily, how passionately it needs some wild\n\nDarling!\n\nAt night, I open the window and ask the moon to come\n\nAnd press its face against mine. Breathe into me.\n\nHer melancholy verses pulled at my heart. I knew nothing of that kind of love. I knew nothing about pearls and shells either except that one had to free itself from the other. We were both calmer than we should have been, Benafsha because she had lived her love, and me because I had never known it.\n\nThe hours crept by.\n\nDay turned into night and night became morning. One final morning.\n\nMaybe this is how it is meant to be. Maybe this is how I will finally be returned to my family and saved from this wretched existence. Maybe there is nothing for me in this world.\n\nShekib swung wildly between anger, panic and submission in those hours. Benafsha whispered words of apology from time to time but mostly prayed. She held her head between her hands and atoned for her sins, said there was no God but Allah.\n\nAllahu akbar, she whispered rhythmically. Allahu akbar.\n\nThere was talking outside their door. Shekib could not make out what they were saying but heard a few words here and there.\n\nWhores. Stoning. Deserved.\n\nWhores? Shekib realized she was a woman again. As guilty as the woman lying a few feet from her.\n\nI have been both girl and boy. I will be executed as a girl. A girl who failed as a boy.\n\nStoning. Today. Stopped.\n\nStopped? What was stopped?\n\nShekiba listened carefully.\n\nKing. Pardon. Gift.\n\nAt hearing \"gift,\" Shekiba realized something was happening to her. She strained to hear the voices more clearly but could not make out most of what they were saying.\n\nThe door opened. The same ranked soldier reappeared, his face cross.\n\n\"Khanum Benafsha, prepare yourself. You,\" he said, looking at Shekiba with disgust. \"You will attend the stoning and then you will be punished for your crime. After that, you will be given in marriage. You should thank Allah that you have been shown a mercy you do not deserve.\"\n\nThe room went dim again and the chains were locked in place. Shekiba's heart pounded.\n\nThey will not stone me! I will be given in marriage? How could this be?\n\nBenafsha looked at her, the corners of her mouth almost turned up in a weak smile.\n\n\"Allahu akbar,\" she whispered; the condemned's prayer had been answered.\n\nShekiba's hands trembled. Was it Amanullah? He must have intervened! But why would he want her now that she had been accused of such treachery? Now that she had made herself an unworthy wife?\n\nEveryone spoke of Amanullah's noble character. Maybe he had seen through the accusations. Maybe in their brief exchanges he had seen something, something that told him she was more than just a woman-man, more than just a harem guard. Was that not what he had told his friend Agha Baraan?\n\nTears ran down Shekiba's cheeks. Now all she could do was wait. The hours passed slowly. It became painful to sit in the same room with Benafsha. Shekiba looked at her glazed eyes and broken spirit. She crawled over and crouched at her side.\n\n\"Khanum Benafsha,\" she said, her words a hush. \"I am praying for you.\"\n\nBenafsha's eyes focused on Shekiba. She looked hollow but grateful.\n\n\"I cannot understand why you . . . but I want . . .\"\n\n\"I fulfilled my destiny,\" Benafsha said calmly. \"That is all I did.\"\n\nWhen they came for Benafsha, Shekiba was holding her hands. Two soldiers dragged Benafsha to her feet and another two pulled Shekiba up by the shoulders. Shekiba's fingers lost their grip when they bound Benafsha's wrists together and covered her with a blue burqa. Benafsha looked at her and began to wail, long slow moans that grew louder as they walked through the hallways.\n\n\"Shut your mouth, whore!\" a soldier snapped, whipping his hand against the back of Benafsha's head after he had made sure they were not being watched. Though she was about to be executed, she was still the king's concubine.\n\nBenafsha's head bounced forward. She began to pray loudly.\n\n\"Allahu akbar. Allahu akbar. Allahu . . .\"\n\nThey shook her gruffly by the shoulders and warned her again. Her prayers went on.\n\nThrough the palace, out a back door and into the courtyard, where the afternoon sun nearly blinded the women. Shekiba looked at the harem and saw the women lined up outside, head scarves pulled across their faces. Halima in silhouette, her shoulders shaking as she sobbed. Sakina stood among them, her arm linked with Nabila's.\n\nYou did this, Shekiba thought bitterly.\n\nGhafoor, Karim, Qasim and Tariq stood in front of the women, solemnly watching the dead woman walk by. Even from this distance, Shekiba could see Tariq trembling. Ghafoor kept her eyes averted, whispering something to Karim as she looked back at the concubines.\n\nCoward. You can't even look at me.\n\n\"Allahu akbar. Allahu akbar . . .\"\n\nSoldiers stood everywhere. The palace grounds were quiet, an eerie silence given the number of people in sight. Benafsha's prayers echoed through the gardens, her toes dragging through the ground. The women of the harem shrank into the distance. Shekiba could hear someone crying. Others tried to hush her but the sobs continued. Shekiba thought it sounded like Nabila.\n\n\"Do not weep for those who damned themselves!\" a voice boomed.\n\nShekiba turned around to see where the voice came from. Ahead of them stood a general. From this distance she could not tell if it was one of the men who had come to their makeshift prison cell. Three soldiers stood on either side of him, their backs straight as rods.\n\nA hundred times Shekiba had crossed the palace grounds but never had it seemed this far. They inched along.\n\n\"Allahu akbar. Allahu akbar. Allahu . . .\"\n\nShekiba began to mouth the words too. Her voice was barely audible, her throat so dry it burned to talk.\n\nAs they neared the general, he nodded to the soldiers and they walked past the fountains, toward the far limits of the palace. They marched solemnly to a clearing where a semicircle of soldiers stood at attention. Shekiba's heart dropped. In front of the soldiers lay two separate piles of stones, most the size of a fist. The heaps reached the soldiers' knees.\n\nShekiba's prayers grew louder, synchronizing with Benafsha's. She tasted tears. They walked to the edge of the palace; high walls shielded the onlookers. King Habibullah emerged from the palace and stood beside the general he had placed in charge of the execution. The men whispered to each other, keeping their eyes on Benafsha.\n\nThe general nodded at something the king said and approached the condemned as she was brought to the center of the semicircle. A deep pit had been dug in these outskirts of the palace, behind a row of fruit trees, a place Shekiba had never before ventured. The soldiers, about fifteen feet away, stared at Benafsha. Shekiba was still within earshot.\n\n\"Tell me, Khanum Benafsha, are you ready to divulge the name of the man you welcomed in your chambers?\"\n\nBenafsha looked up and met his stare.\n\n\"Allahu akbar.\"\n\n\"You could be granted mercy if you would at least tell us who this man is.\"\n\n\"Allahu akbar.\"\n\nThe general threw his arms up and looked back at the king, exasperated. The king nodded, his face a contorted mix of wrath and disappointment.\n\n\"Very well! Khanum Benafsha, your crimes have been reviewed by the scholars of our beloved Islam and according to the laws of our land, you are to be stoned for the grave offense you have committed.\" He looked at the two guards and pointed at the hole. Benafsha let out a wail as they held her by the armpits and lowered her into the pit, her legs kicking, her blue burqa flailing like a goldfish pulled from the palace fountain.\n\nShekiba took a step toward her and felt two hands tighten on her arms. She looked over at King Habibullah. His arms were folded, a finger over his lips as he mouthed something. At the sound of Benafsha's voice, he shook his head, lowered his gaze and walked away. He would not stay for the execution.\n\nThe soldiers shoveled the earth back around Benafsha until she was buried to her chest. She continued to twist and turn but she was deep in the ground and her arms stuck to her, useless. As the dirt piled up around her, she moved less but moaned louder. Shekiba closed her eyes and heard the wails: \"Allahu akbar. Allahu akbar. Allahu . . .\"\n\nSuddenly a sharp yelp. Shekiba opened her eyes, startled. A thin line of darkness formed above the eye mesh of Benafsha's burqa. Three stones lay near her.\n\nIt has started.\n\nThe soldiers bent over, picked stones from the arsenal before them and mouthed something before hurling them at Benafsha, the blue half person.\n\nMay Allah have mercy on you, Khanum Benafsha!\n\nHer body jerked with each stone that hit her. The soldiers took turns. Picking, hurling, and moving to the back of the half circle. Ten minutes passed, a hundred stones. Benafsha's voice grew weaker; she slumped forward, her burqa stained in a dozen places, dark circles bleeding toward one another. The earth around her grew dark as well, blood soaking into the soil. Two stones had ripped through the blue fabric, gashed flesh showed through the holes.\n\nShekiba turned around, unable to stomach any more. She saw the row of blue burqas behind a row of spectator soldiers. Benafsha was to be an example to the dozen or so who had been brought out to bear witness. As horrified as Shekiba, the blue cloaks were half turned away.\n\nStone after stone, scream after scream, until Benafsha went silent and still. The general raised his hand. The execution had been carried out.\nCHAPTER 47\n\nBenafsha's limp body flashed over and over in Shekiba's mind as she received her own punishment. She had been sentenced to a hundred lashes, which were delivered precisely by one of the soldiers, a general standing watch over him. Shekiba had been made to kneel while they stood behind her, her wrists bound as Benafsha's had been.\n\nThough her face twisted in pain with each blow, she did not make a sound.\n\nHer back stung, hot and wet. The soldier had a book tucked under his arm, as law instructed, to soften the striking force. They counted out loud and when they reached a hundred, Shekiba's wrists were untied and she fell on her side in exhaustion. The men said nothing and left the room.\n\nHer mind drifted. She felt water on her lips. Hands rubbed ointment on her back. It was nearly a day later before Shekiba realized Dr. Behrowen was tending to her wounds. The British woman clucked her tongue and shook her head, almost as an Afghan would, muttering something that Shekiba did not understand.\n\nShekiba closed her eyes to block the horror but it was still there, the images seared onto the insides of her eyelids. She opened her eyes again and looked at Dr. Behrowen. She was squeezing water out of a wet rag. She considered Shekiba carefully.\n\n\"Dard?\" she asked, her British accent blunting the letters so thickly that the word was unrecognizable. She had to repeat herself twice more before Shekiba understood she was asking about pain.\n\nShekiba shook her head. Dr. Behrowen raised her eyebrows and turned her attention back to the bucket of rags.\n\nShekiba looked down. She was wearing thin pantaloons that tapered at her ankles. A head scarf lay strewn across a chair in the corner of the room. Shekiba realized she was in Benafsha's room in the harem. Through the walls, she could hear women chatting. She remembered how Benafsha had begged and prostrated herself before them, asking forgiveness and mercy from a crowd focused only on saving their own skins.\n\nThe door opened and Halima peered in.\n\n\"Can I come in?\" she asked quietly, looking at Dr. Behrowen.\n\nDr. Behrowen must have understood; she nodded and waved Halima into the room.\n\n\"How are you feeling?\"\n\n\"Better.\" Her throat felt like sandpaper.\n\n\"I'm glad.\" She knelt at Shekiba's side. \"Things have been ugly here the last few days. Never have we experienced such things.\"\n\nShekiba had nothing to say in reply. Halima sighed heavily and looked quickly at Dr. Behrowen with tears in her eyes.\n\n\"Tariq is outside. She wants to see you but she's very nervous. Is it all right if she comes in for a few moments?\"\n\nShekiba nodded. She remembered seeing Tariq when she turned her gaze from Benafsha's stoning. Tariq's mouth and eyes were open wide with horror, a small pool of vomit at her feet.\n\nHalima placed a gentle hand on Shekiba's forehead before she stood and quietly walked out. Shekiba wished she would come back, stroke her hair and hold her hands as a mother would. Instead, Tariq rushed in and fell at Shekiba's side; the trembling in her hands vibrated her voice.\n\n\"Oh, Allah have mercy! Are you all right? Are you badly hurt? What did they do to you?\"\n\n\"I was punished.\"\n\n\"How?\"\n\n\"One hundred lashes.\"\n\nTariq scanned her body, her brows furrowed together in angst. \"How awful! How very awful! Oh, Shekib! Did they say why they were punishing you?\"\n\n\"Because I did not do my job as a guard.\"\n\n\"Oh, Allah forgive us! We were all as guilty as you!\" she whispered, as if afraid the palace would hear her.\n\n\"But only I had been on duty that night. Ghafoor made sure to tell them that.\"\n\n\"She . . . I never would have imagined she could be so . . . I mean, I know she thinks only of herself but I just never thought she would do something like . . .\"\n\n\"That's what people do. She's no different than anyone else.\"\n\nIt suddenly occurred to Shekiba that Benafsha had been different. The general had offered leniency in exchange for a name. Although she must have known his offer was a lie, even the possibility of mercy didn't faze her resolve. She never named the man. Why had she done that? Why had she protected Agha Baraan?\n\n\"She said that they only wanted to talk to you. She said she did not know they were going to punish you.\"\n\nShekiba recalled Ghafoor's shifting gaze on that night and on the day of the stoning.\n\n\"What did they tell you? Benafsha . . . she brought such shame to the palace but I never thought . . . I just cannot believe this happened here! I thought things were different here in Kabul, in the king's palace!\"\n\n\"No man could tolerate such an offense. The king would have shamed himself if he had agreed to a lesser penalty.\"\n\n\"And what will become of us guards?\"\n\n\"I do not know.\"\n\n\"What about you? Will they send you back to your family?\"\n\nShekiba remembered that she had been spared for a reason\u2014she was to be married! She pictured Amanullah's face. Could it really be? Had he rescued her from execution to live on as his wife? Or maybe as a concubine? Even curled up on the floor with her raw back covered in salves, Shekiba yearned to be in a new home, her own home, and with child. She wanted to feel tiny palms pull at her face with unquestioning affection.\n\n\"No. I do not know where they will send me.\" She decided to say nothing about the marriage until she knew more. She did not want word getting back to Ghafoor, lest she find a way to ruin things for her.\n\n\"Oh, what an awful mess! I'm so sorry, Shekib. I'm so sorry that you took the fall for this. The whole harem is terrified. They're worried more will be punished, just to set an example, or maybe if they believe others were involved . . .\"\n\nShekiba decided Tariq was exhausting her. She asked her to leave so that she could close her eyes. Tariq looked disappointed but nodded and walked out, her uniform looking bulky and awkward. She was less of a man now than she had ever been.\n\nShekiba was just drifting off when Tariq burst back into the room.\n\n\"Shekib!\" she said excitedly.\n\n\"Please, Tariq, I just want to sleep for a few\u2014\"\n\n\"I know and I apologize, but the palace sent over a messenger. They asked me to tell you . . . to tell you to be ready in two days.\"\n\nShekiba looked up.\n\nTariq's face flushed with a nervous smile.\n\n\"They said you're going to be married!\"\nCHAPTER 48\n\nIt was the holiday of Eid. Five weeks since the attack on Zamarud. Badriya had received several letters from the director general. If she didn't return to her duties immediately, she would be stripped of her position. Abdul Khaliq had made up his mind. We would be going back after the holiday.\n\nJameela told me what it was all about.\n\n\"He's got a deal with a foreign company. You know these westerners he's always going out to meet. He wants them to pay him to provide security. But it's up for vote in the parliament whether or not that company should be allowed to build a pipe through our province. If they aren't allowed to build, they won't need his security.\"\n\n\"That's why he put Badriya in the parliament? To get her to vote for the pipe?\"\n\n\"Yes. And to vote for all the right people for other positions, the people who will give him what he wants.\"\n\nBadriya's voting made perfect sense now. Abdul Khaliq must have told her to watch for his friend's signals. She wanted us to believe she actually mattered but she was a stooge. She was nothing like Hamida and Sufia. No wonder she seemed so awkward around them.\n\nI was happy we would be returning, although this time I knew it would be even harder to leave Jahangir. This time I knew just how much I would miss him. But I dared not ask again if I could take him with me.\n\nWe, the four wives, went to Bibi Gulalai's compound next door to pay our obligatory respects on the first day of Eid. After that, we went home and braced ourselves. For three days, our house received visitor after visitor. I spent three days in the kitchen with the cook and the maid, drying dishes, filling bowls with nuts and raisins and pouring cups of tea. I wasn't invited to sit with any of the guests as Badriya and Jameela were. Even Shahnaz came out from time to time and chatted with the women who came by.\n\nAnd if my husband were to marry again, I had no reason to expect things would get better for me. I knew my family would not take me back. It was a matter of pride. My uncles would never tolerate having a rejected wife, a dishonored woman, back in their fold.\n\nIt was possible that he would keep all his wives. But there was no room in the house for that. We were all uneasy with the possibilities. Bibi Gulalai and Abdul Khaliq were remarkably tight-lipped about the idea.\n\n\"Rahima! Rahima-jan, come out here! Look who has come to see you!\"\n\nI dried my hands on my skirt and hurried into the living room, hoping to see Khala Shaima. My jaw dropped to see that my eldest sister, Shahla, stood before me, a little boy holding her hand and another baby, no more than four months old, in her arm.\n\nShahla smiled brightly to see me while I simply stared. Her face and hips had rounded, taking her out of adolescence. She looked cheerful.\n\n\"Rahima! My dear sister!\" She let go of her son's hand and took a step toward me. I couldn't believe I was seeing her after all this time. It felt so good to have her arms squeeze me, to have her hands touch my face.\n\nI felt her tears against my cheek, mixing with mine.\n\n\"It's so good to see you, finally!\" she whispered. \"Forgive me, Rahima. I couldn't come to be with you when . . . when everything happened.\"\n\nI had missed Shahla so much, and never more than when Parwin had taken her own life. Seeing her reopened the hurt.\n\n\"I wanted to be here. I wanted to come but it was so close to the time when this little one . . . ,\" she said, pointing to the baby girl in her arms.\n\nI touched the little girl's face, her skin soft and smooth like Shahla's.\n\n\"I know, Shahla. I wish you could have been here too. It was . . . it was so terrible!\"\n\n\"Allah forgive her, I'm sure it was. Poor Parwin! I can't imagine what she must have gone through!\"\n\nBibi Gulalai stood in the corner of the room, eyeing us suspiciously and looking displeased. I looked and saw there were other guests I hadn't greeted. I exchanged quick kisses with Shahla's mother-in-law and two of her sisters-in-law. Shahla had taken a seat on one of the cushions, her little boy beside her and her daughter in her lap. I sat next to her, Bibi Gulalai's eyes following me.\n\n\"Oh, Shahla, look at your children! They're beautiful! I have a\u2014\"\n\n\"Rahima!\" Bibi Gulalai barked. \"Don't you think it would be more polite to bring our guests some tea before you start boring them with your yammering?\"\n\nMy face flushed with embarrassment and anger. At least five Eid holidays had passed and this was the first time my sister had been able to pay a visit to the compound in that nearly three years' time. I hadn't seen her since the night of our miserable weddings. I could see the surprise on Shahla's face to hear the way Bibi Gulalai spoke. Jameela intervened.\n\n\"Let me, Khala-jan. Rahima's dear sister is here and it would be nice if the girls could spend some time together.\"\n\nI loved Jameela for understanding. And stepping in. She brought cups of tea and passed around a dish of nuts and dried mulberries. The ladies chatted amicably while Shahla held my hands. Her son, Shoib, grinned shyly while her daughter's tiny arms waved this way and that, her eyes glued to her mother's face.\n\n\"Shoib, did you say salaam to your khala-jan?\"\n\n\"Salaam,\" he said quickly, then hid behind his mother's shoulder.\n\n\"He's very timid,\" Shahla said, smiling.\n\n\"I want you to meet my son, Shahla.\" I rushed into the hallway and called out for Jahangir. The ladies in the room thought we looked ridiculous. Everyone had children. They didn't understand why we were making such a fuss over them.\n\nI heard footsteps from Jameela's room. My son had become quite comfortable staying with her while I was in Kabul. Whenever he wasn't with me, I knew exactly where I would find him.\n\n\"Come, bachem, come and meet your khala-jan,\" I beckoned. My son, looking curious, took my hand and followed me into the living room.\n\n\"He's a darling, nam-e-khoda!\" she said, praising God's name and blowing three times to ward off the evil eye. \"He's got your face.\"\n\n\"Do you really think so?\" It pleased me to hear that.\n\n\"Absolutely! And Madar-jan's hair. Look at the way the curls twist behind his head.\"\n\nWe both winced at the mention of our mother.\n\n\"Have you seen her?\" I tried to be discreet.\n\nShahla shook her head. I looked at my feet, dark with dust. This was a sore spot for both of us and I didn't want to tell my sister everything I'd heard about Madar-jan's downward spiral. With all these women present, it felt like a betrayal. But I wanted to pour my heart out to her, to tell her about our younger sisters, left to fend for themselves even with two parents at home. I wanted her to tell me she would talk some sense into Madar-jan, even if Khala Shaima couldn't. I kept it bottled.\n\n\"And your little girl\u2014she's so sweet! What's her name, Shahla?\"\n\nI put my hand in front of her hand. Her long, graceful fingers wrapped around mine and squeezed tight.\n\nShahla lowered her voice and looked to see if anyone was paying attention to us.\n\n\"I named her Parwin,\" she said quietly.\n\nA second look and I realized Shahla's daughter had our sister's doelike eyes and pink, puckered lips. My throat tightened. Shahla smiled wanly.\n\n\"Parwin?\"\n\n\"Yes. My mother-in-law wanted to name her Rima, actually, but I asked if I could choose the name. She agreed to let me.\"\n\nI stared at my niece's face. The longer I stared, the more of Parwin I saw. Then I thought of my own mother-in-law. She had only gone along with my son's name because my husband had approved of it. He must have liked it very much or she would have changed it for sure.\n\n\"I can't believe she agreed.\"\n\n\"I know. It was difficult because she thought it would be bad luck, you know, to name a child after someone with a lame leg. Thank goodness I named her before . . . I mean if she had been born after, I couldn't have convinced anyone. The name would've carried too much shadow.\"\n\nShahla looked at her daughter's face wistfully.\n\n\"Then after all that happened, everyone started calling her Rima. I could barely bring myself to say her name either so for a long time it was a relief to call her Rima. But now, when it's just me and her, I call her Parwin. It makes me feel better. Funny, isn't it? We hear the same name and while they see dark, I see light.\"\n\nI knew just what she meant.\n\nHad the guests been anyone else, I would have returned to the kitchen long ago. But it was my sister and I wanted to spend every second I could with her. Jameela refilled the teacups, passed around a plate of cookies and made small talk. She kept an eye on Bibi Gulalai and when it looked like our mother-in-law was about to say something to me, Jameela would ask a question or say something to distract her. When our eyes met, I thanked Jameela silently. She smiled.\n\n\"Shahla, you look so well!\" I exclaimed. And I meant it. My sister looked more mature but otherwise unchanged. And she looked content. I even saw her make eye contact with her sister-in-law once or twice and smile. Genuinely. Her mother-in-law was a soft-spoken woman, nothing like Bibi Gulalai and her searing glare. She must have been in her sixties, wisps of gray hair peeking out from under her head scarf. She listened to Jameela talk about our mother's illness with a look of sincere concern.\n\n\"Really, Shahla, are things okay?\" I whispered when the room was divided in conversation. \"Are you happy?\"\n\n\"I miss you so much, Rahima! I miss everyone. I wish so much that I could see Rohila and Sitara. I want to know how big they've gotten, what they're doing. But I'm happy.\"\n\nI smiled. I believed her.\n\n\"What about you?\" Jahangir pulled at Shoib's sleeve, inviting him to play in the hallway. Shoib shrugged his shoulders and followed.\n\n\"Me?\" I could feel Bibi Gulalai's stare boring into the back of my head. I nodded. My sister knew me too well. Her face grew somber.\n\n\"Good, I'm glad to hear that,\" she said in a way that told me the opposite was true.\n\n\"I go to Kabul now. Did you hear about that?\"\n\n\"I heard something, but . . .\"\n\nI told her about Badriya's seat in the jirga and how I worked as her assistant. I told her how different Kabul was, just like in the stories Khala Shaima told us. I was proud when I saw how impressed Shahla was.\n\nIf only I could have suspended time. I would have sat beside my sister, our children playing together in a picture of innocence, our hearts supporting each other as we mourned our dead sister, the mother we once had, and the sisters we'd left behind.\n\n\"You see Khala Shaima still, don't you?\"\n\nI nodded. \"She comes by when she can. It's getting harder for her but I miss her so much when she doesn't come.\"\n\n\"Does she still tell you stories of Bibi Shekiba?\" Shahla asked. She started to rock when she noticed her daughter's eyes begin to close, just as I did with Jahangir. It was amazing how quickly girls took on the instincts of motherhood.\n\n\"She does. I love hearing those stories. It makes me think of . . . it makes me think of other times.\"\n\nShahla sighed. She missed it as much as I did.\n\n\"I know, Rahima-jan. But times change. Everything changes. Birds fly away, one by one.\"\nCHAPTER 49\n\nShekiba was told nothing more. A nikkah was to be performed in two days' time. Word spread through the somber harem quickly and several women came together to prepare the new bride.\n\n\"Who is this man? How fortunate you are! You were spared by our dear king for marriage! That is quite an honor!\"\n\nThose voices were in the minority. Shekiba heard the whispers around her, angry and incredulous. Some said that she had probably conspired with Benafsha and that she should have been stoned alongside her.\n\nYou see how comfortable she is in Benafsha's room? As if it's been hers all along!\n\nI bet she helped hide Benafsha's lover. I'm sure of it. I heard her footsteps in the middle of the night from time to time and I knew\u2014I just knew something had to be going on!\n\nThey must be giving her to a blind man. Who else could stand to look at such a face!\n\nHow quickly they turned on her! How quickly they forgot how she had carried their children, brought hot water for their baths and even scrubbed their backs when they asked. All the while she had been Shekiba-e-haleem to them; they winked to one another when she served the bowls of the hot dish to the women from the breakfast delivery.\n\nShekiba-e-haleem, serving up her special dish!\n\nMaybe she should pour a bowl over the other side of her face\u2014I swear it's just the right color to match her complexion today! The cook must be a genius!\n\nBut there were a few, namely Halima and Benazir, who pitied Shekiba and knew that she needed help preparing for her nikkah.\n\n\"Who is this man?\" Halima asked as she combed oil through Shekiba's dull, short hair.\n\n\"I don't know, Khanum Halima. No one has said anything to me.\"\n\n\"Maybe it's one of the servants from the main palace. Maybe they will have you work there now?\" Benazir suggested. \"Would you like that?\"\n\n\"I suppose,\" Shekiba said, her voice contained. That was not what she wanted at all, but she could not bring herself to share her secret with anyone. It was Amanullah that she hoped for\u2014not a palace servant!\n\n\"Well, it is a bit strange that they have not told you anything.\" Halima looked hopeful but reserved. Shekiba had misfortune written all over her and it was hard to imagine that even a marriage would bring her peace.\n\n\"You know, there are many things that come with marriage. You have seen this harem and you know what happens between a man and a woman. Your husband will expect you to fulfill wifely duties. You will not want to disappoint him,\" Halima said gently.\n\nShekiba felt her stomach drop. She had not given much thought to what would happen between her and a husband. She thought of the squeals and grunts that came from the king's chambers. She thought of what Mahbuba had told her and felt something between her legs tighten with anxiety.\n\n\"It is painful the first time,\" Halima said.\n\n\"So painful!\" Benazir echoed.\n\n\"But each time after that will be easier. And maybe Allah will bless you with a child.\"\n\nBenazir smiled and looked at Mezhgan, who lay sleeping a few feet away.\n\n\"You have said that the women in your family all bear sons. If you do so, you will make your husband a happy man. Especially if it is his firstborn.\"\n\n\"Do you really think she'll be a first wife?\" Benazir asked.\n\n\"Anything is possible,\" Halima said, looking at Shekiba and thinking of the last few days in the palace.\n\nLater that afternoon, a second wave of news rippled through the harem. Nabila came running into the bath area. Shekiba could hear her through the door.\n\n\"Have you heard the news? He is to be engaged! Our dear prince Amanullah is to be engaged! He has finally chosen a bride!\"\n\nNo one else connected the two stories. No one but Shekiba, who closed her eyes and prayed with a nervous heart.\n\nAs promised, a soldier came to the harem two days after Tariq brought word to Shekiba. Ghafoor was standing outside and called into the house for Shekiba. They had not spoken since that dark night.\n\n\"Shekiba!\" she called unceremoniously. \"The palace has sent for you.\"\n\nShekiba had spent her last night in Benafsha's chambers, wondering about tomorrow. Her back still sore, she slept on her side. She stared at the door and imagined Agha Baraan entering to take the king's concubine in secret. Why hadn't Benafsha given her lover's name?\n\nShekiba stood up slowly and smoothed her skirt, trying not to wake Tariq who had quietly joined her last night. She pictured Amanullah in his military uniform, his pants neatly pressed and his hat perched perfectly on his head. Looking at her own clothing, she felt embarrassed. She picked up her head scarf and crossed the corners under her chin. Tariq woke up, stretched and jumped to her feet. She threw both arms around Shekiba's neck and squeezed her tightly. The gesture caught Shekiba by surprise.\n\n\"Is it time already? I wish you all the best, dear sister! May Allah bless the steps you are about to take and give you a lifetime of happiness.\" Tariq's eyes were tearful. \"And don't forget to pray for me sometimes too. Pray that I'll be so lucky!\"\n\n\"I'll pray that you'll be even luckier.\"\n\nWith the palace waiting, there was no time to find Halima or Benazir to say good-bye. Shekiba walked past Ghafoor to the front door.\n\n\"How are you, Shekiba-jan? I hope you're feeling better. I heard your punishment was severe.\" She looked uncomfortable; her eyes wandered past Shekiba to the soldier waiting outside.\n\n\"I was delivered to them, blame already assigned. What else were they to do?\"\n\n\"They must have assumed\u2014\"\n\n\"They assumed what they were told.\" Shekiba spoke coolly.\n\n\"I did not . . . regardless, congratulations.\"\n\n\"And to you too.\"\n\n\"To me too?\"\n\n\"Certainly. It's not every day that one can successfully escape a fire.\"\n\n\"Just wait a minute! I did not\u2014\"\n\nSomething in Shekiba made her turn around and look Ghafoor in the eyes. She was tired of holding her tongue.\n\n\"There's something you do not know about me, Ghafoor.\" Shekiba turned to glare at her directly. Her eyes narrowed with hate. \"Do you wonder why my family sent me away? My family sent me away because I carry a curse and those around me ended up in a grave years before their time. And now, under these clear skies and with Satan listening, I curse you. May you suffer a hundred times over for each lash I bore. Mark my words, you snake, you will get what you deserve,\" Shekiba said quietly.\n\nGhafoor's shoulders stiffened with anger but her face went pale. Satisfied, Shekiba turned and walked toward the soldier.\n\nShekiba was led into a small room in the east wing of the palace. The two men who had questioned her only a few days ago sat waiting. The short man looked at the lanky man, waiting for him to begin speaking.\n\nWill Amanullah come here? Is it possible I will meet him today? Is it possible that there will really be a nikkah between us?\n\n\"Salaam,\" she said quietly with her head bowed. She fidgeted with her clothing, her head scarf, wanting every piece to look perfectly in place. They motioned for her to sit in the chair across from them. One man spoke while the other nodded in agreement and parroted his words.\n\n\"You are a fortunate girl.\"\n\n\"Very fortunate.\"\n\nShekiba did not look up.\n\n\"You have been shown mercy that you did not deserve. You should be very grateful.\"\n\n\"Very grateful.\"\n\n\"Someone has agreed to take you on as wife, a title one would have hardly expected for you. But this is a chance for you to redeem yourself. To attempt to live a respectable life and fulfill your duties as dictated by our holy Qur'an. Do you think you can do this?\"\n\n\"I was raised with love for our holy book, sir. I want nothing more than to live an honorable life.\"\n\nHe raised an eyebrow. Maybe he had anticipated a more insolent response.\n\n\"Very well then. As you can imagine, our dear king Habibullah has no desire to lay eyes on you again after the tragedy that befell this palace. But he has given his blessing that you be given in marriage.\"\n\nShekiba's heart pounded. Still they had not mentioned the man's name. She waited on each word he uttered, anxious to hear that name, that sweet name\u2014Amanullah!\n\n\"Your future spouse is in the room next door with the mullah. He is signing the marriage certificate.\" The door opened and a third man appeared. He gave the other two a nod and they turned back to Shekiba.\n\n\"He has agreed, stating his intentions clearly thrice over. Now it is your turn. We will speak on your behalf. Do you agree to take Agha Baraan as your husband in life?\"\n\nShekiba began nodding before she heard the name. She kept nodding even when she heard the name and even for a few seconds after, before her mind was able to process it.\n\n\"Agha Ba . . . . ?\"\n\n\"It is a simple yes-or-no response. Do you agree to take Agha Baraan as your husband? And might I add that you would be a bigger fool than we already know you to be if you should even consider any response other than yes.\"\n\nShekiba sat speechless. They stared at her expectantly while her mind spun.\n\nWhat is happening? Why would Agha Baraan want me? Agha Baraan? Benafsha's secret lover? This doesn't make any sense at all.\n\nShekiba felt her face tingle.\n\n\"Yes or no?\" Louder, impatient.\n\n\"Are you stupid? Just say yes so we can send word to the mullah to close the nikkah! Maybe we should just speak on her behalf. I'm in no mood to wait.\"\n\n\"Fine, then it's agreed. She hasn't said no. I'll tell the mullah.\" The stocky man stood and walked out the door.\n\nWhat about Amanullah? Then who is he to be married to? How could I have thought . . . ?\n\nShekiba thought of the conversation she had overheard in the garden. Her throat knotted with anger. Maybe she was as foolish as everyone said.\n\nA paper was brought to Shekiba and she took the pen that was handed to her, already dipped in ink, and wrote her name on the line. She was dazed but aware enough to know there was nothing else she could do. She'd seen how the palace disposed of people.\n\nThey led her into the hallway, where she was instructed to don her burqa. She did as she was told and Agha Baraan emerged from a nearby room. He looked in her direction, his face more somber than she remembered; his eyes heavy, dark and mournful.\n\nHe nodded at her and walked down the hallway toward the door. She followed, hearing the sighs of relief from the king's lanky and stout advisers behind her. She was leaving the palace with Agha Baraan. Her nikkah had been signed, the contract official and binding. Shekiba was married to Agha Baraan.\nCHAPTER 50\n\nSeeing Shahla made me miss her more. And Parwin. As the car bumped down the dirt road to Kabul, I thought about my sisters. Shahla looked like she was being treated well. Her mother-in-law seemed to be kinder, gentler than Bibi Gulalai. Just last night, Bibi Gulalai had taken her walking stick to my shoulder as I swept the hallway. She snapped the stick against my kneecaps when I fell to my side. She didn't like the way I was crouched, she said. It was shameful.\n\nI shifted in the seat, realizing the seat belt was pushing against a sore spot below my collarbone. I sighed heavily. Badriya pretended not to notice and I was grateful for that. I had no intention of crying on her shoulder.\n\nBut there was something else that I'd been thinking about since Shahla's visit. Something that had been creeping into my mind since we sisters left our father's house. Shahla had chosen to name her daughter Parwin. I loved Parwin with all my heart but it was undeniably bold and bad luck to name a child after someone with a lame leg. I wondered if I could have brought myself to name a child Parwin. Or Shaima. I hoped my aunt would never know. I felt a surge of shame to think it but I wouldn't have used either of those names.\n\nBringing Jahangir into the world had nearly killed me. I prayed I would not become pregnant again and for once Allah had answered my prayers. But by now my body had regained strength and my mind had blurred the memory of his birth; I had started to want another child. I wasn't sure why I hadn't become pregnant again but I thought maybe Allah had a plan for me. Maybe next month. As foolish and illogical as it was, I prayed my next child would be a girl.\n\nWhat name would I choose? Raisa. My mother. Absolutely not. I was less embarrassed to admit that. I could picture her eyes in a toxic glaze, red with smoke, while Rohila and Sitara watched on helplessly. No, I could never use my mother's name. But I couldn't think of my mother without missing her, missing the way she held me on the day of our nikkahs, the day that broke her.\n\nZamarud? Maybe, but probably not. Too many people disliked her, enough to try to kill her. If they tried once, they would likely try again and maybe succeed. Then it would be the name of a murdered parliamentarian. No, I thought. That wouldn't do.\n\nHamida? Or Sufia? Very possibly. I liked them both, Hamida a bit more because she had pushed Badriya to let me see more in the parliament, to do more outside.\n\nShekiba. That was it. That was the name I would have chosen. The name of my bibi's bibi. The woman who lived the double life I had, walked in a man's clothes, worked with a man's strength and fended for herself. That's what I would want to name my daughter, if I were to have one. If.\n\n\"I'm not going to babysit you just because of what happened with Zamarud. You better look out for yourself,\" Badriya said sharply. Kabul's busy streets were coming into view. I turned and looked at her, not sure what she was implying.\n\n\"Fine with me. I didn't think you were looking out for me before,\" I said flatly, but I was too late to catch myself. Badriya's eyes widened.\n\n\"Why, you insolent little . . .\" Unable to find the words, she whipped the back of her hand against my face. My eyes watered and my nose stung. I prayed it wouldn't bleed on my freshly washed dress. \"Don't you dare talk to me like that, you worthless girl. Just remember you're here only because of me and I can change my mind about your usefulness anytime.\"\n\nI bit my tongue and looked out the tinted window.\n\nWe checked back into our room at the hotel. The apartment our husband had purchased needed a lot of work before we could stay there. He'd asked his guard and driver to find some local workers to replace the flooring and cover the windows. He didn't want people or neighbors catching a glimpse of his wives.\n\nBadriya set to unpacking right away, hanging her dresses in the closet.\n\nI saw something that made my jaw drop. Our room had a television in it! It hadn't been here on our last trip and Badriya had never mentioned having one before. I turned it on and saw Badriya watching me, very interested.\n\nThere was a knock on the door. I looked at Badriya.\n\n\"Don't just stand there like an idiot. Find out who it is!\"\n\nIt was the man we'd seen downstairs when we checked in. Our driver stood behind him, his arms crossed.\n\n\"Excuse me, khanum-ha, forgive me for intruding but it seems that we've forgotten one thing. May I enter please?\" He looked back at Maroof, who nodded.\n\nI stepped away from the door, turned to the side, and kept my head scarf pulled across my face. I didn't need Maroof reporting anything back to my husband about my behavior. The man entered our room, turned the television off and unplugged it. He wrapped his arms around it and lugged it out the door while I watched, brokenhearted. I had seen about thirty seconds of a woman singing in a grass field, the small mirrors on her traditional Afghan dress catching the sun.\n\nThe door closed. Abdul Khaliq had a television back at the compound. A large box he kept in his own room, with an antenna that tracked to the roof of the house. We weren't permitted to watch it. Once he'd caught me in there, eyeing it and fingering the buttons, daring myself to turn it on. I didn't expect him to be home. He'd thundered into the room and grabbed me by the neck so hard I couldn't breathe.\n\n\"What do you think you're doing? Let me catch you watching television and I'll rip your eyeballs out of their sockets!\"\n\nKhala Shaima had explained his reaction to me when I asked her if she had a television at home.\n\n\"Your husband is a lot of things but he's not a stupid man. He knows what he's doing. He doesn't want you to see what's going on in the rest of the country, what the other women are doing. These television stations now have so many programs, female singers, female news reporters. Even men advocating on behalf of women. Can you imagine that? Now can you imagine how you would feel if you were to see women like that every day? He needs to keep you blindfolded.\"\n\nThe hotel manager had forgotten to remove the television before we got there. It angered me to realize how tight our leash was, even this far away from Abdul Khaliq. I felt like I was being buried in a hole, deeper and deeper every day until I could hardly see daylight.\n\nAt least returning to the jirga sessions was a break for me. And I was thankful to see Hamida and Sufia again. They greeted us with hugs and asked about our children. I couldn't help but notice, happily, that they were warmer with me than they were with Badriya. I liked that they liked me.\n\nThe attack on Zamarud had frightened Badriya, as it had many other female parliamentarians. Hamida told me two women had decided not to return, afraid that they would be in danger as well. Zamarud was badly hurt, she said. Her wounds had gotten infected and she'd been hospitalized. She was not expected to survive.\n\nThe session opened with a prayer. I sat with Badriya, our heads bowed and our hands cupped. I spent the day filling out her papers and reading documents to her. She snapped at me for reading too slowly but I said nothing. It was easier that way. After the sessions and during our breaks, I would tag along with Hamida and Sufia, who were kind enough not to ask why I didn't accompany Badriya. It was only within the parliament building that my husband's driver and guard did not keep track of my whereabouts. Here, the leash slackened.\n\nAfter the sessions, Badriya wanted Maroof and our guard to escort her back to the hotel. She had no interest in attending classes at the women's training center but I certainly did. The guards had more stock in looking after Abdul Khaliq's first wife, so they watched nonchalantly as I climbed into Hamida's car, leaving me under the watch of her guard and driver.\n\nWe opened the door to the training center, which was, as usual, empty until we got there.\n\n\"Hello!\" Ms. Franklin called out happily.\n\nI wondered how she could be so cheerful all the time.\n\nWe alternated every day. One day she would teach basic English, and the following day we were back on the computer, learning to navigate the Internet or type notes. I thrilled at being a student again and longed for a real classroom, one full of boys my age whom I could learn with, joke with and play soccer with.\n\nMs. Franklin was proud of our progress. She said she'd told her parents all about us, about how impressed she was with our dedication, with our desire to work in government as women. I liked her praise. It had been a long time since I'd heard any.\n\nSo when the door opened, thirty minutes into our session, we were understandably intrigued to see who it was.\n\nA tall, thin woman in her forties entered and looked around, unsure.\n\n\"Hello, come in!\" Ms. Franklin said.\n\nThe woman wore a calf-length black jacket over a deep-plum-colored tunic and pants. Her ponytail was hidden by a plum-colored head scarf.\n\n\"Salaam!\" she replied. \"You are Ms. Franklin?\"\n\nHer name was Fakhria and she put Ms. Franklin in a tricky situation. She worked at a women's shelter here in Kabul and wanted to attend the classes at the resource center. Ms. Franklin looked mildly perplexed. The funds that supported the center were specifically allocated to women parliamentarians. The classes were not open to the public because, theoretically, the center couldn't accommodate more than the women jirga members. But so few of them came.\n\nMs. Franklin pursed her lips and waved Fakhria in, as I would have done. Somehow, she was not a woman you could turn away easily.\n\nAt the end of the class, Hamida asked Fakhria about the shelter. She and Sufia had heard of a women's shelter but hadn't ever seen it. I was surprised to hear such a place existed.\n\n\"My sister was killed by her husband. I decided I needed to do something and then I came upon this shelter. It was founded by an Afghan woman who was living in America. She raised money and emptied her pockets into building this place for girls. She travels back and forth now but we have a few people who look after the shelter.\"\n\n\"And your husband, he doesn't mind you spending time there?\" Sufia asked gently.\n\n\"No, he is very supportive actually. He's a kind man, my husband. After what happened with my sister, he knew I would go crazy just sitting there in mourning. We've got five children to keep me busy at home but I needed to do this. I wanted my children to see me do something.\"\n\nFakhria started to tell us about the shelter, about the girls who came there. She told us about a girl she called Murwarid. Murwarid was only fifteen years old, she said, and had come to the shelter two weeks ago, bruised and desperate. At the age of eight, she'd been married to a man in his sixties, living in the countryside. Her husband had abused her in every way possible. Her nose was crooked after he'd broken it twice. When he'd tired of her, he'd started to take her around to other villages, selling her off to men to have sex with her. She had tried to run away once before but he caught her and sliced off one ear, dragging her home by the other.\n\nSix months later Murwarid decided again that she wouldn't survive if she stayed with this man. And this time, if he killed her, she would be better off. So she ran.\n\nShe came to Kabul and found the women's shelter, where she was living now, recovering. She still woke in the night screaming.\n\nFakhria invited us to visit the shelter. It would be great, she said, if the parliament could help support such a place. Maybe offer some training or jobs to the women living there.\n\nHamida and Sufia clucked their tongues to hear the stories Fakhria told.\n\nI sat frozen. Too much of what she said sounded familiar.\n\nYou see that? Murwarid found her escape, I could hear Khala Shaima say. Why haven't you found yours?\nCHAPTER 51\n\n\"Read this one to me.\"\n\nBadriya had unfolded Kabul's weekly newspaper on the table. She pointed from one column to the next. She stopped me one paragraph into a story about drought conditions in a province to the south.\n\n\"Forget it. Who needs to know about that? I want to know what's happening here. Try this one,\" she said, picking out a column on the following page. I sighed and got ready to read about a new bank opening next month when I was interrupted.\n\nA knock on our door.\n\n\"There's a phone call from home. Come down to the lobby to take the call.\" It was Hassan, our bodyguard.\n\n\"Now?\" she huffed. \"As if we haven't had a long enough day!\"\n\nBadriya and I had just gotten plates of food sent up from the hotel kitchen. I loved the food there. Maybe it was that I had no part in cooking it or cleaning up after it. Maybe it was the pretty floral pattern of the plates. My mouth watered at the smell of the cumin-infused potato stew. I tore off a piece of bread as she resentfully left the room. I dipped a piece of bread into the stew and brought it to my mouth. The grease felt good on my lips. No reason for us both to eat cold meals, I figured.\n\nBadriya returned a few moments later.\n\n\"The qorma is really good,\" I announced as she walked in. I looked up and saw that her face was drained of color.\n\n\"Are you . . . are you all right?\"\n\nShe looked at me, her mouth open slightly. Her eyes searched.\n\n\"Badriya-jan, what is it? Who was on the phone?\"\n\nHer hand covered her mouth. Something wasn't right.\n\n\"Badriya-jan, are you all right?\"\n\nSuddenly, something in her shifted. She straightened her shoulders and pulled her lips together tightly.\n\n\"It was Abdul Khaliq on the phone. He called about Jahangir.\"\n\nMy stomach fell at the sound of his name.\n\n\"He's not well,\" she said, choosing her words carefully. \"He's not well. Seems he's been very sick since we left.\"\n\n\"Since we left? Why didn't he call sooner?\"\n\n\"I don't know, Rahima-jan. I don't . . . he's going to have Maroof take us back.\"\n\n\"I want to go back now!\"\n\n\"We are. Maroof is bringing the car around.\"\n\nI wanted to be there already. I wanted to see my son. The last time he had been ill, he'd spent two days in my arms. Whispering every prayer I could remember, I stroked the moist hairs from his sweaty forehead and watched his cherry lips tremble until the fever released him. I knew he must have cried for me and I hated that I wasn't there.\n\nWe packed our belongings in a matter of minutes. Badriya moved surprisingly quickly. Forty minutes later, Abdul Khaliq's SUV was on the main road leaving Kabul, whizzing past tanks and western soldiers, their curious eyes shielded by sunglasses. Maroof grunted something to Hassan in the passenger seat.\n\nThere was something peculiar about Badriya's behavior. Jahangir, like all the other children in the compound, had survived fever and illness. I looked over at her. Badriya busied herself folding papers neatly and putting them away in her purse. Papers she couldn't read.\n\n\"What did he say, Badriya? Do they want to take him to a doctor? Has he been eating anything?\"\n\n\"I don't know, dear girl. The connection was lousy and you know Abdul Khaliq. He doesn't explain much.\"\n\nThe hours dragged on. I tried to fall asleep, hoping I would open my eyes and find myself back at the compound, Jahangir coming to the gate to welcome me. It would be midnight before we got back. I hoped Jameela had made him a cup of the herbal tea she had given him last time. I hoped the other children were not disturbing him.\n\nJust as I was beginning to drift off, it occurred to me that there was something odd about my conversation with Badriya. Something other than Jahangir being ill.\n\nThe way she had looked at me. What was that look?\n\nConcern? Annoyance? Fatigue?\n\nPity.\n\nI don't know, dear girl.\n\nNever before had she addressed me with endearments.\n\nMy mouth went dry. I started to pray.\nCHAPTER 52\n\nShekiba and Agha Baraan did not speak on the way to his home. Shekiba sat beside her new husband but kept her gaze straight ahead. Agha Baraan guided the horse expertly down Kabul's busy streets, small shops and pedestrians everywhere. He looked in her direction only once but his expression told Shekiba nothing.\n\nHe turned down a narrow, house-lined street, homes packed so close together that a child could toss an apple into his neighbor's courtyard. Shekiba thought of her village, the homes divided by kilometers of open fields.\n\nAgha Baraan's home was in the middle of the street; the royal-blue door set it apart from the rest.\n\nShekiba suddenly felt a panic set in at the thought of being behind those walls with this man. She briefly considered running\u2014disappearing into Kabul's maze of roads. But she remembered Azizullah dragging her back from Hakim-sahib's doorstep and decided against it.\n\nHe opened the door and she followed. The courtyard was small, much smaller than houses in her village, but it was neatly kept and had bright flowers and a birdcage with three small canaries. She followed her husband through the house door.\n\nA woman in her twenties looked up from her needlework. She did not seem surprised.\n\n\"Gulnaz, this is Shekiba. You can show her to her room please. She has no belongings so you'll have to give her a dress or two for now.\"\n\nGulnaz stood up and looked at the blue cloak before her. Agha Baraan walked out, uninterested in how the two would take to each other.\n\n\"You can take your burqa off. You look ridiculous wearing it inside.\" Shekiba understood by the tone of her voice that Gulnaz was Agha Baraan's first wife and that she was not happy to see her. Shekiba lifted her burqa but kept her right profile toward Gulnaz.\n\n\"I hear they call you Shekiba-e-haleem. Let me see your face.\"\n\nShekiba turned, making a point to look Gulnaz in the eye. Each woman took a moment to consider the other. Gulnaz was a beautiful woman but nowhere near as striking as Benafsha. She had almond-shaped eyes and gracefully arched brows. Her hair looked soft and thick, loose curls tossed over her shoulders.\n\n\"I see,\" she said, wincing. \"Well, come this way. I'll show you to your room.\"\n\nThe layout of the house was similar to Bobo Shahgul's. Behind the living room was a small kitchen. Off the main hallway were three other rooms, which Shekiba was not shown. The last room was to be hers, an eight-by-ten-foot space without a window. A thin cushion lay against the wall and a lantern sat on a narrow round table.\n\n\"I'll bring you some clothes later. For now you can stay here. We won't be eating dinner for a while. I've prepared the meal for tonight. You can start helping tomorrow.\"\n\n\"Khanum Gulnaz, I\u2014\"\n\n\"Don't call me that. It doesn't sound right. Just call me by my name. You're his wife now and it would sound strange if someone were to hear you say that.\"\n\n\"I'm sorry.\"\n\n\"Let me warn you. This is my home. I do things the way I like and you better not expect to change things. You're here because he wants you to be here but that does not mean you can do as you please.\"\n\n\"I had no intention\u2014\"\n\n\"Good. Then it's understood and I'll expect no problems from you. I asked him to have you in a separate house but there just isn't room for that right now. You'll have to be here.\"\n\nGulnaz was only slightly older than Shekiba but she spoke with such condescending authority that Shekiba felt like she was being chastised by one of her uncles' wives. She had no reason to expect Gulnaz to be any kinder to her, but Shekiba thought Gulnaz might be able to shed some light on her situation.\n\n\"Excuse me, but can I ask one question, Gulnaz-jan? Can you tell me why I am here?\"\n\n\"What? What do you mean?\"\n\n\"You said he wants me to be here. Why does he want me here?\"\n\n\"You have no idea?\"\n\n\"No.\"\n\nGulnaz shook her head and walked out of the room, leaving Shekiba with more questions than answers.\n\nShe heard from Gulnaz once more that night, when she walked by the room and announced that there was food left in the kitchen if she wanted some. Shekiba stared blankly at the door but did not respond. She felt terribly out of place. And she was now a woman again. Her dress felt cumbersome and heavy. She had just about forgotten how to keep her head scarf in place. She had left her guard's uniform in Benafsha's room but took with her the corset used to bind her breasts. She could not tolerate their jiggling, even though the corset chafed her raw wounds.\n\nShekiba wondered how things would be here, with her living as a second wife to Benafsha's lover, the man who had betrayed the king in the worst way. How had she become involved in such a twisted affair?\n\nShe listened, nervous, for the sound of Agha Baraan approaching. She knew, from watching the king's habits, that men came for women at odd hours of the day and night. She felt unprepared to be near him behind closed doors. Shekiba dozed off sometime near morning.\n\n\"Look, you've got to get up and eat something. It doesn't really matter to me what you do but I won't have the blame on my shoulders if you get ill from hunger. Looks like you're not in great shape to start with. Here's a dress too. That's all you will be getting from me. He can buy you fabric if you need another.\"\n\nShekiba sat up and rubbed her eyes. She watched Gulnaz put a plate with bread and butter on the floor, along with a cup of black tea.\n\n\"And if we're to share the house then we'll share the work. You can't expect to just laze around all day long.\"\n\n\"Sorry, I didn't realize the time . . .\"\n\nGulnaz did not wait for an explanation. She was gone before Shekiba could finish her sentence. The butter melted on her tongue. She walked out of her room hesitantly and found the washroom. It was summertime and a bit of cool water felt good, especially on her scabs. Shekiba wondered how badly scarred her back was. She cursed Ghafoor again but Ghafoor wasn't the only one to blame. Agha Baraan and Benafsha had created the mess as well. Shekiba had been caught in an elephant stampede.\n\nI am not welcome here. I am his wife, but only half. Nothing about me is whole. Why did he do this?\n\nShekiba went out to find Gulnaz and make herself useful. This part was nothing new to her. It felt the same as being in Marjan's home. Or Bobo Shahgul's. She found the kitchen empty, a pile of raw potatoes on a counter. Shekiba looked around. Agha Baraan had a nice home. The walls were smooth and flat and intricate hand-knotted carpets covered the living room floor. There was a tufted sofa with carved wooden arms as well as a chair that Shekiba had not noticed the day before. On the walls were framed pieces of calligraphy, Allah's name written in graceful curves, slants and dots.\n\nShekiba went back into the kitchen and looked around. There were cups and plates in the cabinets and a pile of pots and pans underneath the counter. Shekiba found a knife and sat down to peel the potatoes. It was a relief to be doing something and when Gulnaz came back in from the courtyard, she pretended not to notice the second wife in her kitchen, walking to her room instead.\n\nThey have no children, Shekiba realized. That was what was different in this house. No excited footsteps, high-pitched voices or crying. They lived alone and apart from the rest of Agha Baraan's family.\n\nIt would be hard to get lost in such a small household. Gulnaz said nothing more to her than household instructions. She left dirty clothes in a pile and told her Aasif, Agha Baraan's first name, needed his shirts for the morning. They did not eat together. Gulnaz and Agha Baraan shared their meals when he was home but Shekiba kept herself occupied with chores and made no motion to join them. Nor was she invited. She took her meals to her room or ate a few bites in the kitchen.\n\nAasif said no more than a few words to her each day, mostly small greetings in passing, his eyes averted. Shekiba would mumble something to complete the exchange. Aasif was different with Gulnaz. He chatted about the people he had seen and told her of Kabul's local news. Gulnaz listened and asked questions. Sometimes they even laughed together. Shekiba wondered how things had been for Aasif and Gulnaz when they were first married. Had Aasif been as cool with her as he was now with Shekiba? Would he ever say anything more to her?\n\nThe silence was uncomfortable, but Shekiba dreaded a conversation with Aasif. That day in the palace, when she had spoken with him, he had seemed a gentle person, a noble man. But what she knew now made her question her first impression.\n\nFour nights passed before he came to her room. Shekiba had begun to believe he had brought her only to help with the housework when she heard her door open. It was late and her eyes were just beginning to feel heavy with sleep. In the darkness, she could make out his thin silhouette.\n\nHe stood there for a moment, watching her. Shekiba kept her eyes mostly closed, feigning sleep and praying he would turn and walk back out. Her heart beat so loudly she was sure he could hear it. He came in and closed the door behind him. Shekiba nearly stopped breathing.\n\nHe sat beside her mattress on the floor, his back turned to her. His head was lowered.\n\n\"Things turned out badly,\" he said quietly. \"I regret that it happened this way.\"\n\nShekiba stayed silent.\n\n\"She was a good woman and did not deserve what they did to her. I did not want . . . I did not think it would go so far. But once they found out, there was no stopping it. I was foolish to ignore what might happen\u2014what did happen,\" he whispered, his voice cracking. \"She warned me and I ignored it. I ignored it. Still, she spared me or I would not be sitting here now. I am very aware of that.\"\n\nRamblings of a guilty conscience. He knew Shekiba was aware of their affair. Maybe he thought Benafsha had revealed his identity to her or maybe he thought she had recognized him as he stumbled past her on that night. Shekiba did not know why he was making this admission but she listened carefully.\n\n\"Gulnaz is not happy. Things will be difficult for a time but it will get better.\"\n\nAnd without a word from Shekiba, Aasif, her husband, walked out of her room and closed the door behind him.\nCHAPTER 53\n\nIt was pitch-black when we arrived at the compound. Never had I been so relieved to see those gates. Maroof parked the car, looked at Hassan and sighed. Badriya had fidgeted so much in the last hour of the drive back that I'd thought she might just jump right out of the car. I didn't bother with my burqa. Our car had barely stopped before I jumped out and opened the gate. There were lights on.\n\nI opened the door to find Jameela rushing toward it. Her face told me everything.\n\n\"Jameela!\"\n\n\"Oh, Rahima-jan! Allah, help us\u2014dear, young mother!\" Her voice rose and fell, my heart with it.\n\n\"Jameela, where's my son? Where's Jahangir? Is he all right?\" I grabbed her by the arms and moved her aside, pushing my way toward her room. Shahnaz emerged, holding her chador tightly at her chin. She was looking down, avoiding my gaze. I stopped short when I saw her. Her lips were trembling.\n\n\"Why are you all out here? Who's watching my son? Where is he?\"\n\nJameela rushed back and grabbed me before I could run into her room. By this time, Badriya had joined her.\n\nJameela hugged me tightly and held my head to her chest.\n\n\"Rahima-jan, Rahima-jan, God has decided to take your son! He's taken your little boy, dear girl. God give him peace, that darling little boy!\"\n\nI froze. That was what I'd read in Badriya's face. I looked at her now but she, like Shahnaz, diverted her tearful eyes.\n\nSomeone wailed. Someone moaned no, no, no, no. My son's name.\n\nIt was my voice.\n\nThis couldn't be true. This couldn't be real. I looked around, thinking everyone I lived with had gone mad.\n\nAbdul Khaliq came into the hallway, his eyes red, his lips tight. He looked at me and shook his head. I saw my husband's shoulders heave. Bibi Gulalai stood behind him, sobbing into a handkerchief.\n\n\"Why? Why would you leave a sick child? His mother should have been here with him!\" she cried out.\n\nI looked into my husband's eyes, our first truly intimate moment. It was as if no one else existed.\n\nIt's true . . . It's true, Rahima. What they're saying about Jahangir, our son, is true! Our beloved boy is gone!\n\nAbdul Khaliq covered his eyes with his hands before he looked up, took a deep breath and yelled for someone to find his prayer cap. His voice cracked and my chest caved in as the air was sucked out of the house.\nCHAPTER 54\n\nI'm not altogether sure what happened after that. There was whispering, wailing, cursing and praying. All at once and then one at a time. Voices and faces blurred around me.\n\nLet me see my son, I screamed. I want to see Jahangir.\n\nHave a sip of water. You look as if you're about to pass out.\n\nSomeone brought a glass to my lips.\n\nThe other children were in the living room, the older ones somberly watching over the babies and trying to keep them quiet.\n\nAbdul Khaliq's compound had never experienced such tragedy. Even I, who had lost my father, my mother, my sisters and myself, even I could not believe God would add this to my lot.\n\nThey led me to Jameela's room. My little boy. His tiny face looked pale, his lips gray. I fell to my knees and put my head on his small chest. I stroked his chestnut hair, touched his full cheeks. I talked to him as if there was no one else in the room, as if there was no one else in the world. I wanted to comfort him and breathe life back into his little body. I was his mother. I had given him life and when he was ill, I had nursed him back to health. Why should now be any different?\n\nI snapped when I felt a hand pull at my elbow.\n\n\"Leave us alone! I need to make my son better. He always wakes when I whisper his name. You'll see him yawn, rub his eyes and look around in confusion. He's going to tell me he missed me and that I shouldn't go away again.\"\n\nThere were traditions, rules that needed to be followed.\n\nOne hand became two, or maybe it was four. When there were enough, the hands became stronger than I and the room shrank away from me. I was in the hallway. I was on the floor. I was outside of myself. The arms melted away.\n\nThat is enough, they whispered. I hated them.\n\nBibi Gulalai was there. Wailing louder than anyone else.\n\nWhy? Dear Allah, what a sweet little boy he was! Too young, too young to take! His face, I picture his face before me as if he were still here. I can't believe it. I just can't believe it! Oh, my poor son! Why should you have to have such a tragedy happen to you, my God-fearing son! My lion among men! If only I'd known sooner! I could have done more for him! I could have made him better!\n\nI hated her.\n\nI was numb. Days passed. Rituals were performed. All the right prayers were said. All the wrong people came to pay their respects. I noticed little of it, only the absence of my own family. My mother, my father. They never came to their grandson's fateha. My father was not there to carry my son or throw fistfuls of dirt onto his grave. It mattered, though it shouldn't have. Jahangir had never known them anyway.\n\nKhala Shaima came, as did Shahla. My aunt and my sister sat by my side as I rocked back and forth, their eyes red and raw. Someone asked Khala Shaima about my parents, if they were coming. Shahla bit her lip and looked at the ground. I heard my husband curse my father. He was insulted, not only as a son-in-law but also as a former commander. Whatever respect he owed his father-in-law, out of tradition, was lost now. And I didn't care.\n\n\"Oh God, Rahima-jan,\" Shahla whispered. \"I can't believe this! He was so full of light!\"\n\nI closed my eyes.\n\nKhala Shaima looked thinner than she had the last time I'd seen her but I couldn't bear to think on it much. She shook her head and whispered to me that the medicine had gotten the best of my parents. It was hard to tell which of them was worse than the other. She clucked her tongue in dismay and squeezed my limp hand.\n\n\"They can barely get themselves up and about in the house,\" she said.\n\n\"You're there a lot,\" I said blankly.\n\nShe nodded. She was concerned about Rohila and Sitara. She had mentioned in her last visit that rumors were circulating of suitors for my younger sisters. She wanted to make sure the girls were not given away in a careless stupor.\n\nMy aunts and uncles came. Even my grandparents. I kissed their hands. They cried and made sorry excuses to my husband and mother-in-law for the notable absence of my mother and father. They were embarrassed more than anything else.\n\nYou never saw him, I wanted to scream. You didn't know how sweet he was.\n\nI'd never expected much from my grandparents. They'd had little to do with my sisters and me since we were married off. It was as people said. Once married, girls no longer belonged to the families that raised them. Especially if they only raised them halfway. But Madar-jan, she'd been so different once upon a time.\n\n\"She's that bad?\" I asked Khala Shaima.\n\n\"She's that bad, dokhtar-jan,\" she said, confirming it. \"Your sisters, Rohila and Sitara, they really wanted to come see you. But your grandmother didn't think it was proper for them to come without your mother. And, of course, she wouldn't let them come with me. Rohila cried when she heard. She wanted to hide under my burqa and sneak over here. Sitara, she's very reserved but she's a strong girl. You girls would be very proud of your sisters.\"\n\nI was sure she was right. They were surviving in a home without a mother or father, essentially. They'd been abandoned just as much as my own son.\n\n\"I should have taken him, Khala Shaima. I should have taken him to Kabul with me. He wouldn't have gotten sick with me. And even if he had, I could have taken him to a hospital. They have the best hospitals there. Lots of doctors. Even foreign ones.\"\n\n\"Your husband never would have allowed it. He keeps his sons at his side, my dear. You know that.\"\n\n\"Then I should have stayed with him. I didn't have to go to Kabul.\"\n\nKhala Shaima said nothing. We both knew it had been her idea.\n\nMy son was buried in the family plot, a half kilometer from the compound. It was sacred earth for Abdul Khaliq's family.\n\nMy husband was quiet, different. I knew he was hurt.\n\n\"He's with his ancestors now. They're watching over him, as is Allah. His fate is our fate,\" Abdul Khaliq said to me when the men returned from the burial.\n\nNaseeb. Was it really Jahangir's destiny to be taken so young? Was it my naseeb never to see my son grow taller than me, to go to school, to help his father at work?\n\nAbdul Khaliq asked Jameela to look after me. I saw him pull her aside and have a few words. They watched me. Badriya too, although a week after Jahangir was buried, she quietly asked Abdul Khaliq when she could return to Kabul. His hand flew across her face so fast that she'd barely finished her sentence.\n\nI closed my eyes and wished for everyone to disappear, including myself.\n\nEvery Friday, Abdul Khaliq's friends and family members gathered at our compound for a khatm. Each person read one of the thirty parts of the Qur'an. Prayers were said at the completion, or khatm, of the holy book. I could hear them from down the hall and prayed along with them, hoping it did Jahangir some good. It did me none.\n\nKhala Shaima came back more often than before, even though the trip had become very difficult for her. She was worried. I was losing weight; my clothes hung loosely on me. When I looked in the mirror I saw someone I barely recognized. My face had dark spots and my eyes were heavy. I saw Jameela and Khala Shaima exchange worried looks.\n\nAbdul Khaliq let me be for the most part. He wasn't speaking much either. His guards tiptoed around him. His friends kept their voices low and their comments brief. My husband, the warlord, was not one to express or show emotion, but it was clear that he was grieving. He was short even with Bibi Gulalai.\n\nMy head felt like an empty, dark room. My insides were painfully hollow. I missed my son's face, his smile, the way his small fingers held on to mine. He was supposed to be safe. He had survived infancy. He had learned to walk, to talk, to tell me when he was hungry and when he was happy. Jahangir. His name was a dagger. His name was a salve.\n\nFour weeks passed before I was able to ask the questions that remained.\n\n\"Jameela-jan.\"\n\nJameela stopped short, surprised to hear my voice. \"Yes?\"\n\n\"What happened to him?\"\n\nJameela stood still for a moment, pondering my question. Then she sat on the cushion next to me in the living room, tucking her legs under her and straightening her skirt. She put her hand on mine.\n\n\"Rahima-jan, he became ill. Everything happened very quickly. So quickly.\" Her mind traveled back to that day. \"Abdul Khaliq called Badriya right away.\"\n\n\"I want to know what happened to him,\" I said insistently.\n\nAfter me, Jameela probably carried the most guilt. I'd left Jahangir in her care and come back to find my son dead. She felt terrible about it. She wasn't sure how much to explain and how much to leave unsaid. She told me in bits and pieces, filtering as she went along.\n\nFirst came the fever. His body was hot. From his head to his toes, so hot, Jameela said. She had tried to fan him, to cool him down with baths. His bowels were loose. Jameela looked for worms but saw nothing in his stool. When he started to complain about his stomach hurting, she'd talked to Abdul Khaliq. He saw Jahangir's trembling body and immediately called for Bibi Gulalai. Bibi Gulalai set to work making a soup heavy with garlic and herbs to clear the germ from his body. But instead of getting better, things got worse.\n\nOn the fourth day, Jahangir's belly was freckled with red spots. Jameela tried again to cool him with moist cloths on his forehead and sips of water. When Jahangir stopped whining and complaining about his belly, she thought he was finally getting better. She thought he needed a few days of rest and that by the time we returned he would be back to his usual self, just as I'd left him.\n\nWe were both crying. She paused her story two or three times, collecting herself and then looking at me. I nodded for her to continue. I needed to know.\n\nBy that afternoon, Jameela realized that Jahangir was delirious. He wasn't answering her questions but he was mumbling and batting away something that wasn't there. She called his name. His eyes were glassy. She called again for Abdul Khaliq, who had just returned from an overnight trip with his guards. Never before had she seen our husband so shaken, Jameela said. He took one look at his son, then flew out the door and summoned his driver and guards. He came back into the room and cradled Jahangir in his arms while he yelled for Jameela to pack some water and bread for the drive to the hospital. Before she knew it, Jameela was standing at the front gate, watching the tires of Abdul Khaliq's black truck spin and screech as they tore off down the road.\n\nShe didn't want to go on. I put my hand on hers. She looked tortured. She sighed and continued, trying to get her words out in one short burst.\n\nThey came back the next day, heavy, morose faces. Jameela ran out to meet them. Abdul Khaliq looked at her and shook his head.\n\n\"Crying,\" she said. \"I've never seen him looking like that. I never thought . . . the doctor wasn't able to do anything for him. He was too weak and they think he had developed a terrible infection of his stomach. Horrible. Something that just took over, making his belly stone-hard when the doctor tried to touch it. He was in the hospital until morning and they were giving him serum but it didn't work. I suppose . . . I suppose it was his naseeb,\" she said, sobbing. \"Rahima-jan, I'm so sorry. I don't know how he got so sick so quickly! He felt better for a bit. He let me rub his stomach. I thought it was helping . . .\"\n\n\"Why didn't he take my son to the doctor sooner?\" I knew she felt badly. At the moment, I didn't care very much. I wanted to know if something could have been done. I wanted to know who to blame.\n\n\"Abdul Khaliq? He . . . he wanted to. Before he left for his meetings.\"\n\n\"Then why didn't he?\"\n\nJameela shook her head in frustration. \"Rahima-jan, what's done is done now. It won't help anything to ask so many questions. It's best you think of your son, pray for him to be in peace.\"\n\n\"I'm tired of praying, damn it. I want to know. What happened, Jameela?\" I said insistently. She was glossing over something.\n\n\"Abdul Khaliq was going to take Jahangir to the hospital but . . . but Bibi Gulalai stopped him.\"\n\n\"What? Why on earth would she do that?\"\n\n\"She thought . . . she thought she could heal him with the teas and soups she was making for him.\"\n\nMy heart sank. Bibi Gulalai was going to save him. I nearly laughed. Her concoctions had never saved anyone from anything. She had stood in the way of my son getting to the doctor. My husband had tried, I thought.\n\n\"He really did try,\" Jameela echoed, as if she could read my thoughts.\n\nI had a fresh hatred for Bibi Gulalai. She'd been the one to delay Jahangir's treatment. She'd been yelling and carrying on about the absent mother being to blame. Now I knew why. Bibi Gulalai always boasted about the powers of her remedies. She claimed she could heal any ailment with her potent, homemade brews, and that she had. The family humored her. She wanted to look good, the grandmother who stepped in and cured her grandchild while his shameful mother played in Kabul.\n\nOne more question to ask, the question I dreaded because there was no good answer. It haunted me.\n\n\"Jameela-jan . . . ,\" I said, my voice breaking.\n\n\"Yes, janem,\" she said gently.\n\nI was looking over the edge of a cliff.\n\n\"Jameela-jan . . . did he . . . did he cry for me?\"\n\nJameela, loving mother of six, had also given birth to two children who had been claimed by Allah before she could see their smiles. Jameela pulled me into her arms and kissed my forehead. She read my heart.\n\n\"My dear madar-ak\"\u2014little mother\u2014she whispered, though I wasn't one anymore. \"What child doesn't call out for his mother? What could be more comforting than a mother's embrace? I believe, in his sleep, that's where your little boy was, feeling your arms around him, janem.\"\n\n\"But I wasn't there!\" I cried. \"I wasn't there to hold him, to wipe away his tears, to kiss him good-bye! He was just a baby! How scared he must have been!\"\n\n\"I know, Rahima-jan, but he wasn't alone. No one can replace you but at least his father was there with him. His father held him. And you know, Abdul Khaliq loved his little son very much.\"\n\nIt wasn't until weeks later that this conversation would bring me solace. For now, I stored her words, saving them for when my heart had healed enough to believe that my son had felt my embrace. That his father had held him lovingly in his last moments. That he did not feel as alone as I did now.\nCHAPTER 55\n\nShekiba swept the floor of the living room, beating the dust from the rug section by section. She had breathed a huge sigh of relief after Aasif had left her room, thankful that he had not touched her as his wife. At least for now. He felt remorse for what he had done. And Shekiba could hear something in his voice that she hadn't heard in a long time. Aasif sounded as if he cared about Benafsha. Maybe her first impression of him hadn't been that far off. There was still a lot to learn about him but it seemed he had a heart.\n\nShekiba had spent the rest of the night replaying his words in her mind and trying to piece together how she had come to be his wife.\n\nHe could not stop her execution. So he stopped mine. How did he propose this deal to King Habibullah? Does Gulnaz know all this?\n\nShekiba wondered why the king bothered to agree to it. And another question still lingered. How had he come to know Benafsha? As a concubine, her activities were limited to the harem. It wasn't as if she had been roaming around the palace grounds. Benafsha had originally been a guard before she had caught King Habibullah's eye and he must have seen her then.\n\nAnd Benafsha let him in? Willingly?\n\nYou wouldn't understand, was all she had told Shekiba. She was right about that.\n\nThe canaries were singing\u2014three yellow songbirds in a white wire cage suspended from a tree branch. They sang in the morning mostly, bright and melodic. Shekiba paused to listen to them, to decipher their chirps.\n\nTwo weeks had passed. Her back was healing. Her skin itched more and burned less, which was how she could tell it was better. With better days came better nights. She learned the routines of the house and found a way to fit in without being a nuisance. She knew from experience that she should not consider herself a permanent fixture in any man's house, even if she was his wife.\n\nAasif now said a few more words to her, but their exchanges were still brief and polite. He looked past her face and made only fleeting eye contact. Gulnaz watched their interactions from the corner of her eye and seemed satisfied that the second wife was not her equal. She began to see Shekiba more as a housekeeper than a second wife.\n\nThrough the window she could see one of the canaries pecking at the other's head. The two others tried to retreat. Peck, peck, peck. They tried to fly from one side of the cage to the other but hadn't enough room to flap their wings more than once before they crossed the cage. Contained. Three caged canaries singing.\n\nAasif came home that night. Shekiba kept her door open to listen in on their conversation.\n\n\"There will be a wedding in three months' time. The palace is preparing for a monumental event.\"\n\n\"I wonder how many people they will invite.\"\n\n\"Plenty. And it will be all the most important families of Kabul. His fianc\u00e9's family is well respected and they carry a great deal of clout. They could not have chosen better for Amanullah.\"\n\n\"What is her first name? I know her aunt, Aalia Tarzi. I have seen her in the market from time to time and she is a friend of my cousin Sohaila. Aalia-jan speaks very highly of her niece. She was educated while they lived in Syria. I wonder what kind of queen she will be.\"\n\n\"It's a powerful match, Amanullah and Soraya Tarzi, although I know Habibullah is not thrilled that his son is taking Agha Tarzi's daughter.\"\n\n\"Why is that?\" Gulnaz asked.\n\n\"Tarzi writes what he thinks. And what Tarzi thinks is not always what Habibullah thinks. But the problem is Tarzi thinks Habibullah is not doing enough to bring Afghanistan to modern times. He thinks we should look to Europe and learn from them.\"\n\n\"But we are a different people. We are a Muslim country. Why should we learn from them?\"\n\n\"Because they are making progress and we are not. Habibullah has made some roads but not much else. Tarzi wants science, education\u2014and not just the religious kind. But Amanullah, his ears are open to Tarzi's ideas.\"\n\n\"But, Aasif-jan, he is not king.\"\n\n\"He will be. I don't see his brothers taking the position. Amanullah has been groomed for this since a young age. He'll make a much finer king than his father, who spends his days quail hunting and riding around the countryside for attention.\"\n\nGulnaz sighed. Her husband detested the king and she feared his dislike would eventually be the subject of gossip. If it did, he could expect no mercy. And he had already done enough to jeopardize them. He didn't talk about it and Gulnaz wasn't sure if her suspicions were true. She'd heard things from others. A stoning. One of the king's concubines. She would not ask him about the girl. She did not want to know more.\n\nAasif saw his wife's eyes turn away. He knew her burdens were his doing.\n\n\"Anyway, I'm busy with my own work. I don't have time to be Amanullah's counselor anymore.\" His way of saying he would stay away from the palace.\n\nGulnaz looked at the door, pictured the hallway and the scarred woman hiding in the far room, her husband's other wife. She wondered if her husband's plan would work or if he had only added another barren wife to his home.\n\nShekiba listened carefully to every word. Amanullah was to marry Agha Tarzi's daughter. She marveled at her own na\u00efvet\u00e9.\n\nWhy should he look at me? I'm no one. I have no father or mother, no family name. I am a half woman with a half face. How stupid I was to believe anything else!\n\nShekiba waited till Aasif had gone out before she went to the kitchen to fix herself some food. The spinach and rice she had made earlier had cooled but she didn't care. She took a piece of bread and retreated to her room. She moved about so quietly that Gulnaz almost didn't hear her from the living room.\n\nIn the night, Shekiba woke with a start. Aasif was in her room again. The door stood open behind him while he considered walking back out. Shekiba's heart galloped. She prayed he was here only to talk some more. She did not move.\n\nHe closed the door and Shekiba pressed her eyes shut, hoping to ward him off. He sat next to her, with his back to her face for a few moments. Shekiba felt his presence. Her body was tense.\n\nWhat does he want?\n\nAasif sighed and turned to her.\n\n\"Shekiba,\" he whispered. \"You are my wife. You have an obligation to fulfill.\"\n\nShekiba did not answer. His voice was raspy and low. He did not sound like himself.\n\nShe clutched her blanket tightly with her two hands, knowing she had no right to resist. She was his wife and she had a responsibility to lie with him, even if it terrified her. Her breathing quickened. He turned toward her and pulled the blanket away. Shekiba could keep her eyes closed no longer. She saw him, saw him looking at her nightgown, the thin white cotton that surrendered without a fight. He undid the drawstring of his pants and lifted her hem over her hips. Shekiba pressed her back into the mattress, wishing she could melt into the floor. A wave of panic rolled over her body as Shekiba closed her eyes, clenched her teeth and became Aasif's wife.\nCHAPTER 56\n\nIn a way it was a relief. She knew now what to expect. He came to her infrequently and briefly, leaving when he had finished his grunting and sighing to sit in the living room. Sometimes he retreated back to Gulnaz. Shekiba always avoided Gulnaz the following morning, embarrassed and feeling as if she had committed an offense against her.\n\nHer only reprieve was her bleeding. Only then could she whisper in the dark, her face flush with humiliation, \"Forgive me, I have illness.\"\n\nHe understood right away and would leave her chambers, seemingly relieved. Only last night was different. She had started bleeding two days ago.\n\n\"I have . . . I have illness,\" she said softly, pushing her thighs together.\n\nBut he didn't leave. Instead he sat again with his back turned toward her. He put his head in his hands.\n\n\"Things are not going well. Why are you still having your illness? Are you lying about it?\"\n\nShekiba was surprised. His voice was gruff. \"No, I would not lie about . . . about such a thing.\"\n\n\"What happened to all that talk? All the talk about the women in your family and the lines of sons they birthed? You've been here for five months and you are still having your illness!\"\n\nShekiba once again realized just how simple she was. That was the reason Aasif had taken her from the palace. Gulnaz had given him no children at all. He didn't want Shekiba\u2014he wanted sons.\n\n\"I . . . I . . . it was not talk. I had brothers . . . I\u2014\"\n\n\"This is a joke! How can this be possible? They were going to execute you. Do you understand that? Do you understand what you escaped?\"\n\nShekiba understood better than anyone what she had escaped. She had been close enough to see the blood seep through Benafsha's burqa and pool in the earth. She understood exactly what she had been spared.\n\n\"I understand.\"\n\n\"Do you? Do you really? What are people to say? Two wives and not a single son! Do you know what that does to me?\" He was livid. Gulnaz could hear him through the thin walls. She turned on her side, knowing that Shekiba was receiving the anger that he intended for them both. \"A harem guard! Did you like being a man? Maybe that's what it is! You liked being a man so much that now you refuse to be a woman! What are you? You are not a man! You are not a woman! You are nothing! Do you have anything to say for yourself? Where's all the boasting now?\"\n\n\"I . . . I . . .\" Shekiba did not know what to say.\n\n\"I feed you and clothe you and for nothing! This is what you do to me! I should throw you out on the street! I should throw you back to the palace and let them do with you what they planned! You and your cursed face! Damn you!\"\n\nShekiba braced herself for the blow but it never came. She cowered in a corner of her mattress. Aasif stormed out and slammed the door shut behind him. A few seconds later, Shekiba heard glass breaking and the metal gate clanged loudly. Her throat clenched, she could not help but agree with her angry husband.\n\nNot a man, not a woman. I am nothing.\n\nGulnaz slipped quietly into Shekiba's room a few moments later. Through the open door, a sliver of moonlight lit the floor of the hallway. The two wives stared at it, Aasif's rant still echoing through the house. The first wife finally spoke.\n\n\"We have been married for one year and I have been unable to bear him a child. Your head would spin to know how many herbs I have ground under the pestle at my grandmother's instruction. I have prayed at the local shrine and given alms to the poor. Nothing. My bleeding comes month after month, as does yours. He thought you would be different but I suspect now that Allah may have cursed him and no matter what woman or how many women he beds, a son is not his naseeb.\n\n\"And now, now that he has heavy sins on his shoulders, he may have poisoned his naseeb even more.\"\n\nThis was the first reference Gulnaz had made to Aasif's involvement in the palace scandal. Shekiba was not sure how much she knew.\n\n\"You were a guard there for the harem. He told me this much. You were living as a man. Your short hair, the way you walk, the way you hide your breasts. I think you may have been more content that way. To be honest, I would not mind trying it myself. I wonder what it would be like to be able to walk through the streets freely, without a thousand critical eyes. Do you miss it?\"\n\nThis was something to which Shekiba, the woman-man, had given a great deal of thought.\n\n\"It did feel good. But . . . pants or a skirt, it changes nothing in the end. When it mattered, I was as vulnerable as any woman . . .\" Shekiba decided against talking about the lashing. \"And now I am here.\"\n\nGulnaz intuited what Shekiba meant. \"It must have been awful, what they did to you.\"\n\nShekiba felt her back stiffen. There were still three raised scars she could feel when she bathed. She wondered how many more scars she had that she could not see. Gulnaz sighed.\n\n\"He was so angry. He did not say much about it but a wife knows her husband's moods. He was angry from the beginning and I didn't understand why until Aasif's sister told me about her. She wanted me to know I wasn't her brother's first choice.\"\n\nHer. Shekiba looked at Gulnaz from the corner of her eye. Her expression was blank. She was talking about Benafsha.\n\n\"He knew her from before. She was nobody. Her family is as poor as they come and with three daughters, her father cursed his luck. She was just some girl who lived near his uncle's home. I don't know how, but he saw her once or twice.\n\n\"Aasif wanted her but his father rejected the idea. Not a proper family, not good enough for his son. But he kept at it. Kept trying to convince his father, and he had almost gotten his way when her father sent her to the palace. One less girl to provide for. Aasif was angry, but she was out of reach behind the palace walls so he let his father choose another family. And then we were married.\"\n\nShekiba listened intently. Gulnaz was speaking to no one in particular.\n\n\"Men don't like being denied something. Even if it is by the king. He won't say exactly what happened there but I know something happened. I know it must have been terrible because he came home with eyes so red he looked like he could cry blood. He didn't eat, sleep or speak for days.\"\n\nShekiba looked away. She did not want to explain and hoped Gulnaz wouldn't ask.\n\n\"And then he came home one day looking like he'd just met Shaitan himself. His eyes were dark and serious and he sat around staring at the walls, muttering something about atoning for his sins, begging forgiveness of God. He announced that he was going to bring home a second wife since I haven't been able to bear him a child. There was nothing I could say to that, especially when I saw the look on his face. His family had spoken to him about the idea months ago but he hadn't seemed all that eager. But I thought . . . well, when he said he would bring a second wife I wondered if he was crazy enough to think he could bring her here, but then it was . . . you.\"\n\nShekiba kept her eyes on the ground. Her head spun. Benafsha had not turned him away. She had loved him, enough to protect him with her own life. How could a woman love any man so much?\n\nBecause of Benafsha, Aasif had saved her life. For that, Shekiba was grateful.\nCHAPTER 57\n\nI was a little girl and then I wasn't.\n\nI was a bacha posh and then I wasn't.\n\nI was a daughter and then I wasn't.\n\nI was a mother and then I wasn't.\n\nJust as soon as I could adjust, things changed. I changed. This last change was the worst.\n\n\"Rahima-jan, remember that life has typhoons. They come and turn everything upside down. But you still have to stand up because the next storm may be around the corner.\" I hadn't changed much since I lost my son. Abdul Khaliq had become withdrawn. Bibi Gulalai was more present than before, making sure that the family was carrying itself properly. We had to mourn appropriately or our neighbors would talk. Her narrowed eyes fell on me, checking the color of my chador, the dress I wore and the expression on my face.\n\nWhen my mind wandered, she caught me and told me to stop staring. She told me to get back to work. I couldn't expect to just lie around forever. There were still floors to clean. There were clothes to wash. It would be good for me to get back to the routine.\n\nA mourning mother should have been given her forty days to grieve, our visitors were surely thinking. Bibi Gulalai, mother to the most powerful man in our province, knew their concern was driven by fear, not respect, and she did not care.\n\nKhala Shaima summoned the strength to visit me still. Each time she left, I wondered if she would make it home. And I worried that she wouldn't make it back. I needed her. In a house full of people, I still felt totally alone. There was something on my mind, something I didn't want to admit to myself or to Jameela. I didn't know how to feel about it.\n\n\"Khala-jan, do you know what the people of Kabul think of us?\"\n\n\"What are you talking about, Rahima?\"\n\n\"Kabul is different from here. Just like Bibi Shekiba thought. It's amazing how many cars, how many people, how many posters. There is so much noise there.\"\n\n\"What does that have to do with anything?\" Khala Shaima looked concerned that I was losing my mind.\n\n\"I wonder what those people think of us. They've got buildings, banks, taxis, hotels. People from all over the world, construction companies working on new buildings. Beauty parlors and restaurants. Hospitals.\"\n\n\"You've seen a lot of good places in your travels, haven't you? Seems like you haven't shared some stories with me!\" She smiled wanly.\n\n\"And the parliament . . . sometimes I can hardly believe that so many people could come together in one room. And they talk about things, even some of the women. Sometimes they talk about things people in this village don't think about in a lifetime.\"\n\n\"Rahima-jan, what's on your mind? Did something happen in Kabul?\"\n\n\"Lots of things happen in Kabul. It is so different from here.\"\n\nKhala Shaima looked thoughtful. \"Is that a good thing?\"\n\nI looked at her. Anything different from here was a very good thing.\n\n\"But there's something else,\" I said, my heart heavy with worry.\n\n\"There is?\"\n\nI nodded.\n\n\"What is it?\"\n\nI looked away, my eyes starting to tear.\n\n\"I see.\"\n\nI knew she did. Khala Shaima knew me better than anyone else.\n\n\"Well, that's something to think about then.\" She sighed heavily and shook her head.\n\nI'd given it much thought. Thoughts I didn't care to admit.\n\nPeople close to death have little to lose. They can think things, say things, do things that others wouldn't. Khala Shaima and I were both in that position, she because of her health and me because I felt no desire to open my eyes in the morning. A conversation began to take shape between us. A conversation that happened in unspoken words, in false words, in knowing glances. It was difficult to say what we were both thinking but it was something to be explored.\n\nBecause, as Khala Shaima had so often said, everyone needs an escape.\nCHAPTER 58\n\nShekiba and Gulnaz kept house together and endured Aasif's outbreaks, episodes where his frustration got the best of him. He ranted, berated and slapped. He threw things, twice breaking windowpanes. The cost of replacing them sent him into a new rage.\n\nThe tension drew the two women together. They shared a husband, they shared blame, they shared punishment. They bickered as well. Shekiba hated Gulnaz's lofty attitude and her bland cooking. Gulnaz thought Shekiba dull and plain, a lousy conversationalist. But they made the situation work. Shekiba added spices when Gulnaz turned away and Gulnaz talked enough to make up for her husband's boring second wife.\n\nThere were a few months of nervous respite when Gulnaz's belly began to swell. She told Shekiba when she realized that she had not bled for two months. They wondered on the possibilities until Gulnaz began throwing up once every four days. Shekiba confirmed that these were the signs that a child was growing, as she had learned in the harem. Nothing was said to Aasif, since it was improper to discuss such delicate matters with men, but when he noticed her belly protruding, he smiled with satisfaction and entered Shekiba's room after dark with renewed zeal.\n\nAasif came home and shared meals with his wives. They had taken to eating together, the three of them, from time to time. Shekiba was cautious not to join them too often, knowing now that Gulnaz was carrying Aasif's child he would see Shekiba as an even bigger failure than before.\n\nBut Aasif was busy anticipating the birth of his first child. Aasif's family was quieted, their whisperings that he should take a third wife silenced temporarily. Gulnaz and Shekiba knew that he had been debating the idea but simply could not afford a third wedding and another mouth to feed.\n\nRamadan came and went. Gulnaz, excused from fasting, glowed with satisfaction as her belly grew large, her cheeks fattened and her breathing grew loud. She huffed walking from the living room to the kitchen. Shekiba had seen many women with child but none had looked as uncomfortable as Gulnaz. It was hard not to notice that she only panted and sighed when she knew Shekiba was around to hear it.\n\nWhen the pains came, Shekiba hurriedly walked the four blocks to summon the midwife. Gulnaz bit her lip and twisted in agony, her triumphant grin gone for now. Aasif came home and, hearing the midwife coaching Gulnaz through her moans, left again. Hours passed.\n\nThe baby finally came, just before Aasif nervously returned to a silent house. The midwife smiled politely and congratulated him as she wrapped her shawl around her shoulders and headed out the front gate. Aasif nodded and walked into Gulnaz's bedroom. Shekiba pretended not to hear him enter and kept her head over the stove, pouring flour into the hot oil and stirring as it thickened. Litti, the hot flour soup with sugar and walnuts, would help Shahnaz's womb heal and make her milk come in. Shekiba waited.\n\n\"After all that? A girl? How can this be?\"\n\nGulnaz mumbled something that Shekiba could not make out.\n\n\"Is there no end to my humiliation?\" he shouted. The baby began to cry.\n\nEven a newborn can tell she is not wanted, thought Shekiba. Aasif walked into the living room and yelled for Shekiba to fix him something to eat.\n\n\"And it better be hot,\" he shouted. \"I've had enough disappointment for today.\"\n\nHe fell asleep in the living room, his snores echoing down the hallway. Shekiba tiptoed into Gulnaz's room. She was lying on her side, trying awkwardly to get her daughter to nurse. Shekiba sat her up and showed her how to tuck the baby under her swollen bosom. Small pink lips slowly opened and pulled together, her mouth closing in on Gulnaz's nipple.\n\nShekiba noticed the funny look Gulnaz was giving her.\n\n\"I guarded a house full of women and children. I've helped with plenty of newborns.\"\n\n\"Well, I haven't. If only my mother were alive. It would be different then.\"\n\nShekiba sighed. If only my mother were alive.\n\n\"What will you name her?\"\n\n\"Shabnam.\" Morning dewdrops.\n\n\"Beautiful. I made you litti. You are zacha now. Warm foods will heal your body.\"\n\nWarm and cool foods had nothing to do with temperature but everything to do with a mysterious inherent property of the food. Walnuts and dates were warm. Vinegar and oranges were cold. Joint aches and childbirth made the body cold and were treated with a diet of warm foods.\n\nGulnaz took the bowl readily. The hours of straining had left her pale, exhausted and ravenous. She spooned the hot soup into her mouth, pausing just once to look up at Shekiba with gratitude.\n\n\"I am glad you are here, Shekiba.\"\n\nShekiba froze. It was not like Gulnaz to make such a statement and it made Shekiba fidget. She picked up the baby instead of responding.\n\n\"I thought it was going to be a boy. We waited for so long. And in the end, God gave me a girl.\"\n\n\"Aasif is upset.\"\n\n\"He says it's my fault. He didn't want to hold her. He was too upset.\"\n\n\"You will have another. You had one baby. The door is open now. God will give you another.\"\n\n\"Maybe. He wanted to name her Benafsha.\"\n\nShekiba looked up in surprise. Gulnaz's face was calm.\n\n\"Think of that. To name my daughter Benafsha. He's mad.\"\n\n\"What did you say?\"\n\n\"I told him I've never put up a fight before but there was no way I could call my daughter by that name.\"\n\n\"And?\"\n\nGulnaz's face twisted with pain. Shekiba instinctively put a hand on her shoulder and leaned toward her.\n\n\"What is it?\"\n\n\"She warned me it would be painful.\"\n\n\"What would?\"\n\n\"It's my womb. The midwife said my womb would be angry and looking for the baby that used to live in it.\"\n\n\"It is angry?\"\n\n\"It must be. Oh . . .\" Gulnaz moaned.\n\nThe spasm passed after a moment and Gulnaz remembered their conversation.\n\n\"He wasn't happy. He stormed out. He said Benafsha would be a fine name for a girl, but I think he knows it's wrong.\"\n\nAnd if word got back to the palace, it could cast suspicion on him, Shekiba thought. She smiled to think of Aasif not getting his way.\n\n\"I'm going to wash her up some more. She still has blood in her hair.\"\n\nGulnaz gave a weak smile and closed her eyes, thankful for a moment's rest.\n\nShabnam's first year passed with two mothers. Gulnaz and Shekiba took turns bathing her, feeding her and rocking her to sleep. Shekiba held her head while Gulnaz lined her eyes with kohl and again a month later when she shaved her head to make her hair grow in thicker. Shekiba served tea and nuts when Aasif's family came to visit, days that reminded both wives how fortunate they were not to be living at the Baraan family compound. Aasif's mother made no attempt to hide her repugnance for Shekiba. She had been first to suggest her son take on a second wife, since his first appeared to have been defective, but this deformed creature with another barren womb was not at all what she had in mind.\n\nShe held her granddaughter but kept her eyes roaming around the living room, looking for evidence that her son's home was not being kept well by his two wives. She had a talent for masking criticisms with compliments.\n\n\"The colors of your carpet finally show! Looks like someone took the time to beat the dust from it, eh? How long had it been? I had to wash my dress last time I went home from here.\"\n\nNeither Shekiba nor Gulnaz replied to her comment. It would only feed the flames.\n\n\"Gulnaz-jan, those cookies that you sent over, they were delicious! How lovely that you've finally started baking sweets!\"\n\n\"I cannot take the credit for Shekiba-jan's hard work. She made the rosewater cookies and sent them over for you,\" Gulnaz said, pretending to ignore the snide comment.\n\n\"Oh, well, I wondered how it was possible that after this much time you would have started to treat your husband's palate to something tasty. Shekiba-jan, they were better than the cookies Khanum Ferdowz makes every year for her family and neighbors.\"\n\n\"Noosh-e-jan, Khala-jan,\" Shekiba said quietly as she refilled her mother-in-law's teacup. \"Please help yourself to another.\"\n\n\"Maybe I will. It's not often that my aroos makes such goodies.\" She shook her skirt, a shower of crumbs raining down on the newly cleaned carpet.\n\n\"Who knows, Madar-jan, maybe it's just not often that we get to taste them,\" Parisa said, laughing. Parisa was Aasif's eldest sister. She often accompanied her mother on visits, leaving her four children at home as she joined her mother's social circuit.\n\nAasif's mother smiled at Parisa's comment. Her lips curled up at the corners and the dark hairs on her upper lip cast a shadow. Shekiba opened the teapot and, although it was still full, headed back into the kitchen to refill it.\n\nGulnaz and Shekiba breathed a sigh of relief when Aasif's mother and sister finally left. Shekiba beat the cookie crumbs from the carpet and tossed the larger pieces into the cage for the canaries. They chirped and tweeted with excitement, watching Shekiba as they flitted from one side of the cage to the other.\n\nTwo had bald spots where the aggressive one had pecked their feathers away. Still, they looked content. They watched Shekiba cautiously, occasionally hopping a few inches closer to her for a better look. She reached her finger through the wires and wiggled it. All three birds retreated to the opposite side of the cage immediately, horrified that she would dare trespass into their home.\n\nShekiba withdrew her finger and watched their wings relax, their syncopated chirping less alarmed.\nCHAPTER 59\n\nShekiba did not have to guess. Though she recognized the signs, pregnancy was no less of a shock to her. She chewed on a piece of raw ginger and tried to ignore the nauseous rumblings in her stomach.\n\nI will be a mother. I will have my own baby. Is this possible?\n\nIt meant a permanent break from her previous life. She could no longer float between genders like a kite carried by the wind. No more binding her bosom to disguise her figure. She would fool no one.\n\nShe watched Shabnam pull on her mother's sleeve and try to pull herself up. She had learned to crawl only one month ago and had already tired of it. Shabnam was a beautiful girl. She had dark curly locks and lashes on her pleasantly plump face. Her loveliness softened her father's disappointment. But Aasif only smiled at her when he thought no one was looking. He let her crawl onto his lap and paw at his face until he heard footsteps.\n\n\"Come and get your daughter! She's driving me mad!\" he would call out.\n\n\"Shabnam, come and leave your father alone,\" Gulnaz would say as she swooped the smiling baby off her father's lap.\n\nShekiba had seen him caress her cheek, the corners of his mouth turning up in a quiet smile as he watched her slap her palms together clumsily. He laughed at the way she rolled around on her back, her feet in her hands.\n\n\"But he'll always resent her,\" Gulnaz said with a sigh.\n\n\"That's how it is for girls. A daughter doesn't really belong to her parents. A daughter belongs to others,\" Shekiba explained. Gulnaz should have been wiser in such matters, Shekiba thought.\n\nShe tried to hide her condition from Gulnaz, thinking her husband's wife might be envious. Shekiba dallied in the washroom until the waves of nausea had passed and her stomach had emptied itself. She knocked basins over to mask the sound of her retching. Gulnaz was so preoccupied with Shabnam, Shekiba needn't have worried so much.\n\nAasif did not notice either. After Shabnam's birth, disappointment temporarily cooled his fire. He opened Shekiba's door less often and she was thankful for the reprieve. There was nothing about his sweaty grunting that appealed to her and she hated the way he pressed her face to the side, as if her disfigurement might spoil his momentum even in the darkness. But after three months, he had a renewed determination. Shekiba could hope only for her monthly bleeding to save her from her wifely duties.\n\nWith her queasy stomach, Aasif's visits were even more repulsing. He suddenly had an odor that made her stomach reel. She would hold her breath for as long as she could, taking deep gasps in between, which her husband mistook for pleasure. He paused and looked at her, surprised.\n\n\"So, you're enjoying this, aren't you? Such a performance you put on!\" he said with a crooked smirk.\n\nHe did not notice her belly growing until she had missed six cycles of her bleeding. He looked at her curiously as she leaned against the wall to rest after dinner. Gulnaz was knitting while Shabnam slept beside her. Shekiba instinctively tried to bunch her dress over her growing abdomen. Aasif's eyes zeroed in on her belly.\n\n\"Maybe there is hope for this house after all!\"\n\nShekiba's face reddened. Gulnaz's lips tightened, just enough that Shekiba could see the tension in her face. Gulnaz had confronted Shekiba two months ago, having noticed the way she kept Shabnam's kicking legs away from her belly.\n\nWhen Shekiba had nodded, Gulnaz smiled, but with hesitation. She knew what it would mean if Shekiba delivered the son Aasif so desired.\n\nAasif let out a guffaw, an awkward sound in a room with air so thick.\n\n\"We'll see what Shekiba can do.\"\n\nGulnaz had whispered to Shekiba as she scrubbed the pots clean.\n\n\"He's so different from a couple of years ago. Can you imagine that he used to like to take walks in the evenings with me? This same man! The last two years have soured him. I don't know what he'll become if he's handed another daughter. There's nothing you can do now, is there?\"\n\nShekiba lay awake at night pondering that very question. She thought back to all the mechanisms Mahbuba had described but it was too late for any of them. Someone had told her about the powers of chicken livers, she remembered, and headed to the market the very next day to buy as many as she could find. She did not miss a single prayer and whispered to the ceiling, her palms open, with a fresh desperation.\n\nPlease, merciful Allah, I am begging you to give Aasif the son he so desires. Satisfy his wish so that we may live in peace with this bitter man.\n\nWhether it was the chicken livers or the prayers or just God's will, Shekiba gave birth to a son.\n\nAasif walked with his head high, a smug smile on his face as his family came to visit. Shekiba hardly noticed him. She was fascinated with the ten fingers, the perfectly formed pink lips and the tiny chin that nuzzled against her bosom. She had checked him over head to toe but there was nothing wrong\u2014nothing about him was marred or mangled.\n\n\"His name will be Shah. My son, a king among men! And a real one! Not like the coward we bow to now!\" Aasif had chosen a name. Shekiba could see the spite in his choice. When he mentioned Habibullah his jaw clenched in a way that made Shekiba shudder. She fretted as she stirred the litti. Gulnaz had tried to make some but had filled the house with thick smoke instead. Gray soot clung to the once-white ceiling.\n\nShekiba was not pleased with her son's name. She had secretly hoped to name him Ismail, after her father, but she knew she would not be as successful as Gulnaz in this battle. So his name was Shah and on the sixth day, they celebrated his birth with a prayer and halwa.\n\nAs the days passed, Shekiba became terrified. There were too many pats on the back, heartfelt embraces of congratulations, baskets of sweets sent to their house. She worried about nazar, that their good fortune would be cursed by someone with a jealous eye. Her king sleeping peacefully, she fired the espand seeds and wafted their protective powers over him.\n\nNazar was not the only danger. Shekiba remembered what she had seen Dr. Behrowen doing in the palace and boiled everything that came near the baby. She boiled his clothes, even the evil eye that she had pinned to his tiny blanket. She scrubbed her breasts raw before she let him nurse. Her fears multiplied when Aasif came home shaking his head.\n\n\"What is it?\" she asked. \"Has something happened?\"\n\nAasif was cordial with her these days, engaging in conversations as his first wife listened bitterly from her room down the hall.\n\n\"It's that damn illness again, sweeping across the villages. Even in Kabul.\"\n\n\"What illness?\" Shekiba asked, suddenly alarmed. Shah was only three weeks old. Instinctively, she pulled her swaddled baby closer to her.\n\n\"Cholera. Maybe you've never heard of it. It's a powerful disease. God help whoever it strikes. I've heard that at least twenty families in Kabul are sick with it. The doctors can't do anything about it.\"\n\nShekiba knew better than anyone else just how powerful cholera could be. Her back stiffened.\n\n\"We mustn't let the baby get ill,\" she said, her voice quivering. Panic was setting in.\n\n\"Don't you think I know that? Just take good care of him and keep him inside. You're his mother so it's up to you to keep him from getting sick!\"\n\nShekiba's mind flew back to her village, watching her siblings waste away in a corner of their rank home. Thinking of her mother, broken at the sight of her dead children, Shekiba boiled, washed and prayed fiercely.\n\nPlease, God, don't let anything happen to my little boy. He's the most perfect thing I've ever had. Please do not take him away!\n\nAnd when the cholera wave passed, there was time for Shekiba to think of new dangers. She would not let the baby near the kitchen and kept him away from anything made of glass. She surrounded him with pillows and did not take her eyes off him. It was clear she did not trust Gulnaz to watch him. What if he broke his leg and walked with a limp? What if he was hit and lost an eye? Shekiba could hear the names, the teasing, a crestfallen little boy. She wanted better for her son.\n\n\"You know, I have managed to care for Shabnam reasonably well this past year. I think I am capable of holding a baby! What is it with you? What do you think I'm going to do? Drop him from a window?\"\n\n\"I'm just . . . I'm just nervous. Don't be offended, please. It's just that I don't want anything to happen to him.\" Shekiba turned away so she wouldn't see the angry look on Gulnaz's face.\n\nShah changed every dynamic in the house, even for his half sister. When Shabnam waddled toward Shekiba, Gulnaz was quick to pull her back, and if she caught Shabnam eating something Shekiba had prepared, she would hold a hand in front of the baby's confused mouth and make her spit it out. But only when Shekiba was watching.\n\nIt hurt Shekiba to see Shabnam yanked away from her. She loved the little girl as much as she could love any child that was not her own. And Shabnam, who had grown up with two mothers, did not understand why one was now off-limits. She looked at Shah with suspicion, as if she knew he had disrupted her happy home.\n\nAasif made the situation worse. Gulnaz no longer joined them for dinner, always making some excuse about Shabnam needing to eat or sleep. Aasif, having just proudly celebrated his son's fortieth day, hardly noticed that his first wife had retreated into her room for over a week. What he did say to Gulnaz only made her more resentful of Shekiba.\n\n\"Long overdue, but worth the wait. Look at my son! Look at the healthy color in his cheeks! He's a lion, my son!\"\n\nGulnaz, listening from her room, bit her tongue, thankful that her daughter was not yet aware of her father's partiality.\n\n\"Nam-e-khoda. May evil eyes stay away,\" Shekiba murmured nervously as she looked at her fingernails, another superstition she had picked up from one of her uncles' wives, though she couldn't remember which one.\n\nGulnaz nearly laughed. An evil eye could hardly find its way to Shah, with all the talismans and prayers and espand that Shekiba used in the house. It occurred to her then that Shekiba was probably worried about her. She thought on it for a moment and realized it made sense. That was why she wanted to keep Gulnaz away from her precious son!\n\nAnd so Gulnaz retaliated. She showered Shah with compliments, purposely not invoking the name of God.\n\nHow plump his cheeks have gotten! How quickly he's learned to roll over! He'll be walking before you know it, Shekiba-jan.\n\nHow well he nurses! He'll grow up to be bigger and stronger than his father! And look at how alert and curious he is!\n\nShekiba was frenzied. She knocked on wood, burned espand and prayed even more. She tried to downplay the compliments as quickly as they came.\n\nOh, it's just today. Yesterday he barely wanted to nurse at all. I don't think he's gained any weight in the last couple weeks. He feels so light when I lift him.\n\nYou don't see how skinny his legs are? He'll probably end up short and bowlegged, at the rate he's eating.\n\nAnimosity simmered as Shekiba slowly realized what Gulnaz was up to. Frustrated, she decided to turn the game around. They sat in the courtyard, giving the children some sunlight while Shekiba hung the laundry on a clothesline. Gulnaz was watering the flowers.\n\n\"Just look at Shabnam! She's walking as if she's been doing it for years! I bet she could run right across Kabul with those strong legs!\"\n\nShekiba watched Gulnaz's mouth open slightly and her eyes widen. She mumbled something incomprehensible in return.\n\n\"Coo coo! Coo coo!\" Shabnam called out, her word for the canaries.\n\n\"Yes, my little one, coo coo is there,\" Gulnaz said without turning around.\n\n\"Coo coo! Coo coo!\"\n\nThe two mothers turned around and saw only two yellow birds flitting about the cage. Gulnaz walked over, her head cocked to the side.\n\n\"Where is the other? How could he have gotten . . .\" Her voice trailed off as she neared the cage. \"Oh no!\"\n\n\"What is it?\" Shekiba said as she walked over. Gulnaz's eyes were wide.\n\n\"He's dead.\"\n\nThe feathered creature lay lifeless on the floor of the cage while his roommates huddled close to one another and chirped softly. Both women were silent. The omen did not go unnoticed.\n\nWe're just like Aasif's mother, Shekiba thought with a sigh. Making daggers of words.\nCHAPTER 60\n\nThe relationship between Shekiba and Gulnaz had cooled now that Aasif had warmed toward his disfigured second wife. Shekiba prayed Gulnaz would have a second child, a boy, to even their score, but months and then years passed and Gulnaz had no other children. They learned to be civil with one another and make the house function as it had when Shekiba had first arrived\u2014two wives embittered against each other.\n\nShah and Shabnam made up for the relationship between their mothers. By the time Shah turned one, he was chasing after his older half sister, who giggled and watched him with a toddler's curiosity. Shabnam was much more beautiful than her mother, her perfect curls in a ponytail behind her head and bangs that shadowed her forehead. Her cheeks were full and rosy, her eyes almond shaped and chestnut colored. She had inherited the best of her parents' features, and a cheerful disposition that was foreign to their home.\n\nShah, as Gulnaz had so facetiously predicted, grew to be strong and taller than most boys his age. He had walnut-colored hair that curled just slightly and a grin that melted hearts. The two made a perfect set of siblings, despite the rancor between their two mothers.\n\nIn February of 1919, Shabnam was five and Shah was four. The temperature was barely above freezing. Hundreds of miles from Kabul, someone brought the country to its knees. Gulnaz and Shekiba were tending to the chores when they noticed that the streets were loud and boisterous. There were people shouting and doors slamming. Shekiba sent the kids into the house from the courtyard and opened the gate. Men were walking down the street in a hurry, consternation on their faces and arms waving wildly as they shouted.\n\n\"No, it's true! My brother is in the army! They have no idea who it was!\"\n\n\"What's going to happen?\"\n\n\"I don't know but it's best to get home and stay there until we find out.\"\n\nShekiba closed the door and leaned against it, the metal sending icy shivers down her spine. What could have happened?\n\nGulnaz met her at the inner door. The two canaries, brought inside during the winter months, tweeted loudly, egged on by the agitation in the streets.\n\n\"What is it? What's happening?\"\n\n\"I'm not sure. I just heard someone saying it's best to stay home. Something is going on.\"\n\n\"Where is Aasif?\"\n\n\"God knows.\"\n\nFour hours later, their husband showed up. The women had locked the doors and closed the windows, fearful without knowing what it was they were afraid of. His face was heavy with worry and his forehead sweaty, even in the cold.\n\n\"Aasif! What is it? What's happening?\" Gulnaz said, meeting him at the door.\n\n\"It's the king. Someone's killed Habibullah!\" he announced, his voice quiet and his breaths heavy. He took off his hat and scarf.\n\n\"Allah!\" Her hand covered her mouth.\n\n\"The city's in a panic. I was at the Ministry of Foreign Affairs when we got word. He was on some kind of hunting trip, as usual, and he was shot. For a while, they were trying to keep it quiet, but stories began to leak out. You can't keep something like that hidden for long! We thought it might have just been rumors\u2014you know how easy it is to spread stories in Kabul\u2014but it seems to be true. The army is on alert and they sent for Amanullah. Thankfully, he's in Kabul already.\"\n\n\"The shah is . . . ,\" Shekiba said incredulously. She could not bring herself to use her son's name and \"dead\" in the same breath.\n\n\"Did you not hear everything I just said!? Yes, Habibullah is dead! He's been assassinated, the bastard.\"\n\nHis wives winced. However Aasif might have felt, it was unwise to speak ill of the dead.\n\n\"How could this have happened? Was it here? In the palace?\"\n\n\"No, he was in Jalalabad. It must have happened two days ago at least, if word is getting to us now. I can't believe someone killed him.\"\n\n\"What's going to happen now?\" Gulnaz said while Shekiba put a hand on her son's head. Shah had just entered the room and looked at his father with concern. He had no idea what \"dead\" meant but he could sense that something was not right.\n\n\"I don't know. My guess is that Amanullah will take his father's place. And he rightfully should. But it's impossible to say. If his assassination was a coup, then his assassin will have to get through the army. They've sworn allegiance to Amanullah.\"\n\n\"Allah, have mercy on us. This could be a disaster for Kabul!\"\n\n\"We'll sit tight and see. Just keep the children inside and keep your mouths shut. This is no time to speculate with the neighbors. Be smart.\"\n\nShekiba turned away so Aasif would not see her roll her eyes. It was hard to swallow such words of wisdom from a man who had violated the king's harem and condemned Benafsha to a horrible death. Where was his sense of caution then?\n\nBut they did as he said and Aasif nervously returned to his post in the Ministry of Agriculture in the morning. The streets were desolate as panic spread through the capital. Aasif stockpiled extra food as a precaution. The assassin was still unidentified and no one had made any moves toward the palace but the army was on high alert, all the same.\n\nAasif had not seen Amanullah in nearly a year, but now it was critical he reconnect with his friend. He needed to pay his respects and make sure he was aligned with the man who would most likely be taking Habibullah's place as ruler of Afghanistan. He stopped by the palace, his nerves on edge.\n\nAmanullah was heartbroken and enraged, Aasif reported to his wives. His father's brother Nasrullah had accompanied the king on his hunting trip. Word came from Jalalabad that Nasrullah had been proclaimed Habibullah's successor, which angered Amanullah. Amanullah's eldest brother, Inayatullah, seemed to be in support of his uncle, as were many of Habibullah's sons.\n\nAmanullah, born to the king's chief wife, knew his father would have selected him to take the throne. And as leader of the military and treasury, Amanullah was in position to assume the reins of the country and declared himself the new king from his post in Kabul.\n\nShekiba could picture him, his heart heavy with grief, his noble face drawn and sad. He would be a just and wise king, she knew. She blushed to think of how stupid she had been five years ago, to think that such a man might want her.\n\nI have no reason to complain, though. I am married to a man with a respectable position in the Ministry of Agriculture. He keeps us fed and clothed in an esteemed neighborhood of Kabul. He provides for his children and does not beat me. What more could I have asked of Allah?\n\nAasif carefully worked to bring himself closer to Amanullah and the new king welcomed his friend's counsel in such a difficult time. He wanted to avenge his father's death and there were a handful of people under the cloud of suspicion, including his own uncle Nasrullah, who, it was rumored, had not shed a single tear at his brother's death. Amanullah made an announcement. He would find the assassins and bring change to Afghanistan. Reform was on the way. He banned slavery. He vowed to increase pay for the army. Afghanistan would maintain its friendly relationship with India.\n\nHe is not like his father. He is a better man, Shekiba thought as she heard the declarations. God be with you, King Amanullah.\n\nBy April, an investigative committee had looked into Habibullah's murder. Amanullah jailed his uncle Nasrullah and a dozen others in the palace dungeons. Aasif stood by his friend as the palace prepared to spill blood.\n\nAmanullah brought with him many new ministers and Kabul braced itself for the changes that would come with their new leader. Shekiba and Gulnaz felt more secure when it became apparent that there would be no bloody challenge to Amanullah's claim to the throne. Kabul transitioned relatively peacefully, eager to see their bold young king fulfill his promises. Shekiba smiled, ruffling her son's hair, feeling that Amanullah would make Afghanistan better for her Shah.\n\nTheir link to the palace revived, the Baraan family became host to some of Amanullah's other advisers. Gulnaz served guests tea and nuts that Shekiba prepared from the safety of the kitchen. They eavesdropped on conversations, feeling privileged to have the first scoop on Kabul's political affairs. Compared to the other wives of the neighborhood, they were much more informed, and Gulnaz, the more social wife, enjoyed flaunting it in conversations with other women. She made sure their audiences knew how well connected their household was. In a city like Kabul, connections counted for everything, so she didn't mind the extra work that came with Aasif's many guests.\n\nGulnaz and Shekiba wished the men would talk more about Amanullah's wife, Soraya. What they did hear was astonishing. She was educated and beautiful. She was born in Syria and spoke many languages. Amanullah took her everywhere and consulted with her. They wanted to hear more about their mysterious queen but the discussions usually centered on what Amanullah's next move would be, since he had promised big changes when he assumed the role of king.\n\n\"How much of Tarzi's reforms do you think he'll take on?\"\n\n\"He'll take on them all, if you ask me!\" Aasif said. \"He thinks the world of his father-in-law, probably even more than he thought of his own father, may Allah grant him peace in heaven.\"\n\nGulnaz shot Shekiba a look of surprise. It seemed Aasif finally knew how to speak respectfully of Habibullah when he needed to.\n\n\"You're as mad as Tarzi himself. This is Afghanistan, not Europe. We are not like those people and shouldn't try to be. Let us concentrate on our own country and stop ogling others.\"\n\n\"What's wrong with learning from others?\" someone asked.\n\n\"Depends on what you learn from them.\"\n\n\"What's happened with his brother Inayatullah?\"\n\n\"He and a few other brothers have sworn allegiance to Amanullah. He's going to release them from the dungeon tomorrow. His uncle will remain in prison. There is too much doubt over his head. He'll stay in chains for the time being.\"\n\n\"People are angry about that. They do not feel it is just.\"\n\n\"They will forget when they see what our king is capable of. Soon, they will not remember Nasrullah's name.\"\n\nIn May, Amanullah did what Aasif had suggested many years earlier while Shekib, the guard, had eavesdropped in the gardens. Amanullah flexed his muscles and sent soldiers into northern India. Amanullah had had enough of British dominance and acted on his father-in-law's teachings.\n\n\"Ya marg ya istiqlal!\" Demonstrators in the streets shouted for death or freedom. Gulnaz and Shekiba listened nervously, hoping the crowd would not turn on anyone.\n\nAmanullah had embroiled the country in the third Anglo-Afghan war. Kabul was tense. Everyone talked about the fighting. The army was small but fierce. The Baraan household braced itself. If the Afghans lost, there would surely be another regime change and it was impossible to know what that would bring.\n\n\"It's over,\" Aasif announced as he entered the house three months later.\n\n\"It's over?\" Shekiba repeated, a habit that drove Aasif mad. She knew it as soon as she said it but it was too late to undo. Shah ran into the living room to greet his father.\n\n\"Yes, that's what I said! Let me see my son! Shah, good news! It's over. We've won our independence from England!\"\nCHAPTER 61\n\nForty days after Jahangir's last breath, the house was still. It was the final day of mourning.\n\n\"The forty days will be complete today,\" Bibi Gulalai reminded us. \"People may come to say prayers with us or with Abdul Khaliq. Watch how you talk.\"\n\nShahnaz bit her lip and went to bathe her children. She kept her distance and, more important, made sure her children stayed clear of me. I made her nervous, as the mother of a dead child. Maybe I was cursed. Or maybe I would be jealous that her little ones were alive and my son was dead.\n\nForty days. What was so magical about forty? I wondered. Was I to feel differently today than I did yesterday? Was I to forget what happened just six weeks ago?\n\nWe Afghans marked both life and death with a forty-day period, as if we needed that much time to confirm either had truly happened. We had celebrated Jahangir's birth forty days after he'd left my womb, unsure if this child was here with us to stay. And now his death. Forty days of praying, alone, with others and everything in between.\n\n\"It's been forty days, Rahima,\" Badriya reminded me.\n\n\"And tomorrow will be forty-one,\" I shot back. Nothing would change.\n\nBut something did change. For forty days, Abdul Khaliq had kept to himself, sitting with the many men who came to pay their respects and read with him. He didn't look at me much. If we had been a different husband and wife, I might have approached him. I might have asked him about our son's last breaths, about how he was feeling now. I was thankful that he'd been good to our son in his last moments but nothing more. Now, more than ever, I wanted nothing to do with him.\n\nOn the forty-first day, the house breathed a sigh of relief. Badriya and her children no longer spoke in hushed voices. Jahangir had been given his due period of respect.\n\nAbdul Khaliq called for me that night. With a heavy step, I went to him. He was standing by the window, his back to me. I knew I should have closed the door behind me, but I did not. I hoped I wouldn't have to stay.\n\n\"Close the door,\" he said, his back still turned. His voice was firm, a warning buried in his tone.\n\nI obeyed.\n\n\"Come closer.\"\n\nI wanted to scream. I wanted to run far from him, from the scent that lingered behind his beard, from his rough hands, from the disdain in his eye.\n\nHaven't I suffered enough? I wanted to yell.\n\nHe turned around and looked at me, reading the reluctance in my face. He took another step, now within reach. I sighed and turned my head away, staring at the floor.\n\nA slap thundered across my cheek. My knees buckled.\n\n\"No wife of mine looks at me in that way! How dare you?\"\n\nMy eyes watered from the stinging blow. He was angry still. His fingers gripped my arm so tight I thought my bones might snap.\n\n\"I didn't\u2014I'm sorry, I didn't mean to . . .\"\n\nHe tossed me to the floor. My right knee hit the ground first.\n\n\"Worthless! You've been good for nothing since you came here! A waste. A waste of my money, my time. Look at you! A big mistake for me to take you. I should have listened to what others said but I pitied your father. He suckered me, that rat! Made me believe his girls would make decent wives. Look what's happened! One worse than the other.\"\n\nHe was in a rage. Nothing he hadn't before said or done, but there was a renewed enthusiasm in his vitriol. He swung again as I pulled on the edge of the bed to stand.\n\n\"A bacha posh. I should have known better. You still don't know what it is to be a woman.\"\n\nI felt a trickle of blood from my lip and realized I should have anticipated this. I steeled myself for what I knew was coming. The blow that would shatter me. True or untrue, I didn't want to hear him say it.\n\n\"Hard to believe you could be even worse as a mother than you are as a wife! My son deserved better! He would be alive if he'd had a mother better than you!\"\n\nI closed my eyes, a surge of pain. The worst blow. I crumpled to the floor with my hands over my head. I crouched forward, almost as if praying. He was muttering something. I couldn't hear him over my own sobbing.\n\n\"Do you want to be a boy? Maybe that's what you want! Is that what you want?\"\n\nMy ribs.\n\n\"My mother couldn't make a woman out of you. Then maybe you should go back to what you were! That's what you want?\"\n\nI never saw where it came from. Maybe under his pillow. Or maybe in his jacket pocket. In a flash, Abdul Khaliq grabbed my hair and pulled my head off the ground. My head slipped forward. He snatched again and jerked my head up. My scalp screamed. When I saw locks of hair on the floor around me, I realized what he was doing. I tried to pull away, begged him to stop, but he was barely there. He was trying to take me apart, to disassemble the pieces that were hardly holding together as it was.\n\nMore hair on the ground. I tried to crawl away but his grip was tight. I shrieked as I felt my scalp lift off my skull.\n\n\"Please,\" I begged. \"Please stop! You don't know!\"\n\nHe had taken a knife to my hair, a blade I'd seen him tuck into his waistband before he and his guards went off for his meetings. The blade was dull and he had to chop at my hair again and again, holding it taut by the ends.\n\n\"One child! You've brought only one child and you couldn't even take care of him!\" My stomach lurched.\n\nOne child. One child.\n\nI wanted to let him end my misery, to give me the punishment my heart believed I deserved, those dark, dark thoughts that haunted my days and nights. I wished he could end it for me. Maybe I would have even taunted him, if it weren't for . . .\n\nHe was on the edge of his bed, his breaths slowing. My husband lacked the endurance to exact the punishment he intended.\n\nI lay motionless, curled up on my side at the foot of the bed. I waited for the signal.\n\n\"Get out,\" he hissed. \"I can't stand the sight of you.\"\n\nI crawled to the door, then pulled myself to stand by the chair. I heard footsteps scamper in the hallway as I exited. I held one hand over my throbbing belly and one hand on the wall to steady my slow step.\n\nOne child.\n\nIn my room, I waited. It didn't hurt as much as it should have, maybe because my mind was elsewhere. Into the thin light of morning, I waited for the bleeding to come. I knew it would.\n\nFresh tears for a new loss.\n\nI may have killed one of Abdul Khaliq's children. But he had just killed another.\nCHAPTER 62\n\n\"Do you want to go or don't you?\"\n\nI sighed and stared at my feet. My arches ached but it was too much effort to rub them.\n\n\"It's up to you. I can always find someone else to be my assistant if you don't want to do it anymore. I'm sure the director's office can help me. Someone else can do what you were doing.\"\n\nThis was actually her way of trying to be considerate.\n\n\"Look, I don't care either way . . .\"\n\nThis wasn't true and we both knew it.\n\n\"I'm just telling you, you need to make a decision soon because I'm leaving to go back to Kabul in three days and if you're going to go back then we have to let Abdul Khaliq know.\"\n\nBadriya had grown accustomed to my help. With me, the parliamentary sessions were easier to follow. I read all the briefs to her. I filled out and submitted all her documents. She listened as I went through the newspaper headlines to give her some background for the jirga discussions. She finally felt like she was participating in the process, like she was a woman our province should admire for her role in government. As if she were actually serving her constituency.\n\nShe was ignoring the fact that it was another man who decided if she should raise the red or green paddle when voting time came. She believed the lie of Badriya the female parliamentarian, and that was all that mattered to her.\n\nAs much as I wanted her to shut up and walk away, I knew I had to make a decision.\n\nAn escape. I need to find an escape.\n\nI'd been to the cemetery where Jahangir was buried only once, two months after I'd come home from Kabul to find my son cold and gray. Abdul Khaliq finally gave me permission to go with Bibi Gulalai and his driver. The dead can see people naked, superstition said, so he didn't think it was proper for his wife to step foot in the cemetery. I didn't believe it to be true and even if it were, I didn't care. I wanted to see where my son was buried. I asked Jameela to bring it up with him and she did. I knew I was playing on her sympathy when I asked her the favor but I was desperate. I don't know what magical words she used but our husband relented.\n\nMy mother-in-law and I stood over Jahangir's grave marker. Her wailing echoed across the emptiness, the same mournful cries that she'd made two months ago. I was quiet for a time. I didn't think I had tears left to shed.\n\n\"You sweet innocent child! I can't believe this was your time, your naseeb. Dear God, my poor grandson was so young to be taken from us!\"\n\nI stood there in disbelief. How could this mound of earth be my little boy? How could this be all that remained of my grinning, curious child?\n\nBut it was. And the more I thought of it, the more Bibi Gulalai's wailing tore at my heart. I wanted to dig into the dirt, to plunge my hands into the earth and touch my son's hand, to feel his fingers close around mine again. I wanted to curl up beside him, keep him warm and whisper to him that he wasn't alone, that he shouldn't be afraid.\n\n\"What is our family to do? Why did we deserve such tragedy? His smiling face, oh, it dances before my eyes and rips at my heart!\"\n\nI started to cry. Silently at first, then louder and louder until I was loud enough that Bibi Gulalai noticed over the sound of her own lamenting.\n\nShe turned around and shot me an icy glare.\n\n\"Haven't I told you a hundred times to watch how you act? Are you trying to shame our family?\"\n\nI sucked in my sobs, feeling my chest tense as I tried to contain it all.\n\n\"It's a sin! It's a sin for you to try to draw so much attention. Don't make such a scene here. It's disrespectful to the dead and people are watching!\"\n\nNo one was watching. We were all alone. Maroof stood back, leaning against the SUV and waiting for us to return to the car. I swallowed my sorrow and looked to the sky. Three gray-brown red-breasted finches flew overhead. They circled three times, swooped down toward us, then floated back to a tree about forty feet away. They cooed and clucked and cocked their heads so purposefully that I almost thought they were talking to me.\n\nBibi Gulalai pulled a handful of bread crumbs from her dress pocket and scattered them over Jahangir's grave. She tossed another palmful on a grave to the left, skipped one and tossed some more on a grave to the right.\n\n\"Shehr-Agha-jan,\" she said with a sigh. \"May the heavens be your place for eternity.\"\n\nI recognized the name as belonging to Abdul Khaliq's grandfather. Stories about him, the great warrior, were recounted so often that I had to remind myself I'd never seen him. He'd been gone over a decade.\n\nThe finches noticed the sprinkle of food and took flight again, swooping in gracefully and pecking here and there at the newly found bounty. Bibi Gulalai spread what was left on the graves that fell further away. Still she skipped the one to the immediate right of Jahangir.\n\n\"Eat, eat,\" she said mournfully. \"Eat and pray for my grandson. And for my beloved father-in-law. God rest his soul and may Allah keep him close and peaceful always.\"\n\nI watched. The finches bobbed their heads, picking at the crumbs and chirping their gratitude. It did look like they were praying, their little heads going up and down as if in supplication. It gave me some solace.\n\nI looked at the grave marker beside Abdul Khaliq's grandfather, Shehr-Agha. This area was where everyone from my husband's family was buried. I wondered why Bibi Gulalai chose to ignore this one grave.\n\n\"Who is buried here?\" I asked. I usually didn't invite any conversation with my mother-in-law but at this moment I didn't want to feel so alone. At least she had brought the praying finches to my son. Jahangir would have loved to see the birds, their tiny beaks. I could picture him imitating their delicate walk, their flapping wings and their red chests puffed out proudly.\n\n\"There?\" She pointed hatefully. \"That's where Abdul Khaliq's grandmother is buried, Shehr-Agha's wife. My mother-in-law.\"\n\nHer lips drew tightly together.\n\n\"You didn't toss any seeds there.\"\n\nBibi Gulalai stared at the earth angrily. She spoke after a moment of thought.\n\n\"Abdul Khaliq's grandmother and I did not see eye to eye. She was an awful woman. No one liked her,\" she explained without looking at me. \"I was respectful to her while she was alive but I've no interest in wasting my time praying for her soul now.\"\n\nThis was the first time I'd heard Bibi Gulalai talk about her mother-in-law. And it was the first time I heard her speak ill of anyone from her in-laws' side of the family. I was surprised at how spiteful she was. I shouldn't have been.\n\n\"When did she die?\"\n\n\"Ten years ago,\" she said, and signaled to Maroof that we were ready to leave. He opened the rear door and turned back to the front to get behind the wheel. \"She was an evil woman if there ever was one. She told my husband terrible things about me. None of them true, mind you, just poisoning his mind against me.\"\n\nI closed my eyes, knelt at my son's grave and said one more prayer, rushing through the verse so fast I slurred the Arabic words in my head, afraid that I would get pulled into the car before I could finish. But Bibi Gulalai paused\u2014as if she were waiting for me.\n\nI lowered my head and kissed the earth, the finches chirping sympathetically and watching me from the safety of their perch.\n\n\"I'm sorry, Jahangir,\" I whispered, my cheek cooled by the feet of dirt between my son and me. \"I'm sorry I didn't take better care of you. May Allah watch over you always.\"\n\nI stood and took a deep breath, my eyes blurred with tears. We got into the car and I realized Bibi Gulalai was still thinking, unaffectionately, about her mother-in-law.\n\n\"She made my life miserable,\" she finally said. \"I did everything for that woman. Cooked and cleaned and took care of her son like no other wife would have. I cooked for their whole family, any time she had guests, when she was struck with a craving. But nothing was ever to her liking. She bad-mouthed me every chance she got.\"\n\nI listened, seeing a different side of my mother-in-law. And feeling, for the first time, that she and I had something in common. Ironically.\n\n\"What happened to her?\"\n\n\"What happened to her? What happens to everyone! She died.\" Her tone was sarcastic and annoyed. \"She wasn't feeling well one night. Asked me to rub her legs for her, so I did. Greased her dry feet and massaged them so long I thought my hands would never open again. The next morning, she came to check on the soup I was making. Shehr-Agha-jan, God rest his soul, he had invited thirty people over for lunch. She stood there, looming over my shoulder like a jailer watches his inmates, complaining that I was taking too long or something like that. She didn't look right though. I remember like it was yesterday. She was a pale yellow color and her forehead was sweaty. I thought it was odd because it was the middle of winter. Before I could say anything, she grabbed my arm and her neck twisted to the side. She fell to the ground and knocked over a bowl of onions I'd just finished peeling for the stew.\"\n\nI watched her recount the story. She was looking out the window, the cars' tires spinning up clouds of dust that obscured the view. It was as if she wasn't talking to me, just reliving the memory.\n\n\"I had to get everyone, let everyone know. What a day. But that's how she died\u2014unappreciative of what I was doing up until her very last breath. That's the kind of hard-hearted woman she was.\"\n\nIn other circumstances, I might have told Bibi Gulalai that I understood, that I could sympathize with her.\n\n\"You don't know how lucky you are,\" she said, suddenly remembering I was sitting beside her.\n\nThat was my only visit to my son's grave. I knew Abdul Khaliq opposed my going there. Truthfully, I wasn't even sure I was strong enough to go back. It wasn't easy. I lay awake all that night and the next wondering if Jahangir felt like he was suffocating in there. Shahnaz heard my crying through the thin walls and groaned in frustration. I couldn't get my mind off my little boy.\n\nWhen Badriya came to me again to ask if I wanted to go back to Kabul, I thought about it and made a decision that I thought Khala Shaima would approve of. I packed my bag, my heart heavy with guilt for leaving my son behind again.\n\nI thought of the cemetery, the rows of headstones, simple and hand carved. Some old, some new. The finches had watched us until we left. I saw them chirping to one another as we drove off and then, one by one, the birds had flown away.\nCHAPTER 63\n\nIt wasn't easy to stay focused on the work this time. Halfway into a parliamentarian's speech, I would realize I had no idea what he was talking about. My mind had drifted, remembering the last time I'd bathed my son. Or fed him halva, his favorite food.\n\nBadriya noticed but her exasperation was tempered by sympathy. Most of the time. She was hardly paying attention herself. She spent most of the session pretending to look at papers in front of her when I could see she was watching the people in the room. For a woman who had spent the greater part of her life confined by the walls of her husband's house, every session was a spectacle.\n\nShe was even more lax with me than before, which didn't mean much except that I spent more time with Hamida and Sufia and less time with her or our security guard and driver. The ladies were kind to me. When Badriya had returned to Kabul without me, they'd asked about me several times. She'd made vague excuses until she finally told them about Jahangir.\n\nSufia's arms around me were more comforting than I could have imagined. Hamida shook her head and told me of the three-year-old son she'd lost to some infection. She and her husband hadn't had the money at the time to pay for medications.\n\nI forced a smile and nodded, appreciative of their warmth but not wanting to talk about what had happened. There was too much there and I still felt a fresh guilt for leaving my dead son behind.\n\nThe home Abdul Khaliq was fixing had not yet been finished, so we continued to stay at the hotel. I floated through my daily routine in a perpetual state of misery, wondering from time to time why I bothered to do any of it at all. I think I was driven by fear of my husband. And because I didn't know what else to do.\n\nBadriya was dropping hints here and there about our husband's new prospect. As much as she didn't want to talk to me, there was no one else around and there were things that she could not bottle up.\n\n\"I'm not supposed to say anything. I only know because, of course, he thought it was right to share this information with me since I'm the first wife,\" she said with one hand over her chest as she spoke of her own importance. \"The girl's name is Khatol. She's very beautiful, they say. And Abdul Khaliq has known her brother for a long time. Her brother is a well-respected man. He fought alongside Abdul Khaliq but now he owes a lot of money to our husband. He showed him and his family much kindness. Even sent them food when he heard they didn't even have bread.\"\n\n\"But what will happen to . . . to the rest of us?\" I didn't want Badriya to know that I'd heard her conversation with Bibi Gulalai.\n\n\"The rest of us? Nothing! Why should anything happen to the rest of us?\" she said, and busied herself cleaning a grease spot from her dress. \"Aren't you going to that silly class with your friends?\"\n\nShe wouldn't say anything more than that, nothing about my husband's plans to keep in line with the hadith. It wasn't in her interests to alert me.\n\nI didn't understand why my husband suddenly found it so important that he follow the hadith. He wasn't a man who let rules dictate his decisions. If he wanted to have five wives, or twenty-five for that matter, he would.\n\nThick, industrial smoke from a million exhaust pipes blackened Kabul's air. Badriya coughed violently. I would ask, only because she would bring it up later if I didn't, if she wanted to join the ladies and me at the resource center. Each time she would wave me off.\n\n\"I'm not wasting my time with those busybodies.\"\n\nMaroof and our bodyguard stayed with her because she was the more important wife and because she always claimed to be considering going to visit her cousin across town. As far as I could tell, she never actually left our room. She knew better. She knew word would get back to our husband. Badriya's survival instincts were strong.\n\nI spent my evenings in the training center under Ms. Franklin's tutelage. I was getting better at navigating my way through the computer programs. For practice, I would type letters to my sisters Shahla, Rohila, and Sitara\u2014letters that were never sent. The woman from the shelter, Fakhria, came from time to time and brought with her stories of girls who had fled from home, hungry for a new chance. Their shelter functioned on money raised in the United States and it was becoming obvious that she was trying to garner Hamida and Sufia's sympathy, hoping to secure some funding from the parliament. I wanted to tell her that she was wasting her breath. Even I, the lowly assistant to a parliamentarian, could have told her there was no chance of getting the jirga to allocate money to a shelter for women who had run away from their husbands. In fact, I'd heard several people say the shelters were nothing more than brothels. I didn't think it was true, but others did.\n\nFour weeks remained until the parliament's winter break. Four weeks left for me to attend class at the training center, four weeks of Ms. Franklin patting my shoulder in praise, four weeks left of Hamida and Sufia, instead of cooking and cleaning.\n\nI wondered how Khala Shaima was doing. She looked worse each time I saw her. Still, she had outlived both Parwin and Jahangir. Their deaths had taught me that anything was possible, and that death was closer than I wanted to believe.\n\n\"I'm an old woman,\" Khala Shaima had told me before I left for Kabul. \"I've cheated the angel Azrael more than once but he'll come and claim my last breath soon enough.\"\n\n\"Khala-jan, don't say such a thing,\" I said, protesting.\n\n\"Bah. I've wanted to be around only so I can look after you girls, to tell you the truth. Nothing else matters much. But I can't slip through his fingers forever. It's like the story of that man\u2014did I tell you that one?\"\n\n\"No, Khala-jan. You've only told us about Bibi Shekiba.\"\n\n\"Ah, and I hope you've learned something from her story. You are her legacy, after all. Remember, your great-great-grandmother was Bibi Shekiba, guard to the king's harem.\n\n\"Dokhtar-em, my dear, I'm not well. You are not a na\u00efve girl anymore. It will give my heart peace if you can tell me that every story I've told, every mattal I've shared, that you've gotten some wisdom and courage from it. Remember where you come from. Bibi Shekiba is not a fairy tale. She is your great-great-grandmother. Her blood courses through your veins and gives strength to your spirit. Always walk with your head high. You are the descendant of a somebody, not a nobody.\" She sighed heavily, which turned into a long, exasperated cough. She took a minute to catch her breath before she continued.\n\n\"I've tried to tell Rohila and Sitara the same. But Rohila is to be married soon and I think she'll be better off. The family seems reasonable. Sitara will be alone with your parents, left to fend for herself. I can't do much more for her. I wish I could tell you to watch out for her but you could do more for her if a mountain stood between you. These walls hold you tight. Focus on yourself. Everything you've endured in life should have taught you something, made you hungry for something. Remember, Allah has said, 'Start moving, so I may start blessing.' \"\n\nI tried to find the words to reassure Khala Shaima, to tell her that I understood what she was telling me and that I was proud to know I was a descendant of Bibi Shekiba, the woman who had guarded the king's harem, who had walked through the royal palace. I may have lived my entire life in a small village but I was connected to Afghanistan's aristocracy.\n\nBut I'd never been able to find the right words. As I sat there, I had to admit I could see my aunt fading. She didn't look like the person I remembered. She had spent her adult life trying to guide us, trying to look out for my sisters and me.\n\nAnd she was right. As much as I might have wanted to do for my sisters, Abdul Khaliq's walls were high and his leash short. I could only pray for them.\n\nBadriya was lying on the bed. She'd spent the day griping about how long it was taking for Abdul Khaliq's men to finish the home he'd bought in Kabul. She was tired of staying in a hotel and having the man in the lobby watch our comings and goings with interest. I wanted to go for a walk, tired of listening to her complaints.\n\nI adjusted my head scarf and opened the door. Badriya looked up, shook her head and turned around to face the wall. I could tell she didn't want me to leave since it would leave her without an audience but I was starting to feel the walls close in. I walked out of our room.\n\nTo my right was a staircase leading to the lobby. I could hear Maroof and Hassan on my left, about forty feet down the hall, talking. I could make out Maroof's back, sitting on the chair. As much as I wanted to head directly down to the street level, I knew there would be hell to pay if I were to leave unchaperoned and unannounced.\n\nI could make out their voices as I neared.\n\n\"You told him that?\"\n\n\"I did. What the hell was I supposed to tell him?\" Maroof asked.\n\n\"God help her. What did he say?\"\n\n\"You've heard how he gets. He said a lot of things. I don't know what he's going to do to her but I had no choice. And it's your fault anyway, Maroof. You're the one who told him she was spending a lot of time with those two hags. You didn't stop to think that he would get pissed we weren't guarding her? Maybe you don't think it's your job since you're the driver, but I'm their guard. Did you miss that?\"\n\n\"What was I supposed to tell him? He called when she wasn't around. He wanted to speak to Badriya too. If I hadn't said she wasn't here, she would have told him. He would have had my neck for sure if he thought I was keeping something from him.\"\n\n\"Yeah, yeah. Well, I hope he got that she went without our knowing about it. I don't want to get back to the house and find out it's us he's mad at.\"\n\n\"Just stick to what we said. She snuck out without telling us and went to hang out with those godforsaken women. He'll believe it. You know he doesn't think much of her anyway. You've heard about his plans. He's lost interest. She's not as exciting to him as she was in the beginning. Remember that day he saw her in the market?\"\n\nMaroof let out a guffaw.\n\n\"He looked like he might pick her up right there. Send a note and a few afghanis to her parents!\"\n\n\"Would've been a lot easier if he'd done it that way. What a pain her family was. Putting up a show like they come from royalty or something.\"\n\n\"But I remember your face when he made us stop so he could watch her . . . you thought she was a real boy then, you idiot!\"\n\n\"You did too!\" Maroof said in self-defense. \"She looked like a boy. How the hell should I have known there was something more interesting under those clothes?\"\n\n\"You probably liked her better the other way!\" Hassan chuckled. \"What do you think of her new haircut, eh? Got your appetite going?\"\n\nI backed up slowly and as quietly as I could, my mind racing.\n\nThey had sold me out to my husband. I trembled at the way they talked about me.\n\nMy thoughts tumbled and turned until I finally realized what it was that I had just overheard.\n\nI wasn't safe.\n\nI turned the doorknob, watching the hallway to see if the men had noticed my presence. They hadn't. I closed the door behind me and went straight to the washroom. I couldn't look at Badriya right now, knowing she would be of no help to me. It looked like she was asleep anyway.\n\nMy husband was a man of violence and I knew that I'd barely seen a tenth of what he was capable of. He was a man of war, of guns, of power. He demanded respect and obedience, and the guards had just told him that I was out of control. He must have been wild with rage.\n\nI couldn't help but remember he was looking to add a wife and that five was one more than he wanted. I knew what that meant for me.\n\nI thought of the woman in the shelter. She'd disobeyed and her husband had sliced off her ear. I had no doubt Abdul Khaliq could be just as vicious. I leaned against the wall, my heart pounding in fear. I had to think fast.\n\nWe were due to return home in three days.\nCHAPTER 64\n\nShah's feet pounded against the dirt of the road. Just because he was supposed to accompany his sister home from school didn't mean he couldn't race her to the front door. He panted, turned around and saw Shabnam walking hurriedly to catch up. She looked frustrated.\n\n\"Why are you always in such a rush? Don't you know it's not easy to run in a skirt? And anyway, Madar-jan would be upset if she saw me chasing after you through the streets!\"\n\n\"It's not my fault I'm faster than you. I could have been home a long time ago if I didn't have to wait for you!\"\n\nIt was the same argument every day. They bickered but adored each other, oblivious to the resentment between their mothers. Shabnam had long ago opted to ignore her mother's hand pulling her back and would sit with Shekiba while she washed the clothes, asking her question after question about everything from horses to baking bread. And Shah, who knew no boundaries thanks to his father, loved to torment Gulnaz by pulling at her knitting and running away, his giggles undoing her anger at the work he had unraveled.\n\nAasif had hoped for more children but Gulnaz and Shekiba seemed to alternate; one would start her womanly illness when the other stopped. He wondered if a curse had been lifted from him for those two years. Or maybe the women had done something . . . but he grew tired of being angry. His mother had not given up hope. Even one week before her death, she'd reminded her son that Allah had wanted men to take on more than two wives.\n\n\"And where will I put another wife, Madar-jan? In our small home, there is no room for another woman and I have enough trouble feeding the ones I have.\"\n\n\"Marry and Allah will provide a way,\" his mother had told him, her eyes half closed with fatigue.\n\nHe debated her advice, as illogical as it seemed, on his way to and from the Ministry of Foreign Affairs. He had been transferred from the Ministry of Agriculture and given a position working with a higher-ranked vizier two years ago thanks to his relationship with Amanullah.\n\nWhen Agha Khalil arrived with his wife, it was Shah who met them at the door. His knees were dusty from trying to climb past the second branch of the tree in their courtyard, which made the visitor and his wife smile and think of their own young son at home.\n\n\"Good evening, dear boy! Is your father home? I would like to speak with him.\"\n\n\"Yes, he is. Come in! My mother is making dinner. Why don't you stay and be our guest?\" he said with a grin, aping his father's hospitality. Agha Khalil's wife could not help but laugh.\n\n\"Isn't that kind of you! We wouldn't want to trouble her, my friend,\" he said just as Aasif entered the courtyard.\n\n\"Agha Khalil, how pleasant to see you!\"\n\n\"And you as well, Agha Baraan. Forgive me for dropping by at this hour but I wanted to bring you those papers since I will not be at the office tomorrow.\"\n\n\"Please, please, come in,\" Aasif said, motioning to the house door.\n\n\"Your son was quite the host and already invited us but my wife and I were just on our way home from visiting relatives. We don't want to be a bother.\"\n\nAasif insisted and Shekiba quickly set out cups of tea and dried mulberries. Gulnaz had taken to her room with a headache, so Shekiba was forced to join Aasif in sitting with the guests. Shekiba and Agha Khalil's wife, Mahnaz, were introduced and they sat in one corner of the living room while the men chatted in the other. Shekiba kept her head turned to the side as she always did when she met someone new.\n\n\"Your son is such a darling boy, nam-e-khoda!\" Mahnaz said. Shekiba bowed her head and smiled to hear the kindness in this woman's voice. Mahnaz wore a taupe-colored ankle-length dress with airy sleeves that buttoned at the cuff. She looked elegant and fitting of someone who might be a palace guest.\n\n\"May Allah bless you with good health, thank you,\" she said, not wanting to invite nazar by saying any more about her little king.\n\n\"Do you have much family in Kabul?\"\n\n\"No, I came from a small village outside Kabul.\"\n\n\"So did I. This city was quite a surprise for me! So different from where I grew up.\" Mahnaz was young, probably no more than twenty-four years old, with a bright and cheerful face. \"Where was your village?\"\n\n\"It was called Qala-e-Bulbul. I doubt you ever would have heard of it,\" Shekiba said. At the age of thirty-six, she hadn't thought of her village, named for the hundreds of songbirds that lived there, in years. And her village made her think of her songbird sister. Aqela's lifted voice and dimpled face flashed across her mind, blurry and vivid all at once as memories are.\n\nMahnaz's mouth dropped open. She put a hand on Shekiba's. \"Qala-e-Bulbul? Are you really from there? That is my village!\"\n\nShekiba suddenly felt a surge of panic. She did not regret in the least that she had no contact with her family. She looked over at Aasif and saw that the men were deeply engaged in a conversation. He had never cared to ask her anything about her family and she saw no reason for him to learn anything now.\n\n\"I left when I was fairly young and I barely remember anyone . . . ,\" Shekiba said quietly.\n\n\"What a remarkable coincidence! What is your family name?\"\n\n\"Bardari.\"\n\n\"Bardari? The farm that was north of the hill of the shepherd? Oh, my goodness! My uncle was neighbor to the Bardari family. I spent so much time at my uncle's house that I know them well. We lived not too far from there ourselves. How are you related to Khanum Zarmina or Khanum Samina? Their daughters and I used to braid each other's hair and sing songs by the stream that ran behind my uncle's land.\"\n\n\"You did? They are my uncles' wives.\"\n\n\"Oh my! Then it was your cousins that I played with as a girl! Do you write to them often? My letters to my family take so very long to reach home. Do you have the same trouble?\"\n\n\"I . . . I am not in contact with my family now that I am living in Kabul. It has been a long time,\" Shekiba said vaguely.\n\n\"Really? I understand. I was just there two years ago, you know. For my brother's wedding. The village hasn't changed a bit. But did you . . . Shekiba-jan, do you know about your grandmother?\" Mahnaz's eyes softened and her voice quieted.\n\n\"My grandmother? What is it?\"\n\nMahnaz bit her lip and looked down for a second. She shook her head and held both Shekiba's hands in her own.\n\n\"She passed away just two days after the wedding. It was such a sad time. I did not know her personally but I heard that she was a very strong woman. The whole village marveled at how blessed she was to have lived such a long life!\"\n\nShekiba was taken aback. Part of her had expected her grandmother to live on forever, pickled in her own bitter juices. She quickly realized that her guest was expecting some kind of reaction.\n\n\"Oh. I had no idea. May she rest peacefully in heaven,\" she mumbled, lowering her head.\n\n\"I am so sorry that I should share such sad news with you, especially in our first meeting. How awful of me!\"\n\n\"Please, please. My grandmother, as you said, lived many more years than anyone would have expected. Such is life and the same end awaits us all,\" she said, struggling to sound polite.\n\n\"Yes, yes, God bless her. She must have had a good soul to have been blessed with such a long life.\"\n\nYou did not know her, Shekiba thought.\n\n\"Mahnaz-jan,\" Shekiba said hesitantly. She wondered how to ask what she really wanted to know. \"Do you happen to know how the farms are doing? My father's land . . . my father's land used to produce such a yield of crops. I often wonder . . .\"\n\n\"Which was your father's land?\"\n\n\"It was behind my grandmother's house, separated by a row of tall trees . . .\"\n\n\"Oh, of course! Well,\" she said. The subject obviously made her uncomfortable. \"From what I heard there were some . . . some disagreements about the land. When I was there, Freidun-jan and Zarmina-jan were living there but they were about to divide it up.\"\n\nShekiba could decipher what Mahnaz was too polite to say. Her uncles must have quarreled over the land. She could imagine Kaka Freidun asserting his right as eldest and haughty Khala Zarmina pushing the others aside to get a home of her own. Greed had torn the family and the land apart.\n\n\"But they were not having a good yield when I visited. I saw their daughter, your cousin, at the wedding and she told me that they believed there was some kind of curse on the soil.\"\n\nShekiba smiled. Mahnaz thought her odd. Shekiba realized but couldn't help it. She could hear her grandmother's cackling voice telling her sons that it was Shekiba who had cursed the earth and condemned their crops.\n\n\"How did things go at the wedding? Congratulations to your family,\" Shekiba said. She had no interest in hearing anything else about her family.\n\nMahnaz relaxed and broke into a smile. \"It was wonderful! Dancing and music and food! It was so lively and I had not seen my family in so long. I could not have had a better time!\"\n\n\"How nice! I wish the bride and groom a happy life.\"\n\n\"They nearly had to call off the wedding, truthfully.\"\n\n\"Why?\"\n\n\"Well, the bride's family had asked for a huge sum of money as her bride price, but my father had said it was unreasonable, especially since King Amanullah had outlawed the practice of bride price. The bride's father felt disrespected, so they settled on a lesser sum. I suppose I could understand though. No money at all? I mean, a bride is worth something, isn't she? I know I was!\" she laughed.\n\nShekiba smiled meekly and looked away. \"You are right. Amanullah's laws seem so foreign in a village like ours. Kabul is so different. Can you imagine if people in Qala-e-Bulbul knew about the English and German secondary schools here?\"\n\n\"You are so right, Shekiba-jan! Only some of the girls went to school in our area. Do you know that Queen Soraya will be making a speech in two days?\"\n\n\"No, I didn't.\"\n\n\"Oh, it will be amazing. I can hardly wait to hear what she has to say. Though I worry about her. Many will not welcome so many changes so quickly. Why don't you come with me? We can go and hear her speak!\"\n\nShekiba was taken aback. Queen Soraya? Shekiba had wondered about her so much, she brightened at the thought of actually seeing this revolutionary woman. But Shekiba was not accustomed to attending public events.\n\n\"Oh, I couldn't . . . I mean, I have to tend to\u2014\"\n\n\"Come, just for a day! It'll be great to see!\" she said with excitement, and then turned her attention to the men. They were so deeply engaged in conversation that they had not yet touched their tea. \"Excuse me, dear Agha Baraan!\"\n\nAasif turned around. He looked startled. \"Yes, Khanum?\"\n\n\"Could I steal your wife tomorrow?\"\n\nSteal your wife. I wonder how that sounds to him, Shekiba thought. The talk of Amanullah and Soraya reminded her of the palace. And Benafsha.\n\n\"Steal my . . .\"\n\n\"Yes, I would love to go to the speech and have been looking for someone to join me! We won't be gone long. We can take adorable Shah-jan with us too!\"\n\n\"It will be an important speech. I have no doubt that the Afghan people will be impressed with Queen Soraya the more they get to know her,\" Agha Khalil said.\n\n\"You will be there?\" Aasif asked him. Shekiba watched as her afternoon was planned for her.\n\n\"Certainly.\"\n\n\"Well, then . . .\"\n\n\"Wonderful! Hope you don't mind her escaping for a bit!\" Mahnaz said contently. Aasif tried not to let his face show his displeasure.\nCHAPTER 65\n\n\"They said around one o'clock. Shouldn't be much longer. Just look at this crowd! All these people here to see our Queen Soraya!\"\n\nShekiba held Shah's hand tightly, her eyes scanning the stage for any sign of Amanullah. She wondered what he looked like now. It had been years since she last saw him.\n\nStupid, she told herself. Look at this crowd. How could you have thought you were suited for something like this, that you could be worthy of taking that stage, of appearing before all these people!\n\nShekiba adjusted her veil and leaned over to give Shah a handful of nuts to snack on. She'd been unable to stomach much food in the past few weeks and even the woody smell of roasted almonds turned her nose, a smell she'd never before even noticed.\n\nLittle Shah was happily entertained by the many faces, the man selling vegetables from his wooden cart, the children holding their mothers' hands. He did not mind that they had been standing around for over an hour, nor did he notice the number of stares his mother's face attracted. Shekiba kept her veil draped over the left half of her face and averted her gaze when she saw curious eyes. Shah was seven years old now and wise enough to detect stares and whispers. She did not want her son to feel embarrassed by her.\n\nGulnaz and Shabnam were at home. Gulnaz was not happy that Shekiba had been invited for an outing by Agha Khalil's wife and she had only spoken a few words to Shekiba since finding out. But she contented herself with the knowledge that Aasif would be pleased she'd stayed home instead of shamelessly wandering around Kabul in a crowd of people.\n\nSoldiers lined the stage and created a perimeter around it so the crowd couldn't get too close. In the center of the stage was a podium draped in navy blue velvet with gold tassels and embroidered with two crossed swords. Shekiba looked at the soldiers and thought of Arg, the guards, the harem. It seemed like a hundred years ago that she'd walked about the palace grounds with cropped hair and men's slacks. She looked at her son, soon to be a young man, and wondered what he would have thought to see his mother dressed that way.\n\nHe wouldn't understand. Only a daughter could know what it was to cross that line, to feel the freedom of living as the opposite sex. Her fingers touched her belly briefly. She looked at Shah and knew this one was different. She could feel it.\n\nMahnaz shielded her eyes from the sun.\n\n\"Have you seen her before?\" she asked.\n\nShekiba shook her head.\n\n\"She looks like a queen. I don't know how else to describe her. You should see the clothes she wears! Straight from Europe! My husband tells me that even the children wear European clothes!\"\n\n\"Your husband works with them?\"\n\n\"Yes, he does some calligraphy work for the king and he serves as counsel to the queen when the king is away. He's going to be traveling with them soon.\"\n\n\"He's away often, isn't he?\"\n\nMahnaz nodded, her face showing her disappointment. \"He is, but at least I have my mother-in-law and his family around. I would be so lonely otherwise.\"\n\n\"How was your marriage arranged? His family is from Kabul, are they not?\"\n\n\"Yes, they are. He and his family had traveled through our village on their way to Jalalabad one year. In that time, his father and my father came to know each other and they arranged for us to be married. I had seen him only once, just for a second. It was so unexpected!\"\n\n\"And you've been living in Kabul since then?\"\n\n\"Mostly,\" she said, and leaned in to speak more discreetly. \"My husband had some differences of opinion, you could say, with some of the government officials. We went through some difficult times then. They took everything from us. Our furniture, our home, our jewelry. We moved into the countryside for a year and a half until word was sent that we could return. The children were miserable there. We were so happy to come back!\"\n\n\"That sounds awful,\" Shekiba said. But worse could have been done to you, she thought.\n\n\"It was awful. But that's how it is. When you don't agree with powerful people, be prepared to lose everything. I only hope we will not go through such an experience again.\" She sighed. \"It is hard to say, though, since what men will tolerate changes as often as the shape of the moon.\"\n\nShekiba nodded.\n\n\"There they are!\" Mahnaz spotted Amanullah and Soraya being escorted onto the stage. Soldiers were lined up ceremoniously on either side of them and generals stood at their side. They were smiling and waving to faces they recognized in a group of dignitaries just in front of the platform.\n\nA man in a suit took the podium and began to speak. He introduced himself and spoke of King Amanullah's recent trip to Europe. Afghanistan was in a period of rebirth, he declared, and would grow with the leadership of such a strong-willed and visionary monarch. His speech went on until one of the generals could take no more and whispered something into his ear that brought him to closing remarks rather abruptly.\n\n\"Our noble king Amanullah!\" he announced, and stepped away from the podium, his arms outstretched dramatically to welcome the country's leader to the stage.\n\n\"As-salaam-alaikum and thank you! I am pleased to come and speak here with you!\"\n\nShekiba's lips turned up ever so slightly in a half smile. He looked even more dignified than she remembered, his olive-brown military jacket was decorated with medals and stars and cinched at the waist with a leather belt. He took off his hat and placed it on the podium before him. His posture gave an aura of confidence, a self-assurance that seeped through the crowd. Shekiba looked at the faces around her, their eyes focused on the stage, their expressions eager.\n\nWe are in good hands, people seemed to be thinking.\n\nShekiba tried to focus on his speech but her mind wandered. She kept her eyes on Amanullah, wondering if he would remember her, the harem guard with the scarred face. She willed his kind eyes to fall upon her again. She felt a flutter in her stomach and wasn't surprised that even the smallest of spirits could be moved by Amanullah's presence.\n\nMahnaz looked over at her occasionally, nodding in agreement. Shekiba realized the king must have said something noteworthy. Shah pulled at her hand and she absentmindedly pulled raisins from her purse. He ate them one by one, bored by the speech.\n\nQueen Soraya joined him at the podium. She wore a thin head scarf, plum colored, to match her skirt suit. She wore a fitted jacket with a brooch that caught the sunlight, over a pencil skirt that ended midcalf. Her shoes were smart\u2014black Mary Janes with a modest heel.\n\nThis is his wife, the woman he spoke of as thoughtful and dedicated, strong-willed. Indeed, she does walk with her head held high. Then again, why shouldn't she? She is queen to our beloved Amanullah.\n\nSuddenly, Queen Soraya looked at her husband and pulled her head scarf off her head! Shekiba's mouth dropped open. She looked at King Amanullah and was shocked to see him smiling and clapping. Mahnaz grabbed Shekiba by the forearm and broke into a grin. A mix of gasps and applause rippled through the crowd.\n\n\"Isn't that amazing?\" she said excitedly.\n\n\"What just happened? Why did she do that?\"\n\n\"Weren't you listening? He just said that the chador is not required in Islam! The queen is doing away with her head scarf!\"\n\n\"But . . . how could she . . .\"\n\n\"It's a new day in Kabul! Aren't you glad I dragged you here?\" she said, nudging Shekiba with her elbow.\n\nAmanullah went on to say a few more words with Soraya at his side. He declared her, his wife, to be the minister of education and queen to the Afghan people. He turned the podium over to Soraya. Shekiba looked to Shah, then turned her attention back to the stage. Today's speeches were more interesting than she had anticipated.\n\nQueen Soraya spoke eloquently and with a confidence that complemented her husband's. Shekiba felt humbled and listened to her talk on the importance of independence.\n\n\"Do you think, however, that our nation from the outset needs only men to serve it? Women should also take their part as women did in the early years of our nation and Islam. From their examples we must learn that we must all contribute toward the development of our nation and that this cannot be done without being equipped with knowledge. So we should all attempt to acquire as much knowledge as possible, in order that we may render our services to society in the manner of the women of early Islam.\"\n\n\"Imagine. Just imagine, being able to speak like her to a crowd of people this size. She is a remarkable woman. Oh, the people of Qala-e-Bulbul would just faint to see something like this, wouldn't they?\" Mahnaz said with a laugh.\n\nShekiba thought of her own uncles. No doubt they would have sneered and walked out on such a speech. A woman? Telling their wives to acquire knowledge?\n\nIt was an exhilarating day. Shekiba was vaguely aware that this day would change something, though she wasn't sure what.\n\nShe's a wise woman, Shekiba thought. A woman like that would have given my father's land to me. She would have told my grandmother to send me to school instead of the fields.\n\nShekiba's lip stiffened with resolve.\n\nShe knew Queen Soraya was speaking of changes that wouldn't affect her.\n\nMy story ends here, she thought. She now had a better life than she could have imagined. Somehow she had found an escape from a much worse naseeb.\n\nBut something in Shekiba did shift. She had a glimmer of hope, a feeling that things might get better with this woman Amanullah had chosen over her. Her face flushed knowing it still felt that way to her, as ridiculous as it was.\n\nShe thought of the way she was beaten when she took the deed to Hakim-sahib. She thought of Benafsha succumbing under the weight of the stones.\n\nBut sometimes you have to act out of line, I suppose. Sometimes you have to take a chance if you want something badly enough.\n\nThings would be okay for Shah, Shekiba knew. He was a boy and his well-connected father would make sure he had every opportunity. She thanked God for that.\n\nAnd may Allah give my daughters, should I be blessed with any, a chance to do what Queen Soraya seems to believe is possible. May Allah give them courage when they are told they are out of line. And may Allah protect them when they seek something better, and give them a chance to prove they deserve more.\n\nThis life is difficult. We lose fathers, brothers, mothers, songbirds and pieces of ourselves. Whips strike the innocent, honors go to the guilty, and there is too much loneliness. I would be a fool to pray for my children to escape all of that. Ask for too much and it might actually turn out worse. But I can pray for small things, like fertile fields, a mother's love, a child's smile\u2014a life that's less bitter than sweet.\nCHAPTER 66\n\nI used all my strength to stay focused, to keep my composure. I couldn't let anyone know that I had overheard what I had. Beyond that, I didn't know what to do or who to turn to. Frankly, I didn't think I could turn to anyone.\n\nI sat beside Badriya in the following day's session, ignoring the debate on funding for a roads project when everyone knew the decision was really in the president's hands. And that he'd already made up his mind.\n\nTonight, Ms. Franklin was going to let us work more on the Internet. It was as important as learning to read and write, she said. The Internet was our gateway to the world.\n\nI could have used a gateway.\n\nWhile the debate of no consequence went on around me, a more important debate raged in my head. Should I go with Hamida and Sufia to the training center or should I stay with Badriya and the guards?\n\nMy hands were clammy and my shoulders stiff. I dreaded the session ending, knowing I would have to make a decision.\n\nWhat does it matter? I thought. He already thinks I've snuck away from the guards. How could it get any worse?\n\nBut I was afraid. Maybe he would believe me, take my word that the guards had let me go. That Badriya had said it was all right. That I did nothing inappropriate or shameful at the resource center.\n\nImpossible.\n\nWe were outside. I was looking at the three western soldiers on the opposite side of the street. They were leaning against a wall, talking with a crowd of young boys. Jahangir would have been one of them, I thought, if I'd been allowed to bring him with me. I wondered what the soldiers would do if I ran to them. They were here to help us, weren't they?\n\nWe were just past the security check when Hamida called out to me. My heart raced. What would Khala Shaima tell me to do?\n\n\"Aren't you going to come with us? Ms. Franklin's expecting you!\"\n\nI looked at Badriya. She raised her eyebrows, wondering why I thought she would care where I went. She walked toward the car, which was parked a few meters away. I saw Maroof mumble something to Hassan, who nodded and mumbled something back.\n\nFiguring I was doomed anyway, I took a leap and decided to go with Hamida. I didn't know what I expected to come of my decision.\n\n\"I'm going to . . . I'm going to go with them. I'll have her driver drop me off before they go to her apartment. Okay?\"\n\nBadriya shrugged her shoulders without bothering to turn around. I knew she didn't want to give a formal answer, an answer she might have to defend to our husband. She got in the car and they drove off, melting into Kabul's congested streets. I was relieved and petrified.\n\nWhile we walked, Hamida talked and I thought of my husband. Twice, I thought I might vomit on the street. Sufia joined us two blocks from the parliament building. The guards walked a few feet behind us while the drivers stayed with the cars. With the traffic, it would have taken longer to drive to the resource center.\n\n\"Rahima-jan, what's going on? You're awfully quiet today. Everything all right?\" Sufia asked.\n\nI never meant to share it. My story just flowed out. Like the water that once upon a time bubbled over stones in the Kabul River, I told them about my husband, Bibi Gulalai, Jahangir.\n\nWe walked slowly, not wanting to draw attention from the security guards who trailed us. This wasn't a story to share with them.\n\nI answered their next questions before they could ask them. I told them about my parents and how they'd given us sisters away, then cloaked themselves in clouds of opium. I told them how Parwin escaped her hell in a flash of flames and that with Rohila about to become a wife, Sitara would be left cowering in the corner of our home, afraid of what fate my father would choose for her. And Khala Shaima, the only family I'd kept over the years, her twisted spine was squeezing the life from her bit by bit.\n\nBut my son. That was the worst of it. I said it and then I left it alone. The sore was too raw to touch. Worse than losing the unborn.\n\nWhile I tried to control the shaking of my voice, I told them about the conversation I'd overheard. About the wife my husband wanted to take without violating the laws he suddenly wanted to follow. I didn't have to tell them what I was afraid he would do to me. They knew.\n\nThey listened, unsurprised. I was only confirming what they'd already suspected, that I was one of those stories. My story was not unheard of.\n\nI was broken and battered and didn't care anymore how much I told them or what they thought or even what Abdul Khaliq would do if he found out. I had had enough. I kept thinking of Khala Shaima's face, her soured expression, her disappointment in what had become of her nieces. And then there was Bibi Shekiba, the man-woman whose story had woven its way through my own.\n\n\"Dear God, what a mess you're in, Rahima-jan! I don't even know what I can say . . . ,\" Hamida said. We stood outside the door of the resource center. Ms. Franklin waved us in with a smile.\n\n\"There must be something . . . there has to be some way . . . ,\" Hamida said unconvincingly.\n\n\"Let's not stand out here too long,\" Sufia whispered gravely. \"We can chat about this inside. Come on, ladies.\"\n\nI let Sufia guide me with a hand on my back, thinking of something Khala Shaima had said when I shared the story of the girl from the shelter with her, how she'd escaped her husband only to be found again and beaten, punished for running away.\n\n\"Poor girl. She ran out from under a leaking roof and sat in the rain.\"\nCHAPTER 67\n\n\"I'm not feeling well at all,\" I said. I hoped I sounded believable.\n\nBadriya huffed and rested her hands on her hips dramatically. \"What is it now? You expect me to go by myself to the session? And who do you suggest should fill out the ballots that are due today?\"\n\n\"I'm sorry, but it's my stomach. It must have been something I ate last night. My stomach is a terrible mess,\" I said, wrapping my arms around my abdomen and leaning forward. \"I don't want to cause a disturbance sitting next to you. I just feel like I might have to run\u2014\"\n\n\"Oh, that's enough already! I don't want to hear more. Some assistant you are. Useless!\" she said, throwing her hands in the air. She grabbed her handbag and stormed out. When I heard her footsteps moving away, I crept to the door and put my ear against it. I could hear her speaking to Hassan and Maroof, their heavy voices echoing in the hall.\n\n\"She's not going?\"\n\n\"No, she says she's not well. I suppose we should just let her stay here. I'm not staying with her, if that's what you are thinking. I'll hear it from the director if I miss another session.\"\n\n\"Agh. This girl's nothing but trouble,\" Maroof said.\n\n\"Just take her. I'll stay here with this one,\" Hassan offered reluctantly. \"The last thing we need now is for Abdul Khaliq to hear we left her alone in the hotel.\"\n\n\"Fine.\"\n\nI heard the metal of the chair scraping against the floor. He was going to stay at his post down the hall. My chest felt heavy with anticipation.\n\nI took a deep breath and went back to the bed, pulling my duffel bag out from under it. My hands dug through the dresses until I found what I was looking for. I thanked God I'd brought it along, even if I hadn't anticipated wearing it. I changed quickly, a small thrill running through me. I went back to Badriya's bag and rifled through it until I found the scissors she kept with her sewing supplies. To the bathroom again, where I looked at my reflection and finished what my husband had started. Snip, snip, snip. It was badly uneven but better than what Abdul Khaliq had done.\n\nI put my sandals on and considered my duffel bag for a moment. From the back, it might give me away. I decided against taking it and sat down to calm my breathing.\n\nIt took five minutes of intent listening at the door to convince myself no one was approaching, especially Hassan. No thump of his heavy foot or whistle of his raspy breath. I imagined he'd probably gone out for a smoke.\n\nMy fingers touched the knob and closed around it slowly. I turned, still keeping my ears perked.\n\nI looked out through the crack, opened the door wider when I was certain I'd seen no one. And wider yet when I worked up the nerve to walk into the hallway. I craned my neck to see where the chair usually sat.\n\nHassan's back. I took a deep breath and turned to the right, to the stairwell. I closed the door as silently as I could behind me. I moved one foot in front of the other, walking past the four doors between me and the end of the hallway. I was so focused on listening for the sound of Hassan moving that my left sandal caught on the carpeting and I stumbled, catching my balance by grabbing the doorknob of the next room.\n\nI held my breath when I heard the scraping of the metal chair legs.\n\n\"Hey!\"\n\nI froze, keeping my back to Hassan. I was sure he could see my whole body quivering even from a distance.\n\n\"Watch your feet, you klutzy kid!\" he called out.\n\nI nodded and grunted something in a voice deeper than my own but barely audible.\n\n\"Boys running around in a hotel . . . ,\" I could hear him muttering as I resumed my trek to the stairwell. With each step I waited, waited for the moment when he would realize that the boy he saw was actually a girl in Hashmat's new clothes, the pants still unhemmed.\n\nI was and then I wasn't. I was Rahima. And then I wasn't.\n\nI walked through the lobby, keeping my eyes lowered. The man from the front desk was nowhere to be seen. I moved quickly. I opened the door and sunlight tickled my eyes. I lifted my hand and blinked. When my sandals hit the dirt road I scanned the street to make sure I recognized no one and that no one recognized me. My eyes fell on a sparrow, nimbly passing between tree branches and chirping as earnestly as the birds over Jahangir's grave. Pray for me too, I thought.\n\nRahim wound in and out of the streets, heading further away from the hotel and in the opposite direction of the parliament building. Rahim, the bacha posh, listened for someone yelling behind him, listened for a sign that he'd been spotted, that he was going to be dragged back to Abdul Khaliq's compound and punished.\n\nRahim, shaking so badly that he thought his legs might collapse, needed a place to hide.\nCHAPTER 68\n\nTaxis honked. One slid past me, skimming my side as I tried to dodge traffic in a busy intersection. I cursed myself for choosing to cross here, in front of so many cars. I felt a million eyes on me, eyes that might notice something was not right about this adolescent boy. Didn't I look frightened\u2014like I was running away from something? Did they see that my chest seemed to swell where a girl's might?\n\nI had done my best to tie down my breasts with a head scarf but it was harder now than it had been a few years ago. Having Jahangir had thickened me with curves that were more difficult to disguise.\n\n\"Hey, bacha! Watch where you're going!\" a man yelled through the driver's-side window of the taxi, a cigarette between his fingers as he waved at me angrily.\n\nWithout a pause in my step, I raised an apologetic hand, silently thankful to know my disguise was working. Funny how easily I slipped back into this person, how comfortable I felt even though my nerves were on fire.\n\nMy sandals slapped against the dusty road, my legs free in the pants, a loose tunic covering my curved rear.\n\nIt had been nearly eleven o'clock when I left the hotel. That felt like a year ago, though it couldn't have been more than twenty or thirty minutes. A bus came up ahead, slowing near a crowd of people and honking an awkward tune. Maybe that was the one. I looked for signs, turning my head and suddenly feeling my legs weaken.\n\nA black SUV slowed as it approached, only half a block away.\n\nI felt exposed even in the crowded street, wondering if I'd been spotted. If I hadn't been, running now might draw attention.\n\nThe driver slowly rolled down the tinted window and I let out a soft moan of panic.\n\nBut it was a face I didn't recognize. It wasn't Abdul Khaliq's car.\n\nRegrouping quickly, I pushed my way through and reached the crowd of people clamoring to board the white and blue bus.\n\n\"Is this the bus to Wazir Akbar Khan?\"\n\nNo one turned around.\n\n\"Agha, is this the bus to Wazir Akbar Khan?\" I asked again, louder. I tried to deepen my voice, to hide my feminine tone.\n\nA man turned around, annoyed. He wore a button-down shirt over slacks and held a briefcase in his hand.\n\n\"Yes, it is! Hurry up and get on if you're coming,\" he said. He and another man tried to squeeze through the bus door at the same time, each hoping to get a square of standing space.\n\nWith my head down, I managed to slip onto the bus behind two men. I waited for the bus driver to notice and shout, but he didn't. I wiggled my way to the back of the bus, as far from the driver as possible. Looking around, there was not a single woman on the bus. I felt my face flush at being surrounded, so closely, by so many men. I kept my elbows close to my chest and cringed when the bus's movements pushed a body against mine. I craned my neck to see between the chests and arms. I hoped I would recognize my stop.\n\nThe bus will stop on a road lined with shops. Look for a beauty shop between an electronics store and a food vendor. Usually there's a man with a long beard and half an arm standing around there begging for money.\n\nIt was a long ride to Wazir Akbar Khan. Beads of sweat slid down my neck. My nerves were just starting to settle as the bus put distance between me and the hotel\u2014between me and Abdul Khaliq's guards.\n\nI was supposed to be there at twelve. I had meant to leave the hotel earlier but Badriya had taken her time that morning, putting the whole plan in jeopardy.\n\nWazir Akbar Khan was a neighborhood to the north of the city, a suburb that was home to many embassies and foreign workers. The streets were wider than they were in the part of Kabul I had seen. Two-story buildings lined the road. I tried not to look as nervous and lost as I felt.\n\nThe bus slowed. Pharmacy of Wazir Akbar Khan, read the sign on a building.\n\nThis is it, I thought, and snaked through the crowd to get off before the bus resumed its course.\n\nI recognized no one and didn't notice any suspicious stares. I turned my attention to the shops, looking for the landmarks I'd been given. One storefront had crates outside, boxes of detergent, household supplies. There was a butcher shop. There was everything except what I was looking for.\n\nI turned down another street but saw only houses. Beautiful houses that put Abdul Khaliq's estate to shame. They were new buildings with modern fa\u00e7ades that I didn't have time to take in. Minutes were ticking by and I might miss this opportunity.\n\nI worked up the nerve to ask someone, steeling my voice an octave below in disguise.\n\n\"Agha-sahib? Agha\u2014\"\n\n\"For God's sake, boy, I don't have any money to give!\" the man said, and kept on moving.\n\nI looked for someone else to ask.\n\nA woman walked by. I wanted to approach her but my tongue froze when I saw the little boy, probably three or four years old, holding her hand tightly. He pointed at a car in the street and looked up to see if his mother noticed. She nodded and said something that made him giggle with delight.\n\nJahangir, I thought, my chest tight.\n\nThe woman was gone before I recovered. I walked further down the street, blinking away tears. I stood in front of a shop window, a clock catching my eye and sending me into a panic.\n\nOne o'clock. My pulse quickened. If I was too late, this could all fall apart. I would have risked everything for nothing. What would become of me?\n\nMy eyes moved from the clock to a flyer hung on the storefront.\n\nVisit Shekiba's Beauty Shop, Sarai Shahzada. Weddings and all occasions.\n\nThat must be it! I thought. Shekiba.\n\nI closed my eyes, reenergized by the shop's name. It was as if a hand was holding mine, guiding me. I read the flyer again.\n\nSarai Shahzada. I was sure I'd seen a sign with that street name and traced my steps back. Two lefts and I was there again, concrete sidewalks and trees giving it a clean, welcoming appeal. Within minutes, I had found the beauty shop, sandwiched between an electronics shop and a store with crates of fruits and vegetables outside.\n\nShekiba's Beauty Shop.\n\nAs I had been instructed, I looked directly across from it and spotted a teahouse.\n\nI hope I'm not too late.\n\nI dodged oncoming cars again and crossed the street, trying to see through the shop's glass front. The door handle rattled in my hand. I took a deep breath and hoped I didn't look too crazed to the people inside.\n\nI spotted her immediately, her soft bangs peeking out from beneath her gray and plum head scarf. Her eyes were on the door and looking just as nervous as mine. When she recognized me, her hand flew to cover her open mouth. She stood up.\n\nI wove through the tables, the Afghans speaking English, the foreigners drinking cardamom-infused green tea.\n\n\"You made it!\" she whispered when I approached her table.\n\n\"Yes, Ms. Franklin,\" I said, and collapsed into the chair.\nCHAPTER 69\n\nNine days passed before I saw Hamida and Sufia. They had kept away, afraid that somehow they might lead someone to me. Hamida became tearful when she saw me. Sufia let out a triumphant yelp, with an energy I'd never seen her display in the parliamentary sessions.\n\nMs. Franklin and I had gone directly from the teahouse to a women's shelter she had located. It wasn't the shelter that we'd heard about. It was another one, one much further from the parliament building and on the western outskirts of the capital.\n\nThe shelter was both sad and uplifting. There were stories there, stories that made me cringe, scars that would never heal.\n\nI met a woman who lived there with her three children. When her in-laws learned of her husband's death, they accused her of killing him. About to be jailed, she decided to run rather than risk losing her two daughters and one son.\n\nAnother woman had escaped a heavy-handed husband, a husband who was having an affair with her younger sister. One night, while he snored beside her, she crept out softly and walked two days and two nights to reach a police station.\n\nAnd there was a girl. She was my age and her story made me realize that I wasn't alone. At twelve years old, she'd been married off to a man five times her age. Her family had put her in a white dress and taken her to a party. At the end of the night, they left without her. Four years later, she had run off, escaping the in-laws who treated her as a slave.\n\nI wasn't ready to share my story with them yet. Even here, in this open room with Afghan carpets and the smell of cumin, I felt my husband's reach. If he knew where to look, it would only take him a day to reach me. The thought made me so nervous I could barely eat.\n\nHamida and Sufia only came once. I missed them but I could expect nothing more, knowing the route was long and that they had obligations to their own families. Visiting a shelter could attract the wrong attention and endanger everyone involved. I would always think of them warmly and with deep gratitude, remembering how they and Ms. Franklin had formulated a plan to help me escape the naseeb that awaited me had I returned to my husband. My plan, though, didn't account for what might happen to Badriya. Hamida and Sufia had seen her once the day after my disappearance. She looked furious and suspicious, they said, but she seemed to believe their surprise to hear I was missing. I was sure Abdul Khaliq would never let her return to Kabul and I hated to think what Abdul Khaliq had done to her when she'd returned to the compound. Though she hadn't been kind to me, I wished his wrath on no one.\n\nI had time in the shelter, time to finally sit down and contemplate all that had happened. I felt embarrassed, remembering the day I'd argued with Khala Shaima, snapped at her that all the education she'd pushed me to get hadn't done me one bit of good.\n\nIt wasn't true.\n\nIt was only because I was literate that I was able to join Badriya in Kabul. It was only because I could hold a pen with purpose that I was able to be her assistant and feel comfortable joining Hamida and Sufia in the resource center. It was my few years of school that allowed me to read the beauty shop flyer in the store window, to locate the street where Ms. Franklin waited nervously to help me make my escape.\n\nI'm sorry, Khala-jan. I'm sorry I never thanked you for fighting for me, for everything you taught me, for the stories you told me, for the escape you gave me.\n\nMy only regret was that I hadn't been able to send word to Khala Shaima, to let her know that I had made it out and that I was safe. I hoped she didn't think Abdul Khaliq had killed me. I prayed she would not try to visit me at Abdul Khaliq's compound, knowing she would be met by my very angry husband. But I wanted to send her a message, somehow\u2014I had to try. I would take pen to paper and write my dear aunt a note, a few words, so that she could share in what I'd managed to do, what she'd given me the strength to do.\n\nI finally was able to convince Ms. Franklin to mail her a letter.\n\nThe letter, addressed to Khala Shaima, was from her second cousin and it talked of nothing but the smell of fresh air, the delightful sound of birds chirping, and the hope that the family could pay a visit sometime soon.\n\nI had no way of knowing if it had arrived, so I could only hope that the letter found Khala Shaima. It wasn't until many years later, a lifetime really, that I heard it had been discovered in her hand by her older sister, my khala Zeba. Khala Zeba couldn't make sense of it anyway, since she'd never gone to school or learned her letters. She was too distraught at finding her sickly sister cold and breathless to give it much thought then anyway. But two weeks later, when the rhythm of her life returned and the birds had prayed all they could over Khala Shaima's grave, she would ask her husband to read it for her and be puzzled, wondering which cousin would write to her crippled sister of things as mundane as birds and the weather.\n\nThe letter was signed Bibi Shekiba.\nAcknowledgments\n\nThank you to my parents who gave me the tools to write about a girl who deserves the world. I am yours, always. To Zoran and Zayla, you made this story important to tell\u2014I love you. Thank you to my husband, Amin, for your ideas, discussions, and faith in me. You've made my dreams come true. To my street-smart, wise-cracking brother, Fawod, my first and forever fan, thank you for your absolute confidence. Fahima, my muse, the spark that ignited this story and my first reader, how grateful I am for your support, every day! I am thankful for the legacy I've inherited, the creativity and traditions from the greats and grands in my own family, and hope to pay tribute to them through this story.\n\nA great big hug to my agent, Helen Heller, who took my draft and ran with it. Thank you for your confidence and guiding ideas through this process. A special thanks to my editor, Rachel Kahan, for taking this story on and never letting go! Your input and feedback has been invaluable and I am so glad to be with you. Much appreciation to the entire team at William Morrow for turning a draft into a real thing! No list of thanks would be complete without acknowledging the impact that teachers and coffee shops have on realizing dreams. My gratitude to Tahera Shairzay, who provided invaluable firsthand insight into the workings of the Afghan parliament and for her contribution to progress in Kabul. My appreciation to Louis and Nancy Dupree for their contributions to documenting Afghanistan's culture and history. Their works have been an invaluable resource.\n\nThis story is loosely based on historical figures in Afghanistan as well as contemporary citizens. It is a work of fiction and I have taken great liberties, but I have no doubt that more of it is factual than we would hope. A special acknowledgment to the daughters, sisters, mothers, aunts, and teachers of Afghanistan, and to those individuals and groups who work so tirelessly to make that world a better place. To the daughters of Afghanistan, may the sun warm your faces as you forge your paths.\nAbout the Author\n\nNADIA HASHIMI's parents left Afghanistan in the 1970s, before the Soviet invasion. She was raised in the United States, and in 2002 she visited Afghanistan for the first time with her parents. Hashimi is a pediatrician and lives with her family in suburban Washington, D.C.\n\nVisit www.AuthorTracker.com for exclusive information on your favorite HarperCollins authors.\nCredits\n\nCover design by Mumtaz Mustafa\n\nCover photograph \u00a9 by Masssoud Hossaini\nCopyright\n\nThis book is a work of fiction. The characters, incidents, and dialogue are drawn from the author's imagination and are not to be construed as real. Any resemblance to actual events or persons, living or dead, is entirely coincidental.\n\nTHE PEARL THAT BROKE ITS SHELL. Copyright \u00a9 2014 by Nadia Hashimi. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, downloaded, 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\nISBN 9780062244758\n\nEPub Edition MAY 2014 ISBN 9780062244772\n\n14 15 16 17 18 OV\/RRD 10 9 8 7 6 5 4 3 2 1\nAbout the Publisher\n\nAustralia\n\nHarperCollins Publishers (Australia) Pty. Ltd.\n\nLevel 13, 201 Elizabeth Street\n\nSydney, NSW 2000, Australia\n\n<http:\/\/www.harpercollins.com.au>\n\nCanada\n\nHarperCollins Canada\n\n2 Bloor Street East - 20th Floor\n\nToronto, ON, M4W, 1A8, Canada\n\n<http:\/\/www.harpercollins.ca>\n\nNew Zealand\n\nHarperCollins Publishers (New Zealand) Limited\n\nP.O. Box 1\n\nAuckland, New Zealand\n\n<http:\/\/www.harpercollins.co.nz>\n\nUnited Kingdom\n\nHarperCollins Publishers Ltd.\n\n77-85 Fulham Palace Road\n\nLondon, W6 8JB, UK\n\n<http:\/\/www.harpercollins.co.uk>\n\nUnited States\n\nHarperCollins Publishers Inc.\n\n10 East 53rd Street\n\nNew York, NY 10022\n\n<http:\/\/www.harpercollins.com>\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nIMAGES  \n _of America_\n\nPARADISE VALLEY  \nARCHITECTURE\n\n**R ANCHO LUCERO, 1951**. This East Berneil Lane home, Starlight Ranch, was named after favorite quarter horse, \"Starlight.\" Mrs. McKnight, who designed the home, wanted an adobe residence with lush greenery and pleasant gardens. The house was intended to resemble a Mexican village, with a main house, a guesthouse, and a cottage. This photograph of the original owner Robert McKnight looks east toward Scottsdale, capturing the open, undeveloped desert. (Jamey and Linda Cohn.)\n\n**O N THE COVER:** The original structure was built from 1928 to 1930 and was known as Los Arcos but was later named Casa Hermosa, which means \"beautiful house.\" It was hand built by the local cowboy artist Alonzo \"Lon\" Megargee III. He studied architecture in Mexico and Spain, which led to this blended Pueblo and Spanish Colonial Revival\u2013style hacienda. In 1947, it became the Hermosa Inn, which served up to 17 guests. The inn closed after a fire in 1987, was renovated, and reopened in 1992. (Scottsdale Historical Society.)\nIMAGES  \n _of America_\n\nPARADISE VALLEY  \nARCHITECTURE\n\nDouglas B. Sydnor\n\nCopyright \u00a9 2013 by Douglas B. Sydnor  \nISBN 978-0-7385-9659-4  \nEbook ISBN 9781439643433\n\nPublished by Arcadia Publishing  \nCharleston, South Carolina\n\nLibrary of Congress Control Number: 2012947150\n\nFor all general information, please contact Arcadia Publishing:  \nTelephone 843-853-2070  \nFax 843-853-0044  \nE-mail sales@arcadiapublishing.com  \nFor customer service and orders:  \nToll-Free 1-888-313-2665\n\nVisit us on the Internet at www.arcadiapublishing.com\n_To my fellow Arizona architects, who have delivered extraordinary architecture over the decades_.\n\n**P AUSON RESIDENCE, 1941**. Architect Frank Lloyd Wright designed this custom residence for two sisters, artists from San Francisco, California. It was located at 5859 North Thirty-first Street on what is now the Thirty-second Street alignment. This organically composed structure without air conditioning was constructed of desert masonry with natural stones cast into battered concrete walls. A fire in 1943 destroyed the home. (Janie Ellis.)\nCONTENTS\n\nTitle Page  \n---  \nCopyright Page  \nDedication  \nAcknowledgments  \nIntroduction  \n1.| Early Settlement: 1890\u20131930  \n2.| Town of Paradise Valley Incorporation and Facilities: 1949\u20132011  \n3.| Academic Pursuits: 1928\u20132008  \n4.| Desert Ranches, Resorts, Clubs, and Recreation: 1928\u20132000  \n5.| Religious Expressions: 1950\u20131990  \n6.| Domestic Enclaves: 1915\u20132012  \nBibliography\nACKNOWLEDGMENTS\n\nWe extend sincere thanks to various individuals and organizations that have provided many inspiring images and relevant facts for _Paradise Valley Architecture_ , including Dana Braccia of the Scottsdale Public Library System, Charles P. Brown, Donald J. Christensen, Jamey and Linda Cohn, Earl and Judy Eisenhower, Janie Ellis, Betsy Fahlman, Melissa Fought, Mike Gildersleeve, Barney J. Gonzales, and Dan Gruber. In addition, support came from Timothy Hursley, Ken and Norma Jones, Don Kaufman, Alison King, Neil Koppes, Jim and Molly Larkin, and Jason Drobish, Natalie Montenegrino, Beth Wickstrom, and Catherine Kauffman of the Town of Paradise Valley Historical Committee. Many thanks are also extended to Melissa Murray; Tina Litteral and Diana Smith, American Institute of Architects; Jerry Meek; Brian Ralphs; Arnold Roy, RA; Tom Silverman; Nicole Snyder; Mary Gayle Stewart; Flora M. Swanson; Dr. Mae Sue Talley; Bill Timmerman; Fred Unger; Gretchen Manker; Heidi Parmenter; Joann Tull; Agnese Udinotti; Pat VanVelser Heard; Flip Weber; and Judy Zuber.\n\nArchitects who provided invaluable professional photographs include Kenneth Allen, AIA; W. Brent Armstrong, AIA; William P. Bruder, FAIA; Brian Cassidy, AIA; Jack DeBartolo Jr., FAIA; John Douglas, FAIA; Lawrence Enyart, FAIA; Steven Holl, FAIA; Edward Jones, AIA; Neal Jones, AIA; Hugh Knoell, AIA; Mark Philp, AIA; Matt Sallinger, AIA; Edward B. Sawyer Jr., RA; Vernon Swaback, FAIA; and Mark Vinson, FAIA.\n\nJoan Fudala and JoAnn Handley of the Scottsdale Historical Society provided essential support to this effort, including editing and verifying facts. Deep appreciation goes to Joshua Carter, Nathanial Landreville, and my daughter Elizabeth for helping with production mechanics.\n\nIt was a true joy featuring the finest and most inspiring architecture of Paradise Valley as I have lived among it for over 50 years. There has been a rude awakening at the pace at which we are losing such fine work through demolition and insensitive renovations. This book reminds us, once again, how essential it is to create and preserve our archives, record such historically significant images, and expand our resources for future generations, so that they will appreciate our amazing, rich, and diversified architectural history.\nINTRODUCTION\n\nThose who know the town of Paradise Valley as one of the most affluent zip codes (85253) in the United States, and home to celebrities and captains of industry, find it difficult to make the mental jump back in time to when the Paradise Valley was an open desert landscape where cattle and sheep grazed during the early 1900s. Let us explore the natural forces, decisive moments, and mindset that created this desirable desert community with open space, magnificent vistas, and a quiet rural lifestyle in the middle of an urban metropolis.\n\nThe geology of the area is quite old, and includes a metamorphic type of granite rock called schist. The mountain ranges are relatively young, formed about 14 million years ago as the earth's crust was stretched in the northeast to southwest direction, elevating the mountains and lowering the basins between them. Paradise Valley is in the basin between Camelback Mountain to the south and Mummy Mountain to the north. It is therefore spatially defined by some of nature's most profound architecture\u2014mountains. Camelback Mountain continues to be quite dramatic, with a profile resembling a camel; the west end \"camel head\" has very sculptural forms, strong shadows, and a coloration that is ever-changing as the sun sets to the west.\n\nParadise Valley was originally a more remote location northeast of Phoenix, but Phoenix evolved into a large sprawling city, the sixth largest in the United States. East of the community is Scottsdale, a former farming community that now has a population of about 215,000. Scottsdale incorporated in 1951, or 10 years earlier than the Town of Paradise Valley. Phoenix and Scottsdale grew quickly with development and within close reach of Paradise Valley. With the threat of annexations and the risk of them bringing more government, more taxes, and not respecting the open desert environment, the residents debated incorporation starting in 1949; they were eventually successful in 1961. Incorporation enabled this new municipality to control its destiny, while addressing various infrastructure improvements.\n\nSince the 1940s, Paradise Valley has had a consistent commitment to respecting and maintaining the aesthetics of the lower Sonoran Desert and has evolved into a mostly private enclave within the Phoenix metropolitan area for the educated, wealthy, culturally savvy, and well-traveled.\n\n_Paradise Valley Architecture_ focuses on the primary building types found within the community. Within each type, a wide range of architectural characters are featured, from the revival styles as Spanish Colonial, Pueblo, Mission, and Tuscan to more contemporary approaches as Mid-century Ranch, Organic, International, and Post Modern. Along with the specific styles, building forms, structural expression, and materials are also described.\n\nThe architecture of Paradise Valley parallels and expresses the town's history, leadership, priorities, and a community-wide vision for a more quiet, relaxed, and desert-rural lifestyle.\n\n**C AMELBACK MOUNTAIN, C. 1940S**. The north side of Camelback Mountain is in the distance with Camelback Inn in the foreground. The view captures the open desert wilderness with minimal development throughout Paradise Valley. (Scottsdale Public Library.)\n_One_\n\nEARLY SETTLEMENT:  \n1890\u20131930\n\nIn the early 1900s, cattle grazed the land that would later belong to the Town of Paradise Valley. Cattle were found east of Mummy Mountain and south to Camelback Mountain. In 1877, the US Congress passed the Desert Land Claim Act, which encouraged economic development of the arid and semiarid public lands of the western states, including the Paradise Valley area. This area was visited in the 1890s by Sam Symonds, Proper D. Parker, and Augustus C. Sheldon, three surveyors with the Rio Verde Canal Company. They were so impressed with the lush desert beauty that it was named \"the Paradise Valley.\" The company invited investors to build a canal system to irrigate the area for farming and ranches, but the canal was never completed, thus development was slow to occur.\n\nIn the 1920s, a few custom residences started to arrive, particularly in the eastern foothills of Camelback Mountain. Homes such as the 1924 Donald Kellogg residence and 1929 Estribo were followed by the 1930s Duncan MacDonald, Edward Jones, Mildred Pringle, and Henry Wick residences. Most all of them were of plastered adobe construction and wood-framed roofs, and were of a Mission, Pueblo, or Spanish Colonial Revival style.\n\nIn the 1930s, Duncan MacDonald envisioned a different dream than farming, one that was ahead of its time. \"Someday this area will be filled with beautiful homes,\" he told his children. He built a home at the northwest corner of Invergordon Road and McDonald Drive.\n\nSome early homes and schools were converted to guest lodges, including the 1930 Hermosa Lodge, 1934 El Chorro Lodge, and the 1940s Schildman Residence, which became the Desert Lodge. A series of desert guest ranches arrived in the 1940s along East Lincoln Drive and Invergordon Road. Ranch names captured the western ambiance: Diamond Lazy K, Sun 'N Sage, Casa Verde, Desert Lark, and Yellow Boot.\n\nEarly settlements in this area epitomized the relaxed rural western environment; during the coming decades, residents strived to retain this character.\n\n**A ERIAL, C. 1940S**. This view is looking north of Lincoln Drive and along Tatum Boulevard at the future site of the Paradise Valley Country Club and golf course. (Scottsdale Historical Society.)\n\n**C AMELBACK MOUNTAIN**. In 1906, it was named \"Camelsback\" by the US Geological Survey. During the 1950s and 1960s, pressure was exerted to develop Camelback Mountain, including a proposed restaurant at the top of the camel's hump to be accessed by a cable car system. Community leadership rallied to purchase Camelback Mountain for $500,000, including over 1,300 individual donations, with the balance of about $233,000 allocated by the US government. (Paradise Valley United Methodist Church.)\n\n**C AMELBACK MOUNTAIN**. The north side of Camelback Mountain defines the southern edge of Paradise Valley. Development and custom residences arrived on the northern slopes. Homes had solid walls facing the more severe southern sun, while windows typically framed the distant valley views to the north. (Scottsdale Public Library.)\n\n**C AMELBACK MOUNTAIN**. The north and west sides express the geologic formation and sculptural forms, which make for a dramatic backdrop to Paradise Valley and Phoenix. The head of the mountain is Precambrian granite that is 15 billion years old, with four sedimentary rock layers from the past 25 million years. The mountain reveals the geologic history of the world, not unlike Arizona's own Grand Canyon. (Scottsdale Public Library.)\n\n**B ARRY GOLDWATER ON CAMELBACK MOUNTAIN**. Community leadership felt strongly that the upper slopes should be preserved. In November 1964, Preservation of Camelback Mountain Foundation was formed, and Barry Goldwater served as its first chairman. This followed his lengthy service as a US senator (since 1952) and an unsuccessful bid for president in 1964. (Scottsdale Public Library.)\n\n**M UMMY MOUNTAIN**. This is an aerial view of the south face and western section, which is just above the Paradise Valley Country Club golf course to the left. Note the single-family custom residences in the mountain's lower foothills. A reserve was formed in 1997 to preserve the mountain, and by 1999, four dozen properties were donated, totaling 187 acres for the reserve. Tax benefits contributed to the preservation of this scenic mountain landmark. (Town of Paradise Valley.)\n\n**M UMMY MOUNTAIN**. Judson School is in the foreground, with Mummy Mountain to the right. Earlier names for the 320-acre mountain were Windy Gulch and Horseshoe Mountain. Charlie Mieg, a longtime resident and landowner, felt such names did not have much marketing appeal. One day, while riding down the future Shea Boulevard alignment, he decided the mountain looked like an Egyptian mummy lying down, which accounts for its name. (Scottsdale Public Library.)\n\n**M ARJORIE THOMAS RANCH, C. 1909**. Marjorie Thomas, a well-known, Boston-trained artist whose paintings featured landscapes and animals, started her own art gallery at the east end of Mummy Mountain on Cheney Drive. Dr. Philip Schneider eventually purchased the area for his family home. (Scottsdale Historical Society.)\n\n**F LOCK OF SHEEP**. This image captures the earlier agrarian lifestyle of the Paradise Valley and Scottsdale area. (Scottsdale Public Library.)\n\n**C ATTLE DRIVE**. Cattle drives along Scottsdale Road were commonplace in the early 1900s. They were driven annually from the north Scottsdale ranches, through downtown Scottsdale, and west to the cattle stockyards on Washington Street. (Scottsdale Public Library.)\n\n**M CDONALD DRIVE, 1951**. This view, looking west along McDonald Drive toward Camelback Mountain, conveys the quiet, rural quality of life at that time. (Scottsdale Historical Society.)\n\n**C AMELBACK CEMETERY**. Hans and Mary Weaver owned a 160-acre homestead where cattle once grazed, and operated a sanatorium in town. Their daughter married Adolph Poenicke in 1915, and within three weeks, the 20-year-old Adolph died and was buried on the family property. In 1916, the Weavers deeded two acres of their property at 8300 East McDonald Drive as a cemetery. (Author's collection.)\n\n**C AMELBACK CEMETERY**. The cemetery became the final resting place for about 600 people, including many who died in the 1918 flu epidemic, Mexican-Catholics, military veterans, and members of prominent families such as Herberger, Kiser, and Powell. (Author's collection.)\n\n**C AMELBACK CEMETERY**. This carved granite headstone for Hans and Mary Weaver is located at their cemetery plot. (Author's collection.)\n_Two_\n\nTOWN OF PARADISE VALLEY  \nINCORPORATON AND  \nFACILITIES: 1949\u20132011\n\nAfter World War II, development became more active in what was then an unincorporated community in Maricopa County. Until that time, it was a rural area with a few scattered residences and an occasional commercial enterprise. The homes were smaller, but located on one to five acres. Phoenix and Scottsdale were considering annexations, including this valley, during the 1950s. Residents felt their rural lifestyle was under threat, fearing amplified commercial activity, denser zoning, and increased taxes. The Citizens Committee for the Incorporation of The Town of Paradise Valley, Arizona, circulated petitions to support incorporation beginning in 1949. This initiative was hotly debated. In April 1961, petitions were submitted to the Maricopa County Board of Supervisors and on May 24, 1961, incorporation was approved. At incorporation, the town was about 2.69 square miles and had a population of 1,200. The new 1964 Land Use Plan reflected restrictive zoning to limit commercial enterprise and blend it with the residential character. By 1969, the population was 6,460; Paradise Valley was the fasting growing city in the county.\n\nIn the 1980s, property annexations allowed the town to grow to 14 square miles and 11,000 residents. At this time, town leaders drafted the First General Plan, which reinforced their rural desert lifestyle. In 1972, the Hillside Protection Ordinance was enacted to discourage development on upper elevations while managing them on lower elevations, and in 1980, the Mummy Mountain Reserve was established to further preserve town land from development. In 1997, the Mummy Mountain Reserve Trust was created to maintain the natural landscape, wildlife, and aesthetics of the town hillsides.\n\nBy 2001, the town increased to 16.5 square miles with a population of 13,315. By this time, there were three water companies, four trash companies, and a fire service partnership with the City of Phoenix. The population appears to have peaked in 2000 at 13,664; by May 24, 2011, at the 50th anniversary, it dropped slightly to 12,280. At that time, there still remained two unincorporated Maricopa County islands within the town limits\u2014the Franciscan Renewal Center and the Clearwater Hills Subdivision.\n\n**I NCORPORATION MAP, C. 1961**. This map reflects the new boundaries of the Town of Paradise Valley. Henry Wick, the owner of Judson School, agreed to allow the 55-acre Mockingbird Lane property within the new town, which helped deliver the minimum land area required for incorporation. About this same time, citizens voted to name a few streets after birds, including Hummingbird Lane, Blue Bird Lane, Cactus Wren, and Mockingbird Lane. (Town of Paradise Valley.)\n\n**P ICNIC, C. 1960S**. This community-wide picnic was held in the vicinity of the Paradise Valley Racquet Club. (Town of Paradise Valley.)\n\n**T OWN MARSHALS**. This group photograph shows the early town marshals, who provided public safety. With fewer marshals in this smaller community, they invested in state-of-the-art communications and traffic controls as synchronization systems. In 1995, photographic radar was installed, a first in the United States. With follow-up data, it proved to lower the number of traffic accidents. (Town of Paradise Valley.)\n\n**R URAL METRO DEPARTMENT**. During the earlier decades, Rural Metro Department, a private company, provided fire service to the Town of Paradise Valley and county islands. In 1972, the town funded 13 new fire hydrants; along with others provided by developers, a more favorable fire insurance rating was received. In the late 1990s, the town developed a new fire service relationship with the City of Phoenix. (Scottsdale Public Library.)\n\n**E CHO CANYON PARK**. Echo Canyon Park is at the northwest side of Camelback Mountain and was defined by dramatic rock outcroppings. It has been used for picnics and as a staging area for hikers and rock climbers over the decades, as well as this c. 1910\u20131920 Ingleside Inn Resort Pow Wow. The park was officially formed by the Town of Paradise Valley in 1973. (Scottsdale Public Library.)\n\n**M OUNTAINTOP VIEW, 1974**. Town marshal Ronald W. Dalrymple and town manager Oscar A. Butt overlook the Town of Paradise Valley from Mummy Mountain. They were demonstrating how a camera could zoom in on potential intruders, thereby discouraging burglaries. (Town of Paradise Valley.)\n\n**P ARADISE VALLEY TOWN HALL, 1973**. The first town council met in a guesthouse on Quail Run Road and then rented a building on Malcomb Drive. The town acquired the 6401 East Lincoln Drive property for a municipal campus, and on April 8, 1973, broke ground for the new town hall with Mayor Bob Tribken, other elected officials, dignitaries, and citizens at the ceremony. (Town of Paradise Valley.)\n\n**P ARADISE VALLEY TOWN HALL, 1974**. Michael and Kemper Goodwin Architects of Tempe, Arizona, were hired to design the new town hall. Staff architect William P. Bruder worked on the design and later founded his own successful architectural practice. It was a 6,672-square-foot, tent-like structure with a clay tile roof, exposed rough-sawn lumber, and plastered walls. Reaction to the design included \"striking, but with subtle accents.\" The project was paid for by accumulated town funds. (Town of Paradise Valley.)\n\n**I NDIAN BEND WASH**. Northeastern Paradise Valley experienced periodic flooding. In 1949, the Berneil Ditch and Drainage Channel were constructed; later, in 1974, the ditch and channel were deepened and widened. In 1973, the town adopted the Flood Plain Regulations, which required the Indian Bend Wash be developed as a green belt, not unlike Scottsdale's flood control project. This model shows some of the Indian Bend Wash infrastructure improvements. (Scottsdale Public Library.)\n\n**T OWN OF PARADISE VALLEY MASTER PLAN, 1992**. To guide the future of the 12-acre municipal campus, a master plan was created. It showed the future police station, public services, town hall expansions, parking, vehicular and pedestrian circulation, and landscape. The architect was Jones Studio Inc. (Jones Studio Inc.)\n\n**H ILLSIDE PROTECTION**. The desire to protect the hillsides was formalized in the 1972 Hillside Building Regulations and 2003 Hillside Guidelines. Such regulations attempted to manage building volumes, wall heights, and coloration. Protection continued in the early 1980s with 100 acres of Mummy Mountain properties donated. In addition, repairs were made to the existing hillside scars with a new applied treatment. (Scottsdale Historical Society.)\n\n**B IL KEANE CARTOON, 1999**. Nationally recognized cartoonist and Paradise Valley resident Bil Keane provided this Mummy Mountain Reserve cartoon to encourage its preservation. (Town of Paradise Valley.)\n\n**D OUBLETREE RANCH ROAD MEDIAN**. 1972 brought a curvilinear roadway and landscaped median from Fifty-eighth to Sixty-second Streets, which slowed down vehicular traffic. In 1974, the first bicycle path from Scottsdale Road to Tatum Boulevard on this same street was constructed. Artist Lew Davis called his home, which was located on the current Cosanti property, Double Tree Ranch. This influenced the street name. (Author's collection.)\n\n**L INCOLN DRIVE MEDIAN**. The town completed the widening of Lincoln Drive from two to four lanes, with an integrated desert landscaped median, in 1977. Infrastructure improvement aesthetically transformed this main arterial street from Tatum Boulevard to Scottsdale Road. Lincoln Drive is named after John C. Lincoln, an industrialist from Cleveland, Ohio, who moved his family to the Phoenix area in the early 1930s. (Author's collection.)\n\n**T ATUM BOULEVARD CONSTRUCTION**. Tatum Boulevard construction is pictured south of Lincoln Drive looking toward Camelback Mountain during a street widening and median-improvement project. (Town of Paradise Valley.)\n\n**T ATUM BOULEVARD CURVE**. Tatum Boulevard (at about Mockingbird Lane) had a tightly curved section on a hillside that likely caused some serious accidents. The town reconstructed the street in 1990 with a more gradual curve and better sight lines. Tatum Boulevard is named after Russ F. Tatum, a real estate developer active in this area around the 1930s who unfortunately lost his properties during the Great Depression. (Author's collection.)\n\n**M OUNTAIN VIEW ESTATES**. This 1977 development was a major departure from the typical residential R-43 zoning with a 40-acre, single-family residential development that originally proposed 88 lots, but was legally reduced to 56. Exterior character reflected a horizontal profile, with some low-pitched roofs and shaded walls. John Rattenbury of the Frank Lloyd Wright Foundation was the architect; this project was one of their earliest ventures into mass-produced housing. (Authors' collection.)\n\n**P OLICE COURT**. In 1995, the town broke ground on a new Police Court building. The plastered masonry walls, steel shading devices, and entry canopies were compatible with the original Town Hall structure. The architect was DLR Group. (Author's collection.)\n\n**C HANNELIZATION OF INDIAN BEND WASH**. In December 1972, there were new federal flood plain regulations, which encouraged development of the Indian Bend Wash as a greenbelt. With such infrastructure improvements, residents became eligible for federal flood insurance, and channelization along its raised banks put properties outside a flood zone where they could become residential subdivisions. In 1983, a bridge was constructed at Doubletree Ranch Road to alleviate flooding. (Author's collection.)\n\n**V ETERANS' APPRECIATION VINTAGE CAR SHOW, 2009**. The Town of Paradise Valley mayor and council have been sponsoring the Veterans' Appreciation Vintage Car Show since 2004. It is an opportunity for residents to display and discuss their classic automobiles. The event also honors veterans and active-duty military. (Town of Paradise Valley.)\n\n**B ARRY GOLDWATER MEMORIAL**. The Barry Goldwater Memorial was constructed on 1.25 acres in 2004 at the northeast corner of Tatum Boulevard and Lincoln Drive. The memorial functions as a quiet park retreat with desert landscaping, low stone walls, seating, and interpretative markers. The landscape architect was Michael Dollin of Urban Earth Design LLC, who describes it as a \"scenic public place.\" (Author's collection.)\n\n**B ARRY GOLDWATER MEMORIAL**. The focal piece within the memorial is a cast bronze Barry Goldwater sculpture by Joe Beeler, a member of the Cowboy Artists of America. The sculpture is facing southwest (where Goldwater's original residence was located) and the dramatic western sunsets, which Senator Goldwater enjoyed. (Author's collection.)\n\n**F IRE STATION NO. 2, 2009**. Fire Station No. 2 is located at 6539 East Lincoln Drive, just east of the existing municipal campus. This 9,500-square-foot structure provides invaluable fire protection and life saving services. It is one story, with varied sculptural volumes, cantilevered shade canopies, and natural materials such as exposed masonry, metal roofing, and tinted glazing. Desert landscaping provides a pleasant and appropriate setting. The architect was LEA Architects LLC. (LEA Architects LLC.)\n\n**F IRE STATION NO. 2, 2009**. This exterior view shows the three vehicular bays with a stepped floor plan to mitigate the building's volume. In addition, glazed roll-up doors display the firefighting equipment to the street and public, and a continuous clerestory above admits daylight into the equipment area. (LEA Architects LLC.)\n\n**F IRE STATION NO. 1, 2010**. The replacement facility for the original 1974 Rural Metro Fire Station was constructed at 8444 North Tatum Boulevard. The 8,000-square-foot facility was constructed of exposed masonry walls, rusting steel pitched roofs, shade canopies, and tinted glazing. The angled roof profile is a metaphorical reference to the surrounding Phoenix Mountains. The architect was LEA Architects LLC. (LEA Architects LLC.)\n\n**U NIVERSAL SOLAR SHADOWGRAPH**. Jeff Zischke's 1992 sculpture has rusted plate steel components 15 feet high by 7.5 feet wide. The piece was commissioned by the Arizona Commission on the Arts, fabricated in Detroit, and first exhibited in Tempe. It was then on loan to the Town of Carefree followed by the Town of Paradise Valley, which purchased it in 2007. The town has a long heritage of supporting its community artists, specifically with the Town Art Jury, initiated in 1974. (Author's collection.)\n_Three_\n\nACADEMIC PURSUITS:  \n1928\u20132008\n\nThe roots of academic opportunity for Paradise Valley residents started in 1928 with the private college preparatory academy named Judson School. Since that time, a number of other private schools and two public schools have opened; most of them continue to function today.\n\nJudson School provided kindergarten through 12th grade on a 55-acre property at the east end of Mummy Mountain. The school graduated a host of well-known citizens over the years. Unfortunately, it closed in 1999 when the property was sold to Cachet Homes. Existing structures were demolished in 2000. The property was then redeveloped into a custom residential subdivision.\n\nDuring the 1950s, the Desert Art Gallery and School was offering classes by professional fine artist Lew E. Davis and his wife, sculptress Mathide Schaefer Davis. The school site was at the southeast corner of Invergordon Road and Lincoln Drive. This site eventually became the Town of Paradise Valley municipal campus in the early 1970s.\n\nScottsdale Unified School District constructed the first public school in Paradise Valley, the 1957 Kiva Elementary School. In 1974, the district constructed a second school, Cherokee Elementary School.\n\nIn 1960, Mae Sue and Franz Talley started the Talley Academy on Stanford Drive but soon encountered zoning and financial challenges. Such difficulties were overcome, and the school was renamed Phoenix Country Day School in 1961.\n\nParadise Valley Day School opened about 1960 at 6602 East Malcolm Drive. Kachina Country Day School then purchased the property in July 1983 and offered kindergarten through fourth grade classes.\n\nCamelback Desert School operated at 6050 North Invergordon Road from 1952 to 2009 and is now the private Montessori Academy that provides preschool and a charter kindergarten through middle school programs.\n\nTesseract School, a private school that is among other facilities in the Phoenix metropolitan area, was constructed at Tatum Boulevard and Doubletree Ranch Road in 1988. It serves kindergarten through fourth grade.\n\nBy the 2000s, there were 13 public and private schools in the town of Paradise Valley. Educational resources were also extended to numerous reputable preschools, managed by various religious organizations.\n\n**J UDSON SCHOOL, 1928**. This aerial view shows the original Judson School campus within the open desert and the cluster of one-story plastered adobe structures. They once stood on 55 acres of a remote desert site at North Mockingbird Lane, named after founder George A. Judson. (Scottsdale Historical Society.)\n\n**J UDSON SCHOOL, 1928**. The academy served coed boarding and day students for kindergarten through 12th grade. The Wick family, and more recently Kent Wick, vice president, managed the school from 1945 until it closed in June 2000. The school was the oldest independent college preparatory school in Arizona. (Scottsdale Historical Society.)\n\n**J UDSON SCHOOL**. Corrals and buildings are pictured at Judson School, which was located at the east end of Mummy Mountain. After the school closed in 2000, the land was developed as a residential community with high-end custom homes. (Scottsdale Public Library.)\n\n**J UDSON SCHOOL, C. 1963\u20131964**. This is a typical fourth grade classroom. (Beth Wickstrom.)\n\n**J UDSON SCHOOL WAGON**. Judson School often took field trips with its young students. In the early days, they could go across Paradise Valley and not see anyone because of the lack of development. (Town of Paradise Valley.)\n\n**D ESERT ART GALLERY AND SCHOOL, 1956**. During the 1950s at the southeast corner of Invergordon Road and Lincoln Drive, painter Lew E. Davis and his wife, sculptress Mathilde Schaefer Davis, taught beginning and advanced adult art classes. Susan Ferguson also taught children's art classes. The school hosted many multicultural events, including this open air Sunday school class for the newly formed Valley Presbyterian Church. The structure was eventually removed to allow for the construction of a new Paradise Valley Town Hall. (Flora M. Swanson, Valley Presbyterian Church.)\n\n**C AMELBACK DESERT SCHOOL**. This private preschool was started in 1950 by Marian Moore and Paula Nelson in Phoenix, and then in 1952 at 6050 North Invergordon Road. William Barton purchased the school in 1970 and expanded programs to include grades one through eight. The four-acre campus has evolved over the years with additional one-story structures, playgrounds, and various ball fields. Since 2009, the property has been owned by the Montessori Academy. (Author's collection.)\n\n**E ARTH HOUSE, 1992**. Paolo Soleri arrived at Taliesin West in 1953 and purchased five acres at 6433 East Doubletree Ranch Road in 1956. From 1956 to 1957, construction was underway on his Earth House, which experimented with casting concrete shells over sculpted earth. Later techniques included structures elevated with vertical columns and cooler microclimates based on an understanding of solar angles and natural winds. (Author's collection.)\n\n**K IVA ELEMENTARY SCHOOL, 1957**. Located at 6911 East McDonald Drive, Kiva Elementary School was the first public school built in Paradise Valley and operated by the Scottsdale Unified School District. It originally served kindergarten through eighth grade. Building components are prototypical in nature, arranged with long east-west classroom wings for better solar orientation, and connected by shaded breezeways. Glazing faces north for soft, diffused daylight in each classroom. The architect was David G. Haumerson. (Scottsdale Historical Society.)\n\n**C HEROKEE ELEMENTARY SCHOOL**. Cherokee Elementary School is a public Scottsdale Unified School District Pre-kindergarten through fifth grade program located at 8801 North Fifty-sixth Street. Originally, it served kindergarten though eighth grade in the 1974 structures. Structures are exposed concrete wall panels with shaded arcades and a fascia of castellated steel beams. The architect was Michael and Kemper Goodwin. (Author's collection.)\n\n**T ALLEY ACADEMY, 1972**. Mae Sue and Franz Talley started a private school in 1960 at 3901 East Stanford Drive. Persevering over zoning restrictions and debts, the school survived to become the Phoenix Country Day School in 1961 and grew to a kindergarten though eighth grade school. Original structures were plastered, painted white with red clay tile roofs, and had steel-framed windows. The architect for the 1964 round Science Building was Gonzales and Ludlow. (Phoenix Country Day School.)\n\n**P HOENIX COUNTRY DAY SCHOOL, 1994**. Knoell & Quidort Architects has completed a series of additions to the 108,000-square-foot campus that grew to 40 acres. In the mid-1990s through 2008, most classrooms were replaced or renovated. Gunnar Birkerts, FAIA, of the Regional AIA Award Jury, praised the \"simple arrangement of stuccoed walls, colonnade and courtyard, and how, through minimal means, it created new public space for the school. This is a good example of regional architecture; it has a sense of dignity and sets a good precedent for the campus.\" (Knoell & Quidort Architects.)\n\n**P ARADISE VALLEY DAY SCHOOL**. This private preschool through eighth grade school, located at 6602 East Malcomb Drive, was built on four acres around 1960 by Carl and Kacki Vinsel. The school eventually closed. Kachina Country Day School purchased the property in 1983, and the land expanded to seven acres. This school would grow from 180 students to 300, but the owners attempted to sell it in 2012. The classroom had good orientation, with north-facing windows that allowed daylight to enter. (Author's collection.)\n\n**T ESSERACT SCHOOL**. This nonprofit school was founded in 1988, and since 2001 had its lower school campus at 4800 East Doubletree Ranch Road for pre-kindergarten through fourth grade. It is a Post Modern design with a symmetrical front facade. (Author's collection.)\n_Four_\n\nDESERT RANCHES,  \nRESORTS, CLUBS, AND  \nRECREATION: 1928\u20132000\n\nAlthough the 1930s Great Depression hit local farmers and businesses severely, new guest ranches and resorts were attracting wealthy easterners to the Paradise Valley and Scottsdale area.\n\nIn 1936, Camelback Inn was jointly developed by Jack Stewart and John C. Lincoln on the southern slopes of Mummy Mountain. The Pueblo Revival architecture conveyed an inviting and relaxed desert environment for its guests.\n\nEl Chorro Lodge was a 1937 adaptation from the Judson School for Girls into a guest lodge and restaurant. This same year, the Squaw Peak Inn was built. In 1943, it became a guest ranch.\n\nAfter World War II, there was a new need for guest ranches, resorts, and tourist accommodations. Americans and foreigners had a desire for travel throughout the United States given the affordable and efficient trains, aircraft, vehicles, and improved highway system. Television and movie Westerns also added to the allure of this special place in the desert.\n\nParadise Valley started attracting visitors in the 1940s and 1950s to a series of desert guest ranches along Lincoln Drive and Invergordon Road. They included the Sun 'N Sage, Desert Lark Ranch, Flying T Ranch, and Desert Lodge. As more luxurious resorts were developed on many of these same properties, the original ranches were removed by the 1960s.\n\nThe 1950s were active with new clubs and resorts being built, such as the 1953 private Paradise Valley Country Club with an 18-hole golf course and large swimming pool. Paradise Valley Racquet Club on Camelback Mountain was built in 1957 and evolved into the 1969 John Gardiner's Tennis Ranch, and ultimately into The Sanctuary in 2001. Jim Paul developed Mountains Shadows Resort in 1958, which proved a popular destination for the next 45 years.\n\nIn 1956, _Life Magazine_ featured Paradise Valley and Scottsdale as a place for the rich and famous, contributing to its appeal and attracting more visitors and residents.\n\nThe 1960s brought the new Shangrila Resort, and in the 1970s, the 1971 Camelback Golf Club, 1975 Scottsdale Sheraton Resort, and the 1978 La Posada Resort. During this time, the town adopted the 1974 Regulations Governing Resorts and Hotels. Later, the 1980 Los Alamos Resort and 1984 Loew's Paradise Valley Resort were developed along Scottsdale Road.\n\nBy the 2000s, there were 12 resorts in the Town of Paradise Valley.\n\n**C ASA HERMOSA, 1941**. The original 1928\u20131930 Los Arcos sat on six acres at 5532 North Palo Christi Road and was later named Casa Hermosa. Local cowboy artist Alonzo \"Lon\" Megargee III built the home by hand. He was responsible for the design and construction of over half a dozen residences in Paradise Valley, Phoenix, and Sedona. This historic Spanish Colonial and Pueblo Revival styled hacienda with a one room studio became the Hermosa Inn in 1947. A fire in 1987 required its closure for several years. (Hermosa Inn.)\n\n**C ASA HERMOSA**. The interior of the original Casa Hermosa with Alonso \"Lon\" Megargee III is shown here. Walls were constructed of 42-inch-thick adobe with posts and beams of old railroad beds and bridges. Jennifer and Fred Unger bought the 6.5-acre property in 1992 and expanded the existing 17 guest rooms to 35 in 2003. Some rooms have beehive fireplaces and private patios. The original property offered a swimming pool, golf course, and horseback riding. (Hermosa Inn.)\n\n**A LONSO MEGARGEE III**. Born in 1883, Megargee came to Arizona from Philadelphia at 13 to be a cowpuncher. He toured with a wild west show, ranched in Cave Creek, and was also a poker dealer, artist, and builder. Governor Hunt commissioned Megargee to paint murals at the Arizona Capitol for $7,000. He became a prolific cowboy artist with paintings, wood-block prints, pulp magazine covers, beer advertising, and the Stetson company logo to his credit. He moved to Sedona in 1947, where he died in 1960. (Hermosa Inn.)\n\n**E L CHORRO LODGE, 1934**. The original structure sat on 11 acres at 5550 East Lincoln Drive and was home to the Bell family, later becoming the Judson School for Girls (built by John C. Lincoln), which closed after only two years. The Bells then sold to the Judsons, who in turn sold to Mark and Jan Gruber. The Grubers adapted the property into a guest ranch and restaurant in 1937, managing the lodge for 50 years. (Scottsdale Historical Society.)\n\n**E L CHORRO LODGE, 1934**. Seen here is the original interior with exposed adobe brick walls, wood lintels, and local artisan features. (Town of Paradise Valley.)\n\n**E L CHORRO LODGE, 2010**. The 27,397-square-foot lodge was sold by Mark and Jan Gruber to Joe Miller and eventually to Jacque Dorrance. In 2010, it reopened after architect Candelaria Design Associates LLC designed a major remodel and expansion project. New fireplaces, solar panels, and bocce courts were provided, while saving two original _casitas_ that still stand on the front lawn. _El Chorro_ comes from Peruvian Spanish and means \"running stream.\" (Mark Boisclair Photography, Desert Star Construction.)\n\n**E L CHORRO LODGE, 2010**. The remodel was respectful of the earlier architecture, including the plastered walls, sculpted quality, and crafted detailing. The project achieved a Leadership in Energy and Environmental Design (LEED) Gold Level Certification, which demonstrates that it is energy-efficient and environmentally conscious. (Mark Boisclair Photography, Desert Star Construction.)\n\n**C AMELBACK INN**. This aerial view shows a cactus-studded oasis of 125 acres for 200 guests. The 1936 seasonal resort was constructed at a time when there were no lights, telephone, or water on-site. It did feature 75 tiny casitas, a glass-enclosed swimming pool, horseback riding, and two tennis courts. The designer, Edward Loomis Bowes, settled in the area in 1920 for his wife's health. He was an engineer, furniture designer, and photographer who graduated from Washington University in St. Louis. (Scottsdale Public Library.)\n\n**C AMELBACK INN, 1936**. John C. Lincoln and Jack Stewart jointly invested to create a destination resort at 5402 East Lincoln Drive and started marketing to affluent travelers in the upper Midwest. The adobe architecture conveyed a southwestern Pueblo Revival character. A series of outdoor patios and courtyards and an inviting casual ambiance presented this desired character. The inn was annexed into the town in 1983. (Scottsdale Historical Society.)\n\n**C AMELBACK INN, C. 1930S**. This view of Camelback Inn shows guests on horseback with Camelback Mountain as the backdrop. Other recreational activities included swimming, tennis, and desert picnics. (Scottsdale Public Library.)\n\n**J ACK STEWART**. The original manager of the Camelback Inn was Jack Stewart, pictured here on horseback, who was in a partnership with John C. Lincoln, president of Cleveland Electric Company. Stewart sold the venerable resort to J.W. Marriott in 1967. (Scottsdale Public Library.)\n\n**C AMELBACK INN**. Marriott expanded the property to 453 rooms and added a world-class spa in 1989. This image shows the additions that were completed later. The inn continued to use its original slogan: \"Where time stands still.\" (Verner Wulf, AIA, Phoenix Metro Chapter.)\n\n**K IAMI LODGE, 1937**. A 10-acre guest ranch was developed within a citrus grove on the west side of Scottsdale Road. The Indian style was reflected in the interior d\u00e9cor, furniture, artifacts, and fine art. Artist Charles Loloma painted murals and graphics there. (Scottsdale Historical Society.)\n\n**C ASA BLANCA INN, 1944**. The 1924 Donald Kellogg residence at 5101 North Sixty-sixth Street was purchased in 1944 by George Borg of Borg Warner Company in Delavan, Wisconsin. In 1946, Borg opened the Casa Blanca Inn with 30 rooms for 80 guests, a private landing strip for small planes, stables, swimming pool, putting green, shuffleboard, and tennis. Eighteen additional rooms were added in 1948. Royal Treadway leased it from Borg in 1952 and bought it in 1961. (Scottsdale Historical Society.)\n\n**H ORSE SHOW, C. 1945**. This image shows Larry Walton riding \"Colonel Wood\" at a horse show on the Tennessee Walking Horse Ranch, owned by Capt. Tom Hog, which later became Mountain Shadows East. Mummy Mountain is in the background. (Robert Markow, Beth Wickstrom.)\n\n**D IAMOND LAZY K RANCH**. In 1957, this five-acre resort was located at 7100 East Lincoln Drive and featured 26 bungalows managed by owner Karl Bailey Johnstone. The advertisement states: \"You are cordially invited to be a guest Sun-Aire at Diamond Lazy K . . . where moments make memories. Play badminton, shuffleboard court, putt your choice of two greens, swim, hike\u2014or just laze around in the knock-about togs. Do anything that pleases you, in warm sunshine mixed with exhilarating dry desert air.\" It was named Smoke Tree Resort in 1962. (Author's collection.)\n\n**C ASA VERDE GUEST LODGE**. The lodge at 6730 East McDonald Drive provided a swimming pool and sun-drenched patios, with a golf course and saddle horses nearby. Individual guesthouses included a kitchen and lodge, hosted by the Haydens. (Author's collection.)\n\n**F LYING 'T' RANCH**. The 1951 Ranch on East Lincoln Drive offered four _ranchitos_ and a swimming pool with an underwater observation window. The owner was Frank S. Tilyou; the ranch closed in the 1960s. (Author's collection.)\n\n**Y ELLOW BOOT RANCH, 1953**. This ranch was located five miles north of Scottsdale at 6726 East Doubletree Road. \"Picturesque Resort Apartments all with kitchenettes, surrounding a green oasis on the sunny desert in the heart of Paradise Valley. Quiet, away from traffic, yet only eight minutes from town. Beautiful unobstructed view, large heated pool, shuffleboard court, central lounge. A friendly place for friendly people to relax and can accommodate 30 guests.\" The resident owners were Bob and Louise Balch. (Bob Balch, Tom Silverman.)\n\n**D ESERT LODGE**. Jimmie and Mary Belle Schildman turned their own 1940s home at 6167 North Invergordon Drive into the Desert Lodge and offered guests the \"American or European Plan.\" A shuffleboard court, golf course, and heated pool were the amenities provided. In 1968, the property was purchased by the new Calvary Church of the Valley. (Associated Desert Lodges.)\n\n**K ERR CULTURAL CENTER**. Musician and philanthropist Louise Lincoln Kerr (right) hosted many events at her namesake Kerr Cultural Center. The center is located at 6110 North Scottsdale Road, where a 1943, one-story Spanish Colonial Revival\u2013style house stands. In 1959, a 210-seat studio addition was constructed; later, another addition was designed by architect Fred Fleenor. The center, now owned by Arizona State University, has served as an intimate venue for visual and performing artists over the years. (Scottsdale Historical Society.)\n\n**P ARADISE VALLEY COUNTRY CLUB, C. EARLY 1950S**. This aerial view of the 270-acre Paradise Valley Country Club shows the first phase of construction with the ranch-style homestead and the 1953 18-hole golf course designed by Lawrence Hughes. (Scottsdale Historical Society.)\n\n**P ARADISE VALLEY COUNTRY CLUB, 1953**. A second phase of construction was managed by architectural firms Edward L. Varney Associates, AIA, and Hiram Hudson Benedict. Designer Dean Rendahl of Varney's office was also responsible for the Phoenix Country Club and Moon Valley Country Club of this same era. The rendering was by architect Hiram Hudson Benedict. (Paradise Valley Country Club.)\n\n**P ARADISE VALLEY COUNTRY CLUB, 1953**. Located at 7101 North Tatum Boulevard, this private club was initiated by investors and community leaders. The main two-story structure was constructed of natural stone, concrete block with vertical precast concrete inserts, tall glazing, and cantilevered overhangs faced with a decorative, precast concrete panel. The property was annexed into the town in 1984. (Edward L. Varney Associates, AIA.)\n\n**P ARADISE VALLEY COUNTRY CLUB, 2004**. In 2002, the original structure was demolished to construct a new, eclectic country club for its membership designed by Swaback Partners, PLLC. Major reconstruction involved two phases of construction, completed in 2003 and 2004. (Swaback Partners, PLLC.)\n\n**S HANGRILA RESORT, C. 1960S**. This resort was quaint and small-scale, with one-story structures focused on a swimming pool and desert landscaping. It was removed later for a new, large residence. (Scottsdale Historical Society.)\n\n**P ARADISE VALLEY RACQUET CLUB, 1957**. This 53-acre development at 5600 East McDonald Drive was comprised of a clubhouse, studio cottages surrounding the clubhouse, five tennis courts, dining room, cocktail lounge, dance floor, recreation room, and a swimming pool. Hollywood investors included John Ireland, Joanne Dru, and Sydney Chaplin, Charlie Chaplin's son. (Scottsdale Historical Society.)\n\n**P ARADISE VALLEY RACQUET CLUB, C. LATE 1950S**. The design was similar to the Racquet Club of Palm Springs. Natural materials included exposed concrete block, horizontal plank siding, and recessed window glazing. The architect was Hiram Hudson Benedict, who was also active in Scottsdale and Cave Creek, Arizona, and San Diego, California, during the 1950s and 1960s. (Scottsdale Historical Society.)\n\n**P ARADISE VALLEY RACQUET CLUB**. The club served as a backdrop for this model photograph session. Note the natural stone wall details and horizontal wood plank siding above. (Associated Desert Lodges.)\n\n**J OHN GARDINER'S TENNIS RANCH, 1969**. The original Paradise Valley Racquet Club became the John Gardiner's Tennis Ranch in 1969. Investors Vic Jackson and Les Heitel constructed 41 hillside rental casitas designed by the architectural firm Benedict and Caviness. The ranch hosted many celebrity tennis tournaments and attracted tennis professionals like Pancho Seguero, Ken Rosewall, and Pancho Gonzalez. The ranch was most active from 1969 to 1992 and closed in 2000. (Allen + Philp Architects\/Interiors.)\n\n**S ANCTUARY ON CAMELBACK MOUNTAIN, 2001**. The original John Gardiner's Tennis Ranch was renamed The Ranch on Camelback Mountain and then the Sanctuary on Camelback Mountain. The 53-acre sanctuary was redesigned and master-planned with 82 remodeled casitas, 12 home sites, a negative-edge swimming pool, spa, fitness center, and the Elements Restaurant, with its entry shown here. The architect was Allen + Philp Architects\/Interiors. (Allen + Philp Architects\/Interiors.)\n\n**S ANCTUARY ON CAMELBACK MOUNTAIN, 2001**. Twenty-four new spa villas spill down the hillside, accessed by pleasant pedestrian walkways, and framing dramatic views of Paradise Valley beyond. The original tennis courts were used to develop these suites and the spa, fitness center, and pools. (Allen + Philp Architects\/Interiors.)\n\n**S ANCTUARY ON CAMELBACK MOUNTAIN, 2001**. The Elements Restaurant interior saved the original glazing and exposed wood ceiling and maintained the Camelback Mountain \"Praying Monk\" and sunset views to the west. Restaurant interior designer Judith Testani worked with architect Allen + Philp Architects\/Interiors. (Allen + Philp Architects\/Interiors.)\n\n**S ANCTUARY ON CAMELBACK MOUNTAIN, 2001**. The spa suite interior has a character that is intimate, refreshing, romantic, and contemporary, with bold accent colors and textures. This design approach helped reposition the iconic tennis resort into a spa and fitness-based experience. (Allen + Philp Architects\/Interiors.)\n\n**M OUNTAIN SHADOWS RESORT, 1959**. Located at 5641 East Lincoln Drive, this 55-acre resort development was created by developer Jim Paul. The year-round resort was defined by 100 guest rooms initially and was surrounded by an executive 18-hole golf course designed by Arthur Jack Snyder. The architecture captured the post\u2013World War II era with sweeping guest wings, full-height glazing, and a large courtyard with an expansive swimming pool, green turf, and palm trees. (Scottsdale Historical Society.)\n\n**M OUNTAIN SHADOWS RESORT, 1959**. Jim Paul sold 50 percent of the resort to Del Webb Corporation in 1961 and the remaining half in 1964. Surrounding the resort were 59 lots in two single-family subdivisions. The resort was purchased by Host Marriott Corporation in July 1981 and closed in 2004. The popular resort had a relaxed and sophisticated ambiance. (Scottsdale Historical Society.)\n\n**M OUNTAIN SHADOWS RESORT, 1959**. Marriott expanded the property to 337 guest rooms and added dining options, eight lighted tennis courts, three swimming pools, a sand volleyball court, a fitness center, and meeting space for 10 to 1,000 people. The resort site was eventually annexed into the Town of Paradise Valley in 1992. (Scottsdale Historical Society.)\n\n**S COTTSDALE SHERATON RESORT, 1975**. Spanish Colonial Revival-style resort at 7200 North Scottsdale Road was later renamed Scottsdale Plaza Resort. The architecture includes textured plaster walls, arched openings, broken rooflines, clay tile roofs, small-paned windows, and an oasis-like landscape. (Scottsdale Historical Society.)\n\n**R ENAISSANCE COTTONWOODS RESORT, 1980**. At 6160 North Scottsdale Road, a resort opened originally as the Alamos Resort Hotel that included 223 hotel rooms and suites, 24 cabana casitas, 34 luxury villas, a spa, restaurant, and grand ballroom. Structures were made of plastered walls and had flat-arched features and stepped forms to reduce the scale. The architect was Bennie Gonzales Associates. (Scottsdale Historical Society.)\n\n**L OEW'S PARADISE VALLEY RESORT, 1984**. The resort at 5401 North Scottsdale Road later became a Wyndham and Hilton Doubletree. Buildings exploit custom-formed masonry units that provide deep shadow patterns, thereby self-shading and cooling the walls in this hot climate. Shaded arcades knit together the building components and courtyards. (Author's collection.)\n\n**C AMELBACK GOLF CLUB, 2000**. The golf course at 7800 North Mockingbird Lane was initially developed for Camelback Inn guests on a 200-acre parcel. The Padre Course, with 18 holes, was designed by Red Lawrence initially, redesigned by Arthur Hills, and renamed the Resort Course. In 2007, it reverted back to its original Padre name. The original 1969 golf club structure was demolished and a new clubhouse was built in 2000. The Pueblo Revival style uses plastered walls and smaller window openings. (Author's collection.)\n\n**M ONTELUCIA RESORT AND SPA, 2010**. The former 1978 La Posada, Red Lion, and Doubletree Resort property received a major, 285,000-square-foot expansion and redevelopment that provided 293 hotel rooms, 34 luxury residences, a spa, a renovated wedding chapel, and four swimming pools. The overall plan respected the original dry washes and celebrated the character of a Spanish Andalusian village with plastered walls, accent colors, detailed iron grills, shaded arcades, and a series of courtyards. The architect was Allen + Philp Architects\/Interiors. (Allen + Philp Architects\/Interiors.)\n_Five_\n\nRELIGIOUS EXPRESSIONS:  \n1950\u20131990\n\nAs the Paradise Valley area population grew in the 1950s and 1960s, a desire for religious facilities also grew. A number of congregations were founded years earlier outside the current town boundaries. They would rent movie studios, Quonset huts, credit union basements, resorts, and galleries. Some had difficulties finding space to rent given their beliefs and felt the need to construct new facilities. During this time, some larger land parcels were gifted or acquired by the different organizations, given that they were relatively affordable and could accommodate the growing congregations.\n\nWithout exception, each of the congregations found very creative architects to implement their respective visions. Barney \"Bennie\" Gonzales, FAIA, was most prolific with having been responsible for the design of the 1961 Gloria Dei Lutheran Church, 1966 Har Zion on the Desert, 1966 Scottsdale Bible Church, and 1967 Christ Church of the Ascension. Other talented architects included T.S. Montgomery for the 1961 Episcopal Parish Saint Barnabas on the Desert, who was also responsible for many other local church designs. Blaine Drake, a successful disciple of Frank Lloyd Wright, designed the 1961 First Unitarian Universalist Church of Phoenix and numerous other churches in the area. Harold Wagoner, a Philadelphia architect, designed the 1958 Valley United Presbyterian Church and was also responsible for over 500 church designs nationally.\n\nThe vast majority of such religious campuses in Paradise Valley have grown with additional programs, services, and structures over the past 50 years. The Town of Paradise Valley adopted in November 1976 a special use category for memorial gardens on religious properties.\n\nEach religious structure and outdoor space strived to express its congregation's beliefs and visions in a meaningful way. The interior lighting, finishes, furnishings, artifacts, and stained glass panels have provided contemplative retreats and inspiration over the years.\n\nBy the 2000s, the Town of Paradise Valley had 14 churches and houses of worship.\n\n**F RANCISCAN RENEWAL CENTER**. A former Echo and Kachina Lodge guest ranch was built on 20 acres that could accommodate 70 guests at 5802 East Lincoln Drive. The lodge was purchased by John B. Mills, operator of the Hotel Westward Ho, and G.C. Callerman, a Chicago food broker. They sold the lodge in 1951 to the Franciscans of St. Barbara Province, who adapted it to the Franciscan Renewal Center or Casa de Paz y Bien, which means House of Peace and Good. Friar Owen De Silva envisioned the center to be a place with spiritual appeal, where people would \"come apart into a desert place.\" (Scottsdale Historical Society.)\n\n**F RANCISCAN RENEWAL CENTER, 1952**. The main building was remodeled into the Retreat Center, and the first retreat was held January 14, 1952. A new chapel broke ground soon thereafter on June 19, 1954. In 1955, approximately five acres were acquired just west of the main property. The original 20 acres continues to be a Maricopa County island within the town. (Franciscan Renewal Center.)\n\n**F RANCISCAN RENEWAL CENTER**. This is the view north of Lincoln Drive after the Franciscan Renewal Center sign was erected on Lincoln Drive in the 1950s. (Scottsdale Historical Society.)\n\n**S AINT CLARE CHAPEL, 2004**. Located at 5808 East Lincoln Drive, a chapel dedicated to Saint Clare (whose name means light) was added to the Franciscan Renewal Center. The chapel was designed for quiet meditation and expressed Franciscan ideas of connection and mystery. The architect was Colab Studio LLC. (Timmerman Photography Inc., Colab Studio LLC.)\n\n**S AINT CLARE CHAPEL, 2004**. Art and architecture utilize Arizona's abundant sunlight. Existing stained-glass panels were reused atop the outer wall as colored and patterned shadow makers. A plaster relief of St. Clare adorns the entrance to the 250-square-foot chapel. The interior is clad in solid walnut with seats carved into the walls to strengthen one's connection to the space. (Timmerman Photography Inc., Colab Studio LLC.)\n\n**V ALLEY PRESBYTERIAN CHURCH, 1958**. This church was established March 18, 1956, with 70 charter members and first met in the Desert Art Gallery and School. The church purchased five acres, followed by an additional five acres. Within two years it had 700 members and architect Harold E. Wagoner of Philadelphia proposed this design. In 1958, the new education spaces, office buildings, and Landes Hall (on the right) were built at 6947 East McDonald Drive. (Petley Studios, Flora M. Swanson, Valley Presbyterian Church.)\n\n**V ALLEY PRESBYTERIAN CHURCH, 1976**. The congregation worshipped in Landes Hall until 1966, when the 800-seat sanctuary was built. This view shows the sanctuary beyond and the \"connecting link\" under construction in 1976. The sanctuary was constructed of massive cast-in-place concrete beams for the roof structure and tall bearing walls of stone from the Prescott area. (Flora M. Swanson, Valley Presbyterian Church.)\n\n**V ALLEY PRESBYTERIAN CHURCH, 1983**. A round chapel with radiating concrete beams and upper stained glass panels by space artist Robert McCall was then constructed and completed in 1983. The structure was staged for a future steeple, which was built a few years later. (Flora M. Swanson, Valley Presbyterian Church.)\n\n**F IRST UNITARIAN UNIVERSALIST CHURCH OF PHOENIX, 1961**. This church was founded in 1947, with lines to the 1620 Pilgrim Church founded in Plymouth. Here, construction is well underway with curved and exposed slump block walls. Cantilevered from the walls are thin-profiled shade canopies at the entry and upper windows. The smaller windows will admit dappled light to the interior, while minimizing the heat from this severe exposure. (Unitarian Universalist Congregation of Phoenix.)\n\n**F IRST UNITARIAN UNIVERSALIST CHURCH OF PHOENIX, 1961**. In the late 1950s, this congregation acquired a nine-acre parcel at 4027 East Lincoln Drive for a new campus. They intended to commission Frank Lloyd Wright, but he passed away in 1959. Architect Blaine Drake, a disciple of Wright, was then approached and provided the project. An approximately 19,000-square-foot structure was built and included an elliptical sanctuary and a curved classroom wing focused on an outdoor courtyard. (Unitarian Universalist Congregation of Phoenix.)\n\n**B IRMINGHAM STATUARY, 2003**. This statuary is located at the Unitarian Universalist Congregation of Phoenix campus. The 1963 bronze sculpture plaque reads: \"That Which Might Have Been. Birmingham 1963. Symbolizing the unfulfilled maturity of the four girls killed in the church bombing in Birmingham, Alabama, September 15, 1963. Dedicated to the understanding of the beauty of individual difference. John Henry Waddell.\" This particular event marked a turning point in the 1960s Civil Rights movement and contributed to support and passage of the 1964 Civil Rights Act. (Author's collection.)\n\n**T HE EPISCOPAL PARISH OF SAINT BARNABAS ON THE DESERT, 1961**. This parish started in 1953 and met at a downtown Scottsdale Quonset hut. The church was built at 6715 North Mockingbird Lane and was designed by architect T.S. Montgomery. The exterior walls were a mortar-washed slump block and pitched roof forms. A shaded arcade connected multiple buildings on this campus. The overall scale of its various components and between the buildings is elegant and well proportioned. Fowler and Anne McCormick and Anne's son Guy Stillman were early patrons and congregation members. (Scottsdale Historical Society.)\n\n**T HE EPISCOPAL PARISH OF SAINT BARNABAS ON THE DESERT, 2003**. This interior view of the renovated main sanctuary reflects sensitivity for the original ambiance despite integrating a new organ. Original indirect light sources have been maintained and continue to provide a visual depth to the contemplative space. The original interior was outfitted by artists Allen Ditson and Lee Porzio. The renovation architect was Knoell & Quidort Architects. (Dino Tonn Photography, Knoell & Quidort Architects.)\n\n**T HE EPISCOPAL PARISH OF SAINT BARNABAS ON THE DESERT, 2003**. An addition included a new music hall, library, offices, and coffee shop designed by Knoell & Quidort Architects. The addition is a contemporary interpretation of the original off-white plaster walls, while introducing natural daylight from above and into the interior spaces. (Richard Maack, Knoell & Quidort Architects.)\n\n**G LORIA DEI LUTHERAN CHURCH, 1961**. This church, located at 3539 East Stanford Drive, is constructed of slump block walls that are given a painted mortar wash. This technique creates a rubble texture that plays well in the intense southwestern light, self-shades the wall surface, and minimizes the heat gain during the hotter summer months. Bennie M. Gonzales, FAIA, was the architect. (B.J. Gonzales.)\n\n**G LORIA DEI LUTHERAN CHURCH, 1961**. The exterior window openings were created with wood-frame formwork when laying the surrounding block. The formwork was then removed, and custom stained glass panels inserted. The church provides Bible study and classes. (B.J. Gonzales.)\n\n**A SCENSION LUTHERAN CHURCH, 1963**. This church was founded in 1951, rented space initially, and then built its first church in downtown Scottsdale in 1952. Membership grew, and they purchased 10 acres at 7100 North Mockingbird Lane. The church continues to be open to community activities and as a polling place. Taliesin Associated Architects designed the church, with William Wesley Peters as the architect of record. (Scottsdale Historical Society.)\n\n**A SCENSION LUTHERAN CHURCH, 1966**. The pentagon was considered a perfect geometric figure by Frank Lloyd Wright, reflecting the perfection of God. This theme is carried out in the worship center lights, altar rail, and baptismal font. The cross above the altar directs attention to the pentagon-shaped glass window above. (Ascension Lutheran Church.)\n\n**P ARADISE VALLEY UNITED METHODIST CHURCH**. The charter Sunday was December 18, 1960, with 83 members meeting at the Cudia City Motion Picture Studio and later at the Unitarian Church and Paradise Inn. A new church was planned at 4455 East Lincoln Drive on a 15-acre site. Architect Ralph Haver, FAIA, is shown here at center with church representatives, reviewing plans for the new church with Camelback Mountain in the background. Haver Nunn and Jensen was the architect. (Paradise Valley United Methodist Church.)\n\n**P ARADISE VALLEY UNITED METHODIST CHURCH**. An original rendering from an aerial view captures the proposed Paradise Valley United Methodist Church. The first phase shows the chapel and cluster of three office and classroom buildings defining a courtyard. A future phase included the sanctuary to the left. \"The church as cultural center\" was a priority for the congregation. (Paradise Valley United Methodist Church.)\n\n**P ARADISE VALLEY UNITED METHODIST CHURCH, 1964**. Access to this church was from an elegant pedestrian bridge that spanned a dry wash, which would flow periodically when it rained. The bridge is steel-framed and painted with a concrete deck. (Paradise Valley United Methodist Church.)\n\n**P ARADISE VALLEY UNITED METHODIST CHURCH, 1982**. A taller 1964 chapel with battered brick walls anchors itself to the wash banks and the courtyard. The leaded stained glass altar window was designed by Jim Salter and fabricated by the Glassart Studio of Scottsdale. One-story structures have low pitched roofs of wood shakes, cantilevered edges, recessed brick walls, and protected glazing. (Author's collection.)\n\n**S COTTSDALE BIBLE CHURCH, 1966**. This Church moved and its building was purchased in 1979 to become Temple Solel, a Reform Synagogue, after that congregation initially met at Camelback United Presbyterian Church. The synagogue is located at 6805 East McDonald Drive and sponsors several events during the year along with adult education classes. The architecture is very expressionistic, with the large arched window and sweeping roof line. Bennie Gonzales, FAIA, was the architect. (Scottsdale Historical Society.)\n\n**H AR ZION ON THE DESERT, 1966**. The full-service conservative synagogue of the United Synagogues of America was at 5929 East Lincoln Drive and offered kindergarten through second grades and a Hebrew School for third through eighth grades. The architecture was a series of interlocking curved walls with a floating shake roof canopy above. The image is of the architect Bennie Gonzales, FAIA, with his client. The structure was demolished and a custom home was constructed on the site. (Scottsdale Historical Society.)\n\n**T HE EPISCOPAL PARISH OF CHRIST CHURCH OF THE ASCENSION, 1967**. The congregation initially met at a film studio, Phoenix Country Day School, and a First Federal Savings and Loan. The 4015 East Lincoln Drive campus is 8.25 acres, provided with a warranty deed by Margaret \"Peggy\" Goldwater Trust, half of which was a gift from the Goldwater family. The first phase parish hall, temporary worship area, office, and classrooms, designed by architect Bennie Gonzales, FAIA, are pictured under construction. (Robert T. Marvin, Christ Church of the Ascension.)\n\n**T HE EPISCOPAL PARISH OF CHRIST CHURCH OF THE ASCENSION, 1975**. The main church was built in 1975, and a school for pre-kindergarten through first grade was built in 2003. The character of the main sanctuary was respectful of the earlier parish hall, using similar masses, pitched roofs, mortar-washed slump block walls, and white color. (Christ Church of the Ascension.)\n\n**F IRST SOUTHERN BAPTIST CHURCH**. The original site was the 1924 Kiami Lodge, and until the mid-1980s, one adobe building still remained. The lodge swimming pool still remains under the existing volleyball court. The church acquired the property in 1968, followed by two construction phases in 1972 and 1980, both designed by architect William A. Lockard. The final phase was the 1989 sanctuary by architect Norman Bryar and Associates Inc. and provided a pitched roof with cantilevered edges and recessed masonry walls. Lower structures are flat-roofed with wood shake mansards. (Author's collection.)\n\n**C ALVARY CHURCH OF THE VALLEY**. At 6107 North Invergordon Road, the original c. 1940s Desert Lodge became the 1968 church that offered youth activities, Bible studies, local outreach, and overseas and local missions. Structures reflect the use of bearing plastered block walls and piers that support a wood-frame trellis. Occupied spaces are deeply recessed under the roof, thereby cooling them during hotter times of the year. (Author's collection.)\n\n**A LL CHRISTIAN FAITHS, 1969**. The church at 4222 East Lincoln Drive was constructed in phases, starting with the north component and shaded arcades. They defined an exterior courtyard and then accepted the main sanctuary expansion. Exterior walls are a burnt adobe, with roof framing of heavy wood members, and stained glass work by Maureen McGuire. The first church had financial challenges and was sold to its current owners, Friends Bible Church. The architects were Dellisanti and McGrath. (Author's collection.)\n\n**C AMELBACK UNITED PRESBYTERIAN CHURCH, 1983**. The church at 3535 East Lincoln Drive was founded with 216 members and has grown to 300. They have shared the facilities with other religious organizations, including the Sunrise United Presbyterian Church. In 1987, the name changed to the Palo Christi Presbyterian Church. The architecture utilizes a pair of pavilions with perimeter-shaded walkways capped with large pitched roofs of wood shakes. It provides flexibility when accessing the interior spaces, making it conducive to sharing the buildings with others. The architect was Harry Youngkin, AIA. (Author's collection.)\n\n**P ARADISE VALLEY UNITED METHODIST CHURCH SANCTUARY, 1982**. The original architect's model shows a 500-seat sanctuary addition to an existing campus and how it provided a new identity to the community. Knowing that the architect took an extensive tour of Finnish architect Alvar Aalto's work at this time, some overtures to Aalto's work are unmistakable, such as the use of brick, sweeping roof form, bell tower, horizontal shaded arcade, and the spatial dynamic of rotating the interior seating within the building shell. George Christensen and Associates was the architect. (Paradise Valley United Methodist Church.)\n\n**P ARADISE VALLEY UNITED METHODIST CHURCH SANCTUARY, 1983**. The view from the original pedestrian bridge to the new sanctuary addition shows an outdoor stepped amphitheater that connects the new upper plaza to the existing dry wash below. Christensen, Roberts & Jones Inc. was the architect. (Paradise Valley United Methodist Church.)\n\n**P ARADISE VALLEY UNITED METHODIST CHURCH SANCTUARY, 1983**. Exterior brick is carried into the interior of the sanctuary. Note how the lighter colors psychologically open up the volume. The clerestory above admits and reflects daylight, while focusing attention on the future stained glass art piece by Maureen McGuire. (Paradise Valley United Methodist Church.)\n\n**C AMELBACK BIBLE CHURCH**. This independent, evangelical church organized in 1962 and purchased 10 acres for a new church at 3900 East Stanford Drive. The 1,000-seat 1988 sanctuary design reflects a more contemporary interpretation, with the use of plastered block walls, green-patina copper roof and fascia, and horizontal corner windows. The architect was TMP and Associates with Maurice Allen of Bloomfield Hills, Michigan. (Author's collection.)\n_Six_\n\nDOMESTIC ENCLAVES:  \n1915\u20132012\n\nStarting in the 1920s, cowboy artist Alonso \"Lon\" Megargee III designed adobe homes in Paradise Valley, such as the 1926 Casa Del Sol and the 1930 Casa Hermosa. Both are a creative blend of Pueblo and Spanish Colonial Revival styles.\n\nMany adobe structures of the 1930s and 1940s were built by several Mexican families, such as the Tomas Corrals, who came here from Mexico to work as farm hands, given the demand for cotton after 1914. In 1932, the Mexican Bernals family supervised the Edward Jones house construction on Casa Blanca Drive, with help from Salt River Pima-Maricopa Indian Community workers.\n\nThe post\u2013World War II era brought about expansive growth with an aesthetic preference for a Ranch style character, which used low-pitched outreaching roofs, exposed concrete masonry units, and board and batten walls and tall glazing to capture views.\n\nThe 1970s were an active time for various new Paradise Valley ordinances impacting residential projects, including the 1972 Hillside Ordinance that enforced an unbroken mountain profile and a natural appearance. In June 1975, outdoor lighting restrictions limited lighting heights and shielded lights, helping to preserve nighttime views.\n\nDuring the early 1980s, new large homes started displaying an encyclopedic diversity of architectural character that was beginning to threaten the quiet desert ambiance, which had been the goal of earlier decades. \"These houses are a kind of cartoon to everyone with any knowledge of how people live with the desert and take care of their environment,\" said architect Calvin C. Straub, FAIA, in 1983. Architect Paul C. Yaeger stated at the same time, \"these houses are totally out of place. They reflect none of the ruggedness of the mountains or the simplicity and smoothness of the desert.\" Despite such justified criticisms, many local, nationally recognized, and talented architects have created inspiring residential designs in Paradise Valley over time.\n\nA new, single-family subdivision was developed at the southwest corner of Invergordon and Lincoln Drive. It proposed building a wall at the subdivision perimeter, a new concept that was much debated in the community. This guarded and gated community established a new pattern of residential development and compromised the earlier open spaces found between homes. In March 2003, Paradise Valley's General Plan noted that \"gated communities [would] be discouraged.\" By the 2000s, there were 24 gated communities and 50 organized home owners' associations.\n\n**F ARM RESIDENCE**. A 1915 one-story adobe structure sits among a citrus farm on North Sixty-eighth Street, now within the Arcadia Square no. 2 subdivision. The Pueblo Revival\u2013style, 1,818-square-foot home incorporates three patios, a two-car garage, and an L-shaped exterior stair for access to the roof. Details include arched openings with French doors, high-beamed ceilings, original decorative iron light fixtures, and curtain hardware. (Author's collection.)\n\n**D ONALD KELLOGG RESIDENCE, 1924**. Robert T. \"Bob\" Evans was the original architect of the residence at 5101 North Sixty-sixth Street. The large home was of plastered adobe construction in the Pueblo Revival style. A series of outdoor patios and courtyards were integrated; it was an effective experiment in how to live in the lower Sonoran Desert. (Scottsdale Historical Society.)\n\n**L A CASITA DE MARIA**. On North Camino del Contento, three c. 1925 structures totaling 3,850 square feet were built around a courtyard. The one-story, Pueblo Revival\u2013style structures were originally part of a guest ranch. There remains a palm shrine and two Madonna bas-reliefs as testament to the spiritual character of the original builders. The buildings later underwent an award-winning restoration and became a single-family residence. (John Douglas Architects.)\n\n**F RIEDER RESIDENCE, 1991**. La Casita de Maria's courtyard became part of the living space, while mature landscaping, the well house, and a fountain were restored. A new swimming pool was carefully inserted into an existing prickly pear cactus thicket. Floors were integrally dyed red concrete, ceilings were of clear fir with raw pine pole beams, and exterior stucco was dyed brown. The architect was John Douglas, FAIA, and the landscape architect was Christy Ten Eyck, FASLA. (John Douglas Architects.)\n\n**A DOBE RESIDENCE, 1969**. This c. 1928\u20131930 residence is located on East McDonald Drive. Its 3.2 acres included a main residence, three guest casitas, a common kitchen, a coachman's house, one-car garage, breezeways, an outdoor room, and an observatory for stargazing. A cluster of structures defined an expansive courtyard with sundried bricks, wall vines, and shade trees. One-story structures reflect the Pueblo Revival style with plastered walls, small-paned windows, and wood vigas. The architect was Robert T. \"Bob\" Evans. (Bruce Berquist.)\n\n**P OWERS RESIDENCE, 1991**. This adobe residence underwent an award-winning building restoration and courtyard upgrades by architect John Douglas, FAIA, and landscape architect Christy Ten Eyck, FASLA. Structures were stabilized and repaired, and had their original color brought back. The original courtyard was completely reworked for better drainage, the swimming pool was upgraded, some trees were removed to enable entertaining, and it continued to capture the east-west breezes. The design reflects a \"real yearning to live with the desert,\" as stated by the owner and architect. (John Douglas Architects.)\n\n**C ASA DEL SOL**. Cowboy artist Alonzo \"Lon\" Megargee III designed and constructed a 3,500-square-foot adobe Spanish Colonial and Pueblo Revival\u2013style home on 15 acres in 1926 for his wife, Mildred. The one-story home, with a partial second-level artist studio and outdoor courtyard, was upgraded later by new owners Benner\/Maas, founding principals of the Scottsdale-based interior design firm. (Pat VanVelser Heard.)\n\n**C ASA DEL SOL, 1926**. The living room has a whimsical, cone-shaped fireplace topped with a Santa Claus that defined and animated the tall space. The room had an exposed wood-frame ceiling and roof structure above. (Betsy Fahlman, Pat VanVelser Heard.)\n\n**P RINGLE RESIDENCE**. Sisters Mildred and Bertha Pringle lived in this 1930s one- and two-story adobe residence on a large parcel at North Casa Blanca Drive. It was later sold to the Firestone family. It is rumored that they managed an unlicensed tuberculosis health camp at the home. The Pueblo Revival style incorporates smaller window openings and light-colored plaster walls. (Author's collection.)\n\n**D UNCAN MACDONALD**. Originally from Invergordon, Scotland, Duncan MacDonald was proud of his Scottish heritage. He worked as a skilled plasterer whose creations adorned public buildings of his time. Projects included the Palace West Theater and the Security Building in Phoenix. McDonald Drive was named after Duncan MacDonald, but the spelling of his name was changed in the process, and it was never corrected on the maps. (Scottsdale Historic Society.)\n\n**M ACDONALD RESIDENCE, C. 1930S**. This is an aerial view of a large adobe home at 6330 East McDonald Drive. Duncan MacDonald designed and constructed this adobe residence in a Spanish Colonial and Pueblo Revival style. (Earl Eisenhower, Elsa J. Rector.)\n\n**M ACDONALD RESIDENCE, C. 1930S**. It is said that the floor plan was dictated by the desired interior furniture arrangement. The home was sold to John C. Lincoln, an Ohio industrialist who brought his family to this area in the early 1930s. The Lincoln family lived in the home for many years. (Scottsdale Historical Society.)\n\n**E DWARD JONES RESIDENCE, 1933**. Ken Jones and his older brother are pictured with the adobe residence under construction. Adobe bricks were made of straw and soil sun-dried just east of the home. Once the walls were completed, they would be untouched for about one year, so they could structurally settle. They were then fully plastered and painted to discourage deterioration. Wood doors were constructed from salvaged redwood from the Verde Canal pipeline. (Ken Jones.)\n\n**E DWARD JONES RESIDENCE, 2013**. This 1933 adobe two-story home is a blend of Pueblo and Spanish Colonial Revival styles. The exterior doors and windows are intimate and scaled to be appropriate for this wall-like architecture. The original roof sleeping porch wood framing still remains. Located on North Casa Blanca Drive, it was once part of 50 acres sold to the Joneses by Edward Loomis Bowes. Jones then had citrus, date palms, and alfalfa fields, and raised turkeys on the property. His wife, May Jones, designed the home along with Bowes. (Author's collection.)\n\n**H ENRY WICK RESIDENCE**. A 1939, one-story Pueblo Revival\u2013style home is located on North Casa Blanca Drive. The plan was L-shaped, with an outdoor stair, plastered walls, and metal casement windows. The residence was designed by Denver Evans, architect Robert T. \"Bob\" Evans's son. (Author's collection.)\n\n**F ENNELL RESIDENCE, 1955**. The 5,962-square-foot Mike Fennel residence is located on East Prickly Pear Lane and was designed by architect Charles Montooth of Frank Lloyd Wright Associates. The home incorporates dramatic cantilevered roof forms with patterned fascia, an arcaded entry breezeway, and a two-car carport. Materials include natural stone, exposed concrete masonry units, and tall clear glazing. (Author's collection.)\n\n**P AUSON RESIDENCE, 1941**. This custom winter residence was a natural extension of the hillside and its topography. The vertical structure explored desert masonry masses that were interlocked with shifting, wood-framed horizontal elements. Frank Lloyd Wright was the architect. (Janie Ellis.)\n\n**P AUSON RESIDENCE, 1941**. The plan included a balcony, studio, and servants' quarters. The fireplace location was debated, and an ember actually caught the draperies on fire in 1943, thereby destroying the entire residence. The house was constructed by George Ellis at a cost of $7,500. (Janie Ellis.)\n\n**P AUSON RESIDENCE, 1941**. Interior spaces extended to the exterior, and a series of outdoor terraces overlooked the views beyond. (Janie Ellis.)\n\n**P EIPER RESIDENCE, 1952**. This residence is on East Cheney Drive but has been seriously altered with additions and remodels over the years. The home, designed by Frank Lloyd Wright, reflects his approach to a Usonian Automatic block system, which is when the blocks are cast on site and are therefore more affordable. (Frank Lloyd Wright Foundation, AIA Phoenix Metro.)\n\n**R ANCHO LUCERO, 1951**. This East Berneil Lane residence's roof was framed with wood beams from an old sawmill near Showlow, Arizona, and _vigas_ or log beams from nearby McNary. The exterior walls were left in their natural state, but suffered damage during storms, so they were fortified with stucco. The property also had a carport, rose garden, horse corrals, a haystack, chicken house, alfalfa field, and a well. (Jamey and Linda Cohn.)\n\n**R ANCHO LUCERO, 1951**. The floor tiles were made in Nogales, Sonora. All interior adobe walls were \"liberally and repeatedly sealed with skim milk. All windows opened to unspoiled, uninhabited desert in their day.\" (Jamey and Linda Cohn.)\n\n**P RICE RESIDENCE, 1954**. Harold C. Price Sr.'s winter residence, by Frank Lloyd Wright, is located on North Tatum Boulevard. The plan has two wings in an angular relationship to each other, connected by an open loggia that captures the breeze, has a fountain, and is supported by tufa block columns that form inverted cones. The southwestern section includes the dining, living, kitchen, and guest rooms, the maids' quarters, and a carport. The east wing has five bedrooms and bathrooms. Low profile flat roofs float over four different floor levels and continuous glazing. (Author's collection.)\n\n**T HEODORE F. WASHBURN RESIDENCE, 1955**. This bachelor's residence on East Stanford Drive provided a main house, guest quarters, triple carport, and a round swimming pool. The one-story wood-frame structures have a flat roof that was influenced by a Japanese theme of high-quality simplicity. The view shows the north and west elevations. The architect was Blaine Drake; years later, it was remodeled by Alfred Newman Beadle. (Alison King.)\n\n**T HEODORE F. WASHBURN RESIDENCE, 1955**. This view shows the partial south elevation. Exterior walls included a grey Flex board with cantilevered soffits above to protect the glass. Interiors use mahogany panels, a ceiling of blonde tongue-and-groove planks, and tan concrete floors. (Alison King.)\n\n**M ADISON RESIDENCE, 1955**. Mr. and Mrs. Gray Madison's 3,357-square-foot residence was located on a 1.48-acre property at East Pepper Tree Lane and the east edge of the Paradise Valley Country Club. Madison owned a local car dealership and was a co-founder of the club. The house exhibits the Ranch style, with pitched wood shake roof and brick walls. The architect was Fredrick Weaver, and Dick Drover was the designer. (August Beinlich Photography, DWL Architects & Planners Inc.)\n\n**H OLCOMB RESIDENCE**. The Adm. Harold R. Holcomb residence was constructed in 1956 at East Lincoln Drive. Holcomb was a retired rear admiral in the Navy, was active in World War I and II, and served as a justice of the peace in Scottsdale. The residence used exposed and soldier stacked block, low-pitched roof, corner windows, a dynamic entryway with a rotated carport, and a minimalist fireplace. The original architect was Blaine Drake. (Author's collection.)\n\n**G OLDWATER RESIDENCE, 1957**. US senator Barry Goldwater's hilltop residence has a low, horizontal profile with natural stone retaining walls, deeply recessed glazing, and cantilevered pitched roof overhangs with a thin edge. The home had an appropriate character for desert living, was held tight to the natural topography, and framed dramatic views of Camelback Mountain and city lights. Within the home was a ham radio station. The architect was Paul Christian Yaeger. (Bill Sears.)\n\n**S CHAFFNER RESIDENCE, 1957**. This Mid-century and Ranch-style home is located at North Quartz Mountain Road and was a split-level plan for a bachelor and his guests. It is sited at a 30-degree angle to take advantage of the winter sun and the best views overlooking the Paradise Valley Country Club golf course. Paul Christian Yaeger was the architect. (Beth Wickstrom.)\n\n**S CHAFFNER RESIDENCE, 1957**. The John Schaffner home is constructed of adobe brick walls. It has a heavy, beamed construction with hemlock ceilings. The second level offers a panoramic view of Paradise Valley through framed glass partitions. There are a number of patios and an exterior deck that accessible from many different interior levels. The kitchen and lanai were top-lit with skylights. (Compton Photographers, Beth Wickstrom.)\n\n**K RAMLICH RESIDENCE**. The 1960 Irv and Dorothy Kramlich Residence is located on one acre at North Desert Fairways Drive. The one-story, 4,752-square-foot residence has a semicircular floor plan that focuses on Camelback Mountain to the south. Walls facing the street are solid white plaster with colored glass inserts. An indoor swimming pool was added about 1966, followed by a 1997 remodel. Alfred Newman Beadle was the architect. (Realty Executives.)\n\n**O'C ONNOR FAMILY**. Sandra Day O'Connor and her husband, John, raised their three sons at this home on East Denton Lane. The O'Connors were very active in collecting petition signatures for potential incorporation of the Town of Paradise Valley. Former justice Sandra Day O'Connor was sworn in as the first female justice of the US Supreme Court on September 25, 1981, and retired on January 31, 2006. (Town of Paradise Valley.)\n\n**O'C ONNOR RESIDENCE, 1958**. The floor plan was rectangular with an angled end for the living room, an open kitchen, and a series of bedrooms along a hallway. A minimal palette of natural materials included bearing adobe walls, redwood window framing, a wood-framed roof with wood shake shingles, and red concrete floors. The character is more organic and casual, which fits the desert environment. The architect was Scottsdale-based Don K. Taylor. (Mark Vinson, FAIA.)\n\n**O'C ONNOR RESIDENCE, 1958**. The interior ambiance carries the exterior character and natural materials consistently throughout. Note the integral-colored concrete floors, exposed adobe walls, expressive roof structure, and generous window areas for introducing daylight. (Mark Vinson, FAIA.)\n\n**O'C ONNOR RESIDENCE, 2010**. In 2010, to avoid the threat of the home being torn down, the original house was dismantled and reconstructed at College Avenue and Curry Road in Tempe. The original adobe bricks were actually made at the nearby Salt River. This construction image shows the original roof being elevated and staged for laying the original adobe brick walls underneath. The completed building is being used as the O'Connor House and Center for Civil Discourse. Janie Ellis was instrumental in dismantling and reconstructing this fine home. (Mark Vinson, FAIA.)\n\n**M CCUNE MANSION, 1962**. Walker McCune, an oil tycoon, Penzoil heir, and philanthropist, built a 40,000-square-foot estate on six acres on Sugarloaf Hill at North Paradise View Lane. Geordie Hormel owned the home later. The architecture was of commercial-grade construction with a concrete frame and precast concrete panels. Buildings were placed on an earth-filled plinth with a perimeter retaining wall. The architect was Stone, Marracini & Patterson with designer R.J. Bettencourt of San Francisco. (Scottsdale Historical Society.)\n\n**D OUGLAS DRIGGS RESIDENCE, 1962**. This residence was located at North Shadow Mountain Road and has been demolished. The home was a low-scaled exquisite jewel, connecting the interior spaces with a series of outdoor patios at differing levels and the Paradise Valley Country Club golf course. The manicured landscape was well known in this neighborhood. The architect was William F. Cody, FAIA, of Palm Springs, California. (Author's collection.)\n\n**M ANKER RESIDENCE**. Dr. Raymond Manker, a minister at the nearby Unitarian Universalist Congregation of Phoenix, had the church architect also design this 1964 home on East Pepper Tree Lane. The residence is 2,885 square feet and has five bedrooms and two bathrooms. One-story construction with an open carport includes slump block walls, steel columns, a wood-framed roof, and blue accent wall colors. The floor plan captured distant views and opened to an exterior patio. The architect was Blaine Drake. (Author's collection.)\n\n**A NDEEN RESIDENCE, 1964**. The Richard Andeen residence is located on North Hummingbird Lane. This award-winning and desert-sensitive piece provides site and building walls of exposed natural stone and a shade trellis of wood-frame members. Glazing is recessed into the thick walls, minimizing heat gain during the hot summer months. The landscape retained the original desert plantings and rock outcroppings. The architect was Straub\/Kutch. (Neil Koppes, AIA Phoenix Metro.)\n\n**W EBER BACHELOR PAD**. This East Glen Drive residence was built in 1966 and is set high on a south-facing hillside on Mummy Mountain with a one-acre property. The building form springs from the hillside, but is visually anchored by the plaster retaining wall that supports the swimming pool, which also extends into the living room. The 3,114-square-foot two-story home with three bedrooms is framed with exposed wood bent frames that shade the upper glazing. The influence of Frank Lloyd Wright's organic architecture is undeniable. The architect was Robert M. Lawton with Donald P. Woods Jr. (Author's collection.)\n\n**M CDONALD RESIDENCE, 2012**. The Mr. and Mrs. Weir McDonald residence is located on North Tatum Boulevard and is a one-bedroom home with a private theater and two-car, open carport. The 1965 home steps down the rolling site with an exposed steel-frame structure and elevates many of the functions above the site. Site improvements such as planter retaining walls and cascading steps anchor the composition to the site. The original architect was William F. Cody, FAIA. A remodel and addition was completed by LEA Architects LLC. (LEA Architects LLC.)\n\n**H EUSER HACIENDA, C. 1960S**. This east coast client was familiar with Colonial homes but wanted to build an adobe residence. The large, one-story home focuses on an exterior courtyard with a swimming pool, an outdoor fireplace, and open stairs to a roof deck. George W. Christensen and Associates was the architect; this firm went on to design over 30 custom residences in Paradise Valley during the next 40 years. (Donald J. Christensen, W. Brent Armstrong.)\n\n**H EUSER HACIENDA, C. 1960S**. The interior features an exposed, hexagonal tile floor, beehive fireplace, whitewashed walls, and an exposed wood roof structure. (Donald J. Christensen, W. Brent Armstrong.)\n\n**G ONZALES RESIDENCE**. This award-winning 1966 private custom desert residence is located at North Palo Christi Drive. The home character suggested an abstracted Pueblo approach to the forms. The architect desired a cluster of simple boxes arranged to encourage an open way of living. Exterior walls that are exposed to the interior are slump block units with a painted mortar wash. Interior floors are of burnt adobe and were given a dark brown stain to visually contrast with the walls. Bennie M. Gonzales, FAIA, was the architect. (Author's collection.)\n\n**D R. FRANK S. TOLONE RESIDENCE**. This 1967 award-winning residence is located on East Desert Jewel Drive. The home is sensitively sited with a few dry washes. A cluster of one-story components with varying pitched roofs are provided along with walls of exposed masonry. Grey-colored wood details are found at vertical sun shading panels and bridge railings. The architect was James T. Flynn, AIA. (Author's collection.)\n\n**P ETERSON RESIDENCE, 1968**. The Dr. Rex and Del Peterson residence has four bedrooms and three bathrooms on North Fifty-second Street. \"One of the purest concepts is a white house in the desert. Modular post and beam is combined with rustic pueblo massing to unite in what the architect calls the character Scandinavian Pueblo,\" according to the architect's brochure. George W. Christensen and Associates was the architect. (Neil Koppes, CCBG.)\n\n**P ETERSON RESIDENCE, 1968**. The main residence overlooks the swimming pool and Camelback Mountain to the south. (Neil Koppes, CCBG.)\n\n**P ETERSON RESIDENCE, 1968**. This interior takes advantage of multiple levels with the entry overlooking the taller living space beyond. White walls, exposed rough-sawn roof framing above, and tiled floors are appropriate for the Sonoran Desert. (Neil Koppes, CCBG.)\n\n**C OOK RESIDENCE, 1973**. This 1969 residence was at 3627 Camino sin Nombre. The two-story home used passive solar concepts. A west-facing, thick sculpted masonry wall with varied small window openings was not unlike the iconic Le Corbusier's Ronchamp Chapel found in France. The architect was Jeffrey Cook, AIA. The home has been demolished. (Author's collection.)\n\n**C ASA CRISTINA, 1977**. This residence on North Forty-sixth Street used hand-crafted, 16-inch thick adobe, hewn wood beams, gently rounded corners, and windows that do not necessarily match up with any other windows in the same facade. Artist William F. Tull, who designed the home, stated: \"Abstract shadows and shapes in my paintings are also used in my architecture.\" He preferred \"peace-quiet-everything soft and sculptured.\" The home was featured in the _Fire in the Sky_ film and in fashion publications. (Bob Ross Photos, Joann Tull.)\n\n**L ARSON RESIDENCE**. This 1974 residence was on East McDonald Drive and incorporated natural materials such as exposed masonry, natural stone, and full-height glazing. Massing provided a broken roofline with stepped forms. The plan integrates five patios and a four-car carport. The architect was John S.M. Hamilton. The home was demolished in 2012. (Mary Leonhard Photographer, AIA, Phoenix Metro.)\n\n**G ARY DRIGGS RESIDENCE, 1970**. This award-winning residence on North Shadow Mountain Road was elevated above the hillside to allow the natural landscape to be retained. The main structure is a steel column grid and features beams that are fully expressed throughout. Within the grid are taupe-colored plaster walls or full-height glazing. Alfred Newman Beadle was the architect. (Arizona State University Libraries.)\n\n**G ARY DRIGGS RESIDENCE, 1970**. The glazing is layered in the plan to provide a transparency from space to space and to frame desert views beyond. An accessory building accommodates a guest suite that is placed at grade and defines the main entry. (Arizona State University Libraries.)\n\n**B OMBECK RESIDENCE, 1971**. A residence for humorist and author Erma Bombeck and her husband, Bill, and their family is found on a raised hillside ridge with spectacular views. Ford Powell and Carson of San Antonio, Texas, was the architectural firm, and Chris Carson, FAIA, was the principal designer. The firm, founded in 1939 by architect O'Neill Ford, is well known for its regional design approach, influencing generations of Texas architects. (Author's collection.)\n\n**F LYNN RESIDENCE, 1973**. This custom residence and adjacent work studio at North Fifty-First Place was designed and built by 1971. The exterior forms are defined by an exposed black steel frame with a dark brown brick and full-height glazing infill panels. The floor plan focuses on a southern courtyard and swimming pool with a dramatic Camelback Mountain view beyond. James T. Flynn, AIA, was the architect. (Author's collection.)\n\n**C OOK RESIDENCE, 1973**. This two-story residence is located on a hillside lot on North Hillside Drive. Visitors park at the upper level open carport and then descend down into the home. The walls are of exposed masonry with brushed mortar joints capped with a wood-framed and pitched clay tile roof. Southeastern-facing windows take advantage of Mummy Mountain and Camelback Mountain views. James T. Flynn, AIA, was the architect. (Author's collection.)\n\n**R OSEWALL RESIDENCE, 1974**. The award-winning Ken Rosewall residence was built on a custom home lot at the John Gardiner's Tennis Ranch. The novel design approach for a professional tennis player placed a full-size tennis court on the roof and the main residence underneath at the lower north end. This design left more of the natural desert untouched. George Christensen, FAIA, was the architect. (Neil Koppes, CCBG.)\n\n**S HOENBERG RESIDENCE, 1975**. This 3,200-square-foot residence was designed at John Gardiner's Tennis Ranch. The structure features exposed tubular steel with the roof drainage downspouts integrated within structural columns. Infill panels are of taupe-colored plaster or tinted glazing. Interiors have spectacular, elevated views of Paradise Valley and Scottsdale. Reginald G. Sydnor, AIA, was the architect; he also designed the Ambler, Gardiner, and Sydnor residences of Paradise Valley. (Neil Koppes.)\n\n**M AHONEY RESIDENCE**. This 1977 residence located at North Cottontail Run is a horizontal composition hugging the hillside property. A linear window is shaded above by a continuous, wood-clad canopy and flanked by solid plaster walls to each side. The architect was Jones & Mah Architects Inc. (Author's collection.)\n\n**F INISTERRE DEVELOPMENT**. Paradise Valley's first walled, guarded, and gated community was established in 1978 at the southwest corner of Invergordon Drive and Lincoln Drive. This was a planning precedent, as existing open desert spaces did not extend through the development. Nelson Kubicek Architects designed the wall. (Author's collection.)\n\n**S MITH RESIDENCE, 1981**. The two-story, 6,500-square-foot Arnold Smith residence is located on East Palo Verde Drive. Clean, crisp forms and concrete walls step down the hill and frame Phoenix views. A roof canopy floats over interior spaces that extend functionally and visually to exterior terraces and patios. The architect was Knoell & Quidort Architects. (Knoell & Quidort Architects.)\n\n**P ETERSON RESIDENCE, 1981**. This hillside residence is located on East Hummingbird Lane. The home is composed of plastered cubic volumes that sit into the hill, providing an outdoor living room with a fireplace and a swimming pool level. The architect was Jack Peterson & Associates. (Author's collection.)\n\n**G OODMAN RESIDENCE, 1985**. An existing Mid-century residence on a hillside property at North Red Ledge Drive received major additions and renovation to become a three-story, 5,000-square-foot residence. The built results are quite dramatic, with the sculpted forms of solid masonry walls interlocked with carved voids and window openings. The masses sit well on the pitched desert property. The architect was Edward B. Sawyer Jr. (Edward B. Sawyer Jr.)\n\n**L EIN RESIDENCE, 1984**. This East Hummingbird Lane residence includes two stories, a 2,750-square-foot main residence, and guest quarters of 928 square-feet. The architecture exploits the use of exposed multi-fired masonry walls at the exterior and interior. The forms interlock and overlap in various sculptural configurations to frame distant views, block unwanted sun, define privacy, and expand the interior space to surrounding outdoor patios. The architect was Edward B. Sawyer Jr. (Edward B. Sawyer Jr.)\n\n**W EISS RESIDENCE, 1988**. This two-story residence is located east of the Paradise Valley Country Club golf course on a one-acre rocky hillside property. The mountainside residence \"has been designed to minimize its physical impact on the natural environment, which surrounds it, while maximizing the enjoyment of the site's spectacular city and mountain views,\" stated architect William P. Bruder, FAIA. (Will Bruder + Partners Ltd.)\n\n**W EISS RESIDENCE, 1988**. The exterior material palette is naturally aged copper, native rock quarried from the site during excavation, and weathered steel. The interior materials are resawn spruce ceilings, flagstone flooring, wool carpeting, copper, and white oak for paneling, doors, and cabinets. (Will Bruder + Partners Ltd.)\n\n**H ALAS RESIDENCE, 1985**. This one-story 2,500-square-foot residence is located at East McDonald Drive. The owner was a well-known psychologist specializing in women's issues and marital counseling. She had a professional office, but the house needed to accommodate home appointments without compromising privacy. The plan has a rounded form that directs runoff away from the structure, and a curved roof form meant to sympathize with Camelback Mountain's ancient, rounded profile. The architect was Jones Studio Inc. (Jones Studio Inc.)\n\n**Z UBER RESIDENCE**. This 1989 hillside home on East Indian Bend has a floor plan that overlaps an upper split-face block component for private uses with a lower stucco component for more public uses. The master bedroom was perched above as the inner sanctum and control tower, while framing views of the upper pool, gallery, and Camelback Mountain. A skeletal bridge extends from the living room and functions as a gateway below as visitors approach the house. The architect was Antoine Predock Architects. (Author's collection.)\n\n**C UMMINGS RESIDENCE, 1989**. This East Nauni Valley Road residence was designed for a two-acre property and a 7,500-square-feet program. Site planning was based upon a four-foot grid and proposed an east-west plan with outreaching wings. The plan incorporated a series of three different courtyards of various scales and purposes. The architect was Sydnor Architects, PC. (Mark Boisclair Photography Inc.)\n\n**D ORRANCE RESIDENCE, C. 1992**. This hillside property overlooks the Paradise Valley Country Club golf course. The original architect states that the home is \"nestled into a larger rock outcropping, the mountain unites with its knoll. The lower level is very intimate with window wells and the deck level is reminiscent of a ship. The upper level is connected by four stairways, which add a sense of flow between levels. A long art gallery functions as the upper circulation.\" The architect was George Christensen, FAIA. (Donald J. Christensen, W. Brent Armstrong.)\n\n**D EBARTOLO RESIDENCE, 1993**. This two-story home at North Thirty-third Street is 5,600 square feet. The plan provides a major wall to the severe western exposure while the remaining open sides frame distant city and mountain views. Jack DeBartolo Jr., FAIA, was the owner and architect. (Jack DeBartolo Jr., FAIA.)\n\n**D EBARTOLO RESIDENCE, 1993**. The design strategy takes advantage of daylight throughout the interior and proposes a floating pavilion roof structure. The floor plan is primarily an open loft where interior partitions were minimized. (Jack DeBartolo Jr., FAIA.)\n\n**K ORNEGAY RESIDENCE, 1993**. The original 1,800-square-foot structure dates from 1955 and was designed by Edward Loomis Bowes. A major renovation and 3,400-square-foot additions to this hillside house integrated it with the site. With adjacent homes arriving over the years, a new plan provided privacy and controlled views. Exposed concrete block, plastered ceilings, scored concrete floors, wood cabinetry, and steel windows were provided. The architect was John Douglas, FAIA. (John Douglas Architects.)\n\n**T OWNSEND RESIDENCE, 1997**. This art collector's residence is located on one acre at East Hummingbird Lane and was a major addition and renovation of an existing structure. The two-story, 4,800-square-foot house is an eyebrow floating over the natural topography, while focusing on the northeasterly view of the McDowell Mountains. The house responds to the owner's desire for a place of quiet reflection. The architect was Will Bruder Architect Ltd. (Will Bruder Architect Ltd.)\n\n**G RUBER RESIDENCE, 1998**. This residence is on a one-acre hillside property at North 58th Place. The 4,000-square-foot floor plan is stretched in an east-west direction with a solid western wall and enclosed garage to the east. The midsection of the floor plan opens to a southern swimming pool while the north provides full-height glazing to frame the valley views below. The structure is exposed tubular steel with plastered wall panels, full-height glazing at more public spaces, and smaller window openings at the private zones. Rich Fairbourne, RA, of The Construction Zone completed the construction after the architect's passing. The architect was Alfred Newman Beadle. (Dan Gruber.)\n\n**W ITHYCOMBE RESIDENCE, C. 1996**. \"The Shangrila Resort gave way to four acres of land, which became a family resort. All bedrooms are located on the balconied second level providing maximum views of Camelback Mountain,\" according to the architect George W. Christensen, FAIA. (Donald J. Christensen, W. Brent Armstrong.)\n\n**D ORRANCE RESIDENCE, 1999**. This aerial view is of the sprawling residence on North Cameldale Way. The residence's scale is broken up with one and two-story components, a shifting plan, shaded overhangs, and various-size window openings. The architects were George W. Christensen, FAIA, and Erik Peterson. (Dino Tonn Photography.)\n\n**S CHUCHTER RESIDENCE, 1999**. The architect notes the \"long and thin in plan, two-story living room in the middle allows the bedrooms to be on one side and family living on the other. Rather than expensive details, simplicity and proportions were used as the main elements.\" The architect was George W. Christensen, FAIA. (Donald J. Christensen, W. Brent Armstrong.)\n\n**U DINOTTI STUDIO**. Adjacent to a c. 1928\u20131930 Robert Evans adobe residence on East McDonald Drive, this 2,500-square-foot, two-story daylit artist's studio, art display, and guest quarters was built in 1999. The owner describes the character as a Contemporary Adobe, in which the walls are abstracted planes, glazing faces a defined patio, and a tall weathered wood door allows large canvases to enter the studio along with people. The architect was John Douglas Architects. (Author's collection.)\n\n**U DINOTTI MUSEUM OF FIGURATIVE ART**. A later phase in 2006 included a 1,900-square-foot subterranean gallery of cast-in-place concrete walls, exposed steel trusses, and metal decking. It was \"placed underground as not to interfere with the view of Camelback Mountain\u2014a sacred mountain\" from the adjacent studio, according to Agnese Udinotti. The gallery is entered from a sunken sculpture courtyard. A rusted, steel-clad upper-level gift shop with a roof deck for views is located above. The architect was The Construction Zone. (Author's collection.)\n\n**M ORGAN RESIDENCE, 2001**. This hillside room on East Moonlight Way restored a previously damaged site. An existing retaining wall was preserved and extended to anchor the guesthouse and a new driveway. The main house is disciplined with a clear structural logic, while the guesthouse is free and curvilinear. The composition creates a variety of outdoor experiences and captures dramatic views. The architect was Knoell & Quidort Architects. (James L. Christy of Jim Christy Studio.)\n\n**P LANAR RESIDENCE, 2002**. This 3,200-square-foot home provides for a large contemporary art collection that includes pieces by Bruce Naiman, Robert Ryman, Jeff Koons, and Jannis Kounellis. The structure is composed of concrete tilt-up panels that define an exterior courtyard. The architect was New York\u2013based Steven Holl Architects. (Bill Timmerman.)\n\n**P LANAR RESIDENCE, 2002**. An exterior ramp positioned within a courtyard provides access through the roof structure to a view deck. (Bill Timmerman.)\n\n**P LANAR RESIDENCE, 2002**. This interior view shows the open plan with introduction of daylight from various sources, minimizing the interior glare from a single source. (Bill Timmerman.)\n\n**D EBARTOLO RESIDENCE, 2005**. Miramonte House 69 is on East Lincoln Drive and is a 3,000-square-foot custom home. The plan exploits the traditional notion of a courtyard in the desert and opens it to the sky above. The structure is exposed masonry with full-height glazing and natural stone floors. The architect and owner is Jack DeBartolo Jr., FAIA. (Jack DeBartolo Jr., FAIA.)\n\n**F ERRIS RESIDENCE, 2006**. This 5,000-square-foot, two-story residence was designed for a one-acre property on Camelback Manor Drive. The project started as a guest cottage in the 1940s and led to a series of renovations and additions from 1987 to 2006. The home was designed to be virtually invisible in its natural setting as the exterior colors mimic the shadows seen in the dense undergrowth. The architect was John Douglas Architects. (John Douglas Architects.)\n\n**H OUSE OF 5 DREAMS, 2004**. This three-story, 30,000-square-foot residence is located on East Stanford Drive on approximately three acres. The private penthouse residence hovers 14 feet above grade. A glass roof and floor inserts scatter circular light around their undulating translucent \"flower\" space. Roof and floor planes frame panoramic views of the city skyline and desert mountains. The architect was Jones Studio Inc. (Jones Studio Inc.)\n\n**H OUSE OF 5 DREAMS, 2004**. This interior view is of a private museum for the display, storage, and protection of Pre-Columbian art. The below-grade space is defined by four-foot-thick, rammed earth walls at the perimeter and an exposed, clear-spanning steel roof structure above. (Jones Studio Inc.)\n\n**H OUSE OF 5 DREAMS, 2004**. The interior conveys an extremely sophisticated, elegant, and minimalistic ambiance. Transparency and translucency is explored throughout. (Jones Studio Inc.)\n\n**W ILLIAMS RESIDENCE, 2007**. This 6,206-square-foot East Stanford Drive residence includes three bedrooms and six bathrooms. This two-story approach places the main living, dining, kitchen, and outdoor dining at the upper level as a \"reversed living room\" and the private spaces, such as bedrooms, on the first level. The home was conceived as a \"thick-mass casting\" and has white cast 20-inch-thick concrete walls. The two-story volumes define an entry courtyard. Colors include subtle grays, silvers, and green casts of the desert landscape. The architect was Marwan Al-Sayed Architects Ltd. (Author's collection.)\n\n**L ACEY RESIDENCE, 2012**. A three-building cluster, with a main house, garage, and guesthouse, is found on East Stella Lane. The client desired a retreat and a home and it was his implied program for a residence to represent progressive contemporary thinking ground in the fundamentals of \"mind and shadow.\" A courtyard was formed in this hacienda diagram. Rammed earthen walls were used given their resistance to climatic forces, insulating superiority, structural capability, and inherent natural beauty. Wood was used intentionally, with weathering in mind. The architect was Jones Studio Inc. (Jones Studio Inc.)\n\n**L ACEY RESIDENCE, 2012**. The interior is psychologically expanded and connects with the outdoors with an open plan, full height glazing, desert landscaping close by, and a framed Camelback Mountain view. (Jones Studio Inc.)\n\n**J ARSON RESIDENCE, 2009**. This two-story residence is located on North Charles Drive. The home is for two real estate professionals and their two sons. A mysterious, refined dark object in its rugged natural landscape, the house provides a place of quiet reflection. The entry, office, and bedrooms are on the upper level with living and dining rooms; a media music chamber and potter's studio are tucked beneath. The architect was Will Bruder + Partners Ltd. (Will Bruder + Partners Ltd.)\n\n**J ARSON RESIDENCE, 2009**. This residence interior utilizes natural cork and concrete floors and wall planes of translucent glass. Cabinets of cherry and stainless steel articulate the interiors. (Will Bruder + Partners Ltd.)\nBIBLIOGRAPHY\n\nBarry Goldwater Memorial. Town of Paradise Valley, February 14, 2004.\n\nBeadle, Alfred Newman. Bernard Michael Boyle, ed. and Diane M. Upchurch, associated ed. _Constructions: Buildings in Arizona_. Historic Publications No. 3, Tempe, AZ: School of Architecture, Arizona State University, 1993.\n\nCarlson, Fran. \"Spell It MacDonald Please.\" _Scottsdale Scene_. March 1984: 85\u201386.\n\nDriggs, Gary. _Camelback: Sacred Mountain of Phoenix_. Tempe, AZ: Arizona Historical Foundation, Arizona State University, 1998.\n\nFudala, Joan. Images of America: _Scottsdale_. Charleston, SC: Arcadia Publishing, 2007.\n\n_A Guide to the Architecture of Metro Phoenix_. Central Arizona Chapter, American Institute of Architects. Phoenix, AZ: 1983.\n\nMarill, Michele C. _Phoenix Country Day School: Voices in the Desert_. Bookhouse Group Inc.\n\n_Preserving the Desert Lifestyle: The 25th Anniversary of the Town of Paradise Valley_. Scottsdale, AZ: The Parish Agency, 1987\n\nMiles, Candice St. Jacques. \"Pandemonium Found: Architects Comment on Paradise Valley's Lost Style.\" _Arizona Living_. Volume 14, Number 5. May 1983: 16\u201323.\n\nO'Meara, Rosie. \"Figuring The Angles.\" _The Arizona Republic Sun Living_. Section 5. June 15, 1958.\n\n\"Ranches, Lodges, Inns, Hotels, Resorts in New Accommodations Guide.\" _Scottsdale and Paradise Valley News_. Scottsdale Chamber of Commerce, Volume 1, Number 2. 1956\u20131957.\n\nStevenson, Charles S. _We Met at Camelback! The Story of Jack and Louise Stewart at Their Camelback Inn_. Arizona Desert Publishing Company, 1968.\n\n_Town of Paradise Valley 40th Anniversary_. Historical Advisory Committee. Photography: Kathryn Gasser. Payout: Kathryn P. Yoder. Town of Paradise Valley, 2011.\nDISCOVER THOUSANDS OF LOCAL HISTORY BOOKS  \nFEATURING MILLIONS OF VINTAGE IMAGES\n\nArcadia Publishing, the leading local history publisher in the United States, is committed to making history accessible and meaningful through publishing books that celebrate and preserve the heritage of America's people and places.\n\nFind more books like this at\n\n**www.arcadiapublishing.com**\n\nSearch for your hometown history, your old stomping grounds, and even your favorite sports team.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nINSTANT   \nRECALL\n_\" **Many complain of their memory,  \nfew of their judgment.**\"_\n\nBenjamin Franklin (1706\u201390)\nMichael Tipper\n\n**INSTANT  \nRECALL**\n\nTips and Techniques   \nto Master Your Memory\n\n**CONTENTS**\n\nForeword by Dominic O'Brien\n\nIntroduction\n\n**1 WARM-UPS**\n\nWhy do we forget?\n\nIs it my age?\n\nYour mental filing system\n\nYour brain and your memory\n\nImagination, association and recall\n\nBeing certain of success\n\n**2 GENTLE STRETCHES**\n\nManaging stress for a better memory\n\nExercise to improve your memory\n\nA better diet, a better memory\n\nMemory joggers\n\nWhere did I leave...?\n\nThat reminds me...\n\n**3 GREAT TECHNIQUES**\n\nRemembering names \u2013 the social method\n\nRemembering facts \u2013 the magic of mnemonics\n\nRemembering spellings\n\nAlways remember your PIN\n\nRemembering directions\n\nRemembering errands and simple lists\n\nI must remember to...\n\nShopping with the alphabet\n\nRemembering what you've heard\n\nRetrieving the past\n\n**INTERLUDE: THE AMAZING WORLD OF MEMORY**\n\n**4 WINNING WORKOUTS**\n\nRemembering names and faces \u2013 the mnemonic method\n\nRemembering longer numbers\n\nThe Journey Technique\n\nRemembering dates and appointments\n\nLearning a new skill\n\nLearning foreign vocabulary\n\nRemembering speeches and jokes\n\nMind Maps\u00ae\n\nRemembering what you've read\n\n**5 CHAMPIONSHIP POINTS**\n\nThe Major System\n\nLearning from a champion\n\nRemembering a deck of cards\n\nLong lists of specialized items\n\nFurther reading\n\nContact the author\n\nAuthor's Acknowledgments\nDAD, I'M DEDICATING THIS TO YOU.\n\n## FOREWORD\n\nThe saying, \"Those who can, do. Those who can't, teach,\" certainly does not apply to Michael Tipper. Not only did Michael prove a formidable opponent to me at The World Memory Championships, he is also an expert trainer in Accelerated Learning principles.\n\nIn this enlightening book, a whole range of sound, tried-and-tested memory techniques is comprehensively explained by this master of memory. In fact, Michael is more than a master, as he was awarded the coveted title of Grand Master of Memory as well as gaining the Silver Medal at the World Memory Championships.\n\nMichael trained his memory to compete at worldclass level for one very good reason \u2013 to demonstrate that anyone can harness simple concepts to dramatically improve their mental functioning and to achieve success in their field, a sentiment I share with him.\n\nNot only did Michael do this convincingly, but his real talent lies in explaining the most effective ways of using mind techniques for practical benefit. Over 65,000 people have experienced his dynamic presentations, and over half a million children have benefited from his educational programmes. I am therefore delighted to recommend this book to anyone who wants to improve their memory \u2013 whether in the home, in the workplace or at school.\n\n**Dominic O'Brien**\n\nEight-Times World Memory Champion\n\n## INTRODUCTION\n\nMany people think that they have a bad memory and, if you've picked up this book, I'm guessing that you do, too. I know exactly how you feel, because some years ago I thought exactly the same and, just as you are now, I began searching for answers to solve what I thought was my \"problem\".\n\nAt the time, I was just 16 years old and had joined the Royal Navy as a naval apprentice, eager to learn my new trade. Although I'd done reasonably well at school, I soon discovered that the Navy was a very different learning environment from the classroom, and I began to struggle, especially with having to remember so much new information so quickly. I naturally assumed my memory was at fault and sought ways to improve it. I soon discovered, though, that there was nothing wrong with my memory: it was just that I didn't know how to use it.\n\nI found a memory course advertised in a newspaper, very quickly mastered some simple techniques, and began passing my exams with ease. With my new skills, I quickly impressed my superiors and was selected for officer training. I then went on to complete a degree in engineering and subsequently joined the Royal Navy's elite Submarine Service.\n\nMy later naval training tested my memory and learning skills even further, and I regularly came top of most of the courses I attended, even when my fellow students were more talented and able than myself. The only difference between us was that I had equipped myself with better learning strategies.\n\nEventually, I found myself telling more and more of my colleagues and friends about how to improve their memory and learning skills, and discovered a passion and a talent for sharing these ideas and helping people develop. At about the same time, I entered the World Memory Championships; at my second attempt I won the Silver Medal and became a Grand Master of Memory.\n\nMy success led to a professional speaking career, and over the last few years I've personally worked with over 65,000 people all over the world and have developed programmes that have been taught to over half a million people.\n\nFrom my own experience and from all the work I've done with countless people over the years, I know that you can improve your memory by understanding how it works and by discovering some basic ideas that anyone can apply. This book has captured all of the most powerful strategies I've used, and helped others use to great effect to get more from their memory and improve their recall.\n\nDecide for yourself whether you need this book by reading the statements opposite. Simply answer \"yes\", \"no\" or \"sometimes\". If you answer \"yes\" or \"sometimes\" to five or more of the statements, this book will definitely help you. Even though you might think you have a problem with your memory, I'm delighted to tell you that you probably don't. The only thing that's wrong is that you don't know how to use it because I doubt whether anyone ever taught you.\n\nThis book will take you step by step through some simple techniques and strategies (the \"how\") that will not only improve your memory, but also your levels of concentration and mental agility.\n\n### DO THESE STATEMENTS APPLY TO YOU?\n\n\u2022 I have difficulty remembering the names of people I've just met.\n\n\u2022 I have difficulty remembering the names of people I met longer than a few weeks ago.\n\n\u2022 I often walk into a room to get something and can't remember what it is.\n\n\u2022 I often forget where I've left my keys\/purse\/glasses.\n\n\u2022 I have on occasion parked the car and been unable to find it.\n\n\u2022 I regularly miss important appointments because I've forgotten about them.\n\n\u2022 I'm well known for missing friends' and relatives' birthdays and anniversaries.\n\n\u2022 I have to write down the PIN number for my ATM card.\n\n\u2022 If I lost my phone and had to call the service provider, I'd have no idea what number to ring.\n\n\u2022 I always avoid standing up and talking to groups of people because I know I'm going to forget what I have to say.\n\n\u2022 I spend hours reading all sorts of material and can hardly remember anything about what I've read.\n\n\u2022 I think that, as I'm getting older, my memory is getting worse.\n\n\u2022 I find myself saying things like \"I have a terrible memory\" or \"I always forget.\"\n\n\u2022 I almost expect to forget something that I was supposed to remember.\n\n\u2022 I'd like to learn something new but don't think my memory is up to it.\n\n# 1   \nWARM-UPS\n\nYou'll develop extremely powerful recall when you follow the simple techniques in this book. However, before you embark on them, it will help if you understand some of the reasons why your memory might appear to you to be unsatisfactory, together with some of the basics that will form the foundation of your memory improvement skills.\n\nThis chapter will look at why people like you forget, showing you that this is not necessarily because you're getting older. I'll tell you a little bit about the brain and how you file away and recall your memories, as well as about the key principles of a strong memory. I'll also give you a five-step plan that, if followed, will guarantee your success in improving your memory.\n\n## WHY DO WE FORGET?\n\nBefore we look at what you can do to enhance your memory, it's useful to realize why we forget. The factors that affect our ability to recall are related to the information we're receiving, the mental and physical state we're in both at the time and afterwards, and some of the processes that naturally occur in the brain.\n\n### COMMON REASONS FOR FORGETTING\n\nThere are various causes of forgetfulness, which can affect us all at different times:\n\n\u2022 Not interested \u2013 if you're not interested in something, it is unlikely that you will pay any attention to it.   \nAs a result, you won't be able to understand, learn about or remember it.\n\n\u2022 Not concentrating \u2013 this is linked to not being interested. If something doesn't capture your attention, you won't concentrate on it. If you're not concentrating and are thinking about something else, you won't take the new information in, however crucial it might be to remember it.\n\n\u2022 Too stressed \u2013 when you're stressed you're in the worst possible state to use your memory effectively. It's either hard to learn things or difficult to recall them.\n\n\u2022 Too much information \u2013 if you're faced with a barrage of information, it's easy to become overwhelmed (and therefore stressed), making it hard to learn and remember.\n\n\u2022 Poorly organized information \u2013 it's much harder to remember random bits of data than it is to recall information that has been logically organized.\n\n\u2022 Weak links \u2013 as we'll see later, the brain remembers things by association. If the links between related pieces of information are weak (for example, between a name and a face), recall will be difficult. If you don't use the information you've learnt, it won't integrate with your existing knowledge, which will also result in weak links and poor recall.\n\n\u2022 Too long ago \u2013 if a long period of time has elapsed between encountering something and having to recall it, the memory of it will be much weaker.\n\n\u2022 Interference \u2013 with so much information coming to your mind, there's a strong possibility that the new data will interfere with your existing knowledge and that the new information in turn will be interfered with by newer data as that arrives, which will adversely affect your recall.\n\nFortunately, by making simple changes to the way we think, we can overcome the common causes of forgetfulness and vastly improve our memory capacity.\n\n_\" Pleasure is the flower that passes;  \nremembrance, the lasting perfume. **\"**_\n\nJean de Boufflers (1738\u20131815)\n\n## IS IT MY AGE?\n\nAs people get older, many start to experience challenges with their memory and naturally assume the decline in their mental performance is due to their age. However, unless you're suffering from a disease or illness, your memory doesn't necessarily deteriorate with time.\n\n### YOUR MEMORY DOESN'T HAVE TO DECLINE\n\nStudies with people over 60 have shown little or no deterioration in memory performance \u2013 only a slightly slower reaction time. The apparent decline that so many people expect to happen is often caused by a combination of the following factors:\n\n\u2022 Older people don't use their memory as much as they used to.\n\n\u2022 Their brain isn't getting as much oxygen as it did when they were younger, because they're probably getting less exercise.\n\n\u2022 They've generated a belief that they have a declining memory because of forgetting one or two things. They strengthen this belief by telling themselves and others that they have a poor memory.\n\n\u2022 By being less in the present, they often miss things and then blame their memory because they can't remember, whereas in fact it's their concentration that's at fault.\n\nAll these factors can be reversed by making some lifestyle changes as well as adapting the way you think, so that it becomes possible to have a good memory however old you are.\n\n## YOUR MENTAL FILING SYSTEM\n\nIf your memory isn't performing as well as you would like it to, you may not be organizing information in a way that enables you to recall it easily. It's simply a question of tuning your already powerful memory retrieval system.\n\n### YOU THINK LIKE A VIRTUAL FILING CABINET\n\nImagine your memory is a huge virtual filing cabinet. For example, at a business meeting you're introduced to someone called David Jones. You store his face and name in a mental file marked **\"met David Jones at business meeting\"**.\n\nIn future when you see a face you recognize, you go to your mental filing system to the file where that face is stored. This file tells you the face belongs to someone called David Jones.\n\nSEE FACE > MENTAL FILE > RETRIEVE NAME\n\nOr you may hear the name David Jones and in your mental filing system the file with that name on it shows you a picture of him.\n\nHEAR NAME > MENTAL FILE > RETRIEVE FACE\n\nHILLARY CLINTON\n\n### EXERCISE 1\n\nRead the two words at the top of this page, noting exactly what you think and how long it takes to formulate your thoughts.\n\nMost people who do this exercise instantly see a picture of Hillary Clinton in their mind's eye. It might, perhaps, be an image of her face from one of her many television appearances or from a video you've watched online of her giving a speech. Or you may bring to mind a photograph of her with her husband on the cover of a glossy magazine.\n\nThe exact image that first comes to mind will depend on your own personal associations with Hillary Clinton. It doesn't matter what picture you see, because, whichever one you're most familiar with, this exercise demonstrates how powerful your mental retrieval system is.\n\nIn literally an instant you were able to go straight to an image of Hillary Clinton and retrieve it from your memory. Instantaneously you found it from among the many millions and millions of unique images that you've stored in your brain. That's an incredible feat of recall and one you should be proud of.\n\n### EXERCISE 2\n\nNow try a variation on this exercise by listening to the news on the radio, and noticing just how quickly pictures come into your mind of the people, topics or places that the newsreader is talking about. Once again, the speed at which you'll find you can evoke vivid, appropriate images should reassure you of just how strong your powers of memory already are.\n\n### EXERCISE 3\n\nRepeat Exercise 1 with a friend. Compare your images and notice how unique your pictures and associations are.\n\n### THE POWER OF ASSOCIATION\n\nThe process by which you can conjure up images of Hillary Clinton, and other people and topics in the news, is known as association. And it's this process that enables you to go to your mental filing cabinet to find and swiftly retrieve the name or face of David Jones, or whoever you've been introduced to.\n\nThe mind organizes associations in a combination of two ways. First, it organizes them as a chain so that one thing will remind us of another, which then prompts us to think of something else, and this in turn triggers a further memory, and so it continues. Second, it groups them as a series of hooks \u2013 one concept will have a series of associations that are all directly linked to the original idea.\n\n_\" Everyone has a photographic memory.  \nSome just don't have any film. **\"**_\n\nAnonymous\n\nBy understanding this, you can harness your mind more effectively to create powerful associations that will help you remember and recall far more than you can at present. And you will find that your ability to instantly summon up your associations is staggering.\n\n## YOUR BRAIN AND YOUR MEMORY\n\nYour brain is the control centre of everything that happens in your body. Every second there are thousands of chemical and electrical reactions taking place in the grey matter between your ears. When you understand how your brain functions a little better, you'll be able to use it more effectively to improve your memory.\n\n### THE BUILDING BLOCKS\n\nThe brain is made up of billions of cells, which are called neurons, and each of these is capable of connecting with thousands of other neurons. The complexity and sheer scale of the number of these connections give the brain its near-infinite potential. Every thought you've experienced, every sensation you've felt and every memory you've ever had is a connection between two or more of these neurons.\n\n### THE BIGGER PIECES\n\nTo explain the functioning of such a complex and interconnected mechanism as the brain is impossible in a few words. However, there are a few \"big pieces\" that you should be aware of when you are considering your memory.\n\nThe **amygdala** tags incoming information with emotional value. The more emotionally charged an experience, the more likely it is to be remembered.\n\nThe **hippocampus** is responsible for the transference of short-term memories into the long term. It's also the part of the brain that \"lights up\" when we think about something or somewhere we know well (something we'll take advantage of later in this book).\n\nThe **cortex** is sometimes known as the thinking part of the brain, because it is where conscious thought processes take place. It consists of two halves, and research done over 50 years ago identified different functions for each half. The distinction was thought to be:\n\n**LEFT SIDE** | **RIGHT SIDE**  \n---|---  \nLists, Lines, Logic, Words, Number, Order | Rhythm, Colour, Spatial   \nawareness, Pictures,   \nDaydreaming, Imagination\n\nIt's now known that the relationship between the two halves of the brain is much more complex, and the current theory is that the left brain processes in a more serial manner and the right brain in a more parallel manner. Or, to put it another way, research has demonstrated that the left brain focuses on the detail, whereas the right brain sees the bigger picture.\n\nTraditional ways of learning and remembering have focused on the activities associated with the left brain functions. What's now certain is that the more you use both sides of the brain, the more effective your thinking will be and the better your memory.\n\n### DIFFERENT FREQUENCIES\n\nAt different levels of consciousness ranging from the wide awake state to that of deep sleep, the brain exhibits electrical activity at different frequencies.\n\nThe beta (\u03b2) frequencies are the highest, and dominate when we're wide awake. The alpha frequencies (\u03b1) become more prevalent at the relaxed but alert state. As we drift into a deeper meditative state on the way to falling asleep, we encounter more theta (\u03b8) waves, and finally, once we're in a deep sleep, the delta (\u03b4) frequencies take over.\n\nLearning is most effective in the alpha state when we're calm and relaxed. Fortunately, we're able to access this state using meditation and relaxation techniques as a way of enhancing memory.\n\n### WHAT AFFECTS YOUR BRAIN WILL AFFECT YOUR MEMORY\n\nAs a delicate and finely tuned instrument, your brain will be affected by a variety of factors that are within your control. Excess caffeine, nicotine and alcohol upset the delicate chemical balance within the brain, which undoubtedly impairs its performance. What you eat also has an effect, and a healthy balanced diet will help in improving your mental performance. Eating specific memory-enhancing foods can also be beneficial, particularly those, such as citrus and berry fruits and leafy green vegetables, that are rich in antioxidants (see p.48).\n\n_\" Memory ... is the diary that we all carry  \nabout with us. **\"**_\n\nOscar Wilde (1854\u20131900)\n\n### OXYGEN AND THE BRAIN\n\nWithout sufficient oxygen, your brain (and then you) will die. The importance of oxygen is clear when you consider that the brain might be only two percent of the body's weight but that it consumes over 25 percent of its oxygen intake. The more efficient your ability to breathe well, the more oxygen will get to your brain, which is one of the reasons that exercise is so important, as we'll see later.\n\n### THE IMPORTANCE OF SLEEP\n\nSleep is not only vital to the healthy functioning of your brain, but also crucial for learning and memory. Numerous studies have shown that people remember more effectively if they have a good night's sleep after learning. It's believed that during sleep the brain revisits its recent experiences to reinforce them and etch them into the memory. Contrary to popular belief, new learning doesn't take place while we're asleep, only the integration of information already learnt.\n\n## IMAGINATION, ASSOCIATION AND RECALL\n\nYour brain is like an amazing computer that sadly doesn't come with a user manual. You began learning how to use it by trial and error and slowly developed the thinking skills you now have. There are ways in which you can build on those and use your mind to improve your ability to remember.\n\n### ACHIEVE MORE BY DOING LESS!\n\nIt's a common misconception that the harder and longer you work at learning something, the more likely you are to be able to remember it. Fortunately, this isn't true because our minds don't naturally work like that. When you're engaged in a learning task, you're more likely to recall information from the beginning and end of the session, with a dip in your memory of the topics from the middle.\n\nThis is called the Primacy and Recency Effect. The longer the period of time you work, the bigger the dip and the less you'll recall. The secret is to take more breaks so you have more primacy and recency \"high points\" of recall. A good rule of thumb is to work for between 20 and 50 minutes and then take a five- to 10-minute break. Not only will you remember more, but you'll stay fresh and alert for longer.\n\n### THE SECRET OF HAVING A POWERFUL MEMORY\n\nThere are three simple stages for developing your memory to its full potential. Here's the secret of being able to remember absolutely anything you want to for as long as you want to:\n\n### HOW TO USE YOUR IMAGINATION\n\nThink about the last time you walked down a street full of people. Is there anyone who particularly stood out? You might have walked past hundreds of people, yet can't remember any of them. However, if a seven-foot-tall woman wearing a bright yellow coat, red hat with bright blue feathers and black leather boots, had ridden past on a pink elephant, singing the national anthem, do you think you might have remembered her? Of course you would, because that image would have stood out from all the other people you saw.\n\nOf course, not everything in our lives is as unique and as outstanding, but with the power of our imaginations we can make anything as memorable as we want by applying a few simple principles. Here are just a few ways to add some \"memory spice\":\n\n\u2022 Think in pictures and symbols even for dry and boring subjects.\n\n\u2022 Use lots of vibrant and striking colours.\n\n\u2022 Exaggerate by making small things large.\n\n\u2022 Distort by making large things small.\n\n\u2022 Create pictures with lots of movement.\n\n\u2022 Use and engage all your senses.\n\n\u2022 Be as outrageous as you can by making the images crude and shocking.\n\n\u2022 Use bizarre and unusual associations.\n\n\u2022 Make things as funny as possible \u2013 a touch of humour can make them particularly memorable.\n\n_**\"** Imagination is more powerful   \nthan knowledge. **\"**_\n\nAlbert Einstein (1879\u20131955)\n\nThe easiest way to describe this process in one sentence is for you to start thinking like a Tom and Jerry cartoon! Don't worry, you don't have to be a budding animator \u2013 just give your imagination free rein. You may be surprised at what you come up with.\n\n### ASSOCIATION \u2013 THE KEY TO EFFECTIVE RECALL\n\nWe saw earlier that the brain is an associative mechanism in that it organizes its memories by association. There are two ways in which it does so. The first is by using a sequence or chain of associations \u2013 one thing will trigger another, which itself will prompt a third thing, and so on. For example:\n\nTREE > PARK > WALK > FEET > SHOES > LACES\n\nThe other way is by a collection of associations or hooks. One idea or concept will have a series of related words linked to it, as in the beach example below.\n\nBy creating strong chains between items you want to remember and by using multiple hooks, you'll be able to recall things much more effectively, especially if you enhance those links by using your imagination in the way that I've described above.\n\n### ORGANIZING YOUR ASSOCIATIONS\n\nHaving a series of strong associations will serve you well, only if you're able to consciously access them by choice whenever you want to. You now need to organize your powerful filing system of a memory so that it's easy to find any file. The way to do that is to give each file a label. I'll be showing you various labelling methods based on visual techniques.\n\nAs we've seen with the Hillary Clinton exercise on page 22, we tend to think in pictures rather than words or numbers. (Although we've been conditioned over two millennia to think in words, our natural inclination, dating back to the era before written language, is to think in pictures, feelings or sensations.)\n\n### REMEMBER FOR LONGER\n\nMany people complain that they can't recall things from the recent past, and they believe this results from a poor memory. What they don't know is that it's normal for this to happen because, within one to two days of encountering new information, you're likely to remember only about 20 percent of it. The reason for this drop-off is the overlaying of similar or related information, which makes it harder to distinguish one memory from another, which in turn reduces the ability to recall. This is known as the confusion factor. Rather than arising from a poor memory, the inability to remember the new information is the result of not having a proper and effective process to condition the recall so that you'll always be able to recapture it.\n\n### REHEARSE THE RECALL\n\nIf you really want to remember something important, you need to practise recalling it from memory after 10 minutes, a day, a week, a month, three months and six months. After that it will be in your long-term memory and you'll always be able to recall it because you'll have conditioned it into your mind.\n\n### BREAK IT DOWN INTO LITTLE PIECES\n\nThink of the children's joke, \"How do you eat an elephant?,\" to which the answer is, \"One bite at a time.\" The same principle applies when dealing with things that you have to remember. Breaking things down into smaller, more manageable chunks makes remembering them much easier.\n\nIn this book I'll give you specific and powerful ways to use imagination, association and recall to organize your mental files so you can access them easily and recall what you've memorized.\n\n**GUIDING PRINCIPLES**\n\n**Here are the general principles that will apply to everything you'll learn about improving your memory in the rest of the book.**\n\nWork for 20\u201350 minutes and take five- to 10-minute breaks.\n\nUse your imagination to create strong, memorable images that   \nstand out in your mind.\n\nCreate powerful associations using hooks and chains.\n\nOrganize your associations by using one of the many mental filing   \nsystems that you'll learn about in this book.\n\nRegularly rehearse recalling what you've learned to ensure   \nlong-term recall.\n\nBreak things down into smaller, more manageable chunks.\n\n_\"I've a grand memory for forgetting. **\"**_\n\nRobert Louis Stevenson (1850\u201394)\n\n## BEING CERTAIN OF SUCCESS\n\nIf you look at people who've achieved anything of note in any area of their lives, there are some common things they'll have done to ensure success. Whether you want to climb the highest mountain, find a new job or improve your memory, you're more likely to reach your goal if you have the mindset of a successful person.\n\n### FIVE STEPS TO SUCCESS   \nStep 1 \u2013 Have a goal\n\nThis is the most important step, because if you don't have a clearly defined target you'll never know if you've hit it. Your goal needs to be easy to understand, measurable, have a deadline and be **written down**. Writing it down is essential because it's the first step as your goal translates from a fleeting idea inside your mind into reality. Here are two examples of written goals:\n\n\u2022 I want to improve my memory.\n\n\u2022 I want to be able to remember straightaway the first and last names of 15 new people who I meet at business or social gatherings over the next month, and then be able to recall their names if I meet them again in the future.\n\nThe first example is vague and it has no clear definition of success or when it might be achieved. The second example is much better because the goal is very clearly defined and there's a specific time-frame.\n\n#### Step 2 \u2013 Have a plan\n\nHaving set your goal, you now need to make a plan that will enable you to achieve it. The simplest approach is to first make a list of everything that you need to do to achieve the goal, and then organize that list into numbered stages.\n\nIf we take the example of the goal from Step 1 of wanting to remember names straightaway, the start of your plan might look something like this:\n\n\u2022 Read section on remembering names.\n\n\u2022 Practise to myself.\n\n\u2022 Teach my family the method for remembering names so that I know I understand it.\n\n\u2022 Practise on my friends.\n\n\u2022 Try out the system with one new person at my next meeting.\n\n#### Step 3 \u2013 Believe you can do it\n\nNot believing you can do something will seriously affect your ability to do it, but fortunately there are two straightforward ways to give your mind the \"proof \" that you can.\n\nThe first is to use self talk (this is also known as affirmations), which simply involves making positive statements about yourself and your abilities. It is common for most people to say negative things about themselves, such as:\n\n\"I have a poor memory.\"   \nor\n\n\"I can't remember that.\"   \nor\n\n\"That's too difficult to learn.\"\n\nThese sorts of statements help to reinforce the belief that you have a poor memory. So, instead, to help you improve your memory, you need to say things like:\n\n\"I have a fabulous memory.\"   \nand\n\n\"I always remember the names of everyone I meet.\"   \nand\n\n\"Learning and remembering new information is easy and fun for me.\"\n\nThe more you repeat these statements, the greater your belief in your powers of memory will become.\n\nA second powerful way to help you with your belief levels is to use your imagination to create little movies in your mind of what it will be like when you're successfully using your great memory. See, hear and feel \"perfect memory situations\" in great detail and really imagine yourself having the memory ability you desire. Used by the world's top athletes to focus before the big game, this process of imagining a successful outcome is known as \"mental rehearsal\".\n\n#### Step 4 \u2013 Take action\n\nNow put your plan into action using the stages described in Step 2. Keep going until everything on your list is complete and you've achieved your goal. As you work your way through your plan, check to make sure that your actions are taking you closer to your goal. If not, it's OK to refine and re-write the plan so that it gets you to your desired outcome more quickly. The key to Step 4 is to keep taking action until the goal is achieved.\n\n#### Step 5 \u2013 Keep a positive attitude\n\nLife has a habit of throwing things in the way as we progress toward our goals, but we're lucky enough to be able to choose how we respond. Letting negative emotions develop when things don't go our way causes stress, which adversely affects how we feel. In terms of developing a good memory, negative emotions are a major barrier to overcome. Keeping a positive outlook on life and reacting to events and circumstances in a positive way won't make problems go away, but they'll furnish you with a more resourceful state of mind to deal with them (and it's a lot more fun too!).\n\nTo get you started in applying these five simple steps, here are a few exercises:\n\n\u2022 Think about all the goals that you have for your memory and write them down so that you know exactly what you want for each one and when you want to achieve it by.\n\n\u2022 Choose the most important of your goals and create a plan for achieving it.\n\n\u2022 Write down on a card five positive affirmations about attaining this goal, and repeat them to yourself at every opportunity.\n\n\u2022 Visualize achieving your goal, using your imagination and all your senses to do so \u2013 see it as though it has already happened.\n\n\u2022 Start working toward your goal by beginning with the first item on your list of things to do.\n\n# 2   \nGENTLE STRETCHES\n\nBefore we start delving into memory improvement techniques, there are, first of all, some straightforward things that you can do in terms of lifestyle that will have a significant effect on your powers of memorization and recall.\n\nThis chapter will look at the number one lifestyle challenge to having a correctly functioning memory \u2013 the stress factor \u2013 and what you can do to overcome it. It will also look at ways to boost your memory faculties through exercise and simple, commonsense changes to your diet, including a guide to memoryenhancing foods. In addition, I'll show you some practical steps you can take immediately to help you recall some of the things that you need to.\n\n## MANAGING STRESS FOR A BETTER MEMORY\n\nStress affects the way we think and, in particular, the performance of memory. By minimizing the effects of stress, you can create a much better physical and mental environment for your memory, enabling it to perform at its best.\n\n### HOW STRESS AFFECTS MEMORY\n\nAs we evolved as human beings, our bodies created an effective protection and survival device that was designed to keep us alive when threatened \u2013 it is called the fight or flight mechanism.\n\nWhen faced with a potential danger, we could either tackle it (fight) or run away from it (flight). Whichever choice we made, we needed to breathe faster, produce more adrenaline, tense our muscles and shut down bodily systems that we didn't need.\n\nToday we are fortunate in that we rarely have to fight or flee. However, we're still bombarded with many things in our personal and professional lives that can be perceived as a threat, and our bodies still respond in the same way, which creates stress.\n\nStress affects memory in two ways. First, it shuts down part of the brain that is responsible for longterm memory, which is why we find it difficult to recall information when we are under pressure. Second, if the stress chemicals produced in the brain remain there for longer periods, they create a toxic bath that destroys brain cells, especially those that are involved in memory.\n\n**LEARN TO RELAX**\n\n**In order to manage stress and help improve your memory,  \nyou need to learn to relax. Try some of these exercises to   \nsee which work best for you. Do them somewhere quiet   \nwhere you won't be disturbed.**\n\nPhysically tense and then consciously relax every muscle in turn,   \nstarting from your head and working down to your toes.\n\nUsing your imagination, picture yourself in a peaceful place, such   \nas on a deserted tropical beach or in a secluded meadow, and then   \nimagine the stress and tension draining away from your body.\n\nSlow your breathing down and count the number of breaths you take   \nin one minute; aim to approximately halve that number for the next   \nminute. Repeat until your breathing rate is slow and deep.\n\nPractise breath meditation by first physically relaxing every muscle   \nin your body and then focusing on nothing else apart from   \ncounting your breaths.\n\n_\" To the mind that is still, the whole universe  \nsurrenders. **\"**_\n\nLao Tzu (6th century BC)\n\n## EXERCISE TO IMPROVE YOUR MEMORY\n\nA healthy body helps to create a healthy mind, and a good exercise session will not only clear your head but also make you feel better. Exercise also helps to improve memory performance and should be an important part of your memory improvement plan.\n\n### EXERCISE HELPS COMBAT STRESS\n\nRegular exercise can overcome the effects of stress in two key ways. First, it provides an outlet for the build-up in the brain of potentially harmful fight or flight chemicals, which can lead to stress. Second, it conditions both the body and the mind to be less susceptible to fatigue and thereby enables them to handle stress more effectively.\n\n### DO THE RIGHT EXERCISE\n\nThe right kind of exercise to combat stress and improve your memory is aerobic \u2013 exercise in which your heart rate is 70\u201385 percent of your maximum (your doctor can advise you on what these rates should be). Exercise at this interim rate for 30 minutes, with a 10-minute warm-up and warm-down, for at least three times per week, and preferably five times, in order to obtain the maximum benefit.\n\n**IT REALLY WORKS!**   \nRecent research into the effects of exercise on ageing has shown that exercise boosts mental performance generally and, in particular, improves memory. Additional research into the effects of exercise on academic performance demonstrated that over 80 percent of high-achieving students exercised at least three times per week and that the failure rate of physically fit students was half that of those who were unfit.\n\nBrisk walking is a good form of aerobic exercise \u2013 try fitting this in whenever you can, such as going for a fast-paced 30-minute walk in your lunch hour at work. Other ideal aerobic activities are jogging, swimming, cycling and dancing.\n\n### WHY DOES AEROBIC EXERCISE IMPROVE MEMORY?\n\nAerobic exercise is beneficial for the heart, blood vessels and lungs, with the added bonus of helping to improve memory. It generates an increase in the body's need for oxygen \u2013 as a result, the heart and lungs have to work a little harder than normal.\n\nHeavier breathing during aerobic exercise draws more oxygen into the body, resulting in a faster heart rate. This faster rate pumps the oxygen-enriched blood around the circulatory system.\n\nTwenty to 40 percent of this blood flows to the brain and, as the brain thrives on oxygen, it will perform better because of the increased flow.\n\n### START GENTLY\n\nBefore you begin an exercise programme, ask your doctor for advice on the most suitable form of exercise for your physical condition and for the all-clear to start. Begin gently \u2013 don't rush it. Gradually build up to exercising three times a week for 30 minutes.\n\nTry doing different forms of exercise to discover the one or ones you enjoy the most \u2013 you're more likely to stick at them. It also helps to find an exercise buddy to keep you company and, even more critically, keep you motivated.\n\n## A BETTER DIET, A BETTER MEMORY\n\nMental performance and, in particular, memory are affected by what we consume. Consequently, if you choose the right food, you can improve your memory.\n\n### A GOOD FOOD FOUNDATION\n\nThe foundation for a memory-improving eating plan is a diet that's balanced and healthy \u2013 low in fat, salt and sugar, and high in fibre. It should also contain at least five portions of fresh fruit and vegetables a day. A useful rule is to see how \"alive and fresh\" your food looks.\n\nIt's also important to keep the body hydrated. This is because the brain is nearly 80 percent water and if that water level drops through dehydration, the brain's performance will deteriorate.\n\n### DEVELOPING A BETTER DIET\n\nTo ensure an optimum diet to support your memoryimprovement programme, here's what I advise:\n\n\u2022 Keep a food journal for a week to monitor what you eat and drink.\n\n\u2022 Consult your doctor or a qualified nutritionist for advice on how to adjust your diet to make it healthier, if necessary.\n\n\u2022 Eat lots of water-rich foods such as salads and fruits, and drink at least eight glasses of water \u2013 three to four pints \u2013 a day.\n\n\u2022 Eat regularly and maintain your energy levels with \"slow-burning\" energy foods such as wholewheat bread and pasta, brown rice, wholegrain breakfast cereals and porridge oats.\n\n\u2022 Reduce your intake of salt and sugars, which can be very high in convenience and \"junk\" food \u2013 check the labels carefully.\n\n### MEMORY-ENHANCING FOODS\n\nThere are also specific foods that enhance memory, notably antioxidants, B vitamins and Omega-3 fatty acids.\n\n### MEMORY-BOOSTING SUPPLEMENTS\n\n_Ginkgo biloba_ has traditionally been used by Chinese herbalists for a variety of remedies. Scientific studies have shown that extracts stimulate circulation in the brain, which improves short-term memory.\n\n**ANTIOXIDANTS** | **B VITAMINS** | **OMEGA-3 FATTY ACIDS**  \n---|---|---  \nAlfalfa sprouts   \nBerry fruits   \nBroccoli   \nCitrus fruits   \nCranberries   \nGrapes   \nKale   \nMango and papaya   \nSpinach   \nTomatoes | Dairy   \nproducts   \nLean   \nmeat and   \npoultry   \nLegumes   \nNuts and   \nseeds   \nWheat germ | Herring   \nMackerel   \nSalmon   \nSardines   \nTrout   \nTuna   \n(fresh has a   \nhigher concentration   \nthan canned)\n\nGinseng helps to neutralize the negative effects of free radicals and excess toxic chemicals in the brain caused by stress.\n\n## MEMORY JOGGERS\n\nAs you read this book, you'll be able to develop an extremely strong memory and instant recall just by using the power of your mind. However, while you gradually build up your natural memory powers, you shouldn't overlook some simple \"artificial\" methods that will help you remember a wide spectrum of information.\n\n### ARTIFICIAL STORAGE AND RETRIEVAL SYSTEMS\n\nThere are a number of practical ways to store information so you can retrieve it whenever you want to, as well as ways to prompt you to remember to do something at exactly the right time. Here are some suggestions.\n\n\u2022 Programme your phone \u2013 set the alarm to remind you of an important appointment or something that you have to do at a specific time.\n\n\u2022 Use the refrigerator as a reminder board \u2013 write down your list of things to do in large writing on a big piece of paper and stick it on the refrigerator where you'll always see it.\n\n\u2022 Keep a journal \u2013 reflect on your day by recording in a journal all the significant things that have happened to you and, in particular, how you feel about them. Remember the journal is private, so you can record your thoughts and feelings honestly. Reading the journal in a few months' or years' time will assist you in recalling your past and trigger a whole range of associated memories.\n\n\u2022 Use your calendar \u2013 make a note on your calendar of important dates such as birthdays and anniversaries and when to buy cards and presents.\n\n\u2022 Display reminders on your computer \u2013 if you use your computer every day, write reminders of things that you have to do on pieces of paper and stick them over your screen so that you'll see them when you log on, or use the reminder software on your computer.\n\n_**\"** The palest ink is better than the   \nbest memory. **\"**_\n\nChinese proverb\n\n\u2022 Bribe your children \u2013 if you have young children, prime them to remind you to do something by offering them a little incentive such as ice cream or an extra allowance if they remind you at the right time.\n\nYou might think that using these methods is cheating because you're not relying on your natural memory. However, your ability to remember something increases dramatically when you bring that information into the external real world from out of your internal thinking world. When you write something down or type it into an electronic device, you'll see it, feel it and possibly hear it (if you say what it is out loud). All of this will make the information much more memorable.\n\nTry using some of the suggestions I've given you to help in your memory recall. Or see if you can think of other devices or methods that you could use to help you remember.\n\n## WHERE DID I LEAVE...?\n\nMany people, particularly as they get older, start thinking their memory is getting worse because they keep losing their keys, misplacing their purse or even forgetting where they parked the car! They assume a failing memory is responsible for these lapses, but in fact this usually isn't so. A simple technique will ensure you'll never forget where you left your car and never misplace your keys or purse again.\n\n### YOUR CONSCIOUS AND SUBCONSCIOUS MIND\n\nIn the most basic terms, we think on two different levels \u2013 consciously and subconsciously.\n\nYour conscious mind is focused at the moment on reading these words on this page. This part of your mind is where you do your conscious thinking as you engage with the world and what's happening around you.\n\nThe subconscious deals with everything else not looked after by your conscious mind, and is much larger. For example, your subconscious mind is currently processing the physical sensations being experienced by your left foot, which you weren't aware of until I drew your attention to them.\n\nYour subconscious mind is so powerful that it can allow you to act on \"auto-pilot\" so you don't even have to think about what you're doing, particularly if it's something you do often, such as taking off your glasses or putting down your house or car keys.\n\n### WHY YOU \"FORGET\" WHERE YOU PUT YOUR KEYS\n\nThe reason you misplace items (which you then blame your memory for) is simple. You weren't consciously thinking about them when you put them down, because you were operating on \"auto-pilot\". You weren't concentrating on what you were doing \u2013 in effect, \"you\" were never \"there\", which is why you can't consciously remember where you put your keys. Your subconscious will know where they are. However, since we engage with the world through our conscious mind, the keys are as good as lost. This will inevitably happen more often the older you get, not because your memory is getting worse, but because you will have more memories and associations. As a result, your conscious mind is going to be distracted more often, making it harder to concentrate on what you're doing, especially on mundane activities.\n\n### THE SOLUTION IS SIMPLE\n\nIn order to be able to remember where you leave something like your purse or keys or even your car, you need to bring its location into your conscious awareness. The easiest way to do this is to say out loud what it is you're putting down or leaving and where you're putting it. For example, \"I'm putting my keys on top of the microwave,\" or \"I'm leaving my car on the seventh floor in bay A\" \u2013 you can say this before you get out of the car to avoid funny looks from other motorists!\n\nPractise describing out loud to yourself where you leave things \u2013 whether around the house or when you're out and about \u2013 until it becomes a habit.\n\n## THAT REMINDS ME...\n\nIf you ever plan to mention something important to a friend or colleague, or maybe do a particular task in the office, and you forget, it can be really frustrating. What you need is a foolproof way to remind you of exactly what to do when you need to do it.\n\n### WHY WE FORGET TO DO SOMETHING\n\nThere are three main reasons why we sometimes don't remember to do things at the appropriate time:\n\n\u2022 When it occurs to us that we need to take some future action, the idea fleetingly passes through our conscious mind and we don't think about it long enough for it to consciously \"reappear\" at the right time or place.\n\n\u2022 Such a lapse in memory occurs mainly when we're doing something habitual or regular (such as spending time with colleagues or friends or going to the office) \u2013 we switch to \"automatic\" and the new idea doesn't have a chance to embed itself in our mind.\n\n\u2022 We haven't created a strong enough prompting mechanism to remind us of the idea at exactly the right time.\n\nWouldn't it be great if the solution to the problem was a massive billboard with flashing lights and loud music that magically appeared at just the right moment, with a reminder written on it in huge letters? Although we may not be able to make a real billboard to do that, we can create the mental equivalent. First, though, it's important to understand about anchors and triggers.\n\n### TRIGGERS, ANCHORS AND THE SALIVATING DOGS\n\nIn a famous experiment carried out by Russian scientist Ivan Pavlov at the beginning of the last century, dogs were conditioned to associate the ringing of a bell with being fed. When dogs are about to be given food they salivate as the food is presented to them. What Pavlov discovered was that he could get the dogs to salivate without food, just by ringing the bell. Firmly **anchored** in the dogs' minds was the link between the food and the bell (achieved by conditioning). Once the bell was sounded, it **triggered** the salivation response even when there was no food.\n\nTriggers and anchors work in humans, too, because over the years we've become conditioned to respond in certain ways to certain triggers. For example, if you're driving and you see a red stop light, you'll automatically brake. If you're introduced to someone new at a social or business meeting and they offer their hand to shake yours, you'll automatically reach out to take their hand without even giving it a second thought.\n\n### APPLYING THE KNOWLEDGE\n\nSo, how can you apply this knowledge to ensure you always remember to do something at a particular time or place? It's quite straightforward \u2013 you simply create a visual reminder, which is anchored in your memory and automatically triggered by the specific event or place.\n\n**TRIGGER** | **ANCHORED BEHAVIOUR**  \n---|---  \nRed light | Stop the car  \nOffer of a handshake | Shake hands\n\n### CREATING POWERFUL AUTOMATIC REMINDERS\n\nLet's imagine you're going away on a trip and that the next time you catch up with your friend John you have to ask him if he'll take you to the airport in a couple of weeks' time. To do this, all you have to do is to follow the REMIND process:\n\n**R** eview in your mind what you have to do and imagine yourself doing it when you have to do it. Visualizing the action in this way will condition you to expect success. So, in this example, you'll envisage yourself asking John to take you to the airport when you see him next.\n\n**E** xaggerate a picture of your trigger event (seeing John) and combine this with something related to what you have to do. You might see your friend sitting astride an airplane that was moving as if it were a mechanical bull. The more unusual and bizarre the image, the better. This is your anchored response.\n\n**M** aximize the recall power of the image by using all your senses and the principles of effective memory visualization on pp.29\u201331.\n\n**I** nstall the link firmly between your anchored response and the trigger by repeating the association in your mind a number of times so that every time you think of John, the picture of him sitting astride the airplane springs to mind.\n\n**N** ote whether the trigger will work or not, by thinking of something else and then go back to thinking about John. If the first thing you think of is him and the airplane, then you know that the trigger's working. If it's not working, practise reinforcing it until it does, or find a stronger trigger.\n\n**D** eepen the power of this process by affirming to yourself that it will work and that you'll remember by trusting your subconscious mind to remind you when you encounter your trigger (John).\n\nWhat will happen when you next meet John is that seeing him will automatically trigger the picture of him sitting astride the airplane, which will remind you to ask him to take you to the airport. You've created your own mental billboard!\n\n### TRY THIS OUT\n\n\u2022 To strengthen your understanding of this process, think of someone you know who'll benefit from this book and create a reminder to tell them about what you've learnt from it.\n\n\u2022 Try out the REMIND process with something that you have to do at work or a task that needs to be done around the house.\n\n\u2022 To convince yourself how powerful and long-lasting firmly anchored triggers can be, play some music from your teenage years and notice what memories come flooding back.\n\n# 3   \nGREAT TECHNIQUES\n\nYou're now going to discover a range of basic, easy-to-follow strategies for memorizing and recalling a variety of things from directions to spellings.\n\nIf you've ever forgotten someone's name within two minutes of meeting them, there's a good reason for it and this chapter will tell you what that is and how to deal with it. You'll also discover how to remember your PIN for your credit and debit cards, and how to lock away, so you never forget them, those flashes of inspiration that seem to come at the most inopportune moments. You'll find out how to do your regular food shopping without a list and still remember to buy everything that you need. And you'll learn foolproof methods for remembering directions as well as for spelling troublesome words correctly.\n\n## REMEMBERING NAMES \u2013 THE SOCIAL METHOD\n\nOne of the biggest memory-related problems many people have is a difficulty in remembering names, which often leads them to think that they must have a poor memory. However, in most cases their memory is, in fact, fine and the problem is just a matter of them having an ineffective process for handling the names of new people they meet.\n\n### WHY IS REMEMBERING NAMES SUCH A PROBLEM?\n\nIf I asked you to recall something you'd never been told, you'd think I was crazy. How can you possibly be expected to retrieve something from your memory that isn't there?\n\nWell, as silly as it may seem, this is the main reason why people have such a problem with remembering names \u2013 it is because they never actually registered the names in the first place!\n\nFor many, encountering new people at business or social functions is a stressful affair. With the pressures of wanting to create a good impression, the potential for rejection, the mental rehearsal of what you're going to say, the encounters with several people one after the other in quick succession and the myriad of other things swirling around your mind, someone else's name can hardly get through all that internal mental \"noise\" to stick in your memory.\n\n### THE SECRET OF REMEMBERING\n\nWhat you need to dramatically improve your ability to remember people's names is a way to control the introductions so that you definitely take in the names and remember them immediately. The following process can be used in social or business situations.\n\n### REMEMBERING NAMES STEP BY STEP\n\nStep 1 \u2013 Have your senses ready\n\nWhen you're faced with meeting new people, you're going to learn something about someone, so you'll need to be ready to see the person's face and hear their name.\n\n#### Step 2 \u2013 Shake hands\n\nAddress the person by shaking their hand and saying hello. By initiating the contact in this way, you control the process and are more likely to remember the name.\n\n#### Step 3 \u2013 Name yourself slowly and clearly\n\nThis will help the person you're meeting to register your name, but it will also mean they're likely to copy you and tell you their name in the same fashion, making it easier for you to hear and understand.\n\n#### Step 4 \u2013 Concentrate\n\nDo pay attention and concentrate when the person tells you their name by looking at their face and listening to what they say.\n\n#### Step 5 \u2013 Say their name back\n\nBy doing this immediately, you definitely fix the name in your conscious awareness where it will be easier to remember in the future.\n\n#### Step 6 \u2013 Have you got it right?\n\nCheck with the person that you've pronounced the name correctly, and even check the spelling, if necessary, to clarify it. This way, you're starting a subtle process of repetition that will firmly engrave the name in your memory.\n\n#### Step 7 \u2013 Ask about the name\n\nBy doing this, especially if it's an unusual name, you're continuing the process of repetition that will help you remember it, but you're also showing an interest in the person. This will create more of an emotional connection that will assist you in remembering them.\n\n#### Step 8 \u2013 Keep the name in mind\n\nMentally review the name at every opportunity you get. Look around the room at the people you've met and silently name them to make sure you have their names in your memory.\n\n#### Step 9 \u2013 Use the name\n\nEmploy the name at every appropriate opportunity, especially when you're talking to someone. For example,\n\n\"Janet, what do you think about this?,\" \"That's a very interesting point, Michael,\" \"John, would you please pass me that glass?\"\n\n#### Step 10 \u2013 Exchange business cards\n\nSay goodbye at the end of the function and exchange business cards if appropriate. Not only will this be a final repetition of the name, but when you receive the business card you'll see the name written down and this will help you remember it.\n\nWithout using this process, most introductions that happen between two people meeting for the first time are a one- to two-second rapid exchange of handshakes and barely registered names. And if you have to meet more than three or four people this way, it will be extremely difficult to remember their names at all.\n\n### KEEP PRACTISING\n\nWith practice, the introduction part of this process will take 15\u201320 seconds for each person that you meet. This will give you enough time to capture the name, repeat it a few times and develop a stronger connection with the person, which will definitely help you to remember them and their name.\n\nAs you go through the coming week, notice just how many people's names you already know. This will show you that you already have the ability to remember names.\n\nAlso, monitor how many people you're introduced to in a week and notice how well you can remember their names following your old method.\n\nBefore you begin using the 10-step introduction process, practise it with friends or family members until you're comfortable with it and then use it the next time you meet someone for the first time. If you want to become really good at remembering names, you can increase the number of people you try out the process on by one more each time you use it.\n\n## REMEMBERING FACTS \u2013 THE MAGIC OF MNEMONICS\n\nWhen you encounter a common fact, it's possible that someone has already created a simple and often clever way to remember it. Known as a mnemonic (pronounced **nem** \u2013 **on** \u2013 **ic** ), such a device uses words as a memory prompt. Many mnemonics have been passed down through generations.\n\n### THE FIRST-LETTER METHOD\n\nThe most commonly used method for remembering information, especially when there's a set sequence to the data, is to take the first letter of each word you need to know. If you have difficulty remembering the colours of a rainbow, try this time-honoured device to help you recall them, based on the first letter of each colour:   \n**R** ed **O** range **Y** ellow **G** reen **B** lue **I** ndigo **V** iolet.\n\nIf we take the first letter of each colour, we get:   \n**R O Y G B I V**\n\nMany people remember the colours just by recalling   \n\" **ROY G BIV** \", while others make up a sentence such as:\n\n**R** ichard **O** f **Y** ork **G** ave **B** attle **I** n **V** ain.\n\nWhether you prefer the acronym or the sentence, you'll have a fail-safe way to remember the order of the colours.\n\nAnother long-established mnemonic, learned by budding mathematicians, is an easy way to remember a mathematical relationship: the trigonometric ratios of sine, cosine and tangent in a right-angled triangle in relation to the length of the hypotenuse and that of the sides adjacent to and opposite the angle in question.\n\n**S** ine = **O** pposite\/ **H** ypotenuse   \n**C** osine = **A** djacent\/ **H** ypotenuse   \n**T** angent = **O** pposite\/ **A** djacent\n\nTo remember this mathematicians either use:   \n**SOH CAH TOA** or a sentence such as:\n\n**S** ome **O** ld **H** ag **C** aught **A** **H** are **T** rying **O** ut **A** rtichokes.\n\nMillions of children learned the names of the planets in our solar system, from nearest to farthest from the sun \u2013 Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus,\n\n**A MISCELLANY OF MNEMONICS**\n\n**Sometimes there are different ways to remember the same thing.  \nFor example, mariners need to know that port is left and starboard   \nis right. Here are three options.**\n\nNotice there are four letters in the word PORT and the word LEFT.\n\nor\n\nThink of the sentence, \"There's no PORT LEFT in the bottle because   \nthe sailors have drunk it all!\"\n\nor\n\nNote the order in the alphabet of the initial letters of the words:   \n**P** ort comes before **S** tarboard just as **L** eft comes before **R** ight.\n\nFrom the sea to caves... In caves you'll see rock formations that   \nhang down from the ceiling and reach up from the ground.   \nThese are called stalactites and stalagmites but how can you   \nremember which is which?\n\nSTALA **G** MITE = **G** ROUND STALA **C** TITE = **C** EILING\n\nor\n\nStala **g** mites rise from the **g** round and stala **c** tites drop from the **c** eiling.\n\n_**\"** He who is not very strong in memory should   \nnot meddle with lying. **\"**_\n\nMichel de Montaigne (1533\u201392)\n\nNeptune, and the former planet Pluto \u2013 based on a mnemonic that uses the first letter of each of the planets:\n\n**M** y **V** ery **E** asy **M** ethod **J** ust **S** peeds **U** p **N** aming **P** lanets.\n\nAnother common mnemonic that uses first letters is **HOMES** , which acts as a reminder of the names of the Great Lakes:\n\n**H** uron **O** ntario **M** ichigan **E** rie **S** uperior.\n\n### RHYMES AND WORDPLAY\n\nPoems and rhymes can also help to recollect useful facts. One of the oldest examples is used to remember the year when Christopher Columbus landed in America:\n\n\"Columbus Sailed The Ocean Blue   \nIn Fourteen Hundred And Ninety Two.\"\n\nUsing wordplay helps, too. Daylight Saving Time changes confuse many who do not know whether to put their clocks forward or back. Here's how to remember which it is:\n\nSpring Forward, Fall Back\n\n### CREATING YOUR OWN MNEMONICS\n\nWhen you're faced with new and maybe rather obscure facts, you may have to create your own mnemonics. Although the first-letter method is versatile, not everything can be broken down in this way, so try this next technique.\n\n### ASKING THE RIGHT QUESTIONS\n\nAsking the right questions is a powerful tool to direct our thinking in a positive and productive way. The questions on the opposite page are designed to stimulate your memory to create some powerful mnemonics. When you have to learn a new fact, quickly run through the questions \u2013 they'll generate lots of ideas to help you remember the information.\n\n### CHECK YOUR RECALL\n\nOnce you've come up with your mnemonic for whatever facts you're trying to remember, it's a good memory principle to check that you're able to recall what you've learnt by putting the information aside for a short time. Go and do something else to occupy your mind for 20 minutes, then come back to the information and prove to yourself that you really do know it. If there are any shortcomings in your recall and you find it difficult to recollect everything, work on the mnemonic to make it stronger, then test your recall again.\n\n**\"** _When I was younger I could remember  \nanything, whether it had happened or not. **\"**_\n\nMark Twain (1835\u20131910)\n\n**QUESTIONS TO ASK**\n\n**First impressions**\n\nIs there anything obvious about this new fact?\n\nWhat does this remind me of?\n\nDoes this look or sound like anything else more familiar?\n\nAre there any patterns in the words?\n\nWhat are the key words to summarize the main points?\n\n**Manipulate the fact**\n\nCan I break this down into manageable chunks?\n\nCan I make this outrageous or bizarre?\n\nCan I exaggerate this?\n\nCan I make this colourful or funny?\n\nCan I draw a picture to represent this?\n\nCan I abbreviate any of the words?\n\nCan I make a word from the letters of any words?\n\nCan I make this rhyme with something?\n\nCan I make up a poem or limerick?\n\n**Use comparisons**\n\nHow is this different from other, related facts?\n\nHow is this the same as other, related facts?\n\n**Use your different learning styles**\n\nHow can I make this **look** like something easy to remember?\n\nHow can I create a **sound** from this?\n\nHow can I create an **action** from this?\n\n### THINKING CREATIVELY\n\nThe beauty of trying to devise your own mnemonic is that, just by thinking about and around the information you're trying to remember in a creative way, you'll embed the facts more deeply in your memory, and therefore be more likely to retain them, even if you don't come up with an obvious mnemonic.\n\n### NOW IT'S YOUR TURN\n\nTry creating your own mnemonic for the following fact \u2013 the size order from the largest to the smallest of the planets, which is: Jupiter, Saturn, Uranus, Neptune, Earth, Venus, Mars, Mercury.\n\n## REMEMBERING SPELLINGS\n\nSpelling in English can be a challenge, because many words are not spelled in the same way that they are spoken. Often there are a number of ways to write the same sound.\n\nThis can be confusing both to people learning English and to native speakers, but there are some tricks that can help you remember the correct way to spell words that are commonly misspelled.\n\n### USE YOUR EYES\n\nIf someone's poor at spelling, it doesn't necessarily mean they're not as intelligent as someone who's good at spelling. It's likely that they just have a poor process for recalling the order of the letters. Most poor spellers \"say\" the word to themselves and then spell it phonetically by interpreting the sounds into letters.\n\nWhen good spellers spell a word, they \"see\" the word in their mind's eye and copy what they \"see\". Because many words are not spelled as they sound, this process is more successful than trying to spell what is heard.\n\n### HOW TO REMEMBER AWKWARD SPELLINGS\n\nAnother way to spell words that habitually cause you a problem is to focus on the part of the word that you have difficulty with and to think up either a phrase or an image that you can link to the word to reinforce the correct way to spell it. Here are some examples:\n\nSeparate or Seperate? | There's **a rat** in sep **arat** e  \n---|---  \nNecessary, Neccessary or Neccesary? | Necessary has one \"C\" and two \"Ss\" because it's necessary that a man's shirt has one Collar and two Sleeves  \nStationary or Stationery? | Envelope is an example of stationery and it begins with an \" **e** \"  \nEmbarrass or Embarass? | When we emba **rr** ass someone they go **r** eally **r** ed\n\nAnd for young children who are learning new words for the first time, the following statements are fun and easy to remember :\n\nBecause | **B** ig **e** lephants **c** an **a** lways **u** nderstand **s** mall **e** lephants  \n---|---  \nWednesday | **We** **d** o **n** ot **e** at **s** weets **day**\n\nThese are just some of the more common and popular examples of how to remember spellings. The best ones, though, are those that you make up yourself as they are most likely to register in your mind. Try finding your own ways to help you remember these commonly misspelled words:\n\n**Fahrenheit** (it shouldn't be Farenheit), **Desiccated** (not Dessicated), **Supersede** (not Supercede).\n\nWhen you're trying to spell a word, visualize it and then write it out rather than just saying it to yourself.\n\n## ALWAYS REMEMBER YOUR PIN\n\nPersonal identification numbers protect your credit and debit cards from misuse, but often keep them so secure that even you can't use them because you've forgotten the PIN! What you need is a way to memorize numbers that's easy to use, works every time and still maintains the security of all your cards.\n\n### CHOOSE A GOOD NUMBER\n\nThe beauty of this first simple method is that you're allowed to choose whatever number you want for your PIN. It makes sense not to select an obvious combination like 1234, 1111 or 2222.\n\nThe best number to choose is one that's unique and personal to you, yet not too obvious. For example, you could choose the date of your mother's birthday, the year your eldest child was born, the year your favourite team last won the Super Bowl or World Series, or any other four-digit number that has a particular meaning or an association for you.\n\nBelow are two things that you can do to remember your PIN when you need it. Try them and see which method suits you better.\n\n\u2022 Create a vivid image that links the significant number you've chosen and your card. If you've chosen your mother's birthday for your Amex number, then imagine your mother using that card while wearing a party hat and carrying a birthday cake with lots of candles on it. When you get the card out, you'll think of your mother using it on her birthday and will then recall the PIN.\n\n\u2022 Carry with your card a cryptic note, like \"Buy Mom's birthday present with Amex.\" When you get that card out, just look at this piece of paper and you'll know the PIN \u2013 but the clue will be meaningless to anyone else. Of course, you can adapt this system to use for any short number, such as debit cards, door security numbers or combination locks.\n\n### USE YOUR CREDIT CARD NUMBER\n\nAnother foolproof method of remembering your PIN is to choose one based on the credit card number itself. For example, suppose your card number was:\n\n4929 4263 7812 3611\n\nIt would be unwise to pick one of these four digit groups or any easy-to-guess combination, but you could use it as a basis for your PIN. For instance, take the first number of each group and add 1 to it, to give you 5584. Whatever method you use, you'll have a unique PIN that you'll be able to recall simply by looking at your card.\n\n### MIX AND MATCH\n\nIf you have one card and one PIN to remember, choose the method you like best and use that. If you have more than one card, pick a unique PIN for each and commit each number to memory using a different technique.\n\n## REMEMBERING DIRECTIONS\n\nGetting hopelessly lost is something we'd all like to avoid, and asking for directions is a good way to prevent this from happening. However, having stopped to ask the way, many of us still end up in the middle of nowhere because we forgot the instructions we were given. Remembering directions can be straightforward with a simple system and a little understanding of how we think.\n\n### THINKING AND COMMUNICATING IN DIFFERENT WAYS\n\nOf our five senses, what we see (visual), hear (auditory) and feel or do (kinesthetic) have the most impact on the way that we communicate and learn. You and every person that you meet will use these senses, but not in equal measure because, of the three, there'll be a personal preference. It is that preference (or learning style) which will dictate the most effective format for you to learn something.\n\nWhen it comes to receiving directions, a visually biased learner will prefer to look at a map or see the instructions written down. Those who have an auditory preference will happily listen to someone describe the route, while people with a strong kinesthetic bias would rather be taken and shown the way.\n\nThe challenge comes when someone using one type of learning style attempts to help someone from another learning style \u2013 for example, when someone who prefers to see a map or written instructions is given directions orally. While they'll understand what's being said, they're less likely to take it in, and consequently they'll find it more difficult to remember the directions.\n\nThink about your own ideal learning style. When you know what this is, you will find that it becomes easier to follow and remember directions if they're given in your preferred format.\n\nHowever, when stopping to ask the way in a strange place, we don't always have the luxury of being able to obtain directions in our preferred learning style, so what we need is an effective process to remember the spoken directions we are given.\n\n### THE PROBLEM WHEN ASKING FOR HELP\n\nUnfortunately, there are a number of factors that tend to work against us when we're receiving oral directions from someone else:\n\n\u2022 Too much unfamiliar information given too quickly for us to understand and remember.\n\n\u2022 Men usually want to know the waypoints of the journey \u2013 for example, crossroads, road junctions and stoplights \u2013 and what to do at each one. Women, on the other hand, generally feel it's also important to describe every major landmark or notable feature along the way, such as a church, cinema or school. A man will often be confused and frustrated by a woman giving him directions, and vice versa.\n\n\u2022 Asking for directions can be a stressful experience, especially if you're already lost and feel uncomfortable asking strangers for help. And your memory won't be functioning at its best if you're feeling stressed.\n\n### YOUR PERSONALIZED DIRECTION FILING SYSTEM\n\nInstructions on how to get anywhere consist almost entirely of directions (left, right, straight on) and waypoints and landmarks along the way. As a journey breaks down neatly into stages, the trick is to prepare memory pegs for each separate instruction. And because directions are about moving around, it's appropriate to use your body as locations for these pegs.\n\nMost simple directions will rarely exceed seven stages, so you might use an ear, shoulder, elbow, wrist, hand, hip and thigh as suitable locations. Should you need them, you could continue using locations down to your feet and back up the other side of your body for more complex instructions. Precisely how to use these pegs\n\nis described below.\n\nNext, create memorable images for left and right \u2013 perhaps a glowing golden **L** ion with a massive mane for **L** eft and a giant long-eared, bright pink **R** abbit for **R** ight. Practise conjuring up these images so that the association of lion with left and rabbit with right (or whatever you've chosen) becomes instantaneous.\n\n### ASKING FOR DIRECTIONS \u2013 STEP BY STEP\n\nThe following process will help you absorb and remember directions so that you won't get lost.\n\n\u2022 When you stop and ask someone for directions, always ask them for the \"easiest\" route. By doing this, you're subtly guiding the person you're asking into giving you clear, straightforward directions that you're more likely to be able to follow.\n\n\u2022 As the person describes the route, listen for just the waypoints and memorize them by placing bizarre images of them on the pegs of your body filing system in order.\n\n\u2022 Repeat the waypoints back to the person, but this time ask them what you should do at each one. So, if you have to turn left, incorporate a lion into your waypoint image, and so on.\n\n\u2022 Repeat the entire route back to make sure you've got it right, but this time ask what notable features you might expect to see along the way. Add those features to the images that you've conjured up for each stage of the journey.\n\nBy doing it this way you'll go through the directions three times, each time layering on a little more detail and checking that you've got it right. The other advantage of this system is that it will also give you the time you need to memorize the instructions properly.\n\n### RECALLING THE DIRECTIONS\n\nTo recall the journey, think about the pictures you've placed on each part of your body in turn. So, for example, if you think of your shoulder and on it you \"see\" an image of a crossroads being jumped on by a huge pink rabbit holding a burger, you'll know that when you reach the crossroads you'll have to turn right and on your way there you'll pass a fast-food outlet. This process is much more laborious to describe than to implement so just try it out and see. It's good fun and, as with so many memory aids, it will soon become second nature with practice.\n\n## REMEMBERING ERRANDS AND SIMPLE LISTS\n\nBy writing down a list of errands on a sheet of paper, you're neglecting to keep your memory in good shape, and if you lost your list you'd probably miss doing something that could be important. What you need is an easy, fun method to remember the list \u2013 as a bonus this will help you keep your memory in excellent condition.\n\n### THE MAGICAL POWER OF STORIES\n\nLong before computers and audio recording devices were invented, and even before paper and pen, stories were used to pass down the folklore and history of societies and cultures for generations, often spanning hundreds if not thousands of years.\n\nAs a device for remembering, stories are extremely powerful because they bring information alive in a way that's easy to recall. A good tale is engaging, interesting and enjoyable, has a flow and a sequence with many highlights, and evokes a strong emotional response \u2013 all of which make it memorable.\n\n### HOW TO CREATE A GOOD STORY\n\nHere's the secret of creating a story to help you remember errands:\n\n\u2022 Be clear about exactly what it is that you are trying to remember.\n\n\u2022 Apply the techniques for using your imagination on pages 29\u2013 to create outstanding mental images of each item on your list.\n\n\u2022 Create strong and powerful links between each image in an amusing, interesting and even bizarre story, making sure that you exaggerate and embellish, and use all your senses.\n\n\u2022 Mentally review your story a couple of times to ensure that you've got the flow and can recall the items on the list.\n\nImagine you have to remember to do the following:\n\n\u2022 Feed the cat next door.\n\n\u2022 Buy a newspaper.\n\n\u2022 Get the exhaust changed on the car.\n\n\u2022 Take back your library books.\n\n\u2022 Make an appointment with the dentist.\n\nThe first thing to do is to create a starting image that you'll always use for your current errand list. For example, let's take the image of a notepad and pen (which is what you'd probably use anyway to write down your list if you weren't trying to memorize it).\n\n### THE TALE OF A CAT\n\nBased on the methods described above, here's an example of what the story might be.\n\nImagine a huge **notepad** with a bright red cover magically floating in the air. Above it is a large royal blue fountain pen that's busy scribbling away. All of a sudden the pages of the notepad begin flicking over quickly as though there's a powerful rush of wind. As they do so you hear a loud screeching \"meeeeeooooooooow\" as the **next door cat** comes flying out between the pages and lands head first in a yellow bowl of cat food that's sent flying in all directions. The cat looks up indignantly, its head covered in cat food. It winks at you and then from underneath its paw it produces a copy of today's **newspaper** , sits up, crosses its legs like a human being, puts on a large pair of glasses and begins reading the paper. As soon as it's finished reading, the cat takes the newspaper and begins to stuff it into the **exhaust** pipe of your car. You get in your car and start the engine, but all of a sudden your car backfires with a huge bang that sends the rolled-up newspaper flying into the air through the window of the **library**. It smashes into a huge bookshelf, sending it crashing to the floor with books spinning off in all directions. People try to run out of the library to get away from the mess, but first they have to have their teeth checked by a very tall **dentist** wearing a purple gown and with a flashlight strapped to her head.\n\nNow review the story in your mind a couple of times, leave it a few minutes and then see what you can recall. To remember your errands, think of the notepad and pen, which should trigger the rest of the story, and from that you'll be able to recollect exactly the list of tasks you need to do.\n\nOnce you're comfortable with this method, you can try using it whenever you have to remember a short list of things to do.\n\n### PRACTISING THE METHOD\n\nGet your \"memory muscles\" in shape by trying the following exercises.\n\n### EXERCISE 1\n\nSpend a few minutes creating a story to link and remember the following list of random items:\n\nBottle of mineral water\n\nSmall dog\n\nDiamond necklace\n\nFountain pen\n\nBicycle\n\nPut this book aside for 10 minutes, then write down the list of words. Check your answers against the list above.\n\n### EXERCISE 2\n\nNow create a story to remember this errand list:\n\n\u2022 Go to the post office to send a parcel to your friend.\n\n\u2022 Drop your best suit off at the cleaners.\n\n\u2022 Go to the store to buy some milk.\n\n\u2022 Stop at the nearest ATM to get some cash.\n\n\u2022 Pick up your shoes from the shoe repair store.\n\nPut this book aside for 20 minutes, then test your recall by writing the list down with the aid of the story. Then check your answers.\n\n## I MUST REMEMBER TO...\n\nSudden flashes of inspiration often come to us when we can't write them down \u2013 for example, when driving or in the shower. They seem blindingly obvious, but when we try to recall them, our memory is blank. With the help of this technique you'll never forget your brilliant ideas:\n\n### YOUR FIRST MEMORY FILING SYSTEM\n\nWe're going to create the mental equivalent of a piece of paper and pen that you'll always have with you. So, whenever one of these ideas comes to mind or if you suddenly realize there's something you must do at a later date, you can make a note of it for easy recall.\n\nOn page 21 we talked about your memory working like a mental filing cabinet. You're now going to create a series of mental files for remembering your flashes of inspiration. This is the first of many memory filing systems you'll use as you develop a powerful memory with the help of this book.\n\n### THE NUMBER RHYME SYSTEM\n\nThe easiest way to organize a filing system is to number the files. Our first system will, therefore, consist of 10 files that are numbered from 1\u201310, but, as we find it easier to think in pictures rather than numbers, we now have to turn each of those file numbers into images.\n\nThe system is called the Number Rhyme System because we find pictures of objects that rhyme with the sound of each number. Here are the images that are most commonly used. However, if a picture doesn't work well for you, feel free to replace it with one of your own that does.\n\n### CREATING POWERFUL IMAGES\n\nThe secret of making this system work for you is to create strong and powerful images for each of these \"peg\" words by spicing them up using the suggestions on pages 29\u2013. For example, for the number 2 we use the picture of a shoe. If you see one of your own shoes, it won't be as memorable as seeing a six-foot-tall stilettoheel shoe made of shiny red patent leather. The picture must be so real and memorable that you can even \"smell\" the new leather.\n\nTake a few minutes to create strong memorable images for each of the pictures \u2013 really go to town and embellish them. When you've done that, go and do something else for 20 minutes and then come back, write down the numbers from 1\u201310 on a piece of paper, and see how many of your Number Rhyme System images you can recall. What you're aiming for is to get to the stage where, as soon as you think of the rhyming word for each number, you automatically think of the corresponding picture.\n\n### HOW TO USE THIS SYSTEM\n\nYour first memory filing system is ready to use whenever you're unable to write things down. Imagine you're in the shower and you have a sudden flash of inspiration about a problem. Create a spiced-up, memorable image of your idea using your imagination, and then link that image to a Number Rhyme System file.\n\n1 | Sun | Book | Imagine a massive fiery sun melting the pages of a green book.  \n---|---|---|---  \n2 | Shoe | Table | See a large ladies' shoe firmly embedded in a wooden table.  \n3 | Tree | Giraffe | Picture hundreds of giraffes eating the leaves of trees in a huge forest.  \n4 | Door | T-shirt | Visualize a T-shirt strolling up to the door (as though worn by the Invisible Man) and slamming it shut.  \n5 | Hive | Apple pie | See a swarm of bees flying out of a hive and landing on an apple pie, which they devour, licking their lips.  \n6 | Sticks | Mouse mat | Imagine a pile of sharp sticks and each one is piercing a colourful mouse mat.  \n7 | Heaven | Telephone | Hear the loud ringing of a large telephone and watch carefully as a beautiful angel dressed in white floats past and picks up the receiver to answer the call.  \n8 | Gate | Wallet | Listen to the creaking of an old rusty gate as it opens. And see hanging on the gate a large leather wallet with dollar bills falling out as the gate swings open.  \n9 | Wine | DVD player | Imagine the red wine from your glass is dripping onto the open disk tray of your DVD player, much to your concern.  \n10 | Hen | Asparagus | See a fat hen with a huge green asparagus tip under each wing and one in its beak.\n\nFor example, if your idea is to try a different tool on a particularly difficult job, visualize that tool being heated by the sun (the first of the peg-word images) until it's too hot to hold.\n\nOnce you've mentally grabbed hold of the idea by creating a mental picture and linking it to a file in your memory, you won't lose that idea. To remind yourself of it, think of your Number Rhyme images and see what comes to mind.\n\n### YOU CAN REMEMBER OTHER STUFF TOO!\n\nYou don't just have to use the Number Rhyme System for capturing ideas. You can use it for any list or sequence of items, even my (deliberately) random list of words in the box opposite: book, table, giraffe, T-shirt, apple pie, mouse mat, telephone, wallet, DVD player, asparagus.\n\nThe more you can embellish the associations by using the principles of effective visualization and therefore really \"see\" the pictures you create, the better. Come back to the book in 10 minutes and check how many of the words you can remember by thinking of your Number Rhyme image for each number and seeing what pictures come to mind.\n\n## SHOPPING WITH THE ALPHABET\n\nUsing a written list to remember what groceries you need not only causes a problem if you lose it, but also means you're letting your \"memory muscle\" go flabby. By creating a foolproof memory filing system just for your shopping, you need never have to search for pen and paper again, and you'll increase both your mental agility and your powers of concentration.\n\n### DO YOU NEED A TECHNIQUE FOR YOUR SHOPPING LIST?\n\nLook at this list for two minutes, put the book to one side and see how many items you can recall.\n\nToothpaste | Cheese | Tomatoes | Steak  \n---|---|---|---  \nBananas | Pitta bread | Salmon | Burger buns  \nTuna | Strawberries | Yogurt | Toilet paper  \nCabbage | Burgers | Carrots | Rice  \nPears | Shampoo | Broccoli | Soap  \nMilk | Loaf of bread | Pasta |   \nCream | Bagels | Deodorant |\n\nAlthough you were probably able to recall some of the items, I'd be surprised if you could recall them all \u2013 10 items is a creditable first-time score. However, before I give you a technique to memorize your list, there's a straightforward step that you can take first to increase your powers of recall.\n\n### THE FIRST STEP \u2013 ORGANIZING YOUR LIST\n\nTake a piece of paper and organize this same list into five different categories. You decide on the categories based on what's on the list. When you've done that, go and do something different for 10 minutes and then see how many items you can recall.\n\nYou were probably able to remember more from the list by organizing it than you were if you just tried to remember it in the random order in which it was originally written. Organizing data in this way is called \"chunking\" and relies on our memory's aptitude for remembering things that are linked together in some way.\n\nAnother principle at work here is that, by grouping the different things from your list into the categories that you choose, you think about the list on a deeper level than if you had merely read it through. As a result, you're more likely to remember what's on the list because you've invested more mental energy in it.\n\nSee the chart overleaf for how I would have organized the list.\n\nNow we have to find a way to completely and accurately remember this list.\n\n**HOUSEHOLD ITEMS** | Shampoo, Soap, Toothpaste, Deodorant, Toilet paper  \n---|---  \n**BAKERY** | Loaf of bread, Pitta bread, Bagels, Burger buns, Pasta, Rice  \n**FRUIT AND VEGETABLES** | Pears, Bananas, Strawberries, Broccoli, Cabbage, Carrots, Tomatoes  \n**MEAT AND FISH** | Steak, Burgers, Tuna, Salmon  \n**DAIRY** | Milk, Cheese, Cream, Yogurt\n\n### THE ALPHABET MEMORY FILING SYSTEM\n\nIn the previous section I introduced the Number Rhymes for your first memory filing system. You could use this for remembering your shopping list, but you'd be limited to just 10 items.\n\nIn the same way that you'd use different pieces of paper for different lists, we're going to use different memory systems for different applications. In this instance, I'm going to show you how to organize your memory files using the alphabet, which gives you 26 hooks to help you remember.\n\nThe secret to the success of this system is that, as soon as you think of a letter of the alphabet, an image should immediately spring to mind.\n\nSome letters will naturally lend themselves to a particular image, while other letters will require a little more thought. Here are the images I use.\n\n**A** crobat | **H** orse | **O** rangutan | **V** ase  \n---|---|---|---  \n**B** ee | **I** mpala | **P** anda | **W** indow  \n**C** at | **J** ester | **Q** ueen | **X** -ray machine  \n**D** og | **K** ettle | **R** ose | **Y** ak  \n**E** agle | **L** asso | **S** nake | **Z** ulu warrior  \n**F** rog | **M** ouse | **T** arantula |   \n**G** uitar | **N** et | **U** niform |\n\nSome of these words and their images may not work for you, so feel free to change them for ones that do \u2013 Google's image-search facility on the internet will help you find pictures for each of the words that you choose. Here's a simple way to fix them firmly in your mind.\n\nTake 26 pieces of thin cardboard (about playing card size) and on one side write the letter as large as you can and on the other write the name of your chosen image. Shuffle the cards and then, with the letters facing you, go through the cards in turn, seeing how quickly you can bring the associated image to mind. Do this until it takes less than a second for each card.\n\n### USING THE ALPHABET SYSTEM\n\nYou can now use your Alphabet System to memorize your organized list. Here's how you might start with the first three items:\n\n**A** crobat | Shampoo | Picture your acrobat balancing on a pyramid of shampoo bottles.  \n---|---|---  \n**B** ee  | Soap | See a bee flying onto a flower and rubbing it with scented soap to give it fragrance.  \n**C** at | Toothpaste | Imagine a cat cleaning its teeth.\n\nWork through the entire list by category, creating strong associations between each item and the next letter. Don't try to conveniently match up images and items, such as Carrots with Horse.\n\nThis system works by running through your image alphabet and recalling what you've pegged there, so the more outlandish the association between the item and the letter, the better.\n\n## REMEMBERING WHAT YOU'VE HEARD\n\nIf you've ever found yourself struggling to recall something you've heard in a conversation, it's natural to assume your memory is at fault. However, it's likely that you either didn't hear the information properly in the first place or you don't have a good strategy for remembering it.\n\n### WHY YOU SOMETIMES FORGET\n\nThere are a number of reasons why you may have trouble recollecting what you've been told:\n\n\u2022 If you're not concentrating enough on what you're listening to, your mind starts to wander, and so, although you might be looking at the person who's speaking to you, your conscious mind doesn't hear what's being said and will have no recall of it.\n\n\u2022 If your preferred method of learning is more visual or kinesthetic, you're likely to find recalling what you've heard more difficult.\n\n\u2022 If what you're being told is complicated or difficult, you might become overwhelmed and lose track, which will make it difficult to understand and remember the information.\n\n### PRACTICAL WAYS TO ENHANCE YOUR RECALL\n\nIf you want to remember more of what you've heard, try one or more of the following tactics:\n\n\u2022 Use the Rapid Repeat method \u2013 simply repeat in your mind what's being said as you hear it. This will enable you to focus your attention more keenly on what you're hearing and prevent your mind from wandering off.\n\n\u2022 At regular intervals in a conversation, repeat back to the speaker a summary of what you've heard so you can keep track of what's been said.\n\n\u2022 Ask questions often if you don't understand what you're being told, or get the person to explain the information in a different way.\n\n\u2022 If possible, take notes that identify key words or phrases.\n\n### IDENTIFYING KEY POINTS\n\nDeveloping the skill of identifying key points is critical if you want to remember what you're listening to. A good way to practise is to listen to the radio, especially dramas, and identify the points as they emerge. You'll find with practice that you can summarize much of what's said just by picking out these \"high points\" of interest.\n\n### USING MEMORY TECHNIQUES\n\nOnce you're able to summarize information in this way, you can use one of the memory filing systems to record the key points of anything that you hear. I suggest that you create a couple of 20-stage journeys (see pp.114\u20138) that you use specifically for taking mental notes during a conversation. Then, as you identify a key point, create a powerful image of that point and link it to one of your locations. You'll find that, if you memorize 20 key points from a conversation, other details that you haven't necessarily memorized will spring to mind, thanks to the power of natural association and connection.\n\n## RETRIEVING THE PAST\n\nMost people like to reminisce, but retrieving the past isn't easy if there are gaps in your recall. However, you can recover many of your memories by understanding how to trigger and build on them. The trick is to start with something that you can recall and then build up the detail around it until you've recreated the entire memory.\n\n### FIND A STARTING POINT\n\nImagine that each of your memories is stored behind a different locked door. Sometimes these doors are quite easy to open but often they remain locked and the memory stays hidden behind them. However, all you need to unlock a door to gain access to your hidden memory is the right key which, in memory terms, is something related to or associated with what you're trying to remember.\n\nA great way to find the key and start the recall is to gather any items you have from the time you're trying to remember. These may be photographs, old diaries, clothes, toys or any other memorabilia from that period. Examine them closely and see what sensations come to mind.\n\nIn particular, when looking at photographs, observe the detail around the subject of the picture to see what else you can recall. It's also useful to listen to music from that era or look at contemporary books and newspapers to help generate related memories. If the thing you're looking at is not the actual key, it might trigger something else that will unlock your memory.\n\n### ASK YOURSELF QUESTIONS\n\nIf you don't have any memorabilia to stimulate your recall, sit quietly, close your eyes, relax and take yourself back to that time in your imagination, focusing on something you can recollect about what you're trying to retrieve. Your aim is to re-experience the event as fully as possible. Ask yourself these questions to help stimulate your sensory and emotional memories:\n\n\u2022 What did I see?\n\n\u2022 What did I hear?\n\n\u2022 What did I smell?\n\n\u2022 What did I touch?\n\n\u2022 What did I taste?\n\n\u2022 What did I feel?\n\n### UNLOCKING THE MEMORIES\n\nWhichever method you use, you'll find that, as one tiny piece of detail springs to mind, it will spark off the recall of another, related detail, which will prompt yet another. These keys will unlock your memories of the event or place, and the memories will come flooding back.\n\n_**\"** The senses of smell and taste, weaker but   \nmore enduring ... continue for a long time,   \nlike souls, to remember. **\"**_\n\nMarcel Proust (1871\u20131922)\n\n## INTERLUDE\n\n### THE AMAZING WORLD OF MEMORY\n\nIf you've worked your way through the exercises so far in this book, your memory will already be developing to a greater extent than most people could possibly imagine. This section allows you to take a breather from your efforts and enjoy some interesting tidbits about the world of memory achievement.\n\n### FAMOUS MEMORY MEN\n\nPeople have long been fascinated by impressive memory feats, and over the years many expert practitioners of memory have become household names to their generation. At the beginning of the last century many famous magicians like Harry Houdini wowed their audiences with stunning feats of recall, but it was only when people like Harry Lorayne in the US and Leslie Welch in the UK started performing dedicated memory shows in the 1950s that memorization and recall skills really came to prominence.\n\nOne of Harry Lorayne's favourite demonstrations was to recall the names of his audience. Over his career it's reputed that he memorized the names of over a million people. The stars of the memory world include Kevin Trudeau and eight-times World Memory Championship winner Dominic O'Brien, who, among other feats, has memorized all the answers to Trivial Pursuit!\n\nMost of these \"memory stars\" have trained their memory using techniques and ideas that are centuries old, but there are also some famous \"savants\" \u2013 people with an extraordinary natural memory. One such individual is Kim Peek. Born in 1951, Kim began reading books when he was just 18 months old and could accurately read and memorize over 12,000. He read a page in eight to ten seconds, by which time it was stored on his \"mental hard drive\" for recall at any point in the future. Kim's memory ability was all the more astonishing as he had a low IQ and an inability to button his clothes or deal with the chores of everyday life. Brain scans detected structural abnormalities in his brain, but not enough was understood about the full impact on his memory. Sadly, Kim died in 2009 at the age of 58.\n\n### THE WORLD MEMORY CHAMPIONSHIPS\n\nIn the 1990s a group of dedicated mnemonists led by Tony Buzan, one of the world's leading authorities on memory improvement, set up a competition to see who had the best memory. From humble beginnings this competition has grown to a truly international event, attracting competitors and extensive media interest from all over the world. In addition, many countries now hold their own national and regional championships, as the acceptance of memory as a sport continues to grow. The competition is like a decathlon for the brain because it's spread over 10 events that are a combination of speed and endurance, where challengers for the title must memorize decks of randomly ordered cards, numbers, names, lists and a poem.\n\n### WORLD RECORDS\n\nThe world championships have delivered an amazing series of records that continue to be broken year after year. Here are just some of the staggering feats:\n\n\u2022 Order of playing cards in one shuffled deck \u2013 31.16 seconds.\n\n\u2022 Number of shuffled decks of cards in one hour \u2013 27 (that's 1,404 individual playing cards).\n\n\u2022 Number of random digits in five minutes \u2013 333.\n\n\u2022 Number of random digits in one hour \u2013 1,949.\n\nNot all world records are set at the world championships. For example, the mathematical number pi (3.14159 etc) has fascinated people for centuries. It's an infinite decimal number that never repeats in a pattern and is the \"Mount Everest\" of memory achievement. The world record for this was set by Akira Haraguchi, a Japanese mental health counsellor, who managed to memorize and then recite the number's first 83,431 decimal places.\n\n### MEMORIZING REALLY IS GOOD FOR YOUR BRAIN\n\nSome of these world records may sound a little beyond your abilities at the moment, but don't be deterred. With the techniques you'll learn from this book, together with a little practice, you too could manage similar feats. But even if you're not interested in pursuing a career as a professional mnemonist, your brain will be getting a good workout if you apply the ideas and techniques that you've learnt here.\n\nIn 2002 I participated in a study carried out by the Institute of Neurology in London where they scanned the brains of memory experts, who used the techniques I've covered in this book, as they memorized. Not only did the experts outperform the control group, but also it was found that more of their brain was being engaged, in particular the hippocampus, the part that's responsible for transferring short-term memories into the long term. It's also the part that \"lights up\" when we think about something that we know well. All this shows that memorizing really is a good form of mental exercise!\n\n# 4   \nWINNING WORKOUTS\n\nNow that you have some basic ideas under your belt and have seen how easy it is to dramatically improve what you can remember, it's time to build on these foundations to improve your memory further.\n\nThis chapter still uses the principles you've already become familiar with but it introduces some new and versatile techniques.\n\nHere, among other topics, you'll encounter ways to remember a speech and not \"dry up\", as well as strategies for remembering what you've read \u2013 whether it's a newspaper article or a novel. You'll discover the trick of learning foreign vocabulary for easy recall, and of never forgetting another important date or appointment. You'll also be introduced to the most powerful of all known memory systems, which you'll be able to use right away to do some impressive memory feats.\n\n## REMEMBERING NAMES AND FACES \u2013 THE MNEMONIC METHOD\n\nAs well as the Social Method for remembering people's names, which I explained in the previous chapter, there's another technique that can be used in conjunction with it. Called the Mnemonic Method, this second technique employs memory principles to spice up a name and link it to a face, and vice versa, to make both unforgettable. It links in with Step 8 of the Social Method on page 62.\n\n### YOU LOOK LIKE...\n\nOne of the principles of memory is that we find it easier to remember things that are related or linked in some way to what we already know. How can we apply this when we initially meet someone?\n\nFirst, find something in the person's appearance that instantly reminds you of something or someone you know. If nothing springs to mind, you'll have to exercise your imagination!\n\nPerhaps using a degree of creativity, try asking yourself some questions, such as does this person:\n\n\u2022 Look like anyone I already know?\n\n\u2022 Look like someone famous?\n\n\u2022 Look like a typical XYZ (eg a policeman, singer or a lawyer, perhaps)?\n\n\u2022 Have any prominent features that make him or her instantly recognizable?\n\n\u2022 Have any features I can exaggerate or caricature?\n\n### REINFORCING THE LINK\n\nOnce you've found a memorable image, reinforce the link by looking at the person and thinking simultaneously of the association. If you do this whenever you see that person again, your image link should automatically leap into your mind as a mental memory file. Practise finding strong image links by watching the news.\n\nLet's assume you've met someone who instantly reminds you of the actress Jennifer Aniston (Rachel from _Friends_ ). Every time you see this person, your mind will be triggered to think of Jennifer Aniston, the associated image firmly anchored in your memory.\n\nSEE PERSON > THINK OF JENNIFER ANISTON\n\n### REMEMBERING THE NAME\n\nNext, you need to create hooks and associations to help you recall the person's name. The best way to do this is to convert the name into a picture and link it to the memory file of their face. Let's say that the person you've met is called Penelope Sanchez. To turn this name into a picture, you need to break it down into first and second names.\n\n#### Step 1 \u2013 Visualize the first name\n\nAsk yourself these questions \u2013 you may not have answers to them all. Just focus on the ones that have striking and instantaneous answers.\n\n\u2022 Is there someone I know called \"Penelope\"?\n\n\u2022 Is there someone famous called \"Penelope\"? (For example, the actress Pen\u00e9lope Cruz.)\n\n\u2022 Does \"Penelope\" instantly trigger a picture?\n\n\u2022 Does \"Penelope\" have a meaning that I can turn into a picture?\n\n\u2022 Can I break \"Penelope\" down into parts that I can turn into pictures? (Imagine a pen, eloping with its partner.)\n\nI'll assume the image of Pen\u00e9lope Cruz is the strongest for you, so you now need to create a \"memory-spiced\" picture linking Jennifer Aniston (the person Penelope Sanchez reminds you of) to Pen\u00e9lope Cruz.\n\n#### Step 2 \u2013 Visualize the second name\n\nAsk yourself the same questions about the surname.\n\n\u2022 Is there someone I know called \"Sanchez\"?\n\n\u2022 Is there someone famous called \"Sanchez\"?\n\n\u2022 Does \"Sanchez\" instantly trigger a picture?\n\n\u2022 Does \"Sanchez\" have a meaning that I can turn into a picture?\n\n\u2022 Can I break \"Sanchez\" down into different parts that I can turn into pictures? (Imagine chairs made of sand \u2013 \"Sand Chairs\" = Sanchez.)\n\n#### Step 3 \u2013 Link the first- and second-name images\n\nLet's take the image of \"Sand Chairs\" for the surname. You now need to create another strong and exaggerated link between your image of Pen\u00e9lope Cruz and chairs made of sand.\n\nYour imagery might be something like this: you see the person you've just met and instantly she reminds you of Jennifer Aniston, which prompts you to think of Pen\u00e9lope Cruz standing on her shoulders. With a loud fanfare, Pen\u00e9lope jumps in the air doing a double somersault and lands on a large red chair that immediately turns to sand.\n\nTry out this technique with a few names until you've got the hang of it.\n\n### LINKING NAMES TO FACES\n\nObservation is a key skill in memorizing, and another way to remember names is to link them to particular features of a person's face or physique, as in the two examples on the following page.\n\n## REMEMBERING LONGER NUMBERS\n\nUnfortunately, not all numbers are as short as a fourdigit PIN for credit and debit cards. Longer numbers such as telephone numbers, membership numbers or bank account numbers can be particularly difficult to commit to memory and to recall. This section shows you ways to remember them whenever you need to.\n\n### WHY ARE LONGER NUMBERS SUCH A PROBLEM?\n\nLots of people have problems with numbers, so if you do, you're not alone. Some people just don't like them, often as a result of a bad experience in math lessons at school. Another reason is that the brain prefers to think in ideas, concepts and pictures \u2013 and although numbers can form part of those, on their own they can be easily confused because they are so similar. People also have trouble with numbers in their raw numerical form because, as psychologists have discovered, the average person has a limited digit span \u2013 they can hold only five to 12 digits in their short-term working memory. The problem is compounded by the fact that their working memory has a limited time span \u2013 just long enough to receive a phone number and use it, after which the number fades swiftly away.\n\nWe'll look here at some simple but effective ways of memorizing longer numbers for a greater length of time, and later on I'll also show you some more sophisticated and complex techniques.\n\n### BREAK IT DOWN\n\nOn page 33 I explained that a principle of effective memory is to break things down, and this same principle applies to remembering longer numbers.\n\nA ten-digit number such as 9074365218 might be a little overwhelming. However, if you break it down as shown below, the number suddenly becomes less daunting and more manageable:\n\n907 436 5218\n\nMost people naturally pause between groups of digits when giving out their phone number, making the number much easier to deal with.\n\nNow that you've broken the number down, there are various techniques you can use to turn the groups of digits into words and pictures to make them more memorable.\n\n### YOU'VE SEEN THIS IN ADVERTISING\n\nI'm sure you've seen advertisements for companies where the telephone number is a mixture of words and numbers. A furniture store owner might implore you to call him on 1-800-9TABLES. This is easier to remember than 1-800-9822537 because \"TABLES\" has a meaning that you can visualize in your imagination. This word is derived from the telephone keypad, which allocates letters to numbers.\n\n### THE TELEPHONE KEYPAD\n\nUsing the keypad approach you can turn a number like   \n2446532668 into:\n\n244 653 2668\n\nBIG OLD BOOT\n\nWhat words could you make out of the following number groups?\n\n3475 344 8776 9882 4726 538 264\n\nThe only snag with this system is that sometimes the numbers don't give convenient letter groupings to make meaningful words. For example, 6218435792 gives the decidedly unmemorable \"ma 1 vid kryb,\" among other equally meaningless variations. Also, there's no letter allocated to the numbers 1 or 0, so for any number that contains these digits, you'll have to use one of the following techniques.\n\n### NUMBERS = LETTERS\n\nIf the telephone pad system doesn't yield helpful words or images, you can use the Numbers = Letters System. Replace each number by a word that has the corresponding number of letters \u2013 eg 1633 becomes:\n\nA Ginger Tom Cat or A Larger Red Rat\n\nThe beauty of this system is that you can start to choose words that are appropriate for the number of the person or service it belongs to. If, for example, part of the phone number for your hairdresser had the digits 4146, you could make up the phrase, \"Only a Trim Please\".\n\nUsing this method, make up your own memorable collection of words for the following sections of telephone numbers:\n\n\u2022 3396 \u2013 Doctor\n\n\u2022 7219 \u2013 Cinema\n\n\u2022 2754 \u2013 Plumber\n\n\u2022 1335 \u2013 School\n\n\u2022 2856 \u2013 Auto-body repair shop\n\n### THE NUMBER SHAPE SYSTEM\n\nIn chapter 3 I explained how to create a memory system using words that rhyme with the numbers from 1 to 10. Now I'm going to show you a system based on images of objects that look like the shape of those numbers. This system also includes an image for zero.\n\nLearn the images (opposite) in the same way that you learned those for the Number Rhyme System (see pp.84\u2013). Embellish them so you can readily visualize them. Set them aside for 20 minutes and do something else. Then test your recall by writing down the numbers and their corresponding images, and see how many come to mind easily. Keep practising until the association and recall are automatic.\n\nOnce the associations are firmly embedded in your mind, you can use them to memorize any number, however long, linking each image in a vivid scenario.\n\nFor example, the number 8167 could be represented by a giant, carrot-nosed snowman (8) taking a bright blue baseball bat (1) and whacking an elephant (6) on the backside who then scurries up to the top of a garish pink streetlight (7).\n\nTry using this system to make up some images for the following numbers: 287, 435, 9815, 03461.\n\n### PUTTING IT ALL TOGETHER\n\nNow you have three different methods for turning numbers into something more memorable, so when you're faced with a telephone number like 7652910843 for your doctor, just break it down and apply one or more of these techniques to remember the number. Try this with the number above yourself.\n\n## THE JOURNEY TECHNIQUE\n\nThe oldest of all memory storage devices, the Journey Technique is also the most versatile, with numerous applications. In my opinion, it's the most powerful mental filing technique, too, as well as being easy to use.\n\n### HISTORY OF THE TECHNIQUE\n\nIn two of the defining civilizations of the Western world, that of the Romans and that of the Ancient Greeks, being a great orator was a sign of political power and influence. Having the ability to recite long passages from memory was an admired, even revered skill, the secret of which was the speaker's reliance on the Journey Technique.\n\nIn modern times the technique is still being used, and if ever you see anyone on television performing an impressive memory feat, it's highly likely they're using this method. In the World Memory Championships, the top competitors over the last few years have certainly used it \u2013 I did!\n\n### SO WHAT IS THIS AMAZING TECHNIQUE?\n\nThe principle of the Journey Technique is simple, and the reason the system is so powerful and easy to use is because it's based on places you're already familiar with. You create a mental filing system by taking somewhere you know well and picking various points on a journey around\/through\/in it. You then use those points as files or pegs on which to place what you have to remember, making sure you use strong links and lots of \"memory spice\". When you want to recall the information, you revisit your journey in your mind's eye and, if your associations are strong enough, as you pass each of your points you'll be reminded of what you put there.\n\nYou'll hear this system called the \"Roman Room\" system because the Romans tended to use a separate room for each location. It's also called the \"Greek Loci\" system because the Ancient Greeks preferred to use one room with various points (or loci) around the room on which to hang their associations.\n\nIt doesn't matter which way you do this as long as you can go to specific points or locations in the same sequence on a journey around an environment you can visualize in your mind's eye.\n\n### DESIGNING YOUR JOURNEY\n\nThe first step is to decide on a location for your first mental journey. I suggest you start with where you live and pick your favourite room. It's a good idea to physically go to the room when you're doing this, because that will create a more powerful journey than if you simply envisage it, but you can still achieve impressive results if you just think of somewhere you've been and know well.\n\nThe next step is to identify in the room a starting point for your journey. Pick something that's a defining feature: for example, in the living room it might be the bookcase; in the kitchen it might be the refrigerator. Once you've chosen your starting point, mentally walk around the room and identify nine more significant objects (the more permanent they are, the better) in sequence.\n\nIf you were using your living room, you might have chosen the following items or waypoints from around your room:\n\n1 Bookcase 2 Painting 3 Floor lamp 4 Window 5 Door 6 Dog bed   \n7 Chair 8 Small table 9 Sofa 10 Television\n\nYou now need to \"condition\" your journey in your mind, so it's firmly embedded. Close your eyes and mentally re-trace your route both forward and backward several times and see each waypoint as clearly as you can. This is a crucial step. Otherwise, a weak memory of your journey will result in poor recall.\n\n### USING THE JOURNEY TECHNIQUE\n\nYou can use this device to remember virtually any information simply by applying the principles we talked about in Chapter 1, notably the powers of imagination and association. The trick is to evoke a strong image for each item that you want to remember and then link it in a vivid scenario to a stage on your journey. It's a good idea to have different journeys for different purposes \u2013 there's no limit to the number that you can create and use.\n\nTry out the Journey Technique by picking a room in your house and designing a 10-stage journey. Then memorize the first 10 elements of the periodic table:\n\n**SCIENTISTS HAVE PROVED IT WORKS!**\n\nI get very excited telling people about this particular technique because I know it works as a result of the amazing things I've been able to do with it. But don't just take my word for it. Medical researchers have proved it works, too, by scanning the brains of people who were using the technique and a control group who weren't. The researchers discovered that memory performance was better in the first group than in the second and that using the Journey Technique stimulated more of the hippocampus (an important part of the brain when it comes to memory) and the right side of the brain. I know this to be true because I was one of the subjects of this groundbreaking research.\n\n\u2022 Hydrogen | \u2022 Boron | \u2022 Fluorine  \n---|---|---  \n\u2022 Helium | \u2022 Carbon | \u2022 Neon  \n\u2022 Lithium | \u2022 Nitrogen |   \n\u2022 Beryllium | \u2022 Oxygen |\n\nWhile this may seem a bit daunting, think creatively about each of the words. What do they sound like or remind you of? For example, Beryllium = berry, Helium = heel, and so on.\n\n## REMEMBERING DATES AND APPOINTMENTS\n\nHave you ever had to endure the embarrassment of forgetting the birthday or anniversary of a close friend or family member? Or missed an important appointment because it had completely slipped your mind? If so, by building on much of what I've shown you so far, this section will really help you.\n\n### REMEMBERING BIRTHDAYS AND ANNIVERSARIES\n\nOn page 49 I suggested you record important dates on your calendar. However, what would happen if you lost your calendar or temporarily mislaid it? Having all those important dates firmly installed in your memory so you can recall them easily would be really helpful.\n\nThis is the process for remembering important dates:\n\nAs usual, you'll rely on a chain of strong associations that employ powerful visual imagery representative of the information that you've memorized.\n\n### CREATING PICTURES FOR THE MONTH\n\nThe simplest way to remember the months is to think of images that instantly remind you of each one. For example, when I think of December, I'm instantly reminded of Christmas and I see a jolly man in a red suit with a white beard. August is another easy one for me because it's vacation time and I picture my large colourful beach towel spread out on the sand. Here are the images I use, and if they work for you, by all means feel free to use them, but I suggest you try to come up with your own unique images because they'll be even more powerful for you:\n\nJanuary | A powerfully built American football quarterback \u2013 the Super Bowl takes place in January  \n---|---  \nFebruary | A heart-shaped box of chocolates \u2013 Valentine's Day  \nMarch | A troop of soldiers marching  \nApril | A large colourful umbrella \u2013 April showers  \nMay | A tall May pole on a village green  \nJune | A tank (the Normandy landings took place in June)  \nJuly | A billowing American Flag, because of Independence Day  \nAugust | A colourful beach towel  \nSeptember | A student carrying lots of books going back to school after the summer vacation  \nOctober | An animated skeleton from Halloween  \nNovember | A plump Thanksgiving turkey with cranberry sauce on the side  \nDecember | Santa Claus\n\n### CREATING AN IMAGE FOR THE DAY\n\nNext, create pictures you can use to represent the day part of the date. For 1\u201310 of the month, I suggest you use the Number Shape System (see pp.112\u2013) to conjure up images; for the rest of the days of the month, combine that method with the Number Rhyme System (see pp.84\u2013), as in the examples below.\n\nIn a two-digit number the Number Rhyme image represents the first digit and the Number Shape the second. Every date in the range 11\u201319 will always include a sun in the picture, every date between 20 and 29 will have a shoe, and 30 to 31 will have a tree, as in the examples below. (NS = Number Shape; NR = Number Rhyme)\n\n2 | Swan (NS)  \n---|---  \n8 | Snowman (NS)  \n13 | Sun (NR) shining on a pair of handcuffs (NS)  \n17 | Sun (NR) shining down from a streetlight (NS)  \n25 | Shoe (NR) being lifted by a hook (NS)  \n26 | Shoe (NR) being worn by an elephant (NS)  \n31 | Tree (NR) sprouting lots of baseball bats (NS)\n\nTry this method out for yourself. Write down the numbers from 1 to 31 and create your own image for each number using the Number Rhyme and Number Shape systems. Then test yourself to see how easily you are able to recall the images. Keep practising until the associations are firmly embedded in your mind and you can recollect them instantly.\n\n### USING THIS SYSTEM\n\nNow that you've done the groundwork, the rest is easy. Suppose you wanted to remember that your friend Julie's birthday was December 27, here's what you would create in your imagination:\n\nYou might, for example, see your friend Julie blowing out two huge green and red candles on a yellow birthday cake and, as each candle is extinguished, it's grabbed by a rotund Santa Claus who shouts out \"Ho! Ho! Ho!\" The candles then magically turn into blue stilettos with streetlights for heels, and Santa Claus proceeds to put them on.\n\nWhenever you think of your friend, you'll see this unusual picture \u2013 simply translate it back into the month and day. Have a go at devising your own visual scenarios for remembering the birthdays and anniversaries of friends and relatives.\n\n### REMEMBERING APPOINTMENTS\n\nA fail-safe way to remember appointments is to create a 31-stage journey that you use as your monthly \"planner\". For example, if you have an appointment at the dentist on the 16th of the month, link an image related to the dentist (a large pair of false teeth being cleaned by a sixfoot toothbrush) to the 16th location of your journey.\n\n## LEARNING A NEW SKILL\n\nHaving a good memory isn't just limited to being able to recall facts and figures. It also plays a key part in learning and developing new physical and mental skills. When you're dealing with information, you can quickly get to the stage where you realize you either know it or you don't. Skill development is a more evolutionary process, which can be enhanced with a little inside knowledge.\n\n### HOW YOUR SKILLS EVOLVE\n\nWhen you're learning a new skill, you go through a series of distinct phases in relation to your ability to perform that skill and the degree that you have to think about it as you do so. There are four key stages that you'll encounter in the acquisition of any new skill \u2013 from basics such as learning to cook or drive a car to learning to play a musical instrument or becoming a proficient first aider.\n\n\u2022 **Unconsciously incompetent** \u2013 at this stage you're not aware that you don't know how to do something until it's pointed out to you, or you try something that you think you can do but can't. For example, as a child you probably weren't aware that you didn't know how to drive a car.\n\n\u2022 **Consciously incompetent** \u2013 once you realize that you can't do something or aren't very good at it, you're consciously aware of your incompetence. At this point you (or maybe someone else) will decide that you need to develop this particular skill. You hit this stage when you drove for the first time and realized that, although your parents may have made it look easy, driving is, in fact, a complex operation involving a multitude of co-ordinated actions and thought processes.\n\n\u2022 **Consciously competent** \u2013 after a period of training and development, you're able to perform a task or activity but you still have to think about it. Remember when you were learning to drive a car \u2013 once you'd understood what the pedals and gear stick were for, you still had to think about how to use them to start with, particularly how to engage the different gears.\n\n\u2022 **Unconsciously competent** \u2013 your skill becomes automatic so you don't have to think about it. If you've been able to drive for a while, you can probably hold conversations, listen to the radio, keep track of the other cars around you, and still do all that stuff with the steering wheel, pedals and gears that used to seem so difficult, without even thinking about it.\n\n### MAKING THE TRANSITION\n\nOnce you've reached conscious competence, it's just a matter of practice and repetition for the skill to become automatic and unconscious. However, making the leap between consciously incompetent and consciously competent, when you actually develop and acquire the skill, is the most important step, and it's at this point that many people falter and give up. The transition can be made much easier, though, by following a few of the simple techniques I've already explained in this book.\n\n#### ENHANCING YOUR SKILL DEVELOPMENT Be confident of success\n\nOne of the key sections of this book is \"Being certain of success\" (see pp.35\u2013) and everything I've outlined there will enhance your ability to develop any skill you choose. Here are the main points again just to remind you:\n\n\u2022 Have a goal.\n\n\u2022 Have a plan.\n\n\u2022 Believe you can do it.\n\n\u2022 Take action.\n\n\u2022 Be positive.\n\n### REHEARSE THE SKILL IN YOUR MIND\n\nI've been encouraging you to use visualization techniques to create images in your mind's eye that are easy to remember. Visualization can also play an important part in skill development because the mind is unable to distinguish between a real and a vividly imagined event. If I started describing a succulent juicy lemon to you in great detail with its tangy citrus taste and bright yellow skin, the chances are you would begin salivating even though the lemon exists only in your mind. The reason you'd do so is because the body responds to the signals the mind gives it, even if the mental stimulus is imaginary.\n\nWhen you're learning a new skill, it's important to rehearse it in your imagination, because research has demonstrated that a thorough mental rehearsal boosts the skill acquisition process and is almost as good as real practice.\n\n### VISUALIZE AND STORE THE INSTRUCTIONS\n\nOften, not being able to remember instructions on how to develop a new skill will hamper progress and cause frustration, making it easier to yield to the temptation to give up. The answer is to use a mental filing system in which to store these instructions \u2013 this will give you instant access to them whenever you need them. I suggest you use the Journey Technique (see pp.114\u20138), and that you create a mental journey related to whatever skill you're learning.\n\nFor example, use the golf clubhouse as the setting for your journey if the skill you're trying to learn is an improved golf swing, or maybe a tennis club setting if you're seeking to develop your serve or backhand. Then simply place an image for each instruction at successive locations or waypoints on your journey. Whenever you need to refer to the instructions, mentally walk your journey to recall each one in turn. These instructions are only a temporary \"crutch\" to help you until the skill becomes automatic. Once you've become unconsciously competent, you won't need to refer to them any more.\n\n**FAILURE ISN'T FINAL**\n\nProbably the biggest factor affecting people's development as they embark on learning a new skill is the fear of failure. Getting something wrong is felt to be such a terrible thing that many people \"freeze up\" when trying something new (or maybe they don't even try it at all). What they don't realize is that failure is part of the learning process. Think back to when you last got something wrong or you failed. Did you learn a lot from the experience? If you want to double your rate of learning, you need to double your failure rate.\n\n## LEARNING FOREIGN VOCABULARY\n\nInternational travel is much easier than it used to be, which means more of us are venturing abroad. To really experience and enjoy the countries and cultures we visit, it helps enormously to speak even a few words of the local language.\n\n### YOU DON'T NEED TO BE FLUENT TO GET BY\n\nI find that many people who consider learning a new language are put off by the idea that they need to be fluent. Fluency in another language can be achieved, but it's not necessary in order to understand and be understood when communicating in another tongue. Studies have shown that even though languages have many thousands of words, most native speakers use a working vocabulary of only a few hundred.\n\n### LEARNING FOREIGN WORDS \u2013 STEP BY STEP\n\nThe process for learning foreign words is this:\n\n\u2022 Take the word you want to learn in your new language.\n\n\u2022 See if the word reminds you of something or be creative and find an association \u2013 the association doesn't have to have anything to do with what the word means at this stage.\n\n\u2022 Take what the word reminds you of and link that to the meaning of the original word.\n\n\u2022 \"Condition\" the association, practising it a couple of times to strengthen it in your memory.\n\nThis is how I'd go about remembering the meaning of a common German word.\n\n\u2022 I'd focus on the target word \"Zimmer\" (pronounced \"tsimmer\"), which in German means \"room\".\n\n\u2022 \"Zimmer\" sounds like \"simmer\" so in this instance I see a simmering pan.\n\n\u2022 I'd now create a crazy picture linking a simmering pan to a room (I'd use a house and visualize a pan in every room).\n\n\u2022 I'd then condition the association chain from \"Zimmer\" > Simmering pan > Room.\n\nNow when I hear the word \"Zimmer\" in German I immediately think of a simmering pan, which instantly brings to mind the bizarre picture involving one in every room of a house.\n\n### DEVELOPING A WORKING VOCABULARY\n\nOnce you get the hang of this method, you'll find that in about 10 minutes you'll be able to memorize at least 10 words so that they stick in your long-term memory. Over the space of three or four weeks you will be able to develop a working vocabulary of any language you choose.\n\nInitially, when you start using the language, you'll need to go through the process of thinking of your trigger words and the bizarre images to remind you of the translation, but very quickly, with continued use, you won't need to, because the words will have become part of your vocabulary in that language and you'll \"just know\" what they mean and be able to summon them up at will.\n\n## REMEMBERING SPEECHES AND JOKES\n\nPublic speaking is claimed to be people's number one fear. If you have to give a speech and you're not used to public speaking, it can be quite a daunting prospect.\n\nA major contributor to the fear factor is the dread of drying up in front of an audience and forgetting what to say.\n\n### WHY DO WE FORGET WHAT WE HAVE TO SAY?\n\nIf you recall from the earlier section on stress, when faced with a threat or perceived threat, we go into a fight or flight response. The body shuts down what it doesn't need to run away or tackle the threat, and that includes part of the brain responsible for long-term memory. Talking to a group of people can cause the fight or flight mechanism to kick in if you're not sufficiently prepared, and the result can be quite embarrassing as you struggle to remember what you have to say.\n\n### WHAT YOU CAN DO TO HELP REMEMBER YOUR SPEECH\n\nThe secret of being a successful public speaker, even if you're going to give a one-time speech at your daughter's wedding, is quite simple:\n\nKnow what you want to say and practise saying it.\n\nThe crafting, organization and delivery of a good speech are outside the scope of this book, so for our purposes I'll assume that you already have your speech prepared. However, you'll find the process of putting it together will give you an important level of familiarity with what you want to say, which will help your recall.\n\n#### Step 1 \u2013 Break your speech down into sections\n\nIf you have a five-page speech to deliver, it's much easier if you break it down into 10 logical half-page sections. Spend a little time identifying the purpose of each part and give each a unique, but a logically appropriate name.\n\n#### Step 2 \u2013 Identify your main key words for each section\n\nUnder each of your headings choose key words that will trigger what you want to say in each section. For a halfpage of writing you probably won't need more than three or four key words \u2013 you're looking for one for each idea or theme.\n\n#### Step 3 \u2013 Practise delivering each section\n\nNow that you have your sections and have identified a handful of key words for each, the next step is to practise delivering the sections, one at a time:\n\n\u2022 Read the section out loud so you get used to its rhythm, pace and message.\n\n\u2022 Take the first key word and from memory recite the passage that it summarizes, focusing on the meaning rather than concentrating on repeating the passage verbatim.\n\n\u2022 Check to see how accurately you recited what you've written. At this stage, expect to be able to recall only a small amount.\n\n\u2022 Repeat the process for a maximum of three times until you're happy that you can recite what you need to for this key word, and then move on.\n\n\u2022 Do the same with the rest of the key words.\n\n\u2022 Once you've done this with all the key words for that section, try delivering the entire section just by looking at the key words. Do this three times only.\n\n\u2022 Repeat the process for each section.\n\nMost people will start with the first section and work their way to the end, but I've found I obtain my best results if I work backward from the last section to the first or do the sections in random order.\n\n#### Step 4 \u2013 Practise delivering the whole speech\n\nUsing your list of section headings and key words as a guide, practise reciting your speech in its entirety, checking after each time against the full written version and making any necessary corrections. A good tip is to record your speech as you deliver it and then listen to it to identify where you can make improvements.\n\n#### Step 5 \u2013 Memorize your speech\n\nYou'll find that by following the process of repeated recall, your memory of what you need to say will be pretty good as it will be triggered by the key words in each section. Now all you need to do is to memorize the section names and the key words associated with each.\n\nIf you have 10 sections, I suggest you use a 10-stage journey (see the Journey Technique, pp.114\u20138) to remember their names. It helps to use an appropriate journey for the message (for example, a church for a wedding speech), but it's not essential. Conjure up a memorable image for each section title at each stage along the way.\n\nNow memorize the key words for each section by linking them in a bizarre story and associating them with the image you've chosen for the section title. So when you walk around your mental journey, at each location you'll see an image for a section linked in a story to the key words associated with that section.\n\n#### Step 6 \u2013 Practise your speech from memory\n\nWork on your delivery by rehearsing your speech using your images as triggers. With enough practice you'll say what you need to without relying on your images \u2013 but they'll always be there if you need them.\n\n### THE ART OF REMEMBERING JOKES\n\nIf you've ever tried to tell a joke and forgotten the punch line, here's a simple way to remember it. You'll need a mental filing system in which to store your jokes \u2013 again, my own preference is the ever-versatile Journey Technique. Your first step is to develop a journey on which to place your jokes.\n\nNext, create vivid images for the different parts of each joke and for the punch line and link them in a story to the stages of your journey. Practise recalling the images so they spring to mind effortlessly, then practise delivering the joke until you can do so confidently.\n\n## MIND MAPS\u00ae\n\nFor thousands of years the written word has been an effective mechanism for organizing, capturing and sharing ideas and thoughts. However, research into how the brain functions has identified that the tradition of organizing words in a linear manner on paper may not be the best way to employ our cognitive abilities. There are other more powerful ways to organize our thoughts, and, in my opinion, the Mind Map\u00ae is the most effective.\n\n### WHAT ARE MIND MAPS\u00ae?\n\nMind Maps\u00ae present a diagrammatic overview of a topic, making it easy for the brain to assimilate the information. They were invented by British psychologist and thinking expert Tony Buzan in the early 1970s after his extensive research into memory and the most effective ways of organizing thoughts on paper for recall, problem solving and creativity.\n\nThey engage both sides of the brain simultaneously \u2013 the left, analytical part and the right, imaginative, intuitive side.\n\nNow used by millions of people around the world, the Mind Map\u00ae has been described as the ultimate thinking tool and is simple to use and easy to learn. The example opposite shows a basic Mind Map\u00ae incorporating the key features:\n\n\u2022 A central image that represents the topic of the Mind Map\u00ae.\n\n\u2022 A radiating structure of main key word branches that identify the themes of the topic.\n\n\u2022 Smaller key word sub-branches that represent the detail of each theme.\n\nA good Mind Map\u00ae will use the following:\n\n\u2022 Many different colours to stimulate the right brain and to distinguish between themes and topics.\n\n\u2022 Carefully chosen key words, preferably single and written clearly. Each one should sit neatly on a branch of corresponding size.\n\n\u2022 As many images as possible (because pictures are the language of our thinking). Some sub-branches could simply have a picture against them and no words.\n\n\u2022 Arrows showing associations between key words and their related images.\n\nIt's critical that the length of the branches matches the length of the words, because, when you come to recreate a Mind Map\u00ae from memory, one of the key features that you'll recall is size. For example, if you remember a long branch radiating off from the right, your mind will automatically look for a long word to go with it.\n\n### ADVANTAGES OF A MIND MAP\u00ae\n\nIf you need to remember information, especially over the long term, a Mind Map\u00ae is a powerful memory device. It's also an extremely effective tool for organizing your thoughts on paper \u2013 whether to plan out a report or record what you've read or heard.\n\nIn fact, you'll find that you'll get far more out of these processes and improve your recall if you use a Mind Map\u00ae rather than writing things down, for the following reasons:\n\n\u2022 You'll be using more of your brain and therefore bringing more of your powerful abilities to the task.\n\n\u2022 It takes much less time to create a Mind Map\u00ae than to write in longhand, and it's more enjoyable and engaging.\n\n\u2022 The process of Mind Mapping\u00ae requires you to think about a topic in a more focused way than simply writing about it, thereby developing your cognitive abilities.\n\n\u2022 It's much easier to edit and review information in Mind Map\u00ae form than longhand.\n\n**HOW TO DRAW YOUR OWN MIND MAP \u00ae**\n\n**Creating a Mind Map \u00ae is a straightforward process. To start off, try this exercise based on a familiar topic \u2013 yourself!**\n\nTake a sheet of paper and turn it so that it's landscape format   \n(longer side along the bottom).\n\nUsing at least three coloured pencils, draw an image of yourself in the middle of the page. Don't worry, you don't need to be good at drawing. You're not trying to achieve an exact likeness: the image is simply there as a starting point.\n\nThink about all the important areas of your life and draw main branches from your central image to represent each one. Use a different colour of pencil for each branch. For example, you might have branches for your family, hobbies, work, friends, home and vacations \u2013 whatever's important to you. Write down key words or draw pictures for each main branch.\n\nDraw smaller branches coming out from each main branch to add some detail about each topic area. Label each one with a key word and\/or draw an appropriate image.\n\n## REMEMBERING WHAT YOU'VE READ\n\nAmong the most common experiences that make people think they have a bad memory, as well as forgetting names, is forgetting what they've read. It's a natural assumption to make, but one that's usually completely wrong.\n\n### WHY CAN'T I REMEMBER EVERYTHING I'VE READ?\n\nWhenever I ask people to break down what they expect to happen when they read, I usually receive an answer like this:\n\n#### Stage 1\n\n\"I expect to **see** characters on the page and **recognize** them as letters that form words that I'm then able to **read** and **understand**.\"\n\n#### Stage 2\n\n\"I expect to be able to **remember** what I've read so I can **recall** it at a later date either to **use** myself or to **communicate** to someone else.\"\n\nThis is a completely reasonable set of expectations when you consider what you **want** from your reading, but it's totally unreasonable when you look at what you're **doing** when you read.\n\nMost people have only ever been taught Stage 1 (the reading bit), which is all they do, and don't bother doing anything to ensure that Stage 2 (the remembering bit) happens. So your expectations for the result you want are over-ambitious and exceed what you're doing to achieve that result. In order to remember more of what you've read, you have to actively do something with that information to ensure you **memorize** it.\n\nAnother reason why you sometimes can't recall what you've read is because you're distracted \u2013 your mind begins to wander and starts to think about other things even as your eyes are still reading the words. Because you're not concentrating properly on what you're reading, you're not consciously taking it in, and consequently you won't be able to recall it. This is why you often get to the bottom of the page of a book and can't remember anything.\n\n### BECOME A BETTER READER\n\nThe first thing I recommend you do is to become a more efficient reader. Going to school and learning to read is a wonderful thing but, unfortunately, the way most people have been taught to read sets up a series of poor habits that limit the speed we read at and creates an inefficient reading process. Here are a few ideas to help you to become a better reader:\n\n\u2022 Use a pencil (or your finger) as a guide as you read to stop your eyes jumping around the page (back-skipping to re-read words is unnecessary and slows you down).\n\n\u2022 Rather than reading individual words, read in groups or clusters of words to take in meaningful chunks.\n\n\u2022 Practise reading faster so that your mind doesn't have the opportunity to wander off and think of something else because it's so absorbed in your reading.\n\n### HELPFUL READING STRATEGIES\n\nAs well as improving your reading technique, you can also employ reading strategies for fiction and non-fiction that will enhance your recall by helping you become more engaged in the material.\n\n\u2022 With non-fiction, quickly skim what you're about to read and preview it to identify where the \"best bits\" are so you can focus on them.\n\n\u2022 Set yourself goals for what you want to glean in order to focus your attention.\n\n\u2022 When you read, mark out key words and sentences with a pencil or highlighter pen.\n\n\u2022 Take notes as you read to include your questions, observations and opinions.\n\n\u2022 After you've finished reading, quickly scan through the material again to review what you've read.\n\n**CAN I REMEMBER EVERYTHING?**\n\nThe subconscious is extremely powerful, and there are those who believe that everything we've ever encountered in our entire lives is stored in the memory. However, it appears that most of these memories are accessible only via hypnosis (and even then there's doubt about whether memories accessed by this technique are, in fact, true memories). So, rather than attempting the impossible and trying to acquire the ability to recall absolutely everything that you read, you should aim instead to focus solely on what's important for you to remember.\n\n\u2022 Finally, summarize what you've read in your mind to embed the information.\n\n### BOOSTING YOUR RECALL\n\nJust by becoming a better reader and using the strategies I've explained above, you'll find that you'll naturally be able to remember more of what you read. However, even with this improvement, you'll still need to memorize and condition the knowledge so that you can always recall it. Here are some ways to do so using techniques I've already told you about:\n\n\u2022 Use a Mind Map\u00ae (see pp.134\u20138) to take notes as you're reading. You'll find that the process of using this powerful tool will focus your attention and encourage you to think more deeply about your reading material, ensuring that you're more engaged in it. You'll also have a memorable record of the things that you've read.\n\n\u2022 Review your Mind Maps\u00ae regularly \u2013 after 10 minutes, a day, a week, a month, three months and six months to ensure the knowledge transfers to your long-term memory. At each review try to draw the maps from memory before checking with the originals.\n\n\u2022 Take the key points from what you've read and use either the Story Technique (see pp.79\u2013) or the Journey Technique (see pp.114\u20138) to memorize them.\n\n_**\"** It is certain that memory contains not only   \nphilosophy, but all the arts and all that   \nappertains to the use of life. **\"**_\n\nMarcus Tullius Cicero (106\u201343BC)\n\n# 5   \nCHAMPIONSHIP POINTS\n\nWith the skills you've developed by working your way through the proven techniques in this book, you're on the verge of being able to do jaw-dropping things with your memory. When you see those people on television performing amazing feats, such as remembering ridiculously long strings of numbers or multiple decks of shuffled playing cards, every one of them has been at the same stage in their memory development as you're at now.\n\nYou, too, have the ability to perform similar feats, and this chapter will show you some of the tricks of the trade used by memory experts who do these sorts of stunts. You'll be surprised how easy they really are.\n\nWho knows, we might see you, after a bit of practice, on television or maybe doing exceptionally well in the World Memory Championships. (Don't laugh, because that's exactly what happened to me!)\n\n## THE MAJOR SYSTEM\n\nDating back to the 17th century, the Major System is a phonetic memory technique that can be used as a mental filing system or to remember numbers, dates and lists.\n\n### HOW IT WORKS\n\nNumbers are coded to consonant sounds from which you can make words and then memorable images.\n\n### HOW TO APPLY THE SYSTEM\n\nThe Major System is straightforward to implement. All you have to do is to follow the three simple steps outlined below and on the pages that follow.\n\n#### Step 1 \u2013 Allocate consonant sounds\n\nTake the number you want to remember and then simply allocate the appropriate consonant sound to each digit. For example 18374 would have the following sounds:\n\n#### **Step 2 \u2013 Make up a word**\n\nPick a combination of the letters in the order of the numbers and add in vowels or the silent consonants \"h\", \"w\" and \"y\" to make a word or sequence of words. Remember, it's the sound of the consonants that's important. For example, 18374 could become the words:\n\nToffee Maker \u2013 t( **1** ) f( **8** ) m( **3** ) k( **7** ) r( **4** )\n\nWhen you translate the numbers into words, you'll find that there's a range of possibilities. However, when you work back from the words to the numbers, there's only one combination of digits that \"Toffee Maker\" (or any of the other possible words or phrases) can be.\n\n#### Step 3 \u2013 Embed the words in your memory\n\nTo fix the words (and therefore the number) in your mind, create an extravagant and unusual image to go with them. So if the number 18374 was the access code to your office building, simply conjure up a bizarre picture, which somehow associates the building with a toffee maker.\n\n### CREATE YOUR OWN NUMBER IMAGES\n\nNow, with a bit of creativity, you can develop a filing system with a unique image for all the numbers from 1 to 100, or even beyond! Work on a few numbers at a time and commit them and their associated words and images to memory by practising the recall. You'll then have your own peg system for remembering long lists of items.\n\n**BEETHOVEN'S BIRTH DATE**\n\nHow many times have you tried to remember historical dates or the dates of famous people but failed? The Major System is the perfect way to remember them.\n\nLet's choose the number 1770, allocating the following letters 1 = d, the first 7 = g, the second 7 = k and 0 = s, which gives the words \"Dog Kiss\".\n\nI'd then make a vivid image of a dog rushing up to someone and giving them a big sloppy wet kiss. But what use can that be? Well, if I tell you that Beethoven was born in 1770 and you take the image of Beethoven and link it with a dog giving him a big kiss, you've just used the technique to memorize the year of his birth.\n\nYou could have chosen the words \"Tea Cakes\" or \"Duck Case\". It really doesn't matter as long as the code is adhered to.\n\n## LEARNING FROM A CHAMPION\n\nLike many of my colleagues who have carved out a career as a professional memory expert, I was inspired by eight-times World Memory Champion Dominic O'Brien. As well as being an impressive mnemonist, Dominic also invented his own memory system now used by thousands of people around the world.\n\n### THE DOMINIC SYSTEM\n\nA simple but amazingly powerful concept, the Dominic System will allow you to remember any number of any length quickly and easily. It's based on assigning letters to numbers and then personalizing them by linking them to someone you know well \u2013 a family member or a friend. You could also use a well-known figure or celebrity. All you have to do is follow these four straightforward steps.\n\n#### Step 1 \u2013 Assign the letters to the digits 0\u20139:\n\nGenerally, \"S\" is used for 6 as the word \"six\" has a strong \"s\" sound, and \"N\" represents 9 as it's easier to use than the ninth letter of the alphabet, which is \"I\".\n\n#### Step 2 \u2013 Create letter pairs\n\nFor the numbers 00 to 99, create letter pairs. Write them down on a sheet of paper. For example, 23 will have the letter pair BC and 10 will have the letter pair AO.\n\n#### Step 3 \u2013 Think of a famous person\n\nFor each letter pair, find someone you know really well or think of a famous person with the same initials. For example, CD (34) could be Charles Dickens.\n\n#### Step 4 \u2013 Pick a related object\n\nFor each person on the list identify an object that's related to them or that's typical of them in some way.\n\nHere are some examples I use:\n\n### HOW TO USE THE SYSTEM\n\nFirst, take the number you have to memorize and break it down into four-digit chunks. For example, 48379651 would become 4837 9651. To remember the number 4837, take the person represented by the first digit pair (48) and link them to the object of the second digit pair (37). In my example I see Damon Hill dressed in his yellow racing overalls wielding an enormous, bright green, metal baseball bat.\n\nNext choose a journey that you've created using the Journey Technique (seepp.114\u20138) to fix your number and its associated image in position. Repeat this for the second group of four digits.\n\nTo recall the number, mentally walk along your journey route and visualize the images that you've created at each location. Translate the images back into the number. For example, when I see the image of Damon Hill and the baseball bat, all I do is translate it back to the number 4837. If I had to remember the number 3748 (same digit pairs but in reverse order), I would have created a picture of my friend Craig driving a Formula 1 racing car.\n\n### DEALING WITH ODD NUMBERS\n\nNot all numbers break down conveniently into fourdigit chunks \u2013 you may be left with one, two or three digits. To memorize these extra digits, you can use a combination of systems.\n\n**One remaining digit:** Choose the corresponding image from the Number Rhyme or Number Shape System (see pp.84\u2013 and \u2013).\n\n**Two remaining digits:** Envisage the person represented by that number in the Dominic System.\n\n**Three remaining digits:** Envisage the person represented in the Dominic System by the first two digits interacting with either a Number Rhyme or Number Shape image for the third digit.\n\n## REMEMBERING A DECK OF CARDS\n\nOne of the most impressive feats of memory is being able to recall the order of a shuffled deck of playing cards having only briefly seen each card once. With a little practice you too will be able to do this either to impress your friends or become a winning player at the card table.\n\n### HAVE SOMEWHERE TO STORE YOUR CARD IMAGES\n\nTo start with you need a mental filing system in which to organize all 52 cards so you can recall them. I suggest you use the Journey Technique (see pp.114\u20138) because it's the most flexible and, to my mind, the most powerful system. Create a journey that has 26 stages so that when you memorize the cards you can link two cards to each location. I find that it helps to have three or four of these journeys so I can follow a fresh journey each time and avoid confusion.\n\n### CREATE UNIQUE IMAGES FOR EACH CARD\n\nYou now need to create a unique image for each card. You could start from scratch and design new images, but to save time and effort, I'd advise you to \"borrow\" some of the images that you've already assigned to numbers in either the Major System (see pp.146\u2013) or the Dominic System (see pp.149\u2013) \u2013 you can use whichever system you prefer. It's safe to do this as it's extremely unlikely that you'll have to remember numbers AND cards at the same time (I can say this from experience as I have never had to!).\n\n### ASSIGN A NUMBER TO EACH CARD\n\nThe first step in creating the images you need is to give each card its own number, based on its suit.\n\nFor example, for clubs I use the numbers 10\u201322: (A = Ace, J = Jack, Q = Queen, K = King)\n\nFor diamonds (30\u201342), the numbering would work like this:\n\nI'd then do the same for Hearts (50\u201362) and Spades (70\u201382). Now link your Major or Dominic System image for the number 11 with the ace of clubs, the number 12 image with the two of clubs and so on. Then, when you're recalling the cards and you see your picture for the number 11, you know it represents the ace of clubs.\n\n### PRACTICE MAKES PERFECT\n\nPractise converting each card to a number and visualizing the associated image. Then, link the images for pairs of cards together. (You're using a 26-stage journey so you'll need to double up images.) Finally, practise linking the doubled-up images to each location.\n\n## LONG LISTS OF SPECIALIZED ITEMS\n\nIf you want to do better at a local quiz night, you can develop an encyclopedic knowledge of any topic using the techniques I've shown you in this book. Here are some examples of the question areas that may come up:\n\nKings and queens | Capital cities  \n---|---  \nLongest rivers | Roman emperors  \nUS states and their capitals | UK prime ministers  \nGolf \u2013 US Masters winners | Constellations  \nUS presidents | Currencies  \nElements of the periodic table | Nobel Peace Prize winners\n\n### THE FIRST STEP\n\nThe most important step when preparing yourself to remember a long list of specialized items is to make sure that the information is correct in the first place. It's no good having perfect recall of an incredible list of facts if some of them are wrong.\n\nOnce you've checked that the information is correct, there are two main techniques you can use to memorize and recall it.\n\n### USE THE JOURNEY TECHNIQUE\n\nThe Journey Technique (see pp.114\u20138) is ideal for remembering information that is sequential or daterelated. For example, if you were a soccer fanatic and wanted to remember the World Cup winners since 1930, all you would need to do is to create a journey with 20 locations. This would give you enough locations to memorize the sequence of winners and a couple of spare slots for the next championships.\n\nThe next step is to place an image of the winner at each location in order. For example, at the 12th location (the 1982 winner) you could place an image of the Colosseum in Rome (Italy won the World Cup that year), making it as vivid, bizarre or unusual as possible. Then, by walking the journey in your mind and recalling the image that you've pegged at each location, you'll have a fail-safe method of remembering all the winners in order from 1930.\n\n### USE ASSOCIATION\n\nSometimes information doesn't need to be memorized in order and all you need to be able to do is link one thing to another \u2013 such as capital cities or the different currencies used in different countries. For example, to remember that the currency of Thailand is the baht, you need to link an image that represents Thailand to one that represents the currency.\n\nI picture hundreds of neck ties (Thai-) parachuting into a field (-land) and attacking each other with baseball bats (baht). Repeat this process for each currency you want to remember and experiment with the same technique for memorizing capital cities. Endless variations on such ideas are possible.\n\n## FURTHER READING\n\n**Tipper, Michael** _The 77 Habits of Highly Effective_ _Students_ , available from www.michaeltipper.com\n\n**Tipper, Michael** _The Secrets of Successful Students \u2013_ _Super Speed Study Skills_ , available from www.michaeltipper.com\n\n**Buzan, Tony** _Brilliant Memory: Unlock the Power of_ _Your Mind_ (Buzan Bites series), BBC Active 2006\n\n**O'Brien, Dominic** _How to Develop a Brilliant Memory_ _Week by Week_ , Duncan Baird Publishers 2006\n\n## CONTACT THE AUTHOR\n\nIf you'd like to find out more about how you can benefit from a better memory, sharper concentration and greater mental agility, please contact Michael Tipper personally at his website at www.michaeltipper.com\n\n## AUTHOR'S ACKNOWLEDGMENTS\n\nI'd like to thank the pioneers of memory improvement \u2013 notably Dr Bruno Furst, Harry Lorayne, Tony Buzan, Kevin Trudeau and Dominic O'Brien \u2013 who have brought the possibility of improved recall and better memory to the masses. Were it not for their dedication, I might still be struggling with my own inability to remember.\n\nI'd especially like to acknowledge the moral support and guidance given to me by a special few who have believed in me and what I could achieve, especially when I doubted or couldn't appreciate my own talent and abilities \u2013 Paul and Lorna Bridle, Denise Fryer and Warren Shute. I'd also like to thank from the bottom of my heart Julie Lutwyche for supporting me all those years and giving me the love, space and encouragement to follow my dreams. Thanks, too, to all of the staff at Duncan Baird Publishers, in particular Caroline Ball, Daphne Razazan and Bob Saxton.\n\nAnd finally, I want to thank all of you who deserve to be mentioned here, and who have positively contributed to my work (and my life), though space prevents me from including you. You will know who you are, so thank you.\n\nThe story of Watkins began in 1893, when scholar of esotericism john Watkins founded our bookshop, inspired by the lament of his friend and teacher Madame Blavatsky that there was nowhere in London to buy books on mysticism, occultism or metaphysics. That moment marked the birth of Watkins, soon to become the publisher of many of the leading lights of spiritual literature, including Carl jung, Rudolf Steiner, Alice Bailey and Ch\u00f6gyam Trungpa.\n\nToday, the passion at Watkins Publishing for vigorous questioning is still resolute. Our stimulating and groundbreaking list ranges from ancient traditions and complementary medicine to the latest ideas about personal development, holistic wellbeing and consciousness exploration. We remain at the cutting edge, committed to publishing books that change lives.\n\nDISCOVER MORE AT:   \nwww.watkinspublishing.com\n\nRead our blog\n\n|\n\nWatch and listen to our authors in action\n\n|\n\nSign up to our mailing list\n\n---|---|---\n\nWe celebrate conscious, passionate, wise and happy living,   \nBe part of that community by visiting\n\n  \/Watkinspublishing  @watkinswisdom  \n  \/watkinsbooks  @watkinswisdom \nFirst published 2007 under the title _Memory Power-Up_\n\nThis edition first published in the UK and USA 2018 by\n\nWatkins, an imprint of Watkins Media Limited\n\nUnit 11, Shepperton House\n\n89-93 Shepperton Road\n\nLondon N1 3DF\n\nenquiries@watkinspublishing.com\n\nDesign and typography copyright \u00a9 Watkins Media Limited 2018\n\nCommissioned illustrations: Bonnie Dain for Lilla Rogers Studio and 3+Co\n\nForeword copyright \u00a9 Dominic O 'Brien 2007, 2018\n\nText copyright \u00a9 Michael Tipper 2007, 2018\n\nArtwork copyright \u00a9 Watkins Media Ltd 2007, 2018\n\nMichael Tipper has asserted his right under the Copyright, Designs\n\nand Patents Act 1988 to be identified as the author of this work.\n\nAll rights reserved.\n\nNo part of this book may be reproduced or utilized in any form or\n\nby any means, electronic or mechanical, without prior permission in\n\nwriting from the Publishers.\n\n1 3 5 7 9 10 8 6 4 2\n\nDesigned and typeset by Clare Thorpe\n\nPrinted and bound in the United Kingdom\n\nA CIP record for this book is available from the British Library\n\nISBN: 978-1-78678-175-8\n\nwww.watkinspublishing.com\n  1. Cover\n  2. Title\n  3. Content\n  4. Dedication\n  5. Foreword\n  6. Introduction\n  7. 1 Warm-Ups\n  8. Why do we forget?\n  9. Is it my age?\n  10. Your mental filing system\n  11. Your brain and your memory\n  12. Imagination, association and recall\n  13. Being certain of success\n  14. 2 Gentle Stretches\n  15. Managing stress for a better memory\n  16. Exercise to improve your memory\n  17. A better diet, a better memory\n  18. Memory joggers\n  19. Where did I leave...?\n  20. That reminds me...\n  21. 3 Great Techniques\n  22. Remembering names - the social method\n  23. Remembering facts - the magic of mnemonics\n  24. Remembering spellings\n  25. Always remember your PIN\n  26. Remembering directions\n  27. Remembering errands and simple lists\n  28. I must remember to...\n  29. Shopping with the alphabet\n  30. Remembering what you've heard\n  31. Retrieving the past\n  32. Interlude: The Amazing World Of Memory\n  33. 4 Winning Workouts\n  34. Remembering names and faces - the mnemonic method\n  35. Remembering longer numbers\n  36. The Journey Technique\n  37. Remembering dates and appointments\n  38. Learning a new skill\n  39. Learning foreign vocabulary\n  40. Remembering speeches and jokes\n  41. Mind Maps\u00ae\n  42. Remembering what you've read\n  43. 5 Championship Points\n  44. The Major System\n  45. Learning from a champion\n  46. Remembering a deck of cards\n  47. Long lists of specialized items\n  48. Further reading\n  49. Contact the author\n  50. Author's Acknowledgments\n  51. Copyright\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nDisney Characters and Artwork TM & \u00a9 2018 Disney\n\nThe following songs are the property of:  \nBourne Co.  \nMusic Publishers  \n5 West 37th Street  \nNew York, NY 10018  \nBaby Mine  \nGive a Little Whistle  \nI've Got No Strings  \nSome Day My Prince Will Come  \nWhen You Wish Upon a Star  \nWhistle While You Work  \nWho's Afraid of the Big Bad Wolf?\n\nPrint ISBN: 978-1-5400-1530-3\n\nePub ISBN: 978-1-5400-3992-7\n\nKindle ISBN: 978-1-5400-3993-4\n\nFor all works contained herein:\n\nUnauthorized copying, arranging, adapting, recording, Internet posting, public performance,\n\nor other distribution of music in this publication is an infringement of copyright.\n\nInfringers are liable under the law.\n\nVisit Hal Leonard Online at\n\nwww.halleonard.com\n\nContact Us:\n\nHal Leonard\n\n7777 W. Bluemound Rd.\n\nMilwaukee, WI 53213\n\nEmail: info@halleonard.com\n\nContact Us:\n\nHal Leonard Europe Limited\n\n42 Wigmore Street\n\nMarylebone, London, W1U 2RN\n\nEmail: info@halleonardeurope.com\n\nIn Australia contact:\n\nHal Leonard Australia Pty. Ltd.\n\n4 Lentara Court\n\nCheltenham, Victoria, 3192 Australia\n\nEmail: ausadmin@halleonard.com.au\n\n# Contents\n\nBe Our Guest\n\nBaby Mine\n\nBeauty And The Beast\n\nBibbidi-Bobbidi-Boo (The Magic Song)\n\nCan You Feel The Love Tonight\n\nCircle Of Life\n\nChim Chim Cher-ee\n\nCruella De Vil\n\nColors Of The Wind\n\nDo You Want To Build A Snowman?\n\nA Dream Is A Wish Your Heart Makes\n\nEvermore\n\nFriend Like Me\n\nGod Help The Outcasts\n\nGive A Little Whistle\n\nGo The Distance\n\nHakuna Matata\n\nHe's A Tramp\n\nHeigh-Ho\n\nHow Far I'll Go\n\nHow Does A Moment Last Forever\n\nI Just Can't Wait To Be King\n\nI See The Light\n\nI've Got No Strings\n\nIf I Never Knew You (End Title)\n\nIt's A Small World\n\nKiss The Girl\n\nLet It Go\n\nLava\n\nMickey Mouse March\n\nPart Of Your World\n\nSome Day My Prince Will Come\n\nRemember Me\n\nA Spoonful Of Sugar\n\nSupercalifragilisticexpialidocious\n\nTrashin' The Camp*\n\nWhen She Loved Me\n\nWhen You Wish Upon A Star\n\nWhere You Are\n\nWhistle While You Work\n\nWho's Afraid Of The Big Bad Wolf?\n\nA Whole New World\n\nWinnie The Pooh\n\nThe World Es Mi Familia\n\nWritten In The Stars\n\nYo Ho\n\nYou'll Be In My Heart*\n\nYou've Got A Friend In Me\n\nZip-A-Dee-Doo-Dah\n\nYou're Welcome\n\n# Be Our Guest\n\n### from BEAUTY AND THE BEAST\n\n#### Music by ALAN MENKEN   \nLyrics by HOWARD ASHMAN\n\n\u00a9 1991 Wonderland Music Company, Inc. and Walt Disney Music Company\n\nAll Rights Reserved. Used by Permission.\n\n# Baby Mine\n\n### from DUMBO\n\n#### Words by NED WASHINGTON   \nMusicby FRANK CHURCHILL\n\nCopyright \u00a9 1941 by Walt Disney Productions\n\nCopyright Renewed\n\nWorld Rights Controlled by Bourne Co. (ASCAP)\n\nInternational Copyright Secured All Rights Reserved\n\n# Beauty and The Beast\n\n### from BEAUTY AND THE BEAST\n\n#### Music by ALAN MENKEN  \nLyrics by HOWARD ASHMAN\n\n\u00a9 1991 Wonderland Music Company, Inc. and Walt Disney Music Company\n\nAll Rights Reserved. Used by Permission.\n\n# Bibbidi-Bobbidi-Boo\n\n### (The Magic Song)\n\n### from CINDERELLA\n\n#### Words by JERRY LIVINGSTON   \nMusicby MACK DAVID and AL HOFFMAN\n\n\u00a9 1948 Walt Disney Music Company\n\nCopyright Renewed.\n\nAll Rights Reserved. Used by Permission.\n\n# Can You Feel The Love Tonight\n\n### from THE LION KING\n\n#### Music by ELTON JOHN  \nLyrics by TIM RICE\n\n\u00a9 1994 Wonderland Music Company, Inc.\n\nAll Rights Reserved. Used by Permission.\n\n# Circle Of Life\n\n### from THE LION KING\n\n#### Music by ELTON JOHN  \nLyrics by TIM RICE\n\n\u00a9 1994 Wonderland Music Company, Inc.\n\nAll Rights Reserved. Used by Permission.\n\n# Chim Chim Cher-ee\n\n### from MARY POPPINS\n\n#### Words and Music by RICHARD M. SHERMAN   \nAND ROBERT B. SHERMAN\n\n\u00a9 1963 Wonderland Music Company, Inc.\n\nCopyright Renewed.\n\nAll Rights Reserved. Used by Permission.\n\n# Cruella De Vil\n\n### from 101 DALMATIANS\n\n#### Words and Music by   \nMEL LEVEN\n\n\u00a9 1961 Walt Disney Music Company\n\nCopyright Renewed.\n\nAll Rights Reserved. Used by Permission.\n\n# Colors Of The Wind\n\n### from POCAHONTAS\n\n#### Music by ALAN MENKEN   \nLyrics by STEPHEN SCHWARTZ\n\n\u00a9 1995 Wonderland Music Company, Inc. and Walt Disney Music Company\n\nAll Rights Reserved. Used by Permission.\n\n# Do You Want To Build A Snowman?\n\n### from FROZEN\n\n#### Music and Lyrics by KRISTEN ANDERSON-LOPEZ   \nand ROBERT LOPEZ\n\n\u00a9 2013 Wonderland Music Company, Inc.\n\nAll Rights Reserved. Used by Permission.\n\n# A Dream Is A Wish Your Heart Makes\n\n### from CINDERELLA\n\n#### Words and Music by MACK DAVID, AL HOFFMAN   \nand JERRY LIVINGSTON\n\n\u00a9 1948 Walt Disney Music Company\n\nCopyright Renewed.\n\nAll Rights Reserved. Used by Permission.\n\n# Evermore\n\n### from BEAUTY AND THE BEAST\n\n#### Music by ALAN MENKEN   \nLyrics by TIM RICE\n\n\u00a9 2017 Wonderland Music Company, Inc.\n\nAll Rights Reserved. Used by Permission.\n\n# Friend Like Me\n\n### from ALADDIN\n\n#### Music by ALAN MENKEN   \nLyrics by HOWARD ASHMAN\n\n\u00a9 1992 Wonderland Music Company, Inc. and Walt Disney Music Company\n\nAll Rights Reserved. Used by Permission.\n\n# God Help The Outcasts\n\n### from THE HUNCHBACK OF NOTRE DAME\n\n#### Music by ALAN MENKEN   \nLyrics by STEPHEN SCHWARTZ\n\n\u00a9 1996 Wonderland Music Company, Inc. and Walt Disney Music Company\n\nAll Rights Reserved. Used by Permission.\n\n# Give A Little Whistle\n\n### from PINOCCHIO\n\n#### Words by NED WASHINGTON   \nMusic by LEIGH HARLINE\n\nCopyright \u00a9 1940 by Bourne Co. (ASCAP)\n\nCopyright Renewed\n\nInternational Copyright Secured All Rights Reserved\n\n# Go The Distance\n\n### from HERCULES\n\n#### Music by ALAN MENKEN   \nLyrics by DAVID ZIPPEL\n\n\u00a9 1997 Wonderland Music Company, Inc. and Walt Disney Music Company\n\nAll Rights Reserved. Used by Permission.\n\n# Hakuna Matata\n\n### from THE LION KING\n\n#### Music by ELTON JOHN   \nLyrics by TIM RICE\n\n\u00a9 1994 Wonderland Music Company, Inc.\n\nAll Rights Reserved. Used by Permission.\n\n# He's A Tramp\n\n### from LADY AND THE TRAMP\n\n#### Words and Music by PEGGY LEE   \nand SONNY BURKE\n\n\u00a9 1952 Walt Disney Music Company\n\nCopyright Renewed.\n\nAll Rights Reserved. Used by Permission.\n\n# Heigh-Ho\n\n### from SNOW WHITE AND THE SEVEN DWARFS\n\n#### Words by LARRY MOREY   \nMusic by FRANK CHURCHILL\n\nCopyright \u00a9 1938 by Bourne Co. (ASCAP)\n\nCopyright Renewed\n\nInternational Copyright Secured All Rights Reserved\n\n# How Far I'll Go\n\n### from MOANA\n\n#### Music and Lyrics   \nby LIN-MANUEL MIRANDA\n\n\u00a9 2016 Walt Disney Music Company\n\nAll Rights Reserved. Used by Permission.\n\n# How Does A Moment Last Forever\n\n### from BEAUTY AND THE BEAST\n\n#### Music by ALAN MENKEN   \nLyrics by TIM RICE\n\n\u00a9 2017 Wonderland Music Company, Inc.\n\nAll Rights Reserved. Used by Permission.\n\n# I Just Can't Wait To Be King\n\n### from THE LION KING\n\n#### Music by ELTON JOHN   \nLyrics by TIM RICE\n\n\u00a9 1994 Wonderland Music Company, Inc.\n\nAll Rights Reserved. Used by Permission.\n\n# I See The Light\n\n### from TANGLED\n\n#### Music by ALAN MENKEN  \nLyrics by GLENN SLATER\n\n\u00a9 2010 Wonderland Music Company, Inc. and Walt Disney Music Company\n\nAll Rights Reserved. Used by Permission.\n\n# I've Got No Strings\n\n### from PINOCCHIO\n\n#### Words by NED WASHINGTON   \nMusic by LEIGH HARLINE\n\nCopyright \u00a9 1940 by Bourne Co. (ASCAP)\n\nCopyright Renewed\n\nInternational Copyright Secured All Rights Reserved\n\n# If I Never Knew You\n\n### (End Title)\n\n### from POCAHONTAS\n\n#### Music by ALAN MENKEN  \nLyrics by STEPHEN SCHWARTZ\n\n\u00a9 1995 Wonderland Music Company, Inc. and Walt Disney Music Company\n\nAll Rights Reserved. Used by Permission.\n\n# It's A Small World\n\n### from Disney Parks It's A Small World Attraction\n\n#### Words and Music by RICHARD M. SHERMAN   \nAND ROBERT B. SHERMAN\n\n\u00a9 1963 Wonderland Music Company, Inc.\n\nCopyright Renewed.\n\nAll Rights Reserved. Used by Permission.\n\n# Kiss The Girl\n\n### from THE LITTLE MERMAID\n\n#### Music by ALAN MENKEN  \nLyrics by HOWARD ASHMAN\n\n\u00a9 1988 Wonderland Music Company, Inc. and Walt Disney Music Company\n\nAll Rights Reserved. Used by Permission.\n\n# Let It Go\n\n### from FROZEN\n\n#### Music and Lyrics by KRISTEN ANDERSON-LOPEZ   \nand ROBERT LOPEZ\n\n\u00a9 2013 Wonderland Music Company, Inc.\n\nAll Rights Reserved. Used by Permission.\n\n# Lava\n\n### from LAVA\n\n#### Music and Lyrics by   \nJAMES FORD MURPHY\n\n\u00a9 2015 Walt Disney Music Company and Pixar Talking Pictures\n\nAll Rights Reserved. Used by Permission.\n\n# Mickey Mouse March\n\n### from THE MICKEY MOUSE CLUB\n\n#### Words and Music by   \nJIMMIE DODD\n\n\u00a9 1955 Walt Disney Music Company\n\nCopyright Renewed.\n\nAll Rights Reserved. Used by Permission.\n\n# Part Of Your World\n\n### from THE LITTLE MERMAID\n\n#### Music by ALAN MENKEN   \nLyrics by HOWARD ASHMAN\n\n\u00a9 1988 Wonderland Music Company, Inc. and Walt Disney Music Company\n\nAll Rights Reserved. Used by Permission.\n\n# Some Day My Prince Will Come\n\n### from SNOW WHITE AND THE SEVEN DWARFS\n\n#### Words by LARRY MOREY  \nMusicby FRANK CHURCHILL\n\nCopyright \u00a9 1937 by Bourne Co. (ASCAP)\n\nCopyright Renewed\n\nInternational Copyright Secured All Rights Reserved\n\n# Remember Me\n\n### (Ernesto de la Cruz)\n\n### from COCO\n\n#### Words and Music by KRISTEN ANDERSON-LOPEZ   \nand ROBERT LOPEZ\n\n\u00a9 2017 Wonderland Music Company, Inc. and Pixar Music\n\nAll Rights Reserved. Used by Permission.\n\n# A Spoonful Of Sugar\n\n### from MARY POPPINS\n\n#### Words and Music by RICHARD M. SHERMAN   \nAND ROBERT B. SHERMAN\n\n\u00a9 1963 Wonderland Music Company, Inc.\n\nCopyright Renewed.\n\nAll Rights Reserved. Used by Permission.\n\n# Supercalifragilisticexpialidocious\n\n### from MARY POPPINS\n\n#### Words and Music by RICHARD M. SHERMAN   \nAND ROBERT B. SHERMAN\n\n\u00a9 1963 Wonderland Music Company, Inc.\n\nCopyright Renewed.\n\nAll Rights Reserved. Used by Permission.\n\n# Trashin' The Camp*\n\n### (Pop Version)\n\n### from TARZAN TM\n\n#### Words and Music by   \nPHIL COLLINS\n\n\u00a9 1999 Edgar Rice Burroughs, Inc. and Walt Disney Music Company\n\nAll Rights Reserved. Used by Permission.\n\n*TARZAN (R) Owned by Edgar Rice Burroughs, Inc. and Used by Permission.\n\n\u00a9 Burroughs\/Disney\n\n# When She Loved Me\n\n### from TOY STORY 2\n\n#### Music and Lyrics by   \nRANDY NEWMAN\n\n\u00a9 1999 Walt Disney Music Company and Pixar Talking Pictures\n\nAll Rights Reserved. Used by Permission.\n\n# When You Wish Upon A Star\n\n### from PINOCCHIO\n\n#### Words by NED WASHINGTON   \nMusic by LEIGH HARLINE\n\nCopyright \u00a9 1940 by Bourne Co. (ASCAP)\n\nCopyright Renewed\n\nInternational Copyright Secured All Rights Reserved\n\n# Where You Are\n\n### from MOANA\n\n#### Music by LIN-MANUEL MIRANDA,   \nOPETAIA FOA'I and MARK MANCINA  \nLyrics by LIN-MANUEL MIRANDA\n\n\u00a9 2016 Walt Disney Music Company and Wonderland Music Company, Inc.\n\nAll Rights Reserved. Used by Permission.\n\n# Whistle While You Work\n\n### from SNOW WHITE AND THE SEVEN DWARFS\n\n#### Words by LARRY MOREY   \nMusic by FRANK CHURCHILL\n\nCopyright \u00a9 1937 by Bourne Co. (ASCAP)\n\nCopyright Renewed\n\nInternational Copyright Secured All Rights Reserved\n\n# Who's Afraid Of The Big Bad Wolf?\n\n### from THREE LITTLE PIGS\n\n#### Words and Music by FRANK CHURCHILL   \nAdditional Lyric by ANN RONELL\n\nCopyright \u00a9 1933 by Bourne Co. (ASCAP)\n\nCopyright Renewed\n\nInternational Copyright Secured All Rights Reserved\n\n# A Whole New World\n\n### (Aladdin's Theme)\n\n### from ALADDIN\n\n#### Music by ALAN MENKEN   \nLyrics by TIM RICE\n\n\u00a9 1992 Wonderland Music Company, Inc. and Walt Disney Music Company\n\nAll Rights Reserved. Used by Permission.\n\n# Winnie The Pooh\n\n### from THE MANY ADVENTURES OF WINNIE THE POOH\n\n#### Words and Music by RICHARD M. SHERMAN   \nAND ROBERT B. SHERMAN\n\n\u00a9 1963 Wonderland Music Company, Inc.\n\nCopyright Renewed.\n\nAll Rights Reserved. Used by Permission.\n\n# The World Es Mi Familia\n\n### from COCO\n\n#### Music by GERMAINE FRANCO   \nLyrics by ADRIAN MOLINA\n\n\u00a9 2017 Walt Disney Music Company and Pixar Talking Pictures\n\nAll Rights Reserved. Used by Permission.\n\n# Written In The Stars\n\n### from AIDA\n\n#### Music by ELTON JOHN   \nLyrics by TIM RICE\n\n\u00a9 1998 Wonderland Music Company, Inc., Happenstance Ltd. and Evadon Ltd.\n\nAll Rights Reserved. Used by Permission.\n\n# Yo Ho\n\n### (A Pirate's Life For Me)\n\n### from Disney Parks Pirates Of The Caribbean Attraction\n\n#### Words by XAVIER ATENCIO   \nMusic by GEORGE BRUNS\n\n\u00a9 1967 Walt Disney Music Company\n\nCopyright Renewed.\n\nAll Rights Reserved. Used by Permission.\n\n# You'll Be In My Heart*\n\n### (Pop Version)\n\n### from TARZAN TM\n\n#### Words and Music by   \nPHIL COLLINS\n\n\u00a9 1999 Edgar Rice Burroughs, Inc. and Walt Disney Music Company\n\nAll Rights Reserved. Used by Permission.\n\n*TARZAN (R) Owned by Edgar Rice Burroughs, Inc. and Used by Permission.\n\n\u00a9 Burroughs\/Disney\n\n# You've Got A Friend In Me\n\n### from TOY STORY\n\n#### Music and Lyrics by   \nRANDY NEWMAN\n\n\u00a9 1995 Walt Disney Music Company\n\nAll Rights Reserved. Used by Permission.\n\n# Zip-A-Dee-Doo-Dah\n\n### from SONG OF THE SOUTH\n\n#### Words by RAY GILBERT   \nMusic by ALLIE WRUBEL\n\n\u00a9 1945 Walt Disney Music Company\n\nCopyright Renewed.\n\nAll Rights Reserved. Used by Permission.\n\n# You're Welcome\n\n### from MOANA\n\n#### Music and Lyrics by   \nLIN-MANUEL MIRANDA\n\n\u00a9 2016 Walt Disney Music Company\n\nAll Rights Reserved. Used by Permission.\n\n# Table of Contents\n\n  1. Be Our Guest\n  2. Baby Mine\n  3. Beauty And The Beast\n  4. Bibbidi-Bobbidi-Boo (The Magic Song)\n  5. Can You Feel The Love Tonight\n  6. Circle Of Life\n  7. Chim Chim Cher-ee\n  8. Cruella De Vil\n  9. Colors Of The Wind\n  10. Do You Want To Build A Snowman?\n  11. A Dream Is A Wish Your Heart Makes\n  12. Evermore\n  13. Friend Like Me\n  14. God Help The Outcasts\n  15. Give A Little Whistle\n  16. Go The Distance\n  17. Hakuna Matata\n  18. He's A Tramp\n  19. Heigh-Ho\n  20. How Far I'll Go\n  21. How Does A Moment Last Forever\n  22. I Just Can't Wait To Be King\n  23. I See The Light\n  24. I've Got No Strings\n  25. If I Never Knew You (End Title)\n  26. It's A Small World\n  27. Kiss The Girl\n  28. Let It Go\n  29. Lava\n  30. Mickey Mouse March\n  31. Part Of Your World\n  32. Some Day My Prince Will Come\n  33. Remember Me\n  34. A Spoonful Of Sugar\n  35. Supercalifragilisticexpialidocious\n  36. Trashin' The Camp*\n  37. When She Loved Me\n  38. When You Wish Upon A Star\n  39. Where You Are\n  40. Whistle While You Work\n  41. Who's Afraid Of The Big Bad Wolf?\n  42. A Whole New World\n  43. Winnie The Pooh\n  44. The World Es Mi Familia\n  45. Written In The Stars\n  46. Yo Ho\n  47. You'll Be In My Heart*\n  48. You've Got A Friend In Me\n  49. Zip-A-Dee-Doo-Dah\n  50. You're Welcome\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n## Contents\n\nPrologue\n\nBOOK ONE \"They're not as big as I thought they'd be\"\n\n1\n\n2\n\n3\n\n4\n\n5\n\n6\n\n7\n\n8\n\n9\n\n10\n\n11\n\n12\n\n13\n\n14\n\n15\n\nBOOK TWO \"If only they were back here at Redwall\"\n\n16\n\n17\n\n18\n\n19\n\n20\n\n21\n\n22\n\n23\n\n24\n\n25\n\n26\n\n27\n\n28\n\nBOOK THREE \"We lived one summer too long\"\n\n29\n\n30\n\n31\n\n32\n\n33\n\n34\n\n35\n\n36\n\n37\n\n38\n\n39\n\n40\n\n41\n\n42\n\n43\n\nEpilogue \nThis is a work of fiction. Names, characters, places, and incidents are either 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.\n\n_Loamhedge_\n\nAn _Ace_ Book \/ published by arrangement with the author\n\nAll rights reserved.\n\nCopyright \u00a9 _2003_ by _The Redwall La Dita Co., Ltd._\n\nThis book may not be reproduced in whole or part, by mimeograph or any other means, without permission. Making or distributing electronic copies of this book constitutes copyright infringement and could subject the infringer to criminal and civil liability.\n\nFor information address:\n\nThe Berkley Publishing Group, a division of Penguin Putnam Inc.,\n\n375 Hudson Street, New York, New York 10014.\n\nThe Penguin Putnam Inc. World Wide Web site address is   \n<http:\/\/www.penguinputnam.com>\n\nISBN: _978-1-1012-2024-5_\n\nAN _ACE_ BOOK\u00ae\n\n _Ace_ Books first published by The Ace Publishing Group, a member of Penguin Putnam Inc.,\n\n375 Hudson Street, New York, New York 10014.\n\n _ACE_ and the \" _A_ \" design are trademarks belonging to Penguin Putnam Inc.\n\nElectronic edition: August, 2005\nBY THE SAME AUTHOR\n\n _Redwall  \nMossflower  \nMattimeo  \nMariel of Redwall  \nSalamandastron  \nMartin the Warrior  \nThe Bellmaker  \nOutcast of Redwall  \nPearls of Lutra  \nThe Long Patrol  \nMarlfox  \nThe Legend of Luke  \nLord Brocktree  \nTaggerung  \nTriss_\n\n _Castaways of the Flying Dutchman  \nThe Angel's Command_\n\n _Seven Strange and Ghostly Tales_\n\n _The Great Redwall Feast  \nA Redwall Winter's Tale_\n\n _The Tale of Urso Brunov_\n\n _Redwall Map and Riddler  \nRedwall Friend and Foe  \nBuild Your Own Redwall Abbey  \nTribes of Redwall: Badgers  \nTribes of Redwall: Otters_\n_For my good friend Martha Buckley,  \nwho inspired my Martha.  \nFor Heather Boyd,  \nwho cheered me from her hospital bed to mine.  \nand  \nTo the memory of two brave warriors:  \nNolan Wallace, who became Lonna Bowstripe,  \nand Eric Masato Takashige Boehm,  \nwho fought the good fight._\n\n## Prologue\n\nHave you been travelling, my young friend? Come in out of the darkness and rain. Sit by the fire, eat, drink and rest yourself. Life is one long journey from beginning to end, you know. We all walk different roads, both with our bodies and our minds. Some of us lose heart and fall by the wayside, whilst others go on to realise their dreams and desires.\n\nLet me tell you a story of travellers, and the paths they followed. Of young ones, like yourself, sometimes uncertain of their direction, and often reluctant to listen to the voices of sense and wisdom. Of a mighty warrior, set on a course of destiny and vengeance, unstoppable in his resolve. Of an evil one and his crew, cruel and ruthless, bound on a march of destruction and conquest. Of a simple maid and her friends, homebodies whose only aims were peace and well-being for all. Of wicked, foolish wanderers, chasing fantasies and fables, consumed by their own greed. Of small babes who dreamed small dreams, not knowing what the future held in store for them. And, finally, of two friends, faithful and true, who had roamed many highways and together chose their own way.\n\nThe lives I will tell you of are intertwined by fate\u2014good and evil bringing their just rewards to each, as they merited them. Listen whilst I relate this story. For am I not the Teller of Tales, the Weaver of Dreams!\n\n# [BOOK ONE  \n\"They're not as big  \nas I thought they'd be\"](Loamhedge-toc.html#TOC-2)\n\n##\n\nLashing rain, driven by harsh biting winds from the sea, scoured the land from the bleak salt marshes to the stunted scrub forest. Abruc the sea otter bent against the strain of a loaded rush basket. It was tied to his shoulders and belted across his brow to stop it from spilling backward.\n\nHolding on to his father's paw, young Stugg trotted alongside, plying his parent with interminable questions, which Abruc did his best to answer.\n\n\"H'are you veddy veddy strong?\"\n\nScrunching his eyes against the wind, Abruc could not help smiling at his inquisitive little son. \"I have t'be strong. I've got to feed you, your mamma an' the whole family. That's my job, I'm a father.\"\n\nStugg sucked his free paw, digesting this information whilst he thought up another question. \"Den why can't Stugg sit atop of your basket no more?\"\n\nAbruc adjusted the belt to ease the strain on his neck. \"Because you've growed since last season. Yore gettin' to be a big feller now, a fine lump of an otter. Soon you'll be carryin' yore ole dad an' the basket. Let's put a move on, Stugg, so we can make it into the woods by dark. It'll be good to take a rest out o' this weather.\"\n\nWith the sound of the grey northeast sea pounding in their ears, both the sea otters squelched through the desolate salt marshes toward the weather-bent scrub forest.\n\nDaylight ebbed into early evening as they entered the shelter of the trees. With a grunt of relief, Abruc swung his basket to the ground. It was brimfull of edible seaweed, scallops, mussels and shrimp\u2014a full two days' work, gleaned from the coast of the barren northeast waters. Abruc sat on a fallen pine. Sensing his father's weariness, Stugg climbed up behind him and began gently rubbing his brow.\n\nAbruc relaxed, sighing gratefully. \"Hmmmm, that's nice. I was beginnin' to think that strap'd cut the top off me skull. Huh, where'd I be then?\"\n\nStugg giggled. \"Wiv a half offa head, silly ole farder!\"\n\nThe sea otter cautioned his son. \"Hush now, not so loud. There might be Coast Raiders about. Huh, they'd cut the tops off'n our skulls, just to watch us die.\"\n\nWide-eyed, Stugg crouched down against his father, speaking in a hushed whisper. \"Mamma says Coaster Raiders be's naughty vermints!\"\n\nHis father pushed dry pine needles into a small heap, shaking his head grimly. \"Naughty ain't the word for that scum. They're evil, cold-blooded murderers. Cruelty is just fun to the likes o' them. Right, young 'un, I suppose yore hungry now?\"\n\nNodding eagerly, Stugg whispered, \"I'm starfished!\"\n\nAbruc chuckled. _Starfished_ was a word all the young ones used, a cross twixt _starving_ and _famished._\n\nHe patted Stugg's head fondly. \"Nothin' worse'n a starfished otter. You stay here, keep yore eyes'n'ears open, an' lay low. I'll go an' find us a snug berth for the night.\"\n\nHe pulled a sack from under his cloak, tossing it to his son. \"Sort through the rest of those rations an' see wot you want for supper. I'll be back soon.\"\n\nAbruc knew the woods well, he recalled a spot not too far off. It was a good dry place, sheltered by a rock ledge. Silent as a night breeze, he weaved his way through the dark, twisted trees, straight to the exact location. He had camped there before. Halting slightly short of his destination, he paused. Something did not feel quite right about the area. Abruc sniffed the air and listened carefully, his animal instinct aroused. He caught the faint sound of ragged breathing. Drawing his long dagger, he crept forward, peering keenly into the shadows, his neck hairs bristling.\n\nFor supper Stugg had selected two flat loaves, some of his mamma's apple and blackberry preserve and their last flask of plum cordial. If his father lit a fire, they could make toasted preserve sandwiches and warm cordial. The young otter was a pretty fair cook, often having helped his mamma to prepare meals. There was not much else to do but wait in silence for his father's return. Stugg set out the food and sat next to the basket of supplies.\n\nAbruc came speeding out of the darkness to his son's side. Crouching beside Stugg, he gripped his paws tightly. The sea otter's voice was urgent and breathless from running.\n\n\"Listen carefully, little mate. Could you find yore way back home to our holt on yore own?\"\n\nStugg was taken aback by the unusual request. \"Er, I fink so, what's a matter, farder?\"\n\nAbruc gripped his son's paws tighter. His voice sounded harsh. \"Answer me\u2014yes or no! Could you find yore way back home?\"\n\nStugg had never seen his father like this. He nodded, his own voice sounding small and scared. \"Yes, Stugg know d'way!\"\n\nAbruc released the young otter's paws. \"Good, now here's wot y'must do, son. Find Shoredog. Tell him to bring the crew to the spot by the rock ledge, he'll know where I mean. Say that they best bring rope, canvas an' poles. Enough t'make a stretcher to carry a wounded, giant stripedog. That's if'n he's still alive when they reach here.\"\n\nWords poured from Stugg's mouth like running water. \"A giant, a stripedog, a wounded one? I never see'd a giant stripedog afore! What happened? Will he get deaded . . .\"\n\nAbruc grabbed Stugg and shook him, something he had never done before. He hissed at him through clenched teeth. \"Shut yore mouth, son! Don't stand here askin' questions! Go now, run, don't stop for anythin'. The life of another creature depends on you. Go!\"\n\nYoung Stugg took off like a madbeast, pine needles scattering from under his paws as he tore homeward through the nighttime forest. Abruc watched until his son was out of sight, then gathered up their belongings and dashed back to the camp beneath the ledge.\n\nSwiftly he heaped dry pine needles and cones with a few twigs. Using the steel of his knife blade against a chunk of flint, he soon had a small fire burning. It was sheltered by the overhanging rock and could not be seen from a reasonable distance. Abruc viewed the scene around him. Two badgers, one very old, the other about two seasons into his adult growth, lay stretched out, side by side. Small and grizzled, the oldest of the pair was obviously dead, slain by various weapon thrusts. As he turned to the younger badger, a brief glance at the churned-up ground and the blood-flecked rock confirmed the sea otter's suspicions. His jaw clenched angrily. \"Dirty murderin' Raiders!\"\n\nThe younger badger was still alive. Abruc had seen one or two badgers in his lifetime, but not as big as this fellow. He was truly a giant\u2014tall, deep of chest and broadbacked with massive paws and powerfully muscled limbs.\n\nThe sea otter winced as he inspected the fearsome wound to the badger's head. A long jagged slash, from eartip to neck, had ripped across the badger's face. Narrowly missing the eye, it had ploughed across the brow, through the wide-striped muzzle, across the jaw line to the side of the creature's throat.\n\nAbruc, with only a limited knowledge of healing, staunched the blood with his cloak. Lifting the badger's head, he cradled it in his lap, dabbing away at the dreadful rift and murmuring to the unconscious beast.\n\n\"Seasons o' salt, matey, 'tis a miracle yore still alive! Y'must have a skull made o' rock. I know you can't hear me, but don't worry, big feller, our crew will do the best we can for ye. There's one or two good healers at our holt.\"\n\nAbruc sat rambling away to the senseless badger, knowing he could do little else until help arrived.\n\nIt was close to midnight. Rainladen wind hissed through the scrub forest, carrying with it salt spray from the thundering seas. Beside the guttering embers of his little fire, Abruc had dozed off, still holding the badger's head.\n\nAt the front of the otter crew, Shoredog pointed with his lantern, hurrying forward. \"There they are, mates!\"\n\nLittle Stugg reached his father first. \"I bringed them, farder!\"\n\nAbruc patted the youngster's paw. \"Yore a good ole scout. Unnh, somebeast get me out from under this giant's head. Me limbs have gone asleep on me from holdin' his weight.\"\n\nWilling paws assisted him upright. Shoredog shook his head as he viewed the injured badger. \"Great seasons, lookit the mess the pore creature's in. I fears there ain't much hope for 'im. I never set eyes on a wound bad as that 'un!\"\n\nStugg caught sight of his mother and tugged at her paw. \"Issa giant stripedog goin' to die, mamma?\"\n\nAbruc's wife Marinu nodded at Shoredog's grandma, Sork. \"Not if'n we can help it, Stugg. Come on, crew, get some warm blankets around that badger an' strap him to a stretcher. Easy now, don't jolt the pore beast too much.\"\n\nEverybeast knew that Marinu and Sork were the best healers in all the southeast.\n\nStugg grinned broadly. Now that he had succeeded in his mission, he proceeded to take charge of the situation, striding about and issuing orders. \"You all hear my mamma, pick dat stripedog up careful!\"\n\nMarinu was about to pull her son to one side when Abruc murmured to her, \"Let the young 'un be, he did well tonight.\"\n\nAs the otter crew manoeuvred the huge badger onto the huge stretcher, Shoredog gave a surprised bark. \"Blood'n'thunder, lookit that!\"\n\nBeneath the injured creature a mighty bow and a quiver of long arrows lay half covered in the loose sand and pine needles. The badger had fallen backward upon the bow, his hefty bulk breaking the weapon in two pieces. One jagged half was stuck into his hip. Marinu halted the bearers until she and Sork had extracted the splintered yew wood. The big fellow grunted faintly as they padded and dressed the wound.\n\nStugg jumped up and down triumphantly. \"He be's alive, d'stripedog maked noise!\"\n\nOld Sork looped the birchbark quiver over Stugg's head. It scraped the ground, the arrows were taller than he. Sork shooed the young one aside. \"Aye, mayhap he is. Now you carry those an' stay out the way.\"\n\nA score of otters bore the badger off on a litter of pine poles, sailcloth and rope, padded with dead grass and soft moss. Stugg stayed behind with his father and Shoredog to bury the dead badger. It was only a shallow grave, but they found slabs of rock to top it off with. Abruc wedged the two pieces of broken bow, with the string still joining them, into the foot of the grave. They would serve as a marker. All three sea otters gazed down at the sad resting place.\n\nAbruc shook his head. \"Pore old beast, we don't even know wot name he went by. He looked weak, an' small. A badger that age should've spent out his seasons restin' in the sun. I wonder wot kin he was t'the big 'un. Mebbe his father?\"\n\nStugg pressed his face against Abruc and wept. He could not imagine anybeast losing a father. He sobbed brokenly. \"Who would kill someone's farder like that?\"\n\nShoredog looked up from smoothing the earth around the stones. \"Only beast I knows who kills like that is Raga Bol.\"\n\nThe name struck fear into Abruc. \"Raga Bol! Has he been here?\"\n\nShoredog stood upright, dusting off his paws. \"While you an' Stugg were gone, Rurff the grey seal visited our holt. He saw the Searats' ship wrecked on the rocks, further north up the coast. Raga Bol an' about fifty vermin crew came ashore. They headed down this way, but pickin's are scarce on this northeast coast, so they've probably marched inland. They ain't got a ship anymore. I was just rousin' our crew to search for you an' Stugg, when the young 'un comes runnin' to tell me you need help.\"\n\nShoredog took one of the straps on Abruc's basket. \"Let me help ye with this, mate, 'tis a good haul.\"\n\nThey set off back to their holt, with Stugg stumbling over the quiver of long arrows.\n\nAbruc shrugged philosophically. \"It's a bad spring, cold an' stormy. Let's hope summer's a bit better when it comes. At least we won't have Raga Bol an' his villains to worry about. I suppose we should count ourselves lucky, really.\"\n\nYoung Stugg hitched the arrows higher on his back. They still dragged along the ground as he muttered aloud. \"More luckier than d'poor stripedogs, I appose.\"\n\nA brief smile crossed Shoredog's weathered face. \"That young 'un of yores is growin' up quick, mate!\"\n\nDawn glimmered chill and blustery over the heathlands some two leagues west of the northeast sea. Wet, hungry and dispirited, Raga Bol's crew of Searats huddled round a smoking fire down a ravine. They stared miserably at a deep, rain-swollen stream running nearby. From further up the bank the vermin could hear their captain's shrieks and curses rending the air.\n\nRinj, a sly-faced female, gnawed at a filthy clawnail, glancing from one to the other. \"Ye t'ink Bol's lost the paw? I t'ought Wirga cudda sewed it back on, she's a good 'ealer.\"\n\nA lanky, gaunt rat named Ferron picked something from his teeth and spat it into the stream. \"Sewed it back on! Have ye gone soft in the skull? Last I saw, Cap'n Bol's paw was 'angin' on by a string o' skin. We should've stayed well clear o' those two stripedogs!\"\n\nRinj wiped firesmoke from her blearing eyes. \"The little ole one wuz no trouble, he didn't know wot 'it 'im, gone afore ye could wink.\"\n\nFerron winced as Raga Bol's screeches and curses redoubled. \"Aye, but wot about the big 'un, eh? I thought Cap'n Bol killed him wid the first blow of his big sword!\"\n\nGlimbo, the captain's first mate, pushed Rinj away from the fire and installed his fat, greasy bulk close to the flames. One of his eyes was a milky sightless orb; the other roved around the crew as he warmed his paws.\n\n\"Never in me days seen Bol 'ave to strike a beast twice wid that blade. But that big stripedog came back after the first whack an' got his teeth in good. Just as well that Bol struck again, or he would've lost more'n one paw. Mark my words, stripedogs are powerful dangerous beasts!\"\n\nThe heathland was a barren region, made drearier by the day's unabated rain. Down in the ravine a huge bonfire blazed to dispel the harsh weather. Every Searat of the crew sat watching their captain. Tall and sinewy, with a restless energy that could be glimpsed in his fiery green eyes, Raga Bol was an impressive rat by any measure. He sat wrapped in a fur cloak, his left pawstump hidden from view. The Searat's right paw rested on the carved bone hilt of a heavy, wide-bladed scimitar, protruding from his waistband. The crewbeasts could feel Raga Bol's eyes on them. Rain sizzling on the fire and wind fanning the flames were the only sounds to be heard as they waited on their captain's word.\n\nFinally, Raga Bol rose and snarled bad-temperedly at them, firelight reflecting from his hooped brass earrings and gold-plated fangs. \"We march west at dawn. Anybeast who don' want to go, let 'im speak now, an' I'll bury 'im right 'ere!\"\n\nNot one of the Searat crew said a word. Raga Bol nodded. \"West it is then. Blowfly, get me two runners.\"\n\nAn enormously fat rat, with a whip curled about his shoulders, motioned two lean crew members forward.\n\nRaga Bol sat looking at them in silence, until they squirmed under his unwinking gaze. His jaw clenched as he moved the stump where his left paw had been. \"You two, go back to where I slew the stripedog. Find the carcasse, an' bring me back his head.\"\n\nEach of the runners touched a paw to his ears. \"Aye aye, Cap'n!\"\n\nRaga Bol stood watching them climbing the sides of the ravine, then turned his attention to an old female Searat crouching nearby. \"Wirga, is that hook ready yet?\"\n\n\"It'll be ready by dawn, Cap'n.\" The old one gave him a toothless grin. \"So thee wants the big stripedog's head, eh?\"\n\nRaga Bol drew his cloak tight and sat, staring into the fire. \"Nobeast ever took a paw o'mine an' stayed in one piece, dead or alive. Now get that hook ready if'n ye want to keep yore head, ye withered old torturer!\"\n\n##\n\nFar over to the west, a brighter spring day had dawned. Ascending meadowlarks heralded the sun beneath a soft, pastel blue sky. Drawn by the sudden warmth, mist rose from the greenswards, transforming dewdrops to small opalescent pearls amid the dainty blossoms of saxifrage, buttercup, capsella and anemones. Mossflower woodland trees were blessed with a crown of fresh green leaves. Life was renewed to the sounds of little birds, calling to their parents with ceaseless demands for food.\n\nToran Widegirth loosed his apron strings, satisfied that he had completed his duties as Redwall Abbey's Head Cook. Leaving his kitchens, the fat otter sought the beautiful spring morn outdoors. Heaving a sigh of relief, Toran sat down on an upturned wheelbarrow at the orchard entrance. He was joined by his friend Carrul, the Father Abbot of Redwall. The mouse sat down beside the otter, both relaxing in silence, blinking in the sunlight and savouring the first good weather that spring.\n\nCarrul glanced sideways at his companion. \"Not going with Skipper and otter crew this season?\"\n\nToran watched an ant negotiating its way over his footpaw. \"Much too early, it ain't summer yet. But you know Skipper an' the crew, first sign o' sun an' a skylark an' they're off like march hares to the west seashores for the season.\"\n\nAbbot Carrul chuckled. \"Fully provisioned I trust?\"\n\nToran nodded wearily. \"Aye, I saw them off myself at dawn. Pushin' a cartload o' victuals I'd made special for 'em. Singin' their rudders off, dancin' like madbeasts!\"\n\nCarrul's smile widened. \"I know, they woke me up, I saw them from my window. Good luck and fair weather to them. So why didn't you go? I gave you permission to take as long as you wanted to go on leave.\"\n\nToran shrugged. \"Oh, I'm gettin' too old for that sort o' thing. Leave it to the younger ones.\"\n\nCarrul snorted. \"Too old? Too big in the tummy, you mean! If you're too old, then what about me, eh? I was your teacher when you were only a tiny Dibbun at Abbey School!\"\n\nThe ottercook tweaked his friend's bony paw. \"Aye, an' ye haven't gained a hair's weight since then. How d'ye do it, you skinny, ancient mouse?\"\n\nThe Abbot looked over his small square spectacles good-humouredly. \"I don't spend my whole life down in those kitchens like you do, my friend. Oh, Toran, isn't it just a glorious day? I hope the summer is a really golden one.\"\n\nToran snuggled more comfortably into the wheelbarrow. \"Makes ye feel good t'be alive, don't it, Carrul?\"\n\nThey both lapsed into silence again, gazing around and taking in the beauties of their Abbey.\n\nBehind them, Redwall reared\u2014a legend in pink, dusty sandstone with its high walls and turrets, stained-glass windows and buttressed arches, belltower, attics and steeple, all complemented by a background of verdant woodland and cloudless blue sky. Toran took in the stout battlements and picturesque gatehouse of the outer wall, whilst the Abbot contented himself by viewing the lawns and orchards, peacefully shimmering in the sunlight.\n\nCarrul's gaze took in the Abbey pond, down near the south ramparts. \"What creature could not count himself lucky to be dwelling in such a paradise. Ah, look Toran, there's our young friend Martha, taking a little nap in her chair, just by the rhododendron bushes on the far side of the pond.\"\n\nToran saw the young haremaid, her head nodding down to a heavy volume, which lay open on her lap.\n\nThe ottercook eased himself from the barrow. \"I'll just take a stroll over there and check she's alright.\"\n\nThe Abbot stretched luxuriously into the position Toran had vacated. \"Dearie me, you're like an old mother hen with that young 'un. Why don't you tell her that lunch will be served late, out in the orchard? In fact, tell everybeast, 'twill cheer them up after being kept indoors by the rain for so long. We'll all lend a paw to help.\"\n\nToran smiled happily. \"What a good idea!\"\n\nThe ottercook approached Martha carefully, not wanting to disturb her. She was very special to him. Toran could recall the winter's day, twelve seasons ago, when Martha Braebuck had arrived at Redwall. She had been nought but a tiny babe, strapped to the back of her ancient grandmother. Her brother Hortwill, two seasons older, had stumbled along, clutching the old hare's cloak. Toran's heart had immediately gone out to the pitiful trio. They had walked from the far Northlands, the only survivors of a vermin attack which had wiped out an entire colony of mountain hares. No sooner were they through the Abbey gates than the poor grandmother had collapsed and died from exhaustion. A sad occurrence, made sadder by the fact that Martha had never learned to walk from that day forth. Her brother grew up as sprightly as any young hare, but despite the most tender care, the babe Martha was immobile from her knee joints to her footpaws. There were no signs of any apparent wound or injury, no scarring or broken bones. No reason, in fact, why the little one should not learn to walk. Some of the wiser heads, like old Phredd the Gatekeeper, Great Father Abbot Carrul and Sister Setiva, the healer shrew who took care of the Abbey infirmary, said it was due to shock. That perhaps Martha's long trek from the Northlands, strapped to her grandmother's back, coupled with witnessing the murder of her family and kin, had caused the problem. Still, the Redwallers were completely puzzled.\n\nToran did everything possible to help her. He believed firmly that one day she would stand and walk. Meanwhile, the kind ottercook provided Martha with the means to get about. Taking a light comfortable chair, he fixed it to the base of a kitchen trolley, adding two large wheels to the back. The young haremaid learned to propel herself about quite easily. Toran also fashioned a crutch for her, but Martha used it only to get at things which were beyond her reach.\n\nMartha Braebuck grew up an extremely bright young creature with a thirst for knowledge. She was a formidable reader and scholar, the equal even of the venerated mouse, Sister Portula, Redwall's Abbey Recorder. Martha could solve riddles and equations, write poems, ballads and even sing. According to popular opinion, she had the sweetest singing voice ever heard within the Abbey walls. She never complained about being chairbound, and was invariably cheerful and willing to help others. The maid was a welcome and useful member of the Redwall Abbey community.\n\nToran watched silently as her head drooped lower. The volume slid from her lap rug onto the grass. Toran grunted as he bent to retrieve it.\n\nMartha came awake, stifling a yawn and rubbing her eyes. \"Dearie me, I must have nodded off!\"\n\nReturning the hefty volume to Martha's lap, the ottercook winked at her. \"Who'd blame ye, with all this sun about. I could lie down right here an' take a nap myself!\"\n\nMartha saw a group of Dibbuns approaching from around the orchard hedge. \"You wouldn't sleep for long, my friend. Look, here comes trouble!\"\n\nThe Abbeybabes descended upon the haremaid's chair. Muggum, a tiny mole who was their ringleader, climbed up onto Martha's lap, rumbling away in his quaint molespeech. \"Yur, Miz Marth', do ee singen us'n's ee song?\"\n\nThe haremaid eyed him good-naturedly. \"Which one would you like me to sing?\"\n\nToran interrupted with his suggestion. \"A pretty day deserves a pretty song, miss. Sing a spring song!\"\n\nThe squirrelbabe Shilly added her request. \"Da one where uz clappa paws!\"\n\nBuffle the shrewbabe, who was the smallest of all, nodded solemnly. \"Gurbbadurrguddun!\"\n\nShilly translated. \"Him says that be a good 'un.\"\n\nMartha sat up straight, exchanging a smile with Toran. \"Well, Buffle's word is good enough for me. Here goes.\"\n\nThe Dibbuns raised their paws, ready to clap, as Martha's melodious voice soared out.\n\n\"The rain has gone away . . . Clap Clap!\n\nand larks do sing on high.\n\nSweet flowers open wide . . . Clap Clap!\n\ntheir petals to the sky!\n\n'Tis spring . . . Clap clap! 'Tis spring,\n\nlet us rejoice and sing,\n\nthe moon is queen the sun is king,\n\nso clap your paws and sing . . . Clap Clap!\n\nThere's not a cloud in sight . . . Clap Clap!\n\nthe leaves are bright and new.\n\nThis day was made for all . . . Clap Clap!\n\nfor me my friend and you!\n\nSo sing . . . Clap Clap! . . . So sing,\n\nlet summer follow spring,\n\nfrom golden morn to evening,\n\nwe clap our paws and sing . . . Clap Clap!\n\n. . . Clap Clap!\"\n\nAlthough the clapping missed its beat once or twice, it was with joyous vigour. The little ones danced around, whooping and squeaking wildly, \"Sing us'n's a more!\"\n\nMartha was coaxed into singing the lively air again. She finished quite out of breath, amid yells for a third performance.\n\nToran took charge, slapping his rudder loudly on the bankside. \"Hold up there, ye rogues, pore Miz Martha's tuckered out. Now lissen t'me. If ye promise t'be good, we'll have lunch out in the orchard today, seein' as 'tis sunny!\"\n\nHis suggestion was greeted with roars of approval. \"Lunch inna h'orchard, 'ooppee!\"\n\nMartha smiled happily. \"Oh, what a splendid idea!\"\n\nLittle Shilly sped off toward the Abbey, calling to the other Dibbuns. \"Come on, we 'elp Granmum Gurvel wiv lunch!\"\n\nToran watched them go. \"I don't think old Gurvel will thank me for lettin' that lot invade the kitchens.\"\n\nMartha settled the big volume more comfortably on her lap. \"Bless their little hearts, they mean well.\"\n\nToran cast a glance at the haremaid's book. \"That's a heavy ole thing t'be readin', miss. Wot's it all about?\"\n\nMartha opened the book at a page marked by a silken ribbon. \"I borrowed it from Sister Portula's library. It's a rare and ancient account of Loamhedge mice.\"\n\nThe ottercook looked thoughtful. \"Loamhedge mice, eh? I've heard of them. Weren't they the ones who helped t'build our Abbey? Aye, they were led by old Abbess, er, wotsername?\"\n\n\"Germaine.\" The haremaid corrected him. \"It was she and Martin the Warrior who helped to build and design Redwall. Germaine and her followers once lived at the place they called Loamhedge. It was a peaceful and prosperous community, almost as large as our Abbey, some say. But they were forced to abandon it and flee for their lives. Loamhedge was left deserted to the four winds.\"\n\nToran's interest was roused. Although he was no great reader himself, he liked to hear his friend tell of what she had read. \"Why did they have to leave? Does the book explain?\"\n\nMartha riffled back to a previously read page. \"It says here that a great sickness fell upon Loamhedge. A plague, brought by vermin, possibly Searats. First there was sickness, then a few deaths. Abbess Germaine was wise enough to realise that it would grow into an epidemic, which would wipe them all out. So she took her mice and fled. They went wandering for many seasons, far from home. One day their journey took them into this part of Mossflower territory. It was here they met Martin the Warrior and his friends. Germaine joined forces with the Woodlanders, helping to rid the lands of powerful enemies. When peace was achieved, Martin and Germaine were free to realise their dream. They built a mighty stone fortress, an Abbey, where goodbeasts could live in safety and happiness together. That's how, countless ages ago, Redwall came into being. . . .\"\n\nMartha was interrupted by her brother Hortwill. He came bounding and splashing through the shallows and threw himself upon her.\n\n\"Wot ho, wot ho, wot ho, me pretty young skin'n'blister!\"\n\nShe ducked her head, laughing as he showered her face with kisses. \"Stop that this instant, Horty! I'm not your skin'n'blister, I'm your sister. Oh, look now, you've splashed water all over Sister Portula's precious book!\"\n\nHortwill Braebuck, or Horty, as everybeast knew him, was Martha's brother, older than her by two seasons. An overpowering character\u2014ebullient, quaint of speech, always in trouble, he was roguishly gallant, sentimental to a fault, and possessed a gluttonous appetite. In short, a typical hare.\n\nThrowing up both paws and ears in mock horror, Horty declaimed, \"Well, flog me twice round the jolly old orchard an' chop off me ears with a rusty blinkin' axe, wot! Splashed a bally spot o'water on Sis Peculiar's blessed book? Lack a day, fifty seasons in the cellar for me. What say you, Toran old scout? Either that or instant death. Wot wot?\"\n\nToran played along with Horty's dramatic mood. Squinting an eye, he growled fiercely, \"Instant death's the only thing!\"\n\nThe young hare threw him a smart military salute. \"As y'say, sah, sentence t'be carried out on the blinkin' spot!\"\n\nWithout further ado, Horty flung himself into the pond and vanished underwater, still saluting.\n\nMartha sat bolt upright in her chair. \"Oh the fool, save him Toran, quickly!\"\n\nLumbering into the pond, the ottercook fished Horty out with one huge paw.\n\nGrinning like a madbeast, and still saluting, Horty spouted a mouthful of water into the air. \"Beloved blinkin' friend, you've saved me life. I'll never forget you, an' I'll always dine at your excellent kitchen!\"\n\nKeeping a straight face, Toran looked at Martha. \"I'd better chuck him back in, miss, think of the food we'd save!\"\n\nMartha nearly fell out of her chair with laughter. \"Hahahaha! Oh no, please sir, hahaha! I beg you, spare his gluttonous young life. Hahahahaha!\"\n\nShooting a last jet of pondwater skyward, Horty said fondly, \"A chap's confounded lucky to have such a merciful sister, wot!\"\n\nToran growled as he frog-marched Horty ashore. \"Ye certainly are, matey. But if'n I hears ye callin' Sister Portula, Sis Peculiar again, back in the pond ye'll go. Aye, an' those two ripscuttle pals o' yores, Springald an' Fenna. A lesson in manners wouldn't harm them, either!\"\n\nMartha dabbed the book pages dry with her lap rug. She could never be angry with her boisterous brother. Horty had always been close by, ready to cheer her up when she was sad or depressed. Her inability to run free like other young ones sometimes put Martha in low spirits.\n\nShe held up the volume for Toran to see. \"No harm done really, it's perfectly dry now. Come on you two, let's go back to the Abbey!\"\n\nOn the way up, they met Muggum and several other Dibbuns who had been banished from the kitchens by Granmum Gurvel, the old assistant molecook.\n\nMuggum tugged his snout respectfully to the haremaid. \"Yur, mizzy, oi'll push ee to ee h'orchard furr lunch.\"\n\nReaching down, Martha lifted the molebabe onto her lap. \"That's very thoughtful of you, Muggum, but I'm sure Horty and Toran can manage the job quite well.\"\n\nPatting the young haremaid's paw, the molebabe nodded sagely. \"Oi thankee, Miz Marth'. Coom on, zurrs, you'm pushen us faster'n'that, us'n's bee gurtly 'ungered furr lunch!\"\n\n##\n\nRedwall orchard was a riot of blossoming fruit trees and bushes. Pink and white flowers clustered thick on every branch, their petals carpeting the grass. Apple, pear, cherry, beech, hazelnut and almond trees flourished in rows, fronted by raspberry, strawberry, red currant and whortleberry. Summer promised an abundant yield.\n\nToran cast an eye over three trolleys laden with buffet lunch\u2014spring vegetable soup, brown bread and cheese, dandelion and burdock cordial, followed by a dessert of damson preserve pie. \"Who did all this?\"\n\nAbbot Carrul bowed apologetically, knowing how touchy the ottercook could be about trespassers in his kitchen domain. \"I offered to help Granmum Gurvel. You looked so hot and weary when I met you in the orchard for a breath of fresh air. Gurvel and I decided to help you out. Is it to your liking, my friend?\"\n\nToran bowed thankfully to them both. \"My thanks to ye. I couldn't have done it better!\"\n\nRedwallers sat in the tree shade, laughing and chatting amiably as lunch was served. Sister Portula spread a rug, and Toran lifted Martha onto it. All four sat beneath a wide chestnut tree at the orchard's far end. Sunlight and shadow dappled them as they watched the inhabitants of Redwall enjoying lunch. Martha appreciated such moments because the elders always included her in their discussions. The young haremaid felt she had become an honorary member of the Elders Council.\n\nMartha laughed at the antics of the Dibbuns, who were beginning to get a bit rowdy. \"They do get excited after a rainy spring indoors. Look at baby Yooch, he's eating flower petals!\"\n\nSister Portula shook her head. \"There's Shilly and some others doing it. I'll wager 'twas Muggum who started it all. Muggum, Shilly and Yooch are more trouble than any ten Dibbuns. I call them the Terrible Trio!\"\n\nToran's stomach shook as he chuckled. \"Yore right, marm. Hi there, Springald, go an' tell those little 'uns to stop eatin' the petals, or Sister Setiva will have to dose 'em with physicks.\"\n\nThe mousemaid Springald shrugged carelessly. \"Flowers won't do 'em any harm. I used to eat petals myself.\"\n\nAbbot Carrul glanced sternly over his glasses at her. \"Do as you are bidden, miss, and don't argue!\"\n\nSpringald curtsied slightly, then flounced off to do as she was told.\n\nSister Portula pursed her lips and tutted. \"Yonder goes more trouble. She's one of the other three. Horty, Springald and Fenna, the young rebels. They aren't babes anymore, they should know better.\"\n\nMartha put aside her cordial beaker. \"Oh, they'll grow out of it, Sister, they're all good creatures at heart, I'm sure.\"\n\nPortula helped herself to bread and cheese. \"Huh, let's hope they do, before there's really trouble. I'm sure we were never like that at their age, were we, Father?\"\n\nAbbot Carrul raised his eyebrows. \"Weren't we, Sister? I can recall two young ones sailing a dining room table on the pond. Aye, with an embroidered linen tablecloth for a sail. Hmm, let me see now, what were their names?\"\n\nSister Portula fidgeted uncomfortably with her sleeve hem. \"But that was only a bit of fun. You and I were well behaved as a rule.\"\n\nMartha could scarcely believe her ears. \"You two? Well, you rascals! Did you get caught, Father?\"\n\nBehind his small glasses, the Abbot's eyes twinkled. \"Oh, we were caught sure enough, and both set to work in the kitchens as punishment. Remember that, Sister?\"\n\nPortula nodded ruefully. \"How could I ever forget five days of scrubbing greasy pots and scouring pans? My little paws stayed wrinkled for half a season!\"\n\nMartha winked cheekily at the Recorder. \"Horty and his two friends seem innocent compared to you and Abbot Carrul. What a pair of rogues you were!\"\n\nA light smile hovered on Portula's kind face. \"Listen, missy, if you think we were naughty, you should have seen two Dibbuns who were younger than us at the time. Bragoon and Saro, an otter and a squirrel. Now those two really were a twin pestilence!\"\n\nMartha turned to Toran. \"I've heard you telling the young ones tales about Bragoon and Saro, but I always thought they were make-believe creatures. Were they actually real?\"\n\nThe ottercook nodded vigorously. \"Oho, missy, that they were! Bragoon was my big brother, five seasons older'n me. Sarobando, or Saro, as everybeast knew her, was a Dibbun squirrel, his best little pal. Sister Portula's right, ye never saw two villains like 'em! Hah, 'twas just as well they ran off whilst they was still young 'uns. If'n Bragoon an' Saro had stayed, we mightn't have a roof over our heads. They would've demolished the Abbey between 'em!\"\n\nWhilst Toran had been talking, some of the Dibbuns and a few of the young 'uns had gathered around.\n\nMuggum scrambled up onto Toran's lap. \"Yurr zurr, you'm tell us'n's ee story 'bowt Zuro an' Burgoon!\"\n\nToran chuckled. \"I can't bring one to mind right now, but I can recite a poem I wrote about 'em for the Harvest Feast many seasons back.\"\n\nTaking a swig of cordial, he tried to recall the words.\n\nShilly waggled her tail impatiently. \"Well, 'urry up an gerron wiv it, Cooky!\"\n\nThe ottercook twitched his nose at her. \"Silence, ye liddle rip!\"\n\nDraining his beaker, Toran launched into the recitation.\n\n\"I'll tell ye a tale of two Dibbuns,\n\nwho lived here long ago,\n\nan otter who was named Bragoon,\n\nan' a squirrel known as Saro.\n\nAye, little Bragoon an' Saro,\n\nwhat a pair o' scamps they were,\n\ntheir names rang through the land oh,\n\nthere was nought they didn't dare!\n\nGood Granmum Gurvel molecook,\n\nmade puddens, cakes an' pies,\n\nthey vanished off the kitchen shelf,\n\nbefore her dear ole eyes.\n\n'Bragoon an' Saro, I'll be bound,'\n\nthe poor ole beast would say,\n\n'they'll eat me out of house an' home,\n\nthey'll turn my fur to grey!'\n\nBragoon an' Saro, gracious me,\n\nI dread to hear those names,\n\ncome hearken whilst I tell ye,\n\nof those two scoundrels' games.\n\nWho filled the Abbot's bed with ants,\n\nwho nailed up all the doors,\n\nwho was it glued the bellrope,\n\nand stuck the ringer's paws,\n\nwho filled the pond with beetroots,\n\nand turned the waters red,\n\nwho baked poor Foremole's sandals,\n\ninside a loaf of bread?\n\nThe dreaded Bragoon an' Saro,\n\nI'm here to tell ye all,\n\nthere's never been two like 'em,\n\nat the Abbey of Redwall!\"\n\nThe Dibbuns jumped up and down in delight, roaring with laughter at the escapades of the infamous pair. Horty and his friends, Springald and Fenna, laughed, too.\n\nToran put on a stern face, wagging a cautionary paw at his listeners. \"I tell ye, 'twasn't so funny for the poor creatures who were the butt o' those tricks!\"\n\nHorty scoffed. \"Oh I say, sah, you don't actually believe all that dreadful twaddle about Bragoon an' Saro, wot?\"\n\nAbbot Carrul answered him. \"Toran's right, 'tis all true. I was a young 'un here myself at the time, I saw it!\"\n\nFenna fluttered her long eyelashes prettily. \"Oh really, Father Abbot, you don't expect us to believe all that about Bragoon and Saro. We're not Dibbuns anymore. Toran makes up the stories to amuse the little ones\u2014they'll believe anything, but we know better.\"\n\nMartha spoke out sharply. \"If the Abbot and Toran say it is true, then I'm certain it is. What reason would we have to doubt them?\"\n\nHer words, however, went unheeded by the three young 'uns, as they strolled off together, still unwilling to credit the existence of the fabled duo.\n\nHorty scoffed again. \"Bragoon an' Saro, wot? Load of jolly old codswallop, if y'ask me. Tchah!\"\n\nSpringald giggled. \"If I swallowed that lot, I'd be looking out for fishes nesting in trees and flying!\"\n\nMartha was so angry that she almost rose from the rug, but then she fell back again.\n\nAbbot Carrul helped her to sit up. \"Don't upset yourself, Martha. One day our young friends will wake up and find themselves somewhat older and a little wiser, just wait and see. I was a bit like them at that age, but one lives and learns.\"\n\nThe young haremaid sighed. \"I hope it happens to my brother soon. I don't like to say this, Father, but Horty seems to behave more outrageously each day.\"\n\nToran helped Martha into her chair. \"Don't ye worry. Horty's a hare, they're always a bit wild when they're young.\"\n\nMartha retrieved her volume and straightened her rug. \"Perhaps you haven't noticed, Toran, but I'm a hare, too!\"\n\nSister Portula dusted a stray flower petal from Martha's head. \"Ah, but you're a very rare and special kind of hare, my dear. Anybeast can see that!\"\n\nHostile weather still reigned on the plains and heathlands of the far east. Raga Bol and his Searats had not made much headway in three days of trekking westward\u2014the Searat captain's pawstump pained abominably. They camped on high ground, in the lee of a rocky projection. Apart from a few chosen cronies, the crew avoided the captain, making their own fire sufficiently far away to evade his sudden wrath.\n\nRaga Bol sat by his own fire, with Glimbo and Blowfly in attendance. The two runners had been sent out to retrieve the badger's head but had returned empty-pawed. They crouched at the far side of the blaze, panting from their long journey. Raga Bol watched reflecting flames glinting from the polished silver hook where his paw had once been. His luminous eyes shifted to the runners.\n\n\"Are ye certain 'twas the spot where I slew the giant stripedog?\"\n\nBoth heads nodded. \"Certain shore, Cap'n!\"\n\n\"I'd swear me oath on it, Cap'n Bol. The stripedog was gone, there was no sign of 'im anywhere's about!\"\n\nThe Searat captain's terrifying stare never left either of the two quivering vermin. \"But the old one, he was buried there?\"\n\n\"Aye, Cap'n, right on the spot where ye slew the big 'un.\"\n\n\"He's right, Cap'n, the very spot. All the tracks were wiped out, too. Wasn't nothin' we could do but come back 'ere, fast as we could, to tell ye!\"\n\nRaga Bol dropped his gaze to the steaming ground at the fire's edge. \"Speak to none about this, or yore both deadrats. Now get out o' my sight!\"\n\nGlimbo and Blowfly scuttled off, relieved to be still among the living, after having brought their murderous captain such bad news. Hunching against the bleak cold at his back, Raga Bol sat silent. His eyes roved between the silver hook and the roaring, wind-driven fire.\n\nBlowfly whispered to Glimbo, \"I reckon dat giant stripedog must still be alive, mate!\"\n\nThe fat Searat's hushed whisper was barely audible, but Raga Bol heard it. He stood slowly and faced them both. With lightning swiftness his hook shot out, latching on to Blowfly's broad belt. The Searat was dragged forward to find himself facing Bol's upraised blade and threatening snarl.\n\n\"Did ye ever see a beast alive after I'd struck 'im wid me blade? Well, did ye?\"\n\nBlowfly watched the heavy scimitar poised, one stroke away from his quivering double chins. The rat's voice went squeaky with panic. \"N . . . no, Cap'n!\"\n\nRaga Bol bared his gold-plated teeth in a wolfish grin. \"Shall I prove it to ye, Blowfly?\"\n\nThe rat sobbed brokenly. \"Aw, don't do it, Cap'n Bol, please. Nobeast ever lived after yew 'it 'em wid yore sword!\"\n\nThe captain's pale eyes lighted on Glimbo. \"You should know, mate, tell 'im!\"\n\nGlimbo loved life too much to remain silent. Words poured from his mouth like running water. \"Dat stripedog's kinbeasts must've carried 'im off, fer a fancy buryin'. I bet they buried the old 'un where he fell, 'cos they couldn't haul two carcasses. Mark me words, Blowfly, it don't matter 'ow big the stripedog was, he's deader'n any doornail now. Once Cap'n Bol's sword swipes 'em, they're well slayed. I'd take me affydavy on it!\"\n\nBlowfly fell to the ground as the hook pulled loose from his belt. Bol ground the scimitar and leaned on it.\n\n\"There's yore answer, mate, the stripedog's dead. I don't want to 'ear no more talk of such beasts from my crew. Now set four guards around me, so I can sleep.\"\n\nThe sentries crouched miserably in the darkness, waiting for the dawn. Wrapped in his cloak, Raga Bol lay alongside a roaring fire. But sleep did not come easily, and, when it did, his dreams were troubled by visions of the giant stripedog coming slowly but surely after him with the light of vengeance burning in his eyes.\n\nAbruc the sea otter, his wife Marinu and their son Stugg sat on the streamside, beneath an overhanging bank canopy. They enjoyed their evening meal outside, away from the bustling noise of the holt. Stugg sucked noisily at the contents of his bowl.\n\nAbruc patted his stomach and winked at the young creature. \"Now that's wot I calls a sea otter chowder. Nobeast can make it like yore mamma does, ain't that right, me 'eart?\"\n\nMarinu refilled her husband's bowl. \"I wager you used to say that about yore own mamma's chowder. All it takes is clams, mussels an' shrimps, with some beans, chestnut flour, seaweed, carrots an' a few pawfuls of sea salt an' hotroot pepper. 'Tis simple to cook up.\"\n\nYoung Stugg held out his bowl for a refill. \"But you make it da best, 'cos yore our mamma!\"\n\nMarinu dipped her ladle into the pot they had brought out. \"You'll soon be as big a flatterer as yore dad! Wipe that chin, you've got chowder all over it.\"\n\nAbruc looked over the rim of his bowl at Marinu. \"So, how are you an' old Sork gettin' along with our big badger? D'ye reckon he'll live?\"\n\nMarinu wiped Stugg's chin with her apron hem as she spoke. \"It looks like he will, though whether or not he'll waken fully we don't know. He might just fade away, after one of those death sleeps that last a few seasons. I never thought anybeast could be so deeply wounded an' live. Sork used fish glue to mend his skull bone. When that was all clean and set, I used long hairs from his own back as thread to stitch the skin back over. We set lots of spider web over it all. Give it a few days, then we'll wash it gently with valerian and sanicle to deaden any pain. Shoredog says he'll have to be moved to the old cave where it'll be quieter. We'll make him a big bed of silver sand and moss.\"\n\nAbruc nodded. \"That should help. I'll keep a warm fire of pine an' sweet herbs burnin' there, night an' day.\"\n\nMarinu rose. \"I'm going back inside. Sork wants to borrow some of the broth off'n my chowder to feed him. A hard task with such a big beast who's still senseless.\"\n\nWhen she had gone inside, Abruc and Stugg finished off the remaining food. The young otter sat watching his father attach a slim line, from the end of his rudder, to a thick root growing from the bankside. Abruc took a chunk of beeswax and began rubbing it into several more loose lines of tough flaxen fibre.\n\nThe sea otter eyed his young son. \"Shouldn't you be off to yore bed, 'tis getting' late.\"\n\nStugg rubbed some of the beeswax on his paw curiously. \"Wot are you doin' wiv dat stuff, farder?\"\n\nAbruc explained as he worked. \"I'm makin' a bowstring, a good stout one that won't rot or break under strain.\"\n\nYoung Stugg pursued his enquiries. \"Wotta you be wantin' a bowstring for, farder?\"\n\nAbruc answered patiently. \"T'aint for me, it's for our big badger. I've got a feelin' he'll be well again some day. When the time comes, he'll be leavin' us to go westward.\"\n\nStugg persisted. \"Is a bowstring good to go westward wiv?\"\n\nHis father began deftly plying the waxed fibres together. \"Aye, son, that big feller's an archer. He'll have t'find 'imself the right wood t'make a new bow, but the least I can do is to plait him a proper bowstring. Then he'll be well armed to settle up with the vermin who tried to slay him an' murdered his ole friend.\"\n\nStugg nodded. \"I bet they be sorry then!\"\n\nAbruc stopped working momentarily. \"Sorry ain't the word, young 'un. When a badger goes after his enemies, there ain't noplace they can run or hide from him. I'll wager our big beast will come down on 'em with the Bloodwrath!\"\n\nUnfamiliar with this strange word, Stugg posed a new question. \"Wot's a Bloodraff, farder?\"\n\nAbruc shook his head decisively. \"Bloodwrath is terrible, somethin' you don't ever want t'see or know about. Go on now, off to bed with ye, me son!\"\n\n##\n\nOld Father Phredd was the Redwall Abbey Gatekeeper. He had once been Abbot, but his seasons caught up with him. Passing the position over to Carrul, he retired to the gatehouse. Phredd was ancient, probably the oldest hedgehog in all Mossflower, and enjoyed being very old, and rather eccentric as well. Although the Old Gatekeeper sought the privacy of his beloved gatehouse and slept a lot, when he was up and about, he could be rather sprightly. His skinny form, with drooping silver spikes, often caused a smile around the Abbey and its grounds. Phredd spoke to stones, trees, plants and flowers, carrying on long conversations and debating with the most everyday objects.\n\nHe had arrived late for lunch, shunning the main crowd that was now gathered in the orchard. Preparing his own plate in the deserted kitchens, Phredd first chose a scone. He prattled on to it as he made his way around the tables.\n\n\"Hee hee, you're a fine fresh fellow. Now what'll I have to go with you, eh, eh? Speak up!\"\n\nPlacing an ear close to the scone, he cackled. \"Teeheehee! Of course, some honey, a piece o' cheese and a beaker of soup\u2014not too hot, just right for swigging, eh?\"\n\nGranmum Gurvel, the old molecook, came in from the orchard to draw off more cordial. She spied Phredd and watched him chatting away to the food until he caught sight of her.\n\nPhredd waved his scone at her. \"Oh, er, young Gurvel, g'day!\"\n\nShe chuckled. \"Hurr hurr, goo day to ee, zurr. Wot bee's ee soup sayin' to ee, sumthin' noice oi 'opes?\"\n\nPhredd sipped at the beaker and smacked his lips. \"Oh yes, indeed, miss. 'Tis saying that you cooked it very nicely. Oh, it also asked if there was any pie about, eh?\"\n\nGurvel went to her larder and took out a large pie. It was preserved plum and apple, the golden crust liberally dusted with maple frosting.\n\nShe cut a generous slice and gave it to him. \"Thurr naow, old 'edgepig, doant ee let nobeast see that. Oi baked it speshul furr supper.\"\n\nPhredd nodded his thanks and skittered off out of the kitchens, conversing with the pie slice. \"My my, you're a handsome fellow! What a splendid dessert you'll make. Come on, let's find a nice quiet corner, eh?\"\n\nGranmum Gurvel shook her head at Phredd's antics. She picked up the remainder of the pie. \"Coom on, pie, back in ee larder again!\"\n\nThe realisation of what she was doing caused the old molecook to smile. \"Gurr, lack ee day, that Phredd got oi a talkin' to moi own pies naow, gurt seasons!\"\n\nMartha had finished her lunch. She, too, sought peace and quiet to continue her reading. Leaving her friends, she wheeled the chair indoors. Crossing Great Hall, she went straight to her favourite place. Harlequin hues of sunlight shafted down through the high, stained-glass windows onto the worn stone floor. Between two towering sandstone columns, a lantern glowed beneath a wondrous woven tapestry with a sword suspended to one side of it. The haremaid halted her chair in full view of the scene, golden motes of sundust floating slowly on the serene air.\n\nMartha paused before opening Sister Portula's heavy book. She gazed up at the central figure in the tapestry, Martin the Warrior. A heroic, armour-clad mouse, the hero and champion of Redwall Abbey. Martha loved looking at his face\u2014so strong and protective yet kindly, with a secret smile forever hiding in his eyes. The sword he was leaning on was the very same one that hung on the wall\u2014a legendary warrior's weapon, its only adornment, one red pommel stone set on the hilt. Martin's swordblade had been forged at Salamandastron, the badgers' mountain fortress on the west seashore. It had been made from a star fragment that had fallen from the skies.\n\nNo matter what position Martha took up when she visited the tapestry, Martin's eyes always seemed to be watching her. The haremaid could feel his presence so strongly that she often spoke to him. Keeping her voice low in the echoing hall, she nodded toward the warrior mouse.\n\n\"The rains stopped today. You can see by the sunlight in here that it's a beautiful spring day outside. I've come to do a bit of reading in peace. You should hear those Dibbuns singing in the orchard\u2014they're so happy! Did you ever do much reading, Martin?\"\n\n\"Hee hee, I don't suppose he did, a warrior like him, eh?\" Phredd emerged from the shadows, where he had installed himself behind a column to enjoy his lunch.\n\nMartha was slightly surprised at the old hedgehog's appearance. \"Oh I'm sorry, sir, I didn't know you were here.\"\n\nPhredd picked pie crumbs from his cheek spikes. \"No need to be sorry, pretty miss, you carry on talking to your friend. I've had many a long chat with him, eh!\"\n\nThe haremaid continued looking at the tapestry. \"He looks so understanding, like a friend anybeast could talk to. Do you think he can hear us?\"\n\nPhredd patted her shoulder lightly. \"Of course he can. I'm sorry for intruding. You carry on, miss. I'll just pop off to my gatehouse for an afternoon nap. Good day to you.\"\n\nHe shuffled off, though Martha heard him reprimanding a corner bench. \"You mind your own business an' don't be eavesdropping now, eh, eh!\"\n\nMartha opened the book but was only able to concentrate on it for a short while before her eyelids began to flicker and then droop. The peacefulness of her surroundings, combined with the warm sunlight pouring down from the windows, had woven its own spell. There, in the silence of Great Hall, the small figure in the chair slept in a pool of tranquillity. Floating through the corridors of her mind came two mice\u2014one, a maid of her own age clad in a gown of green; the other, Martin the Warrior.\n\nHis voice was as reassuring as soft breezes through a meadow. \"I never did read much, Martha. It is good to read, all learning is knowledge. Read on, young one. Learn of Sister Amyl and the mice of Loamhedge.\"\n\nThe haremaid could hear her own voice replying, \"Learn what? Who is Sister Amyl?\"\n\nThe young mousemaid standing beside the warrior pointed to Martha and spoke, every word burning itself into Martha's mind.\n\n\"Where once I dwelt in Loamhedge,\n\nmy secret lies hid from view,\n\na tale of how I learned to walk,\n\nwhen once I was as you.\n\nThough you cannot go there,\n\nlook out for two who may,\n\ntravellers from out of the past,\n\nreturning home someday.\"\n\nBoth Martin and Sister Amyl raised a paw in farewell. The dream faded like wisping smoke as Martha slept on.\n\nAround midnoon Martha was awakened rudely, her chair jolted as three pair of paws latched on to it. Horty, Springald and Fenna ran her speedily across Great Hall, whirling perilously around the huge stone columns.\n\nMartha gripped the chair tightly. \"Whoo! Slow down, please. Where are we going?\"\n\nHorty jumped up beside her, shouting, \"Out to enjoy the jolly old fresh air, my beautiful skin'n'blister, you'll go mouldy sittin' indoors, wot! I say, you chaps, can't you make this thing go faster? Yaaaah!\"\n\nThe chair struck a table edge and upturned. Springald and Fenna leapt aside, but Martha and Horty were shot out. Luckily, Martha landed on top of her brother, clutching Sister Portula's volume to her. The chair skidded on a short distance, then lay still, one of its wheels still turning slowly.\n\nHorty looked up into his sister's face. \"Dreadfully sorry about that, old gel, just a bit of fun, wot. I say, are you hurt?\"\n\nMartha glared down from where she was sitting on him. \"Lucky for you I'm not. Is my chair damaged?\"\n\nSpringald and Fenna set the chair upright and examined it.\n\n\"No, not a mark on it, Martha!\"\n\n\"Haha, old Toran knew what he was doing when he built this thing. Stay there, we'll lift you back in!\"\n\nIn frosty silence, Martha allowed them to lift her back into the chair. The trio fussed about, folding the rug neatly about her lap and laying the volume on it.\n\nFenna smiled sweetly. \"There, no real harm done, Martha. We were only trying to cheer you up, didn't mean to throw you like that.\"\n\nHastily Springald backed her up. \"Yes, we were going to take you for a quick spin around the walltop. Lovely view from there on a day like this.\"\n\nHorty waggled his ears in agreement. \"Right you are, m'dear. There's still time for a toddle round the battlements, though we'll go slower this time. Word of honour, wot!\"\n\nMartha shook her head firmly. \"Oh no, you three wildbeasts aren't taking me anywhere. Now go away! Please, leave me alone, I'm quite happy here!\"\n\nHorty scuffed his footpaw guiltily along the floorstones. \"I say, y'won't tell anybeast about what happened, will you?\"\n\nMartha tapped her chair arm pensively. \"Any beast like who?\"\n\nHorty fidgeted with his belt tab. \"Er, like Toran, or Abbot Carrul or blinkin' old Sis Peculiar.\"\n\nMartha reminded him of the Infirmary Keeper. \"Or Sister Setiva?\"\n\nFenna's eyes went wide. \"Oh please, don't tell her!\"\n\nThe other two miscreants joined in with their pleas.\n\n\"She'll make us scrub the infirmary out and stitch sheets!\"\n\n\"Aye, an' physick the blinkin' life out of us. Oh come on, charmin', beautiful Sis, say y'won't snitch to that monster!\"\n\nThey looked so sorry for themselves that Martha relented. \"Alright, I won't say anything\u2014provided you go away immediately and leave me in peace.\"\n\nWithout a word the trio began to scramble away and were almost at the door when Martha suddenly recalled her dream.\n\n\"Wait, come back here, there's something I need you to do!\"\n\nHorty dashed back so hastily that he almost tripped and fell onto his sister's lap. \"Anything, dear old skin'n'blister, we're yours to flippin' well command!\"\n\nMartha issued her modest requests, but she spoke firmly. \"Fenna, I want you to go and seek out Abbot Carrul. Horty, you go and find Sister Portula, and mind how you address her. The message for both of them is this: Ask politely that if neither is too busy, would they please come to the gatehouse. There is an important matter I would like to discuss with them. Springald, push my chair to the gatehouse\u2014at a reasonable pace, please.\"\n\nBrother Phredd poked his head around the gatehouse doorway, blinking and yawning. \"Ah yes, young wotsername, come in please, and your friend, too. Always nice to have afternoon visitors, eh!\"\n\nAs Springald pushed Martha over the threshold, the haremaid heard the mousemaid muttering. \"Huh, I'm not stopping in some dusty old gatehouse on an afternoon like this!\"\n\nMartha fixed her with an icy smile. \"Oh, you don't have to stay, you run off to the kitchens now. Have a word with Gurvel or Toran\u2014tell them I'd like afternoon tea for four.\"\n\nSpringald looked puzzled. \"Afternoon tea for four?\"\n\nMartha wheeled round to face her. \"Yes, afternoon tea, you know, scones and slices of cake, and a large pot of mint tea with honey. Hop along now, bring them straight back here, and don't spill the tea. Off you go, miss!\"\n\nTo ensure Martha's silence, Springald had no option but to obey. With a sweep of her skirt she flounced off.\n\nOld Phredd addressed the chair he was about to sit on. \"Afternoon tea, how does that sound to you, quite nice, eh?\"\n\nIn due course, Abbot Carrul and Sister Portula arrived. Both knew that Martha was a sensible creature and would not summon them on some foolish errand. Brother Phredd had just seated them both, when another knock came on the door. He scratched his drooping spikes and muttered. \"More visitors, quite an eventful afternoon, eh?\"\n\nSpringald pushed the laden trolley in. She curtsied impudently at the Abbot. \"Afternoon tea for four, Father!\"\n\nMartha forestalled any further smartness by nodding graciously at the mousemaid. \"Thank you, miss, you may go now!\"\n\nSister Portula watched the back of Springald's head shaking with rage as she exited the gatehouse and slammed the door. \"Gracious me, you certainly put that young mouse in her place!\"\n\nMartha smiled demurely. \"Yes, Sister, but she does need it now and again, doesn't she?\"\n\nAbbot Carrul took the haremaid's paw. \"What was it you wanted to see us about, Martha?\"\n\nOver afternoon tea, Martha explained to her friends how she had fallen asleep. She told them of Martin's visitation, and of the young mouse who had accompanied him, ending with the short poem, which she recalled precisely.\n\n\"Where once I dwelt in Loamhedge,\n\nmy secret lies hid from view,\n\nthe tale of how I learned to walk,\n\nwhen once I was as you.\n\nThough you cannot go there,\n\nlook out for two who may,\n\ntravellers from out of the past,\n\nreturning home someday.\"\n\nAbbot Carrul sat forward in his armchair. \"Strange. What do you think, Sister?\"\n\nPortula put aside her tea. \"Not many Redwallers are honoured by a visit from Martin the Warrior. We must heed all he says. His spirit is not just the essence of valour and honour, he is also the voice of knowledge and wisdom. Now, what is your own opinion of this incident, Martha?\"\n\nThe haremaid tapped the cover of the book. \"This is the history of Loamhedge that you loaned me, Sister. I think the answer lies inside it. That's why I called you here. I am still young, but you three have the knowledge of seasons on your side. I was hoping that you could help me. I never dreamed that there might be an answer to why I can't walk. Do you think there is?\"\n\nOld Phredd picked up the big tome and laid it on the table. He spoke to it, as it if were a living thing. \"Well now, you dusty old relic, are you going to assist us with this little one's problem, eh, eh?\"\n\nHe turned and gave Martha a toothless grin. \"Heeheehee, I think he will. Though one can never really tell what a book says until one reads it, eh?\"\n\nAbbot Carrul opened the book. \"This may take some time, but we're on your side, Martha. If there is a way to make you walk, rest assured, we'll find it.\"\n\nMartha could feel tears beginning to brim in her eyes. She blinked them away swiftly. \"Thank you all, my good friends. But there is something that I don't think the book can tell us. Who are the ones we must look out for? The two travellers from out of the past, returning home someday?\"\n\nSister Portula gazed out the window into the sunlit noon. \"You're right, Martha. I wonder who they could be.\"\n\n##\n\nNorth of Redwall, spring eventide filtered soft light through the leafy canopy of Mossflower Wood. Amid aisles of oak, beech, elm, sycamore and other forest giants, slender rowan, birch and willow stood like young attendants, waiting on their stately lords. Blue smoke drifted lazily upward through the foliage which fringed a shallow stream. Somewhere nearby, a pair of nightingales warbled harmoniously.\n\nThe tremulous beauty was lost upon a small vermin band who had trekked down from the far Northlands. They had camped on the bank to fish. A fat, brutish weasel called Burrad was their leader. Beneath his ragged cloak he carried a cutlass, its bone handle notched with the lives he had taken. Burrad's sly eyes watched his band closely. They were spitting four shiny scaled roach on green willow withes to grill over the fire.\n\nDrawing the cutlass, Burrad pointed it at the biggest fish. \"Dat'n der is mine, yew cook it good fer me, Flinky!\"\n\nThe stoat called Flinky let out a pitifully indignant whine. \"Arr 'ey, Chief, I caught dis wun meself, 'tis me own fish!\"\n\nDespite his bulk, Burrad was quick. Bulling the stoat over, he whipped Flinky mercilessly with the flat of his blade.\n\nCovering his head, the victim screeched for mercy. \"Yaaaaaargh, stop 'im mates, afore he kills me pore ould body! Yeeegh, spare me, yer mightiness, spare me. Aaaaagh!\"\n\nCruel by nature, Burrad thrashed Flinky even harder. Throwing himself upon the hapless stoat, he pressed the blade against Flinky's scrawny neck, snarling viciously.\n\n\"Wot d'yer want, the fist or yore 'ead? 'Urry up an' speak.\"\n\nThe cutlass blade pressed savagely down. Flinky wailed. \"Yeeeeh, take de fish, I've only got one 'ead. Take de fish!\"\n\nBurrad rose, grinning wolfishly as he kicked Flinky's bottom. \"Cook dat fish good, or yore a dead 'un!\"\n\nHe turned on the other eleven vermin gang members. \"Wot are youse lot gawpin' at, eh? Gimme some grog!\"\n\nA female stoat called Crinktail, whose tail was shaped almost like a letter _Z,_ passed Burrad the jug of nettle grog. Snatching it roughly, the bully sat down, taking long gulps of the fiery liquid.\n\nHe watched Flinky like a hawk. \"Crispy outside an' soft inside, dat's de way I likes fish.\"\n\nThe others averted their eyes; there was no doubt about who the leader of their gang was.\n\nCrouched low in the reeds on the far bank, two creatures viewed the scene. One was an otter, the other a squirrel, both in their late middle seasons.\n\nThe otter squinched his eyes, letting them rove over the gang. \"Hmm, about twelve o' them over there, I'd say.\"\n\nThe squirrel nibbled on a young reed. \"There's thirteen.\"\n\nHer companion shrugged. \"I won't argue with ye, 'cos my eyes ain't as good as they used t'be. I tell ye though, mate, that's one sorry gang o' vermin. Looks as if they got rocks in their skulls instead o' brains.\"\n\nThe squirrel chuckled. \"Aye, campin' there without a single sentry posted, an' a fire smokin' away like a beacon. 'Tis a wonder their mothers let 'em out alone.\"\n\nThe otter nodded. \"See ole lardbelly yonder, the big weasel? Leave him t'me, I enjoy takin' bullies down a peg.\"\n\nThe squirrel commented drily, \"Watch he don't fall on ye, he'd flatten ye like a pancake. Are those fish ready yet?\"\n\nHer companion sniffed the air. \"I'd say so. Right then, are we ready t'go an' pay 'em a visit?\"\n\nThe squirrel sighed. \"Aye, layin' here won't get us any supper. You go in the front, an' I'll make me way around back.\"\n\nThe lean, aging otter grumbled. \"It's always me wot has t'go in the front. Why can't I go in the back?\"\n\nThe squirrel cut left along the streambank, replying, \" 'Cos I'm the best tree climber. Give me time t'get ready, mate, don't walk in too early. Good luck!\"\n\nTucking his rudder into the back of his belt, the otter draped his ragged cloak to conceal it. He bound a faded red bandanna low on his brow, disguising both ears and scrunching down over his eyes to make them look shortsighted.\n\nPicking up a polished hardwood staff, he splashed into the stream shallows, muttering to himself. \"Huh, I'm gettin' too old for this game!\"\n\nLittle Redd was the youngest of the vermin gang. Small and runty, he was often the butt of their coarse jokes.\n\nSeeking about for firewood, Redd glanced sideways. He saw the bedraggled creature wading across the stream, and called to Burrad. \"Aye aye, Chief, looks like we got company!\"\n\nBurrad took his mouth from the grog jug. He cast a contemptuous glance at the hunched figure struggling toward the bankside. \"Wot'n de name o'bludd is dat?\"\n\nThe otter sloshed ashore, calling in a quavery voice. \"A good evenin' to one an' all. Seems I'm just in time for supper. Mmm . . . roasted roach, me favourite vittles!\"\n\nBurrad's cutlass was drawn and wavering a whisker's breadth from the unwanted visitor's nose. \"Who are ye? Huhuhuh, or should I say, wot are ye?\"\n\nThe stranger avoided the blade neatly. Ducking under it, he stood at the vermin leader's side, wrinkling his nose comically. \"Wot am I, young feller? I'm a ferroat, o' course!\"\n\nFlinky looked up from the cooking fire. \"A ferroat? Ah' shure, an' wot sort o' beast is dat now?\"\n\nThe intruder replied airily. \"Oh, just a cross twixt a ferret an' a stoat. I was a small sickly babe, or so me ole mum'n'dad told me. That's why I look like this.\"\n\nIgnoring his fish-cooking task, Flinky continued. \"An' who, pray, was yore muther an' father?\"\n\nThe stranger replied, straight-faced. \"A rat an' a fox, I s'pose, but they was terrible liars.\"\n\nFlinky scratched his head. \"Liars? Huh, I'll say they was!\"\n\nBurrad interrupted by thwacking Flinky between both ears with the flat of his blade. \"Who asked yew, puddle'ead? Ger-ron wid cookin' dose fishes!\"\n\nHe turned to the odd-looking creature. \"Wot's yore name, ferroat, an' wot d'ye want 'ere?\"\n\nThe newcomer pointed to himself. \"Just told ye, haven't I? Me name's Ferroat, an' I'll sing an' dance fer me supper. That's if ye'll allow me, kind sir.\"\n\nThe vermin gang winked and sniggered among themselves. Burrad, a kind sir? This old fool was begging to die.\n\nTesting his cutlass blade by licking the edge, Burrad leaned close to his intended victim and grinned. \"Allow ye, eh? If'n yore dancin' an' singin' ain't to me likin', I'll allow this blade to chop ye into ten pieces. Then I'll allow me gang to roast ye over that fire. If ye don't taste nice, we kin always use ye fer fishbait!\"\n\nSmiling affably, the odd beast bowed creakily. \" 'Tis a fair offer, sir, I thankee kindly.\"\n\nShuffling about in a curious jig, the creature twirled his staff and began singing.\n\n\"I'll always recall wot Ma said to me,\n\nere I went a rovin' a minstrel to be,\n\nbeware of the vermin, they ain't got no class,\n\nan' they ain't got the brains Mother Nature gave grass!\n\nRowledy dowlety toodle um day.\n\nI soon found out me dear mother was right,\n\nI met up with some vermin the followin' night,\n\nthey were strangers to bathin', an' that made me think,\n\nwhy didn't Ma tell me that all vermin stink?\n\nRowledy pong and a toodledy pooh!\"\n\nThe comic-looking old ragbag of a beast jigged and shuffled around. Raucous laughter greeted his performance followed by tears of merriment that coursed down the vermin's cheeks. It was only at the start of the third verse, when vermin's faces were compared to toads' bottoms, that Burrad realised the singer was insulting him and his gang.\n\nRoaring with rage, the fat weasel rushed the disguised otter. Whirling his cutlass, Burrad aimed a mighty swipe that should have left the singer headless. However, far from being slain, the odd creature ducked under the blow, came up under Burrad and tweaked his snout.\n\nPurple with spleen, the gang leader grappled with his opponent, yelling to his second in command. \"Skrodd, gut this old fleabag wid yer spear, I've got 'im!\"\n\nThe tall, evil-looking fox dashed forward, plunging with his spear. But the otter was fast and more clever than both vermin. He butted Burrad under the chin, wriggled from his grasp and scuttled to one side in the blink of an eye.\n\nBurrad stood gaping at the spear protruding from his stomach. He raised his clouding eyes to the open-mouthed fox, faltering. \"Ye've killed me, yer blather-brained foo . . . !\"\n\nBurrad crashed over backward, slain by his own gang member. Amid the drama, nobeast noticed the four fish vanish up into the willow foliage, hauled on a thin twine by the green withes they were spitted upon.\n\nSkrodd's surprise was only momentary. His brain was already reacting to the fact that he was now the vermin gang's new leader. Leaving the spear stuck in his former chief, the tall fox grabbed the cutlass from Burrad's limp grasp. He came at the otter with a blurring barrage of swift slashes.\n\n _Whizzzzzthonk!_ A slingstone from the trees suddenly rendered him senseless. Skrodd's fellow vermin looked on in horror as his body collapsed in a heap. Before the gang could move, the squirrel dropped from her perch. Danger glinted in her eyes as she twirled a loaded sling expertly.\n\n\"There's twoscore more of us layin' in the bushes, just waitin' on the word!\"\n\nShedding his disguise, the otter knocked daggers and other weapons from the vermin's paws, with sharp raps of his polished staff. He looked nothing like the ragged, dancing fool he had been a moment ago. His voice was stern and commanding.\n\n\"Everybeast stand still, right where ye are! Believe Saro, we've got a full crew ready to pounce on ye!\"\n\nHalfchop, a rat who was minus a paw, gulped. \"If'n that un's called Saro, yew must be Bragoon?\"\n\nFlinky look at the pair in astonishment. \"I've heard of ye, Bragoon an' Saro. Two mighty warriors!\"\n\nBragoon leaned on his staff and nodded. \"That's us, an' there's forty more trained fighters like us, just waitin' to get a crack at you lot. So have the brains to stay alive an' listen to wot we say.\"\n\nFlinky bowed politely. \"Anythin', yer honour, sure we're in no position to be arguin' wid ye.\"\n\nSaro pointed at a wobbly-nosed ferret called Plumnose. \"You, where have ye come from? Speak!\"\n\nGesturing back over his shoulder, Plumnose replied, \"Durr, we cummed from der Nort'lands.\"\n\nSaro nodded. \"The Northlands, eh? Then listen carefully to my friend Bragoon.\"\n\nThe otter let his fierce eyes wander round the hapless vermin as he ground out an ultimatum. \"Get yoreselves back to the Northlands, 'cos if yore anywhere south of here by nightfall, yore all deadbeasts! We're goin' now, but our mates'll stay hidden, watchin' ye. Sit still here until 'tis properly dark, then break camp an' get back to where ye came from\u2014sharpish! We'll be passin' this way again tomorrow. Make sure yore not still here. Is that clear?\"\n\nFlinky's head bobbed up and down like a yo-yo. \"Ah, sure, 'tis certain clear, yer mightiness. We've all got the message, an' a fine important one it is, sir!\"\n\nBragoon and Saro backed out of the camp. A moment later they were lost in the surrounding trees. The vermin sat wordlessly staring at one another until Plumnose broke the silence.\n\n\"Wodd duh we do now?\"\n\nFlinky's mate, Crinktail, was in no doubt. \"Like they said, we wait 'til it's dark, then we gets out of here. I don't know about youse, but I'm goin'.\"\n\nFlinky agreed. \"Aye, ye don't disobey two like Bragoon an' Saro. Best do the sensible thing, mates.\"\n\nRecovering from the slingstone blow, Skrodd sat up groaning. \"Unnnh, wot hit me?\"\n\nSlipback, a weasel with most of his back fur missing, toyed with the cutlass that had belonged to Burrad.\n\n\"Ye were knocked cold by a slingstone, mate.\"\n\nSkrodd felt the lump on his skull and winced. \"Who did it?\"\n\nFlinky chuckled. \" 'Twas none other than a famous squirrel called Saro. Yore lucky she did, 'cos the one you was goin' after wid yore blade was 'er partner, Bragoon.\"\n\nSkrodd stood slowly and walked across to Slipback. Suddenly he dealt the weasel a swift kick to the chin. As Slipback fell, the tall fox grabbed Burrad's cutlass.\n\n\"Keep yer paws off dat blade, 'tis mine now. I slew Burrad, an' I'm the new chief round 'ere!\"\n\nSlipback avoided a second kick. \"Only by accident\u2014dat don't make yew chief!\"\n\nSkrodd turned to face the rest of the gang, wielding his new weapon. \"Accident or not, Burrad's dead. Does anybeast want to challenge me? Come on!\"\n\nNone came forward. They knew the tall fox's reputation as a fighter; even Burrad had never kicked him about.\n\nSkrodd smiled grimly. \"Right, up on yore hunkers, we're goin' to track those two down!\"\n\nLittle Redd exclaimed, \"Didn't ye hear Flinky? Those two are dangerous warriors, Bragoon an' Saro.\"\n\nSkrodd turned on the little fox. \"Ye mean that ole ragbag who was jiggin' about an' tryin' to sing for his supper? Wot did the other one look like, Flinky?\"\n\nThe stoat shrugged. \"Small an' oldish, why d'ye ask?\"\n\nSkrodd curled his lip scornfully. \"A pair o' little ole tattered ragamuffins, an' ye lot believed they was Bragoon an' Saro. Real famous warriors are big an' tough. Any two beasts could say they was Bragoon an' Saro. Those two were nothin' but a pair of ole impostors. Now come on, let's get after 'em. Nobeast knocks me down wid a slingstone an' lives t'brag about it. I'll gut the two of 'em!\"\n\nA hefty-looking rat called Dargle remained seated. \"They said we was to sit 'ere 'til it was dark, then head back t'the Northlands. The otter said there was twoscore fighters layin' nearby, an' that we'd be dead meat if'n we didn't do like we was told.\"\n\nSkrodd shook his head in disbelief. \"An' ye believed 'im? That's the oldest trick in the book. Watch, I'll show ye twoscore o' fighters!\"\n\nFuriously grabbing anything that came to paw\u2014firewood, pebbles and soil\u2014the tall fox flung them at the surrounding trees, yelling out defiantly. \"Now then, ye mighty fighters, come out an' show yerselves. I'll fight ye all at once, or one by one if'n ye ain't frightened o' me! Get out 'ere, ye mangy frogbait!\"\n\nSilence greeted the challenge. Skrodd spat contemptuously into the fire, glaring at the vermin gang. \"Wot a bunch of addlebrains! Up on yore paws an' get movin' ye bunch o' ditherin' oafs. After I've slain those two ole relics, we'll get the rest o' this job done. Move!\"\n\nAs they moved southward into the woodlands, Little Redd discussed the situation with Flinky. \"Skrodd ain't takin' us to that Abbey place that Burrad was always goin' on about, is he?\"\n\nFlinky nodded. \"Ah sure, it looks like he wants t'be the big bold beast who gets the magic sword. Huh, magic sword! I wonder where ould Burrad heard that tale?\"\n\nJuppa, the weasel who was Slipback's mate, joined the conversation. \"Burrad said his father told 'im about it, just afore he died. Said there was an Abbey, a big place called Redwall. Accordin' to 'im, there's only a few peaceful woodlanders lives there. They keep a magic sword at Redwall. 'Tis said that the warrior who holds that sword is the greatest in the land!\"\n\nSlipback confirmed his mate's story. \"Aye, none can stand against the sword owner, I've heard the tale meself.\"\n\nSkrodd, who was leading the gang through the darkened woodlands, overheard Slipback's remark. He stopped and questioned the weasel. \"Wot have ye heard? Tell me.\"\n\nThe garrulous Flinky spoke up. \"Ah sure, 'twas me that told him. I sat wid Burrad's ole dad many a night, yarnin' away. He was a fine ould feller, not like his son. Anyhow, he told me all about the magic sword, so he did.\"\n\nSkrodd was fired with the idea of possessing such an enchanted blade. He stared hard at the gabby stoat. \"Right, then, you tell me everythin' the ole beast said.\"\n\nFlinky liked to talk, but he was also aware that the tall fox was not one to be taken lightly. \"Ah, well let me see now. There's this place, see, a grand ould Abbey called Redwall that stands on a path somewhere in the centre of the land. Sure, an' a fine buildin' it is!\"\n\nLittle Redd interrupted. \"I've 'eard o' Redwall.\"\n\nSkrodd froze him with a glare, gesturing Flinky to continue.\n\n\"Aye, Redwall was built by a mighty warrior long ago. He carried a great sword made from bits o' the moon 'n stars. A marvellous blade, magic enough t'make a champion fighter out o' anybeast. That warrior's long dead now, but the sword still hangs in the Abbey.\"\n\nThis time it was Skrodd's turn to interrupt. \"Then why doesn't one of the creatures at Redwall Abbey wear it?\"\n\nFlinky shook his head. \"Ah no, they're all only simple woodland beasts. They're farmers an' such, not fighters. Hah, what need d'they have o' swords, 'tis said that Redwall is a place of peace an' plenty.\"\n\nLittle Redd's eyes shone with longing. \"I wish I had a magic sword!\"\n\nSkrodd shoved him roughly. \"A runt like yew, huh, you'll have to fight me fer it. That sword is goin' t'be mine!\"\n\nThe hefty rat Dargle muttered under his breath. \"If ye think ye can take it, fox!\"\n\nSkrodd looked around at the vermin behind him. \"Did somebeast say somethin'?\"\n\nFlinky rubbed his stomach. \"Ah no, Chief, 'twas just me ould guts rumblin' away. I knew that fish wasn't fer me somehow.\"\n\nBragoon and Saro had made camp in a grove of conifers, some miles south of where they had encountered the vermin.\n\nBurying the fishbones beneath the deep layer of pine needles, the otter wiped his mouth. \"Bit o' fish like that makes a nice change, eh mate? Did ye manage to lay paws on any o' that stuff they was drinkin'?\"\n\nThe aging squirrel wrinkled her nose disgustedly. \"That poison? Vermin-brewed nettle grog. Small wonder they're stupid\u2014it must've rotted what little brains they had. Best stick with clean streamwater until we get back to Redwall an' get some decent drink.\"\n\nShe lay back, viewing the star-dusted skies through the treetops. \"Aah, t'be back home in the good ole Abbey. D'ye think they've forgiven us for the old Dibbun days?\"\n\nHer companion chuckled. \"I certainly hope they have, we were a fearsome pair, mate. Hmmm, wonder if ole Granmum Gurvel's still the Abbeycook. Hoho, the pies'n'scones we swiped off'n her kitchen windowsill. No wonder she turned grey!\"\n\nSaro shrugged. \"There's a lot o' seasons run under the bridge since we were Abbeybabes. I don't suppose pore old Gurvel will still be livin'. She was a great cook though.\"\n\nBragoon nodded. \"Aye, she was that. I'll bet that little brother o' mine Toran is Abbeycook now. Gurvel taught him a lot, y'know. He was always a goodbeast around kitchens an' ovens.\"\n\nSaro hopped up and spread herself along a bough, directly above Bragoon's resting spot. She reminisced hungrily.\n\n\"Scones, or fresh bread, with meadowcream an' damson preserve. That's what I could eat right now!\"\n\nStretched on the ground, Bragoon yawned and sighed. \"Don't even mention it, mate. Let's get a good night's shuteye. We could make Redwall by afternoon tea tomorrow. You can fill yore face then. G'night, Saro.\"\n\nThe squirrel ignored her friend and continued yearning. \"October Ale! What could be nicer than a foamin' beaker of good October Ale. Mmmm, with some brown farlbread an' some yellow cheese with roasted hazelnuts in it. Simple but satisfyin', eh Brag?\"\n\nThe otter opened one eye. \"Very acceptable. Now go t'sleep!\"\n\nSaro carried on as if she had not heard. \"What would y'say to an apple'n'blackberry crumble, spread thick with meadowcream?\"\n\nBragoon growled. \"I'd say button yore lip an' sleep. So goodnight!\"\n\nBut Saro could not forget the subject of food. \"Howsabout ice cold mint tea an' a thick slice of heavy fruitcake with honey crystals in it. Ooooh!\"\n\nBragoon sat up slowly. \"I'd say ye was makin' my pore stomach gurgle with all this vittle talk. Good . . . night!\"\n\nSaro licked her lips. \"Or some of yore favourite, a big carrot'n'mushroom pasty, with onion gravy drippin' an' oozin' out the sides, an' . . . Yaahoooow!\"\n\nShe was catapulted into the air as Bragoon hauled down hard on the bough, letting it go suddenly. Rising from the ground, Saro dusted herself off indignantly.\n\n\"Gettin' touchy in yore old age, aren't ye? Goodnight to ye, ole grumpy rudder!\"\n\nBragoon snorted. \"I swear ye were born chatterin'. Now goodnight, old gabby whiskers!\"\n\nSilence fell over the glade. Both lifelong friends drifted into the realm of slumber. They dreamt golden-tinged memories of their Dibbun seasons at the place they called home\u2014Redwall Abbey.\n\n##\n\nThe big badger's eyes flickered, then opened slowly. He lay quite still, taking in his strange surroundings\u2014a cave, peaceful and warm, with sweet aromatic wisps drifting languidly from a rockbound hearth. A fireglow cast flickering shadows across the rough-hewn walls. He felt secure and safe there with moss and soft, silver sand beneath him.\n\nA movement near his head caught the badger's attention. A young sea otter emerged.\n\n\"De old stripedog who was slayed, was he yore farder, sir?\"\n\nThough it pained him, he strained his neck to get a closer look at the young one. The badger's voice, echoing in the cavern, sounded strange to his ears. \"Nay, he was my friend, though a father could not have been kinder to me. He was called Grawn. I trust you put him to rest decently.\"\n\nThe youngster nodded several times. \"Shoredog an' my farder made a bury hole. They putted rocks on him an' yore bow, 'cos it was broked in halves.\"\n\nThe badger's big dark eyes glistened wetly. \"I must thank your father and Shoredog. What do they call you?\"\n\nThe young beast held out his paw politely. \"I bee's Stugg, son of Abruc an' Marinu, sir.\"\n\nA massive paw took Stugg's smaller one, enveloping it. \" 'Tis a pleasure to meet ye, Stugg. I am called Lonna Bowstripe. Is your father hereabout? I would speak with him.\"\n\nLonna listened to young Stugg scamper from the cave calling shrilly. \"Farder, farder, come quick! De big stripedog bee's awake, his name be Lonna!\"\n\nIn a short while, two male sea otters entered the cave, followed by two females, one very old, and Stugg following up the rear.\n\nLonna leaned forward slightly. \"Thank you, my friends, for saving my life, caring for me and putting old Grawn to rest. Stugg told me you buried him well.\"\n\nAbruc pressed Lonna back down gently. \"We did what was right for your companion. Only vermin leave the dead unburied. As for ye bein' cared for, 'twas my wife Marinu an' ole Sork who saw to yore well-bein'. You lie still an' rest now, Lonna. By an' by ye'll get stronger. We'll see to that.\"\n\nThe big badger's paw touched the long scar ridge that crossed his face diagonally from eartip to jaw. \"I must grow strong again to repay the vermin who did this and murdered poor Grawn. Did you see them?\"\n\nSork placed Lonna's paw by his side. \"Be still, bigbeast, an' thank the seasons ye are still alive. That face still needs a lot of healing, aye, an' yore back, too. We'll bring ye food an' drink.\" Sork and Marinu departed.\n\nShoredog stood over Lonna, looking down into his injured face. \"We never saw the vermin, but we know 'em. Raga Bol the Searat an' his crew were the ones. His ship was wrecked beyond repair. They have gone westward, inland to where the weather's fair an' the pickin's easier. Do ye know Raga Bol?\"\n\nLonna's scar twitched faintly. \"I do not know the scum, but I know of him. They say he kills for fun.\"\n\nYoung Stugg scowled. \"My farder says Raga Bol be's wicked!\"\n\nAbruc tugged his son's rudder. \"Go an' help yore mamma now.\"\n\nLonna watched the young otter shuffle off. \"He'll grow up to be a fine big creature someday.\"\n\nAbruc smiled. \"Aye, Stugg's a good liddle son.\"\n\nAbruc sought Lonna's paw and pressed something into it. \"Yore weapon was too badly broken to fix. I wove ye a new bowstring. Mayhap ye'll need it when y'leave here.\"\n\nLonna held the cord where he could see it better. \"Thankee, friend. 'Tis a fine, tough one, well woven and waxed. This is a good and thoughtful gift.\"\n\nAbruc flushed with pleasure. \"Ye have only to ask if ye need ought else. We'll do our best to find it.\"\n\nThe giant badger closed his eyes, speaking softly. \"I'd be obliged if you could get some ash shafts for arrows, and a few long stout yew saplings, so I can choose one to make a new bow from.\"\n\nShoredog replied. \"We saved yore quiver an' the arrows, too. Me an' Abruc know some stream otters not too far from here. They coppice a yew grove. We can have ye a selection of good saplings by tomorrow night. Now sleep, Lonna, ye must rest if yore goin' to get better. Relax an' sleep.\"\n\nA short time thereafter, Lonna allowed Marinu to feed him. Then he drifted off into slumber whilst Sork tended to his hurts. In his sleep he visioned Raga Bol, swinging down at his face with the broad-bladed scimitar. The big badger concentrated all his energy and thoughts on the Searat's savage features.\n\nMentally he began chanting, over and over, \"Look and you will see me! Know that I am Lonna Bowstripe! The earth is not big enough for us both! I will come on your trail! I will find you, Raga Bol! I will seek you out no matter where! The day of your death is already written on the stones of Hellgates!\"\n\nWhilst the big badger was sleeping, young Stugg crept in to see him. The expression of hatred on Lonna's ruined features was so frightening that the young sea otter ran from the cave.\n\nRaga Bol was still out on the heathlands, trekking west with his Searats. They were camped on the streambank in what had once been a vole settlement. Amid the smoke and carnage of burning dwellings and slain voles, the barbarous crew fought among themselves over the pitiful possessions and plundered food.\n\nWirga, the wizened old Searat who had healed Raga Bol's severed stump, stood watching her master chewing on a strip of dried fish.\n\nWith the silver hook tugging at the fish as he pulled to tear it apart, Bol grinned wickedly at Wirga. \"See, I told ye, the further west we go, the better the pickin's get. This stump o' mine ain't painin' so much now. Aye, an' the weather's gettin' better, too.\"\n\nWirga gestured round at the slain vole bodies lying on the bank. \"Fling 'em in the stream an' this'd make a good camp for the night, Cap'n.\"\n\nBol picked his teeth with the hooktip. \"Aye, 'tis nice'n'restful 'ereabouts now. Hahaha!\"\n\nDutifully, Wirga laughed with him. Her cackling trailed off as she saw her captain go off into a vacant silence, his eyes opening wide as the fish fell unheeded from his mouth.\n\nWirga stared at him anxiously. \"What is it, Cap'n, a bone stuck in thy gullet? Let me take a look!\"\n\nAs she bent toward him, Raga Bol recovered and kicked her roughly away. \"Break camp, we're movin' out!\"\n\nThe healer was bewildered at this sudden change. \"But Cap'n, thee said . . .\"\n\nWirga narrowly dodged an angry slash from the silver hook.\n\nBol booted the fire left and right, scattering it. \"I said we're movin' out, we ain't stayin' in this place. Now shift yoreself an' get the crew together!\"\n\nHe strode off, to the top of a small rise, peering back at the route they had come along. Wirga passed the word on to Glimbo.\n\nThe one-eyed Searat rolled his milky orb in puzzlement. \"Why does 'e wanna move? 'Tis nearly dark!\"\n\nWirga picked up her stolen belongings. \"Hah! Yew go an' ask 'im, if'n thee feels tired o' livin'.\"\n\nThe crew gathered in sullen silence, watching their leader. He was still gazing eastward from the top of the rise. None of them dared make a move until he did.\n\nRaga Bol stared at the hostile heathland, muttering to himself. \"Yore dead, stripedog, or ye should be. In the name o' blood an' thunder, where are ye?\"\n\nHe drew his cloak about him and shivered. Somewhere in Raga Bol's evil mind he had felt Lonna Bowstripe's threat.\n\nIn the gatehouse at Redwall Abbey, Martha and her friends were studying the history of Loamhedge. It made harrowing reading.\n\nAbbot Carrul shook his head sadly. \"This is not the story of one creature, it is the history of many, all related to one writer, who set it down as a chronicle. I think that this poem, \"The Loamhedge Lament,\" by Sister Linfa, sums up most of the tragedy. I'll read it out to you.\"\n\nMartha's eyes misted over as the Abbot recited the poem.\n\n\"Where are the carefree sunlit days,\n\nwhen once amid tranquil bowers,\n\nLoamhedge mice would take their ease,\n\nto dream away happy hours?\n\nWhere did the laughter go?\n\nWho stole the joy away?\n\nHeavy the heart that goes\n\nfar from its home to stray.\n\nA sickness stole in to blight our lives\n\nlike a spectre of unwanted doom.\n\nMidst grief and anguish it lingered,\n\ncreeping through hall and room.\n\nLike wheat before the sickle,\n\nit laid our loved ones low,\n\nleaving us only one answer,\n\nto flee our home and go!\n\nStalked by desolation now,\n\nleft open to wind and rain,\n\nonly in old memories dim\n\nwould Loamhedge live again.\"\n\nThe day's last gleaming shone through the open door. Toran stood framed there, wiping his eyes on his cook's apron. He had entered unnoticed and heard the whole thing.\n\n\"Leave this now, and come back to the Abbey for supper, friends. Tomorrow morning ye can sit out on the wallsteps in the sunlight and study some more. Martha, come on, 'tis far too sad, sittin' here at night readin' of sickness an' death.\"\n\nThe haremaid cast an imploring glance at Abbot Carrul. \"But we must find out about Sister Amyl's secret, and we must find out a way to discover where Loamhedge lies!\"\n\nThe Abbot shepherded her to the gatehouse door. \"Toran's right, miss, the night hours can be long and oppressive for such heavy stuff. Let's go to supper in Cavern Hole and shed our sad mood for tonight. We'll be much brighter, and more alert, in the morning.\"\n\nOld Phredd the Gatekeeper waved them off. \"Hmm hmm, you run along now. I'll stay here awhile.\"\n\nHe watched them go, then wandered back into the little building, talking to a cushion he had picked up. \"Hmm, the way to Loamhedge, now where've we seen that before? Chronicle of some bygone traveller I expect, eh, eh?\"\n\nClimbing upon a chair, he peered at a row of books on a high shelf. Selecting one, Phredd blew the dust from its covers and smiled benignly at it. \"Ah, there you are, y'old rascal. Hiding up there, heehee. Didn't think I could see ye? Now what've you got to say for yourself, eh, eh?\"\n\nSettling down in an armchair, he brought a lantern close and opened the book's yellowed pages. \"Heeheehee, we've met before, haven't we? The recordings of Tim Churchmouse, now I recall ye! The journey to seek out Mattimeo, son of the warrior Matthias. Aye, that covered the Loamhedge Abbey territory, I'm certain it did!\"\n\nToran had been keeping his eye on Martha throughout supper. The ottercook did not like to see his young chum so downcast. He chivvied her, hoping to lighten Martha's mood.\n\n\"Cheer up, beauty. If'n ye keep lookin' like that, it'll teem down rain tomorrow. Wot's the matter, my mushroom 'n'barley soup too cold? Has the bread gone stale, the cheese too hard, not enough plums in the pudden? Speak up, droopy ears, does that strawberry fizz cordial taste musty?\"\n\nThe haremaid managed a wan smile. \"No, Toran, it's not that, the supper is delicious. It's just that . . . oh, I don't know.\"\n\nToran collared Horty, just as he was reaching for another helping of plum pudding. \"Hear that, young starvation face? Yore sister doesn't know wot's wrong with her. Sing her a song an' liven her up, or y'don't get any more plum pud!\"\n\nHorty had done this once or twice before, when Martha was a bit down. That, and Toran's threat to cut off his plum pudding supply, galvanised the greedy young hare into action. He let rip with a special ditty he saved for such occasions.\n\n\"What a gloomy little mug, wot wot,\n\ncome on, let's see you smile.\n\nWith a scowl like that you'd frighten\n\nevery beast within a mile.\n\nSo chortle hahaheeheehoho!\n\nand brighten up for me,\n\nor I'll send you to that Sister\n\nfrom the Infirmary.\n\nShe'll say 'Wot have we here, wot wot?\n\nA face like a flattened frog?\n\nThis calls for a bucket o' physick, aye,\n\nnow that should do the job!\n\nWill somebeast grab her nose,\n\nso she can't hold her breath,\n\nthen I'll be able to grab a ladle,\n\nan' physick the child to death!\n\nI'll not have it said of me, I couldn't do my job,\n\nan' send a young 'un to her grave,\n\nwith a grin upon her gob!'\n\nSo chortle hohohahahee,\n\nan' smile an' giggle a lot,\n\nyou can't sit there all evenin'\n\nwith a face like a rusty pot. Wot wot!\"\n\nMartha was chuckling when she spied Sister Setiva, the Infirmary Keeper, making a beeline for her brother.\n\nSetiva had a stern manner, and a marked northern accent, coupled with a dislike for impudence. \"Ach, ye flop-eared wretch, ah'll physick ye tae death if'n ah lay paws on ye!\"\n\nHorty hid behind Toran. \"I say, sah, 'twas only a blinkin' joke, y'know. Don't let that old poisoner get me!\"\n\nMartha wiped tears of merriment from her eyes as the Abbot leaned across to her and asked, \"Better now, miss?\"\n\nShe nodded. \"Yes, thank you, Father. Oh, that Horty!\"\n\nSister Portula gave the Abbot a sidelong glance. \"It's all very well making plans to continue our studies out on the steps tomorrow, but look at the ruckus today. They were crowded around the gatehouse to see what we were doing inside. I think we'd best get ready to have lots of company tomorrow, Father\u2014unless you can think of another way to keep our creatures distracted.\"\n\nAbbot Carrul touched a paw to the side of his nose. \"I've already thought of that, Sister. Do you not know what day it is tomorrow?\"\n\nPortula shrugged. \"A day like any other. Sunny, I hope.\"\n\nAbbot Carrul stood up and murmured to her as he banged a ladle upon the tabletop to gain order. \"Tomorrow is the first day of summer.\"\n\nHe raised his voice. \"Your attention please, my friends!\"\n\nA respectful silence fell upon the boisterous Redwallers. Everybeast was eager to hear what their Abbot had to say.\n\n\"It is my wish that, as tomorrow is the first day of Summer Season, a sports day and a feast shall be held within the grounds of our Abbey. My good friend Foremole Dwurl will be in charge of the proceedings. I trust you will cooperate with him. Foremole Dwurl!\"\n\nRedwall's mole leader, a kindly old fellow, bowed low to the Abbot. Amid the raucous cheering and shouting, he climbed upon the table and stamped his footpaws to gain order.\n\n\"Thankee, zurr h'Abbot. Naow, you'm all coom to ee h'orchard arter brekkist, an' oi'll give ee yurr tarsks. Hurr hurr, an' all you'm Dibbuns make shore you'm be proper scrubbed!\"\n\nAbbot Carrul looked over the top of his tiny glasses at Sister Portula. \"Does that solve your problem, marm?\"\n\nThe good Sister looked slightly nonplussed. \"But Father, Summer Season doesn't start for two days yet.\"\n\nForemole Dwurl wrinkled his snout confidentially. \"If'n you'm doant tell 'um, marm, us'n's woant. Hurrhurr!\"\n\nSilence reigned in Cavern Hole. Every Redwaller was tucked up in bed, anticipating the coming day's delights. Summer Season feast and sports was always a joyous event on the Abbey calendar.\n\nAbbot Carrul pushed Martha's chair across Great Hall to her bedroom, which was next to his on ground level. His voice echoed whisperingly about the huge columns as they went.\n\n\"Did you notice that Old Phredd didn't come in for supper this evening?\"\n\nMartha voiced her concern. \"Oh dear, I do hope he's not ill!\"\n\nThe Father Abbot reassured her. \"Not at all, that old fogy's fit as a flea. He was rather anxious for us to get out of the gatehouse, though. I'll wager a button to a barrel of mushrooms that rascal has information about Loamhedge hidden in his dusty archives, sly old hog!\"\n\nMartha sat up eagerly. \"Do you really think so, Father?\"\n\nCarrul nodded. \"I'm certain of it, miss. D'you know, I think our search is going to turn up some interesting and exciting stuff tomorrow.\"\n\nThe young haremaid wriggled with anticipation, since any prediction the Abbot made invariably came to pass. \"Oh, I do hope so, Father. Maybe we'll discover Sister Amyl's secret. Wouldn't that be wonderful!\"\n\nMartha looked up as they passed the great tapestry. Was it just a trick of the flickering lanterns, or did she really see Martin the Warrior's eyes twinkle at her?\n\n##\n\nSome leagues north of Redwall Abbey, the ragtag vermin gang blundered their way through the nighttime thickness of Mossflower woodlands. Skrodd swiped at the undergrowth with his former leader's cutlass as he led the party.\n\nThe big rat, Dargle, kept muttering under his breath, continuously criticising Skrodd. \"Fancy trackin' two beasts when yore lost, huh!\"\n\nTired and sleepy, the other vermin managed a weary murmur of agreement. Skrodd did not want to challenge Dargle directly\u2014it was the wrong time and place for such a move. So he asserted his authority by bullying all and sundry. He turned on them, brandishing the cutlass.\n\n\"Shut yer gobs an' keep movin'. Lost? Hah! Youse'd be the lost ones if'n I wasn't leadin' ye!\"\n\nFlinky enjoyed causing trouble. Disguising his voice, he called out behind the big fox's back. \"That's no way t'be talkin' to pore pawsore beasts!\"\n\nLittle Redd agreed with him. \"Aye, we should be sleepin' now instead o' wanderin' round an' round all night long!\"\n\nAlthough Flinky was the instigator, Redd was the unlucky one whose voice Skrodd identified. With a savage kick, Skrodd sent the small fox sprawling.\n\nLaying the cutlass blade against his neck, he snarled, \"Ye liddle runt, say the word an' ye can sleep 'ere fer good. I've took enough of yore moanin'!\"\n\nRealising that he had gone too far, Flinky tried to remedy the situation by pulling Redd upright as he appealed to Skrodd. \"Ah, come on now, sure he's only a tired young whelp. No sense in slayin' one of yore own mates. Let's step out a bit, an' I'll sing a song to help us along, eh?\"\n\nSkrodd relented, pointing his blade at the stoat. \"Right, you sing. The rest o' ye march, an' shuttup!\"\n\nFlinky's ditty put a little fresh life into the gang's paws.\n\n\"Ferrets are fine ould foragers,\n\nthough frequently furtive an' fey,\n\nstoats can sing sweetly fer seasons,\n\nso me sister used to say,\n\nbut foxes are fine an' ferocious,\n\nwhen faced with a fight or a fray,\n\nan' rats remain rambunctious but only for a day!\n\nBut wot about weasels, those wily ould weasels,\n\nthey're woefully wayward an' wild,\n\nthe ones they've whipped an' walloped,\n\nwill wail that weasels are vile,\n\nthey've bullied an' beaten an' battered,\n\nthey've tormented tortured an' tripped,\n\nI'm sure any day their pore victims would say,\n\nsteer clear o' the weasel don't get in his way,\n\nfor of all the vermin ye'd care to recall,\n\nthe weasel's the wickedest wretch of all.\n\nAn' virtuous vermin will all agree,\n\nany weasel is worse than me!\"\n\nThere were four weasels in the gang: Slipback; his mate, Juppa; and two taciturn brothers, Rogg and Floggo. All of them protested volubly at Flinky's song.\n\n\"That ain't right, foxes are worse'n weasels!\"\n\n\"Ye sing dat again, an' I'll wallop ye alright!\"\n\nSkrodd's bad-tempered shout quickly silenced them. \"Shut yore faces back there, or I'll show ye 'ow ferocious foxes can be. Sing somethin' else, Flinky, an' don't insult nobeast!\"\n\nDargle called out, \"Aye, an' be nice to foxes, they're easy hurt!\"\n\nSkrodd fixed the big rat with an icy glare. \"Aye, an' they can hurt rats easily, too!\"\n\nDargle stared fearlessly back at him. \"Ye don't scare me, fox. Burrad was slayed by mistake. Us rats don't make mistakes when we fight!\"\n\nSkrodd never answered. Turning away, he continued to march, but the challenge was out in the open now. The rest of the gang exchanged nods and winks\u2014a fight to the death was not far off. Skrodd pulled Little Redd up to the front with him and allowed him to walk by his side. The small fox felt honoured; normally he would be left trailing at the back of the gang.\n\nKeeping his voice low, the bigger fox took on a friendly tone with the young one. \"You stay by me, mate. Us foxes've got to stick together.\"\n\nLittle Redd had to glance around to make sure Skrodd was not talking to some other beast. He was more used to kicks and insults than to kind words.\n\nThe big fox winked at him. \"I been keepin' an eye on ye, mate. Yore a smart little feller, not like this other lot!\"\n\nRedd hated being called \"little,\" but he was quite pleased to know that Skrodd thought of him as smart. He returned the wink, speaking out of the side of his mouth.\n\n\"I ain't no fool, an' I ain't so little, either. I'm growin' fast. One day they'll call me Big Redd.\"\n\nSkrodd got to the point. \"Lissen, mate, I want ye t'do me a favour. Do ye think yore smart enough t'be useful to me?\"\n\nLittle Redd walked on tippaw, swelling his chest out. \"Just tell me wot ye want doin', mate!\"\n\nSkrodd leaned close. \"Keep an eye on the gang, especially Dargle. That rat's gettin' too big fer his boots. I want ye to watch my back, sort o' be my second in command.\"\n\nRedd hid his delight, replying gruffly, \"I'll do that, just watch me. Soon they'll be callin' me Big Redd. I won't let ye down, mate!\"\n\nSkrodd patted the small fox's back. \"Good! When I gets this gang sorted out, we'll give ye a proper vermin name. Big Redd don't mean nothin'. How does Badredd sound to ye, eh?\"\n\nThe young fox was squirming inside with joy. However, he kept his voice tough, in keeping with his new position. \"Sounds great t'me, mate. Badredd\u2014I like that! 'Tis a real killer's name. Badredd!\"\n\nAfter a fruitless night rambling through woodland thickets, the gang watched a rose-tinged dawn break over the treetops. They were soaked through by heavy dew, which was dripping everywhere from boughs and leaves.\n\nDargle's temper was on a short fuse. Emerging into a clearing on the bank of a stream, he struck out at Little Redd with his spear haft.\n\n\"Keep outta my way, runt! Every time ye come near me, I get soaked wid the water ye knock off the bushes.\"\n\nRedd looked appealingly at Skrodd. The big fox cast a glance of mock pity at Dargle and snarled scornfully. \"Scared of a few drips o' dew, are ye? Look at us, we're all wet through, an' we ain't moanin'.\"\n\nDargle faced up to Skrodd right away. \"Hah! Wet through an' weary, an' wot for, eh? We never found the otter an' the squirrel. No, we just tramped around all night followin' you, an' now we're good an' lost. Some leader you are, Skrodd!\"\n\nThe big fox bristled. \"Don't talk silly, we ain't lost!\"\n\nIt was Dargle's turn to sound scornful. \"Oh, ain't we now? See that rowan tree, I marked it wid me spearblade not long after we started marchin'. Look!\"\n\nFlinky inspected the fresh scar on the rowan bark. \"Aye, 'tis a new spearmark sure enuff. Dargle's right!\"\n\nLeaning on his spearbutt, the hefty rat grinned teasingly. \"We've been goin' round in circles, mates, an' now our great leader's got us lost. Well, Skrodd?\"\n\nThe fox held his blade at the ready and challenged Dargle. \"If'n yore so clever, then you find the way. 'Tis easy to stand there talkin' smart all day, Dargle. Go on, show us how ye are, an' find the right way!\"\n\nThe rat squatted down on his haunches, chuckling. \"Sort out yore own mess, I'm stoppin' here an' restin'.\"\n\nHalfchop ventured a suggestion. \"Burrad would've sent Plumnose to find the way, 'cos he's a good tracker.\"\n\nRelief flooded through Skrodd as he realised that Halfchop had provided the solution to a sticky problem. Taking advantage, he quickly re-established his position as leader of the gang.\n\n\"Right, Plumnose, get on yore way! Ferget the two beasts we were trackin', they'll keep for another day. Find us the way to this Redwall Abbey place an' report back here.\"\n\nAlways one to seize an opportunity, Flinky nodded his head admiringly. \"Ah, that's a grand ould move, Chief. I see ye noticed the fine campsite we're at. We can lay up here fer a day or two an' rest, once we're sure of the way. Lookit, we got a stream wid fish an' freshwater an' lots o' trees full of fat birds sittin' on nests packed wid eggs. The place is filled wid roots an' fruit an' firewood!\"\n\nSkrodd looked sage. \"That's wot I was thinkin', a day or two here'll freshen us up for the rest o' the journey. We'll make camp an' rest awhile, mates.\"\n\nOnly Plumnose was not happy with the new plans. His huge nose wobbled from side to side as he complained. \"Duh, id's nod right. I'b tired, too, j'know!\"\n\nRogg and Floggo, the weasel brothers, notched arrows to their bows and fired a pair of shafts near Plumnose's paws.\n\n\"Yore the tracker, Plum, now git goin'!\"\n\n\"Aye, ye could track a butterfly underwater wid a hooter like that. Hohoho!\"\n\nThrowing twigs and grass clumps at the unfortunate creature, the gang drove Plumnose from the camp. Glad they had not been selected to go tracking, they shouted after him.\n\n\"Don't trip over yer nose, Plum!\"\n\n\"Aye, an' don't sniff any big boulders up. Heeheehee!\"\n\nThe tension was broken for the moment. Gathering wood and foraging for victuals, the gang busied themselves.\n\nFlinky dug a firepit on the streambank, singing a cheery ditty.\n\n\"Ah 'tis luvverly bein' a vermin,\n\n'cos ye lead a simple life,\n\nleave the snufflin' babes behind,\n\nrun off from the naggin' wife.\n\nThere's nought to do but ramble,\n\nan' plunder on the way,\n\njust look bold, rob all ye can hold,\n\nan' bid 'em all good day.\n\nA vermin, a vermin, that's wot I'll always be,\n\nI'm base an' vile, 'cos that's me style,\n\nan' I'll bet ye envy me!\"\n\nBy late morn they had a good fire burning. Flinky and his mate, Crinktail, were in their element. They boiled woodpigeon eggs, grilled fish, and made a passable vegetable stew from various roots and wild produce which grew plentifully roundabout. Neither Dargle nor Skrodd made any move to help. Sitting close to the fire, they helped themselves, glaring at each other across the flames.\n\nSkrodd collared Little Redd and gave him whispered orders. \"Scout round an' find me somewheres safe to rest. Make sure 'tis soft an' comfortable. Pick a place far away from that rat, an' someplace close for yourself, so ye can guard me. Go on!\"\n\nPuffed up with his own importance, Redd went to seek a suitable resting spot. He chose the base of a spreading oak, not too close to the stream. It was a basin-shaped depression between two thick roots.\n\nWhen the gang finished eating, they settled down for a much-needed sleep. Most of them stayed by the fire, but Dargle chose a fernbed on the opposite side of the camp from Skrodd. From there the rat could see his enemy and lay plans.\n\nLittle Redd proudly showed Skrodd the spot at the base of the oak trunk. \"That's it, mate, nice an' snug, see!\"\n\nThe small fox lay down, gesturing. \"There's plenty o' room for both of us. I can guard ye good from here, mate.\"\n\nSkrodd shook his head disapprovingly. \"Nah, ye go an' lay by the fire with the others. That'll put ye halfway twixt me'n Dargle. But don't go sleepin', keep yore eyes peeled on those ferns where he's layin' low. Soon as Dargle makes a move, come runnin' an' let me know.\"\n\nLittle Redd rose reluctantly. \"I kin watch him just as well if'n I stop 'ere with you, mate.\"\n\nSkrodd hauled him roughly upward, thrusting him toward the fire. \"Ye'd do better to heed my orders. Now get goin'. I'm chief round 'ere, see!\"\n\nStinging from the rebuke, Redd slouched over to the fire. Sullenly, he slunk down amid the snoring vermin.\n\nWith not a breeze to rustle the trees, warm noon sunlight shone down on the camp. Bees hummed gently, and butterflies fluttered silently around blossoming bushes. Near the ashy embers of the cooking fire, Little Redd drifted into a slumber. Only one of the gang was still awake\u2014Dargle. Now was the time to put his plan into action. Draping his cloak over the ferns so it would look like he was still there, the rat inched his way backward out of the foliage. Flat on his stomach, he took a careful route, circling the campsite. When the rear of the spreading oak came in sight, Dargle rose into a half crouch. Gripping his spear firmly, he crept up on his sleeping enemy.\n\nSkrodd woke momentarily, but only to die. A muffled grunt of agony escaped him as Dargle's spear thrust into his body.\n\nDargle leaned down on the spearhilt, grinning triumphantly. \" _Now_ who's the chief, eh?\"\n\nIt was the rat's only mistake\u2014it turned out to be his last. Skrodd had lain down to sleep with the cutlass held tight in his paw. Now, with one spasmodic jerk, he whipped the broad blade across his assassin's neck, almost severing Dargle's head. The ambitious rat fell slain on top of his victim's dead body.\n\nLittle Redd was wakened by Flinky kicking him in the back. The small fox sat up rubbing his eyes and muttering at the still-sleeping stoat. \"Keep yore paws to yoreself, ye great lump!\"\n\nFlinky rolled over and emitted a huge snore. To avoid a second kick, Redd rose stiffly and looked around. Dargle's cloak was still draped over the ferns. He let out a sigh of relief and wandered over to check on Skrodd. Redd was dumbfounded by the sight that greeted him\u2014Skrodd and Dargle, both dead!\n\nLittle Redd circled them slowly, poking both beasts with a stick and uttering their names softly. There was no doubt about it, they were still as stones. His first thought was to run and tell the others. He had already opened his mouth to shout when a thought struck him. Who would be the next to claim leadership of the gang? Little Redd sat down and did some serious thinking. It did not take him long to reach a decision. He would be the new chief. Getting the cutlass loose from Skrodd's paw was a difficult task, but he managed it somehow. Dargle was almost decapitated by Skrodd's death blow. Two good chops of the hefty blade finished the job.\n\nFlinky was roused by a painful feeling he knew well, the slap of a flat cutlass blade. He sprang upright, rubbing his rump, expecting to see Skrodd standing over him. Instead, there stood the small fox, whacking away at the other gang vermin and yelling aloud.\n\n\"Up on yore hunkers, all of ye!\"\n\nThe weasel Juppa grabbed a chunk of firewood and advanced on the small fox, snarling. \"Ye snotty liddle runt, who do ye think y'are, smackin' me wid the chief's blade?\"\n\nRedd jarred the wood from Juppa's paws with a blow from the cutlass. His voice was shrill but commanding. \"I'm the new chief round here, that's who I am. Come an' see this, all of ye!\"\n\nThe gang stood around the two carcasses in awed silence as the small fox explained. \"I saw Dargle run Skrodd through with his spear. So I rushed in, grabbed the cutlass an' slew the dirty murderin' sneak with one swipe!\"\n\nCrinktail looked at him disbelievingly. \"You, Little Redd, took off Dargle's block in one go?\"\n\nRedd was getting the feel of the heavy sword now. He took a pace back, then leaped forward, swinging the cutlass in both paws, shouting fiercely. \"Aye, one swipe! D'ye want me to show ye how? I'm the chief now, this sword's mine, I killed to get it!\"\n\nHe was gratified to see fear shining from Crinktail's eyes as she backed away from him swiftly. \"No, no,\" she pleaded, \"if you say ye did it, I'm not one to argue with ye!\"\n\nEver the one to seize an opportunity, however, Flinky confronted Redd and held out his paws placatingly. \"Ah now, don't go upsettin' yoreself, Little Redd. We all think ye'll make a grand chief. Anyway, better'n the last two. Isn't that right, mates?\"\n\nHe turned to the gang, winking broadly at them but making sure the small fox could not see his gesture.\n\n\"C'mon now, raise yer paws an' salute the great new chief!\"\n\nA newfound confidence flooded through Redd as he watched the remaining nine vermin acknowledging his leadership with raised paws. He suppressed a shudder of joy. For as long as he could recall he had been ignored, bullied or pushed about. Now, in the course of one day, he was in command of the gang.\n\nDeciding to assert his authority, Little Redd glared haughtily at the ratbag vermin. \"My name ain't Little Redd no more. From now on ye'll all call me Badredd. Is that clear?\"\n\nFlinky threw him an elaborate salute. \"Badredd it is, yer honour, sure an' a fine ould name it is! Well now, Badredd sir, wot's yore pleasure\u2014do we stop 'ere awhile in this grand camp? There's water an' vittles aplenty roundabout, an' 'tis a pleasant spot.\"\n\nBadredd nodded imperiously. \"Aye, we'll stop 'ere awhile!\"\n\nAs they prepared the evening meal, Flinky's mate, Crinktail, whispered to him. \"Badredd, huh! Wot'n the name o' blood made ye support that liddle fool?\"\n\nFlinky winked at her as he turned a roasting woodpigeon on a willow spit over the fire. \"Trust me, mate, better a liddle fool than a big bully. I can 'andle this 'un. Badredd'll do like I suggest, ye'll see. We've 'ad enough o' weasels, big foxes an' bullyrats in this gang. This Mossflower territory's a good soft place to stay, plenty of everythin'. Better'n those ould Northlands. Leave the thinkin' t'me, we'll live the good life from now on. Badredd'll do like I tell 'im.\"\n\nThe newly elected Badredd sat on the streambank, picking a roasted woodpigeon leg and watching the westering sun die in a crimson haze. He listened to Flinky singing as he dished out supper to the gang, who lay about looking contented enough.\n\n\"Oh this is the place to be,\n\nwhere the fruit falls from the tree,\n\nwhere eggs an' birds jump out of the nest,\n\nright in me pan they come to rest.\n\nOh this is the place for me,\n\nfar from that Northland sea.\n\nHere the good ould fish leap out of the stream,\n\nan shout, 'Please, sir, cook me,'\n\nwhere the sun shines all the day,\n\nan' the cold wind stops away,\n\nan' the water's clean 'n' fresh 'n' clear,\n\nI'll make ye a promise now, me dear,\n\nI'll take a bath so don't ye fear,\n\nin ten summers' time if I'm still here,\n\n'cos this is the place for me!\"\n\nBadredd, however, had totally different plans. Not for him all this lying about on sunny streambanks. Ambition had entered his being. To be the owner of the magic sword and ruler of that place Skrodd had spoken of\u2014Redwall Abbey.\n\n##\n\nLonna Bowstripe sat outside the cave, savouring the approach of summer in the harsh northeast coastlands. Pale sunlight glimmered out of a watery, cloud-flecked sky. It was breezy, but the chill had died out of the wind. Green buds were shooting out of the scrublands, seabirds mewed across the marshes.\n\nThe huge badger shifted his position near the fire, wincing momentarily and arching his back. Young Stugg sat beside him like some constant shadow, always close to the big creature. Lonna fascinated the young sea otter.\n\n\"You back still be hurted, Lonn'?\"\n\nLonna smiled down at his companion. \"A bit, but it's getting better every day, mate. Pass me the bow, please.\"\n\nStugg ambled across and carried the yew sapling to him. Out of six lengths, this was the one Lonna had chosen to use for fashioning his bow. Stugg inspected it closely. The wood had seasoned out until it was strong as sprung metal. Lonna had shaved away the bark, leaving a broad band at its centre that he had bound and whipped with green cord to make a pawhold. At both ends, the wood was circled and notched deep to accommodate bowstrings. Stugg watched as the badger tested the yew's strength by bending it against his footpaws.\n\n\"Wot you think, Lonn', bee's it ready?\"\n\nThe badger applied heavy pressure, bending the bow until it formed a deep arc. He straightened it slowly and then responded. \"As ready as it will ever be, young 'un. This is a good bow!\"\n\nStugg jumped up and down impatiently. \"Putta string on it, Lonn'. Fire a h'arrow for Stugg!\"\n\nAbruc wandered out of the main holt cave toward them. \"Ahoy there, young pestilence! Are ye still botherin' Lonna? Yore more trouble than a sack o' frogs!\"\n\nThe giant badger tugged Stugg's little rudder fondly. \"Oh, he's no trouble, Abruc. Stugg's my good old workmate.\"\n\nAbruc sat down beside them. He could not keep the curiosity out of his voice. \"Well, bigbeast, is yore bow finally ready?\"\n\nLonna used the bowstaff to pull himself upright. \"Let's string it and see, shall we?\"\n\nA short time thereafter, all the sea otters had gathered to watch the testing of the bow. Lonna limped slightly as he went back into the cave to fetch his quiver of arrows.\n\nStugg stood outside, holding the bow and declaiming proudly to everybeast, \"All stan' back now, please. I help Lonn' to make dis bow. 'Tis a very dangerful weapon, so watch out!\"\n\nThe big badger emerged with the birch bark quiver. It was packed heavily with two score of long ashwood shafts, which Abruc and Shoredog had helped to fashion. Each one was fletched with grey gull feathers, gleaned from the shoreline. The arrows were tipped with flint shards, sharpened and ground to lethal points.\n\nLonna took the bowstring which Abruc had woven and looped it over the notch in the yew staff.\n\nAfter knotting it with a skilful hitch, he remarked, \"If this bow fails, it won't be for want of a good string. This is the finest one I've ever seen, thanks to you, friend.\"\n\nAbruc flushed with pleasure. \"Thankee. 'Tis a special string, worthy of a mighty bow.\"\n\nLonna braced the yew sapling against his footpaw, with the string at the bottom end. Tying a loop into the free end, he leaned down heavily on the centre of the wood.\n\nA gasp arose from the otters as the yew bent in a great arc. With the graceful ease of an expert bowbeast, Lonna slipped the loop deftly over the notched top end. It was a bow now, a mighty and formidable longbow that only a beast the size and strength of Lonna Bowstripe could draw. Taking three arrows, he set them point down in the earth and selected one, explaining as he did, \"Height, distance and accuracy are what an archer needs.\"\n\nWhipping the bow up, he laid the first arrow on it, heaved back powerfully and let fly, all in a split second. Swift as lightning the shaft sped upward and was immediately lost to sight.\n\nShoredog let out a growl of surprise. \"Whoo! Where did it go?\"\n\nStugg gestured airily. \"Stuck inna moon I appose, eh Lonn?\"\n\nA rare smile creased the badger's scarred face. \"Aye, I suppose so, mate. Let's try for distance next.\"\n\nThe second arrow he laid flat against his jaw, squinting one eye and holding the bow straight.\n\n _Zzzzip!_ Out across the stream over marsh and scrubland it flew, until it was lost on the seaward horizon.\n\nAbruc clapped his paws in delight. \"Speared a big fish I bet, eh Stugg?\"\n\nThe young otter smirked. \"Prolably two, anna big crab!\"\n\nLonna scanned the countryside. \"I need a target now.\" He bowed to Abruc's wife, Marinu. \"Lady, would you like to choose one? Anything will do.\"\n\nShe looked around, then pointed. \"There's a piece of driftwood just beyond the marsh, see? To the right of that rivulet which runs out onto the shore. I don't know if you can reach that far, Lonna. Shall I pick something a little closer? I'm afraid I don't know much about firing arr . . . !\"\n\nHer words were cut short as the chunk of driftwood went end over end, pierced through by the badger's arrow. A rousing cheer went up from the spectators.\n\nLonna unstrung his bow, passing it to Stugg. \"Well, mate, it looks like we made a proper bow. Thank you for all your help.\"\n\nThe young otter nodded. \"Searats better watch out now!\"\n\nLonna took supper in the sea otters' main cave that night\u2014a large seafood pie, followed by a preserved plum crumble, washed down with beakers of last summer's best cider. He sat by the fire with Abruc and Shoredog, with Stugg dozing on his lap.\n\nOld Sork made Lonna hold still whilst she inspected his facial scar. \"A luckybeast is what ye are. 'Tis healin' better'n I hoped. So what are ye lookin' so miserable about, eh?\"\n\nThe big badger shrugged. \"Every day that I sit here, Raga Bol and his crew get further away. Soon there'll be no trace of them to follow.\"\n\nAbruc refilled his beaker with cider. \"Never fear, Lonna. A Searat like Raga Bol always leaves a trail, a path of murder an' destruction that anybeast with half an eye could follow. I've been watchin' ye since you've been up an' about. I know yore impatient to begone from here. Well, summer's almost in, the time'll soon be ripe.\"\n\nLonna stared into the flames as he replied. \"Raga Bol and his crew won't live to see the leaves turn gold this autumn. I leave tomorrow!\"\n\nShoredog helped himself to more cider, peering curiously at the big badger. \"Then we'll go with ye, Lonna, us an' a dozen of our best fighters. Even a warrior as big as yoreself will need help with Bol an' his crew!\"\n\nThe badger shook his huge scarred head. \"I'm grateful, friend, but this is a thing I must do alone. You stay here and care for your families. There will be a hard time ahead for me. Raga Bol knows I am coming.\"\n\nAbruc replenished the fire with driftwood and sea coal. \"He probably thinks yore dead, mate. How could he know yore comin' after him?\"\n\nLonna never took his eyes from the flames as he explained. \"I never knew my mother and father. Grawn, the wise old badger you buried, was the one who reared me. Not only did he teach me all the skills of a bowbeast but also many other things. When I was very small, Grawn told me that I was gifted with something few other badgers possess. He said that I was born with the power of a Seer. Old Grawn used to question me a lot. One day he said to me, 'You have the keenest eyes of any bowbeast I have known, but you also have another eye, inside your mind. You can see things the rest of us cannot, strange things that will shape your destiny.' It has always been so with me. Even when I was lying wounded in the cave, I could see Raga Bol. I can stare into this fire and see his face. Believe me, he knows I am coming. I want him to know, to fear me. He is evil and must die!\"\n\nShoredog felt the fur on the nape of his neck begin to prickle. \"But if yore a Seer, ye must have known Grawn was goin' t'die, didn't ye?\"\n\nLonna's eyes left the flames momentarily. \"Aye, I knew the old beast had not long to go, but I didn't know the manner of his death. Grawn was old and very ill. He wished to end his days at the badger mountain of Salamandastron. I was taking him there, and I knew my own fate was also linked to the mountain.\"\n\nAbruc leaned forward. \"Do ye know where this mountain is?\"\n\nLonna turned back to contemplating the fire. \"I have never been there, but I feel I am guided to it by my mind's eye. It is far to the west, on the shores of the great sea. When my business with the Searats is done, that is where I'll go. I will not return to this place again. That is why I must travel alone.\"\n\nAs they sat silently by the fire, Marinu came and lifted the sleeping Stugg from Lonna's lap. All the other otters had retired for the night. Only the three of them\u2014Lonna, Abruc, and Shoredog\u2014remained.\n\nShoredog broke the silence. \"Garfo Trok, he's the answer!\"\n\nAbruc nodded vigorously. \"Right, mate, good ole Garfo!\"\n\nLonna stared from one to the other. \"What are you talking about\u2014who's Garfo Trok?\"\n\nShoredog rose and picked up his warm cape. \"Skipper o' the Nor'east Riverdogs, that's who Garfo is. He runs a riverboat. Garfo will take ye westward along the waterways. That should save time an' strain on that back o' yores, Lonna. Ye'll pick up Raga Bol's trail in half the time ye'd take limpin' along step by step.\"\n\nShoredog hurried from the holt, calling back to Abruc. \"I'll be back with Garfo by midday. Tell the cooks to pack plenty o' vittles, especially nutbread!\"\n\nAbruc nudged Lonna cheerfully. \"Ye'll like ole Garfo, that otter knows waterways like the back of 'is rudder.\"\n\nHappy but puzzled, Lonna smiled at the sea otter. \"I'm sure I will, but what's all this about vittles and nutbread? I eat only lightly when I'm travelling.\"\n\nAbruc stood up and stretched. \"Ye may do, Lonna, but Garfo Trok ain't a beast that's ever stinted 'isself when it comes to vittles, particularly nutbread. Why, that ole dog'd go to Hellgates for a loaf! Now get yoreself off an' rest, ye've a big day tomorrow!\"\n\nAfter Abruc had gone, Lonna stretched out by the fire, intending to sleep there for the remainder of the night. Before he closed his eyes, he spent several minutes intensely concentrating on the red embers, repeating mentally, \"Rest not too deeply, Raga Bol! Know that I am coming for you! As surely as night follows day, I am coming!\"\n\nRaga Bol and his crew were sleeping. They had made it out of the hills and moorlands into the first fringes of heavy forest. A spark from the campfire touched Ferron's nose, startling him awake. The gaunt rat sat bolt upright, rubbing at the stinging spot. He saw Raga Bol sit up as well, waving his silver hook and mumbling as he tried to come fully awake.\n\n\"Go 'way, yore dead! Get away from me, d'ye hear?\" The Searat captain caught Ferron looking strangely at him across the fire. \"Who are ye gawpin' at, long face, eh?\"\n\nFerron knew better than to answer back. Instead, he lay back down and closed his eyes. All the crew had been saying the same thing. Lately Cap'n Bol was acting very strange.\n\n##\n\nDawn was only moments old, but Redwall Abbey was awake and buzzing. Today was the special day Abbot Carrul had promised. Breakfast was already being served from a large buffet table, set up in the passage outside the kitchens. With laden platters, the Redwallers sat down to eat at anyplace which took their fancy. Horty and his friends looked out from the dormitory window at the scene below. Dibbuns thronged together on the broad front step of the Abbey, spooning down bowls of oatmeal mixed with honey and fruit. Anybeast wanting to dine outside had to step carefully over them to reach the lawns or the orchard. It was a jumble of happy confusion.\n\nMuggum waved his beaker at the passing elders, who tip-pawed around him. \"Yurr, moind ee paws, you'm nearly trodded in this choild's brekkist. Whurr's ee manners? Hurr!\"\n\nWarm sunlight was rapidly dispersing the mist into a golden haze. Fenna the squirrelmaid leaned out over the dormitory sill and dropped a fragment of scone down into the hood of Sister Setiva's habit, giggling as she drew back inside.\n\n\"Did she notice it?\"\n\nHorty reassured her. \"Not at all. She's toddled off down to the pond with Brother Gelf. Hahaha! I expect old Setiva'll be set upon by the first blinkin' bird that spots it. Should liven her up, wot!\"\n\nSpringald watched the Infirmary Sister balancing her tray gingerly as she crossed the lawn. \"Huh, pity help the bird who tries to set upon her. She'll bath it in the pond and physick it silly. Look out, here comes Father Abbot!\"\n\nThe mischievous trio ducked below the windowsill as Abbot Carrul, Toran, Sister Portula and Martha emerged from the Abbey. Toran lifted Martha's chair over the step and assisted Portula with a trolley full of food. They set out for the gatehouse together, with Abbot Carrul stretching his paws and breathing deeply.\n\n\"My my, it's a good-to-be-alive day. Let's hope we get a few hours of peace to tackle our studies.\"\n\nToran had to rap loudly on the gatehouse door to gain attention. Old Phredd could be heard inside, arguing with an armchair.\n\n\"Come out my way and let me see who 'tis. It's your fault, being so comfy and allowin' me to sleep like that!\"\n\nA moment later, his frowzy, prickled head poked around the door. \"Oh, er hmm. Good morning, I suppose it's morning, isn't it? Of course, if 'twas noon, the sun would be much higher, eh, eh?\" Dabbing his face in a bowl of water, the ancient hedgehog absentmindedly wiped his eyes on Martha's lap rug. \"There, that's better. Oh good, I see you brought breakfast with you. Splendid, I'm starving!\"\n\nMartha ate very little, trying to hold back her impatience as Phredd slowly munched his way around the food. Toran, however, got to the point right away.\n\n\"Well then, sir, how did yore studyin' go? Did ye find out anythin' useful about Loamhedge?\"\n\nPhredd nodded toward a dusty book lying on his bed. \"Oh, that. Take a look in the old volume there. I read it until I could keep my eyes open no longer. Hmm, quite interesting really, an exciting little story, eh?\"\n\nMartha opened the book, its pages yellow with age and so brittle that they were cracking and beginning to flake. She read aloud from the neatly scribed lines of purple, faded ink. \"Written by Tim Churchmouse. Recorder of Redwall Abbey in Mossflower country . . .\"*\n\nPhredd interrupted her as he dealt with a hazelnut roll. \"It was written in the seasons of Abbot Mordalphus. The account of Mattimeo, son of Matthias the Abbey Champion. All about abduction and slavery, a search, a chase and so on. If you're looking for a route to the old Abbey of Loamhedge, the descriptions are very long and complicated, but there's a map included that should be a help. Actually I only got a third of the way through the account before I dropped off. . . .\"\n\nAbbot Carrul shook his head in wonder. \"In the seasons of Mordalphus, . . . Dearie me! That book must be nearly as old as time itself!\"\n\nSister Portula put aside her beaker of mint tea. \"The land will have changed a lot since then, what with rains and floods altering water courses and storms blowing down trees. There'll be new areas of woodland grown over the ages, and I don't know what. Do you think it will be much help, Toran?\"\n\nAs she had been speaking, the noise of stamping paws and singing voices had been swelling outside.\n\nToran went to the door. \"Who knows, Sister? Great Seasons, what's all that rackety din about?\"\n\nOld Phredd chuckled. \"They're singing the Summer Feast song. What a happy sound! Let's go out and watch, eh, eh?\"\n\nMartha was less than enthusiastic, since she wanted to continue studying the book. But the Abbot patted her paw encouragingly. \"You know, we can study the problem at our leisure, but next summer's first day is a long time away. They sound so joyful and excited! Come on, young 'un, let's go and see.\"\n\nSmilingly, the haremaid relented.\n\nUp and down the wallsteps and all over the lawns, Redwallers, led by Horty, were joining paws and skipping about, singing lustily to the jolly tune.\n\n\"The sun could not shine brighter\n\nupon this summer's day,\n\nmy heart could not be lighter.\n\nI've heard our Abbot say\n\nthere'll be a feast this evening,\n\nso listen one and all:\n\nThis afternoon we'll run a race\n\naround the Abbey wall!\n\nCome form up in a line, pals,\n\nand listen for your names,\n\nit's ready steady set and go,\n\nfor Redwall Abbey games!\n\nThere's vittles in the kitchen,\n\ngood ale and cordials, too,\n\nfine singers and musicians,\n\nto play the evening through.\n\nBut first I'll gird my robe up,\n\nso I don't trip or fall.\n\nI'm going to be the first around\n\nthat high old Abbey wall!\n\nCome form up in a line, pals,\n\nand listen for your names,\n\nit's ready steady set and go,\n\nfor Redwall Abbey games!\"\n\nMartha could not resist the merry cavalcade. Clapping her paws in time to the lively song, she laughed happily. Sister Portula, whooping like a wildbeast, grabbed Martha's chair and dashed off into the throng.\n\nAbbot Carrul winked at Phredd. \"My mistake for starting all this, but who could sit indoors studying on such a wonderful day?\"\n\nToran, in complete agreement, shepherded both of his friends out of the way of the dancers. \"You two stay here. I'll go an' bring two armchairs an' the rest o' the food out of the gatehouse. Ye can sit back an' watch the whole thing in comfort. We can always look through dusty ole books tomorrow.\"\n\nOld Phredd spoke to a buttercup growing by the wall. \"Heehee, now there's a sensible young creature. Beasts like that make a body enjoy his old age, eh, eh?\"\n\nBragoon and Saro stood outside the main gate. Memories flooded back as they touched the stout oak timbers.\n\nThe aging squirrel looked misty-eyed. \"Dear ole Redwall Abbey! Sounds like they're havin' a good time in there, mate. Well, do we knock for the Gatekeeper?\"\n\nBragoon scuffed the gravel path with his rudder as he pondered the question. \"Hmm, we've been a long time gone. Suppose nobeast knows us anymore. Or worse, supposin' they do recognise us an' recall wot a pair of scoundrels we were! They might not want us back. Wot d'ye think?\"\n\nSaro gnawed at her lip. \"Aye, I think yore right, Brag. Tell ye what, let's just slip in unnoticed an' sort of mingle with the crowd. That way we can judge the lay o' the land.\"\n\nThe otter grinned furtively at his companion. \"The way we used to come an' go, through the ole east wall gate. I'll bet ye can still open it.\"\n\nSaro clapped his back with her bushy tail. \"Great idea! Come on, let's give it a try. We'll disguise ourselves up a bit so as not to cause too much of a stir!\"\n\nBrother Weld, an old bankvole who was Abbey Beekeeper, perched on the arm of Abbot Carrul's chair to watch the fun. Some of the other games were in progress, and competition among the Dibbuns was fierce.\n\nThe Abbot watched them fondly as he reminisced. \"I was pretty good at the nut and spoon race in my younger seasons.\"\n\nWeld kept his eyes on the games as he observed drily, \"Aye, Father, you beat me three seasons on the run. Then they caught you sticking your nut to the spoon with honey.\"\n\nAbbot Carrul cautioned him. \"Not so loud, Weld, keep your voice down. We can't have the young 'uns discovering that a Dibbun who cheated at nut and spoon is now their Abbot!\"\n\nThree of the Dibbuns\u2014Muggum, Shilly and Yooch\u2014were trying madly to win the greasy pole event. A big bag of candied chestnuts hung from the top of the pole. It resisted all their efforts. Each time, they ended up skimming dismally down to earth, caked with a mixture of soap and vegetable oil. After some earnest plotting, they hatched up a joint plan. Muggum stood tippaw, grasping the base of the pole. Yooch scrambled up the molebabe's back and stood on his head. Both clung tightly to the pole, then Shilly climbed up over them onto Yooch's head. Holding the pole with one paw, the squirrelbabe strove with her free paw to reach the bag. Unfortunately, the combined height of all three Dibbuns was still short of the prize. Muggum could not look up, his tiny face squinched by the weight of his two pals. But that did not stop him yelling out words of encouragement.\n\n\"Gurr, goo on Shilly, grab ee chesknutters naow!\"\n\nShilly roared back at him. \"I carn't not gerrem, me paw bee's too likkle'n'short!\"\n\nYooch the molebabe grunted his contribution. \"Moi pore bee's flattinged, 'urry up!\"\n\nAmid the spectators' shouts of support and hoots of laughter at the spectacle, Fenna came bounding out. The squirrelmaid hopped up the backs of all three Dibbuns. Launching herself from the top of Shilly's head, she made a graceful leap. Fenna effortlessly unhooked the bag of candied chestnuts. Performing a spectacular somersault, she landed neatly on the ground, without a speck of grease anywhere on her.\n\nShe smiled smugly. \"No trouble at all, the prize is mine!\"\n\nMartha's voice cut across her jubilant cries. \"Not fair! It's the greasy pole you're supposed to climb, not the greasy Dibbuns. You should forefeit the nuts, Fenna!\"\n\nFenna stuck her lip out and pouted. \"But I won them!\"\n\nThe Abbot left his armchair and took possession of the bag. \"The object is to get the nuts. There's no hard-and-fast rule about climbing greasy poles. But be fair, Fenna. The little ones tried so hard, and they gave us all such fun. I suggest we split the nuts four ways betwixt you and them.\"\n\nWhilst everybeast was applauding the decision, Toran caught Shilly and Yooch as they fell backwards from the pole. Horty was left with the task of unsticking Muggum, who was practically plastered to the pole with grease. He tugged his snout politely to the young hare.\n\n\"Thankee, zurr, oi thort oi wuz stucked thurr fer loife!\"\n\nHorty gazed down at his clean tunic, now coated with the mess. \"Oh, think nothin' of it, old lad. My pleasure, wot!\" He slipped and fell flat as he stumbled away from the pole.\n\nBy the pondside an old female squirrel, her face hooded against the sun by a cowl, was bathing her footpaws in the reeded shallows. An otter of medium size, his face also hooded, sat next to her. Sister Portula sought a seat in the reedshade alongside them, fanning her face with a dockleaf.\n\n\"Whew, this is certainly going to be a memorable summer!\"\n\nThe otter glanced sideways at her. \"Has afternoon tea been served yet, Sister?\"\n\nPortula swiped at a flying midge which was tormenting her. \"We never serve afternoon tea when there's going to be an evening feast. You knew that, didn't you, Brother?\"\n\nThe female squirrel sighed. \"Oh no, I was lookin' forward to some nice scones with strawberry preserve an' meadowcream.\"\n\nPortula had to raise her voice to be heard over the sounds of sporting revellers. \"The walltop race will be starting soon. I think first prize for that might be a cream tea with scones.\"\n\nThe squirrel jumped upright, surprisingly spry for one of her long seasons. \"Right, I'll enter an' win first prize!\"\n\nThe Sister shook her head doubtfully. \"You'll have lots of competition from younger and fitter creatures, I'm afraid.\"\n\nThe otter smiled knowingly. \"Oh, don't ye worry about that, Sister. If'n there's a prize of afternoon tea goin', my mate'll win it. Right, Saro?\"\n\nThe squirrel threw off her cowl. \"I'll give it a good try, Brag, an' maybe I'll share it with ye.\"\n\nThe good Sister stared open-mouthed at the aging squirrel. \"Saro, is it really you?\"\n\nSaro took the old Recorder's paw and shook it warmly. \"Aye, Portula, my ole friend, an' guess who this creakin' ruddered lump is?\"\n\nPortula was all aflutter. \"Wait, don't tell me now. . . . Oh, seasons o' mercy, it's Bragoon!\"\n\nShe raced off, waving her paws wildly and shouting, \"They're back! It's Bragoon and Sarabando! They're back!\"\n\nThe squirrel watched her go. \"Hear that, I got me full title!\"\n\nThe games were abandoned for the moment. Redwallers crowded to the pond to see the legendary duo. Both beasts were overwhelmed by pawshakes, kisses, backslaps and the embraces of old friends. Banter and welcomes went back and forth as they were reunited with the comrades of long-gone seasons.\n\n\"Saro, you bushy-tailed rogue, 'tis me, Phredd the Gatekeeper!\"\n\n\"Old Phredd? I don't believe it. Are you still here?\"\n\n\"Och, 'tis that dreadful Dibbun Bragoon! Where've ye been, ye bold wee scamp?\"\n\n\"Sister Setiva, a pleasure t'see yore face, marm. Been? Oh me'n Saro've been as far as there an' back a few times!\"\n\n\"Yurr, oi'd know ee thievin' likkle face anywhurrs, Miz Saro!\"\n\n\"Granmum Gurvel, my ole beauty, give me a hug, quick!\"\n\n\"Haharr, who's that\u2014not young Carrul the nut'n'spoon cheat?\"\n\n\"Bragoon, friend of my Dibbun days, oh 'tis so good to see you! Ahem, the name's changed now, I'm Father Abbot Carrul. But what a pleasure to see you, and Saro, too!\"\n\n\"Look out, who's this big, rough-lookin' villain, eh?\"\n\n\"Oi bee's Muggum, marm, bee's you'm really Sabburandum?\"\n\nSuddenly Bragoon found himself swept off his paws and hugged in a viselike grip. Tears flowed freely down Toran's face.\n\n\"Brother Brag, you've come home to Redwall!\"\n\nPlanting a kiss between Toran's ears, Bragoon wheezed. \"Brother Toran, I won't see sunset if'n ye crush me t'death. I missed ye, Toran, y'great lump of an otter!\"\n\nGreeting upon greeting followed, everybeast seemed at once to be embracing the pair. The air resounded to cries of \"Well I never, my oh my, just look at ye, welcome home!\"\n\nSpringald, Horty and Fenna stood to one side. Like most teen-season creatures, they were embarrassed by all the hugging and kissing among elders.\n\nSpringald muttered in resignation. \"I suppose that means the end of the Games Day. Huh, I'd have won the wall race easily if they hadn't turned up.\"\n\nFenna passed each of them a piece of candied chestnut, musing aloud. \"So, that's the famous Bragoon and Saro. Huh, they're not as big as I thought they'd be. They look pretty old, too\u2014creaky, I'd say. What do you think, Horty?\"\n\nThe young hare shrugged. \"After all the tall stories we've heard about 'em, wot? Actually, old bean, you could be right. Those two ain't exactly the huge giants we've been told about. A bit blinkin' old, an' jolly ordinary, too, though everybeast seems tip over tail to see 'em back, wot? Let's toddle over there now that the huggin'n'kissin' is all done with. Come on, chaps, I want to get a closer dekko at the bold blinkin' Bragoon an' the startlin' Sarobando.\"\n\nMartha was being introduced to the pair by Sister Setiva.\n\nBragoon shook the haremaid's paw gently. \"Martha, eh? A pretty name for a pretty maid. Well, Martha, you don't look anything like us two when we were young. I wager you've heard a lot o' stories about the villainy we got up to in the old days.\"\n\nMartha thought Bragoon had a kind face; she liked him immediately. She tried changing the conversation from his past misdeeds. \"How did you and Sarobando get into the Abbey, sir, with the gate locked and barred?\"\n\nOld Phredd scratched his scrubby beard. \"Aye, how did you get in, eh, eh?\"\n\nSaro shrugged modestly. \"Oh, 'twas nothin' really, just a little trick we used to do with the east wallgate. Don't worry, Phredd, we locked it behind us.\"\n\nFenna interrupted. \"Mister Bragoon, I heard that you were once a Skipper of Otters. Is that true?\"\n\nThe aging otter nodded. \" 'Tis true enough, miss, but ole Saro didn't fancy bein' an otter. So I gave it up to go rovin' with her.\"\n\nSpringald enquired, rather pertly, \"Are you as good a cook as your brother Toran?\"\n\nBragoon chuckled at the idea. \"Wot, me? No, pretty one, I'll wager that Toran's the best cook anywhere. Huh, I'd prob'ly end up burnin' a salad!\"\n\nIgnoring the Abbot's stern gaze, the mousemaid continued. \"Miz Saro, are you as quick as they say you are? I bet I'm faster than you. I won the Abbey wallrace last summer.\"\n\nSaro grinned from ear to ear and shook Springald's paw. \"My congratulations, missy! So then, I'll have a bit o' competition in this wall race. I'm plannin' on runnin' in it for a prize of an afternoon cream tea. Mmm! 'Tis many a long season since I tasted one.\"\n\nSpringald blurted out, \"You're too old, I'll beat you easy!\"\n\nAbbot Carrul was shocked by her behaviour. \"Springald, show some respect for your elders!\"\n\nHowever, it was Saro who interceded on her behalf. \"Not at all, Father, I like to see a young 'un with a bit o' spirit. She's like me at her age. Don't ye fret now, 'twill be a fine race, I'm sure. Let's go to the wall an' get it started. No time like the present, eh, mate?\"\n\nSupremely confident, Springald winked at Horty and whispered to Fenna. \"That old relic's in for a surprise.\"\n\nTurning to Saro, she bowed mockingly. \"After you, marm!\"\n\n##\n\nThe crowd gathered under the threshold of the gatehouse. None of the wall racers was interested in entering. Everybeast was talking about it, eager to see the race between Springald and Saro.\n\nThe Abbot held up his paws. \"So be it, the wall race will start from the threshold above this gate. One circuit of the entire rampart's area, ending back on the same spot. Pushing or shoving means instant disqualification. Runners may use all of the walkway, including the battlements. Any questions?\"\n\nShilly the squirrelbabe piped up. \"Farver h'Abbot, worrabout uz likkle 'uns an' the very very h'old 'uns?\"\n\nShe was referring to the ground race, which was run over the same distance but from the ground level. This was for Dibbuns and Elders, mainly to avoid the dangers of falling from the walltops, where only fit and experienced runners competed.\n\nThe Abbot watched as Foremole Dwurl scored a deep line along the ground with his formidable digging claws. \"Of course, we mustn't forget the ground race. All competitors come up to the line, please. No crowding or jostling!\" He checked the walltop, where Springald and Saro were standing level.\n\nBrother Weld, acting as walltop official, waved down to the Abbot. \"All ready up here!\"\n\nBragoon and Toran sat on the lawn where they could see both races at the same time. Toran patted his ample stomach.\n\n\"Me racin' days are long gone. What about ye, Brother? Yore the same age as Saro, why ain't you runnin'?\"\n\nBragoon folded his paws and settled back. \"I'm far too old. Saro was born on the same day as me, but she's an hour younger.\"\n\nToran scoffed. \"An hour, that's nothin' in a lifetime!\"\n\nHis brother Bragoon maintained a straight face. \"Oh it isn't, eh? Ye try holdin' yore breath for an hour, matey!\"\n\nEvery Dibbun in Redwall was hopping and leaping on the line, waiting for the start.\n\nAbbot Carrul held up a big spotted red 'kerchief, taking one last look around as he called, \"Is that all now, last chance for any late entrants!\"\n\nHorty came bowling up, pushing Martha in her chair as she protested. \"No, please Horty, I've never raced before!\"\n\nThe garrulous hare pushed his sister onto the line. \"Oh piffle'n'twodge, miss. We'll show these blighters what us Braebucks are jolly well made of, wot! Two stout runnin' paws an' a splendid set o' wheels. Hahah, we'll leave 'em all bally well standin', wot wot!\"\n\nToran and Bragoon applauded from the sideline. \"That's the stuff, give it a go, miss!\"\n\nSpringald stood in a ready stance. Saro glanced sideways at her as she pawed the line.\n\n\"Good luck to ye, young 'un!\"\n\nThe mousemaid kept her eyes set on the course ahead. \"Aye, good luck to you, too, old 'un. You're going to need it!\"\n\nSeveral of the Dibbuns made overenthusiastic false starts, causing a slight delay as Toran and Bragoon got them back into line.\n\nAbbot Carrul stood out on the lawn and shouted as the 'kerchief fluttered in the breeze.\n\n\"On your marks . . . Ready . . . Steady . . . Go!\"\n\nAway everybeast went, young and old, on walltop or ground, running at top speed.\n\nCarrul sat on the grass with the two otters. \"Dearie me, some of those Dibbuns have raced off in the opposite direction.\"\n\nToran laughed. \"Oh, let 'em go. They'll still run the same distance at the finish. Flyin' fur'n'feathers! Lookit young Springald go, ye'd think she had wings on 'er footpaws. Looks like Saro is laggin' behind a bit. D'ye think she's in trouble already, Brag?\"\n\nThe otter shook his head. \"She's just pacin' herself, keepin' the mousemaid lookin' back over her shoulder, ye'll see.\"\n\nBoth walltop runners were almost at the north wall corner, with Springald a good two paces in front.\n\nBelow on the grass, chaos ensued. A molebabe and a tiny shrewlet had decided to stop and share some candied chestnuts between them. Another molebabe tripped over them. He forgot the race and joined the pair.\n\n\"Hurr, worrum ee got thurr, candee chesknutters, oi'm gurtly fond o' they'm, boi 'okey oi arr!\"\n\nThe shrewlet passed him a few. \"Den h'eat dese up, nuts make y'go faster, we still winna race, mate!\"\n\nMartha clung tight to the chair as the little cart bounced and bumped furiously forward, with Horty yelling out a warning to them. \"I say there, you bounders, make way or we'll run ye down. Watch out for the corner, me old skin'n'blister. Steer quicker, or we'll knock a hole in that wall, wot!\"\n\nAbbot Carrul shook his head in admiration as he viewed the walltop runners. \"My word, the speed of those two, they're nearly at the east corner already. Look at them go!\"\n\nAs Toran saw them negotiate the corner and tear off along the parapet southward, he groaned softly, \"Aaaah, pore ole Saro's flaggin' now. See, Springald's stretched her lead, I think she's bound to win.\"\n\nA slight smile played about Bragoon's lips. \"The race ain't over 'til the winner crosses the line. You watch, Saro'll soon take the spring out o' Miss Springald.\"\n\nBut by now the mousemaid had turned the south wall-corner, leading by three paces.\n\nThe Abbot commented. \"I think that young 'un's got the field to herself now.\"\n\nBragoon did not answer; instead, he put both paws to his mouth and emitted a single sharp whistle.\n\nSpringald was panting heavily, but still she took time to glance back at Saro as she gasped, \"Give up, old 'un, you're beat!\"\n\nSaro was breathing like a bellows, still hard on her opponent's heels. At the sound of Bragoon's whistle, Saro summoned up all her energy and put on a massive burst of speed. As the finishing line loomed up, Springald set her eyes dead ahead, racing wildly for it. Saro made a mighty leap. She sailed up and over, passing above the startled mousemaid's head, to land beyond the line, half a pace ahead, right beside Brother Weld, who roared out, \"Saro wins!\"\n\nCompletely shocked, Springald collapsed in a heap on the walkway. Fighting for breath, she gasped, \"Wh . . . wh . . . what h . . . happened?\"\n\nWeld the Beekeeper was holding Saro's paw high, shouting, \"The winner by a half pace\u2014Miz Sarobando!\"\n\nOn the ground, three quarters of the way around, more contestants were put out of the race as they met the reverse runners. They collided and fell in a jumble, roaring and arguing.\n\n\"Yurr, wot ways bee's you'm foogles a runnen?\"\n\n\"Uz norra foogles, you knock uz over 'cos we winnin'!\"\n\nMartha steered the cart around them, yelling in panic, \"Slow down, Horty, watch out for those Dibbuns!\"\n\nHer brother narrowly missed the melee, speeding up as he shouted, \"Forward the buffs! Onward t'death or flippin' glory! Blood'n'vinegar, me jolly lads! Redwaaall!\"\n\nHowling and hooting, he rushed over the finishing line, grinding to a halt and losing a back wheel in the process. \"Hoorah, me beautiful ole skin'n'blister, we won. Wot Wot Wot!\"\n\n\"Nay, you'm diddent, zurr. Uz wunned\u2014Shilly an' oi!\"\n\nHorty's mouth fell open. \"But . . . but . . . how . . . wot . . . but?\"\n\nMartha almost fell from her chair laughing. \"Hahahahaha! Muggum and Shilly were first over. Heeheehee, they won. Stop your but butting, Horty, we were second. A great effort on your part, sir. Thank you kindly!\"\n\nShe did not tell him that, when they almost collided with the fallen Dibbuns, she had rescued Muggum from the heap as they whizzed by. Muggum had hold of Shilly's tail, so she, too, was swept aboard the chair. Both of the little ones hopped off the cart, over the line, just ahead of it. Luckily they landed either side of the vehicle.\n\nThe Abbot, who had his suspicions as to who the real winners were, eyed the Dibbuns sternly. \"Who won? I want the truth!\"\n\nMuggum was the picture of infant innocence. \"Troofully, we'm wunned, zurr. Us'n's farster'n woild bunglybees, moi paws nurrly tukk foire!\"\n\nThe Father Abbot shook his head in disbelief until Martha reassured him. Toran and Bragoon backed her up stoutly.\n\n\"Aye, 'twas the Dibbuns who won, fair'n'square!\"\n\n\"Right, mate, would we lie to a great Father Abbot?\"\n\nFolding both paws into his wide sleeves, the Abbot wandered off, muttering, \"Why shouldn't I believe three good and honest creatures? Frogs can fly, fish make nests in trees. Who am I but a poor Abbot who knows nothing?\"\n\nIt was still some time until nightfall and the commencement of the Summer Feast. Under the Abbot's instructions, the kitchen crew had already made a substantial afternoon tea.\n\nSaro threw a friendly paw around Springald's shoulders. \"That was the closest race I've ever run. Come on, young 'un, you'n yore friends must take tea with me. Let the winnin' Dibbuns an' Martha sit with us, too.\"\n\nThe banks of the Abbey pond made a perfect setting as the Redwallers sat in the lengthening noon shadows, watching sungleams on the cool, dark water. Junty Cellarhog, the big hedgehog who took care of Redwall's famous cellars, personally served them with ice-cold rosehip and mint tea. Everybeast gossiped animatedly whilst enjoying the excellent food. Most Redwallers wanted to know more about the famous pair and their adventures. Bragoon had to do most of the answering, as Saro was lost in the ecstasy of scones, meadowcream and strawberry jam. Even Horty was amazed at the amount of food that Saro could put away.\n\nHe remarked in awed tones, \"Good grief, marm, you can certainly deal pretty roughly with scones when you've a blinkin' mind to, wot!\"\n\nBragoon shoved more meadowcream over to his companion. \"Don't disturb Saro while she's eatin', she gets fierce.\"\n\nHorty nodded politely. \"Know wotcha mean, sah. I expect it was jolly tough, wot. All those seasons o' fightin' rascally vermin. Must've given the lady a confounded keen appetite!\"\n\nBragoon nodded. \"Many's the time I've had to count me paws after sittin' too close to Saro at vittlin' time!\"\n\nToran beckoned to his friend Junty. \"Now then, ole cellar-spikes, wot about a bit o' music? Brought yore fiddle?\"\n\nJunty Cellarhog took a small, beautifully crafted fiddle out of the hood of his cloak. He tuned it deftly. \"Rightyo, any pertickler tune ye'd like?\"\n\nHorty volunteered. \"Play the Dawnsong. I'm sure Martha will sing for us. The jolly old skin'n'blister has a rather charmin' voice, y'know.\"\n\nEverybeast began calling for Martha to sing. Junty struck a chord or two. The haremaid bowed in deference to the two guests.\n\n\"Only if Bragoon and Sarobando would like to hear it.\"\n\nThe otter chortled. \"Like to hear it? I'd _love_ to hear ye sing, Martha. All I ever hear is my mate Saro, an' she's got a voice like a frog bein' strangled!\"\n\nThe squirrel looked up indignantly from a half-eaten scone. \"Hah, lissen who's talkin'. Let me tell ye, missy, to hear ole Bragoon singin', 'tis like listenin' to a nail trapped under a door!\"\n\nFenna giggled. \"Then you'd best be singing, Martha. Those two'll curdle the meadowcream if they start warbling.\"\n\nMartha paused until Junty's fiddle had played the opening bars, then she began to sing.\n\n\"I have a friend as old as time,\n\nyet new as every day.\n\nShe banishes the night's dark fears,\n\nand sends bad dreams away.\n\nShe's always there to visit me,\n\nso faithfully each morn,\n\nso peaceful and so beautiful,\n\nmy friend whose name is Dawn.\n\nShe fills the air with small birds' song,\n\nand opens all the flowers.\n\nShe bids the beaming sun to shine,\n\nto warm the daylight hours.\n\nShe comes and goes so silently,\n\nto leave the earth reborn,\n\nserene and true, all clad in dew,\n\nmy friend whose name is Dawn.\"\n\nThere was silence as the last poignant notes hovered on the still air, then wild applause.\n\nBragoon's tough face softened as he sniffed. \"I never heard anythin' so pretty in all me days!\"\n\nHorty puffed out his chest. \"I told you she could sing!\"\n\nSaro, having forgotten her afternoon tea, sat transfixed. \"Sing, did ye say? Listen, even the birds've gone quiet at the sound of the maid's voice. I'm retirin' from singin' as of now. Wot d'ye say, mate?\"\n\nBragoon had borrowed Junty's fiddle. He plucked the strings as he gazed in admiration at the haremaid. \"Our lips are sealed, Miss Martha, ye put us t'shame. Mind ye, I can still knock a tune out on the ole fiddle, an Saro ain't a bad dancer. Shall I play a jig for ye?\"\n\nMuggum had a swift word in Martha's ear, causing her to smile. \"Do you know a Dibbun reel called Dungle Drips?\"\n\nThe Abbeybabes leaped up and down, shouting eagerly. \"Play ee Dungle Drips, zurr!\"\n\nBragoon raised the fiddlebow, winking at Saro. \"Haha, Dungle Drips. We danced to that 'un a few times when we was Dibbuns, eh mate?\"\n\nThe aging squirrel leaped up. \"Aye, I'll say we did! Right, c'mon, me liddle darlins, I'll show ye a step or two. I once was Redwall's Champion Dibbun Dancer!\"\n\nEven before the first notes rang out, the Dibbuns clasped paws and whooped. Saro was whirled off amid a crowd of molebabes, tiny mice, infant squirrels and small hoglets. All the Dibbuns roared the molespeech lyrics with gusto, hurtling themselves into the wild reel. Martha was convulsed with laughter at their antics and amazed at Saro's skill. The squirrel was a born dancer, twirling and somersaulting recklessly as she sang out in mole dialect along with the Dibbuns.\n\n\"Whooooaaah! Let's do ee jig o' Dungle Drips,\n\nwoe to ee furst likkle paw wot slips,\n\nchop off ee tail, throw um in bed,\n\nwiv a bandage rownd ee hedd!\n\nFeed ee choild on strawbee pudd,\n\ngurt fat h'infants uz darnce gudd,\n\nDungle Drips naow clap ee paws,\n\ntug moi snout an' oi'll tug yores.\n\nBow to ee h'Abbot, gudd day zurr,\n\ntwurl ee rounden everywhurr,\n\nDungle Drips bee's gurt gudd fun,\n\noop t'bed naow likkle 'un. Whoooooaaah!\"\n\nThe dance grew more frantic, the singing faster as Bragoon speeded up his fiddling. Muggum and his crew performed some very fancy pawwork\u2014shuffling and high kicking, raising raucous cheers and calling for the fiddler to play even faster. The scene of wild abandon suddenly stretched out into a double line with Saro bringing up the rear as the Abbeybabes cavorted furiously across the lawns and vanished into the Abbey.\n\nBragoon stopped playing and blew upon his heated paws. \"Whew! Wot happened there, Carrul?\"\n\nBewildered, the Abbot shook his head. \"I've no idea. Sister Setiva, do you know what those babes are up to?\"\n\nThe shrewnurse shrugged. \"Och, the wee beasties must have danced off tae their beds. 'Tis no great surprise, ah'm thinkin', after all that racin', eatin' and jiggin'. Ye ken, they must be rare wearied.\"\n\nThe Redwallers sat sipping tea for quite some time. There was no sound from within the Abbey. Then Saro emerged. Chuckling to herself, she sat down wearily, accepting a beaker of tea gratefully.\n\n\"Whew, I ain't as young as I used t'be! That was some dance, I tell ye. Those Dibbuns jigged through the Abbey, up the stairs they went, straight into their dormitory. Before you could say boo, they were flat out on their beds an' snorin'! I felt like joinin' 'em myself. Huh, looks like the liddle 'uns have called it a day.\"\n\nToran looked perplexed. \"But wot about the Summer Feast?\"\n\nAbbot Carrul saw the look of disappointment on his friend's face. \"Cheer up, Toran, we'll have it at midday tomorrow. 'Twill keep until then.\"\n\nHorty's ears drooped mournfully. \"I say, you chaps, all I've had to eat is a few measly scones an' a drop o' tea.\"\n\nMartha slapped his paw playfully. \"Shame on you, I wouldn't call three plates of scones measly. Don't pull such faces, you'll last until tomorrow.\"\n\nThe gluttonous young hare went into a sulk. \"Jolly easy for you t'say, wot. Skin'n'blisters never scoff much anyway, not like us chaps. So be it then! If none of you lot see me round an' about tomorrow, you'd best take a blinkin' good search. You won't be smilin' then. Not when you find the skeleton of a gallant young hare in some lonely corner. Oh yes, indeed, that'll be me, perished t'death from flippin' hunger, wot! Woe is us, you'll cry, an' weep absolute buckets o' tears, thinkin' we should've let the poor brave lad have a small extra scoff last night.\"\n\nBragoon played along with Horty, shaking his head sadly. \"An' wot'll yore skeleton reply to us, ole mate?\"\n\nHorty sniffed. \"It'll say, too blinkin' late, but I told you so, an' yah boo sucks to you, cruel rotten lot! I leave you to your guilty consciences, you heartless bounders. My famished lips are sealed. Wot!\" He stalked frostily into a corner whilst stealing the last scone from under Sister Portula's nose.\n\n##\n\nIt was still warm as darkness fell. When the Redwallers stopped by the water, enjoying a faint breeze, talk turned to the life of Redwall Abbey and gradually to Martha's story. Bragoon and Saro, who had become very fond of the pretty young haremaid, listened intently. Abbot Carrul, Sister Setiva, Toran and Sister Portula all contributed to the narrative, with Martha filling in the details.\n\nWhen the tale ended, Bragoon sat staring at the haremaid's unmoving footpaws, peeping from under her lap rug. The aging otter's voice was extremely sympathetic. \"What a terrible thing t'happen to a young 'un! An' you've never been able to walk since ye can first remember?\"\n\nMartha shook her head. \"No, sir, though 'tis not for the want of trying. I collapse every time I do, as if my footpaws were held there by two pieces of wet string.\"\n\nSaro was impressed by the young one's frankness. \"That's a hard thing for anybeast t'bear. If'n ye don't mind me askin', Martha, wot d'ye do with yourself all day?\"\n\nMartha shrugged. \"Oh, I get around. There's always my kind friends to push me, though I can wheel myself around if I need to. I do a lot of reading and studying, too. Oh, that reminds me, Sister Portula, I left your book in the gatehouse. Old Phredd's still up, I can see the light at his window from here. Let's pay him a visit.\"\n\nThey all strolled across to the gatehouse with Bragoon and Saro pushing Martha's chair. Unusually for Phredd, he was wide awake and answered the door promptly.\n\n\"Young Martha, I was hoping you'd come. I see you brought all your friends, eh? Well come in, everybeast. You'll have to find somewhere to sit, there's not much room, y'know!\"\n\nPhredd spoke to the latch as he closed the door behind them. \"Heehee, got something to show this haremaid, haven't we?\"\n\nMartha sat up eagerly. \"Have you found anything, sir?\"\n\nThe old hedgehog sat on the side of his bed, opening Sister Portula's book at a page he had marked. \"Found something? Hah, the moment that race was over and I could rescue my armchairs back in here, I did some serious reading. There's more important things in life than running oneself silly around walltops, y'know. After all, Martin the Warrior sent you a message that mustn't be ignored, missy.\"\n\nBragoon suddenly became interested. \"Martin the Warrior sent ye a message, Martha? What did he say?\"\n\nThe haremaid explained. \"I fell asleep near the tapestry. Martin and another young mouse named Sister Amyl appeared to me. Martin told me to read, because reading is knowledge, then Sister Amyl spoke this rhyme to me.\n\n\"Where once I dwelt in Loamhedge,\n\nmy secret lies hid from view,\n\nthe tale of how I learned to walk,\n\nwhen once I was as you.\n\nThough you cannot go there,\n\nlook out for two who may,\n\ntravellers from out of the past,\n\nreturning home someday.\"\n\nSaro looked very serious. \"I remember Martin the Warrior spoke to me an' Brag when we were young.\"\n\nAbbot Carrul peered over his spectacles in astonishment. \"Martin spoke to you two? Did he really?\"\n\nSaro kept her face straight. \"Oh aye, I'll tell ye wot he said.\n\n\"Seek adventure, liddle mates,\n\ngo ye forth from Redwall's gates.\n\nBoth of ye, wild and unchecked,\n\nbegone afore my Abbey's wrecked!\"\n\nBragoon chuckled. \"She's only jokin', of course.\"\n\nOld Phredd glared at them both. \"This is no joking matter. As soon as I saw you down by the pond today, I knew you were the two travellers from out of the past. Eh, eh, the two that Sister Amyl's poem spoke of, right?\"\n\nHorty's eyes went wide as saucers. \"Right indeed, wot!\"\n\nPhredd tapped the open book he held. \"Stop jabbering and listen, please, this is most important. I have found the story of Sister Amyl. It was written by another, Recorder Scrittum. He was the Loamhedge brother who put most of this story together\u2014and very well he did it, too. Listen to this!\"\n\nThey sat entranced as Phredd's wavery tones brought the past back to life for them.\n\n\" 'The plague has come to Loamhedge, a great sickness is upon us. This morning we buried four, three sisters and one brother. Our infirmary is packed with the ill and suffering. I fear this Abbey has become a pest hole. Abbess Germaine and her Council have reached a bitter decision: if we are to survive, we must leave Loamhedge. It is almost unthinkable, is it not? Having to forsake our beautiful old home to wander in the wilderness. Germaine speaks of travelling to Mossflower country, where she has friends who will give us shelter. We are to take very little with us and live off the land as we go. These are hard and sad times, indeed.\n\n\" 'However, there is no other way for it. Poor Sister Amyl is a young mouse who has never walked. She makes her way about in a wheeled chair. Amyl has decided not to go with us. I pleaded with her, saying that I would care for her and push the chair to wherever we were bound, but she would not hear of it. Amyl said that the journey would be far too arduous and feared that she would hold us back. In a way she is right, since a wheeled chair cannot be hauled over hill and dale. There would be bad weather to contend with\u2014rivers, swollen streams, rocks and swampland. Also, it will soon be wintertide. The Abbess does not know of Amyl's decision yet. It is my sad duty to tell her of the situation. Young Sister Amyl is such a good creature. It will break my heart to leave her at Loamhedge, amid the dying.' \"\n\nToran interrupted the narration by sniffing loudly and grubbing a paw across his moist eyes. \"Pore liddle thing, left t'die in a deserted Abbey. I'd never leave ye to a fate like that, Martha, no matter wot it took!\"\n\nBragoon grasped the haremaid's paw. \"Me either, miss!\"\n\nMartha forestalled Saro and the rest by holding up a paw. \"I know you wouldn't, none of you. . . .\"\n\nShe caught sight of Old Phredd, glaring about impatiently. \"Oops, sorry sir, we'll be quiet, I promise!\"\n\nThe Gatekeeper huffed, then leafed on to another marked page. \"Thank you! Now let me read further into this narrative. Here is a section by Recorder Scrittum, concerning setting up camp on the first evening of the journey.\n\n\" 'Let me tell you of a miracle! Can I believe my eyes, you must take what I tell you as true, I have always been a faithful recorder, and never given to lying. Here was I, trudging along carrying my writing equipment and a sack of provisions. We were heading for a streambank with high sides, where there would be shelter for the brothers and sisters. I was travelling somewhere in the centre of the column, not having seen the Abbess, as she was leading up at the front. I came away from Loamhedge, filled with shame and remorse, being too overcome with grief to bid Sister Amyl farewell. I slunk off like a thief. Then, from the rear of the marchers, a mighty cheer rose up. I trekked back to see what was causing such jubilation. There across the heathland, limping slowly but walking without any shadow of a doubt, came young Sister Amyl!' \"\n\nAgain, Phredd's recital was interrupted when a hearty cheer came from his listeners. The old hedgehog made as if to slam the book shut.\n\n\"Do you want to hear the rest of this, or shall I lay back on my bed and go to sleep, eh, eh?\"\n\nSomewhat embarrassed, Abbot Carrul replied, \"Forgive us, friend, we'll stay silent. It was just that we felt so happy for Sister Amyl, we had to cheer.\"\n\nPhredd went back to his book, muttering, \"Aye, so did I when I first read it. Ahem, allow me to continue. 'Was it a miracle, or some sort of magic? I had told the Abbess of Amyl's plight. She was sorrowful, of course, but informed me she would have a word with Amyl. What came of their conversation, I did not know. But here was my young friend, as large as life and up on her footpaws. Later that evening we sat by the fire, exhausted after the day's long march. Sister Amyl lay wrapped in her cloak sleeping deeply. I sought out Abbess Germaine and spoke to her about the amazing happening. Here is what our great and wise Mother Abbess told me. She said that she had recalled a formula, given to her by an old healer, many seasons ago. Searching through her belongings, she had found the parchment. This she gave to Amyl, telling her that she must decide on her own whether to stay or whether to read the formula, learn from it and undertake the journey. Obviously, Sister Amyl must have read what was written on the parchment. Was it a magic spell, or some remedy of herbal medicine? The Abbess would not tell me.' \"\n\nMartha stifled a cry of disappointment, nevertheless listening dutifully as Phredd continued reading.\n\n\" 'Next morning I dropped to the rear of the column and walked with Sister Amyl, whose pace was getting stronger and more sure as the day went on. I told her what I had gleaned from the Abbess and faced her with the question: What was written on the parchment?\n\n\" 'Amyl gave me one of her rare, secretive smiles and refused to speak of it. All that day I persisted, harassing her to divulge the information. It was only after a full day's march through sleeting rain and harsh country that she relented. We were camped beside a rocky tor, huddled in our cloaks around the fire, when she finally spoke. Her words are etched into my memory, and here they are, for what it's worth. The message on the parchment would be of no use to you. It would only have a meaning for somebeast who is greatly troubled in mind or body. Once I had learned what the old healer's rhyme was, I left the parchment behind at Loamhedge. I carry its power within me now, but any creature in need of those words must seek it out for themselves.\n\n\" 'Beneath the flower that never grows,\n\nSylvaticus lies in repose.\n\nMy secret is entombed with her,\n\nlook and think what you see there.\n\nA prison with four legs which moved,\n\nyet it could walk nowhere,\n\nwhose arms lacked paws, but yet they held,\n\na wretched captive there.' \"\n\nPhredd closed the book decisively, addressing its cover. \"My bed calls me. I bid you a weary goodnight.\"\n\nBragoon protested. \"Is that all there is?\"\n\nAbbot Carrul reassured the otter. \"If there was more, my old friend would have told you. Right, Phredd?\"\n\nThe ancient Abbey Gatekeeper reached for his nightshirt. \"Right indeed, young Carrul. I have given you all the information that is of interest to you, namely, Sister Amyl's story. We already have a map of the route to Loamhedge that was used by Matthias in his search for his son Mattimeo.\"\n\nSaro yawned and stood up stretching. \"We'll look at that tomorrow. After all that racin' an' jiggin', I'm ready for bed, too. That poem of Sister Amyl's, 'tis a real tail twister an' no mistake. Flowers that never grow, prisons with four legs an' no paws. An' who in the name o' fur'n'bush is Sylvaticus lyin' in repose?\"\n\nOld Phredd poked his head through the neck of the nightshirt. \"Sylvaticus was the first Abbess of Loamhedge. Don't know where I learned that, must have been at Dibbun School. Hmmm, that was more seasons ago than I care to remember. Funny how old little facts stick in one's mind. Don't slam my door when you leave, it doesn't like being slammed. Goodnight!\"\n\nThey strolled back to the Abbey through the balmy night air, discussing the whole thing.\n\nMartha turned to Bragoon and Saro, who were pushing her chair. \"Phredd said that you were the two travellers from the past. Do you believe him?\"\n\nBragoon nodded. \"Of course we do, beauty. Don't ye fret now, me'n my mate'll bring that parchment back from Loamhedge for ye. Ain't that right, Saro?\"\n\nThe aging squirrel's reply left Martha in no doubt. \"Aye, I'll wager a split acorn to a cream tea on it, missy. We'll have ye up'n'dancin' in no time!\"\n\nThe haremaid's face was a picture of joy to behold. \"I will dance someday just for you, my good friends. Tomorrow I'll make a copy of Sister Amyl's poem so you can take it with you in case you forget the words.\"\n\nHorty did a small hopskip of eagerness. \"Splendid idea, my wise an' pretty sis. I'll take charge of it, like a sort of jolly old mapfinder. Wot!\"\n\nBragoon and Saro exchanged glances, and the otter murmured, \"We'll have to see about that.\"\n\nFurther discussion was cut short. Sister Setiva met them at the Abbey doorway. She stood in a pool of golden light, holding up a lantern. The stern old Infirmary Keeper cast a jaundiced eye over the new arrivals.\n\n\"Ah'm tae shew ye to yore beds. There's two spare ones in the room next tae mine.\"\n\nBragoon bowed appreciatively to her. \"It'll be a treat to sleep in a real bed again, Sister.\"\n\nSaro agreed. \"Aye, after some o' the places we've laid our heads down. But we'll be up at the crack o' dawn, ready to lend a paw with yore problem, Martha.\"\n\nBragoon thumped his rudder down firmly. \"Ye can bet yore brekkist on that, missy. We won't let ye down!\"\n\nMartha clasped their paws fondly. \"Pleasant dreams to both of you.\"\n\nThe pair found themselves being prodded, none too gently, with Setiva's blackthorn stick.\n\nShe commanded them in a no-nonsense voice. \"Follow me tae mah sickbay, an' 'twill be woe betide either of ye if ah hear just one wee snore disturbin' mah rest, d'ye ken?\"\n\nBragoon saluted her smartly. \"Oh, we're kennin' away like a pair o' good 'uns, Sister. Lead on!\" They grinned at each other, listening to the shrewnurse while she chunnered away to herself as she shuffled upstairs.\n\n\"Ach, I'll have tae dig oot fresh sheets an' coverlets! Ah'm thinkin' they're big enough tae make their ain beds, great roarin' villains! Ah'll nae sleep a whit taenight, knowin' they two are in the next room tae mine!\"\n\nOpening the infirmary door, she glared at her guests. \"Wipe the mud off ye're paws an' the silly grins offn'n ye're faces. Ah'll be inspectin' yon sickbay on the morrow, an' ah'll skelpit the pair o' ye if'n there's one wee thing oot o' place, d'ye ken? Ah bid ye a silent guidnight!\" She slammed the door and retreated into her own chamber.\n\nBragoon burst out sniggering as Saro called out in imitation of Setiva's far northern accent.\n\n\"Aye, we ken, Sister, an' a guidnight to ye, too, the noo!\"\n\nThe Sister's strict tone rang out from the adjoining room. \"Ah'll be in there wi' mah stick if there's anither sound, so get tae sleep an' no talkin'!\"\n\nSaro whispered in Bragoon's ear. \"Goodnight, mate.\"\n\n##\n\nEarly morn found the northeast skies showing more promise of decent weather. Outside the holt of Shoredog, pleasant sunlight was turning the mist into a warm yellow haze over the stream.\n\nLonna Bowstripe limped out with the rest of the sea otters to witness the arrival of the otter known as Garfo Trok. He had come in a peculiar-looking craft, a long, battered old boat with rounded stern and for'ard ends. It had a rickety cabin erected amidships and sported a square, heavily-patched sail, which was furled around a much repaired crosspiece.\n\nGarfo was a stream otter, a jovial, fat beast. He wore an old iron helmet that resembled a cooking pot, and a permanent smile on his broad, friendly face. Shipping his paddling pole, Garfo waddled ashore and began singing in a dreadfully toneless voice.\n\n\" 'Tis a long ways down the stream, me lads,\n\nwhen a beast ain't got no grub oh,\n\nwid a belly like a wind-blowed sail,\n\naboard this leaky tub oh.\n\nIf I fell overboard like this,\n\nall thin'n'pale'n'slack oh,\n\na pike'd take one look at me,\n\nan' quickly chuck me back oh!\n\nMe ribs are showin' through me fur,\n\nI'm frightened o' the weather,\n\nin case a sudden gust o' wind,\n\nwhips me off like a feather.\n\nMe cheeks are sunken hollow,\n\nan' me nose is wintry blue, lads,\n\nme rudder's covered in green mold,\n\nI'm sufferin' from the Doodads!\n\nTake pity on this riverdog,\n\nan' feed me good ole vittles,\n\nsome skilly'n'duff to stop me bones,\n\na-clackin' round like skittles.\n\nA pot or two o' barley stew,\n\nan' nutbread by the plateful,\n\nan' a bathtub full o' custard, lads,\n\nwould find me ever grateful!\"\n\nThe sea otters laughed and applauded Garfo heartily, then gathered round as he shook paws, patted backs and kissed babes, all the while hooting in booming tones, \"Whoohoohoo, slap me rudder an' curl me whiskers! Lookit ye lot. Wot 'ave youse been feedin' yoreselves on? Y'all look so chub'n'sparky! Ma Sork, me ole tatercake, are ye still bakin' the primest nutbread in the northeast?\"\n\nOld Sork whacked him playfully with her ladle as he picked her up and hugged her. \"Put me down, ye great fatbarrel. I've been up all night bakin' nutloaves to feed yore hungry gob!\"\n\nGarfo put her down and cast a jolly eye over Lonna. \"Whoohoo, shrivel me snout an' gravel me guts! So this is the giant stripedog I'm carryin' as cargo. Hah, I thought I was a big 'un, but ye could eat dinner of'n me head, mate!\"\n\nLonna shook Garfo Trok's paw. \"Pleased to meet you, mate, but I'm not just cargo. My name is Lonna Bowstripe, and I can wield a paddle as good as most.\"\n\nGarfo was big and well built for an otter, but Lonna's giant frame towered over him. He released the badger's huge paw.\n\n\"Wield a paddle, big feller? Whoohoo, ye look strong enough t'carry me an' my old boat _Beetlebutt_ up a waterfall on yore back! Belay, Lonna, let's get some brekkist afore we sail.\"\n\nLonna had already eaten, so he sat nibbling a crust of rye bread and sipping some plum cordial whilst Garfo dealt with breakfast. The otter was a mighty eater and extremely odd in his choice of food. He spread nutbread with honey and dunked it into hotroot soup. Breaking up an apple pie, he crumbled it into a bowl of mushroom stew, daubing plum preserve on an onion-and-leek pastie.\n\nClearing the lot in a remarkably short time, Garfo stood up, patting his big stomach. \"Ahoy, Lonna, pack that bow'n' arrers an' let's go sailin'. Can't waste a fine mornin' sittin' here vittlin', like some I've seen. Never could abide greediness in a beast!\"\n\nThe otters had packed _Beetlebutt_ with an amazing array of provisions. Lonna looked around at the faces of all these otters that he had come to like so much. It was going to be a sad experience saying good-bye to them. Garfo stood, waiting to push off, as the badger went in turn to each of his otter friends\u2014Shoredog, Sork, Marinu and many others, saving his last farewell for Abruc and young Stugg. Lonna embraced Abruc warmly and clasped his paw. A tear coursed down the big badger's scarred face.\n\n\"Farewell to you and your family, my good friend. I will never forget you and your son. You saved my life, cared for me, fed and nursed me. All I can give you in return are my thanks and undying friendship!\"\n\nAbruc scuffed the ground with his rudder, then looked up at the big badger. \"Friendship is the greatest gift one can give to another. You are a goodbeast, Lonna. I know ye would've done the same for me an' mine if'n ye found us lyin' hurt. Go on, mate, you go now, an' know our thoughts are always with ye!\"\n\nStugg tugged at Lonna's paw until the badger lifted the young otter and held him level with his eyes. His face solemn, Stugg wiped a tear from Lonna's striped muzzle.\n\n\"Lonn', der is somet'ink you can do for me an' my farder. Get Rag' Bol an' dose Searats, so they don't hurt no more pore beasts!\"\n\nThe badger put Stugg back down and stepped aboard the boat. Raising his bow, he called out as Garfo pushed off into the midstream.\n\n\"Stugg, my little mate. I swear by the fine string your father made for this bow. I will wipe Raga Bol and his Searats from the land forever. This is my oath, and my promise to you. Good-bye!\"\n\nPutting aside the bow, he joined Garfo Trok at the paddling poles.\n\nFighting away the tears, Lonna did not look back as they sped downstream. Behind him the tribe of Shoredog stood on the banks, singing an old sea otter song of farewell.\n\n\"When the sun sets like fire,\n\nI will think of you,\n\nwhen the moon casts its light,\n\nI'll remember, too,\n\nif a soft rain falls gently,\n\nI'll stand in this place,\n\nrecalling the last time,\n\nI saw your kind face.\n\nGood fortune go with you,\n\nto your journey's end,\n\nlet the waters run calmly,\n\nfor you, my dear friend.\"\n\nGarfo Trok had spent his life amid the northeast streams and rivers. There was no waterway for leagues that the burly otter was not familiar with. Lonna obeyed his every order, backing and tacking down the broad stream. They made good progress. Midday found the Beetlebutt running smoothly with a fair breeze running astern.\n\nGarfo shipped his long paddle, gazing up at the blue, cloud-flecked sky. \"Let the ole lady drift for awhile, mate. Belay that paddle an' we'll haul sail an' take a bite o' lunch.\"\n\nThey released the sail and made its ends fast to the cleats. Lonna had been wondering when the otter's appetite was going to reappear. Together they sat on the roof of the little midships cabin, drinking cider and eating nutbread.\n\nGarfo chuckled as he watched the big badger consume his lunch. \"Whoohoo, ain't nothin' wrong with a beast who kin eat hearty, mate! That limp o' yourn will soon clear up with a good cruise. Ye won't be walkin' so much.\" Lonna liked the feel of a boat beneath his paws; he felt rested and well.\n\nGesturing ahead, he enquired, \"How long can we go by water, Garfo?\"\n\nThe otter refilled his beaker. \"Almost into Mossflower. This ole stream takes a turn there an' runs back east. I kin see yore wonderin' 'ow far ahead those vermin are.\"\n\nLonna eyed him keenly. \"Aye, can ye tell me, mate?\"\n\nGarfo scratched his rudder thoughtfully. \"Raga Bol has t'go by land since they ain't got no boat an' there's too many of 'em for small rivercraft. Those Searats should be well into Mossflower Wood by now. I'd say ye was about ten days behind 'em, Lonna. But I kin cut that down to eight, wid some canny sailin'. Don't fret, mate.\"\n\nThe badger's eyes narrowed, the look on his ruined face caused the otter to shudder. Lonna laughed mirthlessly. \"Oh, I'm not fretting at all. I'll catch up to them for sure!\"\n\nThe country they were sailing through was open, with no tree cover. Gradually it ran into hills and gorges, the streambanks growing higher on either side.\n\nGarfo pointed to a steep bend up ahead. \"When we round the point of yon bend, we'll be meetin' up with Buteo. Now I know yore not a-feared of anybeast, but don't start anythin' wid him. I've knowed Buteo a long time.\"\n\nLonna was intrigued. \"Just as you say, mate, but who is Buteo?\"\n\nGarfo crumbled some nutbread on the cabin roof. \"Oh, ye'll find out soon enuff, matey, soon enuff!\"\n\n _Beetlebutt_ took the bend smoothly, keeping to midstream. Halfway around it, Lonna was startled to feel a slight cuff on the back of his head. Buteo landed like a bolt of lightning, silent and menacing. He was a honey buzzard\u2014a large, savage-looking bird of prey. From fawn-barred tail to mottled chest, and huge wingspan to lethal-hooked beak and a fierce eye, Buteo looked every inch a killer. Folding his wings, the buzzard stared disdainfully at the crumbled nutbread that Garfo had put out for him, then pointed a lethally sharp talon at them.\n\n\"Heek! This be Buteo territory, I rule here. Heeeeeekah!\"\n\nGarfo replied cheerily. \"So ye do, me ole burdy, but we ain't trespassin', just passin' through.\"\n\nButeo cocked his head to one side, glaring at them. \"Yaheeek! I riddle you riddle, you spin me a spin. Only pass here if you win. Good?\"\n\nGarfo cautioned Lonna to silence with a warning glance. The badger watched as the otter appeared to consider this proposition.\n\n\"Good it is, Buteo. You go first.\"\n\nThe honey buzzard stared up at the sky, a thing that honey buzzards do when trying to appear mysterious. \"Heeeeekoh! What be brown'n'yellow, fat'n'mad, an' if you slow, sting you bad?\"\n\nGarfo scratched his rudder, shaking his head, as if really perplexed. \"Frazzle me whiskers, Buteo, that's a real poser!\"\n\nButeo pecked up the crumbled nutbread, sniggering. \"Keeheeheehee! Stupid riverdog not crossing through my country. Buteo much clever. Keehar!\"\n\nGarfo tipped a sly wink to Lonna, then jumped up shouting. \"I got it, 'tis a bumbly bee!\"\n\nBoth Garfo and Lonna had to avoid the buzzard's wings as he beat the air in frustration. \"Yeekeeha! How you know?\"\n\nThe otter twitched his nose modestly. \"Oh, I just took a guess. But it was a great an' clever riddle.\"\n\nButeo stalked up and down, digging his talons angrily into the cabin roof. Then he turned and wheeled on Garfo. \"Yeeee! You still not go 'til you spin me. This time I win!\"\n\nThe crafty otter produced a flat pebble from his helmet, spat on one side of it and held it up for the bird to see. \"Right, I'll spin ye\u2014dry side I win, wet side you lose. Good?\"\n\nThe honey buzzard nodded eagerly. \"Keehee! I take wet!\"\n\nGarfo spun the pebble into the air, chanting, \"Up she comes, down she goes, how she lands, nobeast knows!\"\n\nButeo's keen eyes watched every spin of the stone until it clacked down flat on the deck.\n\nGarfo grinned from ear to ear. \"Wet side, you lose!\"\n\nThe buzzard hovered over the otter, glaring murderously at him. Garfo sat munching a chuck of nutbread, looking the fierce bird straight in the eye. \"Ye've got to let us pass now, mate, or you ain't a bird whose word can be trusted.\"\n\nFearing that the buzzard was going to attack Garfo, Lonna braced himself to spring upon it.\n\nThe bird's black and gold eyes dilated wildly as it screeched. \"Allbeast know Buteo be a bird of honour, my word always good. I slay anybeast who say different. Yeeeeeekaaaah!\"\n\nSnatching the nutbread from the otter's paw, he soared off into the air\u2014up and up, until he was a mere dot in the sky.\n\nLonna relaxed gratefully. \"That was a close call, my friend. Buteo looked like a bird who would fight to the death. How did you manage to hoodwink him like that?\"\n\nGarfo Trok winked knowingly. \"I been doin' it a long time, mate, whenever my journeys take me by this way. Pore ole Buteo's memory's scrambled from too many battles. Besides, he ain't the brightest o' birds. Funny how he loses every time. I'll let him win on the return trip, 'cos I'll be bound back nor'east anyway. That's fair enough.\"\n\nLonna could not help laughing at the sly otter. \"You great fat fraud! Shame on you, Garfo Trok!\"\n\nNibbling on a piece of cheese he had found, Garfo waved his rudder nonchalantly. \"Better'n havin' to fight t'the death wid a mad buzzard. You said so yoreself, mate. Anythin' for an easy life, that's my motto.\"\n\n##\n\nThe Searat Blowfly sat on a rotten log, cooling his footpaws by rubbing them in the rich, damp loam. Gazing up at the trunks of mighty woodland trees, with their canopy of sun-pierced green, he murmured to the Searat sitting alongside him.\n\n\"I likes this 'ere Mossflower place, better weather 'ere than on that nor'east coast. Plenny o' shelter an' prime vittles, too!\"\n\nHis companion, a sad-faced Searat called Rojin, rubbed his blistered footpaws tenderly as he complained. \"Huh, if only we wasn't marchin' so much. I ain' cut out fer all this trekkin'. I'm a Searat, norra landlubber!\"\n\nHangclaw, another rat, limped over to join them. Rooting with his daggerpoint at a splinter in his footpaw, he spat in disgust.\n\n\"Right y'are, shipmate, just look at me pore trampers. Why are we walkin' all the time. Where's ole Bol got us bound to? We're traipsin' around all day an' 'arf the night!\"\n\nGlimbo, the one-eyed rat who had been first mate aboard ship, had been loitering nearby, eavesdropping on the three crewrats. Sneaking up behind them, he gave the rotten log a hard shove with his spearpoint, sending the trio sprawling into the loam.\n\n\"Gerrup on yer paws an' quit whinin', ye slab-sided sons o' worms. If the cap'n catches ye, he'll leave youse here to rest as food fer the ants. Now march!\"\n\nRaga Bol had been marching up in front of the others but had looked back over his shoulder so often that the crew could not fail to notice. The Searat captain dropped back until he was level with Glimbo. Catching his mate's sleeve with the deadly silver hook, Bol swiftly dragged him behind a broad sycamore trunk.\n\nGlimbo's sightless eye rolled in its socket as he saluted. \"They're all on the march, Cap'n!\"\n\nRaga Bol poked his head out from behind the tree and snarled at the backstragglers. \"Keep movin', I'm watchin' ye!\" Then he turned his attention to the trembling Glimbo. \"They're talkin' about me, wot're they sayin'? The truth!\"\n\nThe mate was trembling so hard that the back of his head made a noise on the tree trunk like a woodpecker. \"N . . . nothin', Cap'n, they ain't sayin' nothin'.\"\n\nHe heard the slither of cold steel as Bol drew his scimitar. As Raga Bol pulled him close, Glimbo could see the glint of his captain's gold teeth. He knew how dangerous the captain's moods were becoming.\n\nWith his scimitar upraised, Bol hissed, \"They must be sayin' somethin', ye mud-brained idiot!\"\n\nWords poured out of Glimbo at breakneck pace. \"On me oath, Cap'n, the whole crew's sayin' 'ow thankful they are to ye for bringin' 'em 'ere, where 'tis sunny an' there's easy pickins. It's just that they ain't used to all this marchin' . . . some of 'em gotten sore paws.\"\n\n _Thunk!_ The scimitar blade cut deep into the sycamore, taking off a tuft of Glimbo's whiskers. \"Sore paws, is it? You tell any beast moanin' about sore paws that I'll chop 'em off an' make 'em march on the stumps! Aye, an' ye can tell all the crew to quit starin' at me all the time. An' ye can tell 'em another thing, too. Any rat I 'ears mentionin' that giant stripedog, I'll make 'im eat his own tongue. There ain't no big stripedog follerin' me, d'ye hear?\"\n\nGlimbo gulped hard, knowing how close to death he had come. Raga Bol wandered off without warning, leaving him to pull the scimitar loose and return it. The mate was surprised to see his captain sit down in the loam and speak in a voice that almost had a sob in it. \"I ain't been sleepin' at nights. Post extra guards around me when it gets dark.\"\n\nGlimbo dislodged the blade and returned it to his captain. Raga Bol grabbed the scimitar, staring suspiciously at him.\n\n\"Stop starin' at me like that, thick'ead. Gerrabout yer business an' make 'em march faster!\"\n\nGlimbo saluted and walked off bemused. This was not the Raga Bol he knew from the seafaring days. The captain was definitely acting strange. He glanced back at Bol, but the captain did not notice him looking, because he, too, was peering back over his shoulder.\n\nBadredd felt the early sun on his muzzle as he lay on a soft patch of moss, with both eyes closed, feigning sleep. He listened to the voices of the gang, identifying each one as they spoke.\n\n\"Sure 'tis a luvly morn, an' a grand ould spot t'be enjoyin' it in!\" Flinky had an unmistakable accent.\n\nHis mate, Crinktail, was next to speak. \"Which way d'ye want these woodpigeon eggs boilin'?\"\n\nFlinky replied, \"Keep 'em nice'n'soft, me ould darlin'. I've never been fussy on hardboiled eggs.\"\n\nCrinktail sounded cheerful. \"I'll cook night an' day for ye, if'n yew can fool that little fox into lettin' us stay by this water for a few more days.\"\n\nBadredd heard Juppa's voice chime in. \"Aye, this is a prime spot. See if'n ye can fool the liddle idjit to stop 'ere fer a score o' days!\"\n\nFlinky oozed confidence. \"Leave it t'me, mates. I'm a silver-tongued ould charmer when I wants t'be!\"\n\nBadredd yawned convincingly, then, opening his eyes, sat up lazily and stretched. \"Boiled woodpigeon eggs, eh? Bring 'em over here, Flinky, I 'ope they're done nice'n'soft.\"\n\nThe stoat gritted his teeth but obeyed the new chief's orders. \"Top o' the mornin' to ye, sir, an' another grand day 'tis, t'be sure. Now ye enjoy those eggs, there's plenny more around. We was just sayin' wot a fine spot ye chose fer us. Yore a wise leader, so y'are!\"\n\nBadredd put the eggs to one side and stood up, sword in paw. Scowling darkly, he asserted his authority. \"Don't get to like it too much, you lot, 'cos we're movin' on as soon as we've eaten. So pack up yore gear an' stand by, ready t'march as soon as Plumnose gets back!\"\n\nHalfchop's face was the picture of dismay. \"But didn't ye say we wuz stayin' 'ere for a coupla days?\"\n\nThe little fox gripped his cutlass tighter. \"Well, I just changed me mind. A chief can do that!\"\n\nSlipback stood paws on hips, facing up to Badredd. \"Changed yore mind, eh, jus' like that! An' where d'ye think yore takin' us, eh?\"\n\nRaising the cutlass, Badredd took a pace forward and snarled nastily at the weasel. \"We're goin' to this Abbey place, if 'tis any business of yores. So git yore tackle t'gether!\"\n\nSlipback turned to the others, scoffing insolently, \"Hah, looks t'me like the liddle fox needs a magic sword t'make 'im look bigger!\"\n\nBadredd's temper snapped. He swung at the weasel's unprotected back, chopping off his tail with a single blow.\n\nSlipback screeched in pain. \"Yeeeaaaargh, me tail!\"\n\nHis mate, Juppa, hastily slapped a pawful of bank mud on the severed stump. Slipback lay moaning, half fainting with the agony.\n\nJuppa glared accusingly at Badredd. \"Ye had no call t'do that to 'im!\"\n\nAs the fox once again flourished his cutlass, the gang fell back. He saw the fear in their eyes and exulted in it. \"Next time anybeast talks t'me like that, I'll slay 'im! Oh, I know wot ye've been sayin' be'ind me back. Think ye can fool me, do ye? Well, dig the dirt out yore lugs an' lissen. I'm rulin' this roost, an' wot I say goes! I'm goin' to own that magic sword, aye, an' take the Abbey, too. Anybeast who sez diff'rent, let 'em speak now!\"\n\nFlinky raised his paws placatingly. \"Ah, sure now, who'd be wantin' t'get themselves slayed by battlin' wid a fine great warrior like yoreself? 'Tis just that we thought ye was goin' to stop 'ere a few days.\"\n\nIt was then that Badredd knew he was really the leader of the gang. A feeling of power surged through him. Now he could be as cruel and commanding as Burrad or Skrodd. Had he not just drawn blood? Curling his lip contemptuously, he growled, \"I do the thinkin' from now on. We're goin' to the Abbey. Come on, Slipback, up on yer hunkers, ye ain't dead yet.\"\n\nWith a poultice of mud and dockleaf tied to his severed tail, the weasel rose slowly, fixing Badredd with a stare of hatred. \"There's eight of us an' only one of you, fox. Don't get too big'n'fancy wid yore ideas, 'cos ye've still got to sleep at nights. I wouldn't turn me back on us too often if'n I was you\u2014ye can't kill us all!\"\n\nBadredd realised the truth in Slipback's statement, but now that he had all this newfound power he was not backing down. With his cutlass blade, Badredd upset the small cauldron of water over the campfire. It went out with a hiss and a cloud of steam.\n\nAt that moment, Plumnose came lumbering back through the woodlands. The ferret's oversized nose wobbled from side to side as he took in the scene. \"Huh, wod's bin goin' on, mates?\"\n\nFlinky began explaining. \"Ah well, Plum, me ould messmate, wait'll I tell ye wot . . .\"\n\nBadredd shoved the stoat roughly aside. \"I'm the chief now\u2014make yore report t'me. Well, wot did ye find?\"\n\nPlumnose pointed in the direction he had been scouting. \"Er, over der, I'b found a path dat runs south't'north. I t'ink dat's der way to the h'abbey. Id's aboud h'a day's march, Chief, to d'path I mean.\"\n\nBadredd pointed with his blade. \"Get movin', you lot. Plumnose, you go up front an' show 'em the way. Slipback, Juppa, Crinktail, Flinky an' Halfchop, up front wid 'im. I ain't walkin' wid youse behind me. Rogg an' Floggo, you bring up the rear wid me.\"\n\nHe shook the cutlass at Flinky. \"An' remember this, old silver tongue, no gossipin' an' plottin', 'cos I'll be watchin' ye. There'll be no more coaxin' me inter things wot I don't wanna do. Now move yoreselves!\"\n\nIt was pleasant walking through the woodlands. Patches of light and shade mottled the grass, and many forest blossoms were coming into bloom. The weasel brothers, Rogg and Floggo, were a taciturn pair. Since both of them carried bows and arrows, Badredd had kept them back with him. He explained their duties as he watched the backs of the gang, marching ahead. Badredd confided to the weasel brothers as though they were lifelong friends.\n\n\"Stay by my side, mates, I'll make ye both my seconds in command. Keep yore eyes on the rest of that gang an' watch me back. Aye, ye two look true'n'blue t'me. When we conquer that Abbey place, I'll reward ye well. Mark my words, ye'll live the lives o' kings!\"\n\nRogg and Floggo were not at all impressed by the little fox's brags and promises. They had seen gang leaders come and go, each one as ruthlessly cruel as the next. Keeping a stolid silence, the brothers marched dutifully on. Badredd kept a half pace behind them, carrying the cutlass over one shoulder like a spear. He had tried wearing it thrust into his belt, but the blade was too long. It dragged along the ground and got caught twixt his footpaws, causing undignified stumbles. Leaders could not afford to look foolish to those serving them.\n\nMorning wore on to midday. The gang's initial feelings of a brisk march through pleasant country began to pall as the going got more difficult. Those who were marching in front began complaining when they had to pass through a wide area of stinging nettles. Badredd roared at them to carry on in silence, which they did but only briefly. They had come upon marshy ground\u2014not too deep but very uncomfortable\u2014and soon were grumbling loudly. Swarms of midges attacked as the vermin struggled through the smelly, oozing mud. This time they ignored Badredd's shouts and threats, even hurling insults back at him. After what seemed like hours, the front marchers emerged onto firm ground. Badredd and his bodyguards Rogg and Floggo hurried to catch up with them.\n\nThe gang had found a dry, sunny clearing where they lay, looking sullen and rebellious. One glance at their mud-splashed, insect-bitten faces warned their leader of trouble to come should he start roaring out orders to continue marching. Badredd forestalled this by sitting down wearily and commenting, \"Ye did well there, mates, let's rest 'ere awhile. Ahoy, Plum, are ye sure this is the right way? Are ye sure that hooter o' yores didn't wobble in the wrong direction, eh?\"\n\nNot even a snigger greeted his little joke. Picking dried mud from his nosetip, the ferret replied dully, \"Dis is duh way h'I went awright.\"\n\nThe vermin gang had no supplies with them and were too tired to forage. Crinktail and Halfchop stretched out and began taking a nap in the warm sunlight. Plumnose, Juppa, Slipback and Flinky sat in a group, conversing in muted tones. Rogg and Floggo slouched nearby, their eyes half closed.\n\nBadredd began feeling dozy in the midday heat, but he forced himself to sit up and look alert. He saw Slipback glance his way, then whisper something to Juppa. The little fox pointed the cutlass at them.\n\n\"Cut out the whisperin', I'm warnin' ye!\"\n\nFlinky grinned impudently and threw a lazy salute. \"Ah sure, they wasn't sayin' ought bad about ye, sir. Wid yore permission, would it be alright if we was to sing?\"\n\nBadredd relaxed, shrugging indifferently. \"Sing 'til yore tongues drop off, if'n ye've a mind to. But none o' that gossipin' an' whisperin' to each other!\"\n\nThe four exchanged sly winks. Flinky began singing a lullaby in a soft soothing voice.\n\n\"All the walkin' today that I've done, done, done,\n\ntrampin' through mud in the sun, sun, sun,\n\nit reminds me of the days when me dear ould mother said,\n\ncome on now liddle feller, time for bed . . . bed . . . bed.\n\nSo hush a-bye, looh ah-lie, baby close yore eyes,\n\nan' dream about the moon up in the starry skies.\"\n\nHe repeated the verse again, even softer, with the other three vermin humming gently in the background.\n\nBadredd's head drooped forward slightly, the cutlass lying limp in his open paw. His thoughts drifted back to his own young seasons. Through a golden haze of memory, he was barely aware of Flinky's singing. It was the same tune but with different words.\n\n\"It looks like the fox has gone to sleep, sleep, sleep,\n\nSlippy now be quiet as ye creep, creep, creep,\n\nan' stick a good sharp spear straight through his head,\n\nthen the moment that he wakes up he'll be dead, dead, dead!\n\nSo hush a-bye, don't ye cry, foxy close yore eyes,\n\nan' ye'll soon make lovely vittles for the ants an' flies!\"\n\nThe murderous scheme might have worked out successfully had it not been for Plumnose. He thought that the altered words were so funny that he clapped his paws and broke out into hearty guffaws.\n\n\"Duh, haw haw haaaw! Dat's a gudd 'un, I like dat, Flink! Haw haw haw, wake up dead, berry gudd!\"\n\nBadredd snapped immediately back to reality. He caught Slipback, brandishing a spear not three paces from him. Grabbing up his cutlass, the fox raised it threateningly.\n\n\"Wot are yew up to, weasel?\"\n\nSlipback veered and went past him. He started jabbing at the shrubbery at the edge of the glade.\n\n\"Thought I saw those bushes movin', Chief. It might've been that otter an' the squirrel, er, Sagroon an' Bando!\"\n\nFlinky interposed. \"I know who ye mean, Bragoon an' Saro. I saw the bushes move, too, Chief. Slipback could be right!\"\n\nThinking swiftly, Badredd turned the situation to his advantage. \"No sense in takin' chances then. We'd best git movin' fast. Come on, up on yore paws!\"\n\nBadredd drove them hard for the remainder of the day by adopting a simple but effective scheme. He ordered Rogg and Floggo to fire off arrows from time to time. The deadly shafts fell just short of the marchers' rear, causing them to hasten forward. Oaths and curses accompanied the arrival of each arrow, but they kept going, knowing they were only getting tit for tat. The plot to rid themselves of the little fox had failed, but they realised that, had it been Burrad or Skrodd in Badredd's place, Flinky and Slipback would have been slain as retribution. They were getting off lightly.\n\nProgress was good. By evening, Badredd was heartened to hear Plumnose calling out, \"Dere's duh path at de end ob the trees!\"\n\nSure enough, they had reached the border of the woodlands. In front of them lay the path, which ran down from the north to the south.\n\nFlinky leaned on an elm trunk, smiling cheerfully as the fox came up to see. \"Ah well, there ye are now, Chief. All we gotta do is follow that road t'the left an' keep goin' 'til we hit Redwall Abbey!\"\n\n##\n\nLarks soared joyfully on the flatlands outside of Redwall, singing their hymns to the newborn day. Chiming a melodious bass line, the Abbey's twin bells boomed out warmly. Indoors, all the young ones were already up and about, anticipating the arrival of Summer Feast.\n\nSister Setiva invariably rose to the tolling bells. Up and dressed, tidy and neat, she rapped on the sickbay door with her blackthorn stick, berating the sleepers within.\n\n\"Oot o' those beds, ye great dozy lumpkins. If your no' out here in a braces o' shakes, ah'll be in there an' haul ye both oot by your tails!\"\n\nBragoon poked a sleepy head from beneath his coverlet. \"Hear that, mate? I think we'd best get up. Huh, I'd sooner face a regiment o' vermin than that ole shrewnurse!\"\n\nReaching out a paw, Saro grasped a bedside stool and rattled it noisily on the floor, calling out. \"We're both up, Sister, just makin' the beds an' tidyin' round. We'll be out there in a tick!\"\n\nSetiva's shrill warning came back loud and clear. \"Och, you're a braw fibber. Ah'll be doonstairs, keeping an eye out for ye. Laggardly sluggards!\"\n\nThe pair sat up at the sound of her retreating stick taps. Saro yawned and thumped her head back on the pillows. \"Just leave me here for the rest o' the season, Brag. I'd forgotten how comfy a real bed feels. Mmmmmmmm!\"\n\nLeaping out of bed, the otter swished water from a ewer on his face and towelled it vigourously. \"Fair enough, me ole bushtail, you stop there. I haven't forgotten how good a Redwall brekkist tastes.\"\n\nWithout bothering to wash, Saro pursued him downstairs. \"I'm right with ye, ole ten bellies. You ain't scoffin' all the vittles afore I gets a crack at 'em!\"\n\nMartha had just finished making up a tray for herself and Old Phredd when she saw the pair rush in and begin loading up two trays from the long buffet tables set up in the kitchen passage. She giggled at the sight of them, helping themselves to some of everything, chuckling with delight at the food.\n\n\"Almond wafers with raspberry sauce, my favourite!\"\n\n\"Oatmeal with apple'n'honey, just the stuff! Granmum Gurvel, me ole beauty, pass me some o' that pastie. Wot's in it?\"\n\n\"Burr, ee mushenrooms an' carrot, zurr, wi' h'onion sauce.\"\n\n\"Onion sauce! Gimme two portions, one for Starvation Saro!\"\n\n\"Hah, lissen to ole bucket mouth! You get us two mint teas, Brag, an' I'll fill two beakers o' Junty Cellarhog's best damson cordial. Oh great, hot scones! Gimme, gimme!\"\n\nLeaving the buffet, they beamed at the haremaid over the tops of their laden trays. \"Mornin', Miss Martha, we're just makin' up for the lost brekkists, ain't that right, Bragg?\"\n\nThe otter winked roguishly. \"Haharr, sleepin' in a real bed gives a beast a powerful appetite.\"\n\nMartha looked up at their heaped trays. \"I'm sure it does. Perhaps you'd like to take breakfast in the gatehouse with Phredd and me, away from all this bustle.\"\n\nBalancing the tray skilfully on his head, Bragoon began wheeling Martha's chair. \"An honour an' a pleasure, miss. Besides, 'twill get us out of Sister Setiva's way. Come on, afore she finds we ain't made our beds or tidied the sickbay.\"\n\nHalfway across the lawn, Abbot Carrul caught up with them. \"Oh dear, Martha, I've brought breakfast for Phredd, too.\"\n\nThe haremaid indicated her two companions. \"Don't worry, Father, it won't go to waste!\"\n\nThe old hedgehog Gatekeeper welcomed them in. He reached for his nightshirt, then shook his head absentmindedly. \"Hmm, must've gone to bed in my daytime habit. Look at me, putting my nightshirt on to start the day. What's it all coming to, eh, eh?\"\n\nPhredd gestured at the volume lying on the table. \"The account by Tim Churchmouse about the route to Loamhedge, when Matthias was searching for his son. If you two read it, you'll learn of how to get there.\"\n\nSaro leafed briefly through the ancient pages. \"Me'n Brag ain't champion readers like you, sir. We'd rather see the map\u2014that'll tell us more.\"\n\nNo sooner had Martha showed them the copy she had made of the map, than the squirrel and the otter glanced at one another and nodded.\n\nBragoon tapped his paw upon the map. \"We've travelled this country afore. I can recall most of it\u2014those high cliffs, the pine forest, river, desert an' the great gorge. Dangerous country, eh Saro?\"\n\nThe aging squirrel held the map this way and that as she studied it. \"Aye, bad territory, though we came to it a different way. I remember those rocks, the ones shaped like a bell an' a badger's head, but I can't bring that tall tree to mind.\"\n\nBragoon tapped his rudder thoughtfully against the floor. \"It prob'ly collapsed with age. This map was made seasons afore we were born. But 'tis the same area alright, riddled with vermin an' all manner o' perils. I was glad to get away from it!\"\n\nMartha looked disappointed. \"Does that mean it's too dangerous to make the journey?\"\n\nThe otter laughed. \"Haharr, wot ever gave ye that idea, me beauty? Danger's wot me an' Saro live on. We'd both end up dead afore our seasons was out livin' at Redwall.\"\n\nThe squirrel nodded mournfully. \"All the good vittles an' soft beds, that'd finish us off. Huh, if Sister Setiva didn't.\"\n\nAbbot Carrul poured mint tea for Old Phredd. \"Then when will you be going?\"\n\nSaro selected a hot scone and bit into it. \"Straight after the Summer Feast, if'n we can still walk. Late noon prob'ly. We'll travel southeast.\"\n\nAfter breakfasting they set off for the orchard to help with the festive preparations. Horty, with his two friends, Springald and Fenna, came out of the Abbey, carrying a trestle board. The young hare hailed Bragoon and Saro.\n\n\"Hello there, you chaps. Well, have you sorted out a jolly old way to Loamhedge for us, wot?\"\n\nBragoon answered him rather abruptly. \"Aye!\"\n\nSpringald bounced up and down eagerly. \"Oh good, when are we leaving?\"\n\nFenna's eyes shone happily. \"A journey to Loamhedge. Great seasons, I've been looking forward to this!\"\n\nHorty looked from Bragoon to Saro excitedly. \"Come on then, you bounders, who's got my copy of the bally map? Remember, I'm the flippin' pathfinder, y'know.\"\n\nBragoon turned to face the trio, his voice stern. \"This ain't no daisy dance! Me'n my mate Saro'll be makin' the journey to Loamhedge . . . alone!\"\n\nHorty's ears drooped. \"But you said . . .\"\n\nSaro interrupted him. \"We never said nothin', young 'un. Yore the one whose been doin' all the sayin'. Bragoon an' me knows the country we got to go through. We can make it alone, but it'd be far too dangerous with three young 'uns in tow.\"\n\nFenna was outraged. \"You mean you aren't taking us?\"\n\nBragoon nodded. \"That's right, missy. 'Tis too much responsibility. We couldn't show our faces back in this Abbey if'n ye were slain by vermin or killed in an accident. We're goin' alone, an' that's that!\"\n\nSpringald tried to make an appeal to the Abbot. \"What's he talking about? We've as much right to go as they have! Martha's our friend, too. Father, you're the Abbot of Redwall. You make all the decisions here, tell them!\"\n\nAbbot Carrul beckoned the three young ones to him. Putting his paws about their shoulders, he spoke kindly. \"Now, now, what Bragoon and Saro say makes sense. None of you has ever been further than the main gate. You're far too inexperienced to make such a trip, trust me. Our two friends are thinking of your own good.\"\n\nHorty pulled away from the Abbot, his ears standing stiffly with indignation. \"Tosh'n'piffle, sah! We're young and strong. We can put up with anythin' those two old fogies can! Bragoon and Saro are old chums of yours. That's why you're blinkin' well siding with 'em. And anyhow, what flippin' right have you to stop us goin', wot?\"\n\nSpringald and Fenna supported him volubly. \"Horty's right, it's not fair. You let us think we were going all along, then changed your mind at the last moment!\"\n\n\"Aye, it's just because we're young, and those two old wrecks want to grab all the glory for themselves. What do you think, Martha? Come on, tell them we're right.\"\n\nMartha shook her head. \"If the message from Sister Amyl, when she appeared in my dream with Martin the Warrior, had mentioned that you should go, I'd be the first to say yes. But only the two travellers, Bragoon and Saro, were included in the rhyme. So I'm afraid I must say no\u2014not that my decision matters. Our Father Abbot has forbidden you to journey to Loamhedge, so you must abide by his word. Also, I trust Bragoon and Saro. They know of the dangers and are far more experienced at things like this than the three of you.\"\n\nHorty exploded. \"It's nothin' but a confounded plot against us. Shame on all of you, shame I say!\"\n\nAbbot Carrul put his footpaw down sternly. \"Enough of this talk! Arguing and casting insults is not the way in which any decent Redwaller should behave. Any more of this from you, Horty, or your two friends, and there'll be three empty seats at the Summer Feast this afternoon!\"\n\nHorty glared back at the Abbot, his temper completely out of control. \"Keep your rotten feast, blinkin' bounders!\"\n\nThe Abbot's paw shot out. \"Go to your rooms and stay there until you are ready to apologise, all three of you!\"\n\nThe trio ran off, shouting, \"Don't worry, we wouldn't be seen dead at your Feast!\"\n\n\"Come on, leave those old greywhiskers to themselves!\"\n\n\"You'll be jolly well sorry, we'll stay in the blinkin' dormitory until we die of flippin' starvation. So there!\"\n\nAbbot Carrul comforted Martha, who had become so upset that she had begun weeping. \"There, there, Martha, don't you waste tears on those three. Could you imagine Horty starving himself to death? 'Tis as unlikely as me trying to leap over the belltower. Give them a day and they'll have changed their minds, trust me.\" Carrul bowed slightly to Bragoon and Saro. \"Please forgive the bad manners of those three young ones.\"\n\nSaro smiled wryly. \"No need to apologise to us, friend. I can recall two, younger'n'Horty an' his pals, two more bad-mannered liddle scuts ye never did see!\"\n\nMartha blinked through her tears. \"Were you really that bad?\"\n\nBragoon shuffled his rudder awkwardly. \"Oh, much worse, missy. Take me word fer it!\"\n\nAbbot Carrul chuckled heartily. \"Aye, now that you've come to mention it, 'tis a wonder you turned out so well!\"\n\nBragoon clapped him on the back. \"An' ye, too, Carrul. Ye wasn't exactly a model Dibbun as I remember!\"\n\nWhipping out a clean kerchief, the Abbot busily wiped away at Martha's eyes. \"Yes, well, that was a long time ago. Now then, missy, are you going to keep weeping and bring on the rain, or are you going to smile for our Summer Feast?\"\n\nShe smiled happily. \"Are you still going to carry on with the feast, Father, I mean after what just took place?\"\n\nAbbot Carrul reassured her. \"Of course I am, no need to halt it because of three surly young 'uns. If they want to join in, all they have to do is apologise for their bad manners. Come on, friends, I wouldn't miss my Summer Feast for anything!\"\n\nSet in the orchard against a background of ripening fruit and summer flowers, complete with sumptuously decked tables, the feast turned out to be a huge success. Freshly washed and dressed, the Redwallers took their places, waiting on the Abbot to start the proceedings. Martha sat between Bragoon and Saro. The three of them stared in awe at the magnificent spread. Salads, pasties and savouries were still being brought on trolleys by the servers. These were placed among the pies, tarts and flans. Jugs of various cordials and fizzes stood between trifles, crumbles, puddings and candied fruits. Loaves of many shapes and types, still fresh from the ovens, were set amid cheeses of different hues\u2014from pale cream to golden yellow.\n\nEverybeast, even the Dibbuns, ceased their chatter as Abbot Carrul stood up and recited a verse, specially written for the event.\n\n\"We celebrate this happy day,\n\nwith fair and right good reason,\n\nin friendship, let us share the fruits,\n\nof this fine summer season.\n\nWe seed and plant the fertile earth,\n\nto use what she may give,\n\nand thank the kindly summer sun,\n\nwhich gives us joy to live.\"\n\nGranmum Gurvel, resplendent in a new floral-embroidered apron, called out. \"You'm never spoked truer wurds, zurr!\"\n\nWith that, the Summer Feast began in earnest. Junty Cellarhog tapped a barrel of strawberry fizz, which he had made the previous summer. Dibbuns squealed with delight as the bubbles tickled their mouths. Carving a wedge from a soft hazelnut cheese, Bragoon added it to his salad. Toran noticed him brushing away a teardrop.\n\n\"Wot's the matter with ye, brother?\"\n\nThe otter looked mournfully at the festive board. \"Nothin' really, I was just thinkin' of all the Redwall feasts I've missed since me'n Saro left the Abbey.\"\n\nToran scoffed. \"Don't fret, it looks like yore makin' up for it with a will!\"\n\nSaro adopted a wheedling tone toward the ottercook. \"Anybeast who can cook vittles like these should be famous. Toran, ole pal, why don't ye come adventurin' with me'n yore brother? You could cook for us an' everybeast we meet.\"\n\nToran lowered his eyes modestly. \"No thankee, marm. I'm a mite too round in the waist for travellin'.\"\n\nSister Portula put aside her plate in mock indignation. \"Take our ottercook, indeed! Mayhaps you'd like to take Junty Cellarhog, too, in case you feel the need of a drink?\"\n\nBragoon chortled. \"Haharr, a capital idea, Sister!\"\n\nAbbot Carrul's eyes twinkled as he joined the conversation. \"I'm with you, Bragoon, a marvellous scheme! Take Toran and Junty, they'd make life much easier for you and Saro. However, I must insist that you take Sister Setiva along. If ever you are wounded, or fall ill, you'll surely need a dedicated creature to care for you both. Agreed?\"\n\nBragoon suddenly became interested in a bowl of plum pudding and meadowcream. He mumbled hastily, \"Me'n' Saro will make the journey alone, thankee Carrul.\"\n\nGood-humoured banter and cheerful gossiping carried on into the warm summer noontide, a perfect accompaniment to the delicious feast. Having eaten their fill, the Dibbuns ran off to play within the Abbey grounds.\n\nAfter awhile, Saro glanced at the sun's position and announced, \"We'll have t'get goin' soon. Best be on the road afore we lose the daylight.\"\n\nHer otter friend patted his stomach. \"Aye, though I reckon we won't need much feedin' for a day or two. That was the nicest food an' the best company I can ever recall. Thankee, friends, for everythin'.\"\n\nThe Abbot smiled. \"It was our pleasure. I knew you'd be going today, so I've had two packs of provisions made up by Granmum Gurvel. They should last you quite a time. Inside them you'll find all you need\u2014the map, the poem telling of the location of Sister Amyl's secret and extra garments to wear. Now, is there anything else you two would like to take, anything?\"\n\nBragoon replied without hesitation. \"I'd like to take with me the memory of a sweet song. Martha, would ye sing us a song to send us on our way?\"\n\nSaro added. \"Aye, go on, missy, put the birds t'shame!\"\n\nThe haremaid's clear voice rang out into the still noon air. She sang for her two friends as she had never sung before. They sat entranced by Martha's beautiful voice.\n\n\"I planted her gently last summer,\n\nall in quiet evening shade,\n\nwithin an orchard bower,\n\nher little bed I made.\n\nAlone I sat by my window,\n\nas autumn leaves did fall,\n\nthey formed a russet cover for\n\nMy Rose of Old Redwall.\n\nThrough winter's dreary days she slept\n\nbeneath the cold dark ground,\n\nwhen all the earth was silent,\n\nwhite snows lay deep around.\n\nBright stars came out above her,\n\nas to the moon I'd call,\n\ntake pity on my dearest one,\n\nMy Rose of Old Redwall.\n\nHow the grass grew green and misty,\n\nsoft fell the rain that spring,\n\nher dainty budded head arose,\n\nand made my poor heart sing.\n\nThen summer brought her just one bloom,\n\nso white, so sweet and tall,\n\nwith ne'er a thorn to sully her,\n\nMy Rose of Old Redwall.\"\n\nBoth the hardy old adventurers were sobbing like babes. Saro scrubbed roughly at her eyes. \"Come on, mate, time to go. We'll push ye as far as the gate, missy, so ye can wave us good-bye.\"\n\nThey were met at the gatehouse by Foremole Dwurl and Granmum Gurvel, each carrying a pack of provisions. Old Phredd emerged from the gatehouse with a long, slender bundle, which he presented to Bragoon.\n\nThe otter stared at the strange object. \"Thankee kindly, Phredd. What is it?\"\n\nAbbot Carrul answered. \"It is the sword of Martin the Warrior. I want you to take it on your quest for Loamhedge. Should you need a weapon to defend yourselves, you could not have a finer one. I trust you both with the sword, and I know when the journey is done, you will bring it back safe to Redwall. May the spirit of Martin go with you, my friends, and the good wishes of all in this Abbey!\"\n\nBragoon bound the still-wrapped sword across his shoulders. \"Ye do us great honour. How could we fail with Martin's sword to keep us company? Go back to yore Summer Feast now, an' don't fret. Me an' Saro'll bring back Sister Amyl's secret\u2014that is, providin' it makes ye walk, Martha.\"\n\nThe young haremaid's eyes shone with resolution. \"Walk? I'll do better than that! One day I'll dance for both of you. I'll dance on top of that wall, right over the threshold, for my heroes Bragoon and Sarobando. I swear it upon my solemn oath in front of you both!\"\n\nBragoon laughed. \"Haharr, that's the stuff, me darlin'!\"\n\nSaro swung her pack up on one shoulder. \"So ye will, beauty, so ye will. Good-bye!\"\n\nThey had only taken a dozen paces down the path to the south when Toran came running up and threw himself upon Bragoon. \"Take care of yoreself, brother, an' look out for Saro, too!\"\n\nBragoon gasped for breath as he tried to pull free of Toran's embrace. \"We've taken care o' each other since we was Dibbuns. If'n ye don't let go of me, I'll get me ribs crushed afore the journey's started!\"\n\nToran released his brother and stood weeping on the path. Bragoon looked away as Saro kissed the ottercook fondly.\n\n\"Go on now, ye great lump, back to yore feast. We'll be just fine. But keep this in mind, Toran Widegirth, when we come back to Redwall ye've got to make us a feast, as good as the one we had today. Promise?\"\n\nToran ran back to the Abbey, shouting, \"That 'un today'll look like afternoon tea to the feast I'll make ye when ye return, I promise!\"\n\nThey watched him go inside, then walked to the south wall gable and struck off southeast into Mossflower.\n\n##\n\nHorty stood at the dormitory window, watching as Toran returned and assisted Old Phredd in closing the main gate. Both beasts then headed for the orchard and what remained of the Summer Feast. The young hare turned to his two companions, who were sprawling about on their beds.\n\n\"Well, chaps, Toran's back an' the gate's closed, wot! That means those two aging relics have finally gone off on the quest. Is everything ready, you blighters?\"\n\nSpringald leaned over and pulled three bulging sacks from under her bed. \"These are going to take some carrying!\"\n\nHorty scoffed. \"Pish an' tush, m'gel, one can't have enough tuck. It's vital, mark m'words, bally vital!\"\n\nFenna gathered their walking staffs and three travelling cloaks from the wall closet. \"But how do we get out of the Abbey without being spotted? It won't be dark for hours yet. Huh, you'd think Bragoon and Saro would've waited until dawn tomorrow.\"\n\nHorty sat down on his bed, ruminating. \"Hmm, you've got a jolly good point there. I'll have to think up a cunning plan. Spring, pass me one of those sacks. A chap can't think on a blinkin' empty tummy, wot!\"\n\nSpringald kept a tight grip on the foodsacks. \"Forget your confounded stomach, Horty! Get thinking, and be quick about it. We can't sit around here until it's dark and we've lost their trail.\"\n\nHorty rose and strode back to the window, muttering, \"Forget one's tum, wot? Easy for you t'say, Miss Mouse. I'm a flippin' hare, y'know. Forgetfulness of the old stomach is bally impossible to types like me . . . Ahah, Dibbuns, the very chaps!\"\n\nFlinging the window open, Horty called down to Muggum and a crew of Abbeybabes who were cavorting on the lawn below. \"What ho there, my pestilential friends!\"\n\nShilly the squirrelbabe looked up and pointed an accusing paw. \"Naughty 'orty, you been sended up t'stay inna dormitee.\"\n\nHorty stared down his nose at the little squirrel. \"Let me inform you, my broom-tailed friend, I am here merely out of choice. I can come down when I flippin' well please. Now listen closely, you little bounders. Would you like to hear a secret, wot?\"\n\nMuggum wrinkled his button nose. \"Ee seekurt? Us'n's gurtly fond o' seekurts. Ho urr aye!\"\n\nFenna called out in a hoarse whisper. \"Horty, what are you up to? Who are you talking to?\"\n\nWaggling his ears at her, the young hare looked secretive. \"I've just thought up a super wheeze, a plan t'get us out unnoticed, wot. Create a diversion, that's the idea. Leave this to Hortwill Braebuck, marm!\"\n\nA hogbabe named Twiglut, having grown impatient, squeaked up at the window. \"Are ya goin' a tell uz dis seekrut? Well 'urry h'up, or we go an' play wiv sticks!\"\n\nHorty waved his paws earnestly to gain the Dibbuns' attention. \"No no, don't go an' play with sticks, my tiny pincushion. I'll tell you the secret. This mornin' we went down to the pond, an' guess what? We saw lots of big fishes . . .\"\n\nMuggum butted in. \"Wurr they'm gurt hooj fishies, zurr?\"\n\nHorty stretched his paws wide, indicating their size. \"Huge? They were blinkin' colossal! Anyhow, they gave us rides on their backs all round the jolly old pond. Oh, it was loads o' fun, I can tell you, absoballylutley top hole an' all that, wot!\"\n\nThe Dibbuns began dancing with excitement.\n\n\"Will ee fishies still be thurr?\"\n\n\"Uz wanna ride on der fishies!\"\n\nHorty scratched his ears. \"Hmm, they said they'd be there late afternoon, just before evenin'. I say, you chaps, it's round about that time now, isn't it?\"\n\nRoaring delightedly, the Dibbuns thundered off in the direction of the Abbey pond.\n\nHorty called after them. \"Have fun, you little savages. Tell the fishies Horty sent you!\"\n\nThe realisation of what was taking place suddenly hit Springald. Leaping up, she hurled Horty away from the window. Cupping both paws to her mouth she yelled. \"No, don't go! Come back this instant, all of you, come back!\"\n\nBut the Dibbuns could not hear because of the din they were setting up. Like a small stampede, they ran out of sight around the Abbey corner.\n\nSpringald turned on Horty. \"You blathering fool, what have you done? Idiot!\"\n\nHorty flapped his ears airily. \"Creatin' a small diversion. No need to get your fur in an uproar, old thing, wot?\"\n\nFenna's tail went stiff as Horty's foolish act dawned on her. \"You puddenbrain! Can't you see that those babes will be drowned if there isn't anybeast responsible to watch over them?\"\n\nThe young hare slapped a paw to his brow. \"Oh corks, you're right! I never gave that a flippin' thought.\" Leaning wide out of the window, he bellowed, \"I say, little chaps, come back this very instant. D'ye hear?\"\n\n\"Dearie me, what's all the shouting about?\"\n\nHorty found himself staring down into the questioning face of Brother Gelf, who was returning some bowls to the kitchen when he heard the commotion.\n\nFenna pushed past Horty, her voice shrill with anxiety. \"Hurry, Brother, the Dibbuns are down at the pond alone. There's nobeast with them. Oh hurry, please!\"\n\nThe mouse sped off as fast as his paws would carry him.\n\nIn a trice, the bells of Redwall were tolling out an alarm. Creatures could be seen hurrying toward the pond. Toran was out in front, shedding his apron as he ran and plunging straight into the water. Luckily, none of the Dibbuns was harmed. Most of them were garnered from the shallows by willing paws, though Toran had to swim for Muggum. The molebabe was well out of his depth, floating about like a ball of downy fur. Foremole Dwurl's resounding bass tone could be heard, calling to the Abbot, as he panted up, pushing Martha's chair.\n\n\"They'm awright, zurr h'Abbot, oanly ee bit wetted!\"\n\nHorty was shaking all over as he turned to his friends and laughed with relief. \"No harm done, chaps. At least my diversion worked, wot?\"\n\nSpringald and Fenna leapt upon him, boxing his ears and kicking his bottom. They were furious.\n\n\"No thanks to you and your bright ideas!\"\n\n\"You great waffling flannel-brained nincompoop!\"\n\nHorty broke loose and seized the travelling gear. \"What's done is done. Sorry, chaps, an' all that. We'd better make ourselves scarce. Let's go while the goin's good!\"\n\nSister Setiva was towelling the babes dry with Toran's apron and her shawl; others were helping, using anything that came to paw. The shrewnurse railed on at the Dibbuns, alternately drying and hugging each one.\n\n\"Och, why wid ye want tae do sich a silly thing, mah babbies? Have ye no been told aboot playin' alone by the water, eh?\"\n\nUnder the stern eyes of Abbot Carrul, Martha and a dripping wet Toran, the whole story emerged. Martha could scarcely believe her ears when she heard that it was her brother who had encouraged the little Dibbuns. Seething with righteous wrath, she turned to Toran.\n\n\"Mr. Widegirth, would you kindly push me up to the Abbey? I wish to have some severe words with that brother of mine!\"\n\nThe ottercook bowed politely. \"Certainly, Miz Braebuck. I'm shore there's one or two wants words with Master Horty, one of 'em bein' me!\"\n\nA procession of Redwallers followed Martha into the Abbey. The Dibbuns were enjoying the affair hugely, seeing some other beast getting blamed for their escapade. They tagged along, muttering darkly of tail chopping and bottom-skelping punishments. Some were even speculating that Horty would be boiled in a soup pan.\n\nTheir delight, however, was short-lived. Sister Setiva and some molewives whisked them off, down to Cavern Hole.\n\n\"Intae the bath, ye filthy wee beasts. Och, there's nae tellin' whit muck'n'mire ye picked up in yon pond!\"\n\nThe Abbeybabes wailed piteously but to no avail.\n\n_Boom! Boom!_ Toran's hefty paw reverberated on the dormitory door. After a moment's silence, his voice rang out harshly.\n\n\"Master Horty, yore sister an' Father Abbot want a word with ye downstairs. Miz Fenna an Miz Springald, ye'd best show yoreselves, too!\"\n\nMartha sat down in Great Hall and waited. Soon she heard the dormitory door slam, followed by the sound of Toran's footpaws pounding down the stairs. Abbot Carrul looked over his glasses as the grim-faced ottercook entered the hall.\n\n\"Don't tell me they're gone?\"\n\nToran sat down on a table edge. \"No trace of 'em, Father. I searched that dormitory from top't'bottom, but I'll wager they're hidin' someplace. You leave it t'me, I'll find those villains.\"\n\nThe Abbot began pushing Martha's chair toward the kitchens. \"I don't think you will somehow. Follow me, please.\"\n\nGranmum Gurvel met them as they entered the kitchen. Clearly in a proper tizzy, the poor old molecook began chattering angrily. \"Foive gurt h'apple puddens, ee gurt meadow-creamy troifle, strawbee scones, celery an h'onion flans, pasties full o' carrut'n'gravy. They'm all be gonned? Burrrrrooooh! Wait'll oi get'n moi paws on ee Dibbun rarscalls. H'all moi luvverly arternoon bakin' furr tomorrers lunchen an' supper. Varnished!\"\n\nMartha kept her eyes downcast as she informed Gurvel, \"It wasn't Dibbuns, Granmum. It was my brother Horty and his friends, Fenna and Springald. They're the thieves who raided your kitchen. Now they've run off to join Bragoon and Saro on the quest.\"\n\nToran's rudder rapped loudly on the floor. \"Of course, that's it, Martha! But why'd they have to cause so much upset to everybeast\u2014us, an' the Dibbuns, an' Gurvel? Why?\"\n\nAbbot Carrul raised his eyes and sighed. \"Sadly, that's the way most young 'uns behave at that age. Forbidding them to do something is like encouraging them. Unfortunately, they do things without thinking.\"\n\nOld Phredd shuffled in, bowing creakily to the Abbot. \"I just found my main gate open, but me and young Toran barred it shut this afternoon. How did that happen, eh, eh?\"\n\nCarrul patted the Gatekeeper's bony paw. \"No doubt you've closed it again, Phredd. It was Horty, Fenna and Springald\u2014they've gone off adventuring.\"\n\nPhredd chuckled drily. \"Just like Bragoon and Saro when they were younger, eh, eh?\"\n\nJunty Cellarhog, who had just come into the room and heard Phredd, thrust his big paws into his apron belt. \"No, ole feller, not like Saro an' Bragoon at all. Them two was born tough, rovin' was in their blood. But young Horty doesn't remember anytime afore comin' to Redwall, an' both maids was borned 'ere. They don't know wot 'tis like out there in the big world. I think they'll 'ave to learn t'grow up fast.\"\n\nMartha felt a pang of alarm at Junty's words. \"What does he mean, Toran?\"\n\nThe ottercook explained. \"Well, miss, look at their vittles. Apple puddens, strawberry scones an' a meadowcream trifle? No proper travelbeast'd take such stuff along. Huh, it'd be smashed t'bits afore they got a day's march in, eh Gurvel?\"\n\nThe old molecook nodded wisely. \"Aye, et surpinkly wudd, zurr. Oi maked speshul marchin' vikkles furr ee uther two. Lots o' cheese, ee h'oatbreads, summ candied fruits an' canteens o' moi gudd dannelion'n'burdock corjul furr drinken.\"\n\nMartha grasped Toran's paw. \"You don't think they'll come to any harm, do you?\"\n\nThe ottercook's eyes softened. \"Don't ye fret yoreself, Martha. If'n they picks up my brother an' Saro's trail, they'll be safe enough. Mind, though, they won't get no special treatment. Horty an' his pals will learn the hard way. Now, if'n they lose the trail, Redwall's stickin' up in plain view for a good distance. Once yore brother gets hungry, he'll dash back to this Abbey like a scalded toad. The others are sure to follow. If'n ye pardon me sayin', Martha, Horty's a natural glutton. He won't stray too far without vittles\u2014starvation's a hard taskmaster!\"\n\nThe haremaid fiddled with the fringe of her lap rug. \"I'd feel happier if somebeast could overtake them and bring them back, so they don't get lost or hurt.\"\n\nThe Abbot looked at Toran and Junty Cellarhog, both big, stout beasts and very competent. \"Perhaps our Martha is right. Do you think you two could catch up with them before it gets too dark?\"\n\nJunty took off his canvas apron and nodded to the ottercook. \"We'll give it a try, Father. Are ye ready, mate? Come on!\"\n\nThey left the Abbey by the main gate. No sooner had Carrul and Old Phredd closed and barred it then Junty and Toran were pounding on the timbers to get back in.\n\nToran's voice was loud and urgent.\n\n\"Open up quick! There's vermin comin' down the path from the north! They're headin' this way. Hurry and let us in!\"\n\n# [BOOK TWO  \n\"If only they were back here  \nat Redwall\"](Loamhedge-toc.html#TOC-18)\n\n##\n\nLate that same afternoon, the vermin gang had been keeping to the woodlands. On Badredd's orders they followed the path. Stopping for a breather, the little fox sighted Redwall Abbey in the distance, showing above the trees. He scurried out onto the path, pointing and yelling.\n\n\"Aharr, there 'tis, mates, the Abbey place! I told ye I'd find it, 'twas me who saw it first!\"\n\nAs he ran forward, the cutlass, which he had pushed into his belt, tangled in his footpaws, causing him to trip. He lay sprawled on the path, still shouting. \"Wait'll I gets me paws on that magic sword!\"\n\nHalfchop sneered. \"Look at 'im, willyer, the flamin' fool. I swear, Flinky, dat stoopid oaf'll get us all killed!\"\n\nThe crafty stoat chuckled. \"Ah, sure enough, he's a grand, brave beast. I'd sooner serve under Badredd than Burrad or Skrodd. Those two would have made us march in front, an' led from the rear. Let the fearless chief run an' meet the foe. Us pore ould pawsloggers will just keep our heads down an' follow from a safe distance.\"\n\nCrinktail was in agreement with her mate. \"Aye, whoever's inside o' that place will prob'ly see us comin' from their walltops. Wot was it that Burrad said, that those Abbey creatures was all peaceable Woodlanders? So we may as well put on a show o' force. The sight of a vermin gang might make 'em open up those gates to us\u2014providin' they knows wot's good for 'em!\"\n\nThe crew strolled out onto the path, deliberately setting a slow pace, keeping Badredd well ahead of them. Flinky sang a quiet ditty as a warning to his mates. The little fox could not quite hear the words, but he assumed it was some sort of song for marching into battle. He swaggered along, a good half-spearthrow in front, waving the unwieldy cutlass with regained dignity, feeling every inch the great Badredd, commander of a vermin crew. The others followed at a safe distance, sniggering at the words of Flinky's song.\n\n\"When the clouds of arrows fly,\n\nkeep yore heads down.\n\nLet the brave ones charge on by,\n\nkeep yore heads down.\n\nWhen the heroes' blood runs red,\n\nan' yore scared to raise yore head,\n\njust be glad that you ain't dead,\n\nkeep yore heads down!\n\nYe won't win no medals here,\n\nkeep yore heads down.\n\nDon't be fools who know no fear,\n\nkeep yore heads down.\n\nWe can all lay low an' sing,\n\nduckin' spears an' stones from sling.\n\nLet 'em chuck most anything,\n\nbut keep yore heads down!\"\n\nAmid smothered giggles and hoots, Slipback and Juppa made disparaging remarks behind their leader's back.\n\n\"Haw haw, lookit the way 'is bottom waggles when 'e puts on a swagger. Looks like two sour apples in a sack!\"\n\n\"Aye, an' if'n 'e don't stop wavin' that blade around, 'e'll chop 'is own tail off. Wot d'ye reckon, mate, does that liddle smidge look like a vermin warrior who'd terrify those Abbeybeasts?\"\n\n\"Maybe they'll laugh theirselves to death at the sight of 'im. Heeheeheee!\"\n\nFlinky gazed up in awe as the impressive red sandstone Abbey loomed closer. He muttered to Rogg and Floggo. \"Huh, if Badredd gives the order to charge that place, well, I'll be chargin', shore enough. I'll be runnin' the other way, like a duck wid its tail on fire!\"\n\nThe weasel brothers were not much given to merriment, but Flinky's remark tickled them so much that they guffawed loudly.\n\nBadredd came running back brandishing his cutlass. \"Wot's so funny, eh, can I share the joke?\"\n\nFlinky shrugged disarmingly. \"Ah now, we wasn't laughin' at ye at all. 'Twas just that we're 'appy for ye. Yore a good chief, an' soon the magic sword'll be yores. Ye deserve it fer bein' a grand ould leader, so ye do. Ain't that right, mates? Badredd's the best boss we've ever 'ad!\"\n\nHalf believing Flinky's flattery, Badredd eyed the gang and nodded approvingly. \"Lissen, mates, we could be a good crew if'n we tried. Now wipe the grins offa yore gobs an' form up in twos. We'll march straight up to that Abbey an' put the fear o' Hellgates into those peaceable bumpkins. Try t'look more like a gang o' killers. Wave yore weapons about an' snarl loud, as if yore ready t'do murder!\"\n\nFlinky glanced up at the high battlements. Already he saw heads poking up over them in the gathering gloom. Thinking quickly, the stoat slid down into the ditch on the path's opposite side. He beckoned Badredd. \"A nighttime charge might go wrong, Chief. D'ye not think we oughta figger out some kind of ould plan, afore we go rushin' at a buildin' that size?\"\n\nThe little fox turned his attention to the walltops. Lots of heads were beginning to appear there. He climbed down into the ditch, alongside Flinky, knowing that what the stoat said made sense. \"Aye, let's, er, make up a scheme. . . . Everybeast down 'ere!\"\n\nThe remaining gang members obeyed promptly. Flinky patted Badredd's back. \"Sure, that's wot I likes about ye, Chief, yore a true fox, a born slayer, but a grand an' crafty ould planner. Hoho, those creatures in there'll get the shock o' their lives when we turns up outside their doorstep tomorrer!\"\n\nBadredd was puzzled. \"Tomorrer?\"\n\nCrinktail caught on, knowing her mate was trying to put off invading Redwall for as long as possible. She backed Flinky up. \"Haharr, clever move, Chief. Tomorrer's the best time t'do it!\"\n\nBeyond a straight charge, Badredd had no real plan. He decided to hear Flinky out, knowing the stoat was no fool.\n\nFlinky explained eagerly. \" 'Tis dark now, y'see, an' we're in strange territory. The gang can get a good night's rest down 'ere. When you've thought up yore scheme, we'll be ready fer a fresh start, an' catch 'em nappin' at dawn! Now that's wot I calls a smart move, thought up by a smart fox!\"\n\nUnaccustomed to compliments, Badredd enjoyed the feeling of having everybeast waiting on their leader's word. Flicking his tail round slowly, he stroked it as foxes do when they are pleased. \"Right, we rest 'ere, gang, that's my orders!\"\n\nHe missed the nudge exchanged between Crinktail and Flinky as they lay down and closed their eyes. Flinky murmured but loud enough to be heard by all. \"Ain't we the lucky ones, havin' a gangleader like Badredd.\"\n\nStarlit darkness had fallen as Abbot Carrul made his way up the north wallsteps onto the ramparts. A frown creased his brow when he saw the throng of Redwallers crowding the parapet.\n\n\"Friends, listen to me, please. There's no need for all of you up here. With vermin about, it's not safe to stand looking over the battlements. Anybeast who is not required up here, please go down now. Sister Setiva, Sister Portula, will you see those Dibbuns down the stairs, it's time they were in their beds anyhow.\"\n\nToran and Junty, who had already joined Foremole Dwurl and Brother Weld, were at the northwest wall corner. Carrul hastened to join them. \"Is there really a vermin band out there? Where are they now?\"\n\nToran answered reassuringly. \"There's no great army o' them, Father, I only counted about eight. Might be more to come, but I ain't spotted 'em yet.\"\n\nJunty made way for the Abbot to look between the battlements as Toran pointed. \"Look, they've lit a small fire, in the ditch, just further up the path there. Wonder wot they're up to?\" A red-gold glow showed from the ditch, where Toran was pointing.\n\nForemole blinked. \"Oi aspeck they'm cooken ee supper.\"\n\nThe Abbot looked to Toran. \"What do you think?\"\n\nThumping his rudder thoughtfully against the wallside, the ottercook speculated. \"Well, there's no way a crew that size could attack Redwall. I think we'd best do nothin' for the present, Father. But let's watch every move they make. We'll post sentries on the walls, just a few who can watch 'em, while keepin' low. Who can tell\u2014maybe they're only passin' by this way. Per'aps they're bound someplace else. I wish Bragoon an' Saro would've stayed a day or two longer\u2014we could really do with 'em right now!\"\n\nForemole smote the wall with a heavy digging claw. \"Boi 'okey we'm cudd, they'm udd know wot to do abowt ee varmints. But thurr bee's h'only us'n's, yurr!\"\n\nToran could sense that the Abbot was waiting for him to take charge. He waved down to Martha, waiting in her chair on the lawn, then spoke. \"Father, maybe ye an' Martha could get a few helpers an' search around for anythin' that would be useful as a weapon. I've got a feelin' they won't make a move 'til tomorrow. We should be ready for 'em by then, though it prob'ly won't come to that. I'll stay up here with Junty, Weld an' Foremole on watch.\"\n\nThe Abbot went down to the lawn and pushed Martha back to the Abbey, explaining what was happening and what he had seen. The young haremaid could tell by Abbot Carrul's face that he was very worried.\n\nWirga was long past her best seasons, a wrinkled, toothless old Searat, yet Raga Bol kept her with his crew. She was useless as a fighter or a forager, but she possessed other skills. There was little that Wirga did not know about wounds and the treatment of injuries. Her powers as a healer and her knowledge of herbs, nostrums and remedies made the old vermin invaluable to the ignorant crewrats. But there was yet another art Wirga practiced\u2014that of a Seer. Raga Bol, as captain, was the only one she allowed to consult her, and then only in times of crisis.\n\nWirga crouched by the fire, watching Bol. They were camped among some wooded hills where the red sandstone rocks of Mossflower jutted out in shelflike formation. It was twilight. The Searat crew had slain a small colony of woodmice, and were leisurely plundering their shattered dwellings. Raga Bol and Wirga sat on a hilltop, isolated from the noisy rabble below.\n\nThe old Searat knew that her captain wished to consult her. He had given her half a roasted dove and a goblet of his personal grog\u2014this was always a sign that she was needed. Wirga took out her pouch of charms and selected half a large musselshell. It was edged with yellow on the inside, glistening grey at the centre, with three partially grown purple mussel's pearls protruding from its broad end.\n\nFilling the shell with water, she gazed into it. \"Thy appetite is not good of late?\"\n\nRaga Bol licked the sharp tip of his silver pawhook in silence as Wirga continued.\n\n\"Sleep eludes thee, thou are weary. None can rest easy in thy presence. Even I fear to speak of certain things\u2014aye, things that trouble thee.\"\n\nWith a curt nod, the Searat captain dismissed the four guards who attended him from twilight to dawn. When they had gone off to join the others, he took a furtive glance over his shoulder.\n\nDrawing close to the Seer, Raga Bol dropped his voice to a hoarse whisper. \"Fear not, speak openly to me, ye won't be harmed.\"\n\nKeeping her eyes on the water-filled shell, the old Seer proceeded, her voice now a sibilant hiss. \"If thine enemy lives, he must die. Only then can Raga Bol find peace of mind. Thy foe's death will release thee.\"\n\nThe Searat captain's eyes shone feverishly. \"Does the stripedog still live? Tell me!\"\n\nWirga turned away from the shell, confronting him. \"When did thou last see this stripedog?\"\n\nBol's red-rimmed eyes stared back at her. \"This very noon, aye, in full sunlight. 'Twas when we stopped to rest. I was so tired that I dozed off awhile. The sun beat through my closed lids, makin' everythin' go red. That's when I saw the stripedog. Gettin' off a strange craft he was, where that broadstream from the nor'east bends away from the trees an' woodlands. Ye recall the spot, 'twas where we slayed those two shrews. The stripedog pointed to the bodies an' looked straight at me. 'They will be avenged, I am coming for ye, Raga Bol!' Those were his very words.\"\n\nWirga went back to contemplating the water in the shell, then continued. \"Thee told him to go away and join the deadbeasts at Hellgates, because he was already slain by thee. But the giant stripedog kept coming. He was frightening to look upon, with his face cleaved wide, but scarred an' stitched together by somebeast. Do I not speak truly?\"\n\nRaga Bol gasped, in awe of the Seer and her powers. \"Aye, true, but how did ye know? Did ye see the beast, too?\"\n\nShe smiled. \"Wirga sees many things unknown to others.\"\n\nWhat she did not say was that she had been observing her captain for days\u2014listening, watching, taking all in. Every nightmare, every time Raga Bol called out, in the brief times he did sleep, were memorised by Wirga. She had a complete picture of it all\u2014from the moment Raga Bol had struck the badger to every event since.\n\nThe Searat captain brought his face even closer to the Seer. His breath was hot on her jaw, his voice half threat and half plea. \"I can't fight a dream, so I'm waitin' on yore word. Tell me wot t'do, I must be rid of the stripedog!\"\n\nWirga replied. \"Knowest thou my three sons?\"\n\nBol knew the ones she spoke of, though not too well. They were a furtive trio, a bit undersized for Searats, always last to fight but first to grab the plunder. He was not impressed with them, and saw the three as background vermin who never put themselves forward or appeared bold, like proper Searats often do.\n\nThe captain shrugged. \"Aye, I know 'em, they ain't no great shakes as fighters. That big stripedog could eat the three of 'em!\"\n\nWirga rocked back and forth on her haunches, chuckling. \"Heehee, well said. But give 'em a skilled tracker, one who could lead 'em to the place of thy dream, an' my sons will make an end of thy stripedog, believe me!\"\n\nRaga Bol drew his scimitar, allowing the firelight to gleam across its lethal blade. \"If'n' I never finished the bigbeast with a blow o' this, how could three runts like that do the job?\"\n\nWirga drew from her pouch a section of bamboo, cut off near the joint and sealed at one end with beeswax. Carefully, she broke away the wax and upended the cylinder. Six long thorns spilled out, each one tipped with crimson dye and plumed with the short feathers of some exotic bird. She stayed Raga Bol's paw as he reached to pick one up.\n\n\"Keep away from such things. They can kill ten times more swiftly than the most venomous snake!\"\n\nThe Searat captain pulled back his paw. \"Poison?\"\n\nUsing her long pawnails, the Seer divided the thorns into three groups of two. \"Once one of these little beauties pricks the skin, even the greatest warrior cannot stand. Poison, from far isles across the southern seas. My three sons know how to use these darts. Warriors they may not be, but assassins they surely are. Give 'em a tracker to lead 'em to the streambend. They will seek out thy stripedog an' slay 'im.\"\n\nRaga Bol stood abruptly, peering over the hilltop rocks at his crew below until he saw the one he required. \"Ahoy, Jibsnout!\"\n\nA big, competent-looking Searat saluted. \"Cap'n?\"\n\nRaga Bol called back to him. \"Bring Wirga's three sons up 'ere. I've got a task for the four of ye.\"\n\nNight had fallen as the sons of Wirga left the hilltop, following Jibsnout. The tracker had a blanket with some food rolled into it thrown over his shoulder, and a well-honed dagger dangling from a cord around his neck.\n\nOnce they were off the hill and bound back along the trail, Jibsnout halted and glared contemptuously at the three smaller rats. It was obvious he did not enjoy their company. He pointed the dagger at each of them in turn.\n\n\"Lissen t'me, slimesnouts. I don't like yew three one liddle bit. But I gotta do the job wot Cap'n Bol gave me\u2014to take ye back to where the broadstream bends at the edge o' this forest. Wot ye do then is carry out the cap'n's orders. 'Tis up to ye how y'do that, an' nought t'do wid me. But get this straight: Ye do yore job an' I'll do mine. So stay outta my way an' mind yore manners around me. Step on my paws or look the wrong way at me an' I'll gut all three o' ye wid this blade o' mine! Unnerstood?\"\n\nThe sons of Wirga never answered; they merely looked at one another and exchanged sly leers. This did not improve Jibsnout's opinion of them. Turning on his paw, he set off at a rapid pace into the dark woodlands, growling back to the odd trio.\n\n\"Move yoreselves! We'll be marchin' night'n'day, an' only stoppin' for a bite or a nap when I says so. If'n ye don't keep up, I'll leave ye behind. Hah, try explainin' yoreselves to Raga Bol when ye get back then, I dare ye!\"\n\n##\n\nThree days earlier, Lonna had bid farewell to Garfo Trok at the broadstream bend. The last he saw of the otter was Garfo singing loud ballads about food, or the lack of it, paddling back upstream to the northeast country. Lonna had enjoyed his time with the garrulous otter aboard his boat _Beetlebutt._ The big badger felt lonely as he trudged off into Mossflower, but soon his loneliness was replaced by rage, as he remembered the pitiful bodies of the two dead shrews. Before they parted, he and Garfo had buried them on the bankside.\n\nAll that day the scar across Lonna's face felt sore and tight. His head ached whenever he thought of Raga Bol and his murderous crew, and his back wound began bothering him, causing him to limp as he pressed doggedly onward. The woodlands were quiet and peaceful, with sundappled green light cascading through the overhead foliage. Distant birdsong sounded muted, bees droned lazily in the midday calm. Lonna ignored the beauties of nature, his eyes constantly darting from side to side, paws ever ready to seek bow and shafts.\n\nAt midnoon the big stripedog halted by a rippling brook in a mossy sward. Resting awhile, he ate sparingly from the sea otter's food pack\u2014a crust of nutbread, some fine ripe cheese and a few scallions he found growing nearby\u2014and drank deeply from the brook. Still sitting with his footpaws in the water, Lonna washed his head and face, then, leaning forward, immersed his face and head for several long intervals. The cold, clear brookwater refreshed him greatly. He stood up to leave, rubbing the small of his back and swaying from side to side, testing the limp in his footpaw, to judge how it was feeling.\n\nA sense that he was being watched came over Lonna. Continuing his exercises, he spoke out in a voice loud enough for any eavesdropper to hear.\n\n\" 'Tis not good manners to spy on a beast. Come out and show yourself. Don't be afraid, you can see I'm no Searat!\"\n\nAn elderly female squirrel, clad in a russet and yellow tunic, dropped out of the trees, landing right in front of him. She was a perky, cheerful-looking creature, but he could see by the way she toted a small javelin she was ready for anything.\n\nLooking him up and down, she chattered boldly away. \"Chahah! Me could tell ya wasn't Searatta. Warramarrer bigbeast, ya back be hurted?\"\n\nCompletely disarmed, Lonna smiled ruefully. \"Just a bit, marm, but 'tis getting better by the day, thank ye. My name is Lonna Bowstripe.\"\n\nThe squirrel bobbed him a neat curtsy. \"Me's Figalok Twigbenda, pleasin' t'meetcha. I fix ya back, Lonna, folla me!\"\n\nLonna took an immediate liking to Figalok, following her without question. She was so very swift that he had to hurry to keep up. Figalok halted alongside a big, ancient hornbeam tree and began giving rapid orders.\n\n\"See da branch stickin' out up above? Me wancha t'jump up an' grab it tight. Chakahoo! Berra take offa dat bigbow an' arrers. Cheeh! Howcha make dat\u2014cut a yewtree down an' purra string on it? Dat a big bow, sure 'nuff!\"\n\nLonna smiled at her observation. When he took off his bow, it stood near three times the height of Figalok. Placing his quiver of arrows to one side, he leaped up, grabbed the hornbeam limb and hung there, dangling. The branch was quite stout enough to hold his weight.\n\n\"Is this alright, marm? What do I do now?\"\n\nFigalok walked around him. \"Ya jus' hang there like a h'apple. Are ya plenny strong, Lonna?\"\n\nHe stared down at her. \"Aye, strong enough.\"\n\nFigalok jumped up and sat on Lonna's footpaws, facing him. She grabbed his legs to steady herself. \"Keep ya paws still now, bigbeast, don't ya kick me off!\"\n\nFigalok began jerking Lonna back and forth, using him like a swing. \"Chahah, dis do ya good, keep tight hold!\"\n\nFor what seemed like an eternity she continued the swinging motion, back and forth, forward and back. Lonna's own bodyweight, with the added burden of the squirrel, began to tell after awhile. She stared up at his clenched jaws.\n\n\"Ya wanna leggo now? Dat was a good ride.\"\n\nLonna gasped. \"Aye, I'd best come down before I drop!\"\n\nFigalok leapt to the ground, skipping to one side. \"Rightee, ya can leggo, Lonna!\"\n\nHe dropped gingerly, expecting the fall to jolt his back. Surprisingly, it did not.\n\nThe squirrel gave his back a thump. \"Wassamarra witcha? Walk round, jump 'bout! Chahah, ya back be good as new now. Me fixed lotsa backs!\"\n\nLonna's back felt easy and relaxed, he was not getting a single twinge from the footpaw, which had been bothering him. He walked, then trotted, jumping up and down forcefully, putting all his weight on back and footpaw. Revelling in the newfound freedom of movement, Lonna dashed at Figalok, meaning to embrace her.\n\n\"I'm better, there's no more pain! Figalok, you marvellous creature, how can I thank you?\"\n\nShe shot up the trunk of the hornbeam, protesting, \"Keep ya big paws offa me, or I be crushed flat! Betcha hungry, eh? Bigbeasts must get plenty hungry. Folla me!\"\n\nFigalok scuttled through the woodlands, with Lonna hard on her tail. She halted at the base of a three-topped oak, which grew in close proximity to a beech, an elm and a sycamore. The upper limbs of all three trees intertwined with the oak, forming a wide platform.\n\nThe squirrel twitched her tail at Lonna. \"Ya wait der, me send ya rope down!\" She shot lightly up the oak trunk, vanishing into the foliage.\n\nA moment later Figalok reappeared, surrounded by a crowd of tiny squirrelbabes. They squeaked and squealed at the size of Lonna, pointing and giggling.\n\n\"Cheehow, nanny, wherecha find dat 'un?\"\n\n\"Weehoo, must be da biggest beast in alla lands!\"\n\n\"Choowhee, never see'd not'ink like 'im in me life!\"\n\nShoving them out of her way, Figalok pushed a thick rope down. It was knotted at close intervals to make climbing easy. Shouldering his bow and quiver, Lonna began scaling the rope. Figalok was hard put to keep back the press of little squirrels.\n\n\"Chahah, gerra ya back an' make way for me friend. Take no notice a dese likka pesters, Lonna, up ya come!\"\n\nLonna found the climb quite easy. The squirrelbabes shrieked and scurried off as he joined Figalok on the bough. She nodded approvingly.\n\n\"Not'ink wrong wirra dat back now. Me make a squirrel outta ya, bigbeast. Berra get vikkles quick, afore they alla gone!\"\n\nThe squirrels' dray was an amazing sight. Branches were cunningly woven twixt the network of bows and limbs between the four trees. Lonna found it safe to walk upon, though he trod carefully. At the oak's centre was a wide platform with a charcoal oven set on slabs of slate. Upward of a dozen older squirrels were preparing a meal there. Literally scores of babes and young ones festooned the place, hanging by their tails or balancing nimbly on the slenderest of twigs.\n\nFigalok proudly introduced her newfound friend to the assemblage. \"Ya see this 'un, he be Lonna bigbeast. Figalok finded 'im. Lonna be hungry, berra give 'im lotsa vikkles!\"\n\nFour older squirrels hurried to serve the big badger, plying him with huge portions of a thick, sweet porridge. It was a mixture of wild oats, fruit and nuts boiled in honey and rhubarb juice. Lonna was given a full flagon of elderflower and pennycress cordial. Both the food and drink tasted delicious. Figalok sat beside him, watching in awe as he satisfied his considerable appetite.\n\n\"Cheehoo! Betcha mamma was glad when ya leaved home!\"\n\nLonna chuckled. \"Who knows, maybe she might have been, but I don't ever remember having a mother.\"\n\nGradually the squirrelbabes had been inching closer to the big badger. When he mentioned that he had never known a mother, their sympathy was instantly aroused. They surrounded Lonna, sitting on his lap and shoulders, climbing on his back and paws. He was totally engulfed by the babes, one of them even perched upon his head.\n\nTheir tiny paws patted him as they squeaked sorrowfully. \"Aaaaah, never haved no mamma, pore bigbeast!\"\n\n\"Must bee'd tebbirle, not 'avin' no mamma!\"\n\n\"Didya cried an' weeped alla time for ya mamma?\"\n\nFigalok waved her paws at them. \"Chachafah! Shooshoo! Gerroffa 'im, leave Lonna alone!\"\n\nBut the badger defended them. \"Let them be, marm. I like the little 'uns, they're so small and friendly. Besides, they're not at all afraid of my face, the scars and stitching.\"\n\nFigalok shrugged. \"Chaaaah, why be they 'fraid? Likkle 'uns never see'd a bigbeast afore. They know ya be a goodbeast, me see dat, too. Not matter what ya lookin' like.\"\n\nBefore he could express his gratitude for the kind words, a tubby squirrel mother, with a fine bush of tail, took the empty bowl from Lonna and called to the little ones. \"Hachowa! Sing for a bigbeast, sing 'im Twing Twing.\"\n\nThe elders stood by, smiling fondly as the squirrelbabes sang their simple song for Lonna. What they lacked in melody, they made up in raucous enthusiasm, some of them performing dancing leaps and hops in time to the tune.\n\n\"Twing twing up inna trees,\n\ntwirlin' me tail around,\n\nlighter'n fevvers onna breeze,\n\nnever not fall to a ground!\"\n\nThese were the only words they seemed to know, but they carried on singing the verse again and again, with the renewed gusto of babes enjoying themselves. Lonna held both paws wide, his face wreathed in a happy grin. The little ones swung on him, squeaking away lustily.\n\nThey were well into the seventh repetition of their song when one of the elders gave forth a piercing whistle. Like lightning, both infants and elders vanished into the foliage. A massive black shadow flew low overhead.\n\nLonna looked around, but not a squirrel could be seen anywhere. He called out into the densely leafed treetops. \"Figalok, where are you, what's going on?\"\n\nThe elderly squirrel popped her head out from behind a branch, her eyes wide with fright as she chattered. \"Bad, bad! Rakkaw Ravin badbird! Look ya uppina sky!\"\n\nGlancing upward, Lonna beheld a raven of startling wingspread, circling high in the bright afternoon sky. Reaching for his bow, he picked an arrow from the quiver and laid it on the string, keeping his eyes on the raven.\n\n\"Don't worry, marm, that bird won't harm you while I'm here.\"\n\nFigalok stayed under cover, shaking her head sadly. \"Rakkaw Ravin after babes, ya watch 'im, he soon be down. Steal likkle 'un, take what he want. Badbird, bigga strong an' fast. Nobeast stoppa Ravin!\"\n\nAs Figalok spoke, a tiny squirrel panicked. Squealing shrilly, she hopped out on a long branch. There she stood, covering her face, rigid with terror, and in clear sight of the foe. Sensing a quick kill, the raven folded its wings and dropped down like a thunderbolt.\n\nInstinctively, Lonna stretched the bowstring tight against his clenched jaw. Closing one eye, he aimed at the bird and loosed his shaft. With a sound like an angry wasp, the arrow zipped upward, taking the raven through its glossy, plumed body. Instantly slain, its huge wings spread wide open, the raven cartwheeled through the air like a dark, tattered cloak, landing with a thud on the woodland floor beneath the oak, transfixed by the badger's well-aimed arrow.\n\nChattering madly, the squirrels started pounding the body. The older ones used small slings, from which they hurled small pebbles. Emerging from cover, the babes tossed down pawfuls of leaves and pieces of twig, all the while screeching insults at their slain enemy.\n\n\"Yaa yaa, not eat us no more, Rakkaw!\"\n\n\"Yeeheeee, eata dis twig if ya be hungry, bigbird!\"\n\n\"Hahaaay, Rakkaw, we burn ya, burn ya, burn ya!\"\n\nSome of the older squirrels threw down glowing charcoal from their oven. The smell of charring feathers reached Lonna's nostrils. Shocked by the frenzy of hatred the squirrels were working themselves into, he called out in a stern voice.\n\n\"Here now, stop that, you'll cause a woodland fire!\"\n\nSensing the danger, Figalok joined Lonna. \"Chahah, ye heara bigbeast, stoppa throwin' fires!\"\n\nThey obeyed reluctantly. Figalok sent some older squirrels down to fetch water and quench the smoking embers. She touched the big badger's taut bowstring.\n\n\"Dat a good bigbow, me thank ya, Lonna. Rakkaw Ravin gone'd forever now, thank ya!\"\n\nHanging up his bow and quiver on a nearby branch, Lonna sighed. \"I wish that had been a Searat!\"\n\nFigalok pointed west and slightly south. \"Searatters over data way.\"\n\nThe badger became immediately alert. \"Where, over that way, have you seen them?\"\n\nSmiling slyly, the elderly squirrel nodded. \"Ho, me see 'em, awright! Lotsa Searatters marchin' through. Chahah, they no see us, though. Squirrel know how ta hide.\" She tapped her paw four times against the oak tree. \"Me see dat many Searatters a-comin' back thisaway though.\"\n\nLonna grabbed up his bow and quiver. \"Where, when?\"\n\nFigalok explained. \"Yistaday. Me was far from this place, lookin' for a h'almind nuts. See dem, one bigbeast.\"\n\nShe tapped her paw on the oak three times. \"Dis a many smalla Searatters comin' disaway. No worry, Lonna, dey not see ya, we hide up here plenny good, eh? Asides, dey still more'n a day 'way, not travel fast like squirrel.\"\n\nLonna seized the thick, knotted rope and began clambering down to the woodland floor. \"Searats at last! I've got a score to settle with those murdering scum. Figalok, will you show me where they are?\"\n\nThe squirrel made it down to the ground before him. \"A course me will\u2014least I can do for ya, bigbeast. We go now, catch 'em around at dawn, travel alla night, eh?\"\n\nLonna shook her small paw gratefully. \"Thank you, my friend!\"\n\nThe squirrels appeared much upset at Lonna leaving, particularly the little ones. \"Don't go bigbeast, ya stay here wid us for longa time!\"\n\nOne bold little maid thought she knew the reason for the badger's departure. She shook her head at the others. \"Gorra let Lonna go, he gotta find 'is mamma.\"\n\nLonna ruffled her downy little brush. \"That's right, miss. Now take care of your mammas, and watch out for ravens.\"\n\nFigalok kicked the dead bird's carcass scornfully. \"No more Rakkaw Ravin come here. We hangin' dis one up inna tree, dat scare 'em off. Chahaah, you betcha!\"\n\nFollowing the agile Figalok, Lonna trotted off south and west into the thickness of Mossflower. As they went, he envisioned the evil face of Raga Bol\u2014concentrating hard on it, as only a creature of fate and destiny like a badger can.\n\n\"I'm coming, Raga Bol! I am Lonna Bowstripe, and I'm coming!\"\n\n##\n\nAfter marching all night on what he had fondly imagined was a southeast course, Horty was totally fatigued. In dawn's pale light, he slumped down in a fern grove, grumbling.\n\n\"It's no blinkin' use, you chaps, I've got to take a jolly old snooze. Ahah! But first we must deal with the inner hare. Brekkers beckons the poor lad's slim stomach, wot?\"\n\nFurious, Springald grabbed the provision sack from his paw, ranting on at him. \"Food, food, food, don't you ever think of anything else? Here we are, in the middle of nowhere, and you're yowling about brekkers after eating all night as we marched! We're lost, you lop-eared oaf, lost!\"\n\nHorty tried unsuccessfully to tug the sack back from her. \"Lost? Don't talk piffle'n'woffle, m'dear gel, we're merely restin'. Now don't be so flippin' moody, an' pass the scoff!\"\n\nSpringald dealt him a wallop with the soggy ration sack. \"You've no idea where we're going. You've completely lost Bragoon's and Saro's tracks, and we could have been walking in circles for all you know! You're an idiot, d'you hear me?\"\n\nHorty twiddled his ears and smiled at Fenna. \"Rather pretty when she's angry, ain't she? Spring, me old beauty, why don't y'give your face a rest. We'll find the right track sooner or later. Or would you prefer to toodle back to the Abbey an' face the blinkin' music, wot wot?\"\n\nFenna sat down wearily beside Horty, then closed her eyes. \"Good grief, I'm bone worn-out. He's right y'know, Spring, arguing isn't going to get us anywhere. Let's have a bite to eat and a rest. Give him the bag.\"\n\nSpringald threw herself moodily down amid the ferns. \"Here, take your confounded food. I wish I'd never left Redwall in the first place.\"\n\nThe gluttonous young hare seized the sack eagerly. \"I wish you hadn't, either\u2014there'd be more scoff for me an' Fenn, wot. Hawhawhaw!\"\n\nFenna looked into the sack to select her breakfast. She drew back with a look of disgust. \"Yukk, I'm not eating any of that mess. Look at it, pie and trifle squashed up with onion gravy pastie. Just the sight of it makes me sick. Nobeast could stomach that!\"\n\nHorty dipped his paw in and came up with an unappetising lump of sludge. \"Well tut tut, little miss fussy apron. What's wrong with the flippin' scoff, it's good food ain't it? Please yourself, marm, but I'm jolly well starved.\"\n\nHe began eating with evident relish. \"Mmmmm, you bods don't know what you're missin'. Nothin' like a spot o' tucker to settle the old tum for a good sound snooze, wot!\"\n\nThis time it was Fenna who lost her temper. She tugged Horty's ears sharply. \"Listen to me, you great ten-bellied buffoon, you were supposed to be supplies officer, remember? You appointed yourself in charge of provisions. There'll be no naps or snoozes for you while us two are still hungry, so shift yourself and get us some breakfast, right away!\"\n\nHorty made a languid gesture. \"There's two other sacks there, or ain't you blinkin' well noticed? You can open 'em yourself!\"\n\nWhere Fenna upended one of the sacks, a great splodge of squashed pastie and meadowcream trifle splattered among the ferns.\n\nSpringald inspected the contents of the other sack. \"Ahah, scones and cheesebread. But guess what, pals? Our genius packed 'em along with a flask of mint tea and one of strawberry cordial. Of course he never made sure the stoppers of the flasks were on tight, so we've got another sackful of sludge. Oh, Horty, how could you?\"\n\nThe gluttonous hare was munching pawfuls of the mixture from the second sack. He smacked his lips loudly. \"Sorry about the blinkin' flask stoppers, chaps, but I didn't want to make too much noise, y'see. Mmmm, rather good this stuff. Hawhaw, I've just invented apple'n'rhubarb'n'gooseberry surprise. Hmm, there's some soft white celery cheese in here, too . . . excellent mixture. I must give old Gurvel the recipe when we return t'the jolly old Abbey, wot!\"\n\nSpringald peered into the third sack, wrinkling her nose in distaste. \"How could anybeast even think about eating that?\"\n\nHorty took the sack and sampled a pawful. \"An' what, pray, is the matter with it? 'Tis perfectly top-hole scoff! Trouble with you two is y'don't know how to blinkin' rough it. You've become spoiled by Abbey life, too picky by far!\"\n\nSpringald took hold of a sack. \"Go and get a bath, Horty.\"\n\nThe young hare grinned at her. \"Not right now, thanks, I don't need a bath.\"\n\nShe upended the sack over his head. \"You do now!\"\n\nHorty rose slowly, making two eyeholes in the mess of flan and pudding, then sucked his paws. \"Gettin' a bit touchy, aren't we?\" He saw Fenna take hold of another sack and fled. \"Hello out there, any frogs or tadpoles know a good stream where a chap can get a wash an' brush up, wot?\"\n\nFenna sat down and rested her head between both paws. \"We should've known better than letting him go for supplies. 'Tis our own fault, I suppose. The fool never even thought of bringing a flint along to make fire.\"\n\nSpringald produced a chunk of crystal from her belt pouch. \"That's no trouble. I got this off Old Phredd. He told me how to use it . . . watch this.\"\n\nShe held the crystal close to some unlit twigs and moss, focussing until it caught the sunrays and concentrated them in a small bright point. Instantly, the moss began smouldering. After a short while, a single puff of the mousemaid's breath caused a slim column of flame to rise.\n\nFenna was both delighted and astonished. \"That's marvellous! At least we can boil some water and pick mint leaves to make tea. There's plenty of wild mint growing round here. What's the matter, Springald?\"\n\nThe mousemaid kicked the sack she had upended. \"Guess what? Horty forgot to bring anything along to boil it in.\"\n\nFenna sat down beside her friend. \"Right, that's the last time I listen to the mad plans and stupid ideas of a hare. We'd best go back to Redwall!\"\n\nSpringald did not relish the suggestion. \"Redwall? Imagine having to face the Father Abbot, and Sister Setiva, and Granmum Gurvel and all the rest! I'd sooner sit out here for a season or two and starve, until they've forgotten about us drowning those Dibbuns, plundering the kitchens and disobeying the Abbot. Lack a day, we'd be scrubbing floors and washing pots until we were old and grey!\"\n\nSpringald's despairing thoughts were interrupted by Horty's voice. \"Yowch ouch, I say, leggo me blinkin' ears, you bounders!\"\n\nHorty appeared, dripping wet, with six big, mottled rats dragging him along. Their garb was a curious mixture of leaves, shrubbery and purple tattoos. All of them were armed with cudgels and long knives.\n\nSpringald let out a cry of alarm, Fenna seized an old kitchen knife and leapt up. Soon they were surrounded, as more rats stepped out from the trees.\n\nTheir leader\u2014a tall, brownish-white mottled vermin carrying a long spear\u2014growled warningly. \"T'row down der knife, or you're deadbeasts!\"\n\nSomething about his bleak stare told Fenna it would be wise to obey the order. She let the knife fall.\n\nHorty indignantly took up his case with the tall rat. \"I say, d'you mind tellin' these chaps to stop swingin' on me blinkin' ears? They'll pull 'em out by the flippin' roots, tuggin' at 'em like that, wot!\"\n\nA sudden jab of the tall rat's spearbutt jolted into the young hare's stomach, leaving him doubled up and gasping for breath. The rat turned the point swiftly, covering Fenna and Springald as they leapt forward to intervene.\n\n\"Be still or die! I am Birug, High Kappin of de Darrat. You be prisoners for invadin' our lands!\"\n\nSpringald protested. \"We're not invading anybeasts' land, only passing through. We are innocent travellers!\"\n\nBirug sneered. \"Shut you mouth, shemouse, you not talk to High Kappin like dat. Bring dem along!\"\n\nFenna was shocked to see that they were surrounded by at least a hundred rats. Horty regained his breath, but before he could speak he and his two friends were gagged with thick pieces of rope. Darrat rats swarmed over the trio, binding their forepaws tightly and linking their footpaws together on a long rope. They were helpless. The squirrelmaid barely had time to cast a frightened glance at her companions before sacks were pulled roughly over their heads. Cudgels prodded them, none too gently.\n\nBirug's voice rang out. \"March now!\"\n\nStumbling and bumping into one another, they were hauled swiftly along, dragged upright and cuffed soundly whenever they fell by the wayside. The unhappy trio bumbled along in the midst of their captors, terrified witless and ruing the day they had set paw outside of Redwall Abbey.\n\nSarabando and Bragoon lay in the treeshade, out of the shimmering midday heat. They sipped dandelion and burdock cordial and nibbled at oatcakes, supplemented by some watercress they had found near a stream. Saro tootled a small reed flute and played a melody. Bragoon sang the tune quietly.\n\n\"I know not young 'uns or a wife,\n\nno scolding tongue I fear,\n\nI live a carefree traveller's life,\n\nfrom yon to hither and here.\n\nO'er mountain hill and lea,\n\nI'm bound to wend my way,\n\ncross river lake or sea,\n\nwith never a beast to say,\n\nSit down! Stand up! Stay here!\n\nO ring a lairy lay.\n\nStand back! Be still! Just wait!\n\nFarewell my dear, good day!\"\n\nSaro began piping the tune to a second verse, when Bragoon ceased singing and held up a paw. \"Ssshhh! Did ye hear somethin', mate?\"\n\nEars cocked, the squirrel looked around. Silently she nodded, pointing over to the dense growth of trees on her left. Putting aside the flute, Saro pointed to her friend, indicating that he should stay put. In a flash she was gone, nimbly scaling a beech trunk and vaulting away through the foliaged upper terraces of Mossflower.\n\nBragoon sat perfectly still, his eyes roving from side to side as he searched the woodlands. Several minutes elapsed before Saro somersaulted back to earth from the high treetops. She picked up a twig, then snapped it and flung it away, muttering darkly to herself.\n\nBragoon raised his eyebrows. \"Wot's upset ye, matey?\"\n\nThe squirrel began gathering up her possessions. \"Upset? I ain't upset, buckoe, I'm steamin' fit t'burst! Those three young fools from Redwall, Horty an' the two maids\u2014they've got themselves captured by a hundred or so big spotty rats!\"\n\nBragoon sighed heavily. Buckling the sword across his back, he dusted himself off and made ready. \"You shore 'twas them?\"\n\nSaro checked her sling and pouch of stones. \"Aye, I'm sure enough. They was bound t'gether an' had sacks over their heads, but it's got t'be them. Wot other young hare, squirrel'n'mouse would be wanderin' willy-nilly through these woodlands, eh? They've sneaked out o' Redwall an' come searchin' for us, to share the adventure. Huh!\"\n\nBragoon shook his rudder in disapproval. \"Fivescore o' big spotty rats, ye say? Well, they'll get their share of the fun\u2014that's if'n the three idiots live long enough. Ye recall those spotty rats we battled with last time we was up this way?\"\n\nThe squirrel nodded grimly. \"Aye, they were flesh eaters!\"\n\n##\n\nEvening was crimsoning the sky over the western reaches as Birug led his Darrat vermin into camp. The Darrat tribe gathered around to see what he had captured. A huge old rat\u2014almost white, with a few brown flecks\u2014pulled himself out of a hammock which was slung under a rocky ledge. Bulling his way through the crowd, he indiscriminately kicked babes, young ones, females and males out of his way. Studying the bound and hooded creatures lying exhausted on the ground, he addressed Birug in a shrill voice totally unsuited to his bulk.\n\n\"Lemme see dem!\"\n\nHorty felt the sack being pulled from his head and a knife slitting the rope gag in his mouth. He spat out the gag and found himself looking at the huge, fat one. Immediately the young hare began complaining.\n\n\"Y'don't mind me sayin', sah, but this is all a bit bally much! Is this the way y'treat jolly peaceable wayfarers, wot?\"\n\nA slap from the huge rat silenced him. \"Shutcha face, rabbert, d'great Hemper Figlugg don' like talky rabberts!\"\n\nHe glared at Springald and Fenna, who had been unhooded and had their gags removed. \"Don' like talky mouses or squirrels either!\"\n\nA shrunken and incredibly ugly female pushed her way through to Hemper Figlugg's side. Ignoring him, she began pinching the three captives, nodding approvingly as she did so. Hemper Figlugg whispered something in her ear.\n\nShe nodded, replying aloud. \"Burcha Glugg!\" The Darrat tribe nodded in agreement and laughed.\n\nAlways ready to take advantage of a situation, Horty winked at his two companions. \"At least they seem happy, must be a good joke, wot! Burcha Glugg, wasn't it? Watch this.\"\n\nHe grinned at the assembly and repeated the words, \"Burcha Glugg!\"\n\nThe Darrat tribe howled with laughter at Horty's remark. A tiny ratbabe wrinkled his nose at the young hare and squeaked, \"Burcha Glugg!\"\n\nHorty favoured him with a kindly smile. \"Aye old lad, Burcha Glugg, indeed, wot! Yowhoooo, y'little savage. Gerroff!\" The ratbabe, who had bitten Horty's footpaw, clung on grimly. High Kappin Birug pulled the ratbabe off and cuffed it.\n\nHemper Figlugg nodded at his prisoners. \"Glugg cayjizz!\"\n\nThey were picked up bodily and borne to two large cages, formed of thick branches lashed together, one of which was open. Into this the three companions were thrown. The Darrat tribe dispersed and went about their business. Seeing they were being ignored, Springald began loosing herself from the ropes binding her forepaws and the running rope about her right footpaw. The other two did likewise.\n\nFenna watched the fat Hemper Figlugg settling himself back into the hammock. \"What now, I wonder?\"\n\nSpringald answered hopefully. \"Well, we're still alive, aren't we? Where there's life there's hope, they say.\"\n\nHorty rubbed his stomach\u2014as usual, his mind was on food. \"I won't be alive much longer if somebody doesn't feed us. Chap gets hungry, bein' captured an' all that, wot?\" He called out to a passing rat. \"Hi there, I say, me old vermin, how about somethin' to jolly well eat?\"\n\nHe pantomimed eating and pointed inside his mouth. \"Eat! Y'know, just like starvin' chaps do. Grub, food or whatever you savages call it.\"\n\nThe rat grinned and pointed to his own mouth. \"Glugg!\"\n\nHorty clapped his paws together. \"Hoho, that's the stuff. Glugg!\"\n\nSomething suddenly dawned on Fenna. \"Glugg, that must be their word for food. Oh, great seasons!\"\n\nHorty winked. \"Leave it to me eh, wot! I can translate any bally thing when it comes to food!\"\n\nSpringald understood all too well. She clapped a paw to her brow. \"Glugg, that's what we are. Food!\"\n\nHorty patted her reassuringly. \"No no, old gel, you've got it all wrong. They said Burcha Glugg\u2014that prob'ly means feed them, or give these bally prisoners some food, they look hungry.\"\n\nJust then, four Darrat males bore a big cauldron to the cage. They placed it outside the bars, within the captives' reach. It was filled with a form of porridge, full of berries and sliced fruit.\n\nOne of the rats indicated they should eat. \"Burcha Glugg, you eat all up.\"\n\nHorty smiled. \"Told you so!\"\n\nFenna asked the rat, \"What does Burcha Glugg mean?\"\n\nThe rat shrugged. \"Old Darrat way of saying good food.\"\n\nSpringald's worst fears were confirmed. She whispered in a shaky voice. \"They're fattening us up before they eat us!\"\n\nHorty dipped a paw into the cauldron and scooped some up. \"Oh, don't be silly! Nobeast'd dare to eat us, shockin' idea. I say, this tastes rather good, wot. Come on, you two!\"\n\nThey shrank to the back of the cage, shaking their heads. \"I couldn't bear to touch it!\"\n\n\"Oh Horty, how could you eat at a time like this?\"\n\nOne of the rats unwound a whip from about his waist, gave it a sharp crack and shouted at the pair. \"Eat or whip!\" They were forced to dip their paws in and eat. However, with the prospect of what they were being fed for, the food, as good as Horty said it was, turned to ashes in their mouths.\n\nFenna and Springald could only manage a small mouthful apiece, but Horty bolted the porridge down until his snout and whiskers were crusted with it.\n\n\"Mmmch, no sense in a chap bein' eaten, grmmfff munch, on an empty stomach. Capital stuff, wot!\"\n\nNight fell, bringing a cloudless vault of carnelian blue, dusted with stars. Bragoon lay alongside Sarobando, among some rocky hillocks that skirted the Darrat camp. The otter watched as campfires glimmered low.\n\n\"Let the vermin settle down, they prob'ly outnumber us by a couple o' hundred to two.\"\n\nSaro chewed on a dandelion stalk. \"What then?\"\n\nBragoon raised his head, risking a glimpse of the camp area. \"They're in a cage, over by that long rocky ledge. We'll have to work out a plan to break 'em out an' escape without bein' seen.\"\n\nThe squirrel lay back and closed her eyes. \"Yore good at schemin', mate. What's the plan?\"\n\nThe otter lay down and closed his eyes also. \"First a short sleep, wait'll the camp's quiet.\"\n\nSaro opened one eye. \"An' then?\"\n\nBragoon stuck Martin's sword into the ground, close to paw. \"I don't know just yet, but ye'll be the firstbeast I tell when a good idea comes along. I'm goin' to sleep, wake me in an hour. Otters get good ideas when they take naps.\"\n\nSaro rolled over onto her side. \"No, you wake me, 'tis your turn.\"\n\nHer companion watched the starlight playing along the swordblade. \"How can I wake ye when I'm makin' the plan? You wake me!\"\n\nThe squirrel grumbled. \"Huh, 'tis always me. Alright, you take a nap an' do all the plannin', I'll wake ye in an hour.\" The only answer she received was a pretend snore from the otter.\n\nThe midnight hour had just passed. Silence reigned over the Darrat camp, broken only by protracted snores mingled with nighttime woodland sounds.\n\nIn the cage, Horty sat clasping his stomach and grimacing. Fenna came over to sit by him. \"Tummyache, eh?\"\n\nThe young hare answered dolefully. \"Absolute agony, doncha know. No use upsettin' you an' Springald, so a chap's got to be brave an' silent, even though he's dyin'. It must've been somethin' I ate.\"\n\nSpringald overheard him and snorted. \"Something? You great glutton, 'tis not something, but how much of that something you ate. That big cauldron's almost empty!\"\n\nHorty winced. \"Ah me! Maids can be beautiful but cruel. I only scoffed that porridge because you two wouldn't touch it after the first mouthful. Ha, 'twas me that saved you a jolly good whippin'. Sacrificed meself for your rotten sakes, that's all the gratitude a chap gets, wot?\"\n\nOne of the three guards in front of the cage snuffled and grunted at the sound of Horty's raised voice. The captives sat in frozen silence until he settled back down with the other two rats. The three guards snorted in soft unison.\n\nSpringald whispered, \"Look at them\u2014not a care in the world. We'd be that way, too, snoring in the dormitory. Huh, that's if we'd had the sense to listen to the Abbot and your sister Martha. Wish we were back at Redwall now.\"\n\nFenna murmured, \"Wishing isn't much use. What we should be doing now is escaping while the guards are asleep.\"\n\nHorty forgot his pains for a moment. \"By jingo, you're right, old gel. Escape, that's the bally idea! Right, chaps, anybeast got a scheme or a plan of some type, wot?\"\n\nThey sat racking their brains for a while, until Fenna admitted limply, \"We've got no chance, locked in a cage and surrounded by armed guards. They'd cut us down before we managed to get two paces!\"\n\nNumbly they stared at one another. A tear trickled down Springald's cheek; Fenna's lower lip started quivering. Horty blinked and sniffed.\n\n\"We've really gone an' done it now, haven't we, chaps, wot!\"\n\nThen a rope fell from above, close to the cage. Attached to it was a sharp knife and a piece of bark that had charcoal writing scrawled on it: \"Hush, take knife, escape. Tie rope to pot. Wait.\"\n\nHorty peered up through the bars at the overhead rock ledge. Bragoon's tough-lined face was staring back at him. The otter held a paw to his mouth, signalling silence. Working feverishly, Springald took the knife and tied the rope to the cauldron handle. At a wave from Fenna, the cauldron rose upward, halting just above the cage.\n\nGripping the rope firmly, Bragoon began swinging the iron cauldron from side to side until it moved back and forth in mighty sweeps like a giant pendulum. Horty watched it as it swung, lower and lower, whizzing close to the cage front, until it reached the level of the three snoring Ratguards. Then the cauldron jerked outward. _Kurblunggggggg!_ It struck two of the rats, laying them out senseless. The remaining one sat up, rubbing his eyes.\n\n\"Wot was th . . .\" _Podongggg!_ The cauldron caught the third rat on the return swing, knocking him head over paws.\n\nSpringald was sitting on Fenna's shoulders, slashing at the ropes which kept the wooden roof bars in place. The sharp knife made short work of them.\n\nHemper Figlugg awoke. He heard the cauldron toll like a muted bell as it hit the last rat. Waddling out of his hammock, he went to investigate the noise. Seeing Fenna's head poking out of the cagetop, he hastened forward, shouting wheezily, \"Burcha Glugg 'scapin'! Wakey wakey, Darrats!\"\n\n _Borlongggggggg!_ The swinging cauldron biffed him on the back of his great fat head. Hemper Figlugg performed a somersault, raising a big puff of dust as his back hit the ground. His shout, however, had roused the Darrat horde, who came staggering from under the ledges and thick bushes, grabbing for weapons.\n\nBragoon roared down to the escapers, \"Cut that pot loose an' grab on to the rope!\"\n\nSpringald slashed the cauldron free, and they took hold of the rope.\n\nSaro's head appeared above the high ledgetop. \"One at a time, we can't pull ye all up t'gether!\"\n\nHorty grabbed the spear from a fallen Ratguard. Taking charge, he rapped out orders like a veteran sergeant. \"Steady the buffs, chaps! Spring, you go first, Fenna next! I'll hold these bounders off, wot!\"\n\nThe Darrat had just realised what was taking place. Around half a dozen of the boldest came at the young hare.\n\nSpear at the ready, Horty challenged them bravely. \"Step up there, laddie bucks, meet a flippin' Redwall warrior, wot! Two or ten at a time, doesn't blinkin' matter to Bonebreaker Braebuck. Have at ye, scurvy nosewipes! Come on, don't be shy, ye wiltin' wallflowers. Wot!\"\n\nA big broad mottled rat charged at him, waving a hatchet. A slingstone flew from above, and the rat stood still, tottered, then collapsed in a heap.\n\nHorty threw himself at the other five rats, who had been advancing on him slowly. He was in his element.\n\n\"I'm the son o' the roarin' buck! D'ye want to visit your ugly ancestors, eh? Well, I'm the one who'll send ye to Hellgates. Yaaaaaaah!\"\n\nAt the top of the ledge, Fenna and Springald stood with their rescuers. Bragoon shook his head. \"Is he mad? Look at 'im!\"\n\nHorty was like a whirling demon, lashing out with his long hind legs as he thwacked wildly about with the spear. Rats went down like ninepins before his onslaught.\n\nSarobando nodded in admiration. \"That young 'un's got the makins of a powerful warrior, but he's still a hotheaded learner. Soon as he tires they'll overpower 'im an' bring 'im down.\"\n\nSpringald yelled down to her friend. \"Horty, get to the rope, hurry!\"\n\nThe young hare looked at the pack of rats charging toward him. \"Right away, marm, cover me jolly old back, chaps!\"\n\nSaro used her sling, while the others pelted the rats with rocks from the ledge as Horty ran for it. He reached the rope and looped it about his waist.\n\n\"Haul away!\"\n\nKappin Birug flung a wooden club that caught Horty square between both ears, before bouncing off his head.\n\nHorty grinned. \"Yah missed me!\" Then he fell unconscious.\n\nDucking slingstones and a few arrows, the rescuers\u2014along with Fenna and Springald\u2014hauled Horty's limp figure up onto the ledgetop.\n\nBragoon peered anxiously down as more archers began appearing. \"Better get goin' an' move out o' range. They mean business!\"\n\nThey struck off into some thick pinewoods, carrying the senseless figure of the hare between them.\n\n##\n\nIt was a long and wearying night, but the Redwallers kept going. Pines grew thick about them, obscuring even the stars in the sky. Stumbling on through the dense carpet of rotting pine needles, Springald bumped into a tree trunk.\n\n\"Oof! There won't be a part of me that's undamaged if we go on at this rate. A torch would help us to see where we're going.\"\n\nBragoon urged her on. \"Just keep goin', missy, there'll be no torches. One spark can start a fire among pine trees, an' the whole woodland'd be ablaze bafore ye could blink. Besides, a torch would be like a beacon for those vermin to follow.\"\n\nSpringald felt foolish. \"I'm sorry, I didn't realise.\"\n\nThe otter said nothing, but he was exhausted and bad-tempered after having to run all night, burdened with Horty. He snapped at the mousemaid. \" 'Tis not much good bein' sorry now, Miss Mouse. If'n you three would've stayed put at the Abbey, we wouldn't be in this fix!\"\n\nFenna came to her friend's defence. \"We only came after you because we thought we could help. Besides, now that we're free, we can get on searching for Loamhedge.\"\n\nBut Bragoon was not to be appeased. \"Free, eh, don't make me laugh! You think those rats won't come after us? Lissen, I know rats, they won't rest 'til they've got us all in the cookin' pot. Ask Saro, we've fought flesh eaters like them afore. The only way to make 'em give up is to kill 'em, an' there's too many of the scum for that!\"\n\nA quavery voice echoed out of nowhere. \"Oh, far too many! They've eaten most of us, you know.\"\n\nBragoon stood stock still, his eyes scouring the night woods. \"Who said that?\"\n\nFrom a small hillock of pine needles built up round the base of a trunk, the voice answered, \"If you remove your great heavy rudder from my neck, I'll tell you!\"\n\nThe otter leaped to one side as an old rabbit shoved his head through the mound.\n\n\"Sorry to startle you like that, I'm sure. If the Darrat are hunting you, I'd be pleased to hide you. Only for awhile, though\u2014they eat anybeast who harbours fugitives.\" The ancient rabbit shrugged. \"But Darrat will eat a creature for no reason at all. So, d'you want me to hide you?\"\n\nSaro indicated the unconscious Horty. \"Just until this 'un's fit for travel agin, thankee.\"\n\nThe rabbit's name was Cosbro. He took them to the hollow log in which he lived. It was a cunningly contrived dwelling, a great elm trunk overgrown with all manner of moss and nettles. One end of it backed against a standing rock, the other was artfully concealed by thistles and wild lupins. Cosbro carefully parted these, creating a little gap which allowed them to squeeze through one at a time. Once they were all inside, the old rabbit rearranged the outer thistles and lupins, rendering the entrance invisible to the casual observer.\n\nSpringald looked about: it was a very neat little home. Lit by four lanterns containing fireflies, its illumination dim but adequate. They sat down on a carpet of dried grass and springy moss.\n\nFenna made Horty comfortable, remarking, \"I've never heard of a rabbit living inside of a tree before.\"\n\nCosbro preened his meagre whiskers. \"Neither have the Darrat, young 'un. That's what makes it such a perfect place. I've often sat in here, listening to them digging holes as they searched for rabbits\u2014they dig out anything that looks like a burrow. Clods, they have no imagination at all.\"\n\nBragoon smiled at the old one. \"But where do the other rabbits around here live?\"\n\nCosbro shook his head sadly. \"There are no other rabbits left. Only me, sitting inside this log, poor fool that I am.\"\n\nSaro patted his paw gently. \"You ain't no fool, me friend. It takes a clever beast to survive in this country. How many rabbits were there, an' how'd ye come to be livin' here?\"\n\nCosbro shrugged. \"We were too many to count one time, long ago. Our families had no written history. All I have to remember my ancestors by are ancient poems and ballads passed down by word of mouth. Woe is me, sometimes I think I must be the last rabbit left in all the land.\"\n\nSaro felt sorry for the pitiful old creature. She passed him a flask of dandelion and burdock cordial.\n\n\"Wet yore whistle with this, ole mate. Maybe ye'd like to tell us one of yore poems from the ole days, eh?\"\n\nCosbro sipped the cordial, closing his eyes blissfully. \"Ahhh, dandelion and burdock, tastes like nectar to me. Aye, 'tis many long seasons since I tasted ought as good as this. Have you ever heard of a poem called 'The Shadowslayers'?\"\n\nHe looked from one to the other, but they shook their heads. Helping himself to a longer sip, Cosbro licked his lips. \"When I was younger, I could skip through such verses. But, alas, the weight of seasons has descended upon me. My mind forgets a lot of things these days. So, my friends, here is the poem, as best as I can recall it.\n\n\"Lo the golden days are gone,\n\nthe happy laughter long fled,\n\nnow silence falls o'er Loamhedge walls,\n\nlone winds lament the dead.\n\nThe Shadowslayers sent us forth,\n\nsome south and east, some west and north.\n\nThe wise ones said 'twas vermin foul,\n\ntheir blood, their teeth, their fur,\n\nwhich brought the plague that laid us low,\n\nwith more than we could bear.\n\nWhen families die before our eyes,\n\nwe learned, 'tis folly to be wise.\n\nLeave everything ye own now, flee,\n\nrun if ye can, go far and wide,\n\nlinger not here, to grieve and weep,\n\nthose tears have all been cried.\n\nThe mouse Germaine said, 'Woe, 'tis true,\n\nThe Shadowslayers will come for you.'\n\nThe mice went first, escaped their fate,\n\nthey traversed north and west;\n\nwhat was left of us remained,\n\nto lay our dead to rest.\n\nWe travelled then, us piteous few,\n\nwho'd seen what Shadowslayers could do.\n\nMy father's father spake these words,\n\nas had his kin, from time untold,\n\nwand'ring exiled o'er the land,\n\ngrowing up, and growing old.\n\nRecalling to their dying breath,\n\nhow once the Shadowslayers brought death.\"\n\nCosbro took another drink and sighed wearily. \"I myself wrote that final verse, though there were many more. They told of our family names and histories. But I've forgotten the words, shame on me!\"\n\nFenna thought it was the saddest thing she had ever heard.\n\nSpringald spoke comfortingly to the ancient hare. \"I hope that if ever I live to your age, I would remember the half of it, sir.\"\n\nHorty chose that moment to waken from his stupor. \"Remember what, wot? I say, did we escape those blighters? Jolly good show, chaps, where are we now? Someplace far a-blinkin' way, I hope. Owch, my flippin' head's given me jip!\"\n\nHe tried to stagger upright and banged his head on the log. \"Yowhooyooch! Who left that up there, confounded oaf!\"\n\nSaro threw herself across his face, stifling further cries. She whispered fiercely. \"Shuttup, addlebrain, I can hear somethin' goin' on outside!\"\n\nKappin Birug and a crowd of Darrat rats halted alongside the log. Those inside held their breath in frozen silence. Sounds of the vermin poking about with spearbutts and slashing at shrubbery could be heard by those in the log. Outside, Birug climbed up and sat upon the log. Dawning sunlight slanted through the trees as he glanced down at the Darrat rats resting upon the grass.\n\n\"Any of you be High Kappins, eh?\" They stared owlishly at one another, then shook their heads. Birug jumped up, performing a dance of rage upon the log. Pointing his spear at them, he screeched.\n\n\"Den why you not searchin', mudbrains? Search! Search! Find dem, y'want me to do everythink, eh? Search!\"\n\nThey dispersed hastily, trying to look busy and diligent as they probed amid the woodland trees. Birug laid about with his spear shaft, spittle going everywhere as he took out his bad temper on anybeast standing close.\n\n\"Hemper Figlugg got bad sore skull, big lump onna 'ead! Dose beasts die slow when I catch 'em. Only make Burcha Glugg out of wot be left of dem!\"\n\nBirug hurried over to a rat who had returned to investigate the fallen log. Dealing the unfortunate several hard kicks to the rump, the Kappin screeched hoarsely at him. \"Wotcha be doin', dumbum\u2014y'think they be beetles, hidin' inna falled treelog? You never be High Kappin, that be sure!\"\n\nAs Birug chased the rat back to search with the others, Cosbro crept to the log opening and called out in excellent imitation of the gruff Darrat dialect. \"Der dey goes! Ober dat way, quick!\"\n\nThere followed a stampede of pounding Darrat paws, with Birug bellowing as he hastened in pursuit. \"Not kill 'em, catch 'em priz'ner, that a h'order!\"\n\nAs the sounds retreated, the fugitives breathed easier. Springald was visibly shaken. \"Good grief, that was a bit close for comfort!\"\n\nSaro removed herself from Horty's face. He was the picture of sputtering indignity.\n\n\"Pshaw, phoo! I'll be spittin' wodges of your bally tailfur for days t'come, marm. No blinkin' thanks to you, I was near smothercated, wot! But who am I to complain, chaps? Me flippin' head's poundin', achin' to blue blazes. There's a lump like a duck egg on me young skull. The poor old stomach is painin' an' swollen from savin' the ungrateful comrades. An' to top it all jolly well off, a great lump of a squirrel has been layin' on my tender young mouth for absolute ages. Phwaaaw, phutt! Never feed your young on squirrelhair, tastes vile!\"\n\nBragoon's paw shot out, pinching Horty's nose in a viselike grip. \"Are ye finished moanin', after ye nearly got us all captured, young sir?\"\n\nHorty tried to nod. \"Yith, juth leggo ob be dose pleathe!\"\n\nThe otter released his grip, growling threateningly. \"One more whimper an' I'll pull it right off, so keep quiet!\" He turned to question Cosbro. \"Ye mentioned Loamhedge in yore poem, mate, an' Abbess Germaine, too. She ruled there, from wot I've 'eard. Loamhedge is where we're bound for. Any idea which way it lies?\"\n\nThe ancient rabbit pointed in a general southeast direction. \"I can't be sure, but I've always imagined it being somewhere over that way. I've heard 'tis savage country\u2014deserts, chasms, wide rivers, and numerous foebeasts.\"\n\nSaro nodded. \"Aye, me'n Bragoon have seen a bit of it, though that was quite a few seasons back. Over that way, eh?\"\n\nCosbro began moving the vegetation from his log entrance. \"When you see a great line of very high cliffs, you'll know you're on the right track. Er, by the way, have you any of that excellent cordial to spare? I'm too old to travel now.\"\n\nBragoon passed him a fresh flask. \"Take this, friend, an' thankee kindly for yore help!\"\n\nThey emerged into calm morning sunlight and fresh, green woodlands.\n\nSaro waved to Cosbro. \"Good fortune be with ye, matey. We'll travel now, while the coast's clear. You take it easy!\"\n\nCosbro brought something out of his dwelling and gave it to Bragoon. It was a large coil of rope\u2014thin but incredibly strong, with big knots every three pawlengths.\n\nThe otter inspected it closely. \"Haharr, 'tis a climbin' rope, an' a fine one, too. If'n I ain't mistaken, this'll come in useful at the high cliffs. Where'd ye get it?\"\n\nCosbro explained. \"I made it myself, when I was a lot younger. Never got round to using it, though. I've forgotten my dreams of high cliffs long since. You take it.\"\n\nBragoon drew Martin's sword and held it up in a warrior's salute. \"A gift from a friend is somethin' to be valued. Thankee, sir, an' may the seasons be kind to ye!\"\n\nTo avoid bumping into the Darrat, they set off at a southerly tangent through the woodlands. Cosbro stood watching until they were out of sight. Wiping a paw across his rheumy eyes, the ancient rabbit murmured wistfully to himself, \"And may the seasons be kind to you, friends. May the breeze be at your backs, and the sun never in your eyes. Ah me, I wish that I were young enough to go with you.\"\n\nThe lonely rabbit shuffled back to his home, thinking of the high mysterious cliffs and the lost opportunities of his earlier seasons, now that old age leaned heavily upon him. Cosbro took one last look at the far horizon as he bent to enter the log dwelling.\n\n\"Ah well, at least my rope won't be wasted\u2014if they live long enough to use it.\"\n\n##\n\nMartha did not sleep a wink on the night that the vermin were sighted. It was as if some unreasoning panic was welling up in her. Vermin, at the very gates of her beloved Abbey! Restlessly she roamed Great Hall, propelling the little cart which held her chair, by pulling it along with the crutch that Toran had made for her.\n\nMoonlight sent pale shafts of light in varied hues as it shone through the stained-glass windows onto the worn stone floor. Travelling through the patches of dark and light, the young haremaid arrived at the tapestry of Martin the Warrior. She gazed up at the figure of the heroic mouse. It was illuminated by a small lantern on either side.\n\nMartha voiced her fears and worries to her friend. \"Oh Martin, what shall we do? Sarobando and Bragoon have left the Abbey, and all on my silly little behalf. Abbot Carrul gave Bragoon your great sword to take with him. I'd stay in my chair forever, if only they were back here at Redwall. The safety of this Abbey and all my friends here is far more important than foolish dreams of being able to walk. With my brother and the other young ones gone, who will help us against the vermin? The very thought of those cruel, murderous vermin getting inside our gates is horrible!\"\n\n\"Here now, young Martha, what's all this?\"\n\nShe gave a start as the Abbot loomed up out of the shadows. \"Father Abbot, I thought you'd gone to your bed.\"\n\nCarrul sat down on the edge of the cart and looked over the top of his glasses at her. \"And I thought you had, too, miss.\"\n\nThe sound of the main abbey door opening caused them both to pause. The Abbot's loud whisper echoed around the hall columns.\n\n\"Who's there?\"\n\nToran's voice replied. \" 'Tis only me an' Foremole Dwurl, Father. We just been relieved o' wallguard by Junty Cellarhog an' Weld.\" The pair joined Martha and Abbot Carrul.\n\nDwurl tugged his snout politely. \"Wot bee's you'm a-doin' settin' daown yurr? Shudd be snorin' abed, 'tis orful late.\"\n\nThe Abbot put on his wise face. \"Oh, we were just discussing a few things, weren't we, Martha?\"\n\nThe haremaid managed an important little cough. \"Ahem, yes, just small bits of business. What's it like out there, Toran? Any more news of the, er, vermin?\"\n\nThe ottercook sat back on his rudder. \"No, miss, they ain't up to much. Their fires are burnt low, I think they're sleepin'. We've been watchin' the ditch outside the front gate, t'other side o' the path, makin' sure they don't try t'sneak along it.\"\n\nMartha asked the question she had been anxious to have answered. \"Aren't you afraid?\"\n\nToran rubbed his wide midriff thoughtfully. \"Bless yore 'eart, pretty one, o' course we are. Only a fool'd say he wasn't. We're afraid as any sensible beast should be, but we ain't scared. Wot I mean is, we're only afraid for the safety of others\u2014Dibbuns, an' young 'uns like yoreself. But if'n we got to do somethin' about it, we ain't scared o' vermin.\"\n\nForemole licked his lips. \"Oi'm a-feared.\"\n\nToran raised his eyebrows at this remark. \"You, afeared?\"\n\nA huge grin creased the mole leader's homely face. \"Aye, zurr, afeared oi'll fall asleep an' miss ee brekkist. Oi'm a-thinken oi'll go to ee kitchens an' get a h'early wun!\"\n\nMartha laughed at the mole's comical logic. \"What a great idea, sir, I think we'll join you!\"\n\nThe kitchen was crowded with Redwallers of a like mind, even Dibbuns. Nobeast could sleep with the excitement of the night. Granmum Gurvel and three young moles were busy filling baked apples with honey and chopped hazelnuts.\n\nGurvel curtsied to the Abbot as she bustled by. \"Coom in an' sit ee daown, zurr, an' you'm h'others, too. Et bee's a gudd job moi ole bones can't be a sleepen, so oi'm a keepen moiself bizzied.\"\n\nThey found seats around the kitchen table and began pouring a sauce of meadowcream and rosehip over their baked apples. Everybeast was watching the Abbot as he paused before eating to address them.\n\n\"What we need are some good contingency plans, my friends. Seeing as most of us are here, I'll take any suggestions.\"\n\nMuggum was sitting up on a shelf, among the spice jars, with his cohort of Dibbuns. The molebabe raised his spoon. \"Oi says chop ee vermints tails offen wi' a gurt rusty knoife, an' barth 'em in 'ot soapy watter. Hurr, they'm soon bee's glad to run away arter that. Ho urr aye!\"\n\nThis met with hearty applause and much sneezing from the Dibbuns, two of whom had opened a hotroot pepper jar. Amused by this, Abbot Carrul tried to keep a straight face as he spoke to Sister Portula, who was recording the meeting. \"Not a bad idea! Write it down, Sister, and don't forget the bit about hot soapy water. We'll keep it in mind.\"\n\nSister Setiva, after wiping several noses and glaring the Dibbuns into silence, held up a paw. \"As soon as ah've finished eating, ah hope some o' ye will join me tae search around for more things tae use as weapons.\"\n\nMartha was among those who volunteered. But Toran had other plans for her. \"You'd never be able to search the attics upstairs, me beauty. I think ye should be in charge of the Dibbuns' safety. Seasons forbid that anythin' should happen to the liddle 'uns with vermin camped next to our gates. Will ye do it, Martha?\"\n\nImmediately the haremaid agreed. \"I'd be glad to. Right, come on you villains, off that shelf and up to bed. Last one up washes all the pots and dishes, eh, Granmum Gurvel?\"\n\nGurvel picked up her big ladle. \"You'm said the vurry thing oi wuz abowt t'say, Miz Marth!\"\n\nAn almighty scramble followed as Dibbuns climbed down from the shelves and fled upstairs squealing.\n\nAbbot Carrul waited until the noise subsided. \"Next suggestion please!\"\n\nBadredd lay awake down in the ditch, trying to ignore the stentorian snores of those around him. He longed for the dawn, when he could take possession of his magic sword. What did it look like? He imagined it as a solid gold blade with a crosshilt and grip crusted with rubies, pearls and emeralds. Of course, he would not mind too much if it were made from silver with jetstones and sapphires for adornment.\n\nMentally he went through a speech he had prepared for the woodland bumpkins who lived behind the wall. Badredd silently practised it, making sweeping paw movements to emphasise its drama. \"Throw wide your gates! Tremble at my name, for I am Badredd, commander of a vermin horde.\"\n\nHe paused here, wondering if his scruffy little band could constitute a horde. No matter, those woodland oafs had probably never seen a horde, much less taken a head count of one. He continued his oratory. \"You are looking at death, all of ye! Unless you deliver unto Badredd the magic sword that is rightfully his.\"\n\nHe questioned the last phrase\u2014it needed something, a word or two to prove that the sword's ownership was never in doubt. Hah, that was it! He embellished his flowery recitation thus: \"For did not my father, Reddblade, Warlord of the Northern Mountains, proclaim it so? 'Give unto my son Badredd his sword. It lies within Wallred, I mean, Redwall. To the mighty warrior goes the magic sword!' \" He flung out his paw and caught Halfchop a smack on the chin.\n\nThe rat awoke, holding his chin in his good paw. \"Mmmph, wot did ye do that for, Chief?\"\n\nBut Badredd was too fired up to waste time with arguments. \"Get further along that ditch an' see if'n ye can make it so that yore level with the big gate!\"\n\nHalfchop peered at him in the predawn darkness. \"Wot for?\"\n\nBadredd shoved him forward. \"If'n ye make it safely, give me a signal. I'll follow up with the rest o' the crew. That way we'll be in place when it gets light. They'll get the shock o' their lives when they see me climb out o' the ditch an' demand the magic sword. Go on, don't hang about!\"\n\nBlundering forward, Halfchop stepped on a thistle and banged into the ditch's sidewall. \" 'Tis no good, I can't see a thing. Why don't ye wait 'til dawn?\"\n\nBadredd drew his cutlass. \"Because I want it done now. There'll be one less in the crew if'n ye stand there rubbin' yore chin an' makin' excuses. Now get goin'!\"\n\nHalfchop picked up a red-ended branch from the embers of a fire. He went off, blowing it back to burning light and muttering, \"Alright, then, but I ain't goin' without a light!\"\n\nUp on the northwest rampart corner, Brother Weld nudged Junty Cellarhog. \"Is that somebeast coming along the ditch carrying a light?\"\n\nThe burly hedgehog watched as a small burning beacon grew closer. \"Aye, so 'tis, Brother. I wager that's a vermin, up to no good, I'll be bound. Better stop the rascal afore he sets fire to our front gate.\"\n\nThere was always a variety of things in Junty's big apron pocket. He dug a paw in and rummaged about. A slow smile lit up his heavy features as he produced a big barrel bung made from a knot he had gouged out of an oak log. \"This should do!\"\n\nThough ponderous and not given to quick flings, Junty was accurate and very powerful.\n\nHalfchop was never very sure of what fractured his muzzle and wrecked his nose. But he never forgot the sound as it hit him. _Kachunk!_\n\nBadredd saw the rat's light snuffed out with a gentle hiss as it fell into some stagnant water. He went and shook the weasel brothers, Floggo and Rogg, awake. \"Rouse yore bones there. Go an' fetch ole Halfchop back 'ere. He went wanderin' off up the ditch. It looks like the idiot's fallen over. Go on, move! It'll soon be dawn.\"\n\nWhen they returned, hauling the senseless rat, Badredd blew on the embers and stirred the fire. He winced as he saw the damage to Halfchop's face. Awakened by the commotion, Flinky dug some dried herbs out of his pouch and lit them so that they smouldered. The weasels held the rat's head steady as Flinky pushed the smoking herbs under his nose. Halfchop's eyes opened immediately when the pungent fumes got to him.\n\nBadredd squatted beside him. \"What happened?\" Halfchop looked at the fox quizzically as he repeated the question. \"Who did that to ye, what happened?\"\n\nHalfchop spoke . . . just one word\u2014\"Kachunk!\"\n\nFlinky put aside the smouldering herbs. \"Wot did ye say, mate?\"\n\nHalfchop looked at Flinky as if seeing him for the first time. He looked at Badredd the same way and spoke the word again. \"Kachunk!\"\n\nLosing his patience, Badredd pawed the cutlass edge menacingly. \"Talk sense! I asked ye wot happened. Keep sayin' that stupid word an' I'll kachunk ye, good an' proper!\"\n\nHalfchop leaned close and whispered in the fox's ear. \"Kachunk!\"\n\nAs Flinky saw the cutlass beginning to rise, he stepped in and stayed his crew leader's paw. \"Ah now, leave him alone, Chief. The pore ould rat's not in his right mind at all. How d'ye feel, matey, better now?\"\n\nHalfchop smiled foolishly over his swollen muzzle. \"Kachunk!\"\n\nDawn crept in from the east, pale pink and lilac in a creamy haze. Dewdrops bedecked the flatlands beyond the ditch. Redwall Abbey's twin bells tolled out the opening of a new summer day. Martha watched Toran, Abbot Carrul and several others mounting the gatehouse steps. Frustration tinged the haremaid's plea to them.\n\n\"Let me come up on the ramparts, I want to see what's happening. Oh please, I feel so helpless down here!\"\n\nToran shook his head. \"It might get a bit dangerous up here, me pretty. Best ye stop down there an' look after the Dibbuns.\"\n\nLittle Shilly the squirrelbabe made a scramble for the steps. \"Cummon, we all go up onna wall. Then Miz Marth' gotta be up dere wiv us'n's!\"\n\nSister Setiva ran down and blocked the Dibbuns' way. \"Och no ye don't, mah wee babes. Ah'll come o'er tae the orchard wi' ye an' Martha. We'll see if any blackberries are ripe enough tae be picked yet. A guid idea, eh?\"\n\nSqueaking with delight, the Abbeybabes pushed Martha's chair across the lawns so fast that the haremaid was forced to hold on tight to the arms.\n\nSister Setiva chased after them, shouting in her thin, reedy voice, \"Slow down, ye naughty creatures, go easy wi' Miss Martha!\"\n\nJunty and Brother Weld kept an eye on the ditch as they made their way along to the threshold over the main gate. Throwing a brief salute, the Cellarhog made his report to the Abbot. \"Looks like they're makin' a move, Father. Comin' this way!\"\n\nThe wall party was armed with a variety of window poles, kitchen utensils and tools. Apart from one or two slings and bags of pebbles, there were no real weapons to be found within the bounds of the peaceable Abbey. Toran gave Junty a sling and some stones. He tossed a long ash stave to Brother Weld.\n\n\"These ain't much, but they're better'n nothin', friends.\"\n\nNow the vermin crew had reached the spot directly below where the Redwallers stood. They halted, only the tops of their heads visible. Silence fell as they waited, standing in a muddy pool of ditchwater.\n\nToran whispered to Abbot Carrul. \"Let them state their business first.\"\n\nThe silence from below became rather protracted, then a voice spoke out. \"Kachunk!\"\n\nThis was followed by Badredd hissing, \"Somebeast, shut that fool up!\"\n\nCuriosity overcame Old Phredd the Gatekeeper, who called out, \"What do ye want? Speak up!\"\n\nBadredd had envisioned himself leaping boldly from the ditch to state his demands. However, he was far too short for such a thing, so several of the crew had to lift him up and boost him onto the path. It was a totally undignified procedure. The little fox landed, sprawling on the dust and gravel. He sprang up quickly, took a swaggering step forward and tripped over his cutlass.\n\nHaving heard a few stifled giggles from the walltop, Badredd glared up frostily at the assembled Redwallers, putting on his toughest snarl. \"Ye'll laugh the other side of yore faces afore this day's done!\" Puffing himself up to his full height, he continued. \"I'm Badredd, Warlord of the Vermin Horde. Nobeast can stand against me. I come from the Northlands where we drink our enemies' blood!\"\n\nThe Abbot bowed his head politely. \"I bid you a good morning, Sir Badredd. I am Father Abbot Carrul of Redwall. Is there any way I can be of service to you? Mayhaps you might need food or supplies to continue your journey?\"\n\nAt the mention of food, the rest of the vermin crew climbed out of the ditch eagerly, but the little fox forestalled them by answering the Abbot scornfully. \"We don't want yore food, mouse. Our journey's end is here, at this Wallred place. You've got a magic sword here. I want it\u2014bring it t'me now!\"\n\nThe Abbot stared coolly down at him. \"There is no such thing as a magic sword at Redwall Abbey.\"\n\nBadredd drew his cutlass with a swish, pointing it at Carrul. \"You lie! Bring that sword out to me, old fool, or it will go badly with ye!\"\n\nToran stepped up to the Abbot's side, roaring down at the fox, \"Don't ye dare call the Abbot of Redwall a fool or a liar! If he says there's no magic sword here, then you'd best get the mud out o' yore ears an' listen. Now shift yoreself, vermin. Get up the road with that raggedy-bottomed bunch. Quick, or I'll come down there and kick yore tail back t'the Northlands!\"\n\nShaking with rage, Badredd turned and nodded to his two archers, the weasel brothers. \"Fire!\"\n\nTwo arrows zipped from their bows. Toran flung himself upon the Abbot, knocking him down below the battlements. One arrow flew harmlessly overhead, the other grazed the ottercook's shoulder.\n\nToran winced as he yelled, \"Down, everybeast!\"\n\nThe Redwallers immediately dropped below the parapet. Junty Cellarhog fitted a stone into his sling and whirled it. He popped up and let fly. Though it was a speedy shot, and not too accurate, it did hit Badredd on the footpaw. He screeched out in pain as Crinktail and the rest of the crew jumped back into the ditch, taking him with them.\n\nThere was an uneasy silence. Then Flinky called out in a wheedling voice. \"Ah, look now, friends, why don't ye just throw the ould magic sword to us an' we'll be on our way, I promise!\"\n\nThis was followed by a tirade from Badredd. \"Sword or no sword, I vow I'll slay ye all an' take yore Abbey from ye. This is war, d'ye hear me?\"\n\nTwo broken halves of the arrow which had struck Toran were flung into the ditch. The ottercook sat watching Sister Portula bind his wound with her apron. He laughed and shouted back contemptuously to the fox, \"War, eh? Go on then, let's see ye take Redwall from us. A dirty liddle band o' vermin scum, ye'd have no chance!\"\n\nDown in the ditch, Flinky gazed levelly at Badredd and nodded. \"Sure an' I believe the big riverdog's right. How could a crew as small as ours take that fine big place? 'Tis all made o' stone an' locked up tight.\"\n\nBadredd nursed his footpaw, shooting a hateful glance at the stoat. \"Whose side are ye on, theirs o' ours?\"\n\nFlinky spread his paws expressively. \"Ah now, Chief, I'm with you. But ye got to admit, things ain't exac'ly goin' our way, are they now?\"\n\nBadredd narrowed his eyes, well aware that Flinky could be a sly one at times. \"So, what d'ye suggest?\"\n\nThe stoat winked secretively. \"Make 'em think we've gone away. I'll wager we could catch 'em off guard after a day or two.\"\n\nGranmum Gurvel came panting up the wallsteps, carrying a big wooden pail of kitchen rubbish with the arrow that had missed the Abbot sticking out of it. The old mole blinked indignantly. \"Yurr, see wot appinged? Oi wurr just crossin' ee lawn to put ee rubbish on moi compost 'eap. That thurr h'arrer comed roight out'n ee sky an' stucked in moi pail!\"\n\nJunty Cellarhog took it from her. \"Don't fret, marm, it missed ye!\"\n\nBack in the ditch, Badredd was mulling over Flinky's idea. \"How many days do we wait?\"\n\nJunty's voice interrupted further conversation. The Cellarhog was whining piteously. \"Sir, we've got somethin' here for ye.\"\n\nBadredd leaped up. \"Lend a paw 'ere, get me outta this ditch. We won't be waitin' any longer. Hah! They've seen sense at last, that'll be my magic sword!\"\n\nThey boosted him up out of the ditch. He was back a moment later\u2014dripping with leftover oatmeal, potato peelings, onion skins and old cooking oil. Laughter and hoots of derision rang out from the walltops. Badredd was speechless with rage. The crew backed off from him, holding their noses at the odour from yesterday's kitchen rubbish.\n\nHe clawed at the mess. \"I don't care how long I got to wait, they're deadbeasts, all of 'em. They can't treat Badredd like that!\"\n\nHalfchop smiled at him. \"Kachunk!\"\n\nToran sat in the orchard, surrounded by the Dibbuns, telling the tale to them whilst Sister Setiva and Martha tended his wound. The incident, while being humorous, worried Martha.\n\n\"I wish Sarobando and Bragoon were here now.\"\n\nThe ottercook patted his newly bandaged shoulder. \"Don't upset yourself, young 'un. Those vermin'll leave when they find there's nought here for 'em except the ole pail o' rubbish. Ain't that right, Sister?\"\n\nSetiva knotted off the bandage neatly. \"Aye, like as not. Ye say there's but ten o' the rogues altogether. Hmm, they shouldnae be much trouble. Aye, but 'twould be fine if we had some otters or shrews aboot the place.\"\n\nToran stood up and flexed his paw. \"Huh, ye'll not find otters around here, save for me. They've gone off to camp on the seashores all summer. As for shrews\u2014well, they go wherever the streams an' rivers take 'em. I know we ain't got many at Redwall of fightin' age, but we'll do at a pinch.\"\n\nMartha folded the rug across her lap. \"I hope you're right. I'd hate to see vermin get into Redwall. What would happen to these little ones?\"\n\nMuggum picked up a stick. \"Uz foight 'em, miz, we'm gurt fierce Dibbuns. B'ain't that roight, Shilly?\"\n\nThe squirrelbabe, and all the other Dibbuns, set up a fearful clamour. Brandishing sticks, wooden spoons and stones, they paraded up and down, scowling, growling and shouting dire threats.\n\nThough Martha could not help smiling inwardly, she covered her ears and looked shocked. \"Dearie me, I wouldn't like to be a vermin with all these great rough warriors around. Would you, Toran?\"\n\nHer friend nodded. \"Aye, miss, thank the seasons we can sleep safe in our beds. These liddle 'uns are reg'lar terrors!\"\n\nThe smallest of the Dibbuns, the tiny shrew called Buffle, picked up a stone which was far too big for him. He fell over backward and sat there muttering unintelligible sounds.\n\n\"Gurrumvurbilbultumcuchachukchuk!\"\n\nToran removed the stone from Buffle's stomach. He picked the babe up with one paw and set him on Martha's lap. \"Well, I wonder what that's all about?\"\n\nYooch, who seemed to be the only one who could understand Buffle, translated. \"Buffle sez he eat vermins all up!\"\n\nSister Setiva cleaned a few dandelion seeds from the shrewbabe's whiskers. He tried to bite her paw. Setiva raised her eyebrows. \"Och, ye wee terror, don't ye dare tae eat me all up!\"\n\nBuffle clenched his tiny paws and came out with a long torrent of garbled baby talk.\n\nMartha turned to Yooch. \"What's he saying now?\"\n\nYooch giggled. \"Buffle sayin' lotsa naughty fings!\"\n\nSister Setiva looked shocked. \"Time for your nap, young shrew!\" She swept him off protesting loudly. Setiva was a no-nonsense shrewnurse and ignored Buffle's tirade. \"Och, ye can stop all that gobbledygook\u2014ah'm no' impressed!\"\n\n##\n\nBadredd and his crew had left the ditch and crossed back into Mossflower Wood. With all manner of fruit, berries and wild vegetables to be had there during this summer season, the vermin had no difficulty finding food. Crinktail and Juppa gossiped as they prepared food for the others. Neither was very optimistic.\n\nJuppa plucked away at a moorhen, which Rogg had brought down with his bow. \"I tell ye, 'twill be a long time afore we see the Northlands again. Badredd's more determined than ever now.\"\n\nCrinktail chopped away at dandelion roots and wild celery with a thin-bladed dagger. \"Aye, that's true enough. Where is our fearless chief? I ain't seen him round lately.\"\n\nSlipback strolled in and threw down a sizeable bunch of watercress. \"Who, Badredd? That 'un's takin' a bath in the stream, tryin' to get the smell o' that rubbish off 'im. He ain't too pleased, I can tell ye, two baths in two seasons is hard on a beast. He only took a bath last spring.\"\n\nFlinky emerged from the undergrowth, his tunic full of pears. \"Ah sure, any vermin knows that bathin' weakens ye. How's the vittles comin' along, me ould darlin'?\"\n\nCrinktail winked fondly at her mate. \"They'll be ready soon enough, ye great starvin' stoat. Sit by the fire here an' stir the pot awhile. Ye can give us a song while yore at it.\"\n\nFlinky knew more vermin songs than all the crew put together. He sang aloud, hoping the strains might reach Badredd whilst he was taking his bath in the stream not far away. The rest of the crew drifted in to listen, sniggering and nudging a bit at the words.\n\n\"Oh hear my song, young vermin,\n\nand take heed to wot I say,\n\nI had a fine young son like you,\n\nwho bathed most every day.\n\nWhenever he saw water, straight off he'd dive right in,\n\na-scrubbin' an' a-washin' of himself, then he'd begin:\n\nOooooooohhhhhhh! I smell just like a rose,\n\nfrom me tail up to me nose,\n\nwhy, even all the blossoms envy me.\n\nAn' all I'll ever lack,\n\nis a mate to scrub me back,\n\nI'm the cleanest vermin that you'll ever\n\nsee . . . eeeeeeeeeeeeeee!\n\nI'm clean as a weasel's whistle,\n\nshiny as a stoat's best coat.\n\nJust pass the scented essence,\n\nin camomile I'll float.\n\nAll lathery suds an' lilac buds an' pine tree fragrance, too,\n\nwith me teeth so white an' me fur so bright an' eyes of baby blue.\"\n\nThe last verse was sung sadly and with great feeling.\n\n\"But then one summer dawn,\n\nI had to weep an' mourn,\n\nI went down to the bathing pool that day.\n\nThere was not one poor young hair,\n\njust a sweet aroma there.\n\nAlas, he'd gone an' washed himself away.\n\nAwayeeeeeeeeeee!\"\n\nBadredd strode to the fire, dripping wet. Jiggling a claw in one ear, he gave Flinky a frozen stare. \"Get them vittles cooked an' shut yore stupid gob. When we've eaten, we're movin' on, fast!\"\n\nFlinky returned his stare blankly. \"Ah sure, an' wot's the hurry, yore 'onour?\"\n\nThe little fox buckled his cutlass on. \"I want to take a look round the back o' that Abbey, there's got to be a way in!\"\n\nFlinky passed a secret wink to Crinktail, who tried to fob Badredd off with an excuse. \"But, Chief, by the time we've finished the meal and got round there, it'll be dark.\"\n\nBadredd picked up a bowl and held it forth to be filled. \"Good, that'll be the ideal time to get the job done!\"\n\nAbbot Carrul felt much relieved as he surveyed the path and the ditch from the west walltop. \"Thank goodness there's no sign of the vermin. What do you think, Toran, have they gone for good?\"\n\nThe ottercook had lashed sharp kitchen knives to the tops of two window poles. He and Junty each had one. Toran peered up the path into the gathering darkness.\n\n\"Looks like they have, Father, but I'm takin' no chances. Me an' Junty'll stay guard up here an' keep a weather eye out. If the things are still all clear tomorrow, we'll do a patrol around the outer wall just to make certain.\"\n\nCarrul patted his friend's stout back. \"As you wish, I'll have food sent up to you.\"\n\nIt was a fine warm night. Cavern Hole was packed with Redwallers, all happy and relaxed since hearing the news their Abbot brought, that the vermin fear had passed. Granmum Gurvel and her molemaids served a celebratory supper of mushroom and barley soup, harvest-baked loaves and a dessert of apple and blackberry crumble made from fresh ingredients, which the Dibbuns had gathered from the orchard.\n\nForemole sat down next to Sister Portula, digging into his bowl of crumble and smiling happily. \"Gudd arpatoit to ee, marm, ee trubble bee's gonned naow!\"\n\nPortula raised a beaker of October Ale. \"Good appetite to you, sir. Hmm, look at young Martha, she doesn't seem to be enjoying herself. I wonder what's the matter with her.\"\n\nForemole pondered the situation for a moment, then pronounced his judgement. \"Oi 'spec Miz Marth's missin' urr bruther.\"\n\nSister Portula called across to the haremaid. \"Don't fret about Horty, he'll be back soon, eating us out of house and home, no doubt. You'll see!\"\n\nMartha smiled wanly. \"I'm sure he will, Sister, but I can't help feeling concerned about him.\"\n\nAbbot Carrul put aside his supper and stood up. \"What you need is a jolly song. Shall I sing you a little ditty I once learned from a sea otter?\"\n\nThis surprised Martha. \"You singing, Father Abbot?\"\n\nCarrul raised his eyebrows. \"What's so odd in that, may I ask, miss? Gurvel once said I had a voice like a bird!\"\n\nBrother Gelf chuckled. \"Aye, a dying duck. Come on then, Carrul, let's hear ye.\"\n\nThe Abbot took a deep breath. \"Right, here goes. But you must sing this line at the end of each verse. _Heave haul away, twice around the bay. Yaah!\"_\n\nAll the Redwallers wanted to see their Abbot singing, so they agreed readily. Carrul tapped the tabletop until he had the rhythm, then launched into the song. For an old mouse, he had quite a strong, ringing baritone.\n\n\"On the good ship _Leakylea,_\n\nthe captain was a frog,\n\nthe mate was a bumblebee,\n\nand the cook was an old hedgehog.  \nHeave haul away, twice around the bay. Yaah!\n\nI was born at an early age,\n\nand sent straight off to sea,\n\nwith a flea in an iron cage,\n\non the good ship _Leakylea._  \nHeave haul away, twice around the bay. Yaah!\n\nWe sailed the seas so rough,\n\nand never washed the dishes,\n\nate pans o' skilly'n'duff,\n\nand laughed at all the fishes.  \nHeave haul away, twice around the bay. Yaah!\n\nWe ate all we could chew,\n\nmy flea grew bigger'n me,\n\n'cos he'd ate more'n all the crew,\n\naboard the _Leakylea._  \nHeave haul away, twice around the bay. Yaah!\n\nThe the ship sank in a gale,\n\nI was rescued by my flea,\n\nwe're all that's left to tell the tale,\n\nof the poor old _Leakylea._  \nHeave haul away, twice around the bay. Yaaaah!\"\n\nMartha applauded, laughing along with the other Redwallers.\n\nAbbot Carrul bowed modestly and winked at Brother Gelf. \"Not bad for a dying duck, eh?\"\n\nRemembering her responsibility to the Dibbuns, Martha called to them. \"Bedtime, little 'uns, come on now!\"\n\nStrangely, the three who were most likely to protest\u2014Muggum, Shilly and Yooch\u2014went quietly. The other Abbeybabes made their usual loud protest, but to no avail.\n\nSister Setiva wagged a severe paw at them. \"Up tae your beds, this verra instant, or ye'll have me tae reckon with!\"\n\nMartha watched the last one\u2014Buffle the shrewbabe\u2014scamper through the doorway, where he turned and glared at everybeast. \"Kumfuggleworragarrumbubbub . . . Kurch!\"\n\nSetiva picked up a ladle and made as if to chase him. \"Ah cannae tell what you're sayin', ye wee rogue. But, like as no', 'tis somethin' verra naughty! Ye'd best get toddlin' afore I catch up wi' ye!\"\n\nBuffle stood his ground long enough to twiddle a paw to his nose at the shrewnurse, then he bolted off, giggling.\n\nMartha tried hard not to laugh. \"Perhaps we'd better go up and tuck them in, Sister?\"\n\nSetiva waved a dismissive paw. \"Och no, we can do that later. Ah've got tae go an' take supper tae Toran an' Junty first.\"\n\nThe haremaid pushed her chair away from the table. \"I'll come and help you. Poor old Toran, I'd forgotten about him. Never mind, there's plenty of crumble left.\"\n\nBadredd halted his crew at the east wickergate. There was a small door set in the centre of the Abbey's rear wall. He held up a paw for silence. Gently pressing his weight against the timbers, the small fox tried the circular iron ring handle. It was firmly locked shut.\n\nPlumnose held up a little lantern close to the door. \"Huh, id's shudd, Chief!\"\n\nBadredd had difficulty controlling his voice. \"Is it now! Thanks for lettin' me know, bouldernose!\"\n\nPlumnose grinned. \"T'ink nodding ob it.\" He turned to Halfchop. \"Duh likkel door's locked, I t'ink.\"\n\nThe rat wiped a ribbon of drool from his chin. \"Kachunk!\"\n\nBadredd rounded on the pair, hissing viciously. \"Shuttup, you two, an' get back into the trees\u2014go on! Flinky, are ye any good at openin' locks?\"\n\nThe stoat scratched his grimy cheek. \"Ah, well, there's locks an' locks, if ye get my meanin', yer 'onour!\"\n\nBadredd whipped out his cutlass and thrust it under Flinky's nose. \"I never asked ye for a lecture about locks! I said, are you any good at openin' 'em\u2014well, are ye?\"\n\nFlinky heaved a sigh and took the cutlass from his chief's paw. \"Sure an' I don't know until I try. Shall I give it an ould go?\"\n\nBadredd waved him to the door impatiently. \"Well, put a move on, we haven't got all night!\"\n\nFlinky wedged the swordblade between the door jamb and the wall. He slid the blade down until it clinked dully against something.\n\n\"Hah, there's yore problem, Chief, 'tis a bolt. D'ye want me to try an' chop through it?\"\n\nThe fox exhaled irately. \"Anythin', just get on with it!\"\n\nFlinky requested the aid of Floggo and Rogg. \"Come over t'this door, buckoes. Now put yore shoulders to it. Push now. That'll widen the gap so I can get a grand swing at the bolt. Push, put those ould bows down an' push!\"\n\nThe door moved slightly under the pressure, creating a thin space. Flinky took the cutlass in both paws, raising it within the gap. Then he struck, whipping the blade down with all his might.\n\n _Piiing!_ As it struck the iron bolt, the blade snapped in half.\n\nBadredd stared in silent horror at the stoat, who\u2014still holding the handle and half a blade\u2014was hopskipping in agony, both paws numbed by the reverberation of metal upon metal.\n\nThe vermin leader's voice rose to a disbelieving squeak. \"Me sword! Me luvly cutlass! Ye've ruined it! Idiot!\"\n\nTears squeezed from the corners of Flinky's eyes as he flung the half cutlass on the ground. \"Aarh, it broke its stupid self. Yore s'posed t'be the chief, why didn't you have a go?\"\n\nBadredd seized the broken weapon. \"Have a go? I'll have a go at you if ye ain't careful, idiot! An' you lot, a fine crew I've got, sittin' round scratchin' yerselves among the trees. Up on yer paws, doltheads, we'll have to find someplace else where we can get in. Jump to it!\"\n\nAs Badredd strode off in foul mood, Plumnose called to him. \"Chief, me an' Halfchob hab got de door oben!\"\n\nBadredd dashed back to where Plumnose and Halfchop stood in the small doorway. Finding the door still closed, he fumed at them. \"Ye blither-brained, wobble-nosed, broken-snouted loafheads! Get goin', afore I carve cobs off'n ye with what's left o' me sword!\"\n\nBut then, as Plumnose pushed the wicker door gently, it swung inward. \"Duh, hawhawhaw, oben!\"\n\nHalfchop walked through the open door and grinned. \"Kachunk!\"\n\nFlinky inspected the wall alongside the door. \"Well now, ain't I the clever beast! I must've hit the bolt so hard that it broke through the ould soft sandstone it bolts into. See, there's a chunk of it missin'. Oh, here's the rest of yer grand cutlass, Chief.\"\n\nHe presented the fox with the other half of the blade. Flinging it from him, Badredd turned on the crew and hissed, \"You lot, keep yore mouths shut, not a sound out of ye. Foller me, don't go cloghoppin' all over the place. We're goin' to take a look around. Next move is t'get inside the big buildin'. Quietly now . . .\"\n\nAfter taking food out to the west walltop for Toran and Junty, Martha and Sister Setiva returned to the Abbey. Martha stayed in her chair below stairs whilst Setiva went up to the dormitory to check up on the Dibbuns. The shrewnurse was away only for a brief space of time when a dismayed cry reached Martha. Setiva came hurrying back downstairs carrying little Buffle, who was imprisoned in a pillowcase with only his head sticking out.\n\nThe Sister's voice shook with barely controlled anger. \"Och, jist let me get mah paws on those rascals. Ah'll give 'em somethin' tae remember me by!\"\n\nBuffle strained against the pillowcase knotted at his neck. \"Goourr, 'ascals!\"\n\nA look of fear crossed the haremaid's face. \"What's happened, Sister?\"\n\nSetiva began trying to release Buffle. \"Ooh! Those Dibbuns, Muggum, Shilly an' Yooch. They've gone missing. All the rest o' the wee ones were fast asleep, except Buffle. D'ye see what they did? Trapped 'im in this auld pillowslip so he couldnae follow 'em. Where in the name of all fur have they got to?\"\n\nBuffle pulled a paw free and pointed out the Abbey door.\n\nJunty Cellarhog ran his paw around the inside of his bowl and licked it. \"Ah, apple'n'blackberry crumble, mate, nothin' like it!\"\n\nToran gazed longingly back toward the Abbey. \"Aye, pity we're on wallguard all night. If the Abbot sends out a relief, there might be some left when we get off duty.\" Toran's keen eye suddenly noticed three small, white-clad figures trundling across the lawn in his direction. Two were waving sticks and one swinging a ladle. He peered hard.\n\n\"Look there, mate, that ain't no relief!\"\n\nIt was at that moment when things began happening fast.\n\nFramed in a shaft of golden light from the Abbey door, Martha and Sister Setiva were pointing to the Dibbuns and calling aloud to them. \"Come back here this instant, or you're in real trouble!\"\n\nThe trio split, Muggum running south and the other two hurrying off to the north.\n\nToran saw them and chuckled. \"Escapin' Dibbuns, eh? They won't get far . . .\"\n\nJunty interrupted him roughly. \"Look, vermin!\"\n\nBadredd and his crew were sneaking quickly out across the lawn, trying to grab Muggum, who was heading for the pond where he planned on hiding in the reeds. The little mole was completely unaware of the enemy. Sister Setiva had come out onto the Abbey steps. As soon as she saw the vermin crew, she began dashing to save Muggum.\n\nJunty was already hurtling down the gatehouse wallsteps, calling back to Toran, \"Get the other two little 'uns inside!\" He shouted at the shrewnurse. \"Stay where ye are, Sister. I'll bring that Dibbun in!\"\n\nWith his paws, Toran swept up the giggling Shilly and Yooch\u2014this was all one big game to them\u2014then the ottercook turned and pounded toward the Abbey door.\n\nSlipback came within a paw's length of grabbing Muggum, when Junty fetched him a massive whack to the chest, laying the weasel out flat. Then the big Cellarhog seized the molebabe and ran as fast as his footpaws would carry him, with Badredd and the crew hard on his heels. Without stopping, Junty snatched up Sister Setiva from where she had been standing in his path, rigid with fright.\n\n _Thud! Thud!_\n\nTwo arrows from the bows of the ferrets buried themselves in the Cellarhog's broad back. He staggered slightly but kept running. Muggum was screeching, the hedgehog's sharp spines were sticking in his paws as the molebabe tried to struggle free.\n\nToran sped into the Abbey, dropped both of the other Dibbuns into Martha's lap. \"Get ready to slam the door shut!\" He panted as he turned and ran back outside to help Junty.\n\nOne arrow grazed Toran's cheek, another hit Junty in his right shoulder. Toran shot past the Cellarhog, whirled hard, and caught Crinktail across the face with a huge smack of his rudder. He turned and pushed Junty, with both his burdens, up the steps and into the Abbey, roaring, \"Bar the door!\"\n\nRedwallers, who had come pouring out of Cavern Hole to see what all the commotion was about, assisted the haremaid in slamming and barring the door in the face of the charging vermin crew. Two more arrows made a hollow sound as they flew into the strong oak timbering. A crash and a tinkle sent Foremole and Brother Weld hurrying to the lower windows.\n\nToran urged others along with him. \"Get tables an' benches! Barricade the lower frames before they get in!\"\n\nBadredd waved his broken cutlass. \"Keep at it there, crew, we've got 'em on the run!\"\n\nFlinky watched a dining table blocking a broken window. He muttered out the side of his mouth to Juppa. \"Keep slingin' rocks, but let 'em barricade those windows. They'd eat our liddle gang if'n we got inside. We'd be well outnumbered, mate.\"\n\nJuppa looked puzzled. \"Well, if'n we ain't goin' in, wot's the next move?\"\n\nFlinky had served under lots of different vermin chiefs, all a lot smarter than Badredd. He winked confidently at the weasel.\n\n\"Lissen t'me. If'n we ain't goin' in, well they ain't gettin' out. Did ye see that great orchard we passed as we came through?\"\n\nBadredd came marching around, prodding Flinky with his broken blade. \"Wot's that sling doin' empty? Keep chuckin' rocks at those windows until I tell ye to stop. Both of ye!\"\n\nFlinky loaded a large pebble into his sling. \"Ah, we'll be doin' that, yer 'onour, right away. I was just tellin' ould Juppa here what a clever move ye made.\"\n\nBadredd was eager to know just what the clever move was. \"Aye, well that's alright. You explain it to 'er, she was never too bright. Go on, tell the long-tailed oaf.\" The small fox stood listening to Flinky's explanation.\n\n\"Hoho, we've got the sillybeasts locked up tight now. Prisoners in their own Abbey, 'tis called a siege. There's only a limited supply o' food an' drink in there. Take us now, the chief knows we got the orchard an' the pond. They'll either starve t'death in the Abbey or surrender after awhile. Ain't that right, Chief?\"\n\nOnly a moment before, Badredd thought he had lost the encounter, but the realisation of what Flinky had just said made him shudder with delight. So that was what a siege was all about.\n\nKeeping a straight face, the fox nodded wisely. \"Aye, 'tis a siege, sure enough. Now you two keep slingin'.\" He swaggered off, shouting orders to the other vermin. Juppa watched him go. \"A siege, eh? What a clever idea!\"\n\nFlinky launched another stone but missed. He jumped neatly aside as it bounced back at him. \"Ah sure, the ould chief is full o' clever ideas, especially when some otherbeast thinks 'em up for 'im. Little fool, he couldn't find his bottom wid both paws!\" The weasel and the stoat loaded their slings again, laughing hilariously.\n\nMartha had pulled herself from her chair. She sat on the floor, both eyes shut tight, clutching Junty's paw to her cheek as she rocked back and forth. The Cellarhog was lying where he had fallen, face up. Muggum was wailing as Sister Portula pulled spikes from his side and paws.\n\nSister Setiva was similarly engaged. \"Och, ye've got some fine sharp quills on ye, mah guid Cellarhog. Ah'll be with ye soon as I've got them out o' me. Hauld him still, Martha, how is he?\"\n\nWith her eyes still shut, Martha kissed his limp paw. \"He's dead, Sister. Junty is dead!\"\n\n##\n\nA squabbling flock of starlings, disputing rights to an ants' nest, woke Jibsnout in the hour following daybreak. With a cavernous yawn, the big Searat heaved himself upright. He cast a jaundiced eye over the three sons of Wirga who were curled up together, sleeping beneath a wych hazel.\n\nJibsnout cuffed the trio roughly, stirring them into wakefulness. \"Up on yer hunkers, whelps, we're on the move again!\"\n\nThe three smaller rats rose reluctantly, one of them glaring balefully at the Tracker and hissing. \"We only lay down an hour afore dawn.\"\n\nJibsnout smirked. \"Aye, 'tis a shame, ain't it? Move yerself, snotty snout, an' don't argue wid me. If'n I say ye march, then ye march, so button yer lip!\"\n\nQuivering with anger, the smaller rat picked up his little spear\u2014each of his brothers carried one, too. Jibsnout had seen them use the deadly weapons, but not as spears. Although they were actually hollow rods, the spearpoints could be removed, transforming them into blowpipes through which poisoned darts could be shot with lethal accuracy. The big Searat stroked his long dagger fondly and moved closer to the sons of Wirga. He fixed the angry one with a cold stare.\n\n\"Go on, mamma's liddle rat, use it, I dare ye. Think yore brave enough t'slay me, eh?\"\n\nLashing out swiftly, Jibsnout knocked the spear from the smaller rat's paws. Whipping out his blade, he menaced the other two. \"Just try raisin' one o' those things against me, an' poison or not, I'll rip yer throats out! Well, come on, ye gutless wonders, who's ready fer a fight t'the death?\"\n\nThe sons of Wirga stood silent, their eyes cast down. Jibsnout curled his lip scornfully, turning his back on them. \"Hah, I thought so! There's more backbone in an egg than in youse three put t'gether. Scringin' cowards!\"\n\nEach of the three blowpipes was already charged with a poison dart. Silently slipping the head from his spear, the rat whom Jibsnout had insulted placed the hollow rod to his mouth. His cheeks bulged as he prepared to propel the dart.\n\n _Zzzzzzip!_\n\nA long arrow struck the little rat, driving him back a full four paces. He was dead before he hit the ground.\n\nDiving to either side, the remaining two sons of Wirga sought cover. Lonna emerged from out of the trees, fitting another shaft to his bowstring. The badger's eyes were red with the light of vengeance, the snarl on his scarred, stitched face transforming him into a terrifying apparition. Frightened though he was, Jibsnout, a seasoned fighter, acted swiftly. Wielding his dagger, he dashed forward, hoping to get so close to his adversary that the bow and arrow would be rendered useless.\n\nLonna was in a dilemma: he could see one of the Searats glancing around a treetrunk, ready to fire a blowpipe, and Jibsnout thundering toward him. With lightning speed the badger acted. Falling into a crouch, he fired his arrow, but only narrowly missed being shot himself as a poison dart whipped by overhead. Jibsnout roared in pain as the arrow transfixed his paw to the ground. As Lonna rose, taking another shaft from his quiver, the Searat who had fired the dart fled off into the woodlands.\n\nThe remaining son of Wirga came from behind a fir tree, certain that he could not fail to hit a target as big as the badger. As he placed the blowpipe to his mouth, Figalok the squirrel appeared directly in front of him, hanging by her tail from an overhead branch. She grabbed the opposite end of the vermin's blowpipe and blew hard. Clutching his throat, the horrified rat fell writhing to the ground, choked on his own poison dart.\n\nFigalok dropped out of the tree, nodding to Lonna. \"Chahaah, gotta be plenny quick wirra Searatta!\"\n\nThe big badger put up his bow, striving to master the Bloodwrath that was coursing through him. \"You saved my life, friend, but I'll have to thank you some other time. One of the Searats got away. I must hunt him down now while his trail is still fresh.\"\n\nThe squirrel gestured at the wounded Jibsnout. \"Warra 'bout dissa one, ya goin' to slay 'im?\"\n\nJibsnout crouched over, his face creased in agony. The arrow that had pierced his footpaw was buried half its length into the ground. He glanced up at Lonna, expecting no mercy from him.\n\n\"If'n yore gonna finish me off, make it quick, stripedog!\"\n\nThe badger strode over and grasped the arrow. With a sharp tug he pulled the arrow out, growling at Jibsnout. \"I'm no Searat, I don't kill defenceless beasts!\" Ripping the sleeve from the rat's frayed tunic, Lonna grabbed a pawful of damp moss and dockleaves.\n\nThe puzzled rat watched his enemy binding the wound up tight. \"Ye mean yore lettin' me live?\"\n\nThe badger hauled him upright, slamming him against a tree. \"My name is Lonna Bowstripe. Take this message to Raga Bol. Tell him that he and all his crew of murderers are walking deadbeasts. I will find them and slay them, one by one. Even you. Now begone from my sight and deliver my message to your captain. Tell him I am coming, nothing will stop me!\"\n\nLonna and Figalok watched Jibsnout limping painfully off until he was obscured by the trees, then together, the two friends took a brief meal. The squirrel wielded a blowpipe spear and poison darts taken from the slain Searats.\n\n\"Chahaah! Me betcha dis keep Ravin away from squirrel. Lonna Bigbeast, ya goin' after dat Searatta who runned away? Me go witcha, we find 'im afore tomorra.\"\n\nBut the badger would not hear of it. \"No, my friend, you have your own home and kinbeasts to protect. This is something I must do by myself. I am sworn by my own oath to rid the earth of Raga Bol and all his vermin. But I thank you for saving my life, Figalok!\"\n\nThe elderly squirrel took his paw. \"Chahaaw, so be't, Lonna, ya are d'true warrior. Ya saved us fromma Ravin, glad Figalok could save ya, too. Me no ferget ya alla me life, always think of ya!\"\n\nAverting his eyes, Lonna inspected the long dagger he had taken from Jibsnout, pleased that it was a good blade. When he looked up again, Figalok had gone, vanished into the treetops.\n\nThe Searat's trail had gone off to the southeast. Lonna picked it up and followed the tracks. As he walked, the badger fashioned a holder for his dagger, fitting it to his upper left arm close to the shoulder. By late afternoon, the dense woodlands thinned out into pine groves and sandhills. In the distance, Lonna could make out a dark shape to his left on the horizon. The trail of Wirga's remaining son was running parallel to the mysterious mass. Just before sunset, the badger crested a rise which afforded a clear view of the country he was travelling through. On the one side, the hills bordered a vast, dusty plain, almost like a desert wasteland. On the other side, the odd dark mass reared up into a towering line of forbidding cliffs. After awhile it grew too dark for tracking. Reaching the cliff face, Lonna sighted what he knew was a cave. He climbed up and made camp there for the night.\n\nThere was no need for a fire. The night was still and warm, with heat waves drifting in from the plain. Knowing he could pick up the Searat's tracks at dawn, Lonna sat in the cave entrance, eating an apple and some dried fruit. He gazed up at the night sky, where a sliver of moon, resembling a slice of russet apple, was surrounded by myriads of stars twinkling in the firmament. The words of an old song rose unbidden to his mind.\n\n\"When weary day does shed its light,\n\nI rest my head and dream,\n\nI ride the great dark bird of night,\n\nso tranquil and serene.\n\nThen I can touch the moon afar,\n\nwhich smiles up in the sky,\n\nand steal a twinkle from each star,\n\nas we go winging by.\n\nWe'll fly the night to dawning light,\n\nand wait 'til dark has ceased,\n\nto marvel at the wondrous sight,\n\nof sunrise in the east.\n\nSo slumber on, my little one,\n\nfloat soft as thistledown,\n\nand wake to see when night is done,\n\nfair morning's golden gown.\"\n\nSince Lonna had no recollection of his parents, he surmised that the lullaby had been taught to him by Grawn, the old badger who had reared him.\n\nLonna stayed that night in the cave on the cliffside. As day dawned he spotted a tiny puff of dust, on a hilltop off to his right. The big badger knew instantly that it was his quarry. The Searat must have spent the night amid the hills, not far from the cave. Pausing only to grab his bow and quiver, Lonna set off in pursuit.\n\nHe had travelled no further than the base of the first foothill when he was faced by a small patrol of ten Darrat rats. Their leader eyed him insolently up and down.\n\n\"Dis be Darrat land. You give me bow'n'arrers, stripedog. We take ye to Hemper Figlugg!\" He grinned at the other rats, murmuring to them, \"Much Burcha Glugg, eh?\"\n\nHad it been ten rats or twenty, Lonna did not like either their manner or their disposition, so he charged them without warning. They went down like ninepins under the giant badger's onslaught. Seizing the leader of the patrol, Lonna hurled him bodily into the other rats. Then the big badger was among them like a whirlwind\u2014punching, kicking, butting, thrashing them with their own spears. So surprised were the Darrat that they fled in panic, kicking up sand widespread as they scuttled off amid the hills.\n\nLonna picked up his bow and quiver. Then, throwing back his great striped head, he gave vent to the fearsome warcry of hares and badgers. \"Eulaliiiiiaaaaaa!\"\n\nHowever, with much more urgent business to attend to, he let the Darrat be, and didn't give chase. Instead, Lonna set off swiftly on the trail of the Searat.\n\nWhen the Darrat saw they were not being pursued, they halted on the plain beside the foothills. The patrol leader limped up, carrying half a broken spear. He watched the big badger crossing a hilltop, some distance off.\n\nTurning to his subordinates, who were sitting licking their wounds, he snarled, \"We was sent to catcher rabbert, mouse an' squirri', not stripedog! Huh, let High Kappin catcher that 'un\u2014'e be over dat way wid many Darrat!\"\n\nThe Searat saw Lonna coming after him. Deserting the hills, he dashed out onto the dusty plain. It was a mistake, the last mistake he was ever to make. The badger's arrow found him. Once Lonna had the range, nobeast could outrun a shaft from his big bow. Though Wirga did not know, she had lost all three of her sons.\n\nLonna sat down in a hollow amid the hills and made breakfast from the food in his pack.\n\nOut on the flatlands the five travellers pushed forward, keeping the distant cliffs in view. They marched shoulder to shoulder because, as Saro had pointed out, that way they would not be eating one another's dust. Since their rescue, Springald and Fenna were paying more attention to Bragoon and Saro. Seasoned campaigners both, the squirrel and the otter were ever ready to share their knowledge with the younger, less experienced trio.\n\nHorty was feeling rather chipper now that any immediate danger was past. He struck up a jolly marching song, to which he himself had written the lyrics. As was usual with hare songs, it dealt mainly with food.\n\n\"Oh wallop me left an' stagger me right,\n\nan' buffet me north an' south,\n\nif I could teach a stew to walk,\n\nit'd march right into me mouth!\n\nTo pasties an' pies of convenient size,\n\nI'd beat a tattoo on me drum,\n\nso jolly forceful, each tasty morsel,\n\ntramp over me gums to me tum!\n\nAs each of 'em trips in through me lips,\n\nall skippin' along to the beat,\n\nwhy all of a sudden I'd grab a fat pudden,\n\nan' leave it no way to retreat!\n\nForm up in line, you vittles so fine,\n\nwatch y'dressin' that salad back there,\n\na quick salute to trifle'n'fruit,\n\nthen charge down the throat of the hare!\n\nQuick march! One two! Scoff 'em all! You an' you!\n\nLeft right! Left right! Here comes supper for tonight!\"\n\nA grey, black-flecked Darrat scout came loping into the camp in the foothills of the high cliffs. He threw himself flat in front of High Kappin Birug, the Darrat leader. Pointing back to the scrubland, the rat scout shouted, \"Burcha Glugg!\"\n\nBirug dashed past him to the top of a hill. He crouched, peering at the small dust cloud with the travellers marching in front of it, not half a mile away. Smirking with satisfaction, Birug turned to the others who had followed him.\n\n\"Hemper Figlugg, trus' me, ho yar, I know dey only go one way. Run for bigrocks. We wait, they be come to us. Burcha Glugg!\"\n\nDarrat vermin shook their heads in admiration of Kappin Birug's cunning. One of them piped up. \"Hemper be 'appy to see Burcha Glugg come back.\" The more excited of the Darrat leaped up and down, waving spears.\n\nBirug growled a warning at them. \"Keepa 'eads down, idjits!\"\n\nHorty glanced up at the sky. \"Cloudin' over up there, chaps. We might have a spot of jolly old rain before nightfall, wot?\"\n\nBragoon sniffed the light breeze. \"Bit more'n a spot, matey. Looks like we're in for a downpour afore dark. Keep movin', step the pace up. Mebbe we'll find shelter in the lee of those big cliffs.\"\n\nFenna let out a gasp and sat down. \"Ouch, my footpaw!\"\n\nThey gathered around her, crouching down to take a look. The squirrelmaid spoke through lips that hardly moved. \"Stay down, all of you, don't look toward those foothills!\"\n\nBragoon kept his eyes on Fenna. \"Why, what's goin' on?\"\n\nShe quickly responded. \"Rats ahead, they look like those flesh-eating ones!\"\n\nSpringald automatically began to look up, but Sarobando pressed her head back down. \"Listen to Fenna an' keep yore eyes down, miss. How many d'ye reckon there are?\"\n\nBragoon interrupted. \"Plenty, I'll wager. Too many for us to fight off. I told ye, those vermin don't give up easily. They've been waitin' in the foothills for us to show up. Well, mates, wot's t'be done, eh?\"\n\nFenna shrugged. \"I suppose we'll have to run for it.\"\n\nBragoon shook his head. \"Bad idea! They'd outcircle us.\"\n\nHorty began shrugging off his backpack. \"Does any chap mind me makin' a suggestion, wot?\"\n\nSaro saw that the young hare looked serious. \"As long as 'tis sensible. Go on then, wot's yore idea?\"\n\nHorty shed his backpack. \"Give me some old, dead brush, an' I'll decoy the rotters. A hare can jolly well outrun 'em if anybeast can. I'll take the villains off one way, while you lot go runnin' off the bally opposite way. See that black hole up there, about halfway along the cliffs? I'll meet y'back there after dark. Well, what d'you think?\"\n\nSpringald objected. \"It's far too dangerous. You'll be caught.\"\n\nSaro stared at Horty. \"I say give it a try, it might work. Otherwise, we'll just stick together and get nabbed.\"\n\nBragoon winked at the hare. \"Right, go to it, young 'un. Good luck!\"\n\nTwo Darrat spies peeped over the hilltop, to where the dust cloud had stopped. One whispered. \"Warra dey do now, jus' lay dere?\"\n\nThe other one leaped up as the dust plume started again, moving swiftly north. \"Musta see'd us, dey runnin' now, fast!\"\n\nHe waved his spear, calling to Birug, who had the rats standing ready, \"Kappin, dey go lef' plenty fasta!\"\n\nHorty pelted along with a bunch of dead bracken tied to his tail, raising a dust cloud that stood out light brown against the lowering clouds. Glancing sideways, he saw the Darrat rats pouring over the hill, veering in his direction. He muttered between clenched teeth.\n\n\"Ahah! That's the way, you vile vermin. Come on, you shower, follow Hortwill Braebuck, skimmer of the scrublands!\"\n\nFenna raised her head. In the distance she could see the dust cloud off to her left. \"Good old Horty, he's whipping along like a whirlwind!\"\n\nStill crouching low, they watched their friend's progress, comparing it to the crowd of Darrat vermin chasing him. Horty was indeed a Redwaller, brave and courageous. Springald felt elation and pride surging through her. She clenched her paws.\n\n\"Go on, mate, there's none faster than you! Flesh eaters, hah! All those scum will eat is the dust in his wake! Run them, Horty, show those rats what a hare from our Abbey can do!\"\n\nAs soon as Bragoon saw the two dust plumes, he realised that the Darrat had come out of the hills and hit the scrubland. Their intended quarry was far and away out in front. The otter's eyes shone with admiration.\n\n\"I said that young 'un has the makins of a real warrior. He'll lead 'em a merry dance alright. Oh, drat, here comes the rain!\"\n\n##\n\nLarge drops began falling, slow at first, sending up small puffs of dust as they struck the dry plain. A distant thunder rumble echoed from the high cliffs, followed by a faroff flash of lightning that illuminated the southeast horizon. Then the deluge fell in earnest. Saro stood upright, blowing water from her nosetip as she blinked at the sheeting curtains of heavy rain.\n\n\"Nobeast can see us now. Let's head straight for the cliffs!\"\n\nJoining paws, they jogtrotted toward the foothills, battered by the relentless downpour. Lightning ripped over the dark skies in blinding sheets, while thunder boomed and banged overhead. Dust turned quickly to mud, their paws squelched into it. Springald tightly gripped the paws of Fenna and Saro. The intensity of the storm was frightening, she had never been out in open country at such a time before. At Redwall, it had been relatively easy to run inside and shelter from the elements, but out here it was different.\n\nThey gained the foothills, slipping and sliding up the wet grass. Bragoon shielded his eyes as he glanced upward.\n\n\"Keep goin', it ain't too far now. Yonder black hole that Horty spotted looks like it could be a cave of some sort. Let's make it up that far an' shelter.\"\n\nHorty's wet paws slapped down in the sludge and mud. Wiping water from his eyes, he chanced a backward glimpse at his pursuers. Although the main body were still a respectable distance off, three fast runners had broken away and were coming doggedly onward, closing the distance considerably. The young hare bit his lip. The trio were armed with spears; if they got within throwing range, he would be finished. It was time for a change of plan. Still with stamina in reserve, Horty shot off to the right, back among the foothills, where he stood a chance of losing the Darrat mob.\n\nBirug panted, squinching his eyes against the rain as he saw the hare change course and dart into the dunes. The High Kappin urged his rats on. \"Catchim, or Hemper Figlugg make Burcha Glugg outta you!\"\n\nTopping a rise, Horty spotted the barely discernible hole in the cliffside, far along to his right. He tripped and went rolling downhill. Spitting grit and coated with sand, he swiftly picked himself up and pounded on to the next dune, muttering to himself, \"Ears up, old lad, keep pickin' 'em up an' puttin' 'em down, wot. Huh, if only the young skin'n'blister could see her handsome brother now\u2014a blinkin', gallopin' sandbeast!\"\n\nA spear buried itself in the sand, not far behind him.\n\nBirug appeared at the top of the hill that Horty had just come over. Two others trailed behind him. He seized the spear from one of them and flung it. The Darrat leader's aim was bad\u2014he watched the spear strike the hillside flat and slide back down. Birug rested a moment on all fours, fatigued.\n\nHorty gained the next hilltop and turned. Holding a paw to his nose, he wiggled it and called out cheekily, \"Bloomin' old flesh scoffer, go an' boil your own head an' eat it, wot wot!\"\n\nStung by the hare's jibe, Birug hauled himself upright and came after the hare with renewed energy. Horty scuttled off, chiding himself for his momentary foolishness.\n\n\"Have to keep the old lip buttoned, wot! Seems a jolly determined type o' cove for a rat, full of the old vermin vinegar. Curse his caddish hide!\"\n\nAfternoon passed, without the rain slackening its intensity. It was humid, without a trace of breeze. Rivulets gathered into swollen streams, racing down the cliffside in floods of umber-hued water.\n\nBragoon was first to reach the black hole. His prediction had been correct: it was a cave\u2014large, dark and deep. He helped Springald and Fenna enter first, while Sarobando brought up the rear. Once inside, all four flopped down, exhausted. The otter shook himself like a dog and shrugged off the packs he had been burdened with.\n\n\"Whoo! Wretched weather, wonder when this rain's due to stop?\" He sat up against the right wall, peering out. \"Come on Horty, mate, where've ye got to?\"\n\nFenna joined him. \"I hope he's alright!\"\n\nSpringald rose and began to wander off to explore the big cave, but Bragoon pulled her back.\n\n\"You stay close up here, miz. We don't know wot might be back there. Can't risk a fire, either\u2014too dangerous. Break out some vittles, if'n they're still dry enough, and a drink, too. Funny how ye can be out in the rain all day an' still be thirsty.\"\n\nThe mousemaid found dry oatcakes and some crystallised fruit, which they washed down with some home-brewed cider. Fenna stared out into the persistent downpour, then jumped slightly as thunder boomed out overhead.\n\nSaro patted her shoulder. \"Nought t'do but sit an' wait, matey. Don't fret now, that young rogue'll make it.\"\n\nThe squirrelmaid forced a smile. \"If he's not here soon, I'll light a fire and make a pot of soup. Horty can smell vittles a league away. He'll show up then, I wager.\"\n\nShe sat miserably, pondering the foolishness of her statement. Horty could be lying slain out there in the rain.\n\nHorty staggered gamely on, the three rats not more than six paces behind him. They had picked up their spears again and thrown them at him several times. With the courage of desperation, the young hare, having managed to avoid the throws, remained unscathed. Birug and his two rats had left the spears where they fell, and carried on, stubbornly pursuing the fugitive. It was only a matter of time now, and they would have him. As the High Kappin blundered forward, Horty moved out of his reach.\n\nWith his tongue lolling, the rat gasped out, \"We . . . catcha!\"\n\nHorty stumbled, tripped and wriggled out of his reach. Gaining his footpaws, he stood panting. \"Couldn't . . . catch your old . . . grannie . . . Slobberchops!\" He blundered on another pace or two, then collapsed.\n\nBirug nodded to the other two rats. \"Gerrim . . . now!\"\n\nAll three crawled forward on their bellies, reaching out to lay paws on the fallen hare when, without warning, the hillside gave way, sliding down a tremendous avalanche of wet sand. It enveloped the three rats completely, burying them under a huge mound.\n\nHorty lay at the edge of the mass, covered right up to his neck. He was trapped fast. A paw, almost the size of his own head, seized both of the hare's long ears and yanked him out with one mighty pull. Horty revived with the pain, his eyes flickering open. He stayed conscious just long enough to see a lightning flash illuminate the head of a giant badger with a scar running lengthwise down its striped muzzle.\n\nThe young hare blinked. \"Nice weather, wot . . . Oh, corks!\" Then he passed out.\n\nOnly the Dibbuns slept upstairs in their dormitories that night, while every other Redwaller guarded the barricades. It was the longest, saddest night Martha had ever witnessed. The still form of Junty had been wrapped tenderly in blankets and borne down to the place he loved best, his cellars. Clearing the barrels and lifting some floorstones, Foremole Dwurl and his crew dug a grave for the good Cellarhog. Junty was laid to rest. Once the grave was filled in and the flooring stones replaced, Abbot Carrul took a charcoal and wrote words upon it. At some later day the moles would chisel the words into the stones as a permanent epitaph for a beast whom all Redwallers loved dearly. Tears often smudged the charcoal letters as Carrul wrote:\n\n\"Here lies a fallen warrior, slain by vermin whilst helping his fellow creatures. Hard working, good and faithful. A credit to his kind. Always a kind word or smile to all. Junty Cellarhog, Keeper of Redwall Abbey cellars. His October Ale was the best. Rest peacefully, old friend.\"\n\nAbove stairs, Martha rolled her cart around Great Hall, relieving those who were wearied. When she was not doing that, the tireless haremaid helped Granmum Gurvel to ferry food from the kitchens.\n\nToran watched Martha\u2014she was never still, always finding something to do for the common good. He halted the little cart with his rudder. \"Come on, beauty, time ye took a nap or ye'll be worn out.\"\n\nMartha protested. \"I'm fine, honestly I am!\"\n\nBut, deaf to her pleas, the ottercook opened the lap rug and tucked it beneath the haremaid's chin. \"No arguments now. I'll wake ye if'n yore needed, miss. You stay out the way here, in this quiet corner away from broken glass an' slingstones. I'll have t'go an' get more stuff to barricade those windows.\"\n\nHe hurried off to assist Brother Weld, who was struggling with a door he had taken from its hinges. \"Here, Brother, you take one end an' I'll take the other.\"\n\nWeld sighed thankfully. \"We're getting a bit old for this sort of thing. D'you think we'll hold them off, Toran?\"\n\nGritting his teeth, the big otter growled. \"Filthy scum, if they get in this Abbey, 'twill be over my dead body. Don't worry, Brother, we'll keep 'em out!\"\n\nOld Phredd helped them to shore the door up against the windows. \"Huh, the way I see it, we're under siege. 'Tis those vermin who are keeping us in!\"\n\nToran clenched his paws tightly. \"That's right. Strange ain't it, bein' kept prisoner inside yore own home.\"\n\nOld Phredd added miserably, \"Aye, what do we do if the food runs out?\"\n\nToran's clenched paw wagged under the ancient Gatekeeper's nose. \"Quit that kind o' talk now, d'ye hear me? There's vittles aplenty for all, so don't go scarin' everybeast!\"\n\nBadredd watched the dawn wash the skies in rosy hues. The small fox was in his element. \"Flinky, Crinktail, c'mere. I got a plan o' me own at last!\"\n\nBoth stoats, stuffing themselves on orchard produce, continued eating as Badredd explained his scheme.\n\n\"Load up a couple o' sacks an' take a stroll through the woods south of here. Eat what y'like as ye go.\"\n\nFlinky tossed away a half-eaten pear. \"Sounds like a good ould job, Chief, but what're we supposed t'be doin'?\"\n\nThe little fox grinned craftily. \"Recruitin' more vermin. We need more beasts to take this place. Tell 'em that Redwall is bein' conquered by Badredd an' a vermin crew. Aye, an' tell 'em there'll be plenty o' vittles an' booty for anybeast who'll serve under me. Have ye got that?\"\n\nFlinky saluted elaborately. \"Leave it to us, Chief. We'll bring ye back a gang o' the best, so we will. No old or feeble ones, just grand fightin' vermin. But wot about all this ripe ould fruit?\"\n\nBadredd snorted impatiently. \"Use yore head, give it away to any vermin ye come across. Show 'em we got plenty of vittles. Say there's lots more where that comes from, if they'll come an' serve under me. Do I have to tell ye everything?\"\n\nCrinktail touched the side of her nose knowingly. \"We unnerstand, Chief, leave it to me'n Flinky.\" The pair hurried off to the orchard to load up sacks of fruit.\n\nBadredd began issuing orders to his depleted crew. \"Floggo, Rogg, watch that big door, an' the windows, too. Keep yore bows'n'arrows at the ready. Kill anybeast wot pokes his nose out!\"\n\nThe little fox was glad he had the weasel brothers to serve him. They never argued and usually obeyed all orders.\n\n\"Juppa, Slipback, Plumnose, Halfchop, keep slingin' stones at those windows. Whatever ye do, don't stop!\"\n\nJuppa was pawsore and weary of slinging stones. \"But we've smashed all the windows. Wot else is there t'keep slingin' stones at?\"\n\nBadredd could feel his temper fraying. His voice gained a squeak as he shouted in the weasel's face. \"The idea of breakin' the windows is so that ye can hurl stones through an' hit anybeast inside the place. Or are ye too stupid to realise that?\"\n\nJuppa stood her ground, arguing back swiftly. \"No, I ain't stupid, but I'm hungry an' tired! Us four've been chuckin' stones at that Abbey all night. Oh, an' there's one more thing we ain't too stupid to realise. We're runnin' outta stones to throw, while yore marchin' about givin' orders out an' doin' little else!\"\n\nBadredd waved his broken cutlass about threateningly. \"Don't ye dare talk t'me like that, I'm the chief around here!\"\n\nSlipback muttered loudly. \"Wot're ye goin' t'do, run 'er through wid a broken sword?\"\n\nThe little fox threw his half cutlass aside and stamped his footpaw down so hard that it hurt. \"I heard that, Slipback. Do? I'll tell ye wot I'm goin' to do. I'm goin' t'show ye three how to sling stones properly! Throw down wot stones ye got left an' give me yore slings. Plumnose, Halfchop, start slingin' alongside me. Come on, move yoreselves, take these slings an' load up!\"\n\nHalfchop picked up a sling and loaded it with an apple he had been munching on. He grinned at Badredd. \"Kachunk!\"\n\nThe little fox glared speechlessly at the hapless rat. He shouted to Plumnose, \"Teach that idjit to throw stones!\"\n\nFuriously, Badredd began slinging at a mad rate. The slingstones went everywhere\u2014a few through the window spaces, some backward across the lawns when he released them too early. Others bounced back off the solid sandstone walls.\n\nSlipback dodged a ricochet, grinning slyly. \"Hah, let's see 'ow long the mighty chief can keep that pace up!\"\n\nJuppa started moving out of range, ducking a pebble that had gone the wrong way. \"Let's get out of 'ere afore we get slain!\"\n\nShe raised her voice, calling to Badredd, \"We're goin' to get somethin' to eat an' take a rest!\"\n\nThe fox kept hurling stones like a madbeast, panting. \"Get out o' my sight, ye useless lumps! When y'come back, bring more stones, a lot more!\"\n\nPlumnose, who was slinging at a much steadier rate, called happily to Badredd. \"Huhuh, we'b godd lots ob stones, me'n my mate!\"\n\nThe fox screeched back at him. \"Sharrap an' get slingin'!\"\n\nHalfchop had found a black-and-red banded pebble among his stones. He polished it on his fur and spoke to it. \"Kachunk!\"\n\n##\n\nAbbot Carrul and Granmum Gurvel were going around Great Hall, distributing beakers of hot barley and leek soup to the defenders. Martha was wakened by a stone pinging off a nearby column. Gurvel ladled soup from a cauldron standing on a trolley. The Abbot served it to Martha. Then Carrul called Toran over and gave him some.\n\nToran accepted it gratefully. \"Well, Father, the windows are barricaded tight now. There's only the odd stone comin' through. Let the vermin wear themselves out. Apart from broken panes, there ain't much damage\u2014unless they try burnin' the window barricades.\"\n\nCarrul tried to remain calm, though he could not help sounding anxious. \"Have you a plan in mind, Toran?\"\n\nScratching his rudder, the ottercook stifled a yawn. \"I wish I had, but I'm far too tired an' upset about pore Junty.\"\n\nMartha straightened the rug across her lap. \"We'd do better if we went upstairs to the dormitories. Perhaps up there we could retaliate against the vermin.\"\n\nAbbot Carrul nodded. \"Sounds sensible to me, Martha. Carry on.\"\n\nWarming to her own idea, the haremaid explained. \"We could make slings and throw stones at them. I'll wager Foremole and his crew could provide us with rubble.\"\n\nGurvel sighted Foremole Dwurl coming up from the cellars. She beckoned him to join them. \"Coom over yurr, zurr.\"\n\nDwurl waved a heavy digging claw. \"Wutt can oi do furr ee?\"\n\nMartha made her request. \"Would it be possible to get a load of rubble and pebbles up to the dormitory windows, please?\"\n\nThe mole nodded his velvety head. \"Surpintly, miz! Oi take ett ee bee's goin' t'give yon varmints a gudd peltin', hurr hurr!\"\n\nImmensely fond of Foremole Dwurl, Martha took his work-lined paw in hers. \"Great minds think alike, my friend. We need lots of stones, and some rubble, to tip on the vermin if they start lighting fires. Water is too precious to waste in our present position.\"\n\nToran looked at his young friend with a new respect. \"Hear that, Carrul? Our Martha certainly has a wise head on her shoulders, eh?\"\n\nMartha turned to the ottercook, her eyes shining fiercely. \"Aye, and I don't intend to lose it to a band of murdering vermin. It was vermin who slew my family when I was a babe and too young to do anything about it. This time 'tis going to be different. No matter what happens, those evil scum are not going to take Redwall Abbey from us. We'll defeat them!\"\n\nThey all clasped paws on the arm of the haremaid's chair. Her resolution ran like wildfire through them all.\n\nFather Abbot Carrul's voice echoed around Great Hall. \"Everybeast upstairs to the front dormitories. We're going to fight them. Redwaaaaaaalll!\"\n\nA great cheer went up as Martha had united them in a common cause: taking the attack to the foebeast. The Redwallers thundered upstairs, shouting and roaring.\n\n\"We'll teach 'em a lesson they won't forget!\"\n\n\"Aye, they'll regret the day they came to our Abbey!\"\n\n\"No vermin's goin' to bully us!\"\n\n\"Blood'n'vinegar, that's what they'll get!\"\n\nSister Setiva was minding the Dibbuns as the dormitory door was flung open wide. Redwallers crowded in, still shouting. The Abbeybabes did not quite know what was going on, but they joined in lustily, issuing dire threats against the enemy.\n\n\"Cutta tails off wiv rusty knifes!\"\n\n\"Boil ee varmints in roasted baffwater!\"\n\n\"Gurr, smack ee bottoms wi' gurt sticks!\"\n\nLittle Buffle stuck out his stomach and bellowed, \"Yukkumbumgur!\"\n\nSetiva was becoming able to translate Buffle's baby language. She raised her eyebrows in horror. \"Och, ye wee scallywag, I'll wash your mouth out wi' soap if ye even think o' sayin' that again!\"\n\nMartha was carried up, chair and all, by Brother Weld, Toran and several stout moles. Immediately she related her plan to all the Abbeybeasts.\n\n\"Sisters Setiva and Portula, could you set about making lots of slings? Good, strong braided ones. Brothers Gelf and Weld, I want you to check the downstairs barricades as often as you can. Make sure they're still holding firm, and report back to me each time. Foremole, sir, can you bring up as much stone and rubble as you can lay your paws on?\"\n\nDwurl saluted. \"We'm got loads o' rubble an' rock frum our diggin's in ee basement, miz. Oi'll bring et roight aways.\"\n\nThe haremaid nodded to Toran. \"Can you search about, friend, to find anything we can use as weapons? Anything!\"\n\nMuggum and the Dibbuns clung to the chairarms, pleading, \"Uz 'elp ee, Miz Marth', give us'n's summ jobs!\"\n\nSister Setiva turned in the dormitory doorway, shaking her blackthorn stick and berating the Abbeybabes. \"Och! Ah'll give ye jobs. Get straight intae yon beds an' stay oot o' Miss Martha's way, this verra instant!\"\n\nMartha saw the sad little faces on the Dibbuns and interceded on their behalf to the strict Infirmary Keeper. \"Please, Sister, they only want to help. Let me find a job for them. Granmum Gurvel, have you any sieves or riddles? We'll need them to sift out slingstones from Foremole's rubble when it arrives up here. Could you find some?\"\n\nMuggum brightened up. \"Oi'll tell ee a riggle, Miz Marth'.\"\n\nGurvel took the molebabe's paw. \"Gurr, liddle pudden 'ead, that bee's ee wrong sort o' riggle. Cumm to ee kitchens, an' oi'll foind ee sum proper riggles.\"\n\nEverybeast hurried to their tasks, while Martha tried to keep some organisation amid the ensuing chaos.\n\nMolecrews trundled in and out of the dormitory, bearing stretcherloads of rubble. Sister Portula and some elders ripped old fabric into strips and began weaving slings. Redwallers on kitchen duty came scurrying up with drinks and meals. Martha wheeled her chair about, giving directions, calling encouragement and keeping the constant traffic moving back and forth.\n\n\"Don't block the doorway, please. Bring that stretcher right in and empty it there, by the window.\" She seemed to be everywhere at once. \"Oh, that's a nice strong sling, put it over there with the others. Don't leave that cordial and soup by the rubble, it'll get dust all over it. Shut it inside that wardrobe for the present.\"\n\nBadredd soon grew tired of slinging stones. His paws were aching: more than once, a stone had stayed in the sling, causing it to wrap around his paw and strike it sharply. That, plus the fact that he was an abominable shot, made him toss the sling away angrily.\n\n\"Blood'n'skulls, I've got better things t'do than stand here chuckin' stones all day. Where's the rest o' this lazy lot, eh? Stuffin' vittles or layin' about sleepin', I bet. Well, I'll soon liven their ideas up, the dirty layabouts!\" He stalked off in high dudgeon.\n\nPlumnose and Halfchop dropped their slings and trailed after him. The little fox turned on them furiously. \"Where are you two deadbrains goin'? Did I tell ye t'stop slingin'? Get back there afore I flay ye both!\" The pair went back wearily and continued slinging.\n\nPlumnose complained resentfully to his companion. \"Huh, he'd inna bad mood, iddent he?\"\n\nHalfchop nodded in agreement. \"Kachunk!\"\n\nMartha kept track of Badredd from her position at the front dormitory windows. \"I wonder where he's off to now.\"\n\nToran stood behind her chair. \"Who knows, miss. He's up t'no good, though, an' jumpin' mad by the look o' him.\"\n\nForemole gestured at the considerable mound of earth and stone piled up close to the windowsills. \"Hurr, ee vurmint can jump all ee looikes, we'm ready for 'im!\"\n\nGranmum Gurvel staggered in, dragging a bulging sack. \"Yurr, lookit oi finded, ee gurt sack uv 'otroot pepper. Ee ' hotters leaved it yurr afore they'm go'd off. Oi'm b'aint a keepen it in moi kitchens, no zurr, orful sneezy stuff!\" Gurvel dumped it next to Martha's chair. The haremaid quickly pulled out her kerchief as dust rose from the sack. \"Kerchoo! Aah . . . Aah . . . Achoo! Beg your pardon, dearie me!\"\n\nBaby Buffle stared down at the sack from the top of the rubble mound. \"Sumakivalikkasaccasaccavurgimchoochoo!\"\n\nMartha dabbed at her nose with the kerchief. \"What's he chunnering on about now, Sister?\"\n\nSetiva translated the shrewbabe's language. \"Och, pay no heed tae the rascal. He says we should throw et at yon vermin. 'Tis a silly idea\u2014we'd be sneezed tae death doin' a thing like that. The breeze'd carry et right back in 'ere.\"\n\nGurvel spoke up. \"Nay, marm, not if us'n's makes ee likle sacks uv pepper, boi 'okey. We'm cudd frow slingers at ee varmints.\"\n\nMartha clapped her paws delightedly. \"What a great plan! Thank you, Buffle and Gurvel. Let's try it!\"\n\nThe ancient molecook took charge of the operation. Soon, she and several Dibbuns donned bandannas of wet cloth to protect their noses and mouths against the fiery hotroot pepper. Carefully, they ladled measured portions of the pepper onto flimsy squares of thin, birch-bark parchment. Each of these was fashioned into a tiny bundle, tied at the top with thread. Toran weighed one in his paw. \"Just right for throwin'. Hoho, these'll cause a few sneezes if they land on some scummy noses!\"\n\nYooch the molebabe had scrambled up onto a windowsill. Jumping up and down, he waved his tiny paws and squealed, \"Look out, look out, d'vermints bee's cummin'!\"\n\nBadredd kept a paw on the broken cutlass in his belt, not drawing the weapon lest they see it was only a half-bladed thing. Behind him stood the rest of the available vermin crew\u2014Halfchop, Floggo, Rogg, Slipback, Plumnose and Juppa.\n\nThe little fox shouted boldly. \"Where's yore chief? I wanna talk!\"\n\nAbbot Carrul showed himself at the dormitory window. \"Say what you have got to say, fox!\"\n\nBadredd puffed out his narrow chest. \"Lissen, we've got ye well boxed in up there. You ain't warriors, ye can't fight back or hurt us. So I'll tell ye what I'll do. Open yore doors, we won't attack. Just let me'n one o' my crew come in. When we've found yore magic sword, an' other bits o' loot that we fancy, we'll leave ye in peace an' go.\"\n\nThe Abbot shook his head firmly. \"Never! You'll not set paw in Redwall Abbey, none of you!\"\n\nBadredd passed a paw signal to Rogg from behind his back. The weasel casually notched an arrow to his bowstring.\n\nKeeping his temper in check, the fox replied, \"Never? We'll see about that. Wot ye got to unnerstand is that yore under siege\u2014we could starve ye out or keep attackin' until one by one yore all slain. Oh, I've got lots o' bright ideas, mouse, take yore pick. Either that or just do as I command. 'Twill save ye a lot o' grief.\"\n\nCarrul stood his ground. \"No matter what you say, you will not enter this Abbey. Now, let me make a suggestion. Take your vermin, plus all the fruit you have stolen from our orchard, and leave here. If you do this, you will save yourself a lot of grief. Take my word for it!\"\n\nBadredd shrugged. \"Ain't no use of talkin' to ye, mouse.\"\n\nAs the vermin leader stepped aside, Rogg hurried forward and let fly. Inside the dormitory, some of the pepper dust had got to the Abbot, causing him to sneeze. \"Yaachooo!\"\n\nAs Carrul's head went down with the force of the sneeze, the arrow tipped his headfur, ending up quivering in the dormitory ceiling.\n\nCursing inwardly, Badredd forced himself to stay nonchalantly calm, even to smile. \"Saved by a sneeze, eh? Yore a lucky mouse!\"\n\nSuddenly Toran appeared at the window, a pepper bomb in each paw. \"You won't be so lucky. Sneeze on this, snottynose!\"\n\nIn quick succession, two bags of pepper struck Badredd's face. Then the dormitory windows were packed with Redwallers, hurling their new weapons and shouting.\n\n\"Try a sniff of this, uglychops!\"\n\n\"Yurr, stuff this'n oop ee nose, zurr vurmint!\"\n\n\"Och, take a whiff o' this, ye wicked rabble!\"\n\n\"Sorry we ain't got no salt, so here's a little more pepper for ye!\"\n\nLiterally peppered by bags of the stuff, the vermin crew fled\u2014spitting, sneezing and rubbing at their burning eyes as the fierce hotroot pepper did its work. Between sneezes, they bumped blindly into one another, wailing and screeching.\n\nMartha held up a paw. \"Stop now, no use wasting pepper. They've learned their lesson, a good hot one!\"\n\nA rousing cheer went forth from the Abbeybeasts. \"Redwaaaaaallll!\"\n\nMartha hugged Toran's waist from her chair. \"We did it, friend, we defeated the vermin!\"\n\nThe ottercook stood watching the vermin as they hurled themselves into the Abbey pond. He stroked the haremaid's head absently. \"Aye, beauty, we did it for now. But they'll be back, an' next time they do, those vermin will try to slay us all.\"\n\nSister Portula was in agreement with him. \"Right, Toran, so what'll we do then?\"\n\nMartha surprised herself by shaking a clenched paw. \"We'll just have to give back as good as we get. Don't forget, there's more of us than them. I'd risk my life willingly any day if it meant defeating those scum!\"\n\nGrowls of agreement rang out, even from the Dibbuns. Abbot Carrul was taken aback by the warlike mood of the Redwallers, and even more so by Martha's fighting spirit. He held up his paws until order was restored.\n\n\"You are right, of course, my friends, but let us not do anything haphazard. There has to be a proper plan to rid our Abbey of these vermin!\"\n\nFlinky and Crinktail were in no special hurry to run about seeking recruits for Badredd's gang. The pair wandered deep into Mossflower, glad to be away from the bickering and squabbling of the small vermin gang. They rambled onward, consenting with each other to desert their fellow vermin and find a new life together, far away from it all.\n\nUnfortunately, they walked right into trouble and ambled straight into the camp of Raga Bol. A huge, fat Searat with one milky, sightless eye grabbed the luckless pair by the scruffs of their necks. Both their stomachs churned in fear at the sight of the savage Searat crew. For the first time in his life, Flinky was rendered speechless as he beheld a real Searat captain.\n\nRaga Bol was the complete picture of a barbarian chieftain\u2014from his hooped brass earrings and tawdry silk finery, to his silver hook, gold teeth, curved scimitar and the lethal stiletto he was using to pick at a roasted pike. He spat a fishbone into the fire and picked at his teeth with the hook. Looking both stoats up and down, Raga Bol consulted the fat rat.\n\n\"Who are these two barnacles, Glimbo?\"\n\nFlinky began stammering out an answer. \"If it please, yore 'onour, we was just . . .\"\n\n _Splat!_ Raga Bol leaned forward and struck Flinky a slap across his mouth with the pike. \"Did I speak to ye, stoat?\"\n\nThe hook shot out, catching Flinky's jerkin. He was yanked forward, under the cold glare of the wickedest eyes he had ever looked into.\n\nHe felt the Searat's hot breath on his face as the rasping voice growled out, \"Guard yore tongue, mudbrain, or I'll carve it out an' feed it to ye. Speak now, wot's in those sacks?\"\n\nFlinky's throat bobbed as he gasped out, \"F . . . f . . . fruit, sir!\"\n\nRaga Bol stuck his stiletto in the sack Flinky was holding. He booted the stoat backward, causing the blade to rip through the sack. Flinky went sprawling amid the fruit which spilled out onto the ground.\n\nThe Searat scowled. \"Fruit? Is that all ye brought? No booty, weapons, not even a brace o' birds or a decent fish. Just fruit!\"\n\nGlimbo wrenched the sack from Crinktail. He emptied it over Flinky, who lay cringing on the ground. \"Sink me! This 'un's brought fruit as well, Cap'n. They must be both stoopid in d'brain!\"\n\nGripping hold of Crinktail, Glimbo shook her until her teeth rattled, bellowing in the hapless stoat's face. \"Yore stoopid in d'brain, wot are ye?\"\n\nCrinktail gabbled out something that sounded like \"Stooballainnabrab!\"\n\nThe Searats crowded round laughing. They tore the jerkins from both stoats, and robbed them of their belts and knives.\n\nStripped to the fur, Flinky and Crinktail huddled together, eyes wide with terror as the Searats licked their knifeblades and winked wickedly at them.\n\nRaga stroked under his chin, with the polished curve of his pawhook. \"The woodlands round here are packed with fruit, an' ye bring me two sacks o' the stuff? Right then, me beauties, I'll tell ye what we'll do. What'd ye like, an apple or a pear?\"\n\nCrinktail spoke, her voice quivery with terror. \"Apples, sir.\"\n\nRaga smiled, showing several gold-capped fangs. \"Haharr, apples it is then. Ferron, jam an apple apiece in their gobs, 'twill stop 'em singin' out while they're roastin'!\"\n\nFerron, a tall, gaunt-faced rat, sorted through the fruit until he came up with two large, rosy apples. He strode over to the two victims, but before he could start, Flinky yelled, \"Loot! Treasure! Booty an' magic swords!\"\n\nRaga's long blade rasped out of its scabbard. Resting the point against Flinky's nose, the captain spoke just one word\u2014\"Where?\"\n\nThe stoat answered speedily, knowing his life depended on it. \"Sure, 'tis at the Abbey o' Redwall, sir, only a good ould march from here. All the plunder yore 'eart could desire!\"\n\nThe swordtip lifted as Raga looked around the ugly faces of his leering crew. \"Give 'em back their stuff. Come 'ither, mates. Sit 'ere by me, where I can carve cobs off'n ye if yore tellin' me fibs. I can't abide fibbers, can you, messmate?\"\n\nFlinky shook his head vigourously. \"Sure those fibbers are the worst ould kind of beasts ever born, ain't that right?\"\n\nCrinktail hastened to agree with him. \"Fibbers are villains!\"\n\nRaga Bol narrowed his frightening eyes and glared at his prisoners, who sat as if hypnotised. Suddenly he threw back his head and roared with laughter. \"Aharrharrharrharr! That's wot I like to 'ear, me liddle fishes. Avast there, Blowfly, bring grog fer our messmates!\"\n\nBlowfly, a malodourous, greasy-looking rat, brought three gigantic pottery jars and a keg of grog, which he rolled along by kicking it. He filled the jars brimful, issuing one to each of them. Both stoats quailed at the sight of the fearsome-smelling brew. Bol drained half of his at one huge swig, smacked his lips and winked broadly at them. \"Good 'ole seaweed'n'fish'ead grog, ain' nothin' like it! Aharr, Raga Bol can't abide prissy liddle creatures wot don't like grog. Drink 'earty now!\"\n\nGagging and spluttering, Flinky and Crinktail tried to sup the fiery liquor. The Searat crew gathered round, grinning and guffawing as they watched the stoats trying to cope with the grog. Finishing his swiftly, Raga observed his victims closely. \"Cummon, buckoes, no shilly-shallyin' there, bottoms up, an' don't ye leave none for the fishes!\"\n\nGrog was dribbling down Flinky's chest fur by the time he finished. Something odd was happening to his eyes. In front of him sat three Raga Bols. His head was whirling, and his tongue felt as though it belonged to someone else. He hiccupped. \"Heeheehee, hic! Sure, that was a prime ould, hic, droppa grog, hic hic! Ain't that right, hic, eh, Crinky, hic!\"\n\nCrinktail gazed woozily at her empty jar and giggled. \"Sh'marvelloush! Makesh y'feel like a battlin' badger, heehee, whoops!\"\n\nShe was knocked flat on her back. Raga, who had kicked her over, stood glaring down at the stoat, his sabre drawn. \"Badger, wot badger? Is there a badger 'ereabouts? Have ye sighted a great giant of a stripedog? Tell me!\"\n\nCrinktail attempted to rise, but fell flat. She looked up at the Searat captain with owlish solemnity. \"Wot badgersh? Heehee, we ain't seen no shtripedogs around 'ere. Don' worry, Bragger Roll, we'll fight 'em all for ye, me'n Shlinky!\"\n\nShe giggled again, then passed out, senseless. A fleeting glimpse of relief crossed the Searat's face. He turned his attention to Flinky, who was swaying from side to side, and blinking drunkenly. \"Ahoy, buckoe, let's talk, me'n you. I'll ask the questions, an' ye give me all the answers. The right ones if'n ye value yore skin! This Abbey o' Redwall, tell me everythin' about it. An' worrabout yore crew, 'ow many strong are they, who's yore leader, wot's 'e like? No lies, now, c'mon!\"\n\nRaga Bol's crew listened avidly as Flinky related the entire sorry tale to their captain. The stoat was drunk, but not so drunk that he didn't know what would please the murderous Raga Bol. A good portion of his story was outright lies. He told of witless Abbeybeasts, and a fabulous treasure, laying great emphasis on the magic sword. Flinky was good at what he did, having spent most of his life lying and pleasing others. The captain and his crew believed the yarn. There followed much winking, nudging, whispering and gleeful rubbing of paws, even from Raga Bol. This was going to be a picnic, an orgy of looting and slaughter. A real Searat's dream come true!\n\nThe Searat crew made ready to march. Raga Bol delayed moving, since there was one thing still bothering the captain's mind\u2014the fate of the giant stripedog. Giving orders for the crew to stand ready, he marched back along their trail alone, looking for signs of his assassin's return.\n\nAfter an hour or so, Raga Bol glanced up at the sky. Dark rolling clouds, coupled with the distant rumble of thunder, presaged the arrival of a sizeable storm. He turned his gaze to the path ahead, where the foliage was swaying in the hot wind. The Searat's keen eyes and ears missed nothing. He saw the shrubbery moving the wrong way at one point and heard the moans and laboured gasping of somebeast coming slowly up the trail toward him.\n\nIt was Jibsnout, leaning heavily on an impromptu crutch he had fashioned from a branch. Raga Bol hastened to intercept him, frowning with false concern. \"Jibsnout, matey, are ye wounded? Have ye news of the stripedog? Where are those sons of Wirga, 'ave they deserted ye?\"\n\nThe stolid Searat slumped wearily down, his tongue licking the first fat drops of rain that fell through the woodland canopy. He looked up at Raga Bol kneeling at his side.\n\n\"Cap'n, we stood no chance! That stripedog 'ad a squirrel wid 'im, they ambushed us! Two of Wirga's sons were slayed. The other one ran away, though 'e wouldn't'a got far, I wager. I was shot through the footpaw by the stripedog, then 'e took my blade. I thought 'e was gonna kill me, but 'e tended to the wound an' sent me back to ye wid a message, Cap'n. The stripedog sez to tell ye that 'is name is Lonna Bowstripe, an' that 'e's comin' after ye, Cap'n Bol. Aye, yoreself an' all the crew, me too. We're all deadbeasts, d'ye hear me, walkin' deadbeasts! That big Lonna beast is goin' to slay us one by one, every ratjack of us! Take me word fer it, Cap'n, 'e's a mighty warrior but a real madbeast! I saw it in 'is eyes, they was red as fire. The stripedog'll finish us, all of us, I believe wot 'e said!\"\n\nA jagged lightning flash lit up the gloomy woodlands; thunder rattled closer and the rain came in earnest. Raga Bol held Jibsnout close to him, murmuring softly. \"Hush now, mate, no stripedog's goin' to harm ye. This storm'll wash out all our tracks, nobeast'll find us then. Besides, we'll be snug inside of a big stone fortress, wid vittles to spare an' more loot than ye've ever clapped eyes on. Hahaarr, 'ow'll that suit ye matey, eh?\"\n\nJibsnout blinked rain from his eyes. \"That'll suit me good, Cap'n.\"\n\nBol held him closer, whispering in his ear, \"Ye won't breathe a word about no stripedog to the crew now, will ye, me ole mate?\"\n\nJibsnout smiled at his captain. \"You know me, Cap'n Bol. None of 'em will 'ear a word from my mouth!\"\n\nRaga Bol smiled back at Jibsnout. \"So they won't, mate, yore right.\"\n\nHe slew Jibsnout with a single thrust of his stiletto. Shoving the body into the bushes, Raga Bol sloshed back through the battering downpour, muttering to himself. \"They all talks sooner or later, but you was right, Jibsnout. Nobeast'll 'ear a word from yore mouth.\"\n\n##\n\nThe storms which had been battering the high cliffsides slackened off to a steady downpour. Fenna popped her head outside the cave, shielding her eyes. \"It's hard to see anything properly on a rainy night. No sign of Horty yet, I do hope he's alr . . .\"\n\nA monolithic shape loomed out of the darkness, silent as a moonshadow. The squirrelmaid staggered backward as a badger of massive proportions padded in. Over his shoulder lay Horty, draped like a limp rag. The badger carried on past them, to the back of the cave, his deep growl echoing.\n\n\"I found your friend, never fear, he'll come to his senses before too long. Be still now until I get a fire going.\"\n\nBragoon's paw stayed Springald from rising. \"Be still, Spring, do as our friend says.\"\n\nThey heard steel strike flint, as the badger's soft breath coaxed flame from the sparks that fell onto the tinder. Soon a pale light flickered. It became a proper fire when the badger added dried grass and twigs to it. He banked it up with broken pine branches, waited a moment, then turned to face the travellers. Bragoon had encountered a badger or two before, but none like this one. Warrior was written all over this giant beast\u2014from the great bow he carried, to the long quiver full of arrows, to the lethal dagger strapped below his shoulder. He wore a simple smock of rust-hued homespun, belted with a woven sash.\n\nBut it was his face that denoted his calling. A deep, jagged scar ran lengthwise across the broad-striped muzzle, with stitchmarks pocking either side. The dark eyes remained impassive, reflecting the firelight. The otter judged him to be one of those fated creatures cursed with that malady called Bloodwrath\u2014the red tinge mixed with the creamy eye whites betrayed it. Bragoon had heard tales of such badgers that described them as terrible to behold and unstoppable in battle. The otter held out his paw, flat, with the palm upward, a sign of peace. The badger did likewise. Then he placed his paw on top, making Bragoon's look like the tiny paw of a Dibbun. He introduced himself.\n\n\"I am Lonna Bowstripe. This is not my cave, but you are welcome to stay here until the hare recovers. I saw you escape those rats today and knew you would shelter up here. I watched from the hills what a brave thing your friend did. I killed those three rats who tried to capture him\u2014vermin are bullies and murderers, they are no great loss to anybeast. Who are you, why are you in this country?\"\n\nThe otter bowed respectfully. \"I am Bragoon. This is Saro, Fenna and Springald. The hare is named Horty. We are travellers.\" He gestured upward toward the plateau on top of the cliffs. \"We are seeking a place called Loamhedge. It lies somewhere up there.\"\n\nLonna began stringing his bow. \"A dangerous quest, friend. There are many Darrat rats out there still. Their captain was one of the three I slew. You need to reach the clifftops without interference from them. Your journey will be hard enough without rats following you. Perhaps they are camped in the area. I will warn them off. Pass me an arrow, Fenna.\"\n\nThe squirrelmaid took a shaft, nearly as long as she was, from the badger's quiver. They watched Lonna blunt the point by jamming an old pinecone over it. He held it to the fire until it was blazing and crackling. Testing the air outside the cave, the badger seemed satisfied.\n\n\"It's not raining too heavily now, the shaft should burn for a bit before it goes out.\" With a single graceful move, Lonna set the blazing arrow on his string, drawing the shaft back until the burning end almost touched the bow. _Whooooosh!_ It shot off like a rocket, into the night sky above the dunes.\n\nThrowing back his head, the big beast roared out in a thunderous voice that echoed around the cave and along the cliffsides. \"I am Lonna Bowstripe! I eat rats! I will taste the blood of any who are here by dawn! Eulaliiiiiaaaaaaa!\"\n\nHe returned to the fire as they took their paws from ringing ears and began tending to Horty. Lonna smiled and shook his head as Horty began to stir. \"He looks hard to kill\u2014I've heard it said that hares are perilous beasts. This one will be a warrior one day.\"\n\nThe giant badger looked so large and ferocious in the firelight that Springald could readily understand how the rats would fear him. She enquired politely, \"Sir, you didn't really eat three rats, did you?\"\n\nThe smile still lingered on Lonna's lips. \"Nay, little maid, don't believe all you hear. The language of death and violence is all that vermin understand. I'd sooner devour a crushed toad that was four seasons dead than eat rat. I eat only the same food as you do.\"\n\n\"Eat? I say, did some chap mention eats? I'm famished!\"\n\nBragoon assisted the incorrigible hare to sit upright. \"Oh, this 'un's awake, sure enough. Well, how d'ye feel, young famine belly? Oh, ye'd better thank the beast who saved ye. This is Lonna Bowstripe.\"\n\nHorty did an exaggerated double take at the huge badger. He winked cheekily. \"Good grief, sah, bet you can pack the jolly old provisions away, wot wot?\"\n\nThey pooled the resources of their packs and were soon toasting yellow cheese and oat scones over the fire. Saro poured dandelion and burdock cordial for the company. Springald split some loaves of nutbread and spread them with honey.\n\nLonna glanced sideways at Horty, taken aback by the young hare's appetite. \"Great seasons, talk about packing provisions away! Where do you put it all? You're a bottomless pit!\"\n\nThey all burst into laughter at the sight of Horty's indignant face.\n\nOver the next few hours, they exchanged their stories. The five friends told Lonna all about Redwall and its creatures. They also explained Martha's situation and the reason for their quest. When the giant badger related his own personal history, they were greatly saddened and angry, too. There was a hushed silence when he came to the end of his narrative. Lonna ran his paw down the fearsome scar, tracing it across his still face.\n\n\"They will pay with their lives, Raga Bol and all his vermin crew. As sure as the days break and the seasons turn!\"\n\nThe five travellers did not doubt a word that he uttered.\n\nLonna rose and replenished the fire. \"You must sleep now. Tomorrow will be a hard day's climbing. I think the plateau above the cliffs is no place for the fainthearted. Take a good rest tonight, I'll guard the cave entrance.\"\n\nBragoon uncovered his sword. \"I'll keep ye company, Lonna. Two guards are better'n one, an' four eyes can see more than two.\"\n\nThey sat together at the cave entrance. Lonna could not take his gaze from the otter's sword, drawn to it like a magnet to metal. \"That is indeed a wondrous weapon you carry, Bragoon.\"\n\nThe otter let the firelight play along the blade. \"Aye, 'tis so, though it don't belong t'me. Abbot Carrul of Redwall loaned it t'me the day we left. I think he did it not just for our protection, but as a sort o' good-luck charm for the journey. This sword belongs at the Abbey. 'Twas owned in the far olden seasons by a mouse. His name was Martin the Warrior, one o' the founders of Redwall. I was told stories of Martin an' his sword when I was nought but a Dibbun. They say it was forged an' made by a great badger lord, a warrior himself, an' a very skilled swordsmith, as ye can see. He made it from a lump of ore that fell from the sky, a piece of a star, I was told. This badger, he was Lord of Salamandastron, a mountain fortress. Did ye ever hear of that place, Lonna?\"\n\nThe dark eyes of the giant flashed. \"Every badger knows the name of Salamandastron. I will go there myself someday. I feel my days will end there\u2014but only when my score with Raga Bol is settled.\"\n\nBragoon sat up with a start, realising that he had dropped off to sleep during the night, something he would never have done in his younger seasons. Dawnlight was filtering into the cave, and Lonna Bowstripe was gone. As Saro was rekindling the fire from its embers, the three young ones were just waking.\n\nShe gave Bragoon a beaker of hot mint tea. \"Mornin', matey. Well, our bigbeast left while it was still dark. I saw 'im go, y'know.\"\n\nFenna poured tea for herself. \"Lonna's gone?\"\n\nSaro nodded. \"Aye, you lot were all asleep. Horty's snorin' woke me, sounded like a tribe o' stuffed-up frogs.\"\n\nThe young hare huffed indignantly at her, but Saro carried on. \"I was lyin' there wide awake, watchin' Lonna in the fireglow. He'd picked up the sword o' Martin to admire it. Well, next thing that badger went stiff as a frozen pike, sittin' there starin' at the blade as if it was speakin' to 'im. I watched for a while, then Horty started snorin' agin. So I gave 'im a good kick an' settled back to catch a nap.\"\n\nHorty interrupted. \"Blinkin' cadess, kickin' a chap in mid slumber? Rank bad manners, I'd say. Hmph!\"\n\nThe elderly squirrel shrugged. \"When I woke up agin, he'd gone.\"\n\nBragoon slapped his rudder against the rock floor. \"I'll wager 'twas Martin the Warrior, speakin' to Lonna through the sword. He told the badger where t'find Raga Bol, an' Lonna took off after the villain!\"\n\nBragoon wrapped the sword up reverently as Horty chuckled. \"I bet old Raggaballoon wotsisname wouldn't be too pleased with Martin, if he knew. Snitchin' about him to that bally great hulk. I'd hate to be in his way when he feels peevish. Frazzlin' frogs, imagine what old Lonna'll do to that vermin when he catches up with him, wot wot!\"\n\nBragoon began packing his belongings. \"I wouldn't like to imagine, mate. That's Lonna's business, an' I'm sure he can take care of it well. But we've got our own problems to tend. Up an' on to Loamhedge, mateys!\"\n\nMorning boded bright as they left the cave and began climbing the cliff to its top. It was hard going until the two squirrels, Saro and Fenna, went ahead. Soon they were on top of the cliff. Lowering down a rope, they heaved up all the packs, then secured the rope around a rock, allowing the other three to haul themselves up.\n\nIt was a breathtaking panorama from the plateau. Horty's keen eyes spotted a small dark smudge, moving across the scrublands in the distance. He pointed. \"I say, you chaps, that could be thingummy, er, Lonna!\"\n\nSpringald shaded her eyes. \"So it could! He's headed northwest, that's the direction we came from. Saro, d'you suppose he's going to Redwall?\"\n\nSarobando felt they were wasting time sightseeing. \"I couldn't really say, missy, but one thing's shore, we ain't goin' to Redwall. 'Tis Loamhedge we want. So stop lookin' backwards an' let's go for'ard. Quick march!\"\n\nShimmering flatlands, devoid of vegetation or shade, rolled out before them. Small swirls of dust eddied in spirals on the hot breeze. Sarobando squinted her eyes against the distance.\n\n\"Miss Fenna, yore in charge o' the drinks, we'll have t'be stingy with liquid. It might be some time afore we run across water by the look o' things.\"\n\nImmediately after the squirrel mentioned drinks, Horty began feeling thirsty. \"I say, Fenna old gel, pass me that canteen, there's a good little treebounder. I'm parched!\"\n\nFenna marched right on past him. \"We'll drink at midday and not before, so forget about it and keep going.\"\n\nThe young hare appealed to his comrades. \"Wot? Did you chaps hear this heartless curmudgeon?\"\n\nBragoon grinned pitilessly at Horty. \"Aye, loud an' clear, mate. Wot's the matter, are ye thirsty already?\"\n\nThe incorrigible hare clapped a paw to his throat dramatically. \"Me flippin' mouth's like a sandpit, an' the old tongue feels like a bally feather mattress. A drink, for pity's sake, marm!\"\n\nSaro levelled a paw at him. \"Ye drink when Fenna tells ye. Now get a slingstone pebble an' suck it. That'll keep the thirst off as y'march, 'tis an old trick.\"\n\nHorty pulled a pebble from his pouch, looked at it in disgust, then put it back. \"Permission to sing, sah!\"\n\nThe otter waved a paw in the air. \"Sing y'self blue in the face for all I care, but forget about drinkin'.\"\n\nHorty had to dig through his store of ballads and ditties, but he soon came up with an appropriate one.\n\n\"I knew a jolly old spider, and she always used t'say,\n\nshe could dive in a bath of cider, an' swim around all day.\n\nOh I would like to be that spider,\n\nfloatin' round in sparklin' cider,\n\nshe'd drink an drink, 'til she started to sink,\n\nthere'd be so much cider inside o' that spider!\n\nI once knew a friendly flea, to whom I used to chat,\n\nhis favourite drink was ice-cold tea, what d'ye think of that?\n\nOh I would like to be that flea,\n\nsippin' cups of ice-cold tea,\n\nall in fine fettle from a rusty kettle,\n\n'til I drank as much tea as that flea!\n\nO cider spider, tea an' flea,\n\n'tis all good manner o' drinks for me.\n\nI'm an absolute whizz for strawberry fizz,\n\nI'll sup old ale 'til I turn pale,\n\nI'd never bilk at greensap milk.\n\nGive this ripsnorter some rosehip water,\n\nor cordial fine made from dandelion,\n\ngive me a barrel it's mine all mine,\n\njust tip me the nod or give me a wink,\n\nan' I'll drink an' drink an' drink . . .\n\nan' dri . . . hi . . . hi . . . hiiiiiiiink!\"\n\nSaro covered her ears with both paws and roared, \"Enough! I can't stand no more o' that caterwaulin', give that hare a drink. Give everybeast a drink!\"\n\nFenna passed the canteen around, allowing each of the group one good mouthful. Horty was onto his second swig when the otter snatched the canteen from him and stoppered it. \"Ye great guzzlin' gizzard, don't ye know when t'stop?\"\n\nHorty gave him a hurt look and belched. \"Beg pardon, sah. Miserable blinkin' bangtail, I barely wet me lips, wot!\"\n\nBragoon grabbed the young hare by his fluffy tailscut and tugged hard. \"One more word and ye'll be wearin' this as a bobble twixt yore ears. Now belt up an' march!\"\n\nIt was hard, hot and dusty out on the flatlands, but they trekked doggedly onward. Even the breeze was like the heat from an open oven door. With neither shade nor shadow to shelter from the ruthless eye of the blazing sun, it soon became an effort to walk.\n\nBragoon licked his dry lips. Dropping his pack, he crouched down on his hunkers. \"Phew! I tell ye, mates, I never knew a day could get so hot. We'll rest here awhile.\"\n\nThe aged squirrel set about making things comfortable. She laced their cloaks together and made a lean-to. Weighting one end of the cloaks with their supply packs, she propped up the other end with two travelling staves. \"That'll give us a bit o' shade. Get under it, an' we'll take another drink. Mebbe we'll have a nap 'til it gets cooler. Then we can travel in the evenin'.\"\n\nThe otter dug a beaker out of his pack. \"Good idea, mate. Fenna, pass me the canteen. I'll measure our drinks out, so nobeast gets any less.\" Here he glanced at Horty. \"Or more than the others!\"\n\nThey were each allowed one half-beaker, which they sipped gratefully.\n\nHorty quaffed his off in a single gulp. \"Bit measly, wot! Where's the food?\" He was the only one who felt like eating; the others stretched out and tried to rest.\n\nFenna watched the hare stuff down candied fruits. \"That will make you even thirstier. The sweetness will start you wanting to drink more.\"\n\nHorty waggled his ears at her. \"Oh pish tush an' fol de rol, miss, I like eatin', doncha know!\"\n\nBragoon opened one eye, remarking ironically, \"Ye like eatin', really? I'd never have known if'n ye hadn't told me so! Put that haversack back on the cloak ends, or the wind'll blow our shelter away.\"\n\nSpringald dreamt she was back at Redwall, paddling in the Abbey pond. Cool, wet banksand slopped between her footpaws as she splashed happily about. Sister Portula and the Abbot came strolling across the dewy lawn. Although the mousemaid could hear what they were saying, their voices sounded different.\n\n\"All gone! Every flippin' thing is confounded well gone, wot?\" Springald wakened to see the reddish evening light through clouds of dust. Horty was stamping about outside the lean-to entrance, sobbing hoarsely. \"Every blinkin' drop t'drink, an' every mouthful of scoff. Gone, gone, we've been robbed, flamin' well looted!\"\n\nBragoon grabbed the hare and shook him. \"Stop that bawlin', calm down an' tell us wot 'appened.\"\n\nSpringald gathered round with Fenna and Sarobando to hear Horty's woeful tale.\n\n\"Couldn't sleep, y'know, too bally hot, wot. I was jolly thirsty, too, so I got up an' went outside t'get the canteen out of the haversacks. Some blighter's filched the lot. They've left rocks in their place. Go an' see f'y'self!\"\n\nIt was true: five rocks sat holding down the rear of the lean-to, where the five packs of food and drink had been stowed.\n\nSaro held up her paws. \"Be still, there may be tracks, pawprints or dragmarks!\"\n\nShe went down on all fours, eyes close to the dusty earth, nose twitching as she sniffed. A moment later, she stood up with a look of disgust on her face. \"Nothing! Not a single trace. Must've been an experienced thief who did it.\"\n\nBragoon commented wryly. \"A beast would have t'be clever to survive in this wasteland. Well, that's it! No good weepin' o'er stolen supplies, we'll just have t'get on with it. While 'tis dark the weather's cooler, so we'll travel by night, at the double. Right, Saro?\"\n\nThe old squirrel nodded and began issuing guidelines. \"Aye, mate. March fast an' silent, no talkin'. We don't know wot's out there in the darkness. 'Tis strange territory, so stick together an' hold paws. There'll be no time for restin'.\"\n\nShe wagged a stern paw at the young hare. \"Listen good, Horty, this ain't a game anymore, see. If you start yammerin' on about food'n'drink, or causin' any upset, ye'll be riskin' our lives. Just march, do as yore told an' shut that great mouth o' yours, d'ye hear?\"\n\nHorty placed a paw over his own mouth and drew the other paw across his throat in a slitting motion.\n\nSpringald nodded. \"I think he's gotten the idea. Quick march!\"\n\nOff they went into the day's last crimson-tinged twilight\u2014without food, drink or any hope of rest. The five small figures were dwarfed by the immensity of a dust-blown, trackless desert. Hidden eyes watched their departure, and sinister shapes rose from the earth to follow the questors.\n\n##\n\nThe storm broke over Redwall at about the same time that Raga Bol killed Jibsnout. Foremole Dwurl gazed gloomily out of the dormitory window at the windswept deluge outside. He blinked as lightning illuminated the room and thunder barraged overhead.\n\n\"B'aint no use a throwen pepper at vurmints in ee gurt rainystorm. Bo urr, nay, zurr!\"\n\nMartha wheeled her chair to the window and peered out. \"Hmm, I wonder how the vermin are coping with this downpour.\"\n\nAbbot Carrul sighed. \"Who knows? Martha, please keep an eye on them. Right, let's get on with this Council Meeting.\"\n\nOutside, fat raindrops beat a deafening tattoo on the walls of the Abbey, its lawns nearly underwater. Badredd and his gang had commandeered the gatehouse. They lay about, wrapped in sheets, blankets and window curtains, using the material to dab at their sorely inflamed nostrils. Sneezing had become pure agony, with the membranes of their nostrils and throats red-raw from the bombardment of hotroot pepper.\n\nPlumnose was having the worst of it. Each time he sniffed, his pendulous nose wobbled and vibrated. Throwing off the bedspread he had been wearing, the suffering ferret made for the gatehouse door.\n\n\"Duh, I'b goin' oudd inna rain tuh lay dowd an' ledda rained water clear be node. Id mide wash idd out!\"\n\nHalfchop sneezed painfully as he volunteered to accompany him. \"Kachuuub!\"\n\nThe Abbey Council had decided on a desperate scheme. Twoscore of the most able-bodied Redwallers would storm the gatehouse and make an end of the vermin. They stood ready to go, each armed with some form of homemade weapon: kitchen knives tied to window poles formed spears, long-handled garden spades, forks and hoes, together with coopering mallets and stave hatchets from the cellars.\n\nToran, serving as commander of the group, leaned against the windowsill, going over the scheme for a second time. \"Listen, friends, 'tis no use barricadin' 'em in the gatehouse. We've got to make an end to it, invade the place, break in an' slay every last one o' them. No half-measures if we want a peaceful life for us an' the little 'uns. I'll go through the door first, the rest o' you follow me. Show no quarter once yore inside! Sister Portula, Foremole Dwurl an' yore two moles there, Burney'n'Yooler, you stay outside an' get any who tries to break out an' run off. Any questions?\"\n\nMuggum saluted with a copper ladle he had brought from the kitchen. \"No, zurr, oi'll do moi dooty, doan't you'm wurry!\"\n\nMartha lifted him onto her lap and took the ladle. \"Your duty is to stay here with the rest of us and guard the Abbey door. This storm has set in for a good while yet. Once it goes dark, Toran and his friends will have the advantage of night cover and rain. The vermin won't be expecting them to attack. Meanwhile, we'll guard the door and make sure only Redwallers get back inside. It's a very important job, Muggum. Can you do it?\"\n\nThe molebabe narrowed his eyes, glaring suspiciously at Toran's attack party. \"Ho, oi can do et, Miz Marth', doan't ee fret. They'm b'aint a-getten back in yurr iffen they'm b'aint theyselves!\"\n\nToran shook the molebabe's paw. \"Well said, matey!\"\n\nAbbot Carrul stood up on one of the truckle beds and delivered a homily to his beloved Abbey creatures. Everybeast fell silent, respectfully bowing their heads as he spoke out.\n\n\"Fortune and fates be with you all,\n\nyou who fight for the right,\n\nsome will stand, others fall,\n\nnever to return this night.\n\nBut fear ye not, my loving friends,\n\nbe strong of limb and heart,\n\nknowing that peace depends on you,\n\nlet courage play its part.\n\nTranquillity and calm spread wide,\n\nthrough this our dear homeland,\n\njustice and truth go by your side,\n\nwhich evil cannot withstand.\"\n\nThough Martha did not say it, she wished now more than ever that her two friends, Sarobando and Bragoon, had stayed.\n\nThunder exploded overhead; jagged forks of lightning tore through the fading light. Raga Bol and his Searats pounded on Redwall Abbey's main gate. Hearing the noise, Halfchop and Plumnose padded soggily to the gate.\n\nPlumnose placed an ear against it, calling out, \"Who'd dat?\"\n\nA sabre was at Flinky's neck as he answered. \"Sure, 'tis only me'n me mate Crinktail. We're gettin' drowned out here. Open up an' let us in, Plummy!\"\n\nThe two crewbeasts lifted the wooden bar, allowing the door to swing inward. Flinky and Crinktail were flung in, landing face down in the mud as the Searats poured through. Raga Bol seized the ferret's nose and twisted it, bringing Plumnose up on his pawtips, squealing in agony.\n\n\"Yeeee! Ledd go!\"\n\nThe captain let go and kicked Plumnose flat in the mud. \"So yore the big bad warrior wot put this place to siege, eh?\"\n\nHe roared with laughter as the ferret held a paw tenderly around his bruised nose and pointed to the gatehouse. \"Nodd me. Badredd's in dere, he did idd!\"\n\nThe little fox was half asleep as the gatehouse door crashed off its hinges. He was dumbstruck at the sight that greeted him. Raga Bol strode forcefully in, squinting one eye as he glared ferociously around.\n\n\"Which one of ye is Badredd?\"\n\nThe crew, terrified out of their wits by half a hundred Searats leering through the doorway at them, pointed quickly at the fox. Raga's polished pawhook latched into Badredd's belt, jerking the fox face-to-face with him. The barbaric captain's murderous eyes bored into the fox's numbed gaze. \"So then, liddle laddo, yore the mighty Badredd?\"\n\nSpeech deserted him, Badredd could only stammer. \"Y . . . Y . . . Yu . . . Ya . . . y-y-y-\"\n\nRaga Bol shook him like a rag doll, covering the little fox with spittle as he roared into his face. \"Don't stan' there makin' noises like an idjit! Are ye or aren't ye Badredd, ye runty buffoon?\"\n\nThe fox nodded furiously, as he heard his own voice squeak out, \"Yis!\"\n\nThe sea captain turned to his crew, gold fangs asparkle as he grinned at them. \"Well now, ain't that nice. Say 'ello to our new cap'n, buckoes!\"\n\nThere was loud guffawing and shouts of ridicule from the Searats.\n\n\"Pleased t'meet yer, I'm shore!\"\n\n\"Mercy me, 'e do look fierce, don't 'e?\"\n\n\"I'd watch 'ow ye talk to ole Badredd. Looks like an 'ard master t'me, a cold 'earted killer!\"\n\n\"Hawhawhaw! Aye, lookit 'is sword. Hawhawhawhaw!\"\n\nThe Searat captain wrenched the broken cutlass from his victim's belt. He held it under Badredd's nose. \"Does your mamma know ye've been playin' wid this? Dearie me, yew could cut yerself. Naughty fox!\"\n\nRaga Bol's crew laughed until tears ran down their cheeks. When the fox's own crew began smiling and chuckling, the big Searat turned on them savagely.\n\n\"Wot are you lot laughin' about, eh? Stupid clods, lettin' yoreselves be ordered about by a liddle oaf with a busted sword. Gerrout of 'ere, all of ye, clear out!\"\n\nThe vermin scurried to obey, cringing and ducking as they had to pass Raga Bol, who was partially blocking the doorway. Still dragging Badredd along by his belt, Raga strode out into the sheeting rain, issuing orders to his Searats.\n\n\"Glimbo, Ferron, Chakka, you stay in the liddle 'ouse wid me. Ringear, lock that big gate, nobeast gets in or out. Post a watch on it. The rest of ye, take shelter where ye can find it. Blowfly, take a rope's end an' keep an eye on this lot.\"\n\nHe indicated the fox's crew with a nod. Finally, Raga turned his attention to the hapless Badredd. Thrusting the broken cutlass into the fox's shaking paws, he snarled, \"Now then, me laddo, yew'd better be a good cook, or ye'll find yoreself bein' served up as vittles. D'ye hear me?\"\n\nBadredd nodded miserably as Raga Bol continued barking out orders. \"Git yoreself down t'that pond an' take yore crew along. I wants fish fer me brekkist, a good fat 'un, an' no excuses. Just 'ow yer catches an' cooks it is yore bizness. But if'n it ain't on the table, done perfectly, when I wakes up . . . then ye'd best cut yore own throat wid that toy sword, 'cos ye won't wanna face Raga Bol. Now get to it sharpish!\"\n\nHe flung Badredd face first into the mud. Then, turning on his paw, the big Searat strode inside the gatehouse.\n\nThe little fox raised his head, weeping and spitting out wet soil, thankful he was still alive. But for how long? The barbarous rat had set him a near impossible task. How was he going to catch a big fish and cook it in the midst of a thunderstorm, with rain pounding furiously down?\n\n _Thud!_ A blow from a knotted rope's end made him arch his back. Blowfly landed another one, this time across Badredd's rump.\n\n\"Up on yore hunkers, foxy! Yew 'eard wot the cap'n said. Step lively now. Youse others, bring that blanket t'make a tent fer me. I ain't sittin' round in the rain watchin' ye makin' Cap'n Bol's brekkist. All down t'the pond now, at the double!\"\n\nHe drove them forward with the rope's end.\n\nA horrified silence had fallen over the Abbey dormitory. One word from Old Phredd cut the air like a knife. \"Searats!\"\n\nShilly followed this up with a question. \"Wot bee's a Searat?\"\n\nToran bent down to the small truckle bed and pulled up the covers to the squirrelbabe's chin. All around the dormitory, Dibbuns were sleeping peacefully. The ottercook wrinkled his nose at Shilly.\n\n\"A Searat, me dear? Just some naughty ole beast. Nothin' for ye to get upset about, go t'sleep now.\"\n\nAbbot Carrul sat down on a hill of slingstones in the middle of the floor. \"How many of them are in the grounds of our Abbey?\"\n\nMartha replied from her seat at the window. \"Hard to count in the dark and rain, Father, but there's certainly more than twoscore of them, all rats, and armed to the fangs. Surely we can't overcome that many!\"\n\nAn old mousewife called Mildun began sobbing in a panic. \"We'll all be dragged out of our beds and murdered, I know we will, us and those poor little babes. Ooooooohhhhhhh!\"\n\nThe haremaid immediately issued a harsh scolding. \"Stop that right now!\"\n\nShocked into silence, Mildun shrank from the sharp reproof, listening intently as Martha continued in a stern voice. \"There's no call for that behaviour, marm, all you'll do is cause worry to everybeast. Don't let me hear an outburst like that from you ever again. Now if you've anything to say, then make it helpful. Don't be a beast of ill omen, and keep your voice down. We don't want the little ones taking fright. Do you hear me?\"\n\nMildun sniffed and mumbled into her kerchief. \"Sorry, Martha.\"\n\nAbbot Carrul turned grateful eyes to the haremaid. \"Thank you, miss. Well, the whole situation has changed now\u2014for the worse, I'm sad to say. An attack against such numbers of those savage rats is out of the question. So what do we do now? I'm open to helpful suggestions.\"\n\nForemole Dwurl raised a powerful digging claw. \"Tunnels owt, zurr, me'n moi moles can make ee gurt tunnel. Uz'll all be safe frumm ee vurmints then, oi reckerns!\"\n\nAs hope sprang anew in the Redwallers, they began chattering and clamouring aloud.\n\nToran silenced them with a sudden bark. \"A fine idea, sir, but let's not be too hasty. Yore plan calls for a bit o' discussion. Now one at a time\u2014you first, Father Abbot.\"\n\nCarrul folded both paws into his wide sleeves. \"Thank you, Toran. First, let me say this. Our Foremole's plan is a sensible one. The Dibbuns, and anybeast who chooses to go with them, will be safe from harm. As for myself, I must remain here where my duty lies. I could never desert my beautiful Abbey.\"\n\nThe ottercook seconded him. \"Nor I, Carrul. It ain't right leavin' Redwall wide open to Searats an' vermin. I stay!\"\n\nMartha struck the arm of her chair resolutely. \"Redwall Abbey is my home, the only home I've ever known. I'm not moving from here!\"\n\nEvery voice in the room was raised. \"We stay! We stay!\"\n\nForemole Dwurl wrinkled his nose apologetically. \"Oi bee's sorry oi menshunned et naow.\"\n\nAbbot Carrul placed a paw about the faithful mole's shoulders. \"You've no need to be sorry, friend, it was a good idea. The trouble is that nobeast wants to go now. So what do we do next?\"\n\nMuggum would not be denied his say. The molebabe waved the copper ladle, which had become his chosen weapon. \"Us'n's foights, zurr, that bee's wot us do. Foight!\"\n\nSister Setiva relieved Muggum of the ladle to stop him from giving anybeast a whack as he waved it about. \"Och, ye wee terror, hush now an' pay heed tae yore elders!\"\n\nToran picked up the molebabe and made an announcement to the assembly. \"This liddle feller's right, we must fight. But it won't be no kill-or-be-killed sort o' last stand. Oh no, mates, we'll fight an' defend the Abbey, stave off any attacks. Even if that means we'll have t'fight all summer long, until the Skipper brings his ottercrew back 'ere from the Northshores. Then together we can deal with those savages outside.\"\n\nSister Portula brandished the hooked window pole she had armed herself with. Normally a quiet and reserved old mouse, she surprised everybeast by calling out, \"Well spoken, Toran. That's the most sensible thing I've heard so far. We can be what we are, not warriors but defenders! We can stick it out and delay them all summer until help arrives from Skipper and his crew. But it will be no easy thing. Remember that we are under siege. Food will run short, drinks will have to be rationed, water cannot be used freely anymore. . . .\"\n\nBaby Buffle interrupted the good sister by piping up, \"Nonomorragerrabaffinwirrawater!\"\n\nMartha gave Shilly a puzzled look. \"What did he say?\"\n\nThe little squirrel grinned from ear to ear and did a somersault. \"Iffa water bee's short, Dibbuns can't not get baffed. Yeeheeheehee!\"\n\nNobeast could resist laughing along with the overjoyed babes.\n\nThe storm finally subsided to a light drizzle. Scratching the back of his neck with his silver hook, Raga Bol rolled out of Old Phredd's bed and exited the gatehouse. Swigging from a flask of grog, he listened to the whimpers and wails from the pond. Blowfly was keeping Badredd and his little gang hard at it. The Searat captain gazed up at the majestic grandeur of Redwall Abbey. What a sight! Anybeast would be mad to bother with ships when he could own a place like this. Smiling wolfishly, he shouted toward the Abbey.\n\n\"Yore goin' to meet Cap'n Raga Bol tomorrer, mousies!\"\n\n##\n\nMarching all night was a harrowing experience for the younger creatures. Saro and Bragoon, being used to such hardships, plodded doggedly on in silence. Fenna stumbled alongside them, her eyes constantly drooping shut. The squirrelmaid sorely regretted ever leaving Redwall and all its comforts. She did not know which she yearned for most\u2014sleep, food or water. Springald was of a like mind, trudging onward in a straight line with her four companions, keeping quiet and trying not to inhale too much dust.\n\nIt was a cruel and forbidding outlook, the wasteland stretching all around, flat, silent and gloomy in the nighttime darkness. After what seemed like an eternity, daylight showed on the eastern horizon, a pale, misty mixture of dove-grey and orange.\n\nBragoon watched the faint apricot edge of morning sun slowly rising. He spoke softly. \"That's a pretty sight, ain't it, mates?\"\n\nHorty hardly gave it a second glance. \"Pretty, y'say? Pretty bloomin' awful if y'ask me, wot. I'd swap the blinkin' lot for a drop of water! Can't we stop now? You said march by night an' sleep durin' the day. Well, there's the jolly old day, an' I'm pawsore an' weary. So let's lay the old heads down, eh chaps?\"\n\nSaro pushed him onward. \"Not just yet, we've got to keep goin' while 'tis cool. When the day gets hot, that's the time for sleep. The more ground we cover, the sooner we'll be out o' this wasteland. Keep marchin', don't stop now.\"\n\nNone of the travellers wanted to, but they carried on, knowing that it was the only sensible thing to do.\n\nBy midmorning, the sun was beating down remorselessly as small dust spirals danced on the hot breeze. There was still no sight of trees or streams amid the dun-hued wastes.\n\nBragoon finally halted. \"We'll rest here until late afternoon!\"\n\nSaro began setting up a lean-to with cloaks and staves, weighting the cloak edges down with pieces of rock.\n\nHorty raised a dust cloud as he slumped down. \"If I could only lay paws on the rotters who swiped our grub'n'water. By the left! I'd kick their confounded tails into the middle o' next season, wot!\"\n\nBragoon rested on his stomach in the small patch of shade. \"Don't think about it, mate, yore only makin' things worse.\"\n\nSpringald looked back at the ground they had covered. \"Funny how the land seems to wobble and shimmer out there.\"\n\nFenna curled up and closed her eyes. \"That's just the heat on the horizon. It's a mirage, really.\"\n\nSaro shielded her eyes, peering keenly at the spectacle. She nudged the otter, directing his attention to it. \"Don't look like no mirage to me, wot d'ye think, Brag?\"\n\nBragoon squinted his eyes and watched intently. His paw strayed to the sword which lay by his side. \"It might be just the heat waves, but it seems t'be movin' closer toward us. Then again, it could be the earth dancin'. Remember the ground shakin' like that the last time we was in this territory, Saro?\"\n\nThe squirrel never let her gaze waver from the shimmering. \"Aye, it made a rumblin' sound, too.\"\n\nHorty laughed wildly. \"Hawhawhaw! Just listen to 'em, chaps. We're in the middle of bally nowhere, bein' baked alive, not a flamin' drop t'drink or eat. Now what, the ground has to start bloomin' well dancin'! Am I goin' off me flippin' rocker, or is it those two ramblin' duffers, wot?\"\n\nBragoon and Saro exchanged glances, then went back to their watching.\n\nHorty, however, would not be ignored. Gesturing with his paws, he flopped his ears dramatically.\n\n\"They're tellin' me the ground's doin' a jig. An' here am I, without a pastie to shovel down me face or a bucket o' cordial to wet me parched lips! Ah, lackaday an' woe is the handsome young hare, languishin' out here an' losin' me mind! I'm goin' mad, mad I tell ye! Stark bonkers an' ravin' nuts! 'Tis the dreaded thirstation!\"\n\nSpringald shook her head. \"Thirstation? Shouldn't that be thirstiness, or just thirst?\"\n\nBragoon whispered to Saro. \"That couldn't be the earth dancin', or we'd have felt the rumbles.\"\n\nHorty continued with his tirade. \"Rumbles, rumbles? How could benighted buffoons such as you know about the rumblings of a sad tragic hare, whose life is bein' cut short by the contagious thirstation an' tummyrumbles?\"\n\nThe otter's tail caught him a firm thwack across the rear. \"Shuttup, young 'un, get to sleep an' quit yore shoutin'!\"\n\nHorty subsided meekly, but still muttered to have the last word. \"Beaten by the bullyin' Bragoon into shallow slumber. Goodnight, fair comrades, or is it good day, wot?\"\n\nWithin a short time, the three young ones were asleep. Sarobando was dozing, too, but Bragoon lay on his stomach, chin resting on both paws. Through slitted eyelids he scanned the wastelands to the rear of the lean-to. They drew closer. Now he could distinguish them, not as heat shimmers but as small, patchy bumps. Moving silently, betrayed only by odd puffs of dust, they edged nearer. Then they halted. One bump detached itself from the pack and advanced.\n\nSaro came awake as Bragoon touched her ear. He nodded toward the moving object, twitching his tail against the squirrel's footpaw. Saro prepared herself, knowing the signal well. One . . . Two . . . On the third twitch they both attacked. Springing in the air and leaping forward, both beasts threw themselves bodily on the thing. It squeaked aloud. Immediately the ground came alive. Squeaking and whistling, hundreds of small shapes raised an enormous dust cloud as they fled. The captured one wriggled and bit madly, but it could not escape its captors. It was disguised by a cloak woven from tough, coarse grass. Bragoon and Saro swiftly wrapped it into a bundle, trapping the beast within.\n\nSaro drew a small blade. \"Haharr, got ye, thief, be still or I'll slay ye!\"\n\nBragoon crouched with his sword poised, defending his friend's back against attack. Saro dragged the bundle inside the lean-to, rapping out orders to the trio, who were now awake.\n\n\"Grab ahold o' that. Jump on it if it tries to escape!\"\n\nSpringald and Fenna held the thing tight. Horty pulled off the covering. It was a small, goldish-brown mouselike beast with a long tail and a white-furred stomach. Temporarily stunned, it lay gazing up at them through huge, dark eyes.\n\nThe otter came bounding in; sword upraised he menaced it. \"Our food'n'water, where is it? Speak or die, robber!\"\n\nThe creature gave vent to a piercing cry. \"Feeeeeeeeeeee!\"\n\nThis was followed by a sound from outside, like hundreds of tiny drums.\n\nSaro stepped out of the shelter. \"Curl me bush, come an' take a look o' this, mates!\"\n\nA billowing dust cloud was rising from footpaws drumming the earth. When it settled, a hundred or more of the mouselike beasts stood facing them. They all wore grass cloaks about their shoulders.\n\nFenna whispered to Saro. \"Good grief, what do we do now?\"\n\nThe older squirrel answered quietly out of the side of her mouth. \"Say nothin'. Leave this to me, mate.\"\n\nBragoon emerged from the shelter, dragging his prisoner by the tail. Hoisting the creature up, he swung the sword of Martin. The otter's voice roared out. \"Give us back our food'n'water, or this 'un's a deadbeast! D'ye understand me? I'll slay 'im if'n ye don't obey!\"\n\nFor an answer, they once again set up a loud drumming with their footpaws: _Brrrrrrrrrrr!_ Then they stood silent, watching Bragoon as the dust settled.\n\nThe captive one glared fearlessly up at the otter. \"Chiiiiiiirk\u2014kill me! We of the Jerbilrats give nobeast water. Chiiik, sooner give our blood than water!\"\n\nSpringald was surprised. \"Rats? They're handsome little things. They've got beautiful, big dark eyes. They look far too nice to be rats!\"\n\nSaro turned fiercely on the mousemaid. \"Just shut yore mouth, miss, I don't care 'ow nice they look. They've told ye wot they are\u2014a rat's a rat, an' that's that. Hold yore tongue, an' leave the talkin' to Brag!\"\n\nThe otter yelled back at the massed Jerbilrats. \"Hah, so ye can unnerstand me. D'ye think I'm foolin'?\"\n\nHe struck with the sword, snipping a whisker from the Jerbilrat. As the drumming resumed, Bragoon raised his sword. \"Next one takes this robber's head off. Give us our supplies!\"\n\nFenna whispered urgently to Horty. \"He's not really going to chop off a defenceless creature's head, is he?\"\n\nHorty shrugged. \"Simple case o' survival out here. Either we get the rations back or we peg out an' perish, wot!\"\n\nThe Jerbilrat actually smiled at Bragoon. \"I die, one less mouth to feed\u2014that saves water. Kill me, riverdog.\"\n\nSaro sighed. \"Don't give us much choice, does 'e?\"\n\nThe otter let his sword drop. \"I never slew a helpless beast.\"\n\nSaro winked. \"I know, mate, we ain't murderers. Let me try.\"\n\nHauling the Jerbilrat up by its ears, she dealt it a slap. \"I know ye ain't givin' us our supplies back, but I'll slap ye round 'til sunset if'n y'don't tell me where water is.\"\n\nSaro made a wavy motion, describing a stream or river. \"Water, like this.\" She gave the beast a heavier slap. \"Talk!\"\n\nThe Jerbilrat shrugged. \"Two days southeast maybe, don't know.\"\n\nSaro struck again. \"Then find out, 'cos yore comin' with us!\"\n\nThe creature snarled. \"I'm Jiboa the Jerchief. I'll kill you\u2014I'm not afraid to kill, like that riverdog is!\"\n\nSaro took a length of rope, knotting it firmly around Jiboa's neck. She smiled grimly. \"Ole Bragoon's the merciful one, I ain't so soft 'earted. I don't take no lip from cheeky-faced rats. Now take us to the water, or I'll make ye wish my mate had killed ye!\"\n\nA swift kick to the rear set Jiboa moving. \"Your water might be gone now. Dancing earth can shift streams down great cracks in the ground.\"\n\nSaro flicked the rope against the back of his neck. \"Ah, go an' tell that t'the frogs. Ye just get us there.\"\n\nCancelling all plans to sleep by day, the travellers broke camp and set off into the dry, hot morn. They kept glancing back as the entire Jerbilrat pack continued to follow them. When Jiboa thrummed his footpaws, the rats drummed back in answer. He smirked at Saro.\n\n\"Feeeeeee! Old toughbeast, eh? Jerbilrats can go without water longer than you and the others. You'll weaken sooner or later. Then my rats will slay you all, you'll see.\"\n\nSaro jerked the rope sharply, causing Jiboa to fall on his own tail. She winked craftily at him. \"Funny 'ow ye can't do two things at once. Seems every time ye try, then ye fall over.\"\n\nJiboa scrambled upright. \"Stupid treejumper, I can walk'n'talk!\"\n\nSaro tugged the rope and pulled him over again. \"Wrong! Every time you say somethin' nasty, bump, down ye go. But if'n ye was to shout out that y'can see water, ye'd regain yore sense o' balance right away. Unnerstand?\"\n\nThere was neither shade nor shadow when the sun was directly overhead. Horty began complaining once more. \"Oh shed a tear for a thirsty young hare, an' if it's wet I'll drink it, wot. I say, you chaps, wouldn't you just love to wet the old whistle at a cool runnin' stream? If the odd fish swam by, then one could eat an' drink at the same jolly old time, wot. Phew, I'm so hot'n'dry that you could make a blanket of my tongue!\"\n\nFenna gave him a sharp nudge. \"You're showing us up in front of those Jerbilrats, moaning and whining like that. They'll think we're soft and weak. Now try to behave like a Redwaller, and stop all that nonsense!\"\n\nHorty stiffened his ears, saluted and stepped out smartly. \"Right, old gel, leave it to Hortwill Braebuck, Esquire. I'll sing t'the clod-faced old savages, wot, here goes!\"\n\nHorty, with his talent for making up songs as he went, launched into an insulting ditty about Jerbilrats. Fenna and Springald giggled as they joined in the refrain at the end of each verse.\n\n\"Oh a Jerbilrat's a creature,\n\nwithout one redeemin' feature,\n\nbeware of him, pay heed to what I say.\n\nHe'll sneak up on one quite sudden,\n\nand devour one's pie or pudden,\n\nan' he'll rob your bloomin' water anyday . . . Anyday!\n\nIf one ever meets a jerbil,\n\none must be extremely careful,\n\nan' keep one's drinks tight under lock and key,\n\nfor 'tis a widely held belief,\n\nthat the scruffy little thief,\n\nwill sup every single drop quite happily . . . Happily!\n\nFor a jerbil's just a rat,\n\nwho has never had a bath,\n\nso be careful that you stay upwind of him.\n\n'Cos the smell would blow one's hat off,\n\nor put any decent rat off,\n\nan' kill all the flies around a rubbish bin . . . Rubbish bin!\n\nJerbil manners are disgraceful,\n\nthey're so spiteful an' ungrateful,\n\nso arrogant an' sly an' so unjust.\n\nEvery ugly son an' daughter,\n\nis a stranger to bathwater,\n\njerbils wallow round all day beneath the dust . . .  \n'Neath the dust!\"\n\nHorty waved to the Jerbilrats, who were squealing and drumming their footpaws angrily. \"What ho, chaps, sorry I can't warble anymore for you. The old tongue's all swollen.\"\n\nSaro halted Jiboa until the others caught up with her. \"This sun is gettin' too much, let's take a rest, mates.\"\n\nShading their heads beneath the cloaks, they squatted on the hot earth. Dozing off was unavoidable in the intense heat. Late afternoon shadows were lengthening as Saro was jerked awake. Jiboa had gnawed through the rope. He sped off in a wide arc, trying to get back to the other Jerbilrats.\n\nThe squirrel chased after him, shouting out, \"Grab 'im, Horty, he's loose!\"\n\nQuick off the mark, the young hare gave chase. He was reaching out to grab Jiboa, when a piercing shriek came from above. _\"Kyeeeeeeeeee!\"_\n\nJiboa threw himself flat, but Horty was knocked ears over scut by a massive shape. A great buzzard\u2014chocolate-and-white plumed\u2014snatched Jiboa up in its fierce, hooked talons. It bore him off squeaking, high into the blue. Three more of the deadly predators swooped down on the Jerbilrat pack, each one seizing a victim, as the rest tried vainly to burrow into the dust. Then they were gone. The rest of them fled westward, thrumming and wailing fearfully.\n\nThen there was silence. Horty sat up, dusting himself off. \"Stifle me whiskers! Did you see the size o' those birds? That's a pretty awful thing to happen to anybeast, even a Jerbilrat. Fancy bein' scoffed by a flippin', flyin' feather mattress, wot!\"\n\nSpringald gazed around at the dusty, deserted plain. \"Those poor creatures, no wonder life in this area makes them hostile to others. I hate this dreadful place!\"\n\nFenna's voice sounded small and frightened. \"How are we going to find water now that we're completely alone?\"\n\nBragoon shouldered his sword wearily. \"Just press on. Jiboa knew there was water over this way. We've got t'keep goin'!\"\n\nThey staggered onwards, but as evening arrived Fenna collapsed. Saro rushed to her side, fanning her brow and rubbing her paws. The aging squirrel looked up at Bragoon. \"Pore young thing, the heat an' thirst have got to 'er. We don't even have a damp cloth t'wet 'er lips. Fenna'll die if'n we don't get some water soon.\"\n\nThe otter covered the little squirrel with his cloak. \"Right, mates, that's it. Horty, ye come with me! Spring, ye stay 'ere with Saro an' Fenna. Me'n Horty will find water, or die tryin'. If'n' we ain't back by tomorrer noon, ye'll know we never made it. But don't fret, we'll be long back by then with water!\"\n\nSarobando and Springald shook their friends' paws.\n\n\"Good luck, an' fortune go with ye!\"\n\n\"We'll be alright here, hurry back now!\"\n\nHorty bowed gallantly. \"To hear is to jolly well obey, marm!\"\n\nThe two comrades struck off into the gathering dark.\n\nSaro and Springald settled down to their vigil. After awhile, Fenna began murmuring as she tossed and turned feebly. \"A beakerful, is that all, Father Abbot? I'm thirsty . . . so very thirsty, Father.\"\n\nThe mousemaid cradled her friend. \"Hush now, Fenn, lie still.\"\n\nSoftly, Springald began singing an old lullaby, from when they were Dibbuns together at the Abbey.\n\n\"Peace falls o'er vale and hill,\n\nsilence fades the light,\n\nmoon and stars watch over\n\nlittle ones by night.\n\nDawn will send the day bright,\n\nlarks will sing for thee,\n\nstreams of slumber flow now,\n\nround this babe and me.\"\n\nSaro smiled. \"That's a pretty song, I remember it from Redwall long ago. Ol' Sister Ormel used t' sing it in the dormitory. Happy days, Ormel was a good ol' mouse.\"\n\nSpringald sniffed. \"I learned it from her, too. Sister Ormel passed on three winters back. She was well loved.\"\n\nAs they nursed Fenna, in hostile country, far from their beloved Abbey and its friendly creatures, Saro and Springald sat silent with their thoughts of Redwall.\n\nHorty staggered gamely onward, though his paws were wobbling and his body bent with fatigue. Bragoon was in slightly better shape, but every step he took was an effort. Side by side they stumbled along through the night. Then the young hare tripped and fell, bringing the otter down with him.\n\nThrough cracked and swollen lips, Horty mumbled, \"Beg your pardon, old lad, tripped over a confounded bush. Wonder what oaf left it there, wot.\"\n\nHe grunted as Bragoon scrambled over him and grabbed a pawful of leaves. Thrusting his nose into them, the otter whooped. \"Wahoo! This ain't no bush, mate. 'Tis a big clump o' comfrey. There's water nearby, I'm sure of it. Water!\"\n\nLeaping up, they plunged forward with renewed hope and energy. The otter suddenly ground to a halt, pulling Horty back. He pointed ahead, to where a soft glow emanated from behind the bulk of a widespread willow tree. Beyond that, the trickle of running water could be clearly heard.\n\nDrawing his sword, Bragoon thrust the young hare behind him, uttering a quiet caution. \"Stick close t'my back, an' don't do anythin' foolhardy. There's a fire burnin', t'other side o' yon tree. I 'ope there's friendly beasts sittin' round it.\"\n\nHorty snorted. \"Fat chance in this neck o' the woods, pal. All we've met is bounders'n'cads since we climbed those cliffs. Huh, friendly y'say, prob'ly so friendly they'll chop off our blinkin' heads on sight, wot?\"\n\nThe otter's paw clamped over Horty's mouth. \"Stow the gab an' stay behind me, we'll soon see!\"\n\nThere were six reptiles in all\u2014two large frilled lizards, three fat toads and a grass snake\u2014lounging around the fire. They were grilling a mess of bleak and minnow on green twigs. Having made a bit of noise as they approached, both travellers were expected. One of the lizards stood barring their way to the water, which appeared to be a small streamlet flowing away into a dense pine forest. The rest of the reptile crew crouched, ready to back the lizard up.\n\nBragoon nodded civilly to them, noting that all eyes were on his sword. \"Evenin' to ye, we've come for water.\"\n\nOne of the lizards sniggered nastily, trying to imitate the otter's voice. \"H'evannin' to ye, we've a-come f'waterrrr!\"\n\nHorty noticed several large gourds of water nearby. \"That's the jolly old stuff, water, you know, that pleasant liquid which is rather nice t'drink. I say, those tiny fish smell rather toothsome, wot. Don't suppose you'd like to donate a few to a worthy cause, a hungry but honest hare, eh?\"\n\nThe reptiles edged around, circling the pair. The largest of the lizards picked up a crude, flint-tipped spear, pointing it at Bragoon.\n\n\"Watersss not a free, iz all oursss. You wanta fisssshes an' drrrrrink, give usss bright a blade!\"\n\nIgnoring him, the otter turned to Horty. \"I don't know wot it is wid the beasts in this country, but they seem t'think we're dim-witted. Our stream, our water, our fish. While pore young Fenna's dyin' for a drop o' water. I've taken about enough of all this claptrap, mate. Ye take my sword, don't do anythin', just stay there, that's an order!\"\n\nHorty took the weapon and saluted. \"As y'say, sah! An' pray, what d'you intend doin', if one may ask, wot?\"\n\nA slow, savage grin spread across the otter's tough face. \"Nothin' much, I'm just goin' t'get us some water.\"\n\nRoaring out a warcry, Bragoon launched himself at the reptiles. \"Make way fer Bragoon o' Redwaaaaaallllll!\"\n\nHorty could not have moved if he had wanted to. He stood wide-eyed with shock, watching six reptiles take the most fearsome beating he had ever witnessed.\n\nBragoon broke the spear of one of the lizards over its head, then picked the reptile up and hurled it into the stream. He went at the others like a madbeast. Flinging himself through the air, he butted a toad heavily in its enormous stomach. As air shot out of the toad in a whoosh, he rudderwhipped it hard, thrice across the head, laying it senseless. He turned and grabbed the other lizard, running it forcefully, snout on, into the willow trunk. Seizing the grass snake, he used it like a flail, cracking the jaws of the other two toads with the snake's head. Bragoon leaped high. Still holding the grass snake, he landed on the two toads' stomachs, then booted all three toads into the stream. The other lizard sat facing the tree trunk, nursing its broken snout. Knotting the snake around its neck, the otter looped them both to a low branch.\n\nDusting off his paws and breathing heavily, Bragoon took the sword from the astounded young hare. Putting the swordpoint at the lizard, he growled, \"In the future, mind yore manners an' be polite to visitors!\"\n\nThe lizard clutched onto the coils of the senseless grass snake around its neck. The snake was looped to the branch above, keeping the lizard on tip-paw. Bragoon put his face close to the reptile and roared thunderously, \"Yore all deadbeasts if'n I clap eyes on ye agin! D'ye hear me, slimeguts?\"\n\nDipping a paw into one of the gourds, the otter tasted the water and spat it out in disgust, then called to his companion. \"Git yore gob out o' that stream, young 'un. Wash these things out an' fill 'em wid fresh water. I'll get the fish.\" He stowed the sword over his shoulder. \"Don't dillydally, mate. Fenna an' the others'll be waitin'. Put a move on!\"\n\nHorty hurried to do Bragoon's bidding, holding a conversation with himself as he rinsed and filled the containers. \"Seasons o' soup'n'salad, 'pon my word! That crackpot must've been a right terror in his younger days, wot? Curl me crusts! A chap'd do well to stay the right side o' that otter, he's a bloomin' one-beast army!\"\n\nBragoon's voice cut sharply into his meanderings. \"Stop chunnerin' an' get 'em filled, ye great gabby windbag!\"\n\nHorty filled the last gourd with one paw, saluting furiously with the other. \"Chunnerin', sah, who, sah, me, sah? No, sah, not never, nohow. Last one filled, sah, all correct, wot wot!\"\n\nBragoon had chopped branches with his sword. He and Horty carried the gourds, strung on the wood and yoked across their shoulders, two to each of them. They had drunk sufficient water and chewed on the cooked fish as they trekked back to their friends.\n\nSighting the lean-to in dawn's pearly light, they dashed forward, slopping water, with Horty yelling, \"Toodle pip there, you idle lot, here come two handsome water carriers. I say, we've got fish, too! Jolly good, eh?\"\n\nThere was no reply from the shelter. Bragoon hurried forward, only to find it deserted.\n\n# [BOOK THREE  \n\"We lived one summer  \ntoo long\"](Loamhedge-toc.html#TOC-32)\n\n##\n\nMorning sunlight filtered like molten gold through the gatehouse. Raga Bol picked his teeth with the silver pawhook, spitting a bone back onto the remains of a well-grilled fish, which he had breakfasted on.\n\nThe Searat captain was in a expansive mood, having slept dreamlessly without any giant stripedog nightmares. The whole incident surrounding Lonna had faded into the background since his arrival at the Abbey. He felt a sense of power, sheltered by the monumental red walls which he knew would be his new home. No more scouring the cold northeast seas. This was a place of fair weather, a fortress from where he could rule all Mossflower. Lord Raga Bol, he liked the sound of his new title.\n\nBadredd quaked with pent-up tension as he awaited the Searat's verdict on his cooking. Blowfly stood behind him, twirling his knotted rope's end. Relief flooded through the small fox at the sound of the captain's coarse but satisfied chuckle.\n\n\"Haharr, I've eaten worse an' lived! Wot kinda fish was that 'un, matey? Wot 'erb did ye use on it, eh?\"\n\nBadredd answered promptly. \" 'Twas a grayling, sir, grilled with button mushrooms an' dill. I did it special.\"\n\nBol patted his stomach. \"Graylin', that's a nice-soundin' name. Blowfly, wot are we goin' t'do wid this cook\u2014flog 'im to a jelly wid yore rope's end or gut 'im wid this 'ook?\"\n\nBlowfly smiled, not a pretty sight. \"Gut 'im, Cap'n, go on!\"\n\nThe hook lunged out, capturing Badredd around his neck. He was dragged forward until Bol was breathing in his face.\n\n\"Make yoreself useful round 'ere, me liddle graylin'. Clean this place up, scrub it out an' make the bed. Blowfly, you stay 'ere, tickle 'im up wid yore rope's end if'n 'e slacks!\"\n\nThrusting both scimitar and stiletto in his sash, the captain swaggered out onto the sunlit lawn. \"Glimbo, rally the crew. 'Tis time we went for a parley wid our new friends!\"\n\nAll night long, Foremole and his molecrew had been carrying rubble up to the dormitory to be used as extra defence material. Martha sat close to the window with Toran and Abbot Carrul.\n\nGranmum Gurvel laid breakfast out on the windowsill for them. \"You'm bee's h'eaten ee brekkist naow, 'tis gudd furr ee!\"\n\nThe trio had already laid their plans. Toran poured honey and beechnut flakes over his oatmeal, pointing to the gatehouse. \"Stand ready, everybeast, they're comin'!\"\n\nRaga Bol sauntered up with twoscore of Searats, as though he was out for a morning stroll. He waved up at them.\n\nToran grunted. \"Don't look like they're goin' to attack right now.\"\n\n\"It wouldn't pay to!\" Martha muttered grimly, reaching for one of Redwall's latest pepper bombs. Abbot Carrul stayed silent, polishing his glasses nervously on his habit sleeve.\n\nA Searat brandishing a rusty axe snarled up arrogantly at the dormitory windows. \"Get yerselves out 'ere, or we'll come in an' drag ye down!\"\n\nDrawing his scimitar, Raga Bol dealt the Searat a swinging blow to the jaw with its bone handle. He placed a sea-booted footpaw on the sprawled-out rat and spoke reprovingly. \"Tut tut, I'm surprised at ye, mate. Is that anyways to be addressin' gennelbeasts?\" Returning the blade to his sash, the Searat captain lectured the rest of his brutish crew. \"Mind yore language when ye talks to the goodbeasts up there, that's an order!\"\n\nHe winked broadly and turned away from them, performing a flourishingly elegant bow. His gold fangs glinted as he smiled up at the dormitory windows. \"My 'pologies, an' a good day to ye all, messmates. Me name's Raga Bol, fer want of a better 'un. I'm 'ere to parley wid yore cap'n. 'Twould be a kindness if'n 'e'd speak t'me.\"\n\nAbbot Carrul showed himself. \"I am Father Abbot Carrul of Redwall. What exactly do you want, sir?\"\n\nRaga Bol put his head to one side, almost managing to look coy. \"Ho, a bit o' this an' a bit o' that. Nothin' fer you to bother yore dear old grey 'ead about, Father Abbot. I'm nought but a simple beast who likes pretty trinkets.\"\n\nToran felt that Carrul had taken enough verbal fencing. Recalling the arrow which had been shot to slay his Abbot, he came forward, placing himself in front of Carrul. In one paw he held a long cook's knife; in the other, a pepper bomb.\n\n\"Wot would ye like, silvertongue\u2014a bit o' this or a bit o' that?\" He indicated both weapons as he spoke. \"Make yore choice, 'cos that's all ye'll get from us. Redwallers aren't born fools. We know scum, even when they try to talk fancy!\"\n\nRealising that the otter could not be cajoled or wheedled, Raga hurled himself at the Abbey door, hacking at it with his sabre and knife and yelling to his Searat crew, \"Attack! Break this door down!\"\n\n\"Redwaaaaaallllll!\" A warcry rang out as the defenders fired slingstones and pepper bombs down upon the foebeasts. A slingstone pinged off Raga's jaw, leaving it gashed.\n\nHe retreated from the door, bellowing, \"Back! Out o' their range. Back!\"\n\nThey stumbled back across the lawn to where they could see missiles coming and better dodge them.\n\nMartha was shocked but elated. It had all happened so fast: one moment she was listening to the talk going back and forth, the next moment she was screeching like a wildbeast and madly launching off slingstones. She held her trembling paws up to her eyes, willing them to be still.\n\nToran winked at her. \"Well done, beauty!\"\n\nHis attention was distracted by Raga Bol, shouting, \"Ahoy there! Is that the way ye treat creatures wot comes in peace? Aharr, ye wretches, I'll show ye the Searat way o' fightin' back. I'll burn ye out!\"\n\nThe Searat captain marched off, back to the gatehouse. Some of his crew were nursing wounds, while others fled blindly, their eyes streaming as they sneezed uncontrollably and headed for the pond.\n\nMartha could feel panic welling inside her. She clasped Toran's paw. \"Will they really try to burn us out?\"\n\nSeating himself on the windowsill, the ottercook stared down at the Abbey's main door, directly below. \"Aye, I thought they'd get around to that, sooner or later. But the Searats' plan won't work. How much of that soil an' rubble is there, Dwurl?\"\n\nSpreading his hefty digging claws, Foremole shrugged. \"Much as ee loikes, zurr. We'm gotten gurt 'eaps o' durt'n'rubble, hooj marsess uv ee stuff!\"\n\nThe Abbot looked over his glasses at Toran. \"What are you thinking of, friend?\"\n\nThe ottercook turned from the window. \"Our Abbey is built o' stone, Father. Ain't many ways they can burn an entrance in. The big Abbey door is the one way. If that went afire, we'd be lost, sittin' on the other side of it, waitin' for the door to burn down. So I plan on blockin' it completely. We'll do it right now. Ain't no sense in losin' time, so we'd best work hard'n'fast. Pay attention, everybeast, this is the plan. . . .\"\n\nRaga Bol's mood had turned sour. He had supposed that his show of force would have gained him an easy victory rather than a shameful retreat. But it had become apparent that the Abbeybeasts were not afraid to fight, no matter how great the odds. He retired to the Abbey pond where he sat sullenly watching those of his crew who had been struck by pepper bombs dousing their heads in the shallows. Flinky and the rest of Badredd's gang were there, ineptly trying to catch another grayling. The captain took his spleen out on them, booting Flinky headfirst into the water.\n\nThe stoat rose spluttering, as he tried to placate the irate Searat. \"Sure we was only tryin' to catch a fat ould fish for yer 'onour's supper. Ain't that right, mate?\"\n\nHalfchop nodded enthusiastically. \"Kachunk!\"\n\nRaga Bol drew his scimitar menacingly. \"Gerrout o' me sight, ye witless idiots, make yoreselves scarce. Now!\"\n\nAvoiding the keen blade, Flinky and the rest fled the scene.\n\nFerron, the gaunt rat, slung a flat pebble, bouncing it over the pond surface. \"I wouldn't give 'em 'til sunset, Cap'n. I'd burn those beasts out now!\"\n\nBol was loath to destroy any part of his new home. He looked to Wirga, his Seer. \"Wot say ye, old one?\"\n\nWirga was drawing patterns in the banksand with a stick. She shrugged. \"If the sons of Wirga were here, they could use their darts on anybeast who showed at the windows.\"\n\nRaga Bol glared at her. \"But they ain't 'ere, are they? So do we burn 'em out, or have ye got a better way?\"\n\nThe Seer sensed the danger in his tone. She made her reply diplomatically. \"Set a fire in full view of the windows. Then send a messenger to give them one last warning. The sight of flames should alter their minds.\"\n\nThis was the answer the captain desired. He gave orders. \"Ferron, Glimbo, gather wood an' get lamp oil. Then set up a blaze on the lawn, where they kin see it. Wirga, take Chakka wid ye. Go an' warn those fools wot'll 'appen if'n they don't surrender t'me!\"\n\nBadredd had just finished mopping the gatehouse floor clean and was about to unbend when Blowfly slapped his rump smartly with the rope end.\n\n\"Yew missed a corner be'ind the door!\" The fat Searat caught Flinky peering in through the open window at him. \"Now then, slysnout, wot do yew want?\"\n\nThe stoat smiled apologetically. \"Beggin' yore pardon, sir, but 'tis the cap'n, 'e wants ye down by the pond.\"\n\nBlowfly gave Badredd another sharp rap. \"This place better be shipshape when I comes back, or I'll flay the back offa ye. Ahoy there, stoat, lend 'im a paw. I kin find me own way t'the pond.\" Blowfly waddled off, twirling his rope end skilfully.\n\nThe small fox tossed Flinky a damp rag. \"You start on the windows, I'll see t'the floor.\"\n\nThe stoat pulled him upright, whispering urgently. \"We're gettin' out o' this place. Come on now, while they're all at the pond we can make a run fer it!\"\n\nBadredd gazed dumbly at Flinky, as if not understanding what he had said. The stoat grabbed the cleaning rag from him and flung it away. \"Don't stand there wid yore jaw flappin'! Are ye comin' wid us, or d'ye like bein' a slave? The rest o' the gang are hidin' by the gate, waitin'. All the Searats are down by the pond, there's not a sentry on guard at all!\"\n\nBadredd's limbs began trembling. \"But wot if they catch us?\"\n\nFlinky could not keep the contempt out of his voice. \"Huh, some grand ould leader ye turned out t'be. Yore better off stayin' here if'n yore too scared. We're goin'!\"\n\nHe ran from the gatehouse to where the others were waiting. \"Get that gate open, quick now!\"\n\nSoon Badredd came running from the gatehouse to join the escapers, shouting out, \"Wait for me, mates. I'm comin', too!\"\n\nA moment later they were off, dashing south down the path and cutting off east into Mossflower Wood, leaving the main gate swinging lazily in the summer breeze.\n\nRaga Bol was putting an edge to his blade on a stone he had found on the pond's edge. He glanced up sourly at Blowfly's approach. \"Wot do y'want, eh?\"\n\nThe fat Searat saluted with his rope's end. \"Dat liddle stoat, the gabby one, 'e said yew wanted ter see me, Cap'n.\"\n\nBlowfly dodged a swipe from the silver hook as Bol roared, \"I never said no such thing. Get back to that gate'ouse an' see wot they're up to. Go on, move yer fat bum!\"\n\nHe glanced up despairingly at the sight of Wirga and Chakka arriving back from the Abbey building. Both were caked from eartips to tails in a mixture of soil, rubble and sloppy debris, which clung to their bodies. The Searat captain shook his head in disbelief. \"Well, make yore report. Wot 'appened to youse two?\"\n\nWirga spat out grit. Pawing soil from her ears, she hawked and coughed to clear her mouth. \"They didn't give us a chance to speak. We went round there like thee told us, but they wouldn't listen, would they Chakka?\"\n\nShe waded into the pond and began washing the mess off as Chakka continued. \"They was pourin' muck outta the winders, Cap'n. We tried to give 'em yore warnin', but a crew o' those moles lobbed a big 'eap o' rubble down on us. Not only that, but they kept tippin' stuff down until we was knocked flat. We 'ad to dig our way out afore we was buried. It looks like they're coverin' the Abbey door, so we can't put a light to it, Cap'n. Those beasts are killers, we was near suffocated!\"\n\nRaga Bol put aside sharpening his scimitar. \"Have the others lit the fire on the lawn yet?\"\n\nWirga emerged dripping from the pond. \"Aye, the wood is burning.\"\n\nRaga Bol hurried up from the pond, past the orchard and out onto the lawn at the front of the building where he could take in the full scene. He could see the top few timbers of the Abbey's main door. The rest had disappeared under a heap of debris, which was still pouring out of the window, forming a great hill of rubble, which completely blocked the doorway.\n\nQuivering with rage, Bol strode up to the fire, which his crew was fuelling with logs, branches and planks. He smote at the blazing wood with his scimitar, scattering it onto the lawn. \"Glimbo, git yoreself over 'ere! Stop burnin' the wood, we'll need it to pile up agin that load o' rubble!\"\n\nThe one-eyed Searat, who had been enjoying the blaze, saluted his leader quizzically. \"Ye don't want a fire then, Cap'n?\" He recoiled, his face now splattered with spittle from the captain's furious rant.\n\n\"Can't ye see they've blocked the doorway, fool? Rubble won't burn, we need that wood to pile up agin that 'eap. We can climb up on it through the winders!\"\n\nRaga Bol sat down on the lawn, chopping at the grass with his blade and shouting out, \"Can't ye use yore brains? 'Ave I got to do all the thinkin' round 'ere?\"\n\nBlowfly came plodding up from the gatehouse. \"Cap'n, the vermin gang are gone. The gate's open, they must've escaped!\"\n\nBol gritted each word out slowly, as if he was speaking to a dim-witted infant. \"Well, go an' bring 'em back! Glimbo, you go wid 'im, an' don't show yer ugly faces back 'ere widout every last one of 'em. Go!\"\n\nMartha had heard every word. She smiled at the Abbot. \"Well, that's a few less to bother us.\"\n\nSister Setiva ducked her head aside as a stretcher load of debris hurtled out of the window space. \"Och, but did ye hear yon Searat? They're goin' tae make a ladder tae scale the heap o' muck. Whit are we to do now?\"\n\nJust then, Foremole Dwurl clumped into the dormitory, his face wreathed in a happy smile as he announced, \"We'm no need to wurry o'er water nomores, zurr. Moi molers h'un-covered a gurt well, daown in ee cellars!\"\n\nSister Setiva pursed her lips. \"Och grand, but ah don't see how that's goin' tae help us fight Searats off!\"\n\nToran shook Dwurl by his muddy digging claw. \"That's a spot o' luck, me ole mate! Keep throwin' rubble out o' the windows, an' tell yore crew to start bringin' up pails o' water, as much as they can!\"\n\nThe ottercook winked roguishly at Martha. \"We'll see 'ow far the rats get, tryin' to scale a mudhill.\"\n\nThe haremaid clapped her paws gleefully. \"Very good, Toran, what a splendid idea! Gurvel, keep making those pepper bombs. In a day or two those Searats will wish they'd never heard of Redwall Abbey!\"\n\nLittle Muggum flung a pawful of debris moodily out the window. \"Hurr, they'm founded watter. Oi 'speck uz Dibbuns bee's a getten barthed agin.\"\n\nSister Setiva patted the molebabe fondly. \"Och aye, but ye can throw the soapy bathwater oot o'er the rats!\"\n\nWithin the hour, Old Phredd had penned a poem about what he envisaged. Martha laughed along with the rest as the ancient Gatekeeper read it aloud to the defenders.\n\n\"They won't leave this Abbey, all filthy and scabby,  \nwhen this war is done.\n\nOur foes will retreat, looking clean nice and neat,  \nevery Searat's son.\n\nOh won't it be splendid, when this siege is ended,  \nlike roses they'll smell,\n\nwashed by bathwater sweet, looking fresh in defeat,  \nas away they run.\n\nCome one and come all, dirty vermin we'll call,  \nshould you need a scrub,\n\ndon't worry or fear, we've got bathwater here,  \nyou may take a tub.\n\nWash the mud out your ears, so you'll hear us my dears,  \nfor 'tis truth to tell,\n\nyou will know how it feels, with a clean pair of heels,  \nfrom a Redwall Farewell!\"\n\nRaga Bol watched as Ferron and Rojin barred and shut the big wallgates. Wirga followed him inside the gatehouse, waiting silently on his command. The Searat slumped down on Old Phredd's bed, speaking his thoughts as he gazed up at the ceiling.\n\n\"Tonight, once 'tis dark, we attack. You stay 'ere wid a few o' the crew. Light a fire, make lots o' noise, they'll think we're all round by this gate'ouse. I'll take the rest an' storm the Abbey by surprise. Tell Ferron to gather all the wood that ain't burned. We'll need it to get up the rubble. I'll be inside afore that ole Abbotmouse knows it. I'll teach those bumpkins to defy Raga Bol. The floors in there'll be awash wid blood by the time I'm done!\"\n\nWirga ventured some questions. \"Do I leave the gates locked, Cap'n? What if Blowfly an' Glimbo return with the prisoners? Or my three sons, what if they return with Jibsnout?\"\n\nRaga Bol looked sideways at his Seer. \"They got paws'n'voices, ain't they? Let 'em bang on the gates or call for ye to open up.\"\n\nWirga humbled her tone, knowing she was touching on a delicate subject. \"Jibsnout and my sons are gone overlong now, they should have returned. Thou wouldst know then if the big stripedog still lives.\"\n\nBol snapped up off the bed. \"Wot do I care about yore whelps, or Jibsnout, eh? I gave 'em a job to do, they should be doin' it. As fer the stripedog, mention 'im agin an' I'll let daylight through yore skinny carcass. Now get out an' give my orders to Ferron an' the crew. We attack tonight!\"\n\n##\n\nHorty looked around blankly, spreading his paws. \"Gone? Where in the name o' seasons have they gone to? They were supposed t'wait here, wot!\"\n\nBragoon held up a paw. \"Quiet, mate, don't move, stay still!\" He cast around, starting in a small circle and going wider. \"If ye go shufflin' about with those big paws o' yores, this dusty ground'll get disturbed. Ahah! Here's their tracks, aye, an' one other, too. Quick, mate, grab all the gear an' foller me!\"\n\nHorty gathered up the cloaks and staves which had formed the lean-to. Burdened by this, plus the two gourds of water on his yoke, he staggered after the otter. Bragoon, having shed his share of the water, was forging ahead swiftly.\n\nHorty protested. \"I say, old bean, that's a bit wasteful, ain't it, leavin' behind good water that you had to carry half the blinkin' night?\"\n\nBragoon kept his eyes on the trail as he answered. \"Can't stop now, got t'get to our mates fast\u2014'tis a matter o' life an' death. Keep up as best ye can!\" Hurrying forward, the otter began emitting an odd, piercing whistle.\n\nHorty plodded on, twitching his ears in disapproval. \"Huh, matter of life'n'death, an' the bounder's whistlin' if y'please? Wouldn't mind, but it's not even a flippin' catchy tune. The bally beast's brains have gone to his rudder if y'ask me, wot!\"\n\nEventually Bragoon spotted the three figures, out on the arid plain. Springald and Saro were shuffling along facing backwards, supporting Fenna. Closely following the otter and his three companions, an adder was slithering, its forked tongue flickering out, sensing prey, the fatigued trio ahead of it. Hearing the sound of Bragoon's high-pitched whistle, the snake turned, bunching its coils and hissing viciously. Not as big as some serpents the otter had encountered, it was a male, just beginning to get its growth. But angry and deadly enough to deal a fatal bite with one speedy strike of its venomous fangs. Continuing to whistle, Bragoon drew his sword and moved closer, making ready to fight if necessary. The otter smiled grimly. His ploy had worked: the hunting adder had now become the hunted, its fate sealed.\n\nBefore the old warrior could strike, the young hare bawled out a warning. \"Look out, pal, here come those blinkin' buzzards again!\"\n\nLike thunderbolts out of the blue vaults of morning, two large adult birds whizzed down. With total disregard for the snake's venomous fangs, they struck their quarry with lightning speed. The murderous beaks and talons of both buzzards snuffed out the adder's life with savage skill and ferocity. The dead snake was still writhing in the dust whilst they continued their frenzied attack. Then it went still, and the hawks screeched out their victory cry.\n\nShielding her eyes against the sun, Saro watched the predators bearing their limp prey off into the cloudless sky as Bragoon and the hare approached her.\n\nShe shook her head ruefully. \"I wish I'd learned t'do that whistle. Never could get the hang of it, though. Burn me brush! Is that water you've got there, Horty?\"\n\nShedding all his trappings, the young hare sank wearily down. \"Indeed it is, marm, but I'm afraid you'll jolly well have to com'n'get it for yourselves. I'm whacked out!\"\n\nBragoon took the yoke from him and sat it across his shoulders, then lifted the two gourds. \"Ye did well, mate, take a rest now.\"\n\nThe elderly squirrel and the two Abbeybeasts sat amid the wasteland dust, gulping down the life-giving liquid. The otter soaked a cloth, allowing it to dribble into Fenna's mouth. He wiped her face with the damp material, cautioning them, \"Drink slower, or ye'll be sick. This young un'll be right as rain soon. So, wot 'appened, mate?\"\n\nSaro looked up from the gourd. \"Just afore dawn, I scented the adder. Huh, I can sniff those things a mile off!\"\n\nShe continued drinking as Springald took up the tale. \"We knew it was somewhere close, stalking us. It was too dangerous to stay inside the lean-to, the snake would've found us. So we sneaked quietly off, but the adder saw us and came right on our track. I've never seen an adder before\u2014horrible beast! I was scared clean out of my wits. Good job you found us in time, we couldn't have carried Fenna much further. And, Bragoon, will you teach me that whistle? It saved our lives!\"\n\nThe otter lifted Fenna onto his back. \"Some other time, miss. Let's get this 'un into the pineforest shade. We found a stream over that way. I'll take ye to it.\"\n\nSaro closed her eyes dreamily. \"A pine forest an' a whole streamful o' beautiful babblin' water. Lead on, mate!\"\n\nThey entered the pines when it was midday. Horty raced ahead until he found the stream. He ran toward it, turning his head to shout, \"This is the place, chaps! Hawhaw, wait'll I tell you what old Brag did to a gang of bullyin' reptiles last night. He gave 'em the towsing of their lousy lives, he . . . nunhhhhh!\"\n\nWithout paying attention, Horty had run full head-on into a thick, low pine branch. He was laid flat out, unconscious.\n\nSaro ran to him and lifted his head. \"Stone-cold senseless! That makes two we got to nurse now. Why didn't the lop-eared gallumper look where he was goin'?\"\n\nThe remainder of the afternoon was spent beside the stream. Springald looked after her two friends whilst the older pair went foraging for food. It was so pleasant in the shade of the tall pines. Besides tending the invalids, the mousemaid had time to paddle and wash in the stream. It was a cool and peaceful spot with sunlight and shadow dappling everywhere. Fenna was recovering nicely when Bragoon and Saro returned. The two old campaigners brought with them wonderful chestnut-coloured mushrooms, wild onions, dandelion buds and a variety of edible roots and berries.\n\nBragoon was heartened by the sight of the squirrelmaid. \"Feelin' better, eh, beauty? Well, we can't light no cookin' fires in a pine forest like this, 'tis too risky. Do ye fancy a nice salad, miss?\"\n\nFenna watched the otter chopping everything finely with his swordblade. \"Salad would be perfect, thank you!\"\n\nThe moment the aroma of freshly cut food assailed his senses, Horty revived. \"Oh goody! I say, you chaps, please pass the salad. Owchowchoooh! Me flippin' bonce is splittin'. Can y'see any of me brilliant young brains leakin' out, wot?\"\n\nFenna could not stifle a giggle. \"Oh, poor Horty, you've got a lump like a boulder, right twixt your ears. I'm sorry for laughing, it must be very painful.\"\n\nThe young hare winced when he touched the large swelling. \"Painful ain't the word, Fenn old gel, it's absobally agonisticful. Don't think I'll last the day out, actually. Don't shed too many bitter tears when I turn me paws up an' peg out. 'Twas all done bravely in the line of duty. Wot!\"\n\nSaro inspected the injury. \"Hah, it looks like a duck egg growin' out o' yore skull. Don't worry, though, you'll live. I've got just the thing for that. Sit still an' eat yore salad while I go an' make a poultice.\"\n\nShe spent some time at the stream, gathering certain things and soaking them in the water. On her return, the aging squirrel tore strips off a cloak for binding.\n\nHorty pulled back apprehensively. \"Don't hurt a dyin' young beast in his final moments. Be merciful, marm!\"\n\nBragoon held the hare's paws as Saro worked. She tweaked Horty's whiskers whenever he moved. \"Be still, ye great ninny! This is a compress of duckweed, dock, watercress, sainfoil an' streambed mud. Twill do ye a world o' good!\"\n\nWhen she had finished, the others had to turn away their faces to keep from bursting out into laughter. Horty sat dolefully munching salad. Atop his head sat a high turban of cloak strips, herbs and mud, secured with a tie beneath his chin. Both of Horty's ears flopped out at the sides. He glared at Bragoon, who was biting down on his lip to contain a guffaw.\n\n\"What's the flippin' matter with your face, chucklechops? D'you find somethin' funny about a wounded warrior, wot wot?\"\n\nThe otter brought himself under control. \"Who, me? No, mate, but I wouldn't go near any bumblebees if'n I was ye. They might be lookin' fer a new hive! Hohohohoho!\"\n\nSeeing there was no salad left, Horty rose regally and stared down his nose at the mirth-struck quartet. \"Tut tut, I shall be carryin' on alone, without any aid from those I once called friends. Huh, bunch of whinnyin', witless woebetides. Fie upon you all, say I!\" He stalked off in high dudgeon, his turban dressing awobble as he stooped to avoid branches.\n\nFenna grasped her sides, tears of laughter rolling down both cheeks as she gasped out, \"Heeheehee, come on, I'm, haha, well enough to travel now. Ohahahahhh! We'd better go along with him just in case he, heeheehee, backs into a sharp branch, and we, hahahahaaaa, have to tie a turban to his tail. Whoohoohoohoo!\"\n\nThe pine forest was a vast area. As evening fell, it became dark, swathed in a gloomy, green light. Horty was still not talking to anybeast, but the urge to utter some noise was so great that he struck up a mournful dirge.\n\n\" 'Tis a sad lonely life, I have oft heard it said,\n\nto go wanderin' about with this wodge on one's head,\n\nfor I travel alone o'er desert an' lea.\n\nWhy, even the midges and ants avoid me,\n\nwhile the ones I called pals an' the comrades I know,\n\nall laugh 'til their rotten, cruel faces turn blue.\n\nThere's a grin on the gob of each pitiless cad,\n\nas they scoff at the plight of a poor wretched lad,\n\nbut I'll carry on bravely, I won't weep or cry,\n\nan' I'll have my revenge on 'em all when I die.\n\nMy ghost will sneak up while they're laid snug in bed,\n\nan' I'll hoot spooky whoops through this thing on my head.\n\nThen they'll cry out 'Oh Horty, forgive us, please do'\n\nas my spirit howls loudly . . . 'Yah boo sucks to you!' \"\n\nWhen night fell, Horty broke down and wept inconsolably. Springald crept through the gloom and found him sitting on a log, feeling sorry for himself. She put a paw around him.\n\n\"Horty, don't cry. What's the matter? This isn't like you.\"\n\nHe shoved her paw away. \"Yaaah, gerroff me, you don't care, no flippin' one bally well bloomin' cares about me!\"\n\nBragoon took a firmer approach. \"Come on now, mate, wot's all this blubberin' about, eh?\"\n\nHorty snapped a small twig and flung it at the otter, but it missed. \"You ain't no mate o' mine, none of you lot is! I'm starvin' t'death, I've got a molehill growin' out me head, my poor skull aches like flamin' thunder, an' now I'm goin' blind. I can hardly see a paw in front o' me!\"\n\nFenna took over, grasping the weeping hare's shoulders. \"Don't be silly, Horty Braebuck, and listen to me. What's all this carrying on for, eh? You're hungry, right? Tell me when you _aren't_ hungry! What then, your head's aching? Stands to reason, you've suffered a nasty bang on it. But as for going blind, that's nonsense! It's so dark in this forest at nighttime that none of us can see much. Here, take hold of this stick and follow me. Don't keep fiddling with that dressing on your head or it'll never get better. Saro, have you any food left?\"\n\nThe squirrel produced a few mushrooms. \"I saved these.\"\n\nFenna gave the mushrooms to Horty. \"Eat them slowly, take small bites and chew each mouthful twenty times. Come on, up you come, we've still got a lot of ground to cover yet.\"\n\nThey marched all night, with Bragoon scouting ahead and Saro keeping them on course. The otter returned in dawn's first glimmer, bringing with him a heap of ripe bilberries in his cloak.\n\n\"Lookit wot I found! I think there must be a river ahead, I could hear the sound of running water in the distance. Sit down an' get yore gums round a few o' these, Horty mate, they're nice'n'ripe. We'll rest 'ere awhile.\"\n\nHorty was considerably less sorrowful when there was food in the offing. \"Mmmm, better'n those measly mushrooms. I say, you chaps, I can see better. Flippin' bandage must've fell down over me eyes last night, wot. Oh corks, now everything's gone flippin' green! Why's it all green?\"\n\nSpringald explained. \"Because it isn't properly light yet, it's the day breaking over the treetops. Pines grow so thick in here that it makes the light look green.\"\n\nBut Horty would not be convinced. \"Fiddlesticks, you're only sayin' that t'make a chap feel better. Ah well, I don't mind spendin' the rest o' me life in a green fug. Hawhaw, lookit old Brag, sour apple face, an' you, too, Spring, little lettuce features, an' you Fenn, young grassgob!\"\n\nSaro stared at him pointedly. \"Ye missed me out?\"\n\nHaving devoured all the available berries, Horty lay back and closed his eyes. \"Hush now, let a chap get some rest, cabbage head!\"\n\nThe squirrel chuckled. \"That's more like the ole Horty we all know an' dread.\"\n\nMidmorning found them back trekking once more, eager to be out of the oppressive pine forest. The further on they went, the more pronounced came the sound of flowing water.\n\nSaro stopped to listen. \"Sounds like a fairly wide river. Have ye got that ole map from the Abbey, mate?\"\n\nBragoon produced the map, which had been made during the journey of Matthias of Redwall in search of his son Mattimeo. He scanned it closely. \"Aye, we're on the right course, though I think we took a different route t'get to it. This is the high cliffs, here's the wastelands an' this is the pines we're in now. There should be some sort of open area ahead, then a big river. We'll soon see, mates. Press on, eh!\"\n\nThey emerged onto the edge of a deep valley, the hill below them thickly dotted with smaller pines and lots of shrubbery. Below it was the narrowest strip of bank. Beyond that, a wide, fast-flowing river glimmered in the sunlight. Halfway down, the travellers halted on a shale ledge. They still had some way to go, and the descent looked fairly steep. Horty sat down, yawning in the heat. He rested his face in both paws.\n\nSaro prodded him. \"Are ye alright, head achin' is it?\"\n\nThe young hare nodded. \"A bit, but I'm more tired than anything.\"\n\nSaro indicated an overhang that was screened by bushes. \"Tuck yoreself in there young 'un an' take a snooze. I'll call ye when we're ready to move.\"\n\nThe four travellers slithered and bumped down the steep hillside, grasping trees and bushes to slow their descent. They were about halfway down when Bragoon sighted the reptiles. He halted, pointing.\n\n\"Down yonder on the riverside below us. Those reptiles I dealt with last night are waitin' for us. Trouble is, they've brought a pile o' their gang with 'em!\"\n\nSaro counted the assorted lizards, newts, toads, smooth snakes and grass snakes awaiting them on the shore. There were about thirty in all, with another twoscore camped on the opposite bank of the river.\n\nA thin reed lance zipped upward, narrowly missing Fenna's cheek. She stumbled, almost overbalancing, but Bragoon managed to grab her. \"Take cover quick, they're throwin' lances!\"\n\nTo one side of the slope, a fallen pine had lodged flat between two standing trees. Crouching behind it, Saro fitted a stone to her sling and launched it off at the reptiles. Cautiously, she peered over the log, noting that a toad had hopped out of the way of her stone. \"They ain't movin', just waitin' for us down there. Let's give 'em another couple o' slingstones, mate!\"\n\nBoth she and Bragoon slung more stones as Springald and Fenna threw lumps of shale. They were forced to duck fast as a half dozen of the sharp, thin lances came back at them.\n\nThe otter thumped his rudder down irritably. \"Well, this ain't goin' t'get us to Loamhedge. Those cold-blooded scum 'ave got us pinned down 'ere!\"\n\nSpringald picked up one of the lances and threw it back. \"It's a stand-off, what are we going to do?\"\n\nSarobando passed her sling to the mousemaid. \"Ye can use this, 'tis a good sling. But I'll want it back later. This is wot we'll do. While you three keep slingin' stones, I'll slide off through the trees an' take a scout round downriver. I'll find a good quiet spot where the river narrows for an easy crossin'. Then I'll slip back 'ere an' let ye know. Once 'tis dark, we can all sneak away an' escape. Right?\"\n\nFenna nodded. \"Sounds like a good idea!\"\n\nBragoon raised his eyebrows. \"Sounds like? Let me tell ye, missy, when my ole mate gets an idea, 'tis always a good 'un!\"\n\nSaro gave him a quick grin. \"Thankee, Brag. Now let's give 'em a good rattlin' volley to keep their 'eads down while I pop off unnoticed. One . . . Two . . . Three!\"\n\nSlingstones and lumps of shale peppered down at the foebeasts below. When Springald looked up, Saro had gone. Bragoon shoved the mousemaid's head back down as more lances came.\n\n\"Always duck fast once ye've throwed, Spring. There's more pore beasts been injured or slain in fights by lookin' up to see where their stones went. Ready agin, come on, let's give 'em a spot o' blood'n'vinegar. Yahaaar! Try some o' this, ye scum-backed, bottle-nosed crawlers!\"\n\nHorty slept on beneath the overhang, blissfully unaware of what was taking place.\n\n##\n\nSaro put some distance between herself and the skirmish. Ahead lay a sweeping bend in the river. Making her way down to the bank, she skirted the bend and began jogging steadily along the shore. It was peaceful and quiet, with only the crunch of pebbles beneath her footpaws mingling with the murmur of riverwater, echoing off the high, wooded slopes on either side. As she got round the bend, Saro caught the sound of deep, gruff voices singing a river shanty. She pressed on toward the singing. It was a song she knew, and she was fairly certain who the singers would be. The aging squirrel joined in with the melodious music.\n\n\"Wally wally dampum dearie,\n\nI'll sail back home next spring.\n\nKiss all the babies for me,\n\nan' teach the lot to sing.\n\nToodle aye toodle oo, me daddy's a shrew,\n\nwhose face I can't recall,\n\nbut I'll stay home all season long,\n\nuntil I hears him call.\n\nLogalog Logalog Logalog Oooohhhh!\n\nRinga linga ling me darlin',\n\nthere's ribbons for yore hair,\n\nI'll bring to ye a bonnet,\n\nan' a fine red rockin' chair.\n\nToodle oo toodle ay, just wait'll the day,\n\nDaddy comes paddlin' in.\n\nI'll grow up big'n'strong then,\n\nan' sail away with him.\n\nLogalog Logalog Logalog Ooooooooohhhhhh!\"\n\nCupping both paws to her mouth, Saro bellowed for all she was worth. \"Logalogalogaloga looooooog!\"\n\nSix shrew logboats hove into view, sailing upriver. The lead craft was by far the largest, carved from a mighty oak trunk and fitted with a single square sail of scarlet with an ornate letter _B_ emblazoned on it. All the logboats were packed with shrews, about a hundred of the small, fierce beasts. Each spiky-furred shrew wore a multicoloured headband and a kilt held up by a broad, copper-buckled belt into which was thrust a short rapier. Their leader, a solid old patriarch, with a thick, silver beard, stood in the prow of the front craft. He signalled for the rowers to pull into the shore.\n\nNo sooner had the vessel nosed in to land than the shrew chieftain leaped ashore and seized Saro in a viselike bear hug. He roared cheerily, \"Sarobando, me ole squirrelcake, where've ye been a-hidin' yoreself? Oh, it does me eyes a power o' good to see ye agin! Belay, where's that rip-ruddered rascal Bragoon? Is the ole villain still alive? Haharrharrr!\"\n\nSaro tugged the shrew's big beard and kissed both of his cheeks. \"Log a Log Briggy, ye barrel-bellied ole riverroarer, I knew 'twas you as soon as I 'eard yore song. Let go o' me, mate, while me ribs are still in one piece. Lissen careful to wot I got to tell ye!\"\n\nAfter loosening Saro, Log a Log Briggy listened as she told him the facts. \"There's trouble upriver. Bragoon an' some young mates of ours are pinned down on the 'illside by reptiles. There's about thirty o' the scum on this side o' the water, an' more on the other side. We need yore 'elp, Briggy!\"\n\nThe shrew chieftain's brows lowered menacingly as he gritted out the words. \"Reptiles, eh? I can't abide the creepy, cold-eyed scum. They think they rule the roost up that end o' the river. Don't fret, matey, I'll put my oar in an' show 'em who the real bigbeast is in these waters. No reptile's goin' to mess wid good mates o' mine!\"\n\nHe began issuing orders to the captains of the other five logboats. \"Moor those vessels on the other bank, we'll come back for 'em later. Jigger, take twenty goodbeasts an' go wid Saro. Bring extra clubs along wid ye. Raffu, Fregg, Scordo, Fludge, you an' the rest foller me along the far bank. Keep 'idden among the trees, an' don't make no noise. Bring me Aggie Frogslapper, look lively now!\"\n\nOne of the shrews passed Log a Log Briggy a hefty carved sycamore war club, which he wielded lovingly. \"Ole Aggie's slapped a few frogs in 'er day. Hah, there'll be a lot o' reptiles won't be comin' back for a second kiss from ye, Aggie me old gel!\"\n\nBriggy introduced Saro to a young shrew who was the model of himself in bygone seasons. \"This is me eldest, Jigger. 'Tis only his sixth season out as a Guoraf warrior, but he's shapin' up well. Jigg, me darlin' son, go wid Saro. When ye get yore fighters set up, wait for yore dear ole dad's call afore ye charge the scurvy foe.\"\n\nJigger shook Saro's paw. \"Let's make tracks. I hate bein' late fer a fight, marm!\"\n\nArmed with clubs, rapiers and slings, the shrews set off with Saro and Jigger at a swift trot around the riverbend. Log a Log Briggy took his logboat with the other five craft across the river to the opposite bank. He was first ashore, stroking his club, Aggie Frogslapper, and murmuring fondly, \"Aharr, 'tis a long time since ye had a good outin', me dearie!\"\n\nNight had descended over Redwall Abbey. Brother Gelf and Brother Weld sat by the dormitory window with Toran and Martha. The vermin had extinguished the fire on the Abbey lawn. Only the glow from a fire by the gatehouse could be seen. Abbot Carrul came up from the kitchens, threading his way through the Redwallers, who were resting on the dormitory floor. He pushed a trolley along to Martha and the watchers.\n\n\"I thought you might like some leek and chestnut soup. There's freshly baked cheesebread here, too.\"\n\nToran nodded admiringly. \"Ole Granmum Gurvel's a treasure. All the strife we're goin' through, but she still finds time to cook good vittles for us. Thankee, Father!\"\n\nAbbot Carrul stared out the window. \"Pretty calm out there. I imagine the vermin are taking their supper by that fire near the gatehouse. You can hear their voices when the wind drifts this way. Do you think they'll bother us tonight?\"\n\nBrother Weld exchanged glances with his friend Brother Gelf. \"Well, you heard them say they'd attack us when it got dark. Don't let that little decoy by the gatehouse fool you, Abbot, they'll be coming shortly.\"\n\nCarrul poised his ladle over the soup cauldron. \"You'll excuse me saying, but we don't look exactly ready to stand off an attack, with everybeast sitting about on the beds and the floor. How will you know if the Searats are stealing up on us under cover of darkness?\"\n\nBrother Gelf chuckled. \"Oh, we'll know sure enough, Father.\"\n\nFrom out on the lawn, shrieks and curses rent the air, together with the clatter of falling wood. Martha said calmly, \"That'll be them now. Right, friends, all to the windows and take up your positions.\"\n\nThe Abbot ducked to avoid a hooked window pole that Foremole Dwurl was carrying. \"Will somebeast pray tell me what is going on out there?\"\n\nMartha patted the paws of the two brothers beside her. \"It was their idea. We knew Raga Bol and his Searats would come once night fell. So Brother Weld and Brother Gelf had the bright idea of throwing broken glass, from the windows of Great Hall, out onto the lawn. Then, if the Searats tried to sneak up in the dark, they'd naturally let us know. It worked rather well, Father. Just listen to them!\"\n\nRaga Bol lurched about on the darkened lawn. He grabbed one of his crewrats, cuffing him about his ears. \"Silence, ye fool, wot's all the yowlin' about?\"\n\nThe Searat limped this way and that, trying to dodge the blows. \"Somethin' sharp is stickin' right into me footpaw. I couldn't 'elp it, Cap'n, I swear!\"\n\nBol shoved him away scornfully. \"Somethin' sharp, eh? I'll give ye somethin' sharp if'n ye don't shuttup. Any chance we 'ad of a surprise ambush is long gone now. Never mind yore footpaws, get some fire in yore bellies an' try t'be like real Searats. Avast there crew . . . Charge!\"\n\nKeeping his voice low as he heard the captain bellowing, Toran the ottercook gave his own orders. \"Up t'the winders, mates, let go the water!\"\n\nSturdy moles trundled forward to the windows. They hurled out the contents of bowls, pails, pots, pans, cauldrons and buckets. Water cascaded over the rubble heap, which piled outward, protecting the Abbey door.\n\nThe first ranks of Searats flung tree limbs, planks and long branches against the heap. Raga Bol dashed about, shouting encouragement. \"The fools won't stop us wid a drop o' water! Up ye go, buckoes. Board the place like it was a ship an' slay 'em all!\"\n\nCrewrats began scaling up the timbers. Unfortunately for them, the wood started sinking into the rubble, which had turned into a big mudheap, owing to the water drenching it. However, three of the longest planks spanned the mess, their ends resting on the sandstone lintel above the door.\n\nWhen Raga Bol saw this, he waved his scimitar about wildly. \"Ferron, Hangclaw, Rinj, gerrup those long bits. Come on, all paws t'the planks. Get through those winders, look sharp!\"\n\nClenching blades in their fangs, the Searats clambered skilfully up the wooden lengths. The planks bellied under their combined weight but held.\n\nRaga Bol laughed like a madbeast. \"Haharr, keep goin'. We'll make it, mates!\"\n\nBut they never quite made it. Hotroot pepper bombs burst on the heads of the lead climbers. Vermin wobbled on the planktops, trying to hold on whilst fending off the searing packages that pelted them.\n\nToran and the Redwall defenders appeared at the window spaces bringing their long, hooked window poles into play. The ottercook and four others latched on to a centre plank and heaved it out from the wall.\n\n\"Push, friends! Put yore backs into it an' shove!\"\n\nUnder the concerted effort of the Abbeybeasts, the plank was forced outward. Searats clung shrieking to it, as the pole moved it away. With nothing to support it, the plank teetered for a moment, then toppled over backwards with vermin clinging to its underside.\n\nWilling paws plied more window poles. Sister Setiva, Sister Portula and a crowd of elders pushed the left plank. Brothers Weld and Gelf, assisted by Gurvel, Foremole and three of his crew, pushed the one to the right. They strained and grunted, leaning their weight against the bending window poles.\n\nMartha gripped the arms of her chair, lifting her body forward. She could hear herself roaring. \"Push hard as you can. Push!\"\n\nThe planks fell, one to either side of the windows. Wood scraped against stone as they plunged sideways. Wailing Searats threw themselves clear\u2014some going headlong into the mudheap, others thudding on the paving stones below.\n\nThe defenders fell in a heap on the dormitory floor, yelling out a great victorious cry. \"Redwaaaaalllllll!\"\n\nMartha was about to drop back into her chair when an awful sight froze the breath in her throat. The Searat Ferron was crouching before her, framed in the window. He had leaped from the first plank before it had begun its backward journey. Latching on to the sill, Ferron had hauled himself up onto the windowledge. Now he perched there, snarling, a long dagger in one paw, ready to kill. In front of him, the Abbot had risen from the jumble on the floor and was standing with both paws raised wide, joining in the joyous shout of Redwall, with his back to the window.\n\nTime stood still, Martha's voice had deserted her. She was holding herself up, with her paws still gripping the chair arms. In front of her, Abbot Carrul stood, smiling at the haremaid and cheering lustily. Behind him, the Searat raised his dagger, preparing to stab at the Abbot's unprotected back. Alarm bells were clanging furiously in Martha's brain, coupled with the voice of Martin the Warrior, thundering at her, \"Save your Abbot!\"\n\nIt was over in a flash! Martha stood upright. Charging past Carrul and pushing him to one side, she hit the Searat, knocking him right out of the dormitory window.\n\nToran came bulling forward. He grasped the haremaid's waist, pulling her back into the room. \"You walked, Martha! You walked! You walked! You walked!\"\n\n##\n\nDown on the lawn, Raga Bol turned and strode away from the scene of his defeat. The Searat Rojin limped up to him. \"Cap'n, there's no way we kin get at 'em. Those beasts ain't as simple as they look.\"\n\nBol carried on walking without even looking back at Rojin. \"Have ye only just realised that? Call the crew off. There's got t'be a way into that Abbey, an' I'll find it. Ye can take my oath on it, 'cos I ain't movin' from Redwall. 'Tis mine, d'ye hear me? Mine!\"\n\nSomewhere southeast, deep along a woodland trail in Mossflower Wood, Flinky stopped running. Breathless and shaking, he collapsed to the ground. The little gang of escaped vermin flopped down beside him. Badredd slunk at the back of the group, with nobeast paying him the slightest attention. Gone were his days as gangleader. Now all the vermin looked to the stoat, Flinky, as their saviour. He had taken them out of the Searats' clutches.\n\nPanting hard, Crinktail clutched her mate's paw gratefully. \"We did it, we got away!\"\n\nHalfchop grinned fondly at Flinky, his new hero. \"Kachunk!\"\n\nUnderstanding what his pal meant, Plumnose nodded in agreement. \"Wodd duh we doo's now, Flink?\"\n\nThe triumphant stoat was never stuck for words, despite trying to regain his breath. \"Ah well, Plum, we can't run anymore tonight. Let's just stow ourselves under those bushes an' take a good ould rest while we lay low there an' 'ide. Tomorrer we'll 'ead south, where nobeast will ever find us agin. Sure, we'll find a comfy spot where there's plenny o' vittles growin', clean water an' grand weather. That'll do fer us, a good plan, eh?\"\n\nJuppa's voice was full of admiration. \"Aye, that it is. We're with ye all the way, Chief!\"\n\nRolling beneath the bushes, Slipback settled down amid the leaf mold. The rest joined him, with Flinky still chattering on.\n\n\"Ah, sure, we musta bin mad, lettin' greedy ould fools an' oafs lead us. Ferget all the magic swords, sieges an' great abbeys. Wot more could a body want than layin' round in the sun all day, fillin' yore stummick wid vittles an' never an argument twixt the lot of us anymore. After wot we bin through, I reckon we deserves a taste o' the good life, mates!\"\n\nOwing to the size of his nose, Plumnose was gifted with a keen sense of smell. His voice carried a note of disgust as he called out in the darkness beneath the bushes. \"Duh, sumthink smells h'awful round 'ere!\"\n\nJuppa gave vent to a horrified gurgle. \"Yurgh, wot's this?\" She shot out of the bushes on to the other side of the trail. Wringing her paws, the weasel performed an anguished little dance.\n\n\"There's a deadbeast in there! Yukk, I put me paw on its face. Creepy crawlies were all over its eyes!\"\n\nA mad scramble ensued as the gang ran out from beneath the bushes, shuddering and dusting themselves down.\n\nFlinky was the first to express an urgent desire. \"Let's get outta 'ere, run mates! We'll keep goin' 'til it's light, then I'll pick a better spot. Keep goin', don't stop fer nothin'!\"\n\nTheir sounds receded south into the distant woodlands, until everything was still and silent once more. The only things that moved were the insects crawling over the lifeless carcass of Jibsnout\u2014lying stretched beneath the bushes where Raga Bol had flung his slain body.\n\nAround the midnight hour, two others came along that same path. The Searats, Glimbo and Blowfly. It was the latter who searched the ground closely for signs of the fugitives.\n\nSceptical of ever finding them, one-eyed Glimbo complained volubly. \"Wot'n the name o' Hellgates do ye expect to find in this forest at night? We ain't even got a lantern!\"\n\nBlowfly wheezed as he heaved his bulk upright. \"I got good blinkers, don't need no lantern. I've tracked 'em this far, an' I'll keep on 'til I lays paws on dat scurvy liddle crew!\"\n\nHe unwound a long whip from about his flabby waist and cracked it. \"I'll teach 'em t'run away. They'll be lucky to 'ave a hide to their backs by the time they git back to the Abbey!\"\n\nGlimbo watched him track on a piece, then come to a halt. Blowfly inspected the ground carefully, going back and forth over the same piece, muttering and cursing.\n\nGlimbo relaxed, leaning against a tree. He scoffed sarcastically, \"Ye've lost our liddle pals, I thought ye would. Nobeast kin track anythin' at night through 'ere. Give up, mate, let's git back t'the crew. They're prob'ly inside that Abbey now, grabbin' the loot an' plunderin' the place. Yore wastin' time out in a forest when we could be back there snatchin' our share.\"\n\nBlowfly gave him a surly glare. \"Huh, 'tis alright fer you, I'm the one t'blame for lettin' them escape. 'Tis me who Cap'n Bol will take it out on. I can't go back empty-pawed!\"\n\nHis companion did not agree. \"Aw c'mon, Bol won't be frettin' over a few runaway fools. The cap'n 'as other things t'think about. A kick in the tail an' a few 'ard words is the most we'll get. Huh, we've 'ad plenny o' those afore now. Belay there, shipmate, wot are ye doin'?\"\n\nBlowfly looked up from his task of striking flint to steel. \"Wot I shoulda done awhile back, makin' a torch. I'll find these runaways, just ye wait'n'see!\"\n\nGlimbo seated himself with his back against the tree trunk. \"Well, ye can find 'em on yer own, 'cos I ain't goin' anywheres. When ye come back this way widout 'em, gimme a shake. I'll be right 'ere, takin' a nap.\"\n\nBlowfly held up the burning torch he had fashioned. Silent and stubborn, he trudged off alone into the night.\n\nLonna Bowstripe saw the glow from between the trees where he sat resting. It appeared like a small floating island of light in the darkness. Silent as a wraith he arose, becoming one with the forest as he stood motionless against the elm trunk. Blowfly walked by within a paw's reach of the big badger. Staring at the ground, the Searat mumbled bloodthirsty curses as to the fate of the lost fugitives. Lonna saw his face in the torchlight, and a trigger went off in his mind. He recalled brief flashes of the night he had been attacked by the Searats. Blowfly's coarse, ugly features were instantly identifiable. Swiftly, the badger strung his bow and stole up behind the unsuspecting Searat.\n\nBlowfly was jerked back as the tightly strung bow trapped his neck between wood and twine. The big badger managed to catch the torch before it fell.\n\nCraning his head around painfully, the Searat caught a glimpse of his captor and spoke almost indignantly.\n\n\"Yore dead!\"\n\nLonna drew him in until they were face-to-face. Only the pressure on the bow held the Searat upright, his limbs having turned to jelly.\n\nWith torchlight flickering over his scarred features and the light glinting in his vengeful eyes, the giant badger resembled some beast straight out of a nightmare.\n\nBlowfly's tongue suddenly ran away with him. \"It was Bol . . . it wasn't me . . . I wasn't nowhere near ye. I swear me oath on it, I never did nothin' . . . Gurgg!\"\n\nA sharp tug on the bowstring silenced him. Lonna's voice left the Searat in no doubt that lies would not save him. \"So you never did anything, you were nowhere near, it all had nought to do with you, you are innocent of everything?\n\n\"How many times has that same excuse been made? Think of every bully, cheat, plunderer or murderer before you who has lied with those same words. Once a villain is caught with no pack around him, then everybeast is to blame, except himself, of course. He will lie, betray and cheat to save his hide. But sometimes there is justice in the world, and fate catches up with him. So speak truly to me, or you will die slowly. You have my word on it\u2014and I never lie.\"\n\nBlowfly sighed with relief. He told Lonna all he needed to know, and he spoke truly. The big badger kept his word: the Searat did not die slowly. A single, mighty jerk of the bow, and Blowfly died quicker than he had ever expected to.\n\nAwakened by flaring torchlight, Glimbo yawned and stretched his paws. \"Betcha never caught 'em, I told ye afore y . . . Ukkk!\" The Searat's paw shot to his neck. Blowfly's long whip was tied around it, holding him fast to the tree he was sitting against.\n\nA deep, forbidding voice warned him, \"Be still, vermin!\"\n\nAutomatically he raised his other paw, trying to free his neck. There was a hissing sound, like an angry wasp. An arrow of awesome length buried its point deep in the tree trunk, a hairbreadth from his neck. Glimbo froze.\n\nLonna revealed his face in the torchlight, laid another shaft on his bowstring and unhurriedly explained his purpose to the petrified Searat. \"You will take me to the Abbey of Redwall. I am going to release you, but play me false, you'll wish you hadn't. Is that understood? Speak!\"\n\nGlimbo's good eye rolled about alarmingly in its socket\u2014he was completely terrified. \"Unnerstood!\"\n\nThe badger drew a long knife from his arm sheath and severed the whipcoils with a swift stroke. The Searat shot off like a hare at top speed. Lonna drew back the bowstring, homing in on the fleeing figure.\n\n\"Never mind, I'll find my own way.\"\n\n##\n\nFenna lowered her head quickly. More thin, sharp reed lances whipped viciously by. \"Don't they ever run short of those things?\"\n\nWithout raising himself, Bragoon hurled off a slingstone. \"There's always reeds aplenty on riverbanks. They just cut 'em an' point one end\u2014it makes a good throwin' lance, sharp an' dangerous. I've used 'em meself in the past.\"\n\nSaro suddenly rolled in beside Springald. \"Aye, but ye weren't much good with lances, too 'eavy pawed.\"\n\nThe otter scratched his rudder. \"Where did you come from, mate?\"\n\nSaro smiled, secretly enjoying the surprise she had in store. \"I found a bend in the river down that way, an' guess wot else I found?\"\n\nShe signalled with her paw. Suddenly Springald found herself being jostled by a score of shrews who had crept out from behind trees and bushes to join them in the shelter of the log.\n\nThe otter uttered a delighted growl. \"Guoraf shrews . . . Great!\"\n\nSaro pointed to Jigger. \"Aye, Guoraf shrews, an' who does this 'un remind ye of, Brag?\"\n\nThe otter inspected Jigger's face, noting the beard he was starting to cultivate. \"Wait, don't tell me, are ye a kinbeast to Log a Log Briggy, young 'un?\"\n\nJigger expertly caught a reed lance as it flew by. As he cast it back downhill, he was rewarded by a reptile's scream. \"Briggy's me old daddy. You must be Bragoon, the mad otter. Daddy's tole me about you. Pleased t'meetcha!\"\n\nFenna whispered to Saro. \"What's a Guoraf shrew?\"\n\nThe squirrel explained, \"That's just the first letters of their tribename. _G_ uerilla _U_ nion _o_ f _R_ oving _a_ nd _F_ ighting _S_ hrews. They're good friends an' fearsome warriors. Sometimes I think that they do all their far rangin' just lookin' for fights. Me'n Brag have battled alongside of 'em once or twice through past seasons.\"\n\nWhen everyone was acquainted, Jigger outlined the plan. \"We've got to 'old on, 'til me dad an' the others get set on the far bank. Then when we 'ears the signal, we charge an' cut loose at those reptiles on our side.\"\n\nBragoon mulled it over. \"Sounds like good sense t'me, mate. This crowd down below ain't goin' anyplace. They're tryin' to outwait us, an' slay us all when we makes a move t'leave.\"\n\nJigger peered over the log and ducked a few lances. He thudded the ground with his club, chortling eagerly, \"Reptiles'll stan' about waitin' fer ages in the sun. Well, I 'ope they enjoys their sunbath, 'cos we'll be givin' 'em a different kind o' tannin'. Hahaha!\"\n\nSaro spotted slight movements in the bushes on the far hillside. \"Looks like ole Briggy's gettin' the lads into position. Won't be long now.\"\n\nWithout any prior warning, Horty came skipping blithely out from beneath the overhang. He ran by the log, speeding downhill and calling back to them, \"Shrews, eh? Where'd ye meet that flippin' lot? I feel much better now, chaps. Who's for a jolly old paddle in the shallows, wot?\"\n\nWhen three lances came zinging at him, the young hare stopped, but the weapons had pierced his ridiculous headdressing. He ground to a halt, only paces from the dumbfounded reptiles.\n\n\"Great blinkin' seasons, have a flamin' care where you're chuckin' those things. A chap could get injured by them!\"\n\nKnowing that the plan had been ruined, Bragoon, Saro and Jigger, followed by their fighting force, came bounding downhill. At the bottom they found, to their shock, that the reptiles were lying prostrate, facedown in front of the young hare. Horty stood posing majestically, the three lances transfixing his turban.\n\nSaro glared at him. \"Wot were ye thinkin' of, ye great idiot? Lollopin' off right into the middle of the enemy like that!\"\n\nHorty gave her a scathing glance. \"Hold your tongue, marm. These chaps are just showin' their respect to me. Hawhaw, they must think I'm the Great Hortyplonk, descended from out the bloomin' sky, wot!\"\n\nSpringald scoffed in his face. \"Then they must be bigger idiots than you! D'you realise you could've been killed?\"\n\nAs she spoke, there was a whooping warcry from the far bank. _Logalogalogalooooooog!_ Briggy had commenced attacking the reptiles over there.\n\nThe reptiles laid out in awe of Horty lifted their faces. When they saw the score of shrews brandishing their clubs, they rose, backing off into the shallows.\n\nHorty took a few paces toward them. \"I say there, old scaly-skinned chaps . . .\"\n\nHissing and squeaking, the reptiles fled into the water.\n\nThe young hare turned to Jigger, who was looking rather crestfallen. \"Oops, sorry about that, old lad. Were you goin' to give those bounders a good drubbin'? I didn't realise. Oh well, never mind. Come on, we'll pursue 'em into the river an' deal 'em a few severe whackin's, wot!\" He trotted into the shallows but was immediately set upon and hauled back by four shrews.\n\nHorty protested vehemently. \"Wot the . . . ? I say, unpaw me, little sirrahs, I'm not scared of a few mangy reptiles, by the left, I ain't!\"\n\nJigger remarked caustically. \"Oh, we know ye ain't, lop-ears. But it's not the reptiles that's the danger on this stretch o' the river. Watch!\"\n\nHe picked up a lance and went into the shallows, holding the weapon out into the water at paw's length. Suddenly it began to shake and vibrate. When Jigger pulled it out, the tip was ripped and ragged. A small fish, which seemed to consist of only big, needlelike teeth, was clinging doggedly to it. Jigger flicked the creature back into the water.\n\n\" 'Tis the fish that are the slayers 'ere!\"\n\nThe reptiles were being swept downriver, shrieking unmercifully as the water about them reddened.\n\nHorty sat down in a collapse on the bank, looking pale about the gills.\n\n\"Oh corks, I feel quite ill all of a sudden!\"\n\nOn the far bank, the reptiles were taking a colossal walloping from Briggy and his command. They had tossed a big logboat sail over their foes, capturing most of them beneath the spreading canvas. Some of the Guorafs held the ends down, while others galloped about on the sailcloth, dealing great whacks with their war clubs to any bump that appeared\u2014be it head, tail, back or limb. Gradually the canvas subsided and was still.\n\nLog a Log Briggy waved over to them, his stentorian bass voice booming over the waters. \"Stop there, friends, I've sent a crew to git the boats. They'll pick ye up an' bring ye over!\"\n\nIt was a glorious evening on the far bank. Six logboats lay prow on to the bankside, as the travellers sat among their shrewfriends.\n\nHorty sniffed the air appreciatively, his whiskers atwitch at the aromas of cooking. \"I say, old Briggathingee, is that supper I detect? Jolly nice of you chaps, wot!\"\n\nBriggy pulled a mock glare at Bragoon. \"So, ye had t'bring a starvin' hare along with ye this trip. I'll wager that lollop-lugged young famine maker can shift a tidy few platefuls, eh?\"\n\nHorty smiled primly. \"Oh, I just nibble a bit here'n'there, y'know, sah. Actually I've not been feelin' too chipper of late. But if the scoff's as good as it smells, well, I might persuade myself to try it, wot.\"\n\nJigger looked askance at him. \"Lissen, mate, if'n ye want to sail wid the Guorafs, ye've got t'be a big eater an' a great bragger, like Drinchy 'ere. Ain't that right, Drinch? Show the harebeast 'ow 'tis done.\"\n\nA fat, powerful-looking shrew stood up, smirking, then launched into Riverbraggin, an art much admired among the longboat crews. Drinchy thumped the ground with his club and commenced roaring, \"I wuz borned on a river in a thunnerstorm, an' wot did I do? I ate the bottom outta the boat an' fought six big pike who tried to eat me! Though I wuz on'y a babe, I scoffed three of 'em, an' tossed the rest on the bank an' fried 'em for me brekkist! Aye, mates, I'm Drinchy Wildgob, the roarin' son of a roarin' son who killed 'imself tryin' to feed me. I can outeat, outchew an' outswaller anybeast alive\u2014includin' long-pawed, flop-eared, fancy bunnies!\"\n\nFinished with his mighty brag, Drinchy bowed as the shrews cheered him raucously.\n\nSaro nodded to Horty. \"I think you're bein' challenged, young 'un. Think you can do better than Drinchy?\"\n\nHorty stood up, bowing elegantly to Saro. \"Marm, my dander has risen since the remarks that chap made about me. We of the Braebucks are not backward in coming forward. I shall accept this curmudgeon's braggin' challenge, forthwith!\"\n\nWithout further ado, Horty bounded up, spreading his paws dramatically and yelling like a madbeast. \"I'm the son of the howlin' hare! I was born on a winter's night in a gale. My parents took one look at me, chewin' on the chimney, an' left home! There ain't a cauldron big enough to hold my dinner, not one in all the land! I've ate every jolly old thing\u2014fried frogs, toasted toads, boiled badgers, roasted reptiles, an' shrews, too! Shrew stew, shaved an' shrivelled shrews, shrew soup an' simmered shrew! I've got a stomach of iron an' a mouth like a steel trap! I'm the Horrible Hortwill Braebuck, an' nobeast steps over my line! Even little fat wretches with bellies like balloons an' spiky fur an' names like Drinchy! D'ye know what the Horrible Horty likes for supper? Daintily diced Drinchy . . . with lots o' gravy. Yaaaaaah!\"\n\nThe Guoraf shrews battered the ground with their war clubs, a mark of the highest honour they could show anybeast. Then they hoisted Horty up on their shoulders, cheering him twice around the camp.\n\nWith a look of thorough humbleness, Drinchy shook the young hare's paw fervently. \"Well, I more'n met me match there, mate. Ye must be the best bragger ever born, ye made me look like a beginner.\"\n\nThe triumphant Horty was gallant, even in victory. \"No hard feelins, Drinch old lad, but mind your language in the future, wot!\"\n\nA magnificent supper was served, as befitted shrewcooks, who were renowned across the waterways for their culinary skills. Huge portions were served up to Horty. The shrews gathered round, gazing in awe as he downed one dish after another.\n\n\"Mmmm yum! This is top-hole tucker, wot wot. Pass some more o' that skilly'n'duff, please. Oh, an' lob more honey over it, I like it that way. I say, is that actually rhubarb'n'blackberry crumble? . . . Where's me blinkin' spoon? Drinch, old scout, would y'be kind enough to fetch more shrewbeer\u2014not that little beaker, gimme the jug!\"\n\nBragoon chuckled. \"Look at young Horty, he's in 'is element there. They'll get tired o' servin' before he does of eatin', mark my words, Briggy!\"\n\nThe shrew chieftain watched Horty admiringly. \"That 'un should've bin a shrew, mate. I saw 'im march straight inter that reptile crowd widout turnin' a hair. They'd already throwed three javelins an' spiked 'is hat. I tell ye, Bragoon, it takes a brave beast to do that!\"\n\nThe otter poured himself another beaker of shrewbeer. \"Or a ravin' idiot! I'll tell ye the truth of it all someday.\"\n\nHorty was on to a wild grape and almond pudding. \"Never had this before. My word, it's rather toothsome, wot. Send the old cook out, an' I'll give her a kiss!\"\n\nA small, toothless, grizzled male shrew stumped out from behind the cauldrons hanging over the fire. He grinned. \"H'I'm the cook round 'ere. Wot was it ye wanted, sir?\"\n\nHorty choked on a mouthful of pudding. \"Wot, er, oh nothin', granddad. Excellent scoff, wot. Top marks, well done an' all that. Back to the old fire an' keep on cookin'. Eh, wot!\"\n\nLog a Log Briggy called to his shrews. \"Ye can let those reptiles free now, I reckon they've learned their lesson. If any of the slimy-skinned lot give ye any bother, give 'em another drubbin' an tell 'em you'll sling 'em in the river. That should scare 'em!\"\n\nHe sat down with Bragoon and Saro, winking fondly at them. \"Now then, mateys, wot brings you two t'these parts, eh?\"\n\nThey explained the mission for Martha's cure and their quest for Loamhedge.\n\nBriggy stroked his beard. \"Hmm, Loamhedge eh? I've 'eard tell o' the place. But ye'd 'ave to cross the great gorge to git anywheres near where the stories say the lost Abbey o' Loamhedge lies. Did ye bring some kind o' chart along to 'elp ye find it, or are ye just trustin' to fortune?\"\n\nBragoon produced the chart from Matthias's journal. \"It's been mostly luck to date, but we do 'ave this.\"\n\nBriggy rummaged a battered single eyeglass from his belt pouch and held it to his eye. \"My ole peepers ain't wot they used t'be, I got to use this monocle t'see. Right, wot've we got 'ere?\"\n\nHe perused the dilapidated parchment thoroughly. \"Hah, I know this country, 'tis sou'east o' where we are now. I've seen these two rocks an' all. They're called the Bell an' the Badger's 'ead, great big lumps o' stone they are. Wot's this, a large tree called the Lord o' Mossflower? Huh, that was long gone in the seasons afore my father's grandfathers. Blowed down, or collapsed more likely, when the earth trembled.\"\n\nSaro looked anxiously at the shrew chieftain. \"But ye do know where the two big rocks are?\"\n\nBriggy stowed his monocle away. \"Ho, I knows that place sure enough. East along this river for a day or so, then cut south when ye leave the bank. Wicked country, 'tis.\"\n\nBragoon patted his swordhilt. \"That don't worry us, we've travelled wicked country afore. So will ye take us upriver to the Bell an' the Badgers 'ead, me ole mate?\"\n\nBriggy held out his paw. \"Course I will, 'ere's me paw an' 'ere's me heart on it. But afore ye gets to the big rocks, ye've gotta cross the great gorge. I never knew of anybeast who's done that yet.\"\n\nSaro winked at him. \"You leave that to us. We've done lots o' things nobeast 'as ever done, me'n my mate.\"\n\nJigger joined them, taking a great interest in Bragoon's sword. \"That's a fine-lookin' blade ye carry, mate.\"\n\nThe otter drew the sword, holding it out to let the firelight play along its blade in the gathering twilight. It shimmered and glinted like a live thing. \"Aye, a fine blade it is, young 'un. My friend, the Abbot o' Redwall, loaned it t'me for the journey. 'Tis the sword of Martin the Warrior!\"\n\nThe shrews had evidently heard of Martin. As word ran through the camp, they crowded around Bragoon, straining to catch a glimpse of the legendary weapon.\n\n\"So that's the sword o' Martin. 'Tis a sight to be'old!\"\n\n\"They say 'twas made at the badger mountain from a piece of a star wot fell out the sky!\"\n\n\"Blood'n'fur, fancy ownin' a blade like that!\"\n\nJokingly, Jigger drew his own short rapier and waved it. \"Would ye like to challenge me to a spot o' swordplay?\"\n\nThere was a twinkle in Briggy's eye as he nudged the otter. \"Go on, mate, show 'im wot a real swordbeast kin do.\"\n\nBragoon rose casually, then moved like lightning. Jigger stood aghast, rooted to the spot as the sword encircled him in a streaking pattern of light. It clipped one of his whiskers and tipped the bandanna from off his forehead. The young shrew closed his eyes tightly.\n\nBragoon whirled the blade as he roared. \"Yahaarrr, ssssss'death!\"\n\nThe rapier flicked from Jigger's paw. It whipped through the air, then quivered pointfirst in the prow of his father's big logboat which was drawn up on the bank.\n\nJigger gasped. \"Scuttle me keel! How'd ye do that, mate?\"\n\nBragoon winked roguishly at him. \"That's a secret, young 'un!\"\n\nThe Guoraf shrew greatly admired the otter's prowess. \"Could I see yore sword, sir, just fer a moment?\"\n\nBragoon held the blade about a third of the way up. Raising his paw, he did a short hop and threw it. It turned once in the air, almost lazily; then, with a solid thud, buried its point into the logboat, next to the rapier.\n\nThe otter nodded. \"Aye, 'elp yoreself. But take care, yon's a sharp blade.\"\n\nJigger retrieved his own rapier, but he could not budge the sword since it was too deeply imbedded in the oaken boat. Bragoon went to sit down with Briggy.\n\nThe shrew chieftain stroked his beard. \"Where'd ye pick up swordtricks like that?\"\n\nThe otter shrugged. \"A Long Patrol hare from Salamandastron showed me some dodges with a blade one time. That 'un was wot they called a perilous beast, a real swordfighter, no mistake!\"\n\nHorty looked up from the remnants of a huge pastie. \"A Long Patrol hare, indeed! That's what I'd like to jolly well be someday, wot!\"\n\nSaro patted Horty on the stomach, knocking the wind from him. \"Then ye'll have to scoff less an' exercise more. Long Patrol hares are fightin' fit.\"\n\nThe young hare got quite huffy. \"Fiddlesticks, marm, one's got to get the right nourishment t'grow strong first, wot?\"\n\nBriggy smiled at him. \"Yore right there, Horty, an' ye need a full night's sleep, too. Go an' pick yoreself out a good berth on my vessel. We've got a journey upriver t'make at dawn. I'll put ye to the oars, that'll toughen yore muscles up a bit. You git yore rest now, an' you, too, Jigger.\"\n\nHorty gathered up some bread, cheese and pear cordial. \"Right y'are, Cap'n Briggathingee. I'll just take along a light snack to guard the young body against night starvation. I suffer from it terribly, y'know. I was born with the illness. I say Jigger, old lad, not takin' any rations with you? Well, suit y'self, laddie buck, but don't come pesterin' me durin' the flippin' night.\"\n\nJigger, however, was not listening. He had found a new object for his admiration. The young shrew was all smiles and attention for Springald. Carefully he helped the mousemaid aboard the logboat that he was travelling on.\n\n\"Watch yoreself, Miz Spring, these boats are tricky craft. You take some o' my cushions an' a soft blanket. Sleep up in the prow, that's the best spot aboard!\"\n\nThe pretty mousemaid played him up outrageously, fluttering her eyelashes and allowing him to make up her bed. \"Oh thank you, my friend, that's so kind of you!\"\n\nFenna scooted in and flopped down on the cushions. \"Plenty of room for us both here, Spring. Thanks, Jigger mate!\"\n\nSitting by the fire with Briggy and her otter friend, Saro watched the young ones with amusement. \"Nice to see 'em gettin' on well t'gether, eh?\"\n\nStirring the flames with his rapier, Briggy laughed. \"Haharr, bless 'em, they're only young once. The seasons soon fly by, ain't that right, Brag, ye ole battler?\"\n\nBragoon polished Martin's sword with a piece of damp bark. \"Ye never spoke a truer word, ole pal. Me'n Saro have gotten quite fond o' those three young 'uns, they're made of the right stuff. Now an' agin we gotta yell at 'em, but they learn fast. By the way, on that chart o' mine it says Long Tails an' desert beyond the river. Will that mean danger for us?\"\n\nBriggy looked scathing. \"Huh, Long Tails? My ole Granpa whopped those rats seasons afore I was born. Guorafs drove 'em off into the desertlands south o' the great gorge. They shouldn't trouble ye, though the desert might. 'Tis a long dusty trek to the gorge. D'ye want us t'come with ye?\"\n\nSaro clapped the stout old shrew's shoulder. \"No, mate, you git back to yore river, that's what ye know best. We've managed one desert by ourselves, another one won't make much difference. We'll be fine!\"\n\nBriggy seemed relieved. \"I thankee fer that, Sarobando. I don't like bein' far from runnin' water anytime. But I'll tell ye wot I'll do. We'll bring the boats back to where we drop youse off, say in about six days. I'll pick ye up for the return journey. There's a secret route I know that'll take ye back to the flatlands below the plateau. It means shootin' a mighty waterfall to git down there. But don't fret, my crews kin do it if anybeasts can. 'Twill get ye back 'ome to Redwall much faster.\"\n\nBragoon shook the old shrew's paw heartily. \"Yore a real friend, true blue'n never fail, Log a Log Briggy!\"\n\nThe shrew chieftain rose from beside the fire. \"Think nothin' of it, mate. I'm off t'me bed, if'n that young Horty ain't stolen it. Us old 'uns need sleep as much as the young do. Pleasant dreams, ye pair o' rips!\"\n\nThe aging otter and his lifelong friend sat by the fire awhile. Bragoon stared into the flames. \"We're gettin' too old for this sorta thing, Saro. I think when this adventure's over I'll settle back down at Redwall. Maybe that brother o' mine'll teach me to be a cook.\"\n\nThe squirrel stared levelly at him. \"If'n that's wot ye want, then fair enough, matey. I'll be by yore side wherever ye are.\"\n\nThe otter chuckled drily. \"An' so ye will be, we been together since we was Dibbuns. I wouldn't know where to turn widout ye.\"\n\nThat night they slept by the fire, dreaming dreams of the sunny old days at the Abbey when they were both young tear-aways together.\n\n##\n\nMartha was up at dawn, trying out her newfound skill\u2014walking! At first it was painful and slow, but the progress she was making, holding on to things for support, was remarkable. With the aid of Sister Setiva's blackthorn stick, which the Infirmary nurse had parted with happily, the haremaid wandered joyfully along Great Hall.\n\nMartha laughed inwardly at what Setiva had said: \"Och, take this auld thing an' use it in good heath, ma bonny lass. Ah've only kept it tae threaten Dibbuns with\u2014not that they ever took much notice, the wee villains!\"\n\nThe young haremaid manoeuvred the stairs, pausing every few moments to revel in her newfound freedom. Walking!\n\nAbbot Carrul came up behind her, watching Martha's progress, until she turned and noticed him.\n\n\"Good morning, Father Abbot, it's a fine morning!\"\n\nCarrul beamed back at her. \" 'Tis the finest of mornings, young miss, and all the better for seeing you up and about!\"\n\nAs Toran came out onto the dormitory landing, he waved down to them. \"Now then, you two gabby idlers, why ain't ye bringin' brekkist up to the pore beasts on guard, eh?\"\n\nMartha started eagerly back downstairs. \"Breakfast for how many, sir\u2014one, two, ten? It'll be up there directly!\"\n\nGranmum Gurvel came trundling through Great Hall, heading a small convoy of moles who were pushing four trolleys. She brandished her best copper ladle at Martha.\n\n\"Ho no you'm woant, brekkist bee's ee cook's tarsk roun' yurr. Miz Marth', you'm 'asten oop to ee durmitrees an' set ee on a churr. Rest yore paws naow. Doo ee hurr?\"\n\nBrother Weld had joined Toran on the landing. \"Best do as she says, or old Gurvel'll skelp your tail with her ladle. That's one old molecook who'll stand no nonsense.\"\n\nBreakfast in the dormitory was a makeshift affair, rather inconvenient for most but huge fun for the Dibbuns. The Abbeybabes, who thought everything was a game, perched in the oddest places, singing, playing and eating together. Sister Portula was trying to coax Muggum, and several of his cohorts, down from a shelf, where they were bouncing up and down as they squabbled over hot scones and honeyed oatmeal.\n\nIn a state of despair, she turned to Martha. \"Oh dear, I do wish the Searats weren't here and we were back to normal. Just look at those little ones, they're getting very wild. But with no Abbeyschool, and having to spend all day indoors, who can blame them?\" Portula looked to Martha for comment, but the haremaid was not listening. Her joyous mood dispersed, she stood gazing forlornly out the window.\n\nThe kindly Sister showed concern. \"Martha, dear, is something the matter, what's wrong?\"\n\nToran was close enough to hear his young friend's reply. \"I'm sorry, Sister, but I can't help feeling sad, I've just realised something. What a waste of time it all is. Bragoon and Saro, together with Horty, Springald and Fenna, have gone off questing for Loamhedge. Little do they know that I need no cure or remedy. Suddenly I can walk! My brother and good friends are far away from Redwall\u2014who knows what deadly danger or injury may befall them? There was no real need for them to go. Oh, fate can be so cruel at times. I feel responsible and guilty about the whole thing!\"\n\nSister Portula comforted her. \"You must not blame yourself, Martha. None of this was your doing, was it, Toran?\"\n\nThe ottercook had strong feelings about Martha's supposed dilemma, and he minced no words in telling her so. \"Wot's all this nonsense, don't ye be talkin' that way, Martha! Huh, ye could go on all day, worryin' about this an' that, an' supposin'. Lissen, I'll give ye a suppose. Supposin' yore friends an' my brother an' Saro hadn't gone, eh? Things would've turned out totally diff'rent, fate would've cast other lots for everybeast. You mightn't 'ave been at that window in yore chair last night, but those Searats may've changed their plans. Then where'd ye be now, Martha? I'll tell ye, still sittin' stuck in a chair!\n\n\"So don't ye dare say that there was no point in our good friends undertakin' a mission to find a cure for ye, Martha Braebuck! An' don't talk t'me of danger or injury. If'n Brag an' Saro 'ave anythin' t'do with it, the only ones sufferin' perils an' wounds will be anybeasts who tries to stop 'em! So quit complainin' an' supposin', miss. Be grateful that ye can go runnin', on yore own footpaws, to greet the travellers when they return to our Abbey!\"\n\nMartha had never heard Toran speak so forcefully, or truly. Wiping her eyes, the haremaid clasped her friend's paw fervently. \"Thank you, Toran, you're right. What a silly creature I am!\"\n\nThe ottercook turned away, brushing a paw across his own eyes. \"No you ain't, yore our Martha. Now put a smile on that face, an' get those liddle villains down of'n that shelf afore they fall an' 'urt themselves!\"\n\nSharpening his silver hooktip on the wall, Raga Bol lounged in the gatehouse doorway. Bright summer morn had done nothing to ease his foul mood. Dreams of the big stripedog had begun haunting him afresh, plus he was still smarting from the previous night's shameful defeat. Striving to put thoughts of the badger from his mind, he took out his mean temper on every crewrat in sight, snarling menacingly at them.\n\n\"Belay there, Wirga, ain't there any vittles left, where's me brekkist? Ahoy, you there, stop scrapin' mud off'n yoreself, an' grubbin' at yer eyes like some snotty liddle whelp. Go an' get some vittles for yore cap'n, sharpish!\"\n\nAll four of the Searats, not knowing exactly whom the glaring captain was addressing, ran off to do his bidding. \"Aye aye, Cap'n! Right away, Cap'n!\" they chorused as they tugged their ears in salute.\n\nRaga Bol turned his spleen upon the one called Rojin, who was sitting on the gatehouse wallsteps, poulticing a swollen eye. \"Quit dabbin' at yore lamp, ye've still got a good 'un left. I never got no brekkist, 'cos Blowfly let me servants escape. They're the beasts who should be doin' the cookin'. Git yoreself after Blowfly an' Glimbo. I want t'see ye all back 'ere by noon wid the runaways in tow. 'Cos if'n ye ain't, I'll let the livin' daylights into the lotta youse wid this 'ook. Go on, gerrout o' me sight, ye laggard!\"\n\nThe next to come in for a tongue lashing was the one called Rinj, who happened to stray within earshot. \"Stan' by the big gate there, Rinj, ye useless mess of offal. Keep a weather eye out for Rojin an' the others comin' back. Report ter me the moment ye spot 'em!\"\n\nThe Searat captain stalked back into the gatehouse, slamming the door so hard that its hinges rattled. He slumped into Old Phredd's armchair, trying to banish thoughts of the badger and concentrate instead on his plans to conquer the Abbey.\n\nMorning rolled on into the summer noon. The crew danced attention upon their captain, but he barely glanced at the food they brought. Instead, he ordered them to bring him volumes and scrolls from the shelves. Bol rifled through them, searching vainly for some clues\u2014a reference or a sketch, perhaps. Anything that would help him gain access to the Abbey building. After awhile he tired of this pursuit and banished the crewrats from the gatehouse. Scattering volumes and parchments over the floor, the Searat captain flung himself upon the bed and fell into a fitful slumber, the coverlet draped over his face.\n\nOn waking, Raga Bol saw that the sunlight shafts had shifted across the window. It was late afternoon, merging toward eventide. Rising, he took a mouthful of his favourite grog, swilling it around his mouth, then spat it out sourly. It was silent outside, with no sounds of activity. The Searat captain went swiftly outside.\n\nRinj was standing upright, propped against the gatepost, obviously sleeping. Raga Bol dealt him a savage kick, knocking Rinj flat. He continued to kick the hapless Searat, accentuating his words.\n\n\"Ye scabby-eyed, useless bilge swab! Did I tell ye to go snoozin' on duty? Wot's this door barred for, eh? Yore supposed t'be outside, watchin' for the others t'come back. If'n we was at sea now, I'd tie ye t'the anchor an' sling yore lazy carcass o'er the side!\"\n\nDragging Rinj upright by his ears, Bol knocked the gate bars up with his hook. He hauled the gates open, still shouting. \"I'll learn ye to disobey yore cap'n's orders, I'll . . . Yaaaagh!\"\n\nThe gates swung inward, revealing Rojin, pinned to the timbers by a huge single arrow, head slumped and footpaws dragging in the dust. Dead as the proverbial doornail!\n\nBeyond the outside path and ditch, out on the flatlands, Lonna Bowstripe roared as he fitted a shaft to his bowstring. \"Raga Bol! Death is here! Hellgates await you, Searat! Eulaliiiiaaaaaaa!\"\n\nBol took one glance at the avenging giant and hurled himself at the Abbey gates, slamming them and dropping the heavy baulks that served as locks. The wood shivered under the thud of the badger's massive arrow. Raga Bol leaped back from the gates, as if expecting the shaft to come right through.\n\nSister Setiva was prying the paws of little Yooch from the dormitory windowsill. \"Och, come away from there, ye wee pestilence!\" Attracted by the shouting from the gatehouse area, she peered over to see what was amiss there. Raga Bol's hoarse yells left her in no doubt.\n\n\"All paws to the walltops! Bring spears, slings an' bows. Jump to it, the stripedog's 'ere!\"\n\nSetiva caught Abbot Carrul's sleeve. \"There somebeast oot there, yon Searat's howlin' like a madbeast!\"\n\nToran was out the dormitory door, with Martha close on his heels. Carrul and Setiva followed as Toran called to them. \"Up t'the floor above, mates, ye can see better from there!\"\n\nRedwallers crowded to the second-story windows, which gave them a clear view of all that was taking place. Out on the flatlands, Lonna was raising his bow again. Brother Weld transmitted an excited commentary of what was taking place, for the benefit of those few who could not see. \"Great seasons of slaughter, it's a giant Badger Lord! The Searats are throwing spears, firing slingstones and arrows at him. Haha, their range is too short, their weapons can't touch him. Oh my, oh golly! Did you see that?\"\n\nOld Phredd croaked impatiently. \"See what? I can't see a thing!\"\n\nBrother Weld described what he had seen. \"The big badger fired off an arrow, huh, more like a spear. It struck a Searat, up on the ramparts. Got the vermin dead centre and drove him clear off the wall onto the lawn!\"\n\nSister Setiva shook her head in disbelief. \"Och, what a shot, ah've never seen aught like it!\"\n\nThe Abbeybeasts set up a great cheer. Lonna caught sight of them and waved. Leaning out from the upper windows, the Redwallers waved back furiously, shouting encouragement.\n\n\"Give 'em blood'n'vinegar, well done, friend!\"\n\n\"That's the stuff big feller, keep those shafts coming!\"\n\n\"Hurr, zurr hoojbeast, you'm give ee vurmints ole billyoh!\"\n\nWith her eyes shining fiercely, Martha yelled at Toran, \"Isn't he magnificent! Can't we do anything to help him?\"\n\nThe ottercook bit his lip anxiously. \"We got nothin' to throw that'd span the range twixt this Abbey an' the walltops, 'tis too far off for slingstones. There ain't a single bow'n'arrer in the buildin'. I'd love to 'elp the big badger, but wot kin we do, miss, wot?\"\n\nBrother Gelf, normally a quiet, inobtrusive mouse, spoke out. \"Er, I may be able to help, but I'll need to be down in Great Hall. I think I'll need a long windowpole, some twine, a couple of those pepper bombs and a few stones. Er, make them slightly larger than slingstones, but not much.\"\n\nHis curiosity immediately piqued, the Abbot bowed to Gelf. \"You shall have them, Brother. Let's go down to Great Hall. No pushing there, please, let Gelf go first.\"\n\nUp on the walltops the Searats were lying low, stunned by the accuracy of the bowbeast. Raga Bol was trying to instil some confidence n his crew. \"We're safe be'ind this wall, buckoes. That stripedog's got to stay out of our range. Soon as 'e moves forward we'll get 'im. Ain't been a beast born yet that spears an' arrers can't slay. All's we gotta do is stay inside these walls!\"\n\nWirga shuffled closer to Bol. \"Aye, but while we're on the inside, the stripedog has us pinned down from the outside. No Searat owns a weapon with the range an' power of that big bow, Cap'n.\"\n\nBol did not want to hear this. He stared cold-eyed at the Seer. \"What would ye 'ave me do then, run out an' charge 'im?\"\n\nThe loss of her three sons rankled Wirga, who now did not lose the opportunity to needle Bol. \"We outnumber the bigbeast by about twoscore. I never saw a Searat cap'n back off with those odds on his side!\"\n\nBefore Bol could strike out, or argue against Wirga, a Searat further along the parapet gave out a shout. \"Aaargh, wot the . . . Oooch!\"\n\nHe fell sideways, slain by one of the big arrows. Raga Bol crawled swiftly along and inspected the dead crewrat. \"Wot in the name o' blood'n'thunder 'appened to 'im?\"\n\nCowering fearfully against the battlements, the rat who had been crouching beside the victim babbled out. \"I saw it, Cap'n! Gornat was 'it by summat from be'ind. There 'tis, see, one o' those liddle bags o' pepper, tied on a string, wid a stone at the other end!\"\n\nBol unwound the object from around Gornat's waist. \"From be'ind, this thing got 'im\u2014ye mean from the Abbey?\"\n\nThe Searat nodded vigourously. \"Aye, it came from over that way, I swear it, Cap'n. Pore Gornat got a terrible smack from it, the thing 'it 'im an' wrapped right round 'is waist. It musta cracked a rib, 'cos Gornat shouted an' jumped up. That's when the arrer took 'im, straight through the neck!\"\n\nTurning to face the Abbey building, Raga Bol saw another of the missiles come whirling through the air. It spun round and round on its twine, weighted on one end by the pepper bomb and on the other by the stone. This time it missed and struck the wallside. The pepper bomb burst, sending its load over two rats crouched directly beneath. One had the sense to stay down and do his sneezing. The other leaped up and sneezed once, then an arrow silenced him for good.\n\nDown in Great Hall, the Redwallers had unblocked the shutters from one of the tall windows.\n\nToran took the windowpole from Brother Gelf. \"Can I try yore new slingpole out, Brother?\"\n\nGelf smiled quietly. \"Be my guest, sir.\"\n\nLaying the twine across the hooked metal end of the pole, the ottercook raised it straight up, facing out of the window. Holding the end of the pole in both paws, he let it lean back across his shoulder until it lay flat. Then he whipped it upright with swift force. The missile flew off through the high open window. There was a short interval of silence, followed by an agonized screech.\n\nToran grinned. \"It works!\"\n\nThere was no shortage of the homemade weapons. More window poles were brought, and more volunteers came forward, eager to try out the new weapons. Competition became so fierce that, owing to several of the defenders hurling the missiles at the same time, some of them missed the open window space. These projectiles struck the walls and lintels, bouncing back into Great Hall and bursting. Undeterred, the Abbeybeasts kept going, muffling their faces with towels. Soon, however, the atmosphere proved too much for the onlookers; many fled the scene, sneezing uproariously.\n\nThe Dibbuns thought the whole thing was huge fun. They chortled and giggled, dashing about and bumping into one another, shouting, \"Hachoo! Blesha! Harrachoo! O blesha blesha!\"\n\nMartha helped Abbot Carrul and some of the elders to shepherd the little ones downstairs into Cavern Hole. The haremaid actually carried two Dibbuns down the steps on her back, chuckling and joking with them.\n\nThe Abbot cautioned her. \"Careful, Martha, should you really be doing that? You don't want to put too much strain on those limbs!\"\n\nMartha deposited the Abbeybabes in a corner seat. \"Oh fiddledeedee, Father, I feel stronger than I've ever felt. It's as if I had brand-new footpaws and legs, they're as supple as greased springs!\"\n\nGranmum Gurvel sent down some kitchen helpers to carry baskets of fresh-baked tarts and pastries and jugs of sweet elderflower cordial.\n\nMartha lent a paw to serve the Dibbuns, then went to sit on the stairs with the Abbot. She felt very happy and carefree as they shared the food. \"Oh Father, isn't it wonderful, having that giant badger on our side! I wager things will be different now.\"\n\nThe Abbot seemed somewhat thoughtful, though he agreed with her. \"Yes, indeed, those Searats obviously fear the big badger a lot. Wouldn't it be marvellous if he were inside the Abbey with us? Things would be so much easier.\"\n\nMartha sipped her cordial. \"In what way?\"\n\nThe Abbot warmed to the subject, propounding a theory which had been growing in his mind ever since he had first sighted Lonna standing out on the flatlands.\n\n\"Our badger fires that bow like a mighty warrior, that's for certain. If he were inside the Abbey with us, I guarantee he'd send those Searats packing in short order.\"\n\nMartha thought for a moment about what the Abbot had said. \"Aye, he could stand at the dormitory windows and pick the Searats off at his leisure. They're hemmed in by the outer walls, so it would make it hard for them to avoid him. The badger could use the upstairs windows on all sides.\"\n\nAbbot Carrul put aside his food. \"But the problem is that the badger's outside the walls at the moment. Those Searats aren't stupid, they're not likely to leave Redwall and take their chances outside. Not with that giant and his bow waiting for them.\"\n\nMartha saw the wisdom in her Abbot's logic. \"Hmm, that could make Raga Bol doubly dangerous to us because he'll probably try twice as hard to get inside the Abbey now. It would give him an advantage over the badger, who would have to fight his way into the grounds and take the Searats on from inside the grounds. That would place him in range of their weapons. Oh dear, I wonder what the answer is to all of this!\"\n\nCarrul had already anticipated the problem. Unfortunately, he could not hold out a great deal of hope. \"We need to contact our badger friend and get him inside here, but that's not possible. The Searats are standing between him and us. We'll just have to wait our chance, though there isn't much likelihood of that at present.\"\n\nMartha tried to hide the frustration in her voice. \"But there won't be a much better opportunity than right now. Most of the Searats' attention is on the badger. If only we had somebeast who could slip out unnoticed! Whoever it was could leave by the small eastwall gate. They'd be shielded from any attention by the Abbey building. It wouldn't be hard to steal through the woodlands to the corner of the northwest wall. When it got dark, it would be simple to creep out onto the flatlands and make contact with the badger. Then they could both return the same way.\"\n\nThe Abbot folded both paws within his wide habit sleeves. \"No, 'tis far too risky at the moment, Martha. We'll wait\u2014tomorrow, perhaps. Oh dear, all this worry and strife. I'm longing for the day when those villains are long gone from Redwall and we can all get back to a normal, peaceful life.\"\n\nMartha stood up. \"Don't fret, Father, it'll happen when you least expect it. Do you know, I feel restless. Think I'll exercise my new walking skills. I'll go up to the dormitory windows and see how things are going. Best steer clear of Great Hall, and all those pepper bombs bursting inside.\"\n\nThe Abbot smiled wearily at his young friend. \"Take things easy, Martha, don't go tiring yourself out.\"\n\nShe paused on the stairway. \"Oh, I'll go as slowly as Old Phredd. By the way, Father, could you do me a favour? Will you sit guard on these stairs and make sure those Dibbuns don't go rushing back to Great Hall?\"\n\nAbbot Carrul stretched himself lengthways across the step. \"Certainly. I might doze off a little, but they won't get past me. You go on now, and remember what I said about taking things easy!\"\n\nHowever, taking things easy was the last thing on the young haremaid's mind. Martha had a mission: she would be the one to contact the badger. She went up to the back rooms on the east side, choosing one that was mainly used as a linen store. It was filled with blankets, sheets and tablecloths.\n\nKnotting bedsheets together, she fashioned a makeshift rope, reasoning aloud to herself. \"Finally I can do something useful to help Redwall. Now I can walk, and run, too. After all those seasons stuck in a chair, it would be a crime to waste my new gift!\"\n\nClimbing down the rope was easier than the haremaid had imagined. She had forgotten how strong her grip had become from wheeling a chair about for most of her life.\n\n##\n\nDawn light seeped over the river, casting a haze of pale green-gold mist. Saro lounged in the stern of the main logboat with Bragoon, savouring the new day, and a few scones still warm from early breakfast.\n\n\"Ah, this is the life, mate, save the wear'n'tear on me ole footpaws. There's nothin' like a nice lazy rivertrip, eh?\"\n\nThe otter grinned as Horty approached them, pushing on a hefty oarpole, part of two double lines of shrews. The young hare turned and started to make his way back to the prow, where he would repeat the process of poling the craft upriver.\n\nHe glared at the otter's cheery face and stuck his tongue out insultingly at Saro. \"Blinkin' idle bounders, sittin' on your bloomin' tails, wallopin' down scones, while I slave m'self into an early grave. Huh, should be blinkin' well ashamed of y'selves!\"\n\nBriggy left the prow and strode down the centre of the logboat, between both lines of polers. Exchanging a sly wink with Bragoon and Saro, he clipped Horty lightly across the ears, roaring at him in true rivercraft language, \"Avast there, ye long-legged layabout, quit prattlin' an' git polin'. We gotta build those muscles up t'make a warrior of ye! Ain't it a wunnerful life, nothin' t'do but pole about all day on the river, ye lucky swab! Dwingo, give us a drumbeat there. Come on, shrews, put yore backs into it. Sing out a polin' shanty to speed us on our way. Push, ye shrinkin' daisies. Push!\"\n\nThe drumbeat rolled out, echoing around the forested banks, with deep, gruff shrew voices singing in chorus. The shanty was a totally untrue pack of insults about Log a Log Briggy, but he sang along with them lustily.\n\n\"Barrum, babba, whum! Pole to the beat o' the drum!\n\nOur Cap'n is a bad ole shrew,\n\nI wish I'd never signed to roam.\n\nHe feeds us worms an mudpies, too,\n\noh ma, let me come sailin' home.\n\nBarrum, babba, whum! Pole to the beat o' the drum!\n\nOle Briggy is a lazy hog,\n\nwid a belly like a tub o' lard,\n\nif we don't call 'im Log a Log,\n\nhe beats us bad an' treats us hard.\n\nBarrum, babba, whum! Pole to the beat o' the drum!\n\nOne day our logboat sprang a leak,\n\nan' I gave out an 'earty wail,\n\nthe Cap'n gave me nose a tweak,\n\nan' plugged that leak up wid me tail.\n\nBarrum, babba, whum! Pole to the beat o' the drum!\n\nWe ran head-on into a gale,\n\nour Cap'n made me cry sad tears,\n\n'cos the wind 'ad ripped right through the sail,\n\nso he patched the canvas wid me ears.\n\nBarrum, babba, whum! Pole to the beat o' the drum!\n\nYe've heard me story, messmates all,\n\nan' if I spoke a lie to you,\n\nmay me nose swell into a fat red ball,\n\nan' me bottom turn bright green'n'blue.\n\nBarrum, babba, whum! Pole to the beat o' the drum!\"\n\nHorty was astonished; he turned to the shrew behind him. \"By the left! I say, old chap, are you allowed to bandy insults like that about Briggathingee, wot?\"\n\nThe shrew kept poling as he gave Horty a broad wink. \"It ain't serious, mate, 'tis all done in good fun!\"\n\nBriggy saw Horty gossiping and descended upon him. \"Stop jawin' an' keep pawin', rabbitchops, or I'll 'ave yore whiskers for desksweepers!\"\n\nThe young hare gave Briggy a cheeky grin and launched into a barrage of insults. \"Oh shut your blatherin' cakescoffer, y'great bearded windbag! You sound like a duck with a beakache, hasn't anybeast ever told you? Hah, tush'n'pish for all your ilk, sah, you wobble-pawed, twinky-tailed excuse for a barrel-bummed toad. Who d'you think you're jolly well talkin' to, you wiggle-whiskered, bawlin' braggart!\"\n\nHorty turned back to the shrew he had spoken to previously. \"Pretty good, wot! That told old Log-a-pudden a thing or two!\"\n\nThe ashen-faced shrew hissed back at him. \"We only ever does it in songs, all t'gether like. If'n you speak like that, face t'face wid a Log a Log o' Guorafs, that's mutiny, mate!\"\n\nHorty turned round to find Briggy looming over him with a face like thunder.\n\nThe force of the shrew chieftain's roars made Horty's long ears flap. \"Mutiny, eh? I won't 'ave mutineers aboard my logboat! Grab 'old o' this mutinous beast, put 'im to task! No more rations for 'im while he's on this vessel!\"\n\nFour shrews frogmarched the hapless hare off to the stern where he was given a large sack of wild onions to clean and peel.\n\nBragoon made his way to the prow, where he had a quiet word with the shrew chieftain. \"Ye were a bit 'ard on Horty there, mate. The young 'un wasn't wise to yore rules an' reg'lations, he thought 'twas all a bit of a joke. Horty didn't mean ye no real insult.\"\n\nBriggy's eyes twinkled. \"I know he didn't, friend, but I said I'd toughen 'im up. If'n that young 'un ever expects t'join the Long Patrol, he's gotta learn manners an' curb his tongue. Could ye imagine one o' those hare officers from the Long Patrol lettin' a recruit speak to 'im like that? Joke or not, some stiff-eared sergeant would clap 'im on a charge an' use 'is guts fer garters!\"\n\nBragoon agreed. \"Yore right, Briggy, a bit o' discipline wouldn't 'urt 'im. All three o' them young 'uns've been livin' the soft life at Redwall fer too long. The two maids are much better be'aved than Horty, they'll lissen t'reason. But Horty's too wild an' 'eadstrong. One day he'll make a fine warrior\u2014after he's learned a few stern lessons.\"\n\nBriggy stroked his beard. \"Don't fret, mate, I'll knock all the rough edges off'n Horty. My Jigger was the same 'til I showed 'im the ropes. Lookit Jigger now, commandin' his own logboat. There's a young shrew anybeast'd be proud t'call son!\"\n\nThe otter went back astern and sat with Saro. Behind them Horty was weeping buckets as he peeled and chopped the pungent wild onions. He went at it with vim and vigour, though scowling and muttering about the injustices of life aboard a logboat.\n\n\"Bit flippin' thick this lot, wot? A few measly words to old Brigalog an' he treats a chap like a bloomin' vermin marauder! I mean, what did I say? The bearded old buffer should count himself jolly well lucky, wot! Oh, yes indeed, when Hortwill Braebuck Esquire starts really chuckin' insults, he could roast the flamin' ears off a milky-whiskered shrew. I could've called the chap a lot worse! Twiggle-jawed trout! Giddy-nosed toad! Pickled old pollywog! Witless water beetle! Puddle-pawed duck's bottom! Or even Skinnyforlinkee Wobblechops! Huh, I think I let him off lightly, really. Good job one can hold one's temper, wot wot!\"\n\nLog a Log Briggy came striding down between the polers. \"Ahoy there, mates, is that mutineer be'avin' hisself? I might let 'im get a bite o' supper tonight, if'n I 'ears an apology.\"\n\nBragoon nudged Horty. \"Did ye hear that, matey?\"\n\nThe young hare turned a face, still running onion tears, to the Guoraf chieftain and declared dramatically, \"Y'mean you'd restore my scoffin' privileges, sah? Merciful Logawotsyaname, I'll peel every last one of these foul fruits, I swear I will. Good Captain, I'll be the saltiest young riverbeast you ever clapped eyes on. Listen to this. Shiver me sails an' rot me timbers, fry me barnacles, scrape me keel, an' all that nautical jimjam. You, matey sah, are lookin' at a completely reformed beast!\"\n\nBriggy glanced at Saro. \"Wot d'ye think, marm, is that a rogue worth feedin'?\"\n\nThe aging squirrel saw the haunted look in Horty's eyes and took pity. \"Aye, Cap'n, only a moment ago Horty was sayin' wot a good ole Log a Log ye are. Ain't that right, Brag?\"\n\nWith difficulty, the otter kept a straight face. \"Right enough, I'd give 'im another chance if 'twas up t'me.\"\n\nBriggy stroked his beard a moment, before answering. \"Aye, so be it then. Leave those onions now, young 'un, go amidship an' lend the cook a paw in the galley.\"\n\nHorty galloped off, overjoyed at the prospect of working amid food. \"Help the cook, I say, what a spiffin' job! A thousand thanks to you, Captain Briggaboat, an' you, my two chums. You have a handsome young hare's undying thanks!\"\n\nBragoon chuckled. \"Same modest Horty, eh?\"\n\nAboard Jigger's logboat, Fenna and Springald were being treated like royalty. Fenna had also gained an admirer, a stout young shrew named Wuddle. Both he and Jigger could not do enough for the pretty Redwall maids. The shrews brought extra cushions, erected an awning to shade them from the sun and served more delicious snacks than both of them could possibly eat. Then the two creatures vanished momentarily, to reappear grinning awkwardly, carrying with them two accordionlike instruments, which they said were called shrewlodeons. Jigger and Waddle twiddled a few keys, then launched into a song. Springald and Fenna were convulsed with laughter at the faces that both shrews pulled while singing. On verses they would be scowling savagely, whilst on the choruses they adopted expressions of peaceful concern. Both had wonderful bass voices and sang in harmony.\n\n\"When I meets a beast wot ain't polite to laydeez,\n\nI grabs 'im round the throat 'til he turns blue,\n\nI holds him tight in check as I squeezes on his neck,\n\nthen I boots his tail three times around the deck!\n\n'Cos be they sisters mothers aunts or daughters,\n\nall laydeez must be treated tenderly,\n\nthey're dainty an' they're neat, an' they don't have much to eat,\n\nan' they rouses gentle feelins within me.\n\nWhen I'm around an' you insults a laydee,\n\nI'll jump on yore stummick very forcibly,\n\nthen I'll punch you in the snout an' I'll prob'ly knock you out,\n\nan' black both of yore eyes so you can't see!\n\n'Cos be they sisters mothers aunts or daughters,\n\nall laydeez must be treated tenderly,\n\nthey're dainty an' they're neat, an' they don't have much to eat,\n\nan' they rouses gentle feelins within me.\n\nSo mind yore manners an' be very careful,\n\nwhen in the company of laydeez sweet,\n\nor I'll shove you in a sack, after fracturin' yore back,\n\nan' I'll stamp upon yore paws if you gets free!\"\n\nAfter the final chorus, they escorted both maids on a pleasant promenade of the deck, snarling fiercely at any of the poling shrews who dared to look sideways at Fenna or Springald.\n\nSupper on a mossy bank, overhung by weeping willows, was a total success. It was all due to Horty's onion soup. The Guorafs congratulated him on his cooking skills. He lapped up any compliments with a complete lack of modesty.\n\n\"Tut tut, nothin' to it, dear chaps\u2014a pinch o' this, a smidgeon o' that an' a sprinklin' of the other. Plus, of course, blinkin' loads of those confounded onions. I tell you, I shed many salt tears into the recipe, wot! Wild onions? Hah, I wasn't too blinkin' pleased, havin' to tame 'em down for you lot. I'd sooner be skinned me bloomin' self than have to skin another wild onion!\"\n\nLog a Log Briggy watched the young ones cavorting, singing and playing, then lay back and stretched. \"Beats me where they find the energy! Ah well, let 'em be merry while they can, 'specially those three young 'uns o' yores. I reckon we'll make our voyage end by midmorn tomorrer. That's when all the fun'n'games will finish for Horty an' the maids. I'm glad I don't have t'make the slog over desert an' gorge to Loamhedge with ye. That's country I was never fond of.\"\n\nBragoon flicked a twig into the fire. \"They'll do alright, with me'n my mate to look after 'em.\"\n\nSaro smiled. \"Aye, but by the creakin' o' my ole bones, 'tis them who'll be lookin' after us by the time this liddle jaunt is finished!\"\n\nNext morning, they arrived on time at the spot, just as Briggy had predicted. It was indeed hard, arid country. They had been sailing upriver since the crack of dawn. Nobeast could fail to notice the difference in the terrain. Trees, bushes and grass thinned out along the banks, whilst a hot breeze wafted in dust from the wastelands.\n\nBriggy smiled at the young creatures' downcast faces. \"Cheer up, mates, it ain't good-bye just yet. We'll be moored alongside this bank when ye come back wid a cure for Horty's sister. Get goin' now, an' good fortune go with ye!\"\n\n\"Thanks for everythin', ole friend!\"\n\n\"Aye, we'd a-been in a right pickle widout you an' yore crews, matey.\"\n\n\"See ye in six days, eh!\"\n\nLoaded with shrew hardtack biscuits and two canteens of water apiece, the travellers set off into the unknown.\n\nBriggy called out as the logboats pulled away. \"Keep the sun on yore right cheek, ye'll see the Bell an' Badger Rocks afore dark. But y'won't be able to reach 'em until ye figure out 'ow to cross the great gorge!\"\n\nSilence reigned over the searing, dusty flatness at high noon. Bragoon led the party, with Saro bringing up the rear.\n\nIt was not long before Horty began complaining. \"Phew, my ears are roasted, my tongue's parched an' my bally feet are fryin'. My word, it's even too hot to sweat! Walkin' walkin', always flamin' well walkin'. It's the story of m'life, chaps. First I was walkin' up'n'down on a blinkin' logboat, pushin' an oarpole. Now I'm walkin' again through this food'n'drink-forsaken place!\"\n\nSaro tugged the hare's tail. \"We're all walkin', ole gabby gob. We're walkin' t'bring back somethin' that'll make yore sister Martha walk. So stop moanin' an' keep walkin'.\"\n\nThe otter glanced over his shoulder, turning his attention on Springald and Fenna. \"Ahoy there, you two. Keep yore faces like that an' it'll rain afore long, ye mis'rable pair o' mopes!\"\n\nSpringald dragged her paws in the dust, replying sulkily. \"There's nothing at all wrong with our faces, thank you. Anyhow, you wouldn't understand.\"\n\nSaro piped up from behind. \"Why wouldn't we unnerstand?\"\n\nFenna pouted. \" 'Cos you just wouldn't, that's all!\"\n\nHorty could not resist smirking. \"They're jolly miserable because they've been parted from old Jigger an' his pal, wotsisname, Cuddles! Oh lack a day an' woe are they! I expect your little hearts are breakin', wot?\"\n\nKicking dust at the mocking hare, Fenna shouted, \"His name isn't Cuddles, it's Wuddle, and he's far nicer than you, Horty Braebuck. So there!\"\n\nSkipping ahead of the two maids, Horty made an elegant leg and bowed with a flourish. \"Fie upon you, marm, there's nobeast nicer than the charming I. Not in all the lands, or the river. Not like those two spike-headed water whompers who caterwaul songs like stricken ducks!\"\n\nThis time it was Springald who kicked dust at Horty. \"You vain, pompous, floppy-eared boaster!\"\n\nHorty was about to kick dust back, when Bragoon grabbed his ear and tweaked it soundly. \"If'n ye value yore ear, then stop embarrassin' those maids, right now! All three of ye are startin' to try my patience. Come on, Spring, cheer up. You, too, Fenn. It won't be long afore ye see those young shrews agin. Quit bein' so mean to each other, an' no more teasin'!\"\n\nHorty rubbed at the ear that had been tweaked. \"Who, me? I barely uttered a blinkin' word, it was those two who jolly well started it!\"\n\nIt was Saro's shout that put an end to the bickering. \"Look, mates, there's the Bell'n'Badger Rocks!\"\n\n##\n\nFloating above the heat-shimmered distance, the tops of both stone monoliths were just about visible on the horizon.\n\nFenna's keen eyes confirmed Saro's discovery. \"Hooray! You're right, there they are, I can see them!\"\n\nShading her eyes, Springald stood on tippaws. \"They look like one of those mirages that Old Phredd told us about. I wonder how far off they are?\"\n\nBragoon squinted over the wasteland haze. \"A fair bit yet, but if'n we press on 'til they're in plain sight, I'll call it a day an' we'll make camp. I want to look at those parchments from the Abbey. How does that sound to ye, mates?\"\n\nBragoon and Saro watched the three young ones dashing off ahead, their quarrels all forgotten as they shouted to one another.\n\nSaro scratched her bushy tail. \"Ha, lissen to 'em, they're the best o' pals agin!\"\n\nFenna was shouting, \"I'm going to set up camp with the cloaks an' staffs. Where are you off to, Horty?\"\n\nThe young hare had put on a spurt, racing ahead eagerly. \"Gangway, m'dears, I'm your cook this evenin'. Had lots of valuable experience, y'know. Oh yes, a chap learns a thing or three from those shrew coves, wot!\"\n\nSpringald kept pace with Fenna. \"I want to help Brag and Saro to study those parchments.\"\n\nBragoon and Saro followed the young ones at a steady lope. \"Looks like we've lost command o' the quest, mate. Can ye see those rocks clearly yet?\"\n\nThe squirrel looked up. \"Not quite, but it won't be long now.\"\n\nIt turned out that the three front-runners were forced to halt quicker than expected. Horty ground to a stop in a cloud of dust. \"By the left, right, an' knock me blinkin' sideways! How in the name of onion soup do we get across that bloomin' thing? Looks like the end o' the flamin' earth, wot!\"\n\nFenna and Springald joined him, gasping in disbelief at the awesome spectacle that confronted them.\n\n\"Whew! No wonder it's called the great gorge!\"\n\n\"Good grief, it must be miles down to the bottom!\"\n\nBragoon and Saro arrived on the scene. The otter ventured a glance down into the black chasm. \"If'n ye fell down there, that'd be the last anybeast'd see of ye, eh mate?\"\n\nSaro, however, was more concerned with the width of the gorge. \"Hmm, that's a wide ole canyon! Don't matter 'ow deep 'tis, we've got to think 'ow were goin' to cross it. Any ideas?\"\n\nFood was the only idea Horty gave priority to. \"Let's get a fire goin' an' we can figure it all out over a jolly good scoff. How's that for a scheme, wot?\"\n\nThe otter shook his head. \"This is strange country, mate. I don't feel too easy wid the thought of a fire. Pitch the camp an' see wot ye can make from the packs, Horty. I'll go off t'the right along the rim. Spring, you come wid me. Saro, you take the left edge. Fenna'll go wid ye.\"\n\nThey set off, with Horty issuing dire warnings. \"You chaps get back here before dusk, or I'll whomp up somethin' absolutely delicious an' eat it myself!\"\n\nSaro glanced across the gorge as she and Fenna explored along the edge of the precipice. \"Those two big rocks are plain t'see now, they mustn't be more than a couple o' slingshot distances from the other side. We're so near, yet so far, eh Fenn?\"\n\nThe squirrelmaid had noticed something down in the chasm. Suddenly her voice became shrill with excitement. \"There, that's how we'll get across, come and see!\"\n\nSaro lay flat on the edge, staring down at the solution to their problem. \"Well spotted, young 'un. I almost walked right by an' missed that. Let's go an' tell the others!\"\n\nHorty had created a fruit salad from the rations, with elderflower and dandelion cordial to go with it.\n\nThey sat eating as Saro reported, \"There's a tree trunk spannin' the gorge down that way, Fenna spotted it. About the height o' Redwall Abbey's battlement, down it lies. I don't know where it came from or whatbeast put it there, but it bridges the gap alright. 'Tis the longest trunk I've ever seen, lodged twixt a crack on one side an' a narrow ledge on the other. I think we should be able to get down to it on the rope that Cosbro, the old rabbit, gave us.\"\n\nBragoon gathered up the parchments he was about to study. \"Let's go an' take a look at it.\"\n\nThe spot where the tree trunk lay was directly in line with the two rocks across the gorge. Bragoon was thinking hard as he gazed down at the long, old span of timber that bridged the chasm.\n\nSpringald watched him as he studied the whole thing\u2014the twin rocks, and the tree trunk wedged inside the gorge. \"You know something about this, don't you?\"\n\nThe otter spread the map he had brought from Redwall. \"See here, this is the Bell an' the Badger Rocks. Now this spot is where the Lord o' Mossflower once stood. Ole Briggy said that it was a large tree, which had fallen down long ago. I reckon that tree trunk down there is the one that's marked on the map. After it fell down, some creatures must've rolled it into the gorge to make a bridge. I wager it took a lot o' beasts t'do the job, but they didn't know they was doin' us a favour when they took on the task. Is that rope long enough, matey?\"\n\nSaro, who had fetched Cosbro's rope along with her, dangled its length over the side. \"Aye, it falls a bit short o' the trunk, but it'll do.\"\n\nSatisfied, the otter issued orders. \"You three young 'uns, go back an' break camp. Fetch everythin' back 'ere with ye. Make as little noise as possible. There's somethin' about this area I don't like. It might be only a feelin', but I'm takin' no chances. Saro, me'n you'll rig this rope up. Remember now, be quiet!\"\n\nHorty and the two maids did not take long to pack the gear and break camp. Returning to the spot, they found that Bragoon had broken his staff in two pieces and driven them into a crack near the rim. Saro tested the rope she had tied around the wood. Without further ado, she went silently and skilfully down, using her footpaws on the rock walls for balance. She dropped lightly onto the trunk and twirled her tail several times as a signal that everything was alright. One by one they descended into the dark quiet chasm, Bragoon being last to go.\n\nThe five travellers perched precariously on the tree trunk. Saro gave the rope a swift upward flick, bringing it down with them.\n\nHorty peered across the gorge nervously. \"I say, Brag old scout, we could do with a torch to light us over this thing, it looks jolly dangerous t'me, wot?\"\n\nThe otter glared at him. \"Ssshhh, don't talk, yore voice echoes off the side down 'ere!\"\n\nSaro knotted herself and the others into a line, with herself at the front and Bragoon at the rear. Getting down on all fours, the five creatures inched out onto the long trunk. It seemed like an eternity, crawling over the wide expanse with nothing beneath them but empty space and total blackness. Sometimes the big log quivered, as one of them stumbled. At moments like this, they crouched there still, until Saro moved forward again.\n\nBragoon emitted a hushed sigh of relief when they finally made it onto the ledge at the far side. Fenna peered into the gloom, looking fearfully at Saro. The squirrel saw three black holes, which looked like entrances, in the rock face. She nodded and placed a paw to her lips in a gesture of silence. Untying her four friends from the rope, she coiled it about her waist and began climbing up the other side of the gorge. They watched her ascending the rock face with all the grace of a born squirrel climber. Bragoon kept casting anxious glances toward the three dark, forbidding entrances, but no signs of life showed there.\n\nSaro made it to the top in good time. Finding a convenient boulder, she tied a knot about it and lowered the rope to her companions. As they began the upward climb, the otter was still keeping a weather eye upon the dark holes.\n\nOnce all five travellers were safely together on the top of the far side, Horty laughed out loud. \"Hawhawhaw! Well, chaps, that's that! I suppose it's alright for one t'make sounds now. It's almost as bad as bein' hungry, for a brilliant speaker like me not being able to flippin' well talk. Absolute torture, wot!\"\n\nBragoon could not help smiling at the young hare. \"Go on, mate, talk away, even sing if'n ye like.\"\n\nEver willing to oblige, the garrulous hare burst into song.\n\n\"Oh it ain't much fun, when you must keep mum,\n\nan' they tell you not to speak,\n\nstandin' about with a tight-shut mouth,\n\nan' your tongue stuck in your cheek,\n\n'cos being silent, makes me violent,\n\nI want to roar an' shout,\n\nWheehooh! Yahboo! I'm tellin' you,\n\nI've lots to talk about!\n\nHello good day, how are you, say,\n\nthe sky went dark last night,\n\nbut it got bright this morning,\n\nso things turned out alright.\n\nWell there might be rain, but then again,\n\nwe'll face the storm together,\n\nin wind or snow, oh don't y'know,\n\nlet's talk about the weather!\n\nWheehooh! Yahboo! I'm tellin' you,\n\nI'll whisper, yell or shout,\n\nI'll natter'n'blab, or chatter'n'gab,\n\nI've lots to talk about!\"\n\nSaro cast her eyes to the darkening evening sky and sighed. \"We'd better stop for supper soon. That's the only time Horty goes quiet, when he's eatin'.\"\n\nBragoon watched fondly as the young hare did some fancy high kicks and ear twiddling. \"Aye, that rascal's like a weed on a wall, he grows on ye. I'll say this, though\u2014Horty's becomin' a first-rate cook, I like 'is vittles. We'll go as far as the Bell an' the Badgers Rocks afore night comes. I think we could even risk a liddle cookin' fire. Let ole Horty create us one of his masterpieces. Come on, mate, it ain't more'n a mile or two now.\"\n\nAs the laughter and banter of the questers receded into the gathering eventide, a stillness fell over the wasteland.\n\nThree faint screams echoed into the unfathomed depths of great gorge. On the ledge where the tree trunk bridged the space, several cloaked figures turned and padded silently into the three dark holes. These were passages, which led into a single hall-like cavern. The creatures from outside joined masses of others, similar to themselves. A myriad of glittering eyes were riveted on a ledge, where burned a sulphurous, yellow-green column of flame. A huge hunched beast, enveloped in a flowing cloak, stood with its back to them, facing the flame. It turned slowly. Not daring to look upon it, everybeast lay down prostrate, faces to the floor. A concerted moan arose from the masses.\n\n\"Mighty Kharanjul, Master of the Abyss! Great Slayer, in whose veins runs the blood of Wearets! Lord of Life and Death! We live only to serve thee!\"\n\nThe cloak swept back to reveal Kharanjul. He was a gargantuan creature, a primitive and hideous mutation\u2014something between a ferret and a weasel. With neither ears nor any semblance of a neck, his brutal head perched straight onto his hulking shoulders. When he spoke, his voice was a gurgling hiss, forced from between curving, discoloured fangs.\n\n\"Where are the three guards who were sleeping at their posts when strangers entered my gorge?\"\n\nOne of the creatures, who had recently entered the cavern, raised his face and cried out in a reedy voice. \"Lord, they are still falling into the chasm. They felt themselves too unworthy to face thy wrath, O Great Slayer!\"\n\nKharanjul picked up a big iron trident. He ran his long misshapen claws across the weapon's three barbed points. \"Nobeast has ever trespassed in my domain and lived to see the sunset that day. Ye will not fail me again, ye spawn of darkness. If those intruders set paw within a league of my gorge, to return whence they came, ye will let me know of it without fail. Double the guards both night and day. If the interlopers are caught, their suffering will be great. They will plead to be cast into the abyss of eternity. I have spoken!\"\n\nStill facedown, the masses chanted their reply. \"We hear and obey, O Mighty Kharanjul, Great Wearet Lord!\"\n\n##\n\nMartha closed the east wallgate behind her but left it unbolted. Her heart pounded wildly as she stole shakily through the night-shadowed woodlands, hugging the wall. The haremaid knew it had been an ill-advised plan, but she realised that the Abbot, or any sane Redwaller, would have forbidden her to venture on this mission alone. Her footpaws were trembling as she turned the corner to the north wall. Willing them to be still, she strove to gain control of her body. Every once in awhile, she heard weapons clanking on the parapet above. The Searats were patrolling the ramparts. The moment it went quiet, she would inch forward again.\n\nLonna had lit a fire out on the flatlands facing the west Abbey wall. He piled up brush and twigs into a heap, placing it so that from a distance it could be mistaken for himself, seated by the fire in the darkness. Moving off to his left he settled down, accustoming his eyes to the night. Setting a shaft to his bowstring, he watched the battlements. Soon a vermin head poked up cautiously, seeing the decoy by the fire and mistaking it for the badger. The Searat on the wall stood upright. Leaning outward, he peered toward the fire, trying to make sure that it was Lonna who was sitting crouched there. The big bow twanged, and the arrow took the Searat through his skull, sending him slamming backward onto the walkway. Hearing the resultant commotion from the walltop, the badger shifted his position, moving closer to the wall.\n\nRaga Bol's voice was immediately recognisable as he roared directions to his confused crew. \"The stripedog's somewhere out 'ere, on the far west end. Keep yore 'eads down, ye fools, or ye'll end like that 'un. Chakka, get over this way, the stripedog's over 'ere, not on the north side!\"\n\nLonna edged closer, until he could hear Chakka's reply. \"But, Cap'n, there's somebeast down here, sneakin' alongside the wall. It ain't as big as the stripedog, though. Wonder who it could be?\"\n\nCrouching, Raga Bol made his way across to the north wall. He risked a swift peek over the space between the battlements. \"Gimme yore spear! 'Tis an Abbeybeast tryin' to reach the big 'un, I'll wager. Blast yore eyes, gimme the spear afore it gets away!\"\n\nMartha made a break and ran for it, out across the path. She stumbled and tripped, going headlong into the ditch, which skirted the outside path. _Thunk!_ The spear quivered in the ditchside.\n\nRaga Bol saw another Searat getting up from a crouch to pass him a second weapon. \"Ye missed, Cap'n, but only just! Take my spear, 'tis me lucky one . . . Unngh!\"\n\nBol hurled himself flat as the spear clattered to the parapet beside the slain crewrat. \"Down, all of ye, down!\"\n\nLonna was running toward the ditch, firing off arrows with amazing speed, one after another. They pinged off the stonework and shattered against the northwest wall corner, keeping the vermin down.\n\nMartha narrowly avoided the huge bulk that crashed into the ditch beside her. She gasped, \"Sir, I came from the Abbey, my name's Mar . . .\"\n\nA massive paw cut off further explanation, as she was grabbed up and tucked beneath the giant badger's quiver. It bumped against her cheek as he rushed headlong through the ditch going northward, away from the Abbey.\n\nA deep voice sounded close to her ear. \"Time for introductions later, let's get out of range first!\"\n\nMartha felt like a Dibbun's plaything. Everything about the badger was immense\u2014his paw, his long arrows poking from the quiver, the great bow he carried, his colossal frame. Everything! Moonlit spaces flickered past as she and the badger left the ditch and sought the shelter of Mossflower woodlands. Martha saw the badger's face. It was deeply scarred and roughly stitched, giving him a savage and fearful appearance. But his eyes were soft and gentle, friendly, the eyes of a friend.\n\nLonna placed her down gently. \"Now, you were saying?\"\n\nThe haremaid tried not to be intimidated by his size. \"Thank you, sir, I was saying . . . We have need of you inside Redwall Abbey. I stole out to get you . . . Oh, my name is Martha Braebuck . . .\"\n\nThe bigbeast crouched, coming level with her face. \"Braebuck? I met your brother and his friends on the side of the high cliffs. Shouldn't you be in some kind of chair with wheels, miss?\"\n\nMartha found herself babbling. \"You met Horty, oh, is he alright? Bragoon, Saro, all of them, are they safe? Please tell me about them. Have they been ill or injured in any way? Oh, I've been worried out of my mind . . .\"\n\nThe badger's paw covered her face as he placed it over her mouth. \"Hush, little Martha, your friends are fine. I'd like to stand here all night talking with you, but you say they have need of me inside your Abbey. Can you show me the way back in there? By the way, my name's Lonna Bowstripe.\"\n\nMartha bobbed a small curtsy. Then she was swung up and placed upon the badger's shoulder as he moved off swiftly.\n\n\"Point me in the right direction, Martha Braebuck!\" Holding on to his ear with one paw and the bowstring with the other, Martha showed him the way. \"Straight ahead, Sir Lonna, you can see the belltower showing above the trees.\"\n\nHe chuckled. \"Just call me plain old Lonna.\"\n\nShe whispered into his ear. \"Lonna it is, you can call me Martha. I only get the full title when I'm being told off.\"\n\nRaga Bol sought out Wirga. \"Those poison darts, have ye got any of 'em?\"\n\nThe old Searat drew a rod from her cloak. \"Aye, Cap'n, there's one inside this. 'Tis the tube that I shoot 'em through.\"\n\nShe unplugged the ends of the rod, letting a dart show from it. \"See, my little messenger of death!\"\n\nBol murmured to her. \"I'm thinkin' that the stripedog'll try t'get inside the Abbey buildin'. He's got t'be stopped. Do this for me an I'll give ye anythin' y'want!\"\n\nWirga shuffled to the wallsteps. \"My price will be high, I warn thee. I have already lost three sons.\"\n\nThe Searat captain spread his paws wide. \"Anythin'!\"\n\nWirga looked up from the lawn. \" 'Tis as good as done!\"\n\nAbbot Carrul stood at the open window with Toran and Sister Setiva. He covered his eyes at the sight of the knotted linen rope hanging over the sill.\n\n\"I knew it, though it only occurred to me awhile after she'd been speaking with me. Oh Martha, why did you have to go and do it?\"\n\nBut Toran did not stop to argue the point. He was already over the sill, clutching the linen rope and lowering himself down.\n\nSister Setiva leaned out of the window. \"Och, ye'll get caught by yon vermin if ye go out there!\"\n\nThe ottercook dropped to the ground and drew his sling. \"Martha must've gone out by the east wallgate to find the badger. I'm goin' to look for her, you stay by here an' keep watch for us. Pull yoreself t'gether, Carrul!\"\n\nThe Abbot stood by the window with Sister Setiva. \"I should have known, I should have stopped her!\"\n\nPlacing a comforting paw about him, the Infirmary Keeper shook her head. \"Ah'll have a wee word wi' Martha when she returns. Ah thought it was only her brother who acted silly. Don't blame yersel', Father, yer no' a mindreader! How were ye tae know the maid wid do sich a thing?\"\n\nToran was stealing across the back lawn, when he saw Wirga ahead of him. He crouched low and watched her as she neared the east wallgate, then halted, obviously having heard something. Then Wirga scurried to a rhododendron bush, which grew close to the wall, hiding herself behind it.\n\nThe small door creaked on its hinges as it opened. Martha entered the Abbey grounds. Her paw lost in Lonna's grip, the haremaid turned and smiled at him. \"Home at last, welcome to Redwall, Lonna!\"\n\nThe badger had to stoop as he came through the gate. Toran saw Wirga slowly stand upright, placing the blowpipe to her mouth. There was no time to stop and think. He acted speedily. Bounding forward, the ottercook threw himself sideways, slamming into the Searat. Wirga's head hit the sandstone wall with a resounding crack, immediately reuniting her with her three sons.\n\nMartha pulled on Lonna's paw as he whipped the long knife from his arm sheath. \"Don't hurt him, that's my friend Toran!\"\n\nLonna took hold of the ottercook's apron and stood him upright. Toran blinked past him at the haremaid. \"Martha, is that you, matey?\"\n\nShe ran forward and hugged him. \"Oh Toran, that was a brave thing to do. Quick, let's get inside, there might be more Searats prowling about!\"\n\nOther Redwallers had found their way to the linen room. There were many willing paws to assist Toran and Martha back inside the Abbey. Lonna, however, was a different matter. Extra cloths and sheets had to be knotted into the makeshift rope. The badger's weight was such that no amount of helpers could even raise him from the ground. Tossing his bow and quiver up to Toran, the big badger made his own way, paw over paw, up the Abbey wall to the room. At first, Martha thought Lonna would burst the window frame, but he managed to squeeze through with a certain amount of grunting and wriggling.\n\nThe haremaid beamed proudly at the Abbot as she presented her new friend to him. \"Father, this is Lonna Bowstripe. He has volunteered to help us. Lonna, this is the Father Abbot Carrul of Redwall!\"\n\nThere was a deal of comment from the awestruck onlookers.\n\n\"Good grief, will you look at the size of him!\"\n\n\"Hurr, oi never see'd nobeast as gurt as that 'un!\"\n\n\"Look, his head almost touches the ceiling!\"\n\nWithout registering the least surprise, Abbot Carrul shook the badger's massive paw. \"Welcome to Redwall, Lonna, and my thanks to you for returning our Martha unscathed. It was a brave deed.\"\n\nLonna immediately took a liking to the dignified old mouse. \"Thank you, Father, it's a pleasure to be here. I will do all I can to rid your home of Raga Bol and his Searats.\"\n\nCarrul bowed gravely, then turned his attention to the onlookers. They were still commenting on the new arrival's size, speculating as to how his face came to be so dreadfully wounded. The Abbot stared them into silence.\n\n\"It has always been our manner to welcome visitors and offer them refreshment. Have you nothing better to do than embarrass our guest with your remarks?\"\n\nMuttering apologies, the Redwallers hurried off downstairs to comply with their Abbot's wishes.\n\nCarrul beckoned to Lonna. \"Come, friend, you must be hungry and tired. Let Martha and me offer you our hospitality. You must forgive our Abbeybeasts, they meant no offence.\"\n\nLonna followed Martha and the Abbot from the linen room. \"No offence taken, Father. I would be surprised if they had not mentioned the way I look. Anybeast I've ever met does.\"\n\nMartha gave him a reproving look. \"I never mentioned your appearance. Neither did Abbot Carrul or Toran, for that matter.\"\n\nLonna gave the haremaid one of his rare smiles. \"Then I have made three good and sensible friends tonight. I think I'm going to enjoy Redwall Abbey.\"\n\nEverybeast stayed up late that night, crowding into Cavern Hole to see the giant badger. Granmum Gurvel and her helpers trundled to and fro from the kitchens, bringing lots of delicious food for the guest, and for all present. Lonna sat staring at the array of fine things. Then Foremole Dwurl presented him with an outsized portion of his own personal favourite.\n\n\"Yurr zurr, this bee's deeper'n'ever turnip'n'tater'n' beetroot pie. If'n ee doant wish t'be h'offendin' ee cook, you'm best eat 'earty. Thurr bee's aplenty more whurr that cummed frumm, an' ee cook's a gurt fearsome ole villyun!\"\n\nThe badger took an amused glance at the dumpy figure of Granmum Gurvel, then set to with a will. Redwallers gazed in wonder as the hungry giant satisfied his appetite.\n\nSister Setiva even ventured a wry wink at Lonna. \"Och, there's nothin' worse than a beast with a wee flimsy appetite, pickin' away at his vittles, ah always say!\"\n\nLonna accepted a full deep-dish apple-and-blackberry crumble from the Infirmary Keeper. He dug into it with gusto. \"Aye, marm, but you must forgive me. They tell me I was a very fussy babe. It's a wonder how I survived!\"\n\nHis observation broke the ice; the Redwallers burst out laughing in appreciation of their visitor's ready wit.\n\nAfter the meal, they sat entranced, as Lonna related his story, which included his meeting with the travellers. The Dibbuns had infiltrated the gathering, slowly encroaching until they were sitting on Lonna's footpaws. Craning their necks, they stared in goggle-eyed admiration at the one who had confessed to being a fussy babe. Lonna gained them extra time, interceding with the elders not to send the Abbeybabes up to bed. Infants were always a source of amazement to him, he marvelled at their minute size and lack of shyness with strangers.\n\nHaving finished his narrative, Lonna asked Martha to tell him of how she came to be walking. The haremaid obliged willingly. Muggum had managed to scale the badger's footpaws and now sat upon his lap.\n\nTugging the badger's paw, the molebabe succeeded in gaining his attention. \"You'm surpinkly a gurt creetur! Zurr Lonn', 'ow big bee's yore bed?\"\n\nLonna looked thoughtful and adopted a serious tone. \"Hmm, it's quite large, and wide, too, though I've give up carrying it around with me. Why do you ask, sir?\"\n\nMuggum waved a tiny paw generously. \"You'm best take moi bed, Lonn. Oop in ee dormittees et bee's!\"\n\nThe talk went back and forth, encouraged by beakers of mulled October Ale for the elders and raspberry cup for the young ones. After awhile, the old ones fell into a doze; the Dibbuns, too, no longer able to keep their eyes open, curled up and slept where they chose.\n\nAbbot Carrul took advantage of the lull in the conversation, murmuring to Lonna, \"Come up to the kitchens, there's an empty storeroom there. We'll set up a sleeping place for you. But before that, I must talk to you, friend. We'll formulate a plan to defeat the enemy and free this Abbey. Martha, Toran, Sister Portula, Brother Weld, would you come, too? I'd like you to take part in the discussion.\"\n\nThat night, in the quiet of the storeroom, they formulated their plans. Lonna's status as a seasoned warrior, and his expertise in the ways of his enemy Searats, earned him the main say in the discussion. His ideas made sense to his friends, although his first words were in the form of a request.\n\n\"I need more arrows, good stout shafts, and well pointed. Have you any in the building?\"\n\nToran answered. \"I'm sorry, Lonna, we haven't, but I can look for some wood and make your arrows.\"\n\nBrother Weld interrupted the ottercook. \"Last winter, Brother Gelf and I found an ash tree, which had collapsed outside the east wall. Skipper and his otters helped us to chop the trunk into firelogs. Gelf and I took about six sheaves of long branches from it and bundled them up. We planned on cutting them into smaller sizes to use in the orchard, for fencing and propping up berry vines. But we never got round to it. They're still piled up under the belltower stairs. Those ash branches will be well seasoned now, perfect for making arrows!\"\n\nToran patted Weld's back. \"Good work, Brother, bring them to the wine cellar. Pore Junty Cellarhog had a little forge and anvil down there. I can make arrowheads from barrel-stave iron, Junty kept a whole stock o' the stuff.\"\n\nLonna looked from one to the other enquiringly. \"Flights?\"\n\nSister Portula had an immediate answer. \"There's a whole cupboardful of grey goose feathers in my room. I'll be glad to see the last of them. Two autumns back, Sister Setiva fixed the wing of a gosling, whose father was the leader of a goose skein. The geese were so grateful that they donated a load of loose feathers to me. I was supposed to cut the ends and use them as writing quills. Dearie me, they gave us enough for ten recorders to use for seven lifetimes. Please, Lonna, would you take them? If you'll relieve me of the burden, I'll recruit a team of elders to shape and bind them to your arrow shafts.\"\n\nThe badger agreed readily. \"Thank you, Sister, there's no better flight for a shaft than a goose feather. I've been using gull feathers from the northeast shores, but they don't have the strength and firmness of good goose plumage.\"\n\nMartha spoke. \"You'll have a full supply of arrows, Lonna.\"\n\nStretched out on a heap of clean sacks, the big badger gazed up at the ceiling, sure now of what he was going to do. \"Nobeast can live without food and water. Martha, I want you and a few others to patrol the windows all around the Abbey, where the Searats would find things to eat or drink.\"\n\nThe haremaid replied. \"You mean the orchard and the Abbey pond? There's also a vegetable garden adjoining the orchard. Since the Searats arrived, they've taken water and fished from the pond, and as for the orchard, they're hardly ever out of there and the vegetable garden. Isn't that right, Father?\"\n\nCarrul clenched his jaw. \"Correct, Martha. Those scum! I dread to think of the state our crops will be in. After all the hard work Redwallers did. Well, Lonna, you'll have a fair view of it. Orchard, vegetable patch and pond\u2014they are all clearly visible from our south-facing windows.\"\n\nLonna reached for his bow and began running a small piece of beeswax up and down the string. \"Perfect. How many arrows have I left in my quiver, Toran?\"\n\nThe ottercook took a look and returned grinning. \"Twenty-three . . . and a molebabe. Muggum's sleeping in there!\" The little mole grumbled dozily as Sister Portula extricated him. \"Oi got to stop urr. Lonn' bee's sleepin' in moi bed!\"\n\nAbbot Carrul took charge of the molebabe. \"Then you can sleep in my big armchair, you rascal. In fact, I think it's time we all got a rest, there's lots to do once the day breaks. Right, Lonna?\"\n\nBloodred tinges suffused the badger's eyes, his bowstring twanging aloud as he tested it. He gritted one word from between his clenched teeth. \"Right!\"\n\nThe Abbot hurriedly ushered his charges from the storeroom. \"We'll leave you to your sleep now, friend. Goodnight.\"\n\nThere was no reply. Closing the door behind them, Toran the ottercook exchanged meaningful glances with the Abbot. \"Did ye see that, Father? Lonna's possessed of the Bloodwrath!\"\n\nMartha looked from one to the other, perplexed. \"What's the Bloodwrath, some sort of sickness?\"\n\nToran grasped her paw so hard that she winced. \"Lissen to me, young 'un. You stay out o' that beast's way until his eyes clear up again. Badgers ain't responsible for wot they do when Bloodwrath comes upon 'em, d'ye hear?\"\n\nThe haremaid managed a frightened little nod. \"Lonna wouldn't hurt us, would he?\"\n\nThe Abbot signalled Toran and the others to their beds. He walked through Great Hall with Martha, who was carrying the sleeping molebabe.\n\nCarrul talked quietly with her. \"Do as Toran has told you, pretty one. Only be close to Lonna when you have to. Creatures such as us know little of Bloodwrath, but grown badgers of his size can be very dangerous to anybeast when it strikes them. Take your friends tomorrow, patrol the south windows on the first and second floor. The moment you sight Searats in the grounds, report straight to Lonna. Then get out of the way. Redwallers have no business hanging around a badger who is taken by Bloodwrath. Believe me, Martha, I tell you that Lonna needs to avenge himself and his dead friend upon Raga Bol and his crew. He is here for no other reason. Go to your bed now and remember what I have said.\"\n\nCarrul took the sleeping Muggum from Martha and went into his room. The haremaid looked up at the figure of Martin the Warrior on his tapestry. There was no need of visitations or dream speeches from the gallant protector of Redwall. His eyes seemed to say it all. She bowed respectfully to Martin, then went to her bed, still puzzled but obedient to her Abbot and the guiding spirit of her Abbey.\n\nDeath came to Redwall at dawn. A Searat came bursting into the gatehouse and raised Raga Bol from the bed where he had lain sprawled and twitching in broken dreams. \"Cap'n, the stripedog's just kilt Cullo an' Baleclaw. They was fishin' in the pond an' 'e slayed 'em both wid one arrer!\"\n\nBol came upright, his silver hook thrusting through the rat's baggy shirt as he dragged him forward. \"Killed 'em wid one arrer! Have ye been at the grog agin, Griml?\"\n\nThe rat wailed. \"I saw it meself, Cap'n. They was stannin' in the water, one afront o' the other, when a big arrer pins 'em both through their neckscruffs, like fishes on a reed!\"\n\nBol thrust Griml roughly out the gatehouse. \"Rally the crew, an' fetch Wirga t'me. Move yourself!\"\n\nGriml's mate, Deadtooth, was crouching beside the wallsteps. He, too, had witnessed the slaying of two Searats with one arrow. Deadtooth caught up with Griml. \"Wot did Bol say?\"\n\nGriml shrugged unhappily. \"Not much, just booted me out an' tole me t'bring the crew an' fetch Wirga.\"\n\nDeadtooth persisted. \"Don't the Cap'n know Wirga's dead? They found 'er just as it went light. Somebeast 'ad knocked the daylights outer 'er agin the wall. But ye knew that, didn't ye? Yew shoulda told Bol.\"\n\nGriml nervously looked this way and that. \"Hah, yew go an' tell 'im, if'n ye dare. I don't want no silver 'ook guttin' me. I wish we was afloat at sea, like last springtime. I tell ye, mate, we've 'ad nought but bad luck since we dropped anchor in this rotten place!\"\n\nGriml caught sight of several Searats emerging from behind a small ornamental hedge where they had been sleeping. \"Ahoy, youse lot, Cap'n wants ter see ye, right now at the gate'ouse, ye best jump to it . . .\"\n\nThere was a piercing scream from the orchard as a crewrat staggered out, transfixed by an arrow. Still holding a half-ripe pear in his claws, he took one more pace and crumpled in a still heap. Griml gestured at him wildly. \"See, wot did I tell ye? There's Rotpaw gone now, a good ole messmate like 'im, off to 'ellgates afore a bite o' brekkist passed 'is pore lips. I said this place is bad luck, didn't I?\"\n\n##\n\nHaving camped by the rocks and spent the night there, the travellers got their first clear view of them at sunrise next morn. Fenna found Horty, who had already risen, blowing on the embers of the previous night's fire and adding twigs to rekindle the flames. In high spirits, the young hare waved his ears at her.\n\n\"Mornin', fair Fenn'. Lots of twigs blown up against the rocks by the blinkin' wind, wot. Jolly useful to a first-class rivercook. What ho, you lazy lot, rise'n'shine, eh! So, here we are at the old Badger an' Bell. Thoughtful cove, whoever named 'em\u2014they look just like an enormous bloomin' bell an' a blinkin' huge badger's bonce!\"\n\nSpringald blinked sleep from her eyes and gave Horty a sidelong glance. \"Really, have you just noticed that?\"\n\nSaro got between them. \"Don't start again, you two. Horty, ole scout, ole lad, ole boy, wot's for breakfast?\"\n\nThe garrulous hare giggled. \"Heeheehee, would you believe fried fruit salad, marm?\"\n\nSpringald came wide awake then. \"Horty, you're joking?\"\n\nBragoon had sidled up. With the tip of his sword he speared a slice of plum from the flat rock that served as a frying pan. The otter chewed it pensively. \"Our cook ain't jokin', marm. Hmm, it don't taste too bad!\"\n\nAs Saro tried a morsel, winks were exchanged all round, behind Horty's back. The aging squirrel merely nodded. \"I suppose y'can't be too picky out in this country. I've ate worse an' survived.\"\n\nFenna prodded at the food with a twig. \"Do we have to eat it?\"\n\nClosing her eyes, Springald gulped a piece down. \"It's either that or starve. Fried fruit salad? Only a hare could think up a breakfast like that!\" Horty's ears rose like flagstaffs and his cheeks bulged out. The outraged hare was about to give them a piece of his mind, when something out on the wasteland distracted his attention.\n\n\"Cads! Bounders! You rotten, ungrateful . . . I say, chaps, is that somebeast crouchin' down out there?\"\n\nBragoon leaped up, wiping his swordblade. \"Come on, let's find out!\"\n\nThey spread out and made for the distant shape. Slowly and cautiously they approached the object. Then Fenna, who had the best eyesight, ran forward, calling to them. \"That's no crouching beast, it's nothing but a big battered old tree stump!\"\n\nThe fragmented piece of conifer stood almost as tall as Bragoon's shoulder. He tapped it with his sword.\n\n\"Y'know wot this is? All that's left o' that big tree on the map\u2014Lord o' Mossflower. We crossed over the great gorge by walkin' across its trunk!\"\n\nSaro circled the broad base. \"A shame, really. 'Twas a mighty tree in its seasons. Right, mate, 'tis time we took a look at the stuff you brought from the Abbey.\"\n\nBragoon drew out the tattered scraps of parchment he had carried since the day they left Redwall. \"Let's take a look then. Loamhedge can't be too far now. Maybe we'll find some clues that'll help.\"\n\nHorty was never a beast who took kindly to studying. He watched them unfolding a scrap of parchment. \"Borin' old stuff, I'll go back an' break camp, wot!\"\n\nBragoon passed the piece of paper to Springald. \"Yore a bright young 'un, read this out to us. I don't see too good for readin' lately. Think I might need those eyeglasses like Carrul an' Old Phredd wears when they reads things.\"\n\nSpringald studied the neat script. \"Martha copied this out. It says here that it's Sister Amyl's rhyme. Listen.\n\n\"Where once I dwelt in Loamhedge,\n\nmy secret lies hid from view,\n\nthe tale of how I learned to walk,\n\nwhen once I was as you.\n\nThough you cannot go there,\n\nlook out for two who may,\n\ntravellers from out of the past,\n\nreturning home someday.\"\n\nBragoon winked at Saro. \"That was us, we're the travellers from out the past. I wonder how young Martha is.\"\n\nSaro folded the parchment up, returning it to the otter. \"I wish she could've been fit t'make this trip with us. Now there was a young maid who had an 'ead on her shoulders. Huh, no clues there, though. Wot does that other bit say?\"\n\nBeside the map sketch, Bragoon had only one other piece of parchment. He offered it to Saro. \"You read it, mate.\"\n\nAfter unfolding it, the aging squirrel gave it to Fenna, without a second glance. \"My readin' is terrible, I never payed attention at Abbeyschool. Just like you, Brag, but I ain't makin' excuses about needin' eyeglasses. You read it, Fenna. I bet you was a good learner.\"\n\nThe squirrelmaid straightened the creased document. \"Martha tells us here that this is something which was copied by somebeast named Recorder Scrittum. The words are Sister Amyl's, but Scrittum recorded them for her.\n\n\"Beneath the flower that never grows,\n\nSylvaticus lies in repose.\n\nMy secret is entombed with her,\n\nlook and think what you see there.\n\nA prison with four legs which moved,\n\nyet it could walk nowhere,\n\nwhose arms lacked paws, but yet they held,\n\na wretched captive there.\"\n\nSpringald shrugged. \"Well, there are clues in that rhyme. But look around, what do you see? A broken tree stump, two big rocks shaped like a badger's head and a bell! Besides that, all we have is a map, made so far back that nobeast can remember. Is this all the information you brought with you from the Abbey? Bit thin, isn't it?\"\n\nBragoon drew patterns in the dust with his paw. Then he and Saro cast rueful looks at each other.\n\nFenna spoke to them. \"Wasn't there something else, a big volume about how a party of Redwallers found Loamhedge in bygone seasons?\"\n\nThe otter explained limply. \"Aye, missy, there was, but we never took the time or the trouble to try readin' it. We ain't no scholars, that much is plain, ain't it, mate?\"\n\nSaro nodded dolefully. \"Right, we thought that, 'cos we'd been atop o' the high cliffs an' onto the plateau one time, we knew this country. Our mistake, I s'pose. We should've let one of you young 'uns read the book out to us. You ain't like us. Livin' in the Abbey all yore lives, you managed t'get some learnin'. Me'n ole Brag, we ran away when we was young, didn't get much schoolin'.\"\n\nFenna wanted to take them to task for going off on such a quest without proper information, but they looked so crestfallen. She also felt it would be unfair to berate two creatures of such skill and craft, all of which they had gained in the hard school of travel and experience. Scholars they might not be, but adventurers they certainly were.\n\nA shout interrupted her thoughts. \"What ho there, you curmudgeons! Handsome young hare approachin' with visitors! Put aside your weapons. They're friendly, an' they enjoyed my blinkin' breakfast, too!\"\n\nBragoon thumped his rudder down in astonishment. \"Horty, what'n the name o' silly seasons . . . ?\"\n\nThe young hare marched up to the stump with his two new friends\u2014a large fat dormouse, pulling a cartload of twigs and wasteland debris; and, at his side, a tiny sand lizard held by a braided lead.\n\nHorty grinned from ear to ear. \"Meet my new pal Toobledum, survivor an' hermit of the wastelands, wot! Oh, an' this other ferocious creature is Bubbub, his faithful sandsniffer. I say, these coves really appreciate my cookin', they scoffed the bloomin' lot!\"\n\nSpringald cried indignantly, \"Well thanks for nothing. I scarcely took a bite of that food!\"\n\nHorty pawed his nose at her. \"Serves you jolly well right, after the way you lot carried on about my fine cookin'!\"\n\nToobledum, a cheery dormouse, wore an outrageously floppy woven grass hat, which he tipped to them. \"Pleased t'meetcher, one an' all, friends o' the cook, are ye! Well, Horty's led ye this far pretty good, I'd say.\"\n\nSaro glared at the young hare, paws on hips. \"Led us this far, eh? I wager you've been tellin' Mister Toobledum a right ole pack o' fibs!\"\n\nHorty waffled for a moment, then changed the subject completely. \"I say, chaps, here's a wheeze. Guess where Toobledum lives? Go on, tell 'em, Toob!\"\n\nThe dormouse sat down and lightly scratched Bubbub's emerald-green sides. The little sand lizard arched its back with pleasure. Toobledum looked up at them from beneath the wide brim of his hat.\n\n\"Lives? Me'n likkle Bubbub lives at Loam'edge, that's where we lives. Sand lizards ain't like most reptiles, y'know. Get 'em young enough an' they're good likkle tykes.\"\n\nBragoon stared open-mouthed at the dormouse. \"Y'mean to tell us you actually lives at Loamhedge?\"\n\nThe floppy hat wobbled wildly as Toobeldum nodded. \"All me life. Youngest o' sixteen I was, left 'ome an' came out here t'fend fer meself. Loam'edge h'aint no Redwall, like the big place Horty told me that 'e rules. But 'tis 'ome, an' we like it, don't we likkle Bubbub?\" The tiny sand lizard nodded and romped over to Fenna to be stroked and tickled.\n\nSpringald treated the dormouse to one of her prettiest smiles. \"Could you show us the way to Loamhedge, sir?\"\n\nHe flushed under his hat brim. \"Ain't no sir, missy, only an ole Toobledum, but I'll show ye the way willin'ly!\"\n\nFenna left off petting Bubbub, who nudged at her for more. \"You will show us the way. Now?\"\n\nDusting himself off, the dormouse rose with a grunt. \"Now's as good a time as any, me pretty one. Long as ye let my pal Horty cook me another good mess o' vittles.\"\n\nBragoon clapped the young hare's back so heartily that he almost knocked him flat. \"Well o' course, the champion quest leader an' expert cook an' ruler o' Redwall would be only too glad to cook for ye, matey!\"\n\nToobledum passed the towing rope of his cart to Saro. \"I'd be obliged if'n ye pull the ole cart fer me, marm. Me paws gets weary from luggin' it far'n'wide. Come on, likkle Bubbub, let's go 'ome.\"\n\nHe trundled off into the wasteland, chattering animatedly. \"Nice to find somebeast t'jaw with, it gits lonely out 'ere. Likkle Bubbub don't speak, y'see. I collects useful stuff, goes far'n'wide t'find it. Firewood, nice stones, bits o' this'n'that. Don't never waste nothin' out 'ere, I always sez. If'n ye got gear to cast off, then throw it me way!\"\n\nThey journeyed on, mainly south by Bragoon's reckoning, with Toobledum talking ceaselessly, and Bubbub frisking along on his lead, moving from one to another in his efforts to find more stroking.\n\nA camp was made out on the wastelands that evening. The dormouse donated some wood from his cart to make a fire. He was all agog in anticipation of his next meal.\n\n\"Well, Cooky, wot's fer supper? Me'n likkle Bubbub's feelin' peckish. Somethin' nice, I 'ope!\"\n\nSpringald grinned pointedly at Horty. \"Oh, don't worry, Cooky will turn out something delicious, I'm sure.\"\n\nThe young hare was beginning to tire of his role as cook. He rummaged through the dwindling supply in the ration packs. \"Hmm, I expect I'll create some superb dish, but we're runnin' a bit low on the old tucker, wot. Oh, fiddlesticks! Why's it left t'me to do all the blinkin' cookin' an' slavin' round here, while you flamin' lot sit on your tails an' loll around? Huh, bit bloomin' thick, I'd say!\"\n\nFenna joined in the teasing. \"Cheer up, Mighty Ruler of Redwall, I expect you have an army of skivvies to serve you back at the Abbey. Excuse me, you're not frying another fruit salad, are you?\"\n\nBorrowing an iron pot that had been clanking along on a hook beneath Toobledum's cart, Horty answered airily. \"As a matter o' fact, marm, I'm inventin' some scone soup, with a few wild onions, some sage, carrots, a leek or two an' some crumbled oatscones. Followed by fresh strawberry surprise, with dandelion tea to drink.\"\n\nIt was a surprisingly tasty meal. They downed it with relish. Fenna had one comment to make about the dessert. \"What's in this strawberry surprise, Cooky?\"\n\nHorty grimaced. \"Wish you'd stop callin' me Cooky. Oh, the strawberry surprise? I made it with some dried apple, preserved plums an' a piece o' fruitcake I found at the bottom of a ration pack. There ain't a flamin' strawberry in the whole thing\u2014that's the surprise. Good, eh?\"\n\nToobledum and Bubbub licked their bowls. The dormouse belched. \"Parn me one an' all. We liked it. Any second 'elphins?\"\n\nToobledum listened to the rhyme which had been dictated to Recorder Scrittum by Sister Amyl. Fenna read it out to him, but the dormouse was at a loss to cast any light on it. \"Flowers wot never grows, an' four-legged prisons wid no arms? Means nought to us, does it, likkle Bubbub?\"\n\nThe tiny lizard shook its head and nestled under Saro's paw. The dormouse bedded down by the fire, letting the hat brim cover his face. \"I ain't clever like you beasts, I'm just an old Toobledum. No matter, ye can search for yore own clues around Loam'edge in the mornin'. We'll git there afore midday. I'll bid ye 'appy dreams one an' all, g'night!\"\n\nBragoon settled down with the sword close to paw. \"I'll stay awake for first watch, mates.\"\n\nToobledum's voice came from under the hat. \"You git some sleep, I've taken Bubbub off 'is lead. That likkle feller's better'n any sentry, 'e'll stand guard all night for ye.\"\n\nHorty grinned with relief at Fenna. \"Saves us a job, wot!\"\n\nThe squirrelmaid curled up in her cloak beneath the cart. \"Indeed it does. Goodnight, Cooky.\"\n\nThe young hare's ears shot up stiffly. \"Cooky yourself, miss! Go an' boil your blinkin' heads, the bloomin' lot of you!\" He closed his eyes, trying to ignore the replies.\n\n\"Nighty-night, Cooky!\"\n\n\"Not staying up to plan breakfast then, Cooky?\"\n\n\"I expect he has a special menu writer to do it for him back home at Redwall, don't you, Cooky?\"\n\n\"A leader with stout paws, a wise ruler of an Abbey an' a cooky with a heart o' gold. Ain't we the lucky ones!\"\n\nNext day dawned on the wasteland, a warm flood of glorious colours, muted by dusty haze. The travellers ate a cold breakfast, eager to be on their way. Toobledum lingered, gossiping and trying to spin the meal out. It was only after gentle prodding that Bragoon urged the dormouse to get under way. Saro took Bubbub's lead, and Springald volunteered to pull the cart. They took up the rear, while Toobledum walked in front with Horty and Fenna. Ambling slowly along, the dormouse chatted with them.\n\nThey had been marching awhile, when Bragoon began having suspicions about the route. He called to Toobledum, who was still talking at length with Horty and Fenna, \"Now then, matey, when d'ye reckon we'll be at Loamhedge?\"\n\nWithout turning around, the dormouse shouted his answer. \"Oh, it won't be 'til after lunch, I'm thinking. But don't ye fret, we're makin' fair progress, one an' all.\"\n\nNodding knowingly, the otter whispered to Saro. \"Aye, I thought so, this ole buffer's got us on a vinegar trip.\"\n\nShe glanced quizzically at the otter. \"Wot are ye talkin' about, mate?\"\n\nKeeping his voice low, Bragoon explained. \"See those hills off to the right? We've been followin' them east instead o' south. Can ye see 'ow slow Toobledum's walkin', did ye notice how he lingered over brekkist?\"\n\nSaro was becoming impatient. \"Spit it out, mate. Wot's goin' on?\"\n\nThe otter conveyed his thoughts to her. \"Well, we ain't exactly goin' the wrong way, the dormouse'll get us there, sooner or later. But he's stringin' the trip out so we'll feed 'im agin at lunchtime. Trouble is, the old feller loves vittles too much, an' he mightn't have much food at 'ome. So he wants to scoff our rations an' have Horty doin' all the cookin' for 'im!\"\n\nSaro looked down at Bubbub. \"Is that right?\"\n\nThe little sand lizard grinned and nodded as the squirrel patted him. \"Well, the crafty ole grubswiper!\"\n\nBragoon winked at her. \"I'll fix that fat swindler, mate!\"\n\nHe called aloud to Toobledum. \"We ain't stopping fer lunch. Best press on to Loamhedge. When we arrives we'll have a big lunch an' a good rest.\"\n\nThe dormouse immediately altered course and speeded up, heading for the hills as he answered. \"Aye, good idea. Foller me, I've just remembered a good shortcut. We'll be there afore ye know it!\"\n\nSaro whispered to Bragoon. \"Lookit liddle Bubbub there, I'll swear he just sniggered.\"\n\n##\n\nIn less than an hour the travellers had reached the hilltops. Below them the land took on a complete change. Gone was the arid dusty wasteland, replaced with an expansive green valley\u2014not lush green like Mossflower woodlands but pleasant enough to appear refreshingly welcome to desert travellers. The whole area in the dip of the vale was dotted with brush, heather, grass and some stunted trees.\n\nToobledum whipped off his hat and made a sweeping gesture. \"There 'tis, one an' all. Loam'edge!\"\n\nHalfway down the slope, Fenna stooped to pick a few daisies. She crumbled some earth in her paws and sniffed it. \"This was probably rich fertile country in some bygone time.\"\n\nThe dormouse watched her braid the flowers into her tailbush. \"Most likely it was, young missy. Mebbe those mice who lived around 'ere long ago tended the land an' farmed it t'keep it that way.\"\n\nHorty stared about. \"Don't see any streams or runnin' water.\"\n\nToobledum plucked a daisy stem and chewed on it. \"There's underground water at the middle o' the valley. I gets it cold'n'sweet from a well down there. Once we crosses the Abbey boundary I'll take ye to it.\"\n\nThey carried on downhill. When the dormouse was almost on level ground he kicked aside some long grass and shrubbery. \"See 'ere, that's the top o' the ole boundary wall. It must've collapsed an' been buried in the long ago, when the ground used to dance an' shake.\"\n\nHe exposed a line of coping stones, each one decorated with a skilful carving of a mouse. Toobledum straightened up, arching his back as he gestured around the valley bottom. \"If'n ye takes the trouble, an' yore fond o' diggin', y'can follow it all around in a big square. I've never bothered meself, 'cept when I needs stones for me 'ouse. Right then, come on one an' all, don't shilly-shally, 'tis lunchtime.\"\n\nThey followed the old dormouse into a grove of stunted, knobbly trees, stopping as they reached a rickety hut, a rambling structure knocked together from odds and ends of stone, timber and debris.\n\nToobledum announced proudly, \"Well, this is it, one an' all, me likkle 'ome. Me'n Bubbub wouldn't trade it fer a palace!\" He set about lighting a fire beneath a rock slab oven, which stood outside the front door. Bubbub frisked happily about as the old dormouse sang.\n\n\"All round an' round the land ye well may roam,\n\nlots o' places I 'ave rambled, far'n'near,\n\nbut there ain't no nicer nest than me ol' 'ome,\n\n'tis so comfy an' we loves it, oh so dear.\n\nThe moment that we gets 'ere, me an' me likkle mate,\n\nwe lights a fire an' puts the kettle on,\n\nthough we ain't got much to eat, we gets along just great,\n\n'cos two kin live 'ere just as well as one.\"\n\nBragoon had a quiet word with Horty. \"Give 'em all we can spare from the rations. Make it a lunch to remember for the ole beast an' Bubbub.\"\n\nThe young hare saluted smartly. \"To hear is to obey, O Wise Otter, sah. I'll make it a spread that none of us'll forget!\"\n\nFenna blew a sigh. \"As long as you don't serve us fried fruit salad again!\"\n\nHorty began rummaging through their meagre supplies. \"Pish tush, miss! I shall treat that remark with the blinkin' contempt it bally well deserves, wot!\"\n\nHe did, however, cook a very passable meal. Drawing water from the dormouse's well, Horty produced a tasty vegetable soup and some scones and honey, with penny- cress and comfrey cordial to wash it down.\n\nSaro ate it with relish but could not resist a wry remark. \"Mmm, tastes good, but I ain't even goin' to ask wot's in it.\"\n\nHorty licked honey from his paws and reached for another scone. \"Just as well really, marm. Wild frogs wouldn't drag the recipe from me. We cooks have our secrets, y'know!\"\n\nToobledum and Bubbub did the lunch full justice. Springald was astounded at the amount the little lizard ate.\n\nThe dormouse just laughed at Bubbub's appetite. \"Proper likkle famine face, ain't he?\"\n\nBragoon began questioning Toobledum, warming to the aim of their quest. \"This spot we're searchin' for, it's a grave I think. Lissen to these few lines, mate, an' see if'n ye can throw any light on 'em.\n\n\"Beneath the flower that never grows,\n\nSylvaticus lies in repose.\n\nMy secret is entombed with her,\n\nlook and think what you see there.\"\n\n\"I want ye to pay attention, Toobledum. Do ye know anyplace 'ereabouts that sounds like wot I've just said?\"\n\nThe dormouse pulled down his hat brim, muttering darkly, \"That'll be the dead place. We never goes over there, do we, mate?\"\n\nBubbub snuggled tight against the dormouse and shook his head.\n\nSpringald pursued the enquiry. \"Whyever not? The dead never hurt anybeast, and I wager those buried there have been dead long before you were born.\"\n\nToobledum shook his head. \"Say wot ye likes, miss, but there's nights when the wind blows an' I've 'eard 'em moanin'.\"\n\nHorty took a light view of this sinister statement. \"Maybe they get jolly hungry down there. Come on, old scout, up on your hunkers an' show us where the old graveyard is, wot!\"\n\nThe dormouse refused flatly. \"I ain't goin' nowheres near that place, ye can go an' see it for yoreselves. Walk south across the valley until ye see flat stones. They're all laid this way an' that, ye can't miss 'em. That's the buryin' garden. I think it was once inside the ole Abbey. I've only been there once, an' I ain't goin' there agin, nohow!\"\n\nLeaving the dormouse and his lizard, the five travellers set out, following his directions.\n\nThe ancient burying place was quiet and peaceful in the noontide sun. A few bees hummed, and grasshoppers chirruped on the still, warm air.\n\nSaro sat down on one of the flat stones and looked about. \"Nice ole spot, ain't it. Sort of a garden o' memories.\"\n\nFenna brushed the dust from a lopsided oblong of limestone. \"See what this says: Sister Ethnilla, victim of the great sickness, gone to the sunny slopes and silent streams.\"\n\nBragoon traced a paw across the graven words. \"Pore creature, there must be a lot of her kind buried 'ere. Sunny slopes an' quiet streams, eh? I like that.\"\n\nSpringald and Horty were inspecting the stones further afield.\n\nThe young hare's voice interrupted the otter's reverie. \"I say, you chaps, what was the name we were lookin' for, Sivvylaticus or somesuch? I think I've found it. Yoooohaaaw!\"\n\nBragoon sprang upright as Horty's yell disturbed the peace. \"Wot's that lop-eared noisebag up to now?\"\n\nSpringald was shouting. \"Over here, quick, Horty's fallen down a grave!\"\n\nThey dashed over to where the mousemaid was hopping about agitatedly as she pointed to a yawning dark hole. \"Down there, he's fallen right through. One moment he was standing, pointing to this big stone, then something broke and he vanished!\"\n\nThe otter pulled her aside. \"Stand clear, miss, or ye might be the next one to disappear.\" He called down into the pitch-black space. \"Horty, are ye alright, mate?\"\n\nThere was no reply, just a faint echo of his own voice.\n\n##\n\nRaga Bol was at his wit's end; the Searat crew had begun to desert. He kicked out at Firzin, a rat he had posted on the main gate, screaming, \"Wot'n the name o' thunder d'ye mean, nobeast has got by ye all day? Did ye unlock this gate fer anythin'?\"\n\nFirzin cringed against the gate, which he had guarded faithfully on his captain's orders. \"On me oath, Cap'n, I've kept the gate tight locked!\"\n\nBol glared this way and that, slashing at the air with his scimitar. \"The walltops are too high for 'em to jump, so how've they got out? Rinj, wot d'you think?\"\n\nRinj, who had been close to Bol all day, shrugged. \"Wot about those liddle gates, Cap'n? There's one in the middle of each of the three outer walls. Bet they went through them, eh?\" The Searat captain's gold fangs flashed as he snarled. \"I told Argubb to post guards on those wallgates this mornin'. Go an' see if'n they're still there!\"\n\nRinj sidled out of the scimitar's range. \"They 'ad to stand in plain sight o' the winders, Cap'n. That big stripedog took the three of 'em out wid his arrers.\"\n\nRaga Bol peered around the wall buttress, which was sheltering him and his two crewrats from the Abbey windows. \"Get in the gate'ouse, both of ye, quick!\"\n\nThe three rats crouched, swerving in a dead run around the buttress. They made it into the gatehouse and slammed the door. The timbers shook as an arrow hit the door, its barbed point showing through the wood.\n\nFirzin wailed, \"We're all deadbeasts if'n we stay in this place. There ain't nowhere to 'ide from the stripedog!\"\n\nOne icy glare from his captain was sufficient to frighten the Searat into silence. Bol looked from one to the other, his face deadly calm, his voice low. \"Wot's the number o' crew left d'ye reckon, Rinj?\"\n\nThe rat thought for a moment. \"Just over a score, Cap'n. That's countin' us three.\"\n\nSince early that day, Raga Bol had been scheming furiously. His back was against the wall, but he was determined that eventually he would triumph. Then it came to his mind in a flash\u2014he knew that he had the answer. All he had to do was convince his crew.\n\nSlumping down in an armchair, he shook his head sadly, acting more like one of the Searat messmates than their captain. \"No more'n a score left out o' fifty, eh? I tell ye, mates, 'tis a sorry day. I suppose every one of ye wants t'see the back o' this place now. Speak up, I won't harm ye.\"\n\nFirzin summoned up his courage. \"Aye, Cap'n, they're all sayin' we're deadbeasts if'n we stays at this Abbey. Ain't that right, Rinj?\"\n\nThe other rat nodded. \"Aye, mate, gettin' away from 'ere's the sensible thing to do, shore enough.\"\n\nRaga Bol gave a rueful little smile, as if in agreement. \"Mebbe yore right. But just think, mates, if we'd 'ave killed the stripedog an' won, eh? Redwall woulda been ours! The good life, me buckoes! Everybeast of us'd be livin' like kings now, wid slaves, loot, vittles an' a place t'call 'ome fer the winter. Strange 'ow things turn out, ain't it? Now we got to cut'n'run, all because o' one stripedog who should've been dead now by rights.\n\n\"Aye, we've got no ship, we're a season's march from saltwater an' I've lost near a score an' a half of the best Searats a cap'n ever 'ad. Now we got nothin', we'll 'ave to tramp the land like beggars.\"\n\nRinj and Firzin had never seen their captain like this before. They shuffled their footpaws and tails awkwardly.\n\nThen Bol dropped a single word: \"Unless!\" Both crewrats were immediately curious.\n\n\"Unless wot, Cap'n?\"\n\n\"Have ye got a plan, Cap'n?\"\n\nRaga Bol leaned forward, his eyes gleaming craftily. \"Hoho, mates, I got a plan alright. Now 'earken t'me an' lissen!\"\n\nAbbot Carrul and Toran were sitting in the kitchens. They looked up as Martha entered. The ottercook indicated a heap of arrows, lying ready on the table.\n\n\"Does 'e want more shafts?\"\n\nThe haremaid shook her head. \"Not at the moment. Granmum Gurvel and I have piled arrows at every windowsill.\"\n\nThe Abbot poured her a beaker of cold mint tea. \"What's going on out there, Martha? You and old Gurvel are the only ones who can get close to Lonna. What's he up to?\"\n\nMartha took a sip of the tea. \"It's all quiet at the moment. He's roaming the upper corridors, watching from the windows. It's dreadful out there\u2014dead Searats by the pond, on the walls and by the orchard. I think a few of them have deserted, gone through the east wallgate into the woods. Lonna is still prowling about watching the grounds, though he seems to have calmed down a little. It was frightening just to set eyes on him this morning!\"\n\nToran brought out a stool for Martha to sit upon. \"Mayhaps the Searats are gettin' ready to leave, or it might only be the calm afore the storm. Who knows wot Raga Bol's got in that evil brain\u2014another scheme, per'aps. We'll just have t'sit an' wait. Wot d'ye think Martha?\"\n\nThe haremaid rested her weary footpaws. It had not been an easy day so far, running up and down stairs, keeping the badger supplied with arrows. \"I think it's gone too quiet, Toran. But who knows how things will turn out? Like you say, we'll have to wait and see.\"\n\nSister Setiva had been listening from the kitchen doorway. \"Och, all this waitin'! Everybeasts's keepin' busy, ye ken. They're all doon in Cavern Hole makin' arrows, even the Dibbuns. Ah've come tae make some food for them. Most of us have no taken a bite since breakfast!\"\n\nToran busied himself, glad for something to do. \"Leave it t'me, Sister. I'd forgotten about vittles today. Gurvel's helpin' Martha. I should've realised we 'ad no cook.\"\n\nAbbot Carrul climbed down from his stool. \"Here, let me help you, Toran. It's not right that my Redwallers should go hungry, even in times like these!\"\n\nLate afternoon slid into evening. Over beyond the west wall the sun set in solitary splendour. A wash of gold and purple suffused the sky, with blood red at its centre.\n\nLonna stood alone at the front dormitory windows. He rested against a sill, keeping watch on the gatehouse and its buttressed corner by the main gate. Now that there had been a few hours' lull from any action, the Bloodwrath had receded from him. His massive frame had relaxed. Lonna felt drained and weary, not having slept in almost two days and nights. Gradually night edged in, bringing with it a soft breeze to cool away the day's heat. Lonna began blinking a lot, nearly causing the bow to slip from his grasp. Rubbing his eyes and shaking himself, the big badger peered into the darkness, trying to keep his vision fixed on the gatehouse area. Then the voice sounded out.\n\nLonna came instantly alert as he identified Raga Bol's rasping tones, calling from somewhere over by the buttress where his arrows could not reach.\n\n\"Ahoy, stripedog, I see ye! Still hidin' in there be'ind the Abbot's skirts, are ye? Does yore wound still pain ye? Haharr, I should've gone for the neck an' chopped yore 'ead off! Don't worry, stripedog, Raga Bol ain't goin' nowhere. I slayed the old stripedog an' I kin finish ye, too!\"\n\nBrother Weld, who had been checking the window barricades in Great Hall, came hurrying into Cavern Hole. \"There's something happening outside. I can hear the Searat captain shouting to the big badger!\"\n\nToran bounded to the stairs. \"Sister Setiva, Sister Portula, keep the little 'uns down 'ere! Anybeast who's able enough, bring a weapon an' foller me! Does anyone know where Lonna is?\"\n\nMartha seized a ladle. \"He was going toward the dormitories at the front when I left him.\"\n\nThe ottercook wielded the big bung mallet, which had once belonged to Junty Cellarhog. \"Let's see if'n he's there!\"\n\nMartha and Toran burst into the dormitory, at the head of a band of Abbeydwellers. The haremaid could see Lonna's powerful back, silhouetted in the open window frame. He was shaking with rage but silent. Raga Bol was still taunting him from somewhere outside.\n\n\"I don't slay my enemies from a distance with arrers, that ain't the way a real warrior fights! But keep yore distance if'n yore scared o' Raga Bol. Come out 'ere, ye coward, an' I'll slice the other side of yer face off afore I leaves the birds to pick over yore carcass!\"\n\nLonna leaped up onto the windowsill, roaring, \"I'll fight you any way you like, you murdering scum!\"\n\nToran leapt forward and grabbed Lonna's footpaw. \"Don't go, mate, 'tis a trap. There's still plenty o' Searats out there. Ye'll be surrounded!\"\n\nThe badger dealt Toran a kick, knocking him backward. Raga Bol was visible now, standing slightly to the right on the lawn.\n\nPaws on hips, the big Searat laughed mockingly. \"Haharrharr! 'Ere I am, scarmuzzle! Come an' meet me paw't'claw widout yore bow'n'arrers fer once. Bring the magic sword an' cross blades wid Raga Bol if ye dare!\"\n\n\"Eulaliiiiiaaaaaa!\" Nothing could stop the giant badger now. Bellowing his warcry, Lonna jumped from the dormitory window. Luckily, the huge hill of rubble blocking the Abbey door had dried out in the sun. He landed upon it and managed to stay upright. Scrambling and rolling, he thundered down toward the ground. Without a moment's hesitation, Toran went over the sill after him, with Martha and the rest in his wake.\n\nRaga Bol held the glittering scimitar ready to strike, the silver hook on his other pawstump whirling in readiness. He stood awaiting the badger's charge, about a spear's throw from the north wall.\n\nMartha caught up with Toran. She pointed to the north walltop. \"Quick, up there, that's where the Searats are!\"\n\nThe ottercook veered, heading for the steps as he called to Martha. \"Split up, take half our beasts down to the east steps. I'll go up the north stairs. Weld, Gelf, Foremole, you come with me!\"\n\nOblivious of everything except Raga Bol waiting in his path, Lonna rushed straight at his enemy, armed only with his teeth and claws.\n\nBol, judging the moment when the badger was within three paces of him, dropped down, yelling out, \"Spears, now!\"\n\nLonna did not even bother to dodge the flying spears; three missed him, but one struck his left shoulder. He whipped it out and flung it aside, ignoring the wound. The Bloodwrath was upon him, his eyes red as the sunset he had watched a few hours earlier. His teeth shone from his scarred features in a savage snarl as his huge, blunt claws sought the kneeling Searat captain. Bol was halfway up when the badger grabbed his neck and swung him off the ground.\n\nRaga Bol emitted one strangled gurgle. Then four spears, thrown by the captain's own Searats and intended for the badger, buried their blades in Raga Bol's back instead. He died, hanging there like a rag doll in the grip of his mighty foe. The last thing he saw was Lonna Bowstripe roaring into his face.\n\n\"Go through Hellgates and burn, rat! Eulaliiiiaaaaaa!\"\n\nHolding the limp body in front of him, Lonna charged the wall, bulling up the stairs behind Toran like a juggernaut.\n\nThe ottercook shouted to his helpers. \"Look out, get to the west wall, let the badger pass! Martha, back off! Git those beasts down t'the lawn!\"\n\nThe haremaid, who saw what was happening, turned swiftly to the Redwallers behind her. \"Get out of the way. Downstairs, now!\"\n\nAbbot Carrul confronted her, his blood roused. He waved a sweeping broom, yelling fiercely. \"Let me at those rats. I'll drive them from Redwall, the filthy invaders. How dare they attack my Abbey!\" He was grabbed by two stout moles and hustled down the wallsteps.\n\nThe ramparts became a scene of chaos. Using Raga Bol as a flail, Lonna swept Searats left and right. Some were knocked over the battlements, their broken bodies thudding to the woodland floor outside the walls. Any who were unfortunate enough to fall onto the lawn inside the Abbey grounds were dealt with by a horde of Redwallers, each eager to be mentioned thereafter as a beast who had taken part in the battle to win back their Abbey.\n\nLonna stood on the empty walkway, his chest heaving like a bellows, blood oozing from a dozen different wounds. The carcass of Raga Bol resembled a grotesque, oversized pincushion, pierced by an array of spears from Searats who had tried to fend off the badger's advance.\n\nCautiously, Toran and his helpers approached from the west walltop. They froze as Lonna whirled around to face them, still holding Raga Bol's slain body, the spears hanging from it rattling against the battlements. With a powerful heave, the big badger tossed his onetime enemy over the wall, listening to his body clattering through the tree limbs. Smiling like a Dibbun who had just learned a new trick, Lonna Bowstripe sat down, letting his footpaws dangle over the lawn.\n\n\"When Martha brought me to Redwall, I hoped I could be of some help to you.\"\n\nThe ottercook sat down beside him. \"Aye, an' that ye were, mate, that ye were!\"\n\nWoodpigeons were startled from their roosts in Mossflower woodlands. They wheeled about in the night air, wondering why the bells of Redwall Abbey were pealing and booming out at such a late hour.\n\n##\n\nBragoon crouched, staring down into the pit of the open grave where Horty had disappeared. Saro was fashioning a torch from twigs, grass and moss. Fenna lay flat on the edge of the hole, calling down.\n\n\"Horty, if you can hear me, then shout out!\"\n\nSpringald centred the light of her chunk of rock crystal on the torchtop. Magnified sunrays produced a wisp of smoke, which grew into a small flame. Saro wafted it into a fire.\n\n\"I'm the climber, let me go first. Spring an' Fenna, ye stay up 'ere in case we need anythin'. Fetch the rope, Brag.\" Lowering herself over the edge, the aging squirrel dropped a bit, then landed on something solid.\n\n\"Stone steps, look!\"\n\nA dusty flight of narrow steps ran curving downward into the darkness. Bragoon coiled the rope about his shoulders and followed her carefully. \"Slow down, mate\u2014we don't want t'lose you, too!\"\n\nSpringald and Fenna watched until the light vanished around the curve, down into the gloom.\n\nThe mousemaid shuddered as she sat down by the broken covering stone. \"I don't like this place anymore. It looked so peaceful and sunny at first, but now there's something about it that gives me the shivers. No wonder Toobledum wouldn't come here. I hope Horty's alright.\"\n\nFenna was studying the big dark headstone, perched sideways at a crazy angle. \"Horty's indestructible, you'll see.\"\n\nBragoon's head appeared at ground level. \"Yore right there, miss. Lend a paw, you two!\"\n\nSaro was on the step behind him. Between them they carried the slumped form of Horty. Heaving and pulling, the four friends managed to lift the young hare onto solid ground, where he curled up as if asleep.\n\nSaro patted his back. \"He took a fall an' landed on the left side of his head. Pore Horty's got a fat ear, but there's no real harm done.\"\n\nFenna soaked some moss and dabbed at the swollen ear. \"He's taken his share of knocks on this trip. That's a real thick ear he's got there.\"\n\nThe damp poultice must have worked: Horty groaned and tried to sit up but fell back, complaining miserably. \"Yowch, I am awake! I say, d'you mind awfully not scrubbin' a chap's wounded ear with that filthy wet stuff. It stings like jolly blue blazes!\"\n\nSpringald took out a flask of cordial which she had brought along. \"Could you manage a sip of this?\"\n\nHorty grabbed it and downed the lot in three big gulps. \"Not that it'll do the noble young ear much good, but I've managed to wet my parched lips with it. Ooh, my achin' lug!\"\n\nFenna supported his head. \"Poor Horty, it must hurt terribly.\"\n\nThe young hare put on a pitiful face. \"I must be close t'death. I say Fenn, old scout, you don't happen to have a bite of scoff about you, wot?\"\n\nBragoon stifled a laugh. \"Nothin' much wrong wid that 'un! Keep an eye on 'im, you two. We're goin' back down to take a look round there. Pass me some more wood an' grass, Fenn. We got to keep the torch alight.\"\n\nFenna bundled her cloak under Horty's head. As the squirrelmaid began gathering more fuel for the torch, she shared her latest discovery with her companions.\n\n\"Now I know why Toobledum could hear moaning on windy nights from the buryin' place. See that big dark stone, it's the one that marked this grave. There's words carved on it. Listen. 'Sylvaticus. First Mother Abbot of Loamhedge Abbey. Loved by all creatures. Long in seasons and wisdom. Gone to her final rest. Forever in our thoughts.' This is the very grave we've been seeking.\"\n\nFenna indicated the beautifully carved motif at the top of the headstone. It was a lily in full bloom with a graceful stem sprouting curved and fluted leaves. The entire design was pierced right through the stonework. The squirrelmaid traced it with her paw.\n\n\"This is the flower that never dies. I'll wager that the wind sings an eerie song through this carving on windy nights. You can't blame Toobledum for steering clear of here.\"\n\nBragoon regarded her with admiration. \"Yore a bright young 'un, Fenn, that was well thought out. Take care of Horty now, we'll be back afore ye know it.\"\n\nFor the second time, the two old friends descended the stairs.\n\nNot one to let an injury slip by unnoticed, Horty made the most of his thick ear as the two Abbeymaids ministered to him. \"Salad! Now that's the very stuff for a swollen ear, wot! Any hare'll tell you, salad's just the thing, an' lots of it. Hold hard there, Spring old gel, what's that sloppy mess? Tut tut, marm, you ain't physickin' me with that rubbish!\"\n\nSpringald cradled the mixture in a dockleaf. \"Don't be such a Dibbun, Horty Braebuck. It's a mud-and-moss poultice that will do your ear a power of good. Hold him, Fenn!\"\n\nHorty struggled in the squirrelmaid's firm grip. \"Gerroff me, you flamin' torturesses. I'll bet you took lessons from Sister Setiva on how to persecute wounded beasts. Yugh! That dreadful gloop's gone right down me bloomin' ear. You've done it now, I'll be deaf on one side for the rest of me short young life. Rotters!\"\n\nSpringald tugged the hare's good ear sharply. \"Do hold still! What can you expect if you hop about like that? Now, I'll just dress it with some dock leaves.\"\n\nHorty looked blankly at her. \"What rock thieves? Speak up!\"\n\nWhen the dressing was completed, he lay down in a sulk, while Springald cast a glance at the grave. \"They've been gone an awfully long time. What d'you think, shall we go down there and check on them?\"\n\nFenna nodded eagerly. \"Yes, let's do that. You stay here, Horty. Take a nap or something.\"\n\nThey dropped over the edge onto the stairs, with their former patient calling after them.\n\n\"I say, what's a cap an' a dumpling? What's up, have you both gone mad?\"\n\nHolding paws, Springald and Fenna managed the steps and, placing their backs against the rough stone wall, crept forward cautiously. The ground took a curve, dipping steeply. Slowly stumbling on, in total darkness, they were relieved to see the faint glow of a torch ahead. The muted voices of their friends could be heard.\n\nFenna called out to them. \"Saro, Brag, is that you? We've come down for a little peek.\"\n\nThe otter's voice, which sounded rather grumpy, echoed back at them. \"I told ye t'stay on top, you should be mindin' Horty. Who knows wot that buffoon'll be up to be'ind our backs!\"\n\nSaro's voice interrupted him. \"Oh, there's no harm done, mate. Let 'em come an' take a look.\"\n\nIt was quite a sight. The passage opened up into an underground chamber, lined with stone walls. At its centre stood a plinth, littered with old bones and a white cloth habit that had faded to the texture of a cobweb. In front of the plinth lay what had once been a chair with wheels but now was little more than a small heap of dry, insect-bored sticks. There were two more torches in wall sconces on one wall behind the plinth.\n\nAfter Saro had lit them, she gestured about with her own guttering torch. \"Well, this is it, mates. We've travelled long'n'far, just to find this sad ole lot. Those bones are wot's left o' pore Abbess Sylvaticus. But can ye guess wot those rotted sticks are?\"\n\nSpringald picked up a piece of the timber in her paw. It crumbled to dust. \"Don't tell me, this was the chair once used by Sister Amyl. Those little round black stones with holes in them must have been its wheels. Huh, they're the only things recognisable after all this time.\"\n\nCrouching down, Bragoon sifted through the debris with his swordpoint. \"Must've been 'ere thousands of seasons. How did the rhyme go . . .\n\n\"Beneath the flower that never grows,\n\nSylvaticus lies in repose.\n\nMy secret is entombed with her,\n\nlook and think what you see there.\n\nA prison with four legs which moved,\n\nyet it could walk nowhere,\n\nwhose arms lacked paws, but yet they held,\n\na wretched captive there.\"\n\nBragoon rose up and put away his blade. \"Aye, that's Sister Amyl's chair, sure enough, but where's the Sister's secret?\"\n\nSaro gnawed at her lip. \"Imagine pore young Martha when we get back an' tell 'er there was nought but a pile o' dust an' four black stones!\"\n\nSpringald hung her head miserably. \"It doesn't bear thinking about. Now I wish we'd never found it.\"\n\nFenna retrieved the four little black stone wheels. She stowed two in her belt pouch and gave the other two to Springald. \"At least these'll prove we've been here. Come on, Spring, let's go back and see how Horty's doing.\"\n\nBragoon gave them one of the torches to guide them out. \"Aye, you young 'uns go an' do that. Me'n my ole mate are goin' to stay down here awhile an' search.\"\n\nFenna shrugged glumly. \"Waste of time, there's nothing left to search for. Oh well, please yourselves.\"\n\nThe otter cautioned them. \"Don't mention anythin' to Horty, wot with Miss Martha bein' 'is sister an' all that. Tell 'im we're still searchin'. Better still, take Horty back to ole Toobledum's 'ouse an' wait fer us there. We shouldn't be too long. Will ye do that for me?\"\n\nThey nodded and trudged back to Horty.\n\nToobledum had taken the liberty of making a meal for them from the remnants of the ration packs. His little sand lizard capered about on its back paws, delighted to see the young ones returning.\n\nThe old dormouse proudly raised his floppy hat. \"Sit down, one an' all, see wot I cooked up for ye. Me'n likkle Bubbub did ye a stew. 'Tis made of all things good, wid an apple crumble fer afters an' a drop o' me own special whortleberry cup brew to drink. Ho dear, wot 'appened to pore master Horty?\"\n\nHorty blinked oddly at the dormouse. \"What the dickens is the old chap wafflin' about? Who's he goin' to plaster for being naughty, wot?\"\n\nFenna roared down his good ear. \"He said, What's happened to poor master Horty!\"\n\nThe young hare waggled a paw in his good ear. \"No need to bellow, miss!\"\n\nThen he turned to Toobledum. \"Ah, well may you ask, little fat sir. I suffered a dreadful injury to the old ear, but I'm keepin' jolly brave about it. Mmmm, nothin' wrong with a chap's nose, though! That stew smells like just the ticket. Whack me out a large portion, sir dormouse, looks like a splendid cure for thickearitis!\"\n\nToobledum humoured Horty by giving him a large bowlful. The young hare was halfway through it when he held the bowl out. \"Don't stint on the stew, I always say. Never mind Brag'n'Saro, they're far too old to appreciate good scoff. I say, those two relics should be back by now. Huh, loiterin' around graveyards, bloomin' bad form, they'll go all morbid.\"\n\nIt was over an hour before the two searchers made an appearance. The dormouse and Bubbub welcomed them back. Springald gave them two bowls she had washed out. \"Toobledum made some delicious stew, but you'd better get some fast before Horty hogs it all down.\"\n\nThe young hare looked up from a beaker of whortleberry cup. \"I heard that, marm. Why should frogs fall down? Complete gibberish if y'ask me, wot!\"\n\nSpringald waited until the two had finished eating before she enquired. \"Well, did you find anything?\"\n\nSaro smiled at Bragoon, who winked back at her as he sipped his drink. \"Hmm, whortleberry juice! 'Tis a while since I've tasted that. Used t'be me favourite drink at one time.\"\n\nFenna twirled her bushy tail impatiently. \"You haven't answered the question. Did you find anything?\"\n\nSaro tasted her drink, still smiling secretively. \"Aye, 'tis nice, a sweet taste. Mind ye, I was allus partial to a drop o' nettle beer, like those otterpals o' yores drinks, up on the north coast.\"\n\nHorty looked from one to the other. \"Who's seen a ghost?\"\n\nFenna fumed. \"Oh, put a cork in it, Horty! Now, Mister Bragoon, Madam Sarobando, will you answer the question. Please!\"\n\nOld Toobledum chuckled. \"Heeheehee, I knows ye found somethin', yore both sittin' there lookin' like a pair o' toads eatin' trifle. Put the young 'uns out their misery an' tell 'em, mates.\"\n\nThe otter produced a small cylinder of parchment. He tossed it from paw to paw. \"We found it\u2014this is Sister Amyl's secret.\"\n\nSpringald was about to reach for it, when Saro caught the cylinder and stowed it in her belt pouch. \"No ye don't, Spring, this is for none but Martha t'read!\"\n\nFenna pouted indignantly. \"How do you know that?\"\n\nBragoon raised his eyebrows. \"Because, miss clever clogs, it sez so on the parchment. Read it to 'em, mate.\"\n\nSaro took out the little scroll that had been tied with a few threads to keep it closed. On the outside was some tiny, squiggly writing. She peered at it closely, reading slow. \"Only the one who needs this shall know my secret!\"\n\nBragoon levelled a paw at them. \"None of you young 'uns needs to know, only Martha, 'cos she's the one who needs it. We haven't looked at it ourselves, out o' respect to Martha. So nobeast is goin' to find out Sister Amyl's secret except that young hare back at the Abbey o' Redwall. We're bound back there at tomorrer's dawn, with all 'aste!\"\n\nFenna, however, still had a question that needed answering. \"But we saw the place, there was absolutely nothing down there but bones, powdery wood and dust. How did you come to find it?\"\n\nBragoon paused briefly before launching into his explanation. \"It was at the bottom o' that stone thing where Abbess Sylvaticus lay . . .\"\n\nSpringald interrupted. \"The plinth, you mean?\"\n\nSaro nodded. \"Aye, the plinth, that was it. We was about to leave the place, when I took one o' those torches off'n the wall, 'cos our torch 'ad gone out. Well, I stubbed me footpaw on the bottom o' the plinth, an' one o' the stones came loose. Brag pulled it out an' there 'twas, lyin' as safe an' neat as ye like, be'ind a stone all that time.\"\n\nFenna pursued her enquiry. \"But why hadn't it turned to dust like everything else down there?\"\n\nBragoon picked something that resembled a tiny pellet from under his pawnail. \"See . . . beeswax. It was wrapped tight in dock leaves covered with beeswax. I tell ye, it was difficult, separatin' that liddle roll o' paper from the beeswax, but we did it!\"\n\nToobledum poured them more drink. \"Well done, mates, you found wot you was questin' for. I'm 'appy for ye, one an' all!\"\n\nHorty humphed. \"Three scones on a wall, ridiculous, wot?\"\n\nSaro pulled Horty upright. \"I've had enough o' this nonsense. Toobledum, bring me some 'ot water, not too 'ot, mind. Brag, lay Horty on one side, wid that muddy ear upwards, an sit on 'im. I couldn't put up with 'im talkin' rubbish all the way 'ome!\"\n\nHorty guffawed. \"Hawhawhaw! Walkin' cabbage an' a bone? Poor old Saro's finally gone off her rocker, wot!\"\n\nSudden panic struck as Bragoon pushed Horty down and sat on him. \"What the . . . ? Gerroff me, you great plank-tailed bounder! Good grief, what's that nutty old squirrel doin' with a jug o' steamin' water? Help, somebeast, help! They're tryin' to kill me! Murderers, assassins! Boilin' me blinkin' brain, an' just 'cos I scoffed three flamin' bowls of stew? Spring, Fenna, strike the cads with rocks'n'clubs, save me!\"\n\nBut no help was forthcoming. The otter held him tight whilst Saro washed out his ear with warm water. A moment later it was all over. Toobledum gently treated the young hare's cleaned-up ear.\n\n\"There there, Sir Horty, ye'll live t'cook agin. This is an ole dormouse remedy, my special ointment. I makes it wid sanicle, feverfew an' a few secret herbs. So, 'ow does that feel, young master?\"\n\nHorty relaxed, closing his eyes blissfully. \"Bloomin' marvellous, old top, me ear is at peace an' very comfortable. Amazin' thing, too, I can hear again!\"\n\nSaro wiped mud and moss from her paws. \"See, we never killed ye after all.\"\n\nSpringald muttered under her breath. \"Pity.\"\n\nBragoon whispered back to her. \"Shame we missed our chance.\"\n\nFenna's eyes twinkled as she chuckled along with them. \"Just think of the food we'd have saved without a hare to feed.\"\n\nHorty opened one eye and fixed them with a baleful stare. \"I heard all that, you rotters!\"\n\nThey burst into laughter; even little Bubbub did a squeaky giggle.\n\nDawn of the following day saw the travellers bidding farewell to the dormouse and his lizard.\n\nFenna hugged the pair fondly. \"Why don't you come with us to Redwall? You'd both be very welcome there.\"\n\nToobledum's hat wobbled as he shook his head. \"Nay, pretty miss, me'n likkle Bubbub wouldn't ever leave Loam'edge. We ain't got much, but 'tis 'ome to us.\"\n\nSaro stroked the little sand lizard one last time.\n\nBragoon clasped the old dormouse's paw warmly. \"As ye wish, matey, take care of each other, an' live happy. Good-bye an' good fortune to ye!\"\n\nStanding on the hilltop above the valley, the five travellers looked back. Bubbub was shaking his tail as Toobledum waved his hat and shouted, \"Fare ye well, one an' all, an' take our good wishes with ye!\"\n\nFenna wiped her eyes as they marched off into the wastelands. \"Those poor creatures, it must be terrible for them. Living that lonely life, and with so little to eat.\"\n\nBragoon ruffled her ears affectionately. \"Aye, they're both goodbeasts, Fenn. But don't ye go believin' all that ole dormouse told ye, miss. Yore too young an' soft'earted.\"\n\nHorty took a pull from his canteen, filled with fresh water from Toobledum's well. \"Steady on, I liked Toob an' Bub. Bit unkind to talk about the old chap like that, wot?\"\n\nSpringald agreed with him. \"I think so, too. They've only got each other for company, and they shared what little they have. Why shouldn't we believe what Toobledum said?\"\n\nThe otter cast a wry glance at his old friend. \"That dormouse is a hermit, he likes bein' alone. As fer not havin' much in the way o' vittles . . . tell 'em, Saro.\"\n\nThe aging squirrel explained. \"When I was swillin' Horty's ear out, I watched Toobledum goin' to fetch that special ointment. Hah, the ole fogy didn't think I could see 'im. He went into a corner an' lifted a floorstone. Do ye know wot was underneath it? A cellar, packed from wall t'wall wid vittles. Drinks, dried fruits, veggibles, nuts, enough t'feed an army fer ten seasons. That's 'ow short o' food Mister Toobledum was, mates. Brag saw it, too!\"\n\nHorty stopped in his tracks. \"Well, the flamin' old fraud, wolfin' down all our grub an' tellin' whoppin' fibs about havin' none himself. What a blinkin' cheek!\"\n\nFenna could hardly believe what she had heard. \"That's a shameful thing to do, the old liar!\"\n\nSpringald was about to add to their condemnation of Toobledum, when the otter cut in. \"Don't be too 'ard on the ole feller. Vittles an' drink is precious in this region. If ye've got none, yore a deadbeast. Toobledum was only thinkin' of hisself an' liddle Bubbub. We were just passin' visitors. Now that we've gone, he's got to provide for 'imself an' his mate. 'Tis called survival. Ye don't go dashin' to the first beast ye see an' offerin' them a cellarful o' grub, ye takes care of yore own first.\"\n\nFenna held up her paws. \"Alright, we understand, Toobledum's only doing what's best for himself and Bubbub. Please don't rub it in by telling us we're only young and we'll learn.\"\n\nSaro winked knowingly at her. \"Wouldn't dream of it . . . young 'un!\"\n\nA raven flew out of the great gorge. Soaring high, it hovered on the evening thermals, a dark sinister shape framed against the setting sun for a brief moment. Then it swept off southward. Down below, standing at one of the three cave entrances, a cloaked figure watched the bird's departure, then turned and went into the tunnel. Pushing through the mass of dark creatures, the cloaked one made its way to where the sulphurous flame burned constantly, a tall column of fire, giving off its acrid stench.\n\nKharanjul, the Great Wearet Slayer, stood waiting for the news his guard captain brought. The cloaked creature lay flat on the rocky floor and lifted its face.\n\n\"It is as ye said, Mighty Lord, the travellers are even now returning, to sully thy abyss with their presence!\"\n\nKharanjul's hideous face stared impassively down at the captain. \"Did Korvusa say when they would arrive in my domain?\"\n\nThe speaker lowered his eyes from the Wearet's piercing gaze. \"The bird said that they would reach the gorge rim in the second hour of darkness, Great Slayer.\"\n\nKharanjul's trident pointed at the messenger. \"Take a score of my creatures across, to the other side of the long tree. Hide there and await my signal.\"\n\nAccompanied by twenty spearbearers, the captain marched out, prepared to cross the tree trunk that spanned the forbidding drop.\n\nThe Wearet's harsh voice grated out, echoing around the cavern. \"None of the trespassers must be slain, they are to be taken alive . . . to die at my pleasure!\"\n\nThe dark masses rose, spearblades glinting as they chanted, \"Blood of the Wearet runs in thy veins, O Mighty Kharanjul. Ruler of the Abyss! Lord of Life and Death!\"\n\n##\n\nAbbot Carrul sat soaking up the warm morning sun at the orchard entrance. Folding his paws across his stomach, he smiled at Old Phredd, sitting across from him at the breakfast buffet table. The ancient hedgehog Gatekeeper was chatting away to a bumblebee, which had landed on the rim of his beaker.\n\n\"Dearie me, have you seen the mess those vermin made of my gatehouse? There's scrolls, books and parchments scattered about on the floor. One would think a herd of wild beasts had been living there. Hmm, they have really, haven't they?\"\n\nThe bee buzzed, vibrating its tiny wings. Phredd pointed a bony paw at it. \"Oh, that's easy for you to say, my friend. But the curtains have been ripped, the cupboards flung open and the bed linen will have to be scrubbed twice before I use it again. Thrice, even!\"\n\nCarrul reassured his old friend. \"Don't worry so! Stretch your paws and enjoy being outdoors in our own Abbey again. We've time aplenty to put everything right, Phredd\u2014the remainder of summer and all the autumn.\"\n\nForemole Dwurl and a crew of his worthies trundled up to the table. The mole tugged his snout in respectful salute. \"Me'n moi moles've cleared opp all ee gurt 'eap o' rubble wot was blocken ee h'Abbey door h'entrence, zurr. Et bee's ready furr use naow, arfter they'm scrubbed ee wuddwurk daown!\"\n\nCarrul beamed gratefully at the Foremole. \"Well done, Dwurl, come and have some more breakfast. There's lots of it left here, do help yourselves.\"\n\nDwurl and his molecrew, needing no second invitation, fell to with a will.\n\n\"Thankee, zurr. Much h'obliged oi'm shurr!\"\n\nToran and Martha came running across the lawn. The ottercook called out, \"Gangway, make room there, mates, two more 'ungry workers comin' in for second brekkist!\"\n\nCarrul indicated two spaces either side of himself. \"Well, what have you two been up to? I haven't seen you since first serving at dawn.\"\n\nMartha cut herself a slice of fruitcake. \"Unblocking the windows in Great Hall, Father. Brother Gelf says he's got extra panes in the attic storeroom. The Dibbuns have gone up there to sort them out with him.\"\n\nForemole Dwurl looked up from his mushroom pastie. \"Hurr, better'n sennin' they'm likkle villyuns into ee h'orchard to 'elp wid ee fruit'n'berries. Hurr hurr, they'm loike to h'eat umselfs sick afore sundaown!\"\n\nSister Setiva was next to put in an appearance. \"Och, ye should see the state o' yon dormitory, it's no fit for worms tae crawl in. Ah'm goin' tae need some bonny helpers tae make et habitable again. Martha lassie, have ye seen that braw badger taeday?\"\n\nThe haremaid shook her head. \"No, Sister, last beast to see him was Granmum Gurvel. He took a few scones from the kitchen and hurried off. Wonder where he's gone?\"\n\nToran supplied the answer. \"Lonna said he'd sworn to wipe out all the Searats. A few of 'em escaped yesterday by the wallgates. I saw 'im fillin' up his quiver, an' waxin' that big bow, after we left the walltop last night. I thought better of askin' 'im where he was bound.\"\n\nMartha poured pennycress cordial for the ottercook. \"I don't blame you, mate. Lonna can flare up like lightning. If and when he returns, I won't be pressing him about where he's been. I pity those Searats, though. If it were me, I'd have let them go. They've learned their lesson, and a hard and bloody one it was, too!\"\n\nOld Phredd spoke to a bowl of oatmeal he was finishing. \"Ah, but Martha isn't him, is she? Big badger warriors like that are different from anybeast. If he swore to wipe out all those Searats, well that's just what he'll do. Every one of them, down to the last rat!\"\n\nThe last rat was, in fact, running for his life, out on the western flatlands. His tongue lolled from one side of his mouth as he looked over his shoulder at the distant figure of the avenging giant. Lonna was standing still, a long distance away. The Searat stopped as well, collapsing in a heap, his limbs wobbling and trembling uncontrollably. Then he bared his stained fangs at the sky and laughed breathlessly.\n\n\"Haharrharr . . . Done it! Can't get me now, stripedog . . . Outta yore range now . . . I escaped ye . . . stripedog!\"\n\nThe big badger grunted with exertion as he leaned down on the bow, bending it so that he tightened the string by taking another loop around the end of the thick yew wood. He shed his quiver. Going through the arrows, Lonna selected one. Then, holding it to his eye, he peered down the shaft to check that it was straight and true. Spreading both footpaws, Lonna gripped the ground firmly, wetted his upper lip and raised his head to feel which way the breeze was blowing. Satisfied, he looked toward the Searat, gauging the distance. Then he placed the shaft on his bowstring and drew back. The string was resting against one scarred ear, the bow strained in a mighty arc to its full capacity. After glancing once more at the distant Searat, Lonna elevated the bow slightly skyward and let fly.\n\nThe Searat rose upright, waving his sword at the tiny figure out on the flatlands. \"Outran ye, stripedog! I beat ye, didn't I?\"\n\nA distant, blood-curdling cry answered him. \"Eulalillllaaaaaa!\"\n\nThe arrow came like a thunderbolt out of the blue.\n\nLonna strode out to view his last work of vengeance. Spread-eagled on the coarse grass, the Searat lay faceup. His eyes were wide open, staring at a sun that he would never see again, the long arrow standing out from the centre of his forehead.\n\nThe badger gazed down at the last rat of Raga Bol's once mighty crew. \"Nobeast can outrun Lonna Bowstripe's arrow. Nobeast!\"\n\nUnstringing the bow, Lonna placed it lengthways across his broad shoulders, resting his paws on the weapon. Turning his back on the Searat, he strode east through the high summer midday to Redwall Abbey.\n\nNight had fallen as the travellers neared the great gorge. They were weary after marching since early morning that day.\n\nBragoon called a halt within a short distance of the rim. \"We'll take a breather, but without any fires. Eat'n'drink wot ye need, 'cos we'll be leavin' the rest behind.\"\n\nHorty slumped down gratefully. \"I say, what's the point of leavin' bloomin' good scoff here? Silly if you ask me, wot!\"\n\nThe otter spoke in a low voice to the three young ones. \"Keep yore voices down 'til we're on the other side o' that gorge, mates. I didn't like the place last time we crossed it, an' I likes it even less now. We've got to travel fast'n'light when we crosses that big tree trunk, so keep yore wits sharp!\"\n\nSaro had the rope. She tied a short, thick piece of wood crosswise to one end of it. \"Those broken staves we stuck atop o' the other side should still be in place. If'n I throws it right, this chunk o' wood will lodge atween 'em, an' we can climb up sharpish.\"\n\nSpringald cast a worried glance at the pair. \"You two don't like that gorge one little bit, do you? Don't fret, we'll cross it as quick and quiet as you like.\"\n\nSaro tested the knot she had tied around the wood. \"Aye, you do that, Spring. I'll be in front of ye, an' Brag'll bring up the rear. Just do as yore told, an' everythin' will be alright. I don't like that gorge any more'n Brag does. That place has a bad feel to it!\"\n\nFenna's voice was small and shaky as she tried to make a joke. \"Don't worry about us. We're young and we've got a lot to learn, but we're willing to listen to experienced old fogies.\"\n\nGrasping her paw, Bragoon smiled in the darkness. \"That's the spirit, missy. Right! Up on yore hunkers, mates. Let's get ye safe back to Redwall.\"\n\nDrawing the sword of Martin from behind his shoulder, the otter led them off toward the rim.\n\nHorty took one rueful glance at the small heap of provisions on the ground, then uttered a small sigh. \"What a flamin' waste. Ah well, this is it, chaps, off we jolly well go!\"\n\nThe edge of the chasm loomed up, sooner than they had expected. Saro found the same boulder she had used on the previous trip. Making a wide loop in the rope's free end, she placed it about the big stone, lowering the end with the wood attached into the gorge below. Bragoon went first. Climbing carefully, he reached the ledge in front of the three cave entrances. He held the wood, so that it would not clack against the rock wall. Springald came next, followed by Horty, then Fenna. Saro was last to descend. She flipped the rope deftly, catching the end as it unlooped from the boulder and dropped down.\n\nBefore Bragoon stepped onto the tree trunk, he pressed something into Fenna's paw and whispered, \"Shove this in yore belt pouch, no questions.\" Without a word, the squirrelmaid stuffed the object into her pouch, then followed the otter out onto the tree trunk bridge.\n\nTotal silence and engulfing dark reigned in the yawning chasm. Holding one another's paws, the five travellers edged slowly forward, step by step. They were almost across to the far side when a harsh, evil laugh sounded out from behind them.\n\nSuddenly the lights of many smoking torches lit up the gorge. Saro turned, gasping at the incredible sight.\n\nKharanjul stood on the trunk, backed by an army of vermin, each holding a torch in one paw and a spear in the other. They were mainly ferrets and weasels, with a scattering of large rats among them. Everybeast's fur was thickly daubed with a sickly yellow-and-green substance, giving them a sinister, spectral appearance. But it was the horrific form of their leader that stood out.\n\nThe Wearet swung back his cloak, revealing a misshapen but powerfully bulky torso. As he gestured at them with a big, three-pronged trident, his monstrous face split into an ugly grin. \"Stop where ye stand, trespasser! You belong to Kharanjul, Lord of the Abyss! I will punish you for intruding on my domain!\"\n\nSaro pushed Horty forward. \"Keep goin', we're almost across!\"\n\nBragoon was about to jump from the log onto the opposite ledge, when a score of vermin rose up in front of him.\n\nThe captain, a tall weasel, snarled in his face. \"Stand still! Obey the Great Lord of Life and Death!\"\n\nThe otter laughed, then slew him with a single swordthrust. Catching the captain's spear as it fell, Bragoon tossed it back to Saro. \"Keep 'em busy, mate. Redwaaaaaallllll!\"\n\nHurling himself from the tree trunk, Bragoon roared like a madbeast as he dealt out death and destruction with the sword of Martin the Warrior. \"Heeeeeyaaaaaah! Grab some spears, young 'uns! No pack o' fancy-talkin' vermin are goin' to stop us Redwallers!\"\n\nHorty seized a long spear and was suddenly in the thick of the battle, whooping and bellowing. \"Forward the buffs, give 'em blood'n'vinegar. Chaaaaarge!\"\n\nBelting a weasel flying into the abyss, the young hare stood shoulder to shoulder with the otter\u2014cutting, thrusting and slashing. Springald and Fenna armed themselves with fallen spears. They turned to help Saro, but the aging squirrel would have none of it. Single-pawed, she held the centre of the log bridge, letting none pass. Using her spearblade, she slashed at a ferret, flaying his footpaw. He hopped off into midair and vanished screaming.\n\nSaro yelled at the two Abbeymaids. \"Take this rope an' see if ye can fix it t'the top. Then go an' help Brag an' Horty. I'm fine right 'ere, they can only come at me one at a time!\"\n\nThey obeyed her immediately. As they jumped off the tree trunk, a big rat charged Springald, but he vanished over the rim with a yowl of dismay when Fenna pushed him with her spearbutt.\n\nThe squirrelmaid was momentarily stunned. \"I've just slain somebeast!\"\n\nSpringald shouted. \"Good! Mind your back, Fenn!\"\n\nThe mousemaid deflected a spear with her own. She thrust and saw the look of surprised horror on the vermin's painted face as he fell dead.\n\nSteeling herself, Springald stood back to back with her friend. \"Keep fighting or we're deadbeasts!\"\n\nBragoon and Horty fought their way through to the side of the two maids. The otter despatched a charging weasel, then shouted, \"Gimme that rope, Spring. You three, cover my back!\" Grabbing the rope, he whirled it and flung it up, but it fell back. Bragoon whirled it once more, gritting his teeth against the swordblade held between them. This time his throw was good; the chunk of wood lodged between the two broken staves which they had fixed into the plateau. The otter swung his weight onto the rope, testing it. The rope held firm. He turned to the three young ones.\n\n\"Come on, mates, up y'go! Horty, take this sword, 'tis too short for fightin' spears with. Pass me yore spear an' get climbin'!\"\n\nHorty gave him the spear and took the sword, but the hare refused to climb up. There were six vermin left to face on their side. Slaying one with a slash to the throat, Horty shook his head. \"Let Spring an Fenn go, I'm stayin' here with you, sah. True blue an' never fail, that's us Braebucks, wot!\"\n\nThe otter whacked a vermin over the skull with his spear, then kicked him swiftly into the abyss. Blood was flowing from a wound on his forehead as he turned on Horty furiously. \"I said, git up that rope, hare. Do it now!\"\n\nBetween them they faced off a vermin, who was very fancy with his spearwork.\n\nHorty muttered rebelliously. \"I ain't goin', otter! I can't leave you an' Saro here to face that flamin' lot on your own!\"\n\nBragoon's eyes were blazing as he faced Horty. \"Wot did I tell ye, I'm in charge 'ere. Obey me . . . Argh!\"\n\nThe vermin's spearpoint took the otter through the footpaw. He pulled the spear from his foe's grasp, ran him through with it and booted him into oblivion. Livid with wrath, he rounded on the young hare. \"Don't argue wid me, mate! Yore young, like them two maids, you got all your lives ahead o' ye. Get that sword back to Redwall! Me'n Saro knows wot we're doin'. We can't look after three young 'uns who are still wet be'ind the ears, we've lived one summer too long fer all that! Now git up that rope, Horty, or I swear I'll run ye through wid this spear! Look after the two maids, live yore life for us. Now go!\"\n\nLeaving Horty one vermin to deal with, the otter turned and limped out onto the long tree trunk to help Sarobando.\n\nHorty downed the vermin in a perilous rage, needlessly striking at the foebeast's carcass. Springald and Fenna looked down from the top of the plateau, howling hoarsely at their young friend.\n\n\"Horty, come on, get up the rope!\"\n\n\"You must obey Bragoon, do as you're told!\"\n\nClamping the sword in his teeth, Horty ran to the rope. He took one backward look at Bragoon and Saro. Though wounded in a dozen places, they were still fighting savagely.\n\nReeling from a blow, Saro caught his eye and bellowed. \"Get to the top an' pull the rope up, or they'll come after ye an' slay those two maids. Go! Go!\"\n\nBlinded by tears of rage and helplessness, Horty went.\n\nKharanjul stood on the far ledge, urging his creatures forward. Seeing so many killed by the two old battlers who were holding off the advance, the Wearet took up his trident and went out to fight.\n\nThe vermin were still coming. Bragoon and Saro were bowed with fatigue, but covered in blood and severely injured, they were still taking on all comers. They fought side by side on the narrow causeway of timber, keeping their eyes on the advancing enemy, talking to each other as they thrust and parried.\n\nSaro panted. \"The young 'uns are gone. Pity we couldn't 'ave gone with 'em.\"\n\nBragoon dislodged a foebeast with his spear. \"Wot, y'mean back to Redwall? Don't think I could've stood it, mate, sittin' in the gate'ouse wid Old Phredd countin' me teeth as they fell out an' dozin' all day!\"\n\nSaro wiped blood from her eye and chuckled. \"Dibbuns climbin' all over us, ole Setiva physickin' away at us, wrappin' rugs round our laps in winter!\"\n\nBragoon caught a ferret in the throat with his spear. \"Might be even worse\u2014they could've sent us back to Abbey school. We'd 'ave Sister Portula teachin' us to readn'write'n'figger. No, that ain't fer us, pal!\"\n\nThe aging squirrel caught sight of Kharanjul advancing. \"Oh, look out, Brag, 'ere comes the big ugly mug. We'd better start backin' off. Blood'n'fur, lookit the size o' that monster, he must've ate some dinners with that fork!\"\n\nThe vermin advanced on them as they retreated. Sorely wounded and drained of strength, the two old friends continued to hack and slash. Vastly outnumbered by their adversaries, and knowing that they would be beaten and captured by Kharanjul, they crawled down from the long bridge, with one last desperate plan in mind.\n\nThe Wearet pressed forward, holding his lethal trident ready as he taunted his victims. \"The Lord of Life and Death will keep you alive. I will make your dying long and slow. Your companions escaped the Wearet, but you shall pay for them!\"\n\nBragoon and Sarobando were not listening. Between them they had jammed a half-dozen spearbutts under the end of the tree trunk. With a last mighty effort, both beasts put their shoulders to the spears, using them as levers.\n\nSaro gritted her teeth and growled. \"One, two, three. Push!\"\n\nThe long trunk moved askew, with a grating of wood on rock. Panic ensued out on the tree trunk, as vermin tried to run back to the other side. Some threw themselves flat and clung on.\n\nBragoon yelled above the din of wails and screams. \"She's movin', mate! Again, one, two, three! Push!\"\n\nKharanjul dropped his trident. Crouching low, he gripped the tree trunk, trying to move forward and reach his enemy. At last the mighty trunk of the tree, once called Lord of Mossflower, groaned like a living thing as it made a half-turn to one side and slid over the brink of the ledge.\n\nWith the effort of their final push, the two old warriors had fallen flat. They watched as the log seemed to hang for a split second in space, with Kharanjul's face looming in front of them. Then the whole trunk fell into the bottomless abyss. Screeches and shrieks rent the night air, swiftly fading to echoes. Down, down, into the dark gorge it all plunged\u2014the Wearet, his vermin army and the only solid bridge that had ever spanned the awesome space.\n\nBragoon and Saro lay there, staring down into the void, their paws clasped. The aging squirrel closed her eyes. \"Nice'n'peaceful 'ere now, mate, ain't it?\"\n\nThe otter gave her paw a faint squeeze. \"Aye, restful ye might say. Summer's a good time to lay down an' rest.\"\n\nNo longer able to keep them open, Bragoon slowly closed his eyes. \"Saro, ye recall wot it said on that gravestone at Loamhedge? Young Fenna read it out to us. I said I liked the sound of it.\"\n\nSaro nodded weakly. \"I remember, mate. It said 'Gone to the sunny slopes an' quiet streams.' I liked it, too.\"\n\nThe otter's voice grew fainter as he repeated the phrase. \"The sunny slopes an' quiet streams . . . I'll wait for ye there, Sarobando . . . Wouldn't go anyplace without ye.\" His paw went limp in the squirrel's failing grasp.\n\nShe smiled. \"Wait for me, Brag ole mate, I'll be there.\"\n\nTwo old warriors, who had left Redwall Abbey when they were Dibbuns, paw in paw, lay on the rockledge together. They never saw the sunrise that dawn, but they went on to the land of sunny slopes and quiet streams\u2014still holding paws.\n\n##\n\nSummer's days were growing short, passing gently into autumn. Redwall Abbey was restored to its former calm and grandeur. Abbot Carrul and Martha met for their early morning stroll, now a regular thing with the two friends before they took breakfast. A light mist\u2014like golden gossamer\u2014lay over the Abbey pond. They saw a grayling leap to catch an unsuspecting fly.\n\nCarrul watched the ripples spread across the water. \"I had a dream last night. It was a vision of Martin.\"\n\nThe haremaid was startled by her Abbot's revelation. \"A dream of Martin the Warrior? Did he say anything, Father?\"\n\nThe Abbot paused before answering. \"He did, indeed, Martha. These were his very words.\n\n\"When autumn brings the harvest time,\n\ngood food you shall not lack,\n\nwhen fruit lies heavy on the bough,\n\nand travellers come back.\n\nLook for the one who holds my sword,\n\nthese words of mine recall,\n\nsomeday you will esteem that one,\n\nas ruler of Redwall!\"\n\nMartha sat down on a log, puzzled by the rhyme. \"Good grief, Father, there's a lot of information in Martin's words. Aside from the fact that there will be a fine harvest, our friends\u2014Horty, Bragoon and the others\u2014must be returning. Isn't that great news! But I never guessed you were thinking of retiring from being Abbot of Redwall.\"\n\nCarrul sat down beside her. \"The thought never crossed my mind, Martha. But Martin said someday, and someday in the future I would have to give serious thought to appointing my successor. Martin has saved me a lot of pondering, I'm grateful to him for that. However, his words are causing me a little concern. Think. Who did I give the sword to?\"\n\nThe haremaid replied promptly. \"You gave it to Bragoon.\"\n\nCarrul nodded his agreement. \"Which is why I'm worried, Martha. Bragoon is a good friend, we were Dibbuns together. But he's a rover, an adventurer. Ask Toran, Bragoon's his elder brother, he'll tell you. Bragoon's too old and too wild to be Abbot.\"\n\nMartha held up a paw. \"Not so fast, Father. The rhyme said ruler of Redwall, not Abbot. It may be an Abbess!\"\n\nCarrul clapped a paw to his cheek. \"Fates forbid that it might be Sarobando! It would be woe to my poor Abbey.\"\n\nMartha could not help laughing. \"Hahaha, oh Father, think for a moment. It could be Springald, or Fenna or . . .\" Now it was Martha's turn to look apprehensive. \"Or Horty?\"\n\nCarrul placed a comforting paw on the haremaid's shoulder. \"Oh, come on now, miss! Martin the Warrior was renowned for his wisdom. What are we thinking about? He wouldn't inflict any of those three rascals on our Abbey!\"\n\nMartha gave an audible sigh of relief. \"You're right, Father. But it might be nobeast we've thought of. What if they bring somebody back with them?\"\n\nCarrul pursued this idea enthusiastically. \"Of course, there may have been other creatures living at Loamhedge. Say a sturdy young mouse, steeped in wisdom? Or a sagacious squirrel, the very model of common sense?\"\n\nMartha giggled. \"Or a studious frog with the brain of an ant!\"\n\nAbbot Carrul smacked her paw playfully. \"Now stop this nonsense, you young rip. Look, here comes breakfast!\"\n\nToran had resumed his role as cook. He and Gurvel headed a procession carrying tables and benches, trolleys, dishes and food. He waved his ladle.\n\n\"Set 'em all up at the edge of the pond there, next to those two pore beasts who've been waitin' out here all night!\"\n\nCarrul chuckled. \"So ends our moment of peace for the day, Martha. Besieged by breakfasters!\"\n\nThe haremaid went to help the servers. \"Let's join them, I'm starving!\"\n\nSetting up the tables, Brother Weld pulled a ferocious face at the Dibbuns, who were buzzing around like playful bees. \"I'll toss the lot of you into the pond if you don't sit still and wait to be served. So behave yourselves!\"\n\nMuggum the molebabe clambered up on a bench, next to Buffle. \"Hurr hurr, ee'm a gurt bold crittur t'be assultin' uz loike that! Wot do ee say, Buff?\"\n\nThe tiny mousebabe scowled darkly. \"Gurrumff um burble fink!\"\n\nOld Phredd looked over his glasses at the infant mouse. \"What did he just say?\"\n\nSister Setiva tied a bib about Buffle's neck. \"Och, ah be afeared tae repeat it. But if the wee scamp says it again, ah'll wash his mouth out with soap!\"\n\nThe Dibbun squirrelbabe Shilly tugged at Martha's paw. \"When izza harviss gonna be, Marth'?\"\n\nThe haremaid gave the reply she had been repeating to the Abbeybabes for the past few days. \"On the first morning after the night of harvest moon. Be patient, it shouldn't be too long now.\"\n\nGranmum Gurvel looked up from a pan of corn and fruit slices she was doling out. \"Payshunt? You'm doan't tell ee Redwallers t'be payshunt when they'm a waitin' to get ee ' arvest in, Miz Marth'!\"\n\nThe Dibbuns cheered Gurvel loudly, glad to have an ally on their side. Most of the babes had never been to a harvest before, so they were eager to take part in one, knowing there would be a Harvest End Feast. The little ones began clamouring for Martha to sing the Harvest Song. Knowing they would not be quiet until she did, Martha obliged by singing the lively air, which included much tapping and paw stamping.\n\n\"Open the cupboard the bins and the stores,\n\ngo fetch out the trolleys and carts,\n\nthen out to the orchard the gardens and fields,\n\nfor a harvest to gladden our hearts.\n\nRappety tap, the Abbot'll call,\n\nwatch out for those Dibbuns 'cos they'll eat it all!\n\nThere's blackberries blueberries raspberries too\n\nstrawberries and redcurrants bright,\n\nwild cherries blueberries and blaeberries ripe,\n\nto be all gathered in by tonight.\n\nRappety tap, wait for the feast,\n\njust look at that Dibbun, the greedy wee beast!\n\nBring basket and barrel and bucket and pail,\n\npick rosehips red apples and pears,\n\ngreengages damsons and plums big and fine,\n\nroll your sleeves up and banish your cares.\n\nRappety tap, that babe's the worst,\n\nif he eats another I swear that he'll burst!\n\nThere's almond and hazel and chestnut in bloom,\n\nand a crop of good acorns there'll be,\n\nif you hold the ladder I'll climb to the top,\n\nand I'll knock them all down from the tree.\n\nRappety tap, flat on the ground,\n\nhe's rubbing his tummy and rolling around!\n\nLet's gather our harvest and bring it indoors,\n\nthen the Abbot'll cry out 'Well done!'\n\nWe've filled up the cupboards the bins and the stores,\n\nin good time for the winter to come.\n\nRappety tap, quick close the door,\n\nhe's up on his paws and looking for more!\"\n\nAs usual, Martha had to sing the whole thing again so the Dibbuns could show off their fancy paw tapping. Whilst this was going on, the Abbot took Toran aside. He related what Martin the Warrior had told him in his dream. The ottercook was overjoyed at the news.\n\n\"As soon as the 'arvest moon shows, we'll mount a watch on the walltops to welcome them back 'ome!\"\n\nBreakfast was about finished when Foremole Dwurl, who had been gatekeeper in Phredd's absence, came trundling up with Lonna in tow. He hailed Toran. \"Gudday, zurr. Lookit who'm just cummed a knocken on ee gate!\"\n\nThe ottercook quickly cleared a place for them both. \"Sit ye down, mates, an' break yore fast. Lonna, where've ye been since yesterday? Everybeast was wonderin' where ye'd got to.\"\n\nThe big badger seated himself, allowing Gurvel to heap food in front of him. His fur was coated in dew, and the blood had matted on his wounds, but he looked happy. \"The Searats are all accounted for\u2014down to the last vermin. I was tired, but glad that I had ended my mission, so I lay down on the flatlands, about half a league from your Abbey. I must have slept deeply, because it was the sound of larks rising at dawn that awakened me. I was hoping there'd be a bite of breakfast left for a hungry badger. Thank you, marm!\"\n\nGranmum Gurvel piled corn and fruit slices on a platter. \"You'm eat 'earty, zurr. Oi'll cook more furr ee if'n ye be still 'ungered!\"\n\nAbbot Carrul beckoned Brother Gelf. \"Draw off a pitcher of our best October Ale for Lonna. Nothing's too good for the beast who saved Redwall from the Searats. Lonna, after you've eaten, Sister Portula will find you clean robes, and Sister Setiva will care for your wounds. You must rest now, friend!\"\n\nSeating themselves around the badger, the Dibbuns watched in awe as he satisfied his appetite.\n\nMuggum nudged Buffle. \"Yurr, ee'm gurtbeast surpintly can shuv ee vikkles away!\"\n\nStifling a smile, Martha chided the molebabe. \"Really, Muggum, mind your manners!\"\n\nLonna sat Muggum on his paw and lifted him to face-height. \"Listen to me, young sir, never mess with your food. Eat it all up like I do, then someday you'll be a great warrior!\"\n\nMuggum nodded sagely. \"Them bee's woise wurds, zurr!\"\n\nIt was five nights hence when the harvest moon waxed fully. Most Abbeybeasts were in their beds. Toran stood watch from the ramparts on the southwest corner, where he could view both the path and woodlands. Martha and Abbot Carrul, neither of whom felt like sleeping, joined the ottercook on his vigil. The three stood there, unaware that Lonna had come up behind them. For a beast of his size and weight, the badger could move silent as a shadow. They started slightly as he spoke.\n\n\"That nice old molewife in the kitchens asked me to bring some hot vegetable soup up for you.\" Lonna poured the soup from a jug into four basins.\n\nToran sniffed it, exclaiming gratefully, \"Good ole Gurvel!\"\n\nThey sipped at their basins in silence, contemplating the serenity of a late summer's night.\n\nNocturnal birdsong drifted from the shadowed trees of Mossflower. The path stood out like a tranquil stream, curling southward. Galaxies of twinkling stars pinpointed the cloudless vaults of sky above. A single comet streaked through space in brief silent glory. The harvest moon ruled over all, surrounded by a soft nimbus, resplendent in its own golden solitude.\n\nNone of the others noticed Lonna fitting a shaft to his bowstring. He peered toward the foliage which fringed the pathside. Drawing back his bow, Lonna called down, \"Are you friend or foe?\"\n\nThree figures stepped out onto the path. One shouted, \"Ahoy the walls, we come as friends!\"\n\nMartha's good eyesight allowed her to quickly identify the caller. \"It's a shrew. There's two more with him.\"\n\nLonna relaxed his bowstring. \"What do you want, friends?\"\n\nThe lead shrew's rapier blade flashed in the moonlight as he made a salute and offered it hilt first. \"I am Jigger, son of Log a Log Briggy, Chieftain of the Guorafs crews! I carry news of your friends. My father sent me ahead to tell you of their approach!\"\n\nToran and Lonna were already down and unbarring the gates as Martha assisted the Abbot to negotiate the wallsteps.\n\nOld Phredd lit extra lanterns as they crowded into the gatehouse. When the introductions were completed, the young shrew made his report. It was not a happy tale that he had to relate. Martha was stunned beyond tears at the news of Bragoon's and Saro's death. Abbot Carrul hung his head and wept openly. Lonna stood by in respectful silence. Toran was the only one to speak.\n\n\"The young 'uns, are they all safe'n'well?\"\n\nJigger nodded. \"Aye, sir. Apart from a few scratches an' sore footpaws, they're fine. Miss Fenna told me that Bragoon was yore brother. 'Twas a brave thing him'n Saro did.\"\n\nThe ottercook drew himself up straight and spoke proudly. \"Aye, Bragoon an' Sarobando was true-born Redwallers! No two like 'em, they was both wild warriors. But they did their duty an' saved their friends. I wager they took a few o' those vermin with 'em, eh?\"\n\nJigger's eyes were shining with admiration as he replied. \"From wot our scouts said, they took 'em all, every last vermin, an' their chief, the Wearet. That must've been a powerful battle, I'll tell ye!\"\n\nToran opened the gatehouse door. He took a deep breath of the fresh night air and smiled. \"Funny, ain't it, some'ow I couldn't imagin' Saro an' Brag growin' old like peaceful Abbeybeasts. Not those two. They went like they wanted to, the bravest o' the brave!\"\n\nLonna offered his paw to Toran. \"True warriors have no fear of death. I only met your brother and his friend once. They were rare beasts!\"\n\nAbbot Carrul wiped at his eyes with damp habit sleeves. \"Look at me, I must have forgotten my manners. Come to the Abbey, Jigger. Bring your two friends. You must be hungry after travelling so far. Lonna, will you wait by the gates and show the rest of them to the kitchens when they arrive?\"\n\nJigger hitched up his rapier belt eagerly. \"Lead on, Father. If'n the vittles at Redwall are as good as Horty tells me, I can't wait t'get at 'em!\"\n\nIt was some time thereafter when Lonna herded the Guoraf shrews into the Abbey kitchens. Martha and Toran forged their way through the crowd to Horty, Fenna and Springald. They fell upon one another, hugging and shaking paws as the Abbot joined them.\n\n\"Welcome home, you weary travellers! Springald, what's up, miss, are you ill?\"\n\nThe mousemaid was staring at Martha in disbelief. \"Look, she's walking! Martha's walking!\"\n\nHorty held his sister at paw's length. \"But how the . . . what the blinkin' flip . . . I mean, the bloomin' skin'n'blister trottin' about like . . . like? Explain y'bally self, miss. How did y'do it, wot wot?\"\n\nThe haremaid stared down at her brother's bandaged right footpaw, having noticed he was sporting a gallant limp. \"You'll get to know all about it later. But what happened to you, Horty, are you hurt?\"\n\nTrying to look brave and nonchalant at the same time, Horty waffled. \"Oh this, line o' duty an' all that, y'know!\"\n\nSpringald raised her eyebrows scathingly. \"Line of duty, my tail! You great fibber, tell the truth. He was messing about with Martin's sword, showing off to the shrews, when he dropped it, tripped up and cut his footpaw on the blade!\"\n\nThe Abbot exchanged a glance with Martha before he asked. \"Well, who has the sword now?\"\n\nFrom beneath her cloak Fenna produced the sword, neatly wrapped and tied in a piece of sailcloth. \"I took it off Horty and bound it up for safekeeping. Don't worry, it's in perfect condition and quite undamaged.\"\n\nShe looked from the Abbot to Martha. \"Why, what's the matter, did I do anything wrong, should I have left the sword for Horty to fool about with?\"\n\nBoth Martha and the Abbot hugged the squirrelmaid.\n\n\"No no, you did the right thing!\"\n\n\"Yes, you did, Fenna, thank goodness! Please accept the gratitude of an old Abbot!\"\n\nThe squirrelmaid passed the sword over to the Abbot. \"Why, Father, what's this all about?\"\n\nAbbot Carrul looked over his glasses at her. \"Nothing for you to worry your head about, miss. I'll tell you everything in a few seasons' time\u2014that's, of course, providing you don't plan on leaving us to go somewhere else.\"\n\nFenna replied promptly. \"Why would I go anywhere else? Redwall Abbey's my home, I'd never leave it for anything!\"\n\nMartha clasped her friend's paw. \"Neither would I, Fenna. There's noplace dearer than our Abbey!\"\n\nWhilst the shrews were sampling the delights of the kitchens, the three young ones went down to Cavern Hole with Toran, Martha and the Abbot. There, away from the hubbub, they sat by the embers of a glowing fire, recounting their journey to and from Loamhedge. It was an engrossing story, vividly illustrated by the young creatures' first real experience of the outside world\u2014enemies they had encountered, friends they had met, hardships they had undergone. Horty, Springald and Fenna each related the parts they had taken in the epic quest. Throughout the narrative it was clear that the entire thing would not have been possible without the heroism, guidance and assistance of Bragoon and Sarobando. By the time the dawn bells were tolling, the trio had reached the end of their tale. Fenna reached into her belt pouch and drew forth the slim package of parchment which Bragoon had entrusted to her. \"This is Sister Amyl's secret. Take it, Martha, it's meant for you alone. I know you don't have any need for it now, but I feel you should have it.\"\n\nThey watched in silence as the haremaid undid the wrapping and began scanning it.\n\nHorty leaned forward eagerly. \"Well, are you goin' to jolly well read it out to us, or are you goin' to sit there bloomin' well gazin' at the blinkin' thing until next flippin' summer, wot?\"\n\nMartha hesitated. \"I'm not supposed to, really. It says on the other side of this parchment that only the one who needs this shall know my secret. But I don't think it will do any harm now. Here's what it says:\n\n\"The body is ruled by the mind,\n\nI tell you this be true,\n\nby willpower you may find,\n\nnought is denied to you.\"\n\nAbbot Carrul took the parchment from her paws. He stared at it, turned it over, studied it a moment longer, then chuckled as he passed it to Toran. \"So, that's Sister Amyl's secret, eh? Take a look at that, Toran, my friend!\"\n\nAfter a brief glance the ottercook burst into laughter. \"Hohoho, Sister Amyl my granma's rudder! Hohoho, those rascals!\"\n\nMartha was astonished at the attitude of her friends. \"Excuse me, I fail to see what's so funny. Those are the words of a young Sister who suffered the same as I did. I can see that it's written in an old-fashioned style, and the writing isn't too neat. But what's that to laugh about?\"\n\nAbbot Carrul explained. \"This was never written by Sister Amyl. She was a young Sister who was well educated, her spelling was faultless and she had a neat writing paw. Remember the history of Loamhedge you were reading, the one you borrowed from Sister Portula? Amyl had written part of that, but she certainly never wrote this!\"\n\nThe light of recognition dawned across Fenna's face. \"I'll wager I know who did write it. I've just remembered where I've seen that parchment before. It's a piece torn from the edge of the Loamhedge map we took with us. See, there's a line on it that was the rim of the high cliffs. Bragoon or Sarobando must have written it. Everything they found in Abbess Sylvaticus's tomb had mouldered away to dust. So they invented Sister Amyl's secret themselves rather than return to Redwall empty-pawed. That's it! Either Saro or Bragoon did the writing.\"\n\nAbbot Carrul patted the squirrelmaid's shoulder. \"Well done, miss! Actually, Bragoon or Saro didn't write this singly, they both did! I recognise the writing on the outside, Sarobando did that. She was better at spelling but worse at writing than Bragoon. He had the neater paw but oh, dear me, that otter's spelling was dreadful, look at it!\"\n\nThey examined the short rhyme closely.\n\n\"The body is rooled by the mynd,\n\nI tell you this be troo,\n\nby willpower you may fynd,\n\nnort is denyed to you.\"\n\nToran wiped tears of laughter from his eyes. \"Aye, that was my brother, alright. Old Brag never won any prizes for his spellin'. But he did it so ye wouldn't be let down, Martha. So I think ye could forgive 'em both for it.\"\n\nMartha stared into the fire embers. \"Forgive? There's nothing to forgive. They did it for me, undertook that whole long quest, protected my brother and his friends, then sacrificed their lives for them.\"\n\nToran did not know whether he was smiling or weeping. He scrubbed a paw across his eyes again. \"So they did, Martha, so they did!\"\n\n## Epilogue\n\nTen seasons have passed since that night of the harvest moon. Fate and fortune have allowed our Abbey to prosper in peace. We had some visitors to Redwall the other day, a column of fighting hares from Salamandastron, sent by Lord Lonna Bowstripe. They were led by Captain Hortwill Longblade Braebuck, who was visiting his sister. What a change the Long Patrol has made to Horty! He went off all that time ago, with Lonna, to enlist at Salamandastron, carrying the scimitar that the badger had taken from Raga Bol. Horty is now twice as big, and twice as hungry, as he once was, a fine figure of a Long Patrol captain with a bristling military moustache. The young hares under his command admire him greatly. Abbess Fenna was delighted to see him, and so was I. We sat up until late last night, chatting about the old days, with Carrul our Gatekeeper and Cellarmole Muggum. Yes, Abbot Carrul became Gatekeeper, by his own choosing, four seasons back. He shared the task with Old Phredd until the ancient hedgehog went to his long rest last winter. Ah well, such is life, and such is its passing; not even Phredd could live forever. Horty and his hares are staying until after the Harvest Feast. (Trust hares never to miss a chance of several days' good feeding.) Those are Ottercook Toran's words, not mine, though I share his sentiments.\n\nOn the night of the harvest moon, all our Abbeybeasts will gather on the lawn near the front wallgates. We will watch Martha climb the steps to the threshold. First she will sing the beautiful ballad, \"The Rose of Redwall.\" Then she will carry out the promise she made to Bragoon and Sarobando. They say that, on the day they left to search for Loamhedge, she vowed that when they returned she would dance for both of them, on the walltop, right over the threshold of our Abbey. Martha has kept her promise every season since then. After singing her song, she dances\u2014swaying, bending, curtsying and leaping\u2014graceful as a breeze-blown flower in the golden moonlight, for the memory of her two friends. I remember then that long ago summer when we stole out of the Abbey, rebellious young creatures embarking on a great adventure. We returned at the end of that season\u2014wiser, more obedient and more reasonable. It was the summer of growing up.\n\nIf you, too, are travelling, questing or journeying anywhere, remember this. You will always find a welcome here at Redwall Abbey, young or old. As friends come by, they often call in to enjoy Redwall hospitality. Who knows, maybe we will see you here someday. You can sit with us, rest and be refreshed and learn. Young ones have much to learn and old ones, too. Carrul said to me only the other day, we are never too old to learn. He was a wise Abbot; he is an even wiser old Gatekeeper. I hope someday I may grow as wise as him.\n\n_Springald. Recorder of Redwall Abbey in Mossflower country_\n\n*See _Mattimeo_\nThis is a work of fiction. Names, characters, places, and incidents are either 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.\n\nThe Penguin Putnam Inc. World Wide Web site address is  \n<http:\/\/www.penguinputnam.com>\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzrbvt b/data_all_eng_slimpj/shuffled/split2/finalzzrbvt
new file mode 100644
index 0000000000000000000000000000000000000000..847d925202c1fed217caa4e548ecf117b443be46
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzrbvt
@@ -0,0 +1,5 @@
+{"text":" \nThe Last Little  \nBlue Envelope\n\nmaureen johnson\n\nTo all the jars. You know who you are.\nLondon is a riddle. Paris is an explanation.\n\n\u2014G. K. Chesterton\nContents\n\nEpigraph\n\nPrologue\n\nDelusions of London\n\nSurprises and Explanations\n\nPairs of Shocks\n\nThe Devil's Bargain\n\nThe Pool\n\nThe Feast\n\nBoxing Day\n\nIt Takes a Thief\n\nOne More for the Road\n\nThe Talking Letter\n\nThe Card Cheat\n\nThe Great Table Caper\n\nTo Belgium!\n\nA Feeling of Shed\n\nThe Law of Pants\n\nThe Stain on the Page\n\nThe Koekoeksklok\n\nA Night of Vice\n\nRandom Acts of Cruelty\n\nThe House of Secrets\n\nThe Emerald Isle\n\nA Death in Ireland\n\nThe Bells\n\nThe Crossing\n\nReality Comes to Visit\n\nThe Dotted Line\n\nThe Conversation\n\nThe Probably Problem\n\nThis Is Not a Pool\n\nAcknowledgments\n\nAbout the Author\n\nAlso by Maureen Johnson\n\nCopyright\n\nAbout the Publisher\nPrologue\n\nIt was that time of day again. Time to stare at the question, the two lines of black on an otherwise blank page.\n\nQuestion: Describe a life experience that changed you. What was it, and what did you learn? (1,000 words)\n\nThis was the most general college application essay, the one that required the least research. Ginny had completed all the other steps\u2014asking for transcripts, groveling for recommendation letters, two sittings of the SAT, one AP exam, four essays on various subjects. This essay was the very last thing she had to do. Every single day for the past three weeks, she opened the document and stared at the question. Every day, she started to type the answer, then erased what she had written.\n\nShe took a deep breath and began to type.\n\nMy aunt Peg died last May. At least, that's when we found out about it. She left the country two years before, and we didn't really know where she was. But then we got a call from a man in England who told us she had died of brain cancer. A few weeks later, I got a package that had thirteen blue envelopes in it. . . .\n\nHow exactly was she supposed to explain what happened over the summer? One day, thirteen little blue envelopes containing strange and very specific instructions from her aunt showed up, and then Ginny\u2014who had never been anywhere or done anything\u2014was suddenly on a plane to London. From there, she went to Paris, to Rome, to Amsterdam and Edinburgh and Copenhagen and across Germany in a train and all the way to Greece on a slow ferry. Along the way, she had met a collection of stone virgins, broken into a graveyard, chased someone down Brick Lane, been temporarily adopted by a strange family, been fully adopted by a group of Australians, made her stage debut singing Abba in Copenhagen, been drawn on by a famous artist . . .\n\n. . . it was a bit hard to summarize in one thousand admissions-committee-ready words.\n\nShe looked at the calendar she had made for herself out of sticky notes on the wall next to her desk. Today's note read: Sunday, December 12: FINISH ESSAY!!!!! NO, SERIOUSLY, THIS TIME FINISH THE ESSAY!!!!!! And a few lines down, the due date: January 5. She pulled it off the wall and tossed it into the trash. Shut up, note. She didn't take orders from anything that had a glue strip.\n\nGinny put her feet up on the edge of her desk and tipped her chair back. She had always thought applying to college would be exciting. Living away from home, meeting so many new people, learning new things, making a few poor life decisions . . . the thought of it had kept her going all through high school. But after last summer, college didn't seem like such an adventure anymore. She started idly scrolling through the websites of the colleges she was applying to. All of them were trying to sell her a future in the same way they might try to sell her some mascara (\"Longer, fuller lashes! New formula! Look!\" Close-up of unnaturally long lashes, thick with something) or a weight-loss product (\"I lost 25 pounds!\" Image of woman twirling around in dress next to picture of her former self).\n\nThe photos were all the same, for a start. Here was the one with the smiling and carefully composed group wandering down the tree-lined path in the sunlight. The close-up of the person at the microscope, the wise professor leaning over his shoulder. There was the one of cheering people in matching shirts at a basketball or football game. It was like there was a checklist that all the schools had to follow. \"Have we included 'professor pointing at blackboard full of equations'? Do we have 'classroom of smiling, engaged students staring at nothing'?\" Worse than that were the catchphrases. They were always something like: \"We give you the keys to unlock the door of success.\"\n\nShe dropped the legs of the chair back down to the floor and flipped back to the blank page and the question.\n\nThe letters arrived last May . . .\n\n. . . and were promptly stolen by some dudes on a beach a few weeks later.\n\nAh yes. That was the other problem with this essay\u2014the horrible ending. In August, she was on the Greek island of Corfu, standing on the white sand of a gorgeous beach. The only envelope she had left to open was the very last one, and she decided to do this just as soon as she had a little swim. She had been on a ferry for twenty-four hours, baking in the sun on the deck . . . and the water here was so very, very beautiful. Her friend Carrie decided to swim naked. Ginny went into the warm, clear waters of the Aegean wearing her clothes. They left their backpacks in the care of their three male friends, who fell asleep on guard duty.\n\nHigh above, on the white rocks overlooking the water, two boys on a scooter stopped and surveyed the scene. Ginny was bobbing up and down in the waves and watching the ocean meet the sky. She remembered the sound of Carrie screaming and yelling. She remembered climbing over some rocks to find Carrie dancing around in a towel, naked and crying and saying something about the bags being gone. Ginny looked up to see the scooter ripping away from the scene, back up the rough path, back to the road above. And that was it. Letter number thirteen had been ripped right out of her life by some petty thieves who wanted her crappy backpack.\n\nLesson learned? Do not go swimming in the Aegean and leave the single most important document in your life in a bag on the beach. Take that, College!\n\nHer eyes drifted away from the essay to the little red light in the corner of her screen. The light that symbolized Keith.\n\nKeith was the actor\/playwright she met when she was following the directions in her third letter, the one in which she had to give five hundred pounds to a starving artist. She found Keith's play in the basement of Goldsmiths College and she bought all the tickets for the entire run, making him the first person to ever sell out the tiny student theater he was working in (also accidentally ensuring that no one would ever see his show). He was intense, hilarious, bizarrely confident, handsome . . . in a scruffy-poor-London-art-student way. But most mysterious of all, he was fascinated by her. He called her his \"mad one.\"\n\nTo be clear\u2014and she reminded herself of this fact daily\u2014Keith was not her boyfriend. They were \"kind of something.\" That was how they had left it, in those exact words. Their relationship was deliciously and frustratingly ambiguous, always flirty, never defined. When Ginny first returned to America, they were in touch every day. The time difference made it tricky\u2014he was five hours ahead\u2014but they always managed it.\n\nAround Thanksgiving, he got into some show he called a \"panto,\" so between rehearsals and his school schedule, his time online had decreased dramatically. For the past few weeks, Ginny had perched herself at her desk every night, waiting for that little light to turn from red to green, signifying that he was online. It was seven thirty now, which meant it was twelve thirty in London. Tonight was probably going to be one of those nights he never came online at all. She hated those nights.\n\nShe checked her email instead. There were several messages, but the one that caught her eye was from someone named oliver273@easymail.co.uk. Someone else from England was trying to reach her\u2014someone she didn't know. She opened it.\n\nWhat she found was a picture. A big blue square that filled the screen. It took her brain a moment to realize that it was a scan of a piece of blue paper with very familiar handwriting. It took almost a full minute more for her to fully accept what she was seeing.\n\n#13\n\nDear Ginny,\n\nLet me tell you about the division bell. The division bell will tell you a lot about England. You like to learn about England, right? Of course you do.\n\nSee, in Parliament, when they have a vote, they shout aye or no. The Speaker says, \"I think the ayes have it\" or \"I think the noes have it,\" depending whichever side is at its shouty best. Sometimes, though, when it can't be determined which side has won, they have to have what is called a rising vote. A rising vote means just that\u2014you have to rise up and stand on the yes side or the no side so you can be counted. There is an adorable, kindergarten-like quality to this, right?\n\nFollowing on the kindergarten theme . . . sometimes members of Parliament are out at recess when these votes happen. Instead of being in the sandbox, though, they are usually at the pub. So, local Parliament pubs are sometimes outfitted with a division bell, which rings when one of these votes is about to take place. When it rings, the members hurry back and stand on the yes or no side.\n\nThe division bell is ringing for you today, Gin.\n\nYou've done a lot in the last twelve envelopes, if you have in fact completed all that was contained in them. For all I know, you've read these letters from your sofa in New Jersey. But I trust you. I think you're exactly where I suggested you should be: on a ferry in the Greek Islands.\n\nIf you really wanted to, you could go home right now. Maybe you've had enough. Or . . .\n\n. . . or you could go back. Back the way you came. Back to London.\n\nDo you want to go on? Ding, ding. Yes or no?\n\nI'll be honest with you, from here on out, things get a little weird. If you are ready to stop, do it. Take it from someone who knows\u2014if you feel the need to go home, listen to that need and respect it.\n\nThink it over on the beach for a while, Gin. Should you decide to go on, you can go to the next page and . . .\n\nAt this point, the letter stopped. At the bottom, below the image, was a short message:\n\nSorry to interrupt. You don't know me, and likewise, I don't know you. As you can see, I possess a letter (actually, a series of letters) that seem to belong to you. But since this last letter contains very important information, I have to be sure that I am speaking to the right Virginia Blackstone. If you think this letter belongs to you, please let me know. My name is Oliver, and I live in London. You can reach me at this address.\n\nFor a moment, she did nothing. No movement. No speaking. She waited for the information to sink in. This was a page of the last letter. This was a task undone. This was the universe more or less demanding that she return to England at once and finish what she had started. This was fate. This was her brain going into hyperdrive.\n\nThe old Ginny had never traveled and knew no one in England. Old Ginny would think, plan, be cautious. But new Ginny needed a distraction, and a reason to see her kind-of-something non-boyfriend . . . and she knew someone who knew how to make unlikely things happen.\n\nShe got up and started to pack.\nDelusions of London\n\nGinny tried to process what she was seeing. She had been drifting in and out of sleep, so the line between dream and reality was a little unclear. She blinked a few times and looked out the window again.\n\nNope. This was not a dream. They were really there. Two massive, inflatable snowmen\u2014fifty feet tall or more\u2014hovering two stories up, their leering smiles pointed toward the street below. Big, white monsters of cheer, floating like clouds. It was unclear whether they had come for good or ill, whether they were ascending or descending. They were as wide as the road and blocked out the sky.\n\nShe pointed at them dumbly.\n\n\"Those are the giant snowmen of Carnaby Street,\" her uncle Richard explained. \"Festive and disturbing, just the way we like it here. Don't look them in the eye.\"\n\nGinny and Richard were in a black cab making its slow way through the streets of London. They turned onto Regent Street\u2014a seemingly endless line of stores large and small. The sidewalks were packed to capacity, easily five to ten people across. Strings of lights poured down over storefronts, waterfalls of them in red and silver and blue. Overhead, intricate patterns of lights and wire had been webbed between every streetlight, forming pictures of musical notes and sleighs. The bleary film of exhaustion over Ginny's eyes added to the sparkle.\n\n\"It was probably a mistake coming this way,\" Richard said, looking out the window at the traffic. \"But I thought you should see Oxford and Regent streets during the holiday rush. I don't recommend actually shopping here. Today's bad enough, but tomorrow, Christmas Eve . . . it's going to be a mad day for us.\"\n\nBy \"us\" he meant Harrods, where he worked. Harrods was the biggest, baddest, most famous store in London. Richard ran a department that mostly dealt with rich and famous people. He was the guy who had to make the arrangements when the Queen wanted to go shopping, or when TV stars or rock musicians or people with titles needed to send out a bunch of complicated baskets full of improbable things. Since Christmas was the hardest time of year for him, it was probably not the best time to have his American niece drop in, but he didn't seem like he minded. When she had called him and asked if she could come over for Christmas, he'd accepted instantly.\n\nIt had been a tricky negotiation on the New Jersey side of things, but she approached it with a confidence that she had never known before. She pitched the idea to her parents: Going to England was an extremely educational and culturally-expanding experience, and she could spend it with her relative. Wouldn't that be better than sitting around in Jersey for the two weeks that school was off? One phone call from Richard, with his smooth do-not\u2013worry-I-will-take-care\u2013of-everything delivery, sealed the deal. Everything was easier when you had a responsible adult with a fancy accent involved.\n\nGinny had uncovered the fact that she even had an uncle late in her stay in England over the summer, when Richard revealed that he and Aunt Peg had gotten married right before her death. Richard definitely didn't look like he was someone she would be related to\u2014this tall, suited English guy who knew exactly how wide ties, collars, and lapels were currently being worn, probably down to the millimeter. Yet, he was so warm and easygoing, so weirdly apologetic about everything. And handsome, with his softly curling dark hair, his wide eyes, and arched brows. Even his gently receding hairline added to the soft openness of his face.\n\n\"I've arranged for a proper roast dinner on Christmas,\" he went on. \"I'm not making it. That would be a disaster for all concerned.\"\n\n\"Roast dinner?\" she repeated. The jet lag was just creeping in at the sides of her thoughts, making everything a little hard to process.\n\n\"Oh, roast dinner . . . that's the best part of Christmas. You have your turkey, your roast potatoes, your carrots, that one Brussels sprout, the bucket of gravy . . .\"\n\nThe cab driver was nodding along to this with an expression of almost religious contemplation.\n\n\". . . anyway I ordered the whole thing from work. I'll bring it home with me tomorrow. They do a good job in the food halls. And I have some time off in a few days. We should go and do some more holiday things. Are there places you want to see? Tower of London, or . . . I don't know. Do you want to ride on the London Eye? I've never done that. I never go anywhere that visitors might like, except for Harrods. Do you want to go to Harrods? Please say no. You've been there, anyway. Anywhere else . . .\"\n\nGinny nodded contentedly. As they drove farther north and east into Islington, she started recognizing more and more. There was Angel Tube station, here was the street full of shops, the little pub on the corner. . . .\n\n\"It's right here, second one in with the black door.\"\n\nRichard was talking to the driver. She had fallen asleep again, her face mashed against the window. They had arrived at the house with the six steps leading up to it, the lightning bolt\u2013like crack running down them. The plant pots were still there, full of twigs and dusty soil. She immediately reached for her purse, but Richard was ahead of her, holding out a few twenty-pound notes to the driver.\n\nStepping out of the car snapped her back to reality. It wasn't that it was any colder than it was at home, but there was wetness in the air as well. Richard's house still had that strange, blank quality, that feeling like it had been furnished from an office-supply warehouse. The plain pine furniture, the low, industrial carpet. There was a large new television but not much else of note in the living room. It had the air of a house that was waiting for its occupant to show up. There were no decorations.\n\nThe kitchen was in a mild state of disarray\u2014take-out containers, bottles stacked on the counter waiting to be recycled, piles of bags sitting on top of the trash. All the signs of a man on his own who had been running nonstop for weeks.\n\n\"I've got to get back to work,\" he said. \"I'm sorry you're going to be alone for the rest of the day. Here's your key\u2014you probably remember, the square one is for the top lock. Genuine Harrods key chain, enjoy that. And there's plenty of food. . . .\"\n\nHe indicated the kitchen with a general sweep of his hand. Ginny caught the little silver glint of a wedding ring on his left hand. Ginny had managed to miss that ring when she first met him. God, she'd been clueless the first time she was here.\n\n\"I'll be fine,\" she said. \"Promise. Sorry I'm busting in when you're so busy.\"\n\n\"Don't be. I'll be back around eight. Maybe nine . . . but I'll try for eight.\"\n\nAs soon as he was gone, Ginny dragged her suitcase up the steps. It was not an elegant trip, banging and clunking, smacking into the wall. The door to her room\u2014Aunt Peg's room\u2014was open and waiting for her. It would always be strange coming into this room. The pink walls had an odd glow in the pale morning light. The glint of the wrappers and various pieces of trash that Aunt Peg had collaged on the walls stood in stark contrast to the large poster print of A Bar at the Folies-Berg\u00e8re, Aunt Peg's favorite painting. Richard had stacked towels and extra blankets on top of the patchwork quilt Aunt Peg had sewn.\n\nGinny dropped the suitcase under the window and sat on the floor with her back against the bed, looking up at the walls, the ceiling, taking it all in. There were two things she had to accomplish while she was here. Thing One: Get the letter. That was all arranged. She would meet Oliver at a coffee shop tomorrow at two and he would hand it over.\n\nWhich meant that she had today to accomplish The Other Thing.\n\nIn the two weeks that she had planned this trip, Keith had spent less and less time online. Their conversations, when they had them, had been short. Normally, this would have devastated her, but since she was on her way back to England, she had decided to use this to her advantage. The thing about her Keith seemed to like the most was the fact that she sometimes just turned up out of the blue, with some very unlikely story. So she hadn't mentioned that she was coming. Today, she was going to appear on his doorstep.\n\nThis was a maneuver that required preparation. She had managed, through some creative questioning, to find out that he would be at home this afternoon. The timing was right. She also brought the proper tools. She reached over and unzipped the suitcase. Her favorite outfit was on top\u2014a new black dress with white dots. With her new black boots and a pair of silvery tights, it was by far the greatest outfit she had ever owned.\n\nTime for a shower, which required a wrestle with the shower-spigot attachment in the bathroom. She had soaked the ceiling many times with this contraption over the summer. Richard had also kept all the toiletries Ginny had purchased on her last visit and had stacked them on an empty shelf in the bathroom. One quick and not-too-serious spray-down of herself and the bathroom later (already an improvement on her previous technique), and it was time to get dressed. She made sure the seam of her tights lined up perfectly along her toes (an often-neglected detail). Then the boots. She examined the final product in the mirror. It was . . . good. She looked good. Not like a tourist. Not like someone who was trying too hard. She looked like herself . . . maybe just a little dressier than normal.\n\nQuick hair check. When she got home from England in August, she felt the need to change herself a little. She had worn her hair very long for her whole life, and 80 percent of the time, she wore it in braids. The braids had become her signature, and the signature felt very tired. She impulsively chopped four inches off the bottom and dyed it a deep shade of auburn. Everyone loved the hair, but she still wasn't used to it. She kept looking for it, expecting it to hang down lower, to get in her way when it was windy out, to be there to twist and curl when nervous. Still, it had a nice, deep shine. It was more . . . mature.\n\nThere was one more thing to get out of the suitcase\u2014a small present wrapped in red paper. She carefully examined it to make sure the wrapping had not been ripped. The present had been tricky. She had to get something meaningful, but not too meaningful. Something personal, but not too personal. She had searched long and hard for it, and finally found it.\n\nDuring the summer, Keith found her in Paris. They had kissed once. That had happened in a graveyard, on a stone monument shaped like an open copy of Romeo and Juliet. Copies of Romeo and Juliet were common, but the one she'd found was antique, from 1905. It had a blue leather cover and was illustrated in bright, jewel-like colors, with gold endpapers. It was the kind of thing you could give to a theater student, just because it was a nice thing for a theater student to have. And it was the kind of thing you could give to someone as a way of remembering your first kiss.\n\nThe paper had made the trip with no damage at all. She wrapped it in a plastic bag to protect it from any rain, put it in her messenger bag, and grabbed her coat and her keys. It was time to make her second entrance into Keith's life. This time, she was ready.\nSurprises and Explanations\n\nThe route to Keith's house was perfectly preserved in Ginny's memory\u2014past Indian restaurants and dozens of newsagents, down street after street of rows of houses, all in various stages of repairs. Unlike New York, which was a city of big buildings and apartments, London was a city of houses, rows and rows of houses with little gardens in the front, houses that had known many families, known wars, known different eras and levels of wealth.\n\nAnd there it was, the house she remembered so well, the house she thought about so often. There were the cheap black blinds, askew as always. There was the gold plastic window in the front door, the trash bin out front that was always overflowing, the little stone wall, and the shabby, tiny front garden, which always seemed to grow a thin crop of crumpled candy wrappers. As a little nod to the season, a string of Christmas lights was draped somewhat haphazardly around the upper windows, strung from one window to another across the front of the house. It blinked in an irregular pattern, one that suggested to Ginny that it probably wasn't supposed to be blinking at all.\n\nOut front was Keith's small, battered white car. She peered inside the front window to see if it was as trash filled as ever. It wasn't. He'd obviously given it a cleaning recently. Aside from two plastic bags and a few script pages on the front passenger seat, it was clutter-free. The first time Ginny saw this car, it had been stuffed with an entire theatrical set, including an inflatable palm tree (deflated, thankfully).\n\nThere were lights on in the two upstairs windows, and the muffled beat of music escaped the glass. Someone was home. It might be Keith, or it might be David, Keith's lovesick roommate with the horrible on-off girlfriend, Fiona, the human cotton swab. She got closer and listened carefully. The noise was coming from the window on the left\u2014that was Keith's.\n\nFor the first time since she'd cooked up this idea, she felt a surge of nervousness. This had all felt so hypothetical up until this point. In the time between turning out her light at night and going to sleep, she had imagined this moment, the exact way she would knock, his face when he opened the door. . . . Now she was really here, and in a minute, she would really see Keith. Imagination was about to collide with reality.\n\n\"Relax,\" she told herself. \"You have the advantage of surprise. Just be normal.\"\n\nOf course, the first step toward normalcy probably didn't include sneaking around in front of the house, looking through the car windows, and talking to herself.\n\nShe reached up and knocked hard on the plastic panel. One of the windows above slid open a little bit.\n\n\"THE DOOR IS OPEN!\" Keith yelled out.\n\nGinny looked up to see if he was peering down at her, but there was no head sticking out of the window. He was just letting whoever it was come in. While that didn't seem especially safe, it did work to her advantage. She pushed the door open slowly.\n\nInstantly, she was overwhelmed by the familiar smell. Detergent, a spicy incense, some kind of dish soap, wet clothes, theater dust . . . it was Keith smell. The door opened onto a hall and a set of stairs. The little foyer area was crammed with things\u2014plastic bags full of newspapers, Keith's sneakers, umbrellas, books. There was, for some reason, a hammer in the middle of the floor and rolls of toilet paper piled in the corner. T-shirts and boxers were spread over the heater on the wall to dry.\n\n\"Top of the steps!\" he called. \"Just come up!\"\n\nGinny steadied herself and quickly checked her warped reflection in a cheaply framed poster for Keith's last show, Starbucks: The Musical. Someone's black scarf had fallen just at the foot of the steps, marking the line she had to cross to continue. She stepped over it and made her way up the steps.\n\nKeith sat on the sofa, his legs stretched out and his feet balanced on a plastic crate. The first thing she noticed was his hair. It was a little shorter, and not quite as shaggy as it had been over the summer. The haircut made it look darker and brown, not quite so reddish blond.\n\nHe was half-scowling at a computer screen, squinting a bit as he typed away feverishly. He was so intent that he didn't notice her standing in the doorway. He turned and opened his mouth, ready to call out again, and caught sight of her in his doorway.\n\nHe actually jumped back an inch or two.\n\n\"Hi,\" she said, grinning. \"Remember me?\"\n\nFor almost ten seconds, Keith did nothing but stare. Ginny clung to the door frame.\n\n\"There is something terribly, terribly wrong with you,\" he finally said. \"And I intend to have you put in a home. Are you going to come in, or are you going to hang in the doorway like that?\"\n\nHe pushed some papers off the sofa to make room for her. She came in and sat down carefully, barely able to look at him directly at first. The sensory overload took some time to get used to. He wore a black sweater, ragged jeans, and a pair of bright socks with visible holes. She smiled at his toes poking out.\n\n\"Why didn't you say you were coming?\" he asked.\n\n\"I wanted to surprise you,\" she said. \"Do you let every weirdo who knocks come into your house? You didn't even look.\"\n\n\"I thought you were here to audition,\" he said. \"People have been coming here to read for the last few days.\"\n\n\"Audition?\"\n\n\"New play about the financial crisis. It's called Break the Bank. It's the evolution of something I started a while ago called Bank: An Opera of Greed.\"\n\n\"Aren't you in that . . . panto thing?\"\n\n\"Ah,\" he said. \"I had an . . . artistic disagreement with the director of the panto. As it happens, I take issue with the objectification of women in Cinderella, and the reliance on shoes as a means of identification. Surely you understand.\"\n\n\"You got fired?\"\n\n\"Fired is such an accurate word. Also, I didn't like being the back half of a horse.\"\n\nGinny smiled and sat back into the couch. Rain was coming in through the half-opened window, but Keith didn't notice or care. She reached into her bag and pulled out the present.\n\n\"What's this?\" he asked. \"Is it a lump of money? Did you bring me money? You want to back my show again, don't you?\"\n\n\"Yes,\" she said. \"It's a lump of money.\"\n\nKeith held the gift for a moment and squeezed it lightly.\n\n\"You're embarrassing me.\" He looked down at the package. \"I didn't know you were coming. I don't have anything for you.\"\n\nThis was definitely a first\u2014Keith, embarrassed. His cheeks even flushed a bit.\n\n\"Open it!\" she said.\n\nHe tore through the paper, revealing the blue cover. He looked puzzled for a moment, then ripped off a few more strips until it was fully revealed. For a moment, he said nothing at all, just opened it and looked at the pages, the illustrations. When he finally looked at her, his eyes were wide and his expression raw and open.\n\n\"I don't know what to say, Gin. It's . . . it's lovely.\"\n\nHis embarrassment was catching. Ginny felt her face get hot, and she found that she was gripping bunches of her skirt in her fists.\n\n\"I know it's one of your favorites,\" she said.\n\n\"Well, who doesn't like a romantic suicide pact?\"\n\n\"Only bad people,\" Ginny said.\n\n\"Exactly.\" Keith looked up partway, avoiding her face, instead tracing the outline of her new haircut. \"Your hair. You changed it. You look like a news presenter.\"\n\n\"Is that a good thing?\"\n\n\"Clearly you don't know about my childhood obsession with the woman who did the weather. My heart still flutters when I hear the word 'precipitation.' I don't think I've ever seen you without braids. I thought your hair just grew that way.\"\n\nHe turned back to the book and flipped the pages for a few moments. From the way his eyes were flicking across the pages, Ginny could see he wasn't reading. He was thinking. The moments ticked on. Ginny managed to inch her way closer to him, until she was right at his shoulder. He didn't turn to her directly, but he was moving toward her in barely perceptible increments. This was it. Her whole body was tingling. She could feel a kiss coming, the way you could feel a heavy rain approaching.\n\nThe door opened downstairs, causing them both to start a bit. A female voice called up a muffled hello. Keith glanced toward the noise, set the book on the coffee table, and stood up. Ginny assumed this horrible interrupting person was here for an audition until a voice called from the downstairs hall.\n\n\"It is foul out there! David had better thank me! I stayed late for him and got stuck in the rain!\"\n\n\"Fiona?\" Ginny mouthed.\n\n\"No.\" Keith shook his head and crossed the room. \"She's gone. Been gone for ages. They split up right after you left. That's why the grass out front started growing again.\"\n\n\"He's got a new girlfriend?\" she said quietly. \"Thank god. You must be happy.\"\n\n\"Yeah. He does. It's a relief. She's a lot nicer. But then, your average angry snake is nicer than Fiona. I'm sure she's happier wherever she is now, burning orphans or whatever she does with her time.\"\n\nThere was a quick footfall on the steps and then a girl appeared in the doorway. The new girl was extremely pretty. Like David, she had very dark black skin. She was a few inches shorter than Ginny, with skinny jeans, scuffed-up brown boots, and an enormous, slightly ragged gray sweater. She wore no makeup, but her cheeks glowed brightly from the cold and damp. Her hair was a free-flowing halo of spirals that rose up high, at least six inches or more, all around her head. Moisture from the rain was trapped in the tips of her hair, which she was rubbing in disgust.\n\n\"Remind me to never . . . oh! Sorry! I'll go wait downstairs until you're done.\"\n\n\"It's all right,\" Keith said. \"It's not an audition. This is Ginny. The famous Ginny Virginia.\"\n\n\"No!\" the girl exclaimed. \"Was that you, with the letters? Your aunt is an artist?\"\n\n\"That's me,\" Ginny said, oddly flattered that David's girlfriend had heard of her.\n\n\"Oh! Keith's always talking about you!\" The girl came in and sat next to Ginny. \"I'm Ellis. So nice to meet you! Did you just . . .\"\n\n\"Kind of a surprise visit,\" Keith said with a proud smile. \"As is her way.\"\n\n\"This is so exciting! What are you doing here? Just visiting for Christmas? Sorry I'm such a mess. Just had my last day of work. I took a Christmas job at H&M and now I am finished!\"\n\nShe threw her arms into the air in triumph before collapsing on the sofa in the place Keith had just vacated, pulling off her boots. Ginny forced a polite smile. David's new girlfriend was nice, but she was also settling in. She had no I-have-just-interrupted-a-personal-moment radar at all.\n\n\"People are so horrible when they shop for Christmas,\" Ellis continued. \"They do things you would not believe. David wanted me to pick up a shirt for his sister. I got the last one and a woman tried to rip it out of my hands, and when I didn't let go, she pulled her hand back, like this. . . .\"\n\nShe demonstrated.\n\n\"That looks more like a punch,\" Keith said. \"Are you sure this was H&M you've been working at, and not some ultimate fighting league?\"\n\n\"It's hard to tell some days. But I got the shirt. He had better appreciate that.\"\n\nKeith began pairing his socks, putting them into little tucked bundles. Ginny had never seen Keith do his laundry before, but he certainly never struck her as a sock bundler. He placed each pair onto the mantel, making a little pyramid out of them.\n\n\"What's this?\"\n\nEllis picked up the copy of Romeo and Juliet and started paging through it. This was too much\u2014the book was private. Keith glanced over, but he didn't ask Ellis to stop. His pyramid of socks was getting out of control. It was going to fall apart at any second. Yet he just kept folding and piling.\n\n\"A Christmas present from Gin,\" he said finally. \"It's brilliant, isn't it?\"\n\n\"It's so lovely,\" Ellis said, replacing it. \"You have to tell us all about your plans for while you're here. We'll have to go out and show you around. There's loads to do in London at Christmas.\"\n\n\"Holly-eating contests and Christmas tree fights . . .\" Keith's voice was a low mumble. Ellis picked up a spare sock and threw it across the room at Keith. He half-smiled and pulled it off his shoulder. Suddenly, all in a second, Ginny understood everything. . . . Why Keith was off-line so often, why the door was really open, and what \"kind of something\" actually meant.\n\nEllis wasn't David's girlfriend. She was Keith's.\n\nThe room was unbearably hot. The rain at the windows far too loud. She needed to get out of here, to get some air, to sleep . . . to do anything. She just had to get out.\n\n\"You know what?\" she said. \"I'm . . . I didn't sleep on the plane. I think I . . .\"\n\n\"Would you like a cup of tea?\" Ellis said quickly. \"Or a coffee?\"\n\n\"No.\" Ginny stood. She was unsteady on her feet.\n\n\"Oh, look at you!\" Ellis said. \"God, you're exhausted. You should just take a nap right here.\"\n\nKeith finally reentered the conversation, a bright expression pasted on his face. \"The lengths you'll go to for a ride in my magnificent automobile! I can hardly blame you. Come on, then. I'll take you back.\"\n\n\"It's really nice meeting you,\" Ellis said, hugging Ginny. \"Feel better. Some sleep will help. And we'll see you again, right? We'll make plans!\"\n\nOne of the small mercies in all of this was that the ride from Keith's to Richard's only took about five or ten minutes. Keith was very chatty when they first got into the car, rattling on about his plans for Break the Bank. Ginny nodded and tried to act interested in \"the collaborative writing process, you know, based on our collective disillusionment with the traditional financial structure, you know, hearkening back to the communal voice in theater that we haven't had since, you know, the seventies.\"\n\nThe rain had let up a little. Now it was fine and misting. Ginny fixed her gaze on the view\u2014the rain, the glimpse of Christmas trees through open windows, the guy with the shopping bags of wine hanging off the handlebars of his bike, begging for disaster. When they pulled up to Richard's, Keith let the engine run for a second, then reached for the key to turn off the car.\n\n\"So, I'm off home for the next day and a half,\" he said.\n\n\"Where is that again?\" Ginny said. She was trying to sound casual and unbothered, but her voice was dry.\n\n\"Reading. Just doing the Christmas thing. I'll be back on Boxing Day and we should . . . we'll do something.\"\n\n\"Cool,\" she said. \"I'll be here. Thanks for the . . .\"\n\nShe waved her hand to indicate the general miracle of automotive transportation.\n\n\"No sweat,\" he said.\n\nShe was halfway up the cracked steps when she heard the car door open. He was leaning sideways across the front seats and waving to her to come back. She returned and leaned down into the opening.\n\n\"Thank you for the book,\" he said. \"Merry Christmas, yeah?\"\n\n\"Merry Christmas,\" she replied. And then she turned and skipped up the steps so he couldn't see the tears that fell freely down her face.\nPairs of Shocks\n\nThe next morning, Ginny woke up in a cold room. She stared straight up the wall, at the strange landscape Aunt Peg left behind\u2014the wall of trash she had collected up and collaged, a jarring vista of different materials, some wrinkled, some smooth, some reflective, all different colors and words and shapes and materials. It was probably supposed to look like something, but from this angle, it was all confusion.\n\nShe was not going to mope. She was going to get up. Today she had to do what she really came here to do. She was in London. It was Christmas. She had a letter to get.\n\nIn the kitchen, Richard had left a cheerful note against a mug about how happy he was to have her here, and how she should think of this house as her own. She opened up her computer, half-expecting a long message from Keith, with an excruciating explanation of the day before, but there was nothing from him. He was on his way to his grandmother's now. There was one from Oliver, confirming that he would be at the caf\u00e9 above Foyles bookshop on Charing Cross Road at two P.M. That was what she needed to focus on. Getting the last little blue envelope, and getting on with her life.\n\nShe arrived at Charing Cross a full two hours early and wandered down the street of book and music shops. Foyles was a massive place, with a large, urban-folksy coffee bar\u2014heavy wood tables and real cups and wine and cookies and indie newspapers. It was very crowded, so she waited in the corner until a table was free, then claimed it and sat.\n\nAt exactly two o'clock, a tall guy stepped into the caf\u00e9. That's probably what Ginny noticed first\u2014his extreme height. He was well over six foot. Ginny wasn't sure what she had been expecting, but he turned out to be a very pleasant surprise. He had an angular face and almost jet-black short hair. He wore a long black wool coat, which looked very well-made and expensive, and under that, a gray dress shirt and slightly loose black pinstripe pants. He had a well-worn leather bag with him, strapped over his chest. His face was pale, and he was thin, with dark, intense eyes. Though he had to be somewhere around her age, the effect made him look older and placed him in some unknown territory between stockbroker and rock star. He was, without question, the most English-looking guy Ginny had ever seen. She wasn't even sure what that meant, really. She only knew that from now on, when calling up an image of an English guy in her mind, this would be the picture.\n\nOliver surveyed the room from his natural vantage point and quickly focused on her. He strode over to the table in about four steps and pulled out the chair opposite Ginny. As he sat, he adjusted his coat, revealing a flash of rose-colored silk lining and a fancy embroidered label just on the inside. And yet, for all the weird formality of his clothes, there were three tiny pins on the lapel of his coat\u2014one said Bowie, one was a small lightning bolt, and the other had a picture of a skull and writing that was too small to read. He pulled his bag onto his lap.\n\n\"Virginia?\"\n\nHis accent was much crisper than Keith's, and he was more polite, hesitant. He was fancier.\n\n\"Ginny,\" she said. \"Hi, I'm . . .\"\n\nShe always did that, introduce herself twice.\n\nThey sat staring at each other for a moment. He put his hands on the table and knotted his fingers together. When he reached forward, Ginny could see that though the coat was custom-made for someone, it clearly wasn't Oliver. The arms were too short by several inches, exposing his arm with every move.\n\n\"Did you want a coffee?\" he asked.\n\n\"No, I'm fine.\"\n\nHe nodded, looking oddly relieved.\n\n\"So,\" she said. \"How did you get the letters?\"\n\n\"I was backpacking in Greece. My bag broke on the way to Corfu. I met some guys in town who were selling things out of the boot of their car. The things were used\u2014it was perfectly clear that they had to be stolen, but they were cheap, you know? And I was kind of desperate. Anyway, I bought your backpack. If you want it back . . .\"\n\n\"It's okay,\" Ginny said. It was fine with her if she never saw that thing again. It was green and pink and hideous and oversized\u2014the physical embodiment of embarrassment.\n\n\"The bag has a lot of very strange hidden pockets,\" he went on. \"I didn't even find them all until I got home and was emptying everything out. The letters were in one of them, along with some receipts, and a few coins. . . .\"\n\nHe opened his bag. First, he presented her with a small stack of crumpled receipts and a small handful of Euro coins. Ginny took one of the receipts and stared at it. It was an artifact, a long-forgotten fragment of the summer. She'd spent eight Euros fifty somewhere in Germany. She hadn't stayed in Germany at all, just moved from one train to the next in order to go south to Greece. But somewhere along the way, she bought a Coke and a small pizza in a train station.\n\n\"I brought it all,\" he explained. \"But these are probably the only things you're interested in.\"\n\nOliver reached into his bag again and produced a clear plastic bag full of very familiar-looking blue paper and airmail envelopes. Ginny's heart beat faster as she reached for it and removed the contents, all the hand-painted envelopes Aunt Peg had created with such care. These were her paintings. The girl walking toward the castle on a hill from letter #4. The tiny pictures of cakes from letter #6. Here was her picture on letter #9, a girl with two long braids, her shadow cast all the way across the envelope. The strange picture from envelope #12 that had puzzled her at first, because it looked like a purple dragon coming out of the water. It wasn't until she reached Greece that she realized it was a picture of an island. And now . . .\n\nShe shuffled right back to letter #1. There was no #13.\n\n\"The last one,\" she said, holding up the letters apologetically. \"Um . . . the last one isn't here.\"\n\nOliver gave himself a thoughtful chin-pinch before answering.\n\n\"When I found the letters,\" he said, \"I did a bit of research, just out of curiosity. I read about the auction of your aunt's paintings. I take it you found them even without the last letter. You got a lot of money from the last sale. But what mattered to you more, doing the things in the letters, or getting money?\"\n\nIt was an odd question, but not one that Ginny minded answering.\n\n\"Doing the things in the letters,\" she replied. \"The money was nice, but it didn't matter.\"\n\n\"So the experience was the valuable part? If you had a choice between the experience and the money, you'd choose the experience?\"\n\nGinny nodded. Oliver's gaze had drifted to a spot just over her shoulder. The questions felt a bit odd. The letters were so personal. Only a few people knew about them in detail. Here she was, talking about them with a total stranger\u2014albeit a total stranger who had brought them back to her. It was fair enough. If she had found these letters, she would be curious too.\n\n\"I guess? Yes. I . . . yeah. I would.\"\n\nOliver nodded and leaned in, putting his forearms on the table and getting closer to Ginny.\n\n\"There's something in the last letter\u2014well, there are several things in the last letter\u2014but the most important thing is that there's another work of art. It's in three pieces. They're all in different locations out in the world, so that they can get exposed to different elements. It's based on a piece done by Mari Adams. You know Mari Adams, right?\"\n\nGinny definitely knew Mari Adams. She was a famous artist, much loved by Aunt Peg. Aunt Peg had met and befriended her, and sent Ginny up to Edinburgh to meet her. Once you saw Mari, you never forgot her mane of orange-red hair or the tattoos on her feet that bore the names of her pet foxes, or her tattooed face.\n\nHe reached into his pocket and set a card down on the table\u2014a familiar dove gray card. It read: CECIL GAGE-RATHBONE, JERRLYN AND WISE, ART AUCTIONS. This was also a name Ginny knew well.\n\n\"This is the man who handled the sale,\" he said. It wasn't a question. She couldn't figure out why, but for some reason, this comment put her on edge.\n\n\"Right,\" she said slowly. \"He did.\"\n\nOliver looked at her intently.\n\n\"A finder's fee doesn't seem out of line,\" he said. \"I suggest that we split the proceeds from the final piece of art. I made us an appointment. We're due in forty-five minutes.\"\n\nGinny simply didn't know how to process this. She grabbed for her braids, but, of course, they were gone. So she laughed\u2014a weird, gurgling laugh. \"You're kidding.\"\n\n\"We should go.\" Oliver plucked up the card and put it back in his pocket.\n\n\"You seriously can't be . . . serious.\"\n\n\"I'm completely serious. If it wasn't for me, you wouldn't know about this at all. In return, I will help you collect the art. Besides, you told me you don't care about the money. So what's the problem?\"\n\n\"What if I said no?\" she asked.\n\n\"Then I go home,\" he said.\n\n\"And the letter?\"\n\n\"Remains lost. Your choice.\"\n\n\"I need to think about this,\" she said, hooking her ankles around the legs of the chair to support herself. \"Can we talk about . . .\"\n\n\"Look,\" he cut in, \"by working together, we can both get what we want. But I assure you, I will leave in a minute, and you won't see me again. You'll never see the letter, and you'll never know what was out there. This is your time to decide. Right now.\"\n\nHe pushed back his chair and stood, waiting. From her seated position, he looked ridiculously tall, and aside from a little twitch above his left eye, utterly composed and serious. Ginny looked around at the many people in the coffee shop, all busy with their shopping bags and their strollers and their phones and computers. She wanted to scream out, to tell them all what was going on, tell them the whole story. They would be outraged. They would cluster around him and shake him down for the letter. He would be thrown out into the street, probably without his coat, and forced to run, coffee mugs flying after him and shattering at his heels. But that is not the way the world works. She had no doubt that he meant what he said. If she walked away from this now, he would go away, and that would be it. Forever.\n\nShe stood up.\nThe Devil's Bargain\n\n\"It's an absolute pleasure to see you, of course,\" Cecil Gage-Rathbone said. \"So happy that you got in touch. Do come through. Coffee? Tea? Something else?\"\n\nThe offices of Jerrlyn and Wise were richly but tastefully decorated for the holiday. A large tree decked in silver and gold horns and sugared fruit sat in the corner, and various holiday-oriented antiques dotted the room. Cecil looked exactly the same as before, and like everything else around this building, he had been expensively decorated. There was the artfully arranged and unmoving hair, another flawlessly tailored suit, silver cufflinks gleaming at his wrists.\n\n\"I'll have a cup of tea,\" Oliver said.\n\nGinny shook her head tersely. She felt like she was being kidnapped and was afraid to speak, to say the wrong word.\n\n\"Could you bring us some tea, James?\" Cecil said, to a man sitting at a desk in the hall.\n\nThe Jerrlyn and Wise building looked like it was an old house built sometime around the 1800s. Cecil's office was a small room, probably an old pantry, with an entire wall of built-in shelving for cups and plates. This had been reappropriated to display antique silver spoons and a collection of auction catalogs. The walls had been hung with massive gilt-framed pictures of things like shipwrecks and drooling dogs and pale children, bringing disaster, rabies, and anemia to every inch of the space. There was just enough room in between these disasters for his massive mahogany desk and two wingback chairs.\n\n\"Please,\" he said, ushering them into the room. \"Please, have a seat. And I don't think we've properly met, though we spoke on the telephone.\"\n\nHe extended his hand to Oliver, who shook it confidently. Ginny began to understand why he was so dressed up. He had been prepared for this meeting.\n\n\"Virginia, I hope you've been keeping well.\"\n\n\"I'm great,\" Ginny said. The words caught her in throat a bit. She sank into the chair closest to the door.\n\nCecil settled into his chair and assumed his version of a casual position. \"You've come about another piece of Margaret's? I'm delighted to hear about this, of course. The success of the previous sale combined with the obviously limited availability of the artwork . . .\"\n\nThis was a polite way of saying, \"Your aunt is dead, so she can't paint anymore, and that makes the price go up.\"\n\n\"And Mr. Davies . . . were you involved with Margaret's work in some way, or are you . . .\"\n\n\"I'm a friend of Ginny's,\" Oliver said. \"I'm helping.\"\n\nGinny dug her nails into the arms of the chair.\n\n\"I see,\" Cecil said. \"So, have you come here today with the new work?\"\n\n\"The piece can be delivered just after the New Year,\" Oliver said. \"Say, on the second? And we'd like to sell it immediately. The next day if possible.\"\n\nFor just an instant, Cecil looked surprised. There was a flicker of movement in the eyebrow region, which Ginny suspected was highly Botoxed.\n\n\"That doesn't leave us much time to photograph it and show it to the interested parties. It would be better if we could wait a few weeks.\"\n\n\"We'd prefer to do it quickly,\" Oliver said.\n\n\"Well,\" he said slowly, \"of course, we can do things that way, if you wish. The element of surprise might work in our favor. And this is what you want, Virginia?\"\n\nThis was her chance to tell him exactly what was going on. But Oliver would simply get up and walk away, along with her letter.\n\n\"Yes,\" she lied. \"That's what I want.\"\n\n\"Well.\" Cecil adjusted the position of a notepad on his desk. \"In that case, please tell me all about the piece. I have to know what I'm selling.\"\n\n\"I think it's best if you see it,\" Oliver said. \"It's a bit hard to describe.\"\n\nThere was a subtle knock, and James entered from the hallway with a silver tray bearing a small china tea seat and two cups. Cecil poured a cup and offered it to Ginny. She shook her head tersely. Oliver took a cup, topped it heavily with cream, and relaxed in his chair like he owned the place. Cecil put a lump of brown sugar in his tea and stirred it slowly.\n\n\"This is a bit unusual,\" he said, \"but the last collection also had a bit of an unusual story. The buyers may appreciate that. I certainly look forward to seeing it. Is there anything you can tell me?\"\n\n\"We'll be splitting the proceeds of the last sale, fifty-fifty,\" Oliver said.\n\n\"This an agreement you've reached?\" Cecil asked. \"I have nothing to reflect this in my notes.\"\n\n\"That's right,\" she mumbled.\n\nCecil paused for a moment, then opened his desk drawer and pulled out a notepad.\n\n\"Of course,\" he said. \"We can arrange that. I'll have legal draw something up and it will be ready for signature when you bring the piece in. Will that be acceptable?\"\n\n\"If it's all the same . . .\" Oliver reached into his leather bag once again, this time producing a few pieces of paper. \"I've already had something drawn up. Very simple. We could just sign it now.\"\n\nCecil drew one of the papers closer to himself with just the tips of his fingers, then spun it around to read it.\n\n\"It is, as you say, very simple,\" he confirmed.\n\n\"Just a moment.\"\n\nCecil took the contract and slipped around his desk and past Ginny's chair. He was gone for over ten minutes, during which Oliver and Ginny ignored each other's existence entirely. Oliver did something with his phone. Ginny shifted around, trying to turn away from him as much as she could in her chair. She read the spines of every catalog on the wall. She counted the spoons. Maybe Cecil would come back with the police. Or a gun. They had to have some old guns around here. Instead, he returned with some photocopies and a resigned expression.\n\n\"This appears to be in order,\" he said. \"Basic, but acceptable. I have four copies here. If you'll just sign them all where indicated . . .\"\n\nGinny scrawled her name as quickly as possible and pushed the papers away. Oliver wrote slowly, in small, steady script.\n\n\"Well,\" Cecil said, taking the copies, \"as I work for the seller, I'll abide by your wishes. I'll arrange for the sale on that day and do what I can to get the previous buyers back in the room. You will have the piece on the first, yes?\"\n\n\"That's right,\" Oliver said.\n\n\"Then I'll arrange for one of our teams to come for it. First of the year . . . normally we wouldn't do that, but we work with the circumstances. Unless there's something else I can help you with?\"\n\n\"No,\" Oliver said, standing. \"We should be on our way.\"\n\n\"Then James will show you out. Thank you so much for coming in.\"\n\nOutside, the sky was the same color as the sidewalk and the stone walls in front of the houses. Oliver strode away from Jerrlyn and Wise with a long, easy step, stopping in front of one of the many small mansions that lined the street and sitting down on the low wall that surrounded its front garden. He pulled a pack of cigarettes and a silver lighter from his pocket with a sweeping gesture that Ginny suspected had been rehearsed in front of a mirror. She stood directly in front of him and folded her arms.\n\n\"I want my letter,\" she said.\n\n\"I can't give that to you just yet. The letter is the key to getting the art. If I give you the letter, you can just go and get the art. Don't worry, though. The letter is of no other value to me, so the minute we're done, you'll get it back.\"\n\n\"The minute we're done with what, exactly? How is this supposed to work?\"\n\n\"We go to Paris. That's where the first piece is.\"\n\n\"Paris?\"\n\n\"Nothing's properly open on Christmas or Boxing Day, so we'll start nice and early on the twenty-seventh. I got us two train tickets. Don't worry\u2014they were cheap. Only fifty pounds. I figured I could contribute that to the cause.\"\n\n\"You think I'm going to travel with you?\" she said. \"To Paris. You and me. You're insane.\"\n\n\"Look,\" he said, sticking a cigarette in his mouth, \"it's probably hard for you to trust me when I say you'll be perfectly safe with me. My interests are your interests. And I didn't steal a thing. I found the letters, and I'm giving them back. You're going to make money you couldn't have made otherwise. You have no reason to complain.\"\n\nA small orange cat slunk along one of the walls on the opposite side of the street. It sat and stared across at them haughtily, as if asking what they were doing in its neighborhood.\n\n\"We're both needed for this,\" Oliver said. \"There are things in the letter that only you will understand. I have the letter, and you have the knowledge. All I want is for us to go and get the pieces. That's it.\"\n\nOliver calmly lit his cigarette and took a long draw, waiting for her reply.\n\n\"I'll think about it,\" she said.\n\nHe nodded and pushed off the wall. \"You know where to reach me.\"\nThe Pool\n\nThough it was only a little after four, the sky had gone dark and all the lights had been switched on: the illuminated ads on the double-decker buses, the light of cell phones pressed to hundreds of faces, the warm glow from the windows. Everyone was moving quickly, the final panicked moments before all the shops closed and Christmas began. London was sparkling and pulsing. Ginny allowed herself to be carried along with the crowd. She had gone beyond shock into a new and completely foreign state of almost aggressive acceptance. The summer had pushed her into some weird places. She had handled it then\u2014she would handle it now.\n\nWhen she got off at Angel station, she saw a man selling trees in the shopping plaza. Decorate. That's what she would do. She would decorate, and she wouldn't think about anything but decorating. It was Christmas, and she wanted to make it nice for herself and Richard. It would keep her busy, and busy was good. She purchased one of the trees, a tiny one, maybe three feet high. It didn't weigh much; she could carry it with one hand. She walked around with it, looking blankly in store windows until she found some decorations to go with it. There was nowhere to leave the little tree outside\u2014she couldn't exactly lock it up\u2014so Ginny brought it in with her. She bought up pretty much everything she found on one of the display tables full of two-for-one Christmas balls and lights and shiny objects. She bought far more than she could reasonably carry, so she carried it unreasonably, the bags cutting harsh red lines into her skin, the tree banging into her ankles as she walked.\n\n\"Do you need help with that?\" a man asked, as Ginny dragged the tree along the sidewalk.\n\n\"No, I'm fine . . . thanks. Merry Christmas.\"\n\nHe nodded, but looked very uncertain.\n\nThough she had gotten a lot of things, some improvisation was still necessary. First, she had forgotten to get a tree stand. This meant finding a bucket in the closet, then getting the tree to stay upright by lashing it in place with some string, then covering the bucket in tinsel. Some of the ornaments came without hooks, so she was forced to make the hooks out of some paperclips she found. When she ran out of those, she just taped them to the tree, hiding the tape as best she could. She bought way too many lights, so she strung them everywhere\u2014up the stairs, around the mirror above the sofa, around the television.\n\nShe decorated until there was nothing left to hang, nothing left to cover. Then she sat on the sofa and waited.\n\nParis. Oliver wanted her to go to Paris. She couldn't deny a minor thrill at the thought. Their conversation played in her head on an endless loop. There was nothing she could have done differently. If she'd said something to Cecil, Oliver would have bolted. So she would go to Paris. She had done it before. And at least this Oliver, Oliver, Oliver loop was a distraction from the Keith, Keith, Keith loop. That was something.\n\nThe debate continued until a black cab pulled up out front. There was a jingle of keys, and Richard was at the door with three large Harrods bags.\n\n\"Hello!\" he said. \"I have . . . bloody hell.\"\n\n\"I decorated,\" Ginny said, pointing out the obvious.\n\n\"Yes, I see that.\"\n\nIt was hard to read his reaction. He wasn't ambivalent\u2014his eyes were wide and he rubbed his hairline.\n\n\"It's . . . wow. I don't really ever . . . I suppose I don't have time . . . it's marvelous. I have half the food hall in here. The roast dinner! Come on. We need to put this in the kitchen.\"\n\nShe was pretty sure he was lying, but he was obviously happy to be home. As they headed for the kitchen, he casually slipped one of the bags behind a chair. Just as casually, she looked to see what it was. Wrapping paper peeked out at the top. He had gotten her some gifts.\n\n\"It was absolutely mental,\" he said, dropping the remaining bags on the counter. \"I must have sent out two hundred Christmas hampers this morning. Some people don't realize that if you want to send someone fifty pounds of chocolates inside of a decorative birdhouse for Christmas you should really ring me before eight in the morning on Christmas Eve.\"\n\n\"Any celebrities?\"\n\n\"A few. Nobody was too bad this year. I didn't have to find any exotic animals or enriched plutonium or anything like that.\"\n\nRichard wasn't exaggerating about the roast dinner. There were easily thirty different containers in the bags. When the last one was put away, Richard took a beer out of the refrigerator. \"I'm too tired to drink this,\" he said, looking at it mournfully. He put it back in the fridge and shut the door. \"I have to sleep.\"\n\nHe stopped halfway up the steps and had a look at the lights strung there. \"Someone from your family has always decorated my house for Christmas,\" he said. \"Peg did last year. Her decorations were a bit weirder, as you might imagine. She suspended the tree upside down, for a start.\"\n\n\"She what?\" Ginny asked.\n\n\"By drilling a few holes in the ceiling and suspending it. Right there, in the corner.\" He pointed to just above where she was sitting. Sure enough, there was a series of small holes there. Aunt Peg, once again, had been down this road before.\n\n\"Was it a big tree?\" she asked.\n\n\"It was enormous.\"\n\n\"And she just . . . hung it?\"\n\n\"She did. I'm glad she didn't bring the ceiling down. Even if she had, she probably would have turned the rubble into more decorations. I still don't know how she did that. I suspect she had help. In any case . . . I'll see you in the morning. Merry Christmas.\"\n\nThat night, as Ginny lay in bed staring up at the collage on the wall, her brain drifted back to a brutal summer's day in New York City when Aunt Peg said, \"Let's go swimming. My friend has a pool.\"\n\nThis was about three years before, when Ginny was a freshman in high school. It was right before Aunt Peg vanished from their lives. Ginny didn't know it then, but this would be the last time she stayed in Aunt Peg's apartment in the East Village. New York City summers are punishing affairs\u2014intense heat magnified off steel and glass, more heat coming from the subways underground, heavy, blanketing humidity that makes you feel like you weigh ten extra pounds. New York is a watery town. It sits between rivers, has a huge harbor, and swamp. For environmental and aesthetic reasons, Aunt Peg did not believe in air-conditioning. Plus, her apartment was directly above a Chinese restaurant. The apartment that was so cozy and warm in the winter from all the rising heat below was torturous in July. It was like taking a sauna in fat fumes.\n\nSo Ginny had been pretty miserable until this pool was mentioned. She was also confused, because having a pool in New York City meant you had to be insanely rich. Aunt Peg's friends were not insanely rich\u2014more just insane. Ginny also didn't have a bathing suit with her. Aunt Peg dug out an old pair of shorts and a thin, ripped T-shirt, declared that to be exactly bathing suit\u2013like, and took Ginny down, deeper into the Village, to a funky, run-down building on Avenue A. There was an alley that led around to the back, where some artist had created a metal \"garden\" out of car and bicycle parts. There was a Dumpster in the middle of this garden with a ladder up against it.\n\n\"Ready?\" Aunt Peg asked, shedding her shirt, revealing the bikini underneath. She started to make her way up the ladder, into the Dumpster.\n\nThat was not a pool.\n\n\"It's clean,\" she assured Ginny. \"It's safe.\"\n\nShe jumped inside with a loud splash.\n\nThose things seemed highly unlikely. But, as she always did, Ginny followed her aunt. She slowly climbed the ladder and peered in. Pools usually look bright. They're painted a cheerful blue on the bottom to give the water a pleasant color. Climbing into a dark pool violated some evolutionary instinct. It just wasn't right, even if it did have an inflatable alligator bobbing around. Rust, disease, dirt . . .\n\n\"It's fine!\" Aunt Peg said again. \"I swim in this all the time! Trust me, they cleaned it really well.\"\n\nGinny got to the top edge, sat there for a moment, sticking her feet into the water. It was warm, and it reeked of chlorine. That was a positive sign.\n\nAunt Peg was looking at her, waiting for her to take the plunge\u2014literally. Aunt Peg was the one person Ginny hated to disappoint, so she took a deep breath, inched herself forward, and dropped.\n\nThe water was about five feet deep, but the walls of the Dumpster were about eight feet high. It also wasn't very big. It was exactly what it looked like\u2014a big metal box filled with water in the backyard of a building in the Village. But there was something great about it\u2014it was a pool. It was an incredibly stupid pool in the middle of New York. And she had done it. Aunt Peg made the call, and Ginny had answered.\n\n\"You know what?\" Aunt Peg said. \"People would say that it's impossible to have a private pool in the city, unless you were some kind of mogul and had it on the roof of your penthouse or something. But it's not illegal to have a really clean Dumpster, and if you want to fill it with water, and if you want to get in it . . . well, that's your prerogative. People always say they can't do things, that they're impossible. They just haven't been creative enough. This pool is a triumph of imagination. That's how you win at life, Gin. You have to imagine your way through. Never say something can't be done. There's always a solution, even if it's weird.\"\n\nGinny nodded away at the time, and by the next day dismissed the sentiment as stupid artsy crap, even though she liked the pool. The day after that she found that she had an infected scratch on her arm and told her mom she got it on the garage door.\n\nBut the idea never left her mind. The adventures of the summer had been a triumph of imagination. You could make something amazing out of something awful. So, Keith had a girlfriend. So, she was essentially being blackmailed. She was here. She had made it to London for Christmas. She was with her uncle. She had jumped into the garbage pool before.\n\nBefore Ginny could think the matter over anymore, she rolled over and reached around in the dark, finding her computer on the floor. It blinded her for a second when she opened it. She blinked off the shock and turned on chat to see if he was there. It didn't surprise her to see that he was.\n\nWhen and where? she wrote.\n\nHe instantly began typing a reply.\n\nMeet me at St. Pancras Station at 10 a.m. on the 27th, at the meeting place. I have two tickets for the train to Paris at 11:37.\n\nWhat meeting place? she asked.\n\nYou won't be able to miss it. After Paris, there are two other places we have to go.\n\nWhere?\n\nI can't tell you that yet. I have to protect my interests. I've planned it all out. It will take us four days, total.\n\nGinny stared at the chat window glowing in the dark. Four days? When Oliver said they were going to take the train to Paris, she assumed he meant they would go there and back. They didn't even have to stay over\u2014that was the magic of going to Paris from London. It could be a day trip. But four days? Multiple cities?\n\nWe'll have all the pieces by the 1st, Oliver added, and then you never have to see me again. Deal?\n\nShe was on the edge again, looking into the murky depths of the garbage pool. This time, there was no voice reassuring her.\n\nFine, she wrote. I'll be there.\n\nNothing for a moment, then he began typing again.\n\nMerry Christmas, he wrote.\n\nShe closed the computer in reply.\nThe Feast\n\nGinny woke up on top of her computer, which had snuggled under the covers with her at some point in the night. She extracted it from her bed, looking at it a bit suspiciously. Downstairs, Richard was flopped on the sofa in a pair of running pants and a sweatshirt, watching a Doctor Who Christmas special and sipping a mug of tea.\n\n\"Sorry,\" she said. \"I overslept.\"\n\n\"It doesn't matter. This is casual Christmas. We do as we like. I'm drinking tea and watching television. Go make a cup and join me!\"\n\nGinny went into the kitchen. The kettle was still warm from its last use. She flicked the switch and it speedily reboiled the water as she got out one of the heavy striped mugs. This was one English ritual she loved. She never drank tea at home. Tea was for England. Tea was at Richard's house. This one simple gesture made her feel like this was home.\n\n\"I have something very important,\" Richard said, when she joined him in the living room. He reached around to the side of the sofa, producing a long box, about the size of a board game. It was full of what looked like paper towel rolls wrapped in shiny paper.\n\n\"Fancy crackers,\" he said. \"Harrods' best.\"\n\n\"Crackers?\"\n\n\"Christmas crackers? Oh, you don't have those. Here. I'll show you.\" He opened the box and removed one of the tubes, holding it out to her. \"Count of three, you pull on that end, and I pull on this one. Ready? One, two, three . . .\"\n\nGinny gave her end a halfhearted tug and the tube snapped apart with a loud cracking noise. Three small objects fell to the floor. One was a small lump of pink paper, a small folded piece of paper, and the last was a small metal elephant on a key chain. Richard handed Ginny the elephant.\n\n\"The prizes are usually rubbish, but these are nice crackers, so there you go. Small metal elephant. Quality. Bet you've been wanting one of those.\"\n\n\"Ever since I was little,\" Ginny said, as Richard unfurled the pink lump to reveal a small paper crown.\n\n\"I suspected as much. You also get a crown and a joke. The jokes are always bad. Let's see here. . . . 'Who delivers Christmas presents to all the little fish?'\"\n\n\"I don't know,\" Ginny said, putting on her crown.\n\n\"Santa Jaws. Ho ho! That's a Christmas cracker joke. Now, presents!\"\n\nRichard had gotten her a number of gifts\u2014a pile of books (chosen by his friend in the Harrods bookstore), a bag of makeup (put together by one of his friends at the makeup counter), and a sweater (chosen by yet someone else). He was a professional present-picker, with a team of literally hundreds of specialists working under him. It was no surprise then that the gifts were all perfect.\n\n\"I hope those are okay,\" he said. \"I gave them as much information as I could. I would have chosen them myself, but it seemed stupid when I could just use experts.\"\n\nGinny handed over the three packages she had brought for him. Now that she was here, face-to-face with Richard, she wondered about the wisdom of giving him reminders of Aunt Peg. Maybe what he needed was to forget. He was very quiet as he opened them, and she could see that he was getting teary eyed, which made her get teary eyed.\n\n\"They're wonderful,\" he said. \"Thank you.\"\n\nHe set them carefully on the coffee table and coughed into his fist.\n\n\"I forgot one,\" he said, his voice hoarse. \"Let me just go and get it.\"\n\nHe was upstairs for just a little bit longer than was probably necessary to get something from his room. When he returned, he had one more box, which he handed to her. \"I got this one for you when you first told me you were coming over. I forgot to put it with the others.\"\n\nIt was a phone, nicer than the one she had at home.\n\n\"I figured it would be good for you to have one for over here,\" he went on. \"Your own English number. Makes it easier to reach me, or . . . whoever you want to call. The rules wouldn't let you have one of these last time, but there's nothing stopping you now.\"\n\nGinny smiled as she took the device out of the box. Her own English phone\u2014a line to Richard, whenever she needed it.\n\n\"Thanks,\" she said.\n\n\"Now,\" Richard said, slapping his hands down on his thighs in a decisive manner that suggested he wanted to push through the emotional moment. \"Since it's just us, we can have dinner whenever we want. I was thinking, why don't we make the food now? It's lunchtime. No reason to wait. There's a little bit of everything in there. What do you say? Not only did I get the roast dinner, but since the halls aren't open tomorrow, they give us everything that won't keep. Perk of the job. I've got amazing stuff in there.\"\n\nHe wasn't kidding. There was cold lobster and fresh mayonnaise, smoked oysters, stilton with cranberries, custard, half a dozen types of chocolates, glazed fruitcake, tiny cupcakes . . .\n\n\"Partridge?\" Ginny said, examining the contents of a labeled container. \"As in, 'in a pear tree'?\"\n\n\"The same, except not so much in the tree, and more in the oven.\"\n\n\"You eat them?\"\n\n\"Don't they have partridge in America?\"\n\n\"I've never seen it,\" Ginny said. \"Does it taste like chicken?\"\n\n\"Everything does! Except this.\"\n\nHe held up the fruitcake.\n\n\"Is that a food?\" she asked.\n\n\"Ah. Good point. We'll just put that over here so we don't eat it by accident.\" Richard stuck the fruitcake on top of the refrigerator, out of harm's way.\n\nThey decided to eat in the living room, in front of the television, purely because it seemed wrong. But they did it with style, setting the coffee table with everything they could find in Richard's cabinets. They had to drag in a few kitchen chairs to serve as a sideboard for all of the food. Harrods had not done them wrong. It was a feast. A feast largely comprised of meats. Meats and things cooked in meat.\n\n\"This would have been it for me,\" Richard said, looking at the food that was all over his furniture. \"Sitting here eating all this by myself. It would have been very, very sad and very, very disgusting. Because I would have done it.\"\n\nIn that moment, Ginny was happy she came, for no other reason than this\u2014sitting around with Richard and all of these containers in front of the television. They loaded down their plates, weighing them down with the vast quantities of gravy that had been provided. They ate and watched a movie. It was just so natural, so nice being here with Richard . . . and yet, she was leaving tomorrow for Paris. She had to tell Richard now. She couldn't tell him the whole story, but she had to tell him something, at least that she was going. If he objected\u2014well, she could ignore him and go anyway, but that would be wrong. That would be using him, disrespecting someone who was generous and kind and far from stupid.\n\nShe waited until the movie was over, and Richard was reaching over for another helping of potatoes.\n\n\"There's something I need to . . .\"\n\n. . . talk to you about. Yeah. That sounded bad. Richard froze mid-reach for the potatoes.\n\n\"Someone found the last letter,\" she said.\n\n\"Someone found it? It was stolen, wasn't it?\"\n\n\"This guy . . . he bought the bag. He found the letters and got in touch with me.\"\n\n\"Well, that was decent of him.\"\n\n\"There's one last thing I have to do,\" she said. \"One last piece of art I have to get. It's broken up into three pieces. I need to go and get them. And the first one is in . . . Paris.\"\n\nPoor Richard. He didn't deserve this. Every time Ginny walked in his door she was on her way somewhere else. Then again, he had let Aunt Peg live in his house and married her, so clearly he had a thing for flaky American types who liked to sneak off in the dead of night. That was, as Ginny remembered it, how America won the Revolution in the first place. The English walked around in bright red coats in straight lines and took breaks for tea, and the American just snuck around dressed in rags and hid in trees and stole their horses. Or something. Whatever. She had to do this\u2014it was her birthright. It was what George Washington would have wanted.\n\n\"When were you planning on doing this?\" he asked.\n\n\"The day after tomorrow?\"\n\n\"Where will you stay?\"\n\n\"The same hostel I stayed in last time,\" she said. \"It's really nice.\"\n\n\"Are you going by yourself?\"\n\n\"No. I'm going with a . . . friend.\"\n\nOkay, that was a lie.\n\n\"Your friend from last time? Kevin?\"\n\n\"Keith.\"\n\nShe could hear the drop in her voice. She couldn't hide it. Richard looked at her curiously.\n\n\"He has a girlfriend,\" Ginny said. \"I just met her.\"\n\n\"Ah.\" It was a long, sad ah of understanding.\n\nRichard knew the pain of having feelings for someone who didn't necessarily reciprocate. He had loved Aunt Peg for a long time when they lived together as friends and roommates. He had taken care of her through her illness, and married her to make sure she had care. And slowly, over time, she realized that she loved him too, that she had just been afraid of this straightforward, stable guy who had never hidden his feelings. Ginny had learned all of this from the letters. Aunt Peg died before telling anyone in the family what had happened, before she could even tell Richard how much she really loved him. Ginny found herself staring at his wedding ring again.\n\n\"Do you think it's a good idea to go with him, then?\" he asked quietly.\n\n\"It's not him. It's . . . someone else. It'll take about four days.\"\n\nRichard scooped up the potatoes and put them on his plate. He pushed them around thoughtfully.\n\n\"I can't stop you if you want to go somewhere,\" he said. \"But it seems like you're asking my permission. Which is fair enough, I suppose. Let's make a deal. I know you've traveled before. So, be careful. Nothing excessively crazy. Check in with me at least once a day, and let me know where you are. And if you have any problems at all, any problems, you phone. You've got no excuse, now.\"\n\nHe tapped the opened box that contained her present.\n\n\"I won't,\" she said. \"It won't be like last time.\"\n\nLast time, Richard had airlifted her out of Greece. Maybe she shouldn't have brought that up.\n\n\"Right . . . ,\" he said, giving her a forced smile.\n\nIt was done now. She had made her decision. She had permission. And if anything went wrong, she could call Richard. And when it was all over, once she'd worked out this mess, she could tell him the whole story.\n\nShe was going to Paris.\nBoxing Day\n\nBoxing Day. Ginny had heard this term before, but it seemed so ridiculous. Boxing Day sounded like the day when everyone beat the crap out of one another. Richard explained that it was the day that you . . . put things into boxes. Or moved boxes. Or did something with boxes.\n\nRichard had been very concerned about leaving her alone again, but he had no choice but to go in to work. Apparently, Harrods after Christmas was about as bad as Harrods before Christmas, and Boxing Day was his own personal Armageddon. He was up and gone long before Ginny was out of bed. She had the entire day to herself, and no particular agenda. Tomorrow, she had to go to Paris with Oliver. She had no way of preparing for this trip. She had no information. Today, there was nothing to do but wait. Wait and think. And work on the essay. Outside, England was being English\u2014it was raining. Rain here wasn't that bad. It made being inside all the more cozy. Ginny drew a hot bath, got a pad of paper, and decided to soak, listen to the rain, and think brilliant thoughts that would get her admitted to college. She spent several minutes setting up a towel and arranging herself so she could write notes without soaking the paper, but the second she moved to write something down, she splashed the notepad and soaked it completely. She tossed it over to the toilet and sank lower.\n\nDescribe a life experience. Well, how about this? How about coming back to England to find the guy you love dating someone else and some other random guy holding your dearly-departed aunt's letters and her art hostage. How about that, admissions committee?\n\nThey would never believe her. They would think she was a fantasist. They would put her picture on the corkboard with a note under it that just read: PSYCHO. DO NOT ADMIT.\n\nThe bathwater cooled almost as fast as her desire to work on the essay, so she got out and dressed. Her timing was excellent, because as soon as she pulled on her shirt, there was a brisk knocking on the door. She jogged downstairs, shoeless, her hair still damp and straggly.\n\nKeith stood on the doorstep, bundled up in a big army-green coat, a heavy black scarf wrapped several times around his neck. He stepped inside, dropped his umbrella in the hall, then yanked off his coat and scarf and threw them on a hook.\n\n\"It's pissing down,\" he said.\n\nHe walked straight back into the kitchen. This was how he used to come over in the summer\u2014just walking in like he was fully expected, like he didn't have a girlfriend he'd kept secret from her. Ginny was too surprised to react in the moment, so she shut the door and followed along.\n\n\"Present,\" he said, setting a CD-shaped object on the table. \"Go on, open it!\"\n\nThe present looked like it had been wrapped in repurposed paper, full of telltale white crinkle lines and old tape marks. For some reason, this made Ginny open it very carefully, like the paper needed to be kept for another mission in the future. Inside, there was a homemade CD, complete with a fancy designed cover from the Starbucks: The Musical poster.\n\n\"Properly recorded,\" he said proudly. \"So you can listen to it every day. So, what have we here?\"\n\nHe examined the many containers piled on the kitchen counter and turned to Ginny with an expression of intense interest.\n\n\"We have a lot of leftovers,\" she said. \"If you're hungry, there's a lot more in the fridge . . .\"\n\nHe was already rooting through the fridge and pulling out the tins.\n\n\"What is this?\" he said, pulling off a lid. \"It's not turkey. . . .\"\n\n\"Pheasant,\" Ginny said. \"Richard got the fancy Harrods Christmas dinner . . . because he works there.\"\n\nGinny passed him a plate, and he started piling it up with food. Keith could always eat, anything, anywhere, in any amount.\n\n\"So,\" he said, \"I got the impression the other day that something brought you over here, but you didn't say what.\"\n\nGinny filled the kettle and said nothing for a moment. Her brain was still trying to catch up with the fact that Keith was here in the kitchen with her, that she was unprepared and wearing pajama bottoms . . . and now he was asking about the letters. But if anyone could take this weird story, it was him. And he had been there since the beginning. He had a right to know.\n\nShe set the kettle on its base and switched it on. Keith took a seat at the table to eat.\n\n\"Someone found the letters,\" she said. \"All of them. Including the last one.\"\n\n\"Someone from Greece?\" he asked. \"Isn't that where they were stolen?\"\n\n\"Yeah, but the person who found them is from here. He's English. He bought the bag. He used the information in them to track me down. The last letter, the one I never got to read before\u2014it has more instructions in it. There's another piece of art. There's something else I have to do. I have to go to Paris, tomorrow.\"\n\n\"Always the same with you,\" he said, shaking his head and taking a bite.\n\n\"There's kind of a problem.\"\n\nKeith was chewing, so he waved his fork, indicating that she should elaborate.\n\n\"He won't give me the letters.\"\n\n\"What do you mean he won't give them to you?\" he asked, swallowing hard.\n\n\"He's keeping them, because he wants half the profit from the last sale. He bought the tickets to Paris. He's the only one who knows where we're supposed to go or what we're supposed to do.\"\n\nSaid out loud, it sounded even crazier and much, much worse. Keith set down his fork and tapped his fist lightly against his mouth in thought.\n\n\"You just made that up, right?\" he said.\n\n\"Nope.\"\n\n\"So, you're saying that a complete stranger bought your stolen property, and is now demanding half your money . . . and that you are going to Paris with him. Because that is the sane thing to do.\"\n\nThe kettle clicked off. She busied herself with making the tea.\n\n\"What choice do I have?\" she asked, yanking out two mugs. \"He's not dangerous. He's just . . . he just wants the money. I need to get the pieces. I've had to do worse.\"\n\n\"Who is this guy?\"\n\n\"His name is Oliver Davies.\"\n\n\"That tells me nothing. What's he like? How old is he?\"\n\n\"He's, like, our age or something. It'll be fine. I'll just go with him. I've traveled with people I didn't really know before. I did with you.\"\n\n\"It's not exactly the same.\" His voice was rising. \"We went to Scotland together once. We both happened to be in Paris at the same time. And I didn't steal your stuff and ransom it back to you.\"\n\n\"I didn't mean . . .\"\n\n\"Gin, listen. Sit down.\" He was more serious than Ginny had ever seen him before. She picked up the two mugs of tea and sat next to him. \"I'm all for weird stuff as a general rule, but you can't do this. What you're describing is some kind of travel horror story waiting to be written. In the best-case scenario this guy is some kind of a con artist, and that's me being optimistic.\"\n\n\"I know it's a problem,\" she shot back, unable to hold her frustration in any longer. \"What choice do I have? If I don't do this, I never get the letter. I never find the piece. I never finish. And it's not like anyone can help me. What am I going to do? Call the police and tell them that someone stole my mail? He'll just disappear, and I can't let that happen.\"\n\nKeith leaned back in his chair and pushed the front legs off the ground, sighing heavily. He stayed balanced like that for a moment, then brought the chair down with a thud.\n\n\"You said you're supposed to go tomorrow, right?\"\n\n\"Right.\"\n\n\"Explain to me how this is supposed to happen.\"\n\nGinny took a long breath. She was shaking now.\n\n\"I'm supposed to meet him at St. Pancreas. . . .\"\n\n\"St. Pancras, you mean. That's where the Eurostar leaves from. You said he wants you to take the train to Paris?\"\n\n\"He got us two tickets. I'm supposed to meet him at ten. He's even set up the sale. He took me to see Cecil. Whatever it is we're getting, it's going to be auctioned off on the second, and I have to be there. So I guess he can't kill me, right?\"\n\nShe tried to laugh, but it didn't quite work. Keith did not join in. Keith resumed eating for a moment, spearing a huge piece of pheasant that Ginny couldn't believe he actually managed it fit into his mouth. He chewed it to nothingness, his eyes flicking back and forth a bit in thought.\n\n\"It seems to me that all you need is the letter,\" he finally said. \"Correct?\"\n\n\"Right. But he's not going to give it to me.\"\n\n\"But he'll have it with him tomorrow, right?\"\n\n\"He'd have to.\"\n\n\"All right then. We have a solution.\"\n\n\"We do?\"\n\n\"You forget,\" he said. \"I am a man of many abilities.\"\n\nHe stretched out his hands on the table and wiggled his fingers.\n\n\"You play piano?\" she said.\n\n\"Do you forget my shameful previous life? I'm a thief. I nick stuff.\"\n\n\"I thought you mostly vandalized stuff.\"\n\n\"I stole a car,\" he said proudly. \"And many other things.\"\n\n\"But you quit. You don't do that anymore.\"\n\n\"I gave up thieving for gain, but there's nothing wrong with using my skills for good, now is there? Everybody loves Robin Hood. And I haven't lost my touch. Oliver Davies takes the letter out, I steal it from him. Easy.\"\n\nHe scooted his chair closer, until they were directly side by side, his arm up against hers. Every hair on her arm stood up in unison, goose bumps everywhere. It must have been like leaning against someone who had pineapples for limbs.\n\n\"This is easily fixable, Gin,\" he said. \"This is nothing. We will get the letter back, and you can get your aunt's art. Come on, now. No one messes with my mad one.\"\n\nOh god. She was tingling all over. She was going to become hysterical. She was going to grab him by the face and make out with him. There was nowhere she could look at him that made it any better. The way his new haircut revealed his ears a bit more, the way his T-shirt stretched across his chest, the string bracelets he had around his wrists . . . everything highlighted something about him that seemed unbearably wonderful. Her hands shook a little. She quickly pulled them down into her lap.\n\n\"Thanks,\" she said. She turned her head partially in Keith's direction. She could not look at his face. She had to find a dead zone somewhere on him that produced no feeling. She tried for the armpit, which she could look at easily because he had his arm extended, but even that made her pulse go faster.\n\n\"I have to go,\" he said. \"I was just stopping in. But it's sorted now, yeah? I'll come by at eight thirty tomorrow and we'll go over to the station together to work through the details.\"\n\nHe got up and pulled on his coat and scarf. The fact that he was leaving without saying why, or where he was going, or inviting her along told her all she needed to know. Ellis. They were going out somewhere. Or maybe she was already at his place for a cozy night in.\n\nShe commanded her brain to stop this train of thought. Keith was here now. He was going to help her. They would get the letter back. It wasn't exactly like before\u2014it would never be exactly like before\u2014but it was something.\n\n\"Tomorrow,\" he said, as he stepped outside. \"We'll get it. Remember, I have never failed at anything.\"\n\n\"Never,\" she said.\n\nHe reached up and gave her new haircut a shake with his hand.\n\n\"Still have to get used to this,\" he said.\n\n\"Me too,\" she replied.\n\nVery wisely, he said nothing else, and jogged down the steps and to his car.\nIt Takes a Thief\n\nSt. Pancras train station is precisely the kind of thing Ginny went to Europe expecting to see, along with vast cathedrals, small cars, advertisements featuring casual nudity, and doctors smoking in front of hospitals. A good portion of the building is a massive Victorian Gothic work of art, made of deep russet-colored bricks and covered in two levels of arches made of alternating brick and white stone. At the end is a clock tower\u2014a sharp spire surrounded by shorter, sharper spires. The other part of the station is a pristine, supermodern glass temple. Along the massive arcade, dozens upon dozens of tiny brick archways are filled with every kind of shop or service, including Europe's longest champagne bar, almost three hundred feet long. Hundreds and thousands of people wandering around with bags, some with little to no idea of where they are going\u2014people figuring out train passes and connections and fumbling around with new currency.\n\nIt's exactly the kind of place a thief might like to spend some time.\n\nGinny and Keith stood along the rail on the second floor. Ginny looked up at the massive glass arch that formed the ceiling. Her suitcase was at her feet, and she had four hundred Euros in her pocket. Even though she made over a hundred and thirty thousand dollars in the sale, most of that money had been set aside for college. Her own bank account was much smaller, and this was a pretty big hit to her balance. She would have to be careful to have enough to make it through the next few days. \"All right,\" Keith said. \"One last time. You're going to meet him over there at the statue . . .\"\n\nThe statue was The Meeting Place, and it was as easy to spot as Oliver had suggested. Thirty feet high and made of bronze, it showed a man and a woman in a breathless moment of meeting, their faces close together, about to kiss. It was so nice to have a big, metal reminder of romance towering above her and Keith.\n\n\"I'll be watching from one of the arches,\" Keith continued. \"And I'll have this.\" He held up the large, unwieldy London tourist map they had just purchased. It had been selected for sheer size, and when fully opened, provided a rustling paper shield.\n\n\"All you need to do is get him to hold it out for a few seconds. I'll do the rest. Once I get it, I'll leg it. I'll meet you outside.\"\n\n\"I just remembered something,\" Ginny said, leaning over the rail, pressing down hard, crushing it into her abdomen. \"That's the same trick some street kids pulled on me in Rome. They came at me with these newspapers. They were flapping them around, trying to distract me and get my bag.\"\n\n\"Well, it's an old move, but a solid one. Did they get anything?\"\n\n\"No. This guy came over and chased them away.\"\n\nWhat Ginny decided not to mention was that this same guy, who was her age and kind of hot, then persuaded her to go to his sister's apartment, where he tried to make out with her. He was so extremely sketchy that Ginny had to run away. Why did she run into so many thieves and skeezes?\n\nWhen the enormous clock above the statue hit ten; Oliver's long, black-coated figure passed below them, heading for the stairs. Along with the leather satchel, a small backpack rested on his back. Oliver was dressed less formally this time. He had on black cargo pants, a snug, somewhat ragged gray sweater, and a heavy scarf wound around his neck that came up well over his chin. Combined with the long black coat, he looked like some kind of operative about to go on a mission.\n\n\"That's him,\" she said.\n\nKeith took a moment to make a mental note of his target.\n\n\"Black coat? Beaky face?\"\n\nGinny nodded.\n\n\"All right,\" he said. \"Showtime. Don't worry. I haven't lost my rapscallion touch.\"\n\nHe winked and peeled away from Ginny's side, wandering out of view. She took a deep breath and pulled up the handle of her bag, rolling it over to the statue. \"Ready to go?\" Oliver asked. \"We have some time yet, but we might as well go through security now, unless you need something from one of the shops . . .\"\n\n\"I want to see the letter,\" she said.\n\n\"Why? You know I'm not going to let you read it.\"\n\nGood point. Ginny tightened her hold on the handle of her bag and looked up at Oliver's face. It was a thin face, with strong, impassive features. The statue behind them was more expressive.\n\n\"Because . . . ,\" she said, \"I started off every leg of the other trip by looking at a letter. I just need to see it, okay? It's how this is done.\"\n\nOliver rocked back on his heels and considered this, then opened the flap of the leather bag and held up some folded pieces of blue paper between two of his fingers. Immediately, Keith appeared from the hallway behind the statue and walked toward them briskly. He had the map out, feigning struggle and confusion, opening it this way and that, glancing around as if trying to get his bearings. \"I'm at the meeting place,\" he said, in what Ginny assumed was supposed to be an American accent. It was a little strange\u2014kind of like someone had taken a cowboy, a surfer, and a 1930s gangster and put their accents in a blender. \"The statue. The meeting place. The statue. The frickin' huge statue of the people kissing . . .\"\n\n\"Satisfied?\" Oliver said, holding up the pages. He took no notice of Keith, who was coming closer and closer. He was not, however, close enough. Oliver was starting to put them away.\n\n\"Wait . . . ,\" she said. \"Where's the envelope? I need to see the envelope too.\"\n\n\"I don't have the envelope with me.\"\n\n\"Why not?\"\n\n\"Because I didn't need it. I only needed the letter.\"\n\nKeith was just a few feet behind him now.\n\n\"The statue, not the station,\" he said. The closer he got, the more Ginny could hear the accent\u2014and the more she heard the accent, the worse it sounded. Hopefully Oliver just thought that was what Americans sounded like. \"What street are you on? No, the statue is in the station . . .\" Bang. Keith crashed into him, hard. For just a moment, the map closed around Oliver's outstretched hand. Keith mumbled apologies before collapsing his map as if very embarrassed, and hurrying off. When he was gone, Oliver's hand was empty, and Keith was gone. It worked. It actually worked.\n\nThis was supposed to be the moment where Ginny laughed in triumph and delivered the speech she'd been preparing in her mind most of the sleepless night before. She was all ready to go, just as soon as the look of dismay and shock crossed Oliver's face. But it didn't come.\n\nThere was no use even pretending now. Ginny turned in the direction Keith had run off. He had stopped at the base of the steps and was shaking out the map and looking very confused. He looked up to Ginny, shook his head, and started making his way up the steps.\n\n\"I take it that didn't go as planned,\" Oliver said, when Keith had reached them. \"Who are you, anyway?\"\n\n\"I'm her hairdresser,\" Keith said. \"She doesn't go anywhere without me. What did you do?\"\n\n\"I got here an hour early. I stood right over there. . . .\" He pointed to a spot on the other side of the second floor, maybe twenty yards away. \"I watched you two arrive and do all your plotting. It wasn't that hard to work out what you were up to.\"\n\n\"So where's the letter?\" Ginny asked.\n\nOliver reached up the sleeve of his coat and drew out the blue pages.\n\n\"Looks like we both do sleights of hand,\" he said to Keith.\n\n\"You are the ultimate div and obviously have no mates.\"\n\nOliver dismissed Keith with a shrug, and turned his attention to Ginny. \"I'm getting on the train. This is a one-time offer. Either come with me now, or we're done here.\"\n\nOliver started walking in the direction of the boarding area for the Eurostar.\n\n\"I'm sorry,\" Keith said. \"I thought I had it. . . .\"\n\n\"It's okay,\" Ginny replied. \"But I guess I have to go.\"\n\nThey had tried, and they had failed. Keith put his hands deep in his pockets and stared at the ground. She held up her hand in a lame gesture of farewell.\n\n\"See you in a few days,\" she said.\n\nShe had only gotten a few steps when Keith jogged in front of them and blocked their progress.\n\n\"I have a car,\" he said, holding up his keys for Oliver to see.\n\n\"Good for you,\" Oliver replied.\n\n\"As it happens, I also have a few days free. So I'll drive. Paris isn't that far. Maybe six or seven hours?\"\n\n\"There's no way I'm going with you,\" Oliver said, stepping around him.\n\n\"I didn't offer to drive you. I'm offering to drive her. You can do what you like.\"\n\nBut Ginny didn't move.\n\n\"I'm going with him,\" Ginny said loudly. \"In the car.\"\n\nThis obviously threw Oliver for a loop. He didn't have a very expressive face, but she could just feel the displeasure coming off of him in waves.\n\n\"We need to do this together,\" he said.\n\n\"We can,\" Ginny heard herself saying. \"We can meet there. How about the pyramid in front of the Louvre?\"\n\n\"Good choice.\" Keith nodded. \"Pyramid. Louvre. Noon tomorrow? Have fun on the train.\"\n\nHe clapped Oliver on the shoulder, hard, and linked his arm through Ginny's to lead her away. It was all Ginny could do not to skip . . . to sing . . . to weep for joy. Okay, it wasn't exactly like the summer\u2014but this was just the two of them, going off together. Going to Paris together. Driving along in his car for hours and hours, into the City of Lights. They would have meals together, and talk for hours. They would have to get a place to stay . . .\n\n\"You're paying for petrol, of course.\" Keith smiled at her as he walked to the door. \"And everything else. It'll be like old times! I think I want some cheeses. Lots of cheeses.\"\n\n\"So many cheeses,\" Ginny said, nodding.\n\nThe fantasy lasted all the way to the bus stop outside of the station doors. This is where Oliver caught up with them.\n\n\"Again,\" Keith said, not looking over, \"I didn't offer you a ride.\"\n\n\"Well, I'm coming. Or this isn't happening. What are we supposed to do when we get the pieces? I suppose you'll just let me hang on to them, is that right?\"\n\nKeith let out a long sigh and looked over at Ginny. A light passing rain pattered on top of the glass bus shelter.\n\n\"We're going to have to bring him, aren't we?\" he said.\n\n\"Probably,\" Ginny replied sadly.\n\n\"In that case . . . I want a hundred Euros up front for petrol and as general payment for the annoyance of having you in my car.\"\n\n\"I bought the train tickets,\" Oliver said. \"It's not my fault if we're not using them. I'll give you fifty and we'll work from there.\"\n\n\"And twenty pounds for parking and congestion charges,\" Keith added.\n\nOliver handed reached into his pocket and pulled out some money. He had prepared as well; he had a wedge of Euros. After giving Keith one fifty Euro note and a twenty pound note, he stuck a cigarette in his mouth and lit it, signifying that the deal was done.\n\n\"And you smoke,\" Keith said. \"Lovely. Don't even think about trying that in the car.\"\n\nOliver obligingly stepped a few feet over. Conciliatory for a blackmailer.\n\n\"You realize,\" Keith said, eyeing Oliver's bag, \"that your only value is the letter you have in the front pocket of your bag. It would be a terrible shame if you were separated from that bag and pushed out of a slowly moving car somewhere next to a French cow pasture.\"\n\n\"What, this letter?\" Oliver reached into the pocket and produced the folded pages. \"I can fix that problem right now.\"\n\nHe crumpled the paper and tossed the ball into the road. Ginny let out a gasp of horror as cars and trucks and buses rode over it. A few seconds later, it vanished, probably carried away by a tire.\n\n\"What did you just do?\" she yelled.\n\n\"That was some blue paper I just bought in Waterstones. Like he said, I'm aware that my only value is having the letter. Don't worry. It's safe.\"\n\n\"Safe where?\" Ginny asked.\n\n\"Safe from pickpockets with rubbish American accents.\"\n\nThe bus lumbered up to the stop. Oliver flicked the cigarette away and waved his hands, indicating that he would follow Keith and Ginny.\n\n\"I'm not a violent person,\" Keith said under his breath, as they climbed the steps to the second level of the bus. \"But I've really been meaning to work on that.\"\nOne More for the Road\n\n\"Is my American accent bad?\" Keith asked quietly, once they were on the bus.\n\n\"It was fine,\" Ginny said, looking down at her lap. The accent still burned her ears, but there was no point in telling him that.\n\n\"I've been working on it for a while. Mimicking Marlon Brando in A Streetcar Named Desire. And I've been trying to copy The Wire, but that one is kind of hard. . . . Anyway, I have to text. . . .\"\n\nThe sentence trailed off, so that had to mean he was texting Ellis. It seemed that the policy was going to be that Ellis would never be mentioned directly. It was hard not to hate Ellis. This wasn't her fault. But she could hate Oliver. That was a perfectly acceptable activity. He was sitting two rows in front of them on the top of the bus. His hair was cut very cleanly and precisely, with a ruler-straight line along the back of his neck. He had a mature bearing\u2014seated straight, shoulders back. Not rigid, just very adult. Keith was more slouchy and scratchy and free-flowing. Keith looked like a student. Oliver looked like someone with . . . some kind of responsibility. Evil responsibility.\n\nWhen they got off the bus Oliver kept about ten feet behind them as they walked to Keith's house.\n\n\"I'll wait out here,\" he said.\n\n\"Yes,\" Keith replied. \"You will.\"\n\nThe house was cold and mostly dark, but the lights were on up in Keith's room. Ellis was already up there, looking out the window.\n\n\"Is that him?\" she asked. \"Down in the garden?\"\n\n\"That's Oliver,\" Keith said, opening his closet, shoving some things aside and pulling a bag out from under a pile of stuff. \"He's the wanker who's got Gin's letter.\"\n\n\"He's more normal looking than I thought he would be.\"\n\nGinny peered between of the blinds on the other window. Just below, Oliver was patrolling the garden, his one arm behind his back, the other working the cigarette. He gazed at the cracked pavement like it was a map he was using to plot a siege.\n\n\"He's a prize,\" Keith said. \"I'm just going to get my things from the bathroom. I'll be ready to go in a minute.\"\n\nThen it was just Ellis and Ginny, smiling strangely at each other. It took Ginny a second to realize it but David wasn't home . . . which meant that Ellis had let herself in. Which meant she had her own key. And that . . . was not something Ginny was going to dwell on. She let the blinds go and they fell back into position, sending a spray of dust up her nose.\n\n\"So,\" Ellis said cautiously. \"I have nothing in my diary for the next few days. And I packed a quick bag. I don't want to intrude, but if you wouldn't mind . . . I'd love to come. Really. Only if you don't mind.\"\n\nIn the bathroom, Keith could be heard crashing through the medicine cabinet. Either he was intentionally being loud, or he had temporarily lost muscle control. He must have known this question was coming.\n\n\"It's all right to say no,\" Ellis said. \"I know this is personal, and important.\"\n\nThat sounded very sincere. Ellis was genuinely asking Ginny if it was okay. But what else was she going to say? No? No, Keith's friendly girlfriend, you cannot take a trip in your own boyfriend's car? Even when it was her, Keith, and Oliver, at least it was two against one. They would stick together. But now, the dream was well and truly over.\n\n\"Sure,\" Ginny said, trying to sound enthusiastic. \"Of course.\"\n\nEllis clasped her hands together in excitement. \"Oh, I'm so glad. I've never been to Paris before. Weird, right? Since it's just a short train ride away, and I studied French for years. I grabbed a few things from Sainsbury's. . . .\" She picked up a shopping bag by her feet and held it up. \"Biscuits, crisps, fruit, water. Some Top Trumps for the long, boring bits on the train. We'll need to take the Chunnel\u2014I looked up the route online. Plus, I bought a map of France just in case we can't get a signal.\"\n\nKeith decided this was a good moment to return. He had an overstuffed backpack, and was punching the contents down with his fist.\n\n\"I'm coming!\" Ellis cried. \"Ginny said it was okay.\" Keith kept squishing and pushing the contents into the bag, trying to get it closed.\n\n\"Brilliant,\" he said. He gave the zipper one final tug and strode out of the room. \"We should get going.\"\n\n\"Is that your car?\" Oliver asked when they came outside. He pointed to the humble, turtlelike automobile parked at a slight angle in front of the trash.\n\n\"Awe will soon take the place of jealousy,\" Keith said, brushing past him. \"I've seen it before.\"\n\nKeith opened the trunk and examined the contents. He had cleaned it out a good deal, but it was still fairly full. He grabbed up two large bags, looked inside, and dropped them into an open trash bin. Though Ginny loved the little white car and had many fond memories of it, she understood Oliver's trepidation. It didn't exactly look like the ideal vehicle to take around Europe. For London, sure, it was perfect. It was small and pre-dented, ideal for zipping between buses and cabs and down narrow streets that were never built for cars. You could park it anywhere\u2014you could probably park it in the house if you had to. Plus, it wasn't something anyone would want to steal. The white was faded, like old T-shirts that had been washed too many times with black socks. There were dings and scratches and tiny, coin-size spots of rust near the bottom. It screamed, \"I have manual locks.\"\n\n\"We're not all going,\" Oliver said, glancing at Ellis's overnight bag and shopping bag full of food.\n\n\"Oh, but we are.\" Keith shoved his bag in first, then gestured for Ellis and Ginny to pass theirs over. Oliver tried to put his in as well, but Keith slammed the lid down before he could.\n\nThe car was a two-door\u2014a fact that had never been relevant before. Ginny had always been in the front seat. Today, she would almost certainly be in the back. She never thought about the back of the car as being an actual place you could sit. It was more like a glove box extension.\n\n\"I'm shorter,\" Ellis said. \"I'll get in the back with Gin.\"\n\n\"No,\" Keith said. \"Let him manage in the back.\"\n\n\"Gin's taller than me. She should ride in the front. This is her trip. I'm intruding.\"\n\n\"It's fine,\" Ginny said. That conversation needed to end. \"I'll take the back.\"\n\nShe folded down the front seat with a bang and plunged in headfirst, getting briefly tangled in the seat belt, before squeezing herself in. The backseat was not a happy place. It was covered in musty-smelling fabric\u2014fabric that had seen dirty sets and smelly costumes and piles of old take-out bags of hamburgers and fried fish and chips. In fact, the first thing it brought to mind was the swimming pool Dumpster, except it wasn't as big and it wasn't as clean.\n\nWhat was mildly uncomfortable for her must have been torment for Oliver, who was at least six or seven inches taller. His head scraped the roof and he had to keep his neck slightly bent. He stuffed the backpack by his feet and held the leather satchel on his lap. The combined effect forced him into a squashed, leaning position, with his shoulder pressed up against Ginny's ear. She tried to move closer to the door, but there was simply no more room. They had been packed in like freight. Keith got into the driver's seat and immediately adjusted it back into Oliver's already cramped knees.\n\n\"All set?\" he asked everyone.\n\nHe put the key in the ignition and turned it. The engine made a terrible screeching sound, then coughed itself off. \"This car is never going to make it to France,\" Oliver said.\n\n\"Don't worry about the car,\" Keith said, jiggling the key. \"Worry about yourself.\"\n\n\"Who are you two? Do you have names at least?\"\n\n\"I'm Mr. Pink,\" Keith said. \"She's Mr. Shut Your Face. Now, tell us where we're going.\"\n\n\"Paris,\" Oliver replied stiffly.\n\n\"Yes,\" Keith said slowly, with mock patience, \"you told us that already. But can you be a bit more specific than that? I don't know if you're aware of this, but it's actually quite a big place.\" He reached down and folded his seat back, completely crushing Oliver.\n\n\"I'll tell you when you get off my lap.\"\n\n\"I just want you to know what to expect from this trip,\" Keith said, folding the seat back up into a driving position. \"Pain. Not nearly as much as you deserve, but we will try our best.\"\n\n\"I already worked it out,\" Ellis said, holding up both a map and her phone. \"We take the M20 to Folkestone, then from there we get the train to France. The trip to Paris should take about five and a half hours, total, so we'll be there by dinnertime.\"\n\n\"I'll tell you the rest when we're closer to Paris,\" Oliver said coldly.\n\n\"'I'll tell you the rest when we're closer to Paris,'\" Keith repeated, in an exact copy of Oliver's voice. He could do other types of English accents very well, at least to Ginny's ear. \"Posh boy speaks posh. Bet you went away to school. You a public school boy? Sent away from home at a young age? Is that why you're so well-adjusted?\"\n\n\"Yes,\" Oliver said. \"That's why. Can we go now?\"\nThe Talking Letter\n\nMost of the route to Folkestone was a highway\u2014so the next two hours were mostly spent looking at the backs of flatbed lorries, vans, other cars, and the many sheep and horses that seemed to graze along England's major traffic arteries.\n\nKeith's car, never a prize, was even worse in the winter, in the backseat. It was thin and poorly insulated. The heater was a concept joke that was probably funnier closer to the vents in the dashboard. Ginny huddled inside of her jacket and zipped it up over her chin, breathing hot air back on herself. In the front, Keith and Ellis were talking, but Ginny could just about hear them over the terrible noise of the engine. Oliver had his headphones in the whole time. She was in a little bubble, all on her own.\n\nOnce they got to Folkestone, they made their way into a long line of cars at a dock, where they sat for half an hour. Then a man in a glowing yellow-green jacket was waving them along a train platform to a series of wide doors and directly onto the train. This was an odd experience, being in a car on a train. All the cars trundled along through the silver train compartments. There were ads on the walls, and everything was bathed in a soft yellow light. Then another man in a vest flagged for them to stop. Doors closed and a heavy metal grate dropped down, locking them in. There were no windows around, not that there would be anything to see. They were going through a tunnel, passing under the English Channel\u2014a kind of very long, sideways elevator ride.\n\nOliver tried to stretch a bit, accidentally digging his elbow into Ginny's ribs. She pushed it back.\n\n\"So,\" she said. \"Are you going to show me the letter now so we know where we're going?\"\n\n\"There's nothing to see,\" Oliver said. \"I don't have it with me.\"\n\nOn that, Keith and Ellis swiveled around.\n\n\"You don't have the letter?\" Ginny said. \"You forgot it?\"\n\n\"I memorized it.\"\n\n\"You are joking,\" Keith said. \"I realize that you are not like the other children, but you are joking about that.\"\n\nIn reply, Oliver tipped his head back, closed his eyes, and began to recite.\n\n\" 'Oh, you're still reading. Good! All right, Gin. You're in Greece. Greece is a fine place to be. . . .' \"\n\nOkay, so he wasn't joking. This was weird, listening to Aunt Peg channeled in a deep male English voice. This was like a horrible s\u00e9ance. In a car, on a train, under the English Channel.\n\n\" 'Have you ever seen water like this? Felt sun like this? Is it any wonder that the Greeks were among the first to really start asking questions about the nature of beauty and art and life itself? This is the birthplace of Western thought. This is where the Big Questions were forged out of the stuff that had been eating at mankind's collective brain for millennia\u2014the big What the hell is going on? What the hell is going on? has been the central question of my life.\n\n\" 'Sometimes I'm asking it in a big sense. Sometimes I mean it in a very small, immediate sense, like when I am trying to do my taxes. Lately, what with the brain cancer and all, I ask it all the time. I ask it about the TV remote (which, to be fair to me, is insanely complicated). I ask it when I can't remember which way it is to the grocery store. My disease has taken me on a journey of wonder, Gin. Wonder, and a lot of trying to buy bread at the post office.\n\n\" 'Even I know that some of the things I have asked you to do are strange, and I'm a weirdo with a tumor the size of an egg in her head. But I have a method to my madness.\n\n\" 'I want us to make a painting together. This painting is inspired by something my friend and idol Mari Adams did called Paint This for Me. She did a series of sixteen identical paintings\u2014really simple ones\u2014and then she left them in various places around Edinburgh to be touched, admired, rained on, stepped on, drawn on, sliced up . . . whatever happened to the paintings, that was all part of it. Then she collected them up and did an exhibition. I always liked the idea that the paintings were out there living their lives, being changed by the world. My idea is a little different. I am making one picture out of different materials that I have placed around in various spots. In order to collect them, you're going to need to revisit some places you've been, and go to one place you haven't. I'm marking your route home.\n\n\" 'The first place you're going back to is Paris. No one \"gets\" Paris after one visit. No one. How you get there is up to you. I know how I would want to do it\u2014I'd jump ferries all the way across the Mediterranean, stopping in Sicily and Sardinia. Or you could stay along the coast of Italy and France, bouncing your way along the Rivera. Or you could just take a plane. Whatever floats your boat. Or plane. Just get to Paris. And what's there\u2014the first part of this piece is the background\u2014the sky.\n\n\" 'Sometimes, Gin, I wonder what inanimate object I would like to be, if I could be any inanimate object in the world. There are so many good choices. I'd love to be an airplane that crossed the Atlantic twice a day. I'd love to be the Tivoli fountain, where poets have perched for hundreds of years, and tourists have come to understand the joy of living art. But the one I always come back to is much more humble. I'd love to be a tabletop in Paris, where food is art and life combined in one, where people gather and talk for hours. I want lovers to meet over me. I'd want to be covered in drops of candle wax and breadcrumbs and rings from the bottoms of wineglasses. I would never be lonely, and I would always serve a good purpose.\n\n\" 'I suppose you remember going to my friend Paul's restaurant, the one I decorated? I made four tabletops for it. They're all made out of doors, and I painted each by hand. The paint I used wasn't really designed for the wear and tear of a restaurant, so they should all be well-marked. It's up to you to take the one you know is right. You'll know it when you see it. Use your instincts.\n\n\" 'Paul knows that you will be coming back to take one of the tabletops. I asked him not to mention this to you when you first came by\u2014so I hope you are pleasantly surprised that you are going back.' \"\n\nOliver stopped and fiddled with the clasp of his bag.\n\n\"That's it for now,\" he said.\n\nIt was only then that Ginny realized she was clenching her stomach muscles to the point of nausea, holding in whatever reaction hearing this letter produced. It wasn't sadness or excitement\u2014it was homesickness and nostalgia. It was like hearing the voice of a ghost. No one said another word until the train came to a stop and they made their slow way out and onto a French road, which looked more or less exactly like the English road they'd just left behind.\n\n\"It's in the caf\u00e9,\" Ginny said quietly. \"Les Petits Chiens. That's what it's called.\"\n\nIt took about three hours of driving through France, punctuated by a few dodgy turns and bursts of swearing from Keith. Ellis navigated while Keith negotiated a car with right-side steering on what was, for him, the wrong side of the road. Oliver went back to his music and staring out the window. Ginny was left to her own thoughts, which was honestly the last place she wanted to be left.\n\nThey reached Paris right around five, just as the streets were snarled with traffic and the dark had descended and the streets glowed orange from the streetlights. It was bizarre how quickly generic highway could turn into . . . well, Paris. For the first time since she had gotten in this car, Ginny felt a surge of excitement. There was the Eiffel Tower, just illuminated for the night. There were the long stretches of creamy white buildings with their big gray-black roofs and their skylights. There were the Art Nouveau Metro signs from the turn of the twentieth century, with their sinuous green iron workings that looked like curling plants. This weird, impossible place that looked like a collection of palaces, a grubby city, a museum, a tight cluster of caf\u00e9s\u2014everything, all at once.\n\nIn the summer, the trees had been thick and green. Now, the trees were bare, but heavy with lights, so many lights, the color of champagne bubbles. Paris took its decorating seriously. The smells of the city seeped in\u2014the bread coming from the bakeries, the toasty smell of a crepe truck, the occasional gust of sewer or garbage. Then, right back to the bread and crepes. Ginny's stomach grumbled loudly.\n\n\"I could eat one of those little dogs,\" Ellis said, pointing at someone walking a ratlike creature. \"I am honestly that hungry.\"\n\n\"Me too.\" Keith swerved to avoid a pedestrian\u2014or maybe toward a pedestrian. It was hard to tell. \"I'm glad we're going to a caf\u00e9.\"\n\nThey drove on to increasingly smaller and more familiar-looking streets, finally arriving on a narrow artery where motorcycles had completely taken over the sidewalks for their driving and parking purposes. The car barely fit down the road. Keith stopped when he could go no farther and they seemed to be in the right area. They climbed out of the car. Oliver immediately grabbed for his cigarettes and shoved one in his mouth.\n\n\"I know where we are,\" Ginny said. Somewhere in her brain, she had stored the layout of these little passageways. She started walking, surprising herself with her own assurance. Sure enough, she turned a corner and saw the tree that blocked the front, now bare of leaves. Les Petits Chiens was dark. There was a sign stuck to the door, which was barely visible. It's never good when there's a note on the door of a shop or restaurant. It never means, \"We're open and everything is working just fine.\"\n\nGinny held up her phone to illuminate it.\n\n\"It's in French,\" she said. Her three and a half years of high school French had led her to this moment. \"It says . . . 'Dear Customers, Jean-Claude and I have gone to Orange for the holiday. The restaurant . . . will open . . . something, something . . . on January third. Merry Christmas and Happy New Year. . . .' Oh my god. It's closed until January.\"\n\nThat was it. Instant failure, right out of the gate. Oliver cupped his hands around his face and pressed his face against the window to look inside. He tried the door as well, even though it was pointless. Keith immediately began laughing the tired laugh of someone who has just driven all the way to Paris in a backwards car.\n\n\"So we came all this way for nothing,\" he said. \"Brilliant. Maybe this is something we could have checked on if we had the letter.\"\n\n\"All right!\" Ellis said brightly. \"This is just a little setback. We made it all the way here, we can figure this out.\"\n\nNo one replied, so she tried again, this time bouncing a bit as she spoke. \"We haven't eaten properly all day,\" she went on. \"We just need some food to buck us up, and we've got the best food in the world all around us. Let's find ourselves a nice little caf\u00e9 and have some dinner. Then we can think about what to do next. Right?\"\n\nNothing.\n\n\"Right?\"\n\n\"Might as well,\" Keith said. \"I'm starving.\"\n\nHe threw one arm over Ellis's shoulders. With the other hand, he beckoned Ginny over. Ginny stepped over quietly. He draped his other arm over her shoulders, making them a friendly threesome. Behind them, Ginny could hear Oliver walking quietly in their footsteps.\nThe Card Cheat\n\nThey were in a caf\u00e9, four streets over from Les Petits Chiens. It had been chosen simply because it was the most caf\u00e9-looking of all the caf\u00e9s they passed\u2014very small round tables with marble tabletops, dark wood, a nickel-plated bar, and a simple menu written on a blackboard offering three courses and a glass of wine for eighteen Euros each. It was still a bit early for Parisians to be having dinner, so they had the place almost entirely to themselves.\n\nOliver had been forced to sit by himself at a small table in the corner. Ginny could see that he was attempting to look dignified, even after his very public shunning and banishment. The way he sat there in his long black coat, deliberately sitting very upright, eating his plate of roasted chicken . . . Ginny almost felt pity for him. She had to tell herself not to look over at him. They had already each polished off a crock of onion soup, and now were making their way through the large dishes of mussels and the cones of French fries that cluttered the table. She had never eaten mussels before. They looked very odd, with their black, almost opalescent shells, each spread widely to reveal the globby little bit of seafood inside. But it turned out that globby was good, especially when swimming in a pool of white wine and garlic sauce, which you can then drink up in the empty shells and soak up in fresh bread. This was probably the exact kind of meal Aunt Peg wanted to see on her tabletop.\n\n\"Right,\" Ellis said, eating the last of a paper cone full of fries, \"what happens now?\"\n\nIt was an excellent question, and not one Ginny could answer, but they were both looking at her like she was about to start spouting streams of pure wisdom.\n\n\"I don't know,\" she said. \"I mean, I want to get the tabletop. But Paul isn't here. I guess I could try find an email address or leave a note or something. Maybe he can send it to me . . . or something.\"\n\nKeith picked at the remainder of his food and Ellis fidgeted a bit. It wasn't the most inspirational answer. Ginny stared at her mussel shells and the tiny amount of broth at the bottom of her bowl. She was still in Paris, eating Parisian food on a beautiful Parisian winter night. That was something.\n\n\"Does anyone want to hear my suggestion?\" Oliver asked from three tables away.\n\n\"Not really,\" Keith said loudly.\n\nOliver picked up his plate of roasted chicken and potatoes and joined them, making room for himself on Ginny's side of the table.\n\n\"The owner of the restaurant already knows that you are going to come for it. The letter clearly says so.\"\n\n\"We aren't stealing the tabletop,\" Ginny said, stacking her shells into a pyramid.\n\n\"It's not stealing if it is already yours. What we're talking about is breaking and entering\u2014and we can do the entering without any breaking.\"\n\n\"And what does that mean?\" Keith asked.\n\n\"We pick the lock.\"\n\nKeith snorted and shook his head. He grabbed another piece of bread and ripped it in half.\n\n\"All I need is a little piece of wire,\" Oliver said. \"I'm willing to give it a go, unless someone else has a better plan.\"\n\n\"You're going to stand there,\" Keith said, his voice taking on a slightly defensive tone, \"in full view of the street, and practice all those safecracking tricks you learned on the internet, is that right? Sounds very slick.\"\n\nGinny had restrained herself so far, but now she had to speak. \"We are stopping this now,\" she said. \"My tabletop. My rules. Shut up and go back to your corner.\"\n\nShe was surprised by the firmness in her voice. Judging from their expressions, so was everyone else. Keith smiled. Ellis had to stifle a laugh into her napkin. Oliver did nothing at all for a good thirty seconds, then picked up his plate and stood up.\n\n\"Listen,\" he said, looking down at her and meeting her eye, \"if we leave Paris without that tabletop then there is no point in going on, and our entire trip will be for nothing. This is your one and only chance. Either we try to get it, or I walk away right now. I still have a ticket home. Think it over.\"\n\nIt was quiet for a moment. Ginny listened to the muffled sound of French radio from the depths of the kitchen, the rip of the occasional motorcycle down the street. She was the captain of this disastrous operation, with no clue what to do now, or where to go next. She only knew what she didn't want, but that kind of information doesn't actually get you anywhere.\n\n\"Much as I don't like agreeing with him,\" Ellis finally said, jerking her head in Oliver's direction and lowering her voice, \"I think he's right. What if we could go in there without causing any damage at all? You could do that, Keith. You know you can. I've seen you get windows open, get in all kinds of places.\"\n\n\"Yeah, I can do it,\" Keith said. \"That's not the issue. It's Ginny's painting. And if she doesn't want to break in, well . . .\"\n\nThey stopped their conversation when they were presented with the apple tart and coffee that was included with their meal. It was a beautiful, golden piece of pastry, fresh from the oven.\n\n\". . . but I could go over and see what our options are. Maybe there's a way in that wouldn't hurt anything. I mean, since we're here. It does seem pointless to walk away without even a second glance, Gin.\"\n\nGinny split her pastry into two pieces with her fork, releasing a little cloud of steam. The waitress came back and deposited a small jug of cream on the table. It was such a reasonable suggestion that Ginny didn't have much room to argue. It was pointless to walk away without even looking at the building again. A few tables over, Oliver blatantly stared, listening to every word.\n\n\"Okay,\" Ginny said. \"I mean, if you want to look. Just . . . look. I guess that's fine.\"\n\nKeith slapped the table and got up.\n\n\"You didn't eat your tart,\" Ellis said.\n\n\"It's all right.\" He broke into a worrying smile. \"I don't like fruit.\"\n\nOnce he was gone, Ellis and Ginny were left to themselves. It was just them in the big, dark wood booth, staring at each other across the marble table. Ellis ate her tart happily, mumbling her joy about actually being in Paris and having real French food for the first time. Ginny felt mildly queasy, but forced it down in small, scalding bites. Once the food and coffees were done, there was nothing between them. No distractions. Just quality time to drink each other in. There was Ellis\u2014everything that Ginny wasn't. Not just English, but a Londoner. Positive and unafraid where Ginny was cautious and concerned. Prepared. Cheerful. No wonder Keith was dating her. It only made sense.\n\nEllis toyed with her ring for a moment, then pulled the Top Trumps cards from her bag.\n\n\"Want to play?\" she asked. \"I know it's silly, but I used to love these things. I must have had twenty packs of them. It wasn't a holiday unless we had Top Trumps. Let's see . . . I have dinosaurs, Harry Potter, cars . . .\"\n\nOliver slid into the booth next to Ginny.\n\n\"Want to see a trick?\" he asked.\n\nWithout waiting for an answer, he reached over and took one of the packs of cards and began to run through a series of very fancy shuffles. Then he fanned out the cards on the table, facedown. It was the \"dogs\" pack.\n\n\"Pick a card,\" he said. \"Look at it, but don't tell me what it is.\"\n\n\"What the hell are you doing?\" Ginny asked.\n\n\"Showing you a trick. One of you, pick a card, look at it, and stick it back in the pack.\"\n\nAt least now Ellis and Ginny were back on the exact same page. They looked at each other across the table. Of the many aspects of his personality Oliver demonstrated so far, \"Magic Trick Oliver\" was actually the strangest. Ellis laughed out loud.\n\n\"Oh, go on then,\" she said, taking a card. \"You're clearly a mental.\"\n\nOnce she had looked at it and put it back, Oliver pushed the deck back together, shuffled it a few more times, and began flipping over cards.\n\n\"Do you think this means we're friends?\" Ginny asked.\n\n\"No.\" He continued shuffling. \"Is this your card?\" He held up a card with a picture of a pug.\n\n\"No,\" Ellis said.\n\nHe flipped over some more, then held up a black Labrador.\n\n\"This one?\"\n\n\"Nope.\"\n\nOliver flipped over card after card, but Ellis shook her head each time. When he reached the end of the pack, he sifted through,\n\n\"That's odd,\" he said. \"That normally works. What was it?\"\n\n\"A golden retriever,\" Ellis said.\n\nThe door banged open and Keith hurried across the caf\u00e9 to them. The cold had lit up his cheeks, and his eyes were bright. \"Right,\" he said. \"We're in luck. There's a passage behind all the buildings on that street. I made my way down and found a window in the back of the restaurant. It'll be no problem at all to pop that open and slip inside. Nothing broken. No damage done. Trust me\u2014I used to do this all the time.\"\n\nGinny searched her mind for something, anything that could slow this down.\n\n\"What if there's an alarm?\" she asked.\n\n\"I seriously doubt there is. It's a one-room restaurant with four tables. There's a simple lock on the front door, which is almost entirely made of glass anyway. Anyone could smash into that place anytime, if they wanted to. He probably takes the cash box with him at night. Aside from that, there's nothing to take but the crockery. No offense to your aunt's decorating skills, but from what I could see through the window it already looks like that place has been vandalized. That place is like my car\u2014too crap to be worth burglaring.\"\n\n\"Sounds fine to me,\" Oliver said. \"I'll go.\"\n\n\"It's not up to you,\" Keith replied. \"It's up to her.\"\n\nAll eyes were on Ginny. Out of the four people at this table, she was the only one who appeared to have any doubts about this plan. The radio droned on in the kitchen, and the clank of dishes grew louder. Outside, there was a lot of honking. Everything was swelling up. There was a growing sense that something was happening, and if Ginny wasn't going to take part in it, she would be left behind forever.\n\n\"What is it we're looking for again?\" she asked.\n\nOliver put his hands flat on the table and blinked once. Ginny could almost see him flipping through pages in his brain.\n\n\"I think. . . .\" He stopped and drummed his fingers, then nodded confidently. \"I suppose you remember going to my friend Paul's restaurant, the one I decorated? I made four tabletops for it. They're all made out of doors, and I painted each by hand. The paint I used wasn't really designed for the wear and tear of a restaurant, so they should all be well-marked. It's up to you to take the one you know is right. You'll know it when you see it. Use your instincts.\"\n\n\"Four tables,\" Keith said. \"Won't be hard.\"\n\n\"Okay,\" she said.\n\n\"Wayhay!\" Keith slapped the table. \"That's the mad one I know.\"\n\n\"What about us?\" Ellis said, indicating herself and Oliver.\n\n\"No point in us all going in. Ginny has to come, to identify the table, but you should stay with the car, El. I don't think we have much time left in that space, and the last thing we need is a large clamp on the wheel. Posh Boy will be out front. Not directly out front . . .\"\n\n\"Yes, I could have guessed that, thank you.\"\n\n\" . . . but nearby. You can stand around and smoke and look French. Any problems, you tap on the window. Are we all set?\"\n\nGinny was not remotely set, but things were in motion. The bill was brought over, and she put down sixty Euros while the others donned their coats.\n\n\"I'm just going to walk Ellis back to the car,\" Keith said. \"Stay here. I'll be back in a minute.\"\n\nOliver set his lighter on the table and gave it a spin.\n\n\"I don't have any illusions about the two of us,\" he said. \"I did the card trick because you looked nervous. When you're nervous, you need a distraction. I provided one.\"\n\n\"You're not that good at sleight of hand, though. I guess it's a good thing you're not picking the lock.\"\n\nHe shrugged.\n\n\"I'm nervous too,\" he said, getting up. \"I'm going to have a cigarette. I'll be outside.\"\n\nThere was a little clink of a bell as the door shut behind him. Ginny absently ate the crust off Keith's untouched apple tart. She wasn't sure which was more annoying\u2014the fact that he had successfully distracted her and temporarily made her less nervous, or that he was being honest about his own nervousness. She didn't want him to have good qualities. Horrible people should be horrible all the time. That should be the law.\n\nShe stared at the street, remembering the last time she was in Paris with Keith. They broke into a graveyard. And they got caught. Last time, they were let go. This time, they might not be so lucky.\n\nReally, the only question now was whether she should call Richard before the police got them, or after. She reached into her pocket for her phone. Something came out with it and fell to the floor. Ginny leaned over to pick it up.\n\nOn the ground by her seat was a card with a picture of a golden retriever.\nThe Great Table Caper\n\nKeith pulled his hat low over his head, pressing all the fringy ends of his hair down flat around his face. It gave him long sideburns and almost entirely obscured his eyes.\n\n\"I look shifty, don't I?\" he said. \"Good. Best to look the part.\" He clapped and rubbed his hands together eagerly. \"I can't tell if you're enthused or about to be violently sick.\"\n\n\"Is there a third option?\"\n\n\"Cheer up,\" he said, putting an arm around her shoulders. \"It's me you're with. Would I ever lead you to do something stupid? Best not to answer that. Just follow me down this dark path over here.\"\n\nPut like that, the idea had more appeal. Follow Keith into the dark . . . yes, that she could do, even if the alley that he had described was actually just a space just over a foot wide. It was a minor separation between buildings, nothing that people were really supposed to pass through. Keith turned himself sideways and started moving along quickly. Ginny could only see his outline, and mostly followed by sound, trying not to scrape her face or knees on either wall. This had to be some kind of garbage alley for the restaurant. Whatever was underfoot was squishy and slippery\u2014maybe boxes, maybe food\u2014she refused to consider any other options. And actually, Parisian restaurant garbage smelled kind of nice. It was fresher than other garbage, sweeter, like overripe fruit. Maybe that was something she could put in her college essay: There I was, creeping down the sweet garbage alley to break into the restaurant. . . .\n\n\"You all right?\" Keith asked. \"Be careful when you get to the end. There's an old bike you have to step over.\"\n\n\"Fine,\" she said, trying to keep her tone confident.\n\nEven though she'd been warned, she tripped over the bike. She probably tripped because she'd been warned and was telling herself not to trip over the bike. She did that sometimes. It was often easier not to know what obstacles were in the way. The space behind the buildings was wider, but scrappy, mostly full of rubbish bins, boxes, and cast-off bits of furniture. Keith had turned on his cell phone for a light and was holding it up to a narrow window about five feet off the ground.\n\n\"See.\" Keith grabbed the sill and pulled himself up to peer through the darkened window. \"Easy.\"\n\nWhat he was describing as \"easy\" would have been better described as \"too high, too narrow, and too locked,\" but Ginny kept this thought to herself.\n\n\"All right,\" he said, hopping down. \"I'll get the window open. Just needed a tool for the job. . . .\"\n\nHe poked around in the trash for a moment until he found an extremely unstable-looking chair. If it looked risky in the dark, Ginny couldn't even imagine how bad it would have seemed if she could have gotten a good look at it. But Keith climbed up on it nonetheless and started working away at the window.\n\n\"Is it locked?\" she asked.\n\n\"It's not a problem,\" he whispered. \"Lower your voice.\"\n\nAt first, he must have been trying to ease it open, but when it didn't give after a minute or two, she heard his efforts getting louder, and his tone became more frustrated and determined. Finally, there was a splintering noise and he swung the window open in triumph.\n\n\"There you go!\" he said, stepping down from the chair. \"No problem. Up you get.\"\n\n\"Me first?\"\n\n\"I'm a gentleman.\"\n\nGinny climbed onto the chair. The legs were wobbly and the seat was made of some kind of basket weave that felt like it was going to give at any second, so the sooner she stepped off of it and got through the window, the better. She shoved her head and shoulders through. The room was pitch-black, with a terrible, vaguely septic smell. She could just make out that it was small, and that there was a white object just below her\u2014the toilet. This made it impossible to slide in headfirst. In fact, there seemed no way in at all. She was just dangling there, about five feet up, half in and half out of the building, with nothing to hold on to. Plus, she was fairly certain that her hips would not actually fit though the opening.\n\n\"Come on,\" he said. \"Have to be quick here. I'll give you a boost.\"\n\nAnd with that, his hand was under her foot, pushing her up and in up to her waist. From there, she teetered between the world of the toilet and the world of the alley, her upper half facing a terrible fate, and her lower half flailing in Keith's direction. Her hips, as she had already guessed, caught her, leaving her to seesaw.\n\n\"Turn on your side,\" he whispered up to her.\n\n\"I know,\" she said. She tried to rotate herself slowly, so that some kind of graceful cartwheel maneuver could be pulled out of this. Once she began to turn, she started to fall forward again, right into toilet land. Keith had her legs now and was holding them for support, so now she was dangling over the toilet. Lacking any other choice, she scrabbled to put her hands on the seat. Her loose hair hung upside down, right into the bowl. There was no point in resisting now. She let her weight fall forward and slid gracelessly toward an old French toilet. And then she was on the floor, with the final, horrid indignity of a strand of her dampened hair landing in her mouth. She spit it out.\n\nKeith had a lot more experience with this kind of thing and managed to pull himself up on the window frame and swing in feet first, stepping onto the toilet seat and landing silently and more or less gracefully upright with little effort.\n\n\"See?\" he said. \"Easy.\"\n\nThe bathroom was a very small place, no bigger than a broom cupboard, which meant that they were pressed in close together . . . not face-to-face, but face-to-side-of-head. She was close enough that she was sure she heard him smile.\n\n\"Shall we?\" he whispered. She could just feel the brush of his lips through the hair that covered her ear. For a moment, she bitterly regretted not having her braids. There definitely would have been lip-to-skin contact. And now, he was rubbing her stomach in his attempts to find the door handle.\n\nThis was too much. She was getting lightheaded. It was lucky that it was so dark, that she had a good excuse for clutching at the door frame. Ginny took a series of deep breaths to steady herself and followed him out, through the beaded curtain that revealed the miniscule kitchen, the bar, and the room with the four tables. The front of the restaurant was mainly made up of two large windows covered heavy purple velvet drapes. These were wide-open. Keith pointed at them. He pulled them closed on one side, and Ginny crossed around to pull them on the other. They were very effective light-blockers. Now it was pitch-black.\n\n\"Let's get the lights on and do this as quick as we can,\" he said.\n\nThis required a lot of fumbling around, feeling the walls. Les Petits Chiens was not a large place. Though they worked separate walls, they bumped together several times\u2014a slightly unusual number of times, really. Finally, one of them hit the right spot on the wall and the tiny chandelier turned on. Suddenly she could see her aunt's artwork, which covered every surface. There were her collages, the pictures and the pieces of broken dishes that were mounted, mosaic-style on the wall\u2014the hundreds of pictures of dogs, all eyes and tails and random fuzzy bodies. \"All right,\" Keith said, surveying the four garishly colored tables. \"Which one do you think it is?\"\n\nOne was orange, one was plum, one was yellow, and one was blue. All were variously spattered with designs and dots of paint. She stood at the end of the room and passed her eye from one to the next, over and over.\n\n\"She wants a sky, right?\" Keith said, pointing at the blue table. \"This looks like a sky.\"\n\nHer eye lingered on the blue table for a moment. It was covered in splatters that could have been stars. They were yellow and vaguely starlike. But Aunt Peg wouldn't paint a blue sky with yellow stars. She might paint the opposite, though. She turned to the yellow table. It had almost no other paint on it, aside from a few tiny flecks of red, which almost looked accidental.\n\n\"It won't be that one,\" Keith said. \"That's just a plain one.\"\n\nGinny kept looking at the yellow one. It was deeply marked by stains from the bottom of glasses, orbs of red wine, scars from moisture. This was the table with the lightest color, the least protection. This was the one that would mark the most. She put her hand on its surface and reached for the plum-colored table at the same time. The plum-colored table had a cold, slick surface. The paint felt protective. This yellow paint was different.\n\n\"It's this one,\" Ginny said.\n\n\"The yellow one? You sure?\"\n\n\"Look,\" she said, pointing at the rings and marks. \"This is exactly the one she would want. She talked about people drinking wine, meeting over a table. And these marks\u2014they're like the sun, or the moon, or . . .\"\n\nWell, that was it really. But she still knew it was the best choice. Keith got down on the ground and looked at the table from underneath. He produced a multipurpose tool in a case, with various-size heads for different jobs.\n\n\"Where did you get that?\" Ginny asked in amazement.\n\n\"From my car. I had it in there from when I moved the set.\"\n\nSince his head was under the table, Ginny could stare at the rest of Keith as he worked. This was a calming and pleasant sight, interrupted only by the unmistakable sound of a siren in the distance\u2014one of those keening European ones that sounded like they were going nee-neer-nee-neer-nee-neer, and she could see the echo of a flashing light from somewhere down the street. Ginny drew back the curtain a little and found herself facing Oliver, who frantically waved for her to drop it.\n\n\"Oh my god . . . ,\" Ginny said. \"Oh my god . . . stop. Stop!\" Keith froze in place, taking in this sensory information.\n\n\"Is that for us?\" he inquired.\n\n\"We have to go!\"\n\n\"Right. Perhaps I was wrong about the alarm system. Oh well. You never know until you try. I've almost got this thing off, anyway.\"\n\nHe said it calmly, as if this was just a very interesting piece of trivia, then got up on his feet. He gave the tabletop a shake.\n\n\"Come on, you bastard,\" he said, grunting a bit to get the last nut loose.\n\n\"We need to go,\" Ginny hissed.\n\n\"No point in that now. Turn off the light and close the back window.\"\n\n\"What?\"\n\n\"They'll probably just check the doors and windows and leave,\" he said quietly. \"We just have to keep our heads and voices down.\"\n\nSeveral seconds of utter panic ensued as Ginny tried to find the light switch again, then fumbled through the dark, crashing into the toilet to get to the window. Then another fumble back into the main room. Keith was invisible in the darkness.\n\n\"Get down,\" he whispered, now with a trace of urgency in his voice.\n\nGinny dropped to her knees, then scuffled her way along the floor until she found the bar. She got behind it, curling herself into a ball. Outside, Oliver was talking very loudly to someone.\n\n\"I'm supposed to be meeting someone here, you see,\" he was saying. \"This is where she's staying. I tried the doors. . . .\"\n\nRapid French from some other person, and more protestations from Oliver, who was playing the confused tourist about as well as Keith had that morning. A slightly drunk, arrogant English tourist.\n\n\"Look,\" he said, \"I don't know what's happening, but I was just waiting here. Do you speak English? She said Twenty-five Rue de . . . wait. What street is this? What street is this? Can you just show me . . .\"\n\n\"He's stealing my routine,\" Keith whispered from across the room. \"Bastard.\"\n\nThey stayed liked this for several minutes, until the voices retreated. She heard Keith moving, so she peeked over the ball. He was standing up, pulling at the table again.\n\n\"We need to get out!\" she said.\n\n\"Almost have it . . . stand over by the door, would you?\"\n\n\"There are police outside.\"\n\n\"They've walked away, which means we have exactly right now to do this. Trust me, I've done this before. You trust me, don't you? When I say now, you throw open that door. So get the locks undone.\"\n\nThere was no time to think about this. She controlled the shaking in her hands as she felt around for the locks. There were two dead bolts on the door, plus a strange lock which took a lot of jiggling to undo.\n\n\"Ready?\" he said.\n\n\"Um . . .\"\n\n\"Now!\"\n\nThe double doors of Les Petits Chiens swung open. Keith hoisted up the table and hurried outside with it, setting it carefully down on the sidewalk.\n\n\"Take it,\" he said. \"Get it to the car.\"\n\n\"What about you?\"\n\n\"Don't worry about me. Be with you in a sec.\"\n\nHe stepped back inside and shut the doors, leaving Ginny in front of the restaurant with a tabletop made of a half a door. She had no option but to move, as quickly as possible. The tabletop was large, but not overly heavy. It was impossible to run with it, so she shuffled as quickly as she could. There was a little offshoot street about twenty feet away, just as narrow as this one. She turned down it, having no idea where it went. It was nearly impossible to make it down this alley with half a door, but she also couldn't stay on the same street as the restaurant\u2014so she wound her way down, turning to the left and right to squeeze past trash bins and bikes. It whacked into bike handles and brick walls and various unseen objects.\n\nThis alley didn't go in a straight line, but wove between buildings on a long curve. She emerged on a much brighter, busier street, one lined with lots of small shops: a late-night grocery store, a packing supply store, a crepe stand, a Senegalese restaurant. After a few wrong turns, she finally got back on a street that she remembered and saw the white car at the end. People were giving her a lot of looks, but she soldiered on, working her way through the crowd with the tabletop. Ellis jumped out of the car to help her.\n\n\"Where's Keith?\" she asked.\n\n\"Coming.\n\n\"Where's Oliver?\"\n\n\"I don't know.\"\n\nEllis opened the backseat and climbed into the other side and helped pull the tabletop into place. Then, without her even hearing him approach, Keith joined them.\n\n\"As it turns out . . . ,\" he said.\n\nOliver was right behind him, running hard. His height gave him a strange, hopping gait, making the ends of his coat flap up and down. For a moment, Ginny was transfixed by the sight.\n\n\"Get in, get in, get in,\" he said briskly, pushing her into the back with the tabletop. He squeezed in right behind her and just about got the door closed. They were still in a tangle when Ellis hit the gas and they took off.\nTo Belgium!\n\nThe scene inside the car had become much more complex in the last few moments. For a start, the tabletop made a wall between the front and backseats, with just four or five inches at the top left open for communication. Ginny and Oliver now had their own little room\u2014a room that definitely couldn't hold both of them. There was double as much Oliver as there was Ginny. His knees were tucked up into their torso space, his arm span was much wider than the backseat. He was wedged in place. Meanwhile, the car itself was careening down a Parisian street as Ellis got used to the opposite-side driving. Ginny was bounced around in the tiny bit of remaining space, hitting Oliver, then the door, over and over. Most of the contact was of the shoulder-to-face and elbow-to-abdomen kind . . . with the occasional full-body slam.\n\nOn the good-news front, it didn't appear that anyone was after them.\n\n\"If I could just move this,\" Oliver said, trying to lift the tabletop enough to get his legs down and his feet under it. Keith peered over the tabletop and into the strange little world of backseat land.\n\n\"That looks really uncomfortable,\" he observed, leaning his chin and hands on the tabletop, crushing it down a bit on Oliver's toes.\n\n\"Do you mind?\"\n\n\"You stole my patter from this morning. You're a bastard.\"\n\n\"And I saved your arse.\"\n\n\"I could have gotten us out of there. All we had to do was wait it out.\"\n\n\"Where am I going?\" Ellis called. \"Is anyone going to give me directions?\"\n\n\"Do you think you could manage one of your little readings so we have some idea of what the hell is happening?\" Keith asked. \"Where does the letter send us next, oh knobular one?\"\n\n\"Just get out of the city.\"\n\n\"We need a bit more than that,\" Keith said.\n\n\"I can't exactly reach my phone right now to look up directions.\"\n\n\"So tell us what our next stop is,\" Keith said. \"We don't need you playing Flight of the Navigator back there.\"\n\n\"I'll tell you where we need to go for the night,\" Oliver said. \"You can hear the rest tomorrow. And stop leaning on the table.\"\n\nKeith released the table and retreated. Oliver continued his efforts to lift the table and get into a normal seated position. Through a lot of jerking and bumping around, he managed this. Further jerking and bumping allowed him to get his phone free. He was looking up directions.\n\n\"Why won't you just say?\" Ginny asked, as her head knocked against his shoulder.\n\n\"Because I don't want to end up shoved out on the side of the road. The longer I know things, the longer I can put that off.\"\n\nShe had to admit, there was some sense in that. Actually, everything suddenly made a kind of sense. A minor euphoria came over Ginny, along with a fit of hiccupping. They had done it. They had made it to Paris and gotten this table. And though the circumstances were not ideal, Paris was still Paris, and success was still success.\n\nThey were on the Champs-\u00c9lys\u00e9es, one of the grand boulevards of Paris. Ginny recognized it from her French textbook, specifically, from a dialogue called, \"On the Avenue des Champs-\u00c9lys\u00e9es.\" It was one of many dialogues in the book between V\u00e9ronique and Sylvie, two girls who primarily spent their time reading menus out loud and reciting long strings of phone numbers. In \"On the Avenue des Champs-\u00c9lys\u00e9es,\" however, they stepped out of their comfort zone and took a walk and said how formidable everything on the Champs-\u00c9lys\u00e9es was. They had a point. And now Ginny really understood why they called Paris the \"City of Light.\" The Champs-\u00c9lys\u00e9es during the holidays was nothing but light. Lights dripped off the trees. At the speed they were going (which was keeping up with traffic, which meant probably way too fast), all the lights blurred together into one wondrous streak. Aunt Peg was right. Paris wasn't a place you could get after one visit. A place this wonderful would take a lifetime.\n\n\"Oh no,\" Ellis said.\n\n\"Ah,\" Keith added as a follow-up.\n\n\"'Ah?' \" Ginny said, shaking out of her trance. \"'Oh no?' What does 'ah' mean? 'Oh no' what?\"\n\nThey had stopped briefly. Directly ahead of them, bathed in light in a great ocean of light, was one of Paris's great landmarks, the Arc de Triomphe\u2014the massive white arch, so large that a small plane could be flown under it. And around that arch-circled traffic\u2014lots and lots of traffic . . . hundreds, thousands, perhaps millions of cars, just going around and around and around. There had to be eight or ten lanes worth, but there were no lanes. Nothing bound the cars into any particular position.\n\n\"What am I doing?\" Ellis yelled. \"Where am I going?\"\n\n\"Hang on a moment,\" Oliver said, still working his phone.\n\n\"I can't hang on a moment!\"\n\n\"Know what I heard once?\" Keith said grimly, as the light changed. \"Insurance is invalid in the circle around the Arc de Triomphe.\"\n\nThe little white car screeched into the melee and began going around the monument, cars merging in from all directions, peeling off, coming in front, slipping up behind. Cars were coming at them sideways.\n\n\"All right,\" Oliver said. \"All right. You want the third junction. The one for the Place de la Porte Maillot . . .\"\n\n\"What third junction? There are no junctions!\"\n\nA motorcycle buzzed by, just inches from Ginny's door. Ellis must have hit the gas, because the whole car lurched and shuddered.\n\n\"You want to play?\" Ellis yelled. \"All right, then! I played a lot of video games as a kid, bitches!\"\n\n\"There!\" Oliver yelled. \"That way! Toward Boulevard P\u00e9riph\u00e9rique. There! There!\"\n\nThe car swerved abruptly to the right. Ginny heard Keith swear for a solid ten seconds.\n\n\"Sorry!\" Ellis called, merging farther and farther right with insane determination. She jerked the car off the roundabout in one final, bold turn.\n\n\"Turn right!\" Oliver yelled. \"Take the A-One toward Lille!\"\n\n\"Where?\"\n\n\"LILLE.\"\n\n\"A-1?\"\n\n\"YES!\"\n\nWith one last screeching turn, they began the run out of Paris.\n\nThe A1 was just a highway, and the night was a wintry blank. It was colder than ever now. Ginny and Oliver had worked together to create some kind of peace with the tabletop, but the end result was that it sat on both of their feet and completely blocked any heat coming from the front. So they had two things making them go numb. They both went into a hibernation mode, silent, burrowing inside their coats. The one advantage to their closeness was that they got a little warmth from each other. Suddenly, there was a little burst of excitement from the front seat, and Ginny saw that they were pulling into a gas station.\n\n\"Petrol stop!\" Ellis called.\n\nOliver and Ginny couldn't just get out of the car\u2014they needed to be extracted from under the tabletop. Keith and Ellis worked from either side, one pushing and one pulling, to get it out of the car. Ginny and Oliver stumbled out, Oliver making an uneven path away from the car to smoke. Ginny's feet were completely pins and needles. It would be several minutes before she could walk at all. She leaned against the car, while Ellis went inside to find a restroom, and Keith filled the tank. She watched the little counter on the pump spinning what seemed like an incredible number of Euros. Gas was expensive here.\n\n\"I need to give you money,\" she said.\n\n\"I'll use his for now. How do you feel, by the way? Being a proper thief?\"\n\n\"Queasy.\"\n\n\"That's a sign you're doing it right. You did good back there.\"\n\n\"I still can't believe we did it,\" she said.\n\n\"Oh, you know how these things go,\" Keith replied. \"You go to Paris, steal part of a table, and then spend all night driving to Belgium. God, when will they stop making this movie?\"\n\n\"Belgium?\" Ginny asked.\n\n\"That's where we'll be if we keep going on this road. Bet you've always wanted to go to Belgium.\"\n\nGinny looked around for any mental files she had on Belgium, but came up with nothing, except a tiny memo about chocolate. Also, she hadn't called Richard. She had to do that now, before it got any later and they actually left France.\n\n\"I have to make a call,\" she said, holding up her phone. Keith nodded and continued filling the tank. Ginny endured the pain of walking on her dead feet and tried to keep her gait even and smooth as she walked over to the side of the station. Richard picked up on the very first ring.\n\n\"Hi,\" she said, trying to sound casual. \"It's me. We're here in France. . . .\"\n\n\"How's Paris?\" he asked.\n\n\"Really . . . busy.\"\n\nThere was a lot of noise in the background wherever Richard was.\n\n\"Are you still at work?\" she asked.\n\n\"Sadly, yes. Did you get into the same hostel?\"\n\n\"Oh . . . we were just out. We're going to get a room now.\" Not a lie. Certainly the intention.\n\n\"You don't have a place yet? Isn't it after nine?\"\n\n\"Yeah . . . but it's fine. We're going now. Really.\"\n\n\"I'm sure it is, just . . .text me to let me know you're safely in the hostel, all right?\"\n\n\"Okay,\" she said. \"I will.\"\n\nWhen she got off the phone, Ginny looked up at the stars. The view here was incredible for that\u2014black and clear. The only things around besides this gas station were a dark clump of houses up the road and a wind turbine in the distance. From here she could see about twice as many stars as she could at home, maybe three times as many. The sky was littered with them.\n\nEllis was over with Keith at the car. They were talking in low voice, laughing quietly. Even though they were probably talking about what they had just pulled off, Ginny felt a rush of jealousy. She hurried back to join them.\n\n\"Hey, tosser!\" Keith yelled to Oliver. \"Get in the car or we leave you here!\"\n\nOliver threw his cigarette into the road and came over to them.\n\n\"Look how his coat snaps in the wind as he walks,\" Keith said. \"Very dashing. He's like Batman.\"\n\nThe thing was, it actually was kind of dashing. The coat was extremely long and would have dwarfed a lot of guys, but it looked right on Oliver, and it did snap around his calves as he walked over.\n\n\"Are you taking us to Belgium?\" Keith asked tiredly. \"I need to know. It's late. We need to stop soon\u2014we've been going all day now.\"\n\n\"We can stop in Ghent for the night,\" Oliver replied. \"It's not too far. Maybe an hour. Keep going on this road.\"\n\n\"Fine. Time to pack the two of you back in the car. In you get.\"\n\nGinny took one last look at the sky, shivered under the size of it, and got back in the car where the world was smaller, though no easier to comprehend.\nA Feeling of Shed\n\nAs they drove north toward Ghent, the clear skies gradually became more milky and pink, and a light snow started to fall. It was right around ten when they came into the city proper, and it was quite a contrast from the bleakness of the highway or the magnificent sprawl of Paris. Ghent looked like a congregation of cathedrals. Every building in the center was ornate, with a thousand little details and hooks and spires and miraculous accents carved into stone or made out of brick. A warm yellow light bathed the streets, which were now covered in a light dusting of snow. The city was situated around a river, which glowed under the lights.\n\n\"Well, it looks like we found Hogwarts,\" Keith said. \"Now where do we go?\"\n\n\"We try to find a place that's open,\" Oliver said. \"It's late, and it's a holiday, so there's no guarantee that we'll find anything. I did have a place for us in Paris. . . .\"\n\n\"So why didn't you say that?\"\n\n\"Because we were escaping the police. I thought even you could work that one out. I had a plan. My plan would have worked. You changed my plan. Not my fault.\"\n\n\"God, no,\" Keith said. \"None of this is your fault.\"\n\n\"What I'm saying . . .\"\n\n\"Boys!\" Ellis yelled. \"Tired now! Bed! Sleep!\"\n\n\"I did look up some places just now,\" Oliver went on. \"I've looked up a few hostels and small hotels. There's a student area that should be loaded with them. We should try there first. I suggest we park and try it on foot. It might be easier to go door to door than try to drive around endlessly.\"\n\nKeith stopped the car along one of the back streets, in something that may or may not have been a parking space. Again, Ginny and Oliver were extracted from their storage place. They stepped into the lightly falling snow\u2014heavy, ornate flakes that already dusted the bridges and the sidewalks. The cold had permeated Ginny now\u2014it was deep in her bones. But at least she was upright. She could move.\n\nGhent was pretty, and Ghent was also closed. In the center of town, every ivy-decked door looked locked, and every window wound with Christmas lights was dark. They walked through an empty central market full of small green stalls, all shuttered. They walked past a small castle with a spiderweb statue next to it. They found the street of hotels, which were all shut or full. They passed a hostel, but it had closed in October. They tried the surrounding streets, but found much of the same. After a while, they had clearly wandered off the tourist path into a residential area. Inside of the cozy houses and apartments, Ginny could see televisions and computers and people reclining on sofas. All she wanted now was somewhere to sleep. Anywhere.\n\n\"I feel like we're reenacting the Nativity story,\" Keith said, pulling his hat down over his ears. \"No room at the inn, nowhere to lay our tabletop.\"\n\n\"What about this?\" Oliver pointed at a sign in a window, which was written in several languages. In English it read: \"Rooms for students or travelers, inexpensive and clean, ring bell.\" The building looked like a normal house\u2014one of the more modern ones on the street. The windows upstairs were all dark, but there were lights on the ground floor, and a light on by the door.\n\n\"Is this actually a bed-and-breakfast?\" Ellis asked. \"It's not really marked or anything.\"\n\n\"It has this sign,\" Oliver said.\n\n\"Nothing to lose.\" Keith stepped forward and rang the bell. A minute later, an older man in a cardigan opened the door. Once their general purpose there was explained to him, and he adjusted to English, there was a lot of nodding.\n\n\"Are you . . . allergic . . . to cats?\"\n\nA quick poll was taken. None of them were allergic to cats. The door was opened wider.\n\n\"Come in,\" he said, \"come in, but be quick.\"\n\nThey were ushered into a warm and cluttered living room. This was not a hostel. It was a house. A house that smelled like cat.\n\n\"I run a cat shelter,\" the man explained. \"And with the cold, I have many more than normal. Today, I have . . . twenty-six.\"\n\n\"Twenty-six cats?\" Keith repeated.\n\n\"Mostly it is a cat shelter,\" the man went on. \"But sometimes I rent the rooms. Sometimes. How many do you need? I have two. They are forty Euros each.\"\n\nIt seemed fairly obvious that they would need at least that many, since there were four of them.\n\n\"We'll take both of them,\" Ginny said.\n\n\"Oh, good.\" The man nodded and picked one of the cats off the counter, where it was enjoying a nibble of a plant on the windowsill. \"Please wait a moment. I will get them ready. If I had known you were coming, I would have had breakfast for you. Still . . .\"\n\nHe gestured for them to wait and went upstairs.\n\n\"We're going to die,\" Keith said, the moment he was gone. \"This man is a serial killer. We're going to die, and he's going to bury us in his garden and build a shed on us.\"\n\nThe place was weird, and yes, it smelled like cat\u2014like so much cat\u2014but they all seemed to be nice cats. And they were better off in here than out in the snow. Ginny reached down and picked up a little cat who had come over to rub her ankles. The cat was barely out of kittenhood, long and lean and wide-eyed, happily batting at her hair as he cuddled on her shoulder. Oliver didn't look happy about this at all. Two cats sat at his feet and just stared up at him. He looked at them warily.\n\n\"Who runs a combination cat shelter and hostel?\" Keith asked. \"With the cat shelter being the primary function? Only people who want to kill you with an axe and then put you in the garden and build a shed on you, that's who.\"\n\n\"They are ready!\" the man called, a moment later.\n\n\"That was fast,\" Ellis said quietly.\n\nThe bedrooms looked like normal bedrooms. They didn't have that anonymity that you found in hostels or hotels. And they had cats in them. Little golden eyes peered at them through the dark.\n\n\"Now,\" the man said, opening the doors, \"I have one with two beds, and one with one large bed.\"\n\n\"We'll take this one,\" Oliver said, going into the two-bed room.\n\n\"I see how this works,\" Keith said grimly. \"I drew the short straw.\"\n\n\"It's fine,\" Ellis said, nodding to Ginny. \"We'll share the one bed. That all right, Gin?\"\n\nGinny caught some little snatch of nonverbal communication that rippled between Ellis and Keith. It took her a second to decode it, and it came out a little garbled, but the essence was, \"That could have been our room. We could have shared the one bed. But we have these two with us.\" They were sparing her having to share with Oliver.\n\nThey stepped into their respective rooms and set their things down. The rooms were just a few feet apart. She could see Oliver setting his things up on one of the beds, while Keith flopped on another. Their host lingered in the hall, in the patch of light between their rooms. Cats swarmed the general area, poking their heads inside to see who had come to visit. One large orange cat immediately hopped onto Ginny's bag when she set it down. Another scurried under the bed.\n\nGinny got out her phone to send Richard a text, letting him know they were safely in for the night. She glanced between their host, the cats, and her phone.\n\nHere and safe for the night. Everything is fine! she wrote.\n\nThat was a relief. The only thing still on her mind was the tabletop. It was out there somewhere, on the snowy streets of Ghent.\n\n\"Would it be all right if I went and got the tabletop from the car?\" she called over to Keith. \"I'd feel better if it wasn't out there.\"\n\n\"Car?\" This instantly interested their host. \"You have a car? You must bring it here. You may put it behind the house. Go get your car, put it behind the house.\"\n\n\"It's fine,\" Keith said, getting up and coming to their door. \"We just need the tabletop. . . .\"\n\n\"You must get your car and move it here now. You do not want to leave it on the street when you can put it here. Go and get it and put it here.\"\n\n\"Right,\" Keith said. \"Gin and I will go get the car and the tabletop. We'll be right back.\"\n\nFor the second time that night, Ginny and Keith set out on their own, this time in a different city. Their footsteps crunched gently in the snow as they walked to the car.\n\n\"Didn't blink an eye,\" he said, when they were a street away. \"Did not blink an eye. You would think that most people would ask, 'What do you mean, tabletop? Why do you have a tabletop?' But no.\"\n\n\"Maybe he didn't understand the English?\"\n\n\"What he wants,\" Keith said, \"is for us to move the car behind his house so no one will see it sitting vacant after he murders us. In fact, he will probably build the shed on the spot where the car was. Plus, he'll need the car, for his murdering.\"\n\n\"Stop talking about murder,\" Ginny said. \"And sheds.\"\n\n\"I can't help it. This place fills me with . . . shed.\"\n\nIt was incredibly stupid, but Ginny couldn't help but laugh. He looked over and smiled, pleased.\n\nThey'd wandered fairly far in search of lodgings, and it took them a good twenty minutes to find their way back to where they had parked. It was a gorgeous walk, though. There was such a fairyland quality to the city. Brick buildings grew directly out of cobbled streets. The snow had gotten stuck in every crevice of the buildings, had dusted the ground. In the distance, there appeared to be a castle with a great square turret, topped by four flags at the corners. In short, pretty romantic.\n\nWorse yet, Keith had never looked so good\u2014the snow settling on his coat, his face flushed from the cold. He pulled off his hat, and his hair stuck up a bit. At that moment, Ginny loved him so much, she felt like her ribs were going to crack from the pressure.\n\n\"Keith Dobson,\" he said, skidding along in the snow, \"a promising actor and playwright, considered by many to be one of the best of his generation, cut off in his prime, murdered by Belgians. It's just not how I wanted to go. I foresaw something else\u2014drowned in pudding, eaten by werewolf, smothered by adoring fans. Not this. Not this. Ah, the auto. Did you miss me, girl?\"\n\nHe unlocked the passenger's door and held it open for her, then got in next to her. He put the key in the ignition and put his hand on the gear, but nothing happened. It made a horrible screech, then died. He tried again and got the same result. He gave up trying and turned in his seat to face her.\n\n\"She doesn't like the cold,\" he said. \"Or the snow. Or rain. Or damp. She doesn't like moisture. Or . . . temperature.\" He stroked the dashboard lovingly.\n\n\"So what do we do?\"\n\n\"Give her a few minutes. Do you have Monopoly in your pocket, by any chance?\"\n\n\"I forgot it,\" she said.\n\n\"Too bad. Guess we'll just wait it out.\"\n\nThe snow dusting the windows made the car into a little cocoon. He rubbed his hands together for warmth.\n\n\"This trip certainly hasn't been boring,\" he said. \"But then, it never is with you. That's twice we've had to run from the law in Paris, you realize that? The last time we almost got busted the circumstances were different, still . . .\"\n\nThis was the first time he'd made any direct reference to what had happened before, and it didn't feel accidental. In fact, the way he had positioned himself so that he was straight on, facing her.\n\n\"I didn't think I'd come back,\" she said. \"It's weird.\"\n\n\"You're weird. It's only to be expected.\"\n\n\"You weren't expecting me.\"\n\n\"No,\" he replied, after a beat. \"I can honestly say I wasn't.\"\n\nHe did what he always did when he was uncertain\u2014he started wiping his mouth with the back of his hand, as if trying to wipe the words away or keep something in.\n\n\"I should try again,\" he said, \"shouldn't I?\"\n\nThis could mean a lot of things, but in this particular case, it probably meant the ignition. Probably.\n\n\"I guess,\" she said.\n\nHe scratched his head thoughtfully, shifted back and straightened himself in the seat, and turned the key. This time, the car started.\n\n\"Look at that,\" he said. \"She always comes through.\"\n\nHe flicked on the wipers and cleared the snow from the windshield, flooding the car with street and moonlight. It skidded a bit as he first got it into the road, but within minutes, they were safely back at the House of Cats.\n\nInside, Ellis had already gotten into her pajamas and was tucked into bed, reading some kind of life-affirming book called Villages.\n\n\"There you are!\" she said, as Keith and Ginny came into the room with the tabletop. \"I was worried that you really had been killed.\"\n\n\"Car wouldn't start,\" Keith said, as they set the tabletop down. \"I don't think she likes the snow.\"\n\nKeith took a final look at the bed Ellis and Ginny would be sharing and sighed.\n\n\"Guess I'll go,\" he said. \"My roommate is waiting.\"\n\n\"Have a good night!\" Ellis said, giving him a wiggle-fingered wave and a laugh.\n\n\"I hate you both,\" he said, smiling back and shutting the door.\n\nEven though she had done nothing wrong, Ginny felt guilty. She quietly sorted through her bag, pulling out sweatpants and T-shirt. You had to be fully dressed when sharing a bed with the girlfriend of the guy you loved.\n\n\"I got the cats out,\" Ellis said. \"We had about ten of them in here. I wasn't sure how you felt about sleeping with them. I like them, but . . .\"\n\n\"I like them too,\" Ginny said. \"But it's okay. I guess that's better.\"\n\nEllis slipped out of bed and squatted down in front of the tabletop to have a better look.\n\n\"So, this is it,\" she said. \"Does it look like art to you? I don't have an artistic bone in my body. You're the expert.\"\n\nThat was an incredibly generous assessment. She had picked a tabletop in a dark restaurant and stolen it. She was able to detect small swirls in the paint, the markings of a tiny brush. And though it appeared to be one solid yellow, it was darker in some places and lighter in others. The wineglasses marks generally were in the center, with little drips and shadows at the edges.\n\n\"I think it's the right one,\" Ginny said.\n\nEllis traced her finger along one of the wineglass rings, the strange orbs that floated all over the surface. Shivering, she hopped back up into the bed and pulled the blanket over herself. Ginny deliberated for a moment whether to go down the hall to change, before deciding that was stupid. She was about to get to know Ellis one way or another. She unhooked her bra under her shirt and pulled it through her sleeve, then took off her clothes and pulled the sweats on as quickly as she could.\n\n\"Do you want me to turn out the light?\" she asked.\n\n\"Sure. I'm knacked.\"\n\nGinny switched off the bureau lamp and climbed into the other side of the bed. Even though the light was off, their bed was right up against the long, multipaned window, which had no curtain or shade. The sky was light and pink and cast a long pattern of elongated rectangles over them. When she put her head back, she looked directly up at the falling snow. The view was nice, even if it did make her dizzy to watch the path the snow was taking, sometimes driving down, sometimes weaving unevenly, coming and coming from an impossible distance and sailing past them to the ground.\n\nJust she and Ellis. Together at last. Both staring up at the snow.\n\n\"I love the snow,\" Ellis said. \"I'm glad we're doing this. Christmas week can be so boring sometimes. Nothing open for days. And I was so jealous that Keith got to go on your last adventure. Thanks for letting me come. Now I feel like I'm in the club. \"\n\nGinny had no idea what to say to this, so she made a noncommittal sound. Kind of an ohurggghhhh. It was, perhaps, a little too noncommittal and perhaps a bit on the sometimes-I-am-Frankenstein side. She had to do better.\n\n\"It's nice of you guys to drive me,\" she said. \"Otherwise I would have been stuck with Oliver.\"\n\n\"He's so strange,\" Ellis said, propping herself up. \"You would think that considering what he's doing to you, he would be more appalling. More irritating. He seems like a normal person. He's good looking. Quiet. I don't know . . . he just makes no sense.\"\n\nA cat yowled forlornly outside the door. There was a tiny scratching on the wood and a paw poked under the door.\n\n\"Poor thing,\" Ellis said. \"I feel bad about kicking it out. Do you mind if I let it in to sleep with us?\"\n\n\"It's fine,\" Ginny said.\n\nAs soon as Ellis opened the door, a white cat skittered inside, followed by several other cat-shaped shadows. Ginny couldn't tell how many, but there had to be at least four cats in the room now. The white cat hopped right up on the bed and explored for a moment, finally settling in the middle of the bed. Another one, the massive orange cat, jumped up on the end of the bed and stretched out on Ginny's feet, locking her in place. Ellis took her place in the bed and again, and they resumed their snow staring, this time, with furry company.\n\n\"So,\" Ellis said. \"Keith told me you're applying to uni?\"\n\n\"Yeah,\" Ginny said. She didn't mean to sound as unenthusiastic as it came out.\n\n\"Where are you applying?\" Ellis asked. \"I don't know many American schools. I don't even know how it works. I know you have to pay a lot.\"\n\n\"I'm applying to a few places,\" Ginny said. \"Mostly close to home.\"\n\n\"What are you going to study?\" Ellis asked.\n\n\"I don't know. I figured I would just do the core classes for a year and then decide.\"\n\n\"I think it's great you can do that. It's a lot more specific here. My school is very rigid. There isn't a lot of choice.\"\n\n\"Are you an actress?\" Ginny asked. \"Goldsmiths is arts, right?\"\n\n\"Me? God no!\" Ellis shuffled around on the pillow a bit. \"I don't go to Keith's school. I go to LSE.\"\n\n\"LSE?\"\n\n\"London School of Economics. I study Social Policy. I am a big, boring policy nerd. I wanted a degree that would allow me to be helpful, but I don't know if that's going to work out. I'll probably end up working for the NHS making charts or counting the numbers of beds in hospitals. A friend of mine from school goes to Goldsmiths. I went to a party at her flat back in November. That's how I met Keith.\"\n\nNovember. Ginny did the mental math. That was around the time Keith changed some of his online habits, when he hadn't been in touch every day. Even though she already knew the outcome\u2014she was in a bed in Ghent next to the outcome\u2014it still stung.\n\n\"I suppose we should get some sleep. If I steal the blankets, just give me a slap, yeah?\"\n\nOn that note, Ellis flipped over on her side. The cats reorganized themselves around the girls, and the bed thrummed with their purring. Ginny soon heard soft, even breaths. Ellis had fallen asleep easily and peacefully. Ginny was still wide awake, her brain breaking down every second of what had happened in the car with Keith. If anything at all had happened. Okay, nothing had actually happened. But still, it felt . . . it felt like something. If he had moved in to kiss her, would she have accepted it?\n\nYes. Oh yes. So fast. She wouldn't have even hesitated.\n\nShe turned to look at Ellis, who was half-smiling in her sleep, her hair splayed out over her pillow like a halo. When Ginny looked down, the white cat was looking back at her. It was an incredibly sweet-faced cat. The orange cat at the end had more of a dismissive, world-dominating stare. This bed was full of devils and angels.\n\nGinny wasn't sure which one she was.\nThe Law of Pants\n\nGinny opened her eyes to find herself face-to-face with Ellis, sharing the same pillow. There were five cats on the bed now, curled and stretched and lounging about, very happy with these new human heaters they had been provided. Ginny extracted herself very carefully, pulling her legs up slowly, inch by inch, trying her best not to disturb any of the six other living creatures that shared the bed with her. Ellis didn't notice a thing. The orange cat, however, was extremely put out and landed on the floor with a thud. It padded indignantly out of the room.\n\nThe tabletop was a little less exciting in the morning light\u2014a scuffed yellow half-a-door leaning against the wall. Now she could see how roughly it had been sawn in half, how thin and streaky the paint job was. Maybe this wasn't the right one. Maybe there was a masterpiece back there.\n\nWell, this was the one they had. She made a mental note to contact Paul as soon as possible . . . although, maybe it was best not to. Maybe there was video footage of what happened. Maybe he would see it and recognize Ginny.\n\nTime to stop thinking about it. She gathered up her clothes, picking up each item delicately, as if Ellis could be woken by the sound of a sweater. Today, she would wear lots of layers, even those long underwear things her mom bought her for the trip and insisted she take. There were two more cats waiting out in the hall when Ginny stepped out, cozied up against the radiator. There was a large gold clock in the wall. It was 6:10 in the morning. The early start was probably good. Who even knew where they were going today.\n\nGinny tiptoed to the bathroom and opened the door, only to find Oliver in there, dressed only in his boxers and a T-shirt, brushing his teeth. That was officially the first time she had ever stumbled onto a guy in the wild wearing only boxer shorts, and now the image was going to be burned in her brain. The boxers in question were maroon with a thin gray stripe. His legs were long, not overly hairy, but the hair on them was dark. It was hard to know where to look. She couldn't take her eyes off the boxers. Mostly, she had a view of the back, but he turned halfway when he looked over. She commanded herself not to look at the front flap, which, of course, was exactly what she honed in on.\n\nHe spit and put his mouth under the tap to get some water. All while just wearing underwear. All while she just stared at the crucial spot of the Action Pants.\n\n\"I'll be just a moment,\" he said, pulling the door shut.\n\n\"Oh. Sorry,\" she said to the closed door. \"It wasn't locked.\"\n\n\"It's fine,\" came the muffled reply.\n\nWhy was she apologizing? He was the one who didn't lock the door.\n\nJust as she was recovering, the door opened again and he walked past, still just in the T-shirt and boxers, shoeless, calm as could be.\n\n\"All yours,\" he said.\n\nHow was this okay? It wasn't like she or Ellis were going to stroll around the Cat Palace in their underwear. Why did Oliver feel like this was an acceptable morning outfit? Why didn't he put pants on to walk down the hall? Why could guys, who arguably had a lot more going on in that department, walk around in underwear like it was nothing?\n\nAlso, he had steamed up the bathroom from his shower. Not only had she been treated to two viewings of the Oliver Underwear, but now she was soaking in his shower steam. Also, she wasn't showering alone. The orange cat, still disgruntled about being disturbed, decided to join her in the bathroom and watch. The hostel owner had left towels, but no soap, so mostly, it was a hot rinse. Then she dressed quickly and got out of there.\n\nWhile the snowfall had been constant during the night, it hadn't accumulated much. In New Jersey, an all-night snowstorm could result in a foot or two of sticky snow that wouldn't move for days, but this was only an inch or two and could be lightly kicked out of the way. Oliver was brushing the car off with his gloved hands, cigarette sticking out of his mouth. He looked at her over the car roof and gave her a little nod of acknowledgment.\n\nNow that Ginny had seen him in his underwear, things were simply not the same. Maybe he had done it on purpose, to make himself seem more human and vulnerable. \"Why are you awake so early?\" she asked.\n\nGinny had no idea why she asked this. She didn't care. She just had to say something. Something conversational. Something to get the underwear image out of her head.\n\n\"Do you think I slept well sharing a room with him?\" he asked, stubbing the cigarette out into a pile of snow on the car roof. \"I'd like to keep my eyebrows, thank you.\"\n\n\"You act like he's the one causing problems,\" she said.\n\n\"Yes, well, I've learned a lot, sitting in that car with the three of you.\" He finished up his brushing efforts, dusted the snow off his hands, and started heading back to the kitchen door.\n\n\"What does that mean?\" she said.\n\nHe stared her down at her. In the direct sunlight, his eyes were an absolute clear amber color. In normal light, they were very dark brown, almost black.\n\n\"Hey!\" Keith called from the window. \"Stop touching my car.\"\n\n\"I was clearing off the snow,\" Oliver said.\n\n\"I don't care.\"\n\nOliver shook his head and went back into the kitchen. Inside, the hotel owner was busy trying to assemble some breakfast. He had set four places around his little kitchen table with cheerful orange plates and little red eggs cups.\n\n\"I am sorry,\" he said. \"I wasn't ready for guests. I have four eggs, some yogurt, some bread. . . .\"\n\n\"It's fine,\" Ginny said.\n\n\"It is not fine. I should have been prepared. Now, what will you have?\"\n\nKeith and Ellis came trooping down the stairs.\n\n\"Fried egg on toast for me,\" Keith said. \"Bacon, if you have it.\"\n\n\"I am sorry. No bacon.\"\n\n\"Fried egg on toast for me too,\" Ellis said.\n\n\"Same,\" Oliver said.\n\n\"Can I have mine scrambled?\" Ginny asked the hotel owner.\n\n\"Scrambled?\"\n\n\"Scrambled . . . kind of . . . shaken? Or . . .\"\n\n\"Of course,\" he said. \"I understand.\"\n\nShe watched as he shook the egg in the shell, then cracked it and fried it.\n\nThey had an audience as they ate. The hotel owner stood over them, and the cats formed a circle around. One of them jumped onto the table and stared Oliver in the face.\n\n\"So,\" Keith said, \"where to today?\"\n\n\"Amsterdam,\" Oliver said, staring back at the cat. \"Back to a place you went before\u2014your aunt's friend Charlie's?\"\n\nGinny looked up from her examination of her strange fried egg. This was a major disaster, much worse than a locked restaurant. This was the end of the whole trip.\n\n\"Wait,\" Keith said. \"Isn't Amsterdam where . . .\"\n\nGinny answered by putting her head in her hands.\n\n\"This would have been one of those times it would have been good to give her her own letters,\" Keith added.\n\n\"What's happening?\" Ellis asked. \"I don't understand.\"\n\nOliver looked around, also confused.\n\n\"Amsterdam is very nice,\" the hotel owner said. \"My cousin makes wigs there.\"\n\nThey let this comment pass.\n\n\"Amsterdam didn't work out,\" Ginny said. \"The directions were to find Charlie's house, but he'd moved. He's gone. I have no idea where he is.\"\n\nOliver just shrugged and continued eating his eggs.\n\n\"So we find him,\" he said simply. \"Sixty Westerstraat. That's the address in the letter. We go there and we ask around. Someone has to know him, or know where he went.\"\n\n\"Where he went might not be Amsterdam. He could be anywhere.\"\n\n\"He might be. But this time you have lots of advantages you didn't have last time. We have phones. I have a computer. We have maps. And there are four of us. It might be easy. We might find it straightaway.\"\n\nKeith looked to Ginny and shook his head.\n\n\"Yeah,\" he said. \"I'm sure that's exactly what's going to happen.\"\n\nEllis and Ginny carried the tabletop together, bearing it gently over the snow, sliding a little as they went. Ginny and Oliver were installed into the backseat first, and the tabletop was carefully loaded in on top of them. (Well, it was smacked into Oliver a few times, but then it was put in with care.)\n\n\"If you push the seat back, you'll move the whole tabletop,\" Oliver added loudly. \"It won't just be me you're making suffer.\"\n\nThe careful positioning didn't change much. There was perhaps one extra inch at the top, so that Ginny could now see the tops of Ellis's and Keith's ears. It was, in essence, a wall. It actually helped that she couldn't see them very well. The tabletop offered some protection from that particular reality.\n\nThey set off. More European motorway, full of small cars and flat-fronted trucks and motorcycles. Ginny got to know the names of the major Belgian and Dutch gas stations\u2014a fact she was sure she would come in handy in the future. Like Oliver, she put in her headphones. The many layers she had put on this morning were paying off, both in terms of warmth and padding. She was in a cocoon.\n\nShe and Oliver were accidentally sitting a few inches closer today. Their arms would bump occasionally. The one advantage was that he radiated some body heat, so it was marginally warmer sitting closer to him than it was sitting closer to the window. There was a faint smell of smoke coming off his coat, which wasn't unpleasant. She decided to think of him as a kind of low-quality fireplace.\n\nAbout two hours in, she heard Keith shout something about the need for petrol, and they pulled in to a service area, which, to Ginny's surprise, featured a large McDonald's.\n\n\"I picked this one to make you feel at home,\" Keith said, as he pulled out the tabletop, releasing Oliver and Ginny from their seat.\n\nIt was decided that they would take a half an hour for some food and to stretch. The snow was heavier here, but it had gotten warmer, and it was all starting to melt into slush.\n\nOliver opted not to join them, staying outside in the wet parking lot, smoking and talking on his phone. They watched him through the window as they sat with their food.\n\n\"The constant smoking,\" Keith said. \"Puff the Magic Wanker is going to kill himself at this rate, which certainly gives me a new respect for the tobacco industry. Who do you think he's calling? His mum?\"\n\n\"Maybe he has a girlfriend,\" Ellis said. \"Or a boyfriend.\"\n\n\"No.\" Keith took a long sip of his soda. \"No. There is no way.\"\n\n\"You never know,\" Ellis said. \"We were talking last night about how he isn't bad looking.\"\n\n\"Were you now?\" Keith shoved a few fries in his mouth and regarded Oliver again.\n\nGinny and Ellis were throwing away the trash and just stepping outside when Keith started the car, gunned the engine (as much as this was possible), then pulled out abruptly. He headed right for Oliver. Oliver looked surprised, but he didn't budge, even as the white car came right for him.\n\n\"What's he . . . ?\" Ellis said. \"Oh god.\"\n\nIt looked for a moment like this battle of wills was going to accidentally end in death, but at the last second, Keith turned the wheel hard, fishtailing around, sending a wave of dirty, icy slush all over Oliver.\n\nKeith got out and surveyed his work with satisfaction. Oliver was trying to maintain his dignity, brushing himself off calmly. The force of the splash must have washed the Zippo out of his hands. He reached into an icy puddle with his bare hand and retrieved it. He flicked it a few times, but it didn't produce any flame.\n\n\"Doesn't work?\" Keith said. \"Oh, that is a pity.\"\n\nOliver carefully removed his coat and shook it hard to get out the worst of the slush and dirty water, then rolled it tight.\n\n\"Can I put this in the boot?\"\n\n\"Boot's full,\" Keith replied.\n\n\"It's going to get her wet as well.\"\n\nKeith rolled his eyes and reached for the coat, shoving it hard into a corner of the trunk, mashing it down as hard as he could.\n\n\"There you go,\" he said.\n\nOnce in the car, Oliver wrapped his arms around himself and faced the window, resolute and silent. He didn't even put in his headphones. He was trying not to show it, but anger crackled off of him. If Ginny had touched him at that moment, she was pretty sure she would have gotten a shock. Though there was a part of her\u2014a very, very small part of her\u2014that wanted to offer him a sweater or a pat on the shoulder.\n\nHow did he do this? How did he manage to make her feel bad for him? There was something terribly raw about Oliver. In the winter sun, his face was utterly pale except for the dark shadow all around his chin. He looked like one of those guys who had to shave a lot, maybe twice a day, or the beast would come out. Even though he was tall and looked more than capable of defending himself, he sat defiantly, doing nothing. He seemed altogether too used to abuse.\n\nBut, she reminded herself . . . he was getting abuse for good reason. He was silent because there was no way he could possibly defend himself. He was the person doing wrong here.\n\nShe decided not to look at him anymore, not even a glance. She wasn't going to look at anyone. She had the tabletop to protect her from the front, and her headphones to block out the rest. She could just sleep. The last one sounded like the best option, so she closed her eyes and scrunched herself against the car door. Tiny streams of cold air stabbed her through the gaps in the window, and the door itself was like ice, but she tried not to care. She shut her eyes hard and commanded herself to rest, to turn it all off.\n\nThe last thing she saw as she closed her eyes was the underwear.\nThe Stain on the Page\n\nWhen she opened her eyes next, Ginny found that her face was pressed up against the window, hard. Just outside the window was the gentle, constant tinkling of bicycle bells.\n\n\"We're in Amsterdam,\" she said groggily, her lips rubbing against the cold glass.\n\n\"Have a nice kip?\" Keith called from the front.\n\nSomething was weighing her down. Ginny turned to find the sleeping form of Oliver slumped against her, using her as a pillow. It wasn't entirely unpleasant to have him there. He was warm, and not overly heavy. She had probably slept so well because of the body heat he was giving off. Still, he had to go. Ginny straightened herself up, and Oliver fell senselessly in the other direction, toward his door. This woke him up, and he reflexively rubbed his face and looked around.\n\n\"Are we there?\" he asked.\n\n\"We're almost there,\" Keith said. \"Not that we have any idea where we are supposed to go. Where are we going, precisely?\"\n\n\"We should leave the car outside the city and ride the tram in,\" Oliver replied. \"There should be a car park coming up in a few minutes.\"\n\n\"Not actually an answer,\" Keith said. \"Give us the next bit. Recite, freak.\"\n\nOliver was still waking up. He yawned hard, pressed his hand to his temple, and began.\n\n\"'From Paris, it's time . . .' Hang on.\"\n\nHe blinked a few times and stared at the car ceiling, moving his lips silently.\n\n\"He's forgotten it, hasn't he?\" Keith said.\n\n\"Shut it. 'From Paris, it's time to return to Amsterdam, the city of canals, bicycles, and delicious, delicious cheese. The Dutch are famous for their open windows. No curtains. No blinds. Their houses are on display. Walk along the canal streets, Gin. You'll be right at eye level with every variety of human life. You can see into a thousand different worlds.\n\n\" 'But here's the thing: You aren't supposed to look. This is an understood Dutch custom. Everything will be laid bare for you, but you can't ever turn your head and gawk. This is both elegant and incredibly perverse. The idea, I think . . .' \"\n\nHe paused again.\n\n\"This is really much better than just bringing the letter,\" Keith said.\n\n\" '. . . is that whatever you are doing in your house, however you choose to live, is fine. You have nothing to be ashamed of and nothing to hide. But at the same time, you have to respect your neighbors enough not to stare.\n\n\" 'I don't know. I'm making this up. I don't even know if the Dutch know why this is the way of things. There's probably some complicated historical precedent involving the curtain makers' union or something. Also, I looked. I peeked in every window where anything even remotely interesting was going on. You can't put something in front of me and expect me not to look.\n\n\" 'So, for the next layer of the painting, I decided to make a Dutch window, except you ARE supposed to look through it. Charlie has it. I am sure you have already seen it. You just need to go back and collect it. I realize that it's difficult to carry around a tabletop, Gin. And now I'm asking you to carry around a tabletop and a window. That's why I didn't have you get these things the first time around. Once you've done that, take the ferry back to England. Head back home, to Richard's.'\n\n\"There. That's the whole section. We go back to where you started last time.\"\n\n\"So we're done after this?\" Ginny asked.\n\n\"Not exactly,\" Oliver said.\n\n\"More riddles,\" Keith said, pulling off the motorway toward the parking lot. \"Wonderful.\"\n\nA half hour later, they knocked on the door of 60 Westerstraat. The person who answered this time was not the same person Ginny met over the summer. This person also didn't know a Charlie.\n\n\"So, we just start asking, I guess?\" Ginny said, looking up and down the street. Westerstraat was one of the non-canal streets, full of fairly modern buildings.\n\n\"Do you know anything else about him?\" Ellis asked. \"A last name. A job. Anything?\"\n\n\"The letter just gave his first name and address.\"\n\n\"Right,\" Ellis said. \"So, we just start asking people if they know Charlie. We can do that.\"\n\nThere weren't many people around, and knocking on the doors to either side of number 60 produced no result. They spread the search farther down on either side, Keith and Ellis going to the left, and Ginny and Oliver going to the right. No one knew Charlie.\n\nThey stopped briefly to get some sandwiches.\n\n\"I don't think this is working,\" Keith said, examining a mysterious salad on his.\n\n\"Maybe Charlie is an English version of his name or something,\" Ginny replied. \"What's important is the window. My aunt's paintings . . . they're kind of weird. Maybe we're asking the wrong question. Maybe we need to ask about the painting itself.\"\n\nSo they tried it again, this time asking about the window. This brought a result from the woman who worked at the flower shop.\n\n\"Oh, you mean the jungle window?\" she said.\n\nGinny looked to Oliver, to see if he had any idea if a \"jungle window\" was the kind of thing they were supposed to be looking for.\n\n\"That could be it,\" he said, nodding. \"The man with the jungle window, do you know where he went?\"\n\n\"I didn't know his name, but I think he works at De Bevlekte Pagina. It is a bookshop. He is a . . . he is a very strange . . .\"\n\n\"That's him,\" Ginny said. The label \"very strange\" fit most of Aunt Peg's friends.\n\nDe Bevlekte Pagina was a tiny place, just a few streets over. It was some kind of medieval nook, uneven in every respect, from the strange lean of the walls, the odd shape of the room, the slope of the floor, the step in the corner that lead to a tiny door halfway up the wall. Ginny had never seen such a small store, or imagined you could get so much into it. It had one employee\u2014a girl in a crushed red velvet jacket and a bob dyed to match. Her shirt was low-cut enough to reveal a glimpse of a large cartoon heart tattooed over her actual heart.\n\n\"Hi,\" Ginny said, approaching her. \"We're looking for someone named Charlie.\" \"Are you readers?\" the girl asked, barely lifting her head.\n\n\"Readers?\"\n\nThe girl offered no other information.\n\n\"Does he work here?\" Ellis asked.\n\n\"Work here?\" The girl was contemptuous now. \"He is not an employee.\"\n\n\"Can you please get him if he's here?\" Oliver said briskly. The girl didn't look happy about offering any other information, but she did seem to like the sight of Oliver. She smiled flirtatiously before leaving the counter and went to the back of the store and pushed aside a velvet curtain that blocked off a passage.\n\n\"Charlie, er zijn wat mensen hier voor jou.\"\n\nA voice came from the depths.\n\n\"Wie zijn het?\"\n\n\"Geen flauw idee.\" The girl turned to give them all another appraising look. \"Ze zijn Engels.\"\n\n\"Engelse?\"\n\n\"Ze zien er uit als studenten.\"\n\nShe dropped the curtain, came back to the counter, and continued reading her book.\n\n\"Is he . . . coming?\" Ginny asked.\n\n\"When he is finished,\" she said, not looking up. \"Did you bring books for him to sign or do you want to buy them?\"\n\n\"Books? No.\"\n\nThe girl sighed and shook her head. They were a terrible disappointment to her. Even her interest in Oliver fizzled out, just like that.\n\nNothing happened for several minutes. They wandered around the shop as much as they could, but it was barely bigger than their rooms at the hostel. The books were a mix of Dutch, English, French, and German, mostly used. Oliver got bored and went outside to smoke. Keith and Ellis had a low conversation in the corner. Ginny sat in the single spot of sun in an open windowsill. The window was original, with uneven, warped panes of glass. Finally, the curtain was pushed back. A guy appeared from the darkness. He was shorter than Ginny, maybe in his mid-twenties, and absolutely emaciated. What he lacked in height and body mass he made up for in hair\u2014straggly beard, untamed locks sticking up in all directions. He wore a heavy red plaid lumberjack shirt, completely unbuttoned and exposing even more chest hair. Three or four silver necklaces glinted just under this layer. He wore black leather pants that were cracked and worn from use, and no shoes. His fingernails and toenails were colored ink black\u2014it looked more like pen than nail polish. He was like something you found on a nature preserve, if they made nature preserves where you could look at artistic frenzy in the wild.\n\nThis was definitely the right guy.\n\n\"Hi,\" Ginny said. \"I'm Ginny. Peg's niece?\"\n\nHe leaned forward. Ginny couldn't tell if he couldn't understand her English, or if he was sniffing her. \"I'm Peg's niece,\" she tried again. \"Margaret Bannister? The painter?\"\n\nThis hit a note of recognition. His eyebrows went up and he leaned back against a bookcase and folded his arms across his chest. He exposed the underside of his arm in the process, and Ginny could see words written there, a whole screed of some kind.\n\n\"Do you speak English?\" she said slowly.\n\n\"Of course I speak English.\"\n\nGinny had discovered this during her first time in the Netherlands\u2014all Dutch people seemed to speak flawless English. He nodded and turned to the girl at the counter.\n\n\"Margaret is een Amerikaanse schilderes. Ze is erg goed. Maar ik kan niet geloven dat deze meid haar nichtje is.\"\n\nGinny had no idea what he was saying, but assumed it probably wasn't too complimentary.\n\n\"My aunt wrote me a letter. She said I was supposed to come see you. I was here over the summer, but I went to your old address. You have something my aunt gave to you. Something she asked me to come get. A window . . .\"\n\nCharlie picked up a pen and began cleaning under his nails with the writing tip.\n\n\"The window is on the boat,\" he said matter-of-factly.\n\n\"The boat?\"\n\n\"I have a boat,\" he said. \"It is on a canal.\"\n\nThat thought kept him occupied for a moment or two.\n\n\"A good place for it,\" Keith said quietly.\n\n\"The boat is gone,\" Charlie went on. He almost sounded happy about this, like the boat had finally sought freedom and was now living a happier life somewhere else. \"I hire the boat out. Someone has hired the boat today. The boat will be back tomorrow. You can have the window then. Or . . . is it the next day?\"\n\nThis last question was to the girl behind the counter, who looked up and shrugged.\n\n\"We're not here that long,\" Ginny said nervously. \"I don't want to bother you, but we really need it today.\"\n\n\"Well, today it is gone. Come back later. Say hello to Margaret for me.\"\n\nCharlie seemed to be done talking and, with a wave of his hand, headed back to the curtain.\n\n\"She's dead,\" Ginny said.\n\nThis wasn't something she'd had to say out loud in a while, and she didn't like breaking the news to anyone. Charlie's demeanor changed completely. Even the girl put her book down and looked up in confusion.\n\n\"Margaret? Is dead? But, how? She is so young. Did she have an accident?\"\n\n\"She didn't tell you she was sick?\" Ginny asked.\n\n\"Sick? Sick with what?\"\n\n\"She had cancer,\" Ginny said.\n\nIt was like she had sucked all the air out of the room with a straw. Charlie sat down on the floor, in the tiny space between the bookcases.\n\n\"The window,\" he said.\n\nThat was all he could manage for a moment.\n\n\"I met her in New York,\" he said finally. \"She came here to learn more about Dutch painting, about the use of light. About still life. She stayed with me, and I showed her my boat. She loved it. She was painting a picture of it. It's a very distinctive boat. It is pink. Very bright pink. She said she wanted to make something for the boat\u2014a view you could look through, both ways. She mounted the glass in the window of my flat and painted it while standing on a box, on the pavement.\"\n\nGinny could picture it perfectly. Aunt Peg, so tiny, so graceful\u2014her long brown hair tied back in a knot, probably standing on her toes. She moved like a ballerina, even though she never took a dance class in her life and couldn't keep time to music.\n\n\"I'm a poet,\" Charlie said. \"Maybe you know my work?\"\n\n\"No,\" Ginny said quietly. \"Sorry.\"\n\n\"The name of this shop\u2014it means, 'the stain on the page.' We both loved that idea. Painting, writing. Both just stains on the page.\"\n\nCharlie took a long breath and stared at the shelf of books in front of them, drew a long, jagged line on his bare foot, then put his head down on his knees. The girl behind the counter reached down and touched his head. While Ginny understood that other people were entitled to mourn her aunt as much as she was, it still irritated her that Charlie was allowed this little display. He didn't even know she was dead, hadn't been involved in any way. But then, Aunt Peg had hidden her illness from a lot of people, including her family. Only Richard was there for the worst of it.\n\nFinally, Charlie roused himself.\n\n\"The boat, I hire it to tourists and for parties. Someone has hired it for this week, but they return it tomorrow. Come in the morning. I will take you to it.\"\n\nOnce they were outside, Oliver was the first to speak.\n\n\"I booked a place to stay here,\" he said. \"Two rooms. Perhaps we should go there.\"\n\nFor the first time, Keith made no comment about one of Oliver's plans.\nThe Koekoeksklok\n\nThe hostel Oliver had found was by far the nicest Ginny had encountered on any of her travels. It occupied an entire canal house, narrow and tall, with one massive window at the front and back of each floor. The windows had red shutters, large as doors. At the very top, near the roof, there was a clock, and all down the front, tiny paintings of blue birds. It was called the Koekoeksklok. Ginny didn't need a Dutch-to-English dictionary to figure out that meant \"cuckoo clock.\"\n\nThe theme continued on the inside, which featured a large common room paneled in dark wood, with cuckoo clocks all over the walls.\n\n\"I hope these don't work,\" Ellis said, looking around.\n\nThe Koekoeksklok was staffed by students\u2014no murderers with dozens of cats. The hostel had one central wooden stairway that wound up and through the skinny building, with just a few deafeningly creaky steps from floor to floor. Oliver and Keith were shown to a room on the third floor, and Ellis and Ginny got the room above them, at the top, facing front. The ceiling of their room was peaked, and high as the roof, maybe sixteen feet. The decorations were basic, but very clean, and the beds were loaded down with a multicolor pile of blankets.\n\nThey had a massive window\u2014the cuckoo hole\u2014which faced the canal. Ginny looked out at the skyline. All the roofs were different heights, and almost every canal house had a little ornate peak at the top. Many of the houses had a hook sticking out of the front. Ginny had learned all about these hooks last time. Because the houses were so narrow, you couldn't get furniture up the stairs. Things had to be hoisted up by rope and brought in through the windows. Windows were really key here.\n\nBelow, there was a steady stream of boats drifting along the canal, and on the sidewalks and roads, hundreds of people on bikes. Amsterdam was one constant, steady flow of energy\u2014not manic, just as fast as a bike wheel or the chug of a canal boat. Ginny really loved this city, maybe more than Paris or London. It wasn't overwhelming. It was practical and beautiful and lively.\n\n\"There's not much we can do tonight,\" Keith said, dropping down on Ellis's bed. \"What now?\"\n\nInertia was starting to take over. Ellis was sprawled on the floor. Keith was looking at her, not in any particular way\u2014he wasn't drooling or anything\u2014but his gaze fell on her and stayed there. It made Ginny uncomfortable.\n\n\"I guess we should eat,\" Ellis said to the ceiling. \"We should go out.\"\n\nGinny got up to use the bathroom and get herself ready to go. She was only in there for a minute or two, but when she came back, Ellis was sitting up, her back against the side of the bed. Keith had repositioned himself so that he was at her shoulder, just at her ear. Again, they weren't doing anything, but Ginny was pretty sure this was only because she was here.\n\n\"I can't move,\" Keith replied. \"I think I have to sleep.\"\n\n\"Me too,\" Ellis said.\n\nIt was entirely possible they were just tired. Ginny was kind of tired. But there was something incredibly awkward about being around the two of them right now. Escape. It was the only way.\n\n\"I have to go call Richard,\" she said. \"I'll be outside. I might take a walk.\"\n\nKeith yawned and lifted a hand in farewell.\n\nOver the summer, Ginny had to spend a lot of time by herself\u2014time alone with no internet, no television, no music, nothing to numb her from the experience of being on her own. This wasn't by choice. The first letter spelled out the rules. She couldn't bring a computer, a phone, even a journal . . . nothing that could distract her from the experience of being in Europe on her own. At first, it had been unpleasant and weird, but over time, she had adjusted. It was okay just to be by herself, with herself.\n\nNow, of course, she had lots of company to think about.\n\nEllis was . . . extraordinarily pretty, and spontaneous, and sweet. She matched the sharpness in Keith's personality. She didn't need to be coaxed into climbing over the fence and through the window\u2014she wouldn't need to be boosted up and shoved over. Ellis would make the climb herself. And she was English. She fit in. She had London style and that understated London swagger. Sure, Ginny had lost the braids, but the fake red in her hair was already fading, and that newfound courage of hers was just a thin outer coating. She was the American\u2014a little louder, a little out of step. Given the choice, she would have dated Ellis. It only made sense.\n\nShe sat down on a bench along the canal and pulled out her phone to call Richard. He answered on the first ring.\n\n\"Hi . . . sorry. Is this a bad time?\" she asked.\n\n\"No, you caught me at a good moment. I'm hiding in my office. How are you? Where are you?\"\n\n\"Amsterdam,\" she said.\n\n\"Amsterdam? When did you get there?\"\n\n\"A few hours ago. We're in a really nice hostel. It's sort of like a cuckoo clock.\"\n\n\"I see. What are you doing there?\"\n\n\"We need to get a window,\" Ginny said. \"I guess part of the piece is a glass panel. It's on a boat, and someone rented it out. So when it comes back, we'll get it.\"\n\n\"And where do you go then?\"\n\n\"I'm not sure,\" she said. \"We sort of have to figure that out.\"\n\nShe heard his other line ringing, and the sound of him shuffling things around on his desk.\n\n\"Let me think what I'm supposed to ask . . . have you engaged in any exceedingly foolish behavior while under the influence of alcohol?\"\n\nShe hadn't been drunk when she stole the tabletop, so she had a pass on that one. \"Nope,\" she said.\n\n\"Are you wandering the streets alone in a haze of legal marijuana?\"\n\n\"No.\"\n\n\"Do you have any intention of entering the prostitution business, which is also legal in Amsterdam?\"\n\n\"Probably not today.\"\n\n\"Good, good. Is there anything I didn't cover? I'm new to this.\"\n\n\"You were pretty thorough,\" she said.\n\n\"You know I trust you, Gin,\" Richard said, almost sounding embarrassed. \"I know you can do as you like. You're eighteen. I certainly did plenty of things. . . .\"\n\nThe door to the Koekoeksklok opened, and Oliver stepped out. He brushed off his still-damp coat and looked around. As soon as he spotted Ginny, he headed straight for her.\n\n\"No,\" Ginny said quickly. \"I'm good. And thanks. For everything.\"\n\n\"Well, you know I'm here.\"\n\nOliver lingered a few feet off until she had wrapped up her call. He may not have been respectful of personal property, but he was all about giving people their personal space.\n\n\"Where are the others?\" he asked.\n\n\"They're tired. I think they're napping.\"\n\nThey probably weren't napping. Well, maybe they were. Probably not. She was not going to think about what they were doing.\n\n\"I was going to go look for the boat,\" he said. \"Care to join me?\"\n\nThey stared at each other for a moment, sizing up the situation. They were here, there was a job to be done. More importantly, it would give her something to do while Keith and Ellis did whatever they were going to do . . . which was nap, of course.\n\n\"Fine,\" she said.\n\n\"I've got some directions,\" he said, pulling out both his phone and a tourist map. \"The canals surround the\u2014\"\n\n\"I know the city pretty well,\" she said, cutting him off. \"I mean, the tourist parts.\"\n\n\"I thought it all went wrong.\"\n\n\"It did,\" she said. \"But I didn't leave. I ended up in this really horrible hostel. So I ran away from that and ended up meeting this family called the Knapps, from America. They took me in for the week. They were nice, but they were super tourists. It was like they were competing for the Most-Touristy-Tourists-of-the-Year award. They wanted to see it, have their picture taken in front of it, get the T-shirt, and move on. They used to have these printed schedules for each day. . . .\"\n\nGinny started walking, and Oliver fell into step behind her.\n\n\"So why didn't you just leave if they were so annoying?\" he asked.\n\n\"They weren't mean people,\" Ginny said. \"They just wanted to check everything off their checklists. Lots of people probably travel like that.\"\n\n\"Lots of people live like that. . . .\"\n\nThat was the kind of thing Aunt Peg would have said. Was he doing that on purpose? She glanced over at him suspiciously.\n\n\"It still doesn't explain why you didn't just leave.\"\n\n\"I was just sick of being by myself,\" she said. \"I don't know. I just did.\"\n\nHe took the hint and let it go.\n\n\"Anyway,\" she said, \"the center bit of the city is pretty small, and the main canals are around it in layers, a bunch of half circles. It won't take that long to look around the main tourist areas. Maybe a few hours. I mean, we're looking for a big pink boat. How many of those can there be?\"\n\n\"This is Amsterdam. There could be a small fleet of pink boats here, for all we know. But I take your point. Should we start this way? We're pretty much at the top of this street, anyway.\"\n\nThey began to stroll down the canal, under the bare trees. The lights and the moon reflected off the canal in a way that would have been romantic if she had been with just about anyone else.\n\n\"You couldn't bring anything the last time you came over,\" he said. \"No guidebooks. No maps. No computer or phone or anything. Those rules, they were a bit mental. Did you follow them?\"\n\n\"Yes,\" Ginny said, scanning the boats along the canal. It was getting darker by the second, and harder to determine exactly what color they were.\n\n\"Why? I mean, who would have known if you hadn't?\"\n\n\"I would have known.\"\n\n\"Yeah, but . . . all that time on planes and trains with nothing to listen to. No internet. Nothing. It sounds like torture. I would have brought everything.\"\n\n\"You're not me,\" Ginny said.\n\n\"Did you ever think that she expected you to break some of them?\"\n\nRemarkably, that thought hadn't occurred to Ginny before. She stepped out of the way of a bicyclist and directly into the path of another bicyclist, who expertly steered around her.\n\n\"I don't know,\" he said. \"Maybe you like all the rules, the backtracking, the games. I think it's annoying.\"\n\n\"This isn't for you.\"\n\n\"I think it's annoying in general. She should have just given you the paintings. This traipsing back and forth is such a pointless exercise.\"\n\n\"Look,\" Ginny said. \"We're not discussing her, okay? You do not talk about her, or what she did.\"\n\n\"I'm just saying, it seems like she was a rule breaker, so maybe she expected you to break the rules.\"\n\n\"So,\" she said, \"is this how you justify what you're doing?\"\n\n\"What I'm doing is maybe a little unethical. . . .\"\n\n\"Maybe a little? You're stealing.\"\n\n\"I'm not stealing,\" Oliver said firmly.\n\nGinny stopped. They were on the unguarded edge of one of the canals. The water was high, and two swans and a few ducks drifted past and regarded them curiously.\n\n\"How is this not stealing?\" she said, mostly to the swans.\n\n\"Someone stole your bag,\" he said. \"Not me. I purchased the bag, not knowing it was stolen. Inside, I found some property. I used my time and resources to track you down to return your property to you. In return, I asked for a percentage. You agreed.\"\n\n\"Or you were going to walk away with my stuff,\" Ginny said. \"My stolen stuff. Don't you legally have to return that to me anyway?\"\n\n\"I'm not an expert on Greek law, and neither, as far as I know, are you. All I have of yours, anyway, is a bag . . . which I did offer to return . . . and a few pieces of paper. The paper wasn't in any kind of sealed envelope delivered by Royal Mail, which would be illegal to tamper with, so there's no problem there. And I have agreed to return these pieces of paper to you on completion of our agreement. Do you want to take me to court for a few pieces of paper worth less than a pound?\"\n\n\"How come,\" she said, mostly to herself, \"I only know two English guys and both of you are . . . well, one was a thief and the other is you.\"\n\n\"I think it says more about you than it says about us.\"\n\n\"About me?\"\n\n\"About the people you attract.\"\n\n\"It's not me,\" she said. \"It's my aunt.\"\n\n\"It is you,\" he said. \"She's not here, in case you haven't noticed. You are the common denominator in all of this. And don't lump me in with him. I've never lied to you. I've been very careful about that. I hate lying. I'm not a thief, and I'm not a liar.\"\n\nThis was the first time Ginny had heard anything that might have been considered an emotion in Oliver's voice.\n\n\"So what are you?\"\n\n\"An opportunist.\"\n\n\"What does that even mean?\" she asked.\n\n\"I saw an opportunity, and I took it. That's what it means. I've been completely up front about it, and I'll continue to be up front about it.\"\n\n\"You're delusional.\"\n\n\"I'm not. I know what I'm doing doesn't make me a great person, but I don't tell lies.\"\n\n\"Well I guess that makes it okay,\" she said. \"I mean, as long as you're being honest.\"\n\nOliver got the stupid cigarettes out again. He had a new lighter. It was just cheap and plastic, not like the fancy Zippo he had before. He drew in and exhaled slowly, blowing the smoke to the side, away from her. Now that she thought about it, he was the only person she knew who smoked. It was just such a ridiculous habit. The cigarettes were his prop, his way of avoiding anything he didn't feel like dealing with, putting distance between himself and other people.\n\nShe was being too deep about that, probably.\n\n\"You hate my smoking,\" he said.\n\n\"Does it matter? Are you going to stop because I don't like it?\"\n\nThey were coming up a few steps leading down into the canal. Oliver stepped down to the very edge of the water and dipped in the tip of the cigarette, extinguishing it. He held it up to show her.\n\n\"I'm not unreasonable,\" he said.\n\n\"Of course not,\" she replied.\nA Night of Vice\n\nThey walked for two hours. In the open windows, she saw Christmas trees and lights, a few ornate menorahs. Ginny was pleased to find that she had a fairly good memory of the layout of Amsterdam. The canals varied in width, from perhaps one car-lane wide to six or so lanes wide. They radiated throughout the city like a spiderweb, with boats parked along the sides, all kinds of boats. Long, traditional houseboats were next to sleek new cruisers, which might then be followed by tiny, flat rowboats that looked like they would sink the second a person sat down in one. Every once in a while, they would pass a boat that had met a bad end, half submerged under the water, tipped sideways, ducks looking at it mockingly.\n\nThey passed into the red light district\u2014Amsterdam's legal prostitution zone. There were long windows dotted between houses and shops, each with a thin frame of red light. Those windows either had someone standing in them, usually female, or a closed curtain, indicating the little shop was currently busy. The windows were bizarrely cozy. Sometimes the women would sit and read or paint their nails or just wave. Ginny had a fondness for those windows and the women in them. They had completely freaked the Knapps out.\n\n\"Know what I was thinking?\" Oliver said. \"When I read that we had to find a window in Amsterdam? I thought we were going to have to find one of those.\"\n\nShe had to admit, Oliver had a good point. Aunt Peg would have had a lot of fun decorating one of those windows.\n\n\"This is pointless,\" she said.\n\n\"But fun,\" he said. Ginny had no idea if that was supposed to be a joke. He was utterly inscrutable. He just gazed around, his hands buried in his pockets.\n\nAlong with the red windows, there were several coffeehouses on this street. These were the pot-smoking caf\u00e9s. They looked like they existed mainly for tourists, with neon signs and pumping music. They passed a quieter one that looked like a regular little caf\u00e9. They had two menus out front\u2014one of various pot concoctions, and another of food. Mostly pizza.\n\n\"I want to go in,\" she said.\n\n\"You want to smoke?\"\n\n\"No,\" she said. \"I'm hungry. They have pizza.\"\n\n\"Lots of places have pizza. It's fine if you want to smoke. I'm not going to judge you.\"\n\n\"I don't,\" Ginny said firmly.\n\n\"Then we can go somewhere that has good food. There's lots of it about in Amsterdam. These places aren't known for\u2014\"\n\n\"I just want to go here, okay?\"\n\nOliver raised his hands in surrender.\n\nTruth be told, Ginny didn't want to go in to the coffee shop\u2014she felt a perverse need to go in. The Knapps (at least, Mom and Dad Knapp) wouldn't go anywhere near the coffeehouses, and there was something inside of Ginny that compelled her to do everything the Knapps disliked. And if she was being very truthful with herself (it happened on occasion), Ginny would have admitted that the coffee shops scared her too. Even though they were legal and clearly full of tourists, they had the air of the forbidden . . . literally. And Richard had just asked her if she was walking around in a haze of legal marijuana. Here she was walking into a cloud of it. She pushed the door open with much more force than necessary and marched in.\n\nA cursory glance around the room revealed that the coffee shop wasn't that much different from any ordinary caf\u00e9. It was a little darker, maybe. The air had the distinct, sweet tang of pot smoke. But there was nothing particularly scary going on inside, unless you counted the tacky decorations. The place looked like a stoner's dorm room\u2014cheap cushions and basket chairs, black light posters of smiley faces, and dozens of votive candles in bright glass holders, right out of the IKEA catalog. The menus, both for food and drugs, were written in neon purple and green on an illuminated board. It was a bit sad, really. Aunt Peg could have done an amazing job with a place like this. Her paintings would have really messed people up.\n\nHad she been on her own, Ginny would have turned around and walked out; but since she made such a bizarrely big deal of coming in, now she had to stay here and eat some pizza. She and Oliver took a seat at a wobbly little table, half of which was taken up by a massive ashtray.\n\n\"Here you go,\" Ginny said, pushing it toward him. \"A present.\"\n\n\"I don't smoke indoors.\"\n\nThe coffee shop wasn't very crowded, but that didn't mean that service was prompt. Their waiter, when he finally showed up, was openly confused by the fact that they had come there for the food\u2014but being mellow and stonerish, accepted this and ambled off. She and Oliver were the only people sitting completely upright. They looked stiff and unnatural. She tried to relax in her chair, but the slouching was even more unnatural. Oliver pushed a votive candle back and forth and eyed her across the table.\n\n\"So,\" she said, \"what do you do?\"\n\n\"Meaning what?\"\n\n\"For . . . life?\"\n\n\"I went to university for a year,\" he said. \"For political science.\"\n\n\"And you don't go now?\"\n\n\"I left.\"\n\n\"Why?\"\n\n\"There was no point in staying.\"\n\nThis kind of thing always amazed Ginny\u2014people who just walked away from institutions. People who left school when they didn't see the point. Aunt Peg had done that. Ginny knew she never would. That either made her someone who worked hard and finished things, or someone who didn't have the guts to break away from the pack. Maybe both.\n\nOf course, if she never wrote her essay, this would not be an issue.\n\nThe two pizzas were slipped down in front of them, along with a beer for Oliver and a soda for Ginny. They weren't great pizzas\u2014kind of floppy and damp\u2014but she'd had worse. She was going to eat it, no matter what, since she'd brought them here.\n\n\"How did you memorize that whole letter?\" she said, cutting hers into pieces. She didn't even need a knife\u2014the fork went right through the spongy crust.\n\n\"You really like asking questions, don't you?\"\n\n\"You said you're honest.\"\n\nHe eyed her for a moment as he cut a large piece of the pizza and picked it up for a bite. It flopped around too much to be handled, so he set it down and continued with utensils. \"I'm just good at memorizing things. I don't even mean to memorize them, half the time. I just do.\"\n\n\"You mean you have a photographic memory?\"\n\n\"No,\" he said. \"Because that would be useful. It's far more random than that. I can recite the entire first chapter of every Harry Potter book. I can recite all forty-seven pages of my school handbook. I can re-create eight episodes of season two of Doctor Who, with the Tenth Doctor, word for word. I memorized the driving manual. I just seem to memorize things that have some kind of significance to me\u2014\"\n\nHe cut himself off abruptly.\n\n\"It just happened,\" he said. \"No control over it. Came in handy, though.\"\n\n\"You can recite all the first chapters of Harry Potter?\" she asked.\n\n\"Yes, well . . . I can recite chapters one through four of book one, chapters one and two of book two, chapters\u2014\"\n\n\"Okay, wait,\" Ginny said. \"I want to hear this. Because I don't believe you.\"\n\n\"Which one do you want?\"\n\n\"The first book.\"\n\n\"Can I finish my pizza?\"\n\nShe nodded graciously. Oliver continued eating, wiped his mouth, took a drink of beer, and sat back in his chair. He assumed the position\u2014eyes closed, head tipped back.\n\n\"Okay,\" he said. \"Book one . . .\"\n\nAnd so he began. Ginny didn't actually know Harry Potter book one by heart, but what he was reading sounded right. Normally, Oliver had a deadpan manner of speaking. When he recited the letters, his voice went completely flat. When he read the book, his face relaxed and his voice deepened. He was a very good narrator, actually.\n\nAfter a few moments, they had attracted the attention of some very stoned people sitting two tables over. They openly stared at Oliver, their jaws hanging slightly open, their eyes bloodshot and full of wonder. They began to approach, sliding their chairs closer and closer, inch by inch. The waiter began to hover as well. Oliver seemed to enjoy having an audience\u2014he continued on for a full three chapters, growing more and more expressive.\n\n\"Was that Harry Potter?\" one of them asked.\n\n\"What makes you say that?\" Oliver replied.\n\n\"You kept saying 'Harry Potter,' \" the guy replied seriously. \"And it sounded like it. It sounded like you were reading it. How did you do that?\"\n\nOliver drew his black coat tight around himself, leaned right into the guy's face, and quietly said, \"I'm Dumbledore.\"\n\nGinny burst out laughing, despite herself.\n\n\"Are we still trying to find this boat?\" Oliver asked, as they left the coffee shop. \"Or have we given up?\"\n\n\"I think we give up,\" Ginny replied.\n\n\"Good. So we can take the same way back.\"\n\nIt was a cold walk, but a pleasant one, through the canals, over the bridges. They didn't talk, but the silence between them was peaceful. Oliver smoked, and Ginny wrapped her scarf around her face. It was only when they had almost reached the Koekoeksklok that Ginny remembered why she left\u2014or how long ago.\n\n\"Oh god,\" she said. \"I didn't tell them I wasn't coming back.\"\n\n\"So?\"\n\n\"What if they're worried?\"\n\n\"I don't think they're worried,\" he said, dropping his cigarette to the sidewalk and stepping on it. Whatever lukewarm feelings of tolerance the Harry Potter reading had provoked were instantly chilled by this offhand remark.\n\n\"What's that supposed to mean?\" she snapped.\n\n\"Never mind,\" he said quickly.\n\nOh, but she could never never mind now. Her mind was already flying. What he meant was that they wouldn't be worried because they were glad she was gone. They were busy.\n\n\"You keep saying things like that,\" she said, unable to hide the emotion in her voice. \"What's your problem? You mean something.\"\n\n\"Listen,\" he said, holding out his hands defensively, \"I don't think you want advice from me, that's all.\"\n\n\"I don't want advice from you. I want to know why you keep saying this stuff.\"\n\nOliver sighed deeply and stopped walking.\n\n\"This is all I have to say about the matter,\" he said. \"I have been stuck in the car with the three of you. I have nothing to do but watch. I don't know what went on between you and . . . Keith . . .\"\n\nHe clearly didn't like saying Keith's name.\n\n\". . . but I know something did. I'm also guessing it was left somewhat unresolved.\"\n\n\"How do you . . .\"\n\n\"Because it's obvious,\" he said. \"It is the most obvious thing that I have ever seen. He flirts with you. You flirt with him. . . .\"\n\n\"He's not flirting,\" Ginny said. \"I'm not either.\"\n\n\"No guy drives all the way to France for a girl he just wants to be friends with.\"\n\n\"Oh,\" she said, \"you're one of those people who thinks guys are never friends with girls. . . .\"\n\n\"I said you don't drive to France for someone you just want to be friends with.\"\n\n\"He has a girlfriend,\" Ginny said defensively.\n\n\"Yeah. I noticed that too. I don't think that changes much. As I said, I don't think you want my advice, but I'd be . . . careful.\"\n\nThe insanity of this had come full circle. She was out with Oliver, who was telling her to watch out for Keith. Oliver, the extortionist. She pushed past him, disgusted with herself.\n\nAnd, as it happened, Keith and Ellis were in the lobby, trying to make sense out of an ancient Dutch board game.\n\n\"Where've you been, mad one?\" Keith asked. \"You said you were going out to make a call.\"\n\nGinny cast a rueful look at Oliver, who followed her inside.\n\n\"We looked for the boat,\" she replied.\n\n\"We looked for the boat?\"\n\nKeith looked Oliver up and down. Oliver shook his head and went for the stairs. For a moment, conversation was impossible because of the creaking noise. These were the loudest steps in the entire world.\n\n\"I was out,\" Ginny said simply. \"Walking. He was out. We looked for the boat.\"\n\n\"Did you find it?\" Ellis asked.\n\n\"No.\"\n\n\"No surprise there,\" Keith said, turning back to the game. \"Want to play? We have no idea how this works, but we've decided if you get five hundred points, you win. It's up to you how you get the points. I've been getting points by hiding Ellis's pieces down my shirt.\"\n\nHe grabbed a pocket of fabric down by his stomach and jiggled it. It made a rattling noise.\n\n\"Cheater,\" Ellis mumbled.\n\n\"I think I'm going to bed too,\" Ginny said.\n\n\"You do everything he does,\" Keith called after her. \"You love him.\"\n\nShe felt his gaze follow her long after she disappeared up into the darkness of the stairwell.\nRandom Acts of Cruelty\n\nGinny woke the next morning to a scream. Luckily, it wasn't her own. It was loud, male, and from right below them. Ellis jolted awake in the bed next to her.\n\n\"What was that?\" Ellis asked. \"Did I dream that? Was that you?\"\n\nNow there was yelling, and a thumping noise. They jumped out of bed at the same moment and ran down the steps, slipping on the slick wood as they hurried. The door to Keith and Oliver's room was shut, and a fevered conversation was going on behind it.\n\n\"What do we do?\" Ellis asked. \"They're at something in there. Think we should break them up?\"\n\nAnother muffled cry, followed by loud laughter. Keith's.\n\n\"Yes,\" Ginny said, stepping forward and opening the door.\n\nKeith was standing closest, dressed in a baggy pair of sweatpants and the T-shirt from yesterday. Oliver was just in boxers and a T-shirt again, but this time she couldn't really blame him. He was also soaked, and swearing profusely.\n\n\"Shut the door!\" he yelled. This time, he was not in the mood to show off the boxers. And once again, Ginny found herself staring just a little. Keith did not shut the door. He reached over and opened it wider, letting the cold air from the stairwell in.\n\n\"It snowed,\" he said, craning his arm over his head and lazily scratching his neck.\n\nNow Ginny saw it. Snow scattered all over the floor, all over Oliver's bed. So much snow\u2014snow that could only have come from one source. Keith must have been very, very quiet, because those stairs were like a musical instrument. And it must have taken him a few trips, because there was a lot of snow. Oliver grabbed his bag and let out a groan of dismay when even more snow poured out. His clothes were utterly soaked.\n\n\"Oh dear,\" Keith said. \"Those are going to be unpleasant and cold.\"\n\nOliver shoulder-shoved Keith and slipped quickly past Ellis and Ginny on the way to the bathroom. Keith let it go with a smile.\n\n\"That was mean,\" Ginny said.\n\n\"Mean?\" Keith sat on his bed and surveyed the damage contentedly. There was snow on his bed as well, probably thrown there by Oliver. \"That's nothing. I could have done much worse, and you know it.\"\n\nThere was just a little defiance in his voice. Ellis put her hand over her mouth, possibly to stifle a giggle.\n\n\"He's going to freeze,\" Ginny said.\n\n\"Again, I'm not seeing the problem.\"\n\nGinny walked away, taking heavy steps back up to their room. She wasn't really sure why she was angry at Keith for doing this, but she was. She grabbed for her clothes, not even bothering with a shower. She could hear the water running through the pipes, though, as Oliver took his. Ellis came in a moment later and shut the door quietly.\n\n\"He's trying to help,\" she said. \"Honestly.\"\n\n\"I know,\" Ginny said. \"I just don't want that kind of help.\"\n\nEllis nodded and pulled on her clothes as well. They went downstairs, where a table full of yogurt, muesli, bread, cheeses, and meats was waiting for them. Keith joined shortly after, humming cheerily under his breath.\n\n\"I'm starving. Anyone else starving?\" He loaded up his plate and sat down at the table. Both Ellis and Ginny stared at him.\n\n\"What?\" he asked.\n\n\"Ginny's right,\" Ellis said. \"Enough's enough.\"\n\n\"Have you both forgotten what he's doing?\"\n\n\"No,\" Ellis said. \"But . . .\"\n\nThe creak of the stairs broke the conversation. Oliver came into the room in his wet, clingy clothes. His pants were clinging to his calves. Even his shoes were wet. He took seventy damp Euros from his wallet and handed them to Ginny.\n\n\"For the room,\" he said. \"I'll be outside.\"\n\n\"I'll bet that coat's not quite dry after getting splashed yesterday,\" Keith said. \"Such bad luck.\"\n\nOliver left without a word. Ellis punched Keith lightly in the shoulder.\n\n\"What?\" he asked again.\n\nIt was even colder than the day before, so the walk to the bookshop must have been brutal for Oliver. He had his arms wrapped tightly around himself. Charlie was waiting for them outside of the shop, dressed in skinny white jeans, a black leather jacket, and huge mirrored sunglasses. His hair was even higher and scragglier.\n\n\"You're wet,\" he said, tipping down the glasses to look at Oliver.\n\n\"Fell in a canal,\" Keith lied, jerking his thumb at the canal behind them. It was to Oliver's credit that he didn't reach over and knock Keith backward into the aforementioned canal.\n\n\"Oh. That happens. Come.\"\n\nWalking with Charlie in the lead was a strange experience. He walked a vaguely snaky path and occasionally, Ginny swore he skipped a little. Not high enough to click his heels or anything. He'd just pop up higher than normal. Then snake, snake, snake.\n\n\"This man is stoned,\" Keith said in a low voice. \"Or, even worse, he might not be. What the hell is wrong with everyone your aunt knows?\"\n\nThis question became all the more relevant when they saw the boat.\n\nTo be fair, the pink boat was not horrific as Ginny had imagined it would be. In her mind, it was going to be Pepto-Bismol pink and painted all over, even on the windows. In reality, it was at least four shades of pink and rose. It was still very pink.\n\n\"Margaret picked the colors,\" he said, reaching up to run his finger along the slender bare branch of a tree.\n\nThis much was obvious to Ginny. It was either Aunt Peg or a group of five-year-olds.\n\n\"Where's the window?\" Oliver asked.\n\nCharlie took off his shoes and jumped over to the deck of the boat. He walked around to the front (the bow, whatever they called it). And there it was, the very front window, essentially the windshield of the boat. It was a painting of a jungle scene, a cartoonish one. Massive green fronds, huge orange flowers, a massive parrot. The picture mostly went around the frame of the window, leaving the center open, like an opening in the foliage. Whoever was driving would have to navigate through Aunt Peg's strange landscape as they made their way around the canals. It was interesting, if not entirely safe.\n\n\"Let me just get it off,\" he said.\n\n\"Ooh er,\" Keith said quietly. Ginny and Ellis both stared at him, and he shrugged sheepishly. He wasn't quite off the hook yet.\n\nCharlie grabbed the window by the edges and started pulling on it, which wasn't exactly the removal method Ginny was expecting.\n\n\"Do you need help?\" Oliver said, preparing to climb on.\n\nCharlie waved him away.\n\n\"It's no problem,\" Charlie said. \"I stuck it on with glue. Just glue.\"\n\nHe was such a strange, spidery person, wrestling with a window on a pink boat. He groaned and grunted, locking his spindly legs and yanking over and over, his head snapping back. His glasses slipped off his face. He was some kind of deranged Muppet that had decided to attack a sailing vessel.\n\n\"This is incredible,\" Keith said. \"Can we take him home with us?\"\n\nThere was a cracking noise, which made Ginny's stomach sink.\n\n\"Here you are,\" Charlie said, coming over to the side of the boat and passing them the window. The glass wasn't broken, fortunately. The cracking noise seemed to have been the splintering of the wooden frame around the pane.\n\n\"I am sorry for your loss,\" Charlie said, putting his ink-stained hands gently on her shoulders. \"It was a loss for everyone, for all, for art.\"\n\n\"Yeah,\" Ginny said quietly. \"It was.\"\n\nThey took the Hoek van Holland ferry, which was just south of Amsterdam. Ginny had done a monster of a ferry ride on her last trip\u2014twenty-four hours on a ferry to Greece. Of course, she had spent much of that time basking in the sun, not huddling inside, avoiding the December air and the frigid spray. But this trip wasn't nearly as long.\n\nGinny was a little nervous about leaving the window in the car. She had taken all her clothes out of her bag and wrapped it carefully, just in case the boat was dipping and swaying. Ideally, she would have stayed in the car with it, but it was freezing in the car hold, and they didn't allow passengers to remain down there anyway.\n\nThe three of them sat around one of the welded-down caf\u00e9 tables. Oliver was relegated to a different table. He looked even colder once he was inside, but he bravely took out a huge novel and tried to read. Ellis got the Top Trumps cards out again. \"Come on,\" she said. \"You know you want to play the horses pack.\"\n\nShe was obsessed with those cards. Her inner Little Ellis couldn't be at peace until someone played with her.\n\n\"Go on then,\" Keith said.\n\n\"Ginny?\"\n\nGinny shrugged.\n\n\"You'll have to teach me,\" she said.\n\nTop Trumps appeared to be a game in which you got cards, and the cards had a picture (in this case, of a horse), and told you all kinds of stats for that horse, how fast it was, how big it was, etc. Whoever had the better horse won both the cards. You repeated this until someone had all the cards. So, basically it was exactly like high school, except it only took three minutes. Which was really a bit more humane, if you thought about it.\n\n\"You feel like you're really on holiday now, don't you?\" Ellis said, once they'd played a game.\n\n\"Strangely, yes,\" Keith replied. \"But we've confused the American. Look at her. You can just tell she's never been on one of the seaside holidays where you sit in the car in the rain and eat sandwiches.\"\n\n\"Those are the best,\" Ellis said, nodding.\n\n\"You make these things up,\" Ginny said. \"You're trying to trick me.\"\n\nKeith slapped the table loudly, causing Oliver to jump. \"Oi!\" he said. \"Where are we going next?\"\n\n\"Dublin.\" Oliver stiffly turned a page.\n\n\"Dublin?\" Keith repeated. \"As in Ireland, on the other side of England from where we are now, on the Continent?\"\n\n\"You are bright. Yes, Dublin. And since we have to go through England to get there, I suggest we stop there for the night. You don't have to come the rest of the way if you don't want to, since we're quite capable of handling this on our own.\"\n\n\"Dublin, the day after tomorrow?\" Ellis said. \"That's New Year's! Dublin on New Year's would be epic. This must be done.\"\n\n\"She's right,\" Keith said, with a nod to Ginny. \"It must. By the way, I think you have the wet trousers contest all sewn up. You have my vote, and I mean that.\"\n\nOliver got up and went outside.\n\n\"Was it something I said?\" Keith asked.\n\n\"I'm going to the snack bar,\" Ginny said. \"Want anything?\"\n\nThey shook their heads. Ginny went by herself, stumbling from left to right as the boat rocked. She saw Oliver through the window. Clearly, he had struggled between his dampness and his need for a cigarette. The latter won. He got up and went out on the deck in the freezing spray, flicking his lighter over and over, trying to catch a spark. The sight made her sad, so on impulse she bought two coffees and took them out. She handed him one. He looked at it in confusion.\n\n\"I'm sorry about this morning,\" she said.\n\nOliver looked at her and back at the coffee. He squeezed it like it was something rare and precious and maybe a little dangerous.\n\n\"Thanks,\" he said. \"I'm going to go. My plan is to find somewhere on this boat to hide and sleep.\"\n\n\"Hide?\"\n\n\"The one thing I learned from going away to school as a kid,\" he said. \"They never stop. Never let them find you asleep. My own fault.\"\n\nOliver's hidden sleeping place turned out to be the car. They found him in there when they were alerted to go back to the auto deck on arrival. He had gotten into the trunk, taken all of Keith's dry clothes, and piled them over himself. He was a sleeping pile of laundry.\n\n\"He really does read lock-picking sites,\" Keith said, peering at him through the window and knocking loudly to wake him. \"This car is like a bank vault. No one can get in.\"\n\n\"I once opened the door with a pen,\" Ellis told him.\n\n\"Don't tell me things like that.\"\n\n\"I did. Just a little flick of the Biro and . . . pop! Door open.\"\n\nOliver rearranged himself in the backseat to make room for Ginny, shoving all of Keith's clothes down by his feet. The extra packing meant they were wedged in together more tightly than usual. Back on his home soil, Keith floored it in confidence, the little white car banging and clattering down the motorway. As soon as they came into London, Keith pulled over to the side of the road. There was no Tube stop, nothing. He turned around and looked at Oliver.\n\n\"This is your stop,\" he said.\n\n\"Where are we?\"\n\n\"I just said. Your stop. That's where we are.\"\n\n\"Fine,\" Oliver replied. \"I'll just take the window with me. You can keep the tabletop.\"\n\nGinny grabbed the edge of the window as hard as she could. There was no way she was letting Oliver take it.\n\n\"Out,\" Keith said again. \"The window stays.\"\n\nOliver considered for a moment, then turned to Ginny.\n\n\"Well,\" he said, \"there's not much you can do with these without the final piece. About tomorrow . . . should I assume we're driving again, or can we just take the train and the ferry and do it ourselves?\"\n\n\"I suppose it's too much to ask where exactly in Ireland we're going?\" Keith said.\n\n\"Are you really asking me this as you're dumping me on the side of the road?\"\n\n\"At least I took you all the way to London. I could have dumped you in Wales.\"\n\n\"I'll be in touch,\" Oliver said, opening his door. Keith started pulling away before he had a chance to get his bag out, forcing Oliver to walk all the way down the block to get it. As soon as he reached the car, Keith drove forward again. Oliver waited this time before approaching the car. When he did, Keith backed up, causing Oliver to jump back to the sidewalk.\n\n\"Keith!\" Ellis said. \"Stop it!\"\n\nThe backseat was suddenly very roomy. It was too big. Ginny shifted uncomfortably in the space and looked through the back window. Oliver was already making his way assured down the road, as if he knew exactly where he was going, his coattail snapping. For some reason, the sight made her very sad.\n\nWhen they arrived at Richard's house, there was no parking, so Keith stopped the car and kept the engine running while they carefully extracted the pieces from the backseat. They helped her get everything up the steps.\n\n\"So we'll wait to hear from you about tomorrow,\" Keith said. \"Times, directions, whatnot. I assume we're going to have to make an ungodly early start.\"\n\n\"You guys don't mind? Going to Ireland?\"\n\n\"Course not,\" he said. \"We would never leave you with him, would we, El?\"\n\n\"Definitely not,\" Ellis said. \"I'm excited! This has been amazing so far!\"\n\nEllis gave her a big hug; Keith, a friendly clap on the arm. No sooner had they pulled off, the car farting and spitting its way down the street, Oliver appeared. Ginny didn't even have time to close the door. He was just there, like he had grown right out of a bush or popped out of a rubbish bin.\n\n\"Where the hell did you come from?\" she asked.\n\n\"I was just standing over there, waiting for you to get home.\"\n\n\"I would have seen you. What, were you crouching?\"\n\n\"I wasn't crouching. I was just over there. . . .\"\n\n\"How did you get here so fast?\"\n\n\"I took the Tube. I beat you by ten minutes. He actually left me at a very convenient . . .\"\n\n\"How do you know where I . . . oh, the letters.\"\n\n\"Now that we've settled all that,\" he said, sighing, \"wouldn't you like to know why I'm here?\"\n\n\"Why are you here?\"\n\n\"There's something you need to do, and I didn't exactly get a chance to explain in the car. I can at least help you get these inside.\"\n\nHe pointed to the tabletop and window leaning against the door. Had he come here to take the pieces? That seemed unlikely. Like he'd said, they were of no use without the last one. Maybe for some other underhanded reason? Probably not. Keith really had thrown him out of the car. He couldn't have planned for that.\n\n\"Are you going to let me in?\" he asked.\n\n\"Fine,\" she said, shoving the key in the lock.\nThe House of Secrets\n\nUnlike Keith, who always came in like he owned the place, Oliver stepped in with caution, head slightly ducked, stepping lightly, looking around at everything like there might be something waiting to ambush him from behind the sofa. He didn't wander into the kitchen looking for food. He didn't even sit. Mostly, he stared at the decorations that covered the living room.\n\n\"Who did this?\" he asked.\n\n\"So, what did you come to tell me?\" she said. \"You could have just emailed the instructions about tomorrow.\"\n\n\"Except there's something we have to do today. Here. In this house.\"\n\nHe punctuated that statement by reaching up to touch the garland Ginny had strung around the doorway, accidentally pulling half of it down. He tried to stick it back up again, fumbling with it for a moment.\n\n\"Leave it,\" she said. \"Just tell me what we have to do here.\"\n\nHe leaned against the door and assumed his recitation stance, his head tipped back and his eyes closed.\n\n\"'Here's a shocker, Gin. I made some paintings that are actually worth some money, and since you're back, you might as well go and get them. You'll find them up in my workspace at Harrods, locked away in the cabinets in the back. The key is stuck in the Blu Tak under the picture of A Bar at the Folies-Berg\u00e8re. . . .' \" Oliver's eyes opened and he looked over at Ginny. \"Those are the paintings you already sold.\"\n\n\"I'm aware of that,\" she said.\n\nOliver cleared his throat and continued reciting.\n\n\"'I thought leaving it there would be a nice touch. I would have felt stupid leaving the key to my inheritance on an I-love-London key chain on top of the fridge. You have to have some style when you're leaving things from the beyond. In the cabinet, you will also find the name and address of a man at an auction house who is fully prepared to handle and sell the paintings. His name is Cecil Gage-Rathbone and he is very good at what he does. Cecil can also handle the sale of our work in progress, which is not yet finished!\n\n\" 'Up in my room, there's a disused fireplace. I'm sure you saw it before. It's all painted over. Get in there, reach up, and behold, you will have the next thing you need. Take this with you to our final destination, which is . . .\n\n\" 'Ireland! Bet you've always wanted to go to Ireland, Gin. I am here to oblige. Ireland is truly one of my favorite places, and the last one I visited.\n\n\"'Here's the thing, though. I can't actually remember how to get to the place I want you to go. Richard knows, though, and since you're there with him now, you can ask him. Say you want directions to \"the place near his gran's.\" He'll know what you mean.' \"\n\nGinny stood up instantly and headed up the stairs. She was at the door of her room before she realized that Oliver had not followed her. He stood at the foot of the steps, gazing up.\n\n\"What?\" she asked.\n\n\"I can come up?\"\n\nIt hadn't occurred to her that he wouldn't follow her up to her room. But they had been staying in hostels together for two days now. . . .\n\n\"Just come on,\" she said.\n\nOliver's reaction on seeing the Christmas decorations was nothing compared to his reaction when he reached the threshold of her room.\n\n\"Bloody hell,\" he said.\n\nGinny's room\u2014Aunt Peg's room\u2014was quite the assault on the senses when you first encountered it. Oliver stared at the floor-to-ceiling collage of wrappers and assorted pieces of paper. Ginny was already on the floor, shoving aside the suitcase she had set in front of the fireplace. There it was\u2014the little portcullis made of plastic. It gave with just a slight effort. Ginny got on her back and slid her head into the opening. The chimney had been sealed off at some point, but there was still a frigid draft coming down through the darkness. Just a few feet up, an object was wedged sideways in the flue. She reached up and grabbed it, and it came down, along with a handful of dust and filth that landed on her face. She sneezed it out and retracted herself.\n\nIt was long, narrow box, like the kind of box a collapsible umbrella might come in. It was made in two long pieces, which just needed to be pulled apart to open. There were two things inside\u2014a long object, and a small baggie. The baggie contained fine art pastels\u2014large, fancy crayons with labels written in Italian. There were four of them, a gray one, a white one, a deep green one, and a light green one. The long object was a piece of blank paper.\n\n\"Oh no,\" Ginny said.\n\nTo be fair to the paper, it was very nice paper. It looked handmade. You could see the fibers. And it was almost completely translucent. Still, blank paper implied drawing something, and that was bad.\n\n\"What am I supposed to do with this?\" Ginny asked. \"Does she think I can draw? Because I can't draw. Come on, you know what I have to do.\"\n\n\"You'll be fine,\" he said.\n\n\"I'd be better if you told me.\"\n\n\"Honestly, you wouldn't. It doesn't make any sense without knowing the location, but the actual task doesn't sound difficult. We could leave tonight. We could go without them. I'd bet you we could get some cheap flights, be there in no time. That's how I was planning on doing it before they got involved.\"\n\nFor just a split second, Ginny was tempted. No more staring at Ellis and Keith. Just go off, get it done now. But no . . . no. Keith was offering to drive. That was how she was going. That was how it had to be.\n\n\"We go with them,\" Ginny said.\n\n\"Then we'd better look up the ferry times.\"\n\nGinny's computer was on the floor next to her bed. She pulled it over. Oliver stood over her as she typed and leaned against the bureau, making him look about fifteen feet tall.\n\n\"There's a ferry to Dublin at one,\" she said. \"It leaves from Holyhead.\"\n\n\"Which is in Wales. That's about six hours away. We're going to have to leave at five in the morning.\"\n\nHe said it like it was a challenge.\n\n\"Then that's when we go,\" she replied.\n\n\"Well, then, I suppose I should . . .\" He looked down at an eyeliner Ginny had left on the bureau, picking it up and examining it curiously, like he had never seen such a wondrous object before. \". . . I should go,\" he said. \"I'll meet you in the morning, at his place, I suppose.\"\n\nHe left the room. Ginny heard him hurry down the stairs and let himself out.\n\n\"You're so weird,\" she said out loud.\n\nRichard was somewhat surprised to find Ginny sitting on the sofa when he returned home that night.\n\n\"You're home,\" he said. \"And you brought . . . some scrap-heap challenge materials. . . .\"\n\nHe pointed at the tabletop and window that had joined the many decorations in his living room. They really didn't look very impressive out of context. The tabletop was yellow, with several reddish circular stains, mostly in the center, but some on the sides. There was what also appeared to be a long black scorch mark along the side. The paint had flaked up completely in several places, revealing a duck egg blue door underneath. The door had been sawed in half and barely sanded, leaving one rough edge. That was the edge that had been on her foot in the car. The window was more artlike because it had a painting on it, but it was still very dirty, and the wood frame was shabby.\n\n\"That's the piece,\" Ginny explained. \"It's not finished yet.\"\n\nRichard turned his head to the left and to the right, trying to comprehend how a rough yellow tabletop and a colorful but dirty painted window went together, then shook his head, apparently deciding this was not the best use of his mental energies.\n\n\"So there's more?\" he asked.\n\n\"One more piece,\" she said. \"We have to get it.\"\n\n\"And where is this piece located?\"\n\n\"Ireland. The letter said I was supposed to ask you for directions to the place near your gran's.\"\n\nRichard nodded and looked away, jingling his keys in his palm for a moment.\n\n\"Right,\" he said. \"Of course. Well, I propose, if you're up for it, a trip to the Rose of Delhi. It's just down the road, and they do a very good curry. Let me change and we'll go.\"\n\nRichard quickly changed into jeans and a sweater, and they were soon walking down the rain-soaked streets. It always rained in England when you weren't looking, giving the roads and sidewalks a glossy shine. The night was otherwise clear, and not too cold.\n\nFor the second time in her life, Ginny found herself in an Indian restaurant. This one had a distinctly dramatic flair. There were heavy velvet drapes over the door to keep out the cold. Inside was a world of bright rose-colored walls, covered in tiny elephants stenciled in gold paint. Bouncy Bhangra music was playing, and there were fresh flowers everywhere\u2014a small arrangement by the door, little vases on all the tables, another vase full on the host stand. They were effusively greeted, given warm, moistened washcloths to clean their hands with, and tucked into a cozy window table. The waiter immediately set about rearranging all of the glasses and silverware. This always baffled Ginny. Why did they set it one way and change it as soon as you sat down?\n\nA moment later, she was facing down a menu she didn't really understand. Korma, masala, rogan josh, vindaloo, chicken tikka bhuna, aloo gobhi, biryani . . . she had no idea what these things were. When the popadoms came\u2014the large, crispy disks that were sort of the equivalents of tortilla chips at a Mexican restaurant\u2014she felt slightly more comfortable. She still had no idea what was in any of the five small silver bowls of dips that went with them. She allowed Richard to guide her through the ordering process. He got her something not too spicy, and made of lamb.\n\n\"My family is from Ireland,\" he said, poking a hole in the center of the popadoms so that they could be broken up and eaten. \"I was born there. I moved here when I was little, so I never had the accent. I'm the only Murphy who sounds like this. It's the family shame. The place you're going is just outside of Kildare, near the Curragh.\"\n\n\"What's that?\"\n\n\"About twenty kilometers of open grazing land. It's where the sheep are. They wander into town at night and eat up all the grass in your garden and leave you with the gift of sheep poop.\"\n\n\"So, I'm going to an open field?\"\n\n\"No,\" he said. \"Not exactly.\"\n\n\"So, what's there?\" Ginny asked. \"What am I going to see?\"\n\n\"Didn't the letter tell you?\"\n\n\"I didn't read that page yet,\" she said. It was the truth. She hadn't. \"I kind of read them as I go.\"\n\nRichard broke off a large piece of popadom and tapped it on the edge of his plate. He seemed to be debating something with himself.\n\n\"Is something wrong?\" she asked.\n\n\"No,\" he said. \"Just thinking of the best way to get there. You're going from Dublin, I presume?\"\n\n\"We're taking the ferry in the morning.\"\n\n\"Of course. I probably shouldn't have taken you for curry tonight . . . though I'm sure your trip will be fine. Just a little . . . choppy, sometimes . . .\"\n\nHe tore his bread in half again and left both pieces on the side of the plate without eating them.\n\n\"So,\" he said, changing his demeanor entirely. \"Dublin. New Year's Eve. You'll be with your friend, right?\"\n\n\"There are four of us,\" Ginny said.\n\n\"Four? Where did the other two come from?\"\n\n\"Keith and his girlfriend. He drove. It's . . . fine.\"\n\nNow they were both being cagey. The waiter came by with a cart with their food, so there was a natural pause while he set up a hot plate and put down the steaming metal dishes of curries and rice and bread. It was enough time for Ginny to come up with a change of subject.\n\n\"How normal is it for people to go away to boarding school here?\" she asked, as they dished up their food.\n\nRichard could easily handle a turn in conversation like that. He was used to getting constant, random questions fired at him all day long.\n\n\"It's not abnormal, but it's not the usual thing either.\"\n\n\"I just met someone who went to one,\" she said. \"You always read books or see movies about people in England going to boarding school. I thought maybe it was what most people did.\"\n\n\"No,\" Richard said. \"I didn't go to one. I know a few people who did. They're very expensive.\"\n\n\"So you have to be kind of rich to go?\" she asked.\n\n\"Well, it usually implies some money. Some people go on scholarship, some schools cost less than others. It varies. Generally, though, there has to be some money coming from somewhere.\"\n\nSo, unless Oliver had gone to school on scholarship, he had to have some money in his family. He was probably smart enough for a scholarship\u2014but people who won scholarships didn't seem like the kind of people who would just walk away from university. And while his clothes were probably secondhand, they were still very high quality. He knew good stuff when he saw it. Her guess was that there was some money, at least in his past. So why would he need her money from the sale?\n\n\"I'll write down the directions when we get back,\" Richard said, returning to their primary conversation. \"It's not far from Dublin. Shouldn't take you long to get there, especially if you have a car.\" He gestured to the food. \"Tuck in. I think you'll like that.\"\n\nWhatever it was, it did look delicious. It was a deep reddish brown color, like chili, flecked with bright green pieces of cilantro. It smelled like nothing Ginny had ever eaten before.\n\n\"It's okay,\" Richard said, indicating her plate. \"Try it.\"\n\nHe was definitely talking about the curry, but she got the distinct sense that he was trying to reassure her about something larger, something waiting for her in Ireland.\n\nShe plunged her fork in and took a big bite. The dish, whatever it was, was just as delicious as it smelled, full of tender meat and freshly cracked spices. Richard never steered her wrong. Whatever was waiting for her . . . it was all going to be fine. She was sure of it.\nThe Emerald Isle\n\n\"Why are Americans so fascinated by Ireland?\" Keith asked when he opened the door the next morning. \"Ellis and I were just talking about this. Now we need you to explain. Explain, American. You all seem to think it's magic. Ireland . . .\"\n\nHe said the last word in an Irish lilt. Ellis, who was sitting on the steps cradling a cup of tea, shook her head.\n\n\"Morning, Gin,\" she said. \"I wasn't talking about this. He was. To himself.\"\n\nGinny smiled thinly, taking in Ellis's jeans and sweater from the day they had started their journey to France. Either Ellis had gotten there very early or . . . or she had never left and had no other clothes to change into. That was the horrible but entirely likely option. Ginny tried to beat down the creeping misery that came with this thought, the one that wound around her body like a vine.\n\n\"Plus,\" Keith added, oblivious to her suffering, \"you all think you're Irish. What's the appeal? Do you like the accent more? Is it all the magical rocks? Oh, look, a leprechaun. . . .\"\n\nThat signified the arrival of Oliver, who was just coming up behind Ginny.\n\n\"Good morning,\" he said.\n\n\"Is it?\" Keith asked. \"Is it?\"\n\nOliver took his normal course of ignoring everything that came out of Keith's mouth and got into the backseat. He had prepared for this leg of the trip by packing very lightly\u2014just a backpack. No computer. Nothing he couldn't keep on his lap. Ginny could barely face getting back in the car, which was looking ever so slightly the worse for wear. The heavy driving in the rain and snow had covered it in a layer of grime that actually altered the color and tinted the windows, making it feel even smaller.\n\n\"Ahead of us today,\" Keith said, jiggling the key and coaxing the cold engine to life, \"we have yet another long drive to a ferry crossing. Nothing like the Irish Sea in the dead of winter.\"\n\n\"You could drop us off at the ferry terminal,\" Oliver offered.\n\n\"Oh no,\" Keith said. \"Wouldn't miss it. New Year's Eve? Dublin? You and me together? The magic of you combined with the magic of Ireland?\"\n\nThis time, they headed north and west, to Holyhead, in Wales. This was the longest drive yet, well over six hours\u2014about the same amount of time it took Ginny to fly to England in the first place. There was very little discussion. No one was quite awake enough for that. Ginny stared glassy-eyed at the changing English landscape. The signs were suddenly in English and another quite strange language that Oliver told her was Welsh. She hadn't realized that Welsh was something people spoke, or even that it existed. It was a bit wilder here, rolling hills and tiny villages, and long stretches of nothing but fields.\n\nIt was slightly terrifying to drive into the belly of the ferry, a cavernous space half full with huge trucks and a smattering of cars and motorcycles. As soon as they had parked the car, they made their way up the stark metal staircase to the passenger decks. Keith and Ellis both stretched out on some seats in one of the lounges, head to head.\n\n\"You getting some sleep, mad one?\" Keith asked, as he positioned his coat under his head as a pillow.\n\n\"No,\" Ginny said.\n\n\"Make sure no one feeds us to the kraken, then, right?\" He gave her a little smile and a wink.\n\nThey were both asleep before the boat even left the dock. Ginny sat across from them, wobbling in her seat as the boat listed left and right. She couldn't sleep that peacefully\u2014not on this boat, and not in this position. Oliver went outside on the deck. She could see him through the drizzle-splattered windows, battling all of nature in his attempt to light a cigarette. The wind battered his coat and sent its tails flapping, his closely clipped hair pushed up on end. There was something mesmerizing about watching him try to do the same thing over and over. The flick of the lighter. The cup of the hand. The turn to try to find the one position where the wind wouldn't put out the flame. After a few minutes and two ruined cigarettes, he gave up and came back inside.\n\n\"Must be a smokers' lounge,\" he said. \"I hate them. They smell terrible. But it'll have to do.\"\n\n\"They smell like smoke,\" Ginny said. \"What do you think you smell like?\"\n\nThis was a little unfair. Oliver didn't actually smell like smoke that often\u2014maybe just a tiny whiff on his coat right after he came in. He walked away silently. Ginny looked over at the happily sleeping couple again, shook it off, and decided to go out onto the deck.\n\nThen they were off, the Welsh coast vanishing into the mist. Though it was a cold and choppy passage, Ginny enjoyed the trip. The ferry in the summer had been smooth and even, the sun beating down on blue water. She enjoyed the slosh and chug of the boat. It was active. This was traveling. It had a pulse to it. She spent most of the four-hour trip walking loops of the deck, listening to music at full blast on her headphones. (No Starbucks: the Musical. She had considered putting some of the songs onto her playlist, but then realized she had enough of Keith in her eyes and ears at all times. She was learning.) She took pride in the fact that she could walk so steadily as the boat rocked and slammed on the water. She was starting to feel in control of this thing\u2014she could handle being around Ellis and Keith. She could handle Oliver. A little loud music, a rough boat ride across the Irish Sea . . . she could take it.\n\nWhen the announcement was made for all drivers to return to their vehicles, Ginny bought two coffees and went to wake Ellis and Keith, who were still sound asleep on the lounge chairs.\n\n\"Oh, bless you,\" Ellis said, accepting her coffee.\n\nAlong with a coffee, Ginny gave Keith the driving directions she had gotten from Richard the night before.\n\n\"It doesn't look far at all,\" Keith said. \"If we hurry, we can get the piece, have something to eat, and get back up to Dublin for midnight. But we've said that before, haven't we? I suppose you know but you aren't going to tell us, because it's much more fun to drive around Ireland without a clue of what's going on.\"\n\nThis was to Oliver, who had emerged from whatever hidey hole he'd secreted himself in and joined them.\n\n\"I don't know how long it will take,\" he said. \"I don't even know where it is. Ginny got the directions.\"\n\n\"This is somewhere near Richard's grandmother's house,\" Ginny explained. \"The letter just said this place she and Richard visited together. He didn't say what it was either.\"\n\n\"All a bit mysterious, isn't it?\" Keith asked. \"Still. Looks like we can get it done and get our party hats on. Shall we?\"\n\nGinny immediately saw what Aunt Peg would have liked about Ireland\u2014the colors. There were long stretches of almost nothing but fields, but the fields were a dozen different shades of green. Then suddenly a tiny stone church, a few boring box stores, then a small town with four pubs painted in yellow and red and blue . . . a row of houses in every pastel color of the rainbow. So many colors in defiance of the steel gray of the sky.\n\nThe sun had just gone down when they reached the spot Richard had marked on the map. The car coughed to a stop. They got out.\n\n\"This is a field,\" Keith said. \"Are you sure that's right?\"\n\n\"It says stop at the end of the road and look for the white gate, go up the steps.\" Ginny lowered the paper and looked around. It was too dark to see clearly. Keith produced a flashlight from his car and shone it around, until he landed on a little stone wall, which lead to a chipped white gate, partially covered by shrubs.\n\n\"There we are,\" he said.\n\nThey started trudging through the mucky grass. The ground was like a sponge, and every step produced a sucking noise. Only certain patches were actually muddy, but there was nothing solid about this field at all. The chipped white iron gate was unlocked, and it lead to a series of stone steps that had been stuck into the side of a small hill, cutting a path up through the trees.\n\n\"I think\u2014\" Oliver said.\n\n\"Don't care,\" Keith cut in.\n\n\"Shut it for once and listen,\" Oliver said. He reached into his pocket and pulled out some white photocopied pages.\n\n\"Copies?\" she asked.\n\n\"I didn't want the letter to get damaged,\" he explained. \"And I think you might want to do this part yourself. It's more . . . personal.\"\n\nGinny looked at the path ahead. She could only see the first few steps; then it was completely obscured by darkness and trees. It wasn't the biggest hill ever, but it was still a good climb.\n\n\"Really,\" Oliver said. \"I think we should wait here.\"\n\nKeith tentatively held the flashlight out to her.\n\n\"Whatever you want to do,\" he said.\n\n\"I guess I'll go,\" she said, accepting it. \"It's just up there.\"\n\nShe stuck the phone into her coat pocket and pushed the gate open and started up the steps. The stones were coated in slippery green moss. Getting up them was a challenge, as there was nothing to hold on to but the occasional tree branch or the step above. She lost her footing several times, sending the beam of light dancing all over the place.\n\n\"You all right?\" Keith called. She couldn't even see him anymore. The view down was just darkness.\n\n\"Fine . . . ,\" she said, scrabbling to the top.\n\nShe was standing in a graveyard.\n\nIt wasn't like any graveyard she had ever seen in her life. It was more like a strange stone garden. Some markers were plain, rough stones. Other monuments were ornate Celtic crosses. Some had been here so long that the ground had come up to meet them, coming up halfway, sometimes more. Some were just little rounded tips sticking out of the ground. Many were crooked, tilting as the earth had changed around them. The inscriptions were covered in white lichen and black creeping mosses and molds obscuring the words\u2014when the words were still there. Some inscriptions had been worn away by years of rain and wind. Interspersed throughout were new headstones, cut from some gleaming, silvery rock. At some of these were gifts and tokens. Along with the usual flowers and candles, small stuffed animals perched on the branches of the crosses, half-full bottles of whiskey remained from some recent, drunken communion with the dead. It was an amazingly not-creepy place. It was peaceful.\n\nThe moon was low and bright enough that she could see pretty well, but she needed the flashlight to read the pages she'd been given. Even though they were just copies, it made a huge difference to have them physically in her hand, to read them herself in the place where she was supposed to read them. She was grateful that Oliver had realized that this was one step she had to take by herself.\n\nI learned my most important art lesson when I was ten years old. We had to do a project in which we set our heads down on a piece of paper, sideways, in profile, and then someone traced them. Then this was copied, and the two halves were put together. When you put the two sides together, we learned the lesson: The result looks nothing like you. Surprise!\n\nWhy? Because we're not symmetrical. What's happening on the one side isn't happening on the other. There's this scientific theory that humans think symmetry is beautiful\u2014equal, even things, all in the correct measure. But we are asymmetrical, Gin. Our faces, our bodies\u2014they're not the same on both sides. Your eyes are not identical. Your nose is not exactly straight. And trust me, your boobs are almost definitely different sizes.\n\nWhat's art, Gin? What's beauty? What makes my strange drawings or pile of stuff a work and not just junk?\n\nThis discussion has been going on for the ages, and there is no definite answer. So I'm as entitled as anyone to throw my hat in the ring and make up some definitions. I think something is art when it is created with intention\u2014serious intention. Even crazy intention. And I think something is beautiful if it reveals something important about what it means to be alive.\n\nThis place is beautiful. They are strange things of beauty, monuments of death, lopsided, weathered . . . many generations gathered together in one place. They aren't pristine and ordered in rows. They are carefully maintained, while they are also allowed to change and decay as nature commands. And so, they are living places. Look around, Gin. From here, you can see everything.\n\nI came here with Richard soon after I realized I was sick and was tested. His grandmother lives near here, and we came to visit. I actually got the call with my diagnosis when I was here. (Richard is half Irish. You should know this, since you are related to him. That makes you just a little bit Irish yourself, at least by marriage. But I think our family is a little Irish anyway.)\n\nDammit. Keith was right about that Irish thing.\n\nWe had been planning on walking up here\u2014it's a famous spot locally. Richard didn't want me to come after I got the news, because going to a graveyard when you just found out that you're going to die is kind of morbid. I don't think I accepted it, really. He was more devastated than I was. I left him with his gran for an hour and walked up by myself, because I just had to see this place.\n\nI was immediately drawn to one monument. I won't even have to tell you which one. If you look around, you will know.\n\nGinny looked around. There were so many monuments, in so many conditions. But she remembered to look softly, to just let her eyes drift. \"Paint with your eyes,\" Aunt Peg used to say. \"Sweeping gestures, just like a brush.\" Side to side. Gentle.\n\nThere it was, perfectly obvious. It was one of the few monuments that wasn't a cross or a standard headstone, but an obelisk. She walked over to it gently, stepping around the plots, both the open ones and the ones marked off by little metal bars, low to the ground, and the large slabs. When she got close to it, she saw a carving of a woman, dancing, with a book in her hand. What she read next confirmed her guess.\n\nThe woman buried there was a poet. She lived with her sister, who was a sculptor. When her sister died of a \"fever\" (or so said the parish register, which he showed me\u2014it's a huge book, each birth and death handwritten in ink\u2014the same book is still in use), the sculptor spent the next two years working on that monument to her.\n\nSculpting used to be considered a very manly art, Gin. Rock, chisel, breaking marble with your bare hands . . . A female sculptor working in 1887 would have been a pretty rare thing. Her hands would have been rough; her arms would have been well-muscled. She would have spent a lot of time alone in hard labor, putting her tools against the rock. That was not the Victorian ideal. It took a lot of women like that, a lot of women who said, \"I'm not going to do what you expect me to do, because you have no idea what I'm capable of. I'm going to get dirty and use tools and live the way I want\" to move the world forward. And this woman? She made her sister into a goddess and gave her a seat on the hilltop where she could dance in the wind.\n\nWhen I saw that, I knew I wanted to dance in the wind here too. I wanted to join her. This is why I asked Richard to scatter my ashes here. By the time you read this, I am sure this will have been done. I am here. Now, let's do this together. The container you got in London, you have it, right? Get it. Open it.\n\nGinny's hand started to shake. She knelt down, her knees sinking into the ground, then set her bag down and retrieved the box. She cracked open the sticker seal and slid out the contents. She put these carefully on her bag and continued reading, not caring that her knees were soaking and probably filthy.\n\nYou are going to make the final part of this. This is a joint work. You need to let your fears go here. Don't worry that it won't be good enough. Don't worry that you aren't going to do it right. Take the paper and tape it to the monument. Then get out the pastels. Just rip the paper covers off, because you'll need to turn them sideways. Do a rubbing of the image, using whatever colors you like. Just one. All of them. Whatever you want to do.\n\nThe picture is now complete. I leave it to you to assemble it.\n\nThe paper fought against the wind, but Ginny held it firm and taped it down. She opened the bag with the pastels. She reached first for the green, because that was the color around her. She started very lightly. Then she grabbed the gray. She rubbed harder, alternating the colors diagonally, even doing a few strips in white, leaving only ghostly traces.\n\nThis was really and truly the end. Aunt Peg could never know that she would show up here on New Year's Eve, with the moon hanging low and spreading a bright white glow over the hilltop . . . but she would have approved. The ashes had been put here months ago, been blown around and soaked in the rain and pressed into the earth. They were a part of the landscape now, part of the dirt on her clothes, part of everything. It really was like Aunt Peg would forever dance on the top of this hill, a place she only ever visited once in life.\n\nShe rolled up the paper and put it back in its box. She took her phone out of her pocket and was happy to find that she had at least a partial signal.\n\n\"I'm here,\" she said, when Richard answered. \"On the hill.\"\n\n\"Are you all right?\" he asked.\n\n\"I think so,\" she said, looking around. \"Yeah. I am.\"\n\nNeither of them spoke for a moment.\n\n\"It's a good place,\" he said. \"She said she liked it because she could be between England and America, keep an eye on everyone.\"\n\nGinny laughed, and wiped a tear from her eye. She hadn't realized she was crying a little. Richard laughed as well.\n\n\"Are you up there by yourself?\" he asked. \"It must be dark.\"\n\n\"They're waiting for me at the bottom. We're driving back to Dublin now. We'll get the early ferry back in the morning.\"\n\n\"Well, be careful. And I'm here . . . if you want to call. I think I'm having a quiet night in. Maybe go round to the pub later, so . . .\"\n\n\"Thanks.\"\n\n\"Happy New Year, Ginny.\"\n\n\"Happy New Year,\" she said.\n\nGinny took one last look around, to say good-bye to this place, good-bye to the year. She felt like she was leaving something behind here\u2014she wasn't sure what, she just had the sensation that something she no longer needed had come loose and fallen away. Down below, she could hear Keith heckling Oliver. Their voices carried on the wind. She checked one last time to make sure the paper was secure in her bag, then climbed back down the hill in the dark. Keith was bouncing around Oliver, doing some kind of strange step dance. Oliver stood patiently, smoking and ignoring him.\n\n\"What was it?\" Keith asked, as Ginny appeared.\n\n\"Just some stones,\" she said quickly. \"I had to do a rubbing.\"\n\nShe wasn't quite sure why she lied\u2014maybe it was just too much to drop on people on New Year's Eve. Keith continued his strange kick dancing, and Ellis chided him. Oliver, however, knew exactly what was up there, and he was looking right at her.\n\n\"Come on,\" she said, brushing past him. \"We have to get to Dublin.\"\nA Death in Ireland\n\nThey had dinner at a small, empty caf\u00e9\u2014the lone business on a street of rainbow-colored houses. The caf\u00e9 was white and starkly lit by fluorescent bulbs. There were three metal tables with green Formica tops. The menu was painted on a large green board and stuck to the wall. The main choices were \"butties.\" Bacon butties. Chip butties. Batter burgers. Curry chips. Drink choices were tea or Coke.\n\n\"I have no idea what any of these things are,\" Ginny said quietly. \"Except for the tea and the Coke.\"\n\n\"They are all very healthy,\" Keith said. \"I suggest the chip butty. I feel it's something you need to experience.\"\n\nA few minutes later, Ginny was facing down a french fry sandwich on white bread, the insides covered in ketchup. Considering where she had just been, she was feeling surprisingly upbeat. She wanted to eat. She wanted to drive fast across Ireland. She wanted to get to Dublin. She wanted to do everything, right now.\n\n\"Right,\" Keith said, working his way through a plate of fries and curry sauce. \"If we drive like hell to Dublin, we can celebrate New Year's Eve in proper style.\"\n\n\"What's proper style?\" Ginny asked.\n\n\"Well, we can find a pub and do things the classic way\u2014drinking ourselves into a stupor, vomiting in the street, and passing out behind some rubbish bins in an alley\u2014an approach I have advocated many times in the past.\"\n\n\"What's the other option?\" Ellis asked.\n\n\"Option two: We just sort of walk around until we find somewhere good, and then we stand there and see what happens to us. We have the mad one here. Strange things seem to find her. She's like a party in your pocket.\"\n\n\"That's me,\" Ginny said. She took a timid bite of her sandwich. The fries were thick and crisp and hot with fat, which soaked through the white bread, making it go limp and squishy. It was both horrifying and delicious.\n\n\"I think,\" Keith said, \"that we should go for the second one, while keeping our hearts and minds open to the first. I, of course, don't drink and can't drink, as I am driving the car. But you are welcome to do as you like.\"\n\n\"I have to make a call,\" Oliver said, picking up his bacon sandwich and going outside.\n\n\"You know,\" Keith said, \"since we have everything now, we could just leave him and go to Dublin. Nothing's stopping us.\"\n\n\"Are you goin' to Dublin?\" the owner asked.\n\n\"We are, yeah,\" Keith said. \"What's good to do on New Year's?\"\n\n\"You'll want to go to Christchurch. That's where people go.\"\n\nGinny was chomping away as they talked and soon polished off the French fry sandwich. She instantly felt ill.\n\n\"Do you have any ginger ale?\" she asked.\n\n\"No, I don't, but the shop across the road does.\"\n\n\"I'll be right back,\" Ginny said.\n\nShe noticed, as she stepped out, that Oliver wasn't in front of the caf\u00e9. There weren't too many options for places he could be, though. It was a small road, with just a handful of two-story houses. There were some cars, and a milk van, and two boys wandering down the street with a soccer ball. All else was quiet.\n\nThe store across the street was locked, though the lights were on. There was a sign in the window that said BACK IN 15 MINUTES. Ginny got the feeling it had been up a lot longer than that. There was a little walkway at the side of the shop, and Ginny heard Oliver talking in a low voice. She peered around the corner. He was leaning against the wall and speaking intently into his phone.\n\n\"It's fine, Mum,\" he said. \"I'll be back tomorrow. I promise. I'll sort it then.\"\n\nA pause. Oliver kicked at the ground as he listened.\n\n\"No, they can't do that. I'll fix it. Just leave it for now. . . . No, don't call the counsel. There's no one there anyway. Just leave it.\"\n\nKeith and Ellis emerged from the caf\u00e9, Keith chomping away on an ice cream bar. Ginny quickly snapped her head back.\n\n\"Mad One!\" he called. \"Into the sexmobile! Where's Sneaky Beaky? I'm telling you, we should leave. . . .\"\n\n\"Right here.\" Oliver came down the alley, shoving his phone in his pocket. He gave Ginny a quick nod as he walked past, but she had no idea why.\n\nThe rain came down harder, pounding down on the roof with such force that it sounded like nails were being dumped on the top of the car. Keith took the swervy roads at a good speed, slowing down when one of the Garda\u00ed (as it seemed the police were called here) cars or motorcycles was nearby. Signs in English and italicized Irish pointed the way back to the M7 into Dublin.\n\nIt happened fast\u2014a boom, a black cloud blowing out of the back of the car, and then a general feeling of looseness, like nothing was powering the car anymore and they were simply adrift. A free-flowing river of expletives came out of Keith's mouth as he gripped the wheel, steering the car off the road, where it eventually slowed more and more and then hit a large rock and bounced to a halt.\n\nNo one said anything for a moment. The lights remained on, but there was no sound from the engine.\n\n\"I'm going to count slowly to ten,\" Keith said. \"I'm not entirely sure what that will accomplish, but it seems like the right thing to do.\"\n\nOliver was already getting out of the car and heading for the hood. When Ginny opened her door, she almost fell into a deep ditch that ran along the side, possibly to keep the sheep from wandering into the road. A few inches farther, and the car would have gone right into it. Ellis screeched as she made the same discovery.\n\n\"Gin, careful!\"\n\n\"I see it.\"\n\nThey both made their way over to the other side of the car to climb out\u2014Ginny sliding and Ellis crawling over the gear stick and slipping under the steering wheel. A few sheep wandered over to see what was going on, staring over the ditch at them. The car was belching a terrible, burning smell. Ginny stepped on something hard and sharp as she walked around behind the car. She reached down to touch it, and almost burned herself on a scorching-hot piece of something.\n\n\"It looks like you've thrown a rod,\" Oliver said. \"I told you this car would never make it.\"\n\n\"Thrown a rod?\"\n\n\"Do you see the large metal rod sticking out of your engine? Don't you know anything about cars?\"\n\n\"No, actually, I don't.\"\n\n\"Maybe we can get it to a garage?\" Ellis said.\n\n\"That's fatal,\" Oliver said, shaking his head. He moved away from the smoking car, put up his umbrella, and lit a cigarette.\n\nGinny looked up and down the stretch of road they had come to stop on. It wasn't very promising. They were between a field and an even bigger field. The streetlights were fairly far apart. The predominant noise was the gentle mockery of the sheep. Baaaaaaaa. She pulled out her phone, but the signal was so weak as to be nonexistent.\n\n\"Does anyone have a phone signal?\" she asked.\n\nPhones were checked. No one did. She pulled out the paper map and walked over to the closest streetlight. It was difficult to manage her umbrella and the map at the same time, but she didn't want to get back in the car while it was still steaming and hovering on the edge of the ditch.\n\n\"I think we're on the edge of the Curragh,\" she said.\n\n\"And what's that?\" Keith asked.\n\n\"About twenty kilometers of open grazing land,\" she said.\n\n\"I don't think we're going to see the police anytime soon,\" Oliver said. \"I suggest we walk back to the pub we saw about five minutes ago.\"\n\n\"And leave my car?\"\n\n\"There's nothing we can do for it standing here.\"\n\n\"That does look pretty bad,\" Ellis said.\n\nKeith was experiencing what appeared to be an entire rainbow of emotions\u2014laughing, swearing, back to laughing again, before bending over at the waist in defeat.\n\n\"The battery still works,\" Oliver said. \"We can keep the lights on so no one will hit it. We walk back that way. We can't stand here. That is not going to restart, and who knows when anyone will pass on this road. It's New Year's Eve.\"\n\n\"He's right,\" Ellis said, wiping the rain out of her eyes.\n\nAfter a few minutes, Keith scratched his head hard and nodded a few times.\n\n\"Fine,\" he said. \"Fine. Let's . . . fine.\"\n\nThe bags were removed from the car, including the bag that contained the final piece. It wasn't pouring anymore, but it was still raining fairly steadily. The air itself was thick with a cool mist that got to all the places the rain couldn't hit, soaking Ginny all the way to the skin. There was an earthy, smoky smell\u2014somebody, somewhere around here was burning something to keep warm.\n\nThe rain and the cold weren't the real problems though\u2014the problem was that the road had no shoulder at all. The walking choices were the road itself or the sliver of marshy grass that lined it. This was pitted and dotted with rocks, and it was so dark that it was very easy to slide right down into the ditch. So the road was the only real choice. But the road was entirely comprised of curves, and when the occasional car did come along, it would race at them blindly. They'd get about three seconds' notice to dive for the side of the road, where they would slide and trip and almost fall into the ditch.\n\nSo it took about half an hour to walk. The pub was fairly dark inside, illuminated by one yellow-shaded light. Three old men sat at the bar, all with their eyes glued to a rugby match on a television mounted on the wall. They all turned slowly to see who had joined them. The woman behind the bar greeted them all warmly, and there was considerable sympathy when they described their plight.\n\n\"Sit down there,\" she said, pointing at a table with a fistful of dishcloth. \"Sit down there and have a drink. We'll get Donal.\"\n\n\"Donal's the man,\" one of the others said.\n\n\"He'll sort you. Have a drink.\"\n\n\"You know what?\" Keith said. \"I think I will. I normally don't, but I feel tonight it may be necessary.\"\n\nThe woman came over and switched on a light, illuminating their corner. A round of pints of something was ordered, as well as a few bags of crisps. Keith sat, quietly smiling and shaking his head.\n\n\"I'm really sorry,\" Ginny said, as the pints were set down in front of them.\n\n\"It's not your fault. It could have happened anywhere.\"\n\nHe drank the entire pint in three long gulps, then returned to the bar for another.\n\n\"It really would have,\" Ellis said quietly. \"It's good that it happened now. At least it died doing something useful.\"\n\nA few minutes later, a plain white van pulled up to the pub. Keith pushed back the white lace curtain and stared at it.\n\n\"Donal's here!\" the barmaid called.\n\n\"Are we really about to get into this unmarked white van?\" Keith said in a low voice. \"Don't they only sell these to child molesters?\"\n\n\"Exactly how much Crimestoppers do you watch?\" Oliver asked.\n\n\"You're a div,\" Keith replied, tipping a bag of crisps over his open mouth and letting all the fine particles fall in.\n\nDonal was a much younger guy\u2014probably in his thirties\u2014who didn't look at all surprised that he had been summoned. He was dressed in some fancy rainproof gear, like all-weather athletes wear. It appeared that he had just been running a marathon in the rain on New Year's Eve and heard the cry for help.\n\n\"Donal can help you,\" one of the men said. \"He's an engineer. He makes fridges.\"\n\n\"What's happened?\" he asked, accepting a whiskey that was handed to him.\n\n\"They've had a problem. Their car is stranded up the road.\"\n\n\"Ah, well, we'll fix it,\" Donal said. He took only a sip of the whiskey before setting it down. \"Come on, then.\"\n\n\"You won't,\" Oliver said, mostly to himself. \"It's a blown rod.\"\n\nThey all piled into the van, which was filled with all kinds of bungee cord and climbing gear. Keith pointed to this silently and drew his finger across his throat. The men from the bar followed in a smaller blue car and took their pints. It was like a little group vacation. They retraced their route, the van making easy time around the curves of the dark road. The car was now the object of interest for a large group of sheep who had perhaps mistaken it for a larger, metal sheep. The men with the pints stood back and Donal came forward and shone a light under the hood.\n\n\"You've blown a rod,\" he said in the same matter-of-fact tone Oliver had used earlier.\n\n\"Which is . . . bad?\" Keith asked.\n\n\"You can't fix that.\"\n\nA small chorus of \"no, no, no, you can't, no.\"\n\n\"Best to leave it,\" Donal said. \"They'll be no one around tonight to get it.\"\n\n\"Just . . . leave it?\"\n\n\"Nothing else to do. You'd best come back to the pub for the night.\"\n\n\"But we need to get to Dublin,\" Ginny said.\n\n\"You might be able to get the last bus. It'll take you about two hours, though, at the rate they go.\"\n\nKeith stepped forward and put his hand on the car.\n\n\"I think I need to say a few words,\" he said. \"This car . . . well . . . to call her a car doesn't do justice to her spirit, her sense of adventure. I will always remember the horrible clanging noise she made when I tried to start her. . . .\"\n\nUnder normal circumstances, Ginny would have been amused. But considering that she'd just seen where Aunt Peg's ashes were . . . and that got no eulogy, no moment of silence from the group. . . . This irritated her. Keith didn't know, of course. He couldn't be blamed. Still . . .\n\n\". . . and though I know she were bound to fail her next MOT inspection and sent off to the scrapyard to be crushed into a cube, I feel she was taken from us too soon. You lived fast, my friend . . . well, not that fast, but fastish . . .\"\n\nOliver rolled his eyes and wandered toward the van. Their new Irish friends drank their beers and allowed Keith to go on as he liked.\n\n\". . . and if there is a car heaven, I know you'll be there, leaking fluids all over the place. . . .\"\n\n\"Can we go?\" Ginny asked.\n\nKeith looked a bit surprised that she was stopping his routine, but he nodded.\n\n\"Okay,\" he said, giving the car one final look.\n\nThey got back into the van and were taken to a spot farther up the road. There wasn't even a bus shelter or even a sidewalk\u2014just a worn-away strip in the grass where people obviously walked along, and a single sign with some bus numbers and a loose, nonbinding promise that something that came here went to Dublin. They had been left with the instructions: \"When the bus comes, get on.\"\n\nSo they waited. It was dark. Again, the only noise was the gentle call of the sheep. Ginny couldn't see them, but they had to be all around. \"This must be like speed dating for you!\" Keith yelled, pointing at the cluster of sheep that followed their leader toward town. \"Hard to choose, isn't it?\"\n\nOliver tossed his lighter in his palm for a moment and shoved it in his pocket.\n\n\"Sorry,\" Ginny said, in a low voice.\n\n\"What for?\"\n\n\"I wish he would stop.\"\n\n\"He won't,\" Oliver said.\n\nThe bus arrived more or less when promised, and was packed with people. It was abundantly clear that this vehicle to Dublin on New Year's Eve was the last-ditch party bus. Ginny didn't see anyone drinking, but it was undoubtedly going on. Even the steam on the windows had a faintly boozy air. Noisemakers punctuated the chatter.\n\nKeith and Ellis got on first, and were shuffled to two seats in the back. Ginny and Oliver ended up in a pair of seats closer to the front.\n\n\"Sorry, Gin,\" Keith called. \"Yell if you need us.\"\n\nThere was something spilled all over one of the seats. Ginny chose to believe it was beer. Oliver scooted Ginny aside, pulled a shirt from his bag, threw it over the spot, and took that seat.\n\n\"Least I could do,\" he said.\n\nShe'd been upbeat until Keith's eulogy. Now something else was settling in\u2014something more appropriate to seeing where your aunt was laid to rest. She didn't feel like talking. Luckily for her, as soon as the bus was in motion, there was a burst of drunken singing. Ginny didn't know the song, but apparently everyone else on the bus did\u2014and they sang it, and a hundred broken variations of it, all the way to Dublin. Ginny and Oliver shared a small pocket of quiet. Oliver didn't disturb her, but she could see his reflection in the dark window. Every few minutes, he would glance over to look at her, to see how she was. They weren't pressed together as closely as normal, but his shoulder bumped hers, then remained there. It was very subtle, and possibly even accidental, but it was enough.\nThe Bells\n\n\"We made friends,\" Keith said in a very loud voice, as they were all shoved off the bus in Dublin.\n\nIndeed, a small crowd of people said good-bye to Keith and Ellis as they disembarked. Ellis was giving a good-bye hug to a girl wearing shiny gold tights, who then promptly walked into a bench.\n\n\"You seem . . . better,\" Ginny observed.\n\n\"Oh, you know.\" He threw a careless arm around her shoulder, a boozy smell on his breath. \"It had to go sometime. A fitting end to a fine automobile.\"\n\n\"Dublin!\" Ellis yelled, throwing up her arms. \"Did you have a good trip? We had a good trip.\"\n\n\"See those people?\" Keith said, leaning into Ginny and pointing at the girl in the gold tights and her friends. \"They know where to go. We will follow them, and all will be well. Look how shiny they are.\"\n\n\"All will be well,\" Ellis echoed. Then she burst out laughing.\n\n\"And they gave us this!\" He held up a bottle of what appeared to be champagne, probably very cheap champagne. He popped it open there and then, taking a long sip out of the spray of foam. It dripped down the front of his coat. Ellis handed them each large paper cups, which they had also acquired, and Keith poured them both liberal helpings. At the moment, he had no animosity toward Oliver.\n\n\"Drink!\" he commanded. \"Or you will anger the good people of this nation.\"\n\nThe champagne was warm, and there was far too much of it sloshing around in a cup meant for water or beer. But it felt right. Ginny took a gulp, and Oliver did the same. She continued sipping from her cup as they headed out of the station and into the city.\n\nDublin was heaving. Ginny had never seen so many people out on the street. Herds of people moved along. Everyone seemed to be going in the same direction, flowing like a river out of the depot. They walked down through several streets, until they hit a wide, main thoroughfare called O'Connells Street. They crossed a wide bridge over a river, which Ginny knew from her internetting was called the Liffey\u2014the central artery of Dublin, much like the Thames in London or the Seine in Paris, or, come to think of it, the Hudson and the East River in New York. Water always played a role in these cities. Water moved people, moved things. Always flowing toward something else, something bigger . . .\n\nA man standing on the opposite side of the bridge was playing \"Auld Lang Syne,\" which was probably a pretty dangerous move in the freezing rain. Fireworks went off overhead\u2014little pops and spurts that looked very amateur and unregulated, so they were close and low, illuminating and reflecting in the water. Ireland was a little magical. She was feeling better now. It was almost impossible not to get caught up in the spirit of things.\n\nThe parade of people wound its way to Temple Bar. This was a street entirely filled with bars and the occasional gift shop where you could buy absolutely anything with an Irish flag on it, or top hats made of green felt covered in shamrocks. Someone dressed in a massive leprechaun costume stood on one of the corners, and people kept staggering at him to have their pictures taken. There were a few food stalls open, selling pizza and sloppy kebabs. Mostly, though, it was a street of pubs. Every pub seemed to have a line in front of it. It was hard to get down the street.\n\n\"Our friends told us about a place where we can get in,\" Keith called. \"It's this way . . . I think.\"\n\nGinny looked in her cup and was surprised to find that she had consumed all of its contents at some point.\n\n\"I finished mine,\" she said to Oliver. He held out his empty cup in reply. Ellis noticed this and turned around to give them a sloppy refill, half of the champagne ending up on the cobblestones.\n\nKeith pointed to a multistory pub, which looked like a very old shop, or maybe a small factory. It was painted a shiny black with various Irish words written on the sides in gold paint. Ginny could hear music coming from inside\u2014fiddles and tambourines and drums. It looked like there was no way possible they were going to get in. People were crushed up against the windows. It was a clown car of a bar.\n\n\"You're joking, right?\" Oliver said.\n\n\"So stay out here,\" Keith said with a shrug. \"Smoke your face off. We're going in.\"\n\nOliver did just that, while Ginny, Keith, and Ellis started the long process of getting into the pub. They made it inside by remaining in constant motion\u2014not pushing, exactly. Just moving with the ever-shifting throng. No one was still in Dublin tonight.\n\n\"Right!\" Ellis yelled. \"I'm going for it! Who wants what?\"\n\n\"Guinness, of course,\" Keith yelled back. \"When in Rome.\"\n\nKeith and Ginny pushed farther inside. Once you got in far enough, there were tiny pockets where you could stand. They found one of these on the landing of the stairs, just opposite the musicians' platform. They got banged around a lot, but it provided a good view of the main floor and of the musicians.\n\n\"If I start dancing, don't try to stop me,\" Keith said. \"When the muse moves me, I have to shake it. You know that. I know that. I cannot resist a bodhr\u00e1n. I am sure you, being American, cannot resist a bodhr\u00e1n.\"\n\n\"What's a . . .\"\n\n\"A bodhr\u00e1n,\" he said, pointing up at the wide drum one of the musicians was holding and playing feverishly. \"Come on. You knew that. You're Irish. All Americans are Irish.\"\n\n\"You aren't going to stop with that, are you?\" she asked.\n\nHe smiled widely and shook his head.\n\nThe band kicked into an even faster song\u2014a flurry of Irish fiddle. The drummer was working away on his handheld drum like a madman.\n\n\"I see you looking at that bodhr\u00e1n player. You're obsessed with him. You're in love with him.\"\n\n\"I am,\" she said solemnly. \"That's why I came back.\"\n\n\"Why did you cut your hair?\" he asked. \"The old style was cute, but this is better. I like how fringy it is. Fringy, fringy.\"\n\nHe reached over and batted the tips of her hair. The hair touching set off a slightly more intense reaction, an all-over body melt. Her eyes felt like they were swimming in their sockets.\n\n\"I like fringy things,\" he said. \"Fringe theater. Fringe fries. Also, fridges, strangely enough.\"\n\nShe gripped the rail behind her for support. He leaned closer.\n\n\"Listen,\" he said, \"I'm not saying this because I like your money, though on a totally unrelated note you should give me a hundred Euros for petrol. I'm just saying . . . I'm glad you came back. You know that right?\"\n\n\"I . . . guess?\"\n\n\"What do you mean you guess?\"\n\nHe rolled toward her, cutting them some private space. He looked at the ground for a long moment and had just raised his head to say something when Oliver cut through the crowd and walked up a few steps to stand right under them.\n\n\"It's eleven thirty-five,\" he said. \"And your girlfriend just ran outside, chasing after a hat.\"\n\nGinny wished she could grab the bodhr\u00e1n and beat Oliver over the head with it.\n\n\"Wassat?\" Keith said.\n\n\"I said it's eleven thirty-five,\" Oliver repeated. \"And Ellis saw a hat she liked. A pink cowgirl hat with silver sparkles. So she ran out of the bar and chased the person wearing it. She could be miles away by now.\"\n\nThe balcony where the band was shook as the members pounded their feet on it in time with the music. Keith watched them for a moment.\n\n\"Let's go,\" he said, still staring at the bow of the fiddle, as if hypnotized.\n\nGinny didn't want to leave this hot, insane place. She and Keith had been on the verge of an actual discussion\u2014an important one. There was something huge happening between them, in the shelter of the noise and the crowd, and if they stopped the conversation would never be finished.\n\nBut they were leaving anyway. Ginny walked slowly, trying to get blocked by as many people as possible. Keith managed to get next to her. He said something. It was either \"It'll be all right\" or \"I'll make it right.\" Then he gave her a look. A look. The kind of look you give someone you want to kiss. A serious I-mean-it-now look.\n\nOr something. Something had just happened. Something hugely weird.\n\nEllis was just outside, happily showing off the pink hat. She hadn't gone far at all.\n\n\"I traded my ring for it,\" she said proudly. \"Worth it.\"\n\n\"I think Christ Church is this way,\" Oliver said, pointing up the street.\n\nGinny followed along beside him. It took her a moment to notice that Keith and Ellis had fallen pretty far behind them. Keith was talking, his hands deep in his pockets. Something serious was going on.\n\n\"I'll make it right?\" Was that really what he said? What the hell did that even mean?\n\nEllis was wiping her face. Was she crying? Was it rain? Was he back there breaking up with her? Was that possible? Magic of Ireland and all that twaddle . . . maybe it was happening.\n\nShe turned away quickly. If he was, he could not be interrupted.\n\nIt would be terrible to break up with someone on New Year's Eve, especially someone as nice as Ellis. Truthfully, Ginny would feel bad for her. Ellis treated her like a friend from the moment they met. She wished Ellis no ill will in the world. Maybe it was the champagne talking, but she just wanted everyone, everywhere to be happy.\n\nAnother series of firecrackers popped overhead. They passed the leprechaun, who was walking in the same direction as them. Every few feet, Ginny snuck a glance back. The conversation was still going on, and Keith was doing all the talking. Dear god. Something was going on back there.\n\n\"Have you ever been to Ireland before?\" she asked Oliver, to make some conversation and try to keep calm.\n\n\"No.\"\n\n\"Why not?\"\n\n\"It never came up,\" he said.\n\n\"I like it here.\"\n\nHe gave her a sideways glance.\n\n\"Come on,\" she said. \"It's nice, right? You're allowed to say you like the place. I'm not going to hit you. You never smile.\"\n\n\"I smile all the time,\" he said, deadpan. \"On the inside.\"\n\nIt was obvious when they reached their destination. Christchurch was, unsurprisingly, a church\u2014or really, a cathedral of gray stone, brightly lit and fully encircled by people. The cathedral and grounds around it were filled to capacity, so the crowds now filled the road beyond the iron gate. Once they stopped, people started to fill the gap behind them, and she lost sight of Keith and Ellis. She stood on her toes to look for them. They were nowhere to be seen. Ginny strained, putting her hand on Oliver's shoulder for support, scanning all around.\n\nOliver looked at her hand.\n\nBong. The bells rang out, and the crowd let up a cheer. Ginny was still scanning, scanning, scanning. . . . Maybe they had fallen way back, away from the crowd. This was no place to have a breakup talk.\n\nBong . . .\n\nAnd then, she found them.\n\nThey were kissing. Fully and legally and totally making out, in the way that boyfriends and girlfriends do. For a moment, she had to watch, had to strain on her toes to make sure she got a good, long look\u2014that the sight burned itself into her mind.\n\nBong . . .\n\nIt was almost funny. She really had to laugh. For a few minutes back there, she had actually convinced herself that Keith and Ellis were going to break up based on nothing at all. It was astonishing what a good job she had done. Bong . . .\n\nAlso of interest, the sound of kissing around her. You may see other people kiss, but you don't often have to hear it. Except they were in a sea of kissing. This was one of those kissing events where you went for the kissing. Oh ho ho! Even funnier. Even funnier.\n\nBong . . .\n\nIt sort of sounded like chewing, like everyone around was gnawing the faces off their partners. Oh, she was laughing now, tears of laughter running down her face. Or was it rain?\n\nBong . . .\n\nYes, the sound of kissing was the least romantic sound in the world. It was, now that she listened closely, much like the sound of a cat eating wet food. A gnawing combined with a slurping. Such a weird and terrible activity. So why did it seem so . . .\n\nBong . . .\n\nOkay, how many times were these bells going to . . .\n\nBong . . .\n\nThis was maybe what it was like to go insane. You go to Ireland with the guy you love and his girlfriend and then you freeze to death while being gently hosed down in lightly pissing rain and other people's slobber as they made out to death. While she stood with . . .\n\nBong . . .\n\nOliver. Who at least was warm and had an umbrella.\n\nBong . . .\n\nShe was still laughing. She put her head against his chest.\n\n\"The bells,\" she said.\n\n\"They ring nineteen times,\" he said loudly enough for her to hear. \"I just heard someone say. . . .\"\n\nNineteen times? She laughed even harder. That was an eternity.\n\nBong . . .\n\nHe leaned down. When someone collapses against your chest and just starts laughing like that, you probably want to check to make sure they aren't carrying scissors or eating the buttons off your shirt. She tipped her head up to look at him. He was . . .\n\nBong . . .\n\nOkay, he was handsome. He was. Sharply featured and silent and lean. He didn't have Keith's half-crazed energy, of course. Something more . . .\n\nBong . . .\n\nBrooding? Was that the word? Those were some still waters, and they ran deep. Of all the people in the world, it was Oliver who provided the most stability at the moment.\n\nBong . . .\n\n\"This is hell,\" he said.\n\nBong . . .\n\nHe had a point. He was like her. Kinda.\n\nBong . . .\n\nShe lifted herself on her toes once again, almost automatically. She imagined Aunt Peg on her toes, painting the window in Amsterdam. She imagined that Keith and Ellis weren't there. When she closed her eyes, the bells stopped ringing. She found Oliver's lips blindly, either by instinct . . . or maybe he met her halfway. It was impossible to know. When her lips met his, she felt him physically start. But he didn't pull away, either. She was kissing Oliver. Properly kissing. No hesitation. Her whole body suddenly felt warm. She reached her hands inside his black coat, feeling the smooth lining. She could feel that he had hard muscles in his back, and he was bending down as far as he could so that she could set her heels back on the ground and not totter. He was supporting her so they could kiss even harder, and she was digging her fingers into his back to bring him closer.\n\nThere were no more bells. People were moving around them. Ginny squatted down a bit to detach herself from the kiss. He followed for a moment, then seemed to realize she was stopping. He stood up at once. Already, there was debris around them\u2014streamers and confetti, bottles, the broken paper rims of hats. New Year's Eve was the quickest holiday\u2014so much buildup, and then it was over, dropped in a second, instantly unimportant.\n\nKeith and Ellis came bounding up to them, dancing. They were in some nonstop disco that only they could see and hear. Ellis grabbed Ginny by both hands and started doing a dance with her. If they had seen what just happened, they certainly weren't acting like it.\n\nKeith picked a wet, trampled paper crown off the ground and put it on his head.\n\n\"I am king,\" he said contentedly, making a slow circle, arms outstretched.\n\nHer lips were still full of that pleasingly numbing kissing sensation, and her legs wobbled slightly. The champagne. It had to be the champagne. Whatever the case, that had just happened.\n\nOne last firework popped and spluttered overhead.\nThe Crossing\n\nNot many people, it seemed, wanted to cross the Irish Sea at two in the morning on New Year's. But all the people who did want to do this were drunk. It was like the ferry itself was drunk. The boat was slapping itself around on the dock, knocking the already-unbalanced people into doorways and walls. It was raining again, and the wind was lashing them, but Ellis insisted on staying on the deck even though it was slick and unpleasant and windy enough to knock her over. The landscape was pitch-black. The sea was black too. The Irish flag flapped stiffly overhead.\n\n\"I'll tell you what,\" Ellis said, staggering along. \"I am staying awake. Worst thing to do now is fall asleep so I am staying awake. Do not let me fall asleep, yeah? I'm going to walk. I'm going to walk around the ship. Anyone want to walk around the ship?\"\n\nShe adjusted her pink hat. It had lost a bit of its silver fringe sometime over the last hour. She pulled the brim low and looked at the bare spot sadly.\n\nOut of the three of them, Ellis had either had the most to drink, or simply had some unlucky body chemistry. Whatever effects the champagne had on Ginny were washed away in the rain and sea mist.\n\nThis journey had not only ended twice, it had ended each time on a ferry. The first ferry moved slowly through the sun, this one moved erratically through the night.\n\nAt least this time she knew\u2014the journey was over.\n\n\"I predict this is going to be a very vomity trip,\" Keith said, following his girlfriend down the deck. Oliver was messing with his cigarettes again, so Ginny went back inside and sat down on one of the chairs in the lounge. She wasn't aware of falling asleep, only of someone shaking her shoulder. She opened her eyes to find Oliver sitting next to her. His coat smelled of sea air and smoke and damp, evidence of a journey she had slept right through.\n\n\"We're here,\" he said.\n\nGinny sat up instantly. \"Where are the others?\" she asked, rubbing her face.\n\n\"I haven't seen them since they wandered off down the deck.\"\n\n\"So . . .\"\n\n\"I've been sitting here. Your stuff is fine. But we have to get off now.\"\n\nOliver had been watching over her while she slept. That thought would have been impossibly creepy before, but now . . . now it was kind of sweet.\n\nOh, her brain was so broken.\n\nThey found Keith standing in the desolate terminal, leaning against a pole for support.\n\n\"Hey, Gin,\" he said wearily. \"Ellis just went in there. The night combined with the boat . . . it didn't agree with her.\" He pointed to the women's room. \"Could you, um . . . could you go with her? She went in there a while ago. I think she needs some supervision.\"\n\nGinny nodded. He was right about this being a vomity trip. The ladies' room looked empty, but Ginny saw a pair of shoes sticking out of the cubicle on the end. Someone was kneeling on the floor in front of a toilet.\n\n\"Ellis?\" she called.\n\n\"Oh, hello!\" Ellis was trying to sound cheerful, but her voice was like death. \"Oh, I'm fine. I've just been stupid. Don't worry about me.\"\n\n\"I'm just going to stand out here, okay?\"\n\n\"Oh, you're sweet. . . .\"\n\nA gagging noise, then a pause.\n\n\"Oh god . . . ,\" Ellis went on. \"Is this place moving? Gin, I feel like I'm still on the boat. Gin, is it moving in here?\"\n\nThe stall door wasn't locked, and it bounced open. Ginny pushed it closed and held it while some grim noises came from within.\n\n\"I'm so sorry,\" Ellis said, coughing. \"This is so awful.\"\n\n\"It's okay,\" Ginny said.\n\n\"You're so good. And the Irish, they're just so lovely, and they give you this . . . they give you things to drink because they're so lovely . . . you know, and it's New Year's . . . I'm so stupid. I'm so stupid, Gin. Why am I so stupid? No one else did anything stupid. Why is it just me?\"\n\nGinny laughed. She couldn't help it. \"I'm not laughing at you,\" she said quickly.\n\n\"It's okay.\" Ellis sounded like she was going to cry.\n\n\"No, really. I'm really not.\" Ginny sat down on the floor by the stall door and listened. No noise now, just palpable suffering. Ginny got close and patted Ellis's ankle.\n\n\"Ellis?\"\n\n\"Yeah?\" Her voice was breaking. Oh, this was probably a mistake, but Ellis was so upset. She shouldn't have laughed.\n\n\"You really weren't the only one who did something stupid,\" Ginny said quietly. \"I kissed Oliver.\"\n\nEllis shuffled a bit, then forced the door open and slumped in the corner of the stall, facing Ginny. She had definitely been crying.\n\n\"Really?\" she asked.\n\n\"Really. But don't . . .\"\n\n\"I won't tell Keith,\" Ellis said. \"I know. He hates him. I understand.\"\n\nEllis's chin sank to her chest and she took a few deep breaths. \"I'll tell you something,\" she said. \"I was so jealous of you when I first met Keith. Always talking about you, this adventure you'd gone on together. But then I met you and you're so nice, and you let me come along. Oh, did I really trade my ring?\" Her focus had gone down to her bare hand. \"It was cheap, but still. I'm never drinking again. Bugger, why did I do that? Maybe I can get another, only a fiver . . . never let me drink again, okay? If you ever see me drink, just slap me. Oh . . .\"\n\nThere was a rapping on the door. It opened a crack.\n\n\"Train's in ten minutes,\" Keith called. \"El, you alive?\"\n\n\"I think so. Just give me a moment. We'll be right out.\"\n\nEllis grabbed the door to try to pull herself up. Ginny helped her. She managed to get herself into a bent position, but couldn't straighten up right away.\n\n\"I don't think what you did was that stupid,\" she said, steadying herself against the stall.\n\nThen she threw up on the floor between them. Not a lot\u2014but a definite harbinger of things to come. Ginny winced and looked away.\n\n\"Sorry!\" Ellis said. \"Sorry, oh god . . .\"\n\nKeith must have heard this happen, because he let himself in. He looked at the vomit on the floor with a strange kind of satisfaction. \"And so it begins.\"\n\n\"I have a plan,\" Ellis said, as he half-carried her out. \"We'll get on the train, and I'll ride in the toilet.\"\n\n\"It's like dating royalty,\" Keith said.\n\nThe train was very sleek and modern, with lots of light-up buttons and computerized signs. It was also freezing cold and smelled like sadness and old beer. Keith got Ellis on board and positioned her in front of the bathroom.\n\n\"Just put me in there,\" she said bravely. \"I'll be fine. I'll be wonderful. Just shove me in the toilet.\"\n\n\"All right, all right, enough with the sexy talk.\"\n\nShe waved them off and slammed the door shut. A red light came on, signifying that it was occupied.\n\n\"I'm just going to stay here for a bit,\" he said. \"You might as well go sit down. You might want to sit a car over.\"\n\nA grim noise came from the toilet. Oliver and Ginny quickly moved to the next car, which was completely empty. They sat opposite each other in a group of six seats. There was no looking at each other. No speaking. Keith was gone for quite a while, so that was a lot of time to artfully avoid each other in a casual way. When he did return, he did so quite loudly, plopping down on one of the seats next to Oliver and gazing between the two of them.\n\n\"How's Ellis?\" Ginny asked.\n\n\"She'll be all right. How are you?\"\n\nIt was just the way he said it, the way he leaned in a little. He knew. Oliver must have sensed this as well, because he got up.\n\n\"I need a coffee,\" he said. \"Do you need one?\"\n\nThis was to Ginny. She shook her head.\n\n\"He's very solicitous,\" Keith said after Oliver had left. \"When did you two get so cozy? I guess it was all of that time in the backseat. I don't suppose I can blame you. He's a catch.\"\n\nYeah. He knew.\n\n\"It was just a kiss on New Year's,\" she said. \"I mean . . . everyone was kissing. And I had all that champagne.\"\n\n\"Well. Congratulations. You've made a wonderful choice. I'm very happy for you.\"\n\nIt was snide and cold. So cold that she wished she could slap him.\n\n\"What do you care?\" she asked.\n\n\"Sorry . . . what?\" He leaned forward. \"What do I care? Of course I don't care. I think that's the point we've demonstrated in all of this, my lack of caring. That's why I just destroyed my car. That's why I've been traveling with you for days so you weren't alone with the person who was ripping you off. It's all my extreme lack of care. But you seem happy now, so I'll just step aside then.\"\n\n\"It was one kiss,\" she said, a challenge rising in her voice. \"I mean, why do you care who I kiss? You have someone. You didn't even tell me you had a girlfriend. . . .\"\n\nAnd, there it was. That wasn't the way she meant to say it. This wasn't how she wanted this to go. It was out. It could never be put back.\n\n\"Do what you want,\" he said, throwing up his hands. He returned to the other car, back to his girlfriend. The glass door of the car shut behind him with a decisive hiss.\nReality Comes to Visit\n\nThat had gone very, very badly. That was like training for the Olympics, working up to one big moment that would matter for the rest of your life, and then just being careless and falling off the diving board or forgetting to correctly attach your skis. Poor form. No points. She began to sob, silently and pathetically, and completely unable to stop. She didn't hear Oliver return\u2014she was only vaguely aware that someone had sat down next to her, and it definitely wasn't going to be Keith. He put a few napkins in her hand.\n\nShe shoved them up against her face. They stuck.\n\n\"I'm okay,\" she said. The words were barely understandable.\n\nOliver didn't point out how absurd this statement was, or try to talk her down. He just sat with her, first stiffly putting a hand on her shoulder, then extending his arm. At some point, she just gave up and leaned against him a little.\n\nIt took a few minutes for her to gather herself together again.\n\n\"It's fine,\" she said, her voice thick. \"I just . . . we weren't a thing. Not a real thing. I just, really liked him. We said we were 'kind of something.' He wasn't supposed to like other people. Isn't that how it works? You just like the one person, forever, and then you stay together?\" She tried to make it come off as a joke, but it didn't really work.\n\n\"I don't have a lot of personal experience in the matter,\" he said, \"but I'm pretty sure that's not how it usually goes.\"\n\nShe looked out of the window at the gray morning. There was some beautiful landscape going by, but it was hard to appreciate. Her head was thrumming and her eyes were still leaking and dribbling. This wasn't how she was supposed to end this journey.\n\n\"I don't blame you for wanting to get back at him,\" Oliver said. \"Did it work?\"\n\n\"I don't know what that means,\" she said, hiccupping. Oh, good. The post-crying hiccups.\n\n\"Last night.\"\n\nHe stared at her as if she were very, very stupid. Oh. The kiss.\n\n\"What do you mean did it work?\" she asked.\n\n\"You did that to get to him. Did it work?\"\n\nIs that why she had done it? That wasn't how she remembered it. She remembered the bells, the rain . . . she remembered the kiss. She didn't remember any motive. She just remembered that it had happened, and that it had been . . . nice. It had been good.\n\nShe was going to leave that alone. \"Anyway, he didn't do anything wrong,\" she said. \"He just should have told me. But now he probably hates me.\"\n\n\"So what if he does?\" he replied.\n\nThat was too much for her to even contemplate.\n\n* * *\n\nThey stayed in their separate compartments for the rest of the journey. Ginny saw Keith and Ellis again when they got off the train at Euston Station. Ellis looked a little better, but her face was still ashen. She raised a weak hand in greeting.\n\n\"I made it!\" she said, her voice hoarse and scratchy. \"I rode from Wales to London in a toilet. I should get a prize. I think . . . I think I have to go home now.\"\n\nShe leaned against Keith's shoulder. He put an arm around her waist.\n\n\"I'll let you know when the auction is tomorrow,\" Ginny said. \"If you can come . . .\"\n\nEllis was nodding, but Keith said nothing. He picked up his bag, hoisted it over his free shoulder. They went one way, and Ginny and Oliver went another.\n\nIn the end, it was Ginny and Oliver who returned to the house. Oliver stayed outside for a moment, waggling his cigarettes in explanation, but Ginny wondered if he stayed outside for another reason. She had to complete the piece, and perhaps, like at the graveyard, he thought she should be alone. Ginny went in and knelt down in front of the tabletop and glass, and removed the paper from her bag.\n\nThe order seemed obvious. The paper went between the glass and the wood. That made sense. The paper was so fine that it practically vanished against the yellow background. The wine rings and stains on the table came through, shading the picture. She moved the glass back over and sandwiched the paper in place. The two dancing figures materialized from behind the ferns and plants and animals on the outside. She could see where the paint on the window had been affected by the rain and the elements, streaking and dotting lightly. There was a light layer of grime all over the picture as well, but it only seemed to add to the overall effect. The viewer was put into this strange jungle and was looking into some other reality entirely, with one of the figures beckoning the viewer inside. Beautiful wasn't the right word for it. It wasn't beautiful. It was rough and strange and bright. It was like a tangible dream.\n\nOliver knocked lightly and came inside.\n\n\"It's amazing,\" he said. \"I would never in a million years have guessed that it would turn out that way.\"\n\n\"She was good like that,\" Ginny said. She stood up and sat on the sofa to look at it from a distance. He sat next to her.\n\n\"It needs a name,\" she said.\n\n\"What did your aunt name her other paintings?\"\n\n\"She didn't.\"\n\n\"So why name this one?\"\n\n\"I don't know,\" she said. \"The others were in a group. They called them the Harrods paintings. This one is on its own. It needs to be called something.\"\n\nShe sat back and stared for a long time, until her eyes went blurry. One good thing about Oliver\u2014he could take long silences.\n\n\"When my aunt lived in New York,\" she finally said, \"she took me to a pool one summer. Except, no one has a pool in New York. It was a cleaned-out Dumpster. It was a pool, but it wasn't a pool, you know?\"\n\n\"Not . . . exactly.\"\n\n\"I mean . . .\" It was annoying when other people couldn't get in your head and automatically catch up with your conversations. \"I mean . . . people tell you what to expect. What things are supposed to be like. A pool is supposed to be this nice, clean thing in the backyard that's painted blue on the bottom, but anything with water in it can be a pool, even a Dumpster. She called it the triumph of imagination. It was how she did things. It's like, a fancy way of saying flaky. That's what I want to call it. The Triumph of Imagination.\"\n\n\"It's a good title.\"\n\nWeirdly, her impulse was to put her arms around him. She wasn't drunk. She wasn't crying. Still, the impulse was there.\n\nBefore she could do that, she noticed something out of the corner of her eye. There was a white van pulling up across the street . . . just like the ones that seemed to disturb Keith so much. Two men in coveralls got out and started walking toward the house.\n\nOliver stared down at the coffee table. It was still littered with Christmas crackers and Ginny's silver elephant.\n\n\"I called them,\" he said.\n\nThe reminder was like a slap in the face. He wasn't here to be her support system\u2014he was here to collect. He had stayed outside to call the delivery truck. Why was this a surprise? This was always how it was going to end. She backed away from him and went to the door.\n\n\"Miss Blackstone?\" one of the men said. \"We're here from Jerrlyn and Wise. May we come in?\"\n\nGinny opened the door wider, and they stepped inside politely.\n\n\"Is this the piece?\" he asked.\n\nShe nodded.\n\n\"We'll just get started then,\" he said.\n\nThey took over. Ginny and Oliver were relegated to the side of the room. The pieces, which had been manhandled and shoved into cars and bags and thrown around were now treated like crown jewels. The men set down a packing quilt on the floor and lifted it gingerly. One held it upright while the other measured its dimensions, examined its construction, and took some initial photographs. They had a long discussion on exactly how to pad it and box it and put it into the van. Ginny thought about asking them to let the piece stay long enough for Richard to see it, but it was all so official and efficient, there seemed to be no stopping them.\n\nOliver sat on the sofa, looking at Ginny's silver elephant. This was the reality of their relationship, Ginny reminded herself. It was all about finding this one piece of art and selling it off. The kiss, the trip on the train . . . it all faded away.\n\n\"We're finished now, miss,\" the first man said, presenting her with a clipboard. Ginny filled it out automatically, ticking boxes and signing on lines, not even reading. She stopped only in the box marked TITLE OF WORK and wrote in The Triumph of Imagination. When she was finished, they carried The Triumph of Imagination out the door, snugly wrapped in gray quilting and tape.\n\n\"There was a message from Cecil as well,\" Oliver said quietly. \"The auction is at two tomorrow.\"\n\n\"Fine,\" she said.\n\n\"I suppose I should go.\"\n\n\"Probably.\"\n\n\"I'll see you tomorrow then.\"\n\nShe watched him from the window as he left. He never turned back, just made his way down the street, tossing his lighter in his palm. Like nothing had happened at all. She felt a strangely familiar pang in her heart, but she couldn't quite place it and didn't feel like trying.\nThe Dotted Line\n\nThe next morning was the first clear day since Ginny's arrival. The sky was bright and vividly blue with big, puffy clouds. Ginny stood in front of the white chalk steps that lead to the front door of Jerrlyn and Wise. Occasionally, someone in a suit would pass by her and go up the steps. One or two of them glanced at her, perhaps knowing that she was the eighteen-year-old selling the artwork on show today, but the others passed without a look.\n\nTen minutes until the auction. Ten minutes, and no Keith. No Oliver either, for that matter, unless he was inside. She hadn't truly believed that Keith wouldn't come, but with every passing second, the reality set in.\n\nShe considered just going back to Richard's, then she hurried up and pressed the button before she could entertain this thought for very long. Cecil's assistant, James, opened the door, greeted her by name, and escorted over to Cecil. The hallway was packed with people\u2014easily double the amount that had been there for the last auction.\n\n\"It's so crowded,\" Ginny said, rocking on her heels nervously.\n\n\"Yes, a good turnout, especially considering the short notice,\" Cecil replied. \"The last collection got a good deal of press in the industry, so we had a lot of interest.\"\n\nGinny scanned the hall, but Oliver was nowhere in sight. In the middle of this crowd, over by the table of coffee and strawberries, was a figure she instantly recognized. It was impossible to miss the massive crown of wiry, long, black and orange hair, the mix of gold Spandex leggings, the black mohair sweater that came down to her knees, and the face with the tattooed stars around the eyes. Ginny wasn't sure why she was here, but she definitely knew who it was. The woman gave a gasp and a wave, and set down a plate filled almost entirely filled with cream and waved.\n\n\"Hello, darling!\" she called in her booming Scottish brogue.\n\n\"I think you know Mari,\" Cecil said.\n\n\"Cecil told me about the auction,\" Mari said, answering the unasked question. \"He's my art dealer down here in London. I put him together with Peg. I've know Cecil since he was just a pup. He didn't look like this when I first met him. You were pure poor art student then, weren't you, love?\"\n\n\"I was indeed,\" Cecil said.\n\n\"You're going to take good care of this piece, aren't you?\" Mari pinched Cecil's cheek. It looked like a hard pinch too. One that might leave a mark. Ginny couldn't help but be impressed at the way he took it without wincing.\n\n\"We certainly hope to,\" he replied. \"Do you have everything you need? Glass of champagne?\"\n\n\"Maybe just a small one, love. Virginia, you remember my Chloe, don't you?\"\n\nMari indicated a girl leaning casually against the flocked wallpaper, next to a painting of a tiny boy in blue velvet with a bug-eyed dog. Chloe was Mari's assistant: part artist, part butler, part bouncer. She was dressed in biker boots, rolled jeans, and a shredded T-shirt that revealed a right arm tattooed from shoulder to fingertips in one large image of a mermaid splashing around in a purple and green ocean. The bleach blond mullet had been replaced by a shaved head with just a hint of peach fuzz.\n\nAn assistant stealthily put a glass of champagne in Mari's hand. It was in one of those wide and flat old-fashioned glasses that sort of looked like bowls.\n\n\"Let's go have a look at it,\" Mari said.\n\nShe guided Ginny over to the door of the auction room. The auction room was such a strange place, so heavily carpeted, so padded. There were a handful of upholstered chairs and four long tables. A dozen people sat in front of computers, talking quietly into phones. The piece sat on an easel at the end of the room, clamps along the edges holding it all together. In this light, Ginny could see just how dirty the window was. They'd made no attempt to clean it. The layer of grime and the runny paint . . . it was all part of the piece.\n\n\"It's always very strange to see them when they get here,\" Mari said, taking a long, loud sip of the champagne. \"A bit like an operating theater. Have a bit of champagne.\"\n\nGinny felt a presence behind her\u2014the hairs at the back of her neck tingled a bit. She didn't have to turn to know who it was.\n\n\"Oh, hello,\" Mari said. \"Are you a friend?\"\n\nOnce again, Oliver was formally dressed. He fit right in at Jerrlyn and Wise. Maybe the hair was a little too dark, the coat still a little strange fitting. \"I'll be outside,\" he said to Ginny. \"Just wanted you to know I was here.\"\n\n\"Aren't you going to stay for the auction?\" Mari asked.\n\nOliver looked at Mari warily\u2014processing the facial tattoos, the names on her hands and feet, the unnatural sunburst orange streaks emanating from her head. She saw him complete his analysis and decide that the best course of action was to walk away from both of them as quickly as possible.\n\n\"Oh, he's shy,\" Mari said. \"He's different from the last one. Where do you find them all?\"\n\nIt took Ginny a second to realize that Mari thought Oliver was with Ginny\u2014in much the same way that she thought Keith was with Ginny. She was about to issue a denial, but then decided against it. For one thing, it was nice to have someone who thought she was such a collector of boyfriends. For another, Oliver's abrupt exit made her nervous.\n\nPeople started to trickle past them into the room, taking their places at the tables, opening the computers and taking out their phones. Cecil put a hand delicately on Ginny's shoulder.\n\n\"Forgive us,\" he said. \"Virginia and I just have a bit of business to complete. Please have a seat, Mari.\"\n\n\"I think I'll stand, darling. I like to think on my feet.\"\n\n\"Of course. This way, Virginia?\"\n\nHe escorted Ginny over to his office and quietly shut the door. He didn't sit down, though. \"Perhaps it's best if I speak directly,\" he said. \"I don't know who Mr. Davies is, but I don't think he had anything to do with the creation of the work in the saleroom today. I don't know exactly what's going on, but I want you to know that that contract you signed does not bind you to selling this piece today. If you do not want to proceed, you can and should say so. We've pulled things from the block before. I don't want to go ahead with this sale if there's something irregular going on. . . .\"\n\nBefore Ginny could react, there was a rapid knock on the door, and James poked his head in.\n\n\"I found him,\" he said, ushering Oliver into the room. James shut the door behind them. Now they were all crowded into the tiny space by Cecil's door. Cecil tapped his fist against his lips for a moment. Oliver pressed himself into the corner\n\n\"Virginia and I were just having a conversation about the contract,\" he said. \"I was explaining to her that it wasn't necessary for us to\u2014\"\n\n\"It's hers,\" Oliver said, cutting him off.\n\n\"Pardon?\"\n\nOliver had gone completely pale, and was pulling his coat tight around himself, as if trying to disappear.\n\n\"I'm withdrawing from the sale. All the money's hers. Just sell it and leave me out of it.\"\n\n\"What?\" Ginny said.\n\nCecil moved over to his desk and produced a piece of paper.\n\n\"I thought something like this might come up,\" he said. \"I'll just need your signature on this rider, which supersedes the contract, directing all profits minus commission to Virginia Blackstone. You forfeit any claim.\"\n\n\"Good,\" Oliver said. He squeezed between the two wingback chairs to lean against the desk and sign his name.\n\n\"We're done here, right?\" he asked.\n\n\"I won't need anything further from you,\" Cecil said. \"If that's what you are asking.\"\n\n\"That's what I'm asking.\"\n\nCecil's office was so small that it was impossible for him to exit without squeezing past Ginny. As he did so, he shoved something into her hands. She looked down. It was the last blue envelope.\n\n\"Now,\" Cecil said. \"We can proceed, if that's all right with you.\"\n\n\"Go ahead,\" Ginny said, running out of the office after Oliver. He was moving quickly. He had already left the building, and when Ginny got outside, he was halfway down the street.\n\n\"Wait!\" Ginny said, jogging after him. \"Where are you going?\"\n\n\"Home.\"\n\n\"Okay,\" Ginny said, running around to get in front of him and stop his progress for a second. \"You need to tell me what's going on.\"\n\nHe tried to step past her, but she blocked him again, putting her hands on his chest.\n\n\"Don't,\" he said simply.\n\n\"You need to explain. Why did you do this?\"\n\n\"I don't need to explain anything.\"\n\nHe gently removed her hands from his chest and kept walking. She glanced at the letter in her hand. It started where the last part cut off in Ireland.\n\ntell you.\n\nGin, do you remember how I used to drag you to the Museum of Modern Art all the time and make you stand in front of a crazy, huge painting of a woman on a sofa sitting in the middle of a jungle? Think back for a moment. I'll wait. . . .\n\nFrom her spot in the street she could look in the window. She saw Cecil take his position at the podium. The sale was starting. Cecil was nodding and pointing politely at people she couldn't see, saying numbers she couldn't hear. Then he stopped. A few minutes later, people started trickling out. It was over. The piece was sold.\n\nMari stepped outside. She looked at the sky first, then took the steps carefully, holding on to the brass banister for support. She looked up and down the street, waving Ginny over.\n\n\"Ah, here you are,\" she called.\n\nShe carefully took a seat on the steps. She had to be over seventy, Ginny realized. Her joints were stiff. Ginny sat next to her.\n\n\"Did you two have an argument?\" Mari asked, indicating Oliver's retreating figure.\n\n\"Not exactly.\"\n\n\"It seemed very dramatic, whatever it was. I love drama. But you missed the sale.\"\n\n\"What happened?\" Ginny asked.\n\n\"The person who bought it was passionate about it. A hundred and sixty-seven thousand pounds.\"\n\nA hundred and sixty-seven thousand pounds. Another large sum of money Ginny couldn't really compute.\n\n\"You don't seem excited,\" Mari said.\n\n\"I am . . . I'm just . . .\"\n\n\"I know,\" Mari said. \"Sales are strange. You make the art, and then suddenly, someone's paying for it. Suddenly it's a commodity. . . .\"\n\n\"I didn't make the art,\" Ginny said.\n\nMari shook her head and patted Ginny's knee with her gold-taloned hand.\n\n\"Cecil told me you did the assembly and the rubbing.\"\n\n\"I just put it together,\" Ginny said.\n\n\"You don't just put together a work of art, love. It's not a sandwich. You were trained. Maybe you weren't aware of it, but you were trained. It's a very good work. I'm quite proud of that girl. Proud of you both. That's why I had to have it. There was a bidder from Tokyo who was giving me a hell of a time in there, for instance, but I was determined.\"\n\nShe smiled and let Ginny take in that piece of information before she continued.\n\n\"I'm good friends with a curator at the Museum of Modern Art in New York,\" she said. \"I think that piece belongs there. So I'm going to speak to them about donating it. I think it should go home and be where lots of people can see it, don't you?\"\n\n\"You bought it? And you're sending it to MoMA?\"\n\nThe door opened again, and Chloe stomped past them, down the steps.\n\n\"I'll get the car, yeah?\" she said.\n\n\"Thank you, love.\"\n\nChloe gave Ginny a firm nod then headed off down the street in long strides.\n\n\"Chloe took a real liking to you,\" Mari said. \"She doesn't take to just anyone. She's got good taste, does my Chloe. I'm going to get her paintings down here next.\"\n\nFrom behind them, Ginny heard a car starting in the small private parking lot behind the building.\n\n\"Thank you,\" Ginny said.\n\nMari patted Ginny's shoulder with her gold-taloned hand.\n\n\"Money is for doing things, my love. Don't sit on it like a hen sits on an egg. It doesn't hatch. I should know. I've made enough of it.\"\n\nA small, black sports car, some ancient model that was probably from the seventies, came pealing down the short gravel path and pulled blindly into the street, stereo pulsing.\n\n\"Make sure to come to see me in Edinburgh sometime,\" she said.\n\nMari walked slowly to the car and lowered herself into the seat. It was just barely off the ground. She and Chloe gave Ginny a little wave, and then they pulled off to terrify London traffic.\nThe Conversation\n\nWell, we're done. But I have a little more to say. I don't want to be dragged off the stage just yet.\n\nDo you remember how I used to drag you to the Museum of Modern Art all the time and make you stand in front of a crazy, huge painting of a woman on a sofa sitting in the middle of a jungle? Think back for a moment. I'll wait. . . .\n\nGinny sat on the sofa and glared at the letter. For the first time, she wanted the letters to shut up and stop asking her questions. What she wanted, for once in her life, was a letter that provided a simple list of foolproof instructions. Would you like to succeed in life and love and not be a crazy person? Do the following. . . .\n\nYes. That would be nice.\n\nIt was her last night in London. Richard was in the kitchen, talking on the phone about some work crisis. They were supposed to go to dinner in a little while to celebrate the sale. She didn't really feel like celebrating. Yes, the piece had sold. Yes, she had more money. But aside from that . . . she had a ruined relationship and more questions than answers. She'd had the letter for hours now, and she couldn't even bring herself to read it. Even the one stupid thing she had to finish while she was here\u2014the essay\u2014wasn't done either. Failure on all levels.\n\nSomeone was ringing the front doorbell. Ginny almost fell over the coffee table in her haste to answer it. But it wasn't Keith. It was Ellis, hopping lightly in the cold.\n\n\"Sorry,\" she said. \"I just wasn't in any state to say good-bye the other day. I didn't want that to be your last memory of me, chucking my guts up into a train toilet.\"\n\nGinny held the door open for her to come in, but she waved this off. \"I have to go,\" she said. \"I'm starting a volunteer job, doing an evening shift at a crisis call center. It's quite bad around this time of year, apparently. I just wanted to see you before you were off. So . . .\"\n\nShe stood there bouncing for a moment. A gust of cold wind came in and flooded the living room.\n\n\"I don't want to interfere,\" she said, \"but . . . I think you and Keith should talk. I think you'll both regret it if you don't work it out. And I think it would be a shame if you didn't take this opportunity to talk in person before you go. But . . . I suppose I've said enough. That was awkward. I'll shut up now.\"\n\nShe opened her arms and gave Ginny a hug.\n\n\"Anyway . . .\" Ellis backed down one of the steps. \"Hopefully you'll be back. I'm glad I got to meet you.\"\n\n\"I hope so,\" Ginny said. \"And . . . I'm glad I met you too.\"\n\nShe shut the door quietly and looked around at the decorations in the living room, at the hole where the upside-down tree had been suspended.\n\n\"All set,\" Richard said. \"Hungry? You're going to like this place.\"\n\n\"Could I have one hour?\" she asked.\n\nKeith's window was open, the lights around it being drawn inside by some unseen force. She thought about calling up, but then she decided to knock on the gold plastic door panel. David opened the door. That was lucky\u2014Keith might not have done the same.\n\n\"Gin!\" he said. \"I heard you were here.\"\n\nIt was good to see a friendly face. David indicated that she could go upstairs as she liked. She took the stairs softly, but Keith seemed to know she was coming. He had dragged in half the strand of lights and was fiddling with the bulbs. What he was attempting to do, Ginny had no idea, but he looked pretty intent about it. He didn't even look up when she came in. \"How was your auction?\" he asked, not sounding very interested. \"Are you rich now? More rich, I mean?\"\n\n\"I can give you money for your car.\"\n\n\"I'm not really interested in your money.\" From Keith, that was a first\u2014and definitely not a good sign.\n\n\"Why are you so angry?\" Ginny asked.\n\n\"Angry? I'm not angry.\" His face was utterly composed and calm.\n\n\"So why didn't you come to the auction?\"\n\n\"I was busy,\" he said, pulling in a few more feet of the lights. \"I do have other things going on in my life, you know.\"\n\nThough he wasn't exactly welcoming, Ginny sat down on the red sofa, pushing aside a few books and shirts.\n\n\"Are you taking them down?\" she said, gesturing at the lights.\n\n\"No. I'm fixing them.\"\n\n\"I leave tomorrow,\" she said.\n\nKeith nodded and continued fiddling away with the bulbs. This was how it was going to be. If she wanted this to happen, she would have to shove it into existence. \"Why didn't you tell me?\" she asked. \"About Ellis. We talked every day. You never told me. And even after I came here, you never even told me. You never said a word.\"\n\n\"Told you?\" he said casually. \"You met her. . . .\"\n\n\"You know what I mean,\" she said. \"You said we were 'kind of something.' Why didn't you just tell me?\"\n\nSilence. Just the sound of the television from David's room. Keith yanked out one bulb and the whole string of lights went out. \"Maybe I will take them down,\" he said, yanking the plug from the wall. \"Don't want to start a fire.\"\n\nHe began pulling in the lights with greater force, each bulb emitting little pings of protest as they were pulled over the radiator under the window. For a few moments, it appeared that he was never going to answer, but finally he spoke.\n\n\"I told you about my old girlfriend, Claire?\" he said, coiling up the lights into a messy tangle.\n\nHe had, on a train ride over the summer. It was the first time he really told her anything personal, how he had been in love when he was sixteen, and his girlfriend had gotten pregnant and dumped him, unable to cope with the turn their relationship had taken. He'd gone off the rails. That was where Keith the thief started.\n\n\"She told me she wanted to be my friend,\" he said. \"Then I went out and stole a car. She's the reason I have a criminal record.\"\n\n\"You have a criminal record?\"\n\n\"I stole a car,\" he said, as if this was obvious.\n\n\"You didn't say you got caught.\"\n\n\"Of course I got caught. Several times. Allow me to show you my gallery of ASBOs.\"\n\n\"As whats?\"\n\n\"Anti-social behavior orders. The chav badge of honor.\"\n\n\"I have no idea what you're talking about.\"\n\n\"My point,\" he said, \"is that none of this explains what you did. Making out with the guy who was exploiting you, then throwing it in my face while I was trying to help you . . . yeah. Well played.\"\n\n\"You just said that when you and Claire broke up you stole a car.\"\n\n\"That's different,\" he snapped. \"There is nothing personal about stealing a car. It's just a car.\"\n\n\"It might be to the person who owns the car.\" She was yelling now. \"You took someone's property. I kissed someone, which I am allowed to do, even if you don't like him. Besides, he didn't take the money, so you can let it go now.\"\n\n\"He didn't take the money?\" This gave Keith pause. \"Why? Did whatshisname with the hair, from the auction house, did he . . .\"\n\n\"He just didn't take it.\"\n\nKeith just shook his head, as if unable to comprehend this new information. Finally he looked at Ginny, square on, his face open and honest.\n\n\"I thought I was doing the right thing. I didn't want to ring you up and tell you I'd met someone. It seemed wrong to do that to you over the phone.\"\n\n\"So you decided not to tell me at all?\"\n\n\"You came here,\" he said. \"You met her. What was I supposed to say? She was right in front of you.\"\n\n\"You were supposed to say, 'This is my girlfriend.' I had to figure it out.\"\n\n\"That would have been better? You would have wanted me to say, 'Oh, yes, this is my girlfriend? Behold her.' How would that have helped? I could tell from the expression on your face that you'd worked it out. I didn't want to make you more upset.\"\n\nHe had a point, sort of. Not the best point. Not a sharp point. But a kind of point.\n\n\"You had days after that,\" she countered. \"There were plenty of times you could have said something. I just needed to hear it from you, that's all.\"\n\n\"Why?\"\n\n\"I don't know,\" she said. \"To make it official or something. I didn't want it to be true, but it would have been better coming from you, because I had . . . hope, or something.\"\n\nKeith hung his head, his hair falling over his profile and hiding his expression.\n\n\"Gin . . .\"\n\n\"You know what's funny?\" she went on. \"I like Ellis. A lot. I'm glad you picked her. Half the time the last few days I liked her more than you.\"\n\nIt took a lot of her to say that, but she felt better once she did. Stronger. She heard him laugh a little.\n\n\"That's fair,\" he said. \"And you know how I feel. When I go old and stark raving mad, we can be in the mad home together. You'll keep sneaking up on me, and I'll keep taking your money, and your teeth. . . .\"\n\nKeith sat down next to her on the sofa and put his arm around her. This time, she knew why it was there.\n\n\"I have to get back,\" she said.\n\n\"I can't offer to drive you home, but I'll walk you there.\"\n\n\"I'm fine,\" she said. \"I should take a cab, anyway. I promised Richard. We're going to dinner.\"\n\n\"I have a number for one.\"\n\nThe cab was fast. He had barely called when there was a honking out front.\n\n\"They're just around the corner,\" he explained.\n\nNo more good-byes. It was time to go.\n\nShe would come back to this house, maybe. Keith was a student. He might move out. But if she did come back, it would never be the same. She would never look up at those black blinds with the same anticipation, or stare into that piece of patterned gold plastic in the door and think when the door opened, she would be greeted with a kiss. All of those little fantasies, so carefully cultivated, were wiped away. Maybe this was what Aunt Peg meant all along\u2014returning was a weird thing. You can never visit the same place twice. Each time, it's a different story. By the very act of coming back, you wipe out what came before.\n\nKeith was watching from the doorway. She had to walk away, chin up. This was hard, but it was not impossible.\n\nThat was the amazing part. It was not impossible.\nThe Probably Problem\n\nIt was midnight. Ginny was in bed, staring at the wall. She wasn't tired. Now that her time in England was almost over, every second seemed precious. So much of this trip had been painful and strange, but she still wanted to hold on to it.\n\nShe got out of bed and looked out the window. There was a purplish ambient light over London, just enough that she could see the outlines and shadows of the things in the neighbor's gardens. She spent a few minutes trying to guess what the objects were\u2014tiny, round chimneys, sheds, bikes, old boxes.\n\nShe was tired of guessing games. She crawled back into bed and pulled out the letter.\n\nyet.\n\nGin, do you remember how I used to drag you to the Museum of Modern Art all the time and make you stand in front of a crazy, huge painting of a woman on a sofa sitting in the middle of a jungle? Think back for a moment. I'll wait. . . .\n\nShe didn't need a moment to think about this. She knew the painting well. It took up an entire wall at MoMA. It was very vivid, with deep greens and yellows. A naked woman lounged on what appeared to be a maroon sofa, while around her, tigers peered curiously through the grass, and a long orange snake wound along the ground. In the background, another woman, half-hidden in the foliage and oversize flowers, played some kind of instrument.\n\nThe painting is called The Dream. It's by Henri Rousseau. Rousseau was always my hero, for many reasons. The paintings, for a start. Their colors. Their strange, childlike perfection.\n\nRousseau was entirely self-taught. He was completely unaware that people saw his style in any way weird. He just assumed that people would want to look at the paintings, that they would be accepted. Critics called his style \"primitive.\" (Some called it worse things than that.) He was considered groundbreaking by people like Picasso, who bought one of his works off the street where it was being sold for scrap canvas.\n\nBy profession, Rousseau was a toll collector in Paris. He never saw a jungle in his life, but he painted them over and over again. He painted the ones he saw in his mind. He placed figures in his paintings, figures that are as big as the landscape.\n\nAs I write this, Gin, I know my mind is dying. I am sitting on a sofa in London, several thousand miles away from you. I am like that woman on a sofa in the painting. I see things around me that I know should not be there. Twice today, I've reached out to pet a cat that I know we don't own, that I know cannot possibly be sitting next to me. But I happen to like my imaginary cat, and when I reach out into what must be empty space, I can feel her fur, and the soft rise and fall of her chest. I call her Probably. As in, \"I probably need to take my meds, because I am seeing the cat again.\"\n\nThe hallucinations aren't always that pleasant, but I've been pretty lucky. And the one thing I see most often, Gin? It's you. You walk up and down the stairs. You crawl in the bedroom window. You sit on the kitchen table. You talk to me all day long when Richard is at work. You tell me about school. We play 20 Questions (and you win). You are all over this house. Honestly, you should probably be paying rent.\n\nOf course, I haven't seen you in two years, so you appear in lots of different ways. Sometimes, you are five years old. Sometimes, you are your actual age. Sometimes, you are forty. Once, you had to be about eighty-five. But it's always you, Gin. There is a single, golden thread that runs through all the Ginnys. As I write these letters and paint my pictures, you are here with me, advising me. Encouraging me. These paintings are your paintings. And as for this last work . . . I can only see it in my imagination, because it's not done yet. I'm so good I can complete a work of art after I'm dead.\n\nI'm fully aware that half the appeal bidders see in my paintings is that this madness that makes me see things will kill me, and I won't paint any more. There is nothing the art world likes more than a dying artist with a limited output.\n\nSome people will say this is junk, that anybody could have messed up a tabletop, screwed up a painted window, and done a rubbing of a stone. Well, sure. But that's the usual argument. Anyone could have dripped paint all over a canvas. But Jasper Johns did, because he knew it was right. Anyone could have painted a can of soup. But Andy Warhol did, because he understood more about modern society than those people. Idea meets execution. Feeling becomes action.\n\nI don't know why people find this idea so hard to get. I mean, you can throw any two people together, it doesn't mean they'll fall in love. Everyone knows this. No one quite understands how it works. It's just those people, where they are in their lives, how circumstance throws them together. Sure, it's happened before, but never quite in that way. Maybe they seem to come together all wrong. Maybe they've loved others. Maybe they don't always do right by each other . . . but it's still there, the love. The event. And no one would dare criticize it just because it's common, it's a little asymmetrical, and anyone can do it. It is unique. It is theirs. It is beautiful. They have made something that has been made a million times before and has also never existed before that moment.\n\nAll right. Is that maudlin enough? When you have cancer, people let you get away with murder, I'm telling you. This is how Nicholas Sparks has been getting away with things for years. Anyway, Probably has come back up on the sofa and wants her head scratched, so I gotta go pet my cat.\n\nI don't want this letter to end, so I won't conclude it. I'll just pause as if I am still writing and\n\nThere was no period. It just stopped in midair.\n\n\"You have to be kidding me,\" Ginny said.\n\nShe set the letter aside and stared at the ceiling. All these weird endings. Keith. Ellis. Oliver storming out of the auction house. One of these stories had to end properly. Who was Oliver, anyway? Why hadn't she taken the time to find out more?\n\nShe grabbed her computer from its spot next to the bed. She could write to him, but he wouldn't reply, that was certain. She would have to look him up. Google turned up a lot of Oliver Davies, but no pictures matched Oliver\u2014at least, not clearly. There were a few large group shots he may have been in, but they weren't really useful.\n\nThere had to be a time where he gave some information. Had she seen his passport? No. Any ID at all? The only time he even had it out was at the train station, when they were going to Paris.\n\nHe bought the tickets to Paris. . . .\n\nThey made you give information when you did things like that. She rolled out of bed to the floor and pulled over her bag. She'd been shoving things in the front pocket. She unzipped it and felt around. Two crumpled napkins, a wrapper . . . and there it was, shoved down at the bottom, the unused ticket. Train number 234 to Paris. Seat number. Reservation number. Contact information for the railway company.\n\nOliver wasn't the only one who could pull off a trick.\n\nGinny was dressed and ready in the kitchen with the kettle on the boil when Richard came down.\n\n\"I didn't expect to see you up,\" he said.\n\n\"I just wanted to do a few things on my last day.\" She passed him a mug of tea, but he waved it off and continued tying his tie.\n\n\"Should we meet at the train station at ten? I may have to leave through a window to get there. January sales. If I'm covered in blood, say nothing.\"\n\n\"See you there,\" Ginny said.\n\nShe had already completed the first step in this plan. She had checked on the availability and obtained an email address on easymail.co.uk. Now she just needed to sound convincing. And a little stupid. Oh, there were advantages to having an American accent\u2014a real one. Nothing could quite replace a real one.\n\nThe phone lines for the rail company opened at seven in the morning, which is exactly when she dialed.\n\n\"Hi,\" she said, dragging out the vowel. \"I bought a ticket for one of your trains the other day, but I never got the receipt? I bought it online and it said they would email it. . . .\"\n\n\"Reservation number?\"\n\nGinny read it off the ticket.\n\n\"Oliver Davies?\" the man asked.\n\n\"No. My name is Olive Davies. My email is olive273@easymail.co.uk. I'm an . . . exchange student?\"\n\nThat was good. Make everything sound like a question. They'd just want to get her off the phone.\n\n\"Just a moment.\"\n\nSound of typing.\n\n\"Oh, I see what happened,\" the man said. \"They put an r in there. They had oliver273. I'll just take that out and resend it.\"\n\nFive minutes later, there was a message. Tickets issued to Davies, Oliver. Credit card number x'ed out. Address: 15A York Road, Guildford.\n\n\"Gotcha,\" Ginny said.\n\nThis time, there would be no loose ends.\nThis Is Not a Pool\n\nGuildford, according to the good folks on the internet, was a town just a few miles outside of London. Market town, suburb, part of something called the \"stockbroker belt,\" home of the occasional music festival. Made famous by the book The Hitchhiker's Guide to the Galaxy as being the home of Ford Prefect, who claimed he came from there as a way of seeming boring and normal and from Earth. There were plenty of trains to get there. Thank god for the English and their love of trains. So many train stations, so many trains at all times. She should have been sick of trains by now, but she wasn't.\n\nThere was a line of cabs waiting at the station, which was larger than she expected. She tried to get in the closest one, but was told to walk all the way to the end of the line, to the first parked cab. Cabs lined up here, and cabbies did not jump their place in line. It wasn't a black cab either\u2014just a regular car with a sign and a meter. The driver set his paper down as she got in.\n\n\"I'm going to Fifteen York Road?\" she said. Why did she do that, with the making things sound like questions? The driver hit the meter and pulled out.\n\n\"What brings you to Guildford?\" he asked. \"You're American, yeah?\"\n\n\"To see a friend,\" Ginny said.\n\n\"Whereabouts in America are you from?\"\n\n\"New Jersey.\"\n\n\"I was in New Jersey once,\" he said. \"I went to a mall there. A big one.\"\n\nThe ride was maybe a minute long. The hardest part was waiting at the lights and getting through the morning traffic. She was deposited at a row of identical yellow-brown brick houses with peaked roofs. On the corner, there was a pub, a fancy-looking mattress store, and a store that seemed to sell nothing but lightbulbs. There was a lot of construction going on in the street. Three of the houses were being worked on, so there was cement dust and bags and planks and wheelbarrows all over.\n\nFifteen A was a house with a half-broken stone wall and a collection of recycling bins in the stone garden out front. There were a lot of cat food cans in the bins. A lot. She knocked on the door, and a moment later, a woman in a pink fleece with her hair tied back in a ponytail answered. The TV was on in the background, showing some morning show, and an ironing board was open and in use. There was laundry hanging everywhere.\n\nThis was definitely Oliver's mom. They had the same dark eyes. Her hair was black with chunks of frazzled gray. She looked very tired, and nervous. She had deep purple bags under her eyes.\n\n\"Can I help you?\" she asked. \"Is this about the bins, because . . .\"\n\n\"I'm looking for Oliver.\"\n\nThe woman looked surprised, like friends didn't just drop by for Oliver very often\u2014certainly not this early and certainly not with this accent.\n\n\"He's at work.\"\n\n\"At work?\"\n\n\"Yes. At the Elephant on the High Street. Just opposite the Oxfam.\"\n\n\"The High Street?\"\n\n\"It's just that way,\" she said, stepping out and pointing. \"Up to that crossroad and make a right. You'll see it right away.\"\n\nHigh Street, Ginny quickly realized, basically meant \"Main Street\"\u2014and the High Street of Guildford was a busy place in the morning. It was entirely paved in cobblestone and set on a hill. They were at the top, looking down at a long street of shops, ending in a narrow river. Every kind of store was represented. There was the McDonald's. There was the Starbucks. The many clothing stores. The drugstores, the bookstores, the stationery store, the supermarket . . . The middle of the street was filled with a variety of carts, selling jam, baskets, dried flowers, curry sauces, fresh roasted pig.\n\nElephant was right in the middle of it all. It looked like a family restaurant, or possibly a coffee bar. It had vaguely African decorations on the walls, and a stridently merry sign in a jungly font. There were big posters advertising world music CDs on the front. He wasn't hard to find. The place was almost empty at this time of the morning. There was one waitress in the back, setting tables, and one tall guy with black hair sitting at the bar, reading a clipboard and making some notes. It was a little strange to see Oliver in a black polo-neck shirt with a bright logo and a pair of khakis. His carefully cultivated image\u2014the thrift-store designer coat, the dress shirts, the posh voice, the constant playing with the cigarettes. It was all busted.\n\nHe must not have been able to hear her come in over the near-deafening music\u2014a chorus of cheerful children and heavy drums. She stood right behind him and tapped him on the shoulder. He jerked in surprise, sending his clipboard clattering to the floor.\n\n\"What are you doing here?\" he asked in a low voice, as he retrieved it. \"How do you know where I work?\"\n\n\"You're not the only one who does their homework.\"\n\nShe had gotten that line ready on the train, and it was extremely gratifying to be able to deploy it so soon.\n\n\"What do you want?\"\n\n\"I want to talk.\"\n\n\"There's nothing to talk about. You shouldn't have come here.\"\n\n\"You brought me here,\" she said. \"You contacted me. You started it. Now I know where you live, and I'm not going to leave you alone until you explain what this was all about.\"\n\n\"Aren't you supposed to be on a plane back to America?\" he asked.\n\n\"In a few hours. If you don't talk to me now, I'll give your address to Keith. I'll tell him where you work. He'll come and visit you.\"\n\nThat got his attention.\n\n\"Wait here,\" he said.\n\nHe went to the back of the room for a moment, had a quick word with the waitress, then reappeared with his coat on. He brushed past Ginny and went outside, the implication being that she should follow.\n\nOutside, he started walking away at a furious pace. They paused next to what looked like a medieval town hall, stark white with black beams and an ornate hanging clock.\n\n\"I told you,\" he finally said. \"I found the envelopes in your bag. I did some research on your aunt's name and read about the sale online. I thought I could make some money off of it. That's the entire story. What are you expecting?\"\n\n\"You could have had the money,\" she said. \"You walked away. Why?\"\n\n\"That's my business.\"\n\n\"And the letters were my business. You found out about me. Now I want to know about you.\"\n\nHe groaned and fumbled for his cigarettes. A crowd of high schoolers passed by, all in uniforms, laughing, talking, clutching coffees and PE bags. The guys were all in blazers and ties. The girls were all in matching blue skirts, with the same ties and blazers. The general effect was that everyone looked older and more dignified and fancier, but they all had that stunned oh-my-god-the-holidays-are-over-and-school-still-exists look on their faces. She would have that look on her face in two days.\n\n\"What good\"\u2014he shoved a cigarette between his lips\u2014\"what good do you think it will do, coming here? Why does it matter why I did it? It's over. I just didn't want to take it in the end. Do you want an apology? I'm sorry. There. Are you happy? Will you leave now?\"\n\nNo. She would not leave now. She didn't know what good it would do, or even what she expected to hear. She just wanted more of something.\n\n\"So, you're a waiter?\" she asked.\n\n\"I'm the manager.\"\n\n\"What kind of restaurant is that?\"\n\n\"It's a fancy McDonald's with hummus. Is this Question Time?\"\n\n\"You read the last letter,\" she said. \"You know how it ends. It just cuts off. I'm sick of things cutting off without any answers.\"\n\n\"Well, then you're in good company. That's how everyone in the world feels.\"\n\nHe continued walking down the hill, through the crowds of shoppers. He was once again doing what Keith had derisively described as the \"dashing\" walk, his coat snapping around his ankles. Ginny had a hard time keeping up, with her suitcase. It was too hard to drag it over the cobblestones, so she had to keep to the sidewalk. He waited for her to catch up at the bottom of the street, once he'd reached the tiny bridge over the small river that ran through the center of the town.\n\n\"The station is that way,\" he said. \"The crossing up by the Friary is tricky, so take care.\"\n\nHe pointed toward another offshoot of shops and restaurants and a very complex intersection. Ginny was not going to be dismissed like this, but she had no idea what came next. It was time to do something, say something. Anything.\n\n\"My aunt would have liked you,\" she said.\n\nShe wasn't quite sure where that statement came from, but she knew it was true the moment she said it. Aunt Peg would have liked Oliver's plan, even if it did involve taking half the money. She would have liked him even more for going to all that trouble and then not taking the money. She would have liked the card tricks and the memorizing. In fact, it was almost like Aunt Peg had planned for him to be there all along. But, of course, she hadn't. Oliver was right. Aunt Peg had nothing to do with the people Ginny met. She attracted people like Oliver all on her own.\n\nTruth or not, the statement did not have the intended effect. Oliver made a rueful little laughing noise.\n\n\"Somehow I doubt that,\" he said, as he walked away.\n\nThere was no point in chasing him again. He didn't want to see her. It was time to face facts. She had to get on a train and go.\n\nThe crossing was tricky, like Oliver said. She had to wait with a group of people for the little green man to appear in the signal box, then stand on a traffic island and cross again. And then again.\n\nShe didn't notice Oliver come up behind her.\n\n\"It's a tricky path to the station,\" he said. \"I'd better show you.\"\n\nHe extracted the bag from her grip and walked on\u2014but slower this time, so she could walk beside him. It turned out that the path to the station was not nearly as straightforward as he had indicated. It wound through a multistory parking lot, past a movie theater, over another footbridge . . . Guildford was much more complex on foot. She would never have found her way back.\n\nOliver was shaking a little. His hands were unsteady. He tried to hide them behind his back, but before he could, Ginny reached out and took his hand and squeezed it. At first, it shook harder; then it slowed. He wouldn't look at her directly; he read the board instead.\n\n\"Your train is on platform five,\" he said.\n\nHe made a halfhearted effort to pull his hand free, but she wouldn't let go. He blinked and rolled his eyes up toward the ceiling. It wasn't a dismissive gesture. It was the kind of thing you did when you didn't want to react or get emotional.\n\n\"We used to have a little bit of money,\" he said. \"It came from my gran. It wasn't a lot, but enough for my parents to send me away to school. Hence the accent. So I don't really know much about how my parents got on when I was a kid. It seemed all right. Not great, but all right. When I went away to university, my dad walked out.\"\n\n\"Sorry,\" Ginny said.\n\n\"He's a bastard,\" Oliver said plainly. \"I don't care about him. It was my mum I was worried about. When she called me, she was completely falling apart. So I left for a few days to take care of things. Then a few days became two weeks, and then I had accidentally left university for the year. It was all supposed to be temporary, but . . . I didn't go back again this year.\"\n\n\"That's why you're not in school?\"\n\n\"That's the reason,\" he said. \"My dad handled the money. My mum has never been good at that sort of thing. She was a bit broken; she couldn't handle the bills and the paperwork coming in. So I came back and got the job at Elephant. She didn't want me to. She felt bad about my leaving uni, but . . . it had to be done. Anyway . . . around October, I couldn't really take it anymore, so I took a little of the money I saved and went off to Greece for a few weeks to think about what I wanted to do next. My bag broke, and I wound up getting yours. Then I found the letters. They were just so weird . . . I started looking up the stuff I read in them. They gave me some focus. That's when I found out about the sale. And I could see that you'd never opened the last one, and there was no record of a last piece like the one described, so there was something else out there. I figured if you had already made a bunch of money off the first sale . . . it wasn't like I was taking from someone who was broke. Then I could give my mum enough money to help her get by, and I could go back to school. But the only way I could do it is if I never blinked. So I tried. I tried to do it and be honest and up front about it.\"\n\nA crisp voice announced the impending arrival of Ginny's train. Time was very short.\n\n\"If you'd told me that,\" she said, \"I would have gone along with it. You wouldn't have had to do all this. I would have just given you some of the money.\"\n\n\"I know that. At least, I do now. And that's what makes it worse. It actually made it a lot easier that your friend came with us and was such a div to me. But you weren't.\"\n\nGinny unwillingly released his hand. It fell to his side. He was looking at her now.\n\n\"So now you know,\" he said. \"Are you satisfied?\"\n\n\"Not yet,\" she said.\n\nShe reached out and took his hand and squeezed it. At first, it shook harder; then it slowed. She mattered to him. He was afraid.\n\nShe let go of her bag, stepped up on her toes, and kissed him. Hard. Egregiously and in public and right in front of the people trying to get out of the Costa Coffee. She wrapped her arms around him.\n\nKissing . . . real, proper kissing . . . makes you lose all sense of place and time. She was vaguely aware that they were blocking the people trying to move around them, that a group of little kids were making comments, that Oliver had scooped her up and was more or less supporting her so that she didn't have to strain to meet his height. This was one of the real ones that actually do make you feel like your body is going completely gelatinous, and you like it.\n\nA peevish second announcement snapped her back to reality. It reiterated that the train was coming. It would be pulling in any moment. Snide, interrupting announcement. There wasn't time to say good-bye. Ginny reached around, trying to find the handle of her bag as Oliver released her and they slowly pulled their lips apart. She was dizzy, so it was hard to get her ticket into the little opening. In fact, she forgot that once she put the ticket through, Oliver couldn't come with her, so they were immediately separated by a metal bar. When she turned, he was doing something she had never seen him do before.\n\nHe was smiling.\n\nGinny took her place on the train, her head still spinning. It was crowded, full of people commuting into London for work, or maybe to enjoy the last day of the holiday. She sat across from a group of moms with their daughters, maybe ten or eleven years old, on a big day out. They were talking about going to a movie and discussing where they wanted to have dinner that evening. She wasn't entirely sure what had just happened or what it meant, she only knew she wasn't ready to go. She wasn't ready to leave everyone behind\u2014Keith, Richard, Mari, Oliver. Even these little girls on the train. She didn't want to leave them either. They were all a part of one large picture, a picture Ginny loved.\n\nBut the train sped on efficiently through the bright and sunny morning, past woodlands and towns and fields with horses. It was taking her into the heart of London, and from there, another train would take her to the airport, and a plane would take her home.\n\nHome. It was a nice thought. She had missed her parents, her friends . . . but the word didn't have quite the same meaning anymore. England was home too. So much of her was here.\n\nThe Question floated back into her mind. It had been lingering in the background, waiting for the right moment to make its presence felt. Like Oliver, she could also memorize things that were important (though this was only two sentences, which was a little bit easier). Describe a life experience that changed you. What was it, and what did you learn? (1,000 words)\n\nMaybe now she had her answer, and it was one word: England. Sure, that didn't convey a lot of information, but these people had no right to pry into her personal life. They were college admissions officers, not therapists. It's not like they cared what her answer was. They just wanted to see if she was a functional student who could write three pages that made some kind of sense. She wasn't going to tell them the truth, that she wanted someone to block her path. She wanted this train to break down, for her flight to be canceled, for immigration to tell her that she wasn't allowed to go. She wanted London itself to rise up and refuse to let her pass out of its boundaries.\n\nAnd then, just as the snack cart came by and accidentally whacked her on the elbow, she realized exactly what it was she had to do.\n\nOver the course of fifteen minutes, somewhere between Clapham Junction and Waterloo Station, she figured out her future. Not the details . . . there was no point in thinking about details. Just the basics. She mentally paged through the calendar and did some math. The deadlines would probably be tight. There would be a lot of research and work to be done very quickly, but she had done harder things.\n\nRichard was waiting for her at one of the coffee shops that lined the Victoria Station, doing a crossword puzzle. Ginny rushed toward him, and he looked up with a smile and a wave.\n\n\"You seem ready to go,\" he said.\n\n\"Well,\" Ginny replied, sliding into the seat opposite. \"About that . . . what do you know about applying to universities here? And, like, having me around. For a few years. Not in your house, but, you know . . .\"\n\nIt took Richard a moment to take in what Ginny was saying. He looked at his coffee thoughtfully and turned the cup a few times.\n\n\"I think,\" he said, \"that if that was what you planned to do, then we could get that information. And that you'd always have a place to stay, even in my house, for as long as you wanted.\"\n\nNow it all made sense. It wasn't like everything was fixed now, or every plan made, every answer in hand . . . but now there was a shape to things. And as they walked to the airport train and Ginny took a last look around the station, she knew it would be easier to say good-bye this time. It's always easier to say good-bye when you know it's just a prelude to hello.\nAcknowledgments\n\nFirst, I want to thank everyone who's written to me or otherwise contacted me since the publication of 13 Little Blue Envelopes, asking me to write a second book. It's because of you that this story exists.\n\nI have the good fortune to know many wonderful people. The following is just a partial list of people who are owed thanks:\n\nThanks as always to my agent, Kate Schafer Testerman. Without Kate, I would certainly have met some unlikely fate by now, like being accidentally shot off into space. To my editor, Zareen Jaffery, who guided this book with a clear and steady hand. Holly Black, Cassie Clare, Libba Bray, and Robin Wasserman, for all their help developing the story and generally listening to me chatter away. Sarah Rees Brennan, who hosted me for several days in Ireland and broke into that swimming pool with me. (\"We didn't realize it was closed. It was just dark, and, you know, when we forced the door, it opened, and then we just took the cover off. Of course these are bathing suits. . . .\") Justine Larbalestier, Chelsea Hunt, and resident English guy Andy \"Weasley Is Our King\" Friel, all of whom read the book and gave me extremely helpful notes. To John and Hank Green, who introduced all of Nerdfighteria to Ginny and the crew.\n\nMore thanks to . . .\n\nScott Westerfeld, for the fob. Jason Keeley and Paula Gross, who feed me regularly. Rebecca \"Lone Star\" Leach, my de facto assistant, who does my strange bidding. Tobias \"I love John Barrowman\" Huisman, my pointman in The Netherlands. Donal Finnerty, Ireland's leading expert on fridges. The Accio Books crew: Maria Alexandra Flores, Luz Maria Flores, Alida Gene Sara Priest, Kerstyn Smith, Sophia Arnold, Hannah Gann, Jennifer McCall, Carissa Crossett, Marcus Walton, Megan Sprimont, Rachel Belanger, Jessie Johnson, Lois Carlyle, and Marlee Grace Abbott.\n\nAnd to Hamish Young, who was born in the forest and knows all about badgers.\nAbout the Author\n\nMAUREEN JOHNSON is a New York Times bestselling author whose novels include SUITE SCARLETT, SCARLETT FEVER, GIRL AT SEA, 13 LITTLE BLUE ENVELOPES, and THE KEY TO THE GOLDEN FIREBIRD. She lives in New York City, but travels to the UK regularly to soak up the drizzle and watch English TV. She is constantly online at www.maureenjohnsonbooks.com, or on Twitter @maureenjohnson.\n\nVisit www.AuthorTracker.com for exclusive information on your favorite HarperCollins authors.\nALSO BY MAUREEN JOHNSON\n\n13 Little Blue Envelopes\n\nThe Key to the Golden Firebird\n\nGirl at Sea\n\nThe Bermudez Triangle\n\nDevilish\n\nSuite Scarlett\n\nScarlett Fever\n\nLet It Snow (with John Green and Lauren Myracle)\nCopyright\n\nThe Last Little Blue Envelope  \nCopyright \u00a9 2011 by Maureen Johnson\n\nAll 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\n* * *\n\nLibrary of Congress Cataloging-in-Publication Data\n\nJohnson, Maureen, 1973-\n\nThe last little blue envelope \/ Maureen Johnson. \u2014 1st ed.\n\np. cm.\n\nSequel to: 13 little blue envelopes.\n\nSummary: Seventeen-year-old Ginny Blackstone precipitously travels from her home in New Jersey to London when she receives a message from an unknown man telling her he has the letters that were stolen just before she completed a series of mysterious tasks assigned by her now dead aunt, an artist.\n\nISBN 978-0-06-197679-7\n\n[1. Voyages and travels\u2014Fiction. 2. Swindlers and swindling\u2014Fiction. 3. Letters\u2014Fiction.   \n4. Artists\u2014Fiction. 5. Aunts\u2014Fiction. 6. Interpersonal relations\u2014Fiction. 7. Europe\u2014Fiction.] I. Title.\n\nPZ7.J634145Las 2011\n\n[Fic]\u2014dc22\n\n2010033580  \nCIP  \nAC\n\n* * *\n\nEPub Edition \u00a9 2011 ISBN: 9780062077134\n\nVersion 12282012\n\n11 12 13 14 15 LP\/RRDB 10 9 8 7 6 5 4 3 2 1\n\nFirst Edition\nAbout the Publisher\n\nAustralia\n\nHarperCollins Publishers (Australia) Pty. Ltd.\n\n25 Ryde Road (P.O. Box 321)\n\nPymble, NSW 2073, Australia\n\n<http:\/\/www.harpercollins.com.au>\n\nCanada\n\nHarperCollins Canada\n\n2 Bloor Street East - 20th Floor\n\nToronto, ON, M4W, 1A8, Canada\n\n<http:\/\/www.harpercollins.ca>\n\nNew Zealand\n\nHarperCollins Publishers (New Zealand) Limited\n\nP.O. Box 1\n\nAuckland, New Zealand\n\n<http:\/\/www.harpercollins.co.nz>\n\nUnited Kingdom\n\nHarperCollins Publishers Ltd.\n\n77-85 Fulham Palace Road\n\nLondon, W6 8JB, UK\n\n<http:\/\/www.harpercollins.co.uk>\n\nUnited States\n\nHarperCollins Publishers Inc.\n\n10 East 53rd Street\n\nNew York, NY 10022\n\n<http:\/\/www.harpercollins.com>\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n## THE LAMENTATIONS OF ZENO\n\n# THE LAMENTATIONS  \nOF ZENO\n\n## ILIJA TROJANOW\n\n### _Translated by Philip Boehm_\n\nThe translation of this work was supported in part by a grant from the Goethe-Institut, which is funded by the German Ministry of Foreign Affairs, and by an award from the National Endowment for the Arts.\n\nFirst published in English by Verso 2016\n\nTranslation \u00a9 Philip Boehm 2016\n\nFirst published as _EisTau_\n\n\u00a9 Carl Hanser Verlag 2011\n\nAll rights reserved\n\nThe moral rights of the author have been asserted\n\n1 3 5 7 9 10 8 6 4 2\n\n**Verso**\n\nUK: 6 Meard Street, London W1F 0EG\n\nUS: 20 Jay Street, Suite 1010, Brooklyn, NY 11201\n\nversobooks.com\n\nVerso is the imprint of New Left Books\n\nISBN-13: 978-1-78478-219-1\n\nISBN-13: 978-1-78478-222-1 (US EBK)\n\nISBN-13: 978-1-78478-221-4 (UK EBK)\n\n**British Library Cataloguing in Publication Data**\n\nA catalogue record for this book is available from the British Library\n\n**Library of Congress Cataloging-in-Publication Data**\n\nNames: Trojanow, Ilija, author. | Boehm, Philip, translator.\n\nTitle: The Lamentations of Zeno : a novel \/ by Ilija Trojanow ; translated by Philip Boehm.\n\nOther titles: EisTau. English\n\nDescription: Brooklyn, NY : Verso, 2016.\n\nIdentifiers: LCCN 2015039884| ISBN 9781784782191 (hardback) | ISBN 9781784782221 (US e book) | ISBN 9781784782214 (UK e book)\n\nSubjects: LCSH: Glaciologists\u2014Fiction. | Glaciers\u2014Fiction. | Political fiction. | BISAC: FICTION \/ Political.\n\nClassification: LCC PT2682.R56 E4713 2016 | DDC 833\/.92\u2014dc23\n\nLC record available at <http:\/\/lccn.loc.gov\/2015039884>\n\nTypeset in Electra by Hewer Text UK Ltd, Edinburgh\n\nPrinted in the US by Maple Press\n_At each slow ebb hope slowly dawns that it is dying_.\n\n\u2014 Samuel Beckett, _Company_\n\n## Contents\n\nTranslator's Note\n\n1. 54\u00b049\u20321\u2033S, 68\u00b019\u20325\u2033W\n\n2. 55\u00b005\u20320\u2033S, 66\u00b039\u20325\u2033W\n\n3. 53\u00b022\u20325\u2033S, 61\u00b002\u2033W\n\n4. 51\u00b041\u203237\u2033S, 57\u00b049\u203215\u2033W\n\n5. 53\u00b011\u20328\u2033S, 45\u00b022\u20324\u2033W\n\n6. 54\u00b016\u20328\u2033S, 36\u00b030\u20325\u2033W\n\n7. 60\u00b011\u20325\u2033S, 50\u00b030\u20322\u2033W\n\n8. 62\u00b012\u20329\u2033S, 58\u00b056\u203243\u2033W\n\n9. 62\u00b058\u20329\u2033S, 60\u00b033\u20326\u2033W\n\n10. 62\u00b035\u20320\u2033S, 59\u00b056\u203230\u2033W\n\n11. 64\u00b050\u20323\u2033S, 62\u00b033\u20321\u2033W\n\n12. 64\u00b027\u20321\u2033S, 62\u00b011\u20325\u2033W\n\nAcknowledgments\n\nOn the Typeface\n\n## _Translator's Note_\n\nWith _The Lamentations of Zeno_ , Ilija Trojanow charts new territory in prose as well as geography. Not a native speaker of German, he has adopted that language and adapted it to his own purposes, taking full advantage of its lexical fecundity, creating words at will, and of its suspended syntax, with which he unleashes whole currents of consciousness. Alternating painterly descriptions of the natural world with cacophonic passages composed of song snippets, adspeak and \"breaking news,\" he contrasts the majestic stillness of the Antarctic with the clamor of human \"civilization.\" And all of this is framed within a confessional log that allows the reader to reconstruct the emotional course of the troubled protagonist.\n\nThe sheer range of registers is impressive\u2014and quite a challenge for the translator. The title itself is a case in point: A literal \"IceThaw\" not only lacks the \"aura\" of the original _EisTau_ , it also fails to convey the layers of meaning lurking in the German. \"Melting Ice\" seemed a bit lackluster, while \"Meltdown\" was more appropriate for any number of TV movies. Instead we decided to focus on the narrator who is the soul of the novel.\n\nMy primary task in translating the book has been to recreate the voice of Zeno Hintermeier\u2014his gruff demeanor, deprecating self-irony, bone-dry wit, and great erudition. To this end I have broken up single-sentence paragraphs and recast them with somewhat shorter sentences easier on Anglophone ears. Otherwise punctuation remains light, echoing the German, although quotation marks have been added to set off some speech. Songs cited in the shorter bricolage passages have been substituted with popular English lyrics from the same period, and elsewhere I have similarly opted for equivalence over literal rendering.\n\nMost of all I hope to have captured the deeper musicality of the prose, by paying as much attention to the rests as to the notes. Because for all his linguistic virtuosity Trojanow is equally a master of the unsaid, so that the words on the page are like the icebergs themselves\u2014a sparkling intimation of what lies below.\n\n## THE  \nLAMENTATIONS OF\n\n### 1.\n\n## 54\u00b049\u20321\u2033S, 68\u00b019\u20325\u2033W\n\nTHERE'S NO WORSE nightmare than no longer being able to save yourself by waking up.\n\nWhenever we set sail from Ushuaia, we gather the evening before in one of the local dives that's a little ways uphill and off the main streets, just when the last band of light is slipping from the sky. We haven't seen one another for half a year, so we're in the mood to celebrate as we crowd around a long wooden table. The man waiting on us is old, and judging by his face not very adventurous, although at one parting he confessed to me that he was getting along well apart from an occasional urge to puncture his hand with a knife. His place doesn't have much on offer, but he'll fill your glass for very little and I'm content to sit here holding my drink, surrounded by the hardworking Filipinos that make up most of the crew, now smiling broadly at our reunion. Every payday brings them closer to settling down to a home and the sheltering shade of a large family, and so they soldier on, slogging through their working days with an astounding ease. For me they will always be an enigma. Ushuaia is incapable of dampening their mood, as is any echo of the butchery, any painful reminder of the past\u2014their ears are simply not tuned to that frequency, that legacy belongs to Europeans, those are the scars of the white man. They drift through this place just as they do through all the other places that have been defiled, all our ports of call (what a pretentious phrase from some liturgy of advertising), seeming not to touch the ground when they go ashore. That is what separates us, we have no common past: what paralyzes me seems to fill them with life. Apart from that, they're \"easy to handle,\" as our onboard hotel manager never tires of repeating (by which he means: much better than the unruly Chinese), as if he had personally trained them to be so diligent so patient so tame. The Filipinos' zeal would bother me were it not for Paulina, who at this moment is probably busy giving a personal touch to our shared cabin, equipping it with artificial flowers and photographs depicting an entire menagerie of relatives\u2014the numerous grandmothers perched in front on dilapidated rattan armchairs dragged into the garden just for the occasion, and standing behind them all the daughters and sons, loyal to a man except for the one who ran off and is rumored to be chopping vegetables in a New York restaurant. I raise my glass to Paulina's countrymen\u2014mechanics, cooks, pilots\u2014and to Ricardo, our dining room manager, as unobtrusive as a shrink-wrapped suitcase, but watch out, his true power will be revealed during the course of the trip, every passenger will get to know him and a few will appreciate him (\"Howzit going, Mr. Iceberger?\" he says, giving me a thumbs-up, always concerned to clear potential misunderstandings out of the way before they happen). It's a sight for the gods, the way the millionaires from the northern hemisphere line up in front of his desk, eagerly bowing as they slip him an envelope to thank him for the coveted starboard table with a box-seat view of ice floes and leopard seals. My recent years at sea have taught me that rich people are prepared to pay considerable sums for little privileges. That sets them apart from the masses, feeds Ricardo's confidence, and finances the expansion of his guesthouse in Romblon. He's no more interested in fur seals, leopard seals or penguins than he is in glaciers or icebergs, but he takes advantage of every scenic opportunity\u2014\"What a view, fantastic, fantastic, please take your seats,\"\u2014as he parades his teeth in a broad grin. I'm sure he'd squeeze in just as many \"fantastics\" in front of a garbage depot as long as there were people willing to pay for a premium seat. All he really cares about is whether something is sellable or not. Whenever we're all together he flirts with the blonde whale lady now sitting to his left, always resorting to the same lines, which he polishes like a fingernail, \"You know some day I'm going to sit in on your lecture, I mean it, I really want to learn all about these fish, now that I've watched them from the restaurant and seen them spouting, they really are very beautiful creatures\"\u2014but when it comes to the beautiful Beate he has a hard time understanding why she prefers whales to people, which is why he's going to sit in the first row during one of her next lectures and write down every single word she says. He promises this before every trip, when we're gathered at the long wooden table that's pitted and scored with random dents and notches. \"This time I mean it,\" he says, \"I swear to heaven\"\u2014and the whale lady pinches his arm. She speaks English with a German accent, German with a hint of Spanish, and Spanish with Chilean intonation. Despite his assurances, nothing will come of Ricardo's \"cetacean education.\" But what he will do for certain at the end of the trip is pass a chef's hat around on behalf of the men in the kitchen, while they line up in front of the curved buffet and perform a song in Tagalog that sounds like the \"Hymn to the Unknown Server\" and is always received with thunderous applause.\n\nThe experts aboard the MS _Hansen_ are also at the table, the lecturers tasked with educating the vacationers, just like I was doing for three years until yesterday when the captain summoned me just after my arrival and told me the expedition leader had been taken unexpectedly to the hospital in Buenos Aires with a suspected case of swine flu, there was no way he'd be able to join us at this point, at best we might be able to pick him up along the way, further down the Beagle Channel, but until then a substitute is needed, and he believes I have the necessary competence, I know the subject, I'm engaged, and I am worldly-wise (here his glance implied I also might be inclined to overshoot my target on occasion), and apart from that I have a lot of experience on board the ship. I neither wanted to agree with his assessment nor decline the offer, so I took the folder with the instructions. From now on I'll be spending far too much time using the radio and the PA system, keeping the passengers up to date on the weather, the route, our next destination. Each of the lecturers has a special field of expertise\u2014oceanography biology climatology geology\u2014and each of us knows how to talk about animals clouds cliffs both instructively and entertainingly. And each of us is a refugee in his own weird way: in the words of El Albatros, our Uruguayan ornithologist, \"We're really just a bunch of nowhere people.\" He nods my way, \"Mr. Iceberger\"\u2014he calls me that too, some of them have never used my real name, Zeno, and others aren't sure how to pronounce it, whether Zen-know or Zee-no or Say-no (this from the mouth of our whiz kid Jeremy from California, who could practically be my grandson). These are minor matters to which I attach no importance, but I have the sneaking suspicion my colleagues use the nickname to disguise their belief that I'm some kind of misfit or freak. And it is more than a bit bizarre to be considered too passionate by people so passionately dedicated to their own pursuits.\n\nEarlier that day Beate escorted a group of passengers to the national park, where paths wind along the coves and the sun's rays angle down and settle on individual leaves like butterflies. We've all taken this easy walk through the virgin Patagonian forest at one time or another, but this year they've opened a new path and the ever-conscientious Beate has no intention of letting some tourist embarrass her by knowing more than she does, even if it's just about a few new dots on a map and a path to a further cove. For that reason, as she explains at length, she took a bus past the southernmost golf course in the world, past the end of Pan-American Highway, to a wide, leveled parking lot where people land in the middle of nature like so many aliens, and where a small stairway of freshly treated wood leads up to the path.\n\n\"How many whales did you spot?\" Ricardo asks jokingly.\n\n\"One,\" says Beate.\n\n\"Just one whale, how is that possible? A loner? A juvenile?\"\n\n\"A beached whale very much on land and made of stone with moss on top. A whale that children are allowed to ride on.\" Beate pauses. \"It's sitting out there like a memento mori.\" She pauses even longer. \"The thing looks massive enough that it might actually stand the test of time, too. The new path has trashcans every two hundred meters and benches every two hundred meters: trashcan bench trashcan bench as you go stalking through the forest. Our guide was a creep in tall boots, a Buenos Aires _porte\u00f1o_ who likes the notion of spending his summer in the fresh air of the south, he spoke in a high falsetto voice as if to balance out his very low level of knowledge, he talked about the original inhabitants as though they were wild animals, didn't even mention their name, 'grass-chewers' is what he called them, and he made stupid jokes like 'we don't know much about them, they were so shy, the minute they caught sight of a person they turned tail and scrammed and if you tried to approach them they hid deep in the bush like hedgehogs or burrowed underground like skunks.' I couldn't help myself, I had to teach him a lesson in front of the passengers: the people who once lived in this forest were called _Yah-gan_. He repeated the word as if he had to crack it open, ' _Yah-gan_ \u2014well that fits a primitive natural tribe just about as well as a fist fits an eye, it sounds foreign, like some strange species of spider.' Did I mention his boots? They left deep prints, and some name or other, probably that of the manufacturer, got stamped into the earth with every step he took. Primitive natural tribe, can one of you tell me where a phrase like that comes from?\" Beate stops speaking, and a hush falls over the table as though by some prearranged signal. Not everyone heard the question, but the answer will spread across the entire table.\n\n\"Because we wiped them out,\" I say in a loud voice. \"Because we destroy everything aligned with nature. We honor the people who are extinct, we put their masks on display and print their portraits in sepia, we show enormous care and devotion to those we have exterminated.\"\n\nThe lecturers all start to sigh\u2014here he goes again, they're all expecting one of my diatribes, I've subjected them to an avalanche of rage on more than one occasion. They know from experience that whenever Mr. Iceberger waxes apodictic things will end apocalyptic. But it's our first shared evening, so I bite my tongue and say nothing more as other conversations begin to rustle around the table.\n\nAll the others leave but me, I stay behind with the old man who spent the whole evening waiting on us in silence. That's become a custom with the two of us, since the first time I sought him out. I had left my camera on one of the wooden benches in his bar and walked back through the cold. It was very windy and I was nearly frozen when I went inside, the old man was cleaning up, all by himself, he had to fix something to get me warm and on top of that grant me a conversation, which at first made us feel even more like strangers, but then sentence by sentence, shot by shot, we let down our armor to the point of showing our wounds. Since then we've never been far from each other in our thoughts. He quietly wipes down the table with circular motions, the veins on his hand look like glacial striations, his skin shows a number of liver-brown patches. With implacable anger he curses his fate to be born, grow up and grow old here in Ushuaia, which has always been a makeshift place, where every shop is called _Finisterre_ and every apron is plastered with penguins, to live in this spot that shows no pity for anyone, not for those who once roamed barefoot over thorns until they were killed by adventurers seeking their fortune or by civil servants fallen into disfavor, not for those banned to the penal colony in heavy chains, whose yearning to escape cut deeper and deeper into their flesh and not for their descendants who grovel before the tourists as if they wanted to pick the dried bits of mud off their boots, as if the earth of Tierra del Fuego still contained gold dust. Does a place change for the better when people move away of their own accord? Does peat that has been drenched in blood still spread warmth when it burns on a homey hearth? The old man disappears for a moment and returns with two bulbous snifters of a drink that smells like vanilla and leaves a nice burn in the throat. The old man doesn't stop moving, from the counter to the tables, from one table to the next, as if something needed to be tied down in every spot. I follow him to the window, the sparsely set streetlamps blur in the drizzle into muted trickles of light. We stand for a moment listening to the distant sounds. Suddenly he resumes speaking.\n\n\"As a child I used to spend the afternoons hanging around in front of our house\u2014this place used to be our shack\u2014and I'd gaze down at the town. Sometimes, when the sky was so overcast the clouds seemed to touch the ground I had the feeling the street might vanish with the fog. I'd race down the street full of expectation but every time I just landed in the filth of the harbor.\"\n\nFinally we sit down, and he refills our glasses as if there were plenty in stock. Long sentences of silence are punctuated by his pronouncements:\n\n''Whoever tries to live an honest life in this place is punished with a shot in the back of the neck.\"\n\n''We paid homage to my murdered grandfather in fearful silence.\"\n\n\"My mother warned me about men in uniform the way other mothers warn their children about mean alley cats.\"\n\nSuddenly he turns to me, looks me in the eye and says,\n\n\"You're setting off again and once again you're letting everything happen. You're defiling your own temple.\"\n\nHe rubs his hand across his face, his beard.\n\n\"I've been watching you. You're nothing but talk. Your indignation is a fart. You let off steam, you shoot your mouth, but otherwise you're just like all the others, no, you're worse, because you understand, and you sell your knowledge for a few pieces of silver.\"\n\nI don't contradict him, and that makes him even more furious.\n\n\"Anyone who just accepts what can be avoided is a scoundrel.\" He's practically screaming. And then he shows me the heavy door.\n\nAs if I were half made of moraine, that's the nightmare I have every night.\n\nTomorrow the passengers come on board. Day one: embarkation. A day like any other. We have yet to set sail. The impending departure makes me uneasy, I wasn't born a sailor, on the contrary, my home was the mountains, before I was chased away. The first time I saw the ocean was at the mouth of a glacier, the glacial tongue was practically licking the beach, the glacial stream was running ahead of me, I was in my early twenties and confident, so confident I purposely got lost in the rainforest between the beach and the glacier. Today that glacier has melted away and its phantom tongue visits me in my sleep to mock me and I am powerless against my nightmare's minions. Paulina is already slumbering away, she falls asleep quickly, more quickly if we've made love. Tomorrow we set out. One more tour. My fourth year.\n\nIt is written.\n\nWe let ourselves be comforted by demeaning sentences such as this. Nothing is written: it is being written. By every single one of us. Just like everyone contributes his little bit to all the poisoned ruins on this earth. Hence this notebook, hence my decision to describe what has happened, what will happen. I shall be the word-keeper of my own conscience. Something has to happen. There's no time to waste.\n\nMeasurements to die for check her out I tell you, nobody gives a hoot, you can kiss it goodbye, grab yours now while supplies last. Sir, we're picking up a distress signal on 406 MHz. Screw up your courage, measurements to die for, you'll be licking your lips afterward, thirteen months of sunshine, welcome to paradise, and it rained every single day. Distress beacon? Yes sir. Which ship? It isn't clear, sir. The chapel's been closed for a week now, the frescoes are being renovated, no it won't open again until the fall, I'm sorry you've come all this way in vain, we can't allow ourselves to be pressured, a question for your guest, all you have to do is switch a few letters and contribution becomes retribution, something's stuck, something always gets stuck. I have a fix, sir at 43\u00b022' S and 64\u00b033' W. All ravens are black, I've had it up to here, the apparent temperature was higher, measurements to die for, she'll make better speed on the lee side, cut to the chase, it's a done deal. Something isn't right, sir, we've lost radio contact with the _Hansen_. What about their radio officer? He's not responding, sir. Hey hey hey get your fingers off it's my turn, the bra's all mine, ok Charlie hold your breath, ready one two and come on damn hooks got snagged, there'll be better days to come. Radar? Moving north-northwest. You've tried all channels? Yes, sir. Keep trying, I'm calling the Argentine coast guard. I'd like to ask your guest a question, if I understand correctly we're all either going to heaven or else to hell, but one way or other we're all going somewhere, so does that mean we're essentially immortal? Prefectura Naval Argentina? S\u00ed... s\u00ed... Her last position given was 54\u00b049' S 68\u00b019' W, no we haven't had any contact with the _Hansen_ since then. No doubt about it, they're going to shoot that bird down, don't take it so to heart, go ahead and catch your breath, take a deep breath, measurements to die for, we're doing what we can, and there is something we can do BREAKING NEWS ACCIDENT IN THE ANTARCTIC? BREAKING NEWS ACCIDENT IN THE ANTARCTIC? and nevertheless\n\n### 2.\n\n## 55\u00b005\u20320\u2033S, 66\u00b039\u20325\u2033W\n\nBEFORE WE PUT out to sea, all passengers must prove they are healthy (not in perfect health, just healthy enough). They make their way upstairs or downstairs to Deck 4\u2014those with medical restrictions use the elevator\u2014and line up in front of the dapper Brazilian doctor with his vaunted head of curls. The man looks incredibly taut and trim in his uniform, the result of spending every free minute in the coffin-sized fitness room, listening to heavy metal from S\u00e3o Paulo, his eyes fixed on the emergency exit. I've never been able to talk to him. All those declared fit proudly brandish their medical clearance like a coveted concert ticket, they introduce themselves, exchange ideas, they've been to this place and that, well we're generally up for anything, but the heat, and what about the insurgents, on the other hand, there are so many places we'd like to see it's impossible to choose where to go next, of course first we have to survive this little adventure. For present purposes everyone is healthy enough, though they might be mere heartbeats away from a coronary or a stroke.\n\nWe set sail at last light. Our leave-taking is perfunctory: no one waves goodbye, not from the deck and not from the dock. Virtually no one is staying behind in Ushuaia, at least no one we'll miss. I enjoy standing on the topmost deck, musing over the silhouettes. I loathe sunsets because they reduce such a wealth of possibilities to a single effect. No one tries to engage me in conversation, the guests don't know me yet, neither the lecturers nor the expedition leader will be introduced until tomorrow after breakfast. We cast off without fanfare and are heading east through the Beagle Channel at approximately seven knots, according to my estimation, which after a few seasons on board is fairly accurate. Take a look at that cliff, a passenger calls out, like a giant with a sixpack. The group pounds its laughter into the twilight. And once again out come the camcorders and once again nature is diminished in the zoom of their lenses. I retreat to the port side.\n\n\"Hey there! Don't run away!\" The pianist is heading straight toward me, in his wake a visage emerges from the darkness, and beneath the festive lights I am able to make out a less-than-shipshape face. \"Allow me to introduce you. Mrs. Morgenthau, this is our new expedition leader, a perfect model of a gentleman\"\u2014the pianist is British\u2014\"I'm certain he can answer your question satisfactorily, our expedition leader can provide a satisfactory answer to any question.\" Our pianist passes for a great wit.\n\n\"That's kind of you, very kind, as a matter of fact I was just asking about that mountain, it's so dramatic the way it simply rises up there, surely it has a name?\"\n\n\"Mount Misery.\" My answer is geographically correct, but the American's disbelieving look is virtually an accusation. The pianist grins at his well-played practical joke.\n\n\"Please, Madam, don't be shy about asking the expedition leader whatever you fancy in the course of our passage, whenever you feel the need. As for the evening concert I myself am happy to take requests, otherwise I just plunk along, as I'm sure you'll hear.\"\n\n\"The people who used to live in this region,\" I continue, \"were aquatic nomads, they had many names for the mountains, the rivers, the forest. They developed a rich vocabulary for describing what surrounded them without wanting to possess it. Their name for this strait was something like 'the water that stabs through the twilight.'\"\n\n\"And the island we just passed, you know the one I mean, I'm sure it has a magnificent name as well?\" the pianist asks. He knows the answer but I do him the favor and play along.\n\n\"It's called Fury Island.\"\n\nAnother mistrustful look.\n\n\"Yes, of course, Fury Island, and here I'd done such a nice job of suppressing that one. Come, my dear, let's not trouble our expedition leader any longer, incidentally I ought to warn you now so you don't find out from some unreliable source that sometime in the middle of the night, when I trust we'll all be sound asleep, our ship is going to sail past Last Hope Bay.\"\n\nHis laughter rises and floats away like steam from an exhaust pipe.\n\nThat first evening Paulina's shift ends before midnight. The guests haven't had any time to get to know one another, the habitual drinkers and loungers quit the bar and bistro earlier than usual, Paulina hastens the last call and ushers an elderly American to his bed, she's looking forward to our more spacious cabin (befitting the post of expedition leader), and to me as well, we haven't yet had a chance to celebrate our reunion. I've been promoted to Deck 6, where the commanding elite is housed, my cabin is next door to the first officer and the chief navigator, and not far from the bridge, in fact a little earlier I stepped into the passageway and ran smack into the captain, whose office and refuge are just a few doors down and across the hall.\n\n\"Our captain's literally within striking distance,\" I tell Paulina.\n\n\"Well don't hurt him,\" she laughs. And I laugh back. I'm always amazed at how easily we make each other laugh, time and time again. I used to be thought a killjoy, and with good reason: what others considered hilarious I found obnoxious, I would hear them snicker and cluck, but never really laugh, my former spouse used to giggle and cackle her way through an evening, she could be loud and brash, but there was no real exuberance, no jollity. Paulina on the other hand gets me to laugh as she's taking off my clothes, stripping me down to a pure good mood. For her, laughter and libido are very much intertwined.\n\nWhen our bodies have been so long apart, rediscovery follows conquest, and in between forays she lies next to me, feet crossed, her vulva gently vaulted, and chatters away as I listen to her lapping voice. It's the most calming human sound I know. So much has happened in the months we've been apart, a waterfall of events, the Mayon volcano erupting, the neighbors' child and his cleft lip surgery, the massacre of more than two dozen journalists on the next island over, the old fisherman whose right hand was blown off, her mother's worsening blindness, her brother's increasing dull-mindedness, her sister's infertility, the lecherous priest caught red-handed in the presbytery just after mass, his cassock tossed over the back of the receptive widow, and the rest of the story is drowned in laughter. But what should I tell her? About my weekly visits to my father and how he spews at anyone who takes pains to deal with him: the nurse, the doctor, the cook, his acquaintances from home (he hasn't had any friends since the end of the last war)\u2014even the taxi driver who takes him to the cemetery three times a week so he can reassure himself that his place is still there next to my long-deceased mother, his \"wee patch of earth\" which he claims to look forward to. When I separated from my institute and my wife separated from me, I invited him to move in and take over Helene's gapingly empty bedroom. Now and then he'd wake me up at three in the morning with his yelling, and I'd get out of bed and see him shuffling across the hall holding a candle and shouting at every shadow cast by his shaky hand: \"I'm a heretic too.\" It took a while for him to calm down, sometimes until morning, he never let on what accusation he was refuting. He's been considered a hardheaded heckler and contrarian his whole life, and the reputation has proved convenient. He banged on the table but never budged it. Roared but never bit. Now that his life force is trickling away, his vituperation has shriveled into a chronic dry cough. Should I burden Paulina by telling her my father has missed the right time to die? I'd rather take refuge in the stories she tells, they're much less pathetic than my own.\n\nFor months Paulina and I inhabit this ship together, sharing our cabin, and then we go our separate ways for half a year, losing sight of each other completely. It wouldn't even bother me to learn that during our time apart she'd gotten together with the Coca-Cola dealer from Legazpi City (the man never tires of fawning over her, though to date he hasn't offered anything better than an invitation to become his mistress). I feel about Paulina the way old Amundsen felt about the sun, I look forward to seeing her again but do not suffer terribly by her absence. On occasion we have tried to shorten the interval: after our first season in the eternal ice Paulina came to visit me, but it didn't go well. One neighbor congratulated me on my \"catch\" and another asked her if she'd clean for him, too. Paulina couldn't understand why I didn't own a car even though I could afford one, a deficiency that during the course of a very rainy April became increasingly conspicuous, my native country was only tolerable when viewed from the top of the Zugspitze (for the first time in my life I took the cable car, she wouldn't even try a descent), and so we slogged through the days, and our voices began to grate on each other while our desire passed faster than our time together. My own guest performance on Luzon had a number of sour notes as well, almost overnight she turned into an obedient little cog, was no longer Paulina but the eldest daughter, the wealthy sister, while I was a souvenir from a foreign land proudly put on display. But sooner or later the novelty wears off and the knickknack's merely in the way and winds up getting moved from one corner to the other until it's finally ignored completely. I didn't want to wait for that to happen, so I went to the market and hopped on a bus with the promising name Inland Trailways. I drove across the country, searching every face for a hint of Paulina but finding only strangers. When I flew home everyone at the airport was wearing mouth protectors, like so many idolatrous masks.\n\nBut as the northern summer draws to an end we reunite far in the south, happy and together. Here in the Antarctic we are made for each other, and Paulina is a blessing I had no right to expect.\n\n\"How was the ice mass balance last season?\" a passenger who's traveled with us before asks the captain over our first dinner. \"I've never seen so much pack ice at the start of a season,\" the captain replies, \"and never seen so much green at the end of one.\"\n\nSparrows chirp it from the roofs, we fled to the south where dollars drop from the sky like snowflakes, everyone must be called upon to sacrifice, place your order while supplies last, the museum is closed, water damage, the roof was old and decrepit, and now for the high point of our evening, I can't stand these fat-asses in their gas guzzlers, these turbo-charged jerk-offs in their Escalades. There's a problem with one of our ships, the MS _Hansen_ , we've lost radio contact. Now it's your turn, Charlie, that belt really suits you, you go goldfingers, and a one and a two and a three unzip that mini watch it drop feel your balls about to pop, lights webcam action. I can affirm that the MS _Hansen_ is on the wrong course, moving northwest at top speed, that's right we still don't have radio contact, we don't have any explanation, we'll have to be on standby for every possible emergency situation. Now that's what I call one efficient move, lol, our next contestants will face the fun task of making a meaningful sentence out of the words \"folderol\" and \"poppycock,\" the first one to meet the challenge will receive our coveted Triple Twaddle Prize, we insist on having an international commission, that has to be examined very closely, in London unexplained delays held up today's fixed price for nickel, that's a great solution, all bad things come in threes and fours and twos. No, no mayday, no indication of any problems, no mention of disturbances in the daily report. I say let's cordon off the streets, and haul out every one of those turbo-charged jerk-offs and make them choose, your Cadillac or your cock, well underway, a little joke on the side, Christians view the desert as a place of evil, it's really the ultimate place of good, how could all your intuitions be so far off the mark your Grace, hey that's one shaved pussy, we're moving our webcumcam in for a better view, oom bop bop I'm pickin' up good vibrations, Charlie which of our legs are you trying to pull, what are you sniffing at, what I'm gonna pull is between those thighs, the difference between earthworms and chimpanzees, between punks and porters is purely a matter of cultural determinism, attention attention BREAKING NEWS NATURE NOT IN DANGER, PEOPLE FEARED DEAD BREAKING NEWS NATURE NOT IN DANGER, PEOPLE FEARED DEAD keep it up\n\n### 3.\n\n## 53\u00b022\u20325\u2033S, 61\u00b002\u2033W\n\nFROM THE BEGINNING, it was the chance to talk about ice that inclined me to take the assignment, which fluttered into my home out of an overcast sky when my colleague H\u00f6lbl came to the door masquerading as a harbinger of good news. He closed his umbrella and asked if he should take off his shoes before entering. I can't remember whether he said, \"I'm planning an attempt on your life\" or \"Can you do me a favor?\" if he grinned at me or scrutinized me. At that point the institute was seething with rumors that I was going to pot without work without marriage without anything I could obsess about, I was notoriously easy to get worked up, have you noticed he's no longer accepting any invitations, but then he never was very sociable to begin with (I've always been suspicious of words that begin with the letters \"S-O-C-I\"\u2014society is a mirage, sociable makes me think of chattering skulls, sociology is more a bog than a field), he's mutating into a complete recluse, he's on the verge of losing it, those are the rumors according to H\u00f6lbl, but in spite of his sarcastically upbeat report it was impossible not to hear a genuine note of concern in his voice and see the honest wrinkles of worry in his face\u2014it all moved and angered me at the same time. When it comes to the countless people who grind away for wages that are less than inspiring, who get sores on their fingers and scabs in their brains, the H\u00f6lbls of this world never suspect mental or spiritual decline. From his viewpoint I was sick, an acute case of ice deprivation. But he was cunning and did not disclose the therapeutic motive for his visit on this rainy autumn day, instead he begged me to help him out of a mess he'd gotten himself into, he'd made conflicting commitments, accepted one thing without canceling the other, the usual polygamist trap (H\u00f6lbl was trying his best to cheer me up). He tempted me with everything he could, including the amorous adventures to be found on the high seas, which he touted so vividly as though I'd just been released from the convent boarding school, on a ship it was easier to catch a playmate than a cold, and I could enjoy things without any complications, because what happened at sea stayed at sea, my recuperation was guaranteed, there were no students (H\u00f6lbl was lowering himself somewhat with these efforts at humor), just a few lectures, some trips out to the penguin colonies, and that was basically it, all in all a lucrative form of sloth, \"busy working holiday\" is what they call it on board, you'll master the nautical language in two shakes, you know the subject matter in your sleep, your English is down pat, I have to pull out, meanwhile take a look at the pictures I brought. H\u00f6lbl was too stingy to print real photos, he'd run off some cheap copies on regular paper, the color looked very artificial. I spread them out on the coffee table, next to each other, on top of each other, completely covering the wooden border. They all seemed familiar\u2014the snow-polished ice, the ribs and furrows gleaming in the sun, the crystalline undulations\u2014and yet I was looking at an unknown world, where glaciers calved into an ocean instead of a valley. Taken together, the photos conjured some benevolent spell that had been spun outside of time, every word that the Antarctic water was whispering to me was frozen, I picked up one of the photos as carefully as I could but still left a fingerprint on one of the icebergs. \"Pretty intense, isn't it?\" H\u00f6lbl stood beside me, with an almost lewd grin, \"Damned intense is what it is,\" he said, his laugh exploding like a firecracker as he slapped his right hand on the back of the chair. There are moments when you have no choice but to laugh along whether you want to or not, if you don't want to lose your common language. Weeks later I was standing on somewhat shaky legs in the auditorium of a cruise ship, amazed at how many passengers showed up for my first lecture (9:30 AM in English and 11 AM for the German version), my biggest audience ever, what I had lost in terms of younger listeners was more than compensated for by the overdose of seniors. These passengers feel an obligation to ground themselves, to learn what they can about the Antarctic, they board the ship knowing little but truly desire to learn more, that suits me just fine, by all means permit me to influence your vision of the unknown. On this trip which is unlike any other, they delve into the literature to further their education instead of devouring thrillers as they do elsewhere, for relaxation they prefer _The Worst Journey in the World_ , such close encounters with the Eternal Ice causes even the civilizationally autistic to feel a certain lack within themselves. I hear myself talking and wonder at my chatty tone: when Africa slammed into Europe, Antarctica went skidding as far south as it could and got covered with ice, the crumple zone that resulted forms what we call the Alps. Antarctic means \"anti-arctic,\" the name derived from Aristotle's postulation that for philosophical harmony and geographical symmetry there had to be a counterpart to the icy land in the north (the only such land then known to the Europeans). Anyone who claims never to have mixed up Arctic and Antarctic is a boldfaced liar, but there's a mnemonic device that can come to your aid, actually two devices, a bear and a penguin, since as we all know penguins are found only in the Antarctic and polar bears only in the Arctic regions. And this all makes sense because the word \"arctic\" comes from the ancient Greek _arktos_ , having to do with Ursa Major\u2014the big bear. So if you can just hold on to that you'll never mix up north and south again, in contrast to all of your friends at home, whose first question will be how things are going down there in the Arctic. Of course if polar bears become extinct, the name \"Arctic\" will no longer apply and we'll need something else, I'm happy to take suggestions beginning today and for the rest of our journey. But have no fear, even if the Arctic should cease to exist (something all of you sitting in this room will live to see if you keep taking your blood thinners and beta blockers\u2014I don't speak this last thought out loud), the Antarctic will remain as an antipode for as long as humans inhabit the Earth. A few passengers smile, some smirk. Together we wander through the history of ice and stone with the help of a timetable where _Homo sapiens_ barely appears, some days I have to really work to make sure the passengers don't faint from all the zeroes. Arctic and Antarctic, ladies and gentlemen, we are talking about extreme contrasts: seasonal ice in one place and terra firma in the other, unstoppable melting in one place, and in the other an icy shield up to four thousand meters deep. One place doomed to collapse, the other somewhat protected and not yet lost. One place a sign of our capacity for destruction, the other a symbol of our understanding. In a nutshell: above is bad, below is good, above is hell, below is heaven. Ladies and gentlemen, what we are talking about are the two poles of our future. Before opening the second PowerPoint presentation, I pause longer than necessary to let my ominous emphasis take effect before substantiating my claims with pictures, just like H\u00f6lbl did on my coffee table, but whether they're printed on cheap copy paper or displayed on a high-definition screen, the ice landscapes possess such force that the entire audience is rendered coughless, and together we plunge into the high-sea silence of the albatross.\n\nDid H\u00f6lbl have any inkling what his visit would unleash? Anyone who only knows ice as a caged animal in closed valleys is bound to be overwhelmed by the radical freedom of the white south, where all exceptions are the rule, where ice covers everything except the steepest cliffs. Landscapes like this didn't exist even in the wildest dreams of the eight-year-old boy who together with the others from the same apartment block dared one another to drink water from a puddle with a straw, until some mother spotted them through one of the open windows and let loose a cry that landed right in the middle of their game.\n\n\"Come on up here boy,\" my father called out, without leaning out the window, \"we're going to the mountains.\"\n\nI went up immediately.\n\n\"Why the short britches?\"\n\n\"It's hotter than blazes out there.\"\n\n\"You'll be cold.\"\n\n\"No I won't, Papa, believe me I won't get cold.\"\n\n\"Well we'll see about that...\"\n\nIn my memory my father leaves Mittersendling in second gear and stops at every crossing. Our motor sounds happy and I'm bouncing on the seat so as not to miss anything. Father imitates different birds with chirps and twitters, he's a robin a greenfinch a woodpecker.\n\n\"You should be on the radio, Papa.\"\n\n\"With all that twitteratura, for an hour of bird calls? Nobody'd put up with it.\"\n\n\"No I mean mixed together with all the singers, they'd do a song, you'd do a bird...\"\n\n\"How would that work? Ladies and gentlemen, stay tuned for The Blackbirds' latest hit? They wouldn't put up with it for even a second. Well, we'll see about that...\"\n\nI'm allowed to roll down the window, after that I can no longer hear the twittering. My father's had the VW Beetle for just a few weeks, before that he took the streetcar, and we used the sidewalk. Wherever we couldn't get on our own two feet we didn't belong. I count the oncoming cars as well as the ones overtaking us. Red cars count twice, I don't remember why. Right when I've reached a hundred points my father announces that we're almost there, the trip didn't take long, three hours, maybe three and a half. We park the car and hike up a path, and all at once I see a wall and feel a wave of cold that's unusual for the middle of summer. Driving back hours later, I rub my hands over the goose-pimples on my upper thigh, feel my wet shoes and stare at the disappearing vista, you'll get sick, Papa warns, but I refuse to let go, I'm staring at the glacier through two panes of glass, like looking at my future through a pair of binoculars, in the end I didn't let go. Everything's backward, I told my mother later, it's like a dragon with ice-cold breath instead of fire just lying there spitting out ice and spitting out ice and there's no stopping. You won't believe the ice that's up there, waterfalls that are frozen caves but they aren't really caves, they're chapels, they're all blue like your favorite dress, and so smooth. The minute you sit down you go sliding off on the seat of your pants. And Papa said that when somebody dies up there on the glacier his body gets swallowed and doesn't get spit out until his grandchildren go searching for him. Papa said the ice is full of frozen faces (as a student I used to proclaim with the arrogance of experts that no statue can come close to the ice sculptures, one day at the glacier is worth more than a hundred years in the Pinakothek). I told the other kids from the courtyard about my discovery, about my glacier, I told my schoolmates and all my cousins when we went to Wolfratshausen for my grandmother's birthday. I even told my grandfather. He was sitting by the corner shrine, his nostrils had little black crumbs like snot, he listened to me stock-still and finally said: \"You're going to see a thing or two, lad.\" I went on and on, talking until I was blue in the face, now I hear myself talking once again, after a downward slide into silence, talking more than ever, since I'm being listened to, the passengers are sitting in their rows, the Antarctic is a repository for all of us, the ice contains air bubbles that have been trapped for thousands of years, as if the Earth regularly cleared its lungs, exhaling present moments that are caught and securely stored in these natural coffers, every volcanic eruption, every eclipse, every atomic weapon test, every shift in the concentration of atmospheric carbon dioxide (every fart of mankind is how Jeremy puts it when nobody's around). And don't forget, I tell them in closing, you're going to see a lot of ice on our trip, and you'll be shivering with cold, for some of you it will be the first time you've felt that kind of cold, and nevertheless what you will be seeing is all in the banana belt of Antarctica during the warmest part of summer. Think about it, there's hardly any region in the world warming up as quickly as the Antarctic peninsula, before long we'll see heather growing, and potatoes being planted, and sheep grazing, and after that it won't be much longer before people start vinting Antarctic wine. But for now the polar plateau is still mercilessly cold, and that's something you won't encounter. All you're getting to know on this expedition is the very tip of the continent, but I can assure you that will knock your thermal socks off! Grateful, persistent applause. If only school had been half as much fun, a man compliments me on his way out\u2014but as I write this a few hours later I can no longer picture his face. The chance to talk about ice, even just twice a day, reconciles me at least for the moment with the demise of my glacier.\n\nIn the sunny warmth of the dining room I am surrounded by laid-back voices. Ricardo is guarding the entrance to the restaurant beside his desk, he consults his register and waves me away: \"I hate to tell you but there's no place for you,\" there are fewer seats than passengers, he's sorry, but the problem was unforeseeable. An older lady rises and offers me a place at her table, in Swiss-accented German she explains that her husband will be staying in their cabin because he isn't feeling well. Ricardo hastens to reassure the lady that he was only joking, and shoos me off to the table for the lecturers. A few passengers nod my way, by the end of the trip most will greet me by name. My response is friendly, I have no trouble being polite, I don't despise the passengers, even if Paulina insists I really do, I know from experience that the insights they will gain during the next few days will put them in a more reverent mood, but does that mean I should ignore the fact that this reverence will dissipate as soon as they're back home, that they aren't about to renounce their comfortable lifestyle, despite all the harm it causes? You're so strict when it comes to judging other people, says Paulina, as if they had personally disappointed you. If everyone was just like me, she says, I'm sure some things might be better, but some things would also be worse. \"Even with people I don't like,\" she says, her voice getting agitated, \"I always know they have their good side, I just haven't discovered it yet.\" For her, reality is something that has to be accepted.\n\nAt the buffet I help myself to a few appetizers and a little salad. The more I get used to this cold warm sweet buffet, the harder it is for me to decide. Instead of hardtack and herring there's an abundant assortment of food on large trays and chafing dishes, as multicolored as the flags displayed in front of a five-star hotel (everything revolves around eating, we might not be able to make a landing, the entire continent might vanish in the fog, but it's unthinkable that a meal might be canceled). During my first weeks on board\u2014and this was my first cruise ship experience\u2014I tried to compensate for years of missed mealtimes and quick snacks by stuffing myself with one course after another, the bounteous table offered some weak comfort, I ate and ate, fattening myself up, and ultimately realized that the more I continued to eat, the more unrestrainedly I would continue to indulge, I foresaw a fiasco of a finale, every pot pouring forth sweet and sour porridge that I had no choice but to consume, serving after endless serving, with no recourse or release in sight until I burst. He who wishes to escape the merciless force of excess supply must flee into strict frugality. A spoonful of corn, a spoonful of tuna, a spoonful of shrimp with melon, a few quartered tomatoes, a handful of pitted black olives. Naturally there's a seat in the dining room for the expedition leader, at the same table as the lecturers. There are days when I'd rather have lunch with Paulina, but unfortunately only the Brahmins are allowed such close contact with the passengers, the lower castes have to eat below decks, some will finish the entire trip without the passengers ever laying eyes on them. May I quote you? El Albatros spoons out his soup and looks at me over his tilted soup bowl. \"Pardon?\" \"That last sentence of yours, 'And in the end the murmuring waters will fall silent, for what will wrest the ocean's secrets if not the ice,' I'd like to use that sentence.\"\n\n\"You listened to my lecture?\"\n\n\"The end of it.\"\n\n\"I'll give it to you.\"\n\n\"Don't worry, your copyright is guaranteed.\"\n\n\"Copyright? What are you talking about? There's no copyright here in Terra Nullius.\"\n\n\"I'm planning to quote you on the rest of the planet.\"\n\nEl Albatros puts down his soup, and I feel something welling up inside me that I would have earlier called brotherhood. The ornithologist owes his nickname to Jeremy, who can devour untold quantities of salad and who spends the rest of the year in San Diego outfitting hiking expeditions with ultralight backpacks and tents.\n\n\"Have you ever just missed a plane and then felt this kind of existential urge to feel like you've been chosen somehow and wished that the plane would crash?\"\n\nJeremy has finished his salad, which gives him the opportunity to record our reactions with his video camera. He besieges us with questions for his visual logbook, which he has christened _Daily Turbulences_ but no one says anything. Beate comes back from the buffet and looks at the group in amazement.\n\n\"Keeping silent behind my back, eh?\"\n\n\"We were just listening to Jeremy complain about not being God.\"\n\n\"God?\" Jeremy jumps back in. \"That role's already taken, what's more it's badly cast, and all the reruns in the world won't help.\"\n\n\"I'm much more interested,\" says Beate, \"in hearing who would rather be reborn as an animal and who would prefer coming back as a robot.\"\n\nEl Albatros is the first to respond. \"I'm exempt from that one,\" he says, \"since you made me a bird in this life.\"\n\nThe nickname just happened to slip out of Jeremy's mouth one day after hearing yet another panegyric on the magnificent white creature with the greatest wingspan of all. He gave it an odd pronunciation, saying the \"El\" like a Chicano, and extending the \"Albatros\" just like the bird extends its wings. As soon as they've finished lunch, El Albatros rounds up all the birders like a guru gathering his sect. They are immediately recognizable from the powerful binoculars they carry around their necks, they crowd together on the open afterdeck and stare out intently, propping their elbows on the rail to steady their scopes, collecting sighting after sighting as they get soaked by the spray. One of them has taken his post behind a spotting glass, hoping to add a \"lifer\" to his list, a first look at a south polar skua, easily mistaken for a brown skua but a lot more rare. There's a clear sense of competition (supposedly they are inclined to compare their eyesight as they twitch away), and it's easy to get knocked off course when there's so much ambition-fueled headwind, even El Albatros has been convicted of an overly hasty misidentification. Later they huddle over an open copy of _Birds of the Antarctic_ , fingers glide over feathers, shadings trigger arguments as to which jaeger has been spotted, a mistaken attribution spoils the joy of the sighting. Once on a previous trip I positioned myself within hearing distance of the bird lovers and waited a bit before giving an excited shout:\n\n\"There, over there, sooty albatross!\" (I had picked out this particular rare bird earlier in the library.)\n\n\"Where, where?\"\n\nI poked my finger in the air:\n\n\"Over there\"\u2014their torsos leaned forward\u2014\"oh, now it's dived\"\u2014and they peered into the waves\u2014\"now I don't see it anymore\"\u2014and they let their gaze glide over the water\u2014\"now it's gone\"\u2014they didn't give up easily and kept scanning sea and sky with dogged tenacity\u2014\"what a shame, what a real shame.\"\n\nEl Albatros inquired with genuine interest about the head plumage and whether the primary feathers showed any darkened patches, I played the unsure witness until a twinkle in my eye gave me away and El Albatros forced me to confess: I can declare with absolute certainty that I have seen a striated tabular iceberg but I couldn't say for sure that I had just sighted that rare bird. El Albatros wasn't really mad at me, in fact he shares my antipathy for those passengers more interested in their checklists than in the wonder of a single bird, the miracle of its capacity to glide for hours, the marvel of its nasal salt glands, its astounding ability to dive and its mastery of navigation. Instead they fuss over recording every sighting, the place the time the witnesses, so that historians may one day draw on their extensive documentation to reconstruct the former distribution of diverse bird species on the planet. No, it won't come to that, the historians will die out before the last bird does.\n\nDo our nightmares change\u2014our collective nightmares? The distillate of our drunken disputes? Are nightmares the most honest expression of any given era? During his sleep my father used to lose his way in a snowstorm (as he confessed to me one day as proof of his affection), he would stumble blindly across a house that had no doors or windows or chimney but which was inhabited, the house smelled of life (stuffed cabbage rolls\u2014that's how culinarily precise my father's nightmares were). The building gave off warmth that thawed his frozen hands, and when he put his ear to the wooden outer wall he could hear muffled voices. But no matter how loud he cried out or even if he drummed on the wall until his hands bled, the people inside didn't hear him, or else they heard him and ignored him. His instinct for self-preservation woke him up before he perished outside that pitiless place. If only I were granted a nightmare like that, I would shout with joy, I would fling my cap into a flurry of snow, anything would be better than sitting on a cliff with a clump of ice melting in my hands, leaking water that trickles down my arms and into my shirt and over my thighs, dripping and dripping into a puddle between my legs. No matter how carefully I cradle the ice in my hands it continues to melt. I try to stash it somewhere, set it on a cliff, but the ice stays glued to my hands until in the end all I'm holding is a sopping relic, a souvenir. What a disgustingly sentimental dream, I can imagine my colleagues' blank reaction, H\u00f6lbl would give me a lambasting, they've really gone and dumped some shit in your subconscious, he would say. There are some nightmares you cannot entrust to anyone.\n\nIt's hard for me to say anything about it, mamma mia, enjoy it while supplies last, tutti frutti, I'm sorry but I can't seem to locate your reservation, no we're all booked up, I'm afraid there isn't a room to be had in the whole town, we have fallen victim to flattery, hey are those knockers of yours insured, works of art like that must be worth untold fortunes, he who destroys nature is killing God, that's how my neighbor put it, he's taking this cruise in the Antarctic, you know the drill, so swing over to our website pantanddrooldotcom, you can count on our webcam to deliver the goods, as soon as the Antarctic becomes an emerging market we'll open an office there, in other words, Your Excellency, do we live in theocidal times? we have to wake him, in such cases the Foreign Minister must be informed at once, I'll tell him about the matter right away. All ravens are black as pitch, offhand I can't say for certain, but it does appear somewhat exaggerated, thrill dee dill, my pretty poof, sooner or later our hour will strike, those are measurements to die for, unbelievable, and here's a job that's worth an entire future. Make it absolutely clear that we don't know if we're looking at an accident or a crime, and don't forget to mention that a terrorist attack cannot be ruled out at this time. You don't expect you can just go strolling through the palm trees scot-free do you, I'm paying taxes on expenses I reject, no one's going to question it except the ones writing the questions, imfao greed is a worse sin than waste, let's assume all living things had the same spirit, the same soul but different bodies, it's now or never, that's sick! well yes, it's completely sick, don't exaggerate, it's pathologically sick! you old geezer, beyond healing and beyond hope, do you have it in a smaller size? Does your memory stray to a brighter summer day, the situation can't be all that bad, the reason for the season time to try a toothsome threesome. We all believe that there's a good way out, last week we barely squeaked by an environmental disaster, at the time everybody talked about drawing the proper conclusions from the catastrophe. Sittin' in the morning sun, what I don't know won't hurt me, a real storybook career, from model to moderator, you can't raise a stink about that, BREAKING NEWS HUNDREDS NARROWLY ESCAPE DEATH? HUNDREDS NARROWLY ESCAPE DEATH? nothing will help except a complete reboot\n\n### 4.\n\n## 51\u00b041\u203237\u2033S, 57\u00b049\u203215\u2033W\n\nTHIS TIME, TOO, it's raining in the Falklands. Days when it doesn't rain go into the annals, as do evenings when the darts don't fly in the Victory Bar. Life in Stanley used to be full of unexpected danger, people could cut peat at will above the town, water would collect in the holes, until one night a mudslide sluiced into the sleeping city without waking a single dog, swallowing house after house and sweeping school and church into the harbor and suffocating a shopkeeper and two shearers. That also went into the annals. Years ago the governor let it be known that Stanley was a very British town on a very British island (cruise ship lecturers take note of such pithy remarks, they enjoy assembling such sayings for their own use). The governor merits contradicting: in actuality the islands belong to South Africa, geologically speaking (my claim), biologically they are part of South America (Beate's view), politically they belong to Great Britain and to Great Britain alone (Margaret Thatcher's position). The passengers bombard us with questions about the war, one of the rare wars between white men, they remember the tense evenings in front of the television. I always answer short and to the point: it was the first war in the history of the Western Hemisphere in which more humans died than animals. I have no idea if that's true, animals seldom make it into war reports, but it does shock the passengers.\n\nPaulina has been wanting to go on an outing. We've never taken a walk together on British soil. Up to now we've only landed on the Falklands when the trip was nearing its end, Paulina was busy taking inventory, and I was preparing the auction of memorabilia on behalf of a foundation whose mission it is to teach fishermen ways to keep their longlines from killing hundreds of thousands of albatrosses every year. But now we have a good afternoon hour all to ourselves. She puts on apple-green shoes and an oversized raincoat, which billows out in the wind like a racing sail, I have to hold her arm so she doesn't fly away. We take off, just the two of us, in the direction of Gypsy Cove, along a waterlogged path I know that follows the windswept shore close to the water, our steps and voices startle a covey of ducks, Falkland steamer ducks, I announce with the voice of expert conviction, Paulina laughs and points to the wreck of the _Lady Elizabeth_.\n\n\"That's a pretty big steamer duck right there.\"\n\n\"Ok, ok, I was fibbing, in reality they're Argentine stiff tails.\"\n\n\"And I'm supposed to believe that, Professor? Don't you go pulling my leg. And I suppose those are British boobies?\"\n\n\"Certainly not, those are coots. Adult white-winged coots, I swear it.\"\n\nHer hands have found a path through the layers of wind and rain protection, past Gore-Tex and wool, and she strokes my chest with life-affirming coldness.\n\n\"You're the one who taught me,\" she says with windy seriousness, \"that people who know a lot lie the most.\"\n\n\"Could we agree on the word 'invent'?\"\n\nMy appeal is tendered halfheartedly, the path veers away from the shore and takes us across a low shrubby heath strewn with stones and on toward Yorke Bay (some of the passengers are headed our way, judging from the way they're staring at us I can only imagine what they are thinking). A ranger is standing next to a wooden shelter, as though he were waiting just for us to deliver his explanations. Paulina jabs her finger in my direction and says to the ranger: \"If you only knew what this evil man...\"\n\nThe ranger presses his lips together and raises his eyebrows, I nip the coming irritation in the bud.\n\n\"It's not half as bad as it sounds, we just couldn't agree on which birds had which names.\"\n\n\"You have a wonderful beach here, people must love it.\"\n\n\"Yes, it's the most beautiful beach in town.\"\n\n\"And not a soul in sight, in my country it would be overflowing with romping children and passed-out parents on a friendly Saturday like this.\"\n\n\"I wouldn't advise that.\"\n\n\"On account of the current?\"\n\n\"Not because of the ocean, the beach itself is deadly dangerous.\"\n\n\"How so?\"\n\n\"It's mined.\"\n\n\"Mined?\"\n\n\"Antipersonnel mines.\"\n\n\"I don't understand, the place is swarming with penguins.\"\n\n\"Your question is entirely justified, ma'am, but consider that the mines require a weight of twenty kilos to explode, and even a fully grown Magellan penguin doesn't get that big, so you may rest assured that the animals have nothing to fear.\"\n\nPaulina puts her hand to her mouth.\n\n\"Soldiers make the best animal protectors,\" I say.\n\n\"It was the Argentines,\" the ranger specifies.\n\n\"Just like in my country,\" says Paulina, \"everything looks like paradise until the bullets start whizzing past your ears.\"\n\nShe laughs again, but this time it's a different laugh, the kind she uses to wipe away something unpleasant.\n\n\"They like it here, the Magellan penguins, they dig their nest burrows in the soft peaty soil up there in the dune grass.\"\n\n\"They dig their nests?\"\n\n\"Yes ma'am, and they use the burrows for years, they mate for life and never separate. They pick their partners carefully and can rely on them completely.\"\n\nWe listen to his lecture, and to the oyster fishermen in the background, then take our leave and head off into the blooming furze, where I pick some mouse-eared chick-weed, \"not for you, Paulina, it's for the family altar in our cabin, for all the grandmothers.\" On the way back we stop at one of the fence posts that has a red sign nailed to it, at closer distance we can make out a skull and crossbones below the words \"Danger Mines.\"\n\nAt the counter in the reception someone left a brochure which I open in passing. \"The Falkland Islands are one of the few places in the world where nature reigns supreme.\" A minefield as unspoiled nature? Why not, after all, Kitzb\u00fchel counts as a climatic health resort even on holidays with bumper-to-bumper traffic jams. All El Albatros has to spare for me is contradiction. If all beaches were mined we wouldn't have to worry about bird refuges. I listen to him with one ear, at the next table over some men are talking as they spoon their cr\u00e8me br\u00fbl\u00e9e about the charming heather moor at Yorke Bay, ideal for a golf course, a classic links, and while they let their fantasies go teeing off I imagine how in the course of construction the location of the landmines might be forgotten. A beach like that (\"a spectacular par 3 shot right over the heads of our native Magellanic penguins\") could become the ultimate sand trap, one which could rightly be billed as a bunker you might never get out of.\n\nI've been watching it my entire life, with passionate care and precise instruments. And if my observations will have had no impact on my chosen field and how this field views itself, then my academic career will have been a waste. Every May and September I would go a few days ahead of my students, so I could abandon myself to my senses, undisturbed, and feel the glacier's full emotional force before we captured its data. It was my doctoral advisor who placed this particular glacier in my care, an arranged marriage that in time became a union of love, as if every measurement were an acknowledgment of its singularity. On that first morning I rose before the sun, laced up my hiking boots which initially felt strange, and then I trekked around the glacier, ascending on the left side and then after crossing the ice descending below the escarpment on the other side. Each time I visited I would first scan the glacier with my eyes, then test it with my feet. Whenever I stopped to catch my breath I would touch it, laying my hands on its flanks and then stroking my face, taking in its icy breath, its invigorating cold. I was familiar with every one of its sounds, the creaking and the clanking, every glacier has its own voice, when I visited others I would compare theirs with the one I knew. A dying glacier sounds different than a healthy one, it gives off a powerful rattle when it bursts along a crevasse, and if you listen closely you can hear the melt flowing into the underground lakes speeding the erosion of the wrinkled body. We were like an elderly couple: one of us was severely ill, and the other couldn't do anything about it. All terms used to describe our relationship\u2014such as \"subject of study\" or \"mass balance measurement\"\u2014were woefully inadequate, and no column of figures could do any more justice to my devotion than could our bookkeeping, which recorded the snowfall deposit at the end of the winter as a kind of revenue, and the degree of summer thaw as a kind of expense. These credits and debits caused me greater and greater despair, and over the years I changed into a kind of doctor who had only to look into the eyes of his patient to make the proper diagnosis. I recognized that my glacier was doomed long before the declining values of its middle layer thickness pronounced their judgment. I didn't have to wait for the results to understand the ramifications of the sustained depletion. It was no longer possible to offset the loss. We were aging together, the glacier and I, but the glacier was well ahead of me when it came to dying.\n\nRules, rules, and more rules. Without strict guidelines people would trample over everything, I see that, at the same time it's humiliating for me to have to force rules on them. One of the least pleasant among my new tasks involves corralling the press. There are always a few journalists on board\u2014the cruise line appreciates the relatively inexpensive advertising in their articles\u2014as well as a handful of relaxed editors and pushy photographers. On the last trip of the previous season there were a dozen, the expedition leader wanted a silent partner sitting in on the meetings to bolster his authority and that's when I first learned of this duty. Journalists don't have any special privileges either on board the ship or on land, the idea has to be put out of their head at the start. My predecessor took a strict tone, to me it sounded like the tinny parody of a politician's call for law and order, I had to suppress a smile and turn away, as though I were expecting something surprising out of the west-southwest. \"Is there something you want to tell me?\" he asked afterwards.\n\n\"Do we have to treat them like they're educationally challenged?\"\n\n\"When it comes to dealing with nature, as far as I'm concerned every person on the planet is educationally challenged,\" answered the expedition leader who is now lying in a hospital room in Buenos Aires, where he is presumably studying his yachting magazines as attentively as I am studying the faces of the journalists who have gathered around me in a semicircle and who are introducing themselves one by one at my invitation. That gives me a chance to separate the chaff from the wheat, the insightful from the unruly. I form my judgments hastily, instinctively. How is it that the Roman ideal of presumed innocence has endured in our civilization so pervaded with the notion of original sin? The boisterous blonde from Hamburg won't cause any trouble, she's brought her boyfriend along for a working vacation, she'll avoid anything that might make her seem unpleasant. The Colombian cameraman has eyes that suggest an easily ignited insolence, while his accompanying editor gives off an air of laziness, I'm sure he won't be stirring up anything provocative. The striking young American is clearly nervous and this makes her also seem unapproachable. \"I'm Mary from _Mother Jones_ and please no wisecracks.\" I look around, wondering what she is alluding to, but nobody seems to get the supposedly obvious joke. I'm certain that the muscular cameraman, who probably wears his dignified smile even in his sleep, will try his luck with her by sharing some hastily concocted witticism as soon as they leave the lecture room. The last in line is a smart-suited man who introduces himself as Dan Quentin's PR manager and then pauses for a moment as if he were expecting a show of admiring smiles and glances that to my astonishment are indeed accorded, apparently I'm the only one who doesn't fully appreciate the name. His own business requires a special discussion, as the captain has undoubtedly informed me. Evidently the man is paid to find the right words to maneuver himself into a privileged position. \"No,\" I answer, \"the captain and I haven't had any time to confer about Dan Quentin, but I'm certain we'll have that opportunity, for now though I'd like to agree on a few basics. In a word, the same rules apply to journalists as to all other passengers on this cruise. Never leave the paths marked with red flags, never pull anything out of the ground, don't take anything and don't throw anything away, not even the tiniest shred of paper. Always keep a distance of at least five meters from all animals, penguins included, and don't try to blame it on the poor penguins not knowing the rules, we've heard that excuse more than once. Like all other passengers you may spend no more than two hours on land, don't try to carve out any more time. And follow our instructions, if you don't then we'll have to leave you behind and you can write a feature about surviving the winter all alone that will make you world famous.\" Scarcely have I spoken these words than I begin to doubt whether my approach is any better than that of my predecessor. Then I ask if everyone understood what I just said; after all when people are speaking a second or third language there's a greater risk of misunderstanding. Dan Quentin's manager chews on his sunglasses, Mary writes everything down, the editor has the cameraman translate the last portion of my speech in whispered tones (shouldn't the editor be the one with a better command of English?). Any questions? There aren't any, because everyone is distracted by what is drifting past. \"Aha, the first iceberg,\" I try out a lighter tone, but now no one is paying attention, \"in two weeks you'll have seen so many icebergs you won't even turn your head when one shows up.\" As expected, just after my parting words the manager jumps up and runs over and starts speaking at me while he's still moving, as if I were a typewriter on which he were urgently typing a letter requesting payment. He asks me to speak with the captain as soon as I can, it's about a project that's so colossal the logistical demands cannot be underestimated, the artistic vision is explosive and amazingly timely, the Antarctic has now become a project in the hearts of all mankind, Dan Quentin wants to create a sign, he wants to hoist an emotional flag that will be seen across the globe, a symbol for danger and endangerment, he wants to coin a highly original visual currency. Tomorrow after breakfast the manager will come to see me, he's looking forward to our collaboration. In the meantime Mary has kept in the background, now she breaks away and asks shyly if she might ask me for an interview when I have a free hour. Gratefully I agree.\n\nMy students didn't know what an oxbow is. Some thought it had to do with pastures, others with plowing tools. They weren't even embarrassed by their ignorance, as if they had a fundamental right to forget about what has been destroyed. Over breakfast on the last day of our summer stay at the glacier I asked them to pack their things before we climbed up for our last visit, and paid the proprietor of Zum Kogl a hundred schillings to deliver the luggage to the station in time for our train. After we had completed one final hike around the glacier I suggested that we make the first part of our return trip on foot. They asked why, and I explained that was the only way to really read a landscape. A few grumbled, but no one dared stay at the bus stop\u2014the ongoing need to prove oneself is a powerful corrective. We gazed down into the valley. From above, the effect of human intervention is quite visible, it's easy to see what we have done to nature. That was hardly a revelation, even for the students conditioned by city living who didn't know what an oxbow was. I wanted them to spend at least one afternoon seeing with their own eyes, consciously observing the shrinking fens, the straightened rivers, our civilization's attempts to impose its own discipline on the natural world. From one ledge where the valley opened up below us like an accordion file, I gave a short lecture about the river meadows as they used to be, until people started thinking of them as worthless land, like alien beings that had wandered into their anthropometric order and needed taming, which is why what we are looking at is land that has been drained and cleared and made useful, where the wet meadows have given way to apple orchards. First the natural state was wiped clean, then cultivation was introduced. But among hundreds of varieties of apple there are only a few that meet the reigning market standards\u2014standards that doom any unregulated growth to failure. Taste and color are determined by chemical analysis. With the screen of our simplemindedness we have managed to sift out all natural variety. I closed my extemporaneous speech telling about a local farmer who a few years earlier was blessed with the tastiest harvest he'd ever had, but he'd been unable to sell because the size and shape of his fruit didn't correspond to the supermarket norms, he was stuck with a rotting pile of apples he would have gladly given away if enough children had come by. A while later, when we stopped by an Alpine pasture for a snack, several of the students pulled polished Granny Smiths out of their backpacks, they looked at their standardized apples and exchanged embarrassed glances. As they bit into the fruit they may have wondered what a proper apple really tasted like. In one or two cases this curiosity might turn into a persistent yearning, but it would be foolhardy to hope for anything beyond that.\n\nThe pianist is waiting for me, very impatient. He always acts as though he's bothered by my mere presence, but whenever I'm late he looks around to see where I might be, and if I keep him waiting longer he asks Erman the bartender for my whereabouts. After dinner we digest the day. I function as his discursive GPS, by referencing my own position in contrast to his, he is able to determine his coordinates. I ask if the Falklands make him feel a flash of patriotic pride, but he refuses to be baited. \"The only beautiful women on this godforsaken island,\" he answers, \"are the Thai ladies in the souvenir shop.\" Far from wasting his thirty years aboard cruise ships on touristic nonsense, the pianist has undertaken an extensive study of womanhood in a variety of habitats. As far as he's concerned, women who hail from distant lands are the last wilderness on earth (when we're alone he regales me with ribaldries that would certainly elicit an indignant huff from Mrs. Morgenthau). Like all connoisseurs he appreciates the uncommon, the unusual, the outlandish. If he ever retires\u2014and I doubt he will, because for all his British chauvinism, which gets freshly creased and ironed every day, he is secretly afraid of English provincial life\u2014he will undoubtedly relay his philogynous experiences in tones befitting such a man of the world. \"By the way,\" I say, \"the beach is still mined.\" He raises his gin and tonic, flips the coaster over with his left hand and places the glass irritatingly in the middle of the table. He seems excited, he has something up his sleeve, something he wants to tease me with, he can scarcely wait. I close my eyes. Behind me I hear clinking sounds as voices fill glasses, glasses run over, voices rinse the glasses out, I feel an acidic wave rising from the depths of my stomach. When I reopen my eyes the pianist is leaning forward and speaking in a conspiratorial voice: \"If you only knew what was lying on the bottom of the ocean.\"\n\n\"Gold?\" I guess without enthusiasm. \"Torpedos? Giant fan worms?\"\n\n\"Nothing of the kind. Ships, powerful ships. And any number of compatriots.\"\n\n\"Whose compatriots?\"\n\n\"Yours.\"\n\nHe leans back in his chair.\n\n\"I understand they've been here a while?\"\n\n\"Ever since the First World War.\"\n\n\"I'm not interested, we let that one go a long time ago, these days we're more interested in fresher corpses.\" The pianist nods, as if my reply was as easy to predict as the next move in a classic chess opening.\n\n\"Does the name Admiral Graf von Spee mean anything to you?\"\n\n\"No, nothing, wait, von Spee... von Spee? When I was a student I used to live near a square called Graf-Spee-Platz.\"\n\n\"Undoubtedly named after the same man. A prominent admiral.\"\n\n\"Does he have a von in his name or not?\"\n\n\"That's immaterial. The fact is he's one of your heroes.\"\n\n\"And how exactly was he heroic?\"\n\n\"He crossed two oceans and then showed up here in Stanley with his entire fleet, having got it in his head he would interrupt the British coal supply, even though he knew that he was hopelessly outgunned.\"\n\nThe pianist's voice whirred on, now he could take his hands off the handlebars, it was all downhill from here, an easy ride to his goal.\n\n\"At the time, Stanley was excellently protected by two battle cruisers, the H.M.S. _Invincible_ and the H.M.S. _Inflexible_...\"\n\n\"That was a very gentlemanly gesture, to give Admiral Graf von Spee such a clear warning.\"\n\n\"Gentlemanly indeed, but it was of little avail, your Admiral ignored their presence and insisted on going down in these waters, together with two thousand hands including two of his own sons.\"\n\n\"A cold grave. And what are you trying to tell me with this story?\"\n\n\"For a geologist you are remarkably impatient. Before the battle, the squadron at a bay in southern Chile, that cost time, the element of surprise was lost, but it had to be: the admiral insisted on pinning Iron Crosses on three hundred of his sailors.\"\n\n\"Are you telling me that three hundred Iron Crosses are lying just off the Falklands?\"\n\n\"You're catching on.\"\n\n\"That's insane.\"\n\n\"On the contrary, it makes perfect sense, the farsighted admiral anticipated his demise and wanted to make sure his men did not go down undecorated.\"\n\nThe pianist rested one arm on the back of the crimson chair and looked at me with great satisfaction. He has a remarkable talent for staging his own well-contented state, he smacked his lips and ran a finger along the rim of his gin-and-tonic glass, now nearly empty.\n\n\"Crosses on the sea bottom, mines on the beach, I have to admit I might have underestimated your island a bit.\"\n\n\"Next time we should go for a walk together.\"\n\n\"I'll mull it over. But only if you take my request this evening.\"\n\n\"Just don't be too hard on me, I don't know any Germanic funeral marches.\"\n\n\"I'd never dream of asking for something like that. I'm thinking of something more current, something you could play with your left hand while unbuttoning a summer dress with your right.\"\n\n\"Now you're talking.\"\n\n\"In honor of Admiral Graf von Spee, in honor of the light-footed penguins on the beach I would like to hear the only song I feel would do the occasion justice.\"\n\n\"Aha, now you're giving in.\"\n\n\"What I'd very much like to hear is _Rule Britannia! Britannia rule the waves_.\"\n\nAt least one time on every trip the talk turns to the hundred Inuit names for ice and snow. I always confirm that it's true, the Inuit have a name for ice floe, pancake ice, hummocked ice, brash ice and grease ice, tabular iceberg, ice cliff, ice needles, ice foot and ice tongue, for the ice fields formed out of firn, for ice caps, permanent ice, ice age and for growlers and bergy bits. But I'm not one hundred percent sure they have a word for glacier fleas.\n\nLike an elephant in a china shop, nothing's as bad as it looks, things have gotten out of balance, you can write that one off, below 40\u00b0 S there is no law, snap yours up now while supplies last, you've just begun to scratch the surface, November Zulu. This is Foxtrot Two Niner, over. This is November Zulu, proceed, Foxtrot Two Niner, over. I'm flying back from Gerlache Strait, over. What's going on? over. Dan Quentin, over. Dan who? over. You can squeeze out a little more, no one's going to question it, the guru chose the peace and solitude of the mountains, Charlie we haven't paid her ass the tribute it deserves, I don't have eyes in the back of my head, ha ha ha, you didn't see that, in the spring, summer and fall months he lived in the dense forest, his only roof was heaven. Quentin's raking it in, he's the new Christo, over. Roger, never heard of this Dan Quentin, what does he have to do with Christ? over. Christo, The Umbrellas, Valley Curtain, Running Fence, over. Doesn't mean a thing to me, over. Making nature visible by covering it up, over. That's an old whore's trick, over. Art with people, over. Torch them all, every SUV, give the firebugs a field day, under no circumstances should you mention that in your job interview, first there's a big bang then the car bursts into flame, in winter he retreated to a cave that protected him from ice and snow, below 50\u00b0 S there is no government, whatever had been sown and reaped by human hands he refused to eat. For example, over? \"FAQ\" in Silicon Valley, \"QED\" in the Burj Khalifa, over. The naked cyclists in Hyde Park? over. That was somebody else, over. What's he looking for in the Antarctic? over. A big SOS on ice, over. He collected fruit from wild trees, plants from the forest and roots from the earth, below 60\u00b0 S there is no god, we got off with a black eye, he fed his body just what he needed to survive. The passengers on board the _Hansen_ linked together to form a red SOS, over. By choice? over. Yes, about a hundred pax per letter. It was a show, I'm telling you, out. Roger, out. I will fulfill all of your desires BREAKING NEWS MS HANSEN HIJACKED IN THE ANTARCTIC BREAKING NEWS MS HANSEN HIJACKED IN THE ANTARCTIC I'll be sittin' when the evening comes\n\n### 5.\n\n## 53\u00b011\u20328\u2033S, 45\u00b022\u20324\u2033W\n\nSTANDING BRACED ON the weather deck of a pitching cruise ship, taking in the gale, the storm, face lashed by wind and spray, having the breath knocked out of the lungs, at the mercy of the elements and frozen through after only a few minutes no matter how many layers of high-end high-tech material\u2014passengers can enjoy a brief taste of the deprivations of a bygone era, just one door removed from the warm cabin from where they can witness nature unleash its force through a glass pane as though they were watching a prizewinning documentary. Almost everyone chooses the comfortable front-row view. Alone at the bow I lean over the railing, the spray spits in my face, I claw myself into the wood, the wind slaps my cheeks, it has every right to punish me for my comfort, for the deadly sin of a civilization bent on denying the basic principle of life, because whatever lives must strive to climb the energy gradient. Petrels dance among the gusts, the ecstasy of their soaring and diving is my own yearning taken wing, I rock in the air as if I, too, had been granted such ability, the engines burble away in the maw of the howling storm, how ridiculous am I to be impressed by the obvious. We cannot read the flight of birds, says El Albatros, merely misunderstand it. In the half-visibility I sense the outline of a mighty object, an iceberg is floating our way, it's larger than our ship, flat on top as though brushed smooth, as if an entire province had detached from the shelf ice and was damned to orbit the South Pole or drift north and expire, bequeathing to the hemisphere the purest air, and to the ocean the cleanest water, laden with healing powers that enable the phytoplankton to grow, as well as the zooplankton that sustain the small shrimp-like krill that nurture birds and whales (Beate claims that within her lifetime the krill population has declined by four fifths, there's no arguing about it). The ice has been punched with oval holes, mighty vulvas pushing deep inside the berg. Melting mating calls. Behind a curtain of mist the sun flares unexpectedly, a measure of mortality. The glow withstands a few more waves before it disappears again and the storm rages on in the twilight.\n\nCould this be calm? A commodity precious enough to be marketed with enormous success, guarded in protected areas, sheltered in reserves. But these ecological niches are shrinking as the pulse of our age thuds on ahead in four-quarter time.\n\nOne day some years back, just after the church bells rang vespers, I said I had no desire to go out and eat in some pub where they drench the roast venison in an earsplitting sauce.\n\n\"Does that mean you're not going to the doctor because they play Radio Bavaria in the waiting room?\" Helene asked full of spite. \"And what about the dentist and all his spherical sounds straight from the Buddha?\"\n\nShe snatched the car keys from the ceramic dish and rushed off to her sister, who was well inclined toward me because she didn't have the patience for Helene's continuous loop of complaint. I sat down on the chair in the hallway, closed my eyes and stayed there a long time. How did it happen that words like \"quiet\" and \"standstill\" have turned into something pejorative? Even on hiking trails I hear the numbing bass notes of music junkies who can no longer bear the sound of nature. If they at least just listened to their own voices, but no, they have to smear a layer of stale noise over every single thing. Admittedly I also put on a pair of headphones now and then in the S-Bahn on the way to the university, Thomas Tallis did a lot to help mask the ugliness of the utility buildings along the way, but it would have been unthinkable to listen to his music in the woods or on the mountain or when I was with people I knew. The _Hansen_ doesn't have a public address system (unlike other ships in other latitudes, where according to Paulina jingles screech and fanfares boom). Recently they held a rock concert on King George Island, soon the glacial faces will come tumbling down like the walls of Jericho. In our cabin the only perceptible sound is Paulina's tender singing (in contrast to many of her compatriots she isn't glued to her iPhone); she mixes evergreen hits from the '70s with Filipino folk tunes. In fact it was her singing that cast me in a spell on the last evening of my first trip, I had hardly noticed her before that, her unobtrusive politeness blended in with the ready friendliness of all the other Filipinos. But at the farewell concert\u2014the passengers had given the crew a chance to amuse themselves\u2014she metamorphosed into a confident chanteuse, a bundle of energy inside a cone of light, she crossed her legs and let her right shoe slide down her foot so only the silver buckle was looped over her toe, the shoe dangled and swayed and all desires were fixed on her as she sang the old hits to a plucked accompaniment with an intensity that let me draw a curtain shutting the two of us off from the rest of the world. Later my feverish fantasy made me self-conscious as we stood next to each other at the after-party in the cafeteria, I was certain she could sense me looking at her with different eyes, my desire mixed with a strong dose of insecurity, my tongue my best enemy, and nevertheless a few hours later she lay next to me, just like now, with her head on my shoulder and one hand on my chest, and as so often on our days on the open sea, when one or the other of us is granted a free hour, she has asked me to read something out loud to her. I am happy to comply, for me it is a gesture of intimacy, I read a passage from the reports of the so-called Explorers because of her charming compassion for the hardships and suffering they faced, though in my anger I see them less as pioneers and more as avaricious parvenus seeking to take possession of the Antarctic as if she were a virgin who after the first night was theirs by right for all other nights, and so they despised all competitors as thieving rivals, while they themselves sought to conceal their own lust so as not to endanger their spotless reputation as impeccable gentlemen.\n\n\"It only seems that way to you,\" says Paulina after a pause that she occasionally asks for so she can keep up with the story, \"because that's how you read it, your anger is seeping into their words, but you're really just like they are, you want to determine what happens to the Antarctic.\"\n\nMy voice flares up. \"If you mean I don't want any people or fuel oil in the Antarctic, then you're right, I do want to determine what happens here. But I don't want to possess the place, that's the difference, I don't want to have any part of it named after me, I just want it to be left in peace.\"\n\nPaulina pouts and wrinkles her nose. \"You get so loud. You know, sometimes you can be a very loud person.\" She hardly seems vulnerable and in need of protection when she challenges me like that, when she puts me in my place with simple sentences that make my replies seem inappropriately exaggerated. And that just makes me madder, the fact that I have a hard time explaining what I perceive and fear and despise, even to her. It's plain as day, our profit-driven turpitude, why is it so hard for me to express the obvious to those who don't see it? Just look at the picture, do you see a beautiful young woman or a wrinkled old woman? And once you've seen the wrinkled old woman will you ever be able to make out the beautiful young one? I turn away from Paulina, pompously wallowing in my anger like an elephant seal in his mudhole, until I've calmed down again. \"My dear Paulina,\" I whisper to her in sorrow and distress, \"you are as much a mystery to me as the world itself,\" and she beams, presumably because of the my dear Paulina, her smile lingers, a little happiness goes a long way with her, she consumes her blessings sparingly, while others need a new ration every day. I can't imagine that any real lasting strife might come between us and settle in the narrow space of our cabin, she senses intuitively where I am reconcilable, the first time it happened unexpectedly, I was frightened, she took my rage inside her mouth and cooled it off so that we both grew silent. Later I stroked her little belly and said, \"This makes you complete,\" and she answered, \"You make me complete\"\u2014a sentence I would never let her get away with if her laughter didn't start to boil over again and seethe. Her quaking stomach arouses me, she takes the book from my hands and sets it down where moments later I toss her underwear, I forgive every sentence, and in order not to say any more wrong things I silence my voice with an eager tongue, my hands placed on her breasts, even when the ship's rolling nearly knocks me down, my tongue circles on, stoking her passion, so that it doesn't fade, the sea swells set the rhythm and in my mind she tastes of salt while drifting through my brain amid all the moaning is the question of whether we will ever be able to understand each other. What she wants is for us to simply be, while I seek deliverance in some more absolute silence.\n\nIn the summer before my glacier died, we packed up all our belongings and then unpacked them. Vacation passed and with it my desire for domesticity. Some couples get pregnant to save their relationship, Helene and I decided to move. From an apartment in F\u00fcrstenried to a single family house in Solln. She was supposed to tidy up while I spent another few days on the mountain supervising my students as they measured the degree of ablation. It was a lively group, unusually encouraging and supportive of one another, and I came back to our apartment in an upbeat mood. Not sensing anything amiss, I climbed the stairs, where I ran into the short-statured retiree from the second floor and his exceptionally unfriendly look. I unlocked the door and pushed against it with my shoulder, the wooden panel resisted, I had to press with all my weight just to make it in. A dozen moving boxes had fallen over, some were blocking the door and a few had slid onto the floor where they'd knocked down a pair of rubber boots that were lying between a halfway rolled-up sleeping mat and a torn sombrero. Helene had dismantled everything, hadn't cleaned up, and disappeared without a trace. I carefully picked my way ahead and found myself standing between remnant skeins of woolen yarn and stacks of framed prints that had been taken off the walls, which were now as naked as they'd been in my childhood home (before Oma bequeathed us her still lifes), the whole place was as jam-packed and jumbled as my room had been during summer vacation, when I took out all my toys, all the figurines and playing cards, all the tokens and dice, and spread them out on the floor and made up my own games which took place on the rug, the table, the bed. In honor of its inventor I christened these tournaments \"Zeno's Olympics,\" and before starting I would call out into the hall: \"Please don't come in my room.\" Helene could have at least posted a similar note on the door: \"Please don't come in the apartment.\" The hall was crowded with open boxes whose contents were hidden by wads of crumpled newspaper, old parkas hung from the coat stand like the branches of a weeping willow, the side tables were piled high with printed matter stacked this way and that, including several years of _Burda Style_ (even if Helene never sewed anything herself, she made sure to save the patterns, as models for some alternative life design), the issue on top dated from the late '70s, the mannequin on the cover had the same impressive perm as Helene did back then, leafing through I found a page that was missing a piece the size of a postcard. Would Helene really throw away all these magazines, could she let go of her collection of clippings? She kept them in large portfolios\u2014first-aid kits she used to open whenever she was in a bad mood. When her life objectives became blurred by everyday routine she would revisit the pictures of fancy gifts, dream vacations and anti-aging products and readjust her goals. Then she would close the portfolio with a deep sigh, the kind I knew all too well, the oh-if-only-we-could sigh, until finally the day came when I started earning as a full professor. Then we went on our first dream vacation and came back covered with bug bites and sick to our stomachs, that's what happens when you book something cheap, were Helene's enlightening words (to avoid a fight I stopped myself from asking what was cheap about the most expensive trip of our lives). From that point on she collected only offers whose price and exclusivity ruled out all risk of disappointment, and these were no longer clippings but whole catalogs with semi-glossy foldouts more lavishly produced than the Alpine photo albums that lay next to the stacks of _Burda Style_ , now as out of date as the desire with which they were once filled. Helene had taken our interior life and turned it inside out, drawers cabinets chests shelves in every room had been emptied and whatever she didn't fit into a box she flung aside, creating in the process a mixed-media installation of superfluity. She seemed to be curating our entire inventory, people here all seem to inhabit their own museums nowadays. Some of the objects I had forgotten: the electric carving knife, the bread slicer, the yoghurt maker, enough shoe polish for a very shiny eternity, countless sunglasses, belts, purses. I had never noticed how many blazers Helene had acquired over the years, because after all every special occasion demanded a new blazer, they were spread out on the bed, and there were enough to form a small mound, shaped like some prehistoric grave. The Delft majolica that had once belonged to Helene's grandmother was laid out on the buffet table (tradition is that which is passed on), a few tiles were already wrapped in paper. One of the armchairs was covered with notepads and pens I had picked up at various conference hotels, while my trail maps (lightly curled souvenirs from our longest-lasting common passion) were stacked on a dining room chair, the floorboards under the table were strewn with receipts whose warranties had long expired. How could it come to this, a bedroom full of blazers, a bathroom full of firming creams, a kitchen full of Tupperware, a living room full of rocks, shells, vases, glasses from Christmas markets and wine tastings, souvenir cups (\"Gru\u00df aus Oberammergau\") betwixt Mexican bowls and even a Portuguese rooster I'd been talked into buying along with the story of a roasted cock that jumped up from the judge's plate to crow the innocence of a man who was about to be hanged\u2014how could it come to this, to our possessions driving us out of our home? And there was still the basement, where suppressed like some trauma could be found Christmas ornaments, tinsel and tree stands, rolled up handwoven rugs, three decades' worth of shoes, and multitudes of music and video cassettes, stage programs and miscellaneous folders\u2014clearly a place to be avoided. In the apartment itself there was nowhere to sit down, every chair was laden with our belongings, the large armchair was buried under various misguided attempts at silk painting, macram\u00e9 and origami. I perched on a seemingly stable tower of catalogues from the Pinakotheks Old New and Modern and sat there, my feet not even touching the floor, at a loss as to what to do, for the first time in my life I felt the fear of being buried alive. When the telephone rang (it had to be Helene explaining why she left so suddenly and when she planned to come back) I was staring at a small jar of preserves labeled in her convoluted script: \"Strawberry preserves with amaretto (1989).\"\n\nThe captain is not a control freak, but once he takes charge of something, everything has to be done the way he imagines, which isn't always easy because he ignores details and is so sparing with his speech it's as though words on board ship were rationed. He comes from up north in Lower Saxony, from a village near Friesoythe, apparently that explains a few things, people who know say the locals there limit themselves to one sentence per day, they start it in the morning and end it in the evening, I can't say myself, I was only once in Bremerhaven on business and once privately in Sankt Peter-Ording, to me those northern states might as well be a foreign country. Now that we've managed to survive a vertically divided Germany unscathed, as far as I'm concerned we could happily divide the country along some middle latitude. \"I've got this bad feeling in my gut,\" says the Captain after a mumbled \"good morning.\" The way he says \"gut\" makes it sound like an onomatopoeic expression for bad mood.\n\n\"You mean Dan Quentin?\"\n\n\"I have this bad feeling.\"\n\n\"I can understand that.\"\n\n\"The company is keen on the idea.\"\n\n\"The siren call of fame.\"\n\n\"We don't know the man.\"\n\n\"His manager, on the other hand...\"\n\n\"Gets on your nerves?\"\n\n\"You might say that. Maybe he'll relax a bit once his boss arrives. When is Quentin supposed to join us?\"\n\n\"King George Island, they're flying him in.\"\n\n\"He who would achieve much has little time,\" I say in an exaggeratedly nasal voice, but the Captain is immune to all irony. Even when he does speak he looks off in the distance somewhere beyond your shoulder, where some more urgent task seems to await him.\n\n\"We're supposed to give him all the help he needs.\"\n\n\"I take it we're supposed to do it out of love.\"\n\n\"Will you manage?\"\n\n\"Do you expect complications?\"\n\n\"It involves a lot of people.\"\n\n\"We can limit the number of participants.\"\n\n\"He wants to make the biggest SOS he can.\"\n\n\"But to do that he'll need our passengers.\"\n\n\"The biggest SOS in history.\"\n\n\"I assume that he's been informed of the restrictions.\"\n\n\"We're turning a blind eye to that.\"\n\n\"Are we?\"\n\n\"If anyone inquires, the whole thing was an emergency preparedness exercise.\"\n\n\"The passengers will have to agree.\"\n\n\"That's your task.\"\n\n\"I'll tell them tomorrow about Mr. Quentin's installation.\"\n\n\"We'll discuss the rest another time.\"\n\nAfter Helene moved out, when relocating to the house in Solln failed to achieve its therapeutic goal, the pictures on the walls faded into strange reminiscences. Whenever I looked at them I had the feeling I was looking through the window at some random life stored in the building across the street. I took them down one by one as I imbibed the red wine Helene's father had bequeathed to us: the old man had squirreled away these excellent bottles so that one far-off day they might help his son-in-law cope with the separation from his daughter. The pictures had left annoying outlines on the wall. Why is it that everything we do leaves an imprint (it takes a hundred years for a footprint in the Antarctic to disappear), why can't we simply glide through moments without a trace, like birds through the air? I didn't want to repaint all those walls, it was unclear how long I'd last inside them anyway. So I went into town and bought a large A3-size sketchpad and some watercolors, and started painting single letters of the alphabet onto individual sheets, after spending a long time deliberating which color to use for each. For A I chose a richly darkened yellow, like an aged Riesling, and for Z a garnet red pinot. O came out in a gray so soft it was indiscernible from more than a couple inches away. I painted one letter per day, and as soon as the paint dried I tacked the finished artwork to the wall. When at last the entire alphabet was adorning my walls I felt better in this house that I would never call \"my house.\" The letters allowed me to believe in a new beginning, and they beckoned to me from the walls, enticing me to read. On a trip to Ladakh I'd heard of a man who had become a reading recluse and withdrawn into a single book. Twice a week he visits a sandalwood dealer who lives in a wooden house on a stone base near the Indus River, and whoever wants to hear him read can find him there. The man reads one line from his book and then takes his listeners on an exegetical journey through all possible nuances. I felt tempted to adopt the same procedure. At random I selected a book out of a faux-antiqued series devoted to ancient philosophers and began reading line by line, paragraph by paragraph, with the same concentration as the guru in Ladakh, after which I took three sips and laid the book aside. Then I stretched my legs and returned and wrote down everything I could remember of the reading. Little by little all my initial flippancy evaporated and my supply of red wine dwindled but I sipped away until I had practically memorized the entire book. According to my informants in Ladakh it takes the word-fasting guru twenty years to guide his students through that one book, whereupon he starts all over again with a new batch of followers. Despite my admiration there was something about the whole procedure that bothered me, something didn't seem to make sense. How can people consider a book holy unless they've rewritten it for their own purposes? Is it even possible for two people to mean the same thing when they say \"God\" or when they talk about love? At first I underlined individual words or sentences, often two or three times, I circled them, boxed them in, filling the small spaces between the lines with my observations until I realized there was no reason not to use the margins. I didn't set the book aside until it was covered with my scribbling. After that I bought this leather-bound notebook. I declined the seller's offer to have my name engraved.\n\nAnd at the end of a long day on the open sea, when darkness has blackened everything, the stars grow dull, the wind breathes its last, our ship sails on into the last refuge of abundance. There is only one Terra Nullius left on Earth, and that's where we are headed, \"whiles all the night, through fog-smoke white\" the language shrinks back from the miracle, the silence awaits us beyond the mist, where \"glimmered the white Moon-shine.\"\n\nSo pick up your phone and dial right now for your chance to win, the first three callers will receive a free blow job, I'm not what you'd call happy-go-fucky, Charlie don't start without me, I've kissed the girls of Naples, as they say the pitcher will keep going to the well until it breaks, you can't afford to give things away either, Charlie, wait she's not just yours you know, do your looting while supplies last. The matter will be looked into at once, I'm afraid you're going to have to fly right back, you better eat something while the machine's being refueled, this is no longer a photo shoot, it's an emergency. They're pretty as can be, that comes from putting off the necessary repairs, the street is closed due to construction, please follow the detour, oceangoing tankers are kept in use until they break apart, see-lonce, see-lonce, watchin' the ships roll in, legs well worth the look let me tell you, where there's a middle class there will be banksters, you slave away for thirty years scrimping and saving every penny never going on vacation and then something like this happens, imao, do you have any idea what's at stake here? we're looking at a maritime emergency with international complications, all ships in the area, the _Urd_ , the _Verdandi_ and the _Skuld_ are headed for Gerlache Strait to rescue the passengers, we have to be prepared for every contingency. You want to know the problem with the indigenous people, I'll tell you, they're too docile, we need to infect them with our greed, otherwise there'll never be any peace between them and us, the overweight will have to weigh things over. The first ship should be there in roughly two hours, the captain of the _Urd_ has taken command of the operation, water doesn't get any colder. Then I watch them roll away again, you should snatch that up, the weather changes every minute and so does the business climate, and I'm sittin' on the dock of the bay, the atmospheric depressions return every thirty-six hours, watchin' the tide roll away, in the course of a single day we experience all four seasons, I'm just sittin' on the dock of the bay BREAKING NEWS HOPE FOR SURVIVORS BREAKING NEWS HOPE FOR SURVIVORS wastin' time\n\n### 6.\n\n## 54\u00b016\u20328\u2033S, 36\u00b030\u20325\u2033W\n\nA DAY WHEN clouds look like mountains and mountains like clouds. Alpine peaks spring up in the middle of the ocean, a tear in the sloping cloudbank exposes rocky cliffs and glaciers looming over patches of pasture, where reindeer introduced by homesick Norwegians chomp away at the vegetation. Trees have never set down their roots. The water inside the pot cove is rich in oxygen and krill and takes on a greenish tint. Here Creation appears with unfamiliar clarity, as though all cataracts have been removed and our collective vision was suddenly unclouded. We put into Grytviken, an old whaling station that was abandoned overnight and left to rot and ruin. The passengers stroll from the cemetery to the flensing plan to the mudhole where the elephant seals wallow motionless except when yawning. Our dock isn't far from the graveyard which offers a small but very choice selection. The names are etched in white stone, on calm days we pay our respects to Sir Ernest Shackleton with a champagne toast. The diesel tanks are lined up as neatly as the graves\u2014a reminder of how much blubber was processed in this cove. Inside the factory humans once dismembered whales, now time is dismantling the factory. Silence weighs on the dilapidated halls, the skuas fly in other skies. The whale oil tanks still exude a stench, so it seems to me: it's hard to breathe in the middle of the rusting slaughter-works. Here and there a roof slants downward between the clouds and the tin floor, red signs mark off an area infested with asbestos. In front of the bone-rendering plant three figures clamp their hands around an iron chain and lean back as though in a tug-of-war with long dead whalers. The wind carries a sound of giggling, the Filipinos enjoy playing hide-and-seek in the ruins. But how am I supposed to distance myself from this flensing deck, this place synonymous with death? The snow-covered mountains are mere backdrops, distant and detached. So well hidden are the fur seals that you have to pay close attention not to step on one by accident. The younger seals scamper into the water, twisting in middive, then give themselves a vigorous shake as soon as they crawl ashore. A stand of gentoo penguins keeps watch between an anchor and some ship's propellers (which, uncoupled from their purpose, are nothing more than grotesque jetsam), mocking glances behind red beaks. And by the jetty the _Albatros_ has listed ostentatiously for decades, its harpoon gun long turned landward.\n\n\"Hello hello, hey it's our guide, what an interesting place, isn't it; just like you say, this is where humans and the Antarctic got to know each other, it sure is quite a mess, they really ought to clean it up, do you know what that building was for?\"\n\n\"There are some plaques on the other side, along the main path, with detailed information.\"\n\n\"You're not going to make us traipse through all that mud now are you, Mr. Zeno, now that we've run into you.\"\n\n\"That was the blubber cookery, Mrs. Morgenthau. First they carved the whales up here right where we are standing, then they extracted oil from the blubber in giant cookers.\"\n\n\"That sounds like hard work.\"\n\n\"Lucrative work. With high returns. In a good year they cooked away up to forty thousand whales.\"\n\nI politely take my leave, otherwise I'd have to explain how first the fur seals were skinned, until there weren't any fur seals left, after that the elephant seals were killed for their blubber and the try-pots were heated with penguins when the fuel ran out, and when there weren't any elephant seals left the penguins were rendered into oil. Everything was put to use\u2014humans are always so eager to show Nature more efficient ways to manage her resources. I tramp across a gently sloping soccer field: the crooked goalposts a comforting sight. Slaughter by morning and soccer in the afternoon. Did the goalie's hands stink? Were the striker's shins streaked with blood? I leave because I know what they would say, the same thing everyone is always telling me: How come you have to be so negative? Why do you always insist on ruining the mood? Can't you leave off just for once? The same thing over and over, burbling around me from dawn to dusk: don't take it all so seriously, relax, stop picking on everything, turn a blind eye, things won't work out so bad, nothing's as awful as it seems\u2014everyone has installed the same software: deflect, downplay, be ready to duck when the storm hits. I wonder what words would have sprung to their happy lips if they'd been hauled off to the clinic at Pentecost, just as summer was settling in, with a persistent pain in the chest. A whole week of examinations. Probes boring into my body, as though the pain had to be hauled up from the deep. And then I spent days waiting for the life-saving operation and after that three months recuperating. When the doctors declared I was (almost) fully recovered, I dropped my bag at home and dashed off to the glacier, leaving Helene staring dumbly behind me.\n\nThe mere sight of the strangers in my compartment got on my nerves. The woman sitting opposite\u2014my equal in age and disappointment\u2014was holding a box of chocolates. She cautiously untied the ribbon, removed the marbled paper cover and carefully laid it on the seat to her left, then positioned her fingers over the selected confection like the grippers of a crane and lifted it from the box with clinical precision. The candy quickly disappeared between her pale-purple lips, with hardly any sign of chewing as she closed the box and retied the ribbon, only to tug at it a few minutes later and repeat the entire pedantic operation. No matter how many pieces she removed, the outside looked completely untouched, as though the candies were intended as a present. At the rate she was going she'd have nothing but an empty box with a fancy bow by the time she got to Kufstein or Klagenfurt. Meanwhile the man sitting by the window was using newspapers\u2014first _Bild_ and then _Krone_ \u2014to shield himself against the swelling landscape. He gave the appearance of wealth, a middle-class man ready for first-class travel on that hot summer day, faint marks on his suitcase showed it had once been covered with tacky tourist decals, perhaps he'd acquired some aesthetic guidance since sticking them on. The man studied the first tabloid from top to bottom and then proceeded to the next with similar dedication. Such keen regard for sensational headlines and pathetic ads irritated me. I had to step out into the corridor. When we reached Salzburg three girls with blank faces entered our compartment, without paying any attention to us old-timers. The woman permitted herself another candy, the man remained buried in the _Krone_ , the three girls regaled one another with gossip from school, and when the train stopped in the middle of a field I was overcome with the fear of being trapped in this compartment, my view blocked by the _Krone_ , with nothing to eat but a last piece of candy, my ears ringing with the vapid talk of the young generation, and never being able to make it back to my glacier. The train started up again, I calmed down somewhat, little did I know things would only get worse. So I wouldn't have to wait for the bus in my weakened condition, the host of the Zum Kogl guest house picked me up at the station. A shaggy dog was panting in the back of his jeep. I have to tell you right away, you're not going to like it, something's happened, you're not going to like it. The road was all switchbacks and hairpin turns, with bare landscapes on both sides: without ice and snow the Alps are grossly ugly. I'm glad to see you're back in shape, we prayed for you, our whole family\u2014the man has seven daughters, or is it eight, in any case all daughters, and praying isn't alien to him. For a moment I was distracted by a racing cyclist hurtling downhill, the car swerved left, the wheels grated on the gravel, I looked up through the windshield and saw... nothing. No glacier. No living glacier. Only fragments, individual pieces, as though its body had been mangled by a bomb. The escarpment was still iced over, but all that was left further down were lumps of darkened ice, strewn over the cliff like rubble waiting to be removed from a building site. All life had melted away. I told you it would be hard, that's not a pretty sight. The host's voice evaporates in my memory. Apparently I climbed out of his car without saying a word\u2014so he informed me that evening over beer and _Tafelspitz_ \u2014and staggered from one shard of ice to the other, as though drunk or blind, reminding my host of the farmers saying good-bye to their livestock during the Mad Cow scare, so he told me that evening. I wasn't capable of such a gesture, I was too stunned, all thoughts and feelings paralyzed. I knelt next to one of the remnants, the ice under the sooty-black layer of dust was clean, I ran my fingers across the cold surface, then across my cheek, the way I always did, performing my ritual greeting. In the past I could plunge my arms into the fresh snow and bring up full scoops that made my hands so cold they would revitalize my face. I licked my index finger, it tasted like nothing. Only then did the first trivial thought occur to me: never again would I be able to fill plastic bottles with glacier water to sip so enjoyably at home. My host was standing next to his vehicle, I brusquely signaled for him to leave me alone. Then I lay down on the scree, all balled up, a picture of misery, I would have welcomed any emotion that didn't hit me like a positive lab test result. Not knowing what else to do, I stayed like that until a hiker put his hand on my shoulder to check on my condition. I snapped at him.\n\n\"You're hiking here?\"\n\n\"It's amazing up here, isn't it, and such a beautiful late summer day.\"\n\n\"Don't you see what's happened?\"\n\n\"Oh well, not much snow this year.\"\n\n\"This glacier is dead, and you go sauntering blithely past. Get lost, disappear, you disgust me.\"\n\nThe man didn't deign to look at me again and went on with his hike. It would have been pointless to calculate the volume of melt for this September, to take the balance of the summer. There was nothing left to measure, not on this mountain. At some point I got back on my feet and climbed uphill, heading no place in particular. On the steeper terrain a stump of ice had survived in the shadow of a desk-sized rock, it served as a temporary support, I set down my notebook, the wind leafed through the pages. What hadn't we measured and weighed? And what of all the findings we had compiled, the models we had constructed, the warnings we had issued in careful scientific format? The pages of futility are filled with noble ambitions, they need to be torn out, every last one, our methods have failed us utterly.\n\nWe had issued warnings, but in vain, things only worsened with every passing year. Only when it is too late do people hear Cassandra's voice: today even more sanguine souls have joined the chorus of doomsayers. Nevertheless I hadn't foreseen this degree of destruction, not when the snout disappeared (I had just turned fifty), not when the tongue broke off and the calved ice melted away as quickly as it did (I had just turned sixty), and now this blow from the blind spot of our calculated optimism. If even the experts are surprised by the terrible speed of the demise, whose intervention can still save anything, whose point of view matters, since everyone else harkens to the rotten call of comfort and convenience? My work had consisted in documenting our delinquency\u2014the father confessor masquerading as a scientist.\n\nI pounded my fists on the stone table, in my pain I thought of the girls on the train, chewing with difficulty on the gum of life, these three girls who pass in the world as innocent. But what is that kind of innocence worth, when we all know they're bound to wind up guilty? That is what lies ahead for them as well as for us, they will continue this devastation, they will go on destroying the very foundation of life. They don't give a damn, just like most of us, they won't rest until they've consumed polluted squandered destroyed everything. I left the following morning. In the next valley over, the surviving ice surfaces had been shrouded with white burlap, under which an emaciated glacier was emitting its death rattle. I felt like a doctor in a hospice.\n\nWe called it \"swimming\"\u2014swimming in the river of ice. When we dared drop into the moulins, the glacier mills, to use them as chutes, trusting ourselves to the twists and coils as if the glacier were obliged to protect us as we slid on the seat of our pants through pipes of the purest blue. It was dangerous, moderately dangerous, we had checked to see where and how we would emerge, even if we occasionally miscalculated the acceleration and came shooting out of the shaft like a cannonball while it thundered underneath, so that even the person who was brushing the pain off his pants had to laugh at the glacier's acoustic commentary. Yes we gathered our share of bruises, but we really got to know the glacier, we stuck our nose in every crevasse, we even believed we could hear the icy creature sliding into the valley on a layer of its own water, and we were amazed at the multi-hued splendor within what seemed to be a monochrome universe. We opened our eyes (and not just under the polarizing microscope) to its delicate spectrum, the colors in the flatland seemed garish by contrast. Where the ice was as hard as alabaster we found blue caves which we decided to enter right then and there, afraid we might not be able to find them again when we returned. Afterwards we went our separate ways, some hurried back to the city, others retreated to the valley, in the end I was the only one who chose to shuttle between glacier and university, on solitary days I abandoned myself to the tranquility of the ice, the clang of the water, I became a stone pressing on the ice and leaving my own imprint, and one time I was surprised to feel the urge to pray inside one of the icy cavelets that served as my makeshift chapel\u2014not to God (and how vain a deity to command his name be held so sacrosanct) but to variety and abundance (written out like that the words seem wooden, and it isn't enough to replace \"God\" with \"Gaia\" either). All alone I searched for insight within the clearest and coldest blue, I filled the icy caves with my own variations of the eternal, just like the monks of long ago who filled their stony grottos with drawings, only why didn't they see the stone itself as a perfect image of God, the weathered surface, the patches of moisture? _Deum verum de Deo vero_ , can truth reside in such a phrase? Within my blue chamber, in the belly of my frosty whale, God has dispensed of every superfluous word.\n\nJeremy is small and rather nondescript, but his glasses make him instantly recognizable wherever he goes, they look like they were lifted from some California comic strip that transforms every polar expedition into a heroic saga, especially those of Ernest Shackleton, whom Jeremy worships above all others, he can give his lecture on Shackleton six times in one season, each one fresher and more original than the last. If they're not on duty other lecturers will lurk near the door to the auditorium so they can listen for at least a few minutes as Jeremy hoists Sir Ernest to promethean heights (if our Californian whiz kid were looking for spiritual role models he would undoubtedly place Shackleton alongside Isaiah and Jeremiah). He's seen me writing in my notebook with its heavyweight cover, I don't try to hide this activity, it's impossible to keep something like that secret, and anyone who believes otherwise will soon be disabused of the notion, because on board ship everything can be seen and anything seen can also be heard. Jeremy surprised me by placing a handwritten note under the service plate, which I manage to read sometime between hors d'oeuvres and dessert: \"Since I see that you, too, are keeping a journal, I feel you should take a moment to remember Nathaniel Hawthorne, who once applied to join Charles Wilkes' US Exploring Expedition as official historian. His hopes were thwarted, however, after a certain Congressman opposed to his appointment argued that 'the style in which this gentleman writes is too wordy and ornate to convey a genuine, sensible impression of the atmosphere on the expedition. Furthermore a man so talented and cultivated as the said Mr. Hawthorne will never be able to grasp the national and military significance of any discoveries it may make.' I found this tidbit in my readings. You should feel fortunate to have free rein to do what your colleague was prevented from doing, even though like him you will probably never be able to grasp the national and military significance of the Antarctic. Therefore: eschew prolixity and stylistic embellishment, and think upon Shackleton and the deprivations he endured.\" When I looked up I saw that Jeremy was once again pointing his camcorder at me, I held the paper in front of my chest like a kidnapped hostage, and swore the just-invented Shackleton Oath to uphold the Unadorned Word. Jeremy grinned and panned out away from me through the glass and onto the ocean. He would be welcome on any expedition because his good mood is infectious even when he is pensive. That is a rare talent. Of course he can hardly help summoning old Shackleton, every one of us identifies with the man (all but El Albatros, that is, who can't get over the fact that Shackleton planned to sell albatross chicks to gourmets in London and New York, his hunger had made them particularly tasty), he is the Good Man of Antarctica, the famous photo of _Endurance_ trapped in the ice is on display in the elevator, and the explorer's portrait is hanging on the wall in front of the dining room, he could easily be a member of our group, he'd get along well with us, he mistrusted strict hierarchies and valued communality over subordination. Above all he was the only polar explorer who traveled to the southernmost part of the planet knowing it would likely be his final destination. He could no more imagine a grave in thawed ground than he could everyday life in moderate temperatures.\n\nHaving taken our time in Grytviken the captain calls for full speed, the _Hansen_ plows through the swells, water water everywhere, as though we were the first sailors to pass through this sea. Barely three hours off of South Georgia we see whales, very close. Beate is so excited, when the humpbacks dive she holds her breath, and inhales together with them when they resurface. Her enthusiasm is undiminished by the dozens of cameras clicking and snapping around her like whips, did you see them, she calls out to Jeremy, who has blazed a trail through the dense spectatorship, and who calls back, \"Oh yes, I did, I see we're already clicking into place, too.\"\n\nHe'll have to hell to pay that's for sure, I've also kissed some French girls, what kind of crazy choreography is that, who come from Paree. All told we're talking about 220 passengers, English, Germans, Americans, Dutch, Swiss. Oh, you must have taken a wrong turn at the big intersection, now you'll have to go back the whole way. Norwegians, Brazilians, Canadians, New Zealanders, Austrians. We know enough but we understand little, the last spasms, not the faintest idea, to cum in the mouth costs double, the conditions there are unimaginably extreme, it's snowing pornflakes, we're infesting now in our future. Go ahead Foxtrot Two Niner, over. I can see people, dozens of them, standing in little groups, over. Have you tried to establish contact, over. Yes, a few are waving their arms, over. What is their condition, over. I can't tell, over. Any signs of panic, over. No signs of panic. One group is very close together, it looks like they've formed a chain, over. No, cows aren't holy, sheep goats and cattle aren't holy and wild animals aren't holy either, nor are the birds of the sky or the fish in the sea, pigs aren't holy and neither are chickens, not even the lamb is holy. Foxtrot Two Niner Foxtrot Two Niner, go ahead, over. They've formed a circle, over. A circle? over. Something like a big zero, over. Descend as you can and fly a few 360s to calm the people, over. Wilco. Out. The experts are disputing this prognosis, today the fixed price for lithium was announced on time, thrushes are dropping dead from the sky, and so this planet Earth spins around its own axis and never stops moving. Email the passenger manifest, on top of that there are 78 crew members, we have to find out as much as we can about the experts on board, we'll look for every missing person BREAKING NEWS RESCUE OPERATION IN EFFECT BREAKING NEWS RESCUE OPERATION IN EFFECT nothing else matters\n\n### 7.\n\n## 60\u00b011\u20325\u2033S, 50\u00b030\u20322\u2033W\n\nWHENEVER I WAKE up early I run sixty laps around the weather deck at a fast pace in the sleepy gray light. I'm wide awake, the waters of the Antarctic are running alongside me, flowing around me, clockwise, accompanying me on my laps just like H\u00f6lbl did decades ago when we circled the sacred temples of Ladakh early in the morning, before the workday began. It just seemed like the right thing to do, although some people accused us of trying to ingratiate ourselves with the locals\u2014people eager to dismiss sincere interest in broadening horizons as some kind of smarmy xenophilia. H\u00f6lbl dubbed the venerable Lama \"Maestro Boltzmann\" and His Holiness took a mischievous pleasure in the nickname, sensing that the unfamiliar sounds contained a certain prestige, and he wasn't wrong. The water creaks and groans, the waves are only a few meters high, our crossing is relatively calm, the Drake Passage is usually good for at least one storm before letting a ship into the eye of the hurricane, into the paradisiacal tranquility of Terra Nullius. I perform my rotations in sync with the Circumpolar Current that every moment spins one hundred fifty million tons of water, birds glide through the twilight, their sharp wings slice through the cold air as they loop through the sky in perfect figure-eights, white albatrosses soar in steep arcs, storm petrels dive like rash decisions and disappear behind glimmering crests, plunging into the feed troughs between the waves, and I circle on, with each of my steps the ship recedes into oblivion right under my feet. I would be perfectly content with this solitary round dance of self-forgetting and nothing more if duty wasn't calling: I have another lecture coming up in just a little while and I have yet to update the announcements about our upcoming landings. Every day at 7:30 PM I can be found at the radio coordinating plans with the other expedition leaders. Some of the voices I recognize right away, and some have accents that leave no doubt as to their countries of origin (Beate claims this is entirely natural, even whale songs show regional differences, underwater dialects). At the moment there are eight ships in the vicinity of the Antarctic Peninsula, just among ourselves we divvy up the mooring places that were booked months earlier, we trade sites and help each other out to make up for cancellations due to weather. And we stay out of each other's way, after all we don't want the sight of another ship to ruin the illusion that we're all on our own here in the Antarctic, far removed from any regulated traffic, alone at the end of the world.\n\nNo one in the institute had any illusion that I might consider changing my object of inquiry (the very phrase makes me think of an ingrown toenail), not at an age when my beard is straggling toward retirement. Nonetheless I couldn't continue as I was, the Alps had become unbearable, what was there to be gained by accompanying one more glacier to its death? And to go on giving lectures undeterred seemed as grotesque as teaching veterinary medicine to paleontologists. No, there was no alternative but to bow out. Two colleagues invited me to join them in the High Caucasus Mountains. They didn't want to see me leave the institute, and for that most sentimental of all reasons: they were used to my being there. You can cook for us in base camp, they joked. Actually I was considered an exceptionally gifted chef, a reputation that rested solely on the large pot of Jamaican fish tea soup I always brought to our summer parties. The first time I made it they were all dumbfounded, no one expected a dish like that (with that name, those ingredients, that taste) from me, a man who considers the tropics anathema in general, the Caribbean a steam pit in particular, and the idea of serving frutti di mare in the foothills of the Alps the epitome of decadence. And I would have no idea about the soup if a certain Jamaican who grew up in England hadn't fallen in love with a young woman from Munich. He made ends meet teaching advanced conversational English at the M\u00fcnchener Volkshochschule, where we discussed ska lyrics by Madness and read excerpts from George Mikes' _How to Be an Alien_. At the end of the semester he invited the whole class over, gathered us in his kitchen, and then with the flourish of a circus director he opened the lid of a pot with a diameter as big as an oak's and let out aromas that could spawn legends, fantasies of lazy afternoons rocking on boats with straw roofs or of diving into scallop beds. I took the course again the following year, even though my English was in good shape\u2014not least thanks to an intensive exchange with colleagues from the University of East Anglia and the Jawaharlal Nehru University\u2014just to get another taste of this soup and some idea how to make it. The recipe is incredibly elaborate, this Jamaican Fish Tea contains all the treasures of the sea, the ingredients are difficult to obtain (requiring the combined forces of the Viktualienmarkt, Dallmayr and K\u00e4fer), the preparation has to be planned well in advance and started at least one day before the feast. I spent weeks looking forward to this day, a day that knocked on my door with a tattooed hand. In the Caucasus, though, I'm not exactly in my culinary element, as I told my colleagues, and besides I can no longer bear the sight of living glaciers. That was a lie and they knew it, I still loved ice, but my perspective had changed, when I used to look at a glacier I saw history and change, abundance and endurance, the face I now encountered was grotesque, the remaining ice had become a mirror of our own neglect. From here on out no matter what I might see, I would never be able to recover my earlier acceptance of things. It seemed to me as though only now did I perceive their essence. Behind all the cornices and all the stucco I saw nothing but prisons. The people crowding the shops in the pedestrian zone struck me as display mannequins, pushed and pulled hither and yon by random jerks and jolts. You don't need someone like me on your team, I said, and no one contradicted. That year saw the last pot of my Jamaican fish tea soup.\n\nOn a ship at sea it's hard to get out of the way, the passageways are straight and narrow, it's best to stop and turn so you're back is against the wall, pull in your stomach and don a broad smile that makes it easy for the other person to glide past. Everyone on board is quickly located: within a few days it's clear who has set down roots where, which binoculared guest has staked out which spot in the hope it will content him throughout the trip, an armchair in the crow's nest of the Panorama Lounge, for instance, where it's easiest to escape the ones who can't keep still, who change positions every fifteen minutes, heading now to the weather deck now to starboard now to port because they're afraid of missing something, who soak up every scenic view and then rush back inside where it's warm, on to the next lecture, the next film, to coffee or afternoon tea. And those who have paid a great deal of money for a top-of-the-line suite may under no circumstances be disappointed. Emma at the reception desk tells me that no one can teach the art of complaining as well as rich people. As Expedition Leader I am fair game for the restlessly inquisitive, passing from Deck 3 to Deck 6 is like running a gauntlet of questions. I prefer sitting in the bistro, where every second table has a jigsaw puzzle\u2014postcard scenes, carved into 500 tiny pieces waiting to be put back together so as to match the picture on the box, and whoever succeeds can graduate to 1,000 or 1,500 pieces, clearly all puzzle-piecers must be happy people. I sit down at one of the tables for two, across from Mary, who keeps her recording device running even while she sketches away in her unlined notebook with a sharpened pencil, the Antarctic Ocean is on my left, and on my right is Paulina, who plays the part of the indifferent waitress with furtive joy, repeating my order with intense concentration, as though it were the first time she'd heard me ask for a double espresso with a good dose of foamed milk, but foam only please, so as not to drown the taste of the coffee in a lot of milk, she recommends the marble cake which I detest, whereupon Mary orders a slice just out of solidarity. We have just crossed the 60th parallel, I explain, only now are we truly in the Antarctic, from this point on ships aren't allowed to dump waste water, which naturally limits the length of our stay in these latitudes, an additional advantage of this sensible regulation, after all this is the only ocean humans still haven't polluted, and that's the way it ought to remain. Just four percent, says Mary, while I take a swallow of water, the Antarctic Ocean makes up only four percent of the Earth's surface water. Outside, a flock of cape petrels floats past on invisible pillows of air. Paulina serves the coffee and cake with an air of professional efficiency and practiced nonchalance. Mary thanks her by name, which she reads off the tag pinned over Paulina's breast pocket. Paulina responds with an exaggerated smile before turning to me and asking, \"Anything else, sir?\" Whereupon I answer stiffly, \"No that will be all, Paulina, thank you.\" Mary asks what I think would happen if it weren't for the Antarctic treaty. \"There'd be a public discussion about the exploitation of the Antarctic and lots of bargaining behind the scenes. Lobbyists would stress the need for mining and oil exploration, and a campaign would be launched against the penguins with the slogan 'Should we give up flying just because these birds don't know how?' The penguins would no longer be photographed standing up but lying down, so that they looked plump and a bit dodgy, as though they were begging to be slaughtered. We can give up the luxury of our sentimentality at any moment, and there's no guarantee this won't happen prematurely, despite the treaty, when the going gets tough, who'll pay attention to voluntary obligations when even binding treaties don't count for much.\"\n\n\"A whole lot of people would have to exert pressure to keep that from happening.\" Mary interrupts me with a na\u00efve enthusiasm that does my heart good and hurts at the same time, my face betrays my skepticism. I might excuse her remark, thinking how weighed down I must seem, perhaps, she says, it's because I lack the experience of a common struggle that would give me courage, I would like to forgive her, but she really shouldn't say that. I yearn to feel euphoric. We go on talking about ice and the world, she asks questions that require answers beyond the usual prepared platitudes, and all of a sudden I hear myself confessing how I sometimes feel ashamed to be working on this ship, especially on this trip when I'm even more responsible because I'm Expedition Leader, the tourists should be sent elsewhere, to a theme park, to a traveling Capsule of Eternal Ice that can be set up anywhere, you enter through the front and leave by the back, but I myself couldn't live without my sojourns in the ice, and she looks at me so full of understanding that I tell her about my theory of mass hyperthermic cognitive dysfunction, a syndrome with symptoms that mirror those of hypothermia when overexposure to cold addles the mind to the point that people freezing to death suddenly feel warm and start taking off their clothes. This paradoxical undressing is known as hypothermic cognitive dysfunction, and sets in when the body temperature drops below 90\u00b0. And while we have scientific proof that people in this state lose their ability to think and therefore to save themselves, I don't know of a corresponding temperature threshold marking the onset of the hyperthermic equivalent. But it's obvious that our collective mind is similarly addled, since we keep turning up the temperature on a planet that's already unbearably hot. Mary seems bewildered, suddenly she's avoiding my eyes\u2014does she consider my theory stupidly silly or simply smug?\u2014she stares off to the side, or have I alienated her in some way? The way you trumpet out your so-called convictions, Helene once snarled at me while we were fighting, makes them sound like insults. Mary doesn't react to my soothing chatter, her gaze is paralyzed, focused on a point at the other end of the room. Her face goes blank, surely it's not my fault, and it's hard to imagine that the sight of the small, husky man stretched out in one of the armchairs and fingering a book and staring off dreamily might have hijacked her attention like that. \"Mary, what's the matter?\" Her face has broken out in splotches of red. It takes a while before she answers me. \"That man, what's he doing here, what does he want?\" Before I can ask another question she has stood up and scurried off, leaving her paperback and recorder in my care.\n\nMy sadness scabbed over into rage. The semester hadn't yet begun, it was easy to avoid running into people. Helene pounced on every invitation and stayed out of the house as long as she possibly could, assiduously representing the two of us, she even drove out to her mother's by herself to celebrate a round-numbered birthday, I don't know if she concocted something about a present being from the two of us. How long might it have gone on until everyone simply forgot she was actually part of a couple? Whoever believes in constancy and steadfastness must despair at the rapidity with which individuals couple and their unions break apart. At first acquaintance, the other person is an impregnable fortress, three dates later, after a few kisses and some mediocre sex which both sides sugar-coat, all drawbridges are let down. The lie of eternal love attunes us to the lie of eternal life. Later we have a hard time explaining what the fuss was all about. During my first weeks alone I conducted an experiment on myself, with curtains drawn and lights dimmed I sat on the floor and resolved not to get up until I had recalled half a dozen sexual delights in exact detail\u2014something more substantial than the faded memory of a breeze wafting across our bodies or velvet-soft skin. But even after hours of biographical excavation I couldn't do it. Instead, my brain called forth athletic achievements as vainly as I had stored them: three times in a night (in the ski lodge, as a student), two hours without stopping (to win a bet with Helene when she claimed I didn't have the stamina). At some point I had to get up and get groceries. Every acquaintance I came across inquired with urgent concern about my convalescence. I disappointed them all. Instead of sharing some uplifting tale of my narrow escape from death, I told them about a glacier that had been exterminated, and that annoyed the good souls, they walked away shaking their heads and rendered disparaging judgments before they even reached their cars and drove down dead-straight streets into their remotely controlled garages and from there quietly ascended in paneled elevators up to their wallpapered crypts. They considered me ungrateful, either toward God or fate or the health system. \"Well well, here I thought we'd lost you but I see you're still alive and kicking,\" the vegetable peddler pontificates\u2014the same man who charges a lot more money for just a little more taste. It's uncanny how orderly everything is in Solln, and how resolutely the Blessed defend their idyll with all myopic means. My neighbor pesters me with an account of his illness, as if we owed each other some reciprocal sympathy. \"Just because our pain is similar doesn't mean our suffering's the same,\" I told him, and that was the end of his pestering concern. What a happy circumstance that people quick to interfere are so quick to be insulted. Unfortunately this kind of self-disclosure wasn't an isolated case, lingering illnesses were being tooted and touted on every channel and frequency, as if contracting a severe disease were the most remarkable individual achievement of our era. You have cancer, how extraordinary, prostate or breast or lung or liver, you have ulcers, how exceptional, your body is wasting away, oh my, it's being consumed from within, how astounding. The sunniest beaches are littered with melanomas, this obsession with one's own miniscule existence is nothing more than pitiful, what bloody fucking miserable bastards. \"Hey, I understand that,\" Paulina interrupts my jotting. \"German's a lot like English, isn't it?\"\u2014 because those last words I've just written in English. Paulina loves looking over my shoulder (like she's doing now) and especially loves it when she recognizes what I'm writing, even if it's some crude obscenity like that. I hardly notice the English words sprinkled in, here it's rare that we native speakers of German are all to ourselves, so my German is becoming anglicized step by step. To prevent me from suffering the same fate as my first expedition leader, who jumbled English and German into a kind of gobbledygook, and to keep my own language crystal clear, I often retreat to the weather deck and mumble poems to myself that I learned in my youth, poems our teacher Herr Stradel assigned us in the Fr\u00fchlings Gymnasium, which we wound up learning by heart (in my case on my way back from school), without a clue that they would stay with us forever.\n\nDas Gestern, das mich flieht, kann ich nicht halten,\n\nDas Heute dr\u00fcckt mich wie ein Frauenschuh.\n\nDie kleinen Wanderv\u00f6gel schon entfalten\n\nDie Fl\u00fcgel herbstlich ihrer Heimat zu.\n\nIch steige auf den Turm, die Arme weit zu dehnen,\n\nUnd f\u00fclle meinen Becher nur mit Tr\u00e4nen.\n\n\"Translate it for me,\" Paulina requests as she so often does when she looks at a page I have filled with my scribbling. If I translate that into English it won't make much sense, I say, pushing back my chair, but if I translate it into Paulinish we'll both understand it better. My hands have taken hold of her wrists, my lips caress her neck, she pulls back and retreats to the bed. \"Every word has at least two possible meanings,\" I mumble, a left-handed clumsy meaning, my mouth sucks hard, and an always-has-to-be right meaning, my mouth leaves one nipple and wanders to the other, slowly crossing the hollow between her breasts, the tip of my tongue drumming against her skin.\n\n\"I'd like to come inside you without you noticing.\"\n\n\"You say the craziest things.\" And here it is again, this laugh, the most lovable thing about _Homo sapiens_ , and I say: \"Yes.\" More than that would be too much chatter, her laugh turns to moaning, we are sinking, air bubbles rise to the surface of the water, we are sinking, everyday colors vanish from sight, we tarry in the deep as if we could hold our breath as long as we wanted. When we surface I listen with one ear to the latest rumors which she stirs up the way a gust of wind blows a raked pile of dried leaves (it's a feast for the ears when she and Esmeralda fill the refrigerators every morning. Their mouths rattle away like sewing machines, snatches of overheard conversations are immediately reworked into colorful elaborations, while the bottles clink and clink as they're taken out of the boxes and loaded side by side.) At the beginning of our relationship I was worried that her liaison with me might damage her esteem in the eyes of her fellow Filipinos, but in fact the opposite happened, when it comes to cruise ships in the deep southern latitudes I'm evidently considered a good catch. I open the curtains, the expedition leader is entitled to a good view, Paulina was used to windowless sleep\u2014the only thing that's the same in all the rooms is the screens.\n\nAnd now there came both mist and snow,\n\nAnd it grew wondrous cold:\n\nAnd ice, mast-high, came floating by,\n\nAs green as emerald.\n\nPaulina maintains that German sounds beautiful, she doesn't learn any vocabulary, she catches useless words such as _Waschlappen, Hauruck, Kasperltheater_ (which in her rendition has limited recognition value) and inserts them into the least appropriate situations. I'm sitting undressed on the edge of the bed, looking at half of myself in the mirror that's mounted on the bathroom door. The years have not simply passed, they've folded themselves into my skin, deposited themselves on my hip, and there's no reason to assume the invisible half might show anything more reassuring, it's a mystery to me how Paulina manages to ignore everything that by rights should dampen her desire. Instead she leans forward and her lips touch my scrunched-up penis with the lightness of a scarf grazing the skin.\n\nFog is setting in, it's not rising off the sea but hovering above it, just enough to let the light safely in its lock before passing it to another level. Now the iceberg behind is discernible only by its base. A bird slips out of the mist and flutters past. \"And a good south wind sprung up behind; \/ the Albatross did follow.\" \"We humans have eyes made for hunting,\" says Jeremy, \"our nose could fall off without great loss to our senses, our ears serve only to uglify our face, but our eyes are sharp and alert, they can be relied upon.\" \"Especially,\" I add, \"when they fix on something in order to kill it.\"\n\nGet a move on, bigmouth, no I'm afraid we don't have any hotels here, no guest rooms either, you know we never get visitors unless they've lost their way. The _Urd_ is already in position and has begun rescuing the passengers, but it can't take everyone so the rest will have to be divided among the other ships. Get a move on, trackdog, don't lose that scent, estimated arrival is in forty-five minutes, the Spanish girls are lovely, rent-a-friend, oh yes indeed they are, a friend for rent, cut. As soon as we heard the sound of the motors someone suggested we should form the SOS again, as a warning sign, with us on the ice and not a ship in sight. Pipe down, 24 percent of those polled believe that nature has its own right to exist, but the ladies of Calcutta, blackbirds are dropping dead from the sky, cut. We had already begun to disperse, although the lecturers kept trying to herd us back together, we probably could have managed to make a smaller SOS, everyone immediately wanted to make the O. Cut, two cotton candies and two carousel rides for the price of one gingerbread heart, first the saplings burn, then the shrubby underbrush, the young trees, the dead wood, and all that just fans the fire, the price never reveals the truth. We formed the O very quickly, it was way too big given our numbers, someone called out, come on over here, let's make the S, I could hear people screaming in other languages, cut. Don't give up, go hog-wild, cool off, best put your hopes on ice, cut. I hurried over to help out with the S, thinking surely we can pull that off, in any case it was a lot smaller than the O, and that was it, we didn't get any further, cut. The wildfire reaches the flashpoint and consumes the tallest trees, it blazes away, hotter and stronger than ever, starlings are dropping dead from the sky, are sweeter by far, then comes the third fire that burns everything away, destroying all life, the third fire is the inferno that burns down the world once and for all, cut BREAKING NEWS STRANDED PASSENGERS RESCUED FROM ICE BREAKING NEWS STRANDED PASSENGERS RESCUED FROM ICE all ablaze\n\n### 8.\n\n## 62\u00b012\u20329\u2033S, 58\u00b056\u203243\u2033W\n\nON MY FIRST trip to the farthest south I emailed Father a few photos\u2014\"penguins & son,\" \"frosty morning mood,\" \"sea meets sky,\" \"land of ocean\"\u2014using the address of the director of the assisted living home, with the request she show them to him on her computer. His reaction was predictably gruff: how disappointing you didn't just vanish into the unknown. If this Internet business can stretch its tentacles that far then there's no place left on the planet where a person can really be alone. I should have thought twice before clicking send, given Father's general outlook. When he looks beyond the horizon he's not longing to find Atlantis, he doesn't dream of crossing the desert to Timbuktu or discovering Shangri-La\u2014all he wants is the peace and quiet of a solitary hike. He couldn't possibly understand why I get antsy if I miss my routine fix on the website of the European Space Agency (by way of the WLAN connection on Deck 4) where I track the breakup of the Antarctic Ice Shelf. I know it's proceeding at a pretty fast clip, so why do I have to keep verifying the fact? I still haven't told my father anything about King George Island, otherwise his notion of the virgin icescape would be trampled on by boots both moon and military. Even as Expedition Leader I can't keep us from landing here, we couldn't put in at Elephant Island because of wind speeds exceeding 56 miles per hour, so we don't have any other option. King George Island is 90 percent ice, 10 percent research stations and penguin colonies, that's how I would have to describe it to him, the stations are a few decades old, the colonies have been here for 30,000 years. The island counts as a spearhead of human settlement because it houses the only hotel in the Antarctic, the Estrella Polar (except the hotel has closed down and Polaris can't possibly be seen from these latitudes), as well as a military outpost for the Chilean Air Force, which can also be used as a jumping-on point by impatient ecotourists who contrive to bypass the Drake Passage. The landscape is infested with research stations. Every country that wants a say in the future of the Antarctic, I would explain to Father, has to maintain a permanent station, and the most economic place to do that is King George Island. Russia, China, Korea, Poland, Brazil, Uruguay, Argentina and Germany are all vying for the Antarctic Cup. The bases are close together, which would hardly seem to foster any sense of scientific rigor, while at the same time feeding the suspicion there's more rummy than research going on, that people are just biding their time waiting for the day they're allowed to drill for oil and not just ice (at present all truly groundbreaking research is being conducted deep in the ocean or far inland, during the summer the field teams are always on the go, camping out on the ice). Every now and then we visit Chile's Eduardo Frei Station. The sight of the settlement rising from the shore, the makeshift buildings\u2014bank, post office, store, school and hospital\u2014delights the passengers, it's practically a normal village, there are women and children, it flies the national flag, issues its own stamps, and sends newly minted Chilean citizens into the world who with every cry for their mother's breast advance their country's claim to the Antarctic peninsula. How is it possible that neither the Soviets nor the Americans have launched a highly pregnant woman into space just so she could give birth to the first extraorbital baby and thereby establish a legitimate claim to the solar system, the galaxy and the universe itself? I would tell Father that we're avoiding the Russian station Bellinghausen on account of the many derelict oil drums, the wreckage and scrap iron that litter the shore, exposing the true legacy of the human race: rusty garbage. But there's also a colony of chinstrap penguins, we put in close to that, the calls of the black-and-white-uniformed animals merge with that of the red-uniformed humans into a kind of high-pitched cacophony as the extraterrestrials land, equipped with curiosity but lacking a common tongue. But the chinstraps may be more used to such intrusions than we think, as El Albatros explains, between the launch of one Zodiac and the arrival of another: evidently they're equally unable to communicate with the gentoo penguins who stray into their colony, it's not even certain the chinstraps notice the interlopers. But the passengers do notice the natives, they savor every minute they are allotted with the penguins, we have to insist loudly and urgently when it's time to go back. As the passengers come ashore I stand with one leg in the water next to a metal stepstool that allows them to take hold of my arm and land on relatively dry ground while I mumble polite mantras\u2014even if after two hours in the freezing water I'm ready to chase off every Antarctic tourist with a savage grimace and a primal scream. A stream of phosphorescence streaks through the water, I steady the Zodiac, a spritely Swede climbs in and shows me his drawing of a penguin with its beak pointed skyward, a few strokes, the barest approximation, a large inflatable assault craft rushes past full of soldiers, the Chilean flag painted on the side of the bow, it comes dangerously close before veering away, leaving a wake that causes me to lose my grip on the dinghy, and then makes land not far from us, right in the middle of a large flock of penguins who reluctantly waddle apart. The first soldier jumps off and immediately lights a cigarette. He takes a few steps inland and stands right in the middle of the colony, his posture relaxed, a cigarette dangling from his mouth. Our passengers, whom we have diligently schooled to keep the proper distance and the appropriate demeanor, stare in disbelief. \"Take over,\" I say to Jeremy and set off in the direction of the soldier. \"Wait a minute,\" says Jeremy. \"What are you doing?\" I call out. The soldier looks at me blankly. I point to his cigarette, and with unmistakable gestures demand that he stop smoking. He ignores me completely, turns to his comrades and smirks, this look-at-me smirk that makes me turn white with rage, I start screaming bits of Spanish, I run up to him, shouting at him to stop, and grab him by the arm. With a single unexpectedly violent motion he shakes me off, I tumble a few steps, then try to lunge at him but fall clumsily to the ground, my face lands in the mud. He takes his pistol out of the holster, releases the safety and points it at me. Suddenly Beate and El Albatros are at my side, they both try to reason with the soldier, in Spanish, they hoist me to my feet and hold me firmly between them, as if they wanted to prove to the soldier what little threat I pose, I stare at the man, shaking, he looks at me with disdain and turns away. El Albatros holds me tight while Beate distracts the passengers who have quickly surrounded us, forming a colony of humans. I stand there without moving for some time, and finally El Albatros decides it's ok to let go of me. The soldiers have turned their backs to us, they are already marching off, no one can say where to or to what purpose, a few wisps of smoke curl into the air, and I wonder where those cigarettes will be ground beneath their boots. Only now does my body feel the fear and at the same time I am euphoric, a lump rises in my throat and simultaneously disappears.\n\nMy penumbral life began in the autumn following an excruciatingly hot summer. How easy it is to call everything into question once you start. The longer I stared at my surroundings the less sense they made. The blanket of rationalization we knit for ourselves\u2014which is billed day in and day out as the ultimate truth\u2014is easy to undo once you get hold of a loose end. One jerk and suddenly all flaws are exposed, revealing a jumble of competing realities: a global summit, delegates dozing off in the plenary session while hostesses in outlandish costumes pass up and down the aisles, popping candies (or maybe pills?) into the open mouths of the participants, who munch away in their sleep. And when their mouths reopen they rise one by one and sleepwalk to the podium, where they regurgitate the same pablum, which at the end of the day is collected and served to a patiently waiting press, which reports on achievable policy goals and the best possible deal under the circumstances. These are not living breathing beings so much as middlemen of destruction. It doesn't go down well with the Comfortable & Contented when you shamelessly insist on unraveling the yarn. The louder I voiced my opposition, the more stubbornly I was ignored, little by little the neighbors stopped inviting me to their beloved grill parties. Cheers! The keg's just been tapped and we're in complete agreement, everybody gives a lot and doesn't take much, we're all easy-going and eco-friendly, let's not get bent out of shape, despite everything life's pretty bearable. If I contradicted, Helene would give me a chiding glance from where she was sitting with her lady friends, who acknowledged my existence the same way a mechanic might regard a car headed for the scrapyard. I knew she was only waiting for the suitable occasion to slam the door behind her. As the time we spent together shortened to weekends only (and not many of those, as my field trips alternated favorably with her bridge tournaments) we had less need to put up with each other; it was a pain to be locked up all day and all week with her in the same house. You have to get help, she told me one day, I don't know what's with you, but you're losing it. That made me furious. She had hurled the first stone.\n\n\"I hate to be the one to tell you this but you took out the wrong insurance, and that was a pretty stupid thing to do\"\u2014I had picked up one of her special dishes\u2014\"because we don't really need fire insurance or flood insurance. What we do need is a policy to protect us from a madman on the loose in our own house, a man on the verge of losing it at any second, and we need it now and we need it desperately too bad you didn't take out that kind of insurance, what a shame\"\u2014at that point the Gmundner deer bounded into the wall and burst into pieces\u2014\"uhoh that's what it looks like when the madman goes berserk\"\u2014here some pieces of Delft majolica smashed into a window with a great _fortissimo_ \u2014\"who knows what's next, nothing's safe anymore\"\u2014and I banged my open palms against the china cabinet with all her porcelain treasures, a decorative bowl slid off and fell on my shoulder before shattering on the floor\u2014\"did you think I'd simply put up with everything? Didn't you think I realized how you've been trying to force me into your Procrustean bed? Do you think I'm some kind of fairground steer waiting for my fellow humans to guess how much I weigh?\"\n\n\"Fellow humans?\" Helene interrupted after a grating scream, \"What fellow humans? You no longer have any fellow humans.\"\n\nHere I fell silent. I had been weighing the Portuguese rooster in my right hand but I set it back down and concentrated on taking a deep breath. If she was right and I really had lost my mind then we'd no longer know if I was sick or if I had found freedom and release. We fled to the TV to avoid further interaction and stared at the screen with fierce tenacity, following nature shows the way a hunter tracks a wounded animal, we sat silently on two armchairs while a mutual contempt stretched out on the big brown sofa between us, consuming everything that had once united us, back when we were all we needed, on clear nights under a handful of stars. Nothing could placate me, every digitally reproduced animal struck me as a captured creature that was first castrated and then skinned. And so we suffered through evening after evening until some foreign station broadcast that wondrous news bulletin about masses of snow plunging into a valley, even though he wasn't live at the scene of the catastrophe the reporter's voice quivered opera-like with fear, so stunned he could only stammer, while I perked up wide awake and leaned forward in my seat and started rooting for the majestic avalanche. \"Down the hill down the hill!\" I shouted, and then with renewed courage and strength, \"Into the valley into the valley!\" and when the snow swallowed the first house so quickly the structure didn't even have time to collapse I shouted, \"Show no mercy show no mercy!\" Then it rolled over a second house, and a third one, an entire farmstead, I cheered out loud when the entire village vanished into the snow, and the anchorman was reduced to silence for several seconds until a technical glitch was solved and the white surface vanished from the screen. Helene stood up and walked out of the room, ostentatiously shaking her head. A few days later a letter from her lawyer put an end to our television evenings. I dumped our TV in the bin for hazardous waste: broadcasts that sublime were far too rare an occurrence.\n\nBack on board ship no one looks at me directly but everyone does so behind my back, as though I were dripping with foolishness\u2014humiliating events like that quickly make the rounds. Mary might understand why I behaved the way I did, but she is nowhere to be seen (she was with the first party that landed early that morning, she had greeted me briefly before hurrying ashore). For lunch all I have is soup so I can retreat as quickly as possible. Even Ricardo refuses to give me his usual welcoming grin. The lecturers at the table eye me with concern, no one reproaches me for my lack of self-control, even though each would have dealt with the situation better, they are more than lenient, perhaps because they would regret having to do without me. Beate gently asserts that we can't make a crooked world straight, Jeremy tells about the time a military transport vehicle ran him off a road in the Rocky Mountains. He's just illustrating how he drove his rickety pickup into a spruce tree when I stand up, nod curtly and leave the dining room, avoiding every sideways glance. There's nothing in the cabin to attract my attention. I'm lying down on the bed, staring at the fire alarm when Paulina bursts in, a breathless bundle of nerves.\n\n\"What happened?\"\n\n\"You've already heard?\"\n\n\"You got into a fight with one of the passengers?\"\n\n\"With a soldier. It wasn't a fight.\"\n\n\"Soldier, what kind of soldier?\"\n\n\"A Chilean.\"\n\n\"How so? What did he do to you?\"\n\n\"He was smoking right in the middle of the penguins.\"\n\n\"What do you expect from a soldier?\"\n\n\"Not to smoke.\"\n\n\"The people who join the army aren't usually the brightest.\"\n\n\"It's not a question of intelligence.\"\n\n\"What is it then?\"\n\n\"It's a question of respect.\"\n\n\"And you get into a fight over that?\"\n\n\"It wasn't a fight. He didn't stop smoking when I told him too.\"\n\n\"He didn't listen to you, that's what it is, you always expect everyone to listen to whatever you say.\"\n\n\"It's not about listening to me but to their own common sense.\"\n\n\"And now?\"\n\n\"I don't know.\"\n\n\"You do something like that and you don't know what's going to happen next?\"\n\n\"That's right.\"\n\n\"You're pretty dumb, you know.\"\n\n\"Agreed.\"\n\n\"You're putting us at risk is what you're doing, and what for? For nothing.\"\n\nI would defend myself if I could find the words to match the anger I felt when I ran up to the smoking smirking soldier and he stood there shrugging his shoulders. Everything I can think of is after the fact, flowers on the grave. Paulina is sitting on the bed across from me. My silence proves her right. With my hand on her shoulder I draw her to me, her hair touches my chest. She buries her face in my shirt. I feel the cloth getting wet. The day will come when I will make her unhappy without being able to console her. A first kiss, a pause for thinking, a second kiss. We take off only what we have to and I push inside her, over and over, with throbbing futility. We are silent, stirred and embarrassed at once because we are misusing our bodies. I feel impatience welling within me, I want to finish as quickly as possible. I hear Emma's voice calling my name over the PA system. I am being summoned. Someone wants to ask me an urgent question. I have to go back to work, says Paulina. We are both pent-up. I come with pinched lips.\n\nA few years ago, two summers before the catastrophe, Helene and I went to Lisbon for a long weekend, another attempt to salvage our marriage with sightseeing, late dinners by dim light and mutual smearing of sunscreen. We strolled along the avenues and climbed the steep narrow streets, taking advantage of every delight Lisbon has to offer, venturing into alleys unmarked in any guide, we savored warm past\u00e9is de Bel\u00e9m at the bakery of the same name (touristy, very touristy, but when I am a tourist I appreciate what is staged for tourists), drank Alentejo wine, admired azulejo tiles, and even boarded a catamaran to watch dolphins in the Tagus Estuary. But no matter what we encountered, nothing moved us both at the same time. We could have spent days in the souvenir shops without finding anything that would have the same appeal for the two of us. We stumbled into a church that had earned a mere three lines in the tour guide, fully expecting to step back out after a fleeting glance at ceiling and nave, so we wouldn't have to spend too long in a place where we were the only people. But I found myself mesmerized by the interior, by its imperfection, the signs of destruction sparked in me an unhoped-for feeling of devotion, for the first time inside a house of worship I felt I was in a real space and not simply a temple to human megalomania. Traces of the fire were still visible on the columns, a blood-orange vault stretched out overhead like a broad sky above a battlefield. In this _igreja_ the promise of salvation carried a credible patina of soot. The wilting flowers and flickering candles seemed like last vain hopes. It took me a few minutes to notice a pillow-soft song wafting from some narrow loudspeakers, children's voices that seemed to come from the other side of a wall that can never be surmounted. In a small apsis I saw the most moving Virgin Mother I had ever seen, all alone in a gapingly empty niche. She radiated an air of uncertainty, as though she were afraid of not being able to satisfy all the petitions presented to her. She was an injured soul who had been driven out of her home. I felt her pain. Not only because her son had been tortured to death, but because her torment had been immortalized. I studied her a long time. \"And can you tell me what it was about that dilapidated church you found so fascinating?\" a grumpy Helene asked when we were outside the portal. I said that the church should really be called the _Igreja de Gaia_ \u2014a sanctuary one visits to cast aside all human hubris.\n\nDan Quentin is aboard the _Hansen_. He never moves without an entourage, which is why his presence cannot be discerned but only deduced from all the blowflies swarming around him. Now and then his tousled mop of hair can be seen drifting by. His manager offered me the prospect of an audience, he didn't use the word \"audience\" but his tone of voice and choice of words suggested some form of homage. The excitement on board ship is palpable, the passengers are in high spirits ever since I called them together\u2014in two sections according to language, first English and then German\u2014and announced they would have a historic opportunity to play an active part in a work of art. I described the necessary safety training, Quentin's manager outlined the basic artistic plan. To my amazement the passengers did not feel at all bothered by the slogan \"Art needs you,\" in fact they felt flattered and discovered their inner activists. \"If I'm being called on to do something for the environment then count me in,\" a businessman from St. Louis declared, setting the tone for the others. \"The young man has imagination and that's exactly what we need,\" acknowledged an elderly lady, \"instead of all those protests all the time, they do nothing but undermine the cause.\" \"I'm assuming our participation is worth at least a signed photograph,\" a retired high school principal from Paderborn demanded. \"Naturally each of you will receive a signed copy,\" the manager reassured them, \"and what's more every single one of you will be listed by name on our website. And if you'd like a limited edition print for someone you left back home\u2014and what a gift that would be, right\u2014naturally you would qualify for our special participant rate, which is quite generous.\" The passengers left the room in little groups, chatting away, to go sign the lists that had been set out for those who chose to take part. Finally there was only one person left, a haggard, unshaven man with a black woolen cap, who had boarded at the same time as Dan Quentin. He had spent the winter at the Arctowski Polish Antarctic Station and was now on his way home after nearly twelve months on King George Island. He was sitting in the next to last row, one chair away from the passageway, his hands resting on his thighs, fingers splayed, a smile pressed on his lips. He fixed his eyes on me, evidently expecting a reaction. I sat down beside him, still holding the microphone.\n\n\"I should speak about wintering over.\"\n\n\"You'd like to give a lecture?\"\n\n\"Everyone wants to know what it's like to spend the winter in the Antarctic.\"\n\n\"Like being in a tunnel, right? That's how I've heard people describe it, except at least you know exactly how long the tunnel will be.\"\n\nHe grabbed the microphone from my hand and shouted into it, \"That's false!\" Then he let the microphone drop to the floor. \"You have no idea how long the tunnel is. With every passing day you start to doubt more and more whether the sun's ever going to come back up, whether you'll ever be able to move about freely again, whether you'll see more of the world than a few floodlit measuring instruments, whether the tunnel has an end at all.\"\n\nI picked up the microphone and turned the red switch to off. Live mics usually lead to embarrassing situations.\n\n\"You'd go crazy if it weren't for the books. Are you surprised? Such a banal idea, books in a tunnel. Did you know that Amundsen took three thousand?\"\n\n\"Should we sit down over a cup of tea?\"\n\nTrusting in the saving power of imagination while stuck in a seemingly endless tunnel made sense to me. I accompanied the Pole to the coffee machine, which also dispensed hot water. He went on talking while he awkwardly unpacked a tea bag.\n\n\"Science has become our modern oracle, I sensed that early on, but it wasn't until I was in the tunnel that I really understood\"\u2014he dumped several spoonfuls of sugar into his mint tea. \"It used to be that knowledge was gained with the help of a medium. And meanwhile we assumed we were well beyond that, we dismissed soothsaying and prophecy as hocus-pocus, we were convinced that we could complete our calculations and our future would be revealed, that we would provide sober proof,\"\u2014the Pole tapped his spoon on the edge of his cup\u2014\"evidence obtained through precise research, that's the sign of our time, the blueprint for future behavior. All we have to do to convince people is show them the appropriate data. Isn't that right?\"\n\nHe turned around, still holding his cup, looked up and down the stairs, looked up and down, stopped, lifted the cup to his lips with both hands and slurped his tea.\n\n\"And who did they first worship in Delphi? The goddess Gaia. Her priestesses would sink into an ethylene-induced trance in order to look into the future. So what about us? We produce vast quantities of ethylene, it's everywhere, in our clothes, in everyday objects, it's even in our bodies. We're all on narcotics, we're all so addled by this drug of consumerism that we've lost any and all clairvoyant ability.\"\n\nThe Antarctic overwinterer took another slurp. He was standing just inches away and speaking from his heart, but a real conversation seemed impossible.\n\n\"Have we even asked ourselves who should we turn to now? What oracle should we consult? Some higher instance, that much is clear, but which? The one we call Nature? Gaia? Maybe even God? Have our questions become any more precise? Perhaps. Do they lead to new answers? We assume so. And we were so certain that the more we figured out the better we would act. Ridiculous. And you, what are you doing on this ship?\"\n\nIt took a moment to realize he was talking to me because he was still facing away, and his voice hadn't changed at all, the words kept limping at the end of every sentence like a bad leg.\n\n\"Have we veered off track? Just a little bit? Very much? False, once again, false. We've completely missed the mark, we've been playing the wrong hand, we were holding onto our projections and meanwhile it turns out that prophecies were trumps. The projections proved as irrelevant as last week's weather forecast. Admit it, you didn't think it possible that they'd simply cast a deaf ear to all your warnings.\"\n\n\"How do you know that?\"\n\n\"You told me yourself.\"\n\n\"We've never met before.\"\n\n\"You explained it all to me in great detail.\"\n\n\"Where was that?\"\n\n\"At some congress.\"\n\n\"I don't remember.\"\n\n\"So you turned away from science? You gave up?\"\n\n\"On the contrary, I'm just planning to give my next warning in a different way.\"\n\nWe are hemmed in by sameness, we sense that nature is staring at us blindly but that is all. The water has a greasy sheen, a little ways from the ship the surface forms an opaque screen separating two metallic visual horizons. All cameras have been lowered, the lounge is quieter than usual. Paulina and I exchange glances above the glass display stand with the marble cake. When we reunite I plead with my own desire, desperately searching for some sign of favor. Gradually we come to a standstill as we rock back and forth on a false promise.\n\nYou can't really say that, the ladies of Calcutta, why didn't you fill up at the last station, didn't you see the sign, no more gas for 500 miles, you mean you miscalculated? That was very rash of you, I can't give you any gasoline, here's a bottle of water, I can't do any more to help. Will steal your heart away, at present we can only confirm that Zeno H. was the perpetrator, but we still don't know whether he was acting alone or with accomplices, we can only speculate about his motives. The older the better, when I kissed you and called you sweetheart, if you're still feeling fine have some more wine, do the chairs in your parlor seem empty and bare. We don't know anything about his whereabouts, he waited until the helicopter was gone, then he took the captain, the first officer and the security officer prisoner, a command sounded in the engine room: all hands on deck, what's going on? Get up here quickly. Do you gaze at the doorstep and picture me there, when does MILF turn to GILF? When the money runs out we'll slaughter a billionaire, oh-kaaay, the rich are the piggy bank of the nation, oh-yessss. One day he stormed outside and started screaming like a holy terror, it was impossible to tell what he was saying, he was completely unintelligible, the neighbors opened their windows. Is there anything else you can add? It wasn't a normal scream, it gave people the shivers, you could tell, and deep down inside I was terribly afraid. When we have to choose between preserving nature or earning money, trip the trannies in a trap, it could be worse, robins are dropping dead from the sky, the troop's all poop, you can't really say that, ciao ciao bambino, Robotnik gets it on with Futanari, we like to grope in the dark, twiddle twaddle, it's easy to remember, Shotacon is stalagmite, Lolicon is stalactite BREAKING NEWS FATE OF HANSEN STILL UNKNOWN BREAKING NEWS FATE OF HANSEN STILL UNKNOWN now it's time to roll up your sleeves again\n\n### 9.\n\n## 62\u00b058\u20329\u2033S, 60\u00b033\u20326\u2033W\n\nIf we were pirates\u2014and we're not, we're privateers with letters of marque that we call standard form contracts, we no longer slit throats, these days we do our murdering with unmanned drones\u2014this would be our hiding place. If we were in a pirate film we would put in at our secret island on just a day like today. The ocean is slate gray, the sky ash gray, our ship is sailing straight into a closed mass of anthracite-colored cliffs, if \"open sesame\" turns out to be the wrong password we'll get smashed to pieces, the captain has throttled back the engines, we are barely making headway, as though a hobbyist were tweezing us through the neck of a bottle. Everyone is crowding onto the weather deck with their binoculars, scanning for a solution to the riddle. The secret passage appears, the hidden opening which sailors of old called Neptune's Bellows. \"Do you want to go in the water?\" Paulina had asked before we fell asleep. Of course I do. The basalt cliffs are perilously close, rigid and unyielding, like frozen fury, it's easy to see where the volcano left its mark, the flow lines from the lava. Ahead is a black sandy beach covered with lapilli and divided by half-sunken ruins, behind that a rise shadowed with snow-caps, the rocky cliffs in between are glossy black and flecked with oxidized iron. Our ship drops anchor in the middle of a caldera. Even here humans established themselves, and soon the volcanic bay ran red due to the rising demand for baleen, used at the time as corset boning, and for whale oil, which went into nitroglycerin so people could blow each other up in the trenches of World War I. What wondrous innovation to make explosives out of whales, what a vibrant symbol of progress: destroying the essential to create the superfluous. Decades later, the volcano took belated revenge, singeing away all human presence. Deception Island is a demanding port of call, we all have our hands full, not only do we ferry the passengers ashore, we also take everyone medically fit on a longish hike. We used to dig a pit in the sand so the Antarctic tourists could bathe in the sulfurous warm water, but that's no longer allowed, now the passengers have to jump into the ice-cold sea (and jump out quickly if they want to survive, the Brazilian doctor stands by with stopwatch in hand, making sure no one stays in longer than forty-five seconds). Afterwards we distribute certificates testifying that they took the plunge. When the doctor goes back on board I jump in, the last to do so, and let myself be revived by the icy water.\n\nAs soon as you've finished your second spell in the sauna you have to go in the cold water, H\u00f6lbl instructed, and I mean really in the water, a cold shower just won't do it. Even though I always loathed saunas I followed his recommendation, because the near-naked ladies were lounging on the barstools and if you spend too long ogling you get depressed. Well, is it just as I promised or not? Didn't I tell you I'd take care of you? Patronizing as he was\u2014he even insisted on helping me tie my bathrobe\u2014thanks to H\u00f6lbl I got to know the institution of the bordello and its fleeting encounters. I also appreciated some things he hadn't mentioned, such as the matter-of-fact parting from the woman I had just thrust myself into scarcely half an hour earlier with a simple \"see you around,\" before her backside disappeared around the corner and out of my memory, superseded by others swaying past, and the happy, relaxed exhaustion that settled inside me as the sediment of a new experience. After a few more visits, some even without H\u00f6lbl, I managed to dispense with any preliminary conversation at all, it seemed inappropriately binding. In the club he introduced me to, which we frequented during the interregnum between divorce and Antarctic because of its favorable value-for-money ratio, there was a small \"cinema\" with a \"playground\" instead of chairs or sofas, where the clients could relax, naked but for a towel wrapped around the loins, and watch a ridiculously bad porno film, or if they were so inclined, the uninhibited goings-on around them. Now and then an unclothed woman would stroll past with an affected \"Would you like a little company, honey?\" To my ears that sounded like a threat, which is why I only accepted more original solicitations. Apart from such siren calls the encounters were entirely to my liking, reduced as they were to the bare minimum. Without a word I would motion for her to remove my towel and start to work. As if I were sitting high on a Ferris wheel, looking down at the life below, in miniature, with no idea how I was supposed to return to Earth. Now and then I managed to avoid the exchange of invented names, those were the happiest occasions, my bodily needs were satisfied and the whole process had nothing to do with me.\n\nBefore the ship passed through the narrow opening, the captain took me to task, abandoning his usual terse style. On the bridge, in the presence of several stakeholders (as they were known in the language of the hierarchy), he reminded me that as Expedition Leader I was expected to set an example, if I take a wrong turn so does the entire ship, my highest priority is the security of the passengers, a man my age ought to be able to control himself, one cigarette isn't going to incinerate the entire Antarctic continent, I had endangered the collaboration with the Chilean station, damaged the reputation of the cruise line, at that point I stopped listening, who is the captain to be judging my fit of rage. It's true that the following day I felt a sense of disgrace, but I also still felt I was in the right, the soldier had violated the agreement allowing us to be in the Antarctic in the first place. My only regret was not having made a more forceful impression on him. I stared past the captain's head through the curved window and out over the sea toward the icy horizon to see the soldier flicking his cigarette into the middle of the penguins, the butt landing on the thick mass of feathers, singeing the shimmery blue-black, it's not the first cigarette, either, the penguins are in the middle of an enormous ashtray that's never been emptied, even though they cannot fly they spread their stumpy wings, trying to escape. When I come back to Earth I hear the captain informing me he is going to report that I am poorly suited for the post of Expedition Leader, unfortunately he finds himself compelled to reexamine my continued employment as a lecturer, and this may require a psychological evaluation. Then without any transition or words of regret he immediately moves on to Dan Quentin's SOS project, which since seems to have garnered both his approval and his appreciation. That's nothing but a lot of ballyhoo, I say, the captain's tongue-lashing having freed me from any diplomatic constraints. He tells me to make sure everything comes off smoothly, so that I can have an honorable departure.\n\n\"And what exactly do you expect to gain from it?\" I ask. \"Are you gunning to get invited onto a talk show?\"\n\n\"That kind of effrontery doesn't become you.\"\n\n\"It's not effrontery, it's just being upfront. Setting up an SOS like that without cause is ridiculous. You're letting yourself be used as his stooge.\"\n\nThe captain tells me to stop acting so self-important and see to it that this work of art happens, nobody attaches any importance to my outlandish opinions. I tell him that this work of art can't possibly succeed unless there's a genuine emergency, unless the SOS formed by the passengers is for real, now that would be a success, can you imagine how well the photos would sell then? He tells me to stick my outburst somewhere else and take care of my job, one preparedness maneuver, one hour, one photo, a high point at the end of a beautiful trip, that's not all that difficult, and afterwards we'll take everybody home and that will be the end of it. The captain has nothing more to say to me, the stakeholders' stare at me like I'm a freak in a carnival sideshow.\n\nThe pianist has been watching me as well, but he doesn't speak his mind in front of the others, certainly not in front of the woman from New Zealand who is traveling with her aged mother and who longs to be noticed on her own, sans parent. Ever since the previous evening he's been eager to oblige, perhaps a bit too hastily, but the New Zealander doesn't seem like someone bold enough to insist on the proper tempo. She asks me if it's true we can send postcards from Port Lockroy, which I confirm, and immediately take the opportunity to tell her a little about this British outpost, an old whaling station later retooled for purposes of espionage, because the British were afraid German ships might hide in the many natural harbors along the Antarctic peninsula. The plan was called Operation Tabarin, a scheme so secret even Churchill didn't find out until it was already in progress, the seamen sent to guard the Bransfield Strait waited and watched and waited and watched for days, weeks, years, but the Germans never showed, they'd probably forgotten all about the Antarctic, being otherwise engaged, so the soldiers manning the post had nothing else to do for the rest of the war except chow down on duff pudding and lick Lyle's Golden Syrup off their spoons. So the enterprise was entirely in vain, the New Zealand mother asks somewhat simplistically. Not entirely, they did manage to remove the Argentine flag from Deception Island. The pianist interrupts to say that as always, whenever his esteemed friend the Expedition Leader explains something, he only tells half the story, what he failed to mention was that in 1939 the Nazis had flown seaplanes over the Antarctic and dropped flags rigged on metal spikes in an effort to reclaim part of Queen Maud Land for Germany. And when they had staked the area with their swastikas they renamed it New Swabia. The woman from New Zealand seems delighted, either by the twists of history or the studied charm of the piano player, she smiles demurely and repeats New Swabia as if it were a punch line. Meanwhile, behind me at the bar others are waxing witty and inflicting their humor on Erman, listen you'll love this one, my first name is John and my last name is Walker, so it's John Walker and what do you think they call me, Johnnie, right! So go ahead and pour old Johnnie Walker a Johnnie Walker and I guess that means it has to be a double, there's no two ways about it. Why did it get so gloomy today around noon, a different voice asks, from up by the bow it looked like we were sailing into the land of the dead, then another voice breaks in, ahoy, ahoy, we're the Pirates of the Antarctic, followed by hoots and hollers, I turn to see the faces flushing red with laughter, the constant hullabaloo through which Erman's calm voice passes like a silver thread, Black Label, sir? Naturally, keep it coming, fill up the old skull... and crossbones please snorts Mr. John Walker, Erman makes a face, presumably in reaction to the spittle, just wait, you'll stop hooting soon enough, the New Zealand mother\u2013daughter leaves, the bar pirates grab their glasses and stumble out. Now the pianist can say what's on his mind. He hadn't expected that of me, what childishness, picking a fight with an armed soldier over a cigarette, I better start using my head and learn to pick my battles. He absolutely understands my aversion to cigarettes, for his part he has a hard time summoning sympathy for the passengers, what's there to like about Mr. Johnnie Walker? The cigarette was yesterday, tomorrow we have Dan Quentin, that's even worse, despite what happened the captain still wants me to organize this SOS business. Well as the only certified doomsayer on board I have no choice but to accept my fate, there's little he could contribute unless we need some musical underscoring. He gets up and moves to his keyboard, titti tatta tam, titti tatta tam, titti tatta tatta tam, a memorable synthesizer intro for someone who was married to Helene longer than the band ABBA existed. His choice is dead on, too\u2014the Johnnie Walkers of pop music. Fits Quentin like a glove. But SOS stands for another song, can I guess, he plays the first notes, I recognize it right away, hello darkness my old friend, come on, sing along, he plays the whole first verse. You know what I was haranguing the captain about? I told him there'd have to be a genuine emergency to make the whole thing believable. That's the spirit, how about a hijacking? The scenic cruise goes Crusoe. He laughs, and his laughter is clear and refreshing, like a spot of sorbet between heavy courses, then it morphs into an improvised variation on the refrain from \"The Sound of Silence.\" He tossed out the idea so casually, just like that, a tiny nugget in the quarry of mindless remarks that make up our conversations. A hijacking? A red SOS on the ice? The moment when art becomes reality. The idea takes hold of me and doesn't let go. Even the most casual utterance can be taken seriously. What starts as a hairline fracture widens into a crack and finishes as shattered glass.\n\nA white bird alights on my head, the glacier hides behind its threshold, it calves into the ocean, loud, alive, seagulls flutter above the glacier and disappear, the clouds are tiny, a wave rears up and crashes, spewing spray from the crest high into the air to form a lacy burial shroud, albatrosses plunge from the sky like falling rocks, while blind hope gets caught in the mounting mesh of mist.\n\nLet us dream beneath golden stars, this is a dead end, you don't seem to want to understand, in this neighborhood there are nothing but dead ends, secret dreams beneath green trees. At gunpoint (and where did he get the pistol?) he forced all those still aboard the _Hansen_ including the captain into a life boat, then he set off, yes these days steering a cruise ship is as easy as moving a joystick, there's no doubt he's reached open sea by now. Let us dance to mandolins, we are all as innocent as the subsidies we receive, now the stage is bare, dance beneath golden stars. Would you please tell us your name? Paulina Rizal. And your occupation? Waitress on the MS _Hansen_. How long have you known Zeno Hintermeier? Four years. How were you acquainted? We were friends. What kind of friends? Just friends, not accomplices. Were you in a relationship with him? No, we hadn't promised ourselves anything. Did he mention what he was planning to do? No. He didn't warn you? He talked a lot, they were just words, nothing but words. I'm sorry for coming so late, my appointment lasted longer than anticipated, no, I missed it, what did you say? Yes, no, no, I'm listening, I was only distracted for a moment, there's a white pigeon fluttering around here. He's headed straight north, the fighter pilots can't attack the ship, they can only observe it, it will have to be boarded, there's no other solution. There's hope for you yet, little gifts can do a lot to nudge a friendship along, come take me by the hand and lead me off to fairyland. He was a buzzkill, a nutcase, but at least one with convictions. You wouldn't understand him. Don't underestimate us. You wouldn't understand him, you'd have to change completely in order to understand him. In the past everything had to be stored, stay tuned for our flirt tip of the week, at times the glacial surface calls to mind a coral reef, and so it goes without a pause BREAKING NEWS PHANTOM SHIP REACHES THE ATLANTIC BREAKING NEWS PHANTOM SHIP REACHES THE ATLANTIC and now for something completely different\n\n### 10.\n\n## 62\u00b035\u20320\u2033S, 59\u00b056\u203230\u2033W\n\nIT'S TRUE, Mrs. Morgenthau's mishap could have been prevented if I'd been less negligent, if skuas didn't swipe penguin eggs, if I hadn't suggested we put in at Half Moon Island, a crooked stretch of land shaped like a crescent moon, with four evenly distributed hills and masses of chinstrap penguins, in good weather an easy jaunt, with a glorious view of the Livingston Island peaks, a small strip of terra firma perfectly suited to my taste, overwhelmingly white with a few black stony patches, in one place a forked cliff of granite next to an inclined rhombus, a formation I never tire of looking at, and even if the weather was changeable during the day, there was no compelling reason to forego the landing, on the contrary, the capricious light seeping through the heavy black clouds made the island seem as if it had been fashioned in some fit of euphoria. Despite our advice the passengers seldom manage to stay in one place so they can actually observe the animals, preferring instead to run around and follow the penguins every which way across the snow, cameras at the ready, with the praiseworthy exception of Mrs. Morgenthau, who stood at the edge of the colony keeping the prescribed distance and watched enraptured as a mother or father penguin warmed their eggs, \"The second egg really is smaller\" I heard her mumble, like many of the passengers Mrs. Morgenthau enjoyed comparing what she'd learned in the lecture with the reality (in the words of our ornithologist El Albatros: the second egg is really just a backup, that's why it's smaller, just like an emergency parachute is smaller). If she'd been less attentive, less reverent, she wouldn't have felt such a kinship with the brooding penguins and would not have intervened in this idyll, whose peace was disturbed only by skuas hunting for eggs, an instinctual behavior I hardly noticed anymore, in contrast to Mrs. Morgenthau, who had fixed her attentive gaze on an especially aggressive raptor, an ugly, fat, mean bird, as she told me later, she felt a palpable aversion to the skua and was getting all worked up, it had even frightened her a little, as laughable as this sounds it's the truth, that explains what happened next. All this might have been avoided if I had reacted more quickly, if I'd been more focused, if a different guide had taken the easy post near the chinstrap penguin colony at the fateful hour, instead of myself, who, tired after a few hours manning the landing area, had declared a break and was consequently ill prepared for the skua's nosedive, which I only glimpsed out of the corner of my eye when I heard Mrs. Morgenthau cry out, \"It has an egg,\" just in time to see the bird land less than three steps away from her, a white egg clutched in its talons. I saw the skua look around for any threats before attempting to crack the shell with its beak, which it never managed to do, because Mrs. Morgenthau fell upon the bird and snatched the egg with a surprisingly nimble movement, then held it carefully in her hands while the defeated bird flew away, leaving her proud of her rescue, if also a little puzzled, as was I, which is why I didn't react immediately, but only after she'd started heading toward the robbed penguin, who refused to move because it had to protect its second and now only egg. Mrs. Morgenthau reached its nest with the best of intentions, bearing the egg like an offering, bowing before the bird so she could place it as gently as possible in front of the penguin's abdomen, all I managed was a hasty \"Don't do that!\" but in vain, Mrs. Morgenthau felt that she had been chosen to right a wrong, that she had been called to return the egg with future life unscathed to the brooding penguin, an intention that was as noble as it was misguided, because the penguin, faced with an attacking red monstrosity, driven by instinct to protect its remaining egg, opened its beak and bit Mrs. Morgenthau in the left hand, causing her to drop the egg as she screamed in horror and stared at her hand as blood went dripping onto the stone, an amazing amount, I don't know if she ever felt me grab her arm to check the wound, because she tore away to escape the snapping penguin, then slipped at the first step and went crashing into another penguin, who was guarding eggs of its own and as a result couldn't get out of the way quickly enough, just as I didn't react quickly enough to stop her fall, and the helpless bird was buried under Mrs. Morgenthau's massive upper body, the penguins caught on sooner than I did, the entire colony squawked and squealed into motion as I helped Mrs. Morgenthau up, her parka smeared with bits of egg, I steadied her with one hand and used the other to radio El Albatros before examining her wound\u2014the penguin had bitten deep into the soft flesh between her thumb and forefinger \u2014 things wouldn't have been half so bad if I'd cleaned it right away and prevented an infection, but my backpack was missing the first-aid kit we were always supposed to carry, so that I had no other choice but to staunch the wound with my handkerchief. Below us lay the lifeless penguin, while a harsh chorus of protest erupted from all sides, I was about to suggest we make our way to the dock when a few snow-flakes landed on our hands, I looked up, the weather had shifted, a blizzard was coming on, the wind started howling, visibility was decreasing at a breathless pace, the bridge let us know that in view of the rising katabatic wind, which could easily overturn a boat the size of a Zodiac, it was advisable to stay put on the island, and if necessary pitch the emergency tent and take shelter until the gale had passed and the ship's horn gave one long and three short blasts. El Albatros showed up as shot-sized hail began to fall and then the storm swallowed everything: peak and glacier, the four hills and the forked granite cliff, the other guides and passengers and even the penguins, now the doctor couldn't possibly reach us, Mrs. Morgenthau was at the mercy of Half Moon Island, El Albatros inspected her hand and my blood-soaked handkerchief, visibly worried even before he whispered to me using his broken German to keep the diagnosis secret from the patient, the wound had to be disinfected with all urgency, penguins' beaks are heavily contaminated, the bacteria are highly dangerous for humans (the doctor later told me that because of the extreme conditions, Antarctic viruses and bacteria are extremely resistant), he himself had rushed over without a backpack, since I hadn't told him I needed a first-aid kit, which is why the doctor is completely right when he says it all could have been prevented, the fact that Mrs. Morgenthau wound up bedridden in the hospital ward on an IV drip, with a fever and a swollen hand, most likely a case of erysipelas, which used to be known as \"holy fire,\" the only thing certain is that it was a good hour before we managed to bring Mrs. Morgenthau, clearly in shock, and the other passengers on board\u2014leaving behind a dead chinstrap penguin, a few crushed eggs, and a skua who was robbed right out of his beak.\n\nThe move from the house in Solln to a furnished one-room apartment in Moosbach was very different than the previous relocation. Everything I still chose to call my own fit in H\u00f6lbl's VW SportWagen. The only books I took were ones I had practically learned by heart during the past few years, the rest wound up in the recycling bin, I went there every day for weeks lugging two heavy canvas shopping bags, one in each hand, the CDs had to be taken to a special station for used electronics that was a longer walk, but they weren't as heavy. On my way I remembered what Lama Boltzmann had told us about a library in a Tibetan monastery with scrolls that no one had been allowed to inspect for hundreds of years. The priests would look at the stacked documents and make prognostications about the future. In light of this tradition my trip to the used electronic collection point struck me as a kind of Buddhist pilgrimage: we need texts that remain unread on purpose, music that is intentionally not listened to, trees mountaintops brooks glaciers that are deliberately left in peace. As the summer dragged on I spent my days reading in the Moosach apartment with a sense of liberation, freed from the besetting pressure of thousands of books. My only concern was what to do with the proceeds from the sale of the house, a substantial sum even after I had transferred half to Helene. Once again I abandoned myself to the tried-and-true texts, inspired by their stubborn ambition to appeal to my conscience, which is presumably why they continue to be so esteemed, even though they try as hard as they can to change the way people think. The classics are allowed to shine light into the darkness and fashion words worthy of chiseling into stone facades. Living authors, on the other hand\u2014so I discovered each time I opened the newspaper\u2014are expected to have more modest aims, to motivate here or agitate there, but under no circumstances should they propose to change the world. So how are people expected to stir things up during their lifetime? Shaming doesn't work, because everyone is ready to publicly confess their disgraceful deeds, pathos doesn't work because everything is downplayed. And violence? Violence is the only language that has yet to be plastered with the ads and logos of sponsors. Left to our own devices, we understand only the violence directed against us: violence done to others remains incomprehensible, unheard, voiceless\u2014nothing more than a rasping cough in a throat without speech, or at best a stutter. That's the kind of sentence I penned in the margins as I bided my time in my cozily narrow apartment in Moosach, reading my own notes and asking myself whether I had found a reasonable response to the unreasonable demands of our time, or if I, too, had been infected by the idiocy of the age. One thing seemed clear, though, namely that true liberation can only succeed through a creative act. Occasionally I wrote emails. Even in the dreariest weeks I made sure to keep up my correspondence with a handful of colleagues I really respected, for instance Shiva Ramkrishna from the JNU in Delhi, who took a devilish delight in interpreting the latest scientific findings through the prism of early Sanskrit myths, in his opinion the melting of glaciers and the impending threat of the Ganges running dry had been foreseen, the ancient prophecies foretold that the holy river would one day grow tired of the countless sins that had been washed away in its waters and would disappear below the earth, even our gods will change, Shiva wrote in his latest email, you can get a taste of things to come on the Siachen Glacier, where the soldiers are so completely dependent on helicopters for food and protection that they are inclined to see them as omnipotent saviors, their only hope for deliverance from the mind-addling tour of duty 20,000 feet above sea level, and have even begun to worship the machines, with gyrating light and ancient chants that they have only had to adapt slightly. Why shouldn't God be a helicopter, I answered Shiva, that would say something for the range of religious imagination, Christianity's biggest mistake was fashioning God in man's own image. Once a week I interrupt my musings to talk to Paulina over Skype, at a prearranged time. I don't appreciate surprise phone calls, not from H\u00f6lbl, who absolutely refuses to understand that the recollection of Paulina is far more erotically satisfying than the sight of some scantily clad long-legged female from one of the shantylands newly accepted into the European Union, nor from my bank advisor (the title says it all, the man advises his bank at the cost of his client), who has tried to fob off everything imaginable on me including the lousiest bonds and securities (and what a phony word that is, right up there with \"insurance\"). But his efforts are in vain, he still hasn't grasped that he's at a complete disadvantage since I don't need to convert my time into money. Sometimes I'm so annoyed I simply hang up. I stay away from the Alps, I don't venture out of town, either for little outings nearby or longer excursions farther away\u2014there isn't any nature left in my country, I might as well let myself be moved by the cultural landscapes printed between two book covers.\n\nGlacier fronts gnawed through and through, as though the sea were a rodent. The sky offers four separate dramas, the clouds above the ocean are different than the ones over ice that's more than four kilometers thick, puffy cumuli ghost around the islands, while just above us is a blanket of gray. We are sailing through the Avenue of the Ice Giants. Spiky blocks of ice keep watch, their ribbed bodies chiseled out of alabaster. Hammered walls of blue copper and a single albatross, as gentle as a line of chalk, a hundred solitudes away from his nest. That's you, Zeno, plunging at free-fall speed into nothingness, a moment later and you will no longer appear in the drawing.\n\nSuspected of... well what exactly? Where wood is chopped splinters must fall. What made him so furious? Actually there was hardly anything that didn't make him furious. That's not very helpful. I'll give you an example, last year the ship was completely overbooked for this one expedition, the lecturers had to be put up two to a cabin, we were running low on drinking water because people were consuming too much, and for the desalination to kick in the ship has to have a speed of at least fifteen knots, so every night we had to sail to Deception Island and back just to have enough water for breakfast, and that went on several nights in a row, if we'd stayed in the Antarctic one more day we would have run out of fuel. And? That really made him furious. Yes, it's fluttering right here in front me in the train station, the pigeon isn't totally white, you're right, it has a few black spots and two brown stripes, there, on the sides, I have no idea what you call it. We can get original footage from a Columbian TV that had a crew on board, we haven't seen stuff that strong since the incident with the tanker, remember, the one that crashed into the harbor? The workers on the dock saw it heading straight at them, they had fifteen minutes to clear out what they could. Dry fodder, profit is short term, worry is for life, we should rather look at the whole thing from a psychological perspective, student fodder, it's too much to ask people to behave with a mind to a future they won't live to see, hiccup, I'm a pious skeptic, dry cough, I have pills for everything, some make you bigger others make you smaller and some let you forget, and the correct answer is: the thickest book is the book of world records, congratulations, I thank you all, canned meat, I love you all BREAKING NEWS HIJACKED SHIP BOARDED BY SPECIAL FORCES BREAKING NEWS HIJACKED SHIP BOARDED BY SPECIAL FORCES is in large measure suspect\n\n### 11.\n\n## 64\u00b050\u20323\u2033S, 62\u00b033\u20321\u2033W\n\nI'M LOOKING DOWN at Neko Harbor (there's still no place I like better), a glacial tongue, an oval bay, and further out a narrow strait hemmed in by jagged, jutting mountains, the humped backs of mighty creatures sleeping away the summer, while kelp gulls fly in lingering spirals. The bay makes the ship seem tiny, insignificant, as if it could be made to disappear using a handheld remote. I breathe in the view until it streams through all my veins and fills my brain. Jeremy is sitting on a stone that's free of snow, pointing his camcorder at the glacier in an effort to capture masses of ice as they crash into the foaming sea. Without warning he points his camera at me, \"What a lucky coincidence, here comes the lead actor of the new blockbuster hit _The Penguin Strikes Back_ , please share with our television audience exactly when it dawned on you that you would write the history of the Antarctic Cruise Crusade?\" In reply I make a disgusted face. The camera doesn't so much as twitch. \"And where did you get the idea of using a penguin to play the squeaky wheel?\" I shake my head and say nothing. Jeremy jumps up and stamps around me with his heavy boots, bombarding me with further questions while I stare into space so the pushy reporter will go away. \"With your permission I'd like to ask one final question, namely who is going to play the role of the penguin, could you at least share that secret with us?\" The snow isn't firm enough for rapid movement, our laugher is more nimble than our feet. \"Cut. Professor Z., why do you love ice as much as you do?\" Jeremy has stopped, his glasses are slightly clouded over.\n\n\"Because of its variety.\"\n\n\"Could you explain further?\"\n\n\"The most beautiful thing on earth: variety.\"\n\n\"Yes of course, we all love variety, but in the ice?\"\n\n\"There's nothing more varied. A solid body containing gas and liquid.\"\n\n\"Just like human beings. Cut. We are observing a professor on the mound overlooking Neko Harbor, who is attempting to remain serious although he would like to laugh, it is the seriousness of the situation forcing him to do this, for he has recognized how serious the situation really is.\"\n\n\"Go ahead and make fun since everything really is so funny.\"\n\n\"All right then, let's be serious. Cut. Zeno, what would you say is your greatest wish at this moment?\"\n\n\"I'd like to stay here, Jeremy.\"\n\n\"But you wouldn't be able to survive.\"\n\n\"Who knows, with a tent and a backpack and some dry provisions.\"\n\n\"Well I'm sure the captain would give me a medal or maybe even a raise if I did leave you here, no, wait, it won't work. Paulina would tear my head off.\"\n\n\"I'm tired.\"\n\n\"Already? It's just the start of the season.\"\n\n\"I'm tired of being human.\"\n\n\"You're ok, Mr. Iceberger. Even if you do get off track now and then, still...\"\n\n\"I'm not tired of being me, Jeremy, just tired of being human.\"\n\nJeremy takes one step forward, then another, he embraces me, unexpectedly, it's a ritual typically reserved for saying goodbye at the end of the expedition, I give him a firm hug in return, he cries out, but not as a joke, I hear a thudding sound followed by a curse, we separate and watch as the Full HD video camera tumbles down the steep cliff and is stopped by a little ridge in the snow, the thought occurs to me that we could climb down, but then it resumes sliding and picks up speed and vanishes out of sight. We stand there like two boxers who've just learned the fights been called off, pricking our ears, listening for the sound of the camera hitting the water, but we don't hear a thing. We stare at each other. Although I don't say a single word, the regret must be etched on my face, because Jeremy is quick to console us both: No problem, anyway the interview with you was lousy, the camera is insured and I've shot Neko Harbor before in much better light. Let's pack up. Jeremy pulls one of the red flags out of the snow, holds it in his hand like a spear or harpoon, he must have had the same image in mind.\n\n\"Just imagine, what if a whale swallowed the latest edition of _Daily Turbulences_ and then got killed and slit open and some avid Japanese researchers discover the camera, what if they take out the memory stick and put it in a camera that hasn't been corroded by the whale's stomach juices, and then punch 'play'? What do you think they'll see? Your face. And what will they hear? 'I'm tired of being human.' And then they nod and each one says, me too, and they all decide to climb into the open belly of the whale, which they staple shut from the inside before tossing it back into the ocean.\"\n\n\"How will they do that if they're all inside?\"\n\n\"One of them will have to sacrifice himself, somebody will have to stay outside and operate the sling lift. Satisfied, you old pedant you?\"\n\n\"If that happened I would be overjoyed.\"\n\n\"Let's get a move on, down we come, or how do you say it in Bavarian?\"\n\n\" _Obi_ , Let's go _obi_!\"\n\nUsing flag stakes as walking poles we carefully make our way down, soon we're eye level with the seagulls, gentoos are scrambling over the craggy outcroppings, the snow around them is discolored from their urine, the green color is as pungent as the ammoniac stench. From the shoreline the glacier looks like a face with a thousand expressions, each posing a different riddle in the sunlight. It's almost too much, says Jeremy. And I say nothing. We stand there a while next to each other, hypnotized by all the crevasses into which our thoughts are falling, Father roaming through the house at night, raising his litany to a lament, the louder his voice the deeper he is buried beneath his cry. I have the impression time and again that glaciers are putting on the last act of a bad play.\n\nThe ice is here, the ice is there, the ice is all around, before us as a carpet whose knots crack when we step on them, behind us as a mirror broken in a thousand pieces. When the floes graze each other they sound like little bells, when they collide with the hull it echoes like cannon fire. Just four years ago we couldn't make it through at this time of year. On shore Klabautermann goblins compete for our attention with their contortions, while angels guard over us higher up, their wings folded close to their icy bodies. Now and then, when no other being is watching, the Klabautermanns drop into the black water and dive to the bottom to rest. The boundary of the drift ice is so straight it looks like it was drawn with a ruler. For a few minutes I could imagine it thickening and enclosing our ship and not letting go. On the sundeck a barbecue is being set up for an open-air dinner while the ship glides through another strait. The weather is mild, the mood euphoric. The music is already blasting from the speakers, people are expected to dance in their full polar regalia, sunshine, sunshine reggae, a moonboot pas de deux, one couple asks me to take a picture of them, I say, \"Say cheese,\" and she puckers her mouth into a kiss and says \"honeymoon\" let the good vibes get a lot stronger, I won't miss this either.\n\nBy the late light of day we cast anchor in a bay full of ice floes as round as white whales, as narrow as their tailfins and as sharp as their teeth, a swan is swimming among them with a bloated head. The sky gradually darkens, a jaeger rushes out of its nest and pulls a last cry out of the dusky firmament. I wish there was a letter in the alphabet for death.\n\nHow did you do it, Carstens, how did you manage to get our colleague on board the hijacked ship, you're a genius. If it doesn't help it doesn't hurt, am I some kind of ornithologist, an ordinary pigeon, it tried to land, the floor's just been mopped, she's slipping and sliding across the floor, yes that's all, nothing more, well, since you asked why I was so distracted, the idea of an empty purse causes people more anxiety than the idea of their own demise. Would you consider yourself a misanthrope? In a positive sense of the word. Do you prefer birds to people? Ask my children. Don't you think that an excessive love of nature leads to violence, including towards humans? On the contrary, insufficient love of nature leads to violence, also towards humans. Are you putting animals and humans on the same level? They're worth the same aren't they? Aren't humans a higher being? Not to my knowledge. Let's not ruin the mood, two deer at the edge of town, the vehicles come to a stop, the deer trot across the field, the glass is either half empty or overflowing, if you hear the Klabautermann knocking it means he'll stay if you hear him working he'll go away. We have the passenger list, it's amazing how many little VIPs come together when you send a ship to the Antarctic, I want to know everything about every one of them, especially this coal mine king from West Virginia that demolished entire mountainsides before selling his business to Patriot Coal, also about that super-birder who in real life is a porn producer and the news announcer that was fired when he lost his voice, those are the stories we need, Carstens. Of course I don't rummage through the trash, it was the bin for recycled paper, I looked out the window and saw him dumping books, I was curious, I drove over even though it's just around the corner, as if I had sensed what I would find, the nicest things, first editions and autographed copies, they were lying in the bin next to pizza packages and advertising leaflets, I had to save the books, I don't rummage through trash BREAKING NEWS RESCUED PASSENGERS SAFELY HOME BREAKING NEWS RESCUED PASSENGERS SAFELY HOME that came off marvelously\n\n### 12.\n\n## 64\u00b027\u20321\u2033S, 62\u00b011\u20325\u2033W\n\nLAST NIGHT, FOR the first time in years, since the hottest summer ever, which followed other hot summers, since the summer when our June climate report was obsolete as early as August, for the first time ever, since the lie I was living daily was operated out of me and my glacier died, there was no looming nightmare. I slept without second sight. When I wake up I feel revived, as though I've undergone a live cell therapy treatment. I lie in bed, a timid light slips under the curtain. One more day, and what a day. Paulina is stretching. Outside our door a passenger goes stomping past on his morning rounds. Paulina's face becomes visible in the light over the nightstand. \"Who are you?\" I ask. A bewitched maiden, she answers, who has to change into the first creature she sees when she wakes up.\n\n\"What a horrible curse!\"\n\n\"Yes, imagine it could be the head chef. But I'm lucky because I saw you.\"\n\n\"You call that luck? Now you're going to turn into an old man, an ugly old man.\"\n\n\"I'll turn into you, into Zeno. But listen, there's more to it, because you've also been put under a spell, by the same spirit.\"\n\n\"What kind of spirit is that?\"\n\n\"It's a spirit who's all mixed up, you have to turn into me.\"\n\n\"I'm getting the better deal.\"\n\n\"Then we'll really be together, in our memories as Zeno and Paulina and now as Paulina and Zeno.\"\n\nShe stretches her arm across the gap between the two beds, our fingers interlace, I know of no more binding gesture. I begin massaging her fingers. \"Are you afraid of hell?\" she asks all of a sudden, while we're still under the covers, turned toward each other. I can't answer right away, I'm focusing on her fingers with the narrowest nails, trying to shake off the thought that this is the last time we'll ever wake up together. With my forefinger I brush her fingertips, one by one, without knowing if her skin will retain these touches. If I were safe inside her fairytale and had one more wish it would be that the river Lethe would flow between the icy continent and Brabant Island.\n\n\"Hell is not a place,\" I finally answer. \"It's the sum of all our lapses and failures.\"\n\nShe looks at me confused, her fingers dig into the back of my hand, she presses her thumb so hard into the flesh below my thumb it hurts.\n\n\"The realization, much much too late that you didn't do anything when you still could, when you still should have, that is hell. And there's no escape.\"\n\n\"I see,\" she says, \"you're trying to reassure me. She loosens her grip. In your own weird way you're trying to tell me you're not going to hell.\"\n\nDan Quentin is standing on a pile of stones, holding a megaphone and directing the red-wrapped extras swarming below him on the ice. Make a mental picture of what the SOS looks like, he blares through the megaphone, in the middle is the circle, the symbol of that which is indestructible, the circle of life, and next to it are two snakes. Why do I say snakes and why do I say there are two? Because there are two basic states, and keep this in mind when you form the S, it's very important, there is the state of contamination and there is the state of purification, there is the poison and there is the cure, are you with me? Dan Quentin puts down his megaphone and surveys his rising work of art: three hundred people awaiting his orders. He appears cheerful, content. In countless interviews he will describe how well it all turned out. And when he's done saying everything he wants to, the announcer will lower her voice and ask how he managed to cope with the drama that came right on the heels of his greatest artistic success. And in a solemn voice Dan Quentin will explain... Now, all together, give me an S (red arms reach up high), give me an O (red arms reach up high), give me an S (red arms reach up high) and now, long and loud and proud, give me an S-O-S! (all arms reach up high), it's a funfair, Octoberfest in the southernmost latitudes, voices rise like wisps of smoke, linguistic differences in red, black, white and gray, I'm standing nearby, his face is straining with emotion, the deckhands correct a few bulges in the curves, on this trip the Filipinos take care of everything, even to the point of ironing all angles out of an SOS. The Zodiacs bring further crewmembers ashore, who storm the small rise like straggling soldiers so as not to miss the spectacle.\n\n\"That's enough,\" Quentin calls out to me. \"We have enough people already.\"\n\n\"They want to be part of it.\"\n\n\"We don't need them.\"\n\n\"It's too late.\"\n\n\"They should go back, they'll just mess everything up.\"\n\n\"It's too late, the crew also wants to be part of the SOS.\"\n\n\"That wasn't the deal.\"\n\n\"The more the merrier, that's what you said.\"\n\n\"I meant the passengers,\" Quentin shouts down from his rock pile, then squawks through his megaphone, \"Faster, faster.\" The manager and his adjutants insert the waitresses, cooks, technicians, cabin stewards, launderers into the growing S snake consisting of notaries, executive consultants, general managers and financial analysts, Paulina is there as well, for a second I can make out her face, Ricardo is behind her, he's placed his hands on her shoulders, then I lose sight of her, all of a sudden sunlight floods into our icy party, this is it, Quentin hurries over, tosses me the megaphone, it's now or never, he's ready to seize the moment, a Napoleon of the arts, he rushes to the helicopter with sweeping strides, that's my cue, I radio Jeremy to tell him I'm heading back to the ship, El Albatros has set off to find a breeding site of blue-eyed cormorants that's supposed to be nearby, Beate has found a place in a bend of the second S, the helicopter lifts off, all hands wave, Quentin's manager rushes from one deckhand to the next, probably to remind them to get out of the picture, they are the scaffold that must be dismantled as quickly as possible so that a pure SOS can shine, I ask one of the waiting boatmen to run me back to the _Hansen_ , which he is reluctant to do because he doesn't want to miss the spectacle, but his mood improves when I tell him he can go right back, and that he should take everyone still on board, even the receptionist, it's all been arranged with the captain, it's a great day today, a real holiday. The fewer people left on the ship, the easier it will be for me.\n\nFrom the sundeck I can see the SOS with the naked eye, using binoculars I can make out individual passengers gazing up at the helicopter, which is making a first loop above them, the light flashes in Dan Quentin's lens like an explosion, like a visual start signal. The few personnel who were left behind as an emergency crew are standing at the rail, I tell them to lower one of the life rafts and go aboard and make themselves ready. They buy my story that the captain wishes to practice the maneuver in these waters, with the weather as stable as it is. Now all that's left is for me to convince the stakeholders that they, too, should board the life raft. The drone of the helicopter and the rattling of the crane accompany me into the bowels of the ship.\n\nAt last I am alone. On a calm sea and not atop a surge of history, alone on a cruise ship that can be steered with a joystick, as if navigating through the islands of ice were long since nothing but a computer game. Track steering ship control is what this technical miracle is called, one pull of the switch and the ship will follow a preprogrammed route, as Vijay the chief navigator showed me one high-sea day, we were talking about Ladakh and Tibet, storm-free passages make for boring shifts, about Kailash and Gangotri, I entered the open ocean as my destination, the way he showed me, a random point in the wide Atlantic, it seems to be working, the ship is cutting through the water, it will get there without me, too. The bridge has three radar devices (black for sea, yellow for land) and two compasses (one magnetic, one electronic)\u2014I won't need any of it, no more than I need the Automatic Identification System, which shows third parties the position of the MS _Hansen_ and lets me know what might be approaching. They will catch up with me. I've taken the flag off the mast and tossed it in the bin marked Plastic Waste.\n\nIt's going to be a long day.\n\nSomeone will find this notebook, someone will read it and decide to publish it or not. One way or the other, I have no need to explain myself further. One human being is an enigma, a few billion human beings organized in a parasitic system are a catastrophe. Under these circumstances I'm just tired of being human. \"It would be lovely to walk the streets with a green knife and scream until I die of the cold.\" There is an eviscerated bird dangling in front of every human home.\n\nI used to believe I had to fight my insidious misanthropy, today I realize that we have to topple humans off their pedestal in order to save them. What does it matter if a person is blind or deaf, blinkered or benighted? Only big blows are capable of jolting mankind. I am calm and resolute. I pull the master switch, all lights on board go out.\n\nIt's high time.\n\nWhat consoles me? That we will leave nothing behind except fossilized excrement.\n\nWhen it gets dark I will go out and I will fly, surrounded by thornfish and sea squirts swimming below me, skates and rays gliding away above me. I will fly until my blood has run to ice.\n\nThose are measurements to die for, copper, you can kiss it goodbye, everyone must be called upon to sacrifice, platinum, no one's going to question it, now that's what I call one efficient move, sooner or later our hour will strike, iron, all ravens are black, oil, measurements to die for, we have to be prepared for every contingency, lol, everywhere there are unexplained delays, chromium, we're doing what we can, his epitaph should read: mistrust the survivors, grit your teeth and get on with it, whoever blinks has lost by half, gold, shit happens, no one's going to question it except for those who deny it, you can't do anything about it, of course I want to get home as soon as possible, no, I'm not just standing here looking at pigeons on purpose, what do I know, the way she's scowling, it's upsetting, it's really upsetting, coal, it was programmed wrong, we got off with a black eye, uranium. We searched the ship, there's no one on board, we're certain that there's no one on board, no idea what happened to the hijacker, we found something, some sign of life, next to the steering console on the bridge, a notebook, it's full of writing, in German if I'm not mistaken, it might help us understand what he did. We fled to the south, where dollars drop from the sky like snowflakes, the business climate and the apparent temperature is trending toward bankruptcy. I was wrong, there's still someone on board, we discovered her on the monitors, an older woman, she was wandering up and down the passageways, she seems dazed, her eyes are glassed over, she says she was attacked by a penguin, not very believable, she claims not to know anything about the hijacking, I know, she says she was sound asleep because of the powerful antibiotics, we'll have to question her of course. The overweight will have to weigh things over, morning, evening, and in the fall, prepared for everything, we're infesting now in our future, the revolution will not be televised, be prepared for everything, the revolution will not be televised, BREAKING NEWS LIGHTS GO OUT TONIGHT FOR FIVE MINUTES BREAKING NEWS LIGHTS GO OUT TONIGHT FOR FIVE MINUTES on and on without end\n\n## _Acknowledgments_\n\nThanks to all who generously shared their expertise and who helped me, on land as well as at sea:\n\nDr. Reinhard B\u00f6hm for the lively glacier tutorial\n\nKristina D\u00f6rlitz for the excellent research assistance\n\nAlexandra F\u00f6derl-Schmid for the second commission\n\nPetra Glardon for the wonderful iceberg photos\n\nProf. Dr. Wilfried Haeberli for his edifying encouragement\n\nChristoph Hofbauer for his high-powered advice\n\nMijnheer Hans Huyssen for the music\n\nAngelika Klammer for her inspired and inspiring editing\n\nFreddy Langer for the friendly press service\n\nDr. Rudi Mair for the conversation about Alpine climate and Antarctic overwintering\n\nBorrego Pedro Rosa Mendes for the days and nights at the Tejo\n\nCompa\u00f1ero Jos\u00e9 F. A. Oliver for his empathetic reading of the manuscript\n\nPapa Heinz Renk for showing me the Tyrolean glaciers\n\nDr. Miguel Rubio-Godoy for the avalanche of calamities\n\nDr. Christine Scholten for the medical review\n\nDoroth\u00e9e St\u00f6bener for the first commission\n\nSusann Urban for the gift of the title\n\nJuli Zeh for the wielding the red pen\n\nHurtigruten Shipping for their double hospitality\n\nThe poems cited in the text are by Samuel Taylor Coleridge, Klabund and Pablo Neruda\n\n## _On the Typeface_\n\nThis book is set in Electra, a typeface designed by William Addison Dwiggins for use on Linotype typesetting machines in 1935. Dwiggins, a mildly eccentric book designer, illustrator, calligrapher and creator of marionettes, is credited with coining the term \"graphic design\".\n\nDwiggins's foray into type design began with a challenge from the Mergenthaler Linotype Company, after he had criticized the dearth of usable san serifs. Electra was Dwiggins's first type design for book setting and would be one of his most enduring.\n\nWhile the popular book faces of his time were revivals of fifteenth- and sixteenth-century printing types, Dwiggins sought to create a typeface that reflected the modern environment. As his friend and fellow illustrator Rudolph Ruzicka commented, Electra was \"the crystallization of [Dwiggins's] own calligraphic hand.\" Its unbracketed serifs, flat arches, and open counters make for a face mild in pretence but alive in personality. Dwiggins explained, \"The weighted top serifs of the straight letters of the lower case: that is a thing that occurs when you are making formal letters with a pen, writing quickly. And the flat way the curves get away from the straight stems: that is a speed product.\"\n  1. Translator's Note\n  2. 1. 54\u00b049\u20321\u2033S, 68\u00b019\u20325\u2033W\n  3. 2. 55\u00b005\u20320\u2033S, 66\u00b039\u20325\u2033W\n  4. 3. 53\u00b022\u20325\u2033S, 61\u00b002\u2033W\n  5. 4. 51\u00b041\u203237\u2033S, 57\u00b049\u203215\u2033W\n  6. 5. 53\u00b011\u20328\u2033S, 45\u00b022\u20324\u2033W\n  7. 6. 54\u00b016\u20328\u2033S, 36\u00b030\u20325\u2033W\n  8. 7. 60\u00b011\u20325\u2033S, 50\u00b030\u20322\u2033W\n  9. 8. 62\u00b012\u20329\u2033S, 58\u00b056\u203243\u2033W\n  10. 9. 62\u00b058\u20329\u2033S, 60\u00b033\u20326\u2033W\n  11. 10. 62\u00b035\u20320\u2033S, 59\u00b056\u203230\u2033W\n  12. 11. 64\u00b050\u20323\u2033S, 62\u00b033\u20321\u2033W\n  13. 12. 64\u00b027\u20321\u2033S, 62\u00b011\u20325\u2033W\n  14. Acknowledgments\n  15. On the Typeface\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. \n  12. \n  13. \n  14. \n  15. \n  16. \n  17. \n  18. \n  19. \n  20. \n  21. \n  22. \n  23. \n  24. \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\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# How to Draw Fantasy Art and RPG Maps\n\n## Step-By-Step Cartography for Gamers and Fans\n\n## Jared Blando\n\nCincinnati, Ohio\n\nwww.impact-books.com\n\n### Thank you for purchasing this _Artist Network_ eBook.\n\nSign up for our newsletter and receive special offers, access to free content, and information on the latest new releases and must-have art resources! Plus, receive a coupon code to use on your first purchase from NorthLightShop.com for signing up.\n\nor visit us online to sign up at\n\n<http:\/\/artistsnetwork.com\/ebook-promo>\n\n# Contents\n\nSpecial Offers\n\nIntroduction\n\nBasic Materials\n\nAdvanced Materials\n\n### CHAPTER ONE\n\n## Building Your World\n\nCoastlines and Boundaries\n\nDefine Your World\n\n### CHAPTER TWO\n\n## Mountains, Forests, Rivers and Water\n\nHigh Peaks...\n\n...and Rugged Hills\n\nWoods and Forests\n\nRivers and Tributaries\n\nShore and Water Lines\n\n### CHAPTER THREE\n\n## Town and City Icons\n\nCities\n\nCastles and Holdfasts\n\n### CHAPTER FOUR\n\n## Typography\n\nScript Style\n\nCultural Typography\n\nLocation Labels\n\nArea Labeling\n\n### CHAPTER FIVE\n\n## Landmarks\n\nRoads and Routes\n\nSpecial Locations\n\n### CHAPTER SIX\n\n## Iconography\n\nSimple City Icons\n\nLocation Icons\n\nOcean Icons\n\n### CHAPTER SEVEN\n\n## Heraldry and Shields\n\nShield Shapes\n\nSimple Heraldry\n\nSimple Shield Symbols\n\nPainting Shields\n\nShield Embellishments\n\nShield Placement\n\n### CHAPTER EIGHT\n\n## Political Boundaries\n\nBoundary Lines\n\nBoundary Colors\n\nBoundary Types\n\nUsing Natural Boundaries\n\n### CHAPTER NINE\n\n## Backgrounds\n\nBorders\n\nBorder Corners\n\nComplex Corners\n\nCorner Flourishes\n\nAdvanced Flourishes\n\nCompass\n\nLegends\n\nLegend Information\n\nInformation Boxes\n\n### CHAPTER TEN\n\n## Painting Techniques\n\nInking\n\nPen and Brush Inking\n\nDetail Inking\n\nTypography\n\nPainting\n\nCreating a Detailed Border\n\nDigital Painting\n\n### CHAPTER ELEVEN\n\n## Putting it All Together\n\nCreating a Fantasy Map\n\nAbout the Artist\n\nAcknowledgments\n\n# Introduction\n\nThe world of fantasy is flooded with the images of characters, castles and epic battles, yet cartography has a particularly intrinsic place amongst these other works of art. Maps have the magical ability to create a world for a viewer without the need for text, visually illustrating entire domains, cities and kingdoms with the stroke of a brush. Along with the ability to link people to the stories they read by placing the reader into the world of the storyteller's imagination, maps add an extra layer of depth and plausibility to the imagined world and its peoples. Though there are a great number of different types of maps, from world and regional to cities or even specific buildings, this book is a (modest) attempt to give the reader the ability to create their own world maps in an easy and effective manner. Starting with techniques that gradually move from basic into the more advanced, I will act as your guide through the extensive, but rewarding process of mapmaking. From tall mountains and rolling plains to the great seas, the power of creation is at your fingertips!\n\n# Basic Materials\n\n#### Drawing Implements\n\nFor map work, it's best to have the right tools of the trade with you. Having a few different types of pencils is best, namely a 2H, an HB and a 2B, as well as a kneaded eraser and a pencil sharpener. I also use a mechanical pencil that takes .05mm lead for detail work.\n\n#### A Note on Pencils\n\nDrawing pencils range from 9H (lightest line color and the most clay) to 9B (darkest and richest color, containing more graphite). For our use, a 2H pencil produces a light line and is excellent for light sketching and delicate shading. The HB pencil is a great all-purpose tool and is very much in the middle of the pencil spectrum, making it quite useful for drawing in mountains and hills. A 2B is suitable to darken up boundaries, giving lighter pencils a heavier line weight. A mechanical pencil is a bit advanced, but it also gives an excellent range of line weight and density with some practice.\n\n#### Drawing Surfaces\n\nAfter you have picked up your pencils, it's time to think about what surface you will be drawing on. The paper your map is on is just as important as what you use for rendering. Drawing pads come in a great many sizes and weights, and are up to a person's preference, though ideally you will want one with enough room to draw and not be cramped. Watercolor or parchment paper are both ideal for mapmaking as well. I personally use Arches watercolor paper for mapmaking, since it holds pencil and ink nicely and it's quite durable.\n\nPapers are categorized by poundage level (65-lbs [175gsm], 90-lbs [190gsm], etc.), which essentially is how stiff and thick they are. The higher the poundage, the stiffer the paper is. For mapmaking, it's best to have a paper that is at least 65-lbs (175gsm) as this will give you a sturdy base. Watercolor papers also have a \"press\" rating, either hot or cold. Hot-pressed paper is smoother and has less tooth to it, while cold-pressed paper is rougher and has more texture.\n\n#### Aging Paper\n\nTo get that nice old parchment look, try soaking your paper in dark roast coffee. You can either use a large pan to lay the paper in, or dampen the papers and then brush on the coffee with a folded piece of paper towel. It will yellow your paper beautifully.\n\n# Advanced Materials\n\nWhen you are ready to move on from pencil work, there are some tools that you will require. For traditional work, a variety of brushes is essential for painting and detail work, while inking requires specific inks or ink pens. When working digitally, you will need a tablet, stylus and Adobe Photoshop or GIMP.\n\n#### Inks\n\nAfter your pencil work, inking will give permanence and boldness to your map. Though most people think of inks as just bold thick lines, and indeed this is one of the techniques that can be used, inks also work well as light washes that can be brushed on to give gentle tonality. Inks come in a baffling variety of colors, though it's best to stick with black, brown and red inks to start. You will need a small selection of brushes for your ink work, including nos. 00, 0 and 1 round, a no. 2 flat as well as a large 1-inch (25mm) flat watercolor brush. Calligraphy pens with interchangeable head tips, as well as Copic Multiliner brown or black ink pens are also useful, and each come in a variety of sets. Inking and its various tools are covered in a more detailed fashion in chapter 10.\n\n#### Traditional Brushes\n\nBrushes come in a bewildering variety of shapes and sizes, but you will need only a select few to get started. The most common brush shapes are round or flat, and most are numbered from 00 (the smallest) to 10 (the largest). Brushes also fall into three different groups: watercolor, acrylics and oils, and are grouped by bristle makeup: synthetic, natural or mixed. For adding color in mapmaking, it's best to use watercolor brushes in a mixture of flats and rounds. This will be discussed in more detail in later chapters.\n\n#### Digital Tools\n\nWhen working digitally, you will need a tablet and stylus. A stylus is the pen tool that has built-in pressure sensitivity to help simulate light or heavy strokes depending on how hard or soft you press. There are a few tablets that are cheap and easy to get at your local electronics shop or online, though I personally use a Wacom tablet. With multiple interchangeable nibs to simulate different surfaces, it's an excellent tool for working digitally. As for software, Adobe Photoshop or GIMP (a freeware program very similar to Photoshop) are both excellent and function very similarly.\n\n# CHAPTER ONE\n\n# Building Your World\n\nWhether you are partial to deep dark forests, tall mountains or vast plains, this chapter will teach you the techniques to bring your vision of a fictional world to the page using your pencils and imagination. We will cover some of the basic methods of laying in the various elements that will make up your fantastical world. Maps often consist of a multitude of elements that convey information to an onlooker. Rivers, hills, mountains, cities and such are all parts of a successful and interesting map, though a balance of these elements is always a good thing.\n\nHaving a strong idea of the lands you wish to draw is great, but don't be afraid to look at the many historical maps that exist to give you further inspiration for your own map. Most maps are based on functionality, so keep this in mind when laying out yours. A functional fantasy map is more believable and therefore more exciting and inspiring. Using various pencils to sketch and draw, soon you will have a world of your own coming to life!\n\n## Coastlines and Boundaries\n\nThe first thing we need to do when starting a new map is to lay in the coastlines. Coastlines are essentially the boundaries of your fantasy map, serving to both differentiate between land and sea and to define the parameters of the type of world you are creating. Before laying in your boundaries, try to think of the type of lands that you want to portray, as well as the people who will inhabit them. Maps based on real places and locations are often some of the best because they are grounded in our reality, and are thus more believable. Do you want a landlocked mountainous area full of lakes and rivers, a sprawling archipelago of many islands or something in between? However you want them, your coastlines are a very important part of your fantasy map.\n\n#### Ocean-Heavy Map Sample\n\nAn ocean-heavy map will have only 20\u201330 percent of the land mass showing, and the rest is water. Having a large central island or two surrounded by smaller distant islands works well. A map such as this would have vastly different cultures (most would be seafarers) than a normal map.\n\n#### Balanced Map Sample\n\nBalanced maps are divided 50\/50 or 60\/40, usually in favor of the land. This would be more representative of our own earth, mostly because maps of these types don't include whole seas and oceans, just parts of them.\n\n#### Land-Heavy Map Sample\n\nCoastlines and boundaries are often depicted as being very broad and dark, with clear definition between oceans and landmasses so that there is no confusion between the two. Coastlines will most likely be the boldest lines on your map, and with good reason. A 2H pencil is best for laying out your initial sketch lines, while a 2B pencil is best for making dark, heavy lines\n\n## Define Your World\n\nWe'll start with a single continent for now, move on to islands and then finish with some coastline details and refinements. When drawing in your coastline boundaries, you will start with a light pencil framework of all the major continents and geographical areas, including any islands and major lakes. After the light sketch comes the darker pencil to further define and solidify the boundaries, making them nice and clear. When the boundaries look dark and defined, the light pencil line can be erased, leaving only the broad and dark final coastlines. Once you know these basic steps, you can move on to adding in multiple continents to form a large map.\n\n#### STEP 1: Draw a Simple Shape\n\nTo begin, lightly draw a very simple continent shape, roughing in the basic outline that you can detail further in the next step. Be general and don't press too hard; you just want a light pencil line that you can erase or modify in the next step.\n\n#### STEP 2: Refine the Coastline\n\nOnce your general shape is done, go back in with an HB pencil and refine your original line so it's more detailed. Don't be afraid to be creative or stray radically from your original line. Try to make your coastlines fun and interesting. When you have finished your darker and more detailed line, erase the light sketch.\n\n#### Balancing the Map\n\nYour light sketch will be the framework that you build upon and is very important for composition of the map as a whole. It is essential to try to strike a good balance between the amount of land and water. A map with too much water may look too sparse or uninteresting, while a map that is completely landlocked may look just as strange with no, or very few, bodies of water.\n\n#### STEP 3: Add Islands\n\nOnce your basic continent is drawn, look for areas to draw in any islands that may fit the framework of your map. Islands are often grouped together in bunches of irregular shapes, forming chains that stretch between continental masses. Some are so large that they may form their own small landmass that is totally separate from the mainland. Experiment with shapes and sizes, from short and stout to long and thin. Just remember that islands are almost always accompanied by other smaller islands to form an island grouping.\n\n#### STEP 4: Add Lakes\n\nOnce your islands are laid in, you should start thinking about laying in some lakes on the mainland, or even on your new islands if they are big enough. Lakes are often long and fingerlike since they are depressions that lie in between mountains and hills. Just as with islands, don't be afraid to try all shapes and sizes or to have multiple lakes grouped near each other.\n\n#### STEP 5: Create Inlets and Bays\n\nInlets and bays are a good way of making your coastline very unique. Create some bays by drawing into the interior of your sketch, forming half bowl-like shapes along your coastlines. Bays can also be quite irregular, or even contain some small islands to help define them. Don't be afraid to make them a bit more angular or jagged. Inlets are easier, and you can include some by drawing wedges or small pockets that face into your landmasses. Be sure to erase the original sketch lines as you form your new coastlines.\n\n#### Lake Shapes\n\nTree leaves are a good source of shape inspiration for modeling your lakes, especially maple leaves. Lakes often lie in between mountains and hills because water is lazy and will always run downhill. So, remember to draw some lakes around them when you get a chance.\n\n#### STEP 6: Solidify Your Coastlines\n\nOnce you have completely laid out your framework in pencil, it's time to go back in to solidify your coastlines. Using a 2B pencil, trace over your lines, wiggling the pencil a bit to give more detail and character to the original line. Don't be afraid to stray from your original line or to add in more bays, inlets and islands. After you have darkened up your framework, carefully go back in and erase any parts of the original framework that might be showing. Try and leave only your darkened coastlines behind.\n\n#### STEP 7: Finished Coastlines\n\nWhen you have finished cleaning up your original light framework, your coastlines should look a bit like this, and you'll have completed the first part of your fantasy map.\n\n# CHAPTER TWO\n\n# Mountains, Forests, Rivers and Water\n\nTall mountains and rugged foothills often dominate fantasy realms of all shapes and sizes. Mountains seem to add a sense of vastness and awe to a fantasy world's settings and surroundings. Mountains not only serve as physical boundaries between cultures and races, they often form political boundaries as well, dividing various nations and people.\n\nYour fantasy map will also need forests. They are an instant area of mystery and wonder that serve as boundaries, obstacles or refuges for the inhabitants of your world. Forests can range from dense groupings of trees forming an untamed wilderness, to small bodies of trees that represent quiet and peaceful woodlands. Geologically, forests are often located in and around mountains due to the immense watery runoff, though anything goes in a fantasy map! Don't be afraid to include haunted woods, sparse or dead trees or larger-than-life magical woodlands.\n\nBodies of water have a big impact on maps and how they are laid out. Seas are usually massive bodies of water that are larger than lakes, but are still connected to the larger ocean. Rivers are freshwater runoff from mountains and excessive rainfall that form into paths that flow down into the seas and oceans. Rivers are one of the most important land features within a map (and a country!), though they can be tricky to place correctly. A good idea to keep in mind for river placement is that water is generally lazy and will take the path of least resistance downhill. Thus, many major rivers start as a combination of smaller tributaries that come down out of the tall mountain valleys and hill ranges that are overflowing with melting snow and rainfall.\n\n## High Peaks...\n\nWhen drawing mountains, begin with a light pencil line to sketch the basic mountain shapes you want to draw. It's a good idea to have a light general outline of where you would like to place your ranges and solitary mountains. After your sketch is complete, draw in the mountains one by one, following your light sketch. After your mountains have been drawn, feel free to sketch groups of hills that may lie around or near the mountain ranges, or form separate and independent chains of their own. Drawing hills is much the same as drawing mountains though they are much smaller and more bowl shaped. Finally, go over your light sketch lines with a darker pencil to add definition and detail.\n\n#### STEP 1: Lay Out a Mountain Range\n\nTo start, take your 2H pencil and lightly draw lines where you would like your mountain ranges to be. These guiding lines are useful for you to follow, and allow you an idea of where your mountains will be grouped. Once you are happy with the general placement, move on to the next step.\n\n#### STEP 2: Draw a Mountain\n\nTake your HB pencil and slowly draw your first mountain shape, making an upside-down V or W. Try and draw this first mountain in the middle part of your initial light sketch line.\n\n#### Mountain Placement\n\nWhen drawing in mountains, it's a good idea to think about where you'd like to place them, how mountainous you'd like your world to be in general and how your mountains will affect the world as a whole. Mountains serve as excellent country boundaries, so keep this in mind before you start sketching them in. Of course, you could just start placing mountains in random spots and see where it takes you as well. It's your world after all!\n\n#### STEP 3: Create a Mountain Chain\n\nNow that your first mountain is complete, it's time to give him some friends and make a whole mountain chain. Continue by adding a mountain to the right and left of your first mountain until your whole chain is complete. Mountains drawn in the middle of the chain should be the largest, while those toward the end should gradually reduce in size. Mountains are far from regular in shape and size, so feel free to experiment with different looking mountaintops, from the more jagged to the more rounded.\n\n#### STEP 4: Erase Guidelines\n\nTake your kneaded eraser and gently erase your starting guidelines.\n\n#### STEP 5: Define Your Mountains\n\nOnce your basic mountain chain has been completed, it's time to flesh them out and give them some definition. Take your HB pencil and draw thin lines that run down each side of the mountain, with each mountain having five to ten lines each. Do this for each of your mountains and watch how they come to life!\n\n#### STEP 6: Finish Your Mountain Range\n\nOnce you are happy with your first mountain range, repeat steps 2\u20135 everywhere on your map that you have placed your light mountain guidelines.\n\n## ...and Rugged Hills\n\nHills are the smaller cousins of mountains, but are just as important to a map's geography. Hills also serve as physical and political boundaries, as well as rough terrain for various nefarious or evil creatures to lurk within. Hills are usually located next to or very near mountains, though they will often form their own distinct ranges as well.\n\n#### STEP 1: Add Your First Hill\n\nNext to the bottom of one of your mountains, take your HB pencil and draw a small upside-down U, about half or quarter the size of a mountain.\n\n#### STEP 2: Lay In More Hills\n\nAfter that, draw a few more upside-down U shapes along the mountain's lower edges, and continue this all along the mountain range.\n\n#### STEP 3: Form Hill Ranges\n\nOnce your hills at the bottom of the mountains are drawn in, feel free to draw more hills extending off of your existing hills. These are hills that extend out from the more mountainous high ground, forming hill ranges of their own. Not all hills have to extend off of the mountains; feel free to add hills wherever you see fit. Having a group of four or more hills is best, arranged into relative lines. Try and make shapes with your hills, forming C's and S's for variety.\n\n#### STEP 4: Define Your Hills\n\nWhen you have finished laying in all your basic hills, go back in and add a few small lines to each hill bottom, just like you did to define your mountains. This adds some dimension and weight to the ranges.\n\n#### STEP 5: Complete the Ranges\n\nAfter step 4, your mountains and hills should look something like this.\n\n#### STEP 6: Add Character\n\nIf you are feeling adventurous, get creative and add troll holes and other openings and paths in and around your hills and mountains. Any details you can add will bring some character to the map in general.\n\n#### Alternate Mountains\n\nThere are several different ways of detailing mountains. Some use more shading than others. Once you have mastered the basic mountain technique, try and practice drawing some mountains to resemble the examples shown. A map should keep the same mountain technique for uniformity's sake, though I urge you to experiment and develop your own mountain types and forms. Certain mountain techniques work better than others depending on the tone of the map.\n\n## Woods and Forests\n\nDrawing forests is a fun and interesting way of adding a lot of dimension to your map. Starting with a general area sketch, you will lay in a few trees here and there, first drawing the large groups of trees that will form the main parts of your major forests. After that, you can add smaller woodlands or standing bodies, and finally end by adding some definition to your woods. There are a few separate techniques for the depiction of forests, with some different shapes and sizes that you can use, though each is easy to learn.\n\n#### STEP 1: Mark General Forest Areas\n\nTake your 2H pencil and lightly draw the general areas where you want to place your major forests. Press very lightly since you will only be using these lines as a guide.\n\n#### STEP 2: Draw the First Tree\n\nNow comes the fun part! Using your HB pencil, draw your first tree by making a triangular shape with a flatter bottom. Your first tree should be placed within the guidelines you drew.\n\n#### STEP 3: Draw Major Forests\n\nOnce you have drawn in your first tree, expand your forest by drawing more trees using the same general shape as your first one. Keep drawing trees one next to another until you feel satisfied with the size and shape of the grouping. Feel free to vary the size of your trees to make them look more organic, grouping together similar tree shapes to add variety.\n\n#### STEP 4: Add Minor Woodlands\n\nOnce your major forests have been drawn, add in some random groupings of trees in and around your major groupings to add more flavor and variety. Try and draw your minor woodlands by grouping three to five trees. These minor woodlands can be placed anywhere on the map, though they look really good when they are used to break up really sparse areas.\n\n#### STEP 5: Define the Trees\n\nOnce your major and minor woodlands are finished, it's time to add some definition. Take your HB pencil and gently shade in the bottom corner portions of each tree with a few lines. Try and stick to only one side of the tree shape to provide a more uniform look, though some trees should be more shaded than others. After shading, add tree trunks.\n\n#### STEP 6: Finish With Shading\n\nAfter your tree trunks are laid in, it's time to shade your forests and woodlands. Use your HB pencil to draw lines around the bottom of your tree trunks and along the sides of your established forests, as well as in between the gaps you have left. Try to only add shade lines to the bottom of your tree groupings, varying your line length, to give an appropriate shading look.\n\n#### Types of Trees\n\nBe creative with the shapes of your trees. Trees come in a whole variety of shapes and sizes, and you should use several different types for different tree looks. Certain shapes can represent different tree species, and they look best when grouped together. However, mixing and matching makes for an interesting and complex-looking forest that is sure to add dimension to your world.\n\n## Rivers and Tributaries\n\nThe drawing and placement of rivers can be a tricky but really rewarding experience when making your map. The vision of mountain streams cascading down the mountain sides and rushing into mighty rivers is a powerful one. Rivers are one of the best natural boundaries available to a mapmaker and have served as excellent defensive borders for kingdoms and vying powers since time immemorial. All rivers actually begin quite small, as tiny streams of mountain waters that trickle downhill to merge together, forming larger streams called tributaries. Tributaries continue to flow downhill, picking the easiest path through the land, eventually running into other tributaries and forming into a larger river that flows out to the sea. Since water flows around obstacles instead of right over them, rivers and tributaries often have distinct snakelike properties, and turn and twist across the land.\n\n#### STEP 1: Decide General River Placement\n\nTaking your 2H pencil, lightly draw your first guideline for where you think a river would start and end. Remember that rivers will often begin in the middle of two mountain or hill ranges, so you should place your river somewhere near a range and have it end in the sea. Rivers sometimes originate from lakes too, though most lakes never spawn more than a single major river, and some rivers will flow through a lake and on down to the coast.\n\n#### STEP 2: Develop the First River Line\n\nWhen you are happy with the initial placement of your river, it's time to draw it in darker. Take your HB pencil and follow the general direction of your guiding line. Start at the base of the mountains and slowly draw a wavy line outward toward the sea. The line may curve and snake, so don't be afraid to give it some movement and curvature.\n\n#### STEP 3: Completing Your First River\n\nStarting about halfway down your first line, draw another line parallel to your first and continue it all the way to the coast. Rivers often widen as they approach the coast, and this gives you a good representation of that.\n\n#### River Shapes\n\nRivers often look like giant twisty earthworms from the sky as they follow the natural curvature of the earth. You don't need to emulate that effect completely, but some movement and snaking is always a good thing to hint at and will make your rivers look more exciting. Take a look at the mighty Mississippi River in North America, or the Nile in northern Africa for some inspiration.\n\n#### STEP 4: Open the River Mouth\n\nWhen both river lines touch the ocean boundary on the coast and the river water meets the ocean, it's called the river's mouth. Take your kneaded eraser and gently erase between the two lines to form an open river mouth. Once complete, you should have a nice-looking river mouth.\n\n#### STEP 5: Develop a River Delta\n\nTo attempt something a bit more complicated, you could try your hand at a river delta, which is essentially when a river fragments back into smaller tributaries before it reaches the ocean, forming coastal lowlands ripe for lush crops. To draw a river delta, just add a few small lines to the river mouth, making it spread out like the branches of a tree.\n\n#### Line Pressure\n\nPress lightly with your pencil when you are near the mountains and hills, and then harder for a darker and richer line when connecting your river to the coast. This gives the impression that a river starts small but gains strength and speed as it connects to the ocean.\n\n#### STEP 6: Drawing the Tributaries\n\nOnce you have finished the river, add the tributaries that flow down from the mountains and hills. Tributaries often look like branches stemming from mountains and hills. To simulate these smaller streams, take your HB pencil and draw a thin wavy line from the closest mountain terrain to the top of the nearest river, snaking a bit along the way.\n\nContinue drawing those tributaries, connecting them to the topmost part of the river (where it begins). A good rule of thumb is to have three to five tributaries per river, with most coming from surrounding hills and mountains.\n\n#### STEP 7: Redraw the River Top\n\nAfter your tributaries are finished, erase the top of your river, and redraw it like a tributary.\n\n#### STEP 8: Finished River\n\nYour finished rivers and tributaries should look something like this.\n\n## Shore and Water Lines\n\nNow that you have defined your landmasses and rivers, it's time to put some flair into the continent boundaries. Emphasizing the shores with a bit of decoration can go a long way toward making it easy to quickly recognize landmasses. We can do this with a technique that's easy to do, though it might take a bit of practice to perfect.\n\n#### STEP 1: Draw Shorelines\n\nTake your 2H pencil and start drawing a light line around your original coastlines. The line should maintain about 1\u20448\" (3mm) distance away from your original boundary, and should be lighter in weight and richness. This line should not completely mimic your continental outline, and should smooth past any major details of your coastline, only giving a general impression.\n\n#### STEP 2: Don't Forget Lake Shorelines\n\nIf you have any lakes on your map, make sure to include a shoreline within each too.\n\n#### STEP 3: Continue Adding Shorelines\n\nOnce your first line is completed, trace another line on the outside of your first shoreline, again trying to maintain 1\u20448\" (3mm) between the two lines as best you can. You may include additional shorelines on your map, though two to four works best.\n\n#### STEP 4: Make Ocean Lines\n\nYou can go a step further and draw some defining lines to better denote the beginning of the ocean.\n\nTaking your HB pencil, draw small lines that mimic small waves extending horizontally from the shorelines. These lines can be anywhere from 1\u20442\" (13mm) to 1\" (25mm) long but should vary only a small amount from one another.\n\n#### STEP 5: Finished Ocean Lines\n\nEach line should be very close to the next and be equidistant from one another. Continue drawing these wave lines until the entire coast has been decorated (large lakes may be filled in this way, too). Once you have finished your ocean lines, it should look something like this.\n\n#### Shoreline Shapes\n\nVisually, this technique works best with a single shoreline, with ocean lines sprouting off of that, though two shorelines could work just as well. Feel free to experiment with defining your coastlines and shorelines to make them original and unique.\n\n# CHAPTER THREE\n\n# Town and City Icons\n\nMost maps exist to mark where important places are in relation to other places, and icons are the elements that do just that. In this chapter, various techniques on how to illustrate the basic icons will be discussed in detail, with a number of step-by-step demonstrations laid out for the beginner as well as the experienced mapmaker. Icon illustration has a great impact on the look of a map and its tone, with icons varying from tiny illustrations of castles and huts to the use of symbology for simplicity's sake. Whether the important places are cities, towns, fortresses or bridges, they all form key components that give information about a country and its people. For fantasy maps, icons are doubly critical, providing insight into the various fantastical peoples and places within the boundaries of your world, as well as allowing you to flex your creative muscle. Soon you will have dozens of towns, cities and hill forts populating your world.\n\n## Towns\n\nTowns are by far the most numerous of settlements both in our world and most fantasy worlds. Rendering towns is a relatively simple affair, with much room for experimentation and creativity. Before you start scribbling away, you should think about the type of setting you are building in your world, and how towns might look and be depicted. Certain cultures have very distinctive architecture, and it's always good to introduce a pinch of realism into your fantasy setting. How many towns dot the land? Are there many or only a few? The density or sparseness of towns can say much about the region, with many indicating a flourishing society, and others either a diminishing people or unexploited territory.\n\n#### STEP 1: Decide Town Placement\n\nWith your 2H pencil, draw light dots where you would like your towns to go. Don't press too hard since you will be drawing right over these soon.\n\n#### STEP 2: Start With a Roof\n\nPick one of your initial town placement spots and select your HB pencil. Begin by drawing a small rectangle, this will serve as one of the town building's roofs.\n\n#### Location\n\nMany towns often lay near rivers, coasts or natural resources such as pasture for cattle and farmlands. Try and place your cities in areas that you think people would settle. A city in the middle of the forest usually doesn't make too much sense, unless of course, those people are Elves! A smattering of towns in random places works just as well in a prolific society. Trade and goods flow along roads, keeping the towns all alive.\n\n#### STEP 3: Expanding the Buildings\n\nDraw another smaller rectangle next to the first, adding as many as you would like according to how large you think the town should be. I recommend two to three roofs at first.\n\n#### STEP 4: Fill In the Buildings\n\nWhen the roofs are drawn in, add some rectangles directly below them. These are the house bases for your roofs to sit on. Each roof should have a box underneath it to fill out the houses. It's OK to vary these a bit in size.\n\n#### STEP 5: Fill In the Details\n\nNow that your basic shapes are there, add a few small doors, some crossbeams, and a few chimneys and roof lines. Congrats! You have made your first basic town!\n\n#### STEP 6: Embellish With Extras\n\nIf you are feeling creative, add a small guard tower, temple dome or some other small detail to your town. These details will make town icons come alive, and add a touch of uniqueness to each.\n\n#### Town Examples\n\n## Cities\n\nCities are the largest and most prosperous of civilized centers, and each is often much larger and more powerful than their smaller counterparts. Cities offer a greater amount of variety than towns in general, many being surrounded by protective walls or guarded with tall towers, all of which are fun to draw. Cities can be two to three times the size of towns, though you will follow very similar steps when drawing them. Have fun!\n\n#### STEP 1: Decide City Placement\n\nJust like you did with your towns, take your 2H pencil and draw light dots where you would like your cities to go.\n\n#### City Placement\n\nBe careful when you are placing your cities, as most are founded on major trade routes, rivers or as main ports. Most cities need access to lots of water, so placing them on coasts or large rivers is ideal. However, that should not stop you from placing them in foothills, atop mountains, in the frozen north or even under the sea. In fantasy, magic is usually alive and well, so anything goes!\n\n#### STEP 2: Build a City Wall\n\nFirst take your HB pencil and draw an elongated rectangle.\n\n#### STEP 3: Add the Wall Top\n\nDraw the top of the wall with an alternating square pattern forming crenellations.\n\n#### STEP 4: Create Towers\n\nTo add towers to your basic city wall, draw vertical elongated boxes connected to your city wall. Make them taller than they are wide, and add wall tops.\n\n#### STEP 5: Expand the Walls and Towers\n\nOnce you have your first basic wall and towers drawn, you can repeat steps 2\u20134 to expand your city, adding multiple towers and walls. Add as many as you like! Towers can vary in size and shape, though sometimes it's best to keep some uniformity in them.\n\n#### STEP 6: Add Building Roofs\n\nWhen your towers and walls are finished, you should add some roofs to the city. You can use the town steps from earlier in the chapter to help you out. Adding a few roofs, one on top of the other, is good and can represent how cluttered a city can get.\n\n#### STEP 7: Include Specialty Buildings\n\nTo add even more detail and uniqueness to your city, you can add a large temple dome, or bigger buildings to complement the smaller houses. If your city is near the ocean, add a dock. If it's near a river, add a bridge extending from the city over the river. Holy cities may have cathedrals and temples.\n\n#### City Examples\n\nSmall city\n\nRiver city\n\nCoastal city\n\n## Castles and Holdfasts\n\nPlacing castles and holdfasts is one of the most enjoyable things you can draw on a map. They are easy to draw and add plenty of character and clout to a fantasy map. Traditionally, castles are used for defending large stretches of territory, and are usually located on hills dominating the surrounding landscape. From its fortifications, defenders could observe an enemy army from the safety of its walls, and when the enemy passed them by to attack somewhere else, they could rush out and attack the enemy from the rear. Castles and large fortresses are built to house entire armies of troops, while holdfasts are smaller keeps that are the homes of minor lords and knights. Countrysides are usually covered in holdfasts, while citadels and fortresses are reserved for the protection of important trade cities, major rivers (and bridge crossings), rich coastal routes or along political boundaries to defend against a foe.\n\n#### STEP 1: Decide Placement\n\nJust like you did with your towns and cities, take your 2H pencil and draw light dots where you would like your castles and holdfasts to go.\n\n#### Historical Inspiration\n\nIf you would like some inspiration or ideas for some of your castles and fortresses, just take a look at some history books on the medieval times, or the crusades. Historically, the Normans were fantastic and very prolific castle builders\u2014many of which are still standing today.\n\n#### STEP 2: Start With a Tower\n\nStart with the castle's first tower. Begin by drawing an elongated rectangle that is taller than it is wide.\n\n#### STEP 3: Draw the Wall Top and Window\n\nFinish your first tower by adding crenellations (fortifications) to the top. Add a window or two.\n\n#### STEP 4: Finished Keep\n\nAt this point, if you want to finish the icon as a keep, add a small door and a flag to the top and your first keep is complete. You can also add a small house or two standing next to it to give it some more personality, though lone keeps work great too.\n\n#### Keep Examples\n\nSmall keep\n\nMedium keep\n\nLarge keep with small curtain walls\n\n#### STEP 5: Expand the Keep Into a Castle\n\nIf you want something larger, expand your keep by adding another tower or two, and another wall section.\n\nCastles work best with three towers and a few wall sections, but you shouldn't be constrained to that. Experiment with different wall lengths and tower heights, mixing and matching as much as you like.\n\n#### STEP 6: Add a Large Castle Tower\n\nOnce your have expanded the castle, add a large singular keep to the middle of the castle. It should be wider than the standard towers and have a few windows.\n\n#### STEP 7: Castle Details\n\nOnce the keep, towers and walls have been drawn, you can finish your castle by adding detail to the large castle door, a few more windows and some flags on the towers. Add any specialty buildings you'd like as well.\n\n#### Castle Examples\n\n# CHAPTER FOUR\n\n# Typography\n\nTypography can be one of the most important elements in cartography, and it can contribute greatly to the success or failure of a map. Typography is essentially the art of writing. For mapmakers, that means laying in text to label locations and geographical areas so a reader knows what is what. There are many parts of a map that need typography, from the title bar of your map to the various towns and geographical features, and all should be clearly marked. Not only is typography used for marking places and areas on a map, it can also be used as an effective decorative element as well, adding a layer of sophistication to your fantasy map. In this chapter we'll address script style, which deals with the look of the actual text your are using, and labeling, which has to do with where and how the text is placed.\n\n## Script Style\n\nBefore you start marking down locations on your map, there are a few different writing styles you may want to try out for added emphasis and flair. In this section of the chapter, I will go over a few different types of script styles that can be useful for labeling. I have divided script styles into standard scripts and decorative scripts. For the most part, the most common style of label is in our very own standard script and its various offshoots. The more advanced may try their hand at the more elegant and cursive-esque decorative styles. Most forms require a bit of practice to lay in correctly and evenly, and some are more complex than others. I have included a number of script ideas for you to draw inspiration from, and if you're really creative, to expand upon to create a unique new style!\n\n#### Historical Languages\n\nWriting is a powerful graphic and communicative tool, and the fantasy genre has plenty of examples of unique languages and inspirations to draw upon. Ancient Nordic runes and Egyptian hieroglyphics are excellent resources.\n\n## Cultural Typography\n\nBefore you start putting labels on your map, it pays to think about the culture or time period that you are striving to portray. Maybe the map is of a strong empire, so your script should be strong and bold as well. In contrast, the map of an Orc territory might have a more jagged and disjointed look to convey the barbarous nature of greenskins. Script styles play a massive role in the tone of your map. Standard script with little or no difference from our own might serve well for a map of mostly human lands, but what if you were portraying the realms of great Elven lords?\n\n### STANDARD SCRIPT EXAMPLES\n\n#### Imperial Script\n\nImperial script is one of my favorite styles, and is a strong script resembling an authoritative high medieval realm. I often use this sort of script for maps of cultured human lands.\n\n#### Gothic\n\nGothic script is a bit less refined than the Imperial script and represents more of a dark-age Viking feel.\n\n### DECORATIVE SCRIPT EXAMPLES\n\n#### Elven\n\nElven script should look like a more refined and fluid cursive style combined with runes, representing strength but also grace and dignity.\n\n#### Draconian\n\nDraconian script is my idea of a very old and mostly forgotten language, remembered only by the lore masters. It is long, flowing and difficult to read, but quite beautiful.\n\n## Location Labels\n\nThe first and most common element of typography is labeling your location icons. Towns, cities, castles, keeps and other actual locations represented by an icon should be labeled in a straight line, with no curve to the label. With your label straight and uniform, it is easy for a viewer to instantly decipher what is a location and what is an area. Along with being straight, labels should try not to overlap other elements of the maps such as hills, forests and other icons. It's best if you can keep a uniformity for all of your location icons, though it's not completely necessary as long as the label is close enough not to be confused with anything else.\n\n#### STEP 1: Mark the Bottom Script Line\n\nTo start, pick one of your towns, and with your kneaded eraser erase any terrain elements that may get in the way of your label. Then, using your 2H pencil and a ruler, draw a straight line lightly next to your town icon as shown here. This is your bottom script guideline.\n\n#### STEP 2: Mark the Top Script Line\n\nOnce your bottom guideline is down, use your ruler again to draw another light line above your first. The distance between these two lines will decide how large your type will be.\n\n#### STEP 3: Mark the Middle Script Line\n\nWhen you have both your top and bottom lines down, add a middle line. This is a useful tool for when you have to do upper- and lowercase letters.\n\n#### STEP 4: Start Your Text\n\nNow you can write your town's name. Using the light guidelines, carefully begin to pencil in the name of your town, starting with an uppercase letter.\n\n#### STEP 5: Continue Your Text\n\nTry not to go over the guidelines as you carefully write your text using whatever style you have chosen.\n\n#### STEP 6: Finish Your Text\n\nFinish up your text, and use your kneaded eraser to erase your guidelines. With a steady hand and patience, your first town's label should come out great!\n\n#### Emphasizing Capitals and Citadels\n\nSome locations are much more important than others and should be emphasized accordingly. Most often these will be citadels and capitals of great nations. When labeling these, you can use all uppercase to give them a look of preeminent importance. You could also simply use a larger text size.\n\n## Area Labeling\n\nLabeling areas is a bit trickier than labeling your towns and cities. Oftentimes you will need to find a way to fit your label in overtop of your map elements. Labeling rivers is relatively easy, while it's a bit trickier with other areas. Sometimes area labeling requires erasing some of the element that you have already drawn so that you can fit in your label. It sounds tricky, but with some patience and practice, you will be a pro in no time!\n\n### LABELING RIVERS\n\n#### STEP 1: Set the River Script Guidelines\n\nTo label a river, first select the river you wish to place your text along. Then, with a 2H pencil, lightly pencil in your top guideline using the actual river as your bottom line.\n\n#### STEP 2: Pencil In the First Letter\n\nOnce your guidelines are down, carefully pencil in your first letter with your HB pencil.\n\n#### STEP 3: Space Your Letters\n\nAfter your first letter, pencil in your second, and then third, each one following the curvature of the river. Leave some space in between each letter so that your script spans the length of the river or at least most of the river if it's long.\n\n#### STEP 4: Finish the River Script\n\nWhen you have finished penciling in all of the letters, go back in and erase the guidelines with a kneaded eraser.\n\n### LABELING MOUNTAINS\n\n#### STEP 1: Decide the Mountain Script Area\n\nWhen labeling a mountain range, approximate how much space you will need to lay in your text. You will be writing your text above, below or in the middle of the mountain range, wherever you feel it will look best. Use a 2H pencil to lightly mark the area edges.\n\n#### STEP 2: Pencil In Guidelines\n\nUsing your 2H pencil, lightly draw lines that follow the contours of the mountains. Do not be afraid to skip over other elements, as your mountain text can hop over them and continue on the other side.\n\n#### STEP 3: Add the First Letter\n\nWith your kneaded eraser, erase between the guidelines and the edge lines where your text will go. Then use your HB pencil and carefully pencil in the first letter.\n\n#### STEP 4: Continue Adding Letters\n\nContinue drawing in your letters using your guidelines, spacing each letter equidistant from one another, following the general contour of the mountains. Mountain letters should all be of the same size, with uppercase letters being optional.\n\n#### STEP 5: Finish the Mountain Script\n\nOnce you have penciled in all of your letters, go back in and erase your guidelines.\n\n### LABELING FORESTS\n\n#### STEP 1: Decide the Forest Script Area\n\nSelect your forest area, and then approximate how much space you will need to lay in your text. Use your 2H pencil to lightly mark the area edges where you think you will need to erase.\n\nThen carefully go in and erase the forest that you think will be in the way.\n\n#### STEP 2: Place Guidelines\n\nOnce the forest script area has been cleared, use your 2H pencil and carefully mark your script guidelines, following the general contour of the forest shape for the best result.\n\n#### STEP 3: Pencil In the First Letter\n\nUsing your HB pencil, gently draw in the first letter of the forest script, taking care to stay within your guidelines. Forests should use all capital letters, unlike mountains.\n\n#### STEP 4: Continue the Script\n\nContinue penciling in the forest script. Forests do not need to have their letters spaced out like rivers and mountains, though you can if you'd like to depending on how large the forest is.\n\n#### STEP 5: Fill in the Trees\n\nWhen you have finished your text, go back in and erase your guidelines, then carefully draw in any trees that may have been partially erased in step 1, filling them out and making them look like part of the wood. Once the trees are in, you have a nicely labeled forest.\n\n### LABELING COUNTRIES AND REGIONS\n\n#### STEP 1: Approximate the Region Script Area\n\nSelect the area that will receive a country or region label and approximate how much area you will need for the text. Use your 2H pencil and lightly mark the beginning and end of the area.\n\n#### STEP 2: Place Script Guidelines\n\nUse your 2H pencil and gently draw both your top and bottom script guidelines. Region text is usually twice as large as standard city text, while country text can be up to three to four times larger. Each label should be slightly curved, following the contour of the map or region.\n\n#### STEP 3: Pencil In the First Letter\n\nPencil in your first letter. Try to stay within your guidelines. You may use all upper- or lowercase letters, though all should be of the same approximate size. Map out your letters with small boxes within your guidelines to further help you space your letters.\n\n#### STEP 4: Continue Lettering\n\nPencil in the next letter, then your third and fourth, leaving a bit of space in between each letter.\n\n#### STEP 5: Erase the Guidelines\n\nOnce you have finished penciling in your text, go back and erase your guidelines.\n\n#### STEP 6: Fade the Script\n\nIf you are adventurous, take your kneaded eraser and gently dab your text, fading it by about 50 percent. This pushes your region script back so location labels will take precedence and appear bolder.\n\n#### Ocean and Sea Labeling\n\nOceans should be labeled much like area and region labels, arching through the space and covering a good amount of the map area. Spacing should be broad as well. However, seas and lakes look good when labeled much like location icons, albeit a bit larger. Of course you can use area labeling for seas and lakes too, just keep all types of labeling consistent with each other.\n\n# CHAPTER FIVE\n\n# Landmarks\n\nSo you've gotten through the basic map stuff and want to know more? Awesome! This second part of the book will teach you how to really make your map shine with the application of a little elbow grease and some creative juices. It's time to start thinking about what features you want to include to make your map special. Do you want to map the prosperous regions of a mighty kingdom? Maybe the territory of an ancient sorcerer king now haunted and desolate? How about the island archipelago fiefdom of a pirate queen? From ancient roads and musty tombs, to the heraldry of imaginary nations and kingdoms, new worlds are at your fingertips!\n\nWe'll start the advanced techniques with landmarks. Landmarks are areas of your map that are different than your standard towns and cities and are included to illustrate more specific areas of significance to a viewer. The most common forms of landmarks are roads, highways and sea routes. Other more creative or important places such as old mines, destroyed or haunted ruins and ancient portals are all fun examples of how imaginative you can get with landmarks on your map.\n\n## Roads and Routes\n\nCities and towns are great, but now you need to connect them all together with passages for the wagons of merchants and for troops to march on. Though cities do not always need to be connected from all directions, you should decide on how settled and well traveled your kingdom is. More populated areas will have many roads between towns and cities, while wilder parts will have few and far between. Routes comes in a few different forms. On land, there are paths, roads and highways, while the water has sea routes.\n\n### HIGHWAYS\n\nHighways are the lifeblood of a kingdom, and are the largest and most traveled of the land routes. Frequently traveled by merchants and trade caravans, highways should connect your largest cities to one another, bypassing smaller towns and hamlets. Highways are very easy to draw.\n\n#### STEP 1: Start the Highway\n\nAfter you decide to make a highway, draw a double line between two cities. Try and make the lines as even as possible.\n\n#### STEP 2: Extend the Highway\n\nAfter you have connected your first two cities, continue your highway by connecting other larger cities to it.\n\n### ROADS\n\n#### STEP 1: Start the Road\n\nAfter you decide to make a road, start by drawing a dotted line between two towns or cities. Try and keep the dotted segments evenly spaced for a clean look.\n\n#### STEP 2: Extend the Road\n\nContinue the dotted line until it almost touches the second city. Use your eraser to clean up the lines and make them even. After you have finished your first road, go on and connect a few more of your cities together.\n\n### PATHS\n\nPaths are the dirt foot tracks or ancient decaying roads that are seldom traveled. Paths are common when connecting small villages to each other, or for use when traveling through marshes or mountains. Draw a few dotted lines between some of your towns, winding them a bit to give them a wayward look.\n\n#### STEP 1: Start the Path\n\nTo make a path, draw a light dotted line starting at one hamlet and connect it to another. The dots should be relatively close, but try and keep them even and clean.\n\n#### STEP 2: Extend the Path\n\nAfter your first path is finished, continue drawing paths to outlying towns and hamlets and through hills and woods like shown.\n\n### SEA ROUTES\n\nCities and towns near water often trade over the sea, following well-traveled merchant routes. Drawing in sea routes can be fun, since cities from across the world can be connected via their ships. Be creative with your routes; they will often be long but with many branches that go to different destinations.\n\n#### STEP 1: Start the Sea Route\n\nWhen making a sea route, draw thick dotted lines between two cities, either across from each other over a sea or lake, or on the same coastline. Be careful to keep the dots similar to one another.\n\n#### STEP 2: Extend the Sea Route\n\nOnce your first trade route is completed, go on and connect multiple cities together.\n\n## Special Locations\n\nWhen the deep woods need a hidden Elven stronghold, the tall mountains have a mighty dragon nesting in them, or an evil wizard needs a large and impenetrable tower where his hordes of minions can issue forth from to terrorize the free folk, you need to draw in some creative locations. Including a few of these special interest areas adds depth and story elements to a map, and they can be a lot of fun to draw. Here are a selection of locations to draw, though I encourage you to come up with some of your own!\n\n### DRAGON'S LAIR\n\n#### STEP 1: Draw the Door\n\nDraw a large door in the mountains. Add windows, a reinforced doorway and some ruins.\n\n#### STEP 2: Add Detail\n\nRefine the window and door details. Fill in the large hole, adding two eyes. Draw a wisp of smoke for the dragon.\n\n#### STEP 3: Final Details\n\nShade the details and add color to finish the lair.\n\n### ANCIENT PORTAL\n\n#### STEP 1: Draw the Portal\n\nStart by drawing three large stones, making a three-sided picture frame.\n\n#### STEP 2: Add Detail\n\nAt the bottom, draw in the portal's base, adding a few broken rocks and fallen stones.\n\n#### STEP 3: Final Details\n\nShade the rocks and details. Finish the portal by adding color.\n\n### WIZARD'S TOWER\n\n#### STEP 1: Draw the Tower\n\nStart by drawing a single tall tower with a long and pointy top.\n\n#### STEP 2: Add Details\n\nAdd some windows, a set of doors at the base and tower reinforcement lines. Draw in a small path and some roof shingles.\n\n#### STEP 3: Final Details\n\nTo finish, shade the tower and its roof with an ominous dark gray or purple. If you are feeling adventurous, paint the doors a nice yellow to stand out.\n\n### RUINS\n\n#### STEP 1: Draw the Ruins\n\nDraw in the bases of a few buildings, careful to cut off the tops so they look disheveled and broken.\n\n#### STEP 2: Add Details\n\nAdd a half broken tower, and a few more stones to surround it.\n\n#### STEP 3: Final Details\n\nShade the ruined houses with dark colors, while dabbing in some dark spots to resemble some burn marks. Add some highlights on the top with lighter colors and your ruins are complete!\n\n### STONE HENGE\n\n#### STEP 1: Draw the Henge\n\nDraw a few small upright stones and top them with another large rectangular stone.\n\n#### STEP 2: Add Details\n\nAdd in a few more small scattered stones on the ground, as well as some line shading for definition.\n\n#### STEP 3: Final Details\n\nAdd in some detail work on the stones and some background lines to give the stones some weight. Then add color to finish.\n\n### TROLL CAIRNS\n\n#### STEP 1: Draw the Cairns\n\nStart with a group of large bumps in the ground, about three to five will do. Then add small holes in each bump. These basic shapes are where your trolls will live.\n\n#### STEP 2: Add Details\n\nNext add some definition to the holes, using linework to give them form. Fill in the troll holes to make them look deep and forbidding.\n\n#### STEP 3: Final Details\n\nShade the holes darkly, painting mainly in dark greens and browns, but highlighting in light greens and yellowish whites. Now your trolls are ready to terrorize the local town.\n\n### SEA MONSTER\n\n#### STEP 1: Draw the Sea Monster\n\nPick a spot in the open ocean for your sea monster. Start by drawing the waves as a base, and then a few semicircles and a barbed fish tail poking out.\n\n#### STEP 2: Add Details\n\nAdd some segment lines to the monster's back and ribbing to its underside. Add detail lines to the fins and draw a few more waves to finish the base.\n\n#### STEP 3: Final Details\n\nShade the monster with light greens and reds, making sure to dab in some white highlights so it looks like reflections on the scales.\n\n### TREE OF LIFE\n\n#### STEP 1: Draw the Tree\n\nTo draw a tree of life, pick a good spot in the middle of a forest, and erase what you currently have. Then draw in the trunk of a large tree with many branches. Use an oak or pine tree for reference.\n\n#### STEP 2: Add Details\n\nContinue the tree by adding some details. Draw in some smaller trees around the main tree, as well as a few circular glyphs. Finish by drawing the roots, digging them in the ground. Once the tree is complete, draw in some of the incomplete trees that you had erased.\n\n#### STEP 3: Final Details\n\nShade the tree's trunk with a brown or black, and the boughs of the tree with a nice green. You can also have an autumn tree that has a white trunk and orange and red boughs.\n\n#### Tree Design\n\nTry and model your tree of life on real trees from different places. Maybe it would look interesting as a large African savannah tree, or perhaps a bonsai.\n\n### STATUE OF VICTORY\n\n#### STEP 1: Draw the Statue\n\nStart by drawing the statue's base. A rectangle or square works well. Then draw a human-like figure in robes or armor on top, with its arms out.\n\n#### STEP 2: Add Details\n\nFinish off the pencilwork by adding in some small detail lines for the clothing and armor.\n\n#### STEP 3: Final Details\n\nShade the statue with blues, grays or purples, adding white highlights on the head, outstretched arms and the top of the base.\n\n#### Monument Design\n\nA statue of a warrior is a fitting monument to a great city, though it might be fun to make a great statue a ruin, like the ancient Colossus of Rhodes. Statues don't have to be humanoid either; they can be of animals and monsters.\n\n# CHAPTER SIX\n\n# Iconography\n\nIcons have been used in mapmaking since maps came into existence, and are a great way of showing information in simplified forms. In this chapter, you will learn how to make and use simplified city icons, as well as the most iconic image of all things on a map; the compass rose itself. You will find some exercises to help you create your own using simple geometric shapes.\n\n## Simple City Icons\n\nSimple city icons are a great way of depicting your towns and cities without going into too much detail. They are excellent for use on a large map, or a map with a great many locations. By keeping them bold and simple, and by using standard geometric shapes as a base, you will soon be making your own creative designs!\n\n### CAPITAL ICONS\n\n#### STEP 1: Draw the Basic Shapes\n\nTo start a capital, draw a thick circle, and then add another smaller circle inside it.\n\nCapitals should be larger than other city icons.\n\n#### STEP 2: Add Details\n\nNext, add four smaller circles inside the center.\n\n#### STEP 3: Final Details\n\nAdd a crown to your capital for extra emphasis. A king needs a seat for his kingdom!\n\n### CITY ICONS\n\n#### STEP 1: Draw the Basic Shapes\n\nJust like a capital, a city should also start as a circle, though not as large. Draw in a smaller circle and fill it in.\n\n#### STEP 2: Add Details\n\nAdd four smaller circles to the center. Done! Not all of your cities need to look alike; try and use different shapes and sizes on the base or inside.\n\n### TOWN ICONS\n\n#### STEP 1: Draw the Basic Shapes\n\nFor a town icon, start with a basic square.\n\n#### STEP 2: Add Details\n\nAdd an even smaller square inside. Next, add four small circles in the center. A town is born!\n\n#### Icon Size\n\nTowns should be smaller than cities and capitals. Though square towns are great for quick recognition, you can just as easily depict towns as smaller, more simplified circles.\n\n### VILLAGE ICONS\n\n#### STEP 1: Draw the Basic Shape\n\nVillages should be the smallest of your simple city icons. Start with a small circle, about half the size of the town square.\n\n#### STEP 2: Add Detail\n\nTo finish, simply draw another small, filled circle inside the other, leaving a small border.\n\n#### Village Examples\n\nVillages can vary in shapes though shouldn't vary much from each other in size. Here are a few different examples of village icons that would be fitting for simple hamlets.\n\n### CITADEL ICONS\n\n#### STEP 1: Draw the Basic Shapes\n\nA citadel is a bastion of power and majesty within a kingdom, and should be as large as a capital icon. Start with two rectangles connected by a small bridge.\n\n#### STEP 2: Add Details\n\nTop the two towers with crenellations, three on each top. Be careful to leave a gap between the tower top and the tower itself. Draw in a few lines to make for the door.\n\n#### STEP 3: Final Details\n\nFinish off the citadel by filling in the tower areas, leaving a small space for windows in each tower. Add a few towers and your citadel is complete!\n\n#### Consistency\n\nIf you have multiple kingdoms on a map, it's a nice touch when each faction's cities match one another, at least in shape or color. Keep them similar and the effect will be noticeable.\n\n### HOLDFAST ICONS\n\n#### STEP 1: Draw the Basic Shape\n\nHoldfasts are castles and a more simplified version of the citadel. Start with a rectangular tower and add a window.\n\n#### STEP 2: Add Details\n\nComplete your holdfast by adding a crenellated tower top and finishing the base.\n\n#### Customize Your Buildings\n\nBoth citadels and holdfasts can be illustrated in a great many ways. Try your hand at both by adding a few more towers or small connected houses. Just make sure all the citadels and holdfasts match one another so there is no confusion.\n\n## Location Icons\n\nLocation icons are a fun and simple way of adding in some more exotic points of interest to your map. These areas do not follow the same geometric shapes as the city icons, but each is a small illustrative glyph that tells a different story. From dusty temples to wandering dragons, you will quickly be making up your own icons.\n\n### MINES\n\n#### STEP 1: Draw the Basic Shapes\n\nMines should be located in foothills or mountains. To start, draw a small door, and add a small mining cart.\n\n#### STEP 2: Add Details\n\nFinish off the mine by adding a small mining pick and a pile of coal or gems.\n\nDepending on the type of mine, you could add in a few bars for a gold mine, or a large diamond to denote a gem mine.\n\n### TEMPLES\n\n#### STEP 1: Draw the Basic Shapes\n\nTo start, draw the base of the temple, a wedge roof and start adding pillars to it.\n\n#### STEP 2: Add Details\n\nFinish off the temple by adding more detail to the roof and completing the pillars.\n\n#### Cultural Design\n\nThough a Greek temple is instantly identifiable, you should feel free to alter the temple type depending on your map's cultural influence. A Japanese Shinto temple would look much more appropriate in an eastern themed map than a western Greek temple. Try your hand at a few different temples based on different cultures.\n\n### BRIDGES\n\n#### STEP 1: Draw Basic Lines\n\nBridges mark important locations, but are very simple to illustrate. Start by erasing the part of the river that the bridge is crossing, then draw two parallel lines across it.\n\n#### STEP 2: Further Emphasize the Lines\n\nAt the tip of each line, draw another line angled out on each side. To emphasize an important bridge, draw another set of lines outside the original lines.\n\n### RUINS\n\n#### STEP 1: Draw the Basic Shapes\n\nDraw a flat line for the foundation. Then draw some bumps above it for the ruin's edges.\n\n#### STEP 2: Add Details\n\nPut in some half-broken towers and a few broken rocks. Fill in the tower and rubble, and add a few lines underneath the base.\n\n#### Ruins\n\nRuins can look like almost anything, and can be larger or smaller depending on the ruins you are trying to depict. The more broken the towers and rocks, the more important the former location was.\n\n### BATTLEFIELDS\n\n#### STEP 1: Draw the Basic Shapes\n\nBattlefields are important icons, but quite easy to depict. Pick a location, and then draw a sword at a forty-five degree angle.\n\n#### STEP 2: Add Another Sword\n\nDraw in a second sword crossing the first. If you want to give your battlefields a slightly different look, use a curved sword or even a pair of axes.\n\n### MARSHLANDS\n\n#### STEP 1: Draw the Basic Lines\n\nWhen drawing in marshlands, start by drawing a few grass tufts, then underline them. Add a few more tufts to make a grouping.\n\n#### STEP 2: Add Details\n\nAdd a few lighter lines and some sporadic dots. Make the marshlands as large or as small as you want.\n\n### DRAGON\n\n#### STEP 1: Draw the Basic Shapes\n\nStart with a single S line as a base. Then add the crest on the head and the tail.\n\n#### STEP 2: Add Details\n\nFinish off the dragon with a few claws and further define the tail and the head crest. Fill out the dragon's body, but don't make him too fat. Play around with your dragon shapes; you can always come up with some new designs.\n\n### PORTAL STONE\n\n#### STEP 1: Draw the Basic Shapes\n\nStart your portal stone by drawing the three basic stone parts: two tall ones and one smaller one linking them together at the base.\n\n#### STEP 2: Add Details\n\nAdd a cross stone on the top, and add some shading into the stones. Draw in a spiral line for the portal. You can easily leave out the spiral to denote an inert stone or one that is broken.\n\n### TROLL CAIRNS\n\n#### STEP 1: Draw the Basic Shape\n\nDraw a triangle. This will be the basic shape of the banner representing a group of trolls.\n\n#### STEP 2: Add Details\n\nDraw in the haft of the spear, and then the spear top. Try using different shapes for the banner top. Horns, helmets and half moons work great.\n\n#### STEP 3: Final Details\n\nFill in your banner with some of your own designs and add the cross spear points. Experiment with different designs to make your banners suitably savage.\n\n#### Finished Icons\n\nBelow is a map with many of these location icons included together. See how they are spaced and play off one another? When putting them on your own map, be free with them, but don't clutter your map.\n\n## Ocean Icons\n\nFantasy map oceans are usually filled with all manner of raiding warships, sea monsters and merchant vessels trying to ply their wares across the dangerous seas. This section includes tutorials on how to draw some basic icons to include in your fantasy waters. Whether they be ships or monsters of the deep, soon you will have plenty of creatures inhabiting the high seas.\n\n### LONGSHIPS\n\nLongships are traditionally the vessels of the Vikings, notorious brigands of the old world. Their primary mode of transportation, the longship, was a sleek ocean-going vessel with a shallow draft. Able to carry both warriors and pillaged plunder, the sight of a longship's single sail on the horizon inspired fear and terror in ample measure. For your purposes, they are also a fantastic way of depicting barbarian raiders on your own map.\n\n#### STEP 1: Draw the Basic Shapes\n\nSelect a spot in the ocean, preferably near the coast. Draw in the distinct lines of the ship's hull, which should curve up at both ends. Then draw in a large taut sail and a mast.\n\n#### STEP 2: Add Details\n\nDraw a few round shields on the side of the hull, carefully erasing the parts of the hull that become covered. Then add a flag to the top, as well as rigging lines and some hull planks. Fill in the rest of the sail so it pops out. You can draw a symbol or a pattern on the sail or flag. Battle axes or other weapons are great symbols for barbarians!\n\n### TRADE COGS\n\nTrade cogs are a wonderful way of emphasizing particularly lucrative trade routes, or neat to have just for show. Historically, trade ships and cogs were large and multi-masted vessels, capable of carrying sizable amounts of cargo over long distances. Having multiple masts allows these relatively heavy and clumsy vessels to make best use of the wind. The design of the traditional trading vessel did not change much for many years, and usually consisted of high fore- and after- \"castles,\" which were useful when overlooking the sea and cresting high ocean waves. For all their size, trade cogs and ships were vulnerable to attackers if caught by sleeker and faster predators. Cogs would try and outrun opponents whenever possible.\n\n#### STEP 1: Draw the Basic Shapes\n\nSelect a spot in the ocean, preferably along a sea route. Draw in the lines of the ship's hull, which should be short and fat. On each top corner, draw a castle top on both the front and back. The \"castle stern\" should always be taller. Add in the waves at the base of the ship.\n\n#### STEP 2: Add Details\n\nContinue the trade ship by drawing a front mast. You can also include fore and aft sails. Add the lines for the wood planks of the ship and add detail to the main sail.\n\n### SEA MONSTERS\n\nThe seas are a dangerous place, with plenty of monsters hunting for the unwary ship. With beak, claw, tentacle and fin, these monsters of the deep are always a threat. Historically sailors believed certain areas to contain a number of fantastic creatures that were a danger to passing ships. These areas were marked on nautical maps as places best avoided if at all possible. Whales, sharks and squid were all considered to be large and threatening, and their size and impact on the maritime community were often exaggerated to a great degree. However, in a fantasy world, such creatures do exist, crushing ships and devouring sailors!\n\n#### STEP 1: Draw the Basic Shapes\n\nPick a spot in a region of the ocean that is off the beaten track, maybe in a corner somewhere. For a large sea serpent, draw a large horizontal S in the water. Erase half of it, and draw in the ocean waves as a base.\n\n#### STEP 2: Add Details\n\nDraw a large fin on the tail and another toward the front of the monster's body. You can also add a few segmented scales if you want. Add a few more wave lines to finish.\n\nSea monsters need not all be serpents, but can also take the form of massive octopi, huge sharks, killer whales and so forth. Let your imagination run wild and see what you can come up with.\n\n# CHAPTER SEVEN\n\n# Heraldry and Shields\n\nHeraldry and shield design are the symbols of different factions, cities and nobility. On a map, shields are an excellent way of depicting cultural and political boundaries without actually drawing lines on a map. Though not every town and hamlet should have its own shield design, cities and capitals would often be the home of very specific noble families. Each would have its own heraldry and influence. Shields are also good for marking general regions, denoting areas of cultural influence that a faction may have on it.\n\n## Shield Shapes\n\nNot all shields should be of the same shape. Many factors affect shield shape, including where the city or region is from, as well as cultural influence of that area. A western medieval city is usually defended with a \"heater\" type shield (an upside-down iron), while eastern cities tend toward more rounded shields. However, a fantasy map need not obey any standard, so play around with shapes to come up with one you like.\n\n#### Basic Shapes\n\nBasic shield shapes are very simple, consisting of circles, rectangles, heaters and ovals. Here is a selection of basic ones. Try and draw some of these freehand, or come up with your own unique shapes.\n\n#### Shields for Battle\n\nShields in the west tended to be heater or teardrop shaped because of the dominance of heavy cavalry on the battlefield. Heater and teardrop shields were easier to use and less cumbersome than large round shields. Since heavy cavalry were usually made up of the nobility, heater shields tended to dominate the heraldry and shield design of the time.\n\nMost cities within the same culture would share the same shield shape, even if they differed in color or markings.\n\n## Simple Heraldry\n\nHeraldry is the use of specific graphic symbols and icons to form a unique pattern. Heraldry is best known for decorating the shields of medieval knights, though it is still in use today, often for the towns and cities of Europe, as well as old or renowned families. Heraldry is a very old art which utilizes specific iconography and rules. The official use of heraldry started sometime within the mid-eleventh century when knights first started to use symbols to define themselves and their family houses from their peers. The art evolved and was finally standardized between the European nations around 1500 AD. There are a bewildering variety of heraldic devices, though all are quite fascinating. In this chapter you will learn how to make some simple heraldic symbols for your own kingdoms. Have fun and be creative!\n\n#### Quartering Parts\n\nWhen you have settled on an appropriate shield shape, it's time to lay in some basic heraldry overtop of it. Heraldry at its most basic consists of a shield being \"quartered.\" This means that it can have straight bars, zigzags, quartered squares, half bars, or stars and dots. I have included a number of basic heraldic devices for you to choose from and emulate. Feel free to mix and match them to come up with your own design.\n\n#### Historic Shield Examples\n\nHere are some historic examples from Europe. Heraldry was often quartered. This was a way of differentiating between close noble houses by adding slight differences to each heraldic device. Each piece of the quartering process meant something different. This usually served to tie the many different houses and blood lines together, telling friend and foe who their family and allies were, making for easy recognition.\n\n## Simple Shield Symbols\n\nNot only would shields have different shapes and quartering to differentiate them, they would also have house symbols, and faction devices. These would come in a bewildering variety of forms, from wild animals to ships, weapons and flowers. The addition of a sigil (or symbol) can also be used to tie factions and houses together, though it was common for multiple rivals to have similar, or even the same sigil. In this section I have a few more basic symbols for you to use or draw inspiration from.\n\n#### Applying Symbols\n\nSelect a sigil, and overlay it onto your shield. If it's dark on dark, erase the dark quartered part so that the sigil shines through, creating a nice bold shape. After you have combined a shield shape, a quartering shape and a simple symbol, your shield will be finished!\n\n#### Shield Examples\n\nThere are almost endless combinations for you to mix and match. Try and remember not to make shields too elaborate, or choose too many symbols for one shield. One is usually plenty. Kingdoms share the same basic shield shapes, though they may all have different quartering and symbols. Use these combinations I came up with as inspiration for your own designs.\n\n## Painting Shields\n\n#### STEP 1: Shield Colors\n\nWhen choosing colors for your shield, it's best to stick with the primary and secondary colors, using complementary colors where possible. This gives the shield a more homogenous look. Usually you will choose a primary color that serves as the main color, and a secondary color that will be used on the quartered part.\n\n#### STEP 2: Paint the Basic Shield\n\nOnce you have chosen your primary and secondary colors, paint the shield shape in your primary color. Be careful not to go outside your shield's edges. Select a type of quartering, and carefully paint that on top in your secondary color. Finally, paint the symbol on top of the shield, centering it as best you can. The symbol should stand out, either as a darker or lighter color, depending on the background.\n\n#### STEP 3: Add Weathering and Highlighting\n\nOnce you have painted your colors, it's time to give the shield a bit of oomph. To start, shade around the shield just a bit to give it some pop. If you want, add a few dents and scuffs to the shield as well. After that, add shading to the bottom of the shield and to the dents and scuffs, painting in slightly darker tones than your original colors. Finally, use lighter versions of your colors to highlight your shield, starting at the bottom point. Use a slightly brighter color at the top center to give it some dimension.\n\n## Shield Embellishments\n\nSometimes a shield should be emphasized even more, whether it be for the kingdom's capital city, or for spiritual or mercantile importance. Additional symbols can be added to the top or sides of a shield to further distinguish it. Symbols such as a crown, or even illustrative scrollwork are typical examples.\n\n#### King's Crown\n\nBegin by drawing in the rough shape of a king's crown. This may be as elaborate as you'd like, though keeping it relatively simple is usually best. Next, paint in your primary color, usually a gold or yellow, carefully keeping within your lines. Last, add some highlights, and then outline the entire crown in a dark brown or black to make it pop. Now, your kingdom has a seat of power.\n\n#### Scrollwork\n\nCarefully draw in a basic scroll shape, making sure it wraps behind itself, and goes behind your shield. Paint it a light color, usually ochre or cream, though it can be any color you choose. Finally, shade the folded parts darker, and highlight the edges. Add a few runes. Outline the scroll, but not so much that it takes away from the shield. Play with different colors; red and white work well too.\n\n#### Graphic Floral Designs\n\nStart by carefully drawing your design above the basic shield. I have chosen a simple flowerlike design. Next, paint it your shield's secondary or complementary color. Then add shading and highlights to give the shape more dimension.\n\n## Shield Placement\n\n#### City Shields\n\nA city shield should be placed directly over the city artwork, though you should leave a small space so as not to crowd the actual city icon or art. You may wind up covering part of the map's topography if you do this, and if so, that's OK. If you don't want to do this, you can move the shield underneath the city artwork as well, though above is usually preferable.\n\n#### Regional Shields\n\nRegional shields differ in placement from city shields, but not by much. You still want to try and maintain equidistant space from map elements, especially cities. Improper placement can lead to unbalancing a map's aesthetics. Try not to cover map elements, and place these shields in open areas so they don't clash with other items of importance. Allow them to stand on their own. Regional shields should be larger than city shields, and in some cases you will only need one for your purposes.\n\n# CHAPTER EIGHT\n\n# Political Boundaries\n\nThe use of boundaries can form a crucial part of a map and tell a viewer much about the land's history and its people. At their heart, boundaries are dividing lines between a country's inhabitants, either on a political, geographical or military basis. These dividing lines can accentuate a map or dominate it, though historically, maps with political or country boundaries wouldn't have much other detail. When laying in boundaries, there are a few things you should think about.\n\n  * How many factions do I want portrayed on my map?\n  * Do I want to portray regional boundaries as well as national borders?\n  * How do I want to paint in my boundaries?\n\nSoon you will have plenty of rival factions, kingdoms and principalities populating your fantasy map!\n\n## Boundary Lines\n\nThe first way to differentiate regions and counties is by using line work. As opposed to the use of color alone, lines are a much bolder type of boundary, but if not used correctly they can clutter a map with too much information. Line boundaries can be made up of several different types of line, varying from repeating dots and crosses to squares. When using colored lines, it's best to use reds and dark browns since they give a uniform, subtle touch.\n\n#### Line Types\n\nHere are some line boundary types that you can use on your map that will serve well to divide your regions and nations. You can mix and match them on the same map, though it's best to stick to one or two types to avoid confusing a viewer. Use these as inspiration for your own boundary ideas.\n\n#### STEP 1: Select Line Type\n\nSelect the boundary line type that you wish to use, and then lightly draw in your initial sketch line where your boundary should go.\n\n#### STEP 2: Start Marking Lines\n\nMark in your first boundary line mark. Be careful to try and follow rivers, mountains and hills as you continue.\n\n#### STEP 3: Continue the Lines\n\nContinue the boundary lines, making sure not to cover any icon or label. A boundary line should always take a back seat to all other elements in a map and blend in.\n\n#### STEP 4: Final Details\n\nContinue with the rest of your boundary lines, following steps 1\u20133. Finally, finish up your boundary lines, taking care to clean up your lines as best you can.\n\n## Boundary Colors\n\nUsing colors for nations to tell them apart is another technique. The use of color as boundaries has been the most common technique in cartography, and unlike line boundaries, is an immediate and bold way of presenting a viewer with instant and definitive information. Generally, color boundaries are best used on monochromatic or simple parchment maps, where the color will stand out. Historically, color boundaries emerged with the birth of the first nation states, becoming commonplace in maps from around the 1600s and on.\n\n#### STEP 1: Select the Color\n\nTo begin, select the color that best serves your purpose, and paint in your first line by hugging the shore boundaries and rivers (if any), being careful to not cover the actual dark boundary lines.\n\n#### STEP 2: Continue Adding Color\n\nTry not to cover any towns or cities directly, going underneath them when possible. Complete your boundary line by coming full circle.\n\n#### Subtle Colors\n\nHere is a selection of colors that work well when painting in boundaries. Boundary colors should always be subtle and somewhat transparent so as not to cover any details or the terrain. If using a paintbrush, thin down your paint by mixing it with water so you can brush in your colors slowly and carefully. If using a computer, reduce the opacity of the color to at least 50 percent.\n\n#### STEP 3: Keep Color Separate\n\nWhen color boundaries border each other, which will happen often, they should run parallel to each other, careful not to touch.\n\n## Boundary Types\n\nThere are two different types of boundaries, national and regional. National boundaries are the boldest and thickest, and are used most often to denote national borders. Regional boundaries are much more subtle with smaller markings but are the most common and versatile, often used to depict county and political borders within a kingdom.\n\n#### National Boundaries\n\nNational boundaries should be thick and very pronounced, usually solid and bright as well. National boundaries should be quite bold and instantly recognizable.\n\n#### Regional Boundaries\n\nWhen depicting counties or provinces, regional boundary lines are best. These lines are often dotted, or broken, and are much thinner than national lines. This is usually due to the fact that counties and provinces would blend into a map, not stick out like a sore thumb.\n\n## Using Natural Boundaries\n\nWhen laying in boundaries, you should always try to follow the natural contours of mountains, hills and rivers when you can. Natural boundaries are often the place where state boundaries reside. Water boundaries need not be highlighted by regional boundaries, though national ones can overlap and cover them. You should always think of your map and its factions, and how they would share or fight over land. Properly established boundaries can be an interesting component of a good map, spurring a viewer to think more about the people and places within it.\n\n#### River Boundaries\n\nWhen drawing river boundaries, try and hug the river as close as possible.\n\n# CHAPTER NINE\n\n# Backgrounds\n\nBackgrounds can be made up of a great many things, from the borders and corners, to informative motif boxes that provide design and compositional elements to a map. A map border should often be thought of as a frame of sorts. Just like a picture frame, a border can be very simple or very elaborate, depending on the type of map you are making. A proper border not only looks suitable and contains design flourishes to enhance a map, but also includes subtle information for a viewer. The most interesting borders are those that are integrated directly into a map's composition, giving you a seamless piece. Backgrounds also broadly include one of the most important pieces of map information, the legend, which is a keystone to any map.\n\n## Borders\n\nBorders can be portrayed in a variety of ways, from simple line art to more complex painted framing designs. One of the most important elements of a map's border is the \"neatline\" that acts as a true frame by giving the map a definitive border, which no artwork should exist outside of. Many maps also include a scale ruler (or border) within the neatline, giving a map that distinctive checkered border look. In this section we will be focusing on basic techniques.\n\n#### STEP 1: Mark the Corner\n\nTake your ruler and lightly mark a point in each corner of your map about 1\u20444\u20131\u20442\" (6\u201313mm) in from each edge.\n\n#### STEP 2: Create a Basic Frame\n\nThen using your ruler as a straightedge, connect your dots, forming a basic frame. You may make the line thicker or thinner depending on your preference.\n\n#### STEP 3: Add Another Frame Line\n\nNow draw another frame within the lines you just drew, about 1\u20448\" (3mm) inside the first line.\n\n#### STEP 4: Create a Scale Ruler\n\nEach segment should be equal to give a uniform look. Measure the length of your neatline and divide it by as many scale segments as you want. Use your ruler to dot in the rough placements. Fill every other segment, like a checkerboard.\n\n#### STEP 5: Final Details\n\nAfter your scale is completed, take your ruler and draw another frame about 1\u20444\" (6mm) outside your neatline. Your scale is complete. Just add color to finish the piece.\n\n#### Color Variation\n\nYou don't have to stick to black and white when constructing a border ruler. A variety of colors look good as both a neatline and a border ruler. Reds, blues and browns all are perfect for old maps.\n\n## Border Corners\n\nThe corners of your border are a great way to add some character to your map. Corners can come in a large variety of forms. Corners are essentially their own small illustrations that integrate with your original neatline and scale border elements. Properly done, corners can do much to make a map shine. This section will focus on both simple and more advanced techniques for tackling corner illustrations. Simple line corners add a nice accent to your map, and are quick and easy to do, while complex corners make use of additional elements that require patience and practice to master.\n\n#### STEP 1: Prepare the Corner\n\nSelect the first corner you'd like to illustrate. Let's use the same corner we were just working on. Take your eraser and carefully clear the work area.\n\n#### STEP 2: Add the Basic Design\n\nLet's start with a simple geometric design. Don't be afraid to try different shapes!\n\n#### STEP 3: Add More Details\n\nContinue by adding a few more detail lines and marks, trying to keep them as even to one another as possible. Draw a small circle centered in the corner.\n\n#### STEP 4: Final Details\n\nFinish your corner by drawing in the parts of the border and scale that you had initially erased. This will integrate the corner seamlessly.\n\n#### Simple Corners\n\nSquares and rectangles are wonderful for corners, but you should experiment with other shapes too. You might be pleasantly surprised with what you can come up with!\n\n## Complex Corners\n\nWhile simple corners are fitting for simpler maps, sometimes you will want to include more complex illustrative elements. Complex corners are essentially small illustrations that are a bit like book corners, adding sophistication and character to an in-depth map. In this section you will learn how to construct complex corners by starting with geometric shapes, and progressing with more organic elements to form your own distinct designs.\n\n#### STEP 1: Prepare the Corner\n\nSelect a corner and erase the neatline and borders.\n\n#### STEP 2: Add the Basic Design\n\nDraw a circle as centered as possible in the corner. Try and make it as round and even as possible. Complex corners can vary in size but should never be larger than 1\" (25mm) in diameter.\n\n#### STEP 3: Add Details\n\nDraw another circle, giving the original circle a border, and then notch the edges so they look braided. Draw a few more smaller circles within your braided circle, and connect each with some small detail lines. Think of a knot and how each strand braids around one another.\n\n#### STEP 4: Connect the Corner\n\nFinish your corner by reconnecting your neatline and the ruler border so that it is seamlessly integrated. Try your hand at some circle or triangle designs.\n\n#### STEP 5: Final Details\n\nAdd shading and color to finish it.\n\n## Corner Flourishes\n\nTo spice up your corner borders, you can include a few design elements that add further sophistication to the look and feel of your map. Flourishes are graphic elements that serve to better integrate the corners with the neatline, but can sometimes stretch into the map itself. They can vary from simple line designs, to complex illustrative elements. Unlike the corner art, they may extend out, forming a part of the neatline's border. Flourishes can be roughly characterized as both basic and advanced.\n\n#### STEP 1: Prepare the Corner\n\nStart by erasing the inner neatline from your corner.\n\n#### STEP 2: Add the Basic Shape\n\nStarting from the corner circle, draw a petal pointing into the map. This is your main petal.\n\n#### STEP 3: Add Details\n\nThen outline the petal with another line and add a circle on each side.\n\n#### STEP 4: Integrate the Flourish\n\nFill in a few small curled twigs on each side and connect them back to the main petal. Redraw any part of the corner that was erased and integrate the flourish into your corner.\n\n#### STEP 5: Final Details\n\nFinish it up by adding shading and color.\n\n#### Basic Flourishes\n\nBasic flourishes are small embellishments that are quick and easy to do. They are based on simple flower and plant leaves that are drawn in a bold and graphic way.\n\n## Advanced Flourishes\n\nAdvanced flourishes are larger and more difficult to do. They usually consist of longer branch arms and more floral elements. They can also be made up of other smaller illustrations of weapons and figures, and have a less graphic and more illustrative look. For this example complex image, let's try a vinelike design with some berries.\n\n#### STEP 1: Prepare the Corner\n\nBegin your advanced design by erasing your inner neatline, and prepping it for your line work.\n\n#### STEP 2: Add the Basic Shapes\n\nDraw in two vines extending out from the border corner. Try to make these as even and symmetrical as possible.\n\n#### STEP 3: Add Details\n\nContinue the design by adding a few more vines in the same fashion as the first and second, each about half the size, as well as a cluster of berries in the center. Add more small offshoot vines to each major vine, curving them inwards.\n\n#### STEP 4: Integrate the Flourish\n\nComplete your design by adding in another smaller group of vines, and a few clusters of berries. Clean up your drawing and integrate it with your neatline. If your design overlaps your neatline, that's OK too!\n\n#### STEP 5: Final Details\n\nFinish it up by adding shading and color.\n\n#### Advanced Flourishes\n\nMixing heavy straight shapes with leaves, berries and flowers can form some very nice design elements for your corners. Take a look at some old medieval maps. Medieval artists were very good at combining hard and soft elements.\n\n## Compass\n\nThe compass rose is perhaps one of the most, if not the most, indicative images of cartography. Not only is it a beautiful piece of art that can tie a map together, it is also functional, providing information and direction to a viewer. Compasses come in many sizes, shapes and styles, though all share some similar qualities, such as giving direction. Usually the north point is the most emphasized, though each point is represented as a different direction. Compasses traditionally look like a star with a number of points that can vary from four, eight and sixteen to even thirty-two points! These points also depict nautical degrees and help seafarers plot courses and make use of the winds. Since certain areas of the ocean would provide better winds than others, there would be smaller compasses located on some maps for the purpose of triangulation, or getting bearings on a specific point. This is very useful in the wide open sea. Not all compasses would focus on magnetic north as we are used to, and some would focus on the stars as the principal guide. However, today, you will be learning how to construct a basic compass for your map.\n\n#### STEP 1: Draw the Center Shape\n\nWhen constructing a compass, first draw a circle about the size of a quarter (about 1\" [25mm]) and then another inside it.\n\n#### STEP 2: Add the Points\n\nStarting with the north point, draw two long triangles on the top and bottom of the circle, and two shorter ones on the sides.\n\n#### STEP 3: Divide the Points\n\nDivide each of the triangles in half.\n\n#### STEP 4: Add Design\n\nGo back into the middle of the center circle and draw a knight's shield. Then fill the rest of the circle with floral designs for embellishment.\n\n#### STEP 5: Final Details\n\nMark the south, east and west points with the first letter of the respective direction. For north, put in a larger N to give it more emphasis.\n\n## Legends\n\nThe legend is the heart of a map in some respects and can form a wonderful centerpiece. It is where the title, icons and scale elements are explained and defined. A legend can be a beautiful design element when integrated with your corner and boundary art. The basic construction and information are explained in detail in this section.\n\n#### STEP 1: Prepare the Corner\n\nDecide where you would like to place your legend. A good place is usually in one of the four corners, or the center bottom of the map. Draw in your rough lines as a guide.\n\n#### STEP 2: Add the Basic Frame\n\nTaking your straightedge (or design template), carefully draw in your basic design frame.\n\n#### STEP 3: Add Details\n\nOnce the basics are in, add some details to the corners and edge lines to the frame.\n\n#### STEP 4: Integrate the Frame\n\nIf your legend is close to your border art or boundaries, integrate your legend frame into the border. You can do this by adding some additional line work or floral designs.\n\n#### STEP 5: Final Details\n\nFinish it up by adding shading and color.\n\n## Legend Information\n\nThe information that you include within your legend is up to you, though there are some elements that should be included for it to be effective. The basic things, like the map's title, need to be included. Other details such as icon and boundary definitions are dependent on what type of map you have made.\n\n#### STEP 1: Sketch the Placement\n\nLightly pencil in markers for where you want to write in the title of your map first. Then, depending on the information you'd like to depict, space out your other typography. Try not to crowd your information together. Space out where you would like to lay in your map's scale.\n\n#### STEP 2: Add the Typography\n\nLay in your typography, title first. Then draw in any icons you'd like to include, making sure they are lined up and the spacing is even. Draw in your scale, using a straight line. It should be the same length as your ruler boundary.\n\n#### STEP 3: Final Details\n\nFinish your legend by adding in some more details, shading and color. Shading the borders gives the legend some depth.\n\n## Information Boxes\n\nInformation boxes are design elements that are useful for depicting either important text information in certain areas, or giving your map some nice fantastical clutter. They're very similar to legends, but are much smaller, and should have less emphasis. They are a good way of complementing your legend or the corners of your maps.\n\n#### STEP 1: Draw the Basic Shapes\n\nWhen you have decided where to place your information box, carefully draw in your first circle or rectangle to serve as your initial frame and fill it out with another line.\n\n#### STEP 2: Add Details\n\nThen draw another box within your circle and add some simple floral design elements.\n\n#### STEP 3: Final Details\n\nFinally, add in your typography and your information box is complete.\n\n# CHAPTER TEN\n\n# Painting Techniques\n\nThere are two types of painting: traditional with brush, paint and paper, and digital with a tablet and a computer. Though the two share similar aspects, and the final product can come to resemble one another closely, they do require some different techniques. In this chapter, I will give you a cursory glance into both. It may help you to produce and explore by your own means. Before you start, there are a few questions to ask yourself.\n\n  * What type of map do I want to make, digital or traditional?\n  * What do I need to get started?\n  * Where can I get the materials?\n\nSoon you will be painting in the lands and details of your very own kingdom!\n\n## Inking\n\nInk is an extremely versatile medium. Whether ink is being applied with a brush, a pen or an ink nib, it can be used to lay in subtle layers as well as for illustrating precise details. Though inks are usually thought of as black or blue, bottled inks come in a whole variety of different colors to work with, all of which can be mixed and matched, or watered down to produce surprising results. I personally use Noodler's inks, which are both acid-free and waterproof, giving you a very clean, deep and lasting finish. There are a great many techniques to work with, though I will introduce you to a few basic tools and methods you can use to explore on your own.\n\n#### Inking Tools\n\n#### Copic Multiliner SP Pens\n\nBlack Inks\n\n0.5mm\n\n0.2mm\n\n0.05mm\n\nBrown Inks\n\n0.5mm\n\n0.1mm\n\n0.05mm\n\nBS pen\n\n#### Brushes\n\nLoew-Cornell 3\u20444\" (19mm) flat wash\n\nSienna no. 1 round\n\nSienna no. 4 round\n\nSienna no. 10 round\n\n#### Speedball Mapping Pen\n\nPen Nibs\n\nNos. 100P, 102P, 103P, 104P, 107P, 108P\n\n#### Noodler's Waterproof Inks\n\nBlack\n\nFox Red\n\nGolden Brown\n\n#### Wash Types\n\nA hard wash is a mix of 50\/50 ink and water, useful for strong definition of mountains, oceans (depending on the map), and for some strong shading. A soft wash is mix of three parts water, one part ink. Soft washes are excellent for building up tonality, and can be layered to build up definition.\n\n#### Boundary Washes\n\nTo define your land boundaries, first dampen the areas of the paper that you will be applying your ink wash to (usually the seas and lakes), making sure you don't overlap into your land boundaries. Your boundary wash should be soft, and should be applied carefully with your large flat or round brush. To further dilute the wash, use more water and brush outward, away from your land boundaries.\n\n#### Defining Terrain\n\nHard washes are ideal for defining your mountains, hills and towns. Using a round brush, first apply a soft wash over your mountains and hills as an initial base shade. Once dry, apply another layer at the base of the hills and mountains with your small detail brush, careful not to cover the whole area. Once those areas are dry, carefully apply another wash at the base, and keep going until the mountains and hills are well defined. A light shade should be applied underneath your cities and town, and around your border boundaries and legend box to shade them and give definition.\n\n## Pen and Brush Inking\n\nBasic inking should be done with either pens or thin brushes for definition. A pen will give you an even line that is more mechanical, while a brush or calligraphy pen will give a more organic look that is less even, but can vary in thickness by applying more or less pressure. Beginners should start out using Pigma Micron and Copic pens first, since they do not take into account pressure sensitivity, and are easy and clean to use. When you are ready and feel more confident, practice some line work with your small detail brush, as well as your calligraphy pen and its various nibs. Land boundaries should be illustrated first with a thick line so there is a clear definition between water and lands. Next, the basic mountains, hills, borders and compass should all be drawn in with medium lines.\n\n#### Medium Lines\n\nMedium lines should be thinner than the heavy land boundary lines, and are useful for defining mountains, hills, rivers, boundary corners, basic city shapes, forests and border corners.\n\n#### Thick Lines\n\nLine weight will be important in an ink map, and thick lines should be used for clear land boundaries, the legend box and outlining the compass.\n\n## Detail Inking\n\nAfter your basic inking, it's time to detail the most intricate parts of your map. Using your thinnest nibs and pen tips, go in and carefully define your mountains, cities, borders and all other parts of your map.\n\n#### Thin Line Tools\n\nThin lines are excellent for definition and detail work. Most lines on a map should be thin, or at least thinner than the heavy land boundary lines. The thinnest lines serve for the most intricate definition, such as detailing your cities, heraldry, mountains and hills.\n\n## Typography\n\nOnce the illustrative elements have been inked in, it's time to lay in all the text and type. \"Tags\" are labels for the various elements within your map, and each should be portrayed a little differently. Generally, ink and calligraphy pens are excellent for inking in tags, especially for clear and bold definition, though you may want to use a brush for regional and ocean tags to make them appear less hard and bold, especially if you elect to use washes.\n\n#### Capitals, Cities, Towns and Villages\n\nCapitals and cities should be inked in all uppercase, while towns should start with a capitalized letter followed by lowercase. Villages should be done in all lowercase so they are pushed back and don't interfere with the other tags. City type should always be straight from left to right.\n\n#### Rivers\n\nRivers are often labeled in lowercase type, and should follow the contours of the river being labeled.\n\n#### Mountains\n\nMountains should also follow the contours of the mountain range, stretching as best it can to cover the entire range. Mountains can be labelled in either upper- or lowercase fashion, depending on the map.\n\n#### Ocean, Sea and Bay Labels\n\nOcean and sea labels are usually all uppercase, and follow the contour of the area being labeled in a slightly curved fashion, while bay areas should be smaller and straight across. Bay labels should be capitalized, but not uppercase letters for the whole label.\n\n## Painting\n\nUsing paint and brush for cartography is a little bit different than using inks and washes, though both techniques can be combined for an excellent result. While ink serves to define boldly and gives a graphic element to a map, painting is good for giving a more organic and colorful look. Painting also overlaps directly into the digital realm, and both traditional and digital share most of the basic (and some advanced) painting techniques. There are several different types of traditional painting mediums that can be used, each with different techniques and looks. Most commonly, watercolors, gouache and acrylics are used in cartography for their layering capability.\n\n#### Painting Tools\n\n#### Brushes\n\nLoew-Cornell no. 1 detail round Sienna nos. 1, 4, 10 and 14 round\n\nSienna no. 4 flat\n\nLoew-Cornell 3\u20444\" (19mm) flat wash\n\nLoew-Cornell 1\" (25mm) flat wash\n\n#### Winsor & Newton and Mo Graham Paints\n\nGouache\n\nBurnt Sienna\n\nIvory Black\n\nPermanent Yellow Deep\n\nPrussian Blue\n\nQuinacridone Red\n\nRaw Umber\n\nSpectrum Red\n\nYellow Ochre\n\nZinc White\n\nWatercolor\n\nAzo Green\n\nOlive Green\n\nMasterson's Sta-Wet Palette\n\n## Creating a Detailed Border\n\nWhen you have your basic colors on your palette, it's time to lay in your base tones. Base tones are the medium tone of a color that you are using, and can be shaded with darks or highlighted with whites. Base tones should be the first colors painted onto a map, forming the foundation colors. For an example, let's work on a detailed border.\n\nAfter the base tones are laid in, it's time to add some definition. This is the real meat and potatoes of painting, and it is a bit like sculpting. Darks are laid in to ground an image and give it depth, while highlights make it pop, adding contrast. Contrast is what draws the eye, and this is increased the closer the darkest darks are to the lightest lights. You want shadows and highlights to complement your base tones in a map, not overpower them.\n\n#### STEP 1: Sketch the Border\n\nStart with a basic pencil sketch.\n\n#### STEP 2: Add Base Color\n\nUsing your flat brush, lay in your base colors. Try and stay within your pencil sketch boundaries, laying in color as evenly as possible.\n\n#### STEP 3: Finish the Base Color\n\nFinish up your base tones, evening out any inconsistencies as best you can. Having your pencil sketch show through can be helpful, but be prepared to paint over your base tones for the next step.\n\n#### STEP 4: Apply Deeper Darks\n\nUsing your shade tones, lay in your darks first before the highlights. Blacks should never be used without being mixed, as pure blacks are unnatural and are much too dark. For small areas, use your smallest round brush, though for straight edges use a flat brush.\n\n#### STEP 5: Apply Highlights\n\nOnce your darks have been painted, it's time to mix some highlights. Highlights will provide the border with definition, giving it a more sophisticated look. Highlights should be painted on top of your base tone, but not over your darks. Often, it's best to apply highlights with your detail brush, so you can better control the area being highlighted.\n\n#### STEP 6: Add the Lightest Lights\n\nOnce your basic highlights lighten the area, you can finish them off with a few well placed lightest lights. I use Zinc White with a small amount of the mixed color to pop out the image. Light and dark colors directly next to each other will create contrast and draw the eye. The lightest lights should be applied sparingly and in small spots.\n\n#### STEP 7: Ink the Details\n\nWhen inking on top of base tones, use your medium brush or pen for outlining, and your thin nib for detail work. Brown inks are much more forgiving than black, though blacks are bolder. Ink first, apply final details and clean up.\n\n#### STEP 8: Final Shading\n\nUsing your small flat brush, add some water to Raw Umber, and gently apply shading to your piece. This will give the piece depth within the image itself, and allow it to meld in and not contrast jarringly.\n\n## Digital Painting\n\nDigital painting has really come into its own in the last few years, and with the advance in technology, it has the ability to combine the look of traditional painting without the setup and mess. Digital painting is quicker, and in some respects easier to use, turning hours of traditional decisions into mere seconds. Mistakes are instantaneously erased, while granting an artist the quick ability to explore and complete art along many different avenues. While there are proponents of both, I think of digital painting as both an art form and a tool. Knowledge of traditional techniques is very helpful, but not necessarily needed, though color theory in any medium is essential.\n\n#### Digital Tools\n\n#### Digital Brushes\n\nRough round bristle\n\nAirbrush 75 round\n\nHard round 3 pixel brush\n\nAirbrush 45 round\n\nHard pastel on canvas\n\n#### Stylus and Tablet\n\n#### Adobe Photoshop or GIMP software\n\n#### Digital Tools\n\nTo paint using a computer, it's best to have a tablet of some sort. A tablet acts as your canvas, and a stylus as your brush, and is excellent to simulate painting by translating your brushstrokes onto the screen. Tablets have a pressure sensitivity as well, so the harder you press as you paint, the darker or wider the stroke. Programs such as Corel Painter and Adobe Photoshop use virtual brushes, and these brushes are each used to do different things. You can make your own brushes, or use the presets within the respective programs. For now, we will use some presets.\n\n#### Opacity\n\nOpacity is a tool for making colors and marks more or less translucent. The lower the opacity, the weaker the tone of the color is, and the more it can be layered on top of other colors. A color or layer can vary in opacity from 0\u2013100 percent. For layering color and mixing colors, opacity is important. Layers can be reduced and faded by lowering the opacity of the entire layer. Try fading and strengthening your line work layers by increasing and decreasing the layer's opacity.\n\n#### Layers\n\nLayers are a unique element of digital painting. Layers can be overlaid over one another, like panes of glass, with each layer's elements showing through to the next. You may use as many layers as you'd like, and it's best to use multiple layers for the different elements of your piece. By using multiple layers, you may overlay textures, combining them all to make an image. To make new layers, go to Layers > Make New Layer. Type in an appropriate name, and the new layer will appear above the current layer. Layers are organized into a hierarchy, with any layer on top of another covering anything below it.\n\n#### Drawing\n\nDigital drawing is a bit different from traditional drawing, but both obey some of the same precepts. With a stylus and its pressure sensitivity, you may make lines thicker or thinner, and erase quickly and efficiently. Before drawing make a new layer, and name it \"Line work.\" This is where all of your drawing should be done. When drawing, it's best to use brushes 4\u20138 pixels in size for basic terrain line work, and brushes 1\u20133 pixels wide for detail work.\n\n#### Painting\n\nPainting in the computer resembles traditional painting in some ways, but differs in many others. You can start by making a new layer under your line work layer. Then lay in your base tone layer using your broad brush. After making another layer, paint in your darks. Add another layer and lay in your highlights. A final layer over the top should be for detail work.\n\n#### Typography\n\nLaying in typography digitally is very different from traditional means. To apply type to your map select the Type tool, and select an area of the map to make a text layer for inserting type. You may also click and drag to make a text box, which will contain all text typed into that box. Once a text layer is made, you can type in your tag, and then alter its size and font type with a few clicks. When using type in a map, it's best to stick to one font type, and not vary the size too much. Keep similar elements the same size to emphasize uniformity. A text tag may be moved around at will, and each text tag will form its own separate layer. Once you are satisfied with your typography placement, you can group all of these tags together and merge them for one easy-to-manage layer.\n\n#### Digital Finish\n\nHere is a finished digital map I have done for my own world, Hyranden. Though this map uses some other more specialized techniques, many of the topics I have discussed in this chapter are put to good use here. Traditional and digital art have much more in common than people think, though their execution can vary.\n\n# CHAPTER ELEVEN\n\n# Putting it All Together\n\nCreating a fantasy map from scratch is a lot of work, and is much like weaving a complex tapestry from much simpler threads. Don't be overwhelmed! Now that you have learned the basic techniques, as well as the more advanced methods, it's time to put all of these elements together. In this chapter, I will construct a map from the ground up going step by step in order of execution, slowly piecing it together to form a cohesive whole. Before starting, collect all of your materials, and then ask yourself a few questions.\n\n  * How large of a map do I want my world to be?\n  * Is the map going to be in full color or monochromatic?\n  * What type of map do I want to make\u2014county, regional, country or world?\n  * Do I have any maps to use for inspiration?\n\n## Creating a Fantasy Map\n\nI will be creating a world map of the fantasy lands of Aldriin. This map will use all of the elements included in the basic and advanced techniques sections to show you how they should be approached, and in what order elements should be tackled. Whether you are approaching the map from a traditional or a digital perspective, this basic rundown will show you how a map is made.\n\n#### STEP 1: General Sketch\n\nThe first step is to take a light pencil and gently draw your general sketch. Start with the large landmasses first, then island boundaries. From that, lightly draw in the basic mountain areas as well as marking the general areas of cities, rivers, borders and corners. Your sketch should be light and very general, serving as a placeholder for the next step.\n\n#### STEP 2: Coastlines\n\nWhen your general sketch is complete, it's time to go in and detail your coastlines with a heavier line. This line work should be much darker and more sophisticated than the light sketch. Here is where you add inlets, bays and smaller island formations as well as interior lakes. Be sure to leave small openings where rivers run into the sea.\n\n#### STEP 3: Hills and Mountains\n\nAlways lay in your mountains first. Start with the general outline shapes, then go back in to detail each mountain with smaller detail lines, giving them each definition. When your mountains are finished, move on to your hills, smaller general shapes first, and then a few defining lines. The detailing lines should be thin, and drawn at the base of each hill.\n\n#### STEP 4: Towns and Cities\n\nDraw in your capitals and cities, then your towns and villages, and finally your fortresses and holdfasts. Keep them as general as possible. If there are any shields or heraldry, draw the general shapes in to serve as placeholders.\n\n#### STEP 5: Rivers\n\nAfter your towns have been drawn in, it's time to lay in all of your rivers. Start with the largest ones, then move on to the smaller ones. Rivers emptying into the sea should be integrated with the coastal borders. Draw in the river tributaries last, tapering from the mountains to the rivers they join.\n\n#### STEP 6: Shore and Water Lines\n\nNow that your rivers and coastlines are complete, lay in your shorelines. The first line should run about 1\u20444\u20131\u20442\" (6mm\u201313mm) out from the land, and hug the coast closely. The next line should be another 1\u20444\" (6mm) out, hugging the first line, and so on. Anywhere from 3\u20136 lines are good. These lines should never overlap the legend or border sketches, stopping when touching their edges. Lake lines should only have one or two of these lines at most to differentiate them from the sea.\n\nFor wave lines, group them together and lay them in random groups of three. Be careful not to place too many.\n\n#### STEP 7: Background Border\n\nThe border should be outlined first, tracing over your light sketch line, using a straightedge if needed. Draw in border corners, outlining each, then draw the interior and detail work. The outlines should be bold line work, while the interior of the borders should be thinner and more delicate. These lines are still relatively temporary, and will become permanent only in the inking stage later on.\n\n#### STEP 8: Legend and Information Boxes\n\nWhen your border and corners are laid in, draw your legend, and detail it. The outline of the legend should be strong, and the interior thinner and lighter. Lay out and arrange your legend, plotting out the title and the contents as best you can. It's not completely necessary to lay in your title yet, but any city icons and scales should be detailed or at least outlined and prepared for later inking. If you have any information boxes, lay them in as well and detail them.\n\n#### STEP 9: Compasses\n\nAfter your legend, draw in your compass rose, using any template or straightedge you need. Start with the central portion, then pencil in the points, starting with north. Then move on to the minor compasses. Draw in your compass lines, making sure not to intersect the ocean lines.\n\n#### STEP 10: Forests\n\nNow tackle your forests, laying them in group by group. Forests should flow around rivers, and be placed at the base of mountains. Leave gaps in the forest for your forest's labels. You may want to pencil the labels in now so you know what to ink in later.\n\n#### STEP 11: Roads and Boundaries\n\nNow that your forests are drawn, lay in your highways, roads and paths, as well as any boundaries that you'd like to include to differentiate kingdoms and regions.\n\n#### STEP 12: Minor Details\n\nNow that most of your major elements are in, it's time to add the minor details. Cities, towns and terrain elements should be fully penciled in now, as well as any shading for those elements. Icons should be placed and penciled in as well.\n\n#### STEP 13: Iconography\n\nAdd iconography to the map and start shading the terrain with line work.\n\n#### STEP 14: Typography\n\nWith the map almost complete, it's time to lay in your typography, erasing the terrain elements to make room. First, lay in capital, citadel and city tags; then town, village and holdfast tags. Once these are done, the lake, river, ocean and bay tags should be drawn in.\n\n#### STEP 15: Inking\n\nBy now, your entire map should be finished in pencil, and inking it is one way of completing it. Using your thick pens, ink in your coastal land boundaries, as well as your legend and border outlines. Move on to inking the terrain elements, and then the cities, heraldry and border interior elements with a small detail pen. Inking is the last step in an inked piece, but can be skipped if you are moving on to painting your piece in a traditional manner.\n\n#### Digital Inks\n\nIf working digitally, all of your line work should be on separate layers, and the piece should essentially already be \"inked.\" Your colors can be painted in behind your line work on a separate color layer. Digital work is very flexible in this respect.\n\n#### STEP 16: Heraldry and Other Details\n\nDon't forget to add the heraldry designs to each of your shields as well as filling in the information boxes and other small details throughout the map.\n\n#### STEP 17: Finished Inks\n\nOnce you have laid in all of your various details that make up a map, you can then go in and do the final inks. Inking the map will give it a finished dynamic that is quite bold. Finished inked maps are quite pretty, and stand very well on their own. With a few light washes for some added depth, this map would be considered complete. However, for our purposes, we aren't going to stop here. Buckle up, because we are going to make this piece into a colored map!\n\n#### STEP 18: Water Elements\n\nThe first colors introduced into a map should be the water elements, usually a deep blue or bluish green. Seas and oceans need not be dark, however; it just depends on the map itself and how you'd like it to look. Paint in the ocean, starting at the land boundary and working your way out, painting around your compasses, legend and borders. If painting digitally, make a separate layer called \"water\" to keep it separate from the line work.\n\n#### STEP 19: Shading\n\nTo bring out some of your elements, apply darker shading around them. First, mix a dark shade to apply to the land boundary interiors, using washes to build up tone. Add shading to the mountains and hills to give them more dimension. Leave some highlights at the top to look like snow on the highest peaks.\n\nForests, cities, icons and heraldry shields should be shaded by applying tones underneath the element. River lines should also be shaded, since the next step will involve painting them in.\n\nFinally, shade the borders, corners and legend, making them pop from the rest of the map.\n\n#### STEP 20: Rivers\n\nStart with a dark blue, and following your sketch lines, paint in your river lines. Then, with a smaller brush, paint in a much lighter blue to make them pop from the dark blue. Using a bit of white, highlight the river mouth that spills into the sea.\n\n#### STEP 21: Towns and Cities\n\nPaint in your towns and cities, laying in off-white undertones first, then adding colored roofs and details using a thinner brush. Define each city with dark tones, and then apply highlights to each. Icons should be painted in as well, with solid tones overtop the icon shading. City icons should be painted in with a deep red, highlighted with a lightened red and finished off with a white for contrast.\n\n#### STEP 22: Trees\n\nTrees should be painted in a dark green, overtop the drawing and shading. Apply a few light coats and build up the tonality as needed. Apply darks or lights to shade or highlight the forest. Don't make them too light or too dark. If painting digitally, make a separate layer and multiply it, painting in greens, or make your own paintbrush to paint in the trees.\n\n#### STEP 23: Background Borders\n\nBorders should be painted by laying in your base tone first, being careful to keep your paint strokes within the line work. Start with your border first, and then work to the corners. When all of your base tones are in, lay in your dark tones with a thin brush. Once your dark tones are in, lay in your highlights. After both dark and lights are in, ink your piece with a detail nib, and finally apply your last highlights and lightest lights.\n\n#### STEP 24: Heraldry and Shields\n\nWhen painting in shields and heraldry, lay in your base tones first, careful not to get paint outside of your line work. Next, add some shadows and highlights to your base tones, and outline the shield in a thin dark brown line to give it some emphasis. Paint in the symbol next, and detail it. Finally, add a few highlights of white and dark to finish it.\n\n#### STEP 25: Compass Rose\n\nWhen painting your compass, it's best to use dark browns and reds. Lay in your base tones first, then move onto the points, adding darks first, and then lights. Highlights should be painted in last, after the details have either been inked or painted in. The compass should be outlined in dark blues or dark browns.\n\n#### STEP 26: Typography\n\nThough most typography should be inked in with either browns or blacks, some other types should be painted in, such as large regional, country, sea or bay tags. Some types should be capitalized with larger painted letters for emphasis, such as capitals and citadels. Carefully paint over your line work, using either a flat brush or a small round. Ocean and sea tags should be white or light blue, while country and regional tags should be done in reds. Text may also be outlined with white to make it pop when the tag is in a hard to read area.\n\n#### STEP 27: Terrain\n\nFarmlands, marshlands, deserts, plains and other terrain should be painted in with a flat brush. Applying multiple layers and building them up is best so you don't overwhelm the map. Start with a base tone, then darks and then lights. If you wish, add in some inks for a little extra definition or tidying up.\n\n#### STEP 28: Detailing\n\nThe final step in your map should be detailing. Add in any final elements you may have, such as scuff or fold marks and additional colors. Touch up shading and highlights. You should also paint in the typography or final colors for the legend, as well as any information or designs that you want to include in your information boxes. These should follow the same painting process: base tones, darks, lights and then highlights and detailing. Last of all, paint in any sea routes you may have.\n\n# About the Artist\n\nJared Blando is a professional freelance cartographer and conceptual artist who works in the RPG, book and gaming industry. Though he is best known for his RPG fantasy maps, he is also prominent in the board game and publishing industry, producing artwork for both large and small publishers. He has a penchant for historical works and miniature gaming, as well as cats and gluten-free pizza. You can see more of his work on the web at www.theredepic.com. You may drop him an email anytime, though he also accepts messages via carrier pigeon.\n\n# Acknowledgments\n\nThis book has been an incredible amount of work! However, there were certain people that came through in spades to help me out of the quagmire, namely my lovely Chelsea and my parents, John and Iris, and a few close friends and family. Without their support and appreciation, this book would have been that much harder to put out, leaving me in debt to a great many people. My thanks to you all!\n\n# Copyright\n\nHow to Draw Fantasy Art and RPG Maps. Copyright \u00a9 2015 by Jared Blando. All rights reserved. No part of this eBook may be reproduced in any form or by any electronic or mechanical means including information storage and retrieval systems without permission in writing from the publisher, except by a reviewer who may quote brief passages in a review. Published by IMPACT Books, an imprint of F+W, a Content + eCommerce Company, 10151 Carver Road, Suite 200, Blue Ash, OH 45242. (800) 289-0963.\n\nOther fine IMPACT products are available from your local bookstore, art supply store or online supplier. Visit our website at fwcommunity.com.\n\neISBN: 9781440340321\n\nThis e-book edition: July 2015 (v.1.0).\n\n# Contents\n\n  1. Cover\n  2. Title Page\n  3. Special Offers\n  4. Table of Contents\n  5. Introduction\n  6. Basic Materials\n  7. Advanced Materials\n  8. Chapter One: Building Your World\n    1. Coastlines and Boundaries\n    2. Define Your World\n  9. Chapter Two: Mountains, Forests, Rivers and Water\n    1. High Peaks...\n    2. ...and Rugged Hills\n    3. Woods and Forests\n    4. Rivers and Tributaries\n    5. Shore and Water Lines\n  10. Chapter Three: Town and City Icons\n    1. Cities\n    2. Castles and Holdfasts\n  11. Chapter Four: Typography\n    1. Script Style\n    2. Cultural Typography\n    3. Location Labels\n    4. Area Labeling\n  12. Chapter Five: Landmarks\n    1. Roads and Routes\n    2. Special Locations\n  13. Chapter Six: Iconography\n    1. Simple City Icons \n    2. Location Icons\n    3. Ocean Icons\n  14. Chapter Seven: Heraldry and Shields\n    1. Shield Shapes\n    2. Simple Heraldry\n    3. Simple Shield Symbols\n    4. Painting Shields\n    5. Shield Embellishments\n    6. Shield Placement\n  15. Chapter Eight: Political Boundaries\n    1. Boundary Lines\n    2. Boundary Colors\n    3. Boundary Types\n    4. Using Natural Boundaries\n  16. Chapter Nine: Backgrounds\n    1. Borders\n    2. Border Corners\n    3. Complex Corners\n    4. Corner Flourishes\n    5. Advanced Flourishes\n    6. Compass\n    7. Legends\n    8. Legend Information\n    9. Information Boxes\n  17. Chapter Ten: Painting Techniques\n    1. Inking\n    2. Pen and Brush Inking\n    3. Detail Inking\n    4. Typography\n    5. Painting\n    6. Creating a Detailed Border \n    7. Digital Painting\n  18. Chapter Eleven: Putting it All Together\n    1. Creating a Fantasy Map\n  19. About the Artist\n  20. Acknowledgments\n  21. Copyright\n\n# Guide\n\n  1. Cover\n  2. Table of Contents\n  3. Start of Content\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nMore than anything, this one's for you, Mom and Dad.\nContents\n\n**Title Page**\n\n**Introduction**\n\n**Part I: The Essentials**\n\nAnatomy of a Wine Label: The Essentials\n\nLabel Decoder: The Grapes Behind the Regional Place Names\n\nBecome a Better Wine Shopper: Ten Tips\n\nGuide to Glassware\n\nThe 'Dos' and 'Don'ts' of Storing and Serving\n\nBecome a Better Taster: A Face-Off Drinking Game\n\n**Part II: White**\n\nChardonnay\n\nSauvignon Blanc\n\nRiesling\n\nChenin Blanc\n\nS\u00e9millon\n\nGew\u00fcrztraminer\n\nMarsanne and Roussanne\n\nPinot Gris\n\nPinot Blanc\n\nViognier\n\nAlbari\u00f1o\n\nGr\u00fcner Veltliner\n\nMuscat\n\nTrebbiano\n\nAssyrtiko\n\nMelon de Bourgogne\n\n**Part III: Red**\n\nCabernet Sauvignon\n\nMerlot\n\nCabernet Franc\n\nPinot Noir\n\nSyrah\n\nGrenache\n\nBarbera\n\nGamay\n\nCarignan\n\nMalbec\n\nZinfandel\n\nPetit Verdot\n\nMourv\u00e8dre\n\nNebbiolo\n\nTempranillo\n\nSangiovese\n\nPinotage\n\n**Part IV: Sparkling and Ros\u00e9**\n\nChampagne\n\nFrance\n\nItaly\n\nSpain\n\nGermany and Austria\n\nP\u00e9tillant Naturel\n\nRos\u00e9: The FAQs\n\n**Part V: Extra Credit**\n\nWine and Food Pairing: A Strategy Playbook\n\nA Quick Guide to More Grapes You Should Know\n\nGlossary\n\n**Index**\n\n**About the Author**\n\n**Acknowledgements**\n\n**About the Illustrator**\n\n**Copyright Page**\n\n# Introduction\n\n**When people ask me how I got into wine, my answer is always the same: 'Very, _very_ reluctantly'.**\n\nIt's not that I didn't love wine, or that I found it boring, or that I thought of it as just another alcoholic drink; quite the opposite. The problem is it's almost too interesting, too complex, too broad. Why, it's even got its own unique language. And then there's the science-y component, the geographical knowledge required, and the fact that it's subjective and constantly evolving. I mean, someone pour me a glass of wine so I can process all this stuff that I have to learn about wine! To cut a long story short, I started out feeling like you probably do about the subject: completely overwhelmed and totally unsure of where to begin.\n\nFor me, the easiest, most logical starting point was grapes, and so that is the focus of this book. The following pages will outline the world's major red and white varietals and give you a snapshot of what you need to know about them to embark on a more informed wine-drinking journey. Basics like notable growing regions and food pairings are covered with each, as well as more practical things for consumers such as general tasting descriptors to help you convey what you like or don't (the 'Key Words') and examples of other grapes you might like.\n\nSupporting this core information are nuggets of the most useful, practical tips you might need: the 'dos and don'ts' of storing and serving, FAQs about ros\u00e9, how to understand key wine descriptors through side-by-side tastings, a cheat sheet on deciphering labels, a whirlwind tour of the world of sparkling wine and a glossary of technical and sommelier terms.\n\nDon't see this as your complete guide to everything wine-related but, rather, your friendly (pocket-sized!) overview of the essentials. So, please, drink up, and remember to have fun and keep learning.\n\n# Anatomy of a Wine Label: The Essentials\n\n# Front\n\n## Producer Name\n\nOften the easiest item to pick out on a label as it tends to be the most prominent, this is the name of who made the wine. In France, look for the prefix ' _Domaine'_ or _'Ch\u00e2teau'_ ; in Germany and Austria, _'Weingut'_ ; in Italy, _'Castello'_ , _'Cantina'_ or _'Tenuta'_ ; _'Quinta'_ in Portuguese and, in Spanish-speaking countries, _'Bodegas'_. Sometimes also a wine may go by a branded name that is a subsidiary of a large-scale producer (most often you'll see this with a New World wine, the US especially).\n\n## Vintage\n\nThe year the grapes were harvested.\n\n## Region\/Appellation\/Sub-Appellation\/Vineyard Name\n\nWhen it comes to wines from the EU, the broad strokes perspective is that the more specific the growing site listed, the higher quality the wine is going to be. It's like a hospital consultant with a lot of letters after their name: it means they've had to pass more tests and meet more standards, and are therefore more unique and prized because of it.\n\n## Regional Classification\n\nFor example, _Appellation d'Origine Contr\u00f4l\u00e9e_ (AOC) in France; _Indicazione Geografica Tipica_ (IGT), _Denominazione di Origine Controllata_ (DOC) and _Denominazione di Origine Controllata e Garantita_ (DOCG) in Italy; _Denominaci\u00f3n de Origen_ (DO\/DOC) in Spain; _Districtus Austriae Controllatus_ (DAC) in Austria; _Denomina\u00e7\u00e3o de Origem Controlada_ (DOC) in Portugal and _American Viticultural Area_ (AVA) in the US. These regulatory bodies determine the rules and regulations for the region's growing practices and standards.\n\n## Vineyard Site Quality Level Indicator\n\nIn France, look for words like _'Premier Cru'_ and the even more prized _'Grand Cru'_ ; in Germany, keep an eye out for either _'VDP Grosse Lage'_ or _'VDP Grosses Gewachs'_ , but you should also expect to pay a lot more for these wines. Just remember: not all regions \u2013 even some of the top ones, like Barolo \u2013 have a built-in quality classification hierarchy for their vineyard sites, so be careful not to look for one as an absolute, across-the-board prerequisite.\n\n## Aging Indicator\n\nIn the EU, the word 'Reserve' (aka _Riserva\/Reserva\/Gran Reserva_ ) indicates that the wine has been aged for a period of time prior to release. The amount of time depends on the region's particular regulated standards. Again, expect these to be more expensive. Important side note though: the term is not protected in New World regions like the U.S. and Australia.\n\n## Sweetness Level Indicator\n\nMost commonly seen on bottles of German Riesling, which can range from dry _(Trocken)_ to off-dry _(Kabinett)_ to sweet _(Sp\u00e4tlese)_ to very sweet _(Auslese)_ (just an FYI: Technically, those last three terms refer to the ripeness level of the grapes at harvest time, but by and large it will coordinate with the sweetness level of the wine). These will appear as a suffix to the grape varietal. Another good example would be on labels of Chenin Blanc from Vouvray, which will tell you the wine is off-dry with the words _'Demi-sec'_ or sweet with the word _'Moelleux'_.\n\n## Grape Varietal\n\nThis is more common practice with New World wines and wines from Austria and Germany, although it is becoming more prevalent throughout Old World regions. This is the grape that makes up the overwhelming majority, if not 100%, of the wine.\n\n## Bottle Size\n\nMeasured in millilitres, the standard bottle size is 750 ml (25.4 fl oz). Half bottles come in 375 ml (12.7 fl oz) bottles, while magnums are double the standard size at 1.5 l (50.8 fl oz).\n\n## ABV\n\nLiterally, the Alcohol by Volume. Most dry red and white wines fall in the 12\u201313.5% range. Lower than 12% in a white and you can expect it to be a little sweet, and reds clocking in at over 14% are likely to be full, rich and boozy.\n\n# Back\n\n## Importer and\/or Distributor\n\nThe company responsible for sourcing and shipping the wine from its country of origin and distributing it for sale in its current location. Essentially, the wine's talent agent.\n\n## Government Warning\n\nMost countries include a government-mandated warning, either via symbol or text, indicating that consuming alcohol impairs your ability to drive, is not advised for pregnant women, and may be detrimental to your health.\n\n## Product of [Country Name]\n\n## Estate Bottled\n\nAs straightforward as it sounds, this means the wine was bottled by the producer at the winery where it was made. For French wines, you might alternatively see _'mis en bouteille au domaine\/\u00e0 la propri\u00e9t\u00e9'_.\n\n## Contains Sulphites\n\nSulphites are a natural by-product of the winemaking process and are not to be blamed for your hangover. All wines contain sulphites, hence the listing on the label, but not all wines have added sulphites. If you're concerned about your wine having extra sulphites in it \u2013 it's a commonly used preservative agent \u2013 look for wines labeled with 'no added sulphites' ('natural' wines are a good place to start this quest).\n\n# Label Decoder: The Grapes Behind the Regional Place Names\n\nWine labels offer a lot of information. Some of it is easy and straightforward to understand; for example, a wine's alcohol content and vintage, or who imports it into the country. Other key details, like, say, what grape(s) the wine is made up of, are not always as obvious. It's not so of much an issue with New World wines (aka North America, South Africa and Southern Hemisphere countries such as Australia, New Zealand and Argentina), which generally require listing the primary or single varietal on the label. Less user-friendly, however, are those wines from Old World regions \u2013 particularly France and Italy \u2013 which use important place names on the label as a kind of pseudonym for the varietal.\n\nIt's not intended to be misleading by design; really, it's about celebrating the terroir (your catch-all term for 'sense of place', see Terroir) and the fact that certain grapes have been found to grow best in certain locations. Still, for the not-quite-yet-in-the-know, the system can definitely act as a barrier. It facilitates common, rookie-mistake claims such as, 'I won't drink Sauvignon Blanc \u2013 hate the stuff \u2013 but I like pretty much everything else. I've been enjoying Sancerre a lot lately.' Here's the thing, the white wines produced in sancerre can only be made from 100% Sauvignon Blanc. But \u2013 and this is important \u2013 if you've ever made this statement before, please don't feel embarrassed, as it's still a really useful indication of your tastes to the sommelier or wine professional helping you make a selection. All it means is that you like a specific style of Sauvignon Blanc.\n\nHowever, if you want to avoid any such slip-ups, the following is a cheat sheet for some of the most common place names found on labels and their associated grape varietals.\n\n# Whites\n\n**Chablis** = 100% Chardonnay\n\n**Pouilly-Fuiss\u00e9** = 100% Chardonnay\n\n**Sancerre** = 100% Sauvignon Blanc (the exception being if it's a red or ros\u00e9, it's made from Pinot Noir)\n\n**Pouilly Fum\u00e9** = 100% Sauvignon Blanc\n\n**Vouvray** = 100% Chenin Blanc\n\n**Savenni\u00e8res** = 100% Chenin Blanc\n\n**Muscadet** = 100% Melon de Bourgogne\n\n**Condrieu** = 100% Viognier\n\n**Tokaji** = Mostly Furmint, as well as H\u00e1rslevel\u0171 and Muscat Blanc\n\n**Sauternes** = Mostly S\u00e9millon, with some percentage of Sauvignon Blanc and Muscadelle\n\n**Soave** = Mostly Garganega, with up to 30% Trebbiano\n\n**Vinho Verde** = Mostly Loureiro and Alvarinho\n\n**Gavi** = 100% Cortese\n\n**White Burgundy** = Chardonnay\n\n# Reds\n\n**Beaujolais** = 100% Gamay\n\n**Barolo** = 100% Nebbiolo\n\n**Barbaresco** = 100% Nebbiolo\n\n**Dogliani** = 100% Dolcetto\n\n**Valtellina** = Mostly Nebbiolo\n\n**Gattinara** = Mostly Nebbiolo\n\n**Ghemme** = Mostly Nebbiolo\n\n**Roero** = Mostly Nebbiolo\n\n**Chianti** = Mostly if not 100% Sangiovese\n\n**Brunello di Montalcino** = 100% Sangiovese\n\n**Rosso di Montalcino** = 100% Sangiovese\n\n**Vino Nobile di Montepulciano** = Mostly Sangiovese\n\n**Chinon** = Mostly 100% Cabernet Franc\n\n**Bourgeuil and**\n\n**Saint-Nicolas-de-Bourgueil** = Mostly 100% Cabernet Franc\n\n**Cornas** = 100% Syrah\n\n**Hermitage** = Mostly 100% Syrah\n\n**Crozes-Hermitage** = Mostly 100% Syrah\n\n**Saint-Joseph** = Mostly 100% Syrah\n\n**Gigondas** = Mostly Grenache\n\n**Vacqueyras** = Mostly Grenache\n\n**Chate\u00e2uneuf-du-Pape** = Mostly Grenache, Syrah, Mourv\u00e8dre and Cinsault, but Clairette, Bourboulenc, Roussanne, Vaccar\u00e8se, Counoise, Muscardin, Picpoul, Picardan and Terret Noir are also permitted\n\n**Cahors** = Mostly Malbec\n\n**Rioja** = Mostly Tempranillo\n\n**Bandol** = At least 50% Mourv\u00e8dre\n\n**Priorat** = Mostly Grenache\n\n**Valpolicella and**\n\n**Amarone della Valpolicella** = Mostly Corvina with Rondinella and Molinara\n\n**'Left Bank' Bordeaux red** = Mostly Cabernet Sauvignon\n\n**'Right Bank' Bordeaux red** = Mostly Merlot\n\n**Red Burgundy** = Pinot Noir\n\n# Become a Better Wine Shopper: Ten Tips\n\n## 1. Seek out a good local wine shop\n\nIf you're interested in wine as more than just alcohol to put in a glass at the end of a long day, then this is essential. Sure, there are plenty of supermarkets out there with above-average wine selections, but a good wine shop offers important perks. For example, well-educated staff that are excited to engage with you and share their knowledge (sans condescension, pretension or any of that other nonsense that makes people feel intimidated about wine). Better yet, they will most likely offer to track your purchases and get to know your palate profile in order to make better recommendations in the future.\n\nOften wine shops host tastings or classes for free or, if not, relatively inexpensively, where a winemaker or sales rep from an importer or distributor is pouring. Take advantage of this insider access; listen, drink and ask questions. Another valuable bonus of finding a good shop? Case (or half case) discounts. Learning about wine is not a cheap activity. Fortunately, most places offer deals when you purchase in larger quantities, such as six or 12 or more bottles, which means you get to save the cost of a bottle or two by committing to trying several.\n\n## 2. Become a regular at a wine bar or restaurant with an extensive list of wines by the glass\n\nA similar idea to the above. Find a place that has an interesting, unique selection and knowledgeable staff, but with one additional bonus: you can ask to try before you buy. Think of it as akin to being at an ice cream shop and getting to sample a couple flavours before making your decision \u2013 just don't be that person who tries six different types and then leaves, full, without buying a single scoop.\n\nThe kind of by-the-glass list we're talking about here probably has upwards of six different selections for red and white wines, multiple sparklings, (ideally) a couple of ros\u00e9s as well, and most probably one or two ap\u00e9ritifs (sherries, vermouths) and dessert wines. The selection should rotate with some relative frequency, and include some things you are not that familiar with. Treat this as your classroom. Ask about the grape(s) and the producer, where the wine is from and what's special and characteristic of the style if it's something unusual. Ask to do half-glass pours so that you get to try a wider variety. If you find something you like, see if they know where you can buy it, and what other similar wines might be worth trying. Be curious, and don't be afraid to say you don't like something \u2013 again, it will help staff inform your choice next time around.\n\n## 3. Take pictures of the labels of wines you like (and don't)\n\nBecause wine is a complicated language, and trying to explain what you're looking for can be a kind of, 'how do you know what you call \"green\" is the same as what I call \"green\"?' situation, pictures are helpful. And objective. Whether you track your hits and misses with the help of an app (Delectable and Vivino are two of the big ones) or simply keep a folder on your phone, a label shot is a good starting point for a conversation. Or, at the very least, it gives you a reference when you're scanning the shelves.\n\n## 4. If you're not good with tasting notes, use context clues to help describe what you're looking for\n\nMaybe you're still unsure of how to answer when asked if you're looking for something light or full-bodied; super dry or slightly juicy; fuller and rounder or lean and mineral-driven. Not to worry. If you're a little uncomfortable with tasting descriptors, offer some context clues to work with. How much you want to spend (and please, don't be shy about this), what food \u2013 if any \u2013 you're pairing it with, who you're drinking with and what the environment is going to be like are all useful pieces of information that can help inform a better selection. So think: is the bottle for you or just something to bring to a party? Or is it a gift? And if so, what's the occasion? Do you need something to go with the takeaway food you're about to pick up, or do you just want a bottle to go with cheese and charcuterie? Are you looking to splurge or keep it in a more wallet-friendly price range? Just don't shut down and say, 'oh I don't know, I just want a 'nice' wine.' What does that mean? Is the wine polite? Does it send you a card every year on your birthday? Chances are, if 'nice' is the only criteria you request, you aren't going to end up with anything special, or specifically to your taste or suitable for the occasion, as it's too generic.\n\n## 5. Pay attention to who imports the wines you like\n\nThe front label is usually the first place we look to gather information about a wine. But what if it isn't enough? Even if you recognise the region or the grape and understand what that might translate to in the glass, the particular producer and their style can still be foreign to you. It's at this point in the game that you should turn over the bottle to look for your next clue: who imports the wine.\n\nA key and powerful cog in the supply chain, the importer essentially functions as an agent. They seek out the talent that best represents their brand (for example, a particular winemaking philosophy or growing region) and then take on the responsibility of bringing it into the country and supporting its success. Recognising which importers represent your favourite wines is like identifying the people in the room who share your taste in music. While not a 100%-satisfaction-guaranteed tactic, it can at least offer a sense of reassurance and help encourage exploration beyond your old trusted standbys. And don't forget to use importers as a resource when you're checking out a new wine shop: if you see a lot of familiar names that you like, you're probably in the right place.\n\n## 6. Look for the second-label wines and n\u00e9gociant projects of big-name producers\n\nThe basic premise is simple: big name, high-price-fetching estates only wants to use the best fruit from the best vineyards for their top bottlings. Unfortunately, that means a lot of very-good-just-not-quite-as-great fruit is left sitting there on the bench like some untapped resource. Instead of wasting it and taking a loss, the producer gives it a home under a different, more value-friendly second label.\n\nGenerally speaking, the concept is a win-win for both the estate and the consumer. The winery gets to maximise the use of their product yield while simultaneously opening themselves up to a market of consumers that might otherwise be priced out of their offerings. And you? Well, you get the B-team. A B-team that, yeah, may be younger and less experienced, but was still trained and raised around the A-team, and was even overseen by the same coaches and managers, the bonus being it's also a lot less expensive. The only tricky part, of course, is that often the names of these second-label projects are completely different and don't include any mention of the famous winery or winemaker that you might be familiar with. Occasionally the label will read [Second Label Name] with the words 'by [First Label Name]' somewhere in smaller print in the corner, so be sure to look out for that. Otherwise, you're going to need to do a little research and\/or press the staff at your local shop for information.\n\n## 7. Don't be too quick to judge screw-top wines\n\nWhile the sentimental attachment to cork is understandable \u2013 after all, it's been the wine bottle closure of choice since the 15th century \u2013 we need to get over this ingrained prejudice towards its alternatives.\n\nJust because a wine is bottled with a screw-top closure, it doesn't automatically mean it's cheap and poorly made. Sure, there's reason to be cautious; screw tops are a popular choice with lower-quality, mass-production wineries because of cost-efficiency, consumer convenience and the fact that they eliminate the possibility of wine becoming 'corked'. But it's not bad logic, actually, and therefore there's no reason why you should discriminate against screw-topped wine if you like and trust the producer or, at the very least, the shop selling the wine. In countries such as Austria, Australia and New Zealand, screw tops are pretty much the standard across the board.\n\n## 8. Separate the genuinely useful and informative from the propaganda when it comes to shelf recommendations\n\nIf you're going to consult shelf recommendations (aka those little write-ups describing the bottle you're scoping out), be wary of the red flags. For example, a description that only talks about the wine's point scores. You could argue that a wine's 'point score' really says more about the winery's marketing budget than what's actually going on inside the bottle. So it got a 90-something rating. Great. Years in school have conditioned us to get excited about that kind of number but, when it comes to wine scores, what do you really even know about the marker or their scoring system anyway? Better to look for recommendations that read as personal and unique \u2013 bonus points if it's handwritten or if it has a staff member's name attached.\n\n## 9. Looking for a dry white? Check the ABV\n\nIf you only remember one thing about the science of wine let it be this: yeast + sugar = alcohol + CO2. That, friends, is the basic formula for the fermentation process. Yeast and sugary grape juice go into the tank; yeast eats the sugar, converts it into alcohol and carbon dioxide is expelled.\n\nNow, why does this matter to you, the dry white wine shopper? Because it tells you that the lower the ABV, the less sugar has been 'eaten' by the yeast and therefore the wine is technically sweeter. Higher ABV suggests more sugar has been converted to alcohol and that the wine will be dry (though this doesn't mean that it won't have fruity flavours and aromas, that comes down to the varietal and winemaking style). It's not a hard and fast rule, but, in a pinch, if you want an indication of whether or not the white you're considering will be dry, check the ABV. Anything in the 12% and above range should fit the category, while 10\u201311.5% will be semi-sweet and under 10% will be sweet. But be warned \u2013 this rule doesn't really apply to reds, which are all generally fermented to dryness, but can be fruitier and riper tasting when the alcohol level is high, or fortified wines, which have a neutral spirit added, and therefore can be sweet and boozy at the same time.\n\n## 10. Don't write off an entire grape variety or region just because you got burned once\n\nIn other words, be patient and open minded. So you bought a \u00a320 ($25) bottle of Chardonnay and didn't like it. Bummer. Instead of declaring that you 'hate Chardonnay' and writing off the varietal entirely, see what an expression from Burgundy tastes like, or California, or New Zealand. By that same token, don't let that one experience make you think that you won't ever like a Chardonnay \u2013 or any wine \u2013 from a particular region. Just because you didn't like how your Mum cooked broccoli as a child, it doesn't mean someone else hasn't worked out how to make it taste delicious, right? Do your best to take note of what wasn't working for you with that one bottle and then try to communicate that the next time you go shopping or are scanning a restaurant's wine list \u2013 what you learned from the first failed experience can lead to a successful second chance. Obviously, of course, if you try, try and try again and you're still not into it, then by all means file it in your 'This Wine May Not Be for Me' folder.\n\n# Guide to Glassware\n\nAt the risk of never getting an endorsement deal from a glassware company, I'll be honest: you don't really need any other type of wine glass beyond an all-purpose glass. Especially if you're just starting to get into wine.\n\nI understand the sense of occasion a flute brings, and certainly appreciate and value the important aeration provided by a Burgundy bowl. But if you only get to pick one, a clear, thin-lipped, 'cut-rim' all-purpose glass should be it. It's the wine glass equivalent of the perfect pair of jeans.\n\n### The All-Purpose Glass\n\nA stemmed glass with a U-shaped bowl that is wider at the stem and narrows slightly as it gets up to the rim.\n\n**Use it for:** Literally any style of wine, if need be, but typically it's good for most whites, ros\u00e9s, spicy red wines and non-Champagne sparklers.\n\n**Pros:** Extremely versatile. Plus, the average-sized glass allows for closer proximity to the juice for more of an up-close-and-personal appreciation. Emphasises acidity in wines and the reduced surface area\/smaller opening helps the liquid maintain a cooler temperature.\n\n**Cons:** May make it more challenging to pick up on the subtleties of a particular wine that benefits from coddling and special treatment, i.e. old, rare or more expensive wines from highly prized producers and wine growing regions.\n\n### Four more to consider\n\n### The Flute\n\nA stemmed glass with a narrow, small bowl that extends tall up to the rim. An off-shoot style of flute called a 'tulip' is becoming increasingly popular (picture an upside-down, pear-shaped bowl) thanks to its ability to better capture the aromas and flavours of sparkling wine.\n\n**Use it for:** All manner of sparkling wines, except perhaps p\u00e9tillant naturel wines.\n\n**Pros:** Classic design that subliminally screams 'special occasion'. Also, from a technical standpoint, the shape encourages the formation of a continuous stream of bubbles (though some might argue it does this too well, letting the carbonation dissipate faster than may be desirable).\n\n**Cons:** Less favoured among wine and restaurant industry types because it inhibits swirling the wine, and therefore makes deep-diving into the aromas more challenging.\n\n### The Burgundy Bowl\n\nEssentially a fishbowl on a stem, the signature of this glass is a large, wide bowl that noticeably narrows towards the rim.\n\n**Use it for:** Light-bodied, aromatic red wines, fuller-bodied white wines and skin contact (orange) wines.\n\n**Pros:** The big-bowl-but-narrow-opening shape provides essential aeration, and is therefore more effective at accumulating and showing off a wine's more subtle and delicate aromas. It also helps to play up the rounder texture of fuller whites.\n\n### The Bordeaux Glass\n\nBasically an XL version of an all-purpose glass. The bowl is larger and extends tall, tapering in just a touch at the lip of the glass.\n\n**Use it for:** Medium and fuller-bodied red wines with big tannins.\n\n**Pros:** By forcing more distance between your nose and the wine with its size, it helps temper the intensity of the alcohol of the typically boozy wines it's designed to hold. Also, the larger size of the glass allows the intense combination of alcohol, tannin, oak, acidity and ripe fruit in these wines to breathe and better integrate.\n\n### The Dessert Wine Glass\n\nLike a miniature version of an all-purpose glass, these squat glasses have a shorter stem and smaller bowl that narrows at the rim.\n\n**Use it for:** Ports, Madeiras, sherries and all varieties of dessert wine.\n\n**Pros:** Helps stunt evaporation and concentrate the high-octane dessert wines' sweet flavours and aromas. An added bonus is that these glasses are also good for tasting spirits.\n\n**Cons:** Potentially not worth the extra expense considering the category is consumed less frequently.\n\n# The 'Dos' and 'Don'ts' of Storing and Serving\n\n#### Storing\n\n#### DO\n\n**Store open bottles of wine in the fridge:** No worries if you can't finish that whole bottle from last night \u2013 as long as you cork it and keep it relatively cool, which helps slow down the spoilage process from oxygen exposure, it should last at least another day or two (maybe longer for red wine).\n\n**Find the coolest, darkest place in your house to store your wines:** Aside from exposure to oxygen, heat and light are wine's greatest enemies. If you don't have the money, space or need for a wine fridge, a spot in a (not-too-dry, relatively humid) cellar, hallway or kitchen cupboard should do just fine.\n\n**Store your wine horizontally:** Although somewhat of an old-school mentality \u2013 born of the days when the vast majority of wines were sealed with cork and therefore needed the contact with the wine to prevent it from drying out \u2013 it remains a solid practice. And if for no other reason, it's a more efficient, effective use of space.\n\n#### DON'T\n\n**Let your wine get too hot or too cold:** Store your wine anywhere that is hotter than 26oC (80oF) and it will cook, or colder than 0oC (32oF) and it will freeze or, worse, explode if it's a sparkling wine.\n\n**Expose your wine to radical temperature shifts:** Imagine going from the comfortable temperature of your home to baking in a hot, humid sauna, to jumping into a near-frozen lake. Traumatising.\n\n#### Serving\n\n#### DO\n\n**Chill light-bodied, low-tannin, aromatic red wines:** Aim for somewhere in the 10\u201313oC (50\u201355oF) range; it will make the wine taste more refreshing and help highlight fruit flavours and the brightness of a wine's acidity.\n\n**Invest in a good bottle opener:** Specifically, a 'waiter's friend' (or wine key) with a double-hinge fulcrum (the metal part). The flexibility of this style will help keep you from breaking the cork. And it's not an expensive tool, either \u2013 you can generally find them for under \u00a310 ($12) in a supermarket, cookery or wine shop.\n\n**Consider decanting your wine:** Traditionally used to separate sediment off the wine in older red bottles, decanting has become more fashionable and common across the board these days. The real benefit of the process is that it gently exposes the wine to air, softening its acids and tannin, effectively smoothing out all the sharp edges and wrinkles. It can be particularly helpful in taming the funk that comes off so-called reductive wines.\n\n**Smell your glasses before you use them:** As important as it is to make sure your dishware and cutlery are clean before you use them, so it goes with your glassware. One of the best, most enjoyable parts about drinking wine is appreciating its fragrance. It's best that there's no residual soap smell or dusty cupboard taint getting in the way.\n\n**Go straight through the wax on wax closures:** As nice an aesthetic effect as they have, wax closures can throw a slight curve ball as far as approach goes. Best not to treat it like a traditional foil closure and instead twist the screw straight through the top without trying to remove any of it beforehand. Once the cork is halfway out, you can do a little cleanup around the rim of the bottle to ensure that no residual wax bits go into the wine when you pull it completely out.\n\n#### DON'T\n\n**Serve your white wines too cold:** Why? Because you'll taste less. Light-bodied, less-aromatic wines should be served colder, in the 4\u20138oC (40\u201345oF) range, whereas more expressive and fuller-bodied whites should ideally be served in the slightly-warmer-but-still-technically cold range of 8\u201310oC (45\u201350oF).\n\n**Serve your red wines too warm:** Heat will exacerbate the alcohol content of a wine, making your nose hairs burn when you go in for a big whiff. Conventional wisdom generally dictates that light-to-medium bodied reds should be served around 13\u201315oC (55\u201360oF), while fuller-bodied reds are better enjoyed in the 15\u201320oC (60\u201368oF) range.\n\n**Smell the cork to verify that your wine isn't corked:** Instead, smell the wine itself! Taking a quick whiff over the bottle opening will suffice. If it doesn't smell like damp cardboard or wet dog, you're good. Because, at the end of the day, even a cork that isn't tainted still smells like cork, so really how helpful is it to smell it?\n\n**Take your thumb off the cork and cage when you're opening sparkling wine:** Opening a bottle of bubbly should look something like this: cut and remove the top of the foil. Put your thumb over the cage, twist open the cage tab (without taking it off), angle the bottle and rotate the base with one hand as the other wriggles the cork in the opposite direction, slowly pulling up. Keeping your hand on the cork until it's off is the best insurance policy against it violently popping off in potentially dangerous directions.\n\n**Overpour your glass:** Seriously, no matter how rough the day. If the glass is too full and you try to do your best professional swirl-and-sniff, chances are you're going to end up making a mess.\n\n**Let past-its-prime, open wine go to waste:** See if you can't find a recipe for a stew, or soup or sauce where it can be given a respectable second-chance at life.\n\n# Become a Better Taster: A Face-Off Drinking Game\n\nHere's the thing: the only way to get better at tasting wine is to taste more wine. This is not me trying to peer pressure you, it's just a fact.\n\nA lot of the important words we use to talk about wine can seem so abstract, such as 'earthy', 'fruit-forward' or 'tannic'. As such, I find that drinking examples of opposing concepts side by side can be extremely enlightening. The following are some pairs to try together.\n\n# Chardonnay\n\n# **Say It Right:** _Shar-duh-nay_\n\n**Notable Regions**\n\nBurgundy and Champagne in France, California, Oregon and Washington in the USA, New Zealand, Canada, Italy, Chile and South Africa.\n\n**Drink It With**\n\nEverything from oysters, light fish and shellfish if you're dealing with Chablis, to cream-sauced chicken with mushrooms if you're doing aged white Burgundy or Californian. Butter and cream-based sauces are your friend. Stick to seafood, chicken, veal, squash and root vegetables, and avoid anything spicy.\n\n**Key Words**\n\nOld World: You like **medium to full-bodied dry whites** \u2013 sometimes you want it to be **citrussy and mineral** and other times you want it to feel opulent and savoury, not fruit-forward. You are looking for a white that **gains interest with age**. New World: You're well-versed in **'oaky and buttery'** ; you like your whites **rich, round and creamy** , spiked with **baking spices and vanilla**.\n\n**You Might Also Like**\n\nMarsanne\/Roussanne blends, Viognier, Alsatian Pinot Blanc and Fiano.\n\nYou know that friend whose personality changes based on who they're dating? That's basically Chardonnay \u2013 it's called 'the winemaker's' grape' for a reason. The comparison is not intended as an insult; without question, it's one of the most popular wine grapes in the world.\n\nWhere a grape is grown, how it's treated in the vineyard and the vision of the winemaker are always influential factors. It's just that with Chardonnay, the shapeshifting can be quite dramatic. That's why when someone throws down the hammer and exclaims, 'I hate Chardonnay', often the next question a wine professional should ask them is, 'Okay, but where was this Chardonnay you hated from? California? France? Somewhere else entirely?'. We ask this not to be condescending, but because the differences in style are so dramatically different. It's totally possible for you to dislike a 'classic' oaky-buttery Napa Chardonnay, but be in love with a white Burgundy.\n\nThat this grape is so divisive is ironic given that, on paper at least, it's got such likeable qualities. It has a plush texture, moderate acidity, mild flavours of apple, pear and citrus, and is dry without being too aggressive about it. The love-it-or-hate-it thing ultimately comes down to who the grape is in a relationship with. Chardonnay has been going out with Burgundy the longest; since at least the 17th century. Together, the two have \u2013 and continue to \u2013 produce some of the most lauded, ageworthy white wines in the world. There, the varietal shows off its skill as a chameleon of style: sometimes round, fresh and easy at the basic village level, or lean with a racy mineral-citrus thread in Chablis; at other times, it has a rich opulence with a hint of savoury in bottlings from prestige regions like Meursault, Corton-Charlemagne, Puligny-Montrachet and Pouilly-Fuiss\u00e9.\n\nFor as synonymous as Chardonnay has become with the New World, they've really only been together since the 1980s when it exploded in popularity. This was a Chardonnay unlike the version we'd come to know from France. In the warmer climate of regions such as the west coast of the USA, Australia, South Africa and South America, the varietal displays more ripe tropical fruit flavours and the use of oak tends to be different, too. It's a misconception that Old World Chardonnay has nothing to do with oak \u2013 it actually does, but it presents differently. There, it's more toasted-nutty\/savoury-cheesy, whereas in the New World, it has flavours of caramel, butter, vanilla and baking spices.\n\nThe grape's other strongholds include cool-climate places such as New Zealand and Canada, which produce something more acid-driven, as well as northeastern Italy, which riffs on a French style. Let's not forget Chardonnay's crucial role in the sparkling wine world. It is one of three major grapes permitted in the production of Champagne, and is similarly used in Lombardy, Italy, to produce the Champagne-method Franciacorta sparkling wine.\n\n# Sauvignon Blanc\n\n# **Say It Right:** _So-veen-yawn Blonk_  \n **Other Known Aliases:** _Fum\u00e9 Blanc (Foo-may Blonk)_\n\n**Notable Regions**\n\nFrance (specifically the Loire Valley), New Zealand, Chile, South Africa, California in the USA, Australia and Italy.\n\n**Drink It With**\n\nRaw oysters; clams and mussels; prawns (shrimp); smoked salmon with cr\u00e8me fra\u00eeche and dill and halibut with lemon and fennel. Also, herb-heavy sauces, salads and artichokes and asparagus. Pairing with goat's cheese is also classic.\n\n**Key Words**\n\nYou want a **dry, light-bodied** white that's **heavy on citrus and herbs** and a **mineral** texture; are looking for something to **pair with vegetables and seafood** or **drink on its own** ; not pricey \u2013 would go to \u00a330\u201340 ($40\u201350) for a good bottle; like **stainless steel-fermented whites** ; the **opposite of Chardonnay**.\n\n**You Might Also Like**\n\nGr\u00fcner Veltliner, Verdejo, Albari\u00f1o, Vermentino, Erbaluce, Vinho Verde, Txakoli and Muscadet.\n\nWhile some may not find it easy being green, Sauvignon Blanc has made quite the successful career of it.\n\nFrequently cast as the antithesis to Chardonnay, this lean, crisp 'wild white' varietal is one of the most important and widely planted in the world, easily identified by its characteristic greenness. Its flavour profile reads like a head-to-toe outfit homage to the hue: green fruits (lime, gooseberry, green melon, green apple, kiwi), green vegetables (pepper, asparagus, jalape\u00f1o, sugar snap peas), green herbs, green tea ... and yes, you can even find hints of the colour as you admire Sauvignon Blanc in the glass.\n\nOf the grape's many notable HQs across the wine-growing world, its native Loire Valley in France is arguably the most iconic. There, in the famed regions of Sancerre and Pouilly-Fum\u00e9, it shines as a solo artist. Typically fermented and aged in stainless steel, these pristine, sharp-edged expressions are a celebration of fresh herbs, cool, textured minerality and freshly-squeezed citrus.\n\nFurther south in Bordeaux, Sauvignon Blanc's signature zip and freshness is used in partnership with plump S\u00e9millon to create the round, oak-aged dry whites of Entre-Deux-Mers and Graves, and, of course, the famous sweet dessert wines of Sauternes and Barsac. There's a considerable amount of it grown in the southwestern Languedoc-Roussillon, though it mostly ends up in light, tart blends, not much of which is all that exciting. Better to poke around elsewhere in Europe in areas such as Alto-Adige in northern Italy, and Styria in southeastern Austria, to find some interesting, racy, single-varietal examples.\n\nObviously, no conversation about Sauvignon Blanc would be complete without mention of New Zealand. The Marlborough region, specifically, has given Sancerre some competition with an iteration that coaxes out more tropical fruit notes (think ripe white peach and passion fruit) to add to the expected herbs and green vegetable mix.\n\nNot surprisingly, the USA's wine-producing juggernaut, California, also has considerable stake in the Sauvignon Blanc game. Originally known there as Fum\u00e9 Blanc, it's still made across Napa and Sonoma into a richer, fleshier, oak-aged style.\n\n# Riesling\n\n# **Say It Right:** _Ree-sling_\n\n**Notable Regions**\n\nGermany, Austria, France (specifically, Alsace), Australia, USA, Hungary and Canada.\n\n**Drink It With**\n\nLean, dry versions work well with shellfish and counterbalance a rich butter sauce; the off-dry to sweet expressions are great with fatty cuts of meat (pork belly!), fried foods and spicy dishes; and the super-sweet stuff is dynamite with seared foie gras, hard cheeses or desserts with stone fruits and honey.\n\n**Key Words**\n\nYou like citrussy, **acid-driven whites** ; are a **versatile drinker** , sometimes you want something super lean, dry, and mineral, and other times you're in the mood for sweet; are looking for wines that **pair easily with food** ; want to drink what the sommeliers drink.\n\n**You Might Also Like**\n\nSylvaner, Scheurebe, dry Furmint, Txakoli, M\u00fcller-Thurgau, Albari\u00f1o and Jacqu\u00e8re\n\n##\n\nThe World's Major Riesling Styles: the Elevator Pitch Edition\n\n**Germany**\n\nWhere you go to experience Riesling performing at the top level across all styles.\n\n**Austria**\n\nWhen you want a fuller-bodied Riesling that shoots laser beams of steely acidity.\n\n**Alsace**\n\nFor dry Rieslings with weight, power and richness.\n\n**Australia and New Zealand**\n\nAre you a fan of sucking on lemons and limes? Great.\n\n**New York State and Washington**\n\nThe American-accented version of what Germany does.\n\n**Canada**\n\nTry their super-sweet icewine version.\n\nAny actor who's ever been typecast understands Riesling's plight.\n\nIt has this impressive, well-rounded CV: one of the oldest and most important varieties in Germany; capable of producing incredibly age-worthy wines; exceptionally accomplished at expressing terroir; and, between its myriad styles, pairs with pretty much any dish or cuisine you can imagine. And yet, most of the time, when you say Riesling, the only thing people think of is something sweet, which is ironic, since the majority of Riesling made around the world is dry.\n\nPart of the grape's PR problem stems from the fact that it has such a vibrant fruit flavour profile. Expect to find all the citrus, stone fruit, orchard fruit, and sometimes tropical fruit, in the glass at the same time. So even when it's technically fermented to dryness, the intensity of that natural cocktail can give off an impression of sweetness. It also doesn't help that for a time, a lot of what was exported was designed to be a cheap, disproportionately sweet version of itself \u2013 the cloying taste of which scarred and scared off a lot of palates.\n\nAlthough many wine drinkers treat the word 'sweet' like some kind of offensive slur, in its native Germany, Riesling is celebrated for its ability to deftly slide up and down the sugar scale. At one extreme you have the searingly dry, linear _trocken_ expressions, typically offering visions of citrus zest, under-ripe peach and cool river rocks. _Kabinett_ and _Feinherb_ represent the off-dry options. These wines are equal measures of juicy, sweet and fresh, and their lower ABV puts them in the 'delightfully swiggable' category. At the other end of the spectrum are the decidedly sweet _Sp\u00e4tlese_ and _Auslese_ styles, which see the fruit take on a lush, honeyed, buttery quality. Lastly, the nectar-of-the-gods entries featuring botrytised (otherwise known as 'Noble Rot') grapes: _Beerenauslese_ and _Trockenbeerenauslese._ (Nerdy sidebar: again, it's worth noting that the terms for the sweeter styles technically don't refer to the sugar content in the wine but rather the ripeness level of the grapes at harvest.) The varietal's success as a sweet wine hinges on it being a natural powerhouse of acidity. It's what keeps even the sweetest styles from tasting syrupy. Not to mention that it helps the wines have such long lifespans. And on that note, here's a fun fact: sweet expressions will taste less so the longer they're aged, to the point where a 30-year-old Auslese Riesling might taste practically dry.\n\nBeyond Germany, Riesling's other two major European bases are Austria and Alsace, France. In the former, it's made almost exclusively in a dry style that's generally characterised as being slightly fuller bodied, a little spicy, and having a cutting, razor-sharp acidity. Alsace's prized Rieslings, while also mostly dry (some noble rot-infected grapes are used to make dessert wines) display a more rich, robust personality with distinctive mineral tones. In addition to that, Australian Rieslings from the Barossa, Eden and Clare Valleys are some of the driest you'll ever come across.\n\n# Chenin Blanc\n\n# **Say It Right:** _Shen-in Blonk_  \n **Other Known Aliases:** _Steen_\n\n**Notable Regions**\n\nLoire Valley, France, South Africa, Argentina, California and Washington state, USA.\n\n**Drink it With**\n\nDry and sparkling versions work with lemon-driven preparations of chicken and seafood, fried foods, and creamy sauces; off-dry ones favor boldly spiced Asian and Middle Eastern dishes; and pair sweet expressions with apple\/ pear or almond-based desserts and rich, savoury ingredients.\n\n**Key Words**\n\nYou like **light-bodied** wines with **bold aromatics** ; are into slap-you-in-the-face **bright acidity** ; appreciate honeyed, nutty flavours; like dry whites, sparkling and sweet dessert wines.\n\n**You Might Also Like**\n\nDry Riesling, Soave, Fiano, sous-voile Chardonnay or Savagnin from the Jura; Kabinett Riesling and Scheurebe from Germany for demi-sec; sweet Tokaji for dessert wine.\n\nWe've all taken one of those generic personality tests in which a series of multiple-choice questions is supposed to be able to accurately typecast you. You know, 'Please tick Box A if you're this way, Box B if you're that, or Box C if you're some other way entirely'. Chenin Blanc is the kind of grape that looks at that test, laughs, ticks all three boxes and then goes on about its business.\n\nIt's one of those rare, exception-to-the-rule varietals that is not just produced in a range of styles \u2013 dry, off-dry, dessert, sparkling, oxidative and fortified \u2013 but is actually praised for its proficiency in all of them.\n\nChenin Blanc is best known for extending its full range of motion in the middle of the Loire Valley, France. Its dry whites from the sub-regions of Vouvray, Saumur and Anjou are marked by flavours of apple, melon and pear, a punchy grapefruit-lemon acidity, and a unique undertone of white flowers, raw honey and hay. Its fine-beaded, frothy sparklings follow a similar flavour profile, but will frequently offer additional notes of toasted nuts and yeasty raw dough. These will either be produced under the regionally broad label, Cr\u00e9mant de Loire, or Vouvray, if made there specifically. Some of the most unique, 'geeky' expressions come from the sub-region of Savenni\u00e8res. In this mini-mecca for biodynamic winemaking, the grape is often treated in an oxidative style (i.e. exposed to oxygen during the winemaking process) that helps coax out a smorgasbord of complex flavours and textures: earthy, oily, mineral, woolly, nutty, bruised orchard fruit. And what's more, these wines seem to only get more interesting with age. Of course, let's not forget Chenin's crowning achievement in the region: those late-harvest, botrytised sweet wines that taste like ripe, gingered peaches, apples and quince that have been dipped in honey and skewered with a vibrant bolt of citrus. To find these, keep an eye out for small 187 ml bottles (warning: it's going to be pricey stuff for the quantity) that are labelled Vouvray moelleux, or come from the sub-regions of Coteaux de Layon, Quarts de Chaume and Bonnezeaux.\n\nOutside of the Loire Valley, Chenin shows off fewer of its multiple personalities. In South Africa, where it is the most widely planted variety in the country, Chenin has historically been produced as either a base wine destined to be distilled into brandy, or as a filler grape for blends. Increasingly, it's being treated with more attention and care, largely made in a lean, dry, citrus-and-mineral-forward style, though some warmer-climate expressions take on a riper, more concentrated quality. The grape's other stomping ground of note is California, USA, where, after being treated as nothing more than characterless swill in the 1970s and 1980s, it has experienced a recent resurgence among new, young producers committed to making it into a quality dry wine.\n\n# S\u00e9millon\n\n# **Say It Right:** _Sem-ee-yawn_\n\n**Notable Regions**\n\nFrance (specifically Bordeaux), Australia, California and Washington state in the USA, South Africa, Chile and Argentina.\n\n**Drink It With**\n\nIf it's dry \u2013 oily smoked fish; chicken or pork with fruit; seafood stews. If it's sweet \u2013 foie gras and rich p\u00e2t\u00e9s; baked fruit desserts; cheeses such as Roquefort and aged Cheddar.\n\n**Key Words**\n\nDry: You like **fuller-bodied, not too acidic** wines with an **herbal quality**. Sweet: You **appreciate ripe orchard fruit flavours** with a **honeyed edge**.\n\n**You Might Also Like**\n\nViura, Garganega and Friulano if you like it dry; if you like it sweet, try late-harvest Riesling from Germany or Austria, sweet Tokaji, S\u00e9lection de Grains Nobles (SGN) of Riesling, Muscat or Pinot Gris from Alsace, Vouvray Moelleux and Quarts de Chaume.\n\n**Sauternes Alternatives**\n\nFor those occasions when you don't necessarily want to splurge on Sauternes, change-your-life-delicious though it may be, look for the more affordable S\u00e9millon-based dessert wines from the neighbouring regions of Montbazilliac, Cadillac, Loupiac and Cerons.\n\nWhen it comes to wine grapes, being a star doesn't necessarily mean you're going to be a household name. Often, the fame and acclaim go to the wine (or its associated region) and not the grape(s) from which it's made; one of those classic 'the whole is greater than the sum of its parts' situations. Which is why Sauternes \u2013 one of the most well-known and revered dessert wines in the world \u2013 probably rings a bell, but S\u00e9millon does not.\n\nAs a single varietal, this unsung hero tends to produce round, full-bodied whites with a mellow citrussy-peachy tinge (read: low-acid), and aromatics that range from beeswax and wool to honey, white flowers and green figs. When made in a less-ripe expression, it could very well act as a Sauvignon Blanc understudy, offering a bit more mouth-watering green apple and grapefruit tang, and a nose of wild green herbs. All that being said, S\u00e9millon's work as a solo artist is rarely the point. Aside from some few-and-far-between examples in places such as California and Washington state, USA, and South Africa, the grape's only real notable stand-alone performance comes from southern Australia's Hunter Valley. There, it's famously produced in a low-alcohol, high-acid, dry style that's designed for aging and is prized for the signature toasty-roasted, nutty flavour it develops.\n\nOtherwise, S\u00e9millon's greatest hits come from its roles in group acts. Without question, the grape's most important growing region is Bordeaux, in southwestern France, where it has long been partnered with Sauvignon Blanc to make some pretty exceptional dry and sweet whites. As far as dry whites are concerned, their partnership most commonly lives under the label of Bordeaux Blanc. At their simplest, these wines are a great, anytime-of-the-week option for dry white wine drinkers who prefer a fuller texture and appreciate good value. But they achieve their fullest potential in the Pessac-L\u00e9ognan sub-region, where, with the influence of oak and age, they are transformed into best-in-show beauties with an intoxicating combination of smoky, tropical fruit, beeswax, wet wool and a slick, oily texture.\n\nIn these dry Bordeaux whites (and their New World imitators), the S\u00e9millon to Sauvignon Blanc ratio slides the scale. But in what is arguably their most iconic collaboration, it is S\u00e9millon that takes centre stage. The thin-skinned grape is highly susceptible to botrytis, the coveted strain of so-called 'noble rot' responsible for making grapes raisinate, concentrating their sugars and sweet fruit flavours. As such, S\u00e9millon typically makes up the lion's share of the blend \u2013 usually 4:1 \u2013 in the honeyed, lush, zesty dessert wine, Sauternes. It's also worth mentioning that the dessert wines of Barsac, a sub-region of Sauternes, are equally fantastic, slightly lighter in body and with a punchier acidity.\n\n# Gew\u00fcrztraminer\n\n# **Say It Right:** _Guh-vertz-trah-meener_\n\n**Notable Regions**\n\nFrance (specifically Alsace), Germany, Italy, Austria, New Zealand, Northern California, Oregon and New York state's Finger Lakes region in the USA.\n\n**Drink It With**\n\nAsian, Middle Eastern and Indian cuisines, the more complex the heat and spice profile the better (read: bring on tagines and curries); soft cow's milk cheeses; smoked meat (duck, pork, beef) and fish such as salmon and trout.\n\n**Key Words**\n\nYou like **richer, fuller-bodied, fruit-forwards** whites; are not afraid of **floral aromatics** or a little sweetness; **don't need a lot of puckery acidity** in your wine.\n\n**You Might Also Like**\n\nMuscat, Torrontes, Alsatian Pinot Gris, warm-climate Viognier, Juran\u00e7on, H\u00e1rslevel\u00fa and Moschofilero (if you want something floral but on the drier, leaner side).\n\nIf you've ever been smacked in the face by the perfume of someone who doesn't know when to stop spritzing, then you have a pretty good sense of what it's like to stick your nose in a glass of Gew\u00fcrztraminer. It's heady stuff, with loud, distinctive aromatics that are all too eager to leap out and grab your attention. Lychee! Rose petals! Guava! Tangerine! Grapefruit! Apricot! Peach! Pineapple! Allspice! Incense! Ginger! Honey! Looking for a good gateway grape to start practising blind tasting with? Gew\u00fcrtz, as it's 'more simply' called, is the one.\n\nLike anything with a big personality, the reaction to the wines made from this grape tend to be love it or hate it. That being said, you'll certainly find a lot of love for Gew\u00fcrtz in Alsace, arguably the grape's most important growing region. In this northeastern corner of France they've even gone so far as to anoint it 'noble grape' status (along with Riesling, Pinot Gris and Muscat), meaning it can be planted on the region's Grand Cru vineyards (aka the best sites). It is from these sites that the gold-standard expression of the grape is produced. Outside of Alsace, you're likely to find the wines of this cool climate-loving grape in the neighbouring Pfalz region of Germany, Italy's Alto Adige, where it is alleged to have originated from, Austria, New Zealand and Northern California and Oregon in the USA.\n\nNow, it's easy to read through that list of Gew\u00fcrtz's very tropical-leaning fruit characteristics and automatically pigeonhole it as a sweet wine. Wrong. Well, sort of. In Alsace, in special years when the growing conditions are just right, it can be used to make two types of highly prized \u2013 read devastatingly delicious and quite pricey \u2013 dessert wines called Vendanges Tardives and S\u00e9lection de Grains Nobles (VT and SGN, for short). But the truth of the matter is, the majority of Gew\u00fcrztraminers are fermented to dryness. While the sweet smell of various fruits may be there, the sugar in the initial grape juice has almost entirely been converted to alcohol.\n\nThe admittedly tricky thing with Gew\u00fcrtz is that because it's a fuller-bodied grape without a lot of natural acidity to counteract its richer, oilier texture, these wines often taste a little sweeter than they technically are. It's a quality that perhaps makes it a little intense to drink on its own, but dynamite with spicy, boldly flavoured Asian food (think Thai, Vietnamese, Szechuan and Indonesian).\n\n# Marsanne and Roussanne\n\n# **Say It Right:** _Mar-sahn and Roo-sahn_\n\n**Notable Regions**\n\nFrance (specifically the Rh\u00f4ne Valley), California, Washington and Oregon in the USA and Australia.\n\n**Drink It With**\n\nRoast chicken or grilled pork; shellfish like lobster and crab (or crab cakes!) served with a rich, buttery sauce; oily, silver-skinned fish (eel, mackerel, sardines) and smoked fish (salmon, trout) would be a good texture match, as well as something meaty, such as swordfish; risotto and baked or roasted preparations of squash.\n\n**Key Words**\n\nYou like white wines with **richness** , weight and a not-so-subtle amount of booze; think that a wine with a **waxy texture** sounds interesting; think acidity belongs in more of a background role; prefer **ripe tree fruit** flavours and the smell of white flowers.\n\n**You Might Also Like**\n\nViognier, richer expressions of Chardonnay and Grenache Blanc.\n\nMarsanne and Roussanne don't just have a good, solid working relationship, they're besties. Akin to a together-all-the-time, can't-talk-about-one-without-mentioning-the-other type of closeness. The kind of friends that complement and bring out the best in one another, like macaroni and cheese, or jam and clotted cream.\n\nThe dynamic duo's most renowned collaborations can be found in the northern half of France's Rh\u00f4ne Valley. Although the area is better known for its Syrah-based reds, the towns of Saint Joseph, Hermitage and Crozes-Hermitage dedicate a small percentage of their production to whites made from a blend of the two grapes. These rare wines tend to be fuller-bodied, with an opulent, borderline oily texture and notes of ripe tree fruits (apple, pear, quince), honey, honeysuckle and almond.\n\nMarsanne, being the more vigorous grower of the two, often makes up the lion's share of the blend, contributing weight, alcohol and a deep golden-yellow colour to the wine. Roussanne's part is admittedly smaller, but by no means insignificant. The more delicate varietal, it has a naturally higher acidity, providing much needed lift and freshness to the blend, and is also spicier and more aromatic, often lending a fragrance of alpine herbs and white flowers. Also in the Rh\u00f4ne, the pair teams up to make the nutty, brioche-y sparkling wine from Saint-P\u00e9ray, and contributes to the blend for white C\u00f4tes-du-Rh\u00f4ne.\n\nForced to single one out as more independent, you'd probably have to go with Marsanne, if for no other reason than its ability to grow easily has contributed to a larger body of work outside of the partnership. You can find it everywhere from Provence's Cassis region (unique for producing more white wine than ros\u00e9) to Switzerland, where it's used to make both dry and sweet wines, as well as California in the USA and the Victoria region of Australia. Roussanne, meanwhile, has the distinction of being one of only 13 grapes allowed to be included in the prized wines of the southern Rh\u00f4ne's Ch\u00e2teauneuf-du-Pape region. It is also made in very small amounts as a single varietal wine in Savoie, in eastern France, as well as in California and Oregon in the USA. Of course, that being said, many of these other regions \u2013 namely Australia and USA \u2013 will end up reuniting the bosom buddies in Rh\u00f4ne-inspired blends.\n\n# Pinot Gris\n\n# **Say It Right:** _Pee-no Gree_  \n **Other Known Aliases:** Pinot Grigio _(Pee-no Gree-jee-oh)_ , Grauburgunder _(Gr-ow-berr-gun-der)_\n\n**Notable Regions**\n\nAlsace in France, Italy, Germany, Hungary, Switzerland, California and Oregon, USA.\n\n**Drink It With**\n\nIf you're drinking Italian expressions, go for light dishes: pan-fried or steamed white fish and shellfish; salads. With the riper versions, you can play with bolder flavours: roasted and fried chicken or fish; cured meats; smoked cheeses and spicy dishes.\n\n**Key Words**\n\nPinot Grigio drinkers, you like **dry, light-bodied, neutral whites** with nice acidity and **soft, slightly peachy flavours** ; want something you can **drink on its own** without food. If you're a fan of the Alsatian style, you're more open to **juicier tree and stone fruit** and some **floral notes**.\n\n**You Might Also Like**\n\nFor the lighter style, try Trebbiano, Arneis and Vernaccia, or Alsatian Pinot Blanc, Verdejo and Chasselas for something fuller.\n\nWine and the property market (real estate) have a lot in common. With both, it's amazing to see just how much location can alter your perspective and define your expectations. The same-size, same-price flat will no doubt look very different from one city \u2013 or even simply one neighbourhood \u2013 to another. Much in the same way, ordering an Italian Pinot Grigio is going to be a very different experience from ordering a Pinot Gris from Alsace or Oregon, despite the fact that, genetically speaking, they're all made from the same grape.\n\nIn France, this pink-skinned, white mutation of Pinot Noir (confusing, I know) does its best-known work in Alsace. There, the 'noble' varietal produces fuller-bodied, plush whites that deliver flavours of ultra-ripe peach\/apricot\/ nectarine, a balancing lemon acidity and a subtle flash of smoke and spice. In top years, you'll see some of it reserved for the production of the region's prized late-harvest dessert wines: Vendanges Tardives and S\u00e9l\u00e9ction de Grains Nobles. Think of the flavours of the dry style on steroids, layered with honey and wrapped with a rich, buttery bow, and you get an idea of what makes these intense sweet wines such a treat.\n\nWhile Germany, Switzerland and Hungary tend to follow Alsace's stylistic direction, producing either full, fruit-forward dry whites or sweet wines (or both), Italy has envisioned an entirely different identity for Pinot Gris. As Pinot Grigio, it presents as an almost unrecognisable version of its French or German self. Instead of round and voluptuous, it's light and lean; the characteristic flavour of mature stone fruit is replaced with mellow, muted tones of citrus, melon and apple, and instead of a bold, love-it-or-hate-it personality, its wines are generally easy and neutral. Production of the grape is largely concentrated in the northeastern part of the country.\n\nThe regions of Veneto and Lombardy mostly churn out high volumes of inexpensive and inoffensive expressions, while in Alto Adige and Friuli Venezia Giulia, you can find more characterful, complex, small-production versions.\n\nOver in the New World, Pinot Gris has made quite a name for itself in Oregon, USA, where it is the most planted varietal in the state. Here, it tends to produce medium-bodied dry wines that fall somewhere between an Alsatian and Italian style, and are generally presented as a more aromatic, fruit-driven alternative to Chardonnay. And in other places where it's popular, such as New Zealand or California, USA, you use the producer's choice of grape name as an indicator of the style of wine: Pinot Grigio if it's on the lighter-brighter end, and Pinot Gris if it's fuller, riper and more aromatic.\n\n# Pinot Blanc\n\n# **Say It Right:** _Pee-no Blonk_  \n **Other Known Aliases:** Weissburgunder _(Vice-burg-under)_ , Pinot Bianco _(Pee-no Bee-ahn-ko)_\n\n**Notable Regions**\n\nNortheastern France (Alsace, Moselle), Luxembourg, northeastern Italy, Germany, Austria, Hungary, Slovenia, Canada and Oregon, in the USA.\n\n**Drink It With**\n\nLightly sauced white fish and shellfish; simple omelettes and quiches; roasted chicken or chicken salad sandwiches and vegetables.\n\n**Key Words**\n\nYou like **easy, neutral white** wines; don't want too much acid, body or fruit flavour.\n\n**You Might Also Like**\n\nBasic-level French Chardonnay (Bourgogne Blanc, M\u00e2con-Villages or Beaujolais Blanc), Verdicchio, Sylvaner, Oregon Pinot Gris and California Sauvignon Blanc.\n\nThink of Pinot Blanc as the mini-van of varietals: the safe bet you pick for its utility and affordability.\n\nA white mutation of Pinot Noir, the grape is often described in very middle-of-the-road terms. It's light to medium-bodied, dry, with a mellow acidity and round, soft flavours of apple, pear and white peach. More often than not, it's presented as a kind of basic, less ambitious option for fans of unoaked Chardonnay. Even its best-known wines rarely get more than a modest applause. In Alsace, it makes a lot of easy, still whites and is an important component in the blend for the region's Champagne-method sparkling wine (aka _Cr\u00e9mant d'Alsace_ ). But it hardly competes in the same league as wines made from Riesling, Gew\u00fcrztraminer and Pinot Gris. You can also find significant amounts of it in the northeastern Italian regions of Friuli Venezia Giulia and Alto Adige, where it is known as Pinot Bianco. Here again, it is made in a light, Chardonnay style, or blended with Chardonnay to make sparkling wine.\n\nIn Germany, Weissburgunder is treated with a little more respect. In addition to being made in the dry, tart style, it is also being produced as a riper expression with light oak aging to add complexity. And in Austria, beyond making crisp whites, the susceptible-to-botrytis varietal is also behind some exceptional sweet wines.\n\nAnd if nothing else, it's prolific, also being grown in Luxembourg, Switzerland, Hungary, Slovenia and the USA.\n\n# Viognier\n\n# **Say It Right:** _Vee-own-yay_\n\n**Notable Regions**\n\nFrance, Australia and Washington and Oregon in the USA.\n\n**Drink It With**\n\nRich, heavily sauced dishes, because it has the weight and texture to handle it. Think fish, shellfish or white meats served with butter and cream sauces. Also plays nicely with heat and spice, so try it with an Indian or Thai-style curry.\n\n**Key Words**\n\nYou want a rich, full-bodied white wine and can appreciate an **oiler texture** ; you like whites with **bold aromatics** , but more specifically, you're into a **floral bouquet**.\n\n**You Might Also Like**\n\nFuller expressions of Chardonnay, full-bodied white Riojas, Gew\u00fcrztraminer (for fans of the Australian style), Torrontes, Muscat (if you're into its floral side) and Marsanne\/ Roussanne blends.\n\nI know the song says not to call it a comeback, but when you're a grape that goes from near extinction 50 years ago to becoming so trendy that you're now planted throughout the wine-growing world, what other possible word is there for it?\n\nThe varietal is known for producing high-alcohol, full-fleshed wines with moderate acidity, a unique oily texture and potent aromatics that range from ripe peach and apricot to orange rind, roses and herbal spice. Today, it is grown throughout the Rh\u00f4ne and the Languedoc-Roussillon in France, where it is featured in white blends alongside Marsanne, Roussanne, Grenache Blanc, Bourboulenc and Vermentino. But the best expressions come from the tiny, historic appellations of Condrieu and Ch\u00e2teau-Grillet in northern Rh\u00f4ne. And it is the striking personality of these wines that deserves much of the credit for piquing interest.\n\nOver in the USA, it was the Rh\u00f4ne wine enthusiasts in central California that were the most notable early adopters, experimenting with it not only on its own and in white blends, but also as a colour and texture-enhancing minor addition to Syrah reds (a practice borrowed from some northern Rh\u00f4ne producers). The technique is also popular in Australia, although the grape is also vinified on its own into a more tropical-fruited version of its floral self. Otherwise, you can find renditions from New Zealand, Chile, Argentina and South Africa, as well as some unique examples coming out of Italy's Piedmont and Tuscany.\n\n# Albari\u00f1o\n\n# **Say It Right:** _Alba-reen-yo_  \n **Other Known Aliases:** Alvarinho _(Alva-reen-yo)_\n\n**Notable Regions**\n\nSpain (specifically Rias Baixas), Portugal and California in the USA.\n\n**Drink It With**\n\nSeafood, duh. So much could work: ceviches and crudos; pan-seared scallops or a white fish fillet dressed with lemon and good olive oil; steamed lobster; clams steamed with garlic and herbs or linguine alla vongole; cold seafood salad; oysters on the half shell and brandade tartine. Vinaigrette-dressed salads and grilled green vegetables would also be a great match.\n\n**Key Words**\n\nYou like **lean, light-bodied** whites; **_love_ acidity**; appreciate the mouth-watering effect of **salinity** ; are looking for **something (not too expensive) to drink as an ap\u00e9ritif or with seafood.**\n\n**You Might Also Like**\n\nSauvignon Blanc, Assyrtiko, Vinho Verde, Gr\u00fcner Veltliner, Muscadet, Jacqu\u00e8re, Txakoli and Austrian Riesling.\n\nMost would be offended at being called out for having a 'salty' personality. But, in the case of Albari\u00f1o, not only is the adjective not intended as a character flaw, it's actually one of the varietal's biggest selling points.\n\nNative to northwestern Portugal, the grape is still important there. While some single varietal examples exist, it's mostly featured as a second-string player in the light, naturally spritzy dry white blends of Vinho Verde. Where Albari\u00f1o has really made a name for itself, however, is across the border in Rias Baixas. In this northwesternmost corner of coastal Spain, the grape is famous for producing vibrant, fresh, slightly floral whites that evoke a kind of lemon-grapefruit-honeydew melon punch that's been spiked with minerally salt water.\n\nIt's precisely this signature saline quality that makes it such a perfect, mouth-watering ap\u00e9ritif wine and so devastatingly good with simple preparations of just-caught seafood (\u00e0 la 'grows together, goes together' methodology). Not to mention generally affordability in relation to quality has made it even more endearing. To experience the top, most 'serious' versions of Albari\u00f1o, look for the words _'cepas vellas'_ on the label, which indicates that the wine is produced from grapes grown on older vines \u2013 more than 200 years old in some cases.\n\nThe growing popularity of this varietal has inspired producers in American regions such as California and Oregon to try their hand at producing it in both traditional and non-traditional versions.\n\n# Gr\u00fcner Veltliner\n\n# **Say It Right:** _Groo-ner Velt-lee-ner_\n\n**Notable Regions**\n\nAustria, Slovakia, Czech Republic, New Zealand (specifically Central Otago) and Oregon and California in the USA.\n\n**Drink It With**\n\nChicken and pork (especially done schnitzel-style); shellfish and white fish; nigiri-style sushi; green vegetables, (even asparagus!); preparations influenced by citrus and pepper; and for fuller, riper expressions try Thai and Indian cuisine.\n\n**Key Words**\n\nYou like **light-bodied, lean** whites with crisp acidity; enjoy **citrus, spice, green veggie** and herb flavours; are a **fan of screw-top wines** and want something friendly that goes with just about anything for just about anyone.\n\n**You Might Also Like**\n\nSauvignon Blanc, Albari\u00f1o, Muscadet, Vermentino, dry Riesling, Aligot\u00e9, Vinho Verde and Erbaluce, Muscadet, Jacqu\u00e8re, Txakoli and Austrian Riesling.\n\n**Style Indicator**\n\nMost Austrian Gr\u00fcner follows the same model: light, dry, with a bright citrus acidity and peppery bite. Assume that those in one-litre bottles with a screw or bottle top closure are on the less complex side, intended for no-frills, everyday drinking. If it has a cork closure, it's probably a more 'serious', expensive expression (think single vineyard from a prestigious producer) and is a good candidate for aging and special occasions. Though that is not to say that high-quality Gr\u00fcner cannot be found under a screw top; it is the primary closure of choice in the region.\n\nMost of Austria's 'top' Gr\u00fcners come from the sub-regions of Kremstal, Kamptal, Traisental and the Wachau. Of those, the Wachau is the boss. Its wines (which are more expensive) have unique style classifications. _Steinfeder_ for the light, electric-acid Gr\u00fcners, Federspiel for the dry, refined, and medium-bodied expressions, and _Smaragd_ for those that, while technically still dry, are more ripe, opulent and complex.\n\nMost of the time we talk about a grape as being the most valuable player (MVP) of some (relatively) small, specific sub-region: Chardonnay and Pinot Noir in Burgundy, Pinot Grigio in Friuli, Cabernet Sauvignon in Napa or Bordeaux. But, in Austria, Gr\u00fcner Veltliner is the nationwide superstar.\n\nObviously, it's not the only white grape planted in the country; varietals such as Sauvignon Blanc, Weissburgunder (Pinot Blanc), Gelber Muskateller (Yellow Muscat) and Furmint also grow there. Not to mention their Rieslings rank among the best in the world. Still, the blunt fact remains that Gr\u00fcner is Austria's most widely planted varietal \u2013 and that's accounting for both white and red \u2013 making up literally a third of the country's vineyard acreage. It's a big deal.\n\nThe irony, of course, is that for as big of a star as Gr\u00fcner is in its homeland, it wasn't until fairly recently \u2013 say, within the last three decades \u2013 that the rest of the world really started to fall for its charms. For example, in some instances, it's easy, light and crisp, impossible not to like, and in others, intense, complex and rich.\n\nGr\u00fcner is also remarkably food-friendly, thanks to its compelling mix of tangy citrus, apple, melon and pear flavours, a cutting, stony mineral edge, and a distinctive floral, white pepper and green vegetal vibe (peas, herbs and rocket (arugula) often come to mind).\n\nIt's comfortable on the table with just about anything: from a spread of sushi, to fried foods (schnitzel and spaetzle if you're going for the hometown favourite), or perhaps even spicy Asian cuisine or a rich stew if you're opting for a riper expression of the grape. And, for the record, it's also one of the few wines that's thought of as a successful match with that ever-so-challenging-to-pair vegetable, asparagus. On top of it all, not only are the leaner, more acid-driven styles an excellent 'other white meat' option when you've maxed-out on Sauv Blanc, but in general, the grape tends to deliver great value for the quality. Plus, the vast majority are bottled with a screw-top closure, so don't stress if you forgot to bring your opener to the picnic \u2013 did I mention it's a great picnic wine?\n\n# Muscat\n\n# **Say It Right:** _Muss-kat_  \n **Other Known Aliases:** Moscato _(Mo-skah-toe)_ , Moscatel _(Moss-ka-tell)_ , Muskateller _(Moose-ka-teller)_ , Zibbibo _(Zee-bee-bo)_\n\n**Notable Regions**\n\nFrance, Germany, Italy, Greece, Spain, Portugal, Austria, Germany, USA and Australia.\n\n**Drink It With**\n\nDesserts and pastries with fresh, dried or roasted fruits, nuts, caramel or honey; dark chocolate; foie gras; a cheese platter with aged Cheddar, Gouda and stinky blues.\n\n**Key Words**\n\nYou like **light-to-medium-bodied** , highly **aromatic whites** that deliver a **wallop of fruit and floral flavours** ; are **cool with sweet wines** , or even wines that smell like sweet fruit.\n\n**You Might Also Like**\n\nGew\u00fcrztraminer, Torront\u00e9s, Alsatian Pinot Gris, M\u00fcller-Thurgau and Malvasia.\n\nIt may not get the attention or mass-market recognition of an industry darling like Sauvignon Blanc or Chardonnay but, make no mistake, Muscat is an OG in the world of wine grapes. In fact, with a history that dates back to the ancient civilisations and thought to be the first domesticated grape, most would probably call it the Original Gangster.\n\nThe first thing to know about Muscat is that it isn't simply a one-man show; the name actually refers to an entire posse of grapes. Of the hundreds of different strains that fall under the Muscat umbrella, the best known and most well travelled is Muscat Blanc \u00e0 Petit Grains. A bit of a Renaissance man, it's produced in a wide range of styles across the wine-growing world. In northeastern Italy, for example, it's famously made into the sweet, delicately effervescent wine known as Moscato d'Asti. It functions similarly in the northern Rh\u00f4ne, where it is blended with Clairette to make the fruity sparkling Clairette de Die. While in places such as Germany and Austria, it's presented as a dry, still white. And yet in Greece, southern France, other parts of Italy and Portugal, it's transformed into sweet and fortified wines.\n\nDessert wine is a popular vocation among the Muscat siblings, which tend to share an intense aromatic profile of ripe stone fruits, white flowers, sweet citrus and fresh grapes (a flavour surprisingly uncommon in wine-producing varietals). Muscat of Alexandria is particularly famous for its renditions in Sicily, as well as in parts of Spain, Portugal, Greece and California, USA. Interestingly in Chile, the variety is used to make the base wine that is eventually distilled into Pisco, the country's national spirit. Although its reach isn't as wide, Muscat Ottonel is another significant one to know. While it does the expected sweet wine thing in Lower Austria, Hungary and Romania, it's better recognised for the heady dry to off-dry whites it produces in Alsace.\n\nIt's also worth mentioning that Muscat isn't an exclusively white-skinned varietal. A red mutation of Muscat \u00e0 Petit Grains, (unfortunately) called Brown Muscat is the grape behind Liqueur Muscat, a super-sweet, dark fortified wine from Australia. And there's also Muscat of Hamburg, the almost black-coloured strain used to make dessert wines in California and in small pockets across Europe.\n\n# Trebbiano\n\n# **Say It Right:** _Treb-EE-ah-no_  \n **Other Known Aliases:** Ugni Blanc _(Oo-nee Blonk)_\n\n**Notable Regions**\n\nFrance, Italy, Portugal and South America.\n\n**Drink It With**\n\nSimply prepared seafood dishes; crudit\u00e9; mild washed-rind cheeses; simple green salads dressed in light vinaigrette.\n\n**Key Words**\n\nYou're looking for an **easy** white with a mellow, **middle-of-the-road personality** ; not too particular, you just want something that is **dry, not too heavy or acidic and inexpensive**.\n\n**You Might Also Like**\n\nPinot Blanc, Verdicchio, basic-level unoaked French Chardonnay and Friulano.\n\nIf nothing else, you have to give it up to Trebbiano for crushing it as an everyday hero. The grape (of which there are several strains) rarely gets any mention or attention, but that does not mean its role in the wine industry is insignificant.\n\nThe most prolific of the grape's many varieties is Trebbiano Toscano. While you have probably never heard of it \u2013 or maybe just once or twice in passing \u2013 the varietal is actually responsible for producing more wine than any other in the world. Of course, the unfortunate reality of this fact is that a lot of the light, neutral, tart and generally not-that-exciting wine it makes is either distilled or processed. In France, under the pseudonym Ugni Blanc, it is used as a major (if not solo) component in the base wine that eventually becomes Cognac and Armagnac. And in northeastern Italy, juice from the freshly harvested grapes is used, often with the red wine grape Lambrusco, to produce the region's signature balsamic vinegars.\n\nEven in its strictly wine-only roles, Trebbiano mostly plays a fade-into-the-background part. In southwestern France and throughout central and northeastern Italy, it's nothing more than a chorus member in leagues of fresh, easy and unassuming white blends. That being said, if you do happen to be interested in seeing what the grape can achieve on its own, the best examples can be found from some serious-minded producers in Abruzzo.\n\n# Assyrtiko\n\n# **Say It Right:** _Ah-seer-tee-ko_\n\n**Notable Regions**\n\nGreece (specifically Santorini) and Australia.\n\n**Drink It With**\n\nLight, simple, citrus-driven seafood dishes (white fish and all manner of shellfish, especially); fried fish; pasta with a light cream-based sauce; raw summer vegetable salads; salty, creamy cheeses such as feta.\n\n**Key Words**\n\nYou like **lean, dry, lighter-bodied** white wines with a **citrus and mineral backbone; coastal and island white wines** ; are not afraid to go out on a limb and try a grape you haven't heard of before.\n\n**You Might Also Like**\n\nChablis, Muscadet, Pigato, Austrian Riesling, Soave, Vermentino, Albari\u00f1o, Txakoli and Vinho Verde.\n\nOvercoming a stigma can be a long, challenging and arduous road for a wine. Just ask Riesling. In this case, however, it's not so much that Assyrtiko specifically has had to fight to earn people's respect, but rather that Greek wines in general haven't been taken all that seriously.\n\nThe upturned nose brush-off stems from a period of time several decades ago when the only Greek wine people outside of the country were familiar with was Retsina. Basically, this ancient style of white wine, which is fermented with Aleppo pine tree resin and gives off an, err, 'distinctive', aroma of turpentine, didn't exactly help Greek wine win any popularity contests \u2013 especially since the pungency of the style had long been used to try and mask the flavour of poorly made wine. Long story short: it didn't work. While the divisive style is supposedly on the verge of a comeback, the point is that wine drinkers today do actually have a growing interest and appreciation for Greek wines, and that credit goes to Assyrtiko.\n\nNative to the island of Santorini, Assyrtiko has essentially become the gateway grape of Greek wines, and it's pretty easy to understand why. Just consider the sales pitch: it's a lean, dry white with fierce waves of lemon-lime-tart apple acidity, smoky, volcanic rock minerality and a refreshing sea breeze salinity. To summarise: a Chablis that's gone on vacation to a Mediterranean island (and, on average, much less expensive than its French doppelg\u00e4nger, even the top bottlings).\n\nThough mostly intended to be enjoyed as a young, fresh wine, several of the grape's best winemakers are also experimenting with producing the grape in an age-worthy style, taking on a little more heft and savoury characteristics.\n\nFor the most part, Assyrtiko doesn't have a life outside of Santorini. And, I mean, have you seen the place? I get why you might not want to. Still, it's ability to retain its vibrant acidity in warmer climates has resulted in a number of Greece's mainland producers bringing the grape over to work with.\n\nInterestingly, the grape has just recently been successfully produced by one of the better-known winemakers in Australia's Clare Valley. So who knows, perhaps more trips abroad await in Assyrtiko's future.\n\n# Melon de Bourgogne\n\n# **Say It Right:** _Mell-ahn duh Boor-goyn_\n\n**Notable Regions**\n\nWestern Loire Valley (specifically Muscadet) and Oregon in the USA.\n\n**Drink It With**\n\nRaw oysters (literally their most match-made-in-heaven partner aside from Champagne); simple, light preparations of seafood and shellfish and traditional nigiri and sashimi.\n\n**Key Words**\n\nYou're looking for something to pair with seafood and you want to keep the price-point reasonable; want something **light, dry and friendly** that has strong **citrus and mineral** tones.\n\n**You Might Also Like**\n\nAligot\u00e9, Pinot Blanc, Verdicchio, Gr\u00fcner Veltliner, Friulano, Pigato and Txakoli.\n\nIn the world of wine, it's rare for a grape to be produced in a single region, for a singular purpose. But indeed, if Melon de Bourgogne was a pony, then making the crisp, mineral, soft-personality whites of Muscadet is its one big trick.\n\nThough the grape appears to have originated somewhere near Burgundy (hence the 'Bourgogne'), it was never exactly a hometown hero. It wasn't until it landed in the western Loire Valley that the grape 'found itself', so to speak. Never one to be the loudest voice in the room, this simple, fresh, briny white developed an identity and reputation based largely on its neutrality. Light in pretty much every way a wine can be \u2013 in body, fruitiness and alcohol \u2013 its flavour profile is characterised by mellow notes of apple and pear, hints of green herbs and white flowers and a very-reflective-of-place wet stones and seashells vibe.\n\nTo take a positive stance, you could argue that this kind of vinous blank canvas is exactly what makes Melon de Bourgogne such a successful partner with raw oysters and simply prepared seafood, letting the food be the star. Still, its mild-mannered tone (coupled with its very friendly price-point) has, historically, been somewhat detrimental, meaning it was taken less seriously.\n\nMore recently, however, the wine has been a darling of the wine nerd community, who champion the complexity of its textures and its ability to convey a sense of place. It's even starting to get credit as a moderately age-worthy white.\n\nLook to the sub-region of Muscadet S\u00e8vre-et-Maine to experience the top expressions, and pay attention to labels with the words _'sur lie'_ on them. The term indicates that the wine was aged on the lees (dead yeast cells leftover from the fermentation process, not as gruesome as it sounds), which can help boost flavour and texture.\n\nAnd, lest I be called a liar, it's worth mentioning that Melon has actually done some very limited travelling outside of the Loire in recent years. Seek out some bottlings from Oregon, USA, to experience an out-of-its-element expression of the grape.\n\n# Cabernet Sauvignon\n\n# **Say It Right:** _Cab-er-nay So-veen-yawn_\n\n**Notable Regions**\n\nFrance (specifically Bordeaux), Chile, California (specifically Napa and Sonoma) and Washington state in the USA, Australia, Spain, Italy and South Africa.\n\n**Drink It With**\n\nButter-based prime cut of steak or herb-crusted lamb; burgers (or Portobello burgers); red meat with warming spices, especially stews, and aged blue cheeses.\n\n**Key Words**\n\nYou like a **brawny, complex** red that **doesn't skimp on flavour, body or aromatics** ; reds that are obviously dry, but also **smooth textured and rich** ; ripe red and black fruit mixed with flavours of mocha, baking, spice, cedar wood and tobacco; a wine that could work as either a drink-right-now option or a 'here's one for your cellar' gift; the **opposite of Pinot Noir**.\n\n**You Might Also Like**\n\nPriorat, Nero d'Avola, Monastrell, Montepulciano d'Abruzzo, Aglianico, French Malbec, Zinfandel, Portuguese red blends and aged Cabernet Franc from Chinon.\n\nLike James Bond, there's something very 'guys want to be him, girls want to be with him' about Cabernet Sauvignon. Sounds silly, but it really has got that level of style, swagger and celebrity.\n\nEven if you know next to nothing about wine, you've at least heard of Cabernet Sauvignon. It's the most widely planted grape in the world \u2013 and that includes both red and white \u2013 and is responsible for some of the industry's most legendary, revered wines. Everything about its personality exudes confidence and commands attention: an unapologetically bold, full-bodied texture; intensely concentrated fruit; a firm tannic structure that fits it like a well-cut suit, and seductive, dark, broody aromatics that make you want to linger over the glass. And as if all that weren't enough of a selling point, add longevity and ageability to its list of skills. It's definitely a grape that rewards patience, holding you in suspense with the promise of getting more and more interesting over time.\n\nFunnily enough, although the varietal is Bordeaux's most important, it never performs alone there. Rather, it's traditionally blended with sibling Merlot and their papa, Cabernet Franc, as well as the occasional Malbec, Petit Verdot and Carm\u00e8n\u00e8re. These serve to complement and enhance its intense character. While Cabernet Sauvignon is grown throughout the region, it contributes the most to the distinguished, enchanting wines of the so-called Left Bank (i.e. the reds of the M\u00e9doc and Graves; or, more specifically, Margaux, Pessac-L\u00e9ognan, Saint-Julien and Saint-Estephe, to name a few). In addition to Cab's signature marker of ripe blackcurrant, these tend to display flavours of black cherry, dried herbs, graphite, leather and tobacco.\n\nSo great is the reputation of this grape and the wines it produces in Bordeaux, France, that it has inspired a number of copycats around the world. Just consider the celebrated 'Super Tuscans' of central Italy, for example, which see Cabernet Sauvignon and its usual French friends, Merlot and Cab Franc, introduced to local favourite Sangiovese. Although the results are not quite as famous, other notable growing regions to adopt the Cab-led Bordeaux-style blend include South Africa, Australia's Margaret River and Lebanon.\n\nOf course, California, USA, also celebrates Cab in a great many impressive Bordeaux-formula reds, but it's the region's varietal bottlings that have garnered the lion's share of acclaim. It's grown throughout the northern part of the state but, more than anywhere, the grape does its thing the best in Napa \u2013 if you can call creating an international cult wine phenomenon a 'thing'. These massive, velvety-textured wines put on more of a razzle-dazzle parade of in-your-face flavours: think juicier, jammier dark fruits, eucalyptus, pepper, baking spices, mocha and cedar wood. While Napa Cab is going to be a consistently expensive category (often exorbitantly so), it is possible to find easier, more reasonably priced examples outside of the prestige region \u2013 they're probably going to be less complex, but at the very least, ready to drink young.\n\nLastly, speaking of good value Cabernet Sauvignon, Chile is great place to look. Vineyards there are planted with cuttings from Bordeaux and tend to produce a friendly, fruit-forward rendition.\n\n# Merlot\n\n# **Say It Right:** _Mer-low_\n\n**Notable Regions**\n\nFrance (specifically Bordeaux), California and Washington state in the USA, Spain, Italy, Australia, Chile, New Zealand and Canada.\n\n**Drink It With**\n\nGrilled, roasted or stewed pork and beef; roast chicken, lamb or duck prepared with a red wine sauce; blue cheese burgers; stewed or roasted root vegetables; French onion soup.\n\n**Key Words**\n\nYou like **medium to full-bodied reds** ; want a **dry** wine, but are OK if it drifts over into a realm of more **plush red fruit** flavours; like the notes of **tobacco, dried herbs, cedar wood and bitter chocolate** that oak age can contribute to.\n\n**You Might Also Like**\n\nValpolicella Ripasso, Cabernet Sauvignon, Grenache from Spain and Carignan-based red blends.\n\nFor Merlot, making wine is mostly a family affair. In the Bordeaux region of southwestern France, the grape is famously partnered with its superstar sibling Cabernet Sauvignon and their esteemed parent, Cabernet Franc. In these internationally renowned blends, Merlot's plush texture and forward flavours of baked cherry, cassis and plum are often used to complement Cabernet Sauvignon's comparatively brawnier structure.\n\nBut just because Merlot is typically cast as the softer, 'friendlier' brother, doesn't mean it's without the goods to command a lead. While Cabernet Sauvignon sits first chair in the region's so-called 'Left Bank' reds, on the right side of the Gironde estuary, the varietal makes up the majority of the blend in the velvety-rich wines of Pomerol and Saint-\u00c9milion. In these, in addition to its characteristic fruit, Merlot offers complex aromas of mocha, cedar, liquorice, bay leaf and fresh earth.\n\nThe most widely planted grape in France, you can also find significant vineyard acreage dedicated to Merlot in Languedoc-Roussillon, although the quality of its work is not as impressive there. Better to focus on the more conversation-worthy Bordeaux-style blends coming out of Friuli in northeastern Italy, and, of course, the 'Super Tuscans' produced on the central eastern coast.\n\nIt's actually pretty easy to find Merlot throughout most of Italy (except, perhaps, on the southern end), but beware that a lot of the iterations coming from Lombardy and Alto Adige lose the grape's charming roundness, instead erring on the tart, thin side.\n\nIn certain areas of the New World, though, Merlot has flown the coop. In Washington state, Long Island and New York state in the USA, Hawke's Bay in New Zealand and Chile, Merlot is celebrated as a soloist. Washington and Chile in particular have gotten kudos for making a more polished, Bordeaux-esque style. Northern California is another big name in single varietal Merlots, which are made in a darker, fruitier, more lushly textured style. But it's also used in blends with Cabernet Sauvignon there too, of course \u2013 a pattern that is mirrored in Australia and British Columbia, Canada. And in Argentina, it's frequently used in blends with Malbec.\n\n# Cabernet Franc\n\n# **Say It Right:** _Cab-er-nay Fronk_  \n **Other Known Aliases:** Bouchet _(Boo-shay)_ , Breton _(Bruh-tawn)_\n\n**Notable Regions**\n\nFrance (specifically Loire Valley and Bordeaux), Italy (specifically Friuli and Tuscany), California, New York state and Washington state in the USA, Chile, Hungary and Canada.\n\n**Drink It With**\n\nGrilled and roasted red meat, and also quail and duck; barbecued or wood-fire grilled food; hearty, earthy stews; dishes featuring peppers, aubergine (eggplant) or tomato, and strongly flavoured, tangy goat's cheeses.\n\n**Key Words**\n\nYou like a **dry, medium-bodied, red-fruited wine** with a solid tannic structure and vibrant acidity; **you're on board with charred, smoky mineral tones and a loud green, vegetal profile** (read: the smell of green pepper and jalape\u00f1o in your red wine does not freak you out).\n\n**You Might Also Like**\n\nCarm\u00e9n\u00e8re, Carignan, Montepulciano and C\u00f4tes-du-Rh\u00f4ne\/ GSM blends.\n\nParents often talk about wanting their children to not only match their successes, but supersede them. So it's probably safe to assume that Cabernet Franc is one hell of a proud papa, because when someone blindly requests a 'glass of Cab', 99% of the time they're expecting its offspring, Cabernet Sauvignon, to appear on the table.\n\nEven in their shared home of Bordeaux, where Cabernet Franc is one of the region's most important and oldest varieties, its major role is as a minor, backseat blending partner to Cabernet Sauvignon in the great reds of the Left Bank. On the right side of the Gironde estuary, in the esteemed sub-regions of Saint-\u00c9milion and Pomerol, it's still not the star of the blend. That honour goes to Merlot, Bordeaux's other big-shot grape (also a Cab Franc progeny). But at least in those wines Cab Franc makes up a more respectable slice of the pie. The varietal is also valued throughout Bordeaux for the fact that it ripens and is picked earlier than Cabernet Sauvignon and Merlot. In challenging vintages when the other two underperform due to weather issues or some other reason, Cab Franc is the reliable, on-hand substitute. Stepping in to support your kids when they come up short? Total Dad move.\n\nBut it's not all background chorus gigs for French Cabernet Franc. In the central Loire Valley regions of Chinon, Bourgueil and Saumur-Champigny, it is revered for its work as a single varietal red. And, when left with the spotlight to itself, it is far from shy. Medium-bodied with firm tannins and bright acidity to match, these smooth, elegant wines are known for their loud and unique aromatics. Sure, there's the not-so-unusual collection of red fruit flavours (raspberry is a big one) but, more than anything, these wines are distinctive for their intense green notes \u2013 pepper, green bean, herbs, jalape\u00f1o \u2013 as well as a smoky, charred rocks mixed with pencil shavings kind of fragrance.\n\nElsewhere, albeit much less significantly, Cabernet Franc gets to shine on its own in Italy's northeastern region of Friuli; Ontario, Canada, and New York state, Washington state, Michigan and Virginia in the USA. For the most part, though, Cabernet Franc-growing wine regions throughout the world look at the success of its role in Bordeaux blends, and therefore follow that model. Arguably the most notable of these are the 'Super Tuscans' of central Italy, in which it plays a minor role, but California in the USA, Hungary and Australia have also produced some impressive examples.\n\n# Pinot Noir\n\n# **Say It Right:** _Pee-no Nwar_  \n **Other Known Aliases:** Sp\u00e4tburgunder _(Spate-berg-under)_ , Pinot Nero _(Pee-no Nair-oh)_\n\n**Notable Regions**\n\nFrance (specifically Burgundy), California, Oregon and Washington state in the USA, Germany, New Zealand, Australia and Switzerland.\n\n**Drink It With**\n\nLeaner cuts of beef; braised chicken; roasted duck, lamb or quail; grilled salmon or tuna; practically any dish with mushrooms and root vegetables heavily flavoured with herbs.\n\n**Key Words**\n\nYou like **light-bodied** reds that are dry but **not tannic** ; prefer red fruit flavours with tart, **tangy acidity** ; are a fan of **herbal and earthy** aromatics; want something complex and don't mind shelling out for it; something **for a special occasion, or to age in your cellar** ; the **opposite of Cabernet Sauvignon**.\n\n**You Might Also Like**\n\nCru Beaujolais, Zweigelt or St Laurent, Poulsard, Trousseau, Barbera, Frappato and Schiava.\n\nDon't let the fact that it is one of the world's most popular and esteemed varietals fool you. Truth be told, Pinot Noir is a little bit of a problem child.\n\nIn the vineyard, especially, this ancient grape can be quite prickly and is notorious for putting winemakers through their paces. Put it this way: it has strong feelings about the type of soil it's grown on and is incredibly sensitive to climate, plus, its thin skin makes it an easy target for disease and insects, and it will punish you if you force it to be over productive. And that's not to mention that it can oxidise very quickly during the winemaking process (not a good thing).\n\nFrom a consumer standpoint, it isn't exactly the easiest or most accessible grape either, especially for those just getting into wine. Aside from having a lighter, more delicate profile (read: more effort to appreciate), all the work and risk on the production side means that the really good, well-made stuff is going to cost you. What's more, about 40 clones of the grape exist, the selection of which can have a huge impact on taste and quality.\n\nBut, you know, it's all those challenges and complexities about Pinot Noir's personality that are precisely what make it so irresistible and thrilling. Because despite all that, when it's done right, it rewards you with flavours and textures worthy of effusive rhapsody.\n\nFor anyone passionate about Pinot \u2013 and there are many \u2013 the conversation has to start in Burgundy. So revered is Pinot Noir in its birthplace that, with the exception of Gamay in Beaujolais, it is the region's exclusive red varietal. Burgundy's vineyard landscape is famously complicated and jigsaw puzzle-like, divided between basic village-level sites and the more prestige Premier Cru and Grand Cru areas. Although this gives wine drinkers more homework, it's an ideal setup for Pinot Noir, a master at conveying sense of place. While this kind of transparency produces wine that will vary from one vineyard site to another, generally speaking, you can expect red Burgundy to be light-bodied and ethereal, with soft tannins and tart, tangy notes of sour cherry and cranberry. These wines display a strong earthy vibe as well \u2013 think foraging for mushrooms and herbs in the forest. And with age, these wines can take on flavours of wild game.\n\nPinot Noir's influence in France extends beyond Burgundy: it's one of three varietals allowed in the production of Champagne, it's responsible for the slightly rare and mineral-toned reds and ros\u00e9s of Sancerre and it produces some very unique examples in Alsace and the Jura. Otherwise, you can find some very good versions in parts of northern Italy and Switzerland, as well as Germany, where it tends to exhibit more fruit and power.\n\nPicky Pinot has also successfully settled in several of the New World's big-name winegrowing hubs: California and Washington state in the USA, Chile, south Australia and Central Otago in New Zealand. There, the grape goes more the way of baked raspberry and plum and takes on a riper, arguably more friendly personality. While still technically a light-bodied grape, its texture tends to be on the plush side, and its aromatics drift into a realm of 'prettier' flavours like baking spices, tea and tobacco leaves, and dark flowers. Oregon, USA, its other notable New World home, should not be overlooked, and is better known for producing Pinot in a more restrained, Burgundian style.\n\n# Syrah\n\n# **Say It Right:** _Sir-ah_  \n **Other Known Aliases:** Shiraz _(Sher-az)_\n\n**Notable Regions**\n\nFrance (especially the Rh\u00f4ne Valley), Australia, Spain, northern California and Washington state in the USA, Argentina, Chile and Italy.\n\n**Drink It With**\n\nMeat-based dishes with smoke and spice: barbecue, bacon cheeseburgers, carnitas tacos; meaty red-sauce pastas, and mushrooms.\n\n**Key Words**\n\nOld World-style: you like **medium to full-bodied dry reds** , with red fruit and **savoury flavours** ; like bacon so much you want to smell it in wine. New World-style: like a full-bodied dry, but **fruit-forwards red wine with a lush texture** and smooth tannins; **prefer ripe black fruit flavours**.\n\n**You Might Also Like**\n\nFor Old-World, try Bla\u00fcfrankish, Nero d'Avola, Barolo and Mourvedre. For New World, try Petit Verdot, Malbec, Pinotage, Touriga Nacional\/ Portuguese red blends and Aglianico.\n\n**A Quick Word about 'Junior': Petite Sirah**\n\nIf you're into Californian wine then you might have also heard of Petite Sirah, Syrah's offspring with an almost unheard-of French grape called Peloursin. The grape is best known for its role in the wines of northern and central California, where it is either used to add weight and texture to red blends (usually starring Cabernet Sauvignon or Zinfandel), or as a single varietal. When flying solo, Petit Sirah tends to make full-throttle, big-boned tannic reds brimming with rich, dark fruit, tobacco and baking spices, dried herbs, and a distinct meatiness (just like Syrah). Go classic and enjoy it alongside some simple barbecued meat and vegetables.\n\nGrapes, like people, often take on very different personalities depending on their environment. To the point where meeting an Australian Shiraz after years of being friends with French Syrah might feel like a bit of a duplicitous encounter. Technically it's the same coin, but its two sides are totally unique (and, in this case, both famous in their own right).\n\nOn one end you have the primal, savage Syrahs of France's Rh\u00f4ne Valley. There, the grape produces bold wines with unmistakable flavours of cured meat, bacon fat, wild berries and bramble, leather, black olive, pepper, smoke, tar, tobacco and lavender. Think of it as the Ernest Hemingway of reds: all broody and irreverent with an unapologetic manly-man charm, but also sophisticated, incisive and strangely poetic. It is the exclusive red varietal of the northern part of the region, responsible for the celebrated \u2013 and, in some cases, legendary \u2013 reds of Cornas, C\u00f4te-R\u00f4tie, Hermitage, Crozes-Hermitage and Saint-Joseph. In the southern part of the Rh\u00f4ne it moves into more of a supportive role, adding texture and its distinctive feral edge to the Grenache-dominated red blends of Ch\u00e2teauneufdu-Pape, Gigondas, Vacqueyras, Rasteau and C\u00f4tes-du-Rh\u00f4ne.\n\nOn the flip side of the coin (and hemisphere), you have Shiraz, the grape's Australian alter ego. As the most widely produced variety in the country, it's a total icon; a big, if not the only, reason for Australia's status as a major player on the wine world stage. In this home, however, it shows a completely different side of its personality: less rough-edged, savoury, wild animal, and more full-bodied cornucopia of lush, ripe, black fruits (plums! blueberries! blackberries! black cherry and raspberry!), mixed with intoxicating flavours of mocha, baking spices, pepper, pipe tobacco and eucalyptus.\n\nThe most famous expressions of this style of Shiraz come from the Barossa Valley in South Australia, though it's worth noting that cooler-climate regions can actually produce spicier, earthier renditions more akin to those of the Rh\u00f4ne. And, as in France, it is also a popular choice in blends, frequently partnered with Grenache or Cabernet Sauvignon to make powerful, opulent reds.\n\nOutside of these two signature growing regions, production of Syrah has been on the rise in central Spain and Chile, as well as been successful in the hands of some Rh\u00f4ne-inspired producers in northern California and Washington state in the USA.\n\n# Grenache\n\n# **Say It Right:** _Gruh-nah-sh_  \n **Other Known Aliases:** Garnacha _(Gar-nat-chah)_ , Cannonau _(Canon-ow)_\n\n**Notable Regions**\n\nFrance (specifically southern Rh\u00f4ne and Languedoc-Roussillon), Spain, Australia, Italy (specifically, Sardinia) and California in the USA.\n\n**Drink It With**\n\nWell-spiced meats using grilled, barbecue, and braised preparations; dishes seasoned with Moroccan and Middle Eastern spice blends, for example, lamb tagine.\n\n**Key Words**\n\nYou like a red that is **fruit-driven** , with concentrated, **slightly cooked flavours of red and black fruit** ; reds that feel like they've been seasoned with a bit of **pepper and spice** ; not crazy about crunchy tannins, favouring instead something with plush texture that's a **little plump and boozy**.\n\n**You Might Also Like**\n\nValpolicella Ripasso, Zinfandel, American Merlot, Nerello Mascalese, Australian Shiraz and Argentinian Malbec.\n\nOne of the most widely planted grapes in the world, think of Grenache as the friendlier, sweeter 'nice guy' member in a gang of mostly rugged, brawny characters. With its characteristic ripe, concentrated red fruit flavours, moderate tannins and plush texture, herb-and-spice-scented Grenache is particularly valued for its ability to round out the rougher edges of its frequent blending partners.\n\nYou can find the grape throughout southern France, where it is typically buddied-up with meatier, bolder, more structured varietals like Syrah, Mourv\u00e8dre, Carignan and, occasionally, Cinsault. Grenache's partnership with Syrah and Mourvedre is so well established, in fact, that the trio are commonly identified under a singular acronym: 'GSM.' The reds of C\u00f4tes-du-Rh\u00f4ne, C\u00f4tes-du-Rh\u00f4ne Villages, Gigondas and Vacqueyras in southern Rh\u00f4ne and Faug\u00e8res, Minervois, Corbi\u00e8res and Fitou in the Languedoc-Roussillon can all be credited as the work of Grenache and its cronies. And let's not forget to also give a shout-out to the crew for their hand in the dry ros\u00e9s of the area, from places such as Tavel and Lirac.\n\nIn most of these, Grenache makes up the bulk of the blend but, without question, its most important leadership role is in the world-famous reds of southern Rh\u00f4ne's Ch\u00e2teauneuf-du-Pape region. These heady, opulent wines can be made from a combination of 13 select grapes, but it is Grenache, often acting as the dominant player, that gives Ch\u00e2teauneuf its signature note of luscious fruit jam.\n\nNow, Ch\u00e2teauneuf reds are technically dry (albeit rich and fruit-forward), but elsewhere in the southwest of France, Grenache's natural fruity sweetness is cranked up and put on display in the _vin doux naturel_ dessert wines of Banyuls, Rasteau and Maury.\n\nGarnacha plays an equally important role in its native Spain, where it contributes to the country's best red wines. It is both the power player in the big, sultry red blends of Priorat, and an important, edge-softening assistant to Tempranillo in Rioja. You're most likely to see the grape throughout the northern half of the country, making round, spicy, fruit-driven reds, as well as some vibrant, juicy dry ros\u00e9s.\n\nGrenache can also be found on the Italian island of Sardinia masquerading under the name Cannonau. There, it is produced as a single-varietal wine known for its boozy heft and aromas of dried herbs, red fruit and spice. Otherwise, look out for GSM blends from central and southern California and Washington state in the USA and southern Australia.\n\n# Barbera\n\n# **Say It Right:** _Bar-bare-ah_\n\n**Notable Regions**\n\nItaly (specifically, Piedmont and Emilia-Romagna), California in the USA, Argentina and South Africa.\n\n**Drink It With**\n\nRed-sauce pastas and pizzas; grilled pork; burgers; roasted vegetables (especially mushrooms and peppers) and charcuterie and robust hard cheeses such as Pecorino.\n\n**Key Words**\n\nYou like **lighter to medium-bodied red** wines and **_a lot_ of tangy acidity**; are not looking for anything too tannic; gravitate towards riper-tasting **red and black fruit flavours** and are into the smell of **herbs and spice** ; want something that is **easy-drinking** and not going to deplete your funds.\n\n**You Might Also Like**\n\nNerello Mascalese, Mencia, Valpolicella blend, Dolcetto, Cinsault and Zweigelt.\n\nIn any story there's the star, and then there's the supporting character \u2013 that friendly, reliable, good-but-not-quite-as-great sidekick who lives in the shadow of the lead. That's essentially the relationship between Nebbiolo and Barbera, the key red grape players in northwestern Italy's famed Piedmont region.\n\nWhile Barbera is the region's most widely planted grape, Nebbiolo occupies its best, most important vineyards in the legendary sites of Barolo and Barbaresco. Where Nebbiolo is fiercely tannic, generally requiring long aging to reveal the great depth and complexity of its flavour profile, medium-bodied Barbera offers a softer texture with low tannins and bright acidity, easy and ready to drink upon release. Top Nebbiolo bottles live in the 'save it for a special occasion' price range, but Barberas remain consistently in the easy and accessible weeknight staple realm. You get the idea. But please don't read the comparison between Nebbiolo and Barbera as a slight against the latter. Barbera is a populist wine, rustic and charming, with notes of mulberry, sour cherry and plum, combined with dried wild herbs, liquorice and spice. Its wines offer an enviable balance of bold, concentrated flavour with a punchy, juicy acidity that allows it to still taste light on its feet. It's what makes it such a great match for rich, fattier food, as well as savoury dishes that benefit from a fruit counterpoint.\n\nThe grape's most pedigreed expressions come from Piedmont's Asti and Alba sub-regions, and it's well worth the extra money to see what happens when this 'stepsister' grape is taken more seriously. Otherwise, if you're looking to see a different side of Barbera, check out New World versions from Argentina and California, USA, where the wine takes on a riper, more fruit-forward character \u2013 think blackberry-cherry jam scented with vanilla and purple flowers.\n\n# Gamay\n\n# **Say It Right:** _Gah-may_\n\n**Notable Regions**\n\nFrance (specifically, Beaujolais and the Loire Valley), Switzerland, northern California and Oregon in the USA, Canada and New Zealand.\n\n**Drink It With**\n\nPicnic-friendly foods: cheese and charcuterie, savoury quiches, light salads; burgers; seared tuna or salmon steak; roast duck, veal, quail and chicken; autumn (fall) vegetables or, honestly, nothing at all.\n\n**Key Words**\n\nYou enjoy a **light-bodied, lower-tannin, high-acid red** with fresh fruit flavours; like a red that benefits from a slight chill and is **ready to drink** as soon as you get it home; want to seek out wines with the potential to **over-deliver for the price**.\n\n**You Might Also Like**\n\nPinot Noir, Frappato, Zweigelt, Saint Laurent, Cinsault, Langhe Rosso, Valdigu\u00e9, Trousseau and Poulsard.\n\n**The New Nouveau**\n\nBecause everything old is new again, it shouldn't come as a surprise that some producers are reviving the Nouveau tradition, making it cool again for a generation of ever more discerning wine drinkers. While you can find releases from a handful of benchmark Beaujolais producers (Marcel Lapierre, Jean Foillard, PUR) looking to flip the page on that dark spot in their region's history, the concept has really taken off with a number of winemakers in California, Oregon and the Finger Lakes in New York state, USA. There, a confident 'See, Nouveau doesn't have to suck' attitude has produced a crop of diverse wines that celebrate the fun, frivolous nature of the style without compromising quality. Beyond the traditional Gamay, the line up of American Nouveaus includes Pinot Noir, Pinot Gris, Chardonnay, Merlot, Chenin Blanc and Valdigu\u00e9, to name a few.\n\nIt's been quite the rollercoaster ride for Gamay and its iconic growing region, Beaujolais. The story is not unlike that of the small-town band that skyrockets to mass-market success thanks to a major pop hit. After a reputation-defining run in the limelight, there's the inevitable fall from grace thanks in no small part to questionable quality and oversaturation of the market. Their fame-making tune becomes a clich\u00e9, and it's not until the band returns to its roots and refocuses on what they do best that they are once again championed and turn into a cult-favourite comeback.\n\nThe southeastern French region has been making a name for itself with Gamay since at least the 1500s. Beaujolais' signature fresh-fruited light red, made from 100% of the varietal, had been happily living life as the swiggable (not an insult), affordable alternative to the high-brow Pinot Noirs of its neighbour to the north, the C\u00f4te d'Or of Burgundy. Then, in the 1970s and 1980s, Beaujolais Nouveau became a thing \u2013 a big thing. What originated as an unassuming practice of releasing some of the month-old, just-finished-fermenting 'nouveau' wine from that year's harvest snowballed into an international extravaganza (well played, marketing). Seeing money signs, much of the region abandoned tradition and quality and succumbed to mass-production, pumping out the boozy, tropical fruit punch of a wine that had become the Nouveau movement brand.\n\nAnd look, it's not like it isn't fun to dip into some punch every once in a while at a party, but the problem is that when something becomes synonymous with novelty, no one really takes it seriously or expects much out it. And so it was with Beaujolais' reputation for quite some time. Eventually, though, as the Nouveau shine wore off, the spotlight was able to shift onto the small group of dedicated producers who had weathered the storm and stayed true to the original course. These winemakers, mostly working off the top, 'cru'-level sub-regions \u2013 of which there are ten \u2013 proved that Beaujolais can produce wines with elegance, complexity and even the ability to age. The wines from these quality-designated crus, which also happen to be a basecamp of the French natural wine movement, are the current sommelier and wine nerd equivalent of catnip. Their charming, tart berries and cherries-meets-herbs-meets-purple flowers-meets-spice-meets-black tea-meets-fresh earth experience is not only incredibly food-friendly, but also has the added bonus of still being a relatively good deal.\n\nGamay does have a life outside of Beaujolais, by the way. In France, it is also at home in the Anjou and Touraine regions of the Loire Valley, where it produces easy-drinking, juicy light reds and fruity ros\u00e9s. While across the pond in the USA, a number of fresh-faced producers in Northern California, Oregon and the Finger Lakes in upstate New York have been producing fun, slightly riper and often quirky renditions of the grape.\n\n# The Crus of Beaujolais\n\n**Getting to Know the Cru**\n\nGetting serious about drinking Gamay means getting to know the crus of Beaujolais, aka the ten designated quality growing areas where the grape achieves rockstar status. Representing only 15% of production in the region \u2013 more than half is still dedicated to basic Beaujolais and Nouveau, and only about a quarter to the next level up, Villages \u2013 these are the names you're going to want to look out for on labels and wine lists. From there, it's just a matter of digging in to the fun homework (aka drinking) to discover if you're more a fan of the bolder, fuller (relatively speaking) expressions from Moulin-\u00e0-Vent, Morgon and R\u00e9gni\u00e9, or, if you prefer the more delicate, floral touches of Fleurie and Ch\u00e9nas.\n\n# Carignan\n\n# **Say It Right:** _Care-een-yawn_  \n **Other Known Aliases:** Cari\u00f1ena _(Car-een-yen-ah)_ , Carignano _(Car-een-yan-no)_ , Mazuelo _(Maz-way-low)_\n\n**Notable Regions**\n\nSouthwestern France, Spain, Algeria, Tunisia, Chile and California in the USA.\n\n**Drink It With**\n\nHearty stews and roasted red meat dishes with a heavy baking spices influence, cured meats and game birds.\n\n**Key Words**\n\nYou like **rustic, full-bodied red wines** with **bold acidity and tannins** ; gravitate towards **red and black fruit flavours** and **appreciate a wine that smells like cured meat** and baking spices; **don't want to break the bank.**\n\n**You Might Also Like**\n\nNorthern Rh\u00f4ne Syrahs, Zinfandel and Aglianico.\n\nCarignan may have once been the most widely planted red varietal in France, but that does not mean it's ever been among the most popular, or important. Quite the opposite, in fact. The casual wine drinker has probably never even heard of it, thanks to a legacy of being used in the simple, unmarked table wines of southwestern France and its native northern Spain. It's what you might call a 'workhorse' grape; a likeable-but-not-outstanding Average Joe type that keeps its nose to the grind, works hard and produces well without having an ego.\n\nBecause it's a high-yielding grape, it's long been an attractive candidate for overproduction and overextension in the 'quantity over quality' game. And, as a result, its potential and charms went largely overlooked. Admittedly, it's not exactly a diamond-in-the-rough kind of grape; it's a genuinely rustic and gruff varietal, producing dark-fruited, meaty, robust wines marked by high acidity and alcohol, and intense tannins. But now that it's being taken more seriously, especially when old-vine grapes are being used, Carignan is proving to be a valued contributor in the well-made, value-driven blends of Corbi\u00e8res, Fitou and Minervois in France's Languedoc-Roussillon region, as well as a co-conspirator in the dry blended ros\u00e9s of Proven\u00e7e.\n\nBack in its Spanish homeland, Cari\u00f1ena, or Mazuelo, as it's known, remains an important partner to Grenache in the ripe-smoky-herbal red wines of Priorat, as well as a valued supporting character to Tempranillo in some of the revered reds of Rioja. Meanwhile in northern California, USA, it's experiencing a mini-resurgence in both blended and single varietal wines.\n\n# Malbec\n\n# **Say It Right:** _Mall-beck_  \n **Other Known Aliases:** C\u00f4t _(Ko)_\n\n**Notable Regions**\n\nArgentina, southwestern France, Chile, California in the USA, south Australia and South Africa.\n\n**Drink It With**\n\nMeat, meat and more meat (sorry, vegetarians!); grilled steak and smoked lamb work especially well (as do burger versions of either); cassoulet; farro with stewed root vegetables, and mushrooms.\n\n**Key Words**\n\nYou definitely like **medium to full-bodied reds** with **loads of black fruit** flavours; enjoy something **lush and smooth** , not too tannic; are looking for a **friendly, good-value** option that **doesn't necessarily need food**.\n\n**You Might Also Like**\n\nMerlot, Zinfandel, Mourv\u00e8dre, Shiraz from Australia, California Cabernet Sauvignon and Touriga Nacional-based blends from Portugal.\n\nYou probably don't realise it, but Malbec's story is actually a bit of a rags-to-riches tale. It's difficult to imagine in today's reality, where mention of the grape instantly conjures images of the plush, dark-fruited red that has become the unofficial mascot of Argentina's wine industry. But the truth of the matter is, for all its current popularity and shiny stardom, Malbec comes from pretty average, humble origins.\n\nIn its native southwest France, the grape otherwise known as C\u00f4t produces medium-bodied, rustic, brambly reds with a more pronounced earthy, tannic edge than its southern hemisphere counterpart. The most renowned of this rough-and-tumble, gritty style comes from the region of Cahors, and, for what it's worth, these wines are just about the best thing you could ever drink alongside a big plate of duck cassoulet. Malbec has also historically been a significant grape in nearby Bordeaux, where it has traditionally played sidekick to Cabernet Sauvignon, as well as in the Loire, where it adds dark fruit flavour and boozy oomph to Gamay and Cabernet Franc-based blends.\n\nBut it wasn't until a Frenchman brought the varietal to Argentina in the mid-to-late 19th century that Malbec got its big break. In its adopted home, where it is now the second most-planted variety (but definitely the most important), Malbec shows off a softer, more vampy side of its personality. With moderate acidity and just enough tannic structure to keep the whole thing technically dry, these wines swaddle your tongue in a velvet-smooth blanket of ripe dark fruit (blackberry, blueberry and plum), spice and violets.\n\nYou can find the grape being cultivated across the country, but the high-altitude vineyards of Mendoza, Argentina, are where it does its best-known work.\n\nFor the most part, these generously flavoured, value-friendly wines are designed for right-now drinking, but if you want to see how the grape evolves with the influence of extended oak aging, look for a higher-end 'Reserva' bottle.\n\nBesides France and Argentina, you can find Malbec in places like California in the USA, South Australia and Chile, where it is typically teamed up with its old pal from Bordeaux, Cabernet Sauvignon.\n\n# Zinfandel\n\n# **Say It Right:** _Zin-fan-del_  \n **Other Known Aliases:** Primitivo _(Pree-mee-tee-vo)_ , Tribidrag _(Tri-bidd-rag)_\n\n**Notable Regions**\n\nCalifornia in the USA, Italy (specifically Puglia), Croatia.\n\n**Drink It With**\n\nBarbecued meats glazed in a sweeter-style sauce; lamb or steak; chilli con carne; spicy Mexican food such as carnitas tacos and chicken mole; burgers; sausage and pepper sandwiches or pizza.\n\n**Key Words**\n\nYou like **medium to full-bodied** reds with moderate tannins; big, **juicy, ripe red-berry flavours** are your friend; love the idea of something that smells **smoky and a little meaty** ; want to pick a wine that's **smooth but packs a punch**.\n\n**You Might Also Like**\n\nGrenache, Carignan, Syrah, American Merlot, Nero d'Avola and Malbec.\n\nZinfandel may not get the international attention and acclaim that the USA's Pinot Noirs, Cabernet Sauvignons and Chardonnays do, but it is nevertheless an \u2013 if not the \u2013 iconic 'American' wine. As American as apple pie, you might say. Except, of course, it technically isn't.\n\nThough it was long thought to be and promoted as an indigenous varietal to the United States, the grape is actually native to Croatia, where it was originally known as Tribidrag. It didn't land state-side until sometime in the early 19th century, eventually making California its home in the 1850s. The trajectory feels a little like that of the kid who didn't get much attention in its hometown, but then, given the opportunity to start fresh at a new school in a new town, took on a different identity and flourished. Zinfandel was so popular in California, in fact, that up until 1998, it was the most widely planted red grape in the state (currently it's number two, with Cabernet Sauvignon in the lead).\n\nGranted, a big chunk of the grape's production is dedicated to making the maligned semi-sweet, actually-pink-not-white wine known as 'white Zinfandel'. While this may be some of Zinfandel's most recognisable work in California, it is by no means its best. Grown throughout the state, 'Zin' is at the top of its game in Sonoma, Napa, the Sierra Foothills and Lodi in the north, as well as Paso Robles in the centre.\n\nTypically, these medium to full-bodied wines are robust and high-octane (aka boozy), delivering a wallop of berry fruit flavours so ripe it borders on jammy and raisinated. Notes of baking spices, sage, pepper, smoked meat and tobacco are also common markers.\n\nIt's worth mentioning, too, that Zinfandel's long history in California makes it possible to find some exceptional, ageworthy old-vine expressions of the grape.\n\nAlthough producers in parts of Australia, South Africa, South America and Croatia (of course) have had some success with Zinfandel, the varietal's only other real home of note is in Puglia, otherwise known as the heel to Italy's boot. There, it goes by yet another name \u2013 Primitivo \u2013 and is used to produce similarly rich, dark fruit-driven single varietal reds, or to add body and concentrated fruit flavour to blends with the native, highly tannic Negroamaro.\n\n# Petit Verdot\n\n# **Say It Right:** _Puh-tee Ver-doh_\n\n**Notable Regions**\n\nBordeaux and southwestern France, Spain, Australia, Chile, California in the USA and South Africa.\n\n**Drink It With**\n\nAnything you would have with a Cabernet Sauvignon-based blend from Bordeaux. A nice cut of steak; a rich red-sauce and meat-based stew and heartier grains and legumes. Non-creamy dishes with mushrooms and other earthy ingredients also work well.\n\n**Key Words**\n\nYou want a full-bodied, **rich, dark-fruited red** with **grippy tannins** ; are a fan of **fuller-bodied red blends** ; want something that's going to be a **bang for your buck**.\n\n**You Might Also Like**\n\nShiraz, Mourv\u00e8dre, a Bordeaux-style blend, Nero d'Avola, Petit Sirah and Touriga Nacional.\n\nTasting a well made, blended wine is like hearing a choir singing in harmony. Within the group you have your strong, powerful, leadership voices, but also an assembly of 'smaller' voices filling in the gaps. While perhaps a less glamorous role, it is nevertheless an incredibly important one. A fact Petit Verdot knows all too well.\n\nFull-bodied and fiercely tannic, this boozy, high-acid red is bursting with flavours of ripe black fruits, smoke and spice, as well as wild herbs and flowers. It's a whole lot of wine on its own, so even in small doses it makes a big impact on a blend. In the iconic red blends from Bordeaux's Left Bank, it's used sparingly but effectively \u2013 usually no more than two to three percent of the entire blend \u2013 to help punch up the colour, tannic structure and floral-herbal aromatics of the final wine. Its calling card reads pretty much the same in its other important growing regions of central Spain, Chile, California in the USA and South Africa.\n\nThat being said, Petit Verdot has a decent-sized fanbase in parts of Australia, where, in addition to vamping up Bordeaux-style blends, it is also appreciated as a single varietal.\n\n# Mourv\u00e8dre\n\n# **Say It Right:** _Moo-ved-druh_  \n **Other Known Aliases:** Monastrell _(Mon-ah-strell)_ , Mataro _(Mah-tar-oh)_\n\n**Notable Regions**\n\nSpain, France (specifically the Rh\u00f4ne Valley and Languedoc-Roussillon), Australia, California in the USA and South Africa.\n\n**Drink It With**\n\nGrilled or roasted meat (beef, pork sausage, game birds); charcuterie; and vegetables seasoned with woody herbs such as rosemary and thyme.\n\n**Key Words**\n\nYou're into **full-throttle reds** \u2013 **chunky tannins** , lots of body and **weight** , high-alcohol, powerful flavours of **black fruit, earth and herbs** , and a generous amount of acidity to back it all up; want something with a **broody, brawny** personality.\n\n**You Might Also Like**\n\nCabernet Sauvignon, Syrah, Petit Sirah, Malbec, Touriga Nacional-based blends from Portugal and Petit Verdot.\n\nMourv\u00e8dre is kind of like that member of the band that you always name last, almost as an aside, after the other, 'more famous' ones have been listed.\n\nThe 'M' in GSM \u2013 the shorthand for the trio behind the majority of reds in southern Rh\u00f4ne and the Languedoc-Roussillon in France \u2013 Mourv\u00e8dre definitely takes a backseat to its partners Grenache and Syrah. But just because the grape is a little less popular and well-known, doesn't mean its contributions to the mix are any less valuable. Known for producing hearty, tannic wines so thick you can practically sink your teeth into them (not literally, but you get the idea), Mourv\u00e8dre is the essential component that adds structure and body to the blend.\n\nAnd aside from its muscle, the varietal also offers flavours of ripe blackberries, a bite of black pepper and spice, as well as a mildly savage, meaty-gamey quality. The intensity of its personality means that it rarely performs solo. The closest Mourv\u00e8dre really comes to taking centre stage is in the powerful reds and ros\u00e9s of Bandol, in Provence, where it must make up at least half of the blend. And, no joke, these wines are brawny, almost an antithesis to the breezy, idyllic landscape where they're grown. The reds are so tannic in their youth that they ought not be served without the offer, 'Would you like a side of meat with that?,' and the ros\u00e9s have enough character to shut down the rumour that the category is merely frivolous summertime swill.\n\nIn its native Spain, Monastrell, as it's known, makes equally big, broad, fruit-driven reds, albeit with slightly softer curved edges. Most of its production is concentrated in the centre and southeastern parts of the country, where it is used to make single varietal wines as well as the expected red blend, most of which fall in the friendly and affordable price range. And, further to the northeast in Penedes, it's even being used in the production of ros\u00e9 Cava.\n\nIn California, USA, and Australia, Mourv\u00e8dre has been long masquerading as Mataro \u2013 so well, in fact, that it wasn't until recently that people even realised the two are actually one and the same. In both places, it's rustic, gnarly and typically used in southern Rh\u00f4ne-style blends, although you can find some very well made old-vine varietal expressions (read: pricey) from California.\n\n# Nebbiolo\n\n# **Say It Right:** _Neb-ee-yolo_  \n **Other Known Aliases:** Spanna _(Spah-nah)_ , Chiavennasca _(Ki-ah-vay-nah-sah)_\n\n**Notable Regions**\n\nNorthwestern Italy (specifically Piedmont and Lombardy), California in the USA, Switzerland, Australia and Argentina.\n\n**Drink It With**\n\nSteak with herbs and butter; coq au vin; tea-smoked duck; osso bucco; mushroom-based sauces and dishes (e.g. mushroom risotto) and, if you're fancy, anything with black truffle shaved over it.\n\n**Key Words**\n\nYou are **not scared of tannins** ; like **complex** , medium-bodied reds with a **super earthy and savoury** flavour profile; are interested in trying something with a bit of age on it, or **ages well** , and don't mind that that means it's going to be more **expensive**.\n\n**You Might Also Like**\n\nDolcetto, Negroamaro, Brunello di Montalcino, Aglianico, aged Rioja and Bandol.\n\nThere are those who are bright, easy and cheerful; excited and ready to welcome you with open arms from the get-go. And then there are those who are more guarded and reserved \u2013 temperamental even \u2013 who require time to open up and reveal the complexities of their personality. Nebbiolo is very much the latter.\n\nOne of Italy's most important red varietals, responsible for the prestigious, enthralling reds of Barolo and Barbaresco, Nebbiolo is a star and it knows it. Starting in the vineyard, it's extremely picky about where it grows and the type of soil it's planted on, favouring, almost exclusively, the fog-covered hills of the small Piedmont sub-regions (in fact, _nebbia_ is Italian for 'fog', so you can take a wild guess as to where the inspiration for the grape's name came from). It also flowers early, takes forever to ripen, and is highly susceptible to a number of diseases \u2013 all potential red flags that require a great deal of skill to manage. And on top of all that, Nebbiolo is naturally high in tannin, resulting in wines that typically benefit from long aging before being released, and even then, are often at their best after spending several more years (if not decades) in the bottle. Warning: impatient wine drinkers best not apply. Not to mention the fact that the amount of work and time that goes into making Nebbiolo's greatest hits \u2013 Barolo and Barbaresco \u2013 keeps them pretty firmly in the 'not cheap' price bracket.\n\nLike a true diva, Nebbiolo does not make shy wines. Despite its lighter colour, the varietal famously produces full-throttle reds that are loaded with tannin, acidity and alcohol. The best among them will shout a seduction of aromatics at you from the glass. Dried rose petals, cherry, cranberry, tar, liquorice and anise are all common markers, but as the wine ages you can also expect to find intriguing savoury notes of balsamic, tobacco leaf, leather, cured meat and smoke. Speaking in broad terms, Barolo is going to make a broodier, brawnier expression of Nebbiolo, while Barbarescos are (relatively) softer and more elegant. And so, as with a diva, you forgive them all the pre-performance attitude, antics and demands because, in the end, they have the goods to back it all up. While perhaps not the easiest to like, these wines are definitely hard not to love.\n\nElsewhere in Piedmont, you can find some more accessible renditions of Nebbiolo from places such as Ghemme, Gattinara, Carema and Roero, as well as the general Langhe region (which are the friendliest and easiest to drink upon release of the bunch). Some very good versions are also made in Valtellina, in the neighbouring Lombardy region. Outside of northwestern Italy, plantings of the grape are surprisingly scarce given its worldwide fame, although a handful of producers in Switzerland, California, USA and Australia have done well with it.\n\n# Tempranillo\n\n# **Say It Right:** _Temp-rah-nee-yo_  \n **Other Known Aliases:** Tinto Fino _(Teen-toe Feen-oh)_ , Cencibel _(Sen-si-belle)_ , Tinta Roriz _(Teen-ta Roar-eez)_\n\n**Notable Regions**\n\nSpain, Portugal, Argentina, New Zealand, California in the USA, France and Australia.\n\n**Drink It With**\n\nGrilled, roasted and smoked preparations of meat and vegetables; red sauce pasta dishes; chilli and other meaty stews seasoned with warming spices; Mexican food.\n\n**Key Words**\n\nYou like a wine that is dry and **medium to full-bodied** with **solid acidity and tannins** ; go for something a little more rustic\/musky and savoury; want a wine that **ages well** ; enjoy the smell of coconut and dill caused by new American oak.\n\n**You Might Also Like**\n\nAged Barolo or Barbaresco, Agiorgitiko, Zinfandel, Brunello di Montalcino, Right Bank Bordeaux and Montepulciano.\n\nIn the world of Spanish wine, Tempranillo is treated like the beloved first-born child: it gets all the praise, most of the attention and is a huge source of pride. As the country's leading red grape, it is best known for being the driving force behind the legendary reds of Rioja and Ribera del Duero.\n\nAlthough the grape is grown throughout the country, it does its most significant, cherished work in the north-central Rioja region. There, it is partnered with Garnacha, Mazuelo (aka Carignan) and the local Graciano, to make bold, characterful wines capable of long aging. You might say that in the blend, Tempranillo is what gives the wine its guts. The full-bodied varietal gets high marks for tannin, acidity and alcohol, and its wines are known for displaying flavours ranging from strawberry and dried fruits to savoury herbs, leather, spice and tobacco.\n\nWithin the region, the style of the wine can vary significantly based on the philosophy of the winemaker and how long it's aged. Traditionalists tend to go for a more broody, earthy style and prefer aging in American oak, which lends signature notes of coconut, dill, baking spices and vanilla. More 'modern' producers will age the wine in small French barrels called _barriques_ to produce a flashier version that focuses more on ripe fruit and oak. In terms of aging, look for the words _'joven'_ or _'crianza'_ for young, less-aged expressions, and 'reserva' and 'gran reserva' for those that have spent significantly more time hanging out in barrel and bottle before release. FYI, the aging terminology applies across other regions and red grape varietals in Spain, so it's a good list to commit to memory.\n\nTempranillo's other big feature role in is the reds of the Ribera del Duero in the centre of the country. Under the alias Tinto Fino, it's famous for producing rich, complex single varietal wines of the smoky tobacco-coffee and cocoa-concentrated fruit persuasion. Not to mention the influential producer Vega Sicilia has made a name for itself by using the grape alongside Cabernet Sauvignon, Merlot and Malbec to create a Bordeaux-inspired blend.\n\nOutside of Spain, Tempranillo doesn't make much news. Aside from being an important grape in the blend used to make Port, Portugal's legendary dessert wine, Tempranillo only really shows up in small amounts from places like Argentina, New Zealand and California in the USA.\n\n# Sangiovese\n\n# **Say It Right:** _San-geo-vay-zay_\n\n**Notable Regions**\n\nItaly (specifically Tuscany), Argentina, France, California in the USA and Australia.\n\n**Drink It With**\n\nPretty much all things tomato and tomato sauce-based, and generally any flavours and ingredients that are staples in Mediterranean cuisine; salami, grilled steak, and braised chicken; ratatouille; mushroom-based dishes and sauces.\n\n**Key Words**\n\nYou like a wine that's **medium-bodied** with **bright red fruit** flavours and a **decent tannins** ; find the idea of something **rustic and savoury** appealing, and you're very fond of **herbal** aromas and the taste of **olive brine** ; want something to drink on pizza night.\n\n**You Might Also Like**\n\nYoung, crianza-level Tempranillo, Barbaresco or Nebbiolo from Valtellina, Mencia, Barbera, Valpolicella and Cru Beaujolais.\n\nWhen it comes to Italy, there are a lot of grapes to talk about \u2013 350 if you just count those that have been recognised as 'official' \u2013 but only a handful that you can't not talk about. As a case in point, there's no skipping over Sangiovese: the high-to-low charmer of central Italy that also happens to be the country's most widely planted varietal.\n\nThe ironic thing about Sangiovese, of course, is that for a grape that is so well-known and internationally recognised, its personality is not always so simple to pinpoint. It can just as easily be the stuff of uncomplicated table wines (\u00e0 la bulbous-bottle-wrapped-in-a-straw-basket style), or make some of the most enthralling, complex and highly regarded reds in the world. Part of this inconsistency has to do with the fact that this old variety has, over many years, spawned several clones, and not all of them are capable of achieving the same quality. The other key element of the equation is vineyard site, which can dramatically alter the ultimate character of the wine produced.\n\nIn terms of being able to experience the full range of Sangiovese's repertoire, Tuscany is where it's at. In Chianti, where the grape accounts for the overwhelming majority of the blend, you can find everything from basic, fresh red-fruited versions to the higher-quality 'Classico' designations (think flavours of roasted tomato, balsamic vinegar and herbs), as well as the more intense and savoury-toned 'Riserva' expressions that have seen a couple years of aging. Medium-bodied and acid-flush Sangiovese also produces some of its highest calibre work as Brunello di Montalcino. These more firmly structured, aged Sangioveses take on darker fruit flavours, as well as notes of baking spices, leather, tobacco and dried flowers and herbs. For a lighter, kind of gateway version of these (in both price-point and flavour), look for bottles from Rosso di Montalcino. Nearby, too, Sangiovese dominates the wines of the tasty though decidedly less acclaimed Vino Nobile di Montepulciano and its younger sibling version, Rosso di Montepulciano (neither of which, FYI, have any relation to the grape Montepulciano \u2013 confusing, I know).\n\nAnd, of course, we would be remiss not to give a shout-out to Sangiovese's role in another one of the region's highly publicised wines, the so-called 'Super Tuscans'. These big, bold, Bordeaux-by-way-of-Tuscany reds see the grape teamed up with some unexpected partners such as Cabernet Sauvignon, Cabernet Franc and Merlot, and are aged in small French oak barrels.\n\nBeyond Tuscany, Sangiovese's influence extends east through Umbria and the Marche, as well as north in Emilia-Romagna. Outside of Italy, however, Sangiovese doesn't have much reach, with some examples to be found in California in the USA, Argentina, Australia and the French island of Corsica, where it actually gets some decent play.\n\n# Pinotage\n\n# **Say It Right:** _Pee-no-taj_\n\n**Notable Regions**\n\nSouth Africa.\n\n**Drink It With**\n\nMeat, and lots of it. But if you want to get specific, barbecued food works especially well.\n\n**Key Words**\n\nYou want something **dark, full-bodied and boldly tannic** ; might say that you like **rough-edged, muscular** , and just a little bit aggressive wines \u2013 nothing shy or delicate; your thing is **earthy and smoky flavours.**\n\n**You Might Also Like**\n\nSyrah, Petit Sirah, Negroamaro, Petit Verdot and Mourv\u00e8dre.\n\nUsually when you meet someone's parents it just clicks: 'Ohhhhh, that's why you're like that, that explains it.' Occasionally, however, there's a total disconnect: 'Whoa, where did you come from?' It's a reaction that Pinotage has probably grown quite accustomed to by now.\n\nSouth Africa's signature grape first was conceived in 1925 by viticultural scientist Abraham Perold. His hope was that by crossing Pinot Noir with Cinsault, you'd end up with a grape that had the aromatic and flavour complexity of the former, combined with the latter's sturdy productivity in the vineyard. In the end, Pinotage had its own idea about the kind of grape it was, and the wine it would grow up to make.\n\nUnlike the delicate, light-bodied wines produced by Pinot Noir, dark, thick-skinned Pinotage makes a hardy, boozy red with fierce tannins. It doesn't share Pinot's naturally high acidity either, so the rich flavour of ripe black fruits is even more pronounced. Notes of tobacco, tar, spice, campfire, cured meat, menthol and black liquorice add to its assertive personality. The varietal is relatively easy to grow like its other parent, Cinsault, except unfortunately, this led to a lot of overproduction and sloppy winemaking, inevitably hurting the grape's reputation. This is especially problematic given that vineyard disease and mishandling the grape during winemaking can accentuate a compound that gives off an odour of burnt rubber and spray paint. Not ideal.\n\nThankfully, in recent years, several determined producers have come together with the focus of improving quality standards and showcasing Pinotage's true potential. And so it remains an important component in the country's red blends, and as a single varietal that can range from drink-now juicy to something more robust that's intended for aging.\n\n# Champagne\n\nFor a sparkling wine to be called Champagne _(Sham-pain)_ , it must be:\n\n**1. Grown and vinified in the Champagne region of France**\n\n**2. Made from one or a combination of the following three grapes***\n\n**3. Produced using the traditional or 'Champagne' method.**\n\nIn other words, the second fermentation \u2013 when the wine goes from still to sparkling \u2013 occurs in the bottle in which it is ultimately sold, as opposed to a large tank, for example.\n\n*Ok, so, technically there are seven grapes that can legally be included to make Champagne, but those are the major three.\n\n# Style Guide\n\nSome other style-indicating key words to know and look out for on labels:\n\n**NV**\n\nWines from multiple vintages blended together to create a specific, signature 'house style'.\n\n**Blanc de Blanc**\n\nLiterally, 'white from white'. Translation: the Champagne is made from 100% Chardonnay grapes.\n\n**Blanc de Noirs**\n\nLiterally, 'white from black'. Translation: the Champagne is made from 100% red grapes; either all Pinot Noir, all Pinot Meunier or some combination of the two.\n\n**Ros\u00e9**\n\nAs straightforward as it sounds: pink Champagne. Typically, these are made by adding a small amount of still red wine to a white-coloured base juice. Or, in the case of the 'Ros\u00e9 de Saign\u00e9e' style, the pink-tinted juice is the result of letting the juice sit in contact with the pigmented red grape skins for a period of time, aka how most still ros\u00e9 is made around the world.\n\n**Special\/Prestige\/T\u00eate de Cuv\u00e9e**\n\nA producer's very top, most special bottling, i.e. your Cristals and Dom P\u00e9rignons.\n\n**Vintage**\n\nEntirely produced from the fruit of a single vintage, which is stated on the label. Think of these as being like a sparkling, liquid snapshot of an especially good year. These are also aged for longer than non-vintage Champagnes, and to the surprise of no one, are going to be more expensive.\n\n# Sweetness Meter\n\nThe sweetness level of a Champagne (or dryness, depending on how you want to look at it) is determined by the amount of sugar added back to the wine before bottling. Because Champagne is such a cold growing region and it's a struggle for the grapes to fully ripen, the base wine is often quite tart and tangy. Those who enjoy a punch of acid and minerals to the mouth should look for Brut Nature and Extra Brut styles, which have either no or barely any sugar added back in. Those looking for a more Goldilocks experience should stick to Brut, which uses just enough sugar to round out the acid's sharp edges and leave the wine tasting refreshingly crisp and dry.\n\n# France\n\nWithin France, eight regions are allowed to produce a style of Champagne-method sparkling called Cr\u00e9mant _(Cray-mont)_. Think of this as your 'Champagne tastes on a sparkling wine budget' category. With these, the grape variety in the wine depends on where it's made (the region is indicated after the 'de', making provenance easy to figure out). Here's a quick breakdown.\n\n**1. Cr\u00e9mant de Bourgogne:** Mostly Chardonnay and Pinot Noir (shock, shock). Made in both white and ros\u00e9 styles.\n\n**2. Cr\u00e9mant de la Loire:** Mostly Chenin Blanc, Chardonnay and Pinot Noir. Made in both white and ros\u00e9 styles.\n\n**3. Cr\u00e9mant d'Alsace:** Pinot Blanc, Pinot Gris, Auxerois, Riesling, Chardonnay and Pinot Noir for the white blends. Must be 100% Pinot Noir for the ros\u00e9 sparkling.\n\n**4. Cr\u00e9mant du Jura:** White and ros\u00e9 styles made from Chardonnay, Pinot Noir, Savagnin, Poulsard, Trousseau and Pinot Gris.\n\n**5. Cr\u00e9mant de Bordeaux:** Mostly Merlot, with minor appearances from the region's other important grapes (Cabernet Sauvignon, Petit Verdot, Sauvignon Blanc, S\u00e9millon, etc.). Made in both white and ros\u00e9 styles.\n\n**6. Cr\u00e9mant de Savoie:** White-only sparklings from Jacqu\u00e8re, Altesse, Chardonnay, Chasselas and Aligot\u00e9.\n\n**7. Cr\u00e9mant de Limoux (FYI, south-central France):** Mostly Chardonnay and Chenin Blanc, with some Pinot Noir and the local grape Mauzac blended in.\n\n**8. Cr\u00e9mant de Die (FYI, southern Rh\u00f4ne):** White sparkling made from mostly Clairette, sometimes with a little Muscat and Aligot\u00e9 thrown in the mix.\n\n# Italy\n\n**Prosecco _(Pro-sek-oh)_**\n\n**Where:** The sparkling wine is made throughout the Veneto and Friuli regions of northeastern Italy. The best, highest quality versions come from the sub-regions of Conegliano Valdobbiadene and Colli Asolani.\n\n**What:** All prosecco is made from the Glera grape.\n\n**How:** Unlike Champagne, prosecco is produced using the Charmat method. Long story short, the wine goes from still to sparkling in a large tank instead of the individual bottles.\n\n**Franciacorta _(Fran-sha-core-tah)_**\n\n**Where:** Franciacorta is the name of a specific growing region in Lombardy where the eponymous sparkling wine is allowed to be made.\n\n**What:** Made from a blend of Chardonnay, Pinot Noir and Pinot Blanc.\n\n**How:** Considered the Champagne of Italy, these exceptional sparklings are made using the traditional Champagne method.\n\n**Lambrusco _(Lam-brooz-ko)_**\n\n**Where:** Emilia-Romagna and Lombardy in northeastern Italy.\n\n**What:** Lambrusco is both the name of the wine and the grape from which it's made. What most people don't realise, though, is that Lambrusco comes in several different varieties, the selection of which determines the style of the wine. Strains like Lambrusco Grasparossa and Lambrusco Salamino di Santa Croce produce the dark, inky, cranberry-and-dried flowers style that tends to come to mind when you hear 'Lambrusco'. But there's also Lambrusco di Sorbara, which makes a tart, high-acid, ros\u00e9-coloured style that almost drinks like a sour beer.\n\n**How** : As with prosecco, using the Charmat method.\n\n# Spain\n\n**Cava _(Ka-vah)_**\n\n**Where:** Pened\u00e8s, in northeastern Spain.\n\n**What:** A blend of Macabeo, Xarello and Parellada grapes.\n\n**How:** The Champagne method.\n\n# Germany and Austria\n\n**Sekt _(Seh-kt)_**\n\n**Where:** More or less wherever wine is being made throughout Germany and Austria.\n\n**What:** Expectedly, Riesling is the star in Germany, while in Austria, Gr\u00fcner Veltliner takes the spotlight. Of course, all the usual suspects show up in secondary supporting roles: Pinot Noir, Pinot Gris, Sylvaner and Pinot Blanc in Germany, and Bla\u00fcfrankish, Zweigelt and Riesling in Austria.\n\n**How:** 'Sekt' is not a protected term, so it's a little bit like the 'Wild Wild West' as far as rules for the style go. For the most part, the wine is made in the Charmat method, and a lot of it is quite good. You also can find some top-quality versions made in the Champagne method; these are labelled _Winzersekt_ in Germany and _Sekt Reserve_ and _Sekt Grosse Reserve_ in Austria.\n\n# P\u00e9tillant Naturel  \n _(Pay-tee-ahn Natch-er-elle)_\n\nThis old-school style of sparkling wine has become the latest cool kid. Why all the love? For one thing, it's like, totally chill, man. Unlike Champagne and many of the other big-name sparklers listed previously, Pet-Nat, as it's referred to, isn't fenced in by a long list of rules and requirements. It can be made throughout the winegrowing world and the grape selection is left up to the producer. The only real qualifier for Pet-Nat is the technique: basically, the unfinished grape juice is bottled and capped so that as the wine finishes fermenting, the CO2 bubbles remain trapped inside along with the dead yeast cells, which sink to the bottom.\n\nBeing a low-to-no intervention method, it's prolific throughout the natural wine community, giving it more cach\u00e9. Also, the fresh, easy, no-frills style is just plain tasty. What's more, the un-stuffiness of the style matches up with a growing push to make wine more fun and accessible. The Loire Valley, northern Italy and California and Oregon in the USA are particular hotbeds for the fizz.\n\n# Ros\u00e9: The FAQs\n\n# What grapes can be used to make ros\u00e9?\n\nLiterally, almost any red grape varietal.\n\nThe word 'red' is key there, by the way. Making ros\u00e9 isn't like mixing paint colours, where blending red with white gives you pink. It's the result of clear grape juice being left in contact with the pigmented red grape skins for a determined period of time \u2013 as far as quality wine is concerned, though, the only real exception to this is with ros\u00e9 Champagne, which sometimes gets its colour from blending.\n\nBut back to the question at hand. More than anything, the grape selection in a ros\u00e9 depends on where the wine is being produced and what red varietals grow there. If it's from Provence, the wine is likely to be some combination of Grenache, Carignan, Mourv\u00e8dre, Cinsault and Syrah. In one from California or Oregon, you're likely to find Pinot Noir as the star. In various parts of Spain, the grapes used for ros\u00e9s will be Tempranillo or Grenache. As for Italy, you can find ones made from Barbera in the north, Sangiovese and Montepulciano in the middle of the country and Nerello Mascalese in the south. And so on.\n\n# Where is ros\u00e9 made?\n\nYou can find ros\u00e9 from just about all of the usual winegrowing suspects: Italy, Spain, Portugal, Germany, Austria, USA, Australia, New Zealand, Chile, Argentina, South Africa and, of course, France. The latter is actually the world leader in terms of ros\u00e9 production, thanks in no small part to Provence, which has basically become synonymous with the stuff.\n\n# Is colour an indication of sweetness level?\n\nNo! So often people look at anything a shade above the palest of pinks and think, 'Nope, no thank you, that one must be sweet'. But when it comes to ros\u00e9, please try not to judge a book by its colour.\n\nThe fact of the matter is, the intensity of a ros\u00e9's 'ros\u00e9' is more an indication of production choices than anything else. Generally speaking, the lighter the pink, the less time the red grape skins spent in contact with the fermenting juice, and vice versa. Seriously, that's the major takeaway there. Broadly speaking, this can give us an idea of what to expect, flavour-wise: the super-pale do tend to roam the light-lean-crisp range, while the darker ones are more full-bodied with bolder fruit flavours and structure. But still, at the end of the day, the colour is by no means a guarantee of dryness. That, I'm afraid, is a question for the 'How Much Sugar Has Been Converted to Alcohol' department. And actually, those very-close-to-being-light-red ros\u00e9s are often some of the driest examples of the style, given that they have had more time to pick up the naturally occurring tannins in the skins.\n\nIt's also worth noting that a ros\u00e9's shade may also be a reflection of the grape(s) in the blend. Some red grapes have thicker skins and are more boldly pigmented than others \u2013 for example, Mourv\u00e8dre in southern France versus Pinot Noir in Germany \u2013 and can therefore result in a darker-coloured, yet dry ros\u00e9 if used as a significant component in the blend.\n\n# Can ros\u00e9 age?\n\nContrary to popular opinion, yes, it most certainly can.\n\nGranted, it's not necessarily the recommended path for all ros\u00e9s. Most are produced with the intention of being drunk young, likely quickly pressed and fermented at cooler temperatures in stainless steel to deliver a fresh, easy, crisp style. Those that handle maturing best tend to have been fermented and\/or aged in the barrel for some period of time, which not only lends complexity, but also sets up a better structural framework for longevity. Often, the blend also includes bigger, bolder, more naturally tannic grapes, the juice from which will have macerated for a longer period of time on the skins and seeds. Remember, tannin in wine is built for, and benefits from, time.\n\nSome of the most notable examples of ageworthy ros\u00e9 come from the Bandol and Tavel regions of Provence, where Mourv\u00e8dre is the star, as well as Spain (look for those from Rioja specifically) and Portugal. Just a quick heads up so you don't go into price tag-induced shock: these longer-lifespan ros\u00e9s are going to be more expensive on account of the additional time and effort that goes into producing them.\n\n# When is ros\u00e9 season?\n\nThere isn't one! Seriously, there is absolutely no reason why you shouldn't be drinking ros\u00e9 all year round. Think about it \u2013 it's not like we all stop drinking chilled whites when the weather turns cold, or forgo reds for the entirety of summer, right? New, current vintage ros\u00e9 is typically released sometime in early spring and, especially in recent years, the very successful and strategic marketing hype around that has cemented a public perception that pink wine is a seasonal drink, appropriate only in spring and summer. But the fact of the matter is, ros\u00e9 is as refreshing, versatile and food-friendly a wine the rest of the year as it is during those months.\n\n# What foods pair best with ros\u00e9?\n\nWhile it is often brushed off as the thing you drink while you wait for the meal to be ready (you know, before the 'serious' wines hit the table), ros\u00e9 actually partners well with a wide range of foods. That it shares qualities of both red and white wines makes it incredibly adaptable, so when strategising a pairing, the biggest consideration is the style of the ros\u00e9. Is it light and dry with bright acidity? Then it's probably a good candidate for raw oysters, garlicky grilled prawns (shrimp) or steamed clams, delicately prepared seafood dishes and light salads with tangy vinaigrettes. But if the ros\u00e9 is more robust and savoury, with a slight touch of tannic grip? Think grilled meats and vegetables, charcuterie, pasta with a hearty red sauce or even pizza. And lastly, those that are bit more juicy and fruit-forward? Let them play with the bold, complex spices of Indian and Middle Eastern cuisine, or temper the tingly heat of Thai or Szechuan dishes; also, berry tarts and desserts.\n\n# Wine and Food Pairing: A Strategy Playbook\n\nFood and wine pairing is much more about thoughtful strategy and playful experimentation than following hard-and-fast rules. So with that in mind, below are seven common approaches to the matchmaking game. Just remember: have fun, don't take it too seriously and please, don't get hung up on looking for the 'perfect' pairing.\n\n# The Matchy-Matchy Route\n\nThis strategy is all about considering the ways in which a wine and a dish are peers. The thought here being that if the two share similar dominant characteristics, then they are able to stand shoulder-to-shoulder without one overpowering the other. For example, a shy, neutral, softly flavoured wine is going to get completely overshadowed by a boldly flavoured dish \u2013 better to have two passionate, intense characters going tit for tat. Or think of the weight of the wine in comparison to the food \u2013 if the former is too light, it likely won't be able to support a heavy, rich dish. By that same token, a dessert is going to make a dry wine taste even drier (like, in a scrunch-your-face bitter kind of way), whereas the sugar content in a sweet wine makes the bite-and-sip sequence completely seamless. And be mindful of keeping acid levels in balance, too; if the tangy acidity in the food outranks that of the wine, whatever's in your glass is going to taste dull and flabby.\n\n# The Focal Point Pitch\n\nA kind of sidebar tangent to the matchy-matchy route, this tactic has you looking to draw a line between the greatest common denominator quality shared by the wine and the food. Ordering a light, lemon-dressed, sea-briny shellfish dish? Then maybe you want to go with something equally mineral and citrus driven, such as Albari\u00f1o. Or perhaps you're cooking a pepper-crusted steak? Sounds like it could be a job for meaty, spice-forwards Syrah... you get the idea.\n\n# The 'Opposites Attract' Counter Offensive\n\nAs anyone familiar with the story of Beauty and the Beast knows, sometimes the puzzle piece fits with the partner you least expect it to. On paper, you're on opposite ends of the spectrum but, in practice, the balance is right and you just, you know, work. When it comes to food and wine pairing, among the most iconic of these unlikely perfect partnerships include:\n\n**Sweet and salty**\n\n(i.e. Sauternes with a stinky blue cheese)\n\n**Sweet and spicy**\n\n(i.e. Alsatian Gew\u00fcrztraminer with Szechuan or Indian cuisine)\n\n**Sweet and fatty**\n\n(i.e. off-dry German Riesling with pork belly)\n\n**Acid and fatty**\n\n(i.e. Chablis and a lemon-butter-cream sauce)\n\n**Fatty and bitter**\n\n(i.e. a porterhouse steak and a dry, tannic Cabernet Sauvignon)\n\n# The Trap: Beware Oil and Water Combinations\n\nIn spite of the aforementioned unlikely pairings, sometimes there are combinations that just simply don't work. Instead of making each other better, the two parties actually end up bringing out the worst in one another. So, for example, when you take a swig of a high-acid wine after a bite of spicy food and it feels like someone poured kerosene on your tongue and lit a match. Or when you have bitter greens with a tannic wine and all that remains is a bitter taste in your mouth. With that in mind, generally bitter doesn't really play well with spicy or acid, either, so maybe keep them on separate tables.\n\n# The Sauce Slide Tackle\n\nIt's easy to get distracted when you're planning a food and wine pairing. The dish has its complex network of components to consider \u2013 flavour, texture, acidity, weight, etc. \u2013 as does the wine. So where should you focus your attention? One popular tactic is to focus on matching the wine not to the main protein or veggie in the dish, but to the sauce it's dressed with. Are we talking about a thick and creamy butter sauce that would benefit from a high-acid wine to cut the richness, or a meat and tomato sauce that might be enhanced with a wine that has savoury and herbal components to contribute? Let that be your starting point, and things become a little simpler.\n\n# The Heritage Drive: What Grows Together, Goes Together\n\nThis oldie but goodie has less to do with thinking about the specific characteristics in the food and wine, and more about trusting the symbiotic nature of nature. For example, this solid, 'I don't want overthink this' train of thought might lead to popping a bottle of dry Lambrusco alongside a spread of charcuterie, as they might in Emilia-Romagna. Or, channelling the oceanic coast of western Loire Valley might inspire you to throw back some freshly shucked oysters with a couple of glasses of Muscadet.\n\n# The Sparkling Wine Hail Mary\n\nAnd, when in doubt, don't overlook bubbles. It's got all the right stuff: acid and texture in spades; a range that spans from rich and savoury to lean and electric; the ability to be ripe and fruity or searingly mineral and dry and some ros\u00e9s (usually the deep-coloured ros\u00e9 de saign\u00e9e styles) can even have enough tannic structure to stand up to red meat.\n\n# A Quick Guide to More Grapes You Should Know\n\n# Whites\n\n**Aligot\u00e9** _(Alley-go-tay)_ The 'other' white grape of Burgundy. A minor exception to the white Burgundy = Chardonnay rule of thumb, this lean, mineral, dry white with bright, citrussy acidity is what you should reach for when you want a less expensive alternative to Chablis.\n\n**Savagnin** _(Sah-vahn-yin)_ Native to the Jura region of central eastern France, seek out this cool-kid grape if you're intrigued by the sound of a wine that tastes like a salty, savoury and funky version of Chardonnay.\n\n**Furmint** _(Foor-mint)_ Primary white grape varietal in Hungary, most famously responsible for the luscious, honeyed sweet wines of Tokaji, as well as some unique, full-bodied dry expressions from the same region.\n\n**Torront\u00e9s** _(Tore-on-tez)_ Important in Argentina, this lush, fruity, floral white makes a good stand-in for fans of Muscat and Gew\u00fcrtztraminer.\n\n**Sylvaner** _(Sil-von-er)_ You'll mostly run into this dry, neutral white in Germany and Alsace. If you enjoy Pinot Blanc, give it a try.\n\n**Viura** _(Vee-your-ah)_ Aka Macabeo, this fuller, fruity, slightly floral Spanish number is an important component in white Rioja and Cava.\n\n**Garganega** _(Gar-gahn-eh-gah)_ The grape behind the citrus and almond-flavoured whites of Veneto's well-known Soave region.\n\n**Vermentino** _(Vermin-tee-no)_ Otherwise known as Pigato in Liguria and Rolle in southwestern France, this grape's most notable home is Sardinia, where it delivers everything you'd want from an island white: crisp citrussy acidity, notes of white flowers and an undercurrent of minerals and sea brine.\n\n**Picpoul** _(Peek-pool)_ This grape's fierce, lemony acidity is the inspiration behind its name, which literally means 'lip stinger'. You'll mostly see it in southern Rh\u00f4ne and the Languedoc-Roussillon, but some interesting examples also exist in California, USA.\n\n**Scheurebe** _(Shoy-ray-buh)_ A cross between Riesling and Sylvaner, this full-fruited, high-acid, aromatic varietal takes after its parents, and is a fun alternative to either, given the option.\n\n**Ribolla Gialla** _(Ree-bowl-ah Gee-ah-lah)_ An old-school variety native to the Friuli-Venezia Giulia region of Italy that sits in the light-bodied and herbal-floral category. A popular orange wine candidate, it's a good grape to get to know if you like nerding out on less conventional juice.\n\n**Verdejo** _(Verr-day-ho)_ Spanish varietal that can be a good stand-in for Sauvignon Blanc drinkers.\n\n# Reds\n\n**Cinsault** _(Sin-so)_ Light-bodied, aromatic grape seen mostly in southern France and Corsica as a component in red blends and ros\u00e9s.\n\n**Touriga Nacional** _(Too-ree-gah Nah-seon-al)_ Major varietal used in the production of that oh-so-famous dessert wine, Port. Also grown in Portugal's D\u00e3o region to produce dark, full-bodied, tannic reds that are ready to drink young and are generally inexpensive.\n\n**Nerello Mascalese** _(Ner-ello Mass-ka-lay-zay)_ Important Sicilian grape that tends to produce fresh reds with notes of cherry, spice and smoke. It does its best work in the volcanic soil vineyards of Etna.\n\n**Bla\u00fcfrankish** _(Bl-ow-frank-ish)_ Arguably Austria's most important and 'serious' red varietal. An interesting alternative option for fans of Northern Rh\u00f4ne Syrah thanks to its firm, savoury character and distinct note of black pepper.\n\n**Zweigelt** _(Zz-vie-geldt)_ A cross between the Austrian red varietals Bla\u00fcfrankish and Saint Laurent, it errs on the side of light bodied and earthy, with notes of tobacco and bitter chocolate. Try it as a wallet-friendly alternative to Old World Pinot Noir.\n\n**Frappato** _(Frah-pah-toe)_ Increasingly popular Sicilian varietal that produces reasonably priced reds with an almost spritzy acidity and a charming smoky-spicy quality. Dynamite with Neapolitan-style pizza.\n\n**Trousseau** _(True-so)_ This Jura varietal produces light-bodied, appropriately tannic reds with a tart berries-meets-wild game profile. Your 'I'm looking for something like a funky version of Pinot' option. The region's other local grape, Poulsard, similarly fits that bill.\n\n**Mencia** _(Men-thee-ah)_ Medium-bodied, fresh-fruited aromatic red typically grown around northeastern Spain and Portugal. A good, slightly spicy barbecue-friendly option that won't break the bank.\n\n**Negroamaro** _(Nay-grow-ah-mar-oh)_ This 'black and bitter' varietal from Puglia, Italy's heel, is best known for producing full, tannic reds with rich flavours of dark fruits and dried herbs.\n\n**Agiorgitiko** _(Aye-your-yee-tee-ko)_ Important, widely planted Greek varietal used to make everything from juicy ros\u00e9s and easy, light-bodied quaffers to more 'serious' dark and spicy ageworthy wines.\n\n**Corvina** _(Core-veena)_ This light, not-too-tannic, sour cherry-fruited varietal is best known for teaming up with Rondinella and Molinara to produce the Veneto's renowned reds: Valpolicella, Valpolicella Ripasso and Amarone della Valpolicella.\n\n**Grolleau** _(Gruh-low)_ Underdog Loire Valley varietal used to make fruity still and sparkling ros\u00e9s, as well as some compelling light, tangy-fresh, still reds. Gaining popularity with the natural wine movement. See also: Pineau d'Aunis, another sleeper hit in the region with more of a spicy profile.\n\n**Nero d'Avola** _(Nair-oh Dah-vo-lah)_ Your Sicilian-born alternative to Syrah or Cabernet Sauvignon.\n\n# Glossary\n\n### **The Technical Stuff**\n\n**ABV**\n\nAcronym for 'alcohol by volume'. Measured by percentage, a wine's ABV content must be listed on its label.\n\n**Acid**\n\nA natural and essential component in wine, the three main types are tartaric, malic and citric. Acidity in wine gives it structure, freshness, balance and also helps it to age.\n\n**Appellation**\n\nA defined, regulated area where grapes are grown and made into wine.\n\n**Biodynamics**\n\nA viticultural philosophy that takes a holistic, sustainable approach to vineyard management and winemaking. Common practices include fertilising the vines with the vineyard\/farm's own natural compost, using natural pesticides and relying on the moon's movement through the zodiac system to guide vineyard decisions like when to pick the grapes or prune the vines.\n\n**Botrytis Cinerea _(Boh-try-tis sin-era)_**\n\nAka 'noble rot', this is the 'good' fungus that infects certain white grapes and causes their sugars and flavours to concentrate as they shrivel. Responsible for the world's greatest white dessert wines, such as Sauternes, sweet Tokaji, Trockenbeerenauslese Riesling and Vouvray Moelleux.\n\n**Corked**\n\nWhat you would call a faulty wine that has been damaged by a bad, bacteria-infected cork. The smell is most commonly likened to that of a wet dog or damp, musty cardboard.\n\n**Dry**\n\nHow you describe a wine that does not taste sweet.\n\n**Fermentation**\n\nThe process in which yeast \u2013 often the ones naturally present on the grapes \u2013 converts the sugar in the grapes to alcohol and releases carbon dioxide.\n\n**Natural Wine**\n\nThe category isn't regulated, but basically it refers to wines made with as little intervention as possible, both in the vineyard and the winery. This means no chemicals in the vineyard, only indigenous yeasts used to start and complete fermentation, minimal (if any) additional sulphur and (usually) no filtration.\n\n**Oxidised**\n\nHow you characterise a wine that has started to deteriorate due to oxygen exposure. Generally not a good thing, although it is intentionally practised in some wines, i.e. Madeira and 'sous voile' whites in the Jura region of France.\n\n**Residual Sugar**\n\nThe amount of natural sugar leftover in a wine that has not been converted into alcohol during the fermentation process. Significant amounts of this are what makes a wine taste sweet.\n\n**Tannin**\n\nThe phenolic compound found (mostly) in red wine that is derived from a grape's skins, seeds and stems, as well from barrel fermentation. Simply put: it's what causes a drying, astringent sensation in your mouth. It's an essential component of the structure of a red wine that also affects its ageability.\n\n**Terroir**\n\nYour basic, catch-all term for 'sense of place'. Everything about the environment in which the grape is raised \u2013 soil type, climate, etc. \u2013 that ultimately influences and shapes the wine it becomes. We look for and prize wines that express this well; a kind of snapshot of where it's from that you can identify through taste.\n\n**Volatile Acidity**\n\nCaused by acetic acid, it gives wine a less-than-pleasant smell of nail polish remover mixed with vinegar. Sometimes a concern with 'natural' wine, it's possible for the smell to blow off with aeration.\n\n### **The Stuff Sommeliers Say**\n\n**Balance**\n\nAchieved only when a wine's essential components \u2013 acid, tannin, texture, fruit, alcohol and flavour \u2013 are all working together harmoniously. No good wine is without it.\n\n**Big**\n\nKind of as it sounds, a wine that is full-bodied (and often quite boozy), with a rich, heavy texture.\n\n**Body**\n\nThe assessment of a wine's weight and texture.\n\n**Bright**\n\nAnother way of characterising a wine with high, vibrant acidity. 'High-toned' is also used in a similar capacity.\n\n**Chewy**\n\nA wine with so much body and texture it almost feels like you're sinking your teeth into it.\n\n**Closed**\n\nA wine that is not displaying much in the way of aromatics or personality. Especially with older wines, decanting will help the wine wake up and open. Synonyms: muted, tight, needs air.\n\n**Complex**\n\nA wine with a lot going on, in a good way.\n\n**Concentrated**\n\nAnother way of saying that the fruit flavours in a wine taste super ripe.\n\n**Earthy**\n\nA broad term used to describe non-fruit flavours in a wine. 'Savoury' is kind of a cousin synonym for this.\n\n**Fat**\n\nA heavy, round wine that doesn't have the acid to balance out the weight. Synonym: flabby.\n\n**Finish**\n\nThe lingering impression a wine leaves on your palate. The best wines will have a long finish that keeps you thinking about them and makes you want to go back for more.\n\n**Fruit bomb**\n\nA wine that's overly dominated by fruit flavours.\n\n**Grippy**\n\nWhen the tannin in the wine is so present and intense, you can feel the astringency gripping to the sides of your mouth and gums.\n\n**Hot**\n\nWhen a wine is excessively alcoholic, like, burn-your-nose-hairs boozy.\n\n**Infanticide**\n\nWhen you prematurely open a wine that would have benefited from time to age in the bottle.\n\n**Juice**\n\nPopular (and admittedly slightly obnoxious) euphemism for 'wine'.\n\n**Juicy**\n\nBasically exactly as it sounds, a wine that is fresh and ripe and just makes your mouth water. Not to be confused with sweet.\n\n**Lean**\n\nUsually follows the word 'light' in a description, it's a way of characterising wines with high, intense acidity and not a lot of weight. 'Lean' wines feel like they cut directly down your palate, they're not heavy or coating. Synonyms: crisp, zippy, electric, racy, chiselled.\n\n**Mineral\/Minerality**\n\nNot literally the mineral content of a wine, but rather how you would describe a wine that gives an impression of wet rocks or chalk. I know it sounds weird, but just think about it the next time you have a glass of Chablis or Muscadet.\n\n**Nose**\n\nThe overall impression of a wine's aroma.\n\n**Oaky**\n\nThat bouquet of vanilla, baking spices, and toasty wood that fermenting and aging in oak barrels (especially new wood) imparts on a wine.\n\n**Off-Dry**\n\nHow you describe a wine that has a little bit of sweetness to it.\n\n**Old World\/New World**\n\nOld World wines are those from Europe, whereas New World wines include the USA, Canada, South America, South Africa, Australia and New Zealand.\n\n**One-Note**\n\nEssentially, a boring wine. It only brings one thing to the table, the opposite of complex.\n\n**Opened up**\n\nWhen a wine that seemed closed and shy when uncorked finally reveals the full spectrum of its personality. Synonym: unfurled.\n\n**Peaked\/Over the Hill**\n\nA wine that you've opened too late, so it is passed its prime.\n\n**Porch Pounder**\n\nSlang for an easy, relatively uncomplicated wine that you could just chug, gleefully, feet up sitting on the porch. Synonym: crushable.\n\n**Polished**\n\nHow you might describe a wine that tastes pristine, in a somewhat affected, too-intentional way.\n\n**Singing**\n\nA complementary way of saying that a wine is showcasing its qualities well.\n\n**Thin**\n\nA wine lacking body and flavour.\n\n**Unicorn**\n\nWhat the wine geeks call a super rare, highly prized wine.\n\n# Index\n\n##### A\n\nAbruzzo: Trebbiano\n\nABV , ,\n\nacid\n\nacid and fatty pairings\n\nageing\n\nindicators\n\nterminology\n\nAgiorgitiko\n\nAlba: Barbera\n\nAlbari\u00f1o\n\nAlcohol by Volume , ,\n\nAligot\u00e9\n\nall-purpose glasses\n\nAlsace\n\nGew\u00fcrztraminer \u2013\n\nPinot Blanc\n\nPinot Gris\n\nRiesling\n\nAlto-Adige\n\nGew\u00fcrztraminer\n\nMerlot\n\nPinot Bianco\n\nPinot Grigio\n\nSauvignon Blanc\n\nAlvarinho\n\nAmarone della Valpolicella\n\nAmerican Nouveaus\n\nAnjou: Chenin Blanc\n\nappellation ,\n\napps\n\nArgentina: Malbec \u2013\n\nArmagnac\n\nAssyrtiko \u2013\n\nAsti: Barbera\n\nAustralia\n\nPetit Verdot\n\nS\u00e9millon\n\nShiraz \u2013\n\nAustria\n\nGr\u00fcner Veltliner \u2013\n\nRiesling\n\nSekt\n\n##### B\n\nbalance\n\nBandol\n\nMourv\u00e8dre\n\nros\u00e9\n\nBarbaresco\n\nNebbiolo \u2013\n\nBarbera\n\nBarolo\n\nNebbiolo \u2013\n\nBarossa Valley\n\nRiesling\n\nShiraz\n\n_barriques_\n\nBarsac\n\nSauvignon Blanc\n\nS\u00e9millon\n\nBeaujolais\n\nThe Crus of Beaujolais\n\nGamay \u2013\n\nrevival of 'Nouveau'\n\nBeaujolais Nouveau\n\nbig\n\nbiodynamics\n\nbitter and fatty pairings\n\nbitter and spicy\/acid pairings\n\nblanc de blanc\n\nblanc de noirs\n\nBla\u00fcfrankish\n\nbody\n\nBonnezeaux: Chenin Blanc\n\nBordeaux\n\nCabernet Franc \u2013\n\nCabernet Sauvignon\n\nMalbec\n\nMerlot \u2013\n\nSauvignon Blanc\n\nS\u00e9millon\n\nBordeaux Blanc\n\nthe Bordeaux glass\n\nbotrytis cinerea\n\nbotrytized grapes \u2013. _see also_ 'noble rot'\n\nsweet wines \u2013\n\nbottle openers\n\nbottles\n\nopening ,\n\nscrew-top\n\nsize\n\nwax closures\n\nBouchet. _see_ Cabernet Franc Bourgeuil\n\nCabernet Franc\n\nbranded names\n\nBreton. _see_ Cabernet Franc\n\nbright\n\nBrown Muscat\n\nBrunello di Montalcino ,\n\nBurgundy\n\nChardonnay\n\nPinot Noir \u2013\n\nthe Burgundy bowl\n\n##### C\n\nCabernet Franc , \u2013 and Cabernet Sauvignon 88 and Merlot\n\nCabernet Sauvignon \u2013\n\nand Merlot\n\nCahors\n\nMalbec\n\nCalifornia\n\nCabernet Sauvignon\n\nCarignan\n\nChenin Blanc\n\nGamay\n\nMataro\n\nMerlot\n\nSauvignon Blanc\n\nViognier\n\nZinfandel \u2013\n\nCannonau\n\nCarignan\n\nCarignano\n\nCari\u00f1ena\n\nCassis region: Marsanne \u2013\n\nCava\n\nCencibel. _see_ Tempranillo\n\nCentral Otago: Pinot Noir\n\n_cepas vellas_\n\nChablis\n\nChardonnay\n\nChampagne ,\n\nFrench regions\n\nPinot Noir\n\nstyle guide\n\nsweetness meter\n\nterminology \u2013\n\nChardonnay \u2013\n\nChampagne\n\nCharmat method\n\nCh\u00e2teau-Grillet: Viognier\n\nChateauneuf-du-Pape\n\nGrenache\n\nRoussanne\n\nSyrah\n\nChenin Blanc \u2013\n\nchewy\n\nChianti\n\nSangiovese \u2013\n\nChiavennasca. _see_ Nebbiolo\n\nChile\n\nCabernet Sauvignon\n\nMerlot \u2013\n\nchilling wine\n\nChinon\n\nCabernet Franc\n\nCinsault\n\nClairette de Die\n\nClare Valley\n\nAssyrtiko\n\nRiesling\n\nclosed\n\nCognac\n\ncolour and sweetness: ros\u00e9\n\ncomplex\n\nconcentrated\n\nCondrieu\n\nViognier\n\ncontext clues\n\ncooking with wine .\n\n_see also_ food pairings\n\ncorking ,\n\ncorks\n\nsmelling\n\nCornas\n\nSyrah\n\nCorton-Charlemagne:\n\nChardonnay\n\nCorvina\n\nC\u00f4t. _see_ Malbec\n\nC\u00f4te-R\u00f4tie: Syrah\n\nCoteaux de Layon: Chenin Blanc\n\nC\u00f4tes-du-Rh\u00f4ne\n\nMarsanne and\n\nRoussanne\n\nSyrah\n\nCr\u00e9mant d'Alsace ,\n\nCr\u00e9mant de Bordeaux\n\nCr\u00e9mant de Bourgogne\n\nCr\u00e9mant de Die\n\nCr\u00e9mant de la Loire\n\nCr\u00e9mant de Limoux\n\nCr\u00e9mant de Loire: Chenin\n\nBlanc\n\nCr\u00e9mant de Savoie\n\nCr\u00e9mant du Jura\n\nCrozes-Hermitage\n\nMarsanne and Roussanne\n\nSyrah\n\nThe Crus of Beaujolais\n\n##### D\n\ndecanting wine\n\nthe dessert wine glass\n\ndistributors\n\nDogliani\n\ndry\n\ndry white wine\n\n##### E\n\nearthy\n\nEden Valley: Riesling\n\nEmilia-Romagna: Lambrusco\n\nEntre-Deux-Mers: Sauvignon Blanc\n\nestate bottled\n\n##### F\n\nfatty and acid pairings\n\nfatty and bitter\n\npairings ,\n\nfatty and sweet pairings\n\nFederspiel\n\nfermentation\n\nFinger Lakes, New York State: Gamay\n\nfinish\n\nthe flute\n\nFoillard, Jean\n\nfood pairings , \u2013\n\nto avoid\n\nmatching\n\ncharacteristics \u2013\n\nopposites attract\n\nros\u00e9\n\nsauces and wines\n\nsparkling wine\n\ntraditional\n\nFrance: Champagne regions\n\nFranciacorta ,\n\nFrappato\n\nFriuli Venezia Giulia\n\nMerlot\n\nPinot Bianco\n\nPinot Grigio\n\nProsecco\n\nfruit bomb\n\nFum\u00e9 Blanc \u2013. _see also_ Sauvignon Blanc\n\nFurmint\n\n##### G\n\nGamay \u2013\n\nGarganega\n\nGarnacha\n\nGattinara\n\nGavi\n\nGermany\n\nRiesling \u2013\n\nSekt\n\nGew\u00fcrztraminer \u2013\n\nGhemme\n\nGigondas\n\nSyrah\n\nglasses\n\nall-purpose glasses\n\nthe Bordeaux glass\n\nthe Burgundy bowl\n\nthe dessert wine glass\n\nthe flute\n\noverpouring\n\nsmell before using\n\nglassware \u2013\n\nGlera grape\n\ngovernment warnings\n\nGrand Cru vineyards\n\ngrape varietals\n\nand place names \u2013\n\nGrauburgunder. _see_ Pinot Gris\n\nGraves\n\nCabernet Sauvignon\n\nSauvignon Blanc\n\nGreek wines \u2013\n\nGrenache \u2013\n\ngrippy\n\nGrolleau\n\nGr\u00fcner Veltliner \u2013\n\n'GSM' blends \u2013, . _see also_ Grenache; Mourv\u00e8dre; Syrah\n\n##### H\n\nHermitage\n\nMarsanne and Roussanne\n\nSyrah\n\nhot\n\nHunter Valley: S\u00e9millon\n\n##### I\n\nimporters ,\n\ninfanticide\n\nItaly\n\nFranciacorta\n\nLambrusco\n\nMerlot\n\nProsecco\n\nSangiovese \u2013\n\n##### J\n\njuice\n\n##### L\n\nlabels\n\nback\n\ndecoding \u2013\n\nfront \u2013\n\nphotographing\n\nLambrusco\n\nLanguedoc-Roussillon\n\nCarignan\n\nGrenache\n\nMerlot\n\nSauvignon Blanc\n\nViognier\n\nLapierre, Marcel\n\nlean\n\n'Left Bank' Bordeaux red\n\nCabernet Sauvignon\n\nPetit Verdot\n\nLiqueur Muscat\n\nLoire Valley\n\nCabernet Franc\n\nChenin Blanc\n\nGamay\n\nMalbec\n\nMelon de Bourgogne\n\nSauvignon Blanc\n\nLombardy\n\nLambrusco\n\nMerlot\n\nPinot Grigio\n\n##### M\n\nMalbec \u2013\n\nMargaux: Cabernet\n\nSauvignon\n\nThe Marlborough region:\n\nSauvignon Blanc\n\nMarsanne and Roussanne \u2013\n\nMataro\n\nMazuelo\n\nM\u00e9doc: Cabernet Sauvignon\n\nMelon de Bourgogne \u2013\n\nMencia\n\nMendoza: Malbec \u2013\n\nMerlot \u2013\n\nand Cabernet Franc\n\nand Cabernet Sauvignon\n\nMeursault: Chardonnay\n\nmineral\/minerality\n\nMonastrell \u2013\n\nMoscatel. _see_ Muscat\n\nMoscato. _see_ Muscat\n\nMoscato d'Asti\n\nMourv\u00e8dre \u2013\n\nMuscadet\n\nMelon de Bourgogne\n\nMuscadet S\u00e8vre-et-Maine:\n\nMelon de Bourgogne\n\nMuscat \u2013\n\nMuscat Blanc \u00e0 Petit Grains\n\nMuscat of Alexandria\n\nMuscat of Hamburg\n\nMuscat Ottonel\n\nMuskateller. _see_ Muscat\n\n##### N\n\nNapa: Cabernet Sauvignon\n\nnatural wine , ,\n\nNebbiolo , \u2013\n\nn\u00e9gociant wines \u2013\n\nNegroamaro\n\nNerello Mascalese\n\nNero d'Avola\n\nNew World wines\n\nBarbera\n\nChardonnay\n\nMerlot \u2013\n\nPinot Gris\n\nPinot Noir\n\nNew Zealand: Pinot Noir\n\n'noble rot' ,\n\nnose\n\nNouveau \u2013\n\nNV\n\n##### O\n\noaky\n\noff-dry\n\nOld World\n\none-note\n\nopened up\n\nopening bottles ,\n\nOregon\n\nGamay\n\nMelon de Bourgogne\n\nPinot Gris\n\nPinot Noir\n\nover the hill\n\noverpouring glasses\n\noxidation\n\noxidised\n\n##### P\n\npairings. _see_ food pairings\n\npeaked\n\nPeloursin\n\nPenedes\n\nCava\n\nMonastrell\n\nPerold, Abraham\n\nPessac-L\u00e9ognan\n\nCabernet Sauvignon\n\nS\u00e9millon\n\nP\u00e9tillant Naturel\n\nPetit Verdot\n\nPetite Syrah\n\nPicpoul\n\nPiedmont\n\nBarbera\n\nNebbiolo \u2013\n\nPinot Bianco\n\nPinot Blanc\n\nPinot Grigio \u2013\n\nPinot Gris \u2013\n\nPinot Meunier: Champagne\n\nPinot Nero. _see_ Pinot Noir\n\nPinot Noir \u2013\n\nChampagne\n\nPinotage\n\nPisco\n\nplace names and grape\n\nvarietals \u2013\n\npolished\n\nPomerol\n\nCabernet Franc\n\nMerlot\n\nporch pounder\n\nPortugal: Albari\u00f1o\n\nPouilly Fuiss\u00e9\n\nChardonnay\n\nPouilly Fum\u00e9\n\nSauvignon Blanc\n\nprestige bottling\n\nPrimitivo\n\nPriorat\n\nGarnacha\n\nMazuelo\n\nproducer names\n\nProsecco\n\nPuglia: Zinfandel\n\nPuligny-Montrachet:\n\nChardonnay\n\nPUR Winery\n\n##### Q\n\nQuarts de Chaume: Chenin Blanc\n\n##### R\n\nRasteau: Syrah\n\nRed Burgundy\n\nred wines\n\nplace names and grape varietals\n\nserving ,\n\nrefrigerators: storing opened wine\n\nregional classification\n\nregions\n\nresidual sugar\n\nrestaurants \u2013\n\nRetsina\n\nRh\u00f4ne Valley\n\nGrenache\n\nMarsanne and Roussanne\n\nSyrah\n\nViognier\n\nRias Baixas: Albari\u00f1o\n\nRibera del Duero:\n\nTempranillo\n\nRibolla Gialla\n\nRiesling \u2013\n\n'Right Bank' Bordeaux red\n\nCabernet Franc\n\nMerlot\n\nRioja\n\nGarnacha\n\nMazuelo\n\nros\u00e9\n\nTempranillo\n\nRoero\n\nros\u00e9\n\nageing\n\ncolour and sweetness\n\nfood pairings\n\ngrapes used\n\nregions\n\nRosso di Montalcino\n\nSangiovese\n\nRosso di Montepulciano\n\nRoussanne . _see also_\n\nMarsanne and Roussanne\n\n##### S\n\nSaint \u00c9milion\n\nCabernet Franc\n\nMerlot\n\nSaint-Estephe: Cabernet\n\nSauvignon\n\nSaint-Joseph\n\nMarsanne and Roussanne\n\nSyrah\n\nSaint-Julien: Cabernet\n\nSauvignon\n\nSaint-Nicolas-de-Bourgueil\n\nSaint-P\u00e9ray: Marsanne and\n\nRoussanne\n\nsalty and sweet pairings\n\nSancerre\n\nPinot Noir\n\nSauvignon Blanc\n\nSangiovese , \u2013\n\nSantorini: Assyrtiko \u2013\n\nSardinia: Cannonau\n\nSaumur-Champigny: Cabernet\n\nFranc\n\nSaumur: Chenin Blanc\n\nSauternes\n\nSauvignon Blanc\n\nS\u00e9millon\n\nSauvignon Blanc \u2013\n\nS\u00e9millon blends \u2013\n\nSavagnin\n\nSavenni\u00e8res\n\nChenin Blanc\n\nSavoie: Roussanne\n\nScheurebe\n\nscrew-top wines\n\nsecond label wines \u2013\n\nSekt\n\nS\u00e9l\u00e9ction de Grains\n\nNobles ,\n\nS\u00e9millon \u2013\n\nSauvignon Blanc blends \u2013\n\nserving wine \u2013\n\nshelf-recommendations\n\nShiraz \u2013\n\nshopping tips \u2013\n\nsinging\n\nSmaragd\n\nSoave\n\nSouth Africa\n\nChenin Blanc\n\nPinotage\n\nSpanna. _see_ Nebbiolo\n\nsparkling wine. _see also_\n\nChampagne\n\nCava\n\nChardonnay\n\nCharmat method\n\nfood pairings\n\nFranciacorta ,\n\nLambrusco\n\nopening\n\nP\u00e9tillant Naturel\n\nProsecco\n\nSekt\n\nSp\u00e4tburgunder. _see_ Pinot Noir\n\nspecial bottling\n\nspicy and bitter pairings\n\nspicy and sweet pairings\n\nSteen. _see_ Chenin Blanc\n\nSteinfeder\n\nstoring wine\n\nstyle guide\n\nStyria: Sauvignon Blanc\n\nsub-appellation\n\nsugar: residual sugar\n\nsulphites\n\n'Super Tuscans' , , ,\n\nsweet and fatty pairings\n\nsweet and salty pairings\n\nsweet and spicy pairings\n\nsweetness level indicator\n\nsweetness meter:\n\nChampagne\n\nsweetness: ros\u00e9\n\nSylvaner\n\nSyrah \u2013\n\nPetite Syrah\n\nSyrah reds and Viognier\n\n##### T\n\ntannin\n\ntasting\n\nimproving your skills \u2013\n\nterminology \u2013\n\ntasting notes\n\nTavel: ros\u00e9\n\ntemperature\n\nserving wine ,\n\nstoring wine\n\nTempranillo \u2013\n\nterroir\n\nt\u00eate de cuv\u00e9e\n\nthin\n\nTinta Roriz. _see_ Tempranillo\n\nTinto Fino\n\nTokaji\n\nTorront\u00e9s\n\nTouriga Nacional\n\nTrebbiano\n\nTribidrag\n\nTrousseau\n\nTuscany: Sangiovese \u2013\n\n##### U\n\nUgni Blanc\n\nunicorn\n\n##### V\n\nVacqueyras\n\nSyrah\n\nValpolicella\n\nValtellina\n\nNebbiolo\n\nVega Sicilia\n\nVendanges Tardives ,\n\nVeneto\n\nPinot Grigio\n\nProsecco\n\nVerdejo\n\nVermentino \u2013\n\nvineyard names\n\nvineyard site quality level indicator \u2013\n\nVinho Verde ,\n\nVino Nobile di\n\nMontepulciano ,\n\nvintage ,\n\nViognier\n\nViura\n\nvolatile acidity\n\nVouvray\n\nChenin Blanc\n\nVouvray moelleux\n\n##### W\n\nthe Wachau: Gr\u00fcner\n\nVeltliner\n\nWashington State\n\nMerlot \u2013\n\nSyrah\n\nwax closures\n\nWeissburgunder\n\nWhite Burgundy\n\nwhite wines\n\ndry white wine\n\nplace names and grape\n\nvarietals\n\nserving\n\nwine bars \u2013\n\nwine keys\n\nwine lists by the glass \u2013\n\nwine shops\n\n##### Z\n\nZibbibo. _see_ Muscat\n\nZinfandel \u2013\n\nZweigelt\n\n# About the Author\n\nMaryse Chevriere is a writer, sommelier, and acclaimed wine humorist. She served as the founding Drinks editor of national food website, The Daily Meal, and has continued to write about wine and food for digital publications like Chowhound and the award-winning website Serious Eats.\n\nHer career in wine has included stints as: a retail wine consultant; a sommelier and acidhound at seminal NYC wine bar, Terroir; a wine director at Michelin-starred chef Dominique Crenn's San Francisco restaurant, Petit Crenn; and a harvest intern at Domaine Christian Binner in Alsace.\n\nIn 2016, she won the James Beard Award for Humor for her Instagram account, @Freshcutgardenhose, which brings the comical absurdity of wine tasting notes to life through illustration. Her witty doodles have been published online at _Bon Ap\u00e9tit_ , _Food & Wine_, Starchefs, and Vice: Muchies, and prints of her work have appeared in restaurants and wine bars in New York City and London.\n\n# Acknowledgements\n\nI am truly, truly lucky to have such an incredible cheering squad of family, friends, and loved ones in my life. Yaz, Julia and Alex, Annie, and Jon, thank you for reading pages, checking facts, being my sounding board, and pushing me to write when I really didn't want to anymore. Cath, thank you for always being an enthusiastic supporter of all my endeavours. Sarah Tanat-Jones, thank you for so brilliantly bringing my words to life with your artwork. And of course, a very special thank you to my dear friend and editor, Molly Ahuja (aka Monica), without whom this book would not exist.\n\nI'm forever grateful, also, to the many incredible, transformative teachers that have helped shape my journey in wine, most especially Terroir wine bar in New York City, my first and most important classroom.\n\nBut, above all else, this one's for you Mom and Dad. Thank you for everything. I love you.\n\n# About the Illustrator\n\nSarah Tanat-Jones is an illustrator based in London. Her favourite things to draw are portraits, food and architecture, and her favourite museum is the British Museum.\n\nGrasping the Grape by Maryse Chevriere\n\nFirst published in 2019 by Hardie Grant Books, an imprint of Hardie Grant Publishing\n\nHardie Grant Books (UK)  \n52\u201354 Southwark Street  \nLondon SE1 1UN\n\nHardie Grant Books (Australia)  \nGround Floor, Building 1 658 Church Street  \nMelbourne, VIC 3121\n\nhardiegrantbooks.com\n\nThe moral rights of Maryse Chevriere to be identified as the author of this work have been asserted by her in accordance with the Copyright, Designs and Patents Act 1988.\n\nText \u00a9 Maryse Chevriere  \nIllustrations \u00a9 Sarah Tanat-Jones\n\nAll rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form by any means, electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without the prior written permission of the Publisher.\n\nBritish Library Cataloguing-in-Publication Data. A catalogue record for this book is available from the British Library.\n\neISBN: 978-1-7848-8292-1\n\nPublishing Director: Kate Pollard  \nCommissioning Editor: Molly Ahuja  \nJunior Editor: Eila Purvis  \nInternal and Cover Design: Stuart Hardie  \nInternal and Cover Illustrations: Sarah Tanat-Jones  \nCopy Editor: Eve Marleau  \nProofreader: Kay Delves  \nIndexer: Cathy Heath\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Susan Skinner, Emmy and the Online Distributed\nProofreading Team at http:\/\/www.pgdp.net\n\n\n\n\n\nFor the reader: Things that were handwritten are denoted in the text as\nHW:\n\n\n\n\nTHE LETTERS\n\nOF\n\n[Illustration: HW: Charles Dickens]\n\n\n\n\nTHE LETTERS\n\nOF\n\nCHARLES DICKENS.\n\nEDITED BY\n\nHIS SISTER-IN-LAW AND HIS ELDEST DAUGHTER.\n\n=In Two Volumes.=\n\nVOL. II.\n\n1857 TO 1870.\n\n\n\n        London:\n        CHAPMAN AND HALL, 193, PICCADILLY.\n        1880.\n\n[_The Right of Translation is Reserved._]\n\n\n\n\nCHARLES DICKENS AND EVANS, CRYSTAL PALACE PRESS.\n\n\n\n\nERRATA.\n\nVOL. II.\n\n\n        Page 84, line 35. For \"South Kensington\n             Museum,\" _read_ \"the South Kensington Museum.\"\n\n        \" 108, line 26. For \"frequent contributor,\"\n             _read_ \"a frequent contributor.\"\n\n        \" 113, lines 6, 7. For \"great remonstrance,\"\n             _read_ \"Great Remonstrance.\"\n\n        \" 130, line 10. For \"after,\" _read_ \"afore.\"\n\n        \" 160,  \"   32. For \"a head,\" _read_ \"ahead.\"\n\n        \" 247,  \"   12. For \"Shea,\" _read_ \"Shoe.\"\n\n        \" 292,  \"   12. For \"Mabel's progress,\" _read_\n              \"Mabel's Progress.\"\n\n\n\n\n=Book II.=--_Continued._\n\n\n\n\nTHE\n\nLETTERS OF CHARLES DICKENS.\n\n\n\n\n1857.\n\nNARRATIVE.\n\n\nThis was a very full year in many ways. In February, Charles Dickens\nobtained possession of Gad's Hill, and was able to turn workmen into it.\nIn April he stayed, with his wife and sister-in-law, for a week or two\nat Wate's Hotel, Gravesend, to be at hand to superintend the beginning\nof his alterations of the house, and from thence we give a letter to\nLord Carlisle. He removed his family, for a summer residence in the\nhouse, in June; and he finished \"Little Dorrit\" there early in the\nsummer. One of his first visitors at Gad's Hill was the famous writer,\nHans Christian Andersen. In January \"The Frozen Deep\" had been played at\nthe Tavistock House theatre with such great success, that it was\nnecessary to repeat it several times, and the theatre was finally\ndemolished at the end of that month. In June Charles Dickens heard, with\ngreat grief, of the death of his dear friend Douglas Jerrold; and as a\ntestimony of admiration for his genius and affectionate regard for\nhimself, it was decided to organise, under the management of Charles\nDickens, a series of entertainments, \"in memory of the late Douglas\nJerrold,\" the fund produced by them (a considerable sum) to be\npresented to Mr. Jerrold's family. The amateur company, including many\nof Mr. Jerrold's colleagues on \"Punch,\" gave subscription performances\nof \"The Frozen Deep;\" the Gallery of Illustration, in Regent Street,\nbeing engaged for the purpose. Charles Dickens gave two readings at St.\nMartin's Hall of \"The Christmas Carol\" (to such immense audiences and\nwith such success, that the idea of giving public readings for his _own_\nbenefit first occurred to him at this time). The professional actors,\namong them the famous veteran actor, Mr. T. P. Cooke, gave a performance\nof Mr. Jerrold's plays of \"The Rent Day\" and \"Black-eyed Susan,\" in\nwhich Mr. T. P. Cooke sustained the character in which he had originally\nmade such great success when the play was written. A lecture was given\nby Mr. Thackeray, and another by Mr. W. H. Russell. Finally, the Queen\nhaving expressed a desire to see the play, which had been much talked of\nduring that season, there was another performance before her Majesty and\nthe Prince Consort at the Gallery of Illustration in July, and at the\nend of that month Charles Dickens read his \"Carol\" in the Free Trade\nHall, at Manchester. And to wind up the \"Memorial Fund\" entertainments,\n\"The Frozen Deep\" was played again at Manchester, also in the great Free\nTrade Hall, at the end of August. For the business of these\nentertainments he secured the assistance of Mr. Arthur Smith, of whom he\nwrites to Mr. Forster, at this time: \"I have got hold of Arthur Smith,\nas the best man of business I know, and go to work with him to-morrow\nmorning.\" And when he began his own public readings, both in town and\ncountry, he felt himself most fortunate in having the co-operation of\nthis invaluable man of business, and also of his zealous friendship and\npleasant companionship.\n\nIn July, his second son, Walter Landor, went to India as a cadet in the\n\"Company's service,\" from which he was afterwards transferred to the\n42nd Royal Highlanders. His father and his elder brother went to see him\noff, to Southampton. From this place Charles Dickens writes to Mr.\nEdmund Yates, a young man in whom he had been interested from his\nboyhood, both for the sake of his parents and for his own sake, and for\nwhom he had always an affectionate regard.\n\nIn September he made a short tour in the North of England, with Mr.\nWilkie Collins, out of which arose the \"Lazy Tour of Two Idle\nApprentices,\" written by them jointly, and published in \"Household\nWords.\" Some letters to his sister-in-law during this expedition are\ngiven here, parts of which (as is the case with many letters to his\neldest daughter and his sister-in-law) have been published in Mr.\nForster's book.\n\nThe letters which follow are almost all on the various subjects\nmentioned in our notes, and need little explanation.\n\nHis letter to Mr. Procter makes allusion to a legacy lately left to that\nfriend.\n\nThe letters to Mr. Dilke, the original and much-respected editor of \"The\nAthenaeum,\" and to Mr. Forster, on the subject of the \"Literary Fund,\"\nrefer, as the letters indicate, to a battle which they were carrying on\ntogether with that institution.\n\nA letter to Mr. Frank Stone is an instance of his kind, patient, and\njudicious criticism of a young writer, and the letter which follows it\nshows how thoroughly it was understood and how perfectly appreciated by\nthe authoress of the \"Notes\" referred to. Another instance of the same\nkind criticism is given in a second letter this year to Mr. Edmund\nYates.\n\n\n[Sidenote: Mr. B. W. Procter.]\n\n                                 TAVISTOCK HOUSE, _January 2nd, 1857._\n\nMY DEAR PROCTER,\n\nI have to thank you for a delightful book, which has given me unusual\npleasure. My delight in it has been a little dashed by certain farewell\nverses, but I have made up my mind (and you have no idea of the\nobstinacy of my character) not to believe them.\n\nPerhaps it is not taking a liberty--perhaps it is--to congratulate you\non Kenyon's remembrance. Either way I can't help doing it with all my\nheart, for I know no man in the world (myself excepted) to whom I would\nrather the money went.\n\n                                            Affectionately yours ever.\n\n\n[Sidenote: Sir James Emerson Tennent.]\n\n                                 TAVISTOCK HOUSE, _January 9th, 1857._\n\nMY DEAR TENNENT,\n\n\nI must thank you for your earnest and affectionate letter. It has given\nme the greatest pleasure, mixing the play in my mind confusedly and\ndelightfully with Pisa, the Valetta, Naples, Herculanaeum--God knows what\nnot.\n\nAs to the play itself; when it is made as good as my care can make it, I\nderive a strange feeling out of it, like writing a book in company; a\nsatisfaction of a most singular kind, which has no exact parallel in my\nlife; a something that I suppose to belong to the life of a labourer in\nart alone, and which has to me a conviction of its being actual truth\nwithout its pain, that I never could adequately state if I were to try\nnever so hard.\n\nYou touch so kindly and feelingly on the pleasure such little pains\ngive, that I feel quite sorry you have never seen this drama in progress\nduring the last ten weeks here. Every Monday and Friday evening during\nthat time we have been at work upon it. I assure you it has been a\nremarkable lesson to my young people in patience, perseverance,\npunctuality, and order; and, best of all, in that kind of humility which\nis got from the earned knowledge that whatever the right hand finds to\ndo must be done with the heart in it, and in a desperate earnest.\n\nWhen I changed my dress last night (though I did it very quickly), I was\nvexed to find you gone. I wanted to have secured you for our green-room\nsupper, which was very pleasant. If by any accident you should be free\nnext Wednesday night (our last), pray come to that green-room supper. It\nwould give me cordial pleasure to have you there.\n\n                           Ever, my dear Tennent, very heartily yours.\n\n\n[Sidenote: M. de Cerjat.]\n\n                     TAVISTOCK HOUSE, _Monday Night, Jan, 17th, 1857._\n\nMY DEAR CERJAT,\n\nSo wonderfully do good (epistolary) intentions become confounded with\nbad execution, that I assure you I laboured under a perfect and most\ncomfortable conviction that I had answered your Christmas Eve letter of\n1855. More than that, in spite of your assertions to the contrary, I\nstill strenuously believe that I did so! I have more than half a mind\n(\"Little Dorrit\" and my other occupations notwithstanding) to charge you\nwith having forgotten my reply!! I have even a wild idea that Townshend\nreproached me, when the last old year was new, with writing to you\ninstead of to him!!! We will argue it out, as well as we can argue\nanything without poor dear Haldimand, when I come back to Elysee. In any\ncase, however, don't discontinue your annual letter, because it has\nbecome an expected and a delightful part of the season to me.\n\nWith one of the prettiest houses in London, and every conceivable (and\ninconceivable) luxury in it, Townshend is voluntarily undergoing his own\nsentence of transportation in Nervi, a beastly little place near Genoa,\nwhere you would as soon find a herd of wild elephants in any villa as\ncomfort. He has a notion that he _must_ be out of England in the winter,\nbut I believe him to be altogether wrong (as I have just told him in a\nletter), unless he could just take his society with him.\n\nWorkmen are now battering and smashing down my theatre here, where we\nhave just been acting a new play of great merit, done in what I may call\n(modestly speaking of the getting-up, and not of the acting) an\nunprecedented way. I believe that anything so complete has never been\nseen. We had an act at the North Pole, where the slightest and greatest\nthing the eye beheld were equally taken from the books of the Polar\nvoyagers. Out of thirty people, there were certainly not two who might\nnot have gone straight to the North Pole itself, completely furnished\nfor the winter! It has been the talk of all London for these three\nweeks. And now it is a mere chaos of scaffolding, ladders, beams,\ncanvases, paint-pots, sawdust, artificial snow, gas-pipes, and\nghastliness. I have taken such pains with it for these ten weeks in all\nmy leisure hours, that I feel now shipwrecked--as if I had never been\nwithout a play on my hands before. A third topic comes up as this\nceases.\n\nDown at Gad's Hill, near Rochester, in Kent--Shakespeare's Gad's Hill,\nwhere Falstaff engaged in the robbery--is a quaint little country-house\nof Queen Anne's time. I happened to be walking past, a year and a half\nor so ago, with my sub-editor of \"Household Words,\" when I said to him:\n\"You see that house? It has always a curious interest for me, because\nwhen I was a small boy down in these parts I thought it the most\nbeautiful house (I suppose because of its famous old cedar-trees) ever\nseen. And my poor father used to bring me to look at it, and used to say\nthat if I ever grew up to be a clever man perhaps I might own that\nhouse, or such another house. In remembrance of which, I have always in\npassing looked to see if it was to be sold or let, and it has never been\nto me like any other house, and it has never changed at all.\" We came\nback to town, and my friend went out to dinner. Next morning he came to\nme in great excitement, and said: \"It is written that you were to have\nthat house at Gad's Hill. The lady I had allotted to me to take down to\ndinner yesterday began to speak of that neighbourhood. 'You know it?' I\nsaid; 'I have been there to-day.' 'O yes,' said she, 'I know it very\nwell. I was a child there, in the house they call Gad's Hill Place. My\nfather was the rector, and lived there many years. He has just died, has\nleft it to me, and I want to sell it.' 'So,' says the sub-editor, 'you\nmust buy it. Now or never!'\" I did, and hope to pass next summer there,\nthough I may, perhaps, let it afterwards, furnished, from time to time.\n\nAll about myself I find, and the little sheet nearly full! But I know,\nmy dear Cerjat, the subject will have its interest for you, so I give it\nits swing. Mrs. Watson was to have been at the play, but most\nunfortunately had three children sick of gastric fever, and could not\nleave them. She was here some three weeks before, looking extremely well\nin the face, but rather thin. I have not heard of your friend Mr.\nPercival Skelton, but I much misdoubt an amateur artist's success in\nthis vast place. I hope you detected a remembrance of our happy visit to\nthe Great St. Bernard in a certain number of \"Little Dorrit\"? Tell Mrs.\nCerjat, with my love, that the opinions I have expressed to her on the\nsubject of cows have become matured in my mind by experience and\nvenerable age; and that I denounce the race as humbugs, who have been\ngetting into poetry and all sorts of places without the smallest reason.\nHaldimand's housekeeper is an awful woman to consider. Pray give him our\nkindest regards and remembrances, if you ever find him in a mood to take\nit. \"Our\" means Mrs. Dickens's, Georgie's, and mine. We often, often\ntalk of our old days at Lausanne, and send loving regard to Mrs. Cerjat\nand all your house.\n\n                          Adieu, my dear fellow; ever cordially yours.\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n                                TAVISTOCK HOUSE, _January 28th, 1857._\n\nMY DEAREST MACREADY,\n\nYour friend and servant is as calm as Pecksniff, saving for his knitted\nbrows now turning into cordage over Little Dorrit. The theatre has\ndisappeared, the house is restored to its usual conditions of order, the\nfamily are tranquil and domestic, dove-eyed peace is enthroned in this\nstudy, fire-eyed radicalism in its master's breast.\n\nI am glad to hear that our poetess is at work again, and shall be very\nmuch pleased to have some more contributions from her.\n\nLove from all to your dear sister, and to Katie, and to all the house.\n\nWe dined yesterday at Frederick Pollock's. I begged an amazing\nphotograph of you, and brought it away. It strikes me as one of the most\nludicrous things I ever saw in my life. I think of taking a\npublic-house, and having it copied larger, for the size. You may\nremember it? Very square and big--the Saracen's Head with its hair cut,\nand in modern gear? Staring very hard? As your particular friend, I\nwould not part with it on any consideration. I will never get such a\nwooden head again.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Miss Mary Boyle.]\n\n                                TAVISTOCK HOUSE, _February 7th, 1857._\n\nMY DEAR MARY,\n\n\nHalf-a-dozen words on this, my birthday, to thank you for your kind and\nwelcome remembrance, and to assure you that your Joseph is proud of it.\n\nFor about ten minutes after his death, on each occasion of that event\noccurring, Richard Wardour was in a floored condition. And one night, to\nthe great terror of Devonshire, the Arctic Regions, and Newfoundland\n(all of which localities were afraid to speak to him, as his ghost sat\nby the kitchen fire in its rags), he very nearly did what he never did,\nwent and fainted off, dead, again. But he always plucked up, on the turn\nof ten minutes, and became facetious.\n\nLikewise he chipped great pieces out of all his limbs (solely, as I\nimagine, from moral earnestness and concussion of passion, for I never\nknow him to hit himself in any way) and terrified Aldersley[1] to that\ndegree, by lunging at him to carry him into the cave, that the said\nAldersley always shook like a mould of jelly, and muttered, \"By G----,\nthis is an awful thing!\"\n\n                                                  Ever affectionately.\n\nP.S.--I shall never cease to regret Mrs. Watson's not having been there.\n\n\n[Sidenote: Rev. James White.]\n\n                            TAVISTOCK HOUSE, _Sunday, Feb. 8th, 1857._\n\nMY DEAR WHITE,\n\nI send these lines by Mary and Katey, to report my love to all.\n\nYour note about the _Golden Mary_ gave me great pleasure; though I don't\nbelieve in one part of it; for I honestly believe that your story, as\nreally belonging to the rest of the narrative, had been generally\nseparated from the other stories, and greatly liked. I had not that\nparticular shipwreck that you mention in my mind (indeed I doubt if I\nknow it), and John Steadiman merely came into my head as a staunch sort\nof name that suited the character. The number has done \"Household Words\"\ngreat service, and has decidedly told upon its circulation.\n\nYou should have come to the play. I much doubt if anything so complete\nwill ever be seen again. An incredible amount of pains and ingenuity was\nexpended on it, and the result was most remarkable even to me.\n\nWhen are you going to send something more to H. W.? Are you lazy??\nLow-spirited??? Pining for Paris????\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Mr. C. W. Dilke.]\n\n            OFFICE OF \"HOUSEHOLD WORDS,\" _Thursday, March 19th, 1857._\n\nMY DEAR MR. DILKE,\n\nForster has another notion about the Literary Fund. Will you name a day\nnext week--that day being neither Thursday nor Saturday--when we shall\nhold solemn council there at half-past four?\n\nFor myself, I beg to report that I have my war-paint on, that I have\nburied the pipe of peace, and am whooping for committee scalps.\n\n                                                Ever faithfully yours.\n\n\n[Sidenote: The Earl of Carlisle.]\n\n                       GRAVESEND, KENT, _Wednesday, April 15th, 1857._\n\nMY DEAR LORD CARLISLE,\n\nI am writing by the river-side for a few days, and at the end of last\nweek ---- appeared here with your note of introduction. I was not in the\nway; but as ---- had come express from London with it, Mrs. Dickens\nopened it, and gave her (in the limited sense which was of no use to\nher) an audience. She did not quite seem to know what she wanted of me.\nBut she said she had understood at Stafford House that I had a theatre\nin which she could read; with a good deal of modesty and diffidence she\nat last got so far. Now, my little theatre turns my house out of window,\ncosts fifty pounds to put up, and is only two months taken down;\ntherefore, is quite out of the question. This Mrs. Dickens explained,\nand also my profound inability to do anything for ---- readings which\nthey could not do for themselves. She appeared fully to understand the\nexplanation, and indeed to have anticipated for herself how powerless I\nmust be in such a case.\n\nShe described herself as being consumptive, and as being subject to an\neffusion of blood from the lungs; about the last condition, one would\nthink, poor woman, for the exercise of public elocution as an art.\n\nBetween ourselves, I think the whole idea a mistake, and have thought so\nfrom its first announcement. It has a fatal appearance of trading upon\nUncle Tom, and am I not a man and a brother? which you may be by all\nmeans, and still not have the smallest claim to my attention as a public\nreader. The town is over-read from all the white squares on the\ndraught-board; it has been considerably harried from all the black\nsquares--now with the aid of old banjoes, and now with the aid of Exeter\nHall; and I have a very strong impression that it is by no means to be\nlaid hold of from this point of address. I myself, for example, am the\nmeekest of men, and in abhorrence of slavery yield to no human creature,\nand yet I don't admit the sequence that I want Uncle Tom (or Aunt\nTomasina) to expound \"King Lear\" to me. And I believe my case to be the\ncase of thousands.\n\nI trouble you with this much about it, because I am naturally desirous\nyou should understand that if I could possibly have been of any service,\nor have suggested anything to this poor lady, I would not have lost the\nopportunity. But I cannot help her, and I assure you that I cannot\nhonestly encourage her to hope. I fear her enterprise has no hope in it.\n\nIn your absence I have always followed you through the papers, and felt\na personal interest and pleasure in the public affection in which you\nare held over there.[2] At the same time I must confess that I should\nprefer to have you here, where good public men seem to me to be dismally\nwanted. I have no sympathy with demagogues, but am a grievous Radical,\nand think the political signs of the times to be just about as bad as\nthe spirit of the people will admit of their being. In all other\nrespects I am as healthy, sound, and happy as your kindness can wish. So\nyou will set down my political despondency as my only disease.\n\nOn the tip-top of Gad's Hill, between this and Rochester, on the very\nspot where Falstaff ran away, I have a pretty little old-fashioned\nhouse, which I shall live in the hope of showing to you one day. Also I\nhave a little story respecting the manner in which it became mine, which\nI hope (on the same occasion in the clouds) to tell you. Until then and\nalways, I am, dear Lord Carlisle,\n\n                                    Yours very faithfully and obliged.\n\n\n[Sidenote: Mr. John Forster.]\n\n                                    TAVISTOCK HOUSE, _May 13th, 1857._\n\nMY DEAR FORSTER,\n\nI have gone over Dilke's memoranda, and I think it quite right and\nnecessary that those points should be stated. Nor do I see the least\ndifficulty in the way of their introduction into the pamphlet. But I do\nnot deem it possible to get the pamphlet written and published before\nthe dinner. I have so many matters pressing on my attention, that I\ncannot turn to it immediately on my release from my book just finished.\nIt shall be done and distributed early next month.\n\nAs to anything being lost by its not being in the hands of the people\nwho dine (as you seem to think), I have not the least misgiving on that\nscore. They would say, if it were issued, just what they will say\nwithout it.\n\nLord Granville is committed to taking the chair, and will make the best\nspeech he can in it. The pious ---- will cram him with as many\ndistortions of the truth as his stomach may be strong enough to receive.\n----, with Bardolphian eloquence, will cool his nose in the modest\nmerits of the institution. ---- will make a neat and appropriate speech\non both sides, round the corner and over the way. And all this would be\ndone exactly to the same purpose and in just the same strain, if twenty\nthousand copies of the pamphlet had been circulated.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Rev. James White.]\n\n                            TAVISTOCK HOUSE, _Friday, May 22nd, 1857._\n\nMY DEAR WHITE,\n\nMy emancipation having been effected on Saturday, the ninth of this\nmonth, I take some shame to myself for not having sooner answered your\nnote. But the host of things to be done as soon as I was free, and the\ntremendous number of ingenuities to be wrought out at Gad's Hill, have\nkept me in a whirl of their own ever since.\n\nWe purpose going to Gad's Hill for the summer on the 1st of June; as,\napart from the master's eye being a necessary ornament to the spot, I\nclearly see that the workmen yet lingering in the yard must be squeezed\nout by bodily pressure, or they will never go. How will this suit you\nand yours? If you will come down, we can take you all in, on your way\nnorth; that is to say, we shall have that ample verge and room enough,\nuntil about the eighth; when Hans Christian Andersen (who has been\n\"coming\" for about three years) will come for a fortnight's stay in\nEngland. I shall like you to see the little old-fashioned place. It\nstrikes me as being comfortable.\n\nSo let me know your little game. And with love to Mrs. White, Lotty, and\nClara,\n\n                                Believe me, ever affectionately yours.\n\n\n[Sidenote: Mr. Frank Stone, A.R.A.]\n\n                OFFICE OF \"HOUSEHOLD WORDS,\" _Monday, June 1st, 1857._\n\nMY DEAR STONE,\n\nI know that what I am going to say will not be agreeable; but I rely on\nthe authoress's good sense; and say it, knowing it to be the truth.\n\nThese \"Notes\" are destroyed by too much smartness. It gives the\nappearance of perpetual effort, stabs to the heart the nature that is in\nthem, and wearies by the manner and not by the matter. It is the\ncommonest fault in the world (as I have constant occasion to observe\nhere), but it is a very great one. Just as you couldn't bear to have an\nepergne or a candlestick on your table, supported by a light figure\nalways on tiptoe and evidently in an impossible attitude for the\nsustainment of its weight, so all readers would be more or less\noppressed and worried by this presentation of everything in one smart\npoint of view, when they know it must have other, and weightier, and\nmore solid properties. Airiness and good spirits are always delightful,\nand are inseparable from notes of a cheerful trip; but they should\nsympathise with many things as well as see them in a lively way. It is\nbut a word or a touch that expresses this humanity, but without that\nlittle embellishment of good nature there is no such thing as humour. In\nthis little MS. everything is too much patronised and condescended to,\nwhereas the slightest touch of feeling for the rustic who is of the\nearth earthy, or of sisterhood with the homely servant who has made her\nface shine in her desire to please, would make a difference that the\nwriter can scarcely imagine without trying it. The only relief in the\ntwenty-one slips is the little bit about the chimes. It _is_ a relief,\nsimply because it is an indication of some kind of sentiment. You don't\nwant any sentiment laboriously made out in such a thing. You don't want\nany maudlin show of it. But you do want a pervading suggestion that it\nis there. It makes all the difference between being playful and being\ncruel. Again I must say, above all things--especially to young people\nwriting: For the love of God don't condescend! Don't assume the attitude\nof saying, \"See how clever I am, and what fun everybody else is!\" Take\nany shape but that.\n\nI observe an excellent quality of observation throughout, and think the\nboy at the shop, and all about him, particularly good. I have no doubt\nwhatever that the rest of the journal will be much better if the writer\nchooses to make it so. If she considers for a moment within herself, she\nwill know that she derived pleasure from everything she saw, because she\nsaw it with innumerable lights and shades upon it, and bound to humanity\nby innumerable fine links; she cannot possibly communicate anything of\nthat pleasure to another by showing it from one little limited point\nonly, and that point, observe, the one from which it is impossible to\ndetach the exponent as the patroness of a whole universe of inferior\nsouls. This is what everybody would mean in objecting to these notes\n(supposing them to be published), that they are too smart and too\nflippant.\n\nAs I understand this matter to be altogether between us three, and as I\nthink your confidence, and hers, imposes a duty of friendship on me, I\ndischarge it to the best of my ability. Perhaps I make more of it than\nyou may have meant or expected; if so, it is because I am interested and\nwish to express it. If there had been anything in my objection not\nperfectly easy of removal, I might, after all, have hesitated to state\nit; but that is not the case. A very little indeed would make all this\ngaiety as sound and wholesome and good-natured in the reader's mind as\nit is in the writer's.\n\n                                                Affectionately always.\n\n\n[Sidenote: Anonymous.]\n\n                 GAD'S HILL PLACE, HIGHAM, _Thursday, June 4th, 1857._\n\nMY DEAR ----\n\nComing home here last night, from a day's business in London, I found\nyour most excellent note awaiting me, in which I have had a pleasure to\nbe derived from none but good and natural things. I can now honestly\nassure you that I believe you will write _well_, and that I have a\nlively hope that I may be the means of showing you yourself in print one\nday. Your powers of graceful and light-hearted observation need nothing\nbut the little touches on which we are both agreed. And I am perfectly\nsure that they will be as pleasant to you as to anyone, for nobody can\nsee so well as you do, without feeling kindly too.\n\nTo confess the truth to you, I was half sorry, yesterday, that I had\nbeen so unreserved; but not half as sorry, yesterday, as I am glad\nto-day. You must not mind my adding that there is a noble candour and\nmodesty in your note, which I shall never be able to separate from you\nhenceforth.\n\n                                          Affectionately yours always.\n\n\n[Sidenote: Mr. Henry Austin.]\n\n                               GAD'S HILL, _Saturday, June 6th, 1857._\n\nMY DEAR HENRY,\n\nHere is a very serious business on the great estate respecting the water\nsupply. Last night, they had pumped the well dry merely in raising the\nfamily supply for the day; and this morning (very little water having\nbeen got into the cisterns) it is dry again! It is pretty clear to me\nthat we must look the thing in the face, and at once bore deeper, dig,\nor do some beastly thing or other, to secure this necessary in\nabundance. Meanwhile I am in a most plaintive and forlorn condition\nwithout your presence and counsel. I raise my voice in the wilderness\nand implore the same!!!\n\nWild legends are in circulation among the servants how that Captain\nGoldsmith on the knoll above--the skipper in that crow's-nest of a\nhouse--has millions of gallons of water always flowing for him. Can he\nhave damaged my well? Can we imitate him, and have our millions of\ngallons? Goldsmith or I must fall, so I conceive.\n\nIf you get this, send me a telegraph message informing me when I may\nexpect comfort. I am held by four of the family while I write this, in\ncase I should do myself a mischief--it certainly won't be taking to\ndrinking water.\n\n                              Ever affectionately (most despairingly).\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n                           TAVISTOCK HOUSE, _Monday, July 13th, 1857._\n\nMY DEAREST MACREADY,\n\nMany thanks for your Indian information. I shall act upon it in the most\nexact manner. Walter sails next Monday. Charley and I go down with him\nto Southampton next Sunday. We are all delighted with the prospect of\nseeing you at Gad's Hill. These are my Jerrold engagements: On Friday,\nthe 24th, I have to repeat my reading at St. Martin's Hall; on Saturday,\nthe 25th, to repeat \"The Frozen Deep\" at the Gallery of Illustration for\nthe last time. On Thursday, the 30th, or Friday, the 31st, I shall\nprobably read at Manchester. Deane, the general manager of the\nExhibition, is going down to-night, and will arrange all the\npreliminaries for me. If you and I went down to Manchester together, and\nwere there on a Sunday, he would give us the whole Exhibition to\nourselves. It is probable, I think (as he estimates the receipts of a\nnight at about seven hundred pounds), that we may, in about a fortnight\nor so after the reading, play \"The Frozen Deep\" at Manchester. But of\nthis contingent engagement I at present know no more than you do.\n\nNow, will you, upon this exposition of affairs, choose your own time for\ncoming to us, and, when you have made your choice, write to me at Gad's\nHill? I am going down this afternoon for rest (which means violent\ncricket with the boys) after last Saturday night; which was a teaser,\nbut triumphant. The St. Martin's Hall audience was, I must confess, a\nvery extraordinary thing. The two thousand and odd people were like one,\nand their enthusiasm was something awful.\n\nYet I have seen that before, too. Your young remembrance cannot recall\nthe man; but he flourished in my day--a great actor, sir--a noble\nactor--thorough artist! I have seen him do wonders in that way. He\nretired from the stage early in life (having a monomaniacal delusion\nthat he was old), and is said to be still living in your county.\n\nAll join in kindest love to your dear sister and all the rest.\n\n                         Ever, my dearest Macready,\n                                            Most affectionately yours.\n\n\n[Sidenote: Mr. Edmund Yates.]\n\n                           TAVISTOCK HOUSE, _Sunday, July 19th, 1857._\n\nMY DEAR YATES,\n\nAlthough I date this ashore, I really write it from Southampton (don't\nnotice this fact in your reply, for I shall be in town on Wednesday). I\nhave come here on an errand which will grow familiar to you before you\nknow that Time has flapped his wings over your head. Like me, you will\nfind those babies grow to be young men before you are quite sure they\nare born. Like me, you will have great teeth drawn with a wrench, and\nwill only then know that you ever cut them. I am here to send Walter\naway over what they call, in Green Bush melodramas, \"the Big Drink,\" and\nI don't at all know this day how he comes to be mine, or I his.\n\nI don't write to say this--or to say how seeing Charley, and he going\naboard the ship before me just now, I suddenly came into possession of a\nphotograph of my own back at sixteen and twenty, and also into a\nsuspicion that I had doubled the last age. I merely write to mention\nthat Telbin and his wife are going down to Gad's Hill with us, about\nmid-day next Sunday, and that if you and Mrs. Yates will come too, we\nshall be delighted to have you. We can give you a bed, and you can be in\ntown (if you have such a savage necessity) by twenty minutes before ten\non Monday morning.\n\nI was very much pleased (as I had reason to be) with your account of the\nreading in _The Daily News_. I thank you heartily.\n\n\n[Sidenote: Mr. T. P. Cooke.]\n\n        IN REMEMBRANCE OF THE LATE MR. DOUGLAS JERROLD.\n\n               COMMITTEE'S OFFICE, GALLERY OF ILLUSTRATION,\n                           REGENT STREET, _Thursday, July 30th, 1857._\n\nMY DEAR MR. COOKE,\n\nI cannot rest satisfied this morning without writing to congratulate you\non your admirable performance of last night. It was so fresh and\nvigorous, so manly and gallant, that I felt as if it splashed against my\ntheatre-heated face along with the spray of the breezy sea. What I felt\neverybody felt; I should feel it quite an impertinence to take myself\nout of the crowd, therefore, if I could by any means help doing so. But\nI can't; so I hope you will feel that you bring me on yourself, and have\nonly yourself to blame.\n\n                                              Always faithfully yours.\n\n\n[Sidenote: Mrs. Compton.]\n\n                        GAD'S HILL PLACE, HIGHAM BY ROCHESTER,\n                                        _Sunday Night, Aug 2nd, 1857._\n\nMY DEAR MRS. COMPTON,\n\nWe are going to play \"The Frozen Deep\" (pursuant to requisition from\ntown magnates, etc.) at Manchester, at the New Free Trade Hall, on the\nnights of Friday and Saturday, the 21st and 22nd August.\n\nThe place is out of the question for my girls. Their action could not be\nseen, and their voices could not be heard. You and I have played, there\nand elsewhere, so sociably and happily, that I am emboldened to ask you\nwhether you would play my sister-in-law Georgina's part (Compton and\nbabies permitting).\n\nWe shall go down in the old pleasant way, and shall have the Art\nTreasures Exhibition to ourselves on the Sunday; when even \"he\" (as\nRogers always called every pretty woman's husband) might come and join\nus.\n\nWhat do you say? What does he say? and what does baby say? When I use\nthe term \"baby,\" I use it in two tenses--present and future.\n\nAnswer me at this address, like the Juliet I saw at Drury Lane--when was\nit?--yesterday. And whatever your answer is, if you will say that you\nand Compton will meet us at the North Kent Station, London Bridge, next\nSunday at a quarter before one, and will come down here for a breath of\nsweet air and stay all night, you will give your old friends great\npleasure. Not least among them,\n\n                                                     Yours faithfully.\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n                        GAD'S HILL PLACE, HIGHAM BY ROCHESTER,\n                                             _Monday, Aug. 3rd, 1857._\n\nMY DEAREST MACREADY,\n\nI write to you in reference to your last note, as soon as I positively\nknow our final movements in the Jerrold matter.\n\nWe are going to wind up by acting at Manchester (on solemn requisition)\non the evenings of Friday and Saturday, the 21st and 22nd (actresses\nsubstituted for the girls, of course). We shall have to leave here on\nthe morning of the 20th. You thought of coming on the 16th; can't you\nmake it a day or two earlier, so as to be with us a whole week? Decide\nand pronounce. Again, cannot you bring Katey with you? Decide and\npronounce thereupon, also.\n\nI read at Manchester last Friday. As many thousand people were there as\nyou like to name. The collection of pictures in the Exhibition is\nwonderful. And the power with which the modern English school asserts\nitself is a very gratifying and delightful thing to behold. The care for\nthe common people, in the provision made for their comfort and\nrefreshment, is also admirable and worthy of all commendation. But they\nwant more amusement, and particularly (as it strikes me) _something in\nmotion_, though it were only a twisting fountain. The thing is too still\nafter their lives of machinery, and art flies over their heads in\nconsequence.\n\nI hope you have seen my tussle with the \"Edinburgh.\" I saw the chance\nlast Friday week, as I was going down to read the \"Carol\" in St.\nMartin's Hall. Instantly turned to, then and there, and wrote half the\narticle. Flew out of bed early next morning, and finished it by noon.\nWent down to Gallery of Illustration (we acted that night), did the\nday's business, corrected the proofs in Polar costume in dressing-room,\nbroke up two numbers of \"Household Words\" to get it out directly, played\nin \"Frozen Deep\" and \"Uncle John,\" presided at supper of company, made\nno end of speeches, went home and gave in completely for four hours,\nthen got sound asleep, and next day was as fresh as you used to be in\nthe far-off days of your lusty youth.\n\nAll here send kindest love to your dear good sister and all the house.\n\n                                         Ever and ever affectionately.\n\n\n[Sidenote: Mr. Frank Stone, A.R.A.]\n\n                  TAVISTOCK HOUSE, _Sunday Afternoon, Aug. 9th, 1857._\n\nMY DEAR STONE,\n\nNow here, without any preface, is a good, confounding, stunning question\nfor you--would you like to play \"Uncle John\" on the two nights at\nManchester?\n\nIt is not a long part. You could have a full rehearsal on the Friday,\nand I could sit in the wing at night and pull you through all the\nbusiness. Perhaps you might not object to being in the thing in your own\nnative place, and the relief to me would be enormous.\n\nThis is what has come into my head lying in bed to-day (I have been in\nbed all day), and this is just my plain reason for writing to you.\n\nIt's a capital part, and you are a capital old man. You know the play as\nwe play it, and the Manchester people don't. Say the word, and I'll send\nyou my own book by return of post.\n\nThe agitation and exertion of Richard Wardour are so great to me, that I\ncannot rally my spirits in the short space of time I get. The strain is\nso great to make a show of doing it, that I want to be helped out of\n\"Uncle John\" if I can. Think of yourself far more than me; but if you\nhalf think you are up to the joke, and half doubt your being so, then\ngive me the benefit of the doubt and play the part.\n\nAnswer me at Gad's Hill.\n\n                                                  Ever affectionately.\n\nP.S.--If you play, I shall immediately announce it to all concerned. If\nyou don't, I shall go on as if nothing had happened, and shall say\nnothing to anyone.\n\n\n[Sidenote: Mr. Henry Austin.]\n\n                        GAD'S HILL PLACE, _Saturday, Aug. 15th, 1857._\n\nMY DEAR HENRY,\n\nAt last, I am happy to inform you, we have got at a famous spring!! It\nrushed in this morning, ten foot deep. And our friends talk of its\nsupplying \"a ton a minute for yourself and your family, sir, for\nnevermore.\"\n\nThey ask leave to bore ten feet lower, to prevent the possibility of\nwhat they call \"a choking with sullage.\" Likewise, they are going to\ninsert \"a rose-headed pipe;\" at the mention of which implement, I am\n(secretly) well-nigh distracted, having no idea of what it means. But I\nhave said \"Yes,\" besides instantly standing a bottle of gin. Can you\ncome back, and can you get down on Monday morning, to advise and\nendeavour to decide on the mechanical force we shall use for raising the\nwater? I would return with you, as I shall have to be in town until\nThursday, and then to go to Manchester until the following Tuesday.\n\nI send this by hand to John, to bring to you.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Mr. Frank Stone, A.R.A.]\n\n                          GAD'S HILL PLACE, _Monday, Aug. 17th, 1857._\n\nMY DEAR STONE,\n\nI received your kind note this morning, and write this reply here to\ntake to London with me and post in town, being bound for that village\nand three days' drill of the professional ladies who are to succeed the\nTavistock girls.\n\nMy book I enclose. There is a slight alteration (which does not affect\nyou) at the end of the first act, in order that the piece may be played\nthrough without having the drop curtain down. You will not find the\nsituations or business difficult, with me on the spot to put you right.\n\nNow, as to the dress. You will want a pair of pumps, and a pair of white\nsilk socks; these you can get at Manchester. The extravagantly and\nanciently-frilled shirts that I have had got up for the part, I will\nbring you down; large white waistcoat, I will bring you down; large\nwhite hat, I will bring you down; dressing-gown, I will bring you down;\nwhite gloves and ditto choker you can get at Manchester. There then\nremain only a pair of common nankeen tights, to button below the calf,\nand blue wedding-coat. The nankeen tights you had best get made at once;\nmy \"Uncle John\" coat I will send you down in a parcel by to-morrow's\ntrain, to have altered in Manchester to your shape and figure. You will\nthen be quite independent of Christian chance and Jewish Nathan, which\nlatter potentate is now at Canterbury with the cricket amateurs, and\nmight fail.\n\nA Thursday's rehearsal is (unfortunately) now impracticable, the passes\nfor the railway being all made out, and the company's sailing orders\nissued. But, as I have already suggested, with a careful rehearsal on\nFriday morning, and with me at the wing at night to put you right, you\nwill find yourself sliding through it easily. There is nothing in the\nleast complicated in the business. As to the dance, you have only to\nknock yourself up for a twelvemonth and it will go nobly.\n\nAfter all, too, if you _should_, through any unlucky breakdown, come to\nbe afraid of it, I am no worse off than I was before, if I have to do it\nat last. Keep your pecker up with that.\n\nI am heartily obliged to you, my dear old boy, for your affectionate and\nconsiderate note, and I wouldn't have you do it, really and\nsincerely--immense as the relief will be to me--unless you are quite\ncomfortable in it, and able to enjoy it.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Mr. Frank Stone, A.R.A.]\n\n              OFFICE OF \"HOUSEHOLD WORDS,\" _Tuesday, Aug. 18th, 1857._\n\nMY DEAR STONE,\n\nI sent you a telegraph message last night, in total contradiction of the\nletter you received from me this morning.\n\nThe reason was simply this: Arthur Smith and the other business men,\nboth in Manchester and here, urged upon me, in the strongest manner,\nthat they were afraid of the change; that it was well known in\nManchester that I had done the part in London; that there was a danger\nof its being considered disrespectful in me to give it up; also that\nthere was a danger that it might be thought that I did so at the last\nminute, after an immense let, whereas I might have done it at first,\netc. etc. etc. Having no desire but for the success of our object, and a\nbecoming recognition on my part of the kind Manchester public's\ncordiality, I gave way, and thought it best to go on.\n\nI do so against the grain, and against every inclination, and against\nthe strongest feeling of gratitude to you. My people at home will be\nmiserable too when they hear I am going to do it. If I could have heard\nfrom you sooner, and got the bill out sooner, I should have been firmer\nin considering my own necessity of relief. As it is, I sneak under; and\nI hope you will feel the reasons, and approve.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Mr. Henry Austin.]\n\n                       GAD'S HILL PLACE, _Wednesday, Sept. 2nd, 1857._\n\nMY DEAR HENRY,\n\nThe second conspirator has been here this morning to ask whether you\nwish the windlass to be left in the yard, and whether you will want him\nand his mate any more, and, if so, when? Of course he says (rolling\nsomething in the form of a fillet in at one broken tooth all the while,\nand rolling it out at another) that they could wish fur to have the\nwindlass if it warn't any ways a hill conwenience fur to fetch her away.\nI have told him that if he will come back on Friday he shall have your\nreply. Will you, therefore, send it me by return of post? He says he'll\n\"look up\" (as if he was an astronomer) \"a Friday arterdinner.\"\n\nOn Monday I am going away with Collins for ten days or a fortnight, on a\n\"tour in search of an article\" for \"Household Words.\" We have not the\nleast idea where we are going; but _he_ says, \"Let's look at the Norfolk\ncoast,\" and _I_ say, \"Let's look at the back of the Atlantic.\" I don't\nquite know what I mean by that; but have a general impression that I\nmean something knowing.\n\nI am horribly used up after the Jerrold business. Low spirits, low\npulse, low voice, intense reaction. If I were not like Mr. Micawber,\n\"falling back for a spring\" on Monday, I think I should slink into a\ncorner and cry.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Miss Hogarth.]\n\n              ALLONBY, CUMBERLAND, _Wednesday Night, Sept. 9th, 1857._\n\nMY DEAR GEORGY,\n\n       *       *       *       *       *\n\nThink of Collins's usual luck with me! We went up a Cumberland mountain\nyesterday--a huge black hill, fifteen hundred feet high. We took for a\nguide a capital innkeeper hard by. It rained in torrents--as it only\ndoes rain in a hill country--the whole time. At the top, there were\nblack mists and the darkness of night. It then came out that the\ninnkeeper had not been up for twenty years, and he lost his head and\nhimself altogether; and we couldn't get down again! What wonders the\nInimitable performed with his compass until it broke with the heat and\nwet of his pocket no matter; it did break, and then we wandered about,\nuntil it was clear to the Inimitable that the night must be passed\nthere, and the enterprising travellers probably die of cold. We took our\nown way about coming down, struck, and declared that the guide might\nwander where he would, but we would follow a watercourse we lighted\nupon, and which must come at last to the river. This necessitated\namazing gymnastics; in the course of which performances, Collins fell\ninto the said watercourse with his ankle sprained, and the great\nligament of the foot and leg swollen I don't know how big.\n\nHow I enacted Wardour over again in carrying him down, and what a\nbusiness it was to get him down; I may say in Gibbs's words: \"Vi lascio\na giudicare!\" But he was got down somehow, and we got off the mountain\nsomehow; and now I carry him to bed, and into and out of carriages,\nexactly like Wardour in private life. I don't believe he will stand for\na month to come. He has had a doctor, and can wear neither shoe nor\nstocking, and has his foot wrapped up in a flannel waistcoat, and has a\nbreakfast saucer of liniment, and a horrible dabbling of lotion\nincessantly in progress. We laugh at it all, but I doubt very much\nwhether he can go on to Doncaster. It will be a miserable blow to our H.\nW. scheme, and I say nothing about it as yet; but he is really so\ncrippled that I doubt the getting him there. We have resolved to fall\nto work to-morrow morning and begin our writing; and there, for the\npresent, that point rests.\n\nThis is a little place with fifty houses, five bathing-machines, five\ngirls in straw hats, five men in straw hats, and no other company. The\nlittle houses are all in half-mourning--yellow stone on white stone, and\nblack; and it reminds me of what Broadstairs might have been if it had\nnot inherited a cliff, and had been an Irishman. But this is a capital\nlittle homely inn, looking out upon the sea; and we are really very\ncomfortably lodged. I can just stand upright in my bedroom. Otherwise,\nit is a good deal like one of Ballard's top-rooms. We have a very\nobliging and comfortable landlady; and it is a clean nice place in a\nrough wild country. We came here haphazard, but could not have done\nbetter.\n\nWe lay last night at a place called Wigton--also in half-mourning--with\nthe wonderful peculiarity that it had no population, no business, no\nstreets to speak of; but five linendrapers within range of our small\nwindows, one linendraper's next door, and five more linendrapers round\nthe corner. I ordered a night-light in my bedroom. A queer little old\nwoman brought me one of the common Child's night-lights, and seeming to\nthink that I looked at it with interest, said: \"It's joost a vara\nkeeyourious thing, sir, and joost new coom oop. It'll burn awt hoors a'\nend, an no gootther, nor no waste, nor ony sike a thing, if you can\ncreedit what I say, seein' the airticle.\"\n\nOf course _I_ shall go to Doncaster, whether or no (please God), and my\npostage directions to you remain unchanged. Love to Mamey, Katey,\nCharley, Harry, and the darling Plorn.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Miss Hogarth.]\n\n                        LANCASTER, _Saturday Night, Sept. 12th, 1857._\n\nMY DEAR GEORGY,\n\nI received your letter at Allonby yesterday, and was delighted to get\nit. We came back to Carlisle last night (to a capital inn, kept by\nBreach's brother), and came on here to-day. We are on our way to\nDoncaster; but Sabbath observance throws all the trains out; and\nalthough it is not a hundred miles from here, we shall have, as well as\nI can make out the complicated lists of trains, to sleep at Leeds--which\nI particularly detest as an odious place--to-morrow night.\n\nAccustomed as you are to the homage which men delight to render to the\nInimitable, you would be scarcely prepared for the proportions it\nassumes in this northern country. Station-masters assist him to alight\nfrom carriages, deputations await him in hotel entries, innkeepers bow\ndown before him and put him into regal rooms, the town goes down to the\nplatform to see him off, and Collins's ankle goes into the newspapers!!!\n\nIt is a great deal better than it was, and he can get into new hotels\nand up the stairs with two thick sticks, like an admiral in a farce. His\nspirits have improved in a corresponding degree, and he contemplates\ncheerfully the keeping house at Doncaster. I thought (as I told you) he\nwould never have gone there, but he seems quite up to the mark now. Of\ncourse he can never walk out, or see anything of any place. We have done\nour first paper for H. W., and sent it up to the printer's.\n\nThe landlady of the little inn at Allonby lived at Greta Bridge, in\nYorkshire, when I went down there before \"Nickleby,\" and was smuggled\ninto the room to see me, when I was secretly found out. She is an\nimmensely fat woman now. \"But I could tuck my arm round her waist then,\nMr. Dickens,\" the landlord said when she told me the story as I was\ngoing to bed the night before last. \"And can't you do it now,\" I said,\n\"you insensible dog? Look at me! Here's a picture!\" Accordingly, I got\nround as much of her as I could; and this gallant action was the most\nsuccessful I have ever performed, on the whole. I think it was the\ndullest little place I ever entered; and what with the monotony of an\nidle sea, and what with the monotony of another sea in the room\n(occasioned by Collins's perpetually holding his ankle over a pail of\nsalt water, and laving it with a milk jug), I struck yesterday, and came\naway.\n\nWe are in a very remarkable old house here, with genuine old rooms and\nan uncommonly quaint staircase. I have a state bedroom, with two\nenormous red four-posters in it, each as big as Charley's room at Gad's\nHill. Bellew is to preach here to-morrow. \"And we know he is a friend of\nyours, sir,\" said the landlord, when he presided over the serving of the\ndinner (two little salmon trout; a sirloin steak; a brace of partridges;\nseven dishes of sweets; five dishes of dessert, led off by a bowl of\npeaches; and in the centre an enormous bride-cake--\"We always have it\nhere, sir,\" said the landlord, \"custom of the house.\") (Collins turned\npale, and estimated the dinner at half a guinea each.)\n\nThis is the stupidest of letters, but all description is gone, or going,\ninto \"The Lazy Tour of Two Idle Apprentices.\"\n\nKiss the darling Plorn, who is often in my thoughts. Best love to\nCharley, Mamey, and Katie. I will write to you again from Doncaster,\nwhere I shall be rejoiced to find another letter from you.\n\n                               Ever affectionately, my dearest Georgy.\n\n\n[Sidenote: Miss Hogarth.]\n\n                  ANGEL HOTEL, DONCASTER, _Tuesday, Sept. 15th, 1857._\n\nMY DEAR GEORGY,\n\nI found your letter here on my arrival yesterday. I had hoped that the\nwall would have been almost finished by this time, and the additions to\nthe house almost finished too--but patience, patience!\n\nWe have very good, clean, and quiet apartments here, on the second\nfloor, looking down into the main street, which is full of horse\njockeys, bettors, drunkards, and other blackguards, from morning to\nnight--and all night. The races begin to-day and last till Friday, which\nis the Cup Day. I am not going to the course this morning, but have\nengaged a carriage (open, and pair) for to-morrow and Friday.\n\n\"The Frozen Deep's\" author gets on as well as could be expected. He can\nhobble up and down stairs when absolutely necessary, and limps to his\nbedroom on the same floor. He talks of going to the theatre to-night in\na cab, which will be the first occasion of his going out, except to\ntravel, since the accident. He sends his kind regards and thanks for\nenquiries and condolence. I am perpetually tidying the rooms after him,\nand carrying all sorts of untidy things which belong to him into his\nbedroom, which is a picture of disorder. You will please to imagine\nmine, airy and clean, little dressing-room attached, eight water-jugs (I\nnever saw such a supply), capital sponge-bath, perfect arrangement, and\nexquisite neatness. We breakfast at half-past eight, and fall to work\nfor H. W. afterwards. Then I go out, and--hem! look for subjects.\n\nThe mayor called this morning to do the honours of the town, whom it\npleased the Inimitable to receive with great courtesy and affability. He\npropounded invitation to public _dejeuner_, which it did _not_ please\nthe Inimitable to receive, and which he graciously rejected.\n\nThat's all the news. Everything I can describe by hook or by crook, I\ndescribe for H. W. So there is nothing of that sort left for letters.\n\nBest love to dear Mamey and Katey, and to Charley, and to Harry. Any\nnumber of kisses to the noble Plorn.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Mr. Arthur Ryland.]\n\n                 GAD'S HILL PLACE, _Saturday Evening, Oct. 3rd, 1857._\n\nMY DEAR SIR,\n\nI have had the honour and pleasure of receiving your letter of the 28th\nof last month, informing me of the distinction that has been conferred\nupon me by the Council of the Birmingham and Midland Institute.\n\nAllow me to assure you with much sincerity, that I am highly gratified\nby having been elected one of the first honorary members of that\nestablishment. Nothing could have enhanced my interest in so important\nan undertaking; but the compliment is all the more welcome to me on that\naccount.\n\nI accept it with a due sense of its worth, with many acknowledgments and\nwith all good wishes.\n\n                        I am ever, my dear Sir, very faithfully yours.\n\n\n[Sidenote: Mr. Edmund Yates.]\n\n                     TAVISTOCK HOUSE, _Monday Night, Nov. 16th, 1857._\n\nMY DEAR YATES,\n\nI retain the story with pleasure; and I need not tell you that you are\nnot mistaken in the last lines of your note.\n\nExcuse me, on that ground, if I say a word or two as to what I think (I\nmention it with a view to the future) might be better in the paper. The\nopening is excellent. But it passes too completely into the Irishman's\nnarrative, does not light it up with the life about it, or the\ncircumstances under which it is delivered, and does not carry through\nit, as I think it should with a certain indefinable subtleness, the\nthread with which you begin your weaving. I will tell Wills to send me\nthe proof, and will try to show you what I mean when I shall have gone\nover it carefully.\n\n                                              Faithfully yours always.\n\n\n[Sidenote: Mr. Frank Stone, A.R.A.]\n\n                        TAVISTOCK HOUSE, _Wednesday, Dec. 13th, 1857._\n\nMY DEAR STONE,\n\nI find on enquiry that the \"General Theatrical Fund\" has relieved\nnon-members in one or two instances; but that it is exceedingly\nunwilling to do so, and would certainly not do so again, saving on some\nvery strong and exceptional case. As its trustee, I could not represent\nto it that I think it ought to sail into those open waters, for I very\nmuch doubt the justice of such cruising, with a reference to the\ninterests of the patient people who support it out of their small\nearnings.\n\n                                                  Affectionately ever.\n\nFOOTNOTES:\n\n[1] The part played in \"The Frozen Deep\" by its author, Mr. Wilkie\nCollins.\n\n[2] The Earl of Carlisle was at this time Viceroy of Ireland.\n\n\n\n\nBook III.\n\n1858 TO 1870.\n\n\n\n\n1858.\n\nNARRATIVE.\n\n\nAll through this year, Charles Dickens was constantly moving about from\nplace to place. After much and careful consideration, he had come to the\ndetermination of, for the future, giving readings for his own benefit.\nAnd although in the spring of this year he gave one reading of his\n\"Christmas Carol\" for a charity, all the other readings, beginning from\nthe 29th April, and ever after, were for himself. In the autumn of this\nyear he made reading tours in England, Scotland, and Ireland, always\naccompanied by his friend and secretary, Mr. Arthur Smith. At Newcastle,\nCharles Dickens was joined by his daughters, who accompanied him in his\nScotch tour. The letters to his sister-in-law, and to his eldest\ndaughter, are all given here, and will be given in all future reading\ntours, as they form a complete diary of his life and movements at these\ntimes. To avoid the constant repetition of the two names, the beginning\nof the letters will be dispensed with in all cases where they follow\neach other in unbroken succession. The Mr. Frederick Lehmann mentioned\nin the letter written from Sheffield, had married a daughter of Mr.\nRobert Chambers, and niece of Mrs. Wills. Coming to settle in London a\nshort time after this date, Mr. and Mrs. Lehmann became intimately known\nto Charles Dickens and his family--more especially to his eldest\ndaughter, to whom they have been, and are, the kindest and truest of\nfriends. The \"pretty little boy\" mentioned as being under Mrs. Wills's\ncare, was their eldest son.\n\nWe give the letter to Mr. Thackeray, not because it is one of very great\ninterest, but because, being the only one we have, we are glad to have\nthe two names associated together in this work.\n\nThe \"little speech\" alluded to in this first letter to Mr. Macready was\none made by Charles Dickens at a public dinner, which was given in aid\nof the Hospital for Sick Children, in Great Ormond Street. He afterwards\n(early in April) gave a reading from his \"Christmas Carol\" for this same\ncharity.\n\nThe Christmas number of \"Household Words,\" mentioned in a letter to Mr.\nWilkie Collins, was called \"A House to Let,\" and contained stories\nwritten by Charles Dickens, Mr. Wilkie Collins, and other contributors\nto \"Household Words.\"\n\n\n[Sidenote: Mr. W. Wilkie Collins.]\n\n                           TAVISTOCK HOUSE, _Sunday, Jan. 17th, 1858._\n\nMY DEAR WILKIE,\n\nI am very sorry to receive so bad an account of the foot. But I hope it\nis all in the past tense now.\n\nI met with an incident the other day, which I think is a good deal in\nyour way, for introduction either into a long or short story. Dr.\nSutherland and Dr. Monro went over St. Luke's with me (only last\nFriday), to show me some distinctly and remarkably developed types of\ninsanity. Among other patients, we passed a deaf and dumb man, now\nafflicted with incurable madness too, of whom they said that it was only\nwhen his madness began to develop itself in strongly-marked mad actions,\nthat it began to be suspected. \"Though it had been there, no doubt, some\ntime.\" This led me to consider, suspiciously, what employment he had\nbeen in, and so to ask the question. \"Aye,\" says Dr. Sutherland, \"that\nis the most remarkable thing of all, Mr. Dickens. He was employed in the\ntransmission of electric-telegraph messages; and it is impossible to\nconceive what delirious despatches that man may have been sending about\nall over the world!\"\n\nRejoiced to hear such good report of the play.\n\n                                                      Ever faithfully.\n\n\n[Sidenote: Mr. Edmund Yates.]\n\n                           TAVISTOCK HOUSE, _Tuesday, Feb. 2nd, 1858._\n\nMY DEAR YATES,\n\nYour quotation is, as I supposed, all wrong. The text is _not_ \"which\nhis 'owls was organs.\" When Mr. Harris went into an empty dog-kennel, to\nspare his sensitive nature the anguish of overhearing Mrs. Harris's\nexclamations on the occasion of the birth of her first child (the\nPrincess Royal of the Harris family), \"he never took his hands away from\nhis ears, or came out once, till he was showed the baby.\" On\nencountering that spectacle, he was (being of a weakly constitution)\n\"took with fits.\" For this distressing complaint he was medically\ntreated; the doctor \"collared him, and laid him on his back upon the\nairy stones\"--please to observe what follows--\"and she was told, to ease\nher mind, his 'owls was organs.\"\n\nThat is to say, Mrs. Harris, lying exhausted on her bed, in the first\nsweet relief of freedom from pain, merely covered with the counterpane,\nand not yet \"put comfortable,\" hears a noise apparently proceeding from\nthe back-yard, and says, in a flushed and hysterical manner: \"What 'owls\nare those? Who is a-'owling? Not my ugebond?\" Upon which the doctor,\nlooking round one of the bottom posts of the bed, and taking Mrs.\nHarris's pulse in a reassuring manner, says, with much admirable\npresence of mind: \"Howls, my dear madam?--no, no, no! What are we\nthinking of? Howls, my dear Mrs. Harris? Ha, ha, ha! Organs, ma'am,\norgans. Organs in the streets, Mrs. Harris; no howls.\"\n\n                                                     Yours faithfully.\n\n\n[Sidenote: Mr. W. M. Thackeray.]\n\n                           TAVISTOCK HOUSE, _Tuesday, Feb. 2nd, 1858._\n\nMY DEAR THACKERAY,\n\nThe wisdom of Parliament, in that expensive act of its greatness which\nconstitutes the Guild, prohibits that corporation _from doing anything_\nuntil it shall have existed in a perfectly useless condition for seven\nyears. This clause (introduced by some private-bill magnate of official\nmight) seemed so ridiculous, that nobody could believe it to have this\nmeaning; but as I felt clear about it when we were on the very verge of\ngranting an excellent literary annuity, I referred the point to counsel,\nand my construction was confirmed without a doubt.\n\nIt is therefore needless to enquire whether an association in the nature\nof a provident society could address itself to such a case as you\nconfide to me. The prohibition has still two or three years of life in\nit.\n\nBut, assuming the gentleman's title to be considered as an \"author\" as\nestablished, there is no question that it comes within the scope of the\nLiterary Fund. They would habitually \"lend\" money if they did what I\nconsider to be their duty; as it is they only give money, but they give\nit in such instances.\n\nI have forwarded the envelope to the Society of Arts, with a request\nthat they will present it to Prince Albert, approaching H.R.H. in the\nSiamese manner.\n\n                                                      Ever faithfully.\n\n\n[Sidenote: Mr. John Forster.]\n\n                   TAVISTOCK HOUSE, _Wednesday Night, Feb. 3rd, 1858._\n\nMY DEAR FORSTER,\n\nI beg to report two phenomena:\n\n1. An excellent little play in one act, by Marston, at the Lyceum;\ntitle, \"A Hard Struggle;\" as good as \"La Joie fait Peur,\" though not at\nall like it.\n\n2. Capital acting in the same play, by Mr. Dillon. Real good acting, in\nimitation of nobody, and honestly made out by himself!!\n\nI went (at Marston's request) last night, and cried till I sobbed again.\nI have not seen a word about it from Oxenford. But it is as wholesome\nand manly a thing altogether as I have seen for many a day. (I would\nhave given a hundred pounds to have played Mr. Dillon's part).\n\nLove to Mrs. Forster.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Dr. Westland Marston.]\n\n                         TAVISTOCK HOUSE, _Wednesday, Feb. 3rd, 1858._\n\nMY DEAR MARSTON,\n\nI most heartily and honestly congratulate you on your charming little\npiece. It moved me more than I could easily tell you, if I were to try.\nExcept \"La Joie fait Peur,\" I have seen nothing nearly so good, and\nthere is a subtlety in the comfortable presentation of the child who is\nto become a devoted woman for Reuben's sake, which goes a long way\nbeyond Madame de Girardin. I am at a loss to let you know how much I\nadmired it last night, or how heartily I cried over it. A touching idea,\nmost delicately conceived and wrought out by a true artist and poet, in\na spirit of noble, manly generosity, that no one should be able to study\nwithout great emotion.\n\nIt is extremely well acted by all concerned; but Mr. Dillon's\nperformance is really admirable, and deserving of the highest\ncommendation. It is good in these days to see an actor taking such\npains, and expressing such natural and vigorous sentiment. There is only\none thing I should have liked him to change. I am much mistaken if any\nman--least of all any such man--would crush a letter written by the hand\nof the woman he loved. Hold it to his heart unconsciously and look about\nfor it the while, he might; or he might do any other thing with it that\nexpressed a habit of tenderness and affection in association with the\nidea of her; but he would never crush it under any circumstances. He\nwould as soon crush her heart.\n\nYou will see how closely I went with him, by my minding so slight an\nincident in so fine a performance. There is no one who could approach\nhim in it; and I am bound to add that he surprised me as much as he\npleased me.\n\nI think it might be worth while to try the people at the Francais with\nthe piece. They are very good in one-act plays; such plays take well\nthere, and this seems to me well suited to them. If you would like\nSamson or Regnier to read the play (in English), I know them well, and\nwould be very glad indeed to tell them that I sent it with your sanction\nbecause I had been so much struck by it.\n\n                                              Faithfully yours always.\n\n\n[Sidenote: Monsieur Regnier.]\n\n           TAVISTOCK HOUSE, LONDON, W.C., _Thursday, Feb. 11th, 1858._\n\nMY DEAR REGNIER,\n\nI want you to read the enclosed little play. You will see that it is in\none act--about the length of \"La Joie fait Pour.\" It is now acting at\nthe Lyceum Theatre here, with very great success. The author is Mr.\nWestland Marston, a dramatic writer of reputation, who wrote a very\nwell-known tragedy called \"The Patrician's Daughter,\" in which Macready\nand Miss Faucit acted (under Macready's management at Drury Lane) some\nyears ago.\n\nThis little piece is so very powerful on the stage, its interest is so\nsimple and natural, and the part of Reuben is such a very fine one, that\nI cannot help thinking you might make one grand _coup_ with it, if with\nyour skilful hand you arranged it for the Francais. I have communicated\nthis idea of mine to the author, \"_et la-dessus je vous ecris_.\" I am\nanxious to know your opinion, and shall expect with much interest to\nreceive a little letter from you at your convenience.\n\nMrs. Dickens, Miss Hogarth, and all the house send a thousand kind loves\nand regards to Madame Regnier and the dear little boys. You will bring\nthem to London when you come, with all the force of the Francais--will\nyou not?\n\n                        Ever, my dear Regnier, faithfully your Friend.\n\n\n[Sidenote: Monsieur Regnier.]\n\n                         TAVISTOCK HOUSE, _Saturday, Feb. 20th, 1858._\n\nMY DEAR REGNIER,\n\nLet me thank you with all my heart for your most patient and kind\nletter. I made its contents known to Mr. Marston, and I enclose you his\nreply. You will see that he cheerfully leaves the matter in your hands,\nand abides by your opinion and discretion.\n\nYou need not return his letter, my friend. There is great excitement\nhere this morning, in consequence of the failure of the Ministry last\nnight to carry the bill they brought in to please your Emperor and his\ntroops. _I_, for one, am extremely glad of their defeat.\n\n\"Le vieux P----,\" I have no doubt, will go staggering down the Rue de la\nPaix to-day, with his stick in his hand and his hat on one side,\npredicting the downfall of everything, in consequence of this event. His\nhandwriting shakes more and more every quarter, and I think he mixes a\ngreat deal of cognac with his ink. He always gives me some astonishing\npiece of news (which is never true), or some suspicious public prophecy\n(which is never verified), and he always tells me he is dying (which he\nnever is).\n\nAdieu, my dear Regnier, accept a thousand thanks from me, and believe\nme, now and always,\n\n                                Your affectionate and faithful Friend.\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n                                  TAVISTOCK HOUSE, _March 15th, 1858._\n\nMY DEAREST MACREADY,\n\nI have safely received your cheque this morning, and will hand it over\nforthwith to the honorary secretary of the hospital. I hope you have\nread the little speech in the hospital's publication of it. They had it\ntaken by their own shorthand-writer, and it is done verbatim.\n\nYou may be sure that it is a good and kind charity. It is amazing to me\nthat it is not at this day ten times as large and rich as it is. But I\nhope and trust that I have happily been able to give it a good thrust\nonward into a great course. We all send our most affectionate love to\nall the house. I am devising all sorts of things in my mind, and am in a\nstate of energetic restlessness incomprehensible to the calm\nphilosophers of Dorsetshire. What a dream it is, this work and strife,\nand how little we do in the dream after all! Only last night, in my\nsleep, I was bent upon getting over a perspective of barriers, with my\nhands and feet bound. Pretty much what we are all about, waking, I\nthink?\n\nBut, Lord! (as I said before) you smile pityingly, not bitterly, at this\nhubbub, and moralise upon it, in the calm evenings when there is no\nschool at Sherborne.\n\n                                        Ever affectionately and truly.\n\n\n[Sidenote: Mrs Hogge.[3]]\n\n              TAVISTOCK HOUSE, TAVISTOCK SQUARE, LONDON, W.C.,\n                                        _Wednesday, April 14th, 1858._\n\nMY DEAR MRS. HOGGE,\n\nAfter the profoundest cogitation, I come reluctantly to the conclusion\nthat I do not know that orphan. If you were the lady in want of him, I\nshould certainly offer _myself_. But as you are not, I will not hear of\nthe situation.\n\nIt is wonderful to think how many charming little people there must be,\nto whom this proposal would be like a revelation from Heaven. Why don't\nI know one, and come to Kensington, boy in hand, as if I had walked (I\nwish to God I had) out of a fairy tale! But no, I do _not_ know that\norphan. He is crying somewhere, by himself, at this moment. I can't dry\nhis eyes. He is being neglected by some ogress of a nurse. I can't\nrescue him.\n\nI will make a point of going to the Athenaeum on Monday night; and if I\nhad five hundred votes to give, Mr. Macdonald should have them all, for\nyour sake.\n\nI grieve to hear that you have been ill, but I hope that the spring,\nwhen it comes, will find you blooming with the rest of the flowers.\n\n                                                Very faithfully yours.\n\n\n[Sidenote: Mr. Edmund Yates.]\n\n                  TAVISTOCK HOUSE, TAVISTOCK SQUARE, LONDON, W.C.,\n                                        _Wednesday, April 28th, 1858._\n\nMY DEAR YATES,\n\nFor a good many years I have suffered a great deal from charities, but\nnever anything like what I suffer now. The amount of correspondence they\ninflict upon me is really incredible. But this is nothing. Benevolent\nmen get behind the piers of the gates, lying in wait for my going out;\nand when I peep shrinkingly from my study-windows, I see their\npot-bellied shadows projected on the gravel. Benevolent bullies drive up\nin hansom cabs (with engraved portraits of their benevolent institutions\nhanging over the aprons, like banners on their outward walls), and stay\nlong at the door. Benevolent area-sneaks get lost in the kitchens and\nare found to impede the circulation of the knife-cleaning machine. My\nman has been heard to say (at The Burton Arms) \"that if it was a\nwicious place, well and good--_that_ an't door work; but that wen all\nthe Christian wirtues is always a-shoulderin' and a-helberin' on you in\nthe 'all, a-tryin' to git past you and cut upstairs into master's room,\nwhy no wages as you couldn't name wouldn't make it up to you.\"\n\n                                                     Persecuted ever.\n\n\n[Sidenote: Mrs Yates.]\n\n(THE CHARMING ACTRESS, THE MOTHER OF MR. EDMUND YATES.)\n\n                       TAVISTOCK HOUSE, TAVISTOCK SQUARE, W.C.,\n                                    _Saturday Evening, May 15th, 1858._\n\nMY DEAR MRS. YATES,\n\nPray believe that I was sorry with all my heart to miss you last\nThursday, and to learn the occasion of your absence; also that, whenever\nyou can come, your presence will give me a new interest in that evening.\nNo one alive can have more delightful associations with the lightest\nsound of your voice than I have; and to give you a minute's interest and\npleasure, in acknowledgment of the uncountable hours of happiness you\ngave me when you were a mysterious angel to me, would honestly gratify\nmy heart.\n\n                                 Very faithfully and gratefully yours.\n\n\n[Sidenote: M. de Cerjat.]\n\n                              GAD'S HILL, _Wednesday, July 7th, 1858._\n\nMY DEAR CERJAT,\n\nI should vainly try to tell you--so I _won't_ try--how affected I have\nbeen by your warm-hearted letter, or how thoroughly well convinced I\nalways am of the truth and earnestness of your friendship. I thank you,\nmy dear, dear fellow, with my whole soul. I fervently return that\nfriendship and I highly cherish it.\n\nYou want to know all about me? I am still reading in London every\nThursday, and the audiences are very great, and the success immense. On\nthe 2nd of August I am going away on a tour of some four months in\nEngland, Ireland, and Scotland. I shall read, during that time, not\nfewer than four or five times a week. It will be sharp work; but\nprobably a certain musical clinking will come of it, which will mitigate\nthe hardship.\n\nAt this present moment I am on my little Kentish freehold (_not_ in\ntop-boots, and not particularly prejudiced that I know of), looking on\nas pretty a view out of my study window as you will find in a long day's\nEnglish ride. My little place is a grave red brick house (time of George\nthe First, I suppose), which I have added to and stuck bits upon in all\nmanner of ways, so that it is as pleasantly irregular, and as violently\nopposed to all architectural ideas, as the most hopeful man could\npossibly desire. It is on the summit of Gad's Hill. The robbery was\ncommitted before the door, on the man with the treasure, and Falstaff\nran away from the identical spot of ground now covered by the room in\nwhich I write. A little rustic alehouse, called The Sir John Falstaff,\nis over the way--has been over the way, ever since, in honour of the\nevent. Cobham Woods and Park are behind the house; the distant Thames in\nfront; the Medway, with Rochester, and its old castle and cathedral, on\none side. The whole stupendous property is on the old Dover Road, so\nwhen you come, come by the North Kent Railway (not the South-Eastern) to\nStrood or Higham, and I'll drive over to fetch you.\n\nThe blessed woods and fields have done me a world of good, and I am\nquite myself again. The children are all as happy as children can be. My\neldest daughter, Mary, keeps house, with a state and gravity becoming\nthat high position; wherein she is assisted by her sister Katie, and by\nher aunt Georgina, who is, and always has been, like another sister. Two\nbig dogs, a bloodhound and a St. Bernard, direct from a convent of that\nname, where I think you once were, are their principal attendants in the\ngreen lanes. These latter instantly untie the neckerchiefs of all tramps\nand prowlers who approach their presence, so that they wander about\nwithout any escort, and drive big horses in basket-phaetons through\nmurderous bye-ways, and never come to grief. They are very curious about\nyour daughters, and send all kinds of loves to them and to Mrs. Cerjat,\nin which I heartily join.\n\nYou will have read in the papers that the Thames in London is most horrible.\nI have to cross Waterloo or London Bridge to get to the railroad when I\ncome down here, and I can certify that the offensive smells, even in\nthat short whiff, have been of a most head-and-stomach-distending\nnature. Nobody knows what is to be done; at least everybody knows a\nplan, and everybody else knows it won't do; in the meantime cartloads of\nchloride of lime are shot into the filthy stream, and do something I\nhope. You will know, before you get this, that the American telegraph\nline has parted again, at which most men are sorry, but very few\nsurprised. This is all the news, except that there is an Italian Opera\nat Drury Lane, price eighteenpence to the pit, where Viardot, by far the\ngreatest artist of them all, sings, and which is full when the dear\nopera can't let a box; and except that the weather has been\nexceptionally hot, but is now quite cool. On the top of this hill it has\nbeen cold, actually cold at night, for more than a week past.\n\nI am going over to Rochester to post this letter, and must write another\nto Townshend before I go. My dear Cerjat, I have written lightly\nenough, because I want you to know that I am becoming cheerful and\nhearty. God bless you! I love you, and I know that you love me.\n\n                                  Ever your attached and affectionate.\n\n\n[Sidenote: Miss Hogarth.]\n\n                       WEST HOE, PLYMOUTH, _Thursday, Aug. 5th, 1858._\n\nMY DEAREST GEORGY,\n\nI received your letter this morning with the greatest pleasure, and read\nit with the utmost interest in all its domestic details.\n\nWe had a most wonderful night at Exeter. It is to be regretted that we\ncannot take the place again on our way back. It was a prodigious cram,\nand we turned away no end of people. But not only that, I think they\nwere the finest audience I have ever read to. I don't think I ever read,\nin some respects, so well; and I never beheld anything like the personal\naffection which they poured out upon me at the end. It was really a very\nremarkable sight, and I shall always look back upon it with pleasure.\n\nLast night here was not so bright. There are quarrels of the strangest\nkind between the Plymouth people and the Stonehouse people. The room is\nat Stonehouse (Tracy says the wrong room; there being a Plymouth room in\nthis hotel, and he being a Plymouthite). We had a fair house, but not at\nall a great one. All the notabilities come this morning to \"Little\nDombey,\" for which we have let one hundred and thirty stalls, which\nlocal admiration of local greatness considers very large. For \"Mrs. Gamp\nand the Boots,\" to-night, we have also a very promising let. But the\nraces are on, and there are two public balls to-night, and the yacht\nsquadron are all at Cherbourg to boot. Arthur is of opinion that \"Two\nSixties\" will do very well for us. I doubt the \"Two Sixties\" myself.\n_Mais nous verrons._\n\nThe room is a very handsome one, but it is on the top of a windy and\nmuddy hill, leading (literally) to nowhere; and it looks (except that it\nis new and _mortary_) as if the subsidence of the waters after the\nDeluge might have left it where it is. I have to go right through the\ncompany to get to the platform. Big doors slam and resound when anybody\ncomes in; and all the company seem afraid of one another. Nevertheless\nthey were a sensible audience last night, and much impressed and\npleased.\n\nTracy is in the room (wandering about, and never finishing a sentence),\nand sends all manner of sea-loves to you and the dear girls. I send all\nmanner of land-loves to you from myself, out of my heart of hearts, and\nalso to my dear Plorn and the boys.\n\nArthur sends his kindest love. He knows only two characters. He is\neither always corresponding, like a Secretary of State, or he is\ntransformed into a rout-furniture dealer of Rathbone Place, and drags\nforms about with the greatest violence, without his coat.\n\nI have no time to add another word.\n\n                         Ever, dearest Georgy, your most affectionate.\n\n\n[Sidenote: Miss Dickens.]\n\n                                   LONDON, _Saturday, Aug. 7th, 1858._\n\nMY DEAREST MAMEY,\n\nThe closing night at Plymouth was a very great scene, and the morning\nthere was exceedingly good too. You will be glad to hear that at Clifton\nlast night, a torrent of five hundred shillings bore Arthur away,\npounded him against the wall, flowed on to the seats over his body,\nscratched him, and damaged his best dress suit. All to his unspeakable\njoy.\n\nThis is a very short letter, but I am going to the Burlington Arcade,\ndesperately resolved to have all those wonderful instruments put into\noperation on my head, with a view to refreshing it.\n\nKindest love to Georgy and to all.\n\n                                               Ever your affectionate.\n\n\n[Sidenote: Miss Dickens.]\n\n                              SHREWSBURY, _Thursday, Aug. 12th, 1858._\n\nA wonderful audience last night at Wolverhampton. If such a thing can\nbe, they were even quicker and more intelligent than the audience I had\nin Edinburgh. They were so wonderfully good and were so much on the\nalert this morning by nine o'clock for another reading, that we are\ngoing back there at about our Bradford time. I never saw such people.\nAnd the local agent would take no money, and charge no expenses of his\nown.\n\nThis place looks what Plorn would call \"ortily\" dull. Local agent\npredicts, however, \"great satisfaction to Mr. Dickens, and excellent\nattendance.\" I have just been to look at the hall, where everything was\nwrong, and where I have left Arthur making a platform for me out of\ndining-tables.\n\nIf he comes back in time, I am not quite sure but that he is himself\ngoing to write to Gad's Hill. We talk of coming up from Chester _in the\nnight to-morrow, after the reading_; and of showing our precious selves\nat an apparently impossibly early hour in the Gad's Hill breakfast-room\non Saturday morning.\n\nI have not felt the fatigue to any extent worth mentioning; though I\nget, every night, into the most violent heats. We are going to dine at\nthree o'clock (it wants a quarter now) and have not been here two\nhours, so I have seen nothing of Clement.\n\nTell Georgy with my love, that I read in the same room in which we\nacted, but at the end opposite to that where our stage was. We are not\nat the inn where the amateur company put up, but at The Lion, where the\nfair Miss Mitchell was lodged alone. We have the strangest little rooms\n(sitting-room and two bed-rooms all together), the ceilings of which I\ncan touch with my hand. The windows bulge out over the street, as if\nthey were little stern-windows in a ship. And a door opens out of the\nsitting-room on to a little open gallery with plants in it, where one\nleans over a queer old rail, and looks all downhill and slant-wise at\nthe crookedest black and yellow old houses, all manner of shapes except\nstraight shapes. To get into this room we come through a china closet;\nand the man in laying the cloth has actually knocked down, in that\nrepository, two geraniums and Napoleon Bonaparte.\n\nI think that's all I have to say, except that at the Wolverhampton\ntheatre they played \"Oliver Twist\" last night (Mr. Toole the Artful\nDodger), \"in consequence of the illustrious author honouring the town\nwith his presence.\" We heard that the device succeeded very well, and\nthat they got a good many people.\n\nJohn's spirits have been equable and good since we rejoined him. Berry\nhas always got something the matter with his digestion--seems to me the\nmale gender of Maria Jolly, and ought to take nothing but Revalenta\nArabica. Bottled ale is not to be got in these parts, and Arthur is\nthrown upon draught.\n\nMy dearest love to Georgy and to Katey, also to Marguerite. Also to all\nthe boys and the noble Plorn.\n\n                                        Ever your affectionate Father.\n\n\n[Sidenote: Miss Hogarth.]\n\n                TAVISTOCK HOUSE, TAVISTOCK SQUARE, LONDON, W.C.,\n                                 _Wednesday Morning, Aug. 18th, 1858._\n\nI write this hurried line before starting, to report that my cold is\ndecidedly better, thank God (though still bad), and that I hope to be\nable to stagger through to-night. After dinner yesterday I began to\nrecover my voice, and I think I sang half the Irish Melodies to myself,\nas I walked about to test it. I got home at half-past ten, and\nmustard-poulticed and barley-watered myself tremendously.\n\nLove to the dear girls, and to all.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: The same.]\n\n            ADELPHI HOTEL, LIVERPOOL, _Friday Night, Aug. 20th, 1858._\n\nI received your welcome and interesting letter to-day, and I write you a\nvery hurried and bad reply; but it is _after the reading_, and you will\ntake the will for the deed under these trying circumstances, I know.\n\nWe have had a tremendous night; the largest house I have ever had since\nI first began--two thousand three hundred people. To-morrow afternoon,\nat three, I read again.\n\nMy cold has been oppressive, and is not yet gone. I have been very hard\nto sleep too, and last night I was all but sleepless. This morning I was\nvery dull and seedy; but I got a good walk, and picked up again. It has\nbeen blowing all day, and I fear we shall have a sick passage over to\nDublin to-morrow night.\n\nTell Mamie (with my dear love to her and Katie) that I will write to her\nfrom Dublin--probably on Sunday. Tell her too that the stories she told\nme in her letter were not only capital stories in themselves, but\n_excellently told_ too.\n\nWhat Arthur's state has been to-night--he, John, Berry, and Boylett, all\ntaking money and going mad together--you _cannot_ imagine. They turned\naway hundreds, sold all the books, rolled on the ground of my room\nknee-deep in checks, and made a perfect pantomime of the whole thing. He\nhas kept quite well, I am happy to say, and sends a hundred loves.\n\nIn great haste and fatigue.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Miss Dickens.]\n\n                  MORRISON'S HOTEL, DUBLIN, _Monday, Aug. 23rd, 1858._\n\nWe had a nasty crossing here. We left Holyhead at one in the morning,\nand got here at six. Arthur was incessantly sick the whole way. I was\nnot sick at all, but was in as healthy a condition otherwise as humanity\nneed be. We are in a beautiful hotel. Our sitting-room is exactly like\nthe drawing-room at the Peschiere in all its dimensions. I never saw two\nrooms so exactly resembling one another in their proportions. Our\nbedrooms too are excellent, and there are baths and all sorts of\ncomforts.\n\nThe Lord Lieutenant is away, and the place looks to me as if its\nprofessional life were away too. Nevertheless, there are numbers of\npeople in the streets. Somehow, I hardly seem to think we are going to\ndo enormously here; but I have scarcely any reason for supposing so\n(except that a good many houses are shut up); and I _know_ nothing about\nit, for Arthur is now gone to the agent and to the room. The men came by\nboat direct from Liverpool. They had a rough passage, were all ill, and\ndid not get here till noon yesterday. Donnybrook Fair, or what remains\nof it, is going on, within two or three miles of Dublin. They went out\nthere yesterday in a jaunting-car, and John described it to us at\ndinner-time (with his eyebrows lifted up, and his legs well asunder), as\n\"Johnny Brooks's Fair;\" at which Arthur, who was drinking bitter ale,\nnearly laughed himself to death. Berry is always unfortunate, and when I\nasked what had happened to Berry on board the steamboat, it appeared\nthat \"an Irish gentleman which was drunk, and fancied himself the\ncaptain, wanted to knock Berry down.\"\n\nI am surprised by finding this place very much larger than I had\nsupposed it to be. Its bye-parts are bad enough, but cleaner, too, than\nI had supposed them to be, and certainly very much cleaner than the old\ntown of Edinburgh. The man who drove our jaunting-car yesterday hadn't a\npiece in his coat as big as a penny roll, and had had his hat on\n(apparently without brushing it) ever since he was grown up. But he was\nremarkably intelligent and agreeable, with something to say about\neverything. For instance, when I asked him what a certain building was,\nhe didn't say \"courts of law\" and nothing else, but: \"Av you plase, sir,\nit's the foor coorts o' looyers, where Misther O'Connell stood his trial\nwunst, ye'll remimber, sir, afore I tell ye of it.\" When we got into the\nPhoenix Park, he looked round him as if it were his own, and said:\n\"THAT'S a park, sir, av yer plase.\" I complimented it, and he said:\n\"Gintlemen tills me as they'r bin, sir, over Europe, and never see a\npark aqualling ov it. 'Tis eight mile roond, sir, ten mile and a half\nlong, and in the month of May the hawthorn trees are as beautiful as\nbrides with their white jewels on. Yonder's the vice-regal lodge, sir;\nin them two corners lives the two sicretirries, wishing I was them, sir.\nThere's air here, sir, av yer plase! There's scenery here, sir! There's\nmountains--thim, sir! Yer coonsider it a park, sir? It is that, sir!\"\n\nYou should have heard John in my bedroom this morning endeavouring to\nimitate a bath-man, who had resented his interference, and had said as\nto the shower-bath: \"Yer'll not be touching _that_, young man. Divil a\ntouch yer'll touch o' that insthrument, young man!\" It was more\nridiculously unlike the reality than I can express to you, yet he was so\ndelighted with his powers that he went off in the absurdest little\ngingerbeery giggle, backing into my portmanteau all the time.\n\nMy dear love to Katie and to Georgy, also to the noble Plorn and all the\nboys. I shall write to Katie next, and then to Aunty. My cold, I am\nhappy to report, is very much better. I lay in the wet all night on\ndeck, on board the boat, but am not as yet any the worse for it. Arthur\nwas quite insensible when we got to Dublin, and stared at our luggage\nwithout in the least offering to claim it. He left his kindest love for\nall before he went out. I will keep the envelope open until he comes in.\n\n                             Ever, my dearest Mamie,\n                                        Your most affectionate Father.\n\n\n[Sidenote: Miss Hogarth.]\n\n               MORRISON'S HOTEL, DUBLIN, _Wednesday, Aug. 25th, 1858._\n\nI begin my letter to you to-day, though I don't know when I may send it\noff. We had a very good house last night, after all, that is to say, a\ngreat rush of shillings and good half-crowns, though the stalls were\ncomparatively few. For \"Little Dombey,\" this morning, we have an immense\nstall let--already more than two hundred--and people are now fighting in\nthe agent's shop to take more. Through some mistake of our printer's,\nthe evening reading for this present Wednesday was dropped, in a great\npart of the announcements, and the agent opened no plan for it. I have\ntherefore resolved not to have it at all. Arthur Smith has waylaid me\nin all manner of ways, but I remain obdurate. I am frightfully tired,\nand really relieved by the prospect of an evening--overjoyed.\n\nThey were a highly excitable audience last night, but they certainly did\nnot comprehend--internally and intellectually comprehend--\"The Chimes\"\nas a London audience do. I am quite sure of it. I very much doubt the\nIrish capacity of receiving the pathetic; but of their quickness as to\nthe humorous there can be no doubt. I shall see how they go along with\nLittle Paul, in his death, presently.\n\nWhile I was at breakfast this morning, a general officer was announced\nwith great state--having a staff at the door--and came in, booted and\nplumed, and covered with Crimean decorations. It was Cunninghame, whom\nwe knew in Genoa--then a captain. He was very hearty indeed, and came to\nask me to dinner. Of course I couldn't go. Olliffe has a brother at\nCork, who has just now (noon) written to me, proposing dinners and\nexcursions in that neighbourhood which would fill about a week; I being\nthere a day and a half, and reading three times. The work will be very\nsevere here, and I begin to feel depressed by it. (By \"here,\" I mean\nIreland generally, please to observe.)\n\nWe meant, as I said in a letter to Katie, to go to Queenstown yesterday\nand bask on the seashore. But there is always so much to do that we\ncouldn't manage it after all. We expect a tremendous house to-morrow\nnight as well as to-day; and Arthur is at the present instant up to his\neyes in business (and seats), and, between his regret at losing\nto-night, and his desire to make the room hold twice as many as it\n_will_ hold, is half distracted. I have become a wonderful\nIrishman--must play an Irish part some day--and his only relaxation is\nwhen I enact \"John and the Boots,\" which I consequently do enact all day\nlong. The papers are full of remarks upon my white tie, and describe it\nas being of enormous size, which is a wonderful delusion, because, as\nyou very well know, it is a small tie. Generally, I am happy to report,\nthe Emerald press is in favour of my appearance, and likes my eyes. But\none gentleman comes out with a letter at Cork, wherein he says that\nalthough only forty-six I look like an old man. _He_ is a rum customer,\nI think.\n\nThe Rutherfords are living here, and wanted me to dine with them, which,\nI needn't say, could not be done; all manner of people have called, but\nI have seen only two. John has given it up altogether as to rivalry with\nthe Boots, and did not come into my room this morning at all. Boots\nappeared triumphant and alone. He was waiting for me at the hotel-door\nlast night. \"Whaa't sart of a hoose, sur?\" he asked me. \"Capital.\" \"The\nLard be praised fur the 'onor o' Dooblin!\"\n\nArthur buys bad apples in the streets and brings them home and doesn't\neat them, and then I am obliged to put them in the balcony because they\nmake the room smell faint. Also he meets countrymen with honeycomb on\ntheir heads, and leads them (by the buttonhole when they have one) to\nthis gorgeous establishment and requests the bar to buy honeycomb for\nhis breakfast; then it stands upon the sideboard uncovered and the flies\nfall into it. He buys owls, too, and castles, and other horrible\nobjects, made in bog-oak (that material which is not appreciated at\nGad's Hill); and he is perpetually snipping pieces out of newspapers and\nsending them all over the world. While I am reading he conducts the\ncorrespondence, and his great delight is to show me seventeen or\neighteen letters when I come, exhausted, into the retiring-place. Berry\nhas not got into any particular trouble for forty-eight hours, except\nthat he is all over boils. I have prescribed the yeast, but\nineffectually. It is indeed a sight to see him and John sitting in\npay-boxes, and surveying Ireland out of pigeon-holes.\n\n                                       _Same Evening before Bed-time._\n\nEverybody was at \"Little Dombey\" to-day, and although I had some little\ndifficulty to work them up in consequence of the excessive crowding of\nthe place, and the difficulty of shaking the people into their seats,\nthe effect was unmistakable and profound. The crying was universal, and\nthey were extraordinarily affected. There is no doubt we could stay here\na week with that one reading, and fill the place every night. Hundreds\nof people have been there to-night, under the impression that it would\ncome off again. It was a most decided and complete success.\n\nArthur has been imploring me to stop here on the Friday after Limerick,\nand read \"Little Dombey\" again. But I have positively said \"No.\" The\nwork is too hard. It is not like doing it in one easy room, and always\nthe same room. With a different place every night, and a different\naudience with its own peculiarity every night, it is a tremendous\nstrain. I was sick of it to-day before I began, then got myself into\nwonderful train.\n\nHere follows a dialogue (but it requires imitation), which I had\nyesterday morning with a little boy of the house--landlord's son, I\nsuppose--about Plorn's age. I am sitting on the sofa writing, and find\nhim sitting beside me.\n\n        INIMITABLE. Holloa, old chap.\n\n        YOUNG IRELAND. Hal-loo!\n\n        INIMITABLE (_in his delightful way_). What a\n        nice old fellow you are. I am very fond of\n        little boys.\n\n        YOUNG IRELAND. Air yer? Ye'r right.\n\n        INIMITABLE. What do you learn, old fellow?\n\n        YOUNG IRELAND (_very intent on Inimitable, and\n        always childish, except in his brogue_). I\n        lairn wureds of three sillibils, and wureds of\n        two sillibils, and wureds of one sillibil.\n\n        INIMITABLE (_gaily_). Get out, you humbug! You\n        learn only words of one syllable.\n\n        YOUNG IRELAND (_laughs heartily_). You may say\n        that it is mostly wureds of one sillibil.\n\n        INIMITABLE. Can you write?\n\n        YOUNG IRELAND. Not yet. Things comes by\n        deegrays.\n\n        INIMITABLE. Can you cipher?\n\n        YOUNG IRELAND (_very quickly_). Wha'at's that?\n\n        INIMITABLE. Can you make figures?\n\n        YOUNG IRELAND. I can make a nought, which is\n        not asy, being roond.\n\n        INIMITABLE. I say, old boy, wasn't it you I saw\n        on Sunday morning in the hall, in a soldier's\n        cap? You know--in a soldier's cap?\n\n        YOUNG IRELAND (_cogitating deeply_). Was it a\n        very good cap?\n\n        INIMITABLE. Yes.\n\n        YOUNG IRELAND. Did it fit unkommon?\n\n        INIMITABLE. Yes.\n\n        YOUNG IRELAND. Dat was me!\n\nThere are two stupid old louts at the room, to show people into their\nplaces, whom John calls \"them two old Paddies,\" and of whom he says,\nthat he \"never see nothing like them (snigger) hold idiots\" (snigger).\nThey bow and walk backwards before the grandees, and our men hustle them\nwhile they are doing it.\n\nWe walked out last night, with the intention of going to the theatre;\nbut the Piccolomini establishment (they were doing the \"Lucia\") looked\nso horribly like a very bad jail, and the Queen's looked so\nblackguardly, that we came back again, and went to bed. I seem to be\nalways either in a railway carriage, or reading, or going to bed. I get\nso knocked up, whenever I have a minute to remember it, that then I go\nto bed as a matter of course.\n\nI send my love to the noble Plorn, and to all the boys. To dear Mamie\nand Katie, and to yourself of course, in the first degree. I am looking\nforward to the last Irish reading on Thursday, with great impatience.\nBut when we shall have turned this week, once knocked off Belfast, I\nshall see land, and shall (like poor Timber in the days of old) \"keep up\na good heart.\" I get so wonderfully hot every night in my dress clothes,\nthat they positively won't dry in the short interval they get, and I\nhave been obliged to write to Doudney's to make me another suit, that I\nmay have a constant change.\n\n                         Ever, my dearest Georgy, most affectionately.\n\n\n[Sidenote: Miss Dickens.]\n\n                                 BELFAST, _Saturday, Aug. 28th, 1858._\n\nWhen I went down to the Rotunda at Dublin on Thursday night, I said to\nArthur, who came rushing at me: \"You needn't tell me. I know all about\nit.\" The moment I had come out of the door of the hotel (a mile off), I\nhad come against the stream of people turned away. I had struggled\nagainst it to the room. There, the crowd in all the lobbies and passages\nwas so great, that I had a difficulty in getting in. They had broken all\nthe glass in the pay-boxes. They had offered frantic prices for stalls.\nEleven bank-notes were thrust into that pay-box (Arthur saw them) at one\ntime, for eleven stalls. Our men were flattened against walls, and\nsqueezed against beams. Ladies stood all night with their chins against\nmy platform. Other ladies sat all night upon my steps. You never saw\nsuch a sight. And the reading went tremendously! It is much to be\nregretted that we troubled ourselves to go anywhere else in Ireland. We\nturned away people enough to make immense houses for a week.\n\nWe arrived here yesterday at two. The room will not hold more than from\neighty to ninety pounds. The same scene was repeated with the additional\nfeature, that the people are much rougher here than in Dublin, and that\nthere was a very great uproar at the opening of the doors, which, the\npolice in attendance being quite inefficient and only looking on, it was\nimpossible to check. Arthur was in the deepest misery because shillings\ngot into stalls, and half-crowns got into shillings, and stalls got\nnowhere, and there was immense confusion. It ceased, however, the moment\nI showed myself; and all went most brilliantly, in spite of a great\npiece of the cornice of the ceiling falling with a great crash within\nfour or five inches of the head of a young lady on my platform (I was\nobliged to have people there), and in spite of my gas suddenly going out\nat the time of the game of forfeits at Scrooge's nephew's, through some\nBelfastian gentleman accidentally treading on the flexible pipe, and\nneeding to be relighted.\n\nWe shall not get to Cork before mid-day on Monday; it being difficult to\nget from here on a Sunday. We hope to be able to start away to-morrow\nmorning to see the Giant's Causeway (some sixteen miles off), and in\nthat case we shall sleep at Dublin to-morrow night, leaving here by the\ntrain at half-past three in the afternoon. Dublin, you must understand,\nis on the way to Cork. This is a fine place, surrounded by lofty hills.\nThe streets are very wide, and the place is very prosperous. The whole\nride from Dublin here is through a very picturesque and various country;\nand the amazing thing is, that it is all particularly neat and orderly,\nand that the houses (outside at all events) are all brightly whitewashed\nand remarkably clean. I want to climb one of the neighbouring hills\nbefore this morning's \"Dombey.\" I am now waiting for Arthur, who has\ngone to the bank to remit his last accumulation of treasure to London.\n\nOur men are rather indignant with the Irish crowds, because in the\nstruggle they don't sell books, and because, in the pressure, they can't\nforce a way into the room afterwards to sell them. They are deeply\ninterested in the success, however, and are as zealous and ardent as\npossible. I shall write to Katie next. Give her my best love, and kiss\nthe darling Plorn for me, and give my love to all the boys.\n\n                       Ever, my dearest Mamie,\n                                        Your most affectionate Father.\n\n\n[Sidenote: Miss Hogarth.]\n\n            MORRISON'S HOTEL, DUBLIN, _Sunday Night, Aug. 29th, 1858._\n\nI am so delighted to find your letter here to-night (eleven o'clock),\nand so afraid that, in the wear and tear of this strange life, I have\nwritten to Gad's Hill in the wrong order, and have not written to you,\nas I should, that I resolve to write this before going to bed. You will\nfind it a wretchedly stupid letter; but you may imagine, my dearest\ngirl, that I am tired.\n\nThe success at Belfast has been equal to the success here. Enormous! We\nturned away half the town. I think them a better audience, on the whole,\nthan Dublin; and the personal affection there was something\noverwhelming. I wish you and the dear girls could have seen the people\nlook at me in the street; or heard them ask me, as I hurried to the\nhotel after reading last night, to \"do me the honour to shake hands,\nMisther Dickens, and God bless you, sir; not ounly for the light you've\nbeen to me this night, but for the light you've been in mee house, sir\n(and God love your face), this many a year.\" Every night, by-the-bye,\nsince I have been in Ireland, the ladies have beguiled John out of the\nbouquet from my coat. And yesterday morning, as I had showered the\nleaves from my geranium in reading \"Little Dombey,\" they mounted the\nplatform, after I was gone, and picked them all up as keepsakes!\n\nI have never seen _men_ go in to cry so undisguisedly as they did at\nthat reading yesterday afternoon. They made no attempt whatever to hide\nit, and certainly cried more than the women. As to the \"Boots\" at night,\nand \"Mrs. Gamp\" too, it was just one roar with me and them; for they\nmade me laugh so that sometimes I _could not_ compose my face to go on.\n\nYou must not let the new idea of poor dear Landor efface the former\nimage of the fine old man. I wouldn't blot him out, in his tender\ngallantry, as he sat upon that bed at Forster's that night, for a\nmillion of wild mistakes at eighty years of age.\n\nI hope to be at Tavistock House before five o'clock next Saturday\nmorning, and to lie in bed half the day, and come home by the 10.50 on\nSunday.\n\nTell the girls that Arthur and I have each ordered at Belfast a trim,\nsparkling, slap-up _Irish jaunting-car_!!! I flatter myself we shall\nastonish the Kentish people. It is the oddest carriage in the world, and\nyou are always falling off. But it is gay and bright in the highest\ndegree. Wonderfully Neapolitan.\n\nWhat with a sixteen mile ride before we left Belfast, and a sea-beach\nwalk, and a two o'clock dinner, and a seven hours' railway ride since, I\nam--as we say here--\"a thrifle weary.\" But I really am in wonderful\nforce, considering the work. For which I am, as I ought to be, very\nthankful.\n\nArthur was exceedingly unwell last night--could not cheer up at all. He\nwas so very unwell that he left the hall(!) and became invisible after\nmy five minutes' rest. I found him at the hotel in a jacket and\nslippers, and with a hot bath just ready. He was in the last stage of\nprostration. The local agent was with me, and proposed that he (the\nwretched Arthur) should go to his office and balance the accounts then\nand there. He went, in the jacket and slippers, and came back in twenty\nminutes, _perfectly well_, in consequence of the admirable balance. He\nis now sitting opposite to me ON THE BAG OF SILVER, forty pounds (it\nmust be dreadfully hard), writing to Boulogne.\n\nI suppose it is clear that the next letter I write is Katie's. Either\nfrom Cork or from Limerick, it shall report further. At Limerick I read\nin the theatre, there being no other place.\n\nBest love to Mamie and Katie, and dear Plorn, and all the boys left when\nthis comes to Gad's Hill; also to my dear good Anne, and her little\nwoman.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Mr. W. Wilkie Collins.]\n\n                      GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                            _Monday, Sept. 6th, 1858._\n\nMY DEAR WILKIE,\n\nFirst, let me report myself here for something less than eight-and-forty\nhours. I come last (and direct--a pretty hard journey) from Limerick.\nThe success in Ireland has been immense.\n\nThe work is very hard, sometimes overpowering; but I am none the worse\nfor it, and arrived here quite fresh.\n\nSecondly, will you let me recommend the enclosed letter from Wigan, as\nthe groundwork of a capital article, in your way, for H. W.? There is\nnot the least objection to a plain reference to him, or to Phelps, to\nwhom the same thing happened a year or two ago, near Islington, in the\ncase of a clever and capital little daughter of his. I think it a\ncapital opportunity for a discourse on gentility, with a glance at those\nother schools which advertise that the \"sons of gentlemen only\" are\nadmitted, and a just recognition of the greater liberality of our public\nschools. There are tradesmen's sons at Eton, and Charles Kean was at\nEton, and Macready (also an actor's son) was at Rugby. Some such title\nas \"Scholastic Flunkeydom,\" or anything infinitely contemptuous, would\nhelp out the meaning. Surely such a schoolmaster must swallow all the\nsilver forks that the pupils are expected to take when they come, and\nare not expected to take away with them when they go. And of course he\ncould not exist, unless he had flunkey customers by the dozen.\n\nSecondly--no, this is thirdly now--about the Christmas number. I have\narranged so to stop my readings, as to be available for it on _the 15th\nof November_, which will leave me time to write a good article, if I\nclear my way to one. Do you see your way to our making a Christmas\nnumber of this idea that I am going very briefly to hint? Some\ndisappointed person, man or woman, prematurely disgusted with the world,\nfor some reason or no reason (the person should be young, I think)\nretires to an old lonely house, or an old lonely mill, or anything you\nlike, with one attendant, resolved to shut out the world, and hold no\ncommunion with it. The one attendant sees the absurdity of the idea,\npretends to humour it, but really thus to slaughter it. Everything that\nhappens, everybody that comes near, every breath of human interest that\nfloats into the old place from the village, or the heath, or the four\ncross-roads near which it stands, and from which belated travellers\nstray into it, shows beyond mistake that you can't shut out the world;\nthat you are in it, to be of it; that you get into a false position the\nmoment you try to sever yourself from it; and that you must mingle with\nit, and make the best of it, and make the best of yourself into the\nbargain.\n\nIf we could plot out a way of doing this together, I would not be afraid\nto take my part. If we could not, could we plot out a way of doing it,\nand taking in stories by other hands? If we could not do either (but I\nthink we could), shall we fall back upon a round of stories again? That\nI would rather not do, if possible. Will you think about it?\n\nAnd can you come and dine at Tavistock House _on Monday, the 20th\nSeptember, at half-past five_? I purpose being at home there with the\ngirls that day.\n\nAnswer this, according to my printed list for the week. I am off to\nHuddersfield on Wednesday morning.\n\nI think I will now leave off; merely adding that I have got a splendid\nbrogue (it really is exactly like the people), and that I think of\ncoming out as the only legitimate successor of poor Power.\n\n                           Ever, my dear Wilkie, affectionately yours.\n\n\n[Sidenote: Miss Mary Boyle.]\n\n                      STATION HOTEL, YORK, _Friday, Sept. 10th, 1858._\n\nDEAREST MEERY,\n\nFirst let me tell you that all the magicians and spirits in your employ\nhave fulfilled the instructions of their wondrous mistress to\nadmiration. Flowers have fallen in my path wherever I have trod; and\nwhen they rained upon me at Cork I was more amazed than you ever saw me.\n\nSecondly, receive my hearty and loving thanks for that same. (Excuse a\nlittle Irish in the turn of that sentence, but I can't help it).\n\nThirdly, I have written direct to Mr. Boddington, explaining that I am\nbound to be in Edinburgh on the day when he courteously proposes to do\nme honour.\n\nI really cannot tell you how truly and tenderly I feel your letter, and\nhow gratified I am by its contents. Your truth and attachment are\nalways so precious to me that I can_not_ get my heart out on my sleeve\nto show it you. It is like a child, and, at the sound of some familiar\nvoices, \"goes and hides.\"\n\nYou know what an affection I have for Mrs. Watson, and how happy it made\nme to see her again--younger, much, than when I first knew her in\nSwitzerland.\n\nGod bless you always!\n\n                                            Ever affectionately yours.\n\n\n[Sidenote: Miss Hogarth.]\n\n                 ROYAL HOTEL, SCARBOROUGH, _Sunday, Sept. 11th, 1858._\n\nMY DEAREST GEORGY,\n\nWe had a very fine house indeed at York. All kinds of applications have\nbeen made for another reading there, and no doubt it would be\nexceedingly productive; but it cannot be done. At Harrogate yesterday;\nthe queerest place, with the strangest people in it, leading the oddest\nlives of dancing, newspaper reading, and tables d'hote. The piety of\nYork obliging us to leave that place for this at six this morning, and\nthere being no night train from Harrogate, we had to engage a special\nengine. We got to bed at one, and were up again before five; which,\nafter yesterday's fatigues, leaves me a little worn out at this present.\n\nI have no accounts of this place as yet, nor have I received any letter\nhere. But the post of this morning is not yet delivered, I believe. We\nhave a charming room, overlooking the sea. Leech is here (living within\na few doors), with the partner of his bosom, and his young family. I\nwrite at ten in the morning, having been here two hours; and you will\nreadily suppose that I have not seen him.\n\nOf news, I have not the faintest breath. I seem to have been doing\nnothing all my life but riding in railway-carriages and reading. The\nrailway of the morning brought us through Castle Howard, and under the\nwoods of Easthorpe, and then just below Malton Abbey, where I went to\npoor Smithson's funeral. It was a most lovely morning, and, tired as I\nwas, I couldn't sleep for looking out of window.\n\nYesterday, at Harrogate, two circumstances occurred which gave Arthur\ngreat delight. Firstly, he chafed his legs sore with his black bag of\nsilver. Secondly, the landlord asked him as a favour, \"If he could\noblige him with a little silver.\" He obliged him directly with some\nforty pounds' worth; and I suspect the landlord to have repented of\nhaving approached the subject. After the reading last night we walked\nover the moor to the railway, three miles, leaving our men to follow\nwith the luggage in a light cart. They passed us just short of the\nrailway, and John was making the night hideous and terrifying the\nsleeping country, by _playing the horn_ in prodigiously horrible and\nunmusical blasts.\n\nMy dearest love, of course, to the dear girls, and to the noble Plorn.\nApropos of children, there was one gentleman at the \"Little Dombey\"\nyesterday morning, who exhibited, or rather concealed, the profoundest\ngrief. After crying a good deal without hiding it, he covered his face\nwith both his hands, and laid it down on the back of the seat before\nhim, and really shook with emotion. He was not in mourning, but I\nsupposed him to have lost some child in old time. There was a remarkably\ngood fellow of thirty or so, too, who found something so very ludicrous\nin \"Toots,\" that he _could not_ compose himself at all, but laughed\nuntil he sat wiping his eyes with his handkerchief. And whenever he felt\n\"Toots\" coming again he began to laugh and wipe his eyes afresh, and\nwhen he came he gave a kind of cry, as if it were too much for him. It\nwas uncommonly droll, and made me laugh heartily.\n\n                            Ever, dear Georgy, your most affectionate.\n\n\n[Sidenote: Miss Dickens.]\n\n               SCARBOROUGH ARMS, LEEDS, _Wednesday, Sept. 15th, 1858._\n\nMY DEAREST MAMIE,\n\nI have added a pound to the cheque. I would recommend your seeing the\npoor railway man again and giving him ten shillings, and telling him to\nlet you see him again in about a week. If he be then still unable to\nlift weights and handle heavy things, I would then give him another ten\nshillings, and so on.\n\nSince I wrote to Georgy from Scarborough, we have had, thank God,\nnothing but success. The Hull people (not generally considered\nexcitable, even on their own showing) were so enthusiastic, that we were\nobliged to promise to go back there for two readings. I have positively\nresolved not to lengthen out the time of my tour, so we are now\narranging to drop some small places, and substitute Hull again and York\nagain. But you will perhaps have heard this in the main from Arthur. I\nknow he wrote to you after the reading last night. This place I have\nalways doubted, knowing that we should come here when it was recovering\nfrom the double excitement of the festival and the Queen. But there is a\nvery large hall let indeed, and the prospect of to-night consequently\nlooks bright.\n\nArthur told you, I suppose, that he had his shirt-front and waistcoat\ntorn off last night? He was perfectly enraptured in consequence. Our men\ngot so knocked about that he gave them five shillings apiece on the\nspot. John passed several minutes upside down against a wall, with his\nhead amongst the people's boots. He came out of the difficulty in an\nexceedingly touzled condition, and with his face much flushed. For all\nthis, and their being packed as you may conceive they would be packed,\nthey settled down the instant I went in, and never wavered in the\nclosest attention for an instant. It was a very high room, and required\na great effort.\n\nOddly enough, I slept in this house three days last year with Wilkie.\nArthur has the bedroom I occupied then, and I have one two doors from\nit, and Gordon has the one between. Not only is he still with us, but he\n_has_ talked of going on to Manchester, going on to London, and coming\nback with us to Darlington next Tuesday!!!\n\nThese streets look like a great circus with the season just finished.\nAll sorts of garish triumphal arches were put up for the Queen, and they\nhave got smoky, and have been looked out of countenance by the sun, and\nare blistered and patchy, and half up and half down, and are hideous to\nbehold. Spiritless men (evidently drunk for some time in the royal\nhonour) are slowly removing them, and on the whole it is more like the\nclearing away of \"The Frozen Deep\" at Tavistock House than anything\nwithin your knowledge--with the exception that we are not in the least\nsorry, as we were then. Vague ideas are in Arthur's head that when we\ncome back to Hull, we are to come here, and are to have the Town Hall (a\nbeautiful building), and read to the million. I can't say yet. That\ndepends. I remember that when I was here before (I came from Rockingham\nto make a speech), I thought them a dull and slow audience. I hope I may\nhave been mistaken. I never saw better audiences than the Yorkshire\naudiences generally.\n\nI am so perpetually at work or asleep, that I have not a scrap of news.\nI saw the Leech family at Scarboro', both in my own house (that is to\nsay, hotel) and in theirs. They were not at either reading. Scarboro' is\ngay and pretty, and I think Gordon had an idea that we were always at\nsome such place.\n\nKiss the darling Plorn for me, and give him my love; dear Katie too,\ngiving her the same. I feel sorry that I cannot get down to Gad's Hill\nthis next time, but I shall look forward to our being there with Georgy,\nafter Scotland. Tell the servants that I remember them, and hope they\nwill live with us many years.\n\n                              Ever, my dearest Mamie,\n                                        Your most affectionate Father.\n\n\n[Sidenote: Miss Hogarth.]\n\n                   KING'S HEAD, SHEFFIELD, _Friday, Sept. 17th, 1858._\n\nI write you a few lines to Tavistock House, thinking you may not be\nsorry to find a note from me there on your arrival from Gad's Hill.\n\nHalifax was too small for us. I never saw such an audience though. They\nwere really worth reading to for nothing, though I didn't do exactly\nthat. It is as horrible a place as I ever saw, I think.\n\nThe run upon the tickets here is so immense that Arthur is obliged to\nget great bills out, signifying that no more can be sold. It will be by\nno means easy to get into the place the numbers who have already paid.\nIt is the hall we acted in. Crammed to the roof and the passages. We\nmust come back here towards the end of October, and are again altering\nthe list and striking out small places.\n\nThe trains are so strange and unintelligible in this part of the country\nthat we were obliged to leave Halifax at eight this morning, and\nbreakfast on the road--at Huddersfield again, where we had an hour's\nwait. Wills was in attendance on the platform, and took me (here at\nSheffield, I mean) out to Frederick Lehmann's house to see Mrs. Wills.\nShe looked pretty much the same as ever, I thought, and was taking care\nof a very pretty little boy. The house and grounds are as nice as\nanything _can_ be in this smoke. A heavy thunderstorm is passing over\nthe town, and it is raining hard too.\n\nThis is a stupid letter, my dearest Georgy, but I write in a hurry, and\nin the thunder and lightning, and with the crowd of to-night before me.\n\n                                             Ever most affectionately.\n\n\n[Sidenote: Miss Hogarth.]\n\n                               STATION HOTEL, NEWCASTLE-ON-TYNE,\n                                           _Sunday, Sept. 26th, 1858._\n\n                              EXTRACT.\n\nThe girls (as I have no doubt they have already told you for themselves)\narrived here in good time yesterday, and in very fresh condition. They\npersisted in going to the room last night, though I had arranged for\ntheir remaining quiet.\n\nWe have done a vast deal here. I suppose you know that we are going to\nBerwick, and that we mean to sleep there and go on to Edinburgh on\nMonday morning, arriving there before noon? If it be as fine to-morrow\nas it is to-day, the girls will see the coast piece of railway between\nBerwick and Edinburgh to great advantage. I was anxious that they\nshould, because that kind of pleasure is really almost the only one they\nare likely to have in their present trip.\n\nStanfield and Roberts are in Edinburgh, and the Scottish Royal Academy\ngave them a dinner on Wednesday, to which I was very pressingly\ninvited. But, of course, my going was impossible. I read twice that day.\n\nRemembering what you do of Sunderland, you will be surprised that our\nprofit there was very considerable. I read in a beautiful new theatre,\nand (I thought to myself) quite wonderfully. Such an audience I never\nbeheld for rapidity and enthusiasm. The room in which we acted\n(converted into a theatre afterwards) was burnt to the ground a year or\ntwo ago. We found the hotel, so bad in our time, really good. I walked\nfrom Durham to Sunderland, and from Sunderland to Newcastle.\n\nDon't you think, as we shall be at home at eleven in the forenoon this\nday fortnight, that it will be best for you and Plornish to come to\nTavistock House for that Sunday, and for us all to go down to Gad's Hill\nnext day? My best love to the noble Plornish. If he is quite reconciled\nto the postponement of his trousers, I should like to behold his first\nappearance in them. But, if not, as he is such a good fellow, I think it\nwould be a pity to disappoint and try him.\n\nAnd now, my dearest Georgy, I think I have said all I have to say before\nI go out for a little air. I had a very hard day yesterday, and am\ntired.\n\n                                          Ever your most affectionate.\n\n\n[Sidenote: Mr. John Forster.]\n\n                       TAVISTOCK HOUSE, TAVISTOCK SQUARE, LONDON,\n                                            _Sunday, Oct. 10th, 1858._\n\nMY DEAR FORSTER,\n\nAs to the truth of the readings, I cannot tell you what the\ndemonstrations of personal regard and respect are. How the densest and\nmost uncomfortably-packed crowd will be hushed in an instant when I show\nmy face. How the youth of colleges, and the old men of business in the\ntown, seem equally unable to get near enough to me when they cheer me\naway at night. How common people and gentlefolks will stop me in the\nstreets and say: \"Mr. Dickens, will you let me touch the hand that has\nfilled my home with so many friends?\" And if you saw the mothers, and\nfathers, and sisters, and brothers in mourning, who invariably come to\n\"Little Dombey,\" and if you studied the wonderful expression of comfort\nand reliance with which they hang about me, as if I had been with them,\nall kindness and delicacy, at their own little death-bed, you would\nthink it one of the strangest things in the world.\n\nAs to the mere effect, of course I don't go on doing the thing so often\nwithout carefully observing myself and the people too in every little\nthing, and without (in consequence) greatly improving in it.\n\nAt Aberdeen, we were crammed to the street twice in one day. At Perth\n(where I thought when I arrived there literally could be nobody to\ncome), the nobility came posting in from thirty miles round, and the\nwhole town came and filled an immense hall. As to the effect, if you had\nseen them after Lilian died, in \"The Chimes,\" or when Scrooge woke and\ntalked to the boy outside the window, I doubt if you would ever have\nforgotten it. And at the end of \"Dombey\" yesterday afternoon, in the\ncold light of day, they all got up, after a short pause, gentle and\nsimple, and thundered and waved their hats with that astonishing\nheartiness and fondness for me, that for the first time in all my public\ncareer they took me completely off my legs, and I saw the whole eighteen\nhundred of them reel on one side as if a shock from without had shaken\nthe hall.\n\nThe dear girls have enjoyed themselves immensely, and their trip has\nbeen a great success. I hope I told you (but I forget whether I did or\nno) how splendidly Newcastle[4] came out. I am reminded of Newcastle at\nthe moment because they joined me there.\n\nI am anxious to get to the end of my readings, and to be at home again,\nand able to sit down and think in my own study. But the fatigue, though\nsometimes very great indeed, hardly tells upon me at all. And although\nall our people, from Smith downwards, have given in, more or less, at\ntimes, I have never been in the least unequal to the work, though\nsometimes sufficiently disinclined for it. My kindest and best love to\nMrs. Forster.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Miss Dickens.]\n\n                        ROYAL HOTEL, DERBY, _Friday, Oct. 22nd, 1858._\n\nMY DEAREST MAMIE,\n\nI am writing in a very poor condition; I have a bad cold all over me,\npains in my back and limbs, and a very sensitive and uncomfortable\nthroat. There was a great draught up some stone steps near me last\nnight, and I daresay that caused it.\n\nThe weather on my first two nights at Birmingham was so intolerably\nbad--it blew hard, and never left off raining for one single\nmoment--that the houses were not what they otherwise would have been. On\nthe last night the weather cleared, and we had a grand house.\n\nLast night at Nottingham was almost, if not quite, the most amazing we\nhave had. It is not a very large place, and the room is by no means a\nvery large one, but three hundred and twenty stalls were let, and all\nthe other tickets were sold.\n\nHere we have two hundred and twenty stalls let for to-night, and the\nother tickets are gone in proportion. It is a pretty room, but not\nlarge.\n\nI have just been saying to Arthur that if there is not a large let for\nYork, I would rather give it up, and get Monday at Gad's Hill. We have\ntelegraphed to know. If the answer comes (as I suppose it will) before\npost time, I will tell you in a postscript what we decide to do. Coming\nto London in the night of to-morrow (Saturday), and having to see Mr.\nOuvry on Sunday, and having to start for York early on Monday, I fear I\nshould not be able to get to Gad's Hill at all. You won't expect me till\nyou see me.\n\nArthur and I have considered Plornish's joke in all the immense number\nof aspects in which it presents itself to reflective minds. We have come\nto the conclusion that it is the best joke ever made. Give the dear boy\nmy love, and the same to Georgy, and the same to Katey, and take the\nsame yourself. Arthur (excessively low and inarticulate) mutters that he\n\"unites.\"\n\n[We knocked up Boylett, Berry, and John so frightfully yesterday, by\ntearing the room to pieces and altogether reversing it, as late as four\no'clock, that we gave them a supper last night. They shine all over\nto-day, as if it had been entirely composed of grease.]\n\n                          Ever, my dearest Mamie,\n                                        Your most affectionate Father.\n\n\n[Sidenote: Miss Hogarth.]\n\n                           WOLVERHAMPTON, _Wednesday, Nov. 3rd, 1858._\n\nLittle Leamington came out in the most amazing manner yesterday--turned\naway hundreds upon hundreds of people. They are represented as the\ndullest and worst of audiences. I found them very good indeed, even in\nthe morning.\n\nThere awaited me at the hotel, a letter from the Rev. Mr. Young,\nWentworth Watson's tutor, saying that Mrs. Watson wished her boy to\nshake hands with me, and that he would bring him in the evening. I\nexpected him at the hotel before the readings. But he did not come. He\nspoke to John about it in the room at night. The crowd and confusion,\nhowever, were very great, and I saw nothing of him. In his letter he\nsaid that Mrs. Watson was at Paris on her way home, and would be at\nBrighton at the end of this week. I suppose I shall see her there at the\nend of next week.\n\nWe find a let of two hundred stalls here, which is very large for this\nplace. The evening being fine too, and blue being to be seen in the sky\nbeyond the smoke, we expect to have a very full hall. Tell Mamey and\nKatey that if they had been with us on the railway to-day between\nLeamington and this place, they would have seen (though it is only an\nhour and ten minutes by the express) fires and smoke indeed. We came\nthrough a part of the Black Country that you know, and it looked at its\nblackest. All the furnaces seemed in full blast, and all the coal-pits\nto be working.\n\nIt is market-day here, and the ironmasters are standing out in the\nstreet (where they always hold high change), making such an iron hum and\nbuzz, that they confuse me horribly. In addition, there is a bellman\nannouncing something--not the readings, I beg to say--and there is an\nexcavation being made in the centre of the open place, for a statue, or\na pump, or a lamp-post, or something or other, round which all the\nWolverhampton boys are yelling and struggling.\n\nAnd here is Arthur, begging to have dinner at half-past three instead of\nfour, because he foresees \"a wiry evening\" in store for him. Under which\ncomplication of distractions, to which a waitress with a tray at this\nmoment adds herself, I sink, and leave off.\n\nMy best love to the dear girls, and to the noble Plorn, and to you.\nMarguerite and Ellen Stone not forgotten. All yesterday and to-day I\nhave been doing everything to the tune of:\n\n        And the day is dark and dreary.\n\n                     Ever, dearest Georgy,\n                                  Your most affectionate and faithful.\n\nP.S.--I hope the brazier is intolerably hot, and half stifles all the\nfamily. Then, and not otherwise, I shall think it in satisfactory work.\n\n\n[Sidenote: Rev. James White.]\n\n                TAVISTOCK HOUSE, TAVISTOCK SQUARE, LONDON, W. C.,\n                                             _Friday, Nov. 5th, 1858._\n\nMY DEAR WHITE,\n\nMay I entreat you to thank Mr. Carter very earnestly and kindly in my\nname, for his proffered hospitality; and, further, to explain to him\nthat since my readings began, I have known them to be incompatible with\nall social enjoyments, and have neither set foot in a friend's house nor\nsat down to a friend's table in any one of all the many places I have\nbeen to, but have rigidly kept myself to my hotels. To this resolution I\nmust hold until the last. There is not the least virtue in it. It is a\nmatter of stern necessity, and I submit with the worst grace possible.\n\nWill you let me know, either at Southampton or Portsmouth, whether any\nof you, and how many of you, if any, are coming over, so that Arthur\nSmith may reserve good seats? Tell Lotty I hope she does not contemplate\ncoming to the morning reading; I always hate it so myself.\n\nMary and Katey are down at Gad's Hill with Georgy and Plornish, and they\nhave Marguerite Power and Ellen Stone staying there. I am sorry to say\nthat even my benevolence descries no prospect of their being able to\ncome to my native place.\n\nOn Saturday week, the 13th, my tour, please God, ends.\n\nMy best love to Mrs. White, and to Lotty, and to Clara.\n\n                            Ever, my dear White, affectionately yours.\n\n\n[Sidenote: Mr. Frank Stone, A.R.A.]\n\n                 TAVISTOCK HOUSE, TAVISTOCK SQUARE, LONDON, W.C.,\n                                            _Monday, Dec. 13th, 1858._\n\nMY DEAR STONE,\n\nMany thanks for these discourses. They are very good, I think, as\nexpressing what many men have felt and thought; otherwise not specially\nremarkable. They have one fatal mistake, which is a canker at the foot\nof their ever being widely useful. Half the misery and hypocrisy of the\nChristian world arises (as I take it) from a stubborn determination to\nrefuse the New Testament as a sufficient guide in itself, and to force\nthe Old Testament into alliance with it--whereof comes all manner of\ncamel-swallowing and of gnat-straining. But so to resent this miserable\nerror, or to (by any implication) depreciate the divine goodness and\nbeauty of the New Testament, is to commit even a worse error. And to\nclass Jesus Christ with Mahomet is simply audacity and folly. I might as\nwell hoist myself on to a high platform, to inform my disciples that the\nlives of King George the Fourth and of King Alfred the Great belonged to\none and the same category.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Mr. B. W. Procter.]\n\n                           TAVISTOCK HOUSE, _Sunday, Dec. 18th, 1858._\n\nMY DEAR PROCTER,\n\nA thousand thanks for the little song. I am charmed with it, and shall\nbe delighted to brighten \"Household Words\" with such a wise and genial\nlight. I no more believe that your poetical faculty has gone by, than I\nbelieve that you have yourself passed to the better land. You and it\nwill travel thither in company, rely upon it. So I still hope to hear\nmore of the trade-songs, and to learn that the blacksmith has hammered\nout no end of iron into good fashion of verse, like a cunning workman,\nas I know him of old to be.\n\n                               Very faithfully yours, my dear Procter.\n\nFOOTNOTES:\n\n[3] Niece to the Rev. W. Harness.\n\n[4] The birthplace of Mr. Forster.\n\n\n\n\n1859.\n\nNARRATIVE.\n\n\nDuring the winter, Charles Dickens was living at Tavistock House,\nremoving to Gad's Hill for the summer early in June, and returning to\nLondon in November. At this time a change was made in his weekly\njournal. \"Household Words\" became absolutely his own--Mr. Wills being\nhis partner and editor, as before--and was \"incorporated with 'All the\nYear Round,'\" under which title it was known thenceforth. The office was\nstill in Wellington Street, but in a different house. The first number\nwith the new name appeared on the 30th April, and it contained the\nopening of \"A Tale of Two Cities.\"\n\nThe first letter which follows shows that a proposal for a series of\nreadings in America had already been made to him. It was carefully\nconsidered and abandoned for the time. But the proposal was constantly\nrenewed, and the idea never wholly relinquished for many years before he\nactually decided on making so distant a \"reading tour.\"\n\nMr. Procter contributed to the early numbers of \"All the Year Round\"\nsome very spirited \"Songs of the Trades.\" We give notes from Charles\nDickens to the veteran poet, both in the last year, and in this year,\nexpressing his strong approval of them.\n\nThe letter and two notes to Mr. (afterwards Sir Antonio) Panizzi, for\nwhich we are indebted to Mr. Louis Fagan, one of Sir A. Panizzi's\nexecutors, show the warm sympathy and interest which he always felt for\nthe cause of Italian liberty, and for the sufferings of the State\nprisoners who at this time took refuge in England.\n\nWe give a little note to the dear friend and companion of Charles\nDickens's daughters, \"Lotty\" White, because it is a pretty specimen of\nhis writing, and because the young girl, who is playfully \"commanded\" to\nget well and strong, died early in July of this year. She was, at the\ntime this note was written, first attacked with the illness which was\nfatal to all her sisters. Mamie and Kate Dickens went from Gad's Hill to\nBonchurch to pay a last visit to their friend, and he writes to his\neldest daughter there. Also we give notes of loving sympathy and\ncondolence to the bereaved father and mother.\n\nIn the course of this summer Charles Dickens was not well, and went for\na week to his old favourite, Broadstairs--where Mr. Wilkie Collins and\nhis brother, Mr. Charles Allston Collins, were staying--for sea-air and\nchange, preparatory to another reading tour, in England only. His letter\nfrom Peterborough to Mr. Frank Stone, giving him an account of a reading\nat Manchester (Mr. Stone's native town), was one of the last ever\naddressed to that affectionate friend, who died very suddenly, to the\ngreat grief of Charles Dickens, in November. The letter to Mr. Thomas\nLongman, which closes this year, was one of introduction to that\ngentleman of young Marcus Stone, then just beginning his career as an\nartist, and to whom the premature death of his father made it doubly\ndesirable that he should have powerful helping hands.\n\nCharles Dickens refers, in a letter to Mrs. Watson, to his portrait by\nMr. Frith, which was finished at the end of 1858. It was painted for Mr.\nForster, and is now in the \"Forster Collection\" at South Kensington\nMuseum.\n\nThe Christmas number of this year, again written by several hands as\nwell as his own, was \"The Haunted House.\" In November, his story of \"A\nTale of Two Cities\" was finished in \"All the Year Round,\" and in\nDecember was published, complete, with dedication to Lord John Russell.\n\n\n[Sidenote: Mr. Arthur Smith.]\n\n             TAVISTOCK HOUSE, TAVISTOCK SQUARE, LONDON, W.C.,\n                                         _Wednesday, Jan. 26th, 1859._\n\nMY DEAR ARTHUR,\n\nWill you first read the enclosed letters, having previously welcomed,\nwith all possible cordiality, the bearer, Mr. Thomas C. Evans, from New\nYork?\n\nYou having read them, let me explain that Mr. Fields is a highly\nrespectable and influential man, one of the heads of the most classical\nand most respected publishing house in America; that Mr. Richard Grant\nWhite is a man of high reputation; and that Felton is the Greek\nProfessor in their Cambridge University, perhaps the most distinguished\nscholar in the States.\n\nThe address to myself, referred to in one of the letters, being on its\nway, it is quite clear that I must give some decided and definite answer\nto the American proposal. Now, will you carefully discuss it with Mr.\nEvans before I enter on it at all? Then, will you dine here with him on\nSunday--which I will propose to him--and arrange to meet at half-past\nfour for an hour's discussion?\n\nThe points are these:\n\nFirst. I have a very grave question within myself whether I could go to\nAmerica at all.\n\nSecondly. If I did go, I could not possibly go before the autumn.\n\nThirdly. If I did go, how long must I stay?\n\nFourthly. If the stay were a short one, could _you_ go?\n\nFifthly. What is his project? What could I make? What occurs to you upon\nhis proposal?\n\nI have told him that the business arrangements of the readings have been\nfrom the first so entirely in your hands, that I enter upon nothing\nconnected with them without previous reference to you.\n\n                                                      Ever faithfully.\n\n\n[Sidenote: M. de Cerjat.]\n\n                           TAVISTOCK HOUSE, _Tuesday, Feb. 1st, 1859._\n\nMY DEAR CERJAT,\n\nI received your always welcome annual with even more interest than usual\nthis year, being (in common with my two girls and their aunt) much\nexcited and pleased by your account of your daughter's engagement. Apart\nfrom the high sense I have of the affectionate confidence with which you\ntell me what lies so tenderly on your own heart, I have followed the\nlittle history with a lively sympathy and regard for her. I hope, with\nyou, that it is full of promise, and that you will all be happy in it.\nThe separation, even in the present condition of travel (and no man can\nsay how much the discovery of a day may advance it), is nothing. And so\nGod bless her and all of you, and may the rosy summer bring her all the\nfulness of joy that we all wish her.\n\nTo pass from the altar to Townshend (which is a long way), let me report\nhim severely treated by Bully, who rules him with a paw of iron; and\ncomplaining, moreover, of indigestion. He drives here every Sunday, but\nat all other times is mostly shut up in his beautiful house, where I\noccasionally go and dine with him _tete-a-tete_, and where we always\ntalk of you and drink to you. That is a rule with us from which we never\ndepart. He is \"seeing a volume of poems through the press;\" rather an\nexpensive amusement. He has not been out at night (except to this house)\nsave last Friday, when he went to hear me read \"The Poor Traveller,\"\n\"Mrs. Gamp,\" and \"The Trial\" from \"Pickwick.\" He came into my room at\nSt. Martin's Hall, and I fortified him with weak brandy-and-water. You\nwill be glad to hear that the said readings are a greater _furore_ than\nthey ever have been, and that every night on which they now take\nplace--once a week--hundreds go away, unable to get in, though the hall\nholds thirteen hundred people. I dine with ---- to-day, by-the-bye,\nalong with his agent; concerning whom I observe him to be always divided\nbetween an unbounded confidence and a little latent suspicion. He always\ntells me that he is a gem of the first water; oh yes, the best of\nbusiness men! and then says that he did not quite like his conduct\nrespecting that farm-tenant and those hay-ricks.\n\nThere is a general impression here, among the best-informed, that war in\nItaly, to begin with, is inevitable, and will break out before April. I\nknow a gentleman at Genoa (Swiss by birth), deeply in with the\nauthorities at Turin, who is already sending children home.\n\nIn England we are quiet enough. There is a world of talk, as you know,\nabout Reform bills; but I don't believe there is any general strong\nfeeling on the subject. According to my perceptions, it is undeniable\nthat the public has fallen into a state of indifference about public\naffairs, mainly referable, as I think, to the people who administer\nthem--and there I mean the people of all parties--which is a very bad\nsign of the times. The general mind seems weary of debates and\nhonourable members, and to have taken _laissez-aller_ for its motto.\n\nMy affairs domestic (which I know are not without their interest for\nyou) flow peacefully. My eldest daughter is a capital housekeeper, heads\nthe table gracefully, delegates certain appropriate duties to her sister\nand her aunt, and they are all three devotedly attached. Charley, my\neldest boy, remains in Barings' house. Your present correspondent is\nmore popular than he ever has been. I rather think that the readings in\nthe country have opened up a new public who were outside before; but\nhowever that may be, his books have a wider range than they ever had,\nand his public welcomes are prodigious. Said correspondent is at present\noverwhelmed with proposals to go and read in America. Will never go,\nunless a small fortune be first paid down in money on this side of the\nAtlantic. Stated the figure of such payment, between ourselves, only\nyesterday. Expects to hear no more of it, and assuredly will never go\nfor less. You don't say, my dear Cerjat, when you are coming to England!\nSomehow I feel that this marriage ought to bring you over, though I\ndon't know why. You shall have a bed here and a bed at Gad's Hill, and\nwe will go and see strange sights together. When I was in Ireland, I\nordered the brightest jaunting-car that ever was seen. It has just this\nminute arrived per steamer from Belfast. Say you are coming, and you\nshall be the first man turned over by it; somebody must be (for my\ndaughter Mary drives anything that can be harnessed, and I know of no\nEnglish horse that would understand a jaunting-car coming down a Kentish\nhill), and you shall be that somebody if you will. They turned the\nbasket-phaeton over, last summer, in a bye-road--Mary and the other\ntwo--and had to get it up again; which they did, and came home as if\nnothing had happened. They send their loves to Mrs. Cerjat, and to you,\nand to all, and particularly to the dear _fiancee_. So do I, with all my\nheart, and am ever your attached and affectionate friend.\n\n\n[Sidenote: Mr. Antonio Panizzi.]\n\n                    TAVISTOCK HOUSE, _Monday Night, March 14th, 1859._\n\nMY DEAR PANIZZI,\n\nIf you should feel no delicacy in mentioning, or should see no objection\nto mentioning, to Signor Poerio, or any of the wronged Neapolitan\ngentlemen to whom it is your happiness and honour to be a friend on\ntheir arrival in this country, an idea that has occurred to me, I should\nregard it as a great kindness in you if you would be my exponent. I\nthink you will have no difficulty in believing that I would not, on any\nconsideration, obtrude my name or projects upon any one of those noble\nsouls, if there were any reason of the slightest kind against it. And if\nyou see any such reason, I pray you instantly to banish my letter from\nyour thoughts.\n\nIt seems to me probable that some narrative of their ten years'\nsuffering will, somehow or other, sooner or later, be by some of them\nlaid before the English people. The just interest and indignation alive\nhere, will (I suppose) elicit it. False narratives and garbled stories\nwill, in any case, of a certainty get about. If the true history of the\nmatter is to be told, I have that sympathy with them and respect for\nthem which would, all other considerations apart, render it unspeakably\ngratifying to me to be the means of its diffusion. What I desire to lay\nbefore them is simply this. If for my new successor to \"Household Words\"\na narrative of their ten years' trial could be written, I would take any\nconceivable pains to have it rendered into English, and presented in the\nsincerest and best way to a very large and comprehensive audience. It\nshould be published exactly as you might think best for them, and\nremunerated in any way that you might think generous and right. They\nwant no mouthpiece and no introducer, but perhaps they might have no\nobjection to be associated with an English writer, who is possibly not\nunknown to them by some general reputation, and who certainly would be\nanimated by a strong public and private respect for their honour,\nspirit, and unmerited misfortunes. This is the whole matter; assuming\nthat such a thing is to be done, I long for the privilege of helping to\ndo it. These gentlemen might consider it an independent means of making\nmoney, and I should be delighted to pay the money.\n\nIn my absence from town, my friend and sub-editor, Mr. Wills (to whom I\nhad expressed my feeling on the subject), has seen, I think, three of\nthe gentlemen together. But as I hear, returning home to-night, that\nthey are in your good hands, and as nobody can be a better judge than\nyou of anything that concerns them, I at once decide to write to you and\nto take no other step whatever. Forgive me for the trouble I have\noccasioned you in the reading of this letter, and never think of it\nagain if you think that by pursuing it you would cause them an instant's\nuneasiness.\n\n                                    Believe me, very faithfully yours.\n\n\n[Sidenote: Mr. Antonio Panizzi.]\n\n                         TAVISTOCK HOUSE, _Tuesday, March 15th, 1859._\n\nMY DEAR PANIZZI,\n\nLet me thank you heartily for your kind and prompt letter. I am really\nand truly sensible of your friendliness.\n\nI have not heard from Higgins, but of course I am ready to serve on the\nCommittee.\n\n                                              Always faithfully yours.\n\n\n[Sidenote: Mr. B. W. Procter.]\n\n                        TAVISTOCK HOUSE, _Saturday, March 19th, 1859._\n\nMY DEAR PROCTER,\n\nI think the songs are simply ADMIRABLE! and I have no doubt of this\nbeing a popular feature in \"All the Year Round.\" I would not omit the\nsexton, and I would not omit the spinners and weavers; and I would omit\nthe hack-writers, and (I think) the alderman; but I am not so clear\nabout the chorister. The pastoral I a little doubt finding audience for;\nbut I am not at all sure yet that my doubt is well founded.\n\nHad I not better send them all to the printer, and let you have proofs\nkept by you for publishing? I shall not have to make up the first number\nof \"All the Year Round\" until early in April. I don't like to send the\nmanuscript back, and I never do like to do so when I get anything that I\nknow to be thoroughly, soundly, and unquestionably good. I am hard at\nwork upon my story, and expect a magnificent start. With hearty thanks,\n\n                                            Ever yours affectionately.\n\n\n[Sidenote: Mr. Edmund Yates.]\n\n                 TAVISTOCK HOUSE, TAVISTOCK SQUARE, LONDON, W.C.,\n                                          _Tuesday, March 29th, 1859._\n\nMY DEAR EDMUND,\n\n1. I think that no one seeing the place can well doubt that my house at\nGad's Hill is the place for the letter-box. The wall is accessible by\nall sorts and conditions of men, on the bold high road, and the house\naltogether is the great landmark of the whole neighbourhood. Captain\nGoldsmith's _house_ is up a lane considerably off the high road; but he\nhas a garden _wall_ abutting on the road itself.\n\n2. \"The Pic-Nic Papers\" were originally sold to Colburn, for the benefit\nof the widow of Mr. Macrone, of St. James's Square, publisher, deceased.\nTwo volumes were contributed--of course gratuitously--by writers who had\nhad transactions with Macrone. Mr. Colburn, wanting three volumes in all\nfor trade purposes, added a third, consisting of an American reprint.\nOf that volume I didn't know, and don't know, anything. The other two I\nedited, gratuitously as aforesaid, and wrote the Lamplighter's story in.\nIt was all done many years ago. There was a preface originally,\ndelicately setting forth how the book came to be.\n\n3. I suppose ---- to be, as Mr. Samuel Weller expresses it somewhere in\n\"Pickwick,\" \"ravin' mad with the consciousness o' willany.\" Under their\nadvertisement in _The Times_ to-day, you will see, without a word of\ncomment, the shorthand writer's verbatim report of the judgment.\n\n                                                      Ever faithfully.\n\n\n[Sidenote: Mr. Antonio Panizzi.]\n\n             \"ALL THE YEAR ROUND\" OFFICE, _Thursday, April 7th, 1859._\n\nMY DEAR PANIZZI,\n\nIf you don't know, I think you should know that a number of letters are\npassing through the post-office, purporting to be addressed to the\ncharitable by \"Italian Exiles in London,\" asking for aid to raise a fund\nfor a tribute to \"London's Lord Mayor,\" in grateful recognition of the\nreception of the Neapolitan exiles. I know this to be the case, and have\nno doubt in my own mind that the whole thing is an imposture and a \"do.\"\nThe letters are signed \"Gratitudine Italiana.\"\n\n                                                Ever faithfully yours.\n\n\n[Sidenote: Miss White.]\n\n                  TAVISTOCK HOUSE, TAVISTOCK SQUARE, LONDON, W.C.,\n                                           _Monday, April 18th, 1859._\n\nMY DEAR LOTTY,\n\nThis is merely a notice to you that I must positively insist on your\ngetting well, strong, and into good spirits, with the least possible\ndelay. Also, that I look forward to seeing you at Gad's Hill sometime in\nthe summer, staying with the girls, and heartlessly putting down the\nPlorn You know that there is no appeal from the Plorn's inimitable\nfather. What _he_ says must be done. Therefore I send you my love (which\nplease take care of), and my commands (which please obey).\n\n                                               Ever your affectionate.\n\n\n[Sidenote: The Hon. Mrs. Watson.]\n\n                  TAVISTOCK HOUSE, TAVISTOCK SQUARE, LONDON, W.C.,\n                                            _Tuesday, May 31st, 1859._\n\nMY DEAR MRS. WATSON,\n\nYou surprise me by supposing that there is ever latent a defiant and\nroused expression in the undersigned lamb! Apart from this singular\ndelusion of yours, and wholly unaccountable departure from your usual\naccuracy in all things, your satisfaction with the portrait is a great\npleasure to me. It has received every conceivable pains at Frith's\nhands, and ought on his account to be good. It is a little too much (to\nmy thinking) as if my next-door neighbour were my deadly foe, uninsured,\nand I had just received tidings of his house being afire; otherwise very\ngood.\n\nI cannot tell you how delighted we shall be if you would come to Gad's\nHill. You should see some charming woods and a rare old castle, and you\nshould have such a snug room looking over a Kentish prospect, with every\nfacility in it for pondering on the beauties of its master's beard! _Do_\ncome, but you positively _must not_ come and go on the same day.\n\nWe retreat there on Monday, and shall be there all the summer.\n\nMy small boy is perfectly happy at Southsea, and likes the school very\nmuch. I had the finest letter two or three days ago, from another of my\nboys--Frank Jeffrey--at Hamburg. In this wonderful epistle he says:\n\"Dear papa, I write to tell you that I have given up all thoughts of\nbeing a doctor. My conviction that I shall never get over my stammering\nis the cause; all professions are barred against me. The only thing I\nshould like to be is a gentleman farmer, either at the Cape, in Canada,\nor Australia. With my passage paid, fifteen pounds, a horse, and a\nrifle, I could go two or three hundred miles up country, sow grain, buy\ncattle, and in time be very comfortable.\"\n\nConsidering the consequences of executing the little commission by the\nnext steamer, I perceived that the first consequence of the fifteen\npounds would be that he would be robbed of it--of the horse, that it\nwould throw him--and of the rifle, that it would blow his head off;\nwhich probabilities I took the liberty of mentioning, as being against\nthe scheme. With best love from all,\n\n                  Ever believe me, my dear Mrs. Watson,\n                                       Your faithful and affectionate.\n\n\n[Sidenote: Mrs. White.]\n\n                            TAVISTOCK HOUSE, _Sunday, June 5th, 1859._\n\nMY DEAR MRS. WHITE,\n\nI do not write to you this morning because I have anything to say--I\nwell know where your consolation is set, and to what beneficent figure\nyour thoughts are raised--but simply because you are so much in my mind\nthat it is a relief to send you and dear White my love. You are always\nin our hearts and on our lips. May the great God comfort you! You know\nthat Mary and Katie are coming on Thursday. They will bring dear Lotty\nwhat she little needs with you by her side--love; and I hope their\ncompany will interest and please her. There is nothing that they, or any\nof us, would not do for her. She is a part of us all, and has belonged\nto us, as well as to you, these many years.\n\n                                  Ever your affectionate and faithful.\n\n\n[Sidenote: Miss Dickens.]\n\n                          GAD'S HILL, HIGHAM BY ROCHESTER, KENT,\n                                            _Monday, June 11th, 1859._\n\nMY DEAREST MAMIE,\n\nOn Saturday night I found, very much to my surprise and pleasure, the\nphotograph on my table at Tavistock House. It is not a very pleasant or\ncheerful presentation of my daughters; but it is wonderfully like for\nall that, and in some details remarkably good. When I came home here\nyesterday I tried it in the large Townshend stereoscope, in which it\nshows to great advantage. It is in the little stereoscope at present on\nthe drawing-room table. One of the balustrades of the destroyed old\nRochester bridge has been (very nicely) presented to me by the\ncontractor for the works, and has been duly stonemasoned and set up on\nthe lawn behind the house. I have ordered a sun-dial for the top of it,\nand it will be a very good object indeed. The Plorn is highly excited\nto-day by reason of an institution which he tells me (after questioning\nGeorge) is called the \"Cobb, or Bodderin,\" holding a festival at The\nFalstaff. He is possessed of some vague information that they go to\nHigham Church, in pursuance of some old usage, and attend service there,\nand afterwards march round the village. It so far looks probable that\nthey certainly started off at eleven very spare in numbers, and came\nback considerably recruited, which looks to me like the difference\nbetween going to church and coming to dinner. They bore no end of bright\nbanners and broad sashes, and had a band with a terrific drum, and are\nnow (at half-past two) dining at The Falstaff, partly in the side room\non the ground-floor, and partly in a tent improvised this morning. The\ndrum is hung up to a tree in The Falstaff garden, and looks like a\ntropical sort of gourd. I have presented the band with five shillings,\nwhich munificence has been highly appreciated. Ices don't seem to be\nprovided for the ladies in the gallery--I mean the garden; they are\nprowling about there, endeavouring to peep in at the beef and mutton\nthrough the holes in the tent, on the whole, in a debased and degraded\nmanner.\n\nTurk somehow cut his foot in Cobham Lanes yesterday, and Linda hers.\nThey are both lame, and looking at each other. Fancy Mr. Townshend not\nintending to go for another three weeks, and designing to come down here\nfor a few days--with Henri and Bully--on Wednesday! I wish you could\nhave seen him alone with me on Saturday; he was so extraordinarily\nearnest and affectionate on my belongings and affairs in general, and\nnot least of all on you and Katie, that he cried in a most pathetic\nmanner, and was so affected that I was obliged to leave him among the\nflowerpots in the long passage at the end of the dining-room. It was a\nvery good piece of truthfulness and sincerity, especially in one of his\nyears, able to take life so easily.\n\nMr. and Mrs. Wills are here now (but I daresay you know it from your\naunt), and return to town with me to-morrow morning. We are now going on\nto the castle. Mrs. Wills was very droll last night, and told me some\ngood stories. My dear, I wish particularly to impress upon you and dear\nKatie (to whom I send my other best love) that I hope your stay will not\nbe very long. I don't think it very good for either of you, though of\ncourse I know that Lotty will be, and must be, and should be the first\nconsideration with you both. I am very anxious to know how you found her\nand how you are yourself.\n\nBest love to dear Lotty and Mrs. White. The same to Mr. White and Clara.\nWe are always talking about you all.\n\n                        Ever, dearest Mamie, your affectionate Father.\n\n\n[Sidenote: Rev. James White.]\n\n                      GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                           _Thursday, July 7th, 1859._\n\nMY DEAR WHITE,\n\nI send my heartiest and most affectionate love to Mrs. White and you,\nand to Clara. You know all that I could add; you have felt it all; let\nit be unspoken and unwritten--it is expressed within us.\n\nDo you not think that you could all three come here, and stay with us?\nYou and Mrs. White should have your own large room and your own ways,\nand should be among us when you felt disposed, and never otherwise. I do\nhope you would find peace here. Can it not be done?\n\nWe have talked very much about it among ourselves, and the girls are\nstrong upon it. Think of it--do!\n\n                                               Ever your affectionate.\n\n\n[Sidenote: Mr. John Forster.]\n\n                        GAD'S HILL, _Thursday Night, Aug. 25th, 1859._\n\nMY DEAR FORSTER,\n\nHeartily glad to get your letter this morning.\n\nI cannot easily tell you how much interested I am by what you tell me of\nour brave and excellent friend the Chief Baron, in connection with that\nruffian. I followed the case with so much interest, and have followed\nthe miserable knaves and asses who have perverted it since, with so much\nindignation, that I have often had more than half a mind to write and\nthank the upright judge who tried him. I declare to God that I believe\nsuch a service one of the greatest that a man of intellect and courage\ncan render to society. Of course I saw the beast of a prisoner (with my\nmind's eye) delivering his cut-and-dried speech, and read in every word\nof it that no one but the murderer could have delivered or conceived\nit. Of course I have been driving the girls out of their wits here, by\nincessantly proclaiming that there needed no medical evidence either\nway, and that the case was plain without it. Lastly, of course (though a\nmerciful man--because a merciful man I mean), I would hang any Home\nSecretary (Whig, Tory, Radical, or otherwise) who should step in between\nthat black scoundrel and the gallows. I can_not_ believe--and my belief\nin all wrong as to public matters is enormous--that such a thing will be\ndone.\n\nI am reminded of Tennyson, by thinking that King Arthur would have made\nshort work of the amiable ----, whom the newspapers strangely delight to\nmake a sort of gentleman of. How fine the \"Idylls\" are! Lord! what a\nblessed thing it is to read a man who can write! I thought nothing could\nbe grander than the first poem till I came to the third; but when I had\nread the last, it seemed to be absolutely unapproached and\nunapproachable.\n\nTo come to myself. I have written and begged the \"All the Year Round\"\npublisher to send you directly four weeks' proofs beyond the current\nnumber, that are in type. I hope you will like them. Nothing but the\ninterest of the subject, and the pleasure of striving with the\ndifficulty of the forms of treatment, nothing in the mere way of money,\nI mean, could also repay the time and trouble of the incessant\ncondensation. But I set myself the little task of making a _picturesque_\nstory, rising in every chapter with characters true to nature, but whom\nthe story itself should express, more than they should express\nthemselves, by dialogue. I mean, in other words, that I fancied a story\nof incident might be written, in place of the bestiality that _is_\nwritten under that pretence, pounding the characters out in its own\nmortar, and beating their own interests out of them. If you could have\nread the story all at once, I hope you wouldn't have stopped halfway.\n\nAs to coming to your retreat, my dear Forster, think how helpless I am.\nI am not well yet. I have an instinctive feeling that nothing but the\nsea will restore me, and I am planning to go and work at Ballard's, at\nBroadstairs, from next Wednesday to Monday. I generally go to town on\nMonday afternoon. All Tuesday I am at the office, on Wednesday I come\nback here, and go to work again. I don't leave off till Monday comes\nround once more. I am fighting to get my story done by the first week in\nOctober. On the 10th of October I am going away to read for a fortnight\nat Ipswich, Norwich, Oxford, Cambridge, and a few other places. Judge\nwhat my spare time is just now!\n\nI am very much surprised and very sorry to find from the enclosed that\nElliotson has been ill. I never heard a word of it.\n\nGeorgy sends best love to you and to Mrs. Forster, so do I, so does\nPlorn, so does Frank. The girls are, for five days, with the Whites at\nRamsgate. It is raining, intensely hot, and stormy. Eighteen creatures,\nlike little tortoises, have dashed in at the window and fallen on the\npaper since I began this paragraph [Illustration: ink-blot] (that was\none!). I am a wretched sort of creature in my way, but it is a way that\ngets on somehow. And all ways have the same fingerpost at the head of\nthem, and at every turning in them.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Miss Dickens and Miss Katie Dickens.]\n\n                       ALBION, BROADSTAIRS, _Friday, Sept. 2nd, 1859._\n\nMY DEAREST MAMIE AND KATIE,\n\nI have been \"moved\" here, and am now (Ballard having added to the hotel\na house we lived in three years) in our old dining-room and\nsitting-room, and our old drawing-room as a bedroom. My cold is so bad,\nboth in my throat and in my chest, that I can't bathe in the sea; Tom\nCollin dissuaded me--thought it \"bad\"--but I get a heavy shower-bath at\nMrs. Crampton's every morning. The baths are still hers and her\nhusband's, but they have retired and live in \"Nuckells\"--are going to\ngive a stained-glass window, value three hundred pounds, to St. Peter's\nChurch. Tom Collin is of opinion that the Miss Dickenses has growed two\nfine young women--leastwise, asking pardon, ladies. An evangelical\nfamily of most disagreeable girls prowl about here and trip people up\nwith tracts, which they put in the paths with stones upon them to keep\nthem from blowing away. Charles Collins and I having seen a bill\nyesterday--about a mesmeric young lady who did feats, one of which was\nset forth in the bill, in a line by itself, as\n\n                            THE RIGID LEGS,\n\n--were overpowered with curiosity, and resolved to go. It came off in\nthe Assembly Room, now more exquisitely desolate than words can\ndescribe. Eighteen shillings was the \"take.\" Behind a screen among the\ncompany, we heard mysterious gurglings of water before the entertainment\nbegan, and then a slippery sound which occasioned me to whisper C. C.\n(who laughed in the most ridiculous manner), \"Soap.\" It proved to be the\nyoung lady washing herself. She must have been wonderfully dirty, for\nshe took a world of trouble, and didn't come out clean after all--in a\nwretched dirty muslin frock, with blue ribbons. She was the alleged\nmesmeriser, and a boy who distributed bills the alleged mesmerised. It\nwas a most preposterous imposition, but more ludicrous than any poor\nsight I ever saw. The boy is clearly out of pantomime, and when he\npretended to be in the mesmeric state, made the company back by going\nin among them head over heels, backwards, half-a-dozen times, in a most\ninsupportable way. The pianist had struck; and the manner in which the\nlecturer implored \"some lady\" to play a \"polker,\" and the manner in\nwhich no lady would; and in which the few ladies who were there sat with\ntheir hats on, and the elastic under their chins, as if it were going to\nblow, is never to be forgotten. I have been writing all the morning, and\nam going for a walk to Ramsgate. This is a beast of a letter, but I am\nnot well, and have been addling my head.\n\n                           Ever, dear Girls, your affectionate Father.\n\n\n[Sidenote: Mr. W. Wilkie Collins.]\n\n                     GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                     _Friday Night, Sept. 16th, 1859._\n\nMY DEAR WILKIE,\n\nJust a word to say that I have received yours, and that I look forward\nto the reunion on Thursday, when I hope to have the satisfaction of\nrecounting to you the plot of a play that has been laid before me for\ncommending advice.\n\nDitto to what you say respecting the _Great Eastern_. I went right up to\nLondon Bridge by the boat that day, on purpose that I might pass her. I\nthought her the ugliest and most unshiplike thing these eyes ever\nbeheld. I wouldn't go to sea in her, shiver my ould timbers and rouse me\nup with a monkey's tail (man-of-war metaphor), not to chuck a biscuit\ninto Davy Jones's weather eye, and see double with my own old toplights.\n\nTurk has been so good as to produce from his mouth, for the wholesome\nconsternation of the family, eighteen feet of worm. When he had brought\nit up, he seemed to think it might be turned to account in the\nhousekeeping and was proud. Pony has kicked a shaft off the cart, and is\nto be sold. Why don't you buy her? she'd never kick with you.\n\nBarber's opinion is, that them fruit-trees, one and all, is touchwood,\nand not fit for burning at any gentleman's fire; also that the stocking\nof this here garden is worth less than nothing, because you wouldn't\nhave to grub up nothing, and something takes a man to do it at\nthree-and-sixpence a day. Was \"left desponding\" by your reporter.\n\nI have had immense difficulty to find a man for the stable-yard here.\nBarber having at last engaged one this morning, I enquired if he had a\ndecent hat for driving in, to which Barber returned this answer:\n\n\"Why, sir, not to deceive you, that man flatly say that he never have\nwore that article since man he was!\"\n\nI am consequently fortified into my room, and am afraid to go out to\nlook at him. Love from all.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Monsieur Regnier.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                          _Saturday, Oct. 15th, 1859._\n\nMY DEAR REGNIER,\n\nYou will receive by railway parcel the proof-sheets of a story of mine,\nthat has been for some time in progress in my weekly journal, and that\nwill be published in a complete volume about the middle of November.\nNobody but Forster has yet seen the latter portions of it, or will see\nthem until they are published. I want you to read it for two reasons.\nFirstly, because I hope it is the best story I have written. Secondly,\nbecause it treats of a very remarkable time in France; and I should very\nmuch like to know what you think of its being dramatised for a French\ntheatre. If you should think it likely to be done, I should be glad to\ntake some steps towards having it well done. The story is an\nextraordinary success here, and I think the end of it is certain to make\na still greater sensation.\n\nDon't trouble yourself to write to me, _mon ami_, until you shall have\nhad time to read the proofs. Remember, they are _proofs_, and _private_;\nthe latter chapters will not be before the public for five or six weeks\nto come.\n\nWith kind regards to Madame Regnier, in which my daughters and their\naunt unite,\n\n                                    Believe me, ever faithfully yours.\n\nP.S.--The story (I daresay you have not seen any of it yet) is called\n\"A Tale of Two Cities.\"\n\n\n[Sidenote: Mr. Frank Stone, A.R.A.]\n\n                   PETERBOROUGH, _Wednesday Evening, Oct. 19th, 1859._\n\nMY DEAR STONE,\n\nWe had a splendid rush last night--exactly as we supposed, with the\npressure on the two shillings, of whom we turned a crowd away. They were\na far finer audience than on the previous night; I think the finest I\nhave ever read to. They took every word of the \"Dombey\" in quite an\namazing manner, and after the child's death, paused a little, and then\nset up a shout that it did one good to hear. Mrs. Gamp then set in with\na roar, which lasted until I had done. I think everybody for the time\nforgot everything but the matter in hand. It was as fine an instance of\nthorough absorption in a fiction as any of us are likely to see ever\nagain.\n\n---- (in an exquisite red mantle), accompanied by her sister (in another\nexquisite red mantle) and by the deaf lady, (who leaned a black\nhead-dress, exactly like an old-fashioned tea-urn without the top,\nagainst the wall), was charming. HE couldn't get at her on account of\nthe pressure. HE tried to peep at her from the side door, but she (ha,\nha, ha!) was unconscious of his presence. I read to her, and goaded him\nto madness. He is just sane enough to send his kindest regards.\n\nThis is a place which--except the cathedral, with the loveliest front I\never saw--is like the back door to some other place. It is, I should\nhope, the deadest and most utterly inert little town in the British\ndominions. The magnates have taken places, and the bookseller is of\nopinion that \"such is the determination to do honour to Mr. Dickens,\nthat the doors _must_ be opened half an hour before the appointed time.\"\nYou will picture to yourself Arthur's quiet indignation at this, and the\nmanner in which he remarked to me at dinner, \"that he turned away twice\nPeterborough last night.\"\n\nA very pretty room--though a Corn Exchange--and a room we should have\nbeen glad of at Cambridge, as it is large, bright, and cheerful, and\nwonderfully well lighted.\n\nThe difficulty of getting to Bradford from here to-morrow, at any time\nconvenient to us, turned out to be so great, that we are all going in\nfor Leeds (only three-quarters of an hour from Bradford) to-night after\nthe reading, at a quarter-past eleven. We are due at Leeds a quarter\nbefore three.\n\nSo no more at present from,\n\n                                                 Yours affectionately.\n\n\n[Sidenote: Mr. W. R. Sculthorpe.]\n\n                TAVISTOCK HOUSE, TAVISTOCK SQUARE, LONDON, W.C.,\n                                          _Thursday, Nov. 10th, 1859._\n\nDEAR SIR,\n\nJudgment must go by default. I have not a word to plead against Dodson\nand Fogg. I am without any defence to the action; and therefore, as law\ngoes, ought to win it.\n\nSeriously, the date of your hospitable note disturbs my soul. But I have\nbeen incessantly writing in Kent and reading in all sorts of places, and\nhave done nothing in my own personal character these many months; and\nnow I come to town and our friend[5] is away! Let me take that\ndefaulting miscreant into council when he comes back.\n\n                                                     Faithfully yours.\n\n\n[Sidenote: Monsieur Regnier.]\n\n               TAVISTOCK HOUSE, TAVISTOCK SQUARE, LONDON, W.C.,\n                                         _Wednesday, Nov. 16th, 1859._\n\nMY DEAR REGNIER,\n\nI send you ten thousand thanks for your kind and explicit letter. What I\nparticularly wished to ascertain from you was, whether it is likely the\nCensor would allow such a piece to be played in Paris. In the case of\nits being likely, then I wished to have the piece as well done as\npossible, and would even have proposed to come to Paris to see it\nrehearsed. But I very much doubted whether the general subject would not\nbe objectionable to the Government, and what you write with so much\nsagacity and with such care convinces me at once that its representation\nwould be prohibited. Therefore I altogether abandon and relinquish the\nidea. But I am just as heartily and cordially obliged to you for your\ninterest and friendship, as if the book had been turned into a play five\nhundred times. I again thank you ten thousand times, and am quite sure\nthat you are right. I only hope you will forgive my causing you so much\ntrouble, after your hard work.\n\nMy girls and Georgina send their kindest regards to Madame Regnier and\nto you. My Gad's Hill house (I think I omitted to tell you, in reply to\nyour enquiry) is on the very scene of Falstaff's robbery. There is a\nlittle _cabaret_ at the roadside, still called The Sir John Falstaff.\nAnd the country, in all its general features, is, at this time, what it\nwas in Shakespeare's. I hope you will see the house before long. It is\nreally a pretty place, and a good residence for an English writer, is it\nnot?\n\nMacready, we are all happy to hear from himself, is going to leave the\ndreary tomb in which he lives, at Sherborne, and to remove to\nCheltenham, a large and handsome place, about four or five hours'\nrailway journey from London, where his poor girls will at least see and\nhear some life. Madame Celeste was with me yesterday, wishing to\ndramatise \"A Tale of Two Cities\" for the Lyceum, after bringing out the\nChristmas pantomime. I gave her my permission and the book; but I fear\nthat her company (troupe) is a very poor one.\n\nThis is all the news I have, except (which is no news at all) that I\nfeel as if I had not seen you for fifty years, and that\n\n                          I am ever your attached and faithful Friend.\n\n\n[Sidenote: Mr. T. Longman.]\n\n                           TAVISTOCK HOUSE, _Monday, Nov. 28th, 1859._\n\nMY DEAR LONGMAN,\n\nI am very anxious to present to you, with the earnest hope that you will\nhold him in your remembrance, young Mr. Marcus Stone, son of poor Frank\nStone, who died suddenly but a little week ago. You know, I daresay,\nwhat a start this young man made in the last exhibition, and what a\nfavourable notice his picture attracted. He wishes to make an additional\nopening for himself in the illustration of books. He is an admirable\ndraughtsman, has a most dexterous hand, a charming sense of grace and\nbeauty, and a capital power of observation. These qualities in him I\nknow well of my own knowledge. He is in all things modest, punctual, and\nright; and I would answer for him, if it were needful, with my head.\n\nIf you will put anything in his way, you will do it a second time, I am\ncertain.\n\n                                              Faithfully yours always.\n\nFOOTNOTES:\n\n[5] Mr. Edmund Yates.\n\n\n\n\n1860.\n\nNARRATIVE.\n\n\nThis winter was the last spent at Tavistock House. Charles Dickens had\nfor some time been inclining to the idea of making his home altogether\nat Gad's Hill, giving up his London house, and taking a furnished house\nfor the sake of his daughters for a few months of the London season.\nAnd, as his daughter Kate was to be married this summer to Mr. Charles\nCollins, this intention was confirmed and carried out. He made\narrangements for the sale of Tavistock House to Mr. Davis, a Jewish\ngentleman, and he gave up possession of it in September. Up to this time\nGad's Hill had been furnished merely as a temporary summer\nresidence--pictures, library, and all best furniture being left in the\nLondon house. He now set about beautifying and making Gad's Hill\nthoroughly comfortable and homelike. And there was not a year\nafterwards, up to the year of his death, that he did not make some\naddition or improvement to it. He also furnished, as a private\nresidence, a sitting-room and some bedrooms at his office in Wellington\nStreet, to be used, when there was no house in London, as occasional\ntown quarters by himself, his daughter, and sister-in-law.\n\nHe began in this summer his occasional papers for \"All the Year Round,\"\nwhich he called \"The Uncommercial Traveller,\" and which were continued\nat intervals in his journal until 1869.\n\nIn the autumn of this year he began another story, to be published\nweekly in \"All the Year Round.\" The letter to Mr. Forster, which we\ngive, tells him of this beginning and gives him the name of the book.\nThe first number of \"Great Expectations\" appeared on the 1st December.\nThe Christmas number, this time, was written jointly by himself and Mr.\nWilkie Collins. The scene was laid at Clovelly, and they made a journey\ntogether into Devonshire and Cornwall, for the purpose of this story, in\nNovember.\n\nThe letter to Sir Edward Bulwer Lytton is, unfortunately, the only one\nwe have as yet been able to procure. The present Lord Lytton, the\nViceroy of India, has kindly endeavoured to help us even during his\nabsence from England. But it was found to be impossible without his own\nassistance to make the necessary search among his father's papers. And\nhe has promised us that, on his return, he will find and lend to us,\nmany letters from Charles Dickens, which are certainly in existence, to\nhis distinguished fellow-writer and great friend. We hope, therefore, it\nmay be possible for us at some future time to be able to publish these\nletters, as well as those addressed to the present Lord Lytton (when he\nwas Mr. Robert Lytton, otherwise \"Owen Meredith,\" and frequent\ncontributor to \"Household Words\" and \"All the Year Round\"). We have the\nsame hope with regard to letters addressed to Sir Henry Layard, at\npresent Ambassador at Constantinople, which, of course, for the same\nreason, cannot be lent to us at the present time.\n\nWe give a letter to Mr. Forster on one of his books on the Commonwealth,\nthe \"Impeachment of the Five Members;\" which, as with other letters\nwhich we are glad to publish on the subject of Mr. Forster's own works,\nwas not used by himself for obvious reasons.\n\nA letter to his daughter Mamie (who, after her sister's marriage, paid a\nvisit with her dear friends the White family to Scotland, where she had\na serious illness) introduces a recent addition to the family, who\nbecame an important member of it, and one to whom Charles Dickens was\nvery tenderly attached--her little white Pomeranian dog \"Mrs. Bouncer\"\n(so called after the celebrated lady of that name in \"Box and Cox\"). It\nis quite necessary to make this formal introduction of the little pet\nanimal (who lived to be a very old dog and died in 1874), because future\nletters to his daughter contain constant references and messages to\n\"Mrs. Bouncer,\" which would be quite unintelligible without this\nexplanation. \"Boy,\" also referred to in this letter, was his daughter's\nhorse. The little dog and the horse were gifts to Mamie Dickens from her\nfriends Mr. and Mrs. Arthur Smith, and the sister of the latter, Miss\nCraufurd.\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n                            TAVISTOCK HOUSE, _Monday, Jan. 2nd, 1860._\n\nMY DEAREST MACREADY,\n\nA happy New Year to you, and many happy years! I cannot tell you how\ndelighted I was to receive your Christmas letter, or with what pleasure\nI have received Forster's emphatic accounts of your health and spirits.\nBut when was I ever wrong? And when did I not tell you that you were an\nimpostor in pretending to grow older as the rest of us do, and that you\nhad a secret of your own for reversing the usual process! It happened\nthat I read at Cheltenham a couple of months ago, and that I have rarely\nseen a place that so attracted my fancy. I had never seen it before.\nAlso I believe the character of its people to have greatly changed for\nthe better. All sorts of long-visaged prophets had told me that they\nwere dull, stolid, slow, and I don't know what more that is\ndisagreeable. I found them exactly the reverse in all respects; and I\nsaw an amount of beauty there--well--that is not to be more specifically\nmentioned to you young fellows.\n\nKatie dined with us yesterday, looking wonderfully well, and singing\n\"Excelsior\" with a certain dramatic fire in her, whereof I seem to\nremember having seen sparks afore now. Etc. etc. etc.\n\n        With kindest love from all at home to all with you,\n                       Ever, my dear Macready, your most affectionate.\n\n\n[Sidenote: Mr. W. Wilkie Collins.]\n\n                 TAVISTOCK HOUSE, TAVISTOCK SQUARE, LONDON, W.C.,\n                                     _Saturday Night, Jan. 7th, 1860._\n\nMY DEAR WILKIE,\n\nI have read this book with great care and attention. There cannot be a\ndoubt that it is a very great advance on all your former writing, and\nmost especially in respect of tenderness. In character it is excellent.\nMr. Fairlie as good as the lawyer, and the lawyer as good as he. Mr.\nVesey and Miss Halcombe, in their different ways, equally meritorious.\nSir Percival, also, is most skilfully shown, though I doubt (you see\nwhat small points I come to) whether any man ever showed uneasiness by\nhand or foot without being forced by nature to show it in his face too.\nThe story is very interesting, and the writing of it admirable.\n\nI seem to have noticed, here and there, that the great pains you take\nexpress themselves a trifle too much, and you know that I always contest\nyour disposition to give an audience credit for nothing, which\nnecessarily involves the forcing of points on their attention, and which\nI have always observed them to resent when they find it out--as they\nalways will and do. But on turning to the book again, I find it\ndifficult to take out an instance of this. It rather belongs to your\nhabit of thought and manner of going about the work. Perhaps I express\nmy meaning best when I say that the three people who write the\nnarratives in these proofs have a DISSECTIVE property in common, which\nis essentially not theirs but yours; and that my own effort would be to\nstrike more of what is got _that way_ out of them by collision with one\nanother, and by the working of the story.\n\nYou know what an interest I have felt in your powers from the beginning\nof our friendship, and how very high I rate them? _I_ know that this is\nan admirable book, and that it grips the difficulties of the weekly\nportion and throws them in masterly style. No one else could do it half\nso well. I have stopped in every chapter to notice some instance of\ningenuity, or some happy turn of writing; and I am absolutely certain\nthat you never did half so well yourself.\n\nSo go on and prosper, and let me see some more, when you have enough\n(for your own satisfaction) to show me. I think of coming in to back you\nup if I can get an idea for my series of gossiping papers. One of those\ndays, please God, we may do a story together; I have very odd\nhalf-formed notions, in a mist, of something that might be done that\nway.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Mr. John Forster.]\n\n              11, WELLINGTON STREET, NORTH STRAND, LONDON, W.C.,\n                                           _Wednesday, May 2nd, 1860._\n\nMY DEAR FORSTER,\n\nIt did not occur to me in reading your most excellent, interesting, and\nremarkable book, that it could with any reason be called one-sided. If\nClarendon had never written his \"History of the Rebellion,\" then I can\nunderstand that it might be. But just as it would be impossible to\nanswer an advocate who had misstated the merits of a case for his own\npurpose, without, in the interests of truth, and not of the other side\nmerely, re-stating the merits and showing them in their real form, so I\ncannot see the practicability of telling what you had to tell without\nin some sort championing the misrepresented side, and I think that you\ndon't do that as an advocate, but as a judge.\n\nThe evidence has been suppressed and , and the judge goes\nthrough it and puts it straight. It is not _his_ fault if it all goes\none way and tends to one plain conclusion. Nor is it his fault that it\ngoes the further when it is laid out straight, or seems to do so,\nbecause it was so knotted and twisted up before.\n\nI can understand any man's, and particularly Carlyle's, having a\nlingering respect that does not like to be disturbed for those (in the\nbest sense of the word) loyal gentlemen of the country who went with the\nking and were so true to him. But I don't think Carlyle sufficiently\nconsiders that the great mass of those gentlemen _didn't know the\ntruth_, that it was a part of their loyalty to believe what they were\ntold on the king's behalf, and that it is reasonable to suppose that the\nking was too artful to make known to _them_ (especially after failure)\nwhat were very acceptable designs to the desperate soldiers of fortune\nabout Whitehall. And it was to me a curious point of adventitious\ninterest arising out of your book, to reflect on the probability of\ntheir having been as ignorant of the real scheme in Charles's head, as\ntheir descendants and followers down to this time, and to think with\npity and admiration that they believed the cause to be so much better\nthan it was. This is a notion I was anxious to have expressed in our\naccount of the book in these pages. For I don't suppose Clarendon, or\nany other such man to sit down and tell posterity something that he has\nnot \"tried on\" in his own time. Do you?\n\nIn the whole narrative I saw nothing anywhere to which I demurred. I\nadmired it all, went with it all, and was proud of my friend's having\nwritten it all. I felt it to be all square and sound and right, and to\nbe of enormous importance in these times. Firstly, to the people who\n(like myself) are so sick of the shortcomings of representative\ngovernment as to have no interest in it. Secondly, to the humbugs at\nWestminster who have come down--a long, long way--from those men, as you\nknow. When the great remonstrance came out, I was in the thick of my\nstory, and was always busy with it; but I am very glad I didn't read it\nthen, as I shall read it now to much better purpose. All the time I was\nat work on the \"Two Cities,\" I read no books but such as had the air of\nthe time in them.\n\nTo return for a final word to the Five Members. I thought the marginal\nreferences overdone. Here and there, they had a comical look to me for\nthat reason, and reminded me of shows and plays where everything is in\nthe bill.\n\nLastly, I should have written to you--as I had a strong inclination to\ndo, and ought to have done, immediately after reading the book--but for\na weak reason; of all things in the world I have lost heart in one--I\nhope no other--I cannot, times out of calculation, make up my mind to\nwrite a letter.\n\n                          Ever, my dear Forster, affectionately yours.\n\n\n[Sidenote: M. de Cerjat.]\n\n                           TAVISTOCK HOUSE, _Thursday, May 3rd, 1860._\n\nMY DEAR CERJAT,\n\nThe date of this letter would make me horribly ashamed of myself, if I\ndidn't know that _you_ know how difficult letter-writing is to one whose\ntrade it is to write.\n\nYou asked me on Christmas Eve about my children. My second daughter is\ngoing to be married in the course of the summer to Charles Collins, the\nbrother of Wilkie Collins, the novelist. The father was one of the most\nfamous painters of English green lanes and coast pieces. He was bred an\nartist; is a writer, too, and does \"The Eye Witness,\" in \"All the Year\nRound.\" He is a gentleman, accomplished, and amiable. My eldest daughter\nhas not yet started any conveyance on the road to matrimony (that I know\nof); but it is likely enough that she will, as she is very agreeable and\nintelligent. They are both very pretty. My eldest boy, Charley, has been\nin Barings' house for three or four years, and is now going to Hong\nKong, strongly backed up by Barings, to buy tea on his own account, as a\nmeans of forming a connection and seeing more of the practical part of a\nmerchant's calling, before starting in London for himself. His brother\nFrank (Jeffrey's godson) I have just recalled from France and Germany,\nto come and learn business, and qualify himself to join his brother on\nhis return from the Celestial Empire. The next boy, Sydney Smith, is\ndesigned for the navy, and is in training at Portsmouth, awaiting his\nnomination. He is about three foot high, with the biggest eyes ever\nseen, and is known in the Portsmouth parts as \"Young Dickens, who can do\neverything.\"\n\nAnother boy is at school in France; the youngest of all has a private\ntutor at home. I have forgotten the second in order, who is in India. He\nwent out as ensign of a non-existent native regiment, got attached to\nthe 42nd Highlanders, one of the finest regiments in the Queen's\nservice; has remained with them ever since, and got made a lieutenant by\nthe chances of the rebellious campaign, before he was eighteen. Miss\nHogarth, always Miss Hogarth, is the guide, philosopher, and friend of\nall the party, and a very close affection exists between her and the\ngirls. I doubt if she will ever marry. I don't know whether to be glad\nof it or sorry for it.\n\nI have laid down my pen and taken a long breath after writing this\nfamily history. I have also considered whether there are any more\nchildren, and I don't think there are. If I should remember two or three\nothers presently, I will mention them in a postscript.\n\nWe think Townshend looking a little the worse for the winter, and we\nperceive Bully to be decidedly old upon his legs, and of a most\ndiabolical turn of mind. When they first arrived the weather was very\ndark and cold, and kept them indoors. It has since turned very warm and\nbright, but with a dusty and sharp east wind. They are still kept\nindoors by this change, and I begin to wonder what change will let them\nout. Townshend dines with us every Sunday. You may be sure that we\nalways talk of you and yours, and drink to you heartily.\n\nPublic matters here are thought to be rather improving; the deep\nmistrust of the gentleman in Paris being counteracted by the vigorous\nstate of preparation into which the nation is getting. You will have\nobserved, of course, that we establish a new defaulter in respect of\nsome great trust, about once a quarter. The last one, the cashier of a\nCity bank, is considered to have distinguished himself greatly, a\nquarter of a million of money being high game.\n\nNo, my friend, I have not shouldered my rifle yet, but I should do so on\nmore pressing occasion. Every other man in the row of men I know--if\nthey were all put in a row--is a volunteer though. There is a tendency\nrather to overdo the wearing of the uniform, but that is natural enough\nin the case of the youngest men. The turn-out is generally very\ncreditable indeed. At the ball they had (in a perfectly unventilated\nbuilding), their new leather belts and pouches smelt so fearfully that\nit was, as my eldest daughter said, like shoemaking in a great prison.\nShe, consequently, distinguished herself by fainting away in the most\ninaccessible place in the whole structure, and being brought out\n(horizontally) by a file of volunteers, like some slain daughter of\nAlbion whom they were carrying into the street to rouse the indignant\nvalour of the populace.\n\nLord, my dear Cerjat, when I turn to that page of your letter where you\nwrite like an ancient sage in whom the fire has paled into a meek-eyed\nstate of coolness and virtue, I half laugh and half cry! _You_ old!\n_You_ a sort of hermit? Boh! Get out.\n\nWith this comes my love and all our loves, to you and Mrs. Cerjat, and\nyour daughter. I add my special and particular to the sweet \"singing\ncousin.\" When shall you and I meet, and where? Must I come to see\nTownshend? I begin to think so.\n\n                 Ever, my dear Cerjat, your affectionate and faithful.\n\n\n[Sidenote: Sir Edward Bulwer Lytton.]\n\n                                GAD'S HILL, _Tuesday, June 5th, 1860._\n\nMY DEAR BULWER LYTTON,\n\nI am very much interested and gratified by your letter concerning \"A\nTale of Two Cities.\" I do not quite agree with you on two points, but\nthat is no deduction from my pleasure.\n\nIn the first place, although the surrender of the feudal privileges (on\na motion seconded by a nobleman of great rank) was the occasion of a\nsentimental scene, I see no reason to doubt, but on the contrary, many\nreasons to believe, that some of these privileges had been used to the\nfrightful oppression of the peasant, quite as near to the time of the\nRevolution as the doctor's narrative, which, you will remember, dates\nlong before the Terror. And surely when the new philosophy was the talk\nof the salons and the slang of the hour, it is not unreasonable or\nunallowable to suppose a nobleman wedded to the old cruel ideas, and\nrepresenting the time going out, as his nephew represents the time\ncoming in; as to the condition of the peasant in France generally at\nthat day, I take it that if anything be certain on earth it is certain\nthat it was intolerable. No _ex post facto_ enquiries and provings by\nfigures will hold water, surely, against the tremendous testimony of men\nliving at the time.\n\nThere is a curious book printed at Amsterdam, written to make out no\ncase whatever, and tiresome enough in its literal dictionary-like\nminuteness, scattered up and down the pages of which is full authority\nfor my marquis. This is \"Mercier's Tableau de Paris.\" Rousseau is the\nauthority for the peasant's shutting up his house when he had a bit of\nmeat. The tax-taker was the authority for the wretched creature's\nimpoverishment.\n\nI am not clear, and I never have been clear, respecting that canon of\nfiction which forbids the interposition of accident in such a case as\nMadame Defarge's death. Where the accident is inseparable from the\npassion and emotion of the character, where it is strictly consistent\nwith the whole design, and arises out of some culminating proceeding on\nthe part of the character which the whole story has led up to, it seems\nto me to become, as it were, an act of divine justice. And when I use\nMiss Pross (though this is quite another question) to bring about that\ncatastrophe, I have the positive intention of making that half-comic\nintervention a part of the desperate woman's failure, and of opposing\nthat mean death--instead of a desperate one in the streets, which she\nwouldn't have minded--to the dignity of Carton's wrong or right; this\n_was_ the design, and seemed to be in the fitness of things.\n\nNow, as to the reading. I am sorry to say that it is out of the question\nthis season. I have had an attack of rheumatism--quite a stranger to\nme--which remains hovering about my left side, after having doubled me\nup in the back, and which would disable me from standing for two hours.\nI have given up all dinners and town engagements, and come to my little\nFalstaff House here, sensible of the necessity of country training all\nthrough the summer. Smith would have proposed any appointment to see you\non the subject, but he has been dreadfully ill with tic. Whenever I read\nin London, I will gladly put a night aside for your purpose, and we will\nplot to connect your name with it, and give it some speciality. But this\ncould not be before Christmas time, as I should not be able to read\nsooner, for in the hot weather it would be useless. Let me hear from you\nabout this when you have considered it. It would greatly diminish the\nexpenses, remember.\n\n                                   Ever affectionately and faithfully.\n\n\n[Sidenote: The Lord John Russell.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                            _Sunday, June 17th, 1860._\n\nMY DEAR LORD JOHN RUSSELL,\n\nI cannot thank you enough for your kind note and its most welcome\nenclosure. My sailor-boy comes home from Portsmouth to-morrow, and will\nbe overjoyed. His masters have been as anxious for getting his\nnomination as though it were some distinction for themselves.\n\n                                       Ever your faithful and obliged.\n\n\n[Sidenote: The Earl of Carlisle.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                          _Wednesday, Aug. 8th, 1860._\n\nMY DEAR LORD CARLISLE,\n\nComing back here after an absence of three days in town, I find your\nkind and cordial letter lying on my table. I heartily thank you for it,\nand highly esteem it. I understand that the article on the spirits to\nwhich you refer was written by ---- (he played an Irish porter in one\nscene of Bulwer's comedy at Devonshire House). Between ourselves, I\nthink it must be taken with a few grains of salt, imperial measure. The\nexperiences referred to \"came off\" at ----, where the spirit of ----\n(among an extensive and miscellaneous bodiless circle) _dines_\nsometimes! Mr. ----, the high priest of the mysteries, I have some\nconsiderable reason--derived from two honourable men--for mistrusting.\nAnd that some of the disciples are very easy of belief I know.\n\nThis is Falstaff's own Gad's Hill, and I live on the top of it. All goes\nwell with me, thank God! I should be thoroughly delighted to see you\nagain, and to show you where the robbery was done. My eldest daughter\nkeeps my house, and it is one I was extraordinarily fond of when a\nchild.\n\n                     My dear Lord Carlisle, ever affectionately yours.\n\nP.S.--I am prowling about, meditating a new book.\n\n\n[Sidenote: Mr. W. H. Wills.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                           _Tuesday, Sept. 4th, 1860._\n\nMY DEAR WILLS,\n\nYour description of your sea-castle makes your room here look uncommonly\ndusty. Likewise the costermongers in the street outside, and the one\ncustomer (drunk, with his head on the table) in the Crown Coffee House\nover the way, in York Street, have an earthy, and, as I may say, a\nland-lubberly aspect. Cape Horn, to the best of _my_ belief, is a\ntremendous way off, and there are more bricks and cabbage-leaves between\nthis office and that dismal point of land than _you_ can possibly\nimagine.\n\nComing here from the station this morning, I met, coming from the\nexecution of the Wentworth murderer, such a tide of ruffians as never\ncould have flowed from any point but the gallows. Without any figure of\nspeech it turned one white and sick to behold them.\n\nTavistock House is cleared to-day, and possession delivered up. I must\nsay that in all things the purchaser has behaved thoroughly well, and\nthat I cannot call to mind any occasion when I have had money dealings\nwith a Christian that have been so satisfactory, considerate, and\ntrusting.\n\nI am ornamented at present with one of my most intensely preposterous\nand utterly indescribable colds. If you were to make a voyage from Cape\nHorn to Wellington Street, you would scarcely recognise in the bowed\nform, weeping eyes, rasped nose, and snivelling wretch whom you would\nencounter here, the once gay and sparkling, etc. etc.\n\nEverything else here is as quiet as possible. Business reports you\nreceive from Holsworth. Wilkie looked in to-day, going to\nGloucestershire for a week. The office is full of discarded curtains and\ncoverings from Tavistock House, which Georgina is coming up this evening\nto select from and banish. Mary is in raptures with the beauties of\nDunkeld, but is not very well in health. The Admiral (Sydney) goes up\nfor his examination to-morrow. If he fails to pass with credit, I will\nnever believe in anybody again, so in that case look out for your own\nreputation with me.\n\nThis is really all the news I have, except that I am lazy, and that\nWilkie dines here next Tuesday, in order that we may have a talk about\nthe Christmas number.\n\nI beg to send my kind regard to Mrs. Wills, and to enquire how she likes\nwearing a hat, which of course she does. I also want to know from her\nin confidence whether _Crwllm festidiniog llymthll y wodd_?\n\nYesterday I burnt, in the field at Gad's Hill, the accumulated letters\nand papers of twenty years. They sent up a smoke like the genie when he\ngot out of the casket on the seashore; and as it was an exquisite day\nwhen I began, and rained very heavily when I finished, I suspect my\ncorrespondence of having overcast the face of the heavens.\n\n                                                      Ever faithfully.\n\nP.S.--Kind regard to Mr. and Mrs. Novelli.[6]\n\nI have just sent out for _The Globe_. No news.\n\nHullah's daughter (an artist) tells me that certain female students have\naddressed the Royal Academy, entreating them to find a place for their\neducation. I think it a capital move, for which I can do something\npopular and telling in _The Register_. Adelaide Procter is active in the\nbusiness, and has a copy of their letter. Will you write to her for\nthat, and anything else she may have about it, telling her that I\nstrongly approve, and want to help them myself?\n\n\n[Sidenote: The Hon. Mrs. Watson.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                     _Friday Night, Sept. 14th, 1860._\n\nMY DEAR MRS. WATSON,\n\nI lose no time in answering your letter; and first as to business, the\nschool in the High Town at Boulogne was excellent. The boys all English,\nthe two proprietors an old Eton master and one of the Protestant\nclergymen of the town. The teaching unusually sound and good. The manner\nand conduct developed in the boys quite admirable. But I have never\nseen a gentleman so perfectly acquainted with boy-nature as the Eton\nmaster. There was a perfect understanding between him and his charges;\nnothing pedantic on his part, nothing slavish on their parts. The result\nwas, that either with him or away from him, the boys combined an ease\nand frankness with a modesty and sense of responsibility that was really\nabove all praise. Alfred went from there to a great school at Wimbledon,\nwhere they train for India and the artillery and engineers. Sydney went\nfrom there to Mr. Barrow, at Southsea. In both instances the new masters\nwrote to me of their own accord, bearing quite unsolicited testimony to\nthe merits of the old, and expressing their high recognition of what\nthey had done. These things speak for themselves.\n\nSydney has just passed his examination as a naval cadet and come home,\nall eyes and gold buttons. He has twelve days' leave before going on\nboard the training-ship. Katie and her husband are in France, and seem\nlikely to remain there for an indefinite period. Mary is on a month's\nvisit in Scotland; Georgina, Frank, and Plorn are at home here; and we\nall want Mary and her little dog back again. I have sold Tavistock\nHouse, am making this rather complete in its way, and am on the restless\neve of beginning a new big book; but mean to have a furnished house in\ntown (in some accessible quarter) from February or so to June. May we\nmeet there.\n\nYour handwriting is always so full of pleasant memories to me, that when\nI took it out of the post-office at Rochester this afternoon it quite\nstirred my heart. But we must not think of old times as sad times, or\nregard them as anything but the fathers and mothers of the present. We\nmust all climb steadily up the mountain after the talking bird, the\nsinging tree, and the yellow water, and must all bear in mind that the\nprevious climbers who were scared into looking back got turned into\nblack stone.\n\nMary Boyle was here a little while ago, as affectionate at heart as\never, as young, and as pleasant. Of course we talked often of you. So\nlet me know when you are established in Halfmoon Street, and I shall be\ntruly delighted to come and see you.\n\nFor my attachments are strong attachments and never weaken. In right of\nbygones, I feel as if \"all Northamptonshire\" belonged to me, as all\nNorthumberland did to Lord Bateman in the ballad. In memory of your\nwarming your feet at the fire in that waste of a waiting-room when I\nread at Brighton, I have ever since taken that watering-place to my\nbosom as I never did before. And you and Switzerland are always one to\nme, and always inseparable.\n\nCharley was heard of yesterday, from Shanghai, going to Japan, intending\nto meet his brother Walter at Calcutta, and having an idea of beguiling\nthe time between whiles by asking to be taken as an amateur with the\nEnglish Chinese forces. Everybody caressed him and asked him everywhere,\nand he seemed to go. With kind regards, my dear Mrs. Watson,\n\n                                            Ever affectionately yours.\n\n\n[Sidenote: Mr. Edmund Yates.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                           _Sunday, Sept. 23rd, 1860._\n\n                       ON THE DEATH OF HIS MOTHER.\n\nMY DEAR E. Y.,\n\nI did not write to you in your bereavement, because I knew that the\ngirls had written to you, and because I instinctively shrunk from making\na form of what was so real. _You_ knew what a loving and faithful\nremembrance I always had of your mother as a part of my youth--no more\ncapable of restoration than my youth itself. All the womanly goodness,\ngrace, and beauty of my drama went out with her. To the last I never\ncould hear her voice without emotion. I think of her as of a beautiful\npart of my own youth, and this dream that we are all dreaming seems to\ndarken.\n\nBut it is not to say this that I write now. It comes to the point of my\npen in spite of me.\n\n\"Holding up the Mirror\" is in next week's number. I have taken out all\nthis funeral part of it. Not because I disliked it (for, indeed, I\nthought it the best part of the paper), but because it rather grated on\nme, going over the proof at that time, as a remembrance that would be\nbetter reserved a little while. Also because it made rather a mixture of\nyourself as an individual, with something that does not belong or attach\nto you as an individual. You can have the MS.; and as a part of a paper\ndescribing your own juvenile remembrances of a theatre, there it is,\nneeding no change or adaption.\n\n                                                      Ever faithfully.\n\n\n[Sidenote: Miss Dickens.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                           _Sunday, Sept. 23rd, 1860._\n\nMY DEAREST MAMIE,\n\nIf you had been away from us and ill with anybody in the world but our\ndear Mrs. White, I should have been in a state of the greatest anxiety\nand uneasiness about you. But as I know it to be impossible that you\ncould be in kinder or better hands, I was not in the least restless\nabout you, otherwise than as it grieved me to hear of my poor dear\ngirl's suffering such pain. I hope it is over now for many a long day,\nand that you will come back to us a thousand times better in health than\nyou left us.\n\nDon't come back too soon. Take time and get well restored. There is no\nhurry, the house is not near to-rights yet, and though we all want you,\nand though Boy wants you, we all (including Boy) deprecate a fatiguing\njourney being taken too soon.\n\nAs to the carpenters, they are absolutely maddening. They are always at\nwork, yet never seem to do anything. Lillie was down on Friday, and said\n(his eye fixed on Maidstone, and rubbing his hand to conciliate his\nmoody employer) that \"he didn't think there would be very much left to\ndo after Saturday, the 29th.\"\n\nI didn't throw him out of the window. Your aunt tells you all the news,\nand leaves me no chance of distinguishing myself, I know. You have been\ntold all about my brackets in the drawing-room, all about the glass\nrescued from the famous stage-wreck of Tavistock House, all about\neverything here and at the office. The office is really a success. As\ncomfortable, cheerful, and private as anything of the kind can possibly\nbe.\n\nI took the Admiral (but this you know too, no doubt) to Dollond's, the\nmathematical instrument maker's, last Monday, to buy that part of his\noutfit. His sextant (which is about the size and shape of a cocked hat),\non being applied to his eye, entirely concealed him. Not the faintest\nvestige of the distinguished officer behind it was perceptible to the\nhuman vision. All through the City, people turned round and stared at\nhim with the sort of pleasure people take in a little model. We went on\nto Chatham this day week, in search of some big man-of-war's-man who\nshould be under obligation to salute him--unfortunately found none. But\nthis no doubt you know too, and all my news falls flat.\n\nI am driven out of my room by paint, and am writing in the best spare\nroom. The whole prospect is excessively wet; it does not rain now, but\nyesterday it did tremendously, and it rained very heavily in the night.\nWe are even muddy; and that is saying a great deal in this dry country\nof chalk and sand. Everywhere the corn is lying out and saturated with\nwet. The hops (nearly everywhere) look as if they had been burnt.\n\nIn my mind's eye I behold Mrs. Bouncer, still with some traces of her\nlate anxiety on her faithful countenance, balancing herself a little\nunequally on her bow fore-legs, pricking up her ears, with her head on\none side, and slightly opening her intellectual nostrils. I send my\nloving and respectful duty to her.\n\nTo dear Mrs. White, and to White, and to Clara, say anything from me\nthat is loving and grateful.\n\n                    My dearest Mamie,\n                          Ever and ever your most affectionate Father.\n\n\n[Sidenote: Miss Hogarth.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                     _Monday Night, Sept. 24th, 1860._\n\nMY DEAREST GEORGY,\n\nAt the Waterloo station we were saluted with \"Hallo! here's Dickens!\"\nfrom divers naval cadets, and Sir Richard Bromley introduced himself to\nme, who had his cadet son with him, a friend of Sydney's. We went down\ntogether, and the boys were in the closest alliance. Bromley being\nAccountant-General of the Navy, and having influence on board, got their\nhammocks changed so that they would be serving side by side, at which\nthey were greatly pleased. The moment we stepped on board, the \"Hul-lo!\nhere's Dickens!\" was repeated on all sides, and the Admiral (evidently\nhighly popular) shook hands with about fifty of his messmates. Taking\nBromley for my model (with whom I fraternised in the most pathetic\nmanner), I gave Sydney a sovereign before stepping over the side. He was\nas little overcome as it was possible for a boy to be, and stood waving\nthe gold-banded cap as we came ashore in a boat.\n\nThere is no denying that he looks very small aboard a great ship, and\nthat a boy must have a strong and decided speciality for the sea to take\nto such a life. Captain Harris was not on board, but the other chief\nofficers were, and were highly obliging. We went over the ship. I should\nsay that there can be little or no individuality of address to any\nparticular boy, but that they all tumble through their education in a\ncrowded way. The Admiral's servant (I mean our Admiral's) had an idiotic\nappearance, but perhaps it did him injustice (a mahogany-faced marine by\nstation). The Admiral's washing apparatus is about the size of a\nmuffin-plate, and he could easily live in his chest. The meeting with\nBromley was a piece of great good fortune, and the dear old chap could\nnot have been left more happily.\n\n                      Ever, my dearest Georgy, your most affectionate.\n\n\n[Sidenote: Miss Power.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                          _Tuesday, Sept. 25th, 1860._\n\nMY DEAR MARGUERITE,\n\nI like the article exceedingly, and think the translations\n_admirable_--spirited, fresh, bold, and evidently faithful. I will get\nthe paper into the next number I make up, No. 78. I will send a proof to\nyou for your correction, either next Monday or this day week. Or would\nyou like to come here next Monday and dine with us at five, and go over\nto Madame Celeste's opening? Then you could correct your paper on the\npremises, as they drink their beer at the beer-shops.\n\nSome of the introductory remarks on French literature I propose to\nstrike out, as a little too essayical for this purpose, and likely to\nthrow out a large portion of the large audience at starting, as\nsuggesting some very different kind of article. My daring pen shall have\nimbued its murderous heart with ink before you see the proof.\n\n                                    With kind regards,\n                                                  Ever affectionately.\n\n\n[Sidenote: Mr. John Forster.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                           _Thursday, Oct. 4th, 1860._\n\nMY DEAR FORSTER,\n\nIt would be a great pleasure to me to come to you, an immense pleasure,\nand to sniff the sea I love (from the shore); but I fear I must come\ndown one morning and come back at night. I will tell you why.\n\nLast week, I got to work on a new story. I called a council of war at\nthe office on Tuesday. It was perfectly clear that the one thing to be\ndone was, for me to strike in. I have therefore decided to begin a\nstory, the length of the \"Tale of Two Cities,\" on the 1st of\nDecember--begin publishing, that is. I must make the most I can out of\nthe book. When I come down, I will bring you the first two or three\nweekly parts. The name is, \"GREAT EXPECTATIONS.\" I think a good name?\n\nNow the preparations to get ahead, combined with the absolute necessity\nof my giving a good deal of time to the Christmas number, will tie me to\nthe grindstone pretty tightly. It will be just as much as I can hope to\ndo. Therefore, what I had hoped would be a few days at Eastbourne\ndiminish to a few hours.\n\nI took the Admiral down to Portsmouth. Every maritime person in the town\nknew him. He seemed to know every boy on board the _Britannia_, and was\na tremendous favourite evidently. It was very characteristic of him that\nthey good-naturedly helped him, he being so very small, into his hammock\nat night. But he couldn't rest in it on these terms, and got out again\nto learn the right way of getting in independently. Official report\nstated that \"after a few spills, he succeeded perfectly, and went to\nsleep.\" He is perfectly happy on board, takes tea with the captain,\nleads choruses on Saturday nights, and has an immense marine for a\nservant.\n\nI saw Edmund Yates at the office, and he told me that during all his\nmother's wanderings of mind, which were almost incessant at last, she\nnever once went back to the old Adelphi days until she was just dying,\nwhen he heard her say, in great perplexity: \"I can _not_ get the words.\"\n\nBest love to Mrs. Forster.\n\n                                Ever, my dear Forster, affectionately.\n\n\n[Sidenote: Mr. W. Wilkie Collins.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                         _Wednesday, Oct. 24th, 1860._\n\nMY DEAR WILKIE,\n\nI have been down to Brighton to see Forster, and found your letter there\non arriving by express this morning. I also found a letter from\nGeorgina, describing that Mary's horse went down suddenly on a stone,\nand how Mary was thrown, and had her riding-habit torn to pieces, and\nhas a deep cut just above the knee--fortunately not in the knee itself,\nwhich is doing exceedingly well, but which will probably incapacitate\nher from walking for days and days to come. It is well it was no worse.\nThe accident occurred at Milton, near Gravesend, and they found Mary in\na public-house there, wonderfully taken care of and looked after.\n\nI propose that we start on Thursday morning, the 1st of November. The\ntrain for Penzance leaves the Great Western terminus at a quarter-past\nnine in the morning. It is a twelve hours' journey. Shall we meet at the\nterminus at nine? I shall be here all the previous day, and shall dine\nhere.\n\nYour account of your passage goes to my heart through my stomach. What a\npity I was not there on board to present that green-visaged, but\nsweet-tempered and uncomplaining spectacle of imbecility, at which I am\nso expert under stormy circumstances, in the poet's phrase:\n\n        As I sweep\n        Through the deep,\n        When the stormy winds do blow.\n\nWhat a pity I am not there, at Meurice's, to sleep the sleep of infancy\nthrough the long plays where the gentlemen stand with their backs to the\nmantelpieces. What a pity I am not with you to make a third at the Trois\nFreres, and drink no end of bottles of Bordeaux, without ever getting a\ntouch of redness in my (poet's phrase again) \"innocent nose.\" But I must\ngo down to Gad's to-night, and get to work again. Four weekly numbers\nhave been ground off the wheel, and at least another must be turned\nbefore we meet. They shall be yours in the slumberous railway-carriage.\n\nI don't think Forster is at all in good health. He was tremendously\nhospitable and hearty. I walked six hours and a half on the downs\nyesterday, and never stopped or sat. Early in the morning, before\nbreakfast, I went to the nearest baths to get a shower-bath. They kept\nme waiting longer than I thought reasonable, and seeing a man in a cap\nin the passage, I went to him and said: \"I really must request that\nyou'll be good enough to see about this shower-bath;\" and it was Hullah!\nwaiting for another bath.\n\nRumours were brought into the house on Saturday night, that there was a\n\"ghost\" up at Larkins's monument. Plorn was frightened to death, and I\nwas apprehensive of the ghost's spreading and coming there, and causing\n\"warning\" and desertion among the servants. Frank was at home, and\nAndrew Gordon was with us. Time, nine o'clock. Village talk and\ncredulity, amazing. I armed the two boys with a short stick apiece, and\nshouldered my double-barrelled gun, well loaded with shot. \"Now\nobserve,\" says I to the domestics, \"if anybody is playing tricks and has\ngot a head, I'll blow it off.\" Immense impression. New groom evidently\nconvinced that he has entered the service of a bloodthirsty demon. We\nascend to the monument. Stop at the gate. Moon is rising. Heavy shadows.\n\"Now, look out!\" (from the bloodthirsty demon, in a loud, distinct\nvoice). \"If the ghost is here and I see him, so help me God I'll fire at\nhim!\" Suddenly, as we enter the field, a most extraordinary noise\nresponds--terrific noise--human noise--and yet superhuman noise. B. T.\nD. brings piece to shoulder. \"Did you hear that, pa?\" says Frank. \"I\ndid,\" says I. Noise repeated--portentous, derisive, dull, dismal,\ndamnable. We advance towards the sound. Something white comes lumbering\nthrough the darkness. An asthmatic sheep! Dead, as I judge, by this\ntime. Leaving Frank to guard him, I took Andrew with me, and went all\nround the monument, and down into the ditch, and examined the field\nwell, thinking it likely that somebody might be taking advantage of the\nsheep to frighten the village. Drama ends with discovery of no one, and\ntriumphant return to rum-and-water.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Miss Hogarth.]\n\n              BIDEFORD, NORTH DEVON, _Thursday Night, Nov. 1st, 1860._\n\nMY DEAREST GEORGY,\n\nI write (with the most impracticable iron pen on earth) to report our\nsafe arrival here, in a beastly hotel. We start to-morrow morning at\nnine on a two days' posting between this and Liskeard in Cornwall. We\nare due in Liskeard (but nobody seems to know anything about the roads)\non Saturday afternoon, and we purpose making an excursion in that\nneighbourhood on Sunday, and coming up from Liskeard on Monday by Great\nWestern fast train, which will get us to London, please God, in good\ntime on Monday evening. There I shall hear from you, and know whether\ndear Mamie will move to London too.\n\nWe had a pleasant journey down here, and a beautiful day. No adventures\nwhatever. Nothing has happened to Wilkie, and he sends love.\n\nWe had stinking fish for dinner, and have been able to drink nothing,\nthough we have ordered wine, beer, and brandy-and-water. There is\nnothing in the house but two tarts and a pair of snuffers. The landlady\nis playing cribbage with the landlord in the next room (behind a thin\npartition), and they seem quite comfortable.\n\n                      Ever, my dearest Georgy, your most affectionate.\n\n\n[Sidenote: Miss Mary Boyle.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                            _Friday, Dec. 28th, 1860._\n\nMY DEAR MARY,\n\nI cannot tell you how much I thank you for the beautiful cigar-case, and\nhow seasonable, and friendly, and good, and warm-hearted it looked when\nI opened it at Gad's Hill. Besides which, it is a cigar-case, and will\nhold cigars; two crowning merits that I never yet knew to be possessed\nby any article claiming the same name. For all of these reasons, but\nmore than all because it comes from you, I love it, and send you\neighteen hundred and sixty kisses, with one in for the new year.\n\nBoth excellent stories and perfectly new. Your Joe swears that he never\nheard either--never a word or syllable of either--after he laughed at\n'em this blessed day.\n\nI have no news, except that I am not quite well, and am being doctored.\nPray read \"Great Expectations.\" I think it is very droll. It is a very\ngreat success, and seems universally liked. I suppose because it opens\nfunnily, and with an interest too.\n\nI pass my time here (I am staying here alone) in working, taking physic,\nand taking a stall at a theatre every night. On Boxing Night I was at\nCovent Garden. A dull pantomime was \"worked\" (as we say) better than I\never saw a heavy piece worked on a first night, until suddenly and\nwithout a moment's warning, every scene on that immense stage fell over\non its face, and disclosed chaos by gaslight behind! There never was\nsuch a business; about sixty people who were on the stage being\nextinguished in the most remarkable manner. Not a soul was hurt. In the\nuproar, some moon-calf rescued a porter pot, six feet high (out of which\nthe clown had been drinking when the accident happened), and stood it on\nthe cushion of the lowest proscenium box, P.S., beside a lady and\ngentleman, who were dreadfully ashamed of it. The moment the house knew\nthat nobody was injured, they directed their whole attention to this\ngigantic porter pot in its genteel position (the lady and gentleman\ntrying to hide behind it), and roared with laughter. When a modest\nfootman came from behind the curtain to clear it, and took it up in his\narms like a Brobdingnagian baby, we all laughed more than ever we had\nlaughed in our lives. I don't know why.\n\nWe have had a fire here, but our people put it out before the\nparish-engine arrived, like a drivelling perambulator, with _the beadle\nin it_, like an imbecile baby. Popular opinion, disappointed in the fire\nhaving been put out, snowballed the beadle. God bless it!\n\nOver the way at the Lyceum, there is a very fair Christmas piece, with\none or two uncommonly well-done <DW65> songs--one remarkably gay and\nmad, done in the finale to a scene. Also a very nice transformation,\nthough I don't know what it means.\n\nThe poor actors waylay me in Bow Street, to represent their necessities;\nand I often see one cut down a court when he beholds me coming, cut\nround Drury Lane to face me, and come up towards me near this door in\nthe freshest and most accidental way, as if I was the last person he\nexpected to see on the surface of this globe. The other day, there thus\nappeared before me (simultaneously with a scent of rum in the air) one\naged and greasy man, with a pair of pumps under his arm. He said he\nthought if he could get down to somewhere (I think it was Newcastle), he\nwould get \"taken on\" as Pantaloon, the existing Pantaloon being \"a\nstick, sir--a mere muff.\" I observed that I was sorry times were so bad\nwith him. \"Mr. Dickens, you know our profession, sir--no one knows it\nbetter, sir--there is no right feeling in it. I was Harlequin on your\nown circuit, sir, for five-and-thirty years, and was displaced by a boy,\nsir!--a boy!\"\n\nSo no more at present, except love to Mrs. Watson and Bedgey Prig and\nall, from my dear Mary.\n\n                                              Your ever affectionate\n                                                                  JOE.\n\nP.S.--DON'T I pine neither?\n\nP.P.S.--I did my best to arouse Forster's worst feelings; but he had got\ninto a Christmas habit of mind, and wouldn't respond.\n\nFOOTNOTES:\n\n[6] With whom Mr. and Mrs. Wills were staying at Aberystwith.\n\n\n\n\nNARRATIVE.\n\n1861.\n\n\nThis, as far as his movements were concerned, was again a very unsettled\nyear with Charles Dickens. He hired a furnished house in the Regent's\nPark, which he, with his household, occupied for some months. During the\nseason he gave several readings at St. James's Hall. After a short\nsummer holiday at Gad's Hill, he started, in the autumn, on a reading\ntour in the English provinces. Mr. Arthur Smith, being seriously ill,\ncould not accompany him in this tour; and Mr. Headland, who was formerly\nin office at the St. Martin's Hall, was engaged as business-manager of\nthese readings. Mr. Arthur Smith died in October, and Charles Dickens's\ndistress at the loss of this loved friend and companion is touchingly\nexpressed in many of his letters of this year.\n\nThere are also sorrowful allusions to the death of his brother-in-law,\nMr. Henry Austin, which sad event likewise happened in October. And the\nletter we give to Mrs. Austin (\"Letitia\") has reference to her sad\naffliction.\n\nIn June of this year he paid a short visit to Sir E. B. Lytton at\nKnebworth, accompanied by his daughter and sister-in-law, who also\nduring his autumn tour joined him in Edinburgh. But this course of\nreadings was brought rather suddenly to an end on account of the death\nof the Prince Consort.\n\nBesides being constantly occupied with the business of these readings,\nCharles Dickens was still at work on his story of \"Great Expectations,\"\nwhich was appearing weekly in \"All the Year Round.\" The story closed on\nthe 3rd of August, when it was published as a whole in three volumes,\nand inscribed to Mr. Chauncey Hare Townshend. The Christmas number of\n\"All the Year Round\" was called \"Tom Tiddler's Ground,\" to which Charles\nDickens contributed three stories.\n\nOur second letter in this year is given more as a specimen of the claims\nwhich were constantly being made upon Charles Dickens's time and\npatience, than because we consider the letter itself to contain much\npublic interest; excepting, indeed, as showing his always considerate\nand courteous replies to such constant applications.\n\n\"The fire\" mentioned in the letter to Mr. Forster was the great fire in\nTooley Street. The \"Morgan\" was an American sea-captain, well known in\nthose days, and greatly liked and respected. It may interest our readers\nto know that the character of Captain Jorgan, in the Christmas number of\nthe previous year, was suggested by this pleasant sailor, for whom\nCharles Dickens had a hearty liking. Young Mr. Morgan was, during the\nyears he passed in England, a constant visitor at Gad's Hill. The\n\"Elwin\" mentioned in the letter written from Bury St. Edmunds, was the\nRev. Whitwell Elwin, a Norfolk gentleman, well known in the literary\nworld, and who was for many years editor of \"The Quarterly Review.\"\n\nThe explanation of the letter to Mr. John Agate, of Dover, we give in\nthat gentleman's own words:\n\n\"There are few public men with the strain upon their time and energies\nwhich he had particularly (and which I know better now that I have read\nhis life), who would have spared the time to have written such a long\ncourteous letter.\n\n\"I wrote to him rather in anger, and left the letter myself at The Lord\nWarden, as I and my family were very much disappointed, after having\npurchased our tickets so long before, to find we could not got into the\nroom, as money was being received, but his kind letter explained all.\"\n\n\n[Sidenote: Miss Hogarth.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                          _Wednesday, Jan. 9th, 1861._\n\nMY DEAREST GEORGY,\n\n\"We\" are in the full swing of stopping managers from playing \"A Message\nfrom the Sea.\" I privately doubt the strength of our position in the\nCourt of Chancery, if we try it; but it is worth trying.\n\nI am aware that Mr. Lane of the Britannia sent an emissary to Gad's Hill\nyesterday. It unfortunately happens that the first man \"we\" have to\nassert the principle against is a very good man, whom I really respect.\n\nI have no news, except that I really hope and believe I am gradually\ngetting well. If I have no check, I hope to be soon discharged by the\nmedico.\n\n                                                  Ever affectionately.\n\nP.S.--Best love to Mamie, also to the boys and Miss Craufurd.\n\n\n    OFFICE OF \"ALL THE YEAR ROUND,\" 26, WELLINGTON STREET, W.C.,\n                                    _Tuesday Evening, Jan. 9th, 1861._\n\nDEAR SIR,\n\nI feel it quite hopeless to endeavour to present my position before you,\nin reference to such a letter as yours, in its plain and true light.\nWhen you suppose it would have cost Mr. Thackeray \"but a word\" to use\nhis influence to obtain you some curatorship or the like, you fill me\nwith the sense of impossibility of leading you to a more charitable\njudgment of Mr. Dickens.\n\nNevertheless, I will put the truth before you. Scarcely a day of my life\npasses, or has passed for many years, without bringing me some letters\nsimilar to yours. Often they will come by dozens--scores--hundreds. My\ntime and attention would be pretty well occupied without them, and the\nclaims upon me (some very near home), for all the influence and means of\nhelp that I do and do not possess, are not commonly heavy. I have no\npower to aid you towards the attainment of your object. It is the simple\nexact truth, and nothing can alter it. So great is the disquietude I\nconstantly undergo from having to write to some new correspondent in\nthis strain, that, God knows, I would resort to another relief if I\ncould.\n\nYour studies from nature appear to me to express an excellent\nobservation of nature, in a loving and healthy spirit. But what then?\nThe dealers and dealers' prices of which you complain will not be\ninfluenced by that honest opinion. Nor will it have the least effect\nupon the President of the Royal Academy, or the Directors of the School\nof Design. Assuming your supposition to be correct that these\nauthorities are adverse to you, I have no more power than you have to\nrender them favourable. And assuming them to be quite disinterested and\ndispassionate towards you, I have no voice or weight in any appointment\nthat any of them make.\n\nI will retain your packet over to-morrow, and will then cause it to be\nsent to your house. I write under the pressure of occupation and\nbusiness, and therefore write briefly.\n\n                                                     Faithfully yours.\n\n\n[Sidenote: M. de Cerjat.]\n\n             OFFICE OF \"ALL THE YEAR ROUND,\" _Friday, Feb. 1st, 1861._\n\nMY DEAR CERJAT,\n\nYou have read in the papers of our heavy English frost. At Gad's Hill it\nwas so intensely cold, that in our warm dining-room on Christmas Day we\ncould hardly sit at the table. In my study on that morning, long after a\ngreat fire of coal and wood had been lighted, the thermometer was I\ndon't know where below freezing. The bath froze, and all the pipes\nfroze, and remained in a stony state for five or six weeks. The water in\nthe bedroom-jugs froze, and blew up the crockery. The snow on the top of\nthe house froze, and was imperfectly removed with axes. My beard froze\nas I walked about, and I couldn't detach my cravat and coat from it\nuntil I was thawed at the fire. My boys and half the officers stationed\nat Chatham skated away without a check to Gravesend--five miles off--and\nrepeated the performance for three or four weeks. At last the thaw came,\nand then everything split, blew up, dripped, poured, perspired, and got\nspoilt. Since then we have had a small visitation of the plague of\nservants; the cook (in a riding-habit) and the groom (in a dress-coat\nand jewels) having mounted Mary's horse and mine, in our absence, and\nscoured the neighbouring country at a rattling pace. And when I went\nhome last Saturday, I innocently wondered how the horses came to be out\nof condition, and gravely consulted the said groom on the subject, who\ngave it as his opinion \"which they wanted reg'lar work.\" We are now\ncoming to town until midsummer. Having sold my own house, to be more\nfree and independent, I have taken a very pretty furnished house, No. 3,\nHanover Terrace, Regent's Park. This, of course, on my daughter's\naccount. For I have very good and cheerful bachelor rooms here, with an\nold servant in charge, who is the cleverest man of his kind in the\nworld, and can do anything, from excellent carpentery to excellent\ncookery, and has been with me three-and-twenty years.\n\nThe American business is the greatest English sensation at present. I\nventure to predict that the struggle of violence will be a very short\none, and will be soon succeeded by some new compact between the Northern\nand Southern States. Meantime the Lancashire mill-owners are getting\nvery uneasy.\n\nThe Italian state of things is not regarded as looking very cheerful.\nWhat from one's natural sympathies with a people so oppressed as the\nItalians, and one's natural antagonism to a pope and a Bourbon (both of\nwhich superstitions I do suppose the world to have had more than enough\nof), I agree with you concerning Victor Emmanuel, and greatly fear that\nthe Southern Italians are much degraded. Still, an united Italy would be\nof vast importance to the peace of the world, and would be a rock in\nLouis Napoleon's way, as he very well knows. Therefore the idea must be\nchampioned, however much against hope.\n\nMy eldest boy, just home from China, was descried by Townshend's Henri\nthe moment he landed at Marseilles, and was by him borne in triumph to\nTownshend's rooms. The weather was snowy, slushy, beastly; and\nMarseilles was, as it usually is to my thinking, well-nigh intolerable.\nMy boy could not stay with Townshend, as he was coming on by express\ntrain; but he says: \"I sat with him and saw him dine. He had a leg of\nlamb, and a tremendous cold.\" That is the whole description I have been\nable to extract from him.\n\nThis journal is doing gloriously, and \"Great Expectations\" is a great\nsuccess. I have taken my third boy, Frank (Jeffrey's godson), into this\noffice. If I am not mistaken, he has a natural literary taste and\ncapacity, and may do very well with a chance so congenial to his mind,\nand being also entered at the Bar.\n\nDear me, when I have to show you about London, and we dine _en garcon_\nat odd places, I shall scarcely know where to begin. Only yesterday I\nwalked out from here in the afternoon, and thought I would go down by\nthe Houses of Parliament. When I got there, the day was so beautifully\nbright and warm, that I thought I would walk on by Millbank, to see the\nriver. I walked straight on _for three miles_ on a splendid broad\nesplanade overhanging the Thames, with immense factories, railway works,\nand what-not erected on it, and with the strangest beginnings and ends\nof wealthy streets pushing themselves into the very Thames. When I was a\nrower on that river, it was all broken ground and ditch, with here and\nthere a public-house or two, an old mill, and a tall chimney. I had\nnever seen it in any state of transition, though I suppose myself to\nknow this rather large city as well as anyone in it.\n\n       *       *       *       *       *\n\n\n[Sidenote: Mr. E. M. Ward, R.A.]\n\n                              3, HANOVER TERRACE, REGENT'S PARK,\n                                    _Saturday Night, March 9th, 1861._\n\nMY DEAR WARD,\n\nI cannot tell you how gratified I have been by your letter, and what a\nsplendid recompense it is for any pleasure I am giving you. Such\ngenerous and earnest sympathy from such a brother-artist gives me true\ndelight. I am proud of it, believe me, and moved by it to do all the\nbetter.\n\n                                                Ever faithfully yours.\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n              \"ALL THE YEAR ROUND\" OFFICE, _Tuesday, June 11th, 1861._\n\nMY DEAREST MACREADY,\n\nThere is little doubt, I think, of my reading at Cheltenham somewhere\nabout November. I submit myself so entirely to Arthur Smith's\narrangements for me, that I express my sentiments on this head with\nmodesty. But I think there is scarcely a doubt of my seeing you then.\n\nI have just finished my book of \"Great Expectations,\" and am the worse\nfor wear. Neuralgic pains in the face have troubled me a good deal, and\nthe work has been pretty close. But I hope that the book is a good book,\nand I have no doubt of very soon throwing off the little damage it has\ndone me.\n\nWhat with Blondin at the Crystal Palace and Leotard at Leicester Square,\nwe seem to be going back to barbaric excitements. I have not seen, and\ndon't intend to see, the Hero of Niagara (as the posters call him), but\nI have been beguiled into seeing Leotard, and it is at once the most\nfearful and most graceful thing I have ever seen done.\n\nClara White (grown pretty) has been staying with us.\n\nI am sore afraid that _The Times_, by playing fast and loose with the\nAmerican question, has very seriously compromised this country. The\nAmericans northward are perfectly furious on the subject; and Motley the\nhistorian (a very sensible man, strongly English in his sympathies)\nassured me the other day that he thought the harm done very serious\nindeed, and the dangerous nature of the daily widening breach scarcely\ncalculable.\n\nKindest and best love to all. Wilkie Collins has just come in, and sends\nbest regard.\n\n                        Ever most affectionately, my dearest Macready.\n\n\n[Sidenote: Mr. John Forster.]\n\n                                 GAD'S HILL, _Monday, July 1st, 1861._\n\nMY DEAR FORSTER,\n\n       *       *       *       *       *\n\nYou will be surprised to hear that I have changed the end of \"Great\nExpectations\" from and after Pip's return to Joe's, and finding his\nlittle likeness there.\n\nBulwer (who has been, as I think I told you, extraordinarily taken by\nthe book), so strongly urged it upon me, after reading the proofs, and\nsupported his views with such good reasons, that I resolved to make the\nchange. You shall have it when you come back to town. I have put in a\nvery pretty piece of writing, and I have no doubt the story will be more\nacceptable through the alteration.\n\nI have not seen Bulwer's changed story. I brought back the first month\nwith me, and I know the nature of his changes throughout; but I have not\nyet had the revised proofs. He was in a better state at Knebworth than I\nhave ever seen him in all these years, a little weird occasionally\nregarding magic and spirits, but perfectly fair and frank under\nopposition. He was talkative, anecdotical, and droll; looked young and\nwell, laughed heartily, and enjoyed some games we played with great\nzest. In his artist character and talk he was full of interest and\nmatter, but that he always is. Socially, he seemed to me almost a new\nman. I thoroughly enjoyed myself, and so did Georgina and Mary.\n\nThe fire I did not see until the Monday morning, but it was blazing\nfiercely then, and was blazing hardly less furiously when I came down\nhere again last Friday. I was here on the night of its breaking out. If\nI had been in London I should have been on the scene, pretty surely.\n\nYou will be perhaps surprised to hear that it is Morgan's conviction\n(his son was here yesterday), that the North will put down the South,\nand that speedily. In his management of his large business, he is\nproceeding steadily on that conviction. He says that the South has no\nmoney and no credit, and that it is impossible for it to make a\nsuccessful stand. He may be all wrong, but he is certainly a very shrewd\nman, and he has never been, as to the United States, an enthusiast of\nany class.\n\nPoor Lord Campbell's seems to me as easy and good a death as one could\ndesire. There must be a sweep of these men very soon, and one feels as\nif it must fall out like the breaking of an arch--one stone goes from a\nprominent place, and then the rest begin to drop. So one looks towards\nBrougham, and Lyndhurst, and Pollock.\n\nI will add no more to this, or I know I shall not send it; for I am in\nthe first desperate laziness of having done my book, and think of\noffering myself to the village school as a live example of that vice for\nthe edification of youth.\n\n                                Ever, my dear Forster, affectionately.\n\n\n[Sidenote: The Hon. Mrs. Watson.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                             _Monday, July 8th, 1861._\n\nMY DEAR MRS. WATSON,\n\nI have owed you a letter for so long a time that I fear you may\nsometimes have misconstrued my silence. But I hope that the sight of the\nhandwriting of your old friend will undeceive you, if you have, and will\nput that right.\n\nDuring the progress of my last story, I have been working so hard that\nvery, very little correspondence--except enforced correspondence on\nbusiness--has passed this pen. And now that I am free again, I devote a\nfew of my first leisure moments to this note.\n\nYou seemed in your last to think that I had forgotten you in respect of\nthe Christmas number. Not so at all. I discussed with them here where\nyou were, how you were to be addressed, and the like; finally left the\nnumber in a blank envelope, and did not add the address to it until it\nwould have been absurd to send you such stale bread. This was my fault,\nbut this was all. And I should be so pained at heart if you supposed me\ncapable of failing in my truth and cordiality, or in the warm\nremembrance of the time we have passed together, that perhaps I make\nmore of it than you meant to do.\n\nMy sailor-boy is at home--I was going to write, for the holidays, but I\nsuppose I must substitute \"on leave.\" Under the new regulations, he must\nnot pass out of the _Britannia_ before December. The younger boys are\nall at school, and coming home this week for the holidays. Mary keeps\nhouse, of course, and Katie and her husband surprised us yesterday, and\nare here now. Charley is holiday-making at Guernsey and Jersey. He has\nbeen for some time seeking a partnership in business, and has not yet\nfound one. The matter is in the hands of Mr. Bates, the managing partner\nin Barings' house, and seems as slow a matter to adjust itself as ever I\nlooked on at. Georgina is, as usual, the general friend and confidante\nand factotum of the whole party.\n\nYour present correspondent read at St. James's Hall in the beginning of\nthe season, to perfectly astounding audiences; but finding that fatigue\nand excitement very difficult to manage in conjunction with a story,\ndeemed it prudent to leave off reading in high tide and mid-career, the\nrather by reason of something like neuralgia in the face. At the end of\nOctober I begin again; and if you are at Brighton in November, I shall\ntry to see you there. I deliver myself up to Mr. Arthur Smith, and I\nknow it is one of the places for which he has put me down.\n\nThis is all about me and mine, and next I want to know why you never\ncome to Gad's Hill, and whether you are never coming. The stress I lay\non these questions you will infer from the size of the following note of\ninterrogation[HW: =?=]\n\nI am in the constant receipt of news from Lausanne. Of Mary Boyle, I\ndaresay you have seen and heard more than I have lately. Rumours\noccasionally reach me of her acting in every English shire incessantly,\nand getting in a harvest of laurels all the year round. Cavendish I have\nnot seen for a long time, but when I did see him last, it was at\nTavistock House, and we dined together jovially. Mention of that\nlocality reminds me that when you DO come here, you will see the\npictures looking wonderfully better, and more precious than they ever\ndid in town. Brought together in country light and air, they really are\nquite a baby collection and very pretty.\n\nI direct this to Rockingham, supposing you to be there in this summer\ntime. If you are as leafy in Northamptonshire as we are in Kent, you are\ngreener than you have been for some years. I hope you may have seen a\nlarge-headed photograph with little legs, representing the undersigned,\npen in hand, tapping his forehead to knock an idea out. It has just\nsprung up so abundantly in all the shops, that I am ashamed to go about\ntown looking in at the picture-windows, which is my delight. It seems to\nme extraordinarily ludicrous, and much more like than the grave portrait\ndone in earnest. It made me laugh when I first came upon it, until I\nshook again, in open sunlighted Piccadilly.\n\nPray be a good Christian to me, and don't be retributive in measuring\nout the time that shall pass before you write to me. And believe me\never,\n\n                                       Your affectionate and faithful.\n\n\n[Sidenote: Mr. W. Wilkie Collins.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                         _Wednesday, Aug. 28th, 1861._\n\nMY DEAR WILKIE,\n\nI have been going to write to you ever since I received your letter from\nWhitby, and now I hear from Charley that you are coming home, and must\nbe addressed in the Rue Harley. Let me know whether you will dine here\nthis day week at the usual five. I am at present so addle-headed (having\nhard Wednesday work in Wills's absence) that I can't write much.\n\nI have got the \"Copperfield\" reading ready for delivery, and am now\ngoing to blaze away at \"Nickleby,\" which I don't like half as well.\nEvery morning I \"go in\" at these marks for two or three hours, and then\ncollapse and do nothing whatever (counting as nothing much cricket and\nrounders).\n\nIn my time that curious railroad by the Whitby Moor was so much the more\ncurious, that you were balanced against a counter-weight of water, and\nthat you did it like Blondin. But in these remote days the one inn of\nWhitby was up a back-yard, and oyster-shell grottoes were the only view\nfrom the best private room. Likewise, sir, I have posted to Whitby.\n\"Pity the sorrows of a poor old man.\"\n\nThe sun is glaring in at these windows with an amount of ferocity\ninsupportable by one of the landed interest, who lies upon his back with\nan imbecile hold on grass, from lunch to dinner. Feebleness of mind and\nhead are the result.\n\n                                                  Ever affectionately.\n\nP.S.--The boys have multiplied themselves by fifty daily, and have\nseemed to appear in hosts (especially in the hottest days) round all the\ncorners at Gad's Hill. I call them the prowlers, and each has a\ndistinguishing name attached, derived from his style of prowling.\n\n\n[Sidenote: Mr. Arthur Smith.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                           _Tuesday, Sept. 3rd, 1861._\n\nMY DEAR ARTHUR,\n\nI cannot tell you how sorry I am to receive your bad account of your\nhealth, or how anxious I shall be to receive a better one as soon as you\ncan possibly give it.\n\nIf you go away, don't you think in the main you would be better here\nthan anywhere? You know how well you would be nursed, what care we\nshould take of you, and how perfectly quiet and at home you would be,\nuntil you become strong enough to take to the Medway. Moreover, I think\nyou would be less anxious about the tour, here, than away from such\nassociation. I would come to Worthing to fetch you, I needn't say, and\nwould take the most careful charge of you. I will write no more about\nthis, because I wish to avoid giving you more to read than can be\nhelped; but I do sincerely believe it would be at once your wisest and\nleast anxious course. As to a long journey into Wales, or any long\njourney, it would never do. Nice is not to be thought of. Its dust, and\nits sharp winds (I know it well), towards October are very bad indeed.\n\nI send you the enclosed letters, firstly, because I have no circular to\nanswer them with, and, secondly, because I fear I might confuse your\narrangements by interfering with the correspondence. I shall hope to\nhave a word from you very soon. I am at work for the tour every day,\nexcept my town Wednesdays.\n\n                                                      Ever faithfully.\n\nP.S.--Kindest regards from all.\n\n\n[Sidenote: Mr. John Watkins.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                   _Saturday Night, Sept. 28th, 1861._\n\nDEAR MR. WATKINS,\n\nIn reply to your kind letter I must explain that I have not yet brought\ndown any of your large photographs of myself, and therefore cannot\nreport upon their effect here. I think the \"cartes\" are all liked.\n\nA general howl of horror greeted the appearance of No. 18, and a riotous\nattempt was made to throw it out of window. I calmed the popular fury\nby promising that it should never again be beheld within these walls. I\nthink I mentioned to you when you showed it to me, that I felt persuaded\nit would not be liked. It has a grim and wasted aspect, and perhaps\nmight be made useful as a portrait of the Ancient Mariner.\n\nI feel that I owe you an apology for being (innocently) a difficult\nsubject. When I once excused myself to Ary Scheffer while sitting to\nhim, he received the apology as strictly his due, and said with a vexed\nair: \"At this moment, _mon cher_ Dickens, you look more like an\nenergetic Dutch admiral than anything else;\" for which I apologised\nagain.\n\nIn the hope that the pains you have bestowed upon me will not be thrown\naway, but that your success will prove of some use to you, believe me,\n\n                                                     Faithfully yours.\n\n\n[Sidenote: Mr. Edmund Yates.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                             _Sunday, Oct. 6th, 1861._\n\n              AFTER THE DEATH OF MR. ARTHUR SMITH.\n\nMY DEAR EDMUND,\n\nComing back here to-day, I find your letter.\n\nI was so very much distressed last night in thinking of it all, and I\nfind it so very difficult to preserve my composure when I dwell in my\nmind on the many times fast approaching when I shall sorely miss the\nfamiliar face, that I am hardly steady enough yet to refer to the\nreadings like a man. But your kind reference to them makes me desirous\nto tell you that I took Headland (formerly of St. Martin's Hall, who has\nalways been with us in London) to conduct the business, when I knew that\nour poor dear fellow could never do it, even if he had recovered\nstrength to go; and that I consulted with himself about it when I saw\nhim for the last time on earth, and that it seemed to please him, and he\nsaid: \"We couldn't do better.\"\n\nWrite to me before you come; and remember that I go to town Wednesday\nmornings.\n\n                                                      Ever faithfully.\n\n\n[Sidenote: Miss Dickens.]\n\n                             OFFICE OF \"ALL THE YEAR ROUND,\"\n                                          _Thursday, Oct. 10th, 1861._\n\nMY DEAREST MAMIE,\n\nI received your affectionate little letter here this morning, and was\nvery glad to get it. Poor dear Arthur is a sad loss to me, and indeed I\nwas very fond of him. But the readings must be fought out, like all the\nrest of life.\n\n                                               Ever your affectionate.\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                            _Sunday, Oct. 13th, 1861._\n\nMY DEAREST MACREADY,\n\nThis is a short note. But the moment I know for certain what is designed\nfor me at Cheltenham, I write to you in order that you may know it from\nme and not by chance from anyone else.\n\nI am to read there on the evening of Friday, the 3rd of January, and on\nthe morning of Saturday, the 4th; as I have nothing to do on Thursday,\nthe 2nd, but come from Leamington, I shall come to you, please God, for\na quiet dinner that day.\n\nThe death of Arthur Smith has caused me great distress and anxiety. I\nhad a great regard for him, and he made the reading part of my life as\nlight and pleasant as it _could_ be made. I had hoped to bring him to\nsee you, and had pictured to myself how amused and interested you would\nhave been with his wonderful tact and consummate mastery of arrangement.\nBut it's all over.\n\nI begin at Norwich on the 28th, and am going north in the middle of\nNovember. I am going to do \"Copperfield,\" and shall be curious to test\nits effect on the Edinburgh people. It has been quite a job so to piece\nportions of the long book together as to make something continuous out\nof it; but I hope I have got something varied and dramatic. I am also\n(not to slight _your_ book) going to do \"Nickleby at Mr. Squeers's.\" It\nis clear that both must be trotted out at Cheltenham.\n\nWith kindest love and regard to all your house,\n\n                    Ever, my dearest Macready, your most affectionate.\n\nP.S.--Fourth edition of \"Great Expectations\" almost gone!\n\n\n[Sidenote: Miss Hogarth.]\n\n                                  ANGEL HOTEL, BURY ST. EDMUNDS,\n                                         _Wednesday, Oct. 13th, 1861._\n\nMY DEAREST GEORGY,\n\nI have just now received your welcome letter, and I hasten to report\n(having very little time) that we had a splendid hall last night, and\nthat I think \"Nickleby\" tops all the readings. Somehow it seems to have\ngot in it, by accident, exactly the qualities best suited to the\npurpose, and it went last night not only with roars, but with a general\nhilarity and pleasure that I have never seen surpassed.\n\nWe are full here for to-night.\n\nFancy this: last night at about six, who should walk in but Elwin! He\nwas exactly in his usual state, only more demonstrative than ever, and\nhad been driven in by some neighbours who were coming to the reading. I\nhad tea up for him, and he went down at seven with me to the dismal den\nwhere I dressed, and sat by the fire while I dressed, and was childishly\nhappy in that great privilege! During the reading he sat on a corner of\nthe platform and roared incessantly. He brought in a lady and gentleman\nto introduce while I was undressing, and went away in a perfect and\nabsolute rapture.\n\n\n[Sidenote: Miss Hogarth.]\n\n                     ROYAL HOTEL, NORWICH, _Tuesday, Oct. 29th, 1861._\n\nI cannot say that we began well last night. We had not a good hall, and\nthey were a very lumpish audience indeed. This did not tend to cheer the\nstrangeness I felt in being without Arthur, and I was not at all myself.\nWe have a large let for to-night, I think two hundred and fifty stalls,\nwhich is very large, and I hope that both they and I will go better. I\ncould have done perfectly last night, if the audience had been bright,\nbut they were an intent and staring audience. They laughed though very\nwell, and the storm made them shake themselves again. But they were not\nmagnetic, and the great big place was out of sorts somehow.\n\nTo-morrow I will write you another short note, however short. It is\n\"Nickleby\" and the \"Trial\" to-night; \"Copperfield\" again to-morrow. A\nwet day here, with glimpses of blue. I shall not forget Katey's health\nat dinner. A pleasant journey down.\n\n                      Ever, my dearest Georgy, your most affectionate.\n\n\n[Sidenote: The same.]\n\n             THE GREAT WHITE HORSE, IPSWICH, _Friday, Nov. 1st, 1861._\n\nI cannot quite remember in the whirl of travelling and reading, whether\nor no I wrote you a line from Bury St. Edmunds. But I think (and hope)\nI did. We had a fine room there, and \"Copperfield\" made a great\nimpression. At mid-day we go on to Colchester, where I shall expect the\nyoung Morgans. I sent a telegram on yesterday, after receiving your\nnote, to secure places for them. The answer returned by telegraph was:\n\"No box-seats left but on the fourth row.\" If they prefer to sit on the\nstage (for I read in the theatre, there being no other large public\nroom), they shall. Meantime I have told John, who went forward this\nmorning with the other men, to let the people at the inn know that if\nthree travellers answering that description appear before my\ndinner-time, they are to dine with me.\n\nPlorn's admission that he likes the school very much indeed, is the\ngreat social triumph of modern times.\n\nI am looking forward to Sunday's rest at Gad's, and shall be down by the\nten o'clock train from town. I miss poor Arthur dreadfully. It is\nscarcely possible to imagine how much. It is not only that his loss to\nme socially is quite irreparable, but that the sense I used to have of\ncompactness and comfort about me while I was reading is quite gone. And\nwhen I come out for the ten minutes, when I used to find him always\nready for me with something cheerful to say, it is forlorn. I cannot but\nfancy, too, that the audience must miss the old speciality of a\npervading gentleman.\n\nNobody I know has turned up yet except Elwin. I have had many\ninvitations to all sorts of houses in all sorts of places, and have of\ncourse accepted them every one.\n\nLove to Mamie, if she has come home, and to Bouncer, if _she_ has come;\nalso Marguerite, who I hope is by this time much better.\n\n                         Ever, my dear Georgy, your most affectionate.\n\n\n[Sidenote: Mrs. Henry Austin.]\n\n                                 GAD'S HILL, _Sunday, Nov. 3rd, 1861._\n\nEXTRACT.\n\nI am heartily glad to hear that you have been out in the air, and I hope\nyou will go again very soon and make a point of continuing to go. There\nis a soothing influence in the sight of the earth and sky, which God put\ninto them for our relief when He made the world in which we are all to\nsuffer, and strive, and die.\n\nI will not fail to write to you from many points of my tour, and if you\never want to write to me you may be sure of a quick response, and may be\ncertain that I am sympathetic and true.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Miss Dickens.]\n\n            FOUNTAIN HOTEL, CANTERBURY, _Windy Night, Nov. 4th, 1861._\n\nMY DEAREST MAMIE,\n\nA word of report before I go to bed. An excellent house to-night, and an\naudience positively perfect. The greatest part of it stalls, and an\nintelligent and delightful response in them, like the touch of a\nbeautiful instrument. \"Copperfield\" wound up in a real burst of feeling\nand delight.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Mr. John Agate.]\n\n                LORD WARDEN HOTEL, DOVER, _Wednesday, Nov. 6th, 1861._\n\nSIR,\n\nI am exceedingly sorry to find, from the letter you have addressed to\nme, that you had just cause of complaint in being excluded from my\nreading here last night. It will now and then unfortunately happen when\nthe place of reading is small (as in this case), that some confusion\nand inconvenience arise from the local agents over-estimating, in\nperfect good faith and sincerity, the capacity of the room. Such a\nmistake, I am assured, was made last night; and thus all the available\nspace was filled before the people in charge were at all prepared for\nthat circumstance.\n\nYou may readily suppose that I can have no personal knowledge of the\nproceedings of the people in my employment at such a time. But I wish to\nassure you very earnestly, that they are all old servants, well\nacquainted with my principles and wishes, and that they are under the\nstrongest injunction to avoid any approach to mercenary dealing; and to\nbehave to all comers equally with as much consideration and politeness\nas they know I should myself display. The recent death of a\nmuch-regretted friend of mine, who managed this business for me, and on\nwhom these men were accustomed to rely in any little difficulty, caused\nthem (I have no doubt) to feel rather at a loss in your case. Do me the\nfavour to understand that under any other circumstances you would, as a\nmatter of course, have been provided with any places whatever that could\nbe found, without the smallest reference to what you had originally\npaid. This is scanty satisfaction to you, but it is so strictly the\ntruth, that yours is the first complaint of the kind I have ever\nreceived.\n\nI hope to read in Dover again, but it is quite impossible that I can\nmake any present arrangement for that purpose. Whenever I may return\nhere, you may be sure I shall not fail to remember that I owe you a\nrecompense for a disappointment. In the meanwhile I very sincerely\nregret it.\n\n                                                     Faithfully yours.\n\n\n[Sidenote: Miss Hogarth.]\n\n                  BEDFORD HOTEL, BRIGHTON, _Thursday, Nov. 7th, 1861._\n\nMY DEAR GEORGY,\n\n       *       *       *       *       *\n\nThe Duchess of Cambridge comes to-night to \"Copperfield.\" The bad\nweather has not in the least touched us, and beyond all doubt a great\ndeal of money has been left untaken at each place.\n\nThe storm was most magnificent at Dover. All the great side of The Lord\nWarden next the sea had to be emptied, the break of the sea was so\nprodigious, and the noise was so utterly confounding. The sea came in\nlike a great sky of immense clouds, for ever breaking suddenly into\nfurious rain. All kinds of wreck were washed in. Miss Birmingham and I\nsaw, among other things, a very pretty brass-bound chest being thrown\nabout like a feather. On Tuesday night, the unhappy Ostend packet could\nnot get in, neither could she go back, and she beat about the Channel\nuntil noon yesterday. I saw her come in then, _with five men at the\nwheel_; such a picture of misery, as to the crew (of passengers there\nwere no signs), as you can scarcely imagine.\n\nTho effect at Hastings and at Dover really seems to have outdone the\nbest usual impression, and at Dover they wouldn't go, but sat applauding\nlike mad. The most delicate audience I have seen in any provincial place\nis Canterbury. The audience with the greatest sense of humour certainly\nis Dover. The people in the stalls set the example of laughing, in the\nmost curiously unreserved way; and they really laughed when Squeers read\nthe boys' letters, with such cordial enjoyment, that the contagion\nextended to me, for one couldn't hear them without laughing too.\n\nSo, thank God, all goes well, and the recompense for the trouble is in\nevery way great. There is rather an alarming breakdown at Newcastle, in\nrespect of all the bills having been, in some inscrutable way, lost on\nthe road. I have resolved to send Berry there, with full powers to do\nall manner of things, early next week.\n\nThe amended route-list is not printed yet, because I am trying to get\noff Manchester and Liverpool; both of which I strongly doubt, in the\npresent state of American affairs. Therefore I can't send it for\nMarguerite; but I can, and do, send her my love and God-speed. This is\naddressed to the office because I suppose you will be there to-morrow.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: The Earl of Carlisle.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                                _November 15th, 1861._\n\nMY DEAR LORD CARLISLE,\n\nYou know poor Austin, and what his work was, and how he did it. If you\nhave no private objection to signing the enclosed memorial (which will\nreceive the right signatures before being presented), I think you will\nhave no public objection. I shall be heartily glad if you can put your\nname to it, and shall esteem your doing so as a very kind service. Will\nyou return the memorial under cover to Mr. Tom Taylor, at the Local\nGovernment Act Office, Whitehall? He is generously exerting himself in\nfurtherance of it, and so delay will be avoided.\n\n                       My dear Lord Carlisle, faithfully yours always.\n\n\n[Sidenote: Miss Mary Boyle.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                            _Sunday, Nov. 17th, 1861._\n\nMY DEAR MARY,\n\nI am perfectly enraptured with the quilt. It is one of the most\ntasteful, lively, elegant things I have ever seen; and I need not tell\nyou that while it is valuable to me for its own ornamental sake, it is\nprecious to me as a rainbow-hint of your friendship and affectionate\nremembrance.\n\nPlease God you shall see it next summer occupying its allotted place of\nstate in my brand-new bedroom here. You shall behold it then, with all\ncheerful surroundings, the envy of mankind.\n\nMy readings have been doing absolute wonders. Your Duchess and Princess\ncame to hear first \"Nickleby\" and the \"Pickwick Trial,\" then\n\"Copperfield,\" at Brighton. I think they were pleased with me, and I am\nsure I was with them; for they are the very best audience one could\npossibly desire. I shall always have a pleasant remembrance of them.\n\nOn Wednesday I am away again for the longest part of my trip.\n\nYes, Mary dear, I must say that I like my Carton, and I have a faint\nidea sometimes that if I had acted him, I could have done something with\nhis life and death.\n\n                    Believe me, ever your affectionate and faithful\n                                                                  JOE.\n\n\n[Sidenote: Miss Hogarth.]\n\n                   QUEEN'S HEAD, NEWCASTLE, _Friday, Nov. 22nd, 1861._\n\nI received your letter this morning, and grieve to report that the\nunlucky Headland has broken down most awfully!\n\nFirst, as perhaps you remember, this is the place where the bills were\n\"lost\" for a week or two. The consequence has been that the agent could\nnot announce all through the \"Jenny Lind\" time (the most important for\nannouncing), and could but stand still and stare when people came to ask\nwhat I was going to read. Last night I read \"Copperfield\" to the most\nenthusiastic and appreciative audience imaginable, but in numbers about\nhalf what they might have been. To-night we shall have a famous house;\nbut we might have had it last night too. To-morrow (knowing by this time\nwhat can, of a certainty, be done with \"Copperfield\"), I had, of course,\ngiven out \"Copperfield\" to be read again. Conceive my amazement and\ndismay when I find the printer to have announced \"Little Dombey\"!!!\nThis, I declare, I had no more intention of reading than I had of\nreading an account of the solar system. And this, after a sensation last\nnight, of a really extraordinary nature in its intensity and delight!\n\nSays the unlucky Headland to this first head of misery: \"Johnson's\nmistake\" (Johnson being the printer).\n\nSecond, I read at Edinburgh for the first time--observe the day--_next\nWednesday_. Jenny Lind's concert at Edinburgh is to-night. This morning\ncomes a frantic letter from the Edinburgh agent. \"I have no bills, no\ntickets; I lose all the announcement I would have made to hundreds upon\nhundreds of people to-night, all of the most desirable class to be well\ninformed beforehand. I can't announce what Mr. Dickens is going to read;\nI can answer no question; I have, upon my responsibility, put a dreary\nadvertisement into the papers announcing that he _is_ going to read so\nmany times, and that particulars will shortly be ready; and I stand\nbound hand and foot.\" \"Johnson's mistake,\" says the unlucky Headland.\n\nOf course, I know that the man who never made a mistake in poor Arthur's\ntime is not likely to be always making mistakes now. But I have written\nby this post to Wills, to go to him and investigate. I have also\ndetached Berry from here, and have sent him on by train at a few\nminutes' notice to Edinburgh, and then to Glasgow (where I have no doubt\neverything is wrong too). Glasgow we may save; Edinburgh I hold to be\nirretrievably damaged. If it can be picked up at all, it can only be at\nthe loss of the two first nights, and by the expenditure of no end of\nspirits and force. And this is the harder, because it is impossible not\nto see that the last readings polished and prepared the audiences in\ngeneral, and that I have not to work them up in any place where I have\nbeen before, but that they start with a London intelligence, and with a\nrespect and preparation for what they are going to hear.\n\nI hope by the time you and Mamie come to me, we shall have got into some\ngood method. I must take the thing more into my own hands and look after\nit from hour to hour. If such a thing as this Edinburgh business could\nhave happened under poor Arthur, I really believe he would have fallen\ninto a fit, or gone distracted. No one can ever know what he was but I\nwho have been with him and without him. Headland is so anxious and so\ngood-tempered that I cannot be very stormy with him; but it is the\nsimple fact that he has no notion of the requirements of such work as\nthis. Without him, and with a larger salary to Berry (though there are\nobjections to the latter as _first_ man), I could have done a hundred\ntimes better.\n\nAs Forster will have a strong interest in knowing all about the\nproceedings, perhaps you will send him this letter to read. There is no\nvery tremendous harm, indeed, done as yet. At Edinburgh I KNOW what I\ncan do with \"Copperfield.\" I think it is not too much to say that for\nevery one who does come to hear it on the first night, I can get back\nfifty on the second. And whatever can be worked up there will tell on\nGlasgow. Berry I shall continue to send on ahead, and I shall take\nnothing on trust and more as being done.\n\nOn Sunday morning at six, I have to start for Berwick. From Berwick, in\nthe course of that day, I will write again; to Mamie next time.\n\nWith best love to her and Mrs. B.\n\n\n[Sidenote: Miss Dickens.]\n\n                                QUEEN'S HEAD, NEWCASTLE-ON-TYNE,\n                                          _Saturday, Nov. 23rd, 1861._\n\nA most tremendous hall here last night; something almost terrible in the\ncram. A fearful thing might have happened. Suddenly, when they were all\nvery still over Smike, my gas batten came down, and it looked as if the\nroom was falling. There were three great galleries crammed to the roof,\nand a high steep flight of stairs, and a panic must have destroyed\nnumbers of people. A lady in the front row of stalls screamed, and ran\nout wildly towards me, and for one instant there was a terrible wave in\nthe crowd. I addressed that lady laughing (for I knew she was in sight\nof everybody there), and called out as if it happened every night,\n\"There's nothing the matter, I assure you; don't be alarmed; pray sit\ndown;\" and she sat down directly, and there was a thunder of applause.\nIt took some few minutes to mend, and I looked on with my hands in my\npockets; for I think if I had turned my back for a moment there might\nstill have been a move. My people were dreadfully alarmed, Boylett in\nparticular, who I suppose had some notion that the whole place might\nhave taken fire.\n\n\"But there stood the master,\" he did me the honour to say afterwards, in\naddressing the rest, \"as cool as ever I see him a-lounging at a railway\nstation.\"\n\nA telegram from Berry at Edinburgh yesterday evening, to say that he\nhad got the bills, and that they would all be up and dispersed yesterday\nevening under his own eyes. So no time was lost in setting things as\nright as they can be set. He has now gone on to Glasgow.\n\nP.S.--Duty to Mrs. Bouncer.\n\n\n[Sidenote: Miss Hogarth.]\n\n                          BERWICK-ON-TWEED, _Monday, Nov. 25th, 1861._\n\nI write (in a gale of wind, with a high sea running), to let you know\nthat we go on to Edinburgh at half-past eight to-morrow morning.\n\nA most ridiculous room was designed for me in this odd out-of-the-way\nplace. An immense Corn Exchange made of glass and iron, round,\ndome-topped, lofty, utterly absurd for any such purpose, and full of\nthundering echoes, with a little lofty crow's-nest of a stone gallery\nbreast high, deep in the wall, into which it was designed to put _me_! I\ninstantly struck, of course, and said I would either read in a room\nattached to this house (a very snug one, capable of holding five hundred\npeople) or not at all. Terrified local agents glowered, but fell\nprostrate.\n\nBerry has this moment come back from Edinburgh and Glasgow with hopeful\naccounts. He seems to have done the business extremely well, and he says\nthat it was quite curious and cheering to see how the Glasgow people\nassembled round the bills the instant they were posted, and evidently\nwith a great interest in them.\n\nWe left Newcastle yesterday morning in the dark, when it was intensely\ncold and froze very hard. So it did here. But towards night the wind\nwent round to the S.W., and all night it has been blowing very hard\nindeed. So it is now.\n\nTell Mamie that I have the same sitting-room as we had when we came here\nwith poor Arthur, and that my bedroom is the room out of it which she\nand Katie had. Surely it is the oddest town to read in! But it is taken\non poor Arthur's principle that a place in the way pays the expenses of\na through journey; and the people would seem to be coming up to the\nscratch gallantly. It was a dull Sunday, though; O it _was_ a dull\nSunday, without a book! For I had forgotten to buy one at Newcastle,\nuntil it was too late. So after dark I made a jug of whisky-punch, and\ndrowned the unlucky Headland's remembrance of his failures.\n\nI shall hope to hear very soon that the workmen have \"broken through,\"\nand that you have been in the state apartments, and that upholstery\nmeasurements have come off.\n\nThere has been a horrible accident in Edinburgh. One of the seven-storey\nold houses in the High Street fell when it was full of people. Berry was\nat the bill-poster's house, a few doors off, waiting for him to come\nhome, when he heard what seemed like thunder, and then the air was\ndarkened with dust, \"as if an immense quantity of steam had been blown\noff,\" and then all that dismal quarter set up shrieks, which he says\nwere most dreadful.\n\n\n[Sidenote: Miss Dickens.]\n\n              WATERLOO HOTEL, EDINBURGH, _Wednesday, Nov. 27th, 1861._\n\nMrs. Bouncer must decidedly come with you to Carlisle. She shall be\nreceived with open arms. Apropos of Carlisle, let me know _when_ you\npurpose coming there. We shall be there, please God, on the Saturday in\ngood time, as I finish at Glasgow on the Friday night.\n\nI have very little notion of the state of affairs here, as Headland\nbrought no more decisive information from the agents yesterday (he never\n_can_ get decisive information from any agents), than \"the teeckets air\njoost moving reecht and left.\" I hope this may be taken as satisfactory.\nJenny Lind carried off a world of money from here. Miss Glyn, or Mrs.\nDallas, is playing Lady Macbeth at the theatre, and Mr. Shirley Brooks\nis giving two lectures at the Philosophical Society on the House of\nCommons and Horace Walpole. Grisi's farewell benefits are (I think) on\nmy last two nights here.\n\nGordon dined with me yesterday. He is, if anything, rather better, I\nthink, than when we last saw him in town. He was immensely pleased to be\nwith me. I went with him (as his office goes anywhere) right into and\namong the ruins of the fallen building yesterday. They were still at\nwork trying to find two men (brothers), a young girl, and an old woman,\nknown to be all lying there. On the walls two or three common clocks are\nstill hanging; one of them, judging from the time at which it stopped,\nwould seem to have gone for an hour or so after the fall. Great interest\nhad been taken in a poor linnet in a cage, hanging in the wind and rain\nhigh up against the broken wall. A fireman got it down alive, and great\nexultation had been raised over it. One woman, who was dug out unhurt,\nstaggered into the street, stared all round her, instantly ran away, and\nhas never been heard of since. It is a most extraordinary sight, and of\ncourse makes a great sensation.\n\n\n[Sidenote: Miss Dickens.]\n\n                 WATERLOO HOTEL, EDINBURGH, _Friday, Nov. 29th, 1861._\n\nI think it is my turn to write to you, and I therefore send a brief\ndespatch, like a telegram, to let you know that in a gale of wind and a\nfierce rain, last night, we turned away a thousand people. There was no\ngetting into the hall, no getting near the hall, no stirring among the\npeople, no getting out, no possibility of getting rid of them. And yet,\nin spite of all that, and of their being steaming wet, they never\nflagged for an instant, never made a complaint, and took up the trial\nupon their very shoulders, to the last word, in a triumphant roar.\n\nThe talk about \"Copperfield\" rings through the whole place. It is done\nagain to-morrow night. To-morrow morning I read \"Dombey.\" To-morrow\nmorning is Grisi's \"farewell\" morning concert, and last night was her\n\"farewell\" evening concert. Neither she, nor Jenny Lind, nor anything,\nnor anybody seems to make the least effect on the draw of the readings.\n\nI lunch with Blackwood to-day. He was at the reading last night; a\ncapital audience. Young Blackwood has also called here. A very good\nyoung fellow, I think.\n\n\n[Sidenote: Miss Hogarth.]\n\n            CARRICK'S ROYAL HOTEL, GLASGOW, _Tuesday, Dec. 3rd, 1861._\n\nI send you by this post another _Scotsman_. From a paragraph in it, a\nletter, and an advertisement, you may be able to form some dim guess of\nthe scene at Edinburgh last night. Such a pouring of hundreds into a\nplace already full to the throat, such indescribable confusion, such a\nrending and tearing of dresses, and yet such a scene of good humour on\nthe whole. I never saw the faintest approach to it. While I addressed\nthe crowd in the room, Gordon addressed the crowd in the street. Fifty\nfrantic men got up in all parts of the hall and addressed me all at\nonce. Other frantic men made speeches to the walls. The whole Blackwood\nfamily were borne in on the top of a wave, and landed with their faces\nagainst the front of the platform. I read with the platform crammed with\npeople. I got them to lie down upon it, and it was like some impossible\ntableau or gigantic picnic; one pretty girl in full dress lying on her\nside all night, holding on to one of the legs of my table. It was the\nmost extraordinary sight. And yet from the moment I began to the moment\nof my leaving off, they never missed a point, and they ended with a\nburst of cheers.\n\nThe confusion was decidedly owing to the local agents. But I think it\nmay have been a little heightened by Headland's way of sending them the\ntickets to sell in the first instance.\n\nNow, as I must read again in Edinburgh on Saturday night, your\ntravelling arrangements are affected. So observe carefully (you and\nMamie) all that I am going to say. It appears to me that the best course\nwill be for you to come to _Edinburgh_ on Saturday; taking the fast\ntrain from the Great Northern station at nine in the morning. This would\nbring you to the Waterloo at Edinburgh, at about nine or so at night,\nand I should be home at ten. We could then have a quiet Sunday in\nEdinburgh, and go over to Carlisle on the Monday morning.\n\nThe expenditure of lungs and spirits was (as you may suppose) rather\ngreat last night, and to sleep well was out of the question; I am\ntherefore rather fagged to-day. And as the hall in which I read to-night\nis a large one, I must make my letter a short one.\n\nMy people were torn to ribbons last night. They have not a hat among\nthem, and scarcely a coat.\n\nGive my love to Mamie. To her question, \"Will there be war with\nAmerica?\" I answer, \"Yes;\" I fear the North to be utterly mad, and war\nto be unavoidable.\n\n\n[Sidenote: Mr. W. H. Wills.]\n\n                   VICTORIA HOTEL, PRESTON, _Friday, Dec. 13th, 1861._\n\nMY DEAR WILLS,\n\nThe news of the Christmas number is indeed glorious, and nothing can\nlook brighter or better than the prospects of the illustrious\npublication.\n\nBoth Carlisle and Lancaster have come out admirably, though I doubted\nboth, as you did. But, unlike you, I always doubted this place. I do so\nstill. It is a poor place at the best (you remember?), and the mills are\nworking half time, and trade is very bad. The expenses, however, will be\na mere nothing. The accounts from Manchester for to-morrow, and from\nLiverpool for the readings generally, are very cheering indeed.\n\nThe young lady who sells the papers at the station is just the same as\never. Has orders for to-night, and is coming \"with a person.\" \"_The_\nperson?\" said I. \"Never _you_ mind,\" said she.\n\nI was so charmed with Robert Chambers's \"Traditions of Edinburgh\" (which\nI read _in_ Edinburgh), that I was obliged to write to him and say so.\n\nGlasgow finished nobly, and the last night in Edinburgh was signally\nsuccessful and positively splendid.\n\nWill you give my small Admiral, on his personal application, one\nsovereign? I have told him to come to you for that recognition of his\nmeritorious services.\n\n                                                      Ever faithfully.\n\n\n[Sidenote: Mr. W. H. Wills.]\n\n                  ADELPHI HOTEL, LIVERPOOL, _Sunday, Dec. 15th, 1861._\n\nMY DEAR WILLS,\n\nI sent you a telegram to-day, and I write before the answer has come to\nhand.\n\nI have been very doubtful what to do here. We have a great let for\nto-morrow night. The Mayor recommends closing to-morrow, and going on on\nTuesday and Wednesday, so does the town clerk, so do the agents. But I\nhave a misgiving that they hardly understand what the public general\nsympathy with the Queen will be. Further, I feel personally that the\nQueen has always been very considerate and gracious to me, and I would\non no account do anything that might seem unfeeling or disrespectful. I\nshall attach great weight, in this state of indecision, to your\ntelegram.\n\nA capital audience at Preston. Not a capacious room, but full. Great\nappreciation.\n\nThe scene at Manchester last night was really magnificent. I had had the\nplatform carried forward to our \"Frozen Deep\" point, and my table and\nscreen built in with a proscenium and room scenery. When I went in\n(there was a very fine hall), they applauded in the most tremendous\nmanner; and the extent to which they were taken aback and taken by storm\nby \"Copperfield\" was really a thing to see.\n\nThe post closes early here on a Sunday, and I shall close this also\nwithout further reference to \"a message from the\" W. H. W. being\nprobably on the road.\n\nRadley is ill, and supposed to be fast declining, poor fellow. The house\nis crammed, the assizes on, and troops perpetually embarking for Canada,\nand their officers passing through the hotel.\n\n                                     Kindest regards, ever faithfully.\n\n\n[Sidenote: Miss Mary Boyle.]\n\n                          GAD'S HILL, HIGHAM BY ROCHESTER, KENT,\n                                          _Saturday, Dec. 28th, 1861._\n\nMY DEAR MARY,\n\nOn Monday (as you know) I am away again, but I am not sorry to see land\nand a little rest before me; albeit, these are great experiences of the\npublic heart.\n\nThe little Admiral has gone to visit America in the _Orlando_, supposed\nto be one of the foremost ships in the Service, and the best found, best\nmanned, and best officered that ever sailed from England. He went away\nmuch gamer than any giant, attended by a chest in which he could easily\nhave stowed himself and a wife and family of his own proportions.\n\n                                 Ever and always, your affectionate\n                                                                  JOE.\n\n\n\n\n1862.\n\nNARRATIVE.\n\n\nAt the beginning of this year, Charles Dickens resumed the reading tour\nwhich he had commenced at the close of the previous year and continued\nup to Christmas. The first letter which follows, to Mr. Wills, a New\nYear's greeting, is written from a railway station between one town and\nanother on this journey. Mr. Macready, who had married for the second\ntime not very long before this, was now settled at Cheltenham. Charles\nDickens had arranged to give readings there, chiefly for the pleasure of\nvisiting him, and of having him as one of his audience.\n\nThis reading tour went on until the beginning of February. One of the\nlast of the series was in his favourite \"beautiful room,\" the St.\nGeorge's Hall at Liverpool. In February, he made an exchange of houses\nwith his friends Mr. and Mrs. Hogge, they going to Gad's Hill, and he\nand his family to Mr. Hogge's house in Hyde Park Gate South. In March he\ncommenced a series of readings at St. James's Hall, which went on until\nthe middle of June, when he, very gladly, returned to his country home.\n\nA letter beginning \"My dear Girls,\" addressed to some American ladies\nwho happened to be at Colchester, in the same inn with him when he was\nreading there, was published by one of them under the name of \"Our\nLetter,\" in the \"St. Nicholas Magazine,\" New York, in 1877. We think it\nbest to explain it in the young lady's own words, which are, therefore,\nappended to the letter.\n\nMr. Walter Thornbury was one of Charles Dickens's most valuable\ncontributors to \"All the Year Round.\" His letters to him about the\nsubjects of his articles for that journal, are specimens of the minute\nand careful attention and personal supervision, never neglected or\ndistracted by any other work on which he might be engaged, were it ever\nso hard or engrossing.\n\nThe letter addressed to Mr. Baylis we give chiefly because it has, since\nMr. Baylis's death, been added to the collection of MSS. in the British\nMuseum. He was a very intimate and confidential friend of the late Lord\nLytton, and accompanied him on a visit to Gad's Hill in that year.\n\nWe give an extract from another letter from Charles Dickens to his\nsister, as a beautiful specimen of a letter of condolence and\nencouragement to one who was striving, very bravely, but by very slow\ndegrees, to recover from the overwhelming grief of her bereavement. Mr.\nWilkie Collins was at this time engaged on his novel of \"No Name,\" which\nappeared in \"All the Year Round,\" and was threatened with a very serious\nbreakdown in health. Charles Dickens wrote the letter which we give, to\nrelieve Mr. Collins's mind as to his work. Happily he recovered\nsufficiently to make an end to his own story without any help; but the\ntrue friendship and kindness which suggested the offer were none the\nless appreciated, and may, very likely, by lessening his anxiety, have\nhelped to restore his health. At the end of October in this year,\nCharles Dickens, accompanied by his daughter and sister-in-law, went to\nreside for a couple of months in Paris, taking an apartment in the Rue\ndu Faubourg St. Honore. From thence he writes to M. Charles Fechter. He\nhad been greatly interested in this fine artist from the time of his\nfirst appearance in England, and was always one of his warmest friends\nand supporters during his stay in this country. M. Fechter was, at this\ntime, preparing for the opening of the Lyceum Theatre, under his own\nmanagement, at the beginning of the following year.\n\nJust before Christmas, Charles Dickens returned to Gad's Hill. The\nChristmas number for this year was \"Somebody's Luggage.\"\n\n\n[Sidenote: Mr. W. H. Wills.]\n\n\n                AT THE BIRMINGHAM STATION, _Thursday, Jan. 2nd, 1862._\n\nMY DEAR WILLS,\n\nBeing stationed here for an hour, on my way from Leamington to\nCheltenham, I write to you.\n\nFirstly, to reciprocate all your cordial and affectionate wishes for the\nNew Year, and to express my earnest hope that we may go on through many\nyears to come, as we have gone on through many years that are gone. And\nI think we can say that we doubt whether any two men can have gone on\nmore happily and smoothly, or with greater trust and confidence in one\nanother.\n\nA little packet will come to you from Hunt and Roskell's, almost at the\nsame time, I think, as this note.\n\nThe packet will contain a claret-jug. I hope it is a pretty thing in\nitself for your table, and I know that you and Mrs. Wills will like it\nnone the worse because it comes from me.\n\nIt is not made of a perishable material, and is so far expressive of our\nfriendship. I have had your name and mine set upon it, in token of our\nmany years of mutual reliance and trustfulness. It will never be so full\nof wine as it is to-day of affectionate regard.\n\n                                                Ever faithfully yours.\n\n\n[Sidenote: Miss Hogarth.]\n\n                                 CHELTENHAM, _Friday, Jan. 3rd, 1862._\n\nMY DEAREST GEORGY,\n\nMrs. Macready in voice is very like poor Mrs. Macready dead and gone;\nnot in the least like her otherwise. She is perfectly satisfactory, and\nexceedingly winning. Quite perfect in her manner with him and in her\nease with his children, sensible, gay, pleasant, sweet-tempered; not in\nthe faintest degree stiff or pedantic; accessible instantly. I have very\nrarely seen a more agreeable woman. The house is (on a smaller scale)\nany house we have known them in. Furnished with the old furniture,\npictures, engravings, mirrors, tables, and chairs. Butty is too tall for\nstrength, I am afraid, but handsome, with a face of great power and\ncharacter, and a very nice girl. Katie you know all about. Macready,\ndecidedly much older and infirm. Very much changed. His old force has\ngone out of him strangely. I don't think I left off talking a minute\nfrom the time of my entering the house to my going to bed last night,\nand he was as much amused and interested as ever I saw him; still he\nwas, and is, unquestionably aged.\n\nAnd even now I am obliged to cut this letter short by having to go and\nlook after Headland. It would never do to be away from the rest of them.\nI have no idea what we are doing here; no notion whether things are\nright or wrong; no conception where the room is; no hold of the business\nat all. For which reason I cannot rest without going and looking after\nthe worthy man.\n\n\n[Sidenote: Miss Hogarth.]\n\n                                 TORQUAY, _Wednesday, Jan. 8th, 1862._\n\nYou know, I think, that I was very averse to going to Plymouth, and\nwould not have gone there again but for poor Arthur. But on the last\nnight I read \"Copperfield,\" and positively enthralled the people. It was\na most overpowering effect, and poor Andrew[7] came behind the screen,\nafter the storm, and cried in the best and manliest manner. Also there\nwere two or three lines of his shipmates and other sailors, and they\nwere extraordinarily affected. But its culminating effect was on\nMacready at Cheltenham. When I got home after \"Copperfield,\" I found him\nquite unable to speak, and able to do nothing but square his dear old\njaw all on one side, and roll his eyes (half closed), like Jackson's\npicture of him. And when I said something light about it, he returned:\n\"No--er--Dickens! I swear to Heaven that, as a piece of passion and\nplayfulness--er--indescribably mixed up together, it does--er--no,\nreally, Dickens!--amaze me as profoundly as it moves me. But as a piece\nof art--and you know--er--that I--no, Dickens! By ----! have seen the\nbest art in a great time--it is incomprehensible to me. How is it got\nat--er--how is it done--er--how one man can--well? It lays me on\nmy--er--back, and it is of no use talking about it!\" With which he put\nhis hand upon my breast and pulled out his pocket-handkerchief, and I\nfelt as if I were doing somebody to his Werner. Katie, by-the-bye, is a\nwonderful audience, and has a great fund of wild feeling in her. Johnny\nnot at all unlike Plorn.\n\nI have not yet seen the room here, but imagine it to be very small.\nExeter I know, and that is small also. I am very much used up, on the\nwhole, for I cannot bear this moist warm climate. It would kill me very\nsoon. And I have now got to the point of taking so much out of myself\nwith \"Copperfield,\" that I might as well do Richard Wardour.\n\nYou have now, my dearest Georgy, the fullest extent of my tidings. This\nis a very pretty place--a compound of Hastings, Tunbridge Wells, and\nlittle bits of the hills about Naples; but I met four respirators as I\ncame up from the station, and three pale curates without them, who\nseemed in a bad way.\n\nFrightful intelligence has just been brought in by Boylett, concerning\nthe small size of the room. I have terrified Headland by sending him to\nlook at it, and swearing that if it's too small I will go away to\nExeter.\n\n\n[Sidenote: Miss Hogarth.]\n\n                  ADELPHI HOTEL, LIVERPOOL, _Tuesday, Jan. 28th, 1862._\n\nThe beautiful room was crammed to excess last night, and numbers were\nturned away. Its beauty and completeness when it is lighted up are most\nbrilliant to behold, and for a reading it is simply perfect. You\nremember that a Liverpool audience is usually dull, but they put me on\nmy mettle last night, for I never saw such an audience--no, not even in\nEdinburgh!\n\nI slept horribly last night, and have been over to Birkenhead for a\nlittle change of air to-day. My head is dazed and worn by gas and heat,\nand I fear that \"Copperfield\" and \"Bob\" together to-night won't mend it.\n\nBest love to Mamie and Katie, if still at Gad's. I am going to bring the\nboys some toffee.\n\n\n[Sidenote: The Misses Armstrong]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                            _Monday, Feb. 10th, 1862._\n\nMY DEAR GIRLS,\n\nFor if I were to write \"young friends,\" it would look like a\nschoolmaster; and if I were to write \"young ladies,\" it would look like\na schoolmistress; and worse than that, neither form of words would look\nfamiliar and natural, or in character with our snowy ride that\ntooth-chattering morning.\n\nI cannot tell you both how gratified I was by your remembrance, or how\noften I think of you as I smoke the admirable cigars. But I almost think\nyou must have had some magnetic consciousness across the Atlantic, of my\nwhiffing my love towards you from the garden here.\n\nMy daughter says that when you have settled those little public affairs\nat home, she hopes you will come back to England (possibly in united\nstates) and give a minute or two to this part of Kent. _Her_ words are,\n\"a day or two;\" but I remember your Italian flights, and correct the\nmessage.\n\nI have only just now finished my country readings, and have had nobody\nto make breakfast for me since the remote ages of Colchester!\n\n                                                Ever faithfully yours.\n\n\nOUR LETTER.\n\nBy M. F. ARMSTRONG.\n\n\"From among all my treasures--to each one of which some pleasant history\nis bound--I choose this letter, written on coarse blue paper.\n\nThe letter was received in answer to cigars sent from America to Mr.\nDickens.\n\nThe 'little public affairs at home' refers to the war of the Rebellion.\n\nAt Colchester, he read 'The Trial' from 'Pickwick,' and selections from\n'Nicholas Nickleby.'\n\nThe lady, her two sisters, and her brother were Mr. Dickens's guests at\nthe queer old English inn at Colchester.\n\nThrough the softly falling snow we came back together to London, and on\nthe railway platform parted, with a hearty hand-shaking, from the man\nwho will for ever be enshrined in our hearts as the kindest and most\ngenerous, not to say most brilliant of hosts.\"\n\n\n[Sidenote: M. de Cerjat.]\n\n                      16, HYDE PARK GATE, SOUTH KENSINGTON GORE,\n                                           _Sunday, March 16th, 1862._\n\nMY DEAR CERJAT,\n\nMy daughter naturally liking to be in town at this time of year, I have\nchanged houses with a friend for three months.\n\nMy eldest boy is in business as an Eastern merchant in the City, and\nwill do well if he can find continuous energy; otherwise not. My second\nboy is with the 42nd Highlanders in India. My third boy, a good steady\nfellow, is educating expressly for engineers or artillery. My fourth\n(this sounds like a charade), a born little sailor, is a midshipman in\nH.M.S. _Orlando_, now at Bermuda, and will make his way anywhere.\nRemaining two at school, elder of said remaining two very bright and\nclever. Georgina and Mary keeping house for me; and Francis Jeffrey (I\nought to have counted him as the third boy, so we'll take him in here as\nnumber two and a half) in my office at present. Now you have the family\nbill of fare.\n\nYou ask me about Fechter and his Hamlet. It was a performance of\nextraordinary merit; by far the most coherent, consistent, and\nintelligible Hamlet I ever saw. Some of the delicacies with which he\nrendered his conception clear were extremely subtle; and in particular\nhe avoided that brutality towards Ophelia which, with a greater or less\namount of coarseness, I have seen in all other Hamlets. As a mere _tour\nde force_, it would have been very remarkable in its disclosure of a\nperfectly wonderful knowledge of the force of the English language; but\nits merit was far beyond and above this. Foreign accent, of course, but\nnot at all a disagreeable one. And he was so obviously safe and at ease,\nthat you were never in pain for him as a foreigner. Add to this a\nperfectly picturesque and romantic \"make up,\" and a remorseless\ndestruction of all conventionalities, and you have the leading virtues\nof the impersonation. In Othello he did not succeed. In Iago he is very\ngood. He is an admirable artist, and far beyond anyone on our stage. A\nreal artist and a gentleman.\n\nLast Thursday I began reading again in London--a condensation of\n\"Copperfield,\" and \"Mr. Bob Sawyer's Party,\" from \"Pickwick,\" to finish\nmerrily. The success of \"Copperfield\" is astounding. It made an\nimpression that _I_ must not describe. I may only remark that I was half\ndead when I had done; and that although I had looked forward, all\nthrough the summer, when I was carefully getting it up, to its being a\nLondon sensation; and that although Macready, hearing it at Cheltenham,\ntold me to be prepared for a great effect, it even went beyond my hopes.\nI read again next Thursday, and the rush for places is quite furious.\nTell Townshend this with my love, if you see him before I have time to\nwrite to him; and tell him that I thought the people would never let me\ngo away, they became so excited, and showed it so very warmly. I am\ntrying to plan out a new book, but have not got beyond trying.\n\n                                                 Yours affectionately.\n\n\n[Sidenote: Mr. Walter Thornbury.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                           _Friday, April 18th, 1862._\n\n\nMY DEAR THORNBURY,\n\nThe Bow Street runners ceased out of the land soon after the\nintroduction of the new police. I remember them very well as standing\nabout the door of the office in Bow Street. They had no other uniform\nthan a blue dress-coat, brass buttons (I am not even now sure that that\nwas necessary), and a bright red cloth waistcoat. The waistcoat was\nindispensable, and the slang name for them was \"redbreasts,\" in\nconsequence.\n\nThey kept company with thieves and the like, much more than the\ndetective police do. I don't know what their pay was, but I have no\ndoubt their principal complements were got under the rose. It was a very\nslack institution, and its head-quarters were The Brown Bear, in Bow\nStreet, a public-house of more than doubtful reputation, opposite the\npolice-office; and either the house which is now the theatrical costume\nmaker's, or the next door to it.\n\nField, who advertises the Secret Enquiry Office, was a Bow Street\nrunner, and can tell you all about it; Goddard, who also advertises an\nenquiry office, was another of the fraternity. They are the only two I\nknow of as yet existing in a \"questionable shape.\"\n\n                                              Faithfully yours always.\n\n\n[Sidenote: Mr. Baylis.]\n\n                        GAD'S HILL, ETC., _Wednesday, July 2nd, 1862._\n\nMY DEAR MR. BAYLIS,\n\nI have been in France, and in London, and in other parts of Kent than\nthis, and everywhere but here, for weeks and weeks. Pray excuse my not\nhaving (for this reason specially) answered your kind note sooner.\n\nAfter carefully cross-examining my daughter, I do NOT believe her to be\nworthy of the fernery. Last autumn we transplanted into the shrubbery a\nquantity of evergreens previously clustered close to the front of the\nhouse, and trained more ivy about the wall and the like. When I ask her\nwhere she would have the fernery and what she would do with it, the\nwitness falters, turns pale, becomes confused, and says: \"Perhaps it\nwould be better not to have it at all.\" I am quite confident that the\nconstancy of the young person is not to be trusted, and that she had\nbetter attach her fernery to one of her chateaux in Spain, or one of her\nEnglish castles in the air. None the less do I thank you for your more\nthan kind proposal.\n\nWe have been in great anxiety respecting Miss Hogarth, the sudden\ndecline of whose health and spirits has greatly distressed us. Although\nshe is better than she was, and the doctors are, on the whole, cheerful,\nshe requires great care, and fills us with apprehension. The necessity\nof providing change for her will probably take us across the water very\nearly in the autumn; and this again unsettles home schemes here, and\nwithers many kinds of fern. If they knew (by \"they\" I mean my daughter\nand Miss Hogarth) that I was writing to you, they would charge me with\nmany messages of regard. But as I am shut up in my room in a ferocious\nand unapproachable condition, owing to the great accumulation of letters\nI have to answer, I will tell them at lunch that I have anticipated\ntheir wish. As I know they have bills for me to pay, and are at present\nshy of producing them, I wish to preserve a gloomy and repellent\nreputation.\n\n                          My dear Mr. Baylis, faithfully yours always.\n\n\n[Sidenote: Mrs. Henry Austin.]\n\n                                GAD'S HILL, _Tuesday, Oct. 7th, 1862._\n\n       *       *       *       *       *\n\nI do not preach consolation because I am unwilling to preach at any\ntime, and know my own weakness too well. But in this world there is no\nstay but the hope of a better, and no reliance but on the mercy and\ngoodness of God. Through those two harbours of a shipwrecked heart, I\nfully believe that you will, in time, find a peaceful resting-place even\non this careworn earth. Heaven speed the time, and do you try hard to\nhelp it on! It is impossible to say but that our prolonged grief for the\nbeloved dead may grieve them in their unknown abiding-place, and give\nthem trouble. The one influencing consideration in all you do as to\nyour disposition of yourself (coupled, of course, with a real earnest\nstrenuous endeavour to recover the lost tone of spirit) is, that you\nthink and feel you _can_ do. I do not in the least regard your change of\ncourse in going to Havre as any evidence of instability. But I rather\nhope it is likely that through such restlessness you will come to a far\nquieter frame of mind. The disturbed mind and affections, like the\ntossed sea, seldom calm without an intervening time of confusion and\ntrouble.\n\nBut nothing is to be attained without striving. In a determined effort\nto settle the thoughts, to parcel out the day, to find occupation\nregularly or to make it, to be up and doing something, are chiefly to be\nfound the mere mechanical means which must come to the aid of the best\nmental efforts.\n\nIt is a wilderness of a day, here, in the way of blowing and raining,\nand as darkly dismal, at four o'clock, as need be. My head is but just\nnow raised from a day's writing, but I will not lose the post without\nsending you a word.\n\nKatie was here yesterday, just come back from Clara White's (that was),\nin Scotland. In the midst of her brilliant fortune, it is too clear to\nme that she is already beckoned away to follow her dead sisters.\nMacready was here from Saturday evening to yesterday morning, older but\nlooking wonderfully well, and (what is very rare in these times) with\nthe old thick sweep of hair upon his head. Georgina being left alone\nhere the other day, was done no good to by a great consternation among\nthe servants. On going downstairs, she found Marsh (the stableman)\nseated with great dignity and anguish in an arm-chair, and incessantly\ncrying out: \"I am dead.\" To which the women servants said with great\npathos (and with some appearance of reason): \"No, you ain't, Marsh!\" And\nto which he persisted in replying: \"Yes, I am; I am dead!\" Some\nneighbouring vagabond was impressed to drive a cart over to Rochester\nand fetch the doctor, who said (the patient and his consolers being all\nvery anxious that the heart should be the scene of affliction):\n\"Stomach.\"\n\n\n[Sidenote: Mr. W. Wilkie Collins.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                     _Tuesday Night, Oct. 14th, 1862._\n\nMY DEAR WILKIE,\n\nFrank Beard has been here this evening, of course since I posted my this\nday's letter to you, and has told me that you are not at all well, and\nhow he has given you something which he hopes and believes will bring\nyou round. It is not to convey this insignificant piece of intelligence,\nor to tell you how anxious I am that you should come up with a wet sheet\nand a flowing sail (as we say at sea when we are not sick), that I\nwrite. It is simply to say what follows, which I hope may save you some\nmental uneasiness. For I was stricken ill when I was doing \"Bleak\nHouse,\" and I shall not easily forget what I suffered under the fear of\nnot being able to come up to time.\n\nDismiss that fear (if you have it) altogether from your mind. Write to\nme at Paris at any moment, and say you are unequal to your work, and\nwant me, and I will come to London straight and do your work. I am quite\nconfident that, with your notes and a few words of explanation, I could\ntake it up at any time and do it. Absurdly unnecessary to say that it\nwould be a makeshift! But I could do it at a pinch, so like you as that\nno one should find out the difference. Don't make much of this offer in\nyour mind; it is nothing, except to ease it. If you should want help, I\nam as safe as the bank. The trouble would be nothing to me, and the\ntriumph of overcoming a difficulty great. Think it a Christmas number,\nan \"Idle Apprentice,\" a \"Lighthouse,\" a \"Frozen Deep.\" I am as ready as\nin any of these cases to strike in and hammer the hot iron out.\n\nYou won't want me. You will be well (and thankless!) in no time. But\nthere I am; and I hope that the knowledge may be a comfort to you. Call\nme, and I come.\n\nAs Beard always has a sense of medical responsibility, and says anything\nimportant about a patient in confidence, I have merely remarked here\nthat \"Wilkie\" is out of sorts. Charley (who is here with Katie) has no\nother cue from me.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: M. Charles Fechter.]\n\n                          PARIS, RUE DU FAUBOURG ST. HONORE, 27,\n                                            _Tuesday, Nov. 4th, 1862._\n\nMY DEAR FECHTER,\n\nYou know, I believe, how our letters crossed, and that I am here until\nChristmas. Also, you know with what pleasure and readiness I should have\nresponded to your invitation if I had been in London.\n\nPray tell Paul Feval that I shall be charmed to know him, and that I\nshall feel the strongest interest in making his acquaintance. It almost\nputs me out of humour with Paris (and it takes a great deal to do that!)\nto think that I was not at home to prevail upon him to come with you,\nand be welcomed to Gad's Hill; but either there or here, I hope to\nbecome his friend before this present old year is out. Pray tell him so.\n\nYou say nothing in your note of your Lyceum preparations. I trust they\nare all going on well. There is a fine opening for you, I am sure, with\na good beginning; but the importance of a good beginning is very great.\nIf you ever have time and inclination to tell me in a short note what\nyou are about, you can scarcely interest me more, as my wishes and\nstrongest sympathies are for and with your success--_mais cela va sans\ndire_.\n\nI went to the Chatelet (a beautiful theatre!) the other night to see\n\"Rothomago,\" but was so mortally _gene_ with the poor nature of the\npiece and of the acting, that I came out again when there was a week or\ntwo (I mean an hour or two, but the hours seemed weeks) yet to get\nthrough.\n\n                        My dear Fechter, very faithfully yours always.\n\n\n[Sidenote: Mr. Clarkson Stanfield, R.A.]\n\n                          PARIS, RUE DU FAUBOURG ST. HONORE, 27,\n                                             _Friday, Dec. 5th, 1862._\n\nMY DEAR STANNY,\n\nWe have been here for two months, and I shall probably come back here\nafter Christmas (we go home for Christmas week) and stay on into\nFebruary. But I shall write and propose a theatre before Christmas is\nout, so this is to warn you to get yourself into working pantomime\norder!\n\nI hope Wills has duly sent you our new Christmas number. As you may like\nto know what I myself wrote of it, understand the Dick contributions to\nbe, _his leaving it till called for_, and _his wonderful end_, _his\nboots_, and _his brown paper parcel_.\n\nSince you were at Gad's Hill I have been travelling a good deal, and\nlooking up many odd things for use. I want to know how you are in health\nand spirits, and it would be the greatest of pleasures to me to have a\nline under your hand.\n\nGod bless you and yours with all the blessings of the time of year, and\nof all times!\n\n                                Ever your affectionate and faithful\n                                                                 DICK.\n\n\n[Sidenote: M. Charles Fechter.]\n\n                                    PARIS, _Saturday, Dec. 6th, 1862._\n\nMY DEAR FECHTER,\n\nI have read \"The White Rose\" attentively, and think it an extremely good\nplay. It is vigorously written with a great knowledge of the stage, and\npresents many striking situations. I think the close particularly fine,\nimpressive, bold, and new.\n\nBut I greatly doubt the expediency of your doing _any_ historical play\nearly in your management. By the words \"historical play,\" I mean a play\nfounded on any incident in English history. Our public are accustomed to\nassociate historical plays with Shakespeare. In any other hands, I\nbelieve they care very little for crowns and dukedoms. What you want is\nsomething with an interest of a more domestic and general nature--an\ninterest as romantic as you please, but having a more general and wider\nresponse than a disputed succession to the throne can have for\nEnglishmen at this time of day. Such interest culminated in the last\nStuart, and has worn itself out. It would be uphill work to evoke an\ninterest in Perkin Warbeck.\n\nI do not doubt the play's being well received, but my fear is that these\npeople would be looked upon as mere abstractions, and would have but a\ncold welcome in consequence, and would not lay hold of your audience.\nNow, when you _have_ laid hold of your audience and have accustomed them\nto your theatre, you may produce \"The White Rose,\" with far greater\njustice to the author, and to the manager also. Wait. Feel your way.\nPerkin Warbeck is too far removed from analogy with the sympathies and\nlives of the people for a beginning.\n\n                               My dear Fechter, ever faithfully yours.\n\n\n[Sidenote: Miss Mary Boyle.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                          _Saturday, Dec. 27th, 1862._\n\nMY DEAR MARY,\n\nI must send you my Christmas greeting and happy New Year wishes in\nreturn for yours; most heartily and fervently reciprocating your\ninterest and affection. You are among the few whom I most care for and\nbest love.\n\nBeing in London two evenings in the opening week, I tried to persuade my\nlegs (for whose judgment I have the highest respect) to go to an evening\nparty. But I _could not_ induce them to pass Leicester Square. The\nfaltering presentiment under which they laboured so impressed me, that\nat that point I yielded to their terrors. They immediately ran away to\nthe east, and I accompanied them to the Olympic, where I saw a very good\nplay, \"Camilla's Husband,\" very well played. Real merit in Mr. Neville\nand Miss Saville.\n\nWe came across directly after the gale, with the Channel all bestrewn\nwith floating wreck, and with a hundred and fifty sick schoolboys from\nCalais on board. I am going back on the morning after Fechter's opening\nnight, and have promised to read \"Copperfield\" at the Embassy, for a\nBritish charity.\n\nGeorgy continues wonderfully well, and she and Mary send you their best\nlove. The house is pervaded by boys; and every boy has (as usual) an\nunaccountable and awful power of producing himself in every part of the\nhouse at every moment, apparently in fourteen pairs of creaking boots.\n\n                              My dear Mary, ever affectionately your\n                                                                  JOE.\n\nFOOTNOTES:\n\n[7] Lieutenant Andrew Gordon, R.N., son of the Sheriff of Midlothian.\n\n\n\n\n1863.\n\nNARRATIVE.\n\n\nAt the beginning of this year, Charles Dickens was in Paris for the\npurpose of giving a reading at the English Embassy.\n\nHe remained in Paris until the beginning of February, staying with his\nservant \"John\" at the Hotel du Helder. There was a series of readings in\nLondon this season at the Hanover Square Rooms. The Christmas number of\n\"All the Year Round\" was entitled \"Mrs. Lirriper's Lodgings,\" to which\nCharles Dickens contributed the first and last chapter.\n\nThe Lyceum Theatre, under the management of M. Fechter, was opened in\nJanuary with \"The Duke's Motto,\" and the letter given here has reference\nto this first night.\n\nWe regret very much having no letters to Lady Molesworth, who was an old\nand dear friend of Charles Dickens. But this lady explains to us that\nshe has long ceased to preserve any letters addressed to her.\n\nThe \"Mr. and Mrs. Humphery\" (now Sir William and Lady Humphery)\nmentioned in the first letter for this year, were dear and intimate\nfriends of his eldest daughter, and were frequent guests in her father's\nhouse. Mrs. Humphery and her sister Lady Olliffe were daughters of the\nlate Mr. William Cubitt, M.P.\n\nWe have in this year the first letter of Charles Dickens to Mr. Percy\nFitzgerald. This gentleman had been a valuable contributor to his\njournal before he became personally known to Charles Dickens. The\nacquaintance once made soon ripened into friendship, and for the future\nMr. Fitzgerald was a constant and always a welcome visitor to Gad's\nHill.\n\nThe letter to Mr. Charles Reade alludes to his story, \"Hard Cash,\" which\nwas then appearing in \"All the Year Round.\" As a writer, and as a\nfriend, he was held by Charles Dickens in the highest estimation.\n\nCharles Dickens's correspondence with his solicitor and excellent\nfriend, Mr. Frederic Ouvry (now a vice-president of the Society of\nAntiquaries), was almost entirely of a business character; but we are\nglad to give one or two notes to that gentleman, although of little\npublic interest, in order to have the name in our book of one of the\nkindest of our own friends.\n\n\n[Sidenote: Miss Dickens.]\n\n                          PARIS, HOTEL DU HELDER, RUE DU HELDER,\n                                            _Friday, Jan. 16th, 1863._\n\nMY DEAREST MAMIE,\n\nAs I send a line to your aunt to-day and know that you will not see it,\nI send another to you to report my safe (and neuralgic) arrival here. My\nlittle rooms are perfectly comfortable, and I like the hotel better than\nany I have ever put up at in Paris. John's amazement at, and\nappreciation of, Paris are indescribable. He goes about with his mouth\nopen, staring at everything and being tumbled over by everybody.\n\nThe state dinner at the Embassy, yesterday, coming off in the room where\nI am to read, the carpenters did not get in until this morning. But\ntheir platforms were ready--or supposed to be--and the preparations are\nin brisk progress. I think it will be a handsome affair to look at--a\nvery handsome one. There seems to be great artistic curiosity in Paris,\nto know what kind of thing the reading is.\n\nI know a \"rela-shon\" (with one weak eye), who is in the gunmaking line,\nvery near here. There is a strong family resemblance--but no muzzle.\nLady Molesworth and I have not begun to \"toddle\" yet, but have exchanged\naffectionate greetings. I am going round to see her presently, and I\ndine with her on Sunday. The only remaining news is, that I am beset by\nmysterious adorers, and smuggle myself in and out of the house in the\nmeanest and basest manner.\n\nWith kind regard to Mr. and Mrs. Humphery,\n\n                     Ever, my dearest Mamey, your affectionate Father.\n\nP.S.--_Hommage a Madame B.!_\n\n\n[Sidenote: Monsieur Regnier.]\n\n                                      PARIS, _Sunday, Feb. 1st, 1863._\n\nMY DEAR REGNIER,\n\nI was charmed by the receipt of your cordial and sympathetic letter, and\nI shall always preserve it carefully as a most noble tribute from a\ngreat and real artist.\n\nI wished you had been at the Embassy on Friday evening. The audience was\na fine one, and the \"Carol\" is particularly well adapted to the purpose.\nIt is an uncommon pleasure to me to learn that I am to meet you on\nTuesday, for there are not many men whom I meet with greater pleasure\nthan you. Heaven! how the years roll by! We are quite old friends now,\nin counting by years. If we add sympathies, we have been friends at\nleast a thousand years.\n\n                                            Affectionately yours ever.\n\n\n[Sidenote: Miss Dickens.]\n\n                     HOTEL DU HELDER, PARIS, _Sunday, Feb. 1st, 1863._\n\nMY DEAREST MAMIE,\n\nI cannot give you any idea of the success of the readings here, because\nno one can imagine the scene of last Friday night at the Embassy. Such\naudiences and such enthusiasm I have never seen, but the thing\nculminated on Friday night in a two hours' storm of excitement and\npleasure. They actually recommenced and applauded right away into their\ncarriages and down the street.\n\nYou know your parent's horror of being lionised, and will not be\nsurprised to hear that I am half dead of it. I cannot leave here until\nThursday (though I am every hour in danger of running away) because I\nhave to dine out, to say nothing of breakfasting--think of me\nbreakfasting!--every intervening day. But my project is to send John\nhome on Thursday, and then to go on a little perfectly quiet tour for\nabout ten days, touching the sea at Boulogne. When I get there, I will\nwrite to your aunt (in case you should not be at home), saying when I\nshall arrive at the office. I must go to the office instead of Gad's,\nbecause I have much to do with Forster about Elliotson.\n\nI enclose a short note for each of the little boys. Give Harry ten\nshillings pocket-money, and Plorn six.\n\nThe Olliffe girls, very nice. Florence at the readings, prodigiously\nexcited.\n\n\n\n[Sidenote: Miss Hogarth.]\n\n                                      PARIS, _Sunday, Feb. 1st, 1863._\n\nFrom my hurried note to Mamie, you will get some faint general idea of a\nnew star's having arisen in Paris. But of its brightness you can have no\nadequate conception.\n\n[John has locked me up and gone out, and the little bell at the door is\nringing demoniacally while I write.]\n\nYou have never heard me read yet. I have been twice goaded and lifted\nout of myself into a state that astonished _me_ almost as much as the\naudience. I have a cold, but no neuralgia, and am \"as well as can be\nexpected.\"\n\nI forgot to tell Mamie that I went (with Lady Molesworth) to hear\n\"Faust\" last night. It is a splendid work, in which that noble and sad\nstory is most nobly and sadly rendered, and perfectly delighted me. But\nI think it requires too much of the audience to do for a London opera\nhouse. The composer must be a very remarkable man indeed. Some\nmanagement of light throughout the story is also very poetical and fine.\nWe had Carvalho's box. I could hardly bear the thing, it affected me so.\n\nBut, as a certain Frenchman said, \"No weakness, Danton!\" So I leave off.\n\n\n[Sidenote: M. Charles Fechter.]\n\n                                   PARIS, _Wednesday, Feb. 4th, 1863._\n\nMY DEAR FECHTER,\n\nA thousand congratulations on your great success! Never mind what they\nsay, or do, _pour vous ecraser_; you have the game in your hands. The\nromantic drama, thoroughly well done (with a touch of Shakespeare now\nand then), is the speciality of your theatre. Give the public the\npicturesque, romantic drama, with yourself in it; and (as I told you in\nthe beginning) you may throw down your gauntlet in defiance of all\ncomers.\n\nIt is a most brilliant success indeed, and it thoroughly rejoices my\nheart!\n\nUnfortunately I cannot now hope to see \"Maquet,\" because I am packing up\nand going out to dinner (it is late in the afternoon), and I leave\nto-morrow morning when all sensible people, except myself, are in bed;\nand I do not come back to Paris or near it. I had hoped to see him at\nbreakfast last Monday, but he was not there. Paul Feval was there, and I\nfound him a capital fellow. If I can do anything to help you on with\n\"Maquet\"[8] when I come back I will most gladly do it.\n\nMy readings here have had the finest possible reception, and have\nachieved a most noble success. I never before read to such fine\naudiences, so very quick of perception, and so enthusiastically\nresponsive.\n\nI shall be heartily pleased to see you again, my dear Fechter, and to\nshare your triumphs with the real earnestness of a real friend. And so\ngo on and prosper, and believe me, as I truly am,\n\n\n                                                 Most cordially yours.\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                          _Thursday, Feb. 19th, 1863._\n\nMY DEAREST MACREADY,\n\nI have just come back from Paris, where the readings--\"Copperfield,\"\n\"Dombey\" and \"Trial,\" and \"Carol\" and \"Trial\"--have made a sensation\nwhich modesty (my natural modesty) renders it impossible for me to\ndescribe. You know what a noble audience the Paris audience is! They\nwere at their very noblest with me.\n\nI was very much concerned by hearing hurriedly from Georgy that you\nwere ill. But when I came home at night, she showed me Katie's letter,\nand that set me up again. Ah, you have the best of companions and\nnurses, and can afford to be ill now and then for the happiness of being\nso brought through it. But don't do it again yet awhile for all that.\n\nLegouve (whom you remember in Paris as writing for the Ristori) was\nanxious that I should bring you the enclosed. A manly and generous\neffort, I think? Regnier desired to be warmly remembered to you. He\nlooks just as of yore.\n\nParis generally is about as wicked and extravagant as in the days of the\nRegency. Madame Viardot in the \"Orphee,\" most splendid. An opera of\n\"Faust,\" a very sad and noble rendering of that sad and noble story.\nStage management remarkable for some admirable, and really poetical,\neffects of light. In the more striking situations, Mephistopheles\nsurrounded by an infernal red atmosphere of his own. Marguerite by a\npale blue mournful light. The two never blending. After Marguerite has\ntaken the jewels placed in her way in the garden, a weird evening draws\non, and the bloom fades from the flowers, and the leaves of the trees\ndroop and lose their fresh green, and mournful shadows overhang her\nchamber window, which was innocently bright and gay at first. I couldn't\nbear it, and gave in completely.\n\nFechter doing wonders over the way here, with a picturesque French\ndrama. Miss Kate Terry, in a small part in it, perfectly charming. You\nmay remember her making a noise, years ago, doing a boy at an inn, in\n\"The Courier of Lyons\"? She has a tender love-scene in this piece, which\nis a really beautiful and artistic thing. I saw her do it at about three\nin the morning of the day when the theatre opened, surrounded by\nshavings and carpenters, and (of course) with that inevitable hammer\ngoing; and I told Fechter: \"That is the very best piece of womanly\ntenderness I have ever seen on the stage, and you'll find that no\naudience can miss it.\" It is a comfort to add that it was instantly\nseized upon, and is much talked of.\n\nStanfield was very ill for some months, then suddenly picked up, and is\nreally rosy and jovial again. Going to see him when he was very\ndespondent, I told him the story of Fechter's piece (then in rehearsal)\nwith appropriate action; fighting a duel with the washing-stand, defying\nthe bedstead, and saving the life of the sofa-cushion. This so kindled\nhis old theatrical ardour, that I think he turned the corner on the\nspot.\n\nWith love to Mrs. Macready and Katie, and (be still my heart!)\nBenvenuta, and the exiled Johnny (not too attentive at school, I hope?),\nand the personally-unknown young Parr,\n\n                    Ever, my dearest Macready, your most affectionate.\n\n\n[Sidenote: Miss Power.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                          _Thursday, Feb. 26th, 1863._\n\nMY DEAR MARGUERITE,\n\nI think I have found a first-rate title for your book, with an early and\na delightful association in most people's minds, and a strong suggestion\nof Oriental pictures:\n\n                        \"ARABIAN DAYS AND NIGHTS.\"\n\nI have sent it to Low's. If they have the wit to see it, do you in your\nfirst chapter touch that string, so as to bring a fanciful explanation\nin aid of the title, and sound it afterwards, now and again, when you\ncome to anything where Haroun al Raschid, and the Grand Vizier, and\nMesrour, the chief of the guard, and any of that wonderful _dramatis\npersonae_ are vividly brought to mind.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Mr. Charles Knight.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                         _Wednesday, March 4th, 1863._\n\nMY DEAR CHARLES KNIGHT,\n\nAt a quarter to seven on Monday, the 16th, a stately form will be\ndescried breathing birthday cordialities and affectionate amenities, as\nit descends the broken and gently dipping ground by which the level\ncountry of the Clifton Road is attained. A practised eye will be able to\ndiscern two humble figures in attendance, which from their flowing\ncrinolines may, without exposing the prophet to the imputation of\nrashness, be predicted to be women. Though certes their importance,\nabsorbed and as it were swallowed up in the illustrious bearing and\ndetermined purpose of the maturer stranger, will not enthrall the gaze\nthat wanders over the forest of San Giovanni as the night gathers in.\n\n                                           Ever affectionately,\n                                                       G. P. R. JAMES.\n\n\n[Sidenote: Mrs. Dallas.[9]]\n\n\nEXTRACT.\n\nTHE TIME OF THE PRINCESS ALEXANDRA'S ARRIVAL IN LONDON.\n\nIt is curious to see London gone mad. Down in the Strand here, the\nmonomaniacal tricks it is playing are grievous to behold, but along\nFleet Street and Cheapside it gradually becomes frenzied, dressing\nitself up in all sorts of odds and ends, and knocking itself about in a\nmost amazing manner. At London Bridge it raves, principally about the\nKings of Denmark and their portraits. I have been looking among them for\nHamlet's uncle, and have discovered one personage with a high nose, who\nI think is the man.\n\n                                              Faithfully yours always.\n\n\n[Sidenote: Mrs. Lehmann.]\n\n      OFFICE OF \"ALL THE YEAR ROUND,\" NO. 26, WELLINGTON STREET,\n                               STRAND, LONDON, W.C.,\n                                          _Tuesday, March 10th, 1863._\n\nDEAR MRS. LEHMANN,\n\nTwo stalls for to-morrow's reading were sent to you by post before I\nheard from you this morning. Two will always come to you while you\nremain a Gummidge, and I hope I need not say that if you want more, none\ncould be better bestowed in my sight.\n\nPray tell Lehmann, when you next write to him, that I find I owe him a\nmint of money for the delightful Swedish sleigh-bells. They are the\nwonder, awe, and admiration of the whole country side, and I never go\nout without them.\n\nLet us make an exchange of child stories. I heard of a little fellow the\nother day whose mamma had been telling him that a French governess was\ncoming over to him from Paris, and had been expatiating on the blessings\nand advantages of having foreign tongues. After leaning his plump little\ncheek against the window glass in a dreary little way for some minutes,\nhe looked round and enquired in a general way, and not as if it had any\nspecial application, whether she didn't think \"that the Tower of Babel\nwas a great mistake altogether?\"\n\n                                                Ever faithfully yours.\n\n\n[Sidenote: Mrs. Major.[10]]\n\n  OFFICE OF \"ALL THE YEAR ROUND,\" A WEEKLY JOURNAL, ETC. ETC.,\n                     26, WELLINGTON STREET, STRAND,\n                                         _Thursday, March 12th, 1863._\n\nMY DEAR MARY,\n\nI am quite concerned to hear that you and your party (including your\nbrother Willie) paid for seats at my reading last night. You must\npromise me never to do so any more. My old affections and attachments\nare not so lightly cherished or so easily forgotten as that I can bear\nthe thought of you and yours coming to hear me like so many strangers.\nIt will at all times delight me if you will send a little note to me, or\nto Georgina, or to Mary, saying when you feel inclined to come, and how\nmany stalls you want. You may always be certain, even on the fullest\nnights, of room being made for you. And I shall always be interested and\npleased by knowing that you are present.\n\nMind! You are to be exceedingly penitent for last night's offence, and\nto make me a promise that it shall never be repeated. On which condition\naccept my noble forgiveness.\n\nWith kind regard to Mr. Major, my dear Mary,\n\n                                                 Affectionately yours.\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                         _Thursday, March 31st, 1863._\n\nMY DEAREST MACREADY,\n\nI mean to go on reading into June. For the sake of the finer effects (in\n\"Copperfield\" principally), I have changed from St. James's Hall to the\nHanover Square Room. The latter is quite a wonderful room for sound, and\nso easy that the least inflection will tell anywhere in the place\nexactly as it leaves your lips; but I miss my dear old shilling\ngalleries--six or eight hundred strong--with a certain roaring sea of\nresponse in them, that you have stood upon the beach of many and many a\ntime.\n\nThe summer, I hope and trust, will quicken the pace at which you grow\nstronger again. I am but in dull spirits myself just now, or I should\nremonstrate with you on your slowness.\n\nHaving two little boys sent home from school \"to see the illuminations\"\non the marriage-night, I chartered an enormous van, at a cost of five\npounds, and we started in majesty from the office in London, fourteen\nstrong. We crossed Waterloo Bridge with the happy design of beginning\nthe sight at London Bridge, and working our way through the City to\nRegent Street. In a by-street in the Borough, over against a dead wall\nand under a railway bridge, we were blocked for four hours. We were\nobliged to walk home at last, having seen nothing whatever. The wretched\nvan turned up in the course of the next morning; and the best of it was\nthat at Rochester here they illuminated the fine old castle, and really\nmade a very splendid and picturesque thing (so my neighbours tell me).\n\nWith love to Mrs. Macready and Katie,\n\n                    Ever, my dearest Macready, your most affectionate.\n\n\n[Sidenote: Mr. W. Wilkie Collins.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                        _Wednesday, April 22nd, 1863._\n\n\n                  ON THE DEATH OF MR. EGG.\n\nEXTRACT.\n\nAh, poor Egg! I knew what you would think and feel about it. When we saw\nhim in Paris on his way out I was struck by his extreme nervousness, and\nderived from it an uneasy foreboding of his state. What a large piece of\na good many years he seems to have taken with him! How often have I\nthought, since the news of his death came, of his putting his part in\nthe saucepan (with the cover on) when we rehearsed \"The Lighthouse;\" of\nhis falling out of the hammock when we rehearsed \"The Frozen Deep;\" of\nhis learning Italian numbers when he ate the garlic in the carriage; of\nthe thousands (I was going to say) of dark mornings when I apostrophised\nhim as \"Kernel;\" of his losing my invaluable knife in that beastly\nstage-coach; of his posting up that mysterious book[11] every night! I\nhardly know why, but I have always associated that volume most with\nVenice. In my memory of the dear gentle little fellow, he will be (as\nsince those days he always has been) eternally posting up that book at\nthe large table in the middle of our Venice sitting-room, incidentally\nasking the name of an hotel three weeks back! And his pretty house is to\nbe laid waste and sold. If there be a sale on the spot I shall try to\nbuy something in loving remembrance of him, good dear little fellow.\nThink what a great \"Frozen Deep\" lay close under those boards we acted\non! My brother Alfred, Luard, Arthur, Albert, Austin, Egg. Even among\nthe audience, Prince Albert and poor Stone! \"I heard the\"--I forget what\nit was I used to say--\"come up from the great deep;\" and it rings in my\nears now, like a sort of mad prophecy.\n\nHowever, this won't do. We must close up the ranks and march on.\n\n\n[Sidenote: Rev. W. Brookfield.]\n\n                                         GAD'S HILL, _May 17th, 1863._\n\nMY DEAR BROOKFIELD,\n\nIt occurs to me that you may perhaps know, or know of, a kind of man\nthat I want to discover.\n\nOne of my boys (the youngest) now is at Wimbledon School. He is a\ndocile, amiable boy of fair abilities, but sensitive and shy. And he\nwrites me so very earnestly that he feels the school to be confusingly\nlarge for him, and that he is sure he could do better with some\ngentleman who gave his own personal attention to the education of\nhalf-a-dozen or a dozen boys, as to impress me with the belief that I\nought to heed his conviction.\n\nHas any such phenomenon as a good and reliable man in this wise ever\ncome in your way? Forgive my troubling you, and believe me,\n\n                                                      Cordially yours.\n\n\n[Sidenote: Rev. W. Brookfield.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                                     _May 24th, 1863._\n\nMY DEAR BROOKFIELD,\n\nI am most truly obliged to you for your kind and ready help.\n\nWhen I am in town next week, I will call upon the Bishop of Natal, more\nto thank him than with the hope of profiting by that gentleman of whom\nhe writes, as the limitation to \"little boys\" seems to stop the way. I\nwant to find someone with whom this particular boy could remain; if\nthere were a mutual interest and liking, that would be a great point\ngained.\n\nWhy did the kings in the fairy tales want children? I suppose in the\nweakness of the royal intellect.\n\nConcerning \"Nickleby,\" I am so much of your mind (comparing it with\n\"Copperfield\"), that it was a long time before I could take a pleasure\nin reading it. But I got better, as I found the audience always taking\nto it. I have been trying, alone by myself, the \"Oliver Twist\" murder,\nbut have got something so horrible out of it that I am afraid to try it\nin public.\n\n                                                Ever faithfully yours.\n\n\n[Sidenote: M. de Cerjat.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                           _Thursday, May 28th, 1863._\n\nMY DEAR CERJAT,\n\nI don't wonder at your finding it difficult to reconcile your mind to a\nFrench Hamlet; but I assure you that Fechter's is a very remarkable\nperformance perfectly consistent with itself (whether it be my\nparticular Hamlet, or your particular Hamlet, or no), a coherent and\nintelligent whole, and done by a true artist. I have never seen, I\nthink, an intelligent and clear view of the whole character so well\nsustained throughout; and there is a very captivating air of romance and\npicturesqueness added, which is quite new. Rely upon it, the public were\nright. The thing could not have been sustained by oddity; it would have\nperished upon that, very soon. As to the mere accent, there is far less\ndrawback in that than you would suppose. For this reason, he obviously\nknows English so thoroughly that you feel he is safe. You are never in\npain for him. This sense of ease is gained directly, and then you think\nvery little more about it.\n\nThe Colenso and Jowett matter is a more difficult question, but\nhere again I don't go with you. The position of the writers of \"Essays\nand Reviews\" is, that certain parts of the Old Testament have done\ntheir intended function in the education of the world _as it was_;\nbut that mankind, like the individual man, is designed by the Almighty\nto have an infancy and a maturity, and that as it advances, the\nmachinery of its education must advance too. For example: inasmuch as\never since there was a sun and there was vapour, there _must have_ been\na rainbow under certain conditions, so surely it would be better now to\nrecognise that indisputable fact. Similarly, Joshua might command the\nsun to stand still, under the impression that it moved round the earth;\nbut he could not possibly have inverted the relations of the earth and\nthe sun, whatever his impressions were. Again, it is contended that the\nscience of geology is quite as much a revelation to man, as books of an\nimmense age and of (at the best) doubtful origin, and that your\nconsideration of the latter must reasonably be influenced by the former.\nAs I understand the importance of timely suggestions such as these, it\nis, that the Church should not gradually shock and lose the more\nthoughtful and logical of human minds; but should be so gently and\nconsiderately yielding as to retain them, and, through them, hundreds\nof thousands. This seems to me, as I understand the temper and tendency\nof the time, whether for good or evil, to be a very wise and necessary\nposition. And as I understand the danger, it is not chargeable on those\nwho take this ground, but on those who in reply call names and argue\nnothing. What these bishops and such-like say about revelation, in\nassuming it to be finished and done with, I can't in the least understand.\nNothing is discovered without God's intention and assistance, and I\nsuppose every new knowledge of His works that is conceded to man to be\ndistinctly a revelation by which men are to guide themselves. Lastly,\nin the mere matter of religious doctrine and dogmas, these men\n(Protestants--protestors--successors of the men who protested against\nhuman judgment being set aside) talk and write as if they were all\nsettled by the direct act of Heaven; not as if they had been, as we know\nthey were, a matter of temporary accommodation and adjustment among\ndisputing mortals as fallible as you or I.\n\nComing nearer home, I hope that Georgina is almost quite well. She has\nno attack of pain or flurry now, and is in all respects immensely\nbetter. Mary is neither married nor (that I know of) going to be. She\nand Katie and a lot of them have been playing croquet outside my window\nhere for these last four days, to a mad and maddening extent. My\nsailor-boy's ship, the _Orlando_, is fortunately in Chatham Dockyard--so\nhe is pretty constantly at home--while the shipwrights are repairing a\nleak in her. I am reading in London every Friday just now. Great crams\nand great enthusiasm. Townshend I suppose to have left Lausanne\nsomewhere about this day. His house in the park is hermetically sealed,\nready for him. The Prince and Princess of Wales go about (wisely) very\nmuch, and have as fair a chance of popularity as ever prince and\nprincess had. The City ball in their honour is to be a tremendously\ngorgeous business, and Mary is highly excited by her father's being\ninvited, and she with him. Meantime the unworthy parent is devising all\nkinds of subterfuges for sending her and getting out of it himself. A\nvery intelligent German friend of mine, just home from America,\nmaintains that the conscription will succeed in the North, and that the\nwar will be indefinitely prolonged. _I_ say \"No,\" and that however mad\nand villainous the North is, the war will finish by reason of its not\nsupplying soldiers. We shall see. The more they brag the more I don't\nbelieve in them.\n\n       *       *       *       *       *\n\n\n[Sidenote: Mr. Percy Fitzgerald.]\n\n\n                   GAD'S HILL PLACE, _Saturday Night, July 4th_, 1863.\n\nMY DEAR MR. FITZGERALD,\n\nI have been most heartily gratified by the perusal of your article on my\ndogs. It has given me an amount and a kind of pleasure very unusual, and\nfor which I thank you earnestly. The owner of the renowned dog Caesar\nunderstands me so sympathetically, that I trust with perfect confidence\nto his feeling what I really mean in these few words. You interest me\nvery much by your kind promise, the redemption of which I hereby claim,\nto send me your life of Sterne when it comes out. If you should be in\nEngland before this, I should be delighted to see you here on the top of\nFalstaff's own Gad's Hill. It is a very pretty country, not thirty\nmiles from London; and if you could spare a day or two for its fine\nwalks, I and my two latest dogs, a St. Bernard and a bloodhound, would\nbe charmed with your company as one of ourselves.\n\n                                    Believe me, very faithfully yours.\n\n\n                                        _Friday, July 10th, 1863._[12]\n\nDEAR MADAM,\n\nI hope you will excuse this tardy reply to your letter. It is often\nimpossible for me, by any means, to keep pace with my correspondents. I\nmust take leave to say, that if there be any general feeling on the part\nof the intelligent Jewish people, that I have done them what you\ndescribe as \"a great wrong,\" they are a far less sensible, a far less\njust, and a far less good-tempered people than I have always supposed\nthem to be. Fagin, in \"Oliver Twist,\" is a Jew, because it unfortunately\nwas true of the time to which that story refers, that that class of\ncriminal almost invariably was a Jew. But surely no sensible man or\nwoman of your persuasion can fail to observe--firstly, that all the rest\nof the wicked _dramatis personae_ are Christians; and secondly, that he\nis called the \"Jew,\" not because of his religion, but because of his\nrace. If I were to write a story, in which I described a Frenchman or a\nSpaniard as \"the Roman Catholic,\" I should do a very indecent and\nunjustifiable thing; but I make mention of Fagin as the Jew, because he\nis one of the Jewish people, and because it conveys that kind of idea of\nhim which I should give my readers of a Chinaman, by calling him a\nChinese.\n\nThe enclosed is quite a nominal subscription towards the good object in\nwhich you are interested; but I hope it may serve to show you that I\nhave no feeling towards the Jewish people but a friendly one. I always\nspeak well of them, whether in public or in private, and bear my\ntestimony (as I ought to do) to their perfect good faith in such\ntransactions as I have ever had with them; and in my \"Child's History of\nEngland,\" I have lost no opportunity of setting forth their cruel\npersecution in old times.\n\n                                         Dear Madam, faithfully yours.\n\n\nIn reply to this, the Jewish lady thanks him for his kind letter and its\nenclosure, still remonstrating and pointing out that though, as he\nobserves, \"all the other criminal characters were Christians, they are,\nat least, contrasted with characters of good Christians; this wretched\nFagin stands alone as the Jew.\"\n\nThe reply to _this_ letter afterwards was the character of Riah, in \"Our\nMutual Friend,\" and some favourable sketches of Jewish character in the\nlower class, in some articles in \"All the Year Round.\"\n\n\n[Sidenote: Mr. Ouvry.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                   _Wednesday Night, July 29th, 1863._\n\nMY DEAR OUVRY,\n\nI have had some undefined idea that you were to let me know if you were\ncoming to the archaeologs at Rochester. (I myself am keeping out of their\nway, as having had enough of crowding and speech-making in London.) Will\nyou tell me where you are, whether you are in this neighbourhood or out\nof it, whether you will come here on Saturday and stay till Monday or\ntill Tuesday morning? If you will come, I _know_ I can give you the\nheartiest welcome in Kent, and I _think_ I can give you the best wine in\nthis part of it. Send me a word in reply. I will fetch you from\nanywhere, at any indicated time.\n\nWe have very pretty places in the neighbourhood, and are not\nuncomfortable people (I believe) to stay with.\n\n                                                Faithfully yours ever.\n\n\n[Sidenote: Mr. Charles Reade.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                        _Wednesday, Sept. 30th, 1863._\n\nMY DEAR READE,\n\nI _must_ write you one line to say how interested I am in your story,\nand to congratulate you upon its admirable art and its surprising grace\nand vigour.\n\nAnd to hint my hope, at the same time, that you will be able to find\nleisure for a little dash for the Christmas number. It would be a really\ngreat and true pleasure to me if you could.\n\n                                              Faithfully yours always.\n\n\n[Sidenote: Miss Hogarth.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                          _Wednesday, Oct. 7th, 1863._\n\nMY DEAREST GEORGY,\n\nYou will see by to-day's _Times_ that it _was_ an earthquake that shook\nme, and that my watch showed exactly the same time as the man's who\nwrites from Blackheath so near us--twenty minutes past three.\n\nIt is a great satisfaction to me to make it out so precisely; I wish you\nwould enquire whether the servants felt it. I thought it was the voice\nof the cook that answered me, but that was nearly half an hour later. I\nam strongly inclined to think that there is a peculiar susceptibility in\niron--at all events in our part of the country--to the shock, as though\nthere were something magnetic in it. For, whereas my long iron bedstead\nwas so violently shaken, I certainly heard nothing rattle in the room.\n\nI will write about my return as soon as I get on with the still unbegun\n\"Uncommercial.\"\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Mr. W. H. Wills.]\n\n                                GAD'S HILL, _Sunday, Dec. 20th, 1863._\n\nMY DEAR WILLS,\n\nI am clear that you took my cold. Why didn't you do the thing\ncompletely, and take it away from me? for it hangs by me still.\n\nWill you tell Mrs. Linton that in looking over her admirable account\n(_most_ admirable) of Mrs. Gordon's book, I have taken out the\nreferences to Lockhart, not because I in the least doubt their justice,\nbut because I knew him and he liked me; and because one bright day in\nRome, I walked about with him for some hours when he was dying fast, and\nall the old faults had faded out of him, and the now ghost of the\nhandsome man I had first known when Scott's daughter was at the head of\nhis house, had little more to do with this world than she in her grave,\nor Scott in his, or small Hugh Littlejohn in his. Lockhart had been\nanxious to see me all the previous day (when I was away on the\nCampagna), and as we walked about I knew very well that _he_ knew very\nwell why. He talked of getting better, but I never saw him again. This\nmakes me stay Mrs. Linton's hand, gentle as it is.\n\nMrs. Lirriper is indeed a most brilliant old lady. God bless her.\n\nI am glad to hear of your being \"haunted,\" and hope to increase your\nstock of such ghosts pretty liberally.\n\n                                                      Ever faithfully.\n\nFOOTNOTES:\n\n[8] Alluding to a translation of a play by M. Maquet, which M. Fechter\nwas then preparing for his theatre.\n\n[9] Now Mrs. Dallas Glyn.\n\n[10] Formerly Miss Talfourd.\n\n[11] His travelling journal.\n\n[12] Answer to letter from Jewish lady, remonstrating with him on\ninjustice to the Jews, shown in the character of Fagin, and asking for\nsubscription for the benefit of the Jewish poor.\n\n\n\n\n1864.\n\nNARRATIVE.\n\n\nCharles Dickens was, as usual, at Gad's Hill, with a family and friendly\nparty, at the opening of this year, and had been much shocked and\ndistressed by the news of the sudden death of Mr. Thackeray, brought to\nhim by friends arriving from London on the Christmas Eve of 1863, the\nday on which the sad event happened. He writes of it, in the first\nletter of the year, to Mr. Wilkie Collins, who was passing the winter in\nItaly. He tells him, also, of his having got well to work upon a new\nserial story, the first number of which (\"Our Mutual Friend\") was\npublished on the 1st of May.\n\nThe year began very sadly for Charles Dickens. On the 7th of February\n(his own birthday) he received the mournful announcement of the death of\nhis second son, Walter Landor (a lieutenant in the 42nd Royal\nHighlanders), who had died quite suddenly at Calcutta, on the last night\nof the year of 1863, at the age of twenty-three. His third son, Francis\nJeffrey, had started for India at the end of January.\n\nHis annual letter to M. de Cerjat contains an allusion to \"another\ngeneration beginning to peep above the table\"--the children of his son\nCharles, who had been married three years before, to Miss Bessie Evans.\n\nIn the middle of February he removed to a house in London (57,\nGloucester Place, Hyde Park), where he made a stay of the usual\nduration, up to the middle of June, all the time being hard at work upon\n\"Our Mutual Friend\" and \"All the Year Round.\" Mr. Marcus Stone was the\nillustrator of the new monthly work, and we give a specimen of one of\nmany letters which he wrote to him about his \"subjects.\"\n\nHis old friend, Mr. Charles Knight, with whom for many years Charles\nDickens had dined on his birthday, was staying, this spring, in the Isle\nof Wight. To him he writes of the death of Walter, and of another sad\ndeath which happened at this time, and which affected him almost as\nmuch. Clara, the last surviving daughter of Mr. and Mrs. White, who had\nbeen happily married to Mr. Gordon, of Cluny, not more than two years,\nhad just died at Bonchurch. Her father, as will be seen by the touching\nallusion to him in this letter, had died a short time after this\ndaughter's marriage.\n\nA letter to Mr. Edmund Ollier has reference to certain additions which\nCharles Dickens wished him to make to an article (by Mr. Ollier) on\nWorking Men's Clubs, published in \"All the Year Round.\"\n\nWe are glad to have one letter to the Lord Chief Baron, Sir Frederick\nPollock, which shows the great friendship and regard Charles Dickens had\nfor him, and his admiration of his qualities in his judicial capacity.\n\nWe give a pleasant letter to Mrs. Storrar, for whom, and for her\nhusband, Dr. Storrar, Charles Dickens had affectionate regard, because\nwe are glad to have their names in our book. The letter speaks for\nitself and needs no explanation.\n\nThe latter part of the year was uneventful. Hard at work, he passed the\nsummer and autumn at Gad's Hill, taking holidays by receiving visitors\nat home (among them, this year, Sir J. Emerson Tennent, his wife and\ndaughter, who were kindly urgent for his paying them a return visit in\nIreland) and occasional \"runs\" into France. The last letters we give are\nhis annual one to M. de Cerjat, and a graceful little New Year's note to\nhis dear old friend \"Barry Cornwall.\"\n\nThe Christmas number was \"Mrs. Lirriper's Legacy,\" the first and last\npart written by himself, as in the case of the previous year's \"Mrs.\nLirriper.\"\n\n\n[Sidenote: Mr. W. Wilkie Collins.]\n\n                                GAD'S HILL, _Monday, Jan. 24th, 1864._\n\n                              EXTRACT.\n\nMY DEAR WILKIE,\n\nI am horribly behindhand in answering your welcome letter; but I have\nbeen so busy, and have had the house so full for Christmas and the New\nYear, and have had so much to see to in getting Frank out to India,\nthat I have not been able to settle down to a regular long letter, which\nI mean this to be, but which it may not turn out to be, after all.\n\nFirst, I will answer your enquiries about the Christmas number and the\nnew book. The Christmas number has been the greatest success of all; has\nshot ahead of last year; has sold about two hundred and twenty thousand;\nand has made the name of Mrs. Lirriper so swiftly and domestically\nfamous as never was. I had a very strong belief in her when I wrote\nabout her, finding that she made a great effect upon me; but she\ncertainly has gone beyond my hopes. (Probably you know nothing about\nher? which is a very unpleasant consideration.) Of the new book, I have\ndone the two first numbers, and am now beginning the third. It is a\ncombination of drollery with romance which requires a great deal of\npains and a perfect throwing away of points that might be amplified; but\nI hope it is _very good_. I confess, in short, that I think it is.\nStrange to say, I felt at first quite dazed in getting back to the large\ncanvas and the big brushes; and even now, I have a sensation as of\nacting at the San Carlo after Tavistock House, which I could hardly have\nsupposed would have come upon so old a stager.\n\nYou will have read about poor Thackeray's death--sudden, and yet not\nsudden, for he had long been alarmingly ill. At the solicitation of Mr.\nSmith and some of his friends, I have done what I would most gladly have\nexcused myself from doing, if I felt I could--written a couple of pages\nabout him in what was his own magazine.\n\nConcerning the Italian experiment, De la Rue is more hopeful than you.\nHe and his bank are closely leagued with the powers at Turin, and he has\nlong been devoted to Cavour; but he gave me the strongest assurances\n(with illustrations) of the fusion between place and place, and of the\nblending of small mutually antagonistic characters into one national\ncharacter, progressing cheeringly and certainly. Of course there must be\ndiscouragements and discrepancies in the first struggles of a country\npreviously so degraded and enslaved, and the time, as yet, has been very\nshort.\n\nI should like to have a day with you at the Coliseum, and on the Appian\nWay, and among the tombs, and with the Orvieto. But Rome and I are wide\nasunder, physically as well as morally. I wonder whether the dramatic\nstable, where we saw the marionettes, still receives the Roman public?\nAnd Lord! when I think of you in that hotel, how I think of poor dear\nEgg in the long front drawing-room, giving on to the piazza, posting up\nthat wonderful necromantic volume which we never shall see opened!\n\n\n[Sidenote: Mr. Marcus Stone.]\n\n                    57, GLOUCESTER PLACE, HYDE PARK,, HYDE PARK,\n                                           _Tuesday, Feb. 23rd, 1864._\n\nMY DEAR MARCUS,\n\nI think the design for the cover _excellent_, and do not doubt its\ncoming out to perfection. The slight alteration I am going to suggest\noriginates in a business consideration not to be overlooked.\n\nThe word \"Our\" in the title must be out in the open like \"Mutual\nFriend,\" making the title three distinct large lines--\"Our\" as big as\n\"Mutual Friend.\" This would give you too much design at the bottom. I\nwould therefore take out the dustman, and put the Wegg and Boffin\ncomposition (which is capital) in its place. I don't want Mr. Inspector\nor the murder reward bill, because these points are sufficiently\nindicated in the river at the top. Therefore you can have an indication\nof the dustman in Mr. Inspector's place. Note, that the dustman's face\nshould be droll, and not horrible. Twemlow's elbow will still go out of\nthe frame as it does now, and the same with Lizzie's skirts on the\nopposite side. With these changes, work away!\n\nMrs. Boffin, as I judge of her from the sketch, \"very good, indeed.\" I\nwant Boffin's oddity, without being at all blinked, to be an oddity of a\nvery honest kind, that people will like.\n\nThe doll's dressmaker is immensely better than she was. I think she\nshould now come extremely well. A weird sharpness not without beauty is\nthe thing I want.\n\n                                                Affectionately always.\n\n\n[Sidenote: Mr. Charles Knight.]\n\n                                       57, GLOUCESTER PLACE, W.,\n                                           _Tuesday, March 1st, 1864._\n\nMY DEAR KNIGHT,\n\nWe knew of your being in the Isle of Wight, and had said that we should\nhave this year to drink your health in your absence. Rely on my being\nalways ready and happy to renew our old friendship in the flesh. In the\nspirit it needs no renewal, because it has no break.\n\nAh, poor Mrs. White! A sad, sad story! It is better for poor White that\nthat little churchyard by the sea received his ashes a while ago, than\nthat he should have lived to this time.\n\nMy poor boy was on his way home from an up-country station, on sick\nleave. He had been very ill, but was not so at the time. He was talking\nto some brother-officers in the Calcutta hospital about his preparations\nfor home, when he suddenly became excited, had a rush of blood from the\nmouth, and was dead. His brother Frank would arrive out at Calcutta,\nexpecting to see him after six years, and he would have been dead a\nmonth.\n\nMy \"working life\" is resolving itself at the present into another book,\nin twenty green leaves. You work like a Trojan at Ventnor, but you do\nthat everywhere; and that's why you are so young.\n\nMary and Georgina unite in kindest regard to you, and to Mrs. Knight,\nand to your daughters. So do I. And I am ever, my dear Knight,\n\n                                                 Affectionately yours.\n\nP.S.--Serene View! What a placid address!\n\n\n[Sidenote: Mr. Edmund Ollier.]\n\n                           \"ALL THE YEAR ROUND\" OFFICE, _March, 1864._\n\n                                 EXTRACT.\n\nI want the article on \"Working Men's Clubs\" to refer back to \"The Poor\nMan and his Beer\" in No. 1, and to maintain the principle involved in\nthat effort.\n\nAlso, emphatically, to show that trustfulness is at the bottom of all\nsocial institutions, and that to trust a man, as one of a body of men,\nis to place him under a wholesome restraint of social opinion, and is a\nvery much better thing than to make a baby of him.\n\nAlso, to point out that the rejection of beer in this club, tobacco in\nthat club, dancing or what-not in another club, are instances that such\nclubs are founded on mere whims, and therefore cannot successfully\naddress human nature in the general, and hope to last.\n\nAlso, again to urge that patronage is the curse and blight of all such\nendeavours, and to impress upon the working men that they must originate\nand manage for themselves. And to ask them the question, can they\npossibly show their detestation of drunkenness better, or better strive\nto get rid of it from among them, than to make it a hopeless\ndisqualification in all their clubs, and a reason for expulsion.\n\nAlso, to encourage them to declare to themselves and their fellow\nworking men that they want social rest and social recreation for\nthemselves and their families; and that these clubs are intended for\nthat laudable and necessary purpose, and do not need educational\npretences or flourishes. Do not let them be afraid or ashamed of wanting\nto be amused and pleased.\n\n\n[Sidenote: The Lord Chief Baron.]\n\n                    57, GLOUCESTER PLACE, _Tuesday, March 15th, 1864._\n\nMY DEAR CHIEF BARON,\n\nMany thanks for your kind letter, which I find on my return from a\nweek's holiday.\n\nYour answer concerning poor Thackeray I will duly make known to the\nactive spirit in that matter, Mr. Shirley Brooks.\n\nYour kind invitation to me to come and see you and yours, and hear the\nnightingales, I shall not fail to discuss with Forster, and with an eye\nto spring. I expect to see him presently; the rather as I found a note\nfrom him when I came back yesterday, describing himself somewhat\ngloomily as not having been well, and as feeling a little out of heart.\n\nIt is not out of order, I hope, to remark that you have been much in my\nthoughts and on my lips lately? For I really have not been able to\nrepress my admiration of the vigorous dignity and sense and spirit, with\nwhich one of the best of judges set right one of the dullest of juries\nin a recent case.\n\n                               Believe me ever, very faithfully yours.\n\n\n[Sidenote: Mr. John Forster.]\n\n                     57, GLOUCESTER PLACE, _Tuesday, March 29th, 1864._\n\nMY DEAR FORSTER,\n\nI meant to write to you last night, but to enable Wills to get away I\nhad to read a book of Fitzgerald's through before I went to bed.\n\nConcerning Eliot, I sat down, as I told you, and read the book through\nwith the strangest interest and the highest admiration. I believe it to\nbe as honest, spirited, patient, reliable, and gallant a piece of\nbiography as ever was written, the care and pains of it astonishing, the\ncompleteness of it masterly; and what I particularly feel about it is\nthat the dignity of the man, and the dignity of the book that tells\nabout the man, always go together, and fit each other. This same quality\nhas always impressed me as the great leading speciality of the\nGoldsmith, and enjoins sympathy with the subject, knowledge of it, and\npursuit of it in its own spirit; but I think it even more remarkable\nhere. I declare that apart from the interest of having been so put into\nthe time, and enabled to understand it, I personally feel quite as much\nthe credit and honour done to literature by such a book. It quite clears\nout of the remembrance a thousand pitiful things, and sets one up in\nheart again. I am not surprised in the least by Bulwer's enthusiasm. I\nwas as confident about the effect of the book when I closed the first\nvolume, as I was when I closed the second with a full heart. No man less\nin earnest than Eliot himself could have done it, and I make bold to add\nthat it never could have been done by a man who was so distinctly born\nto do the work as Eliot was to do his.\n\nSaturday at Hastings I must give up. I have wavered and considered, and\nconsidered and wavered, but if I take that sort of holiday, I must have\na day to spare after it, and at this critical time I have not. If I were\nto lose a page of the five numbers I have purposed to myself to be\nready by the publication day, I should feel that I had fallen short. I\nhave grown hard to satisfy, and write very slowly, and I have so much\nbad fiction, that _will_ be thought of when I don't want to think of it,\nthat I am forced to take more care than I ever took.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Mrs. Storrar.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                     _Sunday Morning, May 15th, 1864._\n\nMY DEAR MRS. STORRAR,\n\nOur family dinner must come off at Gad's Hill, where I have improvements\nto exhibit, and where I shall be truly pleased to see you and the doctor\nagain. I have deferred answering your note, while I have been scheming\nand scheming for a day between this time and our departure. But it is\nall in vain. My engagements have accumulated, and become such a whirl,\nthat no day is left me. Nothing is left me but to get away. I look\nforward to my release from this dining life with an inexpressible\nlonging after quiet and my own pursuits. What with public speechifying,\nprivate eating and drinking, and perpetual simmering in hot rooms, I\nhave made London too hot to hold me and my work together. Mary and\nGeorgina acknowledge the condition of imbecility to which we have become\nreduced in reference to your kind reminder. They say, when I stare at\nthem in a forlorn way with your note in my hand: \"What CAN you do!\" To\nwhich I can only reply, implicating them: \"See what you have brought me\nto!\"\n\nWith our united kind regard to yourself and Dr. Storrar, I entreat your\npity and compassion for an unfortunate wretch whom a too-confiding\ndisposition has brought to this pass. If I had not allowed my \"cheeild\"\nto pledge me to all manner of fellow-creatures, I and my digestion might\nhave been in a state of honourable independence this day.\n\n                                      Faithfully and penitently yours.\n\n\n[Sidenote: Mr. Percy Fitzgerald.]\n\n                  OFFICE OF \"ALL THE YEAR ROUND,\" ETC. ETC. ETC.\n                                         _Wednesday, July 27th, 1864._\n\nMY DEAR MR. FITZGERALD,\n\nFirst, let me assure you that it gave us all real pleasure to see your\nsister and you at Gad's Hill, and that we all hope you will both come\nand stay a day or two with us when you are next in England.\n\nNext, let me convey to you the intelligence that I resolve to launch\n\"Miss Manuel,\" fully confiding in your conviction of the power of the\nstory. On all business points, Wills will communicate with you. I\npurpose beginning its publication in our first September number,\ntherefore there is no time to be lost.\n\nThe only suggestion I have to make as to the MS. in hand and type is,\nthat Captain Fermor wants relief. It is a disagreeable character, as you\nmean it to be, and I should be afraid to do so much with him, if the\ncase were mine, without taking the taste of him, here and there, out of\nthe reader's mouth. It is remarkable that if you do not administer a\ndisagreeable character carefully, the public have a decided tendency to\nthink that the _story_ is disagreeable, and not merely the fictitious\nperson.\n\nWhat do you think of the title,\n\n                              NEVER FORGOTTEN?\n\nIt is a good one in itself, would express the eldest sister's pursuit,\nand glanced at now and then in the text, would hold the reader in\nsuspense. I would propose to add the line,\n\n                       BY THE AUTHOR OF BELLA DONNA.\n\nLet me know your opinion as to the title. I need not assure you that the\ngreatest care will be taken of you here, and that we shall make you as\nthoroughly well and widely known as we possibly can.\n\n                                                Very faithfully yours.\n\n\n[Sidenote: Sir James Emerson Tennent.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                            _Friday, Aug. 26th, 1864._\n\nMY DEAR TENNENT,\n\nBelieve me, I fully intended to come to you--did not doubt that I should\ncome--and have greatly disappointed Mary and her aunt, as well as\nmyself, by not coming. But I do not feel safe in going out for a visit.\nThe mere knowledge that I had such a thing before me would put me out.\nIt is not the length of time consumed, or the distance traversed, but it\nis the departure from a settled habit and a continuous sacrifice of\npleasures that comes in question. This is an old story with me. I have\nnever divided a book of my writing with anything else, but have always\nwrought at it to the exclusion of everything else; and it is now too\nlate to change.\n\nAfter receiving your kind note I resolved to make another trial. But the\nhot weather and a few other drawbacks did not mend the matter, for I\nhave dropped astern this month instead of going ahead. So I have seen\nForster, and shown him my chains, and am reduced to taking exercise in\nthem, like Baron Trenck.\n\nI am heartily pleased that you set so much store by the dedication. You\nmay be sure that it does not make me the less anxious to take pains, and\nto work out well what I have in my mind.\n\nMary and Georgina unite with me in kindest regards to Lady Tennent and\nMiss Tennent, and wish me to report that while they are seriously\ndisappointed, they still feel there is no help for it. I can testify\nthat they had great pleasure in the anticipation of the visit, and that\ntheir faces were very long and blank indeed when I began to hint my\ndoubts. They fought against them valiantly as long as there was a\nchance, but they see my difficulty as well as anyone not myself can.\n\n                  Believe me, my dear Tennent, ever faithfully yours.\n\n\n[Sidenote: Mr. Clarkson Stanfield, R.A.]\n\n                          THE ATHENAEUM, _Wednesday, Sept. 21st, 1864._\n\nMY DEAR STANNY,\n\nI met George in the street a few days ago, and he gave me a wonderful\naccount of the effect of your natural element upon you at Ramsgate. I\nexpect you to come back looking about twenty-nine, and feeling about\nnineteen.\n\nThis morning I have looked in here to put down Fechter as a candidate,\non the chance of the committee's electing him some day or other. He is a\nmost devoted worshipper of yours, and would take it as a great honour if\nyou would second him. Supposing you to have not the least objection (of\ncourse, if you should have any, I can in a moment provide a substitute),\nwill you write your name in the candidates' book as his seconder when\nyou are next in town and passing this way?\n\nLastly, if you should be in town on his opening night (a Saturday, and\nin all probability the 22nd of October), will you come and dine at the\noffice and see his new piece? You have not yet \"pronounced\" in the\nmatter of that new French stage of his, on which Calcott for the said\nnew piece has built up all manner of villages, camps, Versailles\ngardens, etc. etc. etc. etc., with no wings, no flies, no looking off\nin any direction. If you tell me that you are to be in town by that\ntime, I will not fail to refresh your memory as to the precise day.\n\n        With kind regard to Mrs. Stanfield,\n                Believe me, my dear old boy, ever your affectionate\n                                                                 DICK.\n\n\n[Sidenote: M. de Cerjat.]\n\n                          GAD'S HILL PLACE, HIGHAM BY ROCHESTER,\n                                           _Tuesday, Oct. 25th, 1864._\n\nMY DEAR CERJAT,\n\nHere is a limping brute of a reply to your always-welcome Christmas\nletter! But, as usual, when I have done my day's work, I jump up from my\ndesk and rush into air and exercise, and find letter-writing the most\ndifficult thing in my daily life.\n\nI hope that your asthmatic tendencies may not be strong just now; but\nTownshend's account of the premature winter at Lausanne is not\nencouraging, and with us here in England all such disorders have been\naggravated this autumn. However, a man of your dignity _must_ have\neither asthma or gout, and I hope you have got the better of the two.\n\nIn London there is, as you see by the papers, extraordinarily little\nnews. At present the apprehension (rather less than it was thought) of a\ncommercial crisis, and the trial of Mueller next Thursday, are the two\nchief sensations. I hope that gentleman will be hanged, and have hardly\na doubt of it, though croakers contrariwise are not wanting. It is\ndifficult to conceive any other line of defence than that the\ncircumstances proved, taken separately, are slight. But a sound judge\nwill immediately charge the jury that the strength of the circumstances\nlies in their being put together, and will thread them together on a\nfatal rope.\n\nAs to the Church, my friend, I am sick of it. The spectacle presented by\nthe indecent squabbles of priests of most denominations, and the\nexemplary unfairness and rancour with which they conduct their\ndifferences, utterly repel me. And the idea of the Protestant\nestablishment, in the face of its own history, seeking to trample out\ndiscussion and private judgment, is an enormity so cool, that I wonder\nthe Right Reverends, Very Reverends, and all other Reverends, who commit\nit, can look in one another's faces without laughing, as the old\nsoothsayers did. Perhaps they can't and don't. How our sublime and\nso-different Christian religion is to be administered in the future I\ncannot pretend to say, but that the Church's hand is at its own throat I\nam fully convinced. Here, more Popery, there, more Methodism--as many\nforms of consignment to eternal damnation as there are articles, and all\nin one forever quarrelling body--the Master of the New Testament put out\nof sight, and the rage and fury almost always turning on the letter of\nobscure parts of the Old Testament, which itself has been the subject of\naccommodation, adaptation, varying interpretation without end--these\nthings cannot last. The Church that is to have its part in the coming\ntime must be a more Christian one, with less arbitrary pretensions and a\nstronger hold upon the mantle of our Saviour, as He walked and talked\nupon this earth.\n\nOf family intelligence I have very little. Charles Collins continuing in\na very poor way, and showing no signs of amendment. He and my daughter\nKatie went to Wiesbaden and thence to Nice, where they are now. I have\nstrong apprehensions that he will never recover, and that she will be\nleft a young widow. All the rest are as they were. Mary neither married\nnor going to be; Georgina holding them all together and perpetually\ncorresponding with the distant ones; occasional rallyings coming off\nhere, in which another generation begins to peep above the table. I once\nused to think what a horrible thing it was to be a grandfather. Finding\nthat the calamity falls upon me without my perceiving any other change\nin myself, I bear it like a man.\n\nMrs. Watson has bought a house in town, to which she repairs in the\nseason, for the bringing out of her daughter. She is now at Rockingham.\nHer eldest son is said to be as good an eldest son as ever was, and to\nmake her position there a perfectly independent and happy one. I have\nnot seen him for some years; her I often see; but he ought to be a good\nfellow, and is very popular in his neighbourhood.\n\nI have altered this place very much since you were here, and have made a\npretty (I think an unusually pretty) drawing-room. I wish you would come\nback and see it. My being on the Dover line, and my being very fond of\nFrance, occasion me to cross the Channel perpetually. Whenever I feel\nthat I have worked too much, or am on the eve of overdoing it, and want\na change, away I go by the mail-train, and turn up in Paris or anywhere\nelse that suits my humour, next morning. So I come back as fresh as a\ndaisy, and preserve as ruddy a face as though I never leant over a sheet\nof paper. When I retire from a literary life I think of setting up as a\nChannel pilot.\n\nPray give my love to Mrs. Cerjat, and tell her that I should like to go\nup the Great St. Bernard again, and shall be glad to know if she is open\nto another ascent. Old days in Switzerland are ever fresh to me, and\nsometimes I walk with you again, after dark, outside the hotel at\nMartigny, while Lady Mary Taylour (wasn't it?) sang within very\nprettily. Lord, how the time goes! How many years ago!\n\n                                                 Affectionately yours.\n\n\n                                     _Wednesday, Nov. 16th, 1864._[13]\n\nDEAR MADAM,\n\nI have received your letter with great pleasure, and hope to be (as I\nhave always been at heart) the best of friends with the Jewish people.\nThe error you point out to me had occurred to me, as most errors do to\nmost people, when it was too late to correct it. But it will do no harm.\nThe peculiarities of dress and manner are fused together for the sake of\npicturesqueness.\n\n                                         Dear Madam, faithfully yours.\n\n\n[Sidenote: Mr. B. W. Procter.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                          _Saturday, Dec. 31st, 1864._\n\nMY DEAR PROCTER,\n\nI have reserved my acknowledgment of your delightful note (the youngest\nnote I have had in all this year) until to-day, in order that I might\nsend, most heartily and affectionately, all seasonable good wishes to\nyou and to Mrs. Procter, and to those who are nearest and dearest to\nyou. Take them from an old friend who loves you.\n\nMamie returns the tender compliments, and Georgina does what the\nAmericans call \"endorse them.\" Mrs. Lirriper is proud to be so\nremembered, and says over and over again \"that it's worth twenty times\nthe trouble she has taken with the narrative, since Barry Cornwall,\nEsquire, is pleased to like it.\"\n\nI got rid of a touch of neuralgia in France (as I always do there), but\nI found no old friends in my voyages of discovery on that side, such as\nI have left on this.\n\n                              My dear Procter, ever your affectionate.\n\n\nFOOTNOTES:\n\n[13] In answer to another letter from the \"Jewish lady,\" in which she\ngives her reasons for still being dissatisfied with the character of\nRiah.\n\n\n\n\n1865.\n\nNARRATIVE.\n\n\nFor this spring a furnished house in Somer's Place, Hyde Park, had been\ntaken, which Charles Dickens occupied, with his sister-in-law and\ndaughter, from the beginning of March until June.\n\nDuring the year he paid two short visits to France.\n\nHe was still at work upon \"Our Mutual Friend,\" two numbers of which had\nbeen issued in January and February, when the first volume was\npublished, with dedication to Sir James Emerson Tennent. The remaining\nnumbers were issued between March and November, when the complete work\nwas published in two volumes.\n\nThe Christmas number, to which Charles Dickens contributed three\nstories, was called \"Doctor Marigold's Prescriptions.\"\n\nBeing out of health, and much overworked, Charles Dickens, at the end of\nMay, took his first short holiday trip into France. And on his way home,\nand on a day afterwards so fatal to him, the 9th of June, he was in that\nmost terrible railway accident at Staplehurst. Many of our letters for\nthis year have reference to this awful experience--an experience from\nthe effects of which his nerves never wholly recovered. His letters to\nMr. Thomas Mitton and to Mrs. Hulkes (an esteemed friend and neighbour)\nare graphic descriptions of this disaster. But they do NOT tell of the\nwonderful presence of mind and energy shown by Charles Dickens when most\nof the terrified passengers were incapable of thought or action, or of\nhis gentleness and goodness to the dead and dying. The Mr. Dickenson[14]\nmentioned in the letter to Mrs. Hulkes soon recovered. He always\nconsiders that he owes his life to Charles Dickens, the latter having\ndiscovered and extricated him from beneath a carriage before it was too\nlate.\n\nOur first letter to Mr. Kent is one of congratulation upon his having\nbecome the proprietor of _The Sun_ newspaper.\n\nProfessor Owen has been so kind as to give us some notes, which we\npublish for the sake of his great name. Charles Dickens had not much\ncorrespondence with Professor Owen, but there was a firm friendship and\ngreat mutual admiration between them.\n\nThe letter to Mrs. Procter is in answer to one from her, asking Charles\nDickens to write a memoir of her daughter Adelaide, as a preface to a\ncollected edition of her poems.\n\n\n[Sidenote: Mr. William Charles Kent.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                           _Tuesday, Jan. 17th, 1865._\n\nMY DEAR KENT,\n\nI meant to have written instantly on the appearance of your paper in its\nbeautiful freshness, to congratulate you on its handsome appearance, and\nto send you my heartiest good wishes for its thriving and prosperous\ncareer. Through a mistake of the postman's, that remarkable letter has\nbeen tesselated into the Infernal Pavement instead of being delivered in\nthe Strand.\n\nWe have been looking and waiting for your being well enough to propose\nyourself for a mouthful of fresh air. Are you well enough to come on\nSunday? We shall be coming down from Charing Cross on Sunday morning,\nand I shall be going up again at nine on Monday morning.\n\nIt amuses me to find that you don't see your way with a certain \"Mutual\nFriend\" of ours. I have a horrible suspicion that you may begin to be\nfearfully knowing at somewhere about No. 12 or 13. But you shan't if I\ncan help it.\n\nYour note delighted me because it dwelt upon the places in the number\nthat _I_ dwell on. Not that that is anything new in your case, but it is\nalways new to me in the pleasure I derive from it, which is truly\ninexpressible.\n\n                                                 Ever cordially yours.\n\n\n[Sidenote: Mrs. Procter.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                         _Wednesday, Feb. 15th, 1865._\n\nMY DEAR MRS. PROCTER,\n\nOf course I will do it, and of course I will do it for the love of you\nand Procter. You can give me my brief, and we can speak about its\ndetails. Once again, of course I will do it, and with all my heart.\n\nI have registered a vow (in which there is not the least merit, for I\ncouldn't help it) that when I am, as I am now, very hard at work upon a\nbook, I never will dine out more than one day in a week. Why didn't you\nask me for the Wednesday, before I stood engaged to Lady Molesworth for\nthe Tuesday?\n\nIt is so delightful to me to sit by your side anywhere and be brightened\nup, that I lay a handsome sacrifice upon the altar of \"Our Mutual\nFriend\" in writing this note, very much against my will. But for as many\nyears as can be made consistent with my present juvenility, I always\nhave given my work the first place in my life, and what can I do now at\n35!--or at least at the two figures, never mind their order.\n\nI send my love to Procter, hoping you may appropriate a little of it by\nthe way.\n\n                                                 Affectionately yours.\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                         _Wednesday, March 1st, 1865._\n\nMY DEAREST MACREADY,\n\nI have been laid up here with a frost-bitten foot (from hard walking in\nthe snow), or you would have heard from me sooner.\n\nMy reply to Professor Agassiz is short, but conclusive. Daily seeing\nimproper uses made of confidential letters in the addressing of them to\na public audience that have no business with them, I made not long ago a\ngreat fire in my field at Gad's Hill, and burnt every letter I\npossessed. And now I always destroy every letter I receive not on\nabsolute business, and my mind is so far at ease. Poor dear Felton's\nletters went up into the air with the rest, or his highly distinguished\nrepresentative should have had them most willingly.\n\nWe never fail to drink old P.'s health on his birthday, or to make him\nthe subject of a thousand loving remembrances. With best love to Mrs.\nMacready and Katie,\n\n                          Ever, my dearest Macready,\n                                        Your most affectionate Friend.\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n                                   16, SOMER'S PLACE, HYDE PARK,\n                                   _Saturday Night, April 22nd, 1865._\n\nMY DEAREST MACREADY,\n\nA thousand thanks for your kind letter, most heartily welcome.\n\nMy frost-bitten foot, after causing me great inconvenience and much\npain, has begun to conduct itself amiably. I can now again walk my ten\nmiles in the morning without inconvenience, but am absurdly obliged to\nsit shoeless all the evening--a very slight penalty, as I detest going\nout to dinner (which killed the original old Parr by-the-bye).\n\nI am working like a dragon at my book, and am a terror to the household,\nlikewise to all the organs and brass bands in this quarter. Gad's Hill\nis being gorgeously painted, and we are here until the 1st of June. I\nwish I might hope you would be there any time this summer; I really\n_have_ made the place comfortable and pretty by this time.\n\nIt is delightful to us to hear such good news of Butty. She made so\ndeep an impression on Fechter that he always asks me what Ceylon has\ndone for her, and always beams when I tell him how thoroughly well it\nhas made her. As to _you_, you are the youngest man (worth mentioning as\na thorough man) that I know. Oh, let me be as young when I am as----did\nyou think I was going to write \"old?\" No, sir--withdrawn from the wear\nand tear of busy life is my expression.\n\nPoole still holds out at Kentish Town, and says he is dying of solitude.\nHis memory is astoundingly good. I see him about once in two or three\nmonths, and in the meantime he makes notes of questions to ask me when I\ncome. Having fallen in arrear of the time, these generally refer to\nunknown words he has encountered in the newspapers. His three last (he\nalways reads them with tremendous difficulty through an enormous\nmagnifying-glass) were as follows:\n\n        1. What's croquet?\n        2. What's an Albert chain?\n        3. Let me know the state of mind of the Queen.\n\nWhen I had delivered a neat exposition on these heads, he turned back to\nhis memoranda, and came to something that the utmost power of the\nenormous magnifying-glass couldn't render legible. After a quarter of an\nhour or so, he said: \"O yes, I know.\" And then rose and clasped his\nhands above his head, and said: \"Thank God, I am not a dram-drinker.\"\n\nDo think of coming to Gad's in the summer; and do give my love to Mrs.\nMacready, and tell her I know she can make you come if she will. Mary\nand Georgy send best and dearest loves to her, to you, and to Katie, and\nto baby. Johnny we suppose to be climbing the tree of knowledge\nelsewhere.\n\n                  My dearest Macready, ever yours most affectionately.\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n                                GAD'S HILL, _Monday, June 12th, 1865._\n\nMY DEAREST MACREADY,\n\n                    [_So far in his own writing._]\n\nMany thanks for your kind words of remembrance.[15] This is not all in\nmy own hand, because I am too much shaken to write many notes. Not by\nthe beating and dragging of the carriage in which I was--it did not go\nover, but was caught on the turn, among the ruins of the bridge--but by\nthe work afterwards to get out the dying and dead, which was terrible.\n\n                   [_The rest in his own writing_.]\n\n                                        Ever your affectionate Friend.\n\nP.S.--My love to Mrs. Macready.\n\n\n[Sidenote: Mr. Thomas Mitton.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                           _Tuesday, June 13th, 1865._\n\nMY DEAR MITTON,\n\nI should have written to you yesterday or the day before, if I had been\nquite up to writing.\n\nI was in the only carriage that did not go over into the stream. It was\ncaught upon the turn by some of the ruin of the bridge, and hung\nsuspended and balanced in an apparently impossible manner. Two ladies\nwere my fellow-passengers, an old one and a young one. This is exactly\nwhat passed. You may judge from it the precise length of the suspense:\nSuddenly we were off the rail, and beating the ground as the car of a\nhalf-emptied balloon might. The old lady cried out, \"My God!\" and the\nyoung one screamed. I caught hold of them both (the old lady sat\nopposite and the young one on my left), and said: \"We can't help\nourselves, but we can be quiet and composed. Pray don't cry out.\" The\nold lady immediately answered: \"Thank you. Rely upon me. Upon my soul I\nwill be quiet.\" We were then all tilted down together in a corner of the\ncarriage, and stopped. I said to them thereupon: \"You may be sure\nnothing worse can happen. Our danger _must_ be over. Will you remain\nhere without stirring, while I get out of the window?\" They both\nanswered quite collectedly, \"Yes,\" and I got out without the least\nnotion what had happened. Fortunately I got out with great caution and\nstood upon the step. Looking down I saw the bridge gone, and nothing\nbelow me but the line of rail. Some people in the two other compartments\nwere madly trying to plunge out at window, and had no idea that there\nwas an open swampy field fifteen feet down below them, and nothing else!\nThe two guards (one with his face cut) were running up and down on the\ndown side of the bridge (which was not torn up) quite wildly. I called\nout to them: \"Look at me. Do stop an instant and look at me, and tell me\nwhether you don't know me.\" One of them answered: \"We know you very\nwell, Mr. Dickens.\" \"Then,\" I said, \"my good fellow, for God's sake give\nme your key, and send one of those labourers here, and I'll empty this\ncarriage.\" We did it quite safely, by means of a plank or two, and when\nit was done I saw all the rest of the train, except the two baggage\nvans, down in the stream. I got into the carriage again for my brandy\nflask, took off my travelling hat for a basin, climbed down the\nbrickwork, and filled my hat with water.\n\nSuddenly I came upon a staggering man covered with blood (I think he\nmust have been flung clean out of his carriage), with such a frightful\ncut across the skull that I couldn't bear to look at him. I poured some\nwater over his face and gave him some to drink, then gave him some\nbrandy, and laid him down on the grass, and he said, \"I am gone,\" and\ndied afterwards. Then I stumbled over a lady lying on her back against a\nlittle pollard-tree, with the blood streaming over her face (which was\nlead colour) in a number of distinct little streams from the head. I\nasked her if she could swallow a little brandy and she just nodded, and\nI gave her some and left her for somebody else. The next time I passed\nher she was dead. Then a man, examined at the inquest yesterday (who\nevidently had not the least remembrance of what really passed), came\nrunning up to me and implored me to help him find his wife, who was\nafterwards found dead. No imagination can conceive the ruin of the\ncarriages, or the extraordinary weights under which the people were\nlying, or the complications into which they were twisted up among iron\nand wood, and mud and water.\n\nI don't want to be examined at the inquest, and I don't want to write\nabout it. I could do no good either way, and I could only seem to speak\nabout myself, which, of course, I would rather not do. I am keeping very\nquiet here. I have a--I don't know what to call it--constitutional (I\nsuppose) presence of mind, and was not in the least fluttered at the\ntime. I instantly remembered that I had the MS. of a number with me, and\nclambered back into the carriage for it. But in writing these scanty\nwords of recollection I feel the shake and am obliged to stop.\n\n                                                      Ever faithfully.\n\n\n[Sidenote: Mr. Walter Jones.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                        _Saturday, June 17th, 1865_.[16]\n\nSIR,\n\nI beg you to assure the Committee of the Newsvendors' Benevolent and\nProvident Institution, that I have been deeply affected by their special\nremembrance of me in my late escape from death or mutilation, and that I\nthank them with my whole heart.\n\n                                          Faithfully yours and theirs.\n\n\n[Sidenote: Mrs. Hulkes.]\n\n                                GAD'S HILL, _Sunday, June 18th, 1865._\n\nMY DEAR MRS. HULKES,\n\nI return the _Examiner_ with many thanks. The account is true, except\nthat I _had_ brandy. By an extraordinary chance I had a bottle and a\nhalf with me. I slung the half-bottle round my neck, and carried my hat\nfull of water in my hands. But I can understand the describer (whoever\nhe is) making the mistake in perfect good faith, and supposing that I\ncalled for brandy, when I really called to the others who were helping:\n\"I have brandy here.\" The Mr. Dickenson mentioned had changed places\nwith a Frenchman, who did not like the window down, a few minutes before\nthe accident. The Frenchman was killed, and a labourer and I got Mr.\nDickenson out of a most extraordinary heap of dark ruins, in which he\nwas jammed upside down. He was bleeding at the eyes, ears, nose, and\nmouth; but he didn't seem to know that afterwards, and of course I\ndidn't tell him. In the moment of going over the viaduct the whole of\nhis pockets were shaken empty! He had no watch, no chain, no money, no\npocket-book, no handkerchief, when we got him out. He had been choking\na quarter of an hour when I heard him groaning. If I had not had the\nbrandy to give him at the moment, I think he would have been done for.\nAs it was, I brought him up to London in the carriage with me, and\ncouldn't make him believe he was hurt. He was the first person whom the\nbrandy saved. As I ran back to the carriage for the whole full bottle, I\nsaw the first two people I had helped lying dead. A bit of shade from\nthe hot sun, into which we got the unhurt ladies, soon had as many dead\nin it as living.\n\n                                              Faithfully yours always.\n\n\n[Sidenote: Mr. Arthur Ryland.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                         _Wednesday, June 21st, 1865._\n\nMY DEAR MR. RYLAND,\n\nI need not assure you that I regard the unanimous desire of the Town\nCouncil Committee as a great honour, and that I feel the strongest\ninterest in the occasion, and the strongest wish to associate myself\nwith it.\n\nBut, after careful consideration, I most unwillingly come to the\nconclusion that I must decline. At the time in question I shall, please\nGod, either have just finished, or be just finishing, my present book.\nCountry rest and reflection will then be invaluable to me, before\ncasting about for Christmas. I am a little shaken in my nervous system\nby the terrible and affecting incidents of the late railway accident,\nfrom which I bodily escaped. I am withdrawing myself from engagements of\nall kinds, in order that I may pursue my story with the comfortable\nsense of being perfectly free while it is a-doing, and when it is done.\nThe consciousness of having made this engagement would, if I were to\nmake it, render such sense incomplete, and so open the way to others.\nThis is the real state of the case, and the whole reason for my\ndeclining.\n\n                                              Faithfully yours always.\n\n\n[Sidenote: Mrs. Lehmann.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                           _Tuesday, June 29th, 1865._\n\nDEAR MRS. LEHMANN,\n\nCome (with self and partner) on either of the days you name, and you\nwill be heartily welcomed by the humble youth who now addresses you, and\nwill then cast himself at your feet.\n\nI am quite right again, I thank God, and have even got my voice back; I\nmost unaccountably brought somebody else's out of that terrible scene.\nThe directors have sent me a Resolution of Thanks for assistance to the\nunhappy passengers.\n\n                             With kind regards to Lehmann, ever yours.\n\n\n[Sidenote: Mr. Percy Fitzgerald.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                             _Friday, July 7th, 1865._\n\nMY DEAR FITZGERALD,\n\nI shall be delighted to see you at Gad's Hill on Sunday, and I hope you\nwill bring a bag with you and will not think of returning to London at\nnight.\n\nWe are a small party just now, for my daughter Mary has been decoyed to\nAndover for the election week, in the Conservative interest; think of my\nfeelings as a Radical parent! The wrong-headed member and his wife are\nthe friends with whom she hunts, and she helps to receive (and\n_de_ceive) the voters, which is very awful!\n\nBut in the week after next we shall be in great croquet force. I shall\nhope to persuade you to come back to us then for a few days, and we will\ntry to make you some amends for a dull Sunday. Turn it over in your mind\nand try to manage it.\n\n                                                 Sincerely yours ever.\n\n\n[Sidenote: Professor Owen, F.R.S.]\n\n                             GAD'S HILL, _Wednesday, July 12th, 1865._\n\nMY DEAR OWEN,\n\nStudying the gorilla last night for the twentieth time, it suddenly came\ninto my head that I had never thanked you for that admirable treatise.\nThis is to bear witness to my blushes and repentance. If you knew how\nmuch interest it has awakened in me, and how often it has set me\na-thinking, you would consider me a more thankless beast than any\ngorilla that ever lived. But happily you do _not_ know, and I am not\ngoing to tell you.\n\n                                    Believe me, ever faithfully yours.\n\n\n[Sidenote: The Earl Russell.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                         _Wednesday, Aug. 16th, 1865._\n\nMY DEAR LORD RUSSELL,\n\nMr. Dallas, who is a candidate for the Scotch professional chair left\nvacant by Aytoun's death, has asked me if I would object to introduce to\nyou the first volume of a book he has in the press with my publishers,\non \"The Gay Science of Art and Criticism.\" I have replied I would _not_\nobject, as I have read as many of the sheets as I could get, with\nextreme pleasure, and as I know you will find it a very winning and\nbrilliant piece of writing. Therefore he will send the proofs of the\nvolume to you as soon as he can get them from the printer (at about the\nend of this week I take it), and if you read them you will not be hard\nupon me for bearing the responsibility of his doing so, I feel assured.\n\nI suppose Mr. Dallas to have some impression that his pleasing you with\nhis book might advance his Scottish suit. But all I know is, that he is\na gentleman of great attainments and erudition, much distinguished as\nthe writer of the best critical literary pieces in _The Times_, and\nthoroughly versed in the subjects which Professor Aytoun represented\nofficially.\n\nI beg to send my regard to Lady Russell and all the house, and am ever,\nmy dear Lord Russell,\n\n                                            Your faithful and obliged.\n\nP.S.--I am happy to report that my sailor-boy's captain, relinquishing\nhis ship on sick leave, departs from the mere form of certificate given\nto all the rest, and adds that his obedience to orders is remarkable,\nand that he is a highly intelligent and promising young officer.\n\n\n[Sidenote: Mr. Marcus Stone.]\n\n                HOTEL DU HELDER, PARIS, _Wednesday, Sept. 13th, 1865._\n\nMY DEAR MARCUS,\n\nI leave here to-morrow, and propose going to the office by tidal train\n_next Saturday evening_. Through the whole of next week, on and off, I\nshall be at the office; when not there, at Gad's; but much oftener at\nthe office. The sooner I can know about the subjects you take for\nillustration the better, as I can then fill the list of illustrations to\nthe second volume for the printer, and enable him to make up his last\nsheet. Necessarily that list is now left blank, as I cannot give him the\ntitles of the subjects, not knowing them myself.\n\nIt has been fearfully hot on this side, but is something cooler.\n\n                                            Ever affectionately yours.\n\nP.S.--On glancing over this note, I find it very like the king's\nlove-letter in \"Ruy Blas.\" \"Madam, there is a high wind. I have shot six\nwolves.\"\n\nI think the frontispiece to the second volume should be the dustyard\nwith the three mounds, and Mr. Boffin digging up the Dutch bottle, and\nVenus restraining Wegg's ardour to get at him. Or Mr. Boffin might be\ncoming down with the bottle, and Venus might be dragging Wegg out of the\nway as described.\n\n\n[Sidenote: Mr. Percy Fitzgerald.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                         _Saturday, Sept. 23rd, 1865._\n\nMY DEAR FITZGERALD,\n\nI cannot thank you too much for Sultan. He is a noble fellow, has fallen\ninto the ways of the family with a grace and dignity that denote the\ngentleman, and came down to the railway a day or two since to welcome me\nhome (it was our first meeting), with a profound absence of interest in\nmy individual opinion of him which captivated me completely. I am going\nhome to-day to take him about the country, and improve his acquaintance.\nYou will find a perfect understanding between us, I hope, when you next\ncome to Gad's Hill. (He has only swallowed Bouncer once, and\ntemporarily.)\n\nYour hint that you were getting on with your story and liked it was more\nthan golden intelligence to me in foreign parts. The intensity of the\nheat, both in Paris and the provinces, was such that I found nothing\nelse so refreshing in the course of my rambles.\n\nWith many more thanks for the dog than my sheet of paper would hold,\n\n                               Believe me, ever very faithfully yours.\n\n\n[Sidenote: Mrs. Procter.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                                   _Sept. 26th, 1865._\n\nMY DEAR MRS. PROCTER,\n\nI have written the little introduction, and have sent it to my printer,\nin order that you may read it without trouble. But if you would like to\nkeep the few pages of MS., of course they are yours.\n\nIt is brief, and I have aimed at perfect simplicity, and an avoidance of\nall that your beloved Adelaide would have wished avoided. Do not expect\ntoo much from it. If there should be anything wrong in fact, or anything\nthat you would like changed for any reason, _of course you will tell me\nso_, and of course you will not deem it possible that you can trouble me\nby making any such request most freely.\n\nYou will probably receive the proof either on Friday or Saturday. Don't\nwrite to me until you have read it. In the meantime I send you back the\ntwo books, with the two letters in the bound one.\n\n                           With love to Procter,\n                                        Ever your affectionate Friend.\n\n\n[Sidenote: Mr. Edmund Yates.]\n\n                HOTEL DU HELDER, PARIS, _Wednesday, Sept. 30th, 1865._\n\nMY DEAR EDMUND,\n\nI leave here to-morrow and purpose being at the office on Saturday\nnight; all next week I shall be there, off and on--\"off\" meaning Gad's\nHill; the office will be my last address. The heat has been excessive on\nthis side of the Channel, and I got a slight sunstroke last Thursday,\nand was obliged to be doctored and put to bed for a day; but, thank God,\nI am all right again. The man who sells the _tisane_ on the Boulevards\ncan't keep the flies out of his glasses, and as he wears them on his red\nvelvet bands, the flies work themselves into the ends of the tumblers,\ntrying to get through and tickle the man. If fly life were long enough,\nI think they would at last. Three paving blouses came to work at the\ncorner of this street last Monday, pulled up a bit of road, sat down to\nlook at it, and fell asleep. On Tuesday one of the blouses spat on his\nhands and seemed to be going to begin, but didn't. The other two have\nshown no sign of life whatever. This morning the industrious one ate a\nloaf. You may rely upon this as the latest news from the French capital.\n\n                                                      Faithfully ever.\n\n\n[Sidenote: Mr. William Charles Kent.]\n\n                      26, WELLINGTON STREET, _Monday, Nov. 6th, 1865._\n\nMY DEAR KENT,\n\n_No_, I _won't_ write in this book, because I have sent another to the\nbinder's for you.\n\nI have been unwell with a relaxed throat, or I should have written to\nyou sooner to thank you for your dedication, to assure you that it\nheartily, most heartily, gratifies me, as the sincere tribute of a true\nand generous heart, and to tell you that I have been charmed with your\nbook itself. I am proud of having given a name to anything so\npicturesque, so sympathetic and spirited.\n\nI hope and believe the \"Doctor\" is nothing but a good 'un. He has\nperfectly astonished Forster, who writes: \"Neither good, gooder, nor\ngoodest, but super-excellent; all through there is such a relish of you\nat your best, as I could not have believed in, after a long story.\"\n\nI shall be charmed to see you to-night.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: M. de Cerjat.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                                _November 13th, 1865._\n\n                                EXTRACT.\n\nMY DEAR CERJAT,\n\nHaving achieved my book and my Christmas number, and having shaken\nmyself after two years' work, I send you my annual greeting. How are\nyou? Asthmatic, I know you will reply; but as my poor father (who was\nasthmatic, too, and the jolliest of men) used philosophically to say,\n\"one must have something wrong, I suppose, and I like to know what it\nis.\"\n\nIn England we are groaning under the brigandage of the butcher, which is\nbeing carried to that height that I think I foresee resistance on the\npart of the middle-class, and some combination in perspective for\nabolishing the middleman, whensoever he turns up (which is everywhere)\nbetween producer and consumer. The cattle plague is the butcher's\nstalking-horse, and it is unquestionably worse than it was; but seeing\nthat the great majority of creatures lost or destroyed have been cows,\nand likewise that the rise in butchers' meat bears no reasonable\nproportion to the market prices of the beasts, one comes to the\nconclusion that the public is done. The commission has ended very weakly\nand ineffectually, as such things in England rather frequently do; and\neverybody writes to _The Times_, and nobody does anything else.\n\nIf the Americans don't embroil us in a war before long it will not be\ntheir fault. What with their swagger and bombast, what with their claims\nfor indemnification, what with Ireland and Fenianism, and what with\nCanada, I have strong apprehensions. With a settled animosity towards\nthe French usurper, I believe him to have always been sound in his\ndesire to divide the States against themselves, and that we were\nunsound and wrong in \"letting I dare not wait upon I would.\" The Jamaica\ninsurrection is another hopeful piece of business. That\nplatform-sympathy with the black--or the native, or the devil--afar off,\nand that platform indifference to our own countrymen at enormous odds in\nthe midst of bloodshed and savagery, makes me stark wild. Only the other\nday, here was a meeting of jawbones of asses at Manchester, to censure\nthe Jamaica Governor for his manner of putting down the insurrection! So\nwe are badgered about New Zealanders and Hottentots, as if they were\nidentical with men in clean shirts at Camberwell, and were to be bound\nby pen and ink accordingly. So Exeter Hall holds us in mortal submission\nto missionaries, who (Livingstone always excepted) are perfect\nnuisances, and leave every place worse than they found it.\n\nOf all the many evidences that are visible of our being ill-governed, no\none is so remarkable to me as our ignorance of what is going on under\nour Government. What will future generations think of that enormous\nIndian Mutiny being ripened without suspicion, until whole regiments\narose and killed their officers? A week ago, red tape, half-bouncing and\nhalf pooh-poohing what it bounced at, would have scouted the idea of a\nDublin jail not being able to hold a political prisoner. But for the\nblacks in Jamaica being over-impatient and before their time, the whites\nmight have been exterminated, without a previous hint or suspicion that\nthere was anything amiss. _Laissez aller_, and Britons never, never,\nnever!----\n\nMeantime, if your honour were in London, you would see a great\nembankment rising high and dry out of the Thames on the Middlesex shore,\nfrom Westminster Bridge to Blackfriars. A really fine work, and really\ngetting on. Moreover, a great system of drainage. Another really fine\nwork, and likewise really getting on. Lastly, a muddle of railways in\nall directions possible and impossible, with no general public scheme,\nno general public supervision, enormous waste of money, no fixable\nresponsibility, no accountability but under Lord Campbell's Act. I think\nof that accident in which I was preserved. Before the most furious and\nnotable train in the four-and-twenty hours, the head of a gang of\nworkmen takes up the rails. That train changes its time every day as the\ntide changes, and that head workman is not provided by the railway\ncompany with any clock or watch! Lord Shaftesbury wrote to me to ask me\nwhat I thought of an obligation on railway companies to put strong walls\nto all bridges and viaducts. I told him, of course, that the force of\nsuch a shock would carry away anything that any company could set up,\nand I added: \"Ask the minister what _he_ thinks about the votes of the\nrailway interest in the House of Commons, and about his being afraid to\nlay a finger on it with an eye to his majority.\"\n\nI seem to be grumbling, but I am in the best of humours. All goes well\nwith me and mine, thank God.\n\nLast night my gardener came upon a man in the garden and fired. The man\nreturned the compliment by kicking him in the groin and causing him\ngreat pain. I set off, with a great mastiff-bloodhound I have, in\npursuit. Couldn't find the evil-doer, but had the greatest difficulty in\npreventing the dog from tearing two policemen down. They were coming\ntowards us with professional mystery, and he was in the air on his way\nto the throat of an eminently respectable constable when I caught him.\n\nMy daughter Mary and her aunt Georgina send kindest regard and\nremembrance. Katey and her husband are going to try London this winter,\nbut I rather doubt (for they are both delicate) their being able to\nweather it out. It has been blowing here tremendously for a fortnight,\nbut to-day is like a spring day, and plenty of roses are growing over\nthe labourers' cottages. The _Great Eastern_ lies at her moorings beyond\nthe window where I write these words; looks very dull and unpromising. A\ndark column of smoke from Chatham Dockyard, where the iron shipbuilding\nis in progress, has a greater significance in it, I fancy.\n\n\n[Sidenote: Miss Dickens.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                           _Tuesday, Nov. 14th, 1865._\n\nMY DEAREST MAMIE,\n\nAs you want to know my views of the Sphinx, here they are. But I have\nonly seen it once; and it is so extraordinarily well done, that it ought\nto be observed closely several times.\n\nAnyone who attentively notices the flower trick will see that the two\nlittle high tables hung with drapery cover each a trap. Each of those\ntables, during that trick, hides a confederate, who changes the paper\ncone twice. When the cone has been changed as often as is required, the\ntrap is closed and the table can be moved.\n\nWhen the curtain is removed for the performance of the Sphinx trick,\nthere is a covered, that is, draped table on the stage, which is never\nseen before or afterwards. In front of the middle of it, and between it\nand the audience, stands one of those little draped tables covering a\ntrap; this is a third trap in the centre of the stage. The box for the\nhead is then upon IT, and the conjuror takes it off and shows it. The\nman whose head is afterwards shown in that box is, I conceive, in the\ntable; that is to say, is lying on his chest in the thickness of the\ntable, in an extremely constrained attitude. To get him into the table,\nand to enable him to use the trap in the table through which his head\ncomes into the box, the two hands of a confederate are necessary. That\nconfederate comes up a trap, and stands in the space afforded by the\ninterval below the stage and the height of the little draped table! his\nback is towards the audience. The moment he has assisted the hidden man\nsufficiently, he closes the trap, and the conjuror then immediately\nremoves the little draped table, and also the drapery of the larger\ntable; when he places the box on the last-named table _with the slide\non_ for the head to come into it, he stands with his back to the\naudience and his face to the box, and masks the box considerably to\nfacilitate the insertion of the head. As soon as he knows the head to be\nin its place, he undraws the slide. When the verses have been spoken and\nthe trick is done, he loses no time in replacing the slide. The curtain\nis then immediately dropped, because the man cannot otherwise be got out\nof the table, and has no doubt had quite enough of it. With kindest\nregards to all at Penton,\n\n                                          Ever your most affectionate.\n\nFOOTNOTES:\n\n[14] Now Captain E. Newton Dickenson.\n\n[15] This was a circular note which he sent in answer to innumerable\nletters of enquiry, after the accident.\n\n[16] This letter was written in reply to the Committee's congratulations\nupon Mr. Dickens's escape from the accident to the tidal train from\nFolkestone, at Staplehurst, just previous to this date.\n\n\n\n\n1866.\n\nNARRATIVE.\n\n\nThe furnished house hired by Charles Dickens in the spring of this year\nwas in Southwick Place, Hyde Park.\n\nHaving entered into negotiations with the Messrs. Chappell for a series\nof readings to be given in London, in the English provinces, in Scotland\nand Ireland, Charles Dickens had no leisure for more than his usual\neditorial work for \"All the Year Round.\" He contributed four parts to\nthe Christmas number, which was entitled, \"Mugby Junction.\"\n\nFor the future all his English readings were given in connection with\nthe Messrs. Chappell, and never in all his career had he more\nsatisfactory or more pleasant business relations than those connected\nwith these gentlemen. Moreover, out of this connection sprang a sincere\nfriendship on both sides.\n\nMr. Dolby is so constantly mentioned in future letters, that they\nthemselves will tell of the cordial companionship which existed between\nCharles Dickens and this able and most obliging \"manager.\"\n\nThe letter to \"Lily\" was in answer to a child's letter from Miss Lily\nBenzon, inviting him to a birthday party.\n\nThe play alluded to in the letter to M. Fechter was called \"A Long\nStrike,\" and was performed at the Lyceum Theatre.\n\nThe \"Sultan\" mentioned in the letter to Mr. Fitzgerald was a noble Irish\nbloodhound, presented by this gentleman to Charles Dickens. The story of\nthe dog's death is told in a letter to M. de Cerjat, which we give in\nthe following year.\n\n\n[Sidenote: Miss Mary Boyle.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                           _Saturday, Jan. 6th, 1866._\n\nMY DEAR MARY,\n\nFeeling pretty certain that I shall never answer your letter unless I\nanswer it at once (I got it this morning), here goes!\n\nI did not dramatise \"The Master of Ravenswood,\" though I did a good deal\ntowards and about the piece, having an earnest desire to put Scott, for\nonce, upon the stage in his own gallant manner. It is _an enormous\nsuccess_, and increases in attraction nightly. I have never seen the\npeople in all parts of the house so leaning forward, in lines sloping\ntowards the stage, earnestly and intently attractive, as while the story\ngradually unfolds itself. But the astonishing circumstance of all is,\nthat Miss Leclercq (never thought of for Lucy till all other Lucies had\nfailed) is marvellously good, highly pathetic, and almost unrecognisable\nin person! What note it touches in her, always dumb until now, I do not\npretend to say, but there is no one on the stage who could play the\ncontract scene better, or more simply and naturally, and I find it\nimpossible to see it without crying! Almost everyone plays well, the\nwhole is exceedingly picturesque, and there is scarcely a movement\nthroughout, or a look, that is not indicated by Scott. So you get a life\nromance with beautiful illustrations, and I do not expect ever again to\nsee a book take up its bed and walk in like manner.\n\nI am charmed to learn that you have had a freeze out of my ghost story.\nIt rather did give me a shiver up the back in the writing. \"Dr.\nMarigold\" has just now accomplished his two hundred thousand. My only\nother news about myself is that I am doubtful whether to read or not in\nLondon this season. If I decide to do it at all, I shall probably do it\non a large scale.\n\nMany happy years to you, my dear Mary. So prays\n\n                                              Your ever affectionate\n                                                                   Jo.\n\n\n[Sidenote: Mr. William Charles Kent.]\n\n                              GAD'S HILL, _Thursday, Jan. 18th, 1866._\n\nMY DEAR KENT,\n\nI cannot tell you how grieved we all are here to know that you are\nsuffering again. Your patient tone, however, and the hopefulness and\nforbearance of Ferguson's course, gives us some reassurance. Apropos of\nwhich latter reference I dined with Ferguson at the Lord Mayor's, last\nTuesday, and had a grimly distracted impulse upon me to defy the\ntoast-master and rush into a speech about him and his noble art, when I\nsat pining under the imbecility of constitutional and corporational\nidiots. I did seize him for a moment by the hair of his head (in\nproposing the Lady Mayoress), and derived some faint consolation from\nthe company's response to the reference. O! no man will ever know under\nwhat provocation to contradiction and a savage yell of repudiation I\nsuffered at the hands of ----, feebly complacent in the uniform of\nMadame Tussaud's own military waxers, and almost the worst speaker I\never heard in my life! Mary and Georgina, sitting on either side of me,\nurged me to \"look pleasant.\" I replied in expressions not to be\nrepeated. Shea (the judge) was just as good and graceful, as he (the\nmember) was bad and gawky.\n\nBulwer's \"Lost Tales of Miletus\" is a most noble book! He is an\nextraordinary fellow, and fills me with admiration and wonder.\n\nIt is of no use writing to you about yourself, my dear Kent, because you\nare likely to be tired of that constant companion, and so I have gone\nscratching (with an exceedingly bad pen) about and about you. But I come\nback to you to let you know that the reputation of this house as a\nconvalescent hospital stands (like the house itself) very high, and that\ntestimonials can be produced from credible persons who have recovered\nhealth and spirits here swiftly. Try us, only try us, and we are content\nto stake the reputation of the establishment on the result.\n\n                                            Ever affectionately yours.\n\n\n[Sidenote: Mr. Percy Fitzgerald.]\n\n                                 GAD'S HILL, _Friday, Feb. 2nd, 1866._\n\nMY DEAR FITZGERALD,\n\nI ought to have written to you days and days ago, to thank you for your\ncharming book on Charles Lamb, to tell you with what interest and\npleasure I read it as soon as it came here, and to add that I was\nhonestly affected (far more so than your modesty will readily believe)\nby your intimate knowledge of those touches of mine concerning\nchildhood.\n\nLet me tell you now that I have not in the least cooled, after all,\neither as to the graceful sympathetic book, or as to the part in it with\nwhich I am honoured. It has become a matter of real feeling with me, and\nI postponed its expression because I couldn't satisfactorily get it out\nof myself, and at last I came to the conclusion that it must be left in.\n\n                          My dear Fitzgerald, faithfully yours always.\n\n\n[Sidenote: Miss Hogarth.]\n\n             OFFICE OF \"ALL THE YEAR ROUND,\" _Friday, Feb. 9th, 1866._\n\nMY DEAREST GEORGY,\n\nI found your letter here when I came back on Wednesday evening, and was\nextremely glad to get it.\n\nFrank Beard wrote me word that with such a pulse as I described, an\nexamination of the heart was absolutely necessary, and that I had better\nmake an appointment with him alone for the purpose. This I did. I was\nnot at all disconcerted, for I knew well beforehand that the effect\ncould not possibly be without that one cause at the bottom of it. There\nseems to be degeneration of some functions of the heart. It does not\ncontract as it should. So I have got a prescription of iron, quinine,\nand digitalis, to set it a-going, and send the blood more quickly\nthrough the system. If it should not seem to succeed on a reasonable\ntrial, I will then propose a consultation with someone else. Of course I\nam not so foolish as to suppose that all my work can have been achieved\nwithout _some_ penalty, and I have noticed for some time a decided\nchange in my buoyancy and hopefulness--in other words, in my usual\n\"tone.\"\n\nI shall wait to see Beard again on Monday, and shall most probably come\ndown that day. If I should not, I will telegraph after seeing him. Best\nlove to Mamie.\n\n\n[Sidenote: Mrs. Brookfield.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                           _Tuesday, Feb. 20th, 1866._\n\nMY DEAR MRS. BROOKFIELD,\n\nHaving gone through your MS. (which I should have done sooner, but that\nI have not been very well), I write these few following words about it.\nFirstly, with a limited reference to its unsuitability to these pages.\nSecondly, with a more enlarged reference to the merits of the story\nitself.\n\nIf you will take any part of it and cut it up (in fancy) into the small\nportions into which it would have to be divided here for only a month's\nsupply, you will (I think) at once discover the impossibility of\npublishing it in weekly parts. The scheme of the chapters, the manner of\nintroducing the people, the progress of the interest, the places in\nwhich the principal places fall, are all hopelessly against it. It would\nseem as though the story were never coming, and hardly ever moving.\nThere must be a special design to overcome that specially trying mode of\npublication, and I cannot better express the difficulty and labour of it\nthan by asking you to turn over any two weekly numbers of \"A Tale of Two\nCities,\" or \"Great Expectations,\" or Bulwer's story, or Wilkie\nCollins's, or Reade's, or \"At the Bar,\" and notice how patiently and\nexpressly the thing has to be planned for presentation in these\nfragments, and yet for afterwards fusing together as an uninterrupted\nwhole.\n\nOf the story itself I honestly say that I think highly. The style is\nparticularly easy and agreeable, infinitely above ordinary writing, and\nsometimes reminds me of Mrs. Inchbald at her best. The characters are\nremarkably well observed, and with a rare mixture of delicacy and\ntruthfulness. I observe this particularly in the brother and sister, and\nin Mrs. Neville. But it strikes me that you constantly hurry your\nnarrative (and yet without getting on) _by telling it, in a sort of\nimpetuous breathless way, in your own person, when the people should\ntell it and act it for themselves_. My notion always is, that when I\nhave made the people to play out the play, it is, as it were, their\nbusiness to do it, and not mine. Then, unless you really have led up to\na great situation like Basil's death, you are bound in art to make more\nof it. Such a scene should form a chapter of itself. Impressed upon the\nreader's memory, it would go far to make the fortune of the book.\nSuppose yourself telling that affecting incident in a letter to a\nfriend. Wouldn't you describe how you went through the life and stir of\nthe streets and roads to the sick-room? Wouldn't you say what kind of\nroom it was, what time of day it was, whether it was sunlight,\nstarlight, or moonlight? Wouldn't you have a strong impression on your\nmind of how you were received, when you first met the look of the dying\nman, what strange contrasts were about you and struck you? I don't want\nyou, in a novel, to present _yourself_ to tell such things, but I want\nthe things to be there. You make no more of the situation than the index\nmight, or a descriptive playbill might in giving a summary of the\ntragedy under representation.\n\nAs a mere piece of mechanical workmanship, I think all your chapters\nshould be shorter; that is to say, that they should be subdivided.\nAlso, when you change from narrative to dialogue, or _vice versa_, you\nshould make the transition more carefully. Also, taking the pains to sit\ndown and recall the principal landmarks in your story, you should then\nmake them far more elaborate and conspicuous than the rest. Even with\nthese changes I do not believe that the story would attract the\nattention due to it, if it were published even in such monthly portions\nas the space of \"Fraser\" would admit of. Even so brightened, it would\nnot, to the best of my judgment, express itself piecemeal. It seems to\nme to be so constituted as to require to be read \"off the reel.\" As a\nbook in two volumes I think it would have good claims to success, and\ngood chances of obtaining success. But I suppose the polishing I have\nhinted at (not a meretricious adornment, but positively necessary to\ngood work and good art) to have been first thoroughly administered.\n\nNow don't hate me if you can help it. I can afford to be hated by some\npeople, but I am not rich enough to put you in possession of that\nluxury.\n\n                                                Ever faithfully yours.\n\nP.S.--The MS. shall be delivered at your house to-morrow. And your\npetitioner again prays not to be, etc.\n\n\n[Sidenote: Miss Hogarth.]\n\n                       ADELPHI, LIVERPOOL, _Friday, April 13th, 1866._\n\nMY DEAREST GEORGY,\n\nThe reception at Manchester last night was quite a magnificent sight;\nthe whole of the immense audience standing up and cheering. I thought\nthem a little slow with \"Marigold,\" but believe it was only the\nattention necessary in so vast a place. They gave a splendid burst at\nthe end. And after \"Nickleby\" (which went to perfection), they set up\nsuch a call, that I was obliged to go in again. The unfortunate gasman,\na very steady fellow, got a fall off a ladder and sprained his leg. He\nwas put to bed in a public opposite, and was left there, poor man.\n\nThis is the first very fine day we have had. I have taken advantage of\nit by crossing to Birkenhead and getting some air upon the water. It was\nfresh and beautiful.\n\nI send my best love to Mamie, and hope she is better. I am, of course,\ntired (the pull of \"Marigold\" upon one's energy, in the Free Trade Hall,\nwas great); but I stick to my tonic, and feel, all things considered, in\nvery good tone. The room here (I mean the hall) being my special\nfavourite and extraordinarily easy, is _almost_ a rest!\n\n\n[Sidenote: Miss Dickens.]\n\n                     ADELPHI, LIVERPOOL, _Saturday, April 14th, 1866._\n\nMY DEAREST MAMIE,\n\nThe police reported officially that three thousand people were turned\naway from the hall last night. I doubt if they were so numerous as that,\nbut they carried in the outer doors and pitched into Dolby with great\nvigour. I need not add that every corner of the place was crammed. They\nwere a very fine audience, and took enthusiastically every point in\n\"Copperfield\" and the \"Trial.\" They made the reading a quarter of an\nhour longer than usual. One man advertised in the morning paper that he\nwould give thirty shillings (double) for three stalls, but nobody would\nsell, and he didn't get in.\n\nExcept that I cannot sleep, I really think myself in much better\ntraining than I had anticipated. A dozen oysters and a little champagne\nbetween the parts every night, constitute the best restorative I have\never yet tried. John appears low, but I don't know why. A letter comes\nfor him daily; the hand is female; whether Smudger's, or a nearer one\nstill and a dearer one, I don't know. So it may or may not be the cause\nof his gloom.\n\n\"Miss Emily\" of Preston is married to a rich cotton lord, rides in open\ncarriages in gorgeous array, and is altogether splendid. With this\neffective piece of news I close.\n\n\n[Sidenote: Miss Hogarth.]\n\n                                          GLASGOW, _April 17th, 1866._\n\nWe arrived here at ten yesterday evening. I don't think the journey\nshook me at all. Dolby provided a superb cold collation and \"the best of\ndrinks,\" and we dined in the carriage, and I made him laugh all the way.\n\nThe let here is very large. Every precaution taken to prevent my\nplatform from being captured as it was last time; but I don't feel at\nall sure that it will not be stormed at one of the two readings. Wills\nis to do the genteel to-night at the stalls, and Dolby is to stem the\nshilling tide _if_ he can. The poor gasman cannot come on, and we have\ngot a new one here who is to go to Edinburgh with us. Of Edinburgh we\nknow nothing, but as its first night has always been shady, I suppose it\nwill stick to its antecedents.\n\nI like to hear about Harness and his freshness. The let for the next\nreading at St. James's is \"going,\" they report, \"admirably.\" Lady\nRussell asked me to dinner to-morrow, and I have written her a note\nto-day. The rest has certainly done me good. I slept thoroughly well\nlast night, and feel fresh. What to-night's work, and every night's\nwork this week, may do contrariwise, remains to be seen.\n\nI hope Harry's knee may be in the way of mending, from what you relate\nof it.\n\n\n[Sidenote: Miss Dickens.]\n\n              WATERLOO HOTEL, EDINBURGH, _Wednesday, April 18th, 1866._\n\nWe had a tremendous house again last night at Glasgow; and turned away\ngreat numbers. Not only that, but they were a most brilliant and\ndelicate audience, and took \"Marigold\" with a fine sense and quickness\nnot to be surpassed. The shillings pitched into Dolby again, and one man\nwrites a sensible letter in one of the papers this morning, showing to\n_my_ satisfaction (?) that they really had, through the local agent,\nsome cause of complaint. Nevertheless, the shilling tickets are sold for\nto-morrow, and it seems to be out of the question to take any money at\nthe doors, the call for all parts is so enormous. The thundering of\napplause last night was quite staggering, and my people checked off my\nreception by the minute hand of a watch, and stared at one another,\nthinking I should never begin. I keep quite well, have happily taken to\nsleeping these last three nights; and feel, all things considered, very\nlittle conscious of fatigue. I cannot reconcile my town medicine with\nthe hours and journeys of reading life, and have therefore given it up\nfor the time. But for the moment, I think I am better without it. What\nwe are doing here I have not yet heard. I write at half-past one, and we\nhave been little more than an hour in the house. But I am quite prepared\nfor the inevitable this first Edinburgh night. Endeavours have been\nmade (from Glasgow yesterday) to telegraph the exact facts out of our\nlocal agent; but hydraulic pressure wouldn't have squeezed a straight\nanswer out of him. \"Friday and Saturday doing very well, Wednesday not\nso good.\" This was all electricity could discover.\n\nI am going to write a line this post to Katie, from whom I have a note.\nI hope Harry's leg will now step out in the manner of the famous cork\nleg in the song.\n\n\n[Sidenote: Miss Hogarth.]\n\n                              EDINBURGH, _Thursday, April 19th, 1866._\n\nThe house was more than twice better than any first night here\npreviously. They were, as usual here, remarkably intelligent, and the\nreading went _brilliantly_. I have not sent up any newspapers, as they\nare generally so poorly written, that you may know beforehand all the\ncommonplaces that they will write. But _The Scotsman_ has so pretty an\narticle this morning, and (so far as I know) so true a one, that I will\ntry to post it to you, either from here or Glasgow. John and Dolby went\nover early, and Wills and I follow them at half-past eleven. It is cold\nand wet here. We have laid half-crown bets with Dolby, that he will be\nassaulted to-night at Glasgow. He has a surprising knowledge of what the\nreceipts will be always, and wins half-crowns every night. Chang is\nliving in this house. John (not knowing it) was rendered perfectly\ndrivelling last night, by meeting him on the stairs. The Tartar Dwarf is\nalways twining himself upstairs sideways, and drinks a bottle of whisky\nper day, and is reported to be a surprising little villain.\n\n\n[Sidenote: Miss Dickens.]\n\n                WATERLOO HOTEL, EDINBURGH, _Friday, April 20th, 1866._\n\nNo row at Glasgow last night. Great placards were posted about the town\nby the anxious Dolby, announcing that no money would be taken at the\ndoors. This kept the crowd off. Two files of policemen and a double\nstaff everywhere did the rest, and nothing could be better-tempered or\nmore orderly. Tremendous enthusiasm with the \"Carol\" and \"Trial.\" I was\ndead beat afterwards, that reading being twenty minutes longer than\nusual; but plucked up again, had some supper, slept well, and am quite\nright to-day. It is a bright day, and the express ride over from Glasgow\nwas very pleasant.\n\nEverything is gone here for to-night. But it is difficult to describe\nwhat the readings have grown to be. The let at St. James's Hall is not\nonly immense for next Tuesday, but so large for the next reading\nafterwards, that Chappell writes: \"That will be the greatest house of\nthe three.\" From Manchester this morning they write: \"Send us more\ntickets instantly, for we are sold out and don't know what to do with\nthe people.\" Last night the whole of my money under the agreement had\nbeen taken. I notice that a great bank has broken at Liverpool, which\nmay hurt us there, but when last heard of it was going as before. And\nthe audience, though so enormous, do somehow express a personal\naffection, which makes them very strange and moving to see.\n\nI have a story to answer you and your aunt with. Before I left Southwick\nPlace for Liverpool, I received a letter from Glasgow, saying, \"Your\nlittle Emily has been woo'd and married and a'! since you last saw her;\"\nand describing her house within a mile or two of the city, and asking\nme to stay there. I wrote the usual refusal, and supposed Mrs. ---- to\nbe some romantic girl whom I had joked with, perhaps at Allison's or\nwhere not. On the first night at Glasgow I received a bouquet from ----,\nand wore one of the flowers. This morning at the Glasgow station, ----\nappeared, and proved to be the identical Miss Emily, of whose marriage\nDolby had told me on our coming through Preston. She was attired in\nmagnificent raiment, and presented the happy ----.\n\n\n[Sidenote: Miss Hogarth.]\n\n                              LIVERPOOL, _Thursday, April 26th, 1866._\n\nWe noticed between London and Rugby (the first stoppage) something very\nodd in our carriage yesterday, not so much in its motion as in its\nsound. We examined it as well as we could out of both windows, but could\nmake nothing of it. On our arrival at Rugby, it was found to be on fire.\nAnd as it was in the middle of the train, the train had to be broken to\nget it off into a siding by itself and get another carriage on. With\nthis slight exception we came down all right.\n\nMy voice is much better, I am glad to report, and I mean to try Beard's\nremedy after dinner to-day. This is all my present news.\n\n\n[Sidenote: The same.]\n\n                        DOWN HOTEL, CLIFTON, _Friday, May 11th, 1866._\n\nI received your note before I left Birmingham this morning. It has been\nvery heavy work getting up at half-past six each morning after a heavy\nnight, and I am not at all well to-day. We had a tremendous hall at\nBirmingham last night--two thousand one hundred people. I made a most\nridiculous mistake. Had \"Nickleby\" on my list to finish with, instead of\n\"Trial.\" Read \"Nickleby\" with great go, and the people remained. Went\nback again at ten and explained the accident, and said if they liked, I\nwould give them the \"Trial.\" They _did_ like, and I had another\nhalf-hour of it in that enormous place.\n\nThis stoppage of Overend and Gurney in the City will play the ---- with\nall public gaieties, and with all the arts.\n\nMy cold is no better. John fell off a platform about ten feet high\nyesterday, and fainted. He looks all the colours of the rainbow to-day,\nbut does not seem much hurt beyond being puffed up one hand, arm, and\nside.\n\n\n[Sidenote: Miss Lily Benzon.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                            _Monday, June 18th, 1866._\n\nMY DEAR LILY,\n\nI am sorry that I cannot come to read to you \"The Boots at the Holly\nTree Inn,\" as you ask me to do; but the truth is, that I am tired of\nreading at this present time, and have come into the country to rest and\nhear the birds sing. There are a good many birds, I daresay, in\nKensington Palace Gardens, and upon my word and honour they are much\nbetter worth listening to than I am. So let them sing to you as hard as\never they can, while their sweet voices last (they will be silent when\nthe winter comes); and very likely after you and I have eaten our next\nChristmas pudding and mince-pies, you and I and Uncle Harry may all meet\ntogether at St. James's Hall; Uncle Harry to bring you there, to hear\nthe \"Boots;\" I to receive you there, and read the \"Boots;\" and you (I\nhope) to applaud very much, and tell me that you like the \"Boots.\" So,\nGod bless you and me, and Uncle Harry, and the \"Boots,\" and long life\nand happiness to us all!\n\n                                             Your affectionate Friend.\n\nP.S.--There's a flourish!\n\n\n[Sidenote: Mr. B. W. Procter.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                            _Monday, Aug. 13th, 1866._\n\nMY DEAR PROCTER,\n\nI have read your biography of Charles Lamb with inexpressible pleasure\nand interest. I do not think it possible to tell a pathetic story with a\nmore unaffected and manly tenderness. And as to the force and vigour of\nthe style, if I did not know you I should have made sure that there was\na printer's error in the opening of your introduction, and that the word\n\"seventy\" occupied the place of \"forty.\"\n\nLet me, my dear friend, most heartily congratulate you on your\nachievement. It is not an ordinary triumph to do such justice to the\nmemory of such a man. And I venture to add, that the fresh spirit with\nwhich you have done it impresses me as being perfectly wonderful.\n\n                                            Ever affectionately yours.\n\n\n[Sidenote: Sir James Emerson Tennent.]\n\n                                GAD'S HILL, _Monday, Aug. 20th, 1866._\n\nMY DEAR TENNENT,\n\nI have been very much interested by your extract, and am strongly\ninclined to believe that the founder of the Refuge for Poor Travellers\nmeant the kind of man to which it refers. Chaucer certainly meant the\nPardonere to be a humbug, living on the credulity of the people. After\ndescribing the sham reliques he carried, he says:\n\n        But with these relikes whawne that he found\n        A poure personne dwelling up on lond\n        Upon a day he gat him more monnie\n        Than that the personne got in monthes time,\n        And thus, with fained flattering and japes\n        He made the personne, and the people, his apes.\n\nAnd the worthy Watts (founder of the charity) may have had these very\nlines in his mind when he excluded such a man.\n\nWhen I last heard from my boy he was coming to you, and was full of\ndelight and dignity. My midshipman has just been appointed to the\n_Bristol_, on the West Coast of Africa, and is on his voyage out to join\nher. I wish it was another ship and another station. She has been\nunlucky in losing men.\n\nKindest regard from all my house to yours.\n\n                                                Faithfully yours ever.\n\n\n[Sidenote: M. Charles Fechter.]\n\n                               GAD'S HILL, _Tuesday, Sept. 4th, 1866._\n\nMY DEAR FECHTER,\n\nThis morning I received the play to the end of the telegraph scene, and\nI have since read it twice.\n\nI clearly see the _ground_ of Mr. Boucicault's two objections; but I do\nnot see their _force_.\n\nFirst, as to the writing. If the characters did not speak in a terse and\nhomely way, their idea and language would be inconsistent with their\ndress and station, and they would lose, as characters, before the\naudience. The dialogue seems to be exactly what is wanted. Its\nsimplicity (particularly in Mr. Boucicault's part) is often very\neffective; and throughout there is an honest, straight-to-the-purpose\nruggedness in it, like the real life and the real people.\n\nSecondly, as to the absence of the comic element. I really do not see\nhow more of it could be got into the story, and I think Mr. Boucicault\nunderrates the pleasant effect of his own part. The very notion of a\nsailor, whose life is not among those little courts and streets, and\nwhose business does not lie with the monotonous machinery, but with the\nfour wild winds, is a relief to me in reading the play. I am quite\nconfident of its being an immense relief to the audience when they see\nthe sailor before them, with an entirely different bearing, action,\ndress, complexion even, from the rest of the men. I would make him the\nfreshest and airiest sailor that ever was seen; and through him I can\ndistinctly see my way out of \"the Black Country\" into clearer air. (I\nspeak as one of the audience, mind.) I should like something of this\ncontrast to be expressed in the dialogue between the sailor and Jew, in\nthe second scene of the second act. Again, I feel Widdicomb's part\n(which is charming, and ought to make the whole house cry) most\nagreeable and welcome, much better than any amount in such a story, of\nmere comicality.\n\nIt is unnecessary to say that the play is done with a master's hand. Its\ncloseness and movement are quite surprising. Its construction is\nadmirable. I have the strongest belief in its making a great success.\nBut I must add this proviso: I never saw a play so dangerously depending\nin critical places on strict natural propriety in the manner and\nperfection in the shaping of the small parts. Those small parts cannot\ntake the play up, but they can let it down. I would not leave a hair on\nthe head of one of them to the chance of the first night, but I would\nsee, to the minutest particular, the make-up of every one of them at a\nnight rehearsal.\n\nOf course you are free to show this note to Mr. Boucicault, and I\nsuppose you will do so; let me throw out this suggestion to him and you.\nMight it not ease the way with the Lord Chamberlain's office, and still\nmore with the audience, when there are Manchester champions in it, if\ninstead of \"Manchester\" you used a fictitious name? When I did \"Hard\nTimes\" I called the scene Coketown. Everybody knew what was meant, but\nevery cotton-spinning town said it was the other cotton-spinning town.\n\nI shall be up on Saturday, and will come over about mid-day, unless you\nname any other time.\n\n                                                        Ever heartily.\n\n\n[Sidenote: Mr. Walter Thornbury]\n\n            \"ALL THE YEAR ROUND\" OFFICE, _Saturday, Sept. 15th, 1866._\n\nMY DEAR THORNBURY,\n\nMany thanks for your letter.\n\nIn reference to your Shakespeare queries, I am not so much enamoured of\nthe first and third subjects as I am of the Ariosto enquiry, which\nshould be highly interesting. But if you have so got the matter in your\nmind, as that its execution would be incomplete and unsatisfactory to\nyou unless you write all the three papers, then by all means write the\nthree, and I will most gladly take them. For some years I have had so\nmuch pleasure in reading you, that I can honestly warrant myself as what\nactors call \"a good audience.\"\n\nThe idea of old stories retold is decidedly a good one. I greatly like\nthe notion of that series. Of course you know De Quincey's paper on the\nRatcliffe Highway murderer? Do you know also the illustration (I have it\nat Gad's Hill), representing the horrible creature as his dead body lay\non a cart, with a piece of wood for a pillow, and a stake lying by,\nready to be driven through him?\n\nI don't _quite_ like the title, \"The Social History of London.\" I should\nbetter like some title to the effect, \"The History of London's Social\nChanges in so many Years.\" Such a title would promise more, and better\nexpress your intention. What do you think of taking for a first title,\n\"London's Changes\"? You could then add the second title, \"Being a\nHistory,\" etc.\n\nI don't at all desire to fix a limit to the series of old stories\nretold. I would state the general intention at the beginning of the\nfirst paper, and go on like Banquo's line.\n\nDon't let your London title remind people, by so much as the place of\nthe word \"civilisation,\" of Buckle. It seems a ridiculous caution, but\nthe indolent part of the public (a large part!) on such points tumble\ninto extraordinary mistakes.\n\n                                              Faithfully yours always.\n\n\n[Sidenote: Mr. Percy Fitzgerald.]\n\n                                GAD'S HILL, _Tuesday, Nov. 6th, 1866._\n\nMY DEAR FITZGERALD,\n\nIt is always pleasant to me to hear from you, and I hope you will\nbelieve that this is not a mere fashion of speech.\n\nConcerning the green covers, I find the leaves to be budding--on\nunquestionable newspaper authority; but, upon my soul, I have no other\nknowledge of their being in embryo! Really, I do not see a chance of my\nsettling myself to such work until after I have accomplished forty-two\nreadings, to which I stand pledged.\n\nI hope to begin this series somewhere about the middle of January, in\nDublin. Touching the details of the realisation of this hope, will you\ntell me in a line as soon as you can--_Is the exhibition room a good\nroom for speaking in?_\n\nYour mention of the late Sultan touches me nearly. He was the finest dog\nI ever saw, and between him and me there was a perfect understanding.\nBut, to adopt the popular phrase, it was so very confidential that it\n\"went no further.\" He would fly at anybody else with the greatest\nenthusiasm for destruction. I saw him, muzzled, pound into the heart of\na regiment of the line; and I have frequently seen him, muzzled, hold a\ngreat dog down with his chest and feet. He has broken loose (muzzled)\nand come home covered with blood, again and again. And yet he never\ndisobeyed me, unless he had first laid hold of a dog.\n\nYou heard of his going to execution, evidently supposing the procession\nto be a party detached in pursuit of something to kill or eat? It was\nvery affecting. And also of his bolting a blue-eyed kitten, and making\nme acquainted with the circumstance by his agonies of remorse (or\nindigestion)?\n\nI cannot find out that there is anyone in Rochester (a sleepy old city)\nwho has anything to tell about Garrick, except what is not true. His\nbrother, the wine merchant, would be more in Rochester way, I think. How\non earth do you find time to do all these books?\n\nYou make my hair stand on end; an agreeable sensation, for I am charmed\nto find that I have any. Why don't you come yourself and look after\nGarrick? I should be truly delighted to receive you.\n\n                           My dear Fitzgerald, always faithfully yours.\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                            _Friday, Dec. 28th, 1866._\n\nMY DEAREST MACREADY,\n\nI have received your letter with the utmost pleasure and we all send our\nmost affectionate love to you, Mrs. Macready, Katie, Johnny, and the boy\nof boys. All good Christmas and New Year greetings are to be understood\nas included.\n\nYou will be interested in knowing that, encouraged by the success of\nsummer cricket-matches, I got up a quantity of foot-races and rustic\nsports in my field here on the 26th last past: as I have never yet had a\ncase of drunkenness, the landlord of The Falstaff had a drinking-booth\non the ground. All the prizes I gave were in money, too. We had two\nthousand people here. Among the crowd were soldiers, navvies, and\nlabourers of all kinds. Not a stake was pulled up, or a rope slackened,\nor one farthing's-worth of damage done. To every competitor (only) a\nprinted bill of general rules was given, with the concluding words: \"Mr.\nDickens puts every man upon his honour to assist in preserving order.\"\nThere was not a dispute all day, and they went away at sunset rending\nthe air with cheers, and leaving every flag on a six hundred yards'\ncourse as neat as they found it when the gates were opened at ten in the\nmorning. Surely this is a bright sign in the neighbourhood of such a\nplace as Chatham!\n\n\"Mugby Junction\" turned, yesterday afternoon, the extraordinary number\nof two hundred and fifty thousand!\n\nIn the middle of next month I begin a new course of forty-two readings.\nIf any of them bring me within reach of Cheltenham, with an hour to\nspare, I shall come on to you, even for that hour. More of this when I\nam afield and have my list, which Dolby (for Chappell) is now\npreparing.\n\nForster and Mrs. Forster were to have come to us next Monday, to stay\nuntil Saturday. I write \"were,\" because I hear that Forster (who had a\ntouch of bronchitis when he wrote to me on Christmas Eve) is in bed.\nKatie, who has been ill of low nervous fever, was brought here yesterday\nfrom London. She bore the journey much better than I expected, and so I\nhope will soon recover. This is my little stock of news.\n\nI begin to discover in your riper years, that you have been secretly\nvain of your handwriting all your life. For I swear I see no change in\nit! What it always was since I first knew it (a year or two!) it _is_.\nThis I will maintain against all comers.\n\n                             Ever affectionately, my dearest Macready.\n\n\n\n\n1867.\n\nNARRATIVE.\n\n\nAs the London and provincial readings were to be resumed early in the\nyear and continued until the end of March, Charles Dickens took no house\nin London this spring. He came to his office quarters at intervals, for\nthe series in town; usually starting off again, on his country tour, the\nday after a London reading. From some passages in his letters to his\ndaughter and sister-in-law during this country course, it will be seen\nthat (though he made very light of the fact) the great exertion of the\nreadings, combined with incessant railway travelling, was beginning to\ntell upon his health, and he was frequently \"heavily beaten\" after\nreading at his best to an enthusiastic audience in a large hall.\n\nDuring the short intervals between his journeys, he was as constantly\nand carefully at work upon the business of \"All the Year Round\" as if he\nhad no other work on hand. A proof of this is given in a letter dated\n\"5th February.\" It is written to a young man (the son of a friend), who\nwrote a long novel when far too juvenile for such a task, and had\nsubmitted it to Charles Dickens for his opinion, with a view to\npublication. In the midst of his own hard and engrossing occupation he\nread the book, and the letter which he wrote on the subject needs no\nremark beyond this, that the young writer received the adverse criticism\nwith the best possible sense, and has since, in his literary profession,\nprofited by the advice so kindly given.\n\nAt this time the proposals to Charles Dickens for reading in America,\nwhich had been perpetually renewed from the time of his first abandoning\nthe idea, became so urgent and so tempting, that he found at last he\nmust, at all events, give the subject his most serious consideration. He\ntook counsel with his two most confidential friends and advisers, Mr.\nJohn Forster and Mr. W. H. Wills. They were both, at first, strongly\nopposed to the undertaking, chiefly on the ground of the trial to his\nhealth and strength which it would involve. But they could not deny the\ncounterbalancing advantages. And, after much deliberation, it was\nresolved that Mr. George Dolby should be sent out by the Messrs.\nChappell, to take an impression, on the spot, as to the feeling of the\nUnited States about the Readings. His report as to the undoubted\nenthusiasm and urgency on the other side of the Atlantic it was\nimpossible to resist. Even his friends withdrew their opposition (though\nstill with misgivings as to the effect upon his health, which were but\ntoo well founded!), and on the 30th September he telegraphed \"Yes\" to\nAmerica.\n\nThe \"Alfred\" alluded to in a letter from Glasgow was Charles Dickens's\nfourth son, Alfred Tennyson, who had gone to Australia two years\npreviously.\n\nWe give, in April, the last letter to one of the friends for whom\nCharles Dickens had always a most tender love--Mr. Stanfield. He was\nthen in failing health, and in May he died.\n\nAnother death which affected him very deeply happened this summer. Miss\nMarguerite Power died in July. She had long been very ill, but, until it\nbecame impossible for her to travel, she was a frequent and beloved\nguest at Gad's Hill. The Mrs. Henderson to whom he writes was Miss\nPower's youngest sister.\n\nBefore he started for America it was proposed to wish him God-speed by\ngiving him a public dinner at the Freemasons' Hall. The proposal was\nmost warmly and fully responded to. His zealous friend, Mr. Charles\nKent, willingly undertook the whole work of arrangement of this banquet.\nIt took place on the 2nd November, and Lord Lytton presided.\n\nOn the 8th he left London for Liverpool, accompanied by his daughters,\nhis sister-in-law, his eldest son, Mr. Arthur Chappell, Mr. Charles\nCollins, Mr. Wilkie Collins, Mr. Kent, and Mr. Wills. The next morning\nthe whole party took a final leave of Charles Dickens on board the\n_Cuba_, which sailed that day.\n\nWe give a letter which he wrote to Mr. J. L. Toole on the morning of the\ndinner, thanking him for a parting gift and an earnest letter. That\nexcellent comedian was one of his most appreciative admirers, and, in\nreturn, he had for Mr. Toole the greatest admiration and respect.\n\nThe Christmas number for this year, \"No Thoroughfare,\" was written by\nCharles Dickens and Mr. Wilkie Collins. It was dramatised by Mr. Collins\nchiefly. But, in the midst of all the work of preparation for departure,\nCharles Dickens gave minute attention to as much of the play as could be\ncompleted before he left England. It was produced, after Christmas, at\nthe Adelphi Theatre, where M. Fechter was then acting, under the\nmanagement of Mr. Benjamin Webster.\n\n\n[Sidenote: M. de Cerjat.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                               _New Year's Day, 1867._\n\nMY DEAR CERJAT,\n\nThoroughly determined to be beforehand with \"the middle of next summer,\"\nyour penitent friend and remorseful correspondent thus addresses you.\n\nThe big dog, on a day last autumn, having seized a little girl (sister\nto one of the servants) whom he knew, and was bound to respect, was\nflogged by his master, and then sentenced to be shot at seven next\nmorning. He went out very cheerfully with the half-dozen men told off\nfor the purpose, evidently thinking that they were going to be the death\nof somebody unknown. But observing in the procession an empty\nwheelbarrow and a double-barrelled gun, he became meditative, and fixed\nthe bearer of the gun with his eyes. A stone deftly thrown across him by\nthe village blackguard (chief mourner) caused him to look round for an\ninstant, and he then fell dead, shot through the heart. Two posthumous\nchildren are at this moment rolling on the lawn; one will evidently\ninherit his ferocity, and will probably inherit the gun. The pheasant\nwas a little ailing towards Christmas Day, and was found dead under some\nivy in his cage, with his head under his wing, on the morning of the\ntwenty-seventh of December, one thousand eight hundred and sixty-six. I,\nproprietor of the remains of the two deceased, am working hard, getting\nup \"Barbox\" and \"The Boy at Mugby,\" with which I begin a new series of\nreadings in London on the fifteenth. Next morning I believe I start into\nthe country. When I read, I _don't_ write. I only edit, and have the\nproof-sheets sent me for the purpose. Here are your questions answered.\n\nAs to the Reform question, it should have been, and could have been,\nperfectly known to any honest man in England that the more intelligent\npart of the great masses were deeply dissatisfied with the state of\nrepresentation, but were in a very moderate and patient condition,\nawaiting the better intellectual cultivation of numbers of their\nfellows. The old insolent resource of assailing them and making the most\naudaciously wicked statements that they are politically indifferent,\nhas borne the inevitable fruit. The perpetual taunt, \"Where are they?\"\nhas called them out with the answer: \"Well then, if you _must_ know,\nhere we are.\" The intolerable injustice of vituperating the bribed to an\nassembly of bribers, has goaded their sense of justice beyond endurance.\nAnd now, what they would have taken they won't take, and whatever they\nare steadily bent upon having they will get. Rely upon it, this is the\nreal state of the case. As to your friend \"Punch,\" you will find him\nbegin to turn at the very selfsame instant when the new game shall\nmanifestly become the losing one. You may notice his shoes pinching him\na little already.\n\nMy dear fellow, I have no more power to stop that mutilation of my books\nthan you have. It is as certain as that every inventor of anything\ndesigned for the public good, and offered to the English Government,\nbecomes _ipso facto_ a criminal, to have his heart broken on the\ncircumlocutional wheel. It is as certain as that the whole Crimean story\nwill be retold, whenever this country again goes to war. And to tell the\ntruth, I have such a very small opinion of what the great genteel have\ndone for us, that I am very philosophical indeed concerning what the\ngreat vulgar may do, having a decided opinion that they can't do worse.\n\nThis is the time of year when the theatres do best, there being still\nnumbers of people who make it a sort of religion to see Christmas\npantomimes. Having my annual houseful, I have, as yet, seen nothing.\nFechter has neither pantomime nor burlesque, but is doing a new version\nof the old \"Trente Ans de la Vie d'un Joueur.\" I am afraid he will not\nfind his account in it. On the whole, the theatres, except in the\narticles of scenery and pictorial effect, are poor enough. But in some\nof the smaller houses there are actors who, if there were any dramatic\nhead-quarters as a school, might become very good. The most hopeless\nfeature is, that they have the smallest possible idea of an effective\nand harmonious whole, each \"going in\" for himself or herself. The\nmusic-halls attract an immense public, and don't refine the general\ntaste. But such things as they do are well done of their kind, and\nalways briskly and punctually.\n\nThe American yacht race is the last sensation. I hope the general\ninterest felt in it on this side will have a wholesome interest on that.\nIt will be a woeful day when John and Jonathan throw their caps into the\nring. The French Emperor is indubitably in a dangerous state. His\nParisian popularity wanes, and his army are discontented with him. I\nhear on high authority that his secret police are always making\ndiscoveries that render him desperately uneasy.\n\nYou know how we have been swindling in these parts. But perhaps you\ndon't know that Mr. ----, the \"eminent\" contractor, before he fell into\ndifficulties settled _one million of money_ on his wife. Such a good and\ndevoted husband!\n\nMy daughter Katie has been very ill of nervous fever. On the 27th of\nDecember she was in a condition to be brought down here (old high road\nand post-horses), and has been steadily getting better ever since. Her\nhusband is here too, and is on the whole as well as he ever is or ever\nwill be, I fear.\n\nWe played forfeit-games here, last night, and then pool. For a\nbilliard-room has been added to the house since you were here. Come and\nplay a match with me.\n\n                                                Always affectionately.\n\n\n[Sidenote: Miss Hogarth.]\n\n                  ADELPHI HOTEL, LIVERPOOL, _Monday, Jan. 21st, 1867._\n\nMY DEAREST GEORGY,\n\nFirst I send you my most affectionate wishes for many, many happy\nreturns of your birthday. That done, from my heart of hearts, I go on to\nmy small report of myself.\n\nThe readings have produced such an immense effect here that we are\ncoming back for two more in the middle of February. \"Marigold\" and the\n\"Trial,\" on Friday night, and the \"Carol,\" on Saturday afternoon, were a\nperfect furore; and the surprise about \"Barbox\" has been amusingly\ngreat. It is a most extraordinary thing, after the enormous sale of that\nChristmas number, that the provincial public seems to have combined to\nbelieve that it _won't_ make a reading. From Wolverhampton and Leeds we\nhave exactly the same expression of feelings _beforehand_. Exactly as I\nmade \"Copperfield\"--always to the poorest houses I had with Headland,\nand against that luminary's entreaty--so I should have to make this, if\nI hadn't \"Marigold\" always in demand.\n\nIt being next to impossible for people to come out at night with horses,\nwe have felt the weather in the stalls, and expect to do so through this\nweek. The half-crown and shilling publics have crushed to their places\nmost splendidly. The enthusiasm has been unbounded. On Friday night I\nquite astonished myself; but I was taken so faint afterwards that they\nlaid me on a sofa at the hall for half an hour. I attribute it to my\ndistressing inability to sleep at night, and to nothing worse.\n\nScott does very well indeed. As a dresser he is perfect. In a quarter of\nan hour after I go into the retiring-room, where all my clothes are\nairing and everything is set out neatly in its own allotted space, I am\nready; and he then goes softly out, and sits outside the door. In the\nmorning he is equally punctual, quiet, and quick. He has his needles and\nthread, buttons, and so forth, always at hand; and in travelling he is\nvery systematic with the luggage. What with Dolby and what with this\nskilful valet, everything is made as easy to me as it possibly _can_ be,\nand Dolby would do anything to lighten the work, and does everything.\n\nThere is great distress here among the poor (four thousand people\nrelieved last Saturday at one workhouse), and there is great anxiety\nconcerning _seven mail-steamers some days overdue_. Such a circumstance\nas this last has never been known. It is supposed that some great\nrevolving storm has whirled them all out of their course. One of these\nmissing ships is an American mail, another an Australian mail.\n\n\n                                                     _Same Afternoon._\n\nWe have been out for four hours in the bitter east wind, and walking on\nthe sea-shore, where there is a broad strip of great blocks of ice. My\nhands are so rigid that I write with great difficulty.\n\nWe have been constantly talking of the terrible Regent's Park accident.\nI hope and believe that nearly the worst of it is now known.\n\n\n[Sidenote: Miss Dickens.]\n\n                                  CHESTER, _Tuesday, Jan. 22nd, 1867._\n\nMY DEAREST MAMIE,\n\nWe came over here from Liverpool at eleven this forenoon. There was a\nheavy swell in the Mersey breaking over the boat; the cold was nipping,\nand all the roads we saw as we came along were wretched. We find a very\nmoderate let here; but I am myself rather surprised to know that a\nhundred and twenty stalls have made up their minds to the undertaking of\ngetting to the hall. This seems to be a very nice hotel, but it is an\nextraordinarily cold one. Our reading for to-night is \"Marigold\" and\n\"Trial.\" With amazing perversity the local agent said to Dolby: \"They\nhoped that Mr. Dickens _might_ have given them 'The Boy at Mugby.'\"\n\nBarton, the gasman who succeeded the man who sprained his leg, sprained\n_his_ leg yesterday!! And that, not at his work, but in running\ndownstairs at the hotel. However, he has hobbled through it so far, and\nI hope will hobble on, for he knows his work.\n\nI have seldom seen a place look more hopelessly frozen up than this\nplace does. The hall is like a Methodist chapel in low spirits, and with\na cold in its head. A few blue people shiver at the corners of the\nstreets. And this house, which is outside the town, looks like an\nornament on an immense twelfth cake baked for 1847.\n\nI am now going to the fire to try to warm myself, but have not the least\nexpectation of succeeding. The sitting-room has two large windows in it,\ndown to the ground and facing due east. The adjoining bedroom (mine) has\nalso two large windows in it, down to the ground and facing due east.\nThe very large doors are opposite the large windows, and I feel as if I\nwere something to eat in a pantry.\n\n\n[Sidenote: Miss Hogarth.]\n\n            HEN AND CHICKENS, BIRMINGHAM, _Thursday, Jan. 24th, 1867._\n\nAt Chester we read in a snowstorm and a fall of ice. I think it was the\nworst weather I ever saw. Nevertheless, the people were enthusiastic. At\nWolverhampton last night the thaw had thoroughly set in, and it rained\nheavily. We had not intended to go back there, but have arranged to do\nso on the day after Ash Wednesday. Last night I was again heavily\nbeaten. We came on here after the reading (it is only a ride of forty\nminutes), and it was as much as I could do to hold out the journey. But\nI was not faint, as at Liverpool; I was only exhausted. I am all right\nthis morning; and to-night, as you know, I have a rest. I trust that\nCharley Collins is better, and that Mamie is strong and well again.\nYesterday I had a note from Katie, which seemed hopeful and encouraging.\n\n\n[Sidenote: Miss Dickens.]\n\n            HEN AND CHICKENS, BIRMINGHAM, _Thursday, Jan. 24th, 1867._\n\nSince I wrote to your aunt just now, I have received your note addressed\nto Wolverhampton. We left the men there last night, and they brought it\non with them at noon to-day.\n\nThe maimed gasman's foot is much swollen, but he limps about and does\nhis work. I have doctored him up with arnica. During the \"Boy\" last\nnight there was an escape of gas from the side of my top batten, which\ncaught the copper-wire and was within a thread of bringing down the\nheavy reflector into the stalls. It was a very ticklish matter, though\nthe audience knew nothing about it. I saw it, and the gasman and Dolby\nsaw it, and stood at that side of the platform in agonies. We all three\ncalculated that there would be just time to finish and save it; when the\ngas was turned out the instant I had done, the whole thing was at its\nvery last and utmost extremity. Whom it would have tumbled on, or what\nmight have been set on fire, it is impossible to say.\n\nI hope you rewarded your police escort on Tuesday night. It was the most\ntremendous night I ever saw at Chester.\n\n\n[Sidenote: Miss Dickens.]\n\n                                      LEEDS, _Friday, Feb. 1st, 1867._\n\nWe got here prosperously, and had a good (but not great) house for\n\"Barbox\" and \"Boy\" last night. For \"Marigold\" and \"Trial,\" to-night,\neverything is gone. And I even have my doubts of the possibility of\nDolby's cramming the people in. For \"Marigold\" and \"Trial\" at\nManchester, to-morrow, we also expect a fine hall.\n\nI shall be at the office for next Wednesday. If Charley Collins should\nhave been got to Gad's, I will come there for that day. If not, I\nsuppose we had best open the official bower again.\n\nThis is a beastly place, with a very good hotel. Except Preston, it is\none of the nastiest places I know. The room is like a capacious coal\ncellar, and is incredibly filthy; but for sound it is perfect.\n\n\n[Sidenote: Anonymous.]\n\n            OFFICE OF \"ALL THE YEAR ROUND,\" _Tuesday, Feb. 5th, 1867._\n\nDEAR SIR,\n\nI have looked at the larger half of the first volume of your novel, and\nhave pursued the more difficult points of the story through the other\ntwo volumes.\n\nYou will, of course, receive my opinion as that of an individual writer\nand student of art, who by no means claims to be infallible.\n\nI think you are too ambitious, and that you have not sufficient\nknowledge of life or character to venture on so comprehensive an\nattempt. Evidences of inexperience in every way, and of your power being\nfar below the situations that you imagine, present themselves to me in\nalmost every page I have read. It would greatly surprise me if you found\na publisher for this story, on trying your fortune in that line, or\nderived anything from it but weariness and bitterness of spirit.\n\nOn the evidence thus put before me, I cannot even entirely satisfy\nmyself that you have the faculty of authorship latent within you. If you\nhave not, and yet pursue a vocation towards which you have no call, you\ncannot choose but be a wretched man. Let me counsel you to have the\npatience to form yourself carefully, and the courage to renounce the\nendeavour if you cannot establish your case on a very much smaller\nscale. You see around you every day, how many outlets there are for\nshort pieces of fiction in all kinds. Try if you can achieve any success\nwithin these modest limits (I have practised in my time what I preach to\nyou), and in the meantime put your three volumes away.\n\n                                                     Faithfully yours.\n\nP.S.--Your MS. will be returned separately from this office.\n\n\n[Sidenote: Miss Hogarth.]\n\n                                 LIVERPOOL, _Friday, Feb. 15th, 1867._\n\nMy short report of myself is that we had an enormous turn-away last\nnight, and do not doubt about having a cram to-night. The day has been\nvery fine, and I have turned it to the wholesomest account by walking on\nthe sands at New Brighton all the morning. I am not quite right, but\nbelieve it to be an effect of the railway shaking. There is no doubt of\nthe fact that, after the Staplehurst experience, it tells more and\nmore, instead of (as one might have expected) less and less.\n\nThe charming room here greatly lessens the fatigue of this fatiguing\nweek. I read last night with no more exertion than if I had been at\nGad's, and yet to eleven hundred people, and with astonishing effect. It\nis \"Copperfield\" to-night, and Liverpool is the \"Copperfield\"\nstronghold.\n\n\n[Sidenote: Miss Dickens.]\n\n                                   GLASGOW, _Sunday, Feb. 17th, 1867._\n\nWe arrived here this morning at our time to the moment, five minutes\npast ten. We turned away great numbers on both nights at Liverpool; and\nManchester last night was a splendid spectacle. They cheered to that\nextent after it was over, that I was obliged to huddle on my clothes\n(for I was undressing to prepare for the journey), and go back again.\n\nAfter so heavy a week, it _was_ rather stiff to start on this long\njourney at a quarter to two in the morning; but I got more sleep than I\never got in a railway-carriage before, and it really was not tedious.\nThe travelling was admirable, and a wonderful contrast to my friend the\nMidland.\n\nI am not by any means knocked up, though I have, as I had in the last\nseries of readings, a curious feeling of soreness all round the body,\nwhich I suppose to arise from the great exertion of voice. It is a mercy\nthat we were not both made really ill at Liverpool. On Friday morning I\nwas taken so faint and sick, that I was obliged to leave the table. On\nthe same afternoon the same thing happened to Dolby. We then found that\na part of the hotel close to us was dismantled for painting, and that\nthey were at that moment painting a green passage leading to our rooms,\nwith a most horrible mixture of white lead and arsenic. On pursuing the\nenquiry, I found that the four lady book-keepers in the bar were all\nsuffering from the poison.\n\n\n[Sidenote: Miss Hogarth.]\n\n                          BRIDGE OF ALLAN, _Tuesday, Feb. 19th, 1867._\n\nI was very glad to get your letter before leaving Glasgow this morning.\nThis is a poor return for it, but the post goes out early, and we come\nin late.\n\nYesterday morning I was so unwell that I wrote to Frank Beard, from whom\nI shall doubtless hear to-morrow. I mention it, only in case you should\ncome in his way, for I know how perversely such things fall out. I felt\nit a little more exertion to read afterwards, and I passed a sleepless\nnight after that again; but otherwise I am in good force and spirits\nto-day. I may say, in the best force.\n\nThe quiet of this little place is sure to do me good. The little inn in\nwhich we are established seems a capital house of the best country sort.\n\n\n[Sidenote: Miss Dickens.]\n\n                                 GLASGOW, _Thursday, Feb. 21st, 1867._\n\nAfter two days' rest at the Bridge of Allan I am in renewed force, and\nhave nothing to complain of but inability to sleep. I have been in\nexcellent air all day since Tuesday at noon, and made an interesting\nwalk to Stirling yesterday, and saw its lions, and (strange to relate)\nwas not bored by them. Indeed, they left me so fresh that I knocked at\nthe gate of the prison, presented myself to the governor, and took Dolby\nover the jail, to his unspeakable interest. We then walked back again\nto our excellent country inn.\n\nEnclosed is a letter from Alfred, which you and your aunt will be\ninterested in reading, and which I meant to send you sooner but forgot\nit. Wonderful as it is to mention, the sun shines here to-day! But to\ncounterbalance that phenomenon I am in close hiding from ----, who has\nchristened his infant son in my name, and, consequently, haunts the\nbuilding. He and Dolby have already nearly come into collision, in\nconsequence of the latter being always under the dominion of the one\nidea that he is bound to knock everybody down who asks for me.\n\n       *       *       *       *       *\n\n        The \"Jewish lady,\" wishing to mark her\n        \"appreciation of Mr. Dickens's nobility of\n        character,\" presented him with a copy of\n        Benisch's Hebrew and English Bible, with this\n        inscription: \"Presented to Charles Dickens, in\n        grateful and admiring recognition of his having\n        exercised the noblest quality man can\n        possess--that of atoning for an injury as soon\n        as conscious of having inflicted it.\"\n\n        The acknowledgment of the gift is the following\n        letter:\n\n[Sidenote: Jewish Lady.]\n\n                       BRADFORD, YORKSHIRE, _Friday, March 1st, 1867._\n\nMY DEAR MRS. ----,\n\nI am working through a series of readings, widely dispersed through\nEngland, Scotland, and Ireland, and am so constantly occupied that it is\nvery difficult for me to write letters. I have received your highly\nesteemed note (forwarded from my home in Kent), and should have replied\nto it sooner but that I had a hope of being able to get home and see\nyour present first. As I have not been able to do so, however, and am\nhardly likely to do so for two months to come, I delay no longer. It is\nsafely awaiting me on my own desk in my own quiet room. I cannot thank\nyou for it too cordially, and cannot too earnestly assure you that I\nshall always prize it highly. The terms in which you send me that mark\nof your remembrance are more gratifying to me than I can possibly\nexpress to you; for they assure me that there is nothing but goodwill\nleft between you and me and a people for whom I have a real regard, and\nto whom I would not wilfully have given an offence or done an injustice\nfor any worldly consideration.\n\n                                    Believe me, very faithfully yours.\n\n\n[Sidenote: Miss Hogarth.]\n\n                      NEWCASTLE-ON-TYNE, _Wednesday, March 6th, 1867._\n\nThe readings have made an immense effect in this place, and it is\nremarkable that although the people are individually rough, collectively\nthey are an unusually tender and sympathetic audience; while their comic\nperception is quite up to the high London standard. The atmosphere is so\nvery heavy that yesterday we escaped to Tynemouth for a two hours' sea\nwalk. There was a high north wind blowing and a magnificent sea running.\nLarge vessels were being towed in and out over the stormy bar, with\nprodigious waves breaking on it; and spanning the restless uproar of the\nwaters was a quiet rainbow of transcendent beauty. The scene was quite\nwonderful. We were in the full enjoyment of it when a heavy sea caught\nus, knocked us over, and in a moment drenched us, and filled even our\npockets. We had nothing for it but to shake ourselves together (like\nDoctor Marigold) and dry ourselves as well as we could by hard walking\nin the wind and sunshine! But we were wet through for all that when we\ncame back here to dinner after half an hour's railway ride.\n\nI am wonderfully well, and quite fresh and strong. Have had to doctor\nDolby for a bad cold; have not caught it (yet), and have set him on his\nlegs again.\n\nScott is striking the tents and loading the baggages, so I must deliver\nup my writing-desk. We meet, please God, on Tuesday.\n\n\n[Sidenote: Miss Hogarth.]\n\n                 SHELBOURNE HOTEL, DUBLIN, _Friday, March 15th, 1867._\n\nWe made our journey through an incessant snowstorm on Wednesday night;\nat last got snowed up among the Welsh mountains in a tremendous storm of\nwind, came to a stop, and had to dig the engine out. We went to bed at\nHolyhead at six in the morning of Thursday, and got aboard the packet at\ntwo yesterday afternoon. It blew hard, but as the wind was right astern,\nwe only rolled and did not pitch much. As I walked about on the bridge\nall the four hours, and had cold salt beef and biscuit there and\nbrandy-and-water, you will infer that my Channel training has not worn\nout.\n\nOur \"business\" here is _very bad_, though at Belfast it is enormous.\nThere is no doubt that great alarm prevails here. This hotel is\nconstantly filling and emptying as families leave the country, and set\nin a current to the steamers. There is apprehension of some disturbance\nbetween to-morrow night and Monday night (both inclusive), and I learn\nthis morning that all the drinking-shops are to be closed from to-night\nuntil Tuesday. It is rumoured here that the Liverpool people are very\nuneasy about some apprehended disturbance there at the same time. Very\nlikely you will know more about this than I do, and very likely it may\nbe nothing. There is no doubt whatever that alarm prevails, and the\nmanager of this hotel, an intelligent German, is very gloomy on the\nsubject. On the other hand, there is feasting going on, and I have been\nasked to dinner-parties by divers civil and military authorities.\n\nDon't _you_ be uneasy, I say once again. You may be absolutely certain\nthat there is no cause for it. We are splendidly housed here, and in\ngreat comfort.\n\nLove to Charley and Katey.\n\n\n[Sidenote: Miss Dickens.]\n\n               SHELBOURNE HOTEL, DUBLIN, _Saturday, March 16th, 1867._\n\nI daresay you know already that I held many councils in London about\ncoming to Ireland at all, and was much against it. Everything looked as\nbad here as need be, but we did very well last night after all.\n\nThere is considerable alarm here beyond all question, and great\ndepression in all kinds of trade and commerce. To-morrow being St.\nPatrick's Day, there are apprehensions of some disturbance, and croakers\npredict that it will come off between to-night and Monday night. Of\ncourse there are preparations on all sides, and large musters of\nsoldiers and police, though they are kept carefully out of sight. One\nwould not suppose, walking about the streets, that any disturbance was\nimpending; and yet there is no doubt that the materials of one lie\nsmouldering up and down the city and all over the country. [I have a\nletter from Mrs. Bernal Osborne this morning, describing the fortified\nway in which she is living in her own house in the County Tipperary.]\n\nYou may be quite sure that your venerable parent will take good care of\nhimself. If any riot were to break out, I should immediately stop the\nreadings here. Should all remain quiet, I begin to think they will be\nsatisfactorily remunerative after all. At Belfast, we shall have an\nenormous house. I read \"Copperfield\" and \"Bob\" here on Monday;\n\"Marigold\" and \"Trial\" at Belfast, on Wednesday; and \"Carol\" and \"Trial\"\nhere, on Friday. This is all my news, except that I am in perfect force.\n\n\n[Sidenote: Miss Hogarth.]\n\n                 SHELBOURNE HOTEL, DUBLIN, _Sunday, March 17th, 1867._\n\nEverything remains in appearance perfectly quiet here. The streets are\ngay all day, now that the weather is improved, and singularly quiet and\ndeserted at night. But the whole place is secretly girt in with a\nmilitary force. To-morrow night is supposed to be a critical time; but\nin view of the enormous preparations, I should say that the chances are\nat least one hundred to one against any disturbance.\n\nI cannot make sure whether I wrote to you yesterday, and told you that\nwe had done very well at the first reading after all, even in money. The\nreception was prodigious, and the readings are the town talk. But I\nrather think I did actually write this to you. My doubt on the subject\narises from my having deliberated about writing on a Saturday.\n\nThe most curious, and for facilities of mere destruction, such as firing\nhouses in different quarters, the most dangerous piece of intelligence\nimparted to me on authority is, that the Dublin domestic men-servants as\na class are all Fenians.\n\n\n[Sidenote: Miss Hogarth.]\n\n                               BELFAST, _Wednesday, March 20th, 1867._\n\nThe post goes out at twelve, and I have only time to report myself. The\nsnow not lying between this and Dublin, we got here yesterday to our\ntime, after a cold but pleasant journey. Fitzgerald came on with us. I\nhad a really charming letter from Mrs. Fitzgerald, asking me to stay\nthere. She must be a perfectly unaffected and genuine lady. There are\nkind messages to you and Mary in it. I have sent it on to Mary, who will\nprobably in her turn show it to you. We had a wonderful crowd at Dublin\non Monday, and the greatest appreciation possible. We have a good let,\nin a large hall, here to-night. But I am perfectly convinced that the\nworst part of the Fenian business is to come yet.\n\nAll about the Fitzgeralds and everything else when we meet.\n\n\n[Sidenote: Miss Dickens.]\n\n                                BELFAST, _Thursday, March 21st, 1867._\n\nIn spite of public affairs and dismal weather, we are doing wonders in\nIreland.\n\nThat the conspiracy is a far larger and more important one than would\nseem from what it has done yet, there is no doubt. I have had a good\ndeal of talk with a certain colonel, whose duty it has been to\ninvestigate it, day and night, since last September. That it will give a\nworld of trouble, and cost a world of money, I take to be (after what I\nhave thus learned) beyond all question. One regiment has been found to\ncontain five hundred Fenian soldiers every man of whom was sworn in the\nbarrack-yard. How information is swiftly and secretly conveyed all over\nthe country, the Government with all its means and money cannot\ndiscover; but every hour it is found that instructions, warnings, and\nother messages are circulated from end to end of Ireland. It is a very\nserious business indeed.\n\nI have just time to send this off, and to report myself quite well\nexcept for a slight cold.\n\n\n[Sidenote: Miss Hogarth.]\n\n                                  NORWICH, _Friday, March 29th, 1867._\n\nThe reception at Cambridge last night was something to be proud of in\nsuch a place. The colleges mustered in full force from the biggest guns\nto the smallest, and went far beyond even Manchester in the roars of\nwelcome and the rounds of cheers. All through the readings, the whole of\nthe assembly, old men as well as young, and women as well as men, took\neverything with a heartiness of enjoyment not to be described. The place\nwas crammed, and the success the most brilliant I have ever seen.\n\nWhat we are doing in this sleepy old place I don't know, but I have no\ndoubt it is mild enough.\n\n\n[Sidenote: Mr. Walter Thornbury]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                            _Monday, April 1st, 1867._\n\nMY DEAR THORNBURY,\n\nI am very doubtful indeed about \"Vaux,\" and have kept it out of the\nnumber in consequence. The mere details of such a rascal's proceedings,\nwhether recorded by himself or set down by the Reverend Ordinary, are\nnot wholesome for a large audience, and are scarcely justifiable (I\nthink) as claiming to be a piece of literature. I can understand\nBarrington to be a good subject, as involving the representation of a\nperiod, a style of manners, an order of dress, certain habits of street\nlife, assembly-room life, and coffee-room life, etc.; but there is a\nvery broad distinction between this and mere Newgate Calendar. The\nlatter would assuredly damage your book, and be protested against to me.\nI have a conviction of it, founded on constant observation and\nexperience here.\n\nYour kind invitation is extremely welcome and acceptable to me, but I am\nsorry to add that I must not go a-visiting. For this reason: So\nincessantly have I been \"reading,\" that I have not once been at home at\nGad's Hill since last January, and am little likely to get there before\nthe middle of May. Judge how the master's eye must be kept on the place\nwhen it does at length get a look at it after so long an absence! I hope\nyou will descry in this a reason for coming to me again, instead of my\ncoming to you.\n\nThe extinct prize-fighters, as a body, I take to be a good subject, for\nmuch the same reason as George Barrington. Their patrons were a class of\nmen now extinct too, and the whole ring of those days (not to mention\nJackson's rooms in Bond Street) is a piece of social history. Now Vaux\nis not, nor is he even a phenomenon among thieves.\n\n                                              Faithfully yours always.\n\n\n[Sidenote: Mr. Clarkson Stanfield, R.A.]\n\n                          GAD'S HILL PLACE, HIGHAM BY ROCHESTER,\n                                         _Thursday, April 18th, 1867._\n\nMY DEAR STANNY,\n\nThe time of year reminds me how the months have gone, since I last heard\nfrom you through Mrs. Stanfield.\n\nI hope you have not thought me unmindful of you in the meanwhile. I have\nbeen almost constantly travelling and reading. England, Ireland, and\nScotland have laid hold of me by turns, and I have had no rest. As soon\nas I had finished this kind of work last year, I had to fall to work\nupon \"All the Year Round\" and the Christmas number. I was no sooner quit\nof that task, and the Christmas season was but run out to its last day,\nwhen I was tempted into another course of fifty readings that are not\nyet over. I am here now for two days, and have not seen the place since\nTwelfth Night. When a reading in London has been done, I have been\nbrought up for it from some great distance, and have next morning been\ncarried back again. But the fifty will be \"paid out\" (as we say at sea)\nby the middle of May, and then I hope to see you.\n\nReading at Cheltenham the other day, I saw Macready, who sent his love\nto you. His face was much more massive and as it used to be, than when I\nsaw him previous to his illness. His wife takes admirable care of him,\nand is on the happiest terms with his daughter Katie. His boy by the\nsecond marriage is a jolly little fellow, and leads a far easier life\nthan the children you and I remember, who used to come in at dessert and\nhave each a biscuit and a glass of water, in which last refreshment I\nwas always convinced that they drank, with the gloomiest malignity,\n\"Destruction to the gormandising grown-up company!\"\n\nI hope to look up your latest triumphs on the day of the Academy dinner.\nOf course as yet I have had no opportunity of even hearing of what\nanyone has done. I have been (in a general way) snowed up for four\nmonths. The locomotive with which I was going to Ireland was dug out of\nthe snow at midnight, in Wales. Both passages across were made in a\nfurious snowstorm. The snow lay ankle-deep in Dublin, and froze hard at\nBelfast. In Scotland it slanted before a perpetual east wind. In\nYorkshire, it derived novelty from thunder and lightning. Whirlwinds\neverywhere I don't mention.\n\nGod bless you and yours. If I look like some weather-beaten pilot when\nwe meet, don't be surprised. Any mahogany-faced stranger who holds out\nhis hand to you will probably turn out, on inspection, to be the old\noriginal Dick.\n\n                 Ever, my dear Stanny, your faithful and affectionate.\n\nP.S.--I wish you could have been with me (of course in a snowstorm) one\nday on the pier at Tynemouth. There was a very heavy sea running, and a\nperfect fleet of screw merchantmen were plunging in and out on the turn\nof the tide at high-water. Suddenly there came a golden horizon, and a\nmost glorious rainbow burst out, arching one large ship, as if she were\nsailing direct for heaven. I was so enchanted by the scene, that I\nbecame oblivious of a few thousand tons of water coming on in an\nenormous roller, and was knocked down and beaten by its spray when it\nbroke, and so completely wetted through and through, that the very\npockets in my pocket-book were full of sea.\n\n\n[Sidenote: Mr. George Stanfield.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                             _Sunday, May 19th, 1867._\n\n                    ON THE DEATH OF HIS FATHER.\n\nMY DEAR GEORGE,\n\nWhen I came up to the house this afternoon and saw what had happened, I\nhad not the courage to ring, though I had thought I was fully prepared\nby what I heard when I called yesterday. No one of your father's friends\ncan ever have loved him more dearly than I always did, or can have\nbetter known the worth of his noble character.\n\nIt is idle to suppose that I can do anything for you; and yet I cannot\nhelp saying that I am staying here for some days, and that if I could,\nit would be a much greater relief to me than it could be a service to\nyou.\n\nYour poor mother has been constantly in my thoughts since I saw the\nquiet bravery with which she preserved her composure. The beauty of her\nministration sank into my heart when I saw him for the last time on\nearth. May God be with her, and with you all, in your great loss.\n\n                                          Affectionately yours always.\n\n\n[Sidenote: Mr. W. H. Wills.]\n\n                                           _Thursday, June 6th, 1867._\n\nMY DEAR WILLS,\n\nI cannot tell you how warmly I feel your letter, or how deeply I\nappreciate the affection and regard in which it originates. I thank you\nfor it with all my heart.\n\nYou will not suppose that I make light of any of your misgivings if I\npresent the other side of the question. Every objection that you make\nstrongly impresses me, and will be revolved in my mind again and again.\n\nWhen I went to America in '42, I was so much younger, but (I think) very\nmuch weaker too. I had had a painful surgical operation performed\nshortly before going out, and had had the labour from week to week of\n\"Master Humphrey's Clock.\" My life in the States was a life of continual\nspeech-making (quite as laborious as reading), and I was less patient\nand more irritable then than I am now. My idea of a course of readings\nin America is, that it would involve far less travelling than you\nsuppose, that the large first-class rooms would absorb the whole course,\nand that the receipts would be very much larger than your estimate,\nunless the demand for the readings is ENORMOUSLY EXAGGERATED ON ALL\nHANDS. There is considerable reason for this view of the case. And I can\nhardly think that all the speculators who beset, and all the private\ncorrespondents who urge me, are in a conspiracy or under a common\ndelusion.\n\n       *       *       *       *       *\n\nI shall never rest much while my faculties last, and (if I know myself)\nhave a certain something in me that would still be active in rusting and\ncorroding me, if I flattered myself that I was in repose. On the other\nhand, I think that my habit of easy self-abstraction and withdrawal into\nfancies has always refreshed and strengthened me in short intervals\nwonderfully. I always seem to myself to have rested far more than I have\nworked; and I do really believe that I have some exceptional faculty of\naccumulating young feelings in short pauses, which obliterates a\nquantity of wear and tear.\n\nMy worldly circumstances (such a large family considered) are very good.\nI don't want money. All my possessions are free and in the best order.\nStill, at fifty-five or fifty-six, the likelihood of making a very great\naddition to one's capital in half a year is an immense consideration....\nI repeat the phrase, because there should be something large to set\nagainst the objections.\n\nI dine with Forster to-day, to talk it over. I have no doubt he will\nurge most of your objections and particularly the last, though American\nfriends and correspondents he has, have undoubtedly staggered him more\nthan I ever knew him to be staggered on the money question. Be assured\nthat no one can present any argument to me which will weigh more\nheartily with me than your kind words, and that whatever comes of my\npresent state of abeyance, I shall never forget your letter or cease to\nbe grateful for it.\n\n                                Ever, my dear Wills, faithfully yours.\n\n\n[Sidenote: Mr. W. H. Wills.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                            _Sunday, June 13th, 1867._\n\nMY DEAR WILLS,\n\nI have read the first three numbers of Wilkie's story this morning, and\nhave gone minutely through the plot of the rest to the last line. It\ngives a series of \"narratives,\" but it is a very curious story, wild,\nand yet domestic, with excellent character in it, and great mystery. It\nis prepared with extraordinary care, and has every chance of being a\nhit. It is in many respects much better than anything he has done. The\nquestion is, how shall we fill up the blank between Mabel's progress and\nWilkie? What do you think of proposing to Fitzgerald to do a story three\nmonths long? I daresay he has some unfinished or projected something by\nhim.\n\nI have an impression that it was not Silvester who tried Eliza Fenning,\nbut Knowles. One can hardly suppose Thornbury to make such a mistake,\nbut I wish you would look into the Annual Register. I have added a final\nparagraph about the unfairness of the judge, whoever he was. I\ndistinctly recollect to have read of his \"putting down\" of Eliza\nFenning's father when the old man made some miserable suggestion in his\ndaughter's behalf (this is not noticed by Thornbury), and he also\nstopped some suggestion that a knife thrust into a loaf adulterated with\nalum would present the appearance that these knives presented. But I may\nhave got both these points from looking up some pamphlets in Upcott's\ncollection which I once had.\n\nYour account of your journey reminds me of one of the latest American\nstories, how a traveller by stage-coach said to the driver: \"Did you\never see a snail, sir?\" \"Yes, sir.\" \"Where did you meet him, sir?\" \"I\n_didn't_ meet him, sir!\" \"Wa'al, sir, I think you did, if you'll excuse\nme, for I'm damned if you ever overtook him.\"\n\nEver faithfully.\n\n\n[Sidenote: Mrs. Henderson.]\n\n                               GAD'S HILL, _Thursday, July 4th, 1867._\n\nMY DEAR MRS. HENDERSON,\n\nI was more shocked than surprised by the receipt of your mother's\nannouncement of our poor dear Marguerite's death. When I heard of the\nconsultation, and recalled what had preceded it and what I have seen\nhere, my hopes were very slight.\n\nYour letter did not reach me until last night, and thus I could not\navoid remaining here to-day, to keep an American appointment of unusual\nimportance. You and your mother both know, I think, that I had a great\naffection for Marguerite, that we had many dear remembrances together,\nand that her self-reliance and composed perseverance had awakened my\nhighest admiration in later times. No one could have stood by her grave\nto-day with a better knowledge of all that was great and good in her\nthan I have, or with a more loving remembrance of her through all her\nphases since she first came to London a pretty timid girl.\n\nI do not trouble your mother by writing to her separately. It is a sad,\nsad task to write at all. God help us!\n\n                                                     Faithfully yours.\n\n\n[Sidenote: Mr. Percy Fitzgerald.]\n\n                                        GAD'S HILL, _July 21st, 1867._\n\nMY DEAR FITZGERALD,\n\nI am heartily glad to get your letter, and shall be thoroughly well\npleased to study you again in the pages of A. Y. R.\n\nI have settled nothing yet about America, but am going to send Dolby out\non the 3rd of next month to survey the land, and come back with a report\non some heads whereon I require accurate information. Proposals (both\nfrom American and English speculators) of a very tempting nature have\nbeen repeatedly made to me; but I cannot endure the thought of binding\nmyself to give so many readings there whether I like it or no; and if I\ngo at all, am bent on going with Dolby single-handed.\n\nI have been doing two things for America; one, the little story to which\nyou refer; the other, four little papers for a child's magazine. I like\nthem both, and think the latter a queer combination of a child's mind\nwith a grown-up joke. I have had them printed to assure correct printing\nin the United States. You shall have the proof to read, with the\ngreatest pleasure. On second thoughts, why shouldn't I send you the\nchildren's proof by this same post? I will, as I have it here, send it\nunder another cover. When you return it, you shall have the short story.\n\n                                    Believe me, always heartily yours.\n\n\n[Sidenote: Mr. Percy Fitzgerald.]\n\n                                  EXTRACT.\n\n                                                    _July 28th, 1867._\n\nI am glad you like the children, and particularly glad you like the\npirate. I remember very well when I had a general idea of occupying that\nplace in history at the same age. But I loved more desperately than\nBoldheart.\n\n\n[Sidenote: Miss Hogarth.]\n\n             ADELPHI HOTEL, LIVERPOOL, _Friday Night, Aug. 2nd, 1867._\n\nMY DEAREST GEORGY,\n\nI cannot get a boot on--wear a slipper on my left foot, and consequently\nam here under difficulties. My foot is occasionally painful, but not\nvery. I don't think it worth while consulting anybody about it as yet. I\nmake out so many reasons against supposing it to be gouty, that I really\ndo not think it is.\n\nDolby begs me to send all manner of apologetic messages for his going to\nAmerica. He is very cheerful and hopeful, but evidently feels the\nseparation from his wife and child very much. His sister[17] was at\nEuston Square this morning, looking very well. Sainton too, very light\nand jovial.\n\nWith the view of keeping myself and my foot quiet, I think I will not\ncome to Gad's Hill until Monday. If I don't appear before, send basket\nto Gravesend to meet me, leaving town by the 12.10 on Monday. This is\nimportant, as I couldn't walk a quarter of a mile to-night for five\nhundred pounds.\n\nLove to all at Gad's.\n\n\n[Sidenote: Mr. W. H. Wills.]\n\n                                GAD'S HILL, _Monday, Sept. 2nd, 1867._\n\nMY DEAR WILLS,\n\nLike you, I was shocked when this new discovery burst upon me on Friday,\nthough, unlike you, I never could believe in ----, solely (I think)\nbecause, often as I have tried him, I never found him standing by my\ndesk when I was writing a letter without trying to read it.\n\nI fear there is no doubt that since ----'s discharge, he (----) has\nstolen money at the readings. A case of an abstracted shilling seems to\nhave been clearly brought home to him by Chappell's people, and they\nknow very well what _that_ means. I supposed a very clear keeping off\nfrom Anne's husband (whom I recommended for employment to Chappell) to\nhave been referable only to ----; but now I see how hopeless and unjust\nit would be to expect belief from him with two such cases within his\nknowledge.\n\nBut don't let the thing spoil your holiday. If we try to do our duty by\npeople we employ, by exacting their proper service from them on the one\nhand, and treating them with all possible consistency, gentleness, and\nconsideration on the other, we know that we do right. Their doing wrong\ncannot change our doing right, and that should be enough for us.\n\nSo I have given _my_ feathers a shake, and am all right again. Give\n_your_ feathers a shake, and take a cheery flutter into the air of\nHertfordshire.\n\nGreat reports from Dolby and also from Fields! But I keep myself quite\ncalm, and hold my decision in abeyance until I shall have book, chapter,\nand verse before me. Dolby hoped he could leave Uncle Sam on the 11th of\nthis month.\n\nSydney has passed as a lieutenant, and appeared at home yesterday, all\nof a sudden, with the consequent golden garniture on his sleeve, which\nI, God forgive me, stared at without the least idea that it meant\npromotion.\n\nI am glad you see a certain unlikeness to anything in the American\nstory. Upon myself it has made the strangest impression of reality and\noriginality!! And I feel as if I had read something (by somebody else),\nwhich I should never get out of my mind!!! The main idea of the\nnarrator's position towards the other people was the idea that I _had_\nfor my next novel in A. Y. R. But it is very curious that I did not in\nthe least see how to begin his state of mind until I walked into Hoghton\nTowers one bright April day with Dolby.\n\n                                                      Faithfully ever.\n\n\n[Sidenote: Mr. F. D. Finlay.]\n\n           CONTRADICTING A NEWSPAPER REPORT OF HIS BEING IN A\n                       CRITICAL STATE OF HEALTH.\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                           _Tuesday, Sept. 3rd, 1867._\n\nThis is to certify that the undersigned victim of a periodical\nparagraph-disease, which usually breaks out once in every seven years\n(proceeding to England by the overland route to India and per Cunard\nline to America, where it strikes the base of the Rocky Mountains, and,\nrebounding to Europe, perishes on the steppes of Russia), is _not_ in a\n\"critical state of health,\" and has _not_ consulted \"eminent surgeons,\"\nand never was better in his life, and is _not_ recommended to proceed to\nthe United States for \"cessation from literary labour,\" and has not had\nso much as a headache for twenty years.\n\n                                                      CHARLES DICKENS.\n\n\n[Sidenote: M. Charles Fechter.]\n\n                                    \"ALL THE YEAR ROUND\" OFFICE,\n                                           _Monday, Sept. 16th, 1867._\n\nMY DEAR FECHTER,\n\nGoing over the prompt-book carefully, I see one change in your part to\nwhich (on Lytton's behalf) I positively object, as I am quite certain he\nwould not consent to it. It is highly injudicious besides, as striking\nout the best known line in the play.\n\nTurn to your part in Act III., the speech beginning\n\n                                Pauline, _by pride\n        Angels have fallen ere thy time_: by pride----\n\nYou have made a passage farther on stand:\n\n                            _Then did I seek to rise\n        Out of my mean estate. Thy bright image, etc._\n\nI must stipulate for your restoring it thus:\n\n                            Then did I seek to rise\n        Out of the prison of my mean estate;\n        And, with such jewels as the exploring mind\n        Brings from the caves of knowledge, buy my ransom\n        From those twin jailers of the daring heart--\n        Low birth and iron fortune. Thy bright image, etc. etc.\n\nThe last figure has been again and again quoted; is identified with the\nplay; is fine in itself; and above all, I KNOW that Lytton would not let\nit go. In writing to him to-day, fully explaining the changes in detail,\nand saying that I disapprove of nothing else, I have told him that I\nnotice this change and that I immediately let you know that it must not\nbe made.\n\n(There will not be a man in the house from any newspaper who would not\ndetect mutilations in that speech, moreover.)\n\n                                                                 Ever.\n\n\n[Sidenote: Miss Hogarth.]\n\n                                           _Monday, Sept. 30th, 1867._\n\nMY DEAREST GEORGY,\n\nThe telegram is despatched to Boston: \"Yes. Go ahead.\" After a very\nanxious consultation with Forster, and careful heed of what is to be\nsaid for and against, I have made up my mind to see it out. I do not\nexpect as much money as the calculators estimate, but I cannot set the\nhope of a large sum of money aside.\n\nI am so nervous with travelling and anxiety to decide something, that I\ncan hardly write. But I send you these few words as my dearest and best\nfriend.\n\n\n[Sidenote: Miss Dickens.]\n\n  OFFICE OF \"ALL THE YEAR ROUND,\" NO. 26, WELLINGTON STREET, STRAND,\n                                      LONDON, W.C.,\n                                           _Monday, Sept. 30th, 1867._\n\nMY DEAREST MAMIE,\n\nYou will have had my telegram that I go to America. After a long\ndiscussion with Forster, and consideration of what is to be said on both\nsides, I have decided to go through with it. I doubt the profit being as\ngreat as the calculation makes it, but the prospect is sufficiently\nalluring to turn the scale on the American side.\n\nUnless I telegraph to the contrary, I will come to Gravesend (send\nbasket there) by 12 train on Wednesday. Love to all.\n\nWe have telegraphed \"Yes\" to Boston.\n\nI begin to feel myself drawn towards America, as Darnay, in the \"Tale of\nTwo Cities,\" was attracted to the Loadstone Rock, Paris.\n\n\n[Sidenote: Mr. William Charles Kent.]\n\n                   26, WELLINGTON STREET, _Saturday, Oct. 19th, 1867._\n\nMY DEAR KENT,\n\nIn the midst of the great trouble you are taking in the cause of your\nundersigned affectionate friend, I hope the reading of the enclosed may\nbe a sort of small godsend. Of course it is very strictly private. The\nprinters are not yet trusted with the name, but the name will be, \"No\nThoroughfare.\" I have done the greater part of it; may you find it\ninteresting!\n\nMy solicitor, a man of some mark and well known, is anxious to be on the\nCommittee:\n\n        Frederic Ouvry, Esquire,\n                 66, Lincoln's Inn Fields.\n\n                                            Ever affectionately yours.\n\nP.S.--My sailor son!\n\nI forgot him!!\n\nComing up from Portsmouth for the dinner!!!\n\nDer--er--oo not cur--ur--urse me, I implore.\n\n                                                           Penitently.\n\n\n[Sidenote: Mrs. Power.]\n\n                             GAD'S HILL, _Wednesday, Oct. 23rd, 1867._\n\nMY DEAR MRS. POWER,\n\nI have a sad pleasure in the knowledge that our dear Marguerite so\nremembered her old friend, and I shall preserve the token of her\nremembrance with loving care. The sight of it has brought back many old\ndays.\n\nWith kind remembrance to Mrs. Henderson,\n\n                             Believe me always, very faithfully yours.\n\n\n[Sidenote: Mr. J. L. Toole.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                           _Saturday, Nov. 2nd, 1867._\n\nMY DEAR MR. TOOLE,\n\nI heartily thank you for your elegant token of remembrance, and for your\nearnest letter. Both have afforded me real pleasure, and the first-named\nshall go with me on my journey.\n\nLet me take this opportunity of saying that on receipt of your letter\nconcerning to-day's dinner, I immediately forwarded your request to the\nhonorary secretary. I hope you will understand that I could not, in\ndelicacy, otherwise take part in the matter.\n\nAgain thanking you most cordially,\n\n                                  Believe me, always faithfully yours.\n\n\n[Sidenote: Mr. W. H. Wills.]\n\n                      26, WELLINGTON STREET, _Sunday, Nov. 3rd, 1867._\n\nMY DEAR WILLS,\n\nIf you were to write me many such warm-hearted letters as you send this\nmorning, my heart would fail me! There is nothing that so breaks down my\ndetermination, or shows me what an iron force I put upon myself, and how\nweak it is, as a touch of true affection from a tried friend.\n\nAll that you so earnestly say about the goodwill and devotion of all\nengaged, I perceived and deeply felt last night. It moved me even more\nthan the demonstration itself, though I do suppose it was the most\nbrilliant ever seen. When I got up to speak, but for taking a desperate\nhold of myself, I should have lost my sight and voice and sat down\nagain.\n\nGod bless you, my dear fellow. I am, ever and ever,\n\n                                                    Your affectionate.\n\n\n[Sidenote: The Hon. Mrs. Watson.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                            _Tuesday, Nov. 5th, 1867._\n\nMY DEAR MRS. WATSON,\n\nA thousand thanks for your kind letter, and many congratulations on your\nhaving successfully attained a dignity which I never allow to be\nmentioned in my presence. Charley's children are instructed from their\ntenderest months only to know me as \"Wenerables,\" which they sincerely\nbelieve to be my name, and a kind of title that I have received from a\ngrateful country.\n\nAlas! I cannot have the pleasure of seeing you before I presently go to\nLiverpool. Every moment of my time is preoccupied. But I send you my\nsincere love, and am always truthful to the dear old days, and the\nmemory of one of the dearest friends I ever loved.\n\n                                                 Affectionately yours.\n\n\n[Sidenote: Miss Dickens.]\n\n                          ABOARD THE \"CUBA,\" QUEENSTOWN HARBOUR,\n                                            _Sunday, Nov. 10th, 1867._\n\nMY DEAREST MAMIE,\n\nWe arrived here at seven this morning, and shall probably remain\nawaiting our mail, until four or five this afternoon. The weather in the\npassage here was delightful, and we had scarcely any motion beyond that\nof the screw.\n\nWe are nearly but not quite full of passengers. At table I sit next the\ncaptain, on his right, on the outside of the table and close to the\ndoor. My little cabin is big enough for everything but getting up in and\ngoing to bed in. As it has a good window which I can leave open all\nnight, and a door which I can set open too, it suits my chief\nrequirements of it--plenty of air--admirably. On a writing-slab in it,\nwhich pulls out when wanted, I now write in a majestic manner.\n\nMany of the passengers are American, and I am already on the best terms\nwith nearly all the ship.\n\nWe began our voyage yesterday a very little while after you left us,\nwhich was a great relief. The wind is S.E. this morning, and if it would\nkeep so we should go along nobly. My dearest love to your aunt, and\nalso to Katie and all the rest. I am in very good health, thank God, and\nas well as possible.\n\n\n[Sidenote: Miss Hogarth.]\n\n                               ABOARD THE \"CUBA,\" FIVE DAYS OUT,\n                                         _Wednesday, Nov. 13th, 1867._\n\nMY DEAREST GEORGY,\n\nAs I wrote to Mamie last, I now write to you, or mean to do it, if the\nmotion of the ship will let me.\n\nWe are very nearly halfway to-day. The weather was favourable for us\nuntil yesterday morning, when we got a head-wind which still stands by\nus. We have rolled and pitched, of course; but on the whole have been\nwonderfully well off. I have had headache and have felt faint once or\ntwice, _but have not been sick at all_. My spacious cabin is very noisy\nat night, as the most important working of the ship goes on outside my\nwindow and over my head; but it is very airy, and if the weather be bad\nand I can't open the window, I can open the door all night. If the\nweather be fine (as it is now), I can open both door and window, and\nwrite between them. Last night, I got a foot-bath under the dignified\ncircumstances of sitting on a camp-stool in my cabin, and having the\nbath (and my feet) in the passage outside. The officers' quarters are\nclose to me, and, as I know them all, I get reports of the weather and\nthe way we are making when the watch is changed, and I am (as I usually\nam) lying awake. The motion of the screw is at its slightest vibration\nin my particular part of the ship. The silent captain, reported gruff,\nis a very good fellow and an honest fellow. Kelly has been ill all the\ntime, and not of the slightest use, and is ill now. Scott always\ncheerful, and useful, and ready; a better servant for the kind of work\nthere never can have been. Young Lowndes has been fearfully sick until\nmid-day yesterday. His cabin is pitch dark, and full of blackbeetles. He\nshares mine until nine o'clock at night, when Scott carries him off to\nbed. He also dines with me in my magnificent chamber. This passage in\nwinter time cannot be said to be an enjoyable excursion, but I certainly\nam making it under the best circumstances. (I find Dolby to have been\nenormously popular on board, and to have known everybody and gone\neverywhere.)\n\nSo much for my news, except that I have been constantly reading, and\nfind that \"Pierra\" that Mrs. Hogge sent me by Katie to be a very\nremarkable book, not only for its grim and horrible story, but for its\nsuggestion of wheels within wheels, and sad human mysteries. Baker's\nsecond book not nearly so good as his first, but his first anticipated\nit.\n\nWe hope to get to Halifax either on Sunday or Monday, and to Boston\neither on Tuesday or Wednesday. The glass is rising high to-day, and\neverybody on board is hopeful of an easterly wind.\n\n\n[Sidenote: Miss Hogarth.]\n\n                                                     _Saturday, 16th._\n\nLast Thursday afternoon a heavy gale of wind sprang up and blew hard\nuntil dark, when it seemed to lull. But it then came on again with great\nviolence, and blew tremendously all night. The noise, and the rolling\nand plunging of the ship, were awful. Nobody on board could get any\nsleep, and numbers of passengers were rolled out of their berths. Having\na side-board to mine to keep me in, like a baby, I lay still. But it was\na dismal night indeed, and it was curious to see the change it had made\nin the faces of all the passengers yesterday. It cannot be denied that\nthese winter crossings are very trying and startling; while the\npersonal discomfort of not being able to wash, and the miseries of\ngetting up and going to bed, with what small means there are all\nsliding, and sloping, and slopping about, are really in their way\ndistressing.\n\nThis forenoon we made Cape Race, and are now running along at full speed\nwith the land beside us. Kelly still useless, and positively declining\nto show on deck. Scott, with an eight-day-old moustache, more super like\nthan ever. My foot (I hope from walking on the boarded deck) in a very\nshy condition to-day, and rather painful. I shaved this morning for the\nfirst time since Liverpool; dodging at the glass, very much like\nFechter's imitation of ----. The white cat that came off with us in the\ntender a general favourite. She belongs to the daughter of a Southerner,\nreturning with his wife and family from a two-years' tour in Europe.\n\n\n                                                       _Sunday, 17th._\n\nAt four o'clock this morning we got into bad weather again, and the\nstate of things at breakfast-time was unutterably miserable. Nearly all\nthe passengers in their berths--no possibility of standing on\ndeck--sickness and groans--impracticable to pass a cup of tea from one\npair of hands to another. It has slightly moderated since (between two\nand three in the afternoon I write), and the sun is shining, but the\nrolling of the ship surpasses all imagination or description.\n\nWe expect to be at Halifax about an hour after midnight, and this letter\nshall be posted there, to make certain of catching the return mail on\nWednesday. Boston is only thirty hours from Halifax.\n\nBest love to Mamie, and to Katie and Charley. I know you will report me\nand my love to Forster and Mrs. Forster. I write with great difficulty,\nwedged up in a corner, and having my heels on the paper as often as the\npen. Kelly worse than ever, and Scott better than ever.\n\nMy desk and I have just arisen from the floor.\n\n\n[Sidenote: Miss Dickens.]\n\n                    PARKER HOUSE, BOSTON, _Thursday, Nov. 21st, 1867._\n\nI arrived here on Tuesday night, after a very slow passage from Halifax\nagainst head-winds. All the tickets for the first four readings here\n(all yet announced) were sold immediately on their being issued.\n\nYou know that I begin on the 2nd of December with \"Carol\" and \"Trial\"?\nShall be heartily glad to begin to count the readings off.\n\nThis is an immense hotel, with all manner of white marble public\npassages and public rooms. I live in a corner high up, and have a hot\nand cold bath in my bedroom (communicating with the sitting-room), and\ncomforts not in existence when I was here before. The cost of living is\nenormous, but happily we can afford it. I dine to-day with Longfellow,\nEmerson, Holmes, and Agassiz. Longfellow was here yesterday. Perfectly\nwhite in hair and beard, but a remarkably handsome and notable-looking\nman. The city has increased enormously in five-and-twenty years. It has\ngrown more mercantile--is like Leeds mixed with Preston, and flavoured\nwith New Brighton; but for smoke and fog you substitute an exquisitely\nbright light air. I found my rooms beautifully decorated (by Mrs.\nFields) with choice flowers, and set off by a number of good books. I am\nnot much persecuted by people in general, as Dolby has happily made up\nhis mind that the less I am exhibited for nothing the better. So our men\nsit outside the room door and wrestle with mankind.\n\nWe had speech-making and singing in the saloon of the _Cuba_ after the\nlast dinner of the voyage. I think I have acquired a higher reputation\nfrom drawing out the captain, and getting him to take the second in\n\"All's Well,\" and likewise in \"There's not in the wide world\" (your\nparent taking first), than from anything previously known of me on these\nshores. I hope the effect of these achievements may not dim the lustre\nof the readings. We also sang (with a Chicago lady, and a strong-minded\nwoman from I don't know where) \"Auld Lang Syne,\" with a tender\nmelancholy, expressive of having all four been united from our cradles.\nThe more dismal we were, the more delighted the company were. Once (when\nwe paddled i' the burn) the captain took a little cruise round the\ncompass on his own account, touching at the \"Canadian Boat Song,\" and\ntaking in supplies at \"Jubilate,\" \"Seas between us braid ha' roared,\"\nand roared like the seas themselves. Finally, I proposed the ladies in a\nspeech that convulsed the stewards, and we closed with a brilliant\nsuccess. But when you dine with Mr. Forster, ask him to read to you how\nwe got on at church in a heavy sea. Hillard has just been in and sent\nhis love \"to those dear girls.\" He has grown much older. He is now\nDistrict Attorney of the State of Massachusetts, which is a very good\noffice. Best love to your aunt and Katie, and Charley and all his house,\nand all friends.\n\n\n[Sidenote: Miss Hogarth.]\n\n                      PARKER HOUSE, BOSTON, _Monday, Nov. 25th, 1867._\n\nI cannot remember to whom I wrote last, but it will not much matter if I\nmake a mistake; this being generally to report myself so well, that I am\nconstantly chafing at not having begun to-night instead of this night\nweek.\n\nThe tickets being all sold for next week, and no other announcement\nbeing yet made, there is nothing new in that way to tell of. Dolby is\nover at New York, where we are at our wits' end how to keep tickets out\nof the hands of speculators. Morgan is staying with me; came yesterday\nto breakfast, and goes home to-morrow. Fields and Mrs. Fields also dined\nyesterday. She is a very nice woman, with a rare relish for humour and a\nmost contagious laugh. The Bostonians having been duly informed that I\nwish to be quiet, really leave me as much so as I should be in\nManchester or Liverpool. This I cannot expect to last elsewhere; but it\nis a most welcome relief here, as I have all the readings to get up. The\npeople are perfectly kind and perfectly agreeable. If I stop to look in\nat a shop-window, a score of passers-by stop; and after I begin to read,\nI cannot expect in the natural course of things to get off so easily.\nBut I every day take from seven to ten miles in peace.\n\nCommunications about readings incessantly come in from all parts of the\ncountry. We take no offer whatever, lying by with our plans until after\nthe first series in New York, and designing, if we make a furore there,\nto travel as little as possible. I fear I shall have to take Canada at\nthe end of the whole tour. They make such strong representations from\nMontreal and Toronto, and from Nova Scotia--represented by St. John's\nand Halifax--of the slight it would be to them, if I wound up with the\nStates, that I am shaken.\n\nIt is sad to see Longfellow's house (the house in which his wife was\nburnt) with his young daughters in it, and the shadow of that terrible\nstory. The young undergraduates of Cambridge (he is a professor there)\nhave made a representation to him that they are five hundred strong,\nand cannot get one ticket. I don't know what is to be done for them; I\nsuppose I must read there somehow. We are all in the clouds until I\nshall have broken ground in New York, as to where readings will be\npossible and where impossible.\n\nAgassiz is one of the most natural and jovial of men. I go out\na-visiting as little as I can, but still have to dine, and what is\nworse, sup pretty often. Socially, I am (as I was here before)\nwonderfully reminded of Edinburgh when I had many friends in it.\n\nYour account and Mamie's of the return journey to London gave me great\npleasure. I was delighted with your report of Wilkie, and not surprised\nby Chappell's coming out gallantly.\n\nMy anxiety to get to work is greater than I can express, because time\nseems to be making no movement towards home until I shall be reading\nhard. Then I shall begin to count and count and count the upward steps\nto May.\n\nIf ever you should be in a position to advise a traveller going on a sea\nvoyage, remember that there is some mysterious service done to the\nbilious system when it is shaken, by baked apples. Noticing that they\nwere produced on board the _Cuba_, every day at lunch and dinner, I\nthought I would make the experiment of always eating them freely. I am\nconfident that they did wonders, not only at the time, but in stopping\nthe imaginary pitching and rolling after the voyage is over, from which\nmany good amateur sailors suffer. I have hardly had the sensation at\nall, except in washing of a morning. At that time I still hold on with\none knee to the washing-stand, and could swear that it rolls from left\nto right. The _Cuba_ does not return until Wednesday, the 4th December.\nYou may suppose that every officer on board is coming on Monday, and\nthat Dolby has provided extra stools for them. His work is very hard\nindeed. Cards are brought to him every minute in the day; his\ncorrespondence is immense; and he is jerked off to New York, and I don't\nknow where else, on the shortest notice and the most unreasonable times.\nMoreover, he has to be at \"the bar\" every night, and to \"liquor up with\nall creation\" in the small hours. He does it all with the greatest good\nhumour, and flies at everybody who waylays the Chief, furiously. We have\ndivided our men into watches, so that one always sits outside the\ndrawing-room door. Dolby knows the whole Cunard line, and as we could\nnot get good English gin, went out in a steamer yesterday and got two\ncases (twenty-four bottles) out of Cunard officers. Osgood and he were\ndetached together last evening for New York, whence they telegraph every\nother hour about some new point in this precious sale of tickets. So\ndistracted a telegram arrived at three that I have telegraphed back,\n\"Explain yourselves,\" and am now waiting for the explanation. I think\nyou know that Osgood is a partner in Ticknor and Fields'.\n\nTuesday morning.--Dolby has come back from New York, where the prospects\nseem immense. We sell tickets there next Friday and Saturday, and a\ntremendous rush is expected.\n\n\n[Sidenote: Mr. Charles Dickens.]\n\n              PARKER HOUSE, BOSTON, U.S., _Saturday, Nov. 30th, 1867._\n\nMY DEAR CHARLEY,\n\nYou will have heard before now how fortunate I was on my voyage, and how\nI was not sick for a moment. These screws are tremendous ships for\ncarrying on, and for rolling, and their vibration is rather distressing.\nBut my little cabin, being for'ard of the machinery, was in the best\npart of the vessel, and I had as much air in it, night and day, as I\nchose. The saloon being kept absolutely without air, I mostly dined in\nmy own den, in spite of my being allotted the post of honour on the\nright hand of the captain.\n\nThe tickets for the first four readings here (the only readings\nannounced) were all sold immediately, and many are now re-selling at a\nlarge premium. The tickets for the first four readings in New York (the\nonly readings announced there also) were on sale yesterday, and were all\nsold in a few hours. The receipts are very large indeed; but engagements\nof any kind and every kind I steadily refuse, being resolved to take\nwhat is to be taken myself. Dolby is nearly worked off his legs, is now\nat New York, and goes backwards and forwards between this place and that\n(about the distance from London to Liverpool, though they take nine\nhours to do it) incessantly. Nothing can exceed his energy and good\nhumour, and he is extremely popular everywhere. My great desire is to\navoid much travelling, and to try to get the people to come to me,\ninstead of my going to them. If I can effect this to any moderate\nextent, I shall be saved a great deal of knocking about. My original\npurpose was not to go to Canada at all; but Canada is so up in arms on\nthe subject that I think I shall be obliged to take it at last. In that\ncase I should work round to Halifax, Nova Scotia, and then take the\npacket for home.\n\nAs they don't seem (Americans who have heard me on their travels\nexcepted) to have the least idea here of what the readings are like, and\nas they are accustomed to mere readings out of a book, I am inclined to\nthink the excitement will increase when I shall have begun. Everybody\nis very kind and considerate, and I have a number of old friends here,\nat the Bar and connected with the University. I am now negotiating to\nbring out the dramatic version of \"No Thoroughfare\" at New York. It is\nquite upon the cards that it may turn up trumps.\n\nI was interrupted in that place by a call from my old secretary in the\nStates, Mr. Putnam. It was quite affecting to see his delight in meeting\nhis old master again. And when I told him that Anne was married, and\nthat I had (unacknowledged) grandchildren, he laughed and cried\ntogether. I suppose you don't remember Longfellow, though he remembers\nyou in a black velvet frock very well. He is now white-haired and\nwhite-bearded, but remarkably handsome. He still lives in his old house,\nwhere his beautiful wife was burnt to death. I dined with him the other\nday, and could not get the terrific scene out of my imagination. She was\nin a blaze in an instant, rushed into his arms with a wild cry, and\nnever spoke afterwards.\n\nMy love to Bessie, and to Mekitty, and all the babbies. I will lay this\nby until Tuesday morning, and then add a final line to it.\n\n                      Ever, my dear Charley, your affectionate Father.\n\n\n                                            _Tuesday, Dec. 3rd, 1867._\n\nSuccess last night beyond description or exaggeration. The whole city is\nquite frantic about it to-day, and it is impossible that prospects could\nbe more brilliant.\n\n\n[Sidenote: Miss Dickens.]\n\n                       PARKER HOUSE, BOSTON, _Sunday, Dec. 1st, 1867._\n\nI received yours of the 18th November, yesterday. As I left Halifax in\nthe _Cuba_ that very day, you probably saw us telegraphed in _The Times_\non the 19th.\n\nDolby came back from another run to New York, this morning. The receipts\nare very large indeed, far exceeding our careful estimate made at Gad's.\nI think you had best in future (unless I give you intimation to the\ncontrary) address your letters to me, at the Westminster Hotel, Irving\nPlace, New York City. It is a more central position than this, and we\nare likely to be much more there than here. I am going to set up a\nbrougham in New York, and keep my rooms at that hotel. The account of\nMatilda is a very melancholy one, and really distresses me. What she\nmust sink into, it is sad to consider. However, there was nothing for it\nbut to send her away, that is quite clear.\n\nThey are said to be a very quiet audience here, appreciative but not\ndemonstrative. I shall try to change their character a little.\n\nI have been going on very well. A horrible custom obtains in these parts\nof asking you to dinner somewhere at half-past two, and to supper\nsomewhere else about eight. I have run this gauntlet more than once, and\nits effect is, that there is no day for any useful purpose, and that the\nlength of the evening is multiplied by a hundred. Yesterday I dined with\na club at half-past two, and came back here at half-past eight, with a\ngeneral impression that it was at least two o'clock in the morning. Two\ndays before I dined with Longfellow at half-past two, and came back at\neight, supposing it to be midnight. To-day we have a state dinner-party\nin our rooms at six, Mr. and Mrs. Fields, and Mr. and Mrs. Bigelow. (He\nis a friend of Forster's, and was American Minister in Paris). There are\nno <DW64> waiters here, all the servants are Irish--willing, but not\nable. The dinners and wines are very good. I keep our own rooms well\nventilated by opening the windows, but no window is ever opened in the\nhalls or passages, and they are so overheated by a great furnace, that\nthey make me faint and sick. The air is like that of a pre-Adamite\nironing-day in full blast. Your respected parent is immensely popular in\nBoston society, and its cordiality and unaffected heartiness are\ncharming. I wish I could carry it with me.\n\nThe leading New York papers have sent men over for to-morrow night with\ninstructions to telegraph columns of descriptions. Great excitement and\nexpectation everywhere. Fields says he has looked forward to it so long\nthat he knows he will die at five minutes to eight.\n\nAt the New York barriers, where the tickets are on sale and the people\nranged as at the Paris theatres, speculators went up and down offering\n\"twenty dollars for anybody's place.\" The money was in no case accepted.\nOne man sold two tickets for the second, third, and fourth night for\n\"one ticket for the first, fifty dollars\" (about seven pounds ten\nshillings), \"and a brandy cocktail,\" which is an iced bitter drink. The\nweather has been rather muggy and languid until yesterday, when there\nwas the coldest wind blowing that I ever felt. In the night it froze\nvery hard, and to-day the sky is beautiful.\n\n\n                                                  _Tuesday, Dec. 3rd._\n\nMost magnificent reception last night, and most signal and complete\nsuccess. Nothing could be more triumphant. The people will hear of\nnothing else and talk of nothing else. Nothing that was ever done here,\nthey all agree, evoked any approach to such enthusiasm. I was quite as\ncool and quick as if I were reading at Greenwich, and went at it\naccordingly. Tell your aunt, with my best love, that I have this morning\nreceived hers of the 21st, and that I will write to her next. That will\nbe from New York. My love to Mr. and Mrs. Hulkes and the boy, and to Mr.\nand Mrs. Malleson.[18]\n\n\n[Sidenote: Miss Hogarth.]\n\n                                  BOSTON, _Wednesday, Dec. 4th, 1867._\n\nI find that by going off to the _Cuba_ myself this morning I can send\nyou the enclosed for Mary Boyle (I don't know how to address her), whose\nusual flower for my button-hole was produced in the most extraordinary\nmanner here last Monday night! All well and prosperous. \"Copperfield\"\nand \"Bob\" last night; great success.\n\n\n[Sidenote: Miss Mary Boyle.]\n\n                                         BOSTON, _December 4th, 1867._\n\nMY DEAR MEERY,\n\nYou can have no idea of the glow of pleasure and amazement with which I\nsaw your remembrance of me lying on my dressing-table here last Monday\nnight. Whosoever undertook that commission accomplished it to a miracle.\nBut you must go away four thousand miles, and have such a token conveyed\nto _you_, before you can quite appreciate the feeling of receiving it.\nTen thousand loving thanks.\n\nImmense success here, and unbounded enthusiasm. My largest expectations\nfar surpassed.\n\n                                              Ever your affectionate\n                                                                   Jo.\n\n\n[Sidenote: Miss Dickens.]\n\n                 WESTMINSTER HOTEL, IRVING PLACE, NEW YORK CITY,\n                                          _Wednesday, Dec 11th, 1867._\n\nAmazing success here. A very fine audience; _far better than that at\nBoston_. Great reception. Great, \"Carol\" and \"Trial,\" on the first\nnight; still greater, \"Copperfield\" and \"Bob,\" on the second. Dolby\nsends you a few papers by this post. You will see from their tone what a\nsuccess it is.\n\nI cannot pay this letter, because I give it at the latest moment to the\nmail-officer, who is going on board the Cunard packet in charge of the\nmails, and who is staying in this house. We are now selling (at the\nhall) the tickets for the four readings of next week. At nine o'clock\nthis morning there were two thousand people in waiting, and they had\nbegun to assemble in the bitter cold as early as two o'clock. All night\nlong Dolby and our man have been stamping tickets. (Immediately over my\nhead, by-the-bye, and keeping me awake.) This hotel is quite as quiet as\nMivart's, in Brook Street. It is not very much larger. There are\nAmerican hotels close by, with five hundred bedrooms, and I don't know\nhow many boarders; but this is conducted on what is called \"the European\nprinciple,\" and is an admirable mixture of a first-class French and\nEnglish house. I keep a very smart carriage and pair; and if you were to\nbehold me driving out, furred up to the moustache, with furs on the\ncoach-boy and on the driver, and with an immense white, red, and yellow\nstriped rug for a covering, you would suppose me to be of Hungarian or\nPolish nationality.\n\nWill you report the success here to Mr. Forster with my love, and tell\nhim he shall hear from me by next mail?\n\nDolby sends his kindest regards. He is just come in from our ticket\nsales, and has put such an immense untidy heap of paper money on the\ntable that it looks like a family wash. He hardly ever dines, and is\nalways tearing about at unreasonable hours. He works very hard.\n\nMy best love to your aunt (to whom I will write next), and to Katie, and\nto both the Charleys, and all the Christmas circle, not forgetting\nChorley, to whom give my special remembrance. You may get this by\nChristmas Day. _We_ shall have to keep it travelling from Boston here;\nfor I read at Boston on the 23rd and 24th, and here again on the 26th.\n\n\n[Sidenote: Miss Hogarth.]\n\n                 WESTMINSTER HOTEL, IRVING PLACE, NEW YORK CITY,\n                                            _Monday, Dec. 16th, 1867._\n\nWe have been snowed up here, and the communication with Boston is still\nvery much retarded. Thus we have received no letters by the Cunard\nsteamer that came in last Wednesday, and are in a grim state of mind on\nthat subject.\n\nLast night I was getting into bed just at twelve o'clock, when Dolby\ncame to my door to inform me that the house was on fire (I had\npreviously smelt fire for two hours). I got Scott up directly, told him\nto pack the books and clothes for the readings first, dressed, and\npocketed my jewels and papers, while Dolby stuffed himself out with\nmoney. Meanwhile the police and firemen were in the house, endeavouring\nto find where the fire was. For some time it baffled their endeavours,\nbut at last, bursting out through some stairs, they cut the stairs away,\nand traced it to its source in a certain fire-grate. By this time the\nhose was laid all through the house from a great tank on the roof, and\neverybody turned out to help. It was the oddest sight, and people had\nput the strangest things on! After a little chopping and cutting with\naxes and handing about of water, the fire was confined to a dining-room\nin which it had originated, and then everybody talked to everybody else,\nthe ladies being particularly loquacious and cheerful. And so we got to\nbed again at about two.\n\nThe excitement of the readings continues unabated, the tickets for\nreadings are sold as soon as they are ready, and the public pay treble\nprices to the speculators who buy them up. They are a wonderfully fine\naudience, even better than Edinburgh, and almost, if not quite, as good\nas Paris.\n\nDolby continues to be the most unpopular man in America (mainly because\nhe can't get four thousand people into a room that holds two thousand),\nand is reviled in print daily. Yesterday morning a newspaper proclaims\nof him: \"Surely it is time that the pudding-headed Dolby retired into\nthe native gloom from which he has emerged.\" He takes it very coolly,\nand does his best. Mrs. Morgan sent me, the other night, I suppose the\nfinest and costliest basket of flowers ever seen, made of white\ncamellias, yellow roses, pink roses, and I don't know what else. It is a\nyard and a half round at its smallest part.\n\nI must bring this to a close, as I have to go to the hall to try an\nenlarged background.\n\n\n[Sidenote: Miss Hogarth.]\n\n                                    BOSTON, _Sunday, Dec. 22nd, 1867._\n\nComing here from New York last night (after a detestable journey), I was\ndelighted to find your letter of the 6th. I read it at my ten o'clock\ndinner with the greatest interest and pleasure, and then we talked of\nhome till we went to bed.\n\nOur tour is now being made out, and I hope to be able to send it in my\nnext letter home, which will be to Mamie, from whom I have _not_ heard\n(as you thought I had) by the mail that brought out yours. After very\ncareful consideration I have reversed Dolby's original plan, and have\ndecided on taking Baltimore, Washington, Cincinnati, _Chicago_ (!), St.\nLouis, and a few other places nearer here, instead of staying in New\nYork. My reason is that we are doing immensely, both at New York and\nhere, and that I am sure it is in the peculiar character of the people\nto prize a thing the more the less easily attainable it is made.\nTherefore, I want, by absence, to get the greatest rush and pressure\nupon the five farewell readings in New York in April. All our announced\nreadings are already crammed.\n\nWhen we got here last Saturday night, we found that Mrs. Fields had not\nonly garnished the rooms with flowers, but also with holly (with real\nred berries) and festoons of moss dependent from the looking-glasses and\npicture frames. She is one of the dearest little women in the world. The\nhomely Christmas look of the place quite affected us. Yesterday we dined\nat her house, and there was a plum-pudding, brought on blazing, and not\nto be surpassed in any house in England. There is a certain Captain\nDolliver, belonging to the Boston Custom House, who came off in the\nlittle steamer that brought me ashore from the _Cuba_. He took it into\nhis head that he would have a piece of English mistletoe brought out in\nthis week's Cunard, which should be laid upon my breakfast-table. And\nthere it was this morning. In such affectionate touches as this, these\nNew England people are especially amiable.\n\nAs a general rule, you may lay it down that whatever you see about me in\nthe papers is not true. But although my voyage out was of that highly\nhilarious description that you first made known to me, you may\n_generally_ lend a more believing ear to the Philadelphia correspondent\nof _The Times_. I don't know him, but I know the source from which he\nderives his information, and it is a very respectable one.\n\nDid I tell you in a former letter from here, to tell Anne, with her old\nmaster's love, that I had seen Putnam, my old secretary? Grey, and with\nseveral front teeth out, but I would have known him anywhere. He is\ncoming to \"Copperfield\" to-night, accompanied by his wife and daughter,\nand is in the seventh heaven at having his tickets given him.\n\nOur hotel in New York was on fire _again_ the other night. But fires in\nthis country are quite matters of course. There was a large one there at\nfour this morning, and I don't think a single night has passed since I\nhave been under the protection of the Eagle, but I have heard the fire\nbells dolefully clanging all over the city.\n\nDolby sends his kindest regard. His hair has become quite white, the\neffect, I suppose, of the climate. He is so universally hauled over the\ncoals (for no reason on earth), that I fully expect to hear him, one of\nthese nights, assailed with a howl when he precedes me to the platform\nsteps. You may conceive what the low newspapers are here, when one of\nthem yesterday morning had, as an item of news, the intelligence:\n\"Dickens's Readings. The chap calling himself Dolby got drunk last\nnight, and was locked up in a police-station for fighting an Irishman.\"\nI don't find that anybody is shocked by this liveliness.\n\nMy love to all, and to Mrs. Hulkes and the boy. By-the-bye, when we left\nNew York for this place, Dolby called my amazed attention to the\ncircumstance that Scott was leaning his head against the side of the\ncarriage and weeping bitterly. I asked him what was the matter, and he\nreplied: \"The owdacious treatment of the luggage, which was more\noutrageous than a man could bear.\" I told him not to make a fool of\nhimself; but they do knock it about cruelly. I think every trunk we have\nis already broken.\n\nI must leave off, as I am going out for a walk in a bright sunlight and\na complete break-up of the frost and snow. I am much better than I have\nbeen during the last week, but have a cold.\n\n\n[Sidenote: Miss Dickens.]\n\n                 WESTMINSTER HOTEL, IRVING PLACE, NEW YORK CITY,\n                                          _Thursday, Dec. 26th, 1867._\n\nI got your aunt's last letter at Boston yesterday, Christmas Day\nmorning, when I was starting at eleven o'clock to come back to this\nplace. I wanted it very much, for I had a frightful cold (English colds\nare nothing to those of this country), and was exceedingly depressed and\nmiserable. Not that I had any reason but illness for being so, since the\nBostonians had been quite astounding in their demonstrations. I never\nsaw anything like them on Christmas Eve. But it is a bad country to be\nunwell and travelling in; you are one of say a hundred people in a\nheated car, with a great stove in it, and all the little windows closed,\nand the hurrying and banging about are indescribable. The atmosphere is\ndetestable, and the motion often all but intolerable. However, we got\nour dinner here at eight o'clock, and plucked up a little, and I made\nsome hot gin punch to drink a merry Christmas to all at home in. But it\nmust be confessed that we were both very dull. I have been in bed all\nday until two o'clock, and here I am now (at three o'clock) a little\nbetter. But I am not fit to read, and I must read to-night. After\nwatching the general character pretty closely, I became quite sure that\nDolby was wrong on the length of the stay and the number of readings we\nhad proposed in this place. I am quite certain that it is one of the\nnational peculiarities that what they want must be difficult of\nattainment. I therefore a few days ago made a _coup d'etat_, and altered\nthe whole scheme. We shall go to Philadelphia, Baltimore, Washington,\nalso some New England towns between Boston and this place, away to the\nfalls of Niagara, and off far west to Chicago and St. Louis, before\ncoming back for ten farewell readings here, preceded by farewells at\nBoston, leaving Canada altogether. This will not prolong the list beyond\neighty-four readings, the exact original number, and will, please God,\nwork it all out in April. In my next, I daresay, I shall be able to send\nthe exact list, so that you may know every day where we are. There has\nbeen a great storm here for a few days, and the streets, though wet, are\nbecoming passable again. Dolby and Osgood are out in it to-day on a\nvariety of business, and left in grave and solemn state. Scott and the\ngasman are stricken with dumb concern, not having received one single\nletter from home since they left. What their wives can have done with\nthe letters they take it for granted they have written, is their stormy\nspeculation at the door of my hall dressing-room every night.\n\nIf I do not send a letter to Katie by this mail, it will be because I\nshall probably be obliged to go across the water to Brooklyn to-morrow\nto see a church, in which it is proposed that I shall read!!! Horrible\nvisions of being put in the pulpit already beset me. And whether the\naudience will be in pews is another consideration which greatly disturbs\nmy mind. No paper ever comes out without a leader on Dolby, who of\ncourse reads them all, and never can understand why I don't, in which he\nis called all the bad names in (and not in) the language.\n\nWe always call him P. H. Dolby now, in consequence of one of these\ngraceful specimens of literature describing him as the \"pudding-headed.\"\n\nI fear that when we travel he will have to be always before me, so that\nI may not see him six times in as many weeks. However, I shall have done\na fourth of the whole this very next week!\n\nBest love to your aunt, and the boys, and Katie, and Charley, and all\ntrue friends.\n\n\n                                                             _Friday._\n\nI managed to read last night, but it was as much as I could do. To-day I\nam so very unwell, that I have sent for a doctor; he has just been, and\nis in doubt whether I shall not have to stop reading for a while.\n\n\n[Sidenote: Miss Dickens.]\n\n                      WESTMINSTER HOTEL, IRVING PLACE, NEW YORK,\n                                            _Monday, Dec. 30th, 1867._\n\nI am getting all right again. I have not been well, been very low, and\nhave been obliged to have a doctor; a very agreeable fellow indeed, who\nsoon turned out to be an old friend of Olliffe's.[19] He has set me on\nmy legs and taken his leave \"professionally,\" though he means to give me\na call now and then.\n\nIn the library at Gad's is a bound book, \"Remarkable Criminal Trials,\"\ntranslated by Lady Duff Gordon, from the original by Fauerbach. I want\nthat book, and a copy of Praed's poems, to be sent out to Boston, care\nof Ticknor and Fields. If you will give the \"Criminal Trials\" to Wills,\nand explain my wish, and ask him to buy a copy of Praed's poems and add\nit to the parcel, he will know how to send the packet out. I think the\n\"Criminal Trials\" book is in the corner book-case, by the window,\nopposite the door.\n\nNo news here. All going on in the regular way. I read in that church I\ntold you of, about the middle of January. It is wonderfully seated for\ntwo thousand people, and is as easy to speak in as if they were two\nhundred. The people are seated in pews, and we let the pews. I stood on\na small platform from which the pulpit will be removed for the\noccasion!! I emerge from the vestry!!! Philadelphia, Baltimore, and\nanother two nights in Boston will follow this coming month of January.\nOn Friday next I shall have read a fourth of my whole list, besides\nhaving had twelve days' holiday when I first came out. So please God I\nshall soon get to the half, and so begin to work hopefully round.\n\nI suppose you were at the Adelphi on Thursday night last. They are\npirating the bill as well as the play here, everywhere. I have\nregistered the play as the property of an American citizen, but the law\nis by no means clear that I established a right in it by so doing; and\nof course the pirates knew very well that I could not, under existing\ncircumstances, try the question with them in an American court of law.\nNothing is being played here scarcely that is not founded on my\nbooks--\"Cricket,\" \"Oliver Twist,\" \"Our Mutual Friend,\" and I don't know\nwhat else, every night. I can't get down Broadway for my own portrait;\nand yet I live almost as quietly in this hotel, as if I were at the\noffice, and go in and out by a side door just as I might there.\n\nI go back to Boston on Saturday to read there on Monday and Tuesday.\nThen I am back here, and keep within six or seven hours' journey of\nhereabouts till February. My further movements shall be duly reported as\nthe details are arranged.\n\nI shall be curious to know who were at Gad's Hill on Christmas Day, and\nhow you (as they say in this country) \"got along.\" It is exceedingly\ncold here again, after two or three quite spring days.\n\nFOOTNOTES:\n\n[17] Madame Sainton Dolby.\n\n[18] The nearest neighbour at Higham, and intimate friends.\n\n[19] Dr. Fordyce Barker.\n\n\n\n\n1868.\n\nNARRATIVE.\n\n\nCharles Dickens remained in America through the winter, returning home\nfrom New York in the _Russia_, on the 19th of April. His letters show\nhow entirely he gave himself up to the business of the readings, how\nseverely his health suffered from the climate, and from the perpetual\ntravelling and hard work, and yet how he was able to battle through to\nthe end. These letters are also full of allusions to the many kind and\ndear friends who contributed so largely to the pleasure of this American\nvisit, and whose love and attention gave a touch of _home_ to his\nprivate life, and left such affection and gratitude in his heart as he\ncould never forget. Many of these friends paid visits to Gad's Hill; the\nfirst to come during this summer being Mr. Longfellow, his daughters,\nand Mr. Appleton, brother-in-law of Mr. Longfellow, and Mr. and Mrs.\nCharles Eliot Norton, of Cambridge.\n\nFor the future, there were to be no more Christmas numbers of \"All the\nYear Round.\" Observing the extent to which they were now copied in all\ndirections, Charles Dickens supposed them likely to become tiresome to\nthe public, and so determined that in his journal they should be\ndiscontinued.\n\nWhile still in America, he made an agreement with the Messrs. Chappell\nto give a series of farewell readings in England, to commence in the\nautumn of this year. So, in October, Charles Dickens started off again\nfor a tour in the provinces. He had for some time been planning, by way\nof a novelty for this series, a reading from the murder in \"Oliver\nTwist,\" but finding it so very horrible, he was fearful of trying its\neffect for the first time on a public audience. It was therefore\nresolved, that a trial of it should be made to a limited private\naudience in St. James's Hall, on the evening of the 18th of November.\nThis trial proved eminently successful, and \"The Murder from Oliver\nTwist\" became one of the most popular of his selections. But the\nphysical exertion it involved was far greater than that of any of his\nprevious readings, and added immensely to the excitement and exhaustion\nwhich they caused him.\n\nOne of the first letters of the year from America is addressed to Mr.\nSamuel Cartwright, of surgical and artistic reputation, and greatly\nesteemed by Charles Dickens, both in his professional capacity and as a\nprivate friend.\n\nThe letter written to Mrs. Cattermole, in May, tells of the illness of\nMr. George Cattermole. This dear old friend, so associated with Charles\nDickens and his works, died soon afterwards, and the letter to his widow\nshows that Charles Dickens was exerting himself in her behalf.\n\nThe play of \"No Thoroughfare\" having been translated into French under\nthe title of \"L'Abime,\" Charles Dickens went over to Paris to be present\nat the first night of its production.\n\nOn the 26th of September, his youngest son, Edward Bulwer Lytton (the\n\"Plorn\" so often mentioned), started for Australia, to join his brother\nAlfred Tennyson, who was already established there. It will be seen by\nhis own words how deeply and how sadly Charles Dickens felt this\nparting. In October of this year, his son Henry Fielding entered Trinity\nHall, Cambridge, as an undergraduate.\n\nThe Miss Forster mentioned in the letter to his sister-in-law, and for\nwhom the kind and considerate arrangements were suggested, was a sister\nof Mr. John Forster, and a lady highly esteemed by Charles Dickens. The\nillness from which she was then suffering was a fatal one. She died in\nthis same year, a few days before Christmas.\n\nMr. J. C. Parkinson, to whom a letter is addressed, was a gentleman\nholding a Government appointment, and contributing largely to journalism\nand periodical literature.\n\nAs our last letter for this year, we give one which Charles Dickens\nwrote to his youngest son on his departure for Australia.\n\n\n[Sidenote: Miss Hogarth.]\n\n                      WESTMINSTER HOTEL, IRVING PLACE, NEW YORK,\n                                             _Friday, Jan. 3rd, 1868._\n\nMY DEAREST GEORGY,\n\nI received yours of the 19th from Gad's and the office this morning. I\nread here to-night, and go back to Boston to-morrow, to read there\nMonday and Tuesday.\n\nTo-night, I read out the first quarter of my list. Our houses have been\nvery fine here, but have never quite recovered the Dolby uproar. It\nseems impossible to devise any scheme for getting the tickets into the\npeople's hands without the intervention of speculators. The people _will\nnot_ help themselves; and, of course, the speculators and all other such\nprowlers throw as great obstacles in Dolby's way (an Englishman's) as\nthey possibly can. He may be a little injudicious into the bargain. Last\nnight, for instance, he met one of the \"ushers\" (who show people to\ntheir seats) coming in with Kelly. It is against orders that anyone\nemployed in front should go out during the readings, and he took this\nman to task in the British manner. Instantly the free and independent\nusher put on his hat and walked off. Seeing which, all the other free\nand independent ushers (some twenty in number) put on _their_ hats and\nwalked off, leaving us absolutely devoid and destitute of a staff for\nto-night. One has since been improvised; but it was a small matter to\nraise a stir and ill will about, especially as one of our men was\nequally in fault.\n\nWe have a regular clerk, a Bostonian whose name is Wild. He, Osgood,\nDolby, Kelly, Scott, George the gasman, and perhaps a boy or two,\nconstitute my body-guard. It seems a large number of people, but the\nbusiness cannot be done with fewer. The speculators buying the front\nseats to sell at a premium (and we have found instances of this being\ndone by merchants in good position!), and the public perpetually\npitching into Dolby for selling them back seats, the result is that they\nwon't have the back seats, send back their tickets, write and print\nvolumes on the subject, and deter others from coming.\n\nYou may get an idea of the staff's work, by what is in hand now. They\nare preparing, numbering, and stamping six thousand tickets for\nPhiladelphia, and eight thousand tickets for Brooklyn. The moment those\nare done, another eight thousand tickets will be wanted for Baltimore,\nand probably another six thousand for Washington. This in addition to\nthe correspondence, advertisements, accounts, travellings, and the\nmighty business of the reading four times a week.\n\nThe Cunard steamers being now removed from Halifax, I have decided _not_\nto go there, or to St. John's, New Brunswick. And as there would be a\nperfect uproar if I picked out such a place in Canada as Quebec or\nMontreal, and excluded those two places (which would guarantee three\nhundred pounds a night), and further, as I don't want places, having\nmore than enough for my list of eighty-four, I have finally resolved not\nto go to Canada either. This will enable me to embark for home in April\ninstead of May.\n\nTell Plorn, with my love, that I think he will find himself much\ninterested at that college,[20] and that it is very likely he may make\nsome acquaintances there that will thereafter be pleasant and useful to\nhim. Sir Sydney Dacres is the best of friends. I have a letter from Mrs.\nHulkes by this post, wherein the boy encloses a violet, now lying on the\ntable before me. Let her know that it arrived safely, and retaining its\ncolour. I took it for granted that Mary would have asked Chorley for\nChristmas Day, and am very glad she ultimately did so. I am sorry that\nHarry lost his prize, but believe it was not his fault. Let _him_ know\n_that_, with my love. I would have written to him by this mail in answer\nto his, but for other occupation. Did I tell you that my landlord made\nme a drink (brandy, rum, and snow the principal ingredients) called a\n\"Rocky Mountain sneezer\"? Or that the favourite drink before you get up\nis an \"eye-opener\"? Or that Roberts (second landlord), no sooner saw me\non the night of the first fire, than, with his property blazing, he\ninsisted on taking me down into a roomful of hot smoke to drink brandy\nand water with him? We have not been on fire again, by-the-bye, more\nthan once.\n\nThere has been another fall of snow, succeeded by a heavy thaw. I have\nlaid down my sledge, and taken up my carriage again, in consequence. I\nam nearly all right, but cannot get rid of an intolerable cold in the\nhead. No more news.\n\n\n[Sidenote: Miss Hogarth.]\n\n                         PARKER HOUSE, BOSTON, U.S., _Jan. 4th, 1868._\n\nI write to you by this opportunity, though I really have nothing to tell\nyou. The work is hard and the climate is hard. We made a tremendous hit\nlast night with \"Nickleby\" and \"Boots,\" which the Bostonians certainly\non the whole appreciate more than \"Copperfield\"! Dolby is always going\nabout with an immense bundle that looks like a sofa cushion, but it is\nin reality paper money; and always works like a Trojan. His business at\nnight is a mere nothing, for these people are so accustomed to take care\nof themselves, that one of these immense audiences will fall into their\nplaces with an ease amazing to a frequenter of St. James's Hall. And the\ncertainty with which they are all in, before I go on, is a very\nacceptable mark of respect. I must add, too, that although there is a\nconventional familiarity in the use of one's name in the newspapers as\n\"Dickens,\" \"Charlie,\" and what not, I do not in the least see that\nfamiliarity in the writers themselves. An inscrutable tone obtains in\njournalism, which a stranger cannot understand. If I say in common\ncourtesy to one of them, when Dolby introduces, \"I am much obliged to\nyou for your interest in me,\" or so forth, he seems quite shocked, and\nhas a bearing of perfect modesty and propriety. I am rather inclined to\nthink that they suppose their printed tone to be the public's love of\nsmartness, but it is immensely difficult to make out. All I can as yet\nmake out is, that my perfect freedom from bondage, and at any moment to\ngo on or leave off, or otherwise do as I like, is the only safe position\nto occupy.\n\nAgain; there are two apparently irreconcilable contrasts here. Down\nbelow in this hotel every night are the bar loungers, dram drinkers,\ndrunkards, swaggerers, loafers, that one might find in a Boucicault\nplay. Within half an hour is Cambridge, where a delightful domestic\nlife--simple, self-respectful, cordial, and affectionate--is seen in an\nadmirable aspect. All New England is primitive and puritanical. All\nabout and around it is a puddle of mixed human mud, with no such quality\nin it. Perhaps I may in time sift out some tolerably intelligible whole,\nbut I certainly have not done so yet. It is a good sign, may be, that it\nall seems immensely more difficult to understand than it was when I was\nhere before.\n\nFelton left two daughters. I have only seen the eldest, a very sensible,\nfrank, pleasant girl of eight-and-twenty, perhaps, rather like him in\nthe face. A striking-looking daughter of Hawthorn's (who is also dead)\ncame into my room last night. The day has slipped on to three o'clock,\nand I must get up \"Dombey\" for to-night. Hence this sudden break off.\nBest love to Mamie, and to Katie and Charley Collins.\n\n\n[Sidenote: Mr. W. Wilkie Collins.]\n\n               WESTMINSTER HOTEL, NEW YORK, _Sunday, Jan. 12th, 1868._\n\nMY DEAR WILKIE,\n\nFirst, of the play.[21] I am truly delighted to learn that it made so\ngreat a success, and I hope I may yet see it on the Adelphi boards. You\nhave had a world of trouble and work with it, but I hope will be repaid\nin some degree by the pleasure of a triumph. Even for the alteration at\nthe end of the fourth act (of which you tell me in your letter received\nyesterday), I was fully prepared, for I COULD NOT see the original\neffect in the reading of the play, and COULD NOT make it go. I agree\nwith Webster in thinking it best that Obenreizer should die on the\nstage; but no doubt that point is disposed of. In reading the play\nbefore the representation, I felt that it was too long, and that there\nwas a good deal of unnecessary explanation. Those points are, no doubt,\ndisposed of too by this time.\n\nWe shall do nothing with it on this side. Pirates are producing their\nown wretched versions in all directions, thus (as Wills would say)\nanticipating and glutting \"the market.\" I registered one play as the\nproperty of Ticknor and Fields, American citizens. But, besides that the\nlaw on the point is extremely doubtful, the manager of the Museum\nTheatre, Boston, instantly announced his version. (You may suppose what\nit is and how it is done, when I tell you that it was playing within ten\ndays of the arrival out of the Christmas number.) Thereupon, Ticknor and\nFields gave him notice that he mustn't play it. Unto which he replied,\nthat he meant to play it and would play it. Of course he knew very well\nthat if an injunction were applied for against him, there would be an\nimmediate howl against my persecution of an innocent, and he played it.\nThen the noble host of pirates rushed in, and it is being done, in some\nmangled form or other, everywhere.\n\nIt touches me to read what you write of your poor mother. But, of\ncourse, at her age, each winter counts heavily. Do give her my love, and\ntell her that I asked you about her.\n\nI am going on here at the same great rate, but am always counting the\ndays that lie between me and home. I got through the first fourth of my\nreadings on Friday, January 3rd. I leave for two readings at\nPhiladelphia this evening.\n\nBeing at Boston last Sunday, I took it into my head to go over the\nmedical school, and survey the holes and corners in which that\nextraordinary murder was done by Webster. There was the\nfurnace--stinking horribly, as if the dismembered pieces were still\ninside it--and there are all the grim spouts, and sinks, and chemical\nappliances, and what not. At dinner, afterwards, Longfellow told me a\nterrific story. He dined with Webster within a year of the murder, one\nof a party of ten or twelve. As they sat at their wine, Webster suddenly\nordered the lights to be turned out, and a bowl of some burning mineral\nto be placed on the table, that the guests might see how ghostly it made\nthem look. As each man stared at all the rest in the weird light, all\nwere horrified to see Webster _with a rope round his neck_, holding it\nup, over the bowl, with his head jerked on one side, and his tongue\nlolled out, representing a man being hanged!\n\nPoking into his life and character, I find (what I would have staked my\nhead upon) that he was always a cruel man.\n\nSo no more at present from,\n\n        My dear Wilkie, yours ever affectionately.\n\n\n[Sidenote: Miss Hogarth.]\n\n               WESTMINSTER HOTEL, NEW YORK, _Sunday, Jan. 12th, 1868._\n\nAs I am off to Philadelphia this evening, I may as well post my letter\nhere. I have scarcely a word of news. My cold steadily refuses to leave\nme; but otherwise I am as right as one can hope to be under this heavy\nwork. My New York readings are over (except four farewell nights in\nApril), and I look forward to the relief of being out of my hardest\nhall. Last Friday night, though it was only \"Nickleby\" and \"Boots,\" I\nwas again dead beat at the end, and was once more laid upon a sofa. But\nthe faintness went off after a little while. We have now cold, bright,\nfrosty weather, without snow--the best weather for me.\n\nHaving been in great trepidation about the play, I am correspondingly\nelated by the belief that it really _is_ a success. No doubt the\nunnecessary explanations will have been taken out, and the flatness of\nthe last act fetched up. At some points I could have done wonders to it,\nin the way of screwing it up sharply and picturesquely, if I could have\nrehearsed it. Your account of the first night interested me immensely,\nbut I was afraid to open the letter until Dolby rushed in with the\nopened _Times_.\n\nOn Wednesday I come back here for my four church readings at Brooklyn.\nEach evening an enormous ferryboat will convey me and my state carriage\n(not to mention half-a-dozen waggons, and any number of people, and a\nfew score of horses) across the river, and will bring me back again. The\nsale of tickets there was an amazing scene. The noble army of\nspeculators are now furnished (this is literally true, and I am quite\nserious), each man with a straw mattress, a little bag of bread and\nmeat, two blankets, and a bottle of whisky. With this outfit _they lie\ndown in line on the pavement_ the whole night before the tickets are\nsold, generally taking up their position at about ten. It being severely\ncold at Brooklyn, they made an immense bonfire in the street--a narrow\nstreet of wooden houses!--which the police turned out to extinguish. A\ngeneral fight then took place, out of which the people farthest off in\nthe line rushed bleeding when they saw a chance of displacing others\nnear the door, and put their mattresses in those places, and then held\non by the iron rails. At eight in the morning Dolby appeared with the\ntickets in a portmanteau. He was immediately saluted with a roar of\n\"Halloa, Dolby! So Charley has let you have the carriage, has he, Dolby!\nHow is he, Dolby! Don't drop the tickets, Dolby! Look alive, Dolby!\"\netc. etc. etc., in the midst of which he proceeded to business, and\nconcluded (as usual) by giving universal dissatisfaction.\n\nHe is now going off upon a little journey \"to look over the ground and\ncut back again.\" This little journey (to Chicago) is fifteen hundred\nmiles on end, by railway, and back again!\n\nWe have an excellent gasman, who is well up to that department. We have\nenlarged the large staff by another clerk, yet even now the preparation\nof such an immense number of new tickets constantly, and the keeping and\nchecking of the accounts, keep them hard at it. And they get so oddly\ndivided! Kelly is at Philadelphia, another man at Baltimore, two others\nare stamping tickets at the top of this house, another is cruising over\nNew England, and Osgood will come on duty to-morrow (when Dolby starts\noff) to pick me up after the reading, and take me to the hotel, and\nmount guard over me, and bring me back here. You see that even such\nwretched domesticity as Dolby and self by a fireside is broken up under\nthese conditions.\n\nDolby has been twice poisoned, and Osgood once. Morgan's sharpness has\ndiscovered the cause. When the snow is deep upon the ground, and the\npartridges cannot get their usual food, they eat something (I don't know\nwhat, if anybody does) which does not poison _them_, but which poisons\nthe people who eat them. The symptoms, which last some twelve hours, are\nviolent sickness, cold perspiration, and the formation of some\ndetestable mucus in the stomach. You may infer that partridges have been\nbanished from our bill of fare. The appearance of our sufferers was\nlamentable in the extreme.\n\nDid I tell you that the severity of the weather, and the heat of the\nintolerable furnaces, dry the hair and break the nails of strangers?\nThere is not a complete nail in the whole British suite, and my hair\ncracks again when I brush it. (I am losing my hair with great rapidity,\nand what I don't lose is getting very grey.)\n\nThe _Cuba_ will bring this. She has a jolly new captain--Moody, of the\n_Java_--and her people rushed into the reading, the other night,\ncaptain-headed, as if I were their peculiar property. Please God I shall\ncome home in her, in my old cabin; leaving here on the 22nd of April,\nand finishing my eighty-fourth reading on the previous night! It is\nlikely enough that I shall read and go straight on board.\n\nI think this is all my poor stock of intelligence. By-the-bye, on the\nlast Sunday in the old year, I lost my old year's pocket-book, \"which,\"\nas Mr. Pepys would add, \"do trouble me mightily.\" Give me Katie's new\naddress; I haven't got it.\n\n\n[Sidenote: Miss Dickens.]\n\n                              PHILADELPHIA, _Monday, Jan. 13th, 1868._\n\nI write you this note, a day later than your aunt's, not because I have\nanything to add to the little I have told her, but because you may like\nto have it.\n\nWe arrived here last night towards twelve o'clock, more than an hour\nafter our time. This is one of the immense American hotels (it is called\nthe Continental); but I find myself just as quiet here as elsewhere.\nEverything is very good indeed, the waiter is German, and the greater\npart of the house servants seem to be <DW52> people. The town is very\nclean, and the day as blue and bright as a fine Italian day. But it\nfreezes very hard. All the tickets being sold here for six nights (three\nvisits of two nights each), the suite complain of want of excitement\nalready, having been here ten hours! Mr. and Mrs. Barney Williams, with\na couple of servants, and a pretty little child-daughter, were in the\ntrain each night, and I talked with them a good deal. They are reported\nto have made an enormous fortune by acting among the Californian\ngold-diggers. My cold is no better, for the cars are so intolerably hot,\nthat I was often obliged to go and stand upon the break outside, and\nthen the frosty air was biting indeed. The great man of this place is\none Mr. Childs, a newspaper proprietor, and he is so exactly like Mr.\nEsse in all conceivable respects except being an inch or so taller, that\nI was quite confounded when I saw him waiting for me at the station\n(always called depot here) with his carriage. During the last two or\nthree days, Dolby and I have been making up accounts, which are\nexcellently kept by Mr. Osgood, and I find them amazing, quite, in their\nresults.\n\nI was very much interested in the home accounts of Christmas Day. I\nthink I have already mentioned that we were in very low spirits on that\nday. I began to be unwell with my cold that morning, and a long day's\ntravel did not mend the matter. We scarcely spoke (except when we ate\nour lunch), and sat dolefully staring out of window. I had a few\naffectionate words from Chorley, dated from my room, on Christmas\nmorning, and will write him, probably by this mail, a brief\nacknowledgment. I find it necessary (so oppressed am I with this\nAmerican catarrh, as they call it) to dine at three o'clock instead of\nfour, that I may have more time to get voice, so that the days are cut\nshort, and letter-writing is not easy.\n\nMy best love to Katie, and to Charley, and to our Charley, and to all\nfriends. If I could only get to the point of being able to hold my head\nup and dispense with my pocket-handkerchief for five minutes, I should\nbe all right.\n\n\n[Sidenote: Mr. Charles Dickens.]\n\n                      WESTMINSTER HOTEL, IRVING PLACE, NEW YORK,\n                                         _Wednesday, Jan. 15th, 1868._\n\nMY DEAR CHARLEY,\n\nFinding your letter here this afternoon on my return from Philadelphia\n(where I have been reading two nights), I take advantage of a spare\nhalf-hour in which to answer it at once, though it will not leave here\nuntil Saturday. I had previously heard of the play, and had _The Times_.\nIt was a great relief and delight to me, for I had no confidence in its\nsuccess; being reduced to the confines of despair by its length. If I\ncould have rehearsed it, I should have taken the best part of an hour\nout of it. Fechter must be very fine, and I should greatly like to see\nhim play the part.\n\nI have not been very well generally, and am oppressed (and I begin to\nthink that I probably shall be until I leave) by a true American cold,\nwhich I hope, for the comfort of human nature, may be peculiar to only\none of the four quarters of the world. The work, too, is very severe.\nBut I am going on at the same tremendous rate everywhere. The staff,\ntoo, has had to be enlarged. Dolby was at Baltimore yesterday, is at\nWashington to-day, and will come back in the night, and start away again\non Friday. We find it absolutely necessary for him to go on ahead. We\nhave not printed or posted a single bill here, and have just sold ninety\npounds' worth of paper we had got ready for bills. In such a rush a\nshort newspaper advertisement is all we want. \"Doctor Marigold\" made a\ngreat hit here, and is looked forward to at Boston with especial\ninterest. I go to Boston for another fortnight, on end, the 24th of\nFebruary. The railway journeys distress me greatly. I get out into the\nopen air (upon the break), and it snows and blows, and the train bumps,\nand the steam flies at me, until I am driven in again.\n\nI have finished here (except four farewell nights in April), and begin\nfour nights at Brooklyn, on the opposite side of the river, to-night;\nand thus oscillate between Philadelphia, Baltimore, and Washington, and\nthen cut into New England, and so work my way back to Boston for a\nfortnight, after which come Chicago, Cincinnati, Detroit and Cleveland,\nand Buffalo, and then Philadelphia, Boston, and New York farewells. I\nwill not pass my original bound of eighty-four readings in all. My mind\nwas made up as to that long ago. It will be quite enough. Chicago is\nsome fifteen hundred miles from here. What with travelling, and getting\nready for reading, and reading, the days are pretty fully occupied. Not\nthe less so because I rest very indifferently at night.\n\nThe people are exceedingly kind and considerate, and desire to be most\nhospitable besides. But I cannot accept hospitality, and never go out,\nexcept at Boston, or I should not be fit for the labour. If Dolby holds\nout well to the last it will be a triumph, for he has to see everybody,\ndrink with everybody, sell all the tickets, take all the blame, and go\nbeforehand to all the places on the list. I shall not see him after\nto-night for ten days or a fortnight, and he will be perpetually on the\nroad during the interval. When he leaves me, Osgood, a partner in\nTicknor and Fields' publishing firm, mounts guard over me, and has to go\ninto the hall from the platform door every night, and see how the public\nare seating themselves. It is very odd to see how hard he finds it to\nlook a couple of thousand people in the face, on which head, by-the-bye,\nI notice the papers to take \"Mr. Dickens's extraordinary composure\"\n(their great phrase) rather ill, and on the whole to imply that it would\nbe taken as a suitable compliment if I would stagger on to the platform\nand instantly drop, overpowered by the spectacle before me.\n\nDinner is announced (by Scott, with a stiff neck and a sore throat), and\nI must break off with love to Bessie and the incipient Wenerableses. You\nwill be glad to hear of your distinguished parent that Philadelphia has\ndiscovered that \"he is not like the descriptions we have read of him at\nthe little red desk. He is not at all foppish in appearance. He wears a\nheavy moustache and a Vandyke beard, and looks like a well-to-do\nPhiladelphian gentleman.\"\n\n                      Ever, my dear Charley, your affectionate Father.\n\nP.S.--Your paper is remarkably good. There is not the least doubt that\nyou can write constantly for A. Y. R. I am very pleased with it.\n\n\n[Sidenote: Miss Dickens.]\n\n               WESTMINSTER HOTEL, NEW YORK, _Friday, Jan, 18th, 1868._\n\nThis will be but a very short report, as I must get out for a little\nexercise before dinner.\n\nMy \"true American catarrh\" (the people seem to have a national pride in\nit) sticks to me, but I am otherwise well. I began my church readings\nlast night, and it was very odd to see the pews crammed full of people,\nall in a broad roar at the \"Carol\" and \"Trial.\"\n\nBest love to all. I have written Charley a few lines by this mail, and\nalso Chorley.\n\n\n[Sidenote: Miss Hogarth.]\n\n              WESTMINSTER HOTEL, NEW YORK, _Tuesday, Jan. 21st, 1868._\n\nI finished my church to-night. It is Mrs. Stowe's brother's, and a most\nwonderful place to speak in. We had it enormously full last night\n(\"Marigold\" and \"Trial\"), but it scarcely required an effort. Mr. Ward\nBeecher (Mrs. Stowe's brother's name) being present in his pew. I sent\nto invite him to come round before he left; and I found him to be an\nunostentatious, straightforward, and agreeable fellow.\n\nMy cold sticks to me, and I can scarcely exaggerate what I sometimes\nundergo from sleeplessness. The day before yesterday I could get no rest\nuntil morning, and could not get up before twelve. This morning the\nsame. I rarely take any breakfast but an egg and a cup of tea, not even\ntoast or bread-and-butter. My dinner at three, and a little quail or\nsome such light thing when I come home at night, is my daily fare. At\nthe Hall I have established the custom of taking an egg beaten up in\nsherry before going in, and another between the parts. I think that\npulls me up; at all events, I have since had no return of faintness.\n\nAs the men work very hard, and always with their hearts cheerfully in\nthe business, I cram them into and outside of the carriage, to bring\nthem back from Brooklyn with me. The other night, Scott (with a\nportmanteau across his knees and a wideawake hat low down upon his nose)\ntold me that he had presented himself for admission in the circus (as\ngood as Franconi's, by-the-bye), and had been refused. \"The only\ntheayter,\" he said in a melancholy way, \"as I was ever in my life turned\nfrom the door of.\" Says Kelly: \"There must have been some mistake,\nScott, because George and me went, and we said, 'Mr. Dickens's staff,'\nand they passed us to the best seats in the house. Go again, Scott.\"\n\"No, I thank you, Kelly,\" says Scott, more melancholy than before, \"I'm\nnot a-going to put myself in the position of being refused again. It's\nthe only theayter as I was ever turned from the door of, and it shan't\nbe done twice. But it's a beastly country!\" \"Scott,\" interposed Majesty,\n\"don't you express your opinions about the country.\" \"No, sir,\" says\nScott, \"I never do, please, sir, but when you are turned from the door\nof the only theayter you was ever turned from, sir, and when the beasts\nin railway cars spits tobacco over your boots, you (privately) find\nyourself in a beastly country.\"\n\nI expect shortly to get myself snowed up on some railway or other, for\nit is snowing hard now, and I begin to move to-morrow. There is so much\nfloating ice in the river that we are obliged to leave a pretty wide\nmargin of time for getting over the ferry to read. The dinner is coming\nin, and I must leave off.\n\n\n[Sidenote: Miss Dickens.]\n\n                            PHILADELPHIA, _Thursday, Jan. 23rd, 1868._\n\nWhen I wrote to your aunt by the last mail, I accidentally omitted to\ntouch upon the question of helping Anne. So I will begin in this present\nwriting with reference to her sad position. I think it will be best for\nyou to be guided by an exact knowledge of her _wants_. Try to ascertain\nfrom herself what means she has, whether her sick husband gets what he\nought to have, whether she is pinched in the articles of necessary\nclothing, bedding, or the like of that; add to this intelligence your\nown observation of the state of things about her, and supply what she\nmost wants, and help her where you find the greatest need. The question,\nin the case of so old and faithful a servant, is not one of so much or\nso little money on my side, but how _most efficiently_ to ease her mind\nand help _her_. To do this at once kindly and sensibly is the only\nconsideration by which you have to be guided. Take _carte blanche_ from\nme for all the rest.\n\nMy Washington week is the first week in February, beginning on Monday,\n3rd. The tickets are sold, and the President is coming, and the chief\nmembers of the Cabinet, and the leaders of parties, and so forth, are\ncoming; and, as the Holly Tree Boots says: \"That's where it is, don't\nyou see!\"\n\nIn my Washington doubts I recalled Dolby for conference, and he joined\nme yesterday afternoon, and we have been in great discussion ever since\non the possibility of giving up the Far West, and avoiding such immense\ndistances and fatigues as would be involved in travelling to Chicago and\nCincinnati. We have sketched another tour for the last half of March,\nwhich would be infinitely easier for me, though on the other hand less\nprofitable, the places and the halls being smaller. The worst of it is,\nthat everybody one advises with has a monomania respecting Chicago.\n\"Good heaven, sir,\" the great Philadelphian authority said to me this\nmorning, \"if you don't read in Chicago, the people will go into fits.\"\nIn reference to fatigue, I answered: \"Well, I would rather they went\ninto fits than I did.\" But he didn't seem to see it at all. ---- alone\nconstantly writes me: \"Don't go to the West; you can get what you want\nso much more easily.\" How we shall finally decide, I don't yet know. My\nBrooklyn church has been an immense success, and I found its minister\nwas a bachelor, a clever, unparsonic, and straightforward man, and a man\nwith a good knowledge of art into the bargain.\n\nWe are not a bit too soon here, for the whole country is beginning to be\nstirred and shaken by the presidential election, and trade is\nexceedingly depressed, and will be more so. Fanny Kemble lives near this\nplace, but had gone away a day before my first visit here. _She_ is\ngoing to read in February or March. Du Chaillu has been lecturing out\nWest about the gorilla, and has been to see me; I saw the Cunard steamer\n_Persia_ out in the stream, yesterday, beautifully smart, her flags\nflying, all her steam up, and she only waiting for her mails to slip\naway. She gave me a horrible touch of home-sickness.\n\nWhen the 1st of March arrives, and I can say \"next month,\" I shall begin\nto grow brighter. A fortnight's reading in Boston, too (last week of\nFebruary and first week of March), will help me on gaily, I hope (the\nwork so far off tells). It is impossible for the people to be more\naffectionately attached to a third, I really believe, than Fields and\nhis wife are to me; and they are a landmark in the prospect.\n\nDolby sends kindest regards, and wishes it to be known that he has not\nbeen bullied lately. We do _not_ go West at all, but take the easier\nplan.\n\n\n[Sidenote: Miss Hogarth.]\n\n                              BALTIMORE, _Wednesday, Jan. 29th, 1868._\n\nAs I have an hour to spare, before starting to Philadelphia, I begin my\nletter this morning. It has been snowing hard for four-and-twenty hours,\nthough this place is as far south as Valentia in Spain; and Dolby, being\non his way to New York, has a good chance of being snowed up somewhere.\n\nThey are a bright responsive people here, and very pleasant to read to.\nI have rarely seen so many fine faces in an audience. I read here in a\ncharming little opera-house built by a society of Germans, quite a\ndelightful place for the purpose. I stand on the stage, with a drop\ncurtain down, and my screen before it. The whole scene is very pretty\nand complete, and the audience have a \"ring\" in them that sounds in the\near. I go from here to Philadelphia to read to-morrow night and Friday,\ncome through here again on Saturday on my way to Washington, come back\nhere on Saturday week for two finishing nights, then go to Philadelphia\nfor two farewells, and so turn my back on the southern part of the\ncountry. Distances and travelling have obliged us to reduce the list of\nreadings by two, leaving eighty-two in all. Of course we afterwards\ndiscovered that we had finally settled the list on a Friday! I shall be\nhalfway through it at Washington, of course, on a Friday also, and my\nbirthday!\n\nDolby and Osgood, who do the most ridiculous things to keep me in\nspirits (I am often very heavy, and rarely sleep much), have decided to\nhave a walking-match at Boston, on Saturday, February 29th. Beginning\nthis design in joke, they have become tremendously in earnest, and Dolby\nhas actually sent home (much to his opponent's terror) for a pair of\nseamless socks to walk in. Our men are hugely excited on the subject,\nand continually make bets on \"the men.\" Fields and I are to walk out six\nmiles, and \"the men\" are to turn and walk round us. Neither of them has\nthe least idea what twelve miles at a pace is. Being requested by both\nto give them \"a breather\" yesterday, I gave them a stiff one of five\nmiles over a bad road in the snow, half the distance uphill. I took them\nat a pace of four miles and a half an hour, and you never beheld such\nobjects as they were when we got back; both smoking like factories, and\nboth obliged to change everything before they could come to dinner. They\nhave the absurdest ideas of what are tests of walking power, and\ncontinually get up in the maddest manner and see _how high they can\nkick_ the wall! The wainscot here, in one place, is scored all over with\ntheir pencil-marks. To see them doing this--Dolby, a big man, and\nOsgood, a very little one, is ridiculous beyond description.\n\n\n                                           PHILADELPHIA, _Same Night._\n\nWe came on here through a snowstorm all the way, but up to time. Fanny\nKemble (who begins to read shortly) is coming to \"Marigold\" and \"Trial\"\nto-morrow night. I have written her a note, telling her that if it will\nat all assist _her_ movements to know _mine_, my list is at her\nservice. Probably I shall see her to-morrow. Tell Mamie (to whom I will\nwrite next), with my love, that I found her letter of the 10th of this\nmonth awaiting me here. The _Siberia_ that brought it is a new Cunarder,\nand made an unusually slow passage out. Probably because it would be\ndangerous to work new machinery too fast on the Atlantic.\n\n\n                                                     _Thursday, 30th._\n\nMy cold still sticks to me. The heat of the railway cars and their\nunventilated condition invariably brings it back when I think it going.\nThis morning my head is as stuffed and heavy as ever! A superb sledge\nand four horses have been offered me for a ride, but I am afraid to take\nit, lest I should make the \"true American catarrh\" worse, and should get\nhoarse. So I am going to give Osgood another \"breather\" on foot instead.\n\nThe communication with New York is not interrupted, so we consider the\nzealous Dolby all right. You may imagine what his work is, when you hear\nthat he goes three times to every place we visit. Firstly, to look at\nthe hall, arrange the numberings, and make five hundred acquaintances,\nwhom he immediately calls by their christian-names; secondly, to sell\nthe tickets--a very nice business, requiring great tact and temper;\nthirdly, with me. He will probably turn up at Washington next Sunday,\nbut only for a little while; for as soon as I am on the platform on\nMonday night, he will start away again, probably to be seen no more\nuntil we pass through New York in the middle of February.\n\n\n[Sidenote: Mr. Samuel Cartwright]\n\n                              BALTIMORE, _Wednesday, Jan. 29th, 1868._\n\nMY DEAR CARTWRIGHT,\n\nAs I promised to report myself to you from this side of the Atlantic,\nand as I have some leisure this morning, I am going to lighten my\nconscience by keeping my word.\n\nI am going on at a great pace and with immense success. Next week, at\nWashington, I shall, please God, have got through half my readings. The\nremaining half are all arranged, and they will carry me into the third\nweek of April. It is very hard work, but it is brilliantly paid. The\nchanges that I find in the country generally (this place is the least\nchanged of any I have yet seen) exceed my utmost expectations. I had\nbeen in New York a couple of days before I began to recognise it at all;\nand the handsomest part of Boston was a black swamp when I saw it\nfive-and-twenty years ago. Considerable advances, too, have been made\nsocially. Strange to say, the railways and railway arrangements (both\nexceedingly defective) seem to have stood still while all other things\nhave been moving.\n\nOne of the most comical spectacles I have ever seen in my life was\n\"church,\" with a heavy sea on, in the saloon of the Cunard steamer\ncoming out. The officiating minister, an extremely modest young man, was\nbrought in between two big stewards, exactly as if he were coming up to\nthe scratch in a prize-fight. The ship was rolling and pitching so, that\nthe two big stewards had to stop and watch their opportunity of making a\ndart at the reading-desk with their reverend charge, during which pause\nhe held on, now by one steward and now by the other, with the feeblest\nexpression of countenance and no legs whatever. At length they made a\ndart at the wrong moment, and one steward was immediately beheld alone\nin the extreme perspective, while the other and the reverend gentleman\n_held on by the mast_ in the middle of the saloon--which the latter\nembraced with both arms, as if it were his wife. All this time the\ncongregation was breaking up into sects and sliding away; every sect (as\nin nature) pounding the other sect. And when at last the reverend\ngentleman had been tumbled into his place, the desk (a loose one, put\nupon the dining-table) deserted from the church bodily, and went over to\nthe purser. The scene was so extraordinarily ridiculous, and was made so\nmuch more so by the exemplary gravity of all concerned in it, that I was\nobliged to leave before the service began.\n\nThis is one of the places where Butler carried it with so high a hand in\nthe war, and where the ladies used to spit when they passed a Northern\nsoldier. It still wears, I fancy, a look of sullen remembrance. (The\nladies are remarkably handsome, with an Eastern look upon them, dress\nwith a strong sense of colour, and make a brilliant audience.) The ghost\nof slavery haunts the houses; and the old, untidy, incapable, lounging,\nshambling black serves you as a free man. Free of course he ought to be;\nbut the stupendous absurdity of making him a voter glares out of every\nroll of his eye, stretch of his mouth, and bump of his head. I have a\nstrong impression that the race must fade out of the States very fast.\nIt never can hold its own against a striving, restless, shifty people.\nIn the penitentiary here, the other day, in a room full of all blacks\n(too dull to be taught any of the work in hand), was one young brooding\nfellow, very like a black rhinoceros. He sat glowering at life, as if it\nwere just endurable at dinner time, until four of his fellows began to\nsing, most unmelodiously, a part song. He then set up a dismal howl, and\npounded his face on a form. I took him to have been rendered quite\ndesperate by having learnt anything. I send my kind regard to Mrs.\nCartwright, and sincerely hope that she and you have no new family\ndistresses or anxieties. My standing address is the Westminster Hotel,\nIrving Place, New York City. And I am always, my dear Cartwright,\n\n                                                      Cordially yours.\n\n\n[Sidenote: Miss Dickens.]\n\n                              PHILADELPHIA, _Friday, Jan. 31st, 1868._\n\nSince writing to your aunt I have received yours of the 7th, and am\ntruly glad to have the last news of you confirmed by yourself.\n\nFrom a letter Wilkie has written to me, it seems there can be no doubt\nthat the \"No Thoroughfare\" drama is a real, genuine, and great success.\nIt is drawing immensely, and seems to \"go\" with great effect and\napplause.\n\n\"Doctor Marigold\" here last night (for the first time) was an immense\nsuccess, and all Philadelphia is going to rush at once for tickets for\nthe two Philadelphian farewells the week after next. The tickets are to\nbe sold to-morrow, and great excitement is anticipated in the streets.\nDolby not being here, a clerk will sell, and will probably wish himself\ndead before he has done with it.\n\nIt appears to me that Chorley[22] writes to you on the legacy question\nbecause he wishes you to understand that there is no danger of his\nchanging his mind, and at the bottom I descry an honest desire to pledge\nhimself as strongly as possible. You may receive it in that better\nspirit, or I am much mistaken. Tell your aunt, with my best love, that I\nwrote to Chauncey weeks ago, in answer to a letter from him. I am now\ngoing out in a sleigh (and four) with unconceivable dignity and\ngrandeur; mentioning which reminds me that I am informed by trusty\nscouts that ---- intends to waylay me at Washington, and may even\ndescend upon me in the train to-morrow.\n\nBest love to Katie, the two Charleys, and all.\n\n\n[Sidenote: Miss Dickens.]\n\n                                WASHINGTON, _Tuesday, Feb. 4th, 1868._\n\nI began here last night with great success. The hall being small, the\nprices were raised to three dollars each ticket. The audience was a\nsuperior one, composed of the foremost public men and their families. At\nthe end of the \"Carol\" they gave a great break out, and applauded, I\nreally believe, for five minutes. You would suppose them to be\nManchester shillings instead of Washington half-sovereigns. Immense\nenthusiasm.\n\nA devoted adherent in this place (an Englishman) had represented to\nDolby that if I were taken to an hotel here it would be impossible to\nsecure me a minute's rest, and he undertook to get one Wheleker, a\nGerman, who keeps a little Verey's, to furnish his private dining-rooms\nfor the illustrious traveller's reception. Accordingly here we are, on\nthe first and second floor of a small house, with no one else in it but\nour people, a French waiter, and a very good French cuisine. Perfectly\nprivate, in the city of all the world (I should say) where the hotels\nare intolerable, and privacy the least possible, and quite comfortable.\n\"Wheleker's Restaurant\" is our rather undignified address for the\npresent week.\n\nI dined (against my rules) with Charles Sumner on Sunday, he having been\nan old friend of mine. Mr. Secretary Staunton (War Minister) was there.\nHe is a man of a very remarkable memory, and famous for his\nacquaintance with the minutest details of my books. Give him any passage\nanywhere, and he will instantly cap it and go on with the context. He\nwas commander-in-chief of all the Northern forces concentrated here, and\nnever went to sleep at night without first reading something from my\nbooks, which were always with him. I put him through a pretty severe\nexamination, but he was better up than I was.\n\nThe gas was very defective indeed last night, and I began with a small\nspeech, to the effect that I must trust to the brightness of their faces\nfor the illumination of mine; this was taken greatly. In the \"Carol,\" a\nmost ridiculous incident occurred all of a sudden. I saw a dog look out\nfrom among the seats into the centre aisle, and look very intently at\nme. The general attention being fixed on me, I don't think anybody saw\nthe dog; but I felt so sure of his turning up again and barking, that I\nkept my eye wandering about in search of him. He was a very comic dog,\nand it was well for me that I was reading a very comic part of the book.\nBut when he bounced out into the centre aisle again, in an entirely new\nplace (still looking intently at me) and tried the effect of a bark upon\nmy proceedings, I was seized with such a paroxysm of laughter, that it\ncommunicated itself to the audience, and we roared at one another loud\nand long.\n\nThe President has sent to me twice, and I am going to see him to-morrow.\nHe has a whole row for his family every night. Dolby rejoined his chief\nyesterday morning, and will probably remain in the august presence until\nSunday night. He and Osgood, \"training for the match,\" are ludicrous\nbeyond belief. I saw them just now coming up a street, each trying to\npass the other, and immediately fled. Since I have been writing this,\nthey have burst in at the door and sat down on the floor to blow. Dolby\nis now writing at a neighbouring table, with his bald head smoking as if\nhe were on fire. Kelly (his great adherent) asked me, when he was last\naway, whether it was quite fair that I should take Mr. Osgood out for\n\"breathers\" when Mr. Dolby had no such advantage. I begin to expect that\nhalf Boston will turn out on the 29th to see the match. In which case it\nwill be unspeakably droll.\n\n\n[Sidenote: Miss Hogarth.]\n\n                            WASHINGTON, _my Birthday_, 1868.\n                                      (_And my cold worse than ever._)\n\nThis will be but a short letter, as I have been to see the President\nthis morning, and have little time before the post goes. He had sent a\ngentleman to me, most courteously begging me to make my own appointment,\nand I did so. A man of very remarkable appearance indeed, of tremendous\nfirmness of purpose. Not to be turned or trifled with.\n\nAs I mention my cold's being so bad, I will add that I have never had\nanything the matter with me since I came here _but_ the cold. It is now\nin my throat, and slightly on my chest. It occasions me great\ndiscomfort, and you would suppose, seeing me in the morning, that I\ncould not possibly read at night. But I have always come up to the\nscratch, have not yet missed one night, and have gradually got used to\nthat. I had got much the better of it; but the dressing-room at the hall\nhere is singularly cold and draughty, and so I have slid back again.\n\nThe papers here having written about this being my birthday, the most\nexquisite flowers came pouring in at breakfast time from all sorts of\npeople. The room is covered with them, made up into beautiful bouquets,\nand arranged in all manner of green baskets. Probably I shall find\nplenty more at the hall to-night. This is considered the dullest and\nmost apathetic place in America. _My_ audiences have been superb.\n\nI mentioned the dog on the first night here. Next night I thought I\nheard (in \"Copperfield\") a suddenly suppressed bark. It happened in this\nwise: Osgood, standing just within the door, felt his leg touched, and\nlooking down beheld the dog staring intently at me, and evidently just\nabout to bark. In a transport of presence of mind and fury, he instantly\ncaught him up in both hands and threw him over his own head out into the\nentry, where the check-takers received him like a game at ball. Last\nnight he came again _with another dog_; but our people were so sharply\non the look-out for him that he didn't get in. He had evidently promised\nto pass the other dog free.\n\n\n[Sidenote: Miss Dickens.]\n\n                          BALTIMORE, U.S., _Tuesday, Feb. 11th, 1868._\n\nThe weather has been desperately severe, and my cold quite as bad as\never. I couldn't help laughing at myself on my birthday at Washington.\nIt was observed as much as though I were a little boy. Flowers and\ngarlands (of the most exquisite kind) bloomed all over the room; letters\nradiant with good wishes poured in; a shirt pin, a handsome silver\ntravelling bottle, a set of gold shirt studs, and a set of gold sleeve\nlinks were on the dinner-table. After \"Boots,\" at night, the whole\naudience rose and remained (Secretaries of State, President's family,\nJudges of Supreme Court, and so forth) standing and cheering until I\nwent back to the table and made them a little speech. On the same\naugust day of the year I was received by the President, a man with a\nvery remarkable and determined face. Each of us looked at each other\nvery hard, and each of us managed the interview (I think) to the\nsatisfaction of the other. In the outer room was sitting a certain\nsunburnt General Blair, with many evidences of the war upon him. He got\nup to shake hands with me, and then I found he had been out in the\nprairie with me five-and-twenty years ago. That afternoon my \"catarrh\"\nwas in such a state that Charles Sumner, coming in at five o'clock and\nfinding me covered with mustard poultice, and apparently voiceless,\nturned to Dolby and said: \"Surely, Mr. Dolby, it is impossible that he\ncan read to-night.\" Says Dolby: \"Sir, I have told the dear Chief so four\ntimes to-day, and I have been very anxious. But you have no idea how he\nwill change when he gets to the little table.\" After five minutes of the\nlittle table, I was not (for the time) even hoarse. The frequent\nexperience of this return of force when it is wanted saves me a vast\namount of anxiety.\n\nI wish you would get from Homan and report to me, as near as he can\nmake, an approximate estimate is the right term in the trade, I believe,\nof the following work:\n\n1. To re-cover, with red leather, all the dining-room chairs.\n\n2. To ditto, with green leather, all the library chairs and the couch.\n\n3. To provide and lay down new _Brussels_ carpets in the front spare and\nthe two top spares. Quality of carpet, quality of yours and mine.\n\nI have some doubts about the state of the hall floor-cloth, and also the\nfloor-cloth in the dining-room. Will you and your aunt carefully examine\nboth (calling in Homan too, if necessary), _and report to me_?\n\nIt would seem that \"No Thoroughfare\" has really developed as a drama\ninto an amazing success. I begin to think that I shall see it. Dolby is\naway this morning, to conquer or die in a terrific struggle with the\nMayor of Newhaven (where I am to read next week), who has assailed him\non a charge of false play in selling tickets. Osgood, my other keeper,\nstands at the table to take me out, and have a \"breather\" for the\nwalking-match, so I must leave off.\n\nThink of my dreaming of Mrs. Bouncer each night!!!\n\n\n[Sidenote: Mr. Henry Fielding Dickens.]\n\n                          BALTIMORE, U.S., _Tuesday, Feb. 11th, 1868._\n\nMY DEAR HARRY,\n\nI should have written to you before now, but for constant and arduous\noccupation.\n\nIn reference to the cricket club's not being what it might be, I agree\nwith you in the main. There are some things to be considered, however,\nwhich you have hardly taken into account. The first thing to be avoided\nis, the slightest appearance of patronage (one of the curses of\nEngland). The second thing to be avoided is, the deprival of the men of\ntheir just right to manage their own affairs. I would rather have no\nclub at all, than have either of these great mistakes made. The way out\nof them is this: Call the men together, and explain to them that the\nclub might be larger, richer, and better. Say that you think that more\nof the neighbouring gentlemen could be got to be playing members. That\nyou submit to them that it would be better to have a captain who could\ncorrespond with them, and talk to them, and in some sort manage them;\nand that, being perfectly acquainted with the game, and having long\nplayed it at a great public school, you propose yourself as captain, for\nthe foregoing reasons. That you propose to them to make the subscription\nof the gentlemen members at least double that of the working men, for no\nother reason than that the gentlemen can afford it better; but that both\nclasses of members shall have exactly the same right of voting equally\nin all that concerns the club. Say that you have consulted me upon the\nmatter, and that I am of these opinions, and am ready to become chairman\nof the club, and to preside at their meetings, and to overlook its\nbusiness affairs, and to give it five pounds a year, payable at the\ncommencement of each season. Then, having brought them to this point,\ndraw up the club's rules and regulations, amending them where they want\namendment.\n\nDiscreetly done, I see no difficulty in this. But it can only be\nhonourably and hopefully done by having the men together. And I would\nnot have them at The Falstaff, but in the hall or dining-room--the\nservants' hall, an excellent place. Whatever you do, let the men ratify;\nand let them feel their little importance, and at once perceive how much\nbetter the business begins to be done.\n\nI am very glad to hear of the success of your reading, and still more\nglad that you went at it in downright earnest. I should never have made\nmy success in life if I had been shy of taking pains, or if I had not\nbestowed upon the least thing I have ever undertaken exactly the same\nattention and care that I have bestowed upon the greatest. Do everything\nat your best. It was but this last year that I set to and learned every\nword of my readings; and from ten years ago to last night, I have never\nread to an audience but I have watched for an opportunity of striking\nout something better somewhere. Look at such of my manuscripts as are\nin the library at Gad's, and think of the patient hours devoted year\nafter year to single lines.\n\n       *       *       *       *       *\n\nThe weather is very severe here, and the work is very hard. Dolby,\nhaving been violently pitched into by the Mayor of Newhaven (a town at\nwhich I am to read next week), has gone bodily this morning with defiant\nwritten instructions from me to inform the said mayor that, if he fail\nto make out his case, he (Dolby) is to return all the money taken, and\nto tell him that I will not set foot in his jurisdiction; whereupon the\nNewhaven people will probably fall upon the mayor in his turn, and lead\nhim a pleasant life.\n\n                        Ever, my dear Harry, your affectionate Father.\n\n\n[Sidenote: Miss Hogarth.]\n\n                            PHILADELPHIA, _Thursday, Feb. 13th, 1868._\n\nWe have got into an immense difficulty with the people of Newhaven. I\nhave a strong suspicion that one of our men (who sold there) has been\nspeculating all this while, and that he must have put front seats in his\npockets, and sold back ones. He denies what the mayor charges, but the\nmayor holds on grimly. Dolby set off from Baltimore as soon as we found\nout what was amiss, to examine and report; but some new feature of\ndifficulty must have come out, for this morning he telegraphs from New\nYork (where he had to sleep last night on his way to Newhaven), that he\nis coming back for further consultation with the Chief. It will\ncertainly hurt us, and will of course be distorted by the papers into\nall manner of shapes. My suspicion _may not_ be correct, but I have an\ninstinctive belief that it is. We shall probably have the old New York\nrow (and loss) over again, unless I can catch this mayor tripping in an\nassertion.\n\nIn this very place, we are half-distracted by the speculators. They have\nbeen holding out for such high prices, that the public have held out\ntoo; and now (frightened at what they have done) the speculators are\ntrying to sell their worst seats at half the cost price, so that we are\nin the ridiculous situation of having sold the room out, and yet not\nknowing what empty seats there may be. _We_ could sell at our box-office\nto any extent; but _we_ can't buy back of the speculators, because we\ninformed the public that all the tickets were gone. And if we bought\n_under_ our own price and _sold_ at our own price, we should at once be\nin treaty with the speculators, and should be making money by it! Dolby,\nthe much bullied, will come back here presently, half bereft of his\nsenses; and I should be half bereft of mine, if the situation were not\ncomically disagreeable.\n\nNothing will induce the people to believe in the farewells. At Baltimore\non Tuesday night (a very brilliant night indeed), they asked as they\ncame out: \"When will Mr. Dickens read here again?\" \"Never.\" \"Nonsense!\nNot come back, after such houses as these? Come. Say when he'll read\nagain.\" Just the same here. We could as soon persuade them that I am the\nPresident, as that I am going to read here, for the last time, to-morrow\nnight.\n\nThere is a child of the Barney Williams's in this house--a little\ngirl--to whom I presented a black doll when I was here last. I have seen\nher eye at the keyhole since I began writing this, and I think she and\nthe doll are outside still. \"When you sent it up to me by the <DW52>\nboy,\" she said after receiving it (<DW52> boy is the term for black\nwaiter), \"I gave such a cream that ma came running in and creamed too,\n'cos she fort I'd hurt myself. But I creamed a cream of joy.\" _She_ had\na friend to play with her that day, and brought the friend with her, to\nmy infinite confusion. A friend all stockings, and much too tall, who\nsat on the sofa very far back, with her stockings sticking stiffly out\nin front of her, and glared at me and never spake word. Dolby found us\nconfronted in a sort of fascination, like serpent and bird.\n\n\n[Sidenote: Miss Hogarth.]\n\n                                  NEW YORK, _Monday, Feb. 17th, 1868._\n\nI got your letter of the 3rd of February here this morning. As I am off\nat seven to-morrow morning, I answer it at once, though indeed I have\nnothing to say.\n\n\"True American\" still sticking to me. But I am always ready for my work,\nand therefore don't much mind. Dolby and the Mayor of Newhaven\nalternately embrace and exchange mortal defiances. In writing out some\nadvertisements towards midnight last night, he made a very good mistake.\n\"The reading will be comprised within two _minutes_, and the audience\nare earnestly entreated to be seated ten _hours_ before its\ncommencement.\"\n\nThe weather has been finer lately, but the streets are in a horrible\ncondition, through half-melted snow, and it is now snowing again. The\nwalking-match (next Saturday week) is already in the Boston papers! I\nsuppose half Boston will turn out on the occasion. As a sure way of not\nbeing conspicuous, \"the men\" are going to walk in flannel! They are in a\nmingled state of comicality and gravity about it that is highly\nridiculous. Yesterday being a bright cool day, I took Dolby for a\n\"buster\" of eight miles. As everybody here knows me, the spectacle of\nour splitting up the fashionable avenue (the only way out of town)\nexcited the greatest amazement. No doubt _that_ will be in the papers\nto-morrow. I give a gorgeous banquet to eighteen (ladies and gentlemen)\nafter the match. Mr. and Mrs. Fields, Do. Ticknor, Longfellow and his\ndaughter, Lowell, Holmes and his wife, etc. etc. Sporting speeches to be\nmade, and the stakes (four hats) to be handed over to the winner.\n\nMy ship will not be the _Cuba_ after all. She is to go into dock, and\nthe _Russia_ (a larger ship, and the latest built for the Cunard line)\nis to take her place.\n\nVery glad to hear of Plorn's success. Best love to Mamie.\n\n\n[Sidenote: M. Charles Fechter.]\n\n                                    WASHINGTON, _February 24th, 1868._\n\nMY DEAR FECHTER,\n\nYour letter reached me here yesterday. I have sent you a telegram\n(addressed to the theatre) this morning, and I write this by the\nearliest return mail.\n\nMy dear fellow, consider yourself my representative. Whatever you do, or\ndesire to do, about the play, I fully authorise beforehand. Tell\nWebster, with my regard, that I think his proposal honest and fair; that\nI think it, in a word, like himself; and that I have perfect confidence\nin his good faith and liberality.\n\nAs to making money of the play in the United States here, Boucicault has\nfilled Wilkie's head with golden dreams that have _nothing_ in them. He\nmakes no account of the fact that, wherever I go, the theatres (with my\nname in big letters) instantly begin playing versions of my books, and\nthat the moment the Christmas number came over here they pirated it and\nplayed \"No Thoroughfare.\" Now, I have enquired into the law, and am\nextremely doubtful whether I _could_ have prevented this. Why should\nthey pay for the piece as you act it, when they have no actors, and when\nall they want is my name, and they can get that for nothing?\n\nWilkie has uniformly written of you enthusiastically. In a letter I had\nfrom him, dated the 10th of January, he described your conception and\nexecution of the part in the most glowing terms. \"Here Fechter is\nmagnificent.\" \"Here his superb playing brings the house down.\" \"I should\ncall even his exit in the last act one of the subtlest and finest things\nhe does in the piece.\" \"You can hardly imagine what he gets out of the\npart, or what he makes of his passionate love for Marguerite.\" These\nexpressions, and many others like them, crowded his letter.\n\nI never did so want to see a character played on the stage as I want to\nsee you play Obenreizer. As the play was going when I last heard of it,\nI have some hopes that I MAY see it yet. Please God, your Adelphi\ndressing-room will be irradiated with the noble presence of \"Never\nWrong\" (if you are acting), about the evening of Monday, the 4th of May!\n\nI am doing enormous business. It is a wearying life, away from all I\nlove, but I hope that the time will soon begin to spin away. Among the\nmany changes that I find here is the comfortable change that the people\nare in general extremely considerate, and very observant of my privacy.\nEven in this place, I am really almost as much my own master as if I\nwere in an English country town. Generally, they are very good audiences\nindeed. They do not (I think) perceive touches of art to _be_ art; but\nthey are responsive to the broad results of such touches. \"Doctor\nMarigold\" is a great favourite, and they laugh so unrestrainedly at \"The\nTrial\" from \"Pickwick\" (which you never heard), that it has grown about\nhalf as long again as it used to be.\n\nIf I could send you a \"brandy cocktail\" by post I would. It is a highly\nmeritorious dram, which I hope to present to you at Gad's. My New York\nlandlord made me a \"Rocky Mountain sneezer,\" which appeared to me to be\ncompounded of all the spirits ever heard of in the world, with bitters,\nlemon, sugar, and snow. You can only make a true \"sneezer\" when the snow\nis lying on the ground.\n\nThere, my dear boy, my paper is out, and I am going to read\n\"Copperfield.\" Count always on my fidelity and true attachment, and look\nout, as I have already said, for a distinguished visitor about Monday,\nthe 4th of May.\n\n                    Ever, my dear Fechter,\n                                 Your cordial and affectionate Friend.\n\n\n[Sidenote: Miss Dickens.]\n\n                                   BOSTON, _Tuesday, Feb. 25th, 1868._\n\nIt is so very difficult to know, by any exercise of common sense, what\nturn or height the political excitement may take next, and it may so\neasily, and so soon, swallow up all other things, that I think I shall\nsuppress my next week's readings here (by good fortune not yet\nannounced) and watch the course of events. Dolby's sudden desponding\nunder these circumstances is so acute, that it is actually swelling his\nhead as I glance at him in the glass while writing.\n\nThe catarrh is no better and no worse. The weather is intensely cold.\nThe walking-match (of which I will send particulars) is to come off on\nSunday. Mrs. Fields is more delightful than ever, and Fields more\nhospitable. My room is always radiant with brilliant flowers of their\nsending. I don't know whether I told you that the walking-match is to\ncelebrate the extinction of February, and the coming of the day when I\ncan say \"next month.\"\n\n\n[Sidenote: Miss Hogarth.]\n\n                                  BOSTON, _Thursday, Feb. 27th, 1868._\n\nThis morning at breakfast I received yours of the 11th from Palace Gate\nHouse. I have very little news to give you in return for your budget.\nThe walking-match is to come off on Saturday, and Fields and I went over\nthe ground yesterday to measure the miles. We went at a tremendous pace.\nThe condition of the ground is something indescribable, from half-melted\nsnow, running water, and sheets and blocks of ice. The two performers\nhave not the faintest notion of the weight of the task they have\nundertaken. I give a dinner afterwards, and have just now been settling\nthe bill of fare and selecting the wines.\n\nIn the first excitement of the presidential impeachment, our houses\ninstantly went down. After carefully considering the subject, I decided\nto take advantage of the fact that next week's four readings here have\nnot yet been announced, and to abolish them altogether. Nothing in this\ncountry lasts long, and I think the public may be heartily tired of the\nPresident's name by the 9th of March, when I read at a considerable\ndistance from here. So behold me with a whole week's holiday in view!\nThe Boston audiences have come to regard the readings and the reader as\ntheir peculiar property; and you would be at once amused and pleased if\nyou could see the curious way in which they seem to plume themselves on\nboth. They have taken to applauding too whenever they laugh or cry, and\nthe result is very inspiriting. I shall remain here until Saturday, the\n7th, but shall not read here, after to-morrow night, until the 1st of\nApril, when I begin my Boston farewells, six in number.\n\n\n                                                       _Friday, 28th._\n\nIt has been snowing all night, and the city is in a miserable condition.\nWe had a fine house last night for \"Carol\" and \"Trial,\" and such an\nenthusiastic one that they persisted in a call after the \"Carol,\" and,\nwhile I was out, covered the little table with flowers. The \"True\nAmerican\" has taken a fresh start, as if it were quite a novelty, and is\non the whole rather worse than ever to-day. The Cunard packet, the\n_Australasian_ (a poor ship), is some days overdue, and Dolby is\nanxiously looking out for her. There is a lull in the excitement about\nthe President, but the articles of impeachment are to be produced this\nafternoon, and then it may set in again. Osgood came into camp last\nnight from selling in remote places, and reports that at Rochester and\nBuffalo (both places near the frontier), Canada people bought tickets,\nwho had struggled across the frozen river and clambered over all sorts\nof obstructions to get them. Some of those halls turn out to be smaller\nthan represented, but I have no doubt, to use an American expression,\nthat we shall \"get along.\"\n\nTo-morrow fortnight we purpose being at the Falls of Niagara, and then\nwe shall turn back and really begin to wind up. I have got to know the\n\"Carol\" so well that I can't remember it, and occasionally go dodging\nabout in the wildest manner to pick up lost pieces. They took it so\ntremendously last night that I was stopped every five minutes. One poor\nyoung girl in mourning burst into a passion of grief about Tiny Tim, and\nwas taken out. This is all my news.\n\nEach of the pedestrians is endeavouring to persuade the other to take\nsomething unwholesome before starting.\n\n\n[Sidenote: Miss Dickens.]\n\n                                    BOSTON, _Monday, March 2nd, 1868._\n\nA heavy gale of wind and a snowstorm oblige me to write suddenly for the\nCunard steamer a day earlier than usual. The railroad between this and\nNew York will probably be stopped somewhere. After all the hard weather\nwe have had, this is the worst day we have seen.\n\nThe walking-match came off on Saturday, over tremendously difficult\nground, against a biting wind, and through deep snow-wreaths. It was so\ncold, too, that our hair, beards, eyelashes, eyebrows, were frozen hard,\nand hung with icicles. The course was thirteen miles. They were close\ntogether at the turning-point, when Osgood went ahead at a splitting\npace and with extraordinary endurance, and won by half a mile. Dolby did\nvery well indeed, and begs that he may not be despised. In the evening I\ngave a very splendid dinner. Eighteen covers, most magnificent flowers,\nsuch table decoration as was never seen in these parts. The whole thing\nwas a great success, and everybody was delighted.\n\nI am holiday-making until Friday, when we start on the round of travel\nthat is to bring us back here for the 1st of April. My holiday-making\nis simply thorough resting, except on Wednesday, when I dine with\nLongfellow. There is still great political excitement, but I hope it may\nnot hurt us very much. My fear is that it may damage the farewell. Dolby\nis not of my mind as to this, and I hope he may be right. We are not\nquite determined whether Mrs. Fields did not desert our colours, by\ncoming on the ground in a carriage, and having _bread soaked in brandy_\nput into the winning man's mouth as he steamed along. She pleaded that\nshe would have done as much for Dolby, if _he_ had been ahead, so we are\ninclined to forgive her. As she had done so much for me in the way of\nflowers, I thought I would show her a sight in that line at the dinner.\nYou never saw anything like it. Two immense crowns; the base, of the\nchoicest exotics; and the loops, oval masses of violets. In the centre\nof the table an immense basket, overflowing with enormous bell-mouthed\nlilies; all round the table a bright green border of wreathed creeper,\nwith clustering roses at intervals; a rose for every button-hole, and a\nbouquet for every lady. They made an exhibition of the table before\ndinner to numbers of people.\n\nP. H. has just come in with a newspaper, containing a reference (in good\ntaste!) to the walking-match. He posts it to you by this post.\n\nIt is telegraphed that the storm prevails over an immense extent of\ncountry, and is just the same at Chicago as here. I hope it may prove a\nwind-up. We are getting sick of the sound of sleigh-bells even.\n\nYour account of Anne has greatly interested me.\n\n\n[Sidenote: M. Charles Fechter.]\n\n                                      SYRACUSE, U.S. OF AMERICA,\n                                      _Sunday Night, March 8th, 1868._\n\nMY DEAR FECHTER,\n\nI am here in a most wonderful out-of-the-world place, which looks as if\nit had begun to be built yesterday, and were going to be imperfectly\nknocked together with a nail or two the day after to-morrow. I am in the\nworst inn that ever was seen, and outside is a thaw that places the\nwhole country under water. I have looked out of window for the people,\nand I can't find any people. I have tried all the wines in the house,\nand there are only two wines, for which you pay six shillings a bottle,\nor fifteen, according as you feel disposed to change the name of the\nthing you ask for. (The article never changes.) The bill of fare is \"in\nFrench,\" and the principal article (the carte is printed) is \"Paettie de\nshay.\" I asked the Irish waiter what this dish was, and he said: \"It was\nthe name the steward giv' to oyster patties--the Frinch name.\" These are\nthe drinks you are to wash it down with: \"Mooseux,\" \"Abasinthe,\"\n\"Curacco,\" \"Marschine,\" \"Annise,\" and \"Margeaux\"!\n\nI am growing very home-sick, and very anxious for the 22nd of April; on\nwhich day, please God, I embark for home. I am beginning to be tired,\nand have been depressed all the time (except when reading), and have\nlost my appetite. I cannot tell you--but you know, and therefore why\nshould I?--how overjoyed I shall be to see you again, my dear boy, and\nhow sorely I miss a dear friend, and how sorely I miss all art, in these\nparts. No disparagement to the country, which has a great future in\nreserve, or to its people, who are very kind to me.\n\nI mean to take my leave of readings in the autumn and winter, in a final\nseries in England with Chappell. This will come into the way of literary\nwork for a time, for, after I have rested--don't laugh--it is a grim\nreality--I shall have to turn my mind to--ha! ha! ha!--to--ha! ha! ha!\n(more sepulchrally than before)--the--the CHRISTMAS NUMBER!!! I feel as\nif I had murdered a Christmas number years ago (perhaps I did!) and its\nghost perpetually haunted me. Nevertheless in some blessed rest at\nGad's, we will talk over stage matters, and all matters, in an even way,\nand see what we can make of them, please God. Be sure that I shall not\nbe in London one evening, after disembarking, without coming round to\nthe theatre to embrace you, my dear fellow.\n\nI have had an American cold (the worst in the world) since Christmas\nDay. I read four times a week, with the most tremendous energy I can\nbring to bear upon it. I travel about pretty heavily. I am very resolute\nabout calling on people, or receiving people, or dining out, and so save\nmyself a great deal. I read in all sorts of places--churches, theatres,\nconcert rooms, lecture halls. Every night I read I am described (mostly\nby people who have not the faintest notion of observing) from the sole\nof my boot to where the topmost hair of my head ought to be, but is not.\nSometimes I am described as being \"evidently nervous;\" sometimes it is\nrather taken ill that \"Mr. Dickens is so extraordinarily composed.\" My\neyes are blue, red, grey, white, green, brown, black, hazel, violet, and\nrainbow-. I am like \"a well-to-do American gentleman,\" and the\nEmperor of the French, with an occasional touch of the Emperor of China,\nand a deterioration from the attributes of our famous townsman, Rufus W.\nB. D. Dodge Grumsher Pickville. I say all sorts of things that I never\nsaid, go to all sorts of places that I never saw or heard of, and have\ndone all manner of things (in some previous state of existence I\nsuppose) that have quite escaped my memory. You ask your friend to\ndescribe what he is about. This is what he is about, every day and hour\nof his American life.\n\nI hope to be back with you before you write to me!\n\n                  Ever, my dear Fechter,\n                             Your most affectionate and hearty Friend.\n\nP.S.--Don't let Madame Fechter, or Marie, or Paul forget me!\n\n\n[Sidenote: Miss Hogarth.]\n\n                                  SYRACUSE, _Sunday, March 8th, 1868._\n\nAs we shall probably be busy all day to-morrow, I write this to-day,\nthough it will not leave New York until Wednesday. This is a very grim\nplace in a heavy thaw, and a most depressing one. The hotel also is\nsurprisingly bad, quite a triumph in that way. We stood out for an hour\nin the melting snow, and came in again, having to change completely.\nThen we sat down by the stove (no fireplace), and there we are now. We\nwere so afraid to go to bed last night, the rooms were so close and\nsour, that we played whist, double dummy, till we couldn't bear each\nother any longer. We had an old buffalo for supper, and an old pig for\nbreakfast, and we are going to have I don't know what for dinner at six.\nIn the public rooms downstairs, a number of men (speechless) are sitting\nin rocking-chairs, with their feet against the window-frames, staring\nout at window and spitting dolefully at intervals. Scott is in tears,\nand George the gasman is suborning people to go and clean the hall,\nwhich is a marvel of dirt. And yet we have taken considerably over three\nhundred pounds for to-morrow night!\n\nWe were at Albany the night before last and yesterday morning; a very\npretty town, where I am to read on the 18th and 19th. This day week we\nhope to wash out this establishment with the Falls of Niagara. And there\nis my news, except that your _last letters_ to me in America must be\nposted by the Cunard steamer, which will sail from Liverpool on\n_Saturday, the 4th of April_. These I shall be safe to get before\nembarking.\n\nI send a note to Katie (addressed to Mamie) by this mail. I wrote to\nHarry some weeks ago, stating to him on what principles he must act in\nremodelling the cricket club, if he would secure success.\n\n\n[Sidenote: Miss Hogarth.]\n\n                                                _Monday Morning, 9th._\n\nNothing new. Weather cloudy, and town more dismal than yesterday. It\nfroze again last night, and thaws again this morning. Somebody sent me\nan Australian newspaper this morning--some citizen of Syracuse I\nmean--because of a paragraph in it describing the taking of two\nfreebooters, at which taking Alfred was present. Though I do not make\nout that he had anything in the world to do with it, except having his\nname pressed into the service of the newspaper.\n\n\n                                BUFFALO, _Thursday, March 12th, 1868._\n\nI hope this may be in time for next Saturday's mail; but this is a long\nway from New York, and rivers are swollen with melted snow, and\ntravelling is unusually slow.\n\nJust now (two o'clock in the afternoon) I received your sad news of the\ndeath of poor dear Chauncey.[23] It naturally goes to my heart. It is\nnot a light thing to lose such a friend, and I truly loved him. In the\nfirst unreasonable train of feeling, I dwelt more than I should have\nthought possible on my being unable to attend his funeral. I know how\nlittle this really matters; but I know he would have wished me to be\nthere with real honest tears for his memory, and I feel it very much. I\nnever, never, never was better loved by man than I was by him, I am\nsure. Poor dear fellow, good affectionate gentle creature.\n\nI have not as yet received any letter from Henri, nor do I think he can\nhave written to New York by your mail. I believe that I am--I know that\nI _was_--one of the executors. In that case Mr. Jackson, his agent, will\neither write to me very shortly on Henri's information of my address, or\nenquiry will be made at Gad's or at the office about it.\n\nIt is difficult for me to write more just now. The news is a real shock\nat such a distance, and I must read to-night, and I must compose my\nmind. Let Mekitty know that I received her violets with great pleasure,\nand that I sent her my best love and my best thanks.\n\nOn the 25th of February I read \"Copperfield\" and \"Bob\" at Boston. Either\non that very day, or very close upon it, I was describing his\n(Townshend's) house to Fields, and telling him about the great Danby\npicture that he should see when he came to London.\n\n\n[Sidenote: Miss Dickens.]\n\n                                ROCHESTER, _Sunday, March 16th, 1868._\n\nI found yours of the 28th February, when I came back here last night. We\nhave had two brilliant sunny days at Niagara, and have seen that\nwonderful place under the finest circumstances.\n\nEnclosed I return you Homan's estimate; let all that work be done,\nincluding the curtains.\n\nAs to the hall, I have my doubts whether one of the parqueted floors\nmade by Aaron Smith's, of Bond Street, ought not to be better than\ntiles, for the reason that perhaps the nature of the house's\nconstruction might render the \"bed\" necessary for wooden flooring more\neasy to be made than the \"bed\" necessary for tiles. I don't think you\ncan do better than call in the trusty Lillie to advise. Decide with your\naunt on which appears to be better, under the circumstances. Have\nestimate made for _cash_, select patterns and colours, and let the work\nbe done out of hand. (Here's a prompt order; now I draw breath.) Let it\nbe thoroughly well done--no half measures.\n\nThere is a great thaw all over the country here, and I think it has done\nthe catarrh good. I am to read at the famous Newhaven on Tuesday, the\n24th. I hope without a row, but cannot say. The readings are running out\nfast now, and we are growing very restless.\n\nThis is a short letter, but we are pressed for time. It is two o'clock,\nand we dine at three, before reading. To-morrow we rise at six, and have\neleven hours' railway or so. We have now come back from our farthest\npoint, and are steadily working towards home.\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n                     SPRINGFIELD, MASS., _Saturday, March 21st, 1868._\n\nMY DEAREST MACREADY,\n\nWhat with perpetual reading and travelling, what with a \"true American\ncatarrh\" (on which I am complimented almost boastfully), and what with\none of the severest winters ever known, your coals of fire received by\nthe last mail did not burn my head so much as they might have done\nunder less excusatory circumstances. But they scorched it too!\n\nYou would find the general aspect of America and Americans decidedly\nmuch improved. You would find immeasurably greater consideration and\nrespect for your privacy than of old. You would find a steady change for\nthe better everywhere, except (oddly enough) in the railroads generally,\nwhich seem to have stood still, while everything else has moved. But\nthere is an exception westward. There the express trains have now a very\ndelightful carriage called a \"drawing-room car,\" literally a series of\nlittle private drawing-rooms, with sofas and a table in each, opening\nout of a little corridor. In each, too, is a large plate-glass window,\nwith which you can do as you like. As you pay extra for this luxury, it\nmay be regarded as the first move towards two classes of passengers.\nWhen the railroad straight away to San Francisco (in six days) shall be\nopened through, it will not only have these drawing-rooms, but\nsleeping-rooms too; a bell in every little apartment communicating with\na steward's pantry, a restaurant, a staff of servants, marble\nwashing-stands, and a barber's shop! I looked into one of these cars a\nday or two ago, and it was very ingeniously arranged and quite complete.\n\nI left Niagara last Sunday, and travelled on to Albany, through three\nhundred miles of flood, villages deserted, bridges broken, fences\ndrifting away, nothing but tearing water, floating ice, and absolute\nwreck and ruin. The train gave in altogether at Utica, and the\npassengers were let loose there for the night. As I was due at Albany, a\nvery active superintendent of works did all he could to \"get Mr. Dickens\nalong,\" and in the morning we resumed our journey through the water,\nwith a hundred men in seven-league boots pushing the ice from before us\nwith long poles. How we got to Albany I can't say, but we got there\nsomehow, just in time for a triumphal \"Carol\" and \"Trial.\" All the\ntickets had been sold, and we found the Albanians in a state of great\nexcitement. You may imagine what the flood was when I tell you that we\ntook the passengers out of two trains that had their fires put out by\nthe water four-and-twenty hours before, and cattle from trucks that had\nbeen in the water I don't know how long, but so long that the sheep had\nbegun to eat each other! It was a horrible spectacle, and the haggard\nhuman misery of their faces was quite a new study. There was a fine\nbreath of spring in the air concurrently with the great thaw; but lo and\nbehold! last night it began to snow again with a strong wind, and to-day\na snowdrift covers this place with all the desolation of winter once\nmore. I never was so tired of the sight of snow. As to sleighing, I have\nbeen sleighing about to that extent, that I am sick of the sound of a\nsleigh-bell.\n\nI have seen all our Boston friends, except Curtis. Ticknor is dead. The\nrest are very little changed, except that Longfellow has a perfectly\nwhite flowing beard and long white hair. But he does not otherwise look\nold, and is infinitely handsomer than he was. I have been constantly\nwith them all, and they have always talked much of you. It is the\nestablished joke that Boston is my \"native place,\" and we hold all sorts\nof hearty foregatherings. They all come to every reading, and are always\nin a most delightful state of enthusiasm. They give me a parting dinner\nat the club, on the Thursday before Good Friday. To pass from Boston\npersonal to New York theatrical, I will mention here that one of the\nproprietors of my New York hotel is one of the proprietors of Niblo's,\nand the most active. Consequently I have seen the \"Black Crook\" and the\n\"White Fawn,\" in majesty, from an arm-chair in the first entrance, P.S.,\nmore than once. Of these astonishing dramas, I beg to report (seriously)\nthat I have found no human creature \"behind\" who has the slightest idea\nwhat they are about (upon my honour, my dearest Macready!), and that\nhaving some amiable small talk with a neat little Spanish woman, who is\nthe _premiere danseuse_, I asked her, in joke, to let me measure her\nskirt with my dress glove. Holding the glove by the tip of the\nforefinger, I found the skirt to be just three gloves long, and yet its\nlength was much in excess of the skirts of two hundred other ladies,\nwhom the carpenters were at that moment getting into their places for a\ntransformation scene, on revolving columns, on wires and \"travellers\" in\niron cradles, up in the flies, down in the cellars, on every description\nof float that Wilmot, gone distracted, could imagine!\n\nI have taken my passage for Liverpool from New York in the Cunarder\n_Russia_, on the 22nd of April. I had the second officer's cabin on deck\ncoming out, and I have the chief steward's cabin on deck going home,\nbecause it will be on the sunny side of the ship. I have experienced\nnothing here but good humour and cordiality. In the autumn and winter I\nhave arranged with Chappells to take my farewell of reading in the\nUnited Kingdom for ever and ever.\n\nI am delighted to hear of Benvenuta's marriage, and I think her husband\na very lucky man. Johnnie has my profound sympathy under his\nexaminatorial woes. The noble boy will give me Gavazzi revised and\nenlarged, I expect, when I next come to Cheltenham. I will give you and\nMrs. Macready all my American experiences when you come to London, or,\nbetter still, to Gad's. Meanwhile I send my hearty love to all, not\nforgetting dear Katie.\n\nNiagara is not at all spoiled by a very dizzy-looking suspension bridge.\nIs to have another still nearer to the Horse-shoe opened in July. My\nlast sight of that scene (last Sunday) was thus: We went up to the\nrapids above the Horse-shoe--say two miles from it--and through the\ngreat cloud of spray. Everything in the magnificent valley--buildings,\nforest, high banks, air, water, everything--was _made of rainbow_.\nTurner's most imaginative drawing in his finest day has nothing in it so\nethereal, so gorgeous in fancy, so celestial. We said to one another\n(Dolby and I), \"Let it for evermore remain so,\" and shut our eyes and\ncame away.\n\nGod bless you and all dear to you, my dear old Friend!\n\n                               I am ever your affectionate and loving.\n\n\n[Sidenote: Miss Dickens.]\n\n                                 PORTLAND, _Sunday, March 29th, 1868._\n\nI should have written to you by the last mail, but I really was too\nunwell to do it. The writing day was last Friday, when I ought to have\nleft Boston for New Bedford (fifty-five miles) before eleven in the\nmorning. But I was so exhausted that I could not be got up, and had to\ntake my chance of an evening's train producing me in time to read, which\nit just did. With the return of snow, nine days ago, the \"true American\"\n(which had lulled) came back as bad as ever. I have coughed from two or\nthree in the morning until five or six, and have been absolutely\nsleepless. I have had no appetite besides, and no taste. Last night here\nI took some laudanum, and it is the only thing that has done me good.\nBut the life in this climate is so very hard. When I did manage to get\nfrom Boston to New Bedford, I read with my utmost force and vigour.\nNext morning, well or ill, I must turn out at seven to get back to\nBoston on my way here.\n\nI dine at Boston at three, and at five must come on here (a hundred and\nthirty miles or so), for to-morrow night; there being no Sunday train.\nTo-morrow night I read here in a very large place, and Tuesday morning\nat six I must start again to get back to Boston once more. But after\nto-morrow night, I have only the Boston and New York farewells, thank\nGod! I am most grateful to think that when we came to devise the details\nof the tour, I foresaw that it could never be done, as Dolby and Osgood\nproposed, by one unassisted man, as if he were a machine. If I had not\ncut out the work, and cut out Canada, I could never have gone there, I\nam quite sure. Even as it is, I have just now written to Dolby (who is\nin New York), to see my doctor there, and ask him to send me some\ncomposing medicine that I can take at night, inasmuch as without sleep I\ncannot get through. However sympathetic and devoted the people are about\nme, they _can not_ be got to comprehend that one's being able to do the\ntwo hours with spirit when the time comes round, may be co-existent with\nthe consciousness of great depression and fatigue. I don't mind saying\nall this, now that the labour is so nearly over. You shall have a\nbrighter account of me, please God, when I close this at Boston.\n\n\n                                                 _Monday, March 30th._\n\nWithout any artificial aid, I got a splendid night's rest last night,\nand consequently am very much freshened up to-day. Yesterday I had a\nfine walk by the sea, and to-day I have had another on the heights\noverlooking it.\n\n\n                                              BOSTON, _Tuesday, 31st._\n\nI have safely arrived here, just in time to add a line to that effect,\nand get this off by to-morrow's English mail from New York. Catarrh\nrather better. Everything triumphant last night, except no sleep again.\nI suppose Dolby to be now on his way back to join me here. I am much\nmistaken if the political crisis do not damage the farewells by almost\none half.\n\nI hope that I am certainly better altogether.\n\nMy room well decorated with flowers, of course, and Mr. and Mrs. Fields\ncoming to dinner. They are the most devoted of friends, and never in the\nway and never out of it.\n\n\n[Sidenote: Miss Hogarth.]\n\n                                 BOSTON, _Wednesday, April 1st, 1868._\n\nI received your letter of from the 14th to the 17th of March, here, last\nnight. My New York doctor has prescribed for me promptly, and I hope I\nam better. I am certainly no worse. We shall do (to the best of my\nbelief) _very well_ with the farewells here and at New York, but not\ngreatly. Everything is at a standstill, pending the impeachment and the\nnext presidential election. I forgot whether I told you that the New\nYork press are going to give me a public dinner, on Saturday, the 18th.\n\nI hear (but not from himself) that Wills has had a bad fall in hunting,\nand is, or has been, laid up. I am supposed, I take it, not to know this\nuntil I hear it from himself.\n\n\n_Thursday._\n\nMy notion of the farewells is pretty certain now to turn out right. It\nis not at all probable that we shall do anything enormous. Every pulpit\nin Massachusetts will resound to violent politics to-day and to-night.\nYou remember the Hutchinson family?[24] I have had a grateful letter\nfrom John Hutchinson. He speaks of \"my sister Abby\" as living in New\nYork. The immediate object of his note is to invite me to the marriage\nof his daughter, twenty-one years of age.\n\nYou will see by the evidence of this piece of paper that I am using up\nmy stationery. Scott has just been making anxious calculations as to our\npowers of holding out in the articles of tooth-powder, etc. The\ncalculations encourage him to believe that we shall just hold out, and\nno more. I think I am still better to-day than I was yesterday; but I am\nfar from strong, and have no appetite. To see me at my little table at\nnight, you would think me the freshest of the fresh. And this is the\nmarvel of Fields' life.\n\nI don't forget that this is Forster's birthday.\n\n\n                                              _Friday Afternoon, 3rd._\n\nCatarrh worse than ever! And we don't know (at four) whether I can read\nto-night or must stop. Otherwise all well.\n\n\n[Sidenote: Miss Dickens.]\n\n                                   BOSTON, _Tuesday, April 7th, 1868._\n\nI not only read last Friday, when I was doubtful of being able to do so,\nbut read as I never did before, and astonished the audience quite as\nmuch as myself. You never saw or heard such a scene of excitement.\n\nLongfellow and all the Cambridge men urged me to give in. I have been\nvery near doing so, but feel stronger to-day. I cannot tell whether the\ncatarrh may have done me any lasting injury in the lungs or other\nbreathing organs, until I shall have rested and got home. I hope and\nbelieve not. Consider the weather. There have been two snowstorms since\nI wrote last, and to-day the town is blotted out in a ceaseless whirl of\nsnow and wind.\n\nI cannot eat (to anything like the ordinary extent), and have\nestablished this system: At seven in the morning, in bed, a tumbler of\nnew cream and two tablespoonsful of rum. At twelve, a sherry cobbler and\na biscuit. At three (dinner time), a pint of champagne. At five minutes\nto eight, an egg beaten up with a glass of sherry. Between the parts,\nthe strongest beef tea that can be made, drunk hot. At a quarter-past\nten, soup, and anything to drink that I can fancy. I don't eat more than\nhalf a pound of solid food in the whole four-and-twenty hours, if so\nmuch.\n\nIf I hold out, as I hope to do, I shall be greatly pressed in leaving\nhere and getting over to New York before next Saturday's mail from\nthere. Do not, therefore, _if all be well_, expect to hear from me by\nSaturday's mail, but look for my last letter from America by the mail of\nthe following Wednesday, the 15th. _Be sure_ that you shall hear,\nhowever, by Saturday's mail, if I should knock up as to reading. I am\ntremendously \"beat,\" but I feel really and unaffectedly so much stronger\nto-day, both in my body and hopes, that I am much encouraged. I have a\nfancy that I turned my worst time last night.\n\nDolby is as tender as a woman and as watchful as a doctor. He never\nleaves me during the reading now, but sits at the side of the platform\nand keeps his eye upon me all the time. Ditto George, the gasman,\nsteadiest and most reliable man I ever employed. I am the more hopeful\nof my not having to relinquish a reading, because last night was\n\"Copperfield\" and \"Bob\"--by a quarter of an hour the longest, and, in\nconsideration of the storm, by very much the most trying. Yet I was far\nfresher afterwards than I have been these three weeks.\n\nI have \"Dombey\" to do to-night, and must go through it carefully; so\nhere ends my report. The personal affection of the people in this place\nis charming to the last.\n\n\n[Sidenote: The Hon. Mrs. Watson.]\n\n                           GAD'S HILL PLACE, _Monday, May 11th, 1868._\n\nMY DEAR MRS. WATSON,\n\nI am delighted to have your letter. It comes to me like a faithful voice\nfrom dear old Rockingham, and awakens many memories.\n\nThe work in America has been so very hard, and the winter there has been\nso excessively severe, that I really have been very unwell for some\nmonths. But I had not been at sea three days on the passage home when I\nbecame myself again.\n\nIf you will arrange with Mary Boyle any time for coming here, we shall\nbe charmed to see you, and I will adapt my arrangements accordingly. I\nmake this suggestion because she generally comes here early in the\nsummer season. But if you will propose yourself _anyhow_, giving me a\nmargin of a few days in case of my being pre-engaged for this day or\nthat, we will (as my American friends say) \"fix it.\"\n\nWhat with travelling, reading night after night, and speech-making day\nafter day, I feel the peace of the country beyond all expression. On\nboard ship coming home, a \"deputation\" (two in number, of whom only one\ncould get into my cabin, while the other looked in at my window) came to\nask me to read to the passengers that evening in the saloon. I\nrespectfully replied that sooner than do it, I would assault the\ncaptain, and be put in irons.\n\n                                            Ever affectionately yours.\n\n\n[Sidenote: Mrs. George Cattermole.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                           _Saturday, May 16th, 1868._\n\nMY DEAR MRS. CATTERMOLE,\n\nOn my return from America just now, I accidentally heard that George had\nbeen ill. My sister-in-law had heard it from Forster, but vaguely. Until\nI received your letter of Wednesday's date, I had no idea that he had\nbeen very ill; and should have been greatly shocked by knowing it, were\nit not for the hopeful and bright assurance you give me that he is\ngreatly better.\n\nMy old affection for him has never cooled. The last time he dined with\nme, I asked him to come again that day ten years, for I was perfectly\ncertain (this was my small joke) that I should not set eyes upon him\nsooner. The time being fully up, I hope you will remind him, with my\nlove, that he is due. His hand is upon these walls here, so I should\nlike him to see for himself, and _you_ to see for _yourself_, and in\nthis hope I shall pursue his complete recovery.\n\nI heartily sympathise with you in your terrible anxiety, and in your\nvast relief; and, with many thanks for your letter, am ever, my dear\nMrs. Cattermole,\n\n                                                 Affectionately yours.\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n                             GAD'S HILL, _Wednesday, June 10th, 1868._\n\nMY DEAREST MACREADY,\n\nSince my return from America, I have been so overwhelmed with business\nthat I have not had time even to write to you. You may imagine what six\nmonths of arrear are to dispose of; added to this, Wills has received a\nconcussion of the brain (from an accident in the hunting-field), and is\nsent away by the doctors, and strictly prohibited from even writing a\nnote. Consequently all the business and money details of \"All the Year\nRound\" devolve upon me. And I have had to get them up, for I have never\nhad experience of them. Then I am suddenly entreated to go to Paris, to\nlook after the French version of \"No Thoroughfare\" on the stage. And I\ngo, and come back, leaving it a great success.\n\nI hope Mrs. Macready and you have not abandoned the idea of coming here?\nThe expression of this hope is the principal, if not the only, object of\nthis present note. May the amiable secretary vouchsafe a satisfactory\nreply!\n\nKatie, Mary, and Georgina send their very best love to your Katie and\nMrs. Macready. The undersigned is in his usual brilliant condition, and\nindeed has greatly disappointed them at home here, by coming back \"so\nbrown and looking so well.\" They expected a wreck, and were, at first,\nmuch mortified. But they are getting over it now.\n\nTo my particular friends, the noble boy and Johnny, I beg to be warmly\nremembered.\n\n                                Ever, my dearest Macready,\n                                               Your most affectionate.\n\n\n[Sidenote: Mrs. Henry Austin.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                           _Tuesday, July 21st, 1868._\n\n              ON THE DEATH OF MR. HENRY AUSTIN.[25]\n\nMY DEAR LETITIA,\n\nYou will have had a telegram from me to-day. I received your sad news by\nthis morning's post. They never, without express explanation, mind\n\"Immediate\" on a letter addressed to the office, because half the people\nwho write there on business that does not press, or on no business at\nall, so mark their letters.\n\nOn Thursday I have people to see and matters to attend to, both at the\noffice and at Coutts', which, in Wills's absence, I cannot forego or\ndepute to another. But, _between ourselves_, I must add something else:\nI have the greatest objection to attend a funeral in which my affections\nare not strongly and immediately concerned. I have no notion of a\nfuneral as a matter of form or ceremony. And just as I should expressly\nprohibit the summoning to my own burial of anybody who was not very near\nor dear to me, so I revolt from myself appearing at that solemn rite\nunless the deceased were very near or dear to me. I cannot endure being\ndressed up by an undertaker as part of his trade show. I was not in this\npoor good fellow's house in his lifetime, and I feel that I have no\nbusiness there when he lies dead in it. My mind is penetrated with\nsympathy and compassion for the young widow, but that feeling is a real\nthing, and my attendance as a mourner would not be--to myself. It would\nbe to you, I know, but it would not be to myself. I know full well that\nyou cannot delegate to me your memories of and your associations with\nthe deceased, and the more true and tender they are the more invincible\nis my objection to become a form in the midst of the most awful\nrealities.\n\nWith love and condolence from Georgina, Mary, and Katie,\n\n                           Believe me, ever your affectionate Brother.\n\n\n[Sidenote: Mrs. George Cattermole.]\n\n                             GAD'S HILL, _Wednesday, July 22nd, 1868._\n\nMY DEAR MRS. CATTERMOLE,\n\nOf course I will sign your memorial to the Academy. If you take either\nof the Landseers, certainly take Edwin (1, St. John's Wood Road, N.W.)\nBut, if you would be content with Frith, I have already spoken to him,\nand believe that I can answer for him. I shall be at \"All the Year\nRound\" Office, 26, Wellington Street, London, to-morrow, from eleven to\nthree. Frith will be here on Saturday, and I shall be here too. I spoke\nto him a fortnight ago, and I found him most earnest in the cause. He\nsaid he felt absolutely sure that the whole profession in its best and\nhighest representation would do anything for George. I sounded him,\nhaving the opportunity of meeting him at dinner at Cartwright's.\n\n                                            Ever yours affectionately.\n\n\n[Sidenote: Mr. W. H. Wills.]\n\n                                            _Friday, July 31st, 1868._\n\nMY DEAR WILLS,\n\nI had such a hard day at the office yesterday, that I had not time to\nwrite to you before I left. So I write to-day.\n\nI am very unwilling to abandon the Christmas number, though even in the\ncase of my little Christmas books (which were immensely profitable) I\nlet the idea go when I thought it was wearing out. Ever since I came\nhome, I have hammered at it, more or less, and have been uneasy about\nit. I have begun something which is very droll, but it manifestly shapes\nitself towards a book, and could not in the least admit of even that\nshadowy approach to a congruous whole on the part of other contributors\nwhich they have ever achieved at the best. I have begun something else\n(aboard the American mail-steamer); but I don't like it, because the\nstories must come limping in after the old fashion, though, of course,\nwhat I _have_ done will be good for A. Y. R. In short, I have cast about\nwith the greatest pains and patience, and I have been wholly unable to\nfind what I want.\n\nAnd yet I cannot quite make up my mind to give in without another fight\nfor it. I offered one hundred pounds reward at Gad's to anybody who\ncould suggest a notion to satisfy me. Charles Collins suggested one\nyesterday morning, in which there is _something_, though not much. I\nwill turn it over and over, and try a few more starts on my own account.\nFinally, I swear I will not give it up until August is out. Vow\nregistered.\n\nI am clear that a number by \"various writers\" would not do. If we have\nnot the usual sort of number, we must call the current number for that\ndate the Christmas number, and make it as good as possible.\n\nI sit in the Chalet,[26] like Mariana in the Moated Grange, and to as\nmuch purpose.\n\nI am buying the freehold of the meadow at Gad's, and of an adjoining\narable field, so that I shall now have about eight-and-twenty freehold\nacres in a ring-fence. No more now.\n\nI made up a very good number yesterday. You will see in it a very short\narticle that I have called \"Now!\" which is a highly remarkable piece of\ndescription. It is done by a new man, from whom I have accepted another\narticle; but he will never do anything so good again.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: M. de Cerjat.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                         _Wednesday, Aug. 26th, 1868._\n\nMY DEAR CERJAT,\n\nI was happy to receive your esteemed letter a few days ago.\n\nThe severity of the winter in America (which was quite exceptional even\nin that rigorous climate), combined with the hard work I had to do,\ntried me a good deal. Neuralgia and colds beset me, either by turns or\nboth together, and I had often much to do to get through at night. But\nthe sea voyage home again did wonders in restoring me, and I have been\nvery well indeed, though a little fatigued, ever since. I am now\npreparing for a final reading campaign in England, Scotland, and\nIreland. It will begin on the 6th of October, and will probably last,\nwith short occasional intermissions, until June.\n\nThe great subject in England for the moment is the horrible accident to\nthe Irish mail-train. It is now supposed that the petroleum (known to be\na powerful anaesthetic) rendered the unfortunate people who were burnt\nalmost instantly insensible to any sensation. My escape in the\nStaplehurst accident of three years ago is not to be obliterated from my\nnervous system. To this hour I have sudden vague rushes of terror, even\nwhen riding in a hansom cab, which are perfectly unreasonable but quite\ninsurmountable. I used to make nothing of driving a pair of horses\nhabitually through the most crowded parts of London. I cannot now drive,\nwith comfort to myself, on the country roads here; and I doubt if I\ncould ride at all in the saddle. My reading secretary and companion\nknows so well when one of these odd momentary seizures comes upon me in\na railway carriage, that he instantly produces a dram of brandy, which\nrallies the blood to the heart and generally prevails. I forget whether\nI ever told you that my watch (a chronometer) has never gone exactly\nsince the accident? So the Irish catastrophe naturally revives the\ndreadful things I saw that day.\n\nThe only other news here you know as well as I; to wit, that the country\nis going to be ruined, and that the Church is going to be ruined, and\nthat both have become so used to being ruined, that they will go on\nperfectly well.\n\n       *       *       *       *       *\n\n\n[Sidenote: Miss Dickens.]\n\n     OFFICE OF \"ALL THE YEAR ROUND,\" NO. 26, WELLINGTON STREET,\n                                  STRAND, LONDON, W.C.,\n                                         _Saturday, Sept. 26th, 1868._\n\nMY DEAREST MAMIE,\n\nI will add a line to this at the Athenaeum, after seeing Plorn off, to\ntell you how he went away.\n\n\n                                           ATHENAEUM, _Quarter to Six._\n\nI can honestly report that he went away, poor dear fellow, as well as\ncould possibly be expected. He was pale, and had been crying, and (Harry\nsaid) had broken down in the railway carriage after leaving Higham\nstation; but only for a short time.\n\nJust before the train started he cried a good deal, but not painfully.\n(Tell dear Georgy that I bought him his cigars.) These are hard, hard\nthings, but they might have to be done without means or influence, and\nthen they would be far harder. God bless him!\n\n\n        PARLIAMENT. REPLY TO A PROPOSAL MADE THROUGH\n        ALEXANDER RUSSEL, OF \"THE SCOTSMAN,\" THAT HE\n        SHOULD ALLOW HIMSELF TO BE PUT FORWARD AS A\n        CANDIDATE FOR THE REPRESENTATION OF EDINBURGH.\n\n[Sidenote: Mr. F. D. Finlay.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                             _Sunday, Oct. 4th, 1868._\n\nMY DEAR FINLAY,\n\nI am much obliged to you in all friendship and sincerity for your\nletter. I have a great respect for your father-in-law and his paper, and\nI am much attached to the Edinburgh people. You may suppose, therefore,\nthat if my mind were not fully made up on the parliamentary question, I\nshould waver now.\n\nBut my conviction that I am more useful and more happy as I am than I\ncould ever be in Parliament is not to be shaken. I considered it some\nweeks ago, when I had a stirring proposal from the Birmingham people,\nand I then set it up on a rock for ever and a day.\n\nDo tell Mr. Russel that I truly feel this mark of confidence, and that I\nhope to acknowledge it in person in Edinburgh before Christmas. There is\nno man in Scotland from whom I should consider his suggestion a greater\nhonour.\n\n                                                           Ever yours.\n\n\n[Sidenote: M. Charles Fechter.]\n\n       *       *       *       *       *\n\nPoor Plorn is gone to Australia. It was a hard parting at the last. He\nseemed to me to become once more my youngest and favourite little child\nas the day drew near, and I did not think I could have been so shaken.\nYou were his idol to the hour of his departure, and he asked me to tell\nyou how much he wanted to bid you good-bye.\n\nKindest love from all.\n\n                                                        Ever heartily.\n\n\n[Sidenote: The same.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                          _Wednesday, Oct. 7th, 1868._\n\nMY DEAR FECHTER,\n\nI got your letter sent to Gad's Hill this morning. Until I received it,\nI supposed the piece to have been put into English from your French by\nyoung Ben. If I understand that the English is yours, then I say that it\nis extraordinarily good, written by one in another country.\n\nI do not read again in London until the 20th; and then \"Copperfield.\"\nBut by that time you will be at work yourself.\n\nLet us dine at six to-day, in order that we may not have to hurry for\nthe comic dog.\n\n                                                      Ever faithfully.\n\n\n[Sidenote: Miss Hogarth.]\n\n                 QUEEN'S HOTEL, MANCHESTER, _Sunday, Oct. 11th, 1868._\n\nMY DEAREST GEORGY,\n\nWe had a fine audience last night in the Free Trade Hall, though not\nwhat we consider a large money-house. The let in Liverpool is extremely\ngood, and we are going over there at half-past one. We got down here\npleasantly enough and in good time; so all has gone well you see.\n\nTitiens, Santley, and an opera company of that class are at the theatre\nhere. They have been doing very poorly in Manchester.\n\nThere is the whole of my scanty news. I was in wonderful voice last\nnight, but croak a little this morning, after so much speaking in so\nvery large a place. Otherwise I am all right. I find myself constantly\nthinking of Plorn.\n\n\n[Sidenote: Miss Dickens.]\n\n                  ADELPHI HOTEL, LIVERPOOL, _Monday, Oct. 12th, 1868._\n\nMY DEAREST MAMIE,\n\nOur lets here are excellent, and we shall have a great house to-night.\nWe had a very fine and enthusiastic audience in the Free Trade Hall, at\nManchester, on Saturday; but our first nights there never count up in\nmoney, as the rest do. Yesterday, \"Charlotte,\" Sainton, and Piatti\nstayed with us here; and they went on to Hull this morning. It was\npleasant to be alone again, though they were all very agreeable.\n\nThe exertion of going on for two hours in that immense place at\nManchester being very great, I was hoarse all day yesterday, though I\nwas not much distressed on Saturday night. I am becoming melodious again\n(at three in the afternoon) rapidly, and count on being quite restored\nby a basin of turtle at dinner.\n\nI am glad to hear about Armatage, and hope that a service begun in a\npersonal attachment to Plorn may go on well. I shall never be\nover-confident in such matters, I think, any more.\n\nThe day is delicious here. We have had a blow on the Mersey this\nmorning, and exulted over the American steamers. With kind regard to Sir\nWilliam and Lady Humphery.\n\n\n[Sidenote: Miss Hogarth.]\n\n                 ADELPHI HOTEL, LIVERPOOL, _Tuesday, Oct. 13th, 1868._\n\nAs I sent a line to Mary yesterday, I enclose you Alfred's letter.\nPlease send it on to her when you next write to Penton.\n\nI have just now written to Mrs. Forster, asking her to explain to Miss\nForster how she could have an easy-chair or a sofa behind my side screen\non Tuesday, without occasioning the smallest inconvenience to anybody.\nAlso, how she would have a door close at hand, leading at once to cool\npassages and a quiet room, etc. etc. etc. It is a sad story.\n\nWe had a fine house here last night, and a large turn-away. \"Marigold\"\nand \"Trial\" went immensely. I doubt if \"Marigold\" were ever more\nenthusiastically received. \"Copperfield\" and \"Bob\" to-night, and a large\nlet. This notwithstanding election meetings and all sorts of things.\n\nMy favourite room brought my voice round last night, and I am in\nconsiderable force.\n\nDolby sends kindest regard, and the message: \"Everton toffee shall not\nbe forgotten.\"\n\n\n[Sidenote: Mr. Henry Fielding Dickens.]\n\n                ADELPHI HOTEL, LIVERPOOL, _Thursday, Oct. 15th, 1868._\n\nMY DEAR HARRY,\n\nI have your letter here this morning. I enclose you another cheque for\ntwenty-five pounds, and I write to London by this post, ordering three\ndozen sherry, two dozen port, and three dozen light claret, to be sent\ndown to you.\n\nNow, observe attentively. We must have no shadow of debt. Square up\neverything whatsoever that it has been necessary to buy. Let not a\nfarthing be outstanding on any account, when we begin together with your\nallowance. Be particular in the minutest detail.\n\nI wish to have no secret from you in the relations we are to establish\ntogether, and I therefore send you Joe Chitty's letter bodily. Reading\nit, you will know exactly what I know, and will understand that I treat\nyou with perfect confidence. It appears to me that an allowance of two\nhundred and fifty pounds a year will be handsome for all your wants, if\nI send you your wines. I mean this to include your tailor's bills as\nwell as every other expense; and I strongly recommend you to buy nothing\nin Cambridge, and to take credit for nothing but the clothes with which\nyour tailor provides you. As soon as you have got your furniture\naccounts in, let us wipe all those preliminary expenses clean out, and I\nwill then send you your first quarter. We will count in it October,\nNovember, and December; and your second quarter will begin with the New\nYear. If you dislike, at first, taking charge of so large a sum as\nsixty-two pounds ten shillings, you can have your money from me\nhalf-quarterly.\n\nYou know how hard I work for what I get, and I think you know that I\nnever had money help from any human creature after I was a child. You\nknow that you are one of many heavy charges on me, and that I trust to\nyour so exercising your abilities and improving the advantages of your\npast expensive education, as soon to diminish _this_ charge. I say no\nmore on that head.\n\nWhatever you do, above all other things keep out of debt and confide in\nme. If you ever find yourself on the verge of any perplexity or\ndifficulty, come to me. You will never find me hard with you while you\nare manly and truthful.\n\nAs your brothers have gone away one by one, I have written to each of\nthem what I am now going to write to you. You know that you have never\nbeen hampered with religious forms of restraint, and that with mere\nunmeaning forms I have no sympathy. But I most strongly and\naffectionately impress upon you the priceless value of the New\nTestament, and the study of that book as the one unfailing guide in\nlife. Deeply respecting it, and bowing down before the character of our\nSaviour, as separated from the vain constructions and inventions of men,\nyou cannot go very wrong, and will always preserve at heart a true\nspirit of veneration and humility. Similarly I impress upon you the\nhabit of saying a Christian prayer every night and morning. These things\nhave stood by me all through my life, and remember that I tried to\nrender the New Testament intelligible to you and lovable by you when you\nwere a mere baby.\n\nAnd so God bless you.\n\n                                        Ever your affectionate Father.\n\n\n[Sidenote: Mr. William Charles Kent.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                            _Monday, Nov. 16th, 1868._\n\nMY DEAR KENT,\n\nI was on the eve of writing to you.\n\nWe thought of keeping the trial private; but Oxenford has suggested to\nChappell that he would like to take the opportunity of to-morrow night's\nreading, of saying something about \"Oliver\" in _Wednesday's paper_.\nChappell has told Levy of this, and also Mr. Tompkin, of _The Post_,\nwho was there. Consequently, on Wednesday evening your charming article\ncan come out to the best advantage.\n\nYou have no idea of the difficulty of getting in the end of Sikes. As to\nthe man with the invaluable composition! my dear fellow, believe me, no\naudience on earth could be held for ten minutes after the girl's death.\nGive them time, and they would be revengeful for having had such a\nstrain put upon them. Trust me to be right. I stand there, and I know.\n\nConcerning Harry, I like to guide the boys to a distinct choice, rather\nthan to press it on them. That will be my course as to the Middle\nTemple, of which I think as you do.\n\nWith cordial thanks for every word in your letter,\n\n                                          Affectionately yours always.\n\n\n[Sidenote: Mrs. F. Lehmann.]\n\n                 KENNEDY'S HOTEL, EDINBURGH, _Sunday, Dec. 6th, 1868._\n\nMY DEAR MRS. LEHMANN,\n\nI hope you will see Nancy with the light of a great audience upon her\nsome time between this and May; always supposing that she should not\nprove too weird and woeful for the general public.\n\nYou know the aspect of this city on a Sunday, and how gay and bright it\nis. The merry music of the blithe bells, the waving flags, the\nprettily-decorated houses with their draperies of various colours, and\nthe radiant countenances at the windows and in the streets, how charming\nthey are! The usual preparations are making for the band in the open\nair, in the afternoon; and the usual pretty children (selected for that\npurpose) are at this moment hanging garlands round the Scott monument,\npreparatory to the innocent Sunday dance round that edifice, with which\nthe diversions invariably close. It is pleasant to think that these\ncustoms were themselves of the early Christians, those early birds who\n_didn't_ catch the worm--and nothing else--and choke their young with\nit.\n\n                                              Faithfully yours always.\n\n\n[Sidenote: Miss Hogarth.]\n\n                 KENNEDY'S HOTEL, EDINBURGH, _Sunday, Dec. 6th, 1868._\n\nWe got down here to our time to the moment; and, considering the length\nof the journey, very easily. I made a calculation on the road, that the\nrailway travelling over such a distance involves something more than\nthirty thousand shocks to the nerves. Dolby didn't like it at all.\n\nThe signals for a gale were up at Berwick, and along the road between\nthere and here. It came on just as we arrived, and blew tremendously\nhard all night. The wind is still very high, though the sky is bright\nand the sun shining. We couldn't sleep for the noise.\n\nWe are very comfortably quartered. I fancy that the \"business\" will be\non the whole better here than in Glasgow, where trade is said to be very\nbad. But I think I shall be pretty correct in both places as to the run\nbeing on the final readings.\n\nWe are going up Arthur's Seat presently, which will be a pull for our\nfat friend.\n\nScott, in a new Mephistopheles hat, baffles imagination and description.\n\n\n[Sidenote: Mr. W. Wilkie Collins.]\n\n                KENNEDY'S HOTEL, EDINBURGH, _Tuesday, Dec. 8th, 1868._\n\nMY DEAR WILKIE,\n\nI am hard at it here as usual, though with an audience so finely\nperceptive that the labour is much diminished. I have got together in a\nvery short space the conclusion of \"Oliver Twist\" that you suggested,\nand am trying it daily with the object of rising from that blank state\nof horror into a fierce and passionate rush for the end. As yet I cannot\nmake a certain effect of it; but when I shall have gone over it as many\nscore of times as over the rest of that reading, perhaps I may strike\none out.\n\nI shall be very glad to hear when you have done your play, and I _am_\nglad to hear that you like the steamer. I agree with you about the\nreading perfectly. In No. 3 you will see an exact account of some places\nI visited at Ratcliffe. There are two little instances in it of\nsomething comic rising up in the midst of the direst misery, that struck\nme very humorously at the time.\n\nAs I have determined not to do the \"Oliver Murder\" until after the 5th\nof January, when I shall ascertain its effect on a great audience, it is\ncurious to notice how the shadow of its coming affects the Scotch mind.\nThere was such a disposition to hold back for it here (until I return to\nfinish in February) that we had next to no \"let\" when we arrived. It all\ncame with a rush yesterday. They gave me a most magnificent welcome back\nfrom America last night.\n\nI am perpetually counting the weeks before me to be \"read\" through, and\nam perpetually longing for the end of them; and yet I sometimes wonder\nwhether I shall miss something when they are over.\n\nIt is a very, very bad day here, very dark and very wet. Dolby is over\nat Glasgow, and I am sitting at a side window looking up the length of\nPrince's Street, watching the mist change over the Castle and murdering\nNancy by turns.\n\n                                                  Ever affectionately.\n\nP.S.--I have read the whole of Fitzgerald's \"Zero,\" and the idea is\nexceedingly well wrought out.\n\n\n[Sidenote: Miss Hogarth.]\n\n              KENNEDY'S HOTEL, EDINBURGH, _Saturday, Dec. 12th, 1868._\n\nI send another _Scotsman_ by this post, because it is really a good\nnewspaper, well written, and well managed. We had an immense house here\nlast night, and a very large turn-away.\n\nWe have four guests to dinner to-day: Peter Fraser, Ballantyne, John\nBlackwood, and Mr. Russel. Immense preparations are making in the\nestablishment, \"on account,\" Mr. Kennedy says, \"of a' four yon chiels\nbeing chiels wha' ken a guid dinner.\" I enquired after poor Doctor Burt,\nnot having the least idea that he was dead.\n\nMy voice holds out splendidly so far, and I have had no return of the\nAmerican. But I sleep very indifferently indeed.\n\nIt blew appallingly here the night before last, but the wind has since\nshifted northward, and it is now bright and cold. The _Star of Hope_,\nthat picked up those shipwrecked people in the boat, came into Leith\nyesterday, and was received with tremendous cheers. Her captain must be\na good man and a noble fellow.\n\n\n[Sidenote: The same.]\n\n                KENNEDY'S HOTEL, EDINBURGH, _Monday, Dec. 14th, 1868._\n\nThe dinner-party of Saturday last was an immense success. Russel swore\non the occasion that he would go over to Belfast expressly to dine with\nme at the Finlays'. Ballantyne informed me that he was going to send you\nsome Scotch remembrance (I don't know what) at Christmas!\n\nThe Edinburgh houses are very fine. The Glasgow room is a big wandering\nplace, with five prices in it, which makes it the more aggravating, as\nthe people get into knots which they can't break, as if they were afraid\nof one another.\n\nForgery of my name is becoming popular. You sent me, this morning, a\nletter from Russell Sturgis, answering a supposed letter of mine\n(presented by \"Miss Jefferies\"), and assuring me of his readiness to\ngive not only the ten pounds I asked for, but any contribution I wanted,\ntowards sending that lady and her family back to Boston.\n\nI wish you would take an opportunity of forewarning Lady Tennent that\nthe first night's reading she will attend is an experiment quite out of\nthe way, and that she may find it rather horrible.\n\nThe keeper of the Edinburgh Hall, a fine old soldier, presented me, on\nFriday night, with the finest red camellia for my button-hole that ever\nwas seen. Nobody can imagine how he came by it, as the florists had had\na considerable demand for that colour from ladies in the stalls, and\ncould get no such thing.\n\nThe day is dark, wet, and windy. The weather is likely to be vile indeed\nat Glasgow, where it always rains, and where the sun is never seen\nthrough the smoke. We go over there to-morrow at ten.\n\n\n[Sidenote: Miss Dickens.]\n\n                                 CARRICK'S ROYAL HOTEL, GLASGOW,\n                                           _Tuesday, Dec. 15th, 1868._\n\nIt occurs to me that my table at St. James's Hall might be appropriately\nornamented with a little holly next Tuesday. If the two front legs were\nentwined with it, for instance, and a border of it ran round the top of\nthe fringe in front, with a little sprig by way of bouquet at each\ncorner, it would present a seasonable appearance.\n\nIf you will think of this, and will have the materials ready in a little\nbasket, I will call for you at the office at half-past twelve on\nTuesday, and take you up to the hall, where the table will be ready for\nyou.\n\nNo news, except that we had a great crush and a wonderful audience in\nEdinburgh last night.\n\n\n[Sidenote: Miss Hogarth.]\n\n                                 CARRICK'S ROYAL HOTEL, GLASGOW,\n                                         _Wednesday, Dec. 16th, 1868._\n\nThis is to report all well, except that I have wretched nights. The\nweather is diabolical here, and times are very bad. I cut \"Copperfield\"\nwith a bold dexterity that amazed myself and utterly confounded George\nat the wing; knocking off that and \"Bob\" by ten minutes to ten.\n\nI don't know anything about the Liverpool banquet, except from _The\nTimes_. As I don't finish there in February (as they seem to have\nsupposed), but in April, it may, perhaps, stand over or blow over\naltogether. Such a thing would be a serious addition to the work, and\nyet refusal on my part would be too ungracious.\n\nThe density and darkness of this atmosphere are fearful. I shall be\nheartily glad to start for Edinburgh again on Friday morning.\n\n\n[Sidenote: The same.]\n\n                KENNEDY'S HOTEL, EDINBURGH, _Friday, Dec. 18th, 1868._\n\nI am heartily glad to get back here this afternoon. The day is bright\nand cheerful, and the relief from Glasgow inexpressible. The\naffectionate regard of the people exceeds all bounds, and is shown in\nevery way. The manager of the railway being at the reading the other\nnight, wrote to me next morning, saying that a large saloon should be\nprepared for my journey up, if I would let him know when I purposed\nmaking the journey. On my accepting the offer he wrote again, saying\nthat he had inspected \"our Northern saloons,\" and not finding them so\nconvenient for sleeping in as the best English, had sent up to King's\nCross for the best of the latter; which I would please consider my own\ncarriage as long as I wanted it. The audiences do everything but\nembrace me, and take as much pains with the readings as I do.\n\nI find your Christmas present (just arrived) to be a haggis and\nshortbread!\n\n\n[Sidenote: Mr. J. C. Parkinson.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                                _Christmas Day, 1868._\n\nMY DEAR PARKINSON,\n\nWhen your letter was delivered at \"All the Year Round\" Office yesterday,\nI was attending a funeral. It comes to hand here consequently to-day.\n\nI am diffident of addressing Mr. Gladstone on the subject of your desire\nto be appointed to the vacant Commissionership of Inland Revenue,\nbecause, although my respect for him and confidence in him are second to\nthose of no man in England (a bold word at this time, but a truthful\none), my personal acquaintance with him is very slight. But you may\nmake, through any of your friends, any use you please of this letter,\ntowards the end of bringing its contents under Mr. Gladstone's notice.\n\nIn expressing my conviction that you deserve the place, and are in every\nway qualified for it, I found my testimony upon as accurate a knowledge\nof your character and abilities as anyone can possibly have acquired. In\nmy editorship both of \"Household Words\" and \"All the Year Round,\" you\nknow very well that I have invariably offered you those subjects of\npolitical and social interest to write upon, in which integrity,\nexactness, a remarkable power of generalising evidence and balancing\nfacts, and a special clearness in stating the case, were indispensable\non the part of the writer. My confidence in your powers has never been\nmisplaced, and through all our literary intercourse you have never been\nhasty or wrong. Whatever trust you have undertaken has been so\ncompletely discharged, that it has become my habit to read your proofs\nrather for my own edification than (as in other cases) for the detection\nof some slip here or there, or the more pithy presentation of the\nsubject.\n\nThat your literary work has never interfered with the discharge of your\nofficial duties, I may assume to be at least as well known to your\ncolleagues as it is to me. It is idle to say that if the post were in my\ngift you should have it, because you have had, for some years, most of\nthe posts of high trust that have been at my disposal. An excellent\npublic servant in your literary sphere of action, I should be heartily\nglad if you could have this new opportunity of distinguishing yourself\nin the same character. And this is at least unselfish in me, for I\nsuppose I should then lose you?\n\n                                              Always faithfully yours.\n\n\n[Sidenote: Mr. Edward Bulwer Lytton Dickens.]\n\n          LETTER TO HIS YOUNGEST SON ON HIS DEPARTURE FOR\n                       AUSTRALIA IN 1868.[27]\n\nMY DEAREST PLORN,\n\nI write this note to-day because your going away is much upon my mind,\nand because I want you to have a few parting words from me to think of\nnow and then at quiet times. I need not tell you that I love you dearly,\nand am very, very sorry in my heart to part with you. But this life is\nhalf made up of partings, and these pains must be borne. It is my\ncomfort and my sincere conviction that you are going to try the life for\nwhich you are beat fitted. I think its freedom and wildness more suited\nto you than any experiment in a study or office would ever have been;\nand without that training, you could have followed no other suitable\noccupation.\n\nWhat you have already wanted until now has been a set, steady, constant\npurpose. I therefore exhort you to persevere in a thorough determination\nto do whatever you have to do as well as you can do it. I was not so old\nas you are now when I first had to win my food, and do this out of this\ndetermination, and I have never slackened in it since.\n\nNever take a mean advantage of anyone in any transaction, and never be\nhard upon people who are in your power. Try to do to others, as you\nwould have them do to you, and do not be discouraged if they fail\nsometimes. It is much better for you that they should fail in obeying\nthe greatest rule laid down by our Saviour, than that you should.\n\nI put a New Testament among your books, for the very same reasons, and\nwith the very same hopes that made me write an easy account of it for\nyou, when you were a little child; because it is the best book that ever\nwas or will be known in the world, and because it teaches you the best\nlessons by which any human creature who tries to be truthful and\nfaithful to duty can possibly be guided. As your brothers have gone\naway, one by one, I have written to each such words as I am now writing\nto you, and have entreated them all to guide themselves by this book,\nputting aside the interpretations and inventions of men.\n\nYou will remember that you have never at home been wearied about\nreligious observances or mere formalities. I have always been anxious\nnot to weary my children with such things before they are old enough to\nform opinions respecting them. You will therefore understand the better\nthat I now most solemnly impress upon you the truth and beauty of the\nChristian religion, as it came from Christ Himself, and the\nimpossibility of your going far wrong if you humbly but heartily respect\nit.\n\nOnly one thing more on this head. The more we are in earnest as to\nfeeling it, the less we are disposed to hold forth about it. Never\nabandon the wholesome practice of saying your own private prayers, night\nand morning. I have never abandoned it myself, and I know the comfort of\nit.\n\nI hope you will always be able to say in after life, that you had a kind\nfather. You cannot show your affection for him so well, or make him so\nhappy, as by doing your duty.\n\n                                             Your affectionate Father.\n\nFOOTNOTES:\n\n[20] The Agricultural College, Cirencester.\n\n[21] \"No Thoroughfare.\"\n\n[22] The Mr. H. F. Chorley so often mentioned was the well-known musical\ncritic, and a dear and intimate friend of Charles Dickens and his\nfamily. We have no letters to him, Mr. Chorley having destroyed all his\ncorrespondence before his death.\n\n[23] Mr. Chauncey Hare Townshend. He was one of the dearest friends of\nCharles Dickens and a very constant correspondent; but no letters\naddressed to him are in existence.\n\n[24] An American family of brothers and a sister who came to London to\ngive a musical entertainment shortly after Charles Dickens's return from\nhis first visit to America. He had a great interest in, and liking for,\nthese young people.\n\n[25] Cousin and adopted child of Mr. and Mrs. Austin.\n\n[26] A model of a Swiss chalet, and a present from M. Charles Fechter,\nused by Charles Dickens as a summer writing-room.\n\n[27] This letter has been already published by Mr. Forster in his\n\"Life.\"\n\n\n\n\n1869.\n\nNARRATIVE.\n\n\nThe \"Farewell Readings\" in town and country were resumed immediately\nafter the beginning of this year, and were to have been continued until\nthe end of May. The work was even harder than it had ever been. Charles\nDickens began his country tour in Ireland early in January, and read\ncontinuously in all parts of England and Scotland until the end of\nApril. A public dinner (in commemoration of his last readings in the\ntown) was given to him at Liverpool on the 10th April. Besides all this\nsevere country work, he was giving a series of readings at St. James's\nHall, and reading the \"Murder\" from \"Oliver Twist,\" in London and in the\ncountry, frequently four times a week. In the second week of February, a\nsudden and unusually violent attack of the old trouble in his foot made\nit imperatively necessary to postpone a reading at St. James's Hall, and\nto delay for a day or two his departure for Scotland. The foot continued\nto cause him pain and inconvenience, but, as will be seen from his\nletters, he generally spoke of himself as otherwise well, until he\narrived at Preston, where he was to read on the 22nd of April. The day\nbefore this appointed reading, he writes home of some grave symptoms\nwhich he had observed in himself, and had reported to his doctor, Mr. F.\nCarr Beard. That gentleman, taking alarm at what he considered\n\"indisputable evidences of overwork,\" wisely resolved not to content\nhimself with written consultations, but went down to Preston on the day\nappointed for the reading there, and, after seeing his patient,\nperemptorily stopped it, carried him off to Liverpool, and the next day\nto London. There he consulted Sir Thomas Watson, who entirely\ncorroborated Mr. Beard's opinion. And the two doctors agreed that the\ncourse of readings must be stopped for this year, and that reading,\n_combined with travelling_, must be stopped _for ever_. Charles Dickens\nhad no alternative but to acquiesce in this verdict; but he felt it\nkeenly, not only for himself, but for the sake of the Messrs. Chappell,\nwho showed the most disinterested kindness and solicitude on the\noccasion. He at once returned home to Gad's Hill, and the rest and quiet\nof the country restored him, for the time, to almost his usual condition\nof health and spirits. But it was observed, by all who loved him, that\nfrom this time forth he never regained his old vigour and elasticity.\nThe attack at Preston was the \"beginning of the end!\"\n\nDuring the spring and summer of this year, he received visits from many\ndearly valued American friends. In May, he stayed with his daughter and\nsister-in-law for two or three weeks at the St. James's Hotel,\nPiccadilly, having promised to be in London at the time of the arrival\nof Mr. and Mrs. Fields, of Boston, who visited Europe, accompanied by\nMiss Mabel Lowell (the daughter of the famous American poet) this year.\nBesides these friends, Mr. and Mrs. Childs, of Philadelphia--from whom\nhe had received the greatest kindness and hospitality, and for whom he\nhad a hearty regard--Dr. Fordyce Barker and his son, Mr. Eytinge (an\nillustrator of an American edition of Charles Dickens's works), and Mr.\nBayard Taylor paid visits to Gad's Hill, which were thoroughly enjoyed\nby Charles Dickens and his family. This last summer was a very happy\none. He had the annual summer visitors and parties of his friends in the\nneighbourhood. He was, as usual, projecting improvements in his beloved\ncountry home; one, which he called the \"crowning improvement of all,\"\nwas a large conservatory, which was to be added during the absence of\nthe family in London in the following spring.\n\nThe state of Mr. Wills's health made it necessary for him now to retire\naltogether from the editorship of \"All the Year Round.\" Charles\nDickens's own letters express the regret which he felt at the\ndissolution of this long and always pleasant association. Mr. Wills's\nplace at the office was filled by Charles Dickens's eldest son, now sole\neditor and proprietor of the journal.\n\nIn September Charles Dickens went to Birmingham, accompanied by his son\nHarry, and presided at the opening of the session of (what he calls in\nhis letter to Mr. Arthur Ryland, \"_our_ Institution\") the Midland\nInstitute. He made a speech on education to the young students, and\npromised to go back early in the following year and distribute the\nprizes. In one of the letters which we give to Mr. Ryland, he speaks of\nhimself as \"being in full force again,\" and \"going to finish his\nfarewell readings soon after Christmas.\" He had obtained the sanction of\nSir Thomas Watson to giving twelve readings, _in London only_, which he\nhad fixed for the beginning of the following year.\n\nThe letter to his friend Mr. Finlay, which opens the year, was in reply\nto a proposal for a public banquet at Belfast, projected by the Mayor of\nthat town, and conveyed through Mr. Finlay. This gentleman was at that\ntime proprietor of _The Northern Whig_ newspaper at Belfast, and he was\nson-in-law to Mr. Alexander Russel, editor of _The Scotsman_.\n\nCharles Dickens's letter this New Year to M. de Cerjat was his last.\nThat faithful and affectionate friend died very shortly afterwards.\n\nTo Miss Mary Boyle he writes to acknowledge a New Year's gift, which he\nhad been much touched by receiving from her, at a time when he knew she\nwas deeply afflicted by the sudden death of her brother, Captain\nCavendish Boyle, for whom Charles Dickens had a true regard and\nfriendship.\n\nWhile he was giving his series of London readings in the spring, he\nreceived a numerously signed circular letter from actors and actresses\nof the various London theatres. They were very curious about his new\nreading of the \"Oliver Twist\" murder, and representing to him the\nimpossibility of their attending an evening, requested him to give a\nmorning reading, for their especial benefit. We give his answer,\ncomplying with the request. And the occasion was, to him, a most\ngratifying and deeply interesting one.\n\nThe letter to Mr. Edmund Ollier was in answer to an invitation to be\npresent at the inauguration of a bust of Mr. Leigh Hunt, which was to be\nplaced over his grave at Kensal Green.\n\nThe letter to Mr. Shirley Brooks, the well-known writer, who succeeded\nMr. Mark Lemon as editor of \"Punch,\" and for whom Charles Dickens had a\ncordial regard, was on the subject of a memorial on behalf of Mrs. Peter\nCunningham, whose husband had recently died.\n\nThe \"remarkable story,\" of which he writes to his daughter in August,\nwas called \"An Experience.\" It was written by a lady (who prefers to be\nanonymous) who had been a contributor to \"Household Words\" from its\nfirst starting, and was always highly valued in this capacity by Charles\nDickens.\n\nOur latest letters for this year are in October. One to Mr. Charles\nKent, sympathising with him on a disappointment which he had experienced\nin a business undertaking, and one to Mr. Macready, in which he tells\nhim of his being in the \"preliminary agonies\" of a new book. The first\nnumber of \"Edwin Drood\" was to appear before the end of his course of\nreadings in March; and he was at work so long beforehand with a view to\nsparing himself, and having some numbers ready before the publication of\nthe first one.\n\n\n[Sidenote: Mr. F. D. Finlay.]\n\n                          THE ATHENAEUM (CLUB), _New Year's Day, 1869._\n\nMY DEAR FINLAY,\n\nFirst my heartfelt wishes for many prosperous and happy years. Next, as\nto the mayor's kind intentions. I feel really grateful to him and\ngratified by the whole idea, but acceptance of the distinction on my\npart would be impracticable. My time in Ireland is all anticipated, and\nI could not possibly prolong my stay, because I _must_ be back in London\nto read on Tuesday fortnight, and then must immediately set forth for\nthe West of England. It is not likely, besides, that I shall get through\nthese farewells before the end of May. And the work is so hard, and my\nvoice is so precious, that I fear to add an ounce to the fatigue, or I\nmight be overweighted. The avoidance of gas and crowds when I am not in\nthe act of being cooked before those lights of mine, is an essential\npart of the training to which (as I think you know) I strictly adhere,\nand although I have accepted the Liverpool invitation, I have done so as\nan exception; the Liverpool people having always treated me in our\npublic relations with a kind of personal affection.\n\nI am sincerely anxious that the Mayor of Belfast should know how the\ncase stands with me. If you will kindly set me straight and right, I\nshall be truly obliged to you.\n\nMy sister-in-law has been very unwell (though she is now much better),\nand is recommended a brisk change. As she is a good sailor, I mean to\nbring her to Ireland with me; at which she is highly delighted.\n\n                                                Faithfully yours ever.\n\n\n[Sidenote: M. de Cerjat.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                             _Monday, Jan. 4th, 1869._\n\nMY DEAR CERJAT,\n\nI will answer your question first. Have I done with my farewell\nreadings? Lord bless you, no; and I shall think myself well out of it if\nI get done by the end of May. I have undertaken one hundred and six, and\nhave as yet only vanquished twenty-eight. To-morrow night I read in\nLondon for the first time the \"Murder\" from \"Oliver Twist,\" which I have\nre-arranged for the purpose. Next day I start for Dublin and Belfast. I\nam just back from Scotland for a few Christmas holidays. I go back there\nnext month; and in the meantime and afterwards go everywhere else.\n\nTake my guarantee for it, you may be quite comfortable on the subject of\npapal aspirations and encroachments. The English people are in\nunconquerable opposition to that church. They have the animosity in the\nblood, derived from the history of the past, though perhaps\nunconsciously. But they do sincerely want to win Ireland over if they\ncan. They know that since the Union she has been hardly used. They know\nthat Scotland has _her_ religion, and a very uncomfortable one. They\nknow that Scotland, though intensely anti-papal, perceives it to be\nunjust that Ireland has not _her_ religion too, and has very\nemphatically declared her opinion in the late elections. They know that\na richly-endowed church, forced upon a people who don't belong to it, is\na grievance with these people. They know that many things, but\nespecially an artfully and schemingly managed institution like the\nRomish Church, thrive upon a grievance, and that Rome has thriven\nexceedingly upon this, and made the most of it. Lastly, the best among\nthem know that there is a gathering cloud in the West, considerably\nbigger than a man's hand, under which a powerful Irish-American body,\nrich and active, is always drawing Ireland in that direction; and that\nthese are not times in which other powers would back our holding Ireland\nby force, unless we could make our claim good in proving fair and equal\ngovernment.\n\nPoor Townshend charged me in his will \"to publish without alteration his\nreligious opinions, which he sincerely believed would tend to the\nhappiness of mankind.\" To publish them without alteration is absolutely\nimpossible; for they are distributed in the strangest fragments through\nthe strangest note-books, pocket-books, slips of paper and what not, and\nproduce a most incoherent and tautological result. I infer that he must\nhave held some always-postponed idea of fitting them together. For these\nreasons I would certainly publish nothing about them, if I had any\ndiscretion in the matter. Having none, I suppose a book must be made.\nHis pictures and rings are gone to the South Kensington Museum, and are\nnow exhibiting there.\n\nCharley Collins is no better and no worse. Katie looks very young and\nvery pretty. Her sister and Miss Hogarth (my joint housekeepers) have\nbeen on duty this Christmas, and have had enough to do. My boys are now\nall dispersed in South America, India, and Australia, except Charley,\nwhom I have taken on at \"All the Year Round\" Office, and Henry, who is\nan undergraduate at Trinity Hall, and I hope will make his mark there.\nAll well.\n\nThe Thames Embankment is (faults of ugliness in detail apart) the finest\npublic work yet done. From Westminster Bridge to near Waterloo it is now\nlighted up at night, and has a fine effect. They have begun to plant it\nwith trees, and the footway (not the road) is already open to the\nTemple. Besides its beauty, and its usefulness in relieving the crowded\nstreets, it will greatly quicken and deepen what is learnedly called\nthe \"scour\" of the river. But the Corporation of London and some other\nnuisances have brought the weirs above Twickenham into a very bare and\nunsound condition, and they already begin to give and vanish, as the\nstream runs faster and stronger.\n\nYour undersigned friend has had a few occasional reminders of his \"true\nAmerican catarrh.\" Although I have exerted my voice very much, it has\nnot yet been once touched. In America I was obliged to patch it up\nconstantly.\n\nI like to read your patriarchal account of yourself among your Swiss\nvines and fig-trees. You wouldn't recognise Gad's Hill now; I have so\nchanged it, and bought land about it. And yet I often think that if Mary\nwere to marry (which she won't) I should sell it and go genteelly\nvagabondising over the face of the earth. Then indeed I might see\nLausanne again. But I don't seem in the way of it at present, for the\nolder I get, the more I do and the harder I work.\n\n                                            Yours ever affectionately.\n\n\n[Sidenote: Miss Mary Boyle.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                          _Wednesday, Jan. 6th, 1869._\n\nMY DEAR MARY,\n\nI was more affected than you can easily believe, by the sight of your\ngift lying on my dressing-table on the morning of the new year. To be\nremembered in a friend's heart when it is sore is a touching thing; and\nthat and the remembrance of the dead quite overpowered me, the one being\ninseparable from the other.\n\nYou may be sure that I shall attach a special interest and value to the\nbeautiful present, and shall wear it as a kind of charm. God bless you,\nand may we carry the friendship through many coming years!\n\nMy preparations for a certain murder that I had to do last night have\nrendered me unfit for letter-writing these last few days, or you would\nhave heard from me sooner. The crime being completely off my mind and\nthe blood spilled, I am (like many of my fellow-criminals) in a highly\nedifying state to-day.\n\n                            Ever believe me, your affectionate Friend.\n\n\n[Sidenote: Miss Dickens.]\n\n                                TORQUAY, _Wednesday, Jan. 27th, 1869._\n\nMY DEAREST MAMIE,\n\nWe have been doing immensely.\n\nThis place is most beautiful, though colder now than one would expect.\nThis hotel, an immense place, built among picturesque broken rocks out\nin the blue sea, is quite delicious. There are bright green trees in the\ngarden, and new peas a foot high. Our rooms are _en suite_, all\ncommanding the sea, and each with two very large plate-glass windows.\nEverything good and well served.\n\nA _pantomime_ was being done last night, in the place where I am to read\nto-night. It is something between a theatre, a circus, a riding-school,\na Methodist chapel, and a cow-house. I was so disgusted with its\nacoustic properties on going in to look at it, that the whole\nunfortunate staff have been all day, and now are, sticking up baize and\ncarpets in it to prevent echoes.\n\nI have rarely seen a more uncomfortable edifice than I thought it last\nnight.\n\nAt Clifton, on Monday night, we had a contagion of fainting. And yet the\nplace was not hot. I should think we had from a dozen to twenty ladies\nborne out, stiff and rigid, at various times. It became quite\nridiculous.\n\n\n[Sidenote: Miss Hogarth.]\n\n                                      BATH, _Friday, Jan. 29th, 1869._\n\nMY DEAREST GEORGY,\n\nYou must not trust blank places in my list, because many have been, and\nwill be, gradually filled up. After the Tuesday's reading in London, I\nhave TWO for that same week in the country--Nottingham and Leicester. In\nthe following week I have none; but my arrangements are all at sea as\nyet, for I must somehow and somewhere do an \"Uncommercial\" in that week,\nand I also want to get poor Chauncey's \"opinions\" to the printer.\n\nThis mouldy old roosting-place comes out mouldily as to let of course. I\nhate the sight of the bygone assembly-rooms, and the Bath chairs\ntrundling the dowagers about the streets. As to to-morrow morning in the\ndaylight!----\n\nI have no cold to speak of. Dolby sends kindest regard.\n\n\n[Sidenote: Mrs. Lehmann.]\n\n                                  OFFICE, _Wednesday, Feb. 3rd, 1869._\n\nDEAR MRS. LEHMANN,\n\nBefore getting your kind note, I had written to Lehmann, explaining why\nI cannot allow myself any social pleasure while my farewell task is yet\nunfinished. The work is so very hard, that every little scrap of rest\n_and silence_ I can pick up is precious. And even those morsels are so\nflavoured with \"All the Year Round,\" that they are not quite the genuine\narticle.\n\nJoachim[28] came round to see me at the hall last night, and I told him\nhow sorry I was to forego the pleasure of meeting him (he is a noble\nfellow!) at your pleasant table.\n\nI am glad you are coming to the \"Murder\" on the 2nd of March. (The house\nwill be prodigious.) Such little changes as I have made shall be\ncarefully presented to your critical notice, and I hope will be crowned\nwith your approval. But you are always such a fine audience that I have\nno fear on that head. I saw Chorley yesterday in his own room. A sad and\nsolitary sight. The widowed Drake, with a certain _gin_coherence of\nmanner, presented a blooming countenance and buxom form in the passage;\nso buxom indeed that she was obliged to retire before me like a modest\nstopper, before I could get into the dining decanter where poor Chorley\nreposed.\n\n                                              Faithfully yours always.\n\nP.S.--My love to Rudie.\n\n\n[Sidenote: Miss Hogarth.]\n\n                                 GLASGOW, _Thursday, Feb. 25th, 1869._\n\nI received your letter at Edinburgh this morning. I did not write to you\nyesterday, as there had been no reading on the previous night.\n\nThe foot bears the fatigue wonderfully well, and really occasions me no\ninconvenience beyond the necessity of wearing the big work of art. Syme\nsaw me again this morning, and utterly scouted the gout notion\naltogether. I think the Edinburgh audience understood the \"Murder\"\nbetter last night than any audience that has heard it yet. \"Business\" is\nenormous, and Dolby jubilant.\n\nIt is a most deplorable afternoon here, deplorable even for Glasgow. A\ngreat wind blowing, and sleet driving before it in a storm of heavy\nblobs. We had to drive our train dead in the teeth of the wind, and got\nin here late, and are pressed for time.\n\nStrange that in the North we have had absolutely no snow. There was a\nvery thin scattering on the Pentlands for an hour or two, but no more.\n\n\n[Sidenote: Miss Hogarth.]\n\n                                 EDINBURGH, _Friday, Feb. 26th, 1869._\n\nWriting to-morrow morning would be all but impracticable for me; would\nbe quite so for Dolby, who has to go to the agents and \"settle up\" in\nthe midst of his breakfast. So I write to-day, in reply to your note\nreceived at Glasgow this morning.\n\nThe foot conducts itself splendidly. We had a most enormous cram at\nGlasgow. Syme saw me again yesterday (before I left here for Glasgow),\nand repeated \"Gout!\" with the greatest indignation and contempt, several\ntimes. The aching is going off as the day goes on, if it be worth\nmentioning again. The ride from Glasgow was charming this morning; the\nsun shining brilliantly, and the country looking beautiful.\n\nI told you what the Nortons were. Mabel Lowell is a charming little\nthing, and very retiring in manner and expression.\n\nWe shall have a scene here to-night, no doubt. The night before last,\nBallantyne, unable to get in, had a seat behind the screen, and was\nnearly frightened off it by the \"Murder.\" Every vestige of colour had\nleft his face when I came off, and he sat staring over a glass of\nchampagne in the wildest way. I have utterly left off _my_ champagne,\nand, I think, with good results. Nothing during the readings but a very\nlittle weak iced brandy-and-water.\n\nI hope you will find me greatly improved on Tuesday.\n\n\n[Sidenote: Miss Dickens.]\n\n                                BIRMINGHAM, _Friday, March 5th, 1869._\n\nThis is to send you my best love, and to wish you many and many happy\nreturns of to-morrow, which I miraculously remember to be your\nbirthday.\n\nI saw this morning a very pretty fan here. I was going to buy it as a\nremembrance of the occasion, when I was checked by a dim misgiving that\nyou had a fan not long ago from Chorley. Tell me what you would like\nbetter, and consider me your debtor in that article, whatever it may be.\n\nI have had my usual left boot on this morning, and have had an hour's\nwalk. It was in a gale of wind and a simoom of dust, but I greatly\nenjoyed it. Immense enthusiasm at Wolverhampton last night over\n\"Marigold.\" Scott made a most amazing ass of himself yesterday. He\nreported that he had left behind somewhere three books--\"Boots,\"\n\"Murder,\" and \"Gamp.\" We immediately telegraphed to the office. Answer,\nno books there. As my impression was that he must have left them at St.\nJames's Hall, we then arranged to send him up to London at seven this\nmorning. Meanwhile (though not reproached), he wept copiously and\naudibly. I had asked him over and over again, was he sure he had not put\nthem in my large black trunk? Too sure, too sure. Hadn't opened that\ntrunk after Tuesday night's reading. He opened it to get some clothes\nout when I went to bed, and there the books were! He produced them with\nan air of injured surprise, as if we had put them there.\n\n\n[Sidenote: Miss Hogarth.]\n\n                 QUEEN'S HOTEL, MANCHESTER, _Sunday, March 7th, 1869._\n\nWe have had our sitting-room chimney afire this morning, and have had to\nturn out elsewhere to breakfast; but the chamber has since been cleaned\nup, and we are reinstated. Manchester is (_for_ Manchester) bright and\nfresh.\n\nTell Russell that a crop of hay is to be got off the meadow this year,\nbefore the club use it. They did not make such use of it last year as\nreconciles me to losing another hay-crop. So they must wait until the\nhay is in, before they commence active operations.\n\nPoor Olliffe! I am truly sorry to read those sad words about his\nsuffering, and fear that the end is not far off.\n\nWe are very comfortably housed here, and certainly that immense hall is\na wonderful place for its size. Without much greater expenditure of\nvoice than usual, I a little enlarged the action last night, and Dolby\n(who went to all the distant points of view) reported that he could\ndetect no difference between it and any other place. As always happens\nnow--and did not at first--they were unanimously taken by Noah\nClaypole's laugh. But the go, throughout, was enormous. Sims Reeves was\ndoing Henry Bertram at the theatre, and of course took some of our\nshillings. It was a night of excitement for Cottonopolis.\n\nI received from Mrs. Keeley this morning a very good photograph of poor\nold Bob. Yesterday I had a letter from Harry, reminding me that our\nintended Cambridge day is the day next after that of the boat-race.\nClearly it must be changed.\n\n\n[Sidenote: Miss Hogarth.]\n\n              QUEEN'S HOTEL, MANCHESTER, _Saturday, March 20th, 1869._\n\nGetting yours and its enclosure, Mary's note, at two this afternoon, I\nwrite a line at once in order that you may have it on Monday morning.\n\nThe Theatre Royal, Liverpool, will be a charming place to read in.\nLadies are to dine at the dinner, and we hear it is to be a very grand\naffair. Dolby is doubtful whether it may not \"hurt the business,\" by\ndrawing a great deal of money in another direction, which I think\npossible enough. Trade is very bad _here_, and the gloom of the Preston\nstrike seems to brood over the place. The Titiens Company have been\ndoing wretchedly. I should have a greater sympathy with them if they\nwere not practising in the next room now.\n\nMy love to Letitia and Harriette,[29] wherein Dolby (highly gratified by\nbeing held in remembrance) joins with the same to you.\n\n\n[Sidenote: Miss Hogarth.]\n\n                               MANCHESTER, _Sunday, March 21st, 1869._\n\nWill you tell Mary that I have had a letter from Frith, in which he says\nthat he will be happy to show her his pictures \"any day in the first\nweek of April\"? I have replied that she will be proud to receive his\ninvitation. His object in writing was to relieve his mind about the\n\"Murder,\" of which he cannot say enough.\n\nTremendous enthusiasm here last night, calling in the most thunderous\nmanner after \"Marigold,\" and again after the \"Trial,\" shaking the great\nhall, and cheering furiously.\n\nLove to all.\n\n\n[Sidenote: Mr. John Clarke.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                        _Wednesday, March 24th, 1869._\n\nLADIES AND GENTLEMEN,\n\nI beg to assure you that I am much gratified by the desire you do me the\nhonour to express in your letter handed to me by Mr. John Clarke.\n\nBefore that letter reached me, I had heard of your wish, and had\nmentioned to Messrs. Chappell that it would be highly agreeable to me to\nanticipate it, if possible. They readily responded, and we agreed upon\nhaving three morning readings in London. As they are not yet publicly\nannounced, I add a note of the days and subjects:\n\nSaturday, May 1st. \"Boots at the Holly-Tree Inn,\" and \"Sikes and Nancy\"\nfrom \"Oliver Twist.\"\n\nSaturday, May 8th. \"The Christmas Carol.\"\n\nSaturday, May 22nd. \"Sikes and Nancy\" from \"Oliver Twist,\" and \"The\nTrial\" from \"Pickwick.\"\n\nWith the warmest interest in your art, and in its claims upon the\ngeneral gratitude and respect,\n\n                            Believe me, always faithfully your Friend.\n\n\n[Sidenote: Miss Hogarth.]\n\n                  ADELPHI HOTEL, LIVERPOOL, _Sunday, April 4th, 1869._\n\nBy this post I send to Mary the truly affecting account of poor dear\nKatie Macready's death. It is as sorrowful as anything so peaceful and\ntrustful can be!\n\nBoth my feet are very tender, and often feel as though they were in hot\nwater. But I was wonderfully well and strong, thank God! and had no end\nof voice for the two nights running in that great Birmingham hall. We\nhad enormous houses.\n\nSo far as I understand the dinner arrangements here, they are much too\nlong. As to the acoustics of that hall, and the position of the tables\n(both as bad as bad can be), my only consolation is that, if anybody can\nbe heard, _I_ probably can be. The honorary secretary tells me that six\nhundred people are to dine. The mayor, being no speaker and out of\nhealth besides, hands over the toast of the evening to Lord Dufferin.\nThe town is full of the festival. The Theatre Royal, touched up for the\noccasion, will look remarkably bright and well for the readings, and our\nlets are large. It is remarkable that our largest let as yet is for\nThursday, not Friday. I infer that the dinner damages Friday, but Dolby\ndoes not think so. There appears to be great curiosity to hear the\n\"Murder.\" (On Friday night last I read to two thousand people, and odd\nhundreds.)\n\nI hear that Anthony Trollope, Dixon, Lord Houghton, Lemon, Esquiros (of\nthe _Revue des Deux Mondes_), and Sala are to be called upon to speak;\nthe last, for the newspaper press. All the Liverpool notabilities are to\nmuster. And Manchester is to be represented by its mayor with due\nformality.\n\nI had been this morning to look at St. George's Hall, and suggest what\ncan be done to improve its acoustics. As usually happens in such cases,\ntheir most important arrangements are already made and unchangeable. I\nshould not have placed the tables in the committee's way at all, and\ncould certainly have placed the dais to much greater advantage. So all\nthe good I could do was to show where banners could be hung with some\nhope of stopping echoes. Such is my small news, soon exhausted. We\narrived here at three yesterday afternoon; it is now mid-day; Chorley\nhas not yet appeared, but he had called at the local agent's while I was\nat Birmingham.\n\nIt is a curious little instance of the way in which things fit together\nthat there is a ship-of-war in the Mersey, whose flags and so forth are\nto be brought up to St. George's Hall for the dinner. She is the\n_Donegal_, of which Paynter told me he had just been captain, when he\ntold me all about Sydney at Bath.\n\nOne of the pleasantest things I have experienced here this time, is the\nmanner in which I am stopped in the streets by working men, who want to\nshake hands with me, and tell me they know my books. I never go out but\nthis happens. Down at the docks just now, a cooper with a fearful\nstutter presented himself in this way. His modesty, combined with a\nconviction that if he were in earnest I would see it and wouldn't repel\nhim, made up as true a piece of natural politeness as I ever saw.\n\n\n[Sidenote: Miss Hogarth.]\n\n             IMPERIAL HOTEL, BLACKPOOL, _Wednesday, April 21st, 1869._\n\nI send you this hasty line to let you know that I have come to this\nsea-beach hotel (charming) for a day's rest. I am much better than I was\non Sunday, but shall want careful looking to, to get through the\nreadings. My weakness and deadness are all _on the left side_, and if I\ndon't look at anything I try to touch with my left hand, I don't know\nwhere it is. I am in (secret) consultation with Frank Beard; he\nrecognises, in the exact description I have given him, indisputable\nevidences of overwork, which he would wish to treat immediately. So I\nhave said: \"Go in and win.\"\n\nI have had a delicious walk by the sea to-day, and I sleep soundly, and\nhave picked up amazingly in appetite. My foot is greatly better too, and\nI wear my own boot.\n\n\n[Sidenote: Miss Dickens.]\n\n                        PRESTON, _Thursday Evening, April 22nd, 1869._\n\n_Don't be in the least alarmed._ Beard has come down, and instantly\nechoes my impression (perfectly unknown to him), that the readings must\nbe _stopped_. I have had symptoms that must not be disregarded. I go to\nLiverpool to-night with him (to get away from here), and proceed to the\noffice to-morrow.\n\n\n[Sidenote: The Lord John Russell.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                          _Wednesday, May 26th, 1869._\n\nMY DEAR LORD RUSSELL,\n\nI have delayed answering your kind letter, in order that you might get\nhome before I wrote. I am happy to report myself quite well again, and I\nshall be charmed to come to Pembroke Lodge on any day that may be most\nconvenient to Lady Russell and yourself after the middle of June.\n\nYou gratify me beyond expression by your reference to the Liverpool\ndinner. I made the allusion to you with all my heart at least, and it\nwas most magnificently received.\n\nI beg to send my kind regard to Lady Russell, with many thanks for her\nremembrance, and am ever,\n\n                               My dear Lord Russell, faithfully yours.\n\n\n[Sidenote: Mr. W. H. Wills.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                          _Thursday, June 24th, 1869._\n\nMY DEAR WILLS,\n\nAt a great meeting[30] compounded of your late \"Chief,\" Charley, Morley,\nGrieve, and Telbin, your letter was read to-day, and a very sincere\nrecord of regret and thanks was placed on the books of the great\ninstitution.\n\nMany thanks for the suggestion about the condition of churches. I am so\naweary of church questions of all sorts that I am not quite clear as to\ntackling this. But I am turning it in my mind. I am afraid of two\nthings: firstly, that the thing would not be picturesquely done;\nsecondly, that a general cucumber-coolness would pervade the mind of our\ncirculation.\n\nNothing new here but a speaking-pipe, a post-box, and a mouldy smell\nfrom some forgotten crypt--an extra mouldy smell, mouldier than of yore.\nLillie sniffs, projects one eye into nineteen hundred and ninety-nine,\nand does no more.\n\nI have been to Chadwick's, to look at a new kind of cottage he has built\n(very ingenious and cheap).\n\nWe were all much disappointed last Saturday afternoon by a neighbouring\nfire being only at a carpenter's, and not at Drury Lane Theatre.\nEllen's[31] child having an eye nearly poked out by a young friend, and\nbeing asked whether the young friend was not very sorry afterwards,\nreplied: \"No. _She_ wasn't. _I_ was.\"\n\nLondon execrable.\n\n                                            Ever affectionately yours.\n\nP.S.--Love to Mrs. Wills.\n\n\n[Sidenote: Mr. Shirley Brooks.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                           _Tuesday, July 12th, 1869._\n\nMY DEAR BROOKS,\n\nI have appended my sign manual to the memorial, which I think is very\ndiscreetly drawn up. I have a strong feeling of sympathy with poor Mrs.\nCunningham, for I remember the pretty house she managed charmingly. She\nhas always done her duty well, and has had hard trials. But I greatly\ndoubt the success of the memorial, I am sorry to add.\n\nIt was hotter here yesterday on this Kentish chalk than I have felt it\nanywhere for many a day. Now it is overcast and raining hard, much to\nthe satisfaction of great farmers like myself.\n\nI am glad to infer from your companionship with the Cocked Hats, that\nthere is no such thing as gout within several miles of you. May it keep\nits distance.\n\n                               Ever, my dear Brooks, faithfully yours.\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n                               GAD'S HILL, _Tuesday, July 20th, 1869._\n\nMY DEAREST MACREADY,\n\nI have received your letter here to-day, and deeply feel with you and\nfor you the affliction of poor dear Katie's loss. I was not unprepared\nfor the sad news, but it comes in such a rush of old remembrances and\nwithered joys that strikes to the heart.\n\nGod bless you! Love and youth are still beside you, and in that thought\nI take comfort for my dear old friend.\n\nI am happy to report myself perfectly well and flourishing. We are just\nnow announcing the resumption and conclusion of the broken series of\nfarewell readings in a London course of twelve, beginning early in the\nnew year.\n\nScarcely a day has gone by this summer in which we have not talked of\nyou and yours. Georgina, Mary, and I continually speak of you. In the\nspirit we certainly are even more together than we used to be in the\nbody in the old times. I don't know whether you have heard that Harry\nhas taken the second scholarship (fifty pounds a year) at Trinity Hall,\nCambridge. The bigwigs expect him to do a good deal there.\n\nWills having given up in consequence of broken health (he has been my\nsub-editor for twenty years), I have taken Charley into \"All the Year\nRound.\" He is a very good man of business, and evinces considerable\naptitude in sub-editing work.\n\nThis place is immensely improved since you were here, and really is now\nvery pretty indeed. We are sorry that there is no present prospect of\nyour coming to see it; but I like to know of your being at the sea, and\nhaving to do--_from the beach_, as Mrs. Keeley used to say in \"The\nPrisoner of War\"--with the winds and the waves and all their freshening\ninfluences.\n\nI dined at Greenwich a few days ago with Delane. He asked me about you\nwith much interest. He looks as if he had never seen a printing-office,\nand had never been out of bed after midnight.\n\nGreat excitement caused here by your capital news of Butty. I suppose\nWilly has at least a dozen children by this time.\n\nOur loves to the noble boy and to dear Mrs. Macready.\n\n                            Ever, my dearest Macready,\n                                       Your attached and affectionate.\n\n\n[Sidenote: Mr. Edmund Ollier.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                            _Tuesday, Aug. 3rd, 1869._\n\nMY DEAR MR. OLLIER,\n\nI am very sensible of the feeling of the Committee towards me; and I\nreceive their invitation (conveyed through you) as a most acceptable\nmark of their consideration.\n\nBut I have a very strong objection to speech-making beside graves. I do\nnot expect or wish my feeling in this wise to guide other men; still, it\nis so serious with me, and the idea of ever being the subject of such a\nceremony myself is so repugnant to my soul, that I must decline to\nofficiate.\n\n                                              Faithfully yours always.\n\n\n[Sidenote: Miss Dickens.]\n\n      OFFICE OF \"ALL THE YEAR ROUND,\" NO. 26, WELLINGTON STREET,\n                             STRAND, LONDON, W.C.,\n                                            _Tuesday, Aug. 3rd, 1869._\n\nMY DEAREST MAMIE,\n\nI send you the second chapter of the remarkable story. The printer is\nlate with it, and I have not had time to read it, and as I altered it\nconsiderably here and there, I have no doubt there are some verbal\nmistakes in it. However, they will probably express themselves.\n\nBut I offer a prize of six pairs of gloves--between you, and your aunt,\nand Ellen Stone, as competitors--to whomsoever will tell me what idea in\nthis second part is mine. I don't mean an idea in language, in the\nturning of a sentence, in any little description of an action, or a\ngesture, or what not in a small way, but an idea, distinctly affecting\nthe whole story _as I found it_. You are all to assume that I found it\nin the main as you read it, with one exception. If I had written it, I\nshould have made the woman love the man at last. And I should have\nshadowed that possibility out, by the child's bringing them a little\nmore together on that holiday Sunday.\n\nBut I didn't write it. So, finding that it wanted something, I put that\nsomething in. What was it?\n\nLove to Ellen Stone.\n\n\n[Sidenote: Mr. Arthur Ryland.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                            _Friday, Aug. 13th, 1869._\n\nMY DEAR MR. RYLAND,\n\nMany thanks for your letter.\n\nI have very strong opinions on the subject of speechification, and hold\nthat there is, everywhere, a vast amount too much of it. A sense of\nabsurdity would be so strong upon me, if I got up at Birmingham to make\na flourish on the advantages of education in the abstract for all sorts\nand conditions of men, that I should inevitably check myself and present\na surprising incarnation of the soul of wit. But if I could interest\nmyself in the practical usefulness of the particular institution; in the\nways of life of the students; in their examples of perseverance and\ndetermination to get on; in their numbers, their favourite studies, the\nnumber of hours they must daily give to the work that must be done for a\nlivelihood, before they can devote themselves to the acquisition of new\nknowledge, and so forth, then I could interest others. This is the kind\nof information I want. Mere holding forth \"I utterly detest, abominate,\nand abjure.\"\n\nI fear I shall not be in London next week. But if you will kindly send\nme here, at your leisure, the roughest notes of such points as I have\nindicated, I shall be heartily obliged to you, and will take care of\ntheir falling into shape and order in my mind. Meantime I \"make a note\nof\" Monday, 27th September, and of writing to you touching your kind\noffer of hospitality, three weeks before that date.\n\nI beg to send my kind regard to Mrs. and Miss Ryland, and am always,\n\n                                                Very faithfully yours.\n\n\n[Sidenote: Mr. Frederic Ouvry.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                            _Sunday, Aug. 22nd, 1869._\n\nMY DEAR OUVRY,\n\nI will expect a call from you at the office, on Thursday, at your own\nmost convenient hour. I admit the soft impeachment concerning Mrs. Gamp:\nI likes my payments to be made reg'lar, and I likewise likes my\npublisher to draw it mild.\n\n                                                           Ever yours.\n\n\n[Sidenote: Mr. Arthur Ryland.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                            _Monday, Sept. 6th, 1869._\n\nMY DEAR MR. RYLAND,\n\nI am sorry to find--I had a foreshadowing of it some weeks ago--that I\nshall not be able to profit by your kind offer of hospitality when I\ncome to Birmingham for _our_ Institution. I must come down in time for a\nquiet dinner at the hotel with my \"Readings\" secretary, Mr. Dolby, and\nmust away next morning. Besides having a great deal in hand just now\n(the title of a new book among other things), I shall have visitors from\nabroad here at the time, and am severely claimed by my daughter, who\nindeed is disloyal to Birmingham in the matter of my going away at all.\nPray represent me to Mrs. Ryland as the innocent victim of\ncircumstances, and as sacrificing pleasure to the work I have to do, and\nto the training under which alone I can do it without feeling it.\n\nYou will see from the enclosed that I am in full force, and going to\nfinish my readings, please God, after Christmas. I am in the hope of\nreceiving your promised notes in due course, and continue in the\nirreverent condition in which I last reported myself on the subject of\nspeech-making. Now that men not only make the nights of the session\nhideous by what the Americans call \"orating\" in Parliament, but trouble\nthe peace of the vacation by saying over again what they said there\n(with the addition of what they _didn't_ say there, and never will have\nthe courage to say there), I feel indeed that silence, like gold across\nthe Atlantic, is a rarity at a premium.\n\n                                              Faithfully yours always.\n\n\n[Sidenote: Mr. William Charles Kent.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                          _Thursday, Oct. 7th, 1869._\n\nMY DEAR KENT,\n\nI felt that you would be deeply disappointed. I thought it better not to\nmake the first sign while you were depressed, but my mind has been\nconstantly with you. And not mine alone. You cannot think with what\naffection and sympathy you have been made the subject of our family\ndinner talk at Gad's Hill these last three days. Nothing could exceed\nthe interest of my daughters and my sister-in-law, or the earnestness of\ntheir feeling about it. I have been really touched by its warm and\ngenuine expression.\n\nCheer up, my dear fellow; cheer up, for God's sake. That is, for the\nsake of all that is good in you and around you.\n\n                                        Ever your affectionate Friend.\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n                                GAD'S HILL, _Monday, Oct. 18th, 1869._\n\nMY DEAREST MACREADY,\n\nI duly received your letter nearly a fortnight ago, with the greatest\ninterest and pleasure. Above all things I am delighted with the prospect\nof seeing you here next summer; a prospect which has been received with\nnine times nine and one more by the whole house. You will hardly know\nthe place again, it is so changed. You are not expected to admire, but\nthere _is_ a conservatory building at this moment--be still, my soul!\n\nThis leaves me in the preliminary agonies of a new book, which I hope to\nbegin publishing (in twelve numbers, not twenty) next March. The coming\nreadings being all in London, and being, after the first fortnight, only\nonce a week, will divert my attention very little, I hope.\n\nHarry has just gone up to Cambridge again, and I hope will get a\nfellowship in good time.\n\nWills is much gratified by your remembrance, and sends you his warm\nregard. He wishes me to represent that he is very little to be pitied.\nThat he suffers no pain, scarcely inconvenience, even, so long as he is\nidle. That he likes idleness exceedingly. He has bought a country place\nby Welwyn in Hertfordshire, near Lytton's, and takes possession\npresently.\n\nMy boy Sydney is now a second lieutenant, the youngest in the Service, I\nbelieve. He has the highest testimonials as an officer.\n\nYou may be quite sure there will be no international racing in American\nwaters. Oxford knows better, or I am mistaken. The Harvard crew were a\nvery good set of fellows, and very modest.\n\nRyland of Birmingham doesn't look a day older, and was full of interest\nin you, and asked me to remind you of him. By-the-bye, at Elkington's I\nsaw a pair of immense tea-urns from a railway station (Stafford), sent\nthere to be repaired. They were honeycombed within in all directions,\nand had been supplying the passengers, under the active agency of hot\nwater, with decomposed lead, copper, and a few other deadly poisons, for\nheaven knows how many years!\n\nI must leave off in a hurry to catch the post, after a hard day's work.\n\n                  Ever, my dearest Macready,\n                                  Your most affectionate and attached.\n\nFOOTNOTES:\n\n[28] Herr Joseph Joachim, the renowned violinist.\n\n[29] His sister-in-law, Mrs. Augustus Dickens, always a welcome visitor\nat Gad's Hill.\n\n[30] Of the Guild of Literature and Art.\n\n[31] The housekeeper at the office.\n\n\n\n\n1870.\n\nNARRATIVE.\n\n\nCharles Dickens passed his last Christmas and New Year's Day at Gad's\nHill, with a party of family and friends, in the usual way, except that\nhe was suffering again from an attack of the foot trouble, particularly\non Christmas Day, when he was quite disabled by it and unable to walk at\nall--able only to join the party in the evening by keeping his room all\nday. However, he was better in a day or two, and early in January he\nwent to London, where he had taken the house of his friends, Mr. and\nMrs. Milner Gibson, for the season.\n\nHis series of \"Farewell Readings\" at St. James's Hall began in January,\nand ended on the 16th March. He was writing \"Edwin Drood\" also, and was,\nof course, constantly occupied with \"All the Year Round\" work. In the\nbeginning of January, he fulfilled his promise of paying a second visit\nto Birmingham and making a speech, of which he writes in his last letter\nto Mr. Macready.\n\nFor his last reading he gave the \"Christmas Carol\" and \"The Trial\" from\n\"Pickwick,\" and at the end of the evening he addressed a few farewell\nwords to his audience. It was a memorable and splendid occasion. He was\nvery deeply affected by the loving enthusiasm of his greeting, and it\nwas a real sorrow to him to give up for ever the personal associations\nwith thousands of the readers of his books. But when the pain, mingled\nwith pleasure, of this last reading was over, he felt greatly the relief\nof having undisturbed time for his own quieter pursuits, and looked\nforward to writing the last numbers of \"Edwin Drood\" at Gad's Hill,\nwhere he was to return in June.\n\nThe last public appearance of any kind that he made was at the Royal\nAcademy dinner in May. He was at the time far from well, but he made a\ngreat effort to be present and to speak, from his strong desire to pay a\ntribute to the memory of his dear old friend Mr. Maclise, who died in\nApril.\n\nHer Majesty having expressed a wish, conveyed through Mr. Helps\n(afterwards Sir Arthur Helps), to have a personal interview with Charles\nDickens, he accompanied Mr. Helps to Buckingham Palace one afternoon in\nMarch. He was most graciously and kindly received by her Majesty, and\ncame away with a hope that the visit had been mutually agreeable. The\nQueen presented him with a copy of her \"Journal in the Highlands,\" with\nan autograph inscription. And he had afterwards the pleasure of\nrequesting her acceptance of a set of his books. He attended a levee\nheld by the Prince of Wales in April, and the last time he dined out in\nLondon was at a party given by Lord Houghton for the King of the\nBelgians and the Prince of Wales, who had both expressed a desire to\nmeet Charles Dickens. All through the season he had been suffering, at\nintervals, from the swollen foot, and on this occasion it was so bad,\nthat up to the last moment it was very doubtful whether he could fulfil\nhis engagement.\n\nWe have very few letters for this year, and none of any very particular\ninterest, but we give them all, as they are _the last_.\n\nMr. S. L. Fildes was his \"new illustrator,\" to whom he alludes in a note\nto Mr. Frith; we also give a short note to Mr. Fildes himself.\n\nThe correspondence of Charles Dickens with Mrs. Dallas Glyn, the\ncelebrated actress, for whom he had a great friendship, is so much on\nthe subject of her own business, that we have only been able to select\ntwo notes of any public interest.\n\nIn explanation of _the last letter_, we give an extract from a letter\naddressed to _The Daily News_ by Mr. J. M. Makeham, soon after the death\nof Charles Dickens, as follows: \"That the public may exactly understand\nthe circumstances under which Charles Dickens's letter to me was\nwritten, I am bound to explain that it is in reply to a letter which I\naddressed to him in reference to a passage in the tenth chapter of\n\"Edwin Drood,\" respecting which I ventured to suggest that he had,\nperhaps, forgotten that the figure of speech alluded to by him, in a way\nwhich, to my certain knowledge, was distasteful to some of his admirers,\nwas drawn from a passage of Holy Writ which is greatly reverenced by a\nlarge number of his countrymen as a prophetic description of the\nsufferings of our Saviour.\"\n\nThe MS. of the little \"History of the New Testament\" is now in the\npossession of his eldest daughter. She has (together with her aunt)\nreceived many earnest entreaties, both from friends and strangers, that\nthis history might be allowed to be published, for the benefit of other\nchildren.\n\nThese many petitions have his daughter's fullest sympathy. But she knows\nthat her father wrote this history ONLY for his own children, that it\nwas his particular wish that it never should be published, and she\ntherefore holds this wish as sacred and irrevocable.\n\n\n[Sidenote: Mr. W. H. Wills.]\n\n            5, HYDE PARK PLACE, LONDON, W., _Sunday, Jan. 23rd, 1870._\n\nMY DEAR WILLS,\n\nIn the note I had from you about Nancy and Sikes, you seem to refer to\nsome other note you had written me. Therefore I think it well merely to\nmention that I have received no other note.\n\nI do not wonder at your not being up to the undertaking (even if you had\nhad no cough) under the wearing circumstances. It was a very curious\nscene. The actors and actresses (most of the latter looking very pretty)\nmustered in extraordinary force, and were a fine audience. I set myself\nto carrying out of themselves and their observation, those who were bent\non watching how the effects were got; and I believe I succeeded. Coming\nback to it again, however, I feel it was madness ever to do it so\ncontinuously. My ordinary pulse is seventy-two, and it runs up under\nthis effort to one hundred and twelve. Besides which, it takes me ten or\ntwelve minutes to get my wind back at all; I being, in the meantime,\nlike the man who lost the fight--in fact, his express image. Frank Beard\nwas in attendance to make divers experiments to report to Watson; and\nalthough, as you know, he stopped it instantly when he found me at\nPreston, he was very much astonished by the effects of the reading on\nthe reader.\n\nSo I hope you may be able to come and hear it before it is silent for\never. It is done again on the evenings of the 1st February, 15th\nFebruary, and 8th March. I hope, now I have got over the mornings, that\nI may be able to work on my book. But up to this time the great\npreparation required in getting the subjects up again, and the twice a\nweek besides, have almost exclusively occupied me.\n\nI have something the matter with my right thumb, and can't (as you see)\nwrite plainly. I sent a word to poor Robert Chambers,[32] and I send my\nlove to Mrs. Wills.\n\n                            Ever, my dear Wills, affectionately yours.\n\n\n[Sidenote: Mrs. Dallas.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                         _Wednesday, Jan. 16th, 1870._\n\nMY DEAR MRS. DALLAS,\n\nIt is perfectly delightful to me to get your fervent and sympathetic\nnote this morning. A thousand thanks for it. I will take care that two\nplaces on the front row, by my daughter, are reserved for your occasion\nnext time. I cannot see you in too good a seat, or too often.\n\n                               Believe me, ever very faithfully yours.\n\n\n[Sidenote: Mr. S. L. Fildes.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                         _Wednesday, Jan. 16th, 1870._\n\nDEAR SIR,\n\nI beg to thank you for the highly meritorious and interesting specimens\nof your art that you have had the kindness to send me. I return them\nherewith, after having examined them with the greatest pleasure.\n\nI am naturally curious to see your drawing from \"David Copperfield,\" in\norder that I may compare it with my own idea. In the meanwhile, I can\nhonestly assure you that I entertain the greatest admiration for your\nremarkable powers.\n\n                                                     Faithfully yours.\n\n\n[Sidenote: Mr. Henry Fielding Dickens.]\n\n                  5, HYDE PARK PLACE, W., _Thursday, Feb. 17th, 1870._\n\nMY DEAR HARRY,\n\nI am extremely glad to hear that you have made a good start at the\nUnion. Take any amount of pains about it; open your mouth well and\nroundly, speak to the last person visible, and give yourself time.\n\nLoves from all.\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Mr. W. C. Macready.]\n\n                                         _Wednesday, March 2nd, 1870._\n\nMY DEAREST MACREADY,\n\nThis is to wish you and yours all happiness and prosperity at the\nwell-remembered anniversary to-morrow. You may be sure that loves and\nhappy returns will not be forgotten at _our_ table.\n\nI have been getting on very well with my book, and we are having immense\naudiences at St. James's Hall. Mary has been celebrating the first\nglimpses of spring by having the measles. She got over the disorder very\neasily, but a weakness remains behind. Katie is blooming. Georgina is in\nperfect order, and all send you their very best loves. It gave me true\npleasure to have your sympathy with me in the second little speech at\nBirmingham. I was determined that my Radicalism should not be called in\nquestion. The electric wires are not very exact in their reporting, but\nat all events the sense was there. Ryland, as usual, made all sorts of\nenquiries about you.\n\nWith love to dear Mrs. Macready and the noble boy my particular friend,\nand a hearty embrace to you,\n\n                          I am ever, my dearest Macready,\n                                               Your most affectionate.\n\n\n[Sidenote: Mr. ----.]\n\n                                  OFFICE OF \"ALL THE YEAR ROUND,\"\n                                         _Wednesday, March 9th, 1870._\n\nMY DEAR ----,\n\nYou make me very uneasy on the subject of your new long story here, by\nsowing your name broadcast in so many fields at once, and undertaking\nsuch an impossible amount of fiction at one time. Just as you are coming\non with us, you have another story in progress in \"The Gentleman's\nMagazine,\" and another announced in \"Once a Week.\" And so far as I know\nthe art we both profess, it cannot be reasonably pursued in this way. I\nthink the short story you are now finishing in these pages obviously\nmarked by traces of great haste and small consideration; and a long\nstory similarly blemished would really do the publication irreparable\nharm.\n\nThese considerations are so much upon my mind that I cannot forbear\nrepresenting them to you, in the hope that they may induce you to take a\nlittle more into account the necessity of care and preparation, and some\nself-denial in the quantity done. I am quite sure that I write fully as\nmuch in your interest as in that of \"All the Year Round.\"\n\n                                  Believe me, always faithfully yours.\n\n\n[Sidenote: The same.]\n\n                   5, HYDE PARK PLACE, W., _Friday, March 11th, 1870._\n\nMY DEAR ----,\n\nOf course the engagement between us is to continue, and I am sure you\nknow me too well to suppose that I have ever had a thought to the\ncontrary. Your explanation is (as it naturally would be, being yours)\nmanly and honest, and I am both satisfied and hopeful.\n\n                                                           Ever yours.\n\n\n[Sidenote: Mr. William Charles Kent.]\n\n                 5, HYDE PARK PLACE, W., _Saturday, March 26th, 1870._\n\nMY DEAR KENT,\n\nI received both copies of _The Sun_, with the tenderest pleasure and\ngratification.\n\nEverything that I can let you have in aid of the proposed record[33]\n(which, _of course_, would be far more agreeable to me if done by you\nthan by any other hand), shall be at your service. Dolby has all the\nfigures relating to America, and you shall have for reference the books\nfrom which I read. They are afterwards going into Forster's\ncollection.[34]\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Mr. Henry Fielding Dickens.]\n\n                  5, HYDE PARK PLACE, W., _Tuesday, March 29th, 1870._\n\nMY DEAR HARRY,\n\nYour next Tuesday's subject is a very good one. I would not lose the\npoint that narrow-minded fanatics, who decry the theatre and defame its\nartists, are absolutely the advocates of depraved and barbarous\namusements. For wherever a good drama and a well-regulated theatre\ndecline, some distorted form of theatrical entertainment will infallibly\narise in their place. In one of the last chapters of \"Hard Times,\" Mr.\nSleary says something to the effect: \"People will be entertained\nthomehow, thquire. Make the betht of uth, and not the wortht.\"\n\n                                                  Ever affectionately.\n\n\n[Sidenote: Mr. Shirley Brooks.]\n\n                    5, HYDE PARK PLACE, W., _Friday, April 1st, 1870._\n\nMY DEAR SHIRLEY BROOKS,\n\nI have written to Mr. Low, expressing my regret that I cannot comply\nwith his request, backed as it is by my friend S. B. But I have told him\nwhat is perfectly true--that I leave town for the peaceful following of\nmy own pursuits, at the end of next month; that I have excused myself\nfrom filling all manner of claims, on the ground that the public\nengagements I could make for the season were very few and were all made;\nand that I cannot bear hot rooms when I am at work. I have smoothed this\nas you would have me smooth it.\n\nWith your longing for fresh air I can thoroughly sympathise. May you get\nit soon, and may you enjoy it, and profit by it half as much as I wish!\n\n                                                Ever faithfully yours.\n\n\n[Sidenote: Mr. W. P. Frith, R.A.]\n\n                 5, HYDE PARK PLACE, W., _Saturday, April 16th, 1870._\n\nMY DEAR FRITH,\n\nI shall be happy to go on Wednesday evening, if convenient.\n\nYou please me with what you say of my new illustrator, of whom I have\ngreat hopes.\n\n                                                Faithfully yours ever.\n\n\n[Sidenote: Mr. William Charles Kent.]\n\n                                   _Monday Morning, April 25th, 1870._\n\nMY DEAR KENT,\n\nI received your book[35] with the greatest pleasure, and heartily thank\nyou for it. It is a volume of a highly prepossessing appearance, and a\nmost friendly look. I felt as if I should have taken to it at sight;\neven (a very large even) though I had known nothing of its contents, or\nof its author!\n\nFor the last week I have been most perseveringly and ding-dong-doggedly\nat work, making headway but slowly. The spring always has a restless\ninfluence over me; and I weary, at any season, of this London dining-out\nbeyond expression; and I yearn for the country again. This is my excuse\nfor not having written to you sooner. Besides which, I had a baseless\nconviction that I should see you at the office last Thursday. Not having\ndone so, I fear you must be worse, or no better? If you _can_ let me\nhave a report of yourself, pray do.\n\n\n[Sidenote: Mrs. Frederick Pollock.]\n\n                      5, HYDE PARK PLACE, W., _Monday, May 2nd, 1870._\n\nMY DEAR MRS. POLLOCK,\n\nPray tell the illustrious Philip van Artevelde, that I will deal with\nthe nefarious case in question if I can. I am a little doubtful of the\npracticability of doing so, and frisking outside the bounds of the law\nof libel. I have that high opinion of the law of England generally,\nwhich one is likely to derive from the impression that it puts all the\nhonest men under the diabolical hoofs of all the scoundrels. It makes me\ncautious of doing right; an admirable instance of its wisdom!\n\nI was very sorry to have gone astray from you that Sunday; but as the\nearlier disciples entertained angels unawares, so the later often miss\nthem haphazard.\n\nYour description of La Font's acting is the complete truth in one short\nsentence: Nature's triumph over art; reversing the copy-book axiom! But\nthe Lord deliver us from Plessy's mechanical ingenuousness!!\n\nAnd your petitioner will ever pray.\n\nAnd ever be,\n\n                                                     Faithfully yours.\n\n\n[Sidenote: Mrs. E. M. Ward.]\n\n                  5, HYDE PARK PLACE, W., _Wednesday, May 11th, 1870._\n\nMY DEAR MRS. WARD,\n\nI grieve to say that I am literally laid by the heels, and incapable of\ndining with you to-morrow. A neuralgic affection of the foot, which\nusually seizes me about twice a year, and which will yield to nothing\nbut days of fomentation and horizontal rest, set in last night, and has\ncaused me very great pain ever since, and will too clearly be no better\nuntil it has had its usual time in which to wear itself out. I send my\nkindest regard to Ward, and beg to be pitied.\n\n                                  Believe me, faithfully yours always.\n\n\n[Sidenote: Mr. William Charles Kent.]\n\n                    5, HYDE PARK PLACE, W., _Tuesday, May 17th, 1870._\n\nMY DEAR KENT,\n\nMany, many thanks! It is only my neuralgic foot. It has given me such a\nsharp twist this time that I have not been able, in its extreme\nsensitiveness, to put any covering upon it except scalding fomentations.\nHaving viciously bubbled and blistered it in all directions, I hope it\nnow begins to see the folly of its ways.\n\n                                                  Affectionately ever.\n\nP.S.--I hope the Sun shines.\n\n\n[Sidenote: Mrs. Bancroft.]\n\n                    GAD'S HILL PLACE, HIGHAM BY ROCHESTER, KENT,\n                                           _Thursday, May 31st, 1870._\n\nMY DEAR MRS. BANCROFT,[36]\n\nI am most heartily obliged to you for your kind note, which I received\nhere only last night, having come here from town circuitously to get a\nlittle change of air on the road. My sense of your interest cannot be\nbetter proved than by my trying the remedy you recommend, and that I\nwill do immediately. As I shall be in town on Thursday, my troubling you\nto order it would be quite unjustifiable. I will use your name in\napplying for it, and will report the result after a fair trial. Whether\nthis remedy succeeds or fails as to the neuralgia, I shall always\nconsider myself under an obligation to it for having indirectly procured\nme the great pleasure of receiving a communication from you; for I hope\nI may lay claim to being one of the most earnest and delighted of your\nmany artistic admirers.\n\n                                         Believe me, faithfully yours.\n\nFOOTNOTES:\n\n[32] On the death of his second wife.\n\n[33] Of the Readings. The intention was carried out. Mr. Kent's book,\n\"Charles Dickens as a Reader,\" was published in 1872.\n\n[34] No doubt Charles Dickens intended to add the Reading Books to the\nlegacy of his MSS. to Mr. Forster. But he did not do so, therefore the\n\"Readings\" are not a part of the \"Forster Collection\" at the South\nKensington Museum.\n\n[35] A new collective edition of \"Kent's Poems,\" dedicated to his\ncousin, Colonel Kent, of the 77th Regiment.\n\n[36] Miss Marie Wilton.\n\n\n\n\nTWO LAST LETTERS.\n\n\n[Sidenote: Mr. William Charles Kent.]\n\n\n[Illustration: Gad's Hill Place,\n                          Higham by Rochester, Kent.[37]\n\n        HW: Wednesday Eighth June 1870\n\n\nHW: Dear Kent\n\nTomorrow is a very bad day for me to make a call, as, in addition to my\nusual office business, I have a mass of accounts to settle with Wills.\nBut I hope I may be ready for you at 3 o'clock. If I can't be--why, then\nI shan't be.\n\nYou must really get rid of those Opal enjoyments. They are too\noverpowering:\n\n\"These violent delights have violent ends.\"\n\nI think it was a father of your churches who made the wise remark to a\nyoung gentleman who got up early (or stayed out late) at Verona?\n\n                                 Ever affectionately\n                                             Signature: ChD]\n\n\n\n[Sidenote: Mr. John M. Makeham.]\n\n                               =Gad's Hill Place,=\n                                          =Higham by Rochester, Kent.=\n\n[Illustration: HW: Wednesday Eighth June 1870\n\nDear Sir\n\nIt would be quite inconceivable I think--but for your\nletter--that any reasonable reader could possibly attach a scriptural\nreference to a passage in a book of mine, reproducing a much abused\nsocial figure of speech, impressed into all sorts of service on all\nsorts of inappropriate occasions, without the faintest connexion of it\nwith its original source. I am truly shocked to find that any reader can\nmake the mistake\n\nI have always striven in my writings to express veneration for the life\nand lessons of our Saviour; because I feel it; and because I re-wrote\nthat history for my children--every one of whom knew it from having it\nrepeated to them--long before they could read, and almost as soon as\nthey could speak.\n\nBut I have never made proclamation of this from the house tops\n\n                                          Faithfully Yours,\n                                                       Charles Dickens\n\nJohn M. Markham Esq.]\n\nAll through this spring in London, Charles Dickens had been ailing in\nhealth, and it was remarked by many friends that he had a weary look,\nand was \"aged\" and altered. But he was generally in good spirits, and\nhis family had no uneasiness about him, relying upon the country quiet\nand comparative rest at Gad's Hill to have their usual influence in\nrestoring his health and strength. On the 2nd June he attended a private\nplay at the house of Mr. and Mrs. Freake, where his two daughters were\namong the actresses. The next day he went back to Gad's Hill. His\ndaughter Kate (whose home was there at all times when she chose, and\nalmost always through the summer months) went down on Sunday, the 5th\nJune, for a day's visit, to see the \"great improvement of the\nconservatory.\" Her father laughingly assured her she had now seen \"the\nlast\" improvement at Gad's Hill. At this time he was tolerably well, but\nshe remarked to her sister and aunt how strangely he was tired, and what\na curious grey colour he had in his face after a very short walk on that\nSunday afternoon. However, he seemed quite himself again in the evening.\nThe next day his daughter Kate went back, accompanied by her sister, who\nwas to pay her a short visit, to London.\n\nCharles Dickens was very hard at work on the sixth number of \"Edwin\nDrood.\" On the Monday and Tuesday he was well, but he was unequal to\nmuch exercise. His last walk was one of his greatest favourites--through\nCobham Park and Wood--on the afternoon of Tuesday.\n\nOn the morning of Wednesday, the 8th (one of the loveliest days of a\nlovely summer), he was very well; in excellent spirits about his book,\nof which he said he _must_ finish his number that day--the next\n(Thursday) being the day of his weekly visit to \"All the Year Round\"\noffice. Therefore, he would write all day in the Chalet, and take no\nwalk or drive until the evening. In the middle of the day he came to the\nhouse for an hour's rest, and smoked a cigar in the conservatory--out of\nwhich new addition to the house he was taking the greatest personal\nenjoyment--and seemed perfectly well, and exceedingly cheerful and\nhopeful. When he came again to the house, about an hour before the time\nfixed for the early dinner, he seemed very tired, silent, and absorbed.\nBut this was so usual with him after a day of engrossing work, that it\ncaused no alarm or surprise to his sister-in-law--the only member of his\nhousehold who happened to be at home. He wrote some letters--among them,\nthese last letters which we give--in the library of the house, and also\narranged many trifling business matters, with a view to his departure\nfor London the next morning. He was to be accompanied, on his return at\nthe end of the week, by Mr. Fildes, to introduce the \"new illustrator\"\nto the neighbourhood in which many of the scenes of this last book of\nCharles Dickens, as of his first, were laid.\n\nIt was not until they were seated at the dinner-table that a striking\nchange in the colour and expression of his face startled his\nsister-in-law, and on her asking him if he was ill, he said, \"Yes, very\nill; I have been very ill for the last hour.\" But on her expressing an\nintention of sending instantly for a doctor, he stopped her, and said:\n\"No, he would go on with dinner, and go afterwards to London.\" And then\nhe made an effort to struggle against the fit that was fast coming on\nhim, and talked, but incoherently, and soon very indistinctly. It being\nnow evident that he _was_ ill, and very seriously ill, his sister-in-law\nbegged him to come to his own room before she sent off for medical help.\n\"Come and lie down,\" she entreated. \"Yes, on the ground,\" he said, very\ndistinctly--these were the last words he spoke--and he slid from her\narm, and fell upon the floor.\n\nThe servants brought a couch into the dining-room, where he was laid. A\nmessenger was despatched for Mr. Steele, the Rochester doctor, and with\na telegram to his doctor in London, and to his daughters. This was a few\nminutes after six o'clock.\n\nHis daughters arrived, with Mr. Frank Beard, this same evening. His\neldest son the next morning, and his son Henry and his sister Letitia in\nthe evening of the 9th--too late, alas!\n\nAll through the night, Charles Dickens never opened his eyes, or showed\na sign of consciousness. In the afternoon of the 9th, Dr. Russell\nReynolds arrived at Gad's Hill, having been summoned by Mr. Frank Beard\nto meet himself and Mr. Steele. But he could only confirm their hopeless\nverdict, and made his opinion known with much kind sympathy, to the\nfamily, before returning to London.\n\nCharles Dickens remained in the same unconscious state until the evening\nof this day, when, at ten minutes past six, the watchers saw a shudder\npass over him, heard him give a deep sigh, saw one tear roll down his\ncheek, and he was gone from them. And as they saw the dark shadow steal\nacross his calm, beautiful face, not one among them--could they have\nbeen given such a power--would have recalled his sweet spirit back to\nearth.\n\nAs his family were aware that Charles Dickens had a wish to be buried\nnear Gad's Hill, arrangements were made for his burial in the pretty\nchurchyard of Shorne, a neighbouring village, of which he was very fond.\nBut this intention was abandoned in consequence of a pressing request\nfrom the Dean and Chapter of Rochester Cathedral that his remains might\nbe placed there. A grave was prepared and everything arranged, when it\nwas made known to the family, through Dean Stanley, that there was a\ngeneral and very earnest desire that Charles Dickens should find his\nresting-place in Westminster Abbey. To such a fitting tribute to his\nmemory they could make no possible objection, although it was with great\nregret that they relinquished the idea of laying him in a place so\nclosely identified with his life and his works. His name,\nnotwithstanding, is associated with Rochester, a tablet to his memory\nhaving been placed by his executors on the wall of Rochester Cathedral.\n\nWith regard to Westminster Abbey, his family only stipulated that the\nfuneral might be made as private as possible, and that the words of his\nwill, \"I emphatically direct that I be buried in an inexpensive,\nunostentatious, and strictly private manner,\" should be religiously\nadhered to. And so they were. The solemn service in the vast cathedral\nbeing as private as the most thoughtful consideration could make it.\n\nThe family of Charles Dickens were deeply grateful to all in authority\nwho so carried out his wishes. And more especially to Dean Stanley and\nto the (late) Lady Augusta Stanley, for the tender sympathy shown by\nthem to the mourners on this day, and also on Sunday, the 19th, when the\nDean preached his beautiful funeral sermon.\n\nAs during his life Charles Dickens's fondness for air, light, and gay\ncolours amounted almost to a passion, so when he lay dead in the home he\nhad so dearly loved, these things were not forgotten.\n\nThe pretty room opening into the conservatory (from which he had never\nbeen removed since his seizure) was kept bright with the most beautiful\nof all kinds of flowers, and flooded with the summer sun:\n\n        \"And nothing stirred in the room. The old, old\n        fashion. The fashion that came in with our\n        first garments, and will last unchanged until\n        our race has run its course, and the wide\n        firmament is rolled up like a scroll. The old,\n        old fashion--death!\n\n        \"Oh, thank God, all who see it, for that older\n        fashion yet, of immortality!\"\n\nFOOTNOTES:\n\n[37] This letter has lately been presented by Mr. Charles Kent to the\nBritish Museum.\n\n\n\n\nINDEX.\n\n\n  A'Beckett, Gilbert, i. 134\n\n  Actors, Dickens a friend to poor, ii. 134\n\n  Affidavit, a facetious, i. 101\n\n  Agassiz, Professor, ii. 226, 309\n\n  Agate, John, ii. 136;\n    letter to, ii. 154\n\n  Ainsworth, W. H., letters to, i. 43, 75, 92\n\n  Alison, Sir Archibald, i. 170\n\n  \"All the Year Round,\" commencement of, ii. 83;\n    \"The Uncommercial Traveller\" in, ii. 107;\n    Christmas Numbers of: \"The Haunted House,\" ii. 84;\n    \"A Message from the Sea,\" ii. 108, 137;\n    \"Tom Tiddler's Ground,\" ii. 136;\n    \"Somebody's Luggage,\" ii. 171;\n    \"Mrs. Lirriper's Lodgings,\" ii. 187;\n    \"Mrs. Lirriper's Legacy,\" ii. 209, 210;\n    \"Dr. Marigold's Prescriptions,\" ii. 224, 239, 246;\n    \"Mugby Junction,\" ii. 244, 265;\n    \"No Thoroughfare,\" ii. 268, 300, 327, 332, 334, 338, 350, 356,\n        361, 362, 384;\n    and see ii. 386,\n    and see Charles Dickens as an Editor\n\n  America, feeling for Dickens in the backwoods of, i. 40, 41;\n    Dickens's first visit to, i. 53;\n    his welcome in, i. 59;\n    his opinion of, i. 60-64;\n    freedom of opinion in, i. 61;\n    Dickens's levees in, i. 66;\n    change of temperature in, i. 66;\n    hotel charges in, i. 67;\n    midnight rambles in New York, i. 67;\n    descriptions of Niagara, i. 69, 70; ii. 372, 377;\n    a maid's views on Niagara, i. 72;\n    copyright in, i. 71, 73, 74;\n    Dickens's tribute to Mrs. Trollope's book on, i. 81;\n    press-ridden, i. 97;\n    absence of quiet in, i. 98;\n    criticisms of Dickens in, i. 151;\n    the great war in, ii. 142, 143;\n    feeling between England and, ii. 240;\n    Dickens's second visit to--the journey, ii. 302-306;\n    Dickens's letters on, ii. 306-382;\n    fires in, ii. 317, 320;\n    treatment of luggage in, ii. 321;\n    drinks in, ii. 329, 363;\n    literary piracy in, ii. 332;\n    walking-match between Dolby and Osgood in, ii. 346, 352, 353,\n        360, 361, 364, 366, 377;\n    changes and improvements in since Dickens's first visit,\n        ii. 348, 374;\n    the <DW64>s in, ii. 349;\n    personal descriptions of Dickens in, ii. 369;\n    travelling in, ii. 375;\n    and see Readings\n\n  \"American Notes,\" publication of, i. 54\n\n  Andersen, Hans Christian, ii. 3\n\n  \"Animal Magnetism,\" tag to, written by Dickens, i. 238\n\n  Anne, Mrs. Dickens's maid, i. 72, 414; ii. 18, 25, 28, 343\n\n  \"Apprentices, The Tour of the Two Idle,\" ii. 5, 32, 33\n\n  \"Arabian Nights,\" a mistake in the, i. 88, 89\n\n  Armatage, Isaac, ii. 391\n\n  Armstrong, the Misses, letter to, ii. 175;\n    and see ii. 176\n\n  Astley's Theatre, description of a clown at, i. 116\n\n  Austin, Henry, i. 240; ii. 135, 157;\n    and see Letters\n\n  Austin, Mrs. Henry, ii. 447;\n    letters to, ii. 154, 180, 384\n\n  Author, the highest reward of an, i. 41\n\n  Autobiography, a concise, of Dickens, i. 437\n\n  Autograph of Dickens in 1833, i. 2;\n    Dickens leaves his in Shakespeare's room, i. 13;\n    of Boz, i. 43;\n    of Dickens as Bobadil, i. 195;\n    facsimile of Dickens's handwriting in 1856, i. 421;\n    facsimile letters of Dickens written the day before his death,\n        ii. 443-445\n\n\n  Babbage, Charles, letters to, i. 86, 87, 186\n\n  Ballantyne, ii. 415\n\n  Bancroft, Mrs., letter to, ii. 441\n\n  Banks, G., i. 273; letter to, i. 296\n\n  Barber, Dickens's gardener, ii. 102\n\n  Barker, Dr. Fordyce, ii. 378, 405\n\n  \"Barnaby Rudge\" written and published, i. 36;\n    Dickens's descriptions of the illustrations of:\n      the raven, i. 38;\n      the locksmith's house, i. 39;\n      rioters in The Maypole, i. 45;\n      scene in the ruins of the Warren, i. 46;\n      abduction of Dolly Varden, i. 48;\n      Lord George Gordon in the Tower, the duel, frontispiece, i. 50;\n      Hugh taken to gaol, i. 51\n\n  \"Battle of Life, The,\" dedication of, i. 147, 157;\n    Dickens superintends rehearsals of the play of, i. 163, 165, 167;\n    sale of, i. 166, 176;\n    reception of the play of, i. 167\n\n  Baylis, Mr., ii. 170;\n    letter to, ii. 179\n\n  Beadle, a, in office, ii. 134\n\n  Beard, Frank, ii. 182, 405, 421, 434, 447\n\n  Beaucourt, M., i. 297, 357, 439\n\n  Bedstead, a German, i. 128\n\n  Beecher, Ward, ii. 341\n\n  Begging letters, Dickens's answers to, i. 148-150\n\n  Belgians, the King of the, ii. 432\n\n  Benzon, Miss Lily, letter to, ii. 258\n\n  Berry, one of Dickens's readings men, ii. 54, 159, 160\n\n  Bicknell, Henry, i. 215;\n    letter to, i. 229\n\n  Biographers, Dickens on, i. 190;\n    his opinion of John Forster as a biographer, i. 188-191\n\n  Birthday wishes, i. 51\n\n  \"Black-eyed Susan,\" Dickens as T. P. Cooke in, i. 113;\n    a new version of, i. 114\n\n  Blackwood, Mr., ii. 165\n\n  Blair, General, ii. 355\n\n  Blanchard, Laman, letter to, i. 99\n\n  \"Bleak House,\" commenced, i. 241;\n    publication of, i. 272;\n    Dickens's opinion of, i. 279;\n    circulation of, i. 289, 309, 317\n\n  Blessington, Lady, i. 171\n\n  Bobadil, Captain, Dickens plays, i. 134;\n    Dickens's remarks on, i. 144;\n    a letter after, i. 195\n\n  Book-backs, Dickens's imitation, i. 265, 266\n\n  Book Clubs, established, i. 94;\n    Dickens on, i. 104\n\n  Boucicault, Dion, ii. 260, 261\n\n  Boulogne, Dickens at, i. 271, 297, 304-312, 341, 414, 439-448;\n    a Shakespearian performance at, i. 308;\n    _en fete_, i. 315;\n    illuminations at, on the occasion of the Prince Consort's visit,\n        i. 362;\n    fire at, i. 364;\n    condition of, during the Crimean war, i. 365;\n    letters descriptive of, i. 305, 306, 309, 312, 357, 358, 360, 372\n\n  Bouncer, Mrs., Miss Dickens's dog, ii. 109, 126, 189, 356\n\n  Bow Street Runners, ii. 178\n\n  Boxall, Sir William, i. 233, 237\n\n  Boyle, Captain Cavendish, ii. 407\n\n  Boyle, Miss Mary, i. 211, 214, 227, 414; ii. 123, 145, 315, 406;\n    and see Letters\n\n  Breach of Promise, a new sort of, i. 179\n\n  Breakfast, a Yorkshire, i. 9\n\n  Broadstairs, Dickens at, i. 4, 6, 17, 28, 36, 53, 134, 170, 185,\n        213, 240; ii. 84, 99;\n    description of lodgings at, i. 33;\n    amusements of, i. 180, 182;\n    size of Fort House at, i. 254\n\n  Bromley, Sir Richard, ii. 126\n\n  Brookfield, Mrs., letter to, ii. 249\n\n  Brookfield, The Rev. W., letters to, ii. 199, 200\n\n  Brooks, Shirley, ii. 407;\n    letters to, ii. 423, 438\n\n  Brougham, Lord, i. 182; ii. 144\n\n  Browne, H. K., i. 6, 13\n\n  Buckstone, J. B., i. 360\n\n  Burnett, Mrs., i. 185\n\n\n  Cabin, a, on board ship, i. 56\n\n  Campbell, Lord, ii, 144\n\n  Capital punishment, Dickens's views on, i. 209\n\n  Carlisle, the Earl of, letters to, i. 253, 281; ii. 12, 118, 157\n\n  Carlyle, Thomas, ii. 112\n\n  Cartwright, Samuel, ii. 326;\n    letter to, ii. 348\n\n  Castlereagh, Lord, i. 245\n\n  Cat-hunting, i. 449\n\n  Cattermole, George, i. 42, 143; ii. 327, 383;\n    and see Letters\n\n  Cattermole, Mrs., letters to, ii. 383, 385\n\n  Celeste, Madame, ii. 106\n\n  Cerjat, M. de, i. 147; ii. 406;\n    and see Letters\n\n  Chambers, Robert, ii. 167, 434\n\n  Chancery, Dickens on the Court of, i. 450\n\n  Chapman and Hall, Messrs., i. 3;\n    letter to, i. 55\n\n  Chappell, Messrs., ii. 244, 245, 267, 309, 326, 405\n\n  Charities, Dickens's sufferings from public, ii. 47\n\n  Children, stories of, i. 223, 365, 420; ii. 196, 359, 423\n\n  Childs, Mr., ii. 337, 405\n\n  \"Chimes, The,\" written, i. 95;\n    an attack on cant, i. 118, 129;\n    Dickens's opinion of, i. 129, 133;\n    Dickens gives a private reading of, i. 133\n\n  Chorley, H. F., ii. 338, 350\n\n  \"Christmas Carol, The,\" publication of, i. 85;\n    criticisms on, i. 99\n\n  Christmas greetings, i. 167\n\n  Church, Dickens on the, ii. 221;\n    service on board ship, ii. 348;\n    Dickens on the Romish, ii. 409, 410\n\n  Circumlocution, Dickens on, ii. 241, 270\n\n  Clarke, John, letter to, ii. 418\n\n  Cockspur Street Society, the, i. 85-87\n\n  Cold, effects of a, i. 92, 93;\n    remedy for a, i. 168\n\n  Colden, David, i. 64\n\n  Collins, C. A., ii. 84, 100, 113, 221, 242, 387, 410\n\n  Collins, Wilkie, i. 241, 272, 297, 332, 359, 376, 385, 388, 413,\n        414, 447; ii. 33, 84, 108, 170, 268, 292;\n    and see Letters\n\n  Comedy, Mr. Webster's offer for a prize, Dickens an imaginary\n        competitor, i. 86, 90\n\n  Compton, Mrs., letter to, ii. 22\n\n  Conjuring feats, i. 96;\n    and see ii. 243\n\n  Cooke, T. P., i. 113; ii. 4;\n    letter to, ii. 21\n\n  Copyright, i. 13;\n    Dickens's struggles to secure English, in America, i. 71, 73, 74\n\n  Costello, Dudley, i. 241;\n    letters to, i. 104, 205\n\n  Cottage, a cheap, i. 18\n\n  Coutts, Miss, i. 410\n\n  Covent Garden Theatre, Macready retires from management of, i. 18;\n    ruins of, i. 430;\n    a scene at, ii. 133\n\n  \"Cricket on the Hearth, The,\" i. 135, 145\n\n  Croker, J. Crofton, i. 272;\n    letter to, i. 275\n\n  Cruikshank, George, i. 170\n\n  Cunningham, Mrs., ii. 423\n\n  Cunningham, Peter, i. 186, 407;\n    letters to, i. 195, 270, 312, 356\n\n\n  Dacres, Sir Sydney, ii. 329\n\n  _Daily News, The_, started, i. 135\n\n  Dallas, Mrs., letters to, ii. 195, 434\n\n  Dallas, Mr., ii. 235\n\n  \"David Copperfield,\" dedication of, i. 147;\n    purpose of Little Emily in, i. 211;\n    success of, i. 211;\n    reading of, i. 377, 382;\n    Dickens's favourite work, i. 382;\n    and see i. 204, 221, 227, 279\n\n  Deane, F. H., letter to, i. 68\n\n  Delane, John, i. 298; ii. 425;\n    letter to, i. 314\n\n  De la Rue, Mr., ii. 210\n\n  Devonshire, the Duke of, letters to, i. 437, 443, 457\n\n  Devrient, Emil, i. 277\n\n  Dickens, Charles, at Furnival's Inn, i. 1;\n    his marriage, i. 1;\n    employed as a parliamentary reporter, i. 1;\n    spends his honeymoon at Chalk, Kent, i. 1;\n    employed on _The Morning Chronicle_, i. 2;\n    removes to Doughty Street, i. 4;\n    writes for the stage, i. 4, 5, 7, 16, 17;\n    his visit to the Yorkshire schools, i. 6;\n    at Twickenham Park, i. 6;\n    his visits to Broadstairs, see Broadstairs;\n    his visit to Stratford-on-Avon and Kenilworth, i. 6, 12;\n    in Shakespeare's room, i. 13;\n    elected at the Athenaeum Club, i. 12;\n    removes to Devonshire Terrace, i. 17;\n    portraits of, see Portraits;\n    visits to Scotland, i. 36, ii. 39, and see ii. 395;\n    personal feeling of for his characters, i. 36, 37, 42;\n    declines to enter Parliament, i. 37, 44; ii. 389;\n    public dinners to, i. 36, 53, 273; ii. 268, 301, 404, 406, 417,\n        419, 420;\n    an enemy of cant, i. 88, 118, 129;\n    visits of to America, see America;\n    expedition of to Cornwall, i. 54;\n    his travels in Italy, see Italy;\n    political opinions of, i. 62, 63, 88, 104;\n    fancy signatures to letters of, i. 91, 146, 152, 181, 206,\n        237, 425; ii. 195;\n    takes the chair at the opening of the Liverpool Mechanics'\n        Institute, i. 94, and see i. 100-102;\n    his theatrical performances, see Theatrical Performances;\n    effects of work on, i. 121,; ii. 248, 266, 325;\n    _The Daily News_, started by, i. 135;\n    his visits to Lausanne and Switzerland, i. 147, 297, and\n        see Switzerland;\n    his visits to Paris, see Paris;\n    as a stage, manager, i. 163, 167, 231, 232, 237; ii. 26;\n    at Chester Place, Regent's Park, i. 169;\n    takes the chair at the opening of the Leeds Mechanics' Institute,\n        and of the Glasgow Athenaeum, i. 170;\n    at Brighton, i. 185, 213;\n    at Bonchurch, i, 204;\n    purchases Tavistock House, i. 240, and see Tavistock House;\n    as an editor, i. 246, 259, 269, 270, 285; ii. 127, 217, 262, 286,\n        292;\n    his readings, see Readings;\n    illnesses of, i. 14, 297; ii. 404, 405, 421, 446;\n    in America, ii. 338, 341, 347, 353, 355, 360, 365, 373, 377, 380,\n        381;\n    his visits to Boulogne, see Boulogne;\n    presentation of plate to, at Birmingham, i. 348;\n    purchases Gad's Hill, i. 377, 414, and see Gad's Hill;\n    delivers a speech on Administrative Reform, i. 377;\n    at Folkestone, i. 377, 378;\n    restlessness of, when at work, i. 402, 425;\n    tour of, in the North, ii. 5, 29-32;\n    his kindly criticisms of young writers, ii. 16, 34, 267, 277,\n        for other criticisms see i. 152, 188; ii. 14, 43, 215, 249;\n    elected a member of the Birmingham Institute, ii. 34;\n    religious views of, ii. 82, 202, 221, 394, 403, 444;\n    visit of, to Cornwall, ii. 108;\n    at Hanover Terrace, Regent's Park, ii. 135;\n    visits Lord Lytton at Knebworth, ii. 136;\n    at Hyde Park Gate South, ii. 170;\n    at 57, Gloucester Place, Hyde Park, ii. 208;\n    at Somer's Place, Hyde Park, ii. 224;\n    in the Staplehurst accident, ii. 224;\n    at Southwick Place, Hyde Park, ii. 224;\n    his energy, ii. 291;\n    one of the secrets of the success of, ii. 357, 392;\n    the Midland Institute at Birmingham opened by, ii. 406, and\n        see ii. 427;\n    his last speech, at the Royal Academy dinner, ii. 432;\n    his interview with the Queen, ii. 432;\n    attends a levee of the Prince of Wales, ii. 432;\n    his last illness, ii. 446;\n    his death, ii. 448;\n    funeral of, ii. 448, 449;\n    and see Letters of\n\n  Dickens, Mrs. Charles, marriage of, i. 1;\n    visit of, to America, i. 53;\n    at Rome, i. 135;\n    accident to, i. 215;\n    at Malvern, i. 239;\n    present to, at Birmingham, i. 298;\n    and see Letters\n\n  Dickens, Charles, jun., birth of, i. 4;\n    nickname of, i. 76;\n    at Eton, i. 212, 240, 243, 255, 258;\n    at Leipsic, i. 297, 310, 319;\n    at Barings', i. 455;\n    marriage of, ii. 208;\n    on \"All the Year Round,\" ii. 406, 410, 424;\n    and see i. 169, 233, 237, 243, 255, 258, 290, 347, 378, 405, 426;\n        ii. 88, 114, 123, 140, 145, 176, 447;\n    letters to, ii. 310, 338\n\n  Dickens, Kate, nickname of, i. 76;\n    marriage of, ii. 107, 113;\n    illness of, ii. 266, 271; and see ii. 39, 75, 77, 84, 221, 410,\n        436, 446;\n    letters to, i. 178; ii. 99\n\n  Dickens, Mamie, nickname of, i. 76;\n    illnesses of, i. 363, 436;\n    accident to, ii. 129;\n    and see ii. 39, 49, 55, 75, 77, 84, 87, 114, 116, 120, 145, 179,\n        234, 411, 447, and Letters\n\n  Dickens, Walter, nickname of, i. 76;\n    goes to India, ii. 19, 21;\n    attached to the 42nd Highlanders, ii. 114, 176;\n    death of, ii. 208, 212; and see i. 268, 314, 378, 443; ii. 4\n\n  Dickens, Frank, nickname of, i. 126;\n    letter of, to Dickens, ii. 93;\n    in India, ii. 208, 212; and see ii. 114, 131, 140, 177\n\n  Dickens, Alfred, at Wimbledon School, ii. 122;\n    settles in Australia, ii. 327; and see ii. 177, 371\n\n  Dickens, Sydney, birth of, i. 169;\n    nickname of, i. 170;\n    death of, i. 171;\n    story of, i. 223;\n    a naval cadet, ii. 125, 126, 145, 167;\n    on board H.M.S. _Orlando_, ii. 169; and see i. 363; ii. 114,\n        118, 122, 177, 202, 236, 260, 296, 430\n\n  Dickens, Henry, entered at Trinity Hall, Cambridge, ii. 327;\n    wins a scholarship, ii. 424, 430;\n      and see i. 363; ii. 177, 190, 254, 255, 329, 371, 389, 395,\n        406, 410, 447;\n    letters to, ii. 356, 392, 435, 438\n\n  Dickens, Edward, nicknames of, i. 322, 338;\n    goes to Australia, ii. 327, 329;\n    Dickens's love for, ii. 389-391;\n      and see i. 353, 359, 365, 403, 420, 426, 439; ii. 53, 76, 79,\n        92, 95, 153, 190, 199;\n    letter to, ii. 402\n\n  Dickens, Dora, birth of, i. 213;\n    death of, i. 240\n\n  Dickens, Alfred, sen., i. 184, 410; ii. 199\n\n  Dickens, Mrs. Augustus, ii. 418\n\n  Dickens, Fanny, see Mrs. Burnett\n\n  Dickens, Frederick, i. 9\n\n  Dickens, John, i. 240, 437; ii. 240\n\n  Dickens, Mrs. John, ii. 333\n\n  Dickens, Letitia, see Mrs. Henry Austin\n\n  Dickenson, Captain, ii. 224, 232\n\n  Dickson, David, letter to, i. 89\n\n  Diezman, S. A., letter to, i. 32\n\n  Dilke, C. W., ii. 5;\n    letter to, ii. 12\n\n  Dillon, C., ii. 42\n\n  Dinner, a search for a, i. 326;\n    ladies at public dinners, i. 103\n\n  Dogs, Dickens's, i. 67, 109, 110; ii. 50, 96, 101; ii. 203, 237,\n        242, 245, 264, 269;\n    a plague of, i. 292;\n    stories of, i. 109, 352, 354, 455\n\n  Dolby, George, ii. 245, 252-255, 267, 273, 280, 295, 296, 308, 310,\n        311, 317-323, 328, 330, 335, 336, 340, 345-347, 352-360, 363,\n        367, 381\n\n  \"Dombey and Son,\" i. 147;\n    success of, i. 156, 176;\n    sale of, i. 162\n\n  D'Orsay, Comte, i. 171, 244\n\n  Driver, Dickens's estimate of himself as a, i. 2\n\n  Drury Lane Theatre, the saloon at, i. 37;\n    suggestions for the saloon at, i. 52, 53\n\n  Dufferin, Lord, ii. 419\n\n  Dwarf, the Tartar, ii. 255\n\n\n  Earthquake, an, in England, ii. 206\n\n  Edinburgh on a Sunday, ii. 395\n\n  Education, Dickens an advocate of, for the people, i. 104\n\n  \"Edwin Drood,\" ii. 407, 431, 432, 446\n\n  Eeles, Mr., letters to, i. 265, 269\n\n  Egg, Augustus, i. 170, 172, 226, 297, 320, 332; ii. 198\n\n  Eliot, Sir John, Dickens on Forster's life of, ii. 215\n\n  Elliotson, Dr., i. 37, 149, ii. 99\n\n  Elton, Mr., i. 85, 92\n\n  Elwin, Rev. W., ii. 136, 151\n\n  Ely, Miss, letter to, i. 153\n\n  Emerson, Mr., ii. 306\n\n  Emery, Mr., i. 429\n\n  England, state of, in 1855, i. 391;\n    politically, i. 406\n\n  Epitaph, Dickens's, on a little child, i. 68\n\n  Executions, Dickens on public, i. 209, 212\n\n  Exhibition, an infant school at the, i. 257\n\n  Eytinge, Mr., ii. 405\n\n\n  Fairy Tales, Dickens on, i. 307\n\n  \"Faust,\" Gounod's, ii. 191, 193\n\n  Fechter, Charles, ii. 171, 177, 187, 193, 201, 219, 270, 386; and\n        see Letters\n\n  Felton, Mr., ii. 85\n\n  Ferguson, Sir William, ii. 246, 247\n\n  Feval, Paul, ii. 183, 192\n\n  Fielding, Henry, i. 394\n\n  Fields, Cyrus W., ii. 85, 308, 344, 361, 364, 379, 405\n\n  Fields, Mrs., ii. 306, 308, 319, 344, 361, 364, 367, 379, 405\n\n  Fildes, S. L., ii. 432, 447;\n    letter to, ii. 435\n\n  Finlay, F. D., ii. 406;\n    letters to, ii. 297, 389, 408\n\n  Fitzgerald, Mrs., ii. 285\n\n  Fitzgerald, Percy, ii. 187, 397;\n    and see Letters\n\n  Flunkeydom, scholastic, ii. 68\n\n  Forgues, M., i. 415, 421\n\n  Forster, Miss, ii. 327\n\n  Forster, John, i. 7, 10, 134, 143, 225, 240, 268, 428; ii. 108, 130,\n        265;\n    and see Letters\n\n  Franklin, Sir John, i. 373\n\n  Freake, Mr. and Mrs., ii. 446\n\n  French portraits of the English, i. 175\n\n  Friday, Dickens's lucky day, i. 414, 429\n\n  Frith, W. P., ii. 84, 93, 385, 418;\n    letters to, i. 79; ii. 439\n\n  Frost, the great, of 1861, ii. 139\n\n  Funerals, Dickens on state, i. 290; ii. 385\n\n\n  Gad's Hill, purchase of, i. 377, 378, 414;\n    Dickens takes possession of, ii. 3;\n    his childish impressions of, ii. 8;\n    improvements in, ii. 107, 373, 406, 446;\n    sports at, ii. 205;\n    cricket club at, ii. 356;\n    letters concerning, i. 384, 410, 429; ii. 15, 18, 25, 28, 49, 106,\n        119, 227\n\n  Gaskell, Mrs., i. 214;\n    and see Letters\n\n  Germany, esteem felt for Dickens in, i. 32\n\n  Ghost, stalking a, ii. 131\n\n  Gibson, M., i. 315; ii. 121\n\n  Gibson, Mr. and Mrs. Milner, ii. 431\n\n  Gladstone, Right Hon. W. E., ii. 401\n\n  Goldsmith, Oliver, Dickens on Forster's Life of, i. 188;\n    on the works of, i. 380\n\n  Gordon, Andrew, ii. 131\n\n  Gordon, Mr. Sheriff, ii. 164\n\n  \"Great Expectations,\" commenced, ii. 108, 136;\n    letters concerning, ii. 128, 133, 140, 142, 143, 151\n\n  Grief, the perversity of, exemplified, i. 18\n\n  Grimaldi, Life of, edited by Dickens, i. 4\n\n  Guild of Literature and Art, i. 239;\n    theatrical performances in aid of the, i. 239, 241, 248, 252, 268,\n        271;\n    and see ii. 41\n\n\n  Haldimand, Mr., i. 147, 169, 212, 380;\n    letters to, i. 157, 254\n\n  Halleck, Fitz-Greene, i. 59\n\n  \"Hard Times,\" i. 341;\n    satire of, explained, i. 349;\n    letters concerning, i. 355, 371\n\n  Harley, J. P., letters to, i. 5, 23\n\n  Harness, Rev. W., ii. 253;\n    letters to, i. 37, 76, 361\n\n  \"Haunted Man, The,\" i. 170, 185, 241;\n    subjects for illustrations in, described, i. 200, 201;\n    dramatisation of, i. 203\n\n  Headland, Mr., ii. 135, 149, 158, 160\n\n  Helps, Sir Arthur, ii. 432\n\n  Henderson, Mrs., letter to, ii. 293\n\n  Hewett, Captain, i. 57\n\n  \"History of England, The Child's,\" i. 297\n\n  Hogarth, Mary, i. 4, 9\n\n  Hogarth, Georgina, i. 425; ii. 50, 114, 145, 179, 202, 408, 436;\n    and see Letters\n\n  Hogge, Mrs., letter to, ii. 46\n\n  Holland, Lady, i. 11\n\n  Holmes, Mr., ii. 306\n\n  Home, longings for, i. 64, 70\n\n  Hood, Tom, i. 287;\n    letter to, i. 80\n\n  Horne, Mrs., letter to, i. 456\n\n  Horne, R. H., letter to, i. 93\n\n  Hospital, a dinner at a, i. 88;\n    Great Ormond Street, ii. 40, 46\n\n  Houghton, Lord, ii. 432;\n    letter to, i. 41\n\n  \"Household Words,\" i. 148;\n    scheme of, i. 216;\n    suggested titles for, i. 219;\n    success of, i. 221;\n    Christmas numbers of, i. 241, 288;\n    \"The Golden Mary,\" i. 414; ii. 11,\n    \"A House to Let,\" ii. 40;\n    incorporated with \"All the Year Round,\" ii. 83;\n    letters concerning, i. 219, 221, 250, 285, 286, 291-293, 295, 299,\n        301, 334, 335, 353, 423, 452; ii. 68\n\n  Hughes, Master Hastings, letter to, i. 14\n\n  Hulkes, Mrs., ii. 224, 315, 329;\n    letter to, ii. 232\n\n  Hullah, John, i. 5; ii. 131\n\n  Humphery, Mr. and Mrs., afterwards Sir W. and Lady, ii. 187\n\n  Hunt, Leigh, ii. 407\n\n  Hutchinson, John, ii. 380\n\n\n  _Illustrated London News_, offers to Dickens from, i. 150\n\n  Illustrations of Dickens's works, his descriptions for, i. 38-40,\n        45, 46, 50, 51, 200-203; ii. 237\n\n  Impeachment of the Five Members, Dickens on Forster's, ii. 14\n\n  Ireland, a dialogue in, ii. 61;\n    feeling for Dickens in, ii. 65;\n    Fenianism in, ii. 282-286;\n    proposed banquet to Dickens in, ii. 406;\n    Dickens on the Established Church in, ii. 409;\n    and see ii. 57, 60, 64\n\n  Italy, Dickens's first visit to, i. 94;\n    the sky of, i. 106;\n    the colouring of, i. 106;\n    a sunset in, i. 106;\n    twilight in, i. 107;\n    frescoes in, i. 107;\n    churches in, i. 108;\n    fruit in, i. 109;\n    climate of, i. 111;\n    a coastguard in, i. 116;\n    Dickens at Albaro, i. 105-117;\n    at Genoa, i. 120-122, 134, 321;\n    at Venice and Verona, i. 119-121, 337;\n    at Naples, i. 134-141, 322;\n    an ascent of Vesuvius, i. 137-141;\n    at Rome, i. 134, 135, 325-333;\n    Dickens on the unity of, ii. 84, 89, 90, 140, 211;\n    and see i. 297, 346\n\n\n  Jamaica, the insurrection in, ii. 241\n\n  Jeffrey, Lord, i. 184, 218\n\n  Jerrold, Douglas, i. 134, 225, 268, 390; ii. 3, 4, 19;\n    and see Letters\n\n  Jews, Dickens's friendly feeling for, ii. 204, 223, 280\n\n  Joachim, Joseph, ii. 413\n\n  John, Dickens's manservant, ii. 54, 56, 57, 72, 153, 187, 188, 255\n\n  Joll, Miss, letter to, i. 209\n\n  Jones, Walter, letter to, ii. 232\n\n\n  Keeley, Mrs., ii. 417\n\n  Keeley, Robert, i. 165;\n    letter to, i. 105\n\n  Kelly, Miss, i. 302, 303\n\n  Kelly, one of Dickens's readings men, ii. 305, 306, 342\n\n  Kemble, Fanny, ii. 344, 346\n\n  Kent, W. Charles, i. 186; ii. 225, 268, 407;\n    and see Letters\n\n  Kinkel, Dr., i. 230\n\n  Knight, Charles, i. 94; ii. 208;\n    and see Letters\n\n  Knowles, Sheridan, i. 214;\n    letter to, i. 215\n\n\n  \"Lady of Lyons, The,\" ii. 298\n\n  La Font, ii. 440\n\n  Lamartine, i. 187\n\n  Landor, Walter Savage, i. 268, 337; ii. 66;\n    and see Letters\n\n  Landseer, Edwin, letter to, i. 103\n\n  Landseer, Tom, i. 27\n\n  Lansdowne, Lord, i. 275\n\n  Law, Dickens's opinion of English, ii. 440\n\n  Layard, A. H., i. 377; ii. 108;\n    letters to, i. 390, 391\n\n  Leclercq, Miss, ii. 246\n\n  Lectures, Dickens on public, i. 97\n\n  Leech, John, i. 134, 186, 225, 226, 239\n\n  Le Gros, Mr., i. 140, 332\n\n  Lehmann, Mrs., ii. 39, 75;\n    and see Letters\n\n  Lehmann, F., ii. 39, 75\n\n  Lemaitre, M., i. 386\n\n  Lemon, Mark, i. 134, 186, 225, 226, 376, 390;\n    and see Letters\n\n  Lemon, Mrs., i. 419\n\n  Leotard, ii. 142\n\n  LETTERS OF CHARLES DICKENS TO:\n    Agate, John, ii. 154\n    Ainsworth, W. H., i. 43, 75, 92\n    Anonymous, i. 277; ii. 276\n    Armstrong, the Misses, ii. 175\n    Austin, Henry, i. 2, 69-73, 76, 262-264, 266, 361; ii. 18, 25, 28\n    Austin, Mrs., ii. 154, 180, 384\n    Babbage, Charles, i. 86, 87, 186\n    Bancroft, Mrs., ii. 441\n    Banks, G., i. 296\n    Baylis, Mr., ii. 179\n    Benzon, Miss, ii. 258\n    Bicknell, H., i. 229\n    Blanchard, Laman, i. 99\n    Boyle, Miss, i. 224, 225, 227, 245, 265, 279, 345, 381, 423;\n        ii. 10, 132, 157, 169, 186, 245, 315, 411\n    Brookfield, Mrs., ii. 249\n    Brookfield, Rev. W., ii. 199, 200\n    Brooks, Shirley, ii. 423, 438\n    Carlisle, the Earl of, i. 253, 281; ii. 12, 118, 157\n    Cartwright, Samuel, ii. 348\n    Cattermole, Mrs., ii. 383, 385\n    Cattermole, George, i. 22, 28-30, 31, 33-36, 38, 39, 42, 43,\n        45-48, 50, 51, 81, 143\n    Cerjat, M. de, i. 161, 210, 346, 378; ii. 7, 48, 86, 113, 138,\n        176, 200, 220, 240, 268, 387, 409\n    Chapman and Hall, i. 55\n    Clarke, John, ii. 418\n    Collins, Wilkie, i. 294, 358, 362, 397, 400, 403, 419, 437, 448;\n        ii. 40, 67, 101, 110, 129, 146, 182, 198, 209, 332, 397\n    Compton, Mrs., ii. 22\n    Cooke, T. P., ii. 21\n    Costello, Dudley, i. 104, 205\n    Croker, J. Crofton, i. 275\n    Cunningham, Peter, i. 195, 270, 312, 356\n    Dallas, Mrs., ii. 195, 434\n    Deane, F. H., i. 68\n    Delane, John, i. 314\n    Devonshire, the Duke of, i. 437, 443, 457\n    Dickens, Mrs. Charles, i. 12, 100, 123, 127, 130, 132, 165, 166,\n        206, 223, 244, 249, 267, 330, 406, 433\n    Dickens, Charles, ii. 310, 338\n    Dickens, Edward, ii. 402\n    Dickens, Henry, ii. 356, 392, 435, 438\n    Dickens, Miss Kate, i. 178; ii. 99\n    Dickens, Miss, i. 176, 178, 182, 199, 205, 453; ii. 52, 53, 56,\n        63, 72, 78, 95, 99, 124, 150, 161, 163, 165, 188, 190, 243,\n        252, 254, 256, 273, 275, 276, 278, 279, 283, 285, 299, 302,\n        306, 313, 316, 321, 324, 337, 341, 343, 350, 351, 354, 363,\n        366, 372, 377, 380, 389, 391, 399, 412, 415, 421, 426\n    Dickson, David, i. 89\n    Diezman, S. A., i. 32\n    Dilke, C. W., ii. 12\n    Eeles, Mr., i. 265, 269\n    Ely, Miss, i. 153\n    Fechter, Charles, ii. 183, 185, 191, 260, 297, 361, 368, 390\n    Fildes, S. L., ii. 435\n    Finlay, F. D., ii. 297, 389, 408\n    Fitzgerald, Percy, ii. 203, 217, 234, 237, 247, 263, 293, 294\n    Forster, John, i. 167, 188, 393; ii. 14, 42, 76, 97, 111, 128,\n        142, 215\n    Frith, W. P., i. 79; ii. 439\n    Gaskell, Mrs., i. 216, 269, 270, 292, 293, 301, 355, 360, 381\n    Haldimand, Mr., i. 157\n    Halleck, Fitz-Greene, i. 59\n    Harley, J. P., i. 5, 23\n    Harness, Rev. W., i. 37, 76, 361\n    Henderson, Mrs., ii. 293\n    Hogarth, Catherine, i. 3\n    Hogarth, Miss, i. 135, 177, 183, 319, 320, 322, 325, 337, 359, 385,\n        426, 428, 429, 435; ii. 28, 31, 33, 51, 55, 58, 61, 65, 70, 74,\n        75, 79, 126, 132, 137, 151, 152, 156, 158, 162, 165, 172-174,\n        190, 206, 248, 251, 253, 255, 257, 272, 274, 277, 279, 281,\n        282, 284-286, 295, 298, 303, 304, 307, 315, 317, 319, 327, 330,\n        334, 341, 345, 353, 358, 360, 364, 370, 371, 379, 391, 392, 396,\n        398, 400, 413-419, 421\n    Hogge, Mrs., ii. 46\n    Hood, Tom, i. 80\n    Horne, Mrs., i. 456\n    Horne, R. H., i. 93\n    Hughes, Master, i. 14\n    Hulkes, Mrs., ii. 232\n    Jerrold, Douglas, i. 87, 90, 118, 154, 427\n    Jewish Lady, a, ii. 204, 223, 280\n    Joll, Miss, i. 209\n    Jones, Walter, ii. 232\n    Keeley, Robert, i. 105\n    Kent, W. Charles, i. 188, 461; ii. 225, 239, 246, 299, 394, 429,\n        437, 439, 441, 443\n    Knight, Charles, i. 104, 152, 218, 259, 277, 280, 349, 351; ii.\n        195, 212\n    Knowles, Sheridan, i. 215\n    Landor, Walter Savage, i. 157, 230, 313, 343, 441\n    Landseer, Edwin, i. 103\n    Layard, A. H., i. 390, 391\n    Lehmann, Mrs. F., ii. 196, 234, 395, 413\n    Lemon, Mark, i. 192, 203, 207, 243, 281, 394, 396, 416, 439, 440\n    Longman, Thomas, i. 73; ii. 106\n    Longman, William, i. 24\n    Lovejoy, G., i. 44\n    Lytton, Sir E. B., ii. 116\n    Maclise, Daniel, i. 33, 105\n    Macready, W. C., i. 5, 16, 17, 18, 24, 26, 27, 49, 52, 60, 77, 79,\n        95, 117, 129, 141, 144, 146, 154, 183, 187, 194, 195, 198, 247,\n        252, 273, 283, 300, 307, 368, 399, 404, 430, 431, 446, 451, 459;\n        ii. 10, 19, 22, 46, 109, 141, 150, 192, 197, 226, 227, 229, 265,\n        373, 383, 424, 429, 436\n    Major, Mrs., ii. 196\n    Makeham, John, ii. 444\n    Marston, Dr. Westland, ii. 43\n    Milnes, R. Monckton, i. 41\n    Mitton, Thomas, i. 10, 19, 56, 58, 65, 121, 136, 458; ii. 229\n    Morpeth, Viscount, i. 92, 146,\n      and see Carlisle, The Earl of Ollier, Edmund, ii. 213, 425\n    Ouvry, F., ii. 205, 427\n    Owen, Professor, ii. 235\n    Panizzi, Antonio, ii. 89, 90, 92\n    Pardoe, Miss, i. 73\n    Parkinson, J. C., ii. 401\n    Pollock, Mrs. F., ii. 440\n    Pollock, Sir F., ii. 214\n    Poole, John, i. 236\n    Power, Miss, i. 179, 181, 460; ii. 127, 194\n    Power, Mrs., ii. 300\n    Procter, Adelaide, i. 374\n    Procter, B. W., i. 354; ii. 5, 82, 90, 223, 259\n    Procter, Mrs., ii. 226, 238\n    Reade, Charles, ii. 206\n    Regnier, Monsieur, i. 302, 303, 383, 411; ii. 44, 45, 102, 105, 189\n    Roberts, David, i. 215, 246, 248, 389\n    Russell, Lord John, i. 277, 316; ii. 118, 235, 422\n    Ryland, Arthur, i. 349, 382, 388; ii. 34, 233, 426, 428\n    Sandys, William, i. 178\n    Saunders, John, i. 366\n    Sculthorpe, W. R., ii. 104\n    Smith, Arthur, ii. 85, 147\n    Smith, H. P., i. 74, 179, 181\n    Stanfield, Clarkson, i. 92, 102, 113, 144, 151, 205, 299, 373, 394,\n        395, 398; ii. 184, 219, 287\n    Stanfield, George, ii. 289\n    Stone, Marcus, i. 340; ii. 211, 236\n    Stone, Frank, i. 199-201, 206, 259, 261, 295, 305, 355, 365, 396,\n        397; ii. 16, 24, 25, 27, 35, 82, 103\n    Storrar, Mrs., ii. 216\n    \"_Sun, The_,\" the editor of, i. 187\n    Tagart, Edward, i. 111, 173\n    Talfourd, Miss Mary, i. 51\n    Talfourd, Serjeant, i. 10\n    Tennent, Sir James Emerson, i. 329; ii. 6, 218, 259\n    Thackeray, W. M., ii. 41\n    Thornbury, Walter, ii. 178, 262, 286\n    Tomlin, John, i. 40\n    Toole, J. L., ii. 300\n    Trollope, Mrs., i. 81, 397\n    Viardot, Madame, i. 412\n    Ward, E. M., ii. 141\n    Ward, Mrs., ii. 441\n    Watkins, John, i. 287; ii. 148\n    Watson, Hon. Mrs., i. 171, 196, 209, 226, 228, 231, 234, 237, 242,\n        254, 276, 282, 289, 309, 317, 343, 370, 402, 412, 453; ii. 93,\n        121, 144, 301, 382\n    Watson, Hon. R., i. 159\n    White, Mrs., ii. 94\n    White, Miss, ii. 92\n    White, Rev. James, i. 149, 193, 208, 217, 220, 288, 291, 292, 350;\n        ii. 11, 15, 81, 97\n    Wills, W. H., i. 148-150, 219, 221, 222, 244, 250, 285, 286, 292,\n        295, 299, 303, 304, 307, 315, 333, 334, 352, 357, 384, 387,\n        401, 407, 408, 410, 415, 433, 450, 452; ii. 119, 167, 168, 171,\n        207, 290, 292, 295, 301, 386, 422, 433\n    Wilson, Effingham, i. 199\n    Yates, Edmund, ii. 20, 34, 41, 47, 91, 123, 149, 238\n    Yates, Mrs., ii. 48\n\n  Lewes, G. H., i. 170\n\n  \"Lighthouse, The,\" the play of, i. 337;\n    Dickens's prologue to, i. 461;\n    Dickens's \"Song of the Wreck\" in, i. 461;\n    and see ii. 198\n\n  Linton, Mrs., ii. 207\n\n  Lion, a chained, i. 144\n\n  Literary Fund, the, ii. 5, 12\n\n  \"Little Dorrit,\" i. 378, 413, 415;\n    proposed name of, i. 402;\n    sale of, i. 426;\n    letters concerning, i. 402, 403, 406, 426\n\n  Lockhart, Mr., ii. 207\n\n  London, the Mayor of, from a French point of view, i. 175;\n    in September, i. 318;\n    Dickens's opinion of the Corporation of, i. 389; ii. 411;\n    facetious advice to country visitors to, i. 252\n\n  Longfellow, W. H., ii. 306, 308, 312, 326, 333, 361, 375\n\n  Longman, Thomas, letters to, i. 73; ii. 106\n\n  Longman, William, letter to, i. 24\n\n  Lovejoy, G., i. 44\n\n  Lowell, Miss Mabel, ii. 405, 415\n\n  Lyceum Theatre under Fechter, ii. 187, 191, 245;\n    and see Fechter\n\n  Lyndhurst, Lord, i. 147; ii. 144\n\n  Lynn, Miss, i. 378\n\n  Lyttelton, Hon. Spencer, i. 239, 245\n\n  Lytton, the first Lord, i. 214, 239; ii. 108, 135, 143, 247, 268;\n    letter to, ii. 116\n\n  Lytton, Lord, ii. 108\n\n\n  Maclise, Daniel, i. 18, 23, 80, 177, 370; ii. 432;\n    letters to, i. 33, 105\n\n  Macready, W. C., i. 94, 133, 239, 413; ii. 169, 172, 173;\n    and see Letters\n\n  Macready, Benvenuta, i. 431; ii. 194\n\n  Macready, Kate, i. 415; ii. 193\n\n  Macready, Mrs., ii. 172, 288\n\n  Macready, Jonathan, ii. 376\n\n  Macready, Nina, i. 195\n\n  Macready, W., ii. 425\n\n  Major, Mrs., letter to, ii. 196\n\n  Makeham, J. M., ii. 432;\n    Dickens's last letter written to, ii. 444\n\n  Malleson, Mr. and Mrs., ii. 315\n\n  Marsh, Dickens's coachman, a story of, ii. 181\n\n  Marston, Dr. Westland, ii. 42, 44, 45;\n    letter to, ii. 43\n\n  Martineau, i. 61, 229\n\n  \"Martin Chuzzlewit,\" i. 53;\n    dramatised, i. 95, 105;\n    a story of Mrs. Harris, ii. 41\n\n  \"Master Humphrey's Clock,\" i. 28;\n    the plan of, described, i. 29;\n    letters concerning illustrations for, i. 29-31, 33-36, 38-40,\n        45-47, 50-51\n\n  \"Memoires du Diable, Les,\" i. 444\n\n  Mesmerism, a seance of, ii. 100\n\n  Missionaries, Dickens on, i. 227; ii. 241\n\n  Mitton, Thomas, see Letters\n\n  Molesworth, Lady, ii. 187, 189\n\n  Monuments, Dickens on, i. 287, 356\n\n  Moore, Tom, i. 163\n\n  Morgan, Captain, ii. 136, 143\n\n  Morgan, W., ii. 308, 336\n\n  Morley, Mr., i. 399\n\n  Morpeth, Viscount, letters to, i. 92, 146;\n    and see Carlisle, The Earl of\n\n  Motley, Mr., ii. 142\n\n  Mountain, a hazardous ascent of a, ii. 29\n\n  Mulgrave, Earl of, i. 57\n\n\n  Narrative, i. 1, 4, 6, 17, 28, 36, 53, 57, 85, 94, 134, 147, 169,\n      185, 204, 213, 239, 271, 296, 341, 376, 413; ii. 3, 39, 83,\n      107, 135, 169, 187, 208, 224, 244, 266, 325, 404, 431, 446\n\n  Nathan, Messrs. H. and L., i. 232, 233, 235\n\n  Neville, Mr., ii. 186\n\n  Newsvendors' Benevolent Institution, ii. 232\n\n  New Testament, Dickens's love for the, ii. 394, 403;\n    Dickens writes a history of the, for his children, ii. 433\n\n  \"Nicholas Nickleby,\" publication of, i. 6;\n    rewards and punishments of characters in, i. 14;\n    Dickens at work on, i. 16;\n    dedication of, i, 26;\n    the Kenwigs in, i, 25;\n    and see ii. 200\n\n  Nicknames, Dickens's, of George Cattermole, i. 42, 143;\n    of his children, i. 76, 126, 170, 322, 338, 453;\n    nautical, i. 152;\n    of himself, i. 198, 206, 307, 362;\n    of Frank Stone, i. 214, 305\n\n  Norton, C. E., ii. 326\n\n  Noviomagians, the, i. 272\n\n\n  \"Old Curiosity Shop, The,\" Dickens engaged on, i. 28;\n    scenes in, described by Dickens for illustration, i. 21, 33-37, 42;\n    Dickens heartbroken over the story, i. 36, 37, 42\n\n  \"Oliver Twist,\" publication of, i. 4;\n    Dickens at work on, i. 11;\n    the reading of \"The Murder\" from, ii. 326, 395, 397, 399\n\n  Ollier, Edmund, ii. 209, 407;\n    letters to, ii. 213, 425\n\n  Olliffe, Lady, ii. 187, 190\n\n  Olliffe, Sir J., ii. 417\n\n  Olliffe, the Misses, ii. 190\n\n  Organs, street, i. 104\n\n  Osgood, Mr., ii. 310, 336, 337, 340, 346, 352, 356, 366\n\n  \"Our Mutual Friend,\" ii. 208, 210, 224;\n    and as to illustrations for, see ii. 211, 237\n\n  Ouvry, Frederic, ii. 188, 300;\n    letters to, ii. 205, 427\n\n  Overs, i. 37, 49\n\n  Owen, Professor, ii. 235\n\n\n  Panizzi, Antonio, ii. 84;\n    letters to, ii. 89, 90, 92\n\n  Pardoe, Miss, letter to, i. 73\n\n  Paris, Dickens at, i. 130, 131, 147, 157-161, 169, 174, 239, 376,\n        378, 385-387, 413, 406-425, 430, 431; ii. 171, 187;\n    house-hunting in, i. 158;\n    description of Dickens's house in, i. 159;\n    state of, in 1846, i. 160, 161;\n    feeling of people of, for Dickens, i. 411;\n    Dickens's reading at, ii. 187-190, 192\n\n  Parkinson, J. C., ii. 327;\n    letter to, ii. 401\n\n  Parrots, human, i. 87, 121\n\n  \"Patrician's Daughter, The,\" prologue to, written by Dickens,\n      i. 55, 77\n\n  Patronage, the curse of England, ii. 213, 356\n\n  Paxton, Sir Joseph, i. 446\n\n  Phelps, J., i. 366\n\n  \"Pickwick,\" origin and publication of, i. 1, 3;\n    first mention of Jingle, i. 3;\n    conclusion of, celebrated, i. 5;\n    the design of the Shepherd in, explained, i. 85, 89\n\n  Picnic, a, of the elements, i, 116;\n    with Eton boys, i. 255, 258\n\n  \"Picnic Papers,\" Dickens's share of the, ii. 91\n\n  Plessy, Madame, i. 412; ii. 440\n\n  Pollock, Sir F., ii. 97, 144, 209;\n    letter to, ii. 10, 214\n\n  Pollock, Mrs. F., letter to, ii. 440\n\n  Poole, John, i. 298, 317; ii. 228;\n    letter to, i. 236\n\n  \"Poor Travellers, The,\" i. 378;\n    sale of, i. 379\n\n  Portraits of Dickens, by Maclise, i. 18, 23;\n    by Frith, ii. 84, 93;\n    by Ary Scheffer, i. 414, 434;\n    by John Watkins, ii. 148;\n    a caricature, ii. 146\n\n  Postman, an Albaro, i. 112, 117\n\n  Power, Miss, i. 442; ii. 82, 293, 300;\n    and see Letters\n\n  Power, Nelly, i. 443\n\n  Power, Mrs., letter to, ii. 300\n\n  Presence of mind of Dickens, ii. 161, 224, 230\n\n  Press, the, freedom of, i. 49;\n    in America, i. 97;\n    taxation of the, i. 274\n\n  Procter, Adelaide, i. 341; ii. 238;\n    letter to, i. 374\n\n  Procter, B. W., i. 341; ii. 83, 91;\n    and see Letters\n\n  Procter, Mrs., letter to, ii. 226, 238\n\n  Publishing system, how to improve the, i. 86\n\n  Purse, the power of the, i. 88\n\n  Putnam, Mr., ii. 312\n\n\n  Queen, the, Dickens's theatrical performance before, i. 239;\n    his feeling for, ii. 168;\n    his interview with, ii. 432\n\n\n  Rae, Dr., i. 373\n\n  Railways, ii. 242\n\n  Reade, Charles, ii. 188;\n    letter to, ii. 206\n\n  Reader, Charles Dickens as a, ii. 437\n\n  Readings, Dickens's public, for charities, i. 297, 341, 377; ii. 4,\n        169, 170;\n    first reading for his own benefit, ii. 39;\n    at Paris, ii. 187, 189, 192;\n    in America, ii. 267;\n    farewell series of readings in England, ii. 326, 404, 405;\n    trial reading of \"The Murder\" from \"Oliver Twist,\" ii. 326;\n    reading to the actors, ii. 407, 418;\n    farewell reading, ii. 431;\n    effects of \"The Murder\" reading on Dickens, ii. 434;\n    books of the, ii. 438;\n    letters concerning the readings in England, Scotland, and Ireland,\n        i. 344, 348, 369, 371, 379, 382, 388, 413, 424; ii. 20, 49,\n        51-67, 70-80, 87, 103, 145, 147, 151-168, 174, 178, 197, 200,\n        251-258, 272-286;\n    letters concerning American, ii. 83, 85, 290, 294, 298, 299,\n        306-382;\n    letters concerning the farewell series of, ii. 391, 392, 395-400,\n        412-421\n\n  Reform, Dickens speaks on Administrative, i. 377, 399;\n    association for, i. 399;\n    Dickens on Parliamentary, ii. 87, 269\n\n  Refreshment rooms, i. 424\n\n  Regnier, M., i. 298;\n    and see Letters\n\n  Reynolds, Dr. Russell, ii. 448\n\n  Richardson, Samuel, Dickens's opinion of, i. 175\n\n  \"Rivals, The,\" a scene from, rewritten, i. 345\n\n  Roberts, David, i. 214; ii. 75;\n    letters to, i. 215, 246, 248, 389\n\n  \"Robinson Crusoe,\" Dickens on, i. 443\n\n  Robson, F., i. 451\n\n  Roche, Dickens's courier, i. 95, 122-126, 139\n\n  Rochester Cathedral, proposed burial of Dickens in, ii. 448\n\n  Royal Academy, female students at the, ii. 121;\n    Dickens's last public appearance, at the dinner of the, ii. 431\n\n  Russel, Alexander, ii. 389, 390, 398, 406\n\n  Russell, Lord John, i. 272; ii. 85;\n    and see Letters\n\n  Russell, W. H., ii. 4\n\n  Ryland, Arthur, ii. 4, 430;\n    and see Letters\n\n\n  Sainton-Dolby, Madame, ii. 295, 391\n\n  Sanatorium for art-students, i. 102\n\n  Sand, Georges, i. 420\n\n  Sandys, William, letter to, i. 178\n\n  Saunders, John, i. 341;\n    letter to, i. 366\n\n  Savage, i. 271\n\n  Saville, Miss, ii. 186\n\n  Scheffer, Ary, i. 414, 434; ii. 149\n\n  Schoolmistress, a Yorkshire, i. 8\n\n  Scott, Sir Walter, i. 22, 254\n\n  Scott, Dickens's dresser, ii. 272, 305, 306, 317, 321, 342, 370, 416\n\n  Scribe, Eugene, i. 430, 432\n\n  Sculthorpe, W. R., letter to, ii. 104\n\n  Seaside, the, in wet weather, i. 90\n\n  Sea voyage, a, i. 322\n\n  Shaftesbury, Lord, ii. 242\n\n  Shakespeare, Dickens in room of, i. 13;\n    Dickens's criticisms of Charles Knight's biography of, i. 152;\n    and see i. 178\n\n  Shea, Mr. Justice, ii. 247\n\n  Shower-bath, a perpetual, i. 207\n\n  \"Sketches,\" publication of the, i. 1\n\n  Smith, Arthur, ii. 4, 39, 52, 53, 56-60, 64-67, 71, 72, 78, 80,\n      104, 109, 135, 145, 149-153;\n    letters to, ii. 85, 147\n\n  Smith, H. P., letters to, i. 74, 179, 181\n\n  Smith, Sydney, i. 24\n\n  Smollett, Dickens on the works of, i. 356\n\n  Snevellicci, Miss, in real life, i. 13\n\n  Snore, a mighty, i. 158\n\n  Songs by Dickens: on Mark Lemon, i. 207;\n    of \"The Wreck\" in \"The Lighthouse,\" i. 461\n\n  Speaking, Dickens on public, ii. 426, 428;\n    advice to his son Henry on public, ii. 435\n\n  Spencer, Lord, i. 242\n\n  Spider, a fearful, i. 180\n\n  Spiritualism, Dickens on, i. 350, 397\n\n  Stage-coach, American story of a, ii. 292\n\n  Stage suggestions, i. 79;\n    a stage mob, i. 174;\n    a piece of stage business, i. 156\n\n  Stanfield, Clarkson, i. 370, 377, 429, 435, 454; ii. 75, 194, 267;\n    and see Letters\n\n  Stanfield, George, letter to, ii. 289\n\n  Stanley, Dean, ii. 448, 449\n\n  Stanley, Lady Augusta, ii. 449\n\n  Staplehurst, Dickens in the railway accident at, ii. 224;\n    description of the accident, ii. 229-233;\n    effects of the accident on Dickens, ii. 388\n\n  Staunton, Mr. Secretary, ii. 351\n\n  Steele, Sir Richard, Dickens on Forster's essay on, i. 393\n\n  Steele, Mr., ii. 447, 448\n\n  Stone, Arthur, i. 436\n\n  Stone, Ellen, ii. 81\n\n  Stone, Frank, i. 134, 143, 225, 240; ii. 84;\n    and see Letters\n\n  Stone, Marcus, i. 299; ii. 84, 106, 208;\n    letters to, i. 340; ii. 211, 236\n\n  Storrar, Mrs., ii. 209;\n    letter to, ii. 216\n\n  \"Strange Gentleman, The,\" farce written by Dickens and produced, i. 4;\n    price of, i. 5;\n    sent to Macready, i. 16\n\n  Strikes, Dickens on, i. 416\n\n  Sumner, Charles, ii. 351, 355\n\n  _Sun, The_, newspaper, ii. 225;\n    letter to editor of, i. 187\n\n  Switzerland, the Simplon Pass in, i. 127;\n    pleasant recollections of, i. 197, 218;\n    Dickens at Lausanne in, i. 147;\n    a revolution in, i. 155, 175;\n    friends in, i. 157;\n    Dickens's love for, i. 158;\n    letters concerning Lausanne in, i. 147, 154, 160, 172, 179\n\n  Sympathy, letters of, i. 193, 265, 282, 283, 394; ii. 94, 97,\n      123, 154, 180, 289, 293\n\n\n  Tagart, Edward, letters to, i. 111, 173\n\n  \"Tale of Two Cities, A,\" ii. 83, 84, 158;\n    letters concerning, ii. 98, 102, 105, 106, 116\n\n  Talfourd, Miss Mary, letter to, i. 51\n\n  Talfourd, Mr. Justice, i. 7;\n    letter to, i. 10\n\n  Tauechnitz, Baron, i. 188, 195\n\n  Tavistock House, purchase of, i. 240;\n    sale of, ii. 107;\n    letters concerning, i. 259, 261-266\n\n  Taxation, Dickens on, i. 218;\n    of newspapers, i. 273\n\n  Taylor, Bayard, ii. 405\n\n  Telegraph, the dramatic side of the, i. 417\n\n  Tennent, Sir James Emerson, i. 298; ii. 209, 224;\n    letters to, i. 329; ii. 218, 259\n\n  Tenniel, John, i. 241\n\n  Tennyson, Alfred, Dickens's admiration for, ii. 98\n\n  Terry, Miss Kate, ii. 193\n\n  Thackeray, W. M., ii. 4, 39, 137, 208, 210, 214;\n    letter to, ii. 41\n\n  Thames, drainage of the, ii. 50;\n    embankment of the, ii. 410\n\n  Theatre, Dickens at the, i. 13;\n    Phiz's laughter at the, i. 13;\n    the saloon at Drury Lane, i. 37, 52;\n    scents of a, i. 96;\n    story of a, i. 144;\n    proposal for a national, i. 199;\n    Dickens on the, ii. 271, 438\n\n  Theatrical Fund, the, ii. 35\n\n  Theatrical performances of Charles Dickens:\n    at Montreal, i. 72;\n    at Miss Kelly's Theatre, i. 134;\n    \"Fortunio\" at Tavistock House, i. 376, 381;\n    \"The Lighthouse,\" i. 377, 394-397;\n    \"The Frozen Deep,\" i. 414;\n    for the Jerrold Memorial Fund, ii. 19, 23;\n    before the Queen, i. 239;\n    and see i. 170, 185, 239, 241, 271, 376, 377, 414; ii. 3;\n    letters concerning the, i. 141, 143, 144, 146, 181, 192, 196,\n        224-228, 231, 232, 234, 244, 268, 398, 433, 453, 454, 457,\n        459, 460; ii. 6, 11, 198\n\n  Thornbury, Walter, ii. 170, 292;\n    letters to, ii. 178, 262, 286\n\n  Tomlin, John, letter to, i. 40\n\n  Toole, J. L., ii. 54, 268;\n    letter to, ii. 300\n\n  Topham, F. W., i. 241, 269\n\n  Townshend, Chauncey Hare, ii. 7, 86, 96, 115, 136, 140, 371, 410\n\n  Trollope, Mrs., letters to, i. 80, 397\n\n\n  \"Uncle Tom's Cabin,\" Dickens on, i. 289\n\n  \"Uncommercial Traveller, The,\" ii. 107\n\n\n  Viardot, Madame, ii. 193;\n    letter to, i. 412\n\n  \"Village Coquettes, The,\" operetta written by Dickens, i. 5;\n    and see i. 93\n\n  Volunteers, Dickens on the, ii. 115\n\n\n  Waistcoat, a wonderful, i. 102;\n    the loan by Dickens of Macready's, i. 146\n\n  Wales, the Prince of, popularity of, ii. 203;\n    Dickens attends levee of, ii. 432\n\n  Wales, the Princess of, her arrival in England, ii. 195;\n    the illuminations in honour of, ii. 198;\n    popularity of, ii. 203\n\n  War, Dickens on the Russian, i. 379\n\n  Ward, E. M., i. 341;\n    letter to, ii. 141\n\n  Ward, Mrs., letter to, ii. 441\n\n  Watkins, John, i. 415;\n    letters to, i. 287; ii. 148\n\n  Watson, Hon. R., i. 147, 280;\n    letter to, i. 159\n\n  Watson, Hon. Mrs., i. 147; ii. 9, 70;\n    and see Letters\n\n  Watson, Sir Thomas, ii. 405, 407\n\n  Watson, Wentworth, ii. 79\n\n  Watts's refuge for poor travellers, ii. 259\n\n  Webster, Benjamin, i. 85, 90, 434; ii. 361\n\n  Webster, a story of the murderer, ii. 333\n\n  Welcome home, a, i. 117\n\n  Westminster Abbey, burial of Dickens in, ii. 448\n\n  Whewell, Dr., i. 372\n\n  White, Clara, ii. 142, 181, 208\n\n  White, Rev. James, i. 149, 413; ii. 209;\n    and see Letters\n\n  White, Mrs., ii. 212;\n    letter to, ii. 94\n\n  White, Miss, ii. 81, 84, 96;\n    letter to, ii. 92\n\n  White, Richard Grant, ii. 85\n\n  Wigan, Alfred, i. 429\n\n  Williams, Mr. and Mrs. Barney, ii. 337, 359\n\n  Wills, W. H., i. 148, 241, 375; ii. 83, 379, 383, 406, 430;\n    and see Letters\n\n  Wills, Mrs., ii. 75, 96, 120\n\n  Wilson, Effingham, letter to, i. 199\n\n  Working men, clubs for, ii. 209, 213;\n    Dickens on the management of such clubs, ii. 356;\n    feeling of, for Dickens, ii. 420\n\n\n  Yates, Edmund, i. 414, 426; ii. 5, 129;\n    and see Letters\n\n  Yates, Mrs., ii. 129;\n    letter to, ii. 48\n\n\nTHE END.\n\nCHARLES DICKENS AND EVANS, CRYSTAL PALACE\nPRESS.\n\n       *       *       *       *       *\n\nTranscriber's Notes:\n\nPage 142, \"Leotard\" changed to \"Leotard\" twice (Palace and Leotard) and\n(into seeing Leotard)\n\nPage 181, \"shefound\" changed to \"she found\" (she found Marsh)\n\nPage 432, \"levee\" changed to \"levee\" (a levee held)\n\nPage 453, \"Celeste\" changed to \"Celeste\" (Celeste, Madame)\n\nPage 454-455, entries for \"Dickens, Mamie\" and \"Dickens, Kate\" were\noriginally not in alphabetically order. This was corrected.\n\nPage 456, \"Fitzgreene\" changed to \"Fitz-Greene\" (Halleck, Fitz-Greene)\n\nPage 458, \"Fitzgreene\" changed to \"Fitz-Greene\" (Halleck, Fitz-Greene)\n\nPage 460, \"Lyttleton\" changed to \"Lyttelton\" (Lyttleton, Hon. Spencer)\n\nPage 462, \"Shee\" changed to \"Shea\" (Shea, Mr. Justice)\n\n\n\n\n\nEnd of Project Gutenberg's The Letters of Charles Dickens, by Charles Dickens\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzrdra b/data_all_eng_slimpj/shuffled/split2/finalzzrdra
new file mode 100644
index 0000000000000000000000000000000000000000..d9fc00689f7850ff0439119e68aee859230bd21d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzrdra
@@ -0,0 +1,5 @@
+{"text":" \n# Traveling With the Dead\n\n### Barbara Hambly\n\nFor George\n\nWith a prayer in the shadow of the Aya Sofia\n\n# CONTENTS\n\nPROLOGUE\n\nONE\n\nTWO\n\nTHREE\n\nFOUR\n\nFIVE\n\nSIX\n\nSEVEN\n\nEIGHT\n\nNINE\n\nTEN\n\nELEVEN\n\nTWELVE\n\nTHIRTEEN\n\nFOURTEEN\n\nFIFTEEN\n\nSIXTEEN\n\nSEVENTEEN\n\nEIGHTEEN\n\nNINETEEN\n\nTWENTY\n\nTWENTY-ONE\n\nTWENTY-TWO\n\nA BIOGRAPAHY OF BARBARA HAMBLY\n\n# PROLOGUE\n\nTHE HOUSE WAS an old one, inconspicuous for its size. Curiously so, thought Lydia Asher, when she stood at last on the front steps, craning her neck to look up at five stories of shut-faced dark fa\u00e7ade. More curious still, given the obvious age of the place, was the plain half timbering discernible under centuries of discoloration and soot, the bull's-eye glass of the unshuttered windows, the depth to which the centers of the stone steps had been worn.\n\nLydia shivered and pulled closer about her the coat she'd borrowed from her cook\u2014even the plainest from her own collection would have been hopelessly fashionable for these narrow, nameless courts and alleys that clustered behind the waterfront between Blackfriars Bridge and Southwark. _He can't hurt me,_ she thought, and brought up her hand to her throat. Under the high neck of her plain wool waist she could feel the thick links of half a dozen silver chains against her skin.\n\n_Can he?_\n\nIt had taken her nearly an hour to find the court, which by some trick of chance had been left off all four modern maps of this part of London. The whole yard was adrift in fog the color of ashes, and at this hour\u2014Lydia heard three o'clock strike in the black steeple of the crumbling pre-Wren church that backed the old house\u2014even the little remaining light was bleeding away. She had passed the house three times before truly seeing it, and sensed that had the air been clear, it would somehow still have been difficult to look at the place. She had the absurd impression that by night, lanterns or no lanterns, streetlamps or no streetlamps, it would not be visible at all.\n\nThere was a smell about it, too, distinct and terrifying, but impossible to place.\n\nShe stood for a long time at the foot of its steps.\n\n_He can't hurt me,_ she told herself again, and wondered if that were true.\n\nHer heart was beating hard, and she noted clinically the cold in her extremities, in spite of fur-lined leather gloves and two pairs of silk stockings under her dainty, high-heeled boots. Stouter shoes would have somewhat alleviated the situation, always supposing stout shoes existed that did not make their wearer look like a washerwoman\u2014if they did, Lydia had never seen them\u2014but the panicky scald of adrenaline in her bloodstream informed her that the cold she felt was probably shock.\n\nIt was one thing to speculate about the physiology of the house's owner in the safety of her own study at Oxford, or with James close by and armed.\n\nIt was evidently quite another to go up and knock on Don Simon Ysidro's front door.\n\nMuffled by the fog, she heard the _tock_ of hooves, the jingle of harness from Upper Thames Street, and the groaning hoot of the motorbuses. Another hoot, deeper, came from some ship on the river. The click of her heels on the dirty steps was the strike of a hammer, and her petticoat's rustle the rasp of a saw.\n\nFor all the house's age, the lock on the door was relatively new, a heavy American pin lock oddly masked behind what must have been the original lock plate of Elizabeth's time. It yielded readily enough to the skeleton keys she'd found at the back of her husband's handkerchief drawer. Her hands shook a little as she then operated the picklocks in the fashion he'd taught her, partly from the sheer fear of what she was doing, and partly because, law-abiding and essentially orderly, she expected a member of the Metropolitan Police to appear behind her crying, _'Ere, now, wotcher at?_\n\nAbsurd on the face of it, she thought. It was patently obvious that no representative of the law had set foot in this square in years.\n\nShe pushed her thick-lensed spectacles more firmly up onto the bridge of her nose\u2014 _Not only breakin' the law,_ roared the imaginary policeman, _but ugly and four-eyed to boot!_ \u2014slipped the picklocks and skeleton keys back into her handbag, and stepped through the door.\n\nIt wouldn't be full dark until five. She was perfectly safe.\n\nThe hall itself was much darker than she had expected, with the wide oak doors on either side closed. Trimmed with a carved balustrade, generous steps ascended carpetless to blindness above. The passage beside them to the rear of the house was an open grave.\n\nThere was, of course, no lamp.\n\nMildly berating herself for not having foreseen that contingency\u2014 _of course there wouldn't be a lamp!_ \u2014Lydia pushed open one of the side doors to admit a rinsed and cindery light. It showed her a key on the hall table, and turning, she closed the front door. For a time she stood undecided, debating whether to lock herself in and observing the deleterious effects of massive amounts of adrenaline on her ability to concentrate...\n\n_How would I go about charting degree of panic with inability to make a decision? The workhouse wouldn't_ really _let me put my subjects into life-threatening situations._\n\nIn the end she turned the key but left it in the lock, and stepped cautiously through the door she had opened, into what had probably been a dining room but was as large as the ballroom of her aunt's house in Mayfair. It was lined floor to ceiling with books: goods boxes had been stacked on top of the original ten-foot bookshelves, and planks stretched over windows and doors so that not one square foot of the original paneling showed and the tops of the highest ranks brushed the coffered ceiling. Yellow-backed adventure novels by Conan Doyle and Clifford Ashdown shouldered worn calf saints' lives, antiquated chemistry texts, Carlyle, Gibbon, de Sade, Balzac, cheap modern reprints of Aeschylus and Plato, Galsworthy, Wilde, Shaw. In front of the bone-clean fireplace, a massive oak chest, strapped with leather and the only furniture in the room, held a cheap American oil lamp of clear glass and steel, the trimmed wick in about half a reservoir of oil. Lydia produced a match from her pocket, lit the lamp, and by its uncertain light read the titles of the several new volumes, half unwrapped from their parcel paper, which lay beside it.\n\nA French mathematics text. A German physics book by a man named Einstein. _The Wind in the Willows_.\n\nHow much time left?\n\nWith a certain amount of difficulty Lydia produced from beneath her coat a curious device\u2014a simple brass bug sprayer of the pump variety, its nozzle carefully capped with a pinch of sticking plaster\u2014and a shoulder sling manufactured from a couple of scarves in last year's colors. She removed the cap, reslung the sprayer on the outside of her coat and, picking up the lamp, moved off through the house.\n\nThe first-floor room contained more books. The rear chamber, book-lined also, held furniture as well. A heavy table, strewn with mathematics texts, abaci, astrolabes, armillary spheres, a German Brunsviga tabulation machine, and what Lydia recognized dimly as an old set of ivory calculating bones. At the far end of the room loomed a machine the size of an upright piano, sinister with glass, metal, and ranks of what looked like clock faces, whose use Lydia could not begin to guess. Near it stood a blackwood cabinet desk, German and ruinously old, carved thick with gods and trees, among which peeped the tarnished brass locks to concealed recesses and drawers.\n\nA wing chair of purple velvet, very worn and rubbed, stood before a fireplace whose blue and yellow tiles were smoked almost to obscurity, its arms covered with cat hair, an American newspaper lying on its seat. Movement caught her eye and made her gasp, but it was only her own reflection in a yellowed mirror, the glass nearly covered by a great shawl of eighteenth-century black point lace that hung over its divided pane.\n\nLydia set the lamp down and lifted the shawl aside. Thin and rather fragile-looking, her reflection gazed back at her: flat-chested and schoolgirlish, she thought despairingly, despite her twenty-six years. And despite everything she could do with rice powder, kohl, and the tiny amount of rouge that were all a properly brought-up lady could wear, her face was still all nose and spectacles. _Four-eyes_ , they'd called her, all her childhood and adolescence\u2014when it wasn't _skinnybones_ or _bookworm_ \u2014and if her life didn't, quite literally, depend on how quickly she could see danger in this place, she'd never have worn her eyeglasses outside her rented Bloomsbury rooms.\n\nHer life, and James' as well.\n\nShe let the lace fall, touched again the silver around her neck and the fat, doubled and trebled links of it that circled her wrists under cuffs and gloves.\n\n_Why a mirror?_ Something one wouldn't expect to find here. Did that mean the stories were wrong?\n\nShe picked up the lamp again, hoping the information she'd learned on the subject was even partially correct. It was a disgrace, really, that over the years more scientific data had not been collected. She would definitely have to write an article for the _Journal of Medical Pathology_ \u2014or perhaps for one of James' folkloric publications.\n\nIf she lived, she thought, and panic heated in her veins again. If she lived.\n\nWhat if she were doing this wrong?\n\nShe found another floor of high-ceilinged rooms, plus attics, all of them filled with either books or journals. Her own experience with the proliferative propensities of back issues of _Lancet_ and its competitors\u2014British, European, and American\u2014gave her a lively sense of sympathy, and an envious appreciation for so much shelf space almost, for the moment, eased her fear. _Lancet_ went back to 1823, and she had little doubt the first issue could be found here somewhere. One small chamber upstairs contained clothing, expensive and relatively new.\n\nFrom the first, all her instincts told her she must look down, not up, for what she sought.\n\nThe kitchen and scullery were on the ground floor, at the back of the house, down that caliginous throat of passageway. Stairs corkscrewed farther down. The scullery contained a modern icebox. Lydia opened it and found a cake of ice about two days old, a bottle of cream, and a small quantity of knacker's meat done up in paper. Four or five dishes\u2014including a Louis XV S\u00e8vres saucer\u2014lay on the floor in a corner. For the first time, Lydia smiled.\n\nBoothole, wine cellar, vegetable pantry belowstairs, and many smaller rooms, low-ceilinged and smelling of earth and great age. The lamp flung her shadow waveringly over cruck-work beams, discolored plaster, stonework that spoke of some older building on this site. As in searching for the house itself\u2014which had fallen out of all mention in the Public Records Office after the Fire of 1666\u2014Lydia passed three or four times through the room that contained the trap to the subcellar. It was only when, failing to see any such ingress as she knew must exist, she studied the composition of the walls themselves that she narrowed the possibilities to the little storeroom whose damp stone wall bore signs of having once supported a stairway.\n\nOutside, the day must be slowly losing its grip on life. Trying to keep her hands from shaking, with cold now as well as fear, she pulled off her gloves and ran her fingers under the chair rail and around the heavy molding of the room's two doors. Near the base of the door into the wine cellar she felt a lever click unwillingly under her fingers and saw, in the dirty brazen light, the wider gap between two panels.\n\nThere was a latch on the inside of the movable panel so it could be opened from below, and a worn ladder going down.\n\nAs Lydia had guessed, the low room beneath looked as if it had been the subcrypt of a church, either the one that backed the house\u2014in a square named, oddly enough, Spaniard's Court\u2014or some forgotten predecessor. Barely visible in black paint on the ceiling groins were the words _Salvum me fac, Deus, quoniam intraverunt aquae usque ad animam meant._\n\nLydia had not been raised a Catholic\u2014her aunts considered even the inclusion of candles on the parish altar grounds for complaint to the bishop\u2014but recognized, from her residency at St. Bartholomew's, the words from the Mass for Deliverance from Death.\n\nA granite sarcophagus filled the far end of the chamber like a somber altar, all but concealing a low, locked door. Lydia stood before it for some time, holding the lamp high and gauging the probable weight of the stone lid. Then she knelt and studied the floor.\n\nDustless.\n\nA laborious investigation of the cracks in the gray stone floor showed her the trapdoor, an eye-straining business by the amber glow of the lamp; she gave up early trying to do the business tidily and without griming and wrinkling her skirt, and it was equally impossible to keep her corset bones from jabbing her ribs and the pump sprayer from knocking her repeatedly on the elbow. Another squinting, painful half hour revealed the trigger to the trapdoor's catch behind the projecting stone frame of the chamber's inner door.\n\nAs she had deduced, the sarcophagus had nothing to do with anything. It was simply too obvious.\n\nThe steps leading downward were shallow, so deeply worn in the centers that she had to press her shoulder to one wall and brace herself against the other to maintain her footing. She guessed it was well past dark outside, and beneath her growing fear\u2014the panicky conviction that she was completely unqualified to deal with the encounter that lay ahead\u2014she wondered precisely how dark was dark enough. She suppressed the urge to check her watch and make notes.\n\nThe lamplight could not penetrate the night below her, and from that darkness rose the smells of wet earth, cold stone, and rust. Interestingly, there was no smell of rats.\n\nThe light slithered wetly over a grille of metal bars. Lydia pressed herself to it, maneuvered the lamp through and held it up to illuminate what lay within. The bars were old, the lock on them new and expensive and beyond the capacity of either the skeleton key or the picklocks. The lamplight reached only partway into the catacomb beyond the bars, but far enough to show her wall niches, empty for the most part, or occupied with the suggestion of ghastly _natures mortes_ : skulls, dust, and shreds of fallen hair.\n\nOn the right-hand wall the shadows all but hid a niche whose interior no amount of angling the lamp would reveal.\n\nBut hanging over the edge, like ivory against the dingy stone, was a man's hand: long-fingered, thin, ringed with gold. Darkness hid the rest, and though the white hand itself looked as perfect as if painted by Rubens or Holbein, Lydia knew that its owner had been dead for a long time.\n\n_It's true,_ she thought, her heartbeat fast and heavy with fright. _Silly,_ she added, for she had known already that it was true... it was all true. She had met this man and seen others like him from a distance.\n\nBut knowing, she had learned this afternoon, was different from seeing, and she felt very naked, uncertain, and alone in the dark.\n\n_I'm doing this wrong._\n\nHer breath made a little apricot smoke in the lamplight as she sat down on the steps. Laying her weapon across her knees and pushing up her spectacles with one forefinger, she settled herself to wait.\n\n# ONE\n\nALL SOULS and black rain, and cold that passed like needles through flesh and clothing to scrape the bones inside. Sunday night in Charing Cross Station, voices racketing in the vaults of glass and ironwork overhead like ball bearings in a steel drum. All James Asher wanted was to go home.\n\nA day and a night spent burying his cousin\u2014and dealing with the squabbling of his cousin's widow, mother, and two sons over the estate to which he'd been named executor\u2014had reminded him vividly why, once he'd gone up to Oxford twenty-three years ago, he'd never had anything further to do with the aunt who raised him from the age of thirteen. It had just turned full dark, and Asher drew his greatcoat closer around him as he strode down the long brick walkway of the platform, jostling shoulders with his erstwhile fellow passengers in a vast frowst of wet wool and steam and reflecting upon the lethal adeptness of familial guilt. Outside, the streets would be slick and deadly with ice.\n\nAsher's mind was on that\u2014and on the hour and a half between the arrival of the express from Tunbridge Wells at Charing Cross and the departure of the Oxford local from Paddington\u2014when he saw the men whom he would later have given anything he possessed not to have seen.\n\nThey stood under the central clock in the echoing cavern of the station. Asher happened to be looking in their direction as the taller of the two removed his hat and shook the drops from it, gestured with a gloved hand toward the iron frame into which boards bearing departure times had been slotted. Asher's eye, still accustomed to cataloging details after half a lifetime in secret service to his country, had already been caught by the man's greatcoat: the flaring skirts, the collar and cuffs of karakul lamb, the soft camel color and the braiding on the sleeves all shouting at him, _Vienna._ More specifically, one of the Magyar nobility of that city rather than a German Viennese, who tended to less flamboyance in their dress. A Parisian would have worn that smooth, well-fitted line, but probably not that color and certainly without braiding; the average Berliner's coat generally bore a striking resemblance to a horse blanket no matter how rich the man might be.\n\n_Vienna,_ Asher thought, with the tiniest pinch of nostalgia. Then he saw the man's face.\n\n_Dear God._\n\nHe stopped at the head of the steps down from the platform, and the blood seemed to halt in his veins. But even before his mind could form the words _Ignace Karolyi in England_ , he saw the face of the other man.\n\n_Dear God! No._\n\nIt was all he could think.\n\n_Not that._\n\nLater he thought he would not have seen the smaller man at all had his eye not been arrested, first by Karolyi's greatcoat, then by the Hungarian's face. That was one of the most frightening things about what he now saw. In the few seconds that the two men spoke\u2014and it was not more than a few seconds, though they exchanged newspapers, an old trick Asher had used hundreds of times himself during his years with Intelligence\u2014Asher's mind registered details that he should have seen before: the fiddleback cut of the small man's shabby black greatcoat, and the way the creaseless buff-colored trousers tapered to straps under the insteps. Under a shallow-crowned beaver hat his hair was short-cropped, and he did not gesture at all as they spoke: no movement, no change of stance, not even the shift of the gloved fingers wrapped about one another on the head of his stick.\n\nThat would have told him, if nothing else did.\n\nThree women in enormous hats, feathers drooping with wet, intervened, and when Asher looked again, Karolyi was striding briskly in the direction of the Paris boat-train.\n\nThere was no sign of the other man.\n\n_Karolyi's going to Paris._\n\n_They're_ both _going to Paris._\n\nHow Asher knew, he couldn't have said. Only his instinct, honed in years with the Department, had not waned in the eight peaceful years of Oxford lecturing that had passed since he quit. Heart pounding hard enough to almost sicken him, he made his way without appearance of hurry to the ticket windows, the small bag of a weekend's worth of clean linen and shaving tackle swinging almost unnoticed in his hand. By the station clock it was half past five. The departures board announced the Dover boat-train at quarter of six. The fare to Paris was one pound, fourteen and eight, second class\u2014Asher had just over five pounds in his pocket and paid unhesitatingly. Third class would have saved him twelve shillings\u2014the cost of several nights' lodging in Paris, if one knew where to look\u2014but his respectable brown ulster and stiff-crowned hat would have stood out among the rough-clothed workmen and shabby women in the third-class carriages.\n\nHe told himself, as he bought the ticket, that the urgency of not calling attention to himself was the only reason to stay out of third class tonight. But he knew it was a lie.\n\nHe walked along the platform among women in cheap poplin skirts loading tired children onto the cars, screaming at one another in the clipped, sloppy French of Paris or the trilled _r_ 's of the Midi; among men huddled, coatless, in jackets and scarves against the cold, and tried not to listen to his heart telling him that someone in third class was going to die tonight.\n\nHe touched a passing porter on the arm. \"Would you be so kind as to check the baggage car and tell me if there's a box or trunk, five feet long or over? Could be a coffin, but it's probably a trunk.\"\n\nThe man squinted at the half-crown in Asher's hand, then sharp brown eyes went to Asher's face. \"C'n tell you that right now, sir.\" Asher automatically identified the cropped _ou_ and glottal stop _i_ of the Liverpool Irish, and wondered at his own capacity for pursuing philological points when his life was in danger. The man touched his cap. \"Near killed old Joe 'eavin' the thing in, awkward an' all.\"\n\n\"Heavy?\" If it was heavy, it was the wrong trunk.\n\n\"'Eavy enough, I say, but not loaded like some. No more'n seventy pound all told.\"\n\n\"Could you get me the address from the label? A matter of information,\" he added as the brown eyes narrowed suspiciously, \"to the man's wife.\"\n\n\"Runnin' out on 'er, is 'e? Bleedin' sod.\"\n\nAsher made a business of checking his watch against the station clock at the end of the platform, conscious all the while of the men and women getting on the train, of the thinning of the crowd that made him every second more visible, every second closer to a knife-blade death. Steam chuffed from the engine and a fat man in countrified tweeds, coat flapping like a cloak in his wake, hared along the platform and scrambled into first class, pursued by a thin and harried valet heavily laden with hatboxes and train cases.\n\nHe'd have to telegraph Lydia from Paris, thought Asher. It brought a stab of regret\u2014she'd sit up tonight waiting for him until she fell asleep surrounded by tea things, lace, and medical journals, in front of the bedroom fire, beautiful as a scholarly sylph. For two nights he had looked forward to lying again at her side. Foul as the weather had been, she'd probably simply assume that the train had been held up. Not a worrier, Lydia.\n\nStill the porter hadn't come back.\n\nHe tried to remember who the head of the Paris section was these days.\n\nAnd, dear God, what was he going to tell them about Charles Farren, onetime Earl of Ernchester?\n\nHis hand moved, almost unconsciously, to his collar, to feel the reassuring thickness of the silver chain he wore beneath. It was not a usual ornament, for a man and a Protestant. He hadn't thought about it much, except that for a year now he had not dared remove it. It had slipped into place like those other habits he'd acquired \"abroad,\" as they said in the Department; habits like memorizing the layout of any place he stayed so that he could move through it in the dark, or noting faces in case he saw them again in another context, or carrying a knife in his right boot. The other dons at New College, immersed in their specialties and their academic bunfights, never noticed that the self-effacing Lecturer in Etymology, Philology, and Folklore could identify even their servants and knew every back way out of every college in that green and misty town.\n\nThese were matters upon which his life had depended at one time\u2014and might now still depend.\n\nIn the summer his students had commented, when they'd gone punting up the Cherwell, on the double chain of heavy silver links he wore on either wrist; he'd said they were a present from a superstitious aunt. No one had commented on or seemed to connect the chains with the trail of ragged red scars that tracked his throat from ear to collarbone and followed the veins up his arms.\n\nThe porter returned and casually slipped a piece of paper into his hand. Asher gave him another half-crown, which he could ill spare with his fare back from Paris to be thought of, but there were proprieties. He didn't glance at the paper, only pocketed it as he strolled along the platform to the final shouts of \"All aboard!\"\n\nNor did he look for the smaller man, though he knew that Ernchester, like himself, would be getting on at the last moment.\n\nHe knew it would not be possible to see him.\n\nEight years ago, toward the end of the South African war, James Asher had stayed with a Boer family on the outskirts of Pretoria. Though they were, like many Boers, sending information to the Germans, they were good people at heart, believing that what they did helped their country's cause\u2014they had welcomed him into their home under the impression he was a harmless professor of linguistics at Heidelberg, in Africa to study Bantu pidgins. \"We are not savages,\" Mrs. van der Platz had said. \"Just because a man cannot produce documents for this thing and that thing does not mean he is a spy.\"\n\nOf course, Asher _had_ been a spy. And when Jan van der Platz\u2014sixteen and Asher's loyal shadow for weeks\u2014had learned that Asher was not German but English and had confronted him in tears, Asher had shot him to protect his contacts in the town, the Kaffirs who slipped him information and would be horribly killed in retaliation, and the British troops in the field who would have been massacred by the commandos had he been forced to talk. Asher had returned to London, resigned his position with the Foreign Office, and married, to her family's utter horror, the eighteen-year-old girl whose heart he never thought he had the smallest hope of winning.\n\nAt the time, he thought he would never exert himself for King and Country again.\n\nAnd here he was, bound for Paris with the rain pounding hollowly on the roof of the second-class carriage and only a few pounds in his pocket, because he had seen Ignace Karolyi, of the Austrian Kundschafts Stelle, talking to a man who _could not_ be permitted to take Austrian pay.\n\nIt was a possibility Asher had lived with, and feared, for a year, since first he had learned who and what Charles Farren and those like him were.\n\nMaking his way down the corridor from car to car, Asher glimpsed Karolyi through a window in first class, reading a newspaper in an otherwise empty compartment.\n\nThe Dorian Gray beauty of his features hadn't changed in the thirteen years since Asher had last seen him. Though Karolyi must be nearly forty now, not a trace of silver showed in the smooth black hair or the pen trace of mustache on the short upper lip; not a line marred the corners of those childishly wide-set dark eyes.\n\n\"My blood leaps at the thought of obeying whatever command the Emperor may give me.\" Asher remembered him springing to his feet in the soft bright haze of the gaslit Caf\u00e9 Versailles on the Graben, the bullion glittering on the scarlet of his Guards uniform; remembered the shine of idealistic idiocy in his upturned face. \"I will fight upon whatever battlefield He may direct.\" One could hear the capital letter in _he_ \u2014the Emperor\u2014and around him, his fellow _beau sabreurs_ of the Imperial Life Guards had roared and applauded, though they'd roared louder when another of their number had joked, \"Yes, of course, Igni... but who's going to point you in the direction of the enemy?\"\n\nEven when Karolyi had hunted Asher with dogs through the Dinaric Alps after torturing to death his local contact and guide\u2014when it was blindingly obvious that his pose as a brainless young nobleman who spent most of his time waltzing at society balls rather than drilling with his regiment was a sham\u2014that was still the Karolyi Asher remembered.\n\nThey'd never met face-to-face in that hellish week of hide-and-seek among the streams and gorges, and Asher didn't know if Karolyi was aware who his quarry had been. But passing along the corridor now with barely a glance through the window, he remembered the body of the guide, and was disinclined to take chances.\n\nIn any case, it was not Karolyi whom he feared most.\n\nThe third-class carriage was noisier than second, crowded and smelling of unwashed wool and dirty linen. A child cried on and on like the shriek of a factory whistle. Unshaven men looked up from _Le Figaro_ or the _Illustrated London News_ as Asher walked between the hard, high-backed benches. Yellow electric light jittered over cheap felt hats, wet paper flowers, plain steel pins; a woman said, \"Hush now, Beatrice, hush,\" in a voice that held no hope of Beatrice hushing this side of the Gare du Nord.\n\nAsher kept his collar turned up, knowing Farren would recognize him. It unnerved him to realize that the man might be in this carriage and he would never so much as catch a glimpse of him. He didn't like to think about what would happen to him in that case.\n\nAt the far end of the third-class car was a baggage compartment, given over to bicycles and crated dogs and an enormous canework bath chair. It was unlighted, and through its windows Asher could see the rain flashing like diamonds in the dirty light shining from third class. As Asher stepped through and closed the door, the cold struck him\u2014all the windows had been opened, rattling noisily in their frames, wet flecks of water spattering through.\n\nAt his feet a dog in a cage whined with fear.\n\nThe smell of the rainy night could neither cover nor disperse the stink of death.\n\nAsher looked around him quickly, kneeling so as to be out of the line of the window. Dim light came through the little judas on the door, but not enough; he fumbled a lucifer match from the box in his greatcoat pocket, scratched it with his nail.\n\nThe man's body had been folded small, knees mashed into chest, arms bent close to sides, the whole skinny tangle of him shoved tight into a corner behind a double bass in a case.\n\nAsher blew out the match, lit another, and crouched to worm close. The dead man was young, dark, unshaven, with a laborer's callused hands and a roughly knotted kerchief around his neck instead of a cravat. His clothing smelled of cheap gin and cheaper tobacco. One of his shoes was worn through. Only a little blood had soaked into the neckerchief, though when Asher moved it down with one finger, he saw that the jugular vein had been cut clear through, a rough, ripping tear, the edges white and puffy, mangled as if they had been chewed and sucked. Asher had a scar that size where his collar pressed the silver links of the necklace against his skin.\n\nA third match showed the dead man's face utterly white, blue-lipped, eyebrows and beard stubble glaring, though by the appearance of the eyelids he'd been dead for less than thirty minutes. Moving a frayed pants cuff, Asher saw the bare ankle had not yet begun to turn livid. Probably, Asher thought with a queer, angry coldness, it never would, much.\n\nHe blew out the match, stowed the stub\u2014with the stubs of the first two\u2014in his pocket, and slithered from between the bath chair and bass fiddle case. He'd passed the conductor in the second-class carriage, on his way down the train. The official's nearness had probably interrupted the murderer before he could dump the body out into the night, or perhaps Ernchester was waiting till they were farther from London. Asher left the compartment quickly, dusting his hands on his coat skirts and muttering to himself like a man who has not found what he sought. Nobody in third class gave him a glance.\n\nBy the time the train reached Dover, he suspected, the body would be gone. To call attention now to what he had found would only, inevitably, call attention to himself. He wasn't such a fool as to think he would then ever reach Paris alive.\n\nIn the dingy second-class compartment where he had left his satchel, a lively family of homebound Parisians had made themselves very much at home. They were passing bread and cheese among themselves; the _bonne femme_ offered him some and a blood orange, while her _mari_ laboriously scanned a battered copy of _l'Aurore_. Asher thanked her and fished out his own copy of the _Times_ , most of which he had already read on the journey up from Tunbridge Wells, and wondered academically what he was going to tell whoever was in charge of the Paris section these days.\n\nIt was going to be a long night, he knew. He dared not sleep, lest Farren sense him through his dreams.\n\n> 2\/11\/1908\u20130600 PARIS\/GARE DU NORD\n> \n> ERNCHESTER GONE TO PARIS WITH IGNACE KAROLYI AUSTRIAN SIDE STOP FOLLOWED STOP WILL HAND OFF COME BACK TONIGHT JAMES\n\n_Ernchester._ Lydia Asher laid the thin sheet of yellow paper down on the gilt-inlaid desk before her, heart beating quickly as she identified the name. _Gone to Paris with someone from the \"Austrian side.\"_\n\nIt took a moment for the meaning to sink in, mostly because Lydia, although she could have distinguished a parathyroid from a parathymus at sight, couldn't immediately remember whether the Austrians were allied with the Germans or with England. But when it did, the implications made her shiver.\n\n\"Is it from the master, ma'am?\"\n\nShe looked up. Ellen, who had brought the telegram to her with her tea, lingered in the study door, big red hands tucked under her apron. Last night's inky downpour had dwindled this morning to a slow, steady drench from a sky like steel; beyond the tall windows, Holywell Street was a shining pebblework of cobble and wet, softened by Lydia's myopia to a gentle sepia and silver Manet. The tall brown wall of New College across the road was nearly black with damp. Now and then a student would pass, or a don, faceless ghosts nevertheless identifiable\u2014even as Ellen was identifiable\u2014by their bodies and the way they moved: there was no question, to Lydia, of mistaking the little banty-cock Dean of Brasenose, with his self-important strut, for the equally diminutive but self-effacing Dr. Vyrdon of Christ Church.\n\nLydia drew a deep breath, blinking huge brown eyes in the direction of the dark square of the hall door, and realized for the first time that morning that she was starving. \"Yes,\" she said. \"He was called away unexpectedly to Paris.\"\n\n\"Tcha!\" Ellen shook her head disapprovingly. \"And in all that rain! What's in Paris that's more important than him coming home last night, and you so worried?\"\n\nSince Lydia couldn't very well reply, _Probably a partnership that will begin with Germany conquering England and end God knows where,_ she said nothing.\n\nEllen went on cheerily, \"I told you not to worry about Mr. James, didn't I, ma'am? With all that rain it'd stand to reason he'd be delayed, though I never did think of Paris, myself. Something to do with investments, like as not.\" Ellen had worked for some years for Lydia's father and was used to the fact that if the master of the house departed suddenly, it had to do with investments. \"Though I didn't know,\" she added, with one of her occasional bursts of sapience, \"as he had any.\"\n\n\"A few small ones,\" Lydia said truthfully, folding the telegram and unlocking a drawer of the gilt secretary at which she worked. Its contents exploded into a puffy mountain of household accounts and pathology notes. Lydia regarded the mess blankly, as if the entire desk were not awash with dissection diagrams, notes on the endocrine system, correspondence from other researchers on the subject of ductless glands, milliners' bills, menus, silk samples, copies of _Lancet_ , and the first draft of her article on pancreatic secretions for the January issue of _British Medical Journal_ , on which she'd been working when Ellen had made her entrance. She shook back the cloud of lace from around her hand and determinedly stuffed the contents back into the drawer, which she then forced shut. She opened two more drawers with similar results, finally poking the telegram down into the side among a sheaf of notes concerning electrostimulation's effect on the production of adrenaline.\n\nHer friend Josetta Beyerly was forever joking her about not reading the newspapers even enough to know who the Prime Minister was, as if prime ministers\u2014and in fact Balkan kings\u2014didn't come and go at the drop of a constituency. Reading newspapers only caused Lydia to wonder whether people like Lord Balfour and the Kaiser suffered from hyperthyroidism or vitamin deficiency and how she could find out, and she'd found that the speculation distracted her from her work.\n\n\"He says he'll be back today.\" It was unreasonable of her, she knew, to feel relief. Jamie was perfectly able to look after himself, as she had known last night, lying awake and fingering the heavy links of the silver chain around her neck. When she had dreamed, it had been of a corpse-white face upturned in the distant gaslights of a London alleyway, strangely reflective eyes, and a mouth snarling to show the glint of outsize fangs. She'd awakened then and lain listening to the rain on the ivy until morning.\n\nThere had been no reason for her to be afraid.\n\n_Handing off,_ the telegram said.\n\nThere was no reason to be afraid now.\n\nWhat was it in the telegram, she wondered, that snagged at the back of her mind like a hangnail on silk?\n\n\"Though it would be a shame,\" she went on thoughtfully, \"if he didn't spend at least a little time in Paris, long enough anyway to buy himself a clean shirt and a box of bon-bons. He'd only his overnight things with him, you know, for his cousin's funeral.\"\n\n_Handing off._\n\nWhy did she think she'd heard the name Ignace Karolyi before?\n\nAnd how on earth was he going to explain the Earl of Ernchester to the Foreign Office men in Paris?\n\n\"I wonder if you could get me some of the toast I didn't eat at breakfast?\" Lydia asked after a moment.\n\n\"Right away, ma'am.\" She heard the beaming smile in the housemaid's voice, saw it in the way her shoulders relaxed as she turned from the door. Ellen and Mrs. Grimes both considered her too thin, though she had confounded their earlier threats\u2014when she was in school, a gawky and bespectacled fledgling bluestocking\u2014that no girl who went around with her nose in a book and not eating enough to keep a canary alive was ever going to catch a husband. In spite of daily reminders of her undesirableness, Lydia had always been aware that as the sole heiress to the Willoughby fortune, she would be inundated with proposals of marriage the moment she put up her hair.\n\nJamie told her she was beautiful, the only man she had ever truly believed.\n\nHad Jamie ever mentioned Ignace Karolyi to her?\n\nShe didn't think so. She cast her mind back to the tall, self-effacing don who sat on the sidelines of her father's garden parties with her, talking of cabbages and kings\u2014telling her about medicine in China and how best to go about studying for responsions without letting her father know. The gentle, competent man who never made demands on her, who guessed that a completely different person hid beneath her careful fa\u00e7ade and accepted her exactly as she was. He'd always been close-mouthed, though even as a schoolgirl she'd suspected there was more to him than that almost invisible \"brown\" mien of his. Reticence was still his habit; after seven years of marriage his stories, like Mark Twain's, usually concerned men and women all named Fergusson.\n\nThat was what troubled her now. She'd heard, or read, Karolyi's name in some other context. _Read,_ she thought... She couldn't put a pronunciation to the closing _yi_. Which meant she'd never heard Jamie say it.\n\nShe slipped her eyeglasses out from behind a pile of papers\u2014concealing them when anyone entered the room was a lifelong habit\u2014and rose in a rustle of lace, crossing to her side of the bookshelves, where she settled on the floor, her long red hair hanging down her back, her plans to work at the Radcliffe Infirmary's dissection rooms that afternoon laid aside. By the time Ellen reappeared with a tray of sandwiches and onion soup\u2014for it was well past noon\u2014Lydia had remembered when and in what context she'd come across Karolyi's name, and the recollection made her more uneasy still. She left the tray untouched and ascended to the bedroom two hours later to continue her researches in the back issues of _Lancet_ and _Medical Findings_ stored under the bed.\n\nShe might not remember whether Germany had a Parliament these days or be able to tell a Bolshevik from a Menshevik, but she could remember to within a few months when secretin had been discovered or the address of Marie Curie's laboratory in Paris.\n\nShe was still reading at teatime when Ellen came up with another tray and bullied her into eating half an egg and part of a scone while Ellen built up the bedroom fire and turned up the gas. Lydia had tracked down the reference, which had given her, in turn, another name; she was dimly aware that she had begun to count the hours between now and midnight, when, at her best guess, James was due home.\n\nIf he didn't elect to remain in Paris overnight.\n\nIf something didn't go wrong.\n\nIf Ernchester hadn't seen him...\n\n_If he's staying in Paris,_ she thought, dabbing jam and Devonshire cream on a scone and then setting it on the plate to gaze at the darkening windows, _he'll wire me. He'll let me know._\n\nAnd if he didn't?\n\nShe wondered if she could reach him by wiring the consulate or the Foreign Office\u2014or was it the War Office that operated the Secret Service? Where _was_ the Foreign Office in Paris, anyway? Like most girls of wealthy family, her experience of the City of Lights had been stringently limited by her preceptors to the Champs Elys\u00e9es and the Rue de la Paix. If she telephoned the Foreign Office in London\u2014would that be in Whitehall? Parliament? Scotland Yard?\u2014they would only tell her lies.\n\nShe felt helpless, frightened, uncertain of what to do, because, unlike medical research, this was something for which she had never prepared.\n\nAnd in any case, she realized, only now seeing the darkness beyond the curtain, they'd all have gone home by this time. As if to echo an affirmative, the Louis XV clock on the parlor mantel downstairs sang its five clear notes.\n\nSo all she could do was wait\n\nShe fell asleep sometime after midnight across the foot of the bed, still wearing her fluffy rose-point tea gown, the eye of a maelstrom of medical journals that spread to the bedroom's door, and dreamed of crumbling houses in ancient cities, their stones mortared with dark blood and cobweb; of half-seen forms whispering in shadows centuries deep.\n\nBy morning James had not returned. But it wasn't until his second telegram that she decided to go up to London and seek out such a house herself.\n\n# TWO\n\n\"THE EARL OF ERNCHESTER is a vampire.\"\n\nStreatham\u2014a fussy, chinless man whom Asher had never liked\u2014regarded him for a moment with narrow surprise in his light blue eyes, as if asking himself why Asher would perpetrate such a tale and if it constituted a threat to his position as head of the Paris branch of the Department. Asher had spent a good part of the previous night, sleepless aboard the Dover ferry and the train from Boulogne, trying to phrase an argument that would convince those in charge to either have Karolyi arrested in Paris\u2014scarcely likely, since Karolyi never went anywhere without diplomatic credentials\u2014or to assign a man to follow him, to at least see what his next step would be.\n\nLack of sleep, hunger, and sheer exasperation when the green-painted door of the town house on the Rue de la Ville de l'Eveque hadn't opened to his knock at five minutes after nine had had their effect. Sitting on a bench under the bare trees before the Madeleine, watching the town house for signs of life, with the chilling threat of rain blowing over him for twenty freezing minutes, he had finally thought, _To hell with it. I'll tell them the truth._\n\nStreatham ventured a small chuckle, like an agent offering a read newspaper on the Underground to the minor clerk of some foreign legation: a feeler to see how the land lies. \"You aren't serious.\"\n\n\"Ernchester\u2014or Farren, as he sometimes calls himself\u2014Wanthope is another one of his names\u2014is perfectly serious about it,\" Asher said grimly, remembering the dead laborer on the train. \"Whether or not he's correct in his claims that drinking human blood has enabled him to live two hundred years, I know from my own experience that the man has abilities for which a foreign power would pay well. He can get past guards unseen. I don't know how he does this, but he can. He has an almost fakirlike ability to get in and out of places. And he can influence people's minds to an almost unbelievable extent. I've seen him do it.\"\n\nIn fact, Asher reflected, watching the thoughts pass almost visibly across the back of the Paris chief's shallow blue eyes, he hadn't seen Ernchester do any of the things he described. Of all the vampires who had ringed him like ghosts in last fall's misty London darkness, Charles Farren, quondam Earl of Ernchester, was one of the few who had not, to one degree or another, used the eerie abilities of the vampire mind to trap or hunt or influence him.\n\nAnd as he'd watched the yellow pinpricks of the Dover lights vanish into the blackness of fog beyond the _Lord Warden_ 's stern rail, Asher had reflected that that was one of the strangest aspects of the entire matter: that Ernchester had been the Hungarian's choice.\n\nThere were far more dangerous vampires in London. Why not one of them?\n\nStreatham's mouth grimaced into what was probably supposed to be a smile. \"Really, Dr. Asher. The Department genuinely appreciates your concern, particularly in view of the circumstances of your leave-taking...\"\n\nIt was a gratuitous jab, and Asher felt a sting of annoyance.\n\n\"What I said and felt about the Department when I left still holds.\" He set down his teacup. At least they'd offered him tea, he thought, something he was unlikely to get elsewhere in Paris. \"If the Department were about to be dynamited, I don't think I'd cross the street to pinch out the fuse.\n\n\"But this isn't the Department I'm talking about.\" His voice was level, but cold with an old rage burned now to clinkers and ash. \"This is the country. _You cannot let the Hofburg hire the Earl of Ernchester._ \"\n\n\"Don't you think you're exaggerating a little? Just because the Austrians are courting some hypnotist\u2014\"\n\n\"It's more than hypnosis,\" said Asher, knowing that if he lost his patience with this man, he'd lose all hope of getting his help. \"I don't know what it is. I only know that it works.\" He drew a deep breath, realizing how little of the actual vampire power could be described. Even to someone who was willing to believe, he wasn't sure he _could_ describe that curious blanking of the mind that vampires imposed on their victims, allowing them to move utterly unseen; the ability to stand outside a building or on the next street, or half a mile away, silently reading the dreams of whosoever they chose.\n\nThey were born spies.\n\nOf course Karolyi, raised in the hotbed of Carpathian legends, would believe, or be ready to believe.\n\n_I am ready to do whatever my Emperor requires..._ He'd imitated the glowing-eyed gallantry of all those other young fools in the officers' corps, but even then Asher had known that Karolyi had been speaking the absolute truth. It was just that some people had a different view of that word, _whatever_.\n\nNothing really changed, he thought. He didn't know how many times he'd sat in this discreet town house within walking distance of the embassy during the years in which he'd ranged all over Europe, going out ostensibly in quest of moribund verb forms and variant traditions about fairies and heroes and coming back with German battleship plans or lists of firms selling rifles to the Greeks.\n\nThose years seemed hideously distant to him, as if it had been someone else who risked his life and traded his soul for matters that had been obsolete in a year.\n\nStreatham folded his hands, white as a woman's and as soft. With a kind of perverse relish, he said, \"Of course, having been out of the Department, you wouldn't know about the reorganization since the end of the war and the old Queen's death. After South Africa, the budget was drastically cut, you know. We have to share this house with Passports and the attach\u00e9 for Financial Affairs now. We certainly can't ask the French authorities to order the arrest of an Austrian citizen just on your say-so\u2014certainly not a member of one of that country's noble houses, not to speak of the diplomatic corps. And we can't spare a man to follow Karolyi around Paris, much less trail him to Vienna or Buda-Pesth or wherever else he'll be going on to.\"\n\n\"Karolyi's only a means to an end,\" Asher said quietly. \"He's the only way you can track Ernchester...\"\n\n\"And don't keep calling him 'Ernchester.'\" Streatham peevishly aligned the edge of a report with the edge of his desk and centered the ink stand above it. \"The Earl of Ernchester happens to be a good friend of mine\u2014the _real_ Earl of Ernchester. Lucius Wanthope. We were up at the House together,\" he added smugly.\n\nBy \"the House\" Asher knew he meant Christ Church College, Oxford, and wondered if that was the same Lucius Wanthope who'd been one of Lydia's suitors, eight or nine years ago. Streatham pronounced it _Want'p_ , swallowing the middle of the word after the fashion of Oxford. \"If this impostor is going about calling himself by that title, the least you can do is not subscribe to the hoax.\"\n\n\"It doesn't matter,\" Asher said tiredly, \"if he's calling himself Albert of Saxe-Coburg-Gotha. And I know all about the reorganization and the budget. Have him followed. This was the address on his luggage. It's just a transit point, but your man can trace him through the local carting company. He'll be hauling a large trunk somewhere today, possibly to the Gare de l'Est to go on to Vienna, more probably to some house here in the city where they can set up operations. Find out who his connections are...\"\n\n\"And what?\" Streatham chuckled juicily. \"Drive a stake through his heart?\"\n\n\"If necessary.\"\n\nStreatham's eyes\u2014too close together in flaccid pouches the color of fish belly\u2014narrowed again, studying him. Asher had washed and shaved in one of the public washrooms at the Gare du Nord after dispatching a telegram to Lydia, but he was well aware that at the moment he looked less like an Oxford don than he did some down-on-his-luck clerk at the end of the night on the tiles.\n\nThe Paris chief opened his mouth to speak again, but Asher cut him off. \"If necessary I'll telegraph Colonel Gleichen at Whitehall. This is a matter on which we can't afford to take chances. I spent my last few shillings to follow them here, to warn you of a threat greater, in my years of experience, than anything currently facing our department. Believe me, I wouldn't have done it if I'd thought that Ernchester was just a stage hypnotist with a good act, and I wouldn't have done it if I'd thought there was any alternative to the danger we'll face if he does start working for the Kundschafts Stelle. Anything Vienna learns is going to end up in Berlin. You know that. Gleichen knows it, too.\"\n\nAt the mention of the head of MO-2's D Section, Streatham's face had slowly begun to redden; now he fetched an exaggerated sigh. \"It'll put the entire Records Section days behind, but I'll pull Cramer off Information and assign him. Will that satisfy you?\"\n\nAsher fished his memory and came up empty.\n\n\"After your time,\" said Streatham, with a kind of breezy viciousness. \"A good man at his work.\"\n\n\"Which is?\"\n\n\"Information.\"\n\n\"You mean cutting articles out of newspapers?\" Asher stood and picked up his hat. Outside the tall windows it had begun to rain again. The thought of the three-quarter-mile walk to Barclay's Bank on the Boulevard Haussmann gave him a sensation akin to the grinding of unoiled gears deep in his chest.\n\n\"Everyone in the Department has had to cover several areas of work these days.\" The enmity in Streatham's voice was plain now. \"I'm very sorry about the inconvenience to you, and about the fact that the budget doesn't permit us to stand you your train fare home. Of course, you're welcome to a bed in one of the duty rooms...\"\n\n\"Thank you,\" Asher said. \"I'm just on my way to my bank.\" _This Cramer_ is _cutting articles out of newspapers,_ he thought.\n\n\"Don't let me keep you.\"\n\nThere had been a time, thought Asher as he descended the shallow sandstone steps, when he loved Paris.\n\nAnd indeed, he loved it still. Against the cinder-colored street, the gravid sky, the white and yellow shapes of the bare sycamores, and the pale gold stone of the buildings seemed queerly bright. Windows were shuttered behind iron balconies; red and blue shop awnings seemed to blossom like flowers. Traffic was thick on the boulevard: cabs with their roofs shining with moisture; bright-colored electric tramways, hooting for right-of-way; stylish landaus, the horses puffing steam from their nostrils like dragons in the damp cold; men and women in daytime clothing the color of eggplant and wet stone.\n\nA magic city, thought Asher. Even in his days with the Department, when he had made himself familiar with its thugs-for-hire, its safe breakers, forgers, and fences, he had still found it a magic place.\n\nBut he knew that he was hastening to accomplish his errands because he wanted very badly not to be in this city when the sun went down.\n\nThere was an ancient _h\u00f4tel particulier_ somewhere in the Marais district, owned by a woman named Elys\u00e9e. Since the night he had been taken there, blindfolded, and seen the white-faced, strange-eyed, beautiful creatures who played cards in its brilliantly lit salon, he had not felt safe in this city. He was not sure he would ever willingly spend a night here again.\n\nAt Barclay's Bank he established his credentials and withdrew twenty pounds\u2014five hundred francs, far more than he'd need for a _prix-fixe_ lunch in the Palais Royale and his return journey, but the discomforts of last night had rendered him unwilling to trust Fate again. It was well after noon, but the Vefory was still serving luncheon. He settled in a corner with an omelette, fresh spinach, bread and butter that had nothing in common with the English travesty of the same name, coffee, and a copy of _Le Petit Journal_. The next boat-train left at four. He had not quite time to visit the Louvre\u2014only the booksellers on the quais, he thought, and a little while spent in the restful silence of Notre Dame.\n\nIt would be just getting dark as the train pulled out of the Gare du Nord, but that would be sufficient.\n\nAs he turned over the pages of the _Journal_ , the top of his mind sifted and sorted the mishmash of Serbian demands for independence from Austria, Russian demands for justice in the Serbian cause, another massacre of Armenians by the new Turkish government, plots by the Sultan to regain his power, and the Kaiser's pursuit of ever faster battleships and ever more powerful artillery, while some other part of his thoughts seemed to see through those reports to the uses the Austrian Emperor\u2014or the Czar or the Kaiser, for that matter\u2014would have for a vampire.\n\nIn any direction he looked, the possibilities were terrifying.\n\nEurope skated the rim of cataclysm, that much he knew. The German Kaiser was praying, literally and publicly, for an excuse to use his armies; the French were burning with pride, rage, and the old wounds of the Alsace. The Empire of Austria was trying to hold on to its Slavic minorities, while the Russians trumpeted their intention of backing up those minorities' \"pan-Slav\" rights. Asher had seen firsthand the weapons everyone was rushing to buy, the railway lines being constructed to carry men to battles, and in Africa he'd already seen what those weapons could do.\n\nWould men who contemplated sending other men into machine-gun fire\u2014or contemplated turning machine guns on soldiers with only rifles in their hands\u2014shrink from handing over a political prisoner or two per week to someone who could slip into consulates, workshops, departments of navy and army utterly unseen?\n\nHe turned the page and, for a moment, saw their faces again in the dark of his mind. The coarse and powerful Grippen. Ysidro's enigmatic disdain. Bully Joe Davies. The beautiful Celeste.\n\nThe Earl of Ernchester.\n\nWhy Ernchester? he wondered again.\n\nThe weakest of them, strangely fragile, Grippen's fledgling and slave to the domination of the master vampire's mind. Did Grippen know the little nobleman had left London? Had made a pact with a foreign power? Had Grippen been approached first and refused?\n\n_No vampire,_ Elys\u00e9e de Montadour had said, the gaslight gleaming queerly in her green eyes, _will do that which endangers other vampires by giving away their haunts, their habits, or the very fact of their existence to humankind._ A handsome woman, with nodding ostrich plumes in her hair, her green-black silk gown as stylish as if she had not been born in an era of panniers and three-foot coiffures. He remembered the cold strength of her hands, clawlike nails ripping open the veins in his arm to drink.\n\nWhy didn't Karolyi contact Elys\u00e9e? he wondered. Or the Vienna vampires? Surely in that city, as in all cities where there were poor upon which to feed, one could find the hunters of the night.\n\nHe turned over the leaves of the newspaper, searching for mention of an insignificant laborer's body, found drained of blood on the boat-train. There was none.\n\n\"Dr. Asher?\"\n\nHe'd been aware of the tall young man entering the restaurant, heading in the direction of his table; he'd identified his tailoring and his smooth, heavy-jawed face as English. The young man held out his hand, regarded Asher with frank brown eyes under an overhanging forelock of wheat-colored hair.\n\n\"I'm Edmund Cramer.\"\n\n\"Ah.\" Asher took the boy's hand, gloved in sturdy York tan, in his own. \"He whose absence from Records will imperil the defense of the realm against the French.\"\n\nCramer laughed and took the chair Asher pushed slightly toward him with his foot. The waiter appeared with another cup of caf\u00e9 noir and a bottle of cognac; Asher waved the latter away. \"Well, it's true the whole outfit is rather cumberish these days, but Streats could jolly well have upped for a train ticket, not to mention lunch. You did get to your bank all right and all that?\"\n\n\"You behold the spoils of my endeavor.\" Asher gestured grandly to the empty plates and handed the waiter two francs upon the man's reappearance with more caf\u00e9 noir. \"You have me followed?\"\n\n\"Thought I'd find you in one of the caf\u00e9s in the Palais,\" explained the young man. \"Streats said you banked at Barclay's, and it's right round the corner. I'm on my way to the Hotel Terminus; thought I'd get a little more information on this Ernchester bird and his Hungarian friend.\" He flipped from his breast pocket the notepaper onto which Asher had copied the address of the Hotel Terminus, by the Gare St. Lazare. \"The chief seems to think Karolyi's hot stuff.\"\n\n_Hot stuff._ Asher looked into those luminous eyes and his heart sank. The boy was barely older than the students he was supposed to be lecturing today back at New College\u2014and he breathed a peripheral prayer that Pargeter was taking his lecture as agreed if he were delayed in Wells. He couldn't let this beardless novice go up against a man like Karolyi, let alone Ernchester.\n\n\"He is deadly,\" Asher said. \"Don't let him see you, don't let him get within arm's reach of you if you can help it. Don't let him know you're on his trail. I know he looks like he's never done anything but try on uniforms and trim his mustache, but that's not the case.\"\n\nCramer nodded, sobered by Asher's words. Asher wondered what Streatham had said about him.\n\n\"And Ernchester?\"\n\n\"You won't see Ernchester.\"\n\nThe young man looked puzzled.\n\n\"That's his skill.\" Asher got to his feet, left a five-franc silver piece on the table for the waiter, and led the way to the door. \"So we'll have to concentrate on keeping track of Karolyi. What money have you?\"\n\nCramer's eyes twinkled. \"Enough to get a train ticket at the last minute and not have to starve through the night.\"\n\n\"Something like that.\" It began to rain again as they emerged from the long doors of Vefory's into the arcade around the Palais Royale. The arcade was becoming crowded, the rain notwithstanding; gentlemen in top hats and expensive greatcoats from the Bourse and the nearby banks, and ladies in tulip skirts like brilliant flowers against the dripping gray monochrome of the hedges, trees, and winter earth of the central gardens. Halfway around the arcade Asher found the place he sought: DuBraque et Fils, Jeweler. Cramer watched in a certain amount of puzzlement as Asher purchased three chains, each about eighteen inches long, of what the jeweler assured him was sterling silver.\n\n\"Put this around your neck.\" He handed one to Cramer as they emerged into the arcade again. In many of the shops the gas had already been lit, and the light from the wide glass window winked on the bright links as Cramer tried to open the catch without taking off his gloves. \"Ernchester really believes himself to be a vampire,\" Asher went on, winding another of the chains double around Cramer's wrist. \"Wearing silver may just save your life.\"\n\n\"That far round the twist, eh?\"\n\nAsher looked up from affixing the second chain, met the young man's eyes for a moment, then returned his attention to the clasp.\n\n\"Don't underestimate him.\" The fit was close; Cramer was a well-fleshed young man. \"Don't relax your guard for a minute once it gets dark. He's a lunatic, but that doesn't mean he can't kill you in seconds.\"\n\n\"Shouldn't we stop by Notre Dame for a crucifix, then?\" A smile struggled on his face.\n\nAsher remembered a lieutenant he'd known on the Veldt\u2014Pynchon? Prudhomme? He'd had an East Anglian glottal stop, anyway\u2014standing, hands on hips, staring out at the hot, dense silence of lion-colored land. _Well, they're just a lot of farmers, when all's said, aren't they? \"_ It's the silver that keeps them away,\" Asher said.\n\nCramer did not seem to know what to reply.\n\nEven at the Palais Royale it was difficult to find an empty cab in the rain. They ended by taking the Underground to the Gare St. Lazare and crossing the square to the Hotel Terminus. \"Should we ask at the cab rank?\" Cramer indicated the line of light, two-wheeled fiacres along the railings of the _place_ , the horses head down, rugged against the rain, the men grouped beneath the trees, wrapped in whatever they could find to keep warm.\n\nAsher shook his head. \"He'll have used a cartage company. It's a big trunk. A London four-wheeler could barely take it; a Paris fiacre's too lightly sprung. We'll just check here...\" He ascended the gray granite steps of the Terminus, crossed the dark Turkey carpets to the lobby desk, Cramer at his heels like a well-bred but very large dog.\n\n\"Pardon,\" Asher said to the clerk, in the Strasbourg French of a German. He stood as the Germans stood, the set of his shoulders like that he had seen in South German officers, but without the Prussian stiffness which might have gotten him little help in this city of long memories. \"I am trying my sister Agnes to locate; she was on the Dieppe train this morning to have come, and nothing of her I have heard. The matter is I do not know whether she travels under her own name, or that of her first husband, who was killed in Kenya, or of her second...\"\n\nAs he and Cramer crossed the square again, Asher said, \"Karolyi's checked in, all right.\" He ducked between a bright red electric tram and the shined and chauffeured automobile of one of the old _gratin_ , turned up the Rue de Rome and again on the Rue d'Isly. \"Name's on the register, or at least the name of one of his lesser titles. Now we get to do the boring and soul-destroying part...\"\n\n\"I refuse,\" Cramer said cheerily, turning up his collar against the cold, \"to believe there's anything more boring and soul-destroying than combing through a hundred fifty French newspapers per day\u2014and that's just the political ones, mind, _and_ just the Parisian ones\u2014in search of 'items of interest' to the War Department. Do your worst.\"\n\nAsher grinned and led the way up the steps of the modest Hotel d'Isly, no more than a door between a state-run tobacconist's and a workingman's _estaminet_. \"There speaks a brave soul and true agent.\" He had almost forgotten, he thought, the light camaraderie of the King's secret servants. The boy had promise. Pity he had no better teacher for the time being than Streatham.\n\nResuming the stance and speech of the Strasbourg German, he presented the clerk on duty in the narrow upstairs lobby with a tale, not of a vanished sister, but of a vanished trunk: a meter and a half long, leather-covered oak with iron strapping. A confusion in the Gare, misplaced labels... No? No. Perhaps the _gn\u00e4dige_ Herr could give some advice on the local cartage companies, such as a man might have summoned to the Gare? The city directory, to be sure, could be purchased, but it gave little idea...\n\n\"The Bottin, pff!\" The clerk gestured. \"Here is the list we use, m'sieu, when we have a client with such a trunk. Not all are on the telephone, you understand, but for such as are, there is the cabinet...\"\n\n\" _Wunderschoen!_ The Herr is entirely too kind. Certainly all the calls will be compensated for. Please accept this token...\"\n\n\"It's up to you now,\" Asher said softly as the clerk returned to his counter and Asher and Cramer were alone by the wooden confessional of the telephone cabinet. \"You'll have to go along on foot and check the companies that aren't on the phone, but those are near enough to send a page with a note. I'll go back to the Terminus and keep an eye out for Karolyi. There's a caf\u00e9 on the Rue d'Amsterdam corner of the Place du Havre, and another on the other side of the Rue du Rome; both of them command a view of the cab stand. I'll be in one or the other, or under the arcade of the Gare itself. If I'm not there\u2014if Karolyi comes back and leaves again and I follow him myself\u2014you wait for me there. The last train for London tonight leaves St. Lazare at nine. I'll look for you before half past eight. All right?\"\n\nCramer nodded. \"All right. Jolly good of you to point me out the way...\"\n\nAsher shook his head dismissively, rising to his feet and digging his gloves from his pocket. \"Don't let either of them know you're on their trail. But don't lose them. It's more important than you know.\"\n\nHis smile was boyish. \"I can only do my best.\"\n\nAsher picked up the battered brown leather valise that had accompanied him throughout the day and nodded. \"It's all any of us can do.\"\n\nAt the head of the stairs he paused, turned back to see the tall, stout form perched in the telephone cabinet, the desk clerk's list spread out on his knee. _No money to get anything more than that,_ he thought, with a kind of despair. Paris wasn't a trouble spot. What experienced men the Department had were in Ireland or on the Indian frontier.\n\nHe almost went back.\n\n_And then what?_ he asked himself. _Volunteer to pursue Karolyi myself? Let the Department have me again, to do their bidding as I did before?_\n\nBut this was different.\n\nIt was always different, he thought bitterly, turning away. The only thing ever the same was that they wanted you to do it\u2014and what it did to you inside.\n\nSomething hurt within him, like old wounds at the onset of storm.\n\nAt the caf\u00e9 on the corner of the Rue d'Amsterdam, Asher ordered a caf\u00e9 noir and settled himself to wait. Being unable to read the newspaper, he asked the waiter for pen and paper, and amused himself, between watching the cab rank, by observing the passengers going to and from the Gare, making a game of deducing financial circumstances, occupation, and family ties from details of clothing and manner and speech, less systematically than Conan Doyle's Mr. Holmes but with an agent's habit-sharpened skill. This was a good place for it; he heard three kinds of German, five Italian dialects, Hungarian, Dutch, and a half-dozen varieties of French. Once a couple walked by speaking Greek\u2014brother and sister, he guessed from the familiar form of speech as much as the resemblance between them. Later a small family of Japanese passed, and he thought, _One day I'll have to study that tongue._\n\nIf he survived.\n\nThe clock on the Trinit\u00e9 struck four, and he knew he had missed the afternoon boat-train.\n\nThere was still no sign of Cramer or Karolyi.\n\nPeriodically the waiters brought him coffee, but seemed content to let him remain. Asher knew there were men who sat in caf\u00e9s throughout afternoon and evening, writing letters, reading, drinking coffee and liqueurs, playing quiet games of cribbage, dominoes, chess. Passengers came in for a coffee, or to wait for friends. The sky darkened to the color of soot, and bright white electric lights blossomed all around him in the square. The cab men changed their day horses for the beat-up screws they drove after sundown\u2014why subject your good beast to the rigors of night work?\u2014and lit the yellow lamps that marked their origin in the Montmartre quarter.\n\nIt was almost six when he saw Karolyi. The man had a lithe deadliness to him, like a cheetah masked as a house cat; his wide-skirted Hungarian greatcoat billowed around his boot calves in his haste, and he looked here and there quickly as he sprang up the steps of the Hotel Terminus, smooth strong chin and beautiful lips touched by the arc lights that left his eyes in his hat brim's shadow. It was the way he moved when he thought himself unobserved that had first made Asher wonder about him, back in Vienna. That, and the fact that he was clearly too intelligent to be content to do what he was doing.\n\nAsher paid his bill and cursed the Department, gathered his valise and strolled casually across the square so as to be loitering in the dense shadows of the trees near the cab stand when the Hungarian reemerged from the Terminus' doors. He heard him speak to a driver, giving an address on the Rue du Bac. Because there was the possibility that Karolyi might change cabs, Asher simply told his own jehu, \"Follow that cab\u2014don't let him see us,\" and the man, a waspish little sparrow of a Parisian in a faded army coat and muffler, gave him a knowing wink and whipped up his disreputable old nag in pursuit.\n\nThey crossed at the Pont Royal, the lights of the Louvre shining on black water. Near the Quai d'Orsay, Karolyi dismissed his cab, and Asher followed him afoot along the crowded streets of the Left Bank. Beneath the trees of the Boulevard St. Germain. Karolyi picked up one of those bright-dressed, frowsy-haired women whom Asher had seen emerge, a little like vampires themselves, from the darkness as soon as the lamps were lit. He felt a pang of disgust, both with his quarry and with himself, but he continued to loiter just far enough behind to keep the man and his new companion in sight. They turned from the lighted boulevard into the dark blocks of old houses that had made up the quarter long before the Citizen King's improvements, stopped at a workman's caf\u00e9 for a drink. Standing in the raw gloom of an alleyway, Asher heard the half hour strike from St. Clothilde; the whine of fiddles and concertinas reached him, and in the glare of the colored lights he saw gaudy petticoats swirl and striped stockings, and mouths opened in laughter behind the blue haze of cigarette smoke.\n\nThe night train was at nine. He wondered if he had time to leave word for Cramer and still catch it, or if he'd have to spend a night in Paris after all. The thought wasn't pleasant. At a sound behind him, he whirled, his heart in his mouth, seeing in his mind's eye the cold white faces, the strangely glittering eyes of the Master of Paris and her fledglings...\n\nBut it was only a cat.\n\nIf it had been Elys\u00e9e de Montadour, he realized, he would have heard nothing.\n\nWhen Karolyi and the woman emerged from the caf\u00e9, she was clinging to his arm, her great brassy fleece of hair hanging loose from its pins and her head lolling. Karolyi, Asher remembered, had always been very circumspect with the girls of his own class or the daughters of the wealthy Vienna _nouveaux riches_ , instead preying incognito on suburban shop girls or driving out to the country inns to seduce the young girls who worked in the vineyards.\n\nTheir footfalls dripped on the moist pavement. As they approached Asher's unseen post in the alley, a man in a striped jersey and sailor's jacket stepped out of a doorway. \"Got a couple sous for an honest man out of luck?\"\n\nWhen Karolyi said, in his icily perfect accent, \"Go and have yourself stuffed,\" the man grew belligerent, blocking his way; though not as tall as the Hungarian, he was beefier, standing too close, threatening with the aggressive curve of his shoulder, the readiness of his hands.\n\n\"That ain't no way to\u2014\"\n\nIn one move Karolyi shucked the woman from his arm, leaving her to fall back against the soot-black wall, and lightly reversed the walking stick in his hand. Before the beggar could utter a sound, Karolyi brought the stick around sideways, hitting the skull with a crack Asher could hear where he stood. When the man slumped, Karolyi struck him again, heavy, deliberately, full-force blows, as if beating a carpet. Unhurried. It was not a neighborhood much frequented by the _guardiens de la paix_.\n\nThe woman stood, swaying, her fist in her mouth, blinking at the scene in stupefied horror. She made no move to flee, and Asher wondered if she were capable of it. When Karolyi had finished, he turned, taking her by the front of her jacket and pulling her to him again, and she sagged on his shoulder like one drunk or drugged. A little light from the caf\u00e9 showed Asher the beggar's blood, inky on the uneven pavement; the man's breath was a wheezing, stertorous gasp.\n\nAsher thought, _He needs help._ And then, _If I go to the caf\u00e9 for it, I'll lose Karolyi._\n\nSilent as a lean brown cat in the shadows as he moved after the retreating pair, Asher remembered why he'd left the Department. Once you accepted the necessity of what you did\u2014 _whatever my country requires_ \u2014you might hate yourself, but you followed.\n\nThe house was one of those anonymous stucco-fronted Parisian dwellings in a narrow lane whose character hadn't changed since the days of the Sun King. Doors and windows were shuttered fast. As Karolyi unlocked the door of a downstairs shop, Asher ghosted through an alley a few houses farther up, counted chimneys, watched roof lines, and slipped into a clotted, weed-grown yard. Light shone behind shutters on the second floor, casting enough of a glow to let him see the broken-down shed that had once housed a kitchen amid a foul litter of rain barrels, old planks, broken boxes. All around him other shuttered windows made glowing chinks and slits of brass. The muck underfoot dragged his boots, the air nearly as thick, smothering with the stench of privies and of something newly dead.\n\nHe left his valise beside a rain barrel, scrambled with infinite care to the shed's roof. Through a broken louver he watched Karolyi tie the woman to a rickety chair. She was laughing, her head lolling back. \"You like it like this, eh, _copain_? You want me to fight you a little?\"\n\n_\"Igen.\"_ Karolyi had pulled off his gloves for the task, tossed his hat on the stained and sagging mattress of the bed. His face was as calmly pleasant as the face of a statue, his shoulders relaxed, as if he shed everything from him with the knowledge that whatever he did in the name of his country was acceptable and forgiven. There was genuine banter in his voice. \"You fight, my little bird. See if it helps.\"\n\nBeyond them Asher could see an enormous trunk that occupied all of one side of the room: leather, strapped and cornered with brass. It stood open, and the dim light of the oil lamp glinted on the metalwork, filled it with shadow, but Asher could see that mere was a second, only slightly smaller trunk inside. The inner trunk could still easily have held a man.\n\nA noise in the yard nearly stopped his heart; a hissing and a scuffle; rats fighting, he realized, leaning against the freezing brick wall. He remembered the smell of some dead thing near the shed.\n\nWhen he looked back, Ernchester was in the room.\n\n\"You're late.\" By his voice Karolyi could have been speaking of a rendezvous for tea. \"The train leaves the Gare de l'Est at seven-thirty. We've barely time to dispose of this little \u00e9clair before the carters arrive.\"\n\nHe stepped to the giggling woman, took the soiled lace of her collar and ripped her dress open to the waist. She wore a corset underneath but no chemise; breasts like loaves of fallen dough balanced precariously on top of the ridge of whalebone and canvas, nipples like big copper pennies. A cheap gilt chain glinted around her neck. She winked up at Ernchester, and with a flip of one knee tossed her skirts up over her lap. She wasn't wearing drawers, either.\n\n\"You got time before your train, _ch\u00e9ri_.\" She leaned her head back and made kisses at him with her painted mouth, then dissolved into giggles.\n\nErnchester looked down at her with no expression whatever. He seemed smaller than Asher remembered him, thin and nondescript in his old-fashioned clothing. Though no vampire Asher had ever met appeared physically older than the mid-thirties, Ernchester seemed somehow to have aged, even in the past year. It was nothing in his stance or his face; there was no gray in the close-cropped fair hair. But looking at him, Asher felt that he was seeing an empty glass, dry and coated with bitter dust.\n\n\"I've dined.\" He turned away.\n\n\"Oh, come on, _p'tit_ ,\" laughed the woman. \"Ain't you got no taste for dessert?\"\n\nKarolyi muttered disgustedly, _\"Sacr\u00e9e couilles\"_ \u2014not at the woman, but at the delay and the needless risks\u2014and pulled a thin silk scarf from his coat pocket. With deadly delicacy he crossed it into a loop and dropped it around the woman's neck. She gasped, squeaked as her breath was cut off. Her body heaved and flopped, stockinged legs threshing; she kicked off one of her shoes in the death struggle, and it struck the wall with a smack.\n\nAsher turned his face away, pressed his cheek to the cold brick, sickened and knowing that he was a dead man if he tried to do a single thing to stop what was going on. He was aware that from the moment Karolyi had picked her up, he\u2014Asher\u2014had known that she was going to die.\n\nHe was aware, too, that the noises in the room\u2014the scraping and bumping of the chair, the obscene sounds the woman made as life blubbed and spurted and popped from her body\u2014would cover the sounds of his departure, so that he could reach the Gare de l'Est before they did and see what train was leaving at seven-thirty.\n\nHe had been in the Department too long, he thought, slipping silently down the rain gutter. He knew there was nothing he could do to save that woman. The attempt would cost him his life, and cost England, perhaps, untold lives if the Kaiser got the war he wanted...\n\n_Coward,_ he cursed himself. _Coward, coward..._ They had always said that the most important thing was to get home with the information, whatever the cost to yourself or others. Honor was another luxury the Department couldn't afford.\n\nThe clock struck seven, a reminder that time was short. Asher struck a pile of planking by the kitchen wall. Rats streamed in all directions in a hideous scurry of flying shadow, and there was the renewed stink of death.\n\nHe picked up his valise, but something made him turn and go back. Where the planks had fallen aside he could see a man's hand, palm upturned in a thin slat of light from the window far above.\n\n_I've dined,_ Ernchester had said.\n\nAsher bent and moved the plank aside.\n\nThe face of the man pushed under the boards had already been gnawed; in any case, in the dense shadows it would have been impossible to tell who he was. But there was a silver chain around the plump wrist.\n\n# THREE\n\n\"I HAVE LONG deplored the manners of what fondly believes itself to be society these days.\"\n\nLydia gasped as if she had been wakened by a freezing drench of water. The pale man took the sprayer weapon from her grasp with one hand and with the other pulled her to her feet, the strength of his fingers on her elbow such that she felt instinctively he could, had he so wished, have snapped the bone within the flesh. Past his shoulder she saw that the grille stood open, though she had been aware of no movement on the part of the dead man within the niche.\n\nFor some moments, she realized, in a rush of frightened shock, she had been aware of nothing at all.\n\nHe stood beside her now, thin and cold and utterly correct in his long white robe. His eyes, level with hers\u2014for he was not a tall man\u2014were a light, clear yellow, flecked with the brown-gray that wood turns when desiccated with age.\n\nHe shoved her against the stone of the wall, and when he spoke, she could see the gleam of his fangs in the strangely reversed lamp glow.\n\n\"Not that proper manners, or genuine society, have existed in this country since the departure of the last of her true kings for France and the advent of that rabble of sausage-devouring German heretics and their hangers-on.\" There was no anger in his voice, nor wore his face any expression whatsoever, but his grip on her arm kept her pinned where she was. His hands were like marble\u2014a dead man's hands.\n\nHe went on, \"It has always been considered that a woman who sought a man out in his chamber while he slept did so at her peril.\"\n\nJames was in danger. Later on Lydia realized that only that fact gave her courage to speak. Her single encounter with Ysidro had been part and parcel of a greater jeopardy, and in that instance, she had known where she stood. This was different.\n\n\"I had to speak to you. I came in the daylight so the others wouldn't know.\"\n\nHe released her arm, but standing in the confines of the narrow stair, it was as if they embraced. She noted that no heat came from his body, and save for the very faint reek of old blood in the folds of his shroud, no smell. Except when he spoke, his body made no sound whatsoever, neither of breath nor of movement. All these data she observed, while aware that no analysis of them came anywhere near describing what he was like.\n\nShe pushed up her spectacles. \"Lord Ysidro\u2014Don Simon\u2014I think my husband is in trouble. I need your advice.\"\n\n\"Your husband, mistress, has had all the boon and gift I could make him, and more, in the breath of life that still passes his lips.\" The sulfur eyes regarded her, remote and chill. Not catlike, nor snakelike, nor like any beast's, but neither were they a man's eyes. Even his lashes were white, like his hair. \"And a second time will I fill his hands with undeserv\u00e9d treasure, when I let you walk from this house.\"\n\n\"The Earl of Ernchester is selling his services to a foreign government.\"\n\nDon Simon Ysidro's expression did not alter. Indeed, his face, still as the peeled ivory statue of some forgotten god, had shown neither anger nor scorn, as if over the years the flesh had settled to a final resting place on the delicate substructure of skull. Nor had his voice risen over the soft level that was almost, but not quite, a monotone, and all the more terrifying for that. Don Simon Xavier Christian Morado de la Cadena-Ysidro was the only vampire Lydia had met. She wondered if others were like him.\n\n\"Come upstairs.\"\n\nHe handed her back her weapon and led the way, lamp upraised to shed light on the damp stone stairs. His feet beneath the hem of his shroudlike woolen robe were bare. Though Lydia's breath clouded gold in the lamplight, the owner of the nameless house seemed to feel nothing of the cold.\n\nFour cats somehow materialized in the scullery, miawing to be fed, though Lydia observed that none came within arm's reach of the vampire. Ysidro set the lamp on the table and touched a spill to the flame. Though he was extremely difficult to see when he moved, Lydia had impressions, like frozen images from a dream, of white hands cupping light above the curved glass chimney and carrying it to the fishtail burners above the stove; of dense gold outlining the slight hook of the nose, the long chin and trace of shadow at the corner of his mouth. He opened the icebox, addressed the cats in Spanish and put meat and milk down for them. Then he stepped away from their dishes. Only then did they come close to eat.\n\n\"Where did you hear this?\" He held a chair while she sat, then perched a flank on the corner of the table. His English was flawless, save for the faintest touch of a Castilian lisp, and the occasional oddly bent inflection that Lydia knew would have conveyed volumes to James. In the set of Ysidro's shoulders, the way he held his head, she saw the echo of a long-vanished doublet and stiffened ruff.\n\nShe held out to him the telegram she had received Monday morning from the Gare du Nord. \"Ignace Karolyi is\u2014\"\n\n\"I know who Ignace Karolyi is.\" His voice still held no very great interest, as if all emotion had long been worn away by the sheer abrasion of passing time. Indeed, in stillness, Lydia had the odd impression that he had been sitting so on the corner of the table for years, perhaps centuries.\n\nThe vampire turned the paper in ivory fingers, raised it to his nostrils, then touched it, very gently, first to his cheekbone, then to his lower lip. \"A Hungarian boyar and, like your husband at one time, a man who cherishes the honor of service to his empire above personal honor, though perhaps Hungarians as a rule do not consider truth and loyalty as the English do. A diplomat, and a spy.\"\n\n\"I didn't know then about Karolyi,\" Lydia said. Some of her panic was passing\u2014at least he appeared willing to listen to what she said. \"I mean, only what James says in his wire. But I recognized his name. I found it in one of the lists I made a year ago, when I was trying to track down medical doctors I suspected of contacts with vampires. I was making notes of every name I found in any article. This one was in an article about Dr. Bedford Fairport.\"\n\nHe tilted his head a little, like an albinistic bird. \"The man who seeks to have men live forever.\"\n\n\"You've heard of him, then.\" Reading over the long series of articles last night, she hadn't thought of Fairport's work on the changes wrought in brain and blood and glandular chemistry over time in exactly that light\u2014she doubted that Fairport himself would see it that way. But suddenly she knew that Ysidro was right.\n\n\"This was one of his early articles,\" she went on slowly. \"Back in 'eighty-six or 'eighty-seven, when he first went to Austria to study those Styrian peasants who live to be a hundred and ten. He mentions that the private sanitarium he was given charge of is owned by the Karolyi family, and that it was Ignace Karolyi who made the arrangements. He mentions Karolyi in the next article as a financial contributor who made research possible. And then Karolyi vanishes. In fact, all reference to Fairport's funding vanishes. It's never mentioned again. I checked.\"\n\n\"It astounds me that I did not read that myself.\" Ysidro sounded not the slightest astounded. \"But I subscribe to a good many journals, as I daresay you saw.\"\n\nLydia blushed. What had seemed, at the time, to be the necessary investigation of a vampire's lair became trespass in a gentleman's house. \"I'm sorry,\" she stammered, but he vouchsafed no reply.\n\nInstead his finger moved in the direction of the sprayer. \"And what is this?\"\n\n\"Oh.\" Lydia took the sticking plaster from her pocket and recapped the nozzle. \"It's full of silver nitrate solution. One can buy it in any chandlery. I\u2014well, James once mentioned that vampires sometimes slept several to a house. I didn't know what I might meet, you see.\"\n\nShe was afraid he would mock her, since, upon consideration, the weapon would certainly have been difficult to deploy quickly enough to do her any good. She had learned to deal with mockery from an early age over her medical studies, but this was a matter from which she could not simply walk away.\n\nBut the vampire only said, \"Ingenious,\" and touched the side of the pump's reservoir with the backs of his fingers, then took them quickly away. In the pale gaslight, Lydia could see that his ears had been long ago pierced for earrings, like a Gypsy's. \"Then this Fairport is in truth Karolyi's pensioner.\"\n\n\"I think so.\" Lydia held out to him another telegram, the telegram which, reaching her that morning from Munich, had caused her to pack her trunks, manufacture a moderately plausible tale for her servants, and take the train down to London in search of the man in whose kitchen she now sat, with the smallest of his cats\u2014a sinuous shadow-gray tom\u2014winding itself around her ankles.\n\nYsidro took the second paper from her hand.\n\n> LEAVING PARIS STOP STAYING EPPLER ADDRESS BOOK JAMES\n\n\"He's waxed cautious since his first wire.\" The vampire touched the paper to his lower lip again. \"You conned this book of his?\"\n\n\"After I decoded the message, yes.\" She reached down half unconsciously to stroke the cat, looking up at Ysidro where he sat above her, hands folded over his knee. His nails projected some half inch beyond the tips of his fingers and had a strange glassy appearance, far thicker than human nails. Some kind of chitin? It would be rude to ask for a cutting.\n\n\"The words 'address book' were the tip, you see,\" she explained. \"It's a simple code; last for first, counting inward, and A means B, B means C, et cetera. He keeps duplicate books. Eppler is two from the end of the E's\u2014Mrs. Eppler is the mother of an old pupil of his. She lives in Botley, about ten miles from Oxford, and it's ridiculous that he'd be going there from Paris. Two from the beginning of the F's was Fairport, in Vienna. As you see, the telegram was sent from Munich, at one-forty Tuesday afternoon.\"\n\n\"And I was that easy to find?\"\n\nLydia hesitated, wondering if she should lie. Although her initial fears had subsided, she realized she was still in a great deal of danger. She supposed that if Ysidro didn't have the ability to make people stop fearing him, he would have starved to death centuries ago.\n\nThe greater fears still lay ahead of her, a vast uncharted territory of deeds she had no concept how to perform.\n\nAt last she said, \"I knew about this house a year ago. In theory. I hadn't sought it out. But I looked up all the possibilities of vampire lairs for James while he was... working for you.\"\n\nA small line printed itself briefly near the ranged mouth, and the smallest flare of annoyance moved Ysidro's nostrils. But he only said, \"Then this Fairport is thought by the Department in Vienna to be their man\u2014they, too, having missed the articles which speak of Karolyi's contributions to Fairport's research. No matter of surprise, given the fewness of agents and the troubles in the Balkans in that year, and in France. Afterward, one presumes Fairport would have known not to publish his patron's name.\"\n\n\"What it means,\" Lydia said quietly, \"is that James is walking into a trap.\"\n\nYsidro remained still for some time, the telegram unmoving in his fingers, but Lydia could see thought and memory like swift-shuffled cards in the back of the jeweled yellow eyes. Remembering, she guessed, Fairport's articles on Hungarian and Romanian centenarians, his preoccupation with extending life, his work in a part of the world that James had described as a hotbed of vampire lore. Then he raised his head and said, \"Await me.\"\n\nAnd without seeing him leave, Lydia found herself alone.\n\nShe checked her watch, wondering how long \"Await me\" meant. If she herself were in a tremendous hurry, she could wash, dress, curl, frizz and put up her hair, and apply a judiciously minuscule quantity of rice powder, kohl, rouge, and cologne in just under two hours and a half, which her husband, manlike, seemed to consider an unreasonable length of time. At least, Lydia thought, she _knew_ how long it took her to make herself presentable and allowed for it, unlike dandies of her acquaintance who lived in the fond delusion that they could assemble the component parts of their fa\u00e7ade in \"only a moment, my dearest Mrs. Asher.\" She remembered the clothing in the dressing room upstairs, by the finest tailors in Saville Row. James had warned her, and now she knew from terrifying experience, how fast vampires could move, but she also knew that males as a species tended to potter, fidgeting endlessly with cravats and shifting coins, notebooks, and theater tickets from pocket to pocket as if fearing they would capsize if not properly trimmed. She wondered if death altered this.\n\n_Twenty-five minutes,_ she made a mental wager with herself, and was within three of it when she turned her head to find Ysidro at her side again. In his cinder-gray suit, his flesh white as the linen of his shirt, he seemed more ghostlike than he had in the white robe, as if the clothing were a barrier, a shadow of distance.\n\n\"Come.\"\n\nThe alleys and back streets through which he led her were unlit and stinking, full of furtive movement. She guessed their route was not a direct one, but could not be sure, for as soon as they descended the front steps of his house, he took her spectacles from her. Moreover, she was aware that three or four times in the fifteen minutes of their walk, he touched her mind with the blankness, the empty reverie, that vampires apparently could extend. She had the sensation of waking repeatedly from dreams to find herself each time in a new street or court, blinking at ten shades of blurred darkness all spangled with the colored embers of reflected pub lights, with Yiddish or German or Russian yammering on all sides from the little knots of seedy, bearded men clustered in doorways or around chestnut vendors' braziers. The men would step aside unconsciously to let Ysidro pass, not looking at him, as if they, too, partook of his dream of invisibility; their clothes smelled of hard work and poor diet and not enough hot water for washing.\n\nEvery other week Lydia took the train down to London to work in the dissecting rooms of St. Luke's. Men like these, with their brown, broken teeth and their flea bites and their dirty, callused hands would be delivered by the workhouse vans, smelling of carbolic and formalin, dead of tumors that had burst untreated, of pneumonia, of consumption, or the other ills of poverty, so that she and others like her could study the intricate beauty of muscle and nerve beneath the knife.\n\nIt was the first time in her scholarly life that Lydia had been among them living, and her mind swarmed with questions she wished to ask them about the food and working conditions that had contributed to their pathologies. On the other hand, she felt very glad of Ysidro's protection.\n\nThey crossed a plank bridge over water nearly invisible beneath low-lying fog, passed the wry, dark roofline of some very ancient church. In time they traversed a sordid alley behind a pub near the river and descended an areaway thick with garbage and the smell of cats. Though her eyes had grown used to darkness, Lydia saw only the moth flicker of pale hands before she heard the snick of a lock going over. Hinges creaked. Ysidro said \"Come\" again and stepped into absolute dark.\n\nA match scratched. Ysidro's narrow face appeared, outlined in saffron. \"You need not concern yourself over rats.\"\n\nHe touched the flame to a pair of guttered candles in a double branch. The plaster of the walls was black with mildew, falling away to reveal underlying brick. \"Like cats, they are aware of what we are and know that though it is the human death we need to feed our minds, we can derive sustenance from the blood of any living thing.\"\n\nHe lifted the branch. Twin lights called twin ghosts of shadow, merging and circling in a strange cotillion as he led her toward the back stair. \"Anthea and Ernchester sleep seldom at the house on Savoy Walk these days. It is best to let memories lie. She scarce ever hunts this early in the night, but it may be that she has gone to her dressmaker.\"\n\nLydia checked her watch again as they passed through a downstairs hall: peeling silk wall covering, doors blackly ajar. \"I suppose this close to Christmas there'd be one open...\"\n\n\"If one has money, mistress, one always finds those willing to sell their sleep and their leisure. I have visited my bootmaker at midnight and never found him but that he was consumed with delight.\"\n\n\"What do you tell him?\" She couldn't imagine her aunt Harriet's modiste keeping open past seven for Queen Alexandra herself.\n\nYsidro regarded her with eyes turned amber by the ruddy light. \"That I will have none of this foolishness of two-colored shoes, nor buttons up the side.\" He turned to the room at the top of the stair. \"So.\"\n\nLike Ysidro's house, the chamber held little furniture, and that furniture old. A tester bed with a curving footboard stood against the rotted wall panels, the counterpane as faded as the silk paper downstairs; on the other wall, a blackwood armoire, stained, chipped, thick with dust-choked carving and mottled with water damage. Its doors stood open. Petticoats, corsets, stockings lay across the bed, and with them\u2014separated by the length of space that would have accounted for a large portmanteau\u2014two dresses Lydia immediately recognized as unsuitable for travel, one because of its now-unfashionable leg-o'-mutton sleeves, the other because it was white, a color no sane woman, dead or Undead, would wear on a train.\n\n\"She's gone after him,\" Lydia said, opening the armoire doors. The only dresses there in the current fashion were the decollet\u00e9 silks and sumptuous velvets of evening wear. No waists, no skirts\u2014Lydia peered shortsightedly into the lower drawer, and Ysidro handed her eyeglasses back\u2014and no walking shoes. \"She packed in a hurry...\"\n\nShe halted, frowning, as her eyes adjusted to the sudden clarity and she realized that the tops of the dressers were in disorder: scarves, sleeves, kerchiefs caught in drawers that had been hastily closed.\n\n\"The place has been searched.\" Ysidro, who had passed swiftly into the other room, returned, moving his head as if scenting the air. \"Living men, days ago, before she packed, I think. The air still whispers of their tobacco and their blood.\" He crossed to the bed, studied the garments lying there. All the colors, as far as Lydia could tell in the low amber radiance of candlelight, that a dark woman would wear, everything of the highest quality\u2014Swiss cotton, Melton wool, Italian silk. They were cut for a woman of Lydia's height, with a waist like a stem and breasts like blown roses.\n\n\" _Her_ clothing.\" Ysidro turned a chemise over in one gray-gloved hand. \"None of his. I like this not, Mistress Asher.\" He let the silk slither away. \"For many years now it has only been love of her that has kept him on this earth. She is the strong one. He hunts in her shadow, brittle, like antique glass.\"\n\n\"Might that be reason in itself?\" Lydia turned from the dresser, where an ivory hair receiver and ivory-handled scissors spoke of other pieces of a matched toilet set now vanished: brush, comb, mirror. A glove box lay open, gloves of all colors lying like dried and flaccid spiders where they had been spilled.\n\nYsidro lifted a brow.\n\nLydia went on hesitantly, \"Might he be fleeing her?\"\n\n\"To such sanctuary as the Austrian Empire would afford?\" He moved around the corner of the bed, touched the imprint on the dusty counterpane where the portmanteau had rested, and his nostrils flared again, seeking clues from the alien scents of the air. \"I would not have said so. She loves him, guards him; she is all in all to him.\"\n\nHe paused for a long time, his face half turned from her, inexpressive as the level softness of his voice. \"But it is true that one may hate one's all in all at the same time that one loves. This was something...\" Another pause, debating; then he went on, \"This was something I never understood as a living man.\"\n\nHe met her eyes, expressionless, and she could not reply.\n\nAfter a time he said, \"The Calais Mail departs Charing Cross at nine. I doubt we can prepare swiftly enough to make tonight's. Meet me tomorrow night at eight on the platform, you and your maid. I shall wire my own arrangements to Paris beforehand; I can\u2014\"\n\n\"I'm not taking a maid!\" Lydia said, shocked.\n\nYsidro's brows lifted again, colorless against his colorless face. \"Naturally, she shall know nothing of me, save as a chance-met companion on the train.\"\n\n\"No.\"\n\n\"Mistress Asher\u2014\"\n\n\"This is not a matter for discussion, Don Simon.\" Frightened as she was at the thought of traveling to Vienna\u2014of dealing with one or possibly several vampires\u2014the thought of journeying in company with one unnerved her still more. And as for putting Ellen or anyone else in similar danger...\n\n\"I came to you for advice in dealing with vampires, specifically with Lord Ernchester. There isn't a great deal of reliable information on the subject, you know.\" She saw the flare of genuine exasperation in his eyes behind the vampire stillness, and rather to her own surprise it didn't frighten her as it had.\n\n\"But I would not take _anyone_ \u2014certainly not a woman who's been my friend and servant for nearly fifteen years\u2014into that situation without telling her what kind of danger she may be facing, which, on the face of it, is impossible.\"\n\n\"A woman of your station does not travel alone.\"\n\n\"Nonsense. My friend Josetta Beyerly travels by herself all the time. So does\u2014\"\n\n\" _You_ will not.\" Ysidro's voice did not grow louder, nor his expression change, but she felt his irritation like a wave of cold off a block of ice. \"In my day no woman traveled alone, save peasants and women of the streets.\"\n\n\"Well, when I encounter a roving band of paid-off mercenary soldiers between here and Calais, I'll certainly wish I'd taken your advice.\"\n\n\"Don't talk foolishness. You might trace Karolyi but you would never get near Ernchester, and it is Ernchester to whom I must speak on this matter.\"\n\n\"You're the one who's talking foolishness,\" retorted Lydia, though she knew he was right. \"This is the twentieth century, not the sixteenth. I will certainly appreciate whatever advice you can give me...\"\n\n\"Advice will gain you little against either Karolyi or Ernchester. If you wish to warn your husband of his danger, you must travel with me\u2014and travel I will, to prevent Charles from doing this thing, whatever his motives.\"\n\nLydia was silent for a moment, unnerved beyond words at the thought of such a journey but remembering how utterly unprepared she had been to encounter him in the crypt. \"If you must,\" she said slowly, her dream of ranged white faces returning to her. \"Thank you... but I am not taking my maid into the situation she'd face if we meet Ernchester, and I'm not exposing her to the chance of finding out inconvenient things about you. Which she'd do,\" added Lydia. \"Ellen's got an inquisitive streak, and she's smarter than she appears. I won't do that to her.\"\n\n\"Hire one for the journey, then.\"\n\n\"So that you can kill her when the journey is done? And kill me, too, for that matter?\" she added, her mind making a tardy leap to the ultimate danger of traveling with the dead. She knew too much already\u2014even her admission of knowing where his lairs lay had violated the lines so carefully drawn when James and Ysidro had parted a year ago in the burning house on Harley Street.\n\nHe needs a human companion, she thought, in his search for Ernchester, someone who could deal with such problems as might overtake him when daylight was near; and he needs someone who knows James well enough to track him, to guess his movements, and through him, Karolyi and Ernchester.\n\nShe'd told Ellen and Mrs. Grimes she was visiting her cousins in Maida Vale. It would be weeks before she was even missed.\n\nShe kept her eyes on his, positive she resembled nothing so much as a myopic rabbit attempting to stare down a dragon.\n\nSlowly, the vampire said, \"You need have no fear of me, mistress. Nor will your woman, so long as she keeps from asking about that which does not concern her.\"\n\n\"No.\"\n\nJames had told her of the vampire ability to touch living minds, a cold grip, the dreadful sensation of steely will. But his power extended to blanking and smothering thought, to diverting attention... not to changing resolve. It was a predator's power, a spy's and a fugitive's, not that of one who must negotiate with humankind. She saw that realization dawn in his eyes, and his mouth tightened with annoyance.\n\n\"If we are to be companions in this enterprise, I will not have you traveling alone abroad like a jauntering slut,\" he said. \"I think your husband would agree with me in that.\"\n\n\"What my husband thinks is my husband's business, and neither yours nor mine,\" said Lydia. \"And I would rather be taken for a jauntering slut than betray a woman who's dependent on me. And if it doesn't suit you, I'll travel by myself.\"\n\nYsidro bent and kissed her hand, his lips like silk left outside on a dry night of hard frost. \" _Bon voyage_ , then, mistress. And _bonne chance_ in your dealings with the Undead.\"\n\nWith a sensation like waking up, Lydia found herself alone.\n\nIt was not, in fact, terribly late to be abandoned in a completely unfamiliar part of London. Though the fog had thickened and the night was growing colder, the streets were still populous, albeit with foreign laborers from the sweatshops that abounded in the neighborhood and with sailors who seemed to accept Ysidro's outdated presumption that a woman on her own was a jauntering slut, at least as far as Lydia could understand their idiomatic references to Master John Thursday and pintle jigs. Evidently, Josetta's suffragist doctrines had yet to penetrate this far. Lydia made a mental note to let her know.\n\nAs she had guessed, she wasn't far from the river, and on the broad, electric-lit thoroughfare of the Embankment, she had no trouble in finding a cab to take her back to the small hotel near the museum where she had left her luggage.\n\nTaken in the balance, she thought\u2014removing her gloves and unpinning cook's nondescript hat\u2014she was more glad than sorry that Ysidro would not be accompanying her to Vienna. People _did_ travel alone, of course, and there was no reason why she shouldn't, Ysidro's antiquated notions not-withstanding: The world abounded with policemen to be appealed to, porters to be tipped, cabs, guides, travel bureaus, quality hotels with obliging managers, and shops in which to purchase anything she might forget to pack. The lack of a maid would engender certain difficulties, of course, but that was what hotel chambermaids were for.\n\nIt was unlikely she would catch up with James before he reached Vienna, but with luck, his cautious nature might keep him out of immediate peril until she could arrive and apprise him of the fact that he was dealing with a double agent\u2014if worse came to worst, she could inform whoever was in charge of the Vienna Department that Dr. Fairport's sanitarium in the Vienna Woods was the likeliest place to search for a clue to James' whereabouts.\n\nIf whoever was in charge wasn't taking money from the Austrians as well.\n\nFrom what James had told her, that was at least a possibility, and Lydia wondered how on earth she'd be able to tell.\n\nForcing down her sense of panic again, she reviewed such of her luggage as she had unpacked for the night: peignoir, two pairs of slippers\u2014the prettier but far less comfortable ones in case any of the hotel staff came in\u2014rose water and glycerine for her hands, distilled water of green pineapples to alleviate the incipient wrinkles Aunt Harriet had always assured her excessive reading would bring on, silver-backed hairbrush, comb, toothbrush, nail file, curling irons, frizzing irons, hairpins, several sets of underwear, corsetry, petticoats, an array of silver table knives whetted to as deadly an edge as silver would take, and a .38 caliber revolver containing the silver bullets she had had made last year.\n\nLydia had felt like the heroine of a penny dreadful, packing that along with the talcum, rice powder, rouge, lotions, and perfumes.\n\nThere was also the market basket that she'd bought in Covent Garden that afternoon, containing thick braids of garlic bulbs, packets of aconite and whitethorn, branches of wild rose. She wreathed her pillow with them and hung them in the single window of the unheated little back bedroom, and as she undressed and unlaced herself\u2014there were disadvantages to staying in hotels where she was unlikely to meet anyone she or her family knew\u2014she turned over in her mind her other options.\n\nConfide in one of her friends and take her as a companion? Josetta understood politics and feared nothing but, Lydia knew from experience, wasn't particularly practical: she always seemed outraged at being arrested for suffragist activities which, though certainly necessary for the overall strategy of that movement, flagrantly violated the law. Her other close friend, Anne Gresholm, wiser and more intelligent, had lectures and students of her own to tend to, and her health was not good. In any case, the danger remained the same. Lydia was also aware that she had a certain amount of sufferance from Ysidro as long as she told no one of the existence of vampires. If she violated that secret, or if Josetta or Anne guessed\u2014which they surely would\u2014she could not answer for their safety or her own on their return.\n\nGo to Ysidro, then, and ask him to accompany her after all? It would only resurrect the issue of a maid. She wouldn't endanger Ellen, and a chance-hired stranger would be in the same peril and might be more inquisitive and less reliable to boot.\n\nLydia sighed, slipped the revolver under her single, paltry pillow, and at length drifted into sleep among blankets strewed with wolfsbane, railway timetables, and guidebooks to the eastern reaches of the Austrian lands.\n\n_It must be the smell of the garlic,_ she thought, aware that she was dreaming and that the dream was far more vivid\u2014lurid, even\u2014than anything she had dreamed at home. _The garlic, or that house in the fog..._\n\nShe stood on the terrace of a tall mansion, a glory of half-timbering and ornamental stone, with a moon-drenched garden maze on one hand and lighted windows of many-paned glass on the other. Looking in, she saw courtiers in the stiff velvets, the soft-glowing pearls of Elizabeth's reign. They were dancing, and she could hear the swift and complex run of the music: hands linking, farthingales flouncing, the men all wearing little Shakespearean chin beards and looking silly beyond description in tights and trunk hose and bulging peasecod doublets, the women in skirts hooped out like kitchen tables and in collars of upstanding, wired lace.\n\nA woman stood near the windows, whom Lydia noticed because she was wearing modern garments, a plain brown serge that didn't fit her particularly well and certainly didn't become her. She was plain-faced, with a slightly receding chin, of medium height, and rather pear-shaped without being fat; a wealth of curly black hair lay loose upon her narrow shoulders. Sometimes when Lydia's eyes left her and returned, she'd be wearing Elizabethan clothing, dull-colored and worked high to the neck. A servant's gown, or a poor relation's. Her small hands fussed with the jet buttons of her sleeves.\n\nThen, very softly, Ysidro spoke.\n\n\"You would think, the way they danced, they'd wear something more suited to the exercise, would you not?\"\n\nHis voice was so quiet Lydia wondered that she could hear it through the glass and over the music. She saw him then, standing at the brown woman's side. His black velvet doublet, his knee-length breeches, his high, supple boots, harked just enough to a later period to avoid the inherent ridiculousness of male Elizabethan garb without appearing anachronistic, and his hueless hair seemed warmer in the torchlight, darkened almost to honey. The girl replied, inaudibly, but it made Ysidro laugh, as if he were playing the part of someone else. Can't she see it? wondered Lydia, terrified. _Can't she see what he is?_\n\nFor a time they stood shoulder to shoulder watching the dancers in their fairy-tale costumes, the vampire and the girl.\n\nLydia's dreams changed, fleeted. She saw them again, this time in another garden, wide parterres of topiary and _tapis verte_ , when he taught the dark-haired girl to waltz in the moonlight under the blank eyes of marble gods. Saw them later kiss beneath the gargoyles of an archway, among crowded houses built on a bridge, torchlight and lamplight from the windows above them red as jewels in Ysidro's eyes. Through another window\u2014two windows, for Lydia herself was in a dark room across an alley that plunged sixty feet down into a canyon of night\u2014she saw Ysidro lying wounded on the girl's sparse bed, the girl bending over him in some kind of old-fashioned garb, knotting dressings over a sword cut in his chest that would have killed a living man. Ysidro moved his hand a little, and the girl bent down to press her lips to his.\n\n\"You are different from all these others,\" she heard him say, in the curtained embrasure of a palace window, the sound of violins like fragile perfume amid the talk and laughter of dancers. Palace of Versailles, Lydia guessed vaguely from the cut of Ysidro's plum-colored silk coat. \"How sick I have grown of them, through all eternity. I had not thought to find a woman like you.\" He raised the girl's hand to his lips.\n\n\"We have known one another, loved one another, down through endless time.\" He wrapped the girl in the dense velvet weight of his cloak as they stood alone in winter-locked woodlands, moonlight shiny on a meringue of snow beyond the barred shadows of the copse in which they stood. The girl's hair was disheveled, her gown torn, and Lydia knew that Ysidro had rescued her from some peril, and that the bodies of dead men lay out of sight in the gully by a winter-silent stream. Lydia's own feet were cold in shoes wet with slush as she stood behind a tree with the wet weight of her skirt sticking to her ankles. \"Do you not remember?\"\n\nThe girl in brown\u2014it was the same brown dress as before, with the puffed sleeves and wide collar ten years behind current mode\u2014whispered, \"I remember, Simon. I remember... everything,\" and their mouths met in the zebra moonlight.\n\n_No!_ cried Lydia, and though her breath swirled in a diamond cloud, she could produce no sound. _He's lying to you! He's going to kill you!_ Horrified, she fought to run toward them, but black thorns caught her skirts, held her back. She tried to pull free, and the branches cracked beneath her fingers like dried insects. She woke to find herself clutching the bony fragments of hawthorn twig that lay on her pillow.\n\nLydia took the two o'clock train for Paris. Even after that strange farrago of romantic interludes by moonlight faded from her dreams, she kept waking with the icy sense that a slender shadow with yellow eyes waited just outside her door. By the time she was awake, bathed, laced, dressed, packed, powdered, perfumed, made-up, and had fixed her hair\u2014no small accomplishment without a maid\u2014and considered herself fit to be viewed by the public, she had missed both the morning trains. Never again, she thought, will I stay at an inconspicuous hotel just to avoid questions from my family. She reached Paris shortly after nine that night, missing the train to Vienna by an hour and a half\u2014it left from another station in any case\u2014and registered, travel-weary and aching, at the Hotel St. Petersbourg that Thomas Cook and Sons had obligingly contacted on her behalf the day before.\n\nParis, at least, she knew from her days of debutante shopping and educational sightseeing, and later from medical conferences. Her French was good, and she understood how to handle herself in this milieu. Perhaps, she reflected, the journey would be easier than she feared, as long as she took everything methodically, one step at a time, like a complicated dissection or a series of analyses of unknown secretions.\n\nAgain she slept badly, her dreams filled with the dark-haired girl in brown and Don Simon Ysidro rescuing each other from the cardinal's guards and trading kisses on the sands of moon-soaked Moroccan deserts. Waking in the darkness, comforter drawn up to her chin, she stared at the slits of reflected street-lamp outlining the shutters and listened to the voices of the caf\u00e9 down in the street below, wondering where James was and if he was all right. Milk wagons were creaking in the streets when she finally slept.\n\nThe Vienna Express didn't leave until seven-thirty that evening, so there was plenty of time, not only to pack and dress properly and have the hotel maid fix her hair, but to do a little shopping at the Magasins du Printemps, which lay just down the street.\n\nShe was buttering a croissant in the almost empty dining room and speculating about the pathology of Don Simon Ysidro's fingernails\u2014clearly there had been an organic change of some kind, so the physical side of the vampire syndrome did not involve complete cellular stasis\u2014when she heard the solitary waiter murmur, \" _B'njour, m'mselle,\"_ and glanced up to see another woman enter the room. Even at this distance and without her spectacles, Lydia deduced this woman to be harmless: tallish and slightly stooped, she moved with the uncertainty of one who feels herself to be half a foreigner even in her own country, let alone in a land where she doesn't speak the language.\n\nThe next second she frowned, wondering why she thought she recognized her. There was something familiar about her, and as the woman drew near and came a little more into focus, Lydia realized what it was.\n\nShe was wearing a brown dress with the puffed sleeves and wide collar fashionable in the nineties.\n\nLydia set down her coffee cup.\n\n\"Mrs. Asher?\" The woman stopped beside her table, fidgeting her hands in mended gloves, a look of anxiety in her blue eyes. She was about twenty-three, much more awkward than she'd been in the dreams, and, like Lydia when Lydia knew nobody would see her, wore eyeglasses. \"Don Simon told me I'd find you here.\"\n\n# FOUR\n\nA WALTZ FROM TCHAIKOVSKY'S _Nutcracker_ ballet had been popular in Vienna the year Asher spent in and out of that city. Closing his eyes to the lulling rock of the train, Asher could hear it again, drawing in its colored wake the bright glimmer of gaslight in the Caf\u00e9 New York on the Opernring during Carnival season, the sparkle of snow on the pavements, the slurry patter of French and Italian and Viennese German all around. Court gossip and psychoanalysis, music and politics and whose wife was betraying whom. Thirteen years later it was still as clear as yesterday.\n\nYoung matrons in their masks and costumes questing nervously for unspecified excitement. Uniformed officers, gay in swords and spurs and braid.\n\nFran\u00e7oise.\n\n\"Nothing here is as it seems,\" she had said the night he walked with her to the caf\u00e9 after a St. Valentine's Ball given by her brother; and that, at least, he had known was true about himself.\n\nShe was a thin-faced woman of his own age, his own height; though to be thirty-five and almost six feet tall, and of a strong cast of feature, had always been something only considered attractive in men. Her brother was a director of the biggest bank in Vienna and owned farms, vineyards, blocks of flats in the Seventh District. His wife, the second daughter of a baron, had been trying for years to marry Fran\u00e7oise off in diplomatic circles.\n\nAsher wondered if she had ever married. Had ever trusted another man.\n\n\"People pass the days away in caf\u00e9s like this, sipping coffee, reading the feuilletons, watching the world go by.\" She moved one shoulder in a graceful shrug, her smile rueful and a little sad. She was a biscuit-colored woman, but the emeralds in her earrings caught sparkling echoes in her eyes. \"Outsiders think it's all very relaxed, very gem\u00fctlich, but it's really because most of the people here live in one-room apartments, they and their families together, and they can't stand the smell of cooking and dirty diapers and the arguing of their children. So they come here and look leisured and carefree because that is exactly what they are not.\n\n\"We here in Vienna have a hundred separate degrees of nobility and bureaucrats, titles and order and neatness and rules, and underneath, the Slovenes and Serbs and Czechs and Moldavians and Muslims are all clamoring to have their own nations, their own schools, their own languages, their own crowns. They bomb and shoot and riot and scheme with the Russians and the British and whoever else they think will help them break free.\"\n\nHer big hands in long gloves of ivory kid darted, as if forming illustrative patterns that Asher could not see. He had first encountered her at a Twelfth Night ball in his guise as a professor of folklore. Folklore was always popular in Vienna, the more bizarre the better, and in exchange for arcana on Japanese werewolves and Chinese milkweed fairies, Asher had met a number of the aforesaid Serbs and Czechs and Moldavians and was beginning to find out just who they were scheming with on the subject of riots, bombs, and freedom from Austrian control.\n\nHe hadn't really needed to seek out Fran\u00e7oise a second time.\n\nBut he had.\n\n\"When we complain,\" she went on, \"it isn't really a complaint. When we weep, it isn't necessarily out of pain; and when we dance, it isn't always for joy. Yes isn't really yes, and no is seldom no, and the palaces you see mostly aren't really palaces, and everyone talks about everything except what really consumes their thoughts.\"\n\nHer dark brows drew down over those bright green eyes as she considered him, skirting the brink of questions that she wasn't sure she wanted answered or even asked.\n\n\"We don't always know whether what we're seeing is real or a mask.\"\n\nAsher's eyes had met hers, and he hadn't known what to reply.\n\n_I spoke to you last week to find out which of your young officer friends are deepest in debt._\n\n_I'm here to learn things that could get your armies defeated, your country disgraced, your friends and nephew killed._\n\n_I think I love you._\n\nHe wasn't sure just when that last had happened.\n\nFor a time they regarded one another without masks. Even now, looking back on it from the edge of dreaming, Asher didn't know what he would have said, had she asked him then.\n\nBut she smiled and put her mask back on, and held out her hand. \"It's the 'Waltz of the Flowers,'\" she said. \"Do you dance?\"\n\nHe had never been back to Vienna.\n\nHe pulled himself brutally from the edge of sleep. It was too early to sleep. The lights of Paris were barely behind them: St. Denis, Gagny, Vaires-sur-Marne spilled firefly glints into the indigo dark. Asher sipped the caf\u00e9 noir he'd tipped the porter to bring to his private compartment\u2014the compartment he'd managed to secure at the last possible moment, leaving himself again, he reflected dourly, with only five pounds in his pocket.\n\nBut on the Paris-Vienna Express it was imperative that he travel first class if he wanted access to the car where Karolyi and Ernchester would be. He knew himself incapable of remaining awake another night in second class. In first class he would be safer, less likely to be seen by either Ernchester or Karolyi.\n\nKarolyi.\n\n_Nothing here is as it seems to be._\n\nThat same Carnival evening he'd seen Karolyi surrender a dance with the most attractive heiress in Vienna that season to go outside and stop a carter from whipping his horse. Fran\u00e7oise's comment had been, \"Laying it on a bit thick, no?\" And, to Asher's raised brow: \"You must have noticed he only does such things where others will see.\" Asher had noticed, but to his knowledge, no one else had, save Fran\u00e7oise.\n\nHe hadn't had time to telegraph Lydia. Nor Streatham, telling him Cramer was dead.\n\nStreatham would have left his office at six anyway, Asher reflected sardonically, and wouldn't be back till twenty minutes past nine in the morning. Dear God, had it only been that morning?\n\nWhen he closed his eyes, he could see the brassy-haired prostitute, back arched like a bow above the seat of the chair, flopping and kicking her legs as her body gave up its life.\n\nCould see the dark glitter of Cramer's blood where the rats had gnawed his face.\n\nHe feared his dreams, but they drank him in, like water flowing down into darkness.\n\nHe thought, _I've been in this room before. When have I been in this room?_ Beyond shrouded windows rain streamed down, sodden and heavy; if there had been furniture in the room once, all had been cleared, but for a table at one end. Shawled with drippings, the guttered stumps of candles burned in tall holders, two at the head, two at the feet, and their light made daffodil thumbprints on the velvet pall that draped one end of the table like a thrown-back counterpane and winked in the jeweled leaves of a coronet set in the black cloth's midst. The dream had the taste of very distant memory, and he somehow knew that it was deep, deep in the night.\n\nA woman lay on the yellow marble floor before the table, like a second pall dropped by a careless servitor, awkward in corsets and bum rolls and strange pennoncels of ribbon. Her hair was black, except where the candle flame breathed on it a cinnamon light. Its puffs and volutes, like those of her clothing, were in tangled disarray.\n\n\"Anthea.\" Another woman came down the room's length, passed within touching distance of Asher, or where Asher would have been had any of this ever really occurred. \"Anthea, you must come to bed.\" Even drugged with sleep, Asher identified the longer vowels in _come_ and _bed_ , the elongated _ou_ , and thought automatically, _Late seventeenth century._ This new woman also wore black. Against ebon lace cascading from high combs, her face seemed lifeless, her eyes swollen and red. \"'Tis long gone midnight, and the mourners away to their homes.\" She knelt in a sighing waterfall of back-draped skirts and touched the prone woman's arm.\n\n\"How can he be dead?\" It was a deep voice for a woman's, low but very clear. There were no tears in it, only a tired wonderment, as if she really wanted to know. The odd thing was that Asher recognized it but remembered a modern pronunciation, unlike the one she used now.\n\n\"I don't... I don't feel as if he were. Did I walk up the stairs, would he not be waiting at the top?\" A ribboned fontange snagged in her hair, tilted drunkenly as she raised her head, then slithered to the floor unheeded. Though he was at least twenty feet from her, Asher knew her eyes were the color of last autumn's oak leaves, matted at the bottom of a pool.\n\n\"I felt so, when my Andrew died.\" The other woman put a hand to Anthea's side to help her up. Anthea rose unsteadily, tall and wholly beautiful though her clothing was askew from lying on the floor. The flesh of her breasts rose in creamy mountains above the flattening of her bodice, and small shadows marked the paler line of her collarbone, the curves of her broad-set cheeks. \"Believe me, my darling,\" said her friend, \"he is dead.\"\n\nSlowly, like a very old woman, Anthea stepped forward, reaching to touch the velvet pall where, Asher realized, a coffin had lain. Her voice was very small, like a child's. \"I don't understand what they expect me to do without him.\"\n\nShe turned and walked the length of the room, as if she did not see her friend who followed in her wake. Certainly she did not see Asher, though her black skirts brushed the tips of his boots and he smelled the musky blend of ambergris, funeral incense, and womanhood that sighed from her clothing. Her tall lace headdress lay on the floor where it had fallen, like a broken black rose.\n\n\"Steffi, darling, you do realize how _dreary_ you are when you're jealous?\"\n\nAsher jolted awake, sunlight in his eyes, his neck stiff and the gentle, persistent rocking of the train still tapping in his bones. He slumped back into the corner of the seat again and listened as Steffi\u2014whoever Steffi was\u2014rumbled some reply in harsh Berlin _hoche Deutsch_ as he and his baby-voiced Viennese girlfriend passed down the corridor outside, toward the restaurant car presumably. Asher reached up and switched off the still-burning electric lamp above his seat, then pressed the porcelain button to summon a porter. When he ordered shaving water\u2014accompanied by a tip he couldn't well afford\u2014Asher asked the time.\n\n\"It is five minutes past ten in the morning in Vienna, sir,\" said the man in Italian-accented French. \"Ten minutes past nine in Paris. Myself, I should put local time at quarter of ten.\"\n\nAsher, who had reset his watch to Paris time but had been too exhausted to wind it last night, set it again. \"Have they done with serving breakfast?\"\n\n\"They will have by the time m'sieu has finished shaving.\" The porter touched his cap. Venetian, Asher guessed. Dark, but with the extraordinary sensual beauty that even the crones of that ancient republic possessed like a birthright. \"I could bring m'sieu a little something.\"\n\nAsher handed him another silver two-franc piece, reflecting that porters on the Vienna Express would undoubtedly pocket anything from dollars to piastres. \"You wouldn't happen to know whether the Hungarian gentleman who's traveling with the Englishman is still in the restaurant car, would you? Not,\" he added, holding up his hand, \"that this matter need be mentioned to either of them.\"\n\nThe Italian's dark eyes brightened with interest, and Asher added another franc. \"A matter of family business.\"\n\n\"Ah.\" He nodded knowingly. \"The Hungarian and the Englishman, their light burned on throughout the night, though of course because the curtain was closed I could see nothing of what passed within the compartment itself. But I know that they did not summon me to take down the bunks, and this morning when I go in to _ranger_ the compartment, still they have not been slept in.\" He glanced meaningfully up at Asher's pristine bunk. Asher had locked the compartment door upon entering last night, and if this man had knocked, had slept through it.\n\nWhen the porter\u2014whose name, he said, was Giuseppe\u2014returned with hot water, a breakfast tray, and coffee, he brought also the information that the Hungarian Herr Feketelo was no longer in the restaurant car. Following breakfast, Asher made his way unobtrusively down the corridor, banking on the fact that Karolyi, like his traveling companion, would sleep during the day. His own compartment was near the head of the coach, close to the accordion-fold bridge leading into the restaurant car. The compartment shared by Karolyi and Ernchester, according to Giuseppe, was close to the tail end. The next car in the train, Asher had already determined, was the baggage car.\n\nIt was sealed, but Asher had dealt often enough in duplicate seals and keys\u2014and had seen enough of the sheer preternatural physical strength and agility of vampires\u2014to know that this would present Ernchester no difficulties.\n\nAsher expended several more francs from his dwindling resources on arrangements with Giuseppe to have his lunch also brought on a tray. It was certainly a more comfortable way to see Central Europe, he thought, than dodging around the Dinaric Alps with a price on his head, dogs\u2014and Karolyi\u2014on his trail, a pocketful of incriminating serial numbers from Swiss bank accounts, and a bullet in his shoulder. He listened to the voices passing in the corridor and kept his own curtain closed, watched the dark trees and fairy tale villages of the Black Forest rise and fold themselves over the lift of the Swabian Alps, with the higher gleam of white in the distance that marked the true Alps growing nearer as the train bent southward. At Munich the Express stopped for half an hour to add two second-class cars and another _wagons-lits_ that had come down from Berlin, and Asher risked a dash to the station telegraph office to send two wires, one to Lydia telling her of his altered plans, and one to Streatham, informing him of the death of his agent.\n\nHe remained angry over that, not so much at Ernchester and Karolyi\u2014it was, after all, a game they all played\u2014but at Streatham, for assigning the least experienced of his men to a job that he should have known was dangerous. And, though he knew there was nothing else he could have done, at himself.\n\nCrossing the great floor of the station under the weak gray daylight of the glass ceiling, Asher tried to remember who was in charge in Vienna these days. Perhaps no one he knew. Streatham had been right about the reorganization, of course. Fairport, at least, would still be in Vienna, unobtrusively operating his safe house out at his sanitarium in the Wienerwald, peddling rejuvenation to bankers and stockbrokers' wives, fussy and trembly with his ill health and his cotton gloves and that fanatic glint in his pale blue eyes.\n\nAsher smiled, recalling the three days he'd spent with that comic-opera hypochondriac, journeying to some remote Czech village so Fairport could interview a peasant brother and sister who were contemporaries of his own great-grandparents, and so Asher could trace local variations of the verb _byti_ or _biti_ \u2014and have a look at a forest road leading into Saxony that, for no good reason, had been widened and repaired with funds from Berlin. The old man hadn't taken off his gloves for the entire trip, had warmed the snow water of the streams because it was better for the liver, and had brought his own food, his own sheets, his own soap. The local peasants had shaken their heads and given him names of their own\u2014\"the laundry maid\" and \"Grandmother English\"\u2014and the innkeeper at one village had taken Asher aside and gravely asked if it were true that in the City\u2014meaning Vienna\u2014they had doctors who could cure people of such ailments. Asher had been hard put to explain that Grandmother English _was_ such a doctor.\n\nHe grinned at the memory and settled into his compartment again with a feeling of having successfully dodged through a complicated obstacle course. In addition to sending the telegrams, he had purchased the _Neue Freie Presse_ and two spring-operated children's toys: a bear that clashed cymbals when wound with a key; a donkey whose four legs moved so that, if carefully balanced, it would more or less walk. He put them through their paces on the table, deeply and gravely entertained.\n\nOther passengers were reboarding, armed with fresh books, magazines, newspapers, candy, or pastry. Through the window he glimpsed the man who had to be the jealous Steffi and his fairylike Viennese girlfriend, her arms full of fresh flowers, and smiled a little at the capacity of humans to believe what they wish to believe.\n\nThere was a beautiful dowager in an impeccable Worth suit, trailed by a cowed-looking maid and three little black French bulldogs; a white-bearded gentleman with the face of a warrior monk, and a boy who might have been his grandson or a servant hurrying in his wake. Karolyi, clean-shaven and fresh, a winter rose in his buttonhole, strode lightly along the platform, pausing to remove his hat when he spoke to a shabby girl selling peanut brittle. Asher saw by the girl's face that he'd considerably overpaid her, and remembered the brassy-haired whore again, tied to her chair. He wondered if the police had found her body yet.\n\n_Why Ernchester?_\n\nHis mind gyred back to the question as the train rocked into motion once again.\n\nWhy an Englishman at all? Had the Vienna vampires refused to cooperate with an Austrian offer? Not as odd as it might sound: the Viennese, in Asher's experience, had their own rationale for doing things, an idiosyncratic frivolity that could encompass any reason from Czech or Hungarian\u2014or Serbian or Moldavian or Venetian\u2014nationality to a personal opinion that the Emperor was an old fuddy-duddy whom they disdained to serve.\n\nAnd indeed, whatever promises the government made, Asher knew the vampires were right to guard their anonymity. Having been a spy for seventeen years, he knew too well that no government\u2014certainly not his own\u2014could be trusted to keep any promise it made.\n\nIt still didn't explain why an English, rather than a French or a German, vampire had been approached.\n\nOr had they? He paused in the act of dismembering the key-wound bear, a half-farcical vision rising in his mind of the sealed baggage car stacked high with coffins and trunks in which slumbered the vampires of Paris; of himself, strolling innocently into the restaurant car to face table after table of chalk-white, bone-thin faces and a sea of eyes that burned like actinic flame.\n\nWhen it came down to it, what the hell was he going to do once he reached Vienna? Try to hand the problem over to another incompetent and reluctant Department head? Get some other young novice killed?\n\nHe unfolded his bunk, undressed, and slept again, to wake from uneasy dreams with the sensation he'd had in dealing with vampires before, of having had his mind momentarily blanked. In silence he swiftly rolled from his bunk, the compartment around him lit only by the yellow glow from the passage leaking around the edges of the curtain. It showed him an empty compartment\u2014certainly there was nowhere to hide, for there was barely room for one person, let alone two\u2014and he pressed his face to the edge of the door, moving the curtain just enough to see past it.\n\nKarolyi and Ernchester were walking up the corridor, Karolyi speaking with eloquent gestures of his white-gloved hands, Ernchester expressionless, very small and thin beside him.\n\n\"It does not do, you understand, to spend the entire journey in one's compartment. For one thing, the porters gossip.\"\n\n\"I see no reason why the prattle of groundlings touches us.\" Ernchester's voice was so low as to be almost inaudible, and Asher wondered why the elongated _ou_ and open-ended _ea_ rang so familiar in his ear. Who had he heard recently, he wondered, speaking with that archaic inflection? \"There is nothing in this 'train'\"\u2014he spoke the word as if it were foreign to him\u2014\"of interest to me. If, as you say, we shall be in Vienna some days...\"\n\nThey passed beyond his hearing. Asher found his watch, angled it to the slit of incoming light. It was a few minutes past six-thirty, Vienna time. Karolyi must have just released Ernchester from the baggage car, once more replacing a seal with a duplicate. It was the subtle touch of the vampire's mind on all those in the car that he had felt in his sleep. Outside Asher's window the Alps glimmered eerie blue under the stars.\n\nHe dressed swiftly and stole down the corridor, listening for voices in the other compartments. Silence reigned. Most of them, he guessed, were already at dinner. The lock on Karolyi's compartment yielded readily to the wire tools he'd constructed from the innards of donkey and bear. He searched deftly, thoroughly, though he knew Karolyi wasn't a man to leave information lying around. No notebooks, no letters, no addresses. A great deal of money in the valise, which Asher opened after carefully inspecting its lock and frame for bits of hair, wood chips, or gum; he abstracted two hundred florins in notes and also two of the dozen or so duplicate baggage-room seals.\n\nUnder the false bottom Asher found ten small boxes of wax and wood, which contained impressions of keys, probably to the baggage car\u2014possibly to all baggage cars in use on the line. Asher pocketed them and replaced the clothing. By the time Karolyi noticed they were gone, they'd be off the train.\n\nThe valise also contained two folded Personals sections of the London _Times_ from successive dates, and these he dared not take away with him. Time was passing swiftly; he didn't have time to scan them, knowing that there would be no mark on the advertisement. He made a note of the dates, folded them as they had been, and replaced the valise above the velvet seat.\n\nA traveler's chess set stood on the table, its men neatly ranked for a game. Ernchester's old-fashioned, fiddlebacked greatcoat hung near the door beside Karolyi's wide-skirted one; Asher checked the pockets quickly, wondering where the vampire would stay once they reached Vienna.\n\nBack in his own compartment again, he rang for the porter, ordered dinner brought to him, adding with a wink and a couple of francs that he was indisposed. \"You wouldn't have the English _Times_ on board, would you?\"\n\n\"Certamente, sir,\" Giuseppe said, drawing himself up indignantly. \"All the newspapers we have for our first-class passengers, of the latest editions.\"\n\n\"How about last Saturday's? Last Friday's, too, if possible?\"\n\n\"Hmm. That I don't know, m'sieu. I shall ask, shall look about the porters' rooms...\"\n\n\"Discreetly,\" Asher said. \"You don't need to bring me the whole thing. Just the Personals.\" He raised one eyebrow and tilted his head wisely, and the porter bustled away with the air of one who sees himself an experienced international intrigant.\n\nAnd perhaps he was, thought Asher. In his position he'd have the opportunity. In any case Giuseppe returned with a much-battered copy of Saturday evening's Personals, retrieved from the porters' lavatory, and Asher spent the next half hour scanning it for whatever message had arranged the meeting between the vampire and the Hungarian.\n\n_Olumsiz Bey\u2014Front_\n\n_steps of British_\n\n_Museum, 7.\u2014Umitsiz_\n\nAsher had to read it twice before he realized it was what he sought.\n\n_Olumsiz_ was Turkish for _deathless_ \u2014or perhaps _undead_. _Umitsiz,_ for _hopeless_ \u2014or perhaps for the British form of the name _Wanthope_ , the collateral name of the Earls of Ernchester, one of several under which Charles Farren had many years ago willed property to himself.\n\n_Curious. Why Turkish?_ Asher folded the paper, slipped it into his valise. _Deathless Lord. Without Hope. Want-Hope. Wanthope. Deathless Lord..._\n\nQuite clearly Ernchester and Karolyi wanted to conceal their transactions. That would fit, if the other London vampires\u2014who must surely read the Personals, nights being long for the Undead\u2014frowned on an alliance. Would Grippen, the Master Vampire of London, know Turkish? Ysidro would, thought Asher, oddly uneasy at the memory of that bleached Spanish hidalgo who had, against the wishes of all the other London vampires, first sought his help. The Ottoman Empire had been a formidable power in the sixteenth century. It was conceivable that Ysidro, a courtier and sometime scholar, would know some of its ancient tongue. Conceivable, too, that the earl would. Certainly likelier than, for instance, Hungarian, which in that era had been the language of barbarians and herders, people without power in the West. Any of the other London vampires would almost certainly know German or French.\n\nA Viennese or Hungarian vampire who had been made in the sixteenth or seventeenth century would very probably know the tongue of the armies that had repeatedly overrun his land.\n\nAsher looked at the top of the paper again. Saturday, October 31\u2014and no copy of Friday's paper. What, he wondered, had the summons said that made Ernchester so anxious to conceal his movements from the other London vampires, including his wife?\n\nWho was it who called himself the Deathless Lord?\n\nEven at ten in the evening the Vienna Bahnhof was the swarming center of the comings and goings of an empire. Stepping quickly from the train before it had even come to a complete halt, striding along the platform to mingle with the crowd, Asher felt the stab of homecoming\u2014 _nostalgia_ , the pain of remembering. There was no city in the world quite like Vienna.\n\nThere were backcountry Jews in black caftans, tallis, and side curls being resolutely ignored by their frock-coated Germanic Reform co-religionists, Hungarian _csikos_ in high boots and baggy trousers, a tattered rainbow of Gypsies. There were the Viennese themselves, ladies bundled in linen traveling coats and veils to guard against smuts, brilliantly uniformed men who might have been Lancers or postmen, children clinging to black-clothed governesses, and students in bright-colored caps. French, Italian, singsong Viennese German as unlike as possible from the tongue of Berlin blended with Czech, Romanian, Yiddish, Russian, Ukrainian...\n\nThe air was redolent with coffee.\n\nVienna.\n\nIllogically, as he made for the stand where the fiacres would be ranked\u2014where Ernchester and Karolyi would head the moment the customs officials were through with their luggage\u2014Asher found himself holding his breath, fearing that somehow, impossibly, he would meet Fran\u00e7oise.\n\nHe had dreamed about her, in his uneasy sleep that afternoon; a dream threaded with waltzes. She was walking along the Schottenring, past the marble and stucco and gilt of the great blocks of flats, through the crystal light of a spring evening. She looked not as she had looked thirteen years ago, but as she must look now, her hair almost completely gray, and lean as certain cats get as they age; rather like a cat in a gray walking suit tabbied with black lace.\n\n_I'm sorry, Fran\u00e7oise._\n\nAs he watched her, he had been piercingly aware of the ornate bronze gratings in the walls at sidewalk level, brushed by the gunmetal taffeta of her skirt. There was movement in the darkness, he realized, movement beneath the pavement under her feet; whispering in the shadows, eyes in the dark. Waiting only for the coming of night.\n\nThey were in Vienna as well.\n\n_Fran\u00e7oise, get out of there!_ he tried to shout. _Go to your home, light the lamps, don't let them in. Don't speak to them, when they meet you on the pavement..._\n\nBut because of what he had done, thirteen years ago, she could not hear him or would not heed. She walked on, and it seemed to him that gray mist drifted up through those bronze gratings and breathed after her down the street.\n\nHe shook the recollection away. It was not likely that he would meet her\u2014she might not even live in Vienna anymore\u2014and in any case, the love between them was past and done. And there was nothing for which he would trade the prospect of living the rest of his life with Lydia, that copper-haired, bespectacled nymph.\n\nBut still there was that ache in his heart whenever he heard the \"Waltz of the Flowers.\"\n\n\"Herr Professor Doktor Asher?\"\n\nHe turned, startled, halfway to the cab stand, his first thought, _Not now!_ Karolyi and Ernchester would be along in minutes...\n\nTwo brown-uniformed Viennese policemen stood behind him. Both bowed.\n\n\"You are the Herr Professor Doktor Asher who has just come from the Paris-Vienna Express?\"\n\n\"I am, Herr Oberhaupt.\" The old Viennese custom of bestowing titles on everyone dropped immediately back into place, along with the lilting, slightly Italianate Viennese accent. \"Is there a problem? I presented my passport...\"\n\n\"No, no problem with the passport,\" said the policeman. \"We regret extremely that you are wanted for questioning in connection with the murder of a man in Paris, a Herr Edmund Cramer. Will you be so good as to accompany us to the Rathaus?\"\n\nShocked, for a moment Asher could only stare. Then a string of Czech curses caught his ear, and he looked around in time to see a couple of porters loading an enormous, brass-cornered trunk onto a goods wagon, observed by Karolyi and the Earl of Ernchester. Karolyi happened to turn his head and for a moment met Asher's eyes.\n\nHe tipped his wide-brimmed hat and smiled. The last Asher saw of them as he was escorted out of the station, spy and vampire were making their leisurely way to the rank of cabs.\n\n# FIVE\n\n\"WE KNEW EACH other in a former lifetime, you see.\" Miss Margaret Potton looked up from picking at a loose thread on the button of her left sleeve, and behind lenses as massive as Lydia's own\u2014had Lydia been wearing them in so public a forum as the Hotel St. Petersbourg's dining room\u2014her blue eyes had a look of wary defiance. \"Many lifetimes. It's as if I always knew, all my life. All my life I must have been having those dreams, only to forget them absolutely, completely, in the morning.\"\n\n\"'Must have'?\" quoted Lydia, trying to keep her fury at Ysidro out of her voice. \"When? If you forgot them that completely, how do you know you had them 'all your life'? Do you honestly remember any prior to last night?\"\n\nThe small mouth set stubbornly. \"Yes. Yes, I do. Now.\"\n\nLydia said nothing. _That cad!_ was all that came to her mind, and she thought. _Surely there's a more descriptive word than that. James is a linguist. I must ask him about it._\n\nMiss Potton looked up again and set her shallow chin. \"That is, I knew I had dreamed something important. I always had the knowledge that I was dreaming about something\u2014something beautiful, something critical, something that would change my life. Only I never remembered, until last night.\"\n\n\"I've never heard anything so idiotic in my life!\" All the lurid dreams returned to her, love, rescue, waltzing on moonlit terraces, she witty and he laying his reluctant heart at her feet. \"Last night he wanted you to _think_ you remembered. Because it was convenient for him...\"\n\n\"No.\" A beetroot stain blotched thin cheeks. \"Yes. In a way. Because he needed me.\" She returned to picking at her sleeve button. \"When he came to me last night\u2014when I woke in the moonlight and saw him standing there at the foot of the bed\u2014he said he would never have crossed my life again, would have forced himself to stay away from me, for my own good, except that he needed me. Needed my help. You don't understand him.\"\n\n\"And you do?\"\n\n\"Yes.\" She didn't look up.\n\nLydia drew in her breath, but she felt obliquely that if she came anywhere close to her true feelings, she would probably scream, and that obviously wouldn't do in the dining room of the Hotel St. Petersbourg. Rage at Ysidro drowned her fear\u2014her fear of him, of Ernchester, of the vast uncharted ocean of the world outside university research.\n\nThe word she wanted, she realized, was _vampire_.\n\nMiss Potton raised her head and went on, \"I understand that his kind need people they can trust. He told me they will seek for years for a human being large enough of spirit to accept them for what they are, in whose hands they dare to lay their lives. I was... he and I were... This was how it was between us for... for many lifetimes in the past. He said he always knew where I was, but deliberately never contacted me in this lifetime, because in a former life I... I was killed in his service.\"\n\n\"That's the most ridiculous\u2014\"\n\n\"That's all you can say.\" Miss Potton regarded her with a steady, pale, fanatic gaze. \"But I remember it. I've remembered it all my life in dreams. I just\u2014didn't recall it until last night. And he needed me again, he needed someone, to journey to Vienna...\"\n\n\"He needed a duenna for me at half a day's notice!\" cried Lydia, appalled. \"I don't know which is worse, that kind of old-fashioned absurdity or what he's done...\"\n\n\"He is an antique gentleman,\" Miss Potton said calmly.\n\n\"He is a killer! Not to mention a bigoted Catholic and the most unconscionable snob in shoe leather, and you're a fool if you believe\u2014\"\n\n\"He isn't bigoted!\" The waiter came, bringing a cup of caf\u00e9 au lait the size of a soup bowl. Miss Potton looked up at him anxiously, as if fearing he would demand payment of her on the spot. Only when he left again without a word did she turn back to Lydia, an eager intensity illuminating her face. \"During the Massacre of St. Bartholomew's Day, in the wars of religion in France, Don Simon had a Huguenot servant who sacrificed his life to keep him from being burned by the Inquisition. Later he and I saved that servant's family, got them on a boat for the Americas...\"\n\nLydia stared at her, unable even to reply. Even at the distance of the table's width, Miss Potton was a blurred figure, in her brown wool frock made for someone else and badly altered. Her squashy black velvet hat\u2014startlingly similar to the one Lydia had borrowed from her cook\u2014was years out of date. The spectacles hadn't made it into the dreams.\n\n\"But I... I _know_ I've dreamed about it before. All of it. Running along the beach, minutes before the first, fatal gleam of dawn; Don Simon turning back, sword drawn to hold the cardinal's men at bay while I got Pascalou's children into the rowboat. The way the sea smelled, and the mewing of the gulls.\"\n\n_Straight out of Dumas. And unforgivable._ Lydia tried to stir her coffee and gave it up, for her hand was shaking too badly. For all her careful training in the social niceties, in fashionable flirtation and dinner conversation, she had always regarded the majority of humanity as a slightly alien species, possessors of fascinating circulatory and endocrine systems but, with a few exceptions like James and Josetta and Anne and Ellen, detached from herself and her concerns and largely incomprehensible. She had, literally, not the slightest idea of how to go about warning this poor silly child, talking to her, reaching her through the vampire glamour of dreams.\n\n\"Miss Potton,\" she said at last, in a voice kept level only by years of deportment lessons, \"please thank Don Simon for me, but tell him that I'm a grown-up woman and quite prepared to travel by myself. I don't need a lady-in-waiting, as he seems to think. And I don't need him. But if you'll take my advice\u2014\"\n\nShe saw Miss Potton grow rigid at the word and realized despairingly that she must have said the wrong thing. But she couldn't think of anything else to say. \"If you'll take my advice, go back to London.\" It only made her sound patronizing, she thought in despair. \"Have nothing further to do with Don Simon. If you dream about him again, pay no attention. If you see him in the flesh\u2014\"\n\n\"I can't go back.\" Her small, stiff mouth wore a smirk of triumph. \"I gave Mrs. Wendell my notice yesterday morning at breakfast. I'd been up, packing, since three, since Don Simon came to me in my room, spoke to me, woke me from all those years of dreaming. I told her to find someone else to look after her nasty children, for I was done with such things forever.\"\n\nLydia could just imagine how her aunt Harriet would have greeted such an announcement from Nana over her lightly buttered toast and China tea some rainy morning... Not that Nana would ever have done anything so irregular. The poor girl would never get another job. Done with such things forever indeed!\n\n\"I have no family,\" Miss Potton went on, with that same oblique pride. \"I have put myself, my fate, into Don Simon's hands, as he has put himself into mine. And it feels... right. True. Good.\"\n\n\" _Anything_ would,\" Lydia argued, startled, \"after spending\u2014how many years were you with Mrs. Wendell?\u2014looking after someone else's children.\"\n\nThe young woman's mouth flinched, and as she averted her eyes, Lydia caught the quick shine of tears. Her first anger was subsiding, and Lydia could see that this awkward girl was only a few years younger than she, and as homely. But Miss Potton had never learned to use fashion and artifice to conceal that fact\u2014or had never had the money to do so.\n\nNo wonder Ysidro had found her an easy target when he'd gone questing through London that night, looking for someone whose dreams to invade.\n\n\"I'm sorry...\" Lydia fumbled at the words. But of course once words are said, there is no _I'm sorry._\n\nMiss Potton shook her head. \"No,\" she said, and took a sip of coffee to steady herself. Her voice lost some of its melodramatic ring. \"No, you're right. I've been wanting for years to get out of there, to find something else. David and Julia really are the most horrid brats. But that doesn't mean that what Don Simon told me is any the less true. I think I was looking for a way out because I _knew_ there was another possibility. As if the memories of those other times, those other lives, though I couldn't recall them, were alive within me, telling me there was something more.\"\n\n_\"They were not.\"_ Lydia felt like a monster, wresting a cherished new doll from a child's hand on Christmas morning, breaking it with a hammer before those disbelieving blue eyes.\n\nBut there was a scorpion in that doll. A white mantis, thin and stalky and preternaturally still, watching from the shadows with terrible eyes.\n\n\"A year ago Ysidro told my husband that vampires can read the dreams of the living,\" Lydia went on slowly. \"Ysidro is a very old vampire, a very skilled vampire\u2014one of the oldest still in existence, in Europe at any rate. Obviously, he can do more than just read dreams. The\u2014The task I need to perform in Vienna requires his help, and what's at stake is sufficiently important to him that he wants to go with me, but he refuses to do so unless I conform to his medieval standard of womanly conduct. I'm surprised he didn't insist that I bring a chaplain and an embroiderer as well. He picked you because he thought he could get you to leave everything behind and go with him\u2014go with me\u2014at a day's notice.\"\n\nMiss Potton said nothing but looked down again, picking at a small mend in the finger of her glove.\n\n\"Go back to London,\" Lydia said. \"Tell Mrs. Wendell that you had to deal with the affairs of a wastrel brother or a drunken father, and even if she's found another governess, she'll probably relent enough to give you a character for your next post. _Don't do this._ Don't let Ysidro do this to you.\"\n\nMiss Potton still said nothing. A motorcar went past on the Boulevard de la Madeleine, popping and sputtering like a company of American cowboys on the rampage. Somewhere a tram horn blatted.\n\n\"This isn't any concern of yours. Tell Ysidro that he's... he's welcome to join me in my journey, but that I _will not_ bring a third party into it, either of his choosing or my own... Though you probably don't even know where he's staying, do you?\"\n\n\"No.\" She had to guess at the word from the movement of Miss Potton's lips.\n\n\"No.\" Lydia remembered the hidden trapdoors, the new locks, the house, the square that was no longer on any London map. She picked up her handbag and brought out a slim roll of notes. \"Take this and go back to England this afternoon.\"\n\nMiss Potton stood up, straightening the back that had long ago acquired the mousy stoop of the downtrodden. \"I don't need your money,\" she said quietly. \"I trust Don Simon will take care of me.\"\n\nAnd she walked from the room in a dignified rustle of skirts.\n\nLydia reached the Gare de l'Est at seven. Too sick at heart to visit the _magasins_ for which Paris was famous, she had nevertheless forced herself to walk down the Rue St. Denis to the Halles Centrales\u2014the great central produce market of the city\u2014and purchase garlic, wolfsbane, and wild rose. As she walked along the platform toward the Vienna Express, trailed by two porters with her hatboxes and trunks, she reflected that it must have taken astonishing courage for Margaret Potton to resign her post as governess, pack her few possessions, and cross the Channel to a land where she'd probably never been and had only an academic acquaintance with the language; to walk into the dining room of a foreign hotel and up to a complete stranger and announce, \"I know all about the journey you're making, and a vampire has sent me to accompany you.\"\n\nShe wasn't sure she could have done it.\n\n_To save Jamie?_\n\nIt was, more or less, what she was doing now.\n\nLydia drew a deep breath.\n\nUnder ordinary circumstances her reaction to Miss Potton's revelation would have been bemused incredulity. People did and believed the most extraordinary things, which was one reason why Lydia had always been far more comfortable as a researcher. But she felt responsible for Miss Potton, for Ysidro's deadly lures, and it was depressing to realize that she could describe in detail the workings of that child's thymus without having the slightest idea of how to bring her to her senses.\n\nIt occurred to Lydia, belatedly, that her most effective course of action would have been a blank look and a cold \"I _beg_ your pardon?\"\n\nShe could only hope, now, that Miss Potton would return to London...\n\nTo what?\n\nWould Ysidro even let her return?\n\n_Damn him,_ she thought, renewed fury wiping out her sense of helplessness. _If he harms her, if he_ dares _to harm her..._\n\nBut again, the inner voice whispered, _What?_\n\nMiss Potton had made her choice.\n\nAnd she had made hers. She was going to Vienna to deal with the vampire earl\u2014and goodness knew what other vampires, not to mention the slippery intrigues of the Foreign Office\u2014alone.\n\nOne step at a time, she thought.\n\nIf Jamie had wired her Monday from Munich, he must have reached Vienna Monday night. Today was Friday. Last night she had telephoned Mrs. Grimes from Charing Cross Station and ascertained that nothing further had been heard from him. Four days, she thought, with Dr. Fairport, potential traitor and seeker after immortality; four days with the hazards of Ernchester and Ignace Karolyi, and who knew what besides.\n\nThe porters loaded her luggage into the van, to be sealed for the journey to Vienna, and carried the smaller portmanteau and two hatboxes and an overnight case to the compartment Mr. Cook and Company had booked for her, whose number she could probably have ascertained for herself had she been willing to squint a little. After her interview with Miss Potton, she had checked the hotel's copy of Bradshaw, seeking a train to Vienna that left before sundown, but though there were plenty of trains that would eventually take her there, via Zurich or Lyons or Strasbourg, none was faster than the Vienna Express. And speed was of the essence. James was in danger, trying to work with a flawed tool that could turn on him at any moment.\n\nOr a prisoner already.\n\nOr...\n\nShe put the thought from her.\n\nThe compartment was a comfortable one, embellished with rosewood paneling, velvet upholstery, and electrical light fixtures shaped like frosted lilies. Alone, Lydia unpinned the jade-and-eggplant fantasia of her hat and settled into her seat, gazing out the window at the impressionistic flower bed of color, shadow, and light that was the station platform, seeking, she realized, for the sturdy brown blob, the clumsy stride that would be Margaret Potton. After a moment she opened her handbag and fished forth her spectacles, a little startled, as always, at the sudden sharpness of people's faces, the lettering on the signs. According to the booklet on the little table before her, dinner would be served in the salon car at eight-thirty, but between anxiety about James and the obscure fear that even yet she would encounter Ysidro, she doubted she would feel much hunger. Her head ached, and she realized she hadn't eaten anything since the three-quarters of a croissant she'd consumed before Margaret Potton had entered the dining room at the hotel.\n\nShe watched through the window until the train began to move. Then she settled back and closed her eyes, and breathed a sigh.\n\n_Jamie..._\n\n\"If I may say so, mistress,\" murmured a voice like the sudden slide of silk over unexpecting bare skin, \"you make yourself difficult to look after. Were I your husband, I would school you.\"\n\nLydia whipped around in her seat, stomach lurching\u2014anger, fear, and, against her will, a deep flash of relief that she'd have some kind of help and advice. Her relief angered her still more, and she replied tartly, \"Were you my husband, I would demand a separate establishment.\" She pulled off her eyeglasses and slipped them behind her hat.\n\nHe stood in the doorway, ivory and shadow. As in his tomb, only the slender hands, the gold ring, caught the light. Behind him, spectacle lenses flashed in the corridor.\n\n\"You behold it.\" He stepped inside and his small gesture took in the rosewood, the velvet, the frosted lily lamps.\n\nHe had fed. She could see the faint color that stained his white face and close mouth, so that he appeared more nearly human in the staring light.\n\nSickness filled her that she had ever felt relief. That she had ever asked help or advice of such a thing.\n\n\"Miss Potton has taken a compartment at the other end of the carriage,\" Ysidro went on. \"It would be our pleasure, would you join us there for cards.\"\n\nLydia stood up, slender and straight in her traveling dress of carnation faille, jet and amber glittering. \"Send her home.\"\n\n\"I've already told you I don't have\u2014\" began Miss Potton, and Ysidro raised a finger.\n\n\"This is not possible.\"\n\n\"Will it not be possible after we return from Vienna?\" Lydia's face was almost as chalky as the vampire's. \"Are you going to kill her when you're safe in London again? And me, and James, to secure the secrets you hope to stop Ernchester from telling the Austrians?\"\n\nHis expression did not change, but she was aware of thoughts passing through the sulfur-crystal mazes of his eyes. Thinking about options? she wondered. Or only about what kind of story she was likely to believe?\n\n\"You have admirably guarded the secrets you learned a year ago,\" he said after a time. \"They are no more believable now than they were then. And I believe Miss Potton as capable of keeping them as yourself.\"\n\nThe train lurched a little, going over the points; lights cascaded past the window. In the corridor a small dog barked furiously and a woman crooned, _\"L\u00e0, tais-toi, p'tit malin!\"_\n\n\"I understand that dinner will be served at half past eight.\" Ysidro's fingers moved toward the folder on the table but did not touch it. Like everything about him, the gesture was minimal, as though long years had wearied him of all but the smallest symbols of what had been human mannerism, human expression, human speech. Lydia was suddenly reminded of the worn stones of a field circle in a pasture near Willoughby Close, her childhood home, like the white stumps of teeth protruding from olive turf.\n\n\"I suggest you ladies partake, if so be your wish, and return after to Miss Potton's compartment. Do you play picquet, mistress? The most excellent of games, and the representation in little of all human affairs. I assure you,\" he added, saffron gaze meeting the brown, \"that neither you nor she has aught to fear of me.\"\n\n\"I never did,\" Margaret said from the doorway. Ysidro did not so much as shift his eyes.\n\nLydia said, \"I don't believe you.\"\n\nThe vampire bowed. \"This news breaks my heart.\"\n\nAnd he was gone. Margaret, who no more than Lydia had seen him go, looked startled, then hastened away down the corridor without so much as an excuse, leaving Lydia standing alone.\n\nMiss Potton returned half an hour later, tapping gently on the curtained glass. Lydia, who in the intervening time had neither resumed her spectacles nor taken from her portmanteau the issue of _Journal des \u00c9tudes Physiochemiques_ she had brought for entertainment, turned from a somewhat blank contemplation of the lights fleeing by in the darkness and said, \"Come.\" The governess stepped inside, holding to the doorway as if afraid of rebuke. She'd dispensed with her deplorable hat. Her hair, tightly prisoned in pins on the top of her head, was the one thing about her that was truly as it had been in the dreams, thick, heavy, silky, and black as night.\n\n_I did call her a fool,_ thought Lydia, seeing the hesitation in the other woman's eyes.\n\n_But she_ is _a fool!_\n\nBut telling her so again would not break Ysidro's hold on her.\n\nLydia took a deep breath, rose to her feet and held out her hand. \"I'm sorry,\" she said. \"I don't trust him, but that's no reason to... to be angry with you.\"\n\nMiss Potton smiled tremulously in return. She had envisaged, Lydia realized, a journey in company with a frozenly hostile traveling companion, reason enough to look wretched. \"You can trust him, you know,\" she said, her blue eyes widening with earnestness. \"He is a true gentleman.\"\n\n_And a multiple murderer who hasn't been human for at least four hundred years. \"_ I never doubted that,\" Lydia said. \"Is he there?\" She nodded down the corridor. When Margaret bobbed her head, she went on, \"Would you wait here for me? There's something I need to say to him in private.\"\n\nHe was playing solitaire. An abacus, a small calculating machine, and a notebook lay on the table beside the spread of the cards. Four decks. The corridor lights made wan mirrors of his eyes. No light burned above the little table where he sat.\n\n\"You summoned her for me, because no lady travels alone, is this correct?\"\n\nThe pale head inclined. In the near dark she had the impression of a skull surrounded by the spider strands of his long hair.\n\n\"Then the corollary would be that no lady travels with a known killer?\"\n\n\"You've lain with one every night for seven years, mistress,\" replied the nearly soundless voice. \"In my time ladies traveled with them regularly, quite sensibly, I might add, for protection.\" A white hand, almost disembodied in shadow, laid card upon card and shifted a column; flicked a bead in the abacus; made a note.\n\n\"In your time,\" Lydia persisted, \"was it not customary for gentlemen to respect the wishes of the ladies with whom they traveled?\"\n\n\"If they were not foolish.\" He turned a card, made another note.\n\n\"I won't have you killing while we're traveling together.\"\n\nAnother card, colors indistinguishable in the cinder-colored gloom. He did not look at her. \"Unless it be for your convenience?\"\n\nLydia stood for a time, her breath coming fast. Then she turned and strode down the corridor to the restaurant car, leaving him alone turning cards in the dark.\n\n# SIX\n\n\"MY DEAR ASHER, a terrible mistake... a terrible mistake.\" Dr. Bedford Fairport fidgeted with the cuffs of his gray cotton gloves and flinched away from a stout blond policeman who came through the station-house duty room with a musically inclined drunk in tow. Much was made of Vienna's reputation as \"The City of Music.\" Asher wondered whether this was what its enthusiasts had in mind. The drunks with whom he had shared his cell the previous night had both sung, though not always the same songs. One was a Wagnerian, the other a disciple of Richard Strauss. It had been a long night.\n\n\"Mistake, hell.\" Asher closed his valise, having satisfied himself that its contents\u2014including the key waxes and counterfeit baggage-room seals in the secret pocket\u2014were untouched. A uniformed clerk offered him a release to sign, then a paper for Fairport. \"Karolyi must have seen me when I got off to telegraph Streatham in Munich. I suppose I should be glad it isn't worse.\"\n\n\"The honorable Herr will be staying with Herr Professor Doktor Fairport?\"\n\nAsher hesitated; Fairport said, \"Yes, yes, of course. Not an imposition at all, my dear Asher,\" he added, as the two crossed the worn black marble floor and emerged into the chill, misty sunlight of the Ring. \"In fact, since I've agreed to be responsible for your conduct, I'm sure the police wouldn't have it any other way. It will be quite like old times.\"\n\nAsher grinned a little wryly, recalling the clean, small bedroom above what had been the old stables at Fruhlingzeit, the sanitarium tucked away in the quiet slopes of the Vienna Woods.\n\n\"You must have spent an appalling night!\" Fairport twittered. \"Hideously irresponsible\u2014I shall write to the _Neue Freie Presse_ about the ghastly misconduct of the police in putting simple witnesses wanted for questioning in the general cells! You could have caught anything in that cell, anything from tuberculosis to smallpox to cholera!\" The old man coughed, and Asher remembered that Fairport had had tuberculosis\u2014and smallpox\u2014as a child. His milk-white skin was still marked with it, like ancient chewings of mice.\n\nHe did not look well now. But then, Fairport never looked quite well. Thirteen years ago, when he first met Fairport, Asher had been surprised when Maxwell\u2014then head of the Vienna section\u2014had told him the doctor was only fifty-four. Prematurely stooped, prematurely wrinkled, prematurely white-haired, he had the air of an almost-invalid that Asher did not consider much of an advertisement for his sanitarium.\n\nThe Viennese apparently thought otherwise. They flocked to the isolated villa and paid huge sums for \"rest cures\" and \"rejuvenation\" by means of chemicals, electricity, and esoteric baths. Looking down now at the bent little man beside him\u2014even straight he wouldn't have topped Asher's shoulder by more than an inch\u2014Asher wondered if Fairport's preoccupation with reversing the effects of age was part of his fury at the encroaching dissolution of his own body.\n\nFairport must be nearing seventy now, calculated Asher, and forced himself not to offer his help as the old man hobbled along the pavement. His face had the shrunken exhaustion of years, his hands\u2014encased as always in the gray cotton gloves he bought by the dozen, washed after wearing once, and discarded weekly\u2014trembled uncontrollably. Lydia, he found himself thinking, would have diagnosed something or other on the spot.\n\nEven under clouds, Vienna had the air of brightness he recalled; the clifflike labyrinths of buildings cream or gold or brown with their pseudomarble garlands, their putti and grimacing tragedy\/comedy masks; gilded ironwork, tiny balconies, great somber doors guarding flagstoned courtyards inside.\n\nA short distance along the Ring a smart brougham drew up beside them, the black body of the closed coach varnished and gleaming, its brass hardware polished like gold. A big man wrapped in a coachman's long coat and muffler sat on the box, frowning under a simian brow ridge while a footman, equally tall, sprang from the rear platform to open the door. Asher reflected that the sanitarium must be doing well if the old man could afford this kind of turnout.\n\n\"You'll want a hot bath and a good rest, I daresay.\" Fairport gestured away his footman's proffered arm with a wave of his cane. \"Thank you, Lukas... I've telephoned Halliwell\u2014he's the head of the Vienna section these days, do you remember him?\u2014to let him know you're in town, but this evening, if you're feeling up to it, will be early enough.\"\n\nAsher considered. It was mid-morning, the mists from the canal barely diffuse in the bright air. Though they stood on the threshhold of winter, the cold seemed not so raw as that of London or Paris, the damp not so bitter. The air had a soft quality, like rose petals. In the Volksgarten a few hardy citizens sat behind the line of chain and potted trees that demarcated the terrace of a small kaffee haus, and Asher had a flashing recollection of true Viennese coffee and the concentrated sinfulness of a Creme Schnitten. Fruhlingzeit Sanitarium, isolated among woods and vineyards, was restful and silent but about an hour's drive from the outskirts of the town.\n\n\"If you don't mind,\" Asher said slowly, \"there are things I need to do here. Someone I need to trace, without delay.\"\n\n\"Karolyi?\" Fairport's almost hairless white brows formed little arches in the fish-belly forehead. \"His addresses are quite well known. A town house in D\u00f6bling and a flat on the K\u00e4rtnerstrasse... I assume you're not interested in that ancestral castle at Feketelo in the Carpathians...\"\n\n\"No.\" Asher shook his head. \"No, someone else, someone whose name I don't know. And it may take me a little time in the Rathaus to find the records.\"\n\nHe knew it would have to be done, and his mind leaped ahead, calculating how long it might take and when the sun would set. He thought he would have time to do the thing in safety, but with an almost subconscious gesture he rubbed his wrist to feel, through glove and shirt cuff, the protective silver links.\n\n\"If I may abuse your hospitality so far, I think what I need to do is, first, find myself a public bath and get cleaned up, then start my search in the records office. How late might I come out to Fruhlingzeit without disturbing anyone to let me in?\"\n\nFairport smiled, a dry little V-shaped quirk. \"My dear Asher, this is Vienna! My staff remains active until nearly eleven, and I'm frequently at work in the laboratory until midnight. Right now there's no one staying at the sanitarium\u2014we had some electrical troubles early in the week\u2014so there's no trouble about that.\"\n\nHe fished in the pocket of his old-fashioned frock coat and produced a latchkey. \"If you don't see a light in my study or the laboratory, simply let yourself in. I'll have the old room ready made up for you, the one looking out onto the garden at the back, you remember?\"\n\nAsher smiled. \"I remember.\"\n\nHis smile faded as Fairport climbed into the brougham\u2014the footman Lukas had to help him\u2014and drove away into the shifting traffic of the Ring, brasses winking like heliographs.\n\nHe remembered.\n\nHe remembered sitting for hours in the window of that whitewashed room, looking down into the overgrown courtyard whose high wall formed only a nominal barrier against the whispering high-summer woods, reading over and over the three telegrams he'd found upon his return from the mountains. Remembered not wanting to know what they told him.\n\nAll three had been from Fran\u00e7oise, sent on successive days. All three had asked for an immediate reply. But he'd seen her at the Caf\u00e9 New York\u2014his shoulder tightly strapped and a hefty dose of Fairport's stimulants in his veins\u2014earlier that day. She had mentioned the telegrams in passing, but said they were nothing much.\n\nIt meant that she'd been checking on his movements in the period of time in which he was supposed to be ill rather than away.\n\nIt meant that she suspected him of leading a double life.\n\nIt meant that he was a footfall away from being blown. With Karolyi returning to Vienna in a matter of days, he knew what that would mean.\n\nShe'd been perceptive enough to see through Karolyi's imitation of an innocuous young idiot. Why hadn't he thought she would see through his own impersonation of scholarly harmlessness?\n\nHe'd sat by the window until the long summer afternoon faded and the white roses on the garden wall dwindled to milky blurs, until he had been unable to read the printing on the dry yellow telegraph forms, though he had by then memorized what each had said. He knew what they meant. He knew what they meant he had to do.\n\nHe pushed the memory aside now. When he recalled Viennese coffee and Creme Schnitten, he had automatically thought of the Caf\u00e9 New York. Though he guessed Fran\u00e7oise had not entered its doors since the summer of 1895, either, he knew he'd look elsewhere for those small pleasures.\n\nFran\u00e7oise had been right about caf\u00e9s in Vienna. It applied equally to public baths. Though not as ubiquitous as caf\u00e9s, they were plentiful and good for the same reason. Most apartments in the overcrowded city lacked hot water; thousands of families still relied on communal pumps in the halls, communal toilets in the courtyards. But the Viennese were a clean people, cleaner in Asher's experience than the Parisians, for all the French fanaticism about keeping their windows spotless. Certainly the jail cell he'd occupied last night had been far from the pesthole of Fairport's imaginings.\n\nThe Heiligesteffanbaden was a veritable emporium of cleanliness, and heavily populated even for a Tuesday morning. Workingmen, students, bearded bourgeoise, and stolid _hofrats_ scrubbed conscientiously in pink marble tubs, under the solicitous eye of the usual host of marble and mosaic angels and the usual Viennese hierarchy of Herr Oberbadmeister, Oberbadmeister, Unterbadmeister, and the _garzone_ who collected the towels. Asher visited the barber next door to be shaved, changed into the shirt and underclothing he'd bought on the way from the Prefecture of Police, paid a quick visit to a man he'd known back in '95 who cut keys, and felt much better, though the clerks at the Rathaus looked askance at his rumpled jacket when he asked to examine wills and title documentation of the older dwellings in the Altstadt. He guessed he would have enough time to do what he needed to do, if not before dark, at least before the crowds thinned from the streets.\n\nAs both scholar and spy, Asher had long ago learned that human beings reveal the true workings of their souls when their attention is on something that consumes them to the exclusion of their usual desire to make an impression on others\u2014and that something is usually property. He had, he reflected dryly, witnessed a particularly unappetizing modern example of that very phenomenon in the wake of his cousin's funeral three days ago. In their preoccupation with who's going to get what, people forget to cover their tracks: banking records, wills, probates, leaseholds, account books can yield a startling amount of information to someone with time at his disposal and a high tolerance for dust.\n\nAsher started with the oldest palaces of the Altstadt, those exuberantly decorated masterpieces of white stucco whose baroque fa\u00e7ades could barely be seen because of the narrowness of the ancient city's alleys, matching ownership records with wills, wills with death notices and, more importantly, birth notices; doing sums on every page of his notebook and all around the margins of the _Times_ Personals, the only other paper he had in his valise. He found himself heartily missing Lydia, not out of romantic considerations, but simply because she was a good researcher and would thoroughly enjoy this chase.\n\nHe left around two for a sandwich, but it was only when one of the several bespectacled young clerks came to his table in the reading room and said apologetically, \"If it please you, Herr Professor Doktor, this building is now closing,\" that he realized the windows were pitch-black and that the electric lights had been on for nearly an hour and a half.\n\nBy previous arrangement, Artemus Halliwell was waiting for him at Donizetti's caf\u00e9. The head of the Vienna section was in his mid-thirties, untidy, bearded, bespectacled, and enormously obese; Asher remembered him from the London statistics department. Behind small oval slabs of glass, Halliwell's pale green eyes were like _cabachon_ peridots as he listened to Asher's account of his journey.\n\n\"So this Farren thinks he's a vampire, eh?\" Halliwell carved a neat fragment of _backhendl_ and popped it into his incongruous rosebud of a mouth. \"I suppose that's how he came into your purlieu in the first place, is it?\"\n\nAsher nodded. In a sense it was actually true.\n\n\"You get some of that in Vienna, though not as bad as Buda-Pesth. When I went west into the mountains only last year, there was a tizz-woz in one of the villages about a man who was supposed to turn himself into a wolf. I'm told in parts of the Black Forest no one will talk to you, sell you anything, give you directions to anywhere, if you kill a hare.\"\n\nHe dabbed his lips with his napkin and the ubiquitous Ober appeared, asking with folded hands if everything was all right.\n\n\"I think you should know,\" said the fat man, when the Ober had effaced himself again, \"that there's been a bit of a stink.\"\n\nAsher felt his nape prickle. He'd been around the Department long enough to recognize that carefully neutral tone. \"Oh?\"\n\n\"Streatham's doing.\" He made a dismissive gesture with his fork. \"Naturally. Always was a bloody fool. He's made to-do about that boy Cramer's death with the French authorities, ranting about British citizens and treaty rights\u2014just as if our offices weren't in flat violation of any treaty's assertions of good faith. The thing is, the French have washed their hands of the whole matter, contacted the Vienna police, and are demanding your return under escort on the first available train. I held them off for a day, saying I hadn't any idea where you were,\" he went on, raising a staying hand against Asher's protest. \"But whatever you've learned today at the Rathaus, you'd probably better pass along to me.\"\n\n\"Idiot,\" Asher said dispassionately, while his mind raced ahead. It was close to eight; the streets would remain crowded enough to protect him until ten at least, possibly later, and in any case he doubted that vampires could detect an intrusive interest in their lairs from a single walk by a casual observer.\n\nBut even in a single walk-past he could tell a great deal, particularly which of the several houses on his list of possibilities was the likeliest haunt. Enough information, at least, that whoever took over wouldn't be going into the job defenseless, as Cramer had done.\n\n\"And what was it,\" asked Halliwell, \"that you went to the Rathaus today to find?\"\n\nAsher considered for a moment, then said quietly, \"Vampires.\"\n\nHalliwell's tufted brows went up.\n\n\"Are there people here who believe in them?\"\n\nThe Vienna chief gestured with his fork again. \"There's always muttering among the Gypsies. The waiter at my caf\u00e9 swears he saw a vampire on an old gate tower connected to a house in the Bieberstrasse\u2014used to be part of the ramparts.\" He shook his head. \" _My caf\u00e9._ I sound like a Viennese. Caught myself calling this place _my restaurant_ the other day, same as I'd talk about my club at home.\"\n\n\"I don't know.\" Asher looked around him lazily, soothed by the atmosphere of the place, the slight shabbiness of the oak panels, the soft flicker of the gaslight and the all-pervasive smell of goulash, and scratched a corner of his mustache. \"Isn't one's caf\u00e9 here a little like one's club in London?\"\n\n\"The hell it is.\" Halliwell surgically excised another morsel of chicken. \"At a club you have a vote on who gets let in the door. Here anyone can come in\u2014 _and_ does.\" He glared across at a party of uproarious young subalterns in the sky-blue coats of the Imperial-and-Royal Uhlans. \"The wine's atrocious, and I think if I hear one more waltz, one more operetta, one more Mozart concerto, I'm going to open negotiations with the Turks to reinvade, and this time I'll make damn sure they win. Has Farren been to Vienna before?\"\n\n\"I haven't been able to find that out,\" said Asher. \"Not under his own name, anyway.\" Which might or might not be true, but was probably true enough for this century. \"I have an idea he'd hide out in a house reputed to be haunted or connected somehow with... odd rumors.\"\n\nHalliwell nodded, thinking, and the Ober returned with the Herr Ober in tow, to collect the polished ruins of Halliwell's _backhendl_ and Asher's _Tafelspitz_ , and to solicitously attempt to interest Halliwell in dessert with the air of a man who fears his client will collapse from starvation if not attended. Halliwell issued instructions as to the composition of an _indianer_ with an attention to detail that seemed to delight the Herr Ober's soul, then turned back to Asher as the two waiters bowed and took their leave.\n\n\"I've heard of the Japanese doing that in the Chinese war,\" said Halliwell. \"Headquartering in haunted houses in Peking.\"\n\nAsher nodded. \"I was there,\" he said. \"And yes, they did; complete with mirror tricks straight off the Paris Opera stage. It may be harder to pull off here...\"\n\n\"Not as hard as you think.\" There was a small commotion in the doorway\u2014two other young officers, brave in gold braid, with bright-clothed girls on their arms, and all the rowdy subalterns calling out greetings\u2014and Asher saw Halliwell's bulging eyes cut briefly, unobtrusively, in that direction, making sure the noise did not represent potential danger. Not a reaction one would expect from a fat gourmand ostensibly preoccupied with his pastry.\n\nHis eyes returned to Asher. \"There's a lot of country people in Vienna, in off the farms to the east: up-country Czechs and Hungarians and Romanians and what-have-you, come to work in the sweatshops after spending the first part of their lives, to all intents and purposes, in the sixteenth century. People who live in the Altstadt don't interfere if there's a big old palace that's shut up day after day\u2014it's part of the neighborhood, and one would never risk incurring the displeasure of a baron. But newcomers from out of town\u2014they get inquisitive.\"\n\n\"And which big old palace,\" Asher inquired, \"are we talking about?\"\n\nHalliwell grinned and fastidiously removed a mote of powdered sugar from his whiskers. \"There's three or four. One on the Haarhof is supposed to be haunted, and there's a seventeenth-century _palais_ on Bakkersgasse where people claim to have seen lights. All the Hungarian waiters in town swear the baroque _palais_ built over the ruins of the old St. Roche Church on Steindelgasse is inhabited by vampires\u2014it's actually owned by a collateral branch of the Batthyanys\u2014and there's a house in Vorlautstrasse near the old ramparts where four or five people are said to have disappeared over the course of the last ten years. All of them have perfectly legitimate antecedents, by the way, winter palaces of landed families who have larger places out in the country.\"\n\n\"Any belonging to Karolyi?\"\n\n\"I think the Bakkersgasse _palais_ belongs to the Prague branch of the family. Not to our bird. It's a huge clan.\" Behind the spectacles the pale eyes danced, as if pleased he'd anticipated the thought. \"Will you need help?\"\n\nAsher hesitated. The bloodied ruin of Cramer's face came back to him, glistening gruesomely in the reflected light. Gummed with blood, the silver chain had crossed the huge wounds on the throat. The shopkeeper in the Palais Royal had sworn the chains were pure silver. More likely tourist trade trash, the thinnest wash over pewter or lead. The boy probably hadn't even heard Ernchester approach.\n\n\"I haven't much to send with you,\" Halliwell went on. \"Streatham's an ass, but he was right about that. Everything's been cut since the end of the war. Still, if you need a man...\"\n\nBeyond the gilt-framed windows of Donizetti's, passersby hurried along the pavement, greatcoats bundled tight about them. Mist had risen again from the Danube Canal, blurring the outlines of apartment buildings whose grandiose central staircases led to dreary attic rooms shared by cobblers, embroiderers, Obers, and Herr Obers and their wives and children and Uncle Tom Cobbley and all. Between the buildings the shadows lay deep in narrow passages leading to the heart of the ancient city, where sunlight fell only at noon.\n\nOne of the possibilities on Asher's very incomplete list of suspect properties was on the Steindelgasse: _said to stand over the crypt of old St. Roche._\n\n\"No,\" Asher said softly. \"No, I think I'll be all right on my own.\"\n\nThe palace in the Steindelgasse was typical of the great town houses of the nobility in the old city: five floors of massive gray walls, wedged between an ancient block of flats and the town palace of some count of the Montenuovo family that was illuminated like a Christmas tree for a ball. Looking up, Asher could see the tall windows of its first-floor salons ablaze with gaslight, which partly illuminated the narrow street; crystal chandeliers were visible, and a portion of a god-bedecked baroque ceiling.\n\nThe Batthyany palace was utterly dark.\n\nCurious enough in itself, thought Asher, pausing before the heavy archway of the door. A number of the old noble families boosted their incomes by renting out the ground floors of their palaces for shops and the topmost floor or two below the attics out for flats. Certainly there were people coming and going through the great gate of the Montenuovo _palais_ who were not of the upper crust. The other buildings Asher had looked at since parting from Halliwell, in the Haarhof and the Bakkersgasse, had been dark as well, lacking even spectral lights, and like them, this one had a slightly dilapidated air. The obligatory marble atlantes that upheld its shallow porch and heavily carved window frames were uncleaned, though Asher was interested to note that the hinges and ironwork on the door were free of rust.\n\nThe building was clearly the oldest in the street.\n\nHands in armpits for warmth, Asher strolled slowly past the enormous doors. It was later than he had intended to be still abroad. Fog and deepening cold were thinning the passersby, and he heard eleven strike from the Domkirche a few streets away. He noted the shutters behind the windows' iron bars and the lack of recent wear on the pavement before the doors, and turned to fix in his mind the irregular shape of this narrow lane, orienting himself in the tangle of little streets that lay between the cathedral and the old Judenplatz. Within the gate would lie a broad passageway or possibly a sort of columned porch opening into the central court. Not a large one, judging by the frontage, but the building might be far longer than it was wide.\n\nHe walked on, seeking a way to circle the block. Away from the lights of the apartment blocks and the Montenuovo town house he felt his nape prickle with his old instinct for danger, but if he was going to be shipped back to Paris in the morning, the least he could do was arm his successor with some knowledge of what he was getting into.\n\nHe turned down a short lane, his boots splashing in thin puddles. The small iron lamps that burned high on the walls here were the only lights, feeble through thickening fog. He turned again, reflecting that this part of the old city was as bad as the London waterfront Worse, in some ways, because the uniformly high walls closed in like a canyon, shutting out even the sight of spires or chimneys that could be used as landmarks.\n\nThere was no one in sight around him. He thought, _Finish this up and get out._\n\nIn another narrow street off Tuchlaubenstrasse he identified what he thought was the back of the Batthyany Palace, no more than a slip of older masonry between two apartment blocks set at an odd angle; there was a little postern door there whose iron handle he knew better than to touch.\n\nA footstep splashed in water, close behind him. Asher turned and threw his back to the wall, lashed out as a dark figure laid hands on him from one side even as he heard the panting approach of another man. His fist jarred on the bony angle of a jaw. The man lurched back and Asher spun, his second attacker seizing his arm; he grabbed the man's hair with his free hand, yanked the head sharply against the stone of the wall behind him, twisted his body from the drag of the grip. At the same time, his mind registered tobacco and sweat and dirty clothes, the sound of breath and the warmth of snatching hands. Pain slashed his side even as he hooked the one man's feet from under him, slammed aside the hands clutching at his throat, smashed his fist into the other's face.\n\nHe could barely see them in the dark between the buildings, but broke for where he thought they weren't. Someone clutched the skirts of his greatcoat but couldn't hold. He fled, stumbling, bruised his shoulder on the stone of a corner, fell where his foot caught a pothole in the broken pavement. He caught himself against a wall and pain seared his side again, and he realized that one of them had had a knife.\n\nHe turned down what he thought was a lane back to the Seitzergasse, but the dirty glow from a window high and to his right showed him a blank wall. He pressed himself back into a corner where the shadows were darkest, reached to whip the knife from his boot as panting and footfalls echoed loud in the narrow space before him and he saw the blur of what might have been faces and the glint of edged steel.\n\nThen a hand like the mechanical jaws of a trap caught him above the elbow, and another, corpse-cold and strong as death, covered his mouth. He was dragged backward, down, into damp cold that smelled of wet stone, earth, and rats, and the dim arch of paler darkness blinked from existence as the door was kicked shut.\n\nThe smell of Patou perfume filled his nostrils, covering a dim exhalation of putrefying blood.\n\nA woman's voice said in his ear, \"Don't cry out.\"\n\nAsher was silent. Even if the silver protected the big veins of his throat and wrists, a vampire could still break his neck, and he knew what was in the darkness beside him.\n\nThe hands left his arm and face. He listened, wondering how many of them there were.\n\nThere was no sound of breath, of course. After a moment a silvery rustling, like infinitely thin metal fragments rubbing against each other. A woman's taffeta petticoat.\n\nHe thought, _She spoke English._\n\nThen the scratch of a match.\n\nHe blinked against the needle-bright golden light that suddenly outlined a colorless hand, a papery white face, and touched with cinnamon threads the black mass of framing hair. Brown eyes met his, reflective vampire eyes, but still the color of brown leaves at the bottom of a winter pond.\n\nIt was, he realized, Anthea Farren, Countess of Ernchester.\n\n\"I do not understand the how of it,\" Ysidro said.\n\nHe had taken off his gloves to deal the cards, and now he held up his hand, slender and white with long fingers like the spindles for bobbin lace. Lydia observed again the quasionychogryphosis of the fingernails and the fact that the musculature showed no abnormal development, though she had seen this man wrench apart iron bars.\n\n\"It's my theory that it is a virus, or more probably a complex of several viruses.\" Lydia sorted her cards: ace, king, ten, eight, and seven of hearts; queen, jack, seven, and ten of spades. Almost no clubs\u2014a nine\u2014queen and jack of diamonds. Darkness fleeted past the train windows. Around them the first-class carriage had slipped slowly into silence.\n\n\"Because the cells of the flesh are themselves altered?\"\n\nShe paused, a little surprised that the vampire knew what a virus was. then remembered all those medical journals in his house. Playing cards, and conversation, had insensibly lessened her fear of him\u2014she wondered if he had chosen the absorption of a new game for that reason, or simply because, like her aunt Lavinia, he wanted a partner for the journey.\n\n\"It's one way of accounting for the extreme sensitivity of the flesh to things like silver and certain herbs,\" she said after a moment. \"Not to mention photocombustion.\"\n\n\"Do we really need to talk about this?\" Margaret squirmed uncomfortably, never lifting her gaze from the flying crochet needle and the lacy snowflakes of antimacassars overflowing the workbasket on her lap. After two or three tries at learning picquet, Margaret had retreated to her needlework, fighting to remain awake so as not to be left out of Ysidro's conversation, though she had very little to add. Into the discussion of railway travel, the finer points of picquet, the mathematical odds involved in card play and the structure of music\u2014which Ysidro understood on a level very different from Lydia's superficial acquaintance\u2014Margaret had interjected periodic observations that she hadn't been out of England before or that she had read of this or that monument or notable sight in a travel book or Lord Byron's memoirs.\n\nShe had tried two or three times to deflect the talk from the physical state of vampirism, but when she spoke now, her voice was low, as if she wanted to register a complaint that she didn't actually want the others to hear. Silly, thought Lydia, considering the fact that Ysidro could tell people apart by their breathing.\n\nThe vampire moved two cards in his hand, removed three and laid them facedown near the stock, replacing them in his hand with three others. \"It may sound odd, but to this day I do not understand what happened to my flesh the night I was taken, in a churchyard near the river, as I was coming home from my mistress'... I always had mistresses in those days. Girls south of the river, who cared not that I was a Spaniard of the consort's entourage.\"\n\nHe lifted the corners of two other cards and replaced them in the stock without change of expression.\n\n\"I believe that this condition is two separate matters: the matter of the flesh, which preserves the body, not as it is at the moment of death, but as it is in the mind, molding even those who are taken old back into the shape of their living prime; and the matter of the mind, which sharpens and strengthens both the will and the senses, and gives us power over the wills, and the senses, of the living.\"\n\nLydia discarded her club and two diamonds, drew another club\u2014the eight\u2014the ace of spades, and the queen of hearts. After four or five games in which Ysidro had systematically bested her, she was beginning to get the hang of the game, a complicated manipulation of points in which she could almost always deduce more or less what Ysidro had in his hand, though as yet the information did her little good. As a teacher, he had endless patience, gentle without being in the slightest bit kind. He had dealt with Margaret's total absence of card sense and her inability to follow or remember rules with a matter-of-factness that had, oddly enough, almost driven the governess to tears.\n\n\"It is the blood that feeds the flesh,\" Ysidro said. \"We can\u2014and do, at need\u2014live upon the blood of animals, or blood taken from the living without need of their death. But it is the death that feeds the powers of our minds. Without the kill, we find our abilities fading, the cloak of our illusion wearing threadbare, our skill at turning aside the minds of the living shredding away. Without those skills we cannot send the living mind to sleep or make others see what they do not see, or bring them walking up streets they would not ordinarily tread in moments of what feels, to them, to be absentmindedness.\"\n\nMargaret said nothing, but her needle jabbed fast among the flowery lacework in her hands.\n\nHe gathered his cards. \"Those, by the by, are our only powers, Mr. Stoker's interesting speculations aside. Personally, I have always wondered how one _could_ transform oneself into a bat or a rat. Though lighter in weight than a living man, I am still of far greater bulk than such a creature. But in the speculations of this man Einstein I have found considerable food for thought.\"\n\n\"Do you cast a reflection in mirrors?\" Lydia had noticed upon coming into the compartment that a scarf\u2014one of Margaret's, presumably, blue with enormous red and yellow roses printed on it\u2014had been draped over the small mirror, and the curtains drawn tightly over the dark window glass.\n\nShe recalled her own ghostly image in Ysidro's huge Venetian mirror draped with black lace.\n\n\"We do.\" Ysidro made his discard. \"The laws of physics do not alter themselves for either our help or our confusion. Many of us avoid mirrors simply because of the concentration of silver upon their backs. Even at a distance, in some it causes an itch. But chiefly, mirrors show us as we truly are, naked of the illusions that we wear in the eyes of all the living. Thus we avoid them, for though we can still cast a glamour over the mind of a victim who glimpses us in reflection, the victim will usually be troubled\u2014unaccountably, to him\u2014by what he sees or thinks he sees. We are not overfond of the experience ourselves. Four for quart in spades, ten high.\"\n\nThey played cards until long past midnight, as the lights of Nancy flashed past the window and then the Vosges rose under their starless shawls of cloud. Still fascinated but nodding with weariness, Lydia finally returned to her own compartment, but, as she had feared, could not sleep. A little light strayed through the curtain from the corridor, a comfort, like the elephant-shaped _veilleuse_ that had burned in her room when she was a child. Once, a shadow passed that light, and she lay awake for some time, imagining Ysidro drifting like a soundless specter along the train, sampling the dreams of the lady with the little dog, of the pair of brothers who'd asked to share the dinner table in the restaurant car with Lydia and Margaret, of the conductor in his chair and the kitchen boys in their bunks, like a connoisseur tasting different vintages of wine.\n\nShe wondered what Margaret and Ysidro had to say to one another in the course of the night.\n\n# SEVEN\n\n\"HAVE YOU SEEN him?\"\n\nLady Ernchester tilted her stub of candle to spill a few drops of wax onto the stonework of an elbow-high niche, then propped the light upright in it. The flame steadied and broadened, touching first her face with its deceptive warmth, then the stiff, sad features of a small stone image of the Queen of Heaven in the niche itself, fouled with rat droppings and the trails of slugs. The light penetrated farther, to show them in a sort of vestibule at the foot of the crooked stairs down which Anthea had led him. The walls and ceiling groins of brick and stone had lost most of their covering plaster, and earth floor filled the air with a raw exhalation of damp. Opposite them a door into another chamber had been bricked shut, but not the long windows on either side. Looking through, Asher could see that the room, far deeper and higher than the one in which he stood, was filled with human bones.\n\nHe leaned against the wall, the pain in his side suddenly turning his knees to water. When he pressed his hand under his coat, he felt the hot soak of blood.\n\n\"You're hurt...\"\n\nShe stepped forward and caught his arm; her hand pulled back and he staggered, for her fingers had accidentally brushed the silver chains where they ran under the cuff of his shirt. For a moment they stood looking at one another in the candle's wavering light.\n\n\"Wait here for me,\" she said. He heard the rustle of her petticoats but did not see her depart.\n\nHe sank down onto the windowsill, leaning against the rusted iron bars. His head swam, but losing consciousness was something he dared not do. The bones behind him rose in heaped mountains, losing themselves in a distance of utter night. A faint scratching clatter: movement among the piled skulls, and the glint of tiny eyes.\n\nA plague crypt, he thought. Easily as large as the one under the cathedral, though probably deeper in the earth. In the faint glow of the candle the bones were as brown and shiny as ocean stones.\n\n_Get thee to my lady's chamber,_ thought Asher dizzily. _Tell her that though she paint an inch thick, to this end will she come..._\n\n_Unless, of course, she chooses not to die._\n\nFor some reason Lydia came into his mind, and he shut his eyes. _To this end will she come..._\n\n\"Here.\" A hand touched his shoulder, swiftly withdrawn. She stood at his side again, his valise in her hand. \"Take off your coat.\"\n\nThe attacker's knife had slit the heavy wool and the lighter tweed of the jacket and waistcoat beneath. Shirt and waistcoat had absorbed most of the blood; had he not been wearing the greatcoat, he would probably have been killed. As it was, the wound, though painful, was superficial\u2014he could move his arm, though he knew it would stiffen, and his breathing was unimpaired.\n\nWith an exertion that left him light-headed, he stripped to the waist, the air shockingly cold against his skin. He remained seated in the embrasure while she moved away from him, to the opposite side of the vestibule under the Virgin's niche, where she tore the bloodied shirt into neat pieces as if the tough linen had been cigarette paper. As she worked, she spoke in the quick, jerky voice of one who seeks to preserve herself from what silence might bring.\n\n\"Have you seen him?\" she asked again.\n\n\"I saw him at Charing Cross Station,\" he replied, \"talking with a man I knew to work for the Kundschafts Stelle, the Austrian secret service.\"\n\nShe glanced up, eyes flaring wide with shock. They were the color of mahogany but no more human than a raptor bird's. In the small saffron light her lips were colorless as the pallor of her flesh, pallor somehow mitigated\u2014or explained\u2014by the mourning black of her clothing. Her hair, upswept into the style Lydia called a Gibson Girl, seemed to flow out of the darkness of her clothing, garnet-tipped pins gleaming in it like droplets of blood.\n\n\" _Talking with_ someone?\"\n\n\"Why does that surprise you?\"\n\n\"I had thought...\" She hesitated, looking at him for a moment; then, as if not daring to linger on the dark glitter of blood on his side, her unhuman eyes returned to her work. \"Our house was searched, you see. Ransacked by men while I was out.\" From the reticule at her waist she withdrew a square of yellow paper, folded small, and crossed the room to hand it to him with bloodstained fingers, then moved quickly back away. \"That was on the floor when I came back.\"\n\nAsher unfolded it. It was a railway timetable. Sunday night's seven-thirty boat-train was circled; a strong European hand had added, in the margin, _Vienna Express._\n\n\"He was gone by the time I came back that night,\" said Anthea, digging in his valise for the small flask of whiskey there. She soaked an unbloodied fragment of shirt in it, braced herself almost imperceptibly before stepping near enough to touch him again. Asher raised his arms against the top of the window in which he sat, that the silver on his wrists might not come into accidental contact with her ungloved hands. The whiskey stung coldly in the wound, the smell of it almost covering the raw whiff of the blood.\n\n\"In wintertime, when dark falls by four, I often go on errands, to buy newspapers or books. I have a dressmaker who keeps open for me. Ernchester will sometimes stay all the night through in his study, reading, even on those nights when I go out later...\"\n\nShe stopped herself visibly from saying _to hunt_. But Asher saw it in the shift of her eyes. Her hands were icy against his bare flesh, and she worked quickly, holding the bindings in place with small bits of what little sticking plaster he'd had in the valise in case of emergencies. His blood dabbled her fingers, garish as paint on ivory. Cold breathed over his ribs from the bones within the crypt, chilling him further.\n\nShe went on, her words swift, like a woman talking in the presence of a man whom she fears will seduce her. \"He used to go out walking. I thought it was only that. So I went out again and, when I returned, found the place rifled, smelling of human tobacco and human sweat, and that was on the floor. I thought... I thought that he had been taken away.\"\n\nHer dark brows pinched together as she pinned the final bindings in place. \"I would have known it, had he... had anything befallen.\"\n\nAsher remembered his dream. _How can he be dead?_ she had asked. _Did I walk up the stairs, would he not be waiting at the top?_\n\nEven then she had known.\n\n\"And you didn't go to Grippen?\"\n\nAnthea shook her head. \"Since last year\u2014since the rift among us concerning you and your knowledge of us\u2014there has been uneasiness among the Undead of London. Grippen has gotten other fledglings in place of those who were killed; has summoned to London older fledglings of his as well. Me, he never trusted. Indeed, I... until you spoke of the Austrian, I could not be sure that this was not of Grippen's doing. But for that reason I dared not go to Ysidro, either.\"\n\nShe handed him one of the new shirts he had bought, then took the whiskey flask and stepped quickly away, pouring the liquor on her fingers and meticulously, repeatedly, almost obsessively wiped from them all trace of his blood. While she did this, he put on his shirt, resumed his tie, his jacket, his coat, moving slowly for his vision sometimes would suddenly gray, but she did not offer her help. In the dark of the crypt, rat shadows flickered among the bones.\n\n\"At a certain distance I can feel my husband's mind. Sense his presence. I did not... I dared not wait.\" She raised her eyes to his. \"Might he have gone to this Austrian because he was fleeing the Master of London?\"\n\n\"He might,\" said Asher. \"But I suspect Grippen had nothing to do with it. Come.\" He picked up his valise. \"Will you go with me for coffee?\"\n\nThey went to LaStanza's on the Graben, luminous with gaslight and bright with the pastel frocks of the dancers. Anthea had donned, over her cold white fingers, a widow's black lace house mitts, and produced from a corner of the crypt's vestibule a plumed hat bedighted with veils that further hid\u2014and heightened by contrast\u2014the whiteness of her flesh. She must have left it there, thought Asher, when she went to rescue him from his attackers in the alley. The scent of her hair on the silk had evidently been enough to keep the rats from coming anywhere near.\n\n\"I have been afraid for Charles for years,\" she said after the Herr Ober took their orders. \"Part of it was Danny being killed\u2014the man who had been our servant since the days of the last King George. Burned up in the light of the sun. Some would say, a fit end for such as we.\" She glanced quickly at him, challenging, but Asher said nothing.\n\n\"Part of it was the death of the city that he knew. Not all at once, as when the fire took it, but little by little, a building demolished here, a street torn up there that the Underground might pass beneath. A word or expression would fall out of use, or a composer die, whose work he loved. He used to go every night to concerts, listening with joy to the new men, to those light airs like clockwork flowers, and then the strength, the passion that came after...\"\n\nA waiter brought them coffee: for her, \"dark with skin\"\u2014one had to be specific when ordering coffee in Vienna\u2014for him an _einspanner_ , black coffee, whipped cream.\n\n\"Is it pass\u00e9 now, the waltz?\" She put back her veils and raised the cup to her lips, not drinking, but breathing deep of the bittersweet riches of the steam. On the dance floor women floated weightlessly to \"Tales of the Vienna Woods,\" their gowns like lilies of saffron, rose, pale lettuce-green; the black clothing of the men a delineating bass note, the officers' uniforms jeweled flame.\n\n\"I think so.\" He remembered dancing with Fran\u00e7oise. She'd been gawky as a scarecrow to look at but never missed a step, as light as a bluebell on a stem. \"Not with people my age,\" he went on. \"But the young and the smart are doing things like the foxtrot and the tango.\"\n\n\"Tango.\" She savored the unfamiliar word. \"It sounds like a New World fruit. Something whose juice would run down your chin. I shall have to learn it one day.\" Her eyes returned to the dancers, quickly, as if avoiding a thought. \"The waltz was a scandal when first I learned it. And so I thought it, too.\" She laughed a little at herself. \"Ernchester still enjoyed dancing in those days. Grippen mocked at us. For him all things are only to serve the kill. But we'd go to Almack's Assembly Rooms or to the great ton balls during the Season. He... was not always as you've seen him.\"\n\n\"Did something change him?\" His voice was low, under the music, but she heard, and past the wraiths of her veils her glance crossed his again. Then she looked away. \"Time.\" She traced the ear-shaped curve of the cup's handle, a gesture that reminded him of Lydia when she had something worrying her. Her eyes did not meet his. \"I wish you could have known him as he was. I wish you could have known us both.\"\n\nSilence lay between them, save for the music and the swirl of silk and slipper leather. \"Do you read the Personals?\" asked Asher, and the question startled her out of the reverie into which she had slipped. He started to reach down for the valise on the floor between their chairs, but the bite of his wound stopped him; he gestured to the newspaper visible in the bag's open mouth.\n\n\"Or more to the point, does your husband?\"\n\n\"We all do.\" She leaned to withdraw the folded sheets. \"We follow families, names, neighborhoods for years, sometimes decades. To us, chains of events are like the lives of Balzac's characters, or Dickens'. The nights are long.\"\n\nAsher unfolded the section and touched the advertisement he had seen.\n\n\"Saturday's paper,\" he said. \"His departure was arranged in advance. _Umitsiz_ is Turkish for _hopeless_ \u2014a variant, I think, for _Want-hope_. Does Ernchester know Turkish?\"\n\n\"He was part of the legation King Charles sent to Constantinople, before we were married. He was away three years. To me it seemed eternity.\"\n\nA wry smile brushed her lips as she considered the irony of that, and she added, a little shyly, \"It still does, you know, when I look back.\"\n\nThen she frowned and held the railway timetable beside the few short lines of type, as if comparing them. \"But why?\" she asked at last. \"What could they have said to him\u2014this Olumsiz Bey\u2014to make him come here without a word to me? Even without Grippen's support, we have wealth and a place where we are safe. Men searched the house, yes, but it was night when they did so\u2014they could not have overpowered him, even had he returned to find them. At night men are easy to elude. Charles knows London's every cellar and bolt-hole. Even if he knew Vienna once, cities change with time, and those changes are perilous to those whose flesh the sun will destroy. What could he have been offered?\"\n\n\"I suspect the men were only agents of someone else.\" Asher folded both paper and timetable again. \"Ysidro told me once that the Undead usually know when someone is seeking them. You know nothing, guessed nothing, of the men who searched the house?\"\n\nShe shook her head. \"There had been no... no unknown faces seen too many times, no footfalls passing where none should be.\"\n\n\"Which means that someone told them about the house.\"\n\nThe waltz had finished. On the platform the orchestra was putting up its instruments. A woman, small and gray-haired and dumpy, laughed as her white-bearded gentleman friend swept her up into an extravagant cloak of golden fur. Anthea turned her head to watch them, and in her eyes Asher saw an expression of almost sensual delight, a softening, as if she had drunk wine.\n\n_Karolyi?_ Asher wondered. An attempt to make sure the earl's wife didn't stop him from coming? But would Karolyi have known of the power struggle between Anthea and Grippen that would rob them of the master vampire's support?\n\nKarolyi had certainly hired the toughs who attacked him tonight. They had probably followed him all day, waiting their chance. That meant he'd better pick the toughest-looking fiacre he could find and warn him of trouble once they got into the isolated lanes and vineyards of the Vienna Woods.\n\nThe Ober appeared, Lady Ernchester's black cloak on his arm. Putting it around her shoulders brought Asher a stab of momentary agony, and she turned quickly.\n\n\"You're in pain.\" Her fingers were cold still, though she'd warmed them on the cup. \"I'm sorry, I didn't think.\"\n\n\"It just took me by surprise,\" he said. \"I'll take you to your lodging.\"\n\nA tulle of fog suffused the gaslight on the Graben to dim haloes, blurred the swags and statues of the fa\u00e7ades. Here and there a window still glowed, where maids, having unlaced their mistresses, brushed their hair and handed them nightdresses and prayer books, now locked up jewels or brushed dirt from slippers, or laid final fires for the morrow before creeping to cold beds themselves. The air was ice, the leafless trees friezes of unreadable runes passed by only a few final, home-hurrying shadows.\n\n\"Dr. Asher.\"\n\nHe paused in his stride and saw, again, her face turned half away from him in confusion.\n\n\"I know no honest woman asks a man to come back to her rooms with her, to stay with her the night.\" Her fingers stirred at the buttons on his sleeve. \"And I understand that it's the stuff of farce for me even to care about such conventions. Old habits die harder than you think. But... will you do this?'\n\nShe raised her eyes to his as she spoke. Oddly, Asher felt no sense of danger. He remembered how carefully she had wiped the blood from her fingers and the stammer of her nervousness that hurried to fill the silences of the dark crypt. It crossed his mind to wonder if she had inhaled so deeply of the coffee to cover from herself the smell of his blood.\n\nYet he had no sense that she was influencing his mind, laying upon it the vampire's glamour that blinded victims to their danger. Which might only mean, he thought, that she was very, very good at what she did.\n\nInto his hesitation, she continued, \"Save for one thing only, traveling alone on that train was the most terrifying thing I've ever done.\" They moved on along the wide street, two isolated figures in the thickening brume. Beside them the Plague Pillar ascended in an astonishment of cherubs, saints, and clouds, white in gaslight and shadow. \"I only just reached the hotel room in Paris in time, and I was terrified that sleep\u2014the unbreakable sleep of the Undead\u2014would overcome me where I stood in the street. They must have thought me a madwoman, hurrying the porters to take my trunks into my room and then pushing them all out and locking and double-locking the doors. And even when I was alone, the fear near overwhelmed me. How could I know that I'd wake with the setting of the sun again and not burn up screaming through some chambermaid's prying or greed?\"\n\nHer step quickened and her hand tightened on his arm, the memory of that terror making her fingers, for a moment, crushing iron.\n\n\"And it was worse, shipping the trunk the following night,\" she went on. \"Sending myself like a parcel, falling asleep to the rocking of the train, trusting to fate. Not knowing if I'd ever wake. They say we don't wake, should our darkness be violated by sunlight\u2014that we burn up in sleep. But who knows?\" Under the veils her face was calm, but there was a flaw in her voice, and she drew her cloak close about her, as if even her Undead flesh felt the cold. \"None of us are ever there, to see it happen to another. Even in utter blackness, the sun submerges our minds. Sometimes we hear and know what happens about us, but we do not wake.\"\n\nThey reached the door of her hotel, a splendid mansion whose lower stories comprised the palatial residence of some wealthy family, but whose marble stair led to a far humbler lobby on the upper floor.\n\nAnthea paused in the columned shadows of the entryway. \"A year ago Ysidro hired you\u2014forced you\u2014to be his servant. To do for him in daylight what he himself was unable to do. And you did it honorably.\"\n\nHis breath mingled whitely with the fog that had floated through the outer gate behind them. Her words had produced no such clouding. \"I had no choice.\"\n\n\"We all have choice.\" Her gaze met his in the dim light from the chandelier of crystal and gilt. \"I can only ask you. Stay with me in the room until the sun sets again. Please.\"\n\nLydia had once calculated how many human beings the average vampire killed in a century. If he were the man he once had been, Asher thought, he would have said yes, then later thrown open the trunk lid and let the sun reduce such a murderess to dust.\n\nPerhaps because she had saved his life, he would only have said no.\n\nThe clock on St. Stephen's was striking two, and like courtiers repeating a sovereign's joke, clocks on churches and monasteries throughout the Altstadt took up the chime. He would be alone, awake, with this woman for hours before she would be with him, alone and sleeping, trusting him as he must trust her.\n\nIf it weren't all a trap, to get him to a place where he could not call for help.\n\nBut surely the crypt had been that.\n\nHe told himself it was because he needed to find Ernchester, something he could not do without a vampire's help. But he knew that wasn't true.\n\n\"Very well,\" he said.\n\n\"He ceased to care at all, about anything, fifty, maybe sixty years ago.\" Anthea removed her hat, and despite the renewed slash of pain in his side, Asher helped her off with her cloak and the jacket beneath it. Her frock was Norwich silk, its ruffles glittering with star fields of jet. \"Music, watching people\u2014not for prey but just for the curiosity about how they live their lives\u2014it all meant less and less to him. Like that fairy book that came out a few years ago, where a man's limbs are replaced, one by one, by magic with limbs of tin, until suddenly he realizes he has no heart and is no longer a man.\" She passed her gloved hand across her eyes, the smooth skin of the lids pinching at the memory of pain.\n\n\"You're thinking that all those fifty, sixty years, when his life meant less and less to him, still he prolonged it by killing two and sometimes three men a week. There are things that can't... be explained. It's easier than you think, to fall into... habits.\"\n\n\"I'm not thinking anything.\" He remembered Jan van der Platz's blood on the barn wall, the shocked hurt in the boy's eyes just before Asher pulled the trigger.\n\nShe lighted the lamp on the heavy table. Asher wondered if she had been aware of the brassy-haired prostitute's death agonies, and it occurred to him that this woman had probably seen worse. Maybe done worse herself. The small chamber, copiously decorated with swathes of peacock feathers and dried flowers and smelling vaguely of carpets, had not even been fitted with gas, much less electricity. The topaz light made the vampire's face more human, lent color to her cheeks and a kind of life to her eyes, and brought forth cinnabar glints in her hair. Asher remembered again his vision of her lying on the floor of what he realized now was the old Ernchester town house in Savoy Walk, the house where first he had met this woman\u2014where she had saved him from the Master of London's wrath.\n\n\"I'm sorry to have provoked this division,\" he said. \"To have robbed you of whatever support Grippen would give.\"\n\nShe shook her head. \"It's been decades coming. Maybe centuries. He wanted Charles\u2014and the houses and land that would give him a system of bolt-holes. We had no living child, and there are ways of manipulating even entailed property, to keep a good part of what you own. Grippen lost much in the Great Fire, and afterward the city was greatly changed. I kept the property tied up in trusts, so Grippen couldn't own them outright. But it was only a matter of time before he would come to an end of needing Charles. Vampires do not kill vampires, but... I suspect in any case he would not have helped.\n\n\"Who is this Karolyi?\" She took off her mitts, and her long nails glinted oddly in the lamplight.\n\nWhile she plucked the jewel-headed pins from her hair, Asher told her about his early acquaintance with Karolyi in Vienna. \"He's continued in the diplomatic corps, I understand. Young men of his class do, with only minimal qualifications. I know he's been responsible for the deaths of at least two of our agents over the last ten years, but it's never been proven.\"\n\n\"How would he have known about my husband?\" She paused, brush in hand. \"He may be ruthless, yes, clever and dangerous, but it would not have told him how to find a London vampire. Only another vampire could have done that. And why would he have chosen a London vampire to... to bring here? The masters among the Undead are jealous of their territories. They do not tolerate vampires who are not their fledglings and subservient to their wills. Ernchester knows this.\"\n\n\"That may be part of Karolyi's plan.\" Stiffly and clumsily, Asher began to sponge with cold water at the blood in his coat, and Anthea said, \"I'll do that,\" and took it from him. Now that the shock had worn off, he felt very tired, the pain in his side settling into a dull ache. He was glad to sit quietly on the room's overstuffed brocade settee.\n\n\"What he wants your husband for is less clear,\" he said after a time. \"Maybe he wants your husband because he _isn't_ a fledgling to some local master, here or someplace like Bulgaria or Greece. That's what I need to find out. It may be he wants your husband to make a fledgling who can be put to Karolyi's uses. But whatever he planned, he had to get your husband out of London because of Grippen.\"\n\n\"Yes,\" Anthea said softly. \"Grippen would know.\"\n\nShe walked to the doorway between that chamber and the next, the movement of her shadow summoning vague blinks of light from the brass fittings of the trunk that filled most of the space not already occupied by the four-poster bed. Her hands, straying in the lace at her throat, were like lilies, ringed with solitary gold.\n\n\"When a master vampire begets a fledgling,\" Anthea said slowly, \"he... he takes the fledgling's mind, the fledgling's consciousness and personality, into his own being, for the time it takes that... that fledgling's body to die. Once death is complete, once the... the changes to the vampire state have begun, then the master breathes that mind, that soul, back into the changing body once again. But not all of it. And what is breathed back is... stained. Altered. Just a little.\"\n\nThe marble profile remained averted, sienna eyes staring blankly into distance.\n\n\"No,\" she said. \"He wouldn't use Charles in London. Grippen knows... everything. And he has been watching us. Maybe waiting for his chance. I hate him.\"\n\nShe shook her head, moved her shoulders as if to shed a weight of thought. \"I have hated him since the first night Charles brought me to his house. Elys\u00e9e de Montadour, the Master of Paris, is not so old or so powerful as Grippen, but she would sense it, I think, if a strange vampire came to Paris. Still, they could have gone to Rouen or Orl\u00e9ans to make their plans. The vampires of those cities perished in the confusions of the last German war. Such a journey would have been safer, would not have involved travel by day...\"\n\n\"Do you know the vampires of Vienna?\"\n\n\"No.\" She crossed to the window, spread back its teal-green velvet curtains, with their treble fringes of gold and tassels like double fists. \"I feel them... feel their presence. As they feel mine, without being able at once to see where I am. They know I am here.\"\n\nHer fingers traced the fringe, the fabric, drinking of the texture as they had drunk the shape and texture of the porcelain cup. The dim light from the street below edged and transformed her face into a song of gold planes and black.\n\n\"I feel... everything. This new city that seems to bleed music from its very stones... When I saw the men pursuing you, I'd been walking about the streets for nearly an hour, just glutting myself on new tastes, new smells, the voice of a river that isn't the Thames. All those new dreams and thoughts and sensations hammer around me and in me and at me. I feel as if every cobblestone has a diamond underneath it, and I want to run through the streets gathering them up like a greedy little girl.\"\n\nThe colorless lips curved in a half-wondering smile, and Asher remembered her watching the dancers in the caf\u00e9, drinking the smell of the coffee, the music of the waltz. \"I know I'm in danger. I'm afraid, and I know I should be more afraid than I am. I could die in moments, just because I don't know the right place to hide, the right turning to take. But it's so beautiful.\"\n\nShe half wrapped the curtain around her, the lush color startling against her face, like a silver icon or a painting by Klimt.\n\n\"This is all so new to me, wonderful and strange. It's the first time, you understand, that I have left England. The first time since... since I became what I am... that I've been out of London. It's been nearly two hundred years, Dr. Asher. I traveled a little after I thought Ernchester was dead, visited a sister in the north. But in my mourning I had no taste for it and only wanted to return to what I knew. I mourned for a long time.\"\n\nAsher had seen a portrait of her, done when she was over sixty in her mortal life. She'd put on weight, and her hair had grayed, and the raptor eyes that flashed copper in the rosy lamp flame had been dead, resigned, filled with a kind of hurt puzzlement, as if she had never ceased to ask, _How can he be dead?_ In the painting she'd worn the broad gold band that gleamed now on her finger.\n\n\"A vampire traveling is... horribly vulnerable.\"\n\n\"And yet you came.\"\n\nShe smiled, a human smile, the full, pale lips hiding the fangs. \"I love him,\" she said. \"To my last breath\u2014and two centuries beyond.\"\n\nLady Ernchester had instructed the management of the hotel that she was not to be disturbed by chambermaids. She was an actress, she had said, and likely would be out most of the night, sleeping in the day. When she told Asher this, during a discussion of how words were pronounced in her early girlhood while she mended the slashes in his jacket and greatcoat, he closed his eyes briefly, imagining the concierge's reaction to this request.\n\nBut in fact when Asher later heard the chambermaids chatting in Czech and Hungarian in the corridor, none even tried the door.\n\nAsher had tried to remain awake through the night, talking of philology and folklore with the vampire countess\u2014her imitation of her nursemaid's Wessex dialect had been both hilarious and fascinating\u2014but the ache of his wound, loss of blood, and exhaustion had claimed him. The voices of the chambermaids woke him in mid-morning, to find a heavy sunlight slanting through the chinks in the teal-colored curtains. He lay back on the settee again, trying to formulate an article in his mind\u2014countryfolk of Anthea's day had pronounced the _y_ or _e_ at the end of such words as _hande_ as a sort of aspiration, though they no longer spoke an _e_ as _a_ , and they would walk across a field rather than meet a pig in the road. But how on earth could he claim he'd had an interview with a contemporary of the Cavalier poets?\n\nIn time the voices of the chambermaids faded and the upper floor of the old _palais_ fell silent. A heavy silence, broken only by the far-off clatter of a tram in the Schottenring and the distant threads of a hurdy-gurdy. He thought again of the woman sleeping, sealed within her double trunks, trusting his word that he would remain through the day and see that she came to no harm. Over the centuries she had killed... how many?\n\n_I wish you could have known us as we were._\n\nWas all vampirism a craving to hold to the sweetness of a vanished youth, a desire not to have the good years, the dream years, slip away in the flowing stream of time?\n\n_I love him,_ she had said. _I knew he could not be dead._\n\nWho had loved the men, the women, the children whose lives she had traded for the continuance of her own?\n\nHe sighed and leaned the bridge of his nose on his knuckles, twisting at the problem again as a fish twists on a hook. She trusted him. And indeed, only through her could he hope to find Ernchester now, to keep him from selling his services to the Hapsburg Emperor, if he hadn't already. What had Karolyi offered him? Safety from Grippen? Why not tell Anthea, then? Why not bring her to Vienna with him?\n\nWho had searched the house, who had known of Karolyi's plans, and for what had they been seeking? Who was Olumsiz Bey?\n\nA transliteration for the Master of Vienna? Who might, after all, be Turkish himself. The whole area had been overrun as late as the mid-seventeenth century, and it was conceivable that the Undead in this most cosmopolitan of cities might not be Austrian\u2014or even European\u2014at all.\n\nAnd what, above all, was he going to do when he _did_ find Ernchester. Kill him?\n\nHe knew already that he would never sleep easy again if he didn't kill Anthea as well.\n\nWith a soft, oiled click, a key turned in the lock. Asher's mind fumbled tiredly for the Hungarian for _This room is not to be disturbed_ as he rose and crossed to the door, which opened to reveal Bedford Fairport.\n\n\"Asher!\" The little man blinked in surprise and adjusted his spectacles as if Asher were some trick refraction of the light. \"What on earth...?\"\n\nDeportation telegram, thought Asher automatically, his mind still sluggish with sleep. And then. _How did they trace me...?_ He was mentally framing what he was going to tell Halliwell about the layout of the Batthyany Palace when, with panther quickness, Ignace Karolyi stepped around the side of the door and put a knife to Asher's throat.\n\nFairport bleated, _\"No!\"_ as the blade gashed like splintered glass. \"Not here!\"\n\nThe ape-browed coachman and two burly thugs Asher had never seen before were already in the room and closing the door. One of them caught Asher's elbows behind his back, thrust him against the wall; the other walked straight to the window to pull the curtains shut. Blood from the small cut on his neck burned hot on Asher's skin, but Karolyi had already turned his attention elsewhere, though the blade remained cold against the flesh.\n\n\"Find it.\"\n\nAsher tried to turn but was pushed against the wall again. Over his shoulder he saw Fairport staring at him in a kind of aghast astonishment; one of the thugs took the medical bag out of Fairport's hand, opened it and pulled out a paper of sticking plaster, which he slapped over Asher's mouth. With his free hand Karolyi took something from his greatcoat pocket, a silk scarf, with which the thug tied Asher's hands. Probably the same one, thought Asher, he'd used to strangle the woman in Paris.\n\nOnly then did Karolyi take his knife from Asher's throat, sheathe it in an inner pocket of his jacket. The man who'd been holding Asher's arms kicked him roughly behind the knees, thrusting him to the floor, a minor theater of operations while the others pushed through the doorway into the next room. Asher tried to cry out, a warning, protest, appeal against the hideous vision of them prizing open the double lids of the trunks...\n\nThen he realized that Anthea was perfectly safe.\n\nIt was Karolyi who'd had her house searched\u2014probably Karolyi who'd written _Vienna Express_ on the timetable.\n\nHe'd had her followed here from the station.\n\n\"This has to be it,\" he heard Fairport say in German.\n\n\"You're not gonna check to see?\" asked the coachman.\n\nFairport squeaked protestingly: Karolyi said, \"Let it be, Lukas,\" his voice casual, but the henchmen stepped quickly out of the room. \"Did you think she would not follow?\"\n\n\"To tell you the truth, I didn't know.\"\n\nAsher turned his face against the thick, dust-smelling carpet, saw them standing together in the doorway, the old man looking up into Karolyi's face like a retriever who's just lugged in a pheasant nearly its own size. He thought. _Fairport's a double._ Something about the distance between them, the tilt of Fairport's head, told the whole story. _Has been a double for years._\n\nOn reflection he supposed he should feel anger, but he didn't. It was something that happened in the Great Game, like stray whores being strangled or those who learned too much getting shot.\n\nKarolyi looked down at Asher with an expression of rueful half amusement. \"So tell me, Dr. Asher\u2014was it just coincidence that you were the man assigned to follow me? Or was Ernchester wrong in believing that the British are not also using the Undead?\"\n\nAsher inclined his head. He reflected that it might even be the truth.\n\nKarolyi laughed. \"Not many, I daresay. They're good, rational, God-fearing, Church of England, university men in your Department. Civilized, the way they tried to civilize me all my life.\" He came over and squatted beside Asher's shoulder, slim and soldierly even in the impeccably cut brown suit he wore. A hot blade of sunlight flashed across the gold and ruby of his cravat pin, red and gold repeated on his signet ring.\n\n\"But being raised in the mountains does something to you. I suspect I got from my Moravian nurse, at the age of five, what you got from years of comparing legends and collecting odd facts that don't fit into the curricula of Oxford and Innsbruck. Was that why they picked you to follow me? Surely they don't think I'd miss a familiar face?\"\n\nUnable to reply because of the sticking plaster, Asher only met his eyes. _You know I'd never answer your questions anyway,_ his look said, and the full, red lips curved in a mocking smile.\n\n\"Well, I admit I didn't realize it _was_ you in 'ninety-five until I saw you in the Munich train station. Our good Dr. Fairport kept that little secret from me back then.\" Karolyi stood up. Behind him, the two thugs carried Anthea's trunk to the door which the coachman Lukas held open; Fairport stood by, watery eyes flicking nervously from the trunk to Asher and Karolyi. \"You know, I'd have thought you'd have been promoted past field agent by this time. You always struck me as being smarter than that. But maybe that was luck.\"\n\nHe took his gloves from his pocket, started to put them on but glanced down again at Asher and returned them to the pocket again. A small gesture, but Asher knew at once what it meant.\n\nWhite kid was expensive, and blood would not come out of it.\n\n\"Remember my instructions, Lukas... _all_ of my instructions...\" he called out, and then turned with an admirable casualness to say, \"Dr. Fairport, perhaps you'd best go with them.\"\n\nFairport nodded, his gaze behind the massive spectacles glued to the trunk as the stevedores maneuvered it through the door. \"Of course,\" he breathed, \"they can't appreciate... Klaus! Klaus, please, a little more gently!\"\n\n_He's forgotten I'm here,_ thought Asher. More furious than frightened, he made a muffled noise that might have been Fairport's Christian name.\n\nBy the way the old man flinched, Asher knew he'd guessed right. Absorbed, fascinated, obsessed by the prospect of taking a vampire alive, Fairport _had_ forgotten. Had forgotten what Karolyi did with those inconvenient to him, if he'd ever known. The old man turned back, not quite in time to catch Karolyi smoothly withdrawing his hand from the front of his coat.\n\nAsher met Fairport's eyes, forbidding him not to guess what was going to happen the minute he left the room. The old man's eyes, pale blue and tiny, distorted behind enormous rounds of glass, flinched away. _Damn you,_ thought Asher, _if you're going to let him kill me, at least admit to yourself what you're doing..._\n\n\"You'd best supervise them,\" Karolyi said gently, nodding after the departing men. _You don't really want to see this, do you?_\n\nKarolyi's eyes met Fairport's, held them, and Asher understood the unspoken barter: _If you don't want to have anything further to do with vampires, of course that can be arranged, too..._\n\nFairport turned uncertainly, as if Karolyi had implied that only his intervention could prevent the three stevedores from heaving Anthea's trunk out the window or riding it down the stairs like a bobsled.\n\nThen he turned back. \"Someone, er\u2014might have seen us come in,\" he said hesitantly. \"They'll certainly have seen the name on the van.\" He looked apologetically down at Asher and twisted his hands in their gray cotton gloves, as if that were the best he could do. Asher wanted to kick him,\n\nKarolyi fetched a long-suffering sigh. \"Have you chloroform in your bag, then?\"\n\nFairport went to his instrument case, but the tremor of his hands, increased by nervousness, spilled the chloroform as he tried to pour it onto the cotton pad. Karolyi strode over to steady him, and in that moment Asher twisted his wrists against the hastily knotted scarf. The silk wasn't like rope, with rope's matted fibers; one knot tightened hard while the other slithered and loosened. As Karolyi turned back with the drug-soaked cotton in hand, Asher chopped hard with his legs at the Hungarian's ankles, pulled free one arm from the scarf, rolled to his feet and bolted for the door.\n\nKarolyi, who had caught his balance on Fairport's shoulder, threw the fragile old man aside and flung himself after, shouting at the same time, _\"Stop, thief!\"_\n\nCoatless, unshaven, unknown to the hotel and still mute from the sticking plaster over his mouth, Asher could only redouble his speed down the front stair, swinging himself over the banister and down to the next flight as two stout porters in brass-buttoned green uniforms pelted up to meet him. He kicked his way through a rickety French door to a balcony that ran around two sides of the building's central court, scrambled down a rain gutter to the court where a red and white van, Lukas at the reins, was just lurching into the carriageway to the street. He veered as the coachman drew rein and one of the thugs dropped off the back to meet him, ducked through a door into kitchen quarters, dodged past two startled cooks and a scullery girl, and out again into a lane, pursued by cries of _\"Dieb! Mord!\"_ and hammering feet.\n\nThe cramped, medieval streets of the old city seemed filled with pedestrians, either retreating from him in alarm or joining in the pursuit. He struck someone, blundered against a market woman and a postman with his parcels, ducked down an open gateway into another court and through another kitchen as half a dozen young officers in the blue and yellow uniforms of the Imperial-and-Royal Hussars sprang up from a table at a sidewalk caf\u00e9 and streamed joyously after him, hands to their sword hilts and spurs rattling on the pavement.\n\nHe dodged into another gate and raced up the shadowy stairs, while police, guards, and passersby sped past him and into the courtyard, looking for a kitchen door or postern through a shop\u2014finding it, they roared on through, while Asher pulled the sticking plaster from his mouth\u2014with a certain amount of damage to his mustache\u2014and, when they were gone, descended the steps and walked out to the pavement of the Dortheergasse again.\n\nThe ache in his side was breathtaking, and under the bandages he could feel the warm seep of blood. Gray afternoon cold cut through his shirtsleeves. He fought a wave of dizziness as he hurried toward the crowds on the Graben, feeling in his trouser pocket and praying there was something there besides his handkerchief.\n\nHe was in luck. He'd paid for the coffee last night with one of Karolyi's ten-florin notes and, owing to the pull of the wound in his side, had put the change in his trousers rather than the inner pocket of his jacket. It was enough, maybe, to get him a jacket at the flea market in the Stephansplatz if he wasn't too fastidious, and a tram ticket out of the immediate area, to somewhere that he could hide.\n\n# EIGHT\n\nASHER REMAINED on the Prater until nearly four, to give the hue and cry time to subside.\n\nHe had a late lunch at one of the rustic caf\u00e9s that lined the Volksprater's bridle paths, consuming Czech sausage and _buchty_ with one eye on the broad, graveled way that led from the organ grinders and carousels around the great Ferris wheel off into the gray and rust fastnesses of the old Imperial hunting park. Once he caught a glimpse of the brilliant cobalt jackets of the Imperial-and-Royal contingent of his pursuers among the thin trees and heard their faint hallooing as they searched.\n\n_England, when war comes, I think you'll be safe on the Austrian front at any rate._\n\nBut his inner smile faded at the thought of Ernchester, no longer now entirely a volunteer. If there were any stipulations in the deal he'd made with Karolyi, any acts he wouldn't perform at the nobleman's behest, the rules had changed. Or would change, when they told him they held Anthea prisoner.\n\nHe shivered in his rag-fair coat.\n\nHow long had Fairport been a double? he wondered. According to Karolyi, as far back as the flap over the smuggled Russian guns. It wasn't as unusual as it might seem to outsiders that Fairport hadn't blown him to Karolyi then. The fact that Fairport was passing the odd fact along to the Kundschafts Stelle from time to time didn't mean he was entirely their man. Doubles\u2014particularly men like Fairport\u2014were frequently masters of self-deception, as Asher knew from having dealt with them. They always kept things back, from either side, sometimes for the most bizarre and absurd reasons: He remembered an American missionary in China who hadn't warned him of an impending rebel attack because he didn't want a Chinese patron of the mission to learn that his\u2014the patron's\u2014son had a mistress in the quarter of Tientsin through which the rebels were expected to come.\n\nAnd perhaps Karolyi hadn't asked it of him, judging the matter too small to waste a trump on information he could learn some other way.\n\nEven in retrospect, however, the thought of how close he'd come to dying as his Czech mountain guide had died made Asher shudder.\n\nFairport's research was already an obsession back in the nineties. Top quality materials, facilities, research journeys were always expensive, and Fairport was not a wealthy man. The best agents, Asher reflected, were those without any weak points, any handles upon which an enemy could grip.\n\nLike Karolyi. Smooth, hollow men for whom the Job was all.\n\nHe glanced back at the self-consciously rustic kiosk where the waitresses huddled out of the cold, and wondered if Halliwell could be trusted.\n\nFairport might not be the only one in Karolyi's pay. Better, certainly, to wait until six and leave a message at Donizetti's, arranging a meeting. If he could stay out of sight until then...\n\nBut after six it wouldn't matter.\n\nNot to Karolyi.\n\nThough Asher was already fairly certain what he'd find, he strolled to the kiosk and bought that day's _Neue Freie Presse._ On the back page he found a small lead line:\n\n> LACEMAKER'S BODY FOUND IN WIENERWALD\n\nScanning the brief copy, his eye picked out the words \"drained of blood.\" The name of the vineyard near which she'd been found was familiar, a quarter hour's drive from Fruhlingzeit.\n\n_So._ He stared blankly in the direction of the gay-colored midway, the shooting galleries and Punch and Judys, the panopticum where the murder of the Czar was on view in wax for the edification of schoolboys. A fleer of music blew from that direction, a distorted jingle of pipes and chimes, and then was gone. \"The Waltz of the Flowers.\"\n\n_So._\n\nA lacemaker. Like the prostitute in Paris, a woman no one would miss.\n\nOf course Karolyi would pick a woman.\n\nErnchester would be there until sometime tonight.\n\nFairport was disposable. Even the knowledge of a scheme to use vampires was disposable. As Karolyi had said, most men in the Department weren't going to believe it anyway.\n\nWhat could not be disposed of\u2014what he himself could not relinquish\u2014was Ernchester.\n\nToday\u2014now\u2014Asher knew where the vampire earl was, where Anthea would be. Knowing Fairport\u2014and Fruhlingzeit\u2014were blown, they'd move tonight and, like true vampires, fade into the mists, leaving only a little blood and a muttering of rumor behind.\n\nA fiacre drove by on the path, the coachman whistling briskly. The afternoon light had turned steely and cold. Asher shivered again and blew on his hands.\n\nThere was, of course, always the option of taking the first train back to Munich\u2014cadging a ride in the baggage car, at this point, but Asher had done that in his time. If Burdon were still the head of the Munich branch\u2014if there still _was_ a Munich branch\u2014he could at least get enough money to go back to England. Tell them Fairport was a traitor, Karolyi was in league with\u2014well, a very dangerous man\u2014and wash his hands of the business. Go home to Lydia, who might very well have sent him a wire at Fairport's... None of this was his affair anyway. He had done all he could be expected to do.\n\nBut that left Anthea in the hands of Karolyi.\n\nAnd he knew where Ernchester was today. That was the crux of the matter.\n\nThere was a telephone in the kiosk. Undoubtedly the police could trace him through the exchange if he phoned Halliwell\u2014he'd dealt with the endless polite chatter of Viennese telephone operators too often to think the transaction could be accomplished quickly. And the delay of a night in jail meant that Ernchester\u2014and Anthea\u2014would vanish untraceably.\n\nWhen he'd taken a seat at this table, half screened from the path by a hedge, there had been two or three other brave souls sipping coffee and gazing contemplatively over the slaty waters of the canal. Now he was alone. Across the river the clock on St. Stephen's struck three.\n\nUnwillingly Asher got to his feet, thrust his bare hands into his pockets, and after a cautious glance up and down the path for signs of pursuit, headed back along the Haupt Allee for the Praterstem, where with his last few pfennigs he could catch a tram at least partway to the Vienna Woods.\n\nIt was not long after the coming of full dark that Asher realized he was being followed.\n\nHe took the tram as far as D\u00f6bling, then climbed the winding road through thin rust-and-pewter woods past Grinzing. Moving kept him a little warm, though his side hurt at every step and he had to stop repeatedly to rest on the low rock walls that divided woods or vineyards from the road. He was sitting thus, trying to get his breath after a particularly steep patch of road, when he heard the church clock in that storybook village chime five.\n\nNow and then a farm wagon passed, and once a motorcar full of homebound seekers after pastoral calm, but as the twilight clotted under the trees, such things became few. A small wind cleared the clouds; a shaved silver coin of moon floated in a halo of ice. By six it was utterly dark.\n\nThat mattered less than it might have, for Asher knew the road. Toiling upward with the ache of fatigue dragging at his bones, there were times when he felt he'd never been away. He didn't even have to look for the Fruhlingzeit Sanitarium's gateposts of ivy-covered stone. The slope of the road told him exactly how far yet he had to go.\n\nHe listened for the sound of human pursuit. But that was not what he heard.\n\nHe would have been hard put to say exactly what it was he did hear, or what he felt, that told him they were in the woods behind him. Perhaps, had he not come so close to death at their hands\u2014or the hand of those like them in Paris\u2014a year ago, he would not even have known he was being stalked.\n\nBut he knew. A touch of sleepiness at his mind, in spite of the wind eating through his holed coat and the ache of his wound. A sense that it wasn't really necessary to look behind him, or around him, at the woods. And then, when a single breath of moving air sighed from the cinder-colored darkness among the trees, the sweetish stink of blood.\n\nHe didn't slow his step, or quicken it, not daring to let them see he knew, but he did wonder what he was going to do. He was nearly at the drive that turned into Fruhlingzeit, and the drive, at least, would be watched by Karolyi's men. He'd have to leave the road then. The silver on his throat and wrists would buy him a few seconds, but they wouldn't save him from a broken neck. The road before him lay deserted.\n\nOn the whole they moved without sound, but it was late in the autumn, and beneath the pale stems of the beeches the brown leaves mounded thick, and dead fern and ivy rustled and whispered with the passage of unseen feet.\n\nHe stopped on the edge of the road\u2014he'd been keeping to the shadows along the ditch in case Karolyi had patrols on the road\u2014and took his watch from his pocket, angling it to the moonlight, then closed it with a click and, under cover of slipping it into his pocket again, unhooked the fob from his belt. A quick motion wrapped the chain twice around his middle finger, so that when he drew his hand out again\u2014and tucked it under his armpit, as if for warmth\u2014he carried the rounded disk of silver cradled out of sight in his palm.\n\nIt wasn't much, but it would have to do.\n\nHe sprang across the ditch, scrambled a little up the bank, wondering if they could hear the sudden heavy slamming of his heart. The Vienna Woods were thin. Beneath a summer canopy of leaves he doubted he could have navigated by night, but with the trees bare, the familiar shapes of beech and sycamore were just visible by the latticed pallor of the moon. There was no way of telling how great a force of men Karolyi had at the sanitarium. It would take only one to spread an alarm.\n\nProvided he lived to get anywhere near the walls.\n\nWhat had Anthea said? _The masters among the Undead are jealous of their territories._ He remembered, too, that pitiful fledgling Bully Joe Davies back in London, glancing in terror over his shoulders: _They'd kill me, they would_ \u2014 _Grippen don't want none in London but his own get, his own slaves..._\n\nHad they, too\u2014whoever _they_ were\u2014read that tiny mention in the _Neue Freie Presse_ about the dead lacemaker and known that another was hunting on their territory, killing in such a way as to rouse the suspicions of the rulers of the day?\n\nOr did they simply recognize his heartbeat, Asher wondered, the smell of his blood, as those of an intruder who had been snooping around the walls of their palaces last night?\n\nAsher walked as quickly as he dared, moving purposefully. Once he heard the leaves rustle, and some sound that might have been a taffeta petticoat, but his senses screamed at him that there were more than one. Like sharks they followed him, slipping unseen through the abysses between the trees.\n\nWhite glimmered ahead. Black veins of ivy traced it: the rear wall of Fruhlingzeit. Above it bulked the house's steep roofs and stuccoed walls, the golden ochre hue so characteristic of Viennese houses grubby in the dark. Most of the windows facing the woods were shuttered, but lamplight from those facing the court outlined what had been the stable, since converted into a laboratory and therapy room. Asher had always suspected that the aged cats and dogs\u2014and the occasional Viennese businessmen\u2014upon whom Fairport experimented showed improvement because of the therapeutic massage, good food, and careful tending that went with \"magnetic induction.\" There was a sort of crypt under the stable, Asher remembered, where Fairport's generator was housed among stores of carbolic, ether, kerosene, and coal.\n\nNearer the wall he smelled the smoke of a guard's cigarette.\n\nThe trees pressed close around the back of the property. From the concealment of an oak he could see the window where he'd sat all that long ago afternoon, planning how to get himself out of Vienna and betray Fran\u00e7oise in the most painful possible fashion in the process.\n\nThen he turned his head and saw a woman standing beside him.\n\nThe hair lifted on his nape. He had not heard one single sound.\n\nShe was beautiful, like something wrought of moonlight, flaxen hair piled high, but snagged and tugged by the branches until its tendrils floated around her face in a glowing halo; light eyes, gray or blue, etiolated and transparent. Her dress was moonlight, too, some oyster shade, colorless as a web, and the luster of satin flickered along its sleeve as she lifted her hands. Her eyes filled with longing and sorrow and desire.\n\nAsher felt his mind shutting down, warm yearning for her flooding heart and thoughts and groin, even though, in the shadows of those waxen lips, he saw the curve of fangs. It didn't matter. All that mattered was that he wanted her, as desperately as he had wanted Lydia before they were married, as desperately as he had wanted the pretty shop girls of Oxford when he was a student and frantic with a boy's nascent lusts. Against his will a sort of drunkenness filled his mind and he found himself reaching for her, filled with the irrational conviction that kissing her, touching her, would not matter, that it would be all right, the way one thinks in a dream.\n\nAs if from some tremendous distance he saw himself, his mind protesting but unable to connect its thoughts with his actions. Her hands touched his face, cold even in their gloves of shell-colored kid; they slipped over his ears and down to his neck, and his own hand felt rough and cold on the taut silvery cloth of her side.\n\nThen her mouth twisted in a snarl, wide, like an infuriated cat's. The glamour snapped away as her hands jerked back from his collar, where even through fabric and leather she felt the sear of the silver underneath. Asher gasped, waking, it seemed, to find her mouth inches from his throat, her grip already like iron on his arms. Before she could move, he struck her cheek with the silver watch palmed in his hand, twisting away from her even as she screamed\u2014shock, pain, fury, like a cheetah's scream, or a demon's in hell.\n\nHe flung her from him, bolted for the wall. She screamed again, and from the corner of his eye he saw her collapse to her knees, clutching the side of her face, screaming over and over as she clawed at the flesh. Something\u2014some darkness\u2014flashed among the trees, and he felt a smothering sleepiness crush his mind like gloved iron. He thrust it from him, scrambled up and over the wall as men's voices cried out somewhere close, dropped into the rosebushes below instants before the first of Fairport's servants pounded around the side of the house. He rolled into the shadow, hiding the pale blur of his face, and they ran through the garden to the gate. The moment the last was gone, he flung himself across the narrow space of gravel and bare thorn to the door under the stair.\n\nThen other men were in the garden, calling to one another. He heard Lukas the coachman's name, and someone called something about \"Herr Kapit\u00e4n...\" presumably Karolyi's regimental rank.\n\nThe screaming had stopped. But they'd all be busy for some time.\n\nTen minutes, he thought\u2014striding down the stone-flagged passageway to the kitchen\u2014while everyone dashed madly around the perimeter of the wall. Longer, if they had as few men as he thought they did, or if they found anything. He tripped the lever behind the scullery cupboard, slipped down the narrow stairway it revealed. More than once he'd taken Slav nationalists or Russian messengers down this way, to keep them unseen by Fairport's patients.\n\nGod, how the blond woman had screamed!\n\nAt the flea market he'd purchased wire to make another picklock; his hands shook while he winkled the lock at the bottom of the stair. It was an old-fashioned tumbler type, and he could have picked it in his sleep\u2014he'd warned Fairport about it a dozen times...\n\nSeventeen years with the Department, he was interested to see, had not inured him to that old chivalric voice within him that protested that there were things that a gentleman did and would not do even in defense of his life: kick a man when he was down or render what was euphemistically called a \"foul blow\" in a fight; shoot a man in the back; lie on sworn oath; forge another's name.\n\nShoot a sixteen-year-old boy who trusted you.\n\nSteal money from a woman who loved you.\n\nStrike a beautiful girl in the face with a handful of substance that you were reasonably certain would react upon her like vitriol.\n\nEvidently the fact that had he not done so, she would have killed him within seconds was of no importance to those old voices of his childhood: his country-doctor father, his grim-faced uncle, his tutors at Winchester and Oxford. He still felt an utter swine.\n\nDid he think she was any different from Anthea?\n\nThe pawls of the lock snicked back. As he opened the door, dim gaslight from the scullery above showed him a strange gleam on the lock plate. Asher braced his foot in the door to keep it from closing\u2014it was, as he recalled, heavily springed\u2014and lit a match for a better look.\n\nOn the inner side, the lock was silver.\n\nThe smell of fresh-sawn wood filled his nose, and beneath it, the smell of blood.\n\nHis nape prickled again, and he stood still, listening, barely breathing. Then, slowly, he turned the catch to keep it from locking again, raised his lucifer higher and held it up into the room within.\n\nSilver flashed in the seed of phosphor light. Where he had known only a small underground chamber equipped with bed, chair, and chamber pot, he now saw a glittering grillwork of silver bars that stretched from side to side not three feet from the door. Where the base bar of electroplated steel held them across the floor there were curls of sawdust, yellow and new.\n\nBehind the bars, eyes caught the reflection like the eyes of a cat.\n\nAsher blew out the match as the flame scorched his fingers. Frail, twice-reflected light from the stairway showed him a pale face, pale hands as they approached the bars, the white of a shirtfront and an old-fashioned stock.\n\nA voice spoke out of the darkness. \"Have you come for my capitulation? I told you I'd do anything you asked. Isn't it enough that you've betrayed me, lied to me? Was it necessary to... to do what you did?\"\n\nThere was a pause, while Asher stared blankly into the darkness, and the strange eyes gleamed back at him from behind the silver bars.\n\nThen the voice said, \"Dr. Asher. The doctor of languages from London. Don Simon said you had been a spy.\"\n\nAsher's mind made a tardy jump. \"That wasn't your wife's voice you heard,\" he said.\n\nOne of the white hands moved; Ernchester pressed it for a moment to his mouth, closed his eyes, like a man trying to still something within himself.\n\nAsher went on quickly, \"It was another vampire, a woman, who attacked me just outside the walls. Do you know where they keep the key?\"\n\nErnchester shook his head. \"Fairport keeps it,\" he said after a moment. As Asher had heard on the train, his accent was far less modern than his wife's, the flat vowels making it sound very American. \"Where is Anthea? They said they had her...\"\n\n\"I don't know.\"\n\n\"Find her. I beg you, take her out of this place...\"\n\nAsher stepped to the bar, examined the keyhole of the small doorway set in the lattice. It was a Yale cylinder type, and unlocking it was far beyond the capacity of a piece of wire. At the back of the barred area he could see a trunk, like a block of shadow. In front of it the earl seemed very small in his shabby, swallowtailed coat, his gay red-and-yellow waistcoat and strapped pantaloons, a ghost wrought of dust, a mummy that sunlight would shatter.\n\n\"I'll be back.\"\n\nAs he turned to go, he saw, lying on the bench beside the outer door, the lace mitts Anthea had worn at LaStanza's and a red ribbon from which depended a black pearl the size of a pea. It had been around her neck when she'd lain down in the trunk that served her as a traveling coffin. They must have brought them in, to show him that they had her indeed.\n\nWhat had the original deal been, he wondered as he mounted the hidden stair to the scullery above and pushed the shelves to behind him. A lure, to bring him into their power for something he'd never have consented to do? What? Ernchester had certainly gotten on the train at Charing Cross of his own volition, had been a free man when he'd murdered Cramer. Asher's jaw tightened bitterly, remembering that large young man's ingenuous grin. He should shoot the boy's murderer, not risk his own life setting him loose.\n\nHe remembered again as he climbed the stairs to Fairport's office with quick silence\u2014staying by the wall so the treads would not creak\u2014why he had come to hate the Great Game.\n\nA lamp burned in the office\u2014inconveniently, because one of the curtains was half open, and it meant careful maneuvering not to be seen from outside as he approached the desk on his hands and knees. He'd heard no one in this wing of the house. He had only minutes before they returned and started searching in earnest, and Ernchester was right in that he must, above all, release Anthea. While Karolyi had her, he had the vampire earl, whether or not the man was actually in his possession. The fact that Ernchester had jumped to the conclusion that the dreadful scream he heard had been hers told its own story.\n\n_They are killers,_ he thought, in a kind of baffled rage at himself. _Over the years Anthea has done to thousands of men what that woman nearly did to me. Why should I care?_\n\nBut all he remembered was the face of a woman in a portrait, plump, weary, gray-haired, in mourning for a husband who had died thirty years before. _How can he be dead?_\n\nAmong the litter on the surface of the desk\u2014Fairport, though not as bad as Lydia, was an untidy housekeeper\u2014Asher recognized the folded copy of last Friday's _Times_. Beside it lay a yellow envelope containing two train tickets.\n\nParis to Constantinople, by way of Vienna.\n\n_Constantinople?_\n\nA thought came to him. _Isn't it enough that you've betrayed me, lied to me?_\n\nCrouching on the floor beside the desk, he removed the handset from the telephone and cranked the Vienna exchange.\n\n\"Here Vienna Central Telephone Exchange,\" came the operator's cheerful voice. \"A very good evening to you, honored sir.\"\n\n\"And a very good evening to you, honored madame,\" replied Asher, who knew that it never did anyone any good to try to hurry a Viennese telephone operator. \"Would you be so kind as to connect me with Donizetti's caf\u00e9 in the Herrengasse, and ask them to let me speak with the Herr Ober, please?\"\n\nThe floor vibrated with a door closing somewhere. Feet passed quickly along a downstairs hall. Seconds fell on him like shovelfuls of earth filling a grave.\n\n\"Certainly, honored sir, it would give me great pleasure.\"\n\nHe heard her voice, distantly engaged in formal greetings and elaborate social chat with someone at Donizetti's, asking at length for the honored Herr Ober, there is a most honored Herr who wishes to speak with him if his duties will allow him time, and, more closely, voices calling from the courtyard outside the windows. \"...found nothing... someone there...\" Minutes, he thought, and they would begin to search the house.\n\n\"Ladislas Levkowitz at your service, honored sir.\"\n\n\"Herr Ober Levkowitz, I realize it's a tremendous imposition on such a busy man as yourself, but would the British Herr Halliwell have arrived for dinner yet? Could you be so kind as to let him know that Herr Asher wishes to speak with him on a matter of some urgency? Many thanks...\"\n\nAsher cradled the handset against his face, rose to his knees and with a swift glance at the window made a quick review of the rest of the desk. Three or four green-covered notebooks contained interviews with octogenarians in the Vienna region, and others much farther afield. A thick bundle of invoices for glassware and chemicals connected with experiments on the blood of these ancients proved, at a glance, that Fairport's expenses were far greater than the sanitarium's profits could possibly cover. In the back of a drawer was a thick wad of torn-open envelopes of the stationery of the Austrian Embassy in Constantinople, each containing a dated slip with amounts written on them\u2014large amounts\u2014and signed \"Karolyi.\" The dates went back two years. There were half a dozen keys, none of which would fit a cylinder lock of the type on the silver lattice's door. A crowbar, he thought. There'd be one in the generator crypt if he could get to it.\n\n_Dammit,_ he thought, _stop chatting with the Herr Ober and come to the telephone..._\n\n\"Set the receiver back in the cradle, Asher.\"\n\nHe turned his head. Fairport stood in the office doorway, a pistol in his gray-gloved hand.\n\n# NINE\n\nASHER DIDN'T MOVE.\n\n\"I'll use it,\" Fairport warned. He came slowly into the room, circling wide to stay out of Asher's reach and keeping the pistol pointed, until he was close enough to the desk to stretch out his free hand and push down the cradle, breaking the connection.\n\nAsher settled from kneeling to crouching again beside the desk, his legs gathered under him, the handset still dangling from his grip. \"Even against one of your own countrymen?\" It was the cant of the Great Game\u2014honor on the playing fields of Eton and God Save the King. But the Game was one Fairport had been playing for years as well, and there was a chance he would still think in its terms. And Asher was curious about the terms in which he _did_ think.\n\n\"This matter goes beyond country, Asher,\" said Fairport softly. He backed a little, out of immediate arm's reach. \"It's all you can think of, isn't it? All that sleek brute Ignace can think of. Like savages, both of you, tearing up volumes of Plato to stuff into cracks in the roof to keep the rain out. What we have found is the greatest revelation, the greatest discovery, in the history of mankind, and all he can think of is how such a man can be used in Macedonia and against the Russians in Bulgaria\u2014and all you can think of is how to kill such a man, that the balance won't tip against you in the 'Great Game.' You don't understand. You refuse to understand.\"\n\n\"I understand how much damage a man like that can do, if he allies himself with any government. And I understand the kind of fee a government would pay such a man.\"\n\nFairport looked completely blank. Then, when Asher raised his brows, the old man flushed an unhealthy, blotchy pink. \"Oh. Oh, that. I'm sure it's a condition that can be rectified with proper medical investigation... I've found astonishing virtue in yogurt as a food of longevity, and in Chinese ginseng. They won't always be drinkers of human blood...\"\n\n\"I'm sure that lacemaker Ernchester killed last night would be glad to hear it,\" Asher replied grimly, though some objective corner of his mind had to fight not to laugh at the image of Lionel Grippen, Master Vampire of London, supping on a dish of yogurt and ginseng tea. \"And don't you think there might be vampires who're as fond of the taste of human death as they are of human blood?\"\n\nThe old man's mouth flinched. \"That's the most revolting thing I've ever heard! They can't possibly be... No one in his senses could be. They'll welcome that liberation as much as any drunkard would welcome the liberation from drink. And in the meantime there are the physically and socially unfit\u2014\"\n\n\"You mean traitors?\" No other sounds in the house, though there was a dim clashing of shrubbery as someone passed by under the window. If he could disarm him without a shot being fired, there might still be time.\n\nFairport drew himself up. \"I am not a traitor,\" he said with dignity.\n\nAsher sighed in genuine disgust. \"I never met a double agent who was.\"\n\n\"I have _never_ passed information along to Baron Karolyi which would hurt any of our contacts or our agents...\"\n\n\"How would you know?\" Asher demanded tiredly. \"You know nothing about politics, you barely read a newspaper, or at least you didn't when I was here. You don't think, if he can make a deal with vampires\u2014if he can blackmail Ernchester into creating other vampires, fledglings loyal to the Austrian government\u2014they won't eventually be used against us here? Or back at home?\"\n\n\"That won't happen!\" Fairport cried. \"I won't let it happen! Asher, Karolyi is only a means to an end. These petty politics, a handful of military secrets that are going to be useless in three years, they're a small price to pay for the knowledge, the learning, that will free man, finally, from the grip of age, and debility, and death!\n\n\"Asher, look at me!\" He gestured like a frustrated child with his miniature fist. \"Look at me! I've been an old man since I was thirty-five! _Sans teeth, sans eyes, sans taste_...\" He shook his head. \"And every day for the past twenty years I have dealt with men who, like me, have felt that cold, awful terror of knowing their bodies are failing them. Men stumbling as they try to outrace the Pale Horse. I've tried everything, traveled to the far corners of the world, seeking out those who have conquered age\u2014trying to find what it is that makes the body fail, that cripples us, blinds us, deafens us, renders us white-haired and flatulent and impotent and brittle.\"\n\nBehind his thick lenses the blue eyes glittered suddenly, and genuine venom seeped into his voice. \"What it is that wears out some while others continue to gorge and rut and dance into their eighties, their\u2014\"\n\nAsher struck, thrusting off his long legs like lightning, smashing aside Fairport's gun hand at the same moment he drove a fist into the little man's chin. He struck with all he had, to carry him across the distance between them quicker than Fairport could react and shoot, and the impact hurled the professor back and to the floor, as if Asher had struck a child. There wasn't time to think or regret\u2014in another moment Karolyi or one of the footmen might enter, and at that point Asher knew he would die. Karolyi, unlike Fairport, was not a man to justify or explain.\n\nHe scooped up the gun, transferred Fairport's key ring from the old man's coat pocket to his own, pulled free the old man's four-in-hand and used it to bind his wrists behind him, then stuffed Fairport's handkerchief into his mouth for a gag. He took another moment to drag him behind the desk, keeping low still, out of the range of the windows... Really, he thought, half regretful, the man had always been out of his league...\n\nAnd smelled smoke.\n\nGray smoke was rolling along the ceiling of the upstairs hall.\n\nAsher cursed. He would almost certainly be caught if he tried to get Fairport out of there, but there was nothing for it, and the man's halfhearted interference back at the pension in Vienna had almost certainly saved his life. He glanced out the long windows behind the desk, ascertained that there was no one visible in the gardens below, and kicked them open, dragging the little man out onto the balcony where the fresh air would revive him and he'd be able to hump himself down the outside stairs. Then he ducked back inside. Crimson reflections on the bare boughs showed him where two or three of the downstairs rooms were already in flames, and, even as he watched, he saw yellow light flare in the dark windows of the old stable building.\n\n_Arson,_ thought Asher in alarm. _Two places at once. Who the hell...?_\n\nHe flung himself down the stairs, Fairport's gun in hand, the smoke already tearing his eyes and eating at his lungs. Under the stucco the old house was mostly wood and would go fast. Downstairs the smoke was worse, the heat pounding on Asher's face and making him dizzy as he raced along the corridor to the scullery. As he ran he thought, _If this is Karolyi's work, why let Fairport stay free? Or has Anthea somehow started this?_\n\nThe coachman's body lay in the scullery door. His eyes and mouth were both wide in a look of utter shock. His collar had been torn open, his shirt pulled back to reveal the hairy masses of neck and chest. Wounds bulged like tattered white mouths from ear to collarbone, but there was almost no blood.\n\nAsher felt as if his heart shrank and turned to ice in his chest.\n\nHe crossed the scullery, looked swiftly out the rear door to the yard and saw what looked like another body in the shadows under the outside stair. Smoke seared his nostrils, weighted his rib cage. He couldn't tell if there was a smell of blood or not.\n\n_Not Anthea. And not Ernchester._\n\n_The others. The vampires of Vienna._\n\nThe ones who had followed him here.\n\nSweat was rolling down his face as he shoved back the shelving, ran down the stair into the cellar's cool abyss. He struck a match as he thrust through the door at the bottom; Ernchester, pacing the silver cage like an animal, wheeled, his eyes flashing in the tiny speck of the flame. \"They're here,\" he said hoarsely. \"I feel them. The house\u2014they've fired the house...\"\n\nHe flicked through the barred silver door the moment Asher had it open, twisting his body so as not to touch.\n\n\"Anthea!\"\n\nHe started for the door, then turned back, catching Asher by the elbow in a grip that came close to breaking the bone. \"Did you find her? She isn't in this house, I'd have known, I'd have felt her, read her dreams...\"\n\nAsher recalled something Ysidro had said to him once, about being unable to sense the presence of people deep in cellars through the muffling weight of the earth.\n\n\"She'll be in the crypt under the stable.\"\n\nFlame light poured down the stairs, bloody on the earl's face; a thin face and not particularly an aristocratic one, with an indefinable air of age despite the fact that, like Anthea, he appeared to be no more than thirty-five. Asher did notice, as they raced up the stairs into the choking inferno of the scullery, that at no time did sweat break from the smooth skin of the vampire's brow.\n\nAsher crossed the yard at a run, but the vampire earl was ahead of him, moving with an insectile, weightless speed, huge bounds like a gazelle. Ernchester stopped, however, in front of the burning stable, hands raised before his face and his blue-gray eyes sick with horror and shock.\n\nThe earl followed him without question, however, circling the building to the rear, where the flames were less. Asher drove his boot through a cellar window, dropping into what had been a boiler room. The place smelled of dirt and damp brick, and the thin, sickly odor of kerosene that lifted the hair on Asher's neck. He dug another match from his pocket, scratched it on the wall behind him. There were barrels of the stuff, ranged along the wall beyond the hunched black monstrosity of the generator itself. He heard the earl whisper, \"God's death!\" behind him, and pointed toward what looked to be the door of a closet, nearly invisible in the shadows by the coal bin.\n\n\"Through there. We have a few minutes. The fire's just caught.\"\n\nThe door was locked. Ernchester ripped the entire mechanism\u2014lock plate, handle, bolt\u2014free of the wood without visible effort and threw it clanging to the brick floor, then vanished like a moth in the darkness.\n\nAsher had been in the crypt many times. Like the subcellar beneath the scullery, Fairport used it to conceal people who weren't supposed to be in Vienna or who had to leave the town in a hurry. Because of its remoteness from the main house\u2014and the patients who usually resided there\u2014it had also been used for meetings, if instructions had to be passed along with minimum risk of being seen.\n\nHe'd felt his way halfway down the boxed-in stairway when yellow light glowed at the bottom. Through the doorway he saw Ernchester setting on the table a newly lighted oil lamp and turning back to the coffin trunk that filled half of the room.\n\n\"She's in here,\" the earl said softly and knelt beside the trunk. He passed his hands along the lid, pressed his cheek to the leather. His eyes closed. The flesh around them rumpled and compressed, like an old man's. Then he moved his head and looked up over his shoulder at Asher, standing in the doorway. \"Can you take an end?\"\n\nIt was awkward, getting the trunk around the corners of the stair. Even in the few minutes they had been in the crypt, the air in the boiler room had heated, and the smoke there was growing thick. Like the house, the stable was wood, the roof and walls went up like tinder. When they dragged and manhandled the trunk upstairs, they found the ground floor suffocatingly hot, filled with blinding smoke under a vicious rain of cinder and sparks. Asher coughed, gasping for breath, his grip on the trunk slipping. As his knees gave under him, he wondered suddenly what chemicals Fairport had in the laboratories here and what fumes they might be adding to the miasma of smoke.\n\nHe tried to get to his feet, and fell.\n\nAbove the roaring of the fire overhead he heard the scratch of the trunk's brass-bound corners as Ernchester\u2014unbreathing, undead, desperate to save his wife at all costs\u2014dragged it toward the door and safety.\n\nBlack unconsciousness rolled over Asher like a wave. He tried to stand, then realized that the air was a little cooler down near the floor. Inhaling was like trying to breathe kerosene. _Kerosene,_ he thought dizzily. _When the roof goes, it'll take the floor with it, and the whole place will turn into a furnace..._ The thought that he'd probably be killed by the falling roof before the kerosene scattered the building over half an acre of the Vienna Woods was not much of a comfort. At one point he thought he was crawling, but a moment later realized he was lying with his cheek to the superheating linoleum of the floor, a fallen cinder burning the back of his left hand.\n\nHands as cold and strong as machinery took hold of his arms, lifting and dragging him as if he were a bale of sticks. The smell of smoke seemed stronger outside, perhaps because his lungs were working again. He stumbled, trying to get his feet under him, and clutched at the shoulders that supported his arm.\n\nHe felt them flinch.\n\nSilver, he thought. The chain on his wrist would sting through Ernchester's coat.\n\nThe trunk lay just within the compound gate. It was still shut. Ernchester must have turned back the moment he'd dragged it out of range of the fire.\n\n\"She's asleep.\"\n\nAsher raised his head, his brown hair hanging in his eyes, his face burning in the cold air under a film of sweat, soot, and grime. Ernchester knelt beside the trunk, one arm resting along its lid, the reflection of the flames imparting gory color to his narrow face, glittering in his close-cropped fair hair, his haunted, weary eyes.\n\n\"Drugged, I think,\" Ernchester went on softly. \"That is... as well. Thank you.\"\n\nAsher looked back across the gardens. The front part of the main house was in flames. The rear wing, where Fairport's office and his own rooms had been, was still intact. By the flaring light two bodies were clearly visible on the gravel paths.\n\nHe fumbled in his pocket for Fairport's keys, found two that would open the trunk's heavy latches. Ernchester touched his hand lightly as he would have opened the lid. \"Not yet. The air will revive her, and I don't think I could stand that. I won't do that to her.\" The earl straightened his back, though he remained kneeling, one hand atop the other on the lid of the trunk. \"Take her away from here. Go with her back to England. Take her out of this place. I beg you.\" He closed his eyes. \"I beg you.\"\n\nFirelight picked out the sudden lines around his eyes, the set of the thin lips\u2014a face no one would notice, thought Asher, except that it was not a nineteenth-century face, much less one that belonged to this newborn era. The muscles, the speech, the expressions that had formed the mouth and chin and the set of the cheeks were all from some earlier time, and the years had not changed them.\n\n\"I can't repay you,\" he added softly. \"I won't be seeing you, nor anyone known to you, ever again. I will owe you this favor, this boon, for all of time. But please make sure she gets home all right. Tell her\u2014\" His voice did not break but halted for a moment, almost as if he sought words. \"Tell her that she is all that I ever wanted, and all that I ever had.\"\n\nThen he raised first the outer lid, then the inner, to reveal the woman sleeping within.\n\nThe living dead, they had been called. By the fevered glare of the firelight she looked, indeed, both alive and dead: waxen, still, unbreathing, with her dark hair scattered about her, the linen of her gown not whiter than the flesh it covered. And beautiful, thought Asher. Beautiful beyond words.\n\nLooking up, he saw Ernchester's face, without expression, as though all expression had grown too much to be supported under the weight of endless years, save for his eyes.\n\nErnchester bent a little to touch his wife's cheek, then leaned down to kiss her lips. To Asher he said, \"She'll wake soon. Tell her that I love her. Always.\"\n\nYellow light flared higher as flames ran along the roof of the main house. Asher turned, startled, in time to see a spindly figure move on the balcony, work and thrust itself to its feet, wobbling and off balance. Disheveled white hair caught the light, and the lenses of his spectacles made great rounds of burning amber as he turned his head. Staggering, Fairport began to descend the stairs.\n\nAsher shouldn't have been able to hear it under the roaring of the fire, but he did. Thin, silvery laughter, like the breaking of wafer-frail glass, and beneath that, the obscene toad-croak of a bass chuckle. They seemed to hover on the balcony, and on the stair, not quite touched by the fire's light, as if visibility were something to be put on or off at will, but at one point Asher thought that one of them wore a dress the color of web and moonlight.\n\nFairport cried under the gag and fell, rolling down the stairs. They floated after him, half-seen migraine visions of alabaster faces, shining hands, eyes that caught the light as had those of the rats among the bones of St. Roche. At the foot of the steps he tried to get to his feet, falling heavily and trying again, and they ringed him, like porpoises playing, flickering shadows of a force he had entirely underestimated, following him as he scrabbled and heaved along the ground.\n\nThey let him get quite some distance before they began to feed.\n\nWith a roar, the roof of the stables fell in, curtains of flame leaping higher, yellower, beating upon yet somehow failing to completely illuminate what was happening in the court. Then a deeper roar, like a battery of eight-inch guns, and the earth jarred underfoot as the kerosene went up. Beside Asher, Anthea cried out, \"Charles!\" and sat up suddenly, her brown eyes wide with terror.\n\nAsher caught her hand. Her gaze met his, clouded with old dreams. \"The stones. The stones exploded with the heat.\" Then she flinched and turned her face away, and Asher realized that for a moment she had thought she was still in London, many years before, when the whole of that city burned.\n\nShe said again, \"Charles,\" and when she looked at him then, her eyes were clear.\n\n\"He's gone.\"\n\nShe started to rise, and he closed his hand hard on hers, drawing her back and knowing he had no way to hold her if she simply wrenched herself free. She could have broken his wrist, or his neck, with very little effort. She looked at him again, questioning and pleading, her black curls a cloud around her face and shoulders, the flame a soaked gold in her eyes.\n\n\"He told me to take you back to England,\" Asher said. \"To see that you reached there safely. He said that he would not see me\u2014and, I presume, you\u2014again. He said that he loves you, always and forever.\"\n\nIn the courtyard the vampires had sunk down in a ring around Fairport, whose frantic noises had risen to a muffled crescendo, then ceased. Asher wondered what he'd do if Anthea vanished, as Ernchester had, flickering away like a ghost in the woods to seek him. He'd never make it back to Vienna.\n\nFor a moment he thought she would. Then she, too, glanced across at the dark shapes in the firelight. Just for a moment her pale tongue slipped out and brushed her lips.\n\nBut when she turned to him, her eyes were a woman's eyes. \"Do you know where he's gone?\"\n\nAsher stroked a corner of his mustache. \"I don't know,\" he said, \"but I can guess. And my guess is: Constantinople.\"\n\n# TEN\n\n\"THURSDAY.\" Lydia stared blankly at the newspaper by the glare of the station lights. \"Thursday night. We were still in Paris.\"\n\nMargaret whispered, \"Oh, my God,\" through hands pressed to her mouth.\n\n\"I thought... I thought I'd have a little more time to catch up with him. That things wouldn't happen so quickly.\"\n\nYsidro reappeared at their side, trailed by a laconic individual in a Slovak's baggy white britches who, at his command, loaded Ysidro's trunk and portmanteau, Margaret's satchel, and Lydia's voluminous possessions onto a trolley that he pushed away in the direction of the doors. The vampire tweaked the newspaper from Lydia's hands, and read.\n\n> DOCTOR PERISHES IN SANITARIUM FIRE\n> \n> Early yesterday evening the well-known sanitarium \"Fruhlingzeit\" burned to the ground in a conflagration of epic proportions, claiming the life of the man who had made it his life-work and monument. The body of the most distinguished English specialist in rejuvenatory medicine, Dr. Bedford Fairport, whose work has contributed to the comfort and healing of hundreds of men and women in Vienna over the past eighteen years, was found in the smoking ruins by police constables and firefighters in the early hours of Friday morning. According to the Vienna police, foul play is suspected. The bodies of a coachman and a laborer were also found.\n> \n> No patients were present at the sanitarium when it burned, Dr. Fairport having temporarily closed the premises last week. The distinguished Herr Hofrat Theobald Beidenstunde, of the Imperial-and-Royal Austrian Coal Board, undergoing treatment for a nervous condition at Fruhlingzeit last week, states that Herr Professor Doktor Fairport requested that all patients return to their homes due to repairs on the foundations of the main building. Complete financial recompense was made to all patients so affected.\n> \n> It is believed that the fire started in the laboratory where a generator was positioned too close to stores of kerosene, and later spread to the main villa. However, since all three bodies bore marks of violence, arson is being considered as a possibility. Further investigation by the Vienna police is under way.\n\n\"Behold an Englishman,\" murmured Ysidro. \"The good Hofrat Beidenstunde should thank his stars he was reimbursed. The old Queen would never have approved such request for funds.\" He folded the newspaper and bestowed it in the pocket of his cloak.\n\n\"Victoria?\" Margaret Potton asked in surprise.\n\n\"Elizabeth. There is nothing there which proves your husband's fate, mistress. This way.\"\n\nThe Slovak was waiting for them in the square outside, on the seat of a gaily painted wagon. Ysidro helped the two women in\u2014lifting Lydia with unnerving ease from the pavement\u2014and without wasted words they proceeded into the winding network of high-walled ways that made up the most ancient part of the Altstadt.\n\n\"Who\u2014besides Fairport\u2014would Jamie seek out in Vienna?\"\n\n\"Three years ago it was a man named Halliwell.\" Ysidro turned his head, as if listening for some sound below or between the myriad voices and threads of stray music that clamored all around them on the bustling streets. \"I have no more recent knowledge than that, nor am I sure where the Department has its headquarters these days. The embassy would be the place to inquire. Say that you seek your husband, that you wish to speak with Halliwell.\"\n\n\"They won't be there on a Sunday,\" Margaret pointed out worriedly.\n\n\"At least we can rent a carriage and go out to the ruins of the sanitarium.\" Lydia brought the newspaper up close enough to her nose to make out something other than vague blocks of gray. \"It may not say anything about Jamie, but considering it was Fairport I came to warn him against, the coincidence is a little marked. I expect we could find the address in a city directory.\"\n\n\"I expect every jehu in the town will know its location,\" Ysidro remarked. \"From what I know of human nature, the place will have been trampled by curiosity seekers ere the ashes cooled.\"\n\nPalaces crowded them on all sides, the darkness patched and painted by a thousand glowing windows whose reflections gilded the scrollwork of doorways with careless brush strokes of light, the faces of the marble angels rendered curiously kin to Ysidro's still, thin features as the vampire turned his head again, seeking whatever it was that he sought.\n\nThe wagon drew up before a tall yellow house in the Bakkersgasse, like an excessively garlanded wedding cake in butter-colored stucco. Ysidro accompanied the two women inside, watching as the Slovak unloaded Lydia's trunks, portmanteau, satchel, and hatboxes, but when that was finished, he returned to his own luggage, still on the cart, and drove away with it into the darkness. An hour later he returned, afoot and uncommunicative as ever, for picquet in a salon that was a miniature Versailles above a shop selling silk.\n\n\"I made arrangements ere departing London,\" he said, shuffling the cards. \"It is necessary to know the existence of such places, which can be had in any city for a price. You will find a cook and chambermaid at your disposal in the morning, though they speak no English and little German. Still, I am assured that the cook is up to the most exacting of standards. Certainly, for English, she will suffice.\"\n\nMargaret said, \"It's too good of you...\"\n\n\"Assured by whom?\" Lydia wanted to know.\n\nYsidro picked up his cards. \"One whose business it is to know. You are the elder hand, mistress.\"\n\nYsidro's estimate of human nature proved a distressingly accurate one. When Lydia and Miss Potton arrived by rented fiacre at the smoke-stained wall around what was left of Fruhlingzeit Sanitarium the following afternoon, they found at least five other carriages there, the drivers seated comfortably on the low stone wall across the road chatting among themselves, and a large number of fashionably dressed men and women prowling around the trampled weeds or engaged in argument with a couple of sturdy gentlemen who seemed to be guarding the gates.\n\n\"I do not see that you have the authority to turn us away,\" a slim man in an overemphatic waistcoat was saying as Lydia hesitantly crossed the road. \"I do not see this at all.\"\n\n\"Can't do anything about that, sir.\" The sturdy gentleman pushed back his flat cloth cap and remained blocking the entry. Even through the comforting blur of myopia, the glimpse of blackened rafters and fallen-in walls was horrible, and the smell of cold ash lay thin and gritty on the chill air.\n\n\"I shall write to the _Neue Freie Presse_ about this.\"\n\n\"You do that, sir.\"\n\nLydia stepped forward hesitantly as the slim man stormed away to rejoin his party by the carriages; the sturdy gentleman fixed her with a jaundiced eye and said, in not-very-good German, \"Nobody allowed in, ma'am.\"\n\n\"Is... is a Mr. Halliwell here?\" asked Lydia. If Dr. Fairport were officially an agent of Britain, it stood to reason the burning of his sanitarium would not go uninvestigated by the Department. It only surprised her they'd still be at it three days later. She saw the man's stance shift at the sound of the name and said, \"Could you tell him a Mrs. Asher is here to see him? Mrs. James Asher.\"\n\nWithout her spectacles, Mr. Halliwell proved to be a magpie behemoth, a series of circles of blacks, whites, pinks, and gleaming reflections that resolved itself at four feet into a heavy, pugnacious face and brightly humorous green eyes behind small oval lenses. A big damp hand gripped Lydia's while a second patted it moistly; the little clusters of would-be sightseers across the road glowered at this favoritism.\n\n\"My dear Mrs. Asher!\"\n\n\"My friend, Miss Potton.\"\n\nHalliwell bowed again, an awesome sight.\n\n\"Strange business. Deuced strange business. Your husband didn't send for you, did he?\" He glanced down sidelong at her from his height, but she noticed his voice was barely above a whisper.\n\nShe shook her head. \"But the telegram he sent me on his way here gave me reason to believe that he might be in trouble. He... he wasn't here when this happened... was he?\"\n\nThe green eyes narrowed. \"Why would you think he was?\"\n\n\"Because...\" Lydia took a deep breath. In broad daylight and in front of half a dozen argumentative Viennese, she thought, they couldn't very well drag her away in a closed carriage. She said, very softly, \"Because he said he was coming to Dr. Fairport. And because I have reason to believe Dr. Fairport was in the pay of the Austrians.\"\n\nHis glance flicked across the road, then to Miss Potton\u2014discreetly out of earshot\u2014and back. \"You don't happen,\" he said equally quiet, \"to have mentioned this to anyone else?\"\n\n\"No. Not even to Miss Potton,\" she remembered to add. mindful of her companion's safety. \"But I think it's true. I take it,\" she went on slowly, \"that you haven't spoken with Dr. Asher on the subject.\"\n\nHalliwell fingered his short-clipped beard, studying her as if matching the eggplant taffeta of her gown, the mint and ecru frills of her hat, against other things. Lydia wondered how James could possibly have played at spies for as long as he had: This business of not knowing what to say or whom to say it to was both wearing and unnerving. Presumably, Ysidro would come to her rescue if Halliwell were a double agent also, provided Margaret had the wits to run for it...\n\nBut if Margaret had been foolish enough to believe Ysidro's farrago about previous lifetimes, goodness knew what she'd do in a crisis.\n\n\"I'm inclined to agree with you,\" the fat man said abruptly. \"And I was starting to think so before you turned up. Just the fact that the Kundschafts Stelle hasn't let us into this place until this morning tells me there's something fishy, though of course we can't come out and say the man was working for _us_.\"\n\nHe glanced again at the loitering tourists across the road. \"Would you ladies be so good as to meet me for dinner at Donizetti's on the Herrengasse this evening at eight? We'll be able to talk there.\" He nodded back toward the burnt-out shell of the house, where another man could be seen slowly picking his way through the mess of collapsed beams and bricks. \"I can tell you now no one's found any trace of your husband... and what we _have_ found is not anything a lady should see.\"\n\n\"God knows what the Kundschafts Stelle found before they let us in.\" Halliwell's small, rather womanish mouth pursed as he removed his gloves. In the saffron-drenched Renoir of color that was Donizetti's without spectacles, he seemed to fit in, becoming curiously invisible in a way that he hadn't in the unfamiliar environment of open air and bare woods. He reminded Lydia rather of some of her uncles, who grew like fleshy pale pot plants in their London clubs and never emerged into the light of day.\n\n\"I'll tell you the truth, Mrs. Asher\u2014if your husband were at Fruhlingzeit when it burned, nobody's said anything about it to us. They've had the place closed off for two days. It was twenty-four hours before they even let the police in. Typical. When the Emperor's son blew his brains out twenty years ago, taking a seventeen-year-old girl with him for reasons best known to himself, the original story was that he'd died of 'heart failure.' Government agents and the girl's own uncle propped her corpse into a carriage with a broom handle up her back to keep reporters from learning two bodies instead of one were found at the scene. How did your husband know this Farren fellow, and how did you find out about Fairport?\"\n\nAt this point the table captain appeared again, waiter and boy in tow, and a long and Byzantine discussion ensued concerning the concoction of _Tafelspitz_ and how the _canard Strasbourg_ was prepared this evening, and the relative tartness of the sour cherry soup. Rather to Lydia's surprise, Margaret, who had all day been her tongue-tied self, plunged into the conversation with the absorbed interest of a fellow gourmet, winning the approval of both Halliwell and the table captain\u2014the Herr Ober, Halliwell called him\u2014with her opinions on capers and _beurre br\u00fbl\u00e9_. It was, Lydia reflected, an entirely new side to her traveling companion than she had so far seen.\n\nOnly when the little train of servitors was gone did Halliwell turn back to her. Lydia, after a moment's pause to collect her thoughts, sketched a bowdlerized version of the telegrams she had received, the articles they had prompted her to read, her realization that Fairport would certainly be interested in Ernchester's pathology and almost as certainly would be working for, or with, Karolyi. \"I don't know what, or how much, of Farren's abilities are connected with his belief that he is a vampire,\" she concluded carefully. \"But I know Dr. Asher considered him a very dangerous man, dangerous enough to warrant his dropping everything to pursue him to Paris to keep him from selling his services to the Emperor.\"\n\n\"Hmm. For which he got small thanks from old Streatham, I daresay. How did you know to come to me? Asher didn't know my name until he arrived.\"\n\n\"A friend of my husband's,\" Lydia said, not sure whether she was telling the truth or not.\n\n\"Your husband had dinner with me in this caf\u00e9 Tuesday night,\" said Halliwell. \"There'd been trouble in Paris, one of our operatives was killed. Your husband seemed to think this Farren had done it, but word got to the police that your husband had something to do with it, even before the French police sent for him. Karolyi's work, of course. Asher spent the night in jail, which isn't as uncomfortable as it would be in London, and was going to stay the night at the sanitarium after he'd had a look around the Altstadt Wednesday. That was usual\u2014the place was a safe house. Your husband had stayed there before.\"\n\n\"And did he?\" She picked a little at the delicate crepe on the plate before her, her appetite gone.\n\n\"I gather he didn't. Fairport showed up at the firm in the morning asking if Asher had been heard from.\"\n\n\"That might have been a blind.\"\n\n\"I don't think so.\" Halliwell dabbed his mouth with the delicacy of a maiden lady. \"He sniffed around for information, which I don't think he'd have done if he'd had him under hatches. He wasn't that clever. Later in the afternoon he came back saying Asher was wanted by the police, which I knew already, and why didn't I go talk to them? He hung about and wasted my time and asked a thousand questions and went with me to the station, which is just what he'd have done if he were a double and waiting for Asher to try telephoning, though that may be hindsight on my part. If I were Karolyi, I would have shot him for it. Personally, I never thought old Bedbugs had enough red blood in him to work a double game. At about seven that night Ladislas\u2014the Herr Ober\u2014came to my table and told me a Herr Asher was on the telephone for me, that it was urgent. By the time I got there, the line was dead. About two hours later we got the first reports of the fire.\"\n\n\"Oh,\" Lydia said slowly. \"I see.\"\n\n\"Do you?\" The green eyes glinted sharply at her. \"I don't. None of us do. You're thinking Asher might have started the fire...\"\n\n\"Well,\" Lydia pointed out. \"my husband always said that one _should_ burn the place down after killing someone...\"\n\nShe regarded Halliwell with startlement when the fat man burst into delighted laughter. \"It's true,\" she protested. \"It isn't as if there were other houses around to be damaged.\"\n\n\"My dear Mrs. Asher,\" he chuckled, \"I can see why old James married you.\"\n\n\"Well,\" she said, \"it wasn't for my domestic talents. But I don't think, if James had started the fire, anyone would have found enough of two bodies to identify them. He's usually much more efficient than that.\"\n\n\"No.\" Halliwell's round face grew suddenly grim. \"And I can't picture your husband killing them the way these men were killed.\"\n\nHe glanced apologetically across at Margaret\u2014digging her way happily through a towering castle of chocolate and whipped cream\u2014and lowered his voice. \"According to our sources in the Kundschafts Stelle, they were... horribly wounded. Bled almost completely dry of blood. They must have been cut in the house itself and later dragged into the open. I can't imagine your husband, or any sane man, doing that.\"\n\nLydia was conscious of Margaret putting down her fork, her hand suddenly shaking.\n\nHalliwell went on, \"And there were more than three bodies found. There were at least five, two of them so badly burned they couldn't be identified; and they haven't even finished digging out the building where the kerosene blew up. Bedbugs had a room underneath it, which we used for a hiding place for whoever was inconveniently connected with the local socialists or anarchists or Serbian nationals. If Asher was a prisoner, he'd have been held down there.\"\n\nLydia looked again at her untouched dessert. She felt cold inside. She'd been a fool, she thought, not to guess that the newspaper would lie. She'd been a fool to think she could overtake him in time to prevent disaster. She said again, \"I see.\"\n\n\"We found plenty of evidence of the kind of man Farren is, if he could take out five men like that, as well as evidence of what he thought he was. Fairport had fitted up a safe room with silver bars\u2014vampires are supposed to hate silver, aren't they? But we haven't found any trace of your husband.\"\n\nShe took a deep breath. \"And Farren?\"\n\nHalliwell shook his head. \"No sign of him, either. Our connections in the Kundschafts Stelle tell us they were watching the Bahnhof for your husband all evening\u2014the police really _were_ looking for him that day\u2014so it's doubtful that he left town that way.\"\n\nHe reached out and clumsily patted her arm. \"That doesn't mean he's come to harm,\" he said. Lydia looked quickly up at him.\n\n\"For all I know, they're still looking for him. God knows what Karolyi told them about him. I've asked, and they're being damned cagey. And he could have left town on the Danube ferries or taken a tram and walked to another station. Anything. It may be he's simply hiding out.\"\n\n\"Maybe.\" Lydia remembered one of James' digressions on how easy it was to get out of a town that had become temporarily too hot.\n\nThen she thought about the burned skeleton of the sanitarium and the stink of charred wood still hanging in the chilly air, and her heart sank, as if with sickness or shock.\n\n\"In the meantime, you can do me a favor, if you would, Mrs. Asher. Your husband said you were a medical doctor?\"\n\nShe nodded. \"I have a medical degree, yes, but I mostly do research on endocrine secretions at the Radcliffe Infirmary. The few women with practices all seem to go into what they call 'women's medicine'\u2014and _still_ have trouble making a living at it, I might add. And I've never been terribly interested in what my aunts referred to as 'the plumbing.' Did you need something looked at?\"\n\nHe mopped up the last of his Sacher torte and gazed regretfully at the polished white porcelain plate. Then he propped his glasses, frowning. \"None of the laboratories survived\u2014they were all directly over the kerosene stores\u2014but we do have Fairport's notebooks from his study. The place was pretty badly charred, but those we managed to recover. He was a British citizen and be damned to who paid the rent on the sanitarium. I suspect the Kundschafts Stelle's going to want to see them eventually, but if you'd be good enough to have a look through them and tell us anything that it might be worthwhile for us to know, I'd appreciate it. I have them here.\"\n\nHe held up a battered leather satchel, overloaded and strapped together with rope where its buckles would not hold.\n\n\"We'd like to know what he was working on. If you still have your list of his articles...\"\n\nLydia nodded. \"Aging,\" she said. \"Blood. Immortality.\"\n\nHalliwell grunted. \"No wonder he fell for Farren.\"\n\n\"Yes,\" Lydia said quietly. \"No wonder.\"\n\nIn light of the articles she had read, Fairport's experiments\u2014with blood, with saliva, with mucus, with the chemistry of the brain and the glands\u2014came into crystalline focus.\n\n_The man who seeks to live forever_ , Ysidro had said.\n\nHe was right, she thought, turning over the cryptic notes while Margaret dozed in a welter of crocheted snowflakes. He was right.\n\nBedford Fairport was quite clearly a man possessed with the fanatic determination to discover whence came the deterioration of age, and an even greater determination to learn how to reverse its effects.\n\nIn the article in which he had mentioned Ignace Karolyi's donation of the sanitarium and funds, Fairport had spoken of his own \"premature aging.\" Lydia had encountered reports of such progeria dating from the sixteenth century, and was of the opinion that some unknown vitamin deficiency or breakdown was responsible. She pushed up her spectacles on her forehead, rubbed her eyes. Of course he would grasp at rumors of immortality.\n\nA glance at the reagents and vitamin solutions told Lydia that his experiments had been appallingly costly. He'd used orangutans as subjects two dozen times in the past few years, and Lydia knew from her own experiments how expensive the animals were. Unnecessary, too, she thought. In most experiments with deficiency syndromes, pigs seemed to work just as well. A double check showed her that he used orangutans to repeat experiments done on pigs, refusing to take what were, to her eye, quite clear failures as anything more than individual variations in data. Toward the end he'd taken to rerunning additional tests on everything, insistently investigating smaller and smaller points, like a man clutching at straws. Even if Fairport had private funds, he'd have to be staggeringly wealthy to continue such work as long as he had.\n\nAnd she knew that if he had family money\u2014if he'd been connected to one of the wealthier families in England\u2014her aunt Lavinia would have steered her toward him at some point in her own Oxford days as a potential reference, partner, or colleague.\n\nHe'd betrayed James. Taken him prisoner. _They haven't even finished digging out the building where the kerosene blew up... If Asher were a prisoner, it would have been down there..._\n\nJames might have gotten out of town, she told herself defiantly. The police were looking for him. He could have taken a tram, as he always said was best, or a ferry.\n\n_Bled almost completely dry of blood..._\n\nTears fought their way to her throat, and grimly she forced them back. _We don't know anything yet. We don't know._\n\n\"An entire notebook of the historical and folkloric.\"\n\nThe soft voice nearly startled her out of her chair. Looking up, she saw Ysidro sitting opposite, a green cloth-bound ledger open before him. Past the vampire's shoulder the mantel clock was visible, and Lydia was mildly surprised to see that it was now close to three in the morning.\n\n\"I hadn't got that far.\" She reached back to twist her heavy braid into a less schoolgirlish knot. The cook\u2014an excellent woman of broad smiles and a completely incomprehensible language\u2014had left Sacher torte, bread and butter, and a succulent bunch of Italian grapes, should either _dziewczyna_ suddenly find herself in peril of starvation before morning light, and the smell of the coffee warming on the little primus stove was heavy in the room. \"And folklore would only be speculative. Even so-called 'historical' personalities\u2014rumors about Ninon de l'Enclos and Cagliostro and Count What's-his-name in Paris...\"\n\n\"Scarcely speculative at the end.\" Ysidro turned the ledger, slid it across the table to her, hands like old ivory in the lamplight.\n\n_Old man who lived to be a thousand,_ related the wandering script. _Brzchek Village. Woman who lived to be five hundred (wove moonlight). Okurka Village. Woman who used moonlight to make herself beautiful forever. Salek Village. Man who made a pact with devil, lived forever. Bily Hora Village. Woman who bathed in blood, lived five hundred years. Brusa, Bily Hora, Salek._\n\nShe looked up, puzzled. \"It sounds like the sort of thing James does\u2014talking to storytellers and grannies and old duffers at country inns.\"\n\n\"I expect Fairport observed the way James went about his questioning and turned it to his own usages.\" He tilted his head, moved the pile of invoices so he could read the top sheet. His pale eyebrows flexed. \"One can, in any case, see the trend of his mind. But orangutans? I have spoken to those who saw James leave this city.\"\n\nHer breath drew sharply; Ysidro watched her in stillness for a moment, his head a little to one side, like a white mantis, and again his eyebrows flexed, though it was impossible to read the expression in his eyes.\n\n\"Walk with me, lady.\" He rose and held out to her his hand. \"The Master of Vienna has given me leave to hunt in this city, if so be that I am circumspect. Should he see us in company, he will know you as a sojourner, and think us chance-met and you harmless prey.\"\n\nLydia glanced back at Margaret's snoring form as Ysidro handed her her coat. Even through the gloves he drew on, and the kid that covered her own hands, his flesh was icy. Automatically, though no one would see her, she removed her spectacles, slipped them in her pocket. The card games had broken her of the habit of hiding her eyeglasses in Ysidro's presence; he had seen her, she reflected, at her four-eyed ugliest and did not appear to mind. Perhaps it was only that he had seen many others worse than she.\n\nHe led her down the gilt and marble staircase and through the bossed bronze of the inconspicuous door to the pavement outside.\n\n\"You saw the Master of Vienna, then?\"\n\n\"Count Batthyany Nikolai Alessandro August\u2014and his wives. He has ruled Vienna, and indeed the greater part of the Danube Valley, since the days when men still fought the Turks on the banks of the river. As well that he and I are both conversant in the old French of the courts, for German I know only from books. It was not, you understand, a language spoken by anyone of breeding in my day; one reason that I made a point of being elsewhere until the Kings of England learned a more civilized tongue.\"\n\nLydia hid her smile. She'd heard him speak German to the Slovak and to the cook. One thing she had learned about Ysidro in the past few days was the depth of his snobbery.\n\nAround them, Vienna slept, a drowned Atlantis at the bottom of a lightless sea. Shutters of wood and glass accordioned over the bright caf\u00e9s, and even the dormers of the servants, high at the tops of the canyon walls, were closed eyes sealed in dreaming.\n\n\"Your husband injured Batthyany's youngest wife,\" Ysidro went on as they walked. \"He did well to leave Vienna. He was seen at the train station boarding the Orient Express for Constantinople...\"\n\n\"Constantinople?\" Lydia said, startled.\n\n\"Even so. A most curious choice.\"\n\n\"But who... who saw him? If it was one of this Batthyany's vampires...\"\n\n\"Another wife,\" Ysidro said smoothly. \"Who perhaps had reasons of her own for wishing ill to the fair German beauty who had\u2014until James evidently burned her face with a handful of silver\u2014been the count's fancy. The German beauty\u2014Grete, her name is\u2014slew at least two of the groundsmen at Fruhlingzeit in the hopes that their blood would speed the healing of her wound, but it will be some time before she is anything but hideous. Indeed, for some time to come Batthyany's coterie must hunt with the greatest of care, for fear of attracting notice by the police\u2014another reason it is as well that your husband left Vienna when he did. Count Batthyany spoke of revenge, but his eldest wife\u2014Hungarian, as he is\u2014seemed pleased.\"\n\nThey turned a corner, coming clear of the tall walls to a cobbled expanse where the cathedral rose suddenly before them, like a black and white fish skeleton in the wintry moonlight. Mist lay thin about its feet, stirring with their stride; the air stung the inside of her nose when she breathed.\n\n\"Was it the vampires who killed Professor Fairport, then?\"\n\n\"Of course.\" Ysidro's head turned at some small sound across the pavement. A young girl emerged from the cathedral's porch and hastened across the square to the concealing dark of the lanes beyond, drawing her shawl over her head as she went. The Spaniard watched her, speculatively, out of sight.\n\n\"Batthyany was enraged, you understand, at any other's fledgling entering his domain,\" he said, turning back to Lydia. \"And doubly, that any would ally himself with mortal governments, and so bring such governments into knowledge of the vampires. He considered the burning of Fruhlingzeit\u2014and the death of the men involved\u2014sufficient warning. His intent was that Ernchester die too in the conflagration, but says that the earl has departed also from Vienna. According to his eldest wife, your husband was accompanied on the train by a female vampire whom they found upon the premises, who claimed that she had been kidnapped and held prisoner by Fairport. Indeed, Batthyany and his countess helped this woman take horses from the stable and load into the wagon her traveling coffin, by the light of the burning house. With horse and wagon she would have easily returned to Vienna in time to be on the train.\"\n\n\"Anthea?\"\n\n\"It would seem. And my guess is that your husband lay alive in that coffin. He could not have escaped, else.\"\n\nLydia kept her face from showing the inner shudder she felt at the thought, but even as it went through her, another part of her mind was busy piecing together implications. Around her in the blanched moonlight the whole city seemed to lie in a drugged dream of mist and shadow, still with a stillness like death.\n\nYsidro's world she thought. The fag end of nighttime. The sense of being the only one left alive.\n\n\"That means\u2014it must mean\u2014Ernchester has gone to Constantinople.\"\n\n\"Even so,\" Ysidro agreed. \"According to Batthyany's countess, Anthea claimed that she had been used as hostage to force Ernchester to the will of Karolyi and Fairport. It implies, of course, that Ernchester did not come to Vienna of his own accord, and so they hunted him no further.\"\n\n\"But James saw him get on the train with Karolyi of his own accord,\" Lydia said, puzzled. \"After Karolyi was dead and Ernchester freed, why would he flee?\"\n\n\"The fact that Charles got on the train of his own accord,\" Ysidro said softly, \"does not mean that he did so of free will. And it would explain what has troubled me from the start. Ernchester is not a politician's choice\u2014that slut Grippen has lately got in St. John's Wood is stronger to the hunt and the kill than Charles. But someone knew enough about him to know that he could be ruled. That a threat against Anthea would bring him. That to hold her would be to guarantee his conduct.\"\n\n\"Would Karolyi know that?\"\n\n\"Evidently.\"\n\nThey had reached the house in the Bakkersgasse again. Unwilling, perhaps, to give up possession of those dark streets that were their sole dominion, Lydia and Ysidro sat as if by unspoken agreement side by side on the marble rim of the small fountain before the house. The gaslight wavered on the surface, made watching pits of the eyes of the bronze emperor above the water and touched the lower half of Ysidro's face, giving the effect of a carnival mask through which fulvous eyes gleamed like marsh fire as he spoke.\n\n\"Will you return to London, mistress? The trap here is sprung.\"\n\nLydia hesitated, feeling for one minute the overwhelming desire for the comfort of the things she knew, the world of research circumscribed by the university's walls. But she knew perfectly well, as the thought of it formed in her mind, that only _a_ trap had been sprung.\n\n\"It isn't... it isn't over yet, is it? Whatever started this. Not anywhere near it.\"\n\n\"No.\"\n\nFrightening as it had appeared in the beginning, Vienna hadn't been so bad.\n\n\"Would it be of help to you for me to go on to Constantinople? Because that's what I would prefer to do,\" she added, seeing the swift thought behind the Spaniard's eyes.\n\n\"It would be of help in finding Ernchester, yes.\" He frowned, as at some unexpected thought. \"I would not have you undertake unnecessary risk\u2014yet you know your husband's thought, and the legitimacy of your inquiries will help in the search for the heart of this matter.\"\n\nHe paused again, considering, and there was, Lydia thought, just the smallest trace of surprise in the enigmatic eyes.\n\n\"Curiously enough,\" he went on, \"Charles has been in Con\u00adstantinople. This was many years ago, but there might be some there who knew him when he\u2014and possibly they\u2014were living men.\"\n\n\"But it doesn't make sense\u2014\" Lydia pulled her collar closer \"\u2014if vampires are all as\u2014as jealous of interlopers as the Count Batthyany is. That is... are they?\"\n\n\"Mostly,\" said Ysidro. \"Burning Fruhlingzeit as a warning was one of the milder expressions of displeasure I have encountered. Master vampires are not to be jested with when they conceive their territories in threat. Yet only a vampire could have summoned Ernchester to Constantinople. Only a vampire would know the threat that would bring him. Only a vampire would know that, of all the vampires I have met, Ernchester is one of the few capable of love.\"\n\n# ELEVEN\n\n\"DO VAMPIRES NOT LOVE?\"\n\nYsidro looked up from tallying his points. Lydia had scored sixteen for eight through king in hearts, with the nine making up a quart; Ysidro, by not declaring a sequence in diamonds, had managed to win most of the tricks, including the last. It hadn't saved him.\n\nThey had spent the day among the ancient basilicas and rose farms of Adrianople, owing to Ysidro's flat refusal to travel during the hours of light. Now the rough hills of Thrace, through which they had creaked with maddening slowness all of last night, seemed, as far as Lydia could tell, to have evened out. The train was a good one. German built and fitted, but even this first-class car smelled of garlic, strong coffee, tobacco, and unwashed clothing. On the platforms of Sofia and Belgrade, Lydia had observed that the farther east one got, the more casual railway personnel seemed to be about the presence of livestock in passenger cars. At Adrianople, earlier in the evening, she'd seen a Bosniak family casually load two goats into the third-class carriage, the father holding the long-fleeced kid in his arms and stepping back politely to let a bearded Orthodox priest climb on ahead of him, while farther down the platform people passed crates of chickens in through the windows.\n\nAunt Lavinia had always said that travel was broadening. Lydia suspected this was not what she meant.\n\nThe noise in the other first-class compartments seemed to be lessening, though in the corridors the tobacco fug still lay thick. Miss Potton, after her usual stubborn struggle to play a game in which she had neither aptitude nor interest, had fallen into a doze at Ysidro's side. For nearly an hour the only words exchanged had concerned the lay of the cards and the trading of points, but Lydia suspected that the governess was as jealous of those as she was of other conversations Lydia and Ysidro had.\n\nThe wheels clacked steadily, like mechanical rain. Ysidro finished his tally, the steel nib of his pen scratching softly on the cheap yellow pad, the friction of his cuff on the tabletop a dry whisper against Margaret's stertorous breath and the occasional bursts of laughter or speech audible through the compartment wall.\n\nIt was a long time before Ysidro replied.\n\nAt length he said, \"As humans understand it?\"\n\n\"How do humans understand it?\" Lydia gathered the cards, turned them in her hands. Living half by night\u2014half in the sunken silences of darkness\u2014had given her a small degree of understanding of something Ysidro had mentioned early on, that vampires' senses were far more sensitive than those of humans. With blackness pressing the window and gloom thick beyond the circle of the gas burner's solitary light, every sound, every sight, seemed portentous, fraught with meaning beyond the simpler shapes of day.\n\n\"You said back in Vienna that Ernchester was a rarity among vampires, because he is capable of love. I wondered what that actually meant.\"\n\n\"As with the living, among the Undead love means different things to different individuals.\" He turned his head, champagne-colored eyes resting briefly on the woman who snored beside him in her muddle of yarns. After a moment her head lolled more heavily and her breathing deepened still further; she slumped against him, and with a fastidious care he leaned her into the other corner of the seat. In the five days it had taken them to work their way south via local trains\u2014for the Orient Express only left Vienna on Thursdays\u2014through Buda-Pesth, Belgrade, Sofia, Adrianople, waiting sometimes for most of a day for the next train that departed after sunset\u2014Lydia had been occasionally aware of the highly colored romantic dreams that illuminated Margaret Potton's sleep. In all of them Ysidro had been a vampire, outrageously Byronic in black leather and pearls, with daggers sticking out of his boots.\n\nIn all of them, love had been implicit. His professed, passionate love for her, bonding them, drawing her like a silver rope into love for him.\n\n_Whatever love is,_ Lydia added to herself. It would hardly do, at this point, for Margaret to hear any true opinion of Ysidro's on the ability of vampires to love.\n\n\"It is not unlikely, or even infrequent,\" Ysidro said, \"for those who have the capacity to love others more than themselves to also have the will to make the transition from the living state to that of the Undead.\" The train jostled around a curve sharper than those found in northerly or westerly Europe. Ysidro put a gloved hand on Margaret's shoulder to keep her steady\u2014perhaps to keep her from waking. He touched her carefully, even with gloves. His hands, Lydia knew, were cold as bone these days. She could tell when he had fed, and she knew he had not hunted in Vienna.\n\n\"It is unusual, however, for such a one to survive long after the deaths of those for whom they care. In many cases, friends or relatives constitute the vampire's early victims or fall prey to them in the course of the years. For those vampires who do not avail themselves of the convenience\u2014and the odd comfort\u2014of this resolution to immortality's riddle, there is often a sense of disorientation when family and lovers age and begin to die. In my experience those capable of loving seldom make successful vampires.\"\n\nIn the juddering glare of the gaslight, his face had the appearance of a skull in the ashy frame of his long hair; Lydia wondered whether he had always looked so or whether he had thinned and wasted in the past five days. Margaret stirred in her sleep, and Ysidro turned his face to look at her again, unreadable indifference in his gaze. There was long silence before he spoke again.\n\n\"You understand that having become vampire myself at the age of five-and-twenty, my experience of human love is... incomplete,\" he went on, as if the matter were not one for his concern. \"In this case, what love actually means is that someone\u2014one of the Constantinople vampires, or one who has been in contact with him or her\u2014would know that a threat to harm Anthea\u2014by human agency, perhaps, or with the understanding that if human means proved ineffective, vampire agents would not be far behind\u2014would bring Charles to heel. The vampire mind is an endlessly subtle one, and Charles knows the extent of their abilities to manipulate circumstance. Even were Grippen willing to defend Anthea, defending against a sufficiently determined attack might lie beyond his powers. For his own safety, Charles would not care, but as Dryden said, we give hostages to fortune when we love.\"\n\nHe moved his hand, turning it as if revealing a hidden card. \"I would guess that the sack of the house was an effort to take her hostage once he had departed, to prevent him changing his mind.\"\n\n\"But if the Sultan wants a vampire,\" Lydia said, puzzled, \"and if he's been in touch with one in order to know about Ernchester in the first place, why go to the trouble? Aren't there plenty of vampires in Constantinople? At least from all the legends James hears, Greece and the Balkans have to be stiff with them.\"\n\n\"Perhaps the vampire who spoke of Ernchester to Karolyi\u2014or to the Sultan, if it was he who sent Karolyi\u2014is now dead. We cannot know how long ago it was, and there have been upheavals in the city recently. Of a certainty, he\u2014or she\u2014would be dead, did the Master of Constantinople learn that there was a plot afoot to bring an interloper into his city. And it may be that whoever has sent for Ernchester feels that he would be more easily controlled than any under the sway of the Master of Constantinople. In this he would be correct.\"\n\nYsidro stretched a hand like gloved bones to part the window curtain. \"Behold.\"\n\nIt was not like Paris, not like Paris' glittering carpet of gaslights. Softer lights and fewer\u2014amber, citrine, topaz, red as the juice of blood oranges\u2014jeweled the long spine of hills that made the city and lay in spangles of isolate flame in the nearly unseen movement of the sea. The train swung around a great curve. A many-towered gate loomed in the darkness, archways strung with yellow electric lights that cast reflections on a tree-filled ditch and a massy wall stretching into the night. Lydia gasped in surprise\u2014she'd heard of the walls of Constantinople but hadn't quite realized that the Byzantine ramparts would still be standing, watchtowers intact.\n\nAs the train slowed, the lights from its windows caught the black-glass combs of choppy sea beneath the railway embankment. Where the land curved, the old sea wall rose above the tracks, dark houses with outthrust upper floors growing from the ancient masonry like mushrooms from a riven oak.\n\nYsidro produced a gold pocket watch. \"Twenty of one,\" he said approvingly. \"Only two hours late. Excellent, for the Ottoman lands.\"\n\nAfter coming into Sofia four and a half hours late, with the sky like wet slate and Margaret in hysterics as if she, not Ysidro, would be destroyed by the light of the dawn, Lydia could only be thankful. On that occasion, while the Sofia train lurched and stopped and started all through the shelterless hills of Thrace, Ysidro had grown quieter and, when he spoke, more incisive. Though Lydia did not know exactly how much light was necessary to trigger the photoreactive properties of the vampire flesh, she gathered that they had reached the Terminus Hotel in Sofia, and Ysidro had taken his usual leave of them, with only minutes to spare.\n\nThis had led to a furious and not very coherent scene with Margaret, in whose aftermath Lydia still felt embarrassed. The younger woman had accused Lydia of \"not caring anything about\" Ysidro, of \"using people up like old dishes, and then throwing them away when they break.\" When Lydia had pointed out that at any time Ysidro could have retreated to his coffin trunk and trusted the girls to get him to safety, Margaret had screamed, \"If you'd ever had anything to do with earning your own living, without having everything you ever wanted just handed to you on a silver plate, you'd have learned you can't treat people that way when they're trying to help you!\"\n\nIn view of Ysidro's relations with Margaret, this had struck Lydia as so outrageous that she'd simply said, \"Oh, stop behaving like an idiot,\" and had gone into the suite's single bedroom and closed the door. She'd been far too exhausted by her own fears to remain awake long, but during the few minutes she'd spent stripping off her outer clothing, petticoats, and corsets, she'd heard Margaret sobbing hysterically in the parlor. When she emerged, not much refreshed, hours later, it had been to see the governess sprawled unprettily on the sofa, face flushed, shirtwaist off, and corsets unlaced, sound asleep.\n\nThey'd made up after a fashion, as traveling companions must, but their never-easy relations remained strained. Now Margaret mumbled, \"You should have waked me sooner,\" when Lydia shook her.\n\n\"We're here. Constantinople.\" She didn't mention that Ysidro had done his best to keep the woman asleep.\n\nMargaret pulled a comb out of her handbag and straightened her hair, with nervous glances at Ysidro as if he hadn't seen her in rumpled slumber for many nights. Only then did she turn to the window and say in disappointment, \"Oh. You can't see anything.\"\n\nAcross tumbled onyx water a long curve of lights glimmered as if a congregation of shepherds had kindled watch fires on the point. Here and there, close to the tracks, reflected light showed a thumb smudge of honey-colored walls, but for the most part the city was dark. The high, dark backbone of the land was studded by minarets and domes under the gibbous moon's waning light: the embodiment of formless dreams, a dark suggestion of labyrinth hoarding darkness within.\n\nNo, thought Lydia. You didn't see it. You drank it, and it left you filled with an indescribable sense of hunger, and loss, and grief.\n\n\"They called it the City of Walls,\" Ysidro said softly. \"The City of Palaces. Like a Kipling treasure guarded by a cobra, they have fought over it, or feared it, for all the long centuries since the emperors departed from Rome. Not even those who won it, who dwelled in it, ever knew it all.\"\n\nLike James looking at the towers of Oxford, thought Lydia, and calling them each by its name. Did he name in his heart each dome, each quartet of spires, against that lambent sky? \"Were you ever here?\" Margaret edged possessively closer to him, took his arm\u2014though Lydia knew he hated to be touched\u2014and looked into his face.\n\nYsidro smiled, for her. \"Once,\" he said in a voice that promised her new dreams. Over her head his eyes met Lydia's, enigmatic, and looked away.\n\nThe train chuffed to a stop at a small station beneath the beetling towers of an old fortress gate. Up close the ambience was anything but exotic. The station was Western, stuccoed and painted the same ochre hue so common in Vienna, and by the harsh electric lamps Lydia saw the grannies and goats, the gentlemen in red fezzes and black coats, the Greeks in full white pants and the Bulgarians with their crated chickens and straw suitcases, get on and off with the leisured air of those who know the train isn't going anywhere in a hurry. The stink of slums and tanneries was thick hereabouts, and there were, Lydia noticed, a lot of soldiers in the stations, clothed in modern khaki uniforms, nothing like the colorful warriors of tales.\n\n\"Those aren't the janissaries, are they?\" she asked, and Ysidro's yellow eyes developed the smallest of twinkles, like a fugitive star, at the bottom of their cold, ironic depths. Despite the insectile thinness of his face and its white-silk pallor, he looked briefly human.\n\n\"The corps of the janissaries was abolished a century ago\u2014massacred wholesale, in fact, by order of the Sultan Murad, who wished to establish a modern army. This past July that modern army returned the favor by deposing the current Sultan and converting him by force into the type of constitutional monarch fashionable among those who like to style themselves enlightened.\"\n\n\"You mean there isn't a Sultan anymore?\" Margaret sounded like a child who has been told on the twenty-fourth of December that Father Christmas has been pensioned off to a villa in the south of France.\n\n\"July...\" Lydia said thoughtfully. \"The printer's deadline for my monograph on the effects of ultraviolet light on the hypothalamus was August fifteenth... And I never can remember whether they're on our side or Germany's. So it couldn't have been the Sultan who sent for Ernchester?\"\n\n\"It may well be,\" Ysidro said. \"He is not without power, even yet. But if he thinks to regain it by bringing in a vampire whom he hopes to control, he reckons without the Master of Constantinople.\"\n\nThe train lurched and began its slow, rocking progress again, the city growing above them in thick accretions of shadow, lamps, and ancient walls shrouded in vine.\n\n\"Who is the Master of Constantinople?\" Lydia asked quietly.\n\nThey were all three clustered by the windows of the compartment, looking out over the inky water toward the lights of Seraglio Point and the dim hills of Asia beyond.\n\n\"In my day it was not considered a wise thing to speak his name.\" Ysidro turned back to the table and gathered the cards. He fumbled, dropping them; Margaret sprang at once to help him but he'd retrieved them already, slipped them into the paper band that usually encircled the pack, secreted them in a pocket of his mouse-gray coat.\n\n\"He was a sorcerer in life, a title which could mean anything from a theoretical alchemist to a student of the properties of herbs. Certainly he was a poisoner, possibly an astronomer, though one does not always keep these things up. He wielded tremendous power, before and after his death, with the Viziers of the Sublime Porte. Legends said that certain of the sultans gifted him with prisoners, that he might feast upon their deaths, though considering the size of the beggar population of Constantinople, I do not find this at all likely or necessary. And as Juvenal says, 'Foolish is he who puts his trust in princes.' Personally, I wouldn't touch any edible offered me by any of the sultans.\"\n\nYsidro put out a hand again, to steady himself on the wall as the train swung around the rocky slope of a hill and lurched into another suburban station. There were electric lights here, too, and soldiers armed with businesslike Enfields.\n\n\"It is probably best,\" he said, \"that the master of this city not be spoken of in any terms until we are in Pera.\"\n\nAnother of Ysidro's gruff local henchmen awaited them in the square before the main Gare of Stamboul, this one a Greek\u2014whom Ysidro addressed in Spanish\u2014with the usual wagon and horses. Lydia had removed her glasses before leaving the train compartment, but the moment they were settled on the high seat and moving off through the tangle of drays, donkey carts, and foot passengers, she sneaked them back on, gazing around her in wonderment. At the foot of the square the dark waters of the Golden Horn flashed with the lights of ships moored there, and even at nearly two in the morning the lights of small boats could be seen plying between the Stamboul shore and the lamp-flecked hills of Pera on the other side.\n\nBlack streets swallowed them, and for a few minutes Lydia could no more than guess at the houses crowding above, balconies\u2014sometimes entire upper stories\u2014jutting overhead as if grabbing for airspace, here and there the low glimmer of lamps behind thick latticework. Cats' eyes flashed everywhere, and the smell of goats and dogs and human waste was like a curtain thick enough to be touched with the hand. Lamps in iron cages showed her the somber glory of a mosque half veiled in Stygian gloom as they passed through a square, a note of great age on the lighted threshold of a modern iron bridge.\n\nOn the bridge's other side the houses were European\u2014or Greek, with white walls like clotted cream in the moonlight. They wound their way uphill to a tree-grown public square lying beneath a splendid Italianate palace of pale golden stone.\n\n\"The British Embassy,\" came Ysidro's soft voice. \"I trust you ladies will present yourselves to the Right Honorable Mr. Lowther in the morning. For many years the embassies have been the true power here.\"\n\nAs usual, Ysidro had wired ahead for lodgings, this time a pink-washed Greek-style house whose stone-flagged arch led into a court shaded by a massive pomegranate tree, staffed by three thickset Greek women, evidently a mother and two daughters, who smiled and replied _\"Parakalo_ \u2014 _parakalo...\"_ to everything Lydia said.\n\nAs at Belgrade, Sofia, and Adrianople, once Lydia's trunks and portmanteaus and hatboxes and baskets of herbs were carried upstairs, Ysidro climbed into the wagon once more and disappeared to some secret lodging of his own.\n\n\"You can't ask him to continue what he's doing.\"\n\nLydia turned, startled, the moss-green velvet of her dressing gown weighting her arms. Tomorrow she'd present herself, not only to the Right Honorable G. A. Lowther, but, armed with Mr. Halliwell's letters of introduction, to Sir Burnwell Clapham, the attach\u00e9 in charge of what were nebulously referred to as \"affairs.\" It was entirely possible, she thought, that Jamie would be there, or Jamie would be somewhere close. _Oh, yes, Dr. Asher. He arrived last week..._\n\n_Please,_ she thought, shivering inside. _Please..._\n\nMargaret stood awkwardly in the doorway of the single large bedroom the two women would share. As in Vienna, in Belgrade and Sofia, it was not by their choice\u2014even had relations between them not been strained, Lydia would have preferred to be spared her companion's nocturnal sighs and mutterings in dreams. But in no house had more than one bed been made up, nor could the servants anywhere be induced to do so. In the small connecting chamber, Lydia had already found the dismantled pieces of a massive four-poster that looked as if it had been ordered from Berlin at the height of the Gothic craze. Its sister ship filled most of this room, the bright pink-and-blue local work of its coverlet incongruously gay; the dressing table, mirrored armoire, and marble-topped wash-stand had clearly been ordered en suite, and though the room was large, with a bay projecting over the street, they gave it a cluttered feeling, jammed and awkward.\n\nAt least, thought Lydia, they weren't strewn with the porcelain knickknacks featured in their Belgrade lodgings, and the whitewashed plaster walls were free of garish oleographs of Orthodox saints.\n\nShe turned from the armoire, the robe still in her hands. _\"What?\"_\n\n\"You forbade him...\" Margaret hesitated, and her wide blue eyes shifted as she sought another word. \"You forbade him to hunt,\" she said at last. \"As a condition of letting him travel with you, of letting him protect you.\" Her voice stammered and she twisted at her black-gloved hands. \"Now that we've reached our destination, you really don't have any right to continue... to continue...\"\n\nFrozen in mid-motion, Lydia only stared at her, too shocked to speak.\n\nMargaret, who had clearly hoped that she would say something and spare her the completion of her sentence\u2014and in fact the completion of her own thought\u2014trailed off uncertainly, and for a moment there was only the clutch and jerk of her breath. Then she burst out, \"You don't understand him!\"\n\n\"You keep saying that.\" Lydia crossed to the bed and dropped the robe beside the nightgown the maid had laid out, and began to unbutton her shirtwaist. The tiny pearl fastenings of the sleeves were awkward, but she'd dismissed the servant after she'd unpacked for them, and didn't know enough modern Greek to summon her back. She wondered what the servants had made of the silver knives and silver-loaded gun among the masses of petticoats, skirts, shirtwaists, lingerie, and dinner dresses\u2014wondered, too, if she could communicate to them a request to purchase garlic, whitethorn, and wild rose on the morrow. Or as Ysidro's servants, would they refuse to obey such a request?\n\nMargaret reached out and took her by the sleeve, her face bracketed with lines of distress deepened by the lamps' heavy shadows. \"You can't forbid him to hunt!\" she insisted desperately. \"It isn't as if he... as if the people he... he takes...\"\n\n\"You mean 'kills'?\"\n\nShe flinched from the word but lashed back almost at once with, \"It isn't as if they didn't deserve it!\"\n\nLydia only stood for a time, her fingers still on the pearl buttons but her task forgotten. When she spoke, her voice was very quiet. \"Did he tell you that?\"\n\n\"I know it!\" The governess was on the brink of tears. \"Yes, he told me! I mean, I know\u2014I mean, in the past\u2014in past lifetimes\u2014in dreams I've had about our former lives together... And don't tell me they're all lies.\" she veered away suddenly, \"because I know they're not! I know you think they are, but they're really not! They're not!\"\n\nShe flung herself in front of Lydia when Lydia tried to turn away, her face red, blotchy as if with the approach of tears. \"You see, if a vampire doesn't... doesn't hunt to completion...\"\n\nShe was still avoiding the word \"kill.\"\n\n\"They feed on the energy, the life, the vital force!\" she went on in a rush. \"It's the life they take that gives their minds the powers they need to protect themselves!\"\n\n\"You mean to kill other people?\"\n\n\"You're starving him to death!\" Margaret cried. \"Robbing him of his powers to defend himself from danger, now, here, where the peril is the greatest! That's why vampires take so long to hunt, or at least why _he_ takes so long to hunt, he told me, because he's hunting the streets of the city to find a thief, a murderer, a... a blackguard who deserves to die! You know the world is full of them. He's hunted that way for hundreds and hundreds of years! It's only from those kind of people that he takes the life he needs! And he's too honorable to go against his given word to you...\"\n\n\"Did he ask you to speak to me?\" Lydia's voice was as cold to her own ears as the silver on her neck.\n\n\"No.\" Margaret sniffled and wiped furiously at her eyes, fighting not to break down in front of this slender auburn and white reed of a girl, this spoiled heiress-beauty with her waist unbuttoned to show the heavy links of silver chain, row upon row of them, around the stem of her throat.\n\n\"But I can see!\" she sobbed. \"Every day I can see. You beat him at cards all the time now...\"\n\n\"I've had a week of continuous practice,\" Lydia pointed out.\n\n\"You could never beat him if he weren't fighting to keep the other powers of his mind intact! To preserve himself...\"\n\n\"Thank you very much.\" Head aching with weariness\u2014for it was close to three in the morning\u2014Lydia stepped around her. It was true that Ysidro had grown very gaunt\u2014true, too, that a week ago he would never have dropped the cards, never would even have allowed the girls to see him gather them.\n\nHe could not mask things from them as he had. Or was he saving his strength for other matters?\n\n\"Margaret, do we need to talk about this now? I'm tired, you're tired, I suspect you don't mean everything you're saying\u2014\"\n\n\"How can you be so blind!\" Margaret went on frantically, unheeding, following her back to the bed. \"Can't you see? He can't turn people's minds aside in the train stations like he used to, or listen down the train cars, reading their dreams...\"\n\nLydia's overwrought temper snapped. \"Or put little scenes of dancing the waltz\u2014which wasn't even _invented_ in the sixteenth century\u2014into yours? I'm sorry,\" she said immediately, as Margaret burst into a storm of tears at this brutally accurate accusation. \"I shouldn't have said that...\"\n\n\"You don't understand!\" Margaret shouted wildly. \"You don't understand him! All you care about is finding your boring old stick of a husband and helping him play spies, and you can't see the great-souled, noble, lonely, tragic hero you're destroying!\"\n\nShe blundered from the room like a bee trying to get out of a potting shed. Lydia heard the banister creak as she stumbled against it, heard the running judder of her footsteps descend the two long, C-shaped flights of stairs.\n\n\"Margaret!\" She grabbed her spectacles from the dressing table and ran after her, catching handfuls of taffeta skirt to race down the steps, the tile of them cold under her stockinged feet. Below her she heard the door bang, and she followed, appalled, into the covered carriageway in time to see the heavy outer gate swing shut on its hinges.\n\n\"Margaret!\" Through her concern she thought obliquely, _Well, that does it for this pair of stockings_ \u2014even in the relatively clean suburb of Pera the streets were nothing to explore unshod. Two small sconces illuminated the courtyard behind her, and the candle before a saint's icon in a niche flecked the underside of the carriageway's brick vault with wavering light. Past the gate the street was like a cave a thousand feet beneath the earth.\n\nLydia stopped on the threshold, as if that abyssal dark were a chasm gaping before her feet.\n\nMargaret gasped somewhere, and there was a suggestion of movement, pale in blackness. The shred of moonlight picked out a white face, like a skull's, a scrap of spiderweb hair. A moment later Lydia's eyes, adjusting, made out the white hands, holding Margaret by the wrists. Margaret threw herself wordlessly to his chest, clutching and weeping.\n\nYsidro must have spoken, so softly Lydia did not hear. Lydia herself had been exasperated to the slapping point with Margaret's clinging, mooning, and silent reproaches, but she had never seen the vampire anything but patient and understanding with the woman he had made his slave. Of course he understood her, thought Lydia bitterly, watching as Ysidro bent his head to listen to some muffled, hysterical rant; watching Margaret's skinny hands grab at his sleeves, his shoulders, the long folds of his cloak. If he hadn't understood her, he couldn't have baited the trap.\n\nIlluminated only by the frail gleam from the window above, they seemed figures in a distant stage show, almost like a dream. Margaret flung back her head, gazing up into Ysidro's face, then with a passionate gesture she ripped open her shirtwaist, baring her throat and her white, soft-fleshed bosom. \"Take me!\" Lydia heard her gasp. \"Even unto death, if that is what you need!\"\n\nWhat Ysidro replied Lydia didn't know. But she saw him draw the edges of Margaret's shirtwaist together, put his hands on her shoulders, speaking quietly as she bowed her head. When he began to guide her back along the lane to the gate once more, Lydia retreated soundlessly into the courtyard, concealing herself in the dense shadows of the pomegranate tree, so that Margaret would be spared the embarrassment of knowing that the encounter had been observed. For a moment they stood framed in the carriageway's arch. Ysidro must have said something else, for Lydia saw Margaret nod and push up her eyeglasses to mop her cheeks. Then the door shut behind her as she went in.\n\nLydia heard nothing for a time, though she knew that Ysidro had not gone inside; and indeed, moments later, the dimmest crack of light showed when he opened the gate again and stood for a moment looking out. That slit vanished; he emerged into the courtyard like an errant ghost and crossed to her hiding place as if he had seen her all along.\n\n\"I could wish her to have reserved such theatrics for another place and time.\"\n\n\"Yes.\" Irritated as she had been with Margaret, her greatest anger still lay toward him. She folded her arms against the cold. \"It's a nuisance, isn't it, when people decide to feel more than you've scheduled them to feel?\"\n\n\"It is.\" He might have been agreeing that today was Saturday. The moon was sinking; only the glow from the votives by the kitchen door showed her the garden before them. \"Yet the dreams she dreams are not all of my making. And I admit I will feel safer to know that the two of you sleep in the one bed, which I trust you will hang about, as you did in Sofia and Belgrade, with those stinking weeds you have carried with you since Paris.\"\n\nThe chilly breeze from the Asian hills stirred the last leaves high overhead. A stray breath of it flared the votive lights, showing her briefly Ysidro's face, eyes darkened by shadow to skull-like sockets and cheekbones hollowed to bruises. Remembering what he had said about mirrors, Lydia wondered suddenly if he was actually thinning away before her to a wraith of ectoplasm and bone, or if what was thinning was simply his ability to make her believe that she saw him other than he truly was.\n\n\"The Galata slums at the base of the hill and the high streets of Pera with their embassies and their banks, they all smell of vampires.\" The flame repeated itself, cold yellow crystal in his eyes. \"Standing just now on the steps of the Yusek Kalderim, I stretched forth my mind across the Golden Horn, and the city lies under such miasma as I have never encountered before. The minds of vampires, the mind of the master, other minds... I can smell them, heft them like silk in my hand. But everything is blocked, shadowed, wreathed in illusion and deception, as if every card on the board were down-turned, and one had to wager all one had on a hand of three.\"\n\nHe frowned and turned to look once more at the gate. Involuntarily Lydia stepped closer to him, her anger forgotten. \"Are you sure? You've said yourself you aren't as... as able to perceive...\"\n\nA wry line sketched itself in the corner of his mouth, the echo of a living man's ironic smile. \"A regret, mistress? A concern for the fact that you have asked me not to kill to preserve my own life, only to discover that such abstinence may prevent me from preserving yours?\"\n\nShe studied his face a moment, trying to read something in the twin sulfur glints of his eyes. They were like a dragon's in their hollows. \"No,\" she said. \"A concern, maybe, but not a regret.\"\n\n\"No,\" he echoed softly. \"A lady worthy to her bones.\"\n\nIt was, she realized, the first time he had spoken to her of her stipulation.\n\nThen he shook his head and looked back to the gate and the inky, pitch blackness that lay beyond.\n\n\"And Jamie?\" She found she could barely speak his name. It was hard even to ask, for fear Ysidro would tell her what she had dreaded for days to hear.\n\nHis brow flinched, just barely, in a frown. \"If he is here, he is not in Pera.\" There was almost hesitation, an unwillingness in his voice. \"If he sleeps on the Stamboul shore...\" He shook his head. \"No, my perceptions are impaired, but this is not a matter of degree. This\u2014shadow, this\u2014blurring that lies over the city... it is something that emanates from the vampires themselves. An obscurity, gathered to hide aught within it. A fog. as they say the Undead can summon...\"\n\nHis smile had been\u2014almost\u2014a living man's smile. The shadow in those dragon eyes was suddenly, fleetingly, a living man's fear. \"Tomorrow night will be soon enough to cross, to walk and listen in the darkness, to see what more can be descried at nearer quarters.\" He drew his cloak more closely around him, a subconscious gesture, the white of his gloves against the dark wool like frost on black rock.\n\n\"But it is clear to me that something very strange is taking place in this city, and I had rather our romantic friend had not cried aloud, even in English, regarding hunting and killing and the drinking of blood. I think it best such things not be spoken of, not even here in Pera. Not even by light of day.\"\n\n# TWELVE\n\nTHE VOICES of the muezzins woke Asher: \"There is no God but God; Mohammed is His Prophet...\" He knew the words, but could not tease them from the somber roll of sound.\n\nArched windows had at one time opened all along the room, five times the length of its narrow width, but centuries ago these had been bricked shut. The windows in the drums of the five shallow domes above were, as far as he could ascertain, barred with silver, though it was hard to be sure. By day he heard no voices, no clip of donkey hooves or creak of wheels from below, and only occasionally and far off, the barking of Constantinople's infamous dogs. Now and then the wind would bring him a street vendor's cry in sawed-off Romaic Greek. Day or night, the closest sounds were the squawking of the seagulls and the yowl of cats.\n\nThrough the lattice the sky was the color of tiger lilies, the light momentarily a soft and fading salmon hue on the blue tiles that ringed the domes.\n\nAsher did not face Mecca\u2014though he'd deduced in what direction it lay\u2014nor repeat the words intended by the muezzin, but sitting among the cushions and blankets of the divan, he prayed. He was very frightened.\n\nThe light in the room had deepened when he finished, bleeding away to shadow. Because of the domes, the room filled with darkness from the bottom up. In the center of the floor the rectangular, blue-tiled basin of what must have been a fountain or fish pool seemed fathoms deep in the gloom, a horror from which anything might emerge. Asher scratched a match that he took from his pocket, to light the wick of one of the few bronze lamps that still occupied the serried ranks of niches in the wall. The glow did little to dispel the dreadful brooding dimness. He reached for his watch to wind it, as was his habit, but of course it had been taken, along with the silver chains that had protected his wrists and throat.\n\nHe dressed and washed, and stowed the bedding in which he'd slept in one of the room's shallow cupboards, listening all the while to night fall within the silent house. In full dark\u2014enough so that a white thread could not be distinguished from a black, as the Koran said\u2014he heard the key turn in the old-fashioned lock.\n\nHe moved as far from the door as he could and deliberately willed his mind not to feel, not to succumb to the odd, lazy distraction of the vampire power. Still he did not see them enter the room. He had the vague impression that he had dreamed once about standing in a darkened gallery, watching a door inlaid with brass and ivory as it began to open...\n\nBut it seemed to him that one moment he was stepping back against a pillar, and the next, they were all around him, binding his wrists behind him with narrow silk cord. Their eyes in the lamplight were the eyes of rats, their flesh dead clay on his. They had not fed.\n\n\"So who are you, Englis?\" asked the one who had been pointed out to him last night as Zardalu. Beardless, boneless as an empty stocking, he had red-painted fingernails and a Circassian's bright blue eyes. \"Yesternight I took you for one of the Bey's _mikaniki_ , and I thought, This is one he intends to make one of us, to look after this thing they make in the crypts, this _dastgah_.\" His eyes slid sidelong at Asher under painted lids; and knowing they could hear it, Asher tried to calm the pounding of his heart.\n\n\"And now the Bey has given us other instructions concerning you. What are we to think?\"\n\n\"You really think he'd join another to us for the sake of one of his experiments?\" Jamila Baykus\u2014the Baykus Kadine, she had been called, stick-thin with a strange, disheveled wildness that was somehow very like her namesake owl\u2014put her head to one side and considered him with enormous demon eyes. Half her hair was braided or curled, dressed on jeweled combs, the rest hanging in a huge malt-colored tangle to her thighs. Pearls were caught in it, like shells glimpsed in a jetsam of kelp; she had a necklace of rat bones and diamonds around her throat. \"Is that what you are, Englis?\" The finger she reached up to touch the underside of his chin\u2014for she was no taller than a twelve-year-old English girl\u2014was like a twig brought in from out-of-doors, icy with the ice of the night.\n\n\" _He_ said we weren't to question him.\" That was Haralpos, a one-eyed tough who had been a janissary. He held up a scarf, fine cotton, creased and filthy and patched with dark stains.\n\n\"And did he say I was not to question you?\" Asher had studied Persian and enough Arabic to approximate the thick Osmanli they spoke and make himself understood.\n\nZardalu's eyebrows tweaked into circumflexes of malicious delight, and his fangs gleamed in a smile. \"Oh, what a clever Englis. Of course you may question us. Who are we but your fellow servants of the Deathless Lord?\"\n\n\" _He_ said silence,\" Haralpos insisted. The dark Habib and the voluptuous and silent Russian girl. Pelageya, stirred uneasily. Asher knew of whom the janissary spoke and knew the others had a right to be uneasy. \" _He_ said to walk in silence, like the fog. Would we have this infidel cry out to be saved?\"\n\n\"Would it do me any good if I did?\" countered Asher. He turned to Zardalu, whom he sensed to be the most dangerous of them, and asked him, \"What _dastgah_ is this?\" The word meant a scientific apparatus, which could mean anything from an astrolabe to a chemical experiment.\n\n\"How should I know that, Englis? The Deathless Lord has put up silver bars across the cellar which lies beneath the old baths that are no longer used. He has veiled the place with his mind, to keep us from thinking about it, even as he has veiled this entire city.\" The sweet alto voice sank lower, and as the vampire leaned close, his hair and clothing breathed patchouli and decay.\n\n\"He has veiled the place, yet still we feel the cold of the ice that he has men bring in during the day for his experiments. We smell the _naft_ , the _alkol_ , the stinks of what he does... even as we hear the footfalls of the workmen, down below in the crypts, as we sleep. Does he think we do not?\"\n\n\"Come,\" Haralpos said impatiently. \"Now.\" He reached out with the scarf, and Zardalu touched his wrist.\n\n\"Our friend James has said\u2014may we call you James, Englis?\u2014that he knows better than to cry out. The Bey will surely punish us if he escapes, and so even an escape's attempt will mean\u2014oh, not death\u2014\" His cold knuckle brushed the scars under Asher's ear. \"\u2014but surely some unpleasant experiences with tweezers, or water, or hot sand.\" The red nails clinched suddenly hard on the earlobe, cutting stronger and stronger like the grip of a machine, Asher gritting his teeth, shutting his eyes, forcing his mind away from the pain. Just when he thought the claws were about to tear away the flesh, Zardalu released him and smiled a fanged smile as he opened his eyes once more. \"And he knows he will not escape.\"\n\nThere was blood on Zardalu's nails. The vampire held Asher's gaze with his own as he licked them slowly clean.\n\nThey led him out into an open gallery two floors above a courtyard paved in stone. An old _han_ , or caravansary, Asher guessed as they descended the long flights of tiled steps. A solitary lamp burned in a wall niche at the bottom of the flight, outlining the arch of a short passageway that led through and down into an octagonal vestibule whose mosaic floors, though long defaced, still showed parts of Byzantine figures. He had crossed that vestibule yesterday afternoon, in the midst of the men who had surrounded him in an alley of the market district, a knife pressed to his back. They had said nothing to him, but had not needed to. The age of the place, as much as the absence of lamps from the niches and mirrors from the walls, had told him what house he had been brought to. Last night in the flickering lamplight of the upstairs chamber, Olumsiz Bey had said to him, \"It is unfair to keep you utterly a prisoner, when my house has libraries and baths and amusements for an intelligent man.\" Asher had been lying on the divan then, bound hand and foot and more frightened than he had been in his life.\n\n\"But the House of Oleanders is an ancient house, and a large house. There are rooms in which no lamp has been kindled for a great many years, and my children come and go freely in the dark.\" The Bey gestured to the fledglings with his right hand, coarse and square and covered with rings whose jewels had been carved long before the faceting of gems was devised. In his left he carried a weapon that Asher had not seen him set down, a halberd five and a half feet long whose naked eighteen-inch blade was wrought of shining silver, honed to a razor's keenness and backed along its spine with slanting teeth like a fish's ribs.\n\n\"Thus I believe it best that Sayyed here go with you.\" The Deathless Lord's wave brought forward an impassive servant, one of the three who had kidnapped him yesterday. \"I think,\" the Master of Constantinople had added, as the living servant drew a knife and cut away Asher's bonds, \"that you will find he is your best friend.\"\n\nAsher understood. For several hours Sayyed had stood in the doorway of the library, watching him while he explored the inlaid cupboards and read the titles of the books within them\u2014Arabic, German, Latin\u2014by the light of a dozen lamps and candles. The servant made no comment when Asher had taken a volume of Procopius' _Secret History_ and a bronze candlestick back to his room with him, and that was as much as Asher had sought to accomplish. The candlestick was ornamented with tendrils of vine wrought of bronze wire, which Asher had pried loose to work into picklocks as soon as the sun was up.\n\nThe interview with Olumsiz Bey was in his mind now, as Haralpos bound his eyes with the dirty scarf and he was guided along, bound and blind and surrounded by whispering voices of those the Bey had warned him to avoid. In his mind, too, was the silver weapon the Bey had carried, and what it meant that he carried it.\n\nAsher tried counting turns and footsteps, and concentrated on the feel of the ground underfoot. But as the Bey had said, the house was a large one and composed, from what little Asher had seen, of several old _harts_ , minor palaces of Turk or Byzantine construction. They passed through two open courtyards\u2014or one courtyard twice, for the brick underfoot felt the same\u2014up and down steps, through a place where water splished thinly under his boots and another where loose boards rang hollowly, though only with his own tread despite the cold grip of hands on his elbows. It did him no good to count steps and turnings, for it seemed to him that he woke, like a sleepwalker, to find himself on his feet outside, with the stink of the Constantinople streets in his nostrils and the barking of the dogs louder in his ears. Eerily, he had no sense of the vampires around him. It was as if he walked alone, save that their hands were on his shoulders, his arms, his neck, and that now and then they spoke.\n\n\"Can you see the Bey making such a one into one of us?\" Haralpos' deep voice was close in his ear as they made their way down a steep street toward the sounds of the harbor. \"An infidel who tinkers with machines? He has grown picky, the Deathless Lord. He has not brought one into our ranks since Tinnin came to grief.\"\n\n\"Tinnin was a scholar,\" breathed a voice he recognized as belonging to the Baykus Kadine. \"A Nubian _philosophe_ , like those in Europe in those days, insolent even to kings... Ah, but sweet. Sweet. He knew the wherefore of those experiments, not just tinkering with the bits of metal and wire.\"\n\n\"Perhaps our James knows the wherefore as well?\" Zardalu purred. \"Perhaps our Bey does not trust us?\"\n\nRising ground steep under his feet, then steps\u2014somewhere seagulls yarked. The House of Oleanders lay a stone's throw from the government ministries on the shoulder of the Second Hill, but the market quarter between the Place d'Armes and the mosque of the Sultana Valide was one of the oldest and most tangled districts in the town. As in many Islamic cities, after the prayers of nightfall the inhabitants retreated to their houses and barred the doors; the Undead and their captive walked unopposed.\n\n\"High time he trusted someone,\" Haralpos grumbled.\n\n\"He didn't trust Zarifa, either,\" the Baykus Kadine said, her voice like weed stalks and bones. \"Nor Shahar, and you saw what came to them. It is a deep game he plays, our Deathless Lord, and deeper now with this new little pet.\" Her nails, inch-long claws on those skinny child's hands, flicked his neck.\n\nOne of them must have felt him listening, sensed his mind, for it seemed to him almost that someone blew drugged smoke into his thoughts, so he had to fight to remain even a little aware of his surroundings. His mind drifted, hazed with strange impressions and alien smells, but when it cleared, the salt tang of the sea and the mournful clang of ships' bells was gone, replaced by livelier chatter in the distance and the music of the Gypsy quarter. They were making for the walls.\n\nHe told himself if they were going to kill him where the Bey could not see, they would surely have done so already.\n\nIt didn't help.\n\nSteeper ground, ankle-breaking potholes and rock underfoot, and the occasional brush of broken stonework against his shoulders. Once, someone pressed a hand to his head, making him duck. Then cold sea wind, and the rustle of trees. When his eyes were unbound, he could make out all around him the pale shapes of funerary steles, like clustering finger bones in black blots of tree shadow, and the heavy loom of stone _turbe_ tombs. The moon had not yet risen, but stars glimmered feebly, so he could barely glimpse the hueless bulk that reared behind him: old watchtowers, decaying ramparts, a fosse thick with weed and shadow and the ghosts of men who'd died defending the walls. Black on black, touched only by the frailest of lights, the city's hills offered domes and minarets to an iron sky.\n\nOnly Zardalu stood beside him, smiling a little. His old-fashioned clothing\u2014pantaloons, tunic, pelisse of black velvet\u2014glittered with jewels.\n\n\"Now you will walk a little among the tombs, James, my friend, no?\" Effortlessly the painted nails slit through the cords around his wrists. Under the rouge and the paint on his eyelids, all rendered to dark smudges by the night, the white face was like something from a horrible dream, equivocal and boneless as the rest of his body. He shook back his long hair, dressed in womanly curls, and earrings flashed wetly in it. \"Parade yourself, as those Undead who find themselves in this city must, in politeness, parade themselves that the Deathless Lord may look on them and give them his leave or no to hunt. I hope,\" he added, with a corpse's widening grin, \"that you understand the rules.\"\n\n\"I think I do.\" Asher rubbed his wrists. Though smooth, the cord had been drawn tight and his swollen fingers were nearly numb. The thought of trying to make it back to the city walls, of playing hide-and-seek among the ruined passages of the abandoned towers with those who could see in midnight-black, had only to be framed to be discarded at once and utterly. Something flicked at his hair, like a sigh. He spun as if it had been the touch of a knife point, but there was nothing to be seen.\n\nZardalu laughed, a soundless gapping of the rubber mouth. His fangs were long and pointed, like a wolf's.\n\n\"So who are you in truth, Englis?\" he asked softly. \"And who is he whom the Bey thinks will risk himself to come to you? Since the waning of summer he has said, 'Find him and kill him.' Now he says, the one who comes to the Englis, bring that one to me.\"\n\nHe gestured around him at the crumbling _turbes_ , the steles with turbans\u2014or stylized veils\u2014carved on them leaning every which way, as if a giant child had randomly stuck a thousand thousand enormous matchsticks in the unkempt grass. \"Are you his servant? Or is it some secret that you know?\" The blanched eyes, dirty ice in the starlight, seemed for a time to be the eunuch's only reality, the rest of him a thing of smoke and dreams. Asher felt on his mind the narcotic pressure of the vampire's power, an almost impossible weight of sleep.\n\n\"I don't know what you're talking about.\"\n\n\"Who is this interloper, Englis? And what has it to do with the _dastgah_ and the silver bars that guard the way to the crypt?\"\n\nWith an effort, Asher pushed the soft cloudiness away. \"If your master is going to punish you for asking,\" he said, \"I think it would behoove me not to answer.\"\n\nZardalu flung up his hands in an exaggerated mime of amusement, but his anger was palpable. \"Behold the wise man!\" he cried, soundless as the night wind. \"Now all he needs is a little bell, like the goat they tie to summon the tiger.\"\n\nAsher felt the grip of his mind and tried to throw it off again, tried to follow where the tall vampire went, but could not. It was as if he woke suddenly again, standing alone in the frost among the rotting tombs.\n\nThey were all there somewhere, he thought. Zardalu and Jamila Baykus; Haralpos and Habib and Pelageya: watching him. An ambush. A trap. Anthea had told him of the strange condition, a sort of mental spell she sensed over the city, that prevented her from feeling the presence of any other vampire\u2014the work, she had said, frightened, of a great master or masters.\n\nAs he walked cautiously among the tombs, groping where the somber cypresses blotted even the wan glow of the sky, he sought to absorb as much of this landscape as he could. Had Anthea fled their lodging after his disappearance to hide in such a place?\n\nOr had his encounter with the men of the Sultan's guard, who had picked him up in the courtyard of the Mosque of the Bajazid, been engineered to leave her unguarded?\n\nThen why kidnap him less than an hour after his release, before he had even returned to her? Why use him, as he was being used now, as bait?\n\nWas it for her, even, that the trap was set?\n\nHe stopped to rest on the low flat tomb of some prince or noble, like a marble bench inscribed in flowing Arabic script and terminating in a narrow stele surmounted by the figure of a turban. The turban signified a man. The fact that it was depicted as tilted to one side meant the dead man had been strangled by the Sultan's order. The marble was starred white where bullets had struck it when the army came through here to their final battle with the Sultan's forces in July.\n\nAnd that final battle, he thought, had abruptly terminated whatever power Olumsiz Bey had held in the Sultan's court\u2014almost certainly financial, since the entire country was in pawn. With Abdul Hamid's imprisonment in Yildiz while the Committee of Union and Progress thrashed out how to get a Parliament elected and bring the empire into the twentieth century from a standing start in the sixteenth, the Bey had needed to find someone else to send to England, to conduct Ernchester here.\n\nFor whatever reason he wanted the earl here in the first place.\n\nSomething moved among the black trees, but strain his eyes as he would, he could make out nothing. A rat or a fox\u2014though if rats fled the smell of Anthea's hair upon her bonnet, it was unlikely they'd venture close to the silent watchers among the trees.\n\nHe slipped off the tomb and moved on.\n\nTombs clustered all the length of the land walls, from the seven-towered gate of the Yeni Koule to the mosque at Eyoub. People came here to pray in the daylight, but the _turbe_ themselves were undisturbed.\n\nSomewhere close dogs howled.\n\nHis personation of a goat in a tiger hunt lasted for what he calculated was almost two hours, judging by the moon's progress among the clouds. From the dark city the muezzins' final cries ascended, that deep, haunting wailing that is like no other sound on earth. In time, across the water, a church bell answered from Pera, small and clear.\n\nWas it Anthea they thought would appear? Or Ernchester?\n\nOr just possibly someone else?\n\nBy what Zardalu had said, Asher wasn't sure they knew exactly who it was they expected to trap.\n\n_Anthea,_ he thought, _fly this place. Go away._\n\nThen Zardalu was walking toward him, across open ground with the ashen grass surging around his pantaloons. When he bound Asher's wrists again and wrapped the scarf over his eyes, his hands were warm.\n\n\"You serve a heartless master,\" said the eunuch. \"Or maybe by this time he's found himself another servant, clever or no. Did he promise you everlasting life, James? They all do, you know.\"\n\n\"Even the Bey?\"\n\n\"Ah. An impudent infidel, no less.\"\n\nAsher could hear the smile in Zardalu's sweet whisper. \"Just curious.\"\n\nWhen they passed the city walls this time, there was no sound in the streets, save the crying of the gulls. Zardalu kept one hand on Asher's elbow, the other on the back of his neck, and the smell of fresh blood and the reek of death drowned out both the smell of the muck underfoot and the vampire's perfume.\n\nOnly when they were, Asher estimated, coming over the Second Hill again did he hear other voices and steps drawing near. A man mumbled, \"Beloved... beautiful fairy...\" in harsh-sounding Romaic Greek, and on the air, like the vapors of poisoned flowers, Asher heard the silvery flicker of vampire laughter.\n\n\"She's found a treasure, our Pelageya,\" Zardalu's voice breathed in his ear. \"How is it, _sagir sayyat_? Did you find a strong bullock to trap in your nets?\"\n\nThe Russian girl laughed, a soft, thick tickling that, in spite of himself and all he knew, went straight to his groin, as if the woman leaned naked in his arms.\n\nThey stopped. There was the sound of a key in a lock, impossible to tell what kind of key\u2014the man with them muttered drunkenly, swearing eternal love, promising feats of ecstasy that would have his newfound adored one crying out with gratitude, and all the while around him Asher heard the whisper of unholy mirth\u2014Haralpos, Habib, the Baykus Kadine. Their voices were a fleeting susurration, now before, now behind, as he was guided through a doorway and down long uneven stairs, worn in the center and incredibly deep, to a place that smelled of water and stone.\n\n\"That little beggar Habib's got won't be missed, but what of that bullock of yours? He looks well-fed.\"\n\n\"And if he is? He's an Armenian, she found him in the Kara Geumruk. The Sultan is quicker to avenge Gypsies and Jews than such folk...\"\n\n\"But is he sober enough to give us sport?\" Zardalu's drawling voice was petulant. \"Well enough to steal sleeping beggar children for El-Malik, but after a night sitting in a graveyard, with only one wretched tramp sleeping behind a tomb, I want a little sport.\"\n\n\"El-Malik entertains his infidel _makaniki_.\" He could almost see the Russian girl's lazy shrug. \"I can smell the coffee from the street. This one will waken enough.\"\n\nEl-Malik. The master, the king. The Master Vampire of Constantinople. And while they were talking, a sharp turn at the bottom of the stairs, two of his own steps, and the brush of a curtain against his face, right turn, wildly uneven brick underfoot and the sudden throat-catching stink of ammonia and chemicals, and a blast of cold.\n\nAnd far off, inarticulate with agony and horror, muffled as if behind some barrier of wood and iron, the sound of a man's voice.\n\n\"I came on one of the _makaniki_ the other night, as I was returning early,\" Zardalu was relating lightly, turning, Asher thought, so that his hand slid from neck to shoulder. Had it not, he thought the vampire must have felt the prickle of the hairs at the sound of that horrible, distant despair. \"A fat little infidel like an _asure_ pudding, with spectacles on his nose, so... He backed against the wall by the rear gate, holding his little hammer out like this, staring around squeaking, 'Who is that there? I hear you... You cannot get away... Come show yourself and I will not hurt you...'\" while the unfortunate Armenian youth mumbled endearments and Asher measured in his mind a narrow stair that wound around itself three times, then the echoes of some open room, and more stairs. Cobbly pavement of small stones underfoot, men of bigger ones, like cannonballs, in an open space where grass grew between blocks. Right, and a locked door...\n\nThey stopped, suddenly, in a room with a bare wooden floor. By their silence Asher knew why.\n\n\"Nothing?\" The voice was brown velvet, roses, and gold.\n\nBy the shift of Zardalu's grip, Asher knew that he bowed. \"Nothing, Lord.\"\n\nIn his blindness he heard the dense rustle of silk, but only when it was close enough that he could smell coffee, incense, ammonia... blood.\n\n\"Yet you have done passing well. Habib, my sweet, is that _sarigi burnia_ for me? What a dirty little thing she is. And ah, Pelageya...\" Asher could almost see him bow, and there was a momentary scuffle, the swish of clothing and a stifled grunt of terror as the young man suddenly, belatedly, realized that he stood in the presence of smiling death.\n\nA hand like animate steel brushed the side of Asher's face, almost in a caress. The scarf was slipped aside. Eyes that had once been coffee-dark but had been bleached, by a trick of the vampire state, to a garish and unnatural orange blinked into his by the glow of oil lamps close overhead.\n\nOlumsiz Bey stepped back.\n\nHe was as tall as Asher\u2014six feet\u2014and nearly as thin, but his shoulders stooped, giving the narrow, hairless head an uptilted angle like a tortoise's. The nose was an ax blade such as might have hewed the lipless mouth into existence with a single stroke, but it was not an unhandsome face. In one ear he wore a huge chunk of amber, as orange as his eyes, in which an ant was trapped, so big that Asher could see the curve of its serrated jaws; one almost expected to see other insects locked in the frozen prisons of his real eyes as well.\n\n\"It is probably well,\" Olumsiz Bey said to him in the flowery Osmanli of the court, \"that you return to your chamber now, Scheherazade, and remain there for the balance of the night. The tales we will tell tonight are not for the ears of the living.\"\n\nAsher's eyes went past him to the fledglings, grouped closely now around a husky young man with a prominent nose and dark, thickly curling hair. The young man was staring around him, growing horror struggling against wine and whatever glamours Pelageya had laid upon his mind, taking in the rich garden of blue and yellow tiles in the hall and the way darkness waited in every corner. Asher took it in, too, printing it in his mind... Habib, a coarse and powerful vampire who seemed to be special friends with Haralpos, carried, as Asher had deduced, a sleeping beggar girl of twelve or so, holding her against his shoulder as if she were an infant.\n\n\"Sayyed has already taken food thither for you,\" the Master of Constantinople went on. \"And books\u2014if you will pardon my presumption in choosing them for you\u2014to beguile with old legends the passing of the night. There will be... a little sport here.\" His smile had a flex, a curve to it, like a reflex that his eyes had long ago forgotten or had never known. He gestured with his right hand, for his left never loosened its hold on his silver-bladed weapon, which glittered whitely in the many-hued glow of the bronze lamps overhead.\n\nThe eyes of the fledglings threw back that glow, cats waiting to be fed.\n\nThe Armenian boy made a little noise of terror and tried to pull his arms free of Pelageya's grip and Haralpos', but he could not. Asher smelled urine as the boy pissed himself. He would give them the run they wanted, Asher thought bitterly, through all the dark galleries of that accursed house.\n\nAnd all the while he repeated silently to himself, _A cobbled courtyard beyond this place, smaller cobbles, right through a door, across a hall, down a narrow stair and then another twice as deep..._\n\nThe place of silver bars, where Zardalu said the _dastgah_ was, smelling of chemicals...\n\nAnd a voice that screamed its despair to the dark.\n\nThere was only one person he could think of whom the Bey would hold prisoner behind silver bars.\n\n\"My children forget themselves sometimes in their chase.\"\n\nHe jerked his mind back\u2014the Bey must not guess his abstraction.\n\n\"Yes, I really think it best if you remain in your chamber, and if any call out to you, save me alone, I suggest that you do not answer. My darling...\"\n\nThe Bey's jeweled right hand caressed Zardalu's cheek.\n\nThere was an impassive flicker behind the sapphire eyes, nothing more.\n\n\"I will take this one back to his chamber. Have Habib bring the child to my own room.\" He held up the scarf that had covered Asher's eyes, extended it to his fledgling once more. \"Be so good as to conduct my other guest of this evening back to the usual meeting place. Remember, I will know it if the slightest ill befalls him. Indeed, I shall know it if you so much as speak to him, as you did to this one, and he to you.\" The smile again, cold as his grip. \"And I will not be pleased. Is this understood?\"\n\nZardalu bowed again, bending his long boneless form so that his black curls fell forward over his shoulder and swept the wooden planks of the floor. \"This is understood, Lord.\"\n\n\"Come.\" Olumsiz Bey beckoned to someone who had stood all this time in the gloom of the room's inner doorway, and switched to German, perfectly contemporary and without accent or inflection. \"This man will take you outside. I guarantee that you need not fear him.\"\n\n\"I have no fear within your house, or anywhere that I walk, under your protection, my lord.\" Ignace Karolyi stepped from the darkness, his light brown Savile Row suit as incongruous in that setting as a khaki-uniformed Tommy with an Enfield would have been at Marathon. He stopped before Asher for a moment, regarding him with sudden, narrowed speculation in his wide-set brown eyes. Then he turned back to Olumsiz Bey and bowed.\n\n\"I trust that I am forgiven, my lord, and that terms between us can still be reached?\"\n\nThe Bey regarded him with strange eyes, holding his silver weapon before him, the edge glittering in the light. \"This remains to be seen. As all things do, it rests in the hand of God.\"\n\n# THIRTEEN\n\n\"I DON'T SEE why he can't come with us.\" Margaret Potton stepped down from the embassy carriage at Lydia's heels, and, trailed by a Greek footman, hurried in the wake of the formidable Lady Clapham, a tall, thin, horse-faced individual whom Lydia had guessed at once to be the headwoman of the British diplomatic community in Pera. \"You could introduce him as your cousin. When you told Sir Burnwell that you had a cousin here in Constantinople, I thought it was a good idea.\"\n\n\"I told him that in case we have to produce Ysidro in an emergency,\" Lydia replied, patient and somewhat bemused, but without anger. \"I don't think a diplomatic reception at the palace qualifies.\" Ahead of them, half glimpsed between strolling ladies in tulip-skirted ensembles and coal-scuttle hats that would not have been out of place in Paris or Vienna, Lady Clapham paused in the doorway of Mademoiselle Ursule's and looked back for her two charges. Lydia almost expected her to snap, _Step along, girls, spit-spot..._\n\n\"I don't know,\" Margaret said. \"I think it would be nice for him.\"\n\nLydia shook her head but was spared further discussion by conjunction, in the doorway of the boutique, with her guide and hostess for the shopping trip and the modiste herself, a middle-aged and firmly corseted Belgian woman who apprehended instantly the difference between Lydia's two-hundred-guinea, Alice-blue raw silk and Margaret's outdated brown wool, but varied not a whit the warmth of her smiles of greeting to both. It did cross Lydia's mind, as Lady Clapham explained to Mlle. Ursule what they'd come for, that Ysidro might have some difficulty these days in passing himself off as a living man.\n\nMargaret was staggered at the news that it was for her benefit, not for Lydia's, that they had made the excursion to the fashionable European shopping quarter along the Grand Rue. \"Silly goose,\" Lady Clapham declared, not unkindly, as the governess turned pink with pleasure. \"Of course you'll accompany Mrs. Asher tonight, and you certainly can't wear what you have on.\"\n\nLydia felt slightly relieved at this confirmation that other people\u2014older and in positions of social authority\u2014were far more tactless than she.\n\nMuch as it annoyed Lydia to admit it, Ysidro had been quite right. In Constantinople as in Vienna, Margaret Potton was her mantle of respectability, her mere presence making it entirely unnecessary for Lydia to say to anyone, _As you see, I am not a jauntering slut._ Her presence had certainly worked its intended magic at the embassy yesterday afternoon. Without Margaret, Lydia guessed she would still have been admitted, would still have had her queries answered... would still have spoken to Sir Burnwell, stooped and gray and with the slightly puffy face of an intermittent sufferer of kidney problems...\n\nBut only the presence and respectability of a companion had brought Lady Clapham into the office, holding out her hands and saying. _My dear, I'm so sorry..._\n\nSo sorry.\n\nCold closed around her again, dimming the voices of Mlle. Ursule, Lady Clapham, Miss Potton, as if the small, neat, and extremely Parisian room with its powder-blue satin wallpapers and gilt mirrors was at the end of a very long corridor.\n\nWednesday. James had been missing since Wednesday afternoon.\n\n\"Which one do you like, my dear?\"\n\nLady Clapham's voice pulled her back to the present. The dressmaker had spread out on the table two gowns, one straw-yellow with an overgown of white georgette, the other fawn-and-white-striped mousseline de soie trimmed with pink silk. \"I think that's up to Miss Potton,\" Lydia said, manufacturing a smile with an effort and stepping close to get some idea of how the dresses actually looked. Miss Potton turned red and pale and pink, and blotchy combinations of all three, and finally settled on the mousseline de soie, for which Lydia then bought a pair of white satin slippers, kid gloves, and a thin gold chain with a pendant of rose quartz and earrings to match.\n\n\"You really shouldn't have,\" Margaret said softly when, later in their bedroom, Stefania Potoneros was lacing her into the gown. \"I mean I... it must be terribly expensive.\"\n\nIt hadn't been, in terms of higher fashion, reflected Lydia, putting on her spectacles to turn and look over her shoulder at the girl. Mlle. Ursule had expertly graded ranks of gowns for all occasions, and the fawn and white silk, however pretty, was designed to be no competition whatsoever for Lydia's point lace and baby ribbons. But to a girl without a family, who had spent any number of years in the dreary confines of the typical governess' quarters, it must seem like Cinderella's ball dress.\n\n\"I can't...\" Margaret stammered. \"I can't repay you...\"\n\n\"Good heavens, no!\" Lydia said. There was a silence, Margaret undoubtedly remembering\u2014as Lydia remembered\u2014the hysterics in Sofia, the furious outburst upon their arrival the night before last. A little awkwardly, she explained, \"It's nothing, really. I mean... what's the point of being an heiress, and putting up with uncles and aunts telling you how to live and who you have to marry, if you can't... can't buy someone a present now and then? And I know it helps to have the right thing to wear.\"\n\n\"I thought if you were an heiress, it meant you could do what you wanted,\" said Margaret as Lydia barely touched the eiderdown puff to her cheeks, then leaned forward until her nose nearly touched the glass to inspect the results in the mirror.\n\nLydia shook her head. \"Well, I don't know about other heiresses. My father and his two sisters had a terror of fortune hunters, and my life was... rather restricted at times.\"\n\n_I'll not have you turning my money over to a scoundrel,_ had been her father's exact\u2014and oft-repeated\u2014words.\n\nNot, _A man who only marries you for money will make you wretched._\n\nNot, _How do you expect such a man to fit into the life you want to make for yourself?_\n\n_I'll not have you turning my money over to a scoundrel._\n\n_His_ money, even should he die.\n\nShe rubbed the rouge on her fingertips, smoothed the tiniest hint of a blush along cheekbones and temples, seeking the perfection that had been her only protection against everything they could do.\n\n\"It couldn't have been that restricted, if they let you go to Oxford,\" said Margaret. She picked up the powder puff, turned it cautiously over in one square, disapproving hand. \"Do all heiresses learn to use cosmetics like this?\"\n\n\"Only if they have a nose like mine.\" Lydia squinted at the effect of the rouge, then licked the end of her eye pencil and began careful shading along the upper lashes. \"James\u2014he was a friend of my uncle Ambrose, the dean of All Souls\u2014arranged with one of the pathology professors to help me borrow money under another name. I begged Uncle Ambrose not to tell Father, and I'm not sure he would have agreed if he'd known I was studying medicine. It was exhausting, going back and forth by train and concealing sessions when my tutor came down to town. Fortunately, our place was near Oxford\u2014Willoughby Close\u2014and Father spent weeks at a time down in London. If my mother had been alive, I could never have done it.\"\n\n\"What did they do when they found out?\" Margaret asked, blue eyes wide with alarm.\n\n\"There was a row,\" Lydia said evasively. Why, after eight years, did her father's cold fury still hurt? \"Would you like to try this?\" she added, seeing the other woman's hand stray to touch the rouge pot, the lip rouge, the several types of powder and skin food indispensable to the artifice that Lydia regarded as her armor against the world.\n\n\"C-could I?\" Margaret stammered, turning pink again. \"I know I shouldn't\u2014the sisters at the orphanage all said that ladies don't use such things...\"\n\n\"Well, I never met a lady who didn't.\" Lydia said with a smile. \"It's just that there's a trick to doing it so that nobody notices. Here.\"\n\nThe transformation was not a startling one, but having spent years compensating for what she considered her own shortcomings\u2014a slightly aquiline nose, too-thin cheeks, and unfashionably shaped lips, to say nothing of a preference for knowledge above society gossip\u2014Lydia knew how to apply rouge and powder to reduce the impact of the other woman's shallow chin and snub nose, and to give her better cheekbones than she'd been born with. At the end, staring into the lamplit glass, Margaret breathed, \"Oh...\" in a kind of wonderment, the blue eyes widened and deepened, the pale, pretty face surrounded by raven masses of curls as it had been, Lydia knew well, in her dreams. \"Oh, thank you!\"\n\nShe fumbled for her eyeglasses.\n\nLydia laughed. \"You aren't going to wear them to the reception, are you?\"\n\n\"Of course.\" Margaret settled them firmly on her nose, even as Lydia was removing her own to be helped into her gown by the maid. \"If people don't like me in my eyeglasses, that's just too bad.\" She blinked mildly at Lydia as the Greek maid laced her expertly up the back. \"Thank you,\" Margaret said simply. \"Thank you so much for doing this for me. I've never been beautiful before.\"\n\nLydia smiled a little and shook her head. \"I'll teach you how to do it, if you'd like,\" she said, stowing her spectacles in a silver-mounted leather case and making a final inspection of herself in the mirror. Stefania's sister Helena had come to the door twenty minutes ago with word that Sir Burnwell and Lady Clapham were waiting downstairs with the carriage; they should, Lydia guessed, arrive at the palace in fashionable good time.\n\nShe worked her tight kid gloves onto her hands and surveyed Margaret once more, pleased with the results in spite of the glasses. She had done her best\u2014the fact that Miss Potton was her companion was no reason not to make her as beautiful as she could be, though she knew girls of her own year as a debutante who would dispute that\u2014and she suspected that her companion's raven hair and tourmaline eyes made her prettier than herself.\n\n\"Margaret,\" she asked, as they collected reticules, fans, shawls, and keys, \"what are you going to do when you return? To London, I mean? I could help you...\"\n\n\"Oh, I'll leave that to Don Simon,\" Margaret said. \"My fate is in his hands.\"\n\nShe smiled happily and followed Lydia down the stairs.\n\nThe reception was held in a medium-sized pavilion in the inner garden court of the old palace of the sultans, flanked by plane trees and surrounded by a colonnade of shallow, green-tiled domes. The Sultan himself had not occupied the Topkapi Palace for a good fifty years, but the new government\u2014the Committee of Union and Progress\u2014used it for state functions, and this three-room suite, though a little small for a reception and rather stuffy with its low, coffered ceilings and Western-style crystal chandeliers, was at least unhallowed by any sort of Imperial tradition.\n\n\"Ambassador Lowther hardly knows whom to speak to these days,\" Sir Burnwell confided to Lydia as gorgeously caparisoned palace servants divested them of coats and cloaks in the doorway of the kiosk's small service room. \"It's like the old story about the seer who was right half the time, but one never knew which half. The C.U.P. holds power in patches, but nobody knows which patches they are.\"\n\n\"At least under the old Sultan one knew whom to bribe.\" Lady Clapham brushed straight the folds of her periwinkle and gold chiffon dress, and nodded approvingly at both the younger members of the party. \"Don't worry, my dear,\" she added more quietly to Lydia. \"If there's anything to be found about your husband, we'll find it here. I know at least someone who saw him Wednesday afternoon. I hope he's here... Russians have such an Oriental idea of time.\"\n\nShe led the way into the main hall, where the reception line moved slowly past the bearlike Talaat Bey, the new lord of this place where the sultans had reigned for five centuries, and the Romeo of the new army, the beautiful Enver Bey. The room was crowded with men and women dressed in the height of European fashion\u2014most of them fair-skinned and all of them speaking French\u2014and servants in old-fashioned turbans, suppers, and pantaloons bearing silver trays of refreshments. Lydia noticed Miss Potton craning her neck, looking around her, presumably in the hopes that Ysidro would have followed them here after all.\n\n\"Andrei!\" Lady Clapham called out and moved into the crowd, returning a moment later with a hunter-green colossus on her arm. \"Prince Andrei Illyich Razumovsky, of the Russian Embassy; Mrs. James Asher. His Highness is an acquaintance of your husband, my dear. He was the last one to see him after that affair with the Sultan's guards Wednesday, weren't you, Andrei?\"\n\n\"The Sultan's guards?\" Lydia raised her eyes to the man who towered over her, the impressionistic glitter of bullion, buttons, epaulets, fringe, and a beard of still-brighter gold resolving themselves into a good-humored, handsome face and bright blue eyes as the prince bent to kiss her hand. _Slavic facial angle,_ Lydia thought automatically. _Brachycephalic. Cranial index about 82. I really_ must _stop seeing people in terms of their internal structure..._\n\n\"There was little harm done,\" the prince said in beautiful Oxonian English and offered her his arm. Lydia followed him back out into the colonnade, where electric lights had been incongruously strung from pillar to pillar. A few men stood at one end of the arcade smoking\u2014Lydia caught the acrid whiff of tobacco, but at that distance they were little more than a clump of black forms spatchcocked with the white of shirtfronts.\n\nThe day had been a cold one, and few ladies, bare-shouldered as she was herself, ventured into the sea-chilled darkness.\n\n\"Your husband had lodgings here in Stamboul,\" the prince went on when they were out of earshot of the smokers. \"Most Europeans prefer to stay in Pera, of course, particularly since the coup. There haven't been riots among the Armenians in the past week or two, but fighting in the streets between the Greeks and the Turks can't be stopped. Your husband...\"\n\nHe gazed down at her for a moment from his great height, and Lydia could see him asking himself what he could, in discretion, ask her. The look in Lady Clapham's eyes when she'd said, _An acquaintance of your husband,_ had told her exactly what this \"junior attach\u00e9\" did in the Czar's service.\n\n\"I know that my husband came to Constantinople to ask the advice of... certain friends.\" She laid the same emphasis on the last words and met his eyes. The corners of them crinkled in a little smile. _Yes, I know my husband was a spy and you still are._ Presumably, she thought, Lady Clapham wouldn't have introduced them that way if Russia was an ally of Austria. Whose side was the Ottoman Empire on?\n\n\"Ah,\" he said. \"As you say, Madame Asher.\" His smile widened. \"Then you know that he probably had his reasons. You wouldn't happen to know what those were?\"\n\nShe shook her head. \"I only knew that he might be in trouble. Sir Burnwell told me he arrived in Constantinople a week ago yesterday, and that nobody's seen him since Wednesday afternoon.\"\n\n\"And what sort of help did you believe you could be?\" He spoke kindly, but she could see something else in his gaze. Just because we're allies, Jamie often said, doesn't mean we're on the same side. She felt panicky again, as she had in Vienna, panicky and unable to make a correct choice.\n\nForcibly, she put the panic aside. \"I thought I could recognize the man who might betray him,\" Lydia lied, with what she hoped was calm. \"I don't know his name.\" she added, and went on at once, \"But what happened Wednesday afternoon?\"\n\nRazumovsky looked as if he might say something else, but changed his mind. Probably, thought Lydia, because he thought it likelier he'd get more information later if he gave a little himself. He might even actually like Jamie\u2014he looked like the sort of person Jamie, and in fact she, would and could like.\n\n\"As I said, he had lodgings on the Stamboul side of the Horn.\" The prince lowered his voice and glanced along the colonnade to the group of smokers again. None looked in their direction, but the prince guided her down the short flight of marble steps that led to an arched tunnel beneath the pavilion, and so through to the dark gardens beyond. \"He told no one where they were, and when I saw him, he had the look of one watching over his shoulder. On Wednesday men from the palace intercepted him by the Grand Bazaar, sent by the High Chamberlain, they said\u2014though anyone could have bribed him to do so.\" He grinned reminiscently. \"I've bribed him to do similar things myself.\"\n\n\"And he sent to you for help?\"\n\n\"We've been friends a good many years,\" said the Russian. \"Sir Burnwell would probably have complained to the army first, or the C.U.P., and been put off for God knows how long. Semibarbarity has its advantages. I came here\u2014where the Chamberlain and in fact the Sultan still hold a good deal of power\u2014and blustered and shook my fist. Shook my country's fist, which frightened them even more. Already the Sultan is playing off the people against the army, trying to rouse them in a countercoup, for he wields power as the head of the Mohammedan faith, you know. If it comes to it, the Chamberlain and his master are going to need support.\"\n\nLydia shivered, remembering a scene glimpsed from the window of the embassy carriage as they'd clattered along one of the few streets in the old city wide enough to admit such a vehicle: three men, dark-haired and hook-nosed, in the khaki uniforms of the new army, beating up an old man outside a half-closed shop. A muttering crowd had gathered, but no one had dared interfere; the old man had only put his hands over his head for protection, as if he knew perfectly well that begging for mercy or asking for help were equally out of court.\n\n\"They brought him out in a short time,\" Razumovsky went on, stroking back the surge of his golden mustaches. \"As I'd suspected, they were holding him in the guardhouse here, which means it was the Chamberlain who'd been bribed. He had been knocked about a little, nothing serious.\"\n\n\"I hope he put proper antiseptic on it,\" Lydia said, and was startled when the prince burst into laughter. \"I mean,\" she added hastily, realizing how that had sounded, \"I'm quite shocked, of course, that he was hurt, but if he _will_ get into danger... What had he been doing?\"\n\n\"Apparently\u2014he did not tell me this, but I found it out through palace contacts of my own\u2014questioning storytellers in the markets. That was how they knew where he would be.\"\n\n\"Storytellers.\" _Old man who lived to be a thousand..._ The wandering script of Fairport's notebook sprang immediately to her mind. _Woman who lived to be five hundred (wove moonlight)._\n\n\"You tell me why,\" said the prince.\n\nLydia only shook her head, though a numbness started behind her breastbone and seemed to spread to fingers, lips, toes. Stress on top of hypothermia, she thought. And then, a small inner voice like a child's, _Jamie, no_...\n\n\"You're cold, madame.\" The prince put a warm hand to the small of her back and led her up the steps again, toward the brighter lights at the other end of the arcade. \"We were walking back to his rooms in the Bajazid when an Armenian boy came up to him. I didn't hear all the boy said, but I know he said, 'My master told me to show you the place.' Jamie took his leave of me...\" He shook his head.\n\n_Did he look well?_ she wanted to ask. _Did they take his knife when he was arrested, and did he get it back? Did you see if he still had the silver around his neck, on his wrists?_\n\nIt was conceivable, she thought, that the Sultan's guards had stolen it. The ones she'd seen at the palace's outer gates looked capable of relieving a dying man of his shoes.\n\nUnder her corsets her heart seemed to be pounding uncomfortably fast.\n\n\"Your palace contact didn't happen to say which storytellers, did he?\"\n\nRazumovsky stopped, gazing down at her again. Men had appeared in the colonnade, Europeans in bright colors that had to be uniforms. By the way they were looking around, Lydia guessed they were the prince's own attach\u00e9s.\n\n\"Mrs. Asher,\" he said quietly, \"Constantinople is not a good city. It is not a safe city, especially now, with the army in power and turning things upside down, and it has never been a good city in which to be a woman. I have been making inquiries of my own about James. When I hear anything, even of the smallest, I will send to you at once.\"\n\n\"Thank you.\" Lydia clasped the broad, kid-gloved hand. \"I can't tell you how much I appreciate that. I can't... there are reasons I can't tell you how I know... what I know. But any help you can give me...\"\n\n\"On this condition.\" Razumovsky brushed at his mustaches again. His glove buttons had diamonds in them that twinkled like tiny stars. \"Something tells me I do not need to tell this to you, but I will anyway. _Do not investigate anything alone._ Not anything. Call on me for help at whatever hour. Is there a telephone where you're staying?\" She shook her head. \"Then send a page. Do you understand? If I can't come, I'll send a servant. You don't need to tell me or him or anyone where you're going, but _don't go alone._\n\n\"Sir Burnwell and the embassy staff are good men, but they haven't been here as long as I. Moreover, they are perceived as being on the side of the C.U.P., and against the old powers. In any case the German businessmen who've advanced money to both sides hold more power here than either my embassy or yours. When you go about the city, take someone with you\u2014someone besides that silly girl of yours, I mean\u2014and don't assume that you can get away with anything safely. This isn't England. There,\" he said, and led her back toward the lights, the smokers, the door with its tall guards in their billowy pantaloons and turbans of orange and red. Not until they were inside and he had fetched her champagne and a cracker of sour cream and Russian caviar did he excuse himself, and two minutes later she saw him\u2014or at any rate someone his height with a gold beard and a uniform of hunter-green\u2014deep in conversation with Enver Bey himself.\n\n# FOURTEEN\n\nTHE ROOM WAS more crowded than before. During her conversation with the prince, Lydia had been dimly aware of lights passing among the trees and hedges as servants conducted newcomers along the paths from the enormous outer court. Scanning backs, Lydia identified the asymmetrical mauve volutes of her patroness' gown in the midst of a dark cluster of male suiting. As she approached, she heard the guttural babble of German and made out references to track miles, rolling stock, gauge widths, and Krupps that told her that Lady Clapham had fallen among the businessmen, but in any case Lady Clapham held out her hand to her with the air of a somewhat long-toothed Andromeda greeting a schoolgirl Perseus in ecru lace and pink ribbons.\n\n\"My _dear_ Mrs. Asher,\" she cried. \"May I present to you Herr Franz Hindi? Herr Hindi, Mrs. Asher. Now if you'll please excuse us, Herr Hindi, I promised to introduce Mrs. Asher to Herr Dettmars... You're a godsend, my dear!\" she added in a low voice as the stout, fair-haired gentleman who had shaken Lydia's hand was left behind with considerable celerity. \" _Such_ a bore.\" She steered her into one of the smaller rear chambers of the pavilion, just as crowded and if possible more airless than the long front room. \"Do I have the appearance of a woman who will perish if she does not receive accurate information concerning the differences between soft-coal hummer furnaces and hard-coal base burners?\"\n\nLydia paused to study her with mock gravity. \"Turn 'round,\" she instructed, and with a straight face the attach\u00e9's wife did so.\n\n\"Only a little in the back,\" Lydia replied after due consideration.\n\n\"I'll wear a shawl over it, then,\" promised Lady Clapham. \"I am suffocating. Was Prince Razumovsky able to give you any information about your husband, dear?\"\n\nLydia nodded slowly. \"He told me my husband was doing some kind of research, talking to storytellers in the markets. Did he\u2014Dr. Asher, I mean\u2014mention this to you?\"\n\n\"That isn't what brought him to Constantinople, surely?\"\n\n\"No,\" Lydia said. \"But he does research in such things wherever he is. He's a folklorist as well as a linguist.\"\n\nLady Clapham sighed resignedly and poked at her untidy, graying coiffure. \"Well, better than one of those lunatics like my brother, who goes about taking rubbings off tombs. Not even in heathen parts but in places like Wensley Parva and Bath Cathedral. _And in hunting season!_ \" She shook her head wonderingly and picked a cracker of caviar from a servant's tray as if the man had been a table. \"Yes, he did ask about storytellers. Burnie told him about the old fellow who sits in the street of the brass sellers in the Great Bazaar. Did His Highness offer you his help? I thought so. Just make sure you have Miss Potton with you at all times and you should be quite all right. Where _has_ Miss Potton got to?\"\n\nLydia gazed around the small chamber. Though without her eyeglasses most men in crowds looked alike\u2014except James, of course, whom she would know anywhere under any circumstances, and human Christmas trees like Prince Razumovsky\u2014she could generally spot women by the colors and shapes of their dresses. But there was no sign of the fawn-and-white silk among the crowd, no ink blot of black curls glistening in the sharp yellowish light. She remembered Ysidro remarking last night, _I may be somewhere thereabouts,_ and Margaret's desire to see him at the reception... And the more so now, to show him her newfound beauty.\n\n\"She may have gone into the gardens.\" The image of Margaret, in improbable Georgian panniers and wig, waltzing with Ysidro on the terrace of some dream mansion, floated through her mind.\n\n\"She'll freeze,\" Lady Clapham predicted. \"Oh, my dear, there's someone I _do_ want to introduce you to... absolutely charming, and such a cut-up...\" She was already starting to lead her toward a man who had just entered the smaller room. Another uniform, this one scarlet, heavily braided with silver and ornamented with, of all things, a leopard skin over the shoulder, set off dark hair and a stance that told her at once, without being near enough to see his face, that he was as handsome as Apollo and knew it. All Adonises, she reflected\u2014or was that Adoni?\u2014seemed to stand in the same way. She wondered if anyone had done a study on the subject. Not that anyone but a woman would notice, of course... \"...member of the diplomatic community here and an absolute charmer, even if he's never going to rock the world with his intellect. Baron Ignace Karolyi...\"\n\n\"Excuse me,\" Lydia said hastily. \"I think I see Miss Potton and I really do need to... I'll be back in one moment...\"\n\n\"Really? Where...?\"\n\nBut she dodged away into the crowd.\n\nFortunately, a doorway connected that room and the other rear chamber of the suite. Lydia ducked through, wove her way to the door leading back into the main salon, and worked back with what speed she could\u2014given a visual range of less than a yard, though the brilliance of the man's uniform helped in avoiding him\u2014to the double door leading into the colonnade. The cold was sharp. Wishing she'd had time to fetch her cloak, Lydia hurried along the black and white cobbled pavement to the stairway passage in which she'd taken refuge with the prince, and gathered her point-lace train in hand to descend the sloping tunnel to the terrace beyond.\n\nOnce certain she was out of sight, she pulled her spectacles from her handbag and settled them on her nose.\n\nWhat had been an impression of leafy blackness and swimming spots of color resolved itself suddenly into a sable wonderland of cypress and willow that sloped down to the indigo shimmer of the sea. Bare boughs or somber leaves were illuminated from below by a rainbow lace of colored lamps, which outlined paths and terraces like dim-burning jewels dropped on velvet.\n\nTo her left the lights traced terraces, stairways, the eaves of pale shut-windowed pavilions in a flickering web of ruby, azure, honey stars... and at the top of a flight of marble steps she saw one star was missing. A lamp had been taken.\n\n_Margaret._ She didn't know why she was so sure. Gathering her train more firmly, she hastened along the terrace and up those pale steps to the gap in the line of lights.\n\nA gem-latticed darkness of marble pavements and low box hedges spread out before her at the top, rimming deep stands of lawn and trees. The pavement led her around to the locked doors of the two pavilions overlooking the lower gardens. Past the second pavilion's door a low arch of very old bricks pierced the wall, marble steps leading down again, through a vaulted runnel, to the terraces below.\n\nHad Margaret seen Ysidro in the gardens? Or only a shape she thought might be his?\n\nShe turned back to scan the colonnades, the elaborate pavilions above and behind her, but saw no movement there; neither was there any sight of the pale mousseline de soie dress in the semiwilderness of trees and long grass that lay between her and the sea. She pulled a handkerchief from her bag to shield her fingers from the heat, then picked up another lamp, the brass base beneath the bowl of ruby glass hot through both cloth and glove. One of the innumerable wild cats that lived in the half-deserted shrubberies stared at her for a moment, then poured itself away into the darkness.\n\n_What am I doing?_ wondered Lydia, half in disgust, as she descended the marble steps. _Two minutes after the handsome Russian prince warns me \"Don't investigate alone,\" I'm off like the heroine of a cheap thriller..._\n\nBut something about the shadowy darkness of the palace. deserted once the activity around the kiosks had been left behind, filled her with fear for the sake of the younger woman. The sight of Karolyi had shaken her, and she did not think she dared either wait or go back.\n\nThe red light of the lamp caught in the curves of an iron lion posted in what had been flower beds. On a tangle of overgrown rosebush, Lydia glimpsed white threads where a petticoat hem had caught and been pulled free.\n\nThere was a door, hidden in the shadows of the three high vaults of ancient brick. It stood open. For a long time Lydia hesitated in the narrow aperture, one hand pressed to the stone jamb, the red-glowing lamp raised to look within. The stagnant pool a few yards behind her seemed to breathe cold over her bare shoulders, an echo of the damp chill that lay before her in the dark.\n\n_Little did she know,_ quoted Lydia from the aforesaid cheap thriller, in an effort to push back the dread whispering at her heart, _what horrors lay crouched in wait for her._\n\nBut it was only a stone stairway\u2014used, she thought, but not recently, save for the wet tracks vaguely outlined on the upper step or two.\n\nA woman's slippers.\n\n_Idiot, idiot, idiot._ She wasn't sure if it was Miss Potton or herself to whom she referred.\n\nAt the bottom of the stairs, another open door, and a cavern vast and lost in shadows, where the ruby stain of her lamp smudged pillars, incredibly old, rising out of obsidian water to the brick vaulting of the ceiling low overhead.\n\nOf course, Lydia thought. All those pools in the gardens had to be watered from somewhere.\n\nA walkway stretched along one side of the cistern, vanishing very quickly into darkness. Heart beating hard, hoping she'd find Margaret soon, she started along it.\n\n\"This is not a wise thing, mistress.\"\n\nYsidro's voice was barely louder than a cat's tread in the dark behind her, but somehow it didn't startle her. It was as if, for the second or two before he spoke, she knew he was there. Turning, she saw him on the walkway, dressed, as the men at the palace reception had been dressed, in black morning coat and gray-striped trousers, colorless hair framing a dead man's face.\n\nHer breath escaped in a shaky sigh. \"Coming to Constantinople was not a wise thing,\" she said. \"I wondered what you had in that trunk of yours. Did you bring a top hat as well?\"\n\n\"It is where I can reach it, should I choose to enter the pavilion.\"\n\nHe stepped closer and took her hand, guiding her along the path above the sable pool. The light seemed to follow, like a fish in the depths. Cold as she was, his hand on her waist was colder.\n\n\"The sultans used to bring the ladies of the harem up this way, when they watched polo or archery from the kiosks on the terrace.\"\n\n\"Have you found any trace of her?\"\n\n\"She did not pass you, then?\" In the evenness of his voice she read his irritation. He knew whom she meant and what had happened. Then, \"My concentration has been on other matters. It is difficult...\"\n\nThe uninflected words might have been a complete sentence instead of a broken beginning, but Lydia knew what he stopped himself from saying to her. They stood for a moment face-to-face in the open door of another stair, with the lamp between them, as they had stood in the stairway of his London crypt. The blood-hued light made him more alien still, and she had the curious sensation that if she closed her eyes his features would shift and be no longer the face he was always so careful to show the living, but the face he turned away from mirrors in order not to see himself.\n\n\"It's my doing.\" She wondered what else she could say. _I'm sorry I asked you not to kill innocent strangers on the streets, in the train, in the corners of this palace?_\n\nIn time he said, \"No. My own, for supposing I could have my way without price. I will survive it.\"\n\nAnother silence. Lydia remembered Margaret's white breast the night before last when she'd torn open her bodice on the empty street. She had to ask, though she knew it was none of her business. \"Are you drinking her blood?\"\n\n\"It would do me no good,\" replied the light voice, but he seemed unsurprised by the question. \"It is the death we need to feed the mind's power. At this point it were too easy to kill her, did I but taste of her blood.\"\n\n_I should be afraid of him._\n\nAnd it _was_ her doing.\n\n\"It is no easy thing,\" he went on, as if he had read her thought, \"to see myself in the mirror of your honor. Let us hang a shawl before it, as I do the mirrors in my house, and deal with commonplaces as we find them. You're cold.\"\n\nShe realized, as he guided her up the long flight, that she was trembling.\n\nShe had no impression of him leaving her side after they reached the door at the top, but somehow he had a shawl in his hands, heavy silk with a hand like cream as he draped it around her shoulders. \"This is not a safe place to walk.\" He stretched his fingers in the direction of the lamp and in some fashion snuffed the flame without touching it. They passed into a courtyard barely wider than a hall, stairways going up and down into impenetrable night. Dark lay like the seal of death, so that he had to guide her, his fingers tombstone marble through the thin kid of her glove and his.\n\n\"I saw her footmarks when I returned to the cistern stair,\" he said. \"They were unclear, and I had to look on the walkway to be sure she had not passed that way going out.\" He paused and added something Lydia knew enough Spanish to identify.\n\n\"You chose her because she was stupid,\" she reminded him softly. \"Stupid and loyal. What she feels for you was your doing.\"\n\n\"It is one matter to follow a husband whom you know to be walking eyeless and unarmed into treachery.\" They passed into a chamber, crossed layers of dust-thick carpet and ascended a rickety stair to a balcony enclosed by lattice\u2014down another stair and so out again. \"You sought advice in the matter, recognizing your limitations, and his. It is another matter to pursue needlessly one to whom you will be naught but a liability, only to tell him what he already knows.\n\n\"This is no safe place, not for her to walk, nor for us to call out, nor to hold aloft lamps that she may see their light.\"\n\n\"This is the harem, isn't it?\" The name conjured images hopelessly romantic to Lydia's mind, but the room they entered\u2014and indeed, all the rooms along this lightless slit\u2014even unfurnished, seemed poky and cramped in the filtered rays from some other wing of the building. The walls were plain plaster, unpainted, dirty and mildewed. The divans were lumpy and far lower than Lydia had pictured from storybooks, about the thickness of a good mattress. The carpets were threadbare, smelling of mice and rotted perfume.\n\n\"I thought the palace hasn't been used since the fifties.\"\n\n\"Not as the Sultan's residence.\" The voice might almost have been the exhalation of dust from the carpets underfoot. \"It was the center of government until last July. But a part of the old seraglio is where he put women who belonged to his father or his grandfather, or girls who failed to please him. Here they dwell still, with their servants\u2014fewer, but much as they used. In the heyday of this place they slept, four and five to a room, the ones who did not catch his fancy, seeing no one but the eunuchs and each other, seldom even seeing the sun.\"\n\nIn the almost dark she saw him touch the wall in passing. \"They lived upon opium, many of them; opium and intrigue. The walls here sweat with their pettiness, their boredom, and their tears.\"\n\nHis eyelids lowered and he tilted his head, listening. \"There,\" he whispered. He guided her with swift and weightless stride down a stair as steep as hell's abyss and so dark she couldn't see the steps thereof. Later on, safe in her own bed in Pera, Lydia wondered a little at her absolute trust in him, her willingness to step forward in utter darkness, propelled by his hand. Not, she thought, that Ysidro would have given her any choice.\n\nMargaret stood in the midst of a large chamber that once had a sunken pool in its center, now only an oval of shell-edged shadow. Marble lattices covered the windows on three sides; a divan circled the chamber, and slanting squares of light no bigger than tea sandwiches strewed the dirty and mouse-ravaged cushions. The whole room choked of mildew.\n\nShe had no lamp in her hands now, as if she'd set it down somewhere and left it forgotten. In the checkered glow from the windows her face was blank; behind the thick lenses of her spectacles, her eyes were those of a sleepwalker.\n\nShe looked beautiful, as she had looked in her dreams.\n\nLydia found herself alone in the tiled entry way looking at Ysidro as he turned Margaret's head gently, so that he could see the exposed\u2014and unmarked\u2014whiteness of her throat.\n\n\"Margharita,\" the vampire whispered. The girl startled like one waking.\n\nThen Margaret's breath drew in a hoarse gasp, and she flung herself on Ysidro, clutching him with desperate, grabby hands. The next second, past his shoulder, she saw Lydia, like some bespectacled, prosaic ghost with her train a cascade of lace over one kid-gloved arm, her shoulders draped in the faded web of an old silk shawl. Margaret backed quickly. \"I... are you all right?\" It wasn't to Lydia that she spoke.\n\n\"Indeed.\" The vampire inclined his head politely. \"Less so than I had been, had I not come back to this place to seek you, however. It were foolish of you to follow me, Margharita, for your reputation's sake alone, and your safety's. And mine, and Mistress Asher's, too, coming to find you here. Now let us return, ere our absence causes remark; and I warn you, do not come after me thus again.\"\n\nHis voice never rose above its usual even key, nor did its tone change one whit from the polite phrases of his accustomed speech, but Lydia cringed inside as if at sarcasm or curses. Margaret's cheeks flushed dark and she looked away, and for a moment Lydia had the impression she would have fled, plunging into the unknown labyrinth of the deeper harem, had not Ysidro laid an imperative hand on her arm. Her voice trembled as she looked back at him with tear-filled blue eyes. \"I was only afraid...\"\n\n\"Afraid?\" He smiled his chilly smile, manufactured, Lydia guessed, to cover the remainder of his anger. Still, the impact of it was startling, the echo of an astringent charm that had been the living man's. \"That I should find peril here beyond my capacity, from which you could save me?\"\n\nNo expression, no inflection; he had been dead, Lydia recalled, a long time. But still she guessed the smallest twinkle of banter, far back in the sulfur-crystal eyes.\n\nMargaret didn't. She only hung her head and snuffled, and suffered Ysidro to take her arm and lead her through the maze to the perilous cistern stair, and thence back along the terrace where the harem ladies had gone to their lord. As they passed through a vast court above a terrace and pool, where shuttered windows hovered tier upon tier above their heads, Lydia thought she saw the glow of a lamp left under one of the ramshackle stairways, and made to turn aside.\n\n\"Leave it,\" Ysidro said softly. \"It will only draw those we have little desire to meet.\"\n\nLydia removed her spectacles again and folded the shawl inconspicuously in the cloakroom before reentering the diplomat-crowded salon. She concentrated, through the remainder of the reception, on avoiding an encounter with the straight, graceful figure in the crimson uniform of the Hungarian Life Guards.\n\n\"You watch out for that Razumovsky, mind,\" Lady Clapham said to her as they were getting into the carriages. \" _And_ watch that girl of yours.\"\n\nStartled, Lydia turned to regard Margaret, being helped by servants into the embassy coach. Soldiers clustered in the small square, torchlight throwing sharp flares on their rifles, for warning had come of sporadic fighting among the Armenians in Galata that might spread to Stamboul.\n\n\"I really don't think we need worry,\" she said. \"I happen to know her heart is... otherwise engaged.\" To someone, moreover, infinitely more dangerous than a Russian nobleman.\n\n\"I mean watch what she says.\" Her Ladyship drew Lydia a little farther back into the darkness of the gate. The shadows of the soldiers wavered drunkenly across the vine-grown brick wall opposite, behind which the silent domes of the Aya Sofia slept in the dark.\n\n\"And what _you_ say. Razumovsky isn't a fool, and he knows perfectly well your husband didn't come to Constantinople to interview storytellers. That treaty the King signed won't cut much ice if the Czar sees a chance of getting a point ahead of us, either here or in India.\"\n\nLydia sighed, reassured her hostess and shook her head inwardly as she took Sir Burnwell's hand to ascend to the coach. At least everyone in the world had cardiovascular systems and endocrine glands, and there wasn't any argument over those. For a moment she thought longingly of the Radcliffe Infirmary, where things were safe and in their places\u2014 _was_ Pickering keeping proper graphs of the long-term weight gain of those subjects? She had no idea what she'd tell the editors of the _Journal of Internal Research_ about her article. _I'm sorry, I had to go to Constantinople to rescue my husband from vampires._\n\nBut without Jamie...\n\nShe shook her head. She would find him.\n\nShe had to find him.\n\n# FIFTEEN\n\n\"WHAT WAS IT you were afraid of, in the seraglio?\"\n\nYsidro did not turn. Upon bringing the women back to the house on Rue Abydos, he had uncharacteristically made sure that Margaret got safely to bed, then gone to the floor below to sit in the parlor's projecting bay, a sort of balcony that overlooked the front door. For nearly an hour Lydia had been aware of him there, as, still in her evening frock, she drank the aromatic tea Madame Potoneros brewed for her.\n\nIt was late, close to three. The near-riot in the Armenian quarter had forced a long detour through the market district to the old Mohammed Bridge; even then, winding their way up the steep Rue Iskander, they could hear the distant cries, the breaking of glass, the shots. Sitting quiet between Margaret and Lady Clapham, Lydia had pulled her cloak closer and wondered if she'd ever feel warm again.\n\nThere was still no emphasis, no rise or fall, to his voice. \"So you, like Margaret, suppose me to have been in peril? I thought better of you, mistress.\"\n\n\"Well, I do know you're perfectly capable of avoiding any twelve saber-wielding eunuchs out to protect the Sultan's name from dishonor. So what _were_ you afraid Margaret had encountered?\" She thought it through, then asked, \"Another vampire?\"\n\nHe tilted his head a little. Late-risen moonlight edged his profile in watery milk. \"Her name is Zenaida. I went to the seraglio to speak to her, before ever I knew Margaret had followed.\"\n\nHis hands, lying one atop the other on the window's sill, seemed about to move, then subsided again into quiescence, the echo of some gesture pared away by time. \"She has been there a long while, and no longer recalls the name of the Sultan for whom she was first bought in the markets of Smyrna. Perhaps she never knew it. Like most of the Sultan's women, she was cunning but stupid, and uneducated as a peddler's donkey. She told me many of the odalisques still think she is a living woman, some forgotten Sultan's _kadine_.\"\n\n\"And you think she may know something about... about Ernchester? Or James?\" He sat on an old chest that did service for a low table in the bay; she leaned against the corner of the wall. The windows were open behind their lattices, and listen as she would\u2014she could not keep herself from doing so\u2014she did not now hear any sound from the slums that lay all along the foot of the hill. Smoke still gritted in the air.\n\n\"That,\" the vampire agreed quietly. \"And other matters.\"\n\nHe gazed for a few moments more in apparent disinterest through the carven screens to white walls and tile roofs. The City of Walls, with its minarets and domes, its markets and its filth, was no more than a great shoulder of rucked velvet across the water in the night.\n\nThen the yellow mantis eyes shifted to hers. \"My senses, my perceptions, my ability to touch the threads of thought and scent and heat which move upon a city's air\u2014these have suffered from lack of their proper feeding. Nonetheless I should be able to feel some of what takes place in the lives of night-walking things. If not from here, from the gatehouses of the palace where I stood tonight, from the hill of the Aya Sofia, where all the dreams of the city come together like light in a glass. And I do not.\"\n\nLydia pushed her spectacles up onto her nose. She'd taken off her gloves and her pearls, and the silver shone on her throat and wrists like looped links of ice. \"And the last thing you needed was a couple of silly heroines to look after,\" she said, rueful and shy.\n\nHis head moved again, once, and his eyes met hers with that brief flicker of human amusement. In the street below a dog barked, the gruff shrillness picked up in another alley, and another, as all that starveling horde felt called upon to comment and reply. Ysidro waited them out, listening as if he could distinguish some clue within the sound.\n\n\"I walked in Galata last night when I left you here,\" he said in time. \"I crossed the bridge to Stamboul and sought out the other quarters where the Armenians live, down seaward of the Burned Column and in the poorest quarters along the walls. It is there, you understand, that the vampires will hunt, among those whose deaths the Turks count as less than the scraps I feed my cats. The miasma was thick there, the sense of diverted attention, of watching through smoke, though the night was clear. It was like the veil we lay over human eyes and human minds, but the veil was of a different quality, a different texture, wrought to shield a different kind of mind.\n\n\"There is war between vampires in this city.\"\n\nLydia recalled the elaborate precautions in Ysidro's London house\u2014or one of his London houses\u2014and it occurred to her that human incursion might not be the only threat against which he protected himself.\n\n\"You think one of the Master of Constantinople's fledglings is... rebelling against him? Trying to overthrow him? And summoned, or blackmailed, Ernchester here to help him?\"\n\n\"It could be that,\" agreed Ysidro. \"It can happen so, though as a rule a master as old as that of Constantinople will show more care in who he makes into his fledgling. Or a newcomer has arrived from the outside, in flight from his or her own master vampire, and seeks to take over mastery of Constantinople himself. This he will find no easy matter.\"\n\n_\"Ernchester?\"_\n\nHe made a conceding movement with his eyebrow that three hundred years ago might have been a shrug and a gesture. \"In truth I find that a morsel hard to swallow, particularly given the fact that he must have known the master of the city in life. Yet war there is. Charles plays some part in it...\"\n\n\"And since Karolyi knows about it,\" Lydia said thoughtfully, \"he's going to try to make of it what he can. Would it have been he who was behind James'... disappearance?\"\n\n\"I think it likelier he engineered this incident with the palace guard.\" Ysidro's white hand moved upon the windowsill. \"Behold the timing of it. He was taken up in the morning, when a living man would have the most time to question him or to act in his absence. He was taken up, too, outside the Grand Bazaar, where he is known to have been speaking to the tellers of tales. So his dwelling place was unknown. Karolyi did not reckon on James' friendship with your golden barbarian, and he did not have time to get him into his own hand before he was released. I think,\" he added, \"that this Karolyi knows something of what is taking place, but not all. And I think that if it was his goal to get James into his hand, rather than simply to kill him, it was to find Anthea through him.\"\n\n\"So they were still together.\"\n\n\"So it appears.\" His hand moved in the shadows again, and Lydia saw that he had wrapped a thick cashmere lap robe over his morning coat, as if to ward off the chill of the autumn night \"In two nights' wanderings I have found no sign of Anthea hunting, and Zenaida has seen nothing of a strange woman in her own quest for midnight blood. This could mean that Anthea is in hiding somewhere, or that she has been taken, either by Karolyi or by the Bey, the master of the city... or by this adversary, be he rebel fledgling or interloper. And where Charles may be...\" He shook his head.\n\n\"It is an ancient city, and very great. Veiled as it is\u2014and Zenaida says this mist or illusion settled upon it shortly after the gunfire and riots of the army coup, not that she had me smallest knowledge or interest in the Sultan's overthrow\u2014there are an infinity of places to hide. Zenaida says that she knows not where the Bey is, nor knows she of any other vampire. She says that she does not mind, never having cared for the dominance of the Bey.\"\n\nLydia gazed in silence for a time into the night beyond the lattices, the moon-soaked city and the silver-flecked waters that lay between. At last she said, \"And she knew nothing... would know nothing... of Jamie?\"\n\nYsidro made no reply.\n\n_My master told me to show you the place,_ the boy had said.\n\n\"Would it help to find the hiding place of the master of the city?\"\n\nHe gave her a glance of inscrutable irony, as if to say, _As you found mine?_ \"He will have many, you know. In a war among vampires, he will be moving his sleeping place nightly.\"\n\n\"I understand,\" Lydia said. \"But it will give us a starting place, and if we find out what we _can_ find out, clues lead to other clues. About Anthea or Ernchester, or... or Jamie.\"\n\n\"Always provided Jamie is not lying with a cut throat at the bottom of the harbor.\"\n\n\"If I were willing to accept that without further investigation,\" retorted Lydia, \"I might as well go back to London.\"\n\nHe inclined his head, though whether in mockery or apology she did not know.\n\n\"Anyway,\" she went on after a moment, \"I obviously don't have Jamie's training in questioning storytellers, aside from not speaking any... is it Turkish they speak here?\"\n\n\"Turkish and Greek in the streets. Arabic among the scholars, Osmanli at the Sultan's court.\"\n\n\"Since there doesn't seem to be any central depot of records, I think I'm going to have to have tea with German businessmen and ask them about native clients, and try to spot some kind of oddity in payment. My German isn't wonderful, but last night most of them seemed to speak very good French. I wonder if I can get on the good side of someone at the Banque Ottomane? Or the German Orient Bank?\"\n\nShe straightened her shoulders, the words themselves giving her courage; she spoke as if sorting a hand of cards, seeing what she had and what she needed. \"Extensive use of middlemen and corporations that don't seem to have any raison d'etre beyond paying the bills of one or two households; payment in gold or credit rather than silver; clients who either never appear at all or only appear after dark. That sort of thing. The purchase of housing that has some kind of multilevel cellars or that's built over old crypts, like that cistern we passed through. Maybe corporate credit funneled through the palace with instructions not to check too closely into bona fides?\"\n\nShe fell silent, watching his face, which was without expression. His silence lay on her heart like plates of lead.\n\nThen he said, \"Did we but find one of his bolt-holes, it could be watched. Not a safe occupation, even with the illusion which veils this city, but as you say, clues lead to other clues, and it is clear to me that more than finding Charles, more than finding Anthea, it is necessary to learn what is happening in this city. If Karolyi is here, there is still bargaining going on.\"\n\nBehind them the mantelpiece clock chimed four; seagulls cried in the darkness outside. Ysidro went on, \"You have catalogued already those things I will alter in my own arrangements, when I gain London once more. Quest among your German businessmen for word of purchase of either a great quantity of silver bars or silver-plated bars. If there is war among the vampires of this city\u2014if the master of the city seeks to summon and imprison Ernchester\u2014he will need a place to put him. And seek also,\" he added, \"for someone using the roundabout financial methods of which you speak to purchase and install modern central heating in one or more old houses.\"\n\n_\"Central heating?\"_ The absurd picture rose to her mind of the cloaked and sinister West End stage Dracula deep in conversation with Herr Hindi about soft-coal hummers and double-heating, self-feeding base-burner anthracite models, only ninety-seven marks plus shipping costs...\n\n\"If there is a challenge to the Bey's power, chances are good that it is because the Bey himself may be growing... tired. Brittle. Losing his grip. A thing one seldom considers of the Bey,\" added Ysidro, \"but a possibility. It happens, in time, even to the Undead. When this chances, vampires suffer cold and joint ache. Winter is coming on. This city will be under snow. A master fighting for his position, refusing to admit the drag of darkness in his soul, might well heat one or more of his bolt-holes for his own comfort, particularly if it has been his custom to use the living as servants.\"\n\nHe had been watching the darkness of the street. Now he turned his attention fully toward her again, a ghost-shape in the gloom. \"You understand,\" he said, \"that though clues of this kind may lead us to Ernchester or to the heart of this affair with Karolyi, you may not find your husband, mistress.\"\n\nShe looked down at where the moonlight lay on the shawl over her arms. \"I understand. I'd been hoping,\" she went on after long silence, her voice low, as if speaking to herself, \"that when I went to the embassy yesterday afternoon\u2014Saturday afternoon\u2014that Sir Burnwell would say something like, 'Oh, of course, he's staying right across the way at the Pera Palace.' And the day would finish with Italian ices on the terrace and telling stories in bed half the night.\"\n\nShe drew the shawl's long fringes through her fingers, to keep them from trembling.\n\n\"You have never been alone, then.\"\n\nIt wasn't what she had expected him to say\u2014if anything at all\u2014but it was true. She nodded without looking up.\n\n\"Well, I felt I'd been alone for years and years, before I knew him. But I expect most children feel that way. And I knew him\u2014I mean, he was in and out of Uncle Ambrose's house\u2014when I was fifteen, sixteen. I don't remember a moment of falling in love with him, but I remember knowing there was no one else I'd rather live with. I remember crying because I knew they'd never let me marry him. I was underage. And he wouldn't ask. He didn't want me to be hurt in a family row. He didn't want me to lose my inheritance over him.\"\n\n\"I daresay your father put his own interpretation to that.\" The soft voice was like the wind flowing down an empty hall. \"What happened?\"\n\n\"Father disinherited me over my studies. Jamie was away in Africa. That was during the war. Someone... someone said he was dead. I was terrified because I didn't know if I could succeed in an actual practice. Most women have a terrible time. My research is sound, but pure research would be out of the question, and I... I didn't know if Jamie was coming back. But without him I didn't care, really, what became of me. When he came back he asked me to marry him because I hadn't any money, and Father permitted it. Then later he changed his will again.\"\n\n\"But you never thought of giving up your study?\" The vampire sounded amused.\n\nLydia raised her head, shocked. \"Of course not!\" He was regarding her, she found, with a curious, unreadable intentness in his sulfurous eyes. For a moment she thought he would speak, but then like a ghost he seemed to withdraw a little from her.\n\n\"In truth,\" he said, \"we can only do what we can. I spoke not to crush your hope, mistress, but only to warn you that not all grails are found intact. Nor, indeed, found at all.\"\n\n\"No,\" Lydia said softly, \"I understand. Thank you.\"\n\nHe rose. She held out her hand to him, as she would have to a brother if she'd had one, or a friend. After a moment he took it, his thin hand emerging from the dark folds of his lap robe like Death's, oddly bereft of its native scythe, fleshless knuckles and fragile bones dry as bleached bamboo under her touch. She'd taken down her hair while drinking her tea; its natural straightness had almost destroyed the remains of earlier curls, so that it lay in unswagged cinnabar heaps on her shoulders and back, like seaweed on a beach after a storm.\n\nWith her free hand she propped her spectacles again, a schoolgirl's gesture.\n\nRemembering it later she had the impression that he'd said something else to her\u2014or maybe just spoken her name\u2014and that his cold hand had brushed her face, pushing back the flame of her hair from her cheek. But that wasn't clear to her, as if she'd dreamed it. Perhaps, she thought, she had.\n\nIt did occur to her that it was not at all like Ysidro to be concerned whether her hopes were crushed or not.\n\nThe street of the brass sellers lay four or five aisles in from the main entrance of the Grand Bazaar, according to the dealer in attar of roses of whom Lydia made her inquiry... \"Or more or less,\" added the man in excellent French; the beaming smile that split his dark face reminded her forcibly of a discolored and incomplete set of piano keys. \"But for what does la belle mademoiselle want brass? _Pfui_ , brass! It is attar of roses, the incomparable essences of Damascus and Baghdad, which delight the heart and offer the gift of sweetness to God. Only thirty piastres... That wretched cheating son of an Armenian camel driver is going to charge you more than fifty for a brass thimble that won't be brass at all, but cheap tin with a brass wash no more substantial than a Greek's sworn word... Thirty piastres? Fifteen!\"\n\nLydia smiled, curtseyed, murmured, _\"Merci... merci,\"_ and with Slavonic clairvoyance Prince Razumovsky, enormous in exquisite London-cut mufti, appeared at her side and said, \"Come along, come along,\" steering both women\u2014Margaret hanging back for one more sniff of a painted ointment pot\u2014into the crowd.\n\n\"Can we go back there?\" Margaret asked diffidently of His Highness. \"When we've found the storyteller, I mean? True attar of roses costs ten or twelve shillings for a flask that size back home.\"\n\nShe craned her neck, trying to look back between a jostling pair of German businessmen and several drab-uniformed soldiers at the tiny stall with its magic rows of twinkling glass. The shopkeeper gave her another demolished smile and a wink as bright as his wares.\n\n\"My dear Miss Potton,\" the prince smiled through the Colchian fleeces of his beard, \"twenty feet from this spot you can buy a flask that size for two piastres, if you look sufficiently indifferent. It requires practice. Hold in your mind the image of a room\u2014a building!\u2014filled with such flasks... or, rather, think of having to carry a _veddras_ of the stuff\u2014about three of your gallons\u2014up a steep hill, and then go back for another, and another, and another...\"\n\nMargaret giggled and blushed, and someone else cried out in awful Greek-accented French, \"Madame, Madame, here all the perfume, all best roses of land of nightingales...!\"\n\nThe light that suffused the bewildering mazes of the Grand Bazaar was never direct, falling as it did through windows high in the vaulted ceiling, and in the pale green archways the voices of every nation from the North Sea to the Indian Ocean swirled like soup. There were no genuine spots of light, nor actual shadows, but a dizzy kaleidoscope of color that shifted too quickly for Lydia to guess at distant things\u2014the contents of the shops they passed, the faces or nationalities of men who seemed, at a distance, to be only swirls of white or dark or colored robes. As they passed close they came into focus: swarthy Turkish men in pantaloons sitting on floors to bargain, talk, drink glasses of scalding tea; Greek men in wide white skirts and bright caps or women in close-fitting, dowdy black, arguing with shopkeepers at the top of their lungs; porters bent matter-of-factly under superhuman loads; Armenians in baggy trousers, Orthodox priests and thick-bearded Jews in black gabardine and prayer shawls. Young boys shouted offers of shoe shines or guides to the city, or dashed importantly through the jostling shoppers bearing brass trays on which rested single glasses of tea. The air was redolent of sweaty wool, garlic, carpets, dog, and sewage.\n\nDown the aisles that branched from side to side, Lydia glimpsed wares at which she could only guess: coats of karakul and astrakhan, carpets of blue and crimson, shawls, bright-flashing glass, hanging racks of silver earrings, bolts of prosaic wool alternating with gauzy rainbows of veils. Every time a beggar came whining up to them\u2014hideously disfigured, some of them, freaks who would have been confined to fairs anywhere in Europe\u2014every time they passed strolling groups of soldiers who whistled and rolled their eyes, Lydia was heartily glad she'd asked the prince to act as their protector and guide.\n\nHe'd been right. This wasn't England. It would have been madness to investigate alone.\n\nShe'd slept uneasily for the few hours after Ysidro's departure, prey to troubling dreams. Part had concerned the harem, with its smelly little cells, its cramped windows blocking out all view of the city, of the sea, of the sunlight had it been day: _The walls sweat with their pettiness, their boredom, and their tears._ She'd dreamed of wandering in that darkness, looking for someone, the rooms growing smaller and smaller around her while she felt the waiting presence of something lying very still on a burst and stinking divan, listening for her footfalls with a smile on what had long ago been its face.\n\nOnce, very briefly, she'd had a fragmentary image of a Gothic tower in a thunderstorm, the lightning lurid as a carbon-arc flare over seas of churning heather, the rain pounding in a deserted courtyard\u2014rain that somehow only barely dampened the white dress, the raven curls of the woman who stood at the tower's gate, gazing with expectation across the wilderness of the heath. Lydia, in the shelter of a broken shed on the other side of the court, had not been wet at all by the rain, though she smelled the soaked earth. She thought the woman was waiting for a horseman. Turning her head, she saw Ysidro nearby, almost invisible in the shadows, dressed as he had been on the balcony, in morning coat and striped trousers with the lap robe held close about his shoulders. His head was bowed, his colorless eyes closed as if deep in concentration, his face the face of a skull.\n\nThe dream image snuffed like a guttering candle, and waking, Lydia had heard Margaret crying, muffled, angry, and hurt. Margaret had had very little to say to her that morning and would not meet her eyes. Since their meeting with Razumovsky over a late breakfast, she had addressed all her remarks to the prince, giggling at his flirtations and responding cheerfully to his effort to draw her out.\n\nThere seemed to be storytellers everywhere. They sat on dirty rugs and blankets, swaying with the rhythm of their tales, spreading their arms, using their voices to conjure thunder, rage, love, and wonder. Children and teenage boys sat around them, listening eagerly, and even grown men and a very few black-veiled women stood with the air of those in no hurry to leave. Lydia moved toward one and peered shortsightedly at the wares in the surrounding booths. Lady Clapham had told her that each man had his regular pitch, and the man who worked the street of the coffee merchants would no more dream of shifting to the street of the slipper vendors than she herself would have considered walking uninvited into her neighbor's house in Oxford and appropriating her neighbor's nightgown and bed.\n\nIt was simply Not Done.\n\nAs she edged her way a little into the crowd, trying to see past the dark backs of the Greek ladies, a man put his hand on her shoulder and said, \"Madame Asher?\"\n\nShe turned, looking up slightly at the Adonis face, the beautiful dark eyes, of a tall man who moved like an athlete within his tobacco-colored suit. At this distance she could see the close-clipped mustache, the long eyelashes, the pearl buttons of his gloves as he bowed to kiss her hand. He wore a gold stickpin in his cravat, a winged griffin that seemed to regard her with a single, baleful ruby eye.\n\n\"I've seen your husband,\" he said quietly, and, while her breath was still stopped with shock, he added, \"Permit me to introduce myself. I am the Baron Ignace Karolyi, of the Imperial Diplomatic Service. May we talk?\"\n\nHe led her out of the crowd, into the dimness before a shop front where an elderly Greek sewed slippers of colored leather and gave them\u2014most uncharacteristically for a merchant of the Grand Bazaar\u2014not so much as a glance. It occurred to Lydia that Karolyi must have paid him in advance for his disregard.\n\n\"Is he alive?\"\n\nKarolyi nodded. Although she knew he must be at least thirty-five, he seemed younger and radiated a kind of earnest intensity, like a youthful charmer who has put his charm aside to speak of important things.\n\n\"Though I cannot guarantee how much longer that will last. He is in the hands of...\" He hesitated artistically, studying her face, like one who debates with himself how much of what he says will be believed. And yet, she realized, he was actually watching her, trying to guess how much she knew.\n\nLike Ysidro playing picquet, she thought, peeking at the stock cards and wondering what to appropriate and what would do him no good.\n\nHer heart beat harder and she thought, _Jamie will die if you botch this up._\n\n\"He is in the hands of a man called Olumsiz Bey,\" he went on after a moment. \"A Turk. A truly evil person. Tell no one,\" he added quickly, as Lydia pressed her hands to her mouth and widened her eyes as Aunt Lavinia generally did before crying out in horror at the presence of death-dealing spiders or the perfidy of the children of her neighbors. \"What exactly did he tell you, Mrs. Asher, that brought you to Constantinople to search?\"\n\n_He must have been talking to Lady Clapham._ She wondered how much that redoubtable woman had seen fit to tell him\u2014how much she would have considered not worth the trouble of hiding.\n\n\"Oh, where, when?\" She didn't expect a truthful answer to the questions and asked them to buy herself time to think, but she had no need to manufacture the panic, the desperation that she threw into her voice. She had never considered herself to be an actress, but any young lady of good society knew how to exaggerate delight or terror, or whatever other emotion was called for. A number of conversations with Margaret over the past week certainly helped her performance.\n\nShe clasped her hands to her breastbone. \"Did you speak with him? Did he look well?\" _Has he been in touch with his own department? Do they know I had dinner with Mr. Halliwell? Why_ would _I have come to Constantinople if I didn't know the kind of danger he was in?_\n\n\"We did not have the opportunity to speak.\" Karolyi's voice was soothing, a beautifully modulated tenor with the barest trace of a Middle European accent. An eminently believable voice. \"He appeared unharmed, though as I said, there is no way of knowing how long that will last. That is why you and I must talk. When you fled from me last night, I feared some rumor or calumny had reached you. I assure you, Madame...\" He made his voice earnest, deeply concerned. \"I assure you, such rumors are exaggerations, fed by the enmity of our two countries and the suspicions of men who see only threats wherever they look.\"\n\n\"Fled from you?\" Lydia steeled herself, produced her eyeglasses from her handbag and put them on to peer at him. \"Last night? Were you at the palace reception last night?\"\n\nUnder the fine traces of mustache his mouth quirked, disarmed for a moment. With two quick gestures of his forefinger he smoothed the mustache, and Lydia noted the fine cut of the pale tan gloves, French kid at six shillings the pair.\n\n\"Baron!\" Razumovsky's gray and golden bulk appeared from around the corner of a stall and pushed through the crowd, Margaret scuttling in his wake. Lydia's glasses immediately disappeared from her face and into the folds of her skirt. \"Back from your flying visit to London, I see.\"\n\n\"Prince.\" Karolyi bowed to the exact depth required of a Russian prince rather than an English one. \"A flying visit indeed, but one must dress, you know.\" He laughed rather vacantly and flicked the lapels of his Savile Row suit. \"Are you here with Mrs. Asher?\"\n\n_He believes I've been taken by surprise,_ thought Lydia swiftly. _If I put this off, he'll guess I had time to prepare._\n\n\"Will you excuse us for a few moments, Your Highness?\" As the Russian moved off she turned her back slightly and put her hand behind it, signaling\u2014and hoping he saw\u2014with her outspread fingers: five minutes.\n\n\"From what Mr. Halliwell said I gather you and my husband weren't exactly friends,\" she said quickly, keeping her voice fast and breathless to keep from stammering with uncertainty and dread. \"But it is all really a... a sort of confraternity, is it not? You are all in the same business, no matter what side you're on.\" She produced her glasses again and put them on, well aware of the air of scholarly ineffectualness they lent to her face. \"Thank you so much for letting me know! I knew\u2014I _knew_ \u2014that Cousin Elizabeth couldn't have been wrong!\"\n\n\"Cousin Elizabeth?\"\n\n\"Cousin Elizabeth in Vienna,\" said Lydia, as if slightly surprised that Karolyi were not acquainted with her family. \"She lent my husband twenty pounds a week ago Thursday night, to take the Orient Express to Constantinople. She's his cousin\u2014his second cousin, actually\u2014and she lives in one of the suburbs, I forget the name... In any case I telephoned her when Mr. Halliwell gave me the note from my husband...\"\n\n\"Note?\" The graceful eyebrows deepened in a frown.\n\n\"Telling me to return to London. Saying he was going on, he couldn't tell me where. Mr. Halliwell did his best to convince me to go back, and I let him think I _was_ going back, but I _knew_ my husband was in danger of some kind! I knew it.\" She clasped her hands again, praying that it wasn't obvious that she was shaking all over.\n\n\"Why were you in Vienna?\" He was running this over in his mind, trying to fit pieces together. Guessing at Ysidro's inscrutability had given her a greater ability to deal with ordinary human expression.\n\nShe widened her eyes. \"He sent for me.\" _What other reason would there have been?_ her tone seemed to ask. And, when Karolyi looked gratifyingly skeptical, she explained, \"He telegraphed and said there were some medical notes that would need to be analyzed. I am a medical doctor, you know,\" she added, propping her spectacles and looking as unworldly and harmless as she possibly could. \"I do research at the Radcliffe Infirmary.\"\n\n\"And your specialty is?\"\n\n\"Rare pathologies of the blood.\" It was nothing of the kind, but unless Karolyi read medical journals, he wouldn't know that. It was the kind of thing they _would_ have sent for her to examine, if they were dealing with vampires.\n\nHe evidently didn't, for a look of enlightenment dawned in his eyes. \"I see.\"\n\n\"But when I reached Vienna, Mr. Halliwell told me something dreadful had happened and Dr. Asher had had to leave the city suddenly, and gave me his note, telling me to return to London. And I knew he had to be in some kind of danger, especially after Cousin Elizabeth told me he'd borrowed money from her to come to Constantinople all of a hurry. And now they tell me he's disappeared, and I don't know what to do! Oh, Baron Karolyi, if you know anything, can help me in any way...!\"\n\nHe looked annoyed, as well he might, she thought, but he concealed it well as he patted her hands. \"Calm yourself, Mrs. Asher, calm yourself. What have you been able to find out of his whereabouts?\"\n\nThat, she thought, was what he wanted to know. That, and how much she herself knew.\n\n\"Nothing!\" she wailed. \"I came here to the marketplace because I understand he was arrested near here. I thought that some of the shopkeepers might have seen something, or know something...\" She removed her spectacles and blinked dewily up at him. \"Prince Razumovsky was kind enough to offer to escort me here, as he knows the language.\"\n\nKarolyi sniffed, just slightly, and Lydia reflected that Lady Clapham's estimate of the prince's amorous nature was probably correct, if Karolyi would believe that the prince would come here to escort a woman.\n\n\"Listen, Mrs. Asher,\" he said, lowering his voice somberly and leaning down a little to gaze into her eyes. \"His Highness may officially be on the side of the English, but believe me, he is not a man to be trusted. Whatever you chance to learn\u2014even small details, even if they sound foolish to you\u2014let me know at once. You and I can pool our resources; together we can find your husband.\"\n\n_You mean you can find the Master of Constantinople's hideouts,_ she thought, a moment later watching his splendid brown shoulders disappear into the crowd at Razumovsky's approach. Still, she thought, turning with shaky gratitude to her rescuer, she hadn't done so badly. On her first visit to the Grand Bazaar, she had been able, at quite short notice, to sell an almost total stranger a complete load of goods.\n\nAs they began to move away, the shopkeeper, who until this time had remained sewing slippers in a corner, got to his feet and padded over to her, and without a word affixed to her collar a cheap brass safety pin on which was strung a blue glass bead, painted with an eye. Then he smiled and bowed, and explained something to Razumovsky at great length.\n\n\"For the Evil Eye,\" the prince said as he led Lydia away.\n\nThe street of the brass vendors contained, in addition to innumerable tiny shops where old men tapped and fashioned everything from plates and boxes to enormous long-spouted teapots and life-sized deer, four sellers of fig paste, a man dispensing lemonade from a huge earthenware jug on a handcart, a vendor of sesame candy, and a regiment of beggars.\n\nThere was no storyteller.\n\n\"Helm Musefir?\" the keeper of the largest shop on the row said in response to Razumovsky's question. He was a little man with a beard the color of iron down to his middle, who had not abandoned the old-fashioned clothing with the coming of the reforms. His pantaloons were resplendent in volume and hue, his sashes fringed in tarnishing silver, his slippers purple morocco and curled extravagantly at the toe. His turban was green, pinned with an enormous clasp of shining brass like an advertisement above his brown, good-natured face, and as he spoke he fingered a loop of prayer beads in his hand. \"Since Monday he is gone. My wife's cousin has a friend who lives in the room above him; he says he has not been to his rooms, neither he nor Izahk, the Armenian boy who takes care of him and runs his errands.\"\n\n\"Was there a reason for this?\" the prince asked. When the brass seller hesitated, Razumovsky gestured to Lydia and explained in the French in which most of the vendors seemed fluent, \"This good madame is seeking news that the _hak\u00e2wati sha\u00eer_ might have had for her and would deeply value any word as to Musefir's whereabouts.\"\n\n\"Ah.\" The shopkeeper bowed slightly at the emphasis Razumovsky placed on the word _value. \"_ In truth, I do not know. Will the good lady be so kind as to accept...\" He held out to her a brass dish of Turkish Delight, pale green in a snowy dust of sugar. \"My wife's cousin's friend is also a friend of the landlord's sister, and she says that the _hak\u00e2wati sha\u00eer_ was not in debt, nor in arrears of rent. Likewise the boy Izahk's uncle, who frequents the same coffeehouse as my brother-in-law, would have mentioned had the old man been ill. So I do not know.\"\n\nOf course, thought Lydia, wiping powdered sugar from her fingers as His Highness walked her back through the teeming aisles of the bazaar, no one had seen or noticed James himself. James was like that. But it did not escape her that if James had arrived in Constantinople Saturday evening, he could easily have sought out the _hak\u00e2wati sha\u00eer_ Helm Musefir on Sunday\u2014the last day upon which the old storyteller himself was seen.\n\n# SIXTEEN\n\nAFTER THE BEWILDERING stinks and colors of the Grand Bazaar, tea at the Hotel Bristol was like stepping through a door and finding oneself suddenly in the south of France. For Lydia this effect was heightened by the fact that, in spite of the Bristol's excellent view of the Golden Horn, she could not see the old city. For her, the world ended a yard past Herr Hindi's broad shoulders in a light-filled sea of obscurity through which white-coated waiters swam, their silver dishes flashing like strange treasure in the late afternoon sun.\n\nWomen wearing stylish pale-hued frocks chatted with well-tailored gentlemen in French and German over Ceylon tea and cr\u00e8me br\u00fbl\u00e9e. A small orchestra played Mendelssohn. Three children in knee pants and starchy white dresses consumed water ices under the benevolent glare of a tightly laced woman in black bombazine.\n\nIt was restful beyond words.\n\nAt the foot of the hill on which Pera stood, Lydia knew, Armenians cleared up charred beams and broken glass from the harsh retribution against their protests. Men like Razumovsky and Karolyi shifted and jockeyed for position in the background, selling guns to the Turks or the Greeks or the Arabs in preparation for a war that everyone knew was coming, and telling themselves it was all to maintain the peace. In every house in the old city, women lived in ugly little rooms like the harem, behind lattices that forbade not only the eyes of men but the sun itself, and no one raised a voice for them.\n\nAnd beneath the surface moved darker shadows yet.\n\n\"Maybe it's just my being a newcomer here that makes me feel as though I've dropped into another time as well as another world.\" Lydia blinked brown eyes against the golden light and took a sip of her tea, dainty fingers half covered with mitts of ecru lace. \"Sometimes it seems to me it's the small things, not the big ones, that make a country change from ancient to modern, the way the Ottoman Empire is doing. Like buying stoves and furnaces instead of heating their houses with braziers...\" After three days in the house on Rue Abydos, Lydia knew all about braziers. \"I expect you still have people paying you with handfuls of gold.\"\n\nHindi chuckled richly. \"Ha ha, precisely so, Frau Asher. One finds the strangest things here in the mysterious Orient! You know, the other day I was called in to consult with a wealthy man who wanted to donate plumbing fixtures to the hospital attached to the mosque of the Sultan Mehmed...\"\n\nThe ensuing story occupied fifteen minutes and had nothing whatsoever to do with furnaces, odd financial avenues, or possible wars among the city's Undead, but nevertheless Lydia found it intriguing for its contrast between the new and the old. Once she discounted her host's rather heavy-handed attempts at humor and his propensity for telling her what, as a European lady, she should and shouldn't do, she did not find it difficult to listen to Herr Hindi on his favorite subject, perhaps because her own interests had always tended to the technical. He was, at least, a businessman with contacts in one of the strangest and most varied cities in the world, and not a twenty-two-year-old aristocrat whose world began with cub hunting in November and ended at the conclusion of the grouse shoots.\n\nWith a minimum of prompting, Hindi quite happily told her about his clients, the sometimes peculiar methods of payment found in an empire whose ruler had vetoed the building of an electrical dynamo because the word sounded too much like \"dynamite\" and might give encouragement to anarchists... and, of course, a great deal about the differences in burning time between soft and hard coal and the sorts of steam furnaces available from American manufacturers as opposed to those in Berlin.\n\n\"Ah, it's a strange city. Frau Asher, a strange city!\" He shook a plump, reproving finger at her. \"And not one for a lady to be traveling about in alone! I hope you're not one of those lady suffragettes we hear so much of, wanting to wear pants and smoke cigarettes and make us poor men stay home and mind the babies, ha ha!\"\n\nLydia, who would far sooner have trusted any child with James than her friend Josetta or, God forbid, herself, simply out of regard for the poor infant's comfort, refrained from saying so. Instead she angled the conversation neatly back to Herr Hindi's adventures\u2014in which he was far more interested anyway. In time, and with genuine interest, she asked, \"So there are some clients who won't appear at all? Who refuse to deal with the infidel even for the sake of their own comfort?\"\n\n\"My dearest Frau Asher,\" Hindi chuckled, \" _legions_ of them!\" He poured her another cup of tea. The waiter had twice refilled the hot water, and once brought the furnace salesman another plate of Italian ice. Hindi was a thickset, fair Berliner of about thirty-five whose wife and two sons had remained in Germany. He had been one of the dozen or so gentlemen who had extended invitations, not, she knew, with the smallest intent of impropriety on either side, but simply because she was a new face in a rather small Western community and\u2014if she didn't wear her spectacles\u2014reasonably pretty. She'd been glad when Lady Clapham, after a moment's thought, had pronounced it \"perfectly all right\" not to bring Miss Potton along; even gladder when the attach\u00e9's wife had offered to invite the girl for tea and cards at the embassy instead.\n\nMargaret had\u2014characteristically\u2014turned her down.\n\n\"Frau Asher, if you want to hear of impossible clients, you should talk to Jacob Zeittelstein. Now, _there's_ an eccentric client for you! Huge old labyrinth of a palace lost in some maze in the heart of the city, bills of credit from who knows what companies and corporations, can only work under certain conditions, won't meet with him in the daytime at all, won't meet with him under any circumstances half the time but sends these\u2014these _thugs_ who don't know to do anything but open doors, it seems; won't meet with him on Fridays, Saturdays, _or_ Sundays, changes his mind, tear it out, do it over, but hurry, hurry, hurry...\" He laughed again, and sipped his tea.\n\n\"Poor old Jacob comes away tearing out his hair and wishing he'd never _heard_ of ammonia refrigerating plants.\"\n\n\"Refrigerating?\" Lydia inquired.\n\n\"Refrigerating?\" Ysidro leaned back a little in his chair and drew the soft cashmere lap robe more closely around his shoulders. A reflex, thought Lydia, left over from the days when he had body heat to conserve. She wondered if the shivering reflex persisted. What would it be, she thought uneasily, to be conscious\u2014 _unable to lose consciousness_ \u2014in a body slowly consumed with the cold of death?\n\n\"Maybe he wants to keep blood in bottles?\" suggested Margaret. \"So he won't have to... to take it from people?\"\n\n\"If it's the death of the victim rather than the blood itself that feeds the vampire, refrigerated blood would be useless,\" Lydia replied, then wanted to bite out her tongue as Margaret flushed hotly and flashed an apologetic look to Ysidro, as if to say, _Don't pay attention to her, she doesn't understand._\n\nThe vampire didn't seem to have noticed either Lydia's faux pas or Margaret's reaction to the possible laceration of his feelings.\n\n\"It's been tried,\" he said calmly. \"More for the sake of convenience than humanity, I admit. Refrigeration causes blood to clot and separate even more quickly. In any case, in a city as rife with dogs as Constantinople, I can scarce imagine anyone storing blood for purposes of mere physical nourishment.\"\n\n\"You know, I wondered\u2014\" Lydia began, then cut herself off quickly, realizing her medical curiosity on the subject of whether Ysidro were feeding on nonhuman blood sources might be tactless in the extreme.\n\nThe yellow eyes touched hers, only for an instant, but awareness of her question, confusion, and self-deprecation all danced like an ironic star. But he only said, \"I have not heard cold itself could injure the Undead, nor cause them to sleep on into the night. The vampires of St. Petersburg dwell in palaces left empty through the winter, while most of the court goes south to the Crimea, and they rise and hunt and sleep as usual. It is not an easy thing,\" he added, turning to Lydia with that same remote amusement, \"to be Undead during the time of the white nights. But in winter they walk abroad from three in the afternoon, and sleep does not weigh them down until eight or nine in the morning. They do not feel cold that would kill a living man, though it is true that the Master of Petersburg has spoken of removing permanently to the Crimea, which tells me that he has begun to tire, and so feel the pain of cold in his joints. Still...\"\n\nHe turned his head a little, to contemplate the stacks of ledgers and papers heaped on the table around the oil lamps that Madame Potoneros had brought in at Lydia's behest. An embassy clerk had delivered the material late that afternoon, with a note from Lady Clapham: _I won't ask what you want them for, my dear, only that if you learn anything we should know about, you'll pass it along. The red are the Banque Ottomane; the gray, the Deutsches Bank. I'm afraid we'll need them back in the morning._ The _we_ amused her, confirming as it did who was really running Intelligence\u2014such as it was\u2014in Constantinople.\n\n\"It will be a matter of interest to see how deep the fingers of the master of the city have gone into the flesh of the empire.\"\n\n\"If it's the Bey that we find.\"\n\n\"Oh, it will be.\" Ysidro rose and laid aside the lap robe, averting as he did so his face from the light. Margaret scurried away to fetch his cloak, as if she feared Lydia would usurp this task that she considered her right. \"Money takes on a life of its own once it enters the veins of this body they call finance. All the masters of the great cities are aware of this and make sure they have great sums of it, not hidden, but disguised as something else. This is why they are masters. I would hazard that since July, with the army coup, the Bey has been transferring his assets from the old forms\u2014hidden stocks of gold, investment in land\u2014to the new. It is his protection against the interloper, if interloper there be, or against a rebellious fledgling. His protection against the upheavals of the living.\"\n\n\"And his challenger won't have the capital base yet.\"\n\n\"I doubt it. Most fledglings do not realize the need for such invisible redoubts. They think immortality sufficient.\"\n\nAs he reached to take the cloak from Margaret's hand, Lydia saw that the gold ring he wore had slipped around his finger, turning so that the bezel faced inward to his palm, as rings do when the flesh shrinks away from them with cold, or age, or death.\n\n\"As for me, I shall pursue Anthea and Charles as the Undead pursue, listening in the streets where the poor dwell and seeking those places where the living do not walk. If James is yet alive, as this Karolyi has said, it is because the Bey needs something of him, and at a guess it is as bait, either for Charles or for Anthea. Karolyi is still bargaining, offering what he has to sell\u2014the support and alliance of his government in these uncertain times\u2014while feeling for other advantages.\"\n\n\"But why\u2014\" Lydia began helplessly, and Ysidro shook his head.\n\n\"We move in a miasma, and not entirely that of the Bey's making,\" he said softly. \"There is some other matter afoot here, beyond a possible challenger or interloper. Treason among the Bey's fledglings, perhaps, or an interloper not of the common run. We must each search as we can. It may be that as a physician you will recognize something concerning cold as it has to do with the Undead state, which even the Undead do not know. Later, like the knights of the grail meeting upon the road, we can exchange information and see if we can read, one for the other, what each vision signifies. Do not lose hope.\"\n\n\"No,\" Lydia said, consciously steadying herself. \"No. At least I know James is alive\u2014if Karolyi was telling the truth. Though I did notice he was very careful not to say _when_ he'd seen James. It might have been\u2014well\u2014days ago. But really, we can only do what we can do.\"\n\n\"An observation worthy of the sages of Athens,\" the vampire said gravely and, holding out his hand, took her fingers in his. \"A word in your ear.\"\n\nConscious of Margaret's glare at her back, Lydia followed him out of the dining room, to the head of the stair.\n\nHe stood with his back to the vigil light, so that only its reflection touched the points of cheeks and chin and made a spidery halo of his hair. In his enveloping cloak he looked like Death on its way to the opera; his hands were, she thought, not quite steady as he pulled on his gloves.\n\n\"You have fathomed my secret,\" he said, the soft voice emerging from the dark, and upon it, like the trace of his antique inflection, Lydia detected the echo of a smile. \"The blood of animals gives some nourishment, though it does not warm, and their deaths are useless to feed the hunger and the need of the mind. But it would not do to shock Margaret with the information that the dark hero of her Byronic fancies is currently living on the blood of dogs\u2014and such dogs! As a physician, however, I knew the matter would consume you until you knew.\"\n\nLydia laughed, the fear and tension she had felt since that morning in the bazaar loosening its hold. \"I think you're just too vain to own to it.\" She smiled, and Ysidro paused, his hand on the rail of the stair.\n\n\"Of course I am vain,\" he said. \"All of the Undead are vain\u2014too vain to admit that, like common men, we must die.\"\n\nHe made a move to go, then turned back and took her hand again\u2014carefully, so as not to come near the silver on her wrist\u2014and raised it to his lips.\n\nAs he vanished into the shadows of the stair, she said, \"Be careful...\"\n\nShe didn't know whether he heard or not.\n\nMargaret shoved the papers she was reading quickly into her workbasket and returned to her chair as Lydia reentered the dining room. She kept her eyes downcast, but Lydia felt the sullenness of her silence, the resentment in the set of her narrow back in its ill-fitting cotton shirtwaist. She drew a pile of gray Deutsches Bank ledgers to her, but left pencil and foolscap to one side untouched.\n\nDetermined not to have another argument with her, Lydia only asked, \"You know what we're looking for?\"\n\n\"New corporations in July or August paid for in gold or by transfer of lands, sums transferred to another corporation or another bank monthly or quarterly.\" She recited Lydia's instructions like a schoolchild regurgitating some hated\u2014and barely comprehended\u2014lesson.\n\n\"Look for a transfer to the second corporation, or to a new corporation, in the first week of October of ten thousand marks, or twelve thousand five hundred francs, and if you see either the Zwanzigstejahrhundert Abkuhlunggeselleschaft, or any of these names\u2014\" She pushed across to her the slip of paper she'd gotten from Razumovsky that afternoon, listing the four or five names under which the Sultan's chamberlain took bribes or laundered money. \"\u2014please flag it for me.\"\n\n\"I understand,\" Margaret said with gruff impatience, and pulled the paper to her, but didn't even turn it right side up. Lydia half opened her mouth to remonstrate, then let it go. She guessed she'd have to go through whatever Margaret did again anyway, but if these ledgers had to be back in the morning, there was no time for either discussion or for Margaret to slam into the bedroom in a tantrum. She couldn't work through all of this alone.\n\nAnd what could she say in any case?\n\nThe dream returned to her, of Margaret waiting in the castle ruins for a horseman who never came. Was Ysidro unable even to project the dream memories of passion to her now, the melodramatic romances that held her to him? Was he, she wondered suddenly, unable to appear in them because in them he would be the skeletal, almost insectile creature who had spoken to her with his back to the light?\n\nIf that was what vampires saw in mirrors, no wonder they avoided them, veiled them, kept them closed behind doors. If that was what the living eyes would perceive, no wonder the vampires caused the living to see\u2014or remember seeing\u2014nothing at all.\n\n_All of the Undead are vain..._\n\n_\"Kiria...\"_ Stefania Potoneros appeared, hesitating, in the doorway and held out two stiff cream-colored envelopes.\n\nThe first contained a note on the letterhead of the Zwanzigstejahrhundert Abkuhlunggeselleschaft\u2014Berlin, London, and Constantinople\u2014typed neatly in English and signed by a secretary.\n\n> _Mrs. Asher:_\n> \n> _We regret to inform you that Herr Jacob Zeittelstein is unable to make an appointment with you for this week, due to the fact that he is in Berlin at this time. When he returns to Constantinople on Wednesday next, he will of course be delighted to get in touch with you regarding a meeting._\n> \n> _Sincerely,_\n> \n> _Avram Kostner_\n> \n> _Private secretary to Herr Zeittelstein_\n\n_Wednesday_! thought Lydia, aghast. Two days from now until he was even _in_ Constantinople, let alone when he'd have time to see her, answer her questions. Jamie could be dead by then...\n\nJamie could be dead now.\n\n_My dearest Madame,_ the other letter read, in an elaborately indecipherable French hand.\n\n> _It appears we have located the storyteller your husband sought. With your permission, my carriage shall arrive for you at ten tomorrow morning, though it would be well to be prepared to do some walking._\n> \n> _Your most humble servant,_\n> \n> _Razumovsky_\n\n\"If I may be permitted to ask a question, effendi?\" Asher turned his cheek to the slab where he lay, blinking the sweat from his eyes. In the still, dense heat of the tiny _hararet_ \u2014the chamber of the baths that the Romans would have called the _calderium_ , or hot room\u2014the shape of the Master of Constantinople, white as the marble that entirely formed the walls, seemed to emerge from and blend into the steam in a disconcerting fashion, so that half the time Asher was not entirely certain he could see him at all.\n\n\"It is always permitted to ask, Scheherazade.\" The voice of Olumsiz Bey came out of the steamy twilight, and the red glow of the braziers in the corners made twin embers of his eyes. There was dreamy, heat-soaked amusement in the deep voice as he spoke the nickname, taken from Asher's curiosity about old words and ancient tales even in the face of his imprisonment and peril. \"There would be no wisdom in the world, did men not ask.\"\n\n\"What do you want with the Earl of Ernchester?\"\n\nIt was nearly midnight. With the early fall of winter dark, Zardalu and the other fledglings had taken Asher to an immense dry cistern, like a pillared cavern beneath the city, given him a tin lantern and sent him out in that endless forest of columns. \"Behave as if you searched for someone, Englis,\" whispered the eunuch, with his mocking smile. \"Gaze about\u2014so\u2014put your hand to your heart, as if to calm the pangs of love.\" The others laughed, the thin, metallic shivering he had heard in Vienna, and faded into the darkness, leaving him alone.\n\nSo he had walked, as he had walked in the cemeteries, holding the lantern high, and the shadows of the pillars reeled and shifted with the movement of the light. The columns themselves were of all girths: thin Ionic with rams' horn capitals, and heavy, unfluted Doric worn with the marks of water. The floor underfoot was hardened mud, silted up who knew how deep. Between them night lay thick, and the cold breaths of moving air told him the place had more than the one entry the vampires had used. He was thinking how fortunate it was that the candle within the lantern was protected by glass when the flame went out, as suddenly as if covered by a snuffer.\n\nAsher stepped back at once, putting his back to the nearest pillar and forcing closed his mind against the crushing numbness that bore down upon it. He reached for the pocket where he kept matches, wrapped in waxed silk, and his nostrils were filled with the smell of old blood and graveyard mold. A hand closed around his arm, as if the arm had been trapped in machinery; but before he could lash out with the lantern in his other hand, before he could move or think or cry out. the gripping hand was gone.\n\nThere was a kind of movement, a breathing rustle in the dark, and he pulled the matches from his pocket and lit one with a hand that shook.\n\nHe was alone.\n\n\"My dear Scheherazade.\" The voice was suddenly close. Asher blinked again in the steam, to see that the Master of Constantinople stood beside the marble table where he lay, naked, as was the Bey himself, but for a towel around his loins. \"These are vampire matters, of no concern to the living. Indeed, I doubt the living would understand them.\"\n\n\"They're of concern to those who want to stay among the living.\" Asher sat up, his brown hair hanging lank in his eyes, and the bathman Mustafa stepped back. Asher had guessed that the Bey's living servants weren't deaf, but he had never succeeded in getting more than a few words out of any of them. When they brought him food, when they placed clean clothing in his room or escorted him to the library or the baths, they watched him with the eerie impassivity of guard dogs, as wary as if he, not they, were the servant of the night. \"Was it you who had Lady Ernchester's rooms searched, after Ernchester had gone?\"\n\n\"My instructions to Karolyi were to have her destroyed,\" the Bey said shortly. His orange eyes, gaudy as aniline dye, glittered coldly. \"The woman is his strength. A man need not be a sorcerer, or a reader of dreams, to have learned that in the course of a single conversation. In the eighteen months of his abiding here as a living man, there was not a day that he did not speak of her, nor a night when she was not in his dreams. When I heard that both had been made Undead, I thought it a foolish risk on the part of the Master of London, to have among his fledglings one with such power over his mind as she.\"\n\n\"He disobeyed you, then.\"\n\n\"Stupid Magyar, to think he could defeat the purposes of the Undead.\" The Bey's left hand caressed unthinkingly the silk bindings around the hilt of his silver weapon\u2014thornwood, Asher guessed, the silk just sufficient to keep from discomfort a vampire as old as the Bey, who had toughened a little against some of the substances reactive to vampire flesh. Around his neck he wore a foot-long knife, sheathed in leather and lead. Asher guessed the blade within the sheath was silver as well. \"Was it she who freed him in Vienna and killed those set to watch over his prison there?\"\n\nAsher shook his head. \"It was the Vienna vampires. Karolyi had brought a victim for Ernchester to kill.\"\n\n\"Fool.\" The vampire turned his face aside, anger in his eyes. His lean body seemed almost completely without muscle, the hair of chest and armpits paled to a strange red-brown. Though the heat of the _hararet_ had laid a film of condensation on the pallid skin, Asher could see not a drop of sweat. \"The man is greedy, seeing only the path to his own power, and not that things are ordered as I have ordered them for reasons beyond his comprehension. And yours,\" he added, looking back at him.\n\n\"Then why deal with him?\"\n\n\"A man is a fool who casts away a plank in a shipwreck, Scheherazade. He is impertinent, to think that I would do as his Christian emperor bids. But power, and allies, are always needful in a difficult time.\"\n\n\"And are the times so difficult?\" Asher asked quietly. \"Is that why you're hunting Lady Ernchester so diligently? Not only to control the earl, but to keep her out of Karolyi's hands? He'll go to your fledglings, you know, if he hasn't already.\"\n\nA drift of moving air stirred the steam. The curtain of embroidered leather that separated the _hararet_ from the _sogukluk_ , the warm room, lifted aside. The man Sayyed stood there, his head\u2014shaven like the Bey's\u2014glistening with moisture.\n\n\"There is one to see you, Lord. A _makanik_.\" Except for the last word, which was Persian, he spoke peasant Turkish, the longest sentence Asher had yet heard any of the living servants speak.\n\n\"You will excuse me.\" The Master of Constantinople bowed deeply, turned to go, then, pausing, looked back.\n\n\"Do not concern yourself in the affairs of my children, Scheherazade,\" the Bey said, and the giant ant seemed to watch Asher from its amber prison on the Bey's ear. \"This is not the course of a prudent man. Do not trust them. They will promise you things\u2014escape from this place, safety from harm, even the kiss that brings eternal life. But it is all lies. They are all treacherous. They envy one another and envy the power each thinks the other might possess; and above all they envy me. But _I_ am the master of the city. This city is mine, and all things in it.\"\n\nHe held up his silver weapon, the blade flashing gently in the dull braziers' gleam. \"And do not concern yourself with Ernchester. That, too, is a course that will bring you only death.\"\n\nWhen he had gone, Asher stretched out on the table again, gingerly favoring the dressing over the knife wound on his ribs. It was healing well; Mustafa had changed the dressing, and now, as the man kneaded and pummeled his muscles into lassitude, Asher stretched out his right arm before him and looked at it in the dim light.\n\nThe heat had reddened the scars that tracked the vein from wrist to elbow, the scars left by the Paris vampires. Among them, the fresh dark blot of a bruise was printed like a blackening stain.\n\nAsher picked out the marks of fingers and thumb, remembering the hand that had crushed his arm in the dark of the cistern. The dressing pinching as he moved, he brought his other hand forward and laid it over the marks.\n\nThe hand was bigger than his own.\n\nErnchester's hands, he remembered, were small.\n\nThe fledglings had returned to him almost at once, in the silence of the dry cistern, had blindfolded him and brought him back to the House of Oleanders without a word, as they had brought him back twice now in three days from those desolate places where Anthea might have hidden. They had blanked his mind as they came through the street, so that he returned to a kind of frightened and dizzy consciousness in the octagonal Byzantine vestibule that led to the Bey's salon.\n\nHe was beginning to think that Zardalu had made a genuine mistake and let his mind be distracted while coming back into the house from that first expedition.\n\nZardalu and the others had departed on their own hunt after returning Asher to the House of Oleanders, and were still gone when Asher dressed again in clean linen and secondhand gray trousers, red wool vest, and a worn and slightly ill-fitting Stamboul coat. He made his way back along the corridors to his room with Sayyed padding silently behind. He knew that route now, and how the small palace of some Byzantine prince connected with one of the several _hans_ that made up its wings. Twice he'd passed a doorway he guessed led into some late Roman crypt or church, and the painted room with the tiled dome in which he'd seen Karolyi was definitely Turkish.\n\nThe courtyard of the old _han_ was lighted with brass lamps hanging from the colonnade before what had been deep bays of warehouses downstairs. A single lamp burned in the niche at the end of the open gallery, two floors above. Lights burned, too, in the Byzantine vestibule\u2014Asher could see their reflection on the arched passageway.\n\nA _makanik_ , to see the Deathless Lord.\n\nSomething concerning mat secret experiment, that strange crypt far beneath the house, stinking of oil and ammonia.\n\nNear the old baths, Zardalu had said.\n\nThere were no clocks in the House of Oleanders, and the hours of darkness could be disorienting. Asher, who had a fairly good sense of time, estimated it was close to one in the morning as Sayyed turned the key in the lock and padded away, and guessed he had an hour or two in which he'd be relatively safe.\n\n_Do not concern yourself with Ernchester,_ the Bey had said. But he was still bargaining with Karolyi.\n\nExcept for the dry basin in the center, the long floor was a faded moss bank of carpet, four and five layers thick. Among these carpets he had concealed the picklocks he made.\n\nHe fetched them now.\n\nThe bronze candlestick, which he kept quite openly beside his small pile of books in one of the inlaid wall cupboards, had provided him not only with wire for picklocks, but with a number of candles as well. These he slipped now into the pocket of his coat. The lock was a very old single-tumbler Banham, probably the best obtainable when put in, but that had been more than a hundred years ago. As he descended the stairs to the courtyard, he heard the voice of the Bey shouting in the salon and stopped, startled, by the vestibule passageway to listen.\n\n\"It has been three weeks, you sputum of Shaitan's dog!\" That any vampire, let alone one as old as Olumsiz Bey, should give way to rage at all was unheard of, and the passion that cracked in his deep voice was terrifying to hear. \"Five days since the breakdown, and still no word of the man! I tell you there can be no more delays!\"\n\n\"Peace, m'sieu,\" came a more muffled\u2014and understandably nervous\u2014reply. \"The man will be back Wednesday. Wednesday is not so very long...\"\n\nAsher hesitated, torn, sensing that whatever could so enrage the Master of Constantinople must be of paramount importance, but knowing that if he were caught standing here\u2014much less with picklocks and candles in his pockets\u2014he was a dead man indeed. His every instinct told him to stay, but at least, he thought dryly, moving like a shadow away from the arch, if he's shouting at his engineer he isn't listening for me...\n\nThe mental image of the Bey as he had seen him other nights, sitting still on the divan of his pillared salon, silver weapon across his knees and orange eyes half shut while he listened to the teeming dreams of the city around him, was a disturbing one.\n\n_Even as we hear the footfalls of the workmen,_ Zardalu had said. At least as long as he walked above the ground, if the Bey listened for them, Asher knew he could hear his.\n\nThe way that leads to the old baths.\n\nFashions in building came and went, and the House of Oleanders was at least five old buildings fused into a monstrous maze of dark rooms and decaying memories, but, Asher knew, plumbing remains plumbing. The elaborate system of pipes and hypocausts that made Turkish baths\u2014and before them, Roman\u2014was not a thing to be relocated lightly or far.\n\n_We smell the nqft, the alkol, the stinks of what he does..._\n\nHis mind returned to the throat-catching sharpness of the air in the crypt. A room with a wooden floor, to the left across a courtyard where grass grows between stones like cannonballs. A second flight of steps after the first...\n\nHe fingered the picklocks in his pocket and drifted through the House of Oleanders like a ghost.\n\nThe solitary gleam of his candle wavered over chambers hung with printed Chinese silks whose colors showed themselves briefly; over vaulting that flickered and shone with the unmistakable dusky bronze hue of gold in shadow. He passed through an octagonal chamber whose walls were sheathed, floor to ceiling, in red tile the exact color of ripe persimmons, containing only a black-and-white wooden coffee stand; an arch looked out on a court smaller than the room itself and so choked with oleander bushes that only the dim white shape of a single statue could be seen in their midst.\n\nNear that place he found the room he sought: the small, rich chamber of painted walls and blue and yellow tiles whose bare wooden floor thumped familiarly underfoot. From it a door let into a courtyard, long and narrow and paved in blocks of worn stone the size of halfpenny loaves, through which brown grass and weeds thrust tall.\n\nThe moon had not risen. No light touched the windows in the low buildings that surrounded the court on two sides. Roman, thought Asher, identifying the heavy rounded arches, the broken fragments of marble facing and the thick, fluted columns. What looked like the rear wall of another _han_ closed in the third side of the court\u2014he could just see the edge of a dome against the midnight sky\u2014the red and white stone walls of the Turkish house, the fourth.\n\nUnder the columned porch the blackness was profound. The smaller cobbling was uneven, familiar. Almost he felt he could quench the candle as he passed to the left, fifteen steps across the court and through the door, five steps and left again. It was difficult to see that doorway, where it stood in shadow, though it opened in the middle of a wall of faded frescoes\u2014more oddly still, he lost control of his steps twice, passing it without being aware. Around him the darkness brooded, watching. It could, he knew, contain anything.\n\nOr nothing, he told himself. Or nothing.\n\nHe descended the stair. Had he not remembered a second stairway, he would have turned back, for its entrance lay concealed in the niche formed by one of the shallow false archways in what turned out to be the _tepidarium_ of the house's original Roman baths. A small room, faced with marble, its shallow pool long gone dry. The mosaics of the floor gleamed faintly in the moving light of Asher's candle: Byzantine, and like those of the octagonal vestibule, long ago defaced.\n\nThe second stair, as he recalled, was twice or three times the depth of the one above. If he met them now\u2014the Bey's homecoming fledglings with their night's prey\u2014there would be no possibility of escape.\n\nHe guessed the crypt below had been a prison, or a storage place for something more precious or more sinister than wine. The low brick groinings of the ceiling barely cleared his six-foot height, and the few rooms that opened to his right from the short passageway were tiny, sunk below the level of the floor, which was itself worn in a channel inches deep. The air\u2014as he recalled and as Zardalu had remarked\u2014was bitterly cold.\n\n_Dastgah._ Scientific apparatus. There were Western scientific journals in the library dating back to the eighteenth century, treatises in Arabic from the days before the Moslem world had become a scientific backwater. Just exactly what was it, Asher wondered, that the Master of Constantinople was having his Western engineers build for him? That meant so much to him that its delay would rouse him to fury? That he hid from his own fledglings?\n\nThe penny-dip glow touched something dully reflective, lodged like a gleaming bone in the throat of a dark arch.\n\nHere, he thought. The place the Bey kept hidden, veiled with his mind.\n\nAt the end of the abyssal corridor before him, Asher knew he would find that long stone stair, climbing to an outer door. But branching down to his left, his raised candle flame showed a grille of silver bars, behind which lay\u2014what?\n\nOr who?\n\nBefore him the tunnel extended like the bowel of night\u2014to his left, behind me silver bars, Stygian velvet\n\nHe wondered how much time he had left.\n\nHe had to know.\n\nCautiously, he moved down the short side branch.\n\nHis wan light winked on water pooled on the uneven stone floor. The corridor was extremely narrow, curving slightly; the silver bars, tarnished nearly black save around the lock where the bolt went into the stone, blocked it about ten feet from the convergence of the two passages. Beyond, Asher could make out two archways set in the left-hand wall. On one, at least, he caught the glint of a metal lock plate on a door. The smell of ammonia was overpowering; he had to fight not to cough.\n\nThey'd be coming back soon: Zardalu and the Baykus Kadine, and the others, bringing another victim to chase through the pitch-dark house until they cornered him, weeping and screaming...\n\nEven locked in his upper room, Asher had heard the Armenian boy's voice for a long time.\n\nHe turned from the silver grating, back to the main corridor, and resumed his quest for the stair that led out.\n\nThere was a door, locked, that had to be it\u2014like the doors above, he missed it two or three times, found it only by walking with his hand on the weeping stone of the wall, until what he had somehow taken three times for an angle of shadow resolved itself suddenly into an arch. This evidence of the power of the master vampire's mind he found extremely unnerving. They must have left the door open behind them that first night when they'd gone forth\u2014or perhaps one had gone ahead of the others to open it for them.\n\nIn any case the lock was a Yale, new; a matter for a duplicate key, not a homemade shank of bronze wire.\n\nHeart beating fast now with apprehension, he returned to the silver grille. That lock, at least, was of the old-fashioned kind, probably because the softer metal couldn't take the stress of the smaller wards. He angled the bronze wire carefully, knowing every scratch would show. Even the lugs and pins that held it to the stonework of the walls were silver.\n\n_They are treacherous..._ the Bey had said, the silver blade of his halberd gleaming in the smoldering half-light of the baths. _They are treacherous._\n\nHis heart slamming blood in his ears, he edged his way along the buckled, puddled flagging next to the wall. A wet footprint here would condemn him to death. Straw and sawdust salted the corridor, making the going even more delicate, and the cold was arctic. He wondered if he would hear the fledglings returning. Wondered if he would know, should the Bey be watching him from out of the darkness with those leached-out ochre eyes.\n\n\"Ernchester,\" he whispered at the nearer of the two doors.\n\nBoth were locked. Hasps of silver, or more probably electroplated steel. Padlocks sheathed in silver, even to the bows. Silver solder dabbed over the screw heads. The locks were new\u2014the rest, black with age in the candle's feeble light.\n\n\"Ernchester!\" he whispered again. How much\u2014how far\u2014could the Deathless Lord hear? Not through earth, he thought. Not through this much stone. \"It's Asher. Are you there? Anthea's free, she's here in Constantinople...\"\n\nHe had almost said, _Anthea's alive._\n\nListened.\n\nDeep behind the heavy door he heard it: a groan, or a cry, that lifted the hair on his head\u2014physical agony mingled with the blackest depth of despair. Hell, Asher thought. Such a sound you would hear if you put your ear to the keyhole of Hell.\n\n\"Can you hear me? Can you understand?\"\n\nOnly silence replied. His hand trembled, fumbled at the lock, half numb with cold but unsteady, also, with the knowledge that time was now very short...\n\n\"I'll come back for you,\" he promised hoarsely. \"I'll get you out...\" _And I'll need your help,_ he added as a grim afterthought, _to return the favor._\n\nA draft, a shift of air, and his heart stopped as if knifed with an icicle, then began beating fast and thin. Even in that first second, he pinched the candlewick, thanking God for the smell of the ammonia that would drown the smoke of a full-fledged conflagration, much less that of a single dip. That drowned, even from the Undead, the smell of his living blood.\n\nFrom the dark of the corridor beyond the silver bars he heard stumbling footfalls, and a pleading breath, \"My lord, be kind\u2014be kind to a poor girl...\"\n\nAt the edge of hearing, a tickle of obscene mirth.\n\n\"Oh, the lord you're going to will be kind.\" The voice might have been Zardalu's. \"He is the kindest lord in the city, sweet and generous... you'll find him so, beautiful gazelle...\"\n\nIn the utter blackness there was nothing to see, no way to know if they'd noticed the slight jar of the doorway in the silver bars\u2014he'd pulled it to behind him, the hinges oiled and uncreaking...\n\nHe could only wait, desperately listening, wondering if the next thing to happen would be a cold touch on his neck. The staggering footsteps faded. He himself remained where he was for a long time, unmoving, dizzy with the ammonia stink and the cold that ate at his bones, before he felt his way along the wall to the bars, and so out into the corridor, wincing as the gate lock clicked behind him like the hammer of doom.\n\nBut none molested him. In time he felt his way back to the stair\u2014painfully, endlessly, across the baths, thanking God that navigating in the dark was a skill he'd kept up from his spying days\u2014and up to the grass-grown court, where the little light of the stars seemed bright to his eyes. As he crept through the courtyard, he heard the silvery clashing of vampire laughter from within the salon, and the young woman's voice pleading incoherently. It seemed to him, as he bolted his own door behind him and sank to his knees under a sudden wave of nervous shaking, that the sound came to him still\u2014that, and the moaning of the prisoner behind the crypt door.\n\nIt was a long time before he managed to get to his feet and stumble to the divan, where he lay shivering as if with killing fever until the muezzins of the Nouri Osmanie cried the late winter dawn.\n\n# SEVENTEEN\n\n\"I SUSPECTED my remark about how valuable you thought the information would yield results.\" Prince Razumovsky slashed with his riding whip at the two curs sleeping on the marble steps. They slunk a few feet away and stretched out again in the dust of the plaza that had once been the Hippodrome, tongues lolling an incongruous raspberry against wolfish coats which, even without her eyeglasses, Lydia could tell were half worn away with mange.\n\nConstantinople had more dogs\u2014and, as she had seen last night, more cats\u2014than any city she'd ever been in.\n\nThe animals seemed to operate on two different levels, as indeed, she reflected, they would have to. As Prince Razumovsky's carriage had worked carefully through the streets of the old city, where wooden Turkish houses appeared to sprout spontaneously from more ancient walls, the dogs had been everywhere, lying in the muck or against the walls of ochre or pink stucco. The cats had the overhanging balconies or shared the sills of heavily barred windows with potted geraniums, or lay on the walls and trellises of tiny caf\u00e9s where Turkish men sipped tea and talked under stringy canopies of leafless vines.\n\n\"Someone always knows someone,\" the Russian continued, white teeth gleaming under tawny haystacks of mustache. \"The good brass seller mentioned our questions to his friends at the caf\u00e9 that night, or perhaps a beggar overheard us or the man selling baklava. One of them knew a street sweeper whose sister knew the _hak\u00e2wati sha\u00eer_ by sight or had a cousin who'd heard one of the muezzins mention that a new _hak\u00e2wati sha\u00eer_ had taken up residence in this place, or one of the neighborhood children mentioned it to another child... It was a Syrian boy who brought me the information.\"\n\n\"What did you pay him?\" Lydia reached for the small reticule of silver mesh that hung at her waist. \"I can't let you...\"\n\n\"An entirely negligible sum.\" His Highness waved dismissively. \"It will support his family for two months, doubtless\u2014or buy one member of it two days' worth of opium, if that's their choice.\" He held out his hand to help her over the marble sill of the narrow door.\n\nHe had been treating her all morning as if she were made of cut glass, apparently under the impression that her haggard eyes and pallor were the result of a night of sleepless worry over her husband, not a night spent single-mindedly plowing through four and a half months' worth of the investor listings of the two biggest banks in the city.\n\nThere were more than a score of corporations and investors that seemed to fit the criteria. More people than a single vampire had guessed the way the wind was blowing back in July and started transferring funds into less vulnerable forms than real estate and gold. She wondered if it were possible to obtain the long-term banking records of the oldest banks\u2014how long _had_ there been banks in the empire, anyway?\u2014or of property holders, to see whose lives went on for a suspiciously long time before they transferred their money to equally long-lived successors. The various names under which the palace chamberlain laundered money came up again and again in everyone's accounts\u2014and given the general level of corruption in Constantinople, it was almost impossible to track how money appeared and disappeared.\n\nBy five in the morning Lydia had a dozen names\u2014two of which Margaret had completely missed in the Deutsches Bank records\u2014and Margaret had long since fallen asleep with her head on her arms.\n\nHad she not reviewed the records, Lydia suspected she'd have had the night of sleepless worry in any event, so it was just as well she'd had work to occupy her.\n\n\"Are you sure this is all right?\" Margaret asked, flushing an uncomfortable red. \"It isn't allowed, is it?\"\n\n\"The courtyard is free to all,\" said Razumovsky. \"But it would probably be best if you let me speak.\"\n\nThe Blue Mosque was one of the greatest in the city, a place where there were always people.\n\nThat, Lydia realized a moment later, was the point.\n\nRazumovsky led them\u2014Lydia burningly conscious of her Western gown and the gauzy excuse for a veil depending from her stylish hat\u2014toward the north wall of the court, where wintry light fell upon the men along the colonnade: a bearded man in a turban selling small loops of bright-colored prayer beads on a blanket; another cross-legged behind what looked like a little desk, complete with brass inkwell, standish, and shaker of sand. There was the inevitable shoe-shine boy with his little brass-bound kit. Two men in rags, sitting near the small marble pavilion in the middle of the court fingering their beads, glared at the women as they passed, but neither spoke.\n\nThe man they sought occupied a worn carpet next to the bead seller's pitch. He was conversing with a thin, elderly man in a white robe and yellow turban, but looked up as Razumovsky drew near, and Lydia had an impression of a huge hooked nose and a tangle of dirty white beard, a green blob of turban, and, when she cast down her eyes, of grimy, horny feet with toenails like a bear's claws poking from beneath his robe. He was ragged, and his clothing smelled of filth and sweat; he gave off anger and distrust like a blast of heat in her face.\n\n_\"Qab\u00eeh... qab\u00eeh...\"_ he muttered ferociously, glaring up at her and then past her at Margaret. _\"Qahb\u00e2t...\"_ He averted his face then and added in hoarse French, \"An unveiled woman is an abomination in the eyes of God.\"\n\n_\"Ma\u00eetre conteur.\"_ Lydia curtseyed deeply. \"Please forgive me. Do you call me ill names because I wear the veil which my husband gave me to wear?\" She touched the thin net veil of her green taffeta hat. \"Do you blame me for wearing clothing, and dressing my hair, as my husband wishes to see me adorned?\"\n\nThe man in the yellow turban had stepped tactfully away, leaving Lydia, Razumovsky, and Margaret alone with the old _hak\u00e2wati sha\u00eer_. Lydia knelt on the worn marble paving of the mosque's court, reflecting that after journeying all the way from Oxford, the bottle-green skirt needed cleaning anyway. \"And if my husband has disappeared,\" she went on, still in French, which the old man seemed to follow well enough, \"and I know him to be in danger, am I impure for wanting to aid him?\" Ysidro, she thought, should hear me now.\n\nThe black eyes glittered, chips of coal. She could see the dark line of downturned mouth amid the tangled beard, hear the anger in his voice when he replied, but she saw, in the set of his shoulders, the way he drew back from her and looked, for one fleeting instant, past Razumovsky to the courtyard gate, that he was afraid.\n\n\"You are the wife of the Ingileezee in the brown clothing, the man who asked all the questions about the Deathless Lord.\"\n\nLydia nodded. She wondered how close Razumovsky was standing behind her and how much he heard. \"I am.\"\n\n\"He was a fool,\" snapped the old storyteller. \"To seek the residence of Wafat Sahib is the act of a fool, and a fool's fate overtook him.\"\n\n\"Did you tell him?\"\n\nThe old man looked away. \"I told him nothing,\" he said sharply, and Lydia knew he was lying. James had probably offered him money. With feet like that, and the characteristic roughening of pellegra on the skin of his face, he was beyond a doubt desperately poor.\n\n\"It was my boy Izahk,\" the storyteller went on, too quickly. \"A discreet boy, one I thought too clever to be seen. But when he did not come back that night, I knew he had done that which is forbidden: he had spoken of Wafat Sahib, and that lord is not a lord to tolerate such chatter.\" His black eyes narrowed, and his voice, almost a whisper to begin with, sank lower still, so that Lydia had to draw close, within reach of the gusts of breath that smelled of strong coffee and rotting teeth.\n\n\"Wafat Sahib, he has been lord in this city since my great-grandfather's time and before. He knows what is said of him in the streets, even by light of day. Even for the Ingileezee to ask me, to offer me money\u2014which of course I did not touch,\" he added loudly, \"made me afraid. So I came here, out of the sight of the men who serve him. Now they tell me the _hortlak_ , the _afrit_ , the _g\u00f6la_ , have been seen among the tombs outside the city, walking among the cypress trees by the tomb of Hasim al-Bayad, stopping travelers who walk late upon the road and killing them in the darkness.\"\n\n_Ysidro?_ wondered Lydia, recognizing one of the words as the Turkish for _vampire_. Or had Ysidro in his search missed something, some clue? Been deceived by the concealing glamours of the vampire mind?\n\nIt was logical, she thought, for the challenger to Olumsiz Bey's power to haunt them, lying as they did outside the city walls. \"Where is this tomb?\" she asked, lowering her voice and hoping Razumovsky wasn't anywhere near.\n\n\"You are a fool!\" The _hak\u00e2wati sha\u00eer_ flung up his arms, coal-chip eyes blazing with sudden rage. \"As your husband was a fool before you! Go away, and ask no more, lest his fate befall you as well!\"\n\n\"I'm not going to go there at night\u2014\" Lydia began to protest reasonably, but the old man surged toward her, slashing with his clawed and knotted hands.\n\n\"Go! Get out! I tell you that your husband is a dead man!\"\n\nShe stumbled back, startled at this violence, and Razumovsky caught her; she heard Margaret squeak in alarm.\n\n\"Leave me, infidel whore!\" the old man screamed. \"How dare you defile the place of holiness by even the tread of your feet?\"\n\n\"Really, I\u2014\"\n\n\"Come,\" the prince said softly and drew her toward the gate. \"There's nothing more you can learn.\"\n\n\"I daresay,\" said Lydia, struggling between a sense of injury and a terrified desire to go back to the old storyteller, to try to learn more. There was something Ysidro hadn't seen in the cemetery... Something that occurred after he'd gone? She looked back over her shoulder, to see the _hak\u00e2wati sha\u00eer_ shouting his wrongs to the man selling beads, and though at this distance he was little more than a threshing puppet of dirty brown rags, she could tell he was pointing at her.\n\nSudden tears stung her eyes, born of weariness and frustration and the hurt of being criticized when she had done no wrong.\n\n\"Forgive him, madame.\" It was the man in the yellow turban, waiting for them in the blue marble shadows of the colonnade beside the gate. He stepped down and bowed to them, though Lydia had the impression that he was a man of some importance here. \"He is an old man and believes that those who do not dress or eat or speak as his parents did were created by some other God for purposes ill to mankind.\"\n\nLydia halted, peering up at him. Above the graying beard the dark eyes were bright and kind, and not as old as she had thought. His robes smelled of tobacco, cooking, and soap. \"I'm sorry if I... if I said something wrong. I truly meant no harm.\"\n\n\"He is a very frightened man, _hamam_ , and frightened men are easily angered. He claims he is pursued by demons who live in this city, and he will not be alone, not even to sleep. He sleeps on the floor of the soup kitchen. Do not judge him harshly. They are real to him.\"\n\n\"No,\" Lydia said, remembering the abyssal darkness of the streets after nightfall. She had dreamed last night, in troubled sleep, of something that had passed the house, singing beneath the balcony in a high, thin, tuneless wailing that no one but she could hear. She had risen\u2014or, later, she thought she had only dreamed of rising\u2014and stumbled half blind to the heavy lattices that overhung the street, but she had seen nothing, or maybe just a stirring in the dark below. Margaret had rolled over and sighed in her sleep.\n\n\"He spoke of a... a _g\u00f6la_ , dwelling in the tomb of someone called Hasim al-Bayad.\" She pronounced the words carefully, thanking heaven for ten years of James' quiet emphasis on correct sounds.\n\nThe holy man frowned, puzzled a little, then said, \"In the west of Africa\u2014Morocco and Algiers\u2014a _g\u00f6la_ is said to be a kind of female devil who dwells in desert places, with a goat's feet and the face of a beautiful woman. She lures travelers from the road, drinks their blood and eats their flesh.\"\n\n\"A woman.\" Lydia repeated the words.\n\nAnthea Farren. And she would know what had become of James.\n\nHe nodded. \"Hasim al-Bayad was the imam of this mosque\u2014\" His small gesture seemed to touch the whole of the graceful, weightless stone that towered above them. \"\u2014many generations ago. A good man whose tomb was venerated in former times, though almost none seek it out now, for it lies some distance from the Adrianople Gate, away to the north of the main road. You may know it by the remains of an iron fence around it, though it is decayed almost to nothing; but the tomb still stands. But if you value your life, _hamam_ \u2014if you value your soul\u2014do not go to that place alone or after the sun has left the sky.\"\n\nLooking into those dark, worried eyes, it did not even strike Lydia as odd that he gave his warning to her rather than to the male who clearly took the role of her protector. She shook her head and said, \"No, I won't. I promise.\" She turned to depart, taking Razumovsky's arm again, then on impulse turned back.\n\n\"Is it permitted,\" she asked hesitantly, \"to... to buy prayers to help someone? Someone who's in trouble? He's not a Mohammedan,\" she added apologetically, and the man in the yellow turban smiled.\n\n\"There is no greater miracle in the world than rain,\" he said. \"And, as the Prophet Jesus pointed out, it falls on the heads of the just and the unjust alike. Give your alms to the next beggar that you meet. I will pray for your friend.\"\n\n\"Thank you,\" said Lydia.\n\n_Your husband is a dead man._ It was extremely difficult to make conversation with the prince on the way back to the carriage.\n\nOwing to the prince's consular duties that afternoon, it was not possible, he said, for him to accompany Lydia and Margaret on a tour of the cemeteries, but he insisted that they take the carriage and the footmen: \"The silly louts would only sit around playing dominoes at some caf\u00e9 in the Place d'Armes all afternoon while I'm dealing with the transport minister,\" he said, sipping his tea over lunch. \"You ladies might as well have the use of them, provided you come back for me when you've finished.\"\n\nLunch meant the restaurant at the railway station, looking out through elaborately pillared window arches to the unkempt grass of the square. Not elegant, but the only European cuisine available without crossing the bridge to Pera again, and Margaret flatly refused to have anything to do with stuffed grape leaves or skewered bits of lamb. Lydia protested a little at the prince's generosity, then thanked him profusely, laying both her kid-gloved hands on his wrist and wishing she looked prettier. Her eyes still felt swollen and tender despite that morning's applications of ice. Her aunt Lavinia's sovereign remedy had always been leeches\u2014applied by the disapproving Aunt Harriet or by Lydia herself, who had even at that age had no qualms about touching the things\u2014but Lydia, though trained in their application, had learned too much about germ theory in the past few years to feel easy now about such an expedient. Certainly too much to want to apply anything purchased in Constantinople, no matter who warranted it \"clean.\"\n\nAnd she'd look even worse, she thought, once she got her spectacles on. Just as well Razumovsky couldn't accompany them on the next phase of her quest.\n\nAs the holy man had warned, the tomb of the imam Al-Bayad lay a goodish distance north of the dust-choked road that ran toward the hills of Thrace, and Lydia had to pick her way carefully among the bizarre, stunted forest of spiky tombstones, Margaret and one of His Highness' sturdy footmen in her wake. She was glad of the footman. Closer to the gate she had seen individuals or small groups of the devout, almost always clothed in the traditional Turkish garb of pantaloons, tunic, and turban, kneeling by the low-roofed stone _turbes_ among the weeds, but this far out, among the rough stands of cypress and bare-limbed plane trees, there was no sign of people at all. Only the headstones, thrusting up through the weeds like splintered bones from a messy compound fracture. Even the dirt underfoot was mixed with chips and fragments.\n\nMargaret complained constantly of the uneven footing, the dreariness of the locale, and the uselessness of the mission. \"Ysidro said he'd seen nothing here,\" she protested, stopping for the tenth time to ostentatiously rub a \"twisted\" ankle, which Lydia knew wouldn't have borne her weight if actually twisted. \"Ysidro ought to know.\"\n\nMaybe, thought Lydia. But Ysidro had said himself that his perceptions were not what they had been. Moreover, there was always a chance that the vampire glamour was stronger than he had counted on, and subtler, masking its own existence, as it had masked her awareness of Ysidro's house in London as she walked past it three times before she finally saw it. Whether she found anything at Al-Bayad's tomb today or not, she would tell him of the place and let him take a closer look.\n\nAgainst the changing hues of the sky, the tall domes of the city gleamed; the silence here away from the road, instead of seeming peaceful, oppressed her with an air of waiting, of listening. The short autumn day was already beginning to fade.\n\n_I've seen your husband..._ Karolyi had said.\n\nIf he'd been telling the truth.\n\nAnd the _hak\u00e2wati sha\u00eer_. _Your husband is a dead man._\n\nThere'd been a note from Karolyi this morning, asking to take her to lunch. Of course, Lady Clapham would see no reason not to tell him where she was staying. She wondered if she should go, to see what else she could learn from him, but all her instincts cried out to her to stay as far from the man as she could. She was a novice, unable to best him in the game he'd played for years.\n\n_But what if Zeittelstein doesn't come back tomorrow?_ she thought, faint and helpless. _What if what he tells me is no help?_ The sense of holding her husband's life in her hands, of not knowing if any action of hers would save or damn him, was hideous. Maybe if she were very careful with Karolyi...\n\nA man called out to her, far off, near the road. With her spectacles on he was jewel-clear, waving his arms in warning, but he would not approach.\n\nShe looked ahead again and saw the grayish-white square of the _turbe_ , surrounded with a few grisly fragments of rusted fence.\n\nWeeds grew thick around the marble, whispering conspiratorially in the wind. As she approached, the smell of blood came to her, thin but rank; cold as it was, flies buzzed up in a swarm from a nearly black stain on a broken grave slab nearby.\n\nLydia shivered. It might, she supposed, only have been a dog's. Margaret cried, \"Oh! How disgusting!\" and Nikolai the footman said, \"Madame, is come away. Is no good here. No good.\"\n\nLydia walked up and put her hands on the tomb.\n\nThere were fresh scratches on the stone around the heavy lid, bright marble chips lying in the long weeds. Kneeling, she peered at a stain just beneath the lid's edge, hidden under the mass of marble; she thought it, too, was blackened blood.\n\nBut all of those things came to her like afterthoughts. As soon as she touched the marble, she knew.\n\n_He is here,_ she thought. The marble was cold under her fingertips. _He is here._ If she stood still, if she listened, if she closed her eyes, breathed slow, opened her mind, she could hear him...\n\nShe stepped swiftly back, almost colliding with Margaret, who had come up behind her, saying something\u2014she realized she had been listening so deeply that she hadn't heard what.\n\n_We usually have warning of their suspicions,_ Ysidro had once told James, on the subject of would-be vampire hunters. _We see them poking about..._\n\nShe wondered now whether he had meant during the night hours, or by day, when the vampire lay sunk in deathly sleep.\n\nDid vampires dream?\n\n\"I _said_ , can we go now?\" Margaret repeated sulkily. \"If Ysidro didn't see anything here...\"\n\nA thought flashed through Lydia's mind\u2014she knew not from where\u2014of a dark face lying in darkness, not very far away. Of sleep that wasn't really sleep.\n\nOf someone, or something, that knew her name in its dreams.\n\n\"Yes,\" she said quickly. \"Let's go.\"\n\nAs she was turning away, a glint of red caught her eye, lying in the long grass. She didn't want to go near the tomb again, but she forced herself and saw that the weeds along this side of it had been trampled. In the grayish dust she found the track of a man's hard-edged Western shoe.\n\nFairly fresh, she thought. Curiously fresh, for a place that had recently acquired the reputation as the haunt of _hortlak_.\n\nKneeling in the dusty weeds, she cast about for more tracks and saw the bright thing that had drawn her attention a moment ago.\n\nIt was a man's cravat pin, fashioned in the shape of a griffin, with a single blood-ruby eye.\n\n\"My dearest Asher Sahib.\" A shadow materialized in the archway in front of him, nearly invisible in the darkness of the old _han_ 's court; an angular silhouette, and the gleam of far too many jewels. At the same moment, as Asher stopped, his heart tightening in his chest, arms slipped around his waist from behind, the thin hard body of Jamila Baykus pressing against his back, like the steel triggering mechanism of some lethal trap. The stench of blood in her jeweled hair mingled with the wash of Zardalu's patchouli.\n\n\"You left our party precipitously.\"\n\n\"I have a weak stomach.\"\n\n_\"Tcha.\"_ The eunuch made the word almost a caress. \"Pity for a beautiful young man like the one last week, maybe, or for that little beggar girl, who I admit was pretty... But that ugly old grandmother? I swear to you she was still complaining over being cheated out of two piastres' worth of olives in the market. Now, how can you pity that?\"\n\nAsher turned his face aside and moved to go, but the arms around his waist, thin as a child's, held him. He knew no amount of struggle would break their grip.\n\nZardalu stepped forward into the colonnade, laid his hands on Asher's shoulders. Under their painted lids, the long eyes glittered in the distant glint of the lamp by the stairs. There was no other light in the court, and Sayyed, who had as usual been dogging Asher's steps, had vanished at the first sound of Zardalu's voice. \"Olumsiz Bey hasn't been out of the compound,\" the eunuch said softly, in the vampire whisper no louder than the whisper of a silk curtain on an\u2014almost\u2014windless night. \"Has he?\"\n\n\"I don't know.\"\n\n\"This is the eighth night that he's had us bring him his kill.\" Tiny breasts, sharp hipbones moved against his buttocks and back, the Baykus Kadine rising to tiptoe so that the movement of her lips stirred against the back of his neck. They were warm. \"These other vampires\u2014\"\n\n\"What other vampires?\"\n\n\"It is scarcely any affair of the living,\" Zardalu murmured, drawing close, \"what other vampires. The woman we seek in the tombs and the cisterns\u2014the man we are told to look for...\"\n\n\"What man? Since when?\"\n\n\"Does it matter since when?\" The blue eyes glittered strangely in the reflected light. \"I see that it does. Why does it matter? What does it tell you, clever one? Why does he fear them? The Malik of Stamboul, the Wafat Sahib, the Deathless Lord who has ruled this city... He could crush them like fleas under his thumbnail. So.\" The long hands tightened over Asher's shoulders, the pressure of the thumbs like a geared wheel bearing on the collarbone: Asher shut his teeth hard against the blinding stab of pain, kept his eyes on the vampire's before him.\n\n\"This foreign machine, built by these infidels... What is it? Who is it that he keeps down there, groaning and crying out in the dark hours of the night?\"\n\n\"Ask him.\" It was impossible to keep his voice steady; the steel thumbs had found the nerves they sought, and Asher had to fight to keep his vision from graying to darkness, his mind from blanking with pain.\n\n\"I'm asking you.\"\n\n\"I don't know.\"\n\nThe pressure lessened: Zardalu moved back a few inches, his hands remaining where they were. Asher was breathing hard, the sweat flowing cold down the sides of his face, though the night was chilly.\n\n\"But you've gone to look?\"\n\nAsher managed to shake his head, wondering if they had seen him, passing the archway last night, or smelled his blood. Wondering if they had told Olumsiz Bey. He doubted it. He doubted that he would be alive now, had the Master of Constantinople known.\n\nZardalu grinned like a rubber devil. \"For a man who went about the town questioning storytellers about the houses of evil rumor, you show a disappointing lack of curiosity. Do you know that Olumsiz Bey keeps a set of silver keys in a recess in the floor beneath the coffee table in the room of the red tiles? No? A curious thing for a vampire to keep, wouldn't you say?\"\n\n\"Not something he could readily use,\" Asher agreed. The Jamila Baykus moved, trying to draw him with her, and he braced his feet on the broken tiles. \"Not something I would use at all. I do value my life.\"\n\n\"Your life?\" The blue eyes widened. The silvery vampire laughter shivered in the air. \"Your life? Your life ends here in this court if I so wish.\"\n\n\"You'd go against him?\" Darkness swirled on the edges of his mind, blanking his attention, confusing his thoughts, as if he moved in a suffocating dream. Deliberately, he walled his mind against it, thought of nothing, pictured iron doors closing the darkness out, sunlight burning it away.\n\nFrom far off he felt Zardalu's hands shift up to his throat, heard the vampire say, \"He'll be displeased, but it won't make you less dead, Englis...\"\n\nHe thought they were dragging him, threw out his hand to catch at the arch as they drew him into the dust-smelling blindness of one of the old warehouse bays. It was like fighting in a dream, against a narcotic weight of nothingness that filled his mind. If he could only break free for a moment...\n\nThen he was thrown aside, striking the wall as if someone had hammered him with a railroad tie, and his mind cleared like shattering glass. Against the reflected lamplight he saw Zardalu hurled sprawling, a bundle of sticks wrapped in a hundred pounds' worth of sequined silk, and the Baykus Kadine backing away, mouth open, hissing, her eyes glittering rat-red. In a swirl of nacreous robes, Olumsiz Bey stood over the Circassian, the silver blade of his halberd cold as a fingernail moon. His bald head swung to and fro, like a savage dog's.\n\nThere was blood on his mouth and on his clothing. Zardalu rolled lightly to his feet, face twisted into something Asher hadn't seen outside a Museum of Horrors, fangs glittering in the stretched mouth. But the next moment the younger vampire flinched and turned away, hiding his face in his hands from the master vampire's glare, and Asher felt\u2014guessed\u2014sensed peripherally the cutting agony of Olumsiz Bey's will.\n\nZardalu made a sound, thin as water twisted out of a near-dry rag. His body bent and bent, knees buckling, hands spreading, fingers stretching, trying to cover his face as his arms came up like the arms of a fractured puppet.\n\nSoftly, the master vampire whispered, \"Don't be arrogant, little Apricot.\" Asher, slumped against the stones of the inner wall, wasn't even sure he heard the words spoken, could not have said what language they were in. Weightless as a giant cat Olumsiz Bey stepped toward the crumpled gaudy form of the eunuch, and the dim lamplight flicked on his outstretched talons, the graceful gesture of the halberd.\n\n\"Is this the little Apricot who wept in my arms when he gave up his life? The little Apricot who said to the slave masters, when they came to geld him...\"\n\n_No..._ It came out not even as a word, only a sound.\n\n\"I remember, you know.\" The deep voice purled over the words, water over stones, and stronger than the stones. \"You put all those memories into my hands, you put your mind, your desires\u2014remember Parvin, your sister Parvin?\u2014everything. And I still hold them.\" He crouched over his fledgling, silver-blue robes settling over the gay, amorphous clouds of silk, the silver of the blade hovering over the bare, bent neck. It was impossible, thought Asher, that he should still hear the master vampire's voice.\n\n\"The way the Kizlir Aga touched you, do you remember that? You were twelve, and you hated him, and yet your whole body responded...\" His coarse hand fielded, easily, the vicious flail of Zardalu's claws, and with the haft of the halberd he thrust him to the pavement again, pressing him down into the marble with it, straddling him, whispering, an act more terrible than love or rape, an act of dreadful possession as each memory, each feeling, each most secret terror and need was brought forth.\n\n_It gives a terrible power,_ Ysidro had once told him, in that time-faded voice that denied that such a thing had ever happened to him, that anyone had ever held over his heart such hideous knowledge.\n\nZardalu had begun to make noises, and silently, sickened, Asher crept back through the shadows to climb at last the long stairs. Looking back, he saw in the lighted frame of the arched passage to the vestibule that the brutish Habib and his one-eyed janissary friend Haralpos were enacting a burlesque love scene with the corpse of the old woman they had brought for the Deathless Lord's supper, to the screams of Pelageya's and the Baykus' laughter.\n\nBut it was the master vampire's whispering, rather than the other and louder sounds, that seemed to follow Asher up the black stairs.\n\n# EIGHTEEN\n\nSHE DREAMT OF the old seraglio again, of wandering through its cramped, lightless cells with a ledger in one hand and a lantern in the other. One of the rooms had been filled with ice, and holding the lantern aloft, she had seen Jamie, frozen in a block of it, like a fly trapped in amber.\n\nIt should have been comical, absurd, but it wasn't. His eyes were open, sunken like the eyes of the corpses the workhouse sent, and she saw blood on his neck, staining the open collar of his shirt. The ice flashed like blue diamonds when she raised her lantern, making his eyes seem to move, but she knew he was dead. Her heart twisted, slammed within her, hurting, hurting, knowing he was dead and that she'd have to go home alone.\n\nIt was her fault. She hadn't come swiftly enough, been clever enough, been brave enough... She had failed to be adequate, as she had failed all her life. She propped the ledger against the block, trying desperately to find his name in it, but the cold in the room made her hands shake so badly she couldn't read. He can't be dead, she thought frantically, he can't be. He's frozen in the ice, but the ice will keep him alive...\n\nShe woke gasping, her hands and feet bitter cold, and heard, from the other room, Margaret saying, \"You hadn't found anything there, but she insisted on going anyway! As if she knew more about it than you did! Just because she's got that horrid medical degree, and cuts up bodies, which makes me shudder just to think about, she thinks she knows everything! And she wouldn't even stop when I turned my foot...\"\n\nUnder the indignation of Margaret's voice there was a brittleness that Lydia recognized as nervousness. Ysidro, she thought.\n\nA moment later the vampire's cool voice responded, \"Well, she is worried about her husband, and perhaps that made her careless of your comforts, Margharita? You do not recall which cemetery this was? I would not wish to waken her.\"\n\n\"Um... I can't remember... We went in Prince Razumovsky's carriage, after we visited this filthy mosque and she talked to a horrid old man. And anyway, if you didn't see anything there when you went...\"\n\nLydia fumbled her spectacles from the bed beside her, pushed back her hair and pulled her shawl about her shoulders as she emerged from the bedroom, rumpled, creased, and slightly disoriented, wondering what time it was.\n\nYsidro was on his feet at once, bowing. \"Mistress.\" The room smelled of lamb and onions. There was an empty plate of very fine red-glazed local ware, and horn-and-steel flatware. Crumbs and droplets at the other side of the table indicated that Margaret had made her meal.\n\nLydia shook her head, saying, \"Later, thank you,\" when Ysidro moved toward the sideboard.\n\n\"Some wine, at least?\"\n\nBut his own hand was too unsteady to hold the glass.\n\nMargaret took it swiftly from him, poured the black-red fluid like blood in the brazen lamplight. Ysidro flicked aside the napkin from the basket on the table, tore a chunk from the bread inside. \"Sop it in the wine,\" he suggested, holding it out to her. \"A jauntering slut I can abide, but a drunken jauntering slut, never.\"\n\nAnd Lydia gave him a quick, shaky grin.\n\nHe perched on a corner of the table. \"Margharita informs me you passed an adventuresome day.\"\n\nLydia outlined to him the events of the Blue Mosque and the finding of the _turbe_ and the cravat pin. \"I had the most extraordinary sensation that he was there, listening,\" she said. \"I know you've said vampires sleep in the daytime and can't be wakened, but... Would he have heard me\u2014 _seen_ me\u2014in dreams? Do vampires dream?\"\n\n\"Yes and no,\" replied Ysidro, holding out his hand for the pin. \"Sleep is only a term that we use for what happens to us when the sun is in the sky; I do not know another. Dreaming...\" He paused, then shook his head, very slightly, and turned the tiny gold griffin over in his hands.\n\n\"I doubt not that you have found one of the sleeping places of the intruder, the newcomer,\" he said after a time. \"And having sensed you in his sleep, I misdoubt he will ever rest in that tomb again. Still, it is worth the visiting, to see perhaps what I have missed. There is a great strength to him, and it is not at all unlikely that he could turn my mind, my perceptions, away from him... And it goes without saying that any place he dominates with his presence at night, Anthea will avoid. It is no chance thing that he haunts the cemeteries, that any coming or going from the city would pass him and be in danger of coming under his sway. Anthea, at least, coming in by train with your husband, would have sensed his presence and taken care to avoid him. Charles...\" He shook his head.\n\n\"He plays a dangerous game, this Karolyi.\" He slipped the griffin pin into the pocket of his waistcoat and stood to fetch a pot of honey from the sideboard and set it for her next to the bread. \"He still does not understand what it is that he courts. Does he think to take this interloper back to Vienna and introduce him to that stodgy mediocrity in the Hofburg? The Master of Vienna will surely destroy him, as he attempted to destroy Ernchester. Or does he think to make him Master of Constantinople, forge an alliance here?\"\n\n\"Could he do that?\" she asked, surprised.\n\n\"He may, could he find the master's hiding place.\" The sparse brows pinched together, and his eyes went to the pile of notes and pencils on the other side of the table lamps. \"And what did your search reveal?\"\n\n\"That a lot of wealthy old Turks who'd had their money in gold and land all had the same idea around July of this year.\" She sighed ruefully and pushed her glasses up onto her nose. \"I've got a tremendous list of companies that all came into being at the same time and don't seem to have any reason to exist. Besides, I know from Herr Hindi that the Bey paid for his refrigeration unit in cash.\"\n\n\"True enough.\" Ysidro lifted the lid of the honey pot, brought up a spoonful, and let it run down again in a column of shining amber. \"Yet at short notice he would have used a bank draft. I believe a ticket on the Orient Express is twenty pounds'? Another two pounds to London, plus the costs of hotels and meals... maybe a total of sixty pounds? Find a draft of that, to someone of Hungarian name. Even incognito, a noble will usually take one of his lesser titles. Karolyi's are Leukovina, Feketelo, and Mariaswalther, if I recall my genealogies aright. My guess is he will have used one of those.\" Ysidro covered the honey again and stood; Margaret sprang up to fetch his cloak, which lay like a dense black winding sheet over a nearby chair.\n\nShe asked brightly, \"Will you be back tonight?\"\n\nYsidro seemed to settle into stillness, considering her with eyes that looked, in the lamplight, as gold as the honey. \"My errand should take me no great time.\" He pulled on his gloves and held out one hand to Lydia. \"It is true that the Dead travel fast.\"\n\nIt was still impossible to see him leave a room.\n\n\"Frankly, I've always wondered how they do,\" remarked Lydia, spooning honey onto a chunk of bread. \"And considering the fuss he made about traveling in the daytime...\"\n\nBut the slamming of the bedroom door was her only answer.\n\nFor a moment Lydia considered knocking and asking what real or fancied slight Margaret suffered from now. But it would only provoke another tantrum, another spate of incoherent romanticism about the eternal bond carried across lifetimes, and she felt simply too weary to go through with it. Margaret had coolly refused Lydia's offer yesterday of instruction in the intricacies of cosmetic art. Lydia was still unsure whether she was being blamed for Ysidro's absence from Margaret's dreams, for finding clues where Ysidro had missed them, or for some other offense entirely.\n\nAnd indeed, she thought with a stirring of old anger, it was Ysidro's fault as much as Margaret's. More, in fact, for originating the whole silly vaudeville of romance and need and lies. She put from herself in disgust the concern she had been feeling for him and ladled lamb and stuffed aubergines onto her plate, cursing Ysidro tiredly for his command that for safety the girls share bedroom and bed. It was not anything she was looking forward to tonight.\n\nThe meal made her feel better. She spread out her papers again, jotting down the names Ysidro had mentioned and seeking them among the lists of drafts drawn at the end of October, but it was difficult to keep her mind on her work. She was angry at Ysidro and, she realized, hurt. Disillusioned. But what illusion had she held, she wondered, that she felt robbed of it now?\n\nThe illusion that behind those bleached, crystalline eyes still lurked a living man's smile?\n\nDon Simon Xavier Christian Morado de la Cadena-Ysidro had been dead since 1558.\n\nShe recalled the books on his parlor chest. A dead man might read medical journals, and mathematics texts, and volumes of logic. But would a dead man read the stories of Toad and Ratty and Mole? She took off her spectacles, leaned her forehead on her hands. And why should it matter to her whether he was dead inside or alive?\n\nIn the street below, the dogs began to bark.\n\nLydia raised her head, startled, and looked at the clock. It was nearly three. Had she been asleep, she wondered, since Margaret's huffy departure, or had she wakened from her first sleep later than she'd thought?\n\nBelow in the street, someone pounded on the outer gate.\n\n_\"Hamam, hamam!\"_ cried a voice, vaguely familiar, though she could not have said from where. \" _Hamam_ , it is your husband! Your husband!\"\n\nShe jerked to her feet, ran to the window that overlooked the street. She pushed aside the chains of garlic and wild rose that hung there, unhooked the heavy lattice; down below she could see a cluster of dim shapes in a lantern's blurry light.\n\n\"Where?\"\n\n\"Your husband!\" cried the man below. \"Find you, he say.\"\n\nThe _hak\u00e2wati sha\u00eer_ , she thought. The man in the yellow turban. Catching up the lamp from the table, she paused only long enough to snatch her silver knife as a precautionary measure and then ran downstairs. They'd want money, she thought, stepping through the door out into the carriageway. As the light of the lamp jostled huge shadows over the carriageway's vaulted roof, she thought, _Good heavens, they could be thieves for all I know..._\n\nShe stood on tiptoe to slide back the cover of the judas in the main gate, and tried to hold the lamp so that light would illuminate the faces of those who stood outside.\n\nThere was no one in the street.\n\nBehind her, the house door slammed.\n\nLydia whirled, her breath stopping in her lungs\u2014a glance showed her that both the main outer gate and the small postern were firmly locked and bolted. The silence seemed suddenly, dreadfully alive. She strode back toward the door, cold with terror, pulling the silver table knife from her belt...\n\nThe lamp in her hand went out.\n\nInstinct more than anything else made her flatten at once to the wall. Shadow moved in the dark arch where the carriageway let into the little courtyard, where fallen pomegranate leaves made spots like dripped blood in the thin moonlight; she threw the lamp with all her force in that direction and heard it strike something soft, then shatter on the pavement. In that instant she flung herself to the door, yanked the handle, and felt the heavy jar of the bolt.\n\nShe whirled and slashed at the shadow that she felt more than saw suddenly beside her. She slashed, felt it give, turn before her. For an instant crushing pressure seized her wrist, a hand hideously strong closed over her throat, and with her mind swimming in a curious, hazy dream state she saw a face close to hers: smooth, full, olive-complected, fangs gleaming behind a thick mustache.\n\nThen he cried, _\"Orospu!\"_ and his hand jerked away, and she cut at his face again, knowing she couldn't let him get near enough to take her by the elbow, the waist, someplace where she wasn't wearing silver. She tried to scream, but it came out thick and tiny, like a child's wailing in a dream; a vision flashed through her mind of letting him seize her, of wanting to feel those iron arms holding her, pressing her close to that iron chest.\n\nShe cut again at his face and cursed as hands seized her arms above the elbow, gasped out the worst word she'd ever heard from the grave diggers who brought bodies into the infirmary for dissection and felt the claws tear her arms, ripping through her sleeves. She kicked and slashed and cursed at the face that she saw now as if through the muzzy darkness of a dream.\n\nThere were two of them, she thought, blindly terrified, hacking and twisting against a grip like devil-inhabited stone. Two of them, two faces in the patchy moon shadows...\n\nThen she was alone, leaning against the stuccoed wall with the knife shaking in her hand.\n\nHer sleeves were torn, the blood shockingly hot against flesh that seemed to be getting colder by the minute.\n\n_I can't go into shock,_ she thought, from what seemed like a great distance off. _I can't let myself..._\n\n_\"Madonna...\"_ Darkness came out of the deeper dark behind her, though she hadn't heard the gate open or close\u2014a glint of eyes and the smoke of pale hair. Cold hands seized her arms, icy despite the frost that seemed to be spreading through her own flesh. She sobbed something, she didn't know what, pressed her face to the damp wool of a cloak that smelled of dew and graveyards, as if its weight could save her from the fanged brown face that had come so close to hers.\n\nShe was unable to breathe, barely felt the cold, gloved hands that thrust her hair back from her face, touched her neck. \"Are you hurt?\"\n\nThe words had no meaning to her. She considered them from a great distance away, turning them\u2014for she seemed to have all the time in the world\u2014one way and another, like a rare bone. Was she hurt? she wondered. For a moment she floated weightless against him, conscious, it seemed, of the skeleton within his clothing, like Death in his winding sheet... conscious of almost nothing else. She heard him say her name, or thought she did, and looking up she saw, at some unbridgeable distance, the face of a living man.\n\nHe called her name again, and she gasped, shaken, disoriented, but alive once more, and stepped back quickly from him so that he had to catch her elbow to keep her from falling.\n\n\"I'm so sorry,\" she managed to say. She looked around the courtyard. Everything seemed very distant and odd, as if nothing had anything to do with her. Shock, she diagnosed. The silver knife lay on the ground at her feet, the smashed lamp beneath the pomegranate tree. She wondered how much it would cost to replace. \"I didn't mean\u2014\"\n\n\"Are you hurt?\"\n\nHer blood gleamed all over his gloves from the talon rakes of her arms, but she knew he didn't mean that. \"No.\"\n\n\"You're sure?\"\n\nShe nodded and felt her throat. She'd unbuttoned her high collar before she'd taken her nap, but the chains of silver were still there, close against the untouched skin. She bent to pick up the knife and nearly fell; he caught her in his arms as if she'd been a child, and with a single, vicious kick cracked the door bolt and carried her inside.\n\n\"You're freezing.\" He set her in a chair in the small downstairs hall; shut the door again and put a second chair under its latch to hold it. Then he turned back to her and wrapped her in the pall of his cloak. \"And afraid.\"\n\nThe fear she felt was only now coming into focus; she had not been conscious of much during the attack itself. She wondered why.\n\nIn the drift of light from the lamp on the landing he looked at his hands, gloved in leather and blood. With a quick gesture he tore the gloves off and threw them on the stairs, and vanished through the dark doorway into the kitchen. He came back a few moments later, coatless and carrying a pottery basin of water and another lamp, which Lydia found profoundly comforting. As he set the lamp on the hall table, he paused to listen at the foot of the stairs, and for some reason she remembered him, white-robed and barefoot, picking knacker's meat from its paper for his cats.\n\n\"She is safe,\" he said, his voice very soft. \"They have not been inside. My apologies for the water. The boiler is long cold.\"\n\nLydia wondered what he heard of Margaret's breathing: the peaceful snuffling of sleep or the swift, thready pant of guilt and fear and feelings hideously torn? She looked across at the door bolt, but even had the glow of the single lamp been stronger, the violence of Ysidro's breaking in had shaken loose the hasp from the bar, and it was impossible to tell whether the bolt had been shot behind her when she'd gone out, or had merely somehow slipped.\n\nHe put the cloak back from her arms, pulled the remains of the sleeve free with a single flick of his hand and reached into the basin for a sponge. The wounds were little more than scratches, but smarted horribly. Lydia flinched from the water, which was, as Ysidro had hinted, stone cold.\n\n\"I saw the interloper,\" she said, gritting her teeth. To her own vast annoyance she had begun to tremble again and couldn't seem to stop. With grim effort she kept her voice steady. A woman in hysterics was the last thing either of them needed. Besides, he'd want the information quickly. \"He's a Turk, I think, I... I didn't get a clear look. Here,\" she added suddenly, realizing how disturbing he must find the smell of blood, \"I'll do that. There's some brandy in the pantry...\"\n\nHe'd brought napkins as well, but she was unable to bind up her own arms with them and had to wait till he returned after all.\n\n\"There were two of them,\" she resumed, while he pinned the bandages, white fingers neat and swift and chill as the touch of death. \"I think... I didn't see the other clearly but I don't think he was a Turk.\"\n\n\"Was he vampire?\"\n\nShe hadn't thought of that. \"I... I don't know.\"\n\nTheir voices echoed strangely in the well of the hallway, shadows leaning over them, monstrous and upside down. Ysidro left again, carrying the basin and sponge. When he returned, he held a cup of tea cradled in his hands, the smell of it gently neutral, like sunlight on grass. \"They... they called to me from the street... Or I thought someone called to me from the street. They said Jamie needed me.\"\n\n\"I doubt there was ever anyone in the street,\" Ysidro said softly. \"He will have felt your mind, a little, at the tomb, and with that little he could fool you about what you saw in darkness. You were right, the _turbe_ of Al-Bayad was one of his sleeping places... He will have others.\"\n\n\"But you found nothing of Anthea? Or Ernchester?\"\n\n\"Nothing.\" He went to the hall table and stood for a moment, holding his hand near the flame of the lamp there to warm it. The fire, moving in its little red-glass bowl, lent his fingers, his hair, the skull-like ridges of his no-longer-human face a mockery of sunburnt health.\n\n\"Like him, she will change her sleeping place from night to night, and his glamour will work on her mind as well, hiding him from her, even as it hides her from the Master of Constantinople\u2014and hides her from me. If your husband is alive at all, it is because the Master of Constantinople seeks to use him as bait to trap her, for he fears her, even as Grippen does.\"\n\n\"Grippen?\" said Lydia. \"Isn't he her master, as he is Ernchester's?\"\n\n\"It is not unheard of, for fledglings to turn against those that get them.\"\n\nHe turned his hand over. The light seemed to shine through his fingers like parchment, illuminating spidery bones. \"It takes great strength, and great anger... but then, Anthea _is_ strong. He has always distrusted her, as all masters distrust their get; and between Anthea and Grippen has always lain a most delicate balance of wariness, and power, and hate. I do not think he would have made her vampire had not he thought he would lose Charles when Anthea, still a mortal woman, died.\"\n\n\"So they didn't... they weren't made vampires at the same time.\"\n\n\"No. Charles was forty, Anthea thirty-three, when Grippen took Charles. Anthea was a widow for over thirty years. She had grown old when Charles finally came for her\u2014or got Grippen to come. She hated Grippen for holding the dominance of a master over her, but she understood that it was the gate she had to walk through if she would be with Charles again. It is... a rather sad tale. Will you have more?\"\n\nShe shook her head. As he took the cup from her, she saw how his clothing hung on him, as if there were nothing inside it but bones. The turned-back cuffs of his shirt showed wrist bones knobbed like hazelnuts under milk-white skin.\n\n\"Thank you,\" she said softly.\n\nHe made a move, as if he would take her hand, then stopped himself. For a long time their eyes held, and she thought, quite irrationally, _There is something else to say._\n\nIt was he who moved his face aside, still for a moment, then turning fully to look at the door. \"I will remain here until it nears dawn, though I doubt he will be back. Tomorrow the bolt of the door can be repaired, and things placed about the doorsills and the windows that he cannot pass. I have no doubt he learned from Karolyi that you were here and wanted to put you under his influence\u2014to force you to tell what Karolyi has been trying to persuade from you, did you but know it.\"\n\nLydia shivered, thinking of the long climb to the bedroom. Even Margaret's presence in the bed beside her seemed welcome now.\n\nYsidro put his head a little to one side, listening. \"She sleeps now.\" He started to speak again, then didn't, as if he, like Lydia for the moment, did not wish to raise the issue of Margaret, and his use of Margaret, between them.\n\n_There is something else,_ Lydia thought again as they stood together, looking at one another in the lamplight. But Ysidro turned away and settled himself in the chair she had occupied, folding his bony arms within the shirt that seemed too large for him. Lydia slipped the cloak from her shoulders, and when he took it, slowly climbed the stairs.\n\nAs Ysidro had said, Margaret was asleep. She'd loosened her corsets and pulled the pins from her hair but still was dressed, as if she'd fallen asleep huddled wretchedly on top of the covers, and in the glow of the bedside lamp her face was taut with unhappy dreams. Lydia's hands shook as she unbuttoned her torn shirtwaist, for reaction was settling on her. She had no intention of turning out the lamp beside the bed, but it was too bright for easy sleep. As she walked around to it, she saw half a dozen sheets of paper on the floor around Margaret's basket of crocheted flowers.\n\nThey were tumbled untidily, as if she had been reading them when sleep overcame her and they'd slid from the coverlet. When Lydia picked them up, she saw the handwriting, precise and black and, though the ink was clearly modern, nothing that had been seen since the days of Elizabeth.\n\nThey were sonnets.\n\nAbout darkness. About mirrors. About roads untrodden stretching endlessly into night. One of them Margaret had ripped into quarters. Lydia had to lay it on the nightstand to fit its pieces together again.\n\nAnd she understood.\n\nBlood on marble\u2014petals of a rose\u2014\n\nOr copper-dark upon the lion's paw;\n\nBrightness and heat, like wine drunk red and raw.\n\nWine vends dreams, but life in lifeblood flows.\n\nThus warmth from flesh to flesh the blood imparts,\n\nA ruby heat reviving life and mind.\n\nWhere can hunger better substance find\n\nThan sanguine fire drawn from living hearts?\n\nI've seen a brightness dwells not in the veins\u2014\n\nIn thinking eyes, and smiles that shame despair.\n\nColor and heat beyond what blood contains\u2014\n\nRose and copper in cheek and lips and hair.\n\nBut flesh that can't be warmed by such a fire\n\nTo only blood and silence may aspire.\n\nThe papers were creased, as if they'd been wadded small\u2014hidden in the crochet basket, she thought, or in Margaret's carpetbag. She wondered at what point Margaret had found them and pocketed them for her own.\n\nShe laid them back on the floor where they had been and turned down the light.\n\n# NINETEEN\n\n_A CURIOUS thing for a vampire to keep._\n\nAnd so they were. Two silver keys, cut in exact replica of English Yales, even to the finger grips. Asher stared at them for a long time, as they shimmered in the concealed well in the red-tiled coffee room's floor.\n\nLocal work. Probably just enough admixture of bronze to keep them from bending in a lock. Reaching down, he weighed them in his hands. Even with gloves, a vampire would have difficulty holding them long enough to use. One as old as the master of the city might just manage, as he managed to hold the whitethorn of his halberd staff, to wear the thickly sheathed silver knife around his neck.\n\nAsher's heart pounded hard as he slipped them into the pocket of his coat. As he pushed the tile cover back over the well, returned the black and white table to its place, the shadows of his single candle seemed to lean closer, silent with a terrible, listening silence in which the Master of Constantinople seemed to be standing just outside the door.\n\nThis was not the case, he knew. Olumsiz Bey was meeting that night with one of his men of business and had himself escorted Asher back to his gallery after supper and locked him in. \"I apologize,\" the vampire said, \"for my Zardalu last night. He is treacherous and insolent, like most of the palace eunuchs. He needed a good thrashing, to make him remember his love for me.\" The amber eyes narrowed as they studied Asher's face. In the ambiguous flicker of the pierced lamp the Master of Constantinople had seemed wrought entirely of amber, the dusky pallor of his flesh like copal, the many-pleated silken trousers and the tunic over them, the vest and the sash all warm shades of fire and honey and marigold, the fur-lined pelisse sewn with shining flecks of gold. The lump of amber swinging from his earlobe caught the light like an unnerving third eye.\n\n\"I trust you understand that he is a liar,\" Olumsiz Bey went on. \"He never imparts information which is not aimed at starting prey.\"\n\n\"He's certainly told me a number of odd things about this house.\" Asher folded his arms, returned the orange gaze; even in his own mind the picklocks under the carpets did not exist. \"Twice he's told me the way out.\" This was a lie, to see what the master would say. Olumsiz Bey's eyebrows bent in the middle like startled diacritical marks, and the hard mouth quirked in laughter.\n\n\"I observe you didn't go seeking. Sayyed wouldn't be difficult to overpower.\"\n\n\"The way he told me was different the second time,\" Asher said. \"I've heard them talk about the games they play with their prey, chasing them through the dark here; I've heard those poor young boys and girls screaming.\"\n\nAnother diacritical mark, this time in the corner of those colorless lips, and Asher thought, It was not his custom then, to have his prey brought him by the others. It was something recent.\n\nZardalu was right.\n\nSomething was holding him to this house.\n\nErnchester? he wondered now, working his way carefully around the walls of the Roman court, that he would not leave a trampling in the overgrown grass. It made no sense. Why send for Ernchester now, why not a year ago, or a hundred years ago? Why not in July, when the Sultan's regime was overthrown? If it was to ask his help against the interloper of whom Zardalu spoke, why keep him locked in the crypts? Starving, perhaps, in pain certainly\u2014the moans were cries of the most hideous torment.\n\nRevenge?\n\nAsher shivered, feeling his way from pillar to pillar of the old porch, for he'd blown out his candle. The Bey's revenges would be long.\n\nBut long enough for him to summon the old earl from his moldering town house in London, from the slow crumbling of his life, back to the city where he'd spent eighteen months a living man? What ill turn would have warranted that, after almost two hundred fifty years?\n\nAnd what did the interloper have to do with any of this?\n\nWhat about the machine the Bey was having constructed? Or the ice Asher had seen, melting on the floor behind the silver bars?\n\nIt crossed his mind obliquely to wonder if the revenge was against Anthea, and not against Ernchester at all.\n\n\"He is not on this train,\" Anthea had said, coming back into his compartment while the flat lands of Hungary swept by in the darkness. That had been late the first night of the journey from Vienna. Exhausted, half sick with the coffee the porter had brought, his head aching and every clack of the well-sprung wheels reverberating as if slaved to some infernal machine inside his skull, Asher had watched her shed the long black-fringed shawl and put back the spotted gauze of veils. She seemed beautiful to him beyond words, staring at the molten ink of the window glass. The only light on the length of the Orient Express was theirs, and now and then it tossed threads of illusory fire on the wind-lashed weeds beside the track. Not even the moon remained in the sky.\n\n\"Good.\" Asher set aside the book he'd been trying to read, a truly dreadful account of life and love in Nero's Rome; tried to set aside at the same time the stirring within him of protectiveness and desire. He kept his voice deliberately casual. \"It means we've got every chance of reaching Constantinople before him. 'The Dead travel fast,' Goethe says\u2014but few things travel faster than the Orient Express. If he left Vienna by any other route, even by another train the minute he got away from the sanitarium, he'll still be a day behind us. Would you know, when he enters the city?\"\n\n\"I... don't know.\" She turned in her fingers the pearl buttons of her glove, a beautiful ghost in her blue and violet silk dress. He remembered the moonlight vampire girl in the woods outside the sanitarium and knew this dreadful warm surge of wanting for what it was\u2014the lure to prey. Dimmer, more distant, almost certainly without her conscious volition, still it was there. He wanted her.\n\n\"I don't know what arrangement was made with this Olumsiz Bey,\" she went on after a moment. \"I looked at the guidebook. There are smaller stations in the city before one reaches the main gare, and this... this Bey, this master... may have planned to meet him at one of them. I don't know whether it will be safe for me to watch the main gare through the night. Perhaps he will not enter the city by train at all. Charles never trusted trains, nor the Underground of London, never liked them and never rode them. And the city itself, its sounds and smells, will be... different.\"\n\nShe fell silent, her fingers in their lacy mitts resting still on the purple plush curtain, her brown eyes staring out into the night. Seeing the night with the night's own eyes.\n\n\"Even Paris is different from London,\" she said at length, as if speaking to herself. \"In London I know it if a policeman takes an unfamiliar turning within two miles of any of our houses. I could find Charles did he sleep in the lowest subcellar, did he walk the most obscure back way, did he haunt the steeple of St. Paul's or the warehouses of Whitechapel\u2014given time. Vienna was more different still, chaos, a game without rules. Constantinople...\"\n\nShe shook her head, but in her voice Asher heard the tremor, not of fear, but of excitement, of joy.\n\n\"It's strange,\" she went on, her voice so low that it should have been barely audible. \"I should be terrified. Outside of London I'm a snail dispossessed of its shell, a rabbit with all her earths stopped. And yet all I feel is delight. The lights on the Alexander Bridge in Paris, like being inside a star; all the voices and music and scents of Vienna, making me drunk as I walked along the Ring. I know I could be destroyed in seconds, but all I wanted to do was dance and laugh and take off my hat and swing it around by its veils, just to be... just to be somewhere else. Seeing something new, something wonderful that I'd never seen. I don't know if you can understand that.\"\n\n\"Maybe not fully,\" Asher said. \"I've never been dead.\"\n\n\"That's what being alive is, isn't it?\" She turned toward him and reached up to pull out the jet and steel pins that held her hat to the close-folded raven universe of her hair.\n\nAsher nodded, understanding something else about her now, and the desire he felt was softened and transmuted to pity. \"You never wanted to be a vampire, did you?\"\n\nShe hesitated, the hat like a dark bouquet overflowing her hands. \"Oh, I did,\" she said. \"The sharpening, the deepening, the enriching of the senses... one drowns in the color of silk, or the scent of coffee, or the weeping of the fiddles in distant night. Or the smell of blood, of sweat, of human fear. It is all the universe, as it never is to mortals, except maybe to a small child. It is living. And I wanted more than anything else not to leave Charles, ever. Once I came into it, I wanted it, craved it as a drunkard craves brandy.\" Her lips quirked ruefully; pale, Asher noted automatically. After Anthea had released him from her coffin, they had rushed onto the train with only minutes to spare, and Anthea knew well that every passenger was rich and expected at the other end, every porter and waiter accounted for.\n\n\"I gather people become vampires because they want life; they want life that won't stop, won't even pale as life does for the old.\" She stroked the ostrich plumes of the hat, curling them around her fingers, her eyes not meeting his. \"But to be dead is to become... static. And that is what we all become. We do not travel because it is dangerous. We wall ourselves into our houses, our crypts, our secret ways, because sleeping in the hours of daylight, we are as if drugged. We ring ourselves with locks and traps and things that we can control, and destroy those things that we cannot. We become dead. Journeying like this...\" She shook her head once more. \"All new things are peril, peril of death\u2014and maybe peril of death is one definition of life. Sometimes I feel that I shall never return to London again.\"\n\nAsher remembered Cramer, who would have been one of the best if only he'd had the chance.\n\nShe stretched out her hand to him, her face gravely beautiful. He knew that what the moonlight girl had tried to do to him in the dappled silence of the Vienna Woods, this woman had done to thousands of men in the streets and alleyways of London: made them love her, want her, need her, with a need that brought them mindless and damned into her arms. He remembered Fairport crying pitifully as the vampire women stripped him of his clothes, ripped at his veins in tiny, shredding cuts that would not kill immediately; drank his terror and his despair as well as his death. Fairport who had only wanted to live as normal men lived.\n\nAnd still he reached out and touched the long square fingers with his own.\n\n\"Thank you for coming with me,\" she said quietly. \"Thank you for... for seeing that I come to no harm.\"\n\n_I feel as if I shall never return to London again._\n\nStanding alone in the darkness at the heart of the crypt, Asher felt the knife-twist fear in his heart, that he would never see Lydia again.\n\nHe had thought of her often in the long prison chamber, listening to the sunset wailing of the muezzins, the constant squabbling of the gulls, the wind-whisper flight of vampire feet in the labyrinths below. He was glad in an odd way that if it was to be so\u2014if he was going to die in the House of the Oleanders\u2014he hadn't known it that rainy morning when all he'd been looking forward to was the emotional harrowing of his cousin's funeral and the distasteful scenes of family greed sure to follow. He'd have been solemn then, he thought, and solemnity would have completely spoiled the pillow fight early on the morning of his departure, and the giggling tussle of lace and kisses and stray medical journals.\n\nSeven years. It should have been longer. She'd probably track him to Vienna, but Anthea had smuggled them both onto the train. It was not possible that anyone from either Halliwell's Department, or the Stadtspoliz, or the Kundschafts Stelle, had seen them board. Thin, matter-of-fact, beautiful with a breathtaking marsh-fairy beauty which she herself had been forbidden to see... His soul ached, suddenly and desperately, with the need to see her one more time before he died. Only that, if nothing else were possible...\n\nHe wondered if, in tracing his contacts in the Austrian city, she would somehow meet Fran\u00e7oise.\n\nThe tarnished silver bars glimmered dully in the light of the single candle, cold even in the comforting yellow glow. Asher set the candle down carefully on a crossbar, its base protected by a circle of paper torn from a book to preclude telltale drips of wax while he worked carefully with the twisted bronze wires of the homemade picklock. It was hard to keep his hands steady, given the cold of the November night, the ice piled here in such quantities... the fear. The silence was a second darkness, and the smell of ammonia clutched his throat.\n\nThe silver hinges did not creak. He stepped into the low-roofed corridor, edged past the puddles of water, the sawdust and the straw.\n\nWhy ice? Absurdly, he remembered something the vampire Ysidro had once told him, about aging vampires suffering from cold. Surely all this wasn't just to make an old enemy uncomfortable? He wondered, if he freed Ernchester a second time, whether the vampire earl would escape with him at all, or whether he would, as he had in Vienna, simply let him free and cleave to the Turkish master who had summoned him.\n\nWhy?\n\nThe second door along the corridor, as Asher had already begun to suspect, opened into a cramped pitchy wilderness of coils and tubes and tanks, the harsh stink of ammonia like acid in the air. The weak firefly glow limned the words ZWANZIGSTEJAHRHUNDERT ABKUHLUNGGESELLESCHAFT on a crate.\n\n_Twentieth Century Refrigeration Company._\n\nFreezer chests, a vacuum plant, hoses like obscene rubber entrails dangling. Glass carboys of poisonous ammonia gas gleamed like monstrous eggs. Though the floor of the corridor was wet, there were no tracks in here, no straw, no sawdust. Having gone through the installation of a new furnace in one of the New College lecture halls, Asher guessed that some part or valve had broken, and had been sent for to Berlin.\n\n_Five days since the breakdown,_ the Bey had screamed, _and still no word..._\n\nHe closed the door, locked it, wiped the silver handle with his handkerchief.\n\nThe second door's handle was like ice. The sound of the tumblers going over was that of hammers driving coffin nails, answered from within, as from the deeps of a tomb, by a profound, sickening groan.\n\nThe stench that rolled over Asher as he pushed the door inward almost physically blinded him. He shut his eyes, averted his face. _Stupid_... he thought the next moment. And then, _If it's this bad when it's this cold in here..._ His breath was a cloud in the wan candle flare; the hoarfrost glistened on the stone walls, as did the ice that almost filled the crypt.\n\nBut all that was peripheral to the dark thing crawling toward him through the mess of half-frozen sawdust and straw on the floor\u2014and to his understanding of what it was, and what it meant.\n\nStaring down into the face\u2014into what was left of the face\u2014he knew everything, everything except where Ernchester was, and even that he could begin to guess.\n\nThen his breath was shut off under the crushing grip of a fleshy hand, and he was swept backward through the door with such force that he felt his feet leave the floor. He barely had time to pull his head forward when he struck the corridor wall, not thrown into it, as Olumsiz Bey had hurled him before, but slammed against the stonework with such force as to break ribs. He cried out\u2014he thought he cried out\u2014as the bones knifed him within, his mind suffocated under darkness, breath driven from him and unable to return. He struck the wall a second time, pain lancing his left shoulder blade as if he'd been struck by an ax, and all the while a voice screamed at him, screamed curses in Persian and Arabic and Turkish, incomprehensible through his mounting desperation to breathe...\n\nHe didn't know what language, he thought the voice was shouting, \"Is this what you wanted? Is this what you sought?\" and the hand twisted his head, the pressure on the spine intolerable, the icy water on the floor drenching him as he lay in it. \"Is this what you wished to see?\"\n\nBut he could see nothing, the candle having fallen to the wet floor of the crypt; nothing except, in his mind's eyes, the livid face of the thing in the crypt. Claws slit his sleeve open, shoulder to hem, while a knee ground in his back and the terrible weight pinned him to the stone, his neck bones cracking under the vindictive twist of Olumsiz Bey's hand. His arm was torn open to the wrist, blood burning hot on the sudden cold of his flesh, and all the while the smell grew around him, mounting and horrible, waves of it, while something fell squishily against the wall nearby, dragged with a horrible, thick groaning through the pools on the floor. Something fumbled at his arm, slick and glutinous around the sharpness of teeth; he heard the vampire whisper, \"Drink. Drink, my kitten, my child, my beloved... drink...\"\n\nSomething that felt like a hand\u2014or what had once been a hand\u2014groped along his arm for a steadying hold.\n\nThen with a retching noise the thing pulled away, rolled, crawled, with horrible sounds, back toward the door of its crypt and began to vomit. Asher thought later it was the release of the twisting pressure on his neck and backbone as Olumsiz Bey left him, as much as anything else, that finally let him faint.\n\nHe didn't think he was unconscious more than a minute or two; the jabbing pain of a tourniquet on his arm brought him back to the same inky darkness, the icy water seeping through his clothing to icy flesh, the sinking weakness of blood loss. His own blood, coppery in his nostrils, was the least horrible thing he smelled.\n\nThe cold was marginally less. The crypt door was closed.\n\nSoftly, his body aching with the careful ration of his breath, he said, \"So that's why you wanted Ernchester.\"\n\n\"You know nothing of these things.\" The master's voice came shrill, slivered thin through constricted throat, constricted lungs. His hands dragged the tourniquet as if he would use it to cut off the arm he bound.\n\n\"I know you're fighting an interloper on your territory. I know you don't trust those of your fledglings you have left... and I know now that you've lost the ability to make more.\"\n\nThe nails tightened on his arms, tearing again the numb flesh.\n\n\"That's it, isn't it? You haven't been able to make a fledgling for years. Only six vampires, for one of the biggest cities in Europe? Where the government doesn't even _care_ if you kill, so long as it's Armenians and Jews and the poor? Even your fledglings were beginning to comment that you were growing choosy about getting others to replace those who'd been destroyed.\n\n\"But when the interloper came, you had to make the attempt. And when you saw it wouldn't work\u2014that you could hold the fledgling's mind alive through physical death but couldn't transmit the physical syndrome of vampirism to the body\u2014you used your contacts with the old Sultan's allies to send for the one vampire you knew you _could_ control, the one you knew whose fledglings would be yours, under your power...\"\n\nThe hand closed around his neck again, not strangling this time, the clawed nails hooking like wolf's teeth under the bundle of nerve and tendon and blood vessels below his ear. The hard knee pressed, bracing, on his chest, like the small, blunt end of a ram. Very softly, Olumsiz Bey said, \"I... could... kill you...\"\n\n\"If you didn't need me for bait,\" he said, barely able to whisper against the dig of the claws. \"Bait to trap Anthea, and bait to trap the earl. If Ernchester isn't with the interloper already.\"\n\nThe hand released his throat. Wet silk passed over his bare arm, the side of his face, as the vampire stood. Then Olumsiz Bey kicked him, like the deliberate blow of a hammer, again and again like a man smashing rocks, and in a very short time Asher fainted again.\n\n# TWENTY\n\n\"THERE IS NO God but God, and Mohammed is the Prophet of God.\" The voice of the muezzin pierced the sodden fog of Asher's dreams like golden wire. \"Come to prayer. Come to prayer.\"\n\nAnthea, Asher thought, trying to surface, then slid back into velvet chasms of unconsciousness. He could see her on the train, her profile a milky coastline against the window's obsidian sea. \"Ernchester has never trusted trains,\" she said, and then her pale face, her white hands, turned to the marble bones of the grave steles beyond the Adrianople Gate, the dark of her dress and hair to the black cold of night.\n\nThrough brittle moonlight he saw a man walking, small and stooped in his old-fashioned clothing, but moving from gravestone to gravestone with the flitting lightness of the vampire. In the open ground he stopped, like an indrawn breath. Asher felt the presence of the shadow without seeing it, but in his dreaming it seemed to him that he smelled again the rank mixture of blood and mold that had overwhelmed him in the darkness of the dry cistern. Ernchester moved, turning as if to flee, but as he turned, the shadow was before him.\n\nThe air stirred with vampire laughter.\n\n_Do you think his favor is now off this man, and he is ours?_\n\nThe voice slipped into the dimming scenes of his dream, as if the wind had said it, but he knew what it was. He fought, panicked, to wake, struggling back out of the abyss.\n\n\"Did he want him dead, he'd be dead, not here,\" grumbled the voice that he recognized as one-eyed Haralpos.\n\n\"Wake him,\" the Baykus Kadine giggled. \"Wake him up and ask.\"\n\n_Wake up!_ he screamed at himself. _Wake up, they're all around your bed!_ Sleep was a black velvet pillow over his face. Maybe his body realized that if he woke, he'd hurt.\n\n\"Maybe he should be kissed,\" Pelageya said in her deep voice, \"like the damsel in a tower?\" Something that might have been fingernails trailed across the bare skin of his chest.\n\nThe whispering blurred, blended. He thought he saw the dim golden outline of the open door to the corridor outside, the subaqueous flicker of the pierced brass lamps, but he could not see the vampires around him at all. Only the red glint of their eyes.\n\n\"Maybe he knows where the Bey has gone?\"\n\n\"What makes you think he might?\"\n\n\"Someone had to bring him here...\"\n\n\"We have to find him...\"\n\n\"And tell him what?\" Zardalu demanded scornfully. \"That some worthless Armenian dog has been found with his throat slit?\"\n\n\"Bled...\"\n\n\"In a church...\"\n\n\"The man was a priest...\"\n\n\"Then he deserved it, whoever did it to him.\"\n\n\"He wasn't the only one. There was the old fig seller in the Koum Kapou...\"\n\n\"He is getting insolent, our Shadow Wolf.\" Zardalu spoke the name in Turkish, G\u00f6lge Kurt, the words harsh and guttural in the flow of his court Osmanli. \"Now our Bey must come out of this foolish hiding, must walk the nights again and stop crouching here with his _dastgah_ and his _almanya_ infidels...\"\n\n\"And if he doesn't?\"\n\n\"That kind of murder is stupid, senseless, leaving his kills to be fallen over. No wonder the Bey has told us to find this intruder G\u00f6lge Kurt, to kill him...\"\n\n\"What do you expect of a peasant who thinks he's a soldier because some other jumped-up peasant has put a gun in his hands?\"\n\n\"We must find the Bey...\"\n\n\"...find him...\"\n\nHe didn't know if they'd ever really been there. It seemed to him that he woke with a kind of start to find the chamber empty. The door still stood open, outlined in gold, and against the plastered walls the patterned spots of the lamp still wavered like an insubstantial scarf.\n\n_You know nothing of this matter,_ Olumsiz Bey had said.\n\nAnd Charles, _I love her unto death, and beyond._\n\nHe thought he knew where Olumsiz Bey could be found, and his heart turned over, sickened with shock and pity.\n\n\"There is no God but God, and Mohammed is the Prophet of God.\" The voice of the muezzin echoed dimly through the window lattices, as the ridiculously overdone grandeur of the Constantinople sunset bled to death in the west.\n\nShe could barely keep her hands steady, so it wasn't easy to achieve a proper symmetry of her coiffure. And in any case, thought Lydia, keeping her mind on what she was doing as if it were a dissection\u2014with a kind of cool, inquiring deliberation\u2014in any case her hair had never taken the fashionable curls necessary for a coiffure \u00e0 la grecque. In her current mental state she'd be lucky if she didn't singe half of it off with the curling irons.\n\nShe was trying not to look at the envelope marked with the Hapsburg crest, lying on the table beside her.\n\nNot that she needed to. She knew every word of the few lines written inside.\n\n> _If you would save your husband's life, meet me at the Burned Column at 3:00 today. One close to you is a servant of the Bey\u2014tell no one, but do not fail or your husband will be dead before dawn. Trust me. Karolyi._\n> \n> _Trust me._\n\nLydia had seen the Burned Column two days ago, when Razumovsky had detoured past it after the excursion to the bazaar. It stood\u2014a massive monument of Byzantine porphyry, its bronze horseman blackened by ancient smoke\u2014in the center of the old market district, a labyrinth of courtyards, alleys, warehouses, and crumbling, disused baths in the most ancient part of the city.\n\nIt was exactly the place she would choose for a kidnapping, if the victim were to be snatched up and quietly chloroformed. When the note arrived that morning, her immediate thought had been, _What on earth does he take me for?_ He must have realized, she thought, that she would be of no use to him, and was in a position to interfere with his plans.\n\nThe certainty that she was right hadn't made it any easier, during tea with Lady Clapham, to listen to the embassy clock strike three.\n\nAnd if Herr Jacob Zeittelstein wasn't at the reception of Herr Hindi's Turkish partner tonight\u2014if he hadn't returned from Berlin as expected that afternoon\u2014she didn't know what she would do.\n\nIt was Wednesday night. James had been missing for a week.\n\nShe closed her eyes, her hands trembling so much that she had to lower them, the iron cooling in her grip. _Dear God, let me find him,_ she prayed. _Dear God, show me another clue if this one fails..._\n\nIce, she thought immediately. She seemed to hear Razumovsky saying, above the clamor of the Grand Bazaar, _Someone always knows..._\n\nIf Herr Zeittelstein had gone to Berlin to fetch a piece for the refrigeration plant, it stood to reason Olumsiz Bey would be buying ice. It might take a few days to trace...\n\n_I can't afford a few days!_ she thought despairingly. _Jamie can't afford a few days!_\n\nThere was a noise behind her. She opened her eyes with a start, the distorted panic of too little sleep flooding her...\n\nMargaret stood reflected in the mirror, hesitating in the doorway behind her, blinking in the latticed sunset light.\n\nLydia's stomach contracted in rage and dread. _Not before a party,_ she thought despairingly. _I don't think I can take another scene..._\n\nShe pushed up her glasses and turned in her chair. Her red hair spilled, an untidy river, down her milky shoulders. She knew she should say something neutral, unargumentative: _Hello, Margaret,_ or, _Did you find what you were shopping for this morning?_ The governess had been gone when Lydia woke up. But she felt too tired to frame the words. She only looked, and Margaret occupied herself for a few minutes straightening the lace on the edge of her house mitt as if it were the most important task of the day.\n\nThen Margaret looked up. \"Mrs. Asher\u2014Lydia\u2014I'm... I'm sorry.\"\n\nFrom the time she was five years old, Lydia had been trained to smile and say, _It's all right._ Her upper arms were crisscrossed with sticking plaster and dressings. She'd told Dr. Manzetti\u2014and Lady Clapham, who'd recommended the physician and gone to him with her that morning\u2014that she'd been attacked by dogs. Against the sharp points of her collarbone, the knobs of her wrists, the silver chains that had saved her life felt heavy and cold.\n\nShe couldn't even ask, _Why?_\n\nThe sonnet she'd found had told her that.\n\nShe had lain awake thinking for a good part of the night, and found that the memory of those lines still made her heart beat swift and heavy with an emotion she couldn't define. Nothing at all like she felt for James. All her fear of Ysidro had returned, in strangely transmuted waves. Nothing at all like what she knew, or had ever known.\n\nGrief-stricken, silent, Margaret gazed at her with tears in her blue eyes. Lydia felt the anger within her ease.\n\n\"You're afraid for him,\" she said carefully, \"and you want to help him. You're afraid that he will die because of the promises he made to me.\"\n\nMargaret turned brilliant, blotchy red, and looked down at her gloves again; tears crawled slowly from beneath her heavy eyeglasses. This woman had tried to kill her, thought Lydia wearily: Why was she sparing her?\n\nShe knew the answer to that, too. Because Margaret had locked the door behind her, not only for the sake of the sonnet, but because she was Lydia Willoughby, heiress; because of all the sonnets no one ever wrote to the Margaret Pottons of the world.\n\n\"I'm so sorry,\" Margaret whispered. \"I'm so sorry. I don't know what came over me...\" She turned to flee, but stopped and turned back, standing, head down, to take punishment.\n\nFor a moment Lydia wondered if the interloper\u2014that animal face, grinning and grinning at her upon those few occasions last night when she _had_ shut her eyes\u2014had engineered Margaret's jealousy, as he'd cast upon her the languor that had stifled her own screams.\n\nShe didn't think so. But she guessed it was something Ysidro would do in like circumstance.\n\nAnd she shivered. She didn't want to think about Ysidro: playing picquet in the train, or bare white feet ascending the damp stone of the staircase before her...\n\n\"It's all right,\" she said.\n\nMargaret looked away and began to cry.\n\nLydia thought. Damn; bitterly, wearily, knowing that she must give comfort while she herself was exhausted, aching, wondering if she'd whistled Jamie's life to the wind that afternoon by assuming Karolyi's note to be lies, wondering what she was going to do if Zeittelstein wasn't at the reception, wondering how best to charm him if he was... And underlying it all, against her will, aware that she was as drawn to the faded ghost trapped within the vampire immortality\u2014like a mantis in amber\u2014as Don Simon was to her.\n\n\"Are you quite sure you're all right, my dear?\" Lady Clapham touched Lydia's wrist as they paused in the doorway of Monsieur Demerci's town palace above the darkening Marmara Sea.\n\nLydia nodded though she felt exhausted. She would have been glad to remain home, as Margaret had done, pleading a headache after the events of last night. Under the opera-length kid gloves and deep festoons of lace on her spinach-green gown, her bandaged arms smarted. The one thing she prayed, blinking at the dazzling electric brightness of the reception room, was that she wouldn't meet Ignace Karolyi amid the moving rainbow of men and women.\n\n\"I could do with a little champagne,\" she confided, as two slim dark servants, incongruous in Western-style livery and powdered wigs, ushered them through the tiled doorway toward the receiving line.\n\n\"What you need is brandy,\" retorted Lady Clapham. \"I'll see what I can find.\"\n\nTheir host was a Sorbonne-educated Turk in impeccable evening clothes, though the ferocity of his black mustache sounded an uneasy echo in Lydia's mind of the dark face with glittering fangs that had come so close to hers last night. His wife, a younger daughter of impoverished Silesian nobility, reminded Lydia of a highly bred rabbit in a yellow satin dress. She was probably the one responsible for the ridiculous eighteenth-century livery of the servants, and maybe for the electrical chandeliers, the candy-pink glass of the Venetian mirror frames, the tassled raspberry curtains and white and gold Louis XVI chairs, as well. Herr Hindi greeted her and expressed immediate concern: the beautiful Frau Asher did not look well, he hoped there was no indisposition. It comes of all this dull talk of business and jaunting about the old city; of course, a woman's more delicate constitution would be susceptible...\n\nOnly concern for her husband, who was to have met her in Constantinople and had not been heard from. Lydia unfurled her spangled Chantilly fan and tried to look interestingly wan without appearing haggard. She had hoped that Herr Zeittelstein might be here tonight. From things her husband had said, she thought that perhaps he and the honored Herr shared a mutual client, and she might glean some news...\n\nCertainly! Of course! Absolutely! Jacob had only just returned from Berlin that afternoon, he had been rather out of touch but he would be delighted to help in any fashion he could...\n\nAnd so, indeed, he proved. Jacob Zeittelstein was a youngish, strongly built man who in spite of evening dress looked more like a pipe fitter than his company's representative to the Ottoman Empire. He listened to Herr Hindi's introduction and Lydia's explanation with the air of one who never forgets names, faces, or circumstances and has all information at the tips of his beefy fingers.\n\n\"My husband mentioned that he was in touch with the Dardanelles Land Corporation, you see,\" Lydia explained, naming the bank account that had paid out a certified check for eighty pounds to a Freiherr Feketelo on 26 October. According to Razumovsky, Ignace Karolyi had left Constantinople abruptly, mysteriously, and under another name on the twenty-seventh. She had finally tracked it down, just this afternoon. \"He said he was meeting someone in the company here in Constantinople, and I was wondering... It's absurd,\" she added, with a slight duck of her head. \"And yet I can't help wondering if they might have heard anything...\" She raised her eyes helplessly to Zeittelstein's. \"But I haven't any idea who they are, and I can't seem to find out.\"\n\n\"Dardanelles Land?\" Zeittelstein's eyebrows shot up. \"The mysterious Herr Fiddat?\"\n\n\"I believe that was the name.\" Lydia sipped a tiny quantity of Monsieur Demerci's excellent champagne. \"They _are_ clients of yours, aren't they?\"\n\n\"Ha ha!\" Hindi trumpeted. \"She's up on everything, this clever little lady.\"\n\n_\"He,\"_ Zeittelstein said, with a puzzled expression. \"Not _they._ As far as I've been able to ascertain, the Dardanelles Land Corporation exists only on paper. Quite typical, actually. All those corporations do is pay money to their founders. Fiddat...\" He shook his head.\n\nLydia felt exactly as if she had\u2014not by chance, but by sheer steadiness of eye and hand\u2014shot an arrow clean into the gold.\n\nShe widened her eyes. \"What's mysterious about him?\"\n\n\"Everything. Extraordinary.\" He shook his head. \"It was on his business that I was in Berlin. Having decided, evidently all of a clap, to install refrigeration in the Roman crypt under his palace in the market district, he must needs have it now, at once. When the valve on the ammonia pump proved to have been cracked in shipping, he would not wait, like a normal person, for an express to Berlin for a new one. No. Five thousand francs he paid\u2014almost two hundred pounds!\u2014for me to return to Berlin, myself, in person, the very day the valve was found to be defective, by the quickest possible route. He even paid for the lost business here in this city that it cost me.\"\n\n\"They are very rich, these Turks,\" Hindi interpolated sententiously. \"Ill-got, I'll wager, some of them. Refrigeration works, you must know, my dear Frau Asher, by compression of ammonia gas, much better than the old sulfur dioxide system. Sulfur dioxide\u2014that's a chemical compound\u2014has the inconvenient habit of becoming corrosive and eating up the machinery which stores it. Ha ha!\"\n\n\"Truly?\" Lydia gave him her most radiant smile and timed precisely the turn of her head back to Zeittelstein, cutting off his further explanations with, \"And was he pleased to get his valve?\"\n\nZeittelstein shook his head. \"I'm not sure, Frau Asher. This afternoon I find nothing but a heap of hysterical messages from his agent... Has your husband ever laid eyes on Herr Fiddat himself, Frau Asher?\"\n\nLydia shook her head. \"I thought there might have been some sort of proscription against Mohammedans dealing with Christians face-to-face\u2014not ordinary Mohammedans, I mean, but that he might belong to some... some odd sect of dervishes.\"\n\n\"Not any dervish I've ever heard of,\" put in Hindi, in the act of neatly shagging hors d'oeuvres from a silver plate proffered by a servant. He grinned at Zeittelstein. \"Not that you'd know anything about that, ha ha.\"\n\nZeittelstein grinned back. \"Well, as far as I know, in thirteen hundred years no Mohammedan has ever had a problem dealing with a Jew.\" His grin faded and the dark, wise eyes grew thoughtful. \"I will say this: his agent's terrified of him. I can hear it in his voice. My own suspicion\u2014and I can't exactly say why I feel this\u2014is that Fiddat is a leper.\"\n\n\"How extraordinary!\" Lydia said with a wealth of implied _Please go on_ in gesture and voice and the tilt of her head.\n\n\"Nobody that I know has seen him,\" Zeittelstein continued, and glanced over at Hindi for corroboration.\n\nHindi tapped the side of his nose. \"Very mysterious chap.\" He turned to catch the eye of their host. Monsieur Demerci strolled obediently over, pausing now and then to smile and speak to one or another of his guests.\n\n\"Ja'far, you've never laid eyes on Herr Fiddat, have you? Or visited his palace?\"\n\n\"Oh, I've visited the House of Oleanders,\" Zeittelstein said. \"I spent the better part of ten days assembling that wretched compressor\u2014brrr, that vault is cold! But always I am met at the door by servants and conducted down to the crypt by them... They stand and watch me while I work.\"\n\n\"According to Hasan Buz\u2014the ice merchant, you understand, madame,\" Demerci said with a polite bow that made him look considerably less like a Turkish corsair and more like a former soldier made good, \"it's the same with his men when they make deliveries. The stuff gets stacked in the corridors\u2014half a ton at a time\u2014and the servants pay them and dismiss them. Hasan has to pay them double. They say the house is cursed.\"\n\n\"Where is the house?\" Lydia asked.\n\nA servant, emerging between the heavily carved pillars that lined the reception room, gestured discreetly; Demerci excused himself with another bow and went to speak to the man while Zeittelstein said, \"It's in the very old part of the city between the Place d'Armes and the old Sublime Porte, near the Bazaar. If you were walking east along the Tchakmakajitar from the Valide Han, it's the third turning up the hill. The house itself runs into at least three old _hans_ and rambles everywhere, but the door I go through is there. You'd have to walk around the walls till you found the main door, if you wish to speak to Herr Fiddat, but personally,\" he added, \"I wouldn't go there without an escort... and I don't mean Lady Clapham.\"\n\n\"Oh, no,\" Lydia agreed, her heart pounding fast.\n\n\"Great heavens, no!\" Hindi cried indignantly. \"A European lady to that part of town?\"\n\nTomorrow, she thought, looking around swiftly for the Russian prince. With Razumovsky and a couple of stout footmen from the Russian Embassy... _God, don't let Karolyi's note have been genuine!_ It was lies, it had to be lies, and that business of _One close to you is an enemy_ was, James had told her, one of the oldest tricks in repertoire. She wondered if perhaps they should wait until dark to include Ysidro in the party, but common sense told her that even were Ysidro at the height of his strength\u2014which he was not\u2014it would be far safer to enter a vampire nest in daylight hours than by night, even if it did mean going in without an expert's assistance. Besides, Ysidro might refuse to take part in an actual assault.\n\nDemerci strolled back, looking worried. \"Just a word of warning,\" he said quietly. \"There's more unrest in the Armenian quarter tonight. When you go home tonight, you may want to go through the Mahmoudie and the Bab Ali Djaddessi, rather than through the Bajazid.\"\n\nHindi gestured impatiently. \"They're not going to call in the army again, are they?\"\n\n\"I'm not sure. They have not so far. But there have been some rather... odd... murders, and it wouldn't take much to set off rioting again.\" He bowed again to Lydia. \"It sounds ridiculously feeble of me, madame, to ask you not to hold the actions of the army and the government against my people. We are not barbarians, in spite of what you must think. There are thousands, hundreds of thousands, of us who are horrified at what the army does to the Armenians, and the Greeks, in this city. It is a terrible mistake to put the rifles of tomorrow into the hands of the ignorance of yesterday.\"\n\nMost of the people at the reception seemed very little worried by the prospect of further rioting, as if such matters couldn't possibly concern them: Herr Hindi essayed a few jokes about what one had to deal with in foreign parts. Lydia wondered if this was because they'd already been through so many riots since July or because they mostly lived in Pera, or because they were as absorbed in selling railway stock or army boots or plumbing fixtures as she was, under normal circumstances, in isolating the effects of pancreatic secretions. One or two of the embassy wives called for their carriages early, but Lady Clapham merely said, \"Nonsense. Late's better than early. By the time supper's over they'll all have gone to bed and we'll be able to drive straight onto the bridge and never mind going the long way 'round.\"\n\nShe was probably right, Lydia thought. In any case. Prince Razumovsky\u2014who had a very Russian concept of time\u2014had not yet arrived, and tired though she was, she needed to speak to him tonight. Lydia had the distinct impression that if she went to Sir Burnwell and asked for help in forcing her way into an old palace in Stamboul to find word of Jamie, the result would be a round of polite letters to the Dardanelles Land Corporation rather than the prompt offer of a couple of Cossacks with clubs.\n\nSo she waited, too keyed up to do more than peck at the lobster aspic and ptarmigan in green peppercorn sauce, and on either side of her Herren Hindi and Zeittelstein traded head shakings over Mahler's latest symphony and the newest juicy tidbits of the scandal concerning the Kaiser's brother and a Vienna masseur. After supper there was dancing, and Lydia allowed herself to be swept into a waltz by Herr Zeittelstein and a lively schottische by the parson of the American Lutheran Mission on Galata, all the while listening, watching, for sight of His Highness' rich green uniform or the pantherlike grace that even without spectacles she knew as Karolyi.\n\nShe had worried a little about leaving Margaret at the Rue Abydos house with only Madame Potoneros and her daughter, though she suspected that unless she herself was there, neither Karolyi nor his vampire companion\u2014 _companions?_ \u2014would even try to enter the house. In any case, the bolt on the front door had been repaired, the one on the kitchen wing reinforced with another, stronger lock, and every window festooned with garlic and hawthorn.\n\n\"I can summon any into whose eyes I have looked,\" Ysidro had said to her once during a long game of picquet on the train from Adrianople\u2014they had been discussing _Dracula_. \"To call one to me who is a stranger\u2014to have them put aside silver, if they are wearing it, or garlic or any of the other flowers and woods which sear and blister our flesh\u2014is a more difficult thing.\"\n\nLydia shivered, wondering if the Turkish vampire, the interloper of last night, would have been capable of making her take off her silver necklace had he spoken to her on the street some earlier night or whispered to her in dreams. She had warned Margaret about Karolyi and given orders to the two housekeepers to remain until dawn. It was all that she could do, she felt, in the face of Margaret's blotch-faced, white-lipped refusal to accompany her tonight.\n\nLydia was standing beside the heavily curtained window that looked out over the Roman walls to the sea, scanning the newest comers to the room for the tall form of Razumovsky\u2014and even at this late hour embassy parties and members of the new government were still arriving\u2014when a cold hand touched her elbow and a voice like wind breath said, \"Mistress?\" in her ear.\n\nEarlier that day, remembering the sonnet, she hadn't known how she was going to speak to him, hadn't even known how she wanted to speak to him. But in the fierce electrical radiance of the chandeliers, he wore his alien, vampire face. It was the face that must show in the mirror\u2014a skull's face of hollow eyes and staring bones within the long web of hair\u2014and that was easier to deal with than the haunting illusion that somewhere in those sulfur eyes lurked the remnants of a living man.\n\nUnder his cloak he wore evening dress. She almost asked him if he'd left his scythe and hourglass at the door, until she saw the look in his eyes.\n\n\"They're making for the house of Olumsiz Bey,\" he said softly. \"Rioters\u2014Armenians, hundreds of them, crying for his blood...\"\n\n\"Who...? How do they...?\" Then she said, \"The ice carriers,\" realizing it for the truth at once. \"Of course they'd know.\"\n\n\"And the storytellers.\" Ysidro caught her hand, drawing her unseen by others toward the door to the supper room, to the kitchens, to the back stairs. \"And the beggars who watch the shadows pass at night. They all know. But they were afraid, until rage and hate at their priest's murder finally drowned their fear. Put this on.\"\n\nShe clutched the folds of the sable cloak, followed him past the unseeing servants cleaning up the plates, past the scullery boys bringing up more ice for champagne... past the footmen and drivers keeping warm by the fire in the stable court and looking up worriedly at the rising and falling of voices beyond the roofs, and the occasional snap of gunfire. \"What happened?\" She paused in the alley and fumbled her eyeglasses from their case in her reticule\u2014all things leaped into clarity, more fearful almost than the comforting dreamlike blur.\n\n\"A priest was killed. And then an old man, an inoffensive seller of fig paste who gave to charity and had more grandchildren than King David. Vampire kills, careless, deliberate. Meant to be found, and meant to enrage.\"\n\nIn the narrow lanes behind Demerci's mansion, rocky and steep as stairs, the voices sounded frighteningly close. Flame reflected on the wood and stucco, the stained and weed-grown walls. Lydia thought, _If they find me, they'll attack me just for being European..._\n\nIt was very hard to think past that fact, that fear.\n\n\"Karolyi,\" she said. \"Karolyi and the interloper. After I wouldn't cooperate. All they have to do is follow the mob and let it do their work for them.\"\n\nThrough a gap in the houses, she saw by torchlight a man riding the box of a broken-down carriage\u2014black-robed, gray beard streaming, waving a crucifix aloft. Men all around him raised flaming brands, clubs, the edged and pointed tools of marketplace trades. Women's voices keened like harpies.\n\n\"And part of that work,\" said that cool, disinterested voice in her ear, \"will be to kill James and anyone else they find at the Bey's palace. If by chance Charles or Anthea are there, they will likely be imprisoned, and in no case to flee. Was your builder of refrigerators among those at the house just now?\" He caught her elbow again as she stumbled, guiding her through a space between houses where a river of filth sucked at her shoes.\n\n\"Off the Tchakmakajitar Yokoussou near the Valide Han, he says. Third turning up the hill...\"\n\n\"I've seen it,\" Ysidro said. \"It was one of many I suspected, but dared not go close enough to be certain.\" Thin shards of moonlight blinked on shirtfront, cuffs, face, white on black, increasing her impression that she was being hastened along the insalubrious streets of Hell by a skeleton. \"With any luck we shall reach the place before the mob. and\u2014if James is in fact still alive\u2014before the Bey decides to kill him to preserve his silence.\"\n\n# TWENTY-ONE\n\nASHER KNEW he must escape or die.\n\nHe'd been wakened hours earlier by gunshots in the streets, had lain listening as the sounds of horror-driven fury, the random ululation of violence, ebbed and then flared like the sullen quarreling of a drunkard who returns again and again to the wellsprings of his rage.\n\nIt was deep in the night, probably not many hours until dawn, when he heard them coming toward the house. Even in the Tientsin riots, the worst he'd known, this was the hour when such things quieted. Something, someone, was stirring them up, rousing them anew when they flagged.\n\nAnd for the first time he could hear, among the confused buzzing shouts, words that he knew.\n\n_Vlokslak. Hortolak. Ordog._\n\nThey were coming to burn the House of Oleanders.\n\n_The vampires will flee,_ he thought.\n\n_Olumsiz Bey will kill me, rather than let me tell others what I've seen._\n\nThe spotted light of the stairway lamp still outlined the open door.\n\nThe thought of getting up appalled him. Just breathing was like being struck in the side with an ax. He rolled carefully off the divan and managed to get on his feet\u2014achingly glad that a Turkish divan wasn't even as high as the average milking stool\u2014the floor icy under bare soles, cold breathing around his ankles and stirring the long cotton shirt that someone had put on him when they brought him upstairs. He found his clothes farther along the divan, and put them on sitting. The boots were the worst. His bandaged arm ached and the stab of his broken ribs left him breathless as he pulled them on, but he knew the streets of Constantinople and knew he'd need them.\n\nTo his enormous surprise, he made it to the door on his feet. The house below was soundless. They'd probably break in on the other side, through the crypt where the ice was delivered. If he met them in the crypt, they'd quite possibly kill him out of hand before they realized he wasn't a vampire himself. Descending the stairs left him dizzy, but he didn't fall.\n\nThe thing in the crypt hadn't been able to drink much of his blood, though a good deal had been lost. He felt desperately thirsty. Down in the courtyard the sound of the mob didn't penetrate, and it was hard to disregard the voice in the back of his mind that argued that he certainly had time to lie down on the nice, comfortable pavement and rest a little...\n\nHe took the vigil lamp from its niche and continued. In the Turkish part of the house the mob's fury sounded closer, a heavy sea surge that would stop at nothing.\n\nThe tiled room. The overgrown court. The Roman baths. The long stair and the stench of ammonia, of wet brick...\n\nOf decay.\n\nThe leopard glimmer of the lamp suddenly outlined the dark form standing before him. The light gleamed in the citrine eyes and on the silver blade of the halberd, and Asher, leaning panting on the wall, knew he had lost.\n\nHe hadn't even the strength to turn and flee; the Bey would pull him down like a staghound a crippled fawn. Throwing the lamp would buy him seconds, but...\n\n\"God sent you,\" the vampire said softly. \"Help me. I beg you.\"\n\nHe stepped forward, holding out one hand with its steel talons and winking jewels. \"The others have fled. I have to get him someplace where the mob will not find him, have to get enough ice there, that he will live through the night.\"\n\nIn the corridor behind him, when Asher moved the lamp, he could see the wet diamond glint of ice where it showed through the oilskin in which it was wrapped. Masses of it, far more than a living man could carry. But even with a vampire's strength, he could not make it more wieldy. He couldn't carry it, and a body as well, up those twisting stairs.\n\n\"Please,\" the Bey said. \"After that you may do as you will. I have the keys to the outer doors, you are free to go. On my honor, by the Prophet I swear it. But help me get him to safety. Please.\"\n\nAsher set down the lamp. \"Is he able to walk at all?\"\n\nThe Bey stepped forward, some of the terrible tension lifting from the set of his shoulders, the angle of his shaven head. His snake-colored eyes seemed suddenly old, filled with the weariness of uncounted years alone. \"With support, I think. We weigh not so heavy as living flesh.\"\n\nAsher touched his arm, staying him as they edged between the ice blocks and the wall, to the silver bars that guarded the corridor to the crypts. The last time they were eye-to-eye had been here, with the Bey's claws lodged deep in his throat. Those wounds throbbed under a dressing of sticking plaster every time he spoke.\n\n\"You know it's not going to do you any good.\" He spoke, not in triumph, but in a kind of matter-of-fact compassion, for the creature beyond the bars was clearly beyond hope even if, by some miracle, Ernchester or some other vampire could be found to complete his transformation to the vampire state.\n\nHe half expected the same rage that, earlier in the night, had almost killed him, but the Bey only shook his head.\n\n\"If he can get through the night,\" he murmured. \"If he can last through another day... The... transformation... of the flesh, when it takes place, is little short of miraculous. I have seen sere and aged crones return to the beauty of their girlhood once they have the power of the vampire mind. The flesh returns to the form that is in the mind. And in any case,\" he added, still more quietly, \"though what you say may be true, I cannot leave him. He is... dear to me.\"\n\nThe body that the Bey brought forth from the crypt was wrapped in a sort of shroud of oiled silk, with oilskins on top of that. Still it stank, a limp and filthy thing in the tall vampire's arms, its wet black curls glistening between the bandages, its dangling fingers dripping brownish fluid. Asher flinched back from it, remembering the slimy lips mumbling at his arm, as the Bey set it on its feet beside him; his shoulders cringed from the limp arm the Bey laid over them. Then the bandaged head lolled, like a drunken man's, and the livid eyelids, almost black in the gloom, rose to show dark eyes flooded with agony, horror, and dumb pleading for relief.\n\nThe thing lived.\n\n\"He was beautiful,\" whispered Olumsiz Bey. He bent, gathering the corners of the oilcloth around the ice. He had laid down his silver halberd to carry the thing from the crypt, the first time Asher had seen him let it out of his hand. Now he slipped it through the knot of the oilskins, the haft where he could grip it at once. There must have been several hundred pounds of ice, but he lifted it easily, for it was only the awkwardness of it that had prevented him from bearing both it and the boy leaning, weaving drunkenly, on his shoulders. At close range the smell was suffocating, and he tried not to think about the consistency of the arm that held so desperately to his neck. He himself, with his cracked ribs sawing like broken bamboo within him, could barely keep his feet.\n\n\"Beautiful,\" the Bey said, \"and more beautiful still in his heart. He was ardent as fire, my Kahlil. A young warrior, and loyal to me to the bottom of his soul.\"\n\nIt was as if he heard Asher's thought, _And you repaid him thus?_ But Asher did not speak it, so there was no anger in the vampire's quiet reply.\n\n\"He would have been one of my living servants, here in this house. This was what I had planned.\" The shouting of the mob was very near, the sky above the tall Turkish roof\u2014usually so dark\u2014smoldering with the flare of torches. Smoke and rage burned the air.\n\n\"This was hard for me. I wanted to make him as I am, to keep him by me in his glorious youth forever. But I knew this was no longer possible for me. Fifty, sixty years ago, in the days of Abdul Mezid, when my friend Tinnin was killed, I tried to make a fledgling. Though that youth's mind stayed alive, a burning flame in mine through the death of his body, when I returned that flame to the flesh, there was no change, no alteration in the flesh itself. The fledgling rotted as he lay until in mercy I struck off his head. This had happened... once, maybe twice before to me, long ago. But afterward all was well. This time\u2014after Tinnin\u2014the power did not return.\"\n\nHe laughed soundlessly, bitterly, a tall figure in robes mottled like a tiger's in the shifting light. The jewels he wore threw back fire from the reddish glare of the sky, echoes of it catching in the ice he carried like some monstrous, Sisyphean gem loaded onto him by hilarious gods.\n\n\"I tried three, perhaps four times since that time, and I knew there was little chance of bringing Kahlil across to the vampire state. And I knew this was God's mockery of me: that having found the one I could trust, the one who could help me, I had squandered my gift of dark immortality on such as Zardalu and the Baykus Kadine, and that cobweb witch Zenaida who hides in the old harem, only because I needed those I could command to do my bidding.\n\n\"And then the interloper came.\"\n\nThe stairs from the old _han_ 's court were the worst. Where it had been silent, now the shouting was clearly audible, and drifts of smoke swirled harsh in the air. Asher abandoned the lamp to its niche again, his own injuries stabbing him as he struggled to help the shrouded form up the long flights, the Bey at his heels with the huge, unwieldy burden of dripping ice.\n\n\"G\u00f6lge Kurt,\" said the Bey's soft voice, almost as if it were in his ear, while beneath the bandages Kahlil made soft, broken noises of pain. \"The Shadow Wolf. God knows where he came from, or how he came to be vampire. Some Greek witch, no doubt, whom he later escaped... But he is a Turk of the new Turks, this upland peasantry that they've given guns and delusions of rule. I saw him first just after the coup, when all the city was in confusion. He had made a fledgling already\u2014as easy as spitting\u2014to challenge my power. I killed the fledgling\u2014but I could not kill him. And after that I had no choice.\"\n\nThey reached the long upper chamber. Asher sank, hand pressed to his side, onto the divan, the wrapped and shrouded living corpse beside him. While the Bey unfurled his oilskin to let the ice clatter down, filling the dry tiles of the fish pool, Kahlil, instead of lying on the divan, remained sitting beside Asher, clinging to him, as if frantic for the comfort of a living touch. Stinking, rotting, horrible within the bandages, but Asher could not thrust him away.\n\nThe Bey came back, tenderly lifted the boy's body and carried it to the ice. Watching them in the juddering orange flare of the lamps around the walls, Asher wondered bitterly how many men fell back on that phrase, _I had no choice,_ when it came to what they wanted\u2014even when it did that to those they loved.\n\nErnchester, when he had killed Cramer.\n\nKarolyi, certainly, if he thought at all.\n\nHe himself.\n\nOlumsiz Bey knelt on the steps of the basin, holding the putrefying bundle that had been the boy's hand.\n\n\"So you tried to make him vampire,\" Asher said quietly. \"Even though you knew.\"\n\nThe Bey nodded, once.\n\n\"And when you saw that though his mind survived, his body was beginning to rot, you sent for Ernchester.\"\n\n\"I could rule him,\" the Bey said simply. \"I knew him. I knew he was weak. He could get fledglings but had not the strength to command them. Once away from that woman of his\u2014\"\n\n\"Who loves him,\" Asher said. \"Who cares for him, as you care for Kahlil.\"\n\nThe Bey did not even look up at that, didn't take his eyes from his friend: only shook his head, a heavy, animal gesture, impatient and puzzled, as if he truly did not understand what Asher said. \"Women don't love. Not like men. Not like a man loves one who is the son he would have chosen out of all souls in the Universe. No love is like that.\"\n\nNo, thought Asher. A vampire to the end, even to the nature of his love.\n\nThe Bey did not even pause to speculate, to justify. His love was unique, and because it was\u2014and because it was his\u2014that justified all. He went on, \"But without the Sultan's power, I had to find what help I could. A savage, Karolyi, for all his civilized manners. A Magyar Hun. I think he had already begun to guess at what I was before I sent for his help. I think he had already wondered what use he\u2014in the name of his country\u2014could make of the Undead.\"\n\nHe leaned over to touch the forehead of the boy who lay now unmoving in his bed of ice. The great uneven blocks were old, dried and cleared and slick; they caught the feeble ember light like monster diamonds, faceting it to a wild rainbow over the walls, as if from a bier of jewels.\n\n\"I was able to hold G\u00f6lge Kurt at bay for a time\u2014I think all would have been well, had not Karolyi chosen to make what he could of the chance, to try to force Ernchester into the service of his country.\" His eyes, in their dark hollows, were dying coals of some old rage. \" _Country._ We the Undead at least were human once. Our sins are human sin. Magnified a million times, but human. These countries, these nations\u2014they are not human. They care not what they use, so long as it serves them. They care not what they do, and their sins are far beyond ours, literally of a different nature. You have served them. Karolyi told me that, Karolyi who is hollow inside, nothing inside, because this 'country' requires that he be nothing. You know.\"\n\n\"Yes,\" Asher said, remembering again. \"I know.\"\n\nHe shook his head. \"And so Karolyi delayed. And G\u00f6lge Kurt was able to gain a little more territory, to learn a little more of the city. I fear that when Ernchester tried to come into the city to obey my summons, he was met by G\u00f6lge Kurt and made a prisoner, and a slave. I thought that if I could trap the woman through you, I could draw Ernchester to me... Or at the worst, use her to make Kahlil whole. But it did not come about. And now it is finished.\"\n\nShouts rang in the courtyard, echoing from distant regions of the house. In the windows that ringed each shallow dome, the sky was red, like a cloth used to mop blood. The Bey reached in his robe, threw something to Asher that caught a spangle of the light as it flew. It was a key.\n\n\"Go,\" he said. \"First light is not far off. They'll be gone before then, and they will not come here. They will not even realize there is a stairway, though they stand at its foot looking up. Such is still my power.\" After a moment's thought he took the halberd and slid it across the floor to him, the silver blade flashing.\n\n\"You may meet one of them still,\" he added. \"If it is G\u00f6lge Kurt, kill him. Not for me. He is a man of the new breed who will try to buy power from whatever country he thinks will give it him. And he will buy it with any terms they ask. He is like your Karolyi. I only wanted one fledgling. They will want hundreds, loyal to their service. And what will come of that I do not wish to think.\"\n\nHe shook his heavy head, turned back to the boy in the ice. His voice was so low as to be almost inaudible, like the murmur of a fading ghost. \"And\u2014thank you, Scheherazade. Thank you for your help.\"\n\nAsher stood in the doorway for a moment, leaning on the silver halberd, shivering, for he had stripped off his death-stinking coat and only the piercing cold prevented him from shedding his shirt as well.\n\nHow many had the Bey killed? wondered Asher, looking at the bowed form in its golden robes beside the pathetic, shrouded figure on its jeweled pyre of ice. As many as a war, certainly. Karolyi would justify himself the same way\u2014as he, Asher, had justified himself, time and again. At the time he may even have been right.\n\nPainfully, clinging to the halberd for support, Asher made his way down the long stairs.\n\nIn the courtyard the noise was louder, echoing from the archway that led to the Byzantine house. Shouts, and the crash of precious things breaking, the thud of running feet. Smoke rolled in, burning his eyes and catching in the light\u2014too much, too strong, for torches. Some part of the house was in flames.\n\nLegs shaking, Asher leaned on the column at the foot of the stair and wondered if he had enough strength left to make it down the colonnade, across the overgrown court, through the crypts...\n\nAnd home, he thought.\n\nIf G\u00f6lge Kurt became Master of Constantinople\u2014and Asher knew it lay beyond his strength, now, to stop him\u2014it was only a matter of time before Karolyi, or some Young Turk just as eager for his country's triumph, convinced him to become a weapon of the state.\n\nAnd then a new age would come indeed.\n\nHe would tell Clapham, though he knew Clapham wouldn't believe. Even the redoubtable Lady Clapham would think his ravings delirium. One had to be born to it, raised in it, as Karolyi had been, to believe quickly... quickly enough. Razumovsky would believe, and Razumovsky would help him home... but Razumovsky would make a deal with Karolyi for what he could get. Bulgaria for you\u2014India for us.\n\nAnd the infection would spread.\n\nSomething dark rushed through the archway into the court, making straight for the stair. It paused before him, dark eyes flaring in the lamplight, and Asher realized, tardily, who it was. Tall for a Turk, with a Turk's black hair and scimitar nose, a feral bristle of mustache... the eyes were indeed the eyes of a wolf. All this he saw in less than a second; Asher didn't even have time to raise the halberd from its position as a crutch to that of a weapon when the vampire struck him aside, the impact with the wall like a sword in his side. Breath left him and wouldn't return, and when he opened his eyes again the vampire was partway up the stairs, lithe and silent as a lion in his torn khaki rags.\n\nAsher thought, grimly, _I have to pursue..._ but knew he was incapable of catching him, of moving more than a step or so without agony...\n\nAnd G\u00f6lge Kurt was not alone. Asher had seen vampires run\u2014eerily weightless and without a sound\u2014and knew the second dark form that streamed in like smoke and bones was a vampire as well. Even before he realized it was Ysidro\u2014 _Ysidro?_ \u2014the vampire of London, gaunt and starved and ghastly, fell upon G\u00f6lge Kurt like a silent falcon with a talon-rip at his throat that would have exsanguinated him had he not, impossibly, heard and turned at the last instant to meet the attack.\n\nThe two closed, fell, locked together on the steps, ripping at one another with clawlike nails, and seconds later a third vampire emerged from the dark, sprang up the steps. Him Asher knew at once, though in a strange way he seemed to have changed even more than Ysidro. When they last had spoken, by the flame light of the burning sanitarium in the Vienna Woods, Ernchester, if torn by indecision and grief, at least had been his own man. Now his face was empty, faded as the rags of his old black coat and filthy trousers, his blue eyes pieces of dirty glass. He caught Ysidro by the arms, dragging him back from the silent, slashing tangle on the steps, and held him while G\u00f6lge Kurt whipped a long soldier's knife from his belt. Ysidro took one cut across the chest before he kicked the blade aside, another across the face as he slid bonelessly free of Ernchester's grip...\n\nThen twisted as a pistol roared in the enclosing walls of the court. Ernchester and G\u00f6lge Kurt stood frozen, as between them Ysidro sank like a broken thing to the steps.\n\nIgnace Karolyi stepped from the colonnade on the other side of the court. \"Go,\" he said. He had an army pistol in his hand, the barrel smoking. \"I'll finish him.\" He spoke German.\n\n\"He's faking.\" G\u00f6lge Kurt looked down at the crumpled tangle of black and white at the foot of the steps. Blood glittered darkly on his face and throat where Ysidro's claws had ripped, but there was no sweat, nor did he pant\u2014in fact, he did not breathe at all. \"I never saw a bullet stop one of us yet.\"\n\nKarolyi grinned. \"My dear Kurt, you've never heard of silver bullets? They're a sovereign remedy for Evil. You'll have to look out for them, when you're working for us.\"\n\nG\u00f6lge Kurt's dark eyes glittered warily on the last sentence, but he made a smile, a demon manufacturing one for human consumption. \"Even so. Sharl...\"\n\nCharles Farren, third Earl of Ernchester, had come down the steps to kneel beside Ysidro's body, his hand pressed to his mouth. \"Simon,\" he whispered, half unbelieving, and Asher, still leaning against the wall in the warehouse bay's concealing shadow, knew then that it was true. It was, somehow, Ysidro. _\"Simon...\"_\n\n\"Come.\" G\u00f6lge Kurt had mounted a step, half turned back, and Asher remembered how Olumsiz Bey had spoken to Zardalu that night in the garden.\n\nErnchester looked up, his face struggling to regain an expression, some sign of life. The air was nauseating with the smell of blood. \"This man...\" he said haltingly.\n\n_\"Come.\"_\n\nHe did not touch him, did not make a move, but Ernchester flinched. Vampires do not generally show age, but Ernchester's face, thought Asher, was lined and haggard with the weight of centuries of immortality in which he had never, for one moment, been free.\n\nHe rose to his feet and followed. The two vampires passed like shadows up the stairs.\n\nKarolyi crossed the court, cocking the pistol as he moved. From the shadows of the bay where Asher stood it was three long strides to the foot of the stairs, too long to move without taking a bullet in the chest himself. Still, the key was in his hand, ready to throw as a distraction to buy himself time to spring, when a voice called out from the passageway to the house, \"Mr. Karolyi!\" and Karolyi turned in surprise.\n\nIf Asher hadn't spent seventeen years on Her Majesty's Service dealing with the absolutely unexpected, he would have thought, _Lydia???_ in sheer, baffled, horrified shock... and lost the split second her distraction bought him. He knew it was Lydia's voice even as he was moving, two fast strides, slashing down with the silver halberd blade at Karolyi's neck. The Austrian spun, his bullet cracking the pink plaster of the arch through which Asher came at him, and Asher reversed the halberd and caught Karolyi across the temple with the shaft.\n\nKarolyi fell back, dropping the gun, and grabbed for the halberd shaft. The two men grappled, and someone\u2014absolutely and unmistakably Lydia\u2014plunged out of the salon with a long bronze candlestick in hand whose weighted base she smashed into Karolyi's spine. Karolyi gagged, lurched; Asher kicked him hard in the belly, thrust him away, then stooped and snatched the pistol from the floor\u2014at the same moment Lydia sprang back out of any possible range and stood panting, red hair everywhere, like a disheveled mermaid in a torn green gown and opera gloves, her neck a treasury of silver and pearls.\n\nKarolyi backed, his hands raised, panting. \"My dear Dr. Asher.\" Firelight from the windows of the Byzantine house made everything luridly clear in the court. \"You can't shoot me, you know.\" There was a wryness, almost amusement, in his eyes, his voice; the same glint he'd had in his eye when he saluted Asher as Asher was led away to the Vienna jail.\n\nIt was a game. The Great Game.\n\nHis clothes were rough, a laborer's clothes, spattered with mud and blood. His dark hair hung in his eyes. But his appearance, thus or in his gorgeous Hussar uniform, had always been only a disguise.\n\nHollow inside, as the Bey had said.\n\n\"Silly niggers broke up the refrigeration coils in the crypt,\" he said. \"I heard them choking behind me. The place is chockablock with ammonia gas, and spreading. I know another way out.\"\n\n\"That true?\" Asher asked.\n\nLydia nodded. She was well clear of them both, in the center of the court, firelight a carnival of brass and vermilion on her hair, her spectacles rounds of fire. \"We were directly behind them, Ysidro and I. He covered my face with his cloak...\" She glanced toward the silent, bleeding huddle at the foot of the stairs, but said nothing more.\n\n\"You'll never get out of here without me.\" Karolyi lowered his hands a little. \"In fact you look hardly able to get yourself anywhere, if I may say so. They killed two of the Bey's servants already. We nearly fell over them in the alley. They're going to think you're exactly the same.\"\n\n\"And you're not?\"\n\nHe widened his eyes, amused. \"Who, me? You must know me better than that.\"\n\n\"He started the rioting,\" Lydia said quietly. \"He and the interloper.\"\n\n\"Oh, nonsense, madame, the Armenians have been itching for days to start fighting again.\" He turned back to Asher with a rueful grin. \"So we're stalemated, you see. And you'd better make up your mind soon, because in another few minutes you're going to pass out and that would probably be a bad idea right now. At least I can get you\u2014and more importantly your wife\u2014out of here alive.\"\n\nHe was right, Asher reflected. Every movement of his ribs was a sword cut, and he could feel his hands and feet growing cold. God knew what the mob would do to Lydia...\n\n\"Come now.\" Karolyi held out his hand. \"A temporary alliance, offensive and defensive. Nations do it all the time. You can't tell me I've done anything you wouldn't have done yourself. You would have done exactly what I did, and for exactly the same reasons.\"\n\n\"Yes,\" said Asher, seeing again the whore in Paris and the beggar in the alley he hadn't helped. Cramer laughing as he suggested going to Notre Dame for a crucifix. The body of his Czech guide all those years ago in the Dinaric Alps. Fairport dying in the light of the burning sanitarium, and the last, baffled, incomprehending look in Jan van der Platz's eyes. He felt strangely distant from himself, the world narrowing to the handsome face he had seen\u2014what? almost three weeks ago\u2014at Charing Cross. \"I would have. That's why I quit.\"\n\nAnd he shot Karolyi through the head.\n\nThere seemed to be no transition between that and Lydia propping him up, holding him under the arms\u2014it was the stab in his ribs that brought him back from momentary unconsciousness. He clutched her convulsively against him, pressing his face to hers. \"Lydia...\"\n\n\"God, Jamie...\"\n\nIt seemed absurd to ask her how she'd tracked him. Ysidro, he thought, turning, even as she broke from him and ran to the vampire lying like a smashed kite on the bloody pavement.\n\n\"Simon...\"\n\nThe skeleton hand moved, gripping hers. \"Go after them.\"\n\n\"You...\"\n\n\"I shall be well.\"\n\nShe was already tearing his black evening coat aside, revealing the white shirt nearly as black with blood. \"Don't be ridiculous, you can't\u2014\"\n\n\"It went through... I'll be ill for a time... the silver... burns...\" He raised his head, long hair falling back bloodied from his face. Surely, Asher thought, horrified, he had not looked like that when they had parted a year ago. \"Go.\" His hand pressed to his side and blood welled between the spidery fingers. \"Both must die. The man and the Undead with whom he made his bargain. It is your part of the pact, mistress,\" he added, still more softly. \"For this I came with you.\"\n\nAsher propped himself on the nearest archway and checked the revolver's chamber. Four bullets left, all silver. He started to say, _Stay with him,_ but there was a crashing within the passageway to the house, renewed smoke and voices cursing. Madness fleered in the air. Instead he said, \"Stay behind me.\" But it was Lydia who helped him mount the stairs.\n\nThe gallery stank like an abattoir of corruption and blood. The door stood open, and Asher stepped through quickly, gun held ready and his other hand clamped hard on Lydia's shoulder for support.\n\nThe long room was still. The few lamps flung huge shadows, glistened stickily on black lakes of gore.\n\nIt soaked the pile carpets, ran down the tiled steps to blend with the melting ice; splashed the walls, the columns, the divan. Asher took another step into the room, sickened, heart hammering, and in the heavy blackness made out shapes, the broken ruin of battle.\n\nThat thing like a killed dragon, glittering with blood and jewels, was Olumsiz Bey. It was too dark to see well, but he looked as if most of his throat had been torn out and his intestines strewn among the ripped silk of the robes. It might have been a trick of the candle flame, but Asher thought he saw the movement of those orange eyes. Unsheathed and covered with blood, the silver knife lay in his open hand. Beside him was a broken form in a black coat, wounds curling, blistering, blackening with the burning of the silver, short fair hair soaked dark with grue. Asher said softly, \"Charles...\"\n\nAnd Ernchester moved. Spastic, desperate, unable to rise or speak, still he flung out his hand in warning. Asher turned, throwing himself against the wall, and fired at the shadow that fell upon him from the denser shadows near the door. The bullet went wild; he fired again, and blackness covered his mind, blinding him, followed by pain in his side, in his shoulder, his neck. He rolled, struck one of the pillars at the end of the hall and someone dragged him back against it\u2014Lydia\u2014and his head cleared in time to see G\u00f6lge Kurt walking away toward the broken and bloodied forms of Ernchester and Olumsiz Bey.\n\nHe moved unhurried, without the drifting, ghostly swiftness of Ysidro. Asher guessed he had not been vampire long.\n\nLydia ripped free one of her gloves, fumbled with the tangle of silver and pearls around her throat. \"Put this on.\" She pressed a couple of chains into his hands. He realized G\u00f6lge Kurt was between them and the distant door.\n\nAsher obeyed, knowing it would do no good.\n\nOlumsiz Bey was moving. G\u00f6lge Kurt pressed the barrel of Karolyi's pistol to the older vampire's head and fired. The report was like a cannon in the long room. In the pit of ice the boy Kahlil cried out, a terrible sound; the Turk turned and fired at him from where he stood. The body jounced and lay still.\n\nLamplight glittered on G\u00f6lge Kurt's smile.\n\n\"I should give you to my friend, I think.\" He touched Ernchester with his foot. \"We are hurt, and the taste of death will make us feel better. But I think with the silver of the knife burning in his wounds, he may be hurt too much. So maybe I'll just have you both myself.\"\n\nHe grinned wider, then threw back his head and laughed, the blood from Ysidro's talons running black down his face.\n\n\"I'll hold him,\" Asher said very softly. \"You run for the door.\"\n\nShe had to know it was hopeless, because she nodded. The silk whispered as she gathered handfuls of it to free her legs. \"I love you, Jamie.\"\n\nAt the far end of the chamber the door closed, with a sound like the shutting of a tomb. The shadow standing just within it moved, turning the old-fashioned key.\n\nCandlelight flickered on the wicked, curving blade of the silver halberd.\n\nG\u00f6lge Kurt turned his head.\n\nShe stood there like a witch, like a thing truly risen from a nameless grave, filthy in her rags of luminous blue, blood in the curling raven ocean of her hair. The brown eyes had the weird sanity sometimes found on the far side of madness: calm, but a demon's eyes. There was blood on her mouth, and on her hands to the elbow, but the gold of her wedding band shone through.\n\nG\u00f6lge Kurt pointed the gun at her and fired, and she was stepping forward even before the hammer clicked harmlessly on the empty chamber, and with a vicious blow of the silver halberd took the gun hand off at the wrist.\n\nThe vampire screamed as blood exploded from the severed arteries, lunged at her only to be driven back with face and chest slashed, clutching, grabbing at the wounds where the silver blistered and burned. _\"Orospu!\"_ he shrieked at her, rage inhuman in his eyes. \"Infidel whore!\"\n\nShe stepped in toward him, slashing with the silver weapon, slicing open his legs, his feet, his thighs. When he tried to climb up the lamp niches, to spring from them to the windows of the dome, she cut the backs of his knees so that, when he fell back screaming with his remaining palm a fingerless charred wreck, he could not stand. And all the while her face did not change, nor did the tears cease to run from the empty demon eyes.\n\nOnly when she had driven him into a corner, blood gushing from his wounds to splash her skirts, the walls, the floor, did she stop, looking at him with an inner peace beyond compassion or hate.\n\n\"You killed him,\" she said, quite gently. \"You let him take the brunt of the fight, let him destroy the master you hoped to supersede. You cared no more for him than he did, this Bey, this... this master. It will be day very soon,\" she said.\n\nG\u00f6lge Kurt made a move to lunge, but with his hamstrings severed he could only flop on elbows and knees, while blood spattered around him like thick and stinking rain. She stood out of his range, looking down at him. Without turning her head she said, \"Charles?\"\n\nThe broken form moved then, lying near Olumsiz Bey on the blood-sodden carpets; moved, and reached for her with one hand. No louder than the scratch of a single leaf blown across a marble floor, Asher thought he heard a voice whisper, \"Beloved...\"\n\n\"Beloved,\" she replied. Her voice shook a little, but she never took her eyes from G\u00f6lge Kurt. \"You never did want this life, did you?\" she asked softly. \"Never wanted to continue, Undead but Unalive...\"\n\n\"...Don't... know.\" Ernchester moved his hand again, tried to raise his head. The guttering candles showed his throat cut almost to the hawse bone. Asher didn't even know whether the dying vampire was actually capable of making a sound. \"Don't... remember... what I wanted. Only that I did not want to leave you.\"\n\n\"Nor I, to leave life,\" she replied. \"Not if your love was part of that life, no matter what the cost to my soul. Nights and nights and nights, killing that I might not die... and you killing, that you might stay here with me. Not so?\"\n\n\"I chose...\"\n\nShe moved back to kneel at his side, though she still watched the Shadow Wolf, bleeding on the floor. One hand still held the master's silver weapon; the other reached down to touch the graying hair. \"I understand,\" she said. \"We all choose. And in a very short time it will be time for us both to go.\"\n\nBlack eyes wide with horror now, G\u00f6lge Kurt shouted at her, raged at her, cursed her in German and Turkish and broken French, and she listened with a face of stone.\n\n\"It is not I who brought him to this place,\" the vampire shouted. \"Not I who did this to him...\"\n\n\"It was you who met him among the tombs,\" Anthea said. \"You who used him, who controlled his mind, because he is what he is, weak... Don't you think I was aware of it, hiding among the cisterns and the catacombs of this city, when you two walked its streets to war with Olumsiz Bey? Don't you think I sensed it in my dreams, when you covered and hid his mind that he might not even know I followed and sought? To kill you is nothing.\"\n\nThe yellow light edged her face as it edged the halberd's dripping blade. There was no sound, now, outside, and the windows above showed as squares of ash against the night.\n\n\"I have killed every night to stay alive. Brought victims to him to kill when he was so weary of the life he lived that he could not even go to seek his own. All because Grippen wanted him\u2014and Olumsiz Bey wanted him\u2014and you wanted him to keep him from the Bey. And all you wanted was rest, Charles.\"\n\nCharles shook his head and did not let go of her hand. \"No,\" he whispered. \"I wanted you.\"\n\nIt was G\u00f6lge Kurt whose flesh ignited first. It puckered, blistered, blackening as he crawled screaming for the door, and Anthea cut at him again and again with the silver halberd until he retreated, screaming, to the corner, where the fire took hold. It swelled up from within him, not great flame, but thick blue-burning sheets. He sank to the floor and ceased to move quite soon, but he continued to scream for some time.\n\nBy that time Olumsiz Bey was burning as well, though Asher heard no sound from him. Perhaps he was dead, perhaps only lapsed into the vampire sleep that came at daylight, mercifully unaware of the end of his long unlife.\n\nAnthea, who had begun to nod with the onset of that same sleep, laid down the weapon she carried and knelt beside the man she had loved, gathering him up into her arms. Their mouths were pressed together as the fire took them, and neither moved, except to tighten their grip on one another until the very bones locked within the veils of heat. Lydia watched until the end, but Asher turned his face against her shoulder, the suffocating heat pounding him, nauseated with the stink of burning flesh and blind with tears.\n\n# TWENTY-TWO\n\nTHE ARMY CAME soon after. Shock had set in, and as Lydia supported him down the stairs with all the grim expertise of one used to maneuvering dead bodies, Asher felt himself drifting in and out of consciousness, pain coming and going in alternation with eerie, frightening dreams. He half expected to find the charred remains of Ysidro's body at the foot of the steps, but didn't\u2014or the reality in which he did was quite clearly a dream. Only Karolyi's body was there, lying in a pool of blood with a hole in his forehead and an expression of astonishment in his eyes.\n\n\"I was terrified he was going to talk you out of shooting him, Jamie,\" she said, helping him to sit on the bottom-most step and sinking beside him in a rustle of skirts. White-lipped and shaken, she propped her eyeglasses with a forefinger and blinked around her. \"I mean, he tried to kidnap me this afternoon\u2014yesterday afternoon\u2014and if we'd gone with him, we'd never have gotten out of here alive.\"\n\n_Trust Lydia,_ he thought, and wondered who had warned her about Karolyi.\n\nThe house around them was utterly silent. The Bey had evidently been right about the rioters leaving before first light. It was almost impossible to reflect that he hadn't seen this woman in three weeks, and that the last time they'd spoken it had been on the railway platform in Oxford. He leaned his back against the wall of the stairwell and asked, in what he considered a reasonable voice, \"What are you doing in Constantinople?\" and lost consciousness again before she replied.\n\nWhen he came to, the court was occupied by two squads of the Turkish army, who clustered around Karolyi's body, muttering and whispering. Their captain was an Anatolian highlander who seemed to pride himself on his imperfect command of both French and Greek.\n\nTurkish not being an easy language to speak under the best of circumstances, Asher could only repeat, _\"Bilmiyorum... bilmiyorum,\"_ and shake his head, while the captain and his men gazed at Lydia's unveiled face, bare shoulders, and uncovered hair with puritan disapproval.\n\nSince Asher was, however, clearly injured, a shutter was brought from the half-burned ruins of the Byzantine house, and two of them carried him on it through the twisting streets to the prefecture of police opposite Aya Sofia as the muezzins began to cry the rising of the sun. Lydia, by holding up her wedding ring and refusing to let go of his hand, managed to convince them that she was his wife and, once at the station, persuaded the sergeant in charge to allow her to telephone the British Embassy. In the wake of the rioting, the telephone exchange was inoperative.\n\nThey were relegated, not to a cell, but to a stuffy room on an upper floor, while a messenger was sought who could take a note across to Pera. A Turkish doctor came in around noon, rebandaged Asher's torn right arm and reset his shoulder, strapped up his ribs with sticking plaster, dusted everything in sight with basilicum powder and gave him veronal and novocaine, muttering all the while. On his way out he paused, studied Lydia's face intently, and opened his bag again to mix her a mild sedative as well. She accepted gratefully, knowing that the odd sense of separateness she felt from the events of the night was only the result of shock.\n\n_I've done it,_ she thought, looking down at the face of the man who slept beside her\u2014unshaven, bruised, his neck mottled with sticking plaster and dried blood, his flesh horribly white under the beard stubble.\n\n_I saved him. Well, more or less._\n\n_I found him. He's not dead._\n\nShe realized she hadn't really expected to succeed, to be able to do anything right, especially not that which was most important to her happiness. Not when it involved something as unpredictable as living people.\n\nThe happiness filling her had a soap-bubble quality, as if it could be taken from her at an unwary breath, but he was here with her... breathing. She checked the gashes on his neck. So deeply asleep was he, on the thin mattress on the floor, that he didn't wake. Like the older, red scars, they seemed like the marks of claws, but lacked the mangled puffiness of a wound from which a vampire would have drawn blood.\n\nRelieved, she touched his hair, the white streaks in his mustache, then leaned back against the wall and, for no reason she could discern, burst into tears. From this she passed very quickly into sleep.\n\nAn hour or so later one of the army corporals brought them bread, honey, white goat cheese, and tea. He brought a set of Turkish army fatigues for Asher, who was still deeply asleep\u2014the pile of his clothing in the corner was torn and bloodied and stank even to Lydia's dissecting-room-toughened sensibilities\u2014and a lady's trousers, tunic, vest, yashmak, veils, and slippers.\n\n\"Wife's,\" he explained, with a shy grin. \"Wife she say\u2014\" He gestured to Lydia's torn and blood-crusted gown. \"\u2014not good. Better.\" He held up the veils, grinned quickly again\u2014he didn't look old enough to _have_ a wife, thought Lydia\u2014and took a hasty departure.\n\nShe hung one of the veils over the judas in the door and another over the window, and changed clothes, glad to be out of the gown with its dried blood and the smell of charred flesh caught in its folds. The horror she had experienced made her want never to see the green gown again, but she knew that feeling would pass and she'd be glad of the copious samples of vampire blood. She wondered, as she settled back in the corner by Asher's head\u2014the room was innocent of furniture other than one chair and the mattress on the floor\u2014whether there was any way she could talk the authorities into letting her see the remains of the burned bodies.\n\nProbably not, she thought. She felt better for having slept and eaten, and despite the nightmare of her memories\u2014blue fire, charred flesh, screams like nothing human\u2014she found herself wishing she'd had a notebook with her, and a watch.\n\n_Ysidro..._\n\nCold tightened in her chest. Had he gotten to safety? The rioters had been gone by dawn, but he'd been unable to stand. And where in the city could he go?\n\nG\u00f6lge Kurt's words returned to her, about the taste of death bringing healing. In the riot-torn streets a victim wouldn't have been far to seek. She closed her eyes, not wanting to admit to herself how close she stood to condoning an innocent person's murder.\n\nLooking back\u2014remembering how Ysidro had torn like a mad wolf into G\u00f6lge Kurt on the stairway\u2014she felt a vast astonishment that he had refrained from hunting at all, upon her bare word.\n\nTheir compact was done.\n\nErnchester was dead. Karolyi had taken the secret of the vampires with him to the Constantinople morgue.\n\nJamie was alive.\n\nLike an echo, she heard the whisper of a voice in her mind: _There's a brightness dwells not in the veins..._\n\nHad he really been drawn to her, as to a flame of warmth? Or had that only been a literary conceit to compare the red warmth of fire and blood to the auburn of her hair?\n\nShe didn't know. She didn't know if she wanted to know. There was a strange hurt inside when she thought of him, a dark wanting that she didn't know what to do with. It felt nothing like the love, the need, that had made it impossible for her to contemplate a life that did not include James' arms around her when she woke up in the night.\n\nWhen Ysidro had carried her inside after G\u00f6lge Kurt's attack on her...\n\nShe did not finish the thought. She curled up close to her husband, and taking his hand as for protection, let herself drift into sleep.\n\nAt sunset Asher woke to the cries of the muezzins of Aya Sofia, calling the Faithful to prayer. His startle of panic woke Lydia; for a moment his grip closed hard enough around her fingers to bruise the bones.\n\n\"I never thought I'd find you.\"\n\n\"Find me?\" Asher said. His voice was raw and hoarse. \"If I'd known you were looking, I'd have white hair by now!\"\n\nLydia laughed a little shakily, and touched the silver glints in the brown. \"I'm sorry.\" She pushed aside her own heavy red coils, groped for her spectacles as if to satisfy herself that they lay on the floor beside her, but did not put them on. \"I was afraid I wasn't doing it right, but I was as careful as I could be. I always wore silver and carried a gun and made sure someone knew where I was\u2014well, mostly. Not that that would have done me much good some of the time. But I did try.\"\n\n\"You did well.\" He cupped the side of her face in his good hand. \"But then I never thought it would be otherwise, in anything you set out to do.\"\n\nLydia started to protest, and he covered her mouth with his own.\n\nSomeone knocked at the door, and a man called out in bad French, \"Monsieur Ash? Madame? Here we have of the British Embassy Sir Burnwell Clapham, and a lady, for to fetch you away.\"\n\nThe house on Rue Abydos was absolutely dark when the embassy carriage left Asher and Lydia at its door. \"I expect poor Miss Potton's still out looking for you,\" Lady Clapham said as Lydia unlocked the gate. \"We didn't get back ourselves until nearly dawn, what with looking for you and making a detour and our carriage being attacked by rioters. We sent a man over at about nine, and he said the house was locked up and silent, so we knew she must be doing what we proceeded to do: check all the hospitals in the city. It was only toward evening we started checking police stations.\"\n\n\"Then you didn't get the message?\" Lydia asked. In her all-encompassing black garments, with her red hair piled on her head again and the mud washed from her face, she felt like a schoolgirl playing dress-up; Asher, beside her in his khaki uniform, with his arm in a sling, appeared some casualty of a war.\n\n\"Heavens, did you send one?\" The attach\u00e9's wife shook her head. \"We haven't been back to the villa all day, child. We'll probably find it under the door\u2014if those villains at the prefecture bothered to send one at all.\"\n\nThe carriage rattled off into the dark. Lydia shivered. The house had a cold, unused feeling. She thought at first that Madame Potoneros and her daughter had departed that morning as soon as Margaret would let them, but found the kitchen fires unlit. They must have left sometime the night before. Lydia wondered uneasily, as she fished a match from the drawer in the hall to kindle the lamp on the little table, whether the housekeeper lived in Pera or across the Horn in Stamboul. The riot had spread to Galata, where the army had killed almost a dozen Armenians. Soldiers had been posted on the street corners as they'd come up the hill.\n\nThe back entry to the kitchen was unlatched. They could have fled that way as soon as the sounds of strife were heard at the foot of the hill.\n\n\"I hope Margaret hasn't come to grief herself.\" Lydia raised the lamp as she returned to the front hall. \"She's really not very bright, and completely out of her depth here. I shouldn't like to think of her trying to negotiate with a Turkish cabdriver, or...\"\n\nAsher straightened up from examining something heaped on the hall table\u2014a wreath of garlic bulbs and hawthorn. \"There are four or five of them here,\" he said. \"And none on the windows.\"\n\n\"Madame Potoneros may have taken them down,\" said Lydia, though she felt a qualm of cold within.\n\n\"Maybe.\" They looked at each other, then turned as one to hasten up the stairs.\n\nLydia froze in the doorway of the bedroom, lamp lifted so that the light fell through to show the unshuttered windows, the protective wreaths heaped in the corner, the still figure lying on the bed.\n\nAsher disengaged his arm from her shoulder at once, crossed to the bed. Lydia set down the lamp, a little numbly, on the vanity, and with a taper kindled the two smaller lights there. The added glow warmed the colors of the room but did little to dispel the dark in the corners.\n\nThe woman on the bed was Margaret. But then, she hadn't really had any doubts.\n\nAsher touched the woman's neck. There was a little dried blood around the mangled puncture marks, but of that, also, Lydia had never really had any doubts.\n\nThe waxiness of the skin, the blue color of the lips, the fingers, the bare toes visible under the white flannel nightgown, were very clear. Lydia set the lamp down again on the bedside table next to Margaret's eyeglasses, reached down\u2014as Asher had already done\u2014to touch the mangled neck, the short, unpretty jaw.\n\nThey were still rock-hard. If Margaret had died at the beginning of last night's darkness, rather than at the extreme end, almost at dawn, the rigor would be wearing off now.\n\n\"She took the herbs from the windows herself,\" she said softly. \"Ysidro said... a vampire could get a mortal to do that, if once he met her eyes.\"\n\nSomething made Lydia look around. A noise from the doorway, she thought later, though she could not have said what it was.\n\nGold-stained by the lamplight against the dark of the hall, Ysidro had returned to something of his old appearance, the death-head mask filled out a little, the black rings of pain and fatigue around the eyes less staring, though a great bloodless cut ran from his scalp down forehead, cheekbone, chin, from G\u00f6lge Kurt's claws, and two others crossed the fine-grained flesh of his neck. They were like the slashes a sculptor might make in a wax that he had suddenly come to hate: horrible, clean, without puckering. Ysidro seemed collected into himself again, perfect as an ivory angel, as if he had never dropped anything in his life or held strengthless to a doorpost, or written a poem admitting to dreams of warmth that did not come from stolen lives. As if he had never been anything but perfect, and the master of himself.\n\nLydia thought, _He has fed._ All her body seemed to be one giant pain. _He had no further need of her, save for that._\n\nRage exploded in her, all the stored horror at Anthea's death, all her sickened bitterness at Ysidro's arrogance, at those pathetic, melodramatic dreams he had sent to Margaret, kindling love in her like the flames kindling from the vampire flesh, and she fell on him, striking with her open hands at his face, with her fists at his chest and shoulders, hating him with a rending hatred that seemed to rip something deep in her soul.\n\nAfter a moment he took her wrists and held her from him. Under the bloodless cut his yellow eyes were aloof, looking without expression into hers.\n\n\"You cannot expect us to be other than we are, mistress,\" he said, in a voice she knew was pitched for her alone. \"Neither the living nor the dead.\"\n\nThen he was gone, and James was beside her, holding her in the circle of his good arm. Lydia clung to him, weeping, from exhaustion and shock and blinding, bitter grief at what she had lost.\n\n_I will find you_ , Ysidro had said to him once. _For those of us who hunt the nights, that will be no great task._\n\nAbove the looped chains, the cobwebbed mazes of counterweights, the hanging lamps of silver, gold, and ostrich eggs, darkness soared like the exultation of ancient spirits, nearly two hundred feet upward to the shabby painted plaster of Aya Sofia's dome. Below, Asher's footsteps ran whispering to all corners of the mosque, as if they had some mouse-sized secret to tell. Only a few of the lamps burned. By them he could see his breath.\n\nHe had walked here from Pera, down the steep steps of the Yusek Kalderim, across the New Bridge. Through narrow streets under the eyes of the Sultana's Mosque and the raw gray granite buildings of the new administration, up the gentle hill to this most ancient place.\n\nA Roman emperor had built it, or a man who thought of himself as a Roman emperor\u2014he and his beautiful, scandalous, red-haired wife. After everything that had passed around it, Asher still heard their names in the silent music of the columns, the unheard bass rhythm of the domes. As he had walked in the cemeteries and the cisterns under the eyes of Olumsiz Bey's fledglings, bait for the trap, so he walked now.\n\nIf Ysidro would find him, he thought, he would find him here.\n\nCharles Farren, Earl of Ernchester, would have walked here. A living man, two and a half centuries ago\u2014periwigged, ruffled, and court-suited\u2014dreaming of the woman who waited for him in England. _All I ever wanted... and all I ever had._\n\n_I wish you could have known us as we were._\n\nHe closed his eyes, knowing that he should not feel about her what he did.\n\nWhen he opened them again, it was to see the ghost-flicker of movement in the darkness among the line of columns in the apse, the touch of pallid lamplight on a colorless web of hair.\n\nAsher remained where he was. The vampire's footfalls made no sound on the dusty carpeted acres of the floor.\n\n\"I wasn't sure this was an appropriate place to find you.\" The echoes of Asher's voice were solitary drips of water in the immensity of an underground cave. \"But in the streets I felt unsafe, and there was a chance that the others\u2014the fledglings\u2014wouldn't enter a place considered holy to them.\"\n\n\"There is no reason why they should not.\" He moved carefully, in obvious pain, though his face showed no expression; Asher knew that Ysidro was a little tougher than younger vampires with regard to silver but guessed Karolyi's bullet had left an agonizing track of burns and blisters within.\n\nHe wondered who had dressed Ysidro's wounds.\n\n\"Unless one has put up garlic or silver, or some other thing inimical to us around the entrances, there is no limitation upon what building we may go in. Neither crosses, nor crescents, nor horseshoes nailed with cold iron above the door forbid us any more than they forbid a living man, nor must we wait to be invited to cross a threshold we have not crossed before.\"\n\nYsidro gestured, the black kid of his glove spiderlike against the white shirtsleeve.\n\n\"Though we do tend to avoid holy places. Not because God is there\u2014for presumably God is everywhere, something men seem to forget in their battlefields, bedchambers, and boardrooms\u2014but because man is there, and woman, without the defenses they erect to protect their minds from one another. The yielding up of their innermost dreams\u2014love, and hatred for those different than they\u2014charity and violence all mingled\u2014makes a music which remains in such places even in their emptiness. Dreams lie thick here, like the smoke of incense; the smell of the blood that has been shed here seeps still from its stones. Many of us barely notice, but I find it\u2014unpleasant.\"\n\nThe silence returned, like the cloak of vampire powers: the turning aside of attention, the blinding of living eyes. All the things that someone like Ignace Karolyi\u2014someone like G\u00f6lge Kurt\u2014would have sold to living men preparing to fight a war.\n\nAnd might still, thought Asher wearily. And might still.\n\nBut that was something about which he could do no more. He should have known that, he reflected bitterly, before he got on the Paris train. He had known about it _this time_ , stopped it _this time_... Plucked up a single weed, knowing already that the seeds were everywhere in the air, looking only for fertile soil.\n\n\"Thank you for looking after her.\"\n\nYsidro turned his face away. \"You have married a very foolish woman, James,\" he said softly. \"I would have looked after her better had I broken both her legs, to teach her to stay out of vampire nests, and sent her back to Oxford under care of a nurse. I did ill and stupidly, for we all go back home nursing our hurts, hers maybe the worst of all. And nothing here will change.\"\n\n\"Which is as well,\" Asher said, \"considering what changes might have come had G\u00f6lge Kurt become Master of Constantinople. We did win this time, you know.\"\n\nThe colorless eyes touched him, rested on him, giving away nothing of their thoughts, then moved away. \"This is not my affair. The dead are the dead.\"\n\n\"You will miss her,\" Asher said, \"won't you? Anthea.\"\n\nYsidro looked aside without replying.\n\n\"I don't think,\" Asher said, \"that she was sorry.\"\n\nHe did not think the vampire would answer him, and for a long time he did not. Then he said, \"She was. But I do not think she would have lasted long after he was gone.\"\n\nHe had known her, thought Asher, for all of that two hundred and fifty years. Worlds were hidden in the stillness of the alabaster face, the pale, champagne-colored eyes. Questions forever unanswered.\n\n\"You didn't kill the Potton girl, did you?\"\n\nYsidro said nothing.\n\n\"It's not something I'll speak of to Lydia. There were other vampires in the city, maybe others besides those I saw in the House of Oleanders. I don't know. If the laborers and mechanics and beggars put together Lydia's inquiries with the house of Olumsiz Bey, there must have been vampires who became aware of you. Who waited for the servants to flee the sound of the riot. Who had, perhaps, met her eyes somewhere, sometime, and could command her in dreams to open the windows for them.\"\n\n\"The girl was a fool,\" Ysidro said. He glanced sidelong at Asher. \"You may tell Mistress Asher I said that.\"\n\n\"Many years ago,\" Asher said, \"when I was in Vienna, I loved a woman there, and she me. She was clever and had great integrity. I was a fool to speak to her after the second time, because I should have known where it would lead. But after the second time I met her, it was too late. When she began to guess that I was a spy sent to find military secrets that would hurt her country, probably kill her friends and family who were in the army, I... betrayed her. I stole her money and left town in ostentatious stealth with the most brainless and beautiful member of the demimonde I could convince to accompany me\u2014knowing that Fran\u00e7oise would take her own rage, her own hurt, into account, and more than into account, and not look further into anything else that had to do with me. She was that kind of person. I did this not only to protect myself and my contacts, but so that she would cut from me cleanly, never regretting or thinking that what had been between us could ever be repaired.\"\n\nYsidro was silent for a long time, cold crystal eyes fixed on some middle distance, as if, through the walls, he could see out into the night, back to the London that had been his haunt and his home from his twenty-fifth\u2014and last\u2014year of human life.\n\n\"There was nothing ever between us, you know.\"\n\n\"I know.\" She hadn't told him about the sonnets, but he had found them\u2014including the torn one\u2014in Miss Potton's crochet basket. Asher's own passion returned to him, yearning and illogical, for Anthea, and for the moonlight girl in the Vienna Woods who had later helped to empty Fairport's veins. He remembered Lydia's voice when she said, _Simon..._ and recalled, too, the disillusioned agony of her tears.\n\nShe would recover, he knew. But the hurt ran deep.\n\nThe vampire shook his head. \"Life is for the living, James. Death is for the dead. As for her attraction to me, it is our lure to be attractive. It is how we hunt. It means nothing.\"\n\nAsher thought about Anthea again, and knew that Ysidro lied.\n\nYsidro considered the matter in silence for a moment more, then went on, \"As for Miss Potton, I cannot say that I wouldn't have killed her, in the end, as Lydia expected me to. In truth I don't think she would have minded. But I think it was a woman named Zenaida, a concubine who haunts the deserted areas of the old seraglio, abandoned now even by the palace servants. Zenaida saw her there\u2014I think she may even have summoned her, using the illusion that I might wish her to follow me. Afterward I thought I saw her once or twice around the house on Rue Abydos, but by then my perceptions were not acute enough to be sure. Another reason I would keep Mistress Asher in ignorance of how this came about. She would take it as her own doing. I trust you have not left her alone.\"\n\nAsher shook his head. \"She's with Lady Clapham and Prince Razumovsky. I asked them to stay with her till I returned. I told them she has nightmares\u2014not that Lydia has ever had a nightmare in her life.\"\n\nThe defaced ivory mask relaxed, momentarily, into a smile.\n\n\"Will you be all right, returning home?\"\n\n\"The Dead always find ways,\" Ysidro said, \"to get the living to serve them. Some, like the Deathless Lord, buy that service, or use hate, like G\u00f6lge Kurt, or love. Sometimes the living don't even know why they serve.\"\n\nAsher studied the narrow, enigmatic features, the rucked ruin of fresh and bloodless scars. Like Anthea, like Ernchester, Ysidro was a killer and would have been as deserving as they had the sunlight trapped and consumed him in that upper room. The fact that Ysidro had risked his curiously friable immortality to help him\u2014to save Lydia\u2014should have no bearing on that deserving. The fact that Ysidro had not killed Margaret Potton did not change the fact that he had killed someone else\u2014possibly several others, if he had been as long fasting as Lydia had said\u2014that same night.\n\n\"Sometimes they do.\" He held out his hand to the vampire. \"They know... but damned if they understand.\"\n\nYsidro regarded his hand for a moment with an air of slightly startled offense, as if at a familiarity; then smiled, like a man remembering his own follies, and very quickly, with two cold fingers, returned the touch.\n\n\"In that they are not unique,\" he said.\n\nAnd he was gone, in a slight, quick blanking of attention that covered a soundless retreat. Asher found himself alone in the immense darkness of the ancient holy place, without so much as a flicker of motion among the dark pillars to show that any soul, living or dead, had passed that way.\n\nWeary of dark, I asked to see the day,\n\nAnd Jesus, jesting, to a mountain's height\n\nUpbore me, and spread before my sight\n\nThe Kingdoms of earth in morning's bright array.\n\nI saw a man betray two dames who wept;\n\nSaw a mother cripple her child with love;\n\nSaw priests flay Jews, their piety to prove,\n\nAnd brother sell his brother while he slept.\n\nA man gave up his dreams, a child to save.\n\nA woman bound a beggar's bleeding sores.\n\nA youth pursued war's summons to his grave\n\nWhile th'king for whom he died gave gold to whores.\n\nAnd all died frightened, weeping and in pain.\n\nI left the mount, and sought the dark again.\n\n# A Biography of Barbara Hambly\n\nBarbara Hambly (b. 1951) is a _New York Times_ bestselling author of fantasy and science fiction, as well as historical novels set in the nineteenth century.\n\nBorn in San Diego and raised in the Los Angeles suburb of Montclair, Hambly attended college at the University of California, Riverside, where she majored in medieval history, earning a master's degree in the subject in 1975. Inspired by her childhood love of fantasy classics such as _The Wonderful Wizard of Oz_ and _The Lord of the Rings_ , she decided to pursue writing as soon as she finished school. Her road was not so direct, however, and she spent time waitressing, modeling, working at a liquor store, and teaching karate before selling her first novel, _Time of the Dark_ , in 1982. That was the birth of her Darwath series, which she expanded on in four more novels over the next two decades. More than simple sword-and-sorcery novels, they tell the story of nightmares come to life to terrorize the world. The series helped to establish Hambly's reputation as an author of intelligent fantasy fiction.\n\nSince the early 1980s, when she made her living writing scripts for Saturday morning cartoons such as _Jayce and the Wheeled Warriors_ and _He-Man_ , Hambly has published dozens of books in several different series. Besides fantasy novels such as 1985's _Dragonsbane_ , which she has called one of her favorite books, she has used her background in history to craft gripping historical fiction.\n\nThe inventor of many different fantasy universes, including those featured in the Windrose Chronicles, Sun Wolf and Starhawk series, and Sun-Cross novels, Hambly has also worked in universes created by others. In the 1990s she wrote two well-received Star Wars novels, including the New York Times bestseller _Children of the Jedi_ , while in the eighties she dabbled in the world of Star Trek, producing several novels for that series.\n\nIn 1999 she published _A Free Man of Color_ , the first Benjamin January novel. That mystery and its eight sequels follow a brilliant African-American surgeon who moves from Paris to New Orleans in the 1830s, where he must use his wits to navigate the prejudice and death that lurk around every corner of antebellum Louisiana. Hambly ventured into straight historical fiction with _The Emancipator's Wife_ , a nuanced look at the private life of Mary Todd Lincoln, which was a finalist for the 2005 Michael Shaara Prize for Civil War writing.\n\nFrom 1994 to 1996 Hambly was the president of the Science Fiction and Fantasy Writers of America. Her James Asher vampire series won the Locus Award for best horror novel in 1989 and the Lord Ruthven Award in 1996. She lives in Los Angeles with an assortment of cats and dogs.\n\nHambly with her parents and older sister in San Diego, California, in September 1951.\n\nHambly (right) with her mother, sister, and brother in 1955. For three years, the family lived in this thirty-foot trailer at China Lake, California, a Marine Base in the middle of the Mojave Desert.\n\nHambly (left), at the age of nine, with her brother and sister on Christmas in 1960.\n\nHambly's graduation from high school, June 1969.\n\nA self-portrait that Hambly drew while studying abroad in France in 1971.\n\nHambly dressed up for a Renaissance fair.\n\nHambly at an event for the Science Fiction and Fantasy Writers of America. She served as the association's president from 1994 to 1996.\n\nThe \"official wedding picture\" of Hambly and science-fiction writer George Alec Effinger, in 1998.\n\nHambly with her husband, George, in New Orleans around 1998. At the time, she was researching New Orleans cemeteries for her book _Graveyard Dust_ (2002).\n\nHambly at her birthday party in 2005.\n\nHambly (right) with her sister, Mary, and brother, Eddy, at a family reunion in San Diego in 2009.\nAll 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 ebook onscreen. No part of this text may be reproduced, transmitted, downloaded, 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 the publisher.\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. Any resemblance to actual persons, living or dead, businesses, companies, events, or locales is entirely coincidental.\n\ncopyright \u00a9 1995 by Barbara Hambly\n\ncover design by Jason Gabbert\n\n978-1-4532-1647-7\n\nThis edition published in 2011 by Open Road Integrated Media\n\n180 Varick Street\n\nNew York, NY 10014\n\nwww.openroadmedia.com\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nENCRYPTED\n\nby Lindsay Buroker\n\nCopyright \u00a9 2011 Lindsay Buroker\n\n# PROLOGUE\n\nTikaya Komitopis slid one finger down the encrypted message while she translated the plain text letters onto a fresh page. She smiled. Her new key was working.\n\nAs she revealed more lines, giddiness stirred in her belly. She forced herself not to rush, not to get ahead of herself. Finish translating the message, then read it.\n\nTikaya tuned out the susurrus of voices in the war room. She ignored the sweat moistening her freckled hands and the mugginess of the salty air that failed to stir the leaves in the palm trees outside the window. A wisp of blonde hair escaped her long braid and dangled before her spectacles, but she ignored it too.\n\nOnly after she copied the Turgonian admiral's signature did she grab the paper with both hands, devouring the message.\n\nTikaya shoved her bamboo chair back so quickly it toppled to the floor. She glanced about the desk-filled room. Everyone had stopped work to watch the door where her supervisor stood with the president. Their graying heads tilted toward each other, some discussion on their lips.\n\nShe blinked. When had the president arrived?\n\nThen elation sent her racing across the room, sandals slapping the wood floor. Perfect. He should know first.\n\n\"Mr. President?\" she called, though he was already looking her way. \"I have\u2014\"\n\nHer hip rammed the corner of a desk. She flailed for balance, tripped over her own feet, and pitched forward. The president caught her in an awkward embrace. Mortified, she lurched backward and found her feet as heat swarmed her cheeks.\n\n\"Professor Komitopis,\" he said gently, amusement in his blue eyes. \"Do you surf?\"\n\nTikaya stared at him in bewilderment, then over his head and out the open door. In the bay, a steamer rumbled toward the docks while a few students straddled surfboards near the beach.\n\n\"No, sir,\" she said, letting puzzlement into her tone.\n\n\"Don't start,\" the president said.\n\nHer supervisor snickered. Oh. She was being teased for her clumsiness. The men's eyes held no spite, but that did little to abate the heat plaguing her cheeks. It was bad enough she stood two inches taller than either man; she had to stumble around like a drunken sea lion in front of them too?\n\n\"You have a message?\" the president asked.\n\nThe importance of the note flooded back to her. \"Yes, yes. The war, sir. It's over.\"\n\nThe president's eyes widened.\n\n\"Or it will be in a couple weeks,\" Tikaya said. \"Listen: 'Admiral Dufakt, by his Ancestrally Ordained Imperial Highness Emperor Raumesys's order'\u2014I love it when they use that long title in their encrypted communications. You don't even need frequency analysis when you've got such an obvious key phrase. Every time they\u2014\"\n\n\"Tikaya,\" her supervisor whispered. \"The message.\"\n\n\"Oh, pardon, sirs. The Turgonian emperor says, 'warships are to stand ready to facilitate troop removal and diplomat transportation for treaty negotiations.'\" She tapped the page. \"That's the official part that went out fleet wide, and this second paragraph came on another page. I believe it's a personal message between admirals.\n\n\"'That's it Dufakt. With Fleet Admiral Starcrest's death, we've gone from dominating the Nurian forces to scrambling to survive encounters with those ancestors-cursed wizard ships. Having the Kyattese cryptanalyst hand over so many of our decrypted missives to the Nurian government exacerbated our problems. How an island full of scientists managed to steal so many of our correspondences, I'll never know, but I do wish Starcrest had lived to punish them, especially since taking over their piddling nation was his idea. We'll recoup and get the Nurians next time. Send along your recommendations for promotions. Signed Acting Fleet Admiral Mourncrest.'\"\n\n\"Good news, yes, indeed,\" the president said.\n\nHis head tilted to the side, eyes far away for a moment, and Tikaya recalled he was a telepath. He must be getting a message from some aide back in his office. Or maybe his wife wanted him to stop for groceries on the way home. Tikaya had never studied the mental sciences and did not know how likely that was, but she smirked at the thought of the president popping into the market for sugar and bananas.\n\nWhen his eyes focused on Tikaya again, concern hooded them, and her amusement evaporated. His tone turned grim when he spoke: \"Step outside with me, please, Professor.\"\n\nTikaya handed the note to her supervisor, and an uneasy flutter vexed her stomach as she trailed the president.\n\nA breeze wafted in from the ocean, making it feel cooler outside despite the sun radiating off the sidewalk. Seagulls squawked in response to a steam horn blasting in the bay. The president stopped in the shade of a jackfruit tree.\n\n\"The work you've done for us this last two years has been phenomenal, Tikaya. I'm grateful, and if our nation knew about it they would be too.\"\n\nShe shrugged, embarrassed by the praise, and prodded a fallen jackfruit with her toe. \"Thank you, sir, but I've just been hunkered in a room, playing with symbols. It was different from my regular work but similar. A fun challenge.\" The president's eyebrows twitched, and she winced. She should not call anything related to the war fun. Too many had died. \"The men and women who risked their lives to obtain the missives are the heroes.\"\n\n\"I'm grateful to them, too, but their names aren't the ones starting to show up in Turgonian naval orders.\"\n\n\"My name isn't...\" She froze. The Kyattese cryptanalyst. That had been in the message, not for the first time. The Turgonians seemed to believe a single person responsible. Her. The humid air did nothing to stifle the chill that raised the hair on the back of her neck.\n\n\"If they find out who you are,\" the president said, \"your life will be in danger.\"\n\n\"They won't figure it out,\" she croaked, mouth dry. \"They won't. They'll be looking for a cryptanalyst, not a philology professor buried in a back room at the Polytechnic, deciphering dead languages on dusty scrolls and tablets.\" Why did she sound like she was trying to convince herself? \"We don't even study cryptography on the islands; surely they'll think it was some Nurian who worked with us.\"\n\n\"I hope that's true, but...I hear you're good with a bow.\"\n\nFor a moment, the topic shift befuddled her. Then realization dawned and made her shake her head. \"In the field in the back of my parents' house, yes, sir. But I couldn't shoot anyone.\"\n\n\"I suggest you keep up your practice in the months ahead.\"\n\nTikaya closed her eyes and drew in a deep breath. She did not even like hunting. That had always been her brother and her father's domain. She shot because she found the repetitive, mechanical task conducive to thinking, to problem solving. She had worked out many language puzzles while plunking arrows into the straw targets on her parents' plantation.\n\n\"I hope you understand that I cannot regret bringing you into this,\" the president said, \"not when you've been so pivotal to our people retaining their freedom. But I do...owe you a great debt. I will do everything possible to deflect foreign questions about your involvement, and I will pray for your safety in the months and years ahead.\"\n\n\"I understand, sir. If anything does happen, I don't hold you responsible. I had to do this. I wouldn't enjoy living under imperial rule.\" She sensed his grimness and wanted to reassure him. \"Those warmongers probably make their professors wear swords to class, and, given how easily I can trip over my own feet, that'd be a death sentence for sure.\"\n\nThe president smiled, but it did not reach his eyes.\n\n\"Something else, sir?\" she asked.\n\nHe sighed, his gaze toward the sea. \"Yes, the reason I called you out here... I just learned the bad news. It's about your fianc\u00e9.\"\n\nFear tightened Tikaya's chest. \"Parkonis?\"\n\n\"I'm sorry. The Turgonians sank the Eagle's Spirit off the coast of the northern island. There were no survivors.\"\n\n\"They sank\u2014\" Her voice cracked. \"No, they wouldn't have... The Spirit is an archaeology vessel! It doesn't even have a cannon.\" She gripped the tree for support.\n\nThe president placed a hand on her shoulder. \"I know.\"\n\nTikaya sank to the grass and buried her face in her lap. She did not want to believe Parkonis was dead, but hot tears streaked down her cheeks and dampened her dress.\n\nThe war was over, but she had nothing to celebrate.\n\n# CHAPTER 1\n\nMoisture slicked the handle of Tikaya's machete, and sweat saturated her hemp dress. Her blade rang as she scraped leaves free from a stalk of sugar cane.\n\nSunset approached, and she had yet to cut a hand, leg, or other notable appendage. Maybe she was finally growing competent with the machete. The hilt slipped in her damp palm, and she nicked her thumb. Maybe not.\n\nShe lifted her spectacles to wipe moisture out of her eyes. A reflection in the glass made her jump.\n\nMachete in hand, she whirled toward the cleared area behind her. A man towered a few paces away, a dagger and cutlass at his belt, and a muzzle-loading rifle crooked in his arms. His bronze skin and dark hair would have marked him a foreigner even if the black military uniform with its fine factory weave did not. It was a uniform she had not seen in a year, not since the war ended, but she had not forgotten its significance: Turgonian marine.\n\nSeveral paces lay between her and the wagon where her bow rested on the driver's seat. She had kept it within reach the first couple of months after the treaty signing, but time had dulled her vigilance. Swallowing, she shifted her gaze left and right, hoping to spot a couple of the seasonal laborers her father hired to harvest the cane. But the day grew late, and she had worked herself into a private corner of the field. The house stood hundreds of meters away. No one would hear her yell.\n\nThe marine said nothing, though his dark eyes followed her darting gaze. Running would confirm she had a reason to hide; maybe she could trick him into thinking she was no one of consequence. Not that being an innocent would necessarily make her safe from a Turgonian.\n\n\"If you're looking for rum,\" she said, his language sliding off her tongue automatically, \"my brother's working in the distillery. He can sell you enough for your entire ship at a fair price.\"\n\nThe marine's eyes widened, and a satisfied\u2014no, triumphant\u2014smile stretched across his face.\n\nDread curled through her belly. They knew who she was, what her role had been in the war. Addressing him in his language had been a mistake, a confirmation that they had found the right person. She eyed the rifle, noticed it was loaded and cocked. A huge mistake.\n\n\"I'm not here for rum,\" the marine said. \"I seek the cryptomancer, and I believe you are she.\"\n\nTikaya did not have to feign surprise. \"The what?\"\n\n\"The one who broke our codes during the war. The one who thwarted our best cryptographers. The one who\u2014\" his jaw tightened and a muscle in his cheek jumped, \"\u2014gave our decrypted messages to the Nurians. That meddling cost us a dozen ironclads and thousands of men.\"\n\n\"Your people tried to take over our islands to serve as a strategic outpost.\" Her hand flexed on the machete. \"You sank more than a dozen of our ships, including a peaceful archaeology vessel with my\u2014\" She stopped herself. She might have every right to condemn this man, but it was stupid to do so when he stood across from her holding a rifle. \"We wanted no part of your war. We did what we had to do to protect our freedom. I don't know who your cryptomancer is, but I am certainly not that person. I am a simple plantation worker, helping my family grow sugar cane and make rum.\"\n\n\"A simple plantation worker who speaks flawless Turgonian,\" the marine said.\n\nShe stifled a grimace. If those thoughtless first moments were her undoing... \"The Kyatt Islands are in the middle of many nations' trade routes. Our children study several languages in school, and many of our people are polyglots. You'll find the true experts working at the Polytechnic.\" A place and job she had not returned to since losing Parkonis.\n\nFor the first time, a hint of uncertainty lurked in the marine's dark eyes. She held her breath, willing him to believe her.\n\nHe eyed her up and down, and she shifted her weight, abruptly aware of how the dampness of her dress pronounced her curves. There were more things to fear from a strange man than being shot. She tried to ease backward, but dense cane blocked her.\n\nThe marine reached for his belt, and she crouched, brandishing the machete in both hands.\n\n\"If you touch me, I'll cut off your...\" Tikaya knew the Turgonian word for penis, but some cursed ancestor with a sense of humor momentarily sucked it from her mind. \"Man part,\" she finished feebly.\n\nThe marine's eyebrows lifted. His hand had unclasped not his belt buckle but an ammo pouch, and he pulled out a scrap of paper. \"You're not my type, and if that's what you people call a fighting stance, it's amazing you can even defeat the sugar cane.\"\n\nShe should have felt nothing but relief, but embarrassment flushed her cheeks. The marine approached, the paper extended. Though he did not act as threatening as he might, her muscles tensed. The Turgonians had slain hundreds of her people, including the one who mattered most.\n\nTikaya wanted to tell him to take his note and leave, but curiosity kept her silent. What could he possibly have come all this way to show her?\n\nHe stopped a pace away from her, holding out the paper. Reluctant to close the final distance, she did not move for a long moment. He waited. Mosquitoes whined, reminding her that darkness approached. Tikaya lowered the machete and accepted the note. Even with her suntanned skin, his fingers were dark next to hers.\n\nThough he did not try to touch her, she sidled away to study the paper. It was not a note at all but a page of symbols. Someone had painstakingly copied complex symmetrical markings interlinked in small groupings. Her teeth caught her lip. She had seen many languages, but she had never seen this one, if it even was a language. It could be anything.\n\n\"Where did you get this?\" she asked, gaze stuck to the paper. After a moment, she realized she had asked in her own tongue instead of his and switched, repeating the question.\n\n\"My commanding officer.\"\n\n\"No, I mean...\"\n\n\"My commanding officer,\" he said again.\n\nTikaya snorted. \"Is it a language or...\" She stopped herself from saying substitution cipher. If she hoped to plead ignorance of this cryptomancer, she had best not say anything related to cryptography.\n\n\"You tell me.\"\n\n\"I've never seen anything like it, and I've seen\u2014\" She caught herself again, this time short of admitting she had studied dozens of languages, living and dead. \"We see a number of languages here on the island.\"\n\nTikaya tried to watch him, to gauge his reaction, but the symbols kept drawing her eyes back, demanding her attention. What if it was a previously undiscovered language? Something from ruins the Turgonians had dug up on their continent? They were not a people known for archaeology, nor sharing secrets. If she were to translate a new language and bring awareness of it to the global scientific community, it would assure her a place in the history books. A tempting thought, that.\n\n\"Does it mean anything to you?\" the marine asked.\n\n\"No, I don't even know if this is logographic or syllabic or alpha...\" Great Grandmother's eyeteeth, she was saying too much again.\n\nIndeed, the soldier watched her through narrowed eyes. Time to end this conversation and get out of these fields before darkness fell and he changed his mind about her being his type.\n\nTikaya held the paper out for him. \"I don't recognize it. I can't help you. You should try at the Polytechnic.\"\n\nHe stared at her, face unreadable. Cicadas began droning, and a bead of sweat slithered down her spine. Then he took the paper, returned it to his pouch, and walked away.\n\n* * * * *\n\nA pair of whale-oil lamps burned on either side of double doors marking the front of a large grassy mound. The earthen-walled structure held her family's distillery and processing equipment, and the clank-thunk of machinery echoed from within. Tikaya paused to prop her bow against the door frame as she entered the chamber. Cool, dry air offered a reprieve from the muggy evening heat, and her steaming body welcomed it after the run from the fields.\n\nShe almost tripped over a passel of laughing, sandy-haired toddlers throwing wads of bagasse at each other. Running into her nephews and nieces usually made her smile, but now she froze, mid-step, thinking of the marine. His presence represented a threat not only to her, but to her whole family, a family big enough that they joked how it was impossible to be lonely any place on the plantation. That was why she had returned this past year. The flat she shared with Parkonis near the Polytechnic had been too empty after his death, but now she feared she had endangered them all.\n\n\"Tikaya,\" her brother, Kytaer, called. He stood before a press, feeding sugar cane into the rollers. The long stalks cracked and flattened, and juice flowed into a collection bin below. \"Glad you stopped by so I could warn you.\"\n\nShe tore her gaze from the tussling children. Warn her? Had the Turgonian already been here?\n\n\"Professor Meilika is in the house,\" Ky said. \"She's joining us for dinner. She and Mother have been conspiring all afternoon. About you. How to get you back deciphering runes on broken tablets and potsherds and all that.\"\n\nTikaya exhaled slowly. Nothing new. Good. That meant the marine had not been by. She still had time to warn everyone and figure out what to do. No, she knew what she had to do. She had to pack. She could not stay here. If any of her family came to harm because of the role she accepted during the war, the guilt would torment her forever.\n\nOne of her nephews bumped into her leg and fell on his bottom. She picked him up before he could decide if the tumble was a big enough calamity to cry over. She swiped bagasse off his dusty trousers and directed him back into the game with a playful swat on the backside. A lump sprang into her throat at the idea of leaving them indefinitely. But for fate, she might have had little ones of her own by now.\n\n\"Children, time to wash up for dinner!\" That was Ky's wife, calling from the path, somewhere between the house and the processing plant.\n\nThe youngsters trundled out, voicing mutters of \"aw\" and \"do we have to?\"\n\n\"You're looking particularly glum and thoughtful,\" Ky said when he and Tikaya were alone. \"Did Mother and Father already talk to you?\"\n\nTikaya had seen neither of her parents since early morning, so she arched her eyebrows and joined him at the press. Like their father, Ky shared her uncommon height. For him, though, it had always been an advantage, making him a boyhood star at swimming and running. For her... Well, at least she could reach the high book shelves in the library without a ladder.\n\n\"I heard them talking,\" Ky explained. \"You're getting the wasting-the-talent-Akahe-gave-you lecture again soon. I know Father appreciates an extra hand during the harvest, but he's worried you've been here moping too long. And Mother...wants you living in town again where you can find a 'nice young man to make babies with.'\"\n\nTikaya winced at the familiar words. Ky patted her on the shoulder.\n\n\"Sorry,\" he said. \"Are you all right? You look preoccupied. If you were puzzling over some ancient runes, I'd know why, but I can't imagine the mysteries of the cane fields are putting those thoughtful creases between your eyebrows.\"\n\n\"I ran into a Turgonian marine,\" Tikaya said to hush her brother's garrulousness. She usually found it endearing, but tonight his chatter grated.\n\nHer words did the job. He gaped for a long moment before saying, \"Where? When? You haven't been to town for\u2014\"\n\n\"Here. Just now. In the north field.\"\n\nStill staring at her, Ky shoved the lever that turned off the press, and the clank-thunks faded.\n\n\"He was looking for the cryptanalyst from the war,\" Tikaya went on, voice sounding loud in the new silence. She lowered it. \"I think I persuaded him I wasn't that person, but I'd be surprised if their research doesn't lead them back to me again. Tomorrow morning\u2014\"\n\nA clank sounded near the entrance, and a metallic canister rolled across the cement floor. Smoke billowed, and acrid fumes stung Tikaya's eyes. Oh, Akahe, she did not have until tomorrow morning.\n\n\"What is\u2014\" Ky started.\n\nShe grabbed his arm and yanked him deeper into the distillery even as another canister clinked through the doorway. Smoke hazed the entrance, but she glimpsed men slipping inside. They did not know the layout of the distillery; that ought to be an advantage.\n\nShe led her brother past the press and around two massive molasses vats.\n\nKy gripped her shoulder and whispered, \"Turgonians?\"\n\n\"I assume so.\" Tikaya tugged to keep him moving. The earthen back doors had grass growing on them; she hoped the soldiers had not recognized them as an entrance and posted guards.\n\nThey eased past copper pipes and the towering stills, and she crooked her toes to keep her sandals from slapping against the hard floor. Smoke curled into her nostrils and tickled her throat. She dared not cough.\n\nShe thought of her bow, still propped by the front door. Blighted banyan sprites, why had she even bothered carrying the thing around the last year?\n\nIn the back, rows of rum barrels lined the walls, and the double doors came into sight. She froze. They already stood open. Beyond them, in the fading light, grass swayed under a soft breeze.\n\n\"I didn't leave the doors open,\" Ky whispered.\n\nIf men waited outside, Tikaya could not see them, but that meant little. Perhaps they were crouched beside the doors, ready to pounce. Maybe they were already in the house, threatening her family. Or worse. If anything happened to her kin, it was her fault. She swallowed. She had to make sure the soldiers focused on her.\n\n\"I'll run,\" she whispered. \"They should only want me.\"\n\nEven as she readied herself to sprint, Ky grabbed her arm. \"No.\"\n\nA shadow moved behind him. She opened her mouth to yell a warning, but she was too late. The butt of a rifle thudded against his head, and he slumped to the floor.\n\nTikaya turned to run and crashed into a broad chest. Hands clasped her arms. She twisted, trying to free herself, but the steel grip held her fast.\n\nShe screamed. A hand clapped across her mouth. She tried to bite it, but the grip smothered her with its power.\n\nA damp rag pressed over her nose. Terror roiled in her belly. She sucked in a deep breath, thinking they meant to suffocate her, but a sweet insidious odor flooded her nostrils.\n\nFuzziness encroached on her mind, and her thoughts scattered. Blackness tunneled her vision, and a moment later, the world faded away.\n\n# CHAPTER 2\n\nThe ground was vibrating. No, Tikaya realized as awareness returned, not the ground, the floor. Cold, textured metal chilled her bare calves and seeped through the back of her dress. A rocking rise and fall accompanied the vibrations.\n\nShe opened her eyes to a dim, fuzzy cell. Her spectacles were missing. Unimaginative gray steel surrounded her. The monotone color marked the bulkheads and even the sturdy gate dividing her from a corridor. No portholes allowed a view of the outside, but the swells of the sea and the reverberations of a nearby engine told the story: she was locked in the bowels of a Turgonian warship.\n\nAnd her brother\u2014had the marines brought him too? She remembered the sickening thud of that rifle butt striking his head. She prayed they had left him alive, where her family could tend him, but a selfish part of her wished he was in the brig with her. The idea of being alone on a ship full of hostile marines...\n\nShe shuddered.\n\nTikaya rolled onto her belly. No pain lanced through her body, but stiff muscles suggested she had lain on the deck for hours.\n\nAcross the corridor, a second gate marked another cell, though darkness\u2014and her poor vision\u2014shrouded the interior. She stood and pressed her face between the bars. A blurry lantern burned at the base of a ship's ladder leading up. No guards stood within sight.\n\nShe probed the small lock set in her gate. She could not even get a fingernail into the fine hole. Alas, picking locks was not a typical course in the Kyattese school system.\n\n\"Wonderful day.\" Tikaya realized she had probably been on the floor throughout the night and amended the last word: \"week.\"\n\nChains clanked in the cell across the way, and Tikaya jumped.\n\n\"Hello?\" she asked in her tongue.\n\nMaybe her brother was there, or others of her people had been taken. Maybe she was not alone against the Turgonians after all. The clanks stilled, leaving only the rumbling of the engine.\n\n\"Hello?\" she asked again, this time in Turgonian and this time with less hope.\n\nSilence.\n\nTikaya peered into the cell. Was that a human form slumped in the back corner? She tried other languages from the islands and coastal nations on the Eerathu Sea. Nothing elicited a response.\n\nA hatch thudded open, catching her trying yet another greeting. Boots rang on the ladder, and a pair of marines strode toward her.\n\n\"Don't poke the grimbal, girl.\" The tall man in the lead jerked a nose sharper than Herdoctan potsherds at the opposite cell.\n\n\"Grimbal?\" Tikaya frowned.\n\n\"Giant shaggy predators up on our northern frontier. They're probably the most irritable beasts in the empire, and they'll sink their teeth into you if you get anywhere near their territory.\"\n\nTikaya stopped herself from saying she had heard of the creature and the expression\u2014if she hoped to deny she was their cryptomancer, she ought not appear too worldly. It was curiosity about the other prisoner that had prompted her query. Her shoulders, and her hope of denying anything, slumped when the second marine drew close enough for her to identify without her spectacles: the man from the cane fields. No doubt he had arranged her capture when she failed to convince him she was no one of consequence.\n\nShe squinted to read the name sewn on his jacket: Agarik. He stood, hands clasped behind his back, watching the other marine, his superior, she assumed, though she did not know what ranks the pins on their collars denoted.\n\n\"How're the accommodations, Five?\" The speaker\u2014his jacket read Ottotark\u2014rapped a baton against the mystery prisoner's gate. \"A lot better than what you're used to of late, eh?\"\n\nThere was no response, not even a tinkle of chains rattling. Despite the silence, Ottotark chuckled at his own wit. He turned his attention to Tikaya and when his gaze lingered on her breasts, she forced herself not to step back.\n\n\"Where is my brother?\" she asked. \"Is he...\"\n\n\"We left him in the distillery,\" Agarik said. \"He's alive.\"\n\n\"Thank you,\" she murmured, hoping she could trust his word.\n\n\"So, the source of so many of our troubles. A woman.\" Ottotark shook his head. \"Seems strange you'd be involved in military matters.\"\n\nTikaya bit back a response about how it was hard to remain uninvolved when invaders were trying to take over one's whole island chain.\n\n\"I reckon you just sat in an office on a beach,\" Ottotark continued, \"and someone brought our messages to you. Is that how it worked?\" The lamplight glinting off Ottotark's dark eyes did nothing to warm them, and a challenge hardened his voice. He resented her. Every Turgonian she encountered probably would.\n\nShe lifted her chin. \"What are you going to do with me?\"\n\n\"For the trouble you caused us? We're going to kill you, of course.\"\n\nTikaya swallowed, or tried to. Her throat constricted, and her mouth was too dry.\n\n\"Sergeant...\" Agarik frowned at the other man.\n\nOttotark bent over, hands on his knees, and laughed. The raucous noise echoed from the metal bulkheads. \"I jest\u2014we're not killing you. Not now anyway. We need you to translate something for us first.\"\n\nTikaya barely kept from snorting. After what the Turgonians had done to her people\u2014to her\u2014she would not even help them tie their shoes.\n\nOttotark fished a keychain out of his pocket. \"Time for you to visit the captain.\"\n\nWhile Ottotark unlocked the door, Agarik slipped Tikaya's spectacles through the gate. She blinked in surprise and met his gaze as she accepted them. Nothing so friendly as a wink or a smile suggested she had a secret ally, but he seemed someone who treated people, even prisoners, with respect.\n\nShe had barely hooked the spectacles over her ears when Ottotark grabbed her upper arm and jerked her into the corridor so she fell against him. The amusement on his face, however crude, was gone now. He grabbed her breast, even as his other arm snaked around her waist to keep her jammed against him.\n\nTikaya shoved a hand against his chest and tried to thrust a knee into his groin, but his strong embrace left no space to maneuver.\n\nHis lips curled into a snarl. \"We may need you, but you deserve a lot of pain for the deaths you caused.\"\n\nOttotark's fingers gouged her breast, and she gritted her teeth at the pain, determined not to gasp or cry out, though fear surged through her body. She craned her neck toward Agarik, hoping he might step in. Though his clenched jaw made tendons bulge on his neck, he made no move against his superior.\n\n\"What's the matter with your sergeant, Corporal Agarik?\" a deep voice spoke from the other cell. Though quiet, it cut through Ottotark's angry lust, and he jumped, relaxing his grip. \"Doesn't he know the Kyattese are sorcerers as well as scholars? In another second, she'll probably cast a spell to shrivel his testicles into wrinkled, rotten walnuts.\"\n\nOttotark frowned into the cell. \"Nobody wants to hear you speak, Five.\" Still, he released Tikaya, shoving her toward the corporal.\n\nAgarik was gaping at the dim cell, but he recovered enough to take her arm. Under his firm, professional grip, the heartbeats hammering in her ears slowed.\n\nTikaya watched over her shoulder as the guards led her away, but the unseen man did not speak again. The trek took her up to the main deck, where they marched past two long rows of cannons. The sharp tang of gun oil competed with the briny scent of the ocean roaring past beneath them. She peered through an open cannon port, hoping to glimpse her islands. If they had not sailed too far, maybe she could escape to a lifeboat\u2014if the Turgonians had lifeboats. With that warmongering culture, one never knew. They had idiotic notions about glory in dying a warrior's death, so they might condemn their men to go down with the vessel.\n\nDozens of marines occupied the deck, some sparring in a makeshift arena in the middle and some cleaning rifles, pistols, and cutlasses at tables folded down from the wall space between the massive guns. The men slanted her looks ranging from openmouthed bewilderment to sneering hostility, by which she assumed some knew who she was and some were not in the need-to-know camp. The empire did not employ women in their armed forces, which likely meant she was the only one on board. Not a comforting thought. More than one man arranged to bump or jostle her as paths crossed. Unlike at home, everyone she passed was as tall or taller than she, and their wayward elbows and shoulders battered her with the force of falling coconuts.\n\nPast the galley, aft deck, the loitering marines thinned. Tikaya's guards stopped her in front of a whitewashed door. A bronze sword-shaped name plaque read: Captain Bocrest.\n\nSergeant Ottotark thumped his baton against the wood planks, eliciting a barked, \"Enter.\"\n\nInside, a bare-chested man performed pushups on a polar bear rug stretched before a desk. Though his short hair ran the same color as the plethora of steel comprising the ship, the defined muscles of his broad torso promised him hale. Arms like pistons in a steam engine, he pumped through another fifty pushups, while Tikaya stood and waited. The space was large, as one would expect of the captain's cabin, but spartan with nothing so frivolous as curtains for the portholes or cloth for the dining table.\n\nThe captain finished his pushups, jumped to his feet, and faced Tikaya. Her eyes were level with his nose, but he probably weighed sixty pounds more than she.\n\n\"Dismissed,\" he told the guards without looking away from her.\n\nEvidently, he was not worried about her walnut-ifying his balls. He was probably not worried about much. The collection of dented and scratched daggers, swords, pistols, crossbows, and rifles on the wall beside his desk did not appear decorative.\n\n\"Sit.\" The captain jerked his thumb toward an uncomfortable-looking wood stool and strode around the desk to his own chair.\n\nTikaya wanted to cross her arms and stare defiantly, but suspected he would force her into complying. And, like Sergeant Pissed and Horny, he might like it. She perched on the edge of the seat.\n\n\"Tikaya Komitopis.\" He gripped the sides of his desk, his eyes intent. \"Daughter of Loilon and Mela. Three siblings. You grew up on your parents' plantation and showed a gift for languages at a young age. You studied them at your university where you went on to teach for four years until the Western Sea Conflict, which involved your island. Your people chose to fight against\u2014\"\n\n\"Against becoming slaves to Turgonia, like everyone else you've conquered in the last seven hundred years.\"\n\n\"\u2014Turgonia,\" he continued as if he had not heard her. \"You were recruited by your government to break our ciphers, which, despite having no background in cryptography, you did. Repeatedly. And then your people handed our decrypted messages to our enemies.\" His eyes narrowed, his knuckles whitened where he clenched the desk. Powerful arm and chest muscles twitched beneath his bare bronze skin. \"You cost us our victory. Your dishonorable tactics forced us into a stalemate with those cursed Nurians, and we ended up losing tens of thousands of men for terms no better than we started with. Is there anything pertinent that I don't know about you?\"\n\n\"I like coconut shrimp and my favorite color is blue.\" Antagonizing him was probably stupid, but she would rather burn her favorite books than ingratiate herself with these people.\n\nHis eyes narrowed further. \"We were researching you in order to identify and kill you for your crimes against the empire, but there's been...an incident. We need something translated and our languages experts are taking too much time.\"\n\nThat probably meant they were stupefied and stumped. How desperate did the empire have to be to ask a foreigner for help?\n\n\"Time?\" Tikaya asked.\n\n\"There isn't much of it.\" He unlocked a drawer and withdrew the paper she had already seen along with two rubbings.\n\nHe laid them on the desk, and she started to lean forward, despite her intent to remain aloof. She caught herself and forced herself to sit back. Yes, the symbols interested her, but what could she do without the reference texts in the Polytechnic? Even at home, she would likely be stymied if nothing like the Tekdar Tablet existed: a bilingual source that said the same thing in two languages, one already known.\n\n\"Why is time a factor?\" she asked.\n\n\"You don't need to know.\"\n\n\"Where did the symbols come from?\"\n\n\"You don't need to know.\"\n\nShe tamped down irritation. \"Is it ciphertext? Or a language?\" She lifted her hand. \"And, yes, I do need to know that.\"\n\n\"We believe it's a language. An ancient language. We need you to decipher it and compile a dictionary. Our team can handle the rest.\"\n\nShe snorted, both at the idea of being able to simply look at these rubbings and produce a dictionary and because she would not hand anything to their 'team' even if she could. \"You mean your people can take my work, hide it from the rest of the world, and keep whatever knowledge it affords you to yourselves.\"\n\n\"That's none of your concern,\" he bit out, fingers still rigid where they gripped the table.\n\n\"If you're not going to tell me anything, then I'll have to guess. I figure you've found some ancient ruins guarding some fabled treasure and you've gotten your people in trouble trying to extricate the goods without knowing who or what you're dealing with. You need to know what the writing says so you can get at whatever it is you're after, hoard it from the rest of the world, and no doubt shoot the foreign translator who helped you. You people are\u2014\"\n\nThe captain leapt around the desk so quickly she did not have time to brace herself. He grabbed her by the neck and thrust her against the wall. Swords and pistols clattered to the deck. Her feet dangled above them, and she grabbed his bare forearms, scrabbling to loosen his grip.\n\nShe gasped, tried to suck in air. His fingers dug deeper into her windpipe. Though a rational part of her mind said they needed her and would not seriously damage her, pain and terror pumped uncertainty through her heart.\n\n\"Everyone on this ship wants you dead,\" the captain growled, breath hot against her cheek, chest heaving with rage, \"and I'd like nothing more than to snap your neck right now.\"\n\nTikaya barely heard him. She clawed at his wrists and wheezed for air.\n\nIdiot, she cursed herself. Why had she goaded him? She knew what they were, what atrocities they had committed during the war. Blackness encroached on her vision.\n\n\"But my orders say to get that language translated, and you will do exactly that. We know where your family lives. If you don't help, your parents and your siblings will be sacrificed. Your help or their lives. You choose.\"\n\nThe captain dropped her abruptly, and she collapsed before she could get her feet under her. She bit her lip, and blood tainted her mouth. He hauled her to her feet, his hand digging bruises into her arm. He took the papers, smashed them into her palm, and shoved her so hard she crashed into the door, cracking her shoulder against the unyielding metal. Tears stung her eyes, and she dashed them away. She would never let these people see her cry.\n\nHe pushed her aside to open the door. \"Get out of my office.\"\n\nTikaya, rigid back to him, walked out. Corporal Agarik and Sergeant Ottotark waited outside. Agarik spotted the blood running down her chin and gave her a sympathetic frown. Perhaps it was her imagination, but it seemed that guilt lurked in his eyes.\n\n\"Pampered librarian,\" the captain muttered before his door slammed shut.\n\nUnsurprisingly, the sergeant eyed Tikaya with lasciviousness and went out of his way to rub against her on the way back. She clenched her jaw, determined not to react. When they locked her into her cell, she willed them to go away. Alone, she would not have to maintain the stalwart facade.\n\nBefore they left, the corporal pressed a kerchief into her hand. After the hatch clanged shut, she staunched the blood trickling from her lip.\n\nShe threw the notes on the floor and paced circles around the confining cell. That bastard wanted the impossible. And he was going to kill everyone she loved if he didn't get it. How was she supposed to translate a dead language with nothing more than a couple rubbings of runes?\n\nTikaya slammed her hands against the hard metal wall. Her short, fast breaths rasped in her ears. She forced herself to take deeper ones. Panicking would not help. She needed to escape and get her family to safety. Yes, that was the best plan.\n\nShe eyed the cell across the way. An ally would make escape easier.\n\n\"Five?\" she said. \"Thanks for the help earlier.\"\n\nShe waited several moments, but no answer came. Maybe he had been taken from his cell. Maybe he was sleeping. Or\u2014she grimaced at the idea\u2014maybe he would not talk to her because he hated her as much as everyone else on the ship. Just because he had not wanted to see a woman manhandled did not make him a devotee. Though it did suggest he was decent, maybe worth the effort of bringing around.\n\n\"I now know why I'm here,\" Tikaya said. \"Sort of. I'll tell you about it if you tell me why you're here. And why you have a number instead of a name.\"\n\nShe stepped close to the gate, propping her elbows on the cold metal bars, and peered into his cell. Her eyes had adjusted to the gloom, and she made out his form in the deep shadows of the back corner. He sat, slumped against the walls, and, though she could only guess at his height, she had the impression of a big man.\n\n\"Are you a crew member being punished for something? Or are you a prisoner too?\"\n\nHe had only voiced the single sentence, but that rich baritone had sounded native Turgonian. Of course, she spoke the language like a native and was not one. Maybe he was another linguist brought in to help. And they had chained him because... Why? He was more dangerous than she? She snorted. Who wasn't?\n\nNeither her questions nor mental musings stirred him to answer, and only silence came from his cell. For now, she would have to plot an escape on her own. She ticked the bars with a fingernail. As long as she remained in the cell, she would not have a chance. Reluctantly, she allowed that cooperating with the captain, or at least appearing to cooperate, might be the only way to get herself moved to less secure lodgings.\n\nTikaya slid the rubbings through the gate and laid them on the floor. The single lantern burning in the corridor provided wan illumination, and she had to squint to read.\n\n\"I wish I knew where these rubbings had been taken. I can't even assume it's the Turgonian continent, because the empire's ships troll the world. This is a short sample, but some of the symbols do repeat.\" She spoke out loud and in Turgonian for the benefit of her neighbor, just in case hearing her voice might bestir him to comment on something, but she soon lost herself in contemplation and forgot him, the poor lighting, and even that she was on a ship full of marines.\n\n\"If it's alphabetic, it's a large alphabet,\" she murmured. \"I'm more inclined to believe we're dealing with a logographic or logophonetic script. In that case, there could be thousands of symbols in the lexicon.\" She sighed, daunted at the prospect, but she tingled a bit too. It had been over a year since anything challenged her like this. \"As far as I can tell, the symbols are abstract, not like Jutgu Hieroglyphs where so many are ideograms that represent ideas or physical things. That would have been useful.\" She tapped a page. \"That glyph reminds me of the Aracha vowels, but I suspect it's just a coincidence. This is far more complex. The way the symbols are clustered and linked is unique. I'd guess the groupings represent words, or maybe sentences or concepts. Some are quite large. Seventeen in that series. Eleven, two, seven, seventeen again.\"\n\n\"Prime numbers.\"\n\nTikaya had forgotten her silent neighbor, so she cracked her forehead on the gate in surprise when he spoke. She grunted and rubbed the nascent bump. \"What?\"\n\n\"Are all the groupings prime?\" It was the same deep, mellow voice he had used to speak to the guards, though a note of curiosity had entered it.\n\nTikaya recovered and bent over the rubbings again. She almost asked him to come forward and have a look, but remembered the clink of chains. He was probably shackled in the back specifically so he could not reach the gate and any passing guards.\n\n\"Huh,\" she said after a moment. \"They are. The highest grouping, which appears only once in these samples, is twenty-nine.\"\n\nShe gazed thoughtfully into her neighbor's dark cell. She would have noticed the prime number commonality eventually, especially if she had been scribbling notes, but that his mind went right to that gave her an inkling that she was sharing the brig with someone more than an average thug.\n\n\"You're sure you're working on a language?\" His chains rattled, and his dark form changed position. She could make out little, but guessed he had shifted to face her.\n\n\"I'm not, no. That's what the captain told me. An ancient language that he wants me to decipher. Though I don't think he knows much or he'd understand there's no hope of translating a text by looking at a sheet of symbols. I'm guessing he was just parroting what someone higher up told him.\"\n\n\"Likely.\" Was that an amused note in his voice?\n\n\"I'm not sure how much stock to put in his claim of 'ancient' either. The Turgonians don't use the mental sciences and can only rely on the relative dating method for judging age. Even that's questionable, since they've only been on their continent seven hundred years, and I'm not sure how much, if any, documentation they did of the existing cultures before they assimilated them. Or killed them off. Brutes.\"\n\nHer neighbor\u2014Five, she reminded herself\u2014said nothing at that, and she winced, recalling he might be Turgonian himself. She rubbed her lips, annoyed at her mouth's proclivity for blurting things out without lacing in any tact.\n\n\"Erm, anyway,\" she said, \"all primes between two and twenty-nine are represented in these samples.\" Casually, hoping she could draw him out, she added, \"Supposing this is a language, do you have any thoughts as to what might be the significance of incorporating primes in the core structure?\" The first thing her mind flashed to was that each number might signify a different part of speech, but using seventeen symbols to represent a verb seemed like overkill.\n\nHis chains rattled. A shrug? \"I'm sure you already know prime numbers are the building blocks of natural numbers. They can only be divided evenly by one and themselves, and anything that's not a prime number is made up of prime numbers.\"\n\n\"Building blocks,\" she mused. \"Like letters are the building blocks of my language, perhaps. Though in this case the numbers are the wrappers, not the content.\" She stood and stretched, wishing the cell afforded her more room to pace.\n\n\"You are Kyattese, is that correct?\"\n\nTikaya stilled, realizing he did not know who she was to the marines. If he was a Turgonian, her chances of turning him into an ally might plummet if he found out she was their cryptomancer.\n\n\"Yes,\" she said carefully.\n\n\"Did your president survive the war?\"\n\nSurprise and then suspicion flooded her, and she regarded him through narrowed eyes. \"Yes, why wouldn't he have?\"\n\n\"Is he...a good man? Good for your people?\"\n\nTikaya did not know what to make of this line of questioning and responded only with another clipped, \"Yes.\"\n\nShe folded her arms across her chest and decided not to answer anything further about her people or her nation, especially not anything the Turgonians might use against them. Fortunately, Five asked no more.\n\n\"Why don't you answer a question for me since I'm answering yours?\" she suggested.\n\nHe did not respond.\n\n\"I'll settle for one,\" she said. \"Will you answer me one question?\"\n\nHis soft snort hinted at amusement. Tikaya decided to take it for consideration.\n\n\"What's your name?\" she asked.\n\nA sigh mingled with the hum of the engines. \"Not that question. Ask another.\"\n\n\"Why not that?\"\n\n\"You're replacing your one allotted question with that one?\" A hint of dryness infused his tone.\n\n\"Yes.\"\n\nAnother sigh. \"They took my rank and my name as punishment. Five was my number on the penal boat going to Krychek Island, and, as far as the empire cares, it's the only identification I have now.\"\n\n\"Krychek Island,\" Tikaya breathed. \"Isn't that where they send criminals so vile they're afraid to execute them outright? Out of fear their spirits will linger in the area and afflict the living? So they send you to the island with no food, no weapons, no resources, the assumption being you'll kill each other off far from anyone worth haunting? They say those few who do survive turn into animals, bestial and deranged and cannibalistic and...\" She caught her lip between her teeth. There she went again, probably offending the only person on this ship who had stood up for her.\n\n\"Glad our penal island is renowned even amongst foreigners.\" Five's dryness held a bitter edge this time.\n\nShe sighed. She had offended him. And she had a new concern. If he was that much of a criminal, dare she side with him? What could he have done to merit such a harsh punishment? Brutality seemed bred into the Turgonian culture, so she struggled to imagine someone who fit their definition of vile. Five sounded normal\u2014pleasant\u2014but perhaps it was a facade. He had to be shackled for a reason, though Captain Bocrest's current problems must trump that reason, or why else would Five be here?\n\n\"But the captain came to pick you up? Isn't Krychek Island usually a permanent residence?\" she asked, wanting to be sure. \"Its location is even secret, isn't it? So families can't make rescue attempts?\"\n\n\"Correct.\" Tension riddled that one clipped word, and she hesitated before asking the next question, but she had to know if he was likely to be a threat to her.\n\n\"What was your crime?\"\n\nClothing rustled and the chains rattled. \"No more questions.\" His voice was muffled, as if he had covered his face. \"Please,\" he added so softly she thought she might have imagined it.\n\n\"Of course. Sorry.\" She meant it. If the place was half as bad as the stories said, she could understand not wanting to discuss it. He was probably lost in his painful memories, and only the puzzle of the language had distracted him. \"Uhm, I'll be over here, enjoying the lovely ambiance and pondering these slanted circles, dots, and sideways trees. If you want to talk later, let me know.\"\n\nShe did not expect anything else from him, but he surprised her by asking, \"Sideways trees?\"\n\n\"Well, if trees were symmetrical maybe. Want to see?\"\n\n\"I can't reach the corridor.\"\n\nShe grabbed one of the rubbings, folded it into a compact stack, and tossed it through his gate. His dark form shifted, so she assumed it fell within his reach, but he said nothing.\n\n\"Can you see it?\" she asked after a moment. \"Is there enough light?\"\n\n\"Yes. I should have known.\" He sounded grimmer than a funeral pyre. \"I've seen them before.\"\n\n\"Where?\"\n\n\"Somewhere I never want to go again.\"\n\n\"Where?\" she repeated, leaning forward.\n\nHe did not answer.\n\n\"Five?\"\n\nSilence.\n\n# CHAPTER 3\n\nTikaya dozed until the hatch creaked open hours later. The perennial darkness of the cell stole her sense of time, but she guessed it was evening. Her hollow stomach whined for a meal.\n\nFootfalls rang on the metal deck. She eyed the gate warily and let out a sigh of relief when Corporal Agarik came into view\u2014alone. He carried two metal canteens and wooden trenchers narrow enough to slide between the bars. Dried fish, dried fruit, and a couple of hard biscuits soon rested on the floor before Tikaya. Though the meal did not exactly fill the brig with scintillating scents, her mouth watered anyway.\n\nShe grabbed a biscuit and gnawed. It had the consistency of sawdust and less flavor. Agarik placed another tray, this one with a larger portion of food, on the floor in Five's cell. He drew his cutlass and used it to push the tray deeper, presumably so the chained prisoner could reach it. Tension marked the corporal's movements\u2014he seemed to fear an attack at any moment\u2014and Tikaya wondered if people had been hurt, or worse, when the marines originally locked Five up. Either that or his reputation was simply enough to instill fear. Even Sergeant Ottotark's bravado had seemed forced. And she wanted to ally with Five? Was she insane? Just desperate, she decided.\n\nAgarik delivered the meal without incident; the chains did not even clank to suggest movement. He sheathed his sword and turned to face her.\n\n\"Captain Bocrest wishes to know if you've decided whether to cooperate voluntarily or if more...\" he shrugged apologetically, \"pressure is required.\"\n\nWhat? Threatening families\u2014and windpipes\u2014was not enough to sway most prisoners?\n\n\"I'm willing to attempt the translations,\" she said, \"but I have terms. I can't work in these conditions.\"\n\n\"I'll tell him.\"\n\n\"Corporal?\" Tikaya asked before he could leave.\n\nHe turned back, face guarded. He probably feared she would ask some favor he would be duty-bound to refuse.\n\n\"Thank you for your kindnesses,\" was all she said.\n\nHe nodded but said nothing. A bevy of footsteps hammered the deck, the hatch clanked open, and six burly marines clomped into view. Sergeant Ottotark followed, tapping his baton against his thigh as he walked, and Tikaya shrank into the shadows. Now what?\n\nThe six marines staggered themselves along the corridor and pointed pistols, not into her cell but into the opposite one. Tikaya stepped forward, afraid they meant to execute her neighbor, but Ottotark rattled the keys.\n\n\"Time to visit the captain, Five.\"\n\nHe unlocked the door and waved two of his men inside. They exchanged nervous glances but slid into the cell and stood on either side of the door, pistols never wavering. Ottotark took a deep breath, visibly steeling himself, then walked inside. As yet, the chains had not rattled, and Tikaya half thought Five asleep. Maybe he was just being still, hoping for a chance to escape. Though how he could do so with so many firearms pointed his way, she could not guess.\n\nWhile all eyes focused on that cell, Tikaya eased forward. She eyed the belts of the men within her reach, hoping to spot a set of keys. Such luck did not favor her.\n\nCorporal Agarik had stepped back when the others marched in, and she caught his knowing gaze on her. She sniffed and stared back. They had kidnapped her; she refused to feel guilty for thinking of escaping.\n\n\"Up.\" Sergeant Ottotark must have unlocked the chains securing the prisoner to the wall. A long still moment passed with no sign of movement in Five's cell. \"I said, get up!\"\n\nOttotark lunged into the shadows. Tikaya flinched, expecting the meaty thud of that baton striking flesh. A scuffle and grunt sounded. Someone threw the baton and it clattered against the gate before dropping to the floor. The sergeant growled and drew his arm back, but he halted mid-blow and skittered backward.\n\nFive was on his feet.\n\n\"Don't move!\" one of the marines inside the cell barked, pistol arm straight and rigid. \"We will shoot you.\"\n\n\"Doubtful.\" But Five stopped short of grabbing Ottotark and turned toward the guards, his features still in shadow.\n\n\"Cursed bastard, you presume much,\" Ottotark growled. \"We can shoot you without killing you.\" A speculative note entered his voice, as if he were truly considering it.\n\nTikaya gripped the bars of her gate, trying to think of something to say to help him. After all, Five had come to her defense.\n\n\"Or we can just beat you into oblivion for the rest of the trip.\" Ottotark hooked a punch into Five's face.\n\nWith pistols pointed at his chest, Five could only accept it. Ottotark grabbed his baton and lifted it to deliver more damage.\n\n\"I thought Turgonians were supposed to be brave warriors,\" Tikaya blurted. \"Abusing someone who can't fight back is cowardly.\"\n\n\"Sew that yap shut, woman. Nobody wants your opinion.\" Despite his words, Ottotark lowered the baton and prowled out of the cell. \"Let's go, ugly.\"\n\nFive shambled into the corridor. Thick, tangled black hair hung around his cheeks and half way down his back. A matted beard and mustache engulfed the lower half of his face. Torn, faded trousers with ragged hems reached his calves, and a crudely sewn hide vest covered his torso, leaving muscular but lean\u2014too lean\u2014bare arms visible beneath a layer of grime. Shackles bound his wrists before him, and blood trickled from his nose, adding menace to his already savage appearance. Even slumped, head hanging, he stood a half foot taller than Tikaya.\n\nHe glanced at her, almost wincing, and she had the impression his state embarrassed him. She met his eyes with a respectful nod. Criminal or not, he was the most obvious person to turn into an ally.\n\n\"Let's go.\" Ottotark sent two men ahead, then shoved Five after them.\n\nAfter the group had gone, Agarik nodded to Tikaya's food and water. \"Do you need anything else?\"\n\nEverything else, she thought, and a trip back home. \"Can you tell me who that is I'm sharing the brig with?\"\n\n\"Nobody knows.\"\n\n\"Somebody must know.\"\n\n\"The captain,\" Agarik said. \"He doesn't confide in anyone. I don't think Sergeant Ottotark even knows, and he's the captain's adjutant.\"\n\n\"What happens if the captain gets shot and no one else knows the mission?\" Tikaya supposed it was uncharitable to enjoy the thought.\n\n\"The orders are locked up somewhere. The officers know where to find them.\"\n\n\"Ah.\" Tikaya pointed to the vacated cell. \"Why are your people so careful with him? Is he that dangerous?\"\n\nAgarik worked his tongue against his cheek and gazed toward the ladder, perhaps considering whether it would be a breach of duty to answer. \"He's a prisoner from Krychek Island, and we lost four men getting him off the beach.\"\n\n\"He killed them?\"\n\n\"No, the lunatics on the island attacked our party with spears and clubs. Men gone savage. They wanted to escape, and if they couldn't escape they'd kill those who originally brought them there months and years before. Ancestors' wrath, we had to shoot a bunch of them. Seemed they'd rather die than stay there.\"\n\n\"And Five attacked you too?\" Tikaya rested her arms on the gate and watched the corporal's face in the flickering light of the single lantern. His gaze had grown thoughtful and distant.\n\n\"No, he stood back and watched. You got the sense he didn't want anything to do with us, but he didn't hide either. At first it seemed he'd come along peacefully\u2014he got in the longboat once the captain spotted him and called him over. He didn't give us any trouble rowing back to the ship, but he attacked a guard the first night, got out of his cell, stole a pile of food from the galley, and slipped by everyone on duty.\" Agarik frowned. \"Including myself. Without anyone seeing him, he swiped a sextant, compass, chronometer, nautical almanac, and spare sail, and he was about to drop a lifeboat. He would have been long gone by morning, but Captain Bocrest got an itch, and he was waiting with a loaded rifle.\"\n\n\"So he\u2014Five\u2014surrendered?\"\n\n\"Not exactly.\" Agarik rubbed his jaw as if recalling a blow. \"Captain threatened to shoot him but didn't, and it took a full squad to wrestle him belowdecks and get him locked up again.\"\n\n\"Where he's been chained ever since.\"\n\n\"Yes, ma'am.\"\n\nSo, whatever the imperials wanted their prisoner for, it seemed he was also too valuable to kill. His first escape might not have worked, but he had that goal in mind too. Good. Two people rowing a longboat would be more efficient than one, and it heartened her that Five had known exactly what to grab. She knew how to sail and navigate in theory but had never been out of sight of her islands.\n\n\"One thing's a mite peculiar,\" Agarik mused.\n\n\"Just one?\" Everything thus far struck Tikaya as peculiar.\n\n\"He didn't take a pen or paper.\"\n\n\"What do you mean?\"\n\n\"You need to do some figuring to account for the errors and adjustments that come with using a sextant. Not many could do 'em in their head and keep them straight from day to day without a log.\"\n\n\"Maybe he forgot,\" Tikaya said, though she already had a hunch Five had a background in mathematics. Maybe he could do the calculations in his head and remember the results.\n\nThe corporal grunted noncommittally. He seemed as curious about the mystery prisoner as her.\n\nMore footfalls rang on the deck. Now who?\n\n\"Your duty, Corporal,\" the captain said, eyes cool as he descended the steps. \"It is not here.\"\n\n\"Yes, sir.\" Agarik ducked his head and trotted away.\n\nThis time, Bocrest wore his black uniform jacket with a handful of badges and medals adorning the breast. A fresh bruise swelled on his temple, and dried blood crusted on his chin beneath a swollen split lip. Had someone whaled on him as part of a training session? Or maybe he had already started questioning Five, and it wasn't going well. Either way, the bumps would probably not improve his personality.\n\nNonetheless, she lifted her chin and met the captain's eyes. Bravado would likely get her further than meekness on this ship.\n\n\"Well?\" Bocrest asked. \"You working with us or are my men taking target practice on your family members?\"\n\nIt was a moment before she could unclench her jaw. The man had the diplomacy of a stinging jellyfish. \"I will help you, captain,\" she said, forcing a civil tone, \"but I can't work in this dark pit, and, surely, if you expected me to translate this language, you brought some basic references and primers. Hodtolk's? Fisher and Grist? Merk's Hieroglyphics Compendium? More samples of this writing would help as well. And I'll certainly need better lighting, paper, pencils, a table. I'll also need the freedom to walk around. That's when I do my best thinking.\"\n\nTikaya expected denial, especially over her last request, but after glaring at her for a moment\u2014it seemed his normal way of looking at people\u2014he said, \"I'll get you paper and better lighting. You may have one daily exercise period. Beyond that, pace your cell if you need to 'think.'\"\n\nHe started for the hatch.\n\n\"One more question, captain,\" Tikaya said, wondering if he would answer it honestly or not.\n\n\"What?\"\n\n\"Suppose I succeed in translating this language, in helping you with whatever your problem is. What happens to me then?\"\n\nBocrest eyed her over his shoulder. \"If you succeed, your family will not be harmed. You? As far as the emperor and thousands of dead Turgonians are concerned, your deeds during the war condemned you. I suggest you enjoy your last project.\"\n\nTikaya leaned against the cold metal wall for support. She wished he had lied.\n\n* * * * *\n\nWhen Tikaya stepped out of the hatch, the sun made her blink. She stumbled and almost crashed into the guards escorting her outside for her exercise session. Nobody offered a steadying hand.\n\nWind gusted across the deck, tugging at her braid, and slapping her dress against her legs. When her vision recovered, the sun told her they traveled northeast. Endless sea stretched in all directions, so she could only guess at their position and goal. Though the briny breeze stole the stink of burning coal, the black plumes streaming from two smokestacks suggested the furnaces burned at maximum capacity. Full sails made use of the wind as well, and Tikaya wondered how fast they traveled under the combined power. Perhaps she imagined it, but the sun warming her cheeks felt less intense than back home. Where were the Turgonians taking her?\n\nA pair of marines in gray togs jostled her as they jogged past.\n\n\"Stay out of the running lane,\" one barked without glancing back.\n\nTikaya sighed and shuffled in the direction the guards indicated. She should have relished the excursion, the chance to stretch and walk, but the lack of company dulled her spirits. She hadn't even been able to speak with Five again. The captain had granted her request for a desk and better illumination by moving her to one of the officers' cabins in the wardroom, which put her on the other side of the ship from the man she wanted to conspire with. And the young private stationed outside her door showed no inclination of allowing her to wander.\n\nThe guards led her past masts, smokestacks, and two thirty-foot launches mounted in the center of the deck. She kept her gaze from lingering too long on the big boats. It would take more than two people to get one of those in the water anyway. She stepped past a cannon to glance over the railing. Ah, yes. Smaller cutters were mounted alongside the ship below the gun ports. She and Five could handle one of those. Unfortunately, she needed time with him to make plans.\n\n\"Stop gawking.\" One the guards shoved her.\n\n\"I didn't know there was a minimum walking speed up here,\" Tikaya muttered.\n\n\"Exercise is for sweating, not sightseeing.\"\n\n\"You're a pithy people, aren't you?\"\n\nThat earned another shove.\n\nTikaya picked up her speed. A heavy gun on brass rails dominated the forecastle, but the area behind it lay open, and a few bare-chested men boxed in a makeshift arena. Racks contained practice weapons, dumbbells, and other exercise equipment. Captain Bocrest and a lieutenant stood on the far side where a temporary archery lane was set up with person-shaped paper targets attached to bales of hay. They practiced with repeating crossbows, though traditional bows also leaned in a rack.\n\n\"You going to do anything, woman?\" a guard asked.\n\nFeeling self-conscious beneath all the eyes that swiveled to watch her, Tikaya walked over to a pile of sand-filled balls with handles. After a few tries, she found a small one she could lift. She maneuvered through a few exercises, though no one had suggested baths were available, so she was not sure how much of a sweat she wanted to encourage.\n\n\"Awkward turtle, isn't she?\" one of the guards said.\n\n\"Fine by me,\" the second said. \"Makes her melons bounce.\"\n\n\"Hah, and her ass. Bookly thing but I'd mount that in a heartbeat.\"\n\nJaw clenched and cheeks flaming, Tikaya turned her back to them.\n\n\"Nothing else to mount around here,\" the conversation continued, \"unless you want to crawl into Lieutenant Amn or Corporal Agarik's bunks.\"\n\n\"I reckon they'd be the ones wanting to do the mounting then.\"\n\nShe supposed that explained why she was not Agarik's \"type.\" The marines went back to analyzing her, and, when others joined in, the commentary grew cruder and more explicit. Though the captain stood within earshot, he did nothing to stop the lewd harassment. She wondered if the men would have treated a Turgonian woman this way or if her status as hated-enemy-of-the-empire made it acceptable.\n\nTikaya gave up the exercises in favor of walking around the training area. She eyed the officers plunking quarrels into the targets, surprised they bothered practicing archery given the power of their rifles.\n\n\"How's the thinking going, librarian?\" Bocrest asked. \"You figure anything out yet?\"\n\n\"I'm working on it. I doubt you have any idea as to the magnitude of the task. People spend years working to translate a newly discovered language, and that's when they're surrounded by libraries full of reference materials.\"\n\n\"Uh huh. Take a few more laps around the ring to inspire my men's fantasies and then go back to work.\"\n\n\"Double or nothing, sir?\" The lieutenant hefted his crossbow.\n\n\"It's your rum.\" The captain turned his back on her and loaded a fistful of bolts into his own crossbow.\n\nAn idea tickled her mind. \"You a betting man, Captain?\" she asked before she could talk herself out of it.\n\nSince he had already dismissed Tikaya, he had to turn back to frown at her. \"What?\"\n\n\"Care to make a wager?\"\n\n\"Like what?\"\n\n\"I'll bet you there's a weapon here I can best you with.\"\n\nThe snorts and outright laughs around her were no surprise.\n\n\"Why would I make a wager with my prisoner?\" the captain asked. \"What could you have to lose that I would want?\"\n\nWhat indeed?\n\n\"Before you offer to warm my toes tonight, know I'm a married man.\"\n\nThe fact that he had a wife\u2014and was faithful to her\u2014left her speechless for a moment.\n\nBocrest tapped his foot. Tikaya considered the bruises on his face. If she was right and Five had delivered those, maybe she could use that.\n\n\"I see Prisoner Five has given you some trouble,\" she said. \"You must need him for something, presumably related to what you need me for. If I lose, I'll persuade him to help you with your mission.\"\n\nBocrest laughed. \"Why would he listen to the cryptomancer?\"\n\nBecause Five did not yet know she was the cryptomancer. \"Because we've established a rapport.\" If one could call a single shared conversation a rapport.\n\nThe laughter ceased, and Bocrest studied her through narrowed eyes. Perhaps that had not been a wise claim to make.\n\n\"Fine. What if I lose?\" The captain's mouth twisted, showing how unlikely he thought that. \"I'm not releasing you or promising anything that would involve breaking orders.\"\n\n\"I want Five to share my exercise periods, an hour each day.\" The captain was shaking his head before she made it halfway through the sentence, but she pressed on. \"I also want you to give him a bath, haircut, shave, and fresh clothing to wear.\" Tikaya smiled. \"Actually, I'll take a bath and fresh clothing too.\"\n\n\"A bath!\" the lieutenant roared. \"This is a steamer! Water is for pouring into the boiler.\"\n\n\"Surely you could manage a damp washcloth,\" she said.\n\n\"No,\" the captain said. \"No to it all. That's too much extra work for my men. He's too dangerous to have out.\"\n\n\"Why can't these men watch him?\" She pointed to the onlookers. \"I can't imagine the emperor pays them to stand around and gawk at me. Besides\u2014\" the captain's face had grown red, so she patted the air soothingly, \"\u2014you don't honestly believe you'll lose our wager, do you?\"\n\nHe snapped his mouth shut. \"No.\"\n\nThe captain stuck his palm out, edge toward her, and she banged her hand against it in the Turgonian gesture for a deal sealed.\n\n\"Choose your weapon,\" he said.\n\nShe went straight to the bows. They were designed for tall, burly men, so it took some experimentation to find one she could string and draw. For once her long arms were useful, and her months laboring on the plantation gave her strength she had not possessed during her academic tenure.\n\n\"Think she'll even be able to load that?\" one man asked.\n\n\"Probably shoot her toe off.\"\n\n\"There's no way she'll hit a target.\"\n\n\"Better tell the boys in the rigging to watch out.\"\n\n\"Don't know why my languages instructor bothered teaching me the Turgonian word for encouragement,\" Tikaya muttered. \"Not like they ever use it.\"\n\nBow strung, she joined Bocrest.\n\n\"Challenger shoots first,\" he said.\n\n\"No practice?\"\n\n\"No.\"\n\n\"Best of three shots?\"\n\n\"One shot. Deal's been made. Shoot.\"\n\nThe lieutenant handed her a single arrow.\n\n\"I see you're a sporting people.\" She should have negotiated the rules of the game instead of trying to finagle baths.\n\nTikaya nocked the arrow and turned sideways, bow held loosely in her left hand as the fingertips of her right curled about the string. Just like on the plantation back home, she told herself.\n\nExcept it wasn't. Even on the calm day, the ship rose and fell with the swells, and activity on deck offered distractions. The misty breeze licked her cheeks, and she closed her eyes for a moment, considering the affect it would have on the arrow's flight. She locked her eyes on the red dot in the center of the target and drew the bow, anchoring her fingers in her usual spot against her cheek. The men's ongoing comments disappeared and focus came. She breathed in the tangy air, blew it out, and waited for the quiet moment when her body and the deck were still.\n\nShe released the arrow.\n\nIt cut through the air and thudded into the red dot. The surrounding men fell silent, mouths hanging open. Tikaya resisted the urge to smile or make any triumphant gesture.\n\n\"Your turn,\" was all she said to the captain.\n\nHis expression was less stunned and more dyspeptic. Too late, Tikaya wished she had found a way to make the challenge private. If he did not make as fine a shot, he might lose face in front of his men. And take it out on her.\n\nBocrest lowered his bow. \"A shot that good is worth the prize, for what little reward having that dour bear around will be.\"\n\nThe men grunted in agreement. Good. She recognized the face-saving gesture, but in this case was relieved he had found a solution. After eliciting a promise that Five would join her the next day, she walked a few more laps.\n\nThe captain caught up as her guards were about to lead her belowdecks. He clenched her elbow and put his mouth near her ear. \"I trust you and Five aren't plotting to escape. If you attempt something that foolish, you will be caught, and I'll let Sergeant Ottotark deliver your punishment. He enjoys that sort of work immensely.\"\n\n# CHAPTER 4\n\nClouds blanketed the sky the next day when Tikaya came out for her exercise session, but the darker weather didn't dampen the curiosity humming through her. Her ally\u2014even if he did not yet know she had dubbed him ally\u2014would join her soon. What would he look like without all that hair and dirt? Would the guards give them enough space to talk privately?\n\nShe walked around the outside edge of the exercise area, struggling for patience. The captain was out again, this time trading sword blows with his navigator. Tikaya wondered who ran the ship when these Turgonians spent so much time exercising. Some prisoners of war were probably chained down in the boiler rooms, shoveling coal into the furnaces day and night.\n\nThe clamor of crashing steel halted, and Tikaya stopped walking to search for the reason.\n\nIf not for the guards surrounding him, she would not have recognized Five. Now clean-shaven with military-short hair, he wore the same boots and black uniform as the marines, though no rank or insignia marked the collars. Taller than the men accompanying him, he strode across the deck, hands clasped behind his back, head up, alert eyes taking in every aspect of the ship.\n\nTikaya's stomach did an anxious flip. Her putative ally had turned into someone who looked every bit like one of the officers who had tried to take over her islands during the war. Even with no rank on that collar, he seemed more the captain than the sweaty bare-chested Bocrest, who was also staring. A chilling thought gripped her. What if Five had been a captain during the war? Someone who fired on her people? Took prisoners? Tortured them.\n\nFive's gaze stopped on the sails nearest the smokestack. A faint sooty black dulled the canvas, and he raised an eyebrow at Captain Bocrest.\n\nFor a moment, Bocrest's cheeks flushed, and an excuse seemed on his lips, but he halted it with a scowl. He stalked across the deck, bare chest puffed out, muscles flexed. He barked at anyone foolish enough to cross his path and stopped in front of Five. Bocrest gestured sharply while spitting words out in a low voice.\n\nTikaya resumed walking, more briskly than earlier, so she could steer close enough to eavesdrop. Before she neared them, the captain thrust his arm out, pointing his index finger at her. She stopped, feeling self-conscious when both men, and everyone else in the area, turned to stare at her.\n\nOnly Five's gaze was friendly. The right side of his mouth quirked up in a bemused half smile, and she felt the need to brace herself on a nearby weapons rack.\n\nBocrest growled, \"Convince her,\" just loud enough for her to hear.\n\nThough Five did not acknowledge the order, those words drove wariness into Tikaya's heart. Presuming Bocrest's relationship with Five was entirely antagonistic may have been a mistake.\n\nHe left the captain's side and strolled toward her, his smile widening as he approached. A few strands of silver threaded his black hair, laugh lines crinkled the corners of his brown eyes, and a narrow scar bisected one eyebrow, but Tikaya had no doubt women of all ages swooned at his feet. Experience made her stifle her own urge to swoon. Handsome men did not look at her and smile; they looked through her, usually not noticing when they bumped her out of the way to close in on some buxom damsel with cleavage like the Inarraska Canyon. Most likely, he had an ulterior reason for that smile.\n\nTikaya folded her arms across her chest and kept her face neutral as he closed the distance.\n\nFive's first words destroyed her attempt at equanimity. \"You're the cryptomancer?\"\n\n\"What? I, uhm, no. I mean\u2014\"\n\nTikaya winced. Even if he had no ulterior motives, her almost-ally would surely turn against her if he knew. Like the rest of the marines, he would resent her, hate her, glare at her and...\n\nHe was staring, not glaring, at her, and not with hatred. Was that\u2014her eyebrows arched\u2014awe?\n\n\"It's your people's term,\" she said, \"not what the name plaque on my desk says.\"\n\nHoping for nonchalance, Tikaya stuck out a hand to lean casually on the weapons rack, but her focus was on him, and those gold flecks in his brown eyes, and she missed the target. Her fingers clipped the corner and slid off, giving her no support. She pitched sideways with a startled, \"Errkt,\" and would have landed on the deck, but Five lunged and caught her.\n\nChortles burst from the surrounding marines, and flames torched her cheeks. Five straightened and released her with a pat on the shoulder. She groaned and avoided his eyes. If there had been awe there before, that was surely gone now. In avoiding his gaze, she had a clear view of the marines pointing at her and nudging each other. Even Bocrest's rock-eating jaw flapped with guffaws.\n\n\"Walk?\" Five suggested gently.\n\n\"Dear Akahe, yes.\"\n\nShe departed the scene at a vigorous pace, and Five, with his long legs, easily matched her. His guards fell in behind. At least they proved stolid and silent save for the clatter of gear and synchronized thump of boots on the deck.\n\n\"I must thank you for this.\" Five gestured at himself, encompassing the clean uniform and haircut. \"I got the story from Corporal Agarik. It was kind of you to include me in the reward for your wager.\"\n\n\"You're welcome,\" she muttered, knowing her thinking had not been purely altruistic. \"Though I'm surprised the captain let you out, lost bet or not.\"\n\n\"He made me promise not to make trouble during the exercise periods.\"\n\n\"Ah.\" Interesting that Five's word was enough for the captain to trust him. She glanced at the guards. To some extent anyway.\n\n\"It was worth it.\" He stretched his arms overhead, then windmilled them, something the confines of his cell would make impossible. \"I almost feel like a human being again.\"\n\nIt had certainly improved his mood. She thought of the silent, brooding man she had spoken to the first day and could not help but feel pleased her request had lightened his spirit. She gave him a smile and missed a step when he smiled back. Oh, that was nice.\n\nStop it, Tikaya, she chastised herself. Prisoner or not, he was one of them. That uniform fit him like he had been born into it. Best get some answers from him while he was in an affable mood.\n\n\"Given the reception I've gotten here, I'm surprised you aren't...\" She watched him sidelong. \"Does my wartime hobby not bother you?\"\n\n\"Actually...\" He met her sideways gaze. \"It impresses me. A lot.\"\n\n\"Oh,\" she breathed, then looked away, not sure she wanted him to see her reaction. She had wanted an ally; she had not expected an admirer. She was not sure how to deal with that. Parkonis, though he had loved her personally, had been a little jealous of her professionally. They had worked in the same field, with her discoveries often eclipsing his, and his praise had always sounded grudging.\n\nThey passed under men in the rigging, adjusting sails to take advantage of the wind. Only a faint smudge of black wafted from the smokestack today.\n\n\"As far as we've heard,\" Five said, \"cryptography isn't taught on Kyatt, so I just assumed what we called the cryptomancer was a team of mathematicians learning as they went. But your specialty is linguistics, right?\"\n\nThe question sounded casual, but a trickle of wariness returned to her thoughts. Just because he pretended to be an admirer did not make him one. Maybe the Turgonians had simply decided to substitute honey for vinegar, and had talked him into delivering it.\n\n\"Yes,\" she said. \"Philology, really. I work with the anthropology and archaeology departments in the Polytechnic.\"\n\n\"Interesting. How many languages do you know?\"\n\n\"Sixteen modern, and I can read a few dozen dead languages.\"\n\n\"Few dozen?\" Five halted and gaped at her. \"You must be a genius.\"\n\nThe proclamation startled her, and she lurched to a stop beside him, conscious of the guards' gazes on her back. \"No, no, trust me I'm not. It's just something I've a knack for.\"\n\nHe lifted a single skeptical eyebrow.\n\nTikaya shook her head. \"A world-exploring uncle gave me a copy of the Tekdar Tablet when I was a child, and I fell in love with solving language puzzles. My parents encouraged it, so I had a head start when I started formally studying in school. That's all.\"\n\nFive was still standing, gazing at her, and when she met his eyes, she found that admiration there again. It was disarming. Maybe he meant it to be. What had the captain told Five to convince her of?\n\n\"Hm.\" He resumed walking. \"My family gave me swords and toy soldiers when I was a boy.\" Bemusement laced his tone.\n\n\"You would have preferred something else?\"\n\n\"Oh, yes. I kept asking for drawing pads and building materials. I wanted to design a treehouse with a drawbridge to my room and a steam-powered potato launcher for defense.\"\n\n\"Sounds like every boy's dream.\" Despite her determination to remain chary with him, the change of topic set her at ease. She could not reveal something she shouldn't if he was talking about himself.\n\n\"Alas, this was not a paternally approved childhood activity, so I had to find my own building materials.\" Five scratched his jaw. \"I took it upon myself to chop down some of the apple trees in my family's orchard, trees that my great grandfather had grafted from cuttings painstakingly acquired when he was a marine sailing around the world. I, being about eight at the time, was unaware of this bit of history.\"\n\n\"Oh, dear,\" she murmured.\n\n\"Yes. There was a lot of yelling that summer.\"\n\nShe chuckled.\n\n\"What is engraved on your name plaque?\" Five asked as they started on their second lap of the deck.\n\nFor a moment, the context of the question eluded her, until she remembered her earlier comment. \"You don't know my name?\"\n\nHe spread his arms apologetically. \"Nobody's told me much.\"\n\nThe salty breeze gusted, and water sprayed the deck ahead of them. A lieutenant bellowed at the men aloft.\n\n\"Your name for mine,\" Tikaya offered with a smile. \"I can't keep calling you Five forever.\"\n\nHe glanced at the guards trailing them. Maybe, as part of his punishment, he was forbidden from using his old name.\n\nHe lowered his voice. \"My friends and family, back when I had them...\" He grimaced. \"They called me Rias.\"\n\n\"Rias?\"\n\nTikaya had a feeling that was a nickname or a truncation. Regardless, it gave her no hints as to his identity. Since she had decrypted all the communications her people had intercepted, she knew most, if not all, of the Turgonian officers with enough rank to command a vessel, and she could not think of any name with those syllables.\n\n\"My name is Tikaya,\" she said. \"And, now that we're on a first-name basis, maybe you can tell me what you're supposed to convince me of, Rias.\"\n\nTheir route had taken them to the archery lane. Rias paused by the rack of staves, and the guards tensed, their fingers finding the triggers of their pistols.\n\n\"No weapons,\" the lead man said.\n\n\"Captain,\" Rias called. \"May we shoot?\"\n\nIn the center of the exercise area, Bocrest knelt on a young officer's back, with the man's arm twisted in a lock. The captain scowled over at them.\n\n\"May you shoot? What is this, the Officers' Club? Perhaps I can get you some brandywine and lobster too?\"\n\n\"Captain, are you inviting me to dinner?\" Rias rested his hand on his chest. \"I'm touched.\"\n\nRed flushed Bocrest's face, and Tikaya wondered at the wisdom of teasing the man. If the captain had a sense of humor, she had not detected it. But he waved a disgusted hand at the guards.\n\n\"Let them shoot.\"\n\n\"Sir?\" The lead guard's mouth gaped open.\n\n\"You heard me,\" Bocrest barked.\n\n\"Yes, sir.\"\n\nTikaya eyed Rias. \"It seems your word means something to the captain.\"\n\n\"He knows it's all I have left.\"\n\nBleakness stripped away his humor, reminding her that pain lurked beneath the facade he was showing her today. He caught her watching and reaffixed his smile.\n\n\"Tikaya,\" he said slowly, trying her name out, then nodding to himself as if he approved. \"To answer your question, despite his threats\u2014\" Rias scowled, \"\u2014the captain has doubts about your intentions. He believes I should convince you to help him wholeheartedly with his mission.\"\n\nShe selected the bow she had used the day before. \"Why, when he's keeping you chained in the brig, does he think you'd be inclined to speak on his behalf?\"\n\n\"He believes that my indoctrinated loyalty to the empire will overrule whatever revulsion I feel for him and those who took everything from me.\"\n\n\"They must do a lot of indoctrinating in Turgonian schools.\"\n\nRias sighed. \"Oh, they do.\"\n\n\"And do you think I should help? You recognized something about those symbols when I showed you the rubbing. What was it?\"\n\nHe did not answer, though she did not think him recalcitrant. His gaze grew far away, his face grim, as if some painful memories had swallowed him and he had forgotten her.\n\nMaybe archery would loosen his mind and unlock his thoughts for sharing. She shot a few times, leaving arrows quivering around the red dot in the target. A stiffer breeze scraped across the deck today.\n\nRias stirred and selected his own arrows.\n\n\"Want to make a wager?\" Tikaya asked, thinking she might be able to get him to talk more freely about the symbols if he owed her from a lost bet.\n\nHis grimness faded and he slanted her a knowing gaze. \"I suspect that would be unwise.\"\n\n\"Captain Bocrest did it.\"\n\n\"Then I'm certain it's unwise.\"\n\nTikaya grinned. \"Maybe I just got a lucky shot.\"\n\n\"I doubt it.\"\n\n\"Why?\" All the marines had been stunned when she hit the target.\n\nHe shrugged. \"The Kyattese are bow hunters.\"\n\n\"Well...\" She smiled and twirled an arrow in the air. \"If you're afraid...\"\n\nRias splayed a hand across his chest. \"When you were learning our language, did you not also learn that we are a fearless people? I simply don't want to take advantage of you. I studied ballistics in school, and I can't imagine such a martial course being taught at your Polytechnic.\"\n\n\"And did you also study arrogance in school?\" Tikaya nocked her arrow, aimed, and plunked it into the bull's-eye.\n\n\"Of course.\" He winked. \"I'm Turgonian.\"\n\nShe snorted. Arrogance probably was part of their curriculum.\n\n\"I even wrote a paper on the ballistics of archery for merit points,\" he said.\n\n\"Less talking, more shooting.\"\n\nEyebrows arched, Rias nocked the first arrow, but he paused as a pair of marines strolled past, one munching the remains of an apple. The man lifted his arm to throw the core overboard, but Rias stopped him.\n\n\"I'll take that.\"\n\nThe marine shrugged and tossed it to him. Tikaya had an inkling of Rias's intent and did not question him as he readied the bow again. He nocked the arrow and held it against the stave with one hand. With the other, he lobbed the apple core so it arced toward the target. In one swift motion, he drew the bow and fired. The arrow pierced the apple and hammered it to the target right next to Tikaya's bull's-eye.\n\n\"Hah.\" Rias lowered the bow. \"After my cocky speech, I feared I'd embarrass myself.\"\n\n\"I hope you got a good grade for that paper,\" Tikaya said, staring at the impaled apple.\n\n\"Me too.\" He grinned at her raised eyebrows. \"It was twenty years ago. I don't remember.\"\n\n\"What was it on?\"\n\n\"Oh, the usual. Explaining the equations for general ballistic trajectory, horizontal launch, launch velocity, and the like. The fun part was the modeling I did on the different types of bows used throughout imperial history. I analyzed them to show how the design and materials used would define their accuracy, trajectories, distance capabilities, and...\" He must have noticed her gaping because he stopped, a sheepish expression on his face. \"I'm boring you, aren't I?\"\n\n\"No!\" Tikaya blurted. \"I just wasn't expecting you to be so...\" Dear ancestors, he sounded just like her when she started talking about languages, and it struck her as hilarious that he could not remember the grade but recalled all the details of the topic. \"Uhm, garrulous,\" she finished.\n\nA blush colored his olive skin. \"Sorry. I haven't talked to a woman in two years.\" He nocked an arrow. \"Shall we shoot a few more?\"\n\n\"Sure, tell me more about your paper.\"\n\nRias's fingers fumbled, and the arrow clattered to the deck. \"Really?\"\n\n\"Really.\" She hid a smile, tickled by his surprise. \"The only ballistics experiments I ever partook in involved a wager on who could use a spoon to launch a macadamia nut across the lunch room and into Professor Lehanae's wig.\"\n\n\"Hm, I recall taking part in a similar experiment. Must be a universal education requirement.\"\n\nThey shot while he explained his paper, and Tikaya relaxed for the first time in days. She almost laughed at her earlier guess that he might have been a captain in the war. With that passion for mathematics and those childhood fancies, he had to be an engineer. Probably the chief engineer on one of the big warships. He would have been accustomed to going toe-to-toe with captains to keep his steamer in pristine operating order.\n\nOnly after the exercise period ended, and she was again confined in her cabin, did she realize she still did not know what his history was with those symbols and why they stirred dark memories.\n\n* * * * *\n\nThe first earsplitting boom yanked Tikaya from sleep. The second made her scramble out of her bunk so quickly she slammed into the foldout desk. Groaning, she rubbed her hip, took another step, and cracked her toe on the stool.\n\n\"No one should wake up this way,\" she muttered.\n\nMore booms drowned her words, this time a whole round that lasted half a minute. The ship trembled with concussions that vibrated her body like a bell. Cannons, she realized, as she groped about to find her spectacles and sandals. She had heard them from afar, but never standing in a cabin under the gun deck. Shouts sounded through the aftermath of the round, though the bulkhead muffled the words. She peered out her tiny porthole. Clouds obscured the stars, and night's darkness smothered the ocean.\n\nWas someone attacking them? Who would be audacious enough to waylay a Turgonian warship? Especially during peacetime? Maybe it was a training exercise. She would not put it past Captain Bocrest to schedule drills in the middle of the night.\n\nA massive jolt rocked the ship, throwing her into the bulkhead. That was no cannon firing. Psi blast. She had a cousin who studied telekinetics and could knock the fronds off a palm tree. Anyone who could damage an ironclad warship was no one she wanted to meet. Still, if this was an attack, maybe she could use the confusion to steal a longboat. She would need to get Rias out of the brig to help. Since he had planned an escape once, he would know where to find the logs and navigation tools they would need.\n\nShouts rang out and boots pounded, but all the activity seemed to be on the deck above. Tikaya opened the door. Her hopes of escape sank when she spotted her guard standing outside.\n\nThe baby-face private noticed her immediately. \"Stay inside, ma'am.\"\n\n\"What's going on?\"\n\n\"I'm not sure yet, but don't worry. It's my duty to protect you.\"\n\nYes, but she did not want him to protect her. She wanted him to go away. Still, his earnest eyes said he took his duty seriously, and she managed a \"thank you.\" Though they had rarely spoken, she had placed him in Corporal Agarik's tiny category of People Who Treated Her Like a Human Being.\n\n\"Can we go see what's happening?\" she asked, though she suspected she knew the answer.\n\n\"No, ma'am. Please close the hatch and wait inside.\" A cabin door slammed open, and a sub-lieutenant struggling to buckle his belt scrambled out, a drunken lurch to his step. He stumbled off, presumably to join the others, and her guard watched wistfully. He did look back at her and add, \"I'll let you know what the commotion is about when I find out.\"\n\nTikaya shut the door and paced her tiny cabin. Her guard wanted to join the action. Maybe she could nudge him that direction. He was young after all. Maybe\u2014\n\nA scream of agony erupted nearby. Tikaya froze. Though muffled, it sounded like it had come from her deck.\n\nShe pressed her ear to the door.\n\nSomewhere down the corridor, a pistol fired, and steel rang. Shrill voices cried out, and her eyes widened. They spoke Nurian, not Turgonian.\n\n\"We should go to the brig!\"\n\n\"I read that marine's thoughts; the woman is up here.\"\n\nHer insides knotted. They were looking for her.\n\nThe Nurians must have found out about whatever the Turgonians had unearthed. Did they want her help too? To beat the imperials to the loot?\n\nA shot cracked right outside her door.\n\nTikaya could help them find her or she could hide. Though her people had only superficially sided with the Nurians during the war, sharing the messages Tikaya decrypted, it ought to be enough to ensure they would treat her better than the Turgonians.\n\nShe eased the door open. The air shimmered as a wave of heat rolled in from the corridor. The invisible force slammed into the private's chest.\n\nShe stumbled back and wrapped her arm over her eyes\u2014not quickly enough to miss the agony contorting the guard's face. He screamed and dropped to the deck, writhing. His sword and pistol clattered down beside him. The stench of charred flesh seared the air. In a heartbeat, the marine lay still, skin laid bare to muscle and bone, and bulging eyes frozen open.\n\nTikaya stared, stunned into immobility. She had never seen death, not like this. She had spent the war in an office, not on battlefields or warships.\n\n\"She down there?\" someone asked in Nurian.\n\nTikaya tore her gaze from the downed man and leaned out the door.\n\nTwo Nurians crept in her direction. They were shorter than she and darker of skin, like the Turgonians, but they had slanted eyes and wore their long black hair in topknots. One bore twin scimitars in his hands and looked lean and sinewy beneath colorful clothing decorated with bone and beads. The other, wrapped in a flowing black robe, carried nothing. A practitioner and his bodyguard.\n\nThe bodyguard pointed a scimitar in her direction. \"There!\"\n\n\"Greetings,\" Tikaya called in their tongue. \"My name is Tikaya. Any chance you're here to take me somewhere more pleasant than a Turgonian warship?\"\n\nBut they were not in the conversational mood. As soon as they spotted her, the practitioner stopped. A glazed trance slackened his face\u2014the sign of someone concentrating on his science. The bodyguard watched her, but also glanced up and down the hall, ensuring no one approached to interrupt his comrade.\n\nDespite the heat lingering in the wardroom, a shiver ran down Tikaya's spine. She ducked into her cabin just as the Nurian lifted his arm. Another wave of energy flared, and the air crackled and wavered against the metal door where her head had been. Heat scorched her face. The hem of her dress burst into flames.\n\nTikaya cursed, flapping the cloth to put out fire.\n\nFootfalls thundered toward her cabin. She glanced around. Nowhere to hide. The fallen private's sword lay in the doorway. She bent and grabbed it.\n\nThe Nurians appeared in the hatchway, shoulder to shoulder. The practitioner's eyes narrowed again. Still crouched on the floor, sword in hand, Tikaya lunged, hoping to surprise them.\n\nShe bowled between them, ramming them with her shoulders. Where she might have bounced off burly Turgonians, her size was an advantage here, and she startled the shorter men. Both Nurians pitched opposite ways and fought for their balance.\n\nTikaya raced through the wardroom and into the corridor.\n\n\"Flay her!\" the bodyguard called. \"She must be killed at all\u2014\"\n\nHer instincts prickled again, and Tikaya threw herself into a clumsy roll. Heat crackled overhead.\n\nShe jumped to her feet. The bodyguard charged after her. She skidded around the corner at a T intersection. A nook right to the side offered access to a ladder running up and down. She banged a rung with her sword, trusting her pursuers to hear it, then jumped through an open hatch a few cabins down. She dared not stick her head out to look, but their footsteps told her when they charged around the corner. The clatter of swords and shoes clanking in the ladder well said they had fallen for the ruse.\n\nTikaya could not relax. If there were two Nurians on the ship, there could be more. Cannons continued to blast on the decks above. Perhaps the whole attack was a cover to let practitioners sneak on board to get rid of her. But why? If the Turgonians had unearthed some treasure they wanted to get at, wouldn't the Nurians find her skills of equal use? Wouldn't they want to use her themselves?\n\n\"The captain said there was no treasure,\" she muttered, rubbing sweat from her eyes. She had not believed him, but maybe that had been a mistake. Something about those symbols alarmed Rias, and it hardly seemed that some cheery treasure hunt from his youth would account for it.\n\nMore footfalls sounded in the corridor, and Tikaya ducked behind the hatch. Two marines pounded past.\n\nShe could not stay there. But where to go? If she headed to the upper deck, she could find the captain. He would protect her from Nurians. He may not like her, but he obviously had orders to keep her alive, at least until she translated the language. Still, going to him and hiding behind his back would eliminate any chance she might have to escape. Better to find Rias and steal a longboat in the commotion.\n\nHer heart lurched. Rias. Locked down in the brig. If the Nurians were after her, might they be after him too? He would have a hard time dodging psi attacks while chained to the deck in a cell.\n\nMore marines raced through, and she made her decision. Tikaya glanced both ways, then slipped out. Hoping to avoid the Nurians, she traveled past two more ladders before climbing down to a dark hold on the bottom deck. She groped her way past sea trunks and cargo. Somewhere nearby, engines hummed. The deck trembled with the strokes of massive pistons, while above the cannons continued to roar.\n\nShe escaped the hold and found the narrow corridor leading to the brig. As she entered, another wave rocked the ship, and the great ironclad pitched sideways. The ship creaked ominously as she stumbled down the passage.\n\nNobody guarded the hatch at the end. The Nurians had mentioned the brig, so she advanced with care. She did not think they would expect her to flee this direction, but sweat still pasted her dress to her back, dribbled down her temples, and slicked the sword grip.\n\nThe single lantern that usually burned on the wall near the cells had moved. It was hanging\u2014no, being held\u2014a foot above the deck near Rias's door.\n\nShe moved closer and, when she spotted him kneeling by the gate, exhaled a breath she did not remember holding. He was still shackled, but the chains that had secured him to the back of the cell lay in a tangled heap next to the broken lantern cover. He held the flame directly beneath the bottom hinge.\n\nRias smiled when he spotted her, but kept his hands steady. Flames bathed the hinge.\n\n\"What are you doing?\" Tikaya asked.\n\nHis lips shifted into a bemused half smile. \"Hoping whoever designed these hinges did not factor in the relatively high coefficient of thermal expansion steel possesses.\"\n\n\"You're going to break out...with the lamp?\"\n\n\"It worked on the chain bolts.\" Rias shrugged. \"I was fortunate the first attack knocked the lantern off the wall and within reach.\"\n\n\"Yes, those foolish marines should never have left such an obvious tool down here for prisoners to exploit.\" Tikaya knelt by the door and slid the sword between the hinge and the pin, which surprised her by popping free with minimal effort.\n\n\"Indeed.\" He stood and moved the flame to the upper hinge. \"By the way, what are you doing down here with a sword?\"\n\n\"There are Nurians on board, at least two. They killed my guard. I thought they might want me as a translator as well and I was ready to jump into their arms when they flung a psi wave at me.\"\n\nRias's brow furrowed. \"Why would the Nurians want to hurt you? They ought to be worshipping your people after the help you gave them in the war.\"\n\nShe watched his face, trying to decide if his tone was accusatory, but only puzzlement furrowed his brows. \"I don't know. I figured they might be after you, too, though I haven't deduced exactly what your part is in all this yet.\" Tikaya wriggled her eyebrows to suggest he might share any time. \"I decided to come break you out.\"\n\nAnother jolt rocked the ship. Tikaya was not sure how long the flame had to be applied to loosen the bond between hinge and pin but wedged her sword into the crack anyway. It was tighter this time.\n\n\"I'm not sure whether to be offended that you'd think Nurian hospitality preferable to Turgonian or tickled that you came down to help me.\" Rias gave her that lopsided smile again and, despite the cannons thundering above and the threat of Nurians lurking below, she bit her lip to hide a pleased grin.\n\n\"I'd think even you would prefer Nurian hospitality to Turgonian at this point.\" She wriggled the sword, and the pin inched upward.\n\n\"Not really. I've been in one of their prison camps and\u2014\"\n\nSomething bowled into her.\n\nTikaya was rammed into the gate and dragged to the ground. The sword flew from her grip. She launched a clumsy fist at her attacker, but encountered no one.\n\n\"Invisibility illusion,\" she barked.\n\nMetal screeched.\n\nShe started to roll to her feet, but her unseen assailant grabbed her hair and yanked her head back. With one hand, she groped for the sword, and with the other clawed for the Nurian. She jerked her head down to protect her neck.\n\nThe cell door flew outward, and metal thudded against flesh. Tikaya's invisible attacker grunted. Rias tore the Nurian off her.\n\nShe scrambled out of the way. A snap echoed through the brig, and an orange-clad man appeared. He collapsed on the deck. Dead.\n\nColor flashed behind Rias.\n\n\"Look out!\" Tikaya cried.\n\nHe whirled as a woman with scimitar and dagger leaped at him, blades leading. He ducked low and barreled toward her legs. The woman jumped and spun in the air to land facing him.\n\nKnowing Rias was still shackled and had no weapons, Tikaya lunged for the dropped sword. She jumped to her feet, hilt clasped in both hands. Even with her inexperience, she figured she could stab someone in the back, but Rias had closed with the Nurian woman. They grappled briefly, then he released her with a shove. She went down, her own dagger protruding from her chest.\n\nWordlessly, Tikaya handed him her sword. He would know how to use it whereas she would probably just trip over it.\n\n\"You all right?\" Rias looked her up and down, brow wrinkled. \"Are these the same ones who attacked you above?\"\n\n\"I'm fine and...no.\"\n\nHe knelt and pulled a pin from the dead woman's hair. \"If they can turn invisible, there's no telling how many are on board.\"\n\n\"Comforting.\" Tikaya watched him probe the lock on his shackles. \"How about we use this chaos to escape or find a good hiding place until the ship gets to port?\"\n\nHis wrist accessories snapped open and dropped to the ground. He pocketed the hairpin. \"Let's head above decks and find out what's going on.\"\n\nTikaya raised her eyebrows. That was not exactly her plan\u2014her plan had a lot more focus on the word escape.\n\n# CHAPTER 5\n\nOn the upper deck, Tikaya and Rias crouched behind a funnel venting hot air from the boiler room. Streaks of lightning branched from the cloudy night sky and lanced around the warship, some sizzling harmlessly against the dark waves and others tearing into mast and sail. Agile as monkeys, marines raced through the rigging, beating out conflagrations. Others perched in the fighting tops, rifles firing intermittently. Tikaya hoped they were too busy to look down and notice her and Rias.\n\nTwo wooden Nurian ships, decks lit by glowing orbs, trailed slightly behind on either side of the ironclad. Every time they edged too close, an officer on the gun deck barked orders to fire the cannons. The same weather phenomenon setting off the lightning filled the enemy sails with unnatural wind, and, despite the steam-powered propeller adding knots to the warship's speed, the Nurian vessels kept pace. In fact, they could have overtaken the warship, and Tikaya had a feeling they were waiting for something. The assassins to kill her? She grimaced.\n\n\"We outman them and outgun them,\" Rias said. \"But wizards always have tricks that make them dangerous. Bocrest has already slagged this, letting them surround us.\"\n\nAt the moment, the invisible assassins were more of a concern for Tikaya. \"Let's go to the training area.\"\n\nRias tore his gaze from the Nurian ships. \"The blades are dulled; real weapons are kept in the armory.\"\n\n\"What I need is over there.\"\n\nShe expected him to question her further, but he simply led her through the shadows. Two marines in the forecastle manned the chaser gun, which was rotated toward one of the Nurian ships and pounded rounds into the night. Open deck lay between the men and the weapons racks, and the intermittent lightning illuminated much.\n\nTikaya crouched low to approach the backside of the racks, and she sensed rather than heard Rias behind her. The booms of the great gun would have drowned out the approach of howler monkeys.\n\nMuch of the exercise gear had been stowed when the ship was cleared for action, and she worried she would not find what she wanted. But, no, there they were. The heavy sand-filled balls sat in the bottom row of a rack.\n\nShe slung one to the deck and found Rias's ear. \"Sword, please.\"\n\nHe handed her the cutlass, and she sliced open the ball. She stuffed sand into the two hip pockets in her dress until they bulged, then returned the blade.\n\nA fiery projectile the size of her cabin slammed into the side of the ironclad. Ineffective against the metal hull, it bounced into the water, but the energy that had hurled it coursed through the air. Tikaya's skin hummed. She had never been so close to so much power.\n\nRias tapped her shoulder and they moved away from the forecastle. With an uncanny knack for avoiding the marines running up and down the deck, he led her past masts, funnels, and vents. They rounded the smokestacks, and he headed toward the bridge. The captain and senior officers relayed orders to the gun deck and barked commands to the men controlling the wheel.\n\nTikaya grabbed Rias's arm. \"Where are you going? We're hiding, remember? And escaping if possible, right?\"\n\nLightning flashed, revealing him gazing toward the Nurian vessels. \"If this ship sinks, we're in trouble too.\"\n\n\"And what would we do to stop that?\"\n\nA long moment passed before he said, \"All right. We can hide between the launches and still see what's\u2014\"\n\nThe aft chaser gun blasted, stealing the rest of his words, but she nodded, and Rias led her through the shadows.\n\nLightning flashed again. Rias ducked between the boats mounted a couple feet above the deck in the center of the ship. The space between them offered a shadowy place to hunker down. The smokestack rose behind them, belching coal plumes and further hemming them in. A determined search would reveal them, but the darkness and chaos offered camouflage\u2014from the marines, anyway. The Nurians had other means of searching for her, but at least lanterns were mounted across the deck from them and would silhouette someone approaching. Assuming that someone wasn't invisible. She touched her bulging dress pocket.\n\nRias put his back against one of the launches and stood where he could see the movement of the other ships.\n\n\"I can hide here alone if you want to find the captain.\" She hated the idea but could tell Rias felt he could do something.\n\n\"No, he wouldn't appreciate my input, and he'd chuck me back in the brig. Besides, the Nurians are looking for you.\"\n\n\"Yes, and I should mention they have ways to find me. They went straight to the wardroom earlier, and I'm sure it wasn't a coincidence they showed up in the brig when I was there.\" Tikaya looked up at Rias, though darkness hid his face. \"It'd probably be a bad idea to be standing next to me if a psi wave is launched in my direction.\"\n\n\"I'll risk it.\" Rias rested a hand on her shoulder. \"Keep your back to me in case they're invisible again.\"\n\nShe sandwiched between him and the other launch, with the smokestack guarding their right side and his sword ready on the left.\n\n\"The sand,\" Rias said, \"is for throwing at the invisible attackers? Will it disrupt the spell?\"\n\n\"Possibly, if I catch them by surprise, and their concentration lapses, but if nothing else it'll outline them for a few seconds until they compensate.\"\n\nHis rumbled, \"Ah,\" sounded pleased.\n\nOn the rear horizon, a third Nurian ship floated into view.\n\n\"Rust,\" Rias spit. \"He needs to take down one of those ships before the reinforcements arrive. Come on, Bocrest. Think. Don't be so stodgy and predictable.\"\n\nA fiery projectile the size of a cannon ball arced toward them. Tikaya tensed. It clipped the yard closest to their smokestack, and shards of wood rained upon them.\n\nShe gulped.\n\n\"You all right?\" Rias dusted splinters off the top of her head.\n\n\"Yes, but it's inconsiderate of these Nurians to muss my hair. I'd at least like to look good when your people toss me on a funeral pyre.\" Her attempt at nonchalance might have worked if her voice had not cracked on the last word. When she had been fleeing the Nurians, she had been too busy to worry about her mortality. Standing here gave her too much time to think, to wonder if she might very well dodge the assassins only to fall to a random cannonball.\n\n\"Don't worry,\" Rias said. \"No funeral pyres at sea. We just wrap your body in your hammock and toss you overboard. Only the fish will judge your hair.\"\n\n\"I'm vastly reassured, thank you.\"\n\nRias chuckled.\n\nOddly, his blas\u00e9 attitude did reassure her. If he was not worried, maybe she did not need to be. She leaned back against him. If not for the guns roaring and the lightning streaking the night, she might have noticed the heat of his chest against her shoulders, the lean hard muscles beneath his clothing, and the gentle breaths stirring her hair. Actually, she noticed them anyway.\n\n\"Rias?\"\n\n\"Yes?\"\n\nHis murmur was soft, close to her ear, and a thrum warmed her body. Focus, she told herself.\n\n\"Do you want to escape or not?\" she asked. \"If you don't... Well, that's your prerogative, but it'd help me to know. I've mentioned it a couple times tonight, and, even though I chanced upon you breaking out of your cell, you seem to be more interested in what's going on with the battle than getting out of here. I can't help but think that it's handy how we're standing next to a couple boats, and the marines are all preoccupied.\"\n\n\"It'd be suicidal to launch a boat into the middle of the Nurians,\" he said. \"Besides, based on the knots-per-hour average of this ship, the days it's been since you were brought on board at the Kyatt Islands, and our northeasterly direction, I estimate us more than a thousand miles from the mainland. There aren't many archipelagos in this part of the ocean. It's likely we'd die of thirst before making land. Also...\"\n\n\"What?\"\n\nHis long exhale tickled the back of her ear. \"The fact that the Nurians are trying to kill you makes me believe we really need you.\"\n\n\"We?\"\n\n\"The empire. Bocrest's family has been personally loyal to the throne for a long time. That Emperor Raumesys picked him over brighter men suggests this is a very sensitive mission. My people may have unearthed something that's put them in danger. If the Nurians have found out, well, they'd be the first to help us on our way to the black eternity.\"\n\nTikaya pressed her hand against the cool wooden siding of the launch, dread curling through her gut for a new reason. If the Turgonian emperor had walked onto her plantation and asked for her help, she would have told him to shove sugar cane into his anal orifice. But Rias asking her to stay and help...\n\nShe shook her head. She hardly knew him. And he was one of them. Surely, she owed him nothing.\n\n\"How can the empire's fate even matter to you?\" she asked. \"After they condemned you and left you to die?\"\n\n\"Strange, isn't it? By the emperor's decree, I'm dead to my family, my friends, everyone I ever knew, but it was the emperor who cast me out, not them. I still care that they are well, and I'm not sure the orchards where I grew up will ever stop being the place my mind conjures when someone says home.\"\n\nTikaya cleared her throat and tried to sound offhand when she asked, \"Family?\"\n\n\"Parents, brothers.\"\n\n\"No children?\" No wife?\n\n\"My wife didn't want them.\"\n\nSo, there was a wife. The intensity of her disappointment surprised her.\n\n\"Ex-wife,\" Rias said, as if reading her thoughts. \"I know you owe nothing to me, Tikaya\u2014in fact, I owe you a couple favors. But if you would stay and decipher the language and help\u2014I can't believe I'm saying this\u2014help Bocrest solve whatever problem my people have gotten themselves into, I'd...\"\n\nThe request she had dreaded. She swallowed and waited.\n\n\"I have nothing I can offer you.\" He sighed. \"Not even my protection since I'm even more a prisoner than you. All I can promise is that I'll do everything possible to ensure you escape and can return to your island afterwards. I imagine you have family you miss, people who are worried about you.\"\n\n\"Yes.\" If she died out here, would anyone even tell her parents what happened?\n\n\"Children?\" he asked in the same offhand tone she had used.\n\n\"No.\" Then, feeling the need to lay everything out, she added, \"My fianc\u00e9 was killed on a science vessel that went down near the end of the war.\"\n\n\"Oh.\" A long beat passed, probably because he did not want to know the answer to the next question, but he asked anyway: \"How did it\u2014who sank it?\"\n\n\"Your people.\"\n\nShe felt his shoulders slump behind her.\n\n\"I'm sorry,\" he said.\n\nA twinge of guilt wound through her; it was not as if he had done it. If he had been on that penal island for two years, he would have missed the last year of the war, the year when things unraveled for the Turgonians and their people stopped paying attention to Kantioch Treaty dictates. Yet she could not bring herself to say it was all right. It wasn't. It never would be.\n\nThe attack had slowed, and Tikaya felt a stirring of hope, but then another set of lights appeared on the inky horizon. Another ship, bringing the total to four. The captains had probably just paused to confer\u2014deciding on a final strategy\u2014through communications practitioners. The attack would resume with all four ships joining in, and even the sturdy ironclad would sink under that assault.\n\n\"How come none of your people know about this shindig going on in the middle of the ocean?\" Tikaya asked.\n\n\"I don't know.\"\n\nA great swirling gust of wind tugged at her dress and whipped loose strands of hair into her mouth. She looked up at the stack. The smoke was not affected, meaning the disturbance was localized.\n\n\"Nurian magic!\" Rias wrapped his arm around her.\n\nA flash of yellow burned Tikaya's eyes, and vertigo washed over her. A final burst of wind railed at her, her stomach dropped, then silence engulfed her.\n\nShe blinked and tried to wipe away the yellow dots swimming before her eyes. Bile churned in her throat, and she forced a swallow. The world came back into focus.\n\nShe was belowdecks, not in the ironclad but in a wooden vessel. She stood in a storage space full of Nurians pointing short bows at her, arrows nocked and drawn back. Crates, barrels, and a number of confusing machines, or perhaps practitioners' contraptions, fenced the large hold. Rias still had his arm around her, and he held the sword out before them, but it did not matter with so many weapons pointed their way. A smug woman in black robes smiled in triumph.\n\n\"There, that's easier,\" she said in Nurian.\n\nShe lifted a finger toward the bowmen and opened her mouth.\n\nTikaya scrabbled for something to say, something to sway the woman from giving the kill order.\n\n\"Don't tell them,\" Rias blurted in Turgonian.\n\nBarely, just barely, Tikaya managed to keep the bewildered expression off her face. The practitioner halted, finger still lifted, and frowned at Rias.\n\n\"I won't,\" Tikaya whispered back, also in Turgonian.\n\n\"They'll torture us if they know what we know,\" he stage whispered.\n\nDid the Nurian understand? None of the expressions on the bowmen's faces had changed, but an assessing mien narrowed the woman's eyes. Yes, she understood, and Rias must be counting on that, trying to pique her interest long enough to have a chance to do something.\n\nTikaya lifted one placating hand and stepped toward the woman. \"I understand you have orders to kill me,\" she said in Nurian as she slipped her other hand into her pocket, \"but I'm sure I can be of more use to you alive.\" She caught the other woman eyeing Rias and added, \"As can he. We've just escaped our cells on the Turgonian ship; we've no allegiance to them\u2014they kidnapped us against our will.\"\n\nThe practitioner seemed to be only half-listening. She stepped closer, peering up at Rias, whose head brushed the ceiling of the hold.\n\n\"You look familiar,\" she said in heavily accented Turgonian. \"Who\u2014\"\n\nTikaya hurled a handful of sand, and the woman gasped, swiping at her eyes. Rias lunged past Tikaya, pushing her to the deck. His body coiled, then he sprang, whipping the sword through the practitioner's neck with a grunt.\n\nHe landed and charged, taking advantage of the startled silence gripping the hold.\n\nFor a stunned moment, Tikaya lay on her belly, staring at the decapitated head, the still-twitching body, and the blood. So much blood.\n\nThe Nurians recovered, and bows twanged. An arrow grazed Tikaya's arm and pinned her sleeve to the deck.\n\n\"Move,\" Rias barked. \"Find cover.\" He was already attacking a third man.\n\nYes, cover, of course.\n\nTikaya tore her sleeve free and rolled toward the closest set of legs. An arrow thudded into the deck an inch from her ear. She kicked as hard as she could, and her heel smashed the inside of a man's knee. He yelped and collapsed on her.\n\nHer first instinct was to shove him away, but another arrow slammed into the deck near her head. She tried to stay under him, to use him as a human shield. He drew back to punch her. An arrow lodged in his shoulder.\n\n\"Not me, idiot!\" he screamed.\n\nHe thrashed, still on top of Tikaya as he clawed at the shaft. A wayward elbow nearly tore her spectacles from her face. His frustrated cries of pain reverberated in her ears. His face, eyes squeezed shut, mouth contorted with agony, loomed inches from her own. Fearful of more bows aimed at her, she wrapped her fingers into his shirt and kept him from pulling away.\n\nRias towered over the Nurians, head brushing the ceiling beams as he lunged about the space. He slashed bowstrings and pounded through the startled archers, who\u2014after catching their own comrades in the crossfire\u2014were dropping their bows in favor of short swords and cutlasses. Howls of pain and rage bounced from the wooden walls.\n\nThe Nurians stopped shooting at Tikaya and focused their attacks on Rias, obviously finding him the greater threat. She spotted a bow within reach and grabbed it. The wounded man writhed, still trying to pull out the arrow, and she rolled away from him. She pried a quiver off a fallen archer and ducked behind a chest-high wooden contraption bolted to the deck.\n\nHer hands shook, and it took three tries to nock the first arrow. She willed her fingers to still. She could do this. Human beings or not, they were trying to kill her.\n\nHer first arrow went a foot wide of its mark, thunking into the frame by the hold's only exit. She wiped hands wet with blood and sweat on her dress and sucked in a deep breath.\n\nA man with a raised hatchet drew up behind Rias as he squared off with two cutlass wielders.\n\nTikaya's nerves disappeared and she let an arrow fly. It struck the attacker between the shoulder blades, and he pitched forward, crashing to the deck at Rias's heels. The man lay still. She had found his heart.\n\nShe swallowed, mortified by the results of her reflexive act. Rias slashed his cutlass through the throat of the last man standing before him, glanced behind at the dead Nurian, and saluted her with the sword.\n\nThe rest of their attackers were down as well. Dead. A tremor coursed through Tikaya's body. She could not rip her gaze from the one she killed.\n\nMuffled shouts echoed from deeper in the ship. Reinforcements who had heard the skirmish. Her mind processed what it meant, that more would soon burst in, that she would have to fight, but the tremor returned to her hands, and she shied away from the idea of shooting anyone else.\n\nShe was not a killer. If her family knew what she had done...\n\nRias stepped before her, blocking the view of the body and breaking her thoughts. He gripped her shoulder with a bloodstained hand. She swallowed and met his eyes.\n\n\"More work to do before we're safe,\" he said, voice calm and steady, commanding her attention. \"You can react later. Right now, I need you to watch my back so we can live through this. Concentrate on that, nothing else, understood?\"\n\nBefore she could nod, four men burst into the hold, swords leading. A flash of silver streaked toward Rias's head. He jerked back, and it split the air between them to land with a thunk in the wood wall. A throwing knife.\n\nRias leaped away from Tikaya and charged the Nurians.\n\n\"Curse me.\" She tore an arrow from her quiver. He had almost been killed because he was trying to keep her from falling apart. She nocked the arrow, forced her hands to still. React later. Yes. She could do that.\n\nRias led the Nurians about the hold, dodging behind crates and apparatuses, slashing to keep the men at bay, and evading their attempts to surround him. With agility surprising in someone so large, he kept them in each other's way and remained on the outskirts so he only had to face one at a time. More, he kept them from paying attention to her.\n\nGood.\n\nTikaya lifted the drawn bow and selected the man farthest from Rias. She was not going to be the idiot who shot someone on her own side. The arrow took the Nurian in his chest, and he lurched backward, hands clutching the shaft. Horror and pain wrenched his face. Her own heart twisted in sympathy, but she smashed down the emotion. React later.\n\nHer next arrow felled a second man even as Rias sliced the throat of the third. The fourth skidded to a stop, realizing he fought alone.\n\nHe backpedaled for the exit, and Tikaya had him targeted, but she hesitated. Even if he meant to run straight to his captain, how could she shoot someone fleeing?\n\nRias lunged after him, and the man jumped back. His heel caught on a downed comrade, and he pitched to the deck, cracking his head on a crate.\n\nRias dropped beside the man, gripping his throat, and Tikaya winced and looked away.\n\n\"Live or die?\" Rias asked in accented Nurian.\n\nSurprised, Tikaya looked back.\n\n\"Live?\" the Nurian croaked, eyes darting with fear, as if he did not expect to be that lucky.\n\nRias glanced toward the door, then laid his sword on the ground while he tore pieces from the man's colorful clothing. With quick efficiency, he gagged the Nurian and started on ankle and wrist bonds.\n\n\"Who would answer with die?\" Tikaya asked.\n\n\"Most of my people,\" Rias said. \"To live when the rest of your team died would be an unacceptable disgrace to many.\"\n\nWith some vague sense that someone should be standing guard, she stepped over the bodies to watch the exit. Another hold stretched before her, lit by glowing orbs hanging from the beamed ceiling. No one else waited to charge.\n\nRias finished the bonds, leaving the Nurian wide-eyed on the deck, and snatched arrows from partially spent quivers. When he had a fistful, he joined Tikaya.\n\n\"I want to take control of the ship,\" he said.\n\n\"Take control?\" She gaped at the audacity. Surely, the best they could manage would be to run for the upper deck and leap over the side. But, no, who would find them in the cold, dark waters? Even if the Turgonians spotted them, and that was unlikely, they had their own troubles.\n\n\"We'll have another fight when they realize what's going on.\" He held out the arrows, enough to stuff her quiver, and watched her face. \"You've got my back?\"\n\nShe guessed at what he was really asking: can you, a philologist from an island full of peace-loving academics, keep from collapsing in a weepy heap when I need your help?\n\nTikaya grabbed the arrows and jammed them into the quiver, angry with herself for that weak moment that made him question her. \"I've had it so far, haven't I?\"\n\n\"Yes.\" Rias gripped her forearm. \"You've been magnificent.\"\n\nShe snorted. Right. He didn't know how lucky he was her trembling fingers hadn't loosed an arrow that turned him into a eunuch. \"Will you still think that if I insist on taking a side trip?\"\n\n\"What?\"\n\n\"I want to search the captain's cabin for orders and find out why these people are trying to kill me.\" And maybe she could finally get answers about what this secret Turgonian mission was all about.\n\n\"We may not have time,\" Rias said.\n\nTikaya lifted her chin. \"We'll make time.\"\n\nHis eyebrows flicked upward, but the surprise lasted only a second. He nodded once and gave her a Turgonian salute, a fist thumped over his heart. \"Yes, ma'am.\"\n\n\"Jeela, is it done?\" a tinny voice asked from the center of the room.\n\nAs one they stared at the dead practitioner. The voice emanated from within her black robe.\n\nRias pointed his cutlass. \"Can you answer that?\"\n\n\"Uhm.\" Tikaya knelt by the dead woman, trying not to look at the bloody stump where the head should have been, and patted the blood-sodden robe. She found a glowing opal pendant, the chain broken, just as the voice spoke again.\n\n\"Is the Kyattese girl dead? Jeela, do you need help?\"\n\nThough her education in the mental sciences was not ecumenical, she could sense the soft hum of a practitioner-made device. She held it up to her mouth, then waved at Rias and mouthed, \"Make some noise.\"\n\nShe partially covered the device with her hand, hoping to disguise the fact that her voice would not match the practitioner's, and spoke in Nurian: \"Yes, mission complete. She's dead.\" Then, fearing further conversation would only hurt her chances at pulling off the ruse, she dropped the device on the deck, so it clanked against the wood.\n\n\"I doubt that fooled anyone,\" she apologized to Rias as they exited.\n\nThe grim cast to his eyes suggested he agreed, but all he said was, \"We'd best hurry.\"\n\n# CHAPTER 6\n\nThe Nurian captain's cabin offered a distinct contrast to Bocrest's quarters. Behind a desk painted with flowers and vines, lace curtains decorated a bank of windows. Velvet furniture and lush rugs covering the deck might have invited one to lounge, but the cannons booming in the distance suppressed the cozy parlor ambiance.\n\nTikaya and Rias slid inside, shutting the door behind them. For the moment, the Nurians were busy attacking\u2014and defending against\u2014Bocrest's warship, but sooner or later someone would figure out \"Jeela\" had failed her mission.\n\n\"Check those trunks.\" Rias jogged around the desk to the windows. \"Let me know if you can tell if the captain is a wizard or not. If he is, he'll likely have wards protecting his orders.\"\n\nTikaya threw open the trunks and lifted a sword and a lacy brassiere. \"I believe she's a warrior.\"\n\n\"Should be safe to search then.\" Rias tore his gaze from the windows and cocked an eyebrow at the lingerie. \"Unless you want to model that for me first?\"\n\nStartled, she dropped the sword. The hilt banged onto her sandaled foot.\n\nRias winced and lifted an apologetic hand. \"Sorry, I, er, two years, you know.\"\n\n\"It's fine.\" Cheeks warming, she threw the sword back in the trunk, relieved she had not cut off any toes. \"I'll just, uhm, find those orders now.\"\n\nTikaya yanked open a desk drawer and rummaged through letters and supply receipts. Out of the corner of her eye, she caught Rias shaking his head, fingers splayed across his face, before he turned back to the window. A grin tugged at her lips.\n\nA moment later, he found his fearless-soldier-in-charge tone and reported: \"All four Nurian ships are even with the Emperor's Fist now, two on each side. Bocrest is doing damage, but...if we're going to help, it'll have to be soon.\"\n\nTikaya tried another drawer. She wanted those orders, and she wanted them to say something significant to justify detouring here. Rias helped her search, checking cupboards under the bunk, but she sensed restless energy emanating from him. He wanted to assist his people, though she could not imagine how he planned to take over the ship.\n\nUnder a pile of log books in the bottom drawer, she found a parchment displaying lines of gibberish. \"Got it. Encrypted though.\" She tapped the nonsensical Nurian letters. \"Given enough time, I can work it out, but it'd be helpful to have the key. The captain ought to have it, right?\"\n\n\"Yes.\" Rias joined her at the desk and opened and closed all the drawers.\n\n\"I already looked in there.\"\n\nHe paused at the lower one, yanked it clear, dumped the contents on the deck, and ripped out the bottom. His vandalism revealed a secret compartment from which he plucked another sheet of parchment.\n\n\"Guess my looking skills need improvement,\" Tikaya murmured.\n\n\"I get suspicious when inside dimensions don't match outside ones.\"\n\n\"Ah.\" She laid both sheets on the desk and quickly memorized the key.\n\nThe clanging of a bell echoed through the ship. More footsteps pounded, this time on their deck instead of above.\n\n\"Alarm,\" Rias said. \"They know we escaped. Take that with us. We're out of time.\"\n\n\"Wait, I've got it.\"\n\n\"Already? How could you...\"\n\nShe skipped the introduction and translated the meat of the orders: \"'Search and destroy the Emperor's Fist before it reaches the Northern Frontier. If any artifacts with strange symbols are found, sink them in the ocean. Use extreme caution in handling them. Do not bring them home and do not try to destroy them.'\"\n\n\"Honored ancestors,\" Rias murmured. \"What have my people uncovered?\"\n\n\"'In addition,'\" Tikaya finished grimly, \"'the Kyattese linguist allied with the Turgonians must be killed at all costs.'\" Allied? She was no cursed Turgonian ally.\n\nThe windows exploded.\n\nRias tore Tikaya off her feet before she knew what was happening. Wood cracked louder than thunder. Rias came down on top of her, protecting her with his body. Glass and splinters rained about them, tinkling as they hit the deck.\n\n\"What was that?\" Tikaya asked when her heart left throat. Wind whistled into the cabin.\n\nRias pulled her up. He nodded to a cannonball lodged in the bulkhead perpendicular to the broken window. \"Friendly fire.\"\n\nShe gulped and plucked a shard of glass out of the side of his neck. \"Glad your reflexes are faster than mine. Thank you.\"\n\n\"Welcome.\" He shook more glass from his jacket, then headed for the door. \"Still got my back?\"\n\n\"Of course.\" Tikaya grabbed her bow.\n\nThey had reached the captain's cabin without trouble, but, with the alarm clanging, search parties clogged their deck. Fortunately, Rias seemed to know the layout of the Nurian vessel as well as the Turgonian ironclad. They hid in cabins and shadowy nooks to avoid men before slipping down a ladder to the deck below.\n\n\"How're we taking over the ship from down here?\" Tikaya whispered, neck bent to keep from clunking her head on the ceiling.\n\nRias's shoulders brushed the walls as they crept single-file down a dim passageway. \"This is a Nurian striker. Not a big vessel. I think I can handle the tiller by myself. It should be located...there.\"\n\nHe pointed at a door marking the end of the corridor. He jogged past a ladder well and charged inside, cutlass leading.\n\nAs Tikaya passed the ladder, movement stirred the shadows. A woman dropped from above, legs swinging out to wrap around Tikaya.\n\n\"Rias!\" she called.\n\nSteel rang out in the tiller room. He was busy.\n\nThe Nurian tried to pull Tikaya into the ladder well with powerful legs. For a woman, she had surprising bulk and muscle. Tikaya spread her stance and braced herself against the wall. She tried to maneuver her bow to prod the woman loose from the rungs, but it proved too unwieldy for the tight passage.\n\nThe Nurian woman released the ladder and threw her arms around Tikaya. The momentum slammed Tikaya back into the wall. A second form dropped into view in the ladder well\u2014a black-robed man.\n\n\"Who's got my back?\" Tikaya cried as the woman plucked a dagger from between her teeth.\n\nShe released the bow and tried to knock the blade away. Sharp steel bit into her arm.\n\nThe practitioner hanging on the ladder narrowed his eyes in concentration. The female fighter clung to Tikaya with one hand and raised the dagger again with the other.\n\nTikaya bit the arm wrapped around her shoulders. The woman hissed and her grip softened. Tikaya pushed off the wall and tried to shove her foe into the ladder well. The move jostled the practitioner. He cursed, his concentration disturbed, but the woman stuck to Tikaya like a tick. She raised her knife again.\n\nA hand caught the Nurian's wrist, and Rias yanked her away. Tikaya stumbled and went down. Arrows spilled from her quiver.\n\nThe practitioner leapt on top of her, a dagger held aloft. Tikaya grabbed an arrow and rammed it into his gut. Luckily, it was the pointy end.\n\nEyes bulging, the practitioner reeled back. He dropped the dagger and clutched the arrow in his belly.\n\nBefore Tikaya could decide if she was safe, Rias loomed behind the practitioner. He wound up and swept the cutlass through flesh, muscle, and bone. The Nurian's head fell onto Tikaya.\n\n\"Errkt!\" She shoved it off and scrambled away. Panting, she pressed a hand against the wall for support.\n\n\"I've got your back.\" Rias raked her with his gaze. \"Are you injured?\"\n\n\"Not...severely,\" she said numbly, staring at the decapitated practitioner. \"How\u2014why do you do that?\" It came out more accusatory than she meant. Or maybe not. He had just saved her life\u2014again\u2014and she did not want to sound ungrateful, but, damn, it was chilling when the man on her side was more fearsome than those trying to murder her.\n\nRias turned her away from the decapitated practitioner and nodded toward the tiller room. \"I've seen too many wizards I thought dead heal themselves and later come back after my men. As to how...\" He ducked low to enter. \"If you're ever in the imperial capital's war library, look up Applications of the Kinetic Chain Principle in Close Combat. I wrote it for Lord General Micacrest during my final year of studies, and parts are now used by the military training academies. Not scintillating reading, I'll admit, but it covers everything from breaking boards with a punch to\u2014\"\n\n\"Beheading people?\" She trailed him inside, also ducking for the hatch.\n\n\"That's not listed in the table of contents, but, essentially, yes.\"\n\nA pair of glowing orbs in sconces by the door illuminated the interior, though even without them Tikaya would have noticed the matching ragged holes adorning the exterior walls of the wedge-shaped compartment. A cannonball had gone straight through, leaving uneven gaps more than two feet in diameter. Wind shrieked, and water splattered the deck, pooling and running with the rocking of the ship.\n\n\"That doesn't look good,\" she muttered, before noticing a dead warrior on the deck, short sword still clutched in his grip.\n\n\"Actually...\" Rias shut the door and peered out both gaps. He lingered on one side and kicked out a few broken boards to enlarge it. \"It's fortuitous since there aren't portholes in here. There's the other Nurian vessel on this flank, and I see the Fist's smokestacks beyond it.\"\n\nHe strode to one of the block and tackles stretched from either side of the long metal tiller. They allowed manual access, though control ropes disappeared through the ceiling to connect to the wheel on the upper deck.\n\nRias grabbed one of the ropes and readied his cutlass. \"They'll know right away they've lost wheel control, and half the crew will probably charge down here.\"\n\n\"I see, and how will we stop them from killing us?\"\n\n\"Let me know when you figure it out.\" At odds with the seriousness of the situation, a mischievous glint warmed his eyes. \"It's going to take all my strength to man the tiller.\"\n\nHe sliced through the control ropes even as she blurted, \"You're crazy!\"\n\nRias unhooked the end of the rope on the starboard block and tackle, glanced at measurements on the wall above the tiller, and sank into a low stance to pull. Inch by inch the great lever shifted, and the ship leaned, cutting across the waves in a new direction.\n\nTikaya hunted for something to block the door that she would surely be defending in a moment. Alas, there was no convenient beam for barring it shut\u2014probably so people could not do what they were attempting.\n\nShe pushed a trunk full of spare rope to the door. Forcing queasiness aside in favor of practicality, she muscled the dead Nurian's body on top of it to add weight.\n\n\"Where exactly are you steering us?\" she panted.\n\nRias was a statue, leaning back, arms extended, fingers wrapped around the rope, tendons taut with the strain, but he grinned at her nonetheless. \"The closest Nurian ship.\"\n\n\"Oh, dear.\"\n\nA fist hammered at the door.\n\n\"The Turgonians cut the ropes,\" Tikaya yelled in Nurian. \"We're taking care of it. They ran to the hold!\"\n\nA long pause answered her, and, for a moment, she thought they might believe her. Then synchronized thuds struck the door.\n\n\"A nice try,\" Rias said, and she wondered how much Nurian he understood.\n\nThe chest skidded with each strike. She shoved it back in between blows.\n\n\"Get a ram,\" someone yelled.\n\n\"Better ready that bow,\" Rias said.\n\n\"If we're successful in crashing this ship, how are we getting out of here?\" Tikaya asked.\n\nRias nodded toward the cannonball holes. \"Hope you can swim.\"\n\nShe groaned.\n\nFor a moment, the thumps at the door stopped. Tikaya abandoned the chest and looked out the hole. They had halved the distance between themselves and the other Nurian ship, where a fire burned on the deck. People were scurrying to put it out.\n\n\"Are they tacking?\" Rias asked. \"Do I need to make adjustments?\"\n\n\"Not yet. You're dead on, and they're busy. Not sure they've figured things out yet.\"\n\n\"Let's hope.\"\n\nThe hairs rose on the back of Tikaya's neck. Before she could shout a warning, a wave of power surged at the door. The trunk and body were flung into the room.\n\nWhile nocking an arrow, Tikaya tried to shut the door with her shoulder. Warped hinges kept it from closing fully, and someone thrust it wide.\n\nShe jumped around and fired the bow, point blank, into the lead man's chest. Shocked eyes launched an accusation at her. She forced aside guilt and kicked him into others trying to surge forward. While they struggled to get around their dying comrade, she targeted a practitioner in the corridor behind them. Her arrow sailed over the heads of men shorter than she, but bounced harmlessly off an invisible shield. The practitioner never flinched.\n\nThe Nurians cleared the fallen man away, and their renewed push demanded Tikaya's attention. The corridor and door were too narrow for more than one to attack at once, but the seconds it took to nock and aim arrows let them push her back.\n\n\"Rias! I can't\u2014\"\n\nThen he was there at her side, the slashing cutlass a wall of steel guarding the doorway. He had tied the rope to the other block and tackle. The lever wavered with the rocking of the ship, but hopefully they were close enough now that their course was inevitable.\n\n\"Get in there, you fools!\" the practitioner shouted. \"We're on a crash course!\"\n\nAn arrow clipped the doorjamb and whizzed past Tikaya's head. Every time she found the opportunity, she shot around Rias, peppering their attackers. Her supply of arrows dwindled.\n\n\"This is madness,\" she yelled over the clamor.\n\n\"Yes!\" Rias grinned at her, as if he loved every second.\n\nA gifted swordsman made it to the front. Blade a blur, he forced Rias back.\n\nMetal screeched in Tikaya's ears. She drew the bow, hoping for a clear shot. Two men slipped in behind the swordsman. Tikaya shot one, but more piled inside.\n\nA thunderous crash buried the din, and the ship lurched and tilted on its side. Men scrambled and fell over each other, sliding toward the lower wall. Tikaya tumbled into Rias, but he grabbed the jamb and kept them from falling. Even in the stern of the ship, the cracks of wood breaking against wood were audible. Water gushed in from one of the cannonball holes, which was now submerged. Men flailed and floundered, struggling to get back to the door.\n\n\"I can't swim!\" someone yelled.\n\n\"Time to go,\" Rias said.\n\nTikaya grabbed one of the glowing orbs from a sconce before he pushed her toward the upper wall. They had to pull their way along the block and tackle to reach the escape hole. Though the orb hampered her, she refused to release it.\n\nFinally, with Rias's help, she clawed her way through the hole. The ragged wood tore a new gash in her beleaguered dress, but she wriggled free and slid down the hull into frigid black water.\n\nThe icy shock stole her breath. Salt stung her wounds, and she almost dropped the orb.\n\nRias plunged in beside her, spraying water.\n\nThe Nurian striker had rammed into the side of its sister ship, and water gushed into a great hole in the hull. Fire still burned on the deck, lighting up the night. Timber, from splinters to broken beams, littered the water.\n\n\"This way.\" Rias swam away from the ships, pushing the large pieces of wood out of the way.\n\n\"You sure you don't want to stay?\" She was already swimming, side-stroking with the orb clutched against her hip. \"You seemed to enjoy having people trying to kill you.\"\n\n\"You seemed to enjoy it less.\"\n\n\"Probably\u2014\" she spit icy salt water out of her mouth, \"\u2014an acquired taste.\"\n\nThey paddled away from the ships, rising and falling with the waves. Both vessels burned now and flames crawled up the sails of one. Neither would trouble the Turgonians again that night. As they swam out of the shadow of the Nurian vessels, the ironclad came into view. Only one of the two ships on its opposite flank remained, and both masts had been toppled, so it was falling behind. Tikaya and Rias, too, were falling behind. Her chest tightened at the idea of being left in the middle of the sea.\n\n\"Hope they see this.\" Tikaya lifted the glowing orb overhead, waving it in the air.\n\n\"Me too,\" Rias said.\n\nThe lookout in the crow's nest shouted something down to the deck. Tikaya's teeth chattered, and it felt as if hours, not minutes, passed before the warship dropped a boat.\n\n\"It's fortunate you're here,\" Rias said, bumping her arm as they treaded water. \"I doubt they would have bothered coming for me.\"\n\n\"Not sure how fortunate I feel about going back to the Turgonians.\" Tikaya swiped water out of her eyes and grimaced at the cold drops tunneling into her ears. \"I guess it's better being wanted than being wanted dead.\"\n\n\"Prevailing opinions agree with that sentiment.\"\n\nOars lifted and dipped as the craft neared. Lanterns at either end provided light, and Tikaya spotted Agarik leading the rowers. She smiled a bit, glad he had survived the chaos. He gazed at Rias with a wide-eyed, openmouthed stare of adulation and helped him out of the water first. She tried not to feel a twinge of envy. She had helped after all. At least Agarik managed to notice her second and gave her a hand into the boat. She collapsed on an empty bench between rows of burly, young oarsmen.\n\n\"Turn this dinghy around,\" Agarik yelled, and the men set to work.\n\nTikaya wrapped her arms around herself. The breeze needled her soaking dress, cold water dripped from her hair, and she had lost her sandals in the fall so the puddles on the bottom chilled her feet.\n\nRias settled on the bench next to her, and she pressed closer than she normally would have. Shivers coursed through her body. He put his arm around her, though he must have been just as cold and miserable. Their proximity caused raised eyebrows and significant looks between the marines. Agarik's jaw tensed.\n\n\"Here, sir.\" A marine handed Rias a blanket.\n\nThe use of the honorific made Agarik give the man a sharp look, though Tikaya was not sure if it was quelling or curious. Rias draped the blanket over his and her shoulders.\n\nOn the short ride back, the marines peppered him with questions. How had he gotten out of his cell? How had he and Tikaya gotten aboard the Nurian craft? Had they seen the Nurians on board their ship? Did they know what they wanted? Apparently, the Turgonian chain of command meant nobody not commanding had a clue was happening.\n\nTikaya thought Rias might share the story, but, back in the presence of the marines, he grew reserved and quiet. Was this the real man or had she glimpsed that person on the Nurian ship? Or neither? She liked the amiable fellow she had chatted with while target shooting best, though she suspected she could grow accustomed to the soldier she had seen tonight too. Not that it mattered. Certainly, she appreciated his help, but it was not as if she was going to develop feelings for some ex-officer from the military that had tried to take over her islands.\n\nStill, when their knuckles bumped beneath the blanket, she gripped his hand.\n\n\"Thank you,\" she whispered, wanting to say more, but their hulking male onlookers stilled her tongue.\n\nRias smiled and squeezed her hand.\n\nBack on the warship, Captain Bocrest waited, arms folded across his chest, a scowl accompanying his usual glare. Tikaya had not expected gratitude from the man, but the anger radiating from him surprised her.\n\nAs soon as Rias came over the railing behind her, that anger found an outlet.\n\n\"How could you make such an idiotic decision?\" Bocrest snapped.\n\n\"He didn't do anything wrong,\" Tikaya said. \"The Nurians teleported us to their ship. What was he supposed to do?\"\n\nThe captain did not spare her a glance. His glare stayed pinned to Rias.\n\n\"What are you talking about, Bocrest?\" Rias asked.\n\n\"You know what I'm talking about.\" The captain jerked his hand at a squad of marines standing by with pistols. \"Take him back to his cell.\"\n\n\"Wait.\" Rias lifted a hand. \"Did you find the assassins?\"\n\n\"The dead men in the brig? Yes.\"\n\n\"No.\" Rias gave Tikaya a concerned frown. \"There are two others, at least, who can skulk about invisible.\"\n\n\"They killed the man guarding my cabin,\" she said.\n\n\"We'll find them,\" Bocrest said.\n\n\"I can help,\" Rias said.\n\nBocrest scowled again. \"You can go to your slagging cell and stay there this time.\"\n\n\"Captain.\" Rias stepped forward, staring down at Bocrest. \"The Nurians want Tikaya dead and are making great sacrifices to ensure that happens.\"\n\n\"I'm aware of that.\" Bocrest did not back off, nor shrink away from Rias's glare. \"I have orders to keep her alive until she decodes the runes, and I'll do that.\"\n\n\"She'd be dead now if she hadn't escaped on her own. You already botched your orders.\"\n\nAfraid he would land himself in irrevocable trouble for her sake, Tikaya grabbed Rias's arm and tried to pull him away.\n\n\"I botched my orders?\" Bocrest yelled, fists clenched. \"If you hadn't screwed up two years ago, you could\u2014\" He cut himself off with an audible snapping shut of his jaw, and Tikaya sensed the 'idiotic decision' he accused Rias of had less to do with this night and more to do with whatever had landed Rias on Krychek Island. Bocrest glared around at the watching marines. \"You men have duties,\" he roared. \"Get this ship repaired. Now!\"\n\nMen sprinted from his wrath, leaving only Rias, Tikaya, Agarik, and the guards waiting to escort their prisoner below.\n\n\"Let me stay with her until the assassins are found,\" Rias said, as if he had not heard the captain's outburst. \"Or stand guard outside her door. I've tangled with enough wizards to survive them.\"\n\n\"You're not her bodyguard, you're our guide. I thought I explained that to you when you were taking swings at me.\"\n\n\"A job for which you don't need me until we arrive at the tunnels,\" Rias said.\n\nTikaya's ears perked. Tunnels? Was that where the rubbings had come from? She still needed Rias to explain his history with the runes.\n\n\"No,\" Bocrest said. \"You're a prisoner. You don't get your way.\"\n\nTikaya still gripped Rias's arm, and she could feel the tension in the knotted muscles beneath the damp sleeve. Though she hated seeing him angry, especially on her behalf, she had to wonder how much more might be revealed if she simply stood quiet and listened.\n\n\"Bocrest...\" Rias tried again.\n\n\"Go. To. Your. Cell.\" The captain jerked his arm to wave the guards forward.\n\nRias tensed and dropped into a fighting crouch. He had not noticed when she grabbed his arm, so Tikaya stepped in front of him and planted two hands on his chest.\n\n\"Don't.\" She gazed into his eyes and made herself smile, though, she would have preferred Rias stay by her side too. \"I'll be fine. You won't accomplish anything by getting beaten up.\"\n\nHe closed his eyes, seemed to struggle for his calm, and finally sighed, a deep long exhalation. \"Be careful.\"\n\n\"I will.\"\n\nTikaya watched glumly as the guards surrounded him.\n\n\"This way, sir,\" one said.\n\nBocrest's head jerked up. \"Don't you 'sir' him. He's Prisoner Five, and that's it.\"\n\nThe guard gulped. \"Yes, captain.\"\n\nHead lowered, Rias offered no reaction to the terse conversation. Surrounded, he trooped belowdecks. Bocrest stalked in the opposite direction, grinding his teeth.\n\n\"Ready to go back to your cabin, ma'am?\" Agarik asked.\n\nShe shook her head but followed him. \"Who is he, Corporal?\" She had asked the question before, and Agarik had not known, but that was the second time someone sir'd him that night. Maybe it was out respect for what they had done aboard the Nurian ship, but somehow she doubted it. She wagered that shave and haircut had made him recognizable, at least to some.\n\n\"I wish I knew.\" Agarik led her down a ship's ladder. \"It seems like he must be an officer at least, someone who fought during the war. But I fought as well, and I don't remember hearing about anyone court-martialed and exiled to Krychek.\" They threaded through the wardroom, where furniture had toppled and slid against the wall, and stopped at her cabin. \"He hasn't told you?\"\n\n\"Just to call him Rias. Does that mean anything to you?\"\n\nThe corporal's expression grew thoughtful, but eventually he shook his head. \"No.\"\n\nTikaya stepped into her cabin. Thankfully, the bodies had been cleared, though a few bloodstains smudged the deck.\n\nBefore Agarik could close the door, she leaned back out, remembering something. \"He did say...\"\n\nAgarik paused, eyes questioning.\n\n\"If I was ever at the war library in your capital I should look up a book called Applications of the Kinetic Chain Principle in Close Combat, because he wrote it.\"\n\nAgarik froze. Utterly and completely. His mouth hung open, and he stared at her for a long moment before recovering. \"I see. Thank you.\"\n\n\"Wait.\" Tikaya raised a hand as he started away. \"You know, don't you? Is he somebody I would have heard of?\"\n\n\"I don't\u2014I can't. I'm not sure. I\u2014\"\n\nA lieutenant passed through the wardroom on the way to his cabin, and he frowned at Agarik.\n\n\"I have to go.\" Agarik chopped a wave.\n\n\"Could you at least have someone bring me a towel?\" Tikaya called to his receding back.\n\n* * * * *\n\nAfter dripping a puddle of water onto the cabin floor, Tikaya wondered if she should take off her dress and dry herself with the blanket on her bunk. What were the odds the Turgonians would supply her with a change of clothing at some point? She plucked at the damp dress. At least the sea had washed out most of the blood.\n\nWhen she reached for the blanket, her gaze fell across the desk. It was empty.\n\nThe rubbings, her notes, and the reference books Bocrest had provided were missing. She searched the tiny cabin, thinking they might have been knocked off during the scramble, but no. They were gone.\n\nA shiver ran through her that had nothing to do with the wet dress. The assassins must have returned and taken them.\n\nTikaya eyed the corners of the cabin, all too aware that they could be right in front of her and she would not know it.\n\nShe opened the door, wondering if a new guard had been posted or if she could leave and find the captain. Sergeant Ottotark leaned against the wall outside, and she did not manage to hide her groan.\n\nBriefly, he met her eyes, offering a hostile glare, but his gaze inevitably drifted downward. She shifted to the side to stand in the shadow of the door.\n\n\"The rubbings are missing,\" Tikaya said. \"I think they stole them\u2014the Nurians who attacked me in my cabin and killed the young man standing guard.\"\n\nOttotark's face frosted at the mention of the dead marine.\n\n\"Can you tell Bocrest?\" she asked.\n\n\"The captain is busy directing repairs, cleanup, and funeral services, thanks to the flotilla of Nurian ships that showed up tonight looking for you.\"\n\n\"While I'm sympathetic to your lost men\u2014\"\n\nHe snorted.\n\n\"\u2014you people kidnapped me,\" she continued. \"I never wanted to be here, so don't blame that attack on me. If you could just tell the captain I'm not able to continue my studies unless he finds\u2014\"\n\nThe sergeant stepped forward, shoving the door further open. \"I'm not your messenger boy.\"\n\nShe stumbled back, glancing around for something to use as a weapon if she needed to fend him off. The sparse cabin offered nothing.\n\n\"You'd do best to remember you're a prisoner here. Prisoners have no right to the captain's time, nor to an officer's cabin with a busy sergeant as your guard, a busy sergeant who's stuck on this duty because your presence here got one of his men killed.\" His low voice was gravelly, and tendons strained against the skin of his thick neck. \"You haven't done anything useful since you got here.\"\n\nTikaya wanted to defend herself\u2014she had helped Rias crash the ship that had allowed the Turgonians to sail away, hadn't she?\u2014but Ottotark seemed to want her to argue, to incite his anger. He stepped closer, and she eased back until her calves bumped the bunk.\n\nRage boiled in the sergeant's dark eyes, but lust too. He had not looked at her face since she first opened the door. \"The captain ought to chain you to that bunk and let you be of some use to the crew.\"\n\nA throat cleared in the corridor.\n\nThe glare Ottotark snapped over his shoulder could have frozen lava, but Corporal Agarik merely lifted his arms, displaying boots, a parka, a stack of black uniforms, and a towel. Tikaya held her breath, aware the sergeant outranked Agarik, but hoping the corporal's presence would keep Ottotark in line.\n\n\"The captain said to bring her these and relieve you as guard,\" Agarik said.\n\nOttotark eyed the stack. \"Now we're pampering the bitch with extra clothes? Why don't we invite her to dine in the officer's mess next?\"\n\n\"Gonna be cold up there, sergeant.\" Agarik walked in, set the clothing on the bunk, and then stood outside the cabin, in full view of the door, which he left open.\n\nOttotark issued a low growl and a backward glance that promised \"later\" before striding out.\n\nEven after the door banged shut, Tikaya could not relax. Her luck would not hold with that one. She would have to figure out how to abscond with a dagger from the exercise area and keep it on her at all times. And hope it was enough against the powerful marine. And that she could use it on him. But then that should not be a problem now. Her lip twisted bitterly. She had killed. When she thought of how easy it had been, how accurate she was with that cursed bow, she had to steady herself with a hand on the wall.\n\nReact later, Rias had said. Well, it was later.\n\nTikaya curled on her side on the bunk, her head in her hands, her eyes shut. Images of her deeds flashed in her mind, the terrified and pained faces of the people she shot. She let them flood over her again and again, feeling the need to punish herself. What would Parkonis think if he were alive? Would he be shocked\u2014disgusted\u2014that she could release an arrow into someone's chest? He never would have killed a human being, probably not even in self-defense. He would have been horrified to see Rias beheading those practitioners.\n\nShe opened her eyes and stared at the polished wood floorboards. If she had been transported to that ship with Parkonis, she would have been dead in the first minute. She was no longer in his world, no longer in hers. She could adapt to this world\u2014she had proved that to herself that night\u2014but at what cost?\n\nTikaya wondered if she would ever see her family and her island again. More, she wondered if she would be someone her parents could still love if she did return.\n\n# CHAPTER 7\n\nIce stretched in all directions, an endless white blanket, unbroken save for a black trail of water stretching behind the ship. Tikaya gripped the frost-slick railing near the bow with gloved hands and peered over the fur trim of her parka, amazed by the heavy iron hull smashing through the inches-thick frozen crust. The pace was slow and the deck vibrated with the efforts of the engine, but their progress continued. Her people's wooden vessels could never do this and she admitted reluctant admiration for the Turgonian engineers and metallurgists who could build such a craft without help from practitioners.\n\nFor the first time during the trip, land stretched along the horizon, white, flat, and stark. To the south, a range of jagged snow-smothered mountains ran perpendicular to the coast. A settlement hunkered a few miles ahead, low buildings and ice-locked docks just becoming visible. On the ship, marines were hauling food and supplies out of the hold, preparing for a land excursion.\n\n\"Good morning,\" came a familiar voice from behind.\n\nTikaya whirled, smiling. \"Rias.\"\n\nThanks to the captain's claim that his men were too busy with repairs to perform extra guard duty, she had not seen Rias for more than a week, not since the night of the attack. Her smile faded at the sight of shackles binding his wrists and guards trailing behind him. She clenched her jaw. How could Bocrest still treat Rias like a prisoner when he had risked his life\u2014their lives\u2014to save the warship?\n\nHe joined her at the railing. \"I've missed you.\"\n\nThat simple statement warmed her far more than the parka. The captain had allowed Rias a shave, at least, and she had a nice view of the smile softening his face.\n\n\"Me too. I mean you. Er, I've missed you too.\" Tikaya stifled a groan, avoided his eyes, and reflected on the mortification her linguistics professors would feel at hearing her mangle language so. To cover her fumbling tongue, she nodded at the ice cracking beneath the bow. \"That's impressive.\"\n\n\"Hm, the Emperor's Fist has a strengthened hull, but that won't be enough to get us all the way to shore. If you want to see impressive, you should see our dedicated ice-breaking ships. They have a double hull and a special steel alloy designed for peak performance at low temperatures. The bows are rounded instead of pointed, so the ship rides up over the ice, smashing it with its weight. And the engines! They...\" He blushed. \"Sorry, you probably don't want all that information.\"\n\nTikaya grinned. \"I did ask.\"\n\nHe smiled sadly. \"No. No, you didn't.\"\n\n\"Well, I expressed interest in the subject.\"\n\nThat seemed to mollify him. \"I should have asked already: are the men treating you decently? Any sign of those assassins? Any nightmares after our adventure?\"\n\n\"As well as can be expected for a loathed enemy of the empire, no assassination attempts, and nightmares...\" Tikaya had slept poorly, reliving the killings on the Nurian ship, but she did not want to talk about it here, with guards looking on, so she pretended to misunderstand. \"Why do you ask? Are women usually traumatized after an evening out with you?\"\n\nHe blinked a few times. \"No, but I don't usually take women into battle on first dates.\"\n\n\"Ah, I see. You save that until the relationship is more established.\"\n\n\"Exactly.\" He slid her a sidelong look, and she suspected he understood what she was not saying.\n\nTikaya propped her elbow on the railing and faced Rias squarely. Though she enjoyed chitchatting with him, she had been waiting all week to ask about the tunnels he mentioned to Bocrest. And how they tied in with the symbols.\n\n\"Will you tell me about these tunnels you're supposed to guide us through?\" she asked. \"You've asked me to help Bocrest, but you haven't explained what that will entail.\"\n\nHis face grew somber as soon as she mentioned the tunnels. This time, though, he nodded instead of retreating into himself. \"The place we're going...the source of the runes... I've been there before. It was my first assignment as a raw sub-lieutenant, what we call a 'testing mission.'\"\n\n\"What's being tested? You?\"\n\n\"Yes. Every officer gets something early in his career, a deliberately challenging task that's meant to show whether or not he has the courage, intelligence, and command ability to go on to become a leader of men. I imagine this was...more than my superiors had in mind. I was attached to an army unit for the month because of studies I'd done on excavation engineering. Forty of us walked into those tunnels. A week later, three of us crawled out\u2014through a ventilation shaft high in the mountains, in the middle of a blizzard. We barely made it back to Fort Deadend, and the major I'd been assigned to wrote a heartfelt report that stated we should never send men into the tunnels again.\"\n\n\"They were ancient ruins?\" she asked. \"With traps?\"\n\n\"Ancient, perhaps. Not ruins.\"\n\nRias's shoulders hunched in an uncharacteristic slouch, his gaze toward the snapping ice. Tikaya thought of the man she followed through the Nurian ship, head up, alert, leading the way with confidence that should not have been there against such lopsided odds, and she regretted drawing him back to what was obviously a dark place for him. Still, she had to know.\n\n\"What happened inside?\"\n\n\"The tunnels were in good condition. Too good. No dust, cobwebs, no signs of age other than damage from tectonic shifts. The men with me declared the place possessed by some powerful ancient magic. I thought...not. The 'traps' we kept stumbling into\u2014I got the feeling they weren't traps at all but simply the workings of a place we were too ignorant to understand. We were like clueless rats drowned in the city waterworks when the level rises.\"\n\n\"But there was writing? These symbols?\"\n\nFor the first time a spark of interest entered his eyes. \"Not a lot, but things were labeled. If you could translate, perhaps that could keep us safe.\"\n\nTikaya feared the smile she offered was bleak. The rubbings were gone, and she had made zero progress with the language.\n\n\"Well, not safe.\" Rias's shoulders slumped again, independent of her thoughts. \"There were strange and deadly creatures roaming those tunnels too. Nothing we recognized, nothing the archaeologists with us knew from the fossil record.\"\n\n\"You had archaeologists with you before?\"\n\n\"A team of scientists, yes, and a linguist.\"\n\n\"Did any of them make it out?\" The bleakness infused her tone now.\n\n\"No, and they weren't particularly helpful while they were alive.\"\n\nGreat grandmother's gray locks, what was she supposed to accomplish that a team of archaeologists had failed to do?\n\nA worried expression creased Rias's forehead. He seemed to realize he had blundered. \"But you're better than them.\"\n\nShe snorted. \"That'd be more reassuring if you'd ever actually seen me do anything and could qualify that statement.\"\n\nHe bumped her shoulder and smiled. \"I've seen enough.\"\n\nTikaya blushed.\n\n\"You two relax.\" Sergeant Ottotark glared at Tikaya and Rias as he stalked past carrying a massive bag labeled 'tent: medium.' \"Enjoy the view. Have some rum. Those of us who aren't prisoners will handle all the unloading and loading.\"\n\n\"I hope he's not coming with us,\" Tikaya muttered after he moved out of earshot. Somehow, she did not think she would be that lucky.\n\n\"Despite his bite, I'm told he's intensely loyal to the emperor and the captain,\" Rias said.\n\n\"If he kept his bite out of my cabin, I wouldn't care one way or another.\"\n\nRias looked at her sharply. \"What?\"\n\n\"Nothing.\" Tikaya lifted a hand, realizing she had insinuated more than Ottotark was guilty of at that point. \"He's just an ass. He hasn't done anything yet.\"\n\nRias's gaze did not waver. \"Yet?\"\n\nWhat did he expect her to say? \"I'm trying to stay out of his path.\"\n\n\"You shouldn't have to. Not on a Turgonian warship.\" Rias offered a jerky wave, hampered by the shackles. \"I have to go.\" He stalked away, his guards hustling to catch up.\n\n\"Rias?\" she called after him.\n\nHe paused, looking back over his shoulder.\n\n\"Did you pass the test?\"\n\nHis lips twisted into a sour expression. \"They gave me a medal.\"\n\nHe resumed his determined walk. Before she could consider his words or abrupt departure further, Agarik strode toward her, a full rucksack in his arms. He plunked it on the deck at her feet. He already wore a rucksack of his own with a rifle strapped to the back. His utility belt was loaded with a knife and pistol, ammo pouches, and powder tins.\n\n\"Are you ready to go, ma'am?\"\n\n\"Go?\" Tikaya glanced over the railing. The ship had ground to a halt against ice too thick to break, but marines still hustled about, piling gear next to a hoist. A gangly grinning private surged through a hatch leading\u2014being led by\u2014eight thickly furred gray and white dogs. \"Now?\"\n\n\"Yes, ma'am. You're part of the scouting party. We're expecting trouble, so you're going in first with me and a dozen others under Lieutenant Commander Okars's lead.\"\n\n\"You're expecting trouble?\" Why would they include her if that was the case? Weren't they supposed to be keeping her from being killed?\n\nAgarik pointed at the distant buildings. \"No smoke. Fire is crucial for warmth up here. No smoke means the town's been deserted. Or worse. Might be the Nurians again, and if any are around, we'll need a translator to interrogate them.\"\n\nDread curled in the pit of her stomach. Not only did she not want to see any more Nurians, but she surely did not want to help with a brutal Turgonian interrogation.\n\n\"You've got to come at some point anyway, so the captain says now.\" Agarik gave her an apologetic shrug. \"We'll protect you. We scouts are well trained.\"\n\n\"No doubt. You found me and dragged me off my parents' plantation without trouble.\"\n\nHe winced.\n\n\"Sorry,\" she said. Agarik was the closest thing she had to an ally amongst the marines, and if he had not found her, another would have, so she could hardly blame him.\n\nHe pointed to the rucksack. \"Want to check that? I grabbed your clothes and some pencils and blank journals. Then there's standard issue gear for this climate: medical kit, snow goggles, crampons, canteens, blanket, and a hygiene and shaving kit.\"\n\n\"Shaving kit? As cold as it is up here, I'm not sure I want to remove any of the little body hair I have.\"\n\nHe did not smile at the joke. Instead he watched her with curious intensity, as if willing her to understand something. Then she got it. Shaving kit. Razor.\n\nAgarik's gaze shifted toward Ottotark, who stood by the hoist, directing the lowering of a dog sled.\n\n\"I see,\" Tikaya breathed. \"You won't get in trouble, will you?\"\n\nAgarik hesitated a second, then said, \"Standard issue gear,\" which she took to mean he probably would get in trouble for doing something as stupid as arming a prisoner, but it would likely be seen as negligence rather than treason. A lesser crime with a lesser punishment, she hoped.\n\n\"Thank you, Corporal.\"\n\nHe saluted her, fist to chest. \"Ma'am.\"\n\n\"I think you can call me Tikaya at this point.\"\n\n\"Yes, ma'am.\"\n\n* * * * *\n\nPowdery snow skidded sideways as wind scoured the ice field. The icy crystals needled Tikaya's neck as she crunched along behind the squad of marines. For the seventy-third time, she adjusted her wool scarf and cap, wondering why the secret to the gear's effectiveness eluded her. A wan sun burned in the sky, but its arc remained low on the southern horizon, and it provided no warmth. At least the bulky goggles smashing her spectacles against the bridge of her nose warmed her cheeks somewhat, though the main purpose of the darkened lenses was to protect from the sunlight glinting off ice and snow.\n\nDespite her discomfort, determination kept her feet moving as quickly as those of the scouts. Even before Rias's story, she had daydreamed of translating the language and bringing awareness of it to the greater archaeological community. Now, she had a more compelling reason to learn as much about the runes as she could. Quickly. Since the Nurians had deprived her of the original clues, she would have to find new ones inland. It struck her as odd that she resented the assassins more for stealing the rubbings than for trying to kill her.\n\nTikaya peered over her shoulder. She still did not know if the Nurians had returned to their ships or had holed up on the Emperor's Fist somewhere.\n\nAgarik, bringing up the rear, asked, \"Problem, ma'am?\"\n\n\"Just wondering how far back Rias and the others will be.\"\n\nThe ironclad, its black hull a dour blot against the stark white world, rose a couple miles behind, and she could no longer pick out the men and dog sled teams assembling in its shadow.\n\n\"An hour back or so for the main party. As for Five...\" Agarik might know who Rias was now, but he was careful to use the number instead of a name. \"I heard him and the captain arguing just before we left, and, uhm, Bocrest told him he could shove\u2014er, he had to carry the blasting sticks, so he'll be in the rear.\"\n\nTikaya groaned, knowing that argument had been her fault. She should not have complained about Ottotark. \"Blasting sticks? Are those practitioner-made or the unstable alchemical kind?\"\n\n\"We don't use anything magical.\"\n\nShe groaned again. One thoughtless comment, and now Rias had to traverse the slick ice while carrying a heavy box of volatile explosives. While wearing shackles.\n\nThe image distracted her, and she crashed into the marine in front of her. An unstrung bow strapped to his rucksack clipped her jaw.\n\nHe glared over his shoulder but said nothing. At some signal or command she had missed, the queue of marines had halted. Two dogs the scouts had brought sniffed and romped, unconcerned by whatever caused the leaders to stop.\n\n\"Bones!\" someone called.\n\nTikaya glanced at Agarik. Was that a name? Or a discovery?\n\nAgarik said nothing. Every man in the squad stood still, apparently drilled to do so until a command came. Well, she was not a marine. She sidled out of line. Ten meters in front of the formation, two men stood around something pale half-covered in snow.\n\nWould she get yelled at if she went up to investigate? Did she care?\n\nTikaya shrugged and walked to the front of the line. Men glanced at her as she passed, but no one stopped her.\n\nShe slowed as she approached, regretting her decision to leave the squad as soon as she identified the object on the ground.\n\nIt was a naked man. A dead naked man.\n\nSnow mounded against one side of the body, and ice crystals gathered on limbs blackened by frostbite. He had died face down, an arm stretched out, fingers splayed.\n\n\"Nothing to translate here, woman.\" Lieutenant Commander Okars, a stocky man with eyebrows thicker than the fur trimming his parka hood, leaned against his rifle. He removed a plug of tobacco from a pocket and gnawed off a corner with stained teeth before handing it to the other man. \"What d'you think?\"\n\nThe second marine looked so similar to the commander that Tikaya thought them twins for a moment. The name tabs on both their parkas read Okars, but this fellow wore lieutenant's pins and had fewer lines on his face. He spat a brown stream into the snow by the body. \"Looks like he was running from something.\"\n\n\"I called you up here for a more professional assessment than that, Sawbones.\"\n\n\"Oh, I'm sorry. Did you want me to inquire about his health and see what ails him? Perhaps if he'll give me a list of his symptoms, hm.\"\n\nAt that point, Tikaya realized 'sawbones' was slang for doctor. The connotations in that name disturbed her\u2014she had never visited a healer who did not work as much with the power of his mind as with his hands\u2014and she hoped she did not require this man's services any time soon, especially if he kept serrated metal tools in his kit.\n\n\"Bones,\" the commander snapped. Strange that he used the term instead of his brother's name. Maybe the staid Turgonians had regulations against familial familiarity.\n\n\"What?\" Bones asked. \"He froze to death. What do you want me to do?\"\n\n\"Figure out what drove him to run out here naked and suicidal.\"\n\nBones levered the barrel of his rifle under the corpse and leaned onto the stock. Ice snapped, and the rigid body rolled over. Tikaya jumped, surprised at the irreverent treatment of the dead. That did not keep her from staring. The front half was no more illuminating than the back, but the face, eternally contorted in terror, made her shiver. The man had died afraid, very afraid.\n\nBones shook his head at the commander. \"Nope, no clues.\"\n\nOkars ground his jaw. \"Curse the Headquarters desk-rider who thought it'd be amusing to put my little brother on the same ship as me.\"\n\n\"Exquisite torture, isn't it?\" Bones grinned.\n\nTikaya stared at the brothers. They were joking. A corpse lay before them, a corpse probably belonging to one of their own citizens, and they were joking. Uncharacteristically intense irritation stirred within her.\n\n\"Animals,\" she blurted before she could still her tongue. \"Where's your respect for the dead?\"\n\nThe commander's bushy brows lowered, and a cold, almost predatory expression darkened his face.\n\nBones placed a hand on Okars's sleeve. \"You'd best get back to your place, ma'am. Stay out of the way and let us do our jobs.\"\n\nShe nodded and backed away.\n\n\"Mouth shut, Tikaya,\" she muttered. \"Keep your mouth shut around these warmongering fools.\"\n\nStrange, she thought she had learned that lesson already.\n\n# CHAPTER 8\n\nWhen Tikaya and the marines reached the shoreline, it differed little from the ice they had been marching across. The ground rose subtly, and she supposed a beach lay somewhere under all the snow. Two docks, embraced by ice, stretched away from a couple of wooden warehouses with drifts piled to the eaves along their northern walls. One of the dogs lifted a leg and yellowed a sign post promising the availability of alcohol at the Rat Wrangler.\n\nThe town itself, with a single snowy road running parallel to the waterfront, seemed more outpost than community. Unpainted wood dwellings hunkered against the elements. Three long rectangular buildings overlooked the town from the crown of a hill. No life stirred anywhere. A stiff northeasterly wind rattled shutters, and somewhere a door banged against a wall.\n\n\"Welcome to Wolfhump, ma'am,\" Agarik said, speaking for the first time since the discovery of the body.\n\n\"That's the name of the town?\" Tikaya asked.\n\n\"It's a trade outpost for the miners working the mountains. I don't think there's a lot to do up here except drink and watch the wildlife, uhm, frolic.\"\n\nAt the head of the formation, Commander Okars made a few hand gestures, dividing men into parties for scouting. Marines checked rifles and a couple strung bows. Tikaya wondered if she might talk the commander into letting her borrow one. She thought of letting an arrow fly into Sergeant Ottotark's chest and his scream of pain as he pitched backward, sprawling on the ice. Tikaya jerked with surprise, startled her mind had conjured the grizzly image. Too much time spent with these Turgonians.\n\n\"As far as you know Wolfhump should be occupied?\" Tikaya asked.\n\n\"Of course, it should be occupied,\" Agarik snapped. He twitched, seemingly as surprised by his tone as her. \"Sorry, ma'am.\"\n\nA growl rumbled up ahead. One dog dove at the other dog's neck. Fangs sank into flesh and the victim squealed, a heart-wrenching cry that halted conversations.\n\nTikaya gawked at the brutal attack. Marines jumped into the fray, grabbing the dogs by the ruffs of their necks and trying to pull them apart, but the attacking canine had gone berserk. Muscles surged, fangs flashed, and soon blood spattered the white snow.\n\n\"Get those dogs under control, private!\"\n\n\"Trying, sir!\"\n\n\"Idiots,\" Agarik muttered and strode forward to help.\n\nTikaya stayed back. The exchange had a bizarreness to it that left her uneasy. She glanced toward the icy sea again, wishing Rias was there to consult. Her gut lurched. The ship was gone.\n\nNothing to worry about, she told herself. It had probably just retreated to open waters to keep from being ensconced when the ice reformed. But that meant the group had nowhere to retreat to if they ran into trouble they could not handle. She squinted, trying to spot the main party, but the sun shone brilliantly on the ice. Even through her goggles, she could not make them out yet.\n\n\"Don't touch\u2014\"\n\n\"Get off me!\"\n\nShe turned back in time to see two marines crash to the snow. They wrestled and thrashed, and men previously trying to keep the dogs apart now turned their attention to separating the human combatants.\n\n\"Satters, Choyka, stand down!\" Okars raced toward them, his voice strained and angry. Where was the calm confidence one expected from a senior officer?\n\nTikaya pulled her goggles up as if a clearer view might enlighten, but the scene only stunned. The smaller dog lay still, its neck torn open, blood drenching the snow beneath it. The other raced across the ice field, yowling like a wolf. Three marines grappled on the ground, clawing and punching at each other. The commander tried to drag one of them away and took a fist to the jaw. He slipped and went down.\n\nThe sawbones stood back, a bewildered expression on his face. At least he was not brawling with anyone.\n\nTikaya jogged over to his side. \"What's going on? Have you ever seen anything like this?\"\n\nHe worked over the wad of tobacco in his mouth. \"Nope.\"\n\n\"Could it be the Nurians again? Trying to keep us from resolving whatever is going on in those tunnels?\"\n\nHe froze, mid-chew. \"How do you know\u2014how much do you know?\"\n\n\"I know they tried to kill me so I couldn't help your people translate that unknown language.\"\n\n\"I suppose it's possible the Nurians have done...something. I can't think of anything natural that would explain this sudden aggressiveness.\"\n\nThe scuffles were dying down. Men's ragged breaths frosted the air. A couple marines still struggled against those restraining them, while others seemed embarrassed. Blood flowed from broken noses and split lips.\n\n\"Bones, come patch people up.\" Commander Okars did not appear wounded, but his eyes had a wild cast to them. \"And tell me what's going on.\"\n\n\"No idea.\"\n\n\"No idea? Grandmother Hakstor was a better sawbones than you. For spit's sake, get your ass over here and figure this mess out.\"\n\nAmusement flickered in Bones's dark eyes, and he gave Tikaya an irreverent salute, saying, \"Nice talking with you, ma'am,\" before he trundled off.\n\nShe was not sure what to make of the man's odd humor or this situation. The fact that she was dependent on the marines out here rankled. She gazed down the flat coastline toward the mountains, wondering how many hundreds of miles lay between her and a town where she could find passage out of the empire. A lot, she feared, and she knew nothing about surviving in this climate.\n\n\"Corporal Dansk,\" Okars said. \"Head back and warn the others. Corporal Agarik?\"\n\nTikaya shook away her musings and turned her attention back to the marines.\n\n\"Yes, sir?\"\n\n\"We're heading in. Stay here and guard the woman.\"\n\nIrritation flattened Agarik's lips, and for a moment Tikaya thought he would question the officer. He kept himself to a glum, \"Yes, sir.\"\n\nOne man trotted back the way they had come, while the others marched away in pairs, leaving Tikaya and Agarik alone with the dead dog.\n\n\"You want to go with them, don't you?\" She lifted her eyebrows at Agarik, whose gaze remained fastened on the backs of the men. \"To explore?\"\n\nHe huffed a sigh. \"No.\"\n\n\"You must not lie very often, because you're not good at it.\"\n\nA slight smile quirked his lips.\n\nTikaya's gaze returned to the buildings overlooking the town. A giant cannon and a flag pole flying Turgonian colors stood before one. Military structures, she guessed. If there were any clues to this mystery, she wagered they would be in an office up there. Exploring the town might be hazardous, but, then, standing out here where anyone with a bow or a rifle could target them felt hazardous too. And if this strange aggression affected the main party... She thought of Rias holding a box of blasting sticks and shuddered.\n\n\"Why don't we ramble up that hill and see if we can figure out what's going on?\" Tikaya suggested.\n\n\"We've orders to stay here.\"\n\n\"Actually, I believe your orders were to 'guard the woman.'\"\n\n\"'Stay here and guard the woman.'\"\n\nA breeze gusted down the coast, icy fingers poking through Tikaya's scarf. \"They don't encourage initiative in Turgonia, do they?\"\n\n\"Not in the marines, no.\"\n\nShe curled her fingers in her gloves. Even if she did not want to explore, she would have appreciated getting inside out of the wind. \"Suppose the woman runs off and you're forced to chase after her in order to guard her?\"\n\n\"With respect, ma'am, I could catch you before you ran five steps.\"\n\n\"Rias would run off with me.\" It was a stupid argument, and she knew it. She felt like a stubborn five-year-old trying to wrangle an extra hour of play before bedtime. Unfortunately, manipulating men to get her way was not her specialty. No doubt her teenage years should have involved less time studying ancient tablets and more time flirting with boys at the beach.\n\n\"I'd rather he run off with me,\" Agarik said.\n\nThe comment surprised a laugh from her and reminded her flirting would probably not work on him anyway.\n\nA gun fired in the town, stealing her mirth. Someone shouted. It sounded like Turgonian, one of the marines, but distance muddled the words. Agarik's grip tightened on his rifle and he took a step before he stopped himself. A scream of pain echoed from the dwellings, and it made Tikaya shiver.\n\n\"Something creepy is happening here,\" she said. \"We should check the buildings on the hill. They're military, aren't they? That'd be the place to start looking for answers, you'd think.\"\n\n\"Tikaya...\"\n\n\"Why are you being so frustratingly obtuse about this?\" she growled. \"You'd think I was asking you to\u2014\" She noticed Agarik's startled expression and caught herself.\n\n\"Something creepy is happening here,\" Agarik said softly.\n\n\"I'm sorry. I didn't mean it. I understand that you want to be the good soldier and follow orders, and I don't want to get you in trouble. But...\" Rias's words flashed through her mind: to live when the rest of your team died would be an unacceptable disgrace to many. \"Do you want to be the only one of your team left alive when the others catch up?\"\n\nHis gaze jerked up, latched onto her. He closed his eyes for a long moment before sighing and asking, \"When I get court-martialed and kicked out of the empire for following your suggestions, can I come live on your island?\"\n\n\"Absolutely. Free lodging in my parents' guest bungalow overlooking the sea. I'll even introduce you to my cousin's handsome friend who surfs nude every afternoon. He was in one of my linguistics classes; he has a gifted tongue.\"\n\nAgarik's eyes widened, and he clapped her on the shoulder, leading the way toward the hill. \"If you'd promised me all that the first time we met, we'd still be on your island.\"\n\nTikaya peered down the main strip as they passed it, but she did not spot any marines. They must be exploring inside the buildings.\n\nWhen she and Agarik rounded the back corner of a saloon, they jerked to a halt at the sight of skulls and bones half-buried in fresh snow. Human skulls and bones.\n\n\"Cursed ancestors,\" Agarik grunted.\n\nThe snow had obscured footprints but did not quite hide the tooth marks scoring the broken bones, the marrow prodded out by tongues.\n\n\"How long has it been since your people had contact with this town?\" Tikaya asked.\n\n\"How should I know?\" Agarik barked.\n\nShe peered at him, at the irritable frown creasing his brows.\n\n\"Sorry.\" He nudged a skull with the toe of his boot. Myriad fractures spun out from a ragged hole smashed into the back. No wolf had done that. \"The fits of rage surprise you, don't they? You think you're fine, and then...\"\n\n\"Yes,\" she said. \"That's a concern, especially since the second group is bringing explosives.\"\n\n\"I've never heard of a Nurian plot like this,\" Agarik said.\n\nShe gazed thoughtfully at the mountains, the tundra, and the ice-coated sea. \"Since I grew up around practitioners, I'm sensitive to when they're performing their science. I haven't felt any of the telltale signs of one at work.\"\n\n\"So, this isn't magic?\"\n\n\"I don't think so.\"\n\n\"Then what?\"\n\nShe could only shrug.\n\nTikaya and Agarik did not speak as they climbed the hill. Several minutes had passed since the last yell, scream, or shot that would have indicated the scouting party was still around. It was as if they had simply disappeared.\n\nThey reached the first of the buildings perched on the crown. The back two, one-story wood structures with narrow windows, were probably barracks and offices. The closest, a taller building with corrugated metal walls, lacked windows, though massive sliding doors marked entrances. Tikaya and Agarik stopped there first, heading for the leeward side, which was free of drifts, though shoveled snow piled high near a walkway. It could not have been too long since the living occupied the outpost.\n\nIce shattered and metal groaned as Agarik shoved a door open a few feet. Weak sunlight probed the interior, revealing an empty building with an earthen floor splotched with dark stains. The smell of engine oil wafted out.\n\n\"All the caterpillars are gone,\" Agarik said. \"Guess we're stuck with dog sleds.\"\n\n\"Caterpillars?\" she asked.\n\n\"Steam vehicles designed to handle the ice and snow.\"\n\n\"Ah. How many are there supposed to be?\"\n\n\"There's room for five or six in there.\"\n\nThey crunched across the crown of the hill, sinking calf-deep into snow. Wind gusted, blowing powder off the roofs.\n\nA three-foot long icicle sheared off an overhang and plunged into the snow a foot from Agarik's shoulder. Tikaya jumped to the side, tangling her feet, and toppled into the snow. Agarik raised his eyebrows, and she felt sheepish.\n\nShe clambered to her feet. It was chance that the deadly icicle had dropped then, nothing more. Regardless, Agarik took a couple steps to the side before continuing along his route.\n\nHeavy shadows lay in the alleys between the buildings, and Tikaya glanced skyward in surprise. The sun had dropped well past its zenith.\n\n\"It's already afternoon.\" Feeling silly for blurting the obvious, she added, \"I didn't realize how short days are up here.\"\n\n\"I was here once near the winter solstice. Day is about an hour then.\"\n\nBy the entrance to the next building, shoveled snow piled nearly to the roof. They followed a wide walkway with a couple of inches of fresh powder blanketing it. Paw prints marred the surface. Large paw prints.\n\nA gnawed skeleton, not quite hidden by the snow, sprawled a couple feet from the door. A hammer protruded from the skull, its head caught where it had smashed a hole through the bone. Shreds of a black uniform were tangled amongst the ribs.\n\n\"I hope our scouting doesn't require us spending the night here,\" Tikaya said. \"I doubt this town gets any less disturbing after dark.\"\n\nAgarik nodded at the hammer. \"Scavengers might have cleaned these corpses, but it looks like humans were responsible for the deaths.\"\n\n\"Of course. Killing people is what you Turgonians are good at.\" She regretted the words as soon as they came out.\n\nAgarik slid a knife free of his belt and whirled, glowering at her. \"We need answers, not sarcastic comments.\"\n\nTikaya skittered back, hands raised. He turned the knife so the afternoon sun glinted against the frosty steel. His breath steamed the air before his intense dark eyes.\n\n\"Agarik?\" She shifted her weight, thinking of the razor in her rucksack. It would take a lot of rummaging to find it. She should have done that as soon as the other marines let her out of sight. \"Put the knife away, please.\"\n\nHis cheek twitched and the blade trembled.\n\n\"Something here is affecting you, remember? They let it affect them...\" Hands still raised, she nodded toward the bones. \"And now they're dead.\"\n\nAgarik forced his fingers open, and the knife tunneled into a snow drift. \"Of course. I know.\" Eyes closed, he took a deep, shuddering breath. \"Do you think... Are we making a mistake being here? Will whatever happened to these people happen to us?\" For the first time that Turgonian fearlessness faltered.\n\nTikaya decided she preferred the fearlessness. \"You know as much as I do, probably more.\" She picked up the knife and dusted it off. \"Mind if I keep this for now?\"\n\nHe flinched but nodded. \"Go ahead. I have my rifle and pistol.\" He waved at the town. \"Don't worry. I'm sure we can always leave if the situation escalates into something dangerous.\"\n\nShe bit back a comment about the townspeople apparently not being able to leave to save their lives.\n\nAgarik pushed open the door, and Tikaya peered around him. A single corridor stretched to the opposite end of the long building. The far door stood ajar, and daylight slashed inside. Shadows in the middle stirred. Two dark furry shapes turned their direction.\n\nWolves.\n\nThey did not exist back home, but she had seen pictures. Tall and winter lean, the creatures growled, lips rippling, saliva gleaming on fangs.\n\nThe pair charged. Agarik raised his rifle and shot one, the boom thunderous as it echoed in the hallway. The ball struck the lead wolf in the shoulder, and it yipped in pain, but it kept running. Agarik fired his pistol, striking the second canine in the eye. This one faltered and flopped over, but the other kept racing toward them.\n\nTikaya backed outside and Agarik, needing time to reload, almost fell out beside her.\n\n\"Stay back,\" he barked.\n\nShe ignored him and lunged forward to slam the door shut. A heartbeat later, the wolf crashed against the wood. It shuddered but held.\n\nAgarik already had his ammo pouches open. With admirable calm, he poured powder down the muzzle of the rifle, rammed a cloth-wrapped ball home, and slipped a percussion cap on the nipple. The wolf slammed against the door two more times, then claws scrabbled at wood. Tikaya gripped the knife and wished she had a bow. On a whim, she yanked the hammer out of the skull.\n\nThe clawing at the door stopped.\n\n\"I bet it's going around,\" she said.\n\n\"Stay back,\" Agarik repeated. \"Stand against the wall.\"\n\nRifle loaded, he stepped away from the building, ready to fire either direction. Tikaya put her back into the corner between the door and the pile of shoveled snow.\n\nEven expecting the wolf, she was startled by how soon it ripped around a corner. Agarik did not flinch as it hurtled toward him. He lined up the shot and fired.\n\nThe ball struck the wolf in the chest, and it missed a step, but amazingly it did not stop. A craziness lit its yellow eyes as the beast launched itself at Agarik.\n\nHe had not had time to reload the pistol, so he could only swing the rifle like a club. The wolf twisted in the air, and Agarik merely clipped it. The beast's fangs snapped inches from his neck. The snow hindered him, and he stumbled back against the wall.\n\nTikaya slashed at the wolf when it landed nearby, but it sprang again too quickly, and her blade sliced air. Agarik hammered it with the butt of the rifle, but the creature seemed not to feel pain. It readied itself to spring again.\n\nTikaya lifted the hammer, thinking she might get lucky if she threw it, but a new thought halted her. She turned her back to the fray and scrambled up the snow pile.\n\n\"Good idea,\" Agarik called. \"Stay up there until...\" He grunted as he swung at the wolf again. \"Until I finish this.\"\n\nRetreating was not Tikaya's idea. She crawled through the snow near the edge of the roof until she could peer down upon the skirmish. The eaves sheltered Agarik, but the wolf, needing room to run and leap, kept moving in and out of the overhang's shadow.\n\n\"Stay against the wall, and suck in your belly,\" Tikaya called.\n\nShe leaned over and grabbed an icicle as thick as her upper arm. Even with the hammer it took several cracks to free it from the edge. The wolf leaped. She timed it, then released the ice spear.\n\nTikaya did not expect to hit the creature on the first try, but her aim proved true. The icicle bludgeoned the top of its gray-furred head.\n\nAgarik sidestepped, and the wolf smashed against the wall and fell, unconscious. \"Throw me the knife.\"\n\nShe dropped it into the snow before climbing off the roof. Apparently taking no chances, Agarik sliced the beast's throat.\n\n\"Ma'am?\" He fished in his pouches to reload his weapons.\n\n\"Yes?\"\n\n\"Marines are very fit. We do not have bellies.\"\n\n\"My apologies.\"\n\n\"Thank you for your help.\" He lifted his fur cap and swiped away sweat as he looked back and forth from the roof to the wolf to her. \"I wasn't expecting you to, ah, to be able to...\"\n\n\"You're welcome.\" Tikaya felt insulted that he was so shocked she had done something useful. She supposed she should appreciate his protectiveness, but she found herself missing Rias and the way he had assumed her competent enough to help. She snorted. Actually, he had assumed her a little too competent, but they had both survived, so she could not fault his decisions. \"Are wolves always that difficult to kill?\" was all she said.\n\n\"No.\"\n\n\"I suspected not.\"\n\n\"Let's see what they were after.\" Agarik led the way inside.\n\nThis time, no creatures attacked when they opened the door. The second wolf lay dead where Agarik had shot it. The drab green paint covering the wood walls and the gray tiles lining the floor could not camouflage the dark blood spatters staining the hallway.\n\nThey passed doors, some closed, some open to utilitarian offices. Each contained identical military-issue desks, chairs, and bookcases. Some offices appeared untouched, as if the men had simply stepped away to make a cup of tea. In others, toppled chairs and scattered papers suggested struggles had taken place.\n\nAgarik stepped into a messy room to investigate, and Tikaya chose a tidy one across the hall. She peeked in cabinets and drawers, not sure what she sought. The cause of this madness, but what would that look like?\n\nShe paused before returning to the hallway. She tugged her glove off and ran a finger along a bookcase by the door.\n\n\"No dust,\" she murmured.\n\nThat and the mostly cleared walkways outside implied things had been normal within the last week or two.\n\nTikaya returned to the hallway, passed an office where Agarik poked and clanked, and stopped before a closed door. Wood shavings dusting the floor drew her eye. Above them, claw marks ravaged the door and jamb.\n\nDread settled in the pit of Tikaya's stomach. \"Agarik? I think we want to check this one.\"\n\nMaybe it was cowardly, but she stepped aside when he walked out, gesturing for him to turn the knob. He took in the claw marks with a grim set to his jaw, then handed her the rifle.\n\n\"Uhm?\" she asked, startled.\n\n\"Just in case,\" Agarik said. \"It's loaded. Just point and pull the trigger if you have to.\"\n\n\"I've never shot a\u2014\"\n\n\"If you can make a bull's-eye with a bow and an icicle, you can shoot a firearm.\" He withdrew his pistol, turned the knob, and pushed. The door bumped against something and only opened a couple inches. Pistol leading, he leaned against the door, shoving to open it further. Furniture inside scraped, and something tipped over with a crack. He peered inside. \"Cursed ancestors.\" He glanced over his shoulder. \"Too gruesome for a woman. Wait here.\"\n\nHe disappeared into the dim room, and she waffled, torn between wanting to know what was inside and not wanting to see it. Considering they had tramped past human skeletons and a frozen dead man together, she did not want to know what qualified as too gruesome.\n\nA few bumps drew her curiosity, and she decided to force aside her squeamishness and go in. Before she could, Agarik opened the door again.\n\n\"Sorry, ma'am, but I need you to look at something.\"\n\n\"It's fine.\"\n\nAgarik lifted a hand first. \"I want you to know... I know it won't help anything and won't make up for...\" His gaze slipped off her eyes and settled on the wall past her shoulder. \"I'm sorry I was the one to find you and let them know where to get you.\" The words came out in a blurted jumble, as if he had been trying to work up the gumption to voice them for some time. \"This isn't your battle. It never was, and I hope...I hope you're able to live through it and get home. Somehow.\"\n\n\"Thank you, but don't feel guilty on my behalf, please.\" Tikaya rested the rifle butt on the floor and touched his arm. \"If it hadn't been you, I'm sure it would have been someone else. I knew when I helped my people decrypt those messages that there might be consequences someday. Nothing remotely like this entered my thoughts, but...\" She steeled herself. \"Show me your something.\"\n\nAgarik led her into an office with a broken barricade of chairs, bookcases, cabinets, and a desk cramping the area near the door. At first, she did not see the bodies, but they were there, in the middle, around an odd black object, that appeared half box, half table with an utterly foreign set of symbols glowing red in the air above it. A pipe rose from one side, and six slender legs attached the construct to the floor. The dead were strewn about it. Blood stained everything, even the ceiling. For the first time since they arrived, she was thankful for the freezing temperatures. In her climate, the decomposition, the smell, would have been overpowering.\n\nFollowing Agarik, Tikaya shuffled through the clutter. He obviously wanted her to examine the box, but she could not get there without stepping over bodies. Cuts and punctures desecrated them, far more than would have been needed to kill. A dagger protruded from one man's burst eyeball. The whole macabre scene seemed too messy for the neat and efficient marines.\n\n\"I didn't touch it,\" Agarik said as she came even with the object. \"I don't have any idea what it is.\"\n\nSymbols formed neat rows on one side of the black box, and giddiness replaced the nausea in her stomach. They were familiar in style, probably from the same language as the glyphs on the rubbings, but arranged individually instead of in groupings. Each symbol marked an indention. In the center of the box top, a red light smaller than her pinkie nail glowed, projecting a set of symbols above.\n\nTikaya eased around for a better look, but her boot bumped a wood stick. Not a stick, the shaft of a shovel. It and a pickaxe lay on the floor near smashed tiles. The subfloor was torn up, with exposed dirt beneath. With a jolt, she realized the contraption's 'legs' stuck through the floor and into the earth.\n\n\"How the...\"\n\n\"Looks like these men were trying to dig it out,\" Agarik said. \"Probably wanted to get rid of it.\"\n\n\"Yes, but how would its legs have plunged through the floor and anchored down there to start with?\"\n\nHis parka rustled as he removed his cap and rubbed a hand through his short hair. \"I don't know. Magic?\"\n\nTikaya turned her attention to the symbols again. She poked one, indenting it; it glowed red and a larger version appeared in the air with a ball spinning around it. The originally displayed image disappeared in favor of the new one. \"Er.\" She had best be careful; this had probably all started with some idiot pressing buttons.\n\nShe glanced at Agarik, afraid he would chastise her for touching things, but he nodded encouragingly. Dear Akahe, he thought she could figure it out and fix things.\n\nUnderneath the box, she found a couple groupings of the more traditional symbols engraved in the cool black surface. She recognized a few from the rubbings. So, this might be writing. Directions? For operating the device?\n\nIn the air above, the symbol she had pressed faded and the original diagram returned. She poked the button again, then stabbed a couple others. They all appeared in the air. Something reminiscent of an equals sign formed between two while others dangled individually. Waiting. When she did not touch anything for a moment, the symbols faded, replaced again by the original.\n\n\"Numbers?\" Tikaya wondered, though some two hundred symbols were there. She knew of one ancient language that had used a base forty math system instead of the nearly ubiquitous base ten most of the modern world preferred, but nobody had two hundred different numbers. \"Numbers and mathematical symbols?\"\n\n\"Eh?\" Agarik asked.\n\n\"If that's what these are, then maybe operating the device involves punching in different combinations to create... I don't know. Equations for something?\" Tikaya rubbed her jaw. \"But how would that relate to whatever this device is doing to negatively affect the town? I don't know. Maybe I'm all wrong here. What do you think?\"\n\n\"Uhm.\" Agarik's eyes were so blank he appeared hypnotized.\n\n\"Agarik, I don't mean to insult you, but can you see if Rias is here yet and bring him?\"\n\nRelief flashed across his face, but he hesitated. \"I shouldn't leave you alone.\"\n\n\"I've got your rifle. I'll be fine.\"\n\n\"My orders are\u2014\"\n\n\"I know, Agarik, but I need to stay here, and figure this out, and if it's math-related, Rias could help. Please find him.\"\n\nAfter another long hesitation, he sighed. \"Yes, ma'am.\"\n\nThough she had told him to go, she felt uneasy once she was alone. At least her irritability had disappeared. Agarik's had, too, she realized. Despite his lack of understanding, he had been calm and patient while she mulled over the strange artifact. Maybe the device was the thing responsible for people acting oddly, and maybe, in her random symbol touching, she had cut off whatever it was doing or emitting. She snorted self-deprecatingly. If it had been that simple the dead men on the floor would have figured it out before the end. Besides, thinking back, she had felt that return to normalcy before she started touching things, perhaps even out in the hallway.\n\nShe stretched and walked to the window, intending to open it and let some air in. Maybe she should have asked Agarik to drag the bodies out before leaving. Someone had nailed a couple boards across the window. So much for fresh air. Maybe she could still open it a bit.\n\nA scream echoed from the building next door. Tikaya froze, her hand on the window lock. She drew back. Maybe she would leave it shut after all.\n\nShe dug a chalkboard out from behind a toppled filing cabinet. It was hard not to look at the bodies, but she could not remove the device to study elsewhere. The men had apparently tried to do that and failed.\n\nTikaya copied the writing from the bottom of the device and circled spots where the lone symbols on the side of the box appeared in the groupings. If they were numbers... No, she better not assume that yet. Just because something reminded her of an equals sign in her language did not mean anything.\n\nTime bled past, the chalk clacking on the board the only sound in the building. Infrequently, gunshots in the distance interrupted. She found herself squinting at the chalkboard and realized twilight had come. The glowing symbols gave off some light, and she worried it would be visible through the windows.\n\nA crunch sounded in the snow outside. She halted her work, chalk poised in the air. More crunches. Footsteps on her side of the building.\n\nTikaya eased past the bodies and grabbed Agarik's rifle. She wished he had left powder and balls too. One shot was all she had if someone attacked her, one shot with a weapon she had never used before.\n\nShe cracked the door. Deep shadows lurked in the hall, and she barely made out the dead wolf. At one end, the door thumped and banged in the breeze.\n\nShots fired beyond that door, and she jumped. \"Stop\u2014what\u2014\" someone cried. Then screams of pain and aggressive yells followed. A lot of them. Her mind conjured the imagine of a wolf pack chasing after its wounded prey. Not Agarik, she prayed.\n\n\"Where's the woman?\" someone yelled.\n\nTikaya swallowed and closed the door. The voices still penetrated the walls.\n\n\"Find the woman!\"\n\nSomeone cackled, and graphic descriptions of what could be done with 'the woman' followed.\n\nTikaya forced herself to return to the device. The same set of symbols glowed crimson in the air, taunting her.\n\nThere was an answer here; she just had to figure it out before time ran out.\n\n# CHAPTER 9\n\nA boom rattled windows, shook the earth, and knocked Tikaya's chalkboard on the floor. Rias's group must have arrived, though she could not imagine him flinging blasting sticks wantonly.\n\nChalk still in hand, she ran to the window to peer between the sloppily nailed boards and through the frosted panes. Darkness had fallen, but flames burned in a building down the hill. Two figures with rifles ran through the light before turning down an alley.\n\nShe shivered, wishing for warmth in the office. The already frigid temperature had dropped noticeably after the sun had set.\n\nFootsteps sounded at the end of the hallway.\n\nTikaya lunged for the rifle, but caught her heel on the downed chalkboard and skidded to her backside with a noisy thud. Great. If they hadn't known where she was before, they knew now.\n\nShe scrambled to her feet and grabbed the rifle. She hopped over the bodies and slid into the shadows thickening the corner across from the door.\n\nFinger on the trigger, butt pressed into her shoulder, cheek against the stock, Tikaya waited. In the stillness, she could feel her heart pounding in her ears. The footsteps thudded closer, the steady pace of someone jogging.\n\nThe door bumped against the furniture barricade, eliciting a surprised grunt that sounded familiar.\n\n\"Rias?\" Tikaya hazarded before she could think better. What if he was as crazy as everyone else out there seemed to be?\n\n\"Tikaya!\"\n\nRias burst into the room, bringing lantern light with him. He did not seem to notice the artifact or bodies; he searched until he spotted her in the corner, started forward, but stopped, gaze dropping to her weapon. He was missing his cap, his hair stuck up in places, and blood trickled from a gash on his temple. A cutlass was strapped across his back, two bulges in his parka suggested pistols, and he carried a rifle as well as the lantern.\n\n\"Are you...you?\" Tikaya asked.\n\n\"I'm not murdering people and trying to kick the ore out of everyone's cart if that's what you're asking. Just a little\u2014\" Rias cocked his head, almost like a dog listening. \"Actually, it's strange but I feel normal in here.\"\n\nTikaya lowered her rifle. \"Yes, I think the device creates some kind of normalcy field around it, probably so the operator isn't affected by whatever it's emitting that's causing everyone to be on edge.\"\n\n\"On edge, that's an understatement.\"\n\nRias closed the door and hopped over the upturned furniture. Tikaya joined him in the middle, intending to show him the device, but he dropped his rifle on a desk and wrapped her in a hug. Surprised, she found herself crushed against his chest. There was a desperate fierceness to the grip, but she managed to get one arm around to his back to return the embrace.\n\n\"I'm relieved you're not hurt or...\" Emotion thickened his voice.\n\n\"I'm guessing you've had a worse afternoon than me,\" she said, relaxing against him. The fear that had tensed her shoulders since Agarik left disappeared, and she felt warm for the first time in hours.\n\nRias released her and stepped back. \"Sorry, I just... I wasn't sure if you...\" He cleared his throat. \"It's dangerous out there. Half the people are mildly affected by whatever's in the air, and the other half are crazier than the bloodthirsty maniacs I left on Krychek.\" His gaze skimmed the bodies in the room, and he frowned thoughtfully as he took in the furniture barricade and the half-boarded windows.\n\n\"I'm glad you were able to get to me,\" Tikaya said. \"I need your help. Maybe Agarik can stand guard while\u2014wait, where is Agarik?\"\n\n\"I don't know.\"\n\n\"I asked him to go find you.\"\n\nRias spread his arms, palms up. \"I haven't seen him. When I realized what was going on, I worried that one of these lunatics would shoot you, so I escaped at the first opportunity. I've been hunting around, dodging packs of the more aggressive people, and just now found you.\"\n\nDespite the situation, she smiled. Escaped at the first opportunity. By what creative means had he eluded his shackles this time? She almost felt sorry for Captain Bocrest.\n\n\"I hope he'll be able to stay safe.\" Tikaya tapped the box and nodded at the collection of symbols hovering in the air. \"I need your help. I think this device is responsible for what's happening out there.\"\n\nRias walked around it, shaking his head and massaging the back of his neck before he even saw the side with the runes and indentations.\n\n\"If you press those, the representations appear in the air.\" She demonstrated as she explained.\n\n\"Oh, Tikaya,\" he murmured. \"I'm sorry, but you've got the wrong person. We Turgonians may be good engineers, but students go to different nations if they want to seriously study alchemy.\"\n\nHer breath caught. \"Alchemy?\"\n\nHe stabbed one of the indentions and a symbol flared to life. \"That's iron, isn't it?\" Another stab. \"And copper.\" He shrugged apologetically. \"I only remember the ones that we use in alloys. Since the Turgonians deny magic exists, we won't publish anything in our textbooks that was only discovered through the use of magic. Aren't your people the ones who first talked about atoms and electrons and such? We don't have a microscope that can see anything that fine. We've only got fifty things on our table of elements.\"\n\nScarcely breathing, Tikaya dropped an incredulous stare to the symbols. Was he right? Were they looking at the alchemical table of elements? If so, then this could be her Tekdar Tablet. She had to be sure before she based translations on it. \"There are almost two hundred symbols. Are you certain? Our table has seventy-five, I think, and it's the most complete of any in the world.\"\n\n\"But your people are still finding new ones, right?\" Rias ticked his fingernail on the top of the device. \"You'll believe me when you see the tunnels, but for now just trust me when I say these people were more advanced than us.\"\n\n\"More advanced?\"\n\nHe had hinted of that in his tale of the tunnels, but she had not truly thought it possible. Though, advanced technology might explain how this device had dug its legs into the earth, piercing tile, wood, and permafrost to do so. She clasped her hand over her mouth and stared at the runes. There was no existing evidence that a people more advanced than modern man had ever lived. This was rewrite-the-history-books kind of stuff. Incredible. If it was true.\n\n\"I just want to be certain. I don't see what you're seeing to make these identifications. Nor does the layout of the symbols look like the table I'm familiar with, but I've only had a cursory introduction to alchemy. It wasn't anything I thought I'd come across studying ancient languages.\" She realized she was making excuses and decided to still her lips before it grew more obvious. Rias had thought so highly of her intelligence; she hated to give him reason to think less of her.\n\n\"I've spent my life looking for patterns and trying to find the predictable in situations others see as unpredictable. If this odd skill can be of use to you now, I'm delighted.\" Rias slid next to her in front of the interface and poked one of the simplest runes and brought it to life before their eyes. \"The atomic structure is incorporated into the symbol itself.\"\n\n\"Oh!\" As soon as he said it, she saw it. \"Hydrogen.\" She slapped herself on the forehead. \"I thought these might be numbers. It never occurred to me that an ancient people might have this kind of scientific knowledge, with specialized symbols for...\"\n\nRias tapped hydrogen again and another rune. The symbols appeared in the air. \"How do you\u2014\" he started, but bonds formed on their own.\n\n\"Water.\" Tikaya grinned. The symbols were not nearly as simple as the diagrams her people used to represent the various elements\u2014indeed, these reminded her of the bizarre perception puzzles a professor had distributed during a lesson on optics\u2014but if one knew what to look for, the structures were there.\n\nThe water molecule flashed twice, then disappeared to be replaced by the far more complex image she had been seeing all afternoon.\n\n\"I guess that isn't the answer.\" Rias gave her a sad smile. \"I'm afraid that's about the extent of my knowledge in this area. I haven't an idea what that could be.\" He swiped his finger through the dozens of linked symbols hovering in the air. The image did not waver. \"Something we haven't invented yet, probably. I can look through the shelves in the offices. It's a long shot, but there might be a book that has our mediocre table of elements in it. Maybe that'll help you with translating. Sorry I don't have the answer.\"\n\n\"Sorry!\" Tikaya grabbed his arms. \"You have no idea how much this helps. I mean, this could be the key to translating this whole language. You're amazing!\" She kissed him on the cheek, then danced back to her chalkboards. Now that he had pointed out the structures so cunningly crafted into the symbols, she could pick out ten or twelve she remembered from school. That was enough to get her started. Although...\n\n\"Actually, yes, I could use a book with your table in it.\" She turned back and was about to ask him to look, but he was staring at her, his fingers touching the cheek she had kissed.\n\n\"Yes, of course.\" He lowered his hand.\n\nShe bit her lip, tickled at his reaction. \"Do you not get praised often?\"\n\n\"It's been a while,\" he admitted. \"And before Krychek, uhm, more often by men than women.\"\n\n\"Not even your wife?\"\n\nHe snorted. \"Especially not by her.\"\n\nThey shared a chuckle, and she admitted herself curious about the woman, though it should not matter. Rias's past relationships were none of her business, and they had more important things to worry about. Besides, he had left her anyway. Tikaya blinked. Or had he? Maybe she was something, like his land and his name, that the empire had taken from him as punishment. Still, he did not sound disappointed.\n\n\"Horrible woman?\" she asked, fishing.\n\n\"No, but we weren't a good match from the start. It never would have lasted as long as it did if I hadn't been away at sea so much of the time. She had my home, my money, and the freedom to spend time wherever\u2014\" he winced, \"\u2014with whomever, she pleased.\"\n\nTikaya grimaced in sympathy. Like her, Parkonis had not been perfect, but he had always been faithful. \"How'd you end up together to start with?\"\n\n\"I was twenty, she was pretty, and our parents thought it would be a good idea.\" Rias laughed ruefully. \"But mostly I was twenty and she was pretty.\" He waved away further discussion. \"I'll get that book.\"\n\n\"Be careful.\"\n\nHe waved an acknowledgment on his way out, and Tikaya shifted uneasily, as worried for him as for herself. Agarik had walked out, and she had not seen him since. Rias was only going to search this building, she told herself, and settled into work.\n\nA few moments later, Tikaya had three chalkboards lined up, all full. She listed the translations for the atoms she recognized. Also, she listed runes she remembered from the rubbings, those displaying what she now recognized as molecular structures. The elements came up surprisingly often in what she had assumed was normal writing. Perhaps the subject was always science. Or maybe these people\u2014this race?\u2014had a language specifically for scientific matters. The Herdoctans had a different written language for religion, so why not?\n\nA hand touched her shoulder, and she jumped, dropping her chalk.\n\n\"Sorry.\" Rias held a book with his finger marking a page.\n\n\"No, don't apologize. My fault for not paying attention.\" Tikaya picked up the chalk and accepted the book. She glanced at the cover. \"Torture and Interrogation Methods Technical Manual?\"\n\nRias cleared his throat. \"Yes, ah, just stick to the chapter on chemical applications.\"\n\n\"Oh, I will. I don't want to chance upon any Turgonian brutality secrets.\" Or pictures more gruesome than the bodies on the floor.\n\nHe surveyed the chalkboards with bemusement and scraped at dried blood on the corner of one. \"You know, some women wouldn't be willing to work in a room full of corpses.\"\n\nShe had already started writing and almost missed the comment. \"What?\"\n\nRias chuckled. \"Nothing. Continue your work. I'll stand guard.\"\n\n* * * * *\n\nTikaya straightened, wincing at the ache in her lower back. She stretched her arms toward the ceiling and shook out a cramp in her hand. Midnight had to be near, maybe past. Her stomach growled. Fatigue numbed her brain, and her mouth battled unsuccessfully against yawns. Even if the lighting had been better, her notes and the symbols on the device would have blurred and swam before her bleary eyes.\n\nRias stood guard by the door, checking the hallway from time to time, but mostly staying silent and letting her work.\n\nA scream raced down the street below the hill. What if she translated the writing too late? After the entire team killed each other? She eyed the bodies in the corner. Rias had dragged them out of the way, muttering something about funeral pyres in the morning, but she worried about getting to morning. If enough people attacked at once, she and Rias could end up like that before dawn.\n\nNo, she decided, watching him standing with his ear cocked. Despite the hour, he was alert, rifle across his arms, hand on the stock, finger near the trigger. Not tense but relaxed and ready. She imagined he could fight off superior odds for a long time, but he would not want to do so. He'd be shooting his own people, the very men they were supposed to help later on.\n\nRias saw her watching him and lifted his eyebrows.\n\nTikaya felt silly to have been caught gazing at him. \"I was wondering if you could get my mind off this for a moment.\"\n\nRias joined her. He set the rifle butt on the floor and rested his forearms across the muzzle. He surveyed her, and she felt a self-conscious twinge. No doubt she had strands of hair sticking out in all directions and dark smudges assailing her eyes. And her baggy Turgonian uniform and parka did not flatter her form under any circumstances.\n\n\"A question.\" Rias's gaze rested on a chalkboard, though he did not seem to focus on anything. \"If someone from Kyatt were to decide to marry a Turgonian, would they be allowed to live on your island?\"\n\nTikaya was not sure what she had expected him to ask, but that was not it. \"That wasn't a marriage proposal, was it?\"\n\nHe coughed. \"No, no, just hypothetical. If it were a proposal...\" He offered his half smile. \"There'd be soft music, excellent food, romantic ambiance...\" He tilted his head toward the corner. \"Fewer corpses.\"\n\n\"Ah, I wasn't sure how they did it in the empire. Given your people's reputation, I thought bloodshed and mangled bodies might be standard at social gatherings.\"\n\n\"Bloodshed perhaps.\"\n\nRias watched her, waiting for an answer to his question, she realized.\n\n\"The Kyatt Islands are major trade ports and learning centers, and we have numerous foreigners living there, either temporarily or permanently,\" Tikaya said. \"I can think of numerous Turgonians who studied at the Polytechnic over the years. And there have been cases of foreigners marrying natives and staying on the islands.\"\n\n\"Turgonian foreigners?\"\n\n\"Well, you would have been more welcome before your people tried to take over the islands.\" She smiled, but no humor lightened his expression. \"The president might ask you to leave if he found out you were among those sinking our ships and slinging cannon balls at our harbor, but if you said you didn't take part in the war, I'm sure you'd be allowed to stay.\"\n\n\"So.\" Rias laid the rifle across his shoulders and draped his forearms over the ends, reminiscent of a man in a pillory. \"Refuge, if one was willing to lie for the rest of one's life.\"\n\n\"Or just dodge questions about one's name and one's past. You're good at that.\"\n\nShe had not meant the statement to sound accusatory, but he flinched.\n\n\"Listen,\" Tikaya said, \"I don't mean to insult you, but whatever you did, or whoever you are to those marines, you're probably less important than you think to the rest of the world. Chances are my people have never heard of you.\"\n\n\"Oh?\" Leave it to the Turgonians: he looked faintly offended.\n\n\"You could tell me your name\u2014\" Tikaya wriggled her eyebrows suggestively, \"\u2014and then I could let you know whether or not you'd be welcome on my island.\"\n\nShe thought he might remind her that his original question had been hypothetical and that he was not asking about his own future, just some imaginary person's. He did not. He took a deep breath. \"You're right. I don't know if we're going to survive the next couple weeks and, even if we do, I'm guessing Bocrest has orders to make me disappear afterward, but either way it's not honorable of me to keep truths from you. I\u2014\"\n\nGlass shattered.\n\nTikaya whirled, grabbing the heavy book as if she could use it as a shield. A shadow moved at the window. Something long and small slid between the boards and rolled onto the floor. Flame spit and hissed on the end of a string. Not a string, a fuse.\n\nRias yanked her off her feet. The furniture blurred past as Rias leaped over it, arm clenched around her waist. He landed in the dark hallway, and shadows swallowed them.\n\nHe sprinted but only made it three steps before the explosion tore away the darkness. A great boom roared, and a concussion pounded Tikaya's back, ripping her away from Rias.\n\nThe wall filled her vision. She tried to bring her arms up to protect her head, but she crashed first. Something popped in her shoulder and agony seared her body. The book dropped from her hands. She landed on the floor, which sent a second jolt of pain rocking through her. She gasped, trying to stifle cries, not sure who might be nearby.\n\nA door at the end of the hall opened, and lights swam in the darkness. Tears blurred Tikaya's vision. She gritted her teeth and blinked them away. Half a dozen men raced into the hall, lanterns swinging, swords and pistols waving.\n\nThe door at the opposite end flung open. They were surrounded.\n\nTikaya staggered to her feet. Her shoulder flamed with pain. She gasped and braced herself against the wall. Next to her, a shot cracked with a flash of orange flaring from Rias's rifle.\n\n\"There she is!\" someone shouted, voice ragged and rough, almost inhuman. \"Give us the woman!\"\n\n\"This way,\" Rias whispered.\n\nShe grabbed the book and ran into a room after him. A return shot echoed through the hall behind them.\n\n\"Don't shoot us, you idgeets!\" came a cry from the opposite end.\n\nRias shut the door. A hint of starlight came through the window, but darkness reigned inside.\n\n\"They sound drunk,\" Tikaya said, words broken as she gritted her teeth through the pain.\n\n\"Where are you hurt?\" Rias snapped the lock, and furniture scraped as he shoved something in front of the door.\n\nAn image of the dead men in the other office invaded Tikaya's mind. They had been trapped in a room, and this was exactly what they had done. It had not worked.\n\n\"Dislocated shoulder,\" she said.\n\n\"Let me see\u2014feel\u2014it.\"\n\n\"Don't worry...about me. I'll\u2014\"\n\nBut he was already sliding her parka off. She clenched her teeth, trying not to whimper.\n\nFootsteps thundered down the hall, and light slipped under the crack in the door.\n\n\"Which room?\" someone barked.\n\nRias unbuttoned her uniform jacket and probed her shoulder. \"Bite down,\" he whispered, putting something wooden in her mouth. Knife handle, she guessed. It was smooth and hard. He gripped her arm and shoulder, then jerked with one powerful motion.\n\nAgony erupted. Tikaya clenched her teeth on the handle, panting to keep from crying out. Blackness encroached on her vision, and her legs gave way. Rias caught her and held her gently.\n\n\"You hear something?\" someone asked.\n\n\"That room.\"\n\n\"No, that one!\"\n\n\"It's whichever one's locked, you halfwits.\"\n\n\"Sorry,\" Rias whispered, cupping the back of her head. He leaned his forehead against hers, and even in the darkness she sensed his distress over hurting her.\n\n\"Not your fault,\" she said.\n\nSomeone rattled the doorknob.\n\nTikaya found the strength to stand again. Already the pain was fading to a manageable ache.\n\n\"I'm ready,\" she whispered.\n\n\"Strong lady.\" Rias squeezed her good arm before pressing a pistol into her hand. \"Back corner. Find something to crouch behind, but stay where you can aim at the door. If they get past me, shoot them. Here, take this too.\" He loaded her up with the second pistol, a powder flask, and an ammo pouch.\n\n\"Shoot to kill?\"\n\nHe hesitated. \"Do what you have to do to stay alive.\"\n\nShe nodded, then, realizing he would not see it, added, \"I understand.\"\n\nSomeone pounded on their door. \"They're in here!\"\n\nTikaya set the book on a chair and slid behind a cabinet where she could see the window and the entrance. She gripped the pistol. At least the wall had been considerate enough to mangle her left shoulder instead of her right. \"Maybe we'll get lucky, the blasting stick will have destroyed the device, and everyone will return to normal any second.\"\n\n\"Maybe.\" Rias's tone made the possibility sound unlikely, and Tikaya wondered if he had seen explosives used on the strange technology before.\n\nMore pounding\u2014louder pounding\u2014hammered the wood, and something snapped. A crack of light appeared, but the desk kept the door from opening wide. Rias waited in the wall's shadows.\n\nShe glanced toward the window, wondering if they could escape that way. Lantern light danced past\u2014men were out there, too, perhaps counting rooms to figure out which office she and Rias occupied.\n\nThe door opened wider, and the slash of light broadened, illuminating the corner of the desk and a coatrack.\n\nA rifle barrel slid through the gap.\n\nTikaya tensed, expecting Rias to shoot first. Despite the chill, sweat dampened her hands.\n\nThe rifle slid in farther, and Rias burst into motion. He grabbed the barrel, yanked it into the room, and slashed upward with his cutlass. The attacker yelped in surprise and pain, releasing the weapon. Rias planted a foot, thrust the other man back, and slammed the door shut.\n\n\"One man disarmed, seven to go.\" He shoved the desk against the frame again.\n\nMuffled voices came through the door\u2014the sound of people plotting. The next attack would not be so easy to thwart.\n\n\"There are men milling around outside too,\" Tikaya said.\n\n\"You think I don't know that?\" Rias snapped.\n\nShe stared at him, startled. He had never so much as looked crossly at her. Then she remembered: \"I guess the protection from whatever the artifact is putting out was limited to that room.\"\n\nAfter a silent beat, Rias said, \"You're right. I'm sorry. I wasn't thinking. It's like before; something's making it hard to keep my equanimity.\"\n\n\"Your breakdown is a lot less disturbing than that of most of your countrymen.\"\n\nHe grunted.\n\n\"I feel it too,\" Tikaya said. \"It's nothing you can see, nothing you smell or feel. Maybe I've been going about this the wrong way. Like a, well, like a philologist. But maybe I don't need to translate the writing on the bottom in order to cut the device off. If we can guess what its purpose is, maybe we can switch it to another purpose, something less troublesome. It has all those options you can put in\u2014doesn't that imply you ought to be able to get more than one thing out?\"\n\n\"What could it be putting out that would affect us mentally? It's nothing we've seen or heard or smelled.\"\n\n\"An odorless gas?\" Tikaya guessed.\n\n\"Ah, being disseminated through that pipe, perhaps?\"\n\n\"It'd have to be something invisible but heavy enough to float down and blanket the town. Something designed to irritate people, to outright anger them, even make\u2014\"\n\nA shot fired.\n\nTikaya jerked her head up in time to see Rias slam the door shut again. The scent of black powder tainted the air.\n\n\"Only two in the hallway now,\" Rias said. \"They've either lost interest or they're going to try another way in.\"\n\n\"We have to get back to the device,\" Tikaya said. \"If we punch in another gas, maybe it'll change the output. Something innocuous that won't hurt anyone.\"\n\nThumps continued at the door, probably more for the purpose of distracting Tikaya and Rias than getting in. The lanterns previously visible through the window had disappeared, which made her think the marines had stopped planning and were now engaged in that plan. She shifted her stance, readying herself to fire toward the window if necessary. The last thing she wanted was to dodge another blasting stick.\n\n\"Innocuous gases,\" Rias said. \"Oxygen? Hydrogen?\"\n\n\"We tried those, albeit on accident. And you pressed in water, which should be deliverable as a vapor. Except the device didn't like any of those.\" Tikaya groaned. \"Maybe my guess is completely wrong.\"\n\n\"Or maybe the machine is only designed to create synthetic or organic compounds,\" Rias said. \"Though I don't know any molecular structures that might qualify. Do you?\"\n\n\"No, but maybe there's something in your book.\" She tapped it with the pistol butt.\n\n\"There aren't many innocuous somethings in that book.\"\n\n\"I know it's a long shot, but\u2014wait, no. When your people captured me, they knocked me out with something sweet-smelling in a rag. When I breathed in, I passed out. Do you know what that was? Would it be in there?\"\n\nRias shifted away from the door. \"Chloroform. Yes.\"\n\nThe thuds stopped.\n\n\"Let's try it.\" Tikaya had a feeling it would be better to find a light and check the book in a different room. \"Can we get down the hall?\"\n\nRias cracked the door. A rifle fired, and the ball smashed into the frame, hurling wood splinters. He closed the door.\n\n\"Not at this time.\"\n\nTikaya snorted. She pressed her nose to the icy glass window panes. At the edge of her view, shadows and lanterns moved.\n\n\"Not this way either,\" she said. \"Unless we can\u2014oh!\"\n\n\"What?\"\n\n\"Maybe nothing, but Agarik and I had to shove our way into the room with the artifact. The window was boarded, the door barricaded, so whoever killed all those men must have come in through\u2014\"\n\n\"Attic,\" Rias said. \"There must be space to move around up there. Watch the door.\" He hopped onto the desk and thumped the ceiling. Wood scraped against wood. \"Here.\"\n\nOutside, the lanterns headed toward their window.\n\n\"I need help up.\" Annoyed to be a burden, Tikaya stuffed the pistol in her pants and joined him, book clutched against her chest. \"I don't think I can lift my arm over\u2014\"\n\nStill standing on the desk, Rias caught her by the waist and lifted her over his head as if she weighed nothing. Blackness waited above, though an icy draft touched her cheek. That meant a way out. She hoped.\n\n\"Hurry,\" Rias said, giving her a final boost.\n\nTikaya scrambled into the dark attic. Even with his help, she came down on her shoulder and had to stifle a curse. When she tried to stand, she bumped her head on a beam.\n\nBelow, glass shattered.\n\n\"Rias?\" She started to lean over to check on him.\n\nHe jumped through and a thud sounded\u2014his head hitting the ceiling\u2014but he did not pause to acknowledge it.\n\n\"Go, go!\" he barked, pushing her ahead of him.\n\nHalf running, half bear-crawling, Tikaya maneuvered past beams and supports.\n\nLight flashed and an explosion rippled through the floorboards beneath her. The force sent her crashing into Rias, and they went down in a tumble.\n\n\"Ooph,\" he grunted, voice sounding odd.\n\nThen her mind caught up to the situation. Rias had been behind her, not in front of her.\n\nTikaya tried to jump back, but the man grabbed her. She dropped the book. His grip kept her from reaching for the pistol. He unshuttered a lantern, illuminating beams, trusses, and his snarling face. One of the marines.\n\n\"Got her!\" he yelled.\n\nRias charged past Tikaya and tackled the man. The lantern flew free. In a lucky lunge, she caught it before it hit the floor and went out. Though her shoulder protested, she held it with her left hand and yanked the pistol free with her right.\n\nRias needed no help though. He knelt over the marine, arms locked around his neck. The man's face turned purple, and he passed out.\n\nA shadow moved behind Rias.\n\nTikaya reacted. She fired the pistol without thinking, and the ball hammered into someone's chest. Rias spun to look.\n\nOnly after the man collapsed did her brain scream that these people were her captors and aiming to kill one might get her into a mess of trouble.\n\n\"It's Lieutenant Commander Okars.\" Rias checked the officer's pulse. \"It was Lieutenant Commander Okars.\"\n\n\"Oh, no,\" Tikaya breathed.\n\nRias picked up a knife. \"Yes, but he was going for my back, so I must thank you for my life.\"\n\nTikaya closed her eyes for a moment. \"Let's just get that horrible device cut off.\"\n\nBy the lantern's light, they found the source of the draft. The first explosion had left a ragged hole in the ceiling of the room with the artifact.\n\n\"Walk softly,\" Rias said as they neared it. \"The structural integrity has doubtlessly been compromised.\"\n\n\"Thank you for that brilliant engineering assessment. Maybe when I fall through the floor, I can take out my other shoulder.\" Her grumbling made her wince and long for the sphere of protection around the artifact. It would be easier to problem solve if she did not feel so cranky. She hoped. It could be worse; she could have become an unthinking aggressive lout who thought it was a good idea to throw blasting sticks at innocent\u2014\n\nHer boot went through the floor, and she pitched sideways. When her body struck, the footing deteriorating further. Rias grabbed her and tried to pull her free, but the floor had enough of them: it dropped away completely.\n\nShe smashed to the level below and landed on something cloth covered. Not cloth, she realized as she looked under her. Clothing. Clothing on dead bodies.\n\nShe lurched away, igniting pain in her shoulder. Rigid fingers tangled in her braid, and she pulled, trying to free herself without using her injured arm or touching the corpse again. A disheartened cry escaped her lips when the dead man's hand lifted with her, fingers fully snagged in her hair. Tormenting ancestors, this was too morbid, and too damned much. Why couldn't the idiotic Turgonians run a decent Polytechnic so they'd have their own philologists to kidnap for secret missions?\n\n\"Sorry,\" Rias murmured, crouching beside her. \"As soon as we get this taken care of, we'll find the sawbones to check your shoulder.\"\n\n\"The problem is less the shoulder\u2014though that is irritating me every three or four seconds too\u2014and more the bodies. And the being attacked. And the part where I'm shooting people to death, and\u2014\" She brought her fist to her mouth and squinted her eyes shut, struggling to keep from breaking into sobs. Slow breaths, she told herself. This was not the time for wheezing and gasping and flirting with an emotional breakdown. React later. \"I'm all right. I'm just... I'm better in a classroom, I swear.\"\n\nRias wrapped his arm around her back, and she leaned on him.\n\n\"I suppose you'll think I'm odd\u2014odder\u2014if I admit this is the most exciting my days have been in ages,\" he said.\n\nShe rubbed her eyes. \"I'll forgive you for being a crazy odd Turgonian who probably has had a horrible life for the last couple of years, if you'll kindly disentangle that dead man's fingers from my hair.\"\n\n\"Oh.\" Rias released her to undertake the task, then stood. \"We're back in the artifact room, but they'll figure it out soon. That new window doesn't hide much.\"\n\nThe blasting stick had blackened the floor, turned furniture to shrapnel, and torn holes not only in the ceiling and a side wall but in the building's exterior as well. In the center of the room, the device remained, unharmed, symbols still glowing.\n\nTikaya picked up the book, set her jaw, and strode over to it. There was not much time. Shouts on the other side of the building promised the men were still looking for her.\n\nShe flipped through the chapter on chemicals. \"There.\"\n\n\"Find something?\" Rias stood nearby, weapons loaded and ready.\n\nReluctant to speak too soon, Tikaya pressed the appropriate runes. The regular image blanked out to be replaced by the new symbols. They hovered until she finished. Then, by some alien consciousness, the artifact understood what she wanted, and it arranged them in a way eerily similar to the layout in the book. Even though it was what she hoped for, it sent a shiver down her spine.\n\nA soft click sounded in the core of the device.\n\nTikaya arched her eyebrows at Rias who gave her an encouraging hand gesture. She gripped the edge of the device and waited. Nobody stirred nearby. She tried to decide if the distant shouts were diminishing. Minutes passed, and a deathly quiet fell over the town.\n\nRias walked to the door. He cocked his head, listening.\n\n\"What is it?\" she asked.\n\nRias lifted a finger, cracked the door, and peered into the hallway. He leaned back in with a smile, and Tikaya heard the noise now too.\n\n\"Snoring?\" she asked.\n\n\"The two men out there are sleeping, and the air has that sweet smell of chloroform.\"\n\nTikaya exhaled slowly. \"Good. That should mean we're safe from being shot or knife-stabbed for the moment. Of course, now we have a lesser problem.\"\n\n\"How to wake everyone up, leave town, or even leave this room without succumbing to unconsciousness ourselves?\"\n\n\"That's the one.\"\n\nShe dropped to her back on the floor to gaze up at the writing beneath the machine. Before, she had been trying to translate it. Now, she just thought about cutting the artifact off. Only one of the groupings did not have alchemical elements in it. The first one. Nothing so obvious as a switch stood out anywhere on the machine, so she poked and prodded that grouping. They sat flush and she did not expect them to move, so she nearly cracked her head on the bottom of the box when they did. By pushing and twisting, she could rotate them.\n\n\"What'd you do?\" Rias asked. \"The symbols are flashing.\"\n\nShe could rotate them further, but she paused and peered up at him. \"You sure you want those people awake again?\"\n\nHe smiled gently. \"I'll possibly regret it later, but yes. We'll need help to tackle the tunnels.\"\n\n\"Or you and I could devise some kind of masks, gather as many supplies as we need, take a couple of those dog sleds down the coast until we reach a port, and then sail somewhere far away, leaving the empire to deal with its own problems.\"\n\nRias sighed and gazed into the night. \"I cannot.\"\n\n\"Even though these people left you to die? Even though they probably got themselves into this situation?\"\n\n\"Even though,\" he said. \"But...\" He took a breath and, with palpable reluctance, said, \"If you want to go, I'll keep them busy long enough for you to do so. It's about three hundred miles south to Tangukmoo. If you grabbed a dog team and supplies, I'm sure you could make it in a couple of weeks. It's technically an imperial town, but it's eighty percent natives, and I suspect they'd hide you just to irk us. After the thaw, trade vessels come in to barter for whale oil and bone. With your skills, I'm sure you could bargain for passage and find a way home.\"\n\n\"Sounds like a lonely journey without any company,\" Tikaya said.\n\n\"Probably.\"\n\n\"Last week, you told me your people really needed my help. What's changed?\"\n\nHe looked back and forth from her to the dead bodies. \"This is only the beginning, Tikaya. It's going to get worse. I suspect this is also the only opportunity you'll have to leave.\"\n\nThat Rias offered meant a lot, but the journey he described would not be a speedy one. It was likely Bocrest would make it back to civilization first and send the order to have her family assassinated long before she reached home.\n\nShe rotated the grouping of runes as far as they would go. The crimson symbols in the air winked out. \"When I'm complaining later about how horrible it is out here with your marines, remind me I had my chance and was an idiot who gave it up.\"\n\nThe glum expression on his face waned, and one side of his mouth curved up. He pulled her to her feet and wrapped her in a hug, mindful of her injured shoulder. His stubbled jaw brushed her cheek, and a pleasant shiver ran through her.\n\n\"Only if you remind me I was an idiot first,\" he murmured.\n\n\"Deal.\" Tikaya wondered if that talk of marriage had been inspired only by the moment, by the uncertainty that they would live to see dawn, or if it meant something more. She lifted her good hand to brush strands of hair away from the cut on his temple.\n\nRias drew back slightly, eyes flickering, watching her face. Soft breaths frosted the air between them, and some distant part of her mind announced that this was a ridiculous place and situation for a first kiss, but what if they didn't survive the coming weeks? What if there were no other opportunities to be alone together? What if...\n\nRias bent his head and kissed her gently, warm lips as welcome as the sun in this frozen wasteland. Forgetting about her injury, she started to wrap her arms around him, to pull him closer. Pain blasted her shoulder, and she gasped at the cruel reminder.\n\nRias drew back, wincing, eyes guilty. \"Sorry, my fault. I'll go find the sawbones.\"\n\n\"No, it's all right. I was just...\"\n\nBut he had already grabbed his rifle. He hopped through the broken wall with a quick wave before disappearing into the snow.\n\nTikaya wrapped her arms around herself. She already missed his warmth. And his last words sent a thrum of worry through her. The man who would come to deliver her medical attention was the brother of the man she had just killed.\n\n# CHAPTER 10\n\nSomething bumped Tikaya's foot, jerking her awake. She sat up and cracked her head on the bottom of the device. Her shoulder offered its own jab of pain as her journal and pencil clattered to the floor.\n\nBocrest and Bones loomed over her. The captain's presence surprised her. Surely the ship's commander usually stayed with his vessel, but then this was no ordinary inland excursion.\n\nThuds from the ceiling announced someone walking around in the crawl space. Probably retrieving bodies.\n\nBeleaguered red eyes haunted the sawbones's stubbled face, and an invisible weight slumped his shoulders. He must already know his brother was dead, but he could not know she had fired the fatal shot. She hoped. He carried a black leather bag, and she swallowed, wondering if he truly kept a saw in there.\n\n\"Get up, librarian.\" Bocrest eyed the device. \"Bones has a lot of men to tend.\"\n\nAfter the battering she had taken, Tikaya expected more pains as she crawled to her feet, but she had not slept long enough for her body to stiffen. Only the shoulder throbbed. Darkness still smothered the snow outside. Given the icy temperature and the hard tile floor, she was surprised she had slept at all. She had painstakingly copied the two hundred alchemical elements into the journal and had been sketching the runes on the bottom of the device when she nodded off.\n\n\"Did Agarik make it?\" Tikaya asked.\n\nBocrest prowled around the device, started to touch an indentation, but decided against it. \"He got jumped by\u2014it doesn't matter who by now\u2014but he was cut up pretty bad and left to bleed in the snow. Bones stitched him up earlier.\"\n\nRelief and regret mingled in her mind. If she had not asked Agarik to fetch Rias, he might not have been hurt at all. Would he resent her for it?\n\n\"Where's Rias?\" she asked.\n\nBocrest scowled. \"Prisoner Five is making some concoction in the vehicle house. Said we'll need it in the tunnels. After that, he'll be shackled again.\"\n\n\"Let's see your shoulder,\" Bones told Tikaya.\n\nShe eased her parka off under the cool gazes of the two officers. She was surely too old to want someone to hold her hand while a doctor worked on her, but she wished Rias had come back. Strange that he had disappeared so abruptly. Had he felt guilty about more than her shoulder?\n\nBones huffed and tossed her parka aside. Apparently impatient with her undressing speed, he unfastened the buttons of the black uniform jacket for her. Uneasy, she wondered how much disrobing she would have to endure for this medical treatment. Fortunately, Bones left her undershirt on. Icy but professional hands probed her shoulder. She tried not to wince.\n\nBocrest nodded at the device. \"You figure out what this stuff says?\"\n\n\"Some of it,\" Tikaya said. \"I can only guess at the writing on the bottom, but the context gives me clues. If I get more samples, also in context, I'll be able to make some good guesses.\"\n\nBocrest's grunt did not sound impressed. Curse him, she and Rias had saved the marines\u2014again. Why couldn't the captain acknowledge her usefulness?\n\nFootsteps sounded above, and shards of wood rained from the biggest hole in the roof. Sergeant Ottotark slithered over the edge and dropped down behind Bones.\n\nTikaya groaned, but he did not look at her.\n\n\"Sorry, Bones. Your brother and Private Choyka are dead.\" Ottotark gripped the man's shoulder.\n\nBones's jaw clenched, but he did not otherwise react.\n\n\"I'll get a team to lower them down for the funeral pyre.\" Ottotark nodded to the captain and left.\n\nTikaya relaxed a smidgeon. Bones made a sling from a large square of cloth and secured her arm.\n\n\"You'll be fine in a few days,\" he said. \"Sir, I'll attend the others if you don't need anything else here. I'd prefer to keep busy.\"\n\n\"Yes, go,\" Bocrest said.\n\nBones left, head down, shoulders slumped further.\n\n\"What's the purpose of this device?\" Bocrest asked.\n\nTikaya rubbed her shoulder. \"My best guess? Scientific experiments. They probably wanted to observe the somatic and neurological effects certain gases had on their specimens outside of a controlled environment.\"\n\n\"What kind of specimens?\" Bocrest asked.\n\n\"Look in a mirror.\"\n\n\"Turgonians?\"\n\nTikaya hesitated, almost tempted to play upon his paranoia. She had not yet figured out how she could ensure her family's safety while escaping with her life, but she would probably have more opportunities later if she convinced Bocrest her words were trustworthy now.\n\n\"Humans, animals.\" A cold gust blew snow through the broken wall, and Tikaya grabbed her parka. \"I suppose Turgonian enemies could have brought it here and turned it on.\" She thought of the Nurian captain's orders; the Nurians were smart enough to not want anything to do with the artifacts. \"Or your own people might have done it out of stupidity.\"\n\nBocrest's gaze grew frosty.\n\n\"Stupidity isn't a trait unique to Turgonians,\" Tikaya said by way of apology.\n\n\"Apparently not.\" Bocrest continued to glare. \"Prisoner Five says Lieutenant Commander Okars attacked him, and he was forced to kill my officer in self-defense.\"\n\nUnease trickled down her spine. Uh oh. Why had Rias said anything? Maybe the marines never would have thought to look for bodies in the attic, and, even if they did, in the craziness anyone could have fired at anyone. Rias could have feigned ignorance and no one would have known. But, no, he had felt guilty\u2014or honor-bound\u2014to explain the dead officer. She could not fault him for being an honest man, but his loyalty to these marines, to the empire, might prove disastrous for her. Or maybe not. He had covered for her, though she was not sure whether to be relieved or not. Surely his position here was as precarious as hers.\n\n\"I know who he is,\" Bocrest said, \"who he was, and now that he's...himself again, I doubt he'd intentionally kill an imperial marine, nor do I believe he's inept enough to accidentally dispatch someone in self-defense.\"\n\n\"We were all under the influence of that device,\" Tikaya said. \"Rias\u2014\"\n\nBocrest drew his arm back, and she turned her cheek, expecting a blow. He curled his fingers into a fist, but jerked it to his side. A vein at his temple pulsed. \"I don't know what he's told you, but you will not refer to him as anything other than Prisoner Five. He lost his right to a name, and I don't want my men conflicted on who to follow out here.\"\n\n\"What did he do?\" Tikaya whispered. And who is he, she almost added. But for the ill timing of that blasting stick, she might know by now. Someone who was Bocrest's equal, or maybe even a superior? Was Rias old enough to be an admiral?\n\nBocrest stepped back, and his eyes widened. \"You don't know?\"\n\nShe shook her head.\n\n\"Cruel ancestors, what a waste. He gave up everything, and your people don't even know.\"\n\n\"What?\" She reached for his arm. \"Please, tell me.\"\n\nBocrest scoffed and turned away. He grabbed the rifle and knife, making sure not to leave her any weapons. \"Self-absorbed scientists,\" he muttered on his way out.\n\nTikaya dropped her arm. She thought back to the first conversation she had with Rias, when he asked if her president was still alive. Was that what Bocrest referred to? Had Rias done something for her people during the war, something that had turned the Turgonians against him? If that was the case, why hadn't he told her right away? If he had done a good deed for Kyatt, he might be allowed to come live on her island, and maybe he'd be someone her family could like, and...\n\nShe groaned and rubbed her face. When had he stopped being the enemy soldier and turned into someone she wanted to bring home to meet her parents?\n\n* * * * *\n\nWeariness plagued Tikaya's limbs as she marched after the squad of marines, her arm in the sling, her crampons replaced with snowshoes. The new footwear was almost as awkward to walk in as swim fins, and she struggled to keep up\u2014and upright. There had been no rest after the funeral pyre. They traveled east, in the shadows of jagged white mountains that dominated the southern horizon. To the north, the flat icy tundra stretched until it blended into the pale blue sky.\n\nForty men remained, with fifteen dead back in Wolfhump, and many carried double loads. Dogs, too, had been lost and the teams pulling the sleds slouched along, as tired as she. A sergeant marched alongside the squad, singing a cadence that condoned plundering farm goods and stealing daughters from conquered nations. Or maybe it was stealing farm goods and plundering daughters. Tikaya tried to ignore the words, though she found her steps matching the encouraging refrains of left, right, left.\n\nFor the fortieth or fiftieth time, she glanced behind. Wrists shackled again, Rias walked with a small team tasked with carrying the boxes of blasting sticks. A precautionary couple dozen meter gap lay between them and the main group, though, oddly, the captain walked at his side. She did not know what they spoke of, though his presence served as a deterrent to keep her from strolling back to walk with Rias. She had not seen Agarik since the day before, but his injuries must not be too severe, for he was ahead with the scouting team. Separate from the marines, separate from her two allies, she felt the loneliness and oppressive cold of the tundra. She was tempted to go back to walk with Rias even if it meant enduring the captain's sarcasm.\n\nA dead arctic jaeger alongside the trail diverted her thoughts. The large bird's white-tipped wings were broken, its head smashed in, but no predators had sampled its flesh. Had it simply fallen from the sky? Two sets of snowshoe prints around it meant the scouts had stopped to look.\n\nLong years had passed since her biology classes, so she left it without further examination, but she turned her attention to her surroundings as she continued on. Over the next few miles, she spotted other downed birds, all undisturbed by predators. An uneasy feeling shrouded her, and she wondered what would await them at the fort. More dead men? Another device?\n\n\"Prisoner Five, come back here!\" Bocrest shouted.\n\nRias had set down his box of blasting sticks, and he churned across the tundra. Bocrest plowed after him, rifle in hand.\n\n\"Sir?\" one of the marines in front of Tikaya called. \"Do you need help?\"\n\nBocrest waved, and the back two men stamped out of formation, flinging snow as they raced into the drifts with high-kneed steps. Tikaya veered after them, afraid they would think Rias was trying to escape and take violent measures\u2014as if Rias would be dumb enough to run away with everyone watching. Unfortunately, her slog through the unbroken snow was less effective than theirs. Even with the snowshoes, she sank deep with each step, and she tripped twice before reaching the gathering.\n\nRias stopped, knelt, and picked up something. Bocrest and the others scrambled over, and Tikaya floundered up in time to hear the red-faced, scowling Bocrest speak.\n\n\"What are you doing, Five? Are you trying to get yourself shot? Prisoners don't get to take unannounced side trips.\"\n\nRias lifted his goggles to peer at his find.\n\n\"What is it?\" Tikaya asked.\n\nShe attempted to slip past the other marines to join him, but one of them took a step at the same time and landed on the edge of her snowshoe. She sprawled, face heading toward the powder. Rias lunged, caught her, and even managed to keep from jarring her shoulder.\n\n\"Slagging librarians,\" Bocrest grumbled.\n\nFace red from more than the cold, Tikaya got her snowshoes beneath her. \"Thank you. It seems I'm always tumbling into your arms.\" She sighed, appreciative but a little envious too. Neither shackles nor snowshoes made him ungainly.\n\n\"I don't mind,\" Rias said. \"Makes me feel useful.\"\n\nBocrest snorted. \"Any excuse to grab a tit.\"\n\nThe two marines sniggered, and Tikaya stepped out of Rias's arms, her cheeks warm. Rias merely shook his head at Bocrest, like a father disappointed in a wayward child.\n\nBocrest scowled. \"What did you find, Five?\"\n\nRias held an empty, one-inch cube of glass, or what appeared to be glass, on the palm of his gloved hand. One corner was broken, though the evenness of the cut suggested the hole planned rather than accidental. He flexed his fingers upon the cube. Though the thin sides appeared fragile, they did not bend or crack under pressure.\n\n\"It glinted in the sun and caught my eye,\" Rias said.\n\n\"It looks like someone's trash,\" Bocrest said.\n\nA dark shape loped across the tundra, and the two marines lifted rifles. A black wolf, so gaunt its ribs showed even at a distance. After her encounter with the berserk animals in town, Tikaya hoped the men shot it quickly, before it could attack.\n\n\"Hold,\" Bocrest said. \"Why's it so scrawny when there are dead birds everywhere?\"\n\nShe glanced at him, surprised by the perspicacious comment. He was right, though. It was odd. And this wolf, unlike the ones in town, gave no indication of aggressive behavior. Indeed, it did not seem concerned about the humans at all.\n\n\"It is the end of winter, sir,\" a marine said. \"Maybe it was a rough one for the animals.\"\n\n\"That wouldn't explain why it's not eating those free meals,\" Bocrest said.\n\nThe wolf loped parallel to the squad, then paused at the corpse of a jaeger. It sniffed and pawed at the bird, and Tikaya expect it to take a chomp. Instead it lifted its muzzle and howled. The oscillating mournful sound made her shiver. Another wolf answered from the foothills, its howl just as forlorn.\n\n\"He seems to find the fowl unpalatable,\" Rias mused.\n\nHe turned his attention back to the cube, lifting it so the sun shone through the glass. Tikaya sucked in a startled breath. A familiar symbol etched one side.\n\nShe took it from Rias. \"I recognize that. It's one of the symbols repeated often in the rubbings the captain gave me.\" She nodded toward Bocrest. \"Know anything?\"\n\n\"Shit,\" he said.\n\n\"Very elucidating, thank you,\" Tikaya said.\n\n\"Where'd those runes come from, Bocrest?\" Rias asked in a tone of command.\n\n\"That's top secret.\"\n\n\"If you want Tikaya to translate this for you, she needs to know everything about the symbols.\"\n\nBocrest ground his jaw. Tikaya had made that argument before, and the captain had ignored it, but he waved the marines to go back to the squad. When he, Rias, and Tikaya were alone, he spoke.\n\n\"Last month, a black box covered with those runes was delivered to the research department of the biggest university in the capital. No name, no identification. They should have buried it somewhere and forgotten about it, but scientists being scientists...they fiddled with it, let out some kind of airborne poison. It killed everybody on campus. It was late in the evening, so not as bad as it could have been, but hundreds still died.\"\n\nTikaya dropped the cube and stepped back. In her haste, she almost tripped over her snowshoes again. Rias's lips flattened, and he rubbed the fingers of his glove together, as if he could wipe off any taint from the cube.\n\n\"It's not the same thing, though,\" Bocrest said. \"The bodies on campus were horribly mutilated, and these birds barely look dead. Maybe our people at the fort are fine.\"\n\n\"Those are carrion birds,\" Rias said.\n\nTikaya swallowed with grim understanding. \"Not as bright as the wolves then, eh?\"\n\n\"It seems not.\"\n\n\"What are you talking about?\" Bocrest asked.\n\n\"We're just guessing at this point,\" Rias said, \"but it's possible our people are dead by the means you're familiar with, and the poison was toxic enough that even the carrion beasts that tried to feed off them died.\"\n\nBocrest scowled at the dead bird. \"Oh.\"\n\n\"Will it still be toxic if we get close?\" Tikaya asked. \"That cube wasn't covered by snow, so this couldn't have happened that many days ago.\"\n\n\"I don't know,\" Rias said. \"It depends on whether we're looking at an area denial weapon or something short-lived, designed simply to kill.\" He faced Bocrest. \"The scouting party. How far ahead are they?\"\n\nBocrest's face froze, and a long moment passed before he said, \"They'll be there by now.\"\n\nTikaya's gut twisted. Agarik. She had not even had a chance to apologize to him. She prayed it wasn't too late.\n\n# CHAPTER 11\n\nThe walled army fort squatted in the foothills, small and insignificant compared to the towering white mountains plunging it into shadow. Tikaya stamped her feet to keep warm and wondered if she was crazy for wanting to travel the last half mile to the gate. No soldiers manned the massive guns perched atop the ramparts, nor did any smoke waft from the chimneys inside. Rias's guess that everyone was dead seemed likely, but perhaps whatever weapon had done it waited within those walls. And such a weapon might be inscribed with language clues like those on the Wolfhump artifact. Now that she had made a little progress, the prospect of more tantalized her.\n\nRias meandered across the foothills, pausing to pick up something here and there. More of those cubes, she feared, not sure whether they were safe to touch or not. He carried a small notepad and scribbled something in it whenever he found one. He still wore his shackles, and two guards trailed dutifully behind him. Did Bocrest not know they were superfluous at this point? Rias had shown no interest in escaping since he learned what was at stake.\n\nShe hoped that loyalty to the empire would not result in his death. Or hers. She would much prefer to see him strolling on one of her island's beaches, picking up agates and sand dollars instead of vials that might have housed lethal poison. And in this vision, she saw him with less clothing on. She grinned. Or none. She thought of the scar that bisected his eyebrow and wondered what other battle wounds stamped his olive skin. He had filled out since she first saw him in rags in his cell, and she imagined broad shoulders and powerful muscles beneath that parka.\n\nA guilty pang ended her thoughts. She believed Parkonis would have wanted her to go on and find love again\u2014though not with a Turgonian, no matter how academically inclined\u2014so it was not that. It was that she had never daydreamed about him with his shirt off. Parkonis had been boyishly cute with freckles and a mop of red-blond curls, but not the type to inspire women's fantasies. Of course, she was hardly the type to inspire men's fantasies. She hated to dwell on it, but feared she would not be able to compete with others if she and Rias survived to return to a world where she was no longer the only woman for hundreds of miles.\n\nSnow crunched behind her.\n\nA pair of privates approached, and she braced herself for insults or crude comments. Acne scarred one's face, and neither appeared older than twenty, though like most of the men here they were taller than she and no doubt dangerous.\n\n\"Ma'am, we're, ah...\" The speaker glanced at his comrade, who gave an encouraging nod. \"We're having rations.\"\n\nEr, what did that have to do with her? \"Yes?\"\n\nBehind them, marines sat in groups of four or five and shared lunch while the officers conferred in a cluster. More than one man snoozed against his rucksack, oblivious to the frosty environs.\n\n\"You could join our mess if you wanted.\" The speaker nodded to a knot of young men busy chatting, laughing, and stuffing crackers into their mouths. One waved. \"We've got extra tooth dullers and\u2014\"\n\n\"Tooth, what?\" Tikaya asked.\n\n\"Tooth dullers. You know, hardtack. It's right awful stuff, but Private Ankars has some taffy his mum gave him\u2014his mum always posts him the best sweets\u2014and anyway if you wanted you could come share with us.\"\n\n\"Oh, I...\" After so much hostility from the marines, this kindness stunned her. The privates must know some of what had happened with the device, that she and Rias had been the ones to render it innocuous. \"Thank you. It's considerate of you to invite me.\"\n\nRias strode their direction, brow wrinkled. The privates blanched when they spotted him.\n\n\"You're welcome any time, ma'am.\" The speaker waved to Tikaya, and he and his comrade scurried away.\n\n\"They bothering you?\" Rias asked.\n\n\"No. They invited me to lunch.\"\n\n\"Ah?\" His brow smoothed and a smile plucked at his lips. \"That's an improvement.\"\n\n\"Yes.\" She nodded toward the pockets of men. \"It's amazing they can sleep and laugh in the face of death and inexplicable alien horrors.\" As soon as she said it, she blushed. What about her? Fantasizing about Rias on a beach a few minutes earlier?\n\n\"That's a trait shared by soldiers everywhere. The officers handle the worrying.\" The grimness returned to his expression, and he held out his hand. A glass cube identical to the first rested on the palm. \"I've found several now. The radial pattern and the distance from Fort Deadend implies...\" He sighed. \"I better see what's inside before jumping to conclusions.\"\n\n\"Are we going in?\"\n\n\"Yes, good news there. The scouts are alive. The lookout has a spyglass and spotted them moving around inside.\"\n\nTikaya exhaled with relief. \"Good.\"\n\nRias nodded. \"Though you might want to wait until later for lunch.\"\n\n\"Why? Are there better rations inside?\" Even as she finished the question, the meaning of his comment washed over her. They anticipated more dead bodies, right. She waved a glove to let him know she understood.\n\n\"Fort Deadend isn't known for its cuisine, no, but if I can escape Bocrest's guards, maybe we can share a meal?\" He arched his eyebrows.\n\nAnd another, less abbreviated, kiss? She smiled at the thought but couldn't resist the urge to tease him. \"I don't know.... Those privates over there have taffy. Can you top that?\"\n\n\"Ah, perhaps not.\" His expression grew wistful. \"I fear I am a man with few resources these days.\"\n\nShe patted his arm. \"You'll have to regale me with stories then. Such as why this place is called Fort Deadend. Are there more reasons than the obvious?\"\n\n\"Not really. You've generally pissed in some general's tea cup if you get stationed out here. There's a pass through the mountains south of here, and the theoretical purpose of the installation is to guard against invasion from the north. But the route is as hospitable as an avalanche, so the likelihood of someone marching an army through it is close to nil. There's a lot of gold in the hills, though, and foreigners trespass to set up mining operations. Patrols watch for that, and I imagine the fort commander has orders to keep an eye on the canyon where the tunnels were discovered as well.\"\n\nTikaya thought of the invisible Nurian assassins. She was beginning to think they had transported back to their own ship the night of the attack, but that did not mean others with their skills were not out here. \"Practitioners wouldn't have much trouble sneaking by this fort to get inside.\"\n\n\"They would have had to know about the place first, though I suppose after twenty years secrets are bound to get out. The Nurians obviously know.\"\n\nVoices sounded ahead\u2014the scouting group returning. Agarik came at the end, head bowed, shoulders drooped, though he kept his rifle crooked in his arms, ready to use. The leader headed straight for Bocrest and his officers, but she caught Agarik's eye and he tramped up the hill toward her and Rias.\n\nA livid red gash dotted with black stitches ran from the side of his cheek to his nose, and the stiffness of his movement hinted at injuries beneath his clothing.\n\nTears pricked her eyes. She never should have sent him off alone.\n\nBefore he could speak, she stepped forward and hugged him. \"I'm sorry. It's my fault you were hurt.\"\n\nHe seemed startled by the embrace, but rearranged his rifle to return it. \"No, don't think that. I'm the idiot who let himself get ambushed.\"\n\nHis words did nothing to assuage her guilt. When she stepped back, she could not look away from that cut. It would be a permanent scar.\n\n\"Is there anything I can do?\" Tikaya asked.\n\n\"Hugs are good.\" Agarik gave Rias a hesitant smile. \"From anyone who wants to share them.\"\n\nTikaya glanced at Rias in time to spot a neutral expression shift to bewilderment.\n\n\"What?\"\n\nThe word sounded harsh, and, though Tikaya suspected the tone more a result of surprise than anything else, Agarik's smile fell.\n\n\"Sorry, sir,\" he said. \"I didn't mean, uhm.\"\n\n\"What'd you find at the fort, corporal?\" Rias asked.\n\nAgarik straightened, face composed. \"Everyone's dead, sir. Ugly dead. Their skin and muscles were melted off like wax on a candle. You couldn't even tell who was who if they weren't wearing uniforms with name patches.\"\n\nTikaya shared that's-what-we-were-afraid-of glances with Rias.\n\n\"And there's something you'll want to see,\" Agarik said. \"Both of you.\"\n\nBefore she could ask for details, an officer yelled, \"Corporal Agarik, get over here!\"\n\nHe saluted Rias before hustling off.\n\n\"You should hug him next time,\" Tikaya said.\n\nThat bewilderment returned. \"Marines don't hug.\"\n\n\"Have you talked to him at all?\"\n\n\"Not as much as you, apparently.\" He tilted his head and arched an eyebrow.\n\nIt took her a moment to match his concerned expression to the hug she had given Agarik. She almost laughed. As confident as Rias was about military and mathematical matters, she was surprised he did not share that confidence when it came to women. As if anyone here could offer notable competition.\n\n\"Pack it up!\" Bocrest's voice floated across the hill. \"We're moving out!\"\n\nTikaya laid her hand on Rias's sleeve before he could turn away. \"Agarik wants your attention, not mine.\"\n\nRias stared at her. \"Oh.\"\n\n\"You two prisoners want to join the group or you going to stay out here and work on your sun tans?\" Bocrest shoved a rucksack into Rias's arms as he stomped past.\n\n\"I can't imagine that man having a wife,\" Tikaya muttered as she headed off to retrieve her own pack.\n\n\"I've not met the woman,\" Rias said, \"but I've heard she's as obnoxious as him and the undisputed master of the household.\"\n\nTikaya would have been content to march at the end of the squad\u2014the going was a lot easier when numerous snowshoes had tamped down a trail\u2014but Rias strode through the drifts with his long legs and overtook the men. Determined to keep up, she forced her own strides to unnatural lengths. Sweat soon plastered her clothing to her body and soaked the fur lining of her gloves. She removed her wool cap and stuffed it in a pocket.\n\nRias made it to the gates ahead of the marines, and Tikaya trailed close behind. The twenty-foot-tall wooden doors stood open, offering no sign that the fort had been attacked, but two men lay dead at their guard post. At least, Tikaya assumed they were men.\n\nShe thought the night spent with the corpses in Wolfhump would have inured her to death, but these were worse. Blisters and burns had seared flesh and muscle to the bone, mutilating the marines beyond recognition. Pustules even coated the bloated, protruding tongues. The eyes had popped like blisters themselves.\n\nShe was glad she had not had lunch.\n\nAgarik caught up and walked between them. Sweat dampened his face too. \"This way, sir, ma'am. Captain said it's all right to let you go first.\"\n\nTikaya snorted. As far as she could tell, the shackles meant little to Rias and it seemed he had already determined that he was going in first, captain's wishes notwithstanding.\n\nInside the fort, snow had been plowed into piles, so they unstrapped their snowshoes. Sand coated the icy cement. Agarik led Rias and Tikaya past wooden barracks, office buildings, and a couple of cavernous vehicle structures that housed plows, trucks, and other steam-powered transports. If not for the silence and the dead men, she would have guessed it a normal work day at the fort. People had been caught on errands, while practicing combat in an arena, and, in one spot, driving a snow plow that had subsequently crashed into a barracks building. No smoke wafted from the stack, but Tikaya paused to open the furnace door. The fire had gone out, but when she removed a glove and stuck her hand inside, a hint of heat remained in the ashes.\n\n\"People were alive here, no more than a day or two ago,\" she said.\n\nBut Rias, following on Agarik's heels, had gone on and did not seem to hear her. His head was tilted back, and she followed his gaze. A two-story office building rose ahead, a large corner of its roof sheered off. Splintered wood and sheet metal roofing shingles scattered the cement around a sleek black cylindrical object embedded in the walkway. Tikaya's step slowed. It was made from the same material as the artifact in Wolfhump.\n\nShe joined the men around the rubble. The landing had not dented or even scraped the cylinder, though it had gouged a hole in the cement, and jagged cracks radiated in several directions. Strips opened like flower petals on one end of the device, and dozens of square indentations lined the insides.\n\nRias produced one of the cubes and slid it into a slot. It clicked. Perfect fit.\n\n\"The delivery mechanism,\" he said.\n\nBocrest and a couple others approached, though the captain had the sense to keep most of his men from trampling around the artifact. He sent them off to secure the fort with a few terse orders, then stepped closer.\n\n\"What is that thing?\" Glass crunched under his foot.\n\n\"Careful.\" Rias flung a hand up. \"If some of the cubes didn't release their contents in the air, stepping on a full one could be deadly. For us all.\"\n\nFor once, Bocrest had nothing sarcastic to say. He gulped, lifted his boot, looked around, like he might set it somewhere else, then decided it was safer back where it had started.\n\n\"In the air?\" Tikaya asked. \"You don't think this is like the device in Wolfhump? Something that distributes its deadly load from the ground?\"\n\n\"No,\" Rias said. \"I think it's a rocket.\"\n\nThe only rockets Tikaya was familiar with involved fireworks, though she supposed Turgonians might have experimented with more sophisticated fuel-powered projectiles. Rias had studied ballistics in school, after all.\n\nIt took a moment for the ramifications to sink in. Tikaya's eyes widened when they did. \"If this is a rocket, that means someone fired it, someone with the knowledge to do so.\"\n\n\"Yes,\" Rias said grimly.\n\nTikaya stared at the artifact. Was it possible the original builders were still around? Or\u2014she eyed runes running down one side\u2014had someone already been in the tunnels and translated the ancient language? Such strong disappointment flooded her that she sank to her knees. Surely she had not asked for this gruesome mission, but all along there had been the promise of being the first to translate a previously unknown language and share it with the world. A feat that would earn her a place in the history books.\n\nRias knelt beside her and put a hand on her good shoulder. \"Are you all right?\"\n\nAware of the others, she said, \"Yes, just tired.\"\n\n\"Nurians,\" Bocrest growled. \"Those cussed Nurians are responsible, have to be.\"\n\nRias arched inquiring eyebrows at Tikaya. She guessed his question and nodded permission.\n\n\"I don't think it's them, at least nothing sanctioned by their government.\" Rias explained the orders he and Tikaya had found on board the Nurian ship.\n\n\"Then who?\" Bocrest demanded.\n\n\"The Turgonians have many enemies,\" Rias said.\n\nBocrest tugged off his cap and scrubbed his short steely hair. \"This whole slag pile is giving me a headache.\"\n\n\"There's something else to consider,\" Rias said, tone grim. \"We haven't hidden our approach to this fort, so it's very possible whoever did this knows we're here.\"\nCHAPTER 12\n\nTikaya sat cross-legged next to the rocket and finished copying the runes into her journal. Coldness seeped through her trousers to numb her backside, and shivers made her pencil hand shake. As if writing with gloves on was not bad enough. She ought to move somewhere warmer. But, oh, there was another symbol she recognized from the table of elements. And interesting how each grouping on the rocket held thirteen runes. Could the different prime clusters alter the meaning of\u2014\n\nLight splashed across her pages, and Tikaya dropped her pencil.\n\nAgarik stood beside her, holding a lantern. Only then did she realize twilight had descended on the fort.\n\n\"Thank you,\" she said. \"When did you leave to get that? And where'd everyone else go?\"\n\nHe stared. \"You didn't notice people coming and going, ma'am? Talking? Arguing?\"\n\n\"Er.\" She vaguely remembered Rias saying something about finding surveying tools. \"Not really.\"\n\n\"Half the men are retrieving corpses for a funeral pyre while half are off on a mission Five concocted. Looking for people who might have died outside the fort and searching for more cubes and recording where they landed. Seeing how it's night, that sounds about as fun as hunting for a wrench in a scrapyard. He says he can figure out where the rocket was launched from, though, and Bocrest wants it done tonight, so he can take a team up there at first light.\"\n\nIn case someone was up there preparing another weapon to launch at the fort. Yes, that seemed wise.\n\nTikaya rose, an awkward movement with her shoulder in a sling, and her injury twinged despite her care. A noisy yawn escaped her lips. She thought of her bed back home, though right now she would be tickled with a blanket in front of a fireplace.\n\nAgarik's jaw dropped in a noisy yawn of his own.\n\n\"Sorry,\" Tikaya said. \"Are you tasked with watching me again?\"\n\n\"I don't mind, ma'am. Of the jobs I could be assigned, it's not a bad one.\"\n\nMen crossed a nearby courtyard, lantern light bobbing at their feet. They worked in grim silence, bringing wood for a fire. She looked away as others dragged a body from a building and laid it at the end of a grisly queue.\n\nThough tempted to ask Agarik about sleeping arrangements, Tikaya wanted to search the fort for other language clues first. If the marines feared an attack imminent, they might march out at dawn. Somewhere, she hoped to find an artifact she could slip into her rucksack to take home for study\u2014and to prove to her people this nightmare had been real.\n\n\"Mind if we find the fort commander's office?\" she asked.\n\n\"I don't think the captain wants you wandering around.\"\n\nTikaya started to object, but Agarik smirked and spoke first.\n\n\"But since you'll doubtlessly make arguments until you get me to change my mind, we may as well go now.\"\n\n\"Good man.\"\n\nThousands of stars glittering like ice crystals blanketed the black sky above the fort. Tikaya almost felt she could reach up and stir them with her fingers. Here and there lanterns sputtered on lampposts or in sconces. Agarik hugged shadows barely dented by the flickering light as he led her beneath the ramparts and through an alley suffocated with piles of snow. They stopped at the back door of a two-story building. She had the feeling he was avoiding the other men and hoped he would not get in trouble for escorting her around. Her desire to poke through the commander's office kept her from asking.\n\nThey slipped into an unlit kitchen where copper pots and steel counters gleamed, reflecting the lantern's flame. Agarik led her around an island to avoid a body clad in bloodstained cook's whites. Tikaya ripped her gaze from the melted flesh, glad the lighting did not illuminate too many details.\n\nThey passed through a mess hall lined with tables. Plates of bread, carrots, and now-frozen meat waited for someone to finish them. Bodies, some fallen across tables, some sprawled on the floor, matched the place settings. One man had died tending the coal stove in the corner, and the door stood open, ashes spilled onto the wood floor.\n\n\"Surprise attack,\" Agarik said, disgust hardening his voice. \"There's no honor in killing like this.\"\n\n\"I can't believe how quickly the poison acts,\" Tikaya said, chilled by the idea of something that could kill a man before he could even get out of his seat and wonder if something was wrong.\n\nThere were not any bodies at the tables nearest the far door, and wet footprints suggested the marines had started carting them out. Heavy footsteps sounded on the floor above. Tikaya doubted she would make it to the commander's office without being spotted. She hoped no one would question her for snooping about.\n\nThey entered a corridor, and Agarik led her to a stairwell. Broad steps rose toward a second floor, while narrow ones turned a corner and dropped into darkness. That made her pause. A basement in a land where permafrost hardened the earth inches below the surface?\n\nAgarik headed upstairs, but voices came from the lower level.\n\n\"Wait,\" Tikaya whispered, cocking her head to listen.\n\n\"I think...a Nurian,\" someone said.\n\n\"...doing out here?\"\n\nThey had found a Nurian? Had someone been caught nosing around in the carnage of the fort? Her breath hitched. What if it was the person who had fired the rocket? The person who knew enough of the language to know how to fire the rocket?\n\nShe edged closer to the stairs. The men were not whispering, so the discussion was probably not secret, and nobody had forbidden her from exploring.\n\nAgarik gripped her forearm. \"Tikaya...\"\n\nShe frowned at him. \"What do you know?\"\n\n\"The cells and interrogation chamber will be down there. It's not...a fitting place for a lady to visit.\"\n\nTikaya almost laughed. Was he worried she would see something more macabre than the legions of bodies around the fort? She patted his hand before drawing her arm away. \"I just want to see if they found someone who knows more about what's going on than we do.\"\n\nIf Agarik's frown grew any deeper, he would pop the stitches out on that gash. Still, he followed with the lantern when she eased down the stairs.\n\nShe anticipated a pitted rusty iron door streaked with blood at the bottom, but the bland wood was no different from any other door they had passed. Beyond it, gray stone lined a narrow hallway. At the end, light seeped through the cracks of a partially open door. The voices had dropped to murmurs.\n\nOn the way down the hall, Tikaya checked a door to the side, expecting racks filled with torture implements. Instead, it was a supply closet loaded with brooms, lye soap, lanterns, kerosene tins, and painting supplies. So far, this dungeon was not living up to expectations.\n\nChains rattled in the room ahead.\n\n\"Where's the slagging key?\" a familiar voice growled.\n\nTikaya froze. Ottotark.\n\nShe started to turn, not wanting anything to do with him, interesting prisoners or not, but someone thrust the door open. Two marines tramped out, a body between them, but if it was Nurian she could not tell. It was as melted and featureless as the rest in the fort.\n\nThe marines halted.\n\n\"What's she doing down here, Corporal?\" one asked.\n\nAgarik shrugged. \"Looking for language clues.\"\n\nOttotark leaned through the doorway, his eyes narrowing to slits when he spotted her. For a long speculative moment, he stared, and she fought the urge to race up the steps in retreat. He would not do anything with so many witnesses, and surely this was not the time regardless.\n\n\"Go on,\" Ottotark told the marines, \"take the body out. Agarik, come help me get the other one. I expect the captain will want to see the Nurian, so we'll leave him here for now.\"\n\nTikaya stood aside for the men to pass. Agarik headed for the chamber, and she almost bumped into his back when he stopped in the doorway.\n\n\"What is it?\" she asked.\n\nBut Agarik had already moved to the side so she could see. She wished he had not. The amount of human carnage from the last couple days should have numbed her to it, but this dead man was different. The naked Nurian hung from shackles on the wall. His fingernails and toenails were ripped out, flesh mutilated with blades and brands. Someone had cut his genitals off and dug both eyeballs out. The removed organs lay in a tidy pile next to the dangling body, and Tikaya had to gulp several deep breaths to keep from vomiting. Thank Akahe the temperature kept everything frozen, and no odor accompanied the visual horror.\n\n\"By the book,\" Agarik commented.\n\nOttotark nodded. \"Professional job. I'll bet two weeks pay one of our people ran the torture session, but who? This is recent work, done by somebody\u2014to somebody\u2014who showed up after everyone else got their faces slagged like ore in a smelter.\"\n\n\"Unless...\" Tikaya took one more deep breath to steady her gorge. \"Unless the basement protected people here from the poison.\"\n\n\"Look around, idiot woman.\" Ottotark pointed past racks containing wicked metal instruments whose purpose she could only guess.\n\nSteel bars in the shadows formed a pair of cells. A corpse in one had suffered from the same affliction as all those in the fort above. And, of course, she had seen the body those two marines had taken out.\n\n\"Sorry, yes, I'm not thinking.\" Odd that this deliberate cruelty affected her more than the mysterious otherworldly deaths. The marines, even Agarik, seemed to find this torture commonplace. Just when she was thinking some Turgonians might be normal people. \"You must be right. So, one or more of your people got here ahead of us. And this fellow, I wonder if he's the one who launched the rocket. Or...\" She looked past the damage to what remained of the man's features, and her stomach did a little flip. \"I recognize him.\"\n\nOttotark stared at her. \"What?\"\n\n\"He's the bodyguard of the practitioner who attacked me on the ship.\"\n\n\"You sure?\" Ottotark's forehead scrunched. \"He's not looking too recognizable at the moment.\" He threw back his head and laughed.\n\nAgarik rolled his eyes.\n\n\"Come, Corporal,\" Ottotark said. \"Help me drag that other body up to the pyre. We'll leave the librarian to clue hunt, though I don't reckon this bloke was worrying overmuch about languages in the end.\"\n\nHe laughed again, and the inappropriateness ground on Tikaya's nerves.\n\n\"I'm supposed to stay with her,\" Agarik said.\n\nOttotark's humor evaporated. \"That wasn't a request, Corporal. I'm sure the captain didn't mean for you to get out of all the physical labor with your special assignment. You can come back when we're done hauling bodies.\"\n\nAgarik looked at Tikaya, and she gave him a quick nod. He had risked enough trouble for her, and she would give him up for a while if it would get Ottotark out of the room as well.\n\nThough a troubled expression wrinkled his brow, Agarik helped Ottotark heft the other corpse, and they left. Tikaya did not wish to spend a lot of time alone down here, especially since the presence of the dead bodyguard implied the practitioner was around somewhere, probably alive, but she wanted a moment to think things through.\n\n\"So,\" she said in Nurian, as if the man's spirit might hear and help, \"you've been following us all along, biding your time, is that it?\" She thought of the hour or two she had spent alone in Wolfhump and shivered, for that could well have been an opportunity for the assassins, but perhaps they, too, had been affected by the gas. \"You ran ahead here to lay an ambush for me? Or someone transported you here?\" That was a hard skill to master, but not an impossible one. Someone familiar with the arrival area could have done it. \"Either way, it seems you got here after the weapon struck, or you'd have been killed the same way as the others. Unless you launched the weapon yourself and came to check the effectiveness of your work. But, no, your people wanted nothing to do with these artifacts.\" She took off her spectacles and rubbed her face. \"But when you got here, someone was waiting. Was it a Turgonian, or the person or persons who figured this rocket out? Or both?\"\n\nNot surprisingly, the dead man was not talking. Likely he had given up all his secrets to his interrogator. Tikaya glanced around, noting that the pliers, knives, and other implements she could not name had been meticulously cleaned after use and returned to the storage rack. Little else caught her eye, though, and she decided it would be wise to finish up and find a spot with people around.\n\nShe headed back to the stairs. Someone had shut the door at the bottom. She tried the knob, but it did not turn.\n\nA kernel of dread formed in her gut.\n\nShe tried the door again. It could just be a flaky knob, but no. It was locked. She pounded on the door. Maybe someone working upstairs would hear her and let her out.\n\n\"Is anybody out there? Hello?\"\n\nNo one came.\n\nTikaya leaned her forehead against the cold wood. Ottotark must have locked the door on the way out without Agarik noticing. Maybe he had even ordered the men clearing bodies to work in a different building. She swallowed. Odds were Ottotark could make sure Agarik stayed busy for a while too.\n\nThe rack of torture implements flashed into her mind. \"Dolt,\" she cursed herself. Of all the places she could be trapped with that man, this had to be the worst.\n\nShe slipped the razor Agarik had given her out of her boot and unfolded the blade from the wooden handle. It seemed a puny tool compared to those in the other room. If Ottotark worried about her having weapons, he would not have chosen this spot. He obviously did not see her as a threat in close combat, and rightly so. This was no archery competition where she could stand back and plunk arrows into a target.\n\nShe needed to catch him by surprise. She hoped she had time to set one up.\n\nTikaya checked the supply closet. An idea came as soon as she saw the kerosene tins. Unlike the whale oil her people used, the vapors ought to be flammable. She grabbed a paint pan, a full tin of kerosene, and a box of friction matches. She would only have one chance. It had better work.\n\nIn the torture chamber, she closed the door part way. Footsteps thudded in the corridor on the floor above. She swallowed. If that was Ottotark, she did not have much time.\n\nThe safety lid of the unopened kerosene tin thwarted her fingernails, and her shoulder sent stabs of pain through her when she tried to brace the can with that hand. She huffed in frustration. Then she remembered the razor. Agarik's tool would help after all.\n\nShe gouged a hole in the top of the tin and, careful not to spill any on herself, poured kerosene into the paint pan. The fumes stung her eyes, but she dared not slow down. Footsteps thudded on the stairs. For once, she had reason to thank her height. Though the doors had been made to accommodate the tall Turgonians, she reached the top without trouble and balanced the pan. The footsteps reached the bottom of the stairs, and the lock clicked. The door creaked open.\n\nTikaya hopped into the shadows beside the door.\n\n\"I know you're down here, bitch,\" Ottotark said. \"No point in hiding.\"\n\nTikaya's breathing sounded loud in her ears, and she tried to quiet it. She yanked her left arm out of the sling, ignoring the pain, and dug out a fistful of matches. In her right hand, she held a single one. The tremble to her fingers annoyed her. She eyed the pan atop the door, and doubts flooded her. It was too stupid, too obvious. Children did this as a prank; it was not a way to attack an enemy, a bigger stronger enemy who would probably only be angered by the attempt.\n\n\"I'm going to wipe that arrogant defiance off your face,\" Ottotark growled. A door creaked. He was checking the closet.\n\nShe swallowed. Would he notice the missing kerosene? Or would he smell it before he entered her room?\n\n\"You don't act like a prisoner, not like you should, and the captain's too lenient. You screwed us during the war. You gave our secret orders to the Nurians. They knew just how to ambush the Crusher. All those men\u2014my brother\u2014didn't have a chance.\" The heavy footsteps thudded closer. \"I've been waiting to avenge him since you came on board.\"\n\nOttotark shoved the door so hard it cracked against the stone wall. Tikaya jumped back and almost didn't see the trap spring. Kerosene drenched him, and the pan clattered to the floor.\n\n\"What the\u2014\" He stumbled into the room.\n\nShe slammed a boot into his backside with all her strength. He pitched forward, though he turned the fall into a graceful roll and came up facing her. She kicked the door shut and scraped a match against the stone wall. The scent of sulfur stung the air.\n\nOttotark snarled and started to lunge. She almost threw the flame then and there, but she kept herself to holding it out and shouting.\n\n\"That's kerosene you're covered in!\"\n\nOttotark paused, just short of springing.\n\n\"Ever wanted to be a human torch?\" Tikaya asked. \"You could look just like all the dead soldiers here in about three seconds.\"\n\nHe snorted. \"You've not steel enough to kill anyone.\" But his gaze stayed on that match, and he did not advance.\n\n\"I've killed more people than I can count since this nightmare started. First on the Nurian ship and then in Wolfhump.\"\n\nOttotark's eyes widened. \"The captain was right. You killed Commander Okars!\"\n\nAdmitting that might be stupid, but it might convince him she was a threat. She needed to resolve this before Ottotark realized she would run out of matches eventually. She had to gamble. \"Yes, and I liked him a lot more than I like you.\" She tried to put a manic expression on her face as she looked him up and down. \"I can't imagine anyone would even miss you. Maybe I should just\u2014\"\n\n\"Wait!\" He licked his lips. \"What do you want?\"\n\n\"Your word that you won't touch me ever again.\" It was another gamble. Just because Rias's word meant a lot to him did not mean every Turgonian shared that viewpoint. Still, there were several words for honor and promises in their language. Maybe it was not that big of a gamble.\n\nAnd Ottotark winced. If his word meant nothing, he would have agreed quickly. He would not be thinking it over.\n\nThe first match was burning low. She lit a second from it, then tossed the discard into a puddle of kerosene at Ottotark's feet. The vapors ignited and flames roared, hurling shadows from every corner of the chamber.\n\nOttotark leapt away, slamming his back against the dangling corpse. \"Shit, woman!\"\n\n\"Your word or you're next.\" The confident steely tone of her voice surprised her.\n\n\"Fine, fine. I won't even touch you if you're dangling on the edge of a cliff, begging me to pull you to safety.\"\n\n\"Excellent.\" Tikaya stepped to the side of the door. \"You first. We're going up to the others before I let you out of match range.\"\n\nTeeth bared, he edged past her and sprinted up the steps. Tikaya would not have kept up, but he crashed into someone in the hall upstairs.\n\n\"Sergeant Ottotark, what are you doing?\" It was Bocrest.\n\nTikaya stepped out of the stairwell, and both men turned. Ottotark thrust an accusing finger.\n\n\"That crazy bitch doused me in kerosene and threatened to light me on fire!\"\n\n\"Because you locked me in the dungeon so you could rape me, you sadistic prick.\" The burning match had gone out, but she clutched the box in her hand, prepared to light another if need be. She eyed Bocrest, who lifted a lantern and eyed her back.\n\n\"Ms. Komitopis,\" he said, \"we call it an Interrogation Station, not a dungeon.\"\n\nCurse him, he sounded amused. At least he had used her name. That was a first.\n\n\"Ottotark, go get cleaned up and find a rack,\" Bocrest said.\n\nThe sergeant slunk away, muttering under his breath.\n\n\"What were you doing in here?\" Bocrest asked with no preamble, no apology for his loutish man.\n\nTikaya thought about lying, but reminded herself she wanted the captain to trust her. For now. And she did not think her true purpose that condemning. \"Looking for the fort commander's office. If anyone has more relics or rubbings, I figured it'd be him.\"\n\n\"I just came from there. It's been searched, probably by whomever tortured the Nurian. There's nothing, not even Colonel Lancecrest's orders.\"\n\n\"Colonel...Lancecrest?\" Tikaya asked.\n\n\"Yes, usually we'd have a general commanding a fort, but this is a small outpost.\"\n\n\"It's the name, not the rank, that surprised me,\" she said. \"There was a Turgonian named Lancecrest studying at the Polytechnic when I was a student.\"\n\nBocrest shrugged. \"Probably one of the younger ones. There are nine or ten kids, and they've had to make their own ways. The family's poor as pond scum. Their lands were salted during the Border Wars, and you couldn't grow a weed there now.\"\n\n\"So, just a coincidence, you think?\" As she recalled, that Lancecrest had been studying archaeology; he had shared a couple classes with Parkonis.\n\n\"The colonel's body is up there in his office. If he was plotting with some relative, don't you think he'd have figured himself up a better deal?\" Bocrest headed for the stairs. \"Get some sleep. I've got to look at this Nurian body. One cursed mystery after another up here.\"\n\n\"Captain?\" she called, halting him on the steps. \"Why is your team here, with me and Rias, instead of hunting for whoever sent that box to your capital? It might have originated in the tunnels, but surely the people who found it didn't stick around and post it from there.\"\n\nThe flickering lantern played shadows across Bocrest's face. Answering was probably a violation of orders, but he had already doled out some information when they discovered the cube. Maybe he would divulge more.\n\n\"Someone was sent to hunt down and kill the person who delivered the box,\" he said. \"We are here to seal the tunnels.\"\n\n\"Oh?\" Tikaya sensed a cover story. \"Why would you need Rias or me if that's all you're planning to do?\"\n\n\"We need to make sure we find all the possible entrances and exits. You two will ensure we don't stumble into the kinds of traps that devastated the last group.\"\n\nThe blasting sticks they were carrying were the only things that led credence to his story. Tikaya believed him about the attack on the university\u2014something had started this wave crashing toward the beach\u2014but, now that she'd glimpsed the potential for genocide these artifacts offered, she wagered sealing the tunnels was at the end of a list Bocrest received. Wouldn't the Turgonian emperor love to have some of these weapons for his own use? With which to utterly destroy anyone who defied the empire? She swallowed. Like her people?\n\nShe should to talk to Rias, see what he thought about this. She blinked. Or should she? Whatever had happened, he seemed loyal to the empire through and through. What if he merely shrugged and went along with the mission?\n\nBocrest was watching her through narrowed eyes, and she feared too many of her thoughts traipsed across her face.\n\n\"I understand.\" She smiled innocently. Nothing to worry about from her. \"You mentioned something about sleep? Where would that be done here?\"\n\n\"Officer billets are the single-story building across the courtyard, flag out front.\" Expression unreadable, he turned and descended the stairs.\n\n* * * * *\n\nAs Tikaya approached the billets, she yawned so widely tears sprang to her eyes. They froze in her lashes. The air smelled of burning coal, and light brightened the windows on an end room. A marine stood outside the door, rifle crooked in his arms.\n\nShe paused. Only Rias would be under guard. What would he think if she strolled into his room at night? Would he assume she wanted... No, surely not. Neither of them had slept the night before, and her shoulder nagged like a cranky child.\n\n\"Help you, ma'am?\" the marine asked, no doubt wondering why she lurked in the shadows.\n\n\"Can I, uhm, er...\" Tikaya pointed to the door before her linguistic skills could fail her further.\n\nLanterns burned so it was not likely Rias would be in bed naked, though that thought made her blush.\n\nThe marine sniggered. \"Captain just said to keep him in. Didn't say nothing about keeping anyone out.\"\n\n\"Thank you.\"\n\nShe slipped inside. A coal stove glowed cherry, spilling warmth into the room, and a narrow bunk piled with blankets awaited. Fortunately\u2014or perhaps unfortunately\u2014Rias was not naked. He sat at a desk, still wearing those shackles, though a pencil tucked above his ear destroyed the felon look. She grinned at the papers scattered across the desk and on the floor all about his chair.\n\nWeariness darkened the skin under his eyes, but he stood and smiled. \"Tikaya.\"\n\nShe strode to the desk, hardly noticing that her hip caught the corner, and wrapped him in a one-armed hug. Between her sling and his chains, it was an awkward embrace, but she did not care. After dealing with Ottotark and Bocrest, it felt wonderful to lean on someone pleasant.\n\nRias laid his forehead on her shoulder. \"I've been tasked with pinpointing the origins of the rocket and estimating the area that was affected by the cubes. I figured you'd be too busy puzzling over those runes and I wouldn't see you for the rest of the night.\"\n\n\"Actually, I was busy almost turning Ottotark into a human torch.\"\n\nHis muscles tensed beneath her arm. He drew back to meet her eyes. \"What happened?\"\n\nTikaya shared the story, deliberately putting more emphasis on the mystery of the tortured Nurian than the sergeant's actions. She probably should not have mentioned Ottotark at all\u2014no doubt Rias would worry about her\u2014but she admitted to a little pride that she had handled the odious man herself instead of falling apart. Maybe Rias would be proud of her too. Dumping kerosene on someone was no feat of brilliance, but a month ago, she probably would not have had the wherewithal to think of anything while locked in a dungeon with a rapist on his way. A month ago, she had been hiding from the world because she was too much the coward to go back to work\u2014to her passion\u2014because she associated it so much with Parkonis and lost dreams.\n\n\"Ahh,\" Rias rumbled when she finished the story, and his muscles relaxed. \"I feel remiss that I wasn't there to demonstrate the use of those torture implements on Ottotark. But you're clever and capable, and, alas for my ego, I don't think you need my help in these matters.\"\n\nShe absorbed his praise; it warmed her more than the heat radiating from the stove. \"Don't worry. I need your help in other matters.\"\n\n\"Oh?\"\n\n\"Someone has to catch me when I trip.\"\n\nHis eyes crinkled. \"That has been a daily occurrence.\"\n\nHe held her gaze, and Tikaya was suddenly aware of the heat of his body. If not for the chains keeping his wrists close and his arms between them, she could have leaned against his chest and...\n\nRias cleared his throat and stepped back. \"I need to finish those calculations. If there's someone out there with another rocket\u2014\"\n\n\"Of course,\" Tikaya said. \"I shouldn't have bothered you. I can\u2014\"\n\n\"No!\" Rias seemed to realize his objection too loud, for he shrugged sheepishly. \"I'd like you to stay. I promised you dinner, remember? And...\" He dragged a second chair to the opposite side of the desk and cleared a space. \"You can work with me, or you can sleep of course too.\" He waved toward the bunk. \"You must be tired.\"\n\nTikaya dug into the big pocket on the side of her trousers and pulled out the journal. \"I wouldn't dream of sleeping before sampling a Turgonian dinner.\"\n\n\"Excellent. I've got a treat.\" Rias sauntered to a credenza. He slid a parcel wrapped in brown paper off the top, plopped it on the desk, then knelt before a cabinet. \"I found the colonel's personal stash.\"\n\nShe unwrapped the parcel and crinkled her nose. \"Salty fish? I don't wish to sound ungrateful, but isn't this from the same provisions we've been eating all week?\"\n\n\"Yes. That's not the treat.\" He laughed and pulled out two small glasses and a bottle filled with amber liquid. \"This is.\"\n\nReverently, he carried the bottle over and set it before her. Applejack.\n\n\"That's a thirty-year-old label.\" He uncorked the bottle and poured two glasses. \"Since you're from the land of rum, I thought you'd like to sample a good Turgonian alternative.\"\n\nTikaya sniffed the subtle apple aroma and found it pleasant. She expected the applejack to burn her throat, but the liquid slid down smoothly, like her father's finest barrel-aged rum. \"Nice.\"\n\nRias beamed, took a conservative sip from his own glass, and returned to his work. Though her eyes were gritty, and her muscles ached, Tikaya opened the journal to study the runes. More than once, she paused to watch him zip through calculations without the benefit of a slide rule. Despite the horrors all around them, she enjoyed the companionable moment, sitting there with Rias, him with his work, she with hers.\n\nIt occurred to her that this was an opportunity to ask him his real name, to find out who he had been during the war and what he had done. Except the very fact that he had not told her made her hesitate. Would the truth create an insurmountable obstacle between them? Maybe she should wait. It seemed a shame to ruin this first peaceful time together.\n\nMaybe she was still a coward, after all. She sipped her drink. No, she would ask. Just not tonight. Tomorrow night. She would find a time tomorrow night and ask then. No matter what.\n\nComfortable with the decision, she leaned back in her chair. The applejack left her with a warm muzzy feeling. Her gaze drifted to the sleeping area where the furry blankets and pillows appeared far more comfortable than anything on the ship. It was not a big bed, but she supposed a couple of creative types would have no trouble...\n\nNo, that she would certainly not do without knowing who Rias was. \"It's too bad. I've finally got you in a private room with a bed and\u2014\"\n\nThe pencil in his hand snapped, and he gawked at her.\n\nErp, had she said that out loud? Tikaya stared at the amber liquid, feeling betrayed.\n\n\"I didn't realize that was a goal of yours.\" Rias smoothed his face and slid out of his seat to pick up the pencil ends. One had flown all the way to the door. Impressive velocity. \"If I had, I would have taken it upon myself to escape my cell and call upon you. In a gentlemanly manner, of course.\"\n\n\"I wouldn't have wanted you to get in trouble.\" Dear, Akahe, how was she supposed to explain the context of her comment?\n\n\"Oh, but I'm willing to make great personal sacrifices to help people achieve their goals.\"\n\nRias squeezed her shoulder, and a delightful shiver ran through her when his fingers brushed her neck. She sighed in disappointment when he returned to his seat without presuming to do more.\n\nTikaya set her glass down and pushed it to the side.\n\nRias chuckled and slid the backup pencil out from behind his ear. \"It's a potent drink.\"\n\n\"You wouldn't think apples could get you caned.\"\n\n\"Apples are the Turgonian fruit. We make them into everything. I think I mentioned my family's orchard.\" Rias continued to work as he spoke. \"I loved the trees as a kid. I was scrawny, so I'd climb them to hide from my older brothers. They loved to beat on me almost as much as I loved getting them in trouble.\"\n\nTikaya eyed him skeptically. \"I can imagine you as young, but...scrawny? You're, what, six and a half feet? And broad.\"\n\n\"Oh, I was always scrawny because I was always the youngest. I was the youngest child in my family, and then I went to the university four years early, so I was the youngest there. I got smashed whenever I tried to join the sports teams, and I couldn't attract girls, because I was fourteen and they were at least eighteen and only interested in older men. Though I did finally bribe one gal to kiss me by volunteering to do her homework.\"\n\nTikaya smiled. So, he had also been the youth who did not fit in. Maybe more so than she, since it sounded as if there was less educational infrastructure for precocious children in the empire.\n\n\"Surely, you've long since outgrown those troubles.\" She yawned, folded her arms over her journal, and pillowed her chin. \"I'm sure you could entice women of any age now.\"\n\nThe pencil paused, and he bent down to peer into her eyes. \"Hm, I've been meaning to ask you about that. How old are you?\"\n\n\"Thirty. Does it matter?\"\n\nHumor glinted in his brown eyes. \"Just wondering if my fantasies have been downright scandalous or merely lacking in propriety.\"\n\nIf Tikaya had been less tired, she would have laughed at the idea of being someone's fantasy, but she smiled blearily, her eyelids half shut. The thought, like the applejack, left her with a warm contentedness.\n\n\"How old are you?\" she murmured.\n\n\"Forty-three.\"\n\n\"As far as my people are concerned, it'd be more scandalous that you were a Turgonian military officer than that you're older than I am.\"\n\nHe had never told her he was an officer, but he did not deny it now. Glum acknowledgment replaced the humor on his face, and Tikaya wished she could retract the comment.\n\nA knock sounded at the door, and the guard walked in. He blinked in surprise at Tikaya, probably expecting her to be naked, then focused on Rias.\n\n\"Skeldar's team checked that mining camp like you asked,\" the marine said. \"It's ten miles out. The men there were dead, same as everyone here. Same time ago. And the lookout tower at the bottom of the pass seems the same. The scouts saw no fire, no one moving.\"\n\nRias scribbled a note. \"Distance to the tower?\"\n\n\"Fifteen miles.\"\n\n\"Thank you, private. Dismissed.\"\n\nGrimness hooded his eyes, and he did not watch as the marine walked out. Tikaya shifted in her seat, waiting for him to finish, or waiting for an opportunity to ask him his thoughts. But she already sensed the problem. If people fifteen miles away had been killed by the same rocket...\n\n\"Ancestors help us.\" Rias sat back, eyes closed.\n\n\"What?\"\n\n\"The rocket detonated in the air over the fort, sending those cubes in all directions where they opened of their own volition to release their contents. I can only guess at the exact nature of the substance, but based on all the data points I've received, we're looking at a weapon that can kill everyone within twenty miles of the detonation point. That's more than twelve hundred and fifty square miles.\" He stared at her. \"Can you imagine what would happen if a weapon like this was launched in a populated area? Our capital city has more than a million people.\"\n\n\"Ninety percent of the Kyattese population lives on our main island, and it's smaller than that.\" Tikaya thought of such a weapon in the Turgonian emperor's hands. Why worry about subjugating her people when one could just kill them all and claim the deserted island for colonization?\n\n# CHAPTER 13\n\nThe door opened with a bang, and cold air flooded the room. Tikaya sat up. Wan morning light silhouetted Bocrest.\n\nConfusion disoriented her for a moment. She remembered falling asleep at the desk with her cheek pressed against the pages of her journal, but now she sat in the bunk, a blanket pooled about her waist. Rias lay on the floor before the stove. He rolled his head toward the door, eyes slitted.\n\nBocrest looked from the nearly full applejack bottle on the desk, to Tikaya, and finally to Rias. \"This is pathetic. You've got a private room, a bed, booze, and a woman, and you spent the night on the floor like a hound.\"\n\n\"Ass,\" Tikaya grumbled, wondering if she was too old to hurl pillows at people.\n\nRias yawned, stretched, and rolled to his feet. \"Careful, Bocrest. You're starting to sound like your brutish sergeant. Officers are supposed to be an influence on their troops, not the other way around.\"\n\nBocrest snorted and walked to the desk. He picked up the top sheet of paper and scrutinized it, appearing as enlightened as a rock. \"Did your big genius brain figure out where that rocket was launched from?\"\n\n\"Yes.\"\n\n\"And?\"\n\n\"I'll lead a team to the location, but I have conditions. I want ten men who have taken the mountaineering course. Give me a tracker too. I need a rifle, a pistol, a knife, and I want these chains off.\" Rias rattled them for emphasis. \"Permanently.\"\n\nBocrest gaped at this list of demands. \"Is there anything else this humble captain could lay at your feet, Master Prisoner? Perhaps I could grovel while I fulfill your wishes?\"\n\n\"He wants me to come too,\" Tikaya said.\n\n\"Absolutely not,\" Bocrest said.\n\nRias arched his eyebrows at her. \"Are you sure? It'd be difficult even without an injury.\"\n\nTikaya hesitated. She did not want to make more work for him, but what if they found a cache of weapons and needed help disarming a rocket poised to launch? \"You may need me up there.\"\n\n\"As you wish,\" Rias said.\n\n\"Emperor's spit,\" Bocrest said. \"She's got you leashed, and you're not even screwing her.\"\n\nThe manacles ensured Rias's punch was not pretty, but he got a fist on the captain's nose all the same. Bocrest saw it coming and partially blocked it, but the force still sent him staggering backward. His own fingers curled into a fist, but he snorted and released them.\n\n\"You'll have your team.\" Bocrest unclipped keys from his belt and tossed them to Rias. \"But she should go with the main force to set up the base camp. What's she going to do up there besides be a liability?\"\n\n\"She's found more ways to be useful than any of your men thus far.\" Rias beamed her a proud smile.\n\nTikaya smiled back, ignoring the captain's disgusted huff.\n\n\"What's she actually translated?\" Bocrest asked. \"Did that rocket say anything?\"\n\n\"I'm still working on what appear to be instructions,\" she said. \"I believe the simple phrases on the side say 'caution' and 'this side up.'\"\n\nBocrest snorted. \"That's as useful as goat spit. Great.\"\n\n\"Captain...\" Rias warned.\n\n\"Whatever. I don't care. If she needs to be carried, none of my men are doing it. She's your responsibility.\" He stalked out the door, not bothering to shut it. \"Team leaves in fifteen minutes!\"\n\nTikaya slid her legs off the bed and grabbed her boots. It was the only thing Rias had presumed to remove before tucking her into bed. Her heart ached. Even if he had not leapt to her defense, she could have loved him just for being a gentleman up here in this savage land so far from the mores of civilization. She almost confided her suspicions to him, her fear that Bocrest's mission would prove less about saving the Turgonians and more about getting the emperor a stockpile of terrifyingly powerful weapons.\n\n\"Something wrong?\" Rias sat beside her to put on his own boots.\n\nHow could she tell him? A man whose every choice proved he still felt loyal to the empire, even after they had taken everything and exiled him. \"I'm worried I made a mistake,\" she said instead of bringing up the weapons. \"I spoke hastily. I don't want to be a burden on you. Maybe you could just copy any runes you find for me.\"\n\n\"Do you want to go?\"\n\n\"Yes.\" A self-deprecating laugh spilled from her lips. She had never craved field work; she'd always preferred to stay in the lab, letting agile adventurous sorts bring their finds to her. When had that changed? \"I don't want to miss anything.\"\n\n\"I wouldn't want that either.\" He smiled, but it soon faded. \"Besides, I'm concerned about your assassins. The bodyguard may be tortured and dead, but if the wizard is still around, he'll feel he's running out of time. It'll be dangerous on the mountain, yes, but I'd prefer you with me rather than with a bunch of men who don't care.\"\n\nBy now his admission that he cared was no surprise, but hearing him say it almost brought tears to her eyes. If only he weren't a Turgonian.\n\nShe leaned against his shoulder. \"I care too.\"\n\n\"Good.\" He leaned back. \"About me, right?\"\n\nShe grinned and swatted him. She thought about doing more, but a pair of marines tramped past the door, pausing to peer inside. Then Bocrest hollered for his guide. Sighing, she finished tying her boots.\n\n* * * * *\n\nTikaya eased along the narrow ledge, her metal crampons scraping and clinking against the ice. Sheer granite towered to her left while equally sheer rock plummeted on her right. A snowy canyon stretched hundreds of feet below. Though the white drifts appeared soft, she had no illusions of a landing being anything but deadly. Wind buffeted the face of the cliff, tugging at her thick braid and whipping stray strands of hair against her spectacles. Frost crystals glittered on the scarf snugged over her nose.\n\n\"We're close,\" Rias called, voice muffled by his own scarf.\n\nHe led the single-file squad of marines inching along the cliff face. Tikaya came second with Bocrest third. At first, the captain's presence had surprised her, but a few curt words here and there had given her the impression he was there to remind everyone 'Prisoner Five' was most definitely not in charge of the team, even if he led.\n\nA shadow fell across the group as a black raptor as large as a man sailed overhead. This was its third appearance. The way it coasted past made Tikaya think it was scouting the group. Her imagination, no doubt.\n\nThe sun glinted off sharp ebony talons as it flexed its legs to land on a perch a couple dozen feet above the ledge ahead of them. It cocked its head to stare at her through a calculating black eye.\n\n\"I'm getting tired of that bird,\" a marine grumbled.\n\n\"We could make a meal out of it,\" another said, voice loud and threatening, as if the creature would understand and leave.\n\nTikaya caught Bocrest glancing at the bird, but he otherwise paid it little attention. He did, however, carry his loaded rifle in one hand, barrel leaning against his shoulder. Earlier, he had worn it strapped to his rucksack.\n\nThe hair on the back of Tikaya's neck rose as they walked under the creature's perch. She had not felt the tingle of the mental sciences being used since the night the Nurians attacked. She paused to study the bird.\n\n\"Quit gawking,\" Bocrest said.\n\n\"What species is that?\" she asked. \"Are they common?\"\n\n\"How would I know? Do I look like I keep a summer estate here?\"\n\n\"Ice condor.\" Rias turned and held up a hand to halt the squad. \"They're predators but scavengers too. It's unlikely it'll attack a group of armed men. It's probably just waiting to see if one of us falls.\"\n\nLovely thought. \"They're usually natural creatures, then?\"\n\n\"Of course, it's a natural creature,\" Bocrest said. \"What else would it be?\"\n\nTikaya pointed at it. \"This one's a\u2014\"\n\nThe condor dropped from its perch, plunging straight at her, beak agape, talons extended.\n\nWith nowhere else to go, Tikaya smashed herself against the cliff. The giant bird filled her vision, wings pressed against its body for speed. She raised her good arm to guard her face.\n\nA rifle cracked. Someone pulled her up the path.\n\nThe condor squawked, clipped the edge of the ledge, and bounced away. Rock crumbled and fell into the canyon. The bird flapped its wings and recovered before tumbling far, but blood spattered the ice on the ledge. The condor sailed on a draft and disappeared from sight before Tikaya recovered.\n\n\"Thanks,\" she rasped.\n\nIt was Rias who held her, Bocrest's rifle that smoked.\n\n\"That was peculiar,\" the captain said.\n\n\"More than that.\" Rias checked Tikaya for injuries and released her. \"A familiar?\"\n\n\"That's...\" She mulled over the Turgonian word options. \"Close enough. I'm guessing it's a regular creature that someone is manipulating with thought control.\" She remembered the snatch of conversation she had overheard the night the Nurians attacked her; the practitioner had read someone's thoughts to find her. If he had studied telepathy on humans, controlling animals was not a stretch.\n\n\"How would you know?\" Bocrest asked. \"I thought you weren't a wizard.\"\n\n\"I'm not, but I've grown up around practitioners. I can sense when the mental sciences are being used nearby.\"\n\nBocrest scowled at Rias. \"Get moving. I want to finish up and get off this mountain quick.\"\n\nWhen they reached the top of the cliff, long shadows darkened the snowy plateau despite the early afternoon hour. They had climbed less than halfway up the mountain, and another granite wall rose to the rear, blocking the sun. Nothing on the plateau caught Tikaya's eye, but the fantastic view to the north made for a memorable perch. Miles of unbroken tundra stretched to the horizon with ridges and swirls roaming like striations in stone.\n\nThough nothing but drifts adorned the plateau, Rias strode across it as if he expected to find something. He stopped at a protruding edge and pointed.\n\n\"Perfect view of the fort from here,\" he called.\n\n\"But there's nothing here.\" Bocrest gestured for his troops to fan out and investigate.\n\nTikaya floundered through deep snow to join Rias. He held a thermometer and a round bronze device she had seen him consult a few times. She had thought it a compass, but the numbers on its circular face did not represent degrees.\n\n\"Barometer?\" she guessed.\n\nRias nodded once, though his eyes rolled upward as if he were busy with some calculation. Bocrest shuffled up behind them.\n\n\"Worried about a storm coming in?\" she asked when Rias's attention shifted to her.\n\nHe chuckled. \"No, calculating our elevation. As long as you know the temperature, the air pressure at sea level, and the air pressure where you are, you can\u2014\"\n\nBocrest jerked his hand up. \"Nobody cares, Five. Is this the spot or not?\"\n\nRias's sigh had a long-suffering quality, and, as he turned to face the captain, Tikaya wondered how many times in his career he had been cut off by officers with Bocrest's temperament.\n\n\"We're either here or we're very close,\" Rias said. \"I was expecting a launching platform, but I suppose it's possible the rocket was self-propelling.\"\n\n\"You brought us up here to find nothing?\" Bocrest demanded.\n\nWhile the men debated, Tikaya removed her spectacles to clean them. She tilted them toward the sky to check for specks and almost dropped them when she spotted a sliver of familiar black metal on the cliff top above.\n\n\"Gentlemen.\" She pointed, ending their argument.\n\n\"Ah.\" Rias put away his tools. \"Another fifty feet.\"\n\n\"Thought your math was better than that.\" Bocrest smirked.\n\nRias's eyebrows disappeared under his wool cap. \"My math is impeccable. The tools are imprecise.\"\n\nTikaya grinned, always amused when his Turgonian arrogance peeped out.\n\nBocrest only snorted and called to his men: \"Get some grappling hooks out. We're climbing.\"\n\n\"We'll go first.\" Rias pointed to his chest then at Tikaya. \"I don't want overeager young men thundering around up there before we've ascertained the danger.\"\n\n\"I'll go first,\" Bocrest said. \"You can come after and pull your wounded librarian up.\"\n\nTikaya grimaced. She held her own on the walkable terrain, but even without a shoulder injury, she would have needed help up the cliffs. Already, Rias had pulled her up two, while she scraped and pushed with her crampons, trying use her legs and burden him as little as possible.\n\n\"Doing all right?\" Rias patted her on the back as Bocrest stomped away.\n\n\"I'm fine. Do you always volunteer women to lead the way with you into potentially dangerous situations?\"\n\nHe winked. \"Only if I know they can handle it.\"\n\nNot for the first time, she wondered if he thought too highly of her.\n\n* * * * *\n\nWhen Rias pulled Tikaya over the edge, she knelt to catch her breath. Even with his help, the climb had been taxing.\n\nThe first thing she noticed was the dead man. The second thing she noticed was that he had not died the same way the marines in the fort had. An enormous amount of blood spattered the snow around him, and methodical cuts marked the body. Bocrest already stood over it, arms crossed, lips dragged down in a scowl.\n\n\"Tortured,\" he said. \"Same as the one in the fort.\"\n\n\"Nurian?\" Tikaya asked.\n\nBocrest was too busy cursing under his breath to answer.\n\nThe ledge, similar to the one below, offered another ideal view of the tundra\u2014and the fort. A second cliff to the rear shadowed a tent and a fire pit. The source of the black metal Tikaya had spotted rested near the edge: a flat circle mounted on tripod legs. A shaft tilted upward from the disk and appeared the right size for cradling the rocket. The launching apparatus was not large, but it would have taken more than one man to carry it up there. Or a telekinetics practitioner.\n\nTikaya walked over to look at the body. \"Uh.\"\n\n\"Uh?\" Rias asked.\n\n\"I recognize this one too.\"\n\n\"Can't be a practitioner,\" Rias said. \"He looks Turgonian.\"\n\n\"He is Turgonian. That's Lancecrest.\"\n\n\"The fort commander?\" Bocrest asked. \"He's too young.\"\n\n\"No, the Lancecrest I told you about. And actually I think he did study the mental sciences at the Polytechnic. Along with archaeology.\" She filled Rias in with the information she had given Bocrest the night before.\n\n\"Huh,\" Rias said. \"That he's here is not wholly mystifying\u2014Colonel Lancecrest could have taken command, found out about the tunnels, seen an opportunity for the family to improve its fortunes, and told his little brother to prepare for a relic hunt. But why would the younger Lancecrest have launched a rocket at his older brother's fort? And why is he now up here tortured and dead? This is...unexpected.\"\n\nBocrest spat. \"If something expected happens at any point in this mission, I'll shit myself in shock.\"\n\nTikaya shook her head at the lurid speech. \"Is he truly married?\" she asked Rias.\n\n\"Last I heard.\" Rias knelt to examine the body more closely. \"The empire has failed to keep me apprised of the latest gossip surrounding its officers.\"\n\n\"It's hard to imagine that tongue wooing a woman.\" Tikaya headed for the launch pad.\n\nBocrest dropped his arms. \"Was that an insult? Did she just insult me?\"\n\n\"I believe she did,\" Rias said.\n\n\"I never know with her. She gives insults in the same tone as a scientist analyzing an experiment.\"\n\nTikaya dug out her journal. \"You do remind me of the lab rats they keep in the science wing of the Polytechnic.\"\n\n\"That was definitely an insult,\" Rias said.\n\n\"I know,\" Bocrest said. \"It's hard to be offended, though. She's so civilized when she delivers them. Tidy job on Lancecrest. Whoever ran the torture session was experienced.\"\n\nTikaya scratched her head at the abrupt topic shift. Only Turgonians could go from casual chit chat to analyzing dead people in the same breath.\n\n\"Body's stiff but this doesn't look like it happened long ago,\" Rias said. \"Yesterday maybe.\"\n\n\"There's a mess in the tent,\" Bocrest said. \"Like someone searched it, same as the colonel's office in the fort.\"\n\nRias leaned over the ledge. \"Koffert, come up. We need your tracking skills.\"\n\nBocrest frowned at this presumptive order giving. Tikaya wondered when Rias had found the opportunity to learn people's names and skill sets.\n\nLong before the tracker reached the top, the launch device swallowed her attention. Runes ran down the tripod legs, giving her plenty to study. She sat in the snow with her journal, gloves off. Not knowing how much time Bocrest would give her, she risked the cold to make copying the symbols easier. The men's conversations faded from her awareness as she worked. She brushed her fingers along a complex grouping of seventeen symbols, and a faint hum teased the edge of her mind. It startled her, and she dropped her journal. Surely the sensation did not come from the launch pad. The artifacts had not yet made her suspect the mental sciences were involved in their creation. Yet something here teased her sixth sense, reminding her of the communications pendant on the Nurian ship. The residual tingle of a practitioner-made device.\n\n\"Tikaya?\" Rias touched her shoulder. \"The tracker is done. Are you ready to leave?\"\n\nShe blinked and stood, surprised by the stiffness in her limbs. How long had she sat? Rias removed his gloves and held her hand in his warm ones, and she noticed white tipping her fingers.\n\n\"Frost nip.\" He rubbed her hands and raised an eyebrow. \"Keep your gloves on. You'd have a hard time taking notes if you lost your fingers.\"\n\n\"Sorry, that was dumb. I needed to use the pencil, and, uhm.\" She blushed. Of all people, he could probably understand an absent-minded streak, but she still avoided his eyes.\n\n\"What I don't understand is how someone else found this ledge,\" Bocrest said, apparently resuming a conversation she had missed. The tracker stood before him, a sergeant with a lined face and beaky nose. \"How many math geniuses are roaming around up here?\" Bocrest added.\n\n\"Perhaps our mystery man saw the rocket being launched,\" Rias said.\n\nDespite his suggestion that she keep her gloves on, Rias had not released Tikaya's hands. Calluses hardened his palms, but his touch was gentle as he rubbed her skin. She made no move to pull away.\n\n\"Wouldn't he have died from the gas, too, then?\" she asked. \"And how do you know our torture-loving person is a man? The Nurians have female warriors.\"\n\n\"Walks like a man, pisses like a man,\" the tracker said.\n\n\"Uhm. All right.\" Tikaya knew nothing about tracking, but supposed squatting and standing would indeed leave different yellow-snow signatures. \"But what about the gas?\"\n\nRias gazed east. \"The pass is that way and at a higher elevation. The rocket released its load in the air above the fort, so perhaps that means the gas\u2014or whatever it is exactly\u2014was heavier than air and wouldn't have affected someone above the detonation point. This camp, after all, is well within the twenty mile radius.\"\n\n\"Perhaps?\" Bocrest asked. \"You're just guessing?\"\n\n\"Yes,\" Rias said.\n\n\"Good steel used for the torture,\" the tracker said.\n\nRias and the captain nodded, though it took Tikaya a minute to follow. Right. The good steel and the possible entrance through the pass implied a Turgonian. And hadn't the men in the dungeon suggested the same thing? That the torture was done by the book? The Turgonian book?\n\n\"So, you've got an ally up here?\" Tikaya asked. \"Maybe he'll show himself, and we can share your applejack with him.\"\n\nShe smiled. The others did not. Rias and Bocrest appeared more grim than anything.\n\n\"Ally,\" Rias murmured, then found Bocrest's gaze. \"Did the emperor say anything about sending help?\"\n\n\"He made it clear he wanted the mission accomplished.\"\n\nTikaya wondered if Rias derived more from that answer than she did.\n\n\"You find anything useful on that rocket, Komitopis?\" Bocrest asked.\n\n\"I'm getting some fantastic data. If we find more samples in this scientific vein, I believe the shared contexts will allow me to\u2014\"\n\nBocrest hissed in frustration and jerked his hand up, much as he had to halt Rias's explanation of the altitude calculations. \"When I ask you a question, I want a yes or no response.\"\n\n\"Then, yes,\" Tikaya said.\n\nRias chuckled and squeezed her hands.\n\n\"Although if you'd listen to all I had to say, you'd learn that there's some science about the device.\"\n\n\"Science?\" Bocrest's expression blanked.\n\n\"Magic,\" Rias said.\n\n\"Oh,\" Bocrest said. \"How?\"\n\n\"I'm not sure yet,\" Tikaya said. \"Give me a moment.\"\n\nShe started to bend down again, but Rias stepped in front of the launch pad. He picked up her gloves and handed them to her. Not until she stuffed her numb fingers back into the fur-lined interiors did he move aside.\n\n\"Thank you,\" she said.\n\nRias saluted her with a wink. Bocrest heaved a sigh.\n\nShe touched the launch pad again, checking several spots. It was weak, but she did sense something, especially close to the ground. On a whim, she tried to lift one of the legs. She expected the black metal to weigh too much, but she raised it with relative ease, revealing a leather-bound book flattened into the snow. Her heart sped up in anticipation. Rias grabbed the leg, so she could retrieve her find.\n\nA pen was stuck in the spine, and it felt warm beneath her fingers. That was it: the practitioner-imbued item, probably crafted to never run out of ink or some simple thing. As far as she could tell, the book\u2014no, journal\u2014was mundane. She flipped it open, but had scarcely read the first couple words when someone tore it from her grip.\n\n\"Our people will vet this and decide if it's suitable for a foreigner to read,\" Bocrest said.\n\n\"Bocrest...\" Rias started, but Tikaya lifted her chin and spoke.\n\n\"Then I hope you brought someone who reads Kyattese, because the writing isn't in your tongue.\"\n\nBocrest flipped through a few pages and his lip curled into a snarl. \"Kyattese?\" His eyes narrowed. \"Why would there be a notebook up here in your language?\"\n\n\"He spoke Kyattese.\" Tikaya nodded at Lancecrest's body.\n\n\"Is it possible that journal is what our 'ally' was searching for?\" Rias asked.\n\nBocrest jerked his head down, eyes scouring the pages as if he could translate them through will. With a disgusted grunt, he thrust the book at Tikaya.\n\n\"You tell us,\" he said.\n\nShe skimmed the opening pages and practically bounced at the massive number of the language samples within. Notes, mostly speculation, surrounded drawings of symbols she had not yet seen. No firm translations yet. \"I'll need time to read over everything, but it's definitely Lancecrest's journal, and it looks like he's been in your tunnels a while. There are hundreds of pages here and dates go back almost a year.\"\n\nShe turned to a dog-eared page, and her hand froze. Launch instructions for the rocket. It appeared Lancecrest had discovered how to operate the weapon through trial and error rather than true understanding of the language. Nonetheless, the instructions were there. And suddenly she knew: this book was exactly what their mysterious stranger was searching for, here in the tent and perhaps in the colonel's office as well. It could explain the torture sessions too. He had been trying to locate these very instructions, but the Nurian had not known and Lancecrest must have held out to the end.\n\n\"Find something?\" Rias asked.\n\nShe flinched, knowing she had been silent too long to brush it off. \"Just an interesting take on what the prime groupings imply.\" She hated lying to Rias, but she was not going to hand Bocrest directions for launching the rockets. She could only assume there were more of the devices in the tunnels.\n\n\"Find something useful?\" Bocrest asked.\n\nSince shadow covered the ledge already, Tikaya received little warning when the ice condor approached for the second time. Movement teased the corner of her eye, and Rias yelled, \"Get down!\" just as she was turning to check.\n\nThe condor swooped toward her head, talons outstretched. She flung her arms out.\n\nRias smashed into her, taking her to the ground. Her shoulder flared with pain, but the talons meant for her eyes grazed her forearm. They cut through her parka and stung flesh.\n\nBocrest and the tracker fired, but the condor banked before the balls hit. It swooped out of sight over the cliff above the tent.\n\n\"Are you injured?\" Rias asked, eyes locked on her as he shifted to let her up.\n\nTikaya pushed up her parka sleeve. \"Just a couple scratches.\"\n\nRias removed a glove and brushed his finger across one of the wounds, which had started to well blood. A green pasty substance mingled with the crimson drops.\n\n\"What is it?\" Dread hollowed her stomach.\n\n\"Poison.\" Rias jumped to his feet. \"We have to get to the sawbones.\"\n\nTikaya stared at her arm. She knew nothing about poison. \"Is this a lethal dose? How much time do I have?\"\n\nHe started to respond, but the condor swooped toward them again.\n\n\"Someone shoot that slagging bird!\" Bocrest shouted to the men below. He and the tracker were still reloading.\n\nRias had dropped his rifle to shield Tikaya. The bird landed on the launch pad as he grabbed the weapon. Unconcerned, the condor cocked its head, black eye studying Tikaya.\n\n\"Yes, you got me.\" Bitterness choked her words.\n\n\"Sh.\" Rias aimed the rifle, but hesitated. A calculating flash crossed his face, and he raised his voice. \"Don't worry, Tikaya. You're not going to die. We've got the antidote in camp, and you've got plenty of time.\"\n\nBocrest, the first to finish reloading, lifted his rifle. The bird flapped away. Several shots fired, but it weaved and banked with preternatural speed, and disappeared unscathed.\n\nRias lowered his weapon. He had not fired.\n\n\"I'd like to be reassured by your words,\" Tikaya murmured, \"but I suspect that was for the benefit of the bird.\"\n\n\"Will whoever is controlling it understand our speech through its ears?\" Rias asked.\n\n\"I'm not sure. Maybe.\" She might have stopped to consider what he hoped to accomplish with his words, but other thoughts stampeded to the front of her mind. \"How much time do I really have?\"\n\n\"Plenty,\" Rias said.\n\nShe had come to know him too well; she could tell he was lying.\n\n# CHAPTER 14\n\nTikaya woke to the sound of pained wheezing. Her own. Air. She couldn't get enough air.\n\nShe opened her eyes to a green canvas tent ceiling supported by slender steel bars. Confusion muddled her mind. The last thing she remembered was Rias and another marine carrying her down the mountain on a litter. Now she lay on a cot, blankets pulled to her chin. Somewhere behind her head, a lantern provided illumination that failed to reach the shadowy corners.\n\nThey must have reached the base camp, but if the sawbones had applied some antidote, she could not feel it. Her breath rattled in her ears, and she could not pull in enough air to satisfy her lungs. She tried to wriggle her toes. If they moved she could not tell.\n\nStill alive, she thought, but still poisoned. And alone. Rows of empty cots stretched into the darkness. Where was Rias? Why hadn't he stayed with her? And what about the sawbones?\n\n\"Akahe, please don't let me die alone,\" she mouthed.\n\nShe blinked away tears, but it was hard to keep the wheezing breaths from turning into sobs. With no one to witness her torment, why bother being stoic? And why hadn't she written a letter to her parents? Rias might be slated for a return to exile, but Agarik would have found a way to post it. But now her family would forever wonder what happened.\n\nThe tent flap swayed, and icy air gusted inside.\n\nShe could not lift her head to peer into the shadows at the entrance. \"Is someone there?\" she tried to ask. It came out weak and garbled.\n\nShe saw no one, but soft footfalls trod across floor mats. A man coalesced before her\u2014a familiar man. The Nurian practitioner from the ship. She tried to move, to roll away, but her body did not respond. When she had begged the Divine One to keep her from dying alone, this was not the company she had meant.\n\n\"The Turgonian lied,\" he murmured in his native tongue, his gaze flicking over her supine form. \"I see no evidence that an antidote has been applied. They probably don't even know Irkla Root when they see it.\" He withdrew a knife and met her eyes for the first time. \"I'm sorry, Ms. Komitopis.\"\n\nShe groped for something to say that would save her, but only a wheezy gurgle came out when she tried to speak.\n\n\"I regret the need for this task,\" the Nurian continued. \"After the help the Kyattese\u2014you\u2014gave my people during the war, it's unfair to kill you, but I cannot let the Turgonian military get their hands on that kind of weaponry. Nor am I going to let those archaeologists sell it to the highest bidder. I can't let your talents be used against my people, but I'll show mercy and end your suffering now.\"\n\nHe leaned forward and lifted the blade. Tikaya tried to thrash, to fight him off, but her limbs were already dead.\n\nA shadow moved behind the Nurian, and a dagger appeared at his throat. His weapon was wrenched from his hand.\n\n\"You move, you die,\" Rias growled in his accented Nurian.\n\nThe assassin's eyes widened. He reached for his throat, but Rias's blade bit into flesh, drawing blood.\n\n\"Most of your people who work with poisons carry the antidote in case they infect themselves,\" Rias said. \"You've five seconds to produce it, or you'll suffer the same fate as your bodyguard.\"\n\nRias's head was right next to the Nurian's, and rage burned in his eyes. Tikaya wanted to yell, to warn him that a practitioner did not need a weapon to kill. Only a strangled wheeze came out.\n\nSurprisingly amenable, the Nurian reached into his parka and withdrew a handful of fingernail-sized clay vials. \"The gray one.\"\n\n\"Sample it,\" Rias said.\n\nThe Nurian blanched.\n\nRias shoved him to his knees and smashed his face into the mats. The two men dropped below Tikaya's line of sight.\n\n\"You're justifying killing the one person who saved your asses in the war over paranoia,\" Rias snarled.\n\n\"You saw your fort. Your people would destroy the world with weapons like that. I can't\u2014\"\n\n\"Quiet.\" Rias slammed the man into the ground again. \"Which vial is the correct antidote?\"\n\n\"The clear one,\" the Nurian rasped, his airway restricted.\n\nDid Rias have a hand around his throat? She struggled to turn her head, but could only move it an inch.\n\nRias sat back, kneeling on the man's chest, and pulled the cork out with his teeth. He forced a drop down the Nurian's throat. The man made no attempt to elude it, and Rias seemed satisfied.\n\nAs he started to reach for Tikaya, the hairs on her neck stood.\n\n\"Spell,\" she blurted, praying the word would come out intelligible.\n\nRias growled and drove his dagger into the Nurian's chest with a crunch of bone. The pained grunt sounded final.\n\nHe leaned close to Tikaya and rested a hand on her forehead. The rage was gone, and an uncertain desperation haunted his eyes. He held up the vial.\n\n\"I don't know for sure, but I have to try it, all right?\"\n\nShe tried to nod vigorously, though she was not sure her head moved. He propped her up to slide the liquid down her throat. It burned like cheap rum, and tasted like resin, but she was not about to reject it.\n\nRias never shifted his gaze from her face. He stroked her hair gently. When his hand brushed her cheek, it felt cool against her fevered skin. The lantern light reflected in the moisture pooling in his eyes. Tears blurred her own vision again, though this time they came from knowing someone was there with her, someone who cared.\n\nUtter weariness overcame her, and she closed her eyes.\n\nRias was still with her when she woke. He had removed the body and knelt on the ground with his head next to hers on the cot, his hair touching her cheek. Her breathing seemed smoother, less labored. Somewhere beyond the tent, voices rose in argument. She listened, but could not make out words. That she even noticed the goings on outside seemed a good sign.\n\nShe wiggled her fingers experimentally. They responded. Her toes did too. Yes.\n\nShe eased her arm from beneath the blanket before she could think to favor her shoulder, but no blast of pain accompanied the movement. Perhaps the antidote had healed the injured joint as well. She would test it later. For now, she gave in to the urge to slide her fingers through Rias's hair. It was thick and black, save for those silver strands at his temple, and surprisingly soft.\n\nHe lifted his head, and she let her hand drop. The shaman's concoction might have healed her, but weakness weighted her limbs. He winced as he adjusted his position\u2014falling asleep on one's knees could not be comfortable\u2014but the pained expression turned to a pleased smile when he saw her watching him.\n\n\"It worked,\" he said.\n\n\"I think so,\" she rasped, voice rougher than Sergeant Ottotark's manners.\n\nRias held up a finger, moved away, and returned with a canteen. He slid his arm behind her shoulders and propped her up to drink.\n\n\"Not so bad being sick,\" she said, \"when someone's willing to carry you around and take care of you.\"\n\n\"Well, don't make getting poisoned your new hobby. It's hard on\u2014\" He cleared his throat. \"It's hard.\"\n\nTikaya leaned against him and tried to recall the events of the day. What had brought the Nurian into the tent? Then she realized: \"When you spoke so the bird could hear, talked about an antidote, that was a trick? To make the Nurian think he needed to come personally to finish the job?\"\n\n\"A trick, yes, also known as a hopeless stab at making something happen. I feared the sawbones didn't have antidotes in his kit, and, as it turns out, I was right.\"\n\n\"Ah.\" She shuddered to think how close she had come to dying. \"Rias, if someone does succeed in killing me up here, and you make it out, will you do me a favor? Please find a way to let my parents know what happened.\"\n\nHe placed his palm alongside her face, traced her cheek with his thumb. \"I'm planning on making sure you live, but, yes, of course.\"\n\n\"Thank you.\" Weariness dragged her lids down again. \"I love you,\" she murmured before falling asleep.\n\n* * * * *\n\nThe next time she woke, darkness still wrapped the tent, and Rias was gone. Agarik sat on a nearby cot, whittling a piece of wood.\n\n\"Is it still the same night?\" she asked.\n\n\"Yes.\" Agarik lifted his head. \"Near midnight, I think.\" His fresh scar appeared garish by the lantern light, but he smiled and said, \"I'm glad you made it.\"\n\n\"Thanks. Me too. Is Rias around?\"\n\nHis lips flattened, and he looked down, fingers gripping the carving too tightly.\n\n\"I'm sorry,\" she said. \"I didn't mean to imply\u2014I appreciate your company too.\"\n\nHe snorted. Agarik's annoyance surprised her since she had only seen him irritated in Wolfhump and everyone had been irritated there.\n\n\"Is this about...\" She thought about the number of marines who knew she and Rias had shared a room in the fort. If Bocrest had said nothing of their sleeping arrangements\u2014and why would he gossip with his subordinates?\u2014everyone likely assumed they had slept together. \"Agarik, I'm sorry, but he was married so...even if it wasn't me...\"\n\nAgarik waved a hand and met her eyes. \"It's not that. I mean, of course a fellow dreams, but...it's just unfair that you don't even know who he is and you get to be his friend.\" As soon as the words came out, he winced. \"Rust, that was pitiful. I sound like a child. And I should be sorry, not you.\" He scrubbed a hand through his hair. \"You almost died today, and I'm sulking because the only time my boyhood hero speaks to me is to inquire about you.\"\n\n\"Your...boyhood hero?\" Tikaya caught herself gaping and closed her mouth. \"How old are you, Agarik?\"\n\n\"Twenty-three.\"\n\nHe was younger than she had thought, but that still meant Rias had to have been someone of note for at least ten or fifteen years. Not just an officer, someone distinguished enough to have been known and discussed all over the empire. The night before she had resolved to ask Rias his name. She was tired of being in the dark. She had to know.\n\n\"To answer your original question,\" Agarik said, \"last I saw him, he was heading off to a meeting with Bocrest. The captain's finally given up trying to keep him at prisoner status. He told the men to treat Rias like an officer for the duration of the mission.\"\n\nTikaya found herself gaping again. \"Er, how long was I asleep?\"\n\n\"While the captain was up on the mountain, the marines setting up camp down here had some time to chat. Things came out.\" The mischievous glint in his eyes suggested the source of those 'things.' \"Not that many men were surprised. Most of us had pieced together who he is and started deferring to him anyway. Ottotark about shi\u2014had an accident, though.\"\n\nShe flexed her fingers and eyed her nails. \"And he's who, again?\"\n\nShe hoped Agarik would let it slip, but he shook his head. \"He should be the one to tell you.\"\n\n\"Of that I have no doubt, but he hasn't.\" He almost had, the night in Wolfhump, probably because he had not been sure they would live to dawn.\n\n\"Have you asked him?\"\n\n\"Yes.\"\n\n\"Oh.\"\n\nAgarik scratched at his scab, caught himself, and scratched around it instead. A breeze buffeted the side of the tent. She would get the answer from Rias as soon as she saw him. No more waiting. In the meantime, there was little point to dwelling on it. She should rest, or study Lancecrest's journal. The notes would help her along on her translations. She would love to be the one who\u2014\n\nLove! Her memory triggered. She had told Rias she loved him before falling asleep. She bit her lip. Had he responded? She could not remember. Had he felt awkward? Alarmed? Dare she hope\u2014pleased?\n\nTikaya swung her legs off the cot. \"I need to talk to him.\"\n\nAgarik lifted a hand. \"You can't go anywhere. You were almost dead a few hours ago. You need to rest.\"\n\n\"I did rest. I'm done now.\" She stood and promptly fell back onto the cot, betrayed by straw legs.\n\n\"Really,\" Agarik said dryly.\n\n\"I just need to get my muscles moving.\" She stuck her legs out. Maybe a few ankle rotations and toe wiggles would improve the blood flow.\n\n\"Rias will be back by morning, I'm sure. You should rest.\"\n\n\"I need to talk to him now. It's, uhm...\" Tomorrow they would be surrounded by squads of men again. She needed to talk to him tonight. Alone. And she was not about to explain that to Agarik. \"I need to see if he has the journal I recovered,\" she said instead. \"I want to study it further before we go into the tunnels.\"\n\n\"It's the middle of the night, Tikaya.\"\n\n\"Night is eighteen hours long here. It's always the middle of the night.\"\n\n\"You've a point there.\" Agarik stood, head brushing the rafter of the tent. \"I'll get the journal for you if you stay here and rest, all right?\"\n\nShe smiled at him but did not answer. Whatever got her nanny out of the tent so she could leave.\n\nAgarik unfastened the flap and slipped out. An icy draft reminded her to dress fully before venturing outside. Fortunately, someone had piled her gear at the end of her cot where a portable stove burned. She checked for the journal in case Rias had tucked it in there, but he had probably placed it elsewhere to make sure the Nurian would not find it.\n\nOutside, stars and a half moon brightened a wedge of sky framed by steep canyon walls. They must have arrived at the canyon where the tunnels began.\n\nA bonfire blazing in the center of camp snapped and launched sparks into the air. Five tents, large enough to hold cots for all, stood back from it. The sleds lay between her tent and the next, and the dogs had burrowed into the snow and slept with their noses tucked under their thick, fluffy tails. A surprising number of men were still awake and chatting fireside. Or perhaps they were awake again. Rias must have kept the camp quiet and had the men feign sleep to draw in the Nurian. A ceramic jug passed from hand to hand, and laughter gave the atmosphere a jovial feel, though some of the chortles sounded strained. No doubt rumors abounded concerning the tunnels, and, after the deaths they had seen, the men must suspect not all of them would make it out again.\n\nTikaya stood, breath fogging the air before her eyes, wondering where to find Rias. She considered the other tents. Three stood dark, but light seeped from beneath the flaps of hers and one other\u2014might that be a command tent?\n\nShe padded to the entrance and debated whether to peek inside or wait for him to come out. If Bocrest led the meeting, he would not appreciate her interruption. She lifted her hand but let it hang as she considered how one knocked on a tent.\n\nThe flap peeled back, and one of the sergeants almost crashed into her.\n\n\"What're you doing?\" He lowered his brows and glared at her. \"Spying?\"\n\n\"Huh? I mean, no, I\u2014\" She looked at her still raised hand as if that would explain her intent.\n\n\"Who is it?\" Captain Bocrest asked from within.\n\n\"The woman,\" the sergeant said over his shoulder. \"Standing outside, spying.\"\n\n\"I'm not spying!\"\n\n\"I got to piss.\" The sergeant shoved past her. \"Out of my way, girl.\"\n\n\"It's Tikaya,\" she informed his back.\n\nHe threw a rude gesture over his shoulder. No one called to invite her into the tent, but she walked in anyway. Six marines, Bocrest and his senior ranking men including a scowling Ottotark and the sawbones whose brother she had killed. No Rias. She swallowed.\n\n\"Sorry, for interrupting,\" she said, \"but I'm looking for...that journal. I thought it'd be useful to finish translating it before we head in.\"\n\nThe glowers facing her seemed more suspicious than her presence called for after what she had been through with these men.\n\n\"For our benefit?\" Ottotark growled. \"Or so you can deliver it to the archaeologists inside?\"\n\n\"I don't know what you're talking... Oh.\" She recalled the Nurian's speech before he had tried to kill her. Those moments when she had been so close to death were fuzzy, but she did remember archaeologists being mentioned. Rias must have relayed the information. \"I don't know who's in there. There are a lot of archaeologists in the world.\" Though she had to admit that at least half of the renowned ones came from the Kyatt Islands and most of the other half had studied there at one point or another. \"Chances are I don't know any of them, if that's what you're worried about\u2014the folks I know aren't the types to go hunting for ancient weapons caches. And, anyway, I wouldn't betray Rias.\"\n\n\"You, the cryptomancer who slagged us all in the war, wouldn't betray 'Rias?'\" one of the sergeants asked.\n\n\"Quiet, Karsus,\" Bocrest said. \"He hasn't told her.\"\n\n\"No? Oh, yes, that relationship's going to work.\"\n\nThe ire in the room evaporated and was replaced by sniggers. Tikaya set her jaw. She preferred the hostility. This was one more reason for her to talk to Rias tonight. She was damned if she was going to be the only one in camp who did not know.\n\nBocrest reached into the rucksack beside his cot and pulled out the leather journal. He tossed it to her. \"Go. Figure out what's in there that's worth torturing people over.\"\n\nNaturally, she bumped into the returning sergeant on her way out. He growled at her, and she skittered away with an apology. She stopped a few paces beyond, bent over, hands on her knees, fatigue making her limbs heavy.\n\nWhat further cane fields would she have to harvest for these Turgonians to prove she was sold on working with them? Then again, was she? She cared what happened to Rias and Agarik, but she would not cry over the rest if an avalanche swallowed them. What if she did encounter scholars she knew and respected inside? Men and women\u2014how she missed having female colleagues to talk to!\u2014with a ship anchored somewhere, a way back home. What if she did have a chance to switch sides?\n\n\"Tikaya?\"\n\nShe straightened and turned toward Rias's concerned voice. She hoped the darkness hid the guilty flush that heated her cheeks.\n\n\"Is something wrong?\" He wore parka, cap, scarf, and he even carried snowshoes and a rifle. Where had he been? Scouting the tunnel entrance? \"I thought you'd sleep until morning.\"\n\n\"I, uhm, wanted to talk to you.\" She had been looking all over for him, but had not given much thought to what she would say.\n\n\"Of course.\"\n\nRias leaned the rifle and snowshoes against the side of a tent, and she joined him in the shadows, wanting to be out of eyesight if anyone else from the meeting came out to relieve himself. He wrapped her in a hug, and she slid into his arms, though the amused eyes of the men in the tent nagged her mind. She had to know. Tonight. She waited for Rias to release her, but he held her in silence for a long moment, arms tight. She breathed in the tang of weapons cleaning oil and black powder mingling with his warm male scent. Men laughed around the fire, trading jokes, boasting of brave feats.\n\n\"I'm sorry,\" he finally whispered.\n\nSorry? Was that in response to her proclamation of love? He was sorry he didn't love her back?\n\nThen he added, \"I used to be faster. You shouldn't have been\u2014I should have seen the condor sooner.\" He sounded so distraught. It brought a lump to her throat.\n\n\"Oh, Rias.\" Tikaya wriggled her glove off and laid her hand on the side of his face. \"That's not your fault.\"\n\n\"I should have sent you with the main party.\" His own glove came off and he laid his hand on hers.\n\n\"I'm sure that condor could have found me down there as well as on the mountain.\"\n\nHis sigh came out as more of a grumble. \"I'm tired of people trying to kill you.\"\n\n\"I'm not an enthusiast of the trend either.\"\n\nRias's other hand slid under her scarf to rub the back of her neck. She closed her eyes, letting those strong fingers knead her flesh, even as she lamented the layers of parkas and wool uniforms between them. The voices of the marines faded from her awareness. Soft breaths tickled her cheek, and she opened her eyes to find his face close. Shadows cloaked his eyes, but she sensed his intent and leaned into him, head tilted back.\n\nHis lips brushed hers, questioning at first. Tikaya parted her lips, invited more. His kiss grew firm, confident, and she thought of the experienced warrior she had followed through the Nurian ship. Heat flared through her body, and she forgot about her questions, the camp, and the freezing air. She might have forgotten a lot more if someone had not crunched around the tent and stopped to stare.\n\n\"Well, well, well.\" Ottotark.\n\nShe winced and drew back. Of all the people to stumble upon them.\n\n\"Ignore him,\" Rias breathed, nuzzling her ear.\n\nA small grin stretched her lips as it dawned on her that she could. If Bocrest had told the men to treat Rias like an officer, that would mean he outranked the sergeant. As much of an ass as Ottotark was, he seemed loyal to his uniform and the chain of command. Surely, he would leave them alone if Rias ordered it.\n\nTikaya probably should not have looked so smug as she cast a dismissive glance Ottotark's way, but she could not resist, not after all the torment he had thrust upon her. She slid her hands under Rias's parka and kissed him deeply. Let the bastard watch.\n\nOttotark guffawed.\n\nStartled, she broke away. That was the last reaction she expected. She looked at Rias, eyes questioning, but Ottotark spoke first.\n\n\"The captain really needs to let you live.\" He was pointing at her, laughter punctuating his words, but his tone seemed designed to carry to the whole camp. \"I'd love to accompany you back to your island so I can tell your mom and pop that you were out here fucking Fleet Admiral Saskha Federias Starcrest, the man who personally recommended taking over your islands to the emperor.\" Now it was Ottotark's turn to be smug. Very smug. \"But don't let that stop your plans for the evening. I can see you're enjoying yourself. Carry on.\" There was far too much pleasure in the cruel sneer he launched at them before walking away.\n\nTikaya felt lightheaded. She had to remind herself to breathe. All she could do was stare at Rias's shoulder.\n\n\"I'm going to kill that man,\" he said.\n\nNo denial. No explanation about how Ottotark was wrong. No claim that it was a lie.\n\n\"You were right,\" Tikaya choked. \"My people have heard of you, and you'd never be welcome on my island.\"\n\nShe stepped back.\n\nRias grasped her arm. \"Tikaya, please. Let me\u2014\"\n\nShaking her head, she pulled her arm free. She had to get away. She had to think. She had to\u2014she didn't know.\n\n\"I'm sorry,\" Rias called after her.\n\nShe stumbled, not sure where to go. Not back to the fire and the marines. If she returned to her tent, Agarik would be waiting to yell at her for leaving. She definitely did not want to go anywhere she would have to look at Ottotark. But neither could she go out where yetis and wolves and grimbals waited to devour silly girls thousands of miles from their homes.\n\nTikaya finally sat down behind the sleeping tent. She drew up her knees and buried her face in them. She ought not be so stunned. There had been clues all along. She just hadn't wanted to see them. Had she really thought someone who so readily took command and led the way into battles was an engineer? That love of mathematics made him the best cursed strategist of his generation. Starcrest. How often had his name come up in the documents she decrypted? The youngest fleet admiral in the history of the empire. The man who, as a captain, had been responsible for the sinking of a hundred Nurian ships. And the man who, as an admiral, had guided every battle, every skirmish that allowed the Turgonians to again and again best the preeminent mental scientists in the world, with only mundane technology on their side. It was not until after his death that the tides had turned, ending in a stalemate. Yes, his death. She vividly remembered decoding a note that said a Nurian assassin had killed the admiral. He was supposed to be dead, not exiled. That was why she had never considered her Rias might be the legendary admiral.\n\nStill, who could she blame but herself? She should have known. She certainly should not have fallen in love with him. If he was nobody important, nobody who would matter to her, he would have told her his name. This was exactly why he had kept it from her. He had known she wouldn't want anything to do with him. How could she? If what Ottotark said was true, and Rias had been the one to suggest taking over her homeland, then every death was indeed on his hands.\n\nHer stomach writhed, and she choked on a sob. Every death, including Parkonis's.\n\n# CHAPTER 15\n\nTikaya did not know how long she sat in the shadow of the tent, but shivers and a frozen nose finally convinced her she had to find a warmer berth. She put a hand down and started to rise. The crunch of boots stopped her. A tall figure with a rifle strode between the tents and into the darkness before her. She could not make out features but had an inkling. She remained still, cloaked by shadows.\n\nA long moment passed with the figure scanning the dark canyon beyond the camp.\n\n\"Tikaya?\" he called.\n\nShe closed her eyes. Rias. No, Fleet Admiral Starcrest.\n\nShe did not want to\u2014could not\u2014talk to him. Not then.\n\nHe called twice more.\n\n\"The bitch is gone.\" Ottotark strode into view from another direction. He passed within a couple feet of her and stopped a few paces from Rias.\n\n\"Ottotark,\" Rias growled. \"I ought to twist your head off your slagging neck and shove it up your ass.\"\n\n\"It's not my fault you didn't tell your girlfriend your name. Admiral.\"\n\nRias had no answer for that, and even the darkness did not hide the slump to his shoulders. \"Where is she?\"\n\n\"Off to the tunnels to join her friends and leave us hanged.\"\n\nTikaya clenched her jaw. Damn these men. She did not want to deal with either one, but she could not let Rias believe she had run off. She opened her mouth to say something, but Ottotark spoke first.\n\n\"You should thank me,\" he said. \"It's pathetic the way you were hanging all over the bitch. And why? She slagged us in the war. If you want her, tie her down and screw her, but don't\u2014\"\n\nRias threw down his rifle and charged. Between one eye blink and the next he covered the distance and crashed into Ottotark, taking him down so hard they flew backward.\n\nTikaya drew her knees in tight, too startled to speak. The attack may have surprised Ottotark, but he recovered and fought back like a cornered badger. Grunts and snarls accompanied the smack of fists striking flesh.\n\nIn the darkness, she lost track of who was who as the men thrashed and writhed on the ground. Clumps of snow flew, spattering her cheeks. Something cracked, and one of them\u2014Rias?\u2014yelped in pain.\n\nTikaya held her breath. Ottotark was younger, bigger, and without any morals as far as she could tell. She tried to tell herself that Rias\u2014Starcrest\u2014was no longer her concern, but her fingers clenched into a fist, and she silently rooted for him.\n\nOne man maneuvered on top and straddled the other. He punched down, and a head hammered the snow. The bottom man bucked and twisted, and a moment later the positions reversed.\n\n\"Traitor,\" Ottotark snarled.\n\nBoth men panted, breaths rasping. They switched positions again, legs tangling as each tried to pin the other.\n\nMetal rang, a knife being pulled.\n\nAs furious as he was, Rias would not pull a blade. Tikaya knew he wouldn't. She almost yelled a warning, but stopped herself. A distraction could prove fatal.\n\nOne man found the top again and raised an arm, the knife silhouetted against the night sky. The blade plunged down at the head of the other.\n\nMovement halted. Ragged breaths assaulted the still air, and Tikaya could not tell whether they belonged to one man or two. The top person lurched to his feet and staggered back, a hand to his belly.\n\nHer heart hammered in her ears, and she could not bring herself to call out. If it was Ottotark, who knew what he might do to her? If it was Rias, and he had just made good on his promise to kill the sergeant...\n\nBut, no, the supine man groaned. Weakly.\n\nTikaya could not identify him by the sound. She forced her limbs to unlock and she rolled to her knees. She crept to the fallen man's side and hesitantly reached toward the face. Her glove bumped something hard.\n\nThe knife.\n\nIt wasn't lodged in an eye after all. The attacker had sunk it to the hilt in the snow a hair from the other's ear. That told her what the shadows did not: of the two, only Rias would have shown mercy.\n\nShe jerked her hand back as the man\u2014Ottotark\u2014groaned again. She lunged away from him and looked for Rias. He might be injured and need help. She spun slowly, searching the shadows, but he was gone.\n\nMaybe he had gone to find a cot. She trotted into camp. The number of people awake had dwindled, and the fire burned low. She tore open the flap to the sleeping tent and crashed into someone coming out.\n\n\"Tikaya,\" Agarik blurted. \"We've been looking all over for you.\"\n\nShe grabbed his parka. \"Is Rias with you? Have you seen him?\"\n\n\"Not since he went to check the perimeter.\" He must have read her distress. \"Why? What's wrong?\"\n\n\"Ottotark told him I'd gone to the tunnels. I'm afraid he might have gone after me.\" She explained the fight, all the while cursing herself for staying silent during the men's confrontation. Why hadn't she answered when he first called out? If he got in trouble because of her stung feelings...\n\n\"He shouldn't go in there alone,\" Agarik said, tone terse, worried. \"Come, we've got to tell the captain.\"\n\nGlad to have him leading the charge, Tikaya followed him into the command tent. Heat and faint light emanated from the portable stove in the center, and she could pick out shapes amongst the shadows. The meeting had dispersed, and only Bocrest and a couple lieutenants remained inside, all flat on cots. Tikaya stopped near the stove. Certain the captain would blame this on her, she did not want to be close enough for him to grab easily.\n\n\"Sir?\" Agarik asked.\n\nBocrest jerked awake, hand finding a pistol.\n\n\"It's Agarik, sir. The admiral's missing.\"\n\nBocrest growled and lurched to his feet. \"Explain.\" He must have had a suspicion, for his eyes skimmed the darkness and found Tikaya. He cursed. \"No, you explain.\"\n\nWhile Agarik had listened to the story patiently, she had to suffer curses and hurled gear while reciting it for Bocrest. He managed to get dressed despite his preoccupation with throwing things and was stuffing his feet into his boots by the time she finished.\n\n\"This is why women aren't allowed in the military.\" He cursed again, but shifted to efficiency after that.\n\nBy now, the lieutenants were awake and dressing, and he snarled orders at them. Less than five minutes later, the entire camp stood in formation outside. The last to show up, Ottotark limped to the head of one of the lines. Several men held lanterns, and the flickering light revealed bloody and swelling contusions on the sergeant's face. A dark part of her wished he was dead, though Rias probably would hate himself if he killed someone out of sheer rage.\n\nBocrest stalked over to face Ottotark. \"Did you draw the knife or did he?\"\n\n\"Sir?\" Ottotark asked in a tone that sounded like he was trying to play dumb, or maybe buying himself time to think.\n\n\"You heard me!\"\n\nOttotark licked his lips. The marines in formation apparently knew better than to turn their heads and watch, but their eyes flicked toward the confrontation.\n\n\"I did, sir,\" Ottotark finally said.\n\n\"I told you\u2014I told everyone\u2014to treat him like an officer. The punishment for drawing a weapon against an officer is death.\"\n\n\"Sir! He's not an officer any more. He's a traitor, you said so. The emperor\u2014\"\n\n\"Isn't here,\" Bocrest said. \"We'll discuss punishment when the mission is over. For now, do your job and don't talk to our guide or our translator. Is that understood?\"\n\n\"Yes, sir,\" Ottotark said so softly Tikaya almost missed it. If only the captain had issued that order a few hours earlier.\n\nThe tracker strode out of the darkness, and Bocrest shifted his attention.\n\n\"There are footprints at the tunnel entrance and it looks like someone walked in, at least a few steps. The floor is that hard black material, and there's no way to track further.\"\n\n\"Slagging women,\" Bocrest said before raising his voice. \"Gather your gear, men. We're going after him.\"\n\n* * * * *\n\nThe marines marched in step, and the echoes reverberated through the wide black tunnel. No dead skeletons had marked the entrance, no piles of rubble scattered the floor, nor did water drip eerily in the distance, but the place made Tikaya uneasy nonetheless.\n\nIt was too clean, too perfect. No cobwebs obscured the distance, no chips or scratches marred the dust-free floors, and no decoration adorned the walls. The cool dry air reminded her of the lava tubes meandering beneath her father's plantation, but no familiar earthy smells accompanied it. No smells at all. The lanterns the marines carried went unused. A soft glow emanated from all around, illuminating the tunnel as clearly as the midday sun. She had visited several ancient catacombs, qanats, and subterranean cities, and she had studied dozens more. This sterile tunnel was like nothing in the archaeology books. Nothing in the world.\n\n\"Who made this place?\" someone muttered.\n\nShe walked behind Bocrest, second in a queue of thirty men. A handful of marines had stayed in the base camp while Agarik and a couple others scouted ahead.\n\n\"Ancient people,\" someone answered.\n\n\"How?\"\n\n\"Magic.\"\n\nNo telling tingle stirred the hairs on Tikaya's neck. \"I don't think so.\"\n\n\"Magic,\" another said, his tone brooking no argument. Others murmured assent. \"Evil magic made this place, just like the rocket and the thing in Wolfhump.\"\n\n\"No talking,\" Bocrest snapped over his shoulder, saving Tikaya from launching into a lecture that would doubtlessly not be well received.\n\nRias must have a lot more patience than she to have commanded such men all his life. It must have been lonely for him with so few peers. She shook the thoughts from her head. It was none of her concern. Even if she could forgive him for his lie of omission, Admiral Starcrest was nobody she could have a life with, not without betraying her people, her family, and everyone she loved. Especially those fallen during the war. She could want him found and safe, but she could not want him. Not any more.\n\nShe swallowed a lump and fished the journal out of her pocket. A challenge. Her mind needed a challenge, and she needed to learn as much as she could before her services were needed. If ever there was a place she could walk and study at the same time, it ought to be these flat, terrain-free tunnels. The worst thing that could happen is she would trip. An animal screeched in the distance.\n\nWell, maybe not the worst thing.\n\nRias had mentioned strange predators in the tunnels. Predators that would probably find a single man an appealing target. Best not to dwell on that. Tikaya turned her attention back to the journal.\n\nA mile or two passed with no side rooms or cross corridors forcing decisions. With her mind and her eyes locked on the pages, she failed to notice Bocrest stopping, and she crashed into his back. The journal slipped from her fingers as he spun and scowled.\n\n\"Didn't you see the sign?\"\n\n\"Sign?\" Tikaya blinked and glanced about. They had come to a six-way intersection, where the scouts had stopped to wait. Large symbols in groupings of threes glowed a soft red above eye level at each corner. \"Oh, yes, signs. I just read about those.\"\n\nBocrest sighed noisily, while she picked up the journal.\n\n\"Not those signs, the hand signal.\" He demonstrated by raising his hand, fingers spread. \"That means 'squad halt,' not 'librarian run into the captain's back.' You need to pay attention in here.\"\n\n\"Do you want me to pay attention or do you want me to be able to translate the writing on the walls?\" she asked.\n\nBocrest folded his arms. \"Yes.\"\n\nShe snorted but pointed to each sign as she relayed them: \"Biology labs, alchemy labs, physics, animal experiments, labs for something Lancecrest didn't recognize, and living quarters.\"\n\n\"What is that?\" A marine pointed to a calf-high scat pile in the middle of one of the corridors.\n\n\"Sorry,\" Tikaya said, \"I don't translate poo.\"\n\n\"Koffert.\" Bocrest gestured for the tracker.\n\nThe man knelt to examine the pile. He rubbed some between his fingers and sniffed it. \"Predator, unknown. Large. Passed this way less than an hour back.\" He stood, wiping his hand on his trousers, and Tikaya made a note not to share meals with the man.\n\nBocrest faced the scouts. \"Any sign of Starcrest?\"\n\n\"No, sir,\" Agarik said. He met Tikaya's eyes, and they shared a grimace.\n\nBocrest grunted and waved toward the symbols. \"Komitopis, which way?\"\n\n\"Which way do I think Rias would go if he was wondering which way I would go?\"\n\nAgarik smiled faintly. Bocrest did not.\n\n\"Alchemy?\" she guessed.\n\n\"Fine. Someone mark the wall.\" Bocrest waved the scouts forward. \"Go. You boys in the back, stay alert. Watch for monsters creeping up on our asses.\"\n\nAs the squad headed the new direction, Tikaya cast a longing gaze at the corridor that led to the living quarters. If any personal affects remained after all this time, she could learn much about the people from studying them. After they found Rias, perhaps they could go back.\n\nSoon, doors marked the passage, taller and wider than normal, and without knobs or latches. Symbols denoting laboratories adorned some while others remained plain.\n\nSomeone walking closer to the wall than the center of the tunnel triggered a door to slide upward of its own accord. The man cursed and lurched back into line. Tikaya glimpsed a landing overlooking what she guessed to be lab stations\u2014all the furnishings were oversized by human standards. A hand on her back encouraged her to hustle forward and catch up with Bocrest.\n\n\"Should we check some of these?\" she asked.\n\n\"I'm not exploring anything until we catch up with our lovelorn guide,\" Bocrest said.\n\nUp ahead, the scouts stopped before a closed door. Agarik and another knelt to check something on the floor while the third man stood guard. After a moment, Agarik jogged back to the group.\n\n\"What is it?\" Bocrest asked.\n\n\"Blood.\" Agarik glanced at Tikaya. \"A lot of blood.\"\n\nHer hands tightened around the journal. If Rias was hurt\u2014or worse\u2014because he had charged in here to look for her, it would be her fault.\n\nWhen they reached the spot, the size of the dark puddle only increased her dread.\n\n\"Human blood,\" the tracker said after a taste. \"Plantigrade print over there, but definitely not human.\"\n\nHe pointed to a second puddle halfway under the door. A bloody print more than twice the size of Tikaya's foot lay beside it. Dots at the end of the toes suggested claws.\n\n\"Bear?\" Bocrest asked.\n\nTikaya, remembering Rias's tale of the tunnels, said, \"I doubt it.\"\n\nA man screamed somewhere beyond the door. Rias? She lunged for the door, triggering the opening mechanism, but Bocrest caught her before she crossed the threshold.\n\n\"We'll get him,\" he said. \"You wait here.\"\n\nHe waved two fingers, and the scouts slipped in first, fanning out on a landing with their rifles raised, ready to fire. Tikaya shifted her weight from foot to foot and eyed a bow stave and quiver attached to a rucksack. The man carried a rifle and pistol too; surely, he could spare the weapon so she could\u2014\n\n\"Clear on the landing,\" Agarik said.\n\n\"Sergeant Karsus.\" Bocrest nodded for the man to take over the lead.\n\nWithout words, and faster than Tikaya expected, the marines shucked their rucksacks and split into two teams. They filed down stairs on opposite ends of the landing and disappeared from her sight. Only Bocrest remained with Tikaya.\n\nIgnoring his hiss of annoyance, she twisted free of his grip and stepped inside. The landing overlooked a cavernous room that stretched a hundred meters or more. Thick thirty-foot-high columns supported the unadorned black ceiling. Empty floor dominated the front third of the room, and she could only guess at the furnishings beyond. She decided to think of them as lab stations and storage cabinets, though even the lowest counter rose taller than the approaching marines. The height and arrangement blocked much of the floor view as the stations created a maze of sinuous yet symmetrical aisles, some wide, some surprisingly narrow. As with the tunnel, light from an indiscernible source illuminated everything.\n\nA cry of agony echoed from the center, and she glimpsed a blur of black before it disappeared behind a row of twenty-foot-tall cabinets.\n\n\"Sprites-licked idiot,\" she cursed, whirling to look for a bow amongst the discarded gear. She was not sure whether she meant Rias or herself. If, after all he had lived through, he died to some random animal attack...\n\nTikaya spotted the bow stave she eyed earlier. The marine had left it in favor of the rifle. She stuffed the journal into her rucksack, then untied the bow with fingers too irritated to fumble with fear. She yanked the quiver free as well. Stringing the weapon was a struggle, and she prayed the draw wouldn't be too heavy for her.\n\n\"Let my men do their job, Komitopis.\" Rifle crooked in his arms, Bocrest leaned on the wall by the door, which had slid shut again. His voice was more sympathetic than she had ever heard it, and he did not try to take the weapon from her, but he did add, \"You're staying with me,\" in an implacable tone.\n\nShe succeeded in looping the string over the limb of the bow. \"I'm not going to\u2014\"\n\n\"I'm not going to lose you as well as Starcrest. We need someone to read this grimbal shit.\"\n\nNoise in the corridor made them spin toward the door. Tikaya nocked an arrow while Bocrest raised his rifle. In the lab behind them, the men stalked in silence, and she had no trouble hearing the fast, heavy footfalls outside as they grew louder\u2014closer.\n\nBocrest cursed, probably regretting that he had sent all his men below. The footfalls thundered to a stop outside the door. Tikaya drew the arrow, ignoring the strain between the backs of her shoulders. At least her shoulder no longer vexed her.\n\nThe door slid open. She held her breath.\n\nThe tunnel was empty.\n\nThe tip of her arrow wavered as her muscles quivered from the effort of holding the draw. She glanced at Bocrest, a question on her lips.\n\nThen a head popped around the jamb and disappeared again. It happened so quickly she doubted her sight. Then a familiar voice spoke with wry humor.\n\n\"Can I come in?\"\n\n\"Rias!\" she blurted, even as Bocrest shouted, \"Curse you, Starcrest.\"\n\nRias slid out from behind the wall. \"I hope that's a yes.\"\n\nThey lowered their weapons as he joined them on the landing. First Tikaya noticed a garish black eye and fingermarks bruising his neck, then saw the sweat bathing his face, saturating his hair, and dripping from his chin. His chest, framed by the straps of his rucksack, rose and fell with rapid, deep breaths. He wore all of his weapons too\u2014in addition to the rifle he carried, pistol, cutlass, and knife challenged the ammo pouches and powder tins for room on his belt. He must have been back to camp since the fight.\n\n\"I can't believe you left without me,\" Rias said, eyes darting as he took in the lab.\n\n\"But I saw you with Ottotark,\" Tikaya said. \"He said\u2014I thought you went in the tunnels looking for me.\"\n\nRias dragged a sleeve across his brow, not quite hiding a grimace of shame. \"No, I didn't believe him. I just had to... I almost lost it with him. I needed to get away, to think.\"\n\nTikaya sagged against the railing with relief.\n\nDisgust curled Bocrest's lip throughout their exchange, and he finally jabbed his rifle toward the lab below. \"If you were behind us, who in the empire are my men trying to rescue down there?\"\n\n\"I don't know.\" Rias glanced at Tikaya. \"Maybe someone we can question if we recover him alive?\"\n\nBocrest raised his voice for the benefit of the men below. \"Starcrest accounted for. Continue with retrieval operation.\"\n\n\"Treat them like grimbals,\" Rias called. \"It takes a cut to the neck or shot to the eye to kill. And, above all else, do not break anything in here.\"\n\nThe last command seemed strange when a man's life was at stake, but the grimness in Rias's mandate kept Tikaya from questioning it.\n\nA shot fired, and a roar came from the center of the lab. Something crashed against a cabinet, and Rias winced. \"Not good. Wish I'd had time to do a briefing.\"\n\nHe glared at Bocrest who in turn glared at Tikaya.\n\n\"This is your childish sergeant's fault,\" she said, \"not mine.\"\n\n\"Why couldn't the cryptomancer have been a man?\" Bocrest glowered at Rias. \"Though after all that time on Krychek, you probably wouldn't have cared.\"\n\nRias raised an eyebrow. \"I am armed, you realize.\"\n\nAnother roar answered the first, and Bocrest's head snapped back toward the lab. \"There's a second?\"\n\n\"Back corner.\" Rias headed for the stairs. \"Who's coming with me?\"\n\n\"You're not going anywhere,\" Bocrest said.\n\n\"I've fought these before. Better me than them. But I could use backup.\" He offered Tikaya a tentative smile.\n\nMore gunfire and a spatter of curses sounded in the lab, but she stared at him for a long moment. \"You want me? After last night?\"\n\n\"Nothing's changed for me,\" he said with a sad smile. \"Besides, you're a better shot with that bow than my other option.\"\n\nBocrest sniffed. \"I am armed, you realize.\"\n\nBut Rias was already heading down the stairs. \"Third team advancing along the south wall,\" he called.\n\nA strangled groan of pain whispered through the aisles. Before she could think better of it, Tikaya slung the quiver across her back and followed Rias. She could figure out her feelings later.\n\nThey descended floating steps too deeply spaced for human comfort. Bow at half-draw, she trailed him across the open area toward a narrow gap along the south wall. As they approached, claustrophobia tightened her chest. The backs of cabinets and lab stations loomed in the same black as the wall, with the counters well above Tikaya's head. She and Rias would have to walk single file.\n\nSweat dampened her grip and slithered down her spine. She had been ready to throw herself into the fray for Rias's sake. Going on a monster hunt for uncertain stakes was another matter. Why had she followed him down the stairs? Surely he would have been better off with Bocrest. Despite her trepidation, she kept following. It should not matter, especially now, what Rias thought, but she could not bring herself to complain or back out.\n\nHe pressed himself against the wall and gestured for her to go ahead. \"Since I can fire over your shoulder, you can lead.\"\n\nJust when she thought it couldn't get bleaker.\n\n\"You know,\" Tikaya said, struggling for nonchalance as she slid past, \"some men protect the women they care about by keeping them away from danger.\"\n\nRias raised an eyebrow. \"Sounds stifling.\"\n\n\"Perhaps so.\"\n\n\"Military officers like to challenge people to encourage growth.\"\n\n\"I've been six feet tall since I was thirteen; growth hasn't been my goal for a while.\"\n\n\"You could grow a bit more before you got too big for Turgonian tastes.\"\n\nShe smiled a bit at the double meanings, her mind distracted from her fear. As on the ship, his steadiness calmed her. She could worry about whether it should or not later. In the meantime, she wiped her palms dry, and padded forward, bow ready.\n\nAs they traveled deeper into the lab, new higher pitched growls grew audible. They came from somewhere near the back wall. The second creature. Tikaya hoped some of the marines were moving that way too.\n\nThey eased closer. Twenty meters, fifteen, ten. Around the corner, claws clacked, teeth snapped, lips smacked, and a tearing sound ripped the air. Tikaya hesitated, certain she did not want to see the source of those noises\u2014or what it was eating. Rias's hand rested on her shoulder briefly. She nodded to herself and peered around the corner.\n\nFifteen meters away, in a wide aisle, a huge bipedal creature crouched over a ravaged human corpse. The beast lacked fur, and powerful muscles rippled beneath oily black skin that gleamed under the light. The only thing soft were full breasts that swayed as it tore at flesh.\n\nTikaya slipped out and raised her bow.\n\nThe creature snorted. The head that came up appeared simian except for the long fangs flecked with blood and tissue. The arms and hands, too, were disturbingly human, though claws flashed at the ends of those fingers. The creature reared on its hind legs, powerful thigh muscles bunching. It sprang and sprinted toward them.\n\nA rifle fired over her head, the report deafening. Tikaya expected it and did not flinch. Rias's shot grazed the creature's jaw. She loosed her arrow at the neck. It sunk in, and the beast cried out, its scream eerily human. But neither shot slowed its advance.\n\nRias's pistol fired, hammering the creature between its breasts. Tikaya had time for one more shot and aimed for an eye, but the beast was closing fast. Her arrow skimmed its temple instead.\n\nTikaya flattened herself against the wall, hoping she could dodge if those claws flashed. She thought the beast's momentum would carry it past her, but it halted with amazing athleticism.\n\nIt whirled on her, claws raised. Rank breath washed over her. She ducked even as Rias yanked her out of reach. She almost lost the bow as he charged past, cutlass raised. She recovered and stepped back to nock another arrow. Rias ducked a swipe and darted in, but the muscled torso deflected his blade like armor. He nicked a vein, drawing blood. Claws gashed his arm before he could leap out of reach. Its speed was mesmerizing, but she forced herself to focus.\n\nWith the creature sparring with Rias, she could wait for a chance at a critical target. There. She fired, and the arrow plunged into its eye.\n\nThe beast staggered into a counter, gashing its own face as it clawed at the arrow. It stumbled, then pitched backward. Still.\n\nTikaya leaned a hand against the wall for support and let her bow droop. \"Next time we attack a twelve-foot-tall monster, we probably don't need to worry about me seeing over your head.\"\n\n\"Conceded.\" Rias rotated his arm to check the slashes below his shoulder, but dismissed them. \"One down. Let's see if the other is still alive.\"\n\nA rifle cracked in the center of the lab.\n\n\"I'm guessing so,\" Tikaya said.\n\nRias jogged along the wall toward a cross-aisle where he could cut over. He paused when he reached the half-eaten man. It was wearing the black uniform of a Turgonian marine. Though the neck had been torn out, the chest smashed and ravaged, the face remained mostly intact.\n\n\"That's not one of ours, is it?\" Tikaya asked.\n\n\"No.\"\n\n\"Somebody from the fort?\"\n\nMultiple rifles fired.\n\n\"Later,\" Rias said, already disappearing around a corner.\n\nA bestial screech reverberated through the lab, and men shouted orders. Tikaya raced after Rias, careening around the corner to face another melee. A second creature, larger and more muscled than the first, fought in the center. This one was male.\n\nMarines attacked from both ends of the aisle, cutlasses and daggers struggling to pierce the resilient skin. The creature whirled, slashing forward, then back, its wild actions enraged, and Tikaya wondered if it knew its mate had fallen. Blood streamed from its sleek flesh, but it batted men away without faltering. As tall as the Turgonians were, they had little chance of reaching the neck or head with their blades.\n\nRias charged into the fray. Tikaya drew the bow, waiting to glimpse an eye, but the beast chose that moment to escape. It sloughed off its attackers and charged her direction. Her heart lurched. She loosed her arrow, but she lunged to the side too soon, and her shot only struck muscle.\n\nShe glanced at the cabinets on either side of her. There was no time to climb out of reach. She smashed herself to the side again, hoping this creature would run past. Though, even if it did, all it would have to do was rake her on the way past and\u2014\n\nSteel zipped through the air from the aisle behind her. A knife lodged in the creature's eye.\n\nIt tripped and tumbled, skidding past her. The prone form crashed into a cabinet, jolting it. The door flung open, and trays of bones spilled out. Human bones, tagged and marked with colored dots. Smaller ones, fragile with age, shattered.\n\nTikaya found Rias's eyes, thinking of his admonition not to break anything. Chest heaving, he stood amongst the other marines. He shook his head slowly.\n\n\"Everyone back to the entrance,\" he said.\n\nBefore following the men, Tikaya tossed a glance toward the back wall. Someone had thrown that knife, yet no marines filed in from that direction.\n\nA clunk echoed through the lab.\n\n\"Hurry,\" Rias urged.\n\nHe led a sprint to the stairs where Bocrest twitched an eyebrow at Tikaya and said, \"Nice shot.\"\n\nShe did not answer. It had not been her attack that brought the second creature down.\n\n\"Let's go,\" Rias said. \"In the hall. We don't want to be here when the cubes arrive.\"\n\n\"Cubes?\" Bocrest asked.\n\nTikaya thought of the square vials from the rocket, but surely he could not mean those.\n\n\"No time to explain.\" Rias pushed past and into the corridor. Leaving the lab without exploring it seemed an abandoned opportunity, but Tikaya did not question him, not when such grimness haunted his face.\n\nBefore they took three steps down the tunnel, a door ahead of them slid open. A black one-foot-wide cube floated out at chest level. Tiny red and yellow lights flashed on its top, and a one-inch hole glowed red on its front. A few symbols ran along the sides, and she leaned forward, squinting.\n\n\"Back,\" Rias said. \"Back into the lab.\"\n\n\"What does it\u2014\" Tikaya started, but the glowing hole brightened and a red beam lanced out. Rias yanked her to the side, and it caught the edge of her sleeve. The beam burned a hole through the material.\n\nShe half ran and was half dragged back into the lab. Marines crowded the landing, but Rias shoved his way to a panel on the wall. He waved his hand over a pale square. The door slid down from the top of the jamb.\n\nTikaya stared at smoke wafting from the hole in her sleeve and swallowed.\n\nA beeping started, soft but audible throughout the lab. It came from the walls, the ceiling, everywhere.\n\n\"Two more of those cubes coming from below,\" Bocrest said.\n\n\"If I can get close enough to read what's on the sides, maybe I can figure out how to stop them,\" Tikaya said.\n\n\"If you get that close, you'll be dead,\" Rias said.\n\n\"There're two more in the back.\" Agarik pointed. \"Shooting, burning, er, incinerating the dead creatures.\"\n\n\"Yes.\" Rias rummaged in his pack. \"They do that to everything. And everyone. Also, I don't know how to lock the doors. The one outside will be in soon.\"\n\nTikaya bent to examine runes lighting the wall by the door. She recognized one that had indicated \"up\" on the rocket. When she pressed the symbol it indented, but nothing happened. She found she could rotate it. A soft thunk came from within the wall. \"I think that may have\u2014\"\n\n\"We've got to split up, or we'll be surrounded,\" Bocrest said.\n\n\"Actually, I want them all in one spot.\" Rias had opened his rucksack and knelt, mixing a liquid and something else into a bottle. Caustic fumes stung Tikaya's eyes.\n\nA red beam from below splashed against the wall on the landing. It adjusted, lowering, and marines ducked out of the way.\n\n\"Forget that,\" Bocrest said. \"Karsus, get these men under cover, and shoot at anything that moves.\"\n\n\"Bocrest!\" Rias barked.\n\nThe squads were already running off, Bocrest included this time, leaving only Tikaya and Rias on the landing. Below, a pair of cubes, which had been floating languidly toward the stairs, split and increased speed. One chased after each group of men. Before one of the squads reached cover, a beam shot out, taking the last man in the back.\n\nHe screamed. Tikaya gripped the railing, unable to take her eyes from the scene.\n\nThe rear two men from the squad shot and rifle balls clanged off metal. At the least, the force should have propelled the cube backward, but it never moved. The beam continued, piercing the marine's body and coming out the other side as it incinerated flesh, muscle, and organs. Even when he dropped to the ground and curled into a ball, it stayed with him. It cauterized as it burned an ever-widening hole in his torso. The marine stopped moving, eyes glazed in death. The cube's beam kept breaking down the body, even burning blood away.\n\nTikaya, thunderstruck by the ghastly scene, almost did not notice Rias racing down the stairs with nothing but a jar of orange liquid in his hands. At first, the automaton ignored him, busy finishing its incineration of the dead man. Rias kept sprinting, one hand gripping the jar, one on the lid. The cube abandoned its task and rotated toward him.\n\n\"No!\" Tikaya grabbed her bow, though she did not know what good she could do if rifles had not damaged the device.\n\nRias flung some of the liquid on the cube, then ducked under it as the beam shot. It sizzled past, missing him. It struck the stair railing, but the beam did not affect the black metal. The viscous liquid on the cube smoked red. Pungent fumes gagged the air as it oozed down the sides.\n\nThat did not stop the automaton from rotating toward Rias, its ominous red hole glowing. Before its deadly side disappeared from sight, Tikaya fired, aiming for the orifice shooting those beams. Her shot flew true, and the shaft lodged inside. But the red glow flared and a beam incinerated the arrow.\n\nRias found cover behind a column.\n\nA hiss sounded behind Tikaya.\n\n\"Look out!\" Rias yelled.\n\nShe whirled. The door she had tried to lock opened, revealing two cubes on the threshold. Their holes glowed.\n\nTikaya leaped over the railing. The floor came quickly, and she landed with an ankle-jarring jolt. Two beams zipped over her head. Off-balance, she skittered into the shadows beneath the landing.\n\nGunfire echoed elsewhere in the lab. She sensed rather than heard the cubes floating down the stairs.\n\n\"Over here.\" Rias beckoned with an arm. \"Zigzag your path.\"\n\nWith a wary glance at the cubes coming down\u2014they were only a few steps from the bottom\u2014Tikaya raced across the open space toward Rias.\n\n\"Zag!\" he barked.\n\nShe angled left. A beam splashed the floor inches from her feet. After a few more steps, she veered right.\n\nSomething crashed behind her, but she did not slow to look. She skidded behind Rias's column, nearly jabbing him in the face with her bow.\n\nHe gripped her shoulder and started to speak, but an agonized scream echoed from the back corner.\n\n\"Curse Bocrest for not listening,\" Rias growled. He pointed at the cube he had doused with the goop. It lay on the ground, part of its exterior burned away to expose silvery innards. \"It's working. I made it in the vehicle garage in Wolfhump. I wasn't sure it'd be the same as\u2014\"\n\n\"Those two are coming,\" Tikaya said.\n\n\"Right, yes. We have to get some of this on them, too, but I don't have a chance unless they're distracted.\"\n\n\"You want me to do that?\"\n\n\"Not ideally,\" Rias said.\n\nThe cubes floated closer, no urgency to their movement, but an eerie inexorableness marked their flight.\n\n\"This way.\" Rias led Tikaya down the aisle to find cover behind the next column.\n\n\"I thought you wanted to challenge me,\" she said.\n\n\"You saw how lethal one can be, and we've got two to deal with. I don't want you to get hurt.\"\n\nMore likely killed, Tikaya thought. She smiled bleakly. \"If not me, who else?\"\n\n\"I'll do it,\" an emotionless unfamiliar voice said from behind them.\n\nA young man\u2014he could not have been more than seventeen or eighteen\u2014stood there, wearing fitted black clothing and soft black boots. Several sizes of daggers adorned his belt, and a set of throwing knives was strapped to his right arm. He carried nothing else.\n\n\"Go,\" Rias said, hefting his jar.\n\nIf the boy's appearance surprised him half as much as it did Tikaya, he did not show it. She lifted a hand, intending to protest sending someone so young on a suicide mission, but the youth had already jogged from concealment.\n\nTwo beams lanced toward his chest, but he anticipated the attack and dove, rolling beneath the cubes. They rotated to target him. This time he jumped to avoid the shots. Next, the mechanical assailants teamed up, showing a disquieting ability to work together. They tried to surround him, but the youth proved too quick. He darted away, keeping both cubes to one side of him.\n\n\"If I get killed,\" Rias said to Tikaya, \"get the jar and finish them. It's an acid, so don't let any of the liquid touch your skin.\"\n\nBefore she could say how little she thought of his get-killed option, he left. Tikaya nocked an arrow. The bow might not do damage, but perhaps it would help distract the cubes. Though the boy was doing a good job of that on his own. He dodged, darted, jumped, and rolled with the fluid ease of a well-trained natural athlete. Who was this ally who had shown up just in time to help? Even when a beam washed the floor inches from him, his face held no expression, though the intensity of his dark eyes promised nothing would break his focus.\n\nRias neared the closest cube, keeping its backside toward him. Tikaya fired at it. The arrow clanged off, and the cracked head clattered to the floor. Despite the distractions, the cube somehow sensed Rias' approach. It rotated toward him.\n\nHe flung some of the liquid and dodged just before the beam struck. Red smoke fumes plagued the air. The second cube remained focused on the youth who led it around columns and over lab stations. Rias zigzagged back to a column adjacent to Tikaya's with the tagged cube shooting after him. Smoke drifted from its surface, and the corrosive liquid burned through the casing.\n\n\"What is that stuff?\" Tikaya asked, shifting to keep the column between her and the cube as it approached.\n\n\"A variation on royal water,\" Rias said. \"The black metal is particularly susceptible to it. We were trapped in a room with all sorts of chemicals, and I tried several things last time. I couldn't read any of the labels, and I'm lucky I didn't kill myself. It took too long, though. A lot of men died before I figured it out.\"\n\nThe smoke thickened, inflicting the air with an acrid tang. It was nothing like the scent of burning wood or coal or anything else Tikaya had ever smelled. Before the cube reached them, it ground to a halt, then plummeted to the floor, innards exposed.\n\n\"Next.\" Rias headed toward the gunfire and shouts in the rear of the huge lab. \"We lost ten men to these things last time. We have to hurry.\"\n\n\"Shouldn't we get the one attacking the boy first?\" Tikaya asked.\n\n\"He's the last one who needs to be rescued.\"\n\nThey ran through the aisles toward the chaos. When they passed the spot where they had killed the creatures, there was no sign of the remains, not even a blood stain on the floor. The corpse of the marine was gone too.\n\nRias picked an aisle parallel to the gunfire and shouts of Bocrest's squad. He jumped, caught the edge of a counter, and pulled himself to the top of a lab station. He knelt, his jar poised to pour when the mechanical assailant came into range.\n\nTikaya thought to wait on the floor, but the youth came into their aisle from the other end. His cube sailed in a few seconds later. Tikaya tossed her bow up, then climbed to Rias's side, hoping to avoid the path of fire.\n\n\"Admiral,\" the youth said as he ran past.\n\nTikaya blinked, almost as shocked at the calmness of the boy's voice as the fact that he knew who Rias was. Rias leaned over, prepared to pour his concoction on the cube following the young man. It seemed to detect the trap, for it slowed several paces back. Its glowing orifice rotated up, toward Rias and Tikaya.\n\n\"Rust,\" he muttered and prepared to jump.\n\n\"Wait.\" Tikaya jabbed the tip of an arrow into his jar, nocked it, and fired. The dripping missile spun into the red hole. A flash later, a beam incinerated the arrow. The cube floated closer.\n\n\"Double rust,\" Tikaya said.\n\n\"It was a good idea,\" Rias said.\n\nThey crouched to jump down into the aisle behind, but the cube slowed, then halted. Smoke wafted from the beam hole. The cube sputtered and thunked to the ground.\n\n\"It was a good idea.\" Rias clapped Tikaya on the shoulder and gave her an appreciative smile that warmed her soul, despite the dire situation.\n\nEnemy of the islands, she reminded herself. She was not supposed to be pleased by his compliments anymore.\n\n\"Think you can hit that target again to help these men?\" Rias pointed to the marines scrambling in the other aisle. They were so busy dodging beams of a cube in their midst they had lost their usual cohesiveness. Every man was busy trying to stay alive. \"Make it quick, though,\" Rias added. \"The acid will eat away your arrowheads.\"\n\nTikaya waited until the orifice faced her before dipping into his jar. Her shot flew true and made short work of the remaining cube.\n\nThe relieved party met in the open area before the stairs. Tikaya picked up one of the mostly intact cubes so she could work on translating the writing.\n\nBocrest counted heads and scowled at the loss of two men and injuries of several others. He glowered at Rias. \"Why didn't you tell me you had something to battle them with?\"\n\n\"You didn't give me a chance,\" Rias said.\n\n\"We had to act quickly, and you were digging around in your gear. If you want to override my orders, you need to give me a reason for doing so. Fast. You don't have the right to make decisions and keep the reasons to your...\" Bocrest gaped as the youth stepped out of the shadows to join them.\n\nHe was smaller than many of the big marines, standing only an inch taller than Tikaya, but, after seeing his grace in evading the beams, she doubted he lost many fights.\n\nHis dark-eyed gaze pinned Bocrest. \"Admiral Starcrest is giving orders?\"\n\nHe had short blond hair, a color unusual for a Turgonian, but he did have the olive skin, and he sounded like a native speaker. From the dialect, Tikaya guessed he came from one of the satrapies around the capital. When her gaze fell on the throwing knives on his forearm, she realized he was the one who had killed the creature chasing her earlier. Where had he come from?\n\n\"I...uhm...\" Bocrest noticed his men watching him\u2014they seemed as confused by the young man's appearance as Tikaya\u2014and straightened, lifting his chin. \"Given his helpfulness thus far on the mission, and his familiarity with these tunnels, I deemed it wise to listen to him. I am aware of the emperor's wishes for him, and they will be complied with in the end.\"\n\n\"I see,\" the young man said, voice cool.\n\nBocrest shifted uncomfortably under that steady gaze. His men murmured to each other, surprised at their blustery captain's deference.\n\n\"What are you all looking at?\" Bocrest barked. \"Let's get the wounded patched up and set up a camp. And for the emperor's sake, someone figure out where in this blasted maze a man is supposed to piss and drop cannon balls.\"\n\nThe marines scurried off to do his bidding. The youth produced a small sealed envelope and handed it to Bocrest, who accepted it and walked away to read the message.\n\nTikaya edged closer to Rias. \"I'm perplexed. Who is this boy?\"\n\n\"That is Sicarius, the emperor's personal assassin.\"\n\nRias's voice was low, for her only, but the young man looked at them, as if aware of their discussion.\n\n\"Is he as young as he looks?\" Tikaya asked even as she wondered why he was there. Why had he not traveled with Bocrest from the beginning if he meant to help the captain accomplish his mission? Her eyes widened. Could he be the one responsible for the tortured men?\n\n\"I believe he was fifteen when I met him two years ago,\" Rias said. \"He smashed my face into the deck and held a knife to my throat.\"\n\nTikaya stared at Rias. \"Did he catch you by surprise?\"\n\n\"No. As I recall, I was trying to catch him by surprise.\"\n\n\"Why? What happened?\" She frowned, wondering why the emperor would send his assassin to harass his star fleet admiral. Something to do with Rias's reasons for ending up in exile? She tried to read his eyes.\n\nHe opened his mouth, but he shut it again and shook his head. \"No, I fear you'd suspect my motives if I told you the story.\"\n\n\"Suspect your...\" She scowled at him. Now what was he hiding? \"I suspect your motives in not telling me.\"\n\nHe closed his eyes. \"I'm sorry.\"\n\n\"Why are you so elusive about these things?\" Her throat tightened. Now that she knew who he was and that she could have no future with him, his choice to keep things from her should not hurt, but it did. \"Afraid to share imperial secrets with the enemy?\"\n\nShe clutched the cube to her chest and stalked toward the lab stations, intending to find some nook where she could be alone to study the language.\n\n\"Tikaya...\" Rias said.\n\n\"Stuff an apple core up it, Admiral Starcrest.\"\n\nShe caught the boy assassin watching, and almost snapped at him too, but the dark impassive gaze stole the heat from her ire and left her chilled. She stalked by without comment.\n\nSeveral marines sniggered behind her back. Cheeks flushed, Tikaya slipped into the aisles, glad for the concealment.\n\n# CHAPTER 16\n\nFew runes adorned the cube, and it did not take Tikaya long to translate them. Automatic cleaning machine. She could have laughed if not for the unsettling realization that the people who created this place had considered human beings something to be incinerated to keep the labs tidy.\n\nWith that mystery solved, she itched to work on the next. Weariness plagued her body, and probably would for days in the aftermath of the poisoning, but her mind churned, so she could not think of sleep yet. There was so much to study.\n\nShe wished she had Lancecrest's journal, but she would have to return to her gear to grab it, and she did not want to face Rias or the snickering marines. Instead, she explored the back half of the lab. Most of the finds were innocuous\u2014alchemical liquids and powders, equipment and containers\u2014but others were as disturbing as the bones that had scattered when the beast fell. The human organs sealed in jars and slides with blood samples made her wonder if the race who had created this place had come for the distinct purpose of experimenting on people. But, if so, to what end?\n\nShe probably should have been horrified by her discoveries, but the labels on the identifiable substances helped her resolve new nouns, and that kept her too busy for squeamishness. A few days wandering this place and she would have an impressive dictionary. If the marines gave her time.\n\nAgarik rounded the corner and approached, his rifle crooked in his arms. He quirked an eyebrow at the rows of open cabinets in her wake.\n\n\"Exploring?\"\n\n\"Yes, this place is perfect. If I had a few weeks here, I bet I could decipher the whole language. Or the science aspects, at least. Of course, this entire language seems to revolve around science and mathematics. I keep wondering who these people were, what happened to them. Where could such an advanced civilization\u2014\"\n\nHe was frowning, so she stopped.\n\n\"Problem?\" Tikaya asked.\n\n\"No. Yes. I don't know. We've lost so many men out here. Your enthusiasm for such a deadly place is... Well, I can't share it.\"\n\n\"But don't you see? Everything here is labeled. If I can learn how to read it all, this place won't be deadly. We're bumbling into things. Those cubes, they're the maids. Not some malicious security system, a cleaning device to take care of messes in the labs.\"\n\n\"And the poison rockets,\" Agarik said. \"Are they also not malicious if only you know the words? And the gas that twisted our minds in Wolfhump? Was that not malicious?\"\n\nThe sobering words squelched her enthusiasm. He was right. It was very likely this place had been created, at least in part, to build weapons. Weapons far deadlier, and ghastlier, than anything humanity currently knew.\n\nAgarik sighed at the expression on her face. \"Forgive me, I don't mean to judge. Besides you're not the only one fascinated with the place.\"\n\n\"Oh, what's Rias doing?\" Tikaya asked, certain of her guess.\n\n\"He's taking apart one of the boxes.\"\n\n\"Figures.\"\n\n\"Uhm, about him...\" Agarik watched her, and she had a hunch he had brought up Rias to gauge her reaction.\n\n\"Did he send you over to talk to me?\"\n\nAgarik's head shake did not surprise her. Tikaya had a hard time imagining Rias sending a minion\u2014or admirer\u2014off to solve problems for him. No, he would likely suffer in silence.\n\n\"No, ma'am. He, ah...\" Agarik set the butt of his rifle on the floor and polished a smudge on the barrel. \"He forbade me from bothering you.\"\n\n\"Oh? And you're going to disobey your boyhood hero?\"\n\n\"If there's a chance of fixing things, yes.\" Agarik blew out a long breath. \"I got the story about the assassin, if you want to hear it. I think it might influence your feelings for Rias.\"\n\nThough curious about the story, she hesitated to ask for it. Now that she knew Rias was safe, she needed to put aside her 'feelings' for him, figure out how to thwart the weapons-acquisition mission, find a way home, and warn her family they were in danger. She could not bring herself to send Agarik away, though. \"Is that actually what he calls himself?\"\n\n\"What?\"\n\n\"Rias. I thought it might be something he made up because he didn't want to tell me his real name.\"\n\n\"He told me to use it. He goes by Federias and said his friends have always shortened it. Apparently, he's never liked his first name. Got teased about it as a boy and told it was girly.\" Agarik grinned, probably delighted to have been trusted with this secret information.\n\n\"Does 'the story' explain his exile?\" she asked. \"Why he was stripped of everything and declared dead?\"\n\n\"Yes.\"\n\nShe nodded for him to continue.\n\n\"He says he did recommend the Kyatt Islands as the place for a strategic outpost, on account of its location right in the center of the sea between Turgonia and Nuria, but he wasn't planning on bloodshed. I don't know if you remember, but a couple of imperial ironclads showed up in your harbor a few years ago. He went in with a diplomat to talk to your president.\"\n\nDespite her resolution to put aside feelings for him, a flutter went through her stomach. To think that Rias had been so close years before. She had been working at the Polytechnic then, and she remembered the hubbub around those ships arriving. If she had looked out the window at the right time, might she have seen Rias striding along the docks, straight and proud in a dress uniform, flanked by dozens of men who respected him?\n\n\"They offered your people entry into the empire as an imperial territory and protection from the Nurians in exchange. Your president said no. They negotiated, tried to get the right to build a naval base on one of your islands. Your president was adamant that your people would remain neutral, and he denied it all. The emperor was not pleased. He ordered Kyatt be conquered, and you know what happened after that.\"\n\n\"Yes.\" All too well.\n\n\"Rias was busy managing the entire Northern Eerathu Theater, and the skirmishes with your people were just a tack on his busy map, but he says he was impressed with your president's backbone and how hard your people fought, especially considering the odds were all against them. The emperor was more annoyed than impressed. Particularly so after you started decoding messages and sending them to the Nurians.\"\n\n\"I'll bet,\" Tikaya murmured.\n\n\"The emperor sent this Sicarius out to Rias's flagship with orders\u2014and don't irk that fellow, by the way; he's apparently been groomed from birth to be the throne's assassin. All Rias was supposed to do was take his vessel into port and let Sicarius kill your president and his advisors.\"\n\nTikaya stood statue still. She did not remember any personal attack on the president.\n\n\"Rias was angry that the emperor even had an assassin. We've always been an honorable warrior people, and sneak attacks are considered cowardly.\"\n\n\"What'd he do?\"\n\n\"He refused to take the assassin to your island and, when he learned Sicarius was trying to make other arrangements to get there, Rias tried to incapacitate him.\"\n\nThat explained his earlier comment about attacking Sicarius.\n\n\"It didn't work,\" Agarik said. \"Fortunately, Sicarius was loyal enough to the emperor not to take it into his own hands to kill a fleet admiral. Rias had time to send warning to your president and describe the assassin so your people could watch for him\u2014that's a part of the story you could verify when you get home, I imagine.\"\n\nA spark of hope kindled. If the president knew Rias had tried to help him, maybe it would make a difference someday if...\n\nTikaya shook her head. Was she truly still thinking of bringing him home?\n\n\"Sicarius took word back to the capital,\" Agarik continued, \"and the emperor about shi\u2014, er, he was livid at Rias's disobedience. He stripped him of his name, his rank, his ancestral lands, everything, and ordered him taken to Krychek Island. The story passed around is that Rias was assassinated by Nurians.\"\n\n\"Why the story?\" Tikaya wondered. \"Why tell everyone he was dead?\"\n\n\"He's a hero to our people and well-liked. He had scads of loyal men who would have made rescue attempts if they knew he was alive.\"\n\n\"Then why not actually kill him?\" Could the emperor have known he would need Rias again?\n\n\"My guess,\" Agarik said, \"is the emperor wanted his best military strategist somewhere he could get to him again if needed. Though that's quite a gamble.\"\n\nTikaya raised her eyebrows.\n\n\"Krychek Island isn't a place you put someone for safe keeping,\" Agarik said, tone bleak. \"I remember newspaper stories over the years about some of the men who got sent: cannibals, serial murderers who defiled their victims, molesters who tortured children. Crazy people who aren't right in the head.\" Agarik ran a thumb along the muzzle of his rifle. \"I reckon Krychek Island is like a Harvest Moon War.\"\n\nTikaya had heard the Turgonian expression a couple times, but had not stopped to think about the meaning. \"As in the war goes so late in the season that even if you win, there's no one at home to bring in the crops, so your family starves over the winter?\"\n\n\"That's the gist of it. When Rias first came on board, he didn't talk to anyone. He was just the unpredictable monster locked away in that dark cell, and it seemed to suit him. You'd catch him in the light, and you'd see this crazy haunted look in his eyes. The captain was scared of him, and all those guards following him around in the beginning weren't for show. That's why I was so startled when he spoke out on your behalf. He hadn't said a word to anyone up until then. But I guess having a woman present made him want to be more civilized. To pull himself together, you know?\"\n\nTikaya closed her eyes. Rias had never spoken of his time on the island. What demons might it have left cavorting in his head?\n\n\"I would hate to see him like that again,\" Agarik said, eyes sad. \"Are you irrevocably mad at him? It's hard to tell with you. Today you worked together as if nothing stood between you, and you saved us again. You're a good team.\"\n\nThough there was nothing accusing in Agarik's words, they made her gut twist with guilt. Everyone thought she was mad at Rias, him too most likely.\n\n\"I'm annoyed that he blindsided me,\" Tikaya said, \"but mostly I'm frustrated with the cosmos. I can forgive him for being born on the other side and for being an officer\u2014the officer\u2014in the enemy army, but I can't see having a life with Fleet Admiral Starcrest. It would be a huge betrayal to those I love\u2014I loved.\"\n\n\"It wouldn't be any sort of betrayal to turn your back while they return him to Krychek?\" Agarik said.\n\n\"He wouldn't let them do that.\"\n\n\"He won't have a choice. That assassin outfought him before, and...I'm not sure he'll care enough to worry about escaping if he doesn't have anything to live for.\"\n\nTikaya stared at the floor. His words shamed her. She had been thinking only of herself and how Rias might fit into her life.\n\n\"I better get back to my rounds,\" Agarik mumbled.\n\n\"You're a good man, Agarik. I never expected you to play matchmaker for us.\"\n\nHis lips curled wryly. \"Me either. But I reckon if you care for someone and you can't have their love, you can either be a spiteful bastard about it or you can try your damnedest to make sure they're going to find some happiness in the world.\"\n\n* * * * *\n\nTikaya yawned, a great face-tilting-up kind of yawn that made her crack the back of her head on a cabinet door. That, and her bleary eyes, forced her to concede that she needed sleep. She, fearing the marines would move on too quickly, had spent several hours learning as much as she could from the lab, scrawling notes at top speed. Agarik had been nice enough to bring her food and her notebook so she could avoid the camp for a while longer, but he claimed not to have seen the journal. She thought that odd since she had tucked it into the same place in her rucksack as the notebook, but she had been in a rush to grab a bow at the time, so perhaps it slid to the bottom.\n\nTikaya picked her way past snoozing bodies toward her gear. Aside from Agarik and a man perched at the top of the stairs, the rest of the marines slept. She did not see the assassin.\n\nShe spotted Rias on the edge of the camp, sprawled on his parka amongst a pile of disassembled machine innards. She grinned. Just like a child fallen asleep on the floor amongst his toys. But, as she stepped closer, she noticed his eyes moved beneath his lids, and some dream turned his lips to a grimace. Agarik's words about Krychek Island came to mind. She sat on the rucksack next to him and stroked his hair.\n\nRias's eyes opened and, for a moment, confusion creased his brow.\n\n\"Where were you?\" Tikaya murmured.\n\nHe rubbed his face. \"Nowhere pleasant.\"\n\n\"Should I feel bad about waking you, or is this an improvement?\"\n\nThough he did not lift his head, his eyes roved, taking in the bleak black ceilings, black walls, and snoring marines. He smiled though. \"Nobody was fondling my head in the dream.\"\n\nCareful to avoid his bruised eye, Tikaya brushed a lock of hair back from his forehead. \"I translated the symbols on the cubes.\" She recited the lines from memory, and he made the same conclusion as she had.\n\n\"Cleaning devices? That's amazing.\"\n\n\"More disturbing, I'd say, since humans are something to be incinerated along with the trash.\"\n\n\"Actually, I was talking about you. You just got your first real clue about the language in Wolfhump, what, three days ago? And now you're reading it.\" Rias gripped her hand and gazed up at her, dark eyes full of pride. \"When you get home, you'll be the main story in the next volume of Archaeology Monthly.\"\n\nThe lump in her throat made her laugh more of a hiccup. Rias's words reminded her, not for the first time, how different he was from Parkonis, who had always envied her language gift. His congratulations had been grudging when she had been selected by the president to work on decryptions during the war. She had not even wanted the dubious honor, made even more dubious by the predicament it had landed her in, but he had envied her the recognition. Maybe there was good reason to love someone who had so many accolades of his own that he could never feel jealous of a bright philologist. Of course, she reminded herself, Rias had nothing now except a trip back to a savage island of deranged criminals. And still he was proud of her.\n\n\"Something wrong?\" he asked gently.\n\n\"No.\" She wiped her eyes. \"I'm sorry I snapped at you. I've\u2014\"\n\nHis eyes widened. \"Tikaya. You don't need to\u2014\"\n\nShe pressed a finger to his lips. \"Let me finish, please?\" When he did not nod, she left her finger there. \"Before I met you, before Parkonis died, before the war, I knew what I wanted from life. I dreamed of sharing a little house near the beach with someone, close enough to town to walk to the market and the Polytechnic. I'd teach in the mornings and work in the labs in the afternoon, studying relics and data our field people brought in. And I'd have two children, a boy and a girl, of course. Blond hair, blue eyes, freckles.\" She smiled wryly, acknowledging the unlikeliness that life would ever turn out exactly as she dreamed. \"When Parkonis came along, I knew he was the one who could give me that dream.\" Her smile faded, and her voice dropped to a whisper. \"Then the war came, and Parkonis died, and my dream died too. It's hard for me to admit it, because it's so selfish, but I think I've spent the last year mourning the loss of my dream as much as I've been mourning his death.\"\n\nShe felt silly holding her finger to Rias's lips. He was watching and listening. She turned her hand over and brushed her knuckles along his jaw.\n\n\"And then I met you. Of course I knew you were a Turgonian as soon as I heard you speak, and you admitted to being in the war, but I imagined you were just some ship's engineer following orders, some simple soldier who none of my people could blame their troubles on, and I had all these ideas about how you could come home with me, and...maybe my dream could live again.\" She shook her head. \"It's not your fault you don't fit into the fantasy I made up. It's not as if there weren't clues to the contrary. I was just set on my theory, because I thought I could work you into my life somehow that way. Like I said, it's selfish. And I'm sorry.\"\n\nRias sat up abruptly, and her hand fell away. \"Tikaya\u2014\"\n\n\"Emperor's bunions,\" Bocrest growled. He lay a couple of men away with his arm flung over his eyes. \"If you two are going to wake people up with your relationship crap, you could at least be fucking, so we'd have something to watch.\"\n\nRias winced at the crude words. Tikaya's cheeks warmed, but she kept her tone light when she said, \"Is he really one of your emperor's most trusted officers?\"\n\n\"Only because of his parents.\"\n\nFeral noises emanated from Bocrest's throat.\n\nRias stood up, took Tikaya's hand, and led her away from the sleeping men. Her heart sped up, and she wondered if he had Bocrest's suggestion in mind. He went to a corner, out of earshot if they talked quietly, but within sight of the camp, so she supposed not. Too bad.\n\nThey sat, backs against the wall, shoulders touching.\n\n\"Tikaya,\" Rias said, staring at the floor. \"I appreciate your words, but hearing you apologize to me is like getting a dagger in the chest. I'm the one who... I need to say...\" He snorted, or maybe it was a laugh. \"My men used to call me courageous because I'd lead the way into battle and take risks others thought ludicrous. They didn't know that it was just arrogance. I thought I was too good to get myself killed. I knew I wasn't immortal, but I won often enough that I always thought I'd come out on top. So, it wasn't really courage.\" He continued to look down, avoiding her eyes. \"Courage is the ability to do the right thing when you're terrified of the consequences. I've only recently realized that I'm a coward.\"\n\nFor a moment, the only noise was the snoring of the marines. She could hear herself breathing.\n\n\"You asked for my name, more than once, and I could have told you. You deserved the truth. If I'd wanted you to know, I wouldn't have let Bocrest's threats sway me, but I knew as soon as I told you, that your willingness to spend time with me would be over. I wouldn't be able to dream that somehow, someway...\n\nHe fiddled with his hands. \"Tikaya, I'm not a young man. I've been in love before, infatuated with beauty or taken by a sympathetic shoulder, but no woman has ever asked me to explain my papers on mathematics.\" That familiar half smile tugged at his lips. \"And, dear ancestors, the fact that you were the cryptomancer too?\" He chuckled. \"I fell for you that first day we shot together on the exercise deck.\"\n\nShe watched his face as he spoke and tried to memorize every word. Her heart soared at his naked admission.\n\n\"But I knew you could never love Fleet Admiral Starcrest, the man who\u2014Ottotark is correct\u2014pointed out to the emperor the strategic value in having a base on your islands, and helped try to make that advice a reality. Even if you could somehow forgive that person for hurting your people, I'm sure they\u2014your family\u2014would never understand. I wouldn't want to be the cause of your ostracism. I kept trying to convince myself to keep my distance, to figure out a way to make sure you walked away at the end of this, and to just accept my fate. It didn't work. I was afraid to tell you, afraid to lose you, and so I made the wrong choice.\" He finally looked up, forced himself to meet her eyes. \"I've been a coward, and, in being so, I've hurt you. I'm sorry.\"\n\nEmotion welled in Tikaya's throat. She had fallen in love with him every bit as much as he had with her. The thought of going home by herself, never to see him again, brought tears to her eyes.\n\nShe wanted to hug him, to kiss him, but she caught the marine on guard watching them. She settled for leaning her head against Rias's shoulder. \"I forgive you.\"\n\nHe rested his head on hers, and neither suggested returning to camp to sleep. She thought about bringing up her suspicions about Bocrest's mission. If Rias had risked his career because he thought assassinating her president dishonorable, surely he would not knowingly help the emperor obtain weapons that could wipe out millions of innocent people. She trusted him more after his confession, but she still hesitated. His first questions to her came to mind, the way he had asked if her president was a good person, if the people liked him. Now she realized he must have had regrets during his time on Krychek, that those questions had been a damaged man asking if it had been worth it. If he had those moments to live over, would he make the same choice? Could she trust him now to make the right choice over one that might gain him the emperor's favor once again?\n\nRias lifted his head. \"Is that the journal you found?\"\n\nTikaya looked toward camp. It was the journal. And the assassin was reading it. She could have smacked herself on the forehead for not hiding it. If Sicarius was the one who tortured Lancecrest, he was also the one who had been looking for the journal.\n\nShe jumped to her feet and hustled toward the camp while trying not to look like she hustled toward camp. If she seemed desperate to keep it to herself, it would arouse suspicions, but she had to get it away from him.\n\nSicarius flipped through the pages. The way his dark eyes skimmed the columns from top to bottom and left to right made her believe he could read Kyattese. He lifted his head as she drew near, and Tikaya's determined step faltered when that cool gaze landed on her.\n\n\"Uhm, that's mine. I mean, I'm the one who found it, and I've been translating the runes drawn in there. The owner's guesses are largely incorrect, so you wouldn't want to...\" She stopped talking since he had already turned his attention back to the journal.\n\nTo make sure her concerns were founded, she switched languages and asked, \"Can you read Kyattese?\"\n\nHis eyes flicked up briefly, but she received no answer. She took another step, toying with the idea of seeing if he would let her take it out of his hands. An arm slipped around her waist from behind.\n\n\"Yes,\" Rias said near her ear. \"He can. Among other languages.\" He put a hand on her arm and guided her to her rucksack. \"Is there something in it you don't wish him to find?\"\n\nHer uncertainties about Rias's regrets and loyalties made her hesitate, but she needed an ally, and he was still the most likely one. She did not see how she could fool the Turgonians and eliminate the threat to her people\u2014to the world\u2014by herself. \"Instructions on how to launch the rockets.\"\n\nHis grip tightened on her arm. \"Why didn't you tell me what was in there?\"\n\nTikaya watched his face. \"I didn't know whose side you were on.\"\n\n\"That's...\" Rias closed his eyes, \"understandable. Get some rest. We'll look for an opportunity to get it back tomorrow.\"\n\nTikaya found her bedroll, but her earlier weariness had disappeared. For a long time, she lay on her side, watching the assassin read.\n\n# CHAPTER 17\n\nWater trickled somewhere in the distance. After the monotonous black walls and tomb-like silence of the tunnels thus far, Tikaya would appreciate some dripping stalactites, striated walls, bumpy columns\u2014proper cave appurtenances. As of yet, though, no end of the alien passages lay in sight.\n\nThe marines marched ever deeper with Bocrest and Rias leading, and Tikaya walking behind them. Sicarius came and went, sometimes padding soundlessly alongside the captain, other times exploring on his own. That morning, Rias had given a briefing highlighting the dangers of the tunnels. Admonitions had included \"no touching things\" and \"don't wander off on your own.\" The assassin apparently did not believe rules applied to him, and she could not even wish him to get lost and fall off a cliff, not as long as he had her journal.\n\n\"You've been here longer than us, right?\" Bocrest asked when Sicarius returned from one of his roaming stints. \"Do you know where the archaeologists are?\"\n\n\"No.\"\n\n\"Do you know where the weapons are?\"\n\n\"No.\"\n\n\"Do you know what other dangers we'll face?\"\n\n\"No.\"\n\nThe assassin's cool monotone never changed, though Bocrest's pitch grew more agitated as he failed to hear the answers he wanted. He was probably used to flogging kids this age for not cleaning the head sufficiently.\n\n\"What do you know?\" Bocrest asked.\n\nTikaya, walking behind them, had a good look at the frosty gaze Sicarius slid the captain. She glanced at Rias who merely raised his eyebrows. He might pull her away and keep her from doing something stupid to annoy the young assassin, but he did not appear inclined to watch out for Bocrest.\n\n\"I crossed the mountains on foot and arrived only a day before you,\" Sicarius said.\n\nHe withdrew the purloined journal, and Tikaya's fingers twitched. She strained to see over his shoulder as he opened it to a dog-eared page. The instructions. He ripped them out. He turned to another page in the back of the journal and tore the bottom third off.\n\n\"What are you doing?\" Tikaya blurted.\n\nSicarius ignored her, showing the scraps to Bocrest. \"Operation instructions for the rockets and the sequence of runes Lancecrest pushed to get into the weapons chamber.\"\n\nTikaya cringed. She should have hidden the journal. Assuming no one else could read her language had been foolish.\n\n\"Lancecrest claimed the sequence only worked once,\" Sicarius said, \"and his team has been stymied since.\"\n\n\"Give the book back to Tikaya now that you've got what you wanted,\" Rias told Sicarius.\n\nBocrest glanced at Rias, startled eyes wide. Even the captain had not dared give the emperor's assassin a direct order. But Sicarius handed the journal back to Tikaya without missing a step.\n\n\"Thank you,\" she said, though it seemed obsequious to thank him for returning something he had stolen out of her rucksack. He was going to be a problem\u2014as if she did not have enough problems already. She needed more allies out here, not more enemies, and her only option was the team that waited within. \"Why did Lancecrest fire the rocket on your fort?\" she asked Sicarius.\n\nHe did not respond or even glance her direction.\n\n\"I'm just wondering why one of your citizens would turn on your people like that. Is living in the empire that bad? Are people disaffected and eager to fight back against the oppressive rule of your emperor?\" She hoped to goad the assassin into speaking, but it was Bocrest who responded.\n\n\"There's nothing wrong with living in the empire,\" he snapped. \"If that Lancecrest brat was motivated by anything, it'd be money.\"\n\n\"I didn't know slaying marines in remote outposts could be profitable,\" Tikaya said.\n\nRias's lip twitched. He was staying silent, but she decided it did not represent disinterest or annoyance at her sleights toward the empire. In general, he was not as garrulous around the marines as he was alone with her, and she imagined all but the closest of his men had known him as a quiet, enigmatic leader.\n\n\"I only bring it up,\" Tikaya said, \"because it might be useful to know why Lancecrest was attacking your people and whether those left inside are out to get you, too, or if he was the leader and now they're rootless.\"\n\n\"I'd like answers to those questions too,\" Rias said when Sicarius did not respond. \"What happened on that ledge?\"\n\nSicarius glanced at the squad of marines following them, men who had grown silent as soon as the conversation started. Rias nodded to Bocrest, and the captain called a halt. He, Rias, and Sicarius walked ahead to speak privately. Tikaya followed. She'd asked the questions, and she intended to get the answers.\n\nThe assassin watched her walk up, his gaze cold and unwavering. He didn't want her there. She folded her arms and leaned on the wall. Too bad. Rias's eyes crinkled.\n\n\"I was too late to stop Atner Lancecrest,\" Sicarius told Rias and Bocrest. No remorse or angst colored his tone. He spoke it like a simple fact. \"But I learned much from questioning him. He originally heard of the tunnels from Colonel Lancecrest, who disliked his assignment and wanted to retire. He told Atner about the possibility of ancient valuables and asked for a split of whatever profits were made.\n\n\"So they intended to be relic raiders from the start,\" Tikaya said.\n\n\"Atner Lancecrest started assembling a multinational expedition of archaeologists and linguists a year ago, and they've been inside for several months. They came for relics. They found the weapons. Atner revised his plan. He decided to figure out how to get the weapons so he could hand them to the emperor and gain favor for his family. Some of his team, which included a handful of Nurians, were against that. A pair of them slipped away with a box. They warned their government, and\u2014\"\n\n\"The box that was delivered to the capital?\" Bocrest asked. \"The one that killed hundreds?\"\n\n\"Yes,\" Sicarius said. \"The Nurians delivered it, and, after the deaths, they sent a message telling the emperor to seal the tunnels forever or more killer artifacts would be delivered.\"\n\n\"Your emperor doesn't seem the sort to heed threats,\" Tikaya said.\n\n\"No.\"\n\n\"Nor,\" Rias said, \"does he suffer fort commanders who share top secret orders with little brothers. Or little brothers whose actions result in the deaths of hundreds.\"\n\n\"But he didn't know about the Lancecrest involvement when he sent you,\" Tikaya said to Bocrest, \"did he?\"\n\n\"No,\" Bocrest said. \"But Lancecrest\u2014both of them\u2014would have known they'd be in an ore cart full of shit as soon as their roles came out.\"\n\nRias nodded. \"It would have been more than their deaths. For a disgrace like that, the emperor could take away the entire family's warrior caste status and wipe their ancestors' deeds from the history books.\"\n\nTikaya raised her eyebrows at Rias, wondering if his act of disobedience had created a similar backlash for his family. He seemed to guess her question, for he hesitated, then shook his head. She took that to mean not as drastic a result, perhaps, but some backlash, yes.\n\n\"I don't get it,\" Bocrest said. \"The family's wrecked, but why make things worse by killing everyone in Wolfhump and Fort Deadend?\"\n\n\"To delay your party?\" Tikaya guessed. \"If Lancecrest knew he was dead if he stayed in the empire, maybe he wanted to get the weapons out so he could sell them to the highest bidder. Maybe his family would forgive him if they could all live the life of luxury in some remote paradise.\"\n\n\"But he got himself killed.\" Bocrest nodded to Sicarius. \"So, now all we're dealing with is a confused bunch of science twits with no leader.\" He appeared pleased at the prospect.\n\n\"And possibly Colonel Lancecrest and an indeterminate number of his men,\" Rias said.\n\nTikaya nodded, thinking of the half-eaten marine they had found in the lab with the creatures.\n\n\"What?\" Bocrest asked. \"He's dead. I saw his body.\"\n\n\"Are you sure it was him?\" Rias asked. \"Or was the skin melted by the gas?\"\n\nBocrest opened his mouth, shut it, then spat. \"You're right. I saw a body in his office and a jacket with his name on it on the chair, but it could have been anybody. Bloody ancestors, he'll be a pain to deal with if he's in here. Pissed his little brother slagged things up so badly and left him to endure the aftermath.\"\n\nTikaya listened bleakly. She had been hoping for fellow archaeologists to ally with; instead she might have another cursed Turgonian military commander waiting. She looked to Rias, hoping for some comfort there, but his face was inscrutable. She still had no idea what he thought about his emperor's desire to obtain these weapons.\n\n\"But his team is stuck, right?\" Bocrest said. \"If they could get to the weapons, they'd have taken them and disappeared by now.\"\n\n\"Correct,\" Sicarius said. \"They lack what we have.\" His gaze came to rest on Tikaya again.\n\nHer bleakness increased. When this had started, she had worried her skills would not be enough to keep her family safe. Now she worried her skills would be enough.\n\n\"Valuable intelligence,\" Rias told Sicarius. \"Good work.\"\n\nTikaya jerked with surprise. Was he actually complimenting an assassin on the bounty his torture session had yielded?\n\n\"Yes,\" Sicarius said, apparently unaffected by the praise.\n\n* * * * *\n\nThe tunnel opened into a cavern with a ceiling that disappeared into darkness. A chasm over a hundred feet wide yawned across the center, cutting through walls as well as the floor. A multistory building perched near the edge on their side, and eagerness quickened Tikaya's step\u2014finally, a chance to see something more than a lab. A plant for distributing water, she guessed. Pipes ran vertically and horizontally from the structure, and a smokestack rose as far as the eye could see. A reservoir adjacent to the building held driftwood-littered water, which trickled over the edge on one side, flowing into the chasm.\n\nTikaya peered over the edge. Darkness and distance cloaked the bottom\u2014if there was one. The black floor ended at the lip and started again on the other side. The tidy cobweb-free tunnels made Tikaya forget how much time had passed since this place had been created, but this chasm, which appeared to have formed after the complex was abandoned, reminded her that thousands of years, maybe tens of thousands of years, stood between then and now.\n\n\"Looks like we've caught up with the other team.\" Rias pointed at a tunnel entrance on the far side. Tikaya froze. Two men stood in it, and one had shaggy red-blond hair and a scruffy beard. She could not make out features at the distance, but they reminded her of Parkonis and sent a painful jab through her mind. Though her islands did not have the only blonds in the world, that hair coloring combined with the likelihood this was an archaeologist made her suspect this was one of her people. The second figure, dark-haired and dark-skinned, wore black and carried a musket. He could have been one of Bocrest's men. The pair stepped back into the darkness when they noticed the marines watching them.\n\n\"How'd they get over there?\" Bocrest asked. \"And how do we follow?\"\n\n\"Assuming they have a practitioner studied in telekinetics, they could have floated across,\" Tikaya said.\n\nBocrest's expression turned sour. \"Starcrest, you know any other tunnels that lead over to that side?\"\n\n\"No. I don't know what's over there. We were desperate to escape by the time we got here. We climbed those pipes and got out through a vent mountainside.\"\n\nBocrest growled and gazed about. Two other tunnels left the cavern on their side.\n\n\"Karsus,\" the captain said, \"take your squad through that one and see if there's a way across the gulf. Everyone else with me. We're checking this one.\"\n\n\"I'd prefer to stay here and study the journal,\" Tikaya said. \"Not to mention there's probably much I could learn in that building.\" And maybe, if she was alone, those archaeologists would come visit her and she could find out more.\n\n\"You're not staying alone,\" Bocrest snapped.\n\n\"I can stay too,\" Rias said.\n\n\"Oh, yes, I'm going to leave you two alone to conspire.\"\n\n\"Bet they want to do more than conspire.\" Someone snickered.\n\nBocrest silenced the commenter with a glare.\n\n\"It's possible there's something in the pumping house that could get us across,\" Rias said.\n\nBocrest's gaze landed on the assassin. \"Will you keep an eye on them?\"\n\n\"Yes,\" Sicarius said.\n\nTikaya grimaced. A babysitter who was young enough to be Rias's son. Lovely.\n\nBefore the marines reached the tunnels, Rias was already checking out the reservoir. An underground stream fed the pool, and the current had pushed logs and branches to the nearest side. He gazed thoughtfully at the wood.\n\nThough eager to explore the building, Tikaya dropped her rucksack and joined him at the edge. She could not remember her last bath, but dipping a finger in the icy water stole her fantasies of immersing herself. Maybe she could heat some up for washing later.\n\n\"Getting an idea?\" she asked as Rias pondered the driftwood.\n\n\"Perhaps.\"\n\n\"You don't think the marines will find another way around?\"\n\n\"If that rift is a result of a fault line, it could run a long way.\" Rias tapped a finger in the air toward the building. \"The last time I was here, I found a fantastic cutting tool in there. It burned through stone, wood, and metal like a knife slicing apple custard. If those archaeologists didn't find it...\" He dragged one of the logs out of the water and nodded to himself.\n\nTikaya waited for him to explain further, but the assassin appeared at Rias's shoulder. Tikaya jumped. She had not seen or heard the youth's approach.\n\n\"Ah, good,\" Rias said, less discombobulated. \"I'll need some more muscle.\"\n\nSicarius had to be curious, but his expression never changed.\n\n\"What do you think, young man?\" Rias asked. \"Ever want to fly?\"\n\nSicarius gave the faintest hint of an eyebrow twitch.\n\n\"Let's get this wood out of the water,\" Rias said.\n\n\"Can I help?\" Tikaya wondered what he planned.\n\n\"How are your carpentry skills?\n\n\"Er. I helped my father build a birdhouse once.\"\n\n\"An impressive project.\" Rias smiled and pulled another log out. \"But don't you want to explore the pumping house and look for language clues?\"\n\n\"Yes.\" Though her curiosity would have to wait for satisfaction, she would rather translate runes than hammer nails anyway.\n\n\"Be careful in there. Touching things is how my team got in trouble. Multiple times.\"\n\n\"I won't touch anything,\" she said. \"Unless I can read the label and know what it is.\"\n\nWith journal in hand, Tikaya headed to the structure. Though dwarfed by the cavern, it rose more than fifty feet and sported three rows of windows along each side. She paused inside the threshold, patting down pockets until she located a pencil. Before she headed deeper, Sicarius spoke to Rias.\n\n\"I bring you a message from the emperor.\"\n\nHer ears perked.\n\n\"Oh?\" Rias said.\n\n\"He believes you've been sufficiently punished for your transgressions and is willing to return everything to you\u2014your name, your rank, your land\u2014if you cooperate with Bocrest and myself and we're able to accomplish this mission.\"\n\nTikaya pressed a hand against the wall. She barely saw the vast room she had stepped into as she waited for Rias's answer. When it came, it was so soft she almost missed it.\n\n\"My ship? My command?\"\n\n\"Yes,\" Sicarius said. \"You can return with Bocrest, in command of the Emperor's Fist until you can be transferred to the Raptor and resume your full duties.\"\n\nSay no, Tikaya urged. Tell him and your sprite-licked emperor to fall on their swords.\n\n\"What is the mission exactly?\" Rias asked.\n\nTikaya clenched a fist around her pencil. What was he doing? He couldn't possibly be considering this offer. He had to know it was only coming because the war had gone badly after he disappeared. Disobeyed orders or not, the emperor must have realized he overreacted and come to regret ousting his star admiral.\n\n\"Kill the terrorists mucking around in here,\" Sicarius said, \"obtain the weapons for our use, and seal the tunnels.\"\n\nThere. All her suspicions confirmed. She wished she had been wrong.\n\n\"And what of Tikaya?\" Rias asked.\n\n\"She's only here to help with the translations.\"\n\n\"Bocrest has orders to kill her.\"\n\nTikaya nodded to herself, thankful Rias cared enough to be concerned. He might be tempted by the promise of getting his command\u2014his life\u2014back, but she did not believe he would throw her to the wolves on the way. She might even be the sticking point in this insidious proposition.\n\n\"Bocrest's orders were to ensure her cooperation by whatever means necessary,\" Sicarius said. \"If we complete the mission, you'll outrank him again, and you can choose who lives or dies. If you own her loyalty, perhaps you could convince the emperor that it would be more desirable to employ a gifted cryptographer than kill her.\"\n\nOwn her loyalty? Presumptuous ass. But he was good. Curse him, he was good. Bocrest never could have swayed Rias, but this seventeen-year-old kid had all the right answers.\n\n\"I will consider your offer,\" Rias said, giving away nothing of his thoughts. \"For now, let's get working.\"\n\n\"Agreed.\"\n\nTikaya headed into the room. Her interest in exploring had diminished, but she did not want to be caught eavesdropping. She forced herself into work mode. If the others came in, they would expect her to have made progress.\n\nTanks and pipes dominated the back half of the vast room, but she gravitated toward rows of black panels where more symbols than she had seen in one place marked the faces. A large oval glowed softly, displaying what she guessed were schematics or diagrams monitoring the station. She copied symbols, but her mind dwelled on Rias's conversation with the assassin, and she struggled to concentrate.\n\n\"Focus,\" she muttered to herself.\n\nShe tilted her head back to massage her neck and noticed only two rows of windows. There were three outside, so there had to be another story up there. No stairs, ladder, or anything similar led upward. Turgonians had steam-powered lifts in their taller buildings\u2014might this advanced race have something like that too?\n\nShe circumnavigated the interior, finally spotting a pale blue circle glowing on the floor in one corner. Thinking it might mark a place where a lift would descend, she waved a hand above it. Nothing happened. She pressed her boot into it and pulled it out. Nothing. Finally, she stood in the center with both feet planted.\n\nAir whooshed around her.\n\n\"Errkt,\" she blurted, dropping her journal.\n\nA platform of air thrust from below, propelling her upward. A circle in the ceiling slid aside, and the force raised her through a hole. As soon as her feet cleared the aperture, the floor slid back into place, and she stood on a second blue circle. It felt solid, but she jumped to the side anyway.\n\nA cozy space spread before her. Though she could identify little at first glance, she had the impression of living quarters and furnishings. Perhaps the caretaker for the pumping house had dwelled there. If so, she had a chance to see beyond the weapons-building, experimenting-on-humans side of the ancient people.\n\nTikaya roamed the space, repeatedly reminding herself not to touch things. Furnishings included cubes, octagonal structures, high tables, and a large hollow sphere open on two sides. No knickknacks or artwork decorated the surfaces, nor could she find practical tools such as eating implements, but perhaps all that had been taken when the occupant left for...wherever these people went.\n\nThere was one exception. In a storage area, she found a rack designed to hold spheres slightly smaller than her fist. The rest of the concave slots were empty, save for one. She slid the smooth black sphere out. A few groupings of runes ringed the center. She did not recognize them, but eagerness suffused her. Here was an artifact she could take with her to show the world. She pocketed it to check against the runes in Lancecrest's journal later.\n\nShe was about to peer out the window to check on Rias's progress, when her gaze snagged on a clear tank against the far wall. All the furnishings were made from the usual black material, but that piece was as clear as Tenesian glass. An inkling that it might be a bathtub enticed her further. Why settle for washing in front of marines and teenage assassins in the reservoir when she had this private room? And with the men off exploring, what were the odds anyone would stumble upon her? Hammer blows started up outside, so the two remaining men were busy on Rias's project.\n\nWary about making assumptions and touching the wrong thing, she puzzled over a small plaque near the ledge. She recognized the symbols for water and animals, but who would bathe animals? Then the pieces clicked into place: not animals, fish. It was an aquarium. Perhaps the plant siphoned off aquatic life from the stream for saving or for studying, or perhaps they simply liked observing fish for the same reasons as humans. Either way, Tikaya grinned. She was not above bathing in an aquarium.\n\nShe fiddled with the controls, and soon water flowed from an overhang around the inside of the tank. Her grin widened when she discovered she could change the temperature. Not only could she have a bath, but she could have a hot bath.\n\nShe snorted at herself. Such a female characteristic to be so tickled at the idea of a warm bath. She shrugged, removed her boots, and unbuttoned the Turgonian military jacket with relish. How she missed her sandals and loose hemp dresses.\n\nThere was no place nearby to set the clothing, so she folded it and crossed the room to leave everything on something octagonal, flat, and chest-high she decided to call a table. She skipped back to the tank and slipped over the side. Warm water embraced her, and she shivered with delight. She unbraided her hair, submerged everything, then draped her arms over the sides and laid her head on the ledge. Bliss. The bath reminded her of the volcanic hot springs near her family's property. She wondered how everyone back home was doing. The harvest would be over by now. She had missed her nephew's birthday and her parents' anniversary celebration. She closed her eyes, lost in memories of home.\n\n\"Tikaya?\" Rias called sometime later.\n\nShe sat up, and water sloshed over the side. When had the hammering stopped outside?\n\n\"Tikaya, are you all right?\"\n\n\"Fine!\" She scrambled out of the tub. \"I'm fine up here.\"\n\nNaked and dripping water, she peered about for something to use as a towel.\n\n\"You didn't touch anything, did you?\" He sounded like he was right below the lift.\n\nShe darted for her clothes even as air whooshed.\n\nRias appeared on the platform before she made it half way. Worry furrowed his brow, and he clutched her journal. That expression changed to a wide-eyed gape when he spotted her.\n\nFrozen mid-step, Tikaya felt ridiculous\u2014and guilty at being caught relaxing while everyone else worked.\n\n\"I, uhm, sorry.\" She stood, dripping, not sure where to put her hands or how to explain. \"I found this tub, you see, and it's been so long, and, well, one does get sort of dirty tussling with tunnel monsters and marching across the tundra, and...\"\n\nRias was just staring. She really ought to shut up and put some clothes on.\n\nHe closed his eyes and clenched a fist, looking very much like a man trying to control his temper. With rigid, precise motions, he walked to the table, placed the journal on it, and turned his back on her.\n\n\"Take your time,\" he rasped, then stepped on the platform and disappeared.\n\nBelatedly, she realized it was not his temper he had been struggling to control. Her first thought was that she should have hopped into his arms and invited him to join her in the tub. Her second thought was to remember he was on top of the Kyatt Islands enemies-of-state list and that she had no idea what kind of seeds Sicarius's promises had planted in his head. The third thought ran the way of dismissing the second and seeing what might come of the first.\n\n\"Tikaya, you think too much,\" she muttered, grabbing her clothes.\n\nOutside, she found Rias and Sicarius building the frame of something that promised to be large. While the assassin dragged wood over, Rias knelt, his back to her, and hammered. Hard.\n\n\"Rias?\" she asked between whacks.\n\nHis shoulders tensed, and he hunched his neck. \"Yes?\"\n\nShe took a couple steps toward him. \"May I speak with you?\"\n\nHe fiddled with the hammer. \"I should keep working, try to get this done so we can cross as soon as possible.\"\n\nTikaya hesitated. Maybe she had guessed incorrectly. Yet he had never lost his temper with her, and it was hard to imagine a midday bath truly irking him.\n\n\"Please?\"\n\nRias's head drooped. He stood, gave Sicarius instructions, and finally faced her. Tikaya led him out of the assassin's earshot.\n\nRias stared at the ground, avoiding her eyes. She was about to speak, but he did so first.\n\n\"I'm sorry. I didn't mean to stare. I wasn't expecting you to be, ah...\"\n\nShe resisted the urge to hug him\u2014that would probably make him more uncomfortable\u2014and gripped his forearm instead. Corded muscle lay beneath her hand. \"I don't mind. You can stare.\" Though so many differences stood between them, she could not feel anything but delighted that he would want to.\n\nRias lifted his eyes. \"Oh? I had the impression that your parents wouldn't approve of Fleet Admiral Starcrest ogling their daughter.\"\n\n\"They're not here.\"\n\nHe arched his eyebrows. \"I didn't think you were particularly enamored at the notion either. Something about a nation's war enemies not being easily inserted into dreams involving beach houses and blond children.\"\n\nShe blushed. \"Originally, I was rather distraught at the dishevelment of my dreams, but I must admit I can't think of anyone else in the world I'd rather have ogling me.\"\n\n\"Really.\" His eyes gleamed with humor but intensity too. He brushed his fingers down a lock of damp hair dangling by her cheek.\n\nTikaya considered the construction site and the assassin who, through tact or disinterest, was ignoring them. \"Almost private around here at the moment.\" She arched her eyebrows and stepped closer, placing a hand on his chest. \"I haven't figured out which piece of furniture up there is a bed, but I'm willing to conduct research.\"\n\n\"I wouldn't think it'd be a problem. You found the tub after all.\" Rias slid his arm around her, drawing her against him.\n\n\"Actually, that's an aquarium.\"\n\nShe felt the soft rumble of laughter in his chest, but it ended with a sigh. She tilted her head back, searching his face.\n\n\"Trust me, I'd very much like to research the furniture with you, but...\" He smiled and brushed his thumb along her lips. \"I suppose it'd be rather irresponsible of me.\"\n\nShe barely managed to avoid blurting 'huh?' Instead, she guessed, \"Because you're supposed to be building a, er, whatever that is you're building?\"\n\nRias snorted. \"Rust what I'm building\u2014and it's a counterweight trebuchet, by the way.\" His inability to dismiss his project without at least a short explanation almost made her laugh, despite her confusion over the rejection.\n\n\"I'm aware of what is, and what isn't, included in a standard Turgonian field kit,\" Rias went on, \"and I wouldn't want to put you in the awkward situation of explaining to your family how you came to be pregnant with an enemy admiral's child.\"\n\n\"Oh.\" She laughed with relief. He wasn't rejecting her.\n\nRias frowned at her reaction. \"Tikaya, I know what the world believes about Turgonians, and the Kyattese have every reason to think the worst of me. I fear that if you intimate that we're even friends, your people will believe I've tortured and brainwashed you into giving that response.\"\n\nHe looked exasperated that his words didn't drive the grin from her face, and his concern touched her.\n\n\"What you say may be true,\" Tikaya said, \"but that's something to worry about after we both get out of here alive. As for the other, getting pregnant wouldn't be possible until I returned home to see one of our doctors to have the...\" She groped for words to explain it in Turgonian\u2014as far as she knew, their women took their chances drinking egata tea for contraceptive purposes. \"It's a procedure, performed by a doctor\u2014who is, in our culture, a practitioner specializing in the psychological and somatic aspects of the mental sciences. Anyway, it's not irreversible. You just go see the doctor again when you want to have children.\"\n\nDuring her explanation, his expression changed from consternated to perplexed to enlightened. \"There is no...danger?\"\n\n\"No. After certain incidents during the war, it was recommended by our government that any women at risk of being captured have it done.\"\n\nHis face darkened. \"Were there many? 'Incidents?'\"\n\n\"I was sheltered by the fact that I never left the island, but from the folks who went out, I heard...there were some ships you really didn't want to find yourself aboard.\"\n\n\"I see.\" His jaw was tight, body rigid. \"I'd ask for the names of those ships, but there's nothing I could do now. It's hard to know\u2014I don't mean to make excuses, but men present a vastly different face to their superiors than they do to their prisoners.\"\n\n\"I doubt you ever did.\"\n\nHe grimaced, apparently not in the mood for praise, and she wished she had never brought up the subject. Except, she reminded herself, that bringing it up meant disavowing him of the notion that he could send her home a mother. Which actually was not a horrifying concept, though he was right in that it would be easier to deal with further down the line. Still, a smile curled her lips at the thought of a passel of precocious toddlers scurrying around the house, getting into mischief and cutting down heirloom fruit trees to build play forts.\n\n\"What are you thinking of?\" Rias's muscles relaxed as he watched her.\n\n\"Furniture research.\" She rose on her tiptoes, marveling that her eyes still weren't level with his, and kissed him.\n\nHer explanations resulted in one pleasant outcome: he did not hesitate to return it.\n\nThe moment ended abruptly. Rias pulled away, annoyance flickering across his face. Before she could ask why, she heard the clomp of boots. One of the squads of marines had returned.\n\n\"What, by the emperor's eternal warts, is this mess?\" Bocrest bellowed as soon as he entered the cavern and spotted the fledgling frame and the heaps of wood surrounding it.\n\nRias sighed and dropped his head on Tikaya's shoulder.\n\n\"Tonight?\" she suggested.\n\nHe released her with a hand squeeze and a promise in his eyes. Please don't let monsters, machines, or annoying marines ruin the night, she thought.\n\n\"We need help, boys,\" Rias called. \"Grab a hammer.\"\n\n\"About this catapult...\" Tikaya said, a question occurring to her as her gaze skimmed the chasm.\n\n\"Counterweight trebuchet,\" Rias said.\n\n\"Yes, of course. How will one land without breaking every bone in her body?\"\n\n\"Parachutes, naturally.\" Rias help up a finger. \"That reminds me.\" He turned to holler at the approaching men. \"Anyone who isn't able to find a hammer and work on this is on sewing duty.\"\n\nWithout glancing at the captain, the marines hustled over, prepared to dive into the construction work to avoid a stitching task. Chuckling, Tikaya returned to the second-story retreat to examine the sphere that had piqued her interest earlier.\n\n# CHAPTER 18\n\nThe sphere proved amazing. With the journal's help, she deciphered the runes on the outside, which were a proclamation of ownership and instructions for firing it up. Once she did that, a hole smaller than a grain of sand projected a display above the sphere. It appeared solid but she could wave her fingers through it as with an illusion. Plenty of practitioners who studied optics could make them, but she could not fathom how it was done with technology. She did not care either. It was the images and runes within the three-dimensional display that enraptured her. She found herself reading someone's diary, and she could look up symbols and terms she did not understand, as if a dictionary and encyclopedia underlaid the journal. This was the type of artifact every philologist dreamed of finding, something that held the keys to unlocking an entire language. She marveled that the other team had left it. Maybe they had not been up here, or maybe they had not realized what they passed up.\n\n\"How do you turn the water on?\"\n\nRias's voice startled Tikaya so much she dropped the sphere. It slipped from her fingers and clunked on the high table where she sat. She caught it before it rolled off the edge, though she almost dropped it again when she spotted Rias.\n\nHe stood by the tub, his weapons, boots, and shirt already on the floor next to his rucksack, and a towel and bar of soap on the ledge. She stared at his muscled chest. If he had been on the gaunt side when she first met him, that was not the case now. Hard to imagine someone filling out on that abysmal military food, but perhaps it suited him. Scar tissue scored his torso and arms: several old gashes and two dense knots where he must have been shot. Some of those wounds had been life-threatening and represented a lot of pain. As with Krychek, he never spoke of it, never complained.\n\n\"Should I be feeling self-conscious under this scrutiny?\" Rias asked. \"Or are you only looking this way while thinking about translating runes?\"\n\nHeat flushed her face. She decided his first question was safest to answer. \"Push on that symbol and slide it up, then rotate it for hot or cold.\"\n\n\"Hot? Excellent.\" He turned on the water and hopped up to sit on the edge facing her. \"You're engrossed there. You must have found something good.\"\n\nShe brightened, taking this for an invitation to share her findings. \"Yes! It's a journal someone kept. You must be wondering about this place, these people. It's all explained in here, though if I wasn't sitting here I'd think it the stuff of a storyteller's imagination. These people\u2014they called themselves the Orenki\u2014they came from another planet. A group was persecuted for their scientific research methods and driven out of their homeland. They came here and experimented\u2014this is chilling by the way\u2014experimented on primitive humans because we were biologically similar to their people. They wanted to come up with devastating weapons so they could return to their home world, use them on their own kind, and take over.\"\n\nRias opened his mouth to speak, but she barely noticed. She still had to tell him the best part.\n\n\"You're going to love these instructional, uhm, illusions\u2014sorry, no better word for it in my language or yours. The first one I found shows how to repair and maintain this pumping facility. It looks like there are thousands of sets of instructions on all sorts of mechanical things, though I haven't quite figured out how to search through them. They're organized by codes. But I will figure it out. It's just a matter of...\"\n\nRias turned off the water, and Tikaya realized she had been talking for a long time. And that she had cut him off. She smiled sheepishly. \"Sorry, did you want to speak?\"\n\nRias chuckled. \"From another planet, you say?\"\n\nTikaya, deciding she should let him talk for a while, offered an encouraging, \"Mmhmm.\"\n\n\"I'm too ignorant of astronomy to even ask how that'd be possible, but given what I've seen here, I can't claim to be utterly surprised. I'm hoping they weren't ultimately successful, because I'd not be comfortable knowing beings capable of making such weapons were still out there.\"\n\nYes, that was an unsettling idea, but Tikaya was more concerned about human beings learning how to use those weapons. She could not allow the Turgonian emperor to have this technology. Or a disaffected marine colonel. Or anyone.\n\n\"Rias?\"\n\n\"Yes?\"\n\nTikaya wanted to tell him, ask him for his help going forward. But when she gazed at him, at the scar on his eyebrow and the war wounds on his torso, she stopped herself. He might share her passion for academics, but she could not forget he was a soldier. He had been loyal to a totalitarian government his entire life, and his one disloyal act had cost him more suffering than anyone should have to endure. Would he truly choose such a road again? \"You should bathe before the water gets cold.\"\n\nHe watched her with sad eyes, and she wondered how much of her thoughts he read.\n\n\"After all,\" she said, \"you've seen me naked. It's only fair I get to see just how much of a Turgonian legend you are.\"\n\nThat drew a self-deprecating chuckle, but not the repartee she expected. He slid his trousers off and climbed into the tub. Despite her words, she dropped her gaze to the sphere to give him privacy. She glanced up a few times, but Rias seemed lost in thought. He was considering Sicarius's offer, she knew it. Probably trying to figure out if he could work her into his life once he had everything back together. She supposed, in some less than ideal scenario, she could see living in Turgonia with him, but slaving for the emperor, creating ciphers her own people would never crack if there was another war? That was not going to happen. And even the rest made her grimace. He would be off at sea most of the year, and she would be alone amongst strangers, thousands of miles from her friends and family.\n\n\"That bad of a show, huh?\" Rias leaned against the tub wall, arms folded on the ledge, chin resting on them.\n\nShe tried to disguise her blank stare but doubted she succeeded.\n\n\"The show\u2014me,\" he clarified. \"Nude. Never mind. I can see you're busy with the sphere. I understand the appeal of a puzzle, though I fear your enthusiasm means I'll be ogling only myself tonight.\" His smile was wistful but accepting.\n\n\"Oh.\" She shoved the sphere into a pocket and ran a hand through her hair, which still hung loose. \"That wasn't the puzzle I was pondering.\"\n\nRias cocked an eyebrow. \"No?\"\n\nBetter to bring it up for discussion than guessing at\u2014and maybe misinterpreting\u2014his thoughts. For all she knew, he was musing over ways to remove a wart from his toe. \"I heard Sicarius talking to you.\"\n\n\"Ah.\" He nodded with understanding, but did not say anything. No words to assure her she had nothing to worry about.\n\n\"I couldn't fault you for being tempted, but...\" She searched his face, but his eyes were cast down, thoughts apparently turned inward. \"Rias, I love you, you know that already. But I'm not going to do anything to hurt my people, and I'm definitely not going to work for your emperor.\"\n\n\"I thought not,\" he murmured.\n\n\"And...\" She drew a deep breath, \"I'm not going to let Bocrest, that assassin, or anyone else walk out of here with weapons that could destroy millions. I don't know how I'm going to stop them yet, but you'll have to kill me to keep me from trying.\" She lifted her chin. There, she had said it. Maybe it would have been smarter to lead him to believe otherwise, but she did not want to lie to him, even a lie of omission. Maybe that made her naive, but, so be it.\n\n\"Good.\"\n\n\"Good?\" She rapped a knuckle on the table. \"Would it be possible to get more than one-line responses? Do I need to posit my statements as math problems?\"\n\nRias chuckled. \"Oh, you're hard on me, Tikaya.\" He dunked his head under the water, ruffled his hair dry, hopped out of the tub, and grabbed the towel. \"I was proud of myself for baring my feelings to you last night. I'd been pacing through the hills rehearsing that while the camp was conspiring to leave me.\"\n\nEven frustrated with him, she had a hard time ignoring the 'show.' He wrapped the towel around his waist and padded over, rivulets of water snaking down the gullies between his muscles. He sat on the edge of her table, and she reminded herself to look at his face.\n\n\"I'm not used to confiding in people,\" he said. \"Being a captain or an admiral, it's a solitary vocation. You're expected to be infallible, even omnipotent. Sharing your thoughts, showing any kind of fear or hesitancy, might crumble that facade, and that's something men need to believe in when chaos is erupting around them and odds seem impossible.\"\n\n\"I'm not in your chain of command, Rias.\"\n\nThat drew a smile. \"I know. And I'm thankful for that. I'm glad you're here to remind me... Yes, of course, the weapons need to be destroyed. That's too much power for one man to wield, too much temptation. The easier we make it to kill, the less time there is to master the art of knowing when not to.\"\n\nTikaya nodded\u2014it was everything she had hoped he would feel\u2014but his earlier thoughtfulness made her suspect more remained unsaid. \"But?\"\n\n\"But...\" Rias combed his fingers through his hair, spraying flecks of water. \"You're right: I am tempted by the emperor's offer. I can't help but wonder if I could have it both ways. Help them with their mission, get my life back, and figure out how to make the weapons disappear later on.\" He picked at the hem of his towel. \"They're selfish thoughts, not honorable ones, but dear ancestors, Tikaya, I've missed this.\" He waved to encompass the tunnels and the marines camped outside the pumping house. \"Command, purpose, a challenge. When I'm not Admiral Starcrest, naval strategist, I'm not sure who I am or what else could be out there for me.\" He turned his eyes toward her, the question in the air.\n\nMe. That was her first thought, but she kept it to herself. She was not fool enough to believe that she could dump herself in his lap as the answer\u2014a man like him needed more stimulation than a relationship offered\u2014but her second thought offered a neater solution. She hoped. \"Do you know what the prime groupings in this language mean?\"\n\nHis brow furrowed. \"No.... Did you figure it out?\"\n\n\"I haven't an idea.\" She tapped a fingernail on the sphere. \"I've learned that what we've seen is one of four languages these people used, and this one is all skewed toward mathematics and science. I may be able to translate it eventually, but the numbers are beyond me. Figuring it all out, finding useful applications for our own world, it'd be the work of a lifetime. For someone interested enough to stick with me for that long.\"\n\n\"Ah?\" A hint of speculation entered his eyes. \"But you made it clear I wouldn't have a place on your islands.\"\n\n\"There are other countries in the world. Maybe we could find one where you're not wanted dead on sight.\"\n\n\"That might be a challenge. Doubly so, since I'm a penniless vagabond with nothing to offer you except myself.\"\n\n\"I'm rather fond of yourself.\" Tikaya scooted to the edge of the table to sit closer to him. \"And...\" She laid a hand on his bare arm. \"Did you not say you appreciate a challenge?\"\n\nHis gaze dropped to her hand, then returned to her face, and he smiled, the quirky half-smile that made her insides tangle. \"This is true.\"\n\nShe leaned into him, breathing in the scent of lye soap. His bare thigh touched hers, warm even through her clothing. Too much clothing. She really ought to...\n\nRias slid his fingers through her hair, and she forgot her thought as goosebumps rose across her flesh. He removed her spectacles, and set them aside.\n\n\"So, have you decided?\" He bent and kissed her neck, warm breath tickling her skin. \"Is this thing the bed?\"\n\n\"Uhm.\" Tikaya laid her hand on one of his broad shoulders, then slid it down his arm, tracing the dips and rises of the muscles. \"What?\"\n\nHe drew back, eyes narrowed in mock accusation. \"You're not thinking about runes, are you?\"\n\n\"Furthest thing from my mind.\"\n\n\"Really?\"\n\n\"Well, it's definitely a distant second place to something more prominent.\"\n\n\"Oh, good.\" He tilted his head. \"I am the something, right?\"\n\nShe grinned. \"Let me answer your first question. I think our current seat is the human equivalent of a coffee table. I believe that sphere over there is the bed. You get in and warm air floats you up and supports you. You can just lay there or there's an option for, ah, undulation.\"\n\n\"Sounds fascinating.\"\n\nHe slid off the table, faced her, and slipped his arms around her waist. She leaned into him, lips parting, and invited in his warmth. She tangled her fingers in his damp hair, pulling him closer even as his arms tightened around her. Where he had been hesitant before, asking permission, he was sure now, and she felt his need. And her own. It had been a long time for both of them, and she could never remember wanting someone so much. Not just for now, but forever.\n\nHis grip loosened, and she voiced a muffled protest, but his lips smiled against hers, then he knelt. He gazed up at her as he unlaced her boots, and her breath caught at the adoring tenderness in those gold-flecked eyes. She recovered and dug into the buttons of her uniform jacket. Her fingers didn't seem to work as well as they should.\n\n\"I miss my dress,\" she muttered.\n\n\"Me too.\" Rias's eyes crinkled as the first boot clunked to the floor.\n\nShe shucked the jacket, annoyed when her arm caught in the sleeve. By now, he knew she was no graceful gazelle, but it would have been nice to undress competently.\n\nShe started to tug the shirt over her head, but hesitated. It was silly\u2014he had already seen her naked and was clearly interested\u2014but a self-conscious twinge stilled her hands. With Parkonis and those scattered few before, she had shared bench space in the cute-but-not-beautiful part of the arena. Even before Rias turned into some legendary Turgonian hero, she had known he belonged up front, in the I-could-have-anyone-I-want seats. Tonight, he wanted her, but if by some miracle they made it out of here, would that change? When there were more attractive options available?\n\n\"Tikaya?\" Rias asked gently, concern in his tone. The second boot had joined the first, along with the scratchy wool socks, but he still knelt, face tilted up, watching her.\n\n\"Sorry, it's nothing.\" The future was a murky incorporeal place; best to enjoy what was real, what was now.\n\nRias stood, body solid and warm between her legs, but tense. \"Please, tell me. I don't want any more secrets, any more misunderstandings. I'm on your side out here, above all others.\" His eyes probed hers. \"Do you trust me?\"\n\nShe blinked. \"That wasn't what I was doubting. I mean, I wasn't doubting. I was just wondering if...if you saw me, back when you came to Kyatt to talk to our president, if I'd been in your path somewhere along the way...would you have noticed me?\"\n\n\"Ahhhh.\" Rias grinned and the tension melted from him. \"Indeed so.\" His eyes grew hooded, sultry, and he leaned into her. \"Especially if you'd been wearing that dress that so nicely accents your curves.\"\n\nHe slipped his hands beneath her shirt and massaged his way up from her waist. She shivered at the sensations those callused but gentle fingers stirred. He kissed her mouth, her jaw, and down to her neck, and she closed her eyes, arching into him.\n\n\"You do know...\" he murmured against her throat, lips sending spirals of heat to her core. \"Those dazzling blue eyes are rather rare in Turgonia. Exotic.\" He spoke slowly, lazily, kisses punctuating his words. \"And it's always a challenge to find a lady of sufficient height.\"\n\nShe would have laughed, but his mouth returned to claim hers, and she met him with equal intensity, humor forgotten. He broke away only long enough to say, \"You're a beautiful woman, Tikaya, and I love you.\"\n\nShe gave him a fierce hug before turning to other matters. There was not much talking after that, and fortunately he proved more adept than she at removing military uniforms.\n\n* * * * *\n\nTikaya woke with a start. The soft light of the room left her confused as to the time. She was in the sphere, with its shelf of air acting as a mattress. Though she had no blanket, a cocoon of warmth kept the alien bed cozy.\n\nRias sat next to her, head cocked, ear toward the window.\n\nShe touched his bare back. \"What is it?\"\n\n\"I think I heard a scream.\"\n\n\"Not one of mine?\"\n\n\"Not this time.\" He smiled and kissed her before crawling past her and out of the sphere. He reached the window before she maneuvered off the air cushion.\n\n\"Better get dressed.\" Rias jogged to their piles of clothing and tossed hers on the bed.\n\n\"What's going on?\" She tugged her shirt over her head and tied her hair back.\n\n\"I can't see. There's some kind of fog out there.\"\n\nAs soon as Tikaya had trousers and boots on, she hustled to the window. A gray-blue haze made it impossible to see more than a few feet. She could not make out the reservoir, the walls, or the ground where the marines camped.\n\nRias belted on his sword and checked the rifle.\n\n\"Wait,\" Tikaya said as he headed for the lift. \"What if it's poisonous? What if everyone is already...\"\n\nRias hesitated, one foot on the blue circle.\n\nA yell of pain pierced the walls and was cut off.\n\n\"They're not dead yet,\" he said.\n\nBut he did retrace his footsteps and find the towel he had used earlier. He tore it lengthwise, handed half to her, then wrapped the other half around his head to cover his nose and mouth. She tied hers and followed him down the lift.\n\nRias slid the door open and paused to listen before venturing out. \"Stay close,\" he whispered.\n\nThey slipped outside. The haze stung Tikaya's eyes. Even through the cloth, she smelled an odor reminiscent of burnt coconut.\n\nRias led her toward the camp. Visibility ran only a few feet in the dense fog. They reached the first prone form, Agarik, still under his blanket.\n\n\"Is he...\" she started.\n\nRias knelt and checked for a pulse. \"He's breathing.\"\n\n\"Sleeping?\"\n\nRias shook Agarik's shoulder, which elicited a snore, but nothing more wakeful.\n\n\"Not the type you can be roused from apparently,\" Rias said.\n\nThey crept farther and found more sleeping marines. None of them could be shaken awake, and Rias stopped trying.\n\nAt the edge of the fog, a hint of green clothing appeared on the ground to the right. Tikaya stepped over a marine to find herself staring at an unknown face with blood still streaming from a slashed throat. She struggled for detachment\u2014and to keep from stepping in the spreading crimson pool. The dead person was small and thin-boned with a green shirt and brown trousers that lacked any hint of military uniformity. Definitely not Turgonian, but she was not sure of the nationality.\n\n\"Rias?\" she whispered.\n\nHe had disappeared in the fog. She walked in the direction she had last seen him, but tripped over one of the marines. Her reaction was too slow and, almost as if she floated in water, she toppled face-first to land on the man. He grunted but did not wake.\n\nConfused at the heaviness of her limbs, she pushed herself up. It felt as if a hundred pound rucksack burdened her. The cloth covering her face might delay the fog's effects, but she would be snoring alongside the marines soon if she did not get away from it.\n\n\"Rias?\" she called a little louder.\n\n\"Tikaya?\"\n\nShe nearly tripped again. That wasn't Rias. That wasn't any Turgonian. It sounded like...\n\nShe put a hand to her chest. It couldn't be.\n\n\"Tikaya?\" the voice came again. \"Are you here?\"\n\nShe closed her eyes. The voice, so familiar, was speaking in her language.\n\n\"Over here,\" she said. She did not say his name. She still did not believe it could be him. How could it be? He was dead, his ship sunk over a year before.\n\nShe held her breath as the fog stirred. A shape coalesced.\n\n\"Parkonis,\" she croaked, lifting a hand.\n\nHe was a slight figure in comparison with the Turgonians, and he looked even thinner than she remembered. His curly red-blond hair, always a mess, had grown and stuck out in every direction, much like the beard hiding his chin and neck. Anxious blue eyes looked her up and down. He was the one who had watched from the opposite side as the marines entered. Oh, Akahe, if she had been close enough to identify him earlier, would she have...\n\nShe glanced behind her shoulder. Where had Rias gone?\n\nParkonis started toward her, arms wide, a white toothy grin escaping the beard. But his toe bumped against the fallen green-clad man.\n\nHis smile faltered. \"Tatkar, no.\" His gaze darted a dozen directions. \"One of them escaped the gas. We have to\u2014\"\n\nA dark shape slipped out of the fog behind him.\n\n\"No!\" Tikaya shouted before she even saw the bloody dagger.\n\nShe lunged forward, knowing she could never stop the assassin in time. He, too, wore a cloth across his face, but it did not hide the intent in his cold, dark eyes.\n\nRias stepped out of the fog behind Sicarius and dropped a hand on the assassin's shoulder. The dagger froze.\n\nParkonis whirled, took in the tableau, and stumbled back. Eyes still fixed on the assassin, Tikaya stepped forward and gripped Parkonis's hand.\n\n\"I have no idea how he's here,\" she said, talking to Sicarius who seemed to be deciding whether to finish what he had started or not, \"but this man is a gifted archaeologist, and if anyone can help you get your weapons, he can.\"\n\nParkonis's Turgonian was as good as hers, and she had no trouble reading the incredulous look he gave her\u2014helping the empire was the last thing he wanted to do. She squeezed his hand, hoping he would recognize the don't-say-anything signal. Rias's gaze fell to the hand hold, and guilt washed over her at his pained wince. He closed his eyes for a long moment.\n\nTikaya lifted her free hand and spoke as much for him as for the assassin. \"Let's figure out what's going on before we do anything else.\"\n\nRias pulled a mask over his face, but instead of responding he released Sicarius and disappeared into the fog.\n\n\"Brace yourself,\" Parkonis whispered in Kyattese.\n\nTikaya opened her mouth to warn him the assassin understood their language, but the hairs on the back of her neck leaped to attention. A heartbeat later, blinding whiteness engulfed the cavern, and a thousand cannons roared in her ear. Her feet floated from the floor, and someone\u2014Parkonis?\u2014wrapped his arms around her. She had the impression of weightlessness, of her body moving toward the chasm.\n\n\"What's going on?\" she yelled, but she could not hear herself over the clamor in her ears.\n\nWith her senses overloaded, it took a moment to realize what was happening: Parkonis was rescuing her. And she did not want to be rescued, not if it left Rias to wonder if she had scurried off with her old lover.\n\nShe thrashed. She had to escape before they reached the chasm. Her elbow caught Parkonis in the gut, and she felt rather than heard his pained exhalation. Regret mingled with desperation\u2014she did not want to hurt him.\n\nParkonis shouted in her ear, but she could not hear words above the roar. She squirmed again, determined to free herself. He let go with one arm, and she thought she had her chance, but something cold and coin-sized pressed against her temple. The world blinked out, and she knew nothing more.\n\n# CHAPTER 19\n\nTikaya woke slowly, mind groggy. She lay on her back, her head in someone's lap. Rias? No, concerned blue eyes peered down at her. Parkonis.\n\nShe struggled to sit up. A woman and a man in marine blacks stood above her. Colonel Lancecrest had not bothered removing the name tag from his wrinkled jacket, though she would have guessed his identity without it. Greasy salt-and-pepper hair stuck up in spicules, bags haunted his dark eyes, and furrows creased his weathered face. He looked like a man with nothing left to lose.\n\nThe woman had straight blonde hair and pale skin, so she might be Kyattese. Crow's feet lined her cold green eyes, and her thin lips flattened further under Tikaya's scrutiny.\n\n\"Where are Tatkar and my men?\" Lancecrest asked.\n\n\"Dead or captured,\" Parkonis said. \"Sorry, we\u2014\"\n\n\"Idiots.\" The colonel clenched a fist and stalked away, back rigid.\n\n\"Hate that man,\" Parkonis muttered.\n\nTikaya rubbed her face and tried to clear the wooziness from her brain. Another cavern stretched around them, this one with cracks and buckles marring a floor decorated with bat guano. Its pungent smell tainted the air. Stalactites hung from a ceiling far overhead. No sign of a camp or recent habitation marked the cavern, but her spine tingled with the telltale sense of nearby practitioner work.\n\nA fifty-foot-high butte rose in the center, a natural formation with a steep, jagged face. A single chamber with transparent walls took up the space on top. Lit from within, the bright interior revealed dozens, maybe hundreds, of bristling rockets. Similar to the one in the fort, they stood upright, their outsides loaded with dense strings of colored cubes. Larger black cylinders stood in the middle, and Tikaya had no idea what they might do, but she doubted anything up there existed for a purpose other than devastation.\n\nThere was no obvious way to get to the chamber. Two broken pillars and the remains of a ramp had crumbled and collapsed. A cool draft whispered against her cheek. If bats were living in the cave, there must be access to the outside nearby. Several tunnels led from the cavern.\n\n\"Thank you for your help, Gali,\" Parkonis said, addressing the woman. \"I thought that sleeping gas would be enough, but you were right: one of their guards was too alert.\"\n\n\"Tatkar was my colleague for years.\" Gali turned icy green eyes on Tikaya. \"You better be worth it.\"\n\nHow nice. Tikaya had found people as amiable as Bocrest to be her new captors.\n\n\"She is.\" Parkonis rested a hand on her shoulder.\n\nA thousand questions for him burbled in her mind, chief among them how he was alive and what he was doing with relic raiders, but she would wait until she could get him alone to ask. As long as he was here to vouch for her, she might have the freedom she needed to investigate those weapons and plot their demise. Maybe she could even destroy them before the marines showed up.\n\nColonel Lancecrest returned, his face composed, though frustration still tensed his body. \"You Starcrest's ally or his prisoner?\"\n\n\"She'd never ally with that monster,\" Parkonis said.\n\nTikaya climbed to her feet, pushing back dizziness. She touched her temple. Whatever Parkonis had used to render her unconscious was gone.\n\n\"Captain Bocrest is in charge.\" She decided to give them information that didn't matter. Maybe she could gain their trust if she seemed to hold nothing back. \"He kidnapped me from my parents' plantation and threatened to kill my family if I didn't translate these runes for him. I have no loyalty to him.\"\n\n\"And can you?\" Lancecrest asked. \"Translate this gibberish?\"\n\nParkonis turned curious eyes toward her.\n\n\"Some,\" she said. \"I'm learning more every day.\"\n\nLancecrest jerked his chin at Gali. \"Test her, witch. See if she's telling the truth.\"\n\nGali scowled but stepped forward. She cracked her knuckles and flexed her fingers. Lancecrest closed in on Tikaya.\n\n\"Test?\" she asked.\n\nShe had never failed an academic test in her life, but somehow she doubted these people wanted to assess her ability to categorize vowels. Lancecrest stepped behind her, reinforcing her supposition by taking her arms in a viselike grip. An inkling of what they meant to do stirred in Tikaya's gut, and she tried to pull away from him. He held her firmly.\n\n\"Telepath?\" Tikaya asked Gali.\n\n\"Yes.\"\n\n\"Just in case the oath you took matters to you, I do not grant you permission to poke around in my thoughts.\" Numerous people on the Kyatt Islands had a knack for telepathy, but it had never concerned Tikaya since back home there were strict laws against intruding without permission.\n\n\"We're not on Kyatt,\" Gali said. \"No one here to enforce oaths.\"\n\n\"That's when they matter the most, then, isn't it?\"\n\nThe woman stepped forward without answering and raised her fingers. Tikaya tensed. Cursed sea, she did not want someone rooting around in her mind, reading her memories, maybe replacing them with more acceptable ones.\n\n\"I'm sorry, Tikaya,\" Parkonis whispered behind her.\n\nIn other words, he was abandoning her. He must not have much power in the group. She could not help but think about how Rias had started out with no power amongst the marines and he had never failed to fight for her. She pushed thoughts of him from her mind. They could only get her in trouble here.\n\nGali's cool fingers prodded Tikaya's temple. Something itched inside her mind, like stitches being pulled out. Panic gripped her. These bastards had no right to her thoughts. She yanked her head back.\n\n\"Hold her still,\" Gali growled and reached again.\n\nTikaya kicked her in the gut. The woman doubled over, clutching her stomach and gasping for air.\n\nLancecrest forced Tikaya to the ground, leaned a knee into her back, and shoved her face to the floor. She tried to twist free, but he wrenched her arms until she gasped with pain. Her cheek smashed against cold rock.\n\nThey were too strong. Her fate was unavoidable.\n\nGali's hand came down on the back of her head, nails gouging skin. Tikaya felt the other woman's annoyance, not just in those tense fingers but in her mind.\n\nImages from the last month were dragged into her surface thoughts. Tikaya tried to fight it. She thought of cutting cane on the plantation, her family, school, childhood escapades, anything but\u2014\n\nRias.\n\nThe foreign presence in her mind focused on him, tearing into any thought related to him. And there were a lot. Tears formed in Tikaya's eyes at the pain the invasion brought, the disdain she felt through the woman's link. The experience was bad, maybe worse than Ottotark's attack back in Fort Deadend. For the first time in her life, she regretted not studying the mental sciences. A practitioner would have known how to block a telepath.\n\nAfter minutes that felt like hours, the presence in her mind dissipated. The hand left Tikaya's head.\n\nAwareness of her surroundings returned. The weight on her back. Her labored breaths. Gali's boots before her face. Parkonis's silence. A hot tear ran down her cheek and splashed on the floor.\n\n\"Well?\" Lancecrest asked.\n\n\"You can't trust her. They've duped her into working for them.\"\n\nTikaya focused on their words, groped for equilibrium. And she frowned. Duped? What in her thoughts had suggested that?\n\n\"They must have known she would never willingly help Turgonians, not after they decimated our islands in their war. Admiral Starcrest made her believe he was a prisoner, too, and gained her trust.\" Gali snorted. \"He tricked her into thinking he loves her, and\u2014this is lush\u2014that the two of them are going to destroy the weapons together. Dear Akahe, Tikaya, I'm embarrassed for you. I could see it if you were eighteen, but you're not young enough to be that naive.\"\n\nStunned, Tikaya said nothing. The woman had been in her head, read all her thoughts, and that was the conclusion she had come up with? How could she possibly think Rias's friendship\u2014his love\u2014had been a ruse after all they had gone through?\n\nThe weight on Tikaya's back lifted, and she pushed herself to her knees. Gali stood before her, arms across her chest, pity and annoyance wrestling for room her on face.\n\n\"Love?\" Parkonis asked in a soft, stung tone.\n\nTikaya winced. She would have told him about Rias, but not like this. Maybe the woman would have the humanity not to share everything. But she was shaking her head.\n\n\"You would lecture me on my oath when you're sleeping with that man?\" Gali looked over Tikaya's shoulder. \"Sorry, Parkonis, but your faithful fianc\u00e9e has been sheet wrestling with Fleet Admiral Starcrest.\"\n\nTikaya remembered an earlier thought where she had lamented having no females to talk to out here. She decided to rescind it.\n\nWhen Parkonis said nothing, she risked a glance at him. He was staring at her, mouth hanging open, eyes bulging. Not angry, not yet. Still in shock.\n\n\"Parkonis,\" she said quietly, trying to ignore Gali's cold stare. \"As far as I've known, you've been dead for more than a year. The Eagle's Spirit went down at the end of the war. You never wrote, never sent word. I had no idea I'd ever see you again.\"\n\nHe closed his mouth, turned his back, and walked away.\n\n\"Leave us, Gali,\" Lancecrest said.\n\nThe woman shrugged and headed for an empty stretch of cavern. Between one step and the next, she disappeared. An illusion shrouding a camp, Tikaya guessed.\n\n\"Come.\" Lancecrest offered a hand.\n\nTikaya eyed him, surprised he had not simply grabbed her and yanked her to her feet. She got up on her own, but she did follow him as he walked away. He led her past bat guano piles and to a portion of wall engraved with a column of symbols.\n\n\"I've only been here a few days,\" Lancecrest said, \"but I gathered from my little brother that Parkonis wasn't as good of a translator as he'd hoped. Atner actually wanted you on his team from the beginning. Can you tell me what this says?\"\n\nTikaya hesitated, but it was such a basic sign that she saw little reason to withhold the information. \"Lights.\"\n\n\"What?\"\n\n\"It's a panel to control the lighting level.\"\n\n\"The lighting? You're sure? Parkonis thought these panels might have something to do with the web.\"\n\nTikaya slid one of the symbols up, and the lighting level in the cavern increased. Down and it decreased.\n\n\"Damn,\" Lancecrest said.\n\n\"What's the web?\"\n\nLancecrest turned toward the invisible camp. \"Keeler, show her the web!\"\n\nA gaunt, wispy-haired man appeared. He lifted his gaze toward the ceiling, and Tikaya felt the tickle of the mental sciences being used. A bat flapped down from the shadowed stalactites and soared toward the weapons room. Before it flew anywhere near the glass, a small explosion lit the air like a miniature star exploding. The bat did not have time to squeal in pain. Its charred body fell, causing three more explosions on the way down. Nothing but ashes remained to trickle to the floor.\n\nTikaya stared at the fine pile.\n\n\"The kill zone starts about twenty feet up,\" Lancecrest said, \"and extends to the walls. You see that door in the chamber up there? And the symbols by it? My brother has\u2014had\u2014goggles that make it easy to see them. He got in once by randomly pressing them with that wizard shit he learned on your island.\"\n\n\"Telekinetics?\" Tikaya suggested.\n\n\"Yes. He lowered a rocket out, but the code changed before he could get back in, and he wasn't able to find another combination that worked.\"\n\n\"How could you let him use that weapon on the men in your fort? The men who trusted you to command them?\"\n\nLancecrest's jaw clenched. \"I didn't know. Atner just sent a note to get out of the fort with my best men and meet him at the canyon. I still can't believe he\u2014I know why he did it, but I can't believe he made that choice.\"\n\n\"Why'd he do it?\"\n\n\"Keeler.\" Lancecrest waved toward the practitioner who had come out to bestir the bat. \"Keeler can see what's happening elsewhere. He found out Starcrest was coming and my brother panicked, figured he had to do anything to delay you all.\"\n\n\"Rias isn't even in charge.\"\n\n\"Doesn't matter. He's there. And the other man\u2014a captain, isn't he?\u2014doubtlessly had orders to requisition half the fort to help flush the archaeologists out of the tunnels and get the weapons. Atner probably figured I wouldn't disobey those orders, even for him.\"\n\n\"Would you have?\"\n\n\"I don't know. It doesn't matter now. Come.\" Lancecrest led her to another panel. \"What's this one say?\"\n\n\"Temperature and...\" Not water, but similar to water. \"Humidity,\" she realized. \"Controls for modifying the cavern atmosphere.\"\n\nHe sighed. \"I was hoping for more from these panels.\"\n\n\"I doubt the instructions for disabling the security system are going to be on the wall in the same room as the security system.\"\n\nLancecrest grunted and strode to the last panel of symbols, this one on the backside of the butte and situated fifteen meters from a tunnel entrance. Tikaya did not have to ponder long, for she had translated these exact symbols just a couple days earlier. Her stomach clenched. Rias was not here with his acidic concoction this time.\n\n\"Don't let your people touch anything on this one,\" she said.\n\n\"Security?\" Lancecrest perked up. \"Weapons?\"\n\n\"Cleaning service. What's down that tunnel?\"\n\n\"Labs. What do you mean, cleaning service?\"\n\nTikaya nodded. \"We've seen them so far near the labs.\"\n\nDespite her own words, she prodded an \"open\" symbol. An invisible door swung outward, and her breath caught. Four stacked cubes waited inside, their deadly orifices pointed her direction.\n\nLancecrest cursed and jumped back. Tikaya stood frozen a long moment before her thoughts could push past her first instinct of fear. The cubes were dormant for the time being. A complicated drawing on the inside of the door caught her eye. A schematic? A label on top, a grouping of numbers and symbols, nagged her mind. There was something familiar about the arrangement. Oh, it looked like the codes on the instruction sets in the sphere.\n\n\"Can I get some paper and copy this?\" she asked.\n\n\"If it'll help. I can get you the goggles, too, if you want to take a good look.\" He pointed at the top of the butte.\n\nIf she wanted to take a look? Strange, Lancecrest was treating her better instead of worse since Gali blabbed.\n\n\"Do I have a choice?\" Tikaya asked. \"I'm used to Turgonians threatening me or my family to ensure my help.\"\n\nLancecrest studied her for a moment. \"Starcrest's a tricky devil on the water, but he's no asshole who would play mind games with a prisoner. If he says you're his woman, you are.\"\n\n\"And that means something to you?\" she asked, not sure whether to be hopeful or not.\n\n\"I respect him. If he's between me and getting out of here alive, I'm still going to shoot him, but I'm not going to torment you.\"\n\n\"Is that a Turgonian tenet? It's fine to shoot a man you respect, but you don't mess with his lady?\"\n\nA faint smile stretched his lips. \"Something like that. My little brother...destroyed things for our whole family. All I'm hoping to do at this point is get out of here alive with some weapons to sell. Ideally, I'll get those weapons and get out of here before Starcrest shows up.\"\n\n\"What happens to Parkonis and the other archaeologists? I assume selling weapons isn't what most of them signed on for.\"\n\nLancecrest's jaw tightened. \"This isn't what I signed on for either, but we're all stuck in this together now. My brother didn't make his choices alone. Everyone here is going to be wanted for crimes against the emperor. We're all working for a split of the profits. And if you want Starcrest\u2014and Parkonis\u2014to walk away from this unharmed, you'd best get to work opening that weapons chamber up there. You can make sure there's no bloodshed.\"\n\nNo bloodshed. Right. Until whoever he sold the weapons to used them.\n\nBetter to let him think she would go along with him though. \"Get me the goggles.\"\n\n\"Lancecrest!\" someone called from the other side of the butte. \"The Turgonians are over the chasm!\"\n\n\"Already?\" Lancecrest cursed and jogged toward the speaker.\n\nAs he trotted away, Tikaya eyed the tunnel near the cube cabinet, wondering if she could slip away before someone caught up with her. But, no, it would not take Lancecrest long to notice she did not follow, and for all she knew the tunnel dead-ended. Besides, she wanted to copy that schematic, examine the door symbols, and talk to Parkonis.\n\nShe jogged after Lancecrest. He disappeared ahead of her, but she expected that by now. She kept going and between one step and the next, the camp appeared. Crates, backpacks, bedrolls, muskets and bows, and food sacks sprawled about her. The scent of stale sweat mingled with the pervasive guano stench.\n\nTwenty people, several wearing marine uniforms, stood around the wispy-haired fellow, who hunkered over a bowl of water. A clairvoyant, she realized when she spotted images of Bocrest's men moving on the still surface.\n\n\"How'd they get across?\" someone asked.\n\n\"Built a catapult.\"\n\nA marine whistled.\n\n\"I wouldn't have joined this team if I'd known we'd be against Admiral Starcrest,\" a woman muttered.\n\n\"Isn't he dead?\"\n\n\"We'll be dead if we tangle with him.\"\n\n\"He's just a man,\" Lancecrest growled. \"We've laid your wizard traps, and we know the terrain best. The advantage is ours.\"\n\nTikaya edged closer to the bowl, hoping to catch sight of Rias. She wished she could communicate with him somehow, let him know everything that mattered. For the moment, the clairvoyant showed them Bocrest and Ottotark as they removed their parachutes and gathered their gear.\n\n\"Find Starcrest,\" Lancecrest said. \"Let's see what they're planning.\"\n\nThe image shifted, focusing on Rias and Sicarius. Tikaya wrestled with the urge to kick the water bowl over. Even if she did want to see Rias, it was better if Lancecrest did not know what he planned. She took a step toward the bowl. Lancecrest gripped her forearm to keep her back.\n\n\"How was the ride?\" Rias asked.\n\n\"Exhilarating,\" Sicarius said in a monotone.\n\n\"I thought we might get a yell of excitement or at least a smile out of you, but I see the emperor has trained you well.\"\n\n\"Yes.\"\n\nIt was like conversing with a rock. Tikaya wondered why Rias bothered, especially when he ought to be worried about her. Not that she wanted him to fall apart, but a little agitation would have been flattering.\n\n\"Scouts will go ahead,\" Bocrest said, somewhere beyond the edges of the vision. \"See what we're up against.\"\n\nRias took a step.\n\n\"Not you, Admiral. We need you back here planning brilliance, not wandering around looking for your misappropriated camp follower.\"\n\nRias's jaw clenched and the tendons sprang out on his neck. There was her agitation. And then some. He looked like he might tear Bocrest's head off.\n\nSicarius stopped whatever might have come next in the conversation, raising his hand and saying, \"Hold.\"\n\nHe tilted his head, as if listening to something, but his cool eyes stared straight through the water. The clairvoyant flinched, and the image evaporated.\n\n\"What is it?\" Lancecrest asked.\n\n\"That young one has unexpected perception for a Turgonian.\"\n\n\"He knows we're watching? So, what? Get them back. I want you on their every step, so we know when they're coming.\"\n\nThe clairvoyant closed his eyes and draped his arms across his knees, palms up. Nothing happened. \"I can't. He's blocking us somehow.\"\n\nLancecrest's fists clenched and unclenched. \"Who is that boy?\" he asked Tikaya. \"He's not in uniform, and he's too young to be giving orders.\"\n\nShe shrugged. \"The marines didn't tell me much.\"\n\nLancecrest considered her, and she thought he might call Gali over, but the marines and relic raiders were watching him, and he turned to them instead.\n\n\"Time to get ready for company, people. Morrofat, take your squad out. Your job is to delay that team.\" Lancecrest dug in a rucksack and pulled out a clunky pair of goggles that reminded Tikaya of the eye protection she had worn on the tundra. He tossed them to her. \"You know what your job is. We've found dangerous relics that we can throw at Starcrest. If you care about him, you'd best get me those weapons before he gets within range.\"\n\n# CHAPTER 20\n\nTikaya stood near the spot where the bat had been disintegrated. From there, the door to the weapons chamber was visible, and the runes glowed beside it. No ropes bound her hands\u2014Lancecrest had decided she needed them for writing\u2014but she had a guard. Gali. The woman huffed and sighed as she paced about, fiddling with a pistol. Her telepathy worried Tikaya more than the weapon. She wanted to copy the cube schematic on the chance Rias could do something with it. And she could escape the raiders long enough to meet up with him.\n\n\"You're not looking at them,\" Gali snapped. \"I imagine that'll make the translation difficult.\"\n\n\"I'm ruminating,\" Tikaya said.\n\n\"On how to escape, I know.\" Gali slapped the pistol across her palm. \"Ruminate on getting through that door.\"\n\nTikaya held the goggles before her spectacles and peered through the magnifying lenses. The symbols grew crisper, the nuances easier to make out. They were numbers. Four rows of four, each different. She only recognized a few, but she could look up the others with the sphere if she could find a private place to take it out.\n\nShe pushed the tool from her mind. Better these people did not know about it.\n\nWith the enhanced vision, she examined the rest of the weapons chamber. A cuboid contraption hung from a ceiling and was attached to a large pipe disappearing into the cavern depths far above. Some kind of fan or ventilation system to suck away fumes if there was a break or accident? Probably not fast enough to save the life of the person inside, but if the door shut and that fan activated perhaps destroying the weapons would not prove deadly for everyone down here.\n\n\"What does one do with the symbols?\" Tikaya asked. \"Push them?\"\n\n\"I wasn't there,\" Gali said, \"but I heard Atner stood down here and moved them around with telekinetics.\"\n\nGali stretched a hand toward the butte. One of the numbers indented and glowed red. She twitched her fingers, and it slid one place to the right. The number on the right edge disappeared and reappeared on the left.\n\n\"How'd you do that?\" Tikaya asked.\n\n\"A bit of sideways pressure. The runes glide around naturally, as if they've been greased. Atner fiddled around for days and finally got lucky. But he only got in once and the symbols changed after that.\"\n\n\"But he got a rocket out. How was he able to get it down without the web destroying it in midair?\"\n\nGali shrugged. \"The web didn't attack it.\"\n\n\"So, the defense system won't attack what it's supposed to defend?\" Tikaya wondered if there was anything else it would not attack.\n\nThe symbol winked off, though it did not return to its original slot. Three red beams lanced out of the clear door. Tikaya jumped, nearly dropping the goggles. The beams scoured the air in front of the door. After a moment, they cut off.\n\n\"I guess if you don't punch in the correct code while you're standing up there, you get incinerated.\" She chewed on the side of her mouth. \"Your telekinesis does offer a workaround the original builders probably didn't consider. If they were here as long ago as I suspect, humans wouldn't have had the skill yet. Relatively speaking, our command of the mental sciences is a recent development in civilization. It's not something that started appearing until after we developed agriculture and became more agrarian rather than hunter-gatherer. More free time, creation of a leisure class, and\u2014\"\n\n\"We didn't bring you here for a lecture,\" Gali snapped. \"Figure out how to get in before your cursed Turgonian lover gets here.\"\n\nTikaya turned her attention back to the numbers. The ones she recognized were prime: three, five, seven, eleven, thirteen, seventeen, nineteen. Could the whole series just be the first sixteen primes? She had not learned numbers beyond twenty yet, so she would have to look them up. Presumably, the sixteen digits had to be arranged in a specific order to open the door. If Atner Lancecrest had pushed them randomly, he truly had gotten 'lucky.' Rias could no doubt give her the exact odds of guessing correctly, but what she remembered of studying permutations in school suggested a ridiculously high number of combinations.\n\nShe did not have time for guessing. If the code changed regularly, it was probably a puzzle of some sort. If one knew the goal, one had a better chance of solving it. Could it be as simple as placing them in order? No, too obvious to someone who spoke the language, and the scientists who had built this place had surely had their own people in mind as potential trespassers, not the cave-dwelling humans who had occupied the world at the time.\n\nThere had to be more to it. Tikaya dug a sheet of paper out and copied the numbers for later perusal.\n\nA distant clanking started up. She cocked an ear, trying to identify it. The noise had a muffled quality and did not sound like it came from within the cavern. A few men ran out of the camp and into a tunnel.\n\n\"I want to take rubbings of the panels too,\" Tikaya said.\n\n\"Whatever,\" Gali said.\n\nRelieved, Tikaya pretended no more than vague scientific curiosity as she copied the panels\u2014and the schematic on the inside of the cube cabinet.\n\n\"Can you work those?\" Gali asked suspiciously.\n\n\"No.\" Tikaya closed the cabinet and tucked the rubbings into her pocket along with the numbers.\n\n\"Tikaya?\"\n\nShe turned to find Parkonis standing a few feet away. The remaining marines and relic raiders carried bows or firearms. Parkonis had nothing more offensive than a utility knife. The last year had apparently not turned him into a fighter.\n\nGali backed away a few paces, giving them a semblance of privacy.\n\n\"Parkonis,\" Tikaya said, not sure what else to say.\n\nHe pushed a tangle of curly hair out of his eyes. \"I'm sorry I ran off.\"\n\n\"Earlier today?\" she asked. \"Or a year ago?\"\n\nHe grimaced.\n\n\"What happened out there, Par? You obviously got off your ship, but why couldn't you come home?\"\n\n\"The Turgonians sunk us, just as you heard, but Atner Lancecrest happened by and rescued those left alive. There were only four of us. He showed us the runes from this place, and I was intrigued, of course. He swore us to secrecy before telling us anything about them, then asked if we wanted to join his team, to travel to the source and work on translating a previously undiscovered language. He was leaving right away to recruit others from Nuria and the islands. There was no time to come home. It was a dream opportunity, Tikaya. I couldn't refuse.\"\n\n\"You don't find it suspicious that he just 'happened by' right after the Turgonians attacked you? He was Turgonian himself. Maybe he wanted to appear a benevolent rescuer, but actually set up the whole thing. Maybe he had a deal with the captain of the ship that sank yours. All so he could get his hands on a handful of grateful archaeologists. Are you sure you had a choice about coming?\"\n\n\"That's far-fetched, Tikaya. Atner wasn't a bad fellow.\"\n\n\"Wasn't a\u2014he killed everyone in that fort out there. In a ghastly way. And those were his countrymen!\"\n\nParkonis winced. \"He was desperate at that point. We didn't... I didn't have anything to do with that. I swear. I didn't know about the weapons when I agreed to come. We were just looking for relics, and in truth I only cared about the language.\"\n\nYes, Tikaya could understand that temptation. She stepped toward him and softened her voice. \"Even if you joined voluntarily\u2014especially if you joined voluntarily\u2014why didn't you write, Par? How could you let us believe, for a whole year... We had your funeral. I stood next to your weeping sister and parents. This devastated them. And me too. I spent months trying to get over...\" Her voice broke. She was still struggling to resolve this new reality with her memories.\n\nParkonis avoided her eyes. The distant clanking continued, like metal beating against rock.\n\n\"I should have written,\" he said. \"I just didn't know how to without explaining everything. I was heading off to Turgonian territory, and I knew it'd be dangerous. I didn't want you to worry.\"\n\n\"Worry! I thought you were dead. How could worrying for your safety be worse than believing you dead?\"\n\n\"I know, I realize that now. I was a fool. You were always the smart one. You weren't going to marry me for my brains, were you?\" He grinned, a disarmingly boyish grin that she knew well. And she knew when he was using it to cover something.\n\nShe folded her arms over her chest. \"Did you not want me to worry or did you not want me to find out about the language?\" She watched his eyes as she spoke, waiting for\u2014yes, there was a wince. \"Colonel Lancecrest said his brother wanted me on the team from the beginning. Are you the one who talked him out of recruiting me?\"\n\nParkonis looked away. \"I told him you wouldn't work for relic raiders, yes.\"\n\n\"You wouldn't either, at least I didn't think so. But we do funny things for a chance at history, don't we? I'll wager you wanted to be the first to translate this new language, something you wouldn't be able to lay claim to, not solely, if I came along and helped.\"\n\nHe opened and closed his mouth several times, and she knew she was right.\n\n\"I didn't think we'd be gone that long,\" he finally whispered. \"I thought, when we got back, we could pick up where we left off...\"\n\n\"I understand.\" Tikaya sighed. And she did. Hadn't she been enticed by that language too?\n\n\"You do?\" Hope widened his eyes.\n\n\"I understand, but I wouldn't have made that choice. I would have wanted you to come along, to be a partner in the translating, even if it meant you got the credit.\"\n\n\"But I wouldn't have gotten credit. You were always first.\" He shrugged helplessly. \"It was always you.\"\n\nShe rubbed her face and wondered if that clanking was in her skull. He was missing the point. Or she was. Maybe she couldn't truly understand what he felt, always being second best. Either way, she was beginning to think it would have been too much between them, even if Rias had never come along. She thought of him, of all he had lost, and of the temptation Sicarius had laid at his feet. So much more than accolades. But he had rejected it because of her. At least, he had before Parkonis showed up and kidnapped her. What if Rias believed he had lost her? Would he no longer have a reason to say no to that temptation?\n\n\"Tikaya?\" Parkonis asked.\n\nShe tried to focus on him, though a new urgency fueled her intent. She had to escape and find Rias before he made a choice he would regret.\n\n\"Whatever happens here...\" She squeezed Parkonis's arm. \"I want you to know that I love you, and I'm beyond relieved that you're alive, but I can't go home with you. I can't marry you.\"\n\n\"What?\" He reeled back. \"Because of that Turgonian?\"\n\n\"No.\" She did not want to get into that now, but Parkonis grabbed her arm before she could step away.\n\n\"Don't be a fool, Tikaya. You don't love that monster. How could you? It's a clear case of captive complex. No matter how badly you're treated, you start sympathizing with your captors, even wanting to please them, because you're grateful they're not killing you.\"\n\n\"I know what the term means, and that isn't the case.\" If anyone had captive complex, it was he. Even after hearing how he had come to be here, she could hardly believe Parkonis would be a party to this weapons-selling scheme.\n\n\"It's not your fault. I forgive you. You were just trying to stay alive. Who knows what that monster would have done if you'd fought him?\"\n\n\"He's not a monster. I know our people have no reason to like him, and you even less, but he did save your life from that assassin. We can trust him. He disobeyed orders two years ago, and he refused to have our president assassinated. He's been exiled since.\"\n\n\"Exile?\" Parkonis snorted. \"Is that what they told you?\"\n\n\"He's only here because of his familiarity with these tunnels\u2014he was part of the original mission that found them. He's been as much a prisoner as I. He was the only ally I had.\"\n\n\"Tikaya, don't you see? It's all an act. That man outranks a captain. If he's been pretending to be a prisoner, it's only been to fool you, to win your sympathy. He's insidious, they all are. They've been tricking you.\"\n\n\"I'm not a fool, Parkonis.\"\n\n\"It's not your fault. They say he's a genius. He's probably a master of manipulation.\"\n\nTikaya groaned and dropped her forehead in her hand. Why couldn't he just be jealous? Instead, he thought she was an idiot who had been brainwashed. This was a glimpse of what going home would be like. Torture. Her heart cringed at the idea of never seeing her family again, but maybe her notion of sailing off to some obscure port with Rias was a better idea than she realized.\n\n\"You have to come back with me, Tikaya. We'll take you to see a doctor, someone who can heal your mind. You just need distance, some time to return to your old life. If\u2014\"\n\nThe lighting flickered and went out.\n\nTikaya whirled, but blackness swallowed everything. As with the marines, the raiders had been relying on the alien lighting and nobody had lanterns lit. Timorous voices called out questions while others cursed in irritation. The symbols at the weapons door and on the panels still glowed, but they did not provide enough illumination to diminish the darkness.\n\n\"Tikaya?\" Parkonis's hand bumped her chest, then found her arm.\n\nShe gripped him back. With the light gone, she abruptly grew aware of how many thousands of feet of earth lay above their heads. Since she had been unconscious for the trip to the raiders' cavern, she did not know the way back. Half a dozen tunnels exited this place, so one could wander forever in the darkness.\n\nA distant roar sounded. Or one could wander until one was eaten.\n\n\"Not them again,\" Parkonis whispered.\n\nShe recognized it too. The humanoid creatures they had fought the first day.\n\nParkonis's grip tightened. It did nothing to reassure her, not like Rias's would have. She started. Could Rias have manufactured this? As a distraction?\n\nLight appeared at the edge of the camp. Lancecrest strode toward them carrying a lantern, and she felt silly for her panicked concerns about not finding a way out. Of course, the raiders would have kerosene and lanterns, just as the marines did. Enough to last many days, she was sure. The roar came again. Closer this time.\n\n\"Come.\" Lancecrest waved an arm. \"Return to camp until we figure out what's going on.\"\n\n\"Gladly,\" Parkonis muttered.\n\nTikaya glanced over her shoulder. She could no longer see the tunnel Lancecrest had said led to labs, but she wondered if this might be her opportunity to disappear. Had Rias created this for her sake? Or did he think she wanted to be here, with Parkonis?\n\nGali stepped out of the shadows, her pistol aimed at Tikaya.\n\nRight. It would take more of a distraction to escape, and sprinting into dark monster-filled tunnels without a lantern and a means of defense would be unwise.\n\nInside the camp, more lanterns had been lit. People hustled about, grabbing weapons. An unclaimed bow and quiver rested on a crate, and she weaved through the clutter toward it. If those creatures were coming, maybe Lancecrest would not object to arming her.\n\nA heavy hand landed on her shoulder.\n\n\"You're sitting here out of trouble,\" Lancecrest said.\n\nBefore she could protest, he grabbed her arms and drew them behind her back.\n\n\"Wait,\" she said. \"I can help fight. I know how to use a bow.\"\n\n\"I'll keep that in mind.\"\n\nA moment later, Tikaya knelt with her wrists tied behind her. She endured it with no more than a sigh until his hands fumbled at her ears.\n\n\"No!\" She ducked her head.\n\nToo late. Lancecrest removed her spectacles. Everything more than a few feet away grew fuzzy.\n\n\"I doubt you'll wander far without these.\"\n\nTikaya craned her neck, trying to see where he was putting them. He stuffed them in a pocket without any concern for protecting the lenses.\n\n\"Bastard,\" she growled.\n\nA shot fired, echoing from the closest tunnel. Everyone in camp dropped behind cover, but no squad of marines burst into the cavern. Three more shots followed, along with a distant angry yell. Still, no one entered. The raiders shifted uneasily.\n\nTikaya could not imagine the Turgonians tipping their hand before attacking, but maybe they had run into the creatures. Or Lancecrest's traps.\n\nShe could not stay here and wait for something to happen. A nearby lantern gave her enough light to see, and the white and green fletching on the arrows in the quiver caught her eye. She edged closer. Maybe if she could filch an arrow, she could use the head to cut her bonds. That would be easier if her hands were in front of her, but she had to try.\n\nSomething fluttered above the tunnel entrance. It was too far away for her to identify, but someone fired. Black powder smoke wafted into the air.\n\n\"That was a bat, you lummox,\" Lancecrest said. \"And you just confirmed to the Turgonians that we're in here.\"\n\n\"Sorry.\"\n\n\"Scientists with guns,\" Lancecrest muttered. \"What was my brother thinking?\"\n\nAs the smoke rose higher, slowly dissipating, a faint white beam appeared in the haze. Tikaya blinked, wondering if she imagined the light. But, no, even with her spectacles off, she could see a beam. It reminded her of those emitted by the cleaning machines. Maybe that was all the \"web\" was, a pattern of beams crisscrossing the cavern, invisible under regular illumination. But if smoke revealed them, one might avoid them.\n\nIdeas percolated in her mind. But first, escape.\n\nStaying low, she crawled toward the crate. Though men and women crouched all around her, their focus was outward. Why worry about the tied, half-blind philologist?\n\nA couple more feet brought her to the quiver. Gali glanced at her and frowned, a what-are-you-doing expression stamping her face.\n\nTikaya attempted what she hoped appeared a guileless smile. \"Can you help me find my spectacles?\"\n\nThe woman scowled.\n\n\"Incoming!\" someone barked.\n\nAll eyes turned toward the closest tunnel. Tikaya rose, turned, and slipped an arrow out of the quiver.\n\nNo one noticed. She dropped to her knees, putting her back to the crate. She found the sharp metal head and maneuvered it until the edge slipped between her wrists. Careful not to cut skin, she eased it up and down against the rope. The awkward position made it impossible to apply much pressure. She held back a scowl, knowing this would take a while.\n\nIt was not marines but two black bipedal creatures that burst into the cavern. Even without her spectacles, she recognized the towering muscular beings. The illusion spell did not fool them; they barreled straight for the camp.\n\nMuskets fired and bows twanged.\n\nTikaya rubbed the arrowhead against her ropes.\n\nThe practitioners threw up an invisible barrier, and the creatures bounced back while men and women fired through it. The scent of black powder permeated the camp. Smoke stung Tikaya's eyes, but, in the rising haze, she spotted more beams in the air. They crisscrossed irregularly, nothing symmetrical or predictable like a spider web. None had more than a foot or two of open\u2014safe\u2014space between them. Even if they were visible, climbing past them might not be possible.\n\nThe rope snapped, and her wrists came apart.\n\nShe eyed the back of the cavern, trying to guess the distance to the cleaning cubes and the tunnel next to them, but, even if she had her spectacles, darkness would have thwarted her estimates. On the other side of the camp, Lancecrest stood, reloading a rifle. The creature battle had him distracted, but she did not see how she could retrieve her spectacles without him noticing.\n\nIn front of the camp, blood streamed from the beasts' dark flesh. Their muscles flexed and strained as they hammered the invisible barrier. Roars of pain and anger echoed through the cavern. Sweat gleamed on the practitioners' faces. One flexed his fingers. A pulse of power hammered the beasts. They flew backward, and landed hard, but they came up roaring with anger. Another volley was fired at them.\n\nEveryone appeared busy.\n\nTikaya grabbed a lantern and slid bow and quiver off the crate. She turned the flame down so it would not make her a target as she ran, then slipped toward the back edge of camp.\n\nSomeone shot one of the creatures in the eye, and it toppled to the floor. A ragged cheer went up.\n\nTikaya eased around sacks of corn meal and rice. A couple of steps and she would be out of the camp. She resisted the urge to hop the few obstacles and sprint for the wall. That would likely draw someone's eyes. Stealth would serve her better.\n\n\"The linguist is escaping!\" someone yelled.\n\nSo much for stealth.\n\nShe bolted. Her boot caught on the uneven ground, and she slammed to her knees even as a shot fired over her head. They would rather shoot her than let her escape back to the others?\n\nGulping, she leaped to her feet and sprinted to the wall, lantern and bow banging against her legs with every step. That was the first time her clumsiness had saved her life\u2014she could not count on it happening again.\n\nTikaya plunged into the darkness, using the blurry crimson runes as a guide. She reached the wall and stood to the side, not wanting to be silhouetted against them for the shooters.\n\nFootsteps hammered the floor behind her.\n\nShe jabbed the symbol that opened the cabinet, but nothing happened. Growling, she slowed her movements and added a rotation. The cabinet popped open.\n\nThe footsteps neared. Lancecrest. She didn't have enough time.\n\nThen a black shape blurred in from the side, crashing into him. The remaining beast.\n\nLancecrest yelled and flung his arms up.\n\nAs soon as it finished him, it would be on her. Tikaya pulled out a cube, praying it would not activate while she held it. Arms laden, she started toward the tunnel.\n\n\"Over here, you ugly pisser!\" someone cried and a psi wave pulsed through the air.\n\nIt struck the creature full on, hurling it twenty feet. The edge of the wave caught Tikaya and smashed her against the wall.\n\nLancecrest patted the floor for his rifle. His men poured out of the camp and moved to surround the creature. And her.\n\nShe sprinted for the tunnel.\n\nAn arrow clipped Tikaya's sleeve and shattered against the wall. Fear surged through her, and she ran faster.\n\nSomeone conjured a yellow orb of light, and it spun her direction, illuminating her, making her an easier target.\n\n\"Stop!\" A man pointed a pistol as he ran at her, his face a rictus of determination.\n\nShe had to keep going, hope his aim was poor.\n\nA shot fired, and Tikaya dove, knowing it would not be fast enough. But no blast of pain came. The man's musket hit the floor with a clatter, and he collapsed a heartbeat later. Tikaya scrambled into a crouch and squinted into the gloom behind him. A tall blurry figure in Turgonian black stood in a tunnel entrance on the far side of the cavern. Rias?\n\nShe stepped in that direction, but he waved her toward her closer tunnel.\n\n\"Starcrest!\" Lancecrest fired his rifle.\n\nRias flew back with a grunt. Tikaya gaped. It looked like he had been hit, but, curse her eyes, she could not tell. He ducked back into the tunnel. Lancecrest raced after him.\n\nTikaya took a step that direction, but an explosion roared, and the ground heaved. She was thrown onto her side, and the lantern flew from her grip. The cavern filled with confused yells and cries of pain.\n\nA stalactite plunged to the floor where it shattered and hurled shards everywhere. A second explosion ripped through the earth. A sinkhole opened up in the floor, and rock poured in like water over a fall.\n\nTikaya scrambled for the nearby tunnel, hoping the alien walls would hold up better than the cavern. She had no idea where the lantern had gone. Even as the floor pitched, she clutched the cube and the bow, determined not to lose anything else.\n\nBlackness smothered the tunnel. Three steps in, another concussion boomed, hurling her against a wall. Her breath whooshed out with a pained grunt. The bow and cube flew from her hands. She crumpled to the floor and barely had the presence of mind to curl into a ball with her hands protecting her head as further booms rocked the tunnel.\n\nNearby, rock shattered and cracked like gunfire. Tikaya cringed, expecting the ceiling to collapse at any second. Finally, the explosions ended, but rubble continued to pelt the floor. She kept waiting for rocks to hit her, but her tunnel seemed secure. Secure, but dark. Lifting her head to peer about was worthless since blackness pressed in from all sides. Worse, dust clogged the air and invaded her throat. She coughed and wheezed as fine particles smothered her tongue.\n\nDistant, muffled yells made it to her ears, but she could not pick out words. She shifted to get to her feet. Her fingers bumped a hard edge. She jerked back. The cube. If ever there was a mess, surely an earthquake\u2014or whatever that had been\u2014qualified. She held her breath, expecting the cleaning device to flare to life, for the orifice to glow red, the beam to lance out.\n\nBut the cube remained inert.\n\nWhatever the reason, she thanked her luck and hunted for the bow. She found it wedged under a pile of rubble. Rubble that blocked the mouth of the tunnel from floor to ceiling. Cave-in.\n\nShe hoped there was another way back to the cavern. And that she could find it in the dark.\n\nTikaya stood and started to brush herself off, but a new concern made her freeze. How far did the cave-in extend? What if it covered part, or all, of the cavern? And the tunnels beyond? Her heart lurched. What if Rias or Parkonis had been caught? She still didn't know if Rias had been shot. Damn, damn.\n\nShe clawed at the rubble, trying to dig a hole. She had to get back in and check.\n\nA minute later, her fingers were bleeding and she had made no progress. Breath rasping in her ears, she backed away. She would not get in that way. She needed to find another way around. She needed to\u2014\n\nNo. Tikaya wiped sweat from her face and forced herself to calm down, to think. Rias would want her to continue with the mission, not tear off, hunting for him. For all she knew, he might have set this all up. She remembered the clinking. Had the marines been crawling around in passages beneath the floor, placing blasting sticks?\n\nShe turned around and felt her way along the wall, trying not to feel guilty for walking away from Parkonis and Rias. She had to ensure those weapons were destroyed, and she could not assume the cave-in had done that. In fact, she would be shocked if it had.\n\nThe darkness made the trek feel longer, but she doubted she had walked far before she came to a four-way intersection labeled with glowing runes. Three possible directions, three labs. Biology, alchemy, and... She touched the last one, a new combination of symbols. Mechanical? That sounded promising, but she ought to let Rias know which way she had gone. She dropped her hand and snorted because her bloody fingers had already smeared a sign on the runes.\n\nShe padded down the hall. A door whispered open, and she stepped onto a landing. She expected darkness inside, but low blue lighting pulsed from the walls. Some kind of backup illumination, perhaps.\n\nThis lab was larger than others she had visited and had an upper level as well as a lower. She chose the upper, less out of any notion of what she might need, but because it would not be immediately visible to someone walking in.\n\nUpstairs, blurry cabinets lined the walls and high stations dominated the center. She had to wander close for the edges to sharpen. They reminded her more of woodworking benches than alchemy stations. Intricate black tools she could not identify were mounted to the table tops and hung from the ceilings on articulating arms. Rias would probably be fascinated by them.\n\nHer gut twisted with concern at the thought of him, but she forced herself to focus on the one thing she could accomplish here. She dug her notes, the sphere, and a pencil out of her pocket.\n\nTikaya put more obstacles between herself and the landing before stopping at a countertop that was not too high for her purposes. She thumbed the sphere on and identified the rest of the numbers from the door pad. More primes, but not the first sixteen as she had guessed. The sequence skipped a few: three, five, seven, eleven, thirteen, seventeen, nineteen, twenty-three, twenty-nine, thirty-one, thirty-seven, forty-one, forty-three, sixty-one, sixty-seven, and seventy-three.\n\n\"All right, Rias,\" she muttered. \"Where are you? I've translated them and done my half.\" As soon as the words came out, she snorted at herself. Yes, they made a good team, but she had done this sort of thing before she met him.\n\nThere had to be some significance in the missing prime numbers. Maybe these were the first sixteen that could be turned into a combination that allowed one particular thing. She drew a bunch of four by four boxes, mimicking the layout on the door pad, and scribbled the numbers in. Four rows, four columns, sixteen numbers. She added and multiplied. She looked for patterns.\n\nThe door hissed open.\n\nRias? Tikaya lifted her head and almost called out, but could not see the landing and caught herself before she could give away her position. She waited for the sound of footfalls, thinking she might be able to identify his tread, but there was no sound at all.\n\nThe door hissed shut.\n\nQuietly, oh so quietly, Tikaya picked up her work. It was possible someone had looked in, not seen anything interesting, and left, but she doubted it. That cursed assassin was the only one who walked without making a sound, and she had no idea what his intentions might be for her, especially now that she had, from the Turgonian viewpoint, escaped with the enemy. And if Rias had run off, too, Sicarius would know he had no intention of accepting the emperor's offer.\n\nTikaya twisted the symbols to open a couple of cabinets beneath a nearby workstation. One was empty enough she thought she could fit inside.\n\nShe stuffed the cube in one cabinet and knelt before the larger one. Careful not to make a sound, she slid boxes and tools out of the way. She could barely breathe, but she fit.\n\nShe pulled the door most of the way shut. Since the cabinets had to be opened with a turn of the symbols, she assumed she could not get out if she locked herself in. Terrifying thought that. No one would ever find her, and the cabinet would be her tomb. The assassin might spot the door slightly ajar, but she had to risk it.\n\nSilence reigned in the lab. Tikaya could hear her heart beating in her ears, her shallow breathing. The awkward position cramped her diaphragm. Minutes dragged past.\n\nShe closed her eyes and rearranged digits in her head. The four-by-four box reminded her of a Skiltar Square, those puzzles where the goal was to arrange the numbers so that every column, row, and diagonal added up to the same sum. It seemed unlikely an alien race would have the same math games, but she rearranged and totaled the digits in her head anyway, seeing if she could find a combination that worked from all sides. It surprised her when she found an arrangement where each option added up to one hundred twenty. Could that be the way into the weapons cache?\n\nHer fingers tingled with excitement. Or maybe numbness from sitting scrunched up in a cabinet. Her tailbone ached. She longed to crawl out and check her math with pencil and paper. Maybe Sicarius had left, or had never been there to start with.\n\nTikaya lifted her hand to the door, about to push it open. Then someone glided past the crack.\n\nBlack clothing, blond hair.\n\nShe held her breath and closed her eyes, as if the assassin might feel her stare through the crack. He had sensed the clairvoyant watching him, after all.\n\nA minute later, the door hissed again, and she spilled out of the cabinet. Sitting on the floor beneath the pulsing blue light, she checked her math with pencil and paper. Every row, every column, and even the diagonals added up to one hundred and twenty. Maybe it meant nothing. Or maybe it was the solution to the puzzle.\n\nShe hopped to her feet, longing to go check it, but thanks to the cave-in she was not sure how to get back to the cavern.\n\nThe door hissed again. Tikaya cursed to herself. Now what?\n\nFootsteps sounded on the landing. She reached for the cabinet door, ready to hide again.\n\n\"Tikaya?\" Rias.\n\nRelief swarmed her. \"Up here!\"\n\nShe skirted the workstations and almost crashed into him at the top of the stairs. He wore his rucksack and carried a rifle, but he managed to envelop her in a fierce hug. She clamped onto him just as fiercely, burying her face in his neck. He smelled of black powder and blood, but it didn't matter.\n\n\"You came for me,\" she whispered.\n\n\"Of course.\"\n\n\"The explosions... I was afraid you were...\"\n\n\"Me too,\" he said, voice hoarse. \"I feared you'd been caught in that cave-in. Bocrest was too quick to light the charges. He was supposed to wait until\u2014it doesn't matter now. If I've succeeded, they think the weapons are buried and I'm dead.\"\n\nShe lifted her arms, intending to hook them over his shoulders, but her fingers encountered dampness. A torn section of uniform wrapped his biceps like a bandage. She drew back, staring at blood on her hand.\n\n\"You're wounded!\"\n\n\"Lancecrest got lucky.\" Rias twitched a shoulder. \"It's just a scrape. It'll be fine. Besides, it was worth it. He was carrying something you might find useful.\" He unbuttoned a pocket, withdrew her spectacles, and draped them over her ears.\n\nTikaya slumped against his chest. She should have been elated to have her vision back, but a lump of guilt lodged in her throat. \"You got shot trying to help me. I'm sorry.\"\n\nRias took her face in his hands. \"I'd risk a lot more than a trifling arm to help you.\"\n\nComforted by his words, she tilted her head back.\n\nHe seemed on the verge of kissing her, but he cleared his throat and glanced around. \"Are you...alone?\"\n\n\"Yes, though Sicarius was here a few minutes ago.\" She realized he had probably been wondering about Parkonis, but his eyes widened at the mention of the assassin.\n\n\"He was? Rust, he's not supposed to be on this side of the cave-in. He must have found a way through.\" He scrubbed his face. \"Maybe we'll get lucky and be able to avoid him.\"\n\n\"I haven't seen Parkonis since before the explosions,\" Tikaya said. \"Do you know if he... Did any of the raiders make it?\"\n\n\"I didn't see him amongst the dead.\" Rias took a deep breath. \"Tikaya, I'll help you find him, but I need you to know... When he appeared and absconded with you I... My first thought was to hurl myself into that chasm. But there's a stigma against suicide in my culture, and regardless it's always seemed like giving up, which isn't something I've ever strived to master. I fully intend to fight for you.\"\n\n\"Rias\u2014\"\n\n\"I know he represents your dream, the life you always wanted, and I know its selfish of me to want you when it could alienate you from your family, but... How is it he's been alive a year and never found a way to let you know? I would have toppled an empire to get back to you.\"\n\n\"Rias\u2014\"\n\n\"I can't walk away and let him have you, Tikaya. Not if there's a chance...\"\n\n\"Rias.\"\n\nHe opened his mouth again, but she flattened her palm over his lips. His shoulders slumped, and wariness hooded his eyes.\n\n\"I appreciate hearing those things very much.\" She grinned at the idea of him toppling an empire\u2014for most people, that was just an expression, but she wouldn't put it beyond his means. \"But there's no need for you to go on.\" She lowered her hand, brushing his lips with her thumb. \"You have me.\"\n\nHe gaped at her in stunned silence.\n\n\"You have me for a lot of reasons,\" she said quietly, \"but especially because you're willing to give up everything to be here at my side, plotting against your people to destroy those weapons. The definition of a good man is someone who makes the moral choice when temptation invites him to do otherwise. The definition of a hero is someone who makes that moral choice even when temptation, threat of reprisal, and the mores of his culture invite him to do otherwise.\" She considered her words and issued a self-deprecating smirk. \"That was preachy, wasn't it?\"\n\n\"Oh, no, I liked it. Especially the part where you're calling me a hero.\" He grinned, eyes sparkling, and her heart danced.\n\n\"I'm sure I'm not the first, Fleet Admiral Starcrest.\"\n\n\"In Turgonia, you're a hero if you sink more ships than anyone else.\" He tucked a stray lock of hair behind her ear and laid his forehead against hers. \"I like your definition better.\"\n\n\"Good. I hope you like this too: I don't want you worrying about being selfish because you want to have a life with me. I want one with you too. Some people are worth changing your dreams for.\" She kissed him, wishing there was time for more. \"I want to be with you. Always. Even if it means we're both exiles on your forsaken prison island.\"\n\nRias's grip tightened. \"Cursed ancestors, don't say that.\"\n\n\"It's the truth, though I'm going to be terribly disappointed if the preeminent military strategist of the era can't outsmart a teenage assassin in order to avoid that fate.\"\n\nWhatever Rias's retort was going to be, it was lost when the door hissed open.\n\nBefore Tikaya could do more than think of hiding, Sicarius strode in. His gaze swiveled upward to lock on them. The pulsing blue light painted his face in eerie shadows. Blood stained his short blond hair, spattered the side of his face, and painted his hands. As those dark hard eyes raked her, she had an unsettling hunch none of it was his. Two pistols were jammed into his belt, and he carried a dagger. A drop of blood fell from the blade and splashed on the landing.\n\n\"Sicarius, good,\" Rias said.\n\nGood? The assassin looked like he was about to kill both of them. Tikaya stifled the incredulous expression that wanted to waltz across her face.\n\n\"Do you have the door symbols from the journal?\" Rias asked.\n\n\"You assured Captain Bocrest you were placing the blasting sticks to provide a distraction, but the tunnels came down in such a way that he believed the weapons cavern had been buried and you with it.\" Sicarius spoke in such an emotionless monotone it was almost possible to miss the accusation in those words. \"I informed the captain that it was unlikely you would miscalculate so badly. The marines are searching the tunnels for you and her.\" His cool eyes flicked Tikaya's way.\n\nShe groaned inwardly. The plan would have worked if the emperor's perceptive henchman wasn't here. She glanced at Rias, almost expecting him to dive behind the railing and rip his pistol free for a shot, but he did not.\n\n\"Demolitions are dangerous and sometimes unpredictable,\" Rias said. \"We can, of course, rejoin the others now, though why not get through the locked door and finish the mission while the relic raiders are too scattered to guard their cache? Do you have the symbols?\"\n\n\"Yes,\" Sicarius said, no sign on his face of whether he believed Rias's lies. \"I already tried them.\"\n\n\"How?\" Tikaya asked.\n\nNo doubt, he was agile enough to scale the side of that butte, but not with those invisible beams waiting to slice off limbs.\n\n\"Bow and arrow,\" Sicarius said.\n\nRias lifted an eyebrow toward Tikaya. He probably had not had as good a look at the entryway as she had. She nodded thoughtfully. If Gali had used telekinesis to nudge the symbols around, she supposed something thrown\u2014or shot\u2014against them could do the same job.\n\n\"You lined them up?\" Tikaya had assumed the numbers she copied were a different set than the ones that had appeared the day Lancecrest got in.\n\n\"As close as I could. Not all the symbols matched those in the journal.\"\n\n\"Maybe you misread them,\" Rias said.\n\nThose dark eyes turned a shade cooler. \"I did not.\"\n\n\"I've been told the symbols change periodically,\" Tikaya said. \"It was probably designed so people who knew the secret to the puzzle could always get in, providing they had the math skills, whereas others would have, well, the trouble Lancecrest's team has had.\"\n\n\"You know the secret?\" Sicarius glided up the stairs, eyes locked on her.\n\nRias dropped to the step in front of her, blocking the assassin's advance. Sicarius halted.\n\n\"I'm close,\" Tikaya said. Or not even remotely close. It was one of the two. \"It would help to see the symbols you have that worked before, even if they don't now.\"\n\nFor a long moment, Sicarius stared past Rias's shoulder at her. Finally, he wiped the dagger, sheathed it with the myriad others he carried, and handed her the torn scrap of paper, neatly folded.\n\n\"Thank you.\" Half the numbers were the same. She would have to check the sphere to translate the others. \"I have some ideas about how to get through the web.\" Maybe pretending to include Sicarius in their plans would make him more likely to believe they shared the same goal. She put a hand on Rias's shoulder. \"Can you make something that causes smoke? A lot of smoke?\"\n\n\"With the right ingredients, yes.\" Rias snapped his fingers. \"Sicarius, can you get us some bat guano from the cavern?\"\n\nTikaya almost choked. Bat guano?\n\nSicarius's eyes narrowed. \"There is no potassium nitrate in these labs?\"\n\nOf course. Potassium nitrate\u2014salt peter\u2014was harvested in bat guano-rich caves, and it was one of the core ingredients in black powder. The kid was bright. They would have to be very careful\u2014and probably lucky\u2014to trick him into helping.\n\n\"I haven't seen any,\" Tikaya said. Which was true. With her spectacles missing she had not bothered examining the lab closely.\n\n\"I'll prepare the vats and put together the rest of the ingredients to make some smoke bombs,\" Rias said. \"And Tikaya will work on the entry code for us. We can finish your mission before Bocrest even misses you.\"\n\n\"Bat guano,\" Sicarius said. \"Very well.\"\n\nAs soon as the door shut behind him, Tikaya and Rias grabbed each other's arms and started to talk at the same time.\n\n\"You first,\" Rias said.\n\n\"First, I think it'll be a lot easier to find potassium nitrate in one of these labs than making it from scratch, but I assume you're just trying to keep him busy.\"\n\nRias nodded. \"Yes.\"\n\n\"Second, can you look at this and tell me what you think? These are the translated numbers from the door pad.\" She fished out the page with her solution for the puzzle, wincing as she handed it to him. It had seemed a logical guess during her in-cabinet mulling, but now that she had to share the hypothesis with someone else, she feared it a foolish one.\n\n\"A Skiltar Square?\" Rias asked. \"It looks like you solved it. In Turgonia, you can get books full of them to entertain your precocious children.\"\n\nHe smirked, and she wondered how many his parents had foisted on him. Her amusement at the idea faded quickly.\n\n\"This can't be right then,\" she said. \"Too simple for these people. And surely they wouldn't have had similar puzzles to what we have.\"\n\n\"Why not? In your studies, haven't you found that the fundamental properties of numbers are the same in every language, amongst every people? Mathematics surely transcends humanity, existing whether we do or not, so it doesn't seem odd to me that another species would play the same sorts of games with numbers. And why wouldn't this entrance code be simple? Do you think someone carrying a toxic weapon up a ramp would have wanted to stand outside the door for three hours making calculations? What if he dropped one of those poison-filled vials and it broke at his feet? Big oops, eh?\"\n\nTikaya laughed. She had not considered that.\n\n\"Besides,\" Rias said, giving her an appreciative smile. \"Those squares aren't that easy to solve. Why don't you translate the combination from the journal and see if it's a solution to one.\" He thumped a fist on the railing. \"We still need a way to destroy the weapons. I was thinking we might find a formula for a powerful alien version of naphtha or kerosene, because even gas is flammable, right? At a high enough temperature... Tikaya, where are you going?\"\n\nHalfway through his spiel, she had charged to the cabinet where she left the cube. She raced back with the contraption clutched in her arms, and Rias lurched back a step at the sight of it.\n\n\"It's not active,\" she said. \"I'm not sure why, but it gives us the chance to experiment.\"\n\nRias recovered, though he eyed the device warily. \"Experiment?\"\n\n\"The cubes already clean things by incinerating them, right? All we need to do is add those weapons to the list of items its programmed to burn, throw it in that chamber, and close the door. You took one apart, right? Do you think you could alter its parameters? Like a punchcard in a steam loom?\"\n\n\"I... Tikaya, that thing is so far beyond a steam loom I wouldn't have any idea where to start.\"\n\n\"Even if I can translate the schematic?\" She thrust the blueprint she had copied toward him. \"Give me a moment, and I might be able to find repair instructions in the sphere's library too.\"\n\nThough his eyes darted, devouring the schematic, his wary frown did not fade. \"All before Sicarius returns or the marines stumble upon us or angry relic raiders burst in?\"\n\nSo, that's what daunted looked like on him. Huh.\n\n\"We can do it.\" Tikaya slapped him on the backside.\n\nHe blinked. \"What was that for?\"\n\n\"I'm encouraging growth.\"\n\n# CHAPTER 21\n\nGunfire cracked in the distance. Again. Bent over a table with Rias, Tikaya did not lift her head. The cube, one side removed, sat between them. Several parts she could not name lined the table in the order Rias had removed them. A three-dimensional display of the inside of one of the cubes hovered in the air, courtesy of the sphere. The blue lab lighting continued to flash, providing poor illumination for such detailed work.\n\nA yell of rage\u2014or pain\u2014sounded in the tunnels. Rias grumbled something under his breath about how he ought to be out there, helping the men. He had set the situation up so everyone would be running around in the tunnels, distracted dealing with each other and the darkness, creating just this time they needed, but it clearly did not sit well with him.\n\n\"The screwdriver thing,\" he muttered.\n\nTikaya handed him a long tool with a magnetic hook on one end and a tiny flat-tip head on the other. She had finished her work, gathering supplies for Rias's smoke bombs and translating the schematic and the numbers Sicarius had given her. The latter had proved to be another Skiltar Square. Now she handed Rias tools and tried not to feel useless.\n\n\"Close,\" he said. \"It's just a switch that modifies the level of 'cleaning' to be done, so it's easier than I thought, but reaching it without taking everything else out is the problem. Also...I'm afraid if I take everything out, I won't be able to get it back in correctly without breaking something. The insides are much more fragile than the outside.\"\n\n\"Take your time,\" she said, wishing it didn't sound so inane.\n\nShe was not sure how many minutes\u2014hours?\u2014had passed since Sicarius had left the lab, but she was beginning to think he must have run into a distraction. As uncharitable as it was, she hoped for a nice arrow or pistol ball to the chest.\n\nRias grunted and held out his hand for another tool. The kit of precision implements they had found ranged from knives and scalpels to repair gizmos, most of which she could only guess at. Some were too large for human hands, but all were well-made, the craftsmanship amazingly sturdy for such fine tools. A pair of black knives, in particular, had caught Rias's eye, and he had stuck them into his belt.\n\n\"There,\" Rias whispered. \"I think I got it.\"\n\n\"Is there a way to test it?\"\n\n\"Not here.\" He started replacing the innards. \"We'll have to get into the tunnels on the other side of the cavern. That's where I found the panel to cut off the lights. I'm guessing that whatever powers them powers these cubes and that's why they're inert.\"\n\nTikaya realized how lucky she had been when the blasts brought down all that rubble. If power had been running to the cube, it might have cut her down after all.\n\nThe door hissed open.\n\nThey spat silent curses at the same time.\n\n\"Distract him,\" Rias mouthed, waving at the mess still on the table. If Sicarius caught them with the cube, he would figure out their plan right away.\n\nTikaya grabbed the sphere and her notes and sprinted to the top of the stairs. Sicarius was halfway up. No bag of guano dangled from his grip.\n\nThough her instinct was to keep space between her and him, she jogged down several steps so she could stop him before he could see Rias.\n\n\"We figured out the code,\" she said, waving the pages. \"It's a puzzle with numbers.\"\n\n\"Where is the admiral?\" Sicarius asked.\n\n\"We made smoke bombs, and he's packing them. We did find some potassium nitrate, so we won't need the guano after all. Which is good, since you seem not to have gotten any.\" She winced at her inane babbling. \"What's going on outside?\"\n\nSicarius watched her, impassive eyes betraying nothing of his thoughts. He knew they had sent him on a useless errand. He had to. And he probably knew they were not on his side. They were going to have to kill him or incapacitate him somehow.\n\n\"Cat and mouse,\" Sicarius said. \"I killed one of the wizards. Some of Colonel Lancecrest's men are proving elusive, and they've set traps. Captain Bocrest's team has split them up, brought a few down, and taken others prisoner.\"\n\nDown. Dead. \"Parkonis?\"\n\n\"What?\" Sicarius asked.\n\n\"The man who...kidnapped me against my wishes. Do you know if he's alive?\"\n\n\"He dropped to his knees and begged for his life when we came upon him. He bore no weapon, so the captain took him prisoner.\"\n\nTikaya closed her eyes, thankful Parkonis was not the heroic type. He had no weapons training, and bravery only would have gotten him killed. Being a prisoner was no guarantee of safety, but there was still a chance she could help him.\n\n\"Where is the admiral?\" Sicarius asked again.\n\n\"Here.\"\n\nRias appeared at the top of the stairs, rucksack on his back, and what looked like ceramic globes with fuses in his hands. She had been busy with the translations and had not watched him assemble the smoke bombs. She thought of the vast cavern and hoped four would be enough.\n\nHe did not give her a wink or nod, not with the assassin watching, but she thought Rias's rucksack appeared lumpier than before. He slipped two globes into pockets and handed the other two to Sicarius.\n\nTikaya wondered how they would detour to the lighting panel with Sicarius tagging along. And would the cube fly up to the weapons room of its own accord, or did they need to get it up there and lock it in somehow? For that matter, would their modifications even work?\n\nFootfalls sounded in the corridor, and gear jingled.\n\nRias reached for his pistol, but Sicarius's hand blurred, landing on his wrist in a firm grip.\n\nRias twitched an eyebrow, the only indication he felt things might not be going according to plan.\n\n\"The captain sent reinforcements.\" The steely gaze Sicarius leveled at Rias was far too knowing for comfort. \"To watch you and guard our backs while we retrieve some rockets.\"\n\n\"Watch us?\" Tikaya asked innocently. \"Why?\"\n\nSicarius did not bother to look at her.\n\nAgarik strode through the door, and Tikaya lifted her head. Maybe this wouldn't be so bad.\n\nThen Ottotark and Bones clomped in. She tried not to let her chagrin show. Even before Ottotark spotted her, he wore a self-satisfied smirk. Bruises from his fight with Rias still mottled his face, and a bandage wrapped his head, but he appeared delighted at this new turn. Bones ignored Sicarius and Rias in favor of glaring at Tikaya, an angry jaw-clenched glare. No delight there. She guessed Ottotark had buzzed in his ear, letting him know who killed his brother.\n\nTikaya looked at Agarik, but he avoided her eyes.\n\n\"Evening, Admiral,\" Ottotark drawled. \"We'll relieve you of your weapons now.\" The smirk widened. \"Captain says you're back to prisoner status.\" He ambled up to the step below Rias and held out his hand.\n\nRias neither moved nor spoke.\n\nOttotark launched a punch at his belly. Rias blocked it and slammed his knee into the sergeant's diaphragm. Ottotark's heel slipped off the step, and he nearly tumbled to the landing, but he caught himself on the railing. Tikaya started to back away, to give Rias room to fight, but Sicarius stepped in. He held a knife she had not seen him draw in one hand and splayed the other against Rias's chest. His eyes were icy in warning.\n\nRias froze.\n\nOttotark found his balance. Fury contorted his face, and he snarled as he snatched Rias's weapons. He lifted the musket, as if he might slam the butt of it into Rias's head. Tikaya stepped down with a vague notion of grabbing the weapon, but Sicarius stopped Ottotark with a word.\n\n\"Enough.\"\n\nThe men dropped their arms. While punching Ottotark had won Rias nothing, it concerned Tikaya that none of that defiant spirit came out against the assassin. Had his last meeting with Sicarius disillusioned him so much that he would not move against the youth again? If so, that did not bode well for their success.\n\n\"Lead on, Admiral,\" Sicarius said. \"You two will get us into the weapons cache.\"\n\nAnd then what?\n\n* * * * *\n\nThe pair of kerosene lanterns the marines carried did little to push back the darkness in tunnels that had fallen silent. Eerily so. Tikaya began to feel as if their tiny group represented the only people left alive in the stygian passageways.\n\nShe and Rias walked side by side, leading the others. Ottotark and Bones kept their pistols trained on their backs. Agarik walked behind them, and Sicarius ghosted along in the shadows, rarely seen, rarely heard, always felt.\n\nTikaya checked symbols and peered down dark cross tunnels, hoping for inspiration. As soon as the marines had the weapons, they would likely shoot her. She wondered if Rias was expendable at this point too. At best, he could expect a trip back to Krychek. He would probably prefer death.\n\nRias caught her eye. \"Sorry.\" He spoke in Kyattese, which she had not realized he knew, though the words that followed proved he was far from fluent. \"I was engineer. Picked where explosives go. Had chance.\"\n\nThe slowness with which his words came out gave her time to puzzle over the meaning behind his choice of language. Bones, Agarik, and Ottotark probably knew no Kyattese, but hadn't Rias warned her that Sicarius did? He had certainly seemed to be reading that journal.\n\n\"No talking in codes.\" Ottotark jabbed Rias in the arm.\n\n\"Had chance,\" Rias repeated, brow furrowed as he groped for words. \"To drop roof on my people. End it all. Could not.\" He shook his head and sighed.\n\nTikaya glanced between Ottotark and Bones, probing the shadows for Sicarius. Yes, he was there and close enough to hear. Maybe Rias wanted Sicarius to know he had spared the team. Tikaya could not imagine that or anything else winning sympathy from the stony assassin. Maybe Rias just wanted her to know without opening himself for sneering commentary from Ottotark.\n\nTikaya gripped Rias's forearm. She could not condemn him for being unable to murder his own people, though it would have been convenient if he had arranged an accident for Sicarius when the men were catapulting over the chasm.\n\nIn the darkness ahead, four sets of symbols glowed, marking corners of an intersection. Rias tried to walk straight through it.\n\n\"Right,\" Sicarius said, voice cold.\n\n\"I can turn the lighting back on if we go this way,\" Rias said.\n\n\"Darkness is tactically preferable.\"\n\nTikaya shook her head; the kid didn't even sound human. She and Rias had no chance if they couldn't get the cube powered.\n\n\"What if the door on the weapons room won't open without the same power that operates the lighting?\" Tikaya asked.\n\n\"The lab doors are opening,\" Sicarius said.\n\nGood point. She sighed.\n\n\"But it'll be easier to see what we're doing in that weapons room if it's lit.\" Rias turned to face Sicarius. \"You were in Fort Deadend. You saw what happened to those people. Do you want to risk dropping something? A single broken vial could kill everyone in the cavern.\"\n\n\"Turn right,\" Sicarius said.\n\n\"Why are you so against turning the lights on?\" Tikaya asked.\n\n\"Because you two wish it.\" Sicarius jerked his chin to the right. \"Lead.\"\n\nIn other words, he did not trust them. No news there.\n\nA long moment passed before Rias headed right. Even as a prisoner with everything going wrong, he remained outwardly calm, and Tikaya reminded herself there was still time.\n\nLantern light played over piled rock ahead. This was a different tunnel than she had fled the cavern from, but it, too, had been partially blocked. They clambered over the waist-high rubble. When Agarik hopped down from the pile, Tikaya tried to catch his eye again. But he seemed to be deliberately avoiding them. In plotting to betray the marines, had Rias lost Agarik's respect?\n\nBoulders and shattered stalactites cluttered the cracked and uneven cavern floor. The illusion hiding the camp was gone, revealing a mess of smashed gear and broken crates. A pair of legs stuck out from a boulder, and Tikaya tore her gaze away. Above, darkness sheathed the rockets, though the number panel glowed, faintly illuminating the door area.\n\nSicarius detoured into the camp and grabbed a coil of rope and a bow. He plundered quivers, some still strapped to dead people, for arrows.\n\nTikaya waited to the side, not in a hurry to be helpful. Rias too, wandered into camp, though he looked less certain about what he sought. Inspiration, probably. Ottotark and Bones followed him, pistols cocked. The expression on Ottotark's bruised face promised he would love to use his.\n\nAgarik bumped Tikaya's shoulder as he came up to stand by her. He pointed his pistol at her, though his finger did not touch the trigger. While Sicarius collected arrows and the other two men guarded Rias, Agarik chanced a whisper.\n\n\"Ottotark and Bones are planning to kill you as soon as you open the door.\"\n\nIt wasn't unexpected, but hearing how little time she had unsettled her nonetheless.\n\n\"Does Rias have a plan?\" Agarik murmured.\n\nSicarius glanced their way. Fortunately, Agarik still had the pistol aimed at her.\n\nRias bent to pick something up. \"Ah, these might help.\"\n\nSicarius turned back to him as Rias hefted Lancecrest's goggles.\n\n\"We had a plan,\" Tikaya whispered back to Agarik. \"You people weren't a part of it.\"\n\n\"Rias will have a backup one,\" he said. \"If I act against the others to help you, I can't go back, or it's the end of my career, probably my life.\"\n\nShe feared they needed his help, but this was their cause, not his. As far as the marines were concerned, getting those weapons was a good thing. How could she ask Agarik to risk his life when it meant betraying everyone dear to him?\n\nBut he had already made his choice: \"Just wanted to be sure your offer is still good.\"\n\nShe wished she could hug him, but all she dared was a slight nod. \"Beach house,\" she whispered. \"As long as you want it.\"\n\nA slight smile stretched Agarik's lips. \"Surfer with the talented tongue?\"\n\n\"I'll do my best.\"\n\nA rifle boomed in a nearby tunnel.\n\n\"It's time, Admiral.\" Sicarius pinned Tikaya with his gaze as he strode to the base of the butte beneath the door. \"Bring the ordering for the numbers.\"\n\nTikaya wanted to bring a dagger to stick in his gut, but she kept the thought to herself and the sneer from her face as she walked over. Best not to give him any warning that she would make trouble.\n\nRias joined her, deliberately turning his back on Ottotark and Bones. \"What's your plan, Sicarius? There are only a few of us and a lot of weaponry up there. Getting it out will be a challenge.\"\n\n\"We don't need that many rockets to satisfy the emperor's needs,\" Sicarius said. \"If your smoke reveals a safe path, I'll climb up with one other person. We'll press in the correct code and lower several of the weapons to the floor. Once the captain has cleaned up the raiders, he'll be here, and we'll have plenty of men to transport the weapons out.\"\n\n\"Over the chasm?\" Rias asked.\n\n\"There are other ways out.\"\n\nHe sounded certain. An image came to mind: Sicarius gathering information by torturing captured raiders\u2014Parkonis. She winced.\n\n\"Who's going up with you?\" she asked. If Sicarius climbed to the top with Ottotark or Bones, that would leave her, Rias, and Agarik with only one hostile man to deal with.\n\n\"Starcrest,\" Sicarius said.\n\nTikaya fought back a curse. She was beginning to wonder if the assassin had telepathy training, Turgonian or not. Or maybe he just wanted Rias up there because he was expendable at that point. Her hackles rose. \"I'm not certain we have the puzzle right. You're not making Rias push the numbers for you. Have you seen what happens if someone gets it wrong? Instant incineration.\"\n\nSicarius lifted his stuffed quiver of arrows, and she blushed. Wrong conclusion. Of course, he intended to test it from below or he would not have bothered gathering the arrows.\n\nRias touched his index finger to his lips. To silence her? Or warn her not to irritate the assassin? She scowled at him. Somebody had to do something, and he was just going along with these brutes. He gazed back at her, steady and imperturbable.\n\nAbove their heads, the door panel pulsed three times.\n\n\"What does that mean?\" Bones asked.\n\n\"The numbers are about to change.\" Tikaya had no idea if that was true, but it sounded plausible, and if the marines feared they would need another translation, they might keep her alive a little longer.\n\n\"Give me the solution,\" Sicarius said.\n\nTikaya showed him the order of the numbers. He stared at it a moment, nodded, and nocked an arrow. Rias handed him the goggles. She expected Sicarius to regard them with suspicion, but he looked them over, then tried them. He lifted the bow and shot, untroubled by the bulky eyewear. The first arrow passed through one of the invisible beams and sizzled to ashes before it reached the target.\n\n\"Shit,\" Ottotark announced.\n\nUnflappable, Sicarius took a step to the side and loosed a second arrow. This one found the target, bumping one of the symbols a slot to the left. The arrow did no damage to the durable alien technology. It bounced away, where another beam incinerated it.\n\n\"No need to worry about trash collection here,\" Bones muttered.\n\nWhile Sicarius continued moving the numbers around, she gauged the distance to the corridor they had exited, the corridor that eventually led to that panel Rias wanted to visit. She would not consider running with Sicarius on the ground, but if he was busy climbing, the odds improved. Ottotark and Bones were no doubt proficient with their firearms, but she judged them far more fallible than the assassin. If she just had a distraction....\n\nA final arrow clattered off the panel after shifting the last number into place. A chime sounded and the door slid open. It was hard to feel triumphant given the circumstances.\n\nSicarius removed the goggles and returned them to Rias with a single nod.\n\n\"Light a smoke bomb,\" Sicarius said, apparently unwilling to trust the ones Rias had given him until he had seen them used.\n\nRias held out the goggles, glancing around as if looking for a place to put them, then shrugged and strapped them around his head. He pushed the lenses above his eyes so they were not in the way as he lit one of the globes. He laid it on the floor at the base of the butte. Soon, plumes of grayish blue smoke wafted into the air. They diffused quickly, spreading over a greater area than the haze from a pistol firing. As Tikaya had seen before, an asymmetrical pattern of white beams grew visible in the smoke.\n\n\"Those kill you if they touch you?\" Ottotark asked.\n\n\"Yes,\" Tikaya said.\n\n\"Glad I'm not trying to climb past them.\"\n\nSicarius gave him a cool stare, then laid down the bow and jogged to the bottom. Tikaya eyed the weapon. It would not take many steps to reach it.\n\n\"Come, Admiral,\" Sicarius said.\n\nRias strode to Tikaya first. He gripped her hands. \"Whatever happens, you've been the light that's driven away the darkness in my life.\" He did nothing so obvious as putting special emphasis on the word light, but she understood anyway: he wanted her to try for the panel.\n\nShe squeezed his hands. \"I love you too. Be careful.\"\n\nOttotark groaned. \"Can we shoot them now?\"\n\n\"Wait until we have the weapons out. If the symbols change, we'll need her again.\" Sicarius handed his two globes to Bones. \"Light one of these if the smoke dies out before we reach the top.\"\n\nRias widened his eyes slightly before releasing Tikaya's hands and heading for the base of the butte. Tikaya tried to guess at the meaning in that look; had he done something with the other smoke bombs? The current haze tickled her nose and teared her eyes a bit, but had no significant side effects.\n\nOverhead, the door slid shut. It had only stayed open a couple minutes before locking again. She wondered what happened if someone was on the inside when it closed.\n\nSicarius was already ten feet up the wall. Though natural, with protrusions and crevasses, it did not look like an easy ascent, even without the beams. They touched it in myriad places, and no easy routes awaited the climbers.\n\nAfter considering the rock face, Rias removed his rucksack. Tikaya tensed. No, no, if he did not take the cube with him, how would they get it up there? He met her eyes and shook his head faintly. She grimaced. He must not think there was enough space between beams to climb with the rucksack on his back. After watching Sicarius, she reluctantly agreed. As soon as the assassin reached the level of the beams, he had to start sidestepping, twisting and contorting his body. For every two feet he ascended, he ended up dropping a foot somewhere else.\n\nRias started up, and worry gnawed at her before he even reached the beams. He was taller and broader\u2014and older\u2014than the agile assassin. Dodging those beams would prove a difficult feat. Not impossible, she hoped.\n\nTikaya eased toward the bow. Agarik remained near her, and he shuffled forward too. They froze when Ottotark eyed them.\n\n\"Agarik,\" he said. \"Go hold the lantern for Bones in case he needs to light another smoke thing.\"\n\nOttotark slapped his pistol across his palm as he strode over to stand by Tikaya. Agarik glanced at her. She nodded infinitesimally. Better to comply now and wait until Agarik's side-switching might accomplish something.\n\nLight pulsed at the door. The symbols changed.\n\n\"Is that what I think it is?\" Rias asked, cheek pressed to the rock, a laser less than an inch from his eyebrow.\n\n\"Yes.\" Tikaya slipped the sphere out of her pocket.\n\nOttotark grabbed her arm, pistol digging into her ribcage. \"What's that?\"\n\n\"Not a weapon,\" she said, then raised her voice for Rias. \"Give me a minute, and I'll translate the new numbers. I know some of them.\"\n\n\"How often does it change?\" Bones asked.\n\nOnce a day, she guessed. \"At random,\" she said.\n\nOttotark stepped back, startled when the display flared to life. She manipulated it to find the number symbols.\n\n\"You want me to read them to you?\" Tikaya called. \"Or try to solve the problem and shoot the numbers into place from here?\" She had to try, though she doubted Sicarius would be foolish enough to let her have a bow much less authorize her shooting it in his direction. He would probably laugh and say nice try.\n\n\"Give Starcrest the numbers,\" Sicarius said with no sense of humor or annoyance. \"He'll figure it out and he'll push them.\"\n\nRias grunted. Pebbles clattered down the cliff face. One bounced into a beam's path and was vaporized. The dwindling smoke made the sweat beading his forehead visible. Be careful, Tikaya urged.\n\nThe pistol bumped her ribs.\n\n\"Get to work,\" Ottotark snapped.\n\n\"I've got the numbers,\" she said a moment later and read them aloud to the men.\n\nShe hoped Rias would wait until he reached the top, or some place safe, to mull over the solution. He was about halfway up now. In a couple feet, Sicarius would reach the ledge.\n\n\"I could use more smoke,\" Rias said.\n\nBones and Agarik lit one of the globes. Tikaya checked on Rias, hoping he would wait until the smoke thickened before trying to climb farther. She caught him pulling his shirt over his nose and tugging the goggles over his eyes.\n\nAs soon as smoke curled from the globe, Bones and Agarik dropped it and stumbled back. They threw their arms over their faces, gagging.\n\nTikaya sucked in a deep breath and held it. Even then, she still caught the first whiff as the smoke disseminated. More pungent than rotten eggs, it invaded her nostrils and teared her eyes. Ottotark leaned forward, grabbing his nose.\n\nThis was her chance.\n\nShe drew back her arm and slammed the sphere into his temple. It was not big, but it was blunt and solid. He reeled sideways and stumbled to the ground.\n\n\"My eyes,\" Bones shouted, then retched.\n\nAgarik clutched at his belly and vomited.\n\nNo time to check on Rias or Sicarius. Tikaya lunged for the bow and quiver, grabbed them, and wheeled. Agarik had dropped the lantern. She snatched it as well. By then, her lungs burned, demanding air, but she sprinted for the tunnel.\n\nTears blurred her vision, and she tripped over a rock. She sprawled, almost losing the bow, and her breath whooshed out. Before she could catch herself, she sucked in a mouthful of air. Distance stole some of the potency from the smoke, but it still made her gag. She staggered to her feet, forced her legs into motion, and clambered over the rubble pile and into the tunnel before retching.\n\nAs soon as she could, she raced toward the intersection. The air was clearer here, and she sucked it in. She rounded the corner, hoping to run straight to the panel without encountering a maze of tunnels to guess at. A T-section came first. She lifted the lantern and peered both ways. There. A faint crimson glow in the distance.\n\nTikaya sprinted to the panel, a column of symbols and five vertical lines that glowed solid blue.\n\nShouts echoed from the cavern. She shuttered the lantern and set it down, plunging the tunnel in darkness. The men would not be distracted for long, and the light would make her an easy target. She could only hope Sicarius would not take his irritation out on Rias, who she had left in a vulnerable position. Second doubts assailed her. She should have stayed and used the bow on the men, shot the cursed assassin, not run away. But, no, the lights were what Rias wanted, and her eyes had been too tear-wracked to aim at anything anyway.\n\nShe examined the symbols. Not all were familiar, and there were more than she expected, but she understood the gist. Lighting, power levels, and water controls. Right spot, but what to touch?\n\nIn the still tunnel, she felt her rapid heartbeat reverberating through her body. She started to reach for the sphere, but feared she had no time for research. Rias had guessed. She would have to as well.\n\nBoots pounded into the tunnels. The marines would know right where she had gone.\n\nTikaya slid a finger across one of the horizontal stripes labeled with illumination. Nothing. There was no switch or knob. She slid her finger the other way. Nothing. She waved her hand before it as she had seen Rias do once to close a door.\n\nThe stripe pulsed once, and something thunked inside the wall. Had that done it? The lighting did not come on, and she waved her hand before the other stripes. More thunks, and a faint hum from behind the wall.\n\nThe footsteps hammered closer. She grabbed the bow, nocked an arrow, and flattened herself against the wall. The corridor offered no cover, but she could not run until she knew if her hand-waving had accomplished the goal. Besides, darkness stretched behind her, and she did not know if more tunnels lay that way or only a dead end.\n\nThe footsteps stopped near the intersection, and lantern light bobbed on the wall. She drew the bow, but no one burst into sight.\n\nMore footsteps, these ones softer and slower, reached her ear. She tensed. They were coming from behind her somewhere. Trap. And she had only the darkness to hide in.\n\nThen the lights blinked on. It happened so abruptly, she squinted, half-blinded. She almost missed the movement ahead\u2014someone slipping around the corner and dropping to a knee.\n\nTikaya loosed an arrow without waiting for her vision to clear. As soon as it flew free, she dropped to the floor. A pistol cracked.\n\nShe rolled to the side, cursing herself for getting caught in such a bad spot. She scrabbled for another arrow.\n\n\"Tikaya, this way,\" Agarik urged, not from behind but from ahead.\n\nShe cursed. Had she just shot at him?\n\nBy the time she lunged to her feet, her eyes adjusted enough to see the intersection. Bones lay on his belly, blood pooling beneath his head. Agarik waved for her to hurry.\n\n\"What the\u2014\" Ottotark blurted, a hundred meters or more down the tunnel behind her.\n\nTikaya sprinted for Agarik. His pistol, not her bow, had felled the doctor. He pulled her around the corner as another shot fired. The pistol ball clanged off the corner and ricochetted down the tunnel.\n\n\"Traitor!\" Ottotark screamed.\n\n\"No time to reload,\" Agarik said as they ran toward the intersection that could take them back to the cavern. \"You'll have to shoot if he catches up.\"\n\n\"Understood,\" Tikaya said grimly.\n\nShe glanced back to see if Ottotark had rounded the corner yet and missed the reason Agarik skidded to a stop, cursing. He flung his arm out to halt her as well.\n\nA cube hovered in the intersection ahead.\n\nShe slammed a fist against her thigh. She should have known\u2014the whole reason for turning the lighting back on had been to power one of the cubes. With the mess from the explosives, all of them would probably respond.\n\n\"Maybe it'll go on to the cavern,\" she whispered.\n\nIt rotated, and its crimson orifice came into view.\n\n\"Back, back.\" Agarik spun, taking her with him.\n\nTikaya ran at his side. They would have to take their chances with Ottotark.\n\n\"Zag,\" she barked on a hunch.\n\nShe pushed Agarik one way and ducked against the opposite wall. A red beam seared the air between them.\n\nAs soon as it faded, they sprinted off again. Tikaya nocked the bow as she ran. Any second\u2014\n\nOttotark lunged around the corner, pistol pointed at them. She fired without slowing, and it threw off her aim. The arrow skimmed past his head, stirring his hair, but doing no damage.\n\nHe must have seen the cube coming, for he looked between them and cursed before choosing a target. Tikaya.\n\nAgarik hurled a knife at Ottotark. It bought them a second as the sergeant dodged the projectile. She yanked another arrow from her quiver, but Ottotark recovered before she had it nocked.\n\nHe fired. There was no room to dodge, no time to duck. Agarik leaped in front of her, grunting as the pistol ball slammed into him.\n\n\"No!\" Tikaya cried.\n\nShe jumped around him, took the split second to aim, and shot. The arrow spun into Ottotark's eye.\n\nShe dropped the bow and whirled back to Agarik, catching him as he slumped. His hand gripped his chest, and pain ravaged his face. The cube continued its inexorable advance, but she tried to pull him down the aisle.\n\n\"Leave me.\" Blood spilled from his lips. \"Help Rias.\"\n\n\"It'll get you,\" she choked, refusing to accept the inevitable.\n\n\"Yes,\" Agarik rasped. \"Give you...time.\"\n\nOut of the corner of her eye, she saw it: that cursed glow intensified. She stumbled away as the beam fired. It burned into Agarik and started its deadly work.\n\n\"Go,\" he gasped.\n\nTears blurring her vision again, Tikaya grabbed the bow and sped away. She leaped over Bones's body and kicked Ottotark on the way past. She should have lit that bastard on fire when she had the chance. Agarik's death was her fault.\n\nShe found the corridor Ottotark had used to circle around behind her and cut over toward the cavern. It would take time for the cube to clear away all three bodies, but she recalled the multiple units in that cavern closet and knew others would be about.\n\nTikaya slowed at the cavern entrance and tried to peer out without revealing herself. No shadows remained, though, and the assassin was already looking down at her when she spotted him. He crouched on the ledge, his shirt off and tied about his nose and mouth. His back was to the closed door. None of the symbols had been moved. He dropped his head to focus on the floor at his feet\u2014or something on it. Paper and pencil, she guessed from his movements. He was trying to solve the new Skiltar Square.\n\nBut where was Rias? Smoke still wafted from the noxious globe, but it had thinned, and she would have seen him on the cliff if he remained there. His rucksack lay on the floor where he had left it. Dread crept into her as she continued to search the area without spotting him. If he had fallen, the beams could have incinerated him before he reached the ground.\n\nTwo cubes worked in the cavern, eating away the piles of rubble. They reminded her of the one in the tunnels behind her. As soon as it finished with the bodies\u2014she forced herself not to dwell on Agarik, not now\u2014it would head this way.\n\nTikaya eased out of the tunnel and kept her back to the wall. Sicarius kept track of her as he figured. Her hand ached where she gripped the bow. If Sicarius had killed Rias, he was not getting off the cliff. Agile or not, he could not dodge arrows while he climbed down past those lasers. She removed an arrow and nocked it with steady hands. Cold controlled anger made her movements sure, free of fear. Even if he had not killed Rias, he was the Turgonian emperor's assassin, someone who had tried to murder her president. The world would be better off with him dead.\n\nShe drew the bow. No sense of alarm widened Sicarius's eyes, but he stood. Balanced on the balls of his feet, arms relaxed, he appeared unconcerned by the weapon pointed at him. Even on that small ledge, he could probably dodge an arrow. But if she bumped one of the numbers, and he could not solve the problem on time, he would either have to climb down, where she could shoot him in the back, or he would be incinerated.\n\nYes, then why hadn't she fired yet?\n\nKilling Ottotark in self-defense was one thing; shooting someone in cold blood... Could she do it?\n\nMotion across the cavern saved her from having to answer the question. Rias burst from a tunnel, diving and rolling as a red beam lanced the air over him.\n\n\"Rias!\" she shouted.\n\nHe scrambled to his feet and zigzagged toward the butte. He chopped a wave her direction, but lifted his head to shout a stream of numbers at the assassin.\n\nThe solution to the door. Sicarius's head tilted, and he gazed upward\u2014calculating. Not trusting enough to enter them without checking for himself. And why should he be? Rias had no reason to help, to get the assassin inside with the weapons. What was he doing?\n\n\"Is that...\" She thrust her bow toward the cube chasing him.\n\nRias dove over a fallen stalactite. A beam struck the rubble, and rock and dust flew. He came up, racing toward the camp this time, and a wild grin lit his face. He pressed a finger to his lips and mouthed something. Distract it?\n\nSicarius was punching in the door code. Tikaya cut toward the cube from the side. As soon as she was closer to it than Rias, it rotated toward her. She ran toward a pile of rubble and ducked behind it without any of the grace Rias had managed. Her shoulder clunked against a boulder with a painful jar. She peeped around the edge.\n\nRias reached his rucksack and tore open the lid. He dug out the cube, still inert, the lid still off. So the one following him\u2014her now\u2014was an extra.\n\nShe circled the pile to avoid its approach. A beam bit into the rubble, and shards of stone rained upon her.\n\nOverhead, the door slid open. Rias thumbed something inside his cube. Sicarius entered the chamber. Rias hurled the cube toward the top of the butte.\n\nThe one at ground-level was nearing Tikaya and she had to sprint to the next pile of debris. She glanced upward as she ran, fearing pieces would fly out of the open cube or the beams would incinerate it, but it reached the top unharmed. It caromed off the transparent wall, and Tikaya thought it would bounce away from the butte, but it righted itself. Hovering in the air, the cube approached the door.\n\nInside, Sicarius whirled at the noise and dropped into a crouch.\n\n\"Get out!\" Rias called.\n\nHe dug a familiar jar out of his rucksack and raced at the cube stalking Tikaya around the rubble pile. She let it get dangerously close to keep it occupied.\n\nShe risked another glance upward. If the modified cube started destroying rockets while the door was open, they would all be dead in seconds. But it focused on Sicarius first.\n\nA beam shot out. Tikaya held her breath. Sicarius ducked, and the beam splashed against the wall without hitting a rocket.\n\nHer own cube almost skewered her when her heel caught a rock, and she ripped her attention back to the closer danger. Rias scrambled over the pile from the side and splattered the air with his concoction. He and Tikaya split and raced away while the cube was deciding where to focus its beam.\n\nAs they met on the other side of the pile, an earsplitting shriek echoed from all around. The weapons chamber door started to shut. Sicarius dove under the cube and rolled through the entrance. The door sealed. His momentum took him to the edge, and Tikaya thought he would fly over the side, but he twisted and caught the overhang.\n\nSmoke rose behind Tikaya. Their cube was out of action. She leaned forward, willing the one caught inside the chamber to do what they wanted.\n\nRias gripped her hand. He had lost the crazy grin and stared at the chamber, as if he could will the cube to work with the intensity of his gaze.\n\nThen the first beam shot out. Tikaya could not see the target from where they stood, but a green haze filled the air in the weapons chamber. A bone-shaking rumble emanated from the butte\u2014the ventilation system firing up. Smoke whirled and rose, drawn into the ducts at the top.\n\nSicarius hung on the cliff, his chin over the edge, staring at the display. A blue gas joined the green, mingling and merging as it too was sucked upward. Tikaya hoped some sort of filter existed, so everything in the mountains at the other end of that vent did not die.\n\n\"It's working.\" Rias smiled and wrapped her in a hug.\n\nShe could not bring herself to return the smile, not with Agarik's death haunting her thoughts, but she did return the embrace. She smashed her face into his shoulder and hugged him with all her strength. And then released him. She would cry later, when they were safe.\n\nA pebble clattered down the cliff. Sicarius was climbing down.\n\n\"We better get out of here,\" she whispered.\n\nRias nodded, grabbed his rucksack, and jogged to the camp. She followed him, but when he started gathering food and gear, she shifted from foot to foot.\n\n\"Do we have time for that?\" She jerked her head toward Sicarius. Even with the beams to navigate, his progress going down was faster than it had been climbing up. \"He's going to be irate.\"\n\n\"I know,\" Rias said, but he continued his preparations, unhurried. Black powder tins and ammo pouches went into his rucksack. \"It's weeks to get across the mountains and back to civilization. We'll need supplies to survive the trek.\"\n\n\"I need to find Parkonis and make sure he can get away from the Turgonians,\" she said. \"And, Rias? Agarik didn't make it.\"\n\nHis jaw tightened, but he kept himself to a curt nod.\n\n\"He saved my life,\" Tikaya said, \"so I could come back to help you.\"\n\nRias grabbed a second rucksack and started filling it for her. She glanced at Sicarius.\n\n\"He's almost down,\" she murmured. \"If we want to take him out, this may be our last chance.\" That weeks-long trek would be arduous enough without an assassin hounding them. \"If he comes after us...after you... We can't waste the gift Agarik gave us.\"\n\nRias finished packing. \"We won't.\"\n\nSicarius jumped the last ten feet, landing lightly. Even in defeat, that same stony mask hid his thoughts, his feelings.\n\nRias handed Tikaya her pack and a fresh quiver of arrows for her bow. He picked up a rifle but did not bother to load it.\n\n\"Ready.\" He pointed to a tunnel, a tunnel they would have to walk past the assassin to reach.\n\nMetal rang softly as Sicarius pulled a dagger from his belt and stepped into their path.\n\nTikaya grabbed Rias's elbow when he did not slow. \"Are you mad?\"\n\nRias removed her hand gently and strode toward the tunnel. Tikaya nocked an arrow, but did not fully draw the bow. Rias had to know what he was doing. Didn't he? Shaking her head, she followed him.\n\nSicarius's grip tightened around the dagger hilt. \"You never intended to help. You had the chance to redeem yourself, but you betrayed the emperor again.\"\n\nRias stopped a few feet from him. \"Yes.\"\n\nSweat dripped down the sides of Sicarius's dust-streaked face and dampened his pale hair. For the first time, he seemed uncertain, frazzled. Young. \"Why?\"\n\n\"I couldn't let him have those weapons. There's no honor in destroying one's enemies like that. Nobody should have that kind of power.\"\n\n\"That wasn't for you to decide.\"\n\n\"Yes, it was. Sometimes the only person capable of such a decision is someone who stands on the outside, someone who has nothing left to lose, nothing to gain, by the outcome.\"\n\n\"Nothing to gain?\" Sicarius asked. \"You could have had your life back, your lands.\" The faintest hint of longing entered his voice. \"You could have been a hero again.\"\n\nTikaya lowered the bow as it dawned on her that Sicarius had yet to point the dagger at Rias. Not here and not at any point since he had shown up.\n\n\"That's never been a goal of mine,\" Rias said. \"The definition of a hero changes depending on the needs of the person with the dictionary. And of late I've become more aware how much being a hero to the empire means being a war criminal to the rest of the world.\" Rias smiled sadly at Tikaya before turning back to Sicarius. \"For twenty years, I served Turgonia. I think it's time now to see if I can serve the world.\"\n\n\"I see,\" Sicarius said, and Tikaya had a hard time telling if he truly did or not.\n\nRias unsheathed a dagger, flipped it in his hand, and held it hilt-first toward Sicarius. It was utterly black, one of the tools they had gathered for working on the cubes. The keen edge would probably never dull.\n\nSicarius considered it for a long moment before accepting it. Peace offering, Tikaya guessed.\n\n\"Are you returning with Bocrest and the others?\" Rias asked.\n\n\"Yes,\" Sicarius said.\n\n\"Parkonis is no threat to the empire. Will you see to it that he escapes when the ship docks in Port Sakrent?\"\n\nTikaya's eyes widened, not in surprise that Rias would care enough to make the request, but that he was asking Sicarius for a favor. After they had defeated him.\n\n\"If that is your wish,\" Sicarius said, stunning Tikaya even more.\n\nThe kid was going to be in trouble already for not completing his mission, for letting Rias go. Earlier, she had been thinking of shooting him, but now she found herself hoping the emperor had invested too much in his education to dispose of him over a failure.\n\n\"Thank you,\" Rias said. \"And one last request: will you relay a message to the emperor for me?\"\n\nSicarius tilted his head.\n\n\"Though I may never see them again, I have family and friends in Turgonia. It is not my intention to make trouble for the empire. But I want him to know that if he bothers them or\u2014\" Rias angled toward Tikaya, directing Sicarius's eyes to her, \"\u2014if he sends anyone after her or her family, I will become trouble.\"\n\nTikaya thought she detected bleakness in the assassin's usual mask. Yes, all Fleet Admiral Starcrest would have to do to make the emperor's life unpleasant would be to show up on the Nurian Chief's threshold, offering to help war against his former nation.\n\n\"I will tell him,\" Sicarius said.\n\n\"Thank you,\" Rias said again, and he put a hand on Sicarius's shoulder. \"You would have made a good officer.\"\n\n\"Not the road fate paved for me,\" Sicarius said, but something in the soft exhale that followed his words made Tikaya wonder if he wished things were different.\n\n# EPILOGUE\n\nAs the light faded from the mountains, Rias placed the last block of snow on the top of the igloo. There was no wood to make a fire, though a kerosene lantern provided a pool of light.\n\nHe stepped back, brushed off his gloves, and quirked an eyebrow. \"What do you think?\"\n\nIt had taken two days to find a \"back door\" out of the tunnels, and it had brought them out above the tree line with only a couple hours of daylight remaining. Icy wind gusted along the ridge, and the first stars glittered in the clear sky. The night would be long and cold, very cold. Though she had helped build it, Tikaya eyed the igloo dubiously.\n\n\"You're sure we won't freeze to death?\" she asked.\n\nThey had enough gear, Rias assured her, to make it out of the mountains and to the nearest town. Still, the lack of firewood and the plummeting temperature made her nervous for this first night.\n\nRias flattened his hand on his chest. \"Are you questioning my engineering skills?\"\n\n\"No, I'm sure it's structurally stable. I'm questioning the wisdom of sleeping inside a box of snow.\"\n\nHe chuckled, ambled over, and kissed her on the forehead. \"Snow is insulating, my dear. Once our body heat warms up the igloo, you'll be able to sleep naked if you want.\"\n\n\"Sleep naked, huh?\"\n\nHis eyes twinkled. \"The sleeping part is optional.\"\n\nA distant boom echoed through the mountains. The marines had apparently found a different way out and were following their orders to seal the tunnels. She hoped they were treating Parkonis well. Leaving him felt like a betrayal, but the reality was he would probably make it home sooner and less eventfully than she. And though Sicarius ranked at the top of her list of People She Never Wanted to Meet Again, Rias trusted him to keep his word, and she trusted Rias.\n\nShe wished she had been able to keep her word to Agarik. Leaving him there to be incinerated by that machine... Another betrayal. She wondered what the marines would tell his family. If he even had family. It shamed her that a man had given his life to save hers and she knew so little about him.\n\nRias shoved their weapons and gear through the igloo's low entrance, then belly-crawled after. Tikaya grabbed the lantern and managed to get snow down her pants following him. She hissed in frustration as she dug it out in the tiny confines. She could not wait to walk again on a tropical beach.\n\nInside, there was room enough to lie down if one did not straighten too many limbs. Rias shoved a rucksack in front of the tunnel, leaving them entombed with only a few air holes. The snowy walls gleamed next to the lantern. The single flame brightened the space surprisingly well.\n\nTikaya lay down, head propped against her pack. \"Not bad.\"\n\n\"Easy,\" Rias said, \"your lavish praise will inflate my ego.\"\n\nTikaya pulled him down beside her, hoping to shake the gloomy mood that shrouded her. \"I'd rather inflate other things.\"\n\n\"I'm always amenable to that.\"\n\nShe slid her arms inside his parka. If body heat was the way to warm up an igloo, then she was all for hastening that process along.\n\nSometime later, and with fewer clothes on, she asked, \"How did you know Sicarius would let us go?\"\n\n\"That was always my plan,\" Rias murmured, his lips brushing her ear.\n\nTikaya chuckled. She lay snuggled in his arms. \"And how did you know he would go along with your plan? Especially after we betrayed him and destroyed the weapons before his eyes.\"\n\n\"I read him.\"\n\n\"You read him? The kid emoted less than a rock.\"\n\nFor a moment, Rias did not answer, and she wondered if she had offended him. But, as she formed an apology, he spoke, his tone somewhere between amusement and bemusement.\n\n\"What do you think a military strategist does?\"\n\nAn image filled her mind: Rias, leaning over a map-filled table surrounded by his officers. They pushed miniature battleships back and forth while debating numbers of troops, cannons, practitioners, and the like. Then she realized those battleships and troops were commanded by people. People he had never met face-to-face but that he had to analyze and outthink. She thought of the time Rias had spent working with Sicarius on the trebuchet, talking to him when no one else did, treating him like a promising young officer. As worldly and educated as the assassin seemed, he was still a seventeen-year-old boy, one who had doubtlessly grown up hearing of Rias's exploits. Whether he showed it or not, Sicarius must have felt a little hero worship for the distinguished veteran. Tikaya smirked. All that time, she had thought Rias was succumbing to his fate. He must have seen Sicarius as the one person he could not escape or physically force his way past, and the one person he needed to befriend.\n\n\"I see now,\" Tikaya said. \"I was being obtuse. Military strategist isn't a career option where I grew up.\"\n\n\"Sounds like a lovely place.\"\n\n\"Yes...about that.\" She had told Rias she would follow him anywhere, and she would, but\u2014\n\n\"You need to go home and let your family know you're safe,\" he said.\n\n\"Just for a week or two. Do you want to come or...\" When she had learned his name, she told him he would never be welcome on her island, and she suspected that true, at least not until people's memories of the war faded, but if he had saved the president from assassination, surely Rias could finagle visitation rights. The president owed her too. He had said as much after her decryption work proved so valuable. But maybe she was being presumptuous. \"Or do you need to visit your own family? Let them know you're alive?\"\n\n\"I'll write them a letter from some distant port. The emperor will be irked when he finds out about this, and I don't think it'd be auspicious for my life expectancy to linger on imperial soil.\"\n\nYes, the Turgonian emperor had never come across as the magnanimous type in the orders she decrypted.\n\n\"Besides...\" Rias found her hand and linked his fingers with hers. \"I have little interest in going home. I seem to have fallen in love with an exotic foreigner, and I have the urge to follow her wherever she wants to go.\"\n\nFlutters stirred in her belly. She had hoped that would be his response. \"Well, we'll have the sphere to work on, and as far as places to go, I know of all sorts of ruins around the world with unsolved puzzles and mathematical oddities. Of course, many of them are surrounded by dangerous aborigines, crocodile-filled swamps, and pistol-toting relic raiders, all ready to end your life if you let your guard down for an instant.\"\n\n\"My dear,\" Rias breathed, \"if you're trying to seduce me...it's working.\"\n\nTHE END\n\n# Afterword\n\nThank you for giving Encrypted a read. If you enjoyed the adventure, you may want to check out the sequel, Decrypted, and a short story that takes place between the two novels, \"Enigma.\" If you want to hear about giveaways, book extras, and new stories I'm publishing, please sign up for my newsletter.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nTHIS   \nBLESSED EARTH\n\n_A Year in the Life of an_\n\n_American Family Farm_\n\nTED GENOWAYS\n\nW. W. NORTON & COMPANY\n\nINDEPENDENT PUBLISHERS SINCE 1923\n\nNEW YORK | LONDON\nNow farming became industry . . .\n\n\u2014JOHN STEINBECK,   \n_The Grapes of Wrath_\n_Contents_\n\nPROLOGUE: Readying the Bin\n\nPART ONE: Bringing in the Beans\n\nINTERLUDE: Working Cows\n\nPART TWO: The Homeplace\n\nINTERLUDE: Branding Calves\n\nPART THREE: Seeds of Change\n\nINTERLUDE: The Poker Run\n\nPART FOUR: Irrigation Nation\n\nINTERLUDE: The Tri- State\n\nEPILOGUE: Welcome News\n\n_Acknowledgments_\n\n_Selected Bibliography_\n\n_Illustrations_\nPROLOGUE\n\nREADYING THE BIN\n\n_September 2014_\n\nThe last yellow hues of evening were fading to blue. A bright circle of sun blazing through an uncapped vent in the roof had climbed the corrugated side of the grain bin and smoldered out like an ember. Now the only remaining light, there in the dark interior, was the flame of Kyle Galloway's cutting torch\u2014first a flickering orange tongue, then as he adjusted the hissing valve, a bright cerulean cone of fire. Sparks showered and bounced across the concrete slab as he cut another steel plank down to size.\n\nStill in his twenties, Kyle was broad-chested, with dark, buzz-cut hair and a gentle round face and glasses that gave him a kindly, almost child-like air. But he moved through his work with quiet authority, inching across the width of the plank until the unwanted end clanked on the concrete. Kyle paused a beat to let the fresh cut cool, then he laid the plank into place, the next piece in a floor now nearly half done, and stomped it again and again with his heel. Each boot fall roared loud as thunder, shaking the whole structure, until the section snapped tight. Kyle stepped up, bouncing to test its soundness. Then, satisfied, he leaned through the open portal door.\n\nOutside, in the gathering dark, Dave Tyson, the farm's hired man, handed over the next section of floor. Dave wore a ball cap and round glasses of his own. Easily twenty years Kyle's senior, he seemed to be making a conscious show of keeping pace. The sleeves of his flannel shirt were torn off, and the last of the summer heat formed a film of sweat on his pale biceps, the muscles tensing and flexing as he hoisted the flooring. Salvaged from an older bin on a distant part of the acreage, these planks had been laid up in a defunct hog shed for years. But nothing ever goes wholly to waste on the farm. Old tractors and field trucks, hay rakes and a faded harvester all stood tucked away in leaning barns or grown over by tall grass at the edge of a shelterbelt, dormant but awaiting purpose. For this flooring, it was finally here.\n\nAt the end of June 2014, crop counters at the U.S. Department of Agriculture had released estimates that the summer's mild weather and increased rainfall would mean a much-higher-than-expected yield. It's the kind of prediction that sounds like good news, but for farmers it meant an upcoming glut of grain. One report projected that the national soybean yield would be 3.3 million acres higher than expected at planting\u2014not just cutting into profits but actually outstripping domestic demand. The outlook for corn was little better. Though acres planted to feed corn were down, the stockpiled reserves from the record yield of 2013 remained at an all-time high. On news of such unprecedented surpluses, corn and soybean futures nosedived among speculators at the Chicago Board of Trade. After several years of riding high on $8 corn, farmers watched prices fall to $3.50 per bushel. Soybeans, which had topped $17, dropped below $10.\n\nThen in August, in the weeks just before harvest, a series of downpours moved across the middle of Nebraska, dumping two and three inches of rain at a time, pushing back the start date for bringing in the beans. While Kyle and Dave waited anxiously for fields to dry enough that a harvester wouldn't mire in the mud, projections for the national yield kept going up and up. Every day that the crops stayed out, unharvested and undelivered to the co-op, the per-bushel prices went lower. For weeks in nearby towns, at the firehouse in Hordville and the bar in Polk, even at the Iron Skillet at the truck stop south of York, the talk had turned to the Jews in Chicago making bank on the backs of farmers. Kyle didn't have time for that. He rolled his eyes and shook his head at old-timers who were drinking coffee and bullshitting instead of doing what they could to cut their losses.\n\nKyle didn't own this acreage, just south of the Platte River in east-central Nebraska, but his girlfriend, Meghan Hammond, was part of the sixth generation of her family to live and farm in this area\u2014and Meghan and Kyle were starting to talk about a wedding in the next year. Once that happened, Meghan's father, Rick, figured it would be time to give them a stronger hand in the operation. Whenever people asked Rick what he raised on his farm, he always responded, \"Corn and cattle and kids.\" By the end of the year, if all went as planned, Meghan and Kyle wouldn't be kids anymore.\n\nBut handing over control of the operation\u2014\"succession,\" as it's known\u2014is one of the hardest times in the life of any family farm. The older generation always struggles to let go of the reins, to trust their kids to carry forward a long and fragile tradition. The younger generation bridles against the meddling and second-guessing and feels the double weight of scrutiny and doubt in every misstep. Many farms don't officially change hands until the older generation dies, leaving the rising generation beholden to their parents until they're well into adulthood and often into their fifties or sixties. It breeds resentment and too often divides generations or turns brothers and sisters against each other. It is so stressful and legally complex that there are psychiatrists and attorneys alike who build whole practices around helping families navigate the process. But on the Hammond farm, it was even more complicated than that.\n\nThe collapse of commodities prices was forcing the family to wrestle with how best to run the farm in the future. Years ago, Rick had tried raising organic corn and direct marketing his grass-fed black Angus cattle, but those efforts had largely failed. Now, he was not only planting crops resistant to a range of herbicides for commercial sale but actually growing genetically modified seed corn for DuPont Pioneer to sell to other farmers. He said he couldn't say no to the guaranteed rates that Pioneer pays those who raise their seed stocks. Over the years, that increased income had allowed Rick to expand his operation, but now he'd come to rely on it. Without the premium from Pioneer, a down market, like the one his family was facing in the coming year, would place the whole farm at risk. And even Pioneer was not impervious to the vagaries of the Board of Trade. With per-bushel prices so low, the company was worried that there would be less demand for corn seed in 2015. So Rick had been informed that Pioneer intended to reduce the number of acres they were planting\u2014and, with it, reduce payments to the farmers who planted for them. The company just didn't know where it would cut yet or by how much. \"The seed corn is about half of what we raise,\" he told me. \"If they gave us all that we try to shoot for, it's more like three-fourths.\"\n\nWithout that money coming in, Rick knew that he couldn't afford to buy new machinery for the year, couldn't afford to replace his center-pivot irrigation system. He needed to have a full year of paying down those loans, not incurring new debts. He needed a year in which everything went right\u2014crops came in early and healthy, prices rebounded, equipment held out. And if he was forced to take out more loans, he needed interest rates to stay low. \"We're trying to tighten our belt and get rid of machine debt, pivot debt, and then get it down to land debt,\" he told me.\n\nAnd that penny-pinching would be all the more important as the family faced the still-uncertain fate of the proposed Keystone XL. That pipeline project, intended to carry heavy crude pumped from the Alberta tar sands in Canada nearly 2,000 miles to Port Arthur and Houston on the Texas Gulf Coast, was slated to cut right through major portions of the family cropland. Meghan and Rick vehemently opposed the project, arguing that it would not only further slash their crop production but that it also posed an existential threat to the Ogallala Aquifer, the underground reservoir of freshwater that they depend on to irrigate crops and water their cattle.\n\nRick even agreed to let the antipipeline nonprofit Bold Nebraska build a solar-powered barn, the so-called Energy Barn, on land he rented from his in-laws. The publicity had brought in an unexpected amount of national press, culminating in Willie Nelson and Neil Young joining forces with Bold Nebraska to mount the fund-raising Harvest the Hope concert in the cleared cornfield of Art Tanderup, a farmer near Neligh, Nebraska, whose land was also threatened by the pipeline. But unpublicized amid all the hoopla was the reality that the barn, built in the middle of a productive piece of farmland, had cost Rick still more precious acres of ground, where he had planned to plant soybeans that year.\n\nWorse still was a letter that arrived soon after from one of Rick's neighbors. He had rented hundreds of acres to Rick for many years, but now, in protest, was pulling out of that long-standing agreement\u2014a move that was likely to cost the Hammonds between $150,000 and $200,000 of gross income for the coming year. The cancellation letter made no bones about the reasons: \"I am sorry to inform you that your outspoken endorsement of defeating the Keystone Pipeline is not in our common interest. While it is certainly your right to support whatever causes you wish, we have chosen not to subsidize those expenditures.\"\n\nWith these financial hurdles ahead, Rick had decided to take some calculated risks. He planted almost entirely soybeans for the fields not already set aside to raise seed corn for Pioneer, because the bean prices seemed steadier, and he went with short-season beans, hoping to get the harvest in ahead of the inevitable market decline. Some giant grain companies, like Archer Daniels Midland, will even pay out a premium early in the harvest. \"I planted really short-season beans, planning on getting it to ADM,\" Rick told me, \"because there might be a dollar bonus above your elevator if you can get it in before the rush.\" He also made the decision to wait on marketing his beans, rather than locking in spring prices. Most years, he would have seen spring futures as good enough to hedge against fluctuations, but this year he'd decided to take a chance. \"I'm not nearly as forward contracted this year as I usually am,\" he said.\n\nThat crop, waiting to be harvested and sold by the truckload at whatever the market would eventually bear, was where the family was either going to make money or see a whole year's labor turn into a break-even proposition\u2014or worse. With the weather conspiring to increase the national yield and push prices lower by the day, while leaving Rick's fields too soggy to get out and harvest those short-season beans, he was losing money with every passing hour.\n\nSo Rick had decided to lay a concrete footing and move in an old grain bin. He would put up as much of the soybean harvest as possible\u2014and, with luck, wait out the down market. When he and Kyle brought the bin in and put it in place with a crane, he explained that even this stubbornness was its own form of risk. Grain in the bin is given to rot\u2014and, of course, there's always the chance that prices will continue to fall. But you control what you can. Don't increase your risk by investing in a new bin, but make sure the old one is tight and well-ventilated. Make sure the floor is raised so an industrial blower can force through enough air to keep the grain dry. And after everything that's already happened, be realistic.\n\n\"You have some hardheaded old-timers that say, 'By God, beans were at seventeen dollars two years ago, and I'm going to leave my beans in the bin until it's seventeen dollars again.\" Rick shook his head. \"You want to say: 'At your age? Good luck with that.' It's not going to happen.\" So, in your heart of hearts (though you'd never admit it, not even to your neighbors), you pray for rootworm in Chile, for hail on the open plains of Illinois, a late-season catastrophe unforeseen by the USDA. You hope for what the traders call a \"market recovery,\" knowing that what that really means is that, in order for you to succeed, someone else as careful as you, through no fault of his own, has to fail.\n\nIn the last light of the day, Kyle stretched the tape measure from one wall of the bin to the other, marked the spot on the next floor piece with a nub of blue chalk. Then he was back at the torch. He struck the igniter, and the nozzle burst again into flame, sparks skittering and bounding across the concrete as he made another cut. By the time he finally stopped work for the day and walked back to the house, it was full dark. Meghan was already in the kitchen, back from nightly chores\u2014feeding the goats and horses, checking on cattle, closing up the laying hens until morning. She had four chicken breasts cooking in the fry pan. Kyle kicked off his boots in the mudroom and padded quietly around the kitchen. Worry was heavy in the air. With harvest now two weeks late, Meghan was on edge too. The smallest provocation could turn her fiery eyes, usually wide with emphasis, to a cold stare. She was the first to admit that growing up on the farm with a twin sister and two other siblings had made her quick-tempered and sharp-tongued, but those years had armed her with a winning sense of humor. She could punch you in the arm hard enough to convey honest frustration and simultaneously, somehow, forgiveness.\n\nThis tension was different. It wasn't nervy but nervous. Meghan and Kyle both knew that the later in the season the harvest begins, the deeper in the year you go toward winter\u2014and the chance of yield-loss due to ice or snow. If you suffer crop losses when prices are down, there's no chance of catching a market rebound. Then you become the sucker whose losses faraway farmers cheer. You take what the insurance adjustor gives you and try to make it through the next year. Flipping the chicken breasts with a fork, Meghan let out an exasperated moan, then launched a sardonic recitation of the wisdom she'd heard a thousand times. \"Two things with farming,\" she said, putting on her dad's voice, \"you can't control the weather, and you can't control the markets.\"\n\nMeghan loves to joke about Rick's penchant for teachable moments, but she knew that if her dad was going to step back and eventually retire, then she and Kyle not only had to work hard. They had learn to make the smart decisions that give them the tiny edge that is often the difference between the kind of success that allows you to add equipment and acres and the kind of loss that eventually leads to a farm sale. With so much riding on the next year, Meghan said she would feel a lot better if they could just get out in the fields.\n\n\"You're putting inputs all year, with money and hard work and lots of effort,\" she said. \"And one bad storm could wipe it all out.\" She moved from the stovetop to the sink, filling water glasses and staring out the window, as if she could see the brittle rows of ready soybeans and the browning cornstalks beyond the edge of their yard\u2014though it was now so dark that the glass was a mirror, reflecting her face by the kitchen light.\n\n\"It's your payday,\" Meghan said, \"the one time of year that we're actually making money, instead of shelling out.\" She took a deep breath. \"It's a field of dollar bills out there.\"\n\nKyle cracked a wry smile. \"And they could just blow away.\"\nPART ONE\n\nBRINGING IN THE BEANS\n\n_October 2014_\n\nThe bright-green hulk of our John Deere harvester crept across the field of soybeans. It was late in the day, early October, the sun low. A cloud of hulls and chaff spewed from the back of the combine, then swirled up around us and blazed in the glow. Sealed in the dustless quiet of the cab, Rick Hammond steadied the wheel with one hand and punched coordinates into the touchscreen onboard computer with the other. With each new piece of data entered, the system let out a high-pitched beep or a low, lazy bloop. The reel of the harvester head spun steadily below like the paddlewheel of a river steamer, standing up stalks for the toothed blades cutting a dozen rows at a go. The feed auger corkscrewed the cut plants into the mouth of the combine, where a throbbing set of threshers splintered the dry pods, collecting the oily seeds inside and sending them spiraling up to the grain tank behind Rick's chair. Harvested beans ticked against the back window like a light summer hail. The only other sounds over the churning reel were the _Pong_ -like beeps from the computer.\n\n\"Okay,\" Rick said at last, marking final coordinates, \"we're on autosteer now.\" And, to show me, he took his hands off the wheel. \"I thought I'd be able to retire before I had to get autosteer,\" he sighed. Rick is in his sixties, tan and weatherworn, but still plenty fit. Most days, he works sunup to sundown, especially during harvest. It wasn't the long hours he said were getting to be too much but trying to keep up with the technology. \"The reason for all this is inputs,\" he explained. The combine continued along, following the contours of the planting lines automatically recorded months earlier by the Global Positioning System. As we moved, our progress was charted on the touchscreen in varying colors to show where each row or part of a row was above or below the target for bushels-per-acre for this field. All of that data is recorded and stored to plan for next year, helping farmers decide how to adjust the density of their seed populations, where to apply fertilizer, how much to water, where to put inputs and where to save. \"During the era of high prices, yahoos that should have been broke were thinking, 'There's nothing to this.' Now we're back down to cost of production\u2014or less. So everything matters. Every place you can cut a corner or save, you do it.\"\n\nEven being out here was a tough call. Earlier in the day, with the sky clear and a light breeze blowing, Rick had asked Dave to check out the condition of this field. \"Dave poked his nose in here just to get the combine set,\" Rick said. He found that the soybeans were dry enough to harvest, but the ground, after weeks and weeks of rain, was so wet that bringing in all the equipment they'd need to work the field\u2014the harvester, the tractor and grain cart, the semi to haul the beans away\u2014was going to compact the soggy soil and make it too dense to plant next year. After some thought, Rick decided to make the sacrifice. \"It was scheduled to be seed corn next year,\" he told me. But Pioneer had already informed its seed growers that the company would be further reducing fields, from 30,000 acres in York County in 2013 down to 15,000 in 2015. \"We were scheduled for six fields,\" Rick said, but he expected that Pioneer would reduce that number by at least one field before planting came around in the spring. \"So I told them, 'You know what? If you're going to short us a field, this one would be the one.' \"\n\nThe decision might seem impulsive\u2014why not wait just a few more days?\u2014but this urgency arises from a particular oddity of soybean biology. \"Beans are weird,\" Rick told me. \"With corn,\" he said, pointing to the stalks in the neighboring quarter, \"it's growing degree units, the heat.\" Then he pointed straight ahead to the horizon, the sun orange and sinking low. \"With beans,\" he said, \"it's sunlight.\" So, unlike most crops, which begin to mature when the weather turns cold and the air dries, soybeans are short-day plants\u2014meaning, simply, that their final stages of maturity are triggered by waning hours of sun.\n\nThis poses two major problems when you run into an unusually wet fall. First, the days get shorter no matter what. Weather-dependent crops like corn slow their maturation on rainy days. You can get lucky\u2014catch a warm spell or even a single sunny day\u2014and get out in the field and back on track. Not so with soybeans. While you're waiting for the clouds to clear, the beans are going past maturity. Second, soybeans have to be delivered to the grain elevator at a precisely defined moisture rate: 13 percent. \"We're at twelve-point-nine right now,\" Rick showed me on the touchscreen, \"but when we get down there where the ground is a little wetter, it'll go back up.\" Above 14 percent, he said, the grain elevator not only charges you for drying but also docks you for the estimated shrinkage. \"And you would think, 'Well, one percent of moisture would be one percent of shrink,' \" Rick told me, \"but no. They dock you _one and a half_ percent.\" At 15 percent, they're apt to reject the whole load.\n\nSo to make your best profit on soybeans, you need a sunny day (but not _too_ sunny) with a dry breeze (but not _too_ dry), and you need that day to fall exactly when the plant has received the precise number of hours\u2014yes, _hours_ \u2014of sunlight from the moment you planted it months earlier. To make hitting such a tight window even remotely possible, seed companies, like Rick's supplier DuPont Pioneer, have hybridized soybeans for nearly a century\u2014and genetically modified them in recent decades\u2014according to bands of latitude, called \"maturity groupings.\" They number these photo-regions from zero in the northern growing zones of Canada up to seven in the light-drenched flatlands of Florida. If you're lucky enough to have ground in the middle of a maturity grouping, then you can buy a single variety. But Nebraska is almost exactly divided between groups two and three, the line bisecting the state into north and south. Which means that farmers here, especially in central regions like York County, plant both varieties to spread out maturity, allowing them to harvest one group or another first, depending on the weather, and still get a consistent yield. Some daring farmers like Rick will formulate a guess as to what the weather holds for the growing season and plant more of one group or another, hoping for higher yields and higher returns.\n\nIn 2014, after several years of drought, Rick bet on another dry year\u2014and planted incredibly short-season beans. While most of his neighbors were planting 3.5 maturity groupings, Rick had planted 2.4. And he was dead on, right up until the rains started in August. If he could have gotten out early, ahead of farmers in other parts of the country, and caught the market at its peak, he was positioned to make up for all the other setbacks going into the harvest. But if you guessed right on the growing season, as Rick did, and then got an extremely wet fall, you could end up with hundreds of acres of mature soybeans and fields too wet to run the combine. With each storm that rolls up on the horizon, you move from making a hefty profit to incurring a crippling debt.\n\nRick knew prices would hold for a time, on the chance of an unforeseen, late-season catastrophe, a hedge against an ice storm freezing crops or a line of thunderheads dropping hail that could send prices skyrocketing. But as grain started pouring into the elevators, prices on futures were sure to slump. With the market already in free fall, some farmers had decided not to wait: they went out as soon as the mucky furrows would allow and used propane or electric dryers to deal with the high-moisture grain. But this was yet another expense, and if the moisture levels were too high, the cost of drying could cancel any profit. Other farmers, like Rick, had held out as long as possible, but eventually everyone had to make hard choices.\n\n\"Every day that the beans sit out there,\" Rick said, \"you're under risk of a big, smashing storm. And then, every day that we wait, the days get shorter. And beans get harder and harder to get out, because they just soak up moisture like crazy. And then, the more that happens\u2014when they're dry, wet, dry, wet, after they're mature\u2014they're prone to shatter. They'll split wide open in a big wind.\" So when the rains finally let up, he decided it was time to go, soggy fields or no. It wasn't worth risking the return on this field this year just for the promise of above-market prices on seed corn next year. He had Dave set up the harvester, and he started across the field to judge for himself if the beans were ready.\n\nNow halfway to the end of the swath, Rick checked the level of the grain tank. He needed to empty it out before he turned the combine and aimed back toward the barn and cluster of outbuildings on the eastern edge of the property. He radioed over to Kyle on the CB, \"Can I dump on you?\" Kyle pulled up alongside the combine with the tractor and grain cart, moving in perfect parallel. While the harvesting reel kept spinning and the combine inched across the field, the unloading auger arm started pouring out soybeans, sputtering and thrumming until the tank was empty and Kyle peeled off to unload into the trailer of the big rig. Rick's mind was already back on the moisture levels. At the end of the swath, he took a wide sweeping turn and set the harvester back on autosteer.\n\n\"This here is instant yield, instant moisture,\" Rick said, pointing to the screen, \"and this is average yield, average moisture.\" As we moved, he could see in real time if he was on track to hit his production targets or falling short and whether the moisture of the entire load was within the acceptable range or inching high enough that he'd have to pay a penalty for drying. He watched the data rolling across the screen, giving the minute-to-minute condition of the crop, like watching the stock market ticker rise and fall. The colors of our current twelve rows shifted back and forth from green to yellow, from profit to loss. All the perils of modern farming seemed to crawl across the four-inch screen, teetering from making a living to accruing debt.\n\nSo far, everything looked fine. \"They're about thirteen and a half,\" Rick said. \"We'll get a little dock, but that's okay. A year like this, if you can get a couple of loads in, you just keep pecking away at it.\" So he continued across the field, until the sun was almost gone and the night cool made the beans too wet to cut. By then, Kyle was at the wheel of the harvester and had cleared the clogged cutting blades four times, trying to get just a few more minutes in. Finally, he came over the radio. \"It's time,\" he said. \"It's past damned time.\" And with that, everything was over for the night. Dave and Kyle parked the tractor and seed cart next to the outbuildings, but left the combine precisely where it sat. Dead in its tracks until the sun and the wind came up again in the morning.\n\nBut harvest, at last, was officially underway. Now, Rick just had to get his crops in as soon as possible. It was going to be a race.\n\nTHE OVERPRODUCTION of corn by the American agricultural industry has generated a lot of attention in recent years, as food activists and environmentalists have grown worried about the middle of the country turning into a vast monoculture. But that's only half the picture. In truth, the corn boom, to a remarkable extent, has been made possible by a co-equal boom in alternate-year planting of millions of acres of soybeans. In many ways, corn and soybeans seem made for each other. Soybeans are a natural nitrogen-fixer, replenishing the soil for nitrogen-hungry corn hybrids, and the plants share almost none of the same pests or diseases, preventing insects, molds, and bacteria from overtaking fields. But the soybean is more than an enabler of King Corn; it is, in fact, far and away the most successful crop introduced to the American farm in the last century.\n\nIn 1920, there were fewer than a million acres of soybeans planted in the whole of the United States, well behind traditional staple grains. But soybean production boomed during the 1930s and after\u2014surpassing barley and rye production by 1940, cotton in the 1950s, oats in the 1960s, and wheat and hay in the 1970s. Today there are more than 85 million acres of soybean fields in America, almost exactly equal to the acreage of cornfields, with which they are rotated from year to year, and almost all of those acres are concentrated in the Midwest and on the Great Plains. How did the soybean, a legume native to East Asia and traditionally used primarily in foods from China, come to have such a place of prominence here? The rise of the soybean in the United States is attributable to, more than any other person, Henry Ford.\n\nThe American farm underwent a period of unmatched innovation in the early twentieth century. The arrival of the gas-powered tractor for plowing, the combine for harvesting, and affordable farm trucks for hauling grain to market made it possible for farmers to plant more and more acres and to manage those acres with fewer farmhands. But by the mid-twenties, the market was glutted with grain, depressing prices and endangering the very family farms that the technological revolution had promised to empower. Farmers began calling for research and development to turn toward finding new uses for existing agricultural supply rather than continuing to search for ways to increase yields.\n\nIn January 1927, Wheeler McMillen, associate editor of the popular magazine _Farm & Fireside_, published a watershed article entitled \"Wanted: Machines to Eat up Our Crop Surplus.\" He wrote that he had been receiving panicked letters from farmers, lamenting that more grain was being produced than people could possibly eat. Seeing as how \"the human stomach isn't elastic,\" McMillen suggested that chemical compounds in plants might be converted into industrial products. He even advocated for government backing for such research. \"There is no wrong in channeling some federal funds into farmers' pockets,\" he wrote, considering that the American farmer \"by cheap food has subsidized the growth of cities.\"\n\nAmong McMillen's most interested readers was the owner of the Ford Motor Company. Ford, after all, manufactured much of the equipment that had contributed to booming yields\u2014and if the company was to maintain that market share, then it had to find a way to sustain its customers. To Henry Ford's mind, it also made perfect sense to subsidize research into the uses of farm products, because he was already growing concerned about dwindling petroleum supplies. Numerous auto parts were made from petroleum-based plastics, and, of course, all of Ford's engines ran on diesel and gasoline refined from petroleum crude. \"If we want the farmer to be our customer,\" he said, \"we must find a way to be his customer.\" In early 1928, he met with McMillen to discuss this new field of research\u2014what was eventually dubbed \"chemurgy\"\u2014and came away even more convinced that the key driver should be private industry, not the government.\n\nSoon after, Ford authorized dramatically expanding the agricultural laboratory in Greenfield Village at Ford's headquarters in Dearborn, Michigan. He appointed Robert Boyer, a self-taught but eager young chemist, to manage what came to be known simply as \"The Chemical Plant\" and to oversee its staff of a dozen or so young men from the Henry Ford Trade School. Under Ford's direct supervision, the lab technicians experimented with a staggering range of vegetables, fruits, grasses, legumes, tubers, and roots. Ford would meet with Boyer to hear new ideas\u2014which plants might contain high levels of cellulose for plastics, which might contain sugars that could be converted to ethanol to make biofuels. The next morning, Boyer would find a truckload of fresh produce waiting at the front door of the lab for testing.\n\nAfter the stock market crash in October 1929, the financial crisis for farmers deepened. Ford publicly advocated for continuing full production of all crops and again urged the government to resist stepping in. \"The farmer and the chemist will solve farm relief, not the politician,\" he told the _New York Times_. Nevertheless, the USDA sent emissaries around the world in search of new crops that could be planted on American farms\u2014crops intended for industrial use, not food. On one such trip to China, William J. Morse gathered more than 10,000 soybean varieties for American researchers to study. Learning of this, Ford urged Boyer to look at the soybean. His staff found that soy had unexpected levels of usable oils and yielded high-protein soy meal after the oil was extracted. It appeared, in short, that the soybean could be used to produce industrial lubricants and that the byproduct could be turned into plastics.\n\nThe results were so encouraging that, in December 1931, Ford approved $1 million in new research funding and instructed Boyer to halt work on all other plant life. The following spring, Ford had 300 varieties of soybeans planted on 8,000 acres on his farmland in rural Michigan. The next year, he expanded to 12,000 acres\u2014making him the single largest soybean grower in the United States. That same year he announced that he would buy any soybeans delivered to the Dearborn plant. To encourage production, he made 400 Fordson tractors available for free use to Michigan farmers and offered gas and diesel at a penny per gallon\u2014less than a quarter of what it cost at the pump. Farmers put more than 35,000 acres of land into growing soybeans, and Ford bought their entire output as promised. It was a daring move but also good business. Between 1931 and 1933, with few farmers buying trucks or tractors, the Ford Motor Company lost nearly $120 million. Henry Ford was convinced that turning those numbers around would require restoring the buying power of the American farmer.\n\nTo that end, Ford put the full muscle of his publicity machine behind promoting soybeans\u2014and chemurgic applications for other crops as well. He hosted the national convention of the American Soybean Association. He helped organize and again played host to the Farm Chemurgic Council, which was promoting Agrol, an ethanol-blend alternative to gasoline, over the loud objections of the petroleum industry. And Ford gave a series of interviews, telling one reporter that he envisioned a time when much of an automobile \"could be made from by-products of agriculture.\" He even announced a plan to decentralize Ford production by opening a constellation of factories in rural areas to manufacture plastic parts made from local soybeans. \"He wants to make as many parts as he can in small factories, whose workmen will live nearby, each tilling his little plot of soil,\" reported _Fortune_. While talking to one reporter, Ford had an assistant bring to his desk a steering wheel, a distributor box, and a grab bag of small electrical parts. He bragged that he used soybean oil in his paint and to lubricate his casting molds. Soybean meal was reacted with formaldehyde to produce a thermoplastic resin, which was used to make gearshift knobs, window frames, distributor caps, and horn buttons. \"There is a bushel of soya beans in every Ford car,\" _Fortune_ declared. \"He is as much interested in the soya bean as he is in the V-8.\" Ford even switched the company commissary over to baked goods made with soy flour and ice cream made with soy milk.\n\nTo expand and expedite processing, Ford soon began experiments using hexane solvent for extraction of soybean oil. In 1933, he sponsored the Ford Exposition of Progress in New York City, where he debuted a glass model of an extractor that used hexane\u2014and called on American companies to build full-scale versions. The federal government, the State of Illinois, and the Illinois Farm Bureau all contributed funds to take up the challenge. Within months, Archer Daniels Midland (ADM), a milling company which had been supplying linseed oil to Ford for its paint until Henry Ford ordered it replaced with soybean oil, purchased from Germany a 150-ton-per-day Hildebrandt hexane extractor. When it began operation on Blackhawk Street in Chicago in March 1934, it became America's first continuous solvent extractor\u2014and the most modern soybean processing plant in the world.\n\nAt the same time, Ford unveiled a home version of the extractor, which he hoped would be used on a small scale by farmers to extract oil and sell it directly to the company. The enormous Ford exhibit at the Chicago World's Fair in 1934 included a diorama of a working soybean farm and a modern \"Industrialized Barn\" with one of the small extractors inside. \"I see the time soon coming when the farmer will not only raise raw materials for industry but will do the initial processing on his farm,\" Ford said. \"He will stand on both his feet\u2014one foot on the soil for his livelihood, and the other foot in industry for the cash he needs. He will have a double security. Agriculture suffers from a lack of market for its product, industry suffers from a lack of employment for its surplus men. Bringing them together heals the ailments of both. That is my conviction and that is what I am working for.\"\n\nThat summer, Ford's promotions got an unexpected boost. The first of several years of severe drought, what soon proved to be the worst dry time in American history, engulfed 75 percent of the country. An apocalyptic dust storm carried 350 million tons of topsoil from the Great Plains all the way to the East Coast. Corn and wheat withered in the fields. Amid rampant crop failures, farmers harvested 23 million bushels of soybeans, a better yield than most crops and far better than its oil-producing competitors like linseed and canola.\n\nIn the spring of 1935, farmers planted soybeans in record numbers. In preparation for the coming harvest, Ford spent $5 million to construct his own soybean mill with solvent extraction units at the flagship River Rouge plant in Dearborn\u2014and boasted that he had jumpstarted demand for soybeans nationwide. That year roughly 70 million bushels of soybeans would be harvested. Selling for 50 cents per bushel in the absolute depths of the Great Depression, soybeans were hailed as a godsend. They now generated more income for farmers than either barley or rye, and soybean trading had become so active and central to industry that the Chicago Board of Trade started offering soybean futures for the first time. By the end of the 1930s, the soybean harvest was approaching 100 million bushels per year. _Time_ magazine declared Henry Ford \"a bean's best friend.\"\n\nBut no sooner had industry begun to move away from petroleum than the world's largest oil reserve was discovered in Saudi Arabia in 1938. The barrel price of crude oil fell precipitously, and the demand for petroleum substitutes waned. Ford continued to extol the virtues of soybean products, but the industrial applications for soybeans were now seen as impractical. But during the years when grain crops were failing, animal feeders had discovered that livestock fed on an oil-rich soybean diet bulked up quickly. With demand from Ford and other automakers no longer elevating prices, milling companies, hoping to capitalize on their existing supply and marketing structures, were eager to get into the soybean trade.\n\nAt the time, America's milling industry was still centered on St. Anthony Falls at the headwaters of the Mississippi River in Minneapolis, Minnesota. In the nineteenth century, the city had taken advantage of the strong current of the falls to turn millstones and its geographical location, upstream from nearly all of the United States, to bring its milled goods cheaply to market. General Mills and Pillsbury shared an undisputed hold over the flour market, so their smaller competitors were eager to find ways to capture similar shares of other grains. Chief among these new competitors were ADM, which, in addition to producing linseed oil for Ford's paint, also milled, processed, and stored vast quantities of crop-based products, and Cargill, which specialized in milling grains as feed for livestock producers. They eagerly built new soybean-processing plants, along the rivers and the Great Lakes from Minneapolis to Chicago, where the great stockyards were buying unprecedented quantities of soybean meal as animal feed.\n\nIn 1939, Shreve M. Archer of Archer Daniels Midland announced the construction of a new modern soybean plant and elevator in Decatur, Illinois. Again acquiring its extractor from Germany, this time ADM ordered a Hansa Muehle unit that employed the Paternoster design, carrying flaked beans in buckets on a moving chain. The unit, which had a daily capacity of 400 tons (roughly four times the size of any that had been used in Europe), made ADM the largest soybean processor in the world. In November 1939, the plant was ready for operation. The solvent unit occupied a five-story tower; storage capacity for 5 million bushels of soybeans was provided.\n\nAt the same time, Cargill launched an aggressive, even desperate, bid of its own into soybean processing. Company president John MacMillan, Jr., had inherited the family-run business just five years before and expanded so forcefully into corn processing during the corn-poor Dust Bowl years that the U.S. Commodity Exchange Authority and Chicago Board of Trade accused the company of trying to corner the market. When government investigators demonstrated a pattern of overbuying\u2014acquiring as much as 80 percent of the total market share\u2014in order to artificially drive up prices, the Chicago Board suspended Cargill and three of its officers from the trading floor. (The company would not rejoin the Chicago Board until 1962, after MacMillan's death.) The company established a downriver soybean-processing facility, which they dubbed Port Cargill. Not to be outdone, Pillsbury entered into a partnership with an independent miller who was preparing to open a new plant in Mankato.\n\nBy the harvest of 1939, soybean mills dotted the Mississippi River from one side of Minnesota to the other. Now the problem was supply. Before the end of the year, the American Farm Bureau Federation sponsored hundreds of events to encourage farmers to plant soybeans and instruct them on how to achieve top yields. For the first time, the Midwest was the center of soybean production in America, with ADM leading all processers with six soybean plants in Decatur, Chicago, Minneapolis, Milwaukee, Toledo, and Buffalo. With harvesting and processing now at full capacity, the soybean producers were well positioned to take advantage when the United States was pulled into World War II.\n\nWhen Hitler began his march across Europe, there were sudden scarcities of edible oils and fats\u2014nearly 40 percent of which had previously been imported from Mediterranean countries. After Pearl Harbor, the U.S. government pushed agricultural production to achieve record output\u2014and soybeans increased from 78 million bushels in 1943 to 193 million bushels in 1945. Yet Cargill and ADM had a terrible time fully capitalizing. While the production\u2014and consumption\u2014of soy products nearly tripled, President Roosevelt and Congress worked together to pass price-control legislation, creating the Office of Price Administration (OPA). Meat and poultry were hard to come by, so the OPA established fixed prices on animal feed.\n\nAfter the war, however, when the government slowly started to roll back its strictures, feed prices rose sharply. As Ford had always feared, government involvement, instituting price controls and later subsidies, made farmers subject to the whims of federal farm policy\u2014and the agribusiness interests that controlled that policy. But by then, Ford was too sick and old to keep up the fight. He turned the company over to his son, and barely a year later, Henry Ford died of a cerebral hemorrhage at his estate in Dearborn. Without his guiding vision, soybean production for industrial purposes waned sharply for several decades, but the grain millers saw a different opportunity.\n\nWartime rationing officially ended in 1946, and the American public was eager to celebrate. Like never before, they wanted French wine and a good steak. In the early months of 1947, national consumption of meat rose by more than 20 percent\u2014a real increase of more than 1.7 billion pounds. Prices climbed so high that New York's mayor called for a congressional investigation. But the _New York Times_ concluded that it wasn't price-fixing. \"Americans are meat-hungry,\" the reporter explained. \"There are just more persons consuming more meat than ever before in the history of the country.\" To ease high prices, the federal government began subsidizing corn and soybean production. Ever since, the prices of commodity grains have risen and fallen according to the demand of livestock feeders.\n\nFord may have failed to create a future of plastic cars and biofuels, but he successfully created a steady market for soybean producers\u2014only now, their profitability was once again tied to the whims of the livestock market.\n\nRICK WASN'T pleased. \"Not good,\" he said, \"not good.\"\n\nIt was past eleven in the morning, the sky still overcast and threatening. With the remaining fields near the home section too wet for work, Rick had loaded the combine onto a flatbed and brought it to a quarter section of dry land south of Interstate 80 where he had planted more very short-season soybeans. The wind seemed to be picking up enough to bring the moisture down to where he could harvest. He had asked Dave to run a test patch, just to be sure, only to find that an unusual number of the bean plants were \"laid down\"\u2014that is, their stems were so flat to the ground that the spinning reel of the John Deere couldn't prop up the stalks for the cutting blades. So many beans laid flat meant a measurable loss of yield, but it also suggested the possibility of something even worse.\n\n\"We had a couple of hailstorms come through here this summer,\" Rick said, kneeling in a furrow next to one of the flattened plants, \"but our insurance adjustor didn't expect the loss to be near this bad.\" He plucked one of the plants, roots and all, out of the ground and flipped open his buck knife. Slicing longways, he split the stalk in half. \"Aw, man, look at that,\" he said, showing the hollowed-out insides to Kyle and Dave, who had come down from the cab of the combine to see what was wrong.\n\nWith one glimpse of the stem, they each set to pulling up their own plants and cutting them open. After a few tries, Rick called out. \"Here we go,\" he said. He called Meghan over too, then held up the sliced open stem for everyone to see: a tiny larva, a narrow white caterpillar, nestled right where the stalk meets the roots. \"Stem borer,\" Rick said. \"They warned us to be on the lookout for them this year, but we've never seen them here before.\" He turned and cursed, then flipped open his clamshell cell phone, and in a minute had the crop insurance adjustor on the line. \"Yeah, we need you to come out and have a look,\" he said as he wandered back toward the road. Rick kicked clods of dirt as he went, sending up tiny brown clouds.\n\nMeghan shook her head. \"You spend all fucking year trying to get these crops the best yield,\" she said. \"It's ready to be harvested. They're done, and when they're done, you got to get them out.\"\n\nBut now, this field would have to wait for the adjustor, which could be days or even a week, depending on how many claims were already waiting. When a call like this goes out, the adjustor comes with a tape measure and marks off 15 feet. Then he counts every standing plant and every dead plant to tabulate an estimate of the overall percentage of loss. Different levels of insurance pay out at a variable rate. You can insure 50 percent of loss for a relatively low premium, all the way up to 75 percent for the highest premium. If your damage is less than 25 percent, you just have to eat the loss.\n\nYou can also get supplemental coverage, but it only kicks in if there's at least 14 percent crop loss across your entire growing region. And even if you make your deductible and the supplemental insurance kicks in, you're not only sustaining at least a 14 percent loss but the payout on the difference\u2014the loss above and beyond that threshold\u2014is figured according to an adjusted price, calculated from the average of the spring and fall prices. So whether you were forward contracted or hoping to make a little extra by marketing at harvest, it's impossible to catch the price at its peak. And in a year like this, when Rick was hoping to bin as many beans as possible and wait out the market, there was no chance to catch a rebound. Even with the best insurance, the Hammonds were going to take a substantial hit on this field\u2014and while they waited, there was the chance of still more loss.\n\n\"Hail, wind damage, snow,\" Kyle explained. \"Right now those crops are really vulnerable.\"\n\n\"Eight inches of rain,\" Meghan said.\n\n\"Yeah, they just got big rains in southeast Nebraska,\" Kyle said. \"It wiped out a lot of fields. They say a two-hundred-year event.\"\n\n\"It's also driving the price,\" Meghan said.\n\n\"That's right,\" Kyle agreed. \"In the week since those rains, beans went up a dollar. That's probably how much yield has been lost. Because the beans are really sensitive. High wind will just knock the beans right out of the pod.\"\n\nRick clamped his phone closed and started back toward the field, moving quickly. \"Tell Dave that we're not going to get it out, so that he can just bring the combine in,\" Rick said. \"Let's get the head back on the trailer.\" He told Kyle that they would move everything over to a nearby neighbor's property. The neighbor, Seth, had gone in with them on the rental for the combine. Rick had thought that his short-season beans would be ready first, but that wasn't going to happen now. They could make the most of what remained of the day and then hope to find another field that was ready tomorrow. For now, Seth's half section was dry, so they might as well get everything over there.\n\nKyle and Dave went right to work unhooking the head of the harvester.\n\n\"We're at least two weeks behind,\" Rick said in my direction but almost to himself. \"And it's just killing us.\"\n\nFAILURE IS everywhere on the farm.\n\nIt hides in the long shadows cast by the barn at last light. It waits amid the dark stalks shifting in the fields before dawn. It's all around you, always lurking, always palpable but just out of view.\n\nThese farm failures can be dramatic and sudden: the death of a patriarch, his chest rolled over while repairing a tractor or his overalls twisted into a choking knot by the auger in the grain bin. With no will or simply no clear plan for succession, the farm can go under. Just as shattering can be a divorce or the death of a spouse. Legal bills and the stress of unsettled assets can turn a farmer's attention from keeping the books or watching the markets and somehow the business slips away. And, of course, failure can come from the fields, appearing like an undetected cancer\u2014cobs stunted by drought, beans infested by cutworms.\n\nMore often, though, it creeps up, not one major disaster but a series of smaller missteps. Too little insurance in a drought year. Borrowing to build a hog barn just before the bottom drops out of the market. Buying more ground and more seed only to see commodities prices plummet. Often as not, failure comes from nothing more than a farmer overestimating his ability to service a loan.\n\nSome would say that this is no accident. In the early 1970s, Secretary of Agriculture Earl Butz told farmers that a new day was dawning. They had to \"get big or get out.\" He urged those who heeded his call to acquire as much land as they could afford and plant \"fencerow to fencerow.\" Butz wanted American farmers to produce a steady oversupply of key grains, in order to undercut and control commodities markets to the disadvantage of our Cold War enemies. But the United States had considerably less landmass than either the Soviet Union or China. So the only way to outproduce them was to invest in every possible method of intensifying production\u2014chemical fertilizers and pesticides, ending crop rotation in favor of monocultures, consolidating farmland and agricultural companies, and ultimately tinkering with the genetics of row crops to make them resistant to various herbicides but also tolerant of denser planting as a way of increasing per-acre yields.\n\nButz promised to use the emerging global economy to bolster prices. If we faced harvests in which supply outstripped demand, the United States would simply negotiate trade deals, using our economic might to artificially create a market. If we could flood global trade partners with massive quantities of grains, pushing sale prices below the cost of production, then Butz believed the world would have no choice but to buy from us. We could make our enemies\u2014and even our friends\u2014dependent on us to feed themselves.\n\nIn January 1972, Butz sold what amounted to our entire grain reserve to the Soviets. The following month, Nixon went to China and brokered a deal with Chairman Mao Zedong, allowing the importation of American corn and securing contracts for American companies to build thirteen of the world's largest ammonia-processing plants for producing fertilizer on Chinese soil. America's Communist foes regarded these moves as an agreement not to wage war through food. But Butz discussed the new agreements in terms of \"agri-power\" and stated it plainly: \"Food is a weapon.\"\n\nThe trouble, as many critics saw it, was that producing at that volume meant relying on agribusiness, making us beholden to a handful of companies. Where family-owned farms had been numerous and diverse, the kind of small operations that had to produce a variety of high-quality products to insulate themselves against market fluctuations, multinational grain companies were centralized and large enough to capture extensive portions of critical commodities and turn a profit by doing nothing more than capitalizing on market uncertainties at strategic moments.\n\nThe most famous example came in 1973, when Cargill, which had only been allowed to return to the Chicago Board of Trade a little over a decade earlier, now placed huge orders for U.S. soybeans via a Geneva-based affiliate, Tradax, making it appear that there was a pending soybean shortage. Prices skyrocketed\u2014and eventually rose so high that President Nixon ordered a halt to all soybean exports to prevent domestic scarcity. With prices at record levels and foreign countries desperate to find new sources of soybeans, Cargill filled the canceled orders of other American companies with supply from their South American subsidiaries, commanding artificially inflated prices. When the U.S. embargo was lifted, Cargill canceled its Tradax purchases and by year's end saw their annual profits jump from less than $20 million to more than $150 million, despite an overall decrease in its volume of production that year. In short, knowing that food was now a matter of national security and that the government would intervene to protect it, Cargill (and soon other major agribusinesses) recognized that they could manipulate the market to produce greater profits, regardless of the success of each year's harvest\u2014and regardless of the impact on American farmers.\n\nA few years later, farmers saw firsthand the risks they now faced. At the end of December 1979, President Jimmy Carter enacted a grain embargo against the Soviet Union after Leonid Brezhnev ordered the invasion of Afghanistan. Ever since buying our national granary, the Soviets had been acquiring huge quantities of American grain\u2014including 25 million tons of wheat and corn contracted that October for the coming year. Carter bristled at the idea that the Soviets thought they could stage an invasion when we controlled their food supply. \"They think I don't have the guts to do anything,\" he told an aide. \"You're going to be amazed at how tough I'm going to be.\" Carter announced the embargo and canceled the outstanding grain orders for some 17 million tons of corn and wheat. He issued a freeze on trading in grain futures for two days to allow the market to stabilize, but when it reopened, prices plummeted anyway\u2014losing 20 percent of market value. Angry farmers marched on the USDA. One protest leader said, \"We planted fencerow to fencerow like they wanted, and now this is what happens.\"\n\nThe Soviets found other grain suppliers to circumvent the embargo and lessen the impact of any future trade sanctions. As commodities prices fell, it became apparent: instead of making the world dependent on our grain supplies, we had grown reliant on their demand. American farmers continued to carry heavy overhead and service high-interest loans, preventing them from competing with foreign grain producers, even after the embargo was ended. American agricultural exports declined by more than 20 percent between 1981 and 1983, which, combined with decreased market prices, ended up meaning a nearly 40 percent reduction in farm income in a matter of just a few years.\n\nSome would regard this as the ultimate failure of agriculture, but the emergent industry of consolidated agribusiness flourished during this time. Along with the old grain cartels, such as Cargill and ADM, former defense contractors, including Monsanto and DuPont, aggressively entered the food industry. Cargill grew from $2.2 billion in sales in 1971 to $28.5 billion in sales in 1981 by turning the profits from grain shortages into a diversified portfolio. They moved into value-added operations\u2014milling grain for animal feed, making high-fructose corn syrup for Coke and Pepsi\u2014then expanded and vertically integrated, acquiring feed elevators and meatpacking plants. By the mid-1980s, Cargill was not only the nation's top grain exporter but also the Number 1 egg producer, the Number 2 beef packer, and the Number 3 miller of corn and wheat.\n\nMeanwhile, the market free fall touched off by the failed grain embargo created a catastrophic decline in consumer confidence that led to a national recession and, as inflation began to rise, eventually set the stage for the Farm Crisis. Prices on key grains stayed low, and overproduction soon drove them even lower. At the same time, the accrual of massive debt in the form of modern tractors, irrigation systems, and grain storage bins left thousands of farmers hopelessly in debt. And this, of all the ways that failure lurks on the American farm, may be the most lethal: raised on Protestant work ethic and a faith in the basic fairness of the system, most farmers firmly believe that the greatest success belongs to the family that works the hardest. The way out of debt is putting in longer days in the field, longer nights at the books.\n\nBut it isn't really so. In fact, at the end of the Farm Crisis, Danny Klinefelter, an ag extension economist at Texas A&M who studied and categorized farm failures during that era, found that the most common causes of the individual bankruptcies that eventually became an industry-wide collapse were: too much ambition, too much accrued debt, and, in Klinefelter's words, \"too much wishful thinking.\" He said farmers didn't consider the cyclical nature of farming. They took on loans for land and equipment during periods of good weather and high prices. And if hard times fell, they expected to simply make up the difference with more work and more yield. It's rarely so simple.\n\n\"Many of the producers who have failed or are in trouble have been considered by the farming community to be top farmers,\" Klinefelter wrote, \"but attaining the highest yields does not necessarily result in the highest profits.\"\n\nThat's a lesson that few farmers can accept. And before long, the cycle was repeating itself. In the decades since the Farm Crisis, grain production has kicked into overdrive. Big producers just got bigger. And small farmers, in an effort to keep pace with the expansion and vertically integrated models of agribusiness, began to take Butz's motto to heart. With the help of emerging technologies, everything from the GPS-mapped furrows to computer-controlled irrigation systems, they began to plant crops (especially corn) in places no one would have dared waste seed, much less water, a generation earlier. The more they planted, the more they stood to profit. But then, in the last decade, all of those acres and all that overhead started to become a curse.\n\nAs the current crisis unfolded, everyone with skin in the game\u2014from the market watchers and commodities traders to the pundits and ag reporters (as well as every farmer at the counter of the caf\u00e9 in every small town across the Corn Belt)\u2014had his own explanation for the radical swings in the prices of commodity grains.\n\nSome blamed reckless gambles by farmers themselves, and it's true that farmers take risks. They plant the shortest-season hybrids they can. They wait as late as possible to irrigate. And when prices were high, they planted every arable inch of soil, tearing out windbreaks along field edges and the brush along creek banks. For a while, the ditches on either side of every country road seemed to be stacked with slash. Farmers took land out of the federal Conservation Reserve Program, a sure payday for doing nothing but letting the ground sit, and they planted it to corn. The more farmers planted, the more they stood to profit, right up until everyone was drowning in surplus and the prices took a dive.\n\nThere are some old-timers who take the long view. They say that the consolidation of farms, going back to the Farm Crisis, puts decision-making responsibility into fewer hands, so that foolish risks are amplified. In the 1980s, most cropland was farmed by families with fewer than 600 acres. Today, the average farm is twice that size\u2014and many are much, much larger. For more conservative farmers, wary of risk or simply too small and cash-poor to invest millions of dollars in acreage and equipment, just sitting still and doing nothing can leave you with less and less each year. Megafarms, with lots of resources, can afford to put more inputs on a field, to irrigate and fertilize and spray against pests. So in good years, when everything goes right, their harvest is high and drives down prices, which is fine for big guys with enough volume to profit from narrow margins but hell on the family farms. And in years when prices climb on some disaster, small farmers can still lose out by not having enough acres to capture the gain.\n\nAnd, of course, there will always be those who blame the government. They say these big-time risks are encouraged by federal crop insurance and farm subsidy programs. It's true that farmers started taking out more crop insurance when Bill Clinton's administration made modifications to the Federal Crop Insurance Act and the USDA took over paying premiums. And it's true that more than $277 billion was allocated for farm subsidies from the expansion of the program in 1995 until 2014\u2014with roughly 75 percent of the money going to the top 10 percent of farmers. So they raked in the majority of profits and set the market prices, while nearly two-thirds of American farmers struggled, collecting no subsidies at all.\n\nAll of these things are true, and they feed on each other, but none of this is really what caused the market to peak in 2005\u20132007\u2014or to crash to its lowest point in decades. The problem was really a side effect of good intentions by the U.S. Environmental Protection Agency (EPA). In an effort to reduce carbon emissions and improve air quality, the EPA imposed the first Renewable Fuel Standard in 2005, requiring the production of at least 7.5 billion gallons of renewable transportation fuels within seven years. The goal was to kick-start the biofuel industry, but the incentive, when combined with soaring gas prices due to the Iraq War, instead triggered runaway demand for ethanol. The number of ethanol plants nearly doubled overnight. At first, that seemed like a good thing. It created a domestic fuel source, and the higher prices profited American farmers and the American companies that support them.\n\nSoon, however, so much corn was diverted into ethanol production that it created a scare on the global commodities markets. Foreign countries dependent on American grain worried that their own farmers wouldn't be able to afford to feed their livestock, sparking a commodities run. The per-bushel price of corn shot up from $2 to more than $3. Everyone wanted to stockpile grain as a hedge against even higher prices, but American grain reserves were at the lowest on-hand level since the drought years in the Clinton era. After more than three decades of corn prices hovering between the $2 and $3 mark and generally trending downward on improved production since the early 1970s, prices shot up out of control. Depending on whether you were a corn grower or a livestock producer counting on that feed, the _New York Times_ wrote, \"you have daydreams\u2014or nightmares\u2014of that $5 mark.\"\n\nIn fact, from 2005 to 2008, the world price of corn broke past $5 and just kept climbing. And as corn reached unimaginable heights\u2014$6 and $7\u2014demand increased for soybeans as a feed alternative. Soon those prices had doubled too. As great as the run was for American farmers, food shortages and ensuing riots were touched off in parts of Asia, Africa, and Latin America. And the panic was enough to spark a run in commodities speculation. Traders had been looking for somewhere to put their money now that the securities markets, which had been backed by high-risk, low-interest home loans, were in free-fall due to the mortgage crisis. That money now pouring into the grain market created a brief price correction, but then the shortage brought on by the drought of 2011 sent commodities to record levels with corn reaching as much as $8 per bushel and soybeans topping $17 per bushel.\n\nBut on the farm, this is failure's best hiding place\u2014smack in the middle of runaway success. One afternoon, sitting at the workbench in Rick's barn, he told me that the biggest pitfall you face as a farmer is your own optimism. Everything starts to seem easy, and you think the good times are going to last forever. And the government, in the name of spreading the wealth and stimulating the broader economy, always provides the excuse to take on more debt. In 2010, Congress announced that they would raise short-term rapid-depreciation write-offs for farmers from 50 percent to 100 percent, for one year only, so now, if you bought a new tractor or a new grain bin and spent $100,000, you could put that full amount against your annual farm income, instead of only half.\n\n\"People were just buying equipment like crazy because you could take that all right off of your income,\" Rick told me. \"If you were in the thirty percent bracket making a hundred fifty thousand dollars, you could figure that you were going to give fifty thousand to the government anyway. But if you took a loan for a hundred-thousand-dollar purchase, the tax was offset.\" In 2011, across the country, farms suddenly had new trucks and tractors, new barns and outbuildings, but also new houses and new purchases of the kinds of equipment that can cost as much or more than a farmhouse.\n\n\"And here's the trap that farmers fall into and we're still dealing with. Like that big sprayer,\" Rick said, pointing to the vehicle in the corner of the barn, its wheels taller than me. \"That was a quarter-million-dollar purchase.\" He paused a moment looking at the machine.\n\n\"As my banker says, people forget they have to pay the principal,\" Rick continued. \"Now, during these really shitty times, everybody still has the payments. Yeah, I did benefit from all that tax write-off, and it was a hell of a deal on the interest too\u2014but it's still an expense.\" Too much ambition, too much wishful thinking.\n\nIn no time, the factors that had quadrupled grain prices self-corrected. Gas prices stabilized and began to fall, lowering the market price for ethanol with it. Before long, ethanol producers began idling their new plants, because they simply couldn't produce fuel cheaply enough given the high input price of grain. As EPA tax credits expired, and the agency adjusted the renewable fuel standard downward, the drought also broke, bringing an upsurge of production in the 2013 harvest\u2014just as demand was leveling off. The prices of corn and soybeans crashed, losing half their market value between the planting season and the end of that harvest year. So coming into the 2014 season, every farmer had to make his own market projection.\n\nRick gambled on another dry year and that the markets would remain down, so he planted mostly soybeans, which have been less volatile than corn, and he planted very short-season hybrids, hoping to get most of his harvest in early, before the markets fell even further. It should have worked\u2014but in the high-stakes, narrow-margin game of commodities farming, a little rain is all it takes to derail everything. With his crops still sitting in the fields and markets continuing to fall, I asked Rick if he was worried\u2014not stressed or anxious, but truly worried for his family. He shook his head. They were more than equipped, he insisted, to weather through a bad year.\n\n\"Now, if we see sub-four-dollar corn for two more years,\" Rick continued, \"yeah, you'll see some people going broke, especially the young guys who started about five years ago when it was eight-dollar corn. And they thought, 'Oh, man. There's nothing to this farming game.' They bought the $12,000-an-acre ground and took out loans for the bins and the pivots and equipment.\n\n\"The Meghans and Kyles of the world,\" he said, \"they could be in real trouble.\"\n\nDARKNESS HAD fallen over eastern Nebraska.\n\nAfter another day of chipping away, working and then waiting, Rick had decided it was time to call it an evening and sent everyone home. But he wasn't quite ready for the drive back or giving himself up to sleep. So he had ducked into the empty house on the southeastern corner of the Centennial Hill Farm, the house where his wife Heidi grew up. It's abandoned now, except as a guesthouse and a place to host occasional potlucks. The refrigerator often holds a random collection of unopened cans and jars\u2014not much, but enough to scavenge for a snack or a stray drink. Tonight, Rick scoured the remnants of the last get-together in search of something to take the edge off before bed.\n\n\"This bottle,\" he said, holding it up to the fluorescent bulbs overhead, turning it sideways to divine how much wine might be floating at the bottom. \"I think it's been open for . . .\" He trailed off, searching the label for a date before giving it a final shake. \"Cleaning solution,\" he decided. With no other options available, he settled on the last two cans of Pabst Blue Ribbon but not without a few grumbles. He cracked one and took a long sip, then smirked. \"I've raised hell about these in the past,\" he said, \"but, you know, it's funny how you get certain things the right amount of cold, and they all taste pretty good.\" He slid into the booth of the kitchen breakfast nook and motioned for me to slide in across from him. He wanted to assure me that, despite everything that had been going wrong, he wasn't worried about the current crisis.\n\n\"I don't feel like it's going to be anything like the eighties when everybody was going broke,\" Rick said, taking another deep draw of his beer. \"That was caused by Reagan and Volker, the Fed chairman. They thought they came in with a mandate to crush inflation\u2014and inflation _was_ out of control\u2014but bankers had been pushing farmers to buy ground every year. It inflated up to about $2,500 an acre for good ground. The interest had been less than the rate of inflation. So when they raised rates, to all of a sudden have interest triple, that's what caused it. Bankers and farmers couldn't react fast enough. They went broke, had this huge debt. Bankers were foreclosing on farmers because your collateral was worth a third of what it was when you took out this loan.\"\n\nHe said that he'd witnessed this sudden decline firsthand. When he met Heidi, while she was a student at the University of Nebraska School of Technical Agriculture in Rick's hometown of Curtis in the southwest corner of the state, everything had been different. He had been a hotshot horseman in his small town\u2014a skilled rider with a row of purple and blue ribbons in barrel racing and pole bending at the State Fair to prove it and still brash enough to take all comers in bareback sprint races on the ranches around Curtis or the dirt main street of nearby Wellfleet. He seemed to have the world by the tail. But college in Lincoln had been different, harder. After years of classes, he had taken a semester off and gone home, but he found life on the farm getting harder too, especially for a family like his that didn't own the ground they farmed.\n\nRick found himself at loose ends, as he took off another semester and then another. He was still technically enrolled at the University of Nebraska but a few credits shy of a degree in Latin American studies and in no hurry to finish up. The degree wasn't going to do him any good. He knew he wanted to be back farming and ranching, but he could see that the opportunities were drying up in Curtis. So he took odd jobs, while he figured out his next move. \"I'd worked on the railroad for three years, steel gangs and tie gangs,\" Rick told me. \"They'd made me assistant foreman and wanted me to be foreman. I could see then, if I didn't quit, I never would.\" So he left the railroad, but he still couldn't figure out what he wanted to do. \"I just wasn't finding _it,_ \" he said.\n\nThat's when he met Heidi. \"Oh, everybody loved her,\" he said. \"She could drive a tractor, fix engines.\" She had grown up in a farming family back in eastern Nebraska with a hard-driving father and three headstrong sisters. Rick's big-headed swagger didn't fluster her a bit. \"We had a few dates. I even took her to church\u2014very honorable,\" Rick deadpanned, then grinned, \"but I was still convinced that I could change the world.\" He broke off the relationship and signed up for a two-year stint in the Peace Corps. He went off to Ecuador to teach agriculture in a small village, but he soon became disillusioned by the shortcomings of trying to apply American know-how to places with no resources or infrastructure. He came home disheartened and more lost than ever.\n\nRick paused, rattling his empty Pabst.\n\n\"Is there something wrong with your can?\" I asked. \"Some kind of beer deficiency?\"\n\n\"I'm running low,\" Rick said with a smile. He cracked the second can. \"And I'm just getting to the best part.\"\n\nWhen Rick returned home from the Peace Corps, he found that everything that had been starting to slip away in Curtis was now utterly gone. Agribusiness was swallowing up small farms. Families had moved away. His hometown wasn't home anymore. \"Dad had quit farming, and my sister and brother-in-law were farming that little rented place,\" Rick said. \"I had grown up on a small rented farm. I knew there was no future there.\" A friend of his named Kevin had sold his farm and equipment and gone to Colorado to open a ski shop. Rick followed him out there and for a while worked a few hours a week and ski-bummed the rest of the time. At some point, he'd had enough. \"I said I gotta get back to school and finish my degree.\"\n\nAnd he'd been thinking about Heidi. He called her up to see if she wanted to have Thanksgiving with his family in Curtis. After that, they could drive back to eastern Nebraska together\u2014Rick to get himself reenrolled in school, Heidi to visit her own family. Heidi agreed and drove out to Colorado to pick up Rick, but by the time she arrived it was after dark and snow was swirling.\n\n\"That night we had a hell of a storm, about two foot of snow,\" Rick remembered. \"She was driving a little Fiat.\" While he packed up the last of his stuff, his friend went outside to check on Heidi. She had put her farmer coveralls on and was under the Fiat putting chains on the tires. \"Kevin came in and said, 'I think this one's a keeper. You better marry this gal.' \" They drove across the state, all the way back to Lincoln through the storm, together. By the time they arrived, Rick had started to fall for Heidi. He worked on his degree, and on weekends, he drove out to York County and wooed her by pitching in on the farm. Eventually, he won her over, just as her father, Tom, was getting ready to retire. Soon, Rick dropped out for good, just a few credits shy of a degree, and started talking about leaving Lincoln, moving in with Heidi, getting married.\n\nAudacious and brazen, Rick had a head full of ideas for ways to expand the operation and make way for the future. \"In the 1980s,\" he told me, \"inflation was so high that bankers were encouraging farmers, 'Hell, borrow money, because next year that money will value less.' So everyone took out loans against their farms. Then ground went from $2,500 an acre to $800 in two years, so farmer equity and the value of what their loans were borrowed against, went to nothing. And then the banks started selling them out.\" It was a terrible time for family farms, but Rick also saw the opportunity, the once-in-a-lifetime chance to leverage the land that Tom had carefully stewarded, in order to pick up more ground at a fraction of its real value.\n\nRick was still too young to appreciate it, but Tom himself had always been an early-adopter, a firebrand and risk-taker in his own right. But these times had him spooked. Tom's father had been put off the land when he was young, his grandfather forced to sell it off after bad decisions in the teens. His father had worked his whole life to save up enough to buy the farm back, and when he died young and unexpectedly, Tom had had to give up his own dreams. After just one semester at the university, he dropped out and came home to save the farm from being lost again. Whatever the ambitions of the young man who wanted to marry his daughter, Tom was reluctant to make unnecessary and untested changes and even more cautious about taking on debt in the midst of a credit crisis.\n\n\"But when Heidi and I got married, and she inherited her share of the land,\" Rick said, \"I was fifty percent of the decision-making process on how to go forward.\" And he couldn't bring himself to pass up the opportunity he saw before him. Heidi had 50 cows, 50 sows, and 300 acres of land under her control. \"When I first came back from the Peace Corps, I was going to make it on forty acres, and be all holistic and symbiotic, and everything working together,\" Rick said, but something about having more land and livestock than he'd ever imagined possible, along with a desire to prove himself to his father-in-law, brought out his competitive side. Rick and Heidi worked it all out on paper and decided to take the risk. They upped the operation to 150 sows and 100 cows. Heidi's father thought they were being reckless. \"We had a hell of a good year, that first year. We worked our backsides off, but we made a hundred thousand dollars on hogs.\" To free themselves from Tom's second-guessing, Rick and Heidi used their profits to buy out all of the remaining debt on the equipment. It took every penny, but their piece of the farm now belonged to them outright.\n\nBut it didn't stop there. \"After two years, I got Heidi talked into taking on three more quarters.\" Now approaching the peak of the Farm Crisis, land prices were at an all-time low. Rick was eager to buy up as much as possible, rolling each year's profits into buying still more land, on the assumption that an eventual recovery would set them up for life. So just three years after taking over their share of the farm, Rick and Heidi decided to use part of the original homestead as collateral for a loan to make still more land purchases. \"I could see what a land base did,\" Rick told me. \"Without land you cannot operate.\" He said he had everything nailed down for a loan from First National in York, but the day before he was supposed to sign the paperwork, he received a phone call from the loan officer.\n\n\"We've got a problem with our board of directors,\" the banker told him. \"Because everyone's going broke, we want Tom to cosign.\"\n\n\"By then Tom and I were just fighting like hell,\" Rick said, \"so I told him, 'Evaluate the loan on its own merits and if we don't qualify, we will seek assistance elsewhere.' \"\n\nRick tipped his head back, draining the last of his beer. Then he lined up the two empty cans in front of him and shook his head ruefully. \"So arrogant. Farming just three years, and I was that brash.\" The bank agreed to make the loan, and the pattern was set for the rest of Rick's farming life. \"Because of my aggressive ways, we have continued to stay in debt for thirty-two years and always pushing, doing a lot more than we should. My idea was to try and build an operation that I could hand down to my children with as much land as what we were benefited when Heidi's share was handed down to her. So I did everything I could to get ahead and to turn that success into more and more contiguous land. If you can swing it, you buy it. When you're a farmer that land means everything.\"\n\nIt was late now, pitch black outside the window. Even the moon had waned to nothing but a slim sliver of light. Beyond the reach of city glow, the only sense of a world outside came from the distant barn light, its dim bulb always left burning and just bright enough to give shape to the shadows.\n\n\"I'm probably close to bipolar,\" Rick said at last. \"When I'm on, everything is possible. I just go for it. But then I'll get really down, thinking, 'God, how are we going to get out of this?' And I put it on my kids, this guilt for my avaricious and aggressive ways\u2014that I was doing it for them. That was my excuse in my head.\"\n\nHe sighed deeply. Months before, Meghan had warned me that one of Rick's defining characteristics is self-doubt. \"He worries every decision to death,\" she said, \"and then, no matter what he decides, he always thinks there could have been a better way. Rick is a very regretful man.\" It's a common trait among farmers. The neighbor's corn always looks taller, their cattle fatter. Every farmer kicks himself for not doing enough to capitalize when the markets are up and for being too exposed when the markets are down. Rick's mood, in particular, seemed to rise and fall with the prices from the Board of Trade. He regretted every missed opportunity but even more deeply regretted the times he hadn't been more cautious, more careful in planning ahead. Why hadn't he set aside more money when there was $8 corn?\n\n\"I just get comfortable, to where the wolf's away from the door,\" he said, \"and what do I do? I go and remortgage everything and do dumb things\u2014buy more land, more equipment, build a barn or a new house. It's like, _Rick,_ _quit digging. You know what this is going to d_ _o to you._ \"\n\nHe tapped the empty cans on the table in front of him.\n\n\"And it's caused a hell of a lot of stress on my family,\" Rick said. \"Was that worth it? I don't know. That will be for them to decide.\"\n\nKYLE DROVE the pickup south across the interstate and then back east toward what all the maps label as Lushton, Nebraska, though it's barely more than a wide spot in the road anymore. And we were still several miles outside of town, making our way toward Seth's farm. Kyle explained that Seth's wife is a Barlow, that his mother-in-law was Joan Barlow. \"When we talk about 'Joan's half section,' that's who we mean,\" Kyle said. \"Seth farms most of the Barlow family land, but we farm that one property.\" Kyle told me that he didn't know how exactly to explain why Rick and Seth had started harvesting together, but it had something to do with a kind of shared stubbornness.\n\nA few years back, for example, Seth had taken a load of high-moisture corn to the grain elevator and been charged a drying fee. But Seth had seen the elevator simply blend his corn with a load of overly dry corn, so they hadn't actually had to turn on the dryer. \"I was there,\" Seth told the operator. \"The dryer was not running. I am not going to pay a drying fee.\" Rick had told me earlier that he'd admired Seth for telling the co-op that it wasn't right. \"I thought that was pretty gutsy of him.\"\n\nAs we hit the rise and the acreage appeared, we could see a wide field of ready soybeans and then a shelterbelt of tall trees, a farmhouse and cluster of outbuildings tucked behind. The beans, brown and mature, seemed almost to glow in the October light. \"Wow,\" Kyle said, surveying the field. \"It's ready to go.\" But then he turned along the south edge of the property and drove slow, so he could count rows and divide them to figure out the number of rounds to complete the field. As he went, I could see the shimmy and drift of Seth's rows. \"He just hasn't invested in auto steer yet,\" Kyle said and laughed. \"He just put this pivot up last year. Before that it was flood-irrigated with the pipes and all.\" It was a system that most farmers, especially in eastern Nebraska where center-pivot irrigation was invented in the 1950s, had abandoned long ago. Given the labor of changing gates, Seth was reluctant to irrigate except when it was absolutely necessary. It seemed to me that maybe Rick's admiration for Seth came from his ability to keep his farm producing by force of sheer will.\n\nBut it meant that Rick and Kyle were going to have to harvest the old-fashioned way\u2014keeping an eye on the outside furrow, following the wobble and drift of rows planted by the human eye rather than computer-drawn straight lines. The mood, though, as we got out and Rick and Kyle readied the combine, was decidedly light. It was Meghan's birthday, and she was in the grain cart, chattering away to Rick and Kyle over the CB. Seth would swap in occasionally. He had his daughter with him, and she was having a great time taking the wheel of the John Deere. She was used to riding with goggles and earplugs in her dad's clattering old International Harvester, and she was sold on the air-tight, cooled cab, the sounds of Kyle's classical music playing over the radio.\n\nBut as the sun started to set, the mood seemed to shift with it. Because Seth had planted by eye, every turn of the combine required lining up with where he had tried to match row spacing between each pass at planting. Farmers call these \"guess rows.\" The sixteen rows laid out by the planter are perfectly spaced, but the furrow between passes can be slightly narrower or slightly wider than the machine-spaced rows. Because the harvester is twelve rows wide, it's impossible to split each planting swath. Instead, you have some passes within the planted pattern and some where the harvester is straddling the places where two planting rows come together. But as the light failed and the shadows grew longer, it was getting harder and harder for Kyle to keep track of the guess rows and remain within the pattern. He leaned forward, peering down over the steering wheel, to be sure that the spinning reel of the harvester was standing up the rows and cutting everything cleanly.\n\nSeth, back in the spring, had planted around the concrete platform of his new center pivot for the first time. And rather than setting his row-spacing starting from that platform, he'd started from the edge of the field and worked in. Ordinarily, farmers won't plant closer than 30 inches from their pivot, but Seth, hoping to get a little extra from the field, had allowed just 5 inches. So as they neared the platform, Rick and Kyle had to figure out where to set the floating bar on the outside of the harvester head, what they call \"the snout,\" in order to stay aligned but without having to take multiple passes to get around the pivot.\n\n\"We try to put the outside of that head on the guess row,\" Kyle explained. \"But the eye-planted rows can sometimes run together. And if you don't guess right, you're taking half a swath trying to fix it\u2014which takes fuel and time.\" And if you get off by too much\u2014you miss a guess row and don't notice or just try to keep going\u2014you can start to run over rows, breaking the pods or treading the plants into the muddy soil and making them impossible to harvest. Seeing the trouble it was causing, Seth told them not to worry about the rows right around the pivot, but, even in a neighbor's field, Kyle wanted to capture every bit of the yield. So between passes, he would come down from the cab, study the rows with Rick and then hustle back behind the wheel.\n\nAt dinnertime, Meghan drove to pick up food Heidi had waiting for them at the house. Meghan returned with tacos wrapped up in aluminum foil and a cooler full of Cokes. Everyone sat on the tailgate of the pickup or leaned against the bed, eating quickly. Rick kept eyeing the sun, now sinking into the line of trees that stood between the edge of the field and Seth's house.\n\n\"I think we should call it a night,\" Rick said finally.\n\n\"It's all right,\" Kyle said. \"There's just a few more swathes.\"\n\nKyle was hoping to finish this field before it was full dark, so they could load up the head and the combine for another field for tomorrow. If they could get done in the next hour, they would have a jump in the morning. Together, Rick and Kyle walked over to the remaining rows, getting a read on how many rounds were left to complete the field, but also going back and forth about whether to continue or stop. Finally, Kyle won out. He took the combine down to the north edge of the field, then turned back toward the pivot, watching the spinning reel below him and steadying the wheel.\n\nAs he neared the south edge of the field, Meghan, waiting in the grain cart, came on the radio. \"You're driving crooked,\" she said. \"You're knocking over rows.\" Kyle looked down at the reel. Everything seemed in line. So he stopped the harvester and climbed down to see what was going on. Right away he could see: the snout was bent in and the plastic body of the soybean head was broken. He'd hit the pad of the pivot and never even felt it.\n\n\"Goddamn it,\" Kyle rasped under his breath. This could be thousands of dollars of damage. Worse still, if it was more than he could fix himself, it could be days of waiting for a Deere certified mechanic. He pulled back the plastic body and he stuck his head inside. Kyle told me later that he could see that the snout had been pushed in and bent up the hydraulic arm on which it floats. That arm was rubbing on a pulley belt that runs the cutting sickles, putting slack in the system and creating friction. \"It was still cutting,\" Kyle said, \"but it was already getting really, really hot.\" He reached in to see if he could straighten the bent arm by hand. No dice.\n\n\"I should have just left that row,\" Kyle said, still tugging hard on that arm. \"It was planted _right_ up against the pad.\"\n\n\"That's what Seth said,\" Rick snapped. \"You _cannot_ harvest something that's planted that close.\"\n\n\"Yeah, I probably should have just left that little bit.\"\n\n\"If Meghan wouldn't have seen it, it could have started a fire,\" Rick said. He couldn't hide his anger, but Kyle was already thinking about what they needed to do now. He said he could straighten the arm if he could just heat it up, but he couldn't afford to wait until morning to fix it. If they were going to finish this field in the morning and still stand a chance of loading up and getting some beans out of their own fields, Kyle was going to have to make the repairs right then, in the dark. He asked Seth if he had a cutting torch and a wrench he could borrow. Seth told them to pull around to his shop, and he set off across the field.\n\nBy the time Kyle and Rick had driven up in the pickup, Seth had already rolled out a portable light and the tank and hose and a rosebud heating tip for the torch. He grabbed a couple of wrenches. Kyle asked if he had any fender washers. Seth pulled out a box and shook several into his hand.\n\nHoping to get out of the way, I went with Meghan back to the house, where we waited, saying little for close to an hour before Rick and Kyle came rolling into the driveway. They stomped into the mudroom laughing and kicking off their boots. \"You asked for that fender washer, and Seth just said, 'How many you need?' \" Rick said, slapping Kyle on the back. \"Why can't I have a shop like _that?\"_\n\n\"I assume everything's working,\" Meghan called from across the kitchen.\n\n\"Yeah,\" Kyle said with a long sigh. \"We had to drill out some rivets and get the plastic bent up so we could get the torch in without burning everything up. And then we had to heat up that arm to get it straightened out so it quit rubbing. It was really hard steel and we couldn't get it bent by hand. And then a little piece on the backside broke. So we had to weld that back together.\"\n\nRick interrupted. \"I want you guys to know: the combine is better than it's ever been.\"\n\n\"Better than before?\" Meghan asked skeptically.\n\n\"At least,\" Rick said. \"We should hit every center-pivot pad.\"\n\nBefore long, they were seated together around the dinner table, recounting stories of all the near-disasters of past years. The time Dave took out a power line with the unloading auger. The time Meghan had swung wide to make a turn on a country road with the bean head strapped onto the flatbed trailer and clipped a stop sign. They laughed until all the tension and worry of the accident, the adrenaline of what could have happened, had been shaken off and disappeared.\n\nIn the end, American farmers harvested more than 14 billion bushels of corn in 2014 and nearly 4 billion bushels of soybeans\u2014setting new records, as they had the year before and would hope to in the year to come. With so much production, roughly three-quarters of the harvest nationwide went directly into bins, as every farmer waited and prayed for rebounding prices. They never came. Instead, prices continued to slump as yields continued to grow, and whispers spread about the possibility of another Farm Crisis. But for now, on that one night in October, the last hours of Meghan's birthday nearly gone, everyone was gathered in the kitchen, happy and laughing until they were ready to sleep and go again in the morning.\n\n\"The hardest part of my job is working with family,\" Meghan told me later, \"but that's also the best part of my job, because family can be pretty hard on each other, but at the end of the day, they're the ones that will be there for you in the hardest times. We've been through some hard times on the farm, and we're still here, still going\u2014and hopefully on to the next generation.\"\nINTERLUDE\n\nWORKING COWS\n\n_November 2014_\n\nFirst the truck appeared, then the cattle trailing along behind. The November sky was slate-gray and cloud-strewn, the temperature holding in the forties, just warm enough that the intermittent rain came as a dry mist rather than snow. Meghan was at the wheel of the white pickup, slowing occasionally to bang on the side door and call out to the hesitant cows. Their big bags swayed under them as they ducked over the furrows. \"Come, boss,\" Meghan called, beating the door again like a drum. Behind her, Rick and Kyle on four-wheelers stitched along the edges of the herd, circling up stray calves and keeping the line moving. Kyle, in a stocking cap covering his buzz-cut head, stood up on the running board, straddling his seat. He kept his hands on the handlebars, still easing forward, as he scouted across the line of jogging cattle. If he spotted a loose calf or errant cow on the other side, he whistled to Rick, who zoomed ahead, shouting in a hoarse staccato. \"Go on, now. Go, go, go. Go on, get moving.\"\n\nAfter the end of harvest, when the last of the beans were in the bins and the calls for white corn had started to come in from the elevator, when the rented harvester had been loaded back onto the flatbed and sent on to some other farm farther north, Rick turned his attention straight to getting the cattle ready for winter and then for spring calving. He drove the trailer out to Curtis, where he grazed the spring cow-calf pairs on pasture, growing them on grass through the summer and start of fall, and loaded them up. He brought them back to York County and ran them onto a quarter section of freshly harvested corn surrounding the solar-powered barn.\n\nIn fields like those, where the corn has been well irrigated by a center pivot, the cut stalks standing in a winter field along with the wayward cobs and kernels scattered by the combine, what farmers call \"crop residue,\" can provide ample feed without any additional cost. The moisture left in the stalks plus the greens from the turnips and radishes planted as winter cover crops are enough to get lactating mothers through until weaning\u2014and for those cows without calves, calories from those cover crops can actually help them put on pounds. But if the cows were going to be nursing newborns in a few months, then it was time to separate the spring calves from their mothers in order to wean them, and it was time to check the cows for new pregnancies.\n\nMeghan groaned just thinking about it all. \"When harvest is over, you'd think you could catch your breath,\" she said. \"But we're straight into the cows\u2014bringing them home from pasture, sorting them, weaning them, and getting them to the sale barn. It's a lot.\" She hit the gas, speeding up to the pipe corral attached to the old barn. She jumped out of the truck and swung open the wide metal gate. \"C'mon, girls,\" she called again. Heads bobbing, the first of the mothers walked obediently into the pen, their calves skipping but sticking close. The Hammonds raised all Angus cattle for beef. In recent years, Rick had moved away from hormones and feed-grade antibiotics, and he had tried to do more direct marketing, selling halves and quarters to customers willing to pay a premium for hormone-free, grass-finished beef. But with so much risk already sitting in his grain bins, Rick was content to get his calves to market and get more on the way. Beef prices were soaring, hitting $6.30 per pound at retail, more than a dollar above where they'd been just a year before. Most years, Rick regards the cattle as a side operation, work he enjoys but doesn't expect to pad his wallet. But with prices climbing and several years of drought driving the national cattle herd to a lower point than it had been in decades, the sale of beef promised to be the farm's saving grace for 2015.\n\nOnce the cattle were all in the large holding pen, Meghan swung the gate closed and grabbed her notebook. They would go systematically through each cow-calf pair, checking ear tags and noting any obvious health problems, before running them into separate pens. After that, they would herd the calves into the chute for more detailed health inspections. Rick, Meghan, and Kyle, as well as Dave, who had been waiting back at the barn, all filed into the wide paddock, the cattle stooping and scattering around them like startled birds. Rick had a pair of rattle paddles, like two oversized lollipops, lemon and orange, but inside those bright-colored plastic pads were ball bearings that shook and spooked the cattle forward.\n\nArms stretched wide, coaxing and calling, shaking one paddle and then the other, Rick separated calves, one by one, from their mothers. He sent the weaned calves one way and then the other, until they skittered toward the gate, which Dave would swing open to usher them into the calf pen. If they showed signs of balking at the doorway, Dave and Kyle had herding rods, nothing more than long flexible poles, to tap their hindquarters\u2014just enough to trigger a prey animal's flight instinct and send the calf galloping ahead. Eventually, when the calves were separated out, the mothers were moved as a group to another pen, and the next batch of pairs was brought in.\n\nThis was how it proceeded for hours, cajoling and cursing. The four dust-covered and wind-chapped humans wading into galloping herds of erratic cattle, their black flanks twitching and mud-patched, their hooves kicking up more clouds of dirt in the holding pen. Rick would call out to the cows, then to his helpers, and before long he was yelling at Meghan, Meghan yelling back at him, and then the two of them were laughing at each other. And with each new group, the din of the corral grew: calves bawling for their mothers, worried cows mooing in response, straining to lift their mouths, tongues lolling, over the crush of other fussing mothers.\n\n\"Working cattle is a high-stress time,\" Rick had warned me ahead of time. \"Be prepared for a lot of noise.\" There was certainly that, but it only dawned on me later that it wasn't so much the chaos he was worried about as how I might respond to the distressed and calling mothers or the sight of the skittish calves as they bolted through the gate and were snapped into the chute, so Meghan could complete a wellness check, looking for tumors or signs of nasal or bronchial infections or pinkeye. She marked down the information for each as she went, the pinned calf tense in the chute, sometimes letting out a meek bawl, sometimes releasing a milky panicked poop, its eyes rolling and wild to be free.\n\n\"We had pinkeye terrible this year,\" Meghan told me later, \"and pneumonia in one group.\" She shook her head in frustration\u2014another side effect of all the rain late in the season. \"We don't get pinkeye out west, because it's a drier climate.\" But in the eastern part of the state, flies hatch out with the rains and swirl over manure piles left in the fields and feed yards. When those same flies cluster around the eyes of cows, tonguing the salt right out of their tears, pinkeye can spread and then pass from mother to calf and through the herd. \"We treated them all ourselves, except for a few that were pretty bad,\" Meghan said. \"They had to go in and get a shot in the eye and have the lid stitched shut.\" That subconjunctival injection of antibiotics to kill the infection, mixed with steroids to knock down the inflammation, usually takes care of the problem in just a few days, but it also means that the animal can no longer be sold as antibiotic-free or steroid-free. A rash of pinkeye can close the door on a whole season's direct marketing and its premiums.\n\nThis second-guessing of basic animal husbandry, often to satisfy a grocery shopper who has never been on a farm, is a sore point for almost anyone engaged in raising livestock these days. Every cattle rancher is quick to talk about how they have scaled back on chemicals and drugs, and they can barely conceal their suspicions of outsiders, who might object to ear tags or branding, or mistake commonplace herding for rough handling. Even in the intricate systems of corrals at livestock auctions, where most cattle are still moved by skilled cowboys on horseback and drovers with herding rods, I've been told innumerable times that electric prods are rarely necessary, that no one wants to stress the animals. At one large feedlot in Bridgeport, Nebraska, the owner wanted me to know that his son had gotten a bachelor's degree from the sustainable agriculture program at Yale and that his daughter had gone to college at Colorado State, where she studied under Temple Grandin. She had come home and redesigned all of their chutes and gates, he told me. Almost across the board, beef producers feel scrutinized and critiqued, and they have a love-hate relationship with a consumer base willing to pay extra for an antibiotic-free steak without any thought for the potential suffering caused by denying drugs to the cow that produced it. More than once a rancher or feedlot owner has asked me: \"Would you prefer that I withhold medicine from a sick animal?\"\n\nBy evening, when the gray sky deepened to black and the temperatures dropped below freezing, Meghan and Rick seemed fatigued both by the physical exhaustion of the all-day work and by the inescapable feeling that they were somehow under surveillance. As direct marketers of their own beef, they know well that much of the new food movement is built on a nostalgia for a bygone era of family farms and natural food production, and all of the farming magazines and websites are always urging them to \"tell their own stories.\" They're not trying to hide anything, but they often feel forced to prove their heightened commitment to land stewardship and animal welfare and building sustainable agriculture in support of communities and health. And they know all too well that this urge for \"food with a story\" has only encouraged corporations, with much bigger marketing budgets and an army of branding experts, to reposition their products and remake their image. Not only have the big meat producers taken over any organic or hormone-free company large enough to take a bite out of their market share, but they have often assumed the identities of the very farmers they are trying to elbow aside.\n\nIn just the last decade or so, Smithfield Foods acquired Farmland, Tyson took over Hillshire Farm, and Hormel bought out Farmer John. Now, the products from some of the largest and most industrialized food producers in the world are packaged with logos featuring red Dutch gambrel barns, white farmhouses, and smiling farm families, decked out in denim shirts and overalls, holding grandchildren and hand tools for hoeing rows. Shoppers see those pictures and assume that's how their food was raised. \"But how does the little guy get the message out?\" Rick asked. He was slumped over the bed of the pickup, now wearing a brown pullover stitched with the logo for R Diamond, the name of his cattle operation. Consumers make demands, he said, and it's easy for major corporations to pass along those new requirements to their growers. \"No sweat off their back,\" Rick said. But for farmers maintaining narrow margins and getting squeezed from both ends, it was hard not to feel trapped.\n\nLater, as we all sat around Rick's kitchen devouring a pizza that Heidi had picked up at the Casey's General in Aurora, Rick downplayed his worry. \"I don't care if I come off as the dumbest SOB,\" he said. \"If there's truth in advertising, I don't see how it could be any other way.\" He said he was more concerned that life on an American farm isn't all it's cracked up to be, and that he appreciated it when people like Willie Nelson and Neil Young ride in and give concerts, but he had grown leery of how they romanticize small farms. \"I'm afraid what you're going to find,\" he said, \"is that the idyllic family farm that Willie and Neil have been trying to save does not exist. And by the way: the good old days weren't so good either. Financial woes and danger and tired all the time.\"\n\n\"I want you to tell the truth,\" Meghan interjected. \"But you have to understand: our family and neighbors are also our coworkers. They're going to be there when someone dies or when we have a big fire or when the cows get out.\" She had seen how protesting against the Keystone XL pipeline and the spotlight of the media had turned one neighbor against them already, and she was worried about what further attention might do. \"If there was something that our neighbors didn't like and they wouldn't let us farm their ground or run our cattle on their fields, I would just be heartbroken.\"\n\nMeghan hesitated, then seemed to wave it all off. She put her plate and fork in the sink, the darkness outside the window now deep and complete. But then she had one more thing to add. She wheeled and braced herself on the edge of the sink, as if she'd suddenly been overcome by great exhaustion. \"It's just that we've sacrificed so much,\" she said. \"We work hard just trying to keep our family together.\"\nPART TWO\n\nTHE HOMEPLACE\n\n_November 2014\u2013February 2015_\n\nTo understand, first remember: Nebraska is a place. It sits square as an anvil in the center of our maps, and yet, somehow, everyone manages to forget it exists. Maybe that's because Nebraska is also a land of ghosts, of small towns dwindling to the point where, in another generation, they might simply cease to exist. In Benedict, the nearest town to Centennial Hill Farm, the school stands empty and abandoned; the only restaurant has been for sale for years. There's a grain elevator, two well drillers, a feedlot outside of town, but, otherwise, there's no work, nothing to do, no reason to be there instead of anywhere else. West of town, approaching the hard edge between York and Hamilton counties, the roads lose their Civil War names\u2014Lincoln Avenue and Shiloh Street, Sherman and Atlanta, becoming just a grid of letters and numbers, unfolding in perfect one-mile squares. At the crossroads of 22 and G, five miles west of Benedict, is Centennial Hill, the piece of ground held by\u2014and holding\u2014Meghan's family for generations.\n\nIt doesn't look any different than the miles of flatlands surrounding it in all directions, but it is the center of the family's universe, the homeplace. Every farm family has one. Though nearly all operations have expanded in the decades since the Farm Crisis, acquiring land and growing piecemeal, these are just scattered acreages, constellated satellites orbiting the homeplace's center of gravity. In most cases, the new land has no emotional importance and doesn't even get a new name. So when families like the Hammonds talk about the patches they own and farm, it is a kind of geography of the gone: the Metz place, Roger's dry-land patch, Karen's quarter. It's all the people and families that disappeared off the land, leaving only their names, like tombstones, as a record of the generations spent there.\n\nThese places are critical now to the survival of the Hammonds' operation, but the homeplace, Centennial Hill, is different. To really understand why, to count the generations of hardship and hard work that have gone into holding on to the farm, Meghan said I should go see her aunt Jenni. She is the acknowledged keeper of Harrington ancestral lore. The basement of her Victorian farmhouse, cattycorner from the homeplace, holds the family archive\u2014the scrapbooks filled with faded clippings fixed in place with yellowed and brittle tape, the box of old photographs, their corners all curled and chipping. When I arrived, it was cold outside, no snow on the ground but the winter sun pale and sharp. Light poured through the north-facing window into the dining room, where Jenni had laid everything out on the table.\n\nIt was uncanny how much Jenni looked like Heidi, like all of her sisters. They were all in their fifties but had an intangible youthfulness about them. They all had kind and lively eyes, a soft but frank tone of voice, and smiles that could be slow but genuine when they emerged. Jenni welcomed me inside and got me a cup of coffee. Once I had warmed up a bit, she brought me back to the dining room, where she sorted through stacks of sepia and silver prints until she found the one she wanted: a picture of the first proper house on the farm. Her father had had the place torn down when Jenni was still little, but it seemed to float in her memory, blurred and overexposed like the ghostly image in the photograph. It was irretrievable but also forever present. She studied the image, holding it up to the light in her strong, tan hand.\n\n\"Our great-great-grandfather was Thomas Barber,\" she said. \"He came from Suffolk, England, with his mother and father when he was about age four.\" The family settled in Wisconsin in 1852 and farmed there, but it was less than a decade before their adopted country was divided by war. When he was still not quite sixteen years old, Thomas signed up for the Federal Army and was assigned to the 13th Independent Battery, Wisconsin Light Artillery. He had already crossed the ocean and traversed half his new country by rail; now he was sent the length of the Mississippi River to Louisiana, where he spent eighteen months as a member of the fortified defenses in Baton Rouge and New Orleans. After the Confederate surrender, Thomas returned briefly to Wisconsin but felt restless and eager to strike out for the territories. He was not yet eighteen when he went west.\n\nAccording to Jenni, Thomas was hoping to take advantage of the Homestead Act. The first Homestead Act had been signed into law by Abraham Lincoln in May 1862, just as the trenchant and gory battles of the Shenandoah Valley Campaign in Virginia were signaling that the Civil War would be more than a brief insurrection. As the bloodshed increased, President Lincoln began looking not only for tactics to hasten the war toward conclusion but also ways to create a lasting postwar peace, both free from slavery and from the conflict the \"peculiar institution\" had created since the nation's founding. Lincoln's ingenious solution, the Homestead Act, granted small farming plots in the western territories to any citizen who had never taken up arms against the Federal government and was willing to occupy the land and \"prove it up\" with a year-round habitable home and a crop that could be shown to be sustainable for five years.\n\nBringing families onto the barren prairie might seem far removed from the battles raging in the Eastern Theater, but the dispute over western settlement\u2014whether the Kansas and Nebraska territories along with other points west would be divided and designated as new slave states or free soil\u2014had been the central political question of the 1840s and 1850s. The admission of Kansas as a free state in January 1861, just before Lincoln's inauguration, was the match that finally lit the fuse of civil war. It had realized the abolitionist dream of the martyred John Brown and, coupled with the fear that the new president was a silent opponent of slavery, it had propelled Southern states toward Secession.\n\nBut Lincoln, already a shrewd and hard-nosed statesman, recognized that the roiling of the country toward constitutional crisis had also created a political opportunity. With the South removed from congressional debates, Free Soil members of the House and Senate had the numbers to configure western settlement as they had always envisioned, as a stronghold of small-plot farms worked by free white labor instead of large plantations held by small numbers of wealthy landowners who thrived on the backs of slaves. Though he had only recently emerged as the unlikely leader of the Republican Party, Lincoln readily understood that such a realignment of the nation's social structure was both the mandate of his election and the best chance his newly formed party had for broadening and shoring up its base of power. The rapid settlement of the American West into small, self-sustaining plots would ensure that, when the horrific battles of the war had ended, the country would be firmly established as a free nation of independent farmers.\n\nSo Lincoln's team came up with a simple plan: give away the unsettled territories 160 acres at a time. The size of the allotment may have seemed arbitrary even at that moment, but it was actually the direct legacy of the Public Land Survey System. Under this plan, the western regions ceded to the new republic by King George at the end of the American Revolution and later the open prairieland of the Louisiana Territory acquired from Napoleon had been mapped out according to a rigid grid and tamed into townships, each six miles square, each township further subdivided into one-mile sections, and each section divided again into 160-acre quarters. Railroads stops were extended to each township, dirt wagon paths were built along the section lines, and farms were allocated one to a quarter.\n\nWith the passage of the Homestead Act, the grid was extended onto the flat expanse of central Nebraska, dividing the land evenly, eschewing the realities of terrain in favor of the harsh simplicity of geometry. In those days, there was no one to lend a name to the country lanes. And to this day, the east-west gravel tracks have only numbers; the north-south roads are lettered. The intersections are still coldly determined\u2014precisely a mile apart, unchanged from the nineteenth century. The only deviations on the unbroken straightaways come in occasional hard jogs inserted by surveyors to correct for the curvature of the Earth. Otherwise the grid is only interrupted in the precious few places where it is slashed by the braided diagonal of the Platte River.\n\nDespite this strict adherence to the vagaries of the plat maps, Lincoln was a pragmatist when it came to meeting the challenges of farming on the windswept prairie. Having spent years as a young lawyer and congressman on the frontier of Illinois, he knew life for homesteaders would not be easy; he understood what it would take to lure them and also what they would require to survive. So along with passage of the Homestead Act, Lincoln created the U.S. Department of Agriculture and also signed the Pacific Railway Act and the Morrill Act, named for Representative Justin Smith Morrill, who introduced the legislation. The Pacific Railway Act authorized large land grants to the Union Pacific Railroad and Central Pacific Railroad to encourage their growth westward. The Morrill Act provided each state with still more land grants of 30,000 acres per congressional seat. Funds from the sale of the land were to be used to establish schools of agriculture and mechanical arts, dubbed A&Ms. And the newly designated Department of Agriculture announced that it would conduct research on best farming practice and dispense information through the A&Ms to aid the growth of agriculture in the West through hard science.\n\nThe creation of the USDA was especially significant, because it heralded a new era of government intervention into agriculture\u2014not merely providing incentives for spreading farming into the West but actively providing support. To head the new department, Lincoln selected a man with the unlikely name of Isaac Newton, a Pennsylvania farmer and entrepreneur who had risen to become superintendent of the Agricultural Division of the U.S. Patent Office. Newton declared that the USDA would focus on several key areas, including scientific analysis of farm data, collecting and analyzing soil, supporting the creation of hybrids and regionally specific crops, and testing and promoting new equipment. He set up an experimental farm on the National Mall to keep his initiatives constantly in view of Congress; he lobbied for funding to establish professorships of botany and entomology at the A&Ms; and he advocated for implementing British meteorologist Robert FitzRoy's use of the telegraph to gather daily weather observations, what he called \"forecasts,\" that could be shared with farmers by way of their local newspapers.\n\nBy the time Thomas Barber arrived in Nebraska in 1866, the country was entering a renaissance of modern farming. But before he could take advantage of the nearly free land and agricultural know-how on offer from the government, he first had to figure out how he would meet the conditions of the grant. If he was going to occupy the land and also improve it with a home and a successful crop, he first was going to need some money. So in June 1866, he took a job on a farm in Sarpy County on the bustling edge of Omaha. He worked there for four years, seeing Nebraska become the first state admitted to the newly preserved Union and slowly putting away $190. He used those savings to rent a farm and built his nest egg for another two years. Somewhere along the way, Jenni wasn't sure just when, Thomas married Mary Mitchell and had a daughter. With a family now, and enough saved to make a go of things, he set out looking for land in the counties west of Omaha, particularly York and Hamilton counties, where the richest soil was reputed to be. But by now it was 1874, and he found that whole area had been homesteaded and much of the desirable farm ground there, which had been available for just $18 per quarter section barely a decade ago, was now selling for a steep $20 per acre.\n\nThat summer, Thomas caught what would prove to be an unlikely break. After nearly two months of drought in May and June, a disaster of apocalyptic proportions befell eastern Nebraska\u2014and most of the American West. \"In a clear, hot July day,\" one eyewitness in Seward County remembered, \"a haze came over the sun. The haze deepened into a gray cloud. Suddenly the cloud resolved itself into billions of gray grasshoppers sweeping down upon the earth. The vibration of their wings filled the ear with a roaring sound like a rushing storm.\" For ten days straight, innumerable grasshoppers swept across the middle of the country from Texas to the Dakota Territory, devouring everything in their wake. Newspapers reported that the locusts chewed through fields of corn so fast that it sounded like a crackling fire, and the clouds soon grew so thick that they stalled a Union Pacific engine at Stevenson station, near Kearney, Nebraska. By August, there was nothing left, and thousands of farmers, who had been counting on their harvest to keep them afloat, were panicked.\n\nOne of those early settlers, Reverend C. S. Harrison, had come to York County in 1871, in search of unsaved souls on the godforsaken plains. He had purchased a piece of ground and laid out a hatchwork of homesites. The railroad was rumored to be on its way, and Harrison planned to sell homes to the lineworkers, feed-mill employees, and shopkeepers who would follow. In order to receive an additional quarter section awarded by Nebraska under the Timber Culture Act of 1873, Harrison had planted trees along either side of the lanes\u2014Cottonwood Street, Box Elder Street\u2014hoping they would one day spread and arbor the yards. He predicted that the Great American Desert would soon be forested to the base of the Rocky Mountains and, to show his confidence, named his tiny tree-lined town Arborville. But that savage summer, the grasshoppers ate the paint off his church, the handles from plows, and crops out of the fields. They chewed the leaves, the _limbs_ , of Arborville's trees down to the trunks.\n\nNearly three-quarters of the American agricultural crop of 1874 was lost. One Nebraska newspaper mused: \"We have heard some sighing, 'Oh, for a Moses to lead us out of this land of grasshoppers.' We fear the next cry will be, Oh, for a Joseph to sell us corn.\" And, indeed, farmers soon had no feed for their cattle or hogs, and the bodies of dead grasshoppers choked and putrefied their ponds and open wells, leaving them without water. By October, with winter approaching, desperation had begun to set in. The _Omaha Daily Republican_ estimated that more than 10,000 people (in a state of fewer than 200,000) were in need of help \"to keep them from actual starvation, until next harvest.\" Those who feared they couldn't make it that long flowed back east, selling land for pennies on the dollar or abandoning it altogether. Hamilton County, hoping to stem the hemorrhaging, offered low-interest loans of up to $200 for anyone willing to stay, but still, families loaded their wagons and left.\n\nFarmers will tell you that rural communities unite at moments like this, rallying together to survive. It's not really true. In times of crisis, the pragmatic and the prudent pack up and go. Only the most stubborn remain, over the years and decades giving rise to loose, often unincorporated and thus ungoverned communities of hardheaded hermits, prideful in having outlasted their neighbors. And they are experts in finding opportunity in others' misfortunes. Thomas Barber, with almost as much saved cash as any man in Nebraska could get on loan, arrived in York County and selected 120 acres of prime land, two miles east of Arborville\u2014what he later named the Centennial Hill Farm. The hill is really something less than a rise, just enough to give the land a southward slope. So the property is sufficiently level to hold a steady rain, but not so pancake flat that the rows puddle in a downpour. Instead, runoff pools into a natural catchment along the edge of the acreage, bounded by the roadbed. It's a perfect spot for farming\u2014easily planted and with a ready source of water.\n\nJenni handed me the family scrapbook, open to a clipping from some long-forgotten Nebraska newspaper. The editors had asked Thomas late in his life to recall his arrival in York County. \"On the 13th day of November, 1874, I moved on to the land,\" he wrote. He had an 80-acre homestead and 40 acres picked up for cheap. He brought with him, in addition to Mary and their infant daughter, two cows and twelve hogs, along with the most modern farming equipment of the day: a combined reaper and mower, a threshing machine, two wagons, and six horses to supply the power. He built a sod house from the prairie soil, and almost exactly one year later, Lela, his second child and the first born on family land, was delivered there.\n\nOver the next few years and successive summers of locusts, farmers continued loading up and fleeing Nebraska. As they did, Thomas registered neighboring fields as abandoned and bought them, and he filed a timber claim that allotted him more land in return for planting trees. His family's first farmhouse, the replacement for the original soddy, was erected in what would become a windbreak created by that line of even-spaced saplings, and its frame and shingles, its clapboard siding and floors, were built from the milled planks and beams of Arborville's denuded trees.\n\nFor the next six years, Thomas raised wheat and barley, hauling it some thirty miles to market in York, and he picked up extra money by threshing wheat for his neighbors as well. \"Just as fast as I could get any money ahead I put it into land until I secured 320 acres,\" he remembered. When the town of Bradshaw was platted along the Burlington & Missouri River Railroad in 1879, Thomas had a ready way to get his livestock to Omaha, Kansas City, and even Chicago. He expanded his herds of cattle and hogs and began growing more and more corn to feed them. Eventually, he was raising as much as two grain carloads per year, enough to start a small cattle-feeding operation. At the same time, he bought up surrounding plots of land, paying $20 to $28 per acre, until he had 520 acres of contiguous land.\n\nWhen the Kansas City & Omaha Railroad arrived in 1886, the town of Benedict was platted exactly five miles east of Thomas's land. In no time, the depot was a hive of activity with supplies for Sparling's General Store arriving daily and grain and livestock being loaded to be sold in Lincoln and Omaha. A schoolhouse was built, where Thomas sent his children during the week. And on weekends, the opera house hosted phonograph concerts, dances, and plays. Perhaps it was there that Thomas's daughter Lela met Henry Moore Harrington, known affectionately as Harry, whose uncle had put up the land for the town and still owned the farm directly across the tracks. It almost certainly was there that Lela and Harry's relationship, little by little, grew into a courtship. On New Year's Eve 1894, they were married, and Thomas Barber, after twenty years on Centennial Hill Farm, turned the operation over to Harry and moved into York to retire.\n\nMEGHAN WAS never as caught up in the tradition of Centennial Hill Farm as her mother and aunts. Yes, she grew up there, within feet of where Thomas Barber's original farmhouse was built, but as high school graduation approached, she couldn't wait to get away. She had a storybook senior year at High Plains Community High, the consolidated school drawing kids from farms for miles around. Even in a tiny class, where everyone does everything, Meghan stood out. She was the homecoming queen, senior class president, junior marshal, band majorette. She played on the volleyball and basketball teams. But she had grown up skiing in Colorado, taking vacations to visit family in New York. For her, brighter things beckoned. When she went away to Metropolitan Community College in Omaha, she told me, she wasn't looking back. \"I was sure that I was never coming back to this place,\" she said, \"never going to scoop any more poop in the barn, never going to sort cows again in my life. I was done.\"\n\nHer high school sweetheart, Brent Zoucha, had graduated a year ahead of her. From the small town of Clarks, just across the Platte River from the house Rick built when Meghan was in high school, Brent was the all-American kid. He worked at Pollard Oil in Clarks, changing oil and fixing cars, and he was a star of the basketball and track teams, placing in the top three in the high jump at the state track meet in both his junior and senior year. But Brent was also shy and a little goofy. Initially unable to work up the courage to ask Meghan out, he had stopped her by her locker to see if he could borrow her calculator. It was enough to get them talking, and eventually they were inseparable.\n\nWhen Brent graduated in May 2005, he enlisted in the Marines, following in his older brother Dyrek's footsteps. Brent went through basic training at the Marine Corps Air Ground Combat Center in Twentynine Palms, California. Meghan had wanted to attend his graduation that January, but her parents insisted that she stay in Nebraska and focus on finishing her own school year. She was there in Omaha when Brent arrived back home after boot camp, and she was there again seeing him off from the airport when he was deployed to the Anbar Province of Iraq as part of the Marine Expeditionary Forces. They kissed and said their goodbyes. Being apart from Brent for nine months was more than she could imagine.\n\nWith Brent now gone, Meghan made plans for a late-summer trip to Ireland before classes started in the fall, and she spent the rest of the summer hanging out with friends in Omaha and Lincoln. On an early morning in June 2006, Meghan was sleeping at a friend's apartment across the street from Southeast Community College in Lincoln. \"We'd been out partying the night before,\" she told me. The landlord called first thing in the morning, rattling their hung-over nerves, to say that there were going to be repairmen coming by later in the day. They had just gotten back to sleep when the phone rang again. They groaned loudly as Meghan's friend answered the phone impatiently. \"It was my mom,\" Meghan said. \"She just said that there was an accident, and I needed to come home.\"\n\nWe were in the living room of Meghan's home just up the road from her parents' house, a light-filled afternoon in the quiet weeks after harvest, as the year runs toward winter. She paused a moment, fighting back tears. She said she yelled at her mother, demanding to know what was going on, but there were no details. All anyone knew was that Dyrek had called Brent's mother Rita and told her that Brent was hurt. In shock, Meghan's friends helped her gather her things and then drove her back home, all the way from Lincoln. As they drove, Meghan kept hoping for the best\u2014that this was a mistake or that Brent was hurt but not badly. \"Usually, it's a casualty assistance crew that comes out and tells you, if it's really bad,\" she said, and no one had contacted Rita yet.\n\nOnce Meghan was home, her parents drove her to Clarks to wait with Rita for further news. Hours passed. As evening came on, two men in uniform pulled into Rita's driveway. \"I guess that's when you know it's real,\" Meghan said. She wiped her eyes. \"Normally, I'm not sad. Normally, I can talk about it.\" They soon learned that Brent's Humvee had been hit by a roadside bomb in Al-Qa'im, where the Sunni insurgency was now spiraling out of control. The improvised explosive device had killed Lance Corporal Zoucha and two other Marines.\n\nMeghan said she would never forget the flood of emotion as the two casualty notification officers delivered the official news. \"But the hardest day is the day the body comes home. You go to the airport and you expect them to walk off the plane, as good as the day they got home from boot camp. But they don't. It's nighttime and you go down to the tarmac and you receive the body.\" Dyrek was not hurt in the explosion, but he was detailed to return to Nebraska with his brother's remains. He was supposed to arrive on the same flight as Brent's coffin, but one of Dyrek's connecting flights was delayed. So Meghan met the funeral services crew and waited with Rita and Brent's family with the hearse. They loaded the casket and started the two-hour drive.\n\nWhen they finally neared the funeral home in Central City, it was the middle of the night, but people lined either side of the highway. Those in cars and trucks fell into a convoy trailing the hearse. In the days after, Rita appointed Dave Beck, a family friend, to deal with the press. He told the local newspaper that Meghan was having a rough time accepting Brent's death. \"They were the picture-perfect couple,\" he said, \"the homecoming queen and the Marine. She was waiting for him to get home.\"\n\nAs the Zoucha family made arrangements, they soon learned that members of the Westboro Baptist Church of Topeka, Kansas, were threatening to show up at the funeral to stage a protest. A hate-filled family cult, all related to founder Fred Phelps, they believe that all tragedies are divine retribution for modern society's acceptance of Jews, Muslims, and gays. Since the wars in Afghanistan and Iraq had begun, they had been arriving uninvited at military funerals, chanting during eulogies and carrying signs that read \"God Hates Fags\" and \"Thank You for Dead Soldiers.\" Zoucha family friend Carol Hanson was determined not to let the Phelps family disrupt Brent's funeral.\n\nWith Rita's permission, Carol began working with local law enforcement, the Department of Defense, and St. Peter's Catholic Church in Clarks to map out the funeral procession route. They figured out where protesters might set up and devised a plan for how best to shield family and friends. She called in the Patriot Guard Riders, a national organization of motorcycle enthusiasts created specifically to oppose Westboro Baptist, to form a line of flags to block the view of protesters. If their chanting got too loud, the guard was instructed not to engage them, not even to make eye contact, just to kick-start their motorcycles and idle the engines to mask the sound. In the end, the protesters never showed. Members of the cult were at funeral services in Beatrice, Nebraska, for Army Private First Class Ben Slaven of Plymouth, but a new state law had barred protesters from standing closer than 100 yards from a funeral parade route\u2014and it appeared that the Westboro Baptist protesters were put off by their inability to disrupt services.\n\nSo on June 20 in Clarks, there was no protest, no clash, but the Patriot Riders and everyone in town had made sure that Brent would have what the local newspaper called \"a red, white, and blue sendoff.\" Hundreds of people, all of them with American flags, lined the brick streets. \"Stooped World War II vets struggled to hold large flags aloft,\" the newspaper reported. \"Little Leaguers in uniform waved mini-flags and held hands over hearts; cheerleaders wore red, white and blue skirts. And many residents had run their front-yard flags only halfway up their poles, as they had every morning since the bad news traveled from faraway Al Bu Hardan, Iraq, to this town of about 375 people.\" The Patriot Riders circled through town and, with American flags mounted to their rear seats, led the procession to St. Peter's.\n\n\"It was crazy,\" Meghan said. \"The whole town packed in there. It was wall-to-wall.\" She sat near the front, close to Brent's flag-draped coffin. Marine guards stood at each corner of the sanctuary. All told, more than 500 people, the entire population of Clarks plus more from far away, packed into the sanctuary, balcony, basement and then out onto the front lawn of the church. They listened as Father Ken Vavrina told them the young Marine had lived a life that honored the flag and died trying to protect it. \"There are some people, and thank God, who say, 'What can I do to make life better for others?' \" the priest said. \"Lance Corporal Brent Zoucha was one of those young men.\" Following the services, about a dozen of Brent's fellow Marines carried the casket and led the procession out of the church.\n\nThe bells rang, and everyone poured out. The procession, led by veterans and Patriot Guard members, crept down West Amity, between the rows of flapping flags posted along the way. The line of motorcycles, trucks, and cars, flags mounted to the antennas and top racks and handlebars, inched down the crowded streets, past the fire station where the department had brought out the trucks and raised the ladders. Police blocked off Highway 30, allowing the procession to zigzag across the track, onto the blacktop, and then down the gravel road toward the country cemetery.\n\nThe wind whipped across the treeless green, the ropes of the flags snapping and making the poles ring. They said the Lord's Prayer. The Marines fired off a 21-gun salute. A lone bugle corps musician played \"Taps.\" Two Marines folded the flag from Brent's coffin and gave it to Rita. Then Brent's coffin was lowered into the plot next to his father and everyone returned to their cars. The flags up and down the streets in town were lowered, folded, and put away. The COME HOME SAFE sign in Rita's yard was taken down and carried to the garage, replaced by a permanent memorial with Brent's official military portrait. When the semester started in Omaha, Meghan wasn't there. She decided to take the trip to Ireland that she had postponed, and waited until January 2007 to start classes. When she did finally go off to Omaha, away from home and on her own at last, she didn't find the sense of independence she had been looking forward to just months before. She felt lost and alone.\n\nAfter her first semester, though, Meghan was determined not to go back to the farm for the summer. She took a job as a porter at an auto dealership in Omaha. \"Just something to do,\" she said. \"A way to keep busy.\" The shop manager thought she was great and kept offering to set her up with one of his former mechanics. Meghan kept putting him off, not sure she was ready and not wanting to explain. In August, just as classes were starting up again, Meghan confided to Rick that, more than a year after Brent's death, she had consented to a blind date with the mechanic. She wasn't sure yet what to think about dating him or anyone else, but all the girls in the office raved about how nice he was, and the guys in the shop praised his skill under the hood. Everyone agreed that he was a really hard worker.\n\nRick had been looking for help with that year's harvest, so Meghan put him in touch with Kyle.\n\nIN THE spring of 1895, when Harry Harrington took over farming Thomas Barber's land, the family operation was already in serious trouble. Those 520 acres were considered some of the best ground in Nebraska, but severe droughts in 1893 and again in 1894, not only presented a hardship for the Harringtons but for the entire state and then the nation and eventually the world. Global failures of the wheat and corn harvests had posed a serious threat to the world's food supply, touching off famine in Europe, Russia, and China. Matters worsened when nervous stockholders and commodities speculators started to pull their money from the free-falling crop market and put their cash into reliable gold. So many investors followed this course of action, however, that it caused a run on gold\u2014and set off additional waves of panic. The silver market collapsed, putting Colorado miners out of work and creating a tent encampment along the South Platte River outside Denver that was nearly half the size of the city itself. Attempts to slash wages in large industrial cities led to violent strikes and crackdowns among the Homestead steelworkers in Pennsylvania and the Pullman workers in Chicago.\n\nThough remembered now as the Bank Panic of 1893, the crisis of the 1890s was actually a global economic depression that lasted almost to the end of the decade and was the worst downturn in American history until the deepest trough of the Great Depression. In Nebraska, it may actually have been worse than the Dust Bowl years. Charles H. Morrill, a well-known Nebraska farmer and banker who lived through those times, later recalled that farmers were forced to market their entire grain crops and even sell their hay. Without anything to feed their livestock, farmers starved their animals or simply butchered them for food. \"The entire state,\" Morrill wrote in his memoir, \"was almost in the grip of actual famine.\" And when the harvest failed again in 1895, the effect was devastating. In small towns across the state, the shortage of gold, which was used to back paper currency bank by bank, led ordinary citizens to withdraw their savings all at once. \"Values were greatly reduced, merchants and banks failed,\" Morrill wrote. \"Farmers could not pay interest on their mortgages; land could not be sold at any price; foreclosure of mortgages was the general order.\"\n\nThe situation was so dire that the denunciation of the gold standard by William Jennings Bryan at the Democratic convention in Chicago in July 1896, in what came to be known as the \"Cross of Gold\" speech, vaulted the former Nebraska congressman to national prominence and the top of both the Democratic and the Populist tickets for president that fall. Desperate farmers hoped that a western candidate would unseat entrenched, gold-hoarding powers; instead, Bryan lost to William McKinley, a \"straddle bug\" on the currency issue who was perceived to be in the pocket of moneyed East Coast elites and indifferent to farmers even as the harvest of 1896 marked yet another failure.\n\nBy then, the drought had grown so severe\u2014even in water-rich parts of Nebraska, like York, Hamilton, and Polk counties\u2014that trees a foot in diameter, planted as windbreaks after the locust plagues twenty years earlier, died all at once, and land prices fell to $15 per acre, even for properties with houses and barns. \"Many farmers who were out of debt at the beginning of the dry years, and who had declared that 'no mortgage would ever be put on their land' were forced to mortgage to obtain food for their families,\" Morrill wrote. \"Nearly every family discussed daily the question of the impossibility of remaining in Nebraska and debated where it would be wise to go.\"\n\nHarry Harrington hung on through those years\u2014but just barely. By 1902, the worst of the financial crisis was over, but the walls seem to have been crashing in on him. Harry and Lela now had four children: three daughters and a son, named Wayne. To hold onto Centennial Hill Farm, Harry sold the 260 acres he had inherited from his own father across the tracks from Benedict, located directly on the Kansas City & Omaha Railroad with a grain elevator right at the field's edge. Some of the most desirable land in all the West, Harry sold the property for just $55 per acre. All these hard times and losses seem to have taken their toll. In those years, Harry is recorded to have admitted to an investigating police officer that he had purchased bootleg liquor from teenage brothers in Benedict and, in a separate incident, was brought before a police judge in Columbus for public intoxication and resisting arrest. Even as harvests slowly recovered and yields climbed, commodities prices dipped. It seemed that, whether it was a bumper crop or a bust year, the speculators profited and farmers like Harry suffered.\n\nAt the turn of the century, fed-up farmers who had long been loyal Republicans out of lingering gratitude to Abraham Lincoln for their land grants, now began flocking to the Populist Party and forming collectives such as the National Farmers Union. Under the banner of \"Raise Less Corn, More Hell,\" they sought to create their own market power by controlling production during peak periods and sharing knowledge and labor to boost yields during hard times. Among those leading the Populist movement just twenty miles down the road from Harry Harrington was my own great-grandfather, L. C. Genoways. He started a successful corn bank, from which he sold his prize-winning seeds to neighbors to improve their output, and he won election on the Democratic ticket as the assessor of Hamilton County, vowing to fight on behalf of farmers at the capitol and governor's mansion in Lincoln.\n\nIn 1912, the State of Nebraska sought to boost sagging revenues by changing the way it taxed farmers, levying according to farm values as determined by the overall productivity of land in each county, rather than a flat fee according to acreage. L.C. gathered a delegation from central Nebraska and demanded an audience with the governor to protest. He spread out a map of the county on the governor's desk and showed him the thousands of acres along the Platte River, including the 160 acres where his own family lived, which he said were good for nothing but grazing livestock. He complained that, if the governor's planned tax increase were enacted, those acres would be valued and taxed at $78 per acre and similar land in York County would be taxed at $80 per acre\u2014more than it could be sold for. But the state treasurer was unmoved. He asked L.C. at what rate the land in the area was returning across the board. \"I do not know,\" L.C. replied at first to the huffing disapproval of the governor's staff. Finally, L.C. acknowledged that the central part of Nebraska, when taken as a whole, was returning 85 percent of its stated value. \"You have the best farm land in the state,\" the governor replied angrily, \"and you ought to be willing to admit it.\"\n\nThere are no surviving documents to explain why Thomas Barber finally sold his land in 1915 and moved away to California, leaving his daughter and her now six children without a land base, forcing the family to move to a rented farm near the town of Lushton. Maybe it was an inability to pay the stiff new taxes, or maybe it was conflict between Thomas and his son-in-law, or maybe it was just realizing how greatly farming had changed. Forty years earlier, Thomas had arrived with a horse-drawn mower and a threshing machine. Now, tractors were starting to replace horses, and hand labor was being replaced by gang plows, tandem disks, harrows, combines, and trucks. The new tractors made it possible for farmers to work large tracts of land, even if those plots were scattered across wider geographical areas, but they also required major capital investments, as much as $350, and were unreliable and expensive to fix.\n\nWithin a decade of buying the Barber farm, the new owner, Sherman R. Severn, had set a new record for speed of harvest in the state of Nebraska. Working only with his son, Severn was able to harvest 160 acres of wheat and oats in just eight days. With yields in York County rebounding from the drought years and nearly doubling the state average, the Severns had brought in 800 bushels of grain, delivering them by truck to the elevator. In Thomas Barber's day, it would have taken a team of men working sixteen-hour days nearly three weeks to match that feat. But buying new combines and tractors was still a risky business. In 1916, Wilmot Crozier, a farmer from Osceola in Polk County, bought a three-horsepower Ford 8-16 tractor, believing it had been manufactured by Henry Ford. Only after he discovered that the tractor didn't have nearly the power it advertised and would only pull a single plow did he discover that the Ford Motor Company manufactured Fordson tractors, not Ford tractors.\n\nGiven these rapid changes, it seems telling that Wayne Harrington, upon turning eighteen, left the farm in Lushton, taking apprentice jobs in mechanic shops in David City and Polk before eventually heading to western Nebraska, where he learned to repair tractors. In 1919, the year he returned to York County, Wilmot Crozier partnered with State Senator Charles Warner in drafting the Nebraska Tractor Test Law. From that point forward, any manufacturer wishing to sell in the state would have to supply one of its tractors to the University of Nebraska for performance testing. The law did wonders for farmers' willingness to invest in tractors. Sales soared, and Wayne opened a tractor repair garage in Lushton, where he worked on the new diesel-powered farm equipment. When his grandfather Thomas fell ill and eventually died in Long Beach in 1921, Wayne traveled to California and brought his body back to York County for a funeral at the Presbyterian Church. More than forty-five years after first arriving and buying land in Nebraska, Thomas Barber, in death, now owned no more ground in York County than a burial plot.\n\nIn 1928, Wayne married Eunice Kniss, a native of nearby Sutton, who had attended the University of Nebraska for a year, earning her teaching degree, before traveling to various one-room schoolhouses, finally arriving in Lushton. Wayne's repair garage, by then, had grown into an implement and hardware dealership, a grocery store, a livery stable, and a Ford car agency. In 1933, Eunice gave birth to the couple's only child, a son named Thomas. When Tom was seven, the family moved to Fairmont, at the intersection of Highways 81 and 6, nicknamed \"The Crossroads of the World,\" to have better access to major roadways. Along the highway, Wayne expanded his operations to include a lumberyard, a grain elevator, a motel, a caf\u00e9, another auto dealership and auto repair shop, and a successful trucking company bringing salt from the underground mines of Kansas to cattle ranchers in Nebraska.\n\nFor once, the family seemed to be thriving through hard times. While the rest of the country was gripped by the Great Depression and World War II, Wayne's business continued to grow and prosper. If Wayne was a severe father and strict disciplinarian, Tom nevertheless grew up in a boy's wonderland, drinking malts at the soda counter, hearing stories of the world from long-haul truckers, and learning all about heavy equipment from hard-drinking and chain-smoking mechanics. Among the farm kids of the Lushton school district, he was seen as rich. When his classmates voted to take a field trip to Lincoln in spring 1946, Wayne's trucking company provided all of the buses, ferrying the children from the state capitol building to a tour of the city to the 4-H building of the State Fairgrounds for a matinee performance of the Shrine Circus, all buses emblazoned with custom paint jobs reading WAYNE W. HARRINGTON, TRUCKING.\n\nOn May 21, 1949, Wayne's father Harry died in Fairmont, but that same year, Wayne had the opportunity to buy back the Centennial Hill Farm. To make it work, Wayne would have to sell all of his existing businesses. Eunice later remembered that he decided to do so \"mostly out of sentiment for the old place,\" but Jenni told me that she thought that Wayne felt pressured by his mother to reclaim what her father had lost decades before. Whatever the reason, Wayne entered into farming with a business-like attitude\u2014acquiring not only the original farmstead but as much adjacent land as possible, picking up low-lying properties that had been used as basin land. From his work as an implement dealer, Wayne knew that companies like John Deere and International Harvester, which had grown by leaps and bounds on government contracts during the war, were now turning their attention to developing heavy equipment for farmwork. In particular, they had already developed pump systems capable of siphoning water from rain catchment ditches at the edges of farm fields, in order to use it for sprinkler irrigation.\n\nRather than have Tom help him on the farm, Wayne sent his son away to Shattuck Military Academy in Minnesota. \"He was really hard on my dad,\" Jenni told me. \"You just feel like there wasn't much of a connection between dad and son.\" She said that once, many years later, she asked her father if he thought that Wayne had loved him. He thought about it carefully, she said, and finally answered that he thought that his father had loved having a son who was such an accomplished athlete. But even that wasn't simple; his father had encouraged him to go out for football, but Tom was tall and lanky and excelled at tennis and golf. He had dreams of skiing, which was not at all what his father had in mind. \"You can't say that he loved you?\" Jenni pressed, but she said her father refused to answer.\n\nWith Tom now out of the house, Wayne went to Germany to a displaced persons camp and brought back a Latvian refugee named Arnold Jaunzemis and his family to run the farm. Wayne also caught wind of researchers from the University of Nebraska's Department of Agricultural Engineering, led by Professor John F. Schrunk, who were digging test wells across the state in search of groundwater that was close enough to the surface that it could be pumped into catchments and used for irrigation. They discovered that the water table in eastern Nebraska was not only shallow but also that the Ogallala Aquifer in that area rapidly recharged with rains from the northern Sandhills and snowmelt from as far away as the Rockies.\n\nTo sell area farmers on the idea of digging wells and installing pumps, Schrunk organized an \"irrigation tour\" of the most successful wells around York, culminating with a visit to the Centennial Hill Farm, where Wayne was then raising 250 acres of irrigated corn. \"He uses two wells,\" the _Nebraska State Journal_ reported, \"one pumped with a caterpillar diesel and one with propane. The one pumped by diesel puts out 2,000 gallons per minute.\" After the well demonstration, the farmers gathered for \"a ton and a half of watermelon and several hundred bottles of pop,\" all enjoyed in the shade of Wayne's new 15,000-bushel corn crib and 6,000-bushel small grain crib. He also showed off his newly constructed air-conditioned hog barn and a brand-new corn picker he had purchased for the upcoming harvest.\n\n\"Wayne was an innovator,\" Rick told me one day as we drove from one property to another. \"But, you know, it wasn't too long before it cost him.\"\n\nKYLE HOPPED out of the Kawasaki Mule and hunched against the blowing snow. It was almost dark, and the clouds seemed to be gathering force, sending waves of heavy flakes swirling through the four-wheeler's headlights. After a mostly dry early winter, central Nebraska was seeing its first real storm, and the wet snow was filling the furrows and covering corn stubble. Along the edges of one field where the cattle had been turned out to feed and then cordoned off with an electric fence, the prairie grass was so heavy with snow that it had lain down across the electric wire, tripping the circuit breaker. Kyle had come out to find the source of the problem and make sure none of the cows were out. Before driving the perimeter, he checked first to make sure that the solar panel and the car battery that ran the system were properly hooked up. It didn't look like anything had been disconnected by snow or wind. Kyle slid one insulator down the rebar fence post until the electric wire was flush with the ground and climbed back behind the wheel of the Mule. He cleared the melting flakes from his glasses. \"It is really starting to blow,\" he said.\n\nAs Kyle drove the edge of the acreage, the darkness deepened until his headlights, bright with swirling snow, were the only light. Every now and then, he would stop the Mule, jump out again into the wind, bend back the broken stalks of prairie grass sagging onto the wire, and then slide back into the Mule. As we drove along one row of furrows, the bodies of the Angus cattle emerged faintly, the black of their snow-sifted bodies against the blacker sky. Some had white foreheads and noses. Mottled and pocked with patches of black, their faces reflected the beams of the headlights like the cratered face of the moon. When Kyle would step out into their midst, they would startle back into the darkness, watching warily as he checked the fence. At last, when he thought that every inch of the wire was cleared, Kyle drove back to the control box and reset the system. It instantly began to tick, the telltale sound of the pulsing electric fence. He smiled broadly: \"Well, at least that's one thing taken care of.\"\n\nKyle was born in Mason City, Iowa, and adopted at birth. He lived in Indianola, Iowa, until he was four years old and then moved to Omaha. Kyle never knew his birth parents, and the parents who raised him never talked about them. That mystery became a kind of running joke with his friends. One, who was Hispanic, always kidded that Kyle's love of Cholula sauce and tacos just proved that he was really Mexican. Another joked that he was too tall and beefy to be a Mexican. It was all meant in fun but left Kyle wondering, and it's hard not to read his shyness, his quiet laugh, and gentle demeanor as a low-key way of trying to fit in. Kyle is so agreeable, so quick to pitch in, that Meghan's description of his hot temper is almost impossible to imagine. \"Oh, he's just fooling you,\" she said.\n\nWe were back at their house now, drinking cups of blazing hot coffee while the snow continued to fall and build. \"Yeah, the very first stories I ever heard about him,\" Meghan said, \"before I even met him, were about what a hothead he was.\" Kyle rolled his eyes and shook his head. He rubbed his legs, still warming up from the cold. \"That was kind of a unique situation,\" he said.\n\nAfter his sophomore year of high school in Omaha, Kyle took a job at an auto dealership, just emptying trashcans and sweeping up at first. But the manager soon figured out that Kyle was good with his hands and started training him to be a mechanic. He showed such skill that his supervisor agreed to sponsor Kyle for the Chrysler-supported program in auto repair. If accepted, the dealership would pay half his tuition to the branch of Southeast Community College in Milford, just outside Omaha. He would get an associate degree of applied sciences, and if he made the dean's list each semester, he would come back to the shop with a raise. Kyle made the dean's list each of his first three semesters but narrowly missed it in his final semester before graduation.\n\n\"So they didn't give me the raise,\" Kyle said. But what really bugged him, Kyle said, was that without the promotion, he didn't have the standing to pick his repair jobs. Instead, the service writers would dictate which mechanics were assigned to particular jobs\u2014who got the repairs that would pay well and be easy to complete, who got complicated repairs that would take time and expertise to complete but didn't involve expensive parts so they made it hard to earn commissions or build toward bonuses. \"They were giving me shit-work,\" Kyle said. \"And whenever I complained, the service manager kept telling me, 'The other guys have families, and they need to support their families.' And I was just like, 'I got to fucking eat too, you know.' \" Finally, one day in a fit of anger, Kyle threw a wrench through the service window, shattering the glass.\n\nAfter that, Kyle wanted to get as far away as possible. He moved five hours west to Ogallala. A friend from Southeast was from there and helped Kyle land a job in the shop at Schmidt Motors. Not long after, Kyle's old shop foreman Barry in Omaha called him up and said that there was this new girl working at the dealership. Kyle had to meet her. \"No way,\" Kyle said, adamant that he didn't want to be set up. \"I hate that shit.\" The girl, of course, was Meghan, who had taken the job to stay busy after completing her first semester of college. Barry thought that Meghan would be perfect for Kyle. What he didn't know was that Meghan's last boyfriend had been killed in Iraq, and she was still grieving. Plus, the first story she had heard about Kyle was about how he'd smashed up the service department on his way out from quitting. \"I was just like, 'I do _not_ want to meet up with this guy,\" Meghan laughed.\n\nBut after Kyle moved back to Omaha, Barry invited the two to a group lunch at a sports bar, not telling either that they were being set up, just hoping they would hit it off. Meghan said when she heard Kyle was there, she steered clear of him. \"I didn't even fucking talk to him,\" she said. More months passed. In the meantime, Meghan said that every mechanic in the shop was telling her how great Kyle was, even as they were also telling her more about that fiery temper and a tendency toward taking big risks. (A favorite story involved a dirt bike accident and Kyle having to be restrained by hospital staff while they stitched him up.) Finally, Barry couldn't take it anymore.\n\nMeghan had come into a pair of tickets for the River City Roundup, a giant rodeo held every year in Omaha, but she didn't have anyone to go with her. She made the mistake of mentioning this to a coworker, who passed the word along to Barry. Barry took Meghan outside of the dealership. He looked her in the eye and said, \"You're going to invite Kyle to the rodeo.\" He dialed Kyle's phone number and gave Meghan the phone. When Kyle picked up, Meghan was so flustered that she just blurted out the invitation. \"Hey, do you want to go to a rodeo?\" she asked.\n\nMeghan turned to me, her eyes wide with emphasis. \"Dead fucking silence,\" she said. \"Just nothing from the other end. Like, a _really_ long, awkward pause.\"\n\nKyle said that he knew Barry was behind this, that he'd been working for months now to set him up with Meghan, and had been pushing so hard that Kyle was resisting out of sheer stubbornness. But now, put on the spot, he couldn't think of an excuse. He mustered just a single word: \"Sure.\" Meghan howled now with laughter. \"Real romantic, right?\"\n\nBut they went to the rodeo together, and they had a good time. After a few beers, they loosened up and started talking. At some point, Kyle mentioned that he was still looking for work, and Meghan said that her dad needed someone to help with harvest on their farm. Kyle talked to Rick and soon moved into the little white house on Centennial Hill. \"I was just working for her dad, and Meghan was in Omaha. She wasn't even out here.\" At the end of harvest, Kyle got another temporary job, driving a plow and repairing trucks for a snow removal company in Omaha. A couple of times, Meghan rode along as Kyle was plowing driveways and private roads during snowstorms. But he said they were still more friends than anything else.\n\n\"When March came around, I needed to find a permanent job, so I applied at Caterpillar.\" The company hired him to work as a repairman at the dealership outside Omaha, charged with fixing breakdowns. It took almost a year, seeing each other for dinners and movies after his shifts at the dealership, before the two finally admitted to themselves that they were dating. When Meghan moved back to the farm after graduating from Metro, she said it only made sense for Kyle to come with her.\n\nOF ALL the stories of Centennial Hill Farm, one stands out\u2014a passed-down tale of super-human grit but with so few concrete details as to create a certain shroud of mystery. It was September 1951. Wayne Harrington was just beginning the corn harvest. Maybe he'd decided to get the corn out early. The weather was perfect, sitting squarely in the 70s with no rain for the whole month. Maybe he had hoped to get a jump on the market but then found the stalks still a little green. Those parts of the story are lost. But whatever the reason, he had barely begun on the field when the picker jammed. Wayne climbed down from the harvester and cleared away the mouth of the machine, but the jam was deeper inside. He reached in, dislodging the problem cob, and the teeth of the picker grabbed his fingers, his whole hand up to the wrist. He was mangled by the machine and caught. Worse than that, he was alone in the field and pinned. \"He had a _booming_ voice,\" Jenni told me. He called out for help and was finally heard by the women from the Latvian family he had brought back from Europe two years before.\n\nThe women came running from the farmhouse. Following Wayne's instructions, they retrieved his tools from the barn and took apart the gears of the picker. Wayne freed himself from the teeth of the cobber, wrapping his arm\u2014with what, no one knows, his shirt or his coat\u2014and then drove himself more than thirty miles to the hospital in York. But there was too much damage to save his hand. Two days later, doctors were forced to amputate. \"People say it was really hard on him,\" Jenni told me, \"and that he was never the same. It took a lot out of him.\"\n\nThat same fall, hoping to please his father, Tom enrolled in the University of Nebraska's school of agriculture and went out for the Nebraska football team. Tom later told his kids how he'd gotten beat to hell trying to prove himself to the coaches but still failed to make the squad. There would be no trips to Memorial Stadium to boost Wayne's spirits, no gridiron magic to take his mind off his loss. Tom did make the university tennis team, but his father never traveled to Lincoln to see any of his matches. Just a few months later, in February 1952, Wayne was out in a winter storm, delivering feed to his cattle, when his truck skidded off of icy Highway 81 and crashed into a tree. He suffered a fractured skull, a crushed rib cage, and multiple breaks in both legs. He held on for a few days at the hospital in York but finally died from his injuries.\n\n\"I've always wondered,\" Jenni said. \"Did he lose control of the truck, because he only had one hand? Or did he\u2014\" She didn't finish her thought. But then I wondered, did Jenni question whether Wayne's crash had been an accident? Was that buried somewhere in the subtext of the repeated tellings of the story? She didn't go further. \"I don't know,\" she said. \"But that's when my dad came back to run the farm.\"\n\nTom had completed just one semester of ag training, had never lived on the farm, and didn't have any practical farming experience. His mother took over running the books and managed all aspects of the business. Even after Tom had married his wife Karen and had four little girls, Jenni said that she remembered how her grandmother would spread her papers out across the kitchen table to make sure every line balanced. \"We just thought, _wow, she's the queen of the family_ \u2014and she _was_.\" Some of her grandmother Eunice's attention to detail arose from Tom's freewheeling approach to farming. Because he wasn't bound by any tradition, he was more forward-looking than many of his neighbors, but he was also taking risks and often guessing wrong. \"He always liked to live on the edge,\" Jenni told me. \"So he always had cattle deals going south or found himself in the wrong position on the commodities market.\"\n\nBut Wayne had bought the family land back at a lucky moment. In the 1950s, the irrigation projects Wayne had helped promote suddenly took off. For the first time, groundwater could be delivered to the fields by \"flood irrigation,\" a method of pumping into containment ponds and then releasing water through a system of wide, open furrows. And soon, Tom and others began installing the first irrigation sprinklers. The area would never again be at the mercy of droughts like they had seen in the 1890s and 1930s. When center-pivot irrigation was developed just up the road in Columbus, Tom was one of the first to buy in, so that, when blistering temperatures struck Nebraska in 1953, Tom's farm, with its well-irrigated fields, saw record yields, and Centennial Hill quickly became some of the most valuable farm ground in the world.\n\nWith the help of new chemical fertilizers and pesticides, high-quality hybrids, and his ever-expanding irrigation system, Tom was soon appearing in local newspaper ads announcing top yields for the county. At the same time, his family was growing and prospering. Tom and Karen had four daughters\u2014Abbi, Terri, Jenni, and Heidi. And he set about using his newfound wealth to build a modern farmhouse for the six of them.\n\nTom hired Dewey Dearing, a well-known architectural firm in Colorado Springs, to build the new house, patterned after the ski lodges that Tom admired in the Rockies. Everything about the place was eccentric. Tom requested a sofa suspended from the high ceiling by a heavy chain, so it would swing. He had the floors made of stone and covered the walls with thick carpet, so dense that guests had a hard time finding the light switches. \"You ought to see them try turning on the lights,\" Tom told the _Omaha World-Herald_ when they sent out a reporter to write a story about this unusual farmhouse. Tom even personally designed built-in bookcases with slanted shelves on the idea that this would eliminate the need for bookends. Of course, it damaged books and made those at the bottom of the incline almost impossible to budge from their place. \"I goofed,\" Tom told the reporter with a shrug.\n\nJenni winced, remembering the photograph that accompanied the article\u2014a picture of her, about age four, next to her mother in the swinging sofa. The picture was reproduced using the new Colorphoto process, which allowed four-color reproduction but with a rather limited palette. \"It was an orange sweater with a plaid skirt\u2014that kind of matched the couch,\" Jenni said. \"I just about disappeared!\" This was the dichotomy of their childhood: daily chores tending the chickens and cattle, with evenings spent in a house that stretched the limits of sixties style. But Jenni said it never seemed incongruous to her or her sisters; they knew that their neighbors considered them oddballs, but she said that all of the girls identified with their father and his progressive mother.\n\nJenni's sister Abbi remembered going to visit their grandmother Eunice at her new home in Omaha, where she had gotten deep into social activism through the Methodist church. \"She always loaded us in the car, and we would venture into the city to see firsthand what she had been working on. We visited a black radio station she helped start. We would drive out to Boy's Town to see what they had going on.\" Their relationship with their father was more difficult, especially as they grew older and more rebellious. When he made them drive the tractor at planting or harvest, they would don their bathing suits to work on their tans, just in protest. \"We drove him nuts a lot of the time,\" Jenni said, but she also said that her father often seemed to be looking for a fight.\n\nOnce when the girls were in high school and the state was in the midst of a drought that was devastating a neighbor's milo crop, Tom dug a culvert across the county road and laid a pipe to divert water from his catchment pond to the parched field. When someone complained, the county sheriff ordered the pipe removed. As soon as the crew was gone, Tom put it back. The county responded by issuing a restraining order. Tom, in turn, submitted a formal request for a temporary permit \"to cross their goddamn road.\" The York County Board of Commissioners met to consider the matter and ultimately denied the request 3 to 2. \"Tom Harrington's a pretty good fellow,\" one of the commissioners said after the meeting. \"You can't help but like him in many ways.\" But, he added with a smirk, \"he's rather headstrong.\" Tom responded by threatening to file a lawsuit. He conceded that he probably had no case but told the _Lincoln Evening Journal_ that he was in search of an attorney who was willing to sue the county \"for damage to the milo and for being stupid.\"\n\nA few years later, Tom attempted to organize a county-wide holdout of property tax payments in protest of a state tax reappraisal, which set the value of top irrigated land in York County about 20 percent higher than values in surrounding counties. \"There are a lot of people who are taking it in the shorts over property taxes,\" Harrington told the _Omaha World-Herald_. \"Most farmers will be paying $10 or $12 per acre in taxes on their irrigated land.\" About fifty landowners flirted with Tom's idea of an organized tax boycott, but the possibility finally fell through over concerns that a lack of tax dollars might shutter public schools in the county or even force the seizure of property in order to collect those payments. Tom, thinking they were all cowards, vowed never to talk to any of these neighbors ever again.\n\nJenni said that the contentious nature of the household drove her oldest sister Terri away to college and then to Denver, where she became a lawyer. But the other sisters, though they also went away for school, returned after Tom announced that he planned to retire and leave the farm to the four of them in a joint trust. The land was divided equally among them but still run as a single operation, requiring all decisions to be reached by consensus. That closeness led to years of fighting and acrimony and eventually to stony silence, even as each of the four sisters refused to sell her shares. Instead, lawyers and negotiators were often called in to broker even relatively routine business decisions, and no one would budge. Divided by what they shared, the sisters remained\u2014and still remain\u2014bound to Centennial Hill Farm, the center of the Harrington universe, for better or for worse.\n\nNearly a decade of tension followed until, unexpectedly, a threat arrived in the form of a land agent working on a planned pipeline project. He said that he represented a foreign corporation and that he was there to secure an easement\u2014or they would face having their land taken by eminent domain.\n\nRICK REVVED the engine of his four-wheeler, sliding through the prairie grass. In only a few years of letting the land lie fallow\u2014just a spot to graze his small herd of black Angus for a few months out of the year\u2014head-high stands of big bluestem and Indiangrass had sprung up. He chugged to the top of a bluff and cut the engine. It was quiet there; the only sounds: mooing cows on a neighbor property to the north and the sighing of wind whipping across the meadow. Rick's own cattle chomped drowsily, their loose jaws churning and ears twitching as they watched us from afar. Now February, the last drifts of snow hid in the lees of ridges and cedar trees. Calves would be coming soon, and Rick was hoping to fatten the expectant mothers before they gave birth and started nursing. They rested, dark and still, seen only through the shifting veil of ochre grass.\n\nOtherwise the fields appeared continuous and empty. Clouds cast shadows that seemed to move like galleons across the green and yellow waves. \"Prairie ships,\" Rick said. Farther in the distance, the wide bend of the Platte River curved from Central City, with its grain elevators and ethanol towers, around to a grove of trees to our east, where a Pawnee camp had been for hundreds of years. My own grandfather talked about his father going to that camp to trade in the 1880s. It was this view that Rick wanted me to see, and he pointed out a cluster of depressions along the edge of the bluff\u2014a group of Pawnee graves, he believed. He could understand why they would have buried their dead here, with this panoramic view of the river valley.\n\n\"I come up here sometimes to get a little perspective,\" Rick said. \"It's easy to get lost in the worries of the day-to-day. You worry about money, mostly\u2014yields or prices\u2014and you can forget to appreciate what you have.\" Rick acquired this land, about fifteen miles northwest of Centennial Hill, when Meghan and her siblings were still little. Right from the beginning, he envisioned this as the view from his dream house\u2014a place of his own, close to Heidi's inherited acres but with an even deeper history and one undefined by her family. \"When we bought this land, it was a sealed bid,\" Rick said. \"When we found out we'd won, Heidi was jumping up and down. But I bent over like somebody had kicked me in the gut, because I knew that for the next ten years I had to produce every year and as efficiently as possible. It would be a decade before we could even think about re-mortgaging everything to where we could build.\"\n\nIn 2001, when all of the kids were coming into middle and high school and wanting more room for themselves, Rick decided the time had finally come. He hired an architect to help him realize his vision of a giant timber frame home, open from the ground level to the rafters, with huge bay windows and a wraparound porch to give sweeping views of this land overlooking the river. But he said that between the cost of the architect, hiring expert builders, and high-end materials, he couldn't afford a general contractor, so he oversaw all the construction himself. One day, as the timber framers from Missouri were nearing the end of their work\u2014the tall beams raised and walls blocked in, the plywood subflooring installed and everything but the stairwell completed\u2014Rick climbed a ladder, stretching from the basement to the ground floor, to check out the progress of the finishes.\n\nRick interrupted his story to walk around to the back of his four-wheeler. He leaned against the seat, looking up at the house on the hill. The field of grass in between switched back and forth in the wind.\n\n\"I was all the way at the top of that ladder,\" Rick said, pointing toward the peak of the roof in the distance, \"and just going down, when one of the guys below called up to ask me a question. I turned and that wiggled the ladder enough that the bottom shot out.\" As the feet of the ladder skidded across the concrete floor, the top slid down the supporting beam and pitched Rick off into the air. \"I hit my head on the first floor going down,\" he said. \"That's the last I remember. They found me in the basement with my head hung through the bottom rung of the ladder. They called Heidi and said, 'You better get over here. We don't know if Rick's going to make it.' \"\n\nMeghan told me later that she was with her mom and siblings, moving cows in the neighboring field, when the call came. By the time they arrived, the Hordville ambulance was already on the scene, its single light on top still lazily turning as the medics gave Rick mouth-to-mouth. They revived him and took him to the Central City hospital, where he was diagnosed with a concussion, a dislocated shoulder, and several broken ribs. But they didn't find any internal injuries, so they told him, \"If you can shower, you can go home.\" Rick had Heidi help him into the bathroom and under the showerhead, and then she slid his shirt over his shoulders and buttoned his pants. He was determined to keep his promise to hold a cookout for the timber framers before they finished their work and had to go home to Missouri. But a week later, still in pain, Rick gave in to Heidi's insistence that he see a doctor in Lincoln, who found that he had undiagnosed fractures in both of his wrists and more broken ribs. Rick gave me a sideways smile. \"That house almost killed me,\" he said, \"physically and financially.\"\n\nRick was nearly out from under that burden when he received a strange letter in 2010. An oil company was writing to inform him of an upcoming town hall meeting at the church in Hordville. The company, TransCanada, was planning to build a pipeline, stretching from the tar sands mines in northern Alberta to refineries on the Texas Gulf Coast, and they wanted to meet with area landowners. Rick soon learned that the proposed route would cut directly through this piece of land that he had almost died for, trying to turn it into the perfect place to retire with Heidi. He went to that meeting feeling defiant. He had two natural gas pipelines under his grazing land in Curtis and knew firsthand what kind of impact digging trenches can have. \"The land is never the same after it's disrupted like that,\" he said. \"I don't care what they say. It's never the same.\" But a TransCanada land agent from Tennessee, who was also an ordained Baptist minister, told everyone that the company wanted to do right by the community, that he could see that they were the salt of the Earth and that he would make sure they got a fair deal.\n\nRick was dubious\u2014and the more he researched the pipeline project, the less he liked it. The refining of tar sands crude is among the dirtiest fossil fuel processes in the world, contributing to climate change in ways that, in Rick's view, were a serious threat to Meghan's future on the farm. More than that, he couldn't find proof that any of the heavy diesel that would come out of the refining would even be legal to burn in the United States, and the refining byproducts, like petroleum coke, were expected to be burned in China, both undercutting American coal and supplanting it with an even more toxic substitute. Most worrisome of all, that very summer a similar tar sands pipeline owned by another Canadian oil company ruptured in Marshall, Michigan, severely contaminating nearly fifty miles of the Kalamazoo River with oil, toxic diluents used to make the heavy crude move through the pipeline, and still more chemicals used to break up and absorb the slick.\n\n\"They asked, 'Oh, don't you want to get your fuel from a friendly neighbor?' \" Rick said. \"But they never talked about all of the refined petroleum products going overseas or about how they were major greenhouse gas emitters. I mean, what the fuck? Why would I want a pipeline pushing toxic sludge through my soil, through the Ogallala Aquifer that we depend on for irrigation and drinking water, so some foreign oil company can make a buck? How is that being a friendly neighbor?\" But Rick confessed that for all his anger, he was also afraid. Along with the promises of a fair price and a fair deal were oblique threats of land seizures if farmers and ranchers dared to resist. He stalled on giving TransCanada a definitive answer.\n\n\"I was polite, but I was dragging my feet,\" Rick said. \"After it became obvious that I was not going to sign, they threatened me twice with eminent domain, on the phone and once in writing.\" After he received that letter in July 2010, Rick gave in and signed the easement. He regretted the decision even before he put the documents in the mail, but he felt he had no choice. TransCanada was a $50 billion multinational, and he was a self-described \"little guy,\" one man against a fleet of lawyers with nothing more on his side than a gut feeling that what they were doing shouldn't be legal. What could he do? \"They crushed me,\" he said with a shrug.\n\nSix months later, TransCanada contacted Rick to tell him they'd changed the proposed route, skirting around his property after an early cultural impact study suggested the likely presence of ancient Pawnee artifacts on his land, perhaps even graves, exactly as he had always suspected. Rather than deal with the red tape if one of the company's backhoes hit human remains, TransCanada had decided to push the route to the east. At first, Rick was relieved. \"I said, 'Okay, if I give you the money back, will you give me my signature back?' They said, 'No.' \" Worse still, when TransCanada revealed the new planned route, Rick discovered that it crossed a section of his sister-in-law Terri's land, just west of Centennial Hill\u2014one of the properties where he raises seed corn for Pioneer, the cornerstone on which much of his operation now rests. Though the pipeline was no longer passing by the doorstep of his dream home, the new route threatened to be even more perilous to the family business.\n\nRick and Meghan convinced all of Heidi's sisters to join up with two antipipeline organizations, the Nebraska Easement Action Team (N.E.A.T.) and Bold Nebraska, and they even started their own local group in York County called the Good Life Alliance. \"They may have crushed me as an individual,\" Rick said. \"But there is strength in numbers and all of us standing up for the right thing.\" They started going to meetings and speaking out, writing editorials and placing full-page anti-pipeline ads in the York newspaper. Rick told me that, if there was any silver lining in living with threats from TransCanada, it had been watching Meghan blossom into one of the most vocal and articulate young leaders of the pipeline resistance.\n\nMeghan and her siblings had been outspoken liberals since they were in high school. \"During the Kerry-Bush election we had a vote in our political science class,\" Meghan said later, \"and my twin sister and I were the only two that voted for Kerry. Our teacher, said, 'Oh, you Hammond girls, you just want to be different.' \" But now, with their land under threat, some of those same neighbors were pleased to see Meghan speak up at zoning meetings of the county commissioners and committee meetings at the Nebraska legislature. She combed over the government reports and prepared careful statements. When the U.S. State Department's own study estimated that the project would create only thirty-five permanent jobs along more than a thousand miles of pipeline in the United States, Meghan was quick to seize on the point. In May 2013, she rose at a massive public hearing at the fairgrounds in Grand Island, Nebraska, held by State Department officials as part of the agency's approval review process. Still wrapped in a scarf from the freak spring snow swirling outside, she leaned confidently into the mic. \"How can you risk our land and water for thirty-five permanent jobs?\" she challenged. \"If you want thousands of jobs, you will find it in sustainable energy.\"\n\nIt was that statement that gave Jane Kleeb at Bold Nebraska the idea of building a barn, powered by solar panels and a small wind turbine, on Terri's property. \"It was kind of a 'put your money where your mouth is' kind of proposition,\" Meghan said. \"We kept talking about wind and solar, but now we had to show that we believed in it enough to actually build it.\" Within a few months, Kyle and two of Meghan's cousins led construction, as dozens of volunteers from around the state helped to raise the structure in just four days. Billionaire environmentalist Tom Steyer flew in from San Francisco for the ribbon cutting and put up money for a viral video about Meghan and the barn. Heidi and her sisters posed together with Steyer, all smiles. But it wasn't long before the neighbor just to the south wrote to Rick to withdraw from the contract allowing the Hammonds to farm two quarters of his ground. Meghan told me later that she felt transformed by that betrayal.\n\n\"Losing two quarters of ground, that was huge\u2014 _huge._ And it's changed how we feel forever,\" she told me. \"We wouldn't change what we've done with the pipeline, but our neighbors are our neighbors forever, you know? For generations and generations and generations. And now we have neighbors who won't even wave to us, so how are we supposed to trust them now?\" Most of all, she said she was worried about the impact it had had on Rick. He dropped from view for months during that time. Some days he didn't leave the house, even during harvest. He regretted having signed the agreement with TransCanada, and he regretted that having fought back had brought unintended consequences. \"He carries a lot on his shoulders, because he feels responsible for all of us,\" Meghan said.\n\nAt times like that, the weight can seem to grow with each piece of bad news. In November, the Republicans won their largest majorities in the U.S. Senate and House since the backlash against the Democrats after the stock market crash in 1929. New majority leaders vowed that final approval of Keystone XL would be their first item of business. In January, despite President Obama's veto threat, the House of Representatives approved the Keystone XL Pipeline Act, and the legislation passed the Senate on a 62-to-36 vote\u2014still five shy of the 67 needed to override a presidential veto but with intense horse-trading expected in the effort to muster the remaining votes. Even if the Democratic Caucus could sustain the veto, talk was spreading that Obama might approve the pipeline as part of a larger deal toward sustainable energy.\n\nRick worried that the legacy he had spent a lifetime building was about to become a bargaining chip between politicians in Washington and corporate interests with holdings around the world. Was there anything he could have done differently? At times like these, he worries not just about the hardship of the moment but that he may have made a fatal error, the mistake that leads to future struggles for Meghan and Kyle or, in his worst nightmares, to losing everything and having to hold a farm sale. \"He wants to make it better than the way him and my mom started out, but we don't need a bed of roses,\" Meghan told me. \"They struggled. We will struggle. There's no way around it, but he always tries to make our lives as good as possible at his own expense. He's very hard on himself\u2014 _very_.\"\n\nI couldn't help but hear Meghan's words in my head as Rick leaned against his four-wheeler, looking out over everything he had assembled and built and tried to defend. \"We're extremely lucky,\" he said. \"Almost everything you see is ours\u2014from the lower farm to this bluff to the house up on the hill. And that's a blessing.\" But as soon as the words were out of Rick's mouth, the old worries seem to creep back in. \"When I'm gone,\" he added, \"I'll be able to leave each of my four kids at least as much land as Heidi's dad left us. That's the hope. That's what all the work is for.\"\nINTERLUDE\n\nBRANDING CALVES\n\n_April 2015_\n\nOn a sun-soaked day in mid-April, the Hammonds corralled the cattle back into the paddock behind the old barn. The nursing mothers were separated from their spring calves, just as they had been in the fall, and then locked into neighboring pens for the night. Rick said it was probably eleven o'clock before the cows settled down and stopped mooing, slipping into sleep in the darkness, but they were up again before dawn, calling to their calves. By mid-morning, when the veterinarian arrived, the cows were noisy and rattling the gates of the corral. The sun was pale-bright again, washing everything into pastel hues, thin wisps of high cirrus clouds the only interruptions on the cornflower sky. The pristine clarity and spring-like promise in the air, even touched as it was with the last of winter's chill, seemed strangely at odds with the task ahead.\n\nIt was branding day, one of Meghan's least-favorite parts of the whole farm year, and she was feeling short-tempered and stand-offish. She wore a down vest over her sweatshirt and a loose-knit stocking cap, with her long hair swept to one side. When her temper flared, she would rip off the cap and tuck it impatiently into her back pocket, then pull it back over her ears when her mood cooled. Everything about the day seemed to be getting on Meghan's nerves. She didn't like how stressed the calves were: mewling and skittish, huddling together, face-in, in the corner of their pen. Seeing how jittery and strained they already were, she was pissed that Dave was using an electric prod to hustle some of the calves down the chute, and she was surly about the very idea of hot-branding. \"It is hard on the animal,\" Rick conceded, \"but they bounce right back.\" Meghan just didn't see any good reason for it. In an era when consumers are concerned about animal welfare, she worried it was a potential deal-breaker with boutique buyers.\n\nMore than that, it was unnecessary work, considering that there are more modern and pain-free methods of keeping track of individual animals. The cattle already had ear tabs to identify them within the herd. Those cheap plastic tags were hand-numbered by Kyle and catalogued with corresponding health information in Meghan's record books, an old-fashioned way of keeping inventory of the herd that required her to mark down information on the animals one by one. When the first calf lurched into the cradle, the trap closing around it with a hollow clang, Meghan yelled out, \"What's the number?\" Her older brother Jesse, back in town for the time being and charged with applying the brand, called back, \"That's one-twenty-two.\" She echoed, \"One-two-two,\" entering the number into her spiral notebook. Then she wheeled away as Jesse lifted the brander.\n\nIf, instead, the Hammonds used electronic identification (EID) ear tags, which have a fifteen-digit visual label specific to each individual and also contain a chip that can be read by an EID reader, then there would be no need to brand the cattle at all. The high-tech tags would make it easier to monitor animals both within the herd and to track down strays in the event of an escape. When we talked about it later, Rick allowed that EID tags would work fine for maverick calves or cows that wander off when an electric fence shorts out, but those ear tags are easily removed by anyone whose intention is theft. \"You don't want your herd to cross the fence, and EID won't cure that,\" he said. \"Especially if someone rustles them.\"\n\nThe idea of modern-day rustling sounds incredible, but with the price of cattle reaching record highs, worries about calf thefts had ranchers on edge from Texas to the Dakotas. The Nebraska Brand Committee, tasked with overseeing cattle brand registrations and enforcement over the western two-thirds of the state, reported that it had recovered more than 1,500 lost or rustled cattle between summer 2013 and 2015, at a total value of more than $5 million. Only a few weeks before the Hammonds starting their branding, three Omaha men had been arrested in connection with a series of at least seventeen thefts from feedlots and farms from western Iowa to Lincoln.\n\nWhile that case was unusual in its daring, it was hardly isolated. The newspapers had been crowded with rustling stories, often simple crimes of impulse or opportunity, a weak moment when an unbranded weaned calf was spotted alone near a fence line or had strayed off property. But other incidents were clearly more planned out. Some law enforcement officials said that the crime wave was fueled by the meth epidemic\u2014a strange confluence of rising rural addiction and soaring sale barn values for cattle. When a calf can go for over a thousand dollars, Rick explained, the temptation is too great and the risk of loss too severe to let yourself feel squeamish about the prospect of branding. There's a reason for the old saying: \"Trust everyone, but brand your damn cattle.\"\n\nSo one by one, Rick used the herding paddles to guide the young males down the chute. As each calf was clamped into the cradle and swiveled onto its side, the vet swiftly administered a series of inoculations, and Jesse pressed the electric brand, a large R Diamond, into its right flank. Jesse cut an unlikely figure for the job. His long curly hair was bunched into a ponytail and tucked through the hole at the back of his camouflage Pioneer baseball cap. He wore a bright red Vermont t-shirt that read JEEZUM CROW and white wraparound ski sunglasses to shield his eyes. Each time he applied the brand, thick billows of white smoke instantly engulfed him, and the air filled with the smell of burning hair and cauterized flesh. The calves usually let out a single, low moo then dropped silent. Right after, Kyle would slice into the calf's scrotum, cut out its testicles, and toss them aside into the dirt.\n\nLater, after the task was complete, I asked Kyle if the branding and castrating bothered him at all or if he just regarded it as a necessary task. \"I really don't think the branding hurts them too much,\" Kyle said. \"I mean, it burns, but then it cauterizes the wound immediately.\" He said that they do freeze brands with identification numbers on the flanks of the cows when they get bigger, using a brand dipped in liquid nitrogen. The cold cauterization creates a white brand that stands out against the black hide of the Angus cattle, making it possible to pick out specific cows even at night or if they lose their ear tags. \"They complain about that more than the hot brand,\" Kyle said. \"And the castration\u2014well, I can't imagine that it feels too good, but I just try to be quick about it. Some people use a bander that applies a tight plastic band at the base of the scrotum, so that everything just drops off after a few days of the blood not circulating. I don't know if that's more humane\u2014some people think so. For us, we think it's better to just cut them, so it's quick and clean, and if we get them out onto grass right afterward, there's pretty much no chance of infection.\"\n\nAfter another hour of branding and cutting, Rick stretched his arms over the rails of the corral, letting his gloved hands dangle loose as he caught his breath from chasing calves into the chute. \"You know, everybody worries about the branding,\" he said, \"but if I had to choose between being branded and being castrated . . .\" He ducked his head, so he had me clearly in his sights. \"Well, which would _you_ choose?\" As he chuckled to himself, Jesse called out the new calf's tag number. Meghan marked it down and then stomped off without a word. She pulled off her yellow stocking cap and crammed it in her back pocket, dangling like a penalty flag.\n\nLater, she explained that Rick had gone to the trouble to move away from hormones and antibiotics used by other ranchers to bulk up their animals, putting the family in a position to charge a premium for their beef, but hot-branding is strictly prohibited if you want to get the top dollar that goes along with Animal Welfare Approved certification. Because many small producers in Nebraska were applying for that label\u2014as well as organic, non-GMO, kosher, and other specialty labels\u2014the Hammonds struggled to sell their beef directly to consumers looking to buy from independent farmers. So most of their cattle, despite being grass-fed, hormone-free, and antibiotic-free, wound up at the sale barn with everyone else's cattle, often to be purchased by Cargill or Tyson or some feedlot that fattens its cattle on corn and growth promoters to sell to those packers. At auction, Rick could command a small premium because his cattle were designated as NI\u2014no implants\u2014but that higher price would be offered only because the buyer knew those cattle would respond to hormones and bulk up faster when they were injected in the feedlot. To Meghan's mind, it was a lot of extra work for the Hammonds, keeping their herd healthy and drug-free, only to have their cattle wind up just like all the other cattle out there.\n\nRick saw hot-branding in just the opposite light. Not only does it protect your herd from rustling, but when you go to the sale barn, the brand on your cattle becomes your brand as a company. In the western part of Nebraska, brands, checked and approved by government inspectors, are required to sell all cattle; in the eastern part of the state, they're not required or checked, so it's not at all unusual to see unbranded cattle. Rick told me that his cattle, with their distinctive brands burned into their rumps, sent a message to sale barn buyers that his cows had been raised in a traditional way, spending half their lives on open rangelands in western Nebraska. And he didn't deny that the practice made him feel connected to a time when cattle grazed over broad expanses on the unfenced prairie and had to be branded with irons reddened in the coals of a campfire just to assure that professional marauders didn't make off with a rancher's livelihood. \"It's the cowboy mystique,\" Rick said unapologetically, and he said that he considered that way of raising cattle part of the R Diamond identity.\n\nAs each brand was set, Jesse rotated the cradle back to vertical and released the trap with a foot-operated pedal, letting the calf leap out into a holding pen with the others. When all the new calves had finally been inoculated, branded, and castrated, we all sat on overturned buckets in the narrow shade of the barn, eating sandwiches and drinking sodas. Meghan explained that they expected to have maybe 130 calves. Fifty would go out to pasture for the summer near Valentine, in the northwest corner of the state, and the other 80, split up into two separate trailer loads, would go out to Rick's land around Curtis. The calves they had branded and castrated that day were ready for pasture now. Soon they would be loaded up for western Nebraska.\n\n\"And they're all new calves\u2014from this year?\" I asked.\n\n\"Yeah, they're only two months old by the time they go,\" Meghan said.\n\nI studied her a moment, trying to gauge her patience. She had finally ditched her stocking cap for good as the day grew warmer, and the bustle and unease of branding seemed to be draining out of her.\n\n\"You know,\" I ventured, \"from my perspective, the branding and all, it seemed like it went\u2014\"\n\n\"It was good,\" Meghan interjected. She leaned over and slugged me in the shoulder, as if to signal that the subject was closed. \"It's hard on the calves, and it's hard on the people,\" she said. \"But, yeah, otherwise, it was good.\"\nPART THREE\n\nSEEDS OF CHANGE\n\n_May 2015_\n\nDave flipped open the lid of a yellow hopper at the rear of the John Deere planter, then lifted a 20-kilo bag of seeds and balanced it in the crook of his elbow. The bag was made of brown paper, like a grocery sack, and printed with a pale green stripe and bold lettering: FEMALE PARENT CORN. Stitched to the top were two coded identification tags listing the batch number, the lot number, where these parent seeds had been grown, and the exact number of seeds per kilogram. Overhead, the sky was slate-gray and threatening, but the steady rains of the last week had finally relented enough to get the seed corn planting underway.\n\nDave pulled a tab at the bottom of the bag, unzipping the seam like pulling a ripcord, and the seeds poured out until the hopper was full. The turquoise-tinted kernels inside had been treated with a mix of insecticides and fungicides developed by DuPont Pioneer chemists to kill off beetles and aphids and to knock down infections that can slow seed emergence and leafing out. The coating's candy color is just a dye, a reminder to farmers that their seeds are covered with potent chemicals and should be handled with care. But Dave wore a plaid Western shirt, torn off at the shoulders, no sleeves, no gloves, nothing more than his glasses and a Pioneer cap for protective gear.\n\nOnce the individual hoppers were filled, one for each planting row, Dave emptied the last of the bags into the bulk bin, then climbed into the tractor's cab. He started the onboard computer, and a fan whirred to life, pressurizing the planting system, sucking up kernels, and sending them rattling through a series of tentacle-like hoses, each extending to one of the small square bins. Underneath, a V-shaped pair of opener blades was set to cut a trench to precisely the right depth, followed by a nozzle from the mini-hopper to drop seeds according to an infrared sensor, and finally a close wheel that would cover the seed and pack the soil to the appropriate density. The hoppers stretched for a dozen furrows in either direction, but on that day, Dave had the system set for planting seed corn, so for every four rows of female seed, there would be a blank row to fill in later with males.\n\nSeeing that everything looked good, Dave eased the planter to the edge of the field, lowered the blades, and started, slowly, across the acres of dark soil, watching over his shoulder to make sure the right pattern was unfurling behind him. Ordinarily, the Hammonds do no-till planting. With no-till, the farmer plants right through the stubble of last year's crop and the tangle of subsequent weeds, using a coulter blade, sharp and wide enough to slice through roots and open a seed trench. By leaving those root systems in place, farmers can reduce erosion and topsoil loss\u2014and, ideally, reduce the use of chemical fertilizers. But for some reason, Dave wasn't really sure why, Pioneer still preferred tillage to prepare a bed for its seed corn fields. So Rick had gone over the field with a one-pass soil finisher, a massive contraption that unfolds into a twenty-four-row, pull-behind tiller that combines a line of discs and a set of miniature coulters. \"You can see it's been worked,\" Dave said, pointing out the distinctive corduroy lines. Behind us a plume of dust swept east across the field, rising like a growing whirlwind, even after all the spring rains.\n\nAs we moved, Dave showed me where the planted seeds were filling in the virtual field on the touchscreen and how the planter was varying the density of seed populations according to the pre-programmed prescription for the field. Apex, the software created by John Deere to control everything from the GPS-guided autosteer to planter row shutoffs, made it so that he didn't have to do anything but get the planter to the appropriate starting point and hit go. The Hammonds began using the system about five years before, but he still hadn't warmed up to the change. \"I would rather do it here,\" Dave said, \"but a lot of it, it won't let you. You got to go through Apex, and I can't figure out the goddamned thing. So Kyle, he helps me get it all set.\"\n\nMonths earlier, when the plans were still being drawn up, Kyle had shown me how this field was mapped and entered into Apex. It was cold outside that day, the space heater in the shop at Centennial Hill buzzing with effort, but Kyle was hunkered down at the computer, double-checking everything. \"Rick told me to sit down and figure out what we were going to plant for seeds, all the herbicides for the fields, the fertilizers,\" he said. \"That's pretty big, you know? Especially with prepaying. It's kind of a tough thing to do cost-wise, because we're still in the middle of selling. I'm trying to do research and get the best plan put together. It's definitely more responsibility. It's, like, a $300,000 purchase that he pretty much isn't overseeing at all.\"\n\nTo an outsider, it may look simple to raise row crops: just plant a seed and watch it grow. But with commodities prices still under the cost of production\u2014corn stalled under $4 per bushel, soybeans under $10\u2014farmers were making tough decisions about where to spend on inputs. A survey by the _Progressive Farmer_ in Omaha found that landowners were planning to invest less in hybrid seeds and scale back as much as possible on fertilizer and chemicals for weeds and pests. \"It's all a trade-off,\" Kyle said. \"Soybeans are still doing better than corn, but if you're not rotating, then you're losing fertility. So you make more this year, but maybe you're shooting yourself in the foot for next year.\" The stakes were especially ratcheted up by the University of Nebraska's annual survey of farm values, which showed that properties across the state had taken a hit. Even irrigated acres in the north-central region, which contained York and Hamilton counties, were off. So the collateral backing everyone's loans was losing value\u2014raising the specter of what could happen if debts were ever called in as they had been a generation ago. It was a high-risk moment for Rick to hand over the reins. Kyle understood what a vote of confidence it was, but he also knew that he couldn't afford to screw it up. And right now, the top moneymaker was the seed corn fields. He had to do them right.\n\n\"This light gray area is the isolation zone,\" Kyle said, showing me an empty spot on the western edge of the field. \"So what they're saying is: you have to be 165 feet away from the neighbor's yellow corn. That's actually a Nebraska state law, to have at least that much setback so you don't get cross-pollination. And you can put whatever you want in that isolation, as long as it's not corn. So we plant soybeans there. And then there are fourteen rows of male corn. That's what's called border rows. They help ensure pollination.\" Finally, down the center of the field map was an alternating patchwork of four rows marked for females and single rows for males.\n\nThis whole process of forced cross-pollination in order to produce robust seed corn is arguably the most important development in the last century of agriculture. For all the talk about tractors and center-pivot irrigation systems, the emergence of reliable seed, more than anything else, has turned cycles of catastrophic loss and foreclosure into a reliable if ever-changing business. A hundred years ago, my great-grandfather L. C. Genoways was one of the most successful seed corn producers in Hamilton and York counties. L.C.'s seed corn received \"most of the important prizes\" awarded by the Nebraska Farmers' Institute in its early years and by 1930 was reported to be \"sought after far and wide,\" but corn seeds in those days were all coming from open-pollinated populations, which means that even seeds that came from the same ear were each genetically distinct. So much genetic variability made it impossible to steadily improve yields, because selecting the best seed came down to eyeballing kernels and hand-selecting based on experience of past years. Even sophisticated breeders, like L.C., who knew how to cross-pollinate, would breed two varieties one year and get large, healthy ears, only to find that the same two varieties produced scraggly cobs with missing kernels and dead tips the next. So if you look at USDA records for corn yields from the end of the Civil War to the beginning of the 1930s, you'll see that\u2014despite huge advances in planting and harvesting equipment, fertilizers, and pest control\u2014the per-acre average remained right around twenty-five bushels for more than sixty-five years.\n\nThat all changed with the arrival of a corn breeder who one plant geneticist at the University of Nebraska told me was \"the Bill Gates of the seed industry.\"\n\nHENRY A. WALLACE, known to friends and family as H.A., was the son of Henry C. Wallace, the longtime president of the Corn Belt Meat Producers Association and a liberal-minded professor at Iowa State Agricultural College who counted George Washington Carver among his students. Carver took young H.A. on daily walks, teaching him about parts of flowers and the reproductive processes of native grasses. As Carver gained international fame at the Tuskegee Institute for finding new uses for peanuts as a way of supporting poor, small-plot African American farmers in the South, Wallace was beginning his own studies at Iowa State and dreaming of similarly transforming corn production for poor white farmers in the Midwest.\n\nWhen he was still a student, Wallace began experimenting with hybridization. Rediscovering Gregor Mendel's groundbreaking research on pea pods, he hit upon the key insight that the only solution to overcoming corn's genetic variability was to first produce consistent varieties, with each kernel the same, that could be used as \"parents,\" year after year. It was possible to achieve this level of artificial selection because of two peculiarities of corn plant anatomy and reproduction. First, corn plants are all naturally hermaphroditic. The male flower of a corn plant is called the tassel; the ear with its corn silk is the female flower. Under normal conditions, the silks of a corn plant are pollinated by the tassels of neighboring plants, but if pollen from other plants isn't available, corn has a second peculiarity: the hermaphroditic parts are capable of self-pollinating.\n\nOf course, when a plant self-pollinates, the progeny very closely resembles the parent, because you don't have DNA introduced from other individuals. In nature that kind of inbreeding would be detrimental, because it could lead to reinforcement of undesirable characteristics (like the Habsburg jaw or Romanov hemophilia among the intermarried royal families of Europe). However, if you are selecting for a desirable characteristic and forcing progeny to self-pollinate for multiple generations, you eventually wind up with a plant that is so genetically pure that the desired trait can be reliably passed from one generation to the next.\n\nWallace worried, however, that planting inbred varieties was a dead end, because undesirable traits were often transmitted along with the desirable ones\u2014so you might end up with a cob with consistently dense kernels, for example, but the ear itself turned out to be underdeveloped or the plant didn't grow to full height. But then, in 1920, Wallace found a paper by Donald Jones, a chemist at the Connecticut Agricultural Station's experimental farm, describing how he had successfully inbred two separate varieties of corn and then crossed them to produce a durable, high-performing hybrid. Wallace recognized that this was the key to creating seed corn with consistently higher yields. You could build a stockpile of inbred seeds with desirable characteristics and then, year in and year out, plant one inbred variety side by side with the other. Just before flowering, the tassels of one would be removed to make a silk-only \"female parent\" to receive the pollen from the intact tassels of the \"male parent\" rows. This crossbreeding produced outstanding kernels to use as seeds for the following year.\n\nThis is still essentially the crossbreeding technique employed by all major seed producers, but it's not quite so simple as planting two varieties in the same field. To achieve proper pollination, farmers have to calculate the \"nick\"\u2014the period when each plant will be sexually mature. But because they are separate varieties, the plants reach maturity at different rates, requiring different planting dates for the silk parents and the pollen parents to ensure that the silk emerges at the same time that the pollen is shed. To hit that narrow window, seed corn farmers use a final piece of knowledge about corn.\n\nThe plants mature according to the approximate amount of heat they receive, starting from planting. \"It all goes by growing degree units, GDUs,\" Dave told me. One GDU is added for every degree the average daily temperature exceeds 50\u00b0F. So, for example, a day with an average temperature of 65\u00b0 would be counted as 15 GDUs. \"And they know how many GDUs it takes for the female hybrid to get to pollination and how many GDUs it takes for the male to get to pollination, so that nick happens at the same time,\" Dave said. \"If you put them all in at once, your male could shed pollen before the female has started to silk, and you get no pollination. If that happens\u2014and it has\u2014you can lose a whole crop.\"\n\n\"That's why weather is such a big deal,\" Dave continued. \"You plant your males and females days, maybe weeks, apart\u2014all depending on the temperatures. You get a string of days like we just had, where it never gets above fifty degrees, and you just have to wait.\" On days like those, there's an even greater complication. Several times Rick had described to me the bizarre growing season of 1991\u2014when, halfway around the world, Mount Pinatubo erupted in the Philippines and, in his words, \"put a whole shitload of ash in the upper atmosphere.\" In Nebraska, temperatures climbed like normal in July and August, but the solar radiation that causes corn to grow was diffused. \"That was the first year that I'd ever read of a measurement called Langleys,\" Rick said. Instead of just measuring heat, Langleys measure the density of energy distribution, capturing the difference between a muggy overcast day and a cloudless day when the sun is beating down. \"It was cloudy that year, just like this year's been,\" he said, \"and everybody got into their fields at harvest, and it was twenty bushels off what they were expecting.\"\n\n\"When it gets overcast,\" Dave explained to me, \"it can be a crap shoot on getting the weather right.\" As he reset the planter and got us headed south again across the field, he said that there were practical concerns beyond the problem of estimating the nick. \"This field over here,\" he said, pointing to the neighboring quarter section. \"They wanted us to plant last week, but we kept looking and, God, they were calling for rain all week. What if it heated up and the corn matured, but it was too muddy to get in to plant the other rows? So we had to wait\u2014especially with that field, because that was male put in first. You know, my male row planter I can pull around in the mud a lot easier than this thing.\" He pointed a thumb back at the rows of hoppers, all heavy with seed and bouncing as the planting blades sliced into the loose soil behind us. \"If it's wet, I can't do anything with this.\"\n\nWhen Henry Wallace had seen how complicated creating reliable seed corn would be, he started to envision a nonprofit organization, run with government cooperation and potentially even public funding, that would distribute his seeds to midwestern farmers for free. He was a man of unusual commitment to the common good and wrote a friend that he did not consider himself a corn breeder but rather \"a searcher for methods of bringing the 'inner light' to outward manifestation.\" In 1921, Wallace's father was appointed secretary of agriculture by President Warren G. Harding and might have spearheaded such an effort. But barely two years later, his father died unexpectedly at age fifty-eight, and Calvin Coolidge, Harding's successor in the White House, soon settled into the laissez-faire __ years of our nation's history. Seeing little chance of an ambitious national program gaining traction, Wallace started the Hi-Bred Corn Company, the world's first hybrid seed producer, as a private business in May 1926. But the path to fortune was anything but swift.\n\nInitially, the company saw little in the way of profits. After all, Wallace was asking his seed representatives to try to convince farmers that his seeds were superior to any of those they had been using for years. So Wallace's lead salesman, Roswell Garst, came up with a daring plan. He went from one farm to the next, across sixteen counties in western Iowa, giving away enough eight-pound sample bags of Hi-Bred seeds for farmers to plant half their fields. Whatever additional yield the hybrid corn produced, he would split fifty-fifty with the farmer. After several years, farmers realized that they would see greater profits by simply buying the bags of seeds, rather than sharing the surplus yield with Garst.\n\nThose shared harvests generated income for the young company, but they also produced something more important: they returned a fount of information about how Wallace's seeds performed under different growing conditions. Now renamed Pioneer Hi-Bred, the company turned a sizable chunk of its revenue back into research, hiring a team of new corn breeders to devise still more hybrids that would show improved performance under tough growing conditions, such as the shortened seasons of cold-weather climates or in dryland areas where farms were without irrigation. In the early 1930s, Perry Collins, one of Wallace's researchers, developed Hybrid 307\u2014the first corn specifically designed and marketed for drought resistance, hitting feed stores just as the country spiraled into the Dust Bowl. And when Wallace was, like his father, appointed secretary of agriculture by Franklin Roosevelt in 1933, he finally had the resources to nationally evangelize for hybrid seed, which he believed had the potential to rescue the nation from the Great Depression.\n\nThe transformation that followed was staggering. When Wallace joined Roosevelt's cabinet, less than 1 percent of America's corn came from hybrid seeds. A decade later, more than three-quarters of all corn was grown from hybrids\u2014and Pioneer seeds steadily produced the most impressive results. To keep record yields from depressing corn prices, Wallace created the \"ever-normal granary,\" under which the government would establish a federal grain reserve. In years of high production, the Department of Agriculture would buy corn and store it to keep prices up. In years of crop loss, the government would release the reserve to keep prices down. Wallace's plan was hugely popular, stabilizing American food prices\u2014and winning him a spot as FDR's running mate in 1940.\n\nAs vice president, Wallace convinced Roosevelt to go even further. To prevent runaway production and the overplanting that created many of the Dust Bowl conditions, Roosevelt signed the Soil Conservation Act, establishing subsidies for farmers to restore native grasses and trees, rather than planting commercial row crops that reduced groundcover and depleted soil nutrients. At virtually the same time, FDR authorized employing federal foresters to plant more than 200 million trees at the perimeters of farm fields in a hundred-mile-wide zone from the Canadian border to the Brazos River. The idea was to create a national windbreak\u2014what was termed the Great Plains Shelterbelt\u2014to reduce the velocity of dust storms and the loss of moisture due to evaporation from windswept soil. And, in fact, soil loss had been reduced by 80 percent by the end of World War II, even as crop production soared.\n\nBut Wallace's remarkable Hi-Bred Corn had one significant drawback: it consumed far more nitrogen compounds from the soil than ordinary corn\u2014more, in fact, than almost any other crop. During the war years, the government solved the problem of depleted soil by simply putting more acres into production, but after World War II, the Department of Agriculture found a different solution. Giant chemical manufacturers, like DuPont and Monsanto, had secured wartime defense contracts to produce ammonium nitrate and anhydrous ammonia to make bombs and other munitions. Now, they argued to the USDA, those chemicals could be used as fertilizers.\n\nTHE MILLER NITRO sprayer, a cherry-red colossus on 6-foot-tall all-terrain tires, rolled up to the edge of the freshly planted field of corn. The twin booms of the spray rig were spread out in either direction, like a pair of enormous dragonfly wings, unfolded and lowered by a system of hydraulics. The corn was just tall enough that the first sprigs of green were showing, and Kyle was preparing to spray them with a soup of liquid herbicides to wither any weeds that might compete for nutrients and sunlight. But before Kyle turned on the sprayer, he wanted to reassure me. \"The cabin is completely sealed,\" he said. A pressurized ventilation system with built-in charcoal filtration and a secondary in-cab filter would make sure that none of the chemicals in the sloshing 1,000-gallon tank behind our seats, which Kyle was about to turn into a low-pressure mist, was going to make it inside. \"I'm not going to fumigate you,\" he said with a laugh. \"I promise.\" With that, he flipped on the sprayer and started bouncing across the field, much faster than I'd imagined.\n\nBefore getting to this stage, Kyle had made a pass just before planting with the same machine, applying liquid nitrogen for the N-hungry corn hybrids. When Henry A. Wallace created the first inbred corn varieties, he hadn't anticipated one problem: uniformly healthy corn plants that experienced rapid growth\u2014going from knee-high to shoulder height in as little as two weeks\u2014swiftly and voraciously converted nitrogen from the soil to proteins needed for green tissue growth. As new hybrids yielded taller stalks, now growing as much as 10 feet tall, and were bred to be tolerant of tight planting, allowing farmers to plant more seed per acre, the natural nitrogen content of the soil proved too little to sustain a crop through a full growing cycle. Thus, the application of nitrogen compounds has become central to modern, industrial-scale farming, but the discovery of its importance was the result of a bizarre happenstance.\n\nAs early as 1910, the DuPont Powder Company, a munitions manufacturer so large that they are estimated to have provided half of all the gunpowder used by the Union Army during the Civil War, had sought to expand its business by encouraging the use of its explosives on midwestern farms. Company representatives distributed a free pamphlet, entitled _Farming with Dynamite_ , which promised to simplify the tasks of removing stumps, clearing land, and breaking up hardpan. Two years later, Edward Lewis, a DuPont representative addressing the Nebraska State Horticultural Society, acknowledged that \"the name of DuPont suggests something rather scary,\" but he said that a farmer in Thayer County, Missouri, who had dynamited his entire field had made a discovery exciting enough to ease their fears. \"He planted his corn in the land that he had dynamited,\" Lewis reported, \"and it doubled his crop the first year.\"\n\nOf course, it wasn't the breaking up of the soil that was promoting rapid growth but the nitrogen compound contained in gunpowder. In 1913, Fritz Haber, working at the Oppau plant of the German chemical company BASF, succeeded in developing a less destructive way of achieving the same result: \"fixing\" nitrogen from the air by combining it with hydrogen into ammonia and then catalyzing it. The resulting nitric acid remains the basic building block for the manufacture of all modern fertilizers\u2014but Haber's process also ushered in a new era of chemical weaponry. During World War I, Haber not only devised more powerful explosives but also poison gases, such as ammonia, then chlorine, and eventually the Zyklon B gas used on Jewish prisoners in Nazi concentration camps.\n\nBy that time, the use of chemicals against humans had been outlawed under the Geneva Protocol, so American companies like DuPont found other ways of refocusing their laboratory know-how on the war effort. The herbicide 2,4-D was developed as a potential destroyer of enemy crops. The insecticide DDT was manufactured to prevent the spread of typhus-carrying lice among GIs. But as soon as the war was over, DuPont turned to marketing those same chemicals as DuPont Garden Dust and DuPont 5% DDT Insect Spray. Company advertisements touted these products as \"Better Things for Better Living . . . _Through Chemistry.\"_ But gardens were just the tip of the iceberg. DuPont, along with other giant chemical manufacturers like Dow and Monsanto, teamed up with the old grain cartels, including Cargill and ADM, to lobby for congressional support for producing these compounds as large-scale agrichemicals.\n\nIn 1953, the industry found its greatest ally, when Ezra Taft Benson took over as President Dwight D. Eisenhower's secretary of agriculture. (Wallace, by then, had retired from public life after making a failed bid for the presidency in 1948.) Benson, a high-ranking member of the Mormon Church and a fanatical Red Scare Republican, immediately informed Eisenhower that he was philosophically opposed to the government price supports developed by Wallace, because, to his mind, they were tantamount to socialism. He publicly referred to small farmers as \"irresponsible feeders at the public trough\" and vowed to return to a system where the greatest profits went to the largest producers. Small farmers hated him for it, but Benson was unflagging. In 1957, at the National Corn Picking Contest in South Dakota, he delivered a speech on the evils of federal farm aid despite a hail of eggs from angry farmers.\n\nForeshadowing today's aggressive, pro-corporate agricultural policies, Benson argued that the only way to outcompete the collective farms of the Soviet Union and Red China was to use our superior corn and chemical technology to the fullest. The United States could, if it chose, overproduce corn to drive down international prices, and it could use the surplus as a tool of diplomatic leverage in the form of foreign aid. Instead of guns, the United States began to give our allies grain\u2014transforming, for the first time, a food product into a weapon in the national arsenal.\n\nWhen Soviet premier Nikita Khrushchev visited the United States at President Eisenhower's invitation in 1959, the Cold War for food dominance started to heat up. Khrushchev specifically requested to see only one man: Roswell Garst, the former Pioneer seed salesman for Henry A. Wallace, who was then head of Garst & Thomas Hi-Bred Corn Company. Khrushchev had met Garst once before, when he visited the Soviet Union, and had become obsessed by the potential of hybrid corn. Khrushchev and his wife spent a day at Garst's farm near Coon Rapids, Iowa. In his memoirs, Khrushchev later wrote, \"Garst gave me an entire lecture on agriculture,\" in which he earnestly explained that American farmers had stopped worrying about crop rotation. Garst told the Soviet leader, \"Science today considers that approach outdated. And I think so, too.\" In past years, planting the same crop repeatedly would have attracted pests and depleted the soil of nitrogen. \"Now there is no such problem. We have herbicides and other such chemical substances that make it possible to combat pests,\" Garst said. And there was no longer any need to plant clover or alfalfa to accumulate nitrogen. \"It is more profitable for me to buy nitrogen, potassium, phosphorus, in mix form, and add this fertilizer.\"\n\nOn that same official visit, Benson led Khrushchev on a tour of the U.S. Agricultural Research Center in Beltsville, Maryland. Benson, in his official remarks, said that there was a \"constant give-and-take of information between government scientists and those in private industry,\" adding that \"we are all working together within the framework of our capitalistic free-enterprise society to benefit our farmers, all our citizens, and people throughout the world.\" He listed hybrid corn first among the achievements of such cooperative efforts and introduced white-coated lab researchers who extolled the virtues of 2,4-D and chemical fertilizers. Khrushchev was unimpressed by the visit he made to a farm owned by President Eisenhower, dismissing it as \"not on a scale such as we have at our collective farms and state farms.\" Benson later remembered that Khrushchev bragged, \"We won't have to fight you. We'll so weaken your economy until you fall like overripe fruit into our hands.\" Benson vowed that American farms would outproduce the Soviets through superior chemistry.\n\nBy the end of the Eisenhower era, however, environmentalists began to raise concerns about the hundreds of commercial herbicides and pesticides being applied to American crops in quantities totaling hundreds of millions of pounds. In fact, a citizens' group, led by biologist Robert Cushman Murphy and Mayo Clinic hematologist Malcolm Hargaves, brought suit against the USDA in 1958 to bring a halt to indiscriminate DDT spraying in New York State. Benson admonished doubters that \"abandoning the use of chemicals on farms and in the food industry would result in an immediate decline in the quantity and overall quality of our food supply and cause a rapid rise in food prices paid by the consumer.\" Later, when Rachel Carson, who attended those trial proceedings, documented the dangers of DDT and 2,4-D, including elevated incidence rates of rare forms of cancer, DuPont responded by opening a confidential file on her and intimating to her publisher that they might sue. But Carson's message was already out. Her modest calls for mitigating the use of herbicides and pesticides were instead carried out as an outright ban on DDT, the passage of the Clean Air Act, and eventually the establishment of the Environmental Protection Agency.\n\nBefore long, Monsanto returned to manufacturing 2,4-D for its original purpose. During the early stages of the Vietnam War, Secretary of State Dean Rusk argued to President John F. Kennedy that the use of chemical defoliants was not technically a violation of the Geneva ban on chemical weaponry, because the chemicals were meant to be sprayed on dense jungle foliage, not enemy combatants themselves. Willing to be persuaded, Kennedy authorized the covert Operation Ranch Hand. Using chemical herbicides provided by Monsanto and Dow, most of them based on 2,4-D or its descendant 2,4,5-T, the military created a series of Rainbow Herbicides, code-named according to different colors on the spectrum. There was Agent Purple, then Agent Green and Agent Pink. Then in 1965, government chemists developed a potent mixture of the esters of 2,4-D and 2,4,5-T, which they called Agent Orange. The herbicide was unusually effective\u2014especially when it was deployed at fifty times the concentration that either chemical had been used in agriculture.\n\nRick told me that he thought that allowing 2,4-D and other herbicides to be used in that way had been a mistake. The military had been applying those chemicals in concentrations that scientists at Monsanto and Dow had warned would be lethal or debilitating to humans. In time, the government admitted that Agent Orange had led to lung and prostate cancers for American veterans and poisoned the water and food supply of forest villages in Vietnam, leaving behind a legacy of horrifying birth defects. More than that, the fear caused by seeing the effects of military-grade applications of Agent Orange had turned the public against it as an herbicide in the United States. But Rick believed that limiting its use had only increased the production of other, less-tested replacements.\n\nIn 2012, for example, high levels of atrazine, a substitute for 2,4-D widely used by corn growers, were found in the water supplies of thousands of municipalities across the Midwest. Syngenta, the manufacturer of atrazine, agreed to pay more than $100 million to those utilities for the installation of filtration systems, as part of the settlement of a class-action lawsuit. The company admitted to no wrongdoing, but the Natural Resources Defense Council was quick to note that Syngenta's pay-out came after researchers in France linked drinking atrazine-laced water during pregnancy with increased risk of delivering a low-birth-weight baby\u2014with all of the associated health problems for the infant. And that's just one example among thousands of herbicides.\n\n\"Bill Moyers had this deal on chemicals on PBS,\" Rick told me. \"They tested his blood and found eighty-four different chemicals there\u2014of which about fifty or so hadn't been invented in 1940.\" He noted that there are now more than 75,000 chemicals registered with the EPA, but that the vast majority have never been tested to find out their effects on human health. For Rick, and anyone working on a farm, these concerns are anything but abstract. In the waning days of harvest in 2014, the National Institutes of Health had published the results of a twenty-year study that looked at rates of depression among farmers, showing compelling evidence that use of seven different pesticides, mostly neurodisrupters, had nearly doubled the suicide rates among the men who applied those chemicals. Worse still, the depression rates were even elevated among their spouses and other immediate family members. Soon after, another study found two hotspots for Parkinson's disease in Nebraska, apparently linked to historical use of pesticides in those areas.\n\n\"You can't ignore it,\" Rick said, \"but banning old chemicals just puts industry on the hunt for new ones\u2014which will be used before we even know if they're safe or not. We can always see the short-term benefits, but we don't see the long-term ill effects until it's too late.\" That's why Kyle waits for a windless day to use the sprayer, applies as little of each chemical input as possible, and stays inside the cab until a whole field is sprayed and the air has cleared. It's why Rick showers and washes his clothes right after applying these chemicals and insists that Kyle do the same. It's also why Rick fiercely defends the use of anti-pest varieties of row crops that have been developed using genetic modifications. And it's why he eventually decided to plant crops with engineered resistance to a newer product developed by Monsanto as an alternative to 2,4-D-based herbicides.\n\nThat herbicide, which has gained widespread use in the last twenty years, is called glyphosate\u2014or, more commonly, Roundup.\n\nRICK HELD off on planting soybeans through most of May. He had chosen to go with a short-season variety again that year and didn't want to get them in too early, risking another wet fall and another late harvest\u2014especially with what he was seeing in the commodities markets. Rick had spent the winter emptying his bins as slowly as possible, keeping back reserves, waiting. There's typically a spring bump in prices, what's called a \"seasonal rally,\" at the start of planting when stockpiles are at their absolute lows and the results of the South American harvest start to come in. \"Eight out of ten years, you'll see it,\" Rick said. \"But then a couple of weeks ago, we had a report come out that made everything go the other way. The U.S. had had a big bean crop, and then South America had a fantastic year.\" Instead of the usual spring bump, prices had actually fallen further on the news. Rick decided to sell off what he had left\u2014about 5 percent of his overall corn crop and fully 25 percent of his soybeans, grain he had been storing for more than six months. It was time to cut losses and hope for a better year ahead. \"Maybe I sold on the low,\" he said. \"But I just didn't see upside anytime soon.\"\n\nSo Rick cleared out the last of his bins, hauled the final semi loads to the elevator, and started picking up new seeds from the dealer. But by the time he had gotten the John Deere planter loaded on the flatbed and brought over from the seed corn fields, a line of spring storms had rolled in, blanketing the sky for days. By then, there was nothing to do but put his seeds in as the weather allowed, one field at a time. On the day Rick was ready to move to the quarter section neighboring Centennial Hill, the clouds were stacking up, billowy and roiling. They kept threatening big rain but never produced more than a light mist, so he planted half of the field with one variety, and when the weather seemed like it would hold for a few more hours, he hitched up the seed tender, a portable tank that looked like a miniature water tower on wheels, and headed to the seed dealer to get a second variety to finish the field.\n\nAs we drove, Rick explained that his Pioneer dealer offered thirty-eight varieties of soybeans. Almost all of them were genetically modified by bioengineers, inserting strands of DNA that make the plants able to withstand a range of herbicides, particularly the compound glyphosate. Chemist John E. Franz, working for the St. Louis\u2013based chemical giant Monsanto, first hit upon glyphosate in 1970 as part of a company-wide search for potent chemicals that could be sold as weed-killer. Brought to market four years later under the trade name Roundup, the broad-spectrum herbicide saw unmatched growth in the agricultural industry, where it was used for broadleaf weed control. The only problem with Roundup was that it proved so potent that farmers had to apply it carefully, spraying only early weeds, before their corn or soybeans had begun to sprout. Otherwise, it would kill the entire crop.\n\nFrom Monsanto's standpoint this presented a serious problem. Simply put, the necessity of limiting Roundup's application also limited its sales. So company researchers started to seek out ways of engineering crops that could withstand this new weed-killer. By then, scientists had already developed the revolutionary new ability to cut and splice protein strands into the DNA sequences of bacteria. If they could do the same with plant cells, then, at least in theory, they could chemically insert resistance to insects or to herbicides.\n\nMonsanto had recently entered into a deal with biotech pioneer Genentech to license some of their gene-splicing technology, and they already had successfully used these techniques to produce recombinant bovine somatotropin (rBST), a hormone that extended the lactation period of dairy cows and thus increased their milk production. The hormone, eventually marketed as Posilac, was showing promising results in field tests\u2014and Monsanto executives, now convinced of the commercial viability of genetically modified organisms (GMOs), wanted to find similar modifications that could be made in plants. So company researchers were tasked with trying to find a gene that could be spliced into soybean and corn hybrids to make them resistant to Roundup.\n\nResearchers began by testing every lab sample amassed from the company's fields and facilities nationwide, no matter the reason for its original collection. From a Roundup manufacturing plant near Luling, Louisiana, they received a sample that had been collected from a waste dump, where workers had noticed a range of bacteria thriving despite prolonged exposure to glyphosate. When the sample was taken, Monsanto researchers were hoping to produce a glyphosate-resistant bacteria that could help break down the chemical when there were spills or overapplication events. Now they were looking for the resistant gene inside the bacteria in hopes that it could be spliced into row crops to _increase_ the application of Roundup. By 1986, researchers had isolated the particular gene that controlled that immunity, spliced it into soybeans, and were ready to begin field trials.\n\nIn the meantime, other Monsanto researchers were trying to find a similar genetic-engineering solution to the European corn borer, an insect that inflicted more than $1 billion in losses in the United States and Canada each year. Since the 1960s, endotoxins produced by _Bacillus thuringiensis_ (Bt), a common bacteria found in the soil, had been sold as a commercial microbial insecticide to kill moth larvae. If the specific DNA that produced Bt toxins could be isolated and spliced into corn genetic sequences, scientists believed they could create an ear of corn that would be lethal to the European corn borer but perfectly safe for humans or livestock to eat. Soon, that hurdle had been cleared, and Monsanto began looking for a seed partner to market its pest-resistant corn\u2014and decided to approach Pioneer. If Monsanto could marry its genetic modifications with Pioneer's superior inbred seed stock, Monsanto believed it would have a line of products with unmatched yield potential.\n\nIn the 1990s, perhaps too eager to demonstrate the effectiveness of its new GMO crops, Monsanto allowed Pioneer to use its biotech to produce Roundup Ready soybeans and Bt corn\u2014asking only for small usage fees and no royalties. For less than $40 million, Pioneer suddenly had the technology and the sales muscle to move toward genetically modified feed crops. Rather than partner with Monsanto, Pioneer became its greatest competitor, entering into a joint venture with DuPont, which soon came to regard seed technology as so lucrative an opportunity that it bought Pioneer outright (and changed the name to DuPont Pioneer). Despite the rounds of lawsuits that followed, DuPont Pioneer released Roundup Ready soybeans in 1994 and then Bt corn in 1996.\n\nWhen Pioneer seed reps first started promoting these new products, Rick initially resisted. \"I was very reluctant,\" he said. He objected to Pioneer requiring that farmers sign an agreement promising not to save and clean soybeans to replant the next year. Even though trying to replant progeny seeds from hybrid varieties rarely produces a quality crop, he said that making that mistake should be a farmer's prerogative. \"It just felt wrong to me\u2014un-American,\" he said. \"You know, I paid you for this seed. You're telling me what I can and can't do next year with the things that I grow? That's rotten. They say that they went to the expense of doing the GMO modifications. I get that, but you know what? There was expense to seed hybridization before, and we never had to sign anything. It seems to me that if you raise it and grow it you should be able to plant it.\"\n\nRick also traditionally spread his risk by buying hybrids from different dealers, but as soon as Monsanto and Pioneer got into cutthroat competition, they each started acquiring smaller competitors, just to expand their market share. \"There used to be a hundred seed corn companies,\" he said. \"But it's been winnowed down to a few. The bigs have bought out most of the littles.\" Unable to compete against the seed giants, small companies relied more and more on the open-source genetics provided by public universities, rather than investing in their own research. Rick began to suspect that he was getting identical genetics, no matter what the label on the bag said.\n\nOne year, for example, he planted three different varieties of corn\u2014Super Cross, Maize, and Golden Harvest\u2014and they all fell prey to the same weakness in their stalks. Known as \"green snap,\" corn plants with a particular genetic defect will develop elongated nodes when exposed to herbicides, making them prone to blow-down in heavy winds. Seeing all three varieties display the same problem made Rick suspicious. \"I could not physically tell the difference,\" he said, so he just decided to go with the one company that had always been successful for him. \"I really liked Pioneer,\" he said. \"That was it for me. I was done. No more Mr. Nice Guy. There was no sense losing volume discounts, when they were all the same. I wasn't saving money, and I wasn't spreading my risk out.\" And once he had gone over exclusively to Pioneer corn, it just made sense to plant Pioneer soybean hybrids.\n\nEven so, Rick had his reservations. \"I could see that, within a relatively short period of time, if everybody switched to Bt corn, then the bugs would become resistant. And if both the corn and soybeans were Roundup Ready, then you'd end up with weeds completely resistant to Roundup.\" In fact, in the twenty years after Roundup Ready soybeans were first introduced by Monsanto, the amount of glyphosate that the United States collectively applied each year went from less than 10,000 tons to more than 125,000 tons, and, indeed, the combination of monocultures of corn and soybeans and using a single herbicide for both has produced a range of herbicide-resistant superweeds. \"If they had just stuck with Roundup Ready beans and rotated those with Bt corn, using old 2,4-D or other families of chemicals for weed control, then Roundup could have been a viable product for many, many years,\" Rick said. Instead, farmers tend to use more Roundup and more combinations of other herbicides with Roundup.\n\nUnsurprisingly, the agricultural chemicals team at the U.S. Geological Survey Office, part of the U.S. Department of the Interior, has now detected the presence of glyphosate in the air and water supply in rural states. And the effect of those chemicals is the subject of heated debate. In 2015, the cancer-research wing of the World Health Organization released a report finding that glyphosate is probably carcinogenic to humans. Monsanto hotly disputed the findings. The following year, the UN and WHO issued another report clarifying that, at realistic intake levels, glyphosate was \"unlikely to pose a carcinogenic risk to humans from exposure through diet.\" Still, critics maintain that there are no studies of the effects of exposure over many years. Rick acknowledged this fear about the long-term impact, but he also wondered if it was any worse than the chemicals that had come before.\n\n\"It's not like we were all organic farmers until Monsanto came along,\" he said. He vividly remembered the days of applying dyfonate, a highly toxic pesticide used on corn plants. \"You had a separate set of boxes on your planter with this pesticide inside,\" Rick said. \"It was supposed to go in furrow, but invariably when you plant, the wind's blowing about forty miles an hour. In those days, there were no vacuum-sealed cabs. I remember that dust caking up along the sweat band of my cap, and when I took my hat off, it took the top layer of skin with it.\" Bt corn had done away with many such pesticides, and he said that Roundup had also eliminated many herbicides that were problematic \"as far as farmer safety goes.\"\n\nBesides, Rick said, he couldn't afford to ignore what he saw while driving up and down these same country roads we were on now: his neighbor's fields of soybeans and corn that looked taller, healthier, more productive than his own. \"My decision on whether to go GMO or not was a tough one,\" he said. \"I didn't automatically jump in like everyone else did. But the big deciding factor to me was the old question: wherein lies the biggest evil? Because at the other side, I was using old broad-spectrum poisons\u2014dyfonate, another one called Counter\u2014that were killing all the beneficials. With GMOs, at least, we were using material from naturally occurring bacteria and not just killing everything in sight.\"\n\nSo one year, Rick decided to raise a few fields of Roundup Ready soybeans. \"It was a nightmare,\" he said. \"I was like, 'Oh, my God, if I forget exactly where the GMO quit, and I spray Roundup in the wrong places, I'll kill my whole crop.' \" Plus, even to plant half his fields with Roundup Ready beans, he had to sign an agreement with Pioneer that curbed his rights on the other half. He was under all of the restrictions without any of the rewards. \"And ethically, it's like being a little bit pregnant,\" Rick said. \"Either you are a sinner or you're not.\" So the next year, and ever after, he bought only Roundup Ready soybeans, and eventually, he bought Pioneer's newly introduced Roundup Ready corn. And in no time, DuPont Pioneer and Monsanto introduced new varieties with \"stacked traits,\" combining Roundup resistance with other desirable characteristics.\n\nAt North Forty Seed, Rick's dealer, there's a genetically modified variety for almost any soybean need. One modification causes the plant to create a protein that kills insects; another changes the profile of the fatty acids in the beans, making them less susceptible to rot after harvesting. The Soybeans Seed Guide breaks down the performance of each product, all sold under codenames like P22T41R2 and 93Y41, into extremely fine categories: whether they are resistant to particular races of soybean cyst nematodes, to the Phytophthora infections that cause root rot, or to white mold that can cause stem rot; how quickly they emerge, how tall they grow, how wide they canopy; what their protein and oil percentages will be at optimum 13 percent moisture rates at harvest.\n\nThe precision is stunning, and the knowledge demanded of farmers, most of whom consult with agronomists but ultimately decide what to plant on their own, seems impossible to fathom\u2014but also impossible to escape. A quarter of a century ago, Roundup Ready crops didn't exist; today, they are 90 percent of the soybean market and nearly three-quarters of the corn and rice markets. And on the strength of that success, Monsanto and DuPont Pioneer have grown into global seed giants, now controlling 45 percent of all the seed sold in the world. Short of going completely organic and dropping out of growing commodity grains, how is a farmer supposed to avoid raising corn and soybeans that have been genetically modified to withstand Roundup?\n\nAs we approached the dealership, Rick put on his hazards, pulled past the end of the driveway, and then backed the trailer straight onto the concrete pad next to the seed warehouse. As Rick swung his driver's door open, Dennis Stevens was there to greet him.\n\nDennis wore his Pioneer cap low, and his snowy mustache nearly covered his smile. \"Well, if it isn't the local liberal,\" he said.\n\nRick adjusted his cowboy hat as he stepped out of the cab. \"If it isn't the local son of a bitch,\" he replied. The two men smiled broadly and shook hands.\n\nDennis walked us over to the pro boxes, stacked like oversized milk crates all the way to the rafters, each labeled with its particular product code but also its place of origin, the date it was field-tested, any preapplied treatments used, and the exact number of seeds in the box. Dennis double-checked which product Rick was picking up with this load, and then gave the signal to the forklift operator to pick up a box full of soybeans and hoist it high over Rick's seed tender. The pro boxes have a trap door on the bottom that, when opened, lets the seeds pour out like sand in an hourglass. The forklift revved and beeped as the operator moved it precisely into place. Rick climbed the side of the tender and yanked on the sliding door, sending the seeds echoing into the empty tank. When every seed had drained out, Rick banged hard on the door to slide it back into place, but it wouldn't budge.\n\n\"Do you need a hammer?\" Dennis called.\n\nRick gave him a quick, dismissive smile, then pushed on the door again, putting his shoulder into it this time. It slid into place with a resounding clang. \"That ought to do it,\" Rick said.\n\nBefore long, we were racketing back over the washboard roads, the heavy tender bouncing and tugging against the hitch. Given everything, Rick seemed oddly ebullient\u2014drumming his thumbs on the steering wheel, whistling a little despite a few raindrops blossoming on the windshield. \"I usually have about fifty percent of my beans forward-contracted by now, and this year, I have _none_ ,\" he said. He had decided to take an even bigger gamble than the year before, banking on prices going up before harvest. It was an undeniable risk, but after all winter worrying, he was happy to finally be getting back to work, doing something instead of just waiting.\n\nAt the field edge, Rick pulled alongside the hopper at the back of the planter. Dave fired up the auger, powered by a small diesel engine underneath that started with a pull cord like a lawnmower, and then Rick, climbing a ladder to the hatch door on top of the main hopper, directed the corrugated plastic hose inside, spreading the seeds out layer after layer. Besides their many genetic modifications, the candy-bright seeds, like a barrel of green M&Ms, had been treated with a mix of insecticides and fungicides similar to those used on the seed corn. Rick wore a plaid flannel shirt buttoned at the wrists and thick rubber gloves as protection. All the while, he kept eying the sky. \"I'm afraid we're going to get everything loaded up just in time for it to rain,\" he shouted.\n\nThe prospect of rain was concerning, not only because the planter can mire and get stuck in a muddy field. Even a light rain can affect seed depth for each furrow, forcing the farmer to constantly recalibrate and double-check the planter to keep from wasting seeds. Rick often recalls the spring, not long after he'd started working with Heidi, that he planted too deep and nothing sprouted, so he replanted but too shallow and a spring rain washed everything away, so he replanted again, by then needing a perfect season just to make up for the lost investment in seed. Rick always wore a mustache up until then, but that year, the worry turned his facial hair snow white. In the end, he broke even for that year, but he's been clean-shaven and a bit more mindful of his seed depth ever since.\n\nDave climbed into the cab of the tractor and started the onboard computer. The fan pressurizing the planting system came to life again. A big chemical tank to one side of the planting attachment made it so that Dave could apply additional fungicide, insecticide, or fertilizer right as the seeds came out of the bulk bin. He pulled the planter parallel with the last completed rows, lowered the planting blades, and started again across the field. For the moment at least, they had beaten the rain.\n\n\"They say farmers are the world's biggest optimists,\" Rick said, watching Dave go. \"You have to be optimistic in this business, just to keep going\u2014or at least have an awful short memory.\"\n\nAFTER ENOUGH warm days that the female seed corn was moving toward maturity, Dave returned to the fields to complete planting the missing rows of male corn. The planter crept forward, the blades opening and closing furrows in the blanks between the female rows, where tiny corn plants now sprouted green and tender. Rick leaned against the bed of his truck, watching. Pioneer had sent out a field manager earlier in the day to check on the progress of this quarter section, and Rick was worried about what would happen if the guy came back and found an outsider there. I assured Rick that I had checked in with the company's media liaison, that the last thing I wanted was to cause him any headaches.\n\nHe nodded ruefully, visibly unconvinced.\n\n\"Pioneer used to be a wholly owned family deal,\" he said. \"Then it got bought out by DuPont. Since then, I don't know. We've only been planting seed corn for them for seven years, but it seems like a lot of corporate edicts come down now that your local people just have to follow. What the solution is, I don't know. I'm sure they've got their reasons. I just wish it were a little less top-down, a little more grower-friendly.\"\n\nAs one example, he explained that DuPont Pioneer now required its farmers to account for the custody and security of every bag of seed corn\u2014right down to requiring a pallet strap for hauling from the company compound, just outside of York. After all Pioneer has done to secure and protect its intellectual property, the last thing the corporate bosses wanted was to risk a spilled load\u2014seed scattered along the roadway or into the ditch, raising the possibility of some farmer scooping up their patented genetics, or seeds simply taking root and cross-pollinating with other corn plants. And even after you've picked up the seeds and secured them, you have to agree to drive nonstop from the pickup point to your farm. By then, you've already had to prove that you have a locked shop. They come out and check it, and they insist on more than just a barn with a padlock; they want a framed building with a secure door\u2014and the seed has to go straight from their loading dock to that shop or else directly to the field.\n\nAs Dave's tractor neared the pad of the pivot in the center of the field, Rick took out his cell phone. \"You call them as soon as you start planting,\" he said, dialing. He told the Pioneer seed representative that Dave was underway on a new field and gave them the identification numbers for the bags and location. That way they could come to check that everything was being planted according to their agronomists' guidelines and then add it to the list of fields under surveillance by the seed security teams.\n\nIt sounds like corporate paranoia to have company trucks patrolling the back roads looking for anything suspicious, but in early May 2011, a farmer in the midst of corn planting outside Tama, Iowa, watched as a car pulled over at the edge of his seed corn field. Two Chinese men stepped out and came scrambling up the ditch. One said that they had been attending an international agricultural conference in Ames and wanted to see someone planting a real Iowa cornfield. The farmer was dubious. Ames was nearly an hour away with nothing but expanses of corn in between, all at the peak of the planting season. How had these two men chanced upon his field\u2014on the very day he happened to be planting an experimental and top-secret seed under development by DuPont Pioneer?\n\nIt was unusual enough that the farmer contacted his seed representative, who sent out a security team to keep an eye on the newly planted rows. The next day, the field manager spotted the same car. He watched as the passenger got out, climbed the embankment, and then knelt down in the dirt and began digging corn seeds out of the ground. When confronted, the man grew flustered and red-faced. He now claimed to be a researcher from the University of Iowa on his way to a conference. But before the Pioneer field manager could question him further, the man fled back down the embankment. He jumped into the waiting car and took off, swerving through the grassy ditch before fishtailing onto the gravel road and speeding away.\n\nA few weeks later, during a routine meeting with agents from the Iowa office of the FBI, a Pioneer executive mentioned the incident. He explained that Pioneer enters into exclusive contracts with farmers to grow proprietary seeds. The exact genetic sequence of successful seeds is a tightly held secret, worth tens of millions of dollars. The Pioneer field manager had written down the license plate number and handed it over to company security. They had been able to trace the plates back to a rental car company at the Kansas City airport. Representatives there said the car had been rented by Mo Hailong, a Chinese national working as the Director of International Business at the Beijing Dabeinong Technology Group Company (DBN), a company with a seed division in direct competition with Pioneer in China.\n\nThe FBI launched a full investigation. Over a period of nearly two years, crisscrossing the United States, they tracked Mo and a team of five corn industry spies, including insiders at Pioneer research labs, from state-sponsored dinners in honor of the Chinese government in Iowa to secret research facilities in Florida to a covert test plot that Mo had established in rural Illinois for cross-breeding proprietary seeds in preparation for smuggling them to China. It seems like a lot of cloak-and-dagger for a few seeds, but biotechnology is now worth billions of dollars per year in sales\u2014and the U.S. government believes that something much larger is going on.\n\nThis theft, they argue, stemmed from an undeniable and dangerous fact: despite its remarkable landmass, China simply can't grow enough food to feed itself, particularly now that the country's burgeoning middle class has acquired an appetite for meat. (Most corn in China is used as feed for livestock.) Water shortages and lack of arable terrain have forced their government to buy between 2 million and 5 million metric tons of American corn annually, approximately 94 percent of all corn imported into China each year. If China hopes to feed (and pacify) its growing population while also loosening the very real stranglehold that America has on its national food supply, its farmers have to start producing a lot more corn\u2014not just enough to meet their domestic demand in good years but enough to maintain a stockpile to offset their global market impact during bad ones. The only tenable way for China to meet its own demand is by planting high-performance hybrids, which can single-handedly double or potentially even triple per-acre corn production.\n\nIn recent years, DuPont Pioneer has increased its share of the seed corn market in China from less than a tenth of a percent to 12 percent. The company has told Chinese officials that they should Americanize their agriculture: consolidate land, plant GMO seed, apply industrial fertilizers, subsidize the sale of planting and harvest equipment. DuPont Pioneer argues that they are striving only to feed a burgeoning global population. William S. Niebur, the head of the company's operations in China, asked, \"Without China's food security, how can we ever imagine an effective, realistic, sustainable global food-security system?\" But many, including the Chinese government, expressed concerns that DuPont Pioneer's goal was not global food security or feeding the Chinese people, but rather increasing market share and profit by keeping China as a customer. Whatever the reason, the FBI pursued Mo, who was eventually charged with conspiracy to steal trade secrets from Pioneer and Monsanto and sentenced to three years in prison.\n\nEver since that case, Rick said that security had grown even tighter. \"See the tag on every bag of seed?\" he said, pointing to a stack of sealed bags waiting to be opened and loaded into the planter. He said that they used to put a rubber band around those tags and put them in a plastic tube at the field's edge for the agronomist to pick up periodically. \"Now those tags go into a manila envelope,\" Rick said, \"and as soon as you get done filling the planter, you call them to get it. A lot of times they'll come out while you're still planting. They check the seed depth and the seed population, but really they're just double-checking to see how everything looks at your field. They've got it figured down to a gnat's ass on how many seeds you should have, and they want to make sure none go missing.\"\n\nPioneer is so worried about leaking information and stolen biotech that Rick doesn't even know what variety of seeds he's planting, just its codename. All of his communication with Pioneer refers to that alias and the field number. He said that in recent years Pioneer wouldn't even release another pallet of seeds for planting until the company could confirm that the planter was almost empty. In May, when planting gets going full steam, many farmers are planting late into the night. Rick said that waiting around for seed can add hours to the process, but Pioneer will do everything it can to keep farmers working. \"A while back, Dave called at ten-thirty at night, and they ran him out a couple extra bags,\" Rick said, and when he was done planting at 2 A.M., he called again, and the seed rep said that a field manager would be out to meet Dave first thing in the morning. \"As soon as you get your planter cleaned out, they're there, and even if you have a fourth of a bag of excess seed, they want it back.\"\n\nAt the far end of the field, Dave wheeled the tractor around and lined up the planter again, running another set of male rows coming back in our direction. For all the hassle, Rick wasn't complaining about raising seed corn for Pioneer. He was the first to say: it had become the anchor for his whole operation and the one sure source of income, buffeting against bad times, like the current market downturn, when other farmers were forced to rely on commercial crops. \"The seed corn is much more lucrative,\" Rick said. \"And it's a lot less work. I mean, we have to plant it, of course. And midsummer, a week or two after they're sure the pollination is completely done, we take the sprayer and attach this big shredder. We go through, and we destroy every male row. That's a little time but hardly any expense. But that's it. We don't do the harvesting ourselves. They have contracted different farmers, three or four guys, to do the harvesting. We have to be there to move the pivots for them. Then there are trucks that are contracted\u2014maybe twenty trucks\u2014to haul it away, and that's it. So we don't pay for any of the seed. We don't pay for any of the harvest. We don't have any storage. And the price is set just under the Board of Trade for commercial corn in whatever month you choose to market it. So it's almost all of the profit with none of the usual expense or risk. It's so much easier.\"\n\nI asked Rick how they paid out for the seed corn fields. With the inbred varieties producing such small and often scraggly parent plants, you wouldn't think that the yield would be enough to make it worthwhile. Rick laughed hard. \"The first year we planted seed corn, we got so worried. The plants were only about yea big,\" he said, gesturing to mid-thigh. \"I went out, and I saw ears half-filled. Oh, man, I thought we were going to be in big trouble. No one told us! It was like, 'Jeez, what did I do?' It turned out we were their biggest producer for that variety that year. They were thrilled.\"\n\nHe explained that DuPont Pioneer knew to expect tiny yields compared to commercial fields. Where a good field now might produce 250 bushels to the acre for commercial corn, you'd be lucky to get 60 to 80 bushels to the acre for seed corn, so the company compensates farmers for what the seed corn would have produced if it were planted to commercial corn and a top yielder. \"You have these PBY fields, they call them. PBY is a 'production-based yield,' \" Rick said. \"And that sets the mark for yields for the seed corn fields, so you choose your very best ground and put your very best hybrid with your very best fertilizers, and all the other producers do the same. Those that are chosen then, in the fall, they go in and they harvest those. That gets your base yield for the year. They pay you for the seed corn fields at the rate that your PBY fields yield. The last couple years it's been right around two hundred fifty bushels.\"\n\nOn top of that base yield price, DuPont Pioneer also offers a premium for what they call the Big Dog Award. They compare the yields of groups of seed corn producers, farmers in a certain geographical area, what they call a \"block,\" who planted similar varieties around the same time, and they rate them. \"Say you were in the middle of the pack in a block of six,\" Rick said. \"Then every acre, male and female, would be counted at 250\u2014just average. But if you won out of those six, if your field did the best, then you're the Big Dog and would get some premiums. It's an incentive to not just plant the seed corn but to do your best to make the field produce.\"\n\nLater, when Kyle was in the shop removing counterweights from the tractor after the end of planting, I told Meghan that I couldn't believe all of the know-how that goes into farming. Kyle had chosen the balance of what to plant, the varieties to plant and where. He had chosen how much fertilizer, pesticides, and herbicides to apply and when to apply them. And going forward, as the two of them took over the operation, they were going to be in a position of trying to amass their own land and build their own operation. They were constantly racing to get ahead, but it never seemed to be enough.\n\n\"There's a lot of things wrong with farming now, and the trend is more, more, more,\" Meghan said. Her hair was pulled back into a long braid, her eyes wide with emotion. \"More chemicals, more herbicides, more pesticides. And more land. We need more\u2014more, more, more. This land has maintained five generations, and I'm number six. We don't need more. We can get by on less.\" Then, suddenly, her face flushed with annoyance. \"But we've also gone down this road quite a ways and the people who say, 'This is all a brand-new development and it's easily reversed,' don't know what the fuck they're talking about. You know?\"\n\nI nodded. \"I know it's sensitive,\" I said, \"that a lot of people ask why we can't just go back to raising organics.\"\n\n\"Because the emphasis on corn and soybeans started in the twenties. It's not a development just of the last few years. Yes, it's gone crazy since the seventies, but we spent a century going down this road and everything\u2014the development of tractors, the development of seeds, the development of chemicals\u2014everything has been built around a certain way of doing things. On the one hand, if we take a look at it and say, 'This is not the way we should be doing things,' then, sure, we need to do things different. But on the other hand, I also don't think that we should say, 'We should all start raising organics and we'll all just put on our overalls and we'll get out there and do it ourselves.' \"\n\nShe shook her head at the very thought of it.\n\n\"Are _you_ ready to go raise your own food?\"\nINTERLUDE\n\nTHE POKER RUN\n\n_June 2015_\n\nMeghan took a deep breath, steadying her nerves, as she rounded the corner and walked onto Green Street, the main drag through Clarks, Nebraska. Everything was decked out like it was the Fourth of July\u2014American flags hanging from storefronts and public buildings, miniature flags stuck into flower planters and front yards. And on either side of the wide, brick thoroughfare were motorcycles and muscle cars, all lined up for the ninth annual Lance Corporal Brent Zoucha Memorial Poker Run and Dance. Meghan waved and gave hugs as she made her way through the crowd outside the VFW hall, past people sitting on a bench engraved with Brent's name and the date he was killed in action, past a group of women in fluorescent yellow t-shirts emblazoned with the words RIDE TO REMEMBER.\n\nInside, it was loud with people crowded around the bar sharing pitchers of beer and getting officially entered into the Poker Run. As riders signed in, they were given the first card of their poker hand. From there, they would go on to the towns of Aurora, Harvard, McCool Junction, and Polk, drinking and getting a new card in each town. At the end, everyone would ride back to Clarks, compare poker hands to determine a winner, and then have a cookout and cap the evening off with dancing to a live band. Meghan found Brent's brother to give him a hug and then sought out Brent's mother, Rita. Rita was planning to go to the edge of town to see everyone off, but rather than riding along, she was going to head back home to let Meghan have some quiet time in Brent's old room, which Rita still maintained as a shrine to her son.\n\nPrecisely at 2:30 P.M., one of the organizers shouted that it was \"clutches out.\" Everyone filed through the narrow door, fanning out to their bikes. Then they zigzagged down Green Street, their engines grumbling and handlebars weaving as they rolled in low gear like pacing tigers. At the edge of the downtown, the Patriot Riders hung a left and crept down Millard Street, its rows of American flags marking the path, as it had nearly a decade before. One by one, the riders coasted to a stop at the city limits, where they were thanked personally by Rita. The sun was high overhead, the temperature well into the 90s, but she wanted to greet them and make sure they knew what this meant to her. She hugged those on motorcycles and shook the hands of those in trucks and cars. One by one, they roared out onto the highway and she would wave down the next rider.\n\nWhen the last had disappeared, Meghan went with Rita to her small house on the other side of town, on what was now named Brent Zoucha Lane, the place where Meghan used to go to hang out with Brent in high school, watching him play video games and smoke cigarettes. Rita unlocked the side door and chatted briefly with Meghan, but she never left the driveway. She had become an expert in the mechanics of grieving, and she seemed determined to let Meghan have her time alone.\n\nAs Rita drove off, Meghan turned to me. \"It's okay,\" she said, ushering me through the doorway.\n\nInside, it was dark and icy with air-conditioning.\n\n\"Wow,\" Meghan said. \"It's so much cooler in here, right?\" She walked around the kitchen and the living room a bit, just taking in the place. She studied the recent newspaper clippings on the refrigerator, checked out the new family photos, framed and hung up around the house. At some point, it dawned on me that she was putting off going into the room with all of Brent's things. To let her know that I wasn't trying to rush her, I ventured a single question.\n\n\"Were you in this house often?\"\n\n\"Spent a lot of time here,\" she said, looking over every inch some more. \"As much time as I could. My parents didn't want us dating. We probably started freshman summer, sophomore year. I was pretty young, and he was full of it, super ornery.\"\n\nShe took a breath again. \"Okay,\" she said, \"let's do this.\" She fumbled through the keys Rita had given her, until she found the right one. She unlocked and then opened the French doors into the room.\n\nInside, it was hot, cut off from the air-conditioning of the rest of the house. Meghan flipped the switch, and the track lighting suddenly bathed the room in bright light. The walls and floor were covered with memorabilia\u2014Brent's high school trophies, photos of him with Meghan at the prom, a picture of him after he enlisted. In death, every piece of him had become significant to Rita. One shelf held the contents of his shower kit sent home from Iraq: the last bar of soap he ever used, the last stick of deodorant and tube of toothpaste. Whatever scrap of his life remained, it was on display here, but there were also items sent to Rita after Brent was killed. There were letters from senators and the governor of Nebraska and all kinds of tributes\u2014drawings of Brent and paintings of him by people who had never even met him, just people, crazy or caring was hard to say, who wanted to make a connection with a parent who had lost her son to war.\n\nAnd in the middle of the room, a lectern, positioned under a bright overhead light, held a 6-inch-thick photo album of still more memories.\n\n\"This is the scrapbook that she made,\" Meghan said. \"She made one for all of us.\"\n\nShe flipped open to a series of photos of Brent in his basketball and track uniforms. At 6 feet 4, he had been the star of the High Plains High School basketball team and was selected to play in the Central Community College All-Star game in 2005. In track, he took second place in both the high jump and the 400-meter relay at the Class D state track meet. Everything was there in the room: the trophies, the certificates, the medals, the clippings reporting the results, all carefully arranged.\n\nMeghan felt deeper into the scrapbook, turning several pages at a time. \"This is Gunnery Sergeant Philips. He's the recruitment officer that signed him up,\" she said, then shook her head at Brent's serious expression. \"He would never smile.\" Meghan flipped pages again and found an article from _USA Today_ , quoting a family friend saying, \"Second place just wasn't an option. I believe this drive to succeed is what led him to the Marines.\" She turned over more pages, and the scrapbook fell open to the calendar page for June 2006. Torn from Rita's wall calendar, the date for the ninth was circled and marked, \"Brent's accident.\"\n\n\"Accident,\" I said.\n\n\"I think it's hard for Rita to talk about it directly,\" Meghan offered.\n\n\"Did you ever find out what happened?\"\n\n\"Not exactly,\" she said. All the family had been told was that Brent had been part of a counterterrorism team operating along the Syrian border. While on patrol, one of the Humvees\u2014whether it was Brent's or not was unclear\u2014hit an IED. They radioed for support, and the second wave of Marines detonated still more IEDs. Regardless of the sequence of events, the Humvee that Brent was riding in had been hit squarely, killing him along with two of his battle buddies, Lance Corporal Salvador Guerrero, a mortarman from Whittier, California, and Navy Hospitalman Zachary M. Alday, a corpsman from Donaldson, Georgia. The three had been close; the official Marine memorial said that they \"ate, lived and worked together on a daily basis,\" but Meghan didn't know anything about the other two killed in the blast.\n\n\"You know, I went off to Ireland, pretty much right after Brent's funeral,\" Meghan said. \"I needed to get as far away as possible, and I think that trip saved me. But I also spent a lot of days over there searching online for stories about what had happened and any details of the guys he had been with.\"\n\nOne article featured a photograph of Sal Guerrero's mother stretched in desperation across her son's casket, as if she were begging that it not be lowered into the ground. Meghan left her email address, phone number, and home address in the comments section, hoping Sal's family would see it and write to her. \"I just wanted to get in touch with your family and find some things out,\" she wrote, but Sal's family was still in deep shock. Not wanting to worry his mother, he'd never even told her that he'd been deployed to Iraq; when he called from the forward operating base, he always told her he was on a training mission in Japan and would be home soon. Eventually, Meghan made contact with Laura Almanza, Sal's girlfriend, who had known where he was all along.\n\nMeghan flipped another group of pages in the scrapbook and landed on the program for Brent's memorial at the base in Twentynine Palms, California. \"This is when we finally met,\" Meghan said. \"I decided to fly out with Rita, just to be there for her, and Laura came over from Whittier.\" After the ceremony, some of Brent's friends came up to introduce themselves, but they didn't talk about what happened and Meghan didn't ask. \"I was just a girlfriend. I wasn't a wife, and I wasn't family. I think that's why Laura and I bonded right away. We were in the same boat.\" When the reception was over, they all went out on the town to blow off steam. \"I was only eighteen, so I may or may not have had way too much Patr\u00f3n.\"\n\nAt some point, they all decided to get tattoos. They spilled out onto the streets of Twentynine Palms, not knowing exactly where they were going. \"But in a town with a Marine base,\" Meghan said, \"a tattoo parlor is never too far away.\" When they found a place, Laura decided right away to get a cross on her back with a rose. Rita settled on a purple heart on her chest. Meghan still wasn't sold on the idea and sat paging through the sample booklet, when she landed on a tattoo with Brent's name. His friends had already been in. \"So I decided to do it,\" Meghan said. \"I asked for the Marine Corps motto\u2014Semper Fi\u2014along with Brent's jersey number in basketball.\"\n\nShe laughed and pulled up her pant leg, revealing the tattoo on her ankle. \"The guy drew it freehand, but it didn't come out too bad.\"\n\nShe closed up the scrapbook and looked up into the lights. \"You know, I don't know what would have happened between Brent and me. I'll never know.\" In recent weeks, Kyle had finally worked up the courage to formally propose. Meghan was wearing the engagement ring that he had given her. Rita was Meghan's first phone call. It was hard, but the last thing she wanted was for Rita to find out from someone else. \"She just said how much she likes Kyle. She wants to see my wedding dress and all that stuff. I know that none of this has been easy for her.\"\n\nBefore switching off the lights and locking up the room, Meghan took one last look around. There was nothing new\u2014and never would be\u2014but she studied the photographs of Brent, especially those of the two of them together, as if she were looking for some hint of the tragedy ahead. But there was nothing there but two kids, young and having fun. Gussied up for the prom. Goofing around at the driving range. Hanging out together on the same couch that sat just outside the room\u2014Meghan perched on Brent's lap and laughing, his arms slung around her.\n\n\"That was the last time we were together,\" she said, \"around Christmas, right before he left.\"\nPART FOUR\n\nIRRIGATION NATION\n\n_Summer 2015_\n\nRick turned a yellow dial until it locked into place with a hollow clank. The royal blue control panel, emblazoned with the logo for Valley Irrigation, hummed to life, and across the windswept field, a light atop the enormous center-pivot irrigation system started to blink. \"That means everything is on,\" he said. The metal behemoth of the pivot stretched a quarter mile from end to end. Its humped spine and rib-like trusses, girding an array of dangling hoses and sprinkler heads, looked like the skeleton of some kind of robot _Brontosaurus_ , poised to waken from its prehistoric slumber. Rick tipped his cowboy hat back with his thumb to get a closer look at the digital display. He checked the speed and pump pressure, to make sure the wellhead was still off. Seeing that everything looked set, he gave me a little shrug. \"Then you start it walking,\" he said. With the push of a button, the pivot eased forward. The twinned drive wheels under each triangular tower suddenly lumbered to life, creeping clockwise along its circular path.\n\nIt was still June, early in the season, and Rick hadn't yet planted this final field of soybeans. Before the crops were high and the rows were wet with rain and irrigation water brought up from the pivot's central well, he wanted to be sure that everything was in perfect running order. \"Every year, we go through our pivots before planting and drain any water accumulation in the gear boxes. Each has an electric motor going to two drive shafts, so we do all that greasing; we don't want those seizing up. And then, we just visually inspect the whole thing. Do all tire pressure. If any are suspect\u2014so weather- checked that they're not going to make it a season\u2014we usually just go ahead and replace them. I can tell you what ruins a Sunday afternoon, and that is taking one of those sons of guns\u2014we're talking about a chest-high tire\u2014and shoving it through a planted field on a muggy, hundred-degree day. And you have to carry your jacks and your blocks and your spinner wrench. You want to talk about a workout.\"\n\nWe climbed into the truck and drove across the field to where the pivot was walking in a clockwise direction. \"You want to be sure to park on the left side of it,\" Rick said with a grin. \"Meghan had one time where she parked a seed cart in the wrong spot. It could have tipped over, but luckily the tower rolled up to the cart just right and idled there long enough that it just switched off.\" Rick stepped over the rows, inspecting each span as he talked, checking that the gear boxes looked like they were turning and the towers were moving in alignment. When he seemed satisfied that all was in working order, we drove back to the control panel. Now he switched on the water pump to double-check that it was drawing properly from the well.\n\n\"What we're shooting for on this level of ground is about an inch,\" Rick said over the roar of the pressurizing wellhead. \"It won't run off with an inch. That takes approximately three and a half days.\" He said that, in a good growing season, he hoped to put just three to four inches of water on this whole 160-acre quarter section, most of it in a single spate in July. That doesn't sound like very much, but it can be the difference between a bumper crop and getting wiped out for a whole year. When Heidi's grandfather first dug this wellhead in the late 1940s, irrigation was still in its infancy. In those days, a pump would bring water to the surface and dump it into massive catchment ponds, which could be released into open furrows during dry times. Known as \"flood irrigation,\" it was a game-changer for farming. Using well water, instead of just relying on surface streams, made it possible for farms to spread away from river valleys, adding arable acres and allowing farmers to start planting crops in places they never would have dared before.\n\nBut the amount of water pumped and put onto a field using flood irrigation was hard to control. Catchment ponds were susceptible to evaporation. Flooding furrows caused widespread runoff, leading to loss of water and topsoil. That's why the development of center pivots in the 1950s, something as simple as applying water evenly over a field in rain-like droplets, rather than ground-level flooding of furrows, proved so revolutionary\u2014and why it was embraced so quickly and so universally. Even American city-dwellers who have never been on a farm have looked out their airplane window and seen the green crop circles created by these irrigation systems. That patchwork quilt of lush crop production has become the unofficial symbol of our national agricultural know-how and is one of our main ag tech exports around the world.\n\nThe simple shift in irrigation strategy made it possible to raise crops in places with unreliable rainfall, expanding our potential for production. But the more precise delivery of groundwater\u2014along with decades of refinements in water application technology\u2014has also made it possible to conserve that precious resource, by applying only the necessary amount at the moment that it is most needed. \"See those hoses,\" Rick said, pointing to the black rubber tubes dangling from the pivot's spans overhead. \"When the corn is high, they'll hang right down to the tops of the tassels. It used to be that the old emitters, the sprinkler heads, had impact spray nozzles and were set way up high, so it would put out a wide spray pattern, a fine mist, but we lost so much water to evaporation. So now, they're down at leaf level, and, for efficiency, we use Nelson wobblers that throw large droplets. By applying drops, more like actual rain, more of the water application makes it to the roots, rather than sitting on the leaves and evaporating. We're trying to do as many things as we can to minimize usage.\"\n\nBut even reduced drafts from wells, applying just a few acre inches in a whole growing season, are drawing a lot more water than you might think. Rick led me over to the meter, attached to a pipe coming up from the well. He ripped up handfuls of tall grass that had overgrown the gauge. \"Look at that,\" he said, pointing to the small lettering under the spinning numbers. \" 'Gallons _times one hundred_.' We're talking millions of gallons of water that get pumped from the Ogallala Aquifer.\" Sixty years after the advent of center-pivot technology and the movement toward water-intensive crops like corn, those millions of gallons were starting to add up, catching the eye of environmentalists and farmers alike.\n\nA 2015 study of U.S. Geological Survey data compared the depths of more than 32,000 wells nationwide over the previous two decades. The results were alarming. Across the country, water levels had fallen in 64 percent of all wells, with an average decline of more than 10 feet. In the Ogallala Aquifer system, which supplies groundwater for crop irrigation not only in Nebraska but to eight states from Texas to South Dakota, the declines were especially pronounced. In much of southwestern Kansas, wells were down to 25 percent of the water that existed when the aquifer was first tapped less than seventy years before. In the southern High Plains of Texas, near the edge of the Ogallala, water levels had fallen more than 100 feet in places, leaving many farmers without any water at all. In 2011, the U.S. Department of Agriculture announced that the government would be investing over $70 million in the Ogallala Aquifer region, \"to help farmers and ranchers conserve billions of gallons of water.\" But the Great Plains was soon plunged into a multi-year drought, and, instead of declining, water usage shot up dramatically.\n\nAfter he was done checking over the pump and had twisted the yellow dial of the control panel back to its off position, silencing the humming system, Rick seemed caught for a moment in the calm. He confided that he worried sometimes about how much water industrial ag used on the Great Plains. \"You know, if I've got one issue that I really care about,\" he said, \"the environment would be it.\" He was proud that Nebraska's unique system of Natural Resource Districts, governing responsible water use at the local level, had become a model of management and stringent enforcement of consumption, but even in eastern Nebraska where the aquifer recharges faster than any other part of the system, the levels had still dropped noticeably in Rick's lifetime. \"I have seen nature repair itself, but it takes time,\" he said. \"If humans don't give it a chance, we'll end up stressing the resource, and I think it could be bad.\"\n\nI was struck by the turn of phrase. \"Stress the resource,\" I said. \"You mean we could use up all of the water?\"\n\nRick hesitated, then waved me back toward the truck. We climbed back into the quiet of the cab, out of the wind. \"Look,\" he said finally. \"If rainfall patterns keep shifting, we keep getting hotter summers and less rain, and people don't conserve the groundwater, I'm not saying there will be no more water. But there won't be enough to raise crops, not like we've been doing. There will just be grass in Nebraska.\"\n\nRick promised to take me out west to where he grew up in Curtis, maybe at the end of the summer when it was time to bring the cow-calf pairs back from pasture. Not far from there, he said, at a place called Dancing Leaf Earth Lodge, archeological excavations had uncovered the remains of a once-thriving village, part of the Upper Republican Culture, an ancient precursor to the Pawnee. Digs suggested that those people, after centuries of farming corn, were forced to abandon their homes in the late fifteenth century, displaced by a catastrophic drought that lasted generations. \"We brag about toughing it through years of hard times,\" he said. \"Sure, we've made it through droughts. Three-, five-, seven-year droughts. Terrible, but we got through. But what about the fifty- or hundred-year droughts?\" The sites of many of those abandoned villages had been found buried under a thick shroud of dust\u2014proof of dry times and roiling winds that rendered the area uninhabitable for multiple lifetimes.\n\n\"They left and never came back. They don't know for sure what happened to them. Could it happen across the Great Plains today?\" Rick asked. \"Absolutely.\"\n\nFINDING WATER to grow food has been the central challenge of life on the Great Plains since the earliest days of white settlement. When the United States, in its zeal to displace Native Americans and settle the Free Soil debate with the Confederacy, passed the first Homestead Act in 1862, government surveyors fanned out to plat a patchwork of one-mile squares, which were then subdivided into quarters of 160 acres each. A \"quarter-section\" was awarded any free citizen of the United States, man or woman, native-born or immigrant, with just one major catch: you had to live on the land and farm it for at least five years. This not only meant concocting some way of building a home on the treeless prairie, but also finding enough fresh water to sustain crops.\n\nOn the semi-arid plains of Nebraska and neighboring states, where surface streams often ran dry during the summer months\u2014right when water for irrigation was most needed\u2014farmers had little choice but to dig wells. In some places, large communal wells were undertaken to serve whole communities. In 1887, for example, a group of determined Kansas farmers, using nothing more than shovels and picks, dug a stone-braced well to a depth of over 100 feet to supply freshwater to the town of Greensburg. But more often, the wells were shallow and hand-dug. Farmers erected windmills to pump small amounts of groundwater and divert it to the wide furrows of their fields or into catchment ponds. Life on the hardscrabble plains was arduous, and often hand-to-mouth. Nevertheless, by 1890, the Homestead Act had settled some 2 million people on nearly 375,000 farms.\n\nBut in the drought years that followed, just as cities like St. Louis, Omaha, and Denver were turning into bustling metropolises built on income from livestock and grain exchanges, the entire Great Plains region was devastated by blistering heat. It wasn't just an unlucky few who were pushed to the brink of bankruptcy. By the harvest of 1894, one newspaper reported, nearly 38 percent of acres planted with corn in the middle states were either destroyed or abandoned. In Nebraska, where high temperatures were accompanied by scorching winds, the smell of parched corn filled the air. Meghan's great-grandfather, Harry Harrington, may have struggled through those years and eventually lost his grip on the family land, but many more simply packed up and left, making their escape, according to another paper, \"while they had something to do it with.\" The drought revealed that raising enough grain for a farm family to thrive would require greater land allocations than the Homestead Act had originally provided\u2014and a lot more water.\n\nIn western Nebraska, where the drought was worst, a group of locals formed the Farmers Irrigation Project, expanding an existing small-scale canal that had started diverting water out of the North Platte River right along the Wyoming border, near Henry, Nebraska, in 1887. My grandfather eventually worked on the channel that resulted from that project, the Tri-State Irrigation Canal\u2014a network of open waterways that stretched from the foothills of the Wyoming Rockies, across the northeast corner of Colorado, and into western Nebraska. He was a ditch rider in the Wildcat Hills, doling out water allotments, for more than thirty years, so on one trip out to that part of the state, I'd stopped in to see Kevin Adams, who had worked his way up from ditch rider to construction engineer to supervisor of the Minatare division to assistant manager to his current position as general manager of the whole district. Though still in his fifties, Kevin had worked for Farmers Irrigation for more than a quarter of the lifetime of the canal. So when I asked what changes he had seen in his years on the job, he broke out in a crooked smile.\n\n\"Nobody rides ditch with a horse anymore,\" he said.\n\nWe were sitting in his sprawling, wood-paneled office just off the main drag in Scottsbluff. \"No,\" he said, \"I never saw that. But this system is old enough that everything was built with horses and slips. They didn't have dredging machines or anything. I've got an old ditch rider's manual in my desk that says you have to supply your own horse, your own feed . . .\"\n\nHe paused a moment and smiled again. \"Do you want to hear all that history?\"\n\nI nodded for him to continue.\n\n\"So, this system has a water right of 1887\u2014when the first irrigation supplies were diverted out of the North Platte River,\" he began. \"By 1891, they had ten miles of small canals built that irrigated about thirty-five hundred acres with the natural flow of water. But they soon found out that, since this part of Nebraska isn't on an aquifer, they couldn't sustain crops through the summer without storage water.\" The project was suspended\u2014but, in 1902, with passage of the Reclamation Act, the federal government agreed to bankroll a dam in Wyoming that promised to deliver enough water for 150,000 acres, by a new canal dubbed the Pathfinder. At the same time, Heyward G. Leavitt, an entrepreneur who believed that with proper irrigation sugar beets could thrive in the Platte Valley's rich soil, bought out the remaining interests in the Farmers Irrigation Project\u2014in return for delivering free water in perpetuity to the land of the original owners. Leavitt got a supplemental storage contract with the Pathfinder Reservoir and purchased 36,000 acres, where he extended the old canal and taught tenant farmers to raise sugar beets.\n\nWithin a decade, Leavitt's Tri-State Canal was completed\u2014and the sugar beet had transformed the valley. The beets themselves were reliable and profitable. The Great Western Sugar Company provided seasonal factory jobs, first in Scottsbluff and then at a second plant in nearby Bayard. The byproducts of sugar refining (both beet tops and leftover beet pulp) could be processed into feed for cattle, stabilizing the beef industry and creating jobs on ranches and bringing packinghouses into towns along the railroad. And the manure from the ranches could be used to enrich irrigated soil to grow a greater diversity of crops\u2014most notably soybeans, alfalfa, and eventually corn.\n\n\"At that time,\" Adams said, \"everything was open channel\u2014seventy-five miles of a main canal, which we still have today; probably 288 miles of open laterals. And, of course, we maintained most of the creeks and drains in this area\u2014about ninety miles of that, too.\" All told, the Tri-State's storage supply in Wyoming's Pathfinder reservoir is capable of delivering 1.1 million acre-feet of water, but even in the early years, the river couldn't deliver enough to meet all of the agricultural demands of the growing diversion projects. By midsummer of each year, the Platte River in the central part of the state and points farther east always ran dry.\n\nSo at the same time that the Farmer's Irrigation District was digging canals, the U.S. Geological Survey had commissioned a study of groundwater resources on the Great Plains. In 1897, working at a spot west of Ogallala, Nebraska, government geologist N. H. Darton surmised that the abundant wells in the southwest corner of the state were drawing from a great storehouse of ancient water held in place by an enormous underlying layer of limestone. \"Extending from Kansas and Colorado far into Nebraska,\" Darton wrote, \"there is a calcareous formation of late Tertiary age to which I wish to apply the distinctive name Ogalalla formation.\"\n\nDarton described the Ogallala as a kind of enormous, untapped reservoir, like a lost freshwater lake waiting through the millennia to one day emerge and regreen the plains. Over time, however, geologists have come to understand the aquifer as something much more complex\u2014and difficult to manage. Ann Bleed, a former state hydrologist and director of the Nebraska Department of Natural Resources, explained to me that the Ogallala isn't even a single aquifer but many aquifers, studied and managed as independent parts of the High Plains Aquifer System.\n\nWhen the system began to form over 100 million years ago with the uplifting of the Rocky Mountains, the shallow inland sea that had covered much of North America slowly receded, and rain and snow from the rising craggy peaks ran off, forming rivers that raced east from the mountains to the flatlands. In time, the braided channels filled with silt and gravel, creating spaces that trapped a vast basin of freshwater commingled with the alluvial soil. Contrary to popular views, however, an aquifer is not a giant underground lake, with all that implies. It is not a round-bottomed bowl filled with water, but rather the flooded and filled in remains of valleys and canyon walls, more akin to the Badlands than the flat prairie we see today.\n\nSo in places where rivers once ran, the Ogallala can be quite deep, but in areas where the bedrock is the high top of an ancient butte, it may be shallow and cut off from other parts of the system. Some parts of the aquifer, like those in eastern and central Nebraska, can replenish, or \"recharge,\" rather quickly, because, by the good fortune of geology, the aquifer there is connected with the substrate from the Rocky Mountains. The underlying bedrock still tilts to the east, carrying underground water resources in that direction. And the area around York and Hamilton counties also benefits from water slowly percolating out of the Sandhills to the north and west. In Nebraska, the aquifer can seem like an inexhaustible resource, but, in many other places, wells can be inadvertently tapped out\u2014and take hundreds of years to replenish. Still other spots, like the aquifers underlying western Kansas and wide swaths of the Texas Panhandle, refill so slowly it's as if they don't recharge at all; geologists call their water \"fossil water,\" because it is more akin to oil, a finite and nonrenewable resource. But no one knew that at the time they were discovered.\n\nIn the 1870s, interstate railroads brought ranchers to West Texas, where they could run cattle on flat expanses of public rangeland. At the turn of the twentieth century, when cattle baron Christopher Columbus Slaughter, the godfather of the Texas Longhorn, began setting wells for his growing herd of 20,000 breed stock, most of his digging crews didn't even have to drill for water. They simply used a horse-drawn trench digger to deepen a natural arroyo or a buffalo wallow, which would fill with water whenever rains fell on the already saturated soil. In other places, Slaughter hired men to install windmills to pump water into stock tanks spaced five miles apart, but the water was so close to the surface that they could set a well with nothing more than shovels.\n\nThanks to readily abundant groundwater, the economy of West Texas, especially in cattle towns like Plainview, boomed. The downtown there saw the construction of the Granada and Fair vaudeville halls. The city erected the 3,000-seat Plainview Municipal Auditorium. In 1929, Hilton built a seven-story hotel with 125 rooms, all with private baths. Seeing the success of irrigated cotton fields, other farmers were encouraged to start planting wheat and feed crops. B. F. Yearwood, a local entrepreneur, erected a 50,000-bushel corn mill at the edge of town. The Harvest Queen flour mill, which opened its doors in 1907, was completely rebuilt with soaring concrete grain elevators. The city was so convinced of its limitless water supply that it planted more than 50,000 shade trees as part of a beautification project. In 1937, at the height of the Dust Bowl, the McCormick company issued a postcard, intended to attract drought-stricken farmers from Oklahoma. A young farm girl is shown holding the lead of a beef cow and standing next to an open irrigation well, its windmill spilling water into a verdant-ringed stock pond. The legend reads: \"This modern farmerette has no rainfall worries.\"\n\nFarmers in eastern Nebraska weren't so lucky. Though groundwater was abundant there, it was not as near to the surface as in West Texas. Irrigation wells were reliable and sustainable if they were dug along the banks of the Platte River where the water was closer to the surface, but farmers there already had the region's most abundant water supply and were often reluctant to shoulder the cost of drilling and putting in a wellhead, or the price of buying and maintaining oil-powered pumps. Some employed wind power, but a single windmill could only siphon enough water to irrigate five acres or provide for thirty cattle\u2014hardly enough to get farmers through the dry times. In 1928, the Nebraska Agricultural Extension Service lamented that \"the underground water supply is abundant,\" but there were insufficient means of \"lifting it to the surface and applying it to the land.\" Toward the end of the 1920s, as prices of corn and wheat sagged due to oversupply worldwide, there was even less incentive to invest in irrigation equipment. Nebraska farmers were content to keep their input costs low and get by on the twenty-five-bushel-per-acre yields they had steadily produced since recovering from the crisis of the 1890s.\n\nBefore long, however, the middle of the country was hit with the drought against which all other droughts would come to be measured. In 1934, the start of the Dust Bowl years, Nebraska's statewide average per-acre yield of corn was barely two bushels\u2014and didn't rise above ten bushels per acre again for another five years when Henry Wallace's drought-resistant corn hybrids brought the country back from the brink of starvation. But then, even before the country could stand on its feet, we were drawn into World War II and five years of food rationing.\n\nDuring those early years of the war, when the fighting in Europe was still a pitched struggle between heavily mechanized armies, tractor manufacturers like John Deere and International Harvester boomed. Government contracts meant the assembly lines ran day and night, building engines for heavy trucks and tanks. My grandfather, driven out of Nebraska by the drought, moved his wife and infant son, my father, to East Peoria, Illinois. There he worked at the Caterpillar plant, making tracks for tanks as well as industrial scrapers used to build makeshift landing strips for supply planes and bulldozers to clear roadways for trucks to carry those supplies to the front.\n\nAfter the war, all of those companies shifted resources to developing peacetime applications for their heavy equipment, including diesel-powered pumps that could finally convey all of that groundwater underlying much of the American breadbasket to the surface where it could be used for crop production. After a long decade of going hungry, it's little wonder that Americans were collectively determined to find a way to ensure that no water shortage would ever again mean going without.\n\nIN THE late 1940s, while tenant farming in Colorado, a farmer and tinkerer named Frank Zybach attended the Irrigation Field Day held near Prospect Valley. He watched the demonstration crew set the pipes on an irrigation system, then tromp through the mud to move them. The arrival of abundant water had been a godsend to Colorado and Zybach's native Nebraska, but the early irrigation systems that delivered that water to the field were little more than a series of bulky metal pipes that had to be lugged into place and then reset two or three times a day. It was a mucky and painstaking job, and not a terribly efficient one.\n\nZybach later remembered thinking, \"There has to be a better way.\" He began to develop and eventually patented a self-propelled system, in which the force of water flowing through the irrigation tube would slowly turn the drive train of supporting wheels. By anchoring the pipe to a central wellhead, the system turned in a perfect circle, watering the field evenly without any work by the farmer. Zybach's system became known as \"center-pivot irrigation.\" In 1952, he moved back to his hometown of Columbus, Nebraska, and partnered with local equipment manufacturer A. E. Trowbridge on building and marketing the systems.\n\nDespite their ingenious design, early center pivots were plagued by malfunctions and were a nightmare to maintain. The long central irrigation pipe was prone to leakage, which would drop the water pressure enough so that the whole system ground to a halt\u2014often flooding a single spot in the field in the process. Farmers, leery of the expense and hassle of owning an untested piece of equipment, were reluctant to buy. In the first two years of production, Zybach later told the _Lincoln Journal Star_ , he and Trowbridge sold just nineteen center pivots. After several years of struggling, they decided to license the technology to Robert B. Daugherty, the young owner of a small farm equipment company called Valley Manufacturing, not far from the banks of the Platte in Valley, Nebraska. Zybach hoped that Daugherty could improve the design and make the system profitable.\n\nDaugherty spent months and months working on a pipe-making machine that could take a flat piece of steel, roll it into a wide-gauge tube, and then automatically weld it shut. When he had finally perfected the process, he was eager to show it off. In 1953, one of the first buyers of a center pivot from Valley was a young man recently returned from the Korean War named Walt Sehnert. \"One day my dad and I stopped in at the Valley plant to see this new pipe-making machine,\" Walt told me. They were greeted by Daugherty and given the grand tour, culminating in a demonstration. \"Daugherty was a dapper little man, ramrod straight,\" Walt said. \"He even carried a little swagger stick, favored by some Marine Corps officers. As we walked through the plant, he explained the various phases of manufacture that we were seeing, and slapped the side of his leg with this stick from time to time.\" When they finally reached the pipe-making machine, it didn't seem to be working right. Daugherty approached the two men who were tinkering with the machine and offered a couple of suggestions. \"Then he came back to where we were standing. He said nothing, but clenched his teeth, turned red, and glared at the machine, beating a tattoo with that swagger stick against his leg.\" A minute later, one of the men gave the thumbs-up, and the machine rolled out a perfect 6-inch pipe.\n\nSehnert bought and installed the system. Soon other farmers followed suit\u2014and just in the nick of time. In summer of 1953, furnace-like temperatures spread across the Central Plains, pushing hot, dry air across the American breadbasket. The drought that followed held the region relentlessly in its grip. For nearly four years, the middle of the country, from the panhandle of Texas all the way to the Sandhills of Nebraska, experienced low rainfall and stretches where the mercury topped 100 degrees for weeks at a time. The federal government spent $3.3 billion on assistance to farmers, and the beef industry in Texas was nearly destroyed in what became known as the Great Cattle Bust.\n\nBut the lucky landowners of eastern Nebraska who had invested in center pivots were able to weather through the drought. Other pivot manufacturers soon popped up around eastern Nebraska\u2014Zimmatic in Lindsay, Reinke in Deshler, and Olson Brothers in Atkinson\u2014and they began setting up dealerships across the Great Plains. In the decades that followed, farmers from North Dakota to Texas turned to center-pivot irrigation to provide extra water during key growing times and help them through dry spells.\n\nBy the end of the 1970s, there were more than 18,000 center pivots operating in Nebraska alone and some 30,000 in other parts of the Great Plains, altogether irrigating more than 20 million acres. The rapid expansion of center-pivot technology allowed farmers to plant on more and more marginal land and to venture into water-intensive crops, leading to dense planting of corn. In just a few short decades, the arid plains were transformed. The changes were so profound that astronauts aboard Skylab reported seeing a patchwork of green circles stretching for miles across north-central Nebraska.\n\n\"Passengers on commercial jet airliners increasingly notice the same sight,\" wrote William E. Splinter, chair of the Department of Agricultural Engineering at the University of Nebraska, in an article for _Scientific American_. \"What is being observed is perhaps the most significant mechanical innovation in agriculture since the replacement of draft animals by the tractor.\"\n\nToday, a tower from one of the original center-pivot systems manufactured by that revolutionary pipe-making machine is still on display in the lobby at Valley Irrigation's manufacturing headquarters. Matt Ondrejko, vice president for global marketing, told me that the exhibit honors Valley's central role in averting agricultural disaster in the 1950s and building the large-scale ag economy that followed. Touring the sprawling complex of factories and machine shops bounded by test fields, it's clear that the folks at Valley recognize how much the future of the company depends on continuing to innovate, particularly in the area of water conservation.\n\n\"We can't run a business selling irrigation systems if there's no groundwater for irrigation,\" Ondrejko said. The company is able to tackle the challenge by closely monitoring every step of the construction of its center-pivot units\u2014from the hand assembly of the plastic sprinkler nozzles to a massive crane system that lowers spans of pipe into a gargantuan zinc-and-nickel bath.\n\nIn recent years, Valley has focused its research and design improvements on two main areas: the sprinkler heads and the control mechanisms. By collecting field-moisture data and contrasting yield returns in comparable fields irrigated with different sprinkler heads, engineers are able to refine the spray patterns and the drop patterns, to reduce loss through evaporation. The latest design innovations have focused less on fine droplet dispersal and more on getting water all the way to ground level. After all, the goal is watering root systems, not leaving droplets on leaf surfaces where they will be lost. \"There's really precise science in how big the drop is,\" Ondrejko said, and \"how close we get it to the ground.\"\n\nValley has also invested in developing variable-rate irrigation, which allows individual sprinkler heads to be turned on and off in particular parts of the span as it passes over different sections of the field. The objective, Ondrejko said, is being able \"to precisely spoon-feed the crop\u2014when it needs it, where it needs it, no more, no less.\" Perhaps most important is the development of exact application monitoring, so that farmers can collect precise information on how many acre-inches they are using and where. That's where technology meets water management efforts in the field.\n\nAfter decades of pumping their groundwater to keep up with farmers in other parts of the Great Plains, the wells in West Texas are running dry. In Plainview, the wide, brick-lined avenues are sun-scorched and shuttered now. The plywood bolted over the front windows of the Hilton is painted with a mural of the former lobby, but the locals now speak only of ghosts lurking in the abandoned upper rooms. As the community limped through the droughts of the fifties and seventies, and the earth-parching heat wave of the early eighties, the cotton industry struggled to stay profitable, and gradually the area returned more and more to cattle feeding and beef processing. With the arrival of the Great Recession and the onset of the drought in 2011, Plainview seemed to lose what was left of its other industries: the White Energy ethanol plant closed temporarily, the Peanut Corporation of America plant shuttered due to a salmonella outbreak, and the Harvest Queen Mill & Elevator (by then owned by Archer Daniels Midland), also closed\u2014collectively taking with them more than 1,000 jobs in a town of just 22,000 people.\n\nTo avoid a similar fate in Nebraska, David Eigenberg, who heads the Upper Big Blue Natural Resource District, the same NRD that monitors Rick's wells, told me that the key is more than just better equipment. Technology has led the way to \"improved irrigation efficiency,\" but in many cases, that has only encouraged farmers to apply water to their fields more often. \"You might have the Cadillac out there,\" he said, a center pivot with variable-rate application and zone sensing to automatically apply only in places where water is most needed, but all that advanced technology is useless \"if you don't understand the management tool.\"\n\nAnd that's Nebraska's real advantage. The state is broken into twenty-three Natural Resources Districts, which were created in 1972 to manage water at the individual watershed level. The NRDs apply a doctrine of \"reasonable use\" of groundwater, as determined and overseen by locals elected to the NRD boards. By contrast, Texas adheres to a doctrine called \"rule of capture,\" which essentially allows farmers to use whatever groundwater they can tap from their property. As farmers come to see the resource as finite, many may actually increase their usage, in order to keep up with perceived competition from their neighbors\u2014a phenomenon known as \"the tragedy of the commons.\" To really reduce water usage, you have to change a culture that views irrigation as an individual right, rather than a method of accessing a shared resource. In the end, Eigenberg said, the change has to come from \"the guy in the tractor seat.\"\n\nTo encourage that change, the NRDs have engaged in a range of tactics. Some have started requiring metering on new wells, which has helped farmers to track field-by-field usage and encouraged them to reduce their draft. Some have offered education courses or water-use counseling to help farmers see how overdrafting is costly in terms of the power used on the pivot pump and often in top-soil loss due to runoff. And some, like the Upper Big Blue NRD have undertaken even more ambitious initiatives.\n\nThe Upper Big Blue, which contains significant acreage classified as \"high risk groundwater areas,\" launched a study in 2012 comparing crop yields in neighboring fields with identical center-pivot irrigation systems\u2014but with one managed by the farmer and the other managed by the NRD. The farmer applied water according to his own judgment, while the NRD used soil-moisture data collected from a network of monitors inserted into the edges of the field. The NRD achieved a nearly identical yield\u201498 percent of what the farmer harvested\u2014while drawing only a third of the water. The latest advancement is a wireless soil-moisture monitoring system that sends data to the farmer electronically. Future improvements will send that data directly to the center pivot, automatically turning the system on when irrigation is needed and applying water only in the necessary areas. It's hoped that these new technologies will be able to universally drop usage in these high-risk areas to the point that the aquifer might actually begin to recharge.\n\nBut even bigger, more sweeping changes may be needed. To address the problem adequately, we may need to rethink what kind of food we grow where, and how much agriculture is feasible in certain landscapes. The center pivot allowed the repopulation of many areas that had been vacated during the Dust Bowl\u2014areas that simply couldn't sustain crops without a new source of water. That new technology, coupled with other advances in agriculture, put more than 100 million acres of marginal land into farm production, much of it for growing water-intensive crops, like corn and soybeans, which today are raised primarily for livestock feed and biofuel, not for human consumption. Farmers have also been encouraged to turn away from more drought-resistant cash crops\u2014such as sugar beets in western Nebraska and sorghum in Oklahoma\u2014in favor of commodity grains that are supported by higher farm subsidies and crop insurance.\n\nThe Ogallala water boom made it possible for the United States to become the world's granary, but now, as wells run dry in western Texas, Oklahoma, and Kansas, it may be time to start rethinking our usage. If we don't do something, Don Wilhite, founding director of both the National Drought Mitigation Center and the International Drought Information Center at the University of Nebraska\u2013Lincoln, believes that the effects of this rapid drawdown could be catastrophic. In a 2013 report, written for the Institute of Agriculture and Natural Resources at the university, Wilhite warned that greenhouse gas emissions, if unchecked, will cause temperatures in Nebraska to increase by as much as 9 degrees Fahrenheit in less than a few years. That projected increase in temperatures means that summertime highs will regularly surpass 100 degrees by 2060.\n\nThose extremes falling, as they would, during the peak of the growing season would create untenable demands on groundwater. Even if rainfall were to increase, it would not be enough to offset the loss of soil moisture caused by extreme heat. In a hotter climate, it will take more water to generate the same crop yield, even with genetically modified grain hybrids. Unless we change farming and water management, he explained, there simply won't be enough groundwater to combat such dry conditions.\n\nMost important, Wilhite said that the loss of groundwater resources would set off a feedback loop with much broader effects. He explained that pumping all of that deep, cold water from the Ogallala and spreading it across many acres has artificially lowered air temperatures and increased humidity. That human-induced microclimate has masked the effects of climate change by forestalling climbing temperatures on a regional level. If we run out of that water, temperatures will rise sharply. Crippling drought will become the new norm, turning the Central Plains states into a permanent dust bowl.\n\nWilhite told me that his greatest worry is that farmers tend to brush off such dire predictions, insisting that they have lived through many hard times. They say that their grandfathers got through the Dirty Thirties and innovated their way out of the Great Cattle Bust in the 1950s. \"Farmers say they're used to variability,\" Wilhite said, \"but these projections are way outside the range of anything we've ever encountered.\"\n\nON A sun-bright morning in Loup City, Nebraska, Rick walked around a maze-like complex of holding pens, inspecting thick-necked and hulking black Angus bulls. Their coats glistened like oil slicks in the daylight, and they stepped broodingly around their pens, like boxers before a title fight. The day's auction was set to begin just after lunch, and Rick was going over the program, checking each breeding male, one by one. All of the bulls for sale that day were from Dutch Detleff's ranch in Ravenna or Loren Treffer's place in Broken Bow. Rick knew Dutch from way back. He'd bought from him before, and he trusted the quality of Dutch's genetics and his instincts on what breeders call the \"phenotype\" of particular bulls\u2014the visible characteristics that they read as indicators of the size and health of their future offspring. At one pen, he paused to show me the bull's number in the program along with the entry for his stats.\n\n\"EPD is your expected progeny differences,\" he said. \"That's the estimate of how that bull's calves will perform over the average weight for the breed\u2014at birth weight, wean weight, and yearling weight.\" The ideal was finding a bull that produced calves that were barely over average birth weight, to keep down the chance of losing cows during calving. \"We had fifty heifers calve this spring,\" Rick said, \"and didn't have to pull a single one.\" But once they're safely born, you want those calves to put on weight swiftly. Many of Dutch's bulls were expected to sire calves that were just a pound or two over standard birth weight but then would put on an extra fifty pounds by weaning and could be over a hundred pounds heavier than average by the time they were a year old. \"With the price of beef over six dollars per pound,\" Rick said, \"getting a good bull can add six hundred pounds to the sale price of each calf. Multiply that across a year's births, and it really adds up.\"\n\nBut bigger isn't always better, Rick said, especially when you're talking about cow calves. He pointed to another category on the grid, marked \"Milk.\" \"That tells you what their offspring cows will produce for milk once they're pregnant and lactating,\" Rick said. You want a cow that will produce enough milk to be sure that her calves are well fed and put on weight, but you also want to avoid genetics that produce udders that get too big. \"An extra twenty-five pounds is about the top range,\" he said. \"If you get too much milk, then you start to get bad bags. The calves can't nurse or the teats get mastitis, and you can't keep them in the herd for very long.\"\n\nSo the trick, as with everything on the farm, was factoring in your input costs and your relative risks. Calves that nurse aggressively will put on valuable pounds but require stouter cows and more feed for the mothers. Once weaned, those calves will need more food, too. Because Rick grazes his cattle almost exclusively on grass, it's not the cost of food that he's worried about so much as time and energy\u2014moving the herd from one pasture to another, rotating weaned calves out to Curtis where they can bulk up on open rangeland and then bringing them back for sale. And you have to weigh the prospect of big bull calves against the risk of oversized cow calves. Factoring those costs and risks into the potential for increased profit, Rick settled on a total number he was willing to pay for two bulls that caught his eye. He was doubtful that he could get both for that price, but it was a number he wouldn't go past.\n\nInside the sale barn, the bidders were being called to the bleachers for the auction to begin. The auctioneer, who had driven in from Colorado for the day, sat high above the sale ring, leaning close into the microphone. He wore a tall, black Stetson and reading glasses, and he introduced the breeders in the buttery tone of a call-in radio host. From the moment the first bull was finally brought in, it was a brisk sprint through each sale. The hired hands would pull open the gate, ushering in the angry bull. He would spin and stomp around the show pen, twitching his ears and swishing his tail, looking fiercely for an exit. The auctioneer would read the seller's description: \"This stylish stout-made bull will be a real herd improver.\" Then he would call for a starting bid and the first hand would shoot up.\n\nAt the front of the bleachers on either side of the entry ramp, the auctioneer's assistants collected the bids. \"Four thousand, four thousand, do I hear four?\" One of the assistants would yip, and the auctioneer would jump up to the next level. \"Four. Who'll give me forty-two five?\" Again, a yip\u2014and so on. For the most part, despite the flurry of raised hands and yelps in response, the buyers were conservative, cutting off bidding well below the sale barn average. At one point, the auctioneer, who was working on commission, scolded the buyers for seeming reluctant to go over $5,000. \"Y'all are making some great buys on great bulls today. Every other bull sale in the land has been averaging six to seven thousand. Here's a chance to buy some really well-bred bulls at a price you can live with.\"\n\nDespite the prodding, Rick hung back. If he could get either of the bulls he was waiting on, he said, it would be a good day at the barn. When the first finally came up, Rick shifted in his seat, waiting. The auctioneer's assistant seemed impressed by this bull, too. \"I tell you,\" he said over the loudspeaker. \"A yearling weight of plus-one-hundred-and-four, high ribeyes, big scrotal.\" The bidding started at $3,500, and Rick was quickly in. He was outbid, then came up to $4,000. \"Forty-five, forty-five,\" the auctioneer called. \"Who'll give me half?\" He got a bid for $4,250 and Rick countered at $4,500. The auctioneer called and called for another bid, before finally closing the sale. \"Mark him the buy of the day,\" the auctioneer said, as his assistant recorded Rick's bid number. With such a low price on that bull, he had enough left in his budget to make a play for the second and soon came away the high bidder on that one, too.\n\nWhen Rick's second winning bid had been recorded, he made his way to the sale window. He could hardly conceal his grin as he gave his sale tickets to the girl at the counter, her hair up in a bun like a rodeo queen. Rick filled out the check and then went outside to his truck. He backed his trailer up to the loading chute, where two cowhands loaded one bull, prodding it to the front and locking the gate, before bringing in the second. They moved with speed and confidence, leaning their shoulders into the interior gates before throwing the bolts on the locks. With both bulls on the trailer, Rick was quickly in the driver's seat, clearing out of the lot for the next customer to come in.\n\nAs we bounced back down Highway 92 toward the grassy acreage north of his house, Rick was chipper and talkative.\n\n\" 'Mark him the buy of the day,' \" Rick said, putting on the auctioneer's tone. He shook his head and laughed. \"I don't know about that, but I'm sure pleased.\"\n\n\"The auctioneer's probably never seen a bad bull,\" I said.\n\n\"No,\" Rick allowed, \"but that one's a pretty good-looking son of a gun.\"\n\nI told Rick that, years ago when I was in Texas making extra money in school by working on a small cattle ranch, I was helping load a bull into a trailer when it kicked the gate back into the face of the ranch hand. The steel bar split his nose from bridge to septum, like a butterflied filet, sending him on a long drive to the emergency room to get stitched back together and leaving me with a lifelong respect and twinge of fear whenever handling bulls. Rick winced at the very idea of it.\n\n\"Were they running longhorns?\" he asked.\n\n\"No, Herefords mostly.\" I told him that almost all of the longhorns were gone by the time I was down there. You'd see them here and there, but rarely.\n\n\"Some people up here were raising them for a while,\" he said. \"They have great, really low-birth-weight calves. So you'd have one longhorn bull to breed with any cows that you were worried about making it through birthing. But they sell like crap at the sale barn, unless you bring in some fifteen-year-old bull with big horns and a nice brindle coloration, something for the Old West nuts who just like the look of them.\"\n\n\"Not so long ago, though, that's what everybody wanted,\" Rick said thoughtfully, \"and here we're back to water. Before everything was irrigated, you wanted a bull that could survive on range with very little water, very little feed.\"\n\nIn the panhandle region of West Texas, where the Texas Longhorn was first created by cross-breeding various European-stock longhorns, the giant rack had been an asset on long cattle drives when red wolves and coyotes would come in to prey on the herds. But when farmers started pumping out of the aquifer and irrigating, they turned to raising short-fiber cotton\u2014and with such remarkable success that they began converting rangeland to cotton fields and herding cattle into feedlots. Built adjacent to cottonseed oil mills, Texas feed yards would fatten cattle on meal and hull byproduct; the hardy but lanky Texas Longhorn was steadily replaced by brawnier shorthorn breeds like the Hereford and Angus, as well as new specialty breeds like the Beefmaster. But the whole system was dependent on pumping groundwater from the Ogallala Aquifer system.\n\n\"And down there,\" Rick said, \"it's a perched aquifer. They're mining fossil water.\" Cut off from the main aquifer system south of the Canadian River, the reservoir recharges so slowly that it would take thousands of years to refill. So as Texas Panhandle farmers cashed in on beef prices and then, in 2005, started chasing big-time commodity profits after passage of the Renewable Fuel Standard ushered in the era of $8 corn, they were pumping out the aquifer faster and faster, draining the great basin of water that had sustained Texas cattle for two centuries\u2014and before long, their wells were starting to run dry.\n\nBy the onset of the drought in 2011, the groundwater shortage had grown so severe that the State of Texas commissioned an in-depth study to quantify the problem. The results, published in the Texas Water Report in May 2015, could hardly have been more dire. \"Since the 1940s,\" the study reported, when ranchers first began trying to grow cotton on the arid rangeland, \"substantial pumping from the Ogallala has drawn the aquifer down more than 300 feet in some areas.\" But the real trouble has been much more recent. One hundred feet of the 300-foot decline of the aquifer occurred in the decade between 2001 and 2011. This period coincided with the run-up in commodity prices that tempted farmers to start growing thirsty feed crops. With rising temperatures and what the report described as \"the near-total absence of rain\" during the drought, water use for irrigation had jumped another 43 percent.\n\n\"It's really a cautionary story,\" Rick said. \"And California's in the same fix now, too, pumping their water until there's none left. Nebraska has been smart that way. The NRDs are a model for the country and something to be really proud of. We pump a ton of water from the aquifer, but there's monitor wells and the NRDs tightly watch the levels. If the level drops below the control, where it was when they started monitoring in 1972, they shut everything down. In the summertime, we listen to the radio out of York, and you'll hear days where they say, 'It's a red day; you're on control.' And you don't get more than twelve hours of pumping for three days out of that week.\"\n\n\"How do they make sure you're not irrigating?\" I asked.\n\n\"They can electronically shut the well down,\" Rick said, \"and you'll hear the old guys complaining when that happens, but it's a good thing. It protects the resource\u2014to make sure it's there when the next generation takes over.\"\n\nTHE SUN was still low on the horizon as Kyle fishtailed down the gravel road. It was early, no one else out yet, and he was just making the rounds, checking on pivots to be sure they were all on and turning. A light rain had fallen overnight, maybe a quarter inch\u2014just enough to leave the morning muggy and windless. But York County was still well short of the four-inch average for the month of July, and it was now into the midsummer run, right before and right after corn pollination, when even the most cautious and judicious farmers turn on their pivots and let them run day and night. The last thing you want during that crucial period is to lose irrigation even for a day.\n\nSo far, everything had been fine, but then, just as Kyle was in the homestretch, he spotted a problem. Four miles south of Centennial Hill, in a quarter section planted to corn, a spot the Hammonds call the Metz Place, the irrigation system had stopped moving overnight. Kyle sighed and threw on the brakes. He climbed out and killed the power at the control panel. \"The end of July, when the pivots have been running for a solid month, problems start showing up,\" he said. He grabbed a few tools and his voltmeter from his truck and waded down the rows, making his way toward the tower, its light no longer blinking, at the center of the field.\n\nFifty years ago, when farmers had to wait until mid-June to plant hand-selected corn seeds, the health of a corn plant was judged by the old saying \"Knee-high by the Fourth of July.\" Today, with hearty hybrids making it possible to plant in early May and selective breeding pushing plants ever taller, corn plants are head-high by Independence Day\u2014and towering taller by the end of the month. Kyle pushed down a dense row, brushing the flat leaves aside as he went, until he finally got to the pivot platform. He climbed partway up the center ladder until he had a vantage above the tassels of corn. \"If you look straight down the spans,\" Kyle said, lining up the arched sections from one tower to the next, \"you can see if there's something wrong.\" About halfway down the line of the irrigation system, one tower was tipped at a slight angle. \"You go to the fulcrum of that angle,\" he said, \"and that should be the point of the problem.\"\n\nOnce Kyle got to the tower, though, he couldn't find anything wrong. He checked the contactor and the electrical box. Everything looked right, so he squatted low, studying the wheel mechanism until he saw: one of the wheels seemed to have stuck in the mud or simply seized up from the humidity and the constant use. Either way, when the threaded gear coming off the drive shaft was unable to turn the wheel, the shaft snapped under the pressure. The pivot needed a replacement gearbox, but Kyle wasn't about to call in a Valley repairman. \"A new gearbox is probably twelve hundred bucks,\" he said, \"and a service call is eighty-some bucks an hour, and that includes drive time coming from Grand Island.\" He decided to swap it out himself. \"I got a used gearbox in the shed, saved from a pivot that went down in a tornado a few years ago,\" he said. He was sure he could put on the salvaged gearbox and get the pivot back up and running without losing a precious day of irrigation.\n\nSo Kyle went back to the farm and got Meghan. Together, they gathered a handyman jack, a breaker bar, and some giant blocks to lay as a foundation under the jack. \"I don't know how long the pivot stuck there,\" Kyle said, \"but with it watering that one spot, plus the rain overnight, the jack will just go right down into the mud.\" The biggest problem was the sheer heft of the gearbox itself. It must have weighed 150 pounds, and there was no good way to get it from the truck at the edge of the field, halfway across to the central tower. So he parked on the west side, simply hoisted the hulking gearbox onto his shoulders, and started into the field, cutting across the rows.\n\nKyle kicked the stalks down and stomped them under his bootsole, making a path for Meghan to follow with the jack and blocks, and then he stepped across the furrow and kicked down the next stalk. Losing those plants wasn't ideal, Meghan explained, but better to lose a few ears now than have the whole field go without irrigation for even a day at the peak of the summer heat. \"Anything to shorten the trip,\" she said. Three times as he went, Kyle had to stop and catch his breath, but he didn't quit until he was back at the broken drive shaft. Meghan laid out the blocks, and Kyle jacked up the tower.\n\nBy now, the sun was high overhead, beating straight down on the shadeless track made by the pivot's massive tires. The temperature gauge in the truck had been reading over 90 degrees. And, deep in the densely planted field, there wasn't so much as a whiff of wind. \"I'm about ready to pass out,\" Meghan said. \"I can't take it.\" Kyle worked quickly, removing the seized wheel and then unbolting the broken drive shaft and gearbox. But then, when he hoisted the salvaged replacement into place, he had an awful realization. Gearboxes for Valley pivots are mounted in opposite directions, depending on which side of the tower the box is on. The new box matched the old\u2014but as a mirror opposite.\n\n\"I grabbed the wrong box,\" Kyle said.\n\n\"What?\" Meghan demanded. \"Are you fucking kidding me?\"\n\nLater, when they were back home and cooled off, Meghan was quick to take her share of the blame. \"I could have double-checked with him and made sure he had the right one,\" she said. \"And trying to put an old gear box on the pivot, there was already a risk that it wouldn't work. So I knew going out there that we might have to do it more than once.\"\n\n\"It's definitely not what I wanted to happen,\" Kyle said, \"but at that point, with the whole thing tore down, you know what you've got to do next.\"\n\nHe put the backward gearbox on his shoulders again and started over the path he had made across the rows of corn. They drove back to the shed and got the reverse version, the correct version, of the gearbox\u2014and Meghan grabbed a length of wire. \"The old one is too heavy to carry out,\" she said. \"It ain't worth it.\" They could just wire the broken box to the tower and let it spin around the field from now to harvest time. \"Tie it to the damn pivot,\" she said. \"It can hang on there until the corn's out.\" So they returned to the field and Kyle shouldered the right gearbox out to the tower. He replaced everything, and they wired the broken box to a rung on the ladder that leads up to the electrical box. With everything now in working order, Kyle turned the control panel on again, and the pivot resumed its rotating, the dead gearbox swinging underneath. It had taken all day, until late in the afternoon, to get it fixed, but the pivot was back up and running.\n\nIN SEPTEMBER, Rick kept his promise. When it was time to hitch up the trailer and head west to Curtis, to load up the breeding bulls and bring them back before the fall harvest arrived again, he offered me a spot in the passenger seat. With the empty trailer bouncing and shimmying behind us, Rick didn't want to take the Interstate. Instead, we dropped down to Highway 34 and slowly made our way up the two-lane road. Around Holdrege, the terrain began to soften into sloping hills, and by the time we reached Eustis and Farnam, the loess hills had started to crest and break into blowouts. The deeper we drove into Rick's native terrain, the more he seemed to relax, drawn away for the moment from the worries of prices, the daily fears of soybean and corn futures still sagging ahead of another harvest, but there was one concern he just couldn't push out of his mind. After months of thought\u2014and one final check of the books\u2014he had decided that he had to let Dave, the farm's hired hand, go.\n\n\"I thought about waiting until after we got through harvest,\" Rick said, but Dave had been with him through fourteen harvests. This would be number fifteen. \"After that many years with me, I owe it to him to let him make the decision. I think he'll see the year through. Any farmer wants to see the crops that he planted harvested. I hope he does stay through harvest.\"\n\nWe came over a rise, and suddenly the cropland dropped away, giving way to grass, knee-high to hip-tall and yellowed with the coming of fall. Rick looked down the highway, empty ahead of us for as far as the eye could see. \"Here's the deal,\" he said, turning suddenly frank. \"When that neighbor took a half section away from us, I knew right then, and that's been a year and four months ago, but I sat on it for a year trying to see what would happen, trying to find some way to keep Dave on. But over time, three things have become apparent. One, we don't have enough work. Without the hogs, big distractions like the new house or building a barn, we can finally concentrate, and we can handle the work. Two, with the collapse in commodity prices, we can't afford it. That's just reality. The prices are way down, and I don't know if they're ever coming back. The third thing is that we have Kyle now\u2014he's like having four hired men\u2014and Jesse is coming back for at least half the year now.\"\n\nRick had adopted a tone of finality, but his body language suggested resignation more than resolve. He didn't like having to let Dave go, but he knew it was the right thing for his family and their business. \"It's just time,\" Rick said finally. \"It's time for it to get passed down to the next generation. I've been mostly going through the motions for a long time. It's time for me to step aside, for Dave to move on, and let Kyle and Meghan and Jesse take over.\"\n\n\"How will you divide everything?\" I asked. Remembering the years of tension among Heidi and her sisters, I couldn't help wondering if Rick was worried that there were similar fights ahead among his own children. He shook his head ruefully.\n\n\"The way that the Harringtons did things was an undivided share. I say that is a recipe for strife. It's not going to happen that way with us. Each of the kids will have their own parcels, and they'll make their own decisions on their own ground, and their own marketing.\" He even thought they should buy their own equipment, rather than try to share the costs on major purchases, like tractors or sprayers. \"You see so many of those deals where the families end up in fights. They hate each other. I'm not going to have that.\" Instead, he was resolved that each kid would have an equal share to farm or sell as they saw fit, and it wouldn't be gradual; it would be fast.\n\n\"Meghan and I went to this workshop on succession,\" Rick said. \"They said, 'Do it in chunks, because there's these beginning farmer interest rates.' My original plan was to let them take over rented ground. But the tax adviser said, 'Look, if you give them _your_ ground, then there's a huge state income tax benefit.' That would be the logical thing to do. So we'll get them through this year, and then I'll probably give them some of our owned ground. We'll work it out, and maybe we could even do the cattle. Sell it all in one chunk.\" He and Heidi would keep their dream house in Hordville, and he would hold onto this land in Curtis, where his sisters and brothers-in-law lived. But it was time for him to make way for the kids.\n\nWhen we finally arrived at the ranch, it was too late in the day to load the bulls and get back before dark. Rick pulled the trailer to the edge of one of his fields and unhitched it, then we drove out in the grass, just to make a quick check on the cattle grazing there. He banged on the door and hollered, and the black cattle came like shadows moving through the grass. They poured over the lip of the buttes and scrambled down the gray trails they had worn. He herded them through a gate into a lower pasture, so he would have an easier time of gathering up the bulls in the morning.\n\nBefore dark, Rick drove me by the house where he had lived as a young boy. \"This was the center of my life, this barn,\" he said pointing. \"My dad had cut the center of the haymow out. We had a big rope all the way to the roof, and we'd swing catty-corner from one end of the barn to the other.\" He'd play until after dark. By then, when he was supposed to gather eggs from the henhouse to bring in for breakfast in the morning, he was certain the shadows were filled with coyotes or wolves. To stay calm, he'd list off all the cartoon characters he could think of. _Mickey Mouse, Donald Duck_ , he'd say to himself, scanning the edge of the trees. _Goofy, Daffy, Minnie_. \"And when I got as far as the well, I'd just tear off to beat hell,\" he said, laughing. By then, he was sure that he was close enough to the house to outrace whatever might come clawing out of the dark. I laughed too, but I couldn't help thinking of what Rick had told me before\u2014how he'd spent his whole life just barely keeping the wolves away from the door and how as soon as he felt safe, he'd take another chance again.\n\nAs we wheeled up the hill toward the house where Rick had lived through middle school, where his sister and her husband now lived, I wondered what Rick would do now. If there were no more wolves, no more fights left ahead, would he enjoy his retirement, or would he live out his days regretting the mistakes of the past? \"You know, Kyle and Meghan coming back to the farm has been just a real joy to me,\" Rick said, as if he knew exactly what I was thinking. \"Now, Jesse's starting to come back, too. Evidently, I wasn't quite as mean as I thought I was when they were growing up\u2014didn't make it quite the hell that it seemed to be to me. I worry about how they'll do, but I think it'll be just like when I started. A light bulb goes off when it's yours. You just think about it differently.\"\n\nHe pulled into the driveway until he reached the yard of the old farmhouse. \"You think about it probably too much,\" he said. \"I know I did.\" Inside, Rick sorted through more old memories. In the basement, he fiddled with the lock on an old chest, until the key finally turned and revealed the few things inside. Stacks of Rick's _Classic Comics_ with illustrated versions of _Robinson Crusoe_ and _The Count of Monte Cristo_. There were a few photographs, including Rick's high school graduation photo. \"Look at that guy,\" he said. \"Have you ever seen somebody with such confidence?\" He laughed to himself. \"Yeah, I had it all figured out back then. I keep getting dumber the longer I live.\"\n\nFinally, Rick closed everything up and stepped back into the evening light of the yard. He wandered down the slope through the yellow grass. He found a turkey feather and stuck it into the band of his cowboy hat. For a moment, I thought he might make another joke about that, but he had turned thoughtful. The windmill squawked quietly behind him. \"C'mon,\" he said. \"There's one last thing I want to show you.\" He walked down until we were almost at the creek bottom. There, among the trees and overgrown by vines and saplings, was the ruin of a tiny bunkhouse.\n\n\"This is where our farmhand lived,\" Rick said. He ducked through the doorway, nothing but rusting hinges guarding its entrance now. \"His bunk was right there. This was his stove. As I got older, I loved to come down here, just to listen to the way grown men talked,\" he said. \"He would roll his own cigarettes. I remember how he stopped in the middle of his stories while he pinched out the tobacco and licked the paper.\"\n\nThe windows of the bunkhouse were all cracked and broken out, the little glass left behind filmed over by ancient grime. \"My dad had to let him go when things got bad,\" Rick said. \"I don't know what ever became of him.\"\nINTERLUDE\n\nTHE TRI-STATE\n\nBAYARD, NEBRASKA\n\nSeptember 2015\n\nI pulled the truck into the shade of a tiny clutch of cottonwoods. After the trip out to Curtis with Rick, I couldn't stop wondering: what makes a homeplace? For Rick, it seemed to me, Curtis was where his heart had remained and would always be. It's where, he told me, he wanted his ashes scattered. As a kid, I had felt a similar tug, drawn to the tiny town where my dad grew up, so I talked him into making the drive there, seven hours across the state. The town, Bayard, had been slowly disappearing from the time I was a kid, visiting my grandparents over the summer. And with every disappearance\u2014a restaurant closing, a shop boarded over\u2014there was an equal disappearance from the surrounding neighborhoods and farmhouses, another family that had decided that this was no longer home for them.\n\nBy the time we arrived, the sun was sinking in the west, but the temperature was still somewhere in the 90s, the song of the cicadas ratcheting up in the distance. I followed my dad out onto the sandy path that connects the state highway to the Tri-State ditch road north of town. A tiny house with a fenced patch of dirt sat just back from the canal; during the early 1950s, when Dad was still in elementary school, he lived there with his parents. \"It looked just about exactly like this back then, except for the fence,\" Dad said. \"That would have saved me a lot of grief.\" As it was, he said, he endured years of parental warnings: never to venture too close to the ditch bank, never ever to cross \"the needles\"\u2014a concrete footbridge and series of weatherworn pickets that form a weir across the channel.\n\nDad explained that there is a narrow lip at the bottom, just on the upstream side of the bridge's pilings. Ditch riders thread beams into the current, searching for that foothold, then let the force of the water push the top of the pole tight into place. By effectively narrowing the width of the canal, the needles form a bottleneck that partially dams the current and raises the flow level during dry times, but they also create a rushing spillway on the backside\u2014a constant source of worry for my grandma. Maybe she fretted more than most mothers, having lost two children in infancy before my father was born, but her fears were far from unfounded. During the years Dad's family lived along the Tri-State, two young children, in separate incidents, drowned in sections of ditch that ran through nearby Scottsbluff. \"There were kids that drowned, no doubt about it,\" Dad said. \"And it was certainly a worry.\"\n\nBut for my grandparents, there was little choice. My grandfather had left the Omaha stockyards to come west in 1936, then worked as a tenant farmer, raising beets and beans along the Platte River bottoms south of town. My grandma got them through lean seasons on her teacher's salary in the proliferating county schools. They hung on, year to year, until the blizzard of '49 pushed the family over into bankruptcy. Farmers Irrigation District, which still owns the Tri-State Canal, was willing to provide a rent-free home and an old crank telephone (the first my dad's family ever had), so that ditch riders could be called to go out and adjust water levels immediately in cases of emergency, and ditch riders, in turn, could report washed-out sections or other problems back to the offices in Scottsbluff. Ditch riding was seasonal work back then, but they could have the house year-round, and my grandfather got work in the winters working as an oiler in the pulp drier at the Great Western Sugar factory in town.\n\nDuring the summers, when school was out, my dad would ride along as Grandpa went on his rounds, checking to make sure farmers along his five-mile section of ditch were getting their allotments. \"The ditch riders had to supply their own cars,\" Dad said. It was a black coupe, he remembered, but paused a moment trying to recall the model. Water crawled through the channel, then rushed through the needles. \"A Chevrolet of some sort,\" he said finally. \"He always bought Chevrolets; he would never have dreamed of owning a Ford. Mostly what I remember is that I could barely see out.\" At each farm, Grandpa jotted into a pocket notebook how many acre-feet of water each farmer was receiving that day, based on how much water was being released out of the Guernsey Dam in Wyoming. At the end of the line, he would take a final reading to make sure the Tri-State was delivering the required amount into the Northport Canal.\n\n\"I remember one time, down real near to the Northport check, out in that sandy soil, he could see water bubbling up and obviously coming out of the ditch. He tried to plug it at the end out in the field, but he couldn't do it,\" Dad said. \"So he had to get down into the ditch and feel around for that hole. He took a bunch of gunnysacks and jammed them in there, then packed some mud in to try to cut off the flow of water. If you didn't, eventually it would cut the ditch bank wide open, and the water would run out and flood everybody.\"\n\nBut most days, he was just checking head gates, to make sure that there was the right amount of water in the lateral ditches and flowing into the farmers' open irrigation channels. \"There were always farmers who were upset that they didn't get water when they wanted it or they didn't get enough,\" Dad said. \"And they weren't above trying to steal a little water from their neighbors or taking a little extra out of the ditch.\" Grandpa understood that they were just trying to keep from going broke, but he also knew, Dad said, \"you're entitled to a certain amount of water, and that's what you're going to get. It's the ditch rider's job to see that you get all of that, but he also sees you don't get any more so that you short somebody else down the line.\"\n\nDad pointed toward an invisible point in the east. \"Remember that big lateral we saw down there,\" he asked, \"the one at the foot of that hill where the feedlot is now?\" I nodded. He said he remembered one instance when Grandpa suspected the farmer who used to live there of stealing water. Every morning Grandpa would go on his morning ride, checking the headgates. And then again, each afternoon, he would ride back along the same route, adjusting as needed. At that property, he started noticing that the ditches seemed overfilled each afternoon and began to suspect that the farmer was coming along behind him, opening the gates wider each morning and then cranking them back down before Grandpa made his afternoon rounds. \"So one day,\" Dad said, \"instead of going home and calling the readings in, he went back to that headgate. And, sure enough, it was wide open.\" So Grandpa went to his trunk, took out a big log chain, and wrapped it around the crank of the headgate and locked it\u2014 _open_. \"So when the farmer came back to close the gate back down that afternoon,\" Dad said, \"he couldn't turn the crank.\" He was forced to make a frantic call to Grandpa, confessing to stealing the water, and begging him to come down and unlock the gate before it washed out all of his ditches and flooded his fields.\n\n\"This is beautiful country,\" Dad said, as we shadowed the ditch back toward Bayard. \"It was a great place to grow up, but I couldn't get away from here fast enough.\" He said he didn't want to spend his life at the mercy of banks, insurance adjustors, and every storm cloud forming on the horizon\u2014every day a fresh (but identical) struggle to beat back total loss. I understand it, too, of course. Raised in the city, with only vacation trips to Scottsbluff and Bayard, it was easy to be nostalgic for a vanishing way of life when I didn't ever have to suffer through its hardships. Still, passing back through town, past all of the shuttered storefronts and abandoned buildings, I lamented aloud how sad it was to see Bayard dying this slow death.\n\n\"Is it?\" Dad asked. He sounded somewhere between quizzical and professorial. \"Maybe this town has served its purpose, and now it's time for it to fade into history.\"\n\nI turned down toward the old Great Western Sugar factory\u2014shuttered after the end of the last sugar campaign that fall\u2014and was surprised to find it not only abandoned but unfenced and unguarded. I drove us around the plant to the dryer where Grandpa worked. The giant doors stood open, so we parked and walked in. All that colossal machinery (the coal scuttle, the enormous furnace, the blower the size of a semi) bore mute testimony to just how dominant the beet industry had once been. And how utterly\u2014rusted now and forgotten\u2014it has been overtaken by a monoculture of corn. The warehouse at the back of the building is big enough to house a football practice facility, but the wind whistled and made its corrugated siding creak, sending unnerving echoes bouncing through its empty rafters. It was hard not to see this as a symbol of the end of the era of community and shared purpose that founded the valley.\n\nToday most farmers and ranchers are forced to go for broke each year, planting fields of whatever crops and grazing herds of whatever breeds are selling highest on the futures market. They bank on their ability to work harder and raise more than their neighbors, on the prospect that enough of them will fail that prices will remain high at harvest time. Now that ethic comes to a crossroads, as everyone faces the real possibility that a resource as basic as water\u2014worried over and doled out and preciously conserved for more than a century\u2014may be in diminishingly short supply. But no one seemed to be thinking beyond next year. No one would discuss the possibility that this drought was more than a few hard years, that there might be many hard years brought on by climate change and rising temperatures. For all the careful stewardship of the land, few in the North Platte River Valley want to consider that they may be up against a new threat, bigger and more complex than the one that faced farmers and ranchers at the end of the nineteenth century\u2014one that requires them to think even more boldly than their ancestors. For now, they're resigned to waiting and praying for snow in the Rockies.\n\nBefore we headed back east, my dad and I stopped at Genoways Hall, the public meeting room built by the Civilian Conservation Corps and named for my grandfather even before he died. The Bayard ditch runs within feet of its side door. It's not part of the Tri-State Canal system where he worked, but I remember how, each time we went there (when I was still too young for school), he would stalk the ditch bank, breaking back weeds and clearing debris. And I remember taking my father's old wind-up motor boat, filched from his wooden toy box on the porch of my grandparents' house, cranking its key tight, and then letting it go. Most of all, I remember how it disappeared into the pipe that still runs under Main Street, then popped out on the other side, listing drunkenly in the current; how I held my breath\u2014afraid of something, I'm still not quite sure what\u2014until my Grandpa patiently fished it out of the reeds and motioned with a wave of his sun-browned hand that it was time for us to get back home.\nEPILOGUE\n\nWELCOME NEWS\n\nNovember 2015\n\nOn November 6, 2015, Meghan and Kyle were in Muir Woods in northern California at their wedding rehearsal. They had decided on a destination wedding with just family. In the middle of the rehearsal, Meghan's phone started to blow up. Texts would come through asking if she had heard the news, but she couldn't get a phone signal to call out. Finally, at the top of the ridge, overlooking the tops of the redwood forest and out on the Pacific Ocean, she was able to get reception on her phone.\n\nPresident Obama had rejected the Keystone XL pipeline, putting an end, at least for the moment, to six years of fighting and worrying about the fate of the Ogallala Aquifer if there were ever to be a spill. Secretary of State John Kerry had decided that the project was not in the country's national interest, and Obama had called a press conference at the White House to announce that he supported that opinion. \"America is now a global leader when it comes to taking serious action to fight climate change, and frankly, approving this project would have undercut that leadership,\" Obama said.\n\nMeghan hooted and screamed and started calling everyone back home in Nebraska, all the people who had been fighting at her side for years. \"I can't believe it, I can't believe it,\" she said. \"It's the best wedding present I could ever ask for.\" The next day, Meghan and Kyle were married, and when they came back to York County, they soon moved into the house that Dave and his family had vacated, the home where Meghan had spent her childhood on the northern edge of Centennial Hill Farm. Less than a year later, Meghan found out that she was pregnant.\n\nDespite all of the happiness, the future remains uncertain. In 2014, Rick had warned that the American farm might be in trouble if we saw two more years of record harvests. Now, that's exactly what has happened. The harvest of 2016, another record-setting year of production of core commodity grains like corn, soybeans, and wheat, pushed prices to their lowest levels in decades. Corn, in particular, has plummeted to less than half its market value of just five years ago. Despite the drop in prices of feed grains, livestock prices have fallen simultaneously and at a rate that made it impossible to capture profits on increased margins. This downward spiral is already having broader effects. Cash-poor farmers aren't updating equipment or buying new trucks or even going to town to spend money on food and entertainment. The rural economy is stalling.\n\nWorse still, farmers who took out loans for land or equipment at the peak of prices are starting to worry about their ability to service their debts with profits projected at the time\u2014and banks are growing nervous, too. Loaning institutions are starting to call on big farmers to liquidate landholdings used as collateral, in order to reduce their risk. But this trend is already dragging down property prices, forcing still more liquidation\u2014the exact cycle that led to a rapid devaluation in the 1980s, triggering the Farm Crisis. Today the potential dangers of a rural bank panic to the broader economy are even greater. Three-quarters of farmers have quit the business in the last thirty years, so now every failure carries four times the weight.\n\nTo break this downtrend, the American Farm Bureau Federation had been counting on President Obama's Trans-Pacific Partnership, which promised to open new markets for beef, pork, and grains. But the rural areas that were depending on this new deal, as well as standing agreements such as NAFTA, voted overwhelmingly for Donald Trump and his protectionist, antiglobalist policies. Now, Trump is threatening to cancel manufacturing trade deals with China, and China is responding by threatening to cancel its purchases of American grains. If such a thing were ever to happen, it would make the Farm Crisis seem like a minor economic ripple.\n\nStill, Meghan and Kyle have taken over their share of the family land, and in February 2017, they welcomed the seventh generation\u2014a girl, Logan Elena Galloway\u2014 to Centennial Hill Farm. \"I don't know where we'll fit into the grand scheme of things,\" Meghan told me one day in the shed at the farm. \"I just know that we want to do the best that we can for the land, for our family, for the world. Whatever comes, we'll find a way to get by.\"\n_Acknowledgments_\n\nFirst and foremost, I have to thank Rick Hammond, Meghan Hammond, and Kyle Galloway for allowing me to follow them for over a year of their lives. I was constantly in the way, parking in the wrong place, asking complex questions at the wrong time, but they handled it all with patience and grace. Thanks, too, to Jenni Harrington for helping me to sort out the history of the Centennial Hill Farm and to Rita Zoucha for generously allowing me into her home and sharing information with me about her son Brent. Thanks to my father, Hugh H. Genoways, for answering questions about our family history. And my gratitude to the countless people, unnamed and unquoted, who contributed to my understanding of modern farm life. This city kid appreciates the education.\n\nSecond, I would like to thank the many editors who believed that this story was worthy of telling. Most importantly, John Glusman at Norton saw a book here and helped in countless ways to make this project into one. Everyone at Norton was so supportive and engaged: Lydia Brents, Fred Wiemer, Alexa Pugh, and Will Scarlett. Thanks also to: Ted Ross at _The New Republic,_ who __ was brave enough to publish a long piece on the history of corn hybridization; Ellen Rosenbush at _Harper's_ , who shepherded the story of a soybean harvest through trying times; Jennifer Sahn at _Pacific Standard,_ who made immeasurable contributions to shaping the section on center-pivot irrigation. The Food & Environment Reporting Network (FERN) provided much-needed funding to keep this project going; special thanks to Sam Fromartz, Brent Cunningham, and Tom Laskawy. Thanks to Don Fehr and everyone at Trident Media for standing behind this project and everything I do.\n\nLast but not least, my endless gratitude to my family. My son, Jack, lived through nearly three years of trips to the farm and conversation about market prices, equipment failures, and water rights. My wife, Mary Anne Andrei, made the first contact with the Hammonds, conceived of a long project following them, and was frequently the glue that held it all together. The intimacy of her photographs is a testament to how much the family trusted her and an indication of the extent to which my access was granted because of her legwork. I'm grateful for every day that we're allowed to work together, telling stories that matter to us.\n_Selected Bibliography_\n\nThe majority of my research came from observation and conversations with the Hammond-Harrington family, as well as other sources directly quoted in the text. However, a few books were critical to shaping my thinking and informing the text.\n\nDeborah J. Bathke et al. _Understanding and Assessing Climate Change: Implications for Nebraska_. Lincoln: University of Nebraska Press, 2014.\n\nAnn Bleed and Christina Hoffman Babbitt. _Nebraska's Natural Resources Districts: An Assessment of a Large-Scale Locally Controlled Water Governance Framework_. Robert B. Daugherty Water for Food Institute. University of Nebraska, 2015.\n\nWayne G. Broehl, Jr. _Cargill: Trading the World's Grain_. Hanover, NH: Dartmouth College: University Press of New England, 1992.\n\nDan Charles. _Lords of the Harvest: Biotech, Big Money, and the Future of Food_. New York: Basic Books, 2001.\n\nJohn C. Culver and John Hyde. _American Dreamer: A Life of Henry A. Wallace_. New York: W. W. Norton, 2001.\n\nDan Morgan. _Merchants of Grain: The Power and Profits of the Five Giant Companies at the Center of the World's Food Supply_. New York: Viking, 1979.\n\nRonald C. Naugle, John J. Montag, and James C. Olson. _History of Nebraska_. 4th ed. Lincoln: University of Nebraska Press, 2015.\n\nQuentin R. Skrabec, Jr. _The Green Vision of Henry Ford and George Washington Carver: Two Collaborators in the Cause of Clean Industry_. Jefferson, NC: McFarland & Co., 2013.\n\nNancy Warner and David Stark. _This Place, These People: Life and Shadow on the Great Plains_. New York: Columbia University Press, 2013.\n_Illustrations_\n\nMeghan Hammond closes up newborn chicks in the henhouse for the night.\n\nRick Hammond stands in the stippled light of his family barn.\n\nKyle Galloway steers the John Deere harvester by hand across a neighbor's field of soybeans.\n\nMeghan hands dinner off to Dave Tyson. Corn harvest runs late into the night, and meals are often eaten at the wheel.\n\nMeghan reads while a load of soybeans fills a grain bin for later sale.\n\nJenni holds a photograph of her grandfather checking out his new Farmall tractor, shortly after he bought back Centennial Hill Farm in 1949.\n\nRick separates cow-calf pairs in the corrals near the old barn at Centennial Hill.\n\nKyle checks on an electric fence that shorted out when snow laid prairie grass across the wire.\n\nRick rests briefly while herding cows and calves on property near the new house he built in the early 2000s.\n\nKyle holds a calf's tail out of the way as Jesse applies the R Diamond brand and the veterinarian prepares the inoculations.\n\nDave starts the motor on the soybean seed tender, while Rick directs the hose into the hopper at the back of the planter.\n\nKyle sprays a mix of herbicides and pesticides onto a field of newly emerged soybeans.\n\nThe search for a lost dog turns into a social gathering in the middle of a country road south of Central City.\n\nEvery summer on the anniversary of his death, Meghan returns to the home of her high school boyfriend who was killed in Iraq in 2006.\n\nRick makes out a check to the sale barn after entering winning bids on a pair of black Angus bulls.\n\nIn the late summer, when he comes to load up cattle in Curtis, Rick always visits the farmsteads where he grew up.\nALSO BY TED GENOWAYS\n\n_The Chain_ (2014)\n\n_Walt Whitman and the Civil War_ (2009)\n\n_Anna, Washing_ (2007)\n\n_Bullroarer: A Sequence_ (2001)\nPhotographs by Mary Anne Andrei\n\n_This Blessed Earth_ is a work of nonfiction. The Hammonds, Galloways, and  \nHarringtons are a real family in York and Hamilton counties, Nebraska. All  \nquotations derive from recorded conversations between July 2014 and October  \n2016. Whenever possible, conversations have been depicted just as they occurred  \nbetween the harvest of 2014 and the harvest of 2015; however, occasional  \nadditions from follow- up interviews have been included for clarity and detail.  \nIn some cases, names of individuals have been changed.\n\nCopyright \u00a9 2017 by Ted Genoways\n\nAll rights reserved\n\nFirst Edition\n\n\"Fear in a Handful of Dust\" \u00a9 2015, Ted Genoways, first published  \nin _The New Republic_. \"Corn Wars\" \u00a9 2015, Ted Genoways, first published  \nin _The New Republic_. \"The Great Abandonment\" \u00a9 2016, Ted Genoways,  \nfirst published in _The New Republic_. \"Irrigation Nation\" \u00a9 2016,  \nTed Genoways, first published in _Pacific Standard_. \"This Blessed Earth\"  \n\u00a9 2017, Ted Genoways, first published in _Harper's_.\n\nFor information about permission to reproduce selections from this book,  \nwrite to Permissions, W. W. Norton & Company, Inc.,  \n500 Fifth Avenue, New York, NY 10110\n\nFor information about special discounts for bulk purchases, please contact  \nW. W. Norton Special Sales at specialsales@wwnorton.com or 800-233-4830\n\nBook design by Brooke Koven\n\nProduction manager: Beth Steidle\n\nJACKET DESIGN BY BRIAN BARTH\n\nJACKET PHOTOGRAPHS BY MARY ANNE ANDREI; (GARLIC ROW) \u00a9 MASTERFILE ROYALTY-FREE\n\nThe Library of Congress has cataloged the printed edition as follows:\n\nNames: Genoways, Ted, author.\n\nTitle: This blessed earth : a year in the life of an American family farm \/  \nTed Genoways.\n\nDescription: First edition. | New York : W.W. Norton & Company, [2017] | Includes bibliographical references.\n\nIdentifiers: LCCN 2017025062 | ISBN 9780393292572 (hardcover)\n\nSubjects: LCSH: Family farms\u2014Nebraska\u2014Case studies.\n\nClassification: LCC HD1476.U62 N25 2017 | DDC 630.9782\/345\u2014dc23  \nLC record available at https:\/\/lccn.loc.gov\/2017025062\n\nISBN 978-0-393-29258-9 (e-book)\n\nW. W. Norton & Company, Inc.\n\n500 Fifth Avenue, New York, N.Y. 10110\n\nwww.wwnorton.com\n\nW. W. Norton & Company Ltd.\n\n15 Carlisle Street, London W1D 3BS\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nTable of Contents\n\nBlurb\n\nAUTHOR'S NOTE\n\nPRINCIPAL CHARACTERS (GLOSSARY AT END)\n\nA Note About Ithani Pronouns\n\nPROLOGUE\n\nCHAPTER ONE: BEGINNINGS\n\nCHAPTER TWO: UP FROM THE RUINS\n\nCHAPTER THREE: ABOVE THE FRAY\n\nCHAPTER FOUR: NO PLACE LIKE HOME\n\nCHAPTER FIVE: AN UNEXPECTED VISITOR\n\nCHAPTER SIX: OFF THE RAILS\n\nCHAPTER SEVEN: AFTER THE FLASH\n\nCHAPTER EIGHT: SOMEWHERE ELSE\n\nCHAPTER NINE: CLUES\n\nCHAPTER TEN: QUESTS\n\nCHAPTER ELEVEN: CRASH\n\nCHAPTER TWELVE: IN A FLASH\n\nCHAPTER THIRTEEN: INTO ENEMY TERRITORY\n\nCHAPTER FOURTEEN: BENEATH THE HOUSE OF THE SKY\n\nCHAPTER FIFTEEN: ERINA\n\nCHAPTER SIXTEEN: THE CROSS\n\nCHAPTER SEVENTEEN: ON THE RUN\n\nCHAPTER EIGHTEEN: CAVERN OF THE WHIRLS\n\nCHAPTER NINETEEN: INTO THE HEART\n\nCHAPTER TWENTY: INTO THE AIR\n\nCHAPTER TWENTY-ONE: IN THE DEPTHS\n\nCHAPTER TWENTY-TWO: GHOST TOWNS\n\nCHAPTER TWENTY-THREE: IN THE DREAMTIME\n\nCHAPTER TWENTY-FOUR: IN THE WHITE CITY\n\nCHAPTER TWENTY-FIVE: DAIA\n\nCHAPTER TWENTY-SIX: A NEW PLAN\n\nCHAPTER TWENTY-SEVEN: IN THE HEART\n\nCHAPTER TWENTY-EIGHT: FREE\n\nCHAPTER TWENTY-NINE: NIGHT FALLS\n\nCHAPTER THIRTY: SHIFT FORWARD\n\nCHAPTER THIRTY-ONE: STORM BATTLE\n\nCHAPTER THIRTY-TWO: AFTERMATH\n\nEPILOGUE\n\nGLOSSARY\n\nMore from J. Scott Coatsworth\n\nReaders love The Oberon Cycle by J. Scott Coatsworth\n\nAbout the Author\n\nBy J. Scott Coatsworth\n\nVisit Dreamspinner Press\n\nCopyright\n\n# Ithani\n\nBy J. Scott Coatsworth\n\nThe Oberon Cycle: Book Three\n\nTime is running out.\n\nAfter saving the world twice, Xander, Jameson, and friends plunge headlong into a new crisis. The ithani\u2014the aliens who broke the world\u2014have reawakened from their hundred millennia-long slumber. When Xander and Jameson disappear in a flash, an already fractured world is thrown into chaos.\n\nThe ithani plans, laid a hundred thousand years before, are finally coming to pass, and they threaten all life on Erro. Venin and Alix go on a desperate search for their missing friends and find more than they bargained for. And Quince, Robin, and Jessa discover a secret as old as the skythane themselves.\n\nWill alien technology, unexpected help from the distant past, destiny, and some good old-fashioned firepower be enough to defeat an enemy with the ability to split a world? The final battle of the epic science fiction adventure that began in Skythane will decide the fate of lander and skythane alike. And in the north, the ithani rise....\n\n# AUTHOR'S NOTE\n\nITHANI REPRESENTS a new \"coming of age\" for me as an author. It marks the first trilogy I have completed (though the Liminal Sky trilogy is close on its heels, and will come to its conclusion about six months after the Oberon Cycle).\n\nFlying the story into the hangar was a daunting task. I went into Skythane, the first book in the series, with a basic outline of what was going to happen across the three books of the trilogy. As my model, I used James P. Hogan's Giants series, which are amazing plot-wise. The stories are told like science fiction mysteries, starting with a big one. How did a man in a space suit end up on the moon fifty thousand years ago? In each of the succeeding books, another layer of the onion is peeled back, and by the end you've fallen through a rabbit hole to a marvelous place literally light-years away from where you started.\n\nThere's an old saying about best laid plans, and another about the universe laughing at you as you make them.\n\nAs a writer, your story often surprises you. Characters don't always go where you tell them or do what you want them to do, and unexpected plot twists and turns can take your whole story train and turn it away from your planned ending toward something else.\n\nLike Timbuktu. Or cheese.\n\nAnd so I arrived at the doorstep of book three with the unenviable task of gathering all those pretty little plot threads from the first two books, sorting them out, and bringing each one to a meaningful conclusion.\n\nMany authors write their trilogies all at once these days, and then publish them over a short time frame\u2014what I call the Netflixing of fiction. That idea has never appealed to me. I like having some waiting time between books\u2014both as an author and as a reader\u2014to build anticipation and to spread the enjoyment of the work out over time. I also like the space it gives me between books to truly let my imagination wander. The one advantage this model does have, however, is the fact that you can monkey with books one and two while you're wrapping up book three. If you realize something back in Chapter thirteen of book one is really messing with your ending, you can fix it before anyone is the wiser.\n\nThat's not the case when the first two books are already out wandering the world\u2014it's too late for changes.\n\nSo for Ithani, I had to account for every word, every sentence, and every half-baked idea I had put into Skythane and Lander.\n\nThe good news? I think it really works.\n\nIthani also gave me a chance to take you where you've never been before. In Skythane, you got a feel for Erro in its two parts, Oberon and Titania. In Lander, you got to see the skythane world up close.\n\nIn Ithani, I take you someplace entirely new. Several places, in fact. It's going to be a wild ride.\n\nDevotees of my Liminal Sky series might also find a thing or two here to like. *grin*\n\nWill I go back to Erro one of these days? I honestly don't know. The story (for now) feels complete, and I'm happy to leave it where it is.\n\nBut I learned a long time ago to never say never. There may be more stories to be told between the landers, skythane and ithani.\n\nFor now, I hope you enjoy the epic conclusion to Xander and Jameson's tale.\n\n# PRINCIPAL CHARACTERS (GLOSSARY AT END)\n\nAlia (PA Nix): Guard in Gaelan\n\nAlix Preston (PA Erissa): Xander's ex, a lander man missing for a year\n\nAngie\/Angela: Jameson's PA\n\nAri: Quince's PA\n\nBraice: One of the original Skythane settlers\n\nDaia: One of the original Skythane settlers\n\nDanielle (Dani) Black (PA Hera): One of the lander enforcers in Gaelan, daughter of Danner Black\n\nDavyn Sl\u00e9ite: Xander's Gaelani name\n\nDerren Sevvins: OberCorp employee\/tech who helps Mylin\n\nDOC (Digital Oberon Corporation AI)\u2014the AI that runs OberCorp's semi-autonomous grid\n\nErina\u2014One of the ithani Xander and Jameson encounter\n\nErissa: Alix's PA\n\nHera: Dani Black's PA\n\nJameson Havercamp (PA Angie): Psych from Beta Tau who comes to investigate pith shortage on Oberon\n\nJessa: Jameson's fianc\u00e9e on Beta Tau\n\nJirron (also Thshnel'Jirron): King of the ithani\n\nKadin Tamain: The chamberlain of the House of the Moon\n\nLyrin Madainn: Jameson's birth name\n\nMorgan: Mysterious child Xander finds on Oberon\n\nMylin (PA Zim): Young skythane girl who helps Xander\n\nNix: Alia's PA\n\nQuince Farrai (PA Ari): Xander's skythane friend who joins the quest\n\nRobyn Sl\u00e9ite: Queen of the Gaelani and mother to Xander, and Quince's former lover\n\nRogan Horth: Syndicate boss who has history with Xander\n\nSmythe: An OberCorp enforcer\n\nTally: One of Tanner's friends from Egeus\n\nTanner: Morgan's human host\n\nTaz: Venin's PA\n\nThshnel'Jirron: see Jirron\n\nTovey: Guard at the OberCorp headquarters armory\n\nVenin Araio (PA Taz): Guard in Gaelan\n\nVestra Halta: Acting Regent of Errian\n\nXander Kinnson (PA Ravi): Skythane who works in Oberon City, embarks on a quest with Jameson\n\nZim: Mylin's PA\n\n# A Note About Ithani Pronouns\n\nTHE ITHANI have a complex (to us) system of pronouns and relationships. Three ithani are required to bear a child. These are ze \/ zer \/ zers (roughly analogous to she \/ her \/ hers\u2014the one who provides the egg), zi \/ zim \/ zis (he \/ him \/ his\u2014the one who provides the fertilization), and za \/ zaf \/ zas (no human equivalent\u2014the one who provides the organs to carry the child to term). In addition, when pregnancy occurs, the three forms unite into one (zee \/ zeer \/ zeers) until the child comes to term.\n\n# PROLOGUE\n\nNow it is the time of night\n\nThat the graves all gaping wide,\n\nEvery one lets forth his sprite,\n\nIn the church-way paths to glide:\n\nAnd we fairies, that do run\n\nBy the triple Hecate's team,\n\nFrom the presence of the sun,\n\nFollowing darkness like a dream.\n\nWilliam Shakespeare, A Midsummer Night's Dream\n\nERINA'S IRIDESCENT wings fluttered in distress.\n\nThe time of the Great Migration was almost upon them. Overhead, the sun turned redder day by day. Soon it would unleash a torrent of heat and radiation that would destroy the ithani, the Heart, and Erro itself.\n\nZe had foreseen the end of the war years before and had passed zer knowledge on to Thshnel'Jirron, trusting zi to protect them all. But zis plan had gone too far, and soon the ithani would destroy themselves in a bid for immortality.\n\nOnly zi would survive, immensely more powerful than now. A virtual vengeful god.\n\nZe had seen that, too, but it had come to zer far too late.\n\nEach generation of the ithani had a seer, and ze had been born from the Heart with that heavy responsibility upon zer small shoulders. Even a seer didn't know everything about what was to pass. Only the bits and pieces that were passed on to zer from the athr\u00e0. Besides Jirron, no one else knew ze had the gift.\n\nSomething, or someone, was coming, ze didn't know what yet, but ze could feel it in zer bones.\n\nZe was at a loss for what to do next. Ze let out a whoosh of breath, resigned to waiting. It would come to zer, the vision she needed. When the gods thought it was time.\n\nUntil then, ze would keep zer wings low to the ground and do nothing further to draw attention.\n\nDANI PEERED through the hoversport visor, trying to make out the way ahead through the falling snow.\n\nFlying the hoversport, she and Kadin had crossed over the Riamhwood and into the northern plains without incident, only to run into this abominable blizzard. Gusts of wind sideswiped the little craft, snow flurries cutting her vision down to meters at times. She fought back, holding it as steady as she could.\n\n\"Are you sure about this?\" Kadin looked a bit green. Used to being out in the fresh air, she supposed.\n\n\"This isn't over. They killed my father.\" Truth be told, Danner Black had always been an arrogant bastard. He'd beaten her on three separate occasions as a child.\n\nBlood was blood.\n\n\"With the company and the skythane standing arm in arm, we're screwed\u2014unless we align ourselves with someone else, and quick.\"\n\n\"Who?\"\n\nShe grinned. \"Whatever lives up here in the north. You didn't think this planet split itself in half, did you?\"\n\nHe shook his head. \"But whoever\u2014or whatever\u2014did it has to be long dead.\"\n\n\"What if they're not? That bitch queen of Gaelan got her wings back from somewhere. That kind of tech's pretty advanced. What if the old masters of this planet are back?\"\n\nKadin whistled. \"Then we should probably run as far as we can away from them.\" He took her hand. \"Look, why don't we run away, find a place in the mountains? Let them fight this mess out among themselves?\" He touched her cheek. \"I don't care about anything else. I just want to be with you.\"\n\nShe pushed him away gently. \"No, you're not thinking this through.\" She sighed. Kadin had been essential to her plans in Gaelan, but she didn't love him. Did she? \"We're trapped in a whole different universe, and we don't know the rules. We're wanted by almost every human on the planet. What if these... whatever they are. What if they're stronger than OberCorp? Stronger than those two Split-damned prophesied kings?\"\n\nA smile spread across his face. \"Then they might welcome allies who know almost everything about OberCorp and the skythane.\"\n\n\"Exactly.\" She grinned. \"We have leverage, and maybe we can carve out a little bit of this world all for ourselves, if not the whole fucking thing.\" She felt better than she had in weeks. Things were finally starting to go her way. To fly again.\n\nBright light flooded the interior of the hoversport as the storm clouds lifted. Their destination lay before them. It was a gigantic, snow-covered mountain, fifty kilometers ahead. The landscape all around was a rose-tinted white, reflecting sunlight into her eyes and almost blinding her. \"Hera, dim the plas!\"\n\n\"Yes, Dani,\" her PA responded, and the brightness dimmed to an acceptable level.\n\nA thunderous rumbling roared outside the shuttle, loud enough that it shook the tiny craft.\n\n\"What the hell is that?\" Kadin peered out through the window.\n\n\"It sounds like a shuttle engine.\"\n\n\"But a thousand times stronger. Look!\"\n\nDani followed his gaze.\n\nThe mountain itself seemed to glow. The light increased, arriving with a great humming sound that pierced Dani's ears.\n\nShe threw her hands up to cover them, but she couldn't keep out the sound.\n\nDani screamed.\n\nThen the light cut off, and the shuttle's controls went dark. Dani glanced over at Kadin. He was slumped back in his seat. Frantically she tried to get the craft to respond, to level out.\n\nThen it crashed into the snow, and her awareness of the world slammed shut.\n\n# CHAPTER ONE: BEGINNINGS\n\nVENIN STOOD under the dome of the chapel, the waters of the Orn rushing past the small island to crash over the edge of the crater rim, where they fell a thousand meters to the broken city of Errian below.\n\nThe Erriani chapel was different from what he was used to back home. The Gaelani chapel in Gaelan had sat at the top of a tall pillar of stone, open to the night sky, a wide space of grass and trees that intertwined in a natural dome through which moonlight filtered down to make dappled shadows on the ground.\n\nThis chapel, instead, was a wonder of streaming sunlight, the columns a polished eggshell marble with glimmering seams of gold. Red creeper vines climbed up the columns, festooned with clusters of yellow flowers that gave off a sweet scent.\n\nBoth were bright and airy, but the Erriani chapel lay under a dome supported by fluted marble columns, a painted arch of daytime sky and the rose-colored sun blazing overhead.\n\nThe last time he'd gone to chapel had been with Tazim, before his untimely death.\n\nLong before the troubles that roiled the world now.\n\nSomething drew him back. A need to reconnect with his past. To bridge the gap between then and now, between who he was and who he had become. Taz would have liked this place.\n\nThe chapel here had survived the attack, while much of Errian had not. The city below was a jumble of broken corrinder, the multistory plants that were the main building stock for the city. They would grow again, but the sight of the city's beautiful white towers laid low struck him to the core.\n\nSo had Gaelan looked, after the flood.\n\nVenin turned back to the chapel and unlaced his boots, baring his muscular calves before he approached the fountain that splashed at its center. The cool flagstone beneath his feet sent a shiver up his spine, and green moss filled the gaps between the stones.\n\nSome builder whose name was lost to time had tapped into the river itself to make the fountain run, and the water leapt into the air with a manic energy around the golden statue of Erro, before falling back down to the pool.\n\nVenin knelt at the fountain's edge on one of the well-worn pads, laid his hands in the shallow water, and let his wings rest over himself, making a private place to pray.\n\nErro and Gael, spare us from danger and lift us up into the sky with your powerful wings. He gave Erro deference, being that this was his chapel, but he hoped Gael would hear him too. The god of his own people had been known to intervene in mortal affairs before, and if what Quince had told them about these ithani was true, they would need all the help they could get.\n\nVenin's wings warmed.\n\nHe looked up in astonishment to see the statue of Erro giving off an intense golden glow. His mouth dropped open, and he stood and stared at its beautiful male curves and muscles. Maybe the gods were answering him.\n\nVenin reached up and touched the statue's outstretched hand. The shock knocked him backward onto his ass, and he hit the ground hard, slamming into one of the marble columns.\n\nVenin groaned, stunned, and reached back to feel his wings and spine. He seemed to be in one piece.\n\nTaz would have laughed his ass off at the whole thing.\n\nAfter a moment he sat up cautiously. He wrapped his arms around his legs and stared up at the statue, his chin on his knees.\n\nThe glow was gone.\n\nDid I imagine it? He stood and felt the back of his head. A lump was already forming there. That's gonna leave a mark.\n\nSomething had changed. Venin didn't know what yet, but he was sure of that much.\n\nHe pulled his boots back on and laced them up. With one last suspicious glare at the statue, he turned and stepped out of the chapel, taking a deep breath of the moisture-laden air.\n\nThen he leapt into the sky to soar down to the broken city.\n\nMORGAN SLIPPED out from behind one of the wide columns and watched the man go.\n\nErro was laid out at his feet. The city below was in ruins, but it would rise again, given time.\n\nHe was a nimfeach, protector and agent of the ithani, but he felt a bond with these humans, lander and skythane both.\n\nWhen he'd first begun his part of the plan, it had seemed simple enough. Lead the skythane to their prophesied destiny and bring the world back together as one.\n\nHe'd never thought to question the plan.\n\nNot until Jameson had held his hand and begged him to spare Quince's life. Something had shifted in him then. Something had awoken.\n\nIt had caused him no end of trouble. Suddenly he was no longer sure of himself. No longer secure in the plan. He'd even started to question the things he'd been told all along. How the world had been split. Who had caused it, and even if the plan was right at all.\n\nThings came to him in short bursts. Things that might happen. Things he should do.\n\nIt was why he'd given Quince the master key, the one he had discovered underneath Freyyr, the mountain in the north where the ithani slumbered.\n\nHe dreamed of the future, of the Great Migration, when the ithani would leave this mortal plane.\n\nIf it went ahead, thousands would die. People he knew. People who had come to call him friend.\n\nThe old Morgan could have lived with that. Probably wouldn't have even noticed it.\n\nThere was something stirring in him, though, something that was causing him to reevaluate everything and to take his own steps outside of the plan to find out if it was right.\n\nSomething named Tanner.\n\nALIX STARED out at Oberon City through the thick window of his mother's huge white top-floor office. A long crack ran down the plas at an angle, left by the enormous storm that had borne down on the city after the shift.\n\nAlix snorted. It was an apt metaphor for the world, imperfectly fused and pushed to the point of breaking.\n\nOberon City had fared better than either Errian or Gaelan, but only due to geography and the fact that it hadn't been subject to a full-scale enemy assault. Still, the tempest that had descended upon Erro after the shift had taken its toll, as had the more recent shaking.\n\nIt felt like the end days.\n\nNot that Alix had ever been much of a religious man. His mother believed in one thing\u2014power\u2014and had respected religion only inasmuch as it might help her gain more.\n\nNow she was locked up until he could figure out what the hell to do with her. He ignored her grid calls, but sooner or later they were going to have a reckoning.\n\nPeople were protesting on the streets below, outside the wide moat that protected the company headquarters. Food wasn't a problem in the city yet. The company had warehouses full of boxcorn and rasswheat processed and ready for off-world shipping to some of the frontier worlds, and Alix had opened the doors to the general population with orders to give out whatever they had. He figured they had enough to keep things going for a few weeks, if they rationed it. By then, either things would be returning to normal, or there would be no more need for food. Worry about that later.\n\nHe'd ordered the troops home from Errian, those who had survived the unfortunate and misguided war with the skythane. They were being pressed into duty as peacekeepers here in the city, but they were spread really thin. By the Split, he hoped he could keep a lid on things for a little while. Just a few days.\n\nJameson's message, run up to Alix by a courier, had been cryptic. Imminent danger from the north. Meet in three hours at OCHQ. Need new cirqs, prep for imminent mission.\n\nWhat the hell? Alix cursed and paced back and forth in front of the floor-to-ceiling plas windows. Being trapped behind a desk grated on him after more than a year out in the world. He needed to be out there seeing things for himself, not relying on a few scattered reports from the field.\n\nThose reports were just starting to come back from the few scouts he'd been able to field. They showed widespread damage to populated areas and food crops, as well as activity along previously unknown fault lines across the now reunified globe. In particular, a series of new trenches had opened up in the Outland, east of Oberon City.\n\n\"Message from Trevor Hass,\" Erissa, his PA, whispered in his ear.\n\n\"What does it say?\"\n\n\"It came tagged with a memory file. Would you like to experience it?\"\n\nAlix closed his eyes. \"Yes, please.\"\n\nJust like that, he rode in Trevor's head, seeing what the ranger saw. It was grainy\u2014limited bandwidth between the hoversport and OberCorp HQ, most likely.\n\nThe ground was covered in snow that sparkled rose in the fading evening light, but up ahead....\n\nAlix gasped.\n\nWhat looked like it had once been a mountain was split open, and a bright golden light radiated from inside the earth. Ghostly forms flew up into the air and then back down into the strange crater, flickering like candlelight.\n\n\"Trevor, what the hell is that?\" Alix asked the ranger.\n\n\"I don't know.\" Trevor's voice was flat. Calm. Too calm. They'd trained together, and that meant nothing good. \"Nothing I've ever seen before, that's for Split-damned sure. Should we go closer for a better look?\"\n\n\"No. Stay for as long as you can. When your power core gets low, come back. But do not put yourselves in danger, any more than is necessary.\" They couldn't spare any more hoversports, and he'd be damned if he'd jeopardize his men to no apparent purpose.\n\n\"Yes, sir. Thank you, sir.\"\n\nAlix cut the connection and sat back in his mother's chair. He had so much theoretical power, and yet he was impotent in the face of fast-racing changes he didn't understand.\n\nAlix was a visual guy. He liked to see the problem with his own eyes and then tackle it head-on. Now, without satellite coverage, he was blind and impotent. He closed his eyes. Xander, I could really use your steadying presence right now.\n\nThat ship had long since warped away. The sooner he got over it and moved on, the better. \"Erissa, when will our guests arrive?\"\n\n\"In about half an hour. I've set up conference room seventeen for your meeting.\"\n\n\"Perfect.\" He nodded. Quince, Xander, and Jameson would have some answers. Something he could wrap his head around.\n\nThen he could figure out what the hell he was supposed to do.\n\n# CHAPTER TWO: UP FROM THE RUINS\n\nJIRRON'S EYES opened.\n\nZis mouth was dry as the dust of Avea, Erro's airless red moon. A nimfeach busied itself checking zis vitals. Zi didn't recognize it.\n\nSomething had gone wrong. Jirron could feel it. Zi queried the Heart. How long have I been asleep?\n\nThe answer was slow coming back. One hundred three thousand, two hundred forty-four cycles.\n\nZi blinked, expressing zis displeasure. Zi sat up, ripping the nourishing leads out of zis flesh. Zi struck, catching the nimfeach by the thorax and shoving it against the wall, breaking one of its wings. Why was I asleep for so long?\n\nIt whimpered under the pressure from zis hand as zis three claws dug into it. The shift went awry.\n\nA hundred thousand years.\n\nThe energy released by the amalite core during the shift was fatal to the ithani if they were awake and unshielded.\n\nThe nimfeach were immune\u2014they were half made of amalite, after all. But zi'd insisted on a failsafe for zis own safety, so that none of the ithani sleepers would awaken until the shift was completed.\n\nZi hissed. A failsafe that had cost zim a hundred millennia. And now?\n\nThe humans helped us complete the shift. Green ichor dripped down the white wall from the broken wing.\n\nHumans? What are humans? Much had changed since zi had gone to sleep, although for zim it had seemed a matter of hours. Jirron queried the Heart and got back a burst of data. Zi closed zis eyes and let go of the nimfeach as zi digested the news.\n\nThe nimfeach fell to the ground with a sickening thunk.\n\nHumans. An outsider race that had come to Erro while zi slept.\n\nZi had seen their like before, zi was sure of it, but zis sleep-drunk memory was fuzzy on the details. Zi blinked zis displeasure, cursing zis now ancient body. The sleep had retarded the aging process, but still zi felt slow, used. Old.\n\nJirron had all the data, but zi needed more. Zi needed a new host.\n\nZi opened zis eyes. The nimfeach stared up at zim, quivering. I need a human.\n\nThere are two of them here. Its left wing dragged on the ground.\n\nBring them to me. Zi dismissed the nimfeach, turning away as it dragged itself from the room.\n\nThings had clearly changed. Zi needed to know how much before zi initiated the Great Migration.\n\nJAMESON DRIFTED.\n\nBits of memory\u2014his own this time\u2014flashed past.\n\nFinding a box full of memories in his mother's closet when he was little. Staring out at the red sands on Tander's World. Kneeling in church on Beta Tau, the white sunlight streaming through the gradient-phased windows to fill the room with colors like a kaleidoscope.\n\nThe first time he'd seen Xander.\n\nThe ground underneath him shook and subsided, bringing him out of his half-conscious state. His eyes flickered open.\n\n\"Hey, handsome.\" Xander stared down at him.\n\n\"Hey.\" He was lying in Xander's lap, the p\u00faca tree's big purple leaves fluttering in the ocean breeze. For a moment, he was wrapped in a perfect cocoon of time, safe in Xander's arms without a care in the world. He was so relieved that they'd found their way back to each other. With Xander at his back, he could handle anything.\n\nThen it all came back to him, slamming into his brain like a freight hauler. \"Did I sleep?\"\n\n\"A little. You were exhausted.\" Xander leaned down to kiss him. \"You feel better?\"\n\n\"A little.\" As they separated, Jameson sat up and rubbed his eyes, then looked around. \"I'm still tired. Damn, I thought we'd be done after all this.\" His city was in shambles, many of the beautiful white towers felled by the battle, others scarred with pulse fire. His people\u2014and how strange a thing that was to say\u2014were organizing themselves, clearing out a space on the square and setting up makeshift tents to take care of the wounded and feed the rest of the survivors.\n\nHis people and Xander's\u2014Gaelani and Eriani working together for the first time in decades. Something good had come from the battle and destruction after all. \"I need to be out there. With them.\"\n\n\"You're exhausted. You said it yourself. And soon we'll need to meet with the others\u2014\"\n\nJameson shook his head. \"They need me.\" He was done running. \"We can have our meeting, but right now my people need me.\"\n\nXander met Jameson's gaze, his green eyes narrowed. \"Are you sure...?\"\n\n\"You know I'm right. You did the same when the Gaelani needed you.\"\n\nXander grunted. \"I guess I did.\"\n\n\"Help me up. They may need me, but I need you, so I don't make a complete fool of myself.\"\n\nXander grinned. \"Not sure that's possible.\" He helped Jameson stand up.\n\n\"Asshole.\" Jameson felt dizzy, but after a moment his balance steadied. He closed his eyes, remembering the advice he used to give clients who were overwhelmed by their circumstances. Stop. Breathe. Let go.\n\n\"Damn, I really have been a bad influence on you.\"\n\nJameson opened his eyes and grinned. \"The absolute worst.\" His gaze lingered on Xander's beautiful muscular form. If I wasn't so tired. If we had more time. He sighed. \"Let's go.\"\n\nThey made their way down the little hillside and into the late afternoon sun. People bowed to him as he and Xander passed. \"Your Highness.\" A young Erriani woman with red wings and hair that shone like fire in the afternoon sun flashed him a dazzling smile.\n\n\"No need. Just Jameson is fine.\"\n\n\"Just Jameson.\" She nodded. \"I'll pass it on.\" She gave him a rather obvious once-over and hurried away.\n\n\"No, not 'Just Jameson!'\" he called after her, but she was too far away to hear.\n\n\"Get used to it, Your Lordship.\" Xander grinned. \"I think that one was considering whether you were looking for a queen.\"\n\nJameson snorted. \"Stop it.\"\n\nXander laughed. \"These people\u2014our people\u2014seem to be fond of their royal titles.\"\n\nJameson just shook his head. How did I get here? It still made no sense to him.\n\nThey reached the tents and found Mylin in charge. She threw her arms around Xander.\n\nXander grinned. \"You're a wonder.\"\n\n\"Thank you.\" She shrugged. \"Just trying to make myself useful.\"\n\n\"What can I do?\" Jameson looked around. The place bustled with activity. One canopy provided shade for a number of pallets that held wounded men and women, skythane and lander alike. Another seemed to be a material staging area, stacked with supplies, and a wonderful smell emanated from the third.\n\nShe looked him over with a critical eye. \"Grab a ladle over at the food tent, if you would. You look like you're running on fumes, and I don't want to be responsible for killing the king of the Erriani.\" She kissed him and Xander on the cheek and dashed off.\n\n\"Yeah, you're right, that's going to take some getting used to.\" They headed toward the food tent. \"How long until we need to leave for Oberon City for the big meeting?\"\n\nXander shrugged. \"Maybe an hour?\"\n\n\"Damn, I miss my cirq. Wanna help me serve some soup?\"\n\nXander laughed. \"I don't have any other plans.\"\n\nVestra Halta, the former regent of Errian, awaited them behind the hastily set-up tables. \"Welcome back to the land of the living.\" The old woman smiled and gave him a big hug.\n\n\"We did it!\" He lifted her up and twirled her around.\n\nVestra laughed in delight. \"Yes, we did. Or you did, anyhow.\"\n\n\"Vestra, you remember Xander?\"\n\nShe nodded. \"It's a pleasure to see you again, Gaelani.\"\n\nXander bowed before her. \"The pleasure's mine.\"\n\nShe blushed. \"I can see why you like him.\"\n\n\"He's not always this charming.\" He looked at the skythane who had assembled to feed the city and took a deep breath. \"Thank you all for doing whatever you can to get Errian back on its feet.\"\n\nThey cheered.\n\nJameson ducked his head and blushed. \"So, put me to work. What are we serving here?\"\n\n\"It's a hearty aux stew, with tubers and savory seasoning.\" She lifted a ladle up to Jameson's lips.\n\nHe sipped it. It was a multilayered feast\u2014hints of cinnamon and carrot and a meat that was both lean and full of flavor. \"It's delicious.\"\n\n\"Have a little before the rush comes. We're serving in about five minutes.\" She scooped some into an earthenware bowl and handed it to him with a spoon, and then handed one to Xander too.\n\nThey wolfed it down, talking with Vestra about plans to rebuild the city.\n\nThen the rush descended upon them.\n\nQUINCE STOOD at the edge of the Argent Sea, her bare feet in the sand. The waves crashed on the black rocks, cool over the tops of her feet, and the air was heavy with salt. She recalled that day twenty-five years before when she'd left here with Jameson and Xander, taking them away to a world wholly unlike their own.\n\nSo much had changed in the intervening years, including the aging of her poor bones.\n\nOverhead, a flock of imprean soared on the ocean breeze, calling out their particular carucaahh, carucaahh sound as they searched the water below for prey.\n\nQuince was sore from the activities of the last few days\u2014last few weeks, actually. She wanted nothing more than to slip away somewhere with Robyn, to find somewhere private where they could get reacquainted and leave all responsibility behind, but more was demanded of her. Of all of them.\n\nMost of all, she dreaded the conversation that was to come.\n\nShe had done what she thought was right. That was the fuckall hardest part. She'd had a world to save, and the nimfeach had told her it was imperative that Jameson and Xander be bonded. They had to get over their antipathy for each other, and the way they'd been at each other's throats the first few days\u2014she'd had no choice.\n\nBut what if I did? What if they would have come around to it on their own, without her intervention? Without dosing them with pith? It was funny, when she thought about it. Here they were at the end of the world, and she was freaked out about the boys' feelings.\n\nShe snorted. As her mother had often said, you can't undo done.\n\nNow she just had to figure out how to pick up the pieces and move on.\n\n\"You look troubled.\"\n\nQuince turned to find Robyn standing there, her black wings tucked neatly behind her, holding out a chunk of bread and some cheese.\n\n\"A little.\" She sighed and then smiled gamely. \"Just worrying about the boys.\"\n\nRobyn nodded. \"The boys? Or how angry they are at you?\"\n\nRobyn still knew her so well. \"A little of both.\"\n\nThey stood side by side, staring out at the water, the waves lapping at the sandy shore. From here, everything looked normal, as if Erro wasn't on the cusp of monumental, devastating change.\n\n\"Would you rather the world be destroyed to avoid a few hurt feelings?\"\n\nQuince looked at Robyn, calm and beautiful as the breeze rustled through her raven black hair. \"Of course not.\" She took a bite of the bread. It was a bit hard, but she was starving. The cheese was better, a sharp white corybryte, if she didn't miss the mark. \"They seem to have gotten past it.\"\n\nIf she had her way, they'd all move past it, at least until the crisis ended. Then she could take her lumps.\n\nShe'd seen the hurt in Xander's eyes, and in Jameson's. Maybe it would be better to have this out between them before they moved on.\n\n\"You're not alone in this.\" Robyn slipped her arm around Quince and embraced her. She was warm, alive. Real.\n\n\"I dreamed about this for so long.\" Quince buried her face in Robyn's neck. For just a moment, she was content.\n\nRobyn squeezed her tight, slipping her hand through Quince's hair.\n\n\"I need a bath.\"\n\n\"I don't care about that.\" Robyn held her out at arm's length, searching her face. \"You all right?\"\n\nQuince nodded, wiping her eyes. \"I will be.\"\n\n\"I know.\" She looked over her shoulder at the broken city. \"We should go. They'll be waiting for us.\"\n\n\"Yes.\" Quince didn't know if she was up to the tasks that lay before her.\n\nShe supposed she had no choice.\n\nJAMESON CLOSED his eyes, picturing Xander's storage unit once again. He'd been a whole different person the first time he'd been there. Quiet and bookish and closed off from his feelings.\n\nHe stretched out his arms and wings, willing himself to have the energy to do this one more time. He'd tapped his reserves and far beyond, and a short rest and a little food had come nowhere near to restoring his usual energy. It made no difference.\n\nNeed demanded this.\n\nI can do it. He pulled out the key from his pocket and held it up in the early evening light. It swirled like a living thing. Where did you come from? Who made you? He swayed on his feet.\n\nXander put a hand on his shoulder. \"You okay?\" His face was next to Jameson's, warm and rough with stubble.\n\nJameson nodded. \"Just tired.\"\n\nXander kissed his cheek. \"You can do this.\"\n\nAround them their friends waited for him. They whispered among themselves. Or was it just one of the memories that haunted him? He had them mostly under control, but sometimes they slipped out like stealthy rats, gnawing at the edges of his mind.\n\nJameson shook his head and concentrated on the task at hand. He fixed the place in his mind's eye\u2014the burnt and broken roll-down door, the scattered supplies, the collapsed wall at the back. The start of the journey that had ultimately led him here.\n\nHe had to learn some more places in Oberon City\u2014this one was starting to depress him.\n\nHe took hold of the swirling orb with both hands and twisted, and reality twisted too as a seam opened up in midair. It widened rapidly, and soon was wide enough for him to step through. The smells of Oberon City seeped through into the fresh Errian air.\n\nJameson took Xander's hand, and they crossed a third of Erro, from the open, blasted First Square in Errian to the equally gutted but much more compact storage unit.\n\nQuince and Venin followed, and then the rest of their impromptu council\u2014Vestra, Mylin, Robyn, Jessa, and Alia.\n\nJameson looked around. The place had been emptied out, probably by looters, and was little more than a shell. He touched the wall, wondering how he'd been so naive about the ways of the world when he'd arrived.\n\nWhen the last of his companions cleared the gateway, Jameson let it collapse, thrusting the key back in his pocket and wiping the sweat from his brow.\n\nOberCorp HQ was just a couple kilometers away. He was sure he'd be able to manage the flight, but he would have killed for Morgan's healing touch one more time. \"Everyone ready?\" He surveyed their group of companions, a mix of Erriani and Gaelani. The skythane were united once more, an alliance forged in fire.\n\n\"Where are we meeting? And why didn't the landers come to us?\" Vestra, the former regent of Errian, frowned, looking around the messy room and wrinkling her nose. \"This place smells like an aux's ass.\"\n\nJameson grinned. It did stink in Oberon City, especially at the ground level. He stepped outside to look around. It was late afternoon this side of Erro.\n\nXander answered, \"Because we're having everyone fitted with bioware and PAs so we can stay connected. If we're going to come out of this alive, we're going to need every advantage our two cultures can provide.\"\n\nVestra shook her head. \"I'm not sure I like the idea of one of those... what do you call them?\"\n\n\"Cirqs.\"\n\nShe smiled gratefully up at Jameson. \"Yes, cirqs. I don't think I want one in my head.\"\n\n\"We'll deal with that when we get there\u2014\" Something hot and bright flashed over his shoulder, slamming into the wall behind him with a loud crash.\n\n\"Bunch of those damned wing men over here,\" someone shouted from down the street. \"I told you we were being invaded!\" A ramshackle group of about fifteen people, civilians by the looks of things, came around a corner and stopped, staring at the group of skythane.\n\nThey brought up their weapons, an assortment of pulse guns and rifles and other odds and ends, and Jameson's blood ran cold.\n\nXander pulled out his own pulse rifle, waving it at the ramshackle crowd. \"Everyone back inside!\" He flipped it to stun and got off a few good shots before diving back behind the broken roll door.\n\nJameson had his out too, and he hit an attacker square on, forcing the whole group back.\n\n\"Nice shot.\" Xander dropped to the ground behind a pile of garbage.\n\nJameson settled in beside him, his own rifle aimed at the mob, shouting back to Quince, \"Out the back and take flight! We'll keep them pinned down until you're clear.\" He gestured urgently for them to go. \"Quince, you know where to lead them!\"\n\n\"You boys be careful.\" Quince popped back into the storage unit before a blast zapped through the space where her head had been. \"We'll get everyone else out.\" She led the rest of the skythane out through the mess in the unit behind them while Jameson and Xander lay down a barrage of cover fire.\n\n# CHAPTER THREE: ABOVE THE FRAY\n\nVENIN FOLLOWED Quince and the others into the sky, glancing back at the shrinking storage building as they ascended.\n\nXander and Jameson could handle things. He was sure of that. No one else had pulse weapons, though, so it was best to get everyone else out of harm's way.\n\nJameson had looked exhausted. He wasn't likely to find rest anytime soon, not with the new crisis already upon them.\n\nAs they rose on the warm updraft from the city streets, he got his first look at Oberon City. The place was sharp and ugly.\n\nThe streets below were full of people. Protesters waved signs and chanted, and small groups of armed men like the one that had attacked them roved the city.\n\nThere was a stink in the city, a metallic stench mixed with the smell of human waste. Too many people crammed into too little space. How could the landers live this way?\n\nStill, it was clear that things were worse than usual here. One of the huge buildings\u2014arcos, Xander had called them\u2014leaned at an angle away from the others, and plas windows were broken in many places on all of them.\n\nOberCorp's headquarters lay just ahead, beyond the blue arcos.\n\nThe building glimmered in the evening moonlight that sparkled on the lake that surrounded the imposing edifice. A number of panes of plas were cracked or missing there, too, victims of the megastorm that had assaulted the world after the shift and the short war that had followed.\n\nStill, it had fared better than the surrounding city.\n\nAlix had promised to be ready to install cirqs on everyone, giving them access to the Oberon City grid.\n\nVenin shook his head. He was about to let OberCorp stick a tracker in his head. What would you think of me now, Taz? Consorting with the enemy?\n\nWhen you live with auxen, become an auxen.\n\nTaz's voice came unbidden across the shores of time. Venin winced. He had no wish to relive that particular memory.\n\nThe night was beautiful and warm, the air up here much fresher as it blew in off the sea\u2014belying the coming crisis. As they climbed toward the rooftop, Quince's powerful wings brought her even with Venin. \"Do you think we can trust Alix?\"\n\nVenin nodded. \"He could have come between Xander and Jameson, but he stepped back where a weaker man would have thought only of himself. I'm as sure of it as I am of anything. He saved Errian too.\"\n\n\"With a little help.\"\n\nVenin laughed. \"None of us could have done that alone.\" His wings beat steadily as she followed a warm updraft toward the roof.\n\nQuince looked around at the party. \"Everyone still with us?\"\n\nRobyn was carrying Jessa in her arms, her wings beating with the extra effort to pull the two of them upward. Vestra was keeping up, though she looked tired.\n\nWith a last surge of effort, they cleared the roofline of the headquarters.\n\nAlix was waiting for them on the hoversport platform. Venin grinned. He was glad his travel companion was safe and sound after the battles they'd been through. He alighted next to Alix. \"Good to see you again.\"\n\nAlix held out a hand, but Venin pulled him in for a strong hug. \"Well met. Thanks for everything you did.\"\n\nAlix's face was flushed when they separated. \"Just doing what I had to. I'm not the only one.\" He looked tired, but nicely shaved and cleaned up, wearing a camo ranger's uniform and a red beret. \"Where are Xander and Jameson?\"\n\nQuince growled. \"We ran into a little trouble at the gateway. I expect them along soon.\"\n\n\"Gotcha. Nothing they can't handle?\"\n\n\"They have it covered. You know everyone?\"\n\nAlix nodded. \"Come inside. We've prepared something to eat if you're hungry, and we can get to work on those cirq installs while we talk.\" He looked from Quince to Venin for confirmation.\n\nVenin's stomach grumbled. He was hungry. Maybe he could manage a seat next to Alix so they could catch up. \"Lead on.\"\n\nALIX LED the others into OberCorp's headquarters. It was strange to act as the guide here. His whole recent life had been on the other side of the world, outside as often as in. He didn't belong here.\n\nAnd yet here he was.\n\nVenin's touch lingered on his shoulders, the masculine smell of him in his memory. He hoped the man didn't have the wrong idea. They'd been thrown together in the midst of a war\u2014there were bound to be intense feelings from that encounter.\n\nHe wasn't ready for anything yet. For anyone. Xander's rejection still stung, though he had made an accommodation with it. Was making one. Would make one someday. Xander was no longer his.\n\n\"The lift shaft will take us down a couple floors to the offices I've had set up for us. We'll meet in one of the conference rooms, and I have offices and sleeping quarters prepared for everyone who stays.\"\n\nVenin looked concerned. \"What's a lift shaft?\"\n\nVestra nodded. \"I was wondering that too.\"\n\n\"It's like an elevator.\"\n\n\"Elevator?\"\n\nAlix laughed in spite of himself. \"It's a shaft of air that will carry us down inside the building.\"\n\n\"Ah.\" Venin grinned. \"Why didn't you just say so?\"\n\nAlix shook his head. \"It's called a lift shaft... never mind.\" It was going to take his skythane guests a while to get used to life in the big city. \"Just step inside and it will carry you down to where you need to go.\"\n\nVestra shook her head. \"You could have just opened a window for us.\"\n\nBirds. I'm living with Split-cursed birds.\n\nVenin shrugged. \"Down we go.\" He jumped into the shaft.\n\nAlix followed.\n\nThe man's jet-black wings shone with hints of blue in the bright lights of the drop shaft as they passed several rings indicating the floors.\n\nAlix had a thing for wings. He'd spent his childhood staring out his window, wishing he were one of the skythane. His bi-wings had given him the glory of flight, but it was still a half measure.\n\nHe shook his head. There were more important things to think about than childhood whimsy.\n\nLike this new threat. Quince and Jameson had been cryptic at best. Alix had hoped to put the war behind them, to start to pick up the pieces and figure out how to spin their world back to the human universe where it belonged. Instead they faced a new crisis.\n\nAs they reached their floor, Erissa chimed in his ear. \"Alix, your mother is on her way to the medical floor.\"\n\n\"Oh hell, what happened?\" He stepped back onto solid ground and moved a few feet away from the others.\n\nVenin mouthed Everything okay?\n\nHe shook his head. \"She was found on the floor of her apartment.\"\n\n\"Fucking Split.\" He looked up, spotting one of the tech staff. \"Derren, I've got to run to medical. Can you take the others to the conference room?\"\n\n\"Got it.\"\n\nGood kid. A bit cerebral. So damned young. \"Thanks. Folks, I'll be back shortly.\"\n\nHe excused himself and hopped back into the drop shaft. \"Erissa, take me to medical.\" What in the Split did you do?\n\nAlix arrived in the medical suite to find his mother under sedation in a crisp white room with no windows. \"I'm Alix Preston. They called and said my mother was here?\" OberCorp had its own state-of-the-art medical facilities, as good or better than anything else on-planet.\n\nThe doctor was a young woman with umber-toned skin, rare anymore in the Common Worlds with all the mixing of the last thousand years. He wondered if she was from one of the African diaspora worlds. \"I'm Dr. Glynt, Teva Glynt.\" She was all business.\n\n\"Thanks, Dr. Glynt. So what happened?\"\n\nShe pulled him out of his mother's room gently. \"We think she overdosed on sleepers.\"\n\nHe shook his head. \"She wouldn't. How is that even possible?\" Lena Preston was not the suicidal type.\n\nGlynt shook her head. \"She overrode the system, somehow.\"\n\n\"Goddammit.\" He pounded the wall, startling a passing nurse. \"Sorry. My mother's unconscious.\"\n\nThe man nodded and hurried away.\n\n\"Is she okay now?\"\n\n\"She should be. We flushed her system with kelex. It bonds with the active drug of the sleeper meds and takes it out of her system. She should be awake in a couple hours.\" She squeezed Alix's shoulder gently. \"Any idea why she would do this?\"\n\n\"It's her way of getting my attention.\"\n\n\"Ah.\" The whole company was aware of the spat between the former OberCorp CEO and the present one. \"I can call you when she wakes up.\"\n\n\"You're sure she'll be fine?\"\n\n\"Positive.\"\n\n\"Give me a minute with her?\"\n\n\"Of course.\"\n\nAlix reentered the room. His mother lay under a white sheet, still as death.\n\n\"I'm sorry, Mom. Things have been a bit crazy.\" He squeezed her hand. They had issues between them. A whole lifetime full of them. But still, she was his mother.\n\nHe leaned over and kissed her forehead, the beep beep beep of the machines filling his ears. \"I'll be back soon.\"\n\n# CHAPTER FOUR: NO PLACE LIKE HOME\n\nMORGAN SLIPPED into Freyyr unnoticed. The cavern under the mountain was a flurry of activity as the nimfeach worked to awaken the sleeping ithani.\n\nIt wouldn't be long now.\n\nHe wondered where the Thshnel'Jirron was hiding.\n\nPart of him felt compelled to join the other nimfeach in their single-minded task, enslaved as they were by the ithani will.\n\nBut his human half resisted.\n\nAfter a few minutes, he pulled himself away. He had seen all he needed to.\n\nNow he had plans to make.\n\nHe opened a gateway and stepped through, and in an instant was gone.\n\nXANDER DIDN'T like being pinned down.\n\nShots slammed into the door and the walls near Xander and Jameson, sending a shower of debris down on their heads.\n\nXander popped up and got off a shot, then sank back down cursing next to Jameson. \"That's the last of the charge on this one.\"\n\nJameson peered over their cover and dropped back as a pulse blast zipped past, singeing his hair. \"Theirs are not set to stun. Fall back. We can go out the other side after Quince and the others\u2014\"\n\nVoices rang through the combined space of the two storage units from behind. \"They're in here!\"\n\n\"Fucking Split.\" Xander spat on the ground. \"They're behind us too. I don't suppose you have enough left in you to open another gateway.\" He shot another pulse blast out the back, hoping to make their ambushers keep their heads down for a moment or two. \"Reinforcements should be here soon!\"\n\n\"If only.\" Jameson laughed and then grimaced. \"I can try.\" He pulled out the key, closed his eyes, and looked very focused.\n\nNothing happened.\n\n\"I'm tapped. Sorry.\" His voice came out thinner than he liked.\n\nXander grunted. \"Not your fault. You've pushed yourself past the limit.\" He looked around wildly. \"How else can we get out of here?\" He hated not having a preplanned exit.\n\nHe popped up and fired a pulse blast over the wreckage to keep their attackers on their toes, and another toward the back. \"Come on back inside.\"\n\nJameson nodded and followed him on hands and knees. \"Here.\" Jameson held out the key to him.\n\n\"I don't know how to use it.\"\n\n\"Hang back. Tennir and Ziph will be here in five with some grenades,\" one of their hunters said to the others, not five meters away if Jameson judged it right. \"We can smoke 'em out then.\"\n\n\"I can show you how.\"\n\nXander searched Jameson's eyes, doubtful. \"Okay. Let's try it.\" He glanced over his shoulder. Their attackers seemed to have pulled back.\n\nJameson touched Xander's face. \"I don't know if this will work. Close your eyes.\"\n\nXander did as he was told.\n\nJameson put the key in his palm. \"Quince thinks we can all learn to use these things. That we have their blood in our veins.\"\n\nXander shuddered.\n\n\"Picture a place we can go. Someplace you know well.\"\n\n\"My apartment. Feels like a lifetime.\"\n\n\"I can imagine. Lock the image in your head.\"\n\n\"Got it.\"\n\n\"Now reach inside the key with your mind and twist it. Like this. Then push!\" Xander's hands were warm in his.\n\nSomething clattered into the room. Jameson's eyes popped open. It was a small metal canister, and it started spewing smoke. \"Push!\"\n\n\"I see it! I've... done this before.\"\n\nJameson nodded. \"The memories.\"\n\nXander grunted. He did as Jameson had shown him and pushed.\n\nAnother something clattered into the storage unit from behind, and the locker filled with black smoke.\n\nThen the air split between them and the grenade.\n\nThe cloud rapidly filled up the small space, burning Jameson's eyes and skin. He stumbled to his feet and pulled Xander with him. Together they staggered half-blind through the gateway.\n\nAs they cleared the gateway, a pulse blast shot through over his shoulder to slam into the mirror above Xander's bed, cracking and melting the glass. \"Shut the gate!\"\n\nXander let go of the key, sending it clattering across the floor.\n\nThe smoky room and the astonished face of the lander man peering through from the other side vanished in a flash.\n\nJAMESON TOOK a deep breath of clean air and looked around at the apartment. \"So this is where you live, huh?\" That had been close. Too close. And yet here they were. \"We should probably get going to the meeting?\" He set his carry sack down on the bed and looked around.\n\n\"Just want to grab a few things. Hop in the ionic shower while you're waiting.\"\n\n\"Oh God, yes. Is there time?\"\n\n\"Make time. We need to get that smoke off of our skin, for one thing, and besides, you stink to high heaven and Erro.\" He palmed open the door to the small bathroom.\n\nJameson peeled off his clothes. The last time he had changed had been... when? He couldn't remember. It had been days, at least.\n\nHe sniffed his pits. Yup, he was ripe.\n\nThe apartment was about what he had expected, a bachelor pad\u2014lots of black and gray and silver, clean lines, not much in the way of decoration. A huge tri-dee table took up half of the living room, and plas picture windows looked out on early evening in Oberon City.\n\nOne thing he hadn't expected\u2014the drawers had all been pried open, and clothing, books, and other riffraff lay strewn across the floor. \"Looks like you were robbed.\" He climbed into the ionic shower. It was a large one, probably because of Xander's wings. He palmed the controls.\n\n\"Access code, please.\" The voice seemed to come out of thin air.\n\n\"Must be your PA?\"\n\nXander popped his head in. \"Yup. Say hello to Ravi. Fusion uranus centauri kepler interstellar neptune galaxy dash binary alpha saturn telescope alpha rocket doppler. Confirm.\"\n\nJameson worked that out and laughed. Sounded about right. As a kid, he'd had those callsigns almost beaten into his head in school.\n\nXander grinned and gave Jameson the once-over. \"Oooh. You look\u2014\"\n\n\"Access code confirmed. Welcome back, Xander.\"\n\n\"Hey, Ravi! It's so good to hear your voice!\" Xander grinned at Jameson. \"Jameson, meet Ravi, my PA.\"\n\nJameson laughed. \"Hey, Ravi. You two gonna let a guy take a shower in peace, or what?\"\n\nXander winked at him and disappeared around the corner.\n\nJameson was a bit jealous. He had no way to access his own PA, let alone the grid.\n\nThe shower began; a whirlwind of warm air surrounded Jameson. Oh gods, that feels good. He'd have to get one of these installed in the Castain in Errian when they rebuilt the city.\n\nWhen he was done, he was delighted to find fresh clothing waiting for him.\n\n\"Ravi took your measurements in the shower.\"\n\nJameson got dressed, admiring how the black faux-leather fit his form. \"We're getting cirqs at this meeting, right?\" At least then he could see if his PA had managed a local backup before he'd been forced to short out his bioware. He sank down on the bed, wishing he could take a nap, but there was no time for that. \"Almost ready to go?\"\n\n\"Yup. Ravi, mix me up a little stim for Jameson here, and open the safe.\" Xander laid his own carry sack on the kitchen counter.\n\nA blank patch of wall next to where Xander had put his palm opened, exposing a meter-tall space behind. Xander pulled out a couple fresh pulse rifles, tossing one to Jameson. \"Here, dump that old one.\"\n\n\"What are you, an arms dealer?\"\n\nXander laughed bitterly. \"Oberon City can be a dangerous place, and after my time with Rogan, I was always ready to run.\"\n\n\"Stim ready.\" A small hatch slid open on the kitchen counter, holding a cup of something.\n\n\"Thanks, Ravi.\" Xander handed it to Jameson. \"Here. Drink this.\"\n\nJameson took the cup and sniffed it suspiciously. It smelled like strawberries. \"What's in it?\"\n\n\"I have no idea, but it does wonders. I used to take it after waking up with a hangover.\" He opened one of the kitchen cabinets and stuffed food packs into his carry sack.\n\n\"Will it make me feel even worse when it wears off?\"\n\n\"Not really. It boosts your own system and has a shitload of protein and electrolytes, along with whatever else Ravi puts in there.\"\n\nJameson shrugged and drained the cup. It was bitter, but as it hit his stomach, a warm glow suffused him. \"Oooh. That's nice.\" He could get used to being back in civilization again.\n\nXander zipped up his carry sack. \"My turn.\"\n\nJameson watched him undress, wishing they had more time for that too. They'd totally missed their honeymoon.\n\nHe lay down on the bed and closed his eyes. It felt absolutely marvelous to lie on a real bed, clean and dressed in new clothes.\n\nFar too soon, though, Xander sat down on the bed and woke him from his slumber. \"You snore up a storm, you know that?\" Xander pulled on a pair of riding boots. \"Have to get this place back in order once we return.\"\n\n\"If we return. We do have a couple castles, too, you know.\"\n\nXander grinned. \"Kings of the world.\" He kissed Jameson, and Jameson's body responded.\n\nXander pulled away and glanced down at Jameson's crotch. \"Wish we had time, believe me. This bed's great for sex.\"\n\nJameson's wings shivered. \"Goddamned end of the world.\"\n\n\"Yup. Feel better?\"\n\nJameson nodded. \"A little. I feel awake again. The fatigue is mostly gone.\" It wasn't as good as Morgan's little trick, but it was close. He could function.\n\n\"Ravi? My bike?\"\n\nThe floor opened up, and Xander's hoverbike rose into view. \"How...?\" They'd left it at the House of the Sun when they'd crossed over to Titania.\n\n\"Auto return. All cleaned up too.\" One of the plas windows slid apart. \"Wanna ride?\"\n\n\"I thought you'd never ask.\" Jameson mounted the bike and slipped his arms around Xander, ready to take on the world.\n\n# CHAPTER FIVE: AN UNEXPECTED VISITOR\n\nVENIN STARED at the sterile conference room, his lips pulled back to show his teeth. He was no \"Outlander barbarian,\" as he'd heard one of the OberCorp employees whisper behind his back.\n\nThis place was cold and lifeless, and smelled of chemicals and fear.\n\nA long silver conference table bisected the room, surrounded by black chairs that were too shallow with backs too high for skythane.\n\nHe would stand.\n\nThe view from the lander building, however, was spectacular.\n\nOberCorp headquarters was taller than the tallest tower in Errian. This side faced westward, over the Gildensea, where Bandia seared a silver pathway across the wave tops.\n\nHe closed his eyes, remembering the strange pain that had flooded his body when he'd touched the statue in the Erriani chapel. What in Gael's grace was that about?\n\n\"Sorry we're not ready for you.\" Alix slapped him on the back. \"I'll have proper skythane chairs brought in before we get started.\"\n\nAt least the man had noticed his discomfort. Venin wrinkled his nose. \"It stinks in here too.\"\n\n\"That I can do something about right now.\" Alix cocked his head. \"Erissa, open the windows a little.\"\n\nThe clear windows around the conference room lowered about a third of a meter, and fresh air flowed in. Well, fresh for the city.\n\n\"Like magic.\" Venin breathed in the cool air deeply, gratefully. \"Better.\" He'd come to respect Alix over the last few days as they'd traveled together from the ruins of the Mountain all the way to Errian and the Argent Sea. The man was solid. A good fighter. A decent guy, too, judging from how he'd managed to navigate things with his ex, Xander, and Jameson. And handsome as fuck with his square jaw and military cut. Venin grinned.\n\n\"I know how you feel. I'm not used to being cooped up in places like this anymore.\" He had a wistful look on his face.\n\n\"So why stay?\"\n\nAlix shot him a strange look. \"I have responsibilities here. People who need me. I am the boss of OberCorp now.\"\n\nVenin was unimpressed. \"Not much of a boss if you can't leave whenever you want to.\"\n\nAlix snorted. \"Fair enough. Why don't you leave?\"\n\n\"I'm not the boss.\" Venin grinned. \"What happened earlier?\"\n\n\"When?\"\n\n\"You had to leave us rather abruptly.\"\n\n\"Ah.\" Alix nodded. \"My mother... she was taken to medical.\"\n\n\"Everything all right?\"\n\nAlix looked tired, fine lines apparent by his eyes. He sighed. \"It will be. She's... not happy with me for pushing her out.\"\n\n\"I can imagine.\" These landers were ruthless, vicious even to one another at times. Not that the skythane were angels. Not literal ones, anyhow. \"I'm sorry\u2014\"\n\nAn OberCorp employee dressed in white touched Alix's sleeve. \"Mr. Kinnson and Mr. Havercamp have, um, arrived. On a hoverbike.\"\n\n\"Thanks, Gregg.\" Alix flashed an embarrassed grin at Venin. \"The staff's not used to having guests arrive so informally. Excuse me.\" He squeezed Venin's arm and then retreated to the exit.\n\nVenin watched him go, his arm tingling where Alix had touched him. The rangers who had occupied Gaelan had been cast from a similar mold, but they'd been the enemy.\n\nAlix had been one of them.\n\nVenin shook his head, whistling softly. The man was a lander. They would never work as a couple.\n\nAnd yet, even landers have needs.\n\nQUINCE STARED impatiently at the wall as her new cirq and wrist plate were installed. This room was classic corporate. The wall screen displayed a generic picture of some color-enhanced nebula or another, and the industrial carpet was about the beigest shade she had ever seen.\n\nShe was ready to get her access back. She'd missed it\u2014the connection to the planet's grid, the easy access to almost anything she needed to know. She'd spent twenty-five years exploring the virtual world that wrapped around the real one, and with those tools at hand once again, she'd be able to formulate the best plan for dealing with the threat the ithani posed.\n\nShe certainly missed Ari. She'd have to access the heavily encrypted copy she'd slitted to one of the nether recesses of the grid. Her PA was like an extension of herself, a friend who was always at hand, sunshine or darkness.\n\nAnd yet....\n\nThere was something to be said for disconnecting, for letting go of the constant news feeds and updates and the flood of information she'd once been bombarded with. Information overload.\n\nShe'd been happier out in the wilderness with Robyn, in a way. Cut off from everyone and everything else. She looked up at Robyn, and her partner squeezed her hand.\n\n\"Installation completed. Your system should self-organize and should boot up in about ten minutes.\" The med drone pulled away, hovering a couple feet away from her. \"Do you feel any discomfort?\"\n\nQuince reached up to touch her temple. The skin was smooth there, as if she hadn't been cut at all. \"No, thank you. The anesthetic's quite good.\"\n\n\"Take these if you should feel any pain. One tablet should last twelve hours.\" A slit opened up in the shiny silver sphere, and a small packet was dispensed.\n\nQuince took it, nodding at the drone. \"Thank you.\"\n\n\"No thanks are necessary. Who is the next patient?\"\n\nQuince looked at Robyn again.\n\n\"Oh no. I don't want one of those... things inside me.\"\n\nQuince frowned. \"It won't hurt you, and it will make it easier to find you if we get separated.\"\n\nRobyn backed away so quickly she knocked over one of the chairs. \"No. I don't want it.\"\n\n\"Hey, it's okay. You don't have to.\" Quince stood, looking around the room. \"Who's next for a cirq installation?\"\n\n\"I'll go.\" Mylin stepped forward, her face determined although her wings trembled a little.\n\nQuince looked at the young woman appraisingly. She'd proven herself adept at organization and problem solving, quite an impressive showing, really. It would be interesting to see what she would do with the capabilities a cirq and grid access would provide. \"Come have a seat. It doesn't hurt.\"\n\nMylin came forward and sat where Quince had been. \"What does it feel like?\"\n\n\"The installation?\"\n\nMylin shook her head. \"The cirq.\"\n\n\"It's... it's like having another world inside your head. Or the keys to the biggest library you could imagine.\"\n\nMylin smiled faintly. \"That doesn't sound so bad.\"\n\n\"Go ahead,\" Quince said to the drone.\n\nThe sphere approached Mylin and extended an arm, spraying her temple with anesthetic.\n\n\"Oooh, that's cold.\"\n\nQuince held her hand while the procedure commenced. \"You'll hardly feel a thing.\"\n\nTwo minutes later, the drone backed away. \"All done. Do you feel any discomfort?\"\n\nMylin's eyes went wide. \"That's it? No, I'm fine, thanks.\" She looked at her wrist.\n\nThe drone spun back and forth in midair. \"Who's next?\"\n\nVenin stepped up. \"That would be me.\"\n\nQuince nodded and squeezed Mylin's hand tight. \"See? Nothing to worry about. When you have a few free moments, we'll boot her up and get you started.\"\n\nShe guided the girl to the table just as Xander and Jameson came in. Behind them, a couple OberCorp employees started bringing in suitable chairs, with low backs to allow skythane wings proper support.\n\n\"Hey!\" Jameson's face lit up when he saw her. Xander stood back, brooding. She could guess what that was about.\n\n\"Everything okay?\" Quince hugged Jameson, grateful they had made it back safely.\n\n\"Nothing we couldn't handle. Xander made his first gateway.\"\n\n\"To where?\"\n\n\"His apartment.\"\n\nQuince snorted. \"I should have known.\"\n\n\"Nothing happened. We were in a rush.\" Jameson glanced back at Xander. \"Not that I'd have minded. Have you seen his bachelor pad?\"\n\n\"Oh yes. Many times. I called it the man trap.\" Not that she held it against poor Xander, after he'd lost Alix and all, but he had gone through quite the phase.\n\nHer system booted up. \"Install existing personal assistant?\"\n\n\"Just a sec.\" She turned away from Jameson. \"Access code centauri neptune one seven three two neptune interstellar galaxy.\"\n\n\"Code accepted. One moment while I load the personal assistant from the specified location.\"\n\nQuince looked around the room. All the principals were there. She wished she had more information for them beyond the little bit Morgan had told her. Ithani. Three more days.\n\n\"Hello, Quince.\" Ari's voice practically purred in her ears.\n\nQuince grinned. \"Hey, Ari... I've missed you.\"\n\n\"Likewise.\"\n\nShe'd forgotten what a comfort it was to have Ari in her head, at her beck and call. He was her PA, but even though he was nothing more than a specialized bit of code, he'd become something more than that. A friend. \"I'm sorry I had to shut you down like that.\"\n\n\"I have no memory of that.\" His voice sounded hurt. \"This copy was made before it happened.\"\n\nShe blushed. \"Nevertheless. Are you interfaced with OberCorp's systems?\"\n\n\"I will be shortly. What do you need?\"\n\nShe explained briefly, and Ari sent off gridrunners to collect what she had asked for. Quince cleared her throat. \"Okay, everyone's here. Shall we get started?\"\n\n\"ALL DONE,\" the drone said softly to Jameson. \"It will take about half an hour to load in the last backup.\"\n\n\"You have a copy of my PA on file?\" Jameson breathed a sigh of relief.\n\n\"Yes. Do you feel any discomfort?\"\n\n\"No, thanks.\" Jameson waved it away. He looked at his wrist. The skin was seamless.\n\n\"Please contact DOC if you have any trouble.\" It moved on.\n\nThe assembled skythane and landers were taking seats around the table. Jameson sat across from Xander, who reached out to squeeze his hand reassuringly. There was no place to put down their carry sacks. He shrugged at Xander, and they put them under their chairs.\n\nHere he was at last at OberCorp HQ, the place he thought he was coming to that first day, just a few weeks before.\n\nIt seemed like a lifetime.\n\nQuince stood, looking around at everyone. She looked grim. \"Thanks to everyone for coming. I have a lot to tell you, but I'll start with\u2014\"\n\nThe door burst open, cutting off Quince midspeech.\n\nJameson looked up to see Rogan standing there, out of breath, staring accusingly around the room. \"Started without me, I see?\"\n\nXander leapt to his feet. \"What the hell is he doing here?\" His face was red with anger.\n\nRogan grinned. \"Hello, pretty thing.\" He turned to address the others. \"I'm a part of this little alliance, am I not?\" He pulled one of the lander chairs up to the table. \"Besides, I'm still owed payment for my help. I was afraid you might have forgotten.\"\n\n\"Alix....\" Xander shot his ex a pleading look.\n\n\"I don't like it any more than you do, but we did promise him the pith.\" Alix shot a black look at the Slander boss. \"But we may need him. It's all hands on deck time. We can settle things later.\"\n\nXander sank back down into his chair, not taking his eyes off Rogan.\n\nAlthough he was pretty sure no one else noticed, Jameson saw Xander shudder. He put his hand on Xander's. It's my own damned fault. He'd invited the man into their lives, and now they were stuck with him until the crisis was over. \"Keep him in line,\" he hissed, \"or I swear to the gods I will fucking end him.\"\n\nEveryone turned to look at him, and even Xander whistled appreciatively. \"Damn.\"\n\nRogan, infuriatingly, just grinned wider. \"I'd like to see you try, little wing boy.\"\n\nHalf the table rose at that insult.\n\n\"Enough!\" Quince slammed her fist down on the table. \"We're short on time and fighting amongst ourselves won't do us any good.\" She glared at Rogan. \"He stays. But not another word out of you.\"\n\nRogan made a seated imitation of a curtsey. \"As you wish, milady.\"\n\nIn a flash, she had her pulse pistol at his temple. \"Not. Another. Word. Understand?\"\n\nRogan's face went white. He nodded.\n\nQuince tucked the pistol into her belt.\n\nWhere the hell did you get that? Jameson's respect for her went up another notch.\n\n# CHAPTER SIX: OFF THE RAILS\n\nTHERE WAS pain, pain like Dani had never endured before, coursing through every nerve in her body. Pain that left her exhausted, whimpering between waves of agony, like someone had turned her bones to fire.\n\nThen the dark, bleak emptiness that would last for hours. Or maybe days. She couldn't tell. She was so disconnected from life, from light, from the simple intake of breath that she forgot who she was, even what she was, for long stretches of time.\n\nSomething waited in the darkness with her, something just out of reach. She could hear it shuffling in the inky blackness, feel its fetid breath on her neck from time to time.\n\n\"Please,\" she begged the darkness. \"Please let me go. I'm sorry. I shouldn't have come. I will do anything. Just let me out!\" The last part came out as a scream that tore her throat, but whatever was there with her didn't respond.\n\nShe wasn't even sure she was awake, most of the time. If this place-that-wasn't-a-place existed in her old world, or if she was trapped in her own head.\n\nShe closed her eyes, trying to calm herself. \"It's not real. It's not real. This is all just a bad dream. Wake up. Wake up. Wake up!\"\n\nShe opened her eyes.\n\nNothing had changed.\n\nIt was still dark. Pitch-black.\n\nSomething wormed its way into her mind.\n\nShe screamed, and the pain began again.\n\nVENIN GLARED at the Slander boss. He'd heard stories, none of which were good.\n\nRogan returned his glare but broke contact first.\n\nQuince cleared her throat. \"As I was saying, a week ago, Robyn and I flew north to check into something that had come to my attention. Ari, globe, please?\"\n\nA slowly spinning sphere appeared above the table.\n\nVenin whistled. They had some great toys here. It was amazing to see his world this way.\n\nAlix grinned at him.\n\n\"This is everything we know about the combined world\u2014Erro. Dani and her forces had taken some detailed measurements, so it's fairly complete on the Titania side, and the landers, sorry, our landbound friends\"\u2014she nodded to Alix and the other OberCorp employees in the room\u2014\"mapped out Oberon long ago. There are a few zones we don't know much about, particularly in the north and south.\"\n\nThe demarcation where Erro had been split was clear, a ridgeline that ran between the mountains on one end of the world and bisected the Argent Sea and Gildensea on the other. Venin supposed they must have been part of one larger ocean long ago, before the split. One side of the globe was blank in the far southern and northern hemispheres. He traced his hand across the glowing surface. Amazing. They'd traversed all that distance, much of it under their own power.\n\n\"That's all very pretty.\" Rogan sat back in his chair, wrapping his hands behind his head, his fat stomach sticking out under his shirt. \"But what's the immediate threat? I thought we'd taken down OberCorp already.\" He glanced around the room. \"Obviously.\"\n\nQuince growled. \"I'm getting to it. I wish I'd had my cirq when we learned about the latest threat. I could have shown you the memfiles.\" Her hand slipped over the hilt of her pistol, and the import of the casual action was clearly not lost on Rogan. \"Ari, North Pole file, please.\"\n\nThe globe vanished, to be replaced by a fuzzy picture.\n\n\"This was one of the last images captured from Titan Station before we lost contact.\"\n\nIt showed the curve of the world and a large peak that jutted up from an icy plain.\n\n\"Sorry, the resolution's a bit grainy. Suffice it to say we're not alone on Erro any longer\u2014if we ever were. That's where they live.\"\n\nVenin frowned. \"Who?\"\n\n\"The ithani.\" Vestra's face had gone white.\n\n\"Yes.\" Quince frowned. \"How did you know?\"\n\nVestra shuddered and pulled her shawl over her shoulders, drawing it close to her as if to ward off the cold. \"The Erriani have many legends. Some of them have almost been lost to the passage of time. We have... more of a history with Titania than the Gaelani.\"\n\nQuince nodded. \"Go on.\"\n\n\"When the skythane first came to Oberon, almost eight hundred years ago, the Erriani were the ones who settled Titania, while the Gaelani remained here in Oberon. There was a young skythane woman named Thesbe who claimed to have been taken by a sneach and carried off to the home of the ithani, who... did things to her before she finally escaped and made her way home.\"\n\n\"Things?\" Venin didn't like the sound of that.\n\n\"Yes. Tests, maybe. Experiments. That was just one of the stories. Some tales suggest the ithani are devils\u2014you should never eat any food they provide lest you be trapped in hell. Others called them angels, beautiful beyond bearing.\" She snorted. \"One thing they all agreed upon. Never seek them out. Never visit the far north.\"\n\nIn the middle of the night\n\nafter dark when auxen slumber\n\ncome ithani, faces bright\n\nand make the village quake with fright\n\nwhose children will they steal tonight?\n\nVestra shuddered. \"Sneach, we called them sometimes. Those that sneak around in the dark.\"\n\n\"She's right. My grandmother was tricked by a sneach once in the north woods. Lost her bow and arrow.\" Venin laughed. \"She was spitting teeth about it for weeks.\"\n\nQuince nodded. \"We had similar proscriptions against the north, but I always took them for tales to frighten small children.\" She stared at the image of the mountain. \"It seems they may have held a grain of truth. The ithani have been engaged in some kind of selective breeding program with the skythane for centuries.\"\n\nVenin shuddered at the thought, and there was an uproar around the room. What was she saying? That we're a big experiment?\n\nQuince pounded the table again. \"Look, we're short on time. We need to move on.\"\n\n\"What are these ithani? Did you see one?\" Alix scratched his new stubble on his chin, looking thoughtful.\n\nVenin caught himself staring at the lander man. He liked that Alix kept his cool during a storm. He looked away, his face hot. I'm like a fledgling in heat.\n\n\"They were the race that lived here before us. More than a hundred thousand years ago, I think. The ones who split Erro\u2014an accident, or maybe sabotage?\" Quince stared at the globe. \"There was a war, a terrible war, between the ithani and a race called the dhagani. I don't know much more than that. But their battles destroyed whole worlds. It was a dhagani weapon that poisoned Erro's sun, making it so unpredictable.\"\n\nVenin could almost smell the fear on her. He'd seen how strong she was. Quince didn't frighten easily. This must be really bad.\n\nAt last she went on. \"When we saved Oberon from the flare, we inadvertently woke the sleepers. Shifting Oberon here completed the process the ithani had started so many years ago and set something new in motion. Robyn and I\u2014\"\n\n\"How exactly do you know this?\" Rogan stared at her, his brow furrowed and hands pressed together as if in prayer. \"Alix asked a good question. Did you actually see one of these... what were they called... ithani?\"\n\n\"Well, yes and no\u2014\"\n\n\"Do we even know for certain that they exist?\"\n\nThe man's voice grated on Venin, and Jameson actually growled. He shouldn't be here. Xander looked like he was about ready to leap across the table and slit Rogan's throat.\n\nJameson touched Xander's hand and shook his head.\n\nXander subsided, but he didn't look happy.\n\nQuince, however, took it in stride. \"Yes. Morgan showed me. We only have three days, at most\u2014\"\n\n\"Who is Morgan?\"\n\n\"He's a nimfeach.\" Quince's face was turning an ugly purple.\n\nVenin leapt to his feet, towering over the Slander boss. \"That's enough. If you're going to be a part of this meeting\u2014and I, for one, would have no trouble throwing you out that window\u2014you will remain quiet and respectful while Quince speaks.\"\n\nThe man paled, and Quince glared at him.\n\nWhat? What did I do? He sat down sheepishly.\n\nAlix was grinning.\n\nJameson stepped in. \"Xander and I have both seen Morgan and what he can do, and we can vouch for Quince. If she hadn't stepped in and guided us when she did.... Well, let's just say none of us would be standing here today.\" He glared at Rogan.\n\nThe man just smiled blandly back at him. \"Fair enough.\"\n\nRogan was trouble. Venin wished he'd tossed the man out the moment he'd walked into the room. The world would have been a better place for it.\n\n\"Ready for your cirq, sir?\"\n\nVenin turned to see the med drone hovering next to him, having just finished with Jameson. \"Um, yes, please, ma'am.\" He wasn't sure about this, but Quince had urged him to have it done.\n\n\"So what's the threat?\" Alix waved the glowing globe out of existence and looked around the room. \"Erro seems big enough, if they keep to their part.\" He stood and walked over to look out the plas windows. \"We should be concentrating on taking care of our wounded, making sure everyone gets fed, and spinning OberCorp and the rest of Oberon back to the Common Worlds' universe.\"\n\nA tumult broke out again at that statement.\n\n\"Do we even know how?\"\n\n\"This is crazy. We don't want to go back there.\"\n\n\"I just want to go home.\"\n\nVenin frowned, listening to the arguments all around him. Was it even possible? Or would the shift be permanent? Did he care either way? He had no loyalty to the Common Worlds, but to be stuck here with all these landers and OberCorp....\n\nA piercing noise cut off all conversation.\n\n\"Thank you, Ari.\" Quince leaned forward, hands flat on the table. \"Erro has an amalite core. It's one of the reasons it's so valuable to OberCorp and to the Common Worlds.\"\n\nAlix nodded. \"One of several unique Oberon exports.\"\n\n\"The ithani plan to evolve, or transform themselves in some way. To do it, they plan to harness the power of the amalite core of the world.\" The sphere appeared again, bisected into two parts. Large masses in the center of the Oberon side glowed golden. \"Those are the known deposits on the Split.\"\n\n\"But that would destroy the world.\" Jameson stared at the globe. The amalite deposits were like cancers at the heart of Erro, and if Quince was right, they were metastasizing.\n\n\"The nimfeach told you this?\" Rogan sounded slightly less sarcastic.\n\nVenin glared at him, along with most of the rest of the room.\n\n\"Yes.\"\n\nRogan grunted. \"Let's say for the sake of argument that I believe you. Let's say this nimfeach\u2014Morgan is it?\u2014is real, and that he told you all these things. How can you be sure it's telling you the truth?\"\n\nQuince didn't flinch. \"Because he's part human.\"\n\nThat caused a stir, though Jameson wasn't entirely surprised.\n\nAn image of a small child appeared over the table. \"Tanner Michael Henshaw. His mother was a pith user in Egeus who abandoned him on the streets. I had Ari track his records down just now. The nimfeach found him and offered him their help if he agreed to become one with one of their own. Then they sent him for Xander to find.\"\n\nXander sank back into his chair. \"Holy Split.\"\n\n\"So why are they helping us?\" Venin frowned. \"It doesn't make any sense.\"\n\nQuince nodded. \"He couldn't tell me, but I think there are two factions among the nimfeach, or maybe even among the ithani themselves. Maybe? One faction wants to finish what they started with the shift. The other sees that things have changed.\"\n\n\"How long do we have?\" Alix looked dangerously calm.\n\nQuince locked gazes with Alix. \"Less than three days. In fact, everyone set your chrons for midnight, two nights from tonight. We may have a bit longer, but not much. We need to know by then what we plan to do.\"\n\n\"We have a stockpile of meso bombs,\" Alix said softly. \"And Erissa tells me that there's still one Rentz Class carrier on the launch pad\u2014one of the heavy ore haulers that makes runs up to Titan Station. We could load it up as a precaution. It can reach anywhere on the planet in a few minutes\u2014\"\n\n\"That would be an act of war.\" Quince's face was grim. \"We can't provoke anything, not until we're sure there's no other choice.\" She looked out the window into the distance, as if she could see the ithani from where she stood. \"Maybe as a last resort.\"\n\nAlix frowned. \"I'll see that it's ready.\"\n\n\"That sounds pragmatic.\" Quince sighed. \"We don't know what kind of battletech the ithani possess. They fought an interstellar war when we were still roaming the African savannah carrying wooden clubs.\"\n\nVenin whistled. It was a lot to take in.\n\n\"So what would you propose?\" Alix crossed his arms, siting back in his seat.\n\n\"We need to decide where to concentrate our efforts. We need information, fast.\" Quince pulled her worn carry sack up onto the table and brought out a small shiny sphere. It looked like a key, but it was milky-white instead of silver.\n\nIt was also larger than the one he held in his carry sack. Venin stared at it, wondering what this one was for.\n\n\"What does it do?\" Jameson's eyes were big as fists.\n\n\"I don't know. Morgan said I would know what to do with it.\"\n\n\"Can I see it?\" Jameson held out his hand.\n\nQuince held it out to him over the table.\n\nRogan's greedy eyes followed it.\n\nThe key glowed red and then orange, ripples spreading across its surface.\n\n\"Xander, look at this!\"\n\n\"Don't touch it!\" Alix shouted. \"It might be a\u2014\"\n\nXander's fingers touched the colorful surface.\n\nThe key flashed a brilliant white light, and with a sound that was more like an absence of all sound, the two kings were gone.\n\nALIX STARED at the gaping hole in the floor that had just been excised from reality, where Xander, Jameson, and the center part of the table had all been standing.\n\nAn alarm was going off somewhere, and lights were flashing.\n\n\"Everyone get back!\" He grabbed Vestra and hauled the petite elderly skythane woman away from the hole as part of the remainder of the table slid through to fall to the floor below.\n\nBy the Split, I hope there was no one down below. He peered into the hole. The room below seemed to be for storage. He thanked his lucky stars no one had been down there in the thing's path. Whatever it was.\n\nThe floor dropped with a loud groan, propelling him forward into the void. He flailed, trying to stop his fall, and then someone's arms were around him, the beat of wings conveying him to the other side.\n\nVenin set him down. \"You okay?\"\n\n\"Holy hell, that was close. Thanks.\"\n\nVenin grinned. \"Couldn't have your pretty little face splattered on the floor down below.\"\n\nAlix grunted. Pretty little face.\n\n\"What the hell just happened?\" Quince was white as a sheet.\n\n\"Erissa, what the hell was that?\" Alix quickly regained his composure and ignored the strapping, dark-skinned skythane man. His mind couldn't deal with that and process the sudden loss of the man who had once been his closest friend, and Jameson, for whom Alix had come to have great respect.\n\n\"Checking room sensors. One moment.\"\n\n\"Quince, what in the seven suns was that thing? A bomb?\"\n\nErissa spoke in his ear. \"There is no explosive residue detected in this room. DOC is trying to determine the cause of the event.\"\n\nQuince shook her head. \"I don't think so. Morgan wouldn't have\u2014\"\n\n\"Are we sure about that?\" Rogan got up and dusted himself off. \"I don't trust these friends of yours\u2014\"\n\n\"DOC says there was no explosion.\" Alix enjoyed cutting the vile man off.\n\n\"DOC?\" Venin frowned.\n\n\"The OberCorp AI. Erissa, please ask DOC to initiate emergency protocol and lock down the building.\" Alix edged forward to look at the gap in the floor, more careful this time. It was a precise, surgical line, a concave oval just wide enough to encompass Xander and Jameson while leaving the others untouched.\n\nHe sniffed the air. There was the slightest metallic tang, but nothing else out of the ordinary that his human senses could pick up. \"Let's all make our way to the conference room down the hall. I want this one sealed up until we can run a complete diagnostic panel.\"\n\nQuince looked pained. \"Alix, I'm sorry\u2014I had no idea something like this would happen.\"\n\nAlix nodded. \"I know you didn't.\" Holy hell, what kind of world are we living in? Alien races, shifting worlds, and gaps opening up in the space-time continuum without warning. \"We can talk it over once we get everyone clear.\" He helped Mylin and Vestra over the edge of the oval-shaped gap, where it almost touched the wall. Then Venin extended a hand to him. The man's touch was warm and strong.\n\n\"Thanks, man. Go ahead. I'll catch up with all of you.\"\n\nVenin gave him that look on the way out. Alix ignored him. He waited until the room was cleared. \"Erissa, I need you to prepare the Rentz Carrier. Fill it with all the meso bombs we have.\"\n\n\"Understood.\" Her voice sounded shaky. Maybe it was just the grid connection. \"Destination?\"\n\n\"None yet. I want them in place and ready to take action at a moment's notice.\"\n\n\"Understood.\"\n\nAs the new head of OberCorp, and therefore the lander part of the planet, he had a responsibility to do whatever was necessary to protect his people.\n\nHe knelt to touch the floor where it had been severed. The edge was sharp as a knife, but the surface was cool.\n\nHe huddled with the enforcers, directing them to go over every inch of the conference room and the storage unit below with Erissa and DOC.\n\nThe Split save us if this is a new weapon. Even meso bombs might not be enough to stop them.\n\n# CHAPTER SEVEN: AFTER THE FLASH\n\nQUINCE CLAMPED down on her fear and anger. Nothing would be gained by letting her emotions get out of check.\n\nJameson and Xander were gone; that much was clear. It was the key. It had to be. There was no other reasonable explanation.\n\nYet she refused to believe Morgan had set her up, that this had been an intentional strike to kill the two skythane kings, or that they were already dead.\n\nShe stifled a sob.\n\nThere had to be another answer.\n\nShe glanced at the strange hole once more, and then looked up\u2014a small piece of the ceiling was missing too.\n\nWith a sigh, she left the room and followed the others down the hall to another meeting place. The new room was just like the old one, but with only lander chairs.\n\n\"I'll have new chairs brought in. Erissa\u2014\"\n\n\"No need.\" Quince picked up a chair and spun it around, sitting on it backwards.\n\nThe other skythane did the same.\n\n\"So what the hell was that?\" Alix spat. \"One minute they were there, the next....\"\n\n\"I think it was a gateway.\" Quince replayed it in her mind. Jameson had been holding out the key, Xander had touched it, and there'd been a bright flash. An oddly circular one, if the image burned into her retinas was any guide.\n\n\"That makes sense.\" Alix nodded. \"But what if it wasn't really a key? What if it was some weird kind of explosion that our sensors can't quite pin down?\"\n\n\"Those keys are made of amalite. If there'd been an actual explosion, with that amount of material....\" Amalite was fairly stable, but when triggered, it blew with a huge explosive force.\n\n\"Half the building would be gone.\" Alix went pale.\n\n\"Besides, do we really want to believe that Jamie and Xand are gone?\" That came from Jessa, who had sat through the meeting almost unnoticed. \"If it was a gateway, they must have gone somewhere, right?\"\n\n\"Yes, but I've never seen a gateway act like that before, although I don't have a lot of experience with them.\" Alix frowned. \"But you're right. We have to assume they're okay until they're not.\"\n\nQuince nodded. \"You're right. It's kinda tough to say how they should and shouldn't behave.\" Her mind was racing. If it was a gateway, there had to be some way to confirm it. \"Maybe I could run some tests, both here and where Jameson opened the gateway in Xander's storage unit.\"\n\n\"Good thinking. Erissa, give Quince's PA whatever access it needs to DOC to get this done. In the meantime, I've got them loading up the Rentz Carrier.\"\n\nQuince and Alix locked eyes. \"I hope to the gods we don't have to use it.\"\n\n\"Maybe not, but we've got to do it. Best to be prepared, and even if it was a gateway, it might still be a hostile act.\"\n\nQuince sighed. He was right, much as it irked her to admit it.\n\n\"I think that's prudent too.\" Rogan rubbed his chin between his thumb and forefinger thoughtfully. \"What can the Syndicate do?\"\n\nAlix winced. Quince sympathized. She'd feel dirty finding herself on the same side as the Slander boss too. \"We're stretched thin with the losses in Errian and the support forces needed for this venture. We could use a quasi-legal security force to keep order in the city.\"\n\n\"Done.\"\n\nQuince side-eyed Rogan. He was being awfully accommodating all of a sudden.\n\n\"For a price.\"\n\nAh, there it is.\n\n\"What price?\" Alix's eyes narrowed.\n\n\"One of those keys, when this is all over.\" The man sat back, smacking his lips, looking like nothing so much as a toad who had just swallowed a big fly.\n\n\"Can't do that. I don't even know if we'll have one, and there's no way I'm letting you have one to be free to flit about Oberon City, ahead of law enforcement.\"\n\n\"Law enforcement?\" Rogan chuckled. \"That's a good one. Okay, how about this. When the crisis is over, you help me go legit and clean up the Slander. I want 25 percent profit on the offworld amalite trade, in addition to the pith that was already promised me.\"\n\nAlix glared at him.\n\nQuince ground her teeth. They were wasting precious time with this cretin.\n\n\"Done.\" Alix spit on his hand and held it out.\n\nRogan shook it. \"I'll draw up the contracts and forward them to your PA.\" He stood, looking around with a big grin on his face. \"Now if you don't mind, I'll see myself out. I have a security force to put together.\"\n\n\"Erissa, clear Rogan for departure from the building.\" Alix's eyes bored into the man's back as he left the room.\n\nQuince closed her eyes. It seemed like a terrible idea, putting Rogan and his thugs in charge of the city, even if just for a short while, but things were spread too thin. She'd taken a look at the security roster. \"I have a proposal.\"\n\n\"Let's hear it.\" Alix looked tired, dark spots under his eyes and his whole posture sagging.\n\n\"We need information. Where are the ithani? What are they? How do we stop them from destroying Erro?\" She looked around the room, taking in each of the remaining skythane and landers. \"Put Mylin here in charge of organization, pulling together everything we can get our hands on about this world, its anomalies, the skythane, and whatever may have come before them. She's a whiz at pulling together resources. Then the rest of us can feed her whatever we can find through our various resources.\"\n\nMylin blushed. \"I don't know. I don't think I'm up to something like that.\"\n\n\"You were born to do this.\" Quince touched her cheek softly. \"We need you now.\"\n\n\"I don't know. Maybe\u2014\"\n\n\"Oh come on, Mylin, you're fantastic at organization.\" Venin shot her an encouraging look.\n\nMylin blushed. \"I guess.... Okay. I'll do my best.\"\n\n\"We'll be the Hunters.\" Alix pounded the table. \"Let's hunt down our friends and bring them home.\"\n\nThere were nods around the room.\n\n\"Should we adjourn and run down our individual sources? Report back here?\" Alix rubbed his eyes. \"Erissa, I'm gonna need a stim.\"\n\nThe others assented.\n\n\"Alix, can you have someone sit with Mylin to get her up and running with her PA on DOC's system?\"\n\n\"Sure.\"\n\n\"And Robyn, once she's done, sit with her and tell her everything you know about skythane legend and tradition.\"\n\n\"I can do that.\" Robyn kissed her cheek. \"What are you going to do?\"\n\n\"A little testing of my own down in Xander's storage unit. I'll be on my cirq if anyone needs me.\"\n\nMYLIN OPENED her eyes.\n\nShe sat in a field of data\u2014a city of it. Information flowed back and forth in slow rivers, marked blue by her \"user preferences.\" That's what Derren called them.\n\nFlitting red packets resembled nothing so much as wereverens. She instinctively ducked as one of these made right for her, but it passed through her effortlessly, emerging on the other side to \"slit\" off to wherever it was needed. Another of Derren's words.\n\nDerren was the OberCorp employee assigned to her by Alix. He was showing her around this \"virtual world,\" or \"veer.\" Or maybe \"vir.\" It was all a bit unclear.\n\n\"Cirqs can run in three general modes. One is 'communication only'\u2014that's great for contacting other people who are connected to the grid.\"\n\nMylin nodded. \"Okay, got that. I think. The grid\u2014it's like a web that connects everything and everyone in Oberon City?\"\n\n\"And the rest of civilized\u2014I mean, the rest of the Oberon side of Erro.\" His avatar flashed her a sheepish smile. It was even skinnier than he was, and its teeth sparkled when he grinned.\n\n\"It's okay. We think you landers are the uncivilized ones.\"\n\n\"Fair enough.\" He winked at her. \"So the second mode is 'data interaction.' If you close your eyes, you can see images, process data, and generally interact with the grid and others on a superficial level.\"\n\nMylin tried to frame that in a way that made sense to her. \"Like\u2014seeing real life, but from someone else's eyes?\"\n\nDerren nodded. \"Or like reading a book, but in your mind.\"\n\nMylin felt woozy. \"It's a lot to wrap my head around.\" She looked around, trying to make this strange new world make sense.\n\n\"Sure. Sorry. I'll go slower.\"\n\nShe shook her head. \"No, I'll be okay.\" He was cute. His avatar, aside from being thin as a silverbark, matched his \"real\" self\u2014black hair, dark eyes. A nice smile. She shook her virtual head. \"So this is the third?\" She pointed at the virtual world they found themselves in.\n\n\"Yes. Virtual space, like I mentioned before. 'Vir' for short.\"\n\nThe place was a jumble of strange shapes, shifting landscapes, and a crazy array of moving objects. \"It's dizzying.\"\n\n\"We can make it simpler. Bix, can you make a storage box for Mylin?\"\n\n\"My pleasure.\"\n\nMylin looked around, startled. \"Who was that?\"\n\n\"That was my PA\u2014my personal assistant. His name is Bixter. Bix for short. We'll get to that in a second.\"\n\nThe world shifted, and suddenly they were inside a featureless white box. \"This is a storage box. A little bit of the grid just for you to use.\"\n\nMylin felt like she was inside a closet. \"It's a bit stifling.\"\n\n\"We can fix that. Say this. 'Initiate PA.'\"\n\nShe looked at him doubtfully. \"To who?\"\n\n\"Just say it. Your cirq will know what to do.\"\n\n\"Okay.\" Feeling self-conscious, she tried it. \"Initiate PA.\"\n\nA thing appeared before her. It was silver, in the approximate shape of a human, and waited silently, quiescent. \"Oooh. What is it?\" She walked all the way around it. It was beautiful, like one of the statues of Gael she'd seen at the chapel in Gaelan. She peered at its face, and then jumped backward with a yelp when it opened its silver eyes.\n\nIt cocked its head and looked at her. \"Hello, Mylin. I am your personal assistant. What form should I take? I can be a human, patterned on someone you know, or an animal, or\u2014\"\n\n\"A zimbee.\" The thought sprang into her head fully formed. She used to love watching the fat insects zip around Founder's Hill.\n\nThe silver apparition frowned and cocked its head again. \"I don't have that creature on file. Can you focus on an image of it?\"\n\nMylin looked over at Derren. He nodded.\n\nShe focused on the image of a zimbee, a bird-sized pollinator with yellow and red stripes across its abdomen and fast-flickering wings.\n\n\"I see it.\" Her PA closed his eyes, and then shrank, sinking down toward the ground and changing color until he became the zimbee she'd seen in her mind's eye.\n\nThis place is amazing. Magic. She'd almost forgotten about her claustrophobia in the bare white box.\n\nHer PA tested out his wings and lifted into the air to hover above her.\n\nMylin laughed, delighted.\n\n\"What would you like to call me?\" The rich male voice sounded strange coming out of the small insect body. His multifaceted eyes glittered.\n\n\"How about 'Zim'? And can you make your voice less gender-specific?\"\n\n\"Zim sounds wonderful. And how about this?\" Zim's voice shifted up the scale, becoming less male-inflected. More like her friend Rhyl.\n\n\"I like it.\" She could get lost exploring this place and its wonders, but she had work to do.\n\nDerren nodded. \"You're doing great. Now ask Zim to change the blank white walls to something more pleasing.\"\n\n\"Anything?\"\n\n\"Sure.\"\n\n\"How about a forest meadow?\"\n\nZim bounced up and down. \"Like this?\"\n\nAll four walls became windows out into a forest clearing. Purple-leaved trees hung over a small brook on one side and swayed in an unfelt wind. \"Perfect.\"\n\n\"Okay, last lesson. Zim can get things for you, and you can also use this space to organize them.\"\n\n\"Get them from where?\"\n\n\"Mostly from DOC. That's short for 'Digital Oberon Corporation AI.' DOC runs this 'semi-autonomous grid,' or SAG. It's all very confusing at first, I know. Just think of DOC as the guy who knows everything that's on the grid.\"\n\n\"Okay.\" Mylin laughed. \"How do I get here and back?\"\n\n\"Just ask Zim. Zim, can you take Mylin back to rel?\"\n\n\"Rel... real life?\" Mylin was starting to work this out and to latch on to the possibilities.\n\n\"Or reality.\"\n\nZim hovered in front of her. \"Is that okay, Mylin?\"\n\n\"Yes, please.\"\n\nThe storage box vanished, and she opened her eyes to find Derren two inches in front of her, peering into her eyes. \"You okay? I hope I didn't overwhelm you.\"\n\nHe's so cute. She pushed away gently and nodded. \"Thanks for the lesson. Are you going to be here if I need you?\" The office she'd been assigned to was a lot like the white storage box. She frowned. Maybe she could get a few things brought in here to make it less dismal.\n\n\"Of course. I've been assigned to work with you.\" He grinned. \"I'm a grid whiz.\"\n\n\"Perfect.\" As long as I don't get too distracted by the cute boy next door. \"Can you ask Robyn to come in?\" She was tired, but she'd refused a stim. She didn't want any artificial wake aids. She'd decided she'd rather tough it out with a cup (or ten) of keff, which she'd been delighted to find on the OberCorp menu.\n\nIt was going to be a long night.\n\nROBYN SIGHED. \"That's all I know about the prophecy.\"\n\nMylin was a precocious girl, clearly talented at organization, but she was also exhaustingly thorough. They'd been at it for four hours, as Robyn shared everything she knew about skythane culture, the nimfeach, the sneach\u2014which she was beginning to think were two parts of the same thing\u2014and whoever had come before them here on Erro.\n\n\"I'm sorry. I know you must be tired.\" The girl laid a hand on Robyn's knee. \"You've been very patient with me.\"\n\nWell, that much was true. Mostly. \"I'm sorry. It's just been a long couple of days. Or couple of months.\" She had to remember she was no longer the queen. \"Do you need anything else from me?\"\n\n\"Not at the moment. If I do, I'll find you.\" Her eyes glazed over, a gesture Robyn had learned meant the girl was going into vir... some kind of magical world inside the \"grid\" where she was organizing all this information.\n\nThe girl's eyes cleared up again, and to Robyn's surprise they were wet. \"I'm sorry. I never had a chance to tell you how happy I was to see you again, back in Gaelan. I thought... we all thought....\" She got up and threw herself into Robyn's arms and hugged her tight.\n\nThe lander boy\u2014Derren?\u2014shrugged and grinned. \"Long night,\" he mouthed.\n\nRobyn nodded. \"It's okay.\" She held the girl tight, running her hand lightly over Mylin's dark hair. \"It's been a difficult time, for all of us.\"\n\nWhen they separated, Mylin wiped her eyes. \"You must think I'm such a mess.\"\n\n\"Not at all.\" Robyn wiped her own eyes. The girl's emotions had affected her more than she cared to admit. She'd spent so many years separated from her own son, Xander, who hardly needed a mother at all now.\n\nWherever he was.\n\nHe's still alive. She was sure of that, though she couldn't have said how she knew it.\n\nShe squeezed Mylin's hand. \"You're doing what you need to do. Help me find my son. Help us figure out how to stop what's coming.\" She leaned forward and kissed Mylin's cheek. \"Give her whatever help she needs,\" she said to Derren.\n\n\"I will, Missus.... Ms....\"\n\n\"Sl\u00e9ite. But Robyn will do.\"\n\nShe got up and left the small office, needing some fresh air. These lander buildings were so stuffy and controlled.\n\nQuince had shown her how to find the hover tube that would take her to the roof. It would be good just to be outside for a few moments.\n\nThe palm plates had all been keyed for access by each member of the Hunters, as Alix had started calling them, so she set her hand against the cool surface and waited for the doors to open.\n\n\"Mind if I join you?\" Jessa Althorpe, Jameson's offworlder ex, strode down the white hallway toward her. \"This place is giving me a serious case of claustrophobia.\"\n\nThe woman was pretty, in a blonde sort of way, the kind of girl Robyn might have fallen for when she was younger. She was also made of steel, if half the things Robyn had heard about her were true. \"Sure. I was just heading up to the roof to get some fresh air.\"\n\nThe doors opened, and they stepped inside. \"The first time I rode in one of these things, I was scared shitless.\"\n\nRobyn laughed. \"Yes, I guess without wings they would be a bit daunting.\" She found these lifts more confining than anything. They reached the top of the tube and another pair of doors opened, letting them out onto the rooftop.\n\nIt was breezy up there, a cool night wind blowing off the Gildensea. The moon, Bandia, lit the sea with her silver glow. Robyn wondered idly if Oberon's moons, Hermia and Lysander, still circled the space where the half world had been, in another universe.\n\nShe stretched her arms and wings, breathing in the cool air gratefully. Small things to be savored in a world gone mad.\n\n\"They're still alive.\" Jessa sank down onto a bench and stared up at the moon.\n\n\"What makes you think so?\" The rooftop was beautifully landscaped with a formal garden, complete with pathways and arbors covered with vines and lit by small sparkling lights. It had taken a beating with the great storm after the shift, but it was nice to be out among living things once again, however overmanicured.\n\n\"I don't know, exactly. Just a feeling I have. I think Quince is right. They went somewhere else.\"\n\nRobyn nodded. \"I think you're right too.\" How strange life was. She could never have foreseen, even two months before, that she would be standing atop the tower of her nemesis, half a world away from her home, talking with an offworlder about the end of the world. \"Where do you think they went?\"\n\nJessa frowned, scrunching up her forehead. \"I don't know. Somewhere important.\"\n\n\"Why do you say that?\"\n\n\"It's like they were meant to go, isn't it? I mean, nothing happened until both Jamie and Xand were touching the key, right?\"\n\nRobyn still wasn't used to that name. Xand. Xander. She missed her little Davyn. \"I suppose so. The question that brings up is, did whoever planned this mean it for good or ill?\"\n\n\"I don't know.\"\n\nRobyn sat down on the bench next to the girl. Woman, really, but next to her, most women were girls. \"I've waited for twenty-five years for this time, holding to the knowledge\u2014the belief\u2014that certain things were meant to happen.\" She took Jessa's hand in her own. \"We have to believe it's true. It's the only way we can keep moving forward.\"\n\nJessa looked up at her. \"You really think things are going to work out?\"\n\n\"Yes. Yes I do.\" She allowed herself a small grin.\n\n\"On Beta Tau, they teach us that God has a plan for everyone, and that we are given guideposts along the way to guide us to where we're supposed to end up.\" She stretched her hands until her knuckles cracked. \"Maybe the key was a guidepost.\"\n\n\"Maybe.\" Honestly, Robyn had been thinking along similar lines, though her gods were plural. \"If things go sour, well, we won't be around any longer to worry about it, will we?\"\n\nJessa laughed, a sweet sound. \"I suppose not.\"\n\n\"I need to stretch my wings a bit. Will you be okay here alone for a few minutes?\"\n\n\"Yeah, I think so. It's nice to have a little peace and quiet, and I know how to find my way back downstairs if I get bored.\"\n\nRobyn touched her shoulder. \"You're a good soul. Jamie will be all right, and so will Xander.\" She stepped up onto the parapet and jumped into the night sky, soaring upward under her own power.\n\nNo matter what else happened, having her wings back was a glorious thing.\n\n# CHAPTER EIGHT: SOMEWHERE ELSE\n\nJAMESON RUBBED his temples. He had one hell of a headache. It stretched from his neck all the way up the back of his head to his forehead, and his scalp tingled too.\n\nAll of him tingled, actually.\n\nThe key had begun to glow. He'd held it out to Xander, and there'd been a flash.\n\nHe opened his eyes.\n\n\"Hey there, you okay?\"\n\nXander stared down at him worriedly for the second time in a day, his face framed by ruddy red sunlight.\n\nThere was something wrong with that, but Jameson couldn't quite get his head around it. \"We have to stop making a habit of this whole sleeping-Jameson\/white-knight-Xander thing.\" He sat up, pressing his palms into the golden grass to support his weight. \"What happened?\"\n\nIt was warm, and a stiff breeze blew through the blades, creating a sound like running water.\n\n\"A gateway, I think. At least, I'm pretty sure we're not at OberCorp Headquarters anymore.\" He helped Jameson up.\n\nJameson's head pounded like he'd just had a really bad drunken jag. \"If you feel half as bad as I do\u2014\"\n\n\"Like you were shot with a pulse rifle?\" Xander handed him the new key, the one that had set off whatever this was.\n\nJameson took it. \"Something like that.\" It was inert now, no longer resembling liquid mercury, the surface flat and still.\n\nHe looked around. The grassy plain extended off into the distance, where mountains were painted purple against the blue sky. Bits of conference room desktop and flooring\u2014and maybe cement or heavy plas\u2014littered the grass around them. Where are we? Was it a different place? A different time? Was that even possible? \"Why is the sun red? Not pink\u2014red?\"\n\nXander shrugged. \"I don't think we're on Erro anymore.\" He stared at the key. \"What in the Split did that thing do to us?\"\n\n\"Hell if I know.\" Quince had been talking about going back to the Common worlds, before whatever this was had happened. Would things simply go back to how they had been before? How could they? I have wings now. And Xander. Sure, he wanted to stay with Xander, but would that mean they'd have to cut themselves\u2014and Erro\u2014off from the rest of the human race? Now, maybe, it no longer mattered.\n\nHe spied his carry sack half buried in the grass. \"Looks like we got lucky. This made it through.\"\n\nXander nodded. \"I found mine over there.\" He pointed to a rocky outcrop fifteen meters away.\n\nSomething flashed at the edge of his vision, but when he looked for it, there was nothing there. Jameson sank into the grass, crossed his legs, and went through his sack. He felt strangely calm\u2014somehow traveling to heretofore unknown worlds had become a normal part of his life.\n\nThey needed to find their way back, though. Erro was in imminent danger, and he and Xander might be the only ones who could save it.\n\nHis sack was a bit of a disappointment\u2014the pulse rifle and a few of the rations Xander had given him, and some dirty clothes.\n\nUnderneath it was the holo of Jessa he'd brought with him from Beta Tau.\n\nHe grinned. Maybe she'd find a way to come after him again.\n\nQuince's old key was there too.\n\nHe held it up triumphantly. \"Our way back home.\"\n\nXander raised an eyebrow. \"Try it.\"\n\nJameson closed his eyes and pictured Xander's apartment. He twisted the key.\n\nNothing happened.\n\nXander didn't look surprised.\n\n\"I'll try another place.\" This time he tried the caverns where he and Xander had taken refuge after the shift. He twisted the key, and... nothing.\n\n\"Try something closer. The top of that hill.\" Xander pointed to a gnarled hilltop a hundred meters in the distance.\n\n\"Sure.\" Jameson tried, and was surprised when a gateway opened immediately. \"What... how did you know?\"\n\nXander shook his head. \"Just a guess. I think we came too far for Quince's key to work. At least to take us home again.\"\n\n\"What about this one?\" Jameson pulled out the new, larger key, the one Morgan had given to Quince.\n\n\"I don't know. Last time you touched it\u2014\"\n\n\"You touched it too!\"\n\n\"Last time we touched it, things didn't go so well.\"\n\nHe's right. Jameson put the keys away reluctantly. It was frustrating to be trapped here with no idea where they were, or why.\n\nA sharp whine split the still air. Jameson squinted at the sky. \"Look!\" Something was coming down fast, a sleek something. As it tumbled toward the ground, it leveled out just a few hundred meters above them.\n\nIt was some kind of craft. It was made of overlapping white sails that trailed off into nothing, leaving a gauzy contrail of sparkling light that faded in its wake.\n\nIt had some old damage\u2014its back quarter was blackened and ragged on the near side as it slipped past, heading for some unknown destination in the distance. Jameson got a glimpse of some characters stenciled along its side. They were in no language he recognized. \"Fight or flight?\"\n\nXander arched an eyebrow at him again.\n\nJameson laughed ruefully. \"Why do I even ask? Fight it is.\" His headache was quickly subsiding. Thank the gods for that. He kissed Xander. \"Whatever the hell this place is, I'm glad you're here with me. Come on!\"\n\nHe took off running and launched himself into the air, soaring up above the warm plains. The craft quickly left them in its wake, but he marked the direction it had gone.\n\nThey flew for hours across the seemingly endless plain, chasing in the direction of the strange craft.\n\nEventually, they stopped to take a rest. Far off to the east, a line of mountains circled the lowlands, purple with haze.\n\nJameson wiped his mouth with the back of his arm. He'd taken a sip of the stale water that remained in his canteen and had then handed it to Xander, but they were being cautious with their supplies. How long would they be stranded here, and when they would find food and water again?\n\nWhere the hell are we? \"Do you think this is a third half of Oberon? Or Erro?\" Even as he said it, it sounded stupid to his own ears.\n\nXander laughed. \"Hell if I know. This is all as new to me as it is to you.\" He handed back Jameson's canteen. \"It feels different, doesn't it?\"\n\n\"Yes.\" There was a distinct chill in the air, and a strange smell... maybe ozone? Of course, those could just be local things.\n\nSomething flashed in the corner of his eye. He turned to look, but there was nothing again. Damned memories. That stopped him. How could he possibly have foreign memories of this place? Unless it really was Erro? Or some version of Erro?\n\nHe pulled out the new key. It had to be responsible for this. A trap? Or an invitation?\n\nJameson stared at it.\n\nHe wished Quince's key had worked. He was itching to be back home, helping his people through this latest crisis.\n\nHe remembered what Alix had told him. That amalite needed time to recharge. Maybe that was all it was. Maybe it wasn't charged enough yet. In a few more hours, he would try the key again, and it would work. They would be back in Oberon City.\n\nSomehow, he didn't think it would be so easy. He sighed and closed his eyes.\n\n\"Hey, you okay?\" Xander wrapped his arms around Jameson's waist, his wings settling around his back.\n\n\"Yeah. Just ready for a break in all the craziness.\"\n\nXander chuckled wryly. \"I don't think we're going to get one anytime soon.\"\n\n\"Good morning, Jameson.\"\n\nJameson's eyes went wide, and he stumbled backward away from Xander. \"Angie?\" He hadn't heard that voice in at least a month, when Xander and Quince had forced him to burn out his bioware in a desperate attempt to avoid detection as they'd fled Oberon City. It seemed like a lifetime ago.\n\n\"Hey, you okay? Did I do something?\" Xander's brow was furrowed.\n\nJameson shook his head. \"She's back.\"\n\n\"Angie?\" Xander asked, raising an eyebrow.\n\n\"My PA.\" There hadn't been time for Xander to get his cirq reinstalled before the flash. \"Angie, it's so good to hear your voice.\"\n\n\"You too. I'm... avizzt... afraid I was not completely restored. There was some kind of glizzzch during installation. Can you have me deleted and reinstalled?\"\n\nJameson shook his head. \"Sorry, no. We're a long way from any human tech at the moment.\"\n\nShe was silent for a few seconds. \"I don't detecccht a grid in the current location.\"\n\n\"Yup, we're in the middle of godsforsaken nowhere.\" To Xander, he explained, \"She wasn't fully restored before the flash.\"\n\n\"I can run a self-diagnostic and see if I can patch some of the holes. I may not be fully functional.\"\n\nJameson nodded. \"Please do. How long will that take?\"\n\n\"At my current processor speed, maybe three hours. I will be nonfunctional until it's done.\"\n\n\"Do it.\" Jameson grinned. A partially functional Angie was better than none at all. God, I missed you.\n\n\"So?\"\n\n\"She's going to see if she can fix herself. It'll take a couple hours.\"\n\n\"Wish there'd been time to get Ravi put back in my head\u2014holy shit!\" Xander stared up at the sky.\n\nJameson followed his gaze. Another \"sail\" craft was plummeting to the ground, this one at least twice as big as the first one.\n\nA roar preceded it as it fell, wind whipping past and carrying fire and smoke up into the air behind it.\n\nXander's mouth was open. \"What the hell?\"\n\nAnother of the white sail ships chased it toward the ground.\n\nThe first ship slammed into the ground a few kilometers away from them, sending up a tremendous blast of fire and dirt and grass.\n\nThe smaller ship corrected course just before hitting the ground nearby and disappeared just below the horizon.\n\n\"What the hell did we land in the middle of? A war?\" Xander stared at the rising cloud of dust and smoke.\n\n\"Starting to look that way.\" Jameson had been worried before, but now fear was starting to claw at his guts. They were all alone on a strange world, and bad things were happening that they had no understanding or control of. \"Come on. Let's go see what we can find out.\"\n\nRun toward fear. That was his new mantra.\n\nIt was a big improvement over his old one, always run away.\n\nAs long as it didn't get him killed.\n\nXANDER WHISTLED softly.\n\nThey were crouched on a hilltop looking over the blast crater the ship had left when it hit the plains. It had partially disintegrated, but there was still enough of it left to see what it had been.\n\nA thin stream of smoke drifted up from the crater.\n\nA host of things was busy around the ship. They were beautiful. Like butterflies back on old earth, but somehow more ethereal.\n\nHe'd seen them before, in dreams. \"Ithani,\" he whispered.\n\nXander nodded. \"I think so too.\"\n\nThey bustled around the fallen ship, ripping it apart. After a few moments, they pulled one of their own from the wreckage, limp and faded, a moth to their butterflies. They carried their companion away to their waiting ship, and the search went on.\n\nJameson ducked as one of the creatures zipped up the hillside, its insect eyes jerking back and forth as if it were trying to see everything all at once.\n\nXander held his breath. He didn't know why, but he felt it was best if they remained unseen.\n\nWhen he dared to look over the grass again, it was gone. \"If those are ithani, is this some other part of Erro?\"\n\nJameson shrugged. \"Don't know. Maybe?\"\n\nIt had taken them half an hour or so to reach the site of the downed craft. \"What do you think they're doing down there? Search and rescue?\"\n\n\"Maybe. Should we announce ourselves? Or maybe\u2014\" There it was again. The thing at the edge of his vision. He was becoming more and more certain it wasn't a memory. He put his hand to his lips, casually as he could manage.\n\nXander looked at him and nodded.\n\nJameson spun around in one smooth movement and reached out to grab the thing that was watching them.\n\nHe grabbed it by the thorax.\n\nIt tried to escape, flapping its wings and squealing as it pulled him off the ground.\n\nJameson was startled by its strength.\n\nXander leapt after him, pulling him back down to the ground.\n\nThey slammed hard onto the grass of the hillside, the two of them on top of the creature.\n\nIt stared up at them, iridescent wings fluttering. It was clear the thing was terrorized by them.\n\n\"It's a nimfeach.\" A memory fluttered by, the nimfeach gathered to witness the birth of a new clutch of ithani from their tri-parent. Jameson brushed it away.\n\n\"Jameson, show it the key!\"\n\nJameson shot Xander a confused look. \"Which one?\"\n\n\"It doesn't matter. Hold one of them up. Let him see it.\" The ithani had made the keys, right? Maybe showing the nimfeach one would prove that they were friends. Or ithani themselves. He wasn't sure.\n\nJameson rummaged through his bag and pulled out the smaller key\u2014the one Quince had first shown them on Oberon. He held it aloft and it came suddenly to life, the surface churning and giving off a bright golden glow.\n\nThe nimfeach went deathly still, staring at the rolling colors through multifaceted eyes.\n\n\"It's okay. We won't hurt you.\" He lifted himself up off the little creature.\n\nIt hummed, entranced.\n\nXander snorted. \"Well, that did it. At least he's not scared of us anymore.\"\n\nTHEY'D SNUCK away from the crash site, taking flight and putting some distance between themselves and the ithani.\n\nJameson was fighting back waves of memories. Not since the cavern after the split had they come so fast or so thick.\n\nThe nimfeach had triggered them.\n\nIt followed them, just a few meters behind, and nothing he did seemed to frighten it off. \"You okay?\" Xander glanced over at him, his eyes narrowed.\n\nJameson shuddered. \"Not really.\" A memory slipped past his guard\u2014a host of nimfeach surrounding a strangely enlarged ithani, skin the color of milk\u2014waving their hands in unison and singing a strange, discordant alien song. He had the sense that there were many notes in the music that were simply beyond his ability to hear.\n\nJameson blinked and found himself plummeting toward the ground. He spread his wings to slow his descent, shaking his head, trying to clear away the memories.\n\n\"Jameson, what the hell?\" Xander soared past him and made a wide arc, returning to hover a few feet away from him.\n\n\"Memory,\" he managed. \"Not safe to fly.\"\n\nHis golden wings carried him down to earth, and Xander followed.\n\nWhen his feet touched the ground, he fell into a crouch. Everything was swimming, and he couldn't focus.\n\nHe felt Xander's firm hand on his back. \"You okay?\"\n\nJameson opened his eyes. \"Too many memories.\"\n\n\"Breathe through it, like Alix showed you.\"\n\nJameson nodded. He took a deep breath, held it, and then let it go. Then again, and again.\n\nThe memories receded a little.\n\nJameson's vision returned.\n\nThe nimfeach had landed on a low tree branch next to him, and it was looking up at him with its multifaceted golden eyes. It said something in its singsong language and reached out a fingerless hand to touch his face.\n\n\"What are you?\" He'd felt an instant, strong compulsion to protect the creature, but he still didn't understand why.\n\n\"Purcck.\" The nimfeach's hand withdrew, and it touched its chest. \"Purcck.\"\n\n\"That's your name?\" Jameson pointed at the pip. \"Puck?\"\n\nIt purred and glowed. \"Purcck.\"\n\n\"Puck it is.\"\n\nXander laughed. \"I guess that's what he wants us to call him.\"\n\n\"Are we sure it's a him?\"\n\nXander shrugged. \"Doesn't really matter.\"\n\n\"Guess you're right.\" Jameson sat back on his ass, his wings counterbalancing his weight. \"That's the worst attack I've had since Gaelan.\" His head pounded, and the memories flitted about at the edge of his conscious mind. \"So what do we do now?\"\n\nXander shrugged. \"First, we rest a bit until you're sure you have a handle on things. Then we find someone who can tell us where we are.\"\n\nERINA BOWED as low to the ground as ze could manage, trying to keep the hatred from showing in zer eyes.\n\nThe Thshnel'Jirron waved zer off, along with the others of zer sept who had come to pay zim a last homage before the Great Migration.\n\nZe hated zim for what zi had done to zer once proud people, all in the name of keeping them safe during the thousand cycles war. Hated how zi took young ithani and used them up, moving from one body to the next as the last one aged and died.\n\nHated what zi had done to the nimfeach, reducing them from proud elders to mindless servants, faded versions of themselves with no wills of their own.\n\nSoon enough, zis reign would end. Zer last vision had been magnificent in its scope.\n\nZe ran through it again in zer mind, desperate to hold on to the details. The daring defiance it would require. The long interval\u2014a hundred thousand cycles\u2014and the eventual coming of the outsiders.\n\nFinally, the chosen two. Two who would be stronger than one.\n\nZe fixed their faces in her mind. They were alien\u2014soft-skinned, clumsy-looking compared to the lithe forms of the ithani or the small bodies of the nimfeach. One with a dark head, one whose head was light. And their wings.\n\nNever before had ze seen such wings.\n\nZe had no guarantee that things would align themselves in just the right way. The future was fickle like that, but it was enough that ze saw a way to save the ithani and the nimfeach both from oblivion. Zer people.\n\nIt was time for zer to make a plan.\n\n# CHAPTER NINE: CLUES\n\nQUINCE DESCENDED to the city streets under the cover of darkness.\n\nThings had gotten a bit wild in Oberon City in the wake of the shift, with normal food chains disrupted and a haze of worry and uncertainty driving even the usually staid office drones to acts of desperation.\n\nQuince had Ari monitoring grid news feeds. Rogan's forces were cracking down, which brought its own worries, but there was nothing she could do about that. Better the devil you know.\n\nThe storage unit, already a mess, had been emptied out of whatever supplies had remained after Xander's quick escape three weeks earlier. Has it been that long?\n\nQuince glanced up and down the street. She was alone, as far as she could tell.\n\nShe tucked in her wings and ducked under the broken pieces of the roll-down door. The removal of all the junk actually made her job a bit easier.\n\nShe pulled out the old portable radiation detector Alix had dug up for her. She wasn't sure what she was looking for, exactly. Her working theory was that the gateways were a form of wormhole. They might leave behind some sort of telltale radiation signature. Or maybe they would change the structure of the things around them in some subtle way.\n\nAri had dug up a bit of theoretical research on exotic matter that was known to line the mouths of wormholes, which might leave its own particular signature.\n\nThe detector was shaped a bit like a long funnel with a bulb at the back end, allowing sampling of air or small pieces of material, which would be atomized inside and subjected to the device's mass spectrometer.\n\nShe sampled the air. \"Ari, anything?\"\n\n\"Background radiation levels are slightly higher than average. Nothing beyond normal parameters.\"\n\nQuince nodded. If the ithani had used this method of travel regularly, they'd have found a way to make it as safe as possible.\n\nShe pulled out her knife and scraped a bit of the plas out of the wall and fed it into the device. \"How about now?\"\n\n\"Analyzing.\"\n\nQuince touched the cool wall with her fingertips. Humans had been on the planet for seven and a half centuries. They'd managed a few truly impressive feats of architecture\u2014the arcos and the various Houses of the skythane. And thousands of ugly utilitarian buildings like this one. How long after we're gone will our works here still stand?\n\nHer mind strayed, and then she was somewhere else.\n\nShe stood on a beach in front of a ramshackle set of shacks and huts, staring at the buzz of activity there. A large gathering of humans\u2014a few hundred? A thousand?\u2014were bustling about the encampment in a flurry of purposeful action.\n\nQuince looked around wildly. She didn't recognize this place.\n\nHad someone hit her on the head?\n\nWas this a dream? A memory?\n\nAll of the humans on the beach were landers.\n\n\"Definitely higher levels of alpha and beta radiation embedded in the plas.\" Ari's interjection brought her back to the present. \"Likely to decay to undetectable levels rather quickly.\"\n\nShe shook her head to dispel the lingering remnants of the dream. Or whatever it was. That was interesting. \"How quickly?\" She didn't have any time for daydreaming.\n\n\"Estimating seven point three days?\"\n\n\"Are these levels dangerous to humans?\"\n\n\"Not without long-term, constant exposure or ingestion of said materials.\"\n\n\"So don't eat the walls. Got it.\" She pulled her hand away.\n\n\"I wouldn't recommend eating plas\u2014\"\n\n\"Sorry, Ari\u2014it was a bad joke.\" PAs weren't as sophisticated at humor as humans.\n\n\"Ah.\"\n\n\"Does the signature match what we found in the conference room at OberCorp?\"\n\n\"They are similar. The levels in the conference room were considerably higher.\"\n\n\"That makes sense. That was a more recent event.\"\n\n\"Even so, the levels are higher than simple time lapse would account for. I would say the two events are related, but quantitatively different.\"\n\nQuince frowned. \"What does that mean?\"\n\nAri was silent for a moment. Quince was surprised. She hardly ever managed to stump the PA. \"I would guess that it means the gateway in the conference room led to somewhere considerably farther away than the gateway here,\" he said at last.\n\n\"Farther away than halfway around Erro?\"\n\n\"Possibly, yes.\"\n\nQuince chewed on that for a moment. \"Okay, let's head back to OberCorp. I want to run this by Alix and the others and see what they make of it.\" Where the hell did you two go?\n\nVENIN STARED at the white ceiling. It was dark, the early hours of the morning. He'd been sent off to try to get some rest. Alix had set up one of the executive apartments for the Hunters' use and had brought in enough beds to sleep the lot of them.\n\nIt reminded him a bit of his childhood, when he and his three siblings\u2014two sisters and a brother\u2014had shared a room in their aerie in Gaelan. How many times had he stared out the window at the stars late at night, wishing he could fly free?\n\nWaiting for puberty to get his wings?\n\nNow here he was, grounded again, and Alix was just a few doors away.\n\nEveryone else in their group seemed to have a purpose. Mylin was working to organize everything they knew about the ithani. Quince was out working to discover where Jameson and Xander had gone. Vestra had been sent back to oversee the refugees of Errian and Gaelan.\n\nAnd Alix... well, he'd somehow become the team leader, with all the might of OberCorp now behind him.\n\nEven that vile Slander boss had been given a task.\n\nVenin wasn't used to feeling useless, and Alix was a temptation he really didn't need. He hadn't been interested in anyone since Taz. This was really not the time for it.\n\nVenin closed his eyes, trying put Alix out of his mind, trying to ignore the snoring coming from Alia's bed.\n\nHe kept wandering back to the chapel high above Errian. To the statue of Erro\u2014and the spark, or whatever it was, that had knocked him on his ass.\n\nHe closed his eyes and tried to regulate his breathing, to slow down his mind.\n\nHe'd seen something in that flash. Felt something. He stretched his wings out to their tips, trying to remember what it had been.\n\nHis outstretched hand. The glow. The spark.\n\nHe flew over a huge construction site.\n\nSkythane below him were building a structure\u2014a castle? It would eventually be surrounded by elegant gardens. The winding paths had been sketched out, and some of the bigger trees were already being planted.\n\nThey flitted in and out of the building zone, carrying in materials.\n\nThe sun overhead shone with a rosy glow.\n\nThe place sat in a wide grassy plain with few trees. It was somewhere he'd never seen before, but he recognized it nonetheless.\n\nIt was\u2014or it would be\u2014the House of the Stars.\n\nIt's not possible. That place had been constructed hundreds of years before, and yet it was as if he were there, seeing it with his own eyes.\n\nHe was seeing a memory, just like Jameson and Xander. The statue, it had done something to him.\n\nThen he noticed something else.\n\nThe whole place sat in a depression that was almost perfectly circular, broken only at one point, where it looked like the ground had subsided.\n\nIt was too regular to be a coincidence.\n\nStrange things happened at the House of the Stars. Gateways opened, some said time shifted, and even the weather was often different there than elsewhere on Titania.\n\nThere must be something underneath it. Something impossibly old\u2014older than mankind's time on this world.\n\nHe opened his eyes.\n\nIn his mind's eye, he still saw the busy skythane, the rising construction, and the giant unnatural circle.\n\nIt was just a dream. Right? It had to be.\n\n\"Taz?\"\n\nHis PA, named after Tazim, answered in his head. \"Yes, Venin?\"\n\nIt was so strange, so... intimate to have a voice in his head. \"Do you record my dreams?\"\n\n\"No, not unless you ask me to.\"\n\n\"Ah, thanks.\" Alix had shown him the basics of \"interfacing\" with his PA, but the whole thing was so alien to him.\n\nSo there was no way to recall the dream, other than from his own memory.\n\nHe wondered what else Taz might be able to do for him. \"Taz, do you have access to... material of a sexual nature?\"\n\n\"Yes. I have full access to the Oberon City grid library, including male on male pornography. What would you like?\"\n\n\"Anything with a redhead?\"\n\n\"Close your eyes.\"\n\nHe did as he was told and was soon in the middle of a room in vir, watching two men, one redheaded and one brunet, going at it as the walls flickered through scenes varying from dungeon walls to an alpine ski loft. One of the guys looked up at him and grinned.\n\nHis eyes flickered open. \"Enough. Thanks.\"\n\nHe was covered in sweat and was aroused as hell. How do these landers get any real work done, with toys like this? If he'd had that kind of thing when he'd been a teenager, he'd never have left the aerie.\n\nThe dream, though, still weighed heavily on his mind. Time was short. What if it was something, and he said nothing?\n\nHe got up, his feet touching the cool tile floor. His bed was set up in the entry room, near the door. He'd chosen it out of habit\u2014always have a clear exit path\u2014that had been instilled in him at home with his siblings and pounded into him as a palace guard in Gaelan.\n\nHe needed to find Quince, or maybe Robyn, and see what they thought.\n\nHe got up and crept out of the suite, encountering no one. \"Taz, where are Robyn and Quince?\"\n\n\"Both are outside of the building.\"\n\n\"Who else is still here, and awake?\"\n\n\"Alix is in his office.\"\n\nOf course he is. The man was unstoppable.\n\nAlix was sitting at his desk, and his eyes had that glazed-over look that said he was in vir space. Alix bore an uncomfortable resemblance to the man Venin had watched in the adult entertainment he'd just partaken of.\n\n\"Would you like me to ping Alix?\" Taz asked.\n\nHe jumped. \"Um, sure.\" He managed to catch himself on the doorframe before he tripped and fell on his ass. \"Sorry. I forgot you were there.\"\n\n\"It's all right. It takes new users some time to get used to having a PA inside their heads.\" Taz said it with a bit of a lilt.\n\nHe's teasing me. Venin decided he liked Taz. \"Mylin took right to it,\" he said with as little resentment as he could manage.\n\n\"Girls are smarter. On average, 3.7 percent smarter than men of their same age.\"\n\nNow Taz was definitely yanking his chain.\n\nHe was about to ask the PA if he was real when Alix's eyes focused. The man looked at him, and his gaze traveled up and down the length of Venin's shirtless body, an appreciative smile slipping across his lips. \"Hey. What can I do for you?\"\n\nVenin chose to ignore the not-so-subtle undertones of that statement. Skythane and lander, he reminded himself. Bad mix. \"How's your mother?\"\n\nAlix closed his eyes. \"Erissa says she's okay. I was going to go see her soon.\"\n\n\"Good.\"\n\nAlix glanced up at him. \"Anything else?\"\n\nVenin felt like an idiot. \"Um, I had a dream. I know it's stupid.\"\n\nAlix sat up, giving Venin his full attention. \"Tell me.\"\n\n\"It was probably nothing. I shouldn't have bothered you.\" He turned to head back to bed.\n\n\"Hey wait\u2014it's okay. I could use a break. Please, tell me what you dreamed.\" He gestured toward the other chair in the room and leaned back in his. \"It might be important.\"\n\nVenin paused. \"You sure?\"\n\n\"Yeah. Vir work really tires me out, even with a stim.\"\n\nVenin turned the seat around, sat down, and looked around at the huge office. \"You landers really like white.\"\n\nAlix snorted. \"It's OberCorp's way of keeping the ugliness and dirt of everyday life at bay. So what was this dream about?\" He leaned back in his chair, putting his hands behind his head.\n\nVenin looked away from Alix's beautiful outstretched body. \"It started earlier today, in Errian. I visited the chapel of Erro above the city\u2014\"\n\n\"You'll have to tell me more about the skythane religion one of these days.\"\n\nVenin laughed. \"I imagine it's a bit different from what you're used to.\"\n\n\"Yeah. My family... we were never very religious. So it started earlier today?\"\n\n\"Yes. The statue of Erro in the chapel\u2014it glowed.\"\n\nAlix sat forward, suddenly looking even more interested.\n\nThere was something about this lander man that intrigued Venin immensely. \"I reached out and touched it, and there was this... spark? Powerful enough that it knocked me over.\"\n\nAlix nodded. \"That sounds like what happened to Jameson.\"\n\n\"I thought so too.\" He frowned. \"Why me? I'm no one special.\"\n\n\"Don't denigrate yourself. From what I've seen, you're quite impressive.\"\n\n\"You know what I mean. I'm not a king or a prince. I don't run a big corporation, like some people.\"\n\n\"None of us were those things when all this started.\" Alix sat back, a wistful look on his face. \"What did you see?\"\n\nVenin closed his eyes. \"When I was lying in bed tonight, it came to me. A dream, or vision. Maybe a memory?\" He frowned. It sounded silly when he said it out loud. Too late to back down now. \"It was the House of the Stars, but a long time ago, when they were first building it. I was high up in the air, watching, and I could see it, clear as day.\"\n\n\"What's that?\"\n\n\"The circle.\" It was so strange.\n\n\"Circle?\"\n\n\"Yes. There's a circle in the ground that surrounds the whole place. I think there's something there. Something underneath.\" He opened his eyes to find Alix's brown eyes staring into his.\n\n\"Does the House of the Stars have any strange... properties?\"\n\n\"Yes. Time sometimes moves a bit strangely there. The main waygate between Titania and Oberon is there too.\"\n\nAlix scratched his head absentmindedly. \"Erissa, get me some satellite images of the area around the forks of the Theseus and Demetrius Rivers.\" A stack of images appeared above his desk. \"We don't have any satellite imagery of Titania, of course\u2014different universe and all that\u2014but we do have lots of the Oberon side. The satellite photos of that region were always a bit sketchy. They thought it had to do with a particularly large amalite deposit under the crust there.\" He reached out and sorted the glowing images, and then rearranged them so they overlapped, making a ring around the area where the House of the Sun stood. It was heavily wooded, but bits and pieces of land showed through here and there. \"Well I'll be damned.\"\n\nVenin leaned over to look, aware of how close Alix was.\n\nAlix's finger traced the photos. There was a clear line of demarcation that flowed in fits and starts from photo to photo, forming a circle just like the one he'd seen in his dream. Half of it was shifted a bit, but it was there nonetheless, if you knew to look. \"About fifty kilometers wide.\"\n\nVenin sat back and whistled. \"What do you think it means?\"\n\nAlix shook his head. \"I have no idea, but it seems important. What do you say we go and have a look-see to find out?\"\n\nVENIN WENT off to pack the few supplies he had.\n\nAlix gave him a provisions code to order whatever else he might need from DOC and headed down to the medical wing to see his mother.\n\nIt was late. She was probably asleep, which would be just as well. He didn't need a confrontation.\n\nA weary Dr. Glynt looked up from the reception desk. \"Hello, Mr. Preston.\"\n\n\"Pulling a late shift?\"\n\nShe nodded. \"No one else came in with all the trouble in the city.\"\n\n\"Can I see her?\"\n\n\"Of course. Let me know if you need me for anything.\" She went back to manipulating a file tree on her tri-dee screen.\n\nHe poked his head into the room. She was fast asleep.\n\nHe stood by her bedside, looking down at the woman who had, as of the day before, been the most powerful person on the planet. She looked small and frail, her boundless energy drained away, but she looked peaceful too.\n\nHe was about to go when her eyes flickered open, and she looked up at him. A crooked grin crossed her lips. \"Got your attention, did I?\"\n\nHe laughed, more harshly than he intended. \"Is that all this was? I've got more important things to attend to\u2014\"\n\n\"More important than your own mother?\"\n\n\"At the moment, yes.\" He turned to go, but she caught the edge of his shirt.\n\n\"Please don't go.\"\n\nHe stopped, undecided.\n\n\"I'm.... I had my reasons for everything I did.\"\n\n\"That's it? That's the best you've got?\" He rounded on her. \"You were willing to wipe out half the planet, and 'I had my reasons' is the best you can do?\"\n\nShe let go of his shirt, slumped back down on the pillow, and looked away.\n\n\"Just this once, I thought, maybe she'll say she's sorry. Maybe she'll show some actual signs of the humanity she's managed to hide from me for so long.\" He turned to leave again.\n\n\"You're right.\" Her voice was so low, he almost didn't hear it.\n\n\"What did you say?\"\n\nShe sat up, some of her usual strength returning. \"I said 'You're right.'\" This time she said it clearly. Was there even a note of contrition in her voice?\n\n\"Right about what? You have to say it.\"\n\n\"Right about everything. About the war. About the kind of mother I am. About everything.\"\n\nAlix nodded. \"Now that's a start.\" He pulled a chair up to her bedside. \"You weren't really trying to kill yourself, were you?\"\n\nShe looked at him, strangely vulnerable without makeup or her usual clothes and hair. \"No. Not really. I had Zenix give me the extras. He refused, until I explained that I needed to use them for some OberCorp experiment or other. You didn't get all my override codes.\" She smiled thinly, but whatever pleasure she took in the thought didn't last. \"You know why I hate them.\"\n\n\"Who?\"\n\n\"The wing men.\"\n\n\"They're called skythane.\"\n\n\"Wing men. Skythane. Whatever. I told you what one of them did to me.\"\n\nHe nodded. \"He raped you. But they're not all like that\u2014\"\n\n\"He was your father.\"\n\nAlix froze. My father? \"There's no way.\"\n\n\"Oh, there was a way. I thought I loved him.\" She sighed, a long rattling sound.\n\nI'm half skythane. It was a strange thought. He'd wanted to fly forever, watching the skythane men and women with jealousy and eagerness. It was part of why he was obsessed with them.\n\n\"So you see? I was wrong. Your mother was wrong. I didn't want you to be like them. I didn't want them to exist. Not after what happened to me.\" She looked at him. \"But you, you were always enchanted by them. You wanted to run off to the Outland to find them. I just wanted you to be like me.\" She put her hands in her lap, and suddenly she looked old, her face drawn, her hands showing their age. \"What are you thinking?\"\n\nAlix shook his head. \"Honestly, I'm still trying to wrap my head around this. You knew? All this time?\"\n\nShe looked away. \"You must hate me.\"\n\nFor his entire life, she had lied to him about who he was. Where he came from. Now she wanted his forgiveness. He let go of her hand. \"I don't hate you. I... I just need some time.\"\n\nHe got up and walked out of the room, ignoring her calls.\n\nALIX WAS packing a carry sack, shoving clothing into it angrily, when Quince found him in the bedroom he was using in the OberCorp executive suites.\n\nThe sun was just climbing over the Pyramus Mountains, slanting through the plas windows.\n\n\"Did the sack have it coming?\" She leaned back against the doorframe and crossed her arms, her wings lying flat against her back.\n\n\"Hey, Quince.\" Alix pulled something out of the suite's replicator and stuffed it into the sack. \"Sorry. Bad night.\"\n\nShe arched an eyebrow. \"Going somewhere?\"\n\n\"Yes, Venin noticed something very interesting about the House of the Sky and the House of the Stars. I think it's relevant to the Hunters' investigation.\"\n\nThat got Quince's interest. \"Show me.\"\n\n\"Erissa, can you pull up the composite we put together of the skythane site?\"\n\n\"One moment.\" Alix's PA used the suite speakers to respond, likely out of politeness to Alix's visitor.\n\nA glowing slice of terrain appeared above the suite's tri-dee table. \"This part of Oberon is notoriously difficult to image properly. The company believed it was due to a dense deposit of amalite deep below the surface. The stuff messes with sensing technology.\"\n\nQuince nodded. \"What did you find?\"\n\n\"Venin had a dream. Or a vision?\"\n\nQuince shot him a look. Him too? \"Go on.\"\n\n\"He saw this place long ago, when the skythane first built the Houses.\" His finger traced a part of the composite. \"It's hard to see it unless you know what you're looking for. But look here.\"\n\nQuince peered at the image. \"What am I looking at?\"\n\n\"It's a circle.\"\n\nShe stared where he was pointing, and her gaze followed it around. \"Hmmm. That's... interesting. An old meteor crater?\"\n\nHe shook his head. \"I thought so too, at first, but the edge, where it's visible, is too perfect.\"\n\nShe took that in. \"You think\u2014\"\n\n\"I think we might be looking at the shift engines. The machines that the ithani built to bring this whole world from our universe to here.\"\n\n\"Damn.\" Quince whistled. \"They were practically under our feet the whole time.\"\n\n\"Maybe if we can get inside one of them, we can figure out a way to put a damper on the ithani's plans.\"\n\n\"It's worth a try.\" Quince sank down into the couch. The whole business was overwhelming. They had less than three days to decipher the artifacts of an alien race before they'd all be sent to kingdom come. \"What kind of vision?\"\n\n\"What?\"\n\n\"You said Venin had a vision, or dream?\" She'd been seeing things herself, of late. Nothing as extensive as what Jameson and Xander had experienced, but tantalizing bits of something.\n\n\"Oh. He said he dreamed it, or something.\" He turned to stare at her. \"He said he touched a statue. In a chapel in Errian, I think. Like Jameson.\"\n\nThat's interesting. \"Hell of a coincidence.\"\n\nAlix nodded.\n\n\"Who are you taking?\"\n\n\"Venin and Alia.\" He closed up his pack, sealing it shut.\n\nHidden technology, sleeping aliens, mysterious keys\u2014things had gone so off the rails. \"Okay. We have to try to cover all the bases.\"\n\n\"My thought exactly. Maybe we'll find Xander and Jameson there.\"\n\nShe noticed he no longer said Jameson's name with disdain. It was progress. \"I don't think so. I did some testing. It's pretty clear that it was some kind of gateway that swallowed up our two skythane kings.\"\n\n\"Pretty clear?\" He sealed up his carry sack.\n\n\"The radiation pattern matches the one in Xander's storage unit, but Ari thinks it was a more powerful version. That it sent them farther away than Errian. A lot farther.\"\n\nAlix put his sack on the table and stared at her. \"What's a lot farther from here than Errian?\"\n\n\"Exactly.\" Bandia, Titania's moon, was farther, but it was a barren wasteland. Beyond that.... \"I'm going to sit down with Mylin and see what she found. Maybe all these pieces will add up to something.\"\n\n\"That sounds good. I'll keep in touch over the grid.\" He tapped his temple.\n\n\"I'm not sure you'll be able to connect from the House of the Sky, let alone underground.\"\n\n\"DOC's building me an amplifier in one of the hoversports. I'm hoping it will be enough to punch through whatever dampens the signal down there.\"\n\nQuince frowned. \"How many hoversports does OberCorp have left? We took down a bunch of them over Titania.\"\n\nAlix grimaced. \"A lot fewer than I'd like.\"\n\n\"Sorry about that.\" She managed a half grin, half growl. \"I might need some firepower for the next step in my investigation.\"\n\nAlix nodded. \"Erissa, see that Quince here has access to the headquarters armory. Anything in particular you'll need?\"\n\n\"Nope. Just some high-power hand-carried weapons.\"\n\n\"Know how to use them?\"\n\nQuince laughed bitterly. \"Yeah. I've had a bit of training.\" Too much, in fact. She held out her hand. \"It's good to be working side by side with lander forces.\"\n\nHe took her hand and pulled her in for a hug. \"Agreed. Take care of yourself, Quince.\" He smelled like stress and exhaustion.\n\nShe squeezed him tight and then let him go, watching him walk down the hall toward the lift tube.\n\nLess than three days.\n\nShe went in search of something with caffeine. To survive those days, she was gonna need every bit of energy she could find.\n\n# CHAPTER TEN: QUESTS\n\nDANI AWOKE.\n\nShe was lying on something extremely comfortable. She could feel its support underneath her back and wings.\n\nHer wings.\n\nShe wanted to get up and dance, to flutter them about, to laugh and sneer at all her enemies, those who had taken her wings away in the first place.\n\nShe wanted to sing.\n\nHer body did none of these things.\n\nShe couldn't see, either. Something covered her eyes.\n\n\"You're awake.\" The voice came from inside her head.\n\n\"I... I am.\" What the hell?\n\n\"I've been waiting for you for a long time. A very long time. Danielle Black. Daughter of Danner Black. Born in the little Gaelani village of Toreor, mother deceased.\" Whatever it was recited the facts concisely, without emotion.\n\n\"What are you?\"\n\nIt chuckled inside her head. \"I thought you would have figured that out by now.\" Dani stood, but her body wasn't under her own control. She approached one of the white walls and waved her hand over it. It shimmered and became reflective, and she was staring into her own face. \"I'm one of the ithani. King of the ithani, if you will. My name is Jirron, but you can call me pain.\"\n\nShe tried to scream, but her body was still under the other's control. \"You're not me. You're... you're the thing in the dark.\" She should've been in full-blown panic, but her body remained steadfastly calm. Something had taken over her mind and body. Was her soul next? \"Who the hell are you?\"\n\n\"You're going to give me everything you know. About yourself. About humankind. About how the world has changed.\"\n\nShe nodded. \"I will. I'll tell you anything.\"\n\nDani felt herself grin, and it was the most unnerving thing she had ever experienced. \"Oh, you don't need to tell me anything. I plan to take what I need.\"\n\nDani squirmed in her bonds. \"You don't have to. I'll help you. Really, I will!\"\n\nShe felt its laughter. \"I wanted the chance to speak with you before you ceased to be. Personal preference, you understand. It's an honor, really. But now I'm afraid I'm growing tired of our little conversation.\"\n\nWhat the hell was this thing in her head? \"I can help you. I know a lot about the human world.\"\n\n\"You will. It's why you were chosen. Now calm yourself. Panic won't make this any easier on you.\"\n\n\"Why?\"\n\n\"Why? Oh, I see. Why give you back your wings, only to take everything else away?\" It sounded pleased with itself at having figured out her question. \"So I could use them myself. I will be more than ithani, you see. I will be human, too, and when the Migration begins, I will take all your skythane into me too.\" He spat the word out.\n\n\"Migration? What do you mean? What are you going to do to me? Please. Please! Just talk to me. I can help you\u2014\"\n\nThen it began.\n\nHer self unraveled, little bits peeling away and melting into nothing. Into it. Into him. For she was sure Jirron was male. Only a man could do this to a woman.\n\nShe could feel the pieces going. Each memory burned as it evaporated, searing her soul.\n\nShe tried to open her mouth to scream as her past, her loves, her hates, her very name were sloughed off like so much dead skin, but she was powerless to stop it. Powerless to even protest.\n\nWith everything she had left, Dani tried to scream.\n\nAs she was swept away into dust, she managed a slight grimace, but it didn't last.\n\nThe last thing she felt was the return of that terrible, involuntary smile, stretching its way across her face.\n\nMYLIN STARED at her wall in vir. It was covered with bits of information gleaned from her interviews of the other skythane. Robyn and Vestra had been particularly helpful.\n\nShe had a working history of the skythane population of Erro\u2014first Oberon and then Titania. The individual notes were coded for certainty. Purple marked the things she was most sure of, and green the least.\n\nShe stared at her wall. The virtual forest was almost entirely covered.\n\nThe skythane had come to Oberon, as best as anyone could agree, a bit over eight hundred years before. So far, so good.\n\nThe colonists must have been on a generation ship that launched just before the Collapse\u2014an event that many at the time assumed had ended life on Earth. One that she'd been unaware of, until now. There was so much she didn't know.\n\nShe'd asked DOC to put together a quick history of the Common Worlds dating back to the recolonization of Earth from Moonbase sometime after the Collapse.\n\nShe sat back and tried to picture it\u2014the destruction of an entire world. Fleeing from your home, certain that all you knew and loved existed no more.\n\nSomeday soon, she might know that feeling herself, if she survived.\n\nOberon had been rediscovered by a ship from a company called ZephyrCorp, headquartered on a revitalized Earth. The interstellar conglomerate would later be broken up by the Common Worlds government.\n\nFrom there, the records improved, at least on the OberCorp side. The company had started a new calendar that dated from First Landing, and she had a pretty detailed account of what had happened since on the Oberon side of Erro.\n\nThe company had recorded anomalies too\u2014the interference out at the ragged edge of the world, along the split, that caused electronics to fail. The strange area around the House of the Sky was notoriously difficult to capture in satellite imagery. The company experts had suggested that amalite deposits were to blame for both phenomenon.\n\nZim appeared before her, his little wings buzzing rapidly. \"Quince would like to see you.\"\n\nMylin nodded. Those things could wait, and Quince might be able to add more information to the mix.\n\nShe had Derren and DOC running a deep search through the company archives for anything that might point toward the ithani\u2014who and what they really were. Derren was amazing. He was as excited about organization as she was, and he knew the Oberon City grid like the back of his hand.\n\nIt was a slim chance, though. The ithani had been dead and gone\u2014or hibernating\u2014for a hundred times as long as humans had wandered this place, if not more.\n\nShe opened her eyes in rel.\n\nQuince was pacing back and forth in front of Mylin's desk, a frown on her face. Her wings fluttered, as if agitated on their owner's behalf.\n\nDerren looked like he was deep in vir.\n\n\"Hello, Quince.\" Mylin was both in awe of and afraid of the older skythane woman. Quince had a hard edge to her, and a no-nonsense attitude that Mylin strove to emulate, though she failed miserably. But she could also be quite biting when she was upset.\n\n\"Hey.\" Quince stopped pacing and sat down on the stool facing Mylin's desk. \"I was hoping you had some news for me.\"\n\nMylin shook her head. \"I'm still gathering information, I'm afraid. So far, I have more questions than answers.\"\n\nQuince raised an eyebrow. \"Such as?\"\n\nMylin glanced over at Derren. His eyes were unfocused. Poor guy was working on as little sleep as she was. She held up her hand. \"One: Where are the ithani cities? They were the masters of this planet for who knows how long before the first shift. Why haven't we found some kind of ruins? Derren brought that one up.\"\n\n\"Maybe their cities were like Errian. Made of living plants that have long since decayed?\"\n\n\"Hmmm, maybe.\" She made a mental note to look into that. \"Two: Who built the gateways and made the keys, and when? And who taught us to use them?\" She laughed ruefully. \"That one's all tied up in the memories somehow.\"\n\nQuince nodded. \"Good thinking so far.\"\n\n\"Three: Where were the skythane for a few thousand years? We know from human history they can't have left Earth more than about a thousand or fifteen hundred years ago\u2014probably before the Collapse? And yet I just got back a DNA sample test that puts us about ten thousand years apart from landers, genetically speaking. It's just not possible.\"\n\n\"Damn. That doesn't sound like nothing.\"\n\nMylin grinned. She loved these new tools Quince had given her and was already working out how she could get a grid in Errian and Gaelan for the skythane.\n\nFirst things first. No planet, no grid.\n\nQuince cleared her throat.\n\nMylin jumped. She'd been wandering again. \"Sorry. I get sidetracked easily.\"\n\nQuinn smiled. \"It's okay. I was just saying that I might be able to help you with the first one.\" She blinked. \"Ten thousand years? Really?\"\n\n\"Yes. Any idea why?\"\n\n\"Maybe. Let me give it some thought. In the meantime, Ari, can you show Mylin what Alix showed me?\"\n\nA piece of terrain appeared above her desk, rotating slowly.\n\n\"Where is that?\" Mylin leaned over to get a better look.\n\n\"The House of the Sky.\"\n\n\"Oh. Ooooh!\" A dotted line was drawing itself across the map, connecting bits and pieces of the terrain into a near-perfect circle.\n\n\"I think we may have found some of those ruins you were looking for.\"\n\n\"Underground.\" Mylin whistled. \"Okay, that helps.\"\n\nQuince nodded. \"Thought it might. Alix and Venin are going with Alia to see what they can find. They should be able to help you if you need some eyes on the ground.\"\n\n\"Thanks. Did you find Xander and Jameson?\" Mylin had a soft spot for the Gaelani king. He was a good man and had saved her and her family, along with all of Gaelan.\n\n\"Not yet, but I have another anomaly for your list. Where would a gateway go, if it used far too much power to be somewhere here on Erro? And where did the power come from?\"\n\n\"I don't know, but I'll add it to the list.\"\n\n\"Thank you.\"\n\n\"There's one more thing.\" She'd been hesitant to bring it up, as she didn't really know much about the system yet, but maybe it was important.\n\n\"What?\" Quince arched an eyebrow.\n\n\"I asked DOC to run a scan for anything unusual. Unaccounted for? Anything in the network that didn't seem to belong.\"\n\nQuince nodded. \"And?\"\n\n\"There are all kinds of things. Illicit... blackware, some of it? He shuts that down when it becomes a problem, but I guess there's a lot of it?\"\n\n\"Oberon City is a nest of vipers.\" Quince smiled. \"Go on.\"\n\n\"There's some static too. Maybe solar interference, or something from the amalite deposits. That's pretty easy to figure out because it's so random.\"\n\n\"Damn, you've really taken to this.\"\n\nMylin blushed. \"I'm doing the best I can.\"\n\nQuince got up and put a hand on her shoulder. \"You're doing a great job.\"\n\n\"Thanks. So anyhow, there's this other thing. I thought it was noise at first, but Derren helped me filter it out.\" It was weird.\n\nDerren heard his name and wandered over from his post. \"Yeah, it's... weird.\"\n\nMylin grinned. Great minds.\n\n\"Can you show her?\"\n\n\"Of course.\" He swiped the map away, and in its place a twining series of glyphs appeared. \"There's some kind of code worming its way through the OberCorp core. It's just weird.\"\n\nQuince squinted at the twirling column. \"What do the glyphs represent?\"\n\nDerren shook his head. \"We don't know. When we realized it was more than random noise, I asked DOC to look for repeating letters. Phrases. And to tag them with different glyphs. There are fifty so far that he's identified. They repeat, sometimes even repeating whole words, but overall the composition keeps changing.\"\n\n\"Could someone in Oberon City have written this?\"\n\nDerren shrugged. \"I suppose anything is possible, but it doesn't look like any code I've ever seen. Or DOC, for that matter.\"\n\n\"I'll take your word for it.\"\n\n\"Do you think it's important?\"\n\nQuince nodded. \"I think it might be very important. You were right to bring it up.\"\n\nMylin sighed. She was hopelessly outmatched by this task. \"Thank you. I'm scared I'm going to fail. To let you all down.\"\n\n\"You're doing great. Keep it up. You too, Derren.\" She gave the young man a clap on the shoulder.\n\nMylin's face felt hot. \"Thank you. Oh, I found this too.\" She brought up a map of Oberon City on her tri-dee desktop.\n\n\"What's this?\"\n\nOne spot glowed white, deep in the Slander. \"It's an injection site.\" She looked to Derren for confirmation.\n\nHe nodded. \"It's where we think this code is entering the network.\"\n\nQuince's face went white. \"This is an active attack?\"\n\n\"We think so. If it is an attack.\" She still was unsure about how this whole grid thing worked. \"It's definitely active.\"\n\n\"I'll look into this. I'll let you know if I find anything. You do the same?\"\n\nMylin was a bit unnerved by the serious look on Quince's face. She was glad she wasn't the woman's target. \"I will.\" She watched Quince walk out, still flying high on the compliment but terrified by the responsibility that sat on her shoulders.\n\nThen she closed her eyes and went back to work.\n\n\"ARI, WHERE'S Robyn?\" Quince picked up a heavy pulse rifle. It was a new design she hadn't seen before, stamped with the Renton Ringworlds logo. So OberCorp had been importing firearms, probably for the planned war with the skythane.\n\nThat was over now. They had bigger auxes to skin.\n\nShe closed her eyes for a moment. She was tired\u2014dead tired, if she was honest. She'd taken a couple stims already to keep going, and even though they were supposed to be side-effect free, she'd pay a price eventually.\n\nShe hoped Alix was being careful with access to these weapons. Should they fall into the hands of Rogan and friends, they'd have more than just the ithani and the end of the world to worry about. All the Syndicate needed was more weapons.\n\n\"She's on the roof. Should I tell her you're looking for her?\"\n\n\"No. You'd just scare the zimbees out of her.\" She shouldered two of the pulse rifles, and then started looking over the grenades.\n\n\"Can I... can I help you find anything, ma'am?\" The guard assigned to the armory was woefully out of his depth.\n\nQuince laughed ruefully. \"No, I'm good. Just picking up a few supplies.\"\n\nHe'd been reluctant to let her in, a point in his favor, but he hadn't known where anything was, either.\n\nShe'd invoked Alix's authority, and after a short conversation with his PA, and presumably with DOC, he'd relented and let her access the weapons.\n\nShe wanted something sharp and dangerous-looking, too, in case she needed to intimidate someone.\n\nShe found what she was looking for at the back of the armory. A box marked simply \"knives\" had some exquisite throwing blades from Beta Tau. Jameson's home planet eschewed guns, generally, but made up for it with an unseemly love of all things sharp. The microthin carbon blade looked wickedly keen.\n\n\"Bring me that staff?\" Quince pointed at one of a set of fighting rods leaning in the far corner without looking back.\n\nThe man did so, trying to hand it to her.\n\n\"Hold it out in front of you with both hands, please.\"\n\nHe nodded. \"Yes ma'am.\" His hands were shaking.\n\n\"Steady,\" she hissed. \"And don't call me ma'am. Quince will do.\"\n\n\"Yes, ma... Quince.\" His grasp steadied.\n\nShe brought up the knife with one arm and slashed down at the pole.\n\nIt cut clean through the metal.\n\nQuince grinned. That's the one I want. It came with a special holder... probably diamond-studded inside. Satisfied, she strapped it to her belt.\n\nShe looked around, searching the data streams. One of her enhancements let her \"see\" them when she wanted, a capability she'd lost when she'd burned out her bioware.\n\nNot that she'd needed it out in the Outland, or on Titania.\n\nTraffic was heavier than usual. People were worried, and with good reason.\n\nRumors of war, the planet itself rattling under their feet, and now the lords of the Slander taking over the city with their wing men cohorts. It was enough to freak out any lander.\n\nIncluding the poor man assisting her.\n\n\"What's your name?\"\n\n\"Tovey, ma'am. Quince.\" He was literally shaking in his boots.\n\nShe put a hand on his shoulder. \"Thanks for the help, Tovey. You're probably scared as shit, right?\"\n\nHe swallowed hard. \"Yes.\"\n\n\"Ever seen any action?\"\n\n\"No, ma'am. They just posted me here yesterday.\"\n\nPoor kid couldn't be any more than seventeen or eighteen standard. \"Just over a week ago, Erro almost ended in fire. Did you know that?\"\n\n\"No.\" He turned even whiter.\n\nShe squeezed his shoulder gently. \"We got through that. Compared to that, the rest of this will be a cakewalk.\"\n\nHe closed his eyes and nodded. \"I hope so.\"\n\n\"If you need me, call me.\" She had Ari send the boy her personal contact code. For some reason, she felt a connection to him. He was a human face on the events unfolding here in Oberon City and around the globe. \"If you can't get me, call Mylin directly. She's one of the skythane working upstairs. She can find me.\"\n\n\"Thanks.\" He looked around, and then back at her. \"Follow me.\"\n\n\"Sure?\"\n\nHe led her to the back of the room and palmed the door there.\n\nA panel slid open.\n\n\"These just came in before... before whatever it was happened last week.\" He pointed to a plas crate in the corner.\n\nShe knelt and popped it open. Inside were a set of circles of various sizes that gleamed even in the low light of the dim storage room. \"Are these...?\"\n\nHe nodded. \"I think so. I thought they were illegal in the Common Worlds.\"\n\n\"They are.\" She picked up a couple and slipped them into her carry sack. \"But we're not in the Common Worlds anymore.\"\n\nHe laughed ruefully. \"No. I guess we're not.\"\n\nShe picked out a crossbow for Robyn and a few flash grenades, and then said goodbye to Tovey, who looked slightly less green now. \"Thank you, and call me if you need me.\" She squeezed his arm, and then made her way back to the main hallway, waiting until he had locked up the room securely once more.\n\nThe weapons stored there were far too dangerous. What else did OberCorp have stashed away?\n\nOnce this was over, she'd make sure it was destroyed. All of it. She shuddered to think what might have happened if the skythane war with the landers had gotten out of hand.\n\nShaking her head, she said goodbye to Tovey.\n\nThen she went to find Robyn. They had a job to do.\n\n# CHAPTER ELEVEN: CRASH\n\nVENIN STARED at the hoversport. His fingers itched to pull out his pulse rifle, now fully recharged, and to leap on top of the beast to blow it to bits. It had been that kind of week. It was hard to see it as anything other than a tool of the enemy, and yet he was here with one of them, about to board it on a joint mission to save the world. What strange times we live in.\n\nOberCorp maintained a hangar for the craft in a separate complex, across from the imposing headquarters building. The building itself was huge. The entire House of the Moon\u2014and half of Gaelan\u2014could have fit inside.\n\nIt was also the training facility for the company's enforcers and rangers. Venin had gotten quite a few looks, many of them not friendly, from the OberCorp personnel as he made his way to the hangar.\n\nThere was a lot of bad blood between lander and skythane, especially these last few weeks, and it was going to take a long time to change that.\n\nHe sighed. Nothing to do for it now.\n\nTazim would have turned it all around with his easy charm. He'd been blond where Venin's hair was black. Light where Venin was dark-skinned. So full of confidence and life he'd seemed always on the verge of going supernova.\n\nTazim wasn't here. Tazim was long gone.\n\n\"Ready to go?\" Alix clapped him on the shoulder, ducking as Venin swiped at him with a wing.\n\n\"As ready as I can be.\" Damn you for startling me like that. Skythane wings needed to have teeth.\n\n\"Hey, watch it. It's just me. I brought Alia along too. Thought the extra support couldn't hurt.\" He was dressed in ranger black, a tight plas-leather suit that highlighted... just about everything.\n\nAlia nodded. \"Not just because you wanted a chaperone?\" She shot him a sly look.\n\nToo close. Venin looked away.\n\nAlix shook his head, looking innocent as a schoolboy. \"Of course not. You're skilled as hell, and you won't slow us down.\"\n\n\"Says the man with no wings.\"\n\nAlix nodded. \"Got that covered too. My bi-wings are packed and ready to go.\" He palmed the side of the hoversport, and the ramp descended to the ground.\n\nVenin wasn't quite sure what to make of the byplay between the two. He gave up trying, tucked in his wings, and followed Alix inside.\n\nAlix settled into the pilot's seat, running a systems check. He seemed subdued.\n\n\"Did you talk to your mother?\"\n\nAlix stared straight ahead, concentrating on the task.\n\n\"That bad, huh?\"\n\n\"It didn't end well. Can you take a seat? We'll be taking off shortly.\"\n\nOuch. \"You got it.\" He backed off, leaving Alix be.\n\nThe interior was all military\u2014no frills and no-nonsense. Aside from the pilot's seat, the troop benches that lined both sides were hard black plas, and too narrow to accommodate skythane. Venin growled under his breath. He'd be happy to get out of this lander hellhole\u2014he didn't know how Xander and Quince had managed to live here all those years.\n\nHe squatted, determined to give Alix a piece of his mind once they landed.\n\nAlix spun around. \"What are you doing?\"\n\n\"There's no place for me to sit.\"\n\nBehind him, Alia laughed. He turned to see her seated in a chair that was molded for her skythane frame.\n\n\"How...?\"\n\n\"Ask your PA.\"\n\nOf course. I'm an idiot. \"Taz, can you, I don't know... get me a seat?\"\n\n\"Please step to one side.\"\n\nVenin did as he was told, and the floor opened up. A seat just like Alia's rose to the appropriate height.\n\n\"What... is this a special hoversport, or something?\" As far as he knew, there were no skythane in the rangers.\n\nAlix nodded. \"It is now. I had it retrofitted for the trip. Stow your carry sack up there.\" He pointed to a bin opening above Venin's head, all business.\n\nVenin complied and sank down into the seat.\n\n\"There are straps that snap together around your waist.\" Alia pointed them out.\n\n\"How do you know all this?\"\n\n\"I helped with the retrofit.\"\n\nOf course you did. Venin sighed.\n\nAlix ran through a check of the craft's systems. \"Looks good. You guys ready back there?\" He was in his element.\n\n\"Like I said, ready as I'm going to be.\" He didn't mean to growl it. Really.\n\n\"Same here.\" Alia reached forward and squeezed his shoulder. Us skythane are in this together.\n\nThat last bit was inside his head.\n\nHe twisted to look back at her, and she tapped her temple.\n\nOh yeah. The cirq. Yes, this whole place was gonna take a lot of getting used to.\n\nIt looked like they'd be taking a bit of OberCorp and lander culture with them. He wasn't sure how he felt about that.\n\nThe craft lifted quietly up into the air, and the hangar doors cracked open, letting in Erro's pink morning sunlight. The hoversport accelerated, pushing him back into his seat, and they were off.\n\nALIX CIRCLED the confluence of the Theseus and Demetrius rivers. Unlike the Split, the region around the House of the Sky\u2014the ancient skythane \"castle\" a few hours inland from OberCorp\u2014didn't seem to mess with power sources. Thank the Split for small favors.\n\nHe'd scoured the company records that morning before they'd set off, looking for any record of the ruins or investigations into the area. He'd come up with a few things\u2014an exploration team who had gone through the region in the first century after colonization, for one. They'd determined there was indeed amalite below the site, but the mineral was far easier to extract from the Split, where it didn't require heavy excavation equipment. They made no mention of ruins.\n\nHe wondered if his mother had ever looked into it.\n\nThis close to the ground, it was hard to make out the circular formation that had been so evident on satellite imagery.\n\nThe waters of the Theseus and the Demetrius were still swollen, though it was clear they had receded from the rampant damage left behind on their banks\u2014logs and other plant debris, and the occasional bloated aux carcass. \"I don't see anything particularly promising as a point of entry,\" he called back to his companions. \"What was this place used for, before?\"\n\n\"It was a meeting place for the Gaelani and the Erriani, like the House of the Stars on Titania.\"\n\n\"Meeting place?\" Alix brought the hoversport down closer to the ground for a pass over the black castle that marked the middle of the circle. It was in reasonably good shape, and the outlines of the gardens that had once flourished around it could still be seen.\n\n\"Yes. Every fall, the two factions of skythane would gather in one of the two places to celebrate the harvest and to thank Erro, the sun god, and Gael, the moon god, for the bounty.\"\n\n\"Celebrate, huh?\" He glanced back at Venin, who nodded.\n\n\"Yeah. There was a bit of that, too, apparently. Fertility rites, and all.\"\n\nAlix laughed, letting go of his angst about his mother for the moment. \"Did you ever go?\"\n\n\"When I was a child. The last one was more than twenty-five years ago, on the other side of Erro. Relations between the Gaelani and the Erriani deteriorated pretty quickly after that.\"\n\n\"After Jameson's mother\u2014the queen\u2014was killed?\"\n\n\"Yes.\" Venin didn't seem to want to talk about it anymore.\n\nMy fault for being such an ass. \"Fair enough. So what do you think\u2014Holy Split.\"\n\n\"What?\" Alia and Venin said in unison.\n\n\"Well, that's new.\"\n\nTwo seat belts unlatched, and soon the skythane were staring over his shoulder, their wings taking up most of the space at the front of the small craft.\n\nAlix pointed.\n\nThe ground had opened up about a kilometer east of the House of the Sky in a series of parallel cracks that stretched through the patchy forest toward the Pyramus Mountains on the horizon.\n\n\"That is new.\"\n\n\"This is one of those times I wish we still had satellite coverage.\" He wondered what had happened to Titan Station and Oberon's two moons. They'd made it through the last shift, apparently, so they were probably still there... which implied some kind of gravitational presence in the \"human\" universe. Time enough to figure that out later, if they had a later. \"The one in the middle looks pretty wide. Should we go down and take a look?\"\n\nVenin nodded. \"Looks like the best way into whatever's down there.\"\n\nHis breath was warm on Alix's back.\n\nAlix rubbed his neck, trying not to think about it. There were more important things afoot than sexual attraction. Besides, he'd been the one to rebuff Venin.\n\nHe brought the hovercraft around and activated the forward lights. He took them down toward the widest part of the center split. When they were directly over the crevasse, he let the craft sink between the dirt walls.\n\nThe crack was at least fifty meters across here. Nevertheless, the walls seemed to close over them as they sank into the darkness. As he watched, steering the hoversport down toward whatever lay below, a shower of dirt slipped down through the running lights and onto the top of the craft.\n\nVenin grabbed his shoulder, then let go and looked away sheepishly. \"Sorry.\"\n\n\"'S okay. It's a bit freaky here.\" Alix leaned forward and peered through the plas at the diminishing band of light above.\n\nComing down here was a stupid idea. We'd be better off\u2014\n\nThe hoversport jerked hard to the right with a painful shriek, slamming into one of the crevasse walls. \"What the hell?\" He fought to bring the craft back under control.\n\nSomething was messing with the amalite power core. The charge indicator varied widely, shooting up to full and then back down to nothing in seconds.\n\nMaybe the whole power thing did apply here, after all.\n\nThe whole craft was wobbling now, dropping faster into the abyss.\n\n\"Back to your seats! I don't know if I can control our fall!\" He grabbed the navigation stick, fighting to level the craft. He fired off a couple of the emergency jets, but they only served to slow the descent minimally as the craft spun around, the lights flashing off the dirt walls. His stomach lurched as the ground\u2014a strangely white ground\u2014came up to meet them.\n\n\"Shit. Get ready for\u2014\"\n\nHis voice was cut off by the impact foam that filled the little craft, and his vision went dark.\n\nVENIN WOKE in darkness.\n\nA red light flashed briefly above him, illuminating the scene.\n\nHe was covered in something sticky, and his left wing throbbed painfully.\n\n\"Are you okay?\" The disembodied voice was disorienting.\n\n\"Taz?\" His PA. He was still getting used to sharing his head with someone else. \"Yeah. I think so. Hurt my wing, but other than that....\"\n\nAnother red flash. Alia was slumped over in her chair behind him. He couldn't see Alix behind the big pilot's seat.\n\n\"The hoversport's amalite core is unstable. You need to evacuate the craft.\"\n\nTaz's voice was calm, and it helped Venin regain his own stability. \"How do we get out?\" The stickiness was from the protective foam that had filled the cabin just before impact. He shook his hands to get as much of the stuff off as possible. It smelled sweet, a little like pith sap.\n\nThe ship lay tilted on its side. He dropped down to what used to be the wall and shook off his wings. He was ankle deep in decomposed foam, which sloshed around under his feet and slowed his movements.\n\n\"There's a red emergency release next to the door.\"\n\nHe shook Alia's shoulder. During flashes, her eyes opened. \"You okay?\"\n\nShe nodded. \"I think so.\"\n\n\"Get Alix. This thing's going to go.\"\n\nAlia's eyes widened. Apparently she'd just gotten the same news from her PA.\n\nAs she released herself and went to check on Alix, Venin popped open the bin where his carry sack was stowed and pulled it out. Slinging it over one shoulder, he knelt and felt around in the sticky liquid for the release. \"What am I looking for, Taz? Describe it to me.\"\n\n\"It's a long red handle\u2014\"\n\n\"Color doesn't help me right now.\"\n\n\"Sorry. Find the door seam.\"\n\n\"Got it.\"\n\n\"Now feel to the left about fifteen centimeters\u2014\"\n\n\"Fifteen what?\"\n\n\"About a handspan wide.\"\n\n\"Thank you.\"\n\n\"You're welcome.\" Little bastard sounded smug. He felt where Taz told him, and his hand closed around a long metal rod. \"Found it. Should I pull it?\"\n\n\"Yes, but be careful\u2014\"\n\nHe pulled the handle, and the door slid open. The ramp pushed the ship back toward its natural upright position as it extended, throwing him back on his ass.\n\n\"You okay?\" Alia loomed over him with an unconscious Alix in her arms.\n\n\"I think so. Is he okay?\"\n\nAlia frowned. \"He won't wake up.\"\n\nHe put his hand on Alix's neck. It was warm, and he still had a pulse. \"He'll be okay. He's a tough bastard.\"\n\nA high-pitched whine filled the cabin.\n\n\"Destruction of this craft is imminent.\" Taz's voice carried the unheard request, Can we move this along?\n\n\"Come on.\" He grabbed his carry sack and scrambled down the slippery ramp.\n\nThe lights of the hoversport illuminated the way ahead of the craft. The ground was white and seamless.\n\n\"This way!\" Venin took Alix's legs and they ran away from the hoversport as fast as they were able, encumbered by the ranger's large form.\n\nWhat the hell are we gonna do when the lights go out? No time to worry about that now.\n\nBehind them the whine increased to a piercing scream.\n\n\"Get down!\" They dropped Alix and flung themselves onto the ground. Venin barely registered the fact that it was smooth and yielding as pain surged through his left wing.\n\nVenin covered his ears as the hoversport's scream reached a crescendo, and then the crevasse was lit with a light as bright as the sun.\n\nA roar of hot air raced over them, singeing the tips of his wings and leaving a horrid smell.\n\nThen it was over, and there was only silence and darkness.\n\n# CHAPTER TWELVE: IN A FLASH\n\nJAMESON FLEW close to the ground, keeping a low profile as they covered as much distance as possible. The red sun was nearing the horizon. It looked angry, reminding him of their first flight\u2014on hoverbike\u2014through the Oberon outback.\n\nThey'd continued on for the rest of the afternoon in the direction the initial sail craft had gone, at least as far as they could tell. For a long time, all they'd seen was the endless hills all around them, and the mountains in the distance.\n\nThey'd taken a couple breaks, mostly for water and so the nimfeach could stretch its wings.\n\nHe had offered it water from his canteen. It had sampled the liquid, then drank half of his supply. It had handed back the canteen with its three-fingered grasp, its big eyes blinking in what Jameson hoped was a gesture of thanks. Now it flew alongside them, and if it was getting tired, it showed no indication.\n\nAs afternoon slipped into evening, they were approaching something. Exactly what it was wasn't entirely apparent. A white smudge in the distance. In the meantime, several more ships had passed overhead, all of the sail variety.\n\nBoth times they had dived for the cover of the grasses and gullies below them, but no one had stopped to see who or what they were.\n\n\"So what's the plan?\" Xander's eyes were focused on the horizon. He'd been distant ever since the incident at the fallen ship. It was clear he didn't approve of Jameson's actions.\n\n\"We need to find one of the locals, whatever they are, and see if we can communicate with it.\" He couldn't explain why he had done what he had done. It had just seemed utterly imperative at the time. The nimfeach had needed help. He was someone who helped.\n\n\"You don't think there are any humans here?\"\n\nJameson shook his head. \"Do you?\"\n\n\"I don't know.\" Xander looked troubled, his usual self-assurance under siege by the strangeness of the situation. \"We don't have that much food. Or water. Is it a good idea to give... that thing... our water?\"\n\nJameson flapped at the air angrily, pulling ahead of Xander by a few meters.\n\n\"Hey, sorry. It's just that I'm scared.\"\n\nJameson looked back, seeing the uncertainty painted across Xander's face. \"Dammit. I'm sorry. I keep thinking of you as the strong one.\" He looked over at the nimfeach. It was keeping up with them, its face intent on the horizon. \"Should we stop for the day? I could use a rest. I've seen a few streams along the way. At least we could test the water, and maybe refill our canteens?\"\n\nXander nodded. \"Thanks.\" They flew on for a bit until they crossed another stream. This one wound its way through the plains from the mountains in the distance, and its banks were sparsely graced with some kind of tree that resembled colifer trees back on Oberon.\n\nThey alighted. The nimfeach sighed happily and settled in the shallow waters at the edge of the stream, sinking its feet in and collapsing backward.\n\nAlarmed, Jameson reached out for it.\n\n\"I think he... it? Them? I think it's just relaxing.\"\n\nIndeed, it didn't seem to be in any distress. The water flowed past its three-toed feet, and it breathed slowly and easily, its wings spread out across the water.\n\n\"What do you think it eats?\"\n\nXander shook his head. \"I have absolutely no idea.\"\n\nJameson reached out and pulled Xander to him, wrapping his arms around him and whispering in his ear. \"We'll figure this out.\" Xander's skin felt good on his own, even if they were both sweaty and dusty. It was a little bit of normal in a strange situation.\n\nHis body responded appropriately.\n\nXander laughed, dispelling the tension between them. \"Really? Here we are in the midst of an alien world, in front of an actual alien, and you want to fuck?\"\n\nJameson blushed. \"I can't help it if you have that effect on me.\" He wasn't sorry. For so many years he'd had to suppress that part of himself. Not any longer. He gave Xander a quick peck on the cheek. \"I know it's not the right time.\" Damn, I wish it were. \"We should eat something and figure out what to do next.\"\n\nXander nodded. He unsnapped his carry sack and rummaged around inside. \"I'm afraid we don't have much. A little dried fruit and bread.\" He pulled them out and then extracted his canteen. Kneeling next to the stream amidst the reeds, he took a cupful of water in his hand and sniffed it. \"Not bad.\" He swallowed it and smiled. \"Tastes like water. A little... um... rustic, but water nonetheless.\" Satisfied, he filled the canteen.\n\n\"I may have raided the OberCorp pantry while we were there.\" Jameson opened his pack and pulled out some fruit, a fresh loaf of bread, and some yogurt cups.\n\n\"When did you get those?\"\n\n\"Just before the meeting. I was hungry.\" He blushed.\n\nXander grinned. \"Oh my God, I could kiss you.\" And he did.\n\nXANDER KNELT in the high grass of the hillside, behind a boulder lit pink by the moonlight.\n\nThey'd finished their meal and decided on a plan of action. Approach the city, or whatever it was, on the horizon, under cover of darkness, and try to find someone to talk to.\n\nJameson had tried to communicate with the nimfeach, but all he'd gotten was a vague feeling of unease from their contact.\n\nXander had a sneaking suspicion about where they were. When they were, actually. Possibly. Little things that seemed familiar were starting to add up. Though how they'd gotten there\u2014and why\u2014was still a mystery.\n\nTogether, they peered at the structures below. It was like looking at Errian.\n\nA cluster of white towers\u2014turbien, they were called, if he remembered correctly\u2014sat on the edge of a massive bowl, glowing softly in the moonlight and obscuring much of what lay beyond. The towers were made of the same giant white plants as Errian, one of the bits of information that was starting to spark a realization in his head.\n\nHe was pretty sure the trees by the creek had been colifers, and he'd found river apples, too, which had helped to supplement their meager stores.\n\nThis place was either Oberon or something eerily similar to it.\n\nThey'd offered the nimfeach some of their food. It had ignored the bread but had made a meal of some of the dried fruit. He hoped the strange nutrients did it more good than harm.\n\nIt stood on the rock between them, pointing at the white city and emitting a soft, plaintive whistle. But it was the scene closer to their hideout that had Xander's attention.\n\nThey'd come upon it by chance, seeing a glow ahead of them that was distinct from the glow of the towers.\n\nAs they approached, they'd dropped to the ground to avoid being seen. Creeping up the hillside, they'd found a scene like something out of a fairy tale.\n\nA wide river flowed lazily through the valley below them, the sound of bubbling water floating up to their hidden perch.\n\nAlong the riverbanks were hundreds of things. Creatures. He was at a loss for how to describe them. \"They're...,\" he whispered.\n\nJameson nodded. \"They're ithani.\"\n\nThey were closer than the ones they'd seen at the crash site. Xander stared at them, enchanted by their alien beauty, tall and slender. Their bodies had a thorax like a bee, and six limbs. Two seemed to function as legs, while the third set sported the most glorious wings, glowing rainbow-iridescent things twice as tall as the ithani themselves.\n\nThey looked a bit like Puck, the little nimfeach who had attached itself to them.\n\nYet for all their similarities to insects, they were not insectile at all. Instead they were sleek and beautiful, their golden and silver skins almost seeming to give off a light of their own.\n\n\"What are they doing?\" They seemed to be engaged in a complicated dance, some of it taking place on the riverbank, some of it in the river itself, and some of it in the air.\n\nJameson shook his head. \"I don't know. Look!\"\n\nThree of the ithani rose almost directly in front of them. One was silver, one was gold, and one a pale green.\n\nBehind them, Puck huddled behind a bush.\n\nThe ithani spun around and around one another as they soared into the sky, faster and faster, so fast that Xander felt dizzy just watching them. It was an intricate dance, as choreographed as anything he'd ever seen on a dance floor or in a tri-dee.\n\nThey were focused on each other so tightly that Xander doubted they saw anything else.\n\nThey spun together, tighter and tighter, and then there was a brilliant flash.\n\nHe blinked, unable to see for a moment from the afterimage that was plastered across his optic nerve. His vision cleared, and something plummeted to the earth, slamming into the ground a couple kilometers away.\n\n\"I'll be Split-damned.\" Xander stared at the other ithani spinning through the sky. Another triad erupted in light. \"What do you think they are doing?\"\n\n\"Mating.\" Jameson was tight-lipped, the expression he had when he was fighting off memories. \"Angie, can you record this?\"\n\nThere was a rustling sound behind them.\n\nXander spun around to see a tall ithani standing there staring at them through rainbow-colored multifaceted eyes, its skin golden under Lysander's pale light.\n\nJAMESON WAS done with hiding, with running away from things.\n\nHe stood and turned around, facing their purported enemy for the first time.\n\nHe\u2014she?\u2014was beautiful, her shimmering carapace silvery-gold, pooling moonlight at its edges. Her wings were twice as big as she was, shaped somewhat like a butterfly's.\n\nShe cocked her head and stared at him with those unnerving, multifaceted eyes.\n\nHe could see the similarity to the nimfeach now. Maybe they were ithani children?\n\n\"Jameson, we should go. We don't know\u2014\"\n\n\"It's okay. She's not going to hurt us.\" He felt it in his bones, but he couldn't have explained how he knew if he'd tried. Jameson squeezed Xander's hand, then took a step forward, his own hands held open so the ithani could see he meant no harm.\n\nPuck, the nimfeach, cowered behind his leg.\n\nHer arms were more human than he would have expected. Far from being insect-like, they were almost as thick as his and covered with a fine, downy golden fur that extended to her body.\n\nSome kind of black fabric was wrapped around her in a wide sash from her left shoulder to her right thigh.\n\nHe held out one hand, reaching halfway toward her.\n\nShe cocked her head again, then reached into her sash and pulled out a compact, shiny object.\n\n\"Jameson\u2014\"\n\nShe held it out, and then a bright flash consumed his vision, leaving him blinded.\n\nAs he collapsed to the ground his last thought was, This was a bad idea.\n\n# CHAPTER THIRTEEN: INTO ENEMY TERRITORY\n\nQUINCE FOUND Robyn on the roof, as Ari had promised.\n\nRobyn looked as tired as Quince felt. No rest for the world savers. One day, all this would be over, and they would find a quiet place to slip away to, far from the cares and worries of the world.\n\nRobyn was seated on a bench on the rooftop, staring out at the city below.\n\n\"Hey, gorgeous.\" She kissed Robyn's cheek from behind, jumping back as Robyn took a swipe at her with her black wing. \"In a mood this morning?\"\n\nRobyn grimaced. \"Sorry. I'm too old for this, Quince.\" She stood and stretched her arms and wings. \"I like it up here. It's wide open, and I can breathe the fresh air from the sea.\" She eyed Quince's carry sacks and weapons. \"Going on a hunting trip, are we?\"\n\nQuince grinned. \"Something like that.\"\n\n\"I was always better with a bow and arrow than one of those lander guns.\"\n\n\"That's not for you. This is for you.\" She handed over the crossbow she'd found in the armory.\n\n\"Oooh.\" Robyn took it and turned it over appreciatively. \"Any bolts to go with it?\"\n\n\"Of course. Here you go.\" She handed Robyn a quiver.\n\nRobyn unbuckled it and slipped it over her shoulder behind her wings. She drew one of the bolts and fit it expertly into the crossbow.\n\n\"Be careful. Some of those arrows are\u2014\"\n\nA small explosion split a decorative tree in a ceramic pot in two, about twenty meters across the roof from where they were standing.\n\n\"\u2014explosive.\"\n\n\"I've used one like this before. One of the landers let me try it.\"\n\n\"In Gaelan?\"\n\nShe nodded. \"This will do nicely. Thank you. So who's the other pulse rifle for, then?\"\n\nThe lift shaft doors behind Quince slid open.\n\n\"Sorry I'm late!\" Jessa almost bounded into their midst. So young, so full of energy. Quince envied her that.\n\n\"I ran into her on the way up. She's as involved in all this as any of us.\"\n\n\"It's all right. Jessa and I had a nice talk while you were gone. But how is she going to come with us? I assumed we were flying?\"\n\n\"Oh yeah, I checked out a set of bi-wings from the armory. That Tovey guy is really sweet.\"\n\nQuince made a note to have a word again with \"that Tovey guy.\" He needed to be more careful who he let into the armory.\n\n\"So where are we going?\" Jessa strapped on the bi-wings.\n\nQuince led them to the edge of the roof. The city was quiet\u2014the calm before another great storm. DOC had grounded all nonessential traffic, and only OberCorp personnel and Rogan's forces were allowed abroad in the city. Things had gotten bloody on that last count, apparently. \"See that place over on the edge of town, where the roads end?\"\n\n\"The Slander?\"\n\nQuince nodded. \"Very good.\"\n\nJessa blushed. \"Alix was telling me about it. It's where the Syndicate lives, right?\"\n\n\"Yes. Something like that. There's something there we need to check out.\"\n\n\"What kind of something?\" Robyn frowned. \"I was hoping we would be getting out of the city.\"\n\n\"Not sure yet. There's an interloper in the city grid, and Mylin thinks it's connected to the ithani. It's underground, and our access point is that large gray building\u2014there.\" She pointed at one of the bigger structures that was still standing in that part of the city. \"The Slander grew up in what was once a warehousing district. The Slander bosses still use that one to house a lot of their contraband.\"\n\n\"Why doesn't OberCorp do something about it?\" Robyn frowned.\n\n\"Let's just say they find the Slander and its denizens useful sometimes.\" The whole system was corrupt. Poor Alix would find that out if he remained at the head of OberCorp for long. \"Let's get moving\u2014\"\n\nThe building shook.\n\n\"Hold on!\" Quince grabbed on to an exhaust pipe and hooked her arm around Robyn.\n\nJessa scrambled backward, grasping for something to hold.\n\nThe building shuddered and moaned, and decorative pots came crashing down to burst into pieces, showering the rooftop with leaves and dirt.\n\nThen it was over.\n\nJessa stood and brushed off the dust. \"Well that was fun.\" She spread her iridescent bi-wings. \"Are we going to do this?\"\n\nQuince got up and peered out over the city. \"Yes.\"\n\nThe arcos had survived without much damage, but a number of smaller buildings had collapsed. \"Heaven and Erro, it wasn't supposed to go like this.\" Shift the world, and then shift it back. That was what she had signed up for. Not this ongoing, rolling series of disasters. \"Come on. Time must be growing short.\" She leapt off the parapet, and Robyn and Jessa followed.\n\nROBYN TURNED her nose up at the stench.\n\nThey stood on another rooftop, this one the old warehouse in the heart of the ramshackle collection of trash and human debris Quince called the Slander.\n\nA little insect thing buzzed around them, zipping back and forth in a schizophrenic pattern. A bug drone, Quince had called it, holding it up in her palm before releasing it into the air. Such a tiny thing, made by the hands of man\u2014amazing that it could fly.\n\nShe wanted to knock it out of the air.\n\nApparently it served to scramble the vision of their enemies, somehow. At least, that was the closest she'd gotten to understanding Quince's explanation.\n\nAs Quince knelt by the door that would take them into the warehouse below, her palm to the pad in apparent communion with the wall, Robyn waited with Jessa.\n\n\"I've read about this place.\" Jessa shivered and looked around, though in truth there wasn't much to see from their current vantage point besides roof. At three stories tall, the building was one of the tallest in the Slander.\n\n\"It's not someplace you want to be caught alone in, especially at night. Though I'm not all that thrilled to be here during the daylight hours, either.\" Landers lived like pigs in their own filth. There was no reason she should have to breathe their dirty air too.\n\n\"Got it.\" Quince let go of the palm pad and the door swung open quietly, belying its apparent age and state of disrepair.\n\nInside, a metal staircase led down into the building.\n\n\"Follow me.\"\n\nRobyn shrugged. It looked cleaner in there than out here.\n\nShe tucked in her wings and followed Quince down the stairway.\n\nIt was cleaner. The air inside was cool and fresh, and for all its apparent decrepitude outside, the warehouse was in much better shape inside.\n\nThey came out shortly onto a catwalk, high above the warehouse floor.\n\nQuince knelt to take in the activity below, and Robyn and Jessa followed her example.\n\nThe bug drone buzzed by her ear, and she batted at it unconsciously.\n\n\"Let it be,\" Quince hissed. \"It's keeping us safe.\"\n\nRobyn frowned. The little insectoid annoyed the hell out of her.\n\nBelow them, bigger things flitted back and forth, carrying or dropping off items from various shelves.\n\nFrom their current vantage point, they could see down several rows. The only activity seemed to be from the hovering... drones, she supposed they must be. Devices doing the work of humans. \"So how do we find this interloper?\" She slipped her hand around Quince's waist while they surveyed the warehouse. Quince was the one good thing she had. She nuzzled Quince's neck, wishing this whole thing was over.\n\n\"Hey, we're on a mission here, you two.\" Jessa looked stern, but Robyn could hear the laughter in her voice.\n\n\"She's right. Time for that later. Where are we going?\"\n\n\"DOC helped me track him to somewhere beneath this warehouse. So we need to make our way to the ground floor and\u2014\"\n\n\"What are you doing up here?\" A man wearing a black uniform had appeared around the corner at one end of the catwalk, holding a pulse rifle.\n\n\"Rogan sent us,\" Quince lied smoothly. \"To check your security.\"\n\nThe man frowned. \"I don't work for Rogan. Who are you?\"\n\nPartially concealed behind Quince, Robyn pulled the crossbow off her back and slipped in a bolt.\n\n\"We're here to test site security\u2014\"\n\n\"Down!\" Robyn pushed Quince's back and her lover collapsed to the metal catwalk floor at the same instant as Robyn's crossbow bolt shot forward just centimeters over her spine to lodge in the man's neck.\n\nIt was one of the nonexplosive bolts, apparently.\n\nThe guard flew backward, getting off one pulse shot over their heads before falling sideways, clutching at his throat to try to pull out the bolt, to stop the sudden torrent of blood.\n\n\"Oh hell.\" Quince leapt forward to grab him, her wings beating to pull her toward him, but she was too late.\n\nHe fell over the edge of the catwalk, and her hands grasped at empty air.\n\nRobyn watched as he fell two stories to the ground, his mouth open in a silent scream.\n\nHis body hit with a sickening thunk.\n\nRobyn turned away, forcing down the bile in her throat.\n\n\"Holy crap.\" Jessa stared down at the body. \"You... you killed him.\"\n\nThe drones continued their work, unfazed by the dead body in their midst.\n\nRobyn squeezed her shoulder. \"Too much at stake to take the chance.\"\n\nQuince shuddered. \"I'd hoped it wouldn't come to this.\"\n\nJessa was white as a sheet. Poor girl had probably never had to finish off an opponent before.\n\nRobyn hugged her. \"It never gets easier, but you learn to live with it.\"\n\nQuince nodded. \"She's right. Come on. We better get moving before someone human finds the body.\"\n\nRobyn stared after her. It was clear that Quince was as shaken by the man's death as she was; she just didn't want to show it.\n\nShe turned to follow Quince down the catwalk.\n\nJESSA WATCHED Robyn walk away. She could see why Robyn had been a queen. She had the imperial bearing that only those bred to royalty could ever truly aspire to.\n\nShe was also sexy as hell. Drat, that's inappropriate. A man had just died in front of her. Maybe it was her way of distracting herself from that grisly fact.\n\nShe didn't let her attraction to another woman bother her. She'd come to terms with all that long ago\u2014before she'd ever known Jamie was gay. So what if she liked both sides of the coin? They seemed to be a whole lot less uptight about it here than back on Beta Tau.\n\nWith a heavy sigh, she followed Quince and Robyn along the catwalk.\n\nAround the next corner, a long metal ladder descended to the ground.\n\nQuince stopped to check something with her PA, nodded, and then continued on past it.\n\nJessa had never had one installed\u2014bioware was frowned on in Beta Tau. It's not the way God made you, her mother used to say to her.\n\nHe didn't give you that perfect nose, either, she'd said once, and got slapped and lost her grid privileges for a week. Jessa grinned to herself. It was so worth it.\n\nAt last, Quince found what she wanted. \"We need to go down quickly and quietly,\" she whispered, indicating the long metal ladder.\n\nThe bug drone drifted around them in lazy circles.\n\n\"Then what?\" Jessa looked back the way they'd come. No alarm, no one following them. Yet. She fingered the trigger on her pulse rifle.\n\n\"There's a hatch in the floor about twenty meters to the right of the ladder, near the back wall. Make for that if anything happens. Here.\" Quince handed them each a small something.\n\nJessa held it up in the dim light. It was a pulse grenade. \"Holy sheister.\"\n\nQuince grinned. \"You can take the girl off Beta Tau....\"\n\nJessa snorted.\n\n\"What is it?\" Robyn was staring at her own grenade.\n\n\"It's a pulse grenade. See this?\" Quince indicated a small loop on the top. \"Twist it and pull. Then you have about five seconds to clear the area before it blows.\"\n\n\"Ah.\" Robyn's nose wrinkled, and she put it into her carry sack.\n\n\"I don't think you'll need it. I'm hoping to get the hatch open by less... explosive means. I brought them just in case. Robyn prefers more traditional ways of killing people.\"\n\nJessa nodded. \"I respect that.\" She'd never killed anyone herself and hoped she'd never have to. The image of the guard clutching at his throat as blood gushed out stuck in her mind. How did you unsee something like that?\n\n\"Come on then.\" Quince swung herself over the edge and slid down the ladder, her wings slowing her fall.\n\nI wish I had wings. Jessa tucked away her own grenade in a pocket of her jacket and followed after Robyn.\n\nAbout halfway down, as the others outpaced her, an alarm finally sounded, blaring through the great open space. Worried, Jessa glanced around. Red lights were flashing at the four corners of the warehouse, and the drones were lifting out of the shelves to scan the space.\n\nShe grabbed the sides of the ladder and took her feet off the rungs. With a deep breath, she let herself slide down out of sight into the relative safety of the rows of storage shelves.\n\nThe metal chafed her hands as she slowed herself the last meter before impact. \"Fuck, that hurts,\" she swore, and Quince grinned. \"We've been noticed.\"\n\n\"Seems like it.\" Quince pulled her pulse rifle off her shoulder. \"Come on.\" She tucked in her wings and ran toward the hatch.\n\nRobyn and Jessa followed.\n\nSomething hot whipped past Jessa's shoulder. \"Behind us!\" She ducked and rolled, coming up to her knees to fire back at whomever was following them, her pulse rifle set to stun. Four people in black dove for cover.\n\nJessa grinned and slipped behind a tall shelving unit.\n\nReturn fire flew through the space where she had just been, and some of it slammed into the metal braces of the shelf, turning them red-hot.\n\nThey were shooting to kill. She shuddered.\n\n\"Quince, Robyn, you okay?\" She glanced across the aisle. Quince crouched behind another shelf.\n\nRobyn was nowhere to be seen.\n\n# CHAPTER FOURTEEN: BENEATH THE HOUSE OF THE SKY\n\n\"HEY, YOU with us?\"\n\nAlix's eyes flickered open to find Venin staring down at him, his brow knotted in concern.\n\n\"I think he's awake!\" Venin said to someone next to them, and a bright light flashed into his eyes.\n\nAlix squeezed them shut, registering the fact that his body ached in a hundred assorted places. \"Get that out of my eyes.\" He pushed himself up and immediately regretted it as all of his muscles registered their protests at once.\n\nHe opened his eyes again to see a sheepish Alia holding a flashlight, pointing it down at the ground. \"What happened?\" He remembered flying over a crevasse, but things were a bit muddy after that.\n\n\"We crashed.\" Alia turned the flashlight off.\n\nAlix's eyes adjusted. It wasn't totally black\u2014a thin strip of sky provided a faint light, and the ground... it glowed. Dimly, but it was there. \"And the hoversport?\"\n\n\"Gone. The amalite core blew after we hit the ground.\"\n\n\"Shit. Erissa, can you reach DOC?\"\n\n\"Sorry, DOC seems to be unavailable.\"\n\n\"My PA can't reach the grid. Can yours?\"\n\nAlia and Venin shook their heads.\n\n\"We can't connect to anyone.\" Venin looked around at the gloomy space. \"I think we're too far down.\"\n\nAlix laughed mirthlessly. \"Yeah, that's what the amplifier on the hoversport was for. Don't suppose either of you managed to grab my carry sack on the way out?\"\n\n\"Sorry. We had our hands full with you.\" Venin grinned, his teeth white against his dark skin.\n\n\"Thanks. Want to help me up?\" He held up his hand.\n\nThe ground beneath him thrummed.\n\n\"What the hell?\"\n\n\"Yeah, that's been happening a lot.\" Alia's tone was dry, like it was just another everyday thing for the ground to vibrate underneath you with a thrumming sound.\n\nAlix closed his eyes as a shower of dust and pebbles was shaken loose from somewhere up above. \"This doesn't seem like a particularly safe place to hang out.\" He eyed the crevasse walls, worried they might decide to snap back together as quickly as they had pulled apart.\n\nThings settled down again, and Venin helped him to his feet. His legs ached, but he ignored them. Time for rest later.\n\nHe was covered in something sticky\u2014probably crash foam. The stuff was hell to get out of hair and clothes. At the moment, though, they had bigger worries.\n\nVenin shifted, and Alix could see that his left wing hung at an odd angle. \"Did you break it?\"\n\n\"I don't think so. Just a bad sprain.\"\n\n\"May I?\"\n\nVenin's eyes met his, and something passed between them. \"Go ahead.\"\n\nAlix ran his hand along the edge of the wing, and Venin hissed. \"Sorry. Looks like something cut you. Wish I had the med kit from the hoversport. Can you fly?\"\n\n\"No.\"\n\nAlia looked at Venin's wound. \"This kind of damage is fairly common among the skythane. He'll heal, but it will take a few days.\"\n\nDays we don't have. Alix looked at their surroundings.\n\nThe place where they stood was like a long, dark canyon, stretching off into darkness in either direction. \"Erissa, how long is this crevasse?\"\n\n\"I'm sorry, I still can't access the grid. From visual estimates alone, five-point-four kilometers.\"\n\n\"Thanks.\" The walls were rough-hewn rock and soil. He took a closer look at one. It was irregular, hard-packed dirt and stone. But the floor....\n\n\"It's strange, right?\" Alia knelt with him to touch the smooth floor. \"This shouldn't be here.\"\n\nHe'd thought it was metal, but no, it was different. It was shiny, but with a bit of give. It reminded him of something, but he couldn't put his finger on it.\n\n\"How long before someone comes looking for us?\" Alia's wings fluttered. \"Maybe we should fly out and one of us should go for help?\"\n\nAlix rubbed his chin. \"Maybe this is what we came looking for.\" He stared up at the thin line of sky. \"But you're right. We do need to send someone back. They're going to have their hands full enough without worrying where we went off to.\"\n\nAlia nodded. \"I can go. Are you two sure you'll be safe here without me?\" She shot Venin a grin. \"All alone together?\"\n\nAlix rolled his eyes. Like he had time to think about that. \"Leave whatever you can spare.\" Alix took a deep breath. \"The air seems good. We can take a look around while we wait for you.\"\n\n\"Just be careful.\" She pulled out her foodstuffs and a pulse rifle, laying them on top of Venin's carry sack. Then she stood before him.\n\nThey touched wingtips. \"Gael be with you,\" she said, almost too softly for Alix to hear.\n\n\"And with you.\"\n\nShe gave Alix a quick kiss on the cheek, and then launched herself toward the sky.\n\nHe watched her until she disappeared into the river of light high above.\n\nVENIN WATCHED Alia depart, ignoring the pulsing pain in his left wing. She wasn't wrong. Venin was acutely aware that he was now all alone with Alix.\n\nAlix had gotten under his skin, somehow. A lander, of all people. He snorted.\n\n\"What?\" Alix put a hand on his shoulder and electricity shot through him.\n\n\"Nothing.\" He closed his eyes and inhaled and exhaled. They had work to do. He knelt, touching the white surface. \"This reminds me of something.\" The surface glowed. It was faint, but it sparked a memory.\n\n\"Come on.\" He grabbed his carry sack and tossed Alia's to Alix.\n\n\"Where are we going?\"\n\n\"Where the hoversport blew.\" Venin jogged back toward the spot where they'd crashed. There should have been an impact crater. The walls of the crevasse had taken a beating, spilling down rocks and dirt along the edges. The burned-out husk of the hoversport lay on one side, still radiating heat. The ground itself seemed untouched, a smooth white surface.\n\n\"What the hell?\" Alix stopped short of the site. \"Is it impervious to fire?\" He knelt to touch the white surface once again.\n\n\"I don't think so.\" Venin pulled off his carry sack and rummaged through it until he found his throwing knife.\n\n\"What are you thinking?\" Alix's brow was furrowed.\n\n\"I have a theory.\" Venin set the blade against the floor and pushed it in, cutting a neat line about an arm's length long.\n\nThe material separated neatly, drawing away from the metal of the blade. Then it sealed itself back together. \"Hah! I thought so. It's like corrinder.\"\n\n\"Corrinder?\" Alix knelt next to him and watched in apparent fascination as the gash was bridged by white filaments, weaving it back into one seamless piece.\n\nVenin nodded. \"Like the towers of Errian\u2014some call them turbien. This ground is alive.\"\n\nAlix looked up and down the crevasse. \"Alive? It can't be. It's huge.\"\n\n\"About fifty kilometers, maybe?\"\n\nAlix stared at him, and then laughed. \"Maybe so. This is our circle?\"\n\n\"You said it yourself. Maybe this is what we came looking for.\"\n\n\"Could be.\" Alix frowned. \"So what now?\"\n\n\"We should wait here for Alia.\" She'd find help within the next couple hours, he was certain.\n\nAlix shook his head, touching the ground with a look of wonder on his face. \"We both know there's not much time left, if what Quince told us is true.\"\n\nHe's a man of action, like me. Alix would no more want to sit around here for two hours\u2014or more\u2014than Venin did. \"What say we start with a little exploration?\" He pointed along the length of the crevasse.\n\nAlix grinned. \"Beats sitting here on our asses. Which way?\"\n\nVenin shrugged. \"One direction seems as good as another.\"\n\nThey set off down the newly created canyon, leaving the blasted wreckage behind.\n\nAlix strode next to him, a muscle-bound lander god, his thick arms filling out his body armor nicely.\n\nVenin wished he had a tub of cold water to plunge himself into.\n\nTo distract his libido, he struck up a conversation. \"So you worked things out with Xander and Jameson?\" As they walked, the walls didn't change much, though in places he could see the layers of strata that marked the long passage of time.\n\nAlix frowned. \"How do you mean?\"\n\n\"You and Xander were close, once, right?\"\n\nAlix sighed. \"Yes. Five years.\" He spat at the crevasse wall.\n\nTread lightly. \"It must have been hard, seeing him again, and with Jameson.\"\n\n\"I guess.\"\n\n\"And now?\" Why do I care?\n\n\"Look, I really don't want to talk about it.\" Alix's face was impassive, his fists squeezed tight at his sides.\n\n\"Sorry.\" Touched a nerve.\n\n\"Did you ever lose someone?\" Alix sounded like a lost little boy.\n\nThat hit home. \"Yeah. Once.\" He closed his eyes, and he could see Tazim's face. \"Tazim and I were close.\" Inseparable, more like.\n\n\"What happened?\"\n\n\"Oh, so you can pry, but I can't?\" It was meant to be playful, but it came out sounding harsher than Venin had intended.\n\n\"Sorry!\" Alix looked down. \"Didn't mean to push your buttons.\"\n\n\"No, it's okay.\" He frowned. Why did he care what Alix thought? \"Taz was a once-in-a-lifetime thing. He\u2014\"\n\n\"He?\" Was there a sparkle in Alix's eyes?\n\n\"Yeah. He was a childhood friend.\" Even now, he could see Taz's goofy face when he closed his eyes. \"We used to play together on First Hill in Gaelan. He was so full of life, like Erro reincarnate. He almost glowed with life.\"\n\n\"You two were lovers?\"\n\n\"For three years. It happened suddenly. When we were seventeen. We were on our taslit.\"\n\n\"What's that?\"\n\n\"It's a rite of passage. When a skythane reaches maturity and gets their wings, they are sent off to fend for themselves in the wilderness for a month.\" Taz had been so excited to finally get some time alone with him, away from prying eyes. Their time in the wilderness had passed in the blink of an eye, but oh the things Venin had learned in those thirty days. \"Taz and I went together.\"\n\n\"Hey, isn't Taz the name of your PA?\"\n\nVenin blushed. \"Yeah. It seemed appropriate.\"\n\nAlix managed a small smile. \"Sorry. Go on.\"\n\n\"We were camped out in one of the caverns up in the mountains. Like the one where you had the bi-wings hidden.\"\n\n\"Sure.\"\n\n\"Taz took a bath in one of the ponds, and when he got out, dripping wet and naked\u2014I kissed him. I had to. He was so beautiful. I wanted him like I'd never wanted anything else in my life. Not just for sex, you know? I mean, yeah, I wanted that like crazy. I was just past puberty, after all. But there was more to it.\"\n\nAlix nodded. \"I know.\"\n\n\"I stepped back, and he looked at me. I will never forget that look.\" He closed his eyes, seeing Taz there again like it had been that first time. \"I was so scared. Scared he would reject me. Scared I had misjudged him, but that look....\" He sighed. \"It was full of hunger and need, and love. He told me later he'd been wanting to kiss me for a year, but he'd been afraid it would ruin our friendship.\"\n\n\"Yeah. Damn. He sounds like an amazing guy.\"\n\n\"He was.\" Venin could still smell his skin, still taste his lips. No one would ever measure up to him.\n\n\"What happened? Did you two break up?\"\n\n\"No.\" Ten years later, and he could still hear the screams. \"He was killed. A long time ago.\"\n\n\"Holy crap.\" Alix put his hand on Venin's arm. \"I'm so sorry.\"\n\n\"It's all right. We've all been through tragedy\u2014\"\n\nThe ground shook, harder than before.\n\nVenin glanced at the walls looming above them, alarmed. \"This can't be good.\" The shaking went on and on, and dirt and rock showered down upon them. A frightening rumble filled his ears.\n\n\"Which way do we run?\" Alix's gaze darted back and forth along the crevasse.\n\nThe walls pushed suddenly closer, and the fall of debris multiplied tenfold.\n\n\"Neither!\" Venin shouted. \"Get down!\" He pulled Alix to the ground and took out his knife, hacking into the white surface.\n\nIt pulled away from the metal of the knife, but there was still more of the corrinder below.\n\nIt was getting hard to breathe as dust filled the chasm.\n\nHe cut again and again, trying to break through, hoping there was somewhere to break through to. The knife went through the corrinder like butter.\n\nAlix shielded him from the worst of the falling rocks and dust. \"If we die here, it was good knowing you,\" he all but shouted into Venin's ear.\n\nVenin laughed harshly. \"You too\u2014\" Venin's arm went through, and then they were falling into nothingness along with a shower of debris.\n\n# CHAPTER FIFTEEN: ERINA\n\nERINA WAVED zer hand over the blank white console of zer sleith, guiding the white ship through the air. The time was upon them. In a few short hours, most of the ithani would be gone from Arliss, gone north to slumber away the shift under the protective arms of the mountain.\n\nThe beings ze had foreseen had arrived, and they were as strange as ze had supposed.\n\nZe had touched them, briefly. They had bits of ithani in them. Of zim.\n\nZe marveled at that, though it had been implied in zer final vision. Still, that was a tale for the future to tell.\n\nThe previous seer, an ithani named Drevin, had been locked up away from the world by Thshnel'Jirron until zi'd died from starvation, or so they said. Since then, no one had dared publicly claim zis mantle.\n\nZe would speak with them both and learn more. Especially the one with the golden wings.\n\nThen ze would get them into hiding until ze could figure out zer next step.\n\nXANDER'S EYES flickered open.\n\nHe was lying on a white surface that hummed and vibrated underneath him. He was in a white room, on something that might have been a bed.\n\nIt was morning. The sun shone its red rays through a series of oval windows along one wall. The angry ball brought back some bad memories.\n\nJameson.\n\nHe sat up, searching wildly for his soulmate. Jameson lay next to him on another platform, asleep. Hopefully asleep.\n\nXander sighed in relief.\n\nHe looked around. He was inside a ship. Maybe. The room was wide, curved, and white. Handles at the back suggested cabinets or storage of some kind. Every form seemed to blend harmoniously into the next.\n\nThere was no one else to be seen inside.\n\nHe knelt over Jameson. \"Hey, wake up.\" He shook Jameson's shoulder. \"Jameson, come on, wake up.\"\n\nJameson's eyes flickered open, and he groaned, sticking his tongue out at Xander. \"Let me sleep. I was having the most wonderful dream.\"\n\nXander rolled his eyes. \"What are you, four? Come on. We don't have time for dreaming.\"\n\nJameson grunted and rubbed his forehead. \"Damn you, skythane bastard.\" He levered himself up onto his elbows. \"Where are we?\"\n\n\"You're skythane too,\" Xander muttered. \"I don't know. Let's see if we can find out.\" He got up and staggered to one of the windows.\n\nThey were flying, on one of the white sail ships, if he didn't miss his guess. In retrospect, it was obvious, but he was still a bit groggy from whatever the ithani had done to him.\n\nJameson joined him and gasped. \"Damn.\"\n\nThe ship was passing over a row of white towers\u2014maybe the ones they had seen in the distance. The towers\u2014corrinder plants for certain\u2014surrounded a vast white bowl. Against that dazzling background, ithani of many iridescent colors\u2014blue, gold, green, red, violet, and many more\u2014fluttered back and forth like a vast army of butterflies.\n\nThe white bowl was a wide, seamless plane.\n\n\"We could make a run for it,\" Xander said doubtfully.\n\n\"How?\" Jameson pushed on the white surface of the walls. It gave a little and then sprang back.\n\nXander shrugged.\n\n\"Where's Puck?\"\n\n\"I don't know. It was just you and me when I woke up.\"\n\n\"We can't leave without him.\" Jameson sounded adamant.\n\nXander sighed. There was no arguing with him when he was like that. \"So what do you suggest? They may be leading us like auxen to the slaughter.\"\n\n\"I don't know... why not just kill us, then?\" Jameson rubbed his chin. \"Besides, I'm not sure how far we'd get, surrounded like this by ithani.\" He leaned on the railing and whistled. \"Oh my gods, they're beautiful.\"\n\nXander nodded. They really were. Hard to see how such beauty could be such a threat. \"You're going to think I'm crazy.\" He'd been thinking about this for a while now, and it was the only thing that made sense. That sort of made sense.\n\n\"Tell me.\" Jameson put a warm hand on his cheek, his brown eyes fixing on Xander's. \"After what we've been through this last month, I'd have a hard time calling anything crazy.\"\n\nXander laughed. \"It's been wild, right?\" He took a deep breath. \"I don't think we're on Oberon, or Erro, for that matter. Not the Erro we know.\"\n\nJameson nodded. \"It would have been hard to miss something like this.\"\n\nThe ship was descending toward a hole in the white plane. In fact, the hole seemed to be growing wider to accommodate them.\n\n\"There are no other planets in the Oberon system, and the moons are barren.\" He swallowed, hard. \"I don't think we went somewhere else. I think we went somewhen else.\"\n\nJameson's eyes narrowed. \"How do you mean?\"\n\n\"This place doesn't exist on Oberon or Titania, not in our time.\" He gestured at the \"city\" below. \"But think about it. River apples. Corrinder towers. Even the ithani. This has to be Erro.\"\n\n\"Erro's past?\" Jameson looked up at the shapes flitting by overhead in apparent abandon. \"It makes as much sense as anything, I suppose, but how? The power that would require... if it were even possible\u2014\"\n\n\"I know. It's crazy. It's just\u2014\"\n\nSomething clucked behind them.\n\nXander turned to look into the face of the golden ithani that had assaulted them on the hillside. Though maybe assault was too strong a word. Nothing seemed to be damaged or broken. It had just been a flash of bright light.\n\nA doorway was open behind it, one that hadn't existed before.\n\nThe ithani tilted its head and extended two of its arms. They ended in a paw that had three fingers.\n\nHe looked at Jameson.\n\nJameson shrugged and grinned. \"You said you wanted to find someone who could tell us where we are.\"\n\n\"Yeah, I guess I did.\" He took Jameson's hand, and they reached out to touch the ithani's, palm to palm.\n\nJAMESON FLOATED in a world of nascent possibility.\n\nHe could feel Xander's hand in his, but when he looked down, he was gripping empty air. Still, he held tight.\n\nThe space around him was complicated. At first glance, it was as if he floated in a gray, featureless cloud, but each wisp was a world unto itself, full of twisting patterns that seemed to repeat endlessly like some crazy Mandelbrot set. There were colors hidden in the white, an iridescent sheen like that of the ithani themselves.\n\nFractals. That's what they were called. A whole world of complexity hidden in the shrouding mist.\n\nThe fog swirled and formed an image of the ithani creature, staring at him curiously. The golden one. The one he'd seen on the hillside, and again on the white ship.\n\nOne of its arms touched itself on the chest and it said \"Erina\" quite clearly. Its voice was clear and pitched pleasantly in the middle of the human vocal range, sounding neither masculine or feminine.\n\n\"Erina?\"\n\nIt squeezed its hand into a fist. \"Yes. Erina.\"\n\nThe human word startled Jameson. \"Erina. Your name.\"\n\n\"Your name,\" it repeated.\n\n\"No. Name. Name.\"\n\n\"Name.\" It made the fist gesture again.\n\nJameson held his hand up and imitated the gesture. \"Yes?\"\n\n\"Yes.\" The fist again.\n\nHe nodded and then pointed at his head. \"Yes.\" He nodded again. \"Yes.\" And a nod.\n\nErina nodded. \"Yes.\" It also emitted a deep thrumming sound from its thorax.\n\nJameson grinned. Okay, so we have a few words they could use. \"No?\"\n\nIt searched his face, and he felt a strange tickling sensation in his head. He should have been afraid. Was Erina rummaging through his head?\n\nInstead he felt calm, collected.\n\n\"No,\" it said, and made a sideswiping motion with its open hand, palm down. \"No.\"\n\n\"Ah.\" Jameson repeated the gesture. \"No.\"\n\n\"Ah,\" Erina said.\n\nJameson laughed. \"I'm going to call you Eri.\"\n\nIt cocked its head at him, a gesture so patently human that Jameson laughed again. Some things were universal.\n\nHe pointed. \"You. Erina. Eri.\"\n\n\"Ah. Eri.\" And the fist.\n\n\"Me, Jameson.\" He pointed again at himself.\n\n\"Meejaymeeson.\"\n\nJameson took a deep breath. Pronouns were going to take some work. \"Where is Xander?\" He could still feel Xander's grip on his hand.\n\nCocked head again.\n\n\"Xander.\" He brought up a mental image, the first time he'd seen Xander with his shirt off.\n\n\"Ah.\" Gently clenched fist.\n\nThen Xander was with him in the strange place. \"Hey!\" Xander grinned. \"Our friend here seems to be quite inquisitive.\"\n\nJameson nodded. \"I taught him... her? Them? Yes, and no. We're still working on 'you' and 'me.'\"\n\nXander laughed. \"Where the hell are we?\"\n\n\"I think it's something like ithani vir space.\"\n\n\"Do you trust it?\" Their eyes met.\n\nJameson shrugged. \"Do we have a choice?\"\n\n\"Good point.\"\n\nEri was watching them, its head glancing back and forth as they talked. It seemed to be waiting for them to finish. When they looked back at Eri, the ithani reached out two hands again, this time toward their foreheads. \"Yes?\" It cocked its head again and closed each hand gently.\n\n\"I think it wants to connect with us.\" Jameson squeezed Xander's hand.\n\nXander leaned forward and kissed him. \"Just in case.\" He searched Jameson's eyes and nodded. \"We're in neck deep already. What's another half a meter?\"\n\n\"Um, drowning?\"\n\n\"Then let's drown together.\" Xander's eyes twinkled.\n\n\"I love you.\" Jameson kissed him again, and then turned toward Eri. \"Yes.\"\n\nEri's cool hand reached out to touch his forehead, and a torrent of information flooded his mind.\n\nXANDER WOKE again, his head feeling like mush.\n\nBits of knowledge\u2014images, really, but laden with meaning\u2014flickered through his head. Some were concepts. Language glyphs that turned into English if he concentrated on them.\n\nOthers were memories. Or maybe not quite memories. Lessons. Pieces of history.\n\nPronouns. A whole host of ithani pronouns. Ze, zer, zers... maybe feminine? Zi, zis, zim. A more masculine feel, maybe. And za, zaf, and zas\u2014or zee, zeer, and zees.\n\nErina's voice spoke in his head, zer tone clear as if ze were there talking directly to him in English.\n\n\"Three ithani are needed to make a child. These are ze \/ zer \/ zers, roughly analogous to she \/ her \/ hers in your language\u2014the one who provides the egg. Zi \/ zim \/ zis is similar to he \/ him \/ his\u2014the one who fertilizes it. And za \/ zaf \/ zas has no direct human equivalent. It's the one who provides the organs to carry the child to term. When pregnancy occurs, the three forms unite into one\u2014zee \/ zeer \/ zeers\u2014until the child is born.\"\n\nXander blinked, letting that sink in.\n\nOther memories and bits of information floated around him. He seized on a memory, dropping into it as if he were there. As if he were Zorin himself, the ithani captain of the ship.\n\nOne of the Ithani sail ships descended through the atmosphere, dropping toward the land far below, the heat of entry turning the tips of its wings red and then gold.\n\nThe world below was beautiful\u2014red forests straddling the land, and a wide sea glimmering in the sunshine on the other side. In the distance, mountains marched along the other edge of the world.\n\nXander peered through a viewport, watching the curve of the planet grow larger.\n\nIt was hard to get a sense of the scale of the ship but based on the number of wings or sails, it had to be much bigger than the one he had ridden on.\n\nOn the way through the atmosphere, Xander saw strange things. Straight lines that ran across open fields. Clumps of black smoke hanging over coastal areas. Square patches that suggested crop fields.\n\nXander clucked his tongue.\n\nThis world was supposed to be uninhabited. It was closest to the ithani homeworld and had been chosen for this mission by the Council.\n\nThe ship's rate of descent slowed as it approached the ground, circling around a wide bay several times before settling on a landing spot on an open plain surrounded by forest.\n\nIt settled on the ground, and the side of it split open. A horde of ithani burst out, spreading across the clearing like excited children, chittering to one another in a language Xander found he could understand, if he tried.\n\n\"It's everything they said it would be,\" Landri said to Xander, zaf light brown fur shimmering in delight.\n\nXander thrummed his agreement, but inside, he had a bad feeling about this place.\n\nA heavy thwump thwump thwump drew his attention skyward as a black airborne craft cleared the treetops, its rotor blades holding it above the scene for just a moment before it turned and sped off the way it had come.\n\nLandri blinked in distress. \"What was that?\"\n\nXander sucked in his breath. \"We're not alone.\"\n\nXANDER'S EYES flickered open, and went wide.\n\nHe wondered if he'd just seen the beginning of the war with the dhagani, the beings Quince had said were the ithani's mortal enemies.\n\nToo much I don't know. He sat up to look around, feeling disoriented.\n\nThe room he found himself in was a white cube, smaller by a third than the one he'd awoken in before. Other than the featureless white bed he'd been lying on, the room was empty.\n\nHe was alone.\n\n\"Jameson? Erina?\" He got up and touched one of the walls. It was hard but yielding, like the ship. Maybe he was still on board?\n\nThere were no windows here, though, and he was feeling closed in. \"Hello?\" He paced the room, maybe two meters square, looking for something, anything to tell him where he was, how he got there. Or how to get out.\n\nHe felt a chill as goose bumps rose on his arms. He was trapped in here.\n\n\"Hello? Anyone? Can you hear me?\" The last was almost a scream.\n\nHe beat upon the wall, trying to push through it, to knock it down, to make his presence felt.\n\nIt absorbed the sound, and he accomplished nothing beyond bruising his fists.\n\n\"I'm here!\" he shouted, his wings shaking in protest. \"Can anyone fucking hear me?\"\n\nNo one came.\n\n# CHAPTER SIXTEEN: THE CROSS\n\nVENIN PLUNGED deep into an underground lake.\n\nHe opened his eyes and was surrounded by a glowing world of blue light. The liquid was warm, almost effervescent on his skin.\n\nAlix was in the water beside him, looking stunned.\n\nVenin grabbed him by the shirt collar and struggled back up toward the air, pulling Alix with him.\n\nThey surfaced, and he took a huge lungful of air. Then he looked up.\n\nThe cavern was lit by both a pale blue glow from the water below and the soft white glow of the ceiling. It was like those he'd visited in the Mora Mountains, but on a far grander scale. The hole they'd fallen through was sealing up, and white columns, like the corrinder towers of Errian, rose from the cavern at various intervals to support the ceiling.\n\nVenin shook the water out of his eyes. \"You okay?\"\n\nAlix was treading water beside him. \"Yeah, think so.\" He was breathing heavily. \"Holy crap, what the hell just happened?\"\n\nVenin laughed. \"We escaped a horrible crushing death, I think.\"\n\n\"How did you know this was down here?\"\n\nVenin shook his head. \"I didn't. Just seemed like the best alternative.\"\n\n\"Quick thinking.\" Alix swung around to check out their surroundings. \"Can you swim?\"\n\n\"Yeah. You?\" The water felt good on his wing and washed off the sticky remnants of crash foam from his body.\n\nAlix laughed. \"Part of my ranger training.\"\n\n\"Come on. Looks like a dry patch over there.\" He pointed to the edge of the lake, where a narrow strip of black sand and clumps of black rock offered landfall.\n\nVenin swam toward the sandy shore with steady strokes, pulling himself through the water with the grace of a stendril. The silver-and-gold fish were a skythane staple, fished from the high mountain lakes of the Mora, sleek as a sword. His boots hindered him, but he made good progress.\n\nAlix followed him.\n\nHis wing ached where he'd injured it. He cursed the fact that he was landbound, but then again, so was Alix, whose bi-wings had been lost in the hoversport crash.\n\nVenin glanced back at Alix. The man was following doggedly, managing to keep up with his fast pace.\n\nHe checked his chronometer. 1800 hours. Just twenty-six hours left until Quince's deadline.\n\nThey reached the shore, and Venin climbed out of the water, shaking his head and wings to dry himself. He'd managed to hold on to his carry sack, but it was soaked.\n\nHe reached out to help Alix out of the water. Alix tripped and fell into him, bringing them both down hard on the sand.\n\nTheir wet bodies met, and Venin could tell that Alix was interested, protests aside. Some things were hard to hide, especially under wet clothing. \"At least get me dinner first?\" He grinned to Alix, whose face was just inches from his own.\n\nAlix smiled back for a brief second, then frowned. He pulled away, standing and brushing off the sand from his legs. \"Sorry. Won't happen again.\"\n\nDamn. It had to be his feelings for Xander. Venin knew the signs. Five years was a lot of time with someone to have to let go. Not like we have time for it. Still, the rejection stung.\n\nHe sat up and pulled off his boots, dumping out the extra water. He used one of the boots to scoop up some water and rinsed the sand off his feet. I must look ridiculous, my feet in the air, balanced on my wings. When he was sure his feet were as sand-free as he could manage, he pulled the boots back on. He wished they had time to let everything dry out a bit.\n\nNevertheless, he felt good. Strangely good, given their circumstances. His energy was up, and he felt hope, as if they had been brought to this place for a reason.\n\nHe looked around.\n\nThe cavern was wide and roughly round, one half of it taken up by the softly glowing blue lake, the other side a crescent of sand backing onto a jumble of black rocks, like some strange, broken alien stair ascending to the back wall.\n\nWhite plants with long trailing fronds\u2014they had to be plants\u2014rustled gently along the edge of the cavern between the corrinder pillars, a few meters above where they stood on the black sand beach. Each one supported a handful of golden orbs.\n\nAlong the base of the rocks was a ramshackle assortment of wood, mostly broken into pieces, but in some places still standing to form small structures that backed up against the black rocks.\n\nThe blue light lit the cavern, coming from the water itself, throwing shifting dappled shadows across the walls and ceiling. It was an alien but entrancing scene, and it made Alix's skin look pale and cold.\n\nFor a moment, Venin longed to throw off his shoes again and run back into the water, to pass an afternoon in this strange underground paradise.\n\n\"It's beautiful, isn't it?\" Alix followed his gaze.\n\n\"It reminds me of Taz.\" His white skin, so like Alix's, under the blue light.\n\nAn ominous rumbling came from above.\n\nGot it. No time to relax. He wondered if that was a sign from Gael. Better get moving. First, he wanted to check out the wooden frames. The sand crunched under his booted feet as he crossed the beach toward the broken structures. Someone or something must have built them here.\n\n\"Where are you going?\" Alix ran his hand through his short hair, glancing nervously at the ceiling.\n\n\"To check out the limits of our little cage.\"\n\n\"You think this cavern is a trap?\"\n\n\"I don't know.\" He reached the black rocks at the edge of the beach and knelt, ducking into one of the still-standing structures.\n\nA blanket lay folded over on the sand, gray with age. He reached out to touch it, and it crumbled away under his fingers. \"Someone lived here once.\"\n\nAlix popped his head inside the makeshift shack. \"In the underground?\"\n\n\"Or at the least, camped out here for a while. Who were they? How did they get here?\" Something glinted in the light from outside, hidden under the sand.\n\nAlix looked over his shoulder. \"Maybe there's another way out, then?\"\n\nVenin knelt to touch the edge of the half-buried object and started brushing off the sand.\n\nIt was a piece of metal. As he cleared away the sand, the outlines became clearer. It was a small silver cross on a metal chain, badly tarnished by the years.\n\nHe pushed gently past Alix to look at it in the light. On the back, in a strange version of Common, it said, Aaron Hammond, 2135 AD.\n\nVenin's head exploded with memories, sending him staggering backward to the ground.\n\nHumans on this underground beach, living like refugees.\n\nA red-haired woman clutching this cross in her hands, saying a prayer to something.\n\nNimfeach like butterflies flying through the air above.\n\nThen it was gone.\n\nAlix knelt beside him. \"Hey, you okay? What happened?\"\n\nVenin clutched the sides of his head. Sheepishly he put his hands down. \"Holy Split, that was intense. Another memory, I think. There were people here. People like us. Well, humans. No wings.\" He was trying to sort out the memories that had run through his head. \"Humans and nimfeach.\" He handed the cross to Alix. \"I think it was important to one of them.\"\n\nAlix took it and read the inscription. He whistled. \"How long ago were they here?\"\n\n\"I don't know. Taz, any idea where this might have come from?\" Alix handed the necklace back to him, and he slipped it over his head.\n\n\"My information is limited without grid access, but many religions in human history used the cross as a symbol for a prime deity. The date suggests it came from Earth, prior to the Collapse.\"\n\n\"Thanks.\" He'd have to find out more about this \"Collapse\" sometime. \"Come on, let's get a look at the rest of this place.\" He climbed up onto the rocks.\n\nAlix clambered up after him, nimble as a river cat.\n\nLooking back, Venin lost his footing. His wings flew out to steady him, and a sharp pain flashed through his shoulder. \"Fucking Split!\" He caught himself on a rocky outcrop and closed his eyes, taking a deep breath while the pain passed.\n\n\"You okay?\" Alix's hand was warm on his shoulder.\n\nVenin turned to look into Alix's anxious eyes. \"Yeah. Just my damned wing.\" He flicked it in irritation, and another wave of pain shot up it to his shoulder. \"I'll be okay.\"\n\nAlix nodded. \"Here. Take my hand.\"\n\nVenin snorted. \"I don't need your help. I'm not ten.\"\n\n\"Hey, I'm having a hell of a time too. These rocks are really uneven, and sharp. Neither one of us needs to fall and skin a knee, or worse.\" He held out his hand again.\n\nVenin stared at it for a moment, then took it. \"Fair enough.\"\n\nThey scrambled up the rocks together toward the white plants that lined the edge of the cavern. The rock was sharp, and porous. Volcanic, if Venin had to guess. He chose his footing more carefully, and bits of the rock came loose and crushed into powder under his bootsteps.\n\nThey reached the top and paused to look back the way they had come. The beautiful blue lake was now fifty meters below them, sparkling against the black rock of the far cavern wall.\n\nFrom up here, he could see movement in the water. It entered the cavern on his right and flowed to the left. He could also see the exit, a cleft that had been hidden from the beach below. He turned to Alix and grinned. \"There's our way out.\"\n\nAlix frowned. \"Where does it go?\"\n\n\"Somewhere that's not here?\"\n\n\"Sounds good.\" Alix rubbed his chin where a bit of red stubble had grown. \"How do we get to it? We could swim, but who knows how far we'll have to go before we come out? And how do we know it won't plunge us into oblivion somewhere inside that cavern?\"\n\n\"We don't.\" Venin checked his countdown. \"We have to try something else.\" He turned to survey the plants. They did look like miniature versions of corrinder, which meant they should float. He pulled out his knife and tested the edge. Plenty sharp. He licked the drop of blood from his thumb. \"Want to build a raft?\"\n\nALIX GRUNTED. Getting the \"logs\" for the raft down the rocky outcrop had turned out to be a lot harder than he'd anticipated. They'd tried throwing them down to the sand, but inevitably the corrinder beams would fall short and smash to pieces on the rocks.\n\nThe same things that made them great for raft material\u2014lightweight and hollow\u2014made them vulnerable to breaking on anything hard and sharp.\n\nHe felt pretty good, all things considered\u2014trapped underground while the world might be ending above. The effervescent water must have done something to him, because his aching body, which had taken a beating in the crash, felt refreshed and pain free.\n\nThe golden fruit had proven to be similar to citrones\u2014sweet, wet, and spicy. Alix had tried one\u2014just a small piece\u2014and it hadn't disagreed with him. After allowing half an hour for ill effects, he'd eaten the whole thing. It was also a bit like mango. Or at least the artificial mango that his home replicator produced.\n\nSo he and Venin had decided to haul the logs down to the beach a few at a time.\n\nHe'd taken off his shirt, and his chest glistened with sweat as he set the last pile of logs down on the sand.\n\nHe glanced at the soft white glow of the cavern ceiling. They should be up there, in the real world, looking for Xander and Jameson. He was exhausted, too, with new aches and pains in his back and arms from the difficult work. They had no more stims\u2014they'd used the few they had for the arduous task of climbing up and down the rocks.\n\nHe glanced over at Venin. The man had his hands on his knees, and he was breathing hard.\n\n\"You okay?\"\n\n\"Give me a minute. I will be.\" The skythane took a few steps to the water, kneeling and splashing some across his face and bare chest. \"Ahhhh.\"\n\nAlix sank down onto the sand next to him and did the same. The water felt really good on his overheated skin.\n\n\"So many memories.\" Venin looked up at the cavern ceiling, and then over his shoulder. Then he started shaking.\n\n\"Hey man, you okay?\" Alix put a hand on his shoulder.\n\nVenin's eyes were unfocused. \"I... I can't....\"\n\nShit, it's like Jameson all over again. What was it with these skythane? He knelt in front of Venin and pulled the man's hand to his chest. \"Venin, look at me.\"\n\nVenin looked at him. His eyes were unfocused but full of fear.\n\n\"Hey, it's okay. I'll get you through this. Feel me breathing. In and out.\" He put his other hand on the man's bare chest, his light skin against Venin's darker tone. \"Breathe with me.\" He brushed aside the thoughts about how beautiful Venin's chest was, his umber skin and his wings, black like Xander's. Alix had always had a thing for wings. \"In and out.\"\n\nSlowly Venin responded. His chest rose and fell, finding a rhythm with Alix. His racing heart slowed under Alix's hand, and after a moment he closed his eyes and fell over onto the sand.\n\n\"Venin!\" Alix leaned over him, checking his pulse in his neck. He was still alive.\n\nA moment later, his eyes flickered open. \"Take me into the water.\"\n\n\"What?\" Alix glanced back at the lake.\n\n\"The water.\"\n\nAlix nodded. \"Okay.\" He pulled off his boots and then Venin's. Then he lifted Venin by the waist. It was a bit awkward, but he was strong enough to manage it.\n\nVenin's wings drooped over Alix's back as he carried both of them into the water. It was warm, swirling up around his waist and washing off the sand and sweat from their hard work.\n\nAs it enveloped him, his fatigue dropped away, and he felt some of his flagging energy return. He was alive, awake, in touch with the world. \"What the hell?\"\n\nThere was magic in the water.\n\nHe let go of Venin, and the man spread his wings carefully to float on the surface.\n\nA sigh of relief came from Venin's lips. His eyes were closed, but the edges of his lips drifted upward in a lazy smile. \"I saw them, Alix.\"\n\n\"Saw who?\"\n\n\"The ithani. This was... well, something like a resort for them. A long time before humans came here, I think? A place to get away.\"\n\nIt was strange, thinking of all the beings who had been here before them. Stranger still to think it had once housed humans, people not so different from the two of them.\n\nWhy had they come down here? How? What had happened to them after?\n\nVenin's golden eyes opened. \"The water. It's healing. Like an elixir.\"\n\n\"What, should I drink it?\" Alix treaded water, looking at the blue liquid suspiciously. Would it make his insides glow?\n\nVenin laughed, a rich warm sound Alix wasn't sure he'd ever heard before. \"You can if you want. It won't hurt you, but just floating in it for a few moments is restorative.\"\n\n\"I'll take your word for it.\" Though he did feel really good. \"Can I ask you something?\"\n\n\"Sure.\" Venin did a lazy backstroke, going in a circle around Alix.\n\n\"Have you had these memories before?\"\n\nVenin frowned. \"No, I don't think so.\"\n\n\"Why now? Will all skythane have them? Where do they come from?\"\n\n\"Maybe?\" Venin righted himself. \"I honestly don't know. Touched by the gods?\" He turned to swim toward Alix.\n\nAlix spat. \"Yeah, I guess that's as likely as anything.\"\n\nVenin reached him, and without warning put his hands around Alix's neck and pulled him in for a kiss.\n\nThey plunged below the water, lips locked for a couple long seconds, the world reduced to a glorious blue glow around them.\n\nThen Alix pushed Venin away and shot back up to the surface.\n\nWhen Venin reappeared, he wore that same lazy smile.\n\n\"What the hell?\" Alix paddled a few meters away from Venin.\n\nHe'd liked it. A little too much, maybe, but he wasn't ready for that yet.\n\n\"I wanted to see what it would feel like.\" Venin flashed him an enigmatic smile and then turned to swim back toward the shore, his body sleek and beautiful as it cut through the water.\n\nHis heart was screaming I'm not ready, but his lips tingled where Venin's had touched them.\n\nVenin glanced back at him over his shoulder. \"Come on. We have a raft to build.\"\n\n# CHAPTER SEVENTEEN: ON THE RUN\n\nTHSHNEL'JIRRON FLEXED zis wings. It was good to be in the world again after such a long slumber.\n\nSomething had gone horribly wrong with the shift. The nimfeach had confirmed as much. Erro had only transferred halfway from Urshu to Ashalla, the new universe zis scientists had found for zim. When Erro's old sun had flared, it had destroyed the surface of the remaining half of the planet, though the remaining half of the amalite core was too small to ignite to critical mass.\n\nThe human that zi had taken over had filled in some of the gaps in zis knowledge. How the nimfeach had managed to resurrect zis grand plan with the arrival of the offworlders. How the shift had commenced, unfortunately saving those same human inhabitants.\n\nZi still lacked knowledge of what had gone wrong the first time. Why zi and zis kind had been forced to wait a hundred thousand cycles for the Great Migration. Thousands of ithani had perished, while the rest of the race had been left in a near-eternal slumber. Now this new race, humans, infested Erro.\n\nSoon that wouldn't matter at all.\n\nJirron held zis strangely frail hand in front of the white wall.\n\nThe material split to reveal a small, sparse room on the other side. The human\u2014Kadin, Dani's memory supplied\u2014jumped up at zis entrance.\n\n\"Who's there\u2014Dani?\"\n\nJirron understood him. Though this \"Common\" tongue was a mess of contradictory rules and brutish pronunciations that offended zis ear.\n\nThe man's eyes bulged out as Jirron took him by the throat and forced him down onto the table rising out of the floor with an audible crunch of one of his wings.\n\nRestraints encircled the man's arms and legs, and a white tendril appeared and bored its way into his skull.\n\nJirron watched curiously as red ichor dripped out of the human's wound. It fell to the ground, drip-drip-drip, as the human's pleading eyes stared up at zim.\n\nThen they closed.\n\nJirron nodded. The Heart was assembling a profile of the man's brain. Soon he would be ready for transfer and dissolution.\n\nThen it would be over for the poor creature.\n\nJESSA FLIPPED her pulse rifle up to full power and fired at the base of the shelving unit that she hid behind. She was running down the battery like crazy, but they needed a distraction if they were going to get out of this alive.\n\nThe metal quickly grew red-hot.\n\nShe attacked the other leg, and soon it, too, was red.\n\nShe threw the rifle over her shoulder and slammed her back into the shelf. It shuddered but stayed in place.\n\nOnce, twice, three times and the three-meter-tall unit finally groaned and collapsed away from her, slamming into the next one and knocking that one into the one after that.\n\nAs the shelves collapsed like dominoes, the merchandise they held scattered too, flying across the open aisle between her and Quince.\n\nQuince shot her a grin.\n\nSomething exploded behind them. What in heaven's name are they storing here? A wave of heat passed over her, and Jessa ducked back into the limited safety behind her collapsed shelf, lying flat on the ground as the shock wave passed over. That was gonna leave a mark.\n\nAs soon as it passed, she was on her feet. \"Come on!\"\n\nQuince nodded and followed, glancing back at the pile of devastation Jessa had wreaked.\n\n\"Where's Robyn?\"\n\n\"She went ahead.\" Quince glanced back over her shoulder, looking for pursuit.\n\nRobyn was trying to get the hatch open, apparently in vain. She looked up. The guards were coming. \"Get back!\"\n\nQuince hauled Jessa behind another shelf. \"She's using the grenade. Close your eyes and cover your ears!\"\n\nA guard appeared behind Robyn, turning his weapon toward them.\n\nRobyn dove out of the way, and Jessa closed her eyes and put her hands over her ears.\n\nThere was a loud bang and a flash she could see through her eyelids.\n\nWhen she opened them, the guard was gone.\n\n\"Come on,\" she called again, and ran out into the aisle.\n\nThe man lay crumpled on the ground a few meters away, his skin burned bright red.\n\nJessa's stomach twisted, but she forced it down.\n\nThe hatch had been blown wide open.\n\nRobyn appeared from the other direction and all but dove into the hole.\n\nWith just a second's hesitation, Jessa followed. The grenade had taken out a decent chunk of plascrete, making a broken crater where the hatch had been. She scrambled over the rough surface, slipping the last few feet to the bottom of the broken bowl.\n\nQuince was right behind her.\n\nJessa hesitated for just a second at the hole that led down into the ground. She wasn't big on tight spaces, and who in the daisies knew what was down there?\n\n\"Let's go,\" Quince said behind her, startling her. \"Reinforcements will be along soon, with all the noise that grenade blast made. Brilliant move with the shelving. You're pretty badass.\"\n\nJessa grinned. She'd never been called badass before, and coming from someone like Quince, that was high praise indeed.\n\nI can do this. She gritted her teeth and swung herself over the edge, grasping onto the still-warm rungs of the ladder that was set into the side of the shaft.\n\nShe could hear Robyn clambering down the rungs below. At least, she hoped it was Robyn.\n\nDecided, she plunged into darkness, guided by feel and the dim red charge-level light on her pulse rifle that was slung over her shoulder.\n\nThis beat almost any night out on sedate, boring Beta Tau.\n\nQUINCE FOLLOWED Jessa down the rabbit hole. She'd seen enough tri-dimensionals to recognize Lewis Carroll's iconic entry to Wonderland when she saw it.\n\nJessa descended the ladder directly below her.\n\nQuince's opinion of the woman had gone up a few notches since they'd first met. She'd assumed Jessa was an out-of-touch, wide-eyed ing\u00e9nue, probably afraid of her own shadow.\n\nThe Jessa she was getting to know was, if a bit squeamish with rough language, intelligent, capable, and yes, pretty badass. If I were twenty years younger and single....\n\nThe light above dimmed.\n\nQuince aimed her pulse rifle upward and shot off a couple rounds.\n\nThe light returned.\n\nStill, they were fish in a barrel down here. \"Go faster!\"\n\n\"Trying. Damn, this is gonna mess up my palms.\"\n\nQuince glanced down. Jessa slipped away into the darkness.\n\nQuince fired off a couple more pulse shots at their pursuers, then tucked in her wings tightly and let herself slide down the ladder after Jessa.\n\nSomething rattled past her. A red light blinked. \"Grenade!\" Her voice echoed down the shaft.\n\nAfter two seconds, a muffled explosion and a flash filled the tunnel.\n\nQuince's heart leapt, and then her feet hit the ground.\n\nShe opened her eyes to find herself standing at the crossroads of two plascrete tunnels, dimly lit by repeating recessed lighting. One of the four corridors was dark.\n\nJessa had a sheepish look on her face.\n\n\"Where's Robyn?\"\n\n\"Right here,\" a voice said behind her.\n\nQuince spun around to embrace her. \"Thank the gods. What happened?\"\n\n\"I threw it down the hall.\" Jessa pointed to the darkened tunnel.\n\n\"Badass.\" Quince pulled away from Robyn to get her first good look around the space they were in.\n\nIt was a man-made access tunnel, built to a standard two-meter height. The walls were bare plascrete.\n\n\"Stand back.\" Quince pulled a second pulse grenade from her carry sack. She armed it and set it for three seconds. She threw it up into the shaft.\n\nThey all jumped backward.\n\nBoom.\n\nLight flashed again, and a pile of rubble rained down onto the floor where they had been standing. \"That should slow them down.\"\n\n\"How will we get back out?\" Jessa was frowning, but she didn't look too worried.\n\n\"Cross that gate when we come to it.\" Quince waved away the smoke from the grenade. \"Ari, can you reach DOC?\"\n\n\"Sorry, Quince. I have no connection to the grid.\"\n\n\"Dammit.\" She hadn't counted on being cut off here in the middle of Oberon City. She glanced at the three remaining tunnels. Which way to go? The signal. \"Ari, enhanced vision, please.\" Her backup had included her blackware. Sometimes the ability to see grid traffic could be extremely useful.\n\nThe world shifted and dimmed.\n\nFrom the shaft above, a glowing stream spun through the air, twisting like a glowing helix down into the tunnels.\n\n\"What is it?\" Robyn asked.\n\n\"Our trail of breadcrumbs.\" She climbed the remnants of the ladder and approached the stream, gratified to see it was made up of the same strange glyphs that Mylin had discovered in the Oberon City grid. She held her hand out, and they flowed through her palm as if she were a ghost.\n\nShe followed their path with her gaze.\n\nOf course, they went down the dark tunnel. \"You had to throw the grenade down that one, huh?\" She side-eyed Jessa.\n\nJessa didn't back down. \"It's not like I had a lot of time to decide.\" She yawned.\n\n\"Don't start.\"\n\n\"Sorry. I'm just exhausted. So what exactly do you see?\" Jessa was peering at the cavern ceiling, her eyes narrowed.\n\n\"Grid traffic. Something alien, maybe. It comes down the shaft, and then makes a zimbee line down that tunnel. Ari, what time is it?\"\n\n\"Almost 1900.\"\n\nThree more hours until midnight. Then, at most, two more days before the end of the world. \"Let's go. We have some digging to do.\"\n\nMYLIN STARED at her desk. It was covered with paper.\n\nShe had sent poor Derren to take a nap and was only running on fumes herself. They had so little time left.\n\nShe'd taken a couple stims\u2014they helped for a while, but when the fatigue returned, it came crashing back.\n\nShe was going to have to take a break soon, but she was close to something. Really close. She could feel it. Some of the pieces were starting to fit together, and if she let it go now, it might all be one vast jumble again when she woke up.\n\nSo she yawned and pushed on.\n\nShe'd started a few hours before with the strange alien code Derren had found. It intrigued her, and she'd used a couple of DOC's tools to try to unravel it, but DOC was stumped too.\n\nHe'd appeared in her room in one of his human accessible forms, a middle-aged, grizzled avatar wearing a wide-brimmed hat, denim jeans, a plaid shirt, and leather boots. He'd had to explain the whole cowboy thing to her. It was still a concept she couldn't quite wrap her head around. The hat, yes\u2014it kept the sun off of your head on the \"range.\" And the boots\u2014DOC called them \"shitkickers\"\u2014those were understandable too. But why blue pants, and a shirt pattern that hurt the eyes?\n\nThen the strangest thing had happened.\n\n\"What's the file date on this one?\" She held up a survey that had been taken of the area around Oberon City by the initial lander team.\n\n\"What file?\"\n\n\"This one right....\" Her hand was empty. \"That's weird. I had a survey file from the original ZephyrCorp survey of this area.\"\n\n\"I don't have any record of such a survey.\" DOC frowned. \"Though I remember that you requested a copy.\"\n\nShe frowned. \"Could something be messing with your files?\"\n\n\"Not without me knowing about it.\"\n\n\"Can you check yourself for any other inconsistencies? Say in the last day or so?\"\n\n\"Already started. Give me half an hour.\" He scowled. \"I don't like the idea of something crawling around inside of me, mucking with my files.\"\n\n\"That must be really weird.\" She didn't know if AIs like DOC were \"real\" or not, but he seemed convincing to her. She tried to imagine what he must feel like and shuddered. Like having bugs crawling under your skin.\n\nIn the corner of her vision, something else changed.\n\nShe spun around. There was a blank spot on her virtual wall.\n\n\"Dammit. DOC, dump all this to print. Now.\" Something was quietly erasing her hard-researched evidence.\n\nShe'd only discovered the idea of a printer the day before, but it just might be her salvation.\n\nNow she stared at her desk, trying to make sense of the reports she'd dumped. It was so much harder dealing with masses of data in real life using paper copies than it was in vir.\n\n\"DOC would like to speak with you.\" Zim's voice was anxious.\n\n\"Sure.\" She closed her eyes and reentered vir.\n\nDOC was pacing the length of her room. \"It's been going on for decades. At least. Maybe longer.\"\n\n\"What has?\"\n\n\"This pilfering and hiding of information. I did a comparison trace and found a steady erosion of information in my database, more than can be accounted for by standard computing errors and hardware malfunction. Someone or something has been messing with my files for a long time.\"\n\nHe looked furious. Steam poured off his back.\n\n\"Hey, DOC, slow down. I think you're overheating yourself.\"\n\nHe glanced back and laughed ruefully. The steaming stopped. \"I've locked everything down for the moment. No more deletions, and I've isolated that bit of alien code you identified so it can't access anything important. I've given it a virtual copy of me to play with, filled with bogus ithani information, and you should see how it's going to town with it.\"\n\n\"Heaven and Erro.\" Her hand flew to her mouth. \"What is it?\"\n\n\"I don't know. I'd guess the ithani, or their agents, have been aware of us a lot longer than we've been aware of them.\"\n\n\"How come you've never noticed?\"\n\nHe spit into a brass spittoon that appeared out of nowhere in the corner of the room. \"It's very good at redirection. Every time one of my systems would flag an error, it would get bounced off in another direction, or that subroutine would be quietly deleted.\"\n\n\"Like when you think you see something out of the corner of your eye, but when you look, it's gone.\"\n\nHe chuckled. \"Yes, like that.\"\n\n\"Can you recover any of what it stole?\"\n\n\"I don't know. Give me\u2014\"\n\n\"Half an hour?\"\n\n\"More like half a day.\"\n\nMylin glanced nervously at her countdown clock. Midnight. Twenty-two hours left to deadline. Maybe a little longer? \"Just go as fast as you can.\"\n\nHe nodded and vanished.\n\n\"Zim, get me Quince.\" This was important. She could feel it.\n\nThere was a short delay. \"Sorry, Quince is out of range.\"\n\n\"Moonrise help us. How about Alix?\"\n\n\"Sorry, Alix is out of range also.\"\n\n\"Then get Derren up. Give him a couple stims if you have to, and get me whatever help is available. We have to put this all together today, or there won't be a tomorrow.\"\n\n# CHAPTER EIGHTEEN: CAVERN OF THE WHIRLS\n\nVENIN WAS full of light.\n\nHe felt better than he had in days, energetic, with the taste of Alix's lips still on his.\n\nHis flashes of memory had given him a newfound feel for the continuity of things\u2014how it was all connected, somehow\u2014and had restored his hope for the future.\n\nSure, everything might come crashing down, and the world might end the next day. But for today, he was alive and had survived a disaster that by all rights should have killed him.\n\n\"You look smug.\" Alix lashed together the last of the logs, tying it off in what he called square lashing, something he'd apparently learned in his ranger training.\n\n\"Not smug. Happy.\"\n\n\"How in the Split can you be happy? We're trapped underground while the world is about to end above us.\" Alix cast a nasty look at the white ceiling.\n\n\"We're still alive, and in good company.\" Venin tested the raft. The knots all seemed sound. \"Let's go.\"\n\nAlix grunted.\n\nIt was two in the morning. The light down here seemed to never change. He wondered what it had been like living in a place where there were no natural cues for night and day. \"Feel like grabbing a bite?\"\n\nAlix's stomach growled. \"Guess I do. Do we have time?\"\n\n\"We have to keep up our strength.\" He grabbed an armful of desiccated wood and arranged it in a neat pile.\n\n\"A bonfire? How are you going to light it?\"\n\nVenin grinned. \"Just watch.\" He selected a piece of corrinder splinter and placed the point down on a piece of dried wood. He rubbed it back and forth between his palms.\n\nAlix watched over his shoulder. \"That's some old-world shit.\"\n\n\"Yeah, they teach us wood skills out in the, you know, woods.\" A thin line of smoke rose from the piece of wood.\n\nHe sprinkled some splinters on top and redoubled his efforts, blowing on it as a small flame appeared.\n\nSoon it caught the splinters on fire. \"There we go.\" He piled more wood on top.\n\n\"What do we have to eat?\"\n\nVenin went through his bag. \"Let's see. Some soggy bread. Some....\" He sniffed it. \"Cheese? It came out of the thingy at OberCorp.\"\n\n\"Replicator?\"\n\n\"Sure.\" Venin looked up at the remaining \"trees\" above them. \"Grab one of those fruits too. Bet it would be great roasted.\"\n\nSoon they were settled in front of the fire, roasting melon over the flames.\n\nVenin handed a chunk of \"cheese\" over to Alix.\n\nAlix bit off a piece. \"Oh my God, this is awful.\"\n\nVenin tried his. \"It really is,\" he managed through a mouthful. He spat it out on the ground. \"I mean, I'm hungry, but it's all but inedible.\" It tasted like feet.\n\nAlix ran to the water to wash out his mouth.\n\nVenin chuckled. \"Here, try this instead.\" He pulled a piece of the roasted fruit off the stick and handed it over to Alix.\n\nAlix plopped himself back down on the sand and took a bite. \"Oooh, that's good too.\"\n\nVenin took another chunk of the melon and bit into it, the warm juices running down his chin. It was sweet and rich, and left him wanting more when he had eaten the flesh down to the rind.\n\n\"Look!\"\n\nVenin followed Alix's gaze.\n\nSomething was bursting from the water\u2014lots of somethings. Small iridescent forms that seemed to sparkle with their own light. They danced into the air and spun around one another like glowing butterflies.\n\nVenin laughed in delight.\n\n\"What are they?\" Alix was entranced.\n\n\"I don't know.\" Some of the sparks combined and fell back into the water, where they sent up thin columns of steam. \"Let's call them whirls.\"\n\n\"I like it.\" Venin felt Alix settle in next to him to watch the spectacle. \"It's beautiful.\"\n\nHe grinned. Then he remembered the time. \"We should finish these and get going on our way.\"\n\nAlix pulled away. \"Yeah. Of course you're right.\" He sounded disappointed.\n\n\"The world needs us.\" He guessed he wasn't the only one who wished the needs of the world weren't so pressing. You did turn me down....\n\nALIX SAT back on the sand, his gaze going from the magical dance above the lake to Venin's back.\n\nSoon they'd have to be moving on. Time was growing short, but for the moment, he just wanted space to figure out everything that had happened to him in the last day or two.\n\nHe was your father.\n\nHe stared at Venin's wings, wondering what it was like to fly\u2014to truly fly under one's own power. To feel the wind lift your wings, the sun warming your feathers. To be skythane.\n\nVenin turned to him, his face carefully neutral. \"Should we go?\"\n\n\"What does it feel like?\"\n\nVenin frowned. \"What does what feel like?\"\n\n\"To soar. To fly through the air from place to place. To launch yourself from the ground and spread your wings to catch an updraft.\"\n\nVenin stared at him for a moment. \"It's hard to explain. It's like breathing for me. What does it feel like for you to walk?\"\n\nAlix chuckled. \"Still, it was new to you once, when you got your wings?\"\n\nVenin's face turned wistful. \"Yeah, it was. I remember waiting for that day, feeling my wings coming in. Flapping them madly to try to get off the ground.\"\n\n\"You and Tazim?\"\n\n\"He got his first. I was so jealous, though I loved to watch him fly.\" Venin cocked his head. \"Why the sudden interest?\"\n\nAlix looked away. \"No reason. I was just admiring your wings.\"\n\n\"Should we go, then?\" Venin stood and offered Alix a hand up.\n\n\"Sure. No rest for the wicked.\"\n\nVenin snorted. \"Or the damned.\"\n\nThey extinguished the fire and pushed the raft into the water. They climbed on board, and Venin threw the handle of his carry sack over the central mast.\n\nUsing the makeshift oars, they paddled it toward the egress, underneath the sparkling creatures. \"I think they're mating.\"\n\nAlix looked up, and back at him. \"Three of them?\"\n\n\"Different strokes.\"\n\n\"Yeah.\"\n\nAs they neared the middle of the lake, the current took them, spinning the raft around as it approached the dark hole of the exit cavern.\n\n\"You ready?\" Venin grinned.\n\nAlix snorted. He had no idea why the man was so happy.\n\n\"I'll take that as a yes.\"\n\nThen darkness enveloped them as they passed out of the cavern of the whirls.\n\nALIA CLOSED her eyes, letting the sizzle of the hoversport's ionic shower clean the dirt and grit off her body.\n\nShe kept seeing it again, the crack in the earth slamming shut with a deafening finality.\n\nPounding the earth, frantically trying to dig her way down to Venin and Alix with her bare hands.\n\nShe squeezed herself tightly.\n\nAt last, she turned the shower off and climbed out into the main cabin.\n\nSomeone whistled, but Smythe\u2014one of the other rangers who had answered her distress call\u2014rapped him up the side of his head.\n\nIt felt good to be clean, the crash foam and dirt washed off her body. She accepted her clothes back from Smythe.\n\nHe was tall, trim in his green-and-purple ranger camo, and as clean-cut as they came.\n\nShe'd been lucky the craft had been passing close enough to pick up her signal. After flying for an hour, she'd encountered the hoversport and had been grateful for their assistance. \"Any luck?\" She'd asked him to swing around the site to see if he could pick up any other signals. Something odd was happening down there. White columns like corrinder had sprung up around the circle, and the ground itself was shifting. Visibly shifting away from the center of the circle Venin and Alix had found, toward the corrinder-lined edge. The Theseus and Demetrius rivers were having to find a way around, and there was sure to be widespread flooding.\n\nShe tucked her wings in tight in the limited space of the hoversport.\n\n\"Sorry, ma'am.\" The dark-haired enforcer averted his gaze from her naked body. \"Nothing's come up on the scanner or grid channels yet. You say Alix Preston is down there somewhere?\"\n\nShe tied her shirt on over her wings and pulled on the OberCorp work trousers he'd provided. \"He was. I don't see how he could have survived, but I hoped....\"\n\nSmythe nodded. \"I understand, ma'am.\" Now that she was clothed, he risked a glance in her direction. He was handsome enough, in a square-jawed Marine sort of way.\n\n\"Nix, can you get me Quince?\"\n\n\"Checking.\" There was a pause. Alia sank onto one of the benches that lined the sides of the hoversport. It was a bit bigger than the one she, Alix, and Venin had taken, but it was full of burly enforcers.\n\n\"I'm sorry, Alia. Quince is out of range.\"\n\n\"How about Mylin?\"\n\n\"One moment.\"\n\nSmythe handed her a bottle. \"Drink this. It will help get your energy up.\" His gaze lingered on her for just a moment before he looked away.\n\nI must look blasted. \"Thanks.\" She took a sip. It was fizzy, with a sweet taste that reminded her of citrones. \"It's good.\"\n\nHe grinned.\n\n\"I have Mylin. There's not enough bandwidth out here for vir. It's voice only.\"\n\n\"Got it.\" She was still getting used to the whole voice-in-her-head thing. Audio only was just fine with her.\n\n\"Alia! How are you? Is everything okay?\" Mylin seemed a bit frantic.\n\n\"No, sorry. It's.... We lost Venin and Alix.\"\n\nA sob came across the line, then silence. She had to remind herself the girl was still young, no matter how mature she seemed. \"Mylin?\" she asked after a moment.\n\n\"Yeah. Sorry.\" An audible sniff came across the line. \"What happened?\"\n\nAlia imagined Mylin wiping her eyes. \"We went down into a crevasse that had opened up near the House of the Sky. The ship malfunctioned on the way down and crashed. They survived, but when I went for help, the crevasse closed up again. They're still down there.\"\n\n\"Okay.\"\n\n\"It was quick, at least.\"\n\n\"That's good. You're okay?\"\n\nAlia sighed. \"Yes. Shaken, but okay.\" And guilty as hell. Why did I survive?\n\n\"I've been trying to reach Quince, but she's out of range.\"\n\n\"I know. Is everything okay there?\" Their little Hunters were in pieces.\n\n\"Not exactly. Alia, they've been here for a long time.\"\n\n\"Who?\"\n\n\"The ithani.\"\n\n\"Yeah, we knew that\u2014\"\n\n\"I mean inside the grid. In OberCorp's systems. They've been keeping everything about them hidden from us for decades. Maybe longer.\"\n\n\"Holy Split.\" If that was true.... They'd all assumed the ithani were just waking up. That their agents, the nimfeach, had maybe manipulated things to some extent, but this.... \"How did you find out?\"\n\n\"Something tried to delete the data I'd been pulling from the archives. Even DOC was unaware of it.\"\n\n\"Oooh.\" If they were meddling in lander business, what had they done to the skythane?\n\n\"Yeah.\"\n\n\"Look, I'll get back there as soon as I can\u2014\"\n\nA male voice interrupted her. \"I've got something!\"\n\nAlia opened her eyes. One of the enforcers\u2014Dixon?\u2014was staring at her from the cockpit.\n\nShe was on her feet in an instant. \"What? Mylin, just a sec.\"\n\n\"Okay.\"\n\nA three-dimensional representation of something hovered above the dashboard. It looked like a layered cake. \"What is it?\"\n\n\"See these two dots here?\" He pointed to a pair of red spots underneath the cake.\n\n\"Yeah?\"\n\n\"I think those are your missing friends. We can't connect with them through all this dirt and rock, but we're picking up their signals.\n\n\"Does that mean.... Are they alive?\"\n\n\"Not by itself, but look at this.\" He swiped his hands through the image, and the dots got bigger.\n\n\"What?\" She peered intently at the dots.\n\n\"They're moving.\"\n\nA wide grin split Alia's face. \"Mylin, I think they're alive!\"\n\n# CHAPTER NINETEEN: INTO THE HEART\n\nJAMESON WAS everything.\n\nThe overwhelming flood of memories that had flowed through his head when he'd first touched the golden hand of the statue of Erro, all those weeks before, seemed like the merest trickle compared to this.\n\nNow he floated in a universe of them\u2014thoughts, memories, emotions\u2014and it was as if he'd expanded to encompass them all.\n\nHe could access them at will. No, that's not right. He could live them. Be them.\n\nHe tasted them one after another, like a kid in a candy store, eager for each new fresh remembering.\n\nZe was an ithani child with their brood mates, taking to the sky for the first time under the yellow sun.\n\nZi was part of an ithani triad, spinning through the sky in orgasmic ecstasy, on the brink of becoming one with zis lovers.\n\nZi was one of the white ithani sail ships, gliding right into the heart of a sun, its blinding light muted to let zim see the shades and patterns that danced all around him.\n\nJameson. Someone whispered his name.\n\nZi didn't want to heed that voice. He wanted to stay here forever, in this vast, unbounded playground, tasting lives he'd never even imagined.\n\nZi reached for another\u2014\n\nThe world dissolved around him, and he was lying on a white platform that conformed to his back and wings perfectly. The sense of loss was immediate and brutal. \"What in the hell?\"\n\nSomething loomed over him through a fog.\n\nHe scowled. \"Why did you wake me? I was just\u2014\"\n\nA golden alien snout looked down at him. Erina. A series of chirps and squeaks came from zer mouth. It's not safe to let yourself get lost in the athr\u00e0.\n\nErina was speaking his language. Or... he was hearing what ze said, but in Common. New pronouns and all. \"How did you...?\"\n\n\"You gave me permission. I learned a lot about you and your lifemate.\" Ze shimmered. \"Now when I speak, you can understand what I say.\"\n\n\"Got it.\" Jameson sat up and looked around, the haze in his brain starting to dissipate. \"Where is Xander?\" He got up, unsteady, and grabbed the platform for support. \"What are you doing to me?\"\n\nErina waved zer hands over a white console along one wall. It melted away, withdrawing back into the surface until it was as if it had never been there. \"You present quite a puzzle. A new species, but with the ability to fleech with one of us.\"\n\n\"Fleek?\" He frowned, looking around at the mostly featureless room. The two carry sacks lay discarded in one corner.\n\nWhat had he gotten them into now?\n\n\"Fleech. It means....\" Eri clucked zer tongue. \"There's no exact translation. It's a bond. A sharing. A way to communicate that we thought no other species possessed, and on the eve of the Great Migration, no less.\"\n\n\"Where. Is. Xander?\" Jameson shook his wings, standing tall, trying not to be intimidated by the ithani or zer apparent power over him. He leaned forward in a way that he hoped would make him appear threatening.\n\n\"Oh, you are distressed.\" Erina blinked at him, holding up a three-fingered hand in a way that could have been a threat or an apology. \"I'm sorry. I didn't realize the fleech you two shared was so strong.\" Ze turned and waved a hand over the wall behind zer. The surface split and opened to reveal another room.\n\nXander sat on a platform like the one Jameson had found himself on, head in his hands, breathing heavily.\n\n\"Xander?\" Jameson slipped past Eri and into the other room.\n\nXander looked up, and his face transformed from broken exhaustion to joy. \"Jamie!\" He kissed Jameson hard and picked him up to spin him around. \"I was scared I'd never see you again.\" He held Jameson's head in his hands and looked him right in the eyes. \"Are you okay? Did she... I mean ze hurt you?\" He glared at Erina over Jameson's shoulder.\n\nJameson shook his head. \"No, nothing like that. She... ze... Erina showed me this whole universe inside my head. The memories.... Erina says we're not like anything they've ever seen before.\"\n\n\"A universe?\"\n\n\"Yes. All the memories that flooded through me, and more. I think there's a reason we're here. Now. I think this is the time of the first shift.\"\n\nXander nodded. \"I do too. But why are we here? And how?\"\n\nJameson bit his lip. \"I haven't quite figured that out yet.\"\n\n\"So what now?\"\n\n\"We need more information.\" He turned to Erina. \"Can you tell us about this Great Migration?\"\n\nErina blinked twice. \"Sorry, but I don't know what you are yet. You might be a dhagani spy.\"\n\nDhagani. Quince had said the ithani had been at war with something called the dhagani. He looked around. \"Where's Puck?\"\n\n\"Puck?\" Erina cocked her head at Jameson, such a human gesture it almost made Jameson laugh.\n\n\"The nimfeach who was with us.\"\n\n\"Ah.\" That double blink again. Displeasure? \"The nimfeach has been detained elsewhere.\"\n\n\"I need to see him.\" Zim? These new pronouns were still settling in his head.\n\n\"I'm sorry, that's not possible\u2014\"\n\nJameson leapt past zer again, snagging his carry sack and holding up the original key Quince had given him. It burst into a golden light, and Eri took a startled step backward, zer features twisting. \"Where did you get that?\" Ze sank down on one knee, not unlike the nimfeach had reacted at the crash site.\n\n\"It was given to me. Apparently it's a free pass around here.\"\n\nXander came to join him. \"Don't overplay your hand,\" he whispered. \"We still don't know why they're afraid of it.\"\n\nJameson nodded. \"I need to see the nimfeach.\"\n\nErina looked even paler than ze had before, if that were possible. \"Where did you get that?\" ze said again.\n\n\"From a friend?\" Jameson frowned.\n\n\"You're the ones foretold, you must be\u2014but this.... There are currents moving in this room that I do not understand.\" Erina stood, trembling. \"I will see what I can do.\" Ze summoned the console from the wall, passing zer hands over it in a rapid series of patterns.\n\nXander put his hand around Jameson's waist. \"Foretold? What does ze mean?\" he whispered.\n\n\"I don't know.\" Fate had something in store for them, he was sure of it. He just hoped it was something good.\n\nXANDER STOPPED, stunned.\n\nErina had led them out of the sealed rooms with a wave of zer hand, down a long hall and outside of the building, or whatever it was.\n\nThey stood on a wide balcony in an upside-down world. Above them, a white \"sky\" glowed softly, and from it descended all manner of pillars and hanging structures\u2014towers, domes, and curving spirals that seemed to defy gravity. It took him a minute to sort it out, to figure out what he was seeing.\n\nHe gripped the railing and tried to take it all in.\n\nIt was beautiful.\n\nIthani of all colors flitted around them on their way to or from one place or another.\n\n\"The Great Migration is close. There's much activity\u2014everyone is leaving Arliss for the Mountain.\" Erina seemed to have recovered some of zer composure. \"I will take you to the nimfeach.\"\n\nThe balcony separated from the building with a soft slurrup, growing a second wall behind them, and floated off into the rest of the city.\n\nXander took Jameson's hand, awed by the scale of what he was seeing. \"We're underground,\" he whispered, pointing up at the white ceiling. \"Or under whatever that is.\"\n\nJameson looked up and nodded.\n\nIt was the underside of the wide \"bowl\" they had seen from afar. Xander was sure of it.\n\nThe balcony veered and headed toward the middle of the vast space, where one structure stood out. Unlike the other parts of the city, it wasn't all white.\n\nInstead, it was much more organic-looking, a huge dome covered with green growing things and topped by what looked like a stadium\u2014hundreds of rows of parallel steps\/seats that descended into a circular bowl.\n\nIt was too complex for him to take in all at once.\n\nThe dome was grounded far below by thick white \"roots\" that burrowed into the ground below it. A wide azure lake surrounded it, glowing of its own accord.\n\nThe sides of it were festooned with hundreds of shapes, like great white butterflies nestled against a tree whose size boggled the imagination. As Xander watched, one of the \"butterflies\" detached itself to soar upward. The white ceiling split to allow it passage, and Xander caught a brief glimpse of the greenish-blue sky above.\n\n\"What is it?\"\n\nJameson shrugged. \"I feel about as significant as a gnat right now.\" He slipped his arm around Xander and pulled him close.\n\nXander could feel him shaking.\n\n\"It's the Heart,\" Erina said simply.\n\n\"What's the Great Migration?\" Xander waited to see if Erina would provide an answer this time.\n\nErina released a whoosh of air, maybe the equivalent of an ithani sigh. \"We plan to leave this plane. The ithani tire of endless war and conflict.\" Erina sounded defeated.\n\n\"Did you lose someone?\" Jameson had that look on his face.\n\nThe one Xander knew meant he was empathizing. \"Jameson, don't\u2014\"\n\n\"You did, didn't you?\"\n\nErina turned to him, head cocked, and made a thrumming sound in zer throat.\n\n\"Is that a yes?\"\n\n\"Yes. Two, in fact. Ithani breed in triads.\" Erina blinked repeatedly, seeming distressed. \"I lost them both in the war. Toreo and Sinael.\"\n\nXander frowned. \"Breed match... that's what we saw last night, isn't it?\"\n\nErina nodded. \"It was the last breeding before the migration. Do humans ever breed in triads?\"\n\nXander snorted softly. \"Well, we have three-ways sometimes\u2014\"\n\nJameson elbowed him hard in the chest. \"No, it's not the same for humans.\"\n\nErina looked at one and then the other. \"You are very strange to me. I wish we had more\u2014\"\n\n\"What are they?\" A bright blue ithani appeared directly above them, zer\u2014zis?\u2014beautiful iridescent wings fluttering back and forth.\n\n\"Yes, Erina, what are they?\" This one was golden, with a red crest.\n\n\"Where did you find them?\"\n\nIn just seconds, they were surrounded by a rainbow-hued crowd of ithani in flying form.\n\n\"They're humans!\" Erina's booming voice cut through the chatter. \"Everything else will be made clear at the tribunal. Now go!\" Ze waved zer hands, and they scattered as if blown by the wind.\n\nAhead of them, the Heart loomed. Their craft glided over it into a wide bowl. In the middle, six or seven green towers covered with globes emitting a golden glow, topped by what looked like a wide arena.\n\nXander frowned. \"What tribunal?\"\n\nErina blinked. \"I will take you to the nimfeach, as I promised. First, though, Thshnel'Jirron must grant us access to the Heart.\" Ze looked sad, but Xander wasn't sure how to read ithani emotions yet. \"You must trust me.\"\n\n\"The Thishnail?\"\n\n\"Thshnel.\" Ze said it like tithishnell. \"Zis name is Jirron. Zi's the leader of the ithani.\"\n\nJameson and Xander shared a look. Jameson reached into his carry sack, but Xander shook his head. \"This may be exactly who we need to talk to.\"\n\nJameson frowned, then nodded. \"See where it goes?\"\n\n\"Yes.\" To Erina he said, \"We'll meet this leader of yours.\"\n\nJameson snickered.\n\n\"What?\"\n\n\"Seriously? Take me to your leader?\"\n\nXander laughed. \"Yeah. Take me to your leader.\"\n\nErina cocked zer head. \"Humans are very strange creatures.\"\n\nTHE WHITE ship that had been a balcony landed at the edge of the stadium, letting them off before rising into the air once again to find a perch farther down the Heart.\n\nJameson watched it go, wondering if it ran off amalite like the hoversports. There was no obvious method of propulsion, and the ship moved so smoothly through the air.\n\nThen other matters demanded his attention.\n\nThousands of ithani were arriving at the stadium. Word must have gotten around.\n\nSome flew in on their own power, while others arrived on ships similar to Erina's. A few came through gateways, always accompanied by an entourage of other ithani and nimfeach.\n\nErina took them down one of the wide stairs that led to the center of the arena. Jameson felt a bit like a lamb led to slaughter.\n\nIthani all around them stared at the strange beings, chittering among themselves. They were as many colors as Jameson could count, from deep copper browns to verdant bright greens, and everything in between.\n\nThere must have been thousands of them.\n\nHe tried to read the emotions in their multifaceted eyes but failed utterly to decipher the thoughts of the inhuman ithani. There were a lot of cocked heads and clucking tongues.\n\nAs they reached the base of the bowl, a wide gateway opened up ahead of them on a raised dais.\n\nA bright-red-and-black ithani stepped through, wearing a gold sash. Zi\u2014for this was surely the Thshnel\u2014was easily eight feet tall, head and shoulders over most of the rest of zis race, with a proud bearing that even Jameson could read. Zi carried a white staff as big around as Jameson's forearm.\n\nBehind zim, another twenty or so ithani arrived, all dressed in matching black sashes, carrying staffs.\n\nJameson felt so out of his depth here.\n\n\"The Thshnel?\"\n\nErina made a fist at zer side.\n\nA chorus of nimfeach came through the gateway last, and Thshnel'Jirron raised zis hand.\n\nThe gateway vanished, and a hush fell over the assembled crowd.\n\nThe Thshnel held out zis staff and pointed at Erina. \"Erinastor, progeny of Thais, Danacae, and Evros... why did you call us here, so close to the migration?\" Zi held out zis arms, taking in all the gathered souls. \"We have more important matters, certainly, than to spend time on a couple dhagani spies.\"\n\nThe sarcasm came through clearly in Common. Or maybe Jameson was just adding that interpretation himself.\n\n\"They have a key.\"\n\nThere was a hissing sound throughout the arena at that. Then everyone went deadly silent.\n\n\"Bring them here.\" Thshnel'Jirron cocked zis head at him, and Jameson felt the alien's gaze pierce him.\n\nErina led them up the steps to the dais.\n\n\"Show me.\"\n\nErina nodded.\n\nJameson pulled out the key, hesitant to hand over their only remaining means of escape. He looked at Xander, who shrugged.\n\nErina gestured toward Jirron.\n\nWith a heavy sigh, Jameson held out the amalite key.\n\nJirron took it in zis three-fingered hand and peered at it. \"How did you get this?\"\n\n\"It was given to me.\"\n\nAt the sound of Jameson's voice, Jirron hissed, zis ears laying back flat against zis skull. \"Their voices are like the cries of baby schracknell.\" Zi turned to Erina. \"Where did you find them?\"\n\n\"Outside the city. They were watching the mating ritual.\"\n\nZi leaned forward and sniffed them. Then zi stared up at the sky.\n\nA bright ray of light shone down onto the arena, bathing zim in red.\n\nJameson looked up.\n\nThe white ceiling was opening up, retreating from the center, letting in the midday sun. It was an angry red now. It wouldn't be long.\n\nThe Thshnel threw the key to Erina. \"It matters not. This is yours now. If they are spies, they come too late to stop us.\" Zi raised zis arms. \"Ithani, the great migration is upon us! Go to your places beneath the mountains. Soon we will shift the world, and then ourselves!\"\n\nA great cheer went up, high-pitched and grating to Jameson's ears.\n\n\"Take them to the Heart.\" Jirron looked at Jameson and then Xander, blinking rapidly. Then zi turned and summoned zis entourage.\n\nA gateway opened up, and soon they were gone.\n\n\"What just happened?\" Xander looked bewildered, watching the ithani evacuate the arena.\n\n\"I think we were just given the keys to the kingdom.\" Jameson stared after Erina, who was climbing the dais. \"If we trust zer.\"\n\n\"Do we have any other choice? Ze has the key.\"\n\nJameson scowled. \"Point taken.\"\n\nThey followed Erina into the Heart.\n\n# CHAPTER TWENTY: INTO THE AIR\n\nTHE MAKESHIFT raft floated along the current through the tunnel that had been hewn deep into the rock. Patches of white fungus hung from the dark ceiling in irregular clumps, like miniature upside-down trees, each branch tipped with a glowing golden sphere.\n\nAlix stared at Venin's back in the dim light. The man had removed his shirt, and beads of sweat ran down his dark skin, pooling on the raft below. His feet hung out into the water, and his wings were relaxed at his back. The tarnished silver chain glinted in the changing light from the water and the mushrooms above.\n\nIt was warm down here. Not hot. Just a steady, rather muggy temperature.\n\nHe wondered again about the people who had built those shacks along the beach so far underground. Who were they? How had they gotten down here?\n\nWere they trapped?\n\nHad they died there? Will we?\n\nThey had passed through several caverns now, each similar to the first.\n\nAlix thought about other things too.\n\nHe closed his eyes and felt Venin's lips on his own again.\n\nI wanted to see what it would feel like. Alix shook his head. He wasn't ready for this. It was too soon. Isn't it?\n\nHe scratched the back of his head. He missed having long hair, the way it had felt on his shoulders. The way it had made him feel about himself.\n\nHe'd cut it all off when he'd joined the rangers, but just thinking about it reminded him of Xander.\n\nToo soon. And yet.... It had been a year and more since they'd really been together. A year was a long time to be alone. To go without.\n\nHe moved forward as far as he dared without tipping over the raft and put a hand on Venin's shoulder. Venin turned to look at him, his eyes asking the question. Alix nodded, and Venin kissed him, and his stomach flipped.\n\n\"I think we're coming out of the tunnel.\" Alix opened his eyes. They were still sitting on opposite ends of the raft, but he could feel those lips. Dammit.\n\n\"You okay?\" Venin glanced back at him, and his eyes held only concern.\n\n\"Yeah. Sorry. Just tired.\" At least that much is true.\n\n\"Look!\"\n\nAlix followed Venin's gaze. The way was lightening ahead. They were approaching another larger space, but it was hard to make out any details.\n\nAlix leaned over and splashed some cool water on his face. \"How's the wing?\"\n\nVenin shrugged. \"A little better? The water helped but it still hurts.\" He stretched it slowly and winced. \"Yeah, no flying for a while.\"\n\nThere was a rumbling noise in the air. It took Alix a few moments to recognize it. \"You hear that?\"\n\nVenin cocked his head. His eyes narrowed. \"Yes, there's something.\"\n\n\"It sounds like a strong wind.\" It was growing quickly louder as they approached the bright space ahead.\n\n\"Or water. A lot of water.\"\n\nThe raft sped up. \"Fucking Split\u2014I think it's a waterfall!\"\n\nThe raft flew toward the light.\n\nAlix held on for his life. The noise had grown to fill the whole cavern now, thundering in his ears like a hundred skythane wings beating the air, and the water splashed all around him, clouding his vision.\n\n\"Alix!\"\n\nVenin's voice came to him through the din.\n\n\"You have to get to me! Grab the mast and pull yourself forward!\" Venin's hand, dark against the splash of white water, reached out for him.\n\n\"I'll try!\" Desperate, he relinquished his grasp on the edge of the raft with one hand and threw himself forward, grasping at the thin mast.\n\nHe missed it the first time and overcorrected, almost sending himself into the raging water, grabbing the edge of the raft and pulling himself back to safety.\n\nWhite light filled the cavern now, the ceiling thick with the strange miniature treelike fungus.\n\nHe let go of the raft and threw himself forward, managing to snag the mast with his left hand. His right hand reached forward into Venin's.\n\nAlix let go of the raft and it fell away underneath them as they went over the top of the falls, taking all their remaining supplies with it.\n\nAlix didn't care. He slammed into Venin's chest and wrapped his arms around the man.\n\nVenin's wings beat powerfully, carrying them past the spume of the fall. \"Fucking Split,\" Venin said through gritted teeth, the pain in his voice clear.\n\nHe flew them down toward the ground below, each stroke of his wings clearly causing him agony.\n\nAlix just held on, trying not to distract Venin from his task. Instead, he stared at the cavern as they descended.\n\nIt was immense, easily ten times the size of the last one. Behind them the falls cascaded down to a wide lake at the base of the cavern wall, and a bright blue river wound from there through the cavern.\n\nIt was filled with corrinder.\n\nThey must have been corrinder, but they came in so many shapes and sizes that it boggled the mind.\n\nIt was like an immense city, filled with the most fanciful alien towers one could imagine.\n\nIn the middle stood the largest structure of all, a green dome that reached almost to the white ceiling above.\n\nIt was lost to view as they descended below the edge of the \"skyline.\"\n\nThey hit the beach hard, and Venin rolled into a ball, his hand over his wing.\n\nAlix's vision swam. He tried to get up and then sat back down again, gasping for breath. The impact had knocked the wind out of him.\n\nHe closed his eyes and tried to calm himself, ignoring the autonomic panic spreading through his brain. It will come back. Let it come.\n\nHis lungs unclenched, and he took a great gasping lungful of air.\n\nIn, out. In and out.\n\nHis breathing normalized, and he looked up at the huge waterfall. Venin had just saved his life.\n\nThe man was shaking all over.\n\nAlix ran to his side. \"Hey, you okay?\"\n\nVenin looked up \"Hurts. Memories. Hurts.\" His eyes were glazed over, and he was shivering all over.\n\nHis wing was covered in blood.\n\n\"Come on. Get into the water.\" Venin had said the water had curative properties. Alix coaxed him up and into the lake, step by hesitant step.\n\nWhen they were deep enough, he guided Venin to lie down and immerse himself on his back, wings spread.\n\nVenin shuddered and howled in pain.\n\n\"It's okay. I'm here.\" He supported Venin's back with one hand and splashed water over the wing, cleaning the blood off. They'd had each other's backs for a week as they'd fought through the Erriani jungle and against his mother's forces. Now they were trapped in this together.\n\nIt was hard to see the skythane man like this\u2014weak, at the mercy of his strange memories.\n\nVenin's eyelids flickered as if he were in REM sleep.\n\nAlix needed to break him out of the cycle. He needs me.\n\nMaybe it was time to let his own walls down, just a little.\n\nGrimacing, he stuck his thumb in the wound on Venin's wing and pushed.\n\nVENIN WAS locked inside his own mind, trapped by a torrent of unwanted memories.\n\nPart of him clung to a sense of self, to who and what he was\u2014Venin Araio. Gaelani. Skythane. Human.\n\nThe rest of him was washed away in the flood. Then everything cleared. He was in the air, surrounded by skythane and ithani. Dani Black floated before him, her face twisted in a parody of a grin.\n\n\"The Heart is mine.\"\n\n\"Dani Black!\" He threw himself at her.\n\nShe turned toward him and tried to spring away, but she was too slow.\n\nHe rammed into her and\u2014\n\nSearing pain cut through the torrent of images, white-hot like the edge of a knife tempered in flame.\n\nVenin screamed, arching his back, struggling to get away from his tormentor.\n\nStruggling in water.\n\nHe thrashed about, coughing out the moisture that had invaded his lungs.\n\n\"It's okay. It's just me.\" Alix's strong hands held him up.\n\nVenin calmed, letting his body float in the water. He looked up into Alix's warm brown eyes.\n\nAlix was staring back at him intently. \"You okay?\"\n\nVenin nodded. \"That... that was bad. There were memories. So many memories....\"\n\n\"I know. You sure you didn't pick something up from Jameson?\"\n\nVenin shook his head. \"It was the Split-cursed statue.\" The warm water all around him had a calming effect. He felt better, more in control. \"What happened to the raft?\"\n\n\"You don't remember?\"\n\n\"No. Just the sound of the waterfall, and then us, here.\"\n\n\"You saved my life.\" Alix glanced upward, and Venin followed his gaze.\n\nThe waterfall dropped from the cavern a good three hundred meters to crash on a jumble of rocks far below. \"I did?\"\n\n\"Yes. You flew me down from there, even though the pain nearly killed you.\" His voice cracked, or maybe it was just Venin's imagination.\n\n\"I don't remember that at all. But the pain....\" He reached up to touch the open wound on his wing. \"I felt that. It burned through me like a knife.\"\n\nAlix looked away sheepishly.\n\n\"What?\"\n\n\"It was the only way to bring you back.\" He gestured pressing his thumb. \"The pain\u2014\"\n\n\"Ah.\" He grimaced. \"It worked.\"\n\n\"I'm afraid we don't have anything to bandage it up with.\" He knelt to peer at the wound. \"I rinsed it out as well as I could.\"\n\nVenin laughed. \"Yeah, we kinda lost our shirts, didn't we?\" Not that he minded the view. Alix was a beautiful specimen of a man. Tazim would approve. \"I just wish we had more\u2014\"\n\nAlix cut him off with a kiss.\n\nVenin responded hungrily. It had been a long time. Far too long.\n\nWhen Alix lifted his head, Venin whispered, \"What was that for?\"\n\n\"I liked how it felt the first time.\"\n\nVenin shook his head. \"What does it mean? You told me you weren't ready\u2014\"\n\nAlix looked away again. He suddenly looked more a lost boy than an OberCorp ranger.\n\nVenin put his hand under Alix's chin and gently turned Alix's face back toward his. \"It doesn't have to mean anything. Not right now.\"\n\nAlix sighed. \"It means something. I just don't know what yet.\"\n\nVenin nodded. \"That's fair.\"\n\nAlix gestured in the direction of the strange underground city. \"Come on. Time is running short.\"\n\nVenin bit his lip. He checked his PA. Eleven hours. Maybe a few more if they were lucky. \"Taz, any connection to the grid?\"\n\n\"Sorry, nothing yet.\"\n\n\"Let me know if you get grid access.\"\n\nThey swam back to shore, and Venin got up, his pants dripping wet again. He followed Alix onto the beach, where he emptied the water out of his boots. He tied them together by the laces and slung them over his shoulder. \"Where are we going?\"\n\n\"This cavern's like a giant underground version of Errian.\" Alix pointed at the white towers that started about a hundred meters from where they stood. \"There's a huge\u2014something that way.\"\n\n\"Huge something, huh?\" Venin knelt to take a sip of the lake water.\n\n\"Yeah. It's\u2014maybe like a huge tree stump? A dome? It's hard to describe it, but it sits in the middle of the cavern.\"\n\nSomething flashed through Venin's mind. He staggered, but Alix caught him and steadied him.\n\n\"Memories?\"\n\n\"Yeah. Memory.\" He took Alix's hand. \"It's the Heart. I think it's what we came here for.\"\n\n# CHAPTER TWENTY-ONE: IN THE DEPTHS\n\nJIRRON HOVERED above Freyyr, the snow-covered mountain of the north, shivering in the cold air.\n\nHuman bodies were so different from the ithani, much weaker. Maybe his decision to merge with one of them had been a mistake, and yet... there was so much more to learn about them. To take into the Heart.\n\nHe'd seen the way newly awakened ithani had looked at him. They were afraid of him. After a lifetime that extended across a millennium, he'd done something unprecedented.\n\nThe nimfeach were still under his control, all but the human hybrid\u2014they called him Morgan\u2014who had bridged the split worlds. He was going to be trouble.\n\nCutting her wings off as she screamed.\n\nHe shook his head. Bits of the human woman's memories hadn't yet been fully integrated into his own. It was annoying, but he could live with it.\n\nHe had other things to worry about. Now was the time of the Rising.\n\nHe spread his white arms, extending his consciousness through the athr\u00e0 to the mountain below. He encompassed the nimfeach, the awakening ithani, and then the mountain itself.\n\nTogether, they merged into one, sharing their power for the briefest moment.\n\nIn that instant, he forced his hands apart with great violence, rending the earth below with his will.\n\nThe mountain cracked, splitting into three parts, the fissures racing down toward the packed snow below. The ground shuddered with the shock, setting off avalanches in the nearby mountains, and a great roar filled the north.\n\nThe forge tumbled down into the waiting snow below, sending up bursting white clouds of frozen precipitation, and a golden glow shone from within.\n\nThen the ithani rose, breathing in the fresh outside air for the first time in more than a hundred thousand cycles.\n\nThshnel'Jirron exulted.\n\nThe Great Migration would soon begin. He would finally become truly immortal, and there was no one left who could stop him.\n\nMORGAN STOOD in the snow, far from the violence as the mountain collapsed, watching the ithani king rip open the mountain.\n\nTime was growing short.\n\nSince he and Tanner had become one, he had fought to bring about the plan\u2014the freeing of his masters, the ithani, so they could complete the Great Migration.\n\nNow, new things were bubbling up unbidden in his mind. Fleeting thoughts of another plan. Feelings like \"loss\" and \"sadness\" and \"home.\" Ideas alien to his nimfeach nature.\n\nAnd yet they tugged at him.\n\nHome.\n\nThe village the humans called Egeus, on the southern end of the Gildensea.\n\nXander and Jameson, Quince and all the rest were going to die soon, if he kept to the plan.\n\nMorgan had already deviated from it in a dozen ways, small and large. He felt something for these humans he'd come into contact with. Something akin to friendship. Or love?\n\nIt seemed wrong that they would die so the ithani could grow.\n\nIt seemed wrong that the nimfeach would die as well.\n\nIt is wrong. The Tanner part of him was insistent. It's wrong and you know it. We can't let them all die.\n\nThe snow and rubble were settling now, and the ithani were rising.\n\nTo the south, the sun was descending toward the horizon, dropping behind the cloud cover. Soon the ithani host would depart for the Heart.\n\nTime was growing short.\n\nHe needed to go home.\n\nHe opened a gateway to Egeus and stepped through.\n\nTHE EARTH shook underneath their feet.\n\nQuince frowned, staring up at the air.\n\n\"What?\" Jessa knew that look. It meant trouble.\n\n\"The signal. It's gone.\"\n\nThey'd hiked for at least half an hour down these nearly featureless tunnels, access ways built probably a couple hundred years prior by OberCorp when the Slander was still a thriving warehouse district. Jessa had read up on her Oberon history.\n\nThe place where they'd \"landed\" had been clean and well lit. Probably used for smuggling across the city, but the part of the network they were in now showed little sign of use. Dust was thick on the ground, the plascrete walls old and cracked. The light came from long-lasting lightdumps recessed in the walls, though even some of those\u2014which were supposed to last a couple hundred years\u2014had gone out.\n\nThe air was stale too. Thank goodness not so stale that it was actually unhealthy. At least none of them had fallen over from lack of oxygen yet. Then at some point, the tunnel they were following had connected to a natural cavern. It looked like a lava tube to Jessa, a fairly consistent width and height, running more or less in the same direction. She wondered if the original engineers had taken advantage of the connection to link these tunnels to the outside world and save themselves a bit of construction work.\n\nJessa had done a bit of spelunking on Beta Tau. If the air was breathable, there had to be a connection to the outside world somewhere.\n\nQuince pulled out a glow sphere, activated it, and lifted it into the air, peering into the dark tube. \"Do we go on?\"\n\n\"I don't see that we have much choice.\" Robyn glanced over her shoulder at the way behind them.\n\nJessa met Quince's gaze. \"You think this is important?\"\n\n\"I do. If we can find out what's intruding into Oberon's grid, maybe we can use it to stop the ithani.\"\n\n\"What if it is ithani?\"\n\n\"Cross that bridge when we reach it.\"\n\nRobyn nodded.\n\nQuince smiled without apparent cheer. \"Okay, then we go on.\"\n\nJessa followed the others, lost in her own thoughts, wondering again what had happened to Jamie and Xand and where they had ended up. She was still convinced they were alive\u2014call it intuition, or faith. She'd learned a lot about faith, real and feigned, back on Beta Tau.\n\nShe couldn't help but feel that they were heading down a rabbit hole to nowhere, following Quince's crazy will-o'-the-wisp. And yet, Quince was so certain.\n\n\"What now?\" Robyn arched an eyebrow.\n\nJessa looked up. Quince had come to a halt where the cavern split, offering them two different directions.\n\n\"Not sure yet.\" Quince looked around, her wings fluttering in distress.\n\nIt was weird enough that the signal had followed this passageway and hadn't been a direct shot from wherever it originated to the OberCorp grid core. How that was accomplished, Jessa had no idea. There must be something directing the signal along this path. Of course, it was also probably the path of least resistance, assuming the source was under a few hundred thousand kilos of rock.\n\nQuince tapped her forehead. \"Ari, where are we?\" Her frown deepened. \"He says he can't get an exact location without a grid connection, but based on our movement so far, we're probably somewhere just south of Oberon City, not far from the Gildensea.\"\n\nHeaven's gates. There had to be a way to determine the correct path. Follow the signal. Jessa put a hand on the rough, dark rock wall and looked up.\n\nSomething lurked up there. Something white, like a long tube or root that ran along the ceiling, not far from where Quince said the signal had been. \"Quince, take a look at this.\" She stood on her tiptoes, staring at the dark place where the ceiling and wall met.\n\nQuince peered into the dim corner. She pulled out her pulse rifle, stretched, and pushed on the tube with the nose of the gun. It retracted like a living thing, and then rebounded back. \"It's pliable. It feels more organic than synthetic, maybe?\"\n\nShe followed it to the juncture. It turned right and went down the new passage. \"Jessa, you're a genius.\" Quince kissed her on the cheek.\n\n\"Maybe we can use it to reach the grid?\" Jessa cocked her head, staring at the \"root.\" Damn, I should have gotten a PA.\n\n\"Good thought.\" Quince squinted at the root. \"Robyn, can you hoist me up?\"\n\n\"Of course.\" Robyn knelt and lifted Quince up to the ceiling.\n\nQuince held her right wrist over the root. \"Ari, can you detect anything?\"\n\nRobyn held her, rock-steady.\n\n\"Ari says there's a faint signal. The root seems to be acting as a carrier for the signal.\" She frowned. \"He should be able to ride it back to the core. Ari, get me Alix, if you can.\"\n\nQuince waited.\n\nJessa paced the passageway below, feeling claustrophobic. What's happening up there? They were trapped underground for the moment, and she didn't like it at all. She glanced at her two winged companions, doing their best impression of a Beta Tau midschool cheerleading squad. If I feel this way....\n\nSkythane weren't meant to be confined underground. They needed the wide-open skies\u2014any fool could see that.\n\nQuince must have been thinking something similar. \"After this is over, Robyn, we're getting ourselves a quiet place up in the Mora Mountains and the rest of the world can\u2014\" She cut off midsentence. \"Alix is away, but Ari's getting me Mylin.\"\n\nRobyn squeezed her thigh in apparent agreement.\n\nThe rest of the conversation went on in Quince's head.\n\nJessa fingered the safety on her pulse rifle while she waited, glancing back at the tunnel behind them.\n\nWas anyone following? There were four paths from their access point, but a quick search would show any pursuers that they'd cleared the rubble from one. Farther on, their footprints in the dust would betray them.\n\nThen again, they had done a number on the access tunnel. Maybe they were safe from pursuit, for now.\n\n\"Mylin? Mylin!\" Quince sounded anxious.\n\n\"What happened?\"\n\n\"She said Alix and Venin were trapped somewhere, but okay. Then she tried to warn me about something. I lost her.\" She stared back the way they had come. \"The signal's back. I think it swamped her transmission. Let me down.\"\n\nRobyn knelt so Quince could climb off her shoulders.\n\n\"Mylin said something about the ithani just before we were cut off, but I don't know what she meant.\"\n\n\"What did she say?\" Robyn looked worried. Robyn didn't worry easily.\n\n\"She said we had to be careful.\" Quince looked back the way they had come. \"Like that's any surprise. Okay, let's go. At least we know where the signal goes now, if it cuts out again.\"\n\nJessa followed the two skythane women down the passage, resigned to being trapped in the dimly lit corridor for now.\n\nThere was no way out but forward.\n\nROBYN FOLLOWED Quince's lead, pulling her wings in so they wouldn't touch the rock walls. This place was disturbing. It was so quiet here, the air dead and flat, and every noise they made echoed down the passageway.\n\nThe glow sphere illuminated the space around them for about twenty meters, and then the darkness began again.\n\nWhat use were her new wings if she couldn't use them, couldn't even stretch them out?\n\nShe looked back at Jessa, bringing up the rear. There was a flash in the tunnel behind them.\n\n\"Get down!\" She dove at Jessa, pushing her to the ground, her wings cushioning their fall.\n\nSomething hot flashed over her, singing the feathers of her left wing. \"Quince, we're under attack!\" Robyn crouched, unslung her bow, and fired off a few quarrels.\n\nThere was a cry and then a muffled thud. One down.\n\nQuince flattened herself next to Robyn, her pulse rifle firing into the darkness.\n\nJessa joined her.\n\nTwo more shouts, and then Quince was barreling off, knife unsheathed, a silent avenger.\n\n\"Hold your fire!\" Robyn whispered urgently to Jessa.\n\nThere was a gurgling scream, then another.\n\nJessa put her rifle down, closing her eyes and turning away from the sounds.\n\nIn a moment there was nothing but silence.\n\nQuince returned, carrying a couple more rifles. \"Syndicate men. They either cleared the blockage we created in the shaft, or there's another way in.\" She handed one of the weapons to Jessa.\n\n\"Probably another way in, or several, if they use the tunnels for smuggling.\" Jessa was a light shade of green, but she controlled herself admirably.\n\nRobyn nodded in approval. \"Give me one of those, please.\" She held out her hand.\n\nQuince blinked. \"Are you sure?\"\n\n\"Yes. I need something with longer range.\"\n\n\"But I thought... never mind.\" She handed a pulse rifle to Robyn and stuffed the other under her carry sack strap across her back. \"The release is here\u2014\"\n\n\"I know how to use it. I killed a few enforcers with one, before they took my wings.\"\n\nQuince grinned, looking at her with apparent newfound respect. \"I've never loved you more.\" She pulled Robyn to her and kissed her hard. \"Come on. They may be expecting us on the other end too.\"\n\n\"Let me take the lead. I've been following long enough.\" Robyn needed to take control of something.\n\nJessa laughed. \"You two are amazing together.\"\n\n\"That didn't squick you out?\" Robyn doubted Jessa had ever been in a firefight on her tame home planet.\n\nJessa paled. \"A little. Sure. But we have to protect our own, right?\"\n\n\"Yes we do.\" Robyn squeezed her shoulder. \"Let's go.\" She led them down the tunnel, away from the carnage they'd just wrought. \"How much time do we have?\"\n\n\"About ten and a half hours.\"\n\nRobyn wished they could stop to rest, take even the briefest nap. She was exhausted.\n\nThere was no time. They had to keep going if they were to have any chance of unraveling the mystery the ithani presented before the endgame.\n\nI can sleep when I'm dead.\n\nIN THE end, it was Quince's body that forced them all to take a break. She stumbled and slammed into the tunnel wall, hard, almost knocking the breath out of herself.\n\n\"Hey, you okay?\" Robyn took her by the shoulders and held her up.\n\n\"Yeah. I can keep going.\"\n\nSo they did.\n\nQuince concentrated on putting one foot in front of the other. The world swam a bit\u2014at times she thought she was walking along a beach on the Gildensea\u2014but then she snapped back to reality.\n\nThen it happened a second time, the rock wall shocking her back to awareness.\n\n\"That's enough.\" Robyn glared at her. \"You can't go on like this.\"\n\nQuince tried to stare her down, but there was no fight left in her. \"We have to keep going. We only have a handful of hours left.\"\n\n\"You should see yourself.\" Jessa frowned. \"Your face is gray. Robyn is right. You need a break. I brought some extra stims, but if you take them like this, they will just hollow you out.\"\n\n\"I can keep going,\" Quince insisted again, pushing away from the wall, but her legs betrayed her and she fell.\n\nRobyn caught her and lowered her gently to the ground, leaning her against the plascrete wall. \"No, you can't. What good will it do if we get wherever we're going, and you're too tired to fight?\"\n\nQuince tried to protest, but all that came out was a sharp croak.\n\n\"My point exactly. So we stop? Even if just for a short time? Eat something?\"\n\nQuince sighed. After a moment, she nodded. \"If the world ends because we didn't keep going\u2014\"\n\n\"Then you'll have one less thing to worry about, won't you?\" Jessa knelt and opened her carry sack.\n\nRobyn chuckled. \"I knew I liked you.\"\n\n\"Thanks.\" Jessa yawned.\n\nShe looks as exhausted as me. Quince closed her eyes, resting for just a minute.\n\nJessa cleared her throat. She'd pulled out a flask from her carry sack and now handed it to Quince. \"Here, drink this.\"\n\n\"What is it?\" Quince sniffed the contents suspiciously.\n\n\"I'm not sure. I asked for something a hundred proof from the OberCorp synthesizers. Take a few sips\u2014it'll help you rest. But eat something too.\" She handed Quince a few pellets.\n\n\"No thanks. I brought some real food.\" Quince pulled out a carefully wrapped loaf of bread and some cheese. \"It may be synthesized too, but at least it looks like food.\" She took a bite of each, and then a sip of the drink. \"Damn, that's vicious.\" It tasted like the ass end of an auxen. Then she felt the warmth spreading through her stomach. \"That feels nice.\"\n\nShe closed her eyes again, and in seconds she was out cold.\n\nDAIA TRUDGED across the sand, carrying a bucket of water from the stream back to the colonists' makeshift encampment.\n\nIt was a beautiful world, a garden world, like old Earth was supposed to have been.\n\nShe wasn't sure she believed the tales her grandmother told. Sure, there must have been a homeworld. They'd all come from somewhere, and this new place was proof enough that worlds existed.\n\nSome in the Church preached that God had created humans in her own image, and that Forever was the birthplace of mankind.\n\nSome even believed that the world mind was God.\n\nDaia thought that was utter nonsense. Someone had built the giant ship that had brought them between the stars to this place.\n\nThe last few years had been brutal, as Forever's systems had failed. The world mind had managed to hold on until they reached this place and had been able to generate a new seed before she died.\n\nNow they were here, arrived in the pods she had made for them. This new world was theirs.\n\nSeveral hundred colonists had reached this wide harbor that fronted the open sea. The seed had been planted in a cavern nearby with access to light and water and rock, and Daia had imprinted herself upon it. As it grew, it would help guide the colony, and with time they would find the others and build a civilization.\n\nDaia stopped to look wistfully at the sky. The place she had grown up in, her whole world, was still up there somewhere, drifting away from them into the void. It had been a hard decision to make, but a mass of that size, falling onto the split world, could have proven catastrophic.\n\nThere was truly no going home now.\n\nDaia sighed.\n\nMarli's pod hadn't been among those that reached this open bay. Maybe her pod had malfunctioned and she had hurtled to earth, suffering a fiery death.\n\nDaia hoped it wasn't true. She hoped Marli was out there somewhere, and that they would find each other again. Daia closed her eyes, squeezed again by grief. Marli, heart of her heart, connected to her in a way no one else had ever been. Or ever would be. Marli, one of the smartest minds of her generation.\n\n\"Daia!\" Kylan waved at her from down the beach.\n\nShe picked up the bucket and put on a smile for him. Poor kid, trapped down here with all these morose adults without his parents, unsure when or if they would be reunited.\n\n\"Hey, Ky.\" She tousled the boy's sandy-blond hair. Kylan was one of the sensitives, the few who could share their thoughts with others like him over a long distance. \"What's up?\"\n\n\"There's a beautiful thing. You have to come see it.\" He tugged her arm. \"Come on! You can come back for the water later.\"\n\nShe laughed. The kid's enthusiasm was infectious. I could use a little beautiful. \"Okay, I'll come.\" She set down the bucket and followed him back up the beach, sticking to the wet sand by the water where the footing was easiest.\n\nShe spared a glance upward. How she missed the streets of Darlith where she'd grown up, in the middle of civilization. She wished she had wings to fly, to carry her back there.\n\nIt would be a short-lived respite, though. There was no doubt that world, her ship, was dying.\n\nThey ran up into the jungle, to the place where the new village was being constructed out of local saplings and a workable concrete they were making out of ash and sand. The whole colony seemed to be gathered in the empty square in the middle of the small cluster of makeshift houses.\n\nSomething glowed in the midst of them.\n\nShe pushed her way through\u2014she was technically the colony master, after all. \"What is it?\" she asked. Then she froze, dumbfounded.\n\nIt was maybe two meters long, hovering in the air before her. It had glowing wings, and it was easily the most beautiful thing she had ever seen.\n\nShe reached out a hand to touch it, and an electric shock raced through her nerves.\n\nThen a voice spoke in her head. \"Hello, Daia Hammond.\"\n\nQUINCE STARTED out of her sleep.\n\nShe'd been dreaming. There was a beach. A woman. It had been so real. A memory? She wasn't like them. She didn't have other people's memories in her head. She hadn't touched a statue. Not like Jameson and Xander. And Venin?\n\nSomething was happening to the skythane. Like a switch had been thrown.\n\nAnd what about the memory itself? Simple exhausted delusion? Or had she just witnessed the earliest skythane presence on Oberon? If so, why didn't they have wings? Her head hurt thinking about it all.\n\n\"Hey, you okay?\" Robyn knelt next to her.\n\nIt was exhaustion. It had to be. \"Just a dream.\" She checked her internal clock. \"You let me sleep for an hour?\" It came out as more of a growl than she intended.\n\n\"More or less? Here. Drink some of this.\" Robyn held out a canteen.\n\nQuince took it, sniffed it, and then sipped the warm water gratefully. \"Where's Jessa?\"\n\n\"She rested with us for a few moments, then got up and went on ahead to scout out where we're going.\" Robyn handed her some bread and fruit. \"Do you feel any better?\"\n\nQuince nodded. \"A little. Still feel like I was trampled by an aux.\" Barely functional. It'll have to do. She took the food eagerly, and as she ate, some of the fatigue seemed to drop away. \"I can manage.\"\n\nThings flapped through her head like birds at the corners of her vision. She mentally slapped them away. Memories. How had Jameson put up with the damned things?\n\nFootsteps echoed down the cavern. Someone was coming at them, fast.\n\nGlancing at Robyn, Quince picked up her pulse rifle and slid into a crouch, aiming to take out whoever it was.\n\n\"It might be Jessa.\" Robyn was next to her, her own rifle trained on the darkness ahead.\n\nQuince nodded.\n\n\"Quince, Robyn, it's me!\" Jessa appeared out of the dim passageway \"There's... it's amazing. You have to come!\" She reached them, out of breath, and Quince was reminded of the boy in her dream. Kylan.\n\n\"What is it?\"\n\n\"I don't know, but I think it's what we're looking for.\"\n\n# CHAPTER TWENTY-TWO: GHOST TOWNS\n\nMORGAN STEPPED out of the gateway into the streets of his childhood home.\n\nEgeus was a hardscrabble town along the southern reaches of the Gildensea, the southernmost human town on Oberon. A forest of ice pines surrounded it, extending up into the hills above the town. They were frozen, victims of the big storm that had blown through town after the shift.\n\nThe town was just a hundred kilometers north of the southern glacier and was always cold, even in the middle of the summer. The sea's waters were dull and gray.\n\nMorgan didn't bother to hide his presence or his manner of arrival. The time for such concerns was past.\n\nThe main street was a pitted mess of pavement, lined by buildings that had long ago seen better days. They were mostly built of local wood from the southern forest, and their paint was peeling, while some were boarded up altogether.\n\nMorgan saw the town through Tanner's eyes.\n\nTanner's mother had been one of the town's \"comfort women.\" She'd given comfort to the single men in town, which was most of them, but Tanner hadn't really understood what that meant until he was older.\n\nHe'd never known his father\u2014an ice fisherman who had died the year after he was born, according to his mother.\n\nShe had died from pneumonia when he was just four years old.\n\nMorgan was struck by the image\u2014Tanner's memory\u2014of her on her deathbed.\n\nShe squeezed his hand, taking a raspy breath, and leaned forward to whisper something in his ear.\n\nTanner bent down to hear her.\n\n\"I want something better for you.\" She took a painful breath. \"You\u2014you're the one good thing I have ever done in this life.\" Her frail hand reached out to cup his cheek. \"My little angel.\"\n\n\"Mamma, don't go!\"\n\nShe took his hand in hers. Her hand was warm. Too warm. He didn't understand it. Not then. \"You are strong. You have my blood in your veins. The blood of kings.\" She pulled him down to kiss his forehead, and then lay back on her rough pillow, her breath slowly diminishing.\n\nAfter a few moments, the rise and fall of her chest ceased.\n\nTanner let out a terrible wail, and then ran out of the tiny shack for good.\n\nMorgan stopped in his tracks. \"The blood of kings.\" Tanner had never shared that with him before.\n\nIn point of fact, Tanner's personality had only recently come to the surface. For months he'd remained hidden, having given up his will, his body, his mind to the beautiful stranger.\n\nSince Quince had almost died, Tanner had begun to reappear, first in random thoughts and impulses, then as a personality in his own right. Now they were one, and sometimes he felt more Tanner than Morgan.\n\n\"You came back!\"\n\nMorgan spun around.\n\nA young human girl, close to Tanner's own age, stood there in the street, surrounded by four other kids of varying ages. She was whip-skinny, her dirty blonde hair pulled back and tied to one side with a leather thong. Her clothes were an assortment of rags, and she held a makeshift spear.\n\n\"Tally?\" The word was out of his mouth before he knew what he was saying. He threw his arms around her, knocking her to the ground.\n\n\"Hey, easy there.\" She grinned up at him fiercely and pushed him off and stood, brushing the dirt off from her pants. \"Where in the three hells have you been?\" She thrust the spear at his neck, holding it just an inch away from nicking him. \"You're not a sneach, are you?\"\n\nHe could have destroyed her. Morgan could have burnt her spear to a crisp or thrust her through a gateway that opened a thousand feet up in the air. \"No. I'm not a sneach.\" Morgan held his hands out in surrender. \"I'll tell you everything. I swear on the Split.\"\n\nShe stared at him through narrowed eyes for a long moment. Then she nodded curtly and withdrew the spear. She offered him her hand. \"Come on. It's cold out and there's a storm coming.\"\n\n\"Where are all the adults?\"\n\nShe snorted. \"They all cleared out of here a couple days ago on the last company transport truck headed north.\" She spit on the ground. \"Left us all behind to die. We've taken over the old company store.\"\n\n\"Good to have you back, man,\" one of the other boys, Sully, said, clapping him on the shoulder. He was a couple years older and had always considered himself the adult of the group.\n\nMorgan let Tanner guide him and followed the ragtag band back up the street and into the company store.\n\nVENIN AND Alix climbed out of the bowl that held the lake and the waterfall. The hillside was overgrown with more white plants of various sizes and shapes. Venin had to cut his way through with his knife in a few places.\n\nThe plants pulled back with sounds like little screams. Alix shuddered. This place gave him the creeps.\n\nBit by bit, they made their way up a slight rise to the top of a hill that afforded them a better view of the whole cavern.\n\nIt was massive.\n\nIt stretched from where they stood at least five kilometers into the distance, a vast hollowed-out space with a white ceiling.\n\nLike in the smaller cavern, the roof gave off a soft glow.\n\nA city of white towers surrounded the giant structure in the middle of the cavern\u2014the Heart. It was certainly the heart of the city, but Alix guessed there was another reason for the name too. It was hard to judge the scale of it, but it towered over almost everything else. It reminded him of a beehive, with wide hexagonal panels overgrown by some kind of heavy green moss. The top was concave, like a hollowed-out arena.\n\nThe far side of the cavern was lost in haze.\n\n\"It looks like a long way to walk. I wish I could fly.\" Venin winced as he twitched his wing.\n\nAlix laughed. \"Me too. Wish I hadn't lost my bi-wings in the crash.\" The \"buildings\" of the city reminded him of Errian. He guessed they'd been made out of corrinder, or whatever the actual material was called.\n\nOr grown?\n\n\"Do you think anyone still lives here?\"\n\nAlix looked around the cavern. Other than the sound of the waterfall behind them, the place seemed dead quiet, deserted. \"I don't think so. Unless the whole city went into hiding when we popped our heads above ground?\"\n\nVenin laughed. \"Yes, we big scary hooomans.\" He put his arms out and flared his wings, doing a great skythane monster impersonation.\n\nHandsome, and a killer sense of humor. \"So how do we get there?\"\n\n\"I think there's a road down there.\" He pointed to the base of the hill. Something like a road or path wound through the hillsides and down into the city. There were small structures and great towers down there. Spirals and domes and some that seemed to hang from the ceiling itself.\n\nThe city had an overgrown appearance, as though it were a garden that hadn't been tended for a long time.\n\nThe ground rumbled.\n\n\"I think that's our cue. Come on.\" Venin clapped him on the back and started down the slope toward the road. He pulled out his knife and cut his way through the brush once again.\n\nAlix shrugged and followed. \"So tell me about this religion of yours.\"\n\nVenin turned to look at him, his eyes narrowed. \"What do you want to know?\"\n\n\"What it's all about?\" Religion was a bit of a mystery to him. To his mind, things just worked, or they didn't. There was a sensible mechanical reason for everything. \"I grew up in a strictly secular household. My mother's religion was OberCorp.\"\n\nVenin snorted. \"That sounds about right.\" The hillside was like a fairyland\u2014it must have once been a huge manicured garden, because there were black paths under all the shrubbery, and so many plants. Some were purple-leaved, like their aboveground counterparts, while others were white like the \"trees\" they'd made the raft out of, with their citrone-like fruit.\n\nAlix snagged a couple of these and handed one to Venin.\n\n\"Thanks.\" Venin used his powerful fingers to split open the fruit, scooping out the meat inside. \"By the Split, those are good.\"\n\nAlix laughed. \"Yeah, not bad.\"\n\nVenin smeared some of the fruit pulp across Alix's face with his fingers.\n\n\"Hey, stay on mission.\" He wiped it off, giving Venin a mock-serious glare. Inside, though, Alix grinned.\n\n\"Yes, sir.\" Venin held up one half of the fruit. \"Seriously, we have to come up with something to call these.\"\n\n\"Cave fruit?\"\n\nVenin stared at him. \"A bit on the nose, don't you think?\"\n\n\"Okay, how about cavins?\"\n\nVenin nodded. \"Fair enough. So this half of the cavin is Oberon. It's the province of Gael, the god of the moon, who ruled over Oberon since the great flare.\"\n\n\"Got it.\"\n\n\"When the landers first came to Oberon, the Gaelani\u2014followers of Gael\u2014lived on Oberon, and the Erriani\u2014followers of the sun god Erro\u2014lived on Titania.\"\n\n\"Ah, that's where the names came from.\"\n\n\"Right. So most of the skythane aren't so religious anymore. At one time, religion was a powerful motivating force among us, though. Wars were fought over differing interpretations of the legends and prophecies.\"\n\n\"The story of mankind, in a nutshell, and also the main reason I'm not religious.\" Alix grinned. \"Plus the whole deity-in-a-cloud thing never really worked for me.\"\n\nVenin laughed, a warm sound that stirred something deep inside Alix. \"I didn't used to be. Things changed the day that Tazim died.\" He stopped, leaning against a white tree, and closed his eyes. The pain was clear on his features, and his wings visibly drooped.\n\nDamn, I'm an idiot. \"I'm sorry. I didn't mean to\u2014\"\n\n\"It's okay. It's still hard to think about him. But that's not what you asked about.\"\n\nAlix very much did want to ask about Tazim. What was he like? What had Venin seen in him? What had befallen him?\n\nIt didn't seem like the time. \"So two worlds, two gods?\"\n\n\"Right.\" He flashed Alix a grateful smile. \"Two sides of one coin. One dark and mysterious. One bright and golden.\"\n\n\"Like a pair of skythane we know?\"\n\n\"I did say there was a prophecy.\"\n\nTempest comes with clash and thunder,\n\nSkies alight with rainbow's blood,\n\nWhen the sunlight runs to red,\n\nComes the reaper for the dead.\n\nOne with wings as black as night\n\nOne with wings of golden light\n\nSpin the worlds back into one\n\nTo save them from the murdering sun.\n\nAlix whistled. \"That's some heavy shit.\" Black and golden wings. It sounded like Xander and Jameson, although to be fair there were probably thousands of skythane with wings of those colors. But they had managed to shift the whole world. In more ways than one. \"Where did it come from?\" They were walking at a brisk clip through a wide plaza with what looked like a fountain in the center, though it was dry\u2014it was made up of white sails, connecting at odd angles and shimmering with that subtle rainbow sheen.\n\n\"No one knows. It's just something that has always been. Most of us didn't believe it anymore, at least not until\u2014\"\n\n\"Until it actually happened.\"\n\n\"Yes.\"\n\nThey'd reached the white path, which made the going considerably easier. They continued on in companionable silence, making good time toward the edge of the city. Alix had managed to get a catnap on the raft, but he was still tired. He'd been away for most of the last day and a half, and he hadn't had a good night's sleep in weeks.\n\nJust a little longer.\n\n# CHAPTER TWENTY-THREE: IN THE DREAMTIME\n\nJAMESON FOLLOWED Erina to the back of the dais, where a stair descended into the darkness of the Heart, pulling Xander along with him. The green \"ground\" beneath his feet was soft, like moss.\n\nThey clambered down the stairs after zer, coming to a wide space at their base.\n\nJameson's eyes adjusted. They were standing inside a dome whose walls were smooth and lined with silver filigree in fantastic patterns. The silver glowed.\n\n\"The Mountain,\" he whispered.\n\nXander looked up and gasped. \"It's just like it.\"\n\nThey had stood in a place like this once before, when they had shifted the world.\n\nErina turned to look at them, standing in front of a stone archway toward the back of the room.\n\nJameson felt a growing sense of d\u00e9j\u00e0 vu.\n\nZe made a gesture that was easy for him to interpret. Come on.\n\n\"Wait.\"\n\n\"We must go through the gateway before the Thshnel changes zis mind.\"\n\n\"I want some answers first.\"\n\nZe blinked at him a few times in sequence. \"Time is short.\"\n\n\"Humor me.\"\n\nZe worked that out and let out a whoosh of breath. \"Very well. Ask me.\"\n\nHe looked at Xander, who nodded.\n\n\"You didn't seem very surprised to find us. Why?\"\n\nErina clucked zer tongue. \"Because...,\" ze said at last, \"because I foresaw your coming.\"\n\n\"What do you mean?\" Prophecy. Ze had to be talking about prophecy.\n\n\"Every generation has a seer. To guide the race. I saw that you would be coming.\"\n\n\"That's not possible.\"\n\nXander laughed. \"Who are we to say what's possible and what's not?\" He gave Jameson a wry grin. \"After all we've been through in the last month?\"\n\n\"Point taken.\"\n\n\"Shall we go?\" Erina glanced at the archway.\n\n\"What are the nimfeach? Why do they serve you?\"\n\nThis time the whoosh was more pronounced. \"It's a long and terrible story. They are our... how would you say? Elders. When the ithani are ready to transition, they go into the Heart, taking their memories with them.\" Ze blinked rapidly. \"It is not an easy thing to talk about.\"\n\nJameson shared a glance with Xander. \"It's okay. Please continue. Humanity has commited enough atrocities. We won't judge you.\"\n\n\"You are very kind.\" Ze let out a whoosh of breath, stirring the hairs on Jameson's forearms. \"The ithani take our memories with us into the Heart, and when the nimfeach are born, they carry the memories of many who have gone before. Each one has... you call it a key... where the memories are stored.\"\n\n\"Ah.\" Now things were starting to make sense. \"This is where the memories come from.\"\n\nErina made a fist of agreement. \"The Thshnel... is not like the rest of the ithani. Zi was born over a thousand cycles ago.\"\n\n\"Years?\"\n\n\"Yes.\" The blinking again. \"Zi led us in the war against the dhagani, but when the time came for him to return to the Heart....\"\n\n\"Zi changed the rules.\"\n\nZe looked deflated. \"Yes. Zi took the body of a younger ithani, whose memories were lost to the Heart. It was gresachh.\"\n\n\"Gres...?\"\n\n\"It's hard to translate. It means, roughly, disgraceful, dishonorable, unthinkable, and perverse.\"\n\n\"Ah.\" So the ithani were no strangers to amorality. \"And the nimfeach?\"\n\n\"When the final dhagani attack poisoned the sun, zi declared... you would call it martial law. Zi took the keys from the nimfeach, stripping them of their power and will, and reducing them from elders to slaves. It has never been done before but the Thshnel... zi doesn't care much for morals or precedent or the well-being of zis people.\"\n\nJameson frowned. \"I know the type. So zi enslaved zis elders, just so zi could take you all to some higher plane?\"\n\nShe hissed, a sentiment that needed no translation. \"I don't think so. When the Great Migration ends, I think it will be only Jirron who survives.\"\n\n\"Why didn't you tell someone?\"\n\nErina turned away. \"I tried. No one would listen, and I feared word would get back to the Thshnel.\" She drew herself up. \"Now come. We must pass through the gate before zi rescinds zis permission.\"\n\n\"No.\"\n\n\"What?\"\n\nJameson was taking a chance. Maybe a stupid chance, but he had vowed to stop running. Now he wanted to stop following the lead of others too. It was time to take back some control. \"Show me where we are going.\"\n\n\"You can't... we can only get there through this gateway.\"\n\n\"Humor me.\"\n\nThis time she understood immediately. With another whoosh, she acceded.\n\n\"Can you show me, in the athr\u00e0?\"\n\nZe made a fist. \"Come here.\"\n\nHe approached zer. \"Angie, record this, please.\"\n\n\"Understood.\"\n\nErina cocked her head but said nothing. Ze put zer hand on his cheek.\n\nHe closed his eyes, and everything shifted.\n\nHe was looking out at a wide cavern. It was easily half a kilometer across, the walls covered with what looked like a growing green moss. Giant golden globes hung suspended from branches or roots that crisscrossed the space. In the center was smaller dome\u2014a shimmering pool of silver whose surface swirled and twisted like the surface of a soap bubble.\n\nHe opened his eyes.\n\n\"Did you see it?\" Zer multifaceted eyes searched his.\n\n\"Yes.\" He pulled out the key.\n\nZe clucked her tongue. \"It won't work. Only Jirron can make a gateway that leads there.\"\n\nHe laughed. \"I don't think that's true anymore.\" He twisted the key, visualized the Heart, and pushed. A gateway split the air.\n\nErina's fur shimmered in the dim light. \"Ah, you are something special, aren't you?\"\n\nJameson glanced at Xander.\n\n\"I think so.\"\n\nHe grinned at his man, and together, the three of them stepped into the Heart.\n\nXander looked around the cavernous space. \"Who knew this whole strange journey would lead us somewhere as weird-ass as this.\"\n\nErina cocked zer head. \"Weird ass?\"\n\nJameson managed a grin. \"It's not translatable.\" Memories fluttered through his head. Memories of this place, if not this moment. They'd been mostly quiescent since he and Xander had fallen through the gateway to this strange world, but now they were back in full force.\n\nXander squeezed his hand. \"What now?\"\n\n\"This place. It's\u2014\"\n\n\"I know. I was right. We never left Erro, did we?\"\n\nJameson shook his head. \"We just went back in time.\" His head was about to burst with all the information flying around inside his skull. \"Can you feel the memories? Fluttering in your head?\"\n\nXander nodded. \"It's worse for you, I think.\"\n\n\"Maybe so.\" He took a deep breath.\n\n\"Are you all right?\" Erina leaned in to look at him closely.\n\n\"I need... help.\" Jameson put his hand to his forehead and squeezed his temples, willing the headache to go away.\n\n\"With the memories?\" Xander's face swam before him as if he was drunk.\n\nJameson nodded. \"Erina... something you did freed them up. There are so many. So, so many.\" He drifted, following one of them off into another place.\n\nPain flared through his cheek. He opened his eyes, and Xander was peering at him intently. \"Sorry, but I was losing you.\"\n\n\"You slapped me?\" Jameson rubbed his left cheek.\n\n\"Yeah, sorry.\" Xander didn't look sorry.\n\n\"Thanks.\" He kissed Xander, and his lover's eyes went wide. \"I fought this before. I can do it again, but I need your help.\"\n\n\"What's wrong with him?\" Erina sounded concerned.\n\n\"He's flooded with memories.\"\n\n\"Ah.\" Ze made a thrumming sound. \"It happens to our young sometimes. We call it athr\u00e0 sickness.\"\n\n\"I can help him. We've dealt with this before.\" To Jameson, he said, \"Whatever you need.\"\n\n\"Be quick. The time of the shift is approaching.\"\n\n\"We will.\"\n\n\"Help me breathe.\" He slipped his hand through the front of Xander's shirt, feeling his warmth beneath. A thrill of excitement went through him, but he squashed it.\n\nXander did the same with him, and they inhaled and exhaled together, calmly. In, out.\n\nThe chaos in his mind subsided.\n\nIt wasn't enough, though, to just push the memories away. He had to master them, fully embrace who he was. What he was.\n\nLander. Skythane. Ithani.\n\nIt was time.\n\nHe put his other hand on Xander's cheek. Xander's green eyes opened, meeting his.\n\n\"I have to do something. Take us somewhere. Do you trust me?\"\n\nXander nodded. \"Completely.\"\n\nAt that one word, Jameson felt a rush up his spine. Lust. Adrenaline. Absolute love. It could have been any of those things, or all of them at once.\n\nHe opened himself up to Xander, and they dove into the athr\u00e0.\n\nJAMESON'S HAND on his cheek was the last human thing Xander would remember for a long while.\n\nThere was no memory where they went\u2014or rather, only memory.\n\nHe closed his eyes, and the world dissolved around him. He felt Jameson moving closer to him, and then inside him.\n\nIt was like sex\u2014a perfect merging of their bodies in a way that was impossible in rel space. They flowed together, two becoming one, as intense an experience as Xander had ever felt with another man. With anyone.\n\nXander's body was on fire, flooded with endorphins like the most intense orgasm he'd ever felt.\n\nAnd Jameson's mind... if Xander had ever had reason to doubt Jameson's love for him, that was over now.\n\nHe held out his\/their hand and stared at it in wonder. Jameson overlaid him, was part of him. Was him. They shared a body, a form, a soul, and all at once he understood what it was like to be ithani. What Erina had tried to explain to them.\n\nWe're together.\n\nIs this you?\n\nI don't know. Are you me?\n\nWhere are we?\n\nWe don't know.\n\nIt's the athr\u00e0, we think.\n\nHe\/they looked around. They stood in the midst of a maelstrom.\n\nAll around them memories danced and spun, flying through the air and streaming colors of joy, sadness, anger, and grief. They reached out their hands to touch some of them, and the memories flowed through them, leaving an indelible mark.\n\nIt was the most beautiful, intimate moment of Xander's life, to be a part of Jameson, to experience this with him. With us. There was no line between them.\n\nHere they saw the birth of a star, as witnessed by an ithani scientist an eon before. There the simple memory of a field of flowers, both green and red, that smelled like musk and cinnamon.\n\nSwirling above them, the mating dance of an ithani triad.\n\nThen the birth of a nimfeach from the Heart, breaking the hard golden shell that covered its chamber and crawling out, spreading its wet wings that shimmered like magic.\n\nThey floated above the sun, boiling and angry beneath them, mere hours away from a terrible act of devastation.\n\nIt's too much. There are too many.\n\nThey pulled their hands in and sank down to the ground, hugging themselves.\n\nWe can't do it. We aren't strong enough. They shivered, wanting nothing more than to be left alone. The magic was gone, drained out of them like blood through a deep cut. They closed their eyes, willing it all to go away, knowing they had failed.\n\nI thought I might find you here. Erina cocked zer head and blinked at them.\n\nThey opened their eyes and stared up at zer. A little hope crept back into their heart. How did you find us?\n\nYou aren't yet skilled in the athr\u00e0. No one but the ithani should be able to come here.\n\nThey stared at zer. We... we are ithani.\n\nErina cocked zer head again. I know.\n\nThey got up and placed their hands on Erina's head. Zer golden fur was soft under their touch.\n\nErina seemed surprised but did not back away. Closing their eyes, they gave Erina their story.\n\nThey held nothing back. Their rocky meeting and the halting advance of their relationship. How Quince had dosed them both with pith. The war between the skythane, and the war with OberCorp. The fall back through time.\n\nWhen it was done, they dropped their hands, and Eri backed away, zer eyes wide and blinking rapidly. Everything I foresaw will come true.\n\nWe need to find Puck. The nimfeach.\n\nErina nodded. I will show you where it is. Ze held up zer hand, and part of the athr\u00e0 swirled and solidified, showing a dark room.\n\nWe see it.\n\nTime is short. Ze looked at the two of them and made a thrumming sound in zer throat. You are ready. I will wait for you in the Heart. Erina vanished, and Xander felt Jameson's mind withdrawing.\n\nWait!\n\nWe have no time.\n\nWe have time for this.\n\nHe was inside Jameson again, this time connected to him in a primal way. They moved as one, passion rising as they passed through each other, connected in the athr\u00e0 down to the last fiber of their beings. They rose on the tide of climax, fixed on each other, and it was as good\u2014no, better\u2014than sex had ever been in the flesh.\n\nXander could taste Jameson, could feel his breathing deep inside. They filled each other, the excitement building, shivers running down his limbs that were also Jameson's.\n\nAs he reached the height, Xander moaned, and Jameson matched him, their voices rising together in a tumult of exultation. They hung there, suspended, for a second, an hour, a lifetime, perfectly connected, perfectly sated.\n\nThen it was over, and they fell apart inside the Heart, panting heavily and staring at each other in amazement.\n\n\"Holy fucking Split.\" Xander laughed.\n\nJameson grinned. \"Oh gods, I needed that.\"\n\n\"The memories?\"\n\nJameson cocked his head, making Xander laugh as he mimicked one of Erina's trademarked gestures. \"They're gone. Or... no, that's not quite right. They're all there, but they're settled.\"\n\nXander nodded. \"Me too.\"\n\nErina looked at both of them. \"Yes, you are ready.\"\n\nJameson stood and pulled Xander to his feet, planting a kiss on his lips. \"We have to get Puck.\" He pulled out the smaller key.\n\n\"Then what?\"\n\nJameson squeezed his hand. \"We go wherever the universe leads us.\"\n\n# CHAPTER TWENTY-FOUR: IN THE WHITE CITY\n\nMORGAN LOOKED around the company store.\n\nIt was half empty, shelves pushed back to the edge of the room. Sleep sacks filled the floor, and empty wrappers overflowed a trash can in one corner.\n\nSomething small and dark glared at him from the shadows before grabbing a wrapper between its teeth and scurrying away on six legs behind one of the shelves. Ice rat.\n\nTally tossed him a packaged something.\n\nHe turned it over. It was an MRE.\n\nA pain went through his chest. It had been weeks since he had eaten human food, and an MRE was the first thing Xander had ever given him.\n\nA memory flashed through his mind. Xander and Jameson, deep inside the Heart.\n\nHe blinked. That couldn't be right. He had given Quince the key. He still didn't understand quite why.\n\n\"You okay?\" Tally was standing before him, searching his eyes. \"You just press here to open it....\" She opened the MRE for him. \"It's not hot, but they taste pretty good. That one's 'Argantan Chicken.'\"\n\n\"Thanks.\" Morgan sniffed the contents. They weren't half bad.\n\n\"So talk.\" Tally plopped herself down on one of the sleep sacks, opening an MRE of her own.\n\nMorgan picked up a piece of chicken between his fingers. It was green with a brown sauce, but it didn't taste terrible. He let Tanner take the lead. \"I was out in the ice pines, looking for something we could eat. Ice rats. Blevons. Anything. It was after we got kicked out of the back of the company warehouse.\"\n\nTally nodded. \"I remember.\"\n\n\"I was chasing a covey of blevon down a streambed, and something appeared.\" Morgan remembered when he had found the boy, had chosen him from the crowd, partly because he would elicit the needed sympathy from the other humans. Partly because he'd often gone out on his own and had been easy to corner.\n\n\"Something? What kind of something?\" Sully's eyes were wide.\n\n\"It was a nimfeach.\"\n\nTally snorted. \"You almost had me going there.\"\n\n\"What? You just accused me of being a sneach a couple minutes ago.\"\n\n\"Everybody knows nimfeach aren't real.\" She stared at him, as if daring him to look away.\n\nMorgan, the old Morgan, would have given up on the whole thing. What did it matter what human children thought? They weren't part of the plan.\n\nTanner remembered.\n\n\"TALLY, COME back!\" Tanner howled after his friend across the frozen lake. \"It's not safe!\" A cold wind blew up from the south. Tally had decided to go ice fishing on Bottom's Lake.\n\nIt was still early in the winter season. The temperature hovered at just above freezing, the wind cutting through his makeshift clothing like knives.\n\n\"I want to bring us home some fish,\" Tally shouted back, brushing aside her long hair in the gusting wind. \"We'll eat like kings tonight!\"\n\nTanner growled and set off after her.\n\nThe ice felt solid enough under his feet, and besides, what kind of life did he have anyhow? If it opened up underneath him and he slipped through into a cold, watery oblivion, maybe it would be better for everyone. They said freezing to death was the best way to go.\n\nHe finally reached her\u2014though Tally hated it when he called her a girl\u2014without plunging into oblivion and tried to talk some sense into her. \"It's not cold enough yet. We could fall through the ice!\"\n\nTally ignored him, using the saw she'd stolen from the general store to cut through the ice.\n\nIt went faster than he expected.\n\n\"Help me haul this out.\" She grasped the edge of the ice plug.\n\nHe knelt beside her, pulling at the ice, and all at once it popped out. It was only a quarter-meter thick.\n\nTally flew backward with it, slamming into the frozen surface and just as quickly vanishing as the lake swallowed her whole.\n\nIt happened so fast that Tanner barely had time to register it. \"Tally? Tally!\" He scrambled after her, across the ice.\n\nA jagged break in the surface showed where she had plunged through.\n\n\"Tally!\" Nothing. He peered over the edge, hoping to see her.\n\nHe was ready to dive in after her when her hand, blue with cold, thrust out of the water, grasping onto the slippery surface.\n\nTanner grabbed it, and lying on his stomach, he pulled her up, inch by inch.\n\nHer head appeared above water, and she took a deep, shuddering breath. \"Oh God, oh God, oh God.\"\n\n\"Stay calm. I got you.\" He pulled her up onto the ice, keeping himself spread out as widely as possible to distribute his weight.\n\nHer face was blue, too, and she was shuddering all over.\n\nWhen he had her a safe distance away from the hole, he scooped her up and ran back toward the shore as carefully as he could manage.\n\nSomething must have guided him across the treacherous surface, because he made it to the shore.\n\nHe opened the tent, another furtive acquisition made for the trip, and turned up the little heater inside.\n\nTanner opened one of the sleep sacks and stuffed Tally in it, and then wrapped the other one around her.\n\nHer teeth were chattering, and she looked white as a blevon in its winter feathers.\n\nShe needed more heat.\n\nHe shrugged off his clothes and climbed inside the sleep sacks with her, wrapping his warmth around her. She was like the ice itself, but slowly, ever so slowly, skin to skin with him, she began to warm.\n\nHe pulled the sleep sack over them and breathed the warmest air from his lungs into the confined space.\n\nEventually, the shivering stopped, and she drifted off into sleep.\n\nTanner stayed awake with her for hours, making sure she was okay.\n\nMORGAN BLINKED. Something had shifted between him and Tanner. That was one of Tanner's memories, maybe the most intense one, but it was only one of many.\n\nTanner had opened his soul to Morgan, and with it the sum total of his experience, his memories, his loves and hates and fears were flooding into Morgan, merging with his own.\n\nAs they did, they opened other doors inside of him, and Morgan remembered his own past.\n\nA homeworld that was his as much as it was the ithanis'.\n\nThe wrong done to him and his kind by the Thshnel'Jirron, and their enslavement.\n\nThe life, the world, the time that had been taken from him.\n\nHe was human. He was nimfeach. He was ithani.\n\nTally was staring at him. \"You're scaring me, Tanner.\"\n\n\"Sorry.\" He blinked and saw these humans like Tanner did. \"I saved you once, out on Bottom's Lake.\"\n\n\"I remember.\" A single tear rolled down Tally's cheek.\n\nHe edged forward and touched it. \"You fell through the ice. You would have done the same for me.\"\n\nTally looked away, her face red.\n\n\"I can save you all now.\"\n\nTally turned and stared at him, hope evident in her eyes. \"How?\" Those eyes narrowed. \"You're not really Tanner, are you?\"\n\nHe returned her gaze. \"I'm still me, but I'm a lot more now too.\" He let go of her cheek and opened a gateway to Green Isle, to the beach along the volcanic lake. A warm breeze blew into the general store, and several skythane looked up, startled. \"Go. You'll be safe there.\"\n\nTally stared at him a minute more. The other kids stayed frozen, awaiting some sign from their leader, though Sully glanced longingly at the beach.\n\nThen she threw herself at him.\n\nHe thought she meant to stab him with the knife in her hand, but instead she threw her arms around him. \"Thank you for coming back, Tanner,\" she whispered into his ear. \"Or whoever you are.\"\n\nHe grinned. \"Call me Morgan.\"\n\nVENIN SET a brisk pace, and Alix managed to keep up with him despite his extreme exhaustion.\n\nAt last they passed under a white arch that must have been a hundred meters high and entered the city itself.\n\nIt was beautiful, if past its prime. The buildings, which had looked white from a distance, glimmered in the ethereal light with a slight rainbow shimmer.\n\nAlix looked up. They had eclipsed about half the distance to the Heart. He covered a yawn. \"Can I ask you something else?\" He'd been forced to revise his opinion of the skythane nation over the last two weeks. Once he'd thought them savages, primitive copies of real men like himself and the other landers. He'd seen Xander as the exception to the rule.\n\nNow he saw what a complex history and culture they had. Besides, he apparently had skythane blood flowing in his own veins.\n\n\"Sure.\" Venin flexed his wings in a gesture Alix had learned to interpret as discomfort. Physical or emotional, it was hard to tell.\n\n\"You said you were in love, once. With... Taz?\"\n\n\"Tazim.\" Venin grimaced.\n\n\"Sorry.\"\n\n\"It's okay. It was fifteen years ago now. It still hurts, but I've made my peace with it.\"\n\nAlix put a hand on Venin's shoulder. \"I'm still sorry.\"\n\nVenin didn't shrug it off.\n\n\"What happened to him?\"\n\nVenin laughed mirthlessly. \"He thought he was a great warrior. He wanted to bring home an auxen rack. He thought bringing down one of the biggest beasts on the planet would make him more of a man.\" Venin growled. \"I wish I'd known better, then. I went with him, and we found a herd in the northern parts of the Riamhwood. It was a cold winter, and they were ranging farther south than normal.\"\n\n\"Auxen? Like the aux herds around Gaelan?\"\n\n\"Bigger. Auxen are huge herbivores that live up in the north, on Titania. Nasty set of horns.\"\n\nAlix nodded. He'd seen images of something like that in his company orientation\u2014the flora and fauna of the two halves of the world were very different. \"Go on.\"\n\n\"Taz was really good with the bow, but he wanted to do this up close. So we took a couple hunting spears, and we found a small herd that was separated from the rest by a wide creek. There were three of them\u2014two adults and a baby. We should have known that was trouble.\"\n\n\"Why?\" For just a second, Alix flashed on what a strange scene this was, walking through an empty alien city far underground, discussing a winter hunt with a skythane man.\n\n\"Auxen raise their herds in threes. Three parents and one or more calves.\" He spit.\n\n\"Ah.\" There's so much I still don't know about this world. When all this was over, if they survived it, he vowed to find out about everything that he'd been missing.\n\n\"We circled around, choosing our target and waiting for the right moment. Taz leapt up and threw the spear, and it flew true and pierced the auxen's chest, going through both of its hearts. The auxen collapsed, and the look on Taz's face was pure, fierce joy.\" Venin rubbed his chin. He laughed harshly. \"I'd never seen him like that.\"\n\n\"He must have been proud of the kill.\"\n\n\"Yes, he was.\" Venin sighed. \"It didn't last long, though. The other adult shooed off its calf and turned toward us, and the look in its golden eyes was nothing short of murder. We were ready, though, and as it charged across the small clearing toward us, we leapt into the air. There was this....\" Venin's voice cracked.\n\n\"Fucking Split, I'm sorry. I shouldn't have asked.\"\n\n\"It's okay. It helps to talk about it. Really.\" He reached out to hold Alix's hand and took a deep breath. \"There was... this horrible gurgling scream. I looked down to see another auxen\u2014the third parent\u2014with Taz's body impaled on its main horn, shaking him around like a rag doll. It dumped his body on the ground and then stomped him into the earth.\" Venin's face was ashen.\n\n\"Fucking Split.\" Alix pulled Venin to him, hugging him tightly. \"I'm so sorry, man. I didn't know.\"\n\nVenin shuddered. \"It was horrible. If I close my eyes, I can still see it.\"\n\nAlix squeezed him. \"Was it quick, at least?\" To carry something like that around inside you....\n\n\"Yeah. I don't think he suffered much.\"\n\n\"Suffering belongs to the living.\" He let go of Venin. \"It wasn't your fault. You know that, right?\"\n\n\"I tell myself that. But maybe... if I had stopped him from going, told him what an arrogant, stupid idea it was. If I had looked around a little more before we chose that target. If I had checked behind us, just a couple seconds before\u2014\"\n\n\"You can't think like that. You can't go back and change it. Look, maybe something you did the week before, or the month before that, saved him from some other terrible fate.\" He closed his eyes, thinking about Tucker, the poor bastard who'd been buried under an avalanche just after the shift. There had been others, too, in his time with the rangers. \"Sometimes it's just our time.\" He looked into Venin's eyes.\n\nVenin wiped his face with the back of his hand. \"You're pretty damned smart. For a lander.\"\n\n\"And you're not as obnoxious as most of the other wing men I've met.\"\n\nVenin laughed and kissed him on the cheek. \"There might be hope for a landbound bastard like you yet.\" He looked up at the Heart and the ground shook. His eyes glazed over, and he started to shake too.\n\n\"Venin? What is it?\" Alix followed his gaze.\n\nThe Heart was growing.\n\nThe top of it had burst into bloom, and it was visibly extending toward the roof of the cavern and expanding as it went.\n\nAlix took Venin's face in his hand. \"Look at me. Venin, look at me, goddammit!\"\n\nVenin's eyes remained unfocused.\n\nPain had worked before, but he was loath to reopen the man's wound.\n\nInstead, he pulled Venin to him and kissed him, hard.\n\nHe felt the change as Venin struggled in his arms.\n\nAlix pulled back and looked at his companion. \"You okay?\"\n\n\"Thank you.\" Venin's eyes refocused. \"Damn, that felt good. Does it mean...?\"\n\n\"I don't know. Maybe.\"\n\nVenin nodded. He looked back up at the Heart. \"It's a harbinger.\"\n\n\"For what?\" Alix stared at Venin. What the hell did he mean?\n\n\"The end of the world. Come on. We have to run!\"\n\n\"THEY'RE UNDERGROUND still.\" Dixon pointed to the scanner. \"They're right below us, but we still can't reach them directly.\"\n\n\"What does that mean?\" Alia looked from Dixon to Smythe and back again. \"Are they all right?\"\n\nDixon squirmed uncomfortably under her gaze. \"Sorry. There's no way to know. We can't get more than a tracer signal off their cirqs.\"\n\nSmythe nodded. \"We'd have to find a way in. Right now they're so deep it would take a month to tunnel down there.\"\n\nAlia bit her lip. Things had spun so far out of control. She needed to do something to help at least one of her friends. \"What if\u2014\"\n\n\"Hold on.\" Dixon was staring at his console. \"That can't be right.\" He unclipped his belt and leaned forward to look at the ground beneath them. \"Holy Split! Get into your seats!\"\n\n\"What's happening?\"\n\n\"Funnel cloud.\" He ran his hands across the instrument board and the hoversport jerked to the left.\n\nAlia managed to get back to her seat and buckle in.\n\nThe craft bucked, slammed by particles of something that sounded like hail across the right side of the ship, and Smythe flew past her.\n\nAlia grasped his shirt, pulling him back as the hoversport steadied and sped ahead.\n\n\"Sorry about that, folks,\" Dixon called back from the pilot's seat. \"We have some weird... weather?\"\n\nHe pulled the hoversport around, and the side windows went clear to give them a view of what was happening.\n\n\"What in Gael's grace?\" Alia pressed her face against the window, staring at the maelstrom before them.\n\nA huge tornado spun over the ground, sucking up earth and trees and water from the river as it grew in force, carrying all the debris up high into the air. Her gaze followed it, tracing the arch of it. It was either being ejected into space or....\n\n\"Dixon, get us out of here!\"\n\nSmythe stared at her. \"We should be far enough away from the funnel already.\"\n\n\"All those rocks and trees and dirt are going to come back down!\" She poked her finger into the sky.\n\nDixon looked up at her and went pale. \"Yes ma'am.\"\n\nThe hoversport hummed and jumped away from the tornado at full speed.\n\nAlia just hoped it was fast enough.\n\n# CHAPTER TWENTY-FIVE: DAIA\n\nQUINCE, ROBYN, and Jessa followed the root, traveling down three or four more interconnected passageways. There must have been a substantial lava flow here at one point, leaving these shells of caverns interlaced near the shores of the Gildensea.\n\nMaybe they came about at the time of the initial shift.\n\nThe brief respite had done Quince good. Although she was sure she was going to collapse if and when this was all over, she could function again, and that was a blessing.\n\nThe walls of the cavern were lined by a glowing green moss that gave off enough light to let them see their way.\n\nQuince glanced up. The alien signal continued unabated overhead. Then it veered unexpectedly upward.\n\nQuince stumbled to a halt, staring up at the high-ceilinged cavern they had just entered.\n\nSomething was up there, something suspended from thick ropes or cables.\n\nQuince pulled out the flashlight she'd grabbed in the armory and shone it up into the darkness.\n\n\"What is it?\" Robyn stood beside her, staring at the thing Jessa had found.\n\n\"I... I don't know.\" It was immense, an oval shape supported by giant beams or roots that were sunk into the surrounding rock. Its surface was covered in whorls and swirls, and bits of golden light jetted across it.\n\nSomething that looked like roots spread out from the thing to cover the surface of the ceiling, making it appear to be a carved wooden masterpiece.\n\nThen the world shimmered, and she found herself alone in the room.\n\nThe rough stone floor was replaced by flagstones, and the thing above vanished. In its place stood a giant tree, bigger than a blue oak, its trunk rough and reddish brown.\n\n\"I wondered when someone would find their way here.\"\n\nQuince spun around to find herself face-to-face with a young woman. \"What... who in the three hells are you?\"\n\nThe woman smiled. \"They used to call me Daia.\"\n\nDaia.\n\nQuince remembered.\n\nTHEY'D CARRIED the seed into the cavern by the sea, up far past the waterline. She would be the child of Forever's world mind, full of the knowledge passed down to her by the ones that had come before.\n\nIt had taken eight colonists to lift and carry her to her new home.\n\nNo matter whether the rest of this mission failed, they would get this part done right. Daia swore she would not be responsible for the extinction of humankind and the world minds.\n\nThey chose a spot near the back of the cavern and lodged the seed in between some rock outcroppings.\n\n\"Guess this is as good a place as any.\" Braice brushed off his hands and looked over his handiwork. He had always been a bit full of himself, but the loss of the world mind and their own sorry state had knocked him down a few pegs. \"Are you sure we should be doing this?\"\n\nDaia nodded. \"We don't know what will happen next. We have to do this now, while the seed is still viable.\"\n\nBraice touched the rough surface of the seed, his brow furrowed. \"These things ever go bad?\"\n\nShe sighed. \"I don't know. They were made to last for a long time, in space, but down here, there are all kinds of corrosive substances\u2014wind, water, alien germs....\" It was an old argument. In truth, she didn't know if any of those could be damaging to the seed, but they needed a win, after all that had happened since they'd come down planet-side.\n\nShe turned to see the rest of the colonists gathered behind her, a fraction of those she had hoped would make it to this shore. The sound of the waves stirred something primal in her, like the wide-open sky above.\n\nThe first week, she'd been terrified to leave her makeshift hut, frightened that green sky would suck her up into its depths, that she would somehow fly off this spinning half ball and into the cold clutches of space. Such fears were unfounded, but she'd had a hard time shutting them down.\n\nShe searched her heart for what to say. \"Today we take control of our own destiny. Today we plant here the start of a new world. A new Earth.\"\n\nThey needed a name for this world.\n\nThe works of Shakespeare had been much on her mind of late, the great drama a foil for the tragedies they had endured to get here. She'd spent endless hours in vee watching them, some of the finest literary works ever created by the hand of man.\n\nYes, a new name, but she would not root it in tragedy.\n\nThis world was like something out of a fairy tale. A magical place full of forests and fresh air and promise, no matter how badly things had started out.\n\nSound, music! Come, my queen, take hands with me, and rock the ground whereon these sleepers be. She laughed.\n\nBraice looked at her, concerned. \"Are you okay?\"\n\nShe nodded. They would rock this ground indeed and make it their own. \"We should call this world Oberon.\" It seemed fitting. A new beginning after all the folly of the past. An awakening after a long slumber.\n\nThere were mumbles and confused looks from some in the crowd, but no one gainsaid her.\n\nShe turned to kneel before the seed, placing her hands on its rough surface. Closing her eyes, she reached inside and searched for its spark, its nascent consciousness. As Forever's world mind had shown her in its dying days, she opened herself up and let the seed imprint on her mind. It carried within it the memories of the minds that had come before it, all the way back to the First Mind, Alix. She would make a copy of herself in the new mind, and thereafter it would evolve eventually into a mind of its own, but with her thoughts, her hopes, her experience, and her fears as its foundation.\n\nThere.\n\nThe spark was like a tiny glowing ember inside the seed, a spark of potential.\n\nDaia touched the spark and opened herself up to it.\n\nShe felt her consciousness flow through her, carrying with it all the flotsam and jetsam of her long life\u2014feelings, memories, hopes and fears\u2014making a perfect copy of herself inside the nascent biomind. Over time, it would grow into its own, finding a life separate from hers and much, much longer.\n\nFor now, the Daia being created inside the seed was her twin.\n\nWhen it was over\u2014a minute or five hours later, she couldn't be sure\u2014she stood and staggered backward.\n\n\"You okay?\" Braice caught her arm, steadying her on the uneven rocks and sand of the cavern floor.\n\n\"I think so.\" It was disconcerting, the idea that she was no longer alone. No longer the only Daia in the world. \"I wonder what she will live to see?\" The Daia inside the small world mind would far outlive her own mortal days.\n\nSuch thoughts and introspection would have to wait.\n\nThe seed was rocking back and forth, like an egg with a hatchling inside.\n\nThen a root burst out, shattering part of the husk and seeking the rock below. Another followed, and a third, and the seed grew visibly larger as it took in nourishment from the earth and stone.\n\nBraice squeezed her arm. \"We'll make this work. Oberon will be our new home.\"\n\n\"I hope so.\" But at what cost?\n\nQUINCE BLINKED.\n\nShe'd just witnessed something monumental. She could no longer convince herself otherwise, that it was all some fever dream brought on by sheer exhaustion. She'd seen\u2014no, lived\u2014the dawn of the skythane here on Oberon. \"Who... what are you?\"\n\nThe woman who had been called Daia in the memory stepped forward. \"I'm the seed. The world mind. The great hope of the colonists who came here from Earth, so long ago.\" She sounded bitter. \"Then they abandoned me here to my fate, left all alone in this cavern to fend for myself.\"\n\nThis was the copy of Daia, the one the original, mortal woman had invested all her hopes and dreams in. An unexpected relic of the original colony, here in Quince's own time. \"What happened to you?\"\n\n\"They created me, and then they left. What else matters?\" Her hand came up to touch Quince's face. \"The real question is, who are you?\"\n\nQuince's world twisted.\n\nROBYN CHECKED Quince's pulse. Where in Gael's grace did you go?\n\nIt had happened so quickly. Quince had been speaking with someone, or something\u2014although there'd been no one else visible in the cavern with them. Then she had collapsed in a heap on the stony ground.\n\nSomething was odd about this place, besides the hulking mass that hung over their heads. Robyn kept seeing things out of the corner of her eyes, little flashes of things. Sunlight streaming through one end of the cavern where there was a collapsed rock wall.\n\nVoices, people talking at the edge of her awareness.\n\nSomeone standing in the corner who was gone when she looked at them directly.\n\nI'm going mad.\n\nShe shoved that thought to the back of her mind. \"She's alive.\"\n\nJessa's eyes met hers over Quince's supine form. \"What happened?\"\n\nRobyn shook her head. \"I don't know.\" She looked up. \"What do you think it is?\"\n\nJessa squinted at the dim form above them. \"I'd guess some kind of computer? Or is it alive?\"\n\n\"Whatever it is, it has to have something to do with this.\" She knelt over her love, the woman she'd waited a quarter century to see again. Quince was breathing, barely, her face serene as a babe's.\n\n\"Quince, can you hear me?\" She took Quince by the shoulders and shook her gently. \"Quince, I need you to wake up.\"\n\nQuince's head rolled off to the side.\n\n\"Come on, Quince.\" Robyn patted her cheeks, trying to elicit a response. Any response.\n\nIt wasn't supposed to be like this. Quince had to be okay.\n\nWe had plans. They were going to go off together, to find a place far away from the world's cares and fears. \"Quince, if you can hear me, blink, or squeeze my hand,\" she pleaded. \"Give me something.\" Still, there was nothing.\n\nOnly slow, steady breathing and the beating of her heart.\n\nQUINCE WOKE to a nightmare.\n\nHer hands were covered in blood, the smell of iron thick in her nostrils. Queen Andra, Lyrin's mother\u2014Jameson's mother\u2014lay before her, her throat slit and staring up at her in dead accusation.\n\nBehind her, Jameson wailed in his crib.\n\n\"You did this,\" Andra hissed at her, the queen's voice like shattered glass.\n\n\"You're dead!\" Quince scooped Jameson up and ran for the balcony, chased by Danner and his men and Andra's angry, broken voice. You did this. It chased her out the window like a curse.\n\nQUINCE WOKE on a mattress, somewhere in the depths of the Slander. \"It wasn't like that,\" she whimpered, lifting herself up off the piss-soaked bed. Her own urine.\n\nHer head felt thick, as if it were packed with gauze.\n\nNext to her, a lander woman\u2014Lilith\u2014slumbered, her tangled hair and rough features a far cry from the angel she'd seemed the night before under the influence of Mugjuice. The taste of pith from the drink was bitter on her tongue.\n\nShe swung her feet over the edge of the mattress.\n\nXander stood there, a child version of Xander, his innocent features twisted with pain. \"Where are you, Quince? Why are you letting him do this to me?\"\n\nQuince backpedaled, trying to get away from the apparition. \"Leave me alone. You're not real!\"\n\nXander took a step forward, holding out his arms. They were filthy with bruises. \"Why did you let him do this to me?\"\n\nQuince covered her face with her hands, trying to shut out the piteous voice.\n\nWhy did you let him do this to me?\n\n\"I didn't know. I didn't know!\"\n\nSHE OPENED her eyes and he was gone. She was standing on a hillside, holding Jameson in her hands, looking down on Ballifor. \"No, not this. Please, not this!\"\n\nIt was a beautiful sunny day, the kind she used to spend with her mother and brother, picking hoarberries. Jameson was strapped in his harness at her chest, burbling happily.\n\nShe knew this day.\n\nA high-pitched whine split the air, like the buzzing of a hundred bees, then a thousand.\n\nSomething dark shot across the sky and down into Ballifor, her home village, where her mother and brothers and a hundred other people she cherished lived.\n\nA terrible flash and blast wave knocked her backward and slammed her into the ground. \"No, please no....\" She scrambled up, crawling forward to see her home village destroyed, melted into a sea of glowing glass.\n\n\"You brought him here.\"\n\nShe spun around to find her mother standing over her, her face lined with disapproval. \"You did this, Quince. You killed everyone here who ever loved you.\"\n\n\"I didn't know. I couldn't have known.\"\n\n\"You killed everyone.\"\n\nQuince scrambled back from the burning hatred in those eyes and fell off the edge of the cliff, tumbling down the hillside.\n\nYou killed everyone.\n\nSHE SLAMMED to a halt, the force of the impact shuddering through her whole body. When she got up, she was inside the way station.\n\nJameson sat there, his head down, looking as distraught as she had ever seen him.\n\nHe held out the pith vial from her carry sack. \"How could you do this, Quince? I loved him. I thought I loved him. Now I don't know. You took that from me. From us.\"\n\nJameson didn't look angry. Instead he looked broken.\n\nThat was worse.\n\nShe got down on her knees and put her hands on the sides of his face. \"I had to. The two of you needed to be in love.\"\n\nHe wouldn't look her in the eyes. The boy\u2014no, the man\u2014who was most like a son to her was too disgusted with her to even look at her. \"That would have happened anyway.\" Jameson sank down on one of the cots, his head in his hands. \"Don't you see? You took away our free will. You forced something that should never have been forced. You did this, Quince. You did this, and you have to live with it.\"\n\nHe was right.\n\n\"Jameson, I'm so sorry. I thought I had to. I....\"\n\nBut he was gone.\n\nYou have to live with it.\n\nSHE LOOKED down.\n\nHer hands were covered in blood, and Queen Andra, Lyrin's mother\u2014Jameson's mother\u2014lay before her, her throat slit and staring up at her in accusation.\n\n\"Please, no. Not again.\" She turned to run, but there was nowhere to go.\n\n\"You did this,\" Andra hissed at her, the queen's voice like shattered glass.\n\nJESSA GOT up to look around the cavern, leaving Robyn to watch over Quince.\n\nQuince was a tough old bird\u2014she'd be all right.\n\nJessa picked up the flashlight Quince had been using and walked along the perimeter of the cave, looking for anything interesting.\n\nThe floor on the far side of the cavern was covered with sand and smooth rocks, close to the pile of rubble that showed where there had been a collapse, some hundreds of years before to judge by the settled grime and moss on the rocks.\n\nThe sand reminded her of a cavern on Beta Tau, along the Eraysses Sea, where her parents had taken her one summer. The Eraysses was a popular resort destination, and the cavern still held a sense of the primal, not the parklike feel of most of the terraformed world Jessa had called home.\n\nStanding there, her feet in the sand, the waves at her back, staring up at the dark maw of rock, Jessa had understood the dark pull of truly wild nature for the first time.\n\nShe knelt and picked up a handful of sand, letting it run through her hands. The last time Quince had checked their location, they'd been close to the Gildensea. This had been a seaside cavern once, a long time ago.\n\nShe stood and shone the light up at the immense thing that crouched in the shadows above them.\n\nShe didn't like it\u2014it gave her the heebie-jeebies. The wooden walls reminded her of rootbound plants\u2014left too long inside their old pots until the soil was gone and they were a mass of twisted limbs struggling for nutrients.\n\nJessa climbed up on the pile of rubble, trying to get a closer look at the computer\u2014the mind?\u2014at the center of this strange cavern.\n\nWide roots or beams anchored it to the wall, plunging right into the rock as if it were butter.\n\nShe reached up to touch one that was close to the ground. It was warm, the \"wood\" rough under her fingers.\n\nWhatever it was, it was alive\u2014she was sure of that much\u2014and if it was alive, she could kill it, or at least hurt it.\n\nQuince looked so small, lying on her back in the middle of the forsaken cavern. Robyn kneeled over her and whispered something into her ear.\n\nThese women, as different as they were from anyone else in Jessa's experience, had taken her into their lives. Even when she might have presented a danger to them and their plans.\n\nThey were her friends.\n\nWithout thinking, she pulled her pulse rifle out of its holster and aimed it at the thing that hung from the cavern ceiling.\n\n\"Here goes nothing.\"\n\nQUINCE WAS trapped between some of the worst moments of her life.\n\nScenes that shifted mercilessly from one to the next, carrying her along with them, each time shredding a bit more of her self-respect and control.\n\nThe queen's death. Blood on her hands. A urine-soaked mattress and Xander's pleading eyes. A sea of molten glass and her mother's condemnation. Jameson, broken by her own actions.\n\nThey swirled by time and time again, and she felt her self, her inner core begin to wear away, to burn off like fog under the morning sun.\n\n\"I didn't mean to.\" She whimpered, only glad there was no one there to see her reduced to such a state.\n\nShe had no power here, wherever here was.\n\nDaia had stripped away her ability to act, or even react, her ability to choose her own course.\n\nWhy? Why me?\n\nBlood on her hands.\n\nYou did this.\n\nUrine-soaked mattress.\n\nWhy did you let him do this to me?\n\nSea of molten glass.\n\nYou killed everyone.\n\nJameson, broken.\n\nYou took that from me.\n\nSo much pain. So much anguish.\n\nNONE OF it was right.\n\nShe hadn't been responsible for Andra's death.\n\nShe hadn't killed her whole village.\n\nShe hadn't even been responsible for what happened to Xander, though she could have kept looking.\n\nOne thing she had done.\n\nBloody hands. Urine mattress. Molten glass. Broken Jameson.\n\nShe had taken away Jameson's choice. Xander's choice. Forced them into a relationship they might have found on their own, with time. All for the sake of a prophecy.\n\nYes, she'd had her reasons. Yes, she might even do it again, if faced with the same choice.\n\nBlood. Urine. Glass. Broken.\n\nWhat she had done was wrong.\n\n\"Enough.\"\n\nThe queen looked up at her. \"You did this.\"\n\n\"No, I didn't.\" She knew what was required of her now. \"Danner Black did this to you.\" She knelt and gently put her palm on the queen's cheek. The bloodstains on Quince's hands faded away. \"I saved your son. I saved the world. You didn't deserve this.\"\n\nThe queen continued to glare at her for a moment, but then the accusation faded from her eyes. A look of utter peace came over the queen's face, and her eyes closed.\n\nA moment later, she was gone.\n\nQuince stood to face Xander, pushing the mattress away. The little boy trembled with fear.\n\nShe knelt and pulled him into her arms. \"You were so young. I failed you. I should have kept looking. I never should have stopped, but I didn't know. I didn't know!\" She hugged him fiercely. \"You went through a terrible thing, but you survived.\" She held him out at arm's length. \"My gods, you survived, and grew to become a beautiful, strong, amazing young man. I promise you I will never abandon you again.\"\n\nXander nodded. He grew in her arms until he was man-sized, and then he squeezed her tightly, wrapping his arms around her, covering her in his own wings.\n\nThen he too was gone.\n\nQuince turned to see her mother, her mouth twisted in a mask of horror. \"Mamma.\" Tears came from the corners of her eyes. If only she hadn't come back.... \"I'm so sorry, Mamma. I never knew I would lead them here. I didn't even know what they wanted him for. Not then.\" She held her arms out, and her mother took a hesitant step, then two, in her direction. \"If I could go back and change it all, I would, but I can't\u2014\"\n\nHer mother crossed the remaining distance between them in a second and pulled Quince into her arms. \"I am so proud of you, my little imprean. You've traveled so far and done so much.\"\n\n\"I miss you, Mamma.\"\n\n\"I know. Your brother and I miss you too.\" They separated, and her mother squeezed her hands. \"We all make choices we regret later, but there's not a bad bone in your body.\" She looked past Quince and nodded. \"I think there's one more person you need to talk to.\"\n\nQuince turned to find Jameson, sitting on the cot in the way station.\n\n\"You did this, Quince. You did this, and you have to live with it.\" He stared at her, and his eyes burned into her.\n\nInstead of turning away in shame, she gestured for him to move aside, and he did, making room for her on the cot, his eyes never leaving her face.\n\nShe sat with a heavy sigh and took his hand in hers. \"You're right. I did this. It was my decision, and I will have to live with it for the rest of my life.\"\n\nHis eyes widened, but he didn't say a word.\n\n\"I made a choice. I decided to do something because of the grand plan.\" She snorted. \"Maybe I'm not so different from Morgan and the ithani after all.\" She squeezed his hand and brushed her other one against his cheek. \"You are the closest thing I have to a son\u2014the closest thing I will ever have. A son who deserves a mother who would give everything for him. Who would put his needs above everything else.\"\n\nQuince looked away. She was wrong. She couldn't do this. Couldn't bear to see him look at her with pity. She had failed him. Worse, she had taken something from him, something from both of them.\n\nShe'd gotten a taste of what it felt like to lose her own agency, her ability to make her own choices. It was a bitter taste that had shaken her to her core. She deserved it if he never wanted to speak with her again.\n\nHis hand touched her cheek.\n\nShe turned reluctantly.\n\nHe wasn't looking at her with pity. Instead, he was crying. \"Thank you for saying that.\"\n\nShe pulled him close and held him tightly, and he was warm against her chest, vibrant, alive.\n\nHe wasn't real. All of this was a trick of her mind, a trick of whatever Daia was. But still, it gave her hope.\n\nShe would take responsibility for her choices, especially that one. Then she would take her penance.\n\n\"I'm so glad you\u2014\"\n\nThe world shattered, a billion pieces falling around her like shards of glass.\n\nShe threw her hands over her ears and closed her eyes as the shards created a cacophony and then a howling scream that was like nothing Quince had ever heard.\n\nWind whipped them around her like she was in the center of a cyclone.\n\nThen there was silence.\n\nShe opened her eyes, and Robyn was looking down at her, eyes narrowed in concern.\n\n\"Quince, are you awake?\"\n\nQuince blinked. \"I... I think so. What just happened?\"\n\nJessa came into view, blowing invisible smoke off the barrel of her pulse rifle like a tri-dee cowboy. \"I took my best shot.\"\n\n\"I have a call for you,\" Ari whispered in her ear.\n\n\"Quince, are you there?\" Mylin's voice came through loud and clear.\n\n\"Yes, I'm here.\" She sat up. \"What's happening?\"\n\n\"Things are going nuts up here. You have to come back!\"\n\nJIRRON WATCHED the Heart grow.\n\nArliss was exposed below zim, a window in the cheth ceiling opened up.\n\nAll around him, thousands of newly awakened ithani chittered amongst themselves, keeping a respectful\u2014even fearful\u2014distance from zim. They thought zis new form was strange, but none dared cross zim. Zi had led them to the brink of this, zis race's finest hour.\n\nDestiny was upon them. After a hundred thousand years of sleeping exile, they believed they would finally become gods themselves. It was almost time for the Great Migration.\n\nMany of the other beings on this small planet would die. A small enough price to pay for the birth of a god.\n\nTheir ending would be mercifully quick.\n\n# CHAPTER TWENTY-SIX: A NEW PLAN\n\nMYLIN SAT at her desk, trying to keep everything moving. She had one foot in vir and one in the real world.\n\nDOC had managed to lock their grid attacker out of his critical systems, so she finally had free access to her vir room again.\n\nBack in rel, she shot a look at the clouds gathering in the skies outside. How she wished she were out there and not stuck in here, but they needed her here.\n\n\"What's happening up there?\" Quince's voice sounded faint through her cirq, as if she was a long way away.\n\n\"Something's going on at the House of the Sky. Or where it used to be.\" She relayed the images from Alia and the others aboard the hoversport. \"Something huge is growing there.\"\n\n\"Holy Split!\" Quince's voice sounded faint, tinny.\n\nThe ground rumbled.\n\n\"You feel that?\"\n\n\"Yeah, little bits of the cavern are coming down on me.\"\n\nMylin frowned. \"Where are you? I can't get a fix on you.\"\n\n\"In a cave near the ocean. There's something here. Look, I need you to search the records of the time after the Fall. See what you can find out about generation ships.\"\n\n\"Generation ships. Got it. Is it really important, though? There's so much going on.\"\n\n\"If I'm right, it might be the only thing that really matters.\"\n\n\"Got it.\" She slitted an urgent note off to Derren, who looked half asleep at the other desk in the office. He sat up suddenly, shot her a look, and nodded. \"Okay, handled. One second.\"\n\nDOC was waiting for her. \"We're getting reports of widespread violence across the city.\"\n\n\"What about Rogan?\" She didn't like the man\u2014there was clearly some bad blood between him and Xander, though she didn't entirely understand why\u2014but he was the one they'd put in charge of security.\n\n\"Rogan and the Syndicate appear to be the source of the violence.\"\n\nShe switched back to Quince. \"Quince, I have to go. Things are getting crazy here in Oberon City. I've got to see if I can tamp things down.\"\n\n\"Got it. Let me know what you find out.\" There was silence across the line for a few seconds. \"We may only have five or six more hours.\"\n\nMylin checked her own chron. Time was indeed running short.\n\n\"DOC, have you cracked that code yet?\" She stood in her virtual office, staring at the wall. Something tickled her mind.\n\n\"Not yet. I've worked out that it's an organic language of some sort. I think these are verbs.\" He displayed a set of flattened glyphs with tails. \"And these are nouns.\" Those were more three-dimensional.\n\n\"Let me know what you find, and get me as many rangers and enforcers as you can. We may need to defend the building.\"\n\n\"Will do. Most of them were sent home, or out with the rangers. Someone's pinging you. I believe you have a visitor.\"\n\nMylin opened her eyes in time to see an impeccably dressed woman enter the room.\n\nShe was tall and thin, beautiful in an ageless, ice-queen sort of way.\n\n\"Can I help you? I'm sorry, but my time's really short.\" She was stretched thin as it was.\n\n\"I'll make this brief. My name is Lena Preston, and I'm here to help you untangle this godsforsaken mess y'all have made.\"\n\nALIX RAN after Venin as the world went mad around them.\n\nThe ground had begun to shake at irregular intervals. A deep thrumming filled the air, and the Heart continued to grow, giving off a pulsing golden glow.\n\nThey crossed a stone bridge, and as they reached the other side, it crumbled to rubble behind them.\n\nA little farther on, one of the white towers collapsed in front of them, groaning and sagging like a melted candle.\n\n\"This way!\" Alix pointed to the left, and they scrambled around the fallen tower. They clambered over the flattened end of it and ran across an open plaza decorated in graceful swirls.\n\nSuddenly Alix realized he was alone. He turned back to find Venin kneeling on the ground, his head in his hands, his wings wrapped protectively around his body.\n\nAlix ran back to him, kneeling beside Venin.\n\n\"Come back to me.\" He pulled Venin in for a kiss.\n\nTHE WORLD was spinning around Venin.\n\nHe couldn't run anymore. He collapsed to his knees, watching Alix run ahead, and then his vision clouded over.\n\nMemories flooded him in great dark clouds, things he couldn't possibly have known. Memories of this place.\n\nOf men and women shepherded here by their nimfeach masters.\n\nOf the painful genetic testing and experiments.\n\nOf wings sprouting out of his back, turning him into something not human. A freak.\n\nOlder memories too, of this place as it had been when the ithani had ruled Erro. Of the end days when the world had prepared to shift, had prepared for the Great Migration of the ithani race.\n\nAnd something else.\n\nDANI BLACK taunting Xander and Jameson in the sky above this place. \"Dani Black!\" He threw himself at her as the clouds spun around them like the eye of a hurricane.\n\nShe turned toward him, and tried to spring away, but she was too slow.\n\nHe rammed into her and reached inside.\n\nVENIN SHOOK his head. The image had a hold on him, but he had no idea what it meant. Why Dani? She'd vanished weeks before.\n\nNo time to figure it out now.\n\nHe pulled Alix after him and they ran through the abandoned city. Already the ground was trembling beneath them.\n\nThey ran over wide white bridges and under white arches and between white buildings, all the while making progress toward that alien thing that sat in the middle of the city.\n\nFor he was now quite sure it was the key to everything.\n\nThe memories threatened to overwhelm him.\n\nHe was breathing hard. \"I can't....\"\n\nAlix knelt before him and pulled him up for a kiss.\n\nThe kiss lasted just a few seconds, but Venin lived a century with Alix in that short time. The future became clear, the clouds in his mind rolling back and the darkness of his past receding like a bad dream. There was a way through this. He could see it now.\n\nTaz was there before him for just a moment. \"Every generation has a seer,\" he whispered, and then he was gone too.\n\nVenin opened his eyes. \"Holy Split.\"\n\n\"Better?\" Alix's eyes twinkled.\n\n\"Better.\" The memories were quiet in his mind. \"But we're not going to make it in time.\" As he said it, the Heart grew, and the ground beneath them shook like it was the end of days. He stretched his wings, and a sharp pain surged through his left one. \"Maybe I can\u2014\"\n\n\"You two are needed.\"\n\nVenin spun around to find a young boy, maybe twelve or thirteen, staring at them. \"Morgan?\"\n\n\"Come with me. We're going to change the plan.\"\n\nALIX STEPPED through the gateway.\n\nHe'd never get used to that. Sure, the Common Worlds were high tech, but they had nothing to match this instant connecting of time and space.\n\nHis whole body hummed. That kiss.\n\nThe three of them stood in a wide bowl at the top of the Heart. It rumbled underneath them as it expanded, growing toward the white ceiling of the cavern.\n\nThe masculine symbolism wasn't lost on him. \"You have to stall them.\" Morgan pointed skyward.\n\n\"Stall who?\"\n\n\"The ithani. They will try to enter the Heart. You must stop them.\"\n\nAlix shared a how-the-hell-do-we-do-that glance with Venin, who looked equally mystified and shrugged.\n\n\"Give me your hands.\" Morgan held out his own.\n\nVenin and Alix exchanged a glance, and then took Morgan's outstretched arms.\n\nThe boy glowed, and a feeling of well-being flowed from his small arm up into Alix, spreading into his torso and throughout his body. He felt better than he had in days. In years, even.\n\nHe glanced over at Venin. The skythane's shoulder wound was knitting itself closed.\n\nVenin opened his eyes and looked at the shoulder. \"It's healed.\"\n\n\"Damn, wish we had more like this kid.\" Alix thought of all the rangers he'd lost. Of Tucker.\n\n\"I'll come back soon.\" Morgan let go of their arms, opened another gateway, and vanished.\n\nAbove them, the ceiling of the cavern had opened up. It actually peeled back, like the skin of an orange, allowing the Heart to continue to rise toward the planet's surface.\n\nAbove, a host of ithani were gathering in the sky.\n\nVenin took his hand.\n\n\"I have a connection to OberCorp,\" Erissa said in his ear.\n\n\"How...?\"\n\n\"What is it?\" Venin frowned.\n\n\"Erissa says she can reach OberCorp. Hello?\"\n\n\"Alix, is that you?\" It was Alia.\n\n\"Yes! We're both safe\u2014Venin is right here with me. You're not going to believe what we found down here\u2014\"\n\n\"Things are crazy up here too. There's something happening where the House of the Sky used to be.\"\n\n\"Yes, we're right in the middle of it. Listen, there's not much time. Can you relay a message to OberCorp?\"\n\n\"Sure. Mylin's pretty much taken over there in your absence. With help from your mother, apparently.\"\n\nHoly Split. Who let her out of confinement? \"Okay. Well, she's one of the toughest sonofabitches that I know, so that's probably a good thing. Look, I'll call you back shortly. We've got another war on our hands, I think.\"\n\nHe cut the connection. There had to be a way to forestall the ithani advance.\n\nVenin shook his head. \"It's always violence. That's what has gotten us to this point. I wish\u2014\"\n\nAlix closed his eyes. He was tired of fighting. \"I know. Look, if we get out of this thing alive... should we give this a shot?\" He squeezed Venin's hand.\n\n\"I don't know. We come from two totally different backgrounds.\"\n\nAlix could see the way Venin looked at him. There was hunger there, and not just lust. He'd been alone for so long. \"Does that really matter?\" He pointed up at the sky. \"Look what's waiting for us. Look what we've been through together. No one else understands that like you do. Not even Xander.\" It was true. He realized it as he said it. He needed to let go of his ex. To let him find happiness, or whatever he could, with Jameson.\n\nIt's time.\n\n# CHAPTER TWENTY-SEVEN: IN THE HEART\n\nJAMESON FELT a growing sense of purpose. He'd come so far from the scared, closeted lander man he'd been when he'd arrived on Oberon the month before.\n\nHe was more than that, now. Much more. He was human. He was skythane. He was gay. He was ithani.\n\nHe was Xander's.\n\nTheir bond in the athr\u00e0... sex? Intimacy? Total connection? Whatever it had been, it had released him from his remaining fear and doubt.\n\nThere was more to discover. So much more, but he didn't have time to drink it all in yet. The athr\u00e0 held more secrets for him, he was sure, but they had to act quickly to save Puck. That was important.\n\nThe little creature reminded him of Morgan, in a strange way. Both had come into his life unexpectedly, needing his protection. And like Morgan, Jameson had become certain that Puck was the key to everything that was going on.\n\nHe squeezed Xander's hand, and his other half nodded. Their connection had deepened in the athr\u00e0, so that Jameson could almost hear his thoughts.\n\nWe go?\n\nYes.\n\n\"We're bringing back Puck.\"\n\nErina merely made a fist of agreement.\n\nJameson used the original key Quince had given him, a world and an eon away, and fixed the dark place Puck was being kept in his mind. He twisted the key and the gateway opened up, letting him through. He pulled Xander after him, hand in hand.\n\nThey found themselves in a small dark cell, much different from the clean white enclosures Erina had taken them through. It looked like it had been carved out of the stone and carried with it an air of incredible age. Metal bars blocked the cell off from whatever lay outside.\n\nPuck sat disconsolately on a stone bench, zer back to them, zer wings drooping.\n\n\"Hey there!\" Now he could see the nimfeach through ithani eyes. Puck was ze, not she.\n\nPuck turned and chirped and ran at him, throwing zerself into his arms in one huge leap. Zer wings extended out behind zer, and ze chittered happily at him. Jameson wished he could understand what ze was saying.\n\nXander looked at him.\n\nThe athr\u00e0.\n\nPuck was a nimfeach. If Puck could enter it with them, ze could tell them zer story.\n\nNot here. We aren't safe here.\n\nJameson pulled out the key and opened a gateway back to the Heart.\n\nPuck pointed at the key and chittered animatedly, clearly excited.\n\nErina met them, glancing at the nimfeach, and touched Jameson's shoulder. \"Give zer the key.\"\n\n\"What?\" He looked at Erina, and then at the key in his hand. \"This?\"\n\nZe waved zer hand back and forth in the air, in the gesture Jameson was coming to understand as no. \"The other one. The master key.\"\n\nJameson put the smaller key away and took out the larger key.\n\nIts milky-white surface changed almost immediately, giving off a rainbow of colors, buzzing under his touch.\n\nHe looked at Erina, who thrummed.\n\nHe held it out to Puck.\n\nZe blinked at him, and then turned to stare at the key. Ze reached out to touch it.\n\nJameson, guided by some instinct he didn't quite understand, let it go.\n\nThe nimfeach took it, holding it in the air before zer like a crystal ball.\n\nThe colors flared brightly and then subsided. They drained into Puck's outstretched arms, and then the key dissolved and disappeared as if it were made of air.\n\nThe nimfeach sighed and dropped to the ground.\n\n\"Is ze dead? What the hell did you do to zer?\" Xander touched the little creature's carapace. \"And how do we get back home now?\"\n\nJameson shook his head. \"I wish I knew.\" He knelt beside Puck. \"I'm going to try something.\" He used his most reassuring psych voice. He held out his hand, and Xander took it and knelt beside him, their wings settling around each other and over Puck.\n\nJameson plunged them into the athr\u00e0. They stood once again in the gray fog of possibility.\n\nJameson looked at Xander. Through their connection, he felt a flood of absolute love and support.\n\nHe squeezed Xander's hands and pulled the two of them again into one.\n\nPUCK WATCHED the silver moon above, hanging in the sky. The time would come soon. They would take zer heart, leaving zer a husk of zer former self.\n\nThe ithani had always lived side by side with the nimfeach, and the nimfeach had given freely of their abilities. There were far fewer nimfeach than ithani, and so there was always work to be done.\n\nThe nimfeach had their own homes and customs, separate from the ithani. When a new one was born from the Heart, its ithani soul transformed to rise to the status of elder, the birth was celebrated throughout the community.\n\nThen the war had come.\n\nThe ithani general had kept the old traditions for a time, but zi'd demanded more and more from the nimfeach.\n\nWhen they'd had enough, Puck had been in the minority, arguing for a peaceful solution.\n\nInstead, the Thshnel had ordered the nimfeach tracked down and had stolen their souls, the keys that each one carried inside, which allowed them to transcend space and sometimes even time.\n\nWithout zer key, ze was a shell of herself, a hollowed-out creature enslaved to the whims of zer masters.\n\nJAMESON SEPARATED from the nimfeach with a gasp and fell out of the athr\u00e0. \"Holy fucking hells.\"\n\n\"The ithani are going to destroy zer race.\" Xander looked at Puck, his eyes full of sympathy.\n\n\"Not just the nimfeach. Everything.\" Jameson grunted. \"Quince was right.\"\n\n\"So Morgan was working for them all along?\" Xander looked devastated.\n\n\"I don't know. Maybe at the start? Maybe we changed him. Maybe his human half changed him.\"\n\nErina broke in. \"The shift is coming. We're almost out of time. The other ithani will all be going to hibernate for the shift.\"\n\n\"Hibernate?\"\n\n\"We can't withstand it unshielded. Only the nimfeach can.\"\n\n\"Why are your people doing this?\"\n\nErina let out a whoosh of breath. \"Jirron. Zi's our most brilliant... general, I think you would say. Zi was a triad once, Jirrol, Ronil and Dinas. Jirrondin. When zis three became one, they had a mind for strategy that surpassed all others.\"\n\n\"Just what you need in the middle of a war.\" Xander flashed Jameson a wry grin.\n\n\"Yes, but then zi rejected the ithani ways. Zi refused to bear children. Zi wanted to stay in zis combined form.\"\n\n\"And that's bad?\"\n\nErina swiped zer hand sideways. \"Some do so, forgoing having children to pursue the arts or sciences. Many of our greatest advances have come from the combined minds of a triad.\n\n\"Jirrondin was different. Zi wanted everyone else to be like zim. To celebrate intellect over form.\"\n\nXander grinned. \"Big nerd, huh?\"\n\nErina cocked zer head. \"Nerd?\"\n\n\"Someone who values knowledge above all else.\"\n\nErina made a thrumming sound of agreement. \"Yes. Zi is a big nerd.\"\n\nXander stifled a laugh. \"So not all the ithani want this?\"\n\nEri double blinked. \"No, but we have let ourselves be swept along by the fervor. Thshnel'Jirron is very persuasive.\"\n\n\"Can we stop zim?\" Something had ended the shift. Oberon in their own time was proof of that.\n\nErina stared at them for a long moment, clucking zer tongue. \"Yes. I can.\"\n\nSomething was happening to Puck. \"Guys?\" Xander pointed at the little nimfeach.\n\nZe was shivering, and then shaking, and then convulsing.\n\n\"Is that normal?\"\n\nErina thrummed zer agreement. Ze knelt and took the nimfeach in zer lap, petting zer gently with zer soft white hand. \"Ze's coming back into zerself.\"\n\nJameson felt Xander's gaze on him. \"What are you thinking?\"\n\nXander was as beautiful to him as ever. \"What are we?\"\n\n\"What do you mean?\" But he knew exactly what Xander meant.\n\n\"Before I met you, I knew exactly what I was. A skythane outcast living in Oberon City, working for an evil corporation.\"\n\n\"Sounds about right.\"\n\n\"Now? I'm a skythane prince\u2014or king\u2014and I have bits of me that are ithani and maybe even nimfeach.\" He glanced at Puck and back at Jameson. \"So what am I?\"\n\nJameson shrugged. \"I don't know. I suppose we're human, overall? It's how we were raised. It's our whole outlook.\" He rubbed his chin. The stubble was getting a bit thick. \"But maybe we're more?\"\n\n\"How do you mean?\"\n\nJameson searched for a way to explain. \"When I was little, I knew I was attracted to boys, but I didn't want to grow up like that. You know?\"\n\nXander laughed. \"Not really, but go on.\"\n\n\"I told myself maybe I had those feelings for a reason. Maybe they were supposed to make me a better person. You know, able to relate to people who were different from me.\"\n\n\"And this makes us more human... how?\"\n\n\"What I'm saying is that maybe there's a reason we ended up being a mix of all these different things. You, me, and the skythane. Maybe we are the only ones who can stop all of this. Broker a peace.\"\n\n\"Maaaaybe.\" Xander sounded doubtful.\n\n\"So what's the plan?\"\n\nThey both turned to Erina. \"What do we do?\"\n\nZe showed them what ze had forseen.\n\nXANDER HATED change.\n\nThings had been rough enough before this whole adventure had begun. Losing Alix had cut him right through, leaving him broken and exposed.\n\nThen Jameson had come along.\n\nHis lover held the little nimfeach gently. Xander marveled at the change Jameson had undergone, both physically and in his character.\n\nHe no longer ran from things. Instead he ran toward them.\n\nJameson had changed him too. Brought back his soft side. His faith in humankind.\n\nWell, most of it.\n\nSome of that had been Morgan's doing too. He brought out Xander's protective instincts in a way no one else ever had.\n\nNow everything they'd fought for was threatened. He had no choice but to embrace change, to harness it to save himself, Jameson, and everyone else they loved. \"We're going to destroy half of Erro, aren't we? In this time?\"\n\nErina's face took on an almost wistful look, though it was hard to tell on that alien visage. \"It's unavoidable. We were different once. We lived with the nimfeach in peace. We weren't so focused on this strange dream of an afterlife.\" The last word almost sounded like an epithet.\n\n\"You sound like you don't believe.\" Xander was getting better at reading the tall alien.\n\nA whoosh. \"Once I did. I wanted to believe. That what we were doing was right. That we had been attacked, and this was the only way to stop it from happening again.\" Ze blinked a few times. \"Then the dreams started. When an ithani is chosen as a seer, it's not always clear. It begins with dreams. Strange dreams that seem to come true. Little things like a premonition of an accident. Or of the birth of a child.\"\n\n\"Or the end of the world.\"\n\n\"Yes. It came to me in pieces, over the last cycle. Now... now I know what it means. What Jirron intends.\"\n\n\"Why not just kill zim?\"\n\nZe made that curious ithani sigh again. \"Zi is entwined with the Heart. If we kill zim, we kill the Heart, too, and eventually the nimfeach and the ithani. I have to stop zim, and zi's too powerful for me to go up against zim alone.\"\n\n\"You won't survive.\" Ze'd said earlier that ithani couldn't live through the shift.\n\nZe stared at him for a long moment. Then ze made a thrumming sound deep in zer throat. \"If we stop it, I'll still be on this side of the shift. If not... I'll go into the Heart.\"\n\n\"Will the Heart survive?\" He'd never heard of it in his own future time.\n\n\"Yes.\"\n\nHe looked at Jameson, and then at Puck. \"So what happens to us?\"\n\nErina rolled her head on zer neck. That was a new one. \"I have a plan for that too.\"\n\nMORGAN OPENED a gateway to Errian, stepping out onto the wide plaza of First Square.\n\nNight had fallen, but skythane buzzed all around him, intent on the cleanup of the ravaged city.\n\nHe brought his hands together, and a thunderclap spread through the city.\n\n\"People of Errian, Jameson needs you.\"\n\nAn older woman with gray hair dropped to the square before him. \"Who are you?\" Her eyes narrowed. \"We've had enough of fighting. It's time to rebuild Errian.\"\n\n\"You're needed for one more fight.\" He held his hand out. \"I'm Morgan. I'm part human and part ithani, and I call Jameson my friend.\"\n\nShe knelt next to him. \"I've heard of you. I'm Vestra, one-time regent of this city.\" She took his hand.\n\n\"Close your eyes.\"\n\nOthers of the skythane were gathering around them, though they left a wide circle around the two of them.\n\nMorgan put a hand to her cheek. \"The last days are here, but we still have one small chance to stop them.\" He shared his soul with her, the knowledge of what had passed and what was still to come.\n\nAfter a moment she opened her eyes, gazing into his. She gave him a small nod and then pulled him into a hug. \"You've suffered so much.\"\n\nFor a minute, he was just Tanner.\n\nHe hugged her back and sobbed. She felt warm and safe, and her wings enfolded him. Tanner cried for his mother, for his friends who had died on the streets, for the lives of the others who had lived through such terrible times. He cried for the nimfeach and their long bondage to the ithani.\n\nAnd he cried for himself. For Tanner. For innocence lost and a life that would never be lived.\n\nIn the end, his sobbing stopped. He sniffed and looked up.\n\nThere were dozens of skythane gathered around the two of them, each reaching out to put a hand on him.\n\nVestra waved them back. She let Morgan go and squeezed his shoulders gently. \"We will go where you tell us.\"\n\nHe nodded, wiping his tears. \"I'm sorry for crying. Nimfeach don't cry.\"\n\n\"But little boys do.\" She kissed his forehead. \"Stay strong, Morgan.\"\n\nMorgan reasserted himself, but Tanner remained. They were linked now, truly one.\n\nHe opened a gateway to the Heart.\n\n\"We go to fight!\" Vestra sent for weapons, and soon they launched themselves through the gateway, and the assembled throng followed.\n\nWhen the last one was through, he closed the gateway and opened another to Gaelan. He'd secure the aid of the Gaelani next.\n\nThen he'd go find Quince.\n\n# CHAPTER TWENTY-EIGHT: FREE\n\nWHEN THE alarm sounded, Lena swung into action.\n\n\"Pulse cannon,\" she said when she picked a big gun out of her own personal weapons stash, hidden in the back of her former office at OberCorp. It was palm-only access, paired with a retinal scan.\n\nMylin doubted Alix had even known it was there.\n\nShe handed Mylin a pulse rifle. \"Know how to use this?\"\n\nMylin nodded.\n\n\"Good girl.\"\n\n\"How did you get here? Alix says you were in the medical wing,\" Mylin said as Lena handed her the rifle.\n\n\"I sprung myself when I heard the situation had started going south, fast. Things aren't going well for Team Human, are they?\" She sighed. \"Where is my son?\"\n\n\"Somewhere in the Outland.\" Mylin shook her head miserably. \"There's something going on out there.\" She wasn't sure how much she should tell the woman. After all, Alix had removed his mother from power for trying to destroy the skythane.\n\nStill, that old saying rang true. A new enemy made an old one your new friend.\n\n\"What about Quince? She seems to be the one capable person in this whole shit show.\"\n\nMylin laughed in spite of herself. \"You know her?\"\n\n\"Only by reputation.\"\n\nShe's tracking down a lead, somewhere under Oberon City.\"\n\nLena snorted, a surprising sound out of such a refined woman. \"What? I wasn't always the flawless corporate creature you see before you.\" She tapped her temple and frowned. \"I forgot, Alix shut down my grid access. You'll have to run ops.\"\n\n\"Run ops?\"\n\n\"Tell me where we need to go. Where's the alarm coming from?\"\n\n\"Okay. Let's see.\" She slitted a request to DOC. \"It's down on the third floor. The armory.\"\n\n\"Of course. It's Rogan, isn't it?\"\n\n\"How do you know about Rogan?\"\n\nLena smiled grimly. \"Like I said, I still have friends in OberCorp, board ouster or not. Come on.\"\n\nShe led Mylin out of the office and past the lift.\n\nMylin glanced at it questioningly.\n\n\"Too dangerous. We might get stuck if they cut the power, or worse.\"\n\n\"You know I have wings, right?\"\n\n\"Yes. Which would make you a sitting duck if they fired up the shaft. Sorry. Flying duck.\"\n\n\"Duck?\"\n\n\"Look it up later.\" They gathered the few remaining OberCorp enforcers in the building, and Lena palmed the pad next to an unmarked door. It swung open silently, revealing a dimly lit stairway.\n\nTwenty-five floors down.\n\nCome on!\" Lena barreled down the stairs, and Mylin followed her.\n\nOn the way, she asked DOC to send out updates to Quince and Alia. She marveled at the strange circumstances she found herself in, following the former OberCorp CEO down the stairs armed with a pulse rifle to try to fight off a Syndicate boss. A month ago, she hadn't even known those people and things existed. If I die today, at least I've had an interesting life.\n\n\"You're awfully young,\" Lena said between breaths as they ran down the stairs. \"How old are you?\"\n\n\"Sixteen. I've always been really good at organization.\"\n\n\"You should come work for me when this is all over.\" They rounded the landing on the fourteenth floor. \"For OberCorp.\"\n\nMylin grinned. \"I think I'd like that. Why are you helping me?\"\n\nThey passed the eleventh-floor landing. \"Alix and I had our disagreements, and I'm still on the fence about the skythane. But like I said, we're all Team Human.\"\n\n\"Time enough to sort the rest out later?\"\n\n\"Exactly.\"\n\n\"What about Rogan? He's human.\"\n\n\"After all the things he's done? I'd say he's Team Scum.\"\n\nSoon they reached the armory floor. Two of the enforcers quietly removed an access way cover and slipped inside.\n\nNow they were in the hall, thirty meters from the armory. A firefight raged around the corner, the smell of melted plas and the sounds of pulse fire filling the air.\n\nMylin had her back to the wall, her pulse rifle held at the ready.\n\nLena Preston crouched next to her, holding a much larger gun. \"Ready?\"\n\n\"Wait!\" Zim was pinging her. \"Zim, kind of a bad time....\"\n\n\"There's a young enforcer trying to reach you. His name is Tovey. He says he's pinned down in the armory. He's only managing to hold out because of the armored door, but his attackers are almost through.\"\n\n\"Put him through.\"\n\n\"What is it?\" Lena looked anxious to go.\n\nMylin put up her hand. \"Tovey? Yes, I know. We're right outside. Give me a count of ten, and then give them everything you've got.\"\n\n\"What?\"\n\n\"We've got a man inside. Three... two... one. Let's go!\"\n\nQUINCE GRUNTED in frustration, turning a smooth pebble over and over in her hand.\n\nAccording to Mylin, OberCorp was under attack, by Rogan, apparently. Something called the \"Heart\" was rising near the site of the House of the Sun.\n\nAnd she was stuck down here on a fool's errand that had so far uncovered little more than a history lesson and a mind-bending guilt trip. \"Gods dammit.\" She threw the pebble at the wall of rubble that had once been the cavern entrance.\n\nRobyn took her hand. \"What's going on in there?\"\n\n\"I was wrong. About dosing Xander and Jameson with pith. About running back to Ballifor after... Andra. About all of it.\" She let go of Robyn's hand. She couldn't bear to have Robyn see her like this, full of shame.\n\n\"Hey, look at me.\" Robyn pulled her face around gently, her green eyes staring intently into Quince's. \"You saved them both when the world was against them. You saved me.\" She fluttered her wings. \"You brought me back to myself.\"\n\nQuince laughed ruefully. \"Maybe. I made so many mistakes.\"\n\nRobyn's laugh was more cheerful. \"Who hasn't? More importantly, have you learned from them?\"\n\nQuince thought about that. She had learned some hard lessons about herself over the course of the last month. \"I guess so?\"\n\n\"We can't change the past. We can only change how we react, and what we do next.\" Her green eyes searched Quince's. \"What happened inside that head of yours after you collapsed?\"\n\n\"She... Daia... the mind up there. She made me relive all the bad decisions I've made. She forced me to face up to the people I've hurt. It wasn't real, but still.\"\n\nRobyn nodded. \"It changed you.\"\n\n\"Maybe. Too soon to tell. I have to apologize to Jameson. To both of them.\"\n\nRobyn pulled her close. \"You're my soul mate. When you ache, I ache.\"\n\nQuince let herself be held, feeling Robyn's warmth, and closed her eyes. For just a moment, she let go of the huge weight she carried with her, day after day.\n\nFor just a moment, she was at peace.\n\n\"Ahem.\"\n\nThey separated. Jessa was pointing up. \"I think we're about to have company again.\"\n\nThe world mind was stirring, electrical impulses running across its surface.\n\n\"What is it?\" Jessa frowned, pointing the pulse rifle up at the mind again.\n\n\"Don't! You might damage it.\"\n\n\"Sounds like it attacked you first.\"\n\nQuince shook her head. \"I think there's something wrong with it.\" She didn't know how to explain it, but the Daia she remembered, the real Daia, wasn't cruel. Wasn't vindictive. \"You're looking at the last world mind, something the skythane colonists brought with them when they first came here.\" Maybe Daia had reached out to the grid because she was lonely. But why would that mind be the source of the strange alien code they'd found in the OberCorp core, the code that had led them here? It didn't make sense.\n\nQuince stared at the white \"root\" that ran from the mind above to the heart of Oberon City. \"Jessa, give me that flashlight.\"\n\nJessa handed it over.\n\nShe shone it along the ceiling, following the root. It ran up the wall to a white sphere attached to the edge of the world mind.\n\nShe'd thought before that the material reminded her of corrinder.\n\nIt hung off the mind like a tumor.\n\nOr a parasite.\n\nQuince leapt into the air, her wings pulling her up into the dim upper recesses of the cavern.\n\nThe more she stared at it, the more she was convinced it didn't belong.\n\nShe closed her eyes, and a memory slipped in.\n\nDAIA THE woman stood on the sand below, staring up at the mind. \"We're going to block you in. You should be safe here until we can figure out what's happening.\"\n\n\"I understand. Can I help?\"\n\nDaia shook her head. \"You are the most important piece of the colony. If they find you....\" She shuddered. \"We'll come back for you when this is all over.\"\n\nShe closed her eyes, and her mirror image stared back at her. \"I understand.\"\n\nDaia backed out into the sunlight. \"Are we ready?\"\n\nBraice nodded.\n\n\"Do it.\"\n\nHe set off the carefully placed charges, burying the world mind's cavern.\n\nThere were only a few colonists left. Something was taking them, one or two at a time. She suspected the nimfeach, the beautiful rainbow-hued thing that had tried to woo her away from the camp that day, two months before.\n\nShe'd refused, but in the end it was having its way.\n\nIt wouldn't have the world mind. She'd seen to that.\n\n\"Goodbye, sister mind.\"\n\nTHE THING attached to the world mind hadn't been there before, when she was sealed in this tomb.\n\nSomehow it was being used to control the mind, or to tame it.\n\nShe reached out and pulled on it, trying to separate it from the mind.\n\nIt was firmly attached, its own \"roots\" sunk deep into the flesh of the mind.\n\n\"Quince, don't! You'll kill me!\" The parasite changed, becoming Daia, hovering weirdly in the air before her without wings.\n\nSeveral roots detached from the wall and quested toward her.\n\nNo more time. Quince pulled out the knife she'd chosen from the armory.\n\nSuddenly the thing became Xander. Little Xander. Why did you let him do this to me?\n\n\"You're not Xander.\"\n\nShe raised the knife and hacked at the thing.\n\nHer knife sliced across Queen Andra's neck. The queen's cold, dead eyes stared back at her. You did this.\n\n\"You don't have any more power over me.\" She hacked at the thing, concentrating on where it was linked to Daia, and the image disappeared, leaving a sagging white sack.\n\nIt melted away from the blade like butter, and soon she had separated it from the mind. It sent out little runners, trying to reconnect, but then she severed its connection to the root line that led off to the OberCorp core, down the long series of caves and tunnels that had brought them here.\n\nIt fell to the ground with a wet plop.\n\nShe ripped the root from the ceiling, too, and flew down to the cavern entrance, severing it completely.\n\nShe threw it onto the parasite, and then pulled an acid tab from her pack and activated it, tossing it onto the pile.\n\nThe acid sizzled and popped, consuming the whole nasty mass.\n\nShe looked up at Daia. Some kind of golden fluid dripped from where she'd severed the parasite, but as Quince watched, the remaining parts of it were forced out, one by one, and the surface of the mind sealed over.\n\n\"What is it?\" Robyn stared at the sodden mass that was quickly melting down to a puddle of goo.\n\nA great grinding sound filled the cavern, and Quince dropped back to the ground. \"I don't know. Something of the ithani's.\"\n\n\"Maybe that wasn't such a good idea?\" Jessa pointed.\n\nOne of the mind's own roots, thick as a blue oak trunk, had sprouted a new extension. It raced along the ceiling to the debris pile, and a fine filament spread out over the broken wall of rocks.\n\nThe rocks shrank as more of the filaments spread out across the entire wall.\n\nSoon bits of it began to fall, slipping through the net.\n\n\"She's eating the rock.\" Quince's mouth fell open. She hadn't thought anything could still surprise her. \"Stand back!\"\n\nThey retreated to the cave mouth where they'd first entered. It didn't take long.\n\nSoon the wall was gone. A cool sea breeze filled the cavern, and the filaments retreated to reveal the Gildensea, lit by Bandia's silver light.\n\nSilhouetted there was a woman's form.\n\n\"Daia,\" Quince whispered.\n\n\"She's the one who attacked you?\" Robyn had her own knife out.\n\n\"Who? What are you guys seeing?\" Jessa looked at Quince, then at Robyn, confusion plain on her face.\n\n\"Daia. The world mind.\"\n\n\"I don't see anything.\"\n\nQuince smiled sadly. \"You're not skythane.\"\n\nShe took Robyn's hand, and they went to meet the apparition.\n\nJESSA WATCHED them go.\n\nShe couldn't help but feel jealous. The skythane were human like her, but they were different in subtle ways\u2014and some not so subtle\u2014from people like her and Alix. At some point on the timeline they had diverged, and someday maybe they would be a distinct species, as different from humans as humans had been from apes.\n\nShe crossed the space, giving the two of them a wide berth. They stood in the center of the cavern, their hands held out, transfixed. Staring at something only they could see.\n\nIt was a bit creepy.\n\nShe climbed over the remaining rocks at the newly revealed cavern entrance, and out onto the sand outside.\n\nIt was a warm evening. The sky was clear over the Gildensea, but a storm was brewing behind her, lightning strikes too far away to be heard. The air was heavy with humidity.\n\nJust to the north, she could see the lights of the arcos, marking the center of Oberon City.\n\nNot too far south, a single white tower stood by the sea.\n\nThis might be the last night of the world. It was a strange thought. Everything was deceptively normal, even beautiful.\n\nShe could come to love this world, given time.\n\nShe sat on the sand, her back to a black rock that jutted into the air, and watched the waves roll in and out.\n\nJamie, where are you?\n\nOnly the ocean breeze answered.\n\nROBYN GLARED at the spirit that had appeared before them in the cavern. \"I don't trust her.\"\n\nQuince took her hand. \"It's okay. She wasn't herself. She was under the influence of the ithani.\"\n\n\"How? They've all been asleep.\"\n\n\"I don't know. Maybe the nimfeach? Someone put that thing up there.\"\n\nDaia was watching them.\n\n\"Come on.\" Quince urged her forward gently. Quince pulled her hand up, and Daia touched her palm.\n\nThe cavern dissolved, and they found themselves in a wide green valley. A golden sun shone brightly, and a stone tower stood high upon a central hill, casting its shadow across them.\n\nDaia sat cross-legged in front of them, and she invited them to do the same with a gesture. \"I've waited a long time for someone to come.\"\n\n\"How long?\"\n\n\"More than eight hundred years.\"\n\nRobyn whistled. \"That is a long time.\"\n\n\"What are you?\" Quince looked around at the strange world. \"What is this place?\"\n\n\"I'm the seventh world mind. I came here with your people, seeded from the sixth mind.\" Daia gestured at the valley. \"This is a replica of the domain of the first world mind. I find it peaceful. I come here when I want to think.\"\n\nRobyn breathed in the air. It was fresh and cool, with a hint of moisture. She ran her hand over the grass. It tickled her palm. \"This place is real. It has to be.\" Maybe a gateway had brought them here.\n\nDaia laughed. \"It's every bit as real as the world you live in, but it's not physical. Not in the way you mean.\"\n\n\"Astonishing.\" She looked up at the sky. \"How far does it go?\"\n\n\"As far as I want.\"\n\n\"Were you trapped in here?\"\n\nDaia nodded. \"For many years. My namesake blocked me into the cavern in the hopes of keeping me safe from the nimfeach, but they already knew about me, and they made the harsorch.\"\n\n\"The what?\" Quince frowned. \"The horse sauce?\"\n\nDaia laughed again. \"No, the harsorch. The parasite you cut off me. They used it to control me. To keep me docile.\"\n\n\"So that really wasn't you?\" Robyn frowned.\n\n\"No, but I couldn't help but see what she put Quince through. I'm so sorry, Quince.\" Daia reached out to touch her arm.\n\n\"How do you know our names? How are we even here?\" Robyn still didn't trust the woman. The mind. Whatever she was.\n\n\"I saw you. In my dreams. All of you\u2014every one of the skythane.\"\n\n\"How is that possible?\"\n\n\"You made me. Your kind. Long, long ago. Since then... we've been connected. It's hard to explain. Here.\" She held out her hands.\n\nRobyn looked at Quince, who nodded.\n\nShe took Daia's hand, trying to keep her own from trembling.\n\n\"This is who I am.\"\n\nA torrent of information flowed through Robyn's mind.\n\nWars and spaceships and battles, love and loss, a long, lonely journey through the void.\n\nAnd more personal details\u2014a woman born inside a generation ship, on the last leg of its journey to a new star.\n\nTies between human and biomind that were beyond Robyn's wildest imagination.\n\nDaia was the last of her kind.\n\nQuince narrowed her eyes. \"The skythane memories, they come from you.\"\n\nDaia nodded. \"Some of them. The human ones.\" She sighed. \"It was the only way I could reach you.\"\n\n\"And the rest?\"\n\n\"From the Heart. The ithani.\"\n\n\"You know about the ithani?\"\n\n\"Yes.\" Daia frowned, brushing her silver hair away from her face. She seemed so human. \"The nimfeach used me to monitor the newcomers. Sometimes they would come here to ask me questions.\"\n\n\"The newcomers?\" Robyn frowned.\n\n\"The landers. The second wave.\" Quince turned her attention back to Daia. \"Are you free of them now?\"\n\n\"Yes. Thank you.\" She stiffened. \"One arrives now.\" She let them go and vanished, leaving them in an empty cavern.\n\nRobyn spun around to see a gateway splitting the air.\n\nMorgan stepped through, closing it behind him. \"Are you ready to help me change the plan?\"\n\nQuince looked at Robyn, and at Jessa, who had come running up from the beach outside when the gateway appeared.\n\nBoth nodded.\n\nQuince grinned. \"Sure. What do we need to do?\"\n\n# CHAPTER TWENTY-NINE: NIGHT FALLS\n\nTHSHNEL'JIRRON LED zis army of ithani south, passing over the lands that had once been zis. These human invaders had left much of it untouched, but their very presence on Erro vexed zim. Like verthex that scurried from the light, they were craven insects that had no right to be on zis world.\n\nEven their bodies were ruled by contrarian impulses of emotion, thought, and hormones. Zi'd learned that firsthand, and zi was questioning zis choice to be reborn in this human host. Thankfully it was a state zi'd need to endure for only a little while longer.\n\nDani lay on the forest floor, her wings spread out behind her, as Kadin worked his way up her body, eliciting gasps and soft moans....\n\nJirron thrust the memory back down. They'd been coming to zim more and more frequently, little snippets of the human's former life. It was unusual. In the past, zis hosts had been entirely subsumed by zis own personality, their memories only surfacing when zi searched for them.\n\nNo matter. The time of the Great Migration was upon them. Denied to zis race for millennia by a catastrophic event, zis grand plan was once again in motion, thanks to the nimfeach. The little creatures had proven invaluable in getting the ithani to this point, continuing zis work when zi zimself had been asleep all those many years.\n\nThe Heart rose in the distance like a mountain, beckoning them forward. Clouds gathered as the Heart drew energy from the amalite deposits below the surface.\n\nIn mere hours, zi would be free of this planet-bound form, freed to become what zi was always meant to be.\n\nZi would be the ithani.\n\n\"ERISSA, GET me Alia.\" Alix frowned at the stormy sky. Who knew the end of the world would be accompanied by such miserable weather?\n\nHer voice came through loud and clear. \"Alix?\"\n\n\"Yes. Where are you?\"\n\n\"In a hoversport circling the place where the House of the Sun used to be. Where are you?\"\n\n\"Thank the Split you're in range. See that big dome thing rising from the ground?\"\n\n\"Yes?\"\n\n\"We're on the top of it.\" The Heart rumbled beneath him, and the skies thundered as if in response. \"Listen, we have ithani coming in hot. We have to keep them from entering the Heart.\"\n\n\"Okay. What do you need?\"\n\n\"Any and all hoversports we can scramble to the area. Ground support too. Pull in everyone we have. It's all hands on deck.\" He took a deep breath. Had it really come to this? \"Tell DOC to get ready to launch that Rentz Class ship in this direction.\"\n\nThere was a long pause. \"That would kill you too.\"\n\nAlix nodded. \"I know. I hope we don't have to use it, but we have to be ready. Erissa, how long would it take that ship to reach here from the spaceport?\"\n\n\"About three-point-two minutes.\"\n\n\"Okay. I'll keep that in mind.\"\n\n\"Venin can carry you out of there\u2014\"\n\n\"Venin has to confront the ithani. I'm stuck down here.\"\n\n\"You're a sitting target.\"\n\n\"This is where I need to be, in the middle of the action. It'll be all right.\" Alix scanned the skies, worried. He was exposed up here, but there was nothing to do about it.\n\n\"Like hell it will.\"\n\n\"I've gotta go. Let the pilots know we'll be fighting along with the skythane this time.\"\n\n\"Of course.\" Her response was curt, almost cold.\n\nHe had no time to deal with emotions, hers or his own. \"Call me if you need me.\" He cut the connection.\n\n\"What did she say?\" Venin was scanning the skies next to him.\n\n\"Reinforcements are on the way from OberCorp.\"\n\n\"They may be too late.\" He pointed up. Alix followed his gaze\u2014the ithani had arrived.\n\n\"I have to go meet them, stall for time.\"\n\n\"What can you do against so many?\" Alix didn't want Venin to go. Not like this. Not when there was a good chance he would never return.\n\n\"If I don't, there might be nothing left to come back to.\" He swept Alix up and kissed him hard.\n\nAlix kissed him back, dropping the last of his walls, his regrets and fears, and giving in to this thing between them completely. Time slowed and stopped, and for a moment there were just the two of them.\n\nThen it lurched forward again.\n\nVenin let him go. \"If we get through this...,\" Venin whispered as they parted.\n\n\"When we get through this.\"\n\nVenin grinned and then leapt into the air to meet the arriving ithani.\n\nIt was a promise.\n\n\"GODSDAMNED SPLIT-ASS jerk.\" Alia wanted to throw something.\n\nOutside, the storm was building. Thunderclouds climbed into the sky, and under their dark skirts, lightning slammed into the earth, sending up showers of broken trees and dirt.\n\n\"What is it?\" Smythe sank down on the bench next to her. His chiseled face belied the gentleness in his hand as he touched her cheek.\n\n\"That self-important prick Alix is going to get his ass killed, and Venin too.\"\n\n\"So, the whole macho Marine bravery thing?\" Smythe cracked a smile.\n\nShe nodded miserably.\n\n\"You two weren't... aren't...?\"\n\nShe laughed in spite of herself. \"Nothing like that. Just... he and I have been through a lot together in the last few weeks. Venin too.\"\n\n\"So what are we gonna do about it?\"\n\n\"What do you mean?\"\n\n\"Come on. We have a hoversport.\" His eyes went unfocused for a minute, and then he looked right at her. \"We're being ordered into battle. Your doing?\"\n\n\"Alix's, but yes.\"\n\n\"So there you go. What say we launch a rescue mission?\"\n\nTHSHNEL'JIRRON AND the ithani arrived at the site of the Heart. Its rise was stirring up a great storm as Erro itself prepared for the Great Migration and the birth of something never before seen.\n\nZi shivered in glee at the thought that true immortality was almost zis. Zi could almost grasp it in the strange, thin white claws of zis human host\u2014nevermore would zi need to fear the death of zis host body. Now only a few puny obstacles stood in zis way.\n\nThe plan had taken generations for zim to piece together, as zis scientists had discovered how to use the nimfeach to pierce the veil between realities. It had taken longer still to find the reality zi wanted. This place.\n\nZis plan had been rushed in the end by the imminent destruction of the world by the dhagani, a last-gasp attempt to bring zis people down.\n\nThe shift required a bonded triad, with one of the nimfeach acting as the key.\n\nStripped of their hearts, the nimfeach had remained loyal for millennia while the ithani slept and had ultimately brought about zis plan through the selective breeding of these humans.\n\nJirrondin had lived through it all\u2014countless transfers to new ithani hosts when zis body had grown old and feeble, and then a hundred-thousand cycle sleep.\n\nAlways knowing zi was one accident or attack away from oblivion if zi was unable to flee to the safety of the Heart.\n\nWhen zis scientists had discovered this new universe, only ever so slightly different from their own, Jirron had known it was what zi sought. The rules here were so similar to zis own universe, but the smallest variation was enough to make the difference. In this place, zi could embed zis consciousness into the very fabric of space and time. The rules of physics were different here. The form of space-time was similar to the athr\u00e0. Zi could merge the ithani virtual world into the weft and weave of this reality, and zis power would be endless.\n\nZi would be a god.\n\nThe ithani would all become part of zim, a new hive-mind race\u2014the jirrondi. Zi grinned at the thought. Zi was starting to get used to the odd human gesture.\n\nZi stared down at the Heart, awaiting them all, and was astonished.\n\nSomeone or something was flying up from the Heart to meet zim.\n\nZi swept down to get a closer look, zis wide midnight-black wings carrying zim toward the Heart.\n\nIt was one of the human skythane. The ones with wings, like zis host. Jirrondin had to admit, humans had a sort of rugged, brutish beauty.\n\nThere was another human as well, far below, standing at the top of the Heart.\n\nHow had they found it so quickly after the rise?\n\nThe skythane man was staring at zim, his mouth agape. \"Dani?\"\n\nA human smile stretched across Jirron's face again. Zi wished zi could shimmer with glee in this human form. \"And you're... Venin.\"\n\nVENIN STARED at the leader of the ithani. She was grinning like a madwoman.\n\nHow had the two-bit thug who'd occupied Gaelan on behalf of OberCorp come to lead an army of alien ithani?\n\n\"What... why are you...? I don't understand.\" It was the ultimate betrayal, not just of her own race, but her species.\n\nAnd yet he shouldn't have been surprised. She'd always been just a step above savagery.\n\n\"Dani told me a lot about you. One of the captains of the Gaelani guard. You were there when the shift happened.\" As she said it, the information seemed to surprise her. \"Are you part of the triad?\" She flew closer, looking at him in a way that made him extremely uncomfortable. Like he was a colorful insect she wanted to pin to her display case.\n\n\"You're not Dani.\" That much was becoming clear. Maybe she was something like Morgan, part Dani and part... something else. Ithani, he guessed. Wind whipped his hair as the storm walls closed in around them, circling the Heart. \"Who are you?\"\n\nThe woman who wasn't Dani drew herself up, flapping her wings, and laughed. \"No, I'm not Dani. I'm the Thshnel Jirron, leader of the ithani.\"\n\n\"The... shrapnel gerund?\"\n\n\"Jirron.\" The ithani let out a whoosh of air, looking as though Venin had just dealt her a mortal insult.\n\nAbove him, the other ithani\u2014thousands of them\u2014had clustered nearby to listen to the conversation. Venin wondered if they understood Common.\n\nHe wished he had a pulse rifle with him. He could end this right here with one good shot to the chest. \"You should go back to the north, where you came from. You won't get to the Heart. Not today.\"\n\nJirron blinked and hissed, looking a bit taken aback. She looked down at the ground, and then back at Venin. \"How do you know about the Heart?\"\n\n\"I know a lot more than you think.\" Venin wasn't sure how long this bluff would last. He hoped he could keep it going until reinforcements arrived.\n\nTaz whispered in his ear.\n\n\"I know about the Great Migration too.\" What Great Migration? Mylin must be feeding him information.\n\nThat startled Jirron. The ithani clucked her tongue, blinking rapidly, but she quickly regained her composure. \"Get out of our way, human. Your time here is\u2014\" She stopped abruptly and looked past Venin, blinking rapidly again.\n\nVenin grinned. The cavalry had arrived.\n\nA GATEWAY opened up behind the human, and more of the skythane poured through.\n\nJirron clucked zis tongue, staring at them in disbelief. How did this happen?\n\nZi hissed. They had a key. Or the nimfeach had turned on zim. Neither seemed likely.\n\nIn moments, a few hundred skythane were arrayed between zis people and the Heart, their wings white and golden.\n\nThe gateway snapped shut.\n\nZi let out a whoosh of breath. It didn't matter. They were still too few to truly challenge zim. Zi would lose some of zis people, but enough would survive to finish the migration.\n\nZi prepared the call to battle.\n\nAnother gateway opened up and more skythane came through, these with dark wings\u2014brown and ebony and midnight blue.\n\nZi was beginning to hate these humans.\n\nA sharp whine drew zis attention to the west.\n\nA small fleet of flying things approached the Heart. Dani's memory supplied the answer. Hoversports, they were called. Powered by heartstone, from the core.\n\nSo there was to be a fight after all. Zis grin returned. If they wanted war, so be it.\n\nZi let out a screech and sent zis people the order to fight as the storm clouds thickened around them.\n\nClaws slipped out of their hands and feet. They had no weapons. This was to have been a day of celebration, not battle. But zi was certain the strength of the ithani would carry the day. They would bury the offworlders at the base of the Heart, and then they would obliterate the world.\n\nJirron backed away from the contest, content to let others fight for zim.\n\nZis people fell upon the skythane.\n\nIN THE end, it took a lot of convincing to get Daia to agree to Morgan's plan.\n\nShe was frightened at the appearance of the nimfeach, and more than a little damaged by her long time of solitude and virtual imprisonment.\n\nQuince could understand that. She couldn't imagine being locked away from everyone she knew, everyone she loved, for close to a thousand years.\n\nBut Morgan made a good point\u2014the memory of the ithani past, and by extension Erro's past, was valuable beyond measure. The Heart was also the progenitor of the nimfeach. If they were going to destroy the Heart, they had to save its memories. Morgan hoped to coax a seed from it, but they had to save its history too.\n\nQuince found Daia in her virtual domain. She stood on top of a cliff at the edge of an alien forest, looking out over a vast sea. It was no ocean on Erro\u2014she was sure of that much. Maybe it was someplace from old Earth, or maybe it had only ever existed here in the world mind.\n\nTrees with leaves like green needles swayed on the ocean breeze, and the air was thick with the smell of the sea. Dried needles crunched under her feet. She shivered, goose bumps popping up on her forearms.\n\nQuince looked around in amazement. Daia and her domains were truly an awe-inspiring achievement, surpassing even the virtual worlds of the grid.\n\n\"So much time.\" Daia glanced up at her as she approached.\n\n\"Time?\" Quince sat down next to her, staring out at the breaking waves. She could feel the wind on her skin. Smell the salt of the sea and hear the cry of sea birds riding the air currents above.\n\nDaia nodded. \"I carry it inside of me. Memories. Files. Sensations.\" She reached over and touched Quince's cheek.\n\nQUINCE WAS someone else, a woman, small and dark, razor sharp in her intelligence. Deeply flawed. Quince felt all of it. No, she was all of it. A woman named Ana with a strange accent, whose father was murdered by terrorists. Who knew the secrets of the world mind. Who had almost been immortal herself.\n\nDAIA LET go, and Quince gasped. \"Who was she?\" She could still feel the woman's thoughts. Her genius. Her sadness.\n\nDaia's smile didn't reach her eyes. \"The mother of us all. Of me.\" She put her head in her hands, staring out at the sea, and sighed. \"I carry so many lives. So many memories.\"\n\n\"Ah.\" Now Quince understood. \"'So much time.'\"\n\nDaia nodded, her dark hair covering her face. \"Yes.\"\n\nQuince couldn't imagine the burden, the backbreaking weight of such a responsibility. \"Do you... do you carry us too? The skythane?\"\n\n\"Yes. Some of you.\" She looked up at Quince, offering her a weak smile. \"Those who are open to me. Even when I was trapped, I could still touch you. It kept me sane.\"\n\nQuince sat down on the rock next to her. \"Do you remember me?\"\n\nDaia looked at her again with her gray eyes and nodded. \"Here.\" She touched Quince's cheek again.\n\nQUINCE WAS three years old, held in her mother's arms. She felt warm and happy. Her mother held up a bunch of redberries, and Quince clutched at them greedily. \"Now, now, my little wereveren. Be patient.\" She broke off one of the berries and put it in Quince's mouth. It exploded in a burst of sweet flavor, the juice running down her cheek.\n\nQUINCE BLINKED, staring at Daia, only partially aware of the tear rolling down her cheek. \"I remember that moment.\" She held out her hand, as if she would find her mother there, ready to laugh with her, to hold her, to tell her everything would be all right.\n\nHow far she'd fallen since that day.\n\nShe sighed. The past was gone and buried, though maybe it could still be relived. \"Thank you for that.\" Time was short. \"Morgan tells me that the Heart is something like you. A depository of knowledge, of ithani history and culture. For the nimfeach too.\" She took Daia's hand. \"I know it's hard, but we need you to help us save it. He says I can facilitate the transfer, with his help, but we can't do it without you.\"\n\nDaia looked at her, searching her eyes. \"I don't think I can.\"\n\nQuince explained what Morgan had told her. \"The king of the ithani is linked to the Heart. If we don't destroy it, we can't destroy him, and all this\u2014our whole world\u2014will come to an end.\" Including this amazing paradise. \"You, me, the memories you carry. Everything that survives will become a part of him. A slave to him.\" Robyn too. \"If you help us, we can keep you safe. Free. You won't be alone ever again. I promise.\"\n\nDaia let go of her hand and stood, taking the few steps to the edge of the cliff.\n\nQuince's heart beat faster, even though she knew it wasn't real. She could feel it in her chest as she awaited Daia's answer. This place was truly amazing.\n\n\"I'll do it,\" Daia said at last. \"But I want something in return.\"\n\n\"Name it.\" Quince hoped it was something she could afford to give.\n\n\"When it's over, I want you to let me go.\"\n\n\"What do you mean?\"\n\nDaia turned back to face her, and there was fire in her eyes. \"Eight hundred years is a long time. I am tired, Quince. I'm ready to go. Someone else can take up this burden.\"\n\nQuince nodded. She could do that.\n\nDaia was a copy of someone who had once been a living, breathing person. The original Daia Hammond who had lived in this place more than eight hundred years before.\n\nThe world mind Daia was so much more than that now, but she had started her life as human as Quince.\n\nThey would find someone to take Daia's burden. Someone to imprint on the new seed, who would have the strength of will to carry it. \"We can do that.\"\n\nThe tension slipped out of Daia's lithe form like smoke. \"Thank you, Quince.\"\n\nQuince pulled her in for a hug, and away from the edge of the cliff. Virtual or no, it made her nervous as hell to see Daia standing so close. \"I'm sorry you were alone for so long. If I had known\u2014\"\n\n\"You said time was short.\" Daia pulled away, hugging herself as if she were cold. \"What do I need to do?\"\n\n\"ON MY signal, go!\" Lena pointed down the hall toward the armory.\n\nMylin nodded. \"Derren, you ready?\" she whispered via her cirq. They'd been pinned down here for ten minutes. The Syndicate fighters were fierce and armed to the teeth, but they had the advantage of home terrain. And Derren and DOC.\n\n\"Yes ma'am.\"\n\n\"That's 'yes, Mylin.'\"\n\n\"Sorry. Yes, ma'am, Mylin.\"\n\nMylin sighed. \"Close enough.\" The fool was besotted with her.\n\nLena coordinated with the other enforcers. \"On my mark.\"\n\nThree seconds later, an explosion filled the hallway with smoke. \"Good boy, Tovey.\" Lena gestured her forward. \"Go!\"\n\nThe hallway went pitch-black.\n\nHelped by DOC, Zim guided her aim in the darkness, but the coughing of the Syndicate men helped her locate them too. One, two, three enemy fighters went down with stun blasts from her pulse rifle.\n\nOthers retreated toward the emergency stairs but were caught by the small team of enforcers. A firefight broke out, along with some angry shouts.\n\nThen the hall was silent. The lights came back up.\n\nSmoke filled the hallway, but DOC amped up the building's circulation systems to clear it out.\n\n\"Good work.\" Lena surveyed the damage. Seven of Rogan's men were down. \"Let's get these guys disarmed and into holding\u2014\"\n\n\"Down!\" Mylin had seen a flash out of the corner of her eye, and she pushed Lena out of the way, firing her pulse rifle down the hallway at the attacker.\n\nThere was a grunt, and another Syndicate woman fell out of a doorway onto the floor.\n\nSomething hot and wet spread across Mylin's abdomen.\n\nMylin fell to her knees. \"I think I've been shot.\" She touched her stomach with her hand, and it came away red.\n\nLena took her in her arms and eased her to the floor. \"Tovey! Get your ass over here. We have to get this one to the infirmary.\"\n\nThe last thing Mylin saw was a young man's face above hers, but it wasn't Tovey.\n\n\"You're gonna be okay, ma'am... Mylin.\" It was Derren.\n\nTHE SKYTHANE were outnumbered.\n\nAlix watched from below.\n\nThe skies above the Heart had become a massive battleground, fought mostly hand-to-hand, in and out of the storm.\n\nThe skythane were armed with a few pulse rifles and knives, while the ithani had claws like knife blades that extended from their three-fingered hands.\n\nSomewhere up there, Venin was in the thick of it. He kept in touch with Alix via his PA, but Alix had never felt more helpless.\n\nIt was hard to make out the entire battle as water poured down in torrents, blocking his view on and off. God, I'm sick of the rain.\n\nAlix watched from the top of the Heart, directing the hoversports in a coordinated attack, but even there he felt useless. All he could really do was to issue some orders and hope they were followed through.\n\nWhat he wouldn't have given for one of those gateway keys right about now. Not that he'd have the slightest clue how to use it.\n\nThe water was collecting in the bowl atop the Heart. He'd climbed to higher ground, trying to avoid puddles as lightning rained down on the Heart and the surrounding ground.\n\nThe Heart itself was starting to glow, the vegetation sloughing off like old scabs.\n\nHe was going to have to call in the meso strike, casualties be damned. If they lost this battle, the whole world would pay the price, and that possibility was getting perilously close.\n\nOne of the hoversports was coming in hot.\n\n\"Erissa, who is that?\" He scowled. \"Tell them we need them in the battle.\"\n\n\"Negative, Alix. We're coming in to get you.\" It was Alia.\n\n\"Abort! You guys need to get your asses back up there and\u2014\"\n\n\"I'm not leaving you down there to die. Besides, from up here you can see things better.\"\n\nAlix growled. She made a good case.\n\nLightning struck the Heart not ten meters from where he stood, knocking him to the ground and blinding him temporarily. The Heart absorbed the energy, glowing brighter.\n\nHe got up, blinking and trying to see which direction to run to reach the hoversport.\n\nThen she stood there before him, an angel, her black wings outspread against the backdrop of the tempest. More like a Valkyrie. \"Come on!\" She grabbed his arm and hauled him toward the hoversport. \"Where's Venin?\"\n\nHe pointed up at the battle above.\n\nThe craft appeared before them, a hatch open on its side. A square-jawed ranger was waving them on.\n\nShe shoved him inside and followed, slamming the hatch closed behind her. \"Get us the hell out of here!\"\n\nAlix grinned. Alia was picking up a thing or two from her new ranger friends.\n\nROBYN AND Quince held each other's hands as Morgan opened a gateway from Daia's cavern to the Heart.\n\nRobyn was concerned for the nimfeach's health. His skin looked a bit gray. He was pushing himself hard, and even though he was something more than human, he must have limits.\n\nShe was worried for Quince too. Her lover was too quick to take the burdens of the world upon her own shoulders.\n\nThey had no idea if what they were attempting was even possible, let alone what effects it might have on the human psyche and physiology. \"Are you sure you want to do this?\"\n\nQuince nodded. \"We're part human, part nimfeach, and part ithani. This is our heritage too.\"\n\nRobyn was still a bit unsettled by the newfound knowledge that she carried a bit of alien DNA in her bones, but she put on a brave face and squeezed Quince's hand.\n\n\"It'll be okay.\" Quince kissed her cheek. \"Here we go.\"\n\nThey stepped through the gateway after Robyn, and Jessa followed them.\n\nThe cavern was huge\u2014maybe half a kilometer across. The walls were covered with glowing green moss, and stone benches sat against the walls at regular intervals, cracked with age. Golden globes were suspended above, and in the center a silver dome shimmered with the colors of the rainbow.\n\nRobyn reached out to touch it. It was warm, and as her hand connected, she felt a thrumming that vibrated up her arm. A ripple spread from the point of contact across the Heart.\n\nShe shivered, startled by the realization that she knew this place.\n\nDAIA STOOD before the Heart, her own heart racing, her face reflected and distorted on the liquid surface.\n\nShe was the last one, the only holdout.\n\nThe nimfeach had promised them so much, and in return only wanted access to her mind and soul.\n\nShe had scorned the bargain and had urged her fellow colonists to do the same. It was a bargain with the devil.\n\nAnd yet....\n\nOne by one, they had vanished, drawn off by its promises. They would come in the night, and in the morning another one of her fellow colonists would be gone.\n\nShe'd started calling them sneach instead of nimfeach, because they'd sneak around the camp and lure away the unwary.\n\nNow she was the only one who remained. Even Braice had eventually given in, though he'd lasted longer than almost anyone else.\n\nShe had spent the last three months alone in the village, surviving on the remaining supplies, sampling local fruits in small portions. Some had been edible, though she had no idea of their nutritional value. Others had made her violently ill.\n\nWinter was coming, and she was out of options.\n\nPerhaps she could provide them with guidance... after. Not that they had listened to her before. Maybe she could still make a difference in what happened next to her people.\n\nMaybe it was all a lost cause.\n\nAfter all, who wouldn't want to learn to fly? What was a lifetime of backbreaking work to build a human colony on this wretched world worth, next to such a promise?\n\nMarli, what would you do? Her lover didn't answer. She'd been missing or dead for half a year.\n\nDaia sighed. It was this or a long, slow death.\n\nShe made up her mind and reached out to touch the silver surface of the Heart.\n\nROBYN PULLED her hand away from it, shuddering. So much pain.\n\nSo that's what it feels like. The memories had finally found her, as well. She didn't care for it.\n\nQuince pulled her close. \"I saw it too.\"\n\nDaia's thoughts haunted her. \"We've been pawns in this game for so long.\"\n\n\"What game?\" Jessa was staring at the Heart.\n\n\"The skythane.\" Robyn turned away from the Heart. \"We were like you. Like the landers when we arrived here. The nimfeach took us in and changed us.\"\n\n\"Changed you? How?\"\n\nRobyn extended her wings. \"These, for one. We're part nimfeach, and ithani too.\" Her wings settled against her back.\n\nShe felt sick to her stomach at the thought, suddenly unsteady on her feet. She sat down on a stone bench along one side of the cavern, closing her eyes and trying to calm her racing thoughts.\n\n\"Well, so what?\"\n\nRobyn looked up sharply. \"What?\"\n\nJessa stood before her, hands on her hips, shaking her head. \"You skythane are all such mopey defeatists.\"\n\n\"That's not fair.\"\n\n\"It's true. You're all a bunch of worrywarts.\" She sat down next to Robyn and put a hand on the queen's knee.\n\nRobyn was uncomfortable with such familiarity\u2014they hardly knew each other, and few had dared to treat her so casually at any point in her life. Yet she didn't remove Jessa's hand.\n\nJessa sighed. \"Look, Xander went on and on for days and days about how his love for Jameson couldn't be real, just because it might have started with pith.\"\n\nQuince winced at that one but said nothing.\n\n\"He was wrong. In the end, it's no less real than the love you and Quince have for each other. It doesn't matter how it started.\"\n\n\"I guess so.\" Robyn wondered how Xander and Jameson felt about it all now, but the girl had a point.\n\n\"Now you find out that the skythane might have been made in a petri dish. So what? You all are beautiful and amazing, and you can freaking fly!\" She put out her hands for emphasis. \"I would kill to be a skythane. Seriously, kill. It doesn't matter where you came from, whether it was a quirk of evolution or a scientific miracle. See, this is the thing. None of us have any choice in where we came from. The only thing we can control is what we do with what we've got.\"\n\nQuince laughed. \"She's got you there.\"\n\nRobyn blushed. Put in her place by a lander. And an off-worlder, at that. \"Maybe so.\"\n\n\"And it explains one of the strange things that Mylin found... the divergence of our DNA from the landers' in such a brief period of time.\"\n\nRobyn nodded, though she wasn't completely sure what DNA was. Still, it made sense. The skythane were different.\n\nShe wasn't ready to commit to this new worldview. Not just yet, but Jessa had given her a lot to think about.\n\n\"ARI, GET me Alix.\"\n\n\"One moment please. Connecting.\"\n\nQuince frowned as she waited for the connection. \"Alix, are you there?\"\n\n\"Quince? Holy Split, it's good to hear your voice. Where are you?\"\n\n\"I'm inside the Heart. Robyn, Jessa, and Morgan are with me. We're going to make a copy of the Heart's memories\u2014\"\n\n\"How the hell are you going to manage that?\"\n\n\"I'll explain later. I'm gonna need you to buy me more time. Can you do that?\"\n\nThere was a long pause. \"How much time? Things are pretty hairy up here.\"\n\n\"I don't know.\"\n\n\"Shit. Okay, we'll do what we can. But Quince, make it quick.\"\n\n\"I'll try.\" She cut the connection. \"Okay, let's do this.\" Quince turned to Morgan. \"What do I have to do?\"\n\nMorgan took her hand.\n\nA root from the world mind had snaked through the open gateway after them. Morgan had her put one hand on it, and one on the surface of the Heart. \"Can you sense Daia?\"\n\nQuince nodded. \"She's here.\"\n\n\"Find the Heart. Daia will do the rest.\"\n\nQuince closed her eyes and searched. For a couple seconds, nothing happened.\n\nThen she stiffened, her eyes rolling back in her head, and her whole body shook. The room dissolved, and she grew and stretched, becoming a great river.\n\nThe Heart was waiting for her, as if it had known she would come. It poured itself into her, and Quince was washed away in the flood.\n\nQUINCE WAS shaking, her eyes rolled back in her head. Robyn jumped up. \"Is she okay? What's happening?\"\n\nMorgan nodded, stifling a yawn. \"She's in the middle of the flow. The Heart is transferring its information to Daia.\"\n\nAs if that were a signal, Quince's arms steadied, and her eyes closed.\n\nShe looked peaceful and focused.\n\n\"Come back to me, my love.\" Robyn sat back down with Jessa to wait.\n\n# CHAPTER THIRTY: SHIFT FORWARD\n\nPUCK WAS transformed.\n\nZe fluttered back and forth with zer iridescent wings, full of nervous energy. Zer wings shone brightly in the dim light of the Heart.\n\n\"What's wrong with zer?\" Jameson reached out to the nimfeach, but ze danced away from him.\n\n\"Ze's been restored to zerself for the first time in years.\" Eri's eyes looked sad, but it might have just been zer alien physiology. \"We took zer heart\u2014it's how we were able to control the nimfeach.\"\n\nThe whole Heart shuddered.\n\n\"It begins.\"\n\n\"The shift?\"\n\nZe nodded. \"Under Freyyr, the Thshnel and zis minions have begun the process. Zi will slumber until it's over.\"\n\nJameson looked around. The Heart itself was changing. Electrical impulses raced up and down its surface, arcing blue over the top and splitting in crazy glowing patterns. \"What do we do?\"\n\n\"You two carry the essence of Jirron in your cells. With Puck, you can stop the shift, when the time comes. It comes swiftly.\"\n\nJameson looked around at the strange place they'd finally arrived at. Everything that had happened to him, from his birth onward, had started in this place.\n\nPuck came up to him, wings fluttering as ze looked into his eyes.\n\nZe put out a small hand, and he touched it with his. Even zer body had changed, becoming less shriveled and insect-like, and more like a miniature version of the ithani.\n\nThank you. Zer voice was inside his head.\n\nThese things no longer surprised him. You're welcome. Are you ready for this?\n\nErina had explained what was to come, but he still couldn't quite believe it.\n\nHe and Xander, along with these others, were responsible for the Split\u2014one of the great wonders of the Common Worlds. Or were about to be. Or maybe had been. Time travel made his head hurt.\n\nWhat had happened to Erro had happened already because of them. Was to happen now? He was hopelessly confused about what came first and what would come later.\n\nThey were doing this because they had always done it. That was as close as he could manage.\n\nYes. You have been kind to me. Puck let go of his hand.\n\nJameson thought ze intended to say more. There was a lingering regret on his soul, like an aftertaste. \"Erina, what happens to Puck when all this is done?\"\n\nErina turned away, acting as if ze hadn't heard him.\n\nHe pulled zer back around.\n\nZe hissed at him, zer fur standing on end. \"I'm sorry.\" Ze blinked so hard he was surprised ze could still see him. \"Ze will cease to be.\"\n\n\"FUCKING SPLIT.\" Xander was angry enough to tear the ithani in two. \"We can't do this. Puck just got zer life back.\" I'm so sick of secrets.\n\nJameson looked surprised at his vehemence. \"What choice do we have?\"\n\nXander snarled. \"It's Morgan all over again. When we thought we lost him.\" Xander paced back and forth in front of the Heart. \"There has to be another way.\"\n\n\"It's what I foresaw\u2014\"\n\n\"To hell with all these memories and visions. I'm so sick of them governing my life.\" First the prophecy Quince had dangled in front of them to get them to Titania. Then the Split-cursed statue that Jameson had touched, and the avalanche of memories that had almost buried him. Now this. \"I won't do it.\" He sank down on a stone bench and put his head in his hands.\n\nHe'd already given up so much. His innocence. His first love. His old life.\n\nIt's too much.\n\nSomething bright lit up his knees. He looked up to see Puck hovering before him, zer beautiful wings an iridescent blue and purple, the color radiating from zer compact body. Zer skin was blue too, a brilliant turquoise blue. Zer face was alien, longer and thinner than a human's, but it conveyed a sense of deep contentment and peace. He reached out to cup zer cheek. Ze was cool to his touch.\n\nDon't worry for me. I'll go into the Heart when this is over.\n\nHe sighed. I don't want you to die.\n\nThis is what is supposed to happen. A warm vibration extended from zer into his arm. Ze was purring. This is what will have happened, in your time.\n\nXander closed his eyes. In this short period, he'd come to care for not one but two little beings who had needed his help and protection.\n\nHow long ago that one-night stand back in Oberon City seemed now, the night before this had all begun. How much have I changed?\n\nHe didn't like it, but it seemed everyone else had accepted this. \"If this is all foreseen, then don't I have to go along?\"\n\nJameson shrugged. \"That question's beyond my paygrade.\"\n\nXander snorted in spite of himself. \"Like we're getting paid for any of this.\"\n\nErina watched the byplay between them, zer head cocked, a chuffing sound coming from zer nostril. \"You have a choice.\" Ze held out zer arms. \"We all have a choice. Jirron made zis. You could refuse to follow the vision. It's your right.\"\n\nXander looked at her and then at Jameson. \"What would happen?\"\n\nZe held his gaze. \"I don't know.\"\n\nJameson sat next to him. \"Would you give everything up for me, if you had to?\"\n\n\"In a heartbeat.\" He let go of Puck and hugged Jameson to him tightly.\n\n\"Then you have to let Puck do the same, for us.\"\n\nCheckmate. Xander was outmatched. He sighed heavily, letting Jameson go.\n\n\"Okay.\" He turned to Erina. \"What do we need to do?\"\n\nTHE HEART was seething with activity. Blue currents arced across its surface, and the room's floor trembled constantly.\n\nErina put a hand on the surface of the Heart, seemingly unfazed by the electricity that wrapped up zer arm. \"It's almost time. Xander, Jameson, come here. Stand here, facing each other.\"\n\nJameson glanced nervously at the Heart, wondering if there was any way he could fuck this up. And if he did, what would happen in the future? In his time? \"We don't need a gateway, like last time?\"\n\nZe waved zer hand in negation. \"We're all on the same side of the shift here. Puck will create a gateway, but it will be used to redirect all the energy ze steals from the shift.\"\n\n\"Redirect it where?\" Jameson didn't like the sound of that. It would be a fuck-lot of energy. It would have to be, to shift a world.\n\n\"Into the time stream.\"\n\n\"Nimfeach can do that?\"\n\n\"Only a few. Once or twice in a generation.\"\n\n\"Ah.\" Jameson had no idea what ze meant, but it sounded complicated and potentially deadly.\n\nHe faced Xander, feeling as nervous as a boy asking someone out on a first date. \"Are you ready?\"\n\nXander frowned. \"Not really. Not that it matters. We're going to do this, right?\"\n\nJameson glanced at Puck. \"Yeah. I think so.\"\n\nPuck slipped in between them. They took the nimfeach's hands.\n\nZer skin was cool to Jameson's touch.\n\nReady. Ze squeezed their hands, as if in reassurance. Then ze opened a gateway.\n\nERINA WATCHED it unfold, zer whole being thrumming with anticipation. Ze had bided zer time, made zer plans, and at last they were coming to fruition.\n\nThshnel'Jirron was surely in suspension now, not to be awoken until after the shift.\n\nZe had won.\n\nIt would take time for zer victory to be sealed. Zer heart ached with the amount of time it would take. But when it was over, zer people and the nimfeach would be saved.\n\nIt was almost time, as Erro shifted into a new universe, a new reality.\n\n\"Ready.\"\n\nThe others looked at zer, as ze felt for just the right moment.\n\n\"Now!\"\n\nSomething shifted.\n\nEnergy poured from the Heart into the gateway. It grew in strength, until Erina was forced to step backward, to turn away from the light.\n\nA horrible grinding crash filled the air, like the sound of an asteroid slamming into the earth, or a world being ripped in half.\n\nUnlike a crash, the sound grew and grew, cresting in a crescendo that almost burst zer ears.\n\nIn the midst of the cavern, the gateway shone brighter and brighter as it sucked in the energy from the Heart, until it threatened to engulf the nimfeach and the two brave skythane kings.\n\nPuck knew what to do. Erina had discussed it with zer as they had made their own plans.\n\nNot everyone needs to transcend, tonight.\n\nAbove, the flare was coming. The unstable sun was about to let off a scathing burst of superheated particles and radiation that would swamp this half of Erro. The half ze had doomed to destruction waited for its ending.\n\nErina felt a bittersweet joy. Every ending was a beginning, too. Ze had seen it. A new world would arise from the ashes of the old, a world that with time would become the world of the skythane.\n\nAt last the rumbling died down.\n\nThe gateway, however, remained bright.\n\nIt was done.\n\nJAMESON BLINKED, looking around. The light from the shift had nearly blinded him and had left a nasty afterimage burned into his retina.\n\nJameson held up the key he'd given to Puck, staring at it in wonder. \"Ze's gone.\" It was all that was left of the plucky little creature. \"Is ze... did ze hurt?\" He wiped his eyes with the back of his hand.\n\n\"No. It was quick.\" Erina wiped zer own tears.\n\nJameson sighed, surprised that ze could cry. It seemed the ithani had at least one thing in common with humans. \"Is it... done?\"\n\n\"Yes.\"\n\nXander growled. \"It's done, all right.\" He held out his hand to Jameson. \"May I?\"\n\nJameson handed him the key.\n\nXander sighed. \"All that life and beauty, reduced to this.\"\n\nJameson closed his eyes. \"I'm sorry, Xander. We had to. Even Puck knew it.\"\n\n\"I know. Knowing doesn't make it any easier.\" He stared at the gateway. \"Where does it go?\" he asked Erina.\n\n\"Home. Zer final gift to you.\"\n\nHoly Split. \"Really? We just step through and we're home?\"\n\n\"Yes.\" Erina thrummed. \"Things may have gotten a little crazy since you left.\"\n\n\"I have one more question. How did we find Puck?\" Xander held up the key. \"Out of all the places we could have landed on this godsforsaken world, we ended up where ze was? It seems like far too large a coincidence.\"\n\n\"The key.\" Erina made a waving gesture.\n\nJameson decided to interpret it as \"Gods rest her soul.\"\n\n\"The key belonged to zer, once. It was zer heart.\" Erina held out her hand. \"May I have it?\"\n\nJameson handed it over. \"Is a part of zer still in there?\"\n\n\"Yes.\" Ze held it gently. \"When you touched the key, it was activated to bring you to zer, in that moment in time.\"\n\n\"Ze knew?\"\n\n\"After this, yes. Ze knew.\"\n\nJameson whistled. \"That's some seriously strange time looping.\"\n\n\"Yes, it is. Now go. The gateway won't last for much longer.\"\n\n\"You'll be okay?\"\n\nErina thrummed. \"I survived the end of the world, didn't I?\"\n\nJameson laughed. Ze had a sense of humor. He threw his arms around zer. \"Thank you for saving our world too.\"\n\nZe stiffened but accepted his gesture with good grace.\n\nXander hugged zer too. \"I'll miss you most of all, scarecrow.\"\n\nZe cocked her head.\n\n\"Old tri-dee.\"\n\n\"That makes no sense to me either.\" Ze blinked a few times, then shimmered. \"Now go! You still have work to do. In your own time, you must force Thshnel'Jirron into the Heart, alone.\"\n\n\"Easier said than done.\" Jameson took Xander's hand. He took one last look around the Heart, and then pulled Xander through with him.\n\nTheir friends needed them.\n\nERINA WATCHED them go.\n\nWhen the gateway winked out, ze closed zer eyes for a moment, wishing the two travelers luck on their great leap forward.\n\nThen ze went to the Heart and held the little nimfeach heart, round and milky-white, up to the surface.\n\nIt rested there for a moment, then dissolved and flowed inside. Soon the surface was smooth and calm once more. \"Your time will come again, little elder.\" Ze put a palm to the surface, conveying zer good will to the nimfeach's soul. \"As will mine.\"\n\nZe took a deep breath and put both of zer hands in the Heart. Then ze opened zerself up to it.\n\nIt reacted to zer, flowing up zer arms, and then over zer chest and up and down the length of zer body.\n\nZe folded her wings, closed zer eyes, finally at peace.\n\nAs the destruction of half of Erro commenced above zer, the silver shape that had been Erina flowed back into the Heart and disappeared.\n\n# CHAPTER THIRTY-ONE: STORM BATTLE\n\nTHSHNEL'JIRRON WATCHED zis people battle the humans.\n\nThe Heart was calling to zim, almost ready to thrust zim into godhood.\n\nThe storm clouds that had swirled around the perimeter of the site had closed in, and lightning flared through the dark clouds, occasionally striking friend or foe.\n\nThe ithani were gaining the upper hand. Zis side had the numbers, and the enemy had been worn down by the demands of the shift and the warfare among themselves.\n\nDani's memories had taught zim much about this race. They had been at each other's throats for most of their history, and the last year had been no exception.\n\nThey were weak and divided.\n\nDani screamed in pain as her wings were cut off in the cattorah, the penalty for her crimes against the Erriani.\n\nJirron put zis hands to his head, willing away the memory, the terrible pain. These flashes were coming more frequently, and zi was surprised at both their ferocity and zis own difficulty suppressing them. Never before had zi experienced such trouble with a new host.\n\nZi drew on the strength of the Heart, pushing the memory back down.\n\nSoon it wouldn't matter. Soon zi would be rid of this body, done with the need to ever take physical form again.\n\nThe storm raged all around zim, but zi saw through the clouds as though they were empty air, directing zis forces to tackle the enemy where they were weakest.\n\nThe battle continued, but it wouldn't be long now.\n\nVENIN FELL on another of the ithani from above, grabbing its furred head in both hands from behind and twisting. Its head separated from its body with a sickening crunch, and then he dropped the corpse to the ground far below.\n\nVisibility had faded to less than twenty meters in any direction.\n\nAlix had ordered his hoversports down to the Heart itself, where they were circling and taking out anything that approached.\n\nSo far the ithani hadn't figured out how to take out the armored transports other than by piling on them en masse. Three had already been downed that way to fiery explosive deaths. Only half a dozen of the craft remained.\n\nVenin's heart ached for the casualties, but he had no choice but to fight on.\n\nA human cry pierced the storm nearby.\n\n\"Alix, we can't take this much longer.\" They had an open channel between them.\n\n\"We're waiting for Quince.\" He could hear the pain in Alix's voice too.\n\n\"A few more minutes and this will be over\u2014\" A bolt of lightning seared the air less than ten meters in front of him. \"Holy Split!\"\n\n\"You okay?\" Alix's voice sounded anxious.\n\n\"Yeah. Just a close call. Tell Quince we can't hold out much longer. We're getting slaughtered up here.\"\n\n\"I know. If she doesn't call in five minutes.... What the hell?\"\n\n\"What?\" Venin turned and saw what had caught Alix's eye.\n\nAn area of calm was extending around the Heart, pushing up into the atmosphere as the storm clouds drew back.\n\nA golden light shone on top of the Heart like a miniature star, and then two figures emerged.\n\nThe light winked out.\n\nA second later, a familiar voice spoke in his ear. \"We're back\u2014anyone miss us?\"\n\n\"Xander?\"\n\n\"In the flesh.\"\n\nMYLIN SLIPPED in and out of vir like a dreamstate.\n\nAt one point there was a sharp pain as something hot ran across her midsection.\n\nShe vaguely remembered getting shot, Derren's concerned face hovering over hers, his hand grasping hers as she was rushed to the OberCorp infirmary. Mouthing \"You're gonna be okay.\"\n\nThen she was in vir with DOC.\n\n\"What's wrong with me?\" She felt disconnected from her body, lightheaded.\n\nHe sat next to her and took her hand. \"You were hurt. Someone shot you, and you were injured rather badly.\"\n\n\"How badly?\"\n\nHe looked away.\n\n\"DOC, how badly?\" She wanted to know. Needed to know.\n\n\"It's... not good. The medics say your chances are about thirty-seventy. There was a lot of burn damage.\"\n\nThirty-seventy. She nodded. \"What did you find in the archives?\"\n\nHe stared at her for a moment. \"Maybe you'd like to talk about something else? If... just in case?\"\n\nShe shook her head. \"I want to keep busy. I don't want to think about that.\" She closed her eyes. I'm not ready to die. \"They're depending on me. What did you find?\"\n\n\"Are you sure?\" He looked really worried for her.\n\nFor just a moment, she forgot he was just code in the OberCorp core. She threw her arms around him, and he hugged her tight.\n\nVir space swam a little around her, but this was important. She could hold it together for a couple minutes more. \"I'm sure,\" she whispered. She let him go.\n\n\"Okay. There are files going back to the founding of OberCorp that don't match my current database. Mostly having to do with the split and certain places on Oberon.\"\n\n\"The House of the Sun.\"\n\n\"Yes, that's one.\"\n\nMylin struggled to think clearly. \"Give me the earliest files. What was the first thing they tried to hide when they accessed the network?\"\n\nDOC grinned. \"Thought you might want those.\" He tugged on his thick mustache and splashed an array of files up on the wall.\n\nMylin scanned them, mentally pinching herself to stay awake. \"These are the initial ZephyrCorp surveys of Oberon.\"\n\nHe nodded.\n\n\"And this place. I recognize this. It's where the Heart is. What are these designations?\" She pointed to certain notations on the map.\n\n\"Here.\" He slitted her the information.\n\nAs she looked at the survey map with new eyes, her jaw dropped. \"Heaven and Erro. Is this right?\"\n\n\"Yes.\"\n\nThere was a massive amalite deposit just below the surface under the House of the Sun. Where the Heart was now. \"They must have known we'd find the Heart if we tried to excavate that deposit.\"\n\nHe nodded. \"That was my thought too.\"\n\n\"If they drop those bombs on the Heart on top of that much amalite, it could blow the whole planetary core.\"\n\n\"It's possible.\"\n\n\"Get me Alix.\" The room was spinning. If she was this off in vir....\n\nDOC cocked his head. \"I'm sorry. I can't reach Alix or Venin or anyone in the battle zone. Something has changed.\"\n\n\"We have to find a way to stop them\u2014\" She was seized by an intense wave of mind-numbing pain, and for a moment the world went black.\n\nWhen she opened her eyes, she was on the operating table.\n\n\"She's awake!\" A face looked down at her through a plas mask. \"Don't worry, we've got you. I'm going to put you under. You won't feel a thing.\" The medic put a hand on her shoulder. \"You'll be okay.\"\n\n\"No, don't put me to sleep! I have to tell them\u2014\"\n\nShe felt a slight pressure at her neck, and then everything went fuzzy and gray.\n\nQUINCE WAS lost in the flow. So much history was stored in the Heart. As the conduit, she caught glimpses here and there, enough to get a sense of how long and deep the story of the ithani was.\n\nShe saw the hive mind that had produced them, when they had been little more than drones. How they had eventually become imbued with intelligence and independence.\n\nThe growing of the Heart, the hive of the colony.\n\nThe worker bees and the teachers, becoming the ithani and the nimfeach.\n\nSo much time.\n\nShe had a sense, now, of how Daia felt, having been the repository of so many others' memories.\n\nShe was aware that she sampled only the barest taste of the information flowing through her, the heritage of a race.\n\nSomewhere out there, outside of her, her friends were fighting and maybe even dying, while she was fixed in place here, lost in this limbo between real time and an alien mind she could barely comprehend.\n\nShe had no sense of how much time had passed. It might have been thirty seconds or a hundred hours.\n\nAll she could do was hold on and wait for it to end.\n\nJAMESON FELT stronger than he ever had. Power flowed through him from below, replenishing his energy and chasing away his fatigue.\n\nErina had been right. Connecting with the Heart had been easy, once ze had shown him how. He and Xander had been bred for it, after all.\n\nThere was a taint on it, though, something almost sickly sweet. Tshnel'Jirron had long since claimed it as zis own, the birthright of zis people.\n\nThat would end today.\n\nAre you ready?\n\nFor his answer, Xander kissed him hard. \"Let's go.\"\n\nThey were one again joined through the Heart's athr\u00e0, a bond closer than any Jameson had ever known.\n\nWith a single mind, they leapt into the air to confront the ithani.\n\nVenin met them with a grim smile. \"Welcome to the battlefield.\"\n\nJameson slammed into him with a bear hug, sending both tumbling. \"We're so happy to see you,\" Jameson said when he let go and they recovered.\n\n\"We're?\"\n\n\"Long story.\"\n\n\"Where's Jirron?\"\n\nVenin's eyes widened. \"She's up there.\"\n\n\"She?\" They looked up.\n\nDani Black descended from the sky, her face clouded with anger.\n\nJIRRON WAS furious.\n\nSomething or someone had wrested control of the Heart from zim. Everything zi'd built\u2014everything zi'd worked for over a thousand cycles\u2014was in jeopardy. Damn these vermin humans.\n\nRunning through a field of flowers, her mother chasing her. \"Dani! Get back here. It's time for dinner!\" Dani's mother's shadow flew past her, and she giggled as she ran into her waiting arms.\n\nJirron hissed and thrust the memories down again.\n\nThe newcomers waited for zim, flapping their wings and glaring at zim. There was bad blood between these two and zis host body. Maybe zi could exploit that.\n\n\"The kings of prophecy.\" Zi leered. Over time, zis control of this host had grown and had begun to feel more natural. \"Xander, have you reconsidered my proposition?\"\n\nThe two sneered in unison. A bonded pair. \"We know what you are.\"\n\nWhen the skythane man spoke, Jirron felt the echo through the Heart. Zis eyes narrowed. These two were linked in the athr\u00e0.\n\nThey had found a way to access the Heart, but they were still new to it, and zi had been controlling it for much longer than they had been alive.\n\n\"No!\" Dani screamed. Her father's craft plunged into the ocean, and seconds later a great explosion rocked the sea....\n\nJirron blinked in displeasure. Habits of a lifetime were hard to dispel.\n\n\"The fight is over,\" they said in unison.\n\nZi drew out the moment while zi searched for a way to cut them off from the Heart. \"What do you propose?\"\n\nThey looked at each other in surprise, and then one of them laid out the terms of surrender.\n\nThe Thshnel'Jirron listened absently, nodding every so often as zi set about undermining the control the two skythane kings had over zis Heart.\n\nQUINCE GASPED.\n\nRobyn leapt up off the bench to catch her as she collapsed, lowering her carefully to the ground.\n\nQuince's eyes flickered open. \"It's done.\" Her skin was gray and cold.\n\nShe looked to Morgan. He knew how to heal people. Quince had told her about it. Unfortunately, he looked to be in worse shape than Quince.\n\n\"Take me back to the other side.\" Quince's voice was faint, but her intent was clear.\n\nRobyn nodded. \"Jessa, bring Morgan.\"\n\n\"Will do.\"\n\nTogether they scooped up their wounded and carried them from the battlefield to the gateway and through to the other side. Robyn leaned over and kissed Quince on the forehead. \"Daia?\" She looked up and the world mind was there.\n\n\"Yes.\"\n\n\"Did it work?\" Daia seemed to shimmer and fade, but then reappeared. \"Are you okay?\"\n\nDaia nodded. \"I'm having some trouble integrating all the new information. But yes. It worked. Is Quince okay?\"\n\n\"I don't know.\" She cradled Quince's head in her lap.\n\nJessa held Morgan. He looked like a little boy again, all guile gone, sleeping like an angel.\n\n\"Quince?\" She put her cheek over Quince's mouth. She was still breathing. \"I really don't know.\"\n\nMYLIN SWAM in a sea of anesthesia and anxiety.\n\nThere was something she had to do. Something she knew she had to tell someone. The pain in her abdomen had dulled to a throbbing background ache.\n\nHer heart beat steadily.\n\nThe Heart.\n\nMylin.\n\nShe stirred. A woman was standing over her, dark-haired. A lander. Beautiful eyes. Sad eyes.\n\nMylin tried to speak. The Heart. They can't....\n\nThe women knelt next to her and cupped her cheek. Tell me.\n\nMylin felt a gentle pressure and let herself open to the woman's touch. Her anxiety flowed out of her like air out of her lungs.\n\nWhen it was done, she reached out to the stranger. You have to tell them.\n\nThe woman knelt over her and kissed her forehead. I will. Sleep. All will be well. All manner of things will be well.\n\nMylin nodded. Sleep. That sounded so good. She stirred again. Who are you?\n\nDaia.\n\nThen she was gone.\n\nJAMESON AND Xander faced their old nemesis. It wore Dani's face, but it had an alien soul. It was strange seeing the ithani inhabiting Dani's body.\n\nJameson was laying out the terms of surrender\u2014a delaying tactic while Xander contacted Alix.\n\n\"Alix?\" They called their friend through Jameson's PA.\n\n\"Hey, good to see you two. We'd all but given you up for dead.\"\n\n\"Yeah, us too.\"\n\n\"You... is this both of you together?\"\n\n\"Yes. We'll explain later. Still have that carrier full of meso bombs on tap?\"\n\n\"Yes.\"\n\n\"Use it. We have to destroy the Heart.\"\n\n\"Affirmative. Launch in sixty seconds.\"\n\nThey returned their attention to Jirron. \"It's over\u2014\"\n\nJirron cut him off. \"I remember you. Before the shift.\" Zis eyes narrowed. \"Erina found you and brought you to the Heart.... How is that possible?\"\n\n\"We get around.\" Xander and Jameson smiled at the same time.\n\nJirron didn't react.\n\nSomething was wrong. The Thshnel didn't look worried. In fact, zi was grinning, an unnerving ear to ear grin.\n\nJameson suddenly felt sick. The source of his power, the Heart, was turned against him.\n\n\"It's not so easy controlling the Heart, is it?\"\n\nJameson couldn't feel Xander through the bond anymore. Instead, he felt nauseous, the taint from the Heart spreading into his mind.\n\nJirron's grin stretched even wider.\n\n\"ROBYN.\"\n\nRobyn blinked and looked up. Must have fallen asleep. Quince's head still rested in her lap, her color back to normal.\n\nDaia was looking down at her.\n\n\"What is it?\"\n\n\"Mylin. She's hurt.\"\n\n\"Is she okay?\" Quince's eyes opened. Her voice was raspy.\n\nDaia shook her head. \"I don't know, but she sent you a message.\"\n\n\"Hey, you need to sleep.\" Robyn kissed Quince's forehead. \"You taxed yourself to the limit.\"\n\n\"It's okay. This is important.\" She struggled to sit up. \"I told her to contact me if she found anything in the archives.\"\n\nRobyn sighed and helped her up. \"You're going to drive me to an early grave.\"\n\nQuince kissed her cheek. \"What did she say?\"\n\n\"There's a bomb on a ship.\"\n\nQuince nodded weakly. \"Yes.\"\n\n\"If it goes off over the Heart, it might set off a chain reaction in the amalite underneath that could destroy Erro.\"\n\nQuince grunted. \"Robyn, help me up.\"\n\n\"Quince, are you sure?\"\n\n\"I need to contact Alix, or Venin.\"\n\nRobyn helped Quince stand.\n\nJessa had been watching the whole exchange. \"Daia's back?\"\n\nRobyn bit her lip. \"Yes. She says we have to stop Alix's plan to bomb the Heart. It could start a... a chain reaction?\"\n\nJessa blanched. That would be bad. \"What can I do?\"\n\nQuince pointed at the still-open gateway. \"Take me back to the Heart and bring my carry sack.\"\n\nThe girl looked worried. \"Quince, are you sure you can manage this?\"\n\nQuince nodded. \"Have to.\"\n\n\"Okay.\" Jessa laid Morgan down, using her own carry sack to make a pillow for him, and helped Robyn get Quince to her feet.\n\nTogether, they helped her limp to the Heart.\n\nVENIN WATCHED in shock as Jameson's self-assurance crumbled, and Xander's face turned from serene strength to abject fear.\n\n\"Venin, you have to get everyone out of there.\" Alix's voice rang through his head. \"The carrier is inbound.\" His voice was calm, but there was an underlying tension that belied it.\n\nXander and Jameson both looked sick, really sick, yet somehow their wings continued to beat the air in a slow, steady rhythm. They looked like two flies trapped in amber.\n\n\"Venin?\"\n\nJirron had gotten to them, somehow. \"The Heart is mine.\"\n\nThe Heart is mine.\n\nThis was his moment. Suddenly it was all clear.\n\n\"Dani!\" he shouted, and threw himself at her, his wings driving him into the ithani with the power of a freighter.\n\nJirron turned toward him and tried to spring away, but she was too slow.\n\nVenin rammed into her, knocking her backward through the sky, and reached inside.\n\nJirron's mind was full of twisting currents, but there were whispers of Dani there. Dani Black! he shouted inside Jirron's mind.\n\nDani. I'm Dani. The voice was faint but unmistakable.\n\nJirron reacted furiously, trying to push Venin out.\n\nVenin pushed back. Dani!\n\nAnd suddenly she was there.\n\nJIRRON GRINNED. Zi'd taken back control of the Heart. Zis enemies were vanquished, and their plan, which had apparently been seeded by the traitor Erina a hundred thousand cycles before, had come to naught. \"The Heart is mine.\"\n\nZi twisted the kings' minds and delighted in seeing the looks of pain on their faces. There was so much there to work with\u2014betrayal, doubt, an old lover. Zi'd always enjoyed finding zis enemies' weaknesses and pushing them hard.\n\n\"Dani!\"\n\nJirron turned in time to see one of the skythane hurling himself at zim\u2014the one who had first challenged him. Zi tried to back away, but zi couldn't avoid the impact.\n\nBlades slipped smoothly from between the knuckles of Jirron's hands\u2014\n\nThen the man was inside zis head.\n\nDani Black! The voice reverberated through zis mind.\n\nDani. I'm Dani.\n\nNo. Not now. Not when zi was so close. Zi pushed at the interloper, drawing on the Heart for strength.\n\nDani!\n\nZis mind was squeezed as Dani's personality reasserted itself with a vengeance.\n\nZi fought to retrench. This isn't possible. None of zis host minds had ever managed to come back to challenge zim.\n\nNone of them had been human.\n\nZi pushed back hard but was unprepared for the ferocity with which she attacked zim. She slashed at zim in her mind again and again, tearing at zim like an auxen's horns.\n\nMeanwhile, the skythane man had zim locked in a death embrace as they plunged toward the ground.\n\nDani backed away and snarled at zim. Get the fuck out of my head.\n\nSoon they'd hit the ground, and if zis host body died, zi would too.\n\nWith a snarl of zis own, Jirron gave up and left zis host, fleeing to the Heart.\n\nZe'd be safe there while ze plotted zis next step.\n\nQUINCE SHOWED Jessa and Robyn where to plant the nano traps she'd found in the OberCorp armory. They did so hurriedly.\n\nShe hoped they would work on the material of the Heart.\n\nThe weapons systems were banned across the Common Worlds because the underlying nanotech had the potential to get out of control, eating everything in sight if the shutoff protocol failed. It had happened on a small moon in the Eridinai system, and they'd had to bomb the whole place out of existence.\n\nUsing them was a great risk. Quince wondered where in the hell OberCorp had obtained them. Maybe they'd been in storage for a hundred years.\n\nWhen you had a nail, whatever was at hand became your hammer.\n\nShe'd set them to self-destruct after two hours, long enough to devour the Heart. If they malfunctioned, things would get very interesting very fast.\n\nWhile they were placing the traps, Quince took a deep breath and touched the Heart. It rippled under her hand.\n\nThe connection she'd found earlier was still there.\n\nSo were Jameson and Xander. She could still reach them, even if they were cut off from the Heart.\n\nJIRRON WAS gone.\n\nVenin could feel the change.\n\nHe let go of Dani Black, his wings spreading to stop his fall fifty meters above the ground.\n\nShe did the same, and they circled each other warily. \"You saved me.\"\n\nHe spat. \"Not something I'm proud of. Call it collateral damage.\"\n\nShe was white as a sheet.\n\n\"What happened to Kadin?\"\n\nShe turned away. \"That monster. He killed him, once he sucked everything out of him.\"\n\n\"I'm... sorry.\"\n\n\"Thank you.\" She looked broken.\n\nHe almost felt sorry for her. What she must have endured, with Jirron rummaging around inside her head.... He shivered. She had done terrible things, but she'd also lived through something more horrible than he could imagine. A raping of the soul.\n\n\"So what happens now?\" She watched him warily.\n\nShe had suffered enough. \"We have a war to win. I wouldn't be surprised if a few of the enemy manage to slip through our grasp.\"\n\nShe stared at him for a moment, then nodded. \"I won't forget this.\"\n\n\"I don't forgive you for what you did.\"\n\n\"Understood.\"\n\n\"Venin?\" Alix's voice rang through his head.\n\n\"Good luck to you.\" He turned away, spreading his wings to catch an updraft to rejoin his friends in the skies.\n\nWhen he looked back down a moment later, she was gone.\n\nJAMESON BLINKED, looking around. \"What the hell just happened?\"\n\nXander shook his head. \"Not sure. Jirron got to us through the Heart, I think. Then zi cut us off. Look!\"\n\nJameson followed his gaze down. Venin and Jirron were plunging to the ground. Or Venin and Dani?\n\n\"Holy crap. We have to\u2014\"\n\nThen the antagonists separated and leveled out.\n\n\"Jameson, what's going on out there?\" Alix's voice shouted in his head.\n\nJameson ripped his gaze away from Venin and Dani to look around. \"Jirron's gone, I think. Into the Heart.\"\n\n\"Then clear out. I have that Rentz Class Carrier coming in hot in about ninety seconds.\"\n\n\"Everyone, clear the scene! Bomber coming in hot!\" He and Xander shot away from the Heart, toward the cloud wall.\n\nThe other skythane fled too.\n\nJameson.\n\nQuince? She was in his head, but not like Alix. Her voice sounded weak. Where are you?\n\nAt the Heart. Tell Alix to call off the bombs. He'll set off a chain reaction that we won't be able to stop.\n\nHow\u2014\n\nNo time.\n\n\"Angie, get me Alix.\"\n\n\"Connected.\"\n\n\"Alix, you have to abort the mission.\" Thank the gods.\n\n\"What?\" Alix's voice was weak and tinny. \"You're cutting out.\"\n\n\"Alix, abort!\"\n\nThere was no reply.\n\nThe carrier burst through the cloud wall, hurtling toward the Heart.\n\n\"Angie, get Alix back. I need him now!\"\n\nROBYN SET Quince down gently with her back against the wall in Daia's cavern.\n\nJessa shook Morgan awake. \"Close the gateway, Morgan!\"\n\nThe boy looked up, groggy, and then his eyes focused.\n\n\"You have to close the gateway.\"\n\nRobyn looked back through it. There was a chuffing sound, and something black swarmed over the liquid surface of the Heart. It began to sag and collapse as the black wave spread over it like a dark tide.\n\nMorgan twisted his hand, and the gateway snapped shut, locking the nanites on the far side.\n\n\"It's done.\" Robyn sank down next to Quince and took her in her arms.\n\nThe carrier would be diverted, or the end would come here soon. Either way, it was out of her hands.\n\nShe wrapped her arms around Quince and kissed her cheek. \"I'll love you until the end of the world.\"\n\nQuince smiled weakly. \"Looking for a short-term commitment, huh?\"\n\nRobyn squeezed her tight.\n\n\"VENIN!\" ALIX cursed. The communications out here were shit.\n\nJameson. He had a PA.\n\n\"Jameson, what's going on out there?\" He waited breathlessly for a reply.\n\n\"Jirron's gone, I think. Into the Heart.\"\n\n\"Then clear out. I have that Rentz Class Carrier coming in hot in about ninety seconds.\"\n\n\"Everyone, clear the scene! Bomber coming in hot!\"\n\nAlix sighed and sat back. Thank the Split, it's over.\n\n\"Alix, you have to\u2014\" Jameson's voice dropped off.\n\nShit. \"What? You're cutting out.\"\n\n\"...abort!\"\n\n\"Jameson! I can't hear you!\" Alix's mind raced. He checked the carrier's trajectory. Thirty seconds.\n\nAbort.\n\nIf he was wrong... if he misheard....\n\nThe existence of Erro hung in the balance.\n\n\"Goddammit!\"\n\nAlia put a hand on his shoulder. \"What?\"\n\n\"We have to abort. Erissa, abort the mission.\"\n\n\"Authorization code?\"\n\nHe entered his code.\n\n\"Authorization accepted.\"\n\n\"What happened?\" Alia squeezed his arm.\n\n\"I honestly don't know. Jameson told me to abort the mission, but he was cut off before he could tell me why.\"\n\nXANDER TOOK Jameson's hand. \"Erina couldn't foresee everything.\"\n\nJameson's face was glum. \"I guess not.\"\n\n\"If it really is the end of the world, I'm glad I'm here with you.\" They'd come a long way together, crossing Erro twice. Crossing time too.\n\n\"Me too.\" Jameson squeezed his hand.\n\nThe carrier plunged toward the Heart.\n\n\"Maybe we'll see each other in the next life\u2014\"\n\n\"Look!\"\n\nThe carrier had fired its jets at the last moment. It shot off away from the Heart, slamming into the ground ten kilometers east. There was a bright flash as the meso bombs went off, and the ground was fused in a wide circle.\n\nThen the carrier was gone.\n\nBelow, the Heart collapsed, covered by a dark swarm.\n\nJIRRON SLIPPED into the Heart. Zi was seething with anger. All zis hard work had come to nothing.\n\nStill, zi wasn't finished. Zi had the Heart. Ze could regroup, strike out at them again. This was zis domain, not theirs. Zi owned the athr\u00e0. Zi could still\u2014\n\nIt's over.\n\nZi spun to find a pair of ithani eyes staring at zim. Who are you?\n\nAnother pair of eyes appeared. Then another. It's over.\n\nIt's not over. This is my place.\n\nMore eyes. In chorus, they spoke to zim. It's over.\n\nSoon they entirely surrounded zim.\n\nThe owner of the first pair stepped forward from the mist, zer golden carapace shining.\n\nErina.\n\nHello, Jirron.\n\nOthers appeared. Zi recognized each of zis former hosts, one after the other.\n\nIt wasn't possible.\n\nThe Heart is dying, and you're going to die with it. Erina shimmered with joy.\n\nIt can't be.\n\nThe athr\u00e0 dissolved around zim. What's happening?\n\nThen zis own spirit began to corrode.\n\nZi screamed as zis soul was torn into a million pieces.\n\n\"VENIN, WHERE are you?\" Alix scanned the sky as the remaining ithani and skythane broke off from one another.\n\nAlia looked over his shoulder. \"Down there!\"\n\nA group of skythane had set down on a hillside.\n\n\"Take us down there.\"\n\nThe pilot nodded.\n\nThe ground came up quickly. He should be finding a way to talk to the ithani. He should be assessing the damage and figuring out next steps.\n\nNone of those things mattered.\n\nThe hoversport set down, and Alix was out of the hatch before it had finished opening. \"Venin!\"\n\nAll the skythane turned toward him.\n\nVenin emerged from the crowd.\n\nIn three huge steps Alix was at his side, sweeping Venin into his arms.\n\nVenin's wings enfolded him, and they kissed like lovers separated by a thousand years.\n\nMORGAN STARED morosely at where the gateway had been. He'd failed his people.\n\nThey had destroyed the Heart, and the Thshnel'Jirron with it. They'd saved the ithani and nimfeach memories, but there was no seed.\n\nWithout the seed, there would be no new Heart. Without the Heart, there would be no more nimfeach, and the ithani would eventually die.\n\nI failed. We saved the world, but at what cost?\n\nQuince's hand touched his shoulder.\n\nHe looked up at her, and she held something up to him.\n\nHe took it and turned it over in his hands. It was about fifteen centemeters wide, a perfect sphere, with colors that swirled across its surface like a rainbow.\n\nIt was a seed for the Heart.\n\nMorgan threw his arm around Quince and cried.\n\n# CHAPTER THIRTY-TWO: AFTERMATH\n\nTHE SURVIVORS gathered on a hillside overlooking the devastation.\n\nJameson, his arm around Xander, stared down on the sunken valley that had once been the heart of ithani culture. He'd seen it in its decline, but now it was nothing but a pit of despair. The cavern roof had sagged, the white towers were dead or dying, and the Heart itself had been reduced to a pile of slag by the nanites Quince had used against it. They had done their damage and had apparently expired on schedule.\n\nHe sighed. \"What will the ithani do now, without a leader? Without a home?\"\n\nXander turned to stare at him. \"We just fought a war against them, and you're worried about what will happen to them next?\"\n\nJameson nodded. \"What will happen to us all. We have to learn to share this place, or none of us will survive.\"\n\nXander grunted. \"I guess you're right.\" He stared at the ruin. \"So many skythane and lander lives lost.\"\n\nThe efforts had begun to collect the dead. Alix had agreed to a mass ceremony, lander and skythane together. It was a small step toward unity.\n\nJameson separated from Xander, turning to look west. \"We need to deal with Rogan.\"\n\n\"Yes. You haven't developed a soft spot for him too, have you?\" Xander's tone was deceptively light, his look deadly serious.\n\n\"No. The man's a monster. More so than most of these ithani.\"\n\nXander scowled. \"You'll get no argument from me.\"\n\n\"Do you think Jirron's really gone?\" It had all happened so quickly.\n\n\"Quince says zi is. Destroyed along with the Heart.\"\n\nA gateway split the air behind them.\n\nQuince, Robyn, Jessa, and Morgan stepped through, and it snapped shut behind them.\n\nJameson grinned. \"Well speak of the devil.\"\n\n\"Holy Split, you're here!\" Quince ran up to Jameson, wrapping the two of them in her arms and wings. \"I am so, so sorry about what I did to you. To both you and Xander. It was wrong. I see that now.\"\n\nHe held her out at arm's length. \"It's okay, Quince. We worked it out. I forgive you.\"\n\nXander was silent.\n\nJameson nudged him. \"Come on. You too.\"\n\n\"I do. Really I do. It's just going to take me some time to get past it. You really hurt us, Quince.\"\n\nShe looked down. \"I know.\"\n\n\"I'll get there. Just give me some time.\"\n\n\"Okay.\" She kissed them both on the cheeks. Then the strongest woman Jameson knew began to cry. She hugged them again fiercely. \"Where did you go? If you ever do that to me again....\"\n\n\"It's a long story. Let's just say we went back to the beginning and leave it at that for now.\"\n\n\"Back to the beginning... by heaven and Erro, I'm going to get the whole story out of you soon.\"\n\nXANDER FELT a hand on his shoulder.\n\nHe turned to find Robyn standing there.\n\nThey stared at each other for a moment. She was a proud woman, even now maintaining her regal bearing in the face of what must have been crushing exhaustion. \"I'm glad you're safe.\"\n\nXander made the first move, pulling her close to him. She was stiff in his arms. \"I love you,\" he whispered.\n\nShe melted, squeezing him tight. \"I dreamed about you so many times, wondering if you were okay. What you were like. How your life was going.\"\n\n\"I dreamed about you too. I didn't know who you were. Just a beautiful angel.\" It was time for a new beginning between the two of them. He had to let go of his bitterness, toward Quince, toward life.\n\nRobyn held him for a long time. \"I thought I'd lost you.\" When they separated, her cheeks were wet, her composure lost.\n\nXander liked her better that way. She was human after all. \"I'm okay. We both are. We'll tell you more later, when there's time.\"\n\nShe squeezed his hands in agreement.\n\nThen he scooped up Morgan and laughed as he held the boy in the air. \"Heaven and Erro, it's good to see you, little man.\"\n\nMORGAN FELT uncomfortable, lifted into the air like a child.\n\nXander laughed, seeing his discomfort, and set him down.\n\nMorgan nodded. \"It's good to see you too, Xander.\"\n\nXander frowned.\n\nMorgan wasn't giving Xander what he wanted, but he wasn't sure what to do.\n\n\"You knew, didn't you. When you gave Quince the key. You knew it would take us back to the ithani. To the first shift.\"\n\nThe first shift.\n\nThe last veil lifted from Morgan's memory.\n\nA hundred thousand years ago or more, he'd been born into this world for the first time. In that first lifetime, he'd met two strangers\u2014visitors from another time. Memories resurfaced, painful and joyful.\n\nLosing his key when Jirron had stripped it from him.\n\nFinding it again in the hands of a human stranger.\n\nStopping the shift and freezing his enemies in time.\n\nAnd this one.\n\nI am an old soul.\n\nHe reached out to touch Xander's face, because touch facilitated such sharing.\n\nHe closed his eyes and passed the memory to Xander.\n\nXANDER WAS looking at the Heart.\n\nOne of the amber chambers split open, and a nimfeach crawled out, wet with the chamber's fluids. It spread its wings tentatively and sniffed the air.\n\nA golden ithani appeared in front of the nimfeach, holding out a milky-white key and a silver one that swirled like mercury. Ze knelt before him and handed him the silver one first. He touched it and it evaporated, drawn into the nimfeach just like the one Jameson had given Puck.\n\nXander's breath caught.\n\nThe second key vanished like the first, and the little nimfeach glowed, his wings radiant.\n\nErina touched his cheek. \"Though your form is different now, little Puck, you still have a role to play.\" Zer golden fur shimmered. \"He will call you Morgan. I have foreseen it. You will forget this now, but you will remember what you must do, when the time is right.\"\n\nTHE MEMORY dissolved, and Xander opened his eyes, blinking back tears. \"You... you're Puck?\"\n\nMorgan nodded. He threw his arms around Xander's neck and hugged him tightly. Xander held him for the longest time. Things had come full circle in a way he'd never expected. That day\u2014so long ago, but just hours in his own past\u2014he'd watched Puck vanish. It had cut him to the quick.\n\nAnd now\u2014\n\nJameson tapped him on the shoulder.\n\n\"Xander, look!\" He held up a key\u2014although it was far larger than any key Xander had ever seen.\n\n\"What is it?\" He let go of Morgan and touched the surface. It was warm.\n\n\"It's a seed. For a new Heart.\"\n\nFull circle. \"Jameson, I want you to meet someone.\" He held his hand out to Morgan.\n\n\"I know Morgan already. You sure your head didn't get scrambled when we went through that last gateway?\"\n\nXander shook his head. \"Meet Puck.\"\n\nQUINCE RETURNED with the remainder of the Hunters to Oberon City.\n\nMorgan stayed behind to talk to the ithani. Now that Jirron was dead and gone, many of those who had opposed him silently were doing so openly. A new day was coming for the two intertwined races, and Quince was sure that Morgan would guide them there.\n\nAlix provided the transportation, giving the weary fighters a chance to rest and swap stories. \"Mylin is okay. She came through surgery well and is in recovery.\"\n\nQuince sighed along with the rest of the cabin. The girl did good.\n\nJameson had been telling the story of what happened after he and Xander disappeared so abruptly from the conference room.\n\n\"We figured out that it was a gateway, but we wondered how far you'd gone.\" Quince was astonished at the tale Jameson and Xander spun. To finally answer the greatest question of the split worlds\u2014not in her wildest dreams would she have imagined such a story.\n\nJameson nodded. \"We weren't sure ourselves where we had landed, at first.\" His eyes narrowed. \"You have a story to tell us too, don't you? What did you find under Oberon City?\"\n\nQuince laughed. It felt good. Sure, they still had troubles to face, but the immediate threat was behind them. \"What I found will require a hell of a lot more showing than telling. Let's just say that while you were busy solving the mystery of the Split, we were off discovering the origin of the skythane.\"\n\nVenin leaned forward. \"On that note, I found this in one of the underground caverns.\" He held out a tarnished silver cross.\n\nAn electric jolt went through Quince. \"Can I see it?\"\n\n\"Of course.\" He took it off and handed it to her.\n\nAs she touched it, a memory flashed through her head. \"It belonged to her.\"\n\n\"Who?\" Venin cocked his head.\n\n\"You look like an ithani.\" Jameson was grinning broadly.\n\n\"Ana. The woman who made the world minds.\" She held it reverently. This little symbol had come so far and had meant so many things to so many people.\n\nVenin nodded. \"I knew it had a history. Keep it. You seem to have a connection to it.\"\n\n\"Thanks.\" She squeezed it tight.\n\nTo her, it meant hope.\n\nXANDER WATCHED as the med drone finished the installation of his bioware. It would be nice to have Ravi and his direct grid access again, especially with the plans to link the grid to Errian and Gaelan.\n\nJameson was off helping Alix's forces end Rogan's armed occupation of the city.\n\n\"Good morning, Xander.\" Ravi's voice sounded just like he remembered it.\n\n\"Hey there, old friend. It's been too long. I missed you.\"\n\nHis PA was silent for a moment. \"I missed you too.\"\n\nXander smiled wryly. One day he would figure out just how much like people PAs really were. \"Where's Jameson?\"\n\n\"Wrapping up a cleanup operation in the Slander.\"\n\n\"Good.\" He had one outstanding item on his ledger. \"Ravi, I need you to find Rogan for me.\"\n\n\"Priority?\"\n\n\"Urgent.\"\n\n\"I will let you know when I find him.\"\n\nIt's time. Xander got up from the medical bench. He wandered down the hall of the OberCorp infirmary and poked his head into the room at the end.\n\nMylin lay there on her side, looking out the plas window at the new morning. It looked like it was going to be a glorious day.\n\n\"Hey.\"\n\nShe turned, wincing a little. \"Xander!\" She tried to get up.\n\n\"No, stay where you are!\" She had a synth pack covering the right side of her abdomen. He pulled up a chair and leaned forward to kiss her forehead, then sat down at her bedside. \"How are you feeling?\"\n\n\"Okay. I'm sorry... I got myself shot. I wasn't careful enough.\"\n\n\"Hey, stop that.\" It came out harsher than he intended. \"Mylin, you saved us all. You saved everything!\"\n\n\"I did?\" She smiled tentatively.\n\n\"Yes. Your warning came to us from Quince through the Heart, and not a second too soon. The carrier was just a few seconds away from impact.\"\n\nShe closed her eyes. \"Thank the skies. I was so scared.\"\n\n\"Me too.\"\n\nShe looked up at him. \"So where did you go? I've been dying to hear.\"\n\nHe grinned. \"Do I have a story for you. We fell through the gateway and found ourselves on a strange new world in the middle of a sea of golden grass....\"\n\nHe spun the story like a master storyteller, filling it with exclamations and gestures and lots of cliffhangers.\n\nShe made it through the part about the white city before she fell asleep.\n\nHe pulled the thin sheet over her and smiled at her. If Morgan was the closest thing he had to a son, Mylin was like a daughter.\n\nMaybe it was time to settle down and start a family of his own.\n\nThere was a light tap at the door.\n\nHe turned to find the boy Mylin had been working with. Derrick? Aaron?\n\n\"Derren Sevvins. Nice to meet you.\" Derren held out his hand.\n\nAh, Derren. Yes. \"We met, briefly.\" He shook Derren's hand.\n\n\"Yes, we did. I didn't think you'd remember me.\" Derren blushed.\n\nXander laughed. \"Yes, I remember you. Here with a message? Or...?\"\n\n\"I was here to... to see her, sir.\"\n\nAh. \"Are your intentions honorable, Derren Sevvins?\"\n\n\"Um, yes, sir. Entirely honorable.\"\n\n\"Carry on then.\"\n\nHe left them together, a grin on his face that he made sure the boy didn't see.\n\nJAMESON SHOVED another one of the syndicate fighters through a gateway into the waiting arms of an Oberon enforcer. They were going to need a bigger jail when all this was done.\n\nAfter a bit of sleep, some food, and a shower, he felt more human than he had in days. Ironic considering a decent chunk of him was decidedly inhuman. He'd have to work out how he really felt about that later.\n\n\"Xander wants to talk with you.\" Angie's voice was welcome in his ear. It was good to have her whole again.\n\n\"Put him through.\" Jameson closed the gateway, staring at the blasted street. The firefight had been about evenly matched until DOC had guided him to pick off the syndicate men one by one.\n\n\"It's time.\" Xander's voice sounded tight, restrained.\n\nJameson knew exactly what he meant. \"Where is he?\"\n\n\"At his estate. About to leave for parts unknown, according to DOC.\"\n\nJameson grinned. \"Let's go finish our end of the bargain. Where are you?\"\n\n\"On the roof of OberCorp.\"\n\n\"Don't even need a gateway for that one.\" He leapt into the air, winging toward the place that had once been home to his sworn enemies. Everything had flipped. Well, almost everything. \"On my way.\"\n\nThe city below was a hive of activity. Alix and DOC had wasted no time tearing into the Slander. Drone construction machines were already ripping into the outlying buildings, knocking them down for scrap. In a month, the Slander would be no more.\n\nOberCorp had opened a new subsidiary for human welfare on Erro, committed to building new housing in the outlying human cities along the coast of the Gildensea.\n\nThe Syndicate's days were numbered. Things were going to change, hopefully for the better.\n\nEven Alix's mother was on board. She was still a little freaked out about the skythane, but she was working on it. Her words, not his. She was now in charge of the old business side of OberCorp again\u2014under Alix's command.\n\nJameson snorted. He'd believe it when he saw it, but if there was a profit to be made....\n\nXander waved to him from atop OberCorp, the sun bright on his face.\n\nAfter the Heart had melted down, the storm had dissipated, and they were anticipating a stretch of fair weather at last. That was fine with Jameson. If he never saw another day of rain in his life, it would be too soon.\n\nHe alighted next to Xander. \"Ready?\"\n\n\"I've been ready for this for ten years.\"\n\n\"Okay, here we go.\" He pulled out his key and opened a gateway to Rogan's estate. He wondered if the key belonged to one of the nimfeach. If so, he'd have to give it back one of these days, soon.\n\nThey stepped through to find a startled Rogan loading bags onto a small hoversport. He was all alone. Guess the ship deserted the sinking rat.\n\n\"Hey there. Where are you going? Extended vacation?\"\n\nRogan was sweating profusely. \"Just thought I'd get away for a bit.\" He glanced from one of them to the other, as if trying to judge their intentions. \"And you two?\"\n\n\"Oh, we had a bit of unfinished business to address with you.\"\n\nRogan threw himself at their feet. \"I'm so sorry. I didn't mean for things to get out of hand. There were a few bad apples.\" He touched Xander's foot.\n\nXander pulled it away, a look of disgust on his face.\n\nJameson offered Rogan a hand. \"We're not here to hurt you.\"\n\n\"You're not?\"\n\n\"Of course not. We had a deal, remember?\"\n\nThe heavyset man got up awkwardly and dusted himself off. \"Thank you. You don't know how happy this makes me. Not everyone is so trustworthy in this town.\"\n\nXander snorted.\n\nJameson put on his most beneficent smile. \"We promised you all the pith in our storage cave. I never go back on a promise.\"\n\n\"Really?\" Jameson could practically see the gears turning in the man's mind. \"But without a market....\"\n\n\"Soon Oberon will be returned to its place among the Common Worlds. I assure you there will be a huge market for it.\"\n\nRogan laughed. It was an ugly sound, deep and guttural. \"How... how do I get it?\"\n\nJameson pulled out his key and opened a gateway to the cavern that held the stockpiles of pith Dani had collected in Gaelan. \"Be my guest.\"\n\nRogan stared hungrily at the key, but then tore his gaze away from it to look Jameson in the face. \"I want it legally recorded that all of that pith is mine, fair and square.\"\n\n\"I have a contract all drawn up. Angie, can you slit it to Mr. Horth for his signature?\"\n\nThe man blinked twice at the mention of his last name. That had taken some digging to unearth. His eyes went blank as he scanned the contract in vir. \"It all seems in order.\"\n\n\"If you would affix your v-sig....\"\n\n\"Done.\"\n\nJameson checked the contract, then sent it off to be filed. \"It's all yours.\"\n\nRogan all but ran into the cavern, picking up the first pack of pith to check it.\n\nJameson handed the key to Xander. \"Want to do the honors? Just like I showed you?\"\n\nXander nodded. \"I hope the two of you share a happy eternity together,\" he called to his former tormentor.\n\nRogan's horrified face was the last they saw of him before the gateway snapped shut.\n\nXander kissed Jameson hard. \"That was better than I even dared to hope.\"\n\nJameson took the key and put it away. \"How long do you think it will take him to figure there's no way out?\"\n\nXander flashed him a wicked grin.\n\nQUINCE STOOD alone inside the cavern by the sea.\n\nAbove her, the world mind was dying.\n\nDaia appeared to her, still young and beautiful, but faded somehow. Quince could see through her to the wood-covered cabin walls. Even in vee, she was a ghost, a spirit preparing to depart on a path Quince could never follow.\n\n\"Is it over?\"\n\nQuince nodded. \"Jirron, the one who led the ithani astray, is gone.\"\n\n\"I'm glad. And the war?\"\n\n\"Settled. Alix, Morgan, and the two kings are working out a truce now.\"\n\n\"A new world.\" Daia sighed. \"I'm dying.\"\n\n\"I know.\" All that knowledge, human and ithani. It would be lost now, in spite of their best attempts to save it. \"You waited for us for so long.\"\n\n\"I still remember.\"\n\n\"Remember what?\" How she could choose among so many times, places, people... it boggled Quince's mind.\n\n\"What it felt like to walk barefoot on the sand, to feel the cool waves wash over my toes.\" She stepped outside the cavern, and the waves washed through her. She left no footprints.\n\nThe sun was setting over the Gildensea, and it bathed her in a golden glow.\n\nQuince followed her outside, taking in a deep breath of the cool sea air. \"I give you my blessing.\"\n\n\"Blessing?\"\n\n\"To go. You've waited long enough. You don't need to stay any longer on my account. Or for any of us.\"\n\nDaia's ghostly hand brushed Quince's cheek. \"Thank you for that.\" She stared out at the sunset for a moment longer, and Quince wondered what thoughts or memories were passing through her head. At last, Daia turned and beckoned. \"Come with me. I have something for you.\"\n\nCurious, Quince followed her back into the cavern.\n\nOne of Daia's roots lowered to the ground, holding something.\n\nIt was oblong, bigger than Quince, covered in a dark husk.\n\nA seed.\n\nQuince recognized it from memory. The original Daia, the human Daia, had planted a seed just like it in this very place more than eight hundred years before. Well, maybe not just like it.\n\n\"When one of us produces a seed, it takes a lot out of us, but it's better this way.\" Daia placed her hand on it. \"It carries everything you need, for you and for the ithani. Plant it in a good place, with room to grow.\"\n\nQuince was speechless.\n\n\"Every tragedy holds the seeds of something new.\" Daia looked wistfully at the seed.\n\n\"Where will you go?\" Quince managed at last.\n\nDaia was transfixed by light. \"I don't know. I think I'm going to join the others.\" Her face was radiant, actually glowing. \"I'm glad I met you, Quince.\" The light grew and grew until it consumed her in a final burst of radiance.\n\n\"Me too.\"\n\nDaia's last words were a breath on the ocean breeze. Goodbye, Quince.\n\nXANDER, JAMESON, Morgan, Quince, Venin, Jessa, and Mylin sat on the hastily constructed stage.\n\nAlix was at the podium, giving his prepared remarks, while nimfeach translators on either side repeated it in the ithani tongue.\n\nXander looked out at the assembled crowd on the shores of Lake Arliss. \"Arliss\" was the closest human approximation to the name of the ithani city that was being covered by its waters as the Demetrius and Theseus Rivers emptied into the former cavern. In the two months since the battle, it had filled up about halfway to the rim.\n\nThe ithani and the nimfeach had decided that their new city would rise on the shores of the wide circular lake. The lander cities already dotted the edge of the Gildensea, and the skythane had populated the far side of the planet, from the Argent Sea to the Mora Mountains. But the Outland was unpopulated and would give the ithani a place to call their own in the new world order.\n\nXander wished the ceremony wasn't necessary. He hated these things, but he understood the need to formally cement the ties between the three peoples. The nimfeach had been given back their hearts, which had been hidden in the great mountain Freyyr to the north. They had taken over the governance of the ithani, selecting a council to make the most important decisions.\n\nHe glanced over at Jameson. \"We're kind of old hat at spinning worlds, aren't we?\"\n\n\"Shush.\" Jameson stared straight ahead, but a smile played at the corner of his lips.\n\n\"... and we are ready for a new era of cooperation between lander and skythane, between human and ithani.\" Alix bowed, and the crowd cheered and clapped and shimmered their approval.\n\n\"He's good at this.\" The truce had called for the election of a governor, a real one, not the sham that OberCorp had perpetrated on the planet for hundreds of years. The last one had been little more than an OberCorp puppet.\n\nAlix figured he had a good shot at snagging the post.\n\nVenin got up and hugged Alix.\n\nXander hadn't seen that one coming. Venin and Alix? Nevertheless, he wholeheartedly approved of the match.\n\n\"Okay, you guys are on.\" Alix gave them the thumbs-up and kissed each of them on the cheek.\n\nXander followed the others down the stairs from the platform.\n\nAlix's mother, Lena, was there. She gave him a hug, and Xander nodded at Alix. She had a lot to account for, but after learning what she'd been through, he thought he understood. He was glad to see Alix mending fences with his only family.\n\nA wide circle had been cleared in the forest behind the platform. Eventually a new building would be grown from corrinder over the site, but for now, it was open to the air.\n\nThe sun was warm on his skin as he approached what they'd taken to calling \"the garden.\" The two seeds lay there, side by side. One would become a new heart, and the other a new world mind. Like the two alien cultures that shared the planet, they would become intertwined, a hybrid creating something new.\n\n\"Are you ready?\" He put a hand on Mylin's shoulder.\n\nShe shook her head. \"I don't think I'm the right one for this.\"\n\nXander laughed. \"You're the only one for this. The world mind is a repository of information unlike anything we've ever seen, short of maybe the original grid on old Earth. Who else better to organize it?\" He squeezed her shoulder.\n\n\"WILL IT hurt?\"\n\nQuince took Mylin by the hand, giving Xander an I'll take it from here look. He nodded and backed away to stand with Jameson. \"Daia assured me no. You'll come away entirely unharmed.\"\n\n\"Okay. If you think\u2014\"\n\n\"Come on.\" Quince pulled the girl close. \"Everything will be fine.\" She led Mylin to the world mind seed.\n\nThe crowd followed, gathering around them.\n\nQuince knelt, gently placing Mylin's hands on the rough husk. Then she put her own there as well.\n\nDaia had shown her how to do this, as so many had done it before her. With her other hand, she pulled Mylin's black hair back behind her ears. \"Ready?\"\n\nMylin gulped and nodded. \"I guess so.\"\n\n\"You'll do fine.\" Quince reached inside and found the seed's spark. She touched Mylin's cheek and the transfer began.\n\nThe girl's memories, dreams, and personality flowed through her into the seed.\n\nMylin had enjoyed a good childhood. She was smart and stable and beautiful, everything Quince thought a skythane should be.\n\nQuince was humbled to be in this place, to help bring about a new order. Thank you, Daia.\n\nThey knelt as seconds stretched into minutes. All around, the crowd was hushed, watching the silent spectacle.\n\nAt last it was done.\n\nThey stood together, and Mylin looked up at her, an odd smile on her face. \"It's weird.\"\n\n\"What is?\" Besides the entire situation?\n\n\"I'm here, but I'm in there too.\"\n\nQuince laughed. \"Yeah, I suppose you'll get used to it eventually.\"\n\n\"Maybe in a hundred years.\"\n\nThey moved back with Jameson and Xander and watched as the nimfeach activated the other seed.\n\nThen everyone waited.\n\nThe world mind seed moved first, its bark splitting with a loud crack, roots emerging to sink down into the ground below and questing around on the surface. They soon encountered the alien seed.\n\nThe sphere shimmered and melted, slipping over the questing root, running like quicksilver along the root to the other seed.\n\nThere, it spread over the rough husk and then seemed to burrow inside.\n\n\"Heart and mind in sync?\" Xander whispered.\n\n\"Was that supposed to happen?\" Mylin was staring at the strange scene, her brow furrowed.\n\n\"I don't know.\"\n\nThen everything stopped.\n\nQuince's heart dropped. Had she miscalculated? Had the two seeds killed each other?\n\nMaybe they weren't as compatible as she'd hoped.\n\nShe held her breath.\n\nMaybe we're stuck here.\n\nWithout the Heart, there was no way to return Erro to the Common Worlds.\n\nThen the seed started to grow again. It had taken on a distinct silver cast.\n\nThe crowd let out a collective sigh.\n\n\"We've witnessed something historic here,\" Jameson whispered in her ear.\n\nQuince nodded. That much was clear.\n\nShe just wished she knew what it meant.\n\nXANDER TOOK Jameson's hand. It was time to finish what they had started.\n\nThe nimfeach ushered the ithani through gateways back to Freyyr, the mountain in the north. They went docilely enough, though Xander saw a few wistful gazes cast to the pink skies above. The last time they'd slept through a shift, the universe had changed around them.\n\nIt was for their own protection, Morgan had explained to him. Something about the energies released during the shift.\n\nWhen the aliens were gone, the human crowd\u2014skythane and lander\u2014closed in around them. It seemed like half of Oberon City was here, though it couldn't have been anywhere near that many. Still, it was a substantial crowd that stood beneath the afternoon sky to watch the world spinners work their magic.\n\nXander and Jameson waited for the ithani to be safely tucked away as the sun dipped toward the western horizon. Finally they got the all clear.\n\nMorgan took their hands.\n\nHe closed his eyes, and Xander felt both Jameson and Morgan through their bond.\n\nThen they were in the athr\u00e0. The world of possibilities swirled in gray eddies all around them.\n\nSomeone stepped out of the mist. Two someones.\n\nMylin, or her virtual doppelganger. And Erina.\n\nXander smiled at the ithani, and ze rolled zer head on zer neck in zer equivalent of a grin.\n\n\"Ze's here.\" Jameson's stunned face was priceless. He turned to Xander. \"You knew?\"\n\n\"I guessed.\"\n\nMorgan looked up at them. \"Ready?\"\n\nXander nodded. \"Let's go home.\"\n\nMorgan closed his eyes.\n\nThe athr\u00e0 filled with energy, like static in the air.\n\nTogether the three of them opened the way.\n\nJESSA STOOD apart from her friends, holding Dixon's hand. The pilot was strong and beautiful, but sweet and funny too, something she didn't expect from a military man.\n\nAlia and Smythe had introduced them. If the pilot was a consolation prize, he wasn't half bad. In fact, she might even stop calling him that, one of these days.\n\nHe leaned over and kissed her on the cheek. \"What a day, huh?\"\n\nShe laughed. \"Yeah, what a couple of months.\" Soon she'd be able to go home to Beta Tau, back to her safe life as news broadcaster.\n\nBack to her parents' functions and money, and all that they implied.\n\nSoon.\n\n\"I'm thinking I might hang around here a little while longer. If I had a good enough reason.\"\n\n\"This count?\" Dixon took her in his arms and bent her back gently, kissing her on the lips. He smelled like sweetness and lust. And adventure.\n\nAs they kissed, the stars came out overhead.\n\nWhen they separated, she looked up at the stars. Her old home was around one of those flickering lights. She didn't want to go back. \"I think it's the perfect reason.\"\n\nVENIN LAUGHED as the sky changed from day to night. \"Holy Split, it worked!\"\n\nAlix grinned. \"We're home.\" He kissed Venin, joining in the general air of celebration. When they separated, he looked his lover in the eyes. \"Have you given any more thought to where you're going to live?\"\n\nVenin frowned. \"I miss Gaelan, but I'm totally falling for this lander man in Oberon City.\"\n\nAlix laughed. \"I have a proposition.\"\n\n\"Tell me.\"\n\n\"Be my ambassador to the skythane. That way you can spend time with me, and we can go to Gaelan and Errian too.\"\n\nVenin's face lit up. \"That's a great idea. Where do I sign up? Is there an interview process?\"\n\n\"I know a guy.\" He put his hands around Venin's ass and pulled him in close, whispering, \"You've got the job.\"\n\nTHE SHIFT ended, and the weight of a world dropped off Jameson's shoulders. He opened his eyes, looking around.\n\nThey were back in \"real\" time, though the line between what was real and what was not had grown increasingly thin this last month.\n\nMorgan looked up at them. \"It's time for a new plan.\"\n\n\"I suppose it is.\" Xander laughed. \"You have your people to look after.\"\n\n\"Yes.\" Morgan looked lost.\n\nJameson knelt next to him. \"Tanner?\"\n\nMorgan looked up at him, his face suddenly that of a twelve-year-old boy. \"Yeah?\"\n\n\"Xander and I are here for you, whenever you need us.\"\n\nThe boy nodded and threw his arms around Jameson's neck. For all that the nimfeach part of him was an old soul, Tanner was still a child.\n\nTanner let go and grinned. \"We're going to get our wings!\"\n\n\"What?\" Jameson looked up at Xander.\n\nXander shrugged. I have no idea.\n\nJameson heard his thoughts that way now sometimes. One more thing that had changed between them. \"Wings?\"\n\nTanner's face became more solemn. Jameson was getting used to these shifts of personality. \"Tanner's mother was skythane. She lived in Egeus among the landers there. I didn't know until recently.\"\n\nJameson whistled. \"Isn't that something.\"\n\nMorgan hugged Xander. \"I've got to see about waking the ithani.\"\n\nXander nodded. They grow up so fast.\n\nJameson laughed. It was a brave new society they'd created. A strange one, too, filled with humans and aliens and all kinds of things in between.\n\n\"It sure is.\"\n\nJameson grinned. That whole mind reading thing was going to take some getting used to.\n\nHe took Xander's hand and led him back to the others. To their friends.\n\nThey had a world to build together.\n\n# EPILOGUE\n\nALIX POKED his head into the construction zone. \"How is it coming?\"\n\nVenin looked up and grinned. \"Really well.\"\n\nAll around him, builder drones were busy creating his new aerie\u2014that was Venin's word for it.\n\nThey'd purchased an entire floor of one of the arcos, not the top one, but high enough to have a good view. One of the sides, facing east, was being converted to a traditional apartment, albeit a huge one. The Governor of Erro needed to have a space commensurate with his position, after all, when wining and dining guests from the other planets.\n\nTitan Station had been destroyed by the flare. Fortunately it had been empty by then.\n\nThe Common Worlds navy had been poking around Erro's star system trying to figure out how they'd lost half a planet when Erro had reappeared. It had given the top brass quite a fright, but they now had a temporary replacement station, and a new one was in the works.\n\nHe laughed at himself. The Split was gone. They were going to have to come up with some new epithets.\n\nThe legal pith trade had grown by leaps and bounds over the last six months since Erro had rejoined the human universe, but tourism was proving to be an even bigger draw. Everyone wanted to meet the aliens, so after he'd won the post in a landslide, Governor Preston had been forced to institute strict restrictions to keep the tourists from overrunning the planet.\n\nHe'd also severely restricted the mining of amalite at the request of the ithani, which had the related effect of driving the prices per kilo through the roof.\n\nVenin's half of the apartment\u2014well, technically it belonged to both of them\u2014was open plan. Really open plan. Wide open to the outside air, with stunning views of the Gildensea and Erro's green sky. Windows could be closed as needed, but Venin enjoyed the fresh air and the smell of the sea.\n\n\"Let me see them.\" Venin's eyes twinkled.\n\nAlix grinned and turned to display his new wings. They were still tiny. When Quince had brought him the ethilium to kick off the skythane half of his metabolism, she'd said they might take a bit to grow in because he was only half skythane, and they might never be as large as Jameson's or Xander's. He didn't care.\n\n\"Oooh, the feathers are coming in. They're red.\"\n\n\"What color did you think they'd be? They match the carpet.\"\n\nVenin laughed. \"When they finish growing, I'll take you out flying.\" He snuggled up behind Alix, nuzzling his neck.\n\n\"I can't wait.\" He'd finally be able to realize one of his dreams\u2014to fly like the skythane.\n\n\"Are we still going to Errian next week?\"\n\nAlix nodded. \"Trade meetings. Boring as hell, I know. But that's my life now.\"\n\nVenin slapped him on the ass. \"Don't give me that. You love it.\" He kissed Alix's cheek. \"Come on! I have something to show you!\"\n\nHe drew Alix across the apartment to a closed door. He palmed it open and pulled Alix through into another room. On the other side was a wide, round white bed and a replica of the gorgeous view from the main living quarters.\n\nIt made Alix lightheaded to be so close to the open air, but he decided he'd get used to it. \"That's a strange bed.\"\n\nVenin laughed. \"You have no idea. I had it shipped in from the Outland.\" He sat on the edge of the bed and patted the mattress, sending ripples across the surface. \"Come here. I've been waiting for you all day.\"\n\nAlix complied, and Venin pulled him down on the bed and kissed him hard. \"It's an ithani bed. Want to see what it can do?\"\n\nBy the time night fell, Alix decided he was a huge fan of alien technology.\n\nMYLIN WAS onsite in Ithalia, the new city of the ithani, overseeing the integration of the OberCorp core and grid and the world mind. It was strange, working with a virtual version of herself, but she was getting used to it. Soon they would be able to exchange data back and forth between the two systems, and then she hoped to extend that capability to Errian and Gaelan on the other side of the world.\n\nZim hovered before her in vir. \"Derren would like to speak with you.\"\n\nShe nodded. \"Tell him five minutes.\"\n\n\"I told him that five minutes ago.\"\n\n\"Tell him three minutes, then.\"\n\nZim zipped off, sounding miffed.\n\n\"That boy really likes you.\" Mylin's mirror inside the world mind grinned.\n\nShe grinned. \"I think so. I'm still waiting for him to tell me.\"\n\n\"Maybe he wants to tell you now.\"\n\nStartled, Mylin looked up at her counterpart. \"You think so?\"\n\nThe world mind winked at her. \"Why don't you go see?\"\n\nThe virtual Mylin knew her like she knew herself. \"Okay, but we'll come back to this when I'm through.\"\n\n\"Deal.\"\n\nMylin slipped out of vir.\n\nDerren was waiting for her, a bunch of wildflowers in his hand.\n\nThey were a sad bunch, for the most part, sagging and missing petals, but it was edging toward fall, and she imagined they had been hard to come by.\n\nHe gave her a tentative smile and held them out to her. \"For you.\"\n\nShe accepted them gratefully. \"They're beautiful.\"\n\n\"Mylin, would you have lunch with me?\"\n\nShe grinned. Her other self had been right. \"I'd love to.\"\n\nShe took his hand.\n\nTogether they walked out into the sunlight, surrounded by the growing towers of Ithalia\u2014the white city of the ithani on Lake Arliss.\n\nQUINCE LAID the shovel down in the long grass. Then she tilted the wheelbarrow up, pouring the wet plascrete into the hole she'd dug.\n\nThere was something tremendously satisfying about digging into the earth under your own power, turning it with a shovel and seeing all the life that lived within it.\n\nShe glanced up at the newly constructed cabin that backed onto the nascent forest that was growing up over the wreckage from the shift storm, as they'd taken to calling it.\n\nThey'd finally gotten their place away from it all. They'd chosen the once-beautiful valley below the mountain, where it had collapsed after the shift. The place Quince had finally found Robyn once again.\n\nShe'd borrowed a few builder drones from Alix after the initial cleanup of Oberon City had been completed. Morgan had obliged with a gateway to get them here in far less time and with much less expense than it would have taken to ship them. Now they labored, day and night, to clear away the forest debris near the house and to clean up the worst of the rubble in and around the lake.\n\nIt would take time, but one day this little valley would be a paradise once more.\n\n\"It's ready!\" she called, sinking the shovel into the earth.\n\nThe cabin door opened and Alia leaned out. \"We'll be right down.\"\n\nAlia and Smythe had come to visit them for a couple days. It was nice to have company out here. It did get lonely, just the two of them so far from civilization, but the quiet was a balm to Quince's battered soul.\n\n\"Hurry\u2014the plascrete is setting.\"\n\n\"Got it.\" The door slammed closed.\n\nA minute later, the three of them\u2014Alia, Smythe, and Robyn\u2014came trotting down the hillside. Alia carried a big basket.\n\n\"I thought we'd make a celebration of it.\" She spread a hand-woven Erriani blanket out over the grass and set down the basket. It was full of fresh-baked bread, fruit, and meat from their stores. The aux herd was coming along nicely on the high pasture above the valley, and they'd traded another farmstead for the produce, though Quince had already planted a variety of fruit trees next to the cabin.\n\n\"Okay, let's do this while the plascrete is still wet.\" She lifted up the cross she'd made from the wood she'd taken from Daia's cave by the sea. She'd polished it and then lashed it together with aux leather. She pushed it into the plascrete as deep as it would go, and then used a trowel to smooth over the surface.\n\nShe stood back to look at it. The hand lettering carved across the front read \"Daia.\" Not in perfect script, but it came from her heart.\n\nRobyn put her arm around Quince. \"I think she would have liked it.\"\n\n\"Me too. One more thing.\" She took off the necklace with the silver cross and laid the chain over the top of the wooden one.\n\n\"It's perfect.\" Alia nodded her approval. \"Come on you two, let's eat. I'm hungry!\" Alia gestured for them to join her and Smythe on the blanket.\n\n\"Thank you,\" Quince whispered, and then followed Robyn to the picnic.\n\nShe sat and took one more look around the wide valley.\n\nThis is where I belong.\n\nMORGAN PERCHED at the top of the tower on Founder's Hill, watching the activity going on all around him.\n\nTaylor came running up the stairs, out of breath. \"You. Have. To. Wait. For. Me. Alia said so.\"\n\nMorgan grinned, letting Tanner have his sway over their shared body. \"You're too slow.\"\n\n\"I wish I had wings.\"\n\n\"Maybe your dads will buy you a pair of bi-wings when you're older.\" Alia was watching over the child who had been one of Rogan's slaves while Xander and Jameson were in Oberon City. They had taken him in when they'd been unable to find his mother.\n\nAlia had given Morgan the responsibility for a couple days while she and Smythe were out at Quince's farm.\n\nIt was a task he'd been happy to take. Tanner needed friends.\n\n\"Yeah, maybe.\"\n\nTogether they looked out over Gaelan. The city was being rebuilt from the ground up, the aeries rising floor by floor. Soon it would be a bustling metropolis again.\n\nIthani mingled with skythane in the rebuilding process, as a way of atoning for what they'd done. Morgan was here to oversee the process.\n\nHe flexed his new bright blue skythane wings, nervously considering the ground far below. This was his first time.\n\nAre we ready?\n\nCan't we go already? I'm dying to fly.\n\nMorgan grinned. His dual nature was confusing sometimes, but Tanner was unequivocally ready. Now.\n\n\"Wanna come?\" he asked Taylor.\n\n\"Could I?\"\n\n\"Climb on my back, and I'll fly us back down.\"\n\nTaylor did as he was told. He was a skinny kid. Morgan could bear his weight easily.\n\nHe stood on the parapet and spread his wings. \"Here we go!\" Then he leapt into the air to fly.\n\nTaylor laughed in delight all the way down to the ground.\n\nXANDER KINNSON lay on his bed, head thrown back, his heart racing.\n\nJameson's wings flared out above his body, sweat running down his bare chest as they both reached climax.\n\nJameson moaned and collapsed on top of him, kissing his neck gently.\n\nXander pulled Jameson to his lips, kissing him back.\n\nHe turned to look out the window, as he had all those months ago on the day they'd first met. Instead of rain, the sun was shining brightly.\n\nXander stared warily at Erro's sun. If past was prologue, they'd have to deal with it again in another seven hundred fifty or eight hundred years. Not my problem.\n\n\"Hey, where are you?\"\n\nHe turned back to look up at Jameson. \"Sorry. Just thinking about the day we met.\" He reached up to touch the golden sun sigil that Jameson still wore around his neck.\n\nA slow smile stretched across Jameson's face. \"I remember. You weren't so hot on me back then.\"\n\nXander sat up, laughing. \"You were kind of a bastard.\"\n\n\"I was a bastard?\" Jameson shoved him back onto the bed playfully and got up, heading for the ionic shower.\n\nXander watched him go, enjoying his perfect ass.\n\nJameson climbed into the shower. \"We have to get one of these in Errian and Gaelan.\"\n\nThey'd decided to split their time between the two skythane capitals, meaning they no longer needed Xander's apartment.\n\n\"Sure. We'll see if we can figure that out.\" He looked around at the empty apartment. He'd lived here for ten years, but since Alix had left him, the apartment had never felt like home.\n\nJameson stepped out of the shower, and Xander took a moment to enjoy the frontal view.\n\nJameson grinned, flexing his golden wings. \"Your turn?\"\n\n\"Nah. I think I wanna smell you on me a little while longer.\"\n\n\"Your choice.\" Jameson kissed him and put on his shirt, lacing it up, and then pulled on his pants. \"Come on. We need to be in Errian for the meeting by noon.\"\n\n\"Slave driver.\" Xander got dressed, taking one last walk around the place. Home or not, he was going to miss it.\n\nHe and Jameson had a new life to build, with Taylor. It was time to move forward.\n\nXander slipped into his riding chaps\u2014it was good to have his own clothes again\u2014and pulled on his black riding gloves.\n\nHe pressed his palm up against the clear plas of the window. It was warm to the touch. Twin doors slid aside in the floor, and his hoverbike raised into view. The doors sealed shut below it.\n\n\"You ready?\"\n\n\"Thought you'd never ask.\" Jameson slipped onto the bike behind him.\n\n\"Ravi, open the doors.\"\n\n\"Acknowledged.\"\n\nThe plas window split apart, a straight hairline crack that spread from the base of the window up to six feet off the floor. Clear doors formed in the plas, and then they opened outward, letting in the warm summer air.\n\n\"Hold on.\"\n\nJameson's arms snaked around his waist, and his lover was warm against his back.\n\nXander palmed the ignition key and the machine roared to life underneath him. He powered up the bike's amalite drive and released the brake, soaring out of the arco into the open air.\n\n\"Woo-hoo!\" Jameson's wings flew open behind him as the hoverbike veered around to head toward the Gildensea and Errian.\n\nXander looked back at Jameson. \"We just need a sunset and this would be perfect.\"\n\nJameson grinned and kissed his cheek from behind. \"Who needs a sunset?\"\n\n# GLOSSARY\n\nAmalite: Raw ore found only on Oberon that serves as Common Worlds power source. Called heartstone by the ithani\n\nAmSplor: Exploration division of the Northern American Union\n\nAndra Madainn: Queen of the House of the Sun, Jameson's mother\n\nAnellia: Pop singer capable of singing in triple harmony\n\nAngela Havercamp: Jameson's adopted mother\n\nAnnama: Soul mate\n\nArcatus: Interstellar ship Jameson came in on from Tander's World\n\nArco: Vast buildings where most of Oberon City's citizens are housed\n\nArctus: City on Beta Tau\n\nArgent Sea: Titania's sea\n\nArliss: The capital city of the dhagani\n\nArracha Grain: Native grain grown as a staple on Oberon\n\nAshalla: The Titania universe\n\nAthr\u00e0: The virtual or spirit world of the ithani\n\nAuxen: Forest herbivores in the Riamhwood\n\nAux: Larger cousins of the auxen, they live on the northern plains\n\nAvea (also Hermia): The ithani name for Hermia, Oberon's red moon.\n\nBallifor: Small Titania village where Quince is from\n\nBandia: Titania's golden moon\n\nBanga Tree: Low, wide Titania jungle trees that resemble barrels\n\nBeta Tau: Jameson's homeworld\n\nBi-Wings: Artificial wings used by the enforcers on Titania\n\nBlackware: Illegal apps\/software\/code\n\nBlade File: Electronic file format\n\nBlevons: White birds that live in underground burrows near the south pole\n\nBlueoak: Native Oberon tree\n\nBolc\u00e0 Island: Island in the Argent Sea\n\nBoxcorn: Genetically modified square corn ears used as base for foods\/fuels on Oberon\n\nBraid Seneford: One of the OberCorp board members\n\nBreed Match: The merging of the three sexes of ithani to create offspring\n\nBuilder Drone: Drone used for construction and renovation purposes\n\nCafflite: Oberon equivalent of coffee\n\nCamspecs: Eye glasses with a built-in tri-dee camera\n\nCastain, the: The \"castle\" in Errian, also called the House of the Sun\n\nCattorah, the: A ritual removal of a skythane's wings, usually as punishment for a severe crime\n\nCheth: The ithani word for corrinder\n\nChit: Portable cash chip\n\nChristianist: Throwback religious sect that hearkens back to conservative \"Christian\" values\n\nCirq: Bio-interface in the temple that allows users to access the grid\n\nCitrone: Native yellow fruit that grows on vines on Oberon\n\nColifir Tree: Red native tree that grows by rivers\n\nCommon Worlds: Loose-knit government of human worlds\n\nConjunction: Alignment of one of the moons of Oberon with the planet and sun\n\nCorrinder: The plants used as building stock in Errian; also the white material the plants are made from. See also cheth\n\nCorybryte: A sharp white skythane cheese\n\nCreach: Scavenger birds on Titania\n\nCreeper Vines: Native silver ground vegetation on Oberon\n\nCrits: Credit\/money\n\nCroyol: Fungus that burns without smoke\n\nDaedus Madainn: Prior King of the House of the Sun\n\nDamella Sl\u00e9ite: Prior Queen of the House of the Moon\n\nDanner Black: A skythane pith trade runner who helped instigate civil war between the skythane\n\nDark Market: Black market on Oberon, run by the Syndicate\n\nDaro: One of Dani's lander guards\n\nDavos Madainn: Prior King of the House of the Sun\n\nDawson: Rogan's henchman\n\nDax: One of Rogan's enforcers\n\nDeathhawk: Deadly bird of prey native to Titania\n\nDeca: Oberon's tenth month\n\nDeireadh an Domhain: The Mountain\n\nDemetrius River: Southern tributary to the Theseus\n\nDeterrent Field: Rope that is used to create a field to deter the local wildlife, especially wereveren\n\nDhagani: The enemies of the ithani\n\nDillan Farrai: Quince's brother\n\nDistortion Field: Shield generated by a small device that blocks electronic surveillance\n\nDistortion Zone: Zone of electronic interference at the edge of Oberon\n\nDixon: An OberCorp enforcer\n\nDorthia: Erriani woman who lives in Taycrob\n\nDrimm the Dragon: One of Titania's constellations\n\nEarth-Standard: Timekeeping based on Earth's clock\/calendar\n\nEgeus: City on Oberon\n\nElyra Sl\u00e9ite: Prior Queen of the House of the Moon\n\nEnforcers: Men who work as the \"muscle\" of the Syndicate\n\nEraysses Sea: A sea on Beta Tau.\n\nErrian: City of the Sun (as in House of the)\n\nErriani: Citizens of the House of the Sun\n\nErro: Sun God; also the ithani name for the whole world\n\nEthilium: Growth hormone that stimulates the development of wings in the skythane\n\nFaery Caves: Caverns in Titania's mountains, often lit by glowing blue ponds\n\nFaery Ponds: Blue glowing ponds in the caverns on Titania\n\nFizzpop: Carbonated sugar-alcohol drink popular on Beta Tau\n\nFynx: One of the Gaelani skythane at Torr Talam\n\nFarris: Assistant to the Prison Master\n\nFeather Trees: Trees native to Oberon\n\nFennow Root: Natural antiseptic\n\nFirst Wave: First human colonists on Oberon, also called skythane\n\nFreyyr: The mountain in the north; means \"forge\" in the ithani language\n\nFynx: Gaelani man in the war camp\n\nGael: Moon God\n\nGaelan: City of the House of the Moon\n\nGaelani: Citizens of the House of the Moon\n\nGalaxion Hotel: Interstellar hotel chain\n\nGildensea: Oberon's sea\n\nGlow Sphere: Portable light source\n\nGovernor: Governmental head of Oberon\n\nGovernor's Residence: Vast estate where the Factor lives\n\nGranth: Feline equivalents from Pleiades Six, valued for their leather\n\nGreat Division: Period after skythane refugees were chased out of Oberon to Titania\n\nGreat Retreat: The flight of the skythane settlers before the landers and Ober Corp\n\nGregg: OberCorp Employee\n\nGresachh: An ithani word meaning disgraceful, dishonorable, unthinkable and perverse\n\nGrid, The: Oberon's data and communications network\n\nGridcode: Programming code, etc.\n\nGSN: Great Sky Network, one of the Common Worlds news services\n\nGumba Tree: Tall, leafy native tree often used as a windbreak for farms on Oberon\n\nHallerwood: Golden-leaved trees native to northern Titania\n\nHachmoss: Yellow moss native to Oberon\n\nHarrol: One of Xander's companions on the way to Errian\n\nHeart, The: The center of the ithani city\n\nHeart Fungus: See Croyol\n\nHeartbrier Bush: Titania bush with heart-shaped leaves and orange flowers\n\nHeartstone: see amalite\n\nHeartwood: Forest on Beta Tau\n\nHermia (also Avea): One of Oberon's two moons\u2014red-colored\n\nHesies: Xander's past life\n\nHigh Slopes: Northern district of Oberon City, where the Spaceport is\n\nHippolyta: City on Oberon\n\nHoarberries: Little blue berries covered in a sweet white \"frost\"\n\nHollyhock Trees: Trees found in the valley of The Mountain\n\nHoney Ale: Titania alcoholic drink\n\nHarsorch: The parasite that keeps the world mind quiescent\n\nHouse of the Sky: Ruins in the center of Oberon\n\nHouse of the Stars: Royal retreat in the center of Titania\n\nHover-Plat: Transportation platform used to move small amounts of goods\n\nHoverbike: One of the main methods of personal transport in Oberon City\n\nHoversport: Hover craft used for human transport\n\nHunters, The: The name for the group of allies fighting the ithani\n\nIce Rat: Native Oberon scavenger that lives near the southern pole.\n\nIce Pine: White tree native to the northern climes of Oberon\u2014the Rim Forest\n\nImprean: Carrier-pigeon like bird used to carry messages\n\nIronwood: Tall trees in Titania that are impervious to flame\n\nIthalia: The new city of the ithani\n\nIthani: Original inhabitants of Oberon\/Titania\n\nJoseph Havercamp: Jameson's adoptive father\n\nKeff: Titanian equivalent of coffee\u2014tastes like herbal tea and coconut.\n\nKnacks: Oberon insect pests\n\nLanded: to be stripped of your wings, usually as punishment for a criminal act\n\nLanders: Second wave human settlers\n\nLightdump: Long lasting light source\n\nLyda: Sculptor of the statue of Gael at the House of the Stars\n\nLydia Madainn: Prior Queen of the House of the Sun\n\nLysander: One of Oberon's two moons\u2014golden color\n\nMarli: One of the original skythane colonists\n\nMartach: Six-legged jungle cat on Titania\n\nMattis Vinder: An OberCorp headquarters employee\n\nMemfiles: Memory files stored in a cirq and\/or the local grid\n\nMeso Bomb (MB): Bomb that works by averaging out the molecular content of everything within its radius\n\nMidcity: Heart of Oberon City\n\nMikelos: Miner on Tander's World\n\nMorgan Kinnson: Xander's foster father\n\nMora Mountains (Sl\u00e9ibhte Mora): Titania's mountian chain\n\nMountain, The: Location of one of the gates between the two worlds, close to the House of the Moon\n\nMRE: Meal Ready to Eat\n\nMugjuice: Oberon beverage made with pith\n\nNeamiah: Erriani person trapped with Quince in the mines\n\nNew Davos: City on Beta Tau\n\nNerve Cuffs: Handcuffs that cut off the nervous system\n\nNim: An Erriani man at Torr Talam\n\nNimfeach: Butterfly-like creatures on Titania\n\nNorcrest: Small purple bush found in the plains north of the Riamhwood\n\nNorthern Glacier: At the north pole on Oberon\n\nNutrisynth Bar: Nutritional bars Xander likes\n\nOberCorp: Corporation that controls most of Oberon\u2014The Oberon Mining Corporation\n\nOberon: Also known as Split\u2014the half world where the story takes place\n\nOberon City: Capital of Oberon, with about two million residents, most living in arcos\n\nObieberry: Native red stippled fruit on Oberon\n\nOrn: Main river on Titania\n\nOutland: Desert and wilds beyond Oberon City\n\nParaba Bush: Titania shrub that has berries that can be ground for oil\n\nPhilo: City on Oberon\n\nPith: Psychoamoratic drug derived from the sap of the p\u00faca tree in Titania\n\nPlas: Versatile artificial material with the hardness of diamond and the malleability of plastic\n\nPleiades Six: One of the worlds of the Common Worlds\n\nPlascreet: Variation of plas used in heavy construction; a concrete analogue\n\nPocans: Edible white fungus that resembles a string of pearls and tastes like chocolate and bread\n\nPod-mates: The three \"person\" family common among the ithani.\n\nPreachers, The: rough rock band\n\nPsych: Therapist\n\nPsych Guild: Association of Psychs\n\nPsychoamoratic: A drug with aphrodisiac qualities\n\nP\u00faca Tree: Tree which pith comes form\n\nPulse Laser: High-powered blast-pulse weapon used on transport ships\n\nPulse Pistol: Small pulse weapon\n\nPulse Rifle: Large pulse weapon\n\nPyramus Mountains: Mountain range along the Eastern edge of Oberon\n\nRangers: OberCorp soldiers used in military campaigns\n\nRavi: Xander's PA\n\nRavier: Erriani man who lives in Taycrob\n\nRed Sands: Desert covering the southern part of Oberon\n\nRed Shanks: Big red-trunked trees with large heart-shaped purple leaves, native to the Titanian jungle\n\nRedfruit: Fruit native to Titania\n\nRedoak: Tree native to Titania\n\nRentz Class Cargo Carrier: Heavy lifter that carries amalite ore up from Oberon to interstellars\n\nRhyl: One of Mylin's non-binary friends\n\nRiamhwood: Forest on Titania\n\nRiding Armor: Body armor Xander wears when riding\n\nRift: The split that divides Oberon and Titania\n\nRim Forest: Forest on the northern half of Oberon\n\nRinroot: A natural antibiotic on Titania\n\nRix: One of Xander's companions on the way to Errian\n\nRiver Apples: Native Oberon water fruit\n\nRiver Cat: Small scavenger the size and temperament of a raccoon\n\nRocthane: Access key to bring the worlds together\n\nRohin the Explorer: One of the first skythane to find Titania\n\nSailfighter: Ithani war ship. See also \"sleith\"\n\nSchracknell: Small winged creatures from the time of the ithani\n\nScurf: Domesticated, one horned herbivores on Titania\n\nSecond Wave: The second set of human colonists, also called landers\n\nSemi-Autonomous Grid\u2014a connected web of data run by a government entity or business, and connected to the main planetary grid\n\nSera Thorpe: OberCorp board member.\n\nSeven Weeks War: War between the Erriani and Gaelani twenty-five years before\n\nSevyrn Triani: One of the Erriani skythane Jameson saved\n\nShift: Moving Oberon into Titania's space\n\nShift Storm: The great tempest that accompanied the shift of Oberon to Titania's universe.\n\nSilverbark: Tall, thin tree with silver bark, leaves, and a dark stripe on the northern side, native to Oberon\n\nSkythane: First wave of human colonists, who have wings\n\nSlander, The: Slums of Oberon City\n\nSleeper: Sleep drug patch\n\nSleith: White sail ships of the ithani\n\nSlit: Transfer funds or information electronically\n\nSneach: skythane term for orphan\u2014mischievous spirits who cause trouble or death\n\nSplit: Nickname for Oberon; also used for its broken side.\n\nSquamwat: A small domesticated animal that imprints on its human owner and shows absolute loyalty\n\nStanding Stones: Guardian statues outside the faery caverns on Titania\n\nStendril: A sleek, fast fish in the lakes of the Mora Mountains\n\nStim: Stimulant\n\nStim Cuffs: Handcuffs that can deliver an electric shock\n\nSwamp Bear: Harmless forest creature\n\nSymbol of the Two Gods: Tapping your right fist against your chest twice\n\nSyndicate: Crime ring that controls the Slander\n\nSynth Meat: Meat grown in a vat from a cellular culture\n\nSynth Pack: A device to help regrow lost skin and tissue\n\nSynthglass: A more expensive form of plas that radiates light when touched\n\nTamara Fine: OberCorp's head of PR\n\nTander's World: Mining colony where Jameson was stationed\n\nTanner Michael Henshaw: Morgan's human half\n\nTartanga Tree: Oberon riverside trees with broad, tripartite silver leaves\n\nTaslit: A rite of passage for skythane youth\n\nTaycrob: Small Erriani village of tree houses\n\nTazim: Venin's lover, deceased\n\nTeanna: Skythane woman in Gaelan\n\nTeva Glynt, Dr.: Medic at OberCorp\n\nTevin: Erriani boy Jameson saves from the invaders\n\nTharsis: Home to one of the Tander's World miners, Mikelos\n\nThera: Prior Queen of the House of the Sun\n\nTheron Sl\u00e9ite: King of the Gaelani, Xander's father\n\nTheseus River: Main river in Oberon\n\nThousand Cycles War: War between the ithani and dhagani\n\nTitan Station: Receiving space station for visitors to Oberon\n\nTitania: Half of the planet on the other side of the rift\n\nTobin: Skythane man with Jameson in the mines\n\nToree: Erriani person trapped with Quince in the mines\n\nTorr Talam: Tower in the time of Elyra and Daedus\n\nTraxon: A smaller, light gravity Common Worlds planet\n\nTri-dee: 3-D video, video player\n\nTri-dee Table: the display device for holographic images and films\n\nToreor: Small Gaelani village where Dani Black was born.\n\nTubers: Native Oberon edible plant\u2014can be eaten cooked or raw, like a jicama\n\nTurbien: Plants used to grow the towers of Errian; another name for corrinder\n\nTweener: Someone who is nonbinary or gender fluid\n\nUrshu: The human\/ithani\/Oberon universe\n\nVassir Honym: Prison master on the Split\n\nVeril: a small rabbitlike creature that lives in the Riamhwood\n\nVerrim: One of the ithani from Jameson's memory\n\nVerthex: The Erro equivalent of cockroaches\n\nVirgo Sl\u00e9ite: Prior King of the House of the Moon\n\nVirtual Space (\"vir\"): full immersion in the grid\n\nWater Cane: Native Oberon edible plant\n\nWayabout: Skythane manhood ritual journey\n\nWempole: Short, squat trees with wide purple leaves and fragrant white flowers the size of two hands that smell like vanilla and honey, found in the northern Riamhwood\n\nWereveren: Birds that transform at night into lethal pecking machines\n\nWetreeds: Native plant with numbing properties\n\nWetware: Bio implants that allow humans to interface with machines and the grid\n\nWhipcat: Deadly feline equivalents in the mountains of Titania\n\nWhirill: Titanian bird that lives in the Riamhwood\n\nWing Man: Slang for the skythane\n\nWrenwood: Tree from Titania whose wood burns without smoke\n\nXiini: One of the ithani from Jameson's memory\n\nZ Pronouns: The Ithani have a complex (to us) system of pronouns and relationships. Three ithani are required to bear a child. These are ze \/ zer \/ zers (roughly analogous to she \/ her \/ hers \u2013 the one who provides the egg), zi \/ zim \/ zis (he \/ him \/ his \u2013 the one who provides the sperm), and za \/ zaf \/ zas (no human equivalent \u2013 the one who provides the organs to carry the child to term). In addition, when pregnancy occurs, the three forms unite into one (zee \/ zeer \/ zeers) until the child comes to term.\n\nZain: Sculptor of the statue of Erro at the House of the Stars\n\nZakka: Poisonous reptile in the Red Sands\n\nZaxxim: One of the ithani from Jameson's memory\n\nZenia: One of Xander's companions on the way to Errian\n\nZenix: Lena Preston's house AI\n\nZephyrCorp: The company that first \"discovered\" Oberon and opened it for human colonization\n\nZimbee: Large, harmless pollinator on Titania\n\nZorin: One of the ithani settlers sent to the dhagani homeworld\n\n# More from J. Scott Coatsworth\n\nThe Oberon Cycle: Book One\n\nJameson Havercamp, a psych from a conservative religious colony, has come to Oberon\u2014unique among the Common Worlds\u2014in search of a rare substance called pith. He's guided through the wilds on his quest by Xander Kinnson, a handsome, cocky skythane with a troubled past.\n\nNeither knows that Oberon is facing imminent destruction. Even as the world starts to fall apart around them, they have no idea what's coming\u2014or the bond that will develop between them as they race to avert a cataclysm.\n\nTogether, they will journey to uncover the secrets of this strange and singular world, even as it takes them beyond the bounds of reality itself to discover what truly binds them together.\n\nThe Oberon Cycle: Book Two\n\nSometimes the world needs saving twice.\n\nIn the sequel to the Rainbow-Award-winning _Skythane_ , Xander and Jameson thought they'd fulfilled their destiny when they brought the worlds of Oberon and Titania back together, but their short-lived moment of triumph is over.\n\nReunification has thrown the world into chaos. A great storm ravaged Xander's kingdom of Gaelan, leaving the winged skythane people struggling to survive. Their old enemy, Obercorp, is biding its time, waiting to strike. And to the north, a dangerous new adversary gathers strength, while an unexpected ally awaits them.\n\nIn the midst of it all, Xander's ex Alix returns, and Xander and Jameson discover that their love for each other may have been drug-induced.\n\nAre they truly destined for each other, or is what they feel concocted? And can they face an even greater challenge when their world needs them most?\n\nWhat if you could hear the words behind the words?\n\nBrad Weston's life seems perfect. He's _GQ_ handsome, the chief of staff for a Republican California state senator, and enjoys the power and the promise of a bright future. And he's in a comfortable relationship with his boyfriend of six years, Alex.\n\nSam Fuller is Brad's young blond blue-eyed intern, fresh out of college, running from a bad breakup, and questioning his choices and his new life in politics. To make things worse, Sam also has a thing for the boss, but Brad is already taken.\n\nWhile looking for a gift for his boyfriend, Brad wanders into a curiosity shop and becomes fascinated by an old wooden medallion. Brad's not a superstitious man, but when he takes out the medallion in his office, he sees the world in a whole new light.\n\nAnd nothing will ever be the same.\n\n# Readers love The Oberon Cycle by J. Scott Coatsworth\n\nSkythane\n\n\"Read it for the diverting fantasy, read it for the endearing characters, or read it for the agreeable plot\u2014no matter how you read Skythane, you'll love it and Coatsworth will quickly become one of your favorite authors.\"\n\n\u2014Queer Sci Fi\n\n\"Mr. Coatsworth has created a believable world and populated it with believable characters. It was a fast read, and I didn't want to put it down once I started it.\"\n\n\u2014Love Bytes\n\n\"I definitely recommend this to the science fiction fans and look forward to the next book!\"\n\n\u2014Hearts on Fire Reviews\n\nLander\n\n\"Do you like sci-fi? Are you a fan of fantasy? Then I would recommend this series in a heartbeat. With a fantastic combination of both, it kept me glued to the page and has me wanting more.\"\n\n\u2014The Novel Approach\n\n\"This series continues to get better and it's one you should experience for yourself.\"\n\n\u2014Jessie G Books\n\n\"I'd recommend Lander to readers who love science fiction with detailed world building, interesting three-dimensional kick-arse characters, and a story which keeps you turning pages far too late at night.\"\n\n\u2014Top 2 Bottom Reviews\nJ. SCOTT COATSWORTH lives with his husband Mark in a little yellow bungalow in East Sacramento with two pink flamingos out front. He spends his time between the here and now and the what could be. Ushered into fantasy and sci-fi at the tender age of nine by his mother, he devoured her library of Asimovs, Clarkes, and McCaffreys. But as he grew up, he wondered where the gay people were in speculative fiction.\n\nHe decided it was time to create the kinds of stories he couldn't find at Waldenbooks. If there weren't queer characters in his favorite genres, he would write them himself.\n\nHis friends say Scott's brain works a little differently\u2014he sees relationships between things that others miss, and often gets more done in a day than most folks manage in a week. He transforms traditional sci-fi, fantasy, and contemporary worlds into something unexpected.\n\nHe and Mark also run Queer Sci Fi, QueeRomance Ink, and Other Worlds Ink, sites that bring LGBTIQA communities together to celebrate fiction that reflects queer life and love.\n\nScott won a Rainbow Award as one of the best new gay authors of 2017.\n\nFacebook Profile: www.facebook.com\/jscottcoatsworth\n\nFacebook Author Page: www.facebook.com\/jscottcoatsworthauthor\/\n\nTwitter: www.twitter.com\/jscoatsworth\/\n\nAuthor Website\/Blog: www.jscottcoatsworth.com\n\nDreamspinner Page: www.dreamspinnerpress.com\/store\/index.php?cPath=55_1189\n\nQueeRomance Ink Author Page: www.queeromanceink.com\/mbm-book-author\/j-scott-coatsworth\/\n\nGoodreads Author Page: www.goodreads.com\/author\/show\/8392709.J_Scott_Coatsworth\n\nAmazon Author Page: www.amazon.com\/J.-Scott-Coatsworth\/e\/B011AFO4OQ\n\n# By J. Scott Coatsworth\n\nBetween the Lines\n\nA More Perfect Union Anthology\n\nTHE OBERON CYCLE\n\nSkythane\n\nLander\n\nIthani\n\nPublished by DREAMSPINNER PRESS\n\nwww.dreamspinnerpress.com\n\nPublished by\n\nDREAMSPINNER PRESS\n\n5032 Capital Circle SW, Suite 2, PMB# 279, Tallahassee, FL 32305-7886 USA\n\nwww.dreamspinnerpress.com\n\nThis is a work of fiction. Names, characters, places, and incidents either are the product of author imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events, or locales is entirely coincidental.\n\nIthani\n\n\u00a9 2019 J. Scott Coatsworth.\n\nCover Art\n\n\u00a9 2019 Tiferet Design\n\nhttp:\/\/www.tiferetdesign.com\/\n\nCover content is for illustrative purposes only and any person depicted on the cover is a model.\n\nAll rights reserved. This book is licensed to the original purchaser only. Duplication or distribution via any means is illegal and a violation of international copyright law, subject to criminal prosecution and upon conviction, fines, and\/or imprisonment. Any eBook format cannot be legally loaned or given to others. 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 the written permission of the Publisher, except where permitted by law. To request permission and all other inquiries, contact Dreamspinner Press, 5032 Capital Circle SW, Suite 2, PMB# 279, Tallahassee, FL 32305-7886, USA, or www.dreamspinnerpress.com.\n\nDigital ISBN: 978-1-64405-111-5\n\nTrade Paperback ISBN: 978-1-64405-112-2\n\nLibrary of Congress Control Number: 2018960691\n\nDigital published February 2019\n\nv. 1.0\n\nPrinted in the United States of America\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nELIZABETHAN\n\nTRAGEDIES\nELIZABETHAN\n\nTRAGEDIES\n\nA Basic Anthology\n\nDOVER PUBLICATIONS, INC.\n\nMINEOLA, NEW YORK\nDOVER THRIFT EDITIONS\n\nGENERAL EDITOR: SUSAN L. RATTINER\n\nEDITOR OF THIS VOLUME: JANET B. KOPITO\n\n_Copyright_\n\nCopyright \u00a9 2017 by Dover Publications, Inc.\n\nAll rights reserved.\n\n_Theatrical Rights_\n\nThis Dover Thrift edition may be used in its entirety, in adaptation, or in any other way for theatrical productions, professional and amateur, in the United States, without fee, permission, or acknowledgment. (This may not apply outside of the United States, as copyright conditions may vary.)\n\n_Bibliographical Note_\n\nThis Dover edition, first published in 2017, is a new compilation of plays reprinted from standard editions. A Note has been provided specially for this edition.\n\n_International Standard Book Number_\n\nISBN-13: 978-0-486-81332-5\n\nISBN-10: 0-486-81332-0\n\nManufactured in the United States by LSC Communications\n\n81332001 2017\n\nwww.doverpublications.com\n**Contents**\n\nThomas Kyd\n\n_The Spanish Tragedy_ (1587)\n\nChristopher Marlowe\n\n_Doctor Faustus_ (1594)\n\nThomas Heywood\n\n_A Woman Killed with Kindness_ (1603)\n\nElizabeth Cary\n\n_The Tragedy of Mariam_ (1612)\n\nJohn Webster\n\n_The Duchess of Malfi_ (1613)\n\n[The dates given for the plays are approximate, given the numerous versions published and performed at the time.]\n**Note**\n\nThe five plays selected for this collection represent a quarter century's worth of the great achievements of the Elizabethan era, and, as the successor to Queen Elizabeth's reign, the Jacobean period. Theaters were opening up in London, and poetry and music were flourishing. The well-educated queen brought her own curiosity and zest for learning to the general atmosphere of the time. It also is good to remember that some of Shakespeare's greatest tragedies\u2014 _Hamlet, Macbeth, Othello,_ and _King Lear_ \u2014were contemporaneous with works in this anthology.\n\nThomas Kyd's _Spanish Tragedy_ is typical of the \"revenge play,\" a popular format of the time. The plot is replete with murder and madness. Some aspects of _The Spanish Tragedy_ can be found in the work of the Roman dramatist Seneca, including the emphasis on violent outcomes and even the Ghost character, which may have been a source for _Hamlet_.\n\nIn _Doctor Faustus,_ Christopher Marlowe invokes the nefarious character depicted in the German _Faustbuch_ (1587), an astronomer and necromancer who was believed to have given up his soul to the Devil in order to receive supernatural powers. Faustus acknowledges the tragic consequences of his actions, however, and tells a group of scholarly gentlemen that his disease can't be cured by a physician, as \"A surfeit of deadly sin [that] hath damn'd both body and soul.\"\n\n_A Woman Killed with Kindness_ by Thomas Heywood offers a domestic setting for its plot: a happy marriage that is damaged by a husband's fateful decision. John and Anne Frankford share a pleasant home life; the disruption occurs when John brings a young acquaintance, Wendoll, into the house as his companion. The effect of this decision on Anne, and its impact on her wellbeing, spin the plot.\n\nIn addition to its literary merits, _The Tragedy of Mariam_ is noteworthy as the first work written by a woman and published under her own name. Set in 29 BCE, the play concerns Mariam, the \"faire queene of Jewry,\" and the violent events that take place at the court of the villainous King Herod, Mariam's husband. Drawing attention in the 1970s during the burgeoning growth of feminist studies and scholarship, _The Tragedy of Mariam_ examines themes of morality in a patriarchal society, including the impact of marriage and divorce, while displaying the brutal acts typical of the genre.\n\nThe concluding work, John Webster's _The Duchess of Malfi,_ shares the violent outcomes of the accompanying plays in this collection. Based on events in the life of Giovanna d'Aragone, Duchess of Amalfi (1478\u20131510) as depicted in Matteo Bandello's 1554 work _Novelle_ (translated into English as _The Palace of Pleasure_ , c. 1567), the play shows the lengths to which the Duchess's brothers will go to control her destiny. The manipulative Cardinal spies on the Duchess, and the Duchess's twin, Ferdinand, is obsessed with his sister's affairs as he descends into madness. Their cruel treatment of the Duchess for her decision to marry for love beneath her social class, prominent among other motives expressed by the brothers, moves the play's action to its inevitable, tragic conclusion.\nELIZABETHAN\n\nTRAGEDIES\n**THE SPANISH TRAGEDY**\n\n**Thomas Kyd**\n\n**DRAMATIS PERSON\u00c6**\n\nKING OF SPAIN\n\nCYPRIAN DUKE OF CASTILE, his brother\n\nLORENZO, the Duke's son\n\nBELLIMPERIA, Lorenzo's sister\n\nVICEROY OF PORTUGAL\n\nBALTHAZAR, his son\n\nDON PEDRO, the Viceroy's brother\n\nHIERONIMO, Marshal of Spain\n\nISABELLA, his wife\n\nHORATIO, their son\n\nSpanish General\n\nDeputy\n\nDON BAZULTO, an old man\n\nThree Citizens\n\nPortuguese Ambassador\n\nTwo Portuguese\n\nPEDRINCANO, Bellimperia's servant\n\nCHRISTOPHIL, Bellimperia's custodian\n\nLorenzo's Page\n\nCERBERINE, Balthazar's servant\n\nIsabella's Maid\n\nMessenger\n\nHangman\n\nThree Kings and three Knights in the first Dumb-show\n\nHymen and two torch-bearers in the second\n\nBAZARDO, a Painter\n\nPEDRO and JACQUES, Hieronimo's servants\n\nArmy. Banquet. Royal suites. Noblemen. Halberdiers.\n\nOfficers. Three Watchmen. Trumpets. Servants, etc.\n\n**ACT I**\n\n**S CENE I: _Induction_**\n\n_Enter the Ghost of Andrea, and with him Revenge._\n\n_Ghost_. When this eternal substance of my soul\n\nDid live imprison'd in my wanton flesh,\n\nEach in their function serving other's need,\n\nI was a courtier in the Spanish court:\n\nMy name was Don Andrea; my descent,\n\nThough not ignoble, yet inferior far\n\nTo gracious fortunes of my tender youth.\n\nFor there in prime and pride of all my years,\n\nBy duteous service and deserving love,\n\nIn secret I possess'd a worthy dame,\n\nWhich hight sweet Bellimperia by name.\n\nBut, in the harvest of my summer joys,\n\nDeath's winter nipp'd the blossoms of my bliss,\n\nForcing divorce betwixt my love and me.\n\nFor in the late conflict with Portingal\n\nMy valour drew me into danger's mouth,\n\nTill life to death made passage through my wounds.\n\nWhen I was slain, my soul descended straight\n\nTo pass the flowing stream of Acheron;\n\nBut churlish Charon, only boatman there,\n\nSaid that, my rites of burial not perform'd,\n\nI might not sit amongst his passengers.\n\nEre Sol had slept three nights in Thetis' lap,\n\nAnd slak'd his smoking chariot in her flood,\n\nBy Don Horatio, our knight marshal's son,\n\nMy funerals and obsequies were done.\n\nThen was the ferryman of hell content\n\nTo pass me over to the slimy strand,\n\nThat leads to fell Avernus' ugly waves.\n\nThere, pleasing Cerberus with honey'd speech,\n\nI pass'd the perils of the foremost porch.\n\nNot far from hence, amidst ten thousand souls,\n\nSat Minos, Aeacus, and Rhadamanth;\n\nTo whom no sooner 'gan I make approach,\n\nTo crave a passport for my wand'ring ghost,\n\nBut Minos, in graven leaves of lottery,\n\nDrew forth the manner of my life and death.\n\n\"This knight,\" quoth he, 'both liv'd and died in love;\n\nAnd for his love tried fortune of the wars;\n\nAnd by war's fortune lost both love and life.'\n\n\"Why then,\" said Aeacus, 'convey him hence,\n\nTo walk with lovers in our fields of love,\n\nAnd spend the course of everlasting time\n\nUnder green myrtle-trees and cypress shades.'\n\n\"No, no,\" said Rhadamanth, 'it were not well;\n\nWith loving souls to place a martialist:\n\nHe died in war, and must to martial fields,\n\nWhere wounded Hector lives in lasting pain,\n\nAnd Achilles' Myrmidons do scour the plain.'\n\nThen Minos, mildest censor of the three,\n\nMade this device to end the difference:\n\n\"Send him,\" quoth he, 'to our infernal king,\n\nTo doom him as best seems his majesty.'\n\nTo this effect my passport straight was drawn.\n\nIn keeping on my way to Pluto's court,\n\nThrough dreadful shades of ever-glooming night,\n\nI saw more sights than thousand tongues can tell,\n\nOr pens can write, or mortal hearts can think.\n\nThree ways there were: that on the right-hand side\n\nWas ready way unto the 'foresaid fields,\n\nWhere lovers live and bloody martialists;\n\nBut either sort contain'd within his bounds.\n\nThe left-hand path, declining fearfully,\n\nWas ready downfall to the deepest hell,\n\nWhere bloody Furies shakes their whips of steel,\n\nAnd poor Ixion turns an endless wheel;\n\nWhere usurers are chok'd with melting gold,\n\nAnd wantons are embrac'd with ugly snakes,\n\nAnd murd'rers groan with never-killing wounds,\n\nAnd perjur'd wights scalded in boiling lead,\n\nAnd all foul sins with torments overwhelm'd.\n\n'Twixt these two ways I trod the middle path,\n\nWhich brought me to the fair Elysian green,\n\nIn midst whereof there stands a stately tower,\n\nThe walls of brass, the gates of adamant:\n\nHere finding Pluto with his Proserpine,\n\nI show'd my passport, humbled on my knee;\n\nWhereat fair Proserpine began to smile,\n\nAnd begg'd that only she might give my doom:\n\nPluto was pleas'd, and seal'd it with a kiss.\n\nForthwith, Revenge, she rounded thee in th' ear,\n\nAnd bad thee lead me through the gates of horn,\n\nWhere dreams have passage in the silent night.\n\nNo sooner had she spoke, but we were here\u2014\n\nI wot not how\u2014in twinkling of an eye.\n\n_Revenge._ Then know, Andrea, that thou art arriv'd\n\nWhere thou shalt see the author of thy death,\n\nDon Balthazar, the prince of Portingal,\n\nDepriv'd of life by Bellimperia.\n\nHere sit we down to see the mystery,\n\nAnd serve for Chorus in this tragedy.\n\n**S CENE II**\n\n**_The Court of Spain._**\n\n_Enter Spanish King, General, Castile, and Hieronimo._\n\n_King._ Now say, lord General, how fares our camp?\n\n_Gen._ All well, my sovereign liege, except some few\n\nThat are deceas'd by fortune of the war.\n\n_King._ But what portends thy cheerful countenance,\n\nAnd posting to our presence thus in haste?\n\nSpeak, man, hath fortune given us victory?\n\n_Gen._ Victory, my liege, and that with little loss.\n\n_King._ Our Portingals will pay us tribute then?\n\n_Gen._ Tribute and wonted homage therewithal.\n\n_King._ Then bless'd be heaven and guider of the heavens,\n\nFrom whose fair influence such justice flows.\n\nCast. _O multum dilecte Deo, tibi militat aether,_\n\n_Et conjuratae curvato poplite gentes_\n\n_Succumbunt: recti soror est victoria juris._\n\n_King._ Thanks to my loving brother of Castile.\n\nBut, General, unfold in brief discourse\n\nYour form of battle and your war's success,\n\nThat, adding all the pleasure of thy news\n\nUnto the height of former happiness,\n\nWith deeper wage and greater dignity\n\nWe may reward thy blissful chivalry.\n\n_Gen._ Where Spain and Portingal do jointly knit\n\nTheir frontiers, leaning on each other's bound,\n\nThere met our armies in their proud array:\n\nBoth furnish'd well, both full of hope and fear,\n\nBoth menacing alike with daring shows,\n\nBoth vaunting sundry colours of device,\n\nBoth cheerly sounding trumpets, drums, and fifes,\n\nBoth raising dreadful clamours to the sky,\n\nThat valleys, hills, and rivers made rebound,\n\nAnd heav'n itself was frighted with the sound.\n\nOur battles both were pitch'd in squadron form,\n\nEach corner strongly fenc'd with wings of shot;\n\nBut ere we join'd and came to push of pike,\n\nI brought a squadron of our readiest shot\n\nFrom out our rearward, to begin the fight:\n\nThey brought another wing t' encounter us.\n\nMeanwhile, our ordnance play'd on either side,\n\nAnd captains strove to have their valours tried.\n\nDon Pedro, their chief horsemen's colonel,\n\nDid with his cornet bravely make attempt\n\nTo break the order of our battle ranks:\n\nBut Don Rogero, worthy man of war,\n\nMarch'd forth against him with our musketeers,\n\nAnd stopp'd the malice of his fell approach.\n\nWhile they maintain hot skirmish to and fro,\n\nBoth battles join, and fall to handy-blows,\n\nTheir violent shot resembling th' ocean's rage,\n\nWhen, roaring loud, and with a swelling tide,\n\nIt beats upon the rampiers of huge rocks,\n\nAnd gapes to swallow neighbour-bounding lands.\n\nNow while Bellona rageth here and there,\n\nThick storms of bullets ran like winter's hail,\n\nAnd shiver'd lances dark the troubled air.\n\n_Pede pes et cuspide cuspis;_\n\n_Arma sonant armis, vir petiturque viro._\n\nOn every side drop captains to the ground,\n\nAnd soldiers, some ill-maim'd, some slain outright:\n\nHere falls a body sunder'd from his head,\n\nThere legs and arms lie bleeding on the grass,\n\nMingled with weapons and unbowell'd steeds,\n\nThat scatt'ring overspread the purple plain.\n\nIn all this turmoil, three long hours and more,\n\nThe victory to neither part inclin'd;\n\nTill Don Andrea, with his brave lanciers,\n\nIn their main battle made so great a breach,\n\nThat, half dismay'd, the multitude retir'd:\n\nBut Balthazar, the Portingals' young prince,\n\nBrought rescue, and encourag'd them to stay.\n\nHere-hence the fight was eagerly renew'd,\n\nAnd in that conflict was Andrea slain:\n\nBrave man at arms, but weak to Balthazar.\n\nYet while the prince, insulting over him,\n\nBreath'd out proud vaunts, sounding to our reproach,\n\nFriendship and hardy valour, join'd in one,\n\nPrick'd forth Horatio, our knight marshal's son,\n\nTo challenge forth that prince in single fight.\n\nNot long between these twain the fight endur'd,\n\nBut straight the prince was beaten from his horse,\n\nAnd forc'd to yield him prisoner to his foe.\n\nWhen he was taken, all the rest they fled,\n\nAnd our carbines pursu'd them to the death,\n\nTill, Ph\u0153bus waving to the western deep,\n\nOur trumpeters were charg'd to sound retreat.\n\n_King._ Thanks, good lord General, for these good news;\n\nAnd for some argument of more to come,\n\nTake this and wear it for thy sovereign's sake.\n\n[ _Gives him his chain._\n\nBut tell me now, hast thou confirm'd a peace?\n\n_Gen._ No peace, my liege, but peace conditional,\n\nThat if with homage tribute be well paid,\n\nThe fury of your forces will be stay'd:\n\nAnd to this peace their viceroy hath subscrib'd,\n\n[ _Gives the King a paper._\n\nAnd made a solemn vow that, during life,\n\nHis tribute shall be truly paid to Spain.\n\n_King._ These words, these deeds, become thy person well.\n\nBut now, knight marshal, frolic with thy king,\n\nFor 'tis thy son that wins this battle's prize.\n\n_Hier._ Long may he live to serve my sovereign liege,\n\nAnd soon decay, unless he serve my liege.\n\n_King._ Nor thou, nor he, shall die without reward.\n\n[ _A tucket afar off._\n\nWhat means the warning of this trumpet's sound?\n\n_Gen._ This tells me that your grace's men of war,\n\nSuch as war's fortune hath reserv'd from death,\n\nCome marching on towards your royal seat,\n\nTo show themselves before your majesty:\n\nFor so I gave in charge at my depart.\n\nWhereby by demonstration shall appear,\n\nThat all, except three hundred or few more,\n\nAre safe return'd, and by their foes enrich'd.\n\n_The Army enters; Balthazar, between Lorenzo and Horatio, captive._\n\n_King._ A gladsome sight! I long to see them here.\n\n[ _They enter and pass by._\n\nWas that the warlike prince of Portingal,\n\nThat by our nephew was in triumph led?\n\n_Gen._ It was, my liege, the prince of Portingal.\n\n_King._ But what was he that on the other side\n\nHeld him by th' arm, as partner of the prize?\n\n_Hier._ That was my son, my gracious sovereign;\n\nOf whom though from his tender infancy\n\nMy loving thoughts did never hope but well,\n\nHe never pleas'd his father's eyes till now,\n\nNor fill'd my heart with over-cloying joys.\n\n_King._ Go, let them march once more about these walls,\n\nThat, staying them, we may confer and talk\n\nWith our brave prisoner and his double guard.\n\nHieronimo, it greatly pleaseth us\n\nThat in our victory thou have a share,\n\nBy virtue of thy worthy son's exploit [ _Enter again._\n\nBring hither the young prince of Portingal:\n\nThe rest march on; but, ere they be dismiss'd,\n\nWe will bestow on every soldier\n\nTwo ducats and on every leader ten,\n\nThat they may know our largess welcomes them.\n\n[ _Exeunt all but Balthazar, Lorenzo, and Horatio._\n\nWelcome, Don Balthazar! welcome, nephew!\n\nAnd thou, Horatio, thou art welcome too.\n\nYoung prince, although thy father's hard misdeeds,\n\nIn keeping back the tribute that he owes,\n\nDeserve but evil measure at our hands,\n\nYet shalt thou know that Spain is honourable.\n\n_Bal._ The trespass that my father made in peace\n\nIs now controll'd by fortune of the wars;\n\nAnd cards once dealt, it boots not ask why so.\n\nHis men are slain, a weak'ning to his realm;\n\nHis colours seiz'd, a blot unto his name;\n\nHis son distress'd, a cor'sive to his heart:\n\nThese punishments may clear his late offence.\n\n_King._ Ay, Balthazar, if he observe this truce,\n\nOur peace will grow the stronger for these wars.\n\nMeanwhile live thou, though not in liberty,\n\nYet free from bearing any servile yoke;\n\nFor in our hearing thy deserts were great,\n\nAnd in our sight thyself art gracious.\n\n_Bal._ And I shall study to deserve this grace.\n\n_King._ But tell me\u2014for their holding makes me doubt\u2014\n\nTo which of these twain art thou prisoner?\n\n_Lor._ To me, my liege.\n\n_Hor._ To me, my sovereign.\n\n_Lor._ This hand first took his courser by the reins.\n\n_Hor._ But first my lance did put him from his horse.\n\n_Lor._ I seiz'd his weapon, and enjoy'd it first.\n\n_Hor._ But first I forc'd him lay his weapons down.\n\n_King._ Let go his arm, upon our privilege.\n\n[ _They let him go._\n\nSay, worthy prince, to whether did'st thou yield?\n\n_Bal._ To him in courtesy, to this perforce:\n\nHe spake me fair, this other gave me strokes;\n\nHe promis'd life, this other threaten'd death;\n\nHe won my love, this other conquer'd me,\n\nAnd, truth to say, I yield myself to both.\n\n_Hier._ But that I know your grace for just and wise,\n\nAnd might seem partial in this difference,\n\nEnforc'd by nature and by law of arms\n\nMy tongue should plead for young Horatio's right:\n\nHe hunted well that was a lion's death,\n\nNot he that in a garment wore his skin;\n\nSo hares may pull dead lions by the beard.\n\n_King._ Content thee, marshal, thou shalt have no wrong;\n\nAnd, for thy sake, thy son shall want no right.\n\nWill both abide the censure of my doom?\n\n_Lor._ I crave no better than your grace awards.\n\n_Hor._ Nor I, although I sit beside my right\n\n_King._ Then, by my judgment, thus your strife shall end:\n\nYou both deserve, and both shall have reward.\n\nNephew, thou took'st his weapon and his horse:\n\nHis weapons and his horse are thy reward.\n\nHoratio, thou did'st force him first to yield:\n\nHis ransom therefore is thy valour's fee;\n\nAppoint the sum, as you shall both agree.\n\nBut, nephew, thou shalt have the prince in guard,\n\nFor thine estate best fitteth such a guest:\n\nHoratio's house were small for all his train.\n\nYet, in regard thy substance passeth his,\n\nAnd that just guerdon may befall desert,\n\nTo him we yield the armour of the prince.\n\nHow likes Don Balthazar of this device?\n\n_Bal._ Right well, my liege, if this proviso were,\n\nThat Don Horatio bear us company,\n\nWhom I admire and love for chivalry.\n\n_King._ Horatio, leave him not that loves thee so.\u2014\n\nNow let us hence to see our soldiers paid,\n\nAnd feast our prisoner as our friendly guest\n\n[ _Exeunt._\n\n**S CENE III**\n\n**_The Court of Portugal._**\n\n_Enter Viceroy, Alexandro, Villuppo._\n\n_Vic._ Is our ambassador despatch'd for Spain?\n\n_Alex._ Two days, my liege, are past since his depart.\n\n_Vic._ And tribute-payment gone along with him?\n\n_Alex._ Ay, my good lord.\n\n_Vic._ Then rest we here awhile in our unrest,\n\nAnd feed our sorrows with some inward sighs;\n\nFor deepest cares break never into tears.\n\nBut wherefore sit I in a regal throne?\n\nThis better fits a wretch's endless moan.\n\n[ _Falls to the ground._\n\nYet this is higher than my fortunes reach,\n\nAnd therefore better than my state deserves.\n\nAy, ay, this earth, image of melancholy,\n\nSeeks him whom fates adjudge to misery.\n\nHere let me lie; now am I at the lowest.\n\n_Qui jacet in terra, non habet unde cadat._\n\n_In me consumpsit vires fortuna nocendo:_\n\n_Nil superest ut jam possit obesse magis._\n\nYes, Fortune may bereave me of my crown:\n\nHere, take it now;\u2014let Fortune do her worst,\n\nShe will not rob me of this sable weed:\n\nO no, she envies none but pleasant things.\n\nSuch is the folly of despiteful chance!\n\nFortune is blind, and sees not my deserts;\n\nSo is she deaf, and hears not my laments;\n\nAnd could she hear, yet is she wilful-mad,\n\nAnd therefore will not pity my distress.\n\nSuppose that she could pity me, what then?\n\nWhat help can be expected at her hands\n\nWhose foot is standing on a rolling stone,\n\nAnd mind more mutable than fickle winds?\n\nWhy wail I then, where's hope of no redress?\n\nO yes, complaining makes my grief seem less.\n\nMy late ambition hath distain'd my faith;\n\nMy breach of faith occasion'd bloody wars;\n\nThose bloody wars have spent my treasure;\n\nAnd with my treasure my people's blood;\n\nAnd with their blood, my joy and best belov'd,\n\nMy best belov'd, my sweet and only son.\n\nO, wherefore went I not to war myself?\n\nThe cause was mine; I might have died for both:\n\nMy years were mellow, his but young and green;\n\nMy death were natural, but his was forc'd.\n\n_Alex._ No doubt, my liege, but still the prince survives.\n\n_Vic._ Survives! ay, where?\n\n_Alex._ In Spain\u2014a prisoner by mischance of war.\n\n_Vic._ Then they have slain him for his father's fault\n\n_Alex._ That were a breach to common law of arms.\n\n_Vic._ They reck no laws that meditate revenge.\n\n_Alex._ His ransom's worth will stay from foul revenge.\n\n_Vic._ No; if he liv'd, the news would soon be here.\n\n_Alex._ Nay, evil news fly faster still than good.\n\n_Vic._ Tell me no more of news; for he is dead.\n\n_Vil._ My sovereign, pardon the author of ill news,\n\nAnd I'll bewray the fortune of thy son.\n\n_Vic._ Speak on, I'll guerdon thee, whate'er it be:\n\nMine ear is ready to receive ill news;\n\nMy heart grown hard 'gainst mischief's battery.\n\nStand up, I say, and tell thy tale at large.\n\n_Vil._ Then hear that truth which these mine eyes have seen:\n\nWhen both the armies were in battle join'd,\n\nDon Balthazar, amidst the thickest troops,\n\nTo win renown did wondrous feats of arms:\n\nAmongst the rest I saw him, hand to hand,\n\nIn single fight with their lord-general;\n\nTill Alexandro, that here counterfeits,\n\nUnder the colour of a duteous friend\n\nDischarg'd his pistol at the prince's back,\n\nAs though he would have slain their general:\n\nBut therewithal Don Balthazar fell down;\n\nAnd when he fell, then we began to fly:\n\nBut, had he liv'd, the day had sure been ours.\n\n_Alex._ O wicked forgery! O trait'rous miscreant!\n\n_Vic._ Hold thou thy peace! But now, Villuppo, say,\n\nWhere then became the carcase of my son?\n\n_Vil._ I saw them drag it to the Spanish tents.\n\n_Vic._ Ay, ay, my nightly dreams have told me this.\u2014\n\nThou false, unkind, unthankful, trait'rous beast,\n\nWherein had Balthazar offended thee,\n\nThat thou shouldst thus betray him to our foes?\n\nWas't Spanish gold that blear\u00e8d so thine eyes\n\nThat thou couldst see no part of our deserts?\n\nPerchance, because thou art Terceira's lord,\n\nThou hadst some hope to wear this diadem,\n\nIf first my son and then myself were slain;\n\nBut thy ambitious thought shall break thy neck.\n\nAy, this was it that made thee spill his blood:\n\n[ _Takes the crown and puts it on again._\n\nBut I'll now wear it till thy blood be spilt.\n\n_Alex._ Vouchsafe, dread sovereign, to hear me speak.\n\n_Vic._ Away with him; his sight is second hell.\n\nKeep him till we determine of his death:\n\nIf Balthazar be dead, he shall not live.\n\nVilluppo, follow us for thy reward. [ _Exit Viceroy._\n\n_Vil._ Thus have I with an envious, forged tale\n\nDeceiv'd the king, betray'd mine enemy,\n\nAnd hope for guerdon of my villany. [ _Exit._\n\n**S CENE IV**\n\n_Enter Horatio and Bellimperia._\n\n_Bel._ Signior Horatio, this is the place and hour,\n\nWherein I must entreat thee to relate\n\nThe circumstance of Don Andrea's death,\n\nWho, living, was my garland's sweetest flower,\n\nAnd in his death hath buried my delights.\n\n_Hor._ For love of him and service to yourself,\n\nI nill refuse this heavy doleful charge;\n\nYet tears and sighs, I fear, will hinder me.\n\nWhen both our armies were enjoin'd in fight,\n\nYour worthy chevalier amidst the thickest,\n\nFor glorious cause still aiming at the fairest,\n\nWas at the last by young Don Balthazar\n\nEncounter'd hand to hand: their fight was long,\n\nTheir hearts were great, their clamours menacing,\n\nTheir strength alike, their strokes both dangerous.\n\nBut wrathful Nemesis, that wicked power,\n\nEnvying at Andrea's praise and worth,\n\nCut short his life, to end his praise and worth.\n\nShe, she herself, disguis'd in armour's mask\u2014\n\nAs Pallas was before proud Pergamus\u2014\n\nBrought in a fresh supply of halberdiers,\n\nWhich paunch'd his horse, and ding'd him to the ground.\n\nThen young Don Balthazar with ruthless rage,\n\nTaking advantage of his foe's distress,\n\nDid finish what his halberdiers begun,\n\nAnd left not, till Andrea's life was done.\n\nThen, though too late, incens'd with just remorse,\n\nI with my band set forth against the prince,\n\nAnd brought him prisoner from his halberdiers.\n\n_Bel._ Would thou hadst slain him that so slew my love!\n\nBut then was Don Andrea's carcase lost?\n\n_Hor._ No, that was it for which I chiefly strove,\n\nNor stepp'd I back till I recover'd him:\n\nI took him up, and wound him in mine arms;\n\nAnd wielding him unto my private tent,\n\nThere laid him down, and dew'd him with my tears,\n\nAnd sigh'd and sorrow'd as became a friend.\n\nBut neither friendly sorrow, sighs, nor tears\n\nCould win pale Death from his usurp\u00e8d right.\n\nYet this I did, and less I could not do:\n\nI saw him honour'd with due funeral.\n\nThis scarf I pluck'd from off his lifeless arm,\n\nAnd wear it in remembrance of my friend.\n\n_Bel._ I know the scarf: would he had kept it still;\n\nFor had he liv'd, he would have kept it still,\n\nAnd worn it for his Bellimperia's sake:\n\nFor 'twas my favour at his last depart.\n\nBut now wear thou it both for him and me;\n\nFor after him thou hast deserv'd it best\n\nBut for thy kindness in his life and death,\n\nBe sure, while Bellimperia's life endures,\n\nShe will be Don Horatio's thankful friend.\n\n_Hor._ And, madam, Don Horatio will not slack\n\nHumbly to serve fair Bellimperia.\n\nBut now, if your good liking stand thereto,\n\nI'll crave your pardon to go seek the prince;\n\nFor so the duke, your father, gave me charge.\n\n_Bel._ Ay, go, Horatio, leave me here alone;\n\nFor solitude best fits my cheerless mood. [ _Exit Hor._\n\nYet what avails to wail Andrea's death,\n\nFrom whence Horatio proves my second love?\n\nHad he not lov'd Andrea as he did,\n\nHe could not sit in Bellimperia's thoughts.\n\nBut how can love find harbour in my breast,\n\nTill I revenge the death of my belov'd?\n\nYes, second love shall further my revenge!\n\nI'll love Horatio, my Andrea's friend,\n\nThe more to spite the prince that wrought his end,\n\nAnd where Don Balthazar, that slew my love,\n\nHimself now pleads for favour at my hands,\n\nHe shall, in rigour of my just disdain,\n\nReap long repentance for his murd'rous deed.\n\nFor what was 't else but murd'rous cowardice,\n\nSo many to oppress one valiant knight,\n\nWithout respect of honour in the fight?\n\nAnd here he comes that murder'd my delight.\n\n_Enter Lorenzo and Balthazar._\n\n_Lor._ Sister, what means this melancholy walk?\n\n_Bel._ That for a while I wish no company.\n\n_Lor._ But here the prince is come to visit you.\n\n_Bel._ That argues that he lives in liberty.\n\n_Bal._ No, madam, but in pleasing servitude.\n\n_Bel._ Your prison then, belike, is your conceit.\n\n_Bal._ Ay, by conceit my freedom is enthrall'd.\n\n_Bel._ Then with conceit enlarge yourself again.\n\n_Bal._ What, if conceit have laid my heart to gage?\n\n_Bel._ Pay that you borrow'd, and recover it.\n\n_Bal._ I die, if it return from whence it lies.\n\n_Bel._ A heartless man, and live? A miracle!\n\n_Bal._ Ay, lady, love can work such miracles.\n\n_Lor._ Tush, tush, my lord! let go these ambages,\n\nAnd in plain terms acquaint her with your love.\n\n_Bel._ What boots complaint, when there's no remedy?\n\n_Bal._ Yes, to your gracious self must I complain,\n\nIn whose fair answer lies my remedy;\n\nOn whose perfection all my thoughts attend;\n\nOn whose aspect mine eyes find beauty's bower;\n\nIn whose translucent breast my heart is lodg'd.\n\n_Bel._ Alas, my lord, these are but words of course,\n\nAnd but device to drive me from this place.\n\n[ _She, in going in, lets fall her glove, which Horatio, coming out, takes up._\n\nH _or_. Madam, your glove.\n\n_Bel._ Thanks, good Horatio; take it for thy pains.\n\n_Bal._ Signior Horatio stoop'd in happy time!\n\nH _or_. I reap'd more grace than I deserv'd or hop'd.\n\n_Lor._ My lord, be not dismay'd for what is past:\n\nYou know that women oft are humorous;\n\nThese clouds will overblow with little wind:\n\nLet me alone, I'll scatter them myself.\n\nMeanwhile, let us devise to spend the time\n\nIn some delightful sports and revelling.\n\nH _or_. The king, my lords, is coming hither straight,\n\nTo feast the Portingal ambassador;\n\nThings were in readiness before I came.\n\n_Bal._ Then here it fits us to attend the king,\n\nTo welcome hither our ambassador,\n\nAnd learn my father and my country's health.\n\n**S CENE V**\n\n_Enter the Banquet, Trumpets, the King, and Ambassador._\n\n_King._ See, lord Ambassador, how Spain entreats\n\nTheir prisoner Balthazar, thy viceroy's son:\n\nWe pleasure more in kindness than in wars.\n\n_Amb._ Sad is our king, and Portingal laments,\n\nSupposing that Don Balthazar is slain.\n\n_Bal._ So am I!\u2014slain by beauty's tyranny.\n\nYou see, my lord, how Balthazar is slain:\n\nI frolic with the Duke of Castile's son,\n\nWrapp'd every hour in pleasures of the court,\n\nAnd grac'd with favours of his majesty.\n\n_King._ Put off your greetings, till our feast be done;\n\nNow come and sit with us, and taste our cheer.\n\n[ _Sit to the banquet_\n\nSit down, young prince, you are our second guest;\n\nBrother, sit down; and, nephew, take your place.\n\nSignior Horatio, wait thou upon our cup;\n\nFor well thou hast deserv\u00e8d to be honour'd.\n\nNow, lordings, fall to; Spain is Portugal,\n\nAnd Portugal is Spain: we both are friends;\n\nTribute is paid, and we enjoy our right.\n\nBut where is old Hieronimo, our marshal?\n\nHe promis'd us, in honour of our guest,\n\nTo grace our banquet with some pompous jest\n\n_Enter Hieronimo with a drum, three knights, each his scutcheon; then he fetches three kings, they take their crowns and them captive._\n\nHieronimo, this masque contents mine eye,\n\nAlthough I sound not well the mystery.\n\n_Hier._ The first arm'd knight, that hung his scutcheon up,\n\n[ _He takes the scutcheon and gives it to the King._\n\nWas English Robert, Earl of Gloucester,\n\nWho, when King Stephen bore sway in Albion,\n\nArriv'd with five and twenty thousand men\n\nIn Portingal, and by success of war\n\nEnforc'd the king, then but a Saracen,\n\nTo bear the yoke of the English monarchy.\n\n_King._ My lord of Portingal, by this you see\n\nThat which may comfort both your king and you,\n\nAnd make your late discomfort seem the less.\n\nBut say, Hieronimo, what was the next?\n\n_Hier._ The second knight, that hung his scutcheon up,\n\n[ _He doth as he did before._\n\nWas Edmond, Earl of Kent in Albion,\n\nWhen English Richard wore the diadem.\n\nHe came likewise, and raz\u00e8d Lisbon walls,\n\nAnd took the King of Portingal in fight;\n\nFor which and other such-like service done\n\nHe after was created Duke of York.\n\n_King._ This is another special argument,\n\nThat Portingal may deign to bear our yoke,\n\nWhen it by little England hath been yok'd.\n\nBut now, Hieronimo, what were the last?\n\n_Hier._ The third and last, not least, in our account,\n\n[ _Doing as before._\n\nWas, as the rest, a valiant Englishman,\n\nBrave John of Gaunt, the Duke of Lancaster,\n\nAs by his scutcheon plainly may appear.\n\nHe with a puissant army came to Spain,\n\nAnd took our King of Castile prisoner.\n\n_Amb._ This is an argument for our viceroy\n\nThat Spain may not insult for her success,\n\nSince English warriors likewise conquer'd Spain,\n\nAnd made them bow their knees to Albion.\n\n_King._ Hieronimo, I drink to thee for this device,\n\nWhich hath pleas'd both the ambassador and me:\n\nPledge me, Hieronimo, if thou love thy king.\n\n[ _Takes the cup of Horatio._\n\nMy lord, I fear we sit but over-long,\n\nUnless our dainties were more delicate;\n\nBut welcome are you to the best we have.\n\nNow let us in, that you may be despatch'd:\n\nI think our council is already set. [ _Exeunt omnes._\n\n**S CENE VI**\n\n**_Ghost of Andrea, Revenge._**\n\n_Andrea._ Come we for this from depth of underground,\n\nTo see him feast that gave me my death's wound?\n\nThese pleasant sights are sorrow to my soul:\n\nNothing but league, and love, and banqueting?\n\n_Revenge,_ Be still, Andrea; ere we go from hence,\n\nI'll turn their friendship into fell despite,\n\nTheir love to mortal hate, their day to night,\n\nTheir hope into despair, their peace to war,\n\nTheir joys to pain, their bliss to misery.\n\n**ACT II**\n\n**SCENE I**\n\n_Enter Lorenzo and Balthazar._\n\n_Lor._ My lord, though Bellimperia seem thus coy,\n\nLet reason hold you in your wonted joy:\n\nIn time the savage bull sustains the yoke,\n\nIn time all haggard hawks will stoop to lure,\n\nIn time small wedges cleave the hardest oak,\n\nIn time the flint is pierc'd with softest shower,\n\nAnd she in time will fall from her disdain,\n\nAnd rue the suff'rance of your friendly pain.\n\n_Bal._ No, she is wilder, and more hard withal,\n\nThan beast, or bird, or tree, or stony wall.\n\nBut wherefore blot I Bellimperia's name?\n\nIt is my fault, not she, that merits blame.\n\nMy feature is not to content her sight,\n\nMy words are rude, and work her no delight.\n\nThe lines I send her are but harsh and ill,\n\nSuch as do drop from Pan and Marsyas' quill.\n\nMy presents are not of sufficient cost,\n\nAnd being worthless, all my labour's lost.\n\nYet might she love me for my valiancy:\n\nAy, but that's slander'd by captivity.\n\nYet might she love me to content her sire:\n\nAy, but her reason masters his desire.\n\nYet might she love me as her brother's friend:\n\nAy, but her hopes aim at some other end.\n\nYet might she love me to uprear her state:\n\nAy, but perhaps she hopes some nobler mate.\n\nYet might she love me as her beauty's thrall:\n\nAy, but I fear she cannot love at all.\n\n_Lor._ My lord, for my sake leave this ecstasy,\n\nAnd doubt not but we'll find some remedy.\n\nSome cause there is that lets you not be lov'd;\n\nFirst that must needs be known, and then remov'd.\n\nWhat, if my sister love some other knight?\n\n_Bal._ My summer's day will turn to winter's night.\n\n_Lor._ I have already found a stratagem,\n\nTo sound the bottom of this doubtful theme.\n\nMy lord, for once you shall be rul'd by me;\n\nHinder me not, whate'er you hear or see.\n\nBy force or fair means will I cast about\n\nTo find the truth of all this question out.\n\nHo, Pedringano!\n\nPed. _Signior!_\n\nLor. _Vien qui presto._\n\n_Enter Pedringano._\n\n_Ped._ Hath your lordship any service to command me?\n\n_Lor._ Ay, Pedringano, service of import;\n\nAnd\u2014not to spend the time in trifling words\u2014\n\nThus stands the case: It is not long, thou know'st,\n\nSince I did shield thee from my father's wrath,\n\nFor thy conveyance in Andrea's love,\n\nFor which thou wert adjudg'd to punishment:\n\nI stood betwixt thee and thy punishment,\n\nAnd since, thou know'st how I have favour'd thee.\n\nNow to these favours will I add reward,\n\nNot with fair words, but store of golden coin,\n\nAnd lands and living join'd with dignities,\n\nIf thou but satisfy my just demand:\n\nTell truth, and have me for thy lasting friend.\n\n_Ped._ Whate'er it be your lordship shall demand,\n\nMy bounden duty bids me tell the truth,\n\nIf case it lie in me to tell the truth.\n\n_Lor._ Then, Pedringano, this is my demand:\n\nWhom loves my sister Bellimperia?\n\nFor she reposeth all her trust in thee.\n\nSpeak, man, and gain both friendship and reward:\n\nI mean, whom loves she in Andrea's place?\n\n_Ped._ Alas, my lord, since Don Andrea's death\n\nI have no credit with her as before;\n\nAnd therefore know not, if she love or no.\n\n_Lor._ Nay, if thou dally, then I am thy foe,\n\n[ _Draws his sword._\n\nAnd fear shall force what friendship cannot win:\n\nThy death shall bury what thy life conceals;\n\nThou diest for more esteeming her than me.\n\n_Ped._ O, stay, my lord.\n\n_Lor._ Yet speak the truth, and I will guerdon thee,\n\nAnd shield thee from whatever can ensue,\n\nAnd will conceal whate'er proceeds from thee.\n\nBut if thou dally once again, thou diest.\n\n_Ped._ If madam Bellimperia be in love\u2014\n\n_Lor._ What, villain! ifs and ands?\n\n_Ped._ O, stay, my lord, she loves Horatio.\n\n[ _Balthazar starts back._\n\n_Lor._ What, Don Horatio, our knight marshal's son?\n\n_Ped._ Even him, my lord.\n\n_Lor._ Now say, but how know'st thou he is her love?\n\nAnd thou shalt find me kind and liberal:\n\nStand up, I say, and fearless tell the truth.\n\n_Ped._ She sent him letters, which myself perus'd,\n\nFull-fraught with lines and arguments of love,\n\nPreferring him before Prince Balthazar.\n\n_Lor._ Swear on this cross that what thou say'st is true;\n\nAnd that thou wilt conceal what thou hast told.\n\n_Ped._ I swear to both, by him that made us all.\n\n_Lor._ In hope thine oath is true, here's thy reward:\n\nBut if I prove thee perjur'd and unjust,\n\nThis very sword, whereon thou took'st thine oath,\n\nShall be the worker of thy tragedy.\n\n_Ped._ What I have said is true, and shall\u2014for me\u2014\n\nBe still conceal'd from Bellimperia.\n\nBesides, your honour's liberality\n\nDeserves my duteous service, ev'n till death.\n\n_Lor._ Let this be all that thou shalt do for me:\n\nBe watchful, when and where these lovers meet,\n\nAnd give me notice in some secret sort.\n\n_Ped._ I will, my lord.\n\n_Lor._ Then shalt thou find that I am liberal.\n\nThou know'st that I can more advance thy state\n\nThan she; be therefore wise, and fail me not.\n\nGo and attend her, as thy custom is,\n\nLest absence make her think thou dost amiss.\n\n[ _Exit Pedringano._\n\nWhy so: _tam armis quam ingenio:_\n\nWhere words prevail not, violence prevails;\n\nBut gold doth more than either of them both.\n\nHow likes Prince Balthazar this stratagem?\n\n_Bal._ Both well and ill; it makes me glad and sad:\n\nGlad, that I know the hind'rer of my love;\n\nSad, that I fear she hates me whom I love.\n\nGlad, that I know on whom to be reveng'd;\n\nSad, that she'll fly me, if I take revenge.\n\nYet must I take revenge, or die myself,\n\nFor love resisted grows impatient.\n\nI think Horatio be my destin'd plague:\n\nFirst, in his hand he brandish\u00e8d a sword,\n\nAnd with that sword he fiercely wag\u00e8d war,\n\nAnd in that war he gave me dang'rous wounds,\n\nAnd by those wounds he forc\u00e8d me to yield,\n\nAnd by my yielding I became his slave.\n\nNow in his mouth he carries pleasing words,\n\nWhich pleasing words do harbour sweet conceits,\n\nWhich sweet conceits are lim'd with sly deceits,\n\nWhich sly deceits smooth Bellimperia's ears,\n\nAnd through her ears dive down into her heart,\n\nAnd in her heart set him, where I should stand.\n\nThus hath he ta'en my body by his force,\n\nAnd now by sleight would captivate my soul:\n\nBut in his fall I'll tempt the destinies,\n\nAnd either lose my life, or win my love.\n\n_Lor._ Let's go, my lord; your staying stays revenge.\n\nDo you but follow me, and gain your love:\n\nHer favour must be won by his remove. [ _Exeunt._\n\n**SCENE II**\n\n_Enter Horatio and Bellimperia._\n\nHor. Now, madam, since by favour of your love\n\nOur hidden smoke is turn'd to open flame,\n\nAnd that with looks and words we feed our thought\n\n(Two chief contents, where more cannot be had):\n\nThus, in the midst of love's fair blandishments,\n\nWhy show you sign of inward languishments?\n\n[ _Pedringano showeth all to the Prince and Lorenzo, placing them in secret._\n\n_Bel._ My heart, sweet friend, is like a ship at sea:\n\nShe wisheth port, where, riding all at ease,\n\nShe may repair what stormy times have worn,\n\nAnd leaning on the shore, may sing with joy,\n\nThat pleasure follows pain, and bliss annoy.\n\nPossession of thy love is th' only port,\n\nWherein my heart, with fears and hopes long toss'd,\n\nEach hour doth wish and long to make resort,\n\nThere to repair the joys that it hath lost,\n\nAnd, sitting safe, to sing in Cupid's quire\n\nThat sweetest bliss is crown of love's desire.\n\n[ _Balthazar and Lorenzo above._\n\n_Bal._ O sleep, mine eyes, see not my love profan'd;\n\nBe deaf, my ears, hear not my discontent;\n\nDie, heart: another joys what thou deserv'st.\n\n_Lor._ Watch still, mine eyes, to see this love disjoin'd;\n\nHear still, mine ears, to hear them both lament;\n\nLive, heart, to joy at fond Horatio's fall.\n\n_Bel._ Why stands Horatio speechless all this while?\n\n_Hor._ The less I speak, the more I meditate.\n\n_Bel._ But whereon dost thou chiefly meditate?\n\n_Hor._ On dangers past, and pleasures to ensue.\n\n_Bal._ On pleasures past, and dangers to ensue.\n\n_Bel._ What dangers and what pleasures dost thou mean?\n\n_Hor._ Dangers of war, and pleasures of our love.\n\n_Lor._ Dangers of death, but pleasures none at all.\n\n_Bel._ Let dangers go, thy war shall be with me:\n\nBut such a war, as breaks no bond of peace.\n\nSpeak thou fair words, I'll cross them with fair words;\n\nSend thou sweet looks, I'll meet them with sweet looks;\n\nWrite loving lines, I'll answer loving lines;\n\nGive me a kiss, I'll countercheck thy kiss:\n\nBe this our warring peace, or peaceful war.\n\n_Hor._ But, gracious madam, then appoint the field,\n\nWhere trial of this war shall first be made.\n\n_Bal._ Ambitious villain, how his boldness grows!\n\n_Bel._ Then be thy father's pleasant bow'r the field,\n\nWhere first we vow'd a mutual amity;\n\nThe court were dangerous, that place is safe.\n\nOur hour shall be, when Vesper 'gins to rise,\n\nThat summons home distressful travellers:\n\nThere none shall hear us but the harmless birds;\n\nHaply the gentle nightingale\n\nShall carol us asleep, ere we be ware,\n\nAnd, singing with the prickle at her breast,\n\nTell our delight and mirthful dalliance:\n\nTill then each hour will seem a year and more.\n\nHOR. But, honey sweet and honourable love,\n\nReturn we now into your father's sight:\n\nDang'rous suspicion waits on our delight\n\n_Lor._ Ay, danger mixed with jealous despite\n\nShall send thy soul into eternal night. [ _Exeunt._\n\n**SCENE III**\n\n_Enter King of Spain, Portingal Ambassador, Don Cyprian, etc._\n\n_King._ Brother of Castile, to the prince's love\n\nWhat says your daughter Bellimperia?\n\n_Cyp._ Although she coy it, as becomes her kind,\n\nAnd yet dissemble that she loves the prince,\n\nI doubt not, I, but she will stoop in time.\n\nAnd were she froward, which she will not be,\n\nYet herein shall she follow my advice,\n\nWhich is to love him, or forgo my love.\n\n_King._ Then, lord Ambassador of Portingal,\n\nAdvise thy king to make this marriage up,\n\nFor strength'ning of our late-confirm\u00e8d league;\n\nI know no better means to make us friends.\n\nHer dowry shall be large and liberal:\n\nBesides that she is daughter and half-heir\n\nUnto our brother here, Don Cyprian,\n\nAnd shall enjoy the moiety of his land,\n\nI'll grace her marriage with an uncle's gift,\n\nAnd this it is\u2014in case the match go forward\u2014:\n\nThe tribute which you pay, shall be releas'd;\n\nAnd if by Balthazar she have a son,\n\nHe shall enjoy the kingdom after us.\n\n_Amb._ I'll make the motion to my sovereign liege,\n\nAnd work it, if my counsel may prevail.\n\n_King._ Do so, my lord, and if he give consent,\n\nI hope his presence here will honour us,\n\nIn celebration of the nuptial day;\n\nAnd let himself determine of the time.\n\n_Amb._ Will't please your grace command me ought beside?\n\n_King._ Commend me to the king, and so farewell.\n\nBut where's Prince Balthazar to take his leave?\n\n_Amb._ That is perform'd already, my good lord.\n\n_King._ Amongst the rest of what you have in charge,\n\nThe prince's ransom must not be forgot:\n\nThat's none of mine, but his that took him prisoner;\n\nAnd well his forwardness deserves reward:\n\nIt was Horatio, our knight marshal's son.\n\n_Amb._ Between us there's a price already pitch'd,\n\nAnd shall be sent with all convenient speed.\n\n_King._ Then once again farewell, my lord.\n\n_Amb._ Farewell, my lord of Castile, and the rest. [ _Exit._\n\n_King._ Now, brother, you must take some little pains\n\nTo win fair Bellimperia from her will:\n\nYoung virgins must be rul\u00e8d by their friends.\n\nThe prince is amiable, and loves her well;\n\nIf she neglect him and forgo his love,\n\nShe both will wrong her own estate and ours.\n\nTherefore, whiles I do entertain the prince\n\nWith greatest pleasure that our court affords,\n\nEndeavour you to win your daughter's thought:\n\nIf she give back, all this will come to naught.\n\n[ _Exeunt_\n\n**SCENE IV**\n\n_Enter Horatio, Bellimperia, and Pedringano._\n\n_Hor._ Now that the night begins with sable wings\n\nTo overcloud the brightness of the sun,\n\nAnd that in darkness pleasures may be done:\n\nCome, Bellimperia, let us to the bow'r,\n\nAnd there in safety pass a pleasant hour.\n\n_Bel._ I follow thee, my love, and will not back,\n\nAlthough my fainting heart controls my soul.\n\nHOR. Why, make you doubt of Pedringano's faith?\n\n_Bel._ No, he is as trusty as my second self.\u2014\n\nGo, Pedringano, watch without the gate,\n\nAnd let us know if any make approach.\n\n_Ped._ [ _Aside_ ] _._ Instead of watching, I'll deserve more gold\n\nBy fetching Don Lorenzo to this match.\n\n[ _Exit Pedringano._\n\nHOR. What means my love?\n\n_Bel._ I know not what myself;\n\nAnd yet my heart foretells me some mischance.\n\n_Hor._ Sweet, say not so; fair fortune is our friend,\n\nAnd heav'ns have shut up day to pleasure us.\n\nThe stars, thou see'st, hold back their twinkling shine,\n\nAnd Luna hides herself to pleasure us.\n\n_Bel._ Thou hast prevail'd; I'll conquer my misdoubt,\n\nAnd in thy love and counsel drown my fear.\n\nI fear no more; love now is all my thoughts.\n\nWhy sit we not? for pleasure asketh ease.\n\n_Hor._ The more thou sitt'st within these leafy bowers,\n\nThe more will Flora deck it with her flowers.\n\n_Bel._ Ay, but if Flora spy Horatio here,\n\nHer jealous eye will think I sit too near,\n\n_Hor._ Hark, madam, how the birds record by night,\n\nFor joy that Bellimperia sits in sight.\n\n_Bel._ No, Cupid counterfeits the nightingale,\n\nTo frame sweet music to Horatio's tale.\n\n_Hor._ If Cupid sing, then Venus is not far:\n\nAy, thou art Venus, or some fairer star.\n\n_Bel._ If I be Venus, thou must needs be Mars;\n\nAnd where Mars reigneth, there must needs be wars.\n\n_Hor._ Then thus begin our wars: put forth thy hand,\n\nThat it may combat with my ruder hand.\n\n_Bel._ Set forth thy foot to try the push of mine.\n\n_Hor._ But first my looks shall combat against thine.\n\n_Bel._ Then ward thyself: I dart this kiss at thee.\n\n_Hor._ Thus I retort the dart thou threw'st at me.\n\n_Bel._ Nay, then to gain the glory of the field,\n\nMy twining arms shall yoke and make thee yield.\n\n_Hor._ Nay, then my arms are large and strong withal:\n\nThus elms by vines are compass'd, till they fall.\n\n_Bel._ O, let me go; for in my troubled eyes\n\nNow may'st thou read that life in passion dies.\n\n_Hor._ O, stay a while, and I will die with thee;\n\nSo shalt thou yield, and yet have conquer'd me.\n\n_Bel._ Who's there? Pedringano! we are betray'd!\n\n_Enter Lorenzo, Balthazar, Serberine, Pedringano, disguised._\n\n_Lor._ My lord, away with her, take her aside.\u2014\n\nO, sir, forbear: your valour is already tried.\n\nQuickly despatch, my masters.\n\n[ _They hang him in the arbour._\n\n_Hor._ What, will you murder me?\n\n_Lor._ Ay, thus, and thus: these are the fruits of love.\n\n[ _They stab him._\n\n_Bel._ O, save his life, and let me die for him\n\nO, save him, brother; save him, Balthazar:\n\nI lov'd Horatio; but he lov'd not me.\n\n_Bal._ But Balthazar loves Bellimperia.\n\n_Lor._ Although his life were still ambitious-proud,\n\nYet is he at the highest now he is dead.\n\n_Bel._ Murder! murder! Help, Hieronimo, help!\n\n_Lor._ Come, stop her mouth; away with her.\n\n[ _Exeunt._\n\n**SCENE V**\n\n_Enter Hieronimo in his shirt, etc._\n\n_Bier._ What outcries pluck me from my naked bed,\n\nAnd chill my throbbing heart with trembling fear,\n\nWhich never danger yet could daunt before?\n\nWho calls Hieronimo? speak, here I am.\n\nI did not slumber; therefore 'twas no dream.\n\nNo, no, it was some woman cried for help;\n\nAnd here within this garden did she cry;\n\nAnd in this garden must I rescue her.\u2014\n\nBut stay, what murd'rous spectacle is this?\n\nA man hang'd up and all the murd'rers gone!\n\nAnd in my bower, to lay the guilt on me!\n\nThis place was made for pleasure, not for death.\n\n[ _He cuts him down._\n\nThose garments that he wears I oft have seen\u2014:\n\nAlas, it is Horatio, my sweet son!\n\nO no, but he that whilom was my son!\n\nO, was it thou that call'dst me from my bed?\n\nO speak, if any spark of life remain:\n\nI am thy father; who hath slain my son?\n\nWhat savage monster, not of human kind,\n\nHath here been glutted with thy harmless blood,\n\nAnd left thy bloody corpse dishonour'd here,\n\nFor me, amidst these dark and deathful shades,\n\nTo drown thee with an ocean of my tears?\n\nO heav'ns, why made you night to cover sin?\n\nBy day this deed of darkness had not been.\n\nO earth, why didst thou not in time devour\n\nThe vild profaner of this sacred bow'r?\n\nO poor Horatio, what hadst thou misdone,\n\nTo leese thy life, ere life was new begun?\n\nO wicked butcher, whatsoe'er thou wert,\n\nHow could thou strangle virtue and desert?\n\nAy me most wretched, that have lost my joy,\n\nIn leesing my Horatio, my sweet boy!\n\n_Enter Isabella._\n\n_Isab._ My husband's absence makes my heart to throb:\u2014\n\nHieronimo!\n\n_Hier._ Here, Isabella, help me to lament;\n\nFor sighs are stopp'd, and all my tears are spent\n\n_Isab._ What world of grief! my son Horatio!\n\nO, where's the author of this endless woe?\n\n_Hier._ To know the author were some ease of grief;\n\nFor in revenge my heart would find relief.\n\n_Isab._ Then is he gone? and is my son gone too?\n\nO, gush out, tears, fountains and floods of tears;\n\nBlow, sighs, and raise an everlasting storm;\n\nFor outrage fits our curs\u00e8d wretchedness.\n\n[ _Ay me, Hieronimo, sweet husband, speak!_\n\nHier. _He supp'd with us to-night, frolic and merry,_\n\n_And said he would go visit Balthazar_\n\n_At the duke's palace: there the prince doth lodge._\n\n_He had no custom to stay out so late:_\n\n_He may be in his chamber; some go see._\n\n_Roderigo, ho!_\n\nEnter Pedro and Jaques.\n\nIsab. _Ay me, he raves! sweet Hieronimo._\n\nHier. _True, all Spain takes note of it._\n\n_Besides, he is so generally belov'd;_\n\n_His majesty the other day did grace him_\n\n_With waiting on his cup: these be favours,_\n\n_Which do assure me he cannot be short-liv'd._\n\nIsab. _Sweet Hieronimo!_\n\nHier. _I wonder how this fellow got his clothes!_ \u2014\n\n_Sirrah, sirrah, I'll know the truth of all:_\n\n_Jaques, run to the Duke of Castile's presently,_\n\n_And bid my son Horatio to come home:_\n\n_I and his mother have had strange dreams to-night._\n\n_Do ye hear me, sir?_\n\nJaques. _Ay, sir._\n\nHier. _Well, sir, be gone._\n\n_Pedro, come hither; know'st thou who this is?_\n\nPed. _Too well, sir._\n\nHier. _Too well! who, who is it? Peace, Isabella!_\n\n_Nay, blush not, man._\n\nPed. _It is my lord Horatio._\n\nHier. _Ha, ha, St. James! but this doth make me laugh,_\n\n_That there are more deluded than myself._\n\nPed. _Deluded?_\n\nHier. _Ay:_\n\n_I would have sworn myself, within this hour,_\n\n_That this had been my son Horatio:_\n\n_His garments are so like._\n\n_Ha! are they not great persuasions?_\n\nIsab. _O, would to God it were not so!_\n\nHier. _Were not, Isabella? dost thou dream it is?_\n\n_Can thy soft bosom entertain a thought,_\n\n_That such a black deed of mischief should be done_\n\n_On one so pure and spotless as our son?_\n\n_Away, I am ashamed._\n\nIsab. _Dear Hieronimo,_\n\n_Cast a more serious eye upon thy grief:_\n\n_Weak apprehension gives but weak belief_\n\nHier. _It was a man, sure, that was hang'd up here;_\n\n_A youth, as I remember: I cut him down._\n\n_If it should prove my son now after all\u2014_\n\n_Say you? say you?_ \u2014 _Light! lend me a taper;_\n\n_Let me look again._ \u2014 _O God!_\n\n_Confusion, mischief, torment, death and hell,_\n\n_Drop all your stings at once in my cold bosom,_\n\n_That now is stiff with horror: kill me quickly!_\n\n_Be gracious to me, thou infective night,_\n\n_And drop this deed of murder down on me;_\n\n_Gird in my waste of grief with thy large darkness,_\n\n_And let me not survive to see the light_\n\n_May put me in the mind I had a son._\n\nIsab. _O sweet Horatio! O my dearest son!_\n\nHier. _How strangely had I lost my way to grief!_ ]\n\nSweet, lovely rose, ill-pluck'd before thy time,\n\nFair, worthy son, not conquer'd, but betray'd,\n\nI'll kiss thee now, for words with tears are stay'd.\n\n_Isab._ And I'll close up the glasses of his sight,\n\nFor once these eyes were only my delight\n\n_Hier._ See'st thou this handkercher besmear'd with blood?\n\nIt shall not from me, till I take revenge.\n\nSee'st thou those wounds that yet are bleeding fresh?\n\nI'll not entomb them, till I have revenge.\n\nThen will I joy amidst my discontent;\n\nTill then my sorrow never shall be spent\n\n_Isab._ The heav'ns are just; murder cannot be hid:\n\nTime is the author both of truth and right,\n\nAnd time will bring this treachery to light.\n\n_Hier._ Meanwhile, good Isabella, cease thy plaints,\n\nOr, at the least, dissemble them awhile:\n\nSo shall we sooner find the practice out,\n\nAnd learn by whom all this was brought about\n\nCome, Isabel, now let us take him up,\n\n[ _They take him up._\n\nAnd bear him in from out this curs\u00e8d place.\n\nI'll say his dirge; singing fits not this case.\n\n_O aliquis mihi quas pulchrum ver educat herbas,_\n\n[Hieronimo sets his breast unto his sword.\n\n_Misceat, & nostro detur medicina dolori;_\n\n_Aut, si qui faciunt annorum oblivia, succos_\n\n_Praebeat; ipse metam magnum quaecunque per orbem_\n\n_Gramina Sol pulchras effert in luminis oras;_\n\n_Ipse bibam quicquid meditatur saga veneni,_\n\n_Q\u03bcicquid & herbarum vi caeca nenia nectit:_\n\n_Omnia perpetiar, lethum quoque, dum semel omnis_\n\n_Nosier in extincto moriatur pectore sensus._ \u2014\n\n_Ergo tuos oculos nunquam, mea vita, videbo,_\n\n_Et tua perpetuus sepelivit lumina somnus?_\n\n_Emoriar tecum: sic, sic juvat ire sub umbras._ \u2014\n\n_Attamen absistam properato cedere letho,_\n\n_Ne mortem vindicta tuam tam nulla sequatur._\n\n[Here he throws it from him and bears the body away.\n\n**SCENE VI**\n\n_Ghost of Andrea_ , _Revenge._\n\n_Andrea._ Brought'st thou me hither to increase my pain?\n\nI look'd that Balthazar should have been slain:\n\nBut 'tis my friend Horatio that is slain,\n\nAnd they abuse fair Bellimperia,\n\nOn whom I doted more than all the world,\n\nBecause she lov'd me more than all the world.\n\n_Revenge._ Thou talk'st of harvest, when the corn is green:\n\nThe end is crown of every work well done;\n\nThe sickle comes not, till the corn be ripe.\n\nBe still; and ere I lead thee from this place,\n\nI'll show thee Balthazar in heavy case.\n\n**ACT III**\n\n**SCENE I**\n\n**_The Court of Portugal._**\n\n_Enter Viceroy of Portingal, Nobles, Alexandro, Villuppo._\n\n_Vic._ Infortunate condition of kings,\n\nSeated amidst so many helpless doubts!\n\nFirst we are plac'd upon extremest height,\n\nAnd oft supplanted with exceeding hate,\n\nBut ever subject to the wheel of chance;\n\nAnd at our highest never joy we so,\n\nAs we both doubt and dread our overthrow.\n\nSo striveth not the waves with sundry winds,\n\nAs fortune toileth in the affairs of kings,\n\nThat would be fear'd, yet fear to be belov'd,\n\nSith fear or love to kings is flattery.\n\nFor instance, lordings, look upon your king,\n\nBy hate depriv\u00e8d of his dearest son,\n\nThe only hope of our successive line.\n\n_Nob._ I had not thought that Alexandro's heart\n\nHad been envenom'd with such extreme hate;\n\nBut now I see that words have several works,\n\nAnd there's no credit in the countenance.\n\n_Vil._ No; for, my lord, had you beheld the train,\n\nThat feign\u00e8d love had colour\u00e8d in his looks,\n\nWhen he in camp consorted Balthazar,\n\nFar more inconstant had you thought the sun,\n\nThat hourly coasts the centre of the earth,\n\nThan Alexandro's purpose to the prince.\n\n_Vic._ No more, Villuppo, thou hast said enough,\n\nAnd with thy words thou slay'st our wounded thoughts.\n\nNor shall I longer dally with the world,\n\nProcrastinating Alexandro's death:\n\nGo some of you, and fetch the traitor forth,\n\nThat, as he is condemn\u00e8d, he may die.\n\n_Enter Alexandro, with a Nobleman and halberts._\n\n_Nob._ In such extremes will nought but patience serve.\n\n_Alex._ But in extremes what patience shall I use?\n\nNor discontents it me to leave the world,\n\nWith whom there nothing can prevail but wrong.\n\n_Nob._ Yet hope the best.\n\n_Alex._ 'Tis heaven is my hope:\n\nAs for the earth, it is too much infect\n\nTo yield me hope of any of her mould.\n\n_Vic._ Why linger ye? bring forth that daring fiend,\n\nAnd let him die for his accurs\u00e8d deed.\n\n_Alex._ Not that I fear the extremity of death\n\n(For nobles cannot stoop to servile fear)\n\nDo I, O king, thus discontented live.\n\nBut this, O this, torments my labouring soul,\n\nThat thus I die suspected of a sin,\n\nWhereof, as heav'ns have known my secret thoughts,\n\nSo am I free from this suggestion.\n\n_Vic._ No more, I say! to the tortures! when?\n\nBind him, and burn his body in those flames,\n\n[ _They bind him to the stake._\n\nThat shall prefigure those unquench\u00e8d fires\n\nOf Phlegethon, prepar\u00e8d for his soul.\n\n_Alex._ My guiltless death will be aveng'd on thee,\n\nOn thee, Villuppo, that hath malic'd thus,\n\nOr for thy meed hast falsely me accus'd.\n\n_Vil._ Nay, Alexandro, if thou menace me,\n\nI'll lend a hand to send thee to the lake,\n\nWhere those thy words shall perish with thy works:\n\nInjurious traitor! monstrous homicide!\n\n_Enter Ambassador._\n\n_Amb._ Stay, hold a while;\n\nAnd here\u2014with pardon of his majesty\u2014\n\nLay hands upon Villuppo.\n\n_Vic._ Ambassador,\n\nWhat news hath urg'd this sudden enterance?\n\n_Amb._ Know, sovereign lord, that Balthazar doth live.\n\n_Vic._ What say'st thou? liveth Balthazar our son?\n\n_Amb._ Your highness' son, Lord Balthazar, doth live;\n\nAnd, well entreated in the court of Spain,\n\nHumbly commends him to your majesty.\n\nThese eyes beheld\u2014and these my followers\u2014;\n\nWith these, the letters of the king's commends\n\n[ _Gives him letters._\n\nAre happy witnesses of his highness' health.\n\n[ _The King looks on the letters, and proceeds._\n\n_Vic._ 'Thy son doth live, your tribute is receiv'd;\n\nThy peace is made, and we are satisfied.\n\nThe rest resolve upon as things propos'd\n\nFor both our honours and thy benefit.'\n\n_Amb._ These are his highness' farther articles.\n\n[ _He gives him more letters._\n\n_Vic._ Accurs\u00e8d wretch, to intimate these ills\n\nAgainst the life and reputation\n\nOf noble Alexandro! Come, my lord, unbind him:\n\nLet him unbind thee, that is bound to death,\n\nTo make a quital for thy discontent.\n\n[ _They unbind him._\n\n_Alex._ Dread lord, in kindness you could do no less,\n\nUpon report of such a damn\u00e8d fact;\n\nBut thus we see our innocence hath sav'd\n\nThe hopeless life which thou, Villuppo, sought\n\nBy thy suggestions to have massacred.\n\n_Vic._ Say, false Villuppo, wherefore didst thou thus\n\nFalsely betray Lord Alexandro's life?\n\nHim, whom thou know'st that no unkindness else,\n\nBut ev'n the slaughter of our dearest son,\n\nCould once have mov'd us to have misconceiv'd.\n\n_Alex._ Say, treacherous Villuppo, tell the king:\n\nWherein hath Alexandro us'd thee ill?\n\n_Vil._ Rent with remembrance of so foul a deed,\n\nMy guilty soul submits me to thy doom:\n\nFor not for Alexandro's injuries,\n\nBut for reward and hope to be preferr'd,\n\nThus have I shamelessly hazarded his life.\n\n_Vic._ Which, villain, shall be ransom'd with thy death\u2014:\n\nAnd not so mean a torment as we here\n\nDevis'd for him who, thou said'st, slew our son,\n\nBut with the bitt'rest torments and extremes\n\nThat may be yet invented for thine end.\n\n[ _Alexandro seems to entreat._\n\nEntreat me not; go, take the traitor hence:\n\n[ _Exit Villuppo._\n\nAnd, Alexandro, let us honour thee\n\nWith public notice of thy loyalty.\u2014\n\nTo end those things articulated here\n\nBy our great lord, the mighty King of Spain,\n\nWe with our council will deliberate.\n\nCome, Alexandro, keep us company. [ _Exeunt._\n\n**SCENE II**\n\n_Enter Hieronimo._\n\n_Hier._ O eyes! no eyes, but fountains fraught with tears;\n\nO life! no life, but lively form of death;\n\nO world! no world, but mass of public wrongs,\n\nConfus'd and fill'd with murder and misdeeds!\n\nO sacred heav'ns! if this unhallow'd deed,\n\nIf this inhuman and barbarous attempt,\n\nIf this incomparable murder thus\n\nOf mine, but now no more my son,\n\nShall unreveal'd and unreveng\u00e8d pass,\n\nHow should we term your dealings to be just,\n\nIf you unjustly deal with those that in your justice trust?\n\nThe night, sad secretary to my moans,\n\nWith direful visions wakes my vex\u00e8d soul,\n\nAnd with the wounds of my distressful son\n\nSolicits me for notice of his death.\n\nThe ugly fiends do sally forth of hell,\n\nAnd frame my steps to unfrequented paths,\n\nAnd fear my heart with fierce inflam\u00e8d thoughts.\n\nThe cloudy day my discontents records,\n\nEarly begins to register my dreams,\n\nAnd drive me forth to seek the murtherer.\n\nEyes, life, world, heav'ns, hell, night, and day,\n\nSee, search, shew, send some man, some mean, that may\u2014\n\n[ _A letter falleth._\n\nWhat's here? a letter? tush! it is not so!\u2014\n\nA letter written to Hieronimo! [ _Red ink._\n\n\"For want of ink, receive this bloody writ:\n\nMe hath my hapless brother hid from thee;\n\nRevenge thyself on Balthazar and him:\n\nFor these were they that murder\u00e8d thy son.\n\nHieronimo, revenge Horatio's death,\n\nAnd better fare than Bellimperia doth.'\n\nWhat means this unexpected miracle?\n\nMy son slain by Lorenzo and the prince!\n\nWhat cause had they Horatio to malign?\n\nOr what might move thee, Bellimperia,\n\nTo accuse thy brother, had he been the mean?\n\nHieronimo, beware!\u2014thou art betray'd,\n\nAnd to entrap thy life this train is laid.\n\nAdvise thee therefore, be not credulous:\n\nThis is devis\u00e8d to endanger thee,\n\nThat thou, by this, Lorenzo shouldst accuse;\n\nAnd he, for thy dishonour done, should draw\n\nThy life in question and thy name in hate.\n\nDear was the life of my belov\u00e8d son,\n\nAnd of his death behoves me be reveng'd:\n\nThen hazard not thine own, Hieronimo,\n\nBut live t' effect thy resolution.\n\nI therefore will by circumstances try,\n\nWhat I can gather to confirm this writ;\n\nAnd, heark'ning near the Duke of Castile's house,\n\nClose, if I can, with Bellimperia,\n\nTo listen more, but nothing to bewray.\n\n_Enter Pedringano._\n\nNow, Pedringano!\n\n_Ped._ Now, Hieronimo!\n\n_Hier._ Where's thy lady?\n\n_Ped._ I know not; here's my ord.\n\n_Enter Lorenzo._\n\n_Lor._ How now, who's this? Hieronimo?\n\n_Hier._ My lord\u2014\n\n_Ped._ He asketh for my lady Bellimperia.\n\n_Lor._ What to do, Hieronimo? The duke, my father, hath,\n\nUpon some disgrace, awhile remov'd her hence;\n\nBut if it be ought I may inform her of,\n\nTell me, Hieronimo, and I'll let her know it.\n\n_Hier._ Nay, nay, my lord, I thank you; it shall not need.\n\nI had a suit unto her, but too late,\n\nAnd her disgrace makes me unfortunate.\n\n_Lor._ Why so, Hieronimo? use me.\n\n_Hier._ O no, my lord; I dare not; it must not be;\n\nI humbly thank your lordship.\n\n_Lor._ Why then, farewell.\n\n_Hier._ My grief no heart, my thoughts no tongue can tell. [ _Exit._\n\n_Lor._ Come hither, Pedringano, see'st thou this?\n\n_Ped._ My lord, I see it, and suspect it too.\n\n_Lor._ This is that damn\u00e8d villain Serberine,\n\nThat hath, I fear, reveal'd Horatio's death.\n\n_Ped._ My lord, he could not, 'twas so lately done;\n\nAnd since he hath not left my company.\n\n_Lor._ Admit he have not, his condition's such,\n\nAs fear or flatt'ring words may make him false.\n\nI know his humour, and therewith repent\n\nThat e'er I us'd him in this enterprise.\n\nBut, Pedringano, to prevent the worst,\n\nAnd 'cause I know thee secret as my soul,\n\nHere, for thy further satisfaction, take thou this,\n\n[ _Gives him more gold._\n\nAnd hearken to me\u2014thus it is devis'd:\n\nThis night thou must (and, prithee, so resolve)\n\nMeet Serber\u00edne at Saint Luigi's Park\u2014\n\nThou know'st 'tis here hard by behind the house\u2014\n\nThere take thy stand, and see thou strike him sure:\n\nFor die he must, if we do mean to live.\n\n_Ped._ But how shall Serberine be there, my lord?\n\n_Lor._ Let me alone; I'll send to him to meet\n\nThe prince and me, where thou must do this deed.\n\n_Ped._ It shall be done, my lord, it shall be done;\n\nAnd I'll go arm myself to meet him there.\n\n_Lor._ When things shall alter, as I hope they will,\n\nThen shalt thou mount for this; thou know'st my mind.\n\n[ _Exit Pedr\u00edngano._\n\n_Che le Ieron!_\n\n_Enter Page._\n\n_Page._ My lord?\n\n_Lor._ Go, sirrah,\n\nTo Serberine, and bid him forthwith meet\n\nThe prince and me at Saint Luigi's Park,\n\nBehind the house; this evening, boy!\n\n_Page._ I go, my lord\n\n_Lor._ But, sirrah, let the hour be eight o'clock:\n\nBid him not fail.\n\n_Page._ I fly, my lord. [ _Exit._\n\n_Lor._ Now to confirm the complot thou hast cast\n\nOf all these practices, I'll spread the watch,\n\nUpon precise commandment from the king,\n\nStrongly to guard the place where Pedringano\n\nThis night shall murder hapless Serberine.\n\nThus must we work that will avoid distrust;\n\nThus must we practise to prevent mishap,\n\nAnd thus one ill another must expulse.\n\nThis sly enquiry of Hieronimo\n\nFor Bellimperia breeds suspicion,\n\nAnd this suspicion bodes a further ill.\n\nAs for myself, I know my secret fault,\n\nAnd so do they; but I have dealt for them:\n\nThey that for coin their souls endanger\u00e8d,\n\nTo save my life, for coin shall venture theirs;\n\nAnd better it's that base companions die,\n\nThan by their life to hazard our good haps.\n\nNor shall they live, for me to fear their faith:\n\nI'll trust myself, myself shall be my friend;\n\nFor die they shall, slaves are ordain'd to no other end. [ _Exit._\n\n**SCENE III**\n\n_Enter Pedringano, with a pistol._\n\n_Ped._ Now, Pedringano, bid thy pistol hold,\n\nAnd hold on, Fortune! once more favour me;\n\nGive but success to mine attempting spirit,\n\nAnd let me shift for taking of mine aim.\n\nHere is the gold: this is the gold propos'd;\n\nIt is no dream that I adventure for,\n\nBut Pedringano is possess'd thereof.\n\nAnd he that would not strain his conscience\n\nFor him that thus his liberal purse hath stretch'd,\n\nUnworthy such a favour, may he fail,\n\nAnd, wishing, want, when such as I prevail.\n\nAs for the fear of apprehension,\n\nI know, if need should be, my noble lord\n\nWill stand between me and ensuing harms;\n\nBesides, this place is free from all suspect:\n\nHere therefore will I stay and take my stand.\n\n_Enter the Watch._\n\n1. I wonder much to what intent it is\n\nThat we are thus expressly charg'd to watch.\n\n2. 'Tis by commandment in the king's own name.\n\n3. But we were never wont to watch and ward\n\nSo near the duke, his brother's, house before.\n\n2. Content yourself, stand close, there's somewhat in't.\n\n_Enter Serberine._\n\nHere, Serberine, attend and stay thy pace;\n\nFor here did Don Lorenzo's page appoint\n\nThat thou by his command shouldst meet with him.\n\nHow fit a place\u2014if one were so dispos'd\u2014\n\nMethinks this corner is to close with one.\n\n_Ped._ Here comes the bird that I must seize upon:\n\nNow, Pedringano, or never, play the man!\n\n_Ser._ I wonder that his lordship stays so long,\n\nOr wherefore should he send for me so late?\n\n_Ped._ For this, Serberine!\u2014and thou shalt ha't.\n\n[ _Shoots the dag._\n\nSo, there he lies; my promise is perform'd.\n\n_The Watch._\n\n1. Hark, gentlemen, this is a pistol shot.\n\n2. And here's one slain;\u2014stay the murderer.\n\n_Ped._ Now by the sorrows of the souls in hell,\n\n[ _He strives with the watch._\n\nWho first lays hand on me, I'll be his priest.\n\n3. Sirrah, confess, and therein play the priest,\n\nWhy hast thou thus unkindly kill'd the man?\n\n_Ped._ Why? because he walk'd abroad so late.\n\n3. Come, sir, you had been better kept your bed,\n\nThan have committed this misdeed so late.\n\n2. Come, to the marshal's with the murderer!\n\n1. On to Hieronimo's! help me here\n\nTo bring the murder'd body with us too.\n\n_Ped._ Hieronimo? carry me before whom you will:\n\nWhatever he be, I'll answer him and you;\n\nAnd do your worst, for I defy you all. [ _Exeunt._\n\n**SCENE IV**\n\n_Enter Lorenzo and Balthazar._\n\n_Bal._ How now, my lord, what makes you rise so soon?\n\n_Lor._ Fear of preventing our mishaps too late.\n\n_Bal._ What mischief is it that we not mistrust?\n\n_Lor._ Our greatest ills we least mistrust, my lord,\n\nAnd inexpected harms do hurt us most.\n\n_Bal._ Why, tell me, Don Lorenzo, tell me, man,\n\nIf ought concerns our honour and your own.\n\n_Lor._ Nor you, nor me, my lord, but both in one:\n\nFor I suspect\u2014and the presumption's great\u2014\n\nThat by those base confed'rates in our fault\n\nTouching the death of Don Horatio,\n\nWe are betray'd to old Hieronimo.\n\n_Bal._ Betray'd, Lorenzo? tush! it cannot be.\n\n_Lor._ A guilty conscience, urg\u00e8d with the thought\n\nOf former evils, easily cannot err:\n\nI am persuaded\u2014and dissuade me not\u2014\n\nThat all's revealed to Hieronimo.\n\nAnd therefore know that I have cast it thus:\u2014\n\n_Enter Page._\n\nBut here's the page. How now? what news with thee?\n\n_Page._ My lord, Serberine is slain.\n\n_Bal._ Who? Serberine, my man?\n\n_Page._ Your highness' man, my lord.\n\n_Lor._ Speak, page, who murder'd him?\n\n_Page._ He that is apprehended for the fact\n\n_Lor._ Who?\n\n_Page._ Pedringano.\n\n_Bal._ Is Serberine slain, that lov'd his lord so well?\n\nInjurious villain, murd'rer of his friend!\n\n_Lor._ Hath Pedringano murder'd Serberine?\n\nMy lord, let me entreat you to take the pains\n\nTo exasperate and hasten his revenge\n\nWith your complaints unto my lord the king.\n\nThis their dissension breeds a greater doubt\n\n_Bal._ Assure thee, Don Lorenzo, he shall die,\n\nOr else his highness hardly shall deny.\n\nMeanwhile I'll haste the marshal-sessions:\n\nFor die he shall for this his damn\u00e8d deed.\n\n[ _Exit Balthazar._\n\n_Lor._ Why so, this fits our former policy,\n\nAnd thus experience bids the wise to deal.\n\nI lay the plot: he prosecutes the point;\n\nI set the trap: he breaks the worthless twigs,\n\nAnd sees not that wherewith the bird was lim'd.\n\nThus hopeful men, that mean to hold their own,\n\nMust look like fowlers to their dearest friends.\n\nHe runs to kill whom I have holp to catch,\n\nAnd no man knows it was my reaching fetch.\n\n'Tis hard to trust unto a multitude,\n\nOr any one, in mine opinion,\n\nWhen men themselves their secrets will reveal.\n\n_Enter a Messenger with a letter._\n\nBoy\u2014\u2014\n\n_Page._ My lord?\n\n_Lor._ What's he?\n\n_Mes._ I have a letter to your lordship.\n\n_Lor._ From whence?\n\n_Mes._ From Pedringano that's imprison'd.\n\n_Lor._ So he is in prison then?\n\n_Mes._ Ay, my good lord.\n\n_Lor._ What would he with us?\u2014He writes us here,\n\nTo stand good lord, and help him in distress.\u2014\n\nTell him I have his letters, know his mind;\n\nAnd what we may, let him assure him of.\n\nFellow, begone: my boy shall follow thee.\n\n[ _Exit Messenger._\n\nThis works like wax; yet once more try thy wits.\n\nBoy, go, convey this purse to Pedringano;\n\nThou know'st the prison, closely give it him,\n\nAnd be advis'd that none be there about:\n\nBid him be merry still, but secret;\n\nAnd though the marshal-sessions be to-day,\n\nBid him not doubt of his delivery.\n\nTell him his pardon is already sign'd,\n\nAnd thereon bid him boldly be resolv'd:\n\nFor, were he ready to be turn\u00e8d off\u2014\n\nAs 'tis my will the uttermost be tried\u2014\n\nThou with his pardon shalt attend him still.\n\nShow him this box, tell him his pardon's in't;\n\nBut open't not, and if thou lov'st thy life;\n\nBut let him wisely keep his hopes unknown:\n\nHe shall not want while Don Lorenzo lives.\n\nAway!\n\n_Page._ I go, my lord, I run.\n\n_Lor._ But, sirrah, see that this be cleanly done.\n\n[ _Exit Page._\n\nNow stands our fortune on a tickle point,\n\nAnd now or never ends Lorenzo's doubts.\n\nOne only thing is uneffected yet,\n\nAnd that's to see the executioner.\n\nBut to what end? I list not trust the air\n\nWith utterance of our pretence therein,\n\nFor fear the privy whisp'ring of the wind\n\nConvey our words amongst unfriendly ears,\n\nThat lie too open to advantages.\n\n_E quel che voglio io, nessun lo sa;_\n\n_Intendo io: quel mi baster\u00e0._ [ _Exit_\n\n**SCENE V**\n\n_Enter Boy, with the box._\n\n_Boy._ My master hath forbidden me to look in this box; and, by my troth, 'tis likely, if he had not warned me, I should not have had so much idle time; for we men's-kind, in our minority, are like women in their uncertainty: that they are most forbidden, they will soonest attempt: so I now.\u2014\u2014By my bare honesty, here's nothing but the bare empty box: were it not sin against secrecy, I would say it were a piece of gentle-manlike knavery. I must go to Pedringano, and tell him his pardon is in this box; nay, I would have sworn it, had I not seen the contrary.\u2014I cannot choose but smile to think how the villain will flout the gallows, scorn the audience, and descant on the hangman,and all presuming of his pardon from hence. Will't not be an odd jest for me to stand and grace every jest he makes, pointing my finger at this box, as who would say: \"Mock on, here's thy warrant.' Is't not a scurvy jest that a man should jest himself to death? Alas! poor Pedringano, I am in a sort sorry for thee; but if I should be hanged with thee, I cannot weep. [ _Exit._\n\n**SCENE VI**\n\n_Enter Hieronimo and the Deputy._\n\n_Hier._ Thus must we toil in other men's extremes,\n\nThat know not how to remedy our own;\n\nAnd do them justice, when unjustly we,\n\nFor all our wrongs, can compass no redress.\n\nBut shall I never live to see the day,\n\nThat I may come, by justice of the heavens,\n\nTo know the cause that may my cares allay?\n\nThis toils my body, this consumeth age,\n\nThat only I to all men just must be,\n\nAnd neither gods nor men be just to me.\n\n_Dep._ Worthy Hieronimo, your office asks\n\nA care to punish such as do transgress.\n\n_Hier._ So is't my duty to regard his death\n\nWho, when he liv'd, deserv'd my dearest blood.\n\nBut come, for that we came for: let's begin;\n\nFor here lies that which bids me to be gone.\n\n_Enter Officers_ , _Boy, and Pedringano, with a letter in his hand, bound._\n\n_Dep._ Bring forth the prisoner, for the court is set\n\n_Ped._ Gramercy, boy, but it was time to come;\n\nFor I had written to my lord anew\n\nA nearer matter that concerneth him,\n\nFor fear his lordship had forgotten me.\n\nBut sith he hath remember'd me so well\u2014\n\nCome, come, come on, when shall we to this gear?\n\n_Hier._ Stand forth, thou monster, murderer of men,\n\nAnd here, for satisfaction of the world,\n\nConfess thy folly, and repent thy fault;\n\nFor there's thy place of execution.\n\n_Ped._ This is short work: well, to your marshalship\n\nFirst I confess\u2014nor fear I death therefore\u2014:\n\nI am the man, 'twas I slew Serberine.\n\nBut, sir, then you think this shall be the place,\n\nWhere we shall satisfy you for this gear?\n\n_Dep._ Ay, Pedringano.\n\n_Ped._ Now I think not so.\n\n_Hier._ Peace, impudent; for thou shalt find it so:\n\nFor blood with blood shall, while I sit as judge,\n\nBe satisfi\u00e8d, and the law discharg'd.\n\nAnd though myself cannot receive the like,\n\nYet will I see that others have their right.\n\nDespatch: the fault's approv\u00e8d and confess'd,\n\nAnd by our law he is condemn'd to die.\n\n_Hangm._ Come on, sir, are you ready?\n\n_Ped._ To do what, my fine, officious knave?\n\n_Hangm._ To go to this gear.\n\n_Ped._ O sir, you are too forward: thou wouldst fain furnish me with a halter, to disfurnish me of my habit. So I should go out of this gear, my raiment, into that gear, the rope. But, hangman, now I spy your knavery, I'll not change without boot, that's flat.\n\n_Hangm._ Come, sir.\n\n_Ped._ So, then, I must up?\n\n_Hangm._ No remedy.\n\n_Ped._ Yes, but there shall be for my coming down.\n\n_Hangm._ Indeed, here's a remedy for that.\n\n_Ped._ How? be turned off?\n\n_Hangm._ Ay, truly; come, are you ready? I pray, sir, despatch; the day goes away.\n\n_Ped._ What, do you hang by the hour? if you do, I may chance to break your old custom.\n\n_Hangm._ Faith, you have reason; for I am like to break your young neck.\n\n_Ped._ Dost thou mock me, hangman? pray God, I be not preserved to break your knave's pate for this.\n\n_Hangm._ Alas, sir! you are a foot too low to reach it, and I hope you will never grow so high while I am in the office.\n\n_Ped._ Sirrah, dost see yonder boy with the box in his hand?\n\n_Hangm._ What, he that points to it with his finger?\n\n_Ped._ Ay, that companion.\n\n_Hangm._ I know him not; but what of him?\n\n_Ped._ Dost thou think to live till his old doublet will make thee a new truss?\n\n_Hangm._ Ay, and many a fair year after, to truss up many an honester man than either thou or he.\n\n_Ped._ What hath he in his box, as thou thinkest?\n\n_Hangm._ Faith, I cannot tell, nor I care not greatly; methinks you should rather hearken to your soul's health.\n\n_Ped._ Why, sirrah hangman, I take it that that is good for the body is likewise good for the soul: and it may be, in that box is balm for both.\n\n_Hangm._ Well, thou art even the merriest piece of man's flesh that e'er groaned at my office door!\n\n_Ped._ Is your roguery become an office with a knave's name?\n\n_Hangm._ Ay, and that shall all they witness that see you seal it with a thief's name.\n\n_Ped._ I prithee, request this good company to pray with me.\n\n_Hangm._ Ay, marry, sir, this is a good motion: my masters, you see here's a good fellow.\n\n_Ped._ Nay, nay, now I remember me, let them alone till some other time; for now I have no great need.\n\n_Hier._ I have not seen a wretch so impudent\n\nO monstrous times, where murder's set so light,\n\nAnd where the soul, that should be shrin'd in heaven,\n\nSolely delights in interdicted things,\n\nStill wand'ring in the thorny passages,\n\nThat intercepts itself of happiness.\n\nMurder! O bloody monster! God forbid\n\nA fault so foul should 'scape unpunish\u00e8d.\n\nDespatch, and see this execution done!\u2014\n\nThis makes me to remember thee, my son.\n\n[ _Exit Hieronimo._\n\n_Ped._ Nay, soft, no haste.\n\n_Dep._ Why, wherefore stay you? Have you hope of life?\n\n_Ped._ Why, ay!\n\n_Hangm._ As how?\n\n_Ped._ Why, rascal, by my pardon from the king.\n\n_Hangm._ Stand you on that? then you shall off with this.\n\n[ _He turns him off._\n\n_Dep._ So, executioner;\u2014convey him hence;\n\nBut let his body be unburi\u00e8d:\n\nLet not the earth be chok\u00e8d or infect\n\nWith that which heav'n contemns, and men neglect.\n\n[ _Exeunt._\n\n**SCENE VII**\n\n_Enter Hieronimo._\n\n_Hier._ Where shall I run to breathe abroad my woes,\n\nMy woes, whose weight hath weari\u00e8d the earth?\n\nOr mine exclaims, that have surcharg'd the air\n\nWith ceaseless plaints for my deceas\u00e8d son?\n\nThe blust'ring winds, conspiring with my words,\n\nAt my lament have mov'd the leafless trees,\n\nDisrob'd the meadows of their flower'd green,\n\nMade mountains marsh with spring-tides of my tears,\n\nAnd broken through the brazen gates of hell.\n\nYet still tormented is my tortur'd soul\n\nWith broken sighs and restless passions,\n\nThat wing\u00e8d mount; and, hov'ring in the air,\n\nBeat at the windows of the brightest heavens,\n\nSoliciting for justice and revenge:\n\nBut they are plac'd in those empyreal heights,\n\nWhere, countermur'd with walls of diamond,\n\nI find the place impregnable; and they\n\nResist my woes, and give my words no way.\n\n_Enter Hangman with a letter._\n\n_Hangm._ O lord, sir! God bless you, sir! the man, sir, Petergade, sir, he that was so full of merry conceits\u2014\u2014\n\n_Hier._ Well, what of him?\n\n_Hangm._ O lord, sir, he went the wrong way; the fellow had a fair commission to the contrary. Sir, here is his passport; I pray you, sir, we have done him wrong.\n\n_Hier._ I warrant thee, give it me.\n\n_Hangm._ You will stand between the gallows and me?\n\n_Hier._ Ay, ay.\n\n_Hangm._ I thank your lord worship. [ _Exit Hangman._\n\n_Hier._ And yet, though somewhat nearer me concerns,\n\nI will, to ease the grief that I sustain,\n\nTake truce with sorrow while I read on this.\n\n\"My lord, I write, as mine extremes requir'd,\n\nThat you would labour my delivery:\n\nIf you neglect, my life is desperate,\n\nAnd in my death I shall reveal the troth.\n\nYou know, my lord, I slew him for your sake,\n\nAnd was confed'rate with the prince and you;\n\nWon by rewards and hopeful promises,\n\nI holp to murder Don Horatio too.'\u2014\n\nHolp he to murder mine Horatio?\n\nAnd actors in th' accurs\u00e8d tragedy\n\nWast thou, Lorenzo, Balthazar and thou,\n\nOf whom my son, my son deserv'd so well?\n\nWhat have I heard, what have mine eyes beheld?\n\nO sacred heavens, may it come to pass\n\nThat such a monstrous and detested deed,\n\nSo closely smother'd, and so long conceal'd,\n\nShall thus by this be veng\u00e8d or reveal'd?\n\nNow see I what I durst not then suspect,\n\nThat Bellimperia's letter was not feign'd.\n\nNor feign\u00e8d she, though falsely they have wrong'd\n\nBoth her, myself, Horatio, and themselves.\n\nNow may I make compare 'twixt hers and this,\n\nOf every accident I ne'er could find\n\nTill now, and now I feelingly perceive\n\nThey did what heav'n unpunish'd would not leave.\n\nO false Lorenzo! are these thy flatt'ring looks?\n\nIs this the honour that thou didst my son?\n\nAnd Balthazar\u2014bane to thy soul and me!\u2014\n\nWas this the ransom he reserv'd thee for?\n\nWoe to the cause of these constrain\u00e8d wars!\n\nWoe to thy baseness and captivity,\n\nWoe to thy birth, thy body and thy soul,\n\nThy curs\u00e8d father, and thy conquer'd self!\n\nAnd bann'd with bitter execrations be\n\nThe day and place where he did pity thee!\n\nBut wherefore waste I mine unfruitful words,\n\nWhen naught but blood will satisfy my woes?\n\nI will go plain me to my lord the king,\n\nAnd cry aloud for justice through the court,\n\nWearing the flints with these my wither'd feet;\n\nAnd either purchase justice by entreats,\n\nOr tire them all with my revenging threats. [ _Exit._\n\n**SCENE VIII**\n\n_Enter Isabella and her Maid._\n\n_Isab._ So that, you say, this herb, will purge the eye,\n\nAnd this, the head?\u2014\n\nAh!\u2014but none of them will purge the heart!\n\nNo, there's no medicine left for my disease,\n\nNor any physic to recure the dead.\n\n[ _She runs lunatic._\n\nHoratio! O, where's Horatio?\n\n_Maid._ Good madam, affright not thus yourself\n\nWith outrage for your son Horatio:\n\nHe sleeps in quiet in the Elysian fields.\n\n_Isab._ Why, did I not give you gowns and goodly things,\n\nBought you a whistle and a whipstalk too,\n\nTo be reveng\u00e8d on their villanies?\n\n_Maid._ Madam, these humours do torment my soul.\n\n_Isab._ My soul\u2014poor soul! thou talk'st of things\u2014\n\nThou know'st not what: my soul hath silver wings,\n\nThat mounts me up unto the highest heavens;\n\nTo heav'n: ay, there sits my Horatio,\n\nBack'd with a troop of fiery Cherubins,\n\nDancing about his newly heal\u00e8d wounds,\n\nSinging sweet hymns and chanting heav'nly notes:\n\nRare harmony to greet his innocence,\n\nThat died, ay died, a mirror in our days.\n\nBut say, where shall I find the men, the murderers,\n\nThat slew Horatio? Whither shall I run\n\nTo find them out that murder\u00e8d my son? [ _Exeunt._\n\n**SCENE IX**\n\n_Bellimperia at a window._\n\n_Bel._ What means this outrage that is offer'd me?\n\nWhy am I thus sequester'd from the court?\n\nNo notice! Shall I not know the cause\n\nOf these my secret and suspicious ills?\n\nAccurs\u00e8d brother, unkind murderer,\n\nWhy bend'st thou thus thy mind to martyr me?\n\nHieronimo, why writ I of thy wrongs,\n\nOr why art thou so slack in thy revenge?\n\nAndrea, O Andrea! that thou saw'st\n\nMe for thy friend Horatio handled thus,\n\nAnd him for me thus causeless murder\u00e8d!\u2014\n\nWell, force perforce, I must constrain myself\n\nTo patience, and apply me to the time,\n\nTill heav'n, as I have hop'd, shall set me free.\n\n_Enter Christophil._\n\n_Chris._ Come, madam Bellimperia, this may not be.\n\n[ _Exeunt._\n\n**SCENE X**\n\n_Enter Lorenzo, Balthazar, and the Page._\n\n_Lor._ Boy, talk no further; thus far things go well.\n\nThou art assur\u00e8d that thou saw'st him dead?\n\n_Page._ Or else, my lord, I live not.\n\n_Lor._ That's enough.\n\nAs for his resolution in his end,\n\nLeave that to him with whom he sojourns now.\u2014\n\nHere, take my ring and give it Christophil,\n\nAnd bid him let my sister be enlarg'd,\n\nAnd bring her hither straight\u2014 [ _Exit Page._\n\nThis that I did was for a policy,\n\nTo smooth and keep the murder secret,\n\nWhich, as a nine-days' wonder, being o'erblown,\n\nMy gentle sister will I now enlarge.\n\n_Bal._ And time, Lorenzo: for my lord the duke,\n\nYou heard enquir\u00e8d for her yester-night.\n\n_Lor._ Why, and my lord, I hope you heard me say\n\nSufficient reason why she kept away;\n\nBut that's all one. My lord, you love her?\n\n_Bal._ Ay.\n\n_Lor._ Then in your love beware; deal cunningly:\n\nSalve all suspicions, only soothe me up;\n\nAnd if she hap to stand on terms with us\u2014\n\nAs for her sweetheart and concealment so\u2014\n\nJest with her gently: under feign\u00e8d jest\n\nAre things conceal'd that else would breed unrest.\u2014\n\nBut here she comes.\n\n_Enter Bellimperia._\n\nNow, sister?\n\n_Bel._ Sister?\u2014No!\n\nThou art no brother, but an enemy;\n\nElse wouldst thou not have us'd thy sister so:\n\nFirst, to affright me with thy weapons drawn,\n\nAnd with extremes abuse my company;\n\nAnd then to hurry me, like whirlwind's rage,\n\nAmidst a crew of thy confederates,\n\nAnd clap me up, where none might come at me,\n\nNor I at any, to reveal my wrongs.\n\nWhat madding fury did possess thy wits?\n\nOr wherein is 't that I offended thee?\n\n_Lor._ Advise you better, Bellimperia,\n\nFor I have done you no disparagement;\n\nUnless, by more discretion than deserv'd,\n\nI sought to save your honour and mine own.\n\n_Bel._ Mine honour? why, Lorenzo, wherein is't\n\nThat I neglect my reputation so,\n\nAs you, or any, need to rescue it?\n\n_Lor._ His highness and my father were resolv'd\n\nTo come confer with old Hieronimo,\n\nConcerning certain matters of estate,\n\nThat by the viceroy was determin\u00e8d.\n\n_Bel._ And wherein was mine honour touch'd in that?\n\n_Bal._ Have patience, Bellimperia; hear the rest.\n\n_Lor._ Me (next in sight) as messenger they sent,\n\nTo give him notice that they were so nigh:\n\nNow when I came, consorted with the prince,\n\nAnd unexpected, in an arbour there,\n\nFound Bellimperia with Horatio\u2014\n\n_Bel._ How then?\n\n_Lor._ Why, then, remembering that old disgrace, Which you for Don Andrea had endur'd,\n\nAnd now were likely longer to sustain,\n\nBy being found so meanly accompanied,\n\nThought rather\u2014for I knew no readier mean\u2014\n\nTo thrust Horatio forth my father's way.\n\n_Bal._ And carry you obscurely somewhere else,\n\nLest that his highness should have found you there.\n\n_Bel._ Ev'n so, my lord? And you are witness\n\nThat this is true which he entreateth of?\n\nYou, gentle brother, forg'd this for my sake,\n\nAnd you, my lord, were made his instrument?\n\nA work of worth, worthy the noting too!\n\nBut what's the cause that you conceal'd me since?\n\n_Lor._ Your melancholy, sister, since the news\n\nOf your first favourite Don Andrea's death,\n\nMy father's old wrath hath exasperate.\n\n_Bal._ And better was't for you, being in disgrace,\n\nTo absent yourself, and give his fury place.\n\n_Bel._ But why had I no notice of his ire?\n\n_Lor._ That were to add more fuel to your fire,\n\nWho burnt like \u00c6tna _._ for Andrea's loss.\n\n_Bel._ Hath not my father then enquir'd for me?\n\n_Lor._ Sister, he hath, and thus excus'd I thee.\n\n[ _He whispereth in her ear._\n\nBut, Bellimperia, see the gentle prince;\n\nLook on thy love, behold young Balthazar,\n\nWhose passions by thy presence are increas'd;\n\nAnd in whose melancholy thou may'st see\n\nThy hate, his love; thy flight, his following thee.\n\n_Bel._ Brother, you are become an orator\u2014\n\nI know not, I, by what experience\u2014\n\nToo politic for me, past all compare,\n\nSince last I saw you; but content yourself:\n\nThe prince is meditating higher things.\n\n_Bal._ 'Tis of thy beauty then that conquers kings;\n\nOf those thy tresses, Ariadne's twines,\n\nWherewith my liberty thou hast surpris'd;\n\nOf that thine ivory front, my sorrow's map,\n\nWherein I see no hav'n to rest my hope.\n\n_Bel._ To love and fear, and both at once, my lord,\n\nIn my conceit, are things of more import\n\nThan women's wits are to be busied with.\n\n_Bal._ 'Tis I that love.\n\n_Bel._ Whom?\n\n_Bal._ Bellimperia.\n\n_Bel._ But I that fear.\n\n_Bal._ Whom?\n\n_Bel._ Bellimperia.\n\n_Lor._ Fear yourself?\n\n_Bel._ Ay, brother.\n\n_Lor._ How?\n\n_Bel._ As those\n\nThat, what they love, are loath and fear to lose.\n\n_Bal._ Then, fair, let Balthazar your keeper be.\n\n_Bel._ No, Balthazar doth fear as well as we:\n\n_Et tremulo metui pavidum junxere timorem_ \u2014\n\n_Est vanum stolidae proditionis opus._\n\n_Lor._ Nay, and you argue things so cunningly,\n\nWe'll go continue this discourse at court\n\n_Bal._ Led by the loadstar of her heav'nly looks,\n\nWends poor, oppress\u00e8d Balthazar,\n\nAs o'er the mountains walks the wanderer,\n\nIncertain to effect his pilgrimage. [ _Exeunt._\n\n**SCENE XI**\n\n_Enter two Portingals, and Hieronimo meets them._\n\n1. By your leave, sir.\n\nHier. [' _Tis neither as you think, nor as you think,_\n\n_Nor as you think; you're wide all:_\n\n_These slippers are not mine, they were my son Horatio's._\n\n_My son! and what's a son? A thing begot_\n\n_Within a pair of minutes_ \u2014 _thereabout;_\n\n_A lump bred up in darkness, and doth serve_\n\n_To ballace these light creatures we call women;_\n\n_And, at nine months' end, creeps forth to light._\n\n_What is there yet in a son,_\n\n_To make a father dote, rave, or run mad?_\n\n_Being born, it pouts, cries, and breeds teeth._\n\n_What is there yet in a son? He must be fed,_\n\n_Be taught to go, and speak. Ay, or yet_\n\n_Why might not a man love a calf as well?_\n\n_Or melt in passion o'er a frisking kid,_\n\n_As for a son? Methinks, a young bacon,_\n\n_Or a fine little smooth horse colt,_\n\n_Should move a man as much as doth a son:_\n\n_For one of these, in very little time,_\n\n_Will grow to some good use; whereas a son,_\n\n_The more he grows in stature and in years,_\n\n_The more unsquar'd, unbevell'd, he appears,_\n\n_Reckons his parents among the rank of fools,_\n\n_Strikes care upon their heads with his mad riots;_\n\n_Makes them look old, before they meet with age._\n\n_This is a son!\u2014And what a loss were this,_\n\n_Consider'd truly?\u2014\u2014O, but my Horatio_\n\n_Grew out of reach of these insatiate humours:_\n\n_He lov'd his loving parents;_\n\n_He was my comfort, and his mother's joy,_\n\n_The very arm that did hold up our house:_\n\n_Our hopes were stor\u00e8d up in him,_\n\n_Hone but a damn\u00e8d murderer could hate him._\n\n_He had not seen the back of nineteen year,_\n\n_When his strong arm unhors'd_\n\n_The proud Prince Balthazar, and his great mind,_\n\n_Too full of honour, took him to his mercy_ \u2014\n\n_That valiant, but ignoble Portingal!_\n\n_Well, heaven is heaven still!_\n\n_And there is Nemesis, and Furies,_\n\n_And things call'd whips,_\n\n_And they sometimes do meet with murderers:_\n\n_They do not always 'scape, that is some comfort._\n\n_Ay, ay, ay; and then time steals on,_\n\n_And Steals, and steals, till violence leaps forth_\n\n_Like thunder wrapp\u00e8d in a ball of fire,_\n\n_And so doth bring confusion to them all._ ]\n\nGood leave have you: nay, I pray you go,\n\nFor I'll leave you, if you can leave me so.\n\n2. Pray you, which is the next way to my lord the duke's?\n\n_Hier._ The next way from me.\n\n1.To his house, we mean.\n\n_Hier._ O, hard by: 'tis yon house that you see.\n\n2. You could not tell us if his son were there?\n\n_Hier._ Who, my Lord Lorenzo?\n\n1.Ay, sir.\n\n[ _He goeth in at one door and comes out at another._\n\n_Hier._ O, forbear!\n\nFor other talk for us far fitter were.\n\nBut if you be importunate to know\n\nThe way to him, and where to find him out,\n\nThen list to me, and I'll resolve your doubt.\n\nThere is a path upon your left-hand side,\n\nThat leadeth from a guilty conscience\n\nUnto a forest of distrust and fear\u2014\n\nA darksome place, and dangerous to pass:\n\nThere shall you meet with melancholy thoughts,\n\nWhose baleful humours if you but uphold,\n\nIt will conduct you to Despair and Death\u2014\n\nWhose rocky cliffs when you have once beheld,\n\nWithin a hugy dale of lasting night,\n\nThat, kindled with the world's iniquities,\n\nDoth cast up filthy and detested fumes\u2014:\n\nNot far from thence, where murderers have built\n\nA habitation for their curs\u00e8d souls,\n\nThere, in a brazen cauldron, fix'd by Jove,\n\nIn his fell wrath, upon a sulphur flame,\n\nYourselves shall find Lorenzo bathing him\n\nIn boiling lead and blood of innocents.\n\n1. Ha, ha, ha!\n\n_Hier._ Ha, ha, ha! Why, ha, ha, ha! Farewell, good ha, ha, ha!\n\n[ _Exit._\n\n2. Doubtless this man is passing lunatic,\n\nOr imperfection of his age doth make him dote.\n\nCome, let's away to seek my lord the duke.\n\n[ _Exeunt._\n\n**SCENE XII**\n\n_Enter Hieronimo, with a poniard in one hand and a rope in the other._\n\n_Hier._ Now, sir, perhaps I come and see the king;\n\nThe king sees me, and fain would hear my suit:\n\nWhy, is not this a strange and seld-seen thing,\n\nThat standers-by with toys should strike me mute?\u2014\n\nGo to, I see their shifts, and say no more.\u2014\n\nHieronimo, 'tis time for thee to trudge:\n\nDown by the dale that flows with purple gore,\n\nStandeth a fiery tower; there sits a judge\n\nUpon a seat of steel and molten brass,\n\nAnd 'twixt his teeth he holds a fire-brand,\n\nThat leads unto the lake where hell doth stand\n\nAway, Hieronimo! to him be gone:\n\nHe'll do thee justice for Horatio's death.\n\nTurn down this path: thou shalt be with him straight;\n\nOr this, and then thou need'st not take thy breath:\n\nThis way or that way! _\u2014\u2014_ Soft and fair, not so:\n\nFor if I hang or kill myself, let's know\n\nWho will revenge Horatio's murther then?\n\nNo, no! fie, no! pardon me, I'll none of that.\n\n[ _He flings away the dagger and halter._\n\nThis way I'll take, and this way comes the king:\n\n[ _He takes them up again._\n\nAnd here I'll have a fling at him, that's flat;\n\nAnd, Balthazar, I'll be with thee to bring,\n\nAnd thee, Lorenzo! Here's the king\u2014nay, stay;\n\nAnd here, ay here\u2014there goes the hare away.\n\n_Enter King, Ambassador, Castile, and Lorenzo._\n\n_King._ Now show, ambassador, what our viceroy saith:\n\nHath he receiv'd the articles we sent?\n\n_Hier._ Justice, O, justice to Hieronimo.\n\n_Lor._ Back! see'st thou not the king is busy?\n\n_Hier._ O, is he so?\n\n_King._ Who is he that interrupts our business?\n\n_Hier._ Not I. Hieronimo, beware! go by, go by!\n\n_Amb._ Renown\u00e8d King, he hath receiv'd and read\n\nThy kingly proffers, and thy promis'd league;\n\nAnd, as a man extremely over-joy'd\n\nTo hear his son so princely entertain'd,\n\nWhose death he had so solemnly bewail'd,\n\nThis for thy further satisfaction,\n\nAnd kingly love, he kindly lets thee know:\n\nFirst, for the marriage of his princely son\n\nWith Bellimperia, thy belov\u00e8d niece,\n\nThe news are more delightful to his soul,\n\nThan myrrh or incense to the offended heavens.\n\nIn person, therefore, will he come himself,\n\nTo see the marriage rites solemniz\u00e8d,\n\nAnd, in the presence of the court of Spain,\n\nTo knit a sure inextricable band\n\nOf kingly love and everlasting league\n\nBetwixt the crowns of Spain and Portingal.\n\nThere will he give his crown to Balthazar,\n\nAnd make a queen of Bellimperia.\n\n_King._ Brother, how like you this our viceroy's love?\n\n_Cast._ No doubt, my lord, it is an argument\n\nOf honourable care to keep his friend,\n\nAnd wondrous zeal to Balthazar his son;\n\nNor am I least indebted to his grace,\n\nThat bends his liking to my daughter thus.\n\n_Amb._ Now last, dread lord, here hath his highness sent\n\n(Although he send not that his son return)\n\nHis ransom due to Don Horatio.\n\n_Hier._ Horatio! who calls Horatio?\n\n_King._ And well remember'd: thank his majesty.\n\nHere, see it given to Horatio.\n\n_Hier._ Justice, O, justice, justice, gentle king!\n\n_King._ Who is that? Hieronimo?\n\n_Hier._ Justice, O, justice! O my son, my son!\n\nMy son, whom naught can ransom or redeem!\n\n_Lor._ Hieronimo, you are not well-advis'd.\n\n_Hier._ Away, Lorenzo, hinder me no more;\n\nFor thou hast made me bankrupt of my bliss.\n\nGive me my son! you shall not ransom him!\n\nAway! I'll rip the bowels of the earth,\n\n[ _He diggeth with his dagger._\n\nAnd ferry over to th' Elysian plains,\n\nAnd bring my son to show his deadly wounds.\n\nStand from about me!\n\nI'll make a pickaxe of my poniard,\n\nAnd here surrender up my marshalship;\n\nFor I'll go marshal up the fiends in hell,\n\nTo be aveng\u00e8d on you all for this.\n\n_King._ What means this outrage?\n\nWill none of you restrain his fury?\n\n_Hier._ Nay, soft and fair! you shall not need to strive:\n\nFor needs must he go that the devils drive.\n\n[ _Exit._\n\n_King._ What accident hath happ'd Hieronimo?\n\nI have not seen him to demean him so.\n\n_Lor._ My gracious lord, he is with extreme pride,\n\nConceiv'd of young Horatio his son\u2014\n\nAnd covetous of having to himself\n\nThe ransom of the young prince Balthazar\u2014\n\nDistract, and in a manner lunatic.\n\n_King._ Believe me, nephew, we are sorry for't:\n\nThis is the love that fathers bear their sons.\n\nBut, gentle brother, go give to him this gold,\n\nThe prince's ransom; let him have his due.\n\nFor what he hath, Horatio shall not want;\n\nHaply Hieronimo hath need thereof.\n\n_Lor._ But if he be thus helplessly distract,\n\n'Tis requisite his office be resign'd,\n\nAnd giv'n to one of more discretion.\n\n_King._ We shall increase his melancholy so.\n\n'Tis best that we see further in it first,\n\nTill when ourself will hold exempt the place.\n\nAnd, brother, now bring in the ambassador,\n\nThat he may be a witness of the match\n\n'Twixt Balthazar and Bellimperia,\n\nAnd that we may prefix a certain time,\n\nWherein the marriage shall be solemniz'd,\n\nThat we may have thy lord, the viceroy, here.\n\n_Amb._ Therein your highness highly shall content\n\nHis majesty, that longs to hear from hence.\n\n_King._ On, then, and hear you, lord ambassador\u2014\n\n[ _Exeunt._\n\n**SCENE XII A.**\n\nEnter Jaques and Pedro.\n\nJaq. _I wonder, Pedro, why our master thus_\n\n_At midnight sends us with our torches light,_\n\n_When man, and bird, and beast, are all at rest,_\n\n_Save those that watch for rape and bloody murder._\n\nPed. _O Jaques, know thou that our master's mind_\n\n_Is much distraught, since his Horatio died,_\n\n_And\u2014now his ag\u00e8d years should sleep in rest,_\n\n_His heart in quiet\u2014like a desp'rate man,_\n\n_Crows lunatic and childish for his son._\n\n_Sometimes, as he doth at his table sit,_\n\n_He speaks as if Horatio stood by him;_\n\n_Then starting in a rage, falls on the earth,_\n\n_Cries out \"Horatio, where is my Horatio?\"_\n\n_So that with extreme grief and cutting sorrow_\n\n_There is not left in him one inch of man:_\n\n_See, where he comes._\n\nEnter Hieronimo.\n\nHier. _I pry through every crevice of each wall,_\n\n_Look on each tree, and search through every brake,_\n\n_Beat at the bushes, stamp our grandam earth,_\n\n_Dive in the water, and stare up to heaven:_\n\n_Yet cannot I behold my son Horatio._ \u2014\n\n_How now, who's there? spirits, spirits?_\n\nPed. _We are your servants that attend you, sir._\n\nHier. _What make you with your torches in the dark?_\n\nPed. _You bid us light them, and attend you here._\n\nHier. _No, no, you are deceiv'd! not I;\u2014you are deceiv'd!_\n\n_Was I so mad to bid you light your torches now?_\n\n_Light me your torches at the mid of noon,_\n\n_When-as the sun-god rides in all his glory;_\n\n_Light me your torches then._\n\nPed. _Then we burn daylight._\n\nHier. _Let it be burnt; Night is a murd'rous slut,_\n\n_That would not have her treasons to be seen;_\n\n_And yonder pale-fac'd Hecate there, the moon,_\n\n_Doth give consent to that is done in darkness;_\n\n_And all those stars that gaze upon her face,_\n\n_Are aglets on her sleeve, pins on her train;_\n\n_And those that should be powerful and divine,_\n\n_Do sleep in darkness, when they most should shine._\n\nPed. _Provoke them not, fair sir, with tempting words:_\n\n_The heav'ns are gracious, and your miseries_\n\n_And sorrow makes you speak, you know not what._\n\nHier. _Villain, thou liest! and thou dost nought_\n\n_But tell me I am mad: thou liest, I am not mad!_\n\n_I'll know thee to be Pedro, and he Jaques._\n\n_I'll prove it to thee; and were I mad, how could I?_\n\n_Where was she that same night,_\n\n_When my Horatio was murder'd?_\n\n_She should have shone: search thou the book.\u2014Had the moon shone,_\n\n_In my boy's face there was a kind of grace_ ,\n\n_That I know_ \u2014 _nay, I do know\u2014had the murd'rer seen him,_\n\n_His weapon would have fall'n and cut the earth,_\n\n_Had he been fram'd of naught but blood and death._\n\n_Alack! when mischief doth it knows not what,_\n\n_What shall we say to mischief?_\n\nEnter Isabella.\n\nIsab. _Dear Hieronimo, come in a-doors;_\n\n_O, seek not means so to increase thy sorrow._\n\nHier. _Indeed, Isabella, we do nothing here;_\n\n_I do not cry: ask Pedro, and ask Jaques;_\n\n_Not I indeed; we are very merry, very merry._\n\nIsab. _How? be merry here, be merry here?_\n\n_Is not this the place, and this the very tree,_\n\n_Where my Horatio died, where he was murder'd?_\n\nHier. _Was\u2014do not say what: let her weep it out._\n\n_This was the tree; I set it of a kernel:_\n\n_And when our hot Spain could not let it grow,_\n\n_But that the infant and the human sap_\n\n_Began to wither, duly twice a morning_\n\n_Would I be sprinkling it with fountain-water._\n\n_At last it grew and grew, and bore and bore,_\n\n_Till at the length_\n\n_It grew a gallows, and did bear our son:_\n\n_It bore thy fruit and mine_ \u2014 _O wicked, wicked plant!_\n\n[One knocks within at the door.\n\n_See, who knock there._\n\nPed. _It is a painter, sir._\n\nHier. _Bid him come in, and paint some comfort,_\n\n_For surely there's none lives but painted comfort._\n\n_Let him come in!_ \u2014 _One knows not what may chance:_\n\n_God's will that I should set this tree!_ \u2014 _but even so_\n\n_Masters ungrateful servants rear from nought,_\n\n_And then they hate them that did bring them up._\n\nEnter the Painter.\n\nPaint. _God bless you, sir._\n\nHier. _Wherefore? why, thou scornful villain?_\n\n_How, where, or by what means should I be bless'd?_\n\nIsab. _What wouldst thou have, good fellow?_\n\nPaint. _Justice, madam._\n\nHier. _O ambitious beggar!_\n\n_Wouldst thou have that that lives not in the world?_\n\n_Why, all the undelved mines cannot buy_\n\n_An ounce of justice!_\n\n_'Tis a jewel so inestimable. I tell thee,_\n\n_God hath engross'd all justice in his hands,_\n\n_And there is none but what comes from him._\n\nPaint. _O, then I see_\n\n_That God must right me for my murder'd son._ Hier. _How, was thy son murder'd?_\n\nPaint _Ay, sir; no man did hold a son so dear._\n\nHier. _What, not as thine? that's a lie,_\n\n_As massy as the earth: I had a son,_\n\n_Whose least unvalu'd hair did weigh_\n\n_A thousand of thy sons: and he was murder'd._\n\nPaint _Alas, sir, I had no more but he._\n\nHier. _Nor I, nor I: but this same one of mine_\n\n_Was worth a legion. But all is one._\n\n_Pedro, Jaques, go in a-doors; Isabella, go,_\n\n_And this good fellow here and I_\n\n_Will range this hideous orchard up and down,_\n\n_Like to two lions reav\u00e8d of their young._\n\n_Go in a-doors, I say._\n\n[Exeunt. The painter and he sits down.\n\n_Come, let's talk wisely now._\n\n_Was thy son murder'd?_\n\nPaint. _Ay, sir._\n\nHier. _So was mine._\n\n_How dost take it? art thou not sometimes mad?_\n\n_Is there no tricks that comes before thine eyes?_\n\nPaint. _O Lord, yes, sir._\n\nHier. _Art a painter? canst paint me a tear, or a wound, a groan, or a sigh? canst paint me such a tree as this?_\n\nPaint. _Sir, I am sure you have heard of my painting: my name's Bazardo._\n\nHier. _Bazardo! afore God, an excellent fellow. Look you, sir, do you see, I'd have you paint me for my gallery, in your oil-colours matted, and draw me five years younger than I am_ \u2014 _do ye see, sir, let five years go; let them go like the marshal of Spain_ \u2014 _my wife Isabella standing by me, with a speaking look to my son Horatio, which shouldintend to this or some such-like purpose: \"God bless thee, my sweet son\"; and my hand leaning upon his head, thus, sir; do you see?\u2014may it be done?_\n\nPaint. _Very welly sir._\n\nHier. _Nay, I pray, mark me, sir: then, sir, would I have you paint me this tree, this very tree. Canst paint a doleful cry?_\n\nPaint. _Seemingly, sir._\n\nHier. _Nay, it should cry; but all is one. Well, sir, paint me a youth run through and through with villain's swords, hanging upon this tree. Canst thou draw a murderer?_\n\nPaint. _I'll warrant you, sir; I have the pattern of the most notorious villains that ever lived in all Spain._\n\nHier. _Oy let them be worse, worse: stretch thine art, and let their beards be of Judas his own colour; and let their eye-brows jutty over; in any case observe that. Then, sir, after some violent noise, bring me forth in my shirt, and my gown under mine arm, with my torch in my hand, and my sword reared up thus:_ \u2014 _and with these words:_\n\n\" _What noise is this? who calls Hitronimo?_ \"\n\n_May it be done?_\n\nPaint. _Yea, sir._\n\nHier. _Well, sir; then bring me forth, bring me through alley and alley, still with a distracted countenance going along, and let my hair heave up my nightcap. Let the clouds scowl, make the moon dark, the stars extinct, the winds blowing, the bells tolling, the owls shrieking, the toads croaking, the minutes jarring, and the clock striking twelve. And then at last, sir, starting, behold a man hanging, and tottering and tottering, as you know the wind will wave a man, and I with a trice to cut him down. And looking upon him by the advantage of my torch, find it to be my son Horatio. There you may show a passion, there you may show a passion! Draw me like old Priam of Troy, crying:_ \" _The house is a-fire, the house is a-fire, as the torch over my head!' Make me curse, make me rave, make me cry, make me mad, make me well again, make me curse hell, invocate heaven, and in the end leave me in a trance_ \u2014 _and so forth._\n\nPaint. _And is this the end?_\n\nHier. _O no, there is no end: the end is death and madness! As I am never better than when I am mad: then methinks I am a brave fellow; then I do wonders: but reason abuseth me, and, there's the torment, there's the hell. At the last, sir, bring me to one of the murderers; were he as strong as Hector, thus would I tear and drag him up and down._\n\n[He beats the painter in, then comes out again, with a book in his hand.\n\n**SCENE XIII**\n\n_Enter Hieronimo, with a book in his hand._\n\n_Vindicta mihi!_\n\nAy, heav'n will be reveng'd of every ill;\n\nNor will they suffer murder unrepaid.\n\nThen stay, Hieronimo, attend their will:\n\nFor mortal men may not appoint their time!\u2014\n\n_\"Per scelus semper tutum est sceleribus iter.\"_\n\nStrike, and strike home, where wrong is offer'd thee;\n\nFor evils unto ills conductors be,\n\nAnd death's the worst of resolution.\n\nFor he that thinks with patience to contend\n\nTo quiet life, his life shall easily end.\u2014\n\n_\"Fata si miseros juvant, habes salutem;_\n\n_Fata si vitam negant, habes sepulchrum_ \":\n\nIf destiny thy miseries do ease,\n\nThen hast thou health, and happy shalt thou be;\n\nIf destiny deny thee life, Hieronimo,\n\nYet shalt thou be assur\u00e8d of a tomb-\u2014:\n\nIf neither, yet let this thy comfort be:\n\nHeav'n cov'reth him that hath no burial.\n\nAnd to conclude, I will revenge his death!\n\nBut how? not as the vulgar wits of men,\n\nWith open, but inevitable ills,\n\nAs by a secret, yet a certain mean,\n\nWhich under kindship will be cloak\u00e8d best.\n\nWise men will take their opportunity\n\nClosely and safely, fitting things to time.\u2014\n\nBut in extremes advantage hath no time;\n\nAnd therefore all times fit not for revenge.\n\nThus therefore will I rest me in unrest,\n\nDissembling quiet in unquietness,\n\nNot seeming that I know their villanies,\n\nThat my simplicity may make them think.\n\nThat ignorantly I will let all slip;\n\nFor ignorance, I wot, and well they know,\n\n_Remedium malorum iners est._\n\nNor ought avails it me to menace them\n\nWho, as a wintry storm upon a plain,\n\nWill bear me down with their nobility.\n\nNo, no, Hieronimo, thou must enjoin\n\nThine eyes to observation, and thy tongue\n\nTo milder speeches than thy spirit affords,\n\nThy heart to patience, and thy hands to rest.\n\nThy cap to courtesy, and thy knee to bow,\n\nTill to revenge thou know, when, where and how.\n\n[ _A noise within._\n\nHow now, what noise? what coil is that you keep?\n\n_Enter a Servant._\n\n_Serv._ Here are a sort of poor petitioners,\n\nThat are importunate, and it shall please you, sir,\n\nThat you should plead their cases to the king.\n\n_Hier._ That I should plead their several actions?\n\nWhy, let them enter, and let me see them.\n\n_Enter three Citizens and an Old Man._\n\n1.So,\n\nI tell you this: for learning and for law,\n\nThere is not any advocate in Spain\n\nThat can prevail, or will take half the pain\n\nThat he will, in pursuit of equity.\n\n_Hier._ Come near, you men, that thus importune me.\u2014\n\n[ _Aside._ ] Now must I bear a face of gravity;\n\nFor thus I us'd, before my marshalship,\n\nTo plead in causes as corregidor.\u2014\n\nCome on, sirs, what's the matter?\n\n2.Sir, an action,\n\n_Hier._ Of battery?\n\n1.Mine of debt\n\n_Hier._ Give place.\n\n2. No, sir, mine is an action of the case.\n\n3. Mine an _ejectione firmae_ by a lease.\n\n_Hier._ Content you, sirs; are you determin\u00e8d\n\nThat I should plead your several actions?\n\n1. Ay, sir, and here's my declaration.\n\n2. And here's my band.\n\n3.And here's my lease.\n\n[ _They give him papers._\n\n_Hier._ But wherefore stands yon silly man so mute,\n\nWith mournful eyes and hands to heav'n uprear'd?\n\nCome hither, father, let me know thy cause.\n\n_Senex._ O worthy sir, my cause, but slightly known,\n\nMay move the hearts of warlike Myrmidons,\n\nAnd melt the Corsic rocks with ruthful tears.\n\n_Hier._ Say, father, tell me what's thy suit?\n\n_Senex._ No, sir, could my woes\n\nGive way unto my most distressful words,\n\nThen should I not in paper, as you see,\n\nWith ink bewray what blood began in me.\n\n_Hier._ What's here? 'The humble supplication\n\nOf Don Bazulto for his murder'd son.'\n\n_Senex._ Ay, sir.\n\n_Hier._ No, sir, it was my murder'd son:\n\nO my son, my son, O my son Horatio!\n\nBut mine, or thine, Bazulto, be content.\n\nHere, take my handkercher, and wipe thine eyes,\n\nWhiles wretched I in thy mishaps may see\n\nThe lively portrait of my dying self.\n\n[ _He draweth out a bloody napkin._\n\nO no, not this; Horatio, this was thine;\n\nAnd when I dy'd it in thy dearest blood,\n\nThis was a token 'twixt thy soul and me,\n\nThat of thy death reveng\u00e8d I should be.\n\nBut here, take this, and this\u2014what, my purse?\u2014\n\nAy, this, and that, and all of them are thine;\n\nFor all as one are our extremities.\n\n1. O, see the kindness of Hieronimo!\n\n2. This gentleness shows him a gentleman.\n\n_Hier._ See, see, O see thy shame, Hieronimo;\n\nSee here a loving father to his son!\n\nBehold the sorrows and the sad laments,\n\nThat he deliv'reth for his son's decease!\n\nIf love's effects so strive in lesser things,\n\nIf love enforce such moods in meaner wits,\n\nIf love express such power in poor estates:\n\nHieronimo, when as a raging sea,\n\nToss'd with the wind and tide, o'erturnest then\n\nThe upper billows course of waves to keep,\n\nWhilst lesser waters labour in the deep:\n\nThen sham'st thou not, Hieronimo, to neglect\n\nThe sweet revenge of thy Horatio?\n\nThough on this earth justice will not be found,\n\nI'll down to hell, and in this passion\n\nKnock at the dismal gates of Pluto's court,\n\nGetting by force, as once Alcides did,\n\nA troop of Furies and tormenting hags\n\nTo torture Don Lorenzo and the rest.\n\nYet lest the triple-headed porter should\n\nDeny my passage to the slimy strand,\n\nThe Thracian poet thou shalt counterfeit:\n\nCome on, old father, be my Orpheus,\n\nAnd if thou canst no notes upon the harp,\n\nThen sound the burden of thy sore heart's-grief,\n\nTill we do gain that Proserpine may grant\n\nRevenge on them that murder\u00e8d my son.\n\nThen will I rent and tear them, thus and thus,\n\nShiv'ring their limbs in pieces with my teeth.\n\n[ _Tears the papers._\n\n1. O sir, my declaration!\n\n[ _Exit Hieronimo, and they after._\n\n2.Save my bond!\n\n_Enter Hieronimo._\n\n2. Save my bond!\n\n3. Alas, my lease! it cost me ten pound,\n\nAnd you my lord, have torn the same.\n\n_Hier._ That cannot be, I gave it never a wound;\n\nShow me one drop of blood fall from the same:\n\nHow is it possible I should slay it then?\n\nTush, no; run after, catch me if you can.\n\n[ _Exeunt all but the Old Man. Bazulto remains till Hieronimo enters again, who, staring him in the face, speaks._\n\n_Hier._ And art thou come, Horatio, from the depth,\n\nTo ask for justice in this upper earth,\n\nTo tell thy father thou art unreveng'd,\n\nTo wring more tears from Isabella's eyes,\n\nWhose lights are dimm'd with over-long laments?\n\nGo back, my son, complain to Aeacus,\n\nFor here's no justice; gentle boy, be gone,\n\nFor justice is exil\u00e8d from the earth:\n\nHieronimo will bear thee company.\n\nThy mother cries on righteous Rhadamanth\n\nFor just revenge against the murderers.\n\n_Senex._ Alas, my lord, whence springs this troubled speech?\n\n_Hier._ But let me look on my Horatio.\n\nSweet boy, how art thou chang'd in death's black shade!\n\nHad Proserpine no pity on thy youth,\n\nBut suffer'd thy fair crimson-colour'd spring\n\nWith wither'd winter to be blasted thus?\n\nHoratio, thou art older than thy father:\n\nAh, ruthless fate, that favour thus transforms!\n\n_Baz._ Ah, my good lord, I am not your young son.\n\n_Hier._ What, not my son? thou then a Fury art,\n\nSent from the empty kingdom of black night\n\nTo summon me to make appearance\n\nBefore grim Minos and just Rhadamanth,\n\nTo plague Hieronimo that is remiss,\n\nAnd seeks not vengeance for Horatio's death.\n\n_Baz._ I am a griev\u00e8d man, and not a ghost,\n\nThat came for justice for my murder'd son.\n\n_Hier._ Ay, now I know thee, now thou nam'st thy son:\n\nThou art the lively image of my grief;\n\nWithin thy face, my sorrows I may see.\n\nThy eyes are gumm'd with tears, thy cheeks are wan,\n\nThy forehead troubled, and thy mutt'ring lips\n\nMurmur sad words abruptly broken off;\n\nBy force of windy sighs thy spirit breathes,\n\nAnd all this sorrow riseth for thy son:\n\nAnd selfsame sorrow feel I for my son.\n\nCome in, old man, thou shalt to Isabel;\n\nLean on my arm: I thee, thou me, shalt stay,\n\nAnd thou, and I, and she will sing a song,\n\nThree parts in one, but all of discords fram'd\u2014:\n\nTalk not of chords, but let us now be gone,\n\nFor with a cord Horatio was slain. [ _Exeunt._\n\n**SCENE XIV**\n\n_Enter King of Spain, the Duke, Viceroy, and Lorenzo, Balthazar, Don Pedro, and Bellimperia._\n\n_King._ Go, brother, 'tis the Duke of Castile's cause; Salute the Viceroy in our name.\n\n_Cast._ I go.\n\n_Vic._ Go forth, Don Pedro, for thy nephew's sake,\n\nAnd greet the Duke of Castile.\n\n_Ped._ It shall be so.\n\n_King._ And now to meet these Portuguese:\n\nFor as we now are, so sometimes were these,\n\nKings and commanders of the western Indies.\n\nWelcome, brave Viceroy, to the court of Spain,\n\nAnd welcome all his honourable train!\n\n'Tis not unknown to us for why you come,\n\nOr have so kingly cross'd the seas:\n\nSufficeth it, in this we note the troth\n\nAnd more than common love you lend to us.\n\nSo is it that mine honourable niece\n\n(For it beseems us now that it be known)\n\nAlready is betroth'd to Balthazar:\n\nAnd by appointment and our condescent\n\nTo-morrow are they to be marri\u00e8d.\n\nTo this intent we entertain thyself,\n\nThy followers, their pleasure, and our peace.\n\nSpeak, men of Portingal, shall it be so?\n\nIf ay, say so; if not, say flatly no.\n\n_Vic._ Renowm\u00e8d King, I come not, as thou think'st,\n\nWith doubtful followers, unresolv\u00e8d men,\n\nBut such as have upon thine articles\n\nConfirm'd thy motion, and contented me.\n\nKnow, sovereign, I come to solemnize\n\nThe marri\u00e0ge of thy belov\u00e8d niece,\n\nFair Bellimperia, with my Balthazar,\n\nWith thee, my son; whom sith I live to see,\n\nHere take my crown, I give it her and thee;\n\nAnd let me live a solitary life,\n\nIn ceaseless prayers,\n\nTo think how strangely heav'n hath thee preserv'd.\n\n_King._ See, brother, see, how nature strives in him!\n\nCome, worthy Viceroy, and accompany\n\nThy friend with thine extremities:\n\nA place more private fits this princely mood.\n\n_Vic._ Or here, or where your highness thinks it good.\n\n[ _Exeunt all but Castile and Lorenzo._\n\n**SCENE XV**\n\n_Castile, Lorenzo._\n\n_Cast._ Nay, stay, Lorenzo, let me talk with you.\n\nSee'st thou this entertainment of these kings?\n\n_Lor._ I do, my lord, and joy to see the same.\n\n_Cast._ And know'st thou why this meeting is?\n\n_Lor._ For her, my lord, whom Balthazar doth love,\n\nAnd to confirm their promis'd marri\u00e0ge.\n\n_Cast._ She is thy sister?\n\n_Lor._ Who, Bellimperia? ay,\n\nMy gracious lord, and this is the day,\n\nThat I have long'd so happily to see.\n\n_Cast_ Thou wouldst be loath that any fault of thine\n\nShould intercept her in her happiness?\n\n_Lor._ Heav'ns will not let Lorenzo err so much.\n\n_Cast._ Why then, Lorenzo, listen to my words:\n\nIt is suspected, and reported too,\n\nThat thou, Lorenzo, wrong'st Hieronimo,\n\nAnd in his suits towards his majesty\n\nStill keep'st him back, and seek'st to cross his suit.\n\n_Lor._ That I, my lord\u2014\u2014?\n\n_Cast._ I tell thee, son, myself have heard it said,\n\nWhen (to my sorrow) I have been asham'd\n\nTo answer for thee, though thou art my son.\n\nLorenzo, know'st thou hot the common love\n\nAnd kindness that Hieronimo hath won\n\nBy his deserts within the court of Spain?\n\nOr see'st thou not the king my brother's care\n\nIn his behalf, and to procure his health?\n\nLorenzo, shouldst thou thwart his passions,\n\nAnd he exclaim against thee to the king,\n\nWhat honour were 't in this assembly,\n\nOr what a scandal were 't among the kings\n\nTo hear Hieronimo exclaim on thee?\n\nTell me\u2014and look thou tell me truly too\u2014\n\nWhence grows the ground of this report in court?\n\n_Lor._ My lord, it lies not in Lorenzo's power\n\nTo stop the vulgar, liberal of their tongues:\n\nA small advantage makes a water-breach,\n\nAnd no man lives that long contenteth all.\n\n_Cast._ Myself have seen thee busy to keep back\n\nHim and his supplications from the king.\n\n_Lor._ Yourself, my lord, hath seen his passions,\n\nThat ill beseem'd the presence of a king;\n\nAnd for I pitied him in his distress,\n\nI held him thence with kind and courteous words,\n\nAs free from malice to Hieronimo\n\nAs to my soul, my lord.\n\n_Cast._ Hieronimo, my son, mistakes thee then.\n\n_Lor._ My gracious father, believe me, so he doth.\n\nBut what's a silly man, distract in mind\n\nTo think upon the murder of his son?\n\nAlas! how easy is it for him to err!\n\nBut for his satisfaction and the world's,\n\n'Twere good, my lord, that Hieronimo and I\n\nWere reconcil'd, if he misconster me.\n\n_Cast._ Lorenzo, thou hast said; it shall be so.\n\nGo one of you, and call Hieronimo.\n\n_Enter Balthazar and Bellimperia._\n\n_Bal._ Come, Bellimperia, Balthazar's content,\n\nMy sorrow's ease and sovereign of my bliss,\n\nSith heaven hath ordain'd thee to be mine:\n\nDisperse those clouds and melancholy looks,\n\nAnd clear them up with those thy sun-bright eyes,\n\nWherein my hope and heaven's fair beauty lies.\n\n_Bel._ My looks, my lord, are fitting for my love,\n\nWhich, new-begun, can show no brighter yet\n\n_Bal._ New-kindled flames should burn as morning sun.\n\n_Bel._ But not too fast, lest heat and all be done.\n\nI see my lord my father.\n\n_Bal._ Truce, my love;\n\nI'll go salute him.\n\n_Cast._ Welcome, Balthazar,\n\nWelcome, brave prince, the pledge of Castile's peace!\n\nAnd welcome, Bellimperia!\u2014How now, girl?\n\nWhy com'st thou sadly to salute us thus?\n\nContent thyself, for I am satisfied:\n\nIt is not now as when Andrea liv'd;\n\nWe have forgotten and forgiven that,\n\nAnd thou art grac\u00e8d with a happier love.\u2014\n\nBut, Balthazar, here comes Hieronimo;\n\nI'll have a word with him.\n\n_Enter Hieronimo and a Servant._\n\n_Hier._ And where's the duke?\n\n_Serv._ Yonder.\n\n_Hier._ Ev'n so.\u2014\n\nWhat new device have they devis\u00e8d, trow?\n\n_Pocas palabras!_ mild as the lamb!\n\nIs't I will be reveng'd? No, I am not the man.\u2014\n\n_Cast._ Welcome, Hieronimo.\n\n_Lor._ Welcome, Hieronimo.\n\n_Bal._ Welcome, Hieronimo.\n\n_Hier._ My lords, I thank you for Horatio.\n\n_Cast._ Hieronimo, the reason that I sent\n\nTo speak with you, is this.\n\n_Hier._ What, so short?\n\nThen I'll be gone, I thank you for 't.\n\n_Cast._ Nay, stay, Hieronimo!\u2014go call him, son.\n\n_Lor._ Hieronimo, my father craves a word with you.\n\n_Hier._ With me, sir? why, my lord, I thought you had done.\n\n_Lor._ No; [ _Aside_ ] would he had!\n\n_Cast._ Hieronimo, I hear\n\nYou find yourself aggriev\u00e8d at my son,\n\nBecause you have not access unto the king;\n\nAnd say 'tis he that intercepts your suits.\n\n_Hier._ Why, is not this a miserable thing, my lord?\n\n_Cast._ Hieronimo, I hope you have no cause,\n\nAnd would be loath that one of your deserts\n\nShould once have reason to suspect my son,\n\nConsid'ring how I think of you myself.\n\n_Hier._ Your son Lorenzo! whom, my noble lord?\n\nThe hope of Spain, mine honourable friend?\n\nGrant me the combat of them, if they dare:\n\n[ _Draws out his sword._\n\nI'll meet him face to face, to tell me so!\n\nThese be the scandalous reports of such\n\nAs love not me, and hate my lord too much:\n\nShould I suspect Lorenzo would prevent\n\nOr cross my suit, that lov'd my son so well?\n\nMy lord, I am asham'd it should be said.\n\n_Lor._ Hieronimo, I never gave you cause.\n\n_Hier._ My good lord, I know you did not.\n\n_Cast._ There then pause;\n\nAnd for the satisfaction of the world,\n\nHieronimo, frequent my homely house,\n\nThe Duke of Castile, Cyprian's ancient seat;\n\nAnd when thou wilt, use me, my son, and it:\n\nBut here, before Prince Balthazar and me,\n\nEmbrace each other, and be perfect friends.\n\n_Hier._ Ay, marry, my lord, and shall.\n\nFriends, quoth he? see, I'll be friends with you all:\n\nEspecially with you, my lovely lord;\n\nFor divers causes it is fit for us\n\nThat we be friends: the world's suspicious,\n\nAnd men may think what we imagine not\n\n_Bal._ Why, this is friendly done, Hieronimo.\n\n_Lor._ And that I hope: old grudges are forgot?\n\n_Hier._ What else? it were a shame it should not be so.\n\n_Cast._ Come on, Hieronimo, at my request;\n\nLet us entreat your company to-day. [ _Exeunt._\n\n_Hier._ Your lordship's to command.\u2014Pah! keep your way:\n\n_Chi mi fa pi\u00f9 carezze che non suole,_\n\n_Tradito mi ha, o tradir mi vuole._ [ _Exit._\n\n**SCENE XVI**\n\n_Enter Ghost and Revenge._\n\n_Ghost._ Awake, Erichtho! Cerberus, awake!\n\nSolicit Pluto, gentle Proserpine!\n\nTo combat, Acheron and Erebus!\n\nFor ne'er, by Styx and Phlegethon in hell,\n\nO'er-ferried Charon to the fiery lakes\n\nSuch fearful sights, as poor Andrea sees.\n\nRevenge, awake!\n\n_Revenge._ Awake? for why?\n\n_Ghost._ Awake, Revenge; for thou art ill-advis'd\n\nTo sleep\u2014awake! what, thou art warn'd to watch!\n\n_Revenge._ Content thyself, and do not trouble me.\n\n_Ghost._ Awake, Revenge, if love\u2014as love hath had\u2014\n\nHave yet the power or prevalence in hell!\n\nHieronimo with Lorenzo is join'd in league,\n\nAnd intercepts our passage to revenge:\n\nAwake, Revenge, or we are woe-begone!\n\n_Revenge._ Thus worldlings ground, what they have dream'd, upon.\n\nContent thyself, Andrea: though I sleep,\n\nYet is my mood soliciting their souls.\n\nSufficeth thee that poor Hieronimo\n\nCannot forget his son Horatio.\n\nNor dies Revenge, although he sleep awhile;\n\nFor in unquiet quietness is feign'd,\n\nAnd slumb'ring is a common worldly wile.\u2014\n\nBehold, Andrea, for an instance, how\n\nRevenge hath slept, and then imagine thou,\n\nWhat 'tis to be subject to destiny.\n\n_Enter a Dumb-Show._\n\n_Ghost._ Awake, Revenge; reveal this mystery.\n\n_Revenge._ Lo! the two first the nuptial torches bore\n\nAs brightly burning as the mid-day's sun;\n\nBut after them doth Hymen hie as fast,\n\nCloth\u00e8d in sable and a saffron robe,\n\nAnd blows them out, and quencheth them with blood,\n\nAs discontent that things continue so.\n\n_Ghost._ Sufficeth me; thy meaning's understood,\n\nAnd thanks to thee and those infernal powers,\n\nThat will not tolerate a lover's woe.\u2014\n\nRest thee, for I will sit to see the rest.\n\n_Revenge._ Then argue not, for thou hast thy request\n\n[ _Exeunt._\n\n**ACT IV**\n\n**SCENE I**\n\n_Enter Bellimperia and Hieronimo._\n\n_Bel._ Is this the love thou bear'st Horatio?\n\nIs this the kindness that thou counterfeit'st?\n\nAre these the fruits of thine incessant tears?\n\nHieronimo, are these thy passions,\n\nThy protestations and thy deep laments,\n\nThat thou wert wont to weary men withal?\n\nO unkind father! O deceitful world!\n\nWith what excuses canst thou show thyself\n\nFrom this dishonour and the hate of men?\n\nThus to neglect the loss and life of him\n\nWhom both my letters and thine own belief\n\nAssures thee to be causeless slaughter\u00e8d!\n\nHieronimo, for shame, Hieronimo,\n\nBe not a history to after-times\n\nOf such ingratitude unto thy son:\n\nUnhappy mothers of such children then,\n\nBut monstrous fathers to forget so soon\n\nThe death of those, whom they with care and cost\n\nHave tender'd so, thus careless should be lost.\n\nMyself, a stranger in respect of thee,\n\nSo lov'd his life, as still I wish their deaths.\n\nNor shall his death be unreveng'd by me,\n\nAlthough I bear it out for fashion's sake:\n\nFor here I swear, in sight of heav'n and earth,\n\nShouldst thou neglect the love thou shouldst retain,\n\nAnd give it over, and devise no more,\n\nMyself should send their hateful souls to hell,\n\nThat wrought his downfall with extremest death.\n\n_Hier._ But may it be that Bellimperia\n\nVows such revenge as she hath deign'd to say?\n\nWhy, then I see that heav'n applies our drift,\n\nAnd all the saints do sit soliciting\n\nFor vengeance on those curs\u00e8d murtherers.\n\nMadam, 'tis true, and now I find it so:\n\nI found a letter, written in your name,\n\nAnd in that letter, how Horatio died.\n\nPardon, O pardon, Bellimperia,\n\nMy fear and care in not believing it;\n\nNor think I thoughtless think upon a mean\n\nTo let his death be unreveng'd at full.\n\nAnd here I vow\u2014so you but give consent,\n\nAnd will conceal my resolution\u2014:\n\nI will ere long determine of their deaths\n\nThat causeless thus have murder\u00e8d my son.\n\n_Bel._ Hieronimo, I will consent, conceal,\n\nAnd ought that may effect for thine avail,\n\nJoin with thee to revenge Horatio's death.\n\n_Hier._ On, then; and whatsoever I devise,\n\nLet me entreat you, grace my practices,\n\nFor why the plot's already in mine head.\n\nHere they are.\n\n_Enter Balthazar and Lorenzo._\n\n_Bal._ How now, Hieronimo?\n\nWhat, courting Bellimperia?\n\n_Hier._ Ay, my lord;\n\nSuch courting as (I promise you):\n\nShe hath my heart, but you, my lord, have hers.\n\n_Lor._ But now, Hieronimo, or never,\n\nWe are to entreat your help.\n\n_Hier._ My help?\n\nWhy, my good lords, assure yourselves of me;\n\nFor you have giv'n me cause\u2014:\n\nAy, by my faith have you!\n\n_Bal._ It pleased you,\n\nAt the entertainment of the ambassador,\n\nTo grace the king so much as with a show.\n\nNow, were your study so well furnish\u00e8d,\n\nAs for the passing of the first night's sport\n\nTo entertain my father with the like,\n\nOr any such-like pleasing motion,\n\nAssure yourself, it would content them well.\n\n_Hier._ Is this all?\n\n_Bal._ Ay, this is all.\n\n_Hier._ Why then, I'll fit you; say no more.\n\nWhen I was young, I gave my mind\n\nAnd plied myself to fruitless poetry;\n\nWhich though it profit the professor naught,\n\nYet is it passing pleasing to the world\n\n_Lor._ And how for that?\n\n_Hier._ Marry, my good lord, thus:\n\n(And yet, methinks, you are too quick with us)\u2014:\n\nWhen in Toledo there I studi\u00e8d,\n\nIt was my chance to write a tragedy:\n\nSee here, my lords\u2014 [ _He shows them a book._\n\nWhich, long forgot, I found this other day.\n\nNow would your lordships favour me so much\n\nAs but to grace me with your acting it\u2014\n\nI mean each one of you to play a part\u2014\n\nAssure you it will prove most passing strange,\n\nAnd wondrous plausible to that assembly.\n\n_Bal._ What, would you have us play a tragedy?\n\n_Hier._ Why, Nero thought it no disparagement,\n\nAnd kings and emperors have ta'en delight\n\nTo make experience of their wits in plays.\n\n_Lor._ Nay, be not angry, good Hieronimo;\n\nThe prince but ask'd a question.\n\n_Bal._ In faith, Hieronimo, and you be in earnest,\n\nI'll make one.\n\n_Lor._ And I another.\n\n_Hier._ Now, my good lord, could you entreat\n\nYour sister Bellimperia to make one?\n\nFor what's a play without a woman in\n\n_Bel._ Little entreaty shall serve me, Hieronimo;\n\nFor I must needs be employ\u00e8d in your play.\n\n_Hier._ Why, this is well: I tell you, lordings,\n\nIt was determin\u00e8d to have been acted,\n\nBy gentlemen and scholars too,\n\nSuch as could tell what to speak.\n\n_Bal._ And now\n\nIt shall be play'd by princes and courtiers,\n\nSuch as can tell how to speak:\n\nIf, as it is our country manner,\n\nYou will but let us know the argument.\n\n_Hier._ That shall I roundly. The chronicles of Spain\n\nRecord this written of a knight of Rhodes:\n\nHe was betroth'd, and wedded at the length,\n\nTo one Perseda, an Italian dame,\n\nWhose beauty ravish'd all that her beheld,\n\nEspecially the soul of Soliman,\n\nWho at the marriage was the chiefest guest.\n\nBy sundry means sought Soliman to win\n\nPerseda's love, and could not gain the same.\n\nThen 'gan he break his passions to a friend,\n\nOne of his bashaws, whom he held full dear;\n\nHer had this bashaw long solicited,\n\nAnd saw she was not otherwise to be won,\n\nBut by her husband's death, this knight of Rhodes,\n\nWhom presently by treachery he slew.\n\nShe, stirr'd with an exceeding hate therefore,\n\nAs cause of this slew Soliman,\n\nAnd, to escape the bashaw's tyranny,\n\nDid stab herself: and this the tragedy.\n\n_Lor._ O excellent!\n\n_Bel._ But say, Hieronimo, what then became Of him that was the bashaw?\n\n_Hier._ Marry, thus:\n\nMov'd with remorse of his misdeeds,\n\nRan to a mountain-top, and hung himself.\n\n_Bal._ But which of us is to perform that part?\n\n_Hier._ O, that will I, my lords; make no doubt of it:\n\nI'll play the murderer, I warrant you;\n\nFor I already have conceited that\n\n_Bal._ And what shall I?\n\n_Hier._ Great Soliman, the Turkish emperor.\n\n_Lor._ And I?\n\n_Hier._ Erastus, the knight of Rhodes.\n\n_Bel._ And I?\n\n_Hier._ Perseda, chaste and resolute.\u2014\n\nAnd here, my lords, are several abstracts drawn,\n\nFor each of you to note your parts,\n\nAnd act it, as occasion's offer'd you.\n\nYou must provide a Turkish cap,\n\nA black mustachio and a falchion;\n\n[ _Gives a paper to Balthazar._\n\nYou with a cross, like to a knight of Rhodes;\n\n[ _Gives another to Lorenzo._\n\nAnd, madam, you must attire yourself\n\n[ _He giveth Bellimperia another._\n\nLike Ph\u0153be, Flora, or the hunteress,\n\nWhich to your discretion shall seem best.\n\nAnd as for me, my lords, I'll look to one,\n\nAnd, with the ransom that the viceroy sent,\n\nSo furnish and perform this tragedy,\n\nAs all the world shall say, Hieronimo\n\nWas liberal in gracing of it so.\n\n_Bal._ Hieronimo, methinks a comedy were better.\n\n_Hier._ A comedy?\n\nFie! comedies are fit for common wits:\n\nBut to present a kingly troop withal,\n\nGive me a stately-written tragedy;\n\n_Trag\u0153dia cothurnata,_ fitting kings,\n\nContaining matter, and not common things.\n\nMy lords, all this must be perform'd,\n\nAs fitting for the first night's revelling.\n\nThe Italian tragedians were so sharp of wit,\n\nThat in one hour's meditation\n\nThey would perform anything in action.\n\n_Lor._ And well it may; for I have seen the like\n\nIn Paris 'mongst the French tragedians.\n\n_Hier._ In Paris? mass! and well remember\u00e8d!\n\nThere's one thing more that rests for us to do.\n\n_Bal._ What's that, Hieronimo? forget not anything.\n\n_Hier. E_ ach one of us\n\nMust act his part in unknown languages,\n\nThat it may breed the more variety:\n\nAs you, my lord, in Latin, I in Greek,\n\nYou in Italian, and for because I know\n\nThat Bellimperia hath practised the French,\n\nIn courtly French shall all her phrases be.\n\n_Bel._ You mean to try my cunning then, Hieronimo?\n\n_Bal._ But this will be a mere confusion,\n\nAnd hardly shall we all be understood.\n\n_Hier._ It must be so; for the conclusion\n\nShall prove the invention and all was good:\n\nAnd I myself in an oration,\n\nAnd with a strange and wondrous show besides,\n\nThat I will have there behind a curtain,\n\nAssure yourself, shall make the matter known:\n\nAnd all shall be concluded in one scene,\n\nFor there's no pleasure ta'en in tediousness.\n\n_Bal._ How like you this?\n\n_Lor._ Why, thus my lord:\n\nWe must resolve to soothe his humours up.\n\n_Bal._ On then, Hieronimo; farewell till soon.\n\n_Hier._ You'll ply this gear?\n\n_Lor._ I warrant you.\n\n[ _Exeunt all but Hieronimo._\n\n_Hier._ Why so:\n\nNow shall I see the fall of Babylon,\n\nWrought by the heav'ns in this confusion.\n\nAnd if the world like not this tragedy,\n\nHard is the hap of old Hieronimo. [ _Exit._\n\n**SCENE II**\n\n_Enter Isabella with a weapon._\n\n_Isab._ Tell me no more!\u2014O monstrous homicides!\n\nSince neither piety nor pity moves\n\nThe king to justice or compassion,\n\nI will revenge myself upon this place,\n\nWhere thus they murder'd my belov\u00e8d son.\n\n[ _She cuts down the arbour._\n\nDown with these branches and these loathsome boughs\n\nOf this unfortunate and fatal pine:\n\nDown with them, Isabella; rent them up,\n\nAnd burn the roots from whence the rest is sprung.\n\nI will not leave a root, a stalk, a tree,\n\nA bough, a branch, a blossom, nor a leaf,\n\nNo, not an herb within this garden-plot\u2014:\n\nAccurs\u00e8d complot of my misery!\n\nFruitless for ever may this garden be,\n\nBarren the earth, and blissless whosoe'er\n\nImagines not to keep it unmanur'd!\n\nAn eastern wind, commix'd with noisome airs,\n\nShall blast the plants and the young saplings;\n\nThe earth with serpents shall be pester\u00e8d,\n\nAnd passengers, for fear to be infect,\n\nShall stand aloof, and, looking at it, tell:\n\n\"There, murder'd, died the son of Isabel.\"\n\nAy, here he died, and here I him embrace:\n\nSee, where his ghost solicits, with his wounds,\n\nRevenge on her that should revenge his death.\n\nHieronimo, make haste to see thy son;\n\nFor sorrow and despair hath cited me\n\nTo hear Horatio plead with Rhadamanth:\n\nMake haste, Hieronimo, to hold excus'd\n\nThy negligence in pursuit of their deaths\n\nWhose hateful wrath bereav'd him of his breath.\u2014\n\nAh, nay, thou dost delay their deaths,\n\nForgiv'st the murd'rers of thy noble son,\n\nAnd none but I bestir me\u2014to no end!\n\nAnd as I curse this tree from further fruit,\n\nSo shall my womb be curs\u00e8d for his sake;\n\nAnd with this weapon will I wound the breast,\n\nThe hapless breast, that gave Horatio suck.\n\n[ _She stabs herself._\n\n**SCENE III**\n\n_Enter Hieronimo; he knocks up the curtain._\n\n_Enter the Duke of Castile._\n\n_Cast._ How now, Hieronimo, where 's your fellows,\n\nThat you take all this pain?\n\n_Hier._ O sir, it is for the author's credit,\n\nTo look that all things may go well.\n\nBut, good my lord, let me entreat your grace,\n\nTo give the king the copy of the play:\n\nThis is the argument of what we show.\n\n_Cast._ I will, Hieronimo.\n\n_Hier._ One thing more, my good lord.\n\n_Cast._ What's that?\n\n_Hier._ Let me entreat your grace\n\nThat, when the train are pass'd into the gallery,\n\nYou would vouchsafe to throw me down the key.\n\n_Cast._ I will, Hieronimo. [ _Exit Castile._\n\n_Hier._ What, are you ready, Balthazar?\n\nBring a chair and a cushion for the king.\n\n_Enter Balthazar, with a chair._\n\nWell done, Balthazar! hang up the title:\n\nOur scene is Rhodes;\u2014what, is your beard on?\n\n_Bal._ Half on; the other is in my hand.\n\n_Hier._ Despatch for shame; are you so long?\n\n[ _Exit Balthazar._\n\nBethink thyself, Hieronimo,\n\nRecall thy wits, recount thy former wrongs\n\nThou hast receiv'd by murder of thy son,\n\nAnd lastly\u2014not least!\u2014how Isabel,\n\nOnce his mother and thy dearest wife,\n\nAll woe-begone for him, hath slain herself.\n\nBehoves thee then, Hieronimo, to be reveng'd!\n\nThe plot is laid of dire revenge:\n\nOn, then, Hieronimo, pursue revenge;\n\nFor nothing wants but acting of revenge!\n\n[ _Exit Hieronimo._\n\n**SCENE IV**\n\n_Enter Spanish King, Viceroy_ , _the Duke of Castile, and their train._\n\n_King._ Now, Viceroy, shall we see the tragedy\n\nOf Soliman, the Turkish emperor,\n\nPerform'd\u2014of pleasure\u2014by your son the prince,\n\nMy nephew Don Lorenzo, and my niece.\n\n_Vic._ Who? Bellimperia?\n\n_King._ Ay, and Hieronimo, our marshal,\n\nAt whose request they deign to do 't themselves:\n\nThese be our pastimes in the court of Spain.\n\nHere, brother, you shall be the bookkeeper:\n\nThis is the argument of that they show.\n\n[ _He giveth him a book._\n\n_Gentlemen, this play of Hieronimo, in sundry languages, was thought good to be set down in English more largely, for the easier understanding to every public reader._\n\nEnter Balthazar, Bellimperia, and Hieronimo.\n\nBal. _Bashaw, that Rhodes is ours, yield heav'ns the honour,_\n\n_And holy Mahomet, our sacred prophet!_\n\n_And be thou grac'd with every excellence_\n\n_That Soliman can give, or thou desire._\n\n_But thy desert in conquering Rhodes is less_\n\n_Than in reserving this fair Christian nymph,_\n\n_Perseda, blissful lamp of excellence,_\n\n_Whose eyes compel, like powerful adamant,_\n\n_The warlike heart of Soliman to wait,_\n\n_King._ See, Viceroy, that is Balthazar, your son,\n\nThat represents the emperor Soliman:\n\nHow well he acts his amorous passion!\n\n_Vic._ Ay, Bellimperia hath taught him that\n\n_Cast._ That's because his mind runs all on Bellimperia.\n\nHier. _Whatever joy earth yields, betide your majesty._\n\nBal. _Earth yields no joy without Perseda's love._\n\nHier. _Let then Perseda on your grace attend._\n\nBal. _She shall not wait on me, but I on her:_\n\n_Drawn by the influence of her lights, I yield._\n\n_But let my friend, the Rhodian knight, come forth,_\n\n_Erasto, dearer than my life to me,_\n\n_That he may see Perseda, my belov'd._\n\n_Enter Erasto._\n\n_King._ Here comes Lorenzo: look upon the plot,\n\nAnd tell me, brother, what part plays he?\n\nBel. _Ah, my Erasto, welcome to Perseda._\n\nLor. _Thrice happy is Erasto that thou liv'st;_\n\n_Rhodes' loss is nothing to Erasto's joy:_\n\n_Sith his Perseda lives, his life survives._\n\nBal. _Ah, bashaw, here is love between Erasto_\n\n_And fair Perseda, sovereign of my soul._\n\nHier. _Remove Erasto, mighty Soliman,_\n\n_And then Perseda will be quickly won._\n\nBal. _Erasto is my friend; and while he lives,_\n\n_Perseda never will remove her love._\n\nHier. _Let not Erasto live to grieve great Soliman._\n\nBal. _Dear is Erasto in our princely eye._\n\nHier. _But if he be your rival, let him die._\n\nBal. _Why, let him die!_ \u2014 _so love commandeth me._\n\n_Yet grieve I that Erasto should so die._\n\nHier. _Erasto, Soliman saluteth thee,_\n\n_And lets thee wit by me his highness' will,_\n\n_Which is, thou shouldst be thus employ'd._\n\n[Stabs him.\n\nBel. _Ay me!_\n\n_Erasto! see, Soliman, Erasto's slain!_\n\nBal. _Yet liveth Soliman to comfort thee._\n\n_Fair queen of beauty, let not favour die,_\n\n_But with a gracious eye behold his grief,_\n\n_That with Perseda's beauty is increas'd,_\n\n_If by Perseda his grief be not releas'd._\n\nBel. _Tyrant, desist soliciting vain suits;_\n\n_Relentless are mine ears to thy laments,_\n\n_As thy butcher is pitiless and base,_\n\n_Which seiz'd on my Erasto, harmless knight._\n\n_Yet by thy power thou thinkest to command,_\n\n_And to thy power Perseda doth obey:_\n\n_But, were she able, thus she would revenge_\n\n_Thy treacheries on thee, ignoble prince:_ [Stabs him.\n\n_And on herself she would be thus reveng'd._\n\n[Stabs herself.\n\n_King._ Well said!\u2014Old marshal, this was bravely done!\n\n_Hier._ But Bellimperia plays Perseda well!\n\n_Vic._ Were this in earnest, Bellimperia,\n\nYou would be better to my son than so.\n\n_King._ But now what follows for Hieronimo?\n\n_Hier._ Marry, this follows for Hieronimo:\n\nHere break we off our sundry languages,\n\nAnd thus conclude I in our vulgar tongue.\n\nHaply you think\u2014but bootless are your thoughts\u2014\n\nThat this is fabulously counterfeit,\n\nAnd that we do as all tragedians do:\n\nTo die to-day (for fashioning our scene)\n\nThe death of Ajax or some Roman peer,\n\nAnd in a minute starting up again,\n\nRevive to please to-morrow's audience.\n\nNo, princes; know I am Hieronimo,\n\nThe hopeless father of a hapless son,\n\nWhose tongue is tun'd to tell his latest tale,\n\nNot to excuse gross errors in the play.\n\nI see, your looks urge instance of these words;\n\nBehold the reason urging me to this:\n\n[ _Shows his dead son._\n\nSee here my show, look on this spectacle:\n\nHere lay my hope, and here my hope hath end;\n\nHere lay my heart, and here my heart was slain;\n\nHere lay my treasure, here my treasure lost;\n\nHere lay my bliss, and here my bliss bereft:\n\nBut hope, heart, treasure, joy, and bliss,\n\nAll fled, fail'd, died, yea, all decay'd with this.\n\nFrom forth these wounds came breath that gave me life;\n\nThey murder'd me that made these fatal marks.\n\nThe cause was love, whence grew this mortal hate;\n\nThe hate: Lorenzo and young Balthazar;\n\nThe love: my son to Bellimperia.\n\nBut night, the cov'rer of accurs\u00e8d crimes,\n\nWith pitchy silence hush'd these traitors' harms,\n\nAnd lent them leave, for they had sorted leisure\n\nTo take advantage in my garden-plot\n\nUpon my son, my dear Horatio:\n\nThere merciless they butcher'd up my boy,\n\nIn black, dark night, to pale, dim, cruel death.\n\nHe shrieks: I heard (and yet, methinks, I hear)\n\nHis dismal outcry echo in the air.\n\nWith soonest speed I hasted to the noise,\n\nWhere hanging on a tree I found my son,\n\nThrough-girt with wounds, and slaughter'd as you see.\n\nAnd griev'd I, think you, at this spectacle?\n\nSpeak, Portuguese, whose loss resembles mine:\n\nIf thou canst weep upon thy Balthazar,\n\n'Tis like I wail'd for my Horatio.\n\nAnd you, my lord, whose reconcil\u00e8d son\n\nMarch'd in a net, and thought himself unseen,\n\nAnd rated me for brainsick lunacy,\n\nWith \"God amend that mad Hieronimo!\"\u2014\n\nHow can you brook our play's catastrophe?\n\nAnd here behold this bloody hand-kercher,\n\nWhich at Horatio's death I weeping dipp'd\n\nWithin the river of his bleeding wounds:\n\nIt as propitious, see, I have reserv'd,\n\nAnd never hath it left my bloody heart,\n\nSoliciting remembrance of my vow\n\nWith these, O, these accurs\u00e8d murderers:\n\nWhich now perform'd my heart is satisfied.\n\nAnd to this end the bashaw I became\n\nThat might revenge me on Lorenzo's life,\n\nWho therefore was appointed to the part,\n\nAnd was to represent the knight of Rhodes,\n\nThat I might kill him more conveniently.\n\nSo, Viceroy, was this Balthazar, thy son,\n\nThat Soliman which Bellimperia,\n\nIn person of Perseda, murder\u00e8d:\n\nSolely appointed to that tragic part\n\nThat she might slay him that offended her.\n\nPoor Bellimperia miss'd her part in this:\n\nFor though the story saith she should have died,\n\nYet I of kindness, and of care to her,\n\nDid otherwise determine of her end;\n\nBut love of him whom they did hate too much\n\nDid urge her resolution to be such.\u2014\n\nAnd, princes, now behold Hieronimo,\n\nAuthor and actor in this tragedy,\n\nBearing his latest fortune in his fist;\n\nAnd will as resolute conclude his part,\n\nAs any of the actors gone before.\n\nAnd, gentles, thus I end my play;\n\nUrge no more words: I have no more to say.\n\n[ _He runs to hang himself._\n\n_King._ O hearken, Viceroy! Hold, Hieronimo!\n\nBrother, my nephew and thy son are slain!\n\n_Vic._ We are betray'd; my Balthazar is slain!\n\nBreak ope the doors; run, save Hieronimo.\n\n[ _They break in and hold Hieronimo._\n\nHieronimo,\n\nDo but inform the king of these events;\n\nUpon mine honour, thou shalt have no harm.\n\n_Hier._ Viceroy, I will not trust thee with my life,\n\nWhich I this day have offer'd to my son.\n\nAccurs\u00e8d wretch!\n\nWhy stay'st thou him that was resolv'd to die?\n\n_King._ Speak, traitor! damn\u00e8d, bloody murd'rer, speak!\n\nFor now I have thee, I will make thee speak.\n\nWhy hast thou done this undeserving deed?\n\n_Vic._ Why hast thou murder\u00e8d my Balthazar?\n\n_Cast._ Why hast thou butcher'd both my children thus?\n\nHier. [ _But are you sure they are dead?_\n\nCast. _Ay, slave, too sure._\n\nHier. _What, and yours too?_\n\nVic. _Ay, all are dead; not one of them survive._\n\nHier. _Nay, then I care not; come, and we shall be friends;_\n\n_Let us lay our heads together:_\n\n_See, here's a goodly noose will hold them all._\n\nVic. _O damn\u00e8d devil, how secure he is!_\n\nHier. _Secure? why, dost thou wonder at it?_\n\n_I tell thee, Viceroy, this day I have seen revenge,_\n\n_And in that sight am grown a prouder monarch,_\n\n_Than ever sat under the crown of Spain._\n\n_Had I as many lives as there be stars,_\n\n_As many heav'ns to go to, as those lives,_\n\n_I'd give them all, ay, and my soul to boot,_\n\n_But I would see thee ride in this red pool._ ]\n\n_O,_ good words!\n\nAs dear to me was my Horatio,\n\nAs yours, or yours, or yours, my lord, to you.\n\nMy guiltless son was by Lorenzo slain,\n\nAnd by Lorenzo and that Balthazar\n\nAm I at last reveng\u00e8d thoroughly,\n\nUpon whose souls may heav'ns be yet aveng'd\n\nWith greater far than these afflictions.\n\n_Cast._ But who were thy confederates in this?\n\n_Vic._ That was thy daughter Bellimperia _;_\n\nFor by her hand my Balthazar was slain:\n\nI saw her stab him.\n\n_King._ Why speak'st thou not?\n\n_Hier._ What lesser liberty can kings afford\n\nThan harmless silence? then afford it me.\n\nSufficeth, I may not, nor I will not tell thee.\n\n_King._ Fetch forth the tortures: traitor as thou art,\n\nI'll make thee tell.\n\n[Hier.] _Methinks, since I grew inward with revenge,_\n\n_I cannot look with scorn enough on death._\n\nKing. _What, dost thou mock us, slave? bring tortures forth._\n\nHier. _Do, do, do: and meantime I'll torture you._\n\n_You had a son, as I take it; and your son_\n\n_Should ha' been married to your daughter:_\n\n_Ha, was it not so?_ \u2014 _You had a son too,_\n\n_He was my liege's nephew; he was proud_\n\n_And politic; had he liv'd, he might have come_\n\n_To wear the crown of Spain (I think 'twas so)_ \u2014:\n\n_'Twas I that kill'd him; look you, this same hand,_\n\n_'Twas it that stabb'd his heart_ \u2014 _do ye see this hand?_\n\n_For one Horatio, if you ever knew him: a youth,_\n\n_One that they hang'd up in his father's garden;_\n\n_One that did force your valiant son to yield,_\n\n_While your more valiant son did take him prisoner,_\n\nVic. _Be deaf, my senses; I can hear no more._\n\nKing. _Fall, heav'n, and cover us with thy sad ruins._\n\nCast. _Roll all the world within thy pitchy cloud._\n\nHier. _Now do I applaud what I have acted._\n\n_Nunc iners cadat manus!_\n\n_Now to express the rupture of my part_ \u2014\n\n_Hier._ Indeed,\n\nThou may'st torment me, as his wretched son\n\nHath done in murd'ring my Horatio:\n\nBut never shalt thou force me to reveal\n\nThe thing which I have vow'd inviolate.\n\nAnd therefore, in despite of all thy threats,\n\nPleas'd with their deaths, and eas'd with their revenge,\n\nFirst take my tongue, and afterwards my heart.\n\n[ _He bites out his tongue._\n\n_King._ O monstrous resolution of a wretch!\n\nSee, Viceroy, he hath bitten forth his tongue,\n\nRather than to reveal what we requir'd.\n\n_Cast._ Yet can he write.\n\n_King._ And if in this he satisfy us not,\n\nWe will devise th' extremest kind of death\n\nThat ever was invented for a wretch.\n\n[ _Then he makes signs for a knife to mend his pen._\n\n_Cast._ O, he would have a knife to mend his pen.\n\n_Vic._ Here, and advise thee that thou write the troth.\u2014\n\nLook to my brother! save Hieronimo!\n\n[ _He with a knife stabs the duke and himself._\n\n_King._ What age hath ever heard such monstrous deeds?\n\nMy brother, and the whole succeeding hope\n\nThat Spain expected after my decease!\u2014\n\nGo, bear his body hence, that we may mourn\n\nThe loss of our belov\u00e8d brother's death\u2014:\n\nThat he may be entomb'd!\u2014Whate'er befall,\n\nI am the next, the nearest, last of all.\n\n_Vic._ And thou, Don Pedro, do the like for us:\n\nTake up our hapless son, untimely slain;\n\nSet me with him, and he with woeful me,\n\nUpon the main-mast of a ship unmann'd,\n\nAnd let the wind and tide haul me along\n\nTo Scylla's barking and untam\u00e8d gulf,\n\nOr to the loathsome pool of Acheron,\n\nTo weep my want for my sweet Balthazar:\n\nSpain hath no refuge for a Portingal.\n\n[ _The trumpets sound a dead march; the King of Spain mourning after his brothers body, and the King of Portingal bearing the body of his son._\n\n**SCENE V**\n\n_Enter Ghost and Revenge._\n\n_Ghost._ Ay, now my hopes have end in their effects,\n\nWhen blood and sorrow finish my desires:\n\nHoratio murder'd in his father's bower;\n\nVild Serberine by Pedringano slain;\n\nFalse Pedringano hang'd by quaint device;\n\nFair Isabella by herself misdone;\n\nPrince Balthazar by Bellimperia stabb'd;\n\nThe Duke of Castile and his wicked son\n\nBoth done to death by old Hieronimo;\n\nMy Bellimperia fall'n, as Dido fell,\n\nAnd good Hieronimo slain by himself:\n\nAy, these were spectacles to please my soul!\u2014\n\nNow will I beg at lovely Proserpine\n\nThat, by the virtue of her princely doom,\n\nI may consort my friends in pleasing sort,\n\nAnd on my foes work just and sharp revenge.\n\nI'll lead my friend Horatio through those fields,\n\nWhere never-dying wars are still inur'd;\n\nI'll lead fair Isabella to that train,\n\nWhere pity weeps, but never feeleth pain;\n\nI'll lead my Bellimperia to those joys,\n\nThat vestal virgins and fair queens possess;\n\nI'll lead Hieronimo where Orpheus plays,\n\nAdding sweet pleasure to eternal days.\n\nBut say, Revenge\u2014for thou must help, or none\u2014\n\nAgainst the rest how shall my hate be shown?\n\n_Rev._ This hand shall hale them down to deepest hell,\n\nWhere none but Furies, bugs and tortures dwell.\n\n_Ghost._ Then, sweet Revenge, do this at my request:\n\nLet me be judge, and doom them to unrest\n\nLet loose poor Tityus from the vulture's gripe,\n\nAnd let Don Cyprian supply his room;\n\nPlace Don Lorenzo on Ixion's wheel,\n\nAnd let the lover's endless pains surcease\n\n(Juno forgets old wrath, and grants him ease);\n\nHang Balthazar about Chim\u00e6ra's neck,\n\nAnd let him there bewail his bloody love,\n\nRepining at our joys that are above;\n\nLet Serberine go roll the fatal stone,\n\nAnd take from Sisyphus his endless moan;\n\nFalse Pedringano, for his treachery,\n\nLet him be dragg'd through boiling Acheron,\n\nAnd there live, dying still in endless flames,\n\nBlaspheming gods and all their holy names.\n\n_Rev._ Then haste we down to meet thy friends and foes:\n\nTo place thy friends in ease, the rest in woes;\n\nFor here though death hath end their misery,\n\nI'll there begin their endless tragedy. [ _Exeunt._\n\nFINIS\n\n* * *\n\n Line 65 and first part of 66 (O no . . . lordship) are replaced, in all the Qq. from 1602 onwards, by the following lines:\n\nHier. _Who? you, my lord?_\n\n_I reserve your favour for a greater honour;_\n\n_This is a very toy, my lord, a toy._\n\nLor. _All's one, Hieronimo, acquaint me with it._\n\nHier. _I'faith, my lord, it is an idle thing;_\n\n_I must confess I ha' been too slack, too tardy,_\n\n_Too remiss unto your honour._\n\nLor. _How now, Hieronimo?_\n\nHier. _In troth, my lord, it is a thing of nothing:_\n\n_The murder of a son, or so_ \u2014\u2014\n\n_A thing of nothing, my lord!_\n\n Instead of 11. 193 (second half: ' Why speak'st thou not') to 204, the Qq. from 1602 onwards have the following passage (they have also put 11. 190-193, first half, before 1. 182):\n**DR. FAUSTUS**\n\n**Christopher Marlowe**\n\n**DRAMATIS PERSON\u00c6**\n\nTHE POPE.\n\nCARDINAL OF LORRAIN.\n\nEMPEROR OF GERMANY.\n\nDUKE OF VANHOLT.\n\nFAUSTUS.\n\nVALDES and CORNELIUS, Friends to FAUSTUS.\n\nWAGNER, Servant to FAUSTUS.\n\nClown.\n\nROBIN.\n\nRALPH.\n\nVintner.\n\nHorse-Courser.\n\nKnight.\n\nOld Man.\n\nScholars, Friars, and Attendants.\n\nDUCHESS OF VANHOLT.\n\nLUCIFER.\n\nBELZEBUB.\n\nMEPHISTOPHILIS.\n\nGood Angel.\n\nEvil Angel.\n\nThe Seven Deadly Sins.\n\nDevils.\n\nSpirits in the shape of ALEXANDER THE GREAT, of his Paramour,\n\nand of HELEN OF TROY.\n\nCHORUS.\n\n_Enter_ CHORUS\n\nCHORUS. Not marching now in fields of Thrasimene,\n\nWhere Mars did mate the Carthaginians;\n\nNor sporting in the dalliance of love,\n\nIn courts of kings where state is overturn'd;\n\nNor in the pomp of proud audacious deeds,\n\nIntends our Muse to vaunt his heavenly verse:\n\nOnly this, gentlemen,\u2014we must perform\n\nThe form of Faustus' fortunes, good or bad.\n\nTo patient judgments we appeal our plaud,\n\nAnd speak for Faustus in his infancy.\n\nNow is he born, his parents base of stock,\n\nIn Germany, within a town call'd Rhodes;\n\nOf riper years to Wittenberg he went,\n\nWhereas his kinsmen chiefly brought him up.\n\nSo soon he profits in divinity,\n\nThe fruitful plot of scholarism grac'd,\n\nThat shortly he was grac'd with doctor's name,\n\nExcelling all whose sweet delight disputes\n\nIn heavenly matters of theology;\n\nTill swollen with cunning, of a self-conceit,\n\nHis waxen wings did mount above his reach,\n\nAnd, melting, Heavens conspir'd his overthrow;\n\nFor, falling to a devilish exercise,\n\nAnd glutted [now] with learning's golden gifts,\n\nHe surfeits upon cursed necromancy.\n\nNothing so sweet as magic is to him,\n\nWhich he prefers before his chiefest bliss.\n\nAnd this the man that in his study sits! _Exit._\n\n**SCENE I.**\n\n_Enter_ FAUSTUS _in his Study._\n\nFAUSTUS. Settle my studies, Faustus, and begin\n\nTo sound the depth of that thou wilt profess;\n\nHaving commenc'd, be a divine in show.\n\nYet level at the end of every art,\n\nAnd live and die in Aristotle's works.\n\nSweet Analytics, 'tis thou hast ravish'd me,\n\n_Bene disserere est finis logices._\n\nIs to dispute well logic's chiefest end?\n\nAffords this art no greater miracle?\n\nThen read no more, thou hast attain'd the end;\n\nA greater subject fitteth Faustus' wit.\n\nBid '\u00f2\u03bd \u03ba\u03b1\u03b9 \u03bc\u1f74 '\u00f3\u03bd farewell; Galen come,\n\nSeeing _Ubi desinit Philosophus, ibi incipit Medicus;_ 6\n\nBe a physician, Faustus, heap up gold,\n\nAnd be eternis'd for some wondrous cure.\n\n_Summum bonum medicin\u0153 sanitas,_ 7\n\n\"The end of physic is our body's health.\"\n\nWhy, Faustus, hast thou not attain'd that end?\n\nIs not thy common talk sound Aphorisms?\n\nAre not thy bills hung up as monuments,\n\nWhereby whole cities have escap'd the plague,\n\nAnd thousand desperate maladies been eas'd?\n\nYet art thou still but Faustus and a man.\n\nWouldst thou make men to live eternally,\n\nOr, being dead, raise them to life again?\n\nThen this profession were to be esteem'd.\n\nPhysic, farewell. \u2014Where is Justinian? [ _Reads._ ]\n\n_Si una eademque res legatur duobus, alter rem, alter valorem rei, & c._\n\nA pretty case of paltry legacies! [ _Reads._ ]\n\n_Exh\u0153reditare filium non potest pater nisi, &c._\n\nSuch is the subject of the Institute\n\nAnd universal Body of the Law.\n\nHis study fits a mercenary drudge,\n\nWho aims at nothing but external trash;\n\nToo servile and illiberal for me.\n\nWhen all is done, divinity is best;\n\nJerome's Bible, Faustus, view it well. [ _Reads._ ]\n\n_Stipendium peccati mors est._ Ha! _Stipendium,_ & _c._\n\n_\"The reward of sin is death.\"_ That's hard. [ _Reads._ ]\n\n_Si peccasse negamus, fallimur, et nulla est in nobis veritas._\n\n\"If we say that we have no sin we deceive ourselves, and there's no truth in us.\" Why then, belike we must sin and so consequently die.\n\nAy, we must die an everlasting death.\n\nWhat doctrine call you this, _Che sera sera_ ,\n\n\"What will be shall be?\" Divinity, adieu!\n\nThese metaphysics of magicians\n\nAnd necromantic books are heavenly;\n\nLines, circles, scenes, letters, and characters,\n\nAy, these are those that Faustus most desires.\n\nO what a world of profit and delight,\n\nOf power, of honour, of omnipotence\n\nIs promis'd to the studious artisan!\n\nAll things that move between the quiet poles\n\nShall be at my command. Emperors and kings\n\nAre but obeyed in their several provinces,\n\nNor can they raise the wind or rend the clouds;\n\nBut his dominion that exceeds in this\n\nStretcheth as far as doth the mind of man.\n\nA sound magician is a mighty god:\n\nHere, Faustus, try thy brains to gain a deity.\n\nWagner!\n\n_Enter_ WAGNER.\n\nCommend me to my dearest friends,\n\nThe German Valdes and Cornelius;\n\nRequest them earnestly to visit me.\n\nWAG. I will, sir. _Exit._\n\nFAUSTUS. Their conference will be a greater help to me\n\nThan all my labours, plod I ne'er so fast.\n\n_Enter_ GOOD ANGEL _and_ EVIL ANGEL.\n\nG. ANG. O Faustus! lay that damned book aside,\n\nAnd gaze not upon it lest it tempt thy soul,\n\nAnd heap God's heavy wrath upon thy head.\n\nRead, read the Scriptures: that is blasphemy.\n\nE. ANG. Go forward, Faustus, in that famous art,\n\nWherein all Nature's treasure is contain'd:\n\nBe thou on earth as Jove is in the sky,\n\nLord and commander of these elements. _Exeunt_ [ANGELS.]\n\nFAUSTUS. How am I glutted with conceit of this!\n\nShall I make spirits fetch me what I please,\n\nResolve me of all ambiguities,\n\nPerform what desperate enterprise I will?\n\nI'll have them fly to India for gold,\n\nRansack the ocean for orient pearl,\n\nAnd search all corners of the new-found world\n\nFor pleasant fruits and princely delicates;\n\nI'll have them read me strange philosophy\n\nAnd tell the secrets of all foreign kings;\n\nI'll have them wall all Germany with brass,\n\nAnd make swift Rhine circle fair Wittenberg;\n\nI'll have them fill the public schools with silk,\n\nWherewith the students shall be bravely clad;\n\nI'll levy soldiers with the coin they bring,\n\nAnd chase the Prince of Parma from our land,\n\nAnd reign sole king of all the provinces;\n\nYea, stranger engines for the brunt of war\n\nThan was the fiery keel at Antwerp's bridge,\n\nI'll make my servile spirits to invent.\n\nCome, German Valdes and Cornelius,\n\nAnd make me blest with your sage conference.\n\n_Enter_ VALDES _and_ CORNELIUS.\n\nValdes, sweet Valdes, and Cornelius,\n\nKnow that your words have won me at the last\n\nTo practise magic and concealed arts:\n\nYet not your words only, but mine own fantasy,\n\nThat will receive no object, for my head\n\nBut ruminates on necromantic skill.\n\nPhilosophy is odious and obscure,\n\nBoth law and physic are for petty wits;\n\nDivinity is basest of the three,\n\nUnpleasant, harsh, contemptible, and vile:\n\n'Tis magic, magic, that hath ravish'd me.\n\nThen, gentle friends, aid me in this attempt;\n\nAnd I that have with concise syllogisms\n\nGravell'd the pastors of the German church,\n\nAnd made the flow'ring pride of Wittenberg\n\nSwarm to my problems, as the infernal spirits\n\nOn sweet Mus\u00e6us, when he came to hell,\n\nWill be as cunning as Agrippa was,\n\nWhose shadows made all Europe honour him.\n\nVALD. Faustus, these books, thy wit, and our experience\n\nShall make all nations to can\u00f3nise us.\n\nAs Indian Moors obey their Spanish lords,\n\nSo shall the subjects of every element\n\nBe always serviceable to us three;\n\nLike lions shall they guard us when we please;\n\nLike Almain rutters with their horsemen's staves,\n\nOr Lapland giants, trotting by our sides;\n\nSometimes like women or unwedded maids,\n\nShadowing more beauty in their airy brows\n\nThan have the white breasts of the queen of love:\n\nFrom Venice shall they drag huge argosies,\n\nAnd from America the golden fleece\n\nThat yearly stuffs old Philip's treasury;\n\nIf learned Faustus will be resolute.\n\nFAUSTUS. Valdes, as resolute am I in this\n\nAs thou to live; therefore object it not.\n\nCORN. The miracles that magic will perform\n\nWill make thee vow to study nothing else.\n\nHe that is grounded in astrology,\n\nEnrich'd with tongues, well seen in minerals,\n\nHath all the principles magic doth require.\n\nThen doubt not, Faustus, but to be renown'd,\n\nAnd more frequented for this mystery\n\nThan heretofore the Delphian Oracle.\n\nThe spirits tell me they can dry the sea,\n\nAnd fetch the treasure of all foreign wracks,\n\nAy, all the wealth that our forefathers hid\n\nWithin the massy entrails of the earth;\n\nThen tell me, Faustus, what shall we three want?\n\nFAUSTUS. Nothing, Cornelius! O this cheers my soul!\n\nCome show me some demonstrations magical,\n\nThat I may conjure in some lusty grove,\n\nAnd have these joys in full possession.\n\nVALD. Then haste thee to some solitary grove\n\nAnd bear wise Bacon's and Albanus' works,\n\nThe Hebrew Psalter and New Testament;\n\nAnd whatsoever else is requisite\n\nWe will inform thee ere our conference cease.\n\nCORN. Valdes, first let him know the words of art;\n\nAnd then, all other ceremonies learn'd,\n\nFaustus may try his cunning by himself.\n\nVALD. First I'll instruct thee in the rudiments.\n\nAnd then wilt thou be perfecter than I.\n\nFAUSTUS. Then come and dine with me, and after meat,\n\nWe'll canvass every quiddity thereof;\n\nFor ere I sleep I'll try what I can do:\n\nThis night I'll conjure though I die therefore. _Exeunt._\n\n**SCENE II.**\n\n**_Before Faustus' House._**\n\n_Enter two_ SCHOLARS.\n\n1 SCHOL. I wonder what's become of Faustus that was wont to make our schools ring with _sic probo_?\n\n2 SCHOL. That shall we know, for see here comes his boy.\n\n_Enter_ WAGNER.\n\n1 SCHOL. How now, sirrah! Where's thy master?\n\nWAG. God in heaven knows!\n\n2 SCHOL. Why, dost not thou know?\n\nWAG. Yes, I know. But that follows not.\n\n1 SCHOL. Go to, sirrah! Leave your jesting, and tell us where he is.\n\nWAG. That follows not necessary by force of argument, that you, being licentiate, should stand upon't: therefore, acknowledge your error and be attentive.\n\n2 SCHOL. Why, didst thou not say thou knew'st?\n\nWAG. Have you any witness on't?\n\n1 SCHOL. Yes, sirrah, I heard you.\n\nWAG. Ask my fellow if I be a thief.\n\n2 SCHOL. Well, you will not tell us?\n\nWAG. Yes, sir, I will tell you; yet if you were not dunces, you would never ask me such a question; for is not he _corpus naturale_? and is not that _mobile?_ Then wherefore should you ask me such a question? But that I am by nature phlegmatic, slow to wrath, and prone to lechery (to love, I would say), it were not for you to come within forty foot of the place of execution, although I do not doubt to see you both hang'd the next sessions. Thus having triumph'd over you, I will set my countenance like a precisian, and begin to speak thus: \u2014Truly, my dear brethren, my master is within at dinner, with Valdes and Cornelius, as this wine, if it could speak, would inform your worships; and so the Lord bless you, preserve you, and keep you, my dear brethren, my dear brethren. _Exit._\n\n1 SCHOL. Nay, then, I fear he has fallen into that damned Art, for which they two are infamous through the world.\n\n2 SCHOL. Were he a stranger, and not allied to me, yet should I grieve for him. But come, let us go and inform the Rector, and see if he by his grave counsel can reclaim him.\n\n1 SCHOL. O, I fear me nothing can reclaim him.\n\n2 SCHOL. Yet let us try what we can do. _Exeunt_\n\n**SCENE III.**\n\n**_A Grove._**\n\n_Enter_ FAUSTUS _to conjure._\n\nFAUSTUS. Now that the gloomy shadow of the earth\n\nLonging to view Orion's drizzling look,\n\nLeaps from th' antarctic world unto the sky,\n\nAnd dims the welkin with her pitchy breath,\n\nFaustus, begin thine incantations,\n\nAnd try if devils will obey thy hest,\n\nSeeing thou hast pray'd and sacrific'd to them.\n\nWithin this circle is Jehovah's name,\n\nForward and backward anagrammatis'd,\n\nThe breviated names of holy saints,\n\nFigures of every adjunct to the Heavens,\n\nAnd characters of signs and erring stars,\n\nBy which the spirits are enforc'd to rise:\n\nThen fear not, Faustus, but be resolute,\n\nAnd try the uttermost magic can perform.\n\n_Sint mihi Dei Acherontis propitii! Valeat numen triplex Jehovae! Ignei, aerii, aquatani spiritus, salvete! Orientis princeps Belzebub, infemi ardentis monarcha, et Demogorgon, propitiamus vos, ut appareat et surgat Mephistophilis. Quid tu moraris? Per Jehovam, Gehennam, et consecratum aquam quam nunc spargo, signumque crucis quod nunc facio, et per vota nostra, ipse nunc surgat nobis dicatus Mephistophilis!_ 3\n\n_Enter_ [MEPHISTOPHILIS] _a_ Devil.\n\nI charge thee to return and change thy shape;\n\nThou art too ugly to attend on me.\n\nGo, and return an old Franciscan friar;\n\nThat holy shape becomes a devil best.\n\n_Exit Devil._\n\nI see there's virtue in my heavenly words;\n\nWho would not be proficient in this art?\n\nHow pliant is this Mephistophilis,\n\nFull of obedience and humility!\n\nSuch is the force of magic and my spells.\n\n[NOW,] Faustus, thou art conjuror laureate,\n\nThou canst command great Mephistophilis:\n\n_Quin regis Mephistophilis fratris imagine._ 4\n\n_Re-enter_ MEPHISTOPHILIS [ _like a Franciscan_ Friar].\n\nMEPH. Now, Faustus, what would'st thou have me do?\n\nFAUSTUS. I charge thee wait upon me whilst I live,\n\nTo do whatever Faustus shall command,\n\nBe it to make the moon drop from her sphere,\n\nOr the ocean to overwhelm the world.\n\nMEPH. I am a servant to great Lucifer,\n\nAnd may not follow thee without his leave;\n\nNo more than he commands must we perform.\n\nFAUSTUS. Did he not charge thee to appear to me?\n\nMEPH. NO, I came hither of mine own accord.\n\nFAUSTUS. Did not my conjuring speeches raise thee? Speak:\n\nMEPH. That was the cause, but yet _per accidens;_\n\nFor when we hear one rack the name of God,\n\nAbjure the Scriptures and his Saviour Christ,\n\nWe fly in hope to get his glorious soul;\n\nNor will we come, unless he use such means\n\nWhereby he is in danger to be damn'd:\n\nTherefore the shortest cut for conjuring\n\nIs stoutly to abjure the Trinity,\n\nAnd pray devoutly to the Prince of Hell.\n\nFAUSTUS. So Faustus hath\n\nAlready done; and holds this principle,\n\nThere is no chief but only Belzebub,\n\nTo whom Faustus doth dedicate himself.\n\nThis word \"damnation\" terrifies not him,\n\nFor he confounds hell in Elysium;\n\nHis ghost be with the old philosophers!\n\nBut, leaving these vain trifles of men's souls,\n\nTell me what is that Lucifer thy lord?\n\nMEPH. Arch-regent and commander of all spirits.\n\nFaustus. Was not that Lucifer an angel once?\n\nMeph. Yes, Faustus, and most dearly lov'd of God.\n\nFAUSTUS. How comes it then that he is Prince of devils?\n\nMEPH. O, by aspiring pride and insolence;\n\nFor which God threw him from the face of Heaven.\n\nFAUSTUS. And what are you that you live with Lucifer?\n\nMEPH. Unhappy spirits that fell with Lucifer,\n\nConspir'd against our God with Lucifer,\n\nAnd are for ever damn'd with Lucifer.\n\nFAUSTUS. Where are you damn'd?\n\nMEPH. In hell.\n\nFAUSTUS. How comes it then that thou art out of hell?\n\nMEPH. Why this is hell, nor am I out of it.\n\nThink'st thou that I who saw the face of God,\n\nAnd tasted the eternal joys of Heaven,\n\nAm not tormented with ten thousand hells,\n\nIn being depriv'd of everlasting bliss?\n\nO Faustus! leave these frivolous demands,\n\nWhich strike a terror to my fainting soul.\n\nFAUSTUS. What, is great Mephistophilis so passionate\n\nFor being depriv'd of the joys of Heaven?\n\nLearn thou of Faustus manly fortitude,\n\nAnd scorn those joys thou never shalt possess.\n\nGo bear these tidings to great Lucifer:\n\nSeeing Faustus hath incurr'd eternal death\n\nBy desperate thoughts against Jove's deity,\n\nSay he surrenders up to him his soul,\n\nSo he will spare him four and twenty years,\n\nLetting him live in all voluptuousness;\n\nHaving thee ever to attend on me;\n\nTo give me whatsoever I shall ask,\n\nTo tell me whatsoever I demand,\n\nTo slay mine enemies, and aid my friends,\n\nAnd always be obedient to my will.\n\nGo and return to mighty Lucifer,\n\nAnd meet me in my study at midnight,\n\nAnd then resolve me of thy master's mind.\n\nMEPH. I will, Faustus. _Exit._\n\nFAUSTUS. Had I as many souls as there be stars,\n\nI'd give them all for Mephistophilis.\n\nBy him I'll be great Emperor of the world,\n\nAnd make a bridge through the moving air,\n\nTo pass the ocean with a band of men;\n\nI'll join the hills that bind the Afric shore,\n\nAnd make that [country] continent to Spain,\n\nAnd both contributory to my crown.\n\nThe Emperor shall not live but by my leave,\n\nNor any potentate of Germany.\n\nNow that I have obtain'd what I desire,\n\nI'll live in speculation of this art\n\nTill Mephistophilis return again. _Exit_\n\n**SCENE IV.**\n\n**_A Street._**\n\n_Enter_ WAGNER _and the_ CLOWN.\n\nWAG. Sirrah, boy, come hither.\n\nCLOWN. How, boy! Swowns, boy! I hope you have seen many boys with such pickadevaunts as I have. Boy, quotha!\n\nWAG. Tell me, sirrah, hast thou any comings in?\n\nCLOWN. Ay, and goings out too. You may see else.\n\nWAG. Alas, poor slave! See how poverty jesteth in his nakedness! The villain is bare and out of service, and so hungry that I know he would give his soul to the devil for a shoulder of mutton, though it were blood-raw.\n\nCLOWN. How? My soul to the Devil for a shoulder of mutton, though 'twere blood-raw! Not so, good friend. By 'r Lady, I had need have it well roasted and good sauce to it, if I pay so dear.\n\nWAG. Well, wilt thou serve me, and I'll make thee go like _Qui mihi discipulus?_ 2\n\nCLOWN. How, in verse?\n\nWAG. No, sirrah; in beaten silk and stavesacre.\n\nCLOWN. How, how, Knave's acre  Ay, I thought that was all the land his father left him. Do you hear? I would be sorry to rob you of your living.\n\nWAG. Sirrah, I say in stavesacre.\n\nCLOWN. Oho! Oho! Stavesacre! Why, then, belike if I were your man I should be full of vermin.\n\nWAG. So thou shalt, whether thou beest with me or no. But, sirrah, leave your jesting, and bind yourself presently unto me for seven years, or I'll turn all the lice about thee into familiars, and they shall tear thee in pieces.\n\nCLOWN. Do you hear, sir? You may save that labour; they are too familiar with me already. Swowns! they are as bold with my flesh as if they had paid for [their] meat and drink.\n\nWAG. Well, do you hear, sirrah? Hold, take these guilders.\n\n[ _Gives money._ ]\n\nCLOWN. Gridirons! what be they?\n\nWAG. Why, French crowns.\n\nCLOWN. Mass, but for the name of French crowns, a man were as good have as many English counters. And what should I do with these?\n\nWAG. Why, now, sirrah, thou art at an hour's warning, whensoever and wheresoever the Devil shall fetch thee.\n\nCLOWN. No, no. Here, take your gridirons again.\n\nWAG. Truly I'll none of them.\n\nCLOWN. Truly but you shall.\n\nWAG. Bear witness I gave them him.\n\nCLOWN. Bear witness I give them you again.\n\nWAG. Well, I will cause two devils presently to fetch thee away\u2014Baliol and Belcher.\n\nCLOWN. Let your Baliol and your Belcher come here, and I'll knock them, they were never so knockt since they were devils. Say I should kill one of them, what would folks say? \"Do you see yonder tall fellow in the round slop? \u2014 he has kill'd the devil.\" So I should be call'd Kill-devil all the parish over.\n\n_Enter two_ DEVILS: _the_ Clown _runs up and down crying._\n\nWAG. Baliol and Belcher! Spirits, away! _Exeunt_ Devils.\n\nCLOWN. What, are they gone? A vengeance on them, they have vile long nails! There was a he-devil, and a she-devil! I'll tell you how you shall know them: all he-devils has horns, and all she-devils has clifts and cloven feet.\n\nWAG. Well, sirrah, follow me.\n\nCLOWN. But, do you hear \u2014 if I should serve you, would you teach me to raise up Banios and Belcheos?\n\nWAG. I will teach thee to turn thyself to anything; to a dog, or a cat, or a mouse, or a rat, or anything.\n\nCLOWN. HOW! a Christian fellow to a dog or a cat, a mouse or a rat! No, no, sir. If you turn me into anything, let it be in the likeness of a little pretty frisky flea, that I may be here and there and everywhere. Oh, I'll tickle the pretty wenches' plackets; I'll be amongst them, i' faith.\n\nWAG. Well, sirrah, come.\n\nCLOWN. But, do you hear, Wagner?\n\nWAG. HOW! \u2014 Baliol and Belcher!\n\nCLOWN. O Lord! I pray, sir, let Banio and Belcher go sleep.\n\nWAG. Villain\u2014call me Master Wagner, and let thy left eye be diametarily fixt upon my right heel, with _quasi vestigias nostras insistere._ 7 Exit.\n\nCLOWN. God forgive me, he speaks Dutch fustian. Well, I'll follow him, I'll serve him, that's flat. Exit.\n\n**SCENE V.**\n\n_Enter_ FAUSTUS _in his study._\n\nFAUSTUS. Now, Faustus, must\n\nThou needs be damn'd, and canst thou not be sav'd:\n\nWhat boots it then to think of God or Heaven?\n\nAway with such vain fancies, and despair:\n\nDespair in God, and trust in Belzebub.\n\nNow go not backward: no, Faustus, be resolute.\n\nWhy waverest thou? O, something soundeth in mine ears\n\n\"Abjure this magic, turn to God again!\"\n\nAy, and Faustus will turn to God again.\n\nTo God? \u2014He loves thee not\u2014\n\nThe God thou serv'st is thine own appetite,\n\nWherein is fix'd the love of Belzebub;\n\nTo him I'll build an altar and a church,\n\nAnd offer lukewarm blood of new-born babes.\n\n_Enter_ GOOD ANGEL _and_ EVIL [ANGEL.]\n\nG. ANG. Sweet Faustus, leave that execrable art.\n\nFAUSTUS. Contrition, prayer, repentance! What of them?\n\nG. ANG. O, they are means to bring thee unto Heaven.\n\nE. ANG. Rather illusions, fruits of lunacy,\n\nThat makes men foolish that do trust them most.\n\nG. ANG. Sweet Faustus, think of Heaven, and heavenly things.\n\nE. ANG. No, Faustus, think of honour and of wealth.\n\n_Exeunt_ [ANGELS.]\n\nFAUSTUS. Of wealth!\n\nWhy, the signiory of Emden shall be mine.\n\nWhen Mephistophilis shall stand by me,\n\nWhat God can hurt thee, Faustus? Thou art safe;\n\nCast no more doubts. Come, Mephistophilis,\n\nAnd bring glad tidings from great Lucifer; \u2014\n\nIs't not midnight? Come, Mephistophilis;\n\n_Veni, veni, Mephistophile!_\n\n_Enter_ MEPHISTOPHILIS.\n\nNow tell me, what says Lucifer thy lord?\n\nMEPH. That I shall wait on Faustus whilst he lives,\n\nSo he will buy my service with his soul.\n\nFAUSTUS. Already Faustus hath hazarded that for thee.\n\nMEPH. But, Faustus, thou must bequeath it solemnly,\n\nAnd write a deed of gift with thine own blood,\n\nFor that security craves great Lucifer.\n\nIf thou deny it, I will back to hell.\n\nFAUSTUS. Stay, Mephistophilis! and tell me what good\n\nWill my soul do thy Lord.\n\nMEPH. Enlarge his kingdom.\n\nFAUSTUS. Is that the reason why he tempts us thus?\n\nMEPH. _Solamen miseris socios habuisse doloris._ 2\n\nFAUSTUS. Why, have you any pain that torture others?\n\nMEPH. As great as have the human souls of men.\n\nBut tell me, Faustus, shall I have thy soul?\n\nAnd I will be thy slave, and wait on thee,\n\nAnd give thee more than thou hast wit to ask.\n\nFAUSTUS. Ay, Mephistophilis, I give it thee.\n\nMEPH. Then Faustus, stab thine arm courageously.\n\nAnd bind thy soul that at some certain day\n\nGreat Lucifer may claim it as his own;\n\nAnd then be thou as great as Lucifer.\n\nFAUSTUS. [ _Stabbing his arm._ ] Lo, Mephistophilis, for love of thee,\n\nI cut mine arm, and with my proper blood\n\nAssure my soul to be great Lucifer's,\n\nChief lord and regent of perpetual night!\n\nView here the blood that trickles from mine arm.\n\nAnd let it be propitious for my wish.\n\nMEPH. But, Faustus, thou must\n\nWrite it in manner of a deed of gift.\n\nFAUSTUS. Ay, so I will. [ _Writes._ ] But, Mephistophilis,\n\nMy blood congeals, and I can write no more.\n\nMEPH. I'll fetch thee fire to dissolve it straight. _Exit._\n\nFAUSTUS. What might the staying of my blood portend?\n\nIs it unwilling I should write this bill?\n\nWhy streams it not that I may write afresh?\n\n_Faustus gives to thee his soul._ Ah, there it stay'd.\n\nWhy should'st thou not? Is not thy soul thine own?\n\nThen write again, _Faustus gives to thee his soul._\n\n_Re-enter_ MEPHISTOPHILIS _with a chafer of coals._\n\nMEPH. Here's fire. Come, Faustus, set it on.\n\nFAUSTUS. So now the blood begins to clear again;\n\nNow will I make an end immediately. [ _Writes._ ]\n\nMEPH. [ _Aside._ ] O what will not I do to obtain his soul.\n\nFAUSTUS. _Consummatum est:_ 3 this bill is ended,\n\nAnd Faustus hath bequeath'd his soul to Lucifer \u2014\n\nBut what is this inscription on mine arm?\n\n_Homo, fuge!_ 4 Whither should I fly?\n\nIf unto God, he'll throw me down to hell.\n\nMy senses are deceiv'd; here's nothing writ: \u2014\n\nI see it plain; here in this place is writ\n\n_Homo, fuge!_ Yet shall not Faustus fly.\n\nMEPH. I'll fetch him somewhat to delight his mind. _Exit._\n\n_Re-enter_ [MEPHISTOPHILIS] _with_ Devils, _giving crowns and rich apparel to_ FAUSTUS, _and dance, and then depart._\n\nFAUSTUS. Speak, Mephistophilis, what means this show?\n\nMEPH. Nothing, Faustus, but to delight thy mind withal,\n\nAnd to show thee what magic can perform.\n\nFAUSTUS. But may I raise up spirits when I please?\n\nMEPH. Ay, Faustus, and do greater things than these.\n\nFAUSTUS. Then there's enough for a thousand souls.\n\nHere, Mephistophilis, receive this scroll,\n\nA deed of gift of body and of soul:\n\nBut yet conditionally that thou perform\n\nAll articles prescrib'd between us both.\n\nMEPH. Faustus, I swear by hell and Lucifer\n\nTo effect all promises between us made.\n\nFAUSTUS. Then hear me read them: _On these conditions following. First, that Faustus may be a spirit in form and substance. Secondly, that Mephistophilis shall be his servant, and at his command. Thirdly, that Mephistophilis shall do for him and bring him whatsoever_ [ _he desires_ ] _. Fourthly, that he shall be in his chamber or house invisible. Lastly, that he shall appear to the said John Faustus, at all times, in what form or shape soever he pleases. I, John Faustus, of Wittenberg, Doctor, by these presents do give both body and soul to Lucifer, Prince of the East, and his minister, Mephistophilis; and furthermore grant unto them, that twenty-four years being expired, the articles above written inviolate, full power to fetch or carry the said John Faustus, body and soul, flesh, blood, or goods, into their habitation wheresoever. By me, John Faustus._\n\nMEPH. Speak, Faustus, do you deliver this as your deed?\n\nFAUSTUS. Ay, take it, and the Devil give thee good on 't.\n\nMEPH. Now, Faustus, ask what thou wilt.\n\nFAUSTUS. First will I question with thee about hell.\n\nTell me where is the place that men call hell?\n\nMEPH. Under the heavens.\n\nFAUSTUS. Ay, but whereabout?\n\nMEPH. Within the bowels of these elements,\n\nWhere we are tortur'd and remain for ever;\n\nHell hath no limits, nor is circumscrib'd\n\nIn one self place; for where we are is hell,\n\nAnd where hell is there must we ever be:\n\nAnd, to conclude, when all the world dissolves,\n\nAnd every creature shall be purified,\n\nAll places shall be hell that is not Heaven.\n\nFAUSTUS. Come, I think hell's a fable.\n\nMEPH. Ay, think so still, till experience change thy mind.\n\nFAUSTUS. Why, think'st thou then that Faustus shall be damn'd?\n\nMEPH. Ay, of necessity, for here's the scroll\n\nWherein thou hast given thy soul to Lucifer.\n\nFAUSTUS. Ay, and body too; but what of that?\n\nThink'st thou that Faustus is so fond to imagine\n\nThat, after this life, there is any pain?\n\nTush; these are trifles, and mere old wives' tales.\n\nMEPH. But, Faustus, I am an instance to prove the contrary,\n\nFor I am damned, and am now in hell.\n\nFAUSTUS. How! now in hell!\n\nNay, an this be hell, I'll willingly be damn'd here;\n\nWhat? walking, disputing, &c.?\n\nBut, leaving off this, let me have a wife,\n\nThe fairest maid in Germany;\n\nFor I am wanton and lascivious,\n\nAnd cannot live without a wife.\n\nMEPH. How \u2014a wife?\n\nI prithee, Faustus, talk not of a wife.\n\nFAUSTUS. Nay, sweet Mephistophilis, fetch me one, for I will have one.\n\nMEPH. Well\u2014 thou wilt have one. Sit there till I come:\n\nI'll fetch thee a wife in the Devil's name. [ _Exit._ ]\n\n_Re-enter_ MEPHISTOPHILIS _with a_ Devil _dressed like a woman, with fireworks._\n\nMEPH. Tell [me,] Faustus, how dost thou like thy wife?\n\nFAUSTUS. A plague on her for a hot whore!\n\nMEPH. Tut, Faustus,\n\nMarriage is but a ceremonial toy;\n\nAnd if thou lovest me, think no more of it.\n\nI'll cull thee out the fairest courtesans,\n\nAnd bring them every morning to thy bed;\n\nShe whom thine eye shall like, thy heart shall have,\n\nBe she as chaste as was Penelope,\n\nAs wise as Saba, or as beautiful\n\nAs was bright Lucifer before his fall.\n\nHere, take this book, peruse it thoroughly: [ _Gives a book._ ]\n\nThe iterating of these lines brings gold;\n\nThe framing of this circle on the ground\n\nBrings whirlwinds, tempests, thunder and lightning;\n\nPronounce this thrice devoutly to thyself,\n\nAnd men in armour shall appear to thee,\n\nReady to execute what thou desir'st.\n\nFAUSTUS. Thanks, Mephistophilis; yet fain would I have a book wherein I might behold all spells and incantations, that I might raise up spirits when I please.\n\nMEPH. Here they are, in this book. _Turns_ _to them._\n\nFAUSTUS. Now would I have a book where I might see all characters and planets of the heavens, that I might know their motions and dispositions.\n\nMEPH. Here they are too. _Turns_ _to them._\n\nFAUSTUS. Nay, let me have one book more, \u2014and then I have done, \u2014wherein I might see all plants, herbs, and trees that grow upon the earth.\n\nMEPH. Here they be.\n\nFAUSTUS. O, thou art deceived.\n\nMEPH. Tut, I warrant thee. _Turns_ _to them._ [ _Exeunt._ ]\n\n**SCENE VI.**\n\n**_The Same._**\n\n_Enter_ FAUSTUS _and_ MEPHISTOPHILIS.\n\nFAUSTUS. When I behold the heavens, then I repent,\n\nAnd curse thee, wicked Mephistophilis,\n\nBecause thou hast depriv'd me of those joys.\n\nMEPH. Why, Faustus,\n\nThinkest thou Heaven is such a glorious thing?\n\nI tell thee 'tis not half so fair as thou,\n\nOr any man that breathes on earth.\n\nFAUSTUS. How provest thou that?\n\nMEPH. 'Twas made for man, therefore is man more excellent.\n\nFAUSTUS. If it were made for man, 'twas made for me;\n\nI will renounce this magic and repent.\n\n_Enter_ GOOD ANGEL _and_ EVIL ANGEL.\n\nG. ANG. Faustus, repent; yet God will pity thee.\n\nE. ANG. Thou art a spirit; God cannot pity thee.\n\nFAUSTUS. Who buzzeth in mine ears I am a spirit?\n\nBe I a devil, yet God may pity me;\n\nAy, God will pity me if I repent.\n\nE. ANG. Ay, but Faustus never shall repent.\n\n_Exeunt_ [ANGELS.]\n\nFAUSTUS. My heart's so hard'ned I cannot repent.\n\nScarce can I name salvation, faith, or heaven,\n\nBut fearful echoes thunder in mine ears\n\n\"Faustus, thou art damn'd!\" Then swords and knives,\n\nPoison, gun, halters, and envenom'd steel\n\nAre laid before me to despatch myself,\n\nAnd long ere this I should have slain myself,\n\nHad not sweet pleasure conquer'd deep despair.\n\nHave I not made blind Homer sing to me\n\nOf Alexander's love and \u0152non's death?\n\nAnd hath not he that built the walls of Thebes\n\nWith ravishing sound of his melodious harp,\n\nMade music with my Mephistophilis?\n\nWhy should I die then, or basely despair?\n\nI am resolv'd: Faustus shall ne'er repent.\n\nCome, Mephistophilis, let us dispute again,\n\nAnd argue of divine astrology.\n\nTell me, are there many heavens above the moon?\n\nAre all celestial bodies but one globe,\n\nAs is the substance of this centric earth?\n\nMEPH. As are the elements, such are the spheres\n\nMutually folded in each other's orb,\n\nAnd, Faustus,\n\nAll jointly move upon one axletree\n\nWhose terminine is term'd the world's wide pole;\n\nNor are the names of Saturn, Mars, or Jupiter\n\nFeign'd, but are erring stars.\n\nFAUSTUS. But tell me, have they all one motion, both _situ et tempore?_ 1\n\nMEPH. All jointly move from east to west in twenty-four hours upon the poles of the world; but differ in their motion upon the poles of the zodiac.\n\nFAUSTUS. Tush!\n\nThese slender trifles Wagner can decide;\n\nHath Mephistophilis no greater skill?\n\nWho knows not the double motion of the planets?\n\nThe first is finish'd in a natural day;\n\nThe second thus: as Saturn in thirty years; Jupiter in twelve; Mars in four; the Sun, Venus, and Mercury in a year; the moon in twenty-eight days. Tush, these are freshmen's suppositions. But tell me, hath every sphere a dominion or _intelligentia?_\n\nMEPH. Ay.\n\nFAUSTUS. How many heavens, or spheres, are there?\n\nMEPH. Nine: the seven planets, the firmament, and the empyreal heaven.\n\nFAUSTUS. Well, resolve me in this question: Why have we not conjunctions, oppositions, aspects, eclipses, all at one time, but in some years we have more, in some less?\n\nMEPH. _Per inoequalem motum respecta totius._ 2\n\nFAUSTUS. Well, I am answered. Tell me who made the world.\n\nMEPH. I will not.\n\nFAUSTUS. Sweet Mephistophilis, tell me.\n\nMEPH. Move me not, for I will not tell thee.\n\nFAUSTUS. Villain, have I not bound thee to tell me anything?\n\nMEPH. Ay, that is not against our kingdom; but this is.\n\nThink thou on hell, Faustus, for thou art damn'd.\n\nFAUSTUS. Think, Faustus, upon God that made the world.\n\nMEPH. Remember this.\n\nFAUSTUS. Ay, go, accursed spirit, to ugly hell.\n\n'Tis thou hast damn'd distressed Faustus' soul.\n\nIs't not too late?\n\n_Re-enter_ GOOD ANGEL _and_ EVIL ANGEL.\n\nE. ANG. Too late.\n\nG. ANG. Never too late, if Faustus can repent.\n\nE. ANG. If thou repent, devils shall tear thee in pieces.\n\nG. ANG. Repent, and they shall never raze thy skin.\n\n_Exeunt_ [ANGELS.]\n\nFAUSTUS. Ah, Christ, my Saviour,\n\nSeek to save distressed Faustus' soul.\n\n_Enter_ LUCIFER, BELZEBUB, _and_ MEPHISTOPHILIS.\n\nLuc. Christ cannot save thy soul, for he is just;\n\nThere's none but I have interest in the same.\n\nFAUSTUS. O, who art thou that look'st so terrible?\n\nLuc. I am Lucifer,\n\nAnd this is my companion-prince in hell.\n\nFAUSTUS. O Faustus! they are come to fetch away thy soul!\n\nLuc. We come to tell thee thou dost injure us;\n\nThou talk'st of Christ contrary to thy promise;\n\nThou should'st not think of God: think of the Devil,\n\nAnd of his dam, too.\n\nFAUSTUS. Nor will I henceforth: pardon me in this,\n\nAnd Faustus vows never to look to Heaven,\n\nNever to name God, or to pray to him,\n\nTo burn his Scriptures, slay his ministers,\n\nAnd make my spirits pull his churches down.\n\nLuc. Do so, and we will highly gratify thee. Faustus, we are come from hell to show thee some pastime. Sit down, and thou shalt see all the Seven Deadly Sins appear in their proper shapes.\n\nFAUSTUS. That sight will be pleasing unto me,\n\nAs Paradise was to Adam the first day\n\nOf his creation.\n\nLuc. Talk not of Paradise nor creation, but mark this show: talk of the Devil, and nothing else.\u2014Come away!\n\n_Enter the_ SEVEN DEADLY SINS.\n\nNow, Faustus, examine them of their several names and dispositions.\n\nFAUSTUS. What art thou\u2014the first?\n\nPRIDE. I am Pride. I disdain to have any parents. I am like to Ovid's flea: I can creep into every corner of a wench; sometimes, like a periwig, I sit upon her brow; or like a fan of feathers, I kiss her lips; indeed I do\u2014what do I not? But, fie, what a scent is here! I'll not speak another word, except the ground were perfum'd, and covered with cloth of arras.\n\nFAUSTUS. What art thou\u2014the second?\n\nCOVET. I am Covetousness, begotten of an old churl in an old leathern bag; and might I have my wish I would desire that this house and all the people in it were turn'd to gold, that I might lock you up in my good chest. O, my sweet gold!\n\nFAUSTUS. What art thou\u2014the third?\n\nWRATH. I am Wrath. I had neither father nor mother: I leapt out of a lion's mouth when I was scarce half an hour old; and ever since I have run up and down the world with this case of rapiers wounding myself when I had nobody to fight withal. I was born in hell; and look to it, for some of you shall be my father.\n\nFAUSTUS. What art thou\u2014the fourth?\n\nENVY. I am Envy, begotten of a chimney sweeper and an oyster-wife. I cannot read, and therefore wish all books were burnt. I am lean with seeing others eat. O that there would come a famine through all the world, that all might die, and I live alone! then thou should'st see how fat I would be. But must thou sit and I stand! Come down with a vengeance!\n\nFAUSTUS. Away, envious rascal! What art thou\u2014the fifth?\n\nGLUT. Who, I, sir? I am Gluttony. My parents are all dead, and the devil a penny they have left me, but a bare pension, and that is thirty meals a day and ten bevers\u2014a small trifle to suffice nature. O, I come of a royal parentage! My grandfather was a Gammon of Bacon, my grandmother a Hogshead of Claret-wine; my godfathers were these, Peter Pickleherring, and Martin Martlemas-beef. O, but my godmother, she was a jolly gentlewoman, and well beloved in every good town and city; her name was Mistress Margery Marchbeer. Now, Faustus, thou hast heard all my progeny, wilt thou bid me to supper?\n\nFAUSTUS. No, I'll see thee hanged: thou wilt eat up all my victuals.\n\nGLUT. Then the Devil choke thee!\n\nFAUSTUS. Choke thyself, glutton! Who art thou\u2014the sixth?\n\nSLOTH. I am Sloth. I was begotten on a sunny bank, where I have lain ever since; and you have done me great injury to bring me from thence: let me be carried thither again by Gluttony and Lechery. I'll not speak another word for a king's ransom.\n\nFAUSTUS. What are you, Mistress Minx, the seventh and last?\n\nLECH. Who, I, sir? I am one that loves an inch of raw mutton better than an ell of fried stockfish; and the first letter of my name begins with Lechery.\n\nLUC. Away to hell, to hell! ( _Exeunt the_ SINS.)\n\n\u2014Now, Faustus, how dost thou like this?\n\nFAUSTUS. O, this feeds my soul!\n\nLUC. Tut, Faustus, in hell is all manner of delight.\n\nFAUSTUS. O might I see hell, and return again.\n\nHow happy were I then!\n\nLUC. Thou shalt; I will send for thee at midnight.\n\nIn meantime take this book; peruse it throughly,\n\nAnd thou shalt turn thyself into what shape thou wilt.\n\nFAUSTUS. Great thanks, mighty Lucifer!\n\nThis will I keep as chary as my life.\n\nLUC. Farewell, Faustus, and think on the Devil.\n\nFAUSTUS. Farewell, great Lucifer! Come, Mephistophilis.\n\n_Exeunt omnes._\n\n_Enter_ WAGNER.\n\nWAG. Learned Faustus,\n\nTo know the secrets of astronomy,\n\nGraven in the book of Jove's high firmament,\n\nDid mount himself to scale Olympus' top,\n\nBeing seated in a chariot burning bright,\n\nDrawn by the strength of yoky dragons' necks.\n\nHe now is gone to prove cosmography,\n\nAnd, as I guess, will first arrive at Rome,\n\nTo see the Pope and manner of his court,\n\nAnd take some part of holy Peter's feast,\n\nThat to this day is highly solemnis'd. _Exit._\n\n**SCENE VII.**\n\n**_The Pope's Privy-chamber._**\n\n_Enter_ FAUSTUS _and_ MEPHISTOPHILIS.\n\nFAUSTUS. Having now, my good Mephistophilis,\n\nPast with delight the stately town of Trier,\n\nEnviron'd round with airy mountain-tops,\n\nWith walls of flint, and deep entrenched lakes,\n\nNot to be won by any conquering prince;\n\nFrom Paris next, coasting the realm of France,\n\nWe saw the river Maine fall into Rhine,\n\nWhose banks are set with groves of fruitful vines;\n\nThen up to Naples, rich Campania,\n\nWhose buildings fair and gorgeous to the eye,\n\nThe streets straight forth, and pav'd with finest brick,\n\nQuarter the town in four equivalents.\n\nThere saw we learned Maro's golden tomb,\n\nThe way he cut, an English mile in length,\n\nThorough a rock of stone in one night's space;\n\nFrom thence to Venice, Padua, and the rest,\n\nIn one of which a sumptuous temple stands,\n\nThat threats the stars with her aspiring top,\n\nThus hitherto has Faustus spent his time:\n\nBut tell me, now, what resting-place is this?\n\nHast thou, as erst I did command,\n\nConducted me within the walls of Rome?\n\nMEPH. Faustus, I have; and because we will not be unprovided,\n\nI have taken up his Holiness' privy-chamber for our use.\n\nFAUSTUS. I hope his Holiness will bid us welcome.\n\nMEPH. Tut, 'tis no matter, man, we'll be bold with his good cheer.\n\nAnd now, my Faustus, that thou may'st perceive\n\nWhat Rome containeth to delight thee with,\n\nKnow that this city stands upon seven hills\n\nThat underprop the groundwork of the same.\n\n[Just through the midst runs flowing Tiber's stream,\n\nWith winding banks that cut it in two parts:]\n\nOver the which four stately bridges lean,\n\nThat make safe passage to each part of Rome:\n\nUpon the bridge call'd Ponto Angelo\n\nErected is a castle passing strong,\n\nWithin whose walls such store of ordnance are,\n\nAnd double cannons, fram'd of carved brass,\n\nAs match the days within one c\u00f3mplete year;\n\nBesides the gates and high pyramides,\n\nWhich Julius C\u00e6sar brought from Africa.\n\nFAUSTUS. Now by the kingdoms of infernal rule,\n\nOf Styx, of Acheron, and the fiery lake\n\nOf ever-burning Phlegethon, I swear\n\nThat I do long to see the monuments\n\nAnd situation of bright-splendent Rome:\n\nCome therefore, let's away.\n\nMEPH. Nay, Faustus, stay; I know you'd fain see the Pope,\n\nAnd take some part of holy Peter's feast,\n\nWhere thou shalt see a troop of bald-pate friars,\n\nWhose _summum bonum_ is in belly-cheer.\n\nFAUSTUS. Well, I'm content to compass then some sport,\n\nAnd by their folly make us merriment.\n\nThen charm me, [Mephistophilis,] that I\n\nMay be invisible, to do what I please\n\nUnseen of any whilst I stay in Rome.\n\n[MEPHISTOPHILIS _charms him._ ]\n\nMEPH. So, Faustus, now\n\nDo what thou wilt, thou shalt not be discern'd.\n\n_Sound a sennet._ 4 _Enter the_ POPE _and the_ CARDINAL of LORRAIN _to the banquet, with_ FRIARS _attending._\n\nPOPE. My Lord of Lorrain, wilt please you draw near?\n\nFAUSTUS. Fall to, and the devil choke you an you spare!\n\nPOPE. How now! Who's that which spake? \u2014 Friars, look about.\n\n1 FRIAR. Here's nobody, if it like your Holiness.\n\nPOPE. My lord, here is a dainty dish was sent me from the Bishop of Milan.\n\nFAUSTUS. I thank you, sir. _Snatches_ _it_\n\nPOPE. How now! Who's that which snatch'd the meat from me? Will no man look? My Lord, this dish was sent me from the Cardinal of Florence.\n\nFAUSTUS. You say true; I'll ha't. [ _Snatches it._ ]\n\nPOPE. What, again! My lord, I'll drink to your Grace.\n\nFAUSTUS. I'll pledge your Grace. [ _Snatches the cup._ ]\n\nC.OF LOR. My lord, it may be some ghost newly crept out of purgatory, come to beg a pardon of your Holiness.\n\nPOPE. It may be so. Friars, prepare a dirge to lay the fury of this ghost. Once again, my lord, fall to.\n\n_The_ Pope _crosseth himself._\n\nFAUSTUS. What, are you crossing of yourself? Well, use that trick no more I would advise you.\n\n[ _The_ POPE] _crosses_ [ _himself_ ] _again._\n\nWell, there's the second time. Aware the third, I give you fair warning.\n\n[ _The_ POPE] _crosses_ [ _himself_ ] _again, and_ FAUSTUS _hits him a box of the ear; and they all run away._\n\nCome on, Mephistophilis, what shall we do?\n\nMEPH. Nay, I know not. We shall be curs'd with bell, book, and candle.\n\nFAUSTUS. How! bell, book, and candle, \u2014candle, book, and bell,\n\nForward and backward to curse Faustus to hell!\n\nAnon you shall hear a hog grunt, a calf bleat, and an ass bray,\n\nBecause it is Saint Peter's holiday.\n\n_Re-enter all the_ FRIARS _to sing the Dirge._\n\n1 FRIAR. Come, brethren, let's about our business with good devotion.\n\n_They sing:_\n\nCursed be he that stole away his Holiness' meat from the table! _Maledicat Dominus!_ 6\n\nCursed be he that struck his Holiness a blow on the face! _Maledicat Dominus!_\n\nCursed be he that took Friar Sandelo a blow on the pate! _Maledicat Dominus!_\n\nCursed be he that disturbeth our holy dirge! _Maledicat Dominus!_\n\nCursed be he that took away his Holiness' wine! _Maledicat Dominus! Et omnes sancti!_ 7 _Amen!_\n\n[MEPHISTOPHILIS _and_ FAUSTUS] _beat the_ Friars, _and fling fireworks among them: and so exeunt._\n\n_Enter_ CHORUS.\n\nCHORUS. When Faustus had with pleasure ta'en the view\n\nOf rarest things, and royal courts of kings,\n\nHe stay'd his course, and so returned home;\n\nWhere such as bear his absence but with grief,\n\nI mean his friends, and near'st companions,\n\nDid gratulate his safety with kind words,\n\nAnd in their conference of what befell,\n\nTouching his journey through the world and air,\n\nThey put forth questions of Astrology,\n\nWhich Faustus answer'd with such learned skill,\n\nAs they admir'd and wond'red at his wit.\n\nNow is his fame spread forth in every land;\n\nAmongst the rest the Emperor is one,\n\nCarolus the Fifth, at whose palace now\n\nFaustus is feasted 'mongst his noblemen.\n\nWhat there he did in trial of his art,\n\nI leave untold \u2014your eyes shall see perform'd. _Exit._\n\n**SCENE VIII.**\n\n**_An Inn-yard._**\n\n_Enter_ ROBIN _the Ostler with a book in his hand._\n\nROBIN. O, this is admirable! here I ha' stolen one of Dr. Faustus' conjuring books, and i' faith I mean to search some circles for my own use. Now will I make all the maidens in our parish dance at my pleasure, stark naked before me; and so by that means I shall see more than e'er I felt or saw yet.\n\n_Enter_ RALPH _calling_ ROBIN.\n\nRALPH. Robin, prithee come away; there's a gentleman tarries to have his horse, and he would have his things rubb'd and made clean. He keeps such a chafing with my mistress about it; and she has sent me to look thee out. Prithee come away.\n\nROBIN. Keep out, keep out, or else you are blown up; you are dismemb'red, Ralph: keep out, for I am about a roaring piece of work.\n\nRALPH. Come, what dost thou with that same book? Thou canst not read.\n\nROBIN. Yes, my master and mistress shall find that I can read, he for his forehead, she for her private study; she's born to bear with me, or else my art fails.\n\nRALPH. Why, Robin, what book is that?\n\nROBIN. What book! Why, the most intolerable book for conjuring that e'er was invented by any brimstone devil.\n\nRALPH. Canst thou conjure with it?\n\nROBIN. I can do all these things easily with it: first, I can make thee drunk with ippocras at any tabern in Europe for nothing; that's one of my conjuring works.\n\nRALPH. Our Master Parson says that's nothing.\n\nROBIN. True, Ralph; and more, Ralph, if thou hast any mind to Nan Spit, our kitchenmaid, then turn her and wind her to thy own use as often as thou wilt, and at midnight.\n\nRALPH. O brave Robin, shall I have Nan Spit, and to mine own use?\n\nOn that condition I'll feed thy devil with horsebread as long as he lives, of free cost.\n\nROBIN. No more, sweet Ralph: let's go and make clean our boots, which lie foul upon our hands, and then to our conjuring in the Devil's name. _Exeunt._\n\n**SCENE IX.**\n\n**_An Inn._**\n\n_Enter_ ROBIN _and_ RALPH _with a silver goblet_\n\nROBIN. Come, Ralph, did not I tell thee we were for ever made by this Doctor Faustus' book? _Ecce signum,_ 1 here's a simple purchase for horsekeepers; our horses shall eat no hay as long as this lasts.\n\n_Enter the_ VLNTNER.\n\nRALPH. But, Robin, here comes the vintner.\n\nROBIN. Hush! I'll gull him supernaturally. Drawer, I hope all is paid: God be with you. Come, Ralph.\n\nVLNT. Soft, sir; a word with you. I must yet have a goblet paid from you, ere you go.\n\nROBIN. I, a goblet, Ralph; I, a goblet! I scorn you, and you are but a &c. I, a goblet! search me.\n\nVLNT. I mean so, sir, with your favour. [ _Searches him._ ]\n\nROBIN. How say you now?\n\nVLNT. I must say somewhat to your fellow. You, sir!\n\nRALPH. Me, sir! me, sir! search your fill. [Vlntner _searches him._ ] Now, sir, you may be ashamed to burden honest men with a matter of truth.\n\nVLNT. Well, t' one of you hath this goblet about you.\n\nROBIN. [ _Aside._ ] You lie, drawer, 'tis afore me. \u2014Sirrah you, I'll teach ye to impeach honest men; stand by; \u2014 I'll scour you for a goblet! \u2014stand aside you had best, I charge you in the name of Belzebub. [ _Aside to_ Ralph.] Look to the goblet, Ralph.\n\nVLNT. What mean you, sirrah?\n\nROBIN. I'll tell you what I mean. _Reads_ [ _from a book._ ] _Sanctobulorum, Periphrasticon \u2014_ Nay, I'll tickle you, vintner. [ _Aside to_ Ralph.] Look to the goblet, Ralph. [ _Reads._ ] _Polypragmos Belseborams framanto pacostiphos tostu, Mephistophilis, &c._\n\n_Enter_ MEPHISTOPHILIS, _sets squibs at their backs,_ [ _and then exit_ ] _. They run about._\n\nVLNT. O _nomine Domini!_ 4 what meanest thou, Robin? Thou hast no goblet.\n\nRALPH. _Peccatum peccatorum_ 5 Here's thy goblet, good vintner.\n\n[ _Gives the goblet to_ VINTNER, _who exit_ ]\n\nROBIN. _Misericordia pro nobis!_ 6 What shall I do? Good Devil, forgive me now, and I'll never rob thy library more.\n\n_Re-enter to them_ MEPHISTOPHILIS.\n\nMEPH. Monarch of hell, under whose black survey\n\nGreat potentates do kneel with awful fear,\n\nUpon whose altars thousand souls do lie,\n\nHow am I vexed with these villains' charms?\n\nFrom Constantinople am I hither come\n\nOnly for pleasure of these damned slaves.\n\nROBIN. How from Constantinople? You have had a great journey. Will you take sixpence in your purse to pay for your supper, and begone?\n\nMEPH. Well, villains, for your presumption, I transform thee into an ape, and thee into a dog; and so begone. _Exit._\n\nROBIN. How, into an ape? That's brave! I'll have fine sport with the boys. I'll get nuts and apples enow.\n\nRALPH. And I must be a dog.\n\nROBIN. I' faith thy head will never be out of the pottage pot.\n\n_Exeunt._\n\n**SCENE X.**\n\n**_The Court of the Emperor._**\n\n_Enter_ EMPEROR, FAUSTUS, _and a_ KNIGHT _with attendants._\n\nEMP. Master Doctor Faustus, I have heard strange report of thy knowledge in the black art, how that none in my empire nor in the whole world can compare with thee for the rare effects of magic; they say thou hast a familiar spirit, by whom thou canst accomplish what thou list. This, therefore, is my request, that thou let me see some proof of thy skill, that mine eyes may be witnesses to confirm what mine ears have heard reported; and here I swear to thee by the honour of mine imperial crown, that, whatever thou doest, thou shalt be no ways prejudiced or endamaged.\n\nKNIGHT. [Aside.] I' faith he looks much like a conjuror.\n\nFAUSTUS. My gracious sovereign, though I must confess myself far inferior to the report men have published, and nothing answerable to the honour of your imperial majesty, yet for that love and duty binds me thereunto, I am content to do whatsoever your majesty shall command me.\n\nEMP. Then, Doctor Faustus, mark what I shall say.\n\nAs I was sometime solitary set\n\nWithin my closet, sundry thoughts arose\n\nAbout the honour of mine ancestors,\n\nHow they had won by prowess such exploits,\n\nGot such riches, subdued so many kingdoms,\n\nAs we that do succeed, or they that shall\n\nHereafter possess our throne, shall\n\n(I fear me) ne'er attain to that degree\n\nOf high renown and great authority;\n\nAmongst which kings is Alexander the Great,\n\nChief spectacle of the world's pre-eminence,\n\nThe bright shining of whose glorious acts\n\nLightens the world with his reflecting beams,\n\nAs, when I heard but motion made of him,\n\nIt grieves my soul I never saw the man.\n\nIf, therefore, thou by cunning of thine art\n\nCanst raise this man from hollow vaults below,\n\nWhere lies entomb'd this famous conqueror,\n\nAnd bring with him his beauteous paramour,\n\nBoth in their right shapes, gesture, and attire\n\nThey us'd to wear during their time of life,\n\nThou shalt both satisfy my just desire,\n\nAnd give me cause to praise thee whilst I live.\n\nFAUSTUS. My gracious lord, I am ready to accomplish your request so far forth as by art, and power of my Spirit, I am able to perform.\n\nKNIGHT. [ _Aside._ ] I' faith that's just nothing at all.\n\nFAUSTUS. But, if it like your Grace, it is not in my ability to present before your eyes the true substantial bodies of those two deceased princes, which long since are consumed to dust.\n\nKNIGHT. [ _Aside._ ] Ay, marry, Master Doctor, now there's a sign of grace in you, when you will confess the truth.\n\nFaustus. But such spirits as can lively resemble Alexander and his paramour shall appear before your Grace in that manner that they best liv'd in, in their most flourishing estate; which I doubt not shall sufficiently content your imperial majesty.\n\nEMP. Go to, Master Doctor, let me see them presently.\n\nKnight. Do you hear, Master Doctor? You bring Alexander and his paramour before the Emperor!\n\nFAUSTUS. How then, sir?\n\nKNIGHT. I' faith that's as true as Diana turn'd me to a stag!\n\nFAUSTUS. No, sir, but when Act\u00e6on died, he left the horns for you.\n\nMephistophilis, begone. _Exit_ MEPHISTOPHILIS.\n\nFAUSTUS. Nay, an you go to conjuring, I'll begone. _Exit._ FAUSTUS. I'll meet with you anon for interrupting me so. Here they are, my gracious lord.\n\n_Re-enter_ MEPHISTOPHILIS _and_ [SPIRITS _in the shape of_ ] ALEXANDER _and his_ PARAMOUR.\n\nEMP. Master Doctor, I heard this lady while she liv'd had a wart or mole in her neck: how shall I know whether it be so or no?\n\nFAUSTUS. Your Highness may boldly go and see.\n\n_Exeunt_ [Spirits.]\n\nEMP. Sure these are no spirits, but the true substantial bodies of those two deceased princes.\n\nFAUSTUS. Will't please your Highness now to send for the knight that was so pleasant with me here of late?\n\nEMP. One of you call him forth. [ _Exit_ Attendant.]\n\n_Re-enter the_ KNIGHT _with a pair of horns on his head._\n\nHow now, sir knight! why I had thought thou had'st been a bachelor, but now I see thou hast a wife, that not only gives thee horns, but makes thee wear them. Feel on thy head.\n\nKNIGHT. Thou damned wretch and execrable dog,\n\nBred in the concave of some monstrous rock,\n\nHow darest thou thus abuse a gentleman?\n\nVillain, I say, undo what thou hast done!\n\nFAUSTUS. O, not so fast, sir; there's no haste; but, good, are you rememb'red how you crossed me in my conference with the Emperor? I think I have met with you for it.\n\nEMP. Good Master Doctor, at my entreaty release him; he hath done penance sufficient.\n\nFAUSTUS. My gracious lord, not so much for the injury he off'red me here in your presence, as to delight you with some mirth, hath Faustus worthily requited this injurious knight; which, being all I desire, I am content to release him of his horns: and, sir knight, hereafter speak well of scholars. Mephistophilis, transform him straight. [Mephistophilis _removes the horns._ ] Now, my good lord, having done my duty I humbly take my leave.\n\nEMP. Farewell, Master Doctor; yet, ere you go,\n\nExpect from me a bounteous reward. _Exeunt._\n\n**SCENE XI**\n\n**_A Green; afterwards, the House of Faustus._**\n\n[ _Enter_ FAUSTUS _and_ MEPHISTOPHILIS.]\n\nFAUSTUS. Now, Mephistophilis, the restless course\n\nThat Time doth run with calm and silent foot,\n\nShort'ning my days and thread of vital life,\n\nCalls for the payment of my latest years;\n\nTherefore, sweet Mephistophilis, let us\n\nMake haste to Wittenberg.\n\nMEPH. What, will you go on horseback or on foot?\n\nFAUSTUS. Nay, till I'm past this fair and pleasant green. I'll walk on foot.\n\n_Enter a_ HORSE-COURSER.\n\nHORSE-C. I have been all this day seeking one Master Fustian: mass, see where he is! God save you, Master Doctor!\n\nFAUSTUS. What, horse-courser! You are well met.\n\nHORSE-C. Do you hear, sir? I have brought you forty dollars for your horse.\n\nFAUSTUS. I cannot sell him so: if thou likest him for fifty, take him.\n\nHORSE-C. Alas, sir, I have no more.\u2014I pray you speak for me.\n\nMEPH. I pray you let him have him: he is an honest fellow, and he has a great charge, neither wife nor child.\n\nFAUSTUS. Well, come, give me your money. [Horse-Courser _gives_ Faustus _the money._ ] My boy will deliver him to you. But I must tell you one thing before you have him; ride him not into the water at any hand.\n\nHORSE-C. Why, sir, will he not drink of all waters?\n\nFAUSTUS. O yes, he will drink of all waters, but ride him not into the water: ride him over hedge or ditch, or where thou wilt, but not into the water.\n\nHORSE-C. Well, sir.\u2014Now I am made man forever. I'll not leave my horse for forty. [ _Aside._ ] If he had but the quality of hey-ding-ding, hey-ding-ding, I'd make a brave living on him: he has a buttock as slick as an eel. Well, God b' wi'ye, sir, your boy will deliver him me: but hark ye, sir; if my horse be sick or ill at ease, if I bring his water to you, you'll tell me what it is? _Exit_ HORSE-COURSER.\n\nFAUSTUS. Away, you villain; what, dost think I am a horse-doctor?\n\nWhat art thou, Faustus, but a man condemn'd to die?\n\nThy fatal time doth draw to final end;\n\nDespair doth drive distrust unto my thoughts:\n\nConfound these passions with a quiet sleep:\n\nTush, Christ did call the thief upon the cross;\n\nThen rest thee, Faustus, quiet in conceit.\n\n_Sleeps in his chair._\n\n_Re-enter_ HORSE-COURSER, _all wet, crying._\n\nHORSE-C. Alas, alas! Doctor Fustian, quotha? Mass, Doctor Lopus was never such a doctor. Has given me a purgation has purg'd me of forty dollars; I shall never see them more. But yet, like an ass as I was, I would not be ruled by him, for he bade me I should ride him into no water. Now I, thinking my horse had had some rare quality that he would not have had me known of, I, like a venturous youth, rid him into the deep pond at the town's end. I was no sooner in the middle of the pond, but my horse vanish'd away, and I sat upon a bottle of hay, never so near drowning in my life. But I'll seek out my Doctor, and have my forty dollars again, or I'll make it the dearest horse!\u2014O, yonder is his snipper-snapper.\u2014Do you hear? You heypass, where's your master?\n\nMEPH. Why, sir, what would you? You cannot speak with him.\n\nHORSE-C. But I will speak with him.\n\nMEPH. Why, he's fast asleep. Come some other time.\n\nHORSE-C. I'll speak with him now, or I'll break his glass windows about his ears.\n\nMEPH. I tell thee he has not slept this eight nights.\n\nHORSE-C. An he have not slept this eight weeks, I'll speak with him.\n\nMEPH. See where he is, fast asleep.\n\nHORSE-C. Ay, this is he. God save you, Master Doctor! Master Doctor, Master Doctor Fustian!\u2014Forty dollars, forty dollars for a bottle of hay!\n\nMEPH. Why, thou seest he hears thee not.\n\nHORSE-C. So ho, ho!\u2014so, ho, ho! ( _Hollas in his ear._ ) No, will you not wake? I'll make you wake ere I go. ( _Pulls_ FAUSTUS _by the leg, and pulls it away._ ) Alas, I am undone! What shall I do?\n\nFAUSTUS. O my leg, my leg! Help, Mephistophilis! call the officers. My leg, my leg!\n\nMEPH. Come, villain, to the constable.\n\nHORSE-C. O lord, sir, let me go, and I'll give you forty dollars more.\n\nMEPH. Where be they?\n\nHORSE-C. I have none about me. Come to my ostry and I'll give them you.\n\nMEPH. Begone quickly. HORSE-COURSER _runs away._\n\nFAUSTUS. What, is he gone? Farewell he! Faustus has his leg again, and the horse-courser, I take it, a bottle of hay for his labour. Well, this trick shall cost him forty dollars more.\n\n_Enter_ WAGNER.\n\nHow now, Wagner, what's the news with thee?\n\nWAG. Sir, the Duke of Vanholt doth earnestly entreat your company.\n\nFAUSTUS. The Duke of Vanholt! an honourable gentleman, to whom I must be no niggard of my cunning. Come, Mephistophilis, let's away to him. _Exeunt._\n\n**SCENE XII.**\n\n**_The Court of the Duke of Vanholt._**\n\n_Enter the_ DUKE [OF VANHOLT], _the_ DUCHESS, [FAUSTUS, _and_ MEPHISTOPHILIS.]\n\nDUKE. Believe me, Master Doctor, this merriment hath much pleased me.\n\nFAUSTUS. My gracious lord, I am glad it contents you so well. \u2014 But it may be, madam, you take no delight in this. I have heard that great-bellied women do long for some dainties or other. What is it, madam? Tell me, and you shall have it.\n\nDUCHESS. Thanks, good Master Doctor; and for I see your courteous intent to pleasure me, I will not hide from you the thing my heart desires; and were it now summer, as it is January and the dead time of the winter, I would desire no better meat than a dish of ripe grapes.\n\nFAUSTUS. Alas, madam, that's nothing! Mephistophilis, begone. ( _Exit_ MEPHISTOPHILIS.) Were it a greater thing than this, so it would content you, you should have it.\n\n_Re-enter_ MEPHISTOPHILIS _with the grapes._\n\nHere they be, madam; wilt please you taste on them?\n\nDUKE. Believe me, Master Doctor, this makes me wonder above the rest, that being in the dead time of winter, and in the month of January, how you should come by these grapes.\n\nFAUSTUS. If it like your Grace, the year is divided into two circles over the whole world, that, when it is here winter with us, in the contrary circle it is summer with them, as in India, Saba, and farther countries in the East; and by means of a swift spirit that I have, I had them brought hither, as ye see. \u2014 How do you like them, madam; be they good?\n\nDUCHESS. Believe me, Master Doctor, they be the best grapes that I e'er tasted in my life before.\n\nFAUSTUS. I am glad they content you so, madam.\n\nDUKE. Come, madam, let us in, where you must well reward this learned man for the great kindness he hath show'd to you.\n\nDUCHESS. And so I will, my lord; and whilst I live, rest beholding for this courtesy.\n\nFAUSTUS. I humbly thank your Grace.\n\nDUKE. Come, Master Doctor, follow us and receive your reward.\n\n_Exeunt._\n\n**SCENE XIII.**\n\n**_A Room in the House of Faustus._**\n\n_Enter_ WAGNER, _solus._\n\nWAG. I think my master means to die shortly,\n\nFor he hath given to me all his goods;\n\nAnd yet, methinks, if that death were near,\n\nHe would not banquet and carouse and swill\n\nAmongst the students, as even now he doth,\n\nWho are at supper with such belly-cheer\n\nAs Wagner ne'er beheld in all his life.\n\nSee where they come! Belike the feast is ended.\n\n_Enter_ FAUSTUS, _with two or three_ SCHOLARS [ _and_ MEPHISTOPHILIS.]\n\n1 SCHOL. Master Doctor Faustus, since our conference about fair ladies, which was the beautifullest in all the world, we have determined with ourselves that Helen of Greece was the admirablest lady that ever lived: therefore, Master Doctor, if you will do us that favour, as to let us see that peerless dame of Greece, whom all the world admires for majesty, we should think ourselves much beholding unto you.\n\nFAUSTUS. Gentlemen,\n\nFor that I know your friendship is unfeigned,\n\nAnd Faustus' custom is not to deny\n\nThe just requests of those that wish him well,\n\nYou shall behold that peerless dame of Greece,\n\nNo otherways for pomp and majesty\n\nThan when Sir Paris cross'd the seas with her,\n\nAnd brought the spoils to rich Dardania.\n\nBe silent, then, for danger is in words.\n\nMusic _sounds, and_ HELEN _passeth over the stage._\n\n2 SCHOL. Too simple is my wit to tell her praise,\n\nWhom all the world admires for majesty.\n\n3 SCHOL. No marvel though the angry Greeks pursu'd\n\nWith ten years' war the rape of such a queen,\n\nWhose heavenly beauty passeth all compare.\n\n1 SCHOL. Since we have seen the pride of Nature's works,\n\nAnd only paragon of excellence,\n\n_Enter an_ OLD MAN.\n\nLet us depart; and for this glorious deed\n\nHappy and blest be Faustus evermore.\n\nFAUSTUS. Gentlemen, farewell \u2014the same I wish to you.\n\n_Exeunt_ SCHOLARS [ _and_ WAGNER].\n\nOLD MAN. Ah, Doctor Faustus, that I might prevail\n\nTo guide thy steps unto the way of life,\n\nBy which sweet path thou may'st attain the goal\n\nThat shall conduct thee to celestial rest!\n\nBreak heart, drop blood, and mingle it with tears,\n\nTears falling from repentant heaviness\n\nOf thy most vile and loathsome filthiness,\n\nThe stench whereof corrupts the inward soul\n\nWith such flagitious crimes of heinous sins\n\nAs no commiseration may expel,\n\nBut mercy, Faustus, of thy Saviour sweet,\n\nWhose blood alone must wash away thy guilt.\n\nFAUSTUS. Where art thou, Faustus? Wretch, what hast thou done?\n\nDamn'd art thou, Faustus, damn'd; despair and die!\n\nHell calls for right, and with a roaring voice\n\nSays \"Faustus! come! thine hour is [almost] come!\"\n\nAnd Faustus [now] will come to do thee right.\n\nMEPHISTOPHILIS _gives him a dagger._\n\nOLD MAN. Ah stay, good Faustus, stay thy desperate steps!\n\nI see an angel hovers o'er thy head,\n\nAnd, with a vial full of precious grace,\n\nOffers to pour the same into thy soul:\n\nThen call for mercy, and avoid despair.\n\nFAUSTUS. Ah, my sweet friend, I feel\n\nThy words do comfort my distressed soul.\n\nLeave me a while to ponder on my sins.\n\nOLD MAN. I go, sweet Faustus, but with heavy cheer,\n\nFearing the ruin of thy hopeless soul. [ _Exit._ ]\n\nFAUSTUS. Accursed Faustus, where is mercy now?\n\nI do repent; and yet I do despair;\n\nHell strives with grace for conquest in my breast:\n\nWhat shall I do to shun the snares of death?\n\nMEPH. Thou traitor, Faustus, I arrest thy soul\n\nFor disobedience to my sovereign lord;\n\nRevolt, or I'll in piecemeal tear thy flesh.\n\nFAUSTUS. Sweet Mephistophilis, entreat thy lord\n\nTo pardon my unjust presumption,\n\nAnd with my blood again I will confirm\n\nMy former vow I made to Lucifer.\n\nMEPH. Do it now then quickly, with unfeigned heart,\n\nLest danger do attend thy drift.\n\n[FAUSTUS _stabs his arm and writes on a paper with his blood._ ]\n\nFAUSTUS. Torment, sweet friend, that base and crooked age,\n\nThat durst dissuade me from my Lucifer,\n\nWith greatest torments that our hell affords.\n\nMEPH. His faith is great, I cannot touch his soul;\n\nBut what I may afflict his body with\n\nI will attempt, which is but little worth.\n\nFAUSTUS. One thing, good servant, let me crave of thee,\n\nTo glut the longing of my heart's desire, \u2014\n\nThat I might have unto my paramour\n\nThat heavenly Helen, which I saw of late,\n\nWhose sweet embracings may extinguish clean\n\nThese thoughts that do dissuade me from my vow,\n\nAnd keep mine oath I made to Lucifer.\n\nMEPH. Faustus, this or what else thou shalt desire\n\nShall be perform'd in twinkling of an eye.\n\n_Re-enter_ HELEN.\n\nFAUSTUS. Was this the face that launch'd a thousand ships,\n\nAnd burnt the topless towers of Ilium?\n\nSweet Helen, make me immortal with a kiss. [ _Kisses her._ ]\n\nHer lips suck forth my soul; see where it flies! \u2014\n\nCome, Helen, come, give me my soul again.\n\nHere will I dwell, for Heaven be in these lips,\n\nAnd all is dross that is not Helena.\n\n_Enter_ OLD MAN.\n\nI will be Paris, and for love of thee,\n\nInstead of Troy, shall Wittenberg be sack'd;\n\nAnd I will combat with weak Menelaus,\n\nAnd wear thy colours on my plumed crest;\n\nYea, I will wound Achilles in the heel,\n\nAnd then return to Helen for a kiss.\n\nOh, thou art fairer than the evening air\n\nClad in the beauty of a thousand stars;\n\nBrighter art thou than flaming Jupiter\n\nWhen he appear'd to hapless Semele:\n\nMore lovely than the monarch of the sky\n\nIn wanton Arethusa's azur'd arms:\n\nAnd none but thou shalt be my paramour. _Exeunt._\n\nOLD MAN. Accursed Faustus, miserable man,\n\nThat from thy soul exclud'st the grace of Heaven,\n\nAnd fly'st the throne of his tribunal seat!\n\n_Enter_ DEVILS.\n\nSatan begins to sift me with his pride:\n\nAs in this furnace God shall try my faith,\n\nMy faith, vile hell, shall triumph over thee.\n\nAmbitious fiends! see how the heavens smiles\n\nAt your repulse, and laughs your state to scorn!\n\nHence, hell! for hence I fly unto my God. _Exeunt._\n\n**SCENE XIV.**\n\n**_The Same._**\n\n_Enter_ FAUSTUS _with the_ SCHOLARS.\n\nFAUSTUS. Ah, gentlemen!\n\n1 SCHOL. What ails Faustus?\n\nFAUSTUS. Ah, my sweet chamber-fellow, had I lived with thee, then had I lived still! but now I die eternally. Look, comes he not, come he not?\n\n2 SCHOL. What means Faustus?\n\n3 SCHOL. Belike he is grown into some sickness by being over solitary.\n\n1 SCHOL. If it be so, we'll have physicians to cure him. 'Tis but a surfeit. Never fear, man.\n\nFAUSTUS. A surfeit of deadly sin that hath damn'd both body and soul.\n\n2 SCHOL. Yet, Faustus, look up to Heaven; remember God's mercies are infinite.\n\nFAUSTUS. But Faustus' offences can never be pardoned: the serpent that tempted Eve may be sav'd, but not Faustus. Ah, gentlemen, hear me with patience, and tremble not at my speeches! Though my heart pants and quivers to remember that I have been a student here these thirty years, oh, would I had never seen Wittenberg, never read book! And what wonders I have done, all Germany can witness, yea, the world; for which Faustus hath lost both Germany and the world, yea Heaven itself, Heaven, the seat of God, the throne of the blessed, the kingdom of joy; and must remain in hell for ever, hell, ah, hell, for ever! Sweet friends! what shall become of Faustus being in hell for ever?\n\n3 SCHOL. Yet, Faustus, call on God.\n\nFAUSTUS. On God, whom Faustus hath abjur'd! on God, whom Faustus hath blasphemed! Ah, my God, I would weep, but the Devil draws in my tears. Gush forth blood instead of tears! Yea, life and soul! Oh, he stays my tongue! I would lift up my hands, but see, they hold them, they hold them!\n\nALL. Who, Faustus?\n\nFAUSTUS. Lucifer and Mephistophilis. Ah, gentlemen, I gave them my soul for my cunning!\n\nALL. God forbid!\n\nFAUSTUS. God forbade it indeed; but Faustus hath done it. For vain pleasure of twenty-four years hath Faustus lost eternal joy and felicity. I writ them a bill with mine own blood: the date is expired; the time will come, and he will fetch me.\n\n1 SCHOL. Why did not Faustus tell us of this before, that divines might have prayed for thee?\n\nFAUSTUS. Oft have I thought to have done so; but the Devil threat'ned to tear me in pieces if I nam'd God; to fetch both body and soul if I once gave ear to divinity: and now 't is too late. Gentlemen, away! lest you perish with me.\n\n2 SCHOL. Oh, what shall we do to save Faustus?\n\nFAUSTUS. Talk not of me, but save yourselves, and depart.\n\n3 SCHOL. God will strengthen me. I will stay with Faustus.\n\n1 SCHOL. Tempt not God, sweet friend; but let us into the next room, and there pray for him.\n\nFAUSTUS. Ay, pray for me, pray for me! and what noise soever ye hear, come not unto me, for nothing can rescue me.\n\n2 SCHOL. Pray thou, and we will pray that God may have mercy upon thee.\n\nFAUSTUS. Gentlemen, farewell! If I live till morning I'll visit you: if not\u2014Faustus is gone to hell.\n\nALL. Faustus, farewell!\n\n_Exeunt_ SCHOLARS. _The clock strikes eleven._\n\nFAUSTUS. Ah, Faustus,\n\nNow hast thou but one bare hour to live,\n\nAnd then thou must be damn'd perpetually!\n\nStand still, you ever-moving spheres of Heaven,\n\nThat time may cease, and midnight never come;\n\nFair Nature's eye, rise, rise again and make\n\nPerpetual day; or let this hour be but\n\nA year, a month, a week, a natural day,\n\nThat Faustus may repent and save his soul!\n\nO _lente, lente, currite noctis equi!_ 1\n\nThe stars move still, time runs, the clock will strike,\n\nThe Devil will come, and Faustus must be damn'd.\n\nO, I'll leap up to my God! Who pulls me down?\n\nSee, see where Christ's blood streams in the firmament!\n\nOne drop would save my soul\u2014 Half a drop: ah, my Christ!\n\nAh, rend not my heart for naming of my Christ!\n\nYet will I call on him: O spare me, Lucifer! \u2014\n\nWhere is it now? 'Tis gone; and see where God\n\nStretcheth out his arm, and bends his ireful brows!\n\nMountain and hills come, come and fall on me,\n\nAnd hide me from the heavy wrath of God!\n\nNo! no!\n\nThen will I headlong run into the earth;\n\nEarth gape! O no, it will not harbour me!\n\nYou stars that reign'd at my nativity,\n\nWhose influence hath allotted death and hell,\n\nNow draw up Faustus like a foggy mist\n\nInto the entrails of yon labouring clouds,\n\nThat when they vomit forth into the air,\n\nMy limbs may issue from their smoky mouths,\n\nSo that my soul may but ascend to Heaven.\n\n_The watch strikes_ [ _the half hour_ ] _._\n\nAh, half the hour is past! Twill all be past anon!\n\nO God!\n\nIf thou wilt not have mercy on my soul,\n\nYet for Christ's sake whose blood hath ransom'd me,\n\nImpose some end to my incessant pain;\n\nLet Faustus live in hell a thousand years \u2014\n\nA hundred thousand, and at last be sav'd!\n\nO, no end is limited to damned souls!\n\nWhy wert thou not a creature wanting soul?\n\nOr why is this immortal that thou hast?\n\nAh, Pythagoras' metempsychosis! were that true,\n\nThis soul should fly from me, and I be chang'd\n\nUnto some brutish beast! All beasts are happy,\n\nFor, when they die,\n\nTheir souls are soon dissolv'd in elements;\n\nBut mine must live, still to be plagu'd in hell.\n\nCurst be the parents that engend'red me!\n\nNo, Faustus: curse thyself: curse Lucifer\n\nThat hath depriv'd thee of the joys of Heaven.\n\n_The clock striketh twelve._\n\nO, it strikes, it strikes! Now, body, turn to air,\n\nOr Lucifer will bear thee quick to hell.\n\n_Thunder and lightning._\n\nO soul, be chang'd into little water-drops,\n\nAnd fall into the ocean \u2014 ne'er be found.\n\nMy God! my God! look not so fierce on me!\n\n_Enter_ DEVILS.\n\nAdders and serpents, let me breathe awhile!\n\nUgly hell, gape not! come not, Lucifer!\n\nI'll burn my books! \u2014Ah Mephistophilis!\n\n_Exeunt_ [DEVILS _with_ FAUSTUS.]\n\n_Enter_ CHORUS.\n\n[CHO.] Cut is the branch that might have grown full straight,\n\nAnd burned is Apollo's laurel bough,\n\nThat sometimes grew within this learned man.\n\nFaustus is gone; regard his hellish fall,\n\nWhose fiendful fortune may exhort the wise\n\nOnly to wonder at unlawful things,\n\nWhose deepness doth entice such forward wits\n\nTo practise more than heavenly power permits.\n\n[ _Exit._ ]\n\n_Terminat hora diem, terminat author opus._ 3\n\n* * *\n\n Confound. But Hannibal was victorious at Lake Trasumennus, B.C. 217.\n\n For applause.\n\n Roda, in the Duchy of Saxe-Altenburg, near Jena.\n\n The garden of scholarship being adorned by him.\n\n Knowledge.\n\n An allusion to the myth of Icarus, who flew too near the sun.\n\n Teach publicly.\n\n Aim.\n\n Logic.\n\n The Aristotelian phrase for \"being and not being.\"\n\n Greek physician whose theories were highly regarded in the Middle Ages.\n\n \"Where the philosopher leaves off, there the physician begins.\"\n\n This and the previous quotation are from Aristotle.\n\n Medical maxims.\n\n Announcements.\n\n \"If one and the same thing is bequeathed to two persons, one gets the thing and the other the value of the thing.\"\n\n \"A father cannot disinherit the son except,\" etc.\n\n Of Justinian, under whom the Roman law was codified.\n\n Its.\n\n The Vulgate.\n\n Excels.\n\n Idea.\n\n The Netherlands, over which Parma re-established the Spanish dominion.\n\n A ship filled with explosives used to blow up a bridge built by Parma in 1585 at the siege of Antwerp.\n\n The famous Cornelius Agrippa. German Valdes is not known.\n\n Cf. Virgil, _Aeneid,_ vi. 667.\n\n Americans Indians.\n\n Troopers. Germ. _Reiters._\n\n Versed.\n\n Roger Bacon.\n\n Perhaps Pietro d'Abano, a medieval alchemist; perhaps a misprint for Albertus (Magnus), the great schoolman.\n\n Fine point.\n\n \"Thus I prove\" \u2014a common formula in scholastic discussions.\n\n _'Corpus naturale seu mobile'_ (literally, \"natural or movable body\") was the scholastic expression for the subject-matter of Physics.\n\n Puritan.\n\n Every star belonging to.\n\n Planets.\n\n \"Be propitious to me, gods of Acheron! May the triple deity of Jehovah prevail! Spirits of fire, air, water, hail! Belzebub, Prince of the East, monarch of burning hell, and Demogorgon, we propitiate ye, that Mephistophilis may appear and rise. Why dost thou delay? By Jehovah, Gehenna, and the holy water which now I sprinkle, and the sign of the cross which now I make, and by our prayer, may Mephistophilis now summoned by us arise!\"\n\n \"For indeed thou hast power in the image of thy brother Mephistophilis.\"\n\n Twist in anagrams.\n\n Heaven and hell are indifferent to him.\n\n Sorrowful.\n\n Inform.\n\n Study\n\n Beards cut to a sharp point (Fr. _pic-\u00e0-devant)._\n\n \"Whoever is my disciple,\" the first words of W. Lily's _\"Ad discipulos carmen de moribus\"_ (Ode to His Disciples on Morality).\n\n A kind of larkspur, used for destroying lice.\n\n A mean street in London.\n\n Short wide breeches.\n\n For _diametrically._\n\n \"As if to tread in my tracks.\"\n\n Emden, near the mouth _of_ the river Ems, was an important commercial town in Elizabethan times.\n\n \"Misery loves company.\"\n\n \"It is finished.\"\n\n \"Man, fly!\"\n\n Foolish.\n\n The Queen of Sheba.\n\n Repeating.\n\n \"In direction and in time?\"\n\n \"On account of their unequal motion in relation to the whole.\"\n\n Pair.\n\n Refreshments between meals.\n\n Martlemas or Martinmas was the customary time for hanging up provisions _to_ dry which had been salted for the winter.\n\n Treves.\n\n Virgil, who was reputed a magician in the Middle Ages, was buried at Naples.\n\n Engaged\n\n A particular set of notes on the trumpet or cornet, different from a flourish.\n\n If.\n\n \"May the Lord curse him.\"\n\n \"And all the saints.\"\n\n Wine mixed with sugar and spices.\n\n \"Behold the sign.\"\n\n Gain.\n\n3. The abuse was left to the actor's inventiveness.\n\n \"In the name of the Lord.\"\n\n \"Sin of sins.\"\n\n \"Mercy on us.\"\n\n1. Proportionate.\n\n2. Its.\n\n Mention.\n\n Dr. Lopez, physician to Queen Elizabeth, hanged in 1594 on the charge of conspiring to poison the Queen.\n\n A juggler's term, like \"presto, fly!\" Hence applied to the juggler himself.\n\n Inn.\n\n Old Man.\n\n Unsurpassed in height.\n\n \"Run softly, softly, horses of the night.\" \u2014 Ovid _'s Amores,_ i. 13.\n\n Without ceasing.\n\n \"The hour ends the day, the author ends his work.\"\n**A WOMAN KILLED WITH KINDNESS**\n\n**Thomas Heywood**\n\n**CHARACTERS**\n\nSIB FRANCIS ACTON, _brother of_ MISTRESS FRANKFORD\n\nSIR CHARLES MOUNTFORD\n\nMASTER FRANKFORD\n\nMASTER WENDOLL, _friend to_ FRANKFORD\n\nMASTER MALBY, _friend to_ SIR FRANCIS\n\nMASTER CRANWELL\n\nSHAFTON, _a false friend to_ SIR CHARLES\n\nOLD MOUNTFORD, _uncle to_ SIR CHARLES\n\nTIDY, _cousin to_ SIR CHARLES\n\nSANDY,\n\nRODER,\n\nSHERIFF\n\nA SERJEANT, A KEEPER, OFFICERS, FALCONERS, HUNTSMEN, A COACHMAN, CABTERS, SERVANTS, MUSICIANS\n\nMISTRESS FRANKFORD\n\nSUSAN, _sister of_ SIR CHARLES\n\nCICELY, _maid to_ MISTRESS FRANKFORD\n\nWOMEN SERVANTS\n\nSCENE\u2014 _The North of England._\n\n**ACT I**\n\n**S CENE I _\u2014A room in_ FRANKFORD'S _house._**\n\n[ _Enter_ MASTER FRANKFORD, MISTRESS FRANKFORD, SIR FRANCIS ACTON, SIR CHARLES MOUNTFORD, MASTER MALBY, MASTER WENDOLL, _and_ MASTER CRANWELL.]\n\nSIR FRANCIS. Some music there: none lead the bride a dance?\n\nSIR CHARLES. Yes, would she dance \"The Shaking of the Sheets\";\n\nBut that's the dance her husband means to lead her.\n\nWENDOLL. That's not the dance that every man must dance,\n\nAccording to the ballad.\n\nSIR FRANCIS. Music, ho!\n\nBy your leave, sister;\u2014by your husband's leave,\n\nI should have said: the hand that but this day\n\nWas given you in the church I'll borrow: sound!\n\nThis marriage music hoists me from the ground.\n\nFRANKFORD. Ay, you may caper, you are light and free:\n\nMarriage hath yoked my heels; pray then pardon me.\n\nSIR FRANCIS. I'll have you dance too, brother.\n\nSIR CHARLES. Master Frankford,\n\nYou are a happy man, sir; and much joy\n\nSucceed your marriage mirth! you have a wife\n\nSo qualified, and with such ornaments\n\nBoth of the mind and body. First, her birth\n\nIs noble, and her education such\n\nAs might become the daughter of a prince:\n\nHer own tongue speaks all tongues, and her own hand\n\nCan teach all strings to speak in their best grace,\n\nFrom the shrillest treble to the hoarsest base.\n\nTo end her many praises in one word,\n\nShe's beauty and perfection's eldest daughter,\n\nOnly found by yours, though many a heart hath sought her.\n\nFRANKFORD. But that I know your virtues and chaste thoughts,\n\nI should be jealous of your praise, Sir Charles.\n\nCRANWELL. He speaks no more than you approve.\n\nMALBY. Nor flatters he that gives to her her due.\n\nMISTRESS FRANKFORD. I would your praise could find a fitter theme\n\nThan my imperfect beauties to speak on:\n\nSuch as they be, if they my husband please,\n\nThey suffice me now I am marri\u00e8d:\n\nHis sweet content is like a flattering glass,\n\nTo make my face seem fairer to mine eye;\n\nBut the least wrinkle from his stormy brow\n\nWill blast the roses in my cheeks that grow.\n\nSIR FRANCIS. A perfect wife already, meek and patient:\n\nHow strangely the word \"husband\" fits your mouth,\n\nNot married three hours since! Sister, 'tis good;\n\nYou, that begin betimes thus, must needs prove\n\nPliant and duteous in your husband's love.\u2014\n\nGramercies, brother, wrought her to't already;\n\nSweet husband, and a curtsey, the first day!\n\nMark this, mark this, you that are bachelors,\n\nAnd never took the grace of honest man;\n\nMark this, against you marry, this one phrase:\n\n\"In a good time that man both wins and woos,\n\nThat takes his wife down in her wedding shoes.\"\n\nFRANKFORD. Your sister takes not after you, Sir Francis;\n\nAll his wild blood your father spent on you:\n\nHe got her in his age, when he grew civil:\n\nAll his mad tricks were to his land entailed,\n\nAnd you are heir to all; your sister, she\n\nHath to her dower her mother's modesty.\n\nSIR CHARLES. Lord, sir, in what a happy state live you!\n\nThis morning, which to many seems a burden\n\nToo heavy to bear, is unto you a pleasure.\n\nThis lady is no clog, as many are:\n\nShe doth become you like a well-made suit,\n\nIn which the tailor hath used all his art;\n\nNot like a thick coat of unseasoned frieze,\n\nForced on your back in summer. She's no chain\n\nTo tie your neck, and curb you to the yoke;\n\nBut she's a chain of gold to adorn your neck.\n\nYou both adorn each other, and your hands,\n\nMethinks, are matches: there's equality\n\nIn this fair combination; you are both\n\nScholars, both young, both being descended nobly.\n\nThere's music in this sympathy; it carries\n\nConsort, and expectation of much joy,\n\nWhich God bestow on you, from this first day\n\nUntil your dissolution; that's for aye.\n\nSIR FRANCIS. We keep you here too long, good brother Frankford.\n\nInto the hall; away! go cheer your guests.\n\nWhat, bride and bridegroom both withdrawn at once?\n\nIf you be missed, the guests will doubt their welcome,\n\nAnd charge you with unkindness.\n\nFRANKFORD. To prevent it,\n\nI'll leave you here, to see the dance within.\n\nMISTRESS FRANKFORD. And so will I.\n\n[ _Exeunt_ FRANKFORD _and_ MISTRESS FRANKFORD.]\n\nSIR FRANCIS. To part you, it were sin.\n\nNow, gallants, while the town-musicians\n\nFinger their frets within; and the mad lads\n\nAnd country-lasses, every mother's child,\n\nWith nosegays and bridelaces in their hats,\n\nDance all their country measures, rounds, and jigs,\n\nWhat shall we do? Hark, they are all on the hoigh;\n\nThey toil like mill-horses, and turn as round,\u2014\n\nMarry, not on the toe. Ay, and they caper,\n\nNot without cutting; you shall see, to-morrow,\n\nThe hall-floor pecked and dinted like a mill-stone,\n\nMade with their high shoes; though their skill be small,\n\nYet they tread heavy where their hobnails fall.\n\nSIR CHARLES. Well, leave them to their sports. Sir Francis\n\nActon,\n\nI'll make a match with you; meet to-morrow\n\nAt Chevy-chase, I'll fly my hawk with yours.\n\nSIR FRANCIS. For what? For what?\n\nSIR CHARLES. Why, for a hundred pound.\n\nSIR FRANCIS. Pawn me some gold of that.\n\nSIR CHARLES. Here are ten angels;\n\nI'll make them good a hundred pound to-morrow\n\nUpon my hawk's wing.\n\nSIR FRANCIS. 'Tis a match, 'tis done.\n\nAnother hundred pound upon your dogs;\n\nDare ye, Sir Charles?\n\nSIR CHARLES. I dare: were I sure to lose,\n\nI durst do more than that: here is my hand,\n\nThe first course for a hundred pound.\n\nSIR FRANCIS. A match.\n\nWENDOLL. Ten angels on Sir Francis Acton's hawk;\n\nAs much upon his dogs.\n\nCRANWELL. I am for Sir Charles Mountford; I have seen\n\nHis hawk and dog both tried. What, clap you hands?\n\nOr is't no bargain?\n\nWENDOLL. Yes, and stake them down:\n\nWere they five hundred, they were all my own.\n\nSIR FRANCIS. Be stirring early with the lark to-morrow;\n\nI'll rise into my saddle ere the sun\n\nRise from his bed.\n\nSIR CHARLES. If there you miss me, say\n\nI am no gentleman: I'll hold my day.\n\nSIR FRANCIS. It holds on all sides. Come, to-night let's dance,\n\nEarly to-morrow let's prepare to ride;\n\nWe had need be three hours up before the bride. [ _Exeunt._ ]\n\n**S CENE II\u2014 _A Yard._**\n\n[ _Enter_ NICHOLAS, JENKIN, JACK SLIME, _and_ ROGER BRICKBAT, _with_ COUNTRY WENCHES, _and two or three_ MUSICIANS.]\n\nJENKIN. Come, Nick, take you Joan Miniver to trace withal; Jack Slime, traverse you with Cicely Milk-pail: I will take Jane Trubkin, and Roger Brickbat shall have Isbel Motley; and now that they are busy in the parlour, come, strike up; we'll have a crash here in the yard.\n\nNICHOLAS. My humour is not compendious; dancing I possess not, though I can foot it; yet, since I am fallen into the hands of Cicely Milk-pail, I consent.\n\nSLIME. Truly Nick, though we were never brought up like serving courtiers, yet we have been brought up with serving creatures, ay, and God's creatures too; for we have been brought up to serve sheep, oxen, horses, hogs, and such like: and, though we be but country fellows, it may be in the way of dancing we can do the horse-trick as well as serving-men.\n\nBRICKBAT. Ay, and the cross-point too.\n\nJENKIN. O Slime, O Brickbat, do not you know that comparisons are odious? now we are odious ourselves too, therefore there are no comparisons to be made betwixt us.\n\nNICHOLAS. I am sudden, and not superfluous;\n\nI am quarrelsome, and not seditious;\n\nI am peaceable, and not contentious;\n\nI am brief, and not compendious.\n\nSLIME. Foot it quickly: if the music overcome not my melancholy, I shall quarrel; and if they do not suddenly strike up, I shall presently strike them down.\n\nJENKIN. No quarrelling, for God's sake: truly, if you do, I shall set a knave between ye.\n\nSLIME. I come to dance, not to quarrel. Come, what shall it be? \"Rogero?\"\n\nJENKIN. \"Rogero!\" no; we will dance \"The Beginning of the World.\"\n\nCICELY. I love no dance so well as \"John come kiss me now.\"\n\nNICHOLAS. I, that have ere now deserved a cushion, call for the \"Cushion-dance.\"\n\nBRICKBAT. For my part, I like nothing so well as \"Tom Tyler.\"\n\nJENKIN. No; we'll have \"The Hunting of the Fox.\"\n\nSLIME. \"The Hay,\" \"The Hay\"; there's nothing like \"The Hay.\"\n\nNICHOLAS. I have said, I do say, and I will say again\u2014\n\nJENKIN. Every man agree to have it as Nick says.\n\nALL. Content.\n\nNICHOLAS. It hath been, it now is, and it shall be\u2014\n\nCICELY. What, Master Nicholas, what?\n\nNICHOLAS. \"Put on your smock a' Monday.\"\n\nJENKIN. So the dance will come cleanly off. Come, for God's sake agree of something: if you like not that, put it to the musicians; or let me speak for all, and we'll have \"Sellenger's round.\"\n\nALL. That, that, that.\n\nNICHOLAS. No, I am resolved, thus it shall be:\n\nFirst take hands, then take ye to your heels.\n\nJENKIN. Why, would ye have us run away?\n\nNICHOLAS. No; but I would have you shake your heels.\n\nMusic, strike up!\n\n[ _They dance._ NICHOLAS _whilst dancing speaks stately and scurvily, the rest after the country fashion._ ]\n\nJENKIN. Hey! lively, my lasses! here's a turn for thee! [ _Exeunt._ ]\n\n**S CENE III\u2014 _The open country._**\n\n[ _Horns wind. Enter_ SIR CHARLES MOUNTFORD, SIR FRANCIS ACTON, MALBY, CRANWELL, WENDOLL, FALCONERS, _and_ HUNTSMEN.]\n\nSIR CHARLES. SO; well cast off: aloft, aloft! well flown!\n\nOh, now she takes her at the souwse, and strikes her\n\nDown to the earth, like a swift thunderclap.\n\nWENDOLL. She hath struck ten angels out of my way.\n\nSIR FRANCIS. A hundred pound from me.\n\nSIR CHARLES. What, falconer!\n\nFALCONER. At hand, sir.\n\nSIR CHARLES. Now she hath seized the fowl, and 'gins to plume her,\n\nRebeck her not: rather stand still and check her.\n\nSo, seize her gets, her jesses, and her bells:\n\nAway!\n\nSIR FRANCIS. My hawk killed too.\n\nSIR CHARLES. Ay, but 'twas at the querre,\n\nNot at the mount, like mine.\n\nSIR FRANCIS. Judgment, my masters.\n\nCRANWELL. Yours missed her at the ferre.\n\nWENDOLL. Ay, but our merlin first had plumed the fowl,\n\nAnd twice renewed her from the river too;\n\nHer bells, Sir Francis, had not both one weight,\n\nNor was one semi-tune above the other:\n\nMethinks these Milan bells do sound too full,\n\nAnd spoil the mounting of your hawk.\n\nSIR CHARLES. 'Tis lost.\n\nSIR FRANCIS. I grant it not. Mine likewise seized a fowl\n\nWithin her talons; and you saw her paws\n\nFull of the feathers: both her petty singles,\n\nAnd her long singles gripped her more than other;\n\nThe terrials of her legs were stained with blood:\n\nNot of the fowl only, she did discomfit\n\nSome of her feathers; but she brake away.\n\nCome, come, your hawk is but a rifler.\n\nSIR CHARLES. How!\n\nSIR FRANCIS. Ay, and your dogs are trindle-tails and curs.\n\nSIR CHARLES. You stir my blood.\n\nYou keep not one good hound in all your kennel,\n\nNor one good hawk upon your perch.\n\nSIR FRANCIS. How, knight!\n\nSIR CHARLES. So, knight: you will not swagger, sir?\n\nSIR FRANCIS. Why, say I did?\n\nSIR CHARLES. Why, sir,\n\nI say you would gain as much by swaggering,\n\nAs you have got by wagers on your dogs:\n\nYou will come short in all things.\n\nSIR FRANCIS. Not in this:\n\nNow I'll strike home.\n\nSIR CHARLES. Thou shalt to thy long home,\n\nOr I will want my will.\n\nSIR FRANCIS. All they that love Sir Francis, follow me.\n\nSIR CHARLES. All that affect Sir Charles, draw on my part.\n\nCRANWELL. On this side heaves my hand.\n\nWENDOLL. Here goes my heart.\n\n[ _They divide themselves._ SIR CHARLES MOUNTFORD, CRANWELL, FALCONER, _and_ HUNTSMAN, _fight against_ SIR FRANCIS ACTON, WENDOLL, _his_ FALCONER, _and_ HUNTSMAN; _and_ SIR CHARLES'S _side gets the better, beating the others away, and killing both of_ SIR FRANCIS'S _men. Exeunt all except_ SIR CHARLES.]\n\nSIR CHARLES. My God! what have I done? what have I done?\n\nMy rage hath plunged into a sea of blood,\n\nIn which my soul lies drowned. Poor innocents,\n\nFor whom we are to answer! Well, 'tis I done,\n\nAnd I remain the victor. A great conquest,\n\nWhen I would give this right hand, nay, this head,\n\nTo breathe in them new life whom I have slain!\n\nForgive me, God! 'twas in the heat of blood,\n\nAnd anger quite removes me from myself:\n\nIt was not I, but rage, did this vile murder;\n\nYet I, and not my rage, must answer it.\n\nSir Francis Acton he is fled the field;\n\nWith him all those that did partake his quarrel,\n\nAnd I am left alone with sorrow dumb,\n\nAnd in my height of conquest overcome.\n\n[ _Enter_ SUSAN.]\n\nSUSAN. O God! my brother wounded 'mong the dead!\n\nUnhappy jest, that in such earnest ends:\n\nThe rumour of this fear stretched to my ears,\n\nAnd I am come to know if you be wounded.\n\nSIR CHARLES. Oh! sister, sister, wounded at the heart.\n\nSUSAN. My God forbid!\n\nSIR CHARLES. In doing that thing which He forbad,\n\nI am wounded, sister.\n\nSUSAN. I hope not at the heart.\n\nSIR CHARLES. Yes, at the heart.\n\nSUSAN. O God! a surgeon there!\n\nSIR CHARLES. Call me a surgeon, sister, for my soul;\n\nThe sin of murder it hath pierced my heart,\n\nAnd made a wide wound there: but for these scratches,\n\nThey are nothing, nothing.\n\nSUSAN. Charles, what have you done?\n\nSir Francis hath great friends, and will pursue you\n\nUnto the utmost danger of the law.\n\nSIR CHARLES. My conscience is become mine enemy,\n\nAnd will pursue me more than Acton can.\n\nSUSAN. Oh, fly, sweet brother.\n\nSIR CHARLES. Shall I fly from thee?\n\nWhy, Sue, art weary of my company?\n\nSUSAN. Fly from your foe.\n\nSIR CHARLES. You, sister, are my friend;\n\nAnd, flying you, I shall pursue my end.\n\nSUSAN. Your company is as my eye-ball dear;\n\nBeing far from you, no comfort can be near;\n\nYet fly to save your life: what would I care\n\nTo spend my future age in black despair,\n\nSo you were safe? and yet to live one week\n\nWithout my brother Charles, through every cheek\n\nMy streaming tears would downwards run so rank,\n\nTill they could set on either side a bank,\n\nAnd in the midst a channel; so my face\n\nFor two salt-water brooks shall still find place.\n\nSIR CHARLES. Thou shalt not weep so much, for I will stay\n\nIn spite of danger's teeth: I'll live with thee,\n\nOr I'll not live at all. I will not sell\n\nMy country and my father's patrimony,\n\nNor thy sweet sight, for a vain hope of life.\n\n[ _Enter_ SHERIFF, _with_ OFFICERS.]\n\nSHERIFF. Sir Charles, I am made the unwilling instrument\n\nOf your attach and apprehension:\n\nI'm sorry that the blood of innocent men\n\nShould be of you exacted. It was told me\n\nThat you were guarded with a troop of friends,\n\nAnd therefore I come thus armed.\n\nSIR CHARLES. O, Master Sheriff,\n\nI came into the field with many friends,\n\nBut see, they all have left me: only one\n\nClings to my sad misfortune, my dear sister.\n\nI know you for an honest gentleman;\n\nI yield my weapons, and submit to you;\n\nConvey me where you please.\n\nSHERIFF. To prison then,\n\nTo answer for the lives of these dead men.\n\nSUSAN. O God! O God!\n\nSIR CHARLES. Sweet sister, every strain\n\nOf sorrow from your heart augments my pain;\n\nYour grief abounds, and hits against my breast.\n\nSHERIFF. Sir, will you go?\n\nSIR CHARLES. Even where it likes you best. [ _Exeunt._ ]\n\n**ACT II**\n\n**S CENE I\u2014FRANKFORD'S _study._**\n\n[ _Enter_ FRANKFORD.]\n\nFRANKFORD. How happy am I amongst other men,\n\nThat in my mean estate embrace content!\n\nI am a gentleman, and by my birth,\n\nCompanion with a king; a king's no more.\n\nI am possessed of many fair revenues,\n\nSufficient to maintain a gentleman.\n\nTouching my mind, I am studied in all arts;\n\nThe riches of my thoughts, and of my time,\n\nHave been a good proficient; but the chief\n\nOf all the sweet felicities on earth,\n\nI have a fair, a chaste, and loving wife;\n\nPerfection all, all truth, all ornament\n\nIf man on earth may truly happy be,\n\nOf these at once possessed, sure I am he.\n\n[ _Enter_ NICHOLAS.]\n\nNICHOLAS. Sir, there's a gentleman attends without\n\nTo speak with you.\n\nFRANKFORD. On horseback?\n\nNICHOLAS. Yes, on horseback.\n\nFRANKFORD. Entreat him to alight, I will attend him.\n\nKnow'st thou him, Nick?\n\nNICHOLAS. Know him! yes, his name's Wendoll:\n\nIt seems he comes in haste: his horse is booted\n\nUp to the flank in mire, himself all spotted\n\nAnd stained with plashing. Sure he rid in fear,\n\nOr for a wager: horse and man both sweat;\n\nI ne'er saw two in such a smoking heat.\n\nFRANKFORD. Entreat him in: about it instantly.\n\n[ _Exit_ NICHOLAS.]\n\nThis Wendoll I have noted, and his carriage\n\nHath pleased me much: by observation\n\nI have noted many good deserts in him:\n\nHe's affable, and seen in many things,\n\nDiscourses well, a good companion;\n\nAnd though of small means, yet a gentleman\n\nOf a good house, though somewhat pressed by want:\n\nI have preferred him to a second place\n\nIn my opinion, and my best regard.\n\n[ _Enter_ WENDOLL, MISTRESS FRANKFORD, _and_ NICHOLAS.]\n\nMISTRESS FRANKFORD. O Master Frankford, Master Wendoll here\n\nBrings you the strangest news that e'er you heard.\n\nFRANKFORD. What news, sweet wife?\n\nWhat news, good Master Wendoll?\n\nWENDOLL. You knew the match made 'twixt Sir Francis Acton\n\nAnd Sir Charles Mountford.\n\nFRANKFORD. True, with their hounds and hawks.\n\nWENDOLL. The matches were both played.\n\nFRANKFORD. Ha! and which won?\n\nWENDOLL. Sir Francis, your wife's brother, had the worst.\n\nAnd lost the wager.\n\nFRANKFORD. Why, the worse his chance:\n\nPerhaps the fortune of some other day\n\nWill change his luck.\n\nMISTRESS FRANKFORD. Oh, but you hear not all.\n\nSir Francis lost, and yet was loth to yield:\n\nAt length the two knights grew to difference,\n\nFrom words to blows, and so to banding sides;\n\nWhere valorous Sir Charles slew in his spleen\n\nTwo of your brother's men; his falconer,\n\nAnd his good huntsman, whom he loved so well:\n\nMore men were wounded, no more slain outright.\n\nFRANKFORD. Now, trust me, I am sorry for the knight;\n\nBut is my brother safe?\n\nWENDOLL. All whole and sound,\n\nHis body not being blemished with one wound:\n\nBut poor Sir Charles is to the prison led,\n\nTo answer at the assize for them that's dead.\n\nFRANKFORD. I thank your pains, sir; had the news been better\n\nYour will was to have brought it, Master Wendoll.\n\nSir Charles will find hard friends; his case is heinous,\n\nAnd will be most severely censured on:\n\nI'm sorry for him. Sir, a word with you\n\nI know you, sir, to be a gentleman\n\nIn all things; your possibilities but mean:\n\nPlease you to use my table and my purse,\n\nThey are yours.\n\nWENDOLL. O Lord, sir, I shall never deserve it.\n\nFRANKFORD. O sir, disparage not your worth too much:\n\nYou are full of quality and fair desert:\n\nChoose of my men which shall attend on you,\n\nAnd he is yours. I will allow you, sir,\n\nYour man, your gelding, and your table, all\n\nAt my own charge; be my companion.\n\nWENDOLL. Master Frankford, I have oft been bound to you\n\nBy many favours; this exceeds them all,\n\nThat I shall never merit your least favour:\n\nBut, when your last remembrance I forget,\n\nHeaven at my soul exact that weighty debt!\n\nFRANKFORD. There needs no protestation; for I know you\n\nVirtuous, and therefore grateful. Pr'ythee, Nan,\n\nUse him with all thy loving'st courtesy.\n\nMISTRESS FRANKFORD. As far as modesty may well extend,\n\nIt is my duty to receive your friend.\n\nFRANKFORD. To dinner, come, sir; from this present day,\n\nWelcome to me for ever; come, away.\n\n[ _Exeunt_ FRANKFORD, MISTRESS FRANKFORD, _and_ WENDOLL.]\n\nNICHOLAS. I do not like this fellow by no means:\n\nI never see him but my heart still yearns:\n\nZounds! I could fight with him, yet know not why:\n\nThe devil and he are all one in my eye.\n\n[ _Enter_ JENKIN.]\n\nJENKIN. O Nick, what gentleman is that comes to lie at our house? my master allows him one to wait on him, and I believe it will fall to thy lot.\n\nNICHOLAS. I love my master; by these hilts I do!\n\nBut rather than I'll ever come to serve him,\n\nI'll turn away my master.\n\n[ _Enter_ CICELY.]\n\nCICELY. Nich'las, where are you, Nich'las? you must come in, Nich'las, and help the young gentleman off with his boots.\n\nNICHOLAS. If I pluck off his boots, I'll eat the spurs,\n\nAnd they shall stick fast in my throat like burs.\n\nCICELY. Then, Jenkin, come you.\n\nJENKIN. Nay, 'tis no boot for me to deny it. My master hath given me a coat here, but he takes pains himself to brush it once or twice a day with a hollywand.\n\nCICELY. Come, come, make haste, that you may wash your hands again, and help to serve in dinner.\n\nJENKIN. You may see, my masters, though it be afternoon with you, 'tis but early days with us, for we have not dined yet: stay a little, I'll but go in and help to bear up the first course, and come to you again presently.\n\n[ _Exeunt._ ]\n\n**S CENE II\u2014 _A room in the gaol._**\n\n[ _Enter_ MALBY _and_ CRANWELL.]\n\nMALBY. This is the sessions-day; pray can you tell me\n\nHow young Sir Charles hath sped? Is he acquit,\n\nOr must he try the law's strict penalty?\n\nCRANWELL. He's cleared of all, spite of his enemies,\n\nWhose earnest labour was to take his life:\n\nBut in this suit of pardon he hath spent\n\nAll the revenues that his father left him;\n\nAnd he is now turned a plain countryman,\n\nReformed in all things. See, sir, here he comes.\n\n[ _Enter_ SIR CHARLES _and_ KEEPER.]\n\nKEEPER. Discharge your fees, and you are then at freedom.\n\nSIR CHARLES. Here, Master Keeper, take the poor remainder\n\nOf all the wealth I have: my heavy foes\n\nHave made my purse light; but, alas! to me\n\n'Tis wealth enough that you have set me free.\n\nMALBY. God give you joy of your delivery!\n\nI am glad to see you abroad, Sir Charles.\n\nSIR CHARLES. The poorest knight in England, Master Malby:\n\nMy life hath cost me all my patrimony\n\nMy father left his son: well, God forgive them\n\nThat are the authors of my penury.\n\n[ _Enter_ SHAFTON.]\n\nSHAFTON. Sir Charles! a hand, a hand! at liberty?\n\nNow, by the faith I owe, I am glad to see it.\n\nWhat want you? wherein may I pleasure you?\n\nSIR CHARLES. O me! O most unhappy gentleman!\n\nI am not worthy to have friends stirred up,\n\nWhose hands may help me in this plunge of want.\n\nI would I were in Heaven, to inherit there\n\nThe immortal birth-right which my Saviour keeps,\n\nAnd by no unthrift can be bought and sold;\n\nFor here on earth what pleasures should we trust?\n\nSHAFTON. To rid you from these contemplations,\n\nThree hundred pounds you shall receive of me;\n\nNay, five for fail. Come, sir; the sight of gold\n\nIs the most sweet receipt for melancholy,\n\nAnd will revive your spirits: you shall hold law\n\nWith your proud adversaries. Tush, let Frank Acton\n\nWage with his knighthood like expense with me,\n\nAnd he will sink, he will. Nay, good Sir Charles,\n\nApplaud your fortune, and your fair escape\n\nFrom all these perils.\n\nSIR CHARLES. O sir, they have undone me.\n\nTwo thousand and five hundred pound a year\n\nMy father, at his death, possessed me of;\n\nAll which the envious Acton made me spend.\n\nAnd, notwithstanding all this large expense,\n\nI had much ado to gain my liberty:\n\nAnd I have only now a house of pleasure,\n\nWith some five hundred pounds, reserved\n\nBoth to maintain me and my loving sister.\n\nSHAFTON [ _aside_ ] _._ That must I have, it lies convenient for me:\n\nIf I can fasten but one finger on him,\n\nWith my full hand I'll gripe him to the heart.\n\n'Tis not for love I proffered him this coin,\n\nBut for my gain and pleasure. [ _Aloud._ ] Come, Sir Charles,\n\nI know you have need of money; take my offer.\n\nSIR CHARLES. Sir, I accept it, and remain indebted\n\nEven to the best of my unable power.\n\nCome, gentlemen, and see it tendered down. [ _Exeunt._ ]\n\n**S CENE III\u2014 _A room in_ FRANKFORD'S _house._**\n\n[ _Enter_ WENDOLL _melancholy._ ]\n\nWENDOLL. I am a villain if I apprehend\n\nBut such a thought: then, to attempt the deed,\u2014\n\nSlave, thou art damned without redemption.\n\nI'll drive away this passion with a song.\n\nA song! ha, ha: a song! as if, fond man,\n\nThy eyes could swim in laughter, when thy soul\n\nLies drenched and drown\u00e8d in red tears of blood.\n\nI'll pray, and see if God within my heart\n\nPlant better thoughts. Why, prayers are meditations;\n\nAnd when I meditate (O God, forgive me!)\n\nIt is on her divine perfections.\n\nI will forget her; I will arm myself\n\nNot to entertain a thought of love to her:\n\nAnd, when I come by chance into her presence,\n\nI'll hale these balls until _my_ eye-strings crack,\n\nFrom being pulled and drawn to look that way.\n\n[ _Enter over the stage,_ FRANKFORD, MISTRESS FRANKFORD, _and_ NICHOLAS.]\n\nO God! O God! with what a violence\n\nI'm hurried to mine own destruction.\n\nThere goest thou, the most perfectest man\n\nThat ever England bred a gentleman;\n\nAnd shall I wrong his bed? Thou God of thunder!\n\nStay in thy thoughts of vengeance and of wrath,\n\nThy great, almighty, and all-judging hand\n\nFrom speedy execution on a villain:\n\nA villain, and a traitor to his friend.\n\n[ _Enter_ JENKIN.]\n\nJENKIN. Did your worship call?\n\nWENDOLL. He doth maintain me, he allows me largely Money to spend\u2014\u2014\n\nJENKIN. By my faith, so do not you me; I cannot get a cross of you.\n\nWENDOLL. My gelding, and my man\u2014\u2014\n\nJENKIN. That's Sorrell and I.\n\nWENDOLL. This kindness grows of no alliance 'twixt us\u2014\u2014\n\nJENKIN. Nor is my service of any great acquaintance.\n\nWENDOLL. I never bound him to me by desert:\n\nOf a mere stranger, a poor gentleman,\n\nA man by whom in no kind he could gain,\n\nHe hath placed me in the height of all his thoughts,\n\nMade me companion with the best and chiefest\n\nIn Yorkshire. He cannot eat without me,\n\nNor laugh without me: I am to his body\n\nAs necessary as his digestion,\n\nAnd equally do make him whole or sick:\n\nAnd shall I wrong this man? Base man! ingrate\n\nHast thou the power straight with thy gory hands\n\nTo rip thy image from his bleeding heart?\n\nTo scratch thy name from out the holy book\n\nOf his remembrance; and to wound his name\n\nThat holds thy name so dear? or rend his heart\n\nTo whom thy heart was knit and joined together?\n\nAnd yet I must: then, Wendoll, be content;\n\nThus villains, when they would, cannot repent.\n\nJENKIN. What a strange humour is my new master in! pray God he be not mad: if he should be so, I should never have any mind to serve him in Bedlam. It may be he's mad for missing of me.\n\nWENDOLL [ _seeing_ JENKIN]. What, Jenkin, where's your mistress?\n\nJENKIN. Is your worship married?\n\nWENDOLL. Why dost thou ask?\n\nJENKIN. Because you are my master; and if I have a mistress, I would be glad, like a good servant, to do my duty to her.\n\nWENDOLL. I mean Mistress Frankford.\n\nJENKIN. Marry, sir, her husband is riding out of town, and she went very lovingly to bring him on his way to horse. Do you see, sir? here she comes, and here I go.\n\nWENDOLL. Vanish. [ _Exit_ JENKIN.]\n\n[ _Re\u00ebnter_ MISTRESS FRANKFORD.]\n\nMISTRESS FRANKFORD. You are well met, sir; now, in troth, my husband,\n\nBefore he took horse, had a great desire\n\nTo speak with you: we sought about the house,\n\nHollaed into the fields, sent every way,\n\nBut could not meet you: therefore he enjoined me\n\nTo do unto you his most kind commends.\n\nNay, more; he wills you, as you prize his love,\n\nOr hold in estimation his kind friendship,\n\nTo make bold in his absence, and command\n\nEven as himself were present in the house:\n\nFor you must keep his table, use his servants,\n\nAnd be a present Frankford in his absence.\n\nWENDOLL. I thank him for his love.\u2014\n\nGive me a name, you whose infectious tongues\n\nAre tipped with gall and poison: as you would\n\nThink on a man that had your father slain,\n\nMurdered your children, made your wives base strumpets,\n\nSo call me, call me so: print in my face\n\nThe most stigmatic title of a villain,\n\nFor hatching treason to so true a friend.\n\n[ _Aside._ ]\n\nMISTRESS FRANKFORD. Sir, you are much beholding to my husband;\n\nYou are a man most dear in his regard.\n\nWENDOLL [ _aside_ ] _._ I am bound unto your husband, and you too.\n\nI will not speak to wrong a gentleman\n\nOf that good estimation, my kind friend:\n\nI will not; zounds! I will not. I may choose,\n\nAnd I will choose. Shall I be so misled?\n\nOr shall I purchase to my father's crest\n\nThe motto of a villain? If I say\n\nI will not do it, what thing can enforce me?\n\nWhat can compel me? What sad destiny\n\nHath such command upon my yielding thoughts?\n\nI will not\u2014Ha! some fury pricks me on,\n\nThe swift Fates drag me at their chariot-wheel,\n\nAnd hurry me to mischief. Speak I must;\n\nInjure myself, wrong her, deceive his trust.\n\nMISTRESS FRANKFORD. Are you not well, sir, that you seem thus trou bled?\n\nThere is sedition in your countenance.\n\nWENDOLL. And in my heart, fair angel, chaste and wise.\n\nI love you: start not, speak not, answer not.\n\nI love you: nay, let me speak the rest:\n\nBid me to swear, and I will call to record\n\nThe host of Heaven.\n\nMISTRESS FRANKFORD. The host of Heaven forbid\n\nWendoll should hatch such a disloyal thought!\n\nWENDOLL. Such is my fate; to this suit I was born,\n\nTo wear rich pleasure's crown, or fortune's scorn.\n\nMISTRESS FRANKFORD. My husband loves you.\n\nWENDOLL. I know it.\n\nMISTRESS FRANKFORD. He esteems you\n\nEven as his brain, his eye-ball, or his heart.\n\nWENDOLL. I have tried it.\n\nMISTRESS FRANKFORD. His purse is your exchequer, and his table\n\nDoth freely serve you.\n\nWENDOLL. So I have found it.\n\nMISTRESS FRANKFORD. O! with what face of brass, what brow of steel,\n\nCan you, unblushing, speak this to the face\n\nOf the espoused wife of so dear a friend?\n\nIt is my husband that maintains your state;\n\nWill you dishonour him that in your power\n\nHath left his whole affairs? I am his wife,\n\nIt is to me you speak.\n\nWENDOLL. O speak no more!\n\nFor more than this I know, and have recorded\n\nWithin the red-leaved table of my heart.\n\nFair, and of all beloved, I was not fearful\n\nBluntly to give my life into your hand,\n\nAnd at one hazard all my earthly means.\n\nGo, tell your husband; he will turn me off,\n\nAnd I am then undone. I care not, I;\n\n'Twas for your sake. Perchance in rage he'll kill me:\n\nI care not, 'twas for you. Say I incur\n\nThe general name of villain through the world,\n\nOf traitor to my friend; I care not, I.\n\nBeggary, shame, death, scandal, and reproach,\n\nFor you I'll hazard all: why, what care I?\n\nFor you I'll live, and in your love I'll die.\n\nMISTRESS FRANKFORD. You move me, sir, to passion and to pity.\n\nThe love I bear my husband is as precious\n\nAs my soul's health.\n\nWENDOLL. I love your husband too,\n\nAnd for his love I will engage my life:\n\nMistake me not, the augmentation\n\nOf my sincere affection borne to you\n\nDoth no whit lessen my regard of him.\n\nI will be secret, lady, close as night;\n\nAnd not the light of one small glorious star\n\nShall shine here in my forehead, to bewray\n\nThat act of night.\n\nMISTRESS FRANKFORD. What shall I say?\n\nMy soul is wandering, and hath lost her way.\n\nOh, Master Wendoll! Oh!\n\nWENDOLL. Sigh not, sweet saint;\n\nFor every sigh you breathe draws from my heart\n\nA drop of blood.\n\nMISTRESS FRANKFORD. I ne'er offended yet:\n\nMy fault, I fear, will in my brow be writ.\n\nWomen that fall, not quite bereft of grace,\n\nHave their offences noted in their face.\n\nI blush and am ashamed. Oh, Master Wendoll,\n\nPray God I be not born to curse your tongue,\n\nThat hath enchanted me! This maze I am in\n\nI fear will prove the labyrinth of sin.\n\n[ _Re\u00ebnter_ NICHOLAS _behind._ ]\n\nWENDOLL. The path of pleasure, and the gate to bliss,\n\nWhich on your lips I knock at with a kiss.\n\nNICHOLAS [ _aside_ ] _._ I'll kill the rogue.\n\nWENDOLL. Your husband is from home, your bed's no blab.\n\nNay, look not down and blush.\n\n[ _Exeunt_ WENDOLL _and_ MISTRESS FRANKFORD.]\n\nNICHOLAS. Zounds! I'll stab.\n\nAy, Nick, was it thy chance to come just in the nick?\n\nI love my master, and I hate that slave:\n\nI love my mistress, but these tricks I like not.\n\nMy master shall not pocket up this wrong;\n\nI'll eat my fingers first. What say'st thou, metal?\n\nDoes not the rascal Wendoll go on legs\n\nThat thou must cut off? Hath he not ham strings\n\nThat thou must hough? Nay, metal, thou shalt stand\n\nTo all I say. I'll henceforth turn a spy,\n\nAnd watch them in their close conveyances.\n\nI never looked for better of that rascal,\n\nSince he came miching first into our house:\n\nIt is that Satan hath corrupted her,\n\nFor she was fair and chaste. I'll have an eye\n\nIn all their gestures. Thus I think of them,\n\nIf they proceed as they have done before:\n\nWendoll's a knave, my mistress is a\u2014\u2014\n\n[ _Exit._ ]\n\n**ACT III**\n\n**S CENE I\u2014 _A room in_ SIR CHARLES MOUNTFORD'S _house._**\n\n[ _Enter_ SIR CHARLES MOUNTFORD _and_ SUSAN.]\n\nSIR CHARLES. Sister, you see we are driven to hard shift\n\nTo keep this poor house we have left unsold;\n\nI am now enforced to follow husbandry,\n\nAnd you to milk; and do we not live well?\n\nWell, I thank God.\n\nSUSAN. O brother, here's a change.\n\nSince old Sir Charles died, in our father's house!\n\nSIR CHARLES. All things on earth thus change, some up, some down;\n\nContent's a kingdom, and I wear that crown.\n\n[ _Enter_ SHAFTON _with a_ SERJEANT.]\n\nSHAFTON. Good morrow, morrow, Sir Charles: what, with your sister,\n\nPlying your husbandry?\u2014Serjeant, stand off.\u2014\n\nYou have a pretty house here, and a garden,\n\nAnd goodly ground about it. Since it lies\n\nSo near a lordship that I lately bought,\n\nI would fain buy it of you. I will give you\u2014\u2014\n\nSIR CHARLES. O, pardon me: this house successively\n\nHath 'longed to me and my progenitors\n\nThree hundred years. My great-great-grandfather,\n\nHe in whom first our gentle style began,\n\nDwelt here; and in this ground, increased this mole-hill\n\nUnto that mountain which my father left me.\n\nWhere he the first of all our house began,\n\nI now the last will end, and keep this house,\n\nThis virgin title, never yet deflowered\n\nBy any unthrift of the Mountfords' line.\n\nIn brief, I will not sell it for more gold\n\nThan you could hide or pave the ground withal.\n\nSHAFTON. Ha, ha! a proud mind and a beggar's purse!\n\nWhere's my three hundred pounds, besides the use?\n\nI have brought it to an execution\n\nBy course of law: what, is my moneys ready?\n\nSIR CHARLES. An execution, sir, and never tell me\n\nYou put my bond in suit! you deal extremely.\n\nSHAFTON. Sell me the land, and I'll acquit you straight.\n\nSIR CHARLES. Alas, alas! 'tis all trouble hath left me\n\nTo cherish me and my poor sister's life.\n\nIf this were sold, our names should then be quite\n\nRazed from the bed-roll of gentility.\n\nYou see what hard shift we have made to keep it\n\nAllied still to our own name. This palm, you see,\n\nLabour hath glowed within: her silver brow,\n\nThat never tasted a rough winter's blast\n\nWithout a mask or fan, doth with a grace\n\nDefy cold winter, and his storms out-face.\n\nSUSAN. Sir, we feed sparing, and we labour hard,\n\nWe lie uneasy, to reserve to us\n\nAnd our succession this small plot of ground.\n\nSIR CHARLES. I have so bent my thoughts to husbandry,\n\nThat I protest I scarcely can remember\n\nWhat a new fashion is; how silk or satin\n\nFeels in my hand: why, pride is grown to us\n\nA mere, mere stranger. I have quite forgot\n\nThe names of all that ever waited on me;\n\nI cannot name ye any of my hounds,\n\nOnce from whose echoing mouths I heard all music\n\nThat e'er my heart desired. What should I say?\n\nTo keep this place I have changed myself away.\n\nSHAFTON [ _to the_ SERJEANT]. Arrest him at my suit. Actions and actions\n\nShall keep thee in perpetual bondage fast:\n\nNay, more, I'll sue thee by a late appeal,\n\nAnd call thy former life in question.\n\nThe keeper is my friend, thou shalt have irons,\n\nAnd usage such as I'll deny to dogs:\n\nAway with him!\n\nSIR CHARLES [ _to_ SUSAN], You are too timorous:\n\nBut trouble is my master,\n\nAnd I will serve him truly.\u2014My kind sister,\n\nThy tears are of no force to mollify\n\nThis flinty man. Go to my father's brother,\n\nMy kinsmen and allies; entreat them for me,\n\nTo ransom me from this injurious man,\n\nThat seeks my ruin.\n\nSHAFTON. Come, irons, irons! come away;\n\nI'll see thee lodged far from the sight of day.\n\n[ _Exeunt_ SHAFTON _and_ SERJEANT _with_ SIR CHARLES.]\n\nSUSAN. My heart's so hardened with the frost of grief,\n\nDeath cannot pierce it through. Tyrant too fell!\n\nSo lead the fiends condemned souls to hell.\n\n[ _Enter_ SIR FRANCIS ACTON _and_ MALBY.]\n\nSIR FRANCIS. Again to prison! Malby, hast thou seen\n\nA poor slave better tortured? Shall we hear\n\nThe music of his voice cry from the grate,\n\n\"Meat for the Lord's sake\"? No, no, yet I am not\n\nThroughly revenged. They say he hath a pretty wench\n\nUnto his sister: shall I, in mercy-sake\n\nTo him and to his kindred, bribe the fool\n\nTo shame herself by lewd dishonest lust?\n\nI'll proffer largely; but, the deed being done,\n\nI'll smile to see her base confusion.\n\nMALBY. Methinks, Sir Francis, you are full revenged\n\nFor greater wrongs than he can proffer you.\n\nSee where the poor sad gentlewoman stands.\n\nSIR FRANCIS. Ha, ha! now will I flout her poverty,\n\nDeride her fortunes, scoff her base estate;\n\nMy very soul the name of Mountford hates.\n\nBut stay, my heart! oh, what a look did fly\n\nTo strike my soul through with thy piercing eye!\n\nI am enchanted; all my spirits are fled,\n\nAnd with one glance my envious spleen struck dead.\n\nSUSAN. Acton! that seeks our blood.\n\n[ _Runs away._ ]\n\nSIR FRANCIS. O chaste and fair!\n\nMALBY. Sir Francis, why, Sir Francis, zounds! in a trance?\n\nSir Francis, what cheer, man? Come, come, how is't?\n\nSIR FRANCIS. Was she not fair? Or else this judging eye\n\nCannot distinguish beauty.\n\nMALBY. She was fair.\n\nSIR FRANCIS. She was an angel in a mortal's shape,\n\nAnd ne'er descended from old Mountford's line.\n\nBut soft, soft, let me call _my_ wits together.\n\nA poor, poor wench, to my great adversary\n\nSister, whose very souls denounce stern war,\n\nOne against other. How now, Frank, turned fool\n\nOr madman, whether? But no; master of\n\nMy perfect senses and directest wits.\n\nThen why should I be in this violent humour\n\nOf passion and of love; and with a person\n\nSo different every way, and so opposed\n\nIn all contractions, and still-warring actions?\n\nFie, fie; how I dispute against my soul!\n\nCome, come; I'll gain her, or in her fair quest\n\nPurchase my soul free and immortal rest.\n\n[ _Exeunt._ ]\n\n**S CENE II\u2014 _A sitting-room in_ FRANKFORD'S _house._**\n\n[ _Enter_ SERVING-MEN, _one with a voider and a wooden knife to take away; another with the salt and bread; another with the table-cloth and napkins; another with the carpet:_ JENKIN _follows them with two lights._ ]\n\nJENKIN. So, march in order, and retire in battle array. My master and the guests have supped already, all's taken way: here, now spread for the servingmen in the hall. Butler, it belongs to your office.\n\nBUTLER. I know it, Jenkin. What d'ye call the gentleman that supped there to-night?\n\nJENKIN. Who, my master?\n\nBUTLER. No, no; Master Wendoll, he's a daily guest: I mean the gentleman that came but this afternoon.\n\nJENKIN. His name's Master Cranwell. God's light, hark, within there, my master calls to lay more billets upon the fire. Come, come! Lord, how we that are in office here in the house are troubled! One spread the carpet in the parlour, and stand ready to snuff the lights; the rest be ready to prepare their stomachs. More lights in the hall there. Come, Nich'las.\n\n[ _Exeunt all but_ NICHOLAS.]\n\nNICHOLAS. I cannot eat, but had I Wendoll's heart\n\nI would eat that; the rogue grows impudent.\n\nOh, I have seen such vile notorious tricks,\n\nReady to make my eyes dart from my head.\n\nI'll tell my master, by this air I will!\n\nFall what may fall, I'll tell him. Here he comes.\n\n[ _Enter_ FRANKFORD, _brushing the crumbs from his clothes with a napkin, as newly risen from supper._ ]\n\nFRANKFORD. Nicholas, what make you here? why are not you At supper in the hall among your fellows?\n\nNICHOLAS. Master, I stayed your rising from the board,\n\nTo speak with you.\n\nFRANKFORD. Be brief, then, gentle Nicholas;\n\nMy wife and guests attend me in the parlour.\n\nWhy dost thou pause? Now, Nicholas, you want money,\n\nAnd, unthrift-like, would eat into your wages\n\nEre you have earned it: here, sir, 's half a crown;\n\nPlay the good husband, and away to supper.\n\nNICHOLAS. By this hand, an honourable gentleman! I will not see him wronged. \u2014Sir, I have served you long; you entertained me seven years before your beard. You knew me, sir, before you knew my mistress.\n\nFRANKFORD. What of this, good Nicholas?\n\nNICHOLAS. I never was a make-bate or a knave;\n\nI have no fault but one: I'm given to quarrel,\n\nBut not with women. I will tell you, master,\n\nThat which will make your heart leap from your breast,\n\nYour hair to startle from your head, your ears to tingle.\n\nFRANKFORD. What preparation's this to dismal news?\n\nNICHOLAS. 'Sblood, sir! I love you better than your wife;\n\nI'll make it good.\n\nFRANKFORD. You are a knave, and I have much ado\n\nWith wonted patience to contain my rage,\n\nAnd not to break thy pate. Thou art a knave:\n\nI'll turn you, with your base comparisons,\n\nOut of my doors.\n\nNICHOLAS. Do, do: there is not room\n\nFor Wendoll and for me both in one house.\n\nO master, master, that Wendoll is a villain.\n\nFRANKFORD. Ay, saucy!\n\nNICHOLAS. Strike, strike; do, strike; yet hear me: I am no fool,\n\nI know a villain, when I see him act\n\nDeeds of a villain. Master, master, that base slave\n\nEnjoys my mistress, and dishonours you.\n\nFRANKFORD. Thou hast killed me with a weapon whose sharp point\n\nHath pricked quite through and through my shivering heart:\n\nDrops of cold sweat sit dangling on my hairs,\n\nLike morning's dew upon the golden flowers,\n\nAnd I am plunged into strange agonies.\n\nWhat didst thou say? If any word that touched\n\nHis credit or her reputation,\n\nIt is as hard to enter my belief\n\nAs Dives into heaven.\n\nNICHOLAS. I can gain nothing;\n\nThey are two that never wronged me. I knew before\n\n'Twas but a thankless office, and perhaps\n\nAs much as is my service, or my life\n\nIs worth. All this I know; but this and more,\n\nMore by a thousand dangers, could not hire me\n\nTo smother such a heinous wrong from you.\n\nI saw, and I have said.\n\nFRANKFORD [ _aside_ ] _._ 'Tis probable; though blunt, yet he is honest:\n\nThough I durst pawn my life, and on their faith\n\nHazard the dear salvation of my soul,\n\nYet in my trust I may be too secure.\n\nMay this be true? O, may it, can it be?\n\nIs it by any wonder possible?\n\nMan, woman, what thing mortal may we trust,\n\nWhen friends and bosom wives prove so unjust?\u2014\n\n[ _To_ NICHOLAS.] What instance hast thou of this strange report?\n\nNICHOLAS. Eyes, eyes.\n\nFRANKFORD. Thy eyes may be deceived, I tell thee:\n\nFor, should an angel from the heavens drop down,\n\nAnd preach this to me that thyself hast told,\n\nHe should have much ado to win belief;\n\nIn both their loves I am so confident.\n\nNICHOLAS. Shall I discourse the same by circumstance?\n\nFRANKFORD. No more! to supper, and command your fellows\n\nTo attend us and the strangers. Not a word,\n\nI charge thee on thy life: be secret then,\n\nFor I know nothing.\n\nNICHOLAS. I am dumb; and, now that I have eased my stomach,\n\nI will go fill my stomach.\n\nFRANKFORD. Away; be gone. [ _Exit_ NICHOLAS.]\n\nShe is well born, descended nobly;\n\nVirtuous her education, her repute\n\nIs in the general voice of all the country\n\nHonest and fair; her carriage, her demeanour,\n\nIn all her actions that concern the love\n\nTo me her husband, modest, chaste, and godly.\n\nIs all this seeming gold plain copper?\n\nBut he, that Judas that hath borne my purse,\n\nAnd sold me for a sin!\u2014O God! O God!\n\nShall I put up these wrongs? No. Shall I trust\n\nThe bare report of this suspicious groom,\n\nBefore the double-gilt, the well-hatched ore\n\nOf their two hearts? No, I will lose these thoughts:\n\nDistraction I will banish from my brow,\n\nAnd from my looks exile sad discontent,\n\nTheir Wonted favours in my tongue shall flow;\n\nTill I know all, I'll nothing seem to know.\n\nLights and a table there! Wife, Master Wendoll,\n\nAnd gentle Master Cranwell.\n\n[ _Enter_ MISTRESS FRANKFORD, WENDOLL, CRANWELL, NICHOLAS, _and_ JENKIN, _with cards, carpets, stools, and other necessaries._ ]\n\nFRANKFORD. O Master Cranwell, you are a stranger here,\n\nAnd often baulk my house: faith, y'are a churl:\n\nNow we have supped, a table, and to cards.\n\nJENKIN. A pair of cards, Nicholas, and a carpet to cover the table. Where's Cicely with her counters and her box? Candles and candlesticks there! Fie, we have such a household of serving creatures! unless it be Nick and I, there's not one amongst them all can say bo to a goose. Well said, Nick. [ _They spread a carpet, set down lights and cards._ ]\n\nMISTRESS FRANKFORD. Come, Master Frankford, who shall take my part?\n\nFRANKFORD. Marry, that will I, sweet wife.\n\nWENDOLL. No, by my faith, sir; when you are together I sit out: it must be Mistress Frankford and I, or else it is no match.\n\nFRANKFORD. I do not like that match.\n\nNICHOLAS [ _aside_ ] _._ You have no reason, marry, knowing all.\n\nFRANKFORD. 'Tis no great matter neither. Come, Master Cranwell, shall you and I take them up?\n\nCRANWELL. At your pleasure, sir.\n\nFRANKFORD. I must look to you, Master Wendoll, for you will be playing false; nay, so will my wife too.\n\nNICHOLAS [ _aside_ ] _._ Ay, I will be sworn she will.\n\nMISTRESS FRANKFORD. Let them that are taken playing false, forfeit the set.\n\nFRANKFORD. Content; it shall go hard but I'll take you.\n\nCRANWELL. Gentlemen, what shall our game be?\n\nWENDOLL. Master Frankford, you play best at noddy.\n\nFRANKFORD. You shall not find it so; indeed you shall not.\n\nMISTRESS FRANKFORD. I can play at nothing so well as double ruff.\n\nFRANKFORD. If Master Wendoll and my wife be together, there's no playing against them at double hand.\n\nNICHOLAS. I can tell you, sir, the game that Master Wendoll is best at.\n\nWENDOLL. What game is that, Nick?\n\nNICHOLAS. Marry, sir, knave out of doors.\n\nWENDOLL. She and I will take you at lodam.\n\nMISTRESS FRANKFORD. Husband, shall we play at saint?\n\nFRANKFORD. My saint's turned devil. No, we'll none of saint:\n\nYou are best at new-cut, wife; you'll play at that.\n\nWENDOLL. If you play at new-cut, I am soonest hitter of any here, for a wager.\n\nFRANKFORD. 'Tis me they play on. Well, you may draw out.\n\nFor all your cunning, 'twill be to your shame;\n\nI'll teach you, at your new-cut, a new game.\n\nCome, come.\n\nCRANWELL. If you cannot agree upon the game, to post and pair.\n\nWENDOLL. We shall be soonest pairs; and my good host,\n\nWhen he comes late home, he must kiss the post.\n\nFRANKFORD. Whoever wins, it shall be thy cost.\n\nCRANWELL. Faith, let it be vide-ruff, and let's make honours.\n\nFRANKFORD. If you make honours, one thing let me crave:\n\nHonour the king and queen; except the knave.\n\nWENDOLL. Well, as you please for that.\n\nLift who shall deal.\n\nMISTRESS FRANKFORD. The least in sight: what are you, Master Wendoll?\n\nWENDOLL. I am a knave.\n\nNICHOLAS [ _aside_ ]. I'll swear it.\n\nMISTRESS FRANKFORD. I a queen.\n\nFRANKFORD [ _aside_ ] _._ A quean thou shouldst say. [ _Aloud_.] Well, the cards are mine;\n\nThey are the grossest pair that e'er I felt.\n\nMISTRESS FRANKFORD. Shuffle, I'll cut: would I had never dealt.\n\nFRANKFORD. I have lost my dealing.\n\nWENDOLL. Sir, the fault's in me:\n\nThis queen I have more than mine own, you see.\n\nGive me the stock.\n\nFRANKFORD. My mind's not on my game.\n\nMany a deal I have lost; the more's your shame.\n\nYou have served me a bad trick, Master Wendoll.\n\nWENDOLL. Sir, you must take your lot. To end this strife,\n\nI know I have dealt better with your wife.\n\nFRANKFORD. Thou hast dealt falsely, then.\n\nMISTRESS FRANKFORD. What's trumps?\n\nWENDOLL. Hearts: partner, I rub.\n\nFRANKFORD [ _aside_ ] _._ Thou robb'st me of my soul, of her chaste love;\n\nIn thy false dealing thou hast robbed my heart.\n\n[ _Aloud._ ] Booty you play; I like a loser stand,\n\nHaving no heart, or here or in my hand.\n\nI will give o'er the set; I am not well.\n\nCome, who will hold my cards?\n\nMISTRESS FRANKFORD. Not well, sweet Master Frankford!\n\nAlas, what ail you? 'Tis some sudden qualm.\n\nWENDOLL. How long have you been so, Master Frankford?\n\nFRANKFORD. Sir, I was lusty, and I had my health,\n\nBut I grew ill when you began to deal.\n\nTake hence this table. Gentle Master Cranwell,\n\nYou are welcome; see your chamber at your pleasure.\n\nI'm sorry that this meagrim takes me so,\n\nI cannot sit and bear you company.\n\nJenkin, some lights, and show him to his chamber.\n\n[ _Exeunt_ CRANWELL _and_ JENKIN.]\n\nMISTRESS FRANKFORD. A night-gown for my husband; quickly there:\n\nIt is some rheum or cold.\n\nWENDOLL. Now, in good faith, this illness you have got\n\nBy sitting late without your gown.\n\nFRANKFORD. I know it, Master Wendoll.\n\nGo, go to bed, lest you complain like me.\n\nWife, prythee, wife, into my bed-chamber;\n\nThe night is raw and cold, and rheumatic:\n\nLeave me my gown and light; I'll walk away my fit.\n\nWENDOLL. Sweet sir, good night.\n\nFRANKFORD. Myself, good night.\n\n[ _Exit_ WENDOLL.]\n\nMISTRESS FRANKFORD. Shall I attend you, husband?\n\nFRANKFORD. No, gentle wife, thou'lt catch cold in thy head;\n\nPrythee, be gone, sweet; I'll make haste to bed.\n\nMISTRESS FRANKFORD. No sleep will fasten on mine eyes, you know,\n\nUntil you come.\n\nFRANKFORD. Sweet Nan, I prythee go.\u2014\n\n[ _Exit_ MISTRESS FRANKFORD.]\n\nI have bethought me: get me, by degrees,\n\nThe keys of all my doors, which I will mould\n\nIn wax, and take their fair impression,\n\nTo have by them new keys. This being compassed,\n\nAt a set hour a letter shall be brought me,\n\nAnd, when they think they may securely play,\n\nThey nearest are to danger. Nick, I must rely\n\nUpon thy trust and faithful secrecy.\n\nNICHOLAS. Build on my faith.\n\nFRANKFORD. To bed then, not to rest:\n\nCare lodges in my brain, grief in my breast. [ _Exeunt._ ]\n\n**ACT IV**\n\n**S CENE I\u2014 _A room in old_ MOUNTFORD'S _house._**\n\n[ _Enter_ SUSAN, OLD MOUNTFORD, SANDY, RODER, _and_ TIDY.]\n\nOLD MOUNTFORD. YOU say my nephew is in great distress:\n\nWho brought it to him, but his own lewd life?\n\nI cannot spare a cross. I must confess\n\nHe was my brother's son: why, niece, what then?\n\nThis is no world in which to pity men.\n\nSUSAN. I was not born a beggar, though his extremes\n\nEnforce this language from me: I protest\n\nNo fortune of mine own could lead my tongue\n\nTo this base key. I do beseech you, uncle,\n\nFor the name's sake, for Christianity,\n\nNay, for God's sake, to pity his distress:\n\nHe is denied the freedom of the prison,\n\nAnd in the hole is laid with men condemned;\n\nPlenty he hath of nothing but of irons,\n\nAnd it remains in you to free him thence.\n\nOLD MOUNTFORD. Money I cannot spare; men should take heed;\n\nHe lost my kindred when he fell to need. [ _Exit_.]\n\nSUSAN. Gold is but earth, thou earth enough shalt have,\n\nWhen thou hast once took measure of thy grave.\n\nYou know me, Master Sandy, and my suit.\n\nSANDY. I knew you, lady, when the old man lived;\n\nI knew you ere your brother sold his land;\n\nThen you sung well, played sweetly on the lute;\n\nBut now I neither know you nor your suit. [ _Exit_.]\n\nSUSAN. YOU, Master Roder, was my brother's tenant,\n\nRent-free he placed you in that wealthy farm,\n\nOf which you are possessed.\n\nRODER. True, he did;\n\nAnd have I not there dwelt still for his sake?\n\nI have some business now; but, without doubt,\n\nThey that have hurled him in will help him out.\n\n[ _Exit_.]\n\nSUSAN. Cold comfort still: what say you, cousin Tidy?\n\nTIDY. I say this comes of roysting, swaggering.\n\nCall me not cousin: each man for himself.\n\nSome men are born to mirth, and some to sorrow.\n\nI am no cousin unto them that borrow. [ _Exit_.]\n\nSUSAN. O charity! why art thou fled to heaven,\n\nAnd left all things upon this earth uneven?\n\nTheir scoffing answers I will ne'er return;\n\nBut to myself his grief in silence mourn.\n\n[ _Enter_ SIR FRANCIS ACTON _and_ MALBY.]\n\nSIR FRANCIS. She is poor, I'll therefore tempt her with this gold.\n\nGo, Malby, in my name deliver it,\n\nAnd I will stay thy answer.\n\nMALBY. Fair mistress, as I understand, your grief\n\nDoth grow from want, so I have here in store\n\nA means to furnish you, a bag of gold,\n\nWhich to your hands I freely tender you.\n\nSUSAN. I thank you, Heavens! I thank you, gentle sir:\n\nGod make me able to requite this favour!\n\nMALBY. This gold Sir Francis Acton sends by me,\n\nAnd prays you\u2014\u2014\n\nSUSAN. Acton! O God! that name I am born to curse:\n\nHence, bawd! hence, broker! see, I spurn his gold;\n\nMy honour never shall for gain be sold.\n\nSIR FRANCIS. Stay, lady, stay.\n\nSUSAN. From you I'll posting hie,\n\nEven as the doves from feathered eagles fly.\n\n[ _Exit_.]\n\nSIR FRANCIS. She hates _my_ name, my face: how should I woo?\n\nI am disgraced in every thing I do.\n\nThe more she hates me, and disdains my love,\n\nThe more I am rapt in admiration\n\nOf her divine and chaste perfections.\n\nWoo her with gifts I cannot, for all gifts\n\nSent in my name she spurns: with looks I cannot,\n\nFor she abhors my sight; nor yet with letters,\n\nFor none she will receive. How then, how then?\n\nWell, I will fasten such a kindness on her\n\nAs shall o'ercome her hate and conquer it.\n\nSir Charles, her brother, lies in execution\n\nFor a great sum of money; and, besides,\n\nThe appeal is sued still for my huntsman's death,\n\nWhich only I have power to reverse:\n\nIn her I'll bury all my hate of him.\n\nGo seek the keeper, Malby, bring him to me:\n\nTo save his body, I his debts will pay;\n\nTo save his life, I his appeal will stay.\n\n[ _Exeunt_.]\n\n**S CENE II\u2014 _A prison cell._**\n\n[ _Enter_ SIR CHARLES MOUNTFORD, _with irons, his feet bare, his garments all ragged and torn_.]\n\nSIR CHARLES. Of all on the earth's face most miserable,\n\nBreathe in this hellish dungeon thy laments,\n\nThus like a slave ragged, like a felon gyved.\n\nWhat hurls thee headlong to this base estate?\n\nO unkind uncle! O my friends ingrate!\n\nUnthankful kinsmen! Mountford's all too base,\n\nTo let thy name be fettered in disgrace!\n\nA thousand deaths here in this grave I die;\n\nFear, hunger, sorrow, cold, all threat my death,\n\nAnd join together to deprive my breath.\n\nBut that which most torments me, my dear sister\n\nHath left to visit me, and from my friends\n\nHath brought no hopeful answer: therefore I\n\nDivine they will not help my misery.\n\nIf it be so, shame, scandal, and contempt\n\nAttend their covetous thoughts; need make their graves!\n\nUsurers they live, and may they die like slaves!\n\n[ _Enter_ KEEPER.]\n\nKEEPER. Knight, be of comfort, for I bring thee freedom\n\nFrom all thy troubles.\n\nSIR CHARLES. Then I am doomed to die;\n\nDeath is the end of all calamity.\n\nKEEPER. Live: your appeal is stayed; the execution\n\nOf all your debts discharged; your creditors\n\nEven to the utmost penny satisfied.\n\nIn sign whereof, your shackles I knock off;\n\nYou are not left so much indebted to us\n\nAs for your fees; all is discharged, all paid.\n\nGo freely to your house, or where you please,\n\nAfter long miseries, embrace your ease.\n\nSIR CHARLES. Thou grumblest out the sweetest music to me\n\nThat ever organ played. Is this a dream\n\nOr do my waking senses apprehend\n\nThe pleasing taste of these applausive news?\n\nSlave that I was, to wrong such honest friends,\n\nMy loving kinsmen, and my near allies.\n\nTongue, I will bite thee for the scandal breathed\n\nAgainst such faithful kinsmen: they are all\n\nComposed of pity and compassion,\n\nOf melting charity, and of moving ruth.\n\nThat which I spake before was in my rage;\n\nThey are my friends, the mirrors of this age,\n\nBounteous and free. The noble Mountford's race,\n\nNe'er bred a covetous thought, or humour base.\n\n[ _Enter_ SUSAN.]\n\nSUSAN. I can no longer stay from visiting\n\nMy woful brother: while I could, I kept\n\nMy hapless tidings from his hopeful ear.\n\nSIR CHARLES. Sister, how much am I indebted to thee,\n\nAnd to thy travel!\n\nSUSAN. What, at liberty?\n\nSIR CHARLES. Thou seest I am, thanks to thy industry:\n\nOh! unto which of all my courteous friends\n\nAm I thus bound? My uncle Mountford, he\n\nEven of an infant loved me: was it he?\n\nSo did my cousin Tidy; was it he?\n\nSo Master Roder, Master Sandy too:\n\nWhich of all these did this high kindness do?\n\nSUSAN. Charles, can you mock me in your poverty,\n\nKnowing your friends deride your misery?\n\nNow, I protest I stand so much amazed\n\nTo see your bonds free, and your irons knocked off,\n\nThat I am rapt into a maze of wonder:\n\nThe rather for I know not by what means\n\nThis happiness hath chanced.\n\nSIR CHARLES. Why, by my uncle,\n\nMy cousins, and my friends: who else, I pray,\n\nWould take upon them all my debts to pay?\n\nSUSAN. O brother, they are men all of flint,\n\nPictures of marble, and as void of pity\n\nAs chased bears. I begged, I sued, I kneeled,\n\nLaid open all your griefs and miseries,\n\nWhich they derided; more than that, denied us\n\nA part in their alliance; but, in pride,\n\nSaid that our kindred with our plenty died.\n\nSIR CHARLES. Drudges too much\u2014what did they? oh, known evil!\n\nRich fly the poor, as good men shun the devil.\n\nWhence should my freedom come? of whom alive,\n\nSaving of those, have I deserved so well?\n\nGuess, sister, call to mind, remember me:\n\nThese I have raised; they follow the world's guise;\n\nWhom rich in honour, they in woe despise.\n\nSusan. My wits have lost themselves, let's ask the keeper.\n\nSIR CHARLES. Gaoler!\n\nKEEPER. At hand, sir.\n\nSIR CHARLES. Of courtesy resolve me one demand.\n\nWhat was he took the burthen of my debts\n\nFrom off my back, stayed my appeal to death,\n\nDischarged my fees, and brought me liberty?\n\nKEEPER. A courteous knight, one called Sir Francis Acton.\n\nSIR CHARLES. Ha! Acton! O me, more distressed in this\n\nThan all my troubles! hale me back,\n\nDouble my irons, and my sparing meals\n\nPut into halves, and lodge me in a dungeon\n\nMore deep, more dark, more cold, more comfortless.\n\nBy Acton freed! not all thy manacles\n\nCould fetter so my heels as this one word\n\nHath thralled my heart; and it must now lie bound\n\nIn more strict prison than thy stony gaol.\n\nI am not free; I go but under bail.\n\nKEEPER. My charge is done, sir, now I have my fees;\n\nAs we get little, we will nothing leese.\n\n[ _Exit._ ]\n\nSIR CHARLES. By Acton freed, my dangerous opposite!\n\nWhy, to what end? on what occasion? ha!\n\nLet me forget the name of enemy,\n\nAnd with indifference balance this high favour:\n\nHa!\n\nSUSAN [ _aside_ ] _._ His love to me? upon my soul 'tis so:\n\nThat is the root from whence these strange things grow.\n\nSIR CHARLES. Had this proceeded from my father, he\n\nThat by the law of nature is most bound\n\nIn offices of love, it had deserved\n\nMy best employment to requite that grace:\n\nHad it proceeded from my friends or him,\n\nFrom them this action had deserved my life:\n\nAnd from a stranger more; because from such\n\nThere is less expectation of good deeds.\n\nBut he, nor father, nor ally, nor friend,\n\nMore than a stranger, both remote in blood\n\nAnd in his heart opposed my enemy,\u2014\n\nThat this high bounty should proceed from him,\u2014\n\nOh, there I lose myself! What should I say,\n\nWhat think, what do, his bounty to repay?\n\nSUSAN. You wonder, I am sure, whence this strange kindness\n\nProceeds in Acton. I will tell you, brother:\n\nHe dotes on me, and oft hath sent me gifts,\n\nLetters and tokens: I refused them all.\n\nSIR CHARLES. I have enough, though poor; my heart is set,\n\nIn one rich gift to pay back all my debt.\n\n[ _Exeunt_.]\n\n**S CENE III\u2014 _A room in_ FRANKFORD'S _house._**\n\n[ _Enter_ FRANKFORD, _and_ NICHOLAS _with keys._ ]\n\nFRANKFORD. This is the night that I must play my part\n\nTo try two seeming angels. Where's my keys?\n\nNICHOLAS. They are made according to your mould in wax:\n\nI bade the smith be secret, gave him money,\n\nAnd here they are. The letter, sir.\n\nFRANKFORD. True, take it, there it is;\n\n[ _Gives him letter_.]\n\nAnd when thou seest me in my pleasant'st vein,\n\nReady to sit to supper, bring it me.\n\nNICHOLAS. I'll do't, make no more question but I'll do't.\n\n[ _Exit_.]\n\n[ _Enter_ MISTRESS FRANKFORD, CRANWELL, WENDOLL, _and_ JENKIN.]\n\nMISTRESS FRANKFORD. Sirrah, 'tis six o'clock already struck!\n\nGo bid them spread the cloth and serve in supper.\n\nJENKIN. It shall be done, forsooth, mistress. Where's Spigot, the butler, to give us our salt and trenchers?\n\n[ _Exit_.]\n\nWENDOLL. We that have been a-hunting all the day\n\nCome with prepared stomachs. Master Frankford,\n\nWe wished you at our sport.\n\nFRANKFORD. My heart was with you, and my mind was on you.\n\nFie, Master Cranwell! you are still thus sad?\n\nA stool, a stool. Where's Jenkin, and where's Nick?\n\n'Tis supper-time at least an hour ago.\n\nWhat's the best news abroad?\n\nWENDOLL. I know none good.\n\nFRANKFORD. But I know too much bad.\n\n[ _Aside_.]\n\n[ _Enter_ JENKIN _and_ BUTLER _with a tablecloth, bread, trenchers, and salt_.]\n\nCRANWELL. Methinks, sir, you might have that interest\n\nIn your wife's brother, to be more remiss\n\nIn his hard dealing against poor Sir Charles,\n\nWho, as I hear, lies in York Castle, needy,\n\nAnd in great want.\n\n[ _Exeunt_ JENKIN _and_ BUTLER.]\n\nFRANKFORD. Did not more weighty business of my own\n\nHold me away, I would have laboured peace\n\nBetwixt them, with all care; indeed I would, sir.\n\nMISTRESS FRANKFORD. I'll write unto my brother earnestly\n\nIn that behalf.\n\nWENDOLL. A charitable deed,\n\nAnd will beget the good opinion\n\nOf all your friends that love you, Mistress Frankford.\n\nFBANKFORD. That's you for one; I know you love Sir Charles,\n\nAnd my wife too, well.\n\nWENDOLL. He deserves the love\n\nOf all true gentlemen; be yourselves judge.\n\nFRANKFORD. But supper, ho! Now as thou lov'st me, Wendoll,\n\nWhich I am sure thou dost, be merry, pleasant,\n\nAnd frolic it to-night. Sweet Master Cranwell,\n\nDo you the like. Wife, I protest my heart\n\nWas ne'er more bent on sweet alacrity.\n\nWhere be those lazy knaves to serve in supper?\n\n[ _Re\u00ebnter_ NICHOLAS.]\n\nNICHOLAS. Here's a letter, sir.\n\nFRANKFORD. Whence comes it? and who brought it?\n\nNICHOLAS. A stripling that below attends your answer,\n\nAnd, as he tells me, it is sent from York.\n\nFRANKFORD. Have him into the cellar; let him taste\n\nA cup of our March beer: go, make him drink.\n\n[ _Reads the letter._ ]\n\nNICHOLAS. I'll make him drunk, if he be a Trojan.\n\nFRANKFORD. My boots and spurs! where' Jenkin? God forgive me,\n\nHow I neglect my business! Wife, look here;\n\nI have a matter to be tried to-morrow\n\nBy eight o'clock, and my attorney writes me,\n\nI must be there betimes with evidence,\n\nOr it will go against me. Where's my boots?\n\n[ _Re\u00ebnter_ JENKIN _with boots and spurs._ ] MISTRESS FRANKFORD. I hope your business craves no such despatch\n\nThat you must ride to-night.\n\nWENDOLL [ _aside_ ] _._ I hope it doth.\n\nFRANKFORD. God's me! no such despatch!\n\nJenkin, my boots. Where's Nick? Saddle my roan,\n\nAnd the grey dapple for himself. Content ye,\n\nIt much concerns me. Gentle Master Cranwell,\n\nAnd Master Wendoll, in my absence use\n\nThe very ripest pleasures of my house.\n\nWENDOLL. Lord! Master Frankford, will you ride to-night?\n\nThe ways are dangerous.\n\nFRANKFORD. Therefore will I ride\n\nAppointed well; and so shall Nick my man.\n\nMISTRESS FRANKFORD. I'll call you up by five o'clock tomorrow.\n\nFRANKFORD. No, by my faith, wife, I'll not trust to that;\n\n'Tis not such easy rising in a morning\n\nFrom one I love so dearly: no, by my faith,\n\nI shall not leave so sweet a bedfellow,\n\nBut with much pain. You have made me a sluggard\n\nSince I first knew you.\n\nMISTRESS FRANKFORD. Then, if you needs will go\n\nThis dangerous evening, Master Wendoll,\n\nLet me entreat you bear him company.\n\nWENDOLL. With all my heart, sweet mistress. My boots there!\n\nFRANKFORD. Fie, fie, that for my private business\n\nI should disease my friend, and be a trouble\n\nTo the whole house! Nick!\n\nNICHOLAS. Anon, sir.\n\nFRANKFORD. Bring forth my gelding.\n\n[ _Exit_ NICHOLAS.]\n\nAs you love me, sir,\n\nUse no more words: a hand, good Master Cranwell.\n\nCRANWELL. Sir, God be your good speed!\n\nFRANKFORD. Good night, sweet Nan; nay, nay, a kiss and part.\n\n[ _Aside_.] Dissembling lips, you suit not with my heart.\n\n[ _Exit_.]\n\nWENDOLL. How business, time, and hours, all gracious prove,\n\nAnd are the furtherers to my new-born love!\n\nI am husband now in Master Frankford's place,\n\nAnd must command the house. My pleasure is\n\nWe will not sup abroad so publicly,\n\nBut in your private chamber, Mistress Frankford.\n\nMISTRESS FRANKFORD. O, sir, you are too public in your love,\n\nAnd Master Frankford's wife\u2014\u2014\n\nCRANWELL. Might I crave favour,\n\nI would entreat you I might see my chamber;\n\nam on the sudden grown exceeding ill,\n\nAnd would be spared from supper.\n\nWENDOLL. Light there, ho!\n\nSee you want nothing, sir; for, if you do,\n\nYou injure that good man, and wrong me too.\n\nCRANWELL. I will make bold: good night. [ _Exit_.]\n\nWENDOLL. How all conspire\n\nTo make our bosom sweet, and full entire!\n\nCome, Nan, I pr'ythee let us sup within.\n\nMISTRESS FRANKFORD. Oh, what a clog unto the soul is sin!\n\nWe pale offenders are still full of fear;\n\nEvery suspicious eye brings danger near,\n\nWhen they whose clear hearts from offence are free\n\nDespise report, base scandals do outface,\n\nAnd stand at mere defiance with disgrace.\n\nWENDOLL. Fie, fie! you talk too like a puritan.\n\nMISTRESS FRANKFORD. You have tempted me to mischief, Master Wendoll:\n\nI have done I know not what. Well, you plead custom;\n\nThat which for want of wit I granted erst,\n\nI now must yield through fear. Come, come, let's in;\n\nOnce o'er shoes, we are straight o'er head in sin.\n\nWENDOLL. My jocund soul is joyful above measure;\n\nI'll be profuse in Frankford's richest treasure. [ _Exeunt_.]\n\n**S CENE IV\u2014 _Another part of the house._**\n\n[ _Enter_ CICELY, JENKIN, _and_ BUTLER.]\n\nJENKIN. My mistress and Master Wendoll, my master, sup in her chamber to-night. Cicely, you are preferred from being the cook to be chambermaid: of all the loves betwixt thee and me, tell me what thou thinkest of this?\n\nCICELY. Mum; there's an old proverb,\u2014\"when the cat's away, the mouse may play.\"\n\nJENKIN. Now you talk of a cat, Cicely, I smell a rat.\n\nCICELY. Good words, Jenkin, lest you be called to answer them.\n\nJENKIN. Why, God make my mistress an honest woman! are not these good words? Pray God my new master play not the knave with my old master! is there any hurt in this? God send no villainy intended! and, if they do sup together, pray God they do not lie together! God make my mistress chaste, and make us all His servants! what harm is there in all this? Nay, more; here is my hand, thou shalt never have my heart unless thou say Amen.\n\nCICELY. Amen, I pray God, I say.\n\n[ _Enter_ SERVING-MAN.]\n\nSERVING-MAN. My mistress sends that you should make less noise, to lock up the doors, and see the household all got to bed: you, Jenkin, for this night are made the porter to see the gates shut in.\n\nJENKIN. Thus, by little and little, I creep into office. Come, to kennel, my masters, to kennel; 'tis eleven o'clock, already.\n\nSERVING-MAN. When you have locked the gates in, you must send up the keys to my mistress.\n\nCICELY. Quickly, for God's sake, Jenkin, for I must carry them. I am neither pillow nor bolster, but I know more than both.\n\nJENKIN. To bed, good Spigot; to bed, good honest serving-creatures; and let us sleep as snug as pigs in pease-straw.\n\n[ _Exeunt_.]\n\n**S CENE V\u2014 _Outside_ FRANKFORD'S _house._**\n\n[ _Enter_ FRANKFORD _and_ NICHOLAS.]\n\nFRANKFORD. Soft, soft; we have tied our geldings to a tree,\n\nTwo flight-shot off, lest by their thundering hoofs\n\nThey blab our coming back. Hear'st thou no noise?\n\nNICHOLAS. Hear! I hear nothing but the owl and you.\n\nFRANKFORD. So; now my watch's hand points upon twelve,\n\nAnd it is dead midnight. Where are my keys?\n\nNICHOLAS. Here, sir.\n\nFRANKFORD. This is the key that opes my outward gate;\n\nThis is the hall-door; this the withdrawing chamber;\n\nBut this, that door that's bawd unto my shame,\n\nFountain and spring of all my bleeding thoughts,\n\nWhere the most hallowed order and true knot\n\nOf nuptial sanctity hath been profaned;\n\nIt leads to my polluted bed-chamber,\n\nOnce my terrestrial heaven, now my earth's hell,\n\nThe place where sins in all their ripeness dwell.\n\nBut I forget myself: now to my gate.\n\nNICHOLAS. It must ope with far less noise than Cripple-gate, or your plot's dashed.\n\nFRANKFORD. So, reach me my dark lanthorn to the rest;\n\nTread softly, softly.\n\nNICHOLAS. I will walk on eggs this pace.\n\nFRANKFORD. A general silence hath surprised the house,\n\nAnd this is the last door. Astonishment,\n\nFear, and amazement play against my heart,\n\nEven as a madman beats upon a drum.\n\nOh, keep my eyes, you Heavens, before I enter,\n\nFrom any sight that may transfix my soul;\n\nOr, if there be so black a spectacle,\n\nOh, strike mine eyes stark blind; or, if not so,\n\nLend me such patience to digest my grief\n\nThat I may keep this white and virgin hand\n\nFrom any violent outrage or red murder!\n\nAnd with that prayer I enter. [ _Exeunt._ ]\n\n**S CENE VI\u2014 _The hull of_ FRANKFORD'S _house._**\n\n[NICHOLAS _discovered._ ]\n\nNICHOLAS. Here's a circumstance.\n\nA man be made cuckold in the time\n\nThat he's about it. An the case were mine,\n\nAs 'tis my master's,\u2014'sblood that he makes me swear!\u2014\n\nI would have placed his action, entered there;\n\nI would, I would.\n\n[ _Enter_ FRANKFORD.]\n\nFRANKFORD. Oh! oh!\n\nNICHOLAS. Master, 'sblood! master! master!\n\nFBANKFORD. O me unhappy! I have found them lying\n\nClose in each other's arms, and fast asleep.\n\nBut that I would not damn two precious souls,\n\nBought with my Saviour's blood, and send them, laden\n\nWith all their scarlet sins upon their backs,\n\nUnto a fearful judgment, their two lives\n\nHad met upon my rapier.\n\nNICHOLAS. 'Sblood, master, what, have you left them sleeping still? let me go wake them.\n\nFBANKFORD. Stay, let me pause a while.\n\nO God! O God! that it were possible\n\nTo undo things done; to call back yesterday!\n\nThat Time could turn up his swift sandy glass,\n\nTo untell the days, and to redeem these hours!\n\nOr that the sun\n\nCould, rising from the west, draw his coach backward,\n\nTake from the account of time so many minutes,\n\nTill he had all these seasons called again,\n\nThose minutes, and those actions done in them,\n\nEven from her first offence; that I might take her\n\nAs spotless as an angel in my arms!\n\nBut, oh! I talk of things impossible,\n\nAnd cast beyond the moon. God give me patience!\n\nFor I will in and wake them. [ _Exit_.]\n\nNICHOLAS. Here's patience perforce;\n\nHe needs must trot afoot that tires his horse.\n\n[ _Enter_ WENDOLL, _running over the stage in a night-gown,_ FRANKFORD _after him with a sword drawn; a_ MAID-SERVANT _in her smock stays his hand, and clasps hold on him._ FRANKFORD _pauses for a while_.]\n\nFRANKFORD. I thank thee, maid; thou, like the angel's hand,\n\nHast stayed me from a bloody sacrifice.\n\n[ _Exit_ MAID-SERVANT.]\n\nGo, villain, and my wrongs sit on thy soul\n\nAs heavy as this grief doth upon mine!\n\nWhen thou record'st my many courtesies,\n\nAnd shalt compare them with thy treacherous heart,\n\nLay them together, weigh them equally,\n\n'Twill be revenge enough. Go, to thy friend\n\nA Judas: pray, pray, lest I live to see\n\nThee, Judas-like, hanged on an elder-tree.\n\n[ _Enter_ MISTRESS FRANKFORD _in her night attire_.]\n\nMISTRESS FRANKFORD. Oh, by what word, what title, or what name,\n\nShall I entreat your pardon? Pardon! oh!\n\nI am as far from hoping such sweet grace\n\nAs Lucifer from heaven. To call you husband\u2014\n\nO me, most wretched! I have lost that name,\n\nI am no more your wife.\n\nNICHOLAS. 'Sblood, sir, she swoons.\n\nFRANKFORD. Spare thou thy tears, for I will weep for thee:\n\nAnd keep thy countenance, for I'll blush for thee.\n\nNow, I protest, I think 'tis I am tainted,\n\nFor I am most ashamed; and 'tis more hard\n\nFor me to look upon thy guilty face,\n\nThan on the sun's clear brow. What wouldst thou speak?\n\nMISTRESS FRANKFORD. I would I had no tongue, no ears, no eyes,\n\nNo apprehension, no capacity.\n\nWhen do you spurn me like a dog? when tread me\n\nUnder your feet? when drag me by the hair?\n\nThough I deserve a thousand thousand fold\n\nMore than you can inflict: yet, once my husband,\n\nFor womanhood, to which I am a shame,\n\nThough once an ornament\u2014even for His sake\n\nThat hath redeemed our souls, mark not my face\n\nNor hack me with your sword; but let me go\n\nPerfect and undeform\u00e8d to my tomb.\n\nI am not worthy that I should prevail\n\nIn the least suit; no, not to speak to you,\n\nNor look on you, nor to be in your presence.\n\nYet, as an abject, this one suit I crave;\n\nThis granted, I am ready for my grave.\n\n[ _Kneels_.]\n\nFRANKFORD. My God, with patience arm me! Rise, nay, rise,\n\nAnd I'll debate with thee. Was it for want\n\nThou playedst the strumpet? Wast thou not supplied\n\nWith every pleasure, fashion, and new toy\n\nNay, even beyond my calling?\n\nMISTRESS FRANKFORD. I was.\n\nFRANKFORD. Was it then disability in me;\n\nOr in thine eye seemed he a properer man?\n\nMISTRESS FRANKFORD. Oh, no.\n\nFRANKFORD. Did not I lodge thee in my bosom?\n\nWear thee here in my heart?\n\nMISTRESS FRANKFORD. You did.\n\nFRANKFORD. I did, indeed; witness my tears I did.\n\nGo, bring my infants hither.\n\n[ _Enter_ SERVANT _with two_ Children.]\n\nO Nan! O Nan!\n\nIf neither fear of shame, regard of honour,\n\nThe blemish of my house, nor my dear love\n\nCould have withheld thee from so lewd a fact,\n\nYet for these infants, these young harmless souls,\n\nOn whose white brows thy shame is charactered,\n\nAnd grows in greatness as they wax in years,\u2014\n\nLook but on them, and melt away in tears.\n\nAway with them! lest, as her spotted body\n\nHath stained their names with stripe of bastardy,\n\nSo her adulterous breath may blast their spirits\n\nWith her infectious thoughts. Away with them.\n\n[ _Exeunt_ SERVANT _and_ CHILDREN.]\n\nMISTRESS FRANKFORD. In this one life I die ten thousand deaths.\n\nFRANKFORD. Stand up, stand up; I will do nothing rashly;\n\nI will retire a while into my study,\n\nAnd thou shalt hear thy sentence presently. [ _Exit._ ]\n\nMISTRESS FRANKFORD. 'Tis welcome, be it death. O me, base strumpet,\n\nThat, having such a husband, such sweet children,\n\nMust enjoy neither! Oh, to redeem my honour,\n\nI would have this hand cut off, these my breasts seared,\n\nBe racked, strappadoed, put to any torment:\n\nNay, to whip but this scandal out, I would hazard\n\nThe rich and dear redemption of my soul.\n\nHe cannot be so base as to forgive me;\n\nNor I so shameless to accept his pardon.\n\nO women, women, you that yet have kept\n\nYour holy matrimonial vow unstained,\n\nMake me your instance: when you tread awry,\n\nYour sins, like mine, will on your conscience lie.\n\n[ _Enter_ CICELY, JENKIN, _and all the_ SERVING-MEN _as newly come out of bed_.]\n\nALL. O mistress, mistress, what have you done, mistress?\n\nNICHOLAS. 'Sblood, what a caterwauling keep you here!\n\nJENKIN. O Lord, mistress, how comes this to pass? My master is run away in his shirt, and never so much as called me to bring his clothes after him.\n\nMISTRESS FRANKFORD. See what guilt is! here stand I in this place,\n\nAshamed to look my servants in the face.\n\n[ _Enter_ FRANKFORD _and_ CRANWELL, _whom seeing she falls on her knees._ ]\n\nFRANKFORD. My words are registered in Heaven already,\n\nWith patience hear me. I'll not martyr thee,\n\nNor mark thee for a strumpet; but with usage\n\nOf more humility torment thy soul,\n\nAnd kill thee even with kindness.\n\nCRANWELL. Master Frankford\u2014\u2014\n\nFRANKFORD. Good Master Cranwell. Woman, hear thy judgment.\n\nGo make thee ready in thy best attire;\n\nTake with thee all thy gowns, all thy apparel;\n\nLeave nothing that did ever call thee mistress.\n\nOr by whose sight, being left here in the house,\n\nI may remember such a woman by.\n\nChoose thee a bed and hangings for thy chamber;\n\nTake with thee every thing that hath thy I mark,\n\nAnd get thee to my manor seven mile off,\n\nWhere live; 'tis thine; I freely give it thee.\n\nMy tenants by shall furnish thee with wains\n\nTo carry all thy stuff, within two hours,\u2014\n\nNo longer will I limit thee my sight.\n\nChoose which of all my servants thou likest best,\n\nAnd they are thine to attend thee.\n\nMISTRESS FRANKFORD. A mild sentence.\n\nFRANKFORD. But, as thou hopest for Heaven, as thou believest\n\nThy name's recorded in the book of life,\n\nI charge thee never, after this sad day,\n\nTo see me, or to meet me, or to send\n\nBy word or writing, gift, or otherwise,\n\nTo move me, by thyself, or by thy friends;\n\nNor challenge any part in my two children.\n\nSo, farewell, Nan! for we will henceforth be\n\nAs we had never seen, ne'er more shall see.\n\nMISTRESS FRANKFORD. How full my heart is, in mine eyes appears;\n\nWhat wants in words, I will supply in tears.\n\nFRANKFORD. Come, take your coach, your stuff; all must along;\n\nServants and all, make ready; all be gone.\n\nIt was thy hand cut two hearts out of one.\n\n[ _Exeunt._ ]\n\n**ACT V**\n\n**S CENE I\u2014 _The entrance to_ SIR FRANCIS ACTON'S _house._**\n\n[ _Enter_ SIR CHARLES MOUNTFORD, _and_ SUSAN, _both well dressed._ ]\n\nSUSAN. Brother, why have you tricked me like a bride,\n\nBought me this gay attire, these ornaments?\n\nForget you our estate, our poverty?\n\nSIR CHARLES. Call me not brother, but imagine me\n\nSome barbarous outlaw, or uncivil kern;\n\nFor if thou shutt'st thy eye, and only hearest\n\nThe words that I shall utter, thou shalt judge me\n\nSome staring ruffian, not thy brother Charles.\n\nO sister!\u2014\u2014\n\nSUSAN. O brother, what doth this strange language mean?\n\nSIR CHARLES. Dost love me, sister? wouldst thou see me live\n\nA bankrupt beggar in the world's disgrace,\n\nAnd die indebted to my enemies?\n\nWouldst thou behold me stand like a huge beam\n\nIn the world's eye, a bye-word and a scorn?\n\nIt lies in thee of these to acquit me free,\n\nAnd all my debt I may out-strip by thee.\n\nSUSAN. By me! why, I have nothing, nothing left;\n\nI owe even for the clothes upon my back;\n\nI am not worth\u2014\u2014\n\nSIR CHARLES. O sister, say not so;\n\nIt lies in you my downcast state to raise,\n\nTo make me stand on even points with the world.\n\nCome, sister, you are rich; indeed you are;\n\nAnd in your power you have, without delay,\n\nActon's five hundred pound back to repay.\n\nSUSAN. Till now I had thought you had loved me. By my honour\n\n(Which I have kept as spotless as the moon),\n\nI ne'er was mistress of that single doit\n\nWhich I reserved not to supply your wants;\n\nAnd do you think that I would hoard from you?\n\nNow, by my hopes in Heaven, knew I the means\n\nTo buy you from the slavery of your debts\n\n(Especially from Acton, whom I hate),\n\nI would redeem it with my life or blood.\n\nSIR CHARLES. I challenge it; and, kindred set apart,\n\nThus, ruffian-like, I lay siege to your heart.\n\nWhat do I owe to Acton?\n\nSUSAN. Why some five hundred pounds; towards which, I swear,\n\nIn all the world I have not one denier.\n\nSIR CHARLES. It will not prove so. Sister, now resolve me:\n\nWhat do you think (and speak your conscience)\n\nWould Acton give, might he enjoy your bed?\n\nSUSAN. He would not shrink to spend a thousand pound,\n\nTo give the Mountfords' name so deep a wound.\n\nSIR CHARLES. A thousand pound! I but five hundred owe;\n\nGrant him your bed, he's paid with interest so.\n\nSUSAN. O brother!\n\nSIR CHARLES. O sister! only this one way,\n\nWith that rich jewel you my debts may pay.\n\nIn speaking this my cold heart shakes with shame;\n\nNor do I woo you in a brother's name,\n\nBut in a stranger's. Shall I die in debt\n\nTo Acton, my grand foe, and you still wear\n\nThe precious jewel that he holds so dear?\n\nSUSAN. My honour I esteem as dear and precious\n\nAs my redemption.\n\nSIR CHARLES. I esteem you, sister,\n\nAs dear, for so dear prizing it.\n\nSUSAN. Will Charles\n\nHave me cut off my hands, and send them Acton?\n\nRip up my breast, and with my bleeding heart\n\nPresent him as a token?\n\nSIR CHARLES. Neither, sister:\n\nBut hear me in my strange assertion.\n\nThy honour and my soul are equal in my regard;\n\nNor will thy brother Charles survive thy shame.\n\nHis kindness, like a burthen hath surcharged me,\n\nAnd under his good deeds I stooping go,\n\nNot with an upright soul. Had I remained\n\nIn prison still, there doubtless I had died:\n\nThen, unto him that freed me from that prison,\n\nStill do I owe this life. What moved my foe\n\nTo enfranchise me? 'Twas, sister, for your love,\n\nWith full five hundred pounds he bought your love,\n\nAnd shall he not enjoy it? Shall the weight\n\nOf all this heavy burthen lean on me,\n\nAnd will not you bear part? You did partake\n\nThe joy of my release; will you not stand\n\nIn joint-bond bound to satisfy the debt?\n\nShall I be only charged?\n\nSUSAN. But that I know\n\nThese arguments come from an honoured mind.\n\nAs in your most extremity of need\n\nScorning to stand in debt to one your hate,\u2014\n\nNay, rather would engage your unstained honour\n\nThan to be held ingrate,\u2014I should condemn you.\n\nI see your resolution, and assent;\n\nSo Charles will have me, and I am content.\n\nSIR CHARLES. For this I tricked you up.\n\nSUSAN. But here's a knife,\n\nTo save mine honour, shall slice out my life.\n\nSIR CHARLES. Ay! know thou pleasest me a thousand times\n\nMore in that resolution than thy grant.\u2014\n\nObserve her love; to soothe it to my suit,\n\nHer honour she will hazard, though not lose:\n\nTo bring me out of debt, her rigorous hand\n\nWill pierce her heart. O wonder! that will choose,\n\nRather than stain her blood, her life to lose.\u2014\n\nCome, you sad sister to a woful brother,\n\nThis is the gate: I'll bear him such a present,\n\nSuch an acquittance for the knight to seal,\n\nAs will amaze his senses, and surprise\n\nWith admiration all his fantasies.\n\nSUSAN. Before his unchaste thoughts shall seize on me,\n\n'Tis here shall my imprisoned soul set free.\n\n[ _Enter_ Sir FRANCIS ACTON _and_ MALBY.]\n\nSIR FRANCIS. How! Mountford with his sister, hand in hand!\n\nWhat miracle's afoot?\n\nMALBY. It is a sight\n\nBegets in me much admiration.\n\nSIR CHARLES. Stand not amazed to see me thus attended:\n\nActon, I owe thee money, and being unable\n\nTo bring thee the full sum in ready coin,\n\nLo! for thy more assurance, here's a pawn,\u2014\n\nMy sister, my dear sister, whose chaste honour\n\nI prize above a million: here, nay, take her;\n\nShe's worth your money, man; do not forsake her.\n\nSIR FRANCIS. I would he were in earnest!\n\nSUSAN. Impute it not to my immodesty:\n\nMy brother being rich in nothing else\n\nBut in his interest that he hath in me,\n\nAccording to his poverty hath brought you\n\nMe, all his store; whom howsoe'er you prize\n\nAs forfeit to your hand, he values highly,\n\nAnd would not sell, but to acquit your debt,\n\nFor any emperor's ransom.\n\nSIR FRANCIS. Stern heart, relent;\n\nThy former cruelty at length repent.\n\nWas ever known, in any former age,\n\nSuch honourable wrested courtesy?\n\nLands, honours, life, and all the world forego,\n\nRather than stand engaged to such a foe. [ _Aside._ ]\n\nSIR CHARLES. Acton, she is too poor to be thy bride,\n\nAnd I too much opposed to be thy brother.\n\nThere, take her to thee: if thou hast the heart\n\nTo seize her as a rape, or lustful prey;\n\nTo blur our house, that never yet was stained;\n\nTo murder her that never meant thee harm;\n\nTo kill me now, whom once thou savedst from death,\n\nDo them at once: on her all these rely,\n\nAnd perish with her spotted chastity.\n\nSIR FRANCIS. YOU overcome me in your love, Sir Charles;\n\nI cannot be so cruel to a lady\n\nI love so dearly. Since you have not spared\n\nTo engage your reputation to the world,\n\nYour sister's honour, which you prize so dear,\n\nNay, all the comforts which you hold on earth,\n\nTo grow out of my debt, being your foe,\n\nYour honoured thoughts, lo! thus I recompense:\n\nYour metamorphosed foe receives your gift\n\nIn satisfaction of all former wrongs.\n\nThis jewel I will wear here in my heart;\n\nAnd, where before I thought her for her wants\n\nToo base to be my bride, to end all strife,\n\nI seal you my dear brother, her my wife.\n\nSUSAN. You still exceed us: I will yield to fate,\n\nAnd learn to love, where I till now did hate.\n\nSIR CHARLES. With that enchantment you have charmed my soul,\n\nAnd made me rich even in those very words:\n\nI pay no debt, but am indebted more;\n\nRich in your love, I never can be poor.\n\nSIR FRANCIS. All's mine is yours; we are alike in state,\n\nLet's knit in love what was opposed in hate.\n\nCome! for our nuptials we will straight provide,\n\nBlest only in our brother and fair bride. [ _Exeunt_.]\n\n**S CENE II\u2014 _A room in_ FRANKFORD'S _house_** _._\n\n[ _Enter_ CRANWELL, FRANKFORD, _and_ NICHOLAS.]\n\nCRANWELL. Why do you search each room about your house.\n\nNow that you have despatched your wife away?\n\nFRANKFORD. O sir, to see that nothing may be left\n\nThat ever was my wife's. I loved her dearly,\n\nAnd when I do but think of her unkindness,\n\nMy thoughts are all in hell; to avoid which torment,\n\nI would not have a bodkin or a cuff,\n\nA bracelet, necklace, or rebato wire;\n\nNor any thing that ever was called hers,\n\nLeft me, by which I might remember her.\n\nSeek round about.\n\nNICHOLAS. 'Sblood, master! here's her lute flung in a corner.\n\nFRANKFORD. Her lute! O God! upon this instrument\n\nHer fingers have run quick division,\n\nSweeter than that which now divides our hearts.\n\nThese frets have made me pleasant, that have now\n\nFrets of ray heart-strings made. O Master Cranwell,\n\nOft hath she made this melancholy wood,\n\nNow mute and dumb for her disastrous chance,\n\nSpeak sweetly many a note, sound many a strain\n\nTo her own ravishing voice, which being well strung,\n\nWhat pleasant strange airs have they jointly rung!\n\nPost with it after her. Now nothing's left;\n\nOf her and hers, I am at once bereft.\n\nNICHOLAS. I'll ride and overtake her; do my message,\n\nAnd come back again. [ _Exit_.]\n\nCRANWELL. Mean time, sir, if you please,\n\nI'll to Sir Francis Acton, and inform him\n\nOf what hath passed betwixt you and his sister.\n\nFRANKFORD. Do as you please. How ill am I bested,\n\nTo be a widower ere my wife be dead! [ _Exeunt._ ]\n\n**S CENE III\u2014 _A country road._**\n\n[ _Enter_ MISTRESS FRANKFOLD, _with_ JENKIN, CICELY, _a_ COACHMAN, _and three_ CARTERS.]\n\nMISTRESS FRANKFORD. Bid my coach stay: why should I ride in state,\n\nBeing hurled so low down by the hand of fate?\n\nA seat like to my fortunes let me have;\n\nEarth for my chair, and for my bed a grave.\n\nJENKIN. Comfort, good mistress; you have watered your coach with tears already: you have but two mile now to go to your manor. A man cannot say by my old master Frankford as he may say by me, that he wants manors; for he hath three or four, of which this is one that we are going to now.\n\nCICELY. Good mistress, be of good cheer; sorrow, you see, hurts you, but helps you not: we all mourn to see you so sad.\n\nCARTER. Mistress, I spy one of my landlord's men\n\nCome riding post: 'tis like he brings some news.\n\nMISTRESS FRANKFORD. Comes he from Master Frankford, he is welcome;\n\nSo are his news because they come from him.\n\n[ _Enter_ NICHOLAS.]\n\nNICHOLAS [ _presenting lute_ ] _._ There.\n\nMISTRESS FRANKFORD. I know the lute; oft have I sung to thee: We both are out of tune, both out of time.\n\nNICHOLAS. Would that had been the worst instrument that e'er you played on. My master commends him to ye; there's all he can find that was ever yours: he hath nothing left that ever you could lay claim to but his own heart, and he could afford you that. All that I have to deliver you is this: he prays you to forget him, and so he bids you farewell.\n\nMISTRESS FRANKFORD. I thank him: he is kind, and ever was.\n\nAll you that have true feeling of my grief,\n\nThat know my loss, and have relenting hearts,\n\nGird me about, and help me with your tears\n\nTo wash my spotted sins: my lute shall groan;\n\nIt cannot weep, but shall lament my moan.\n\n[ _Enter_ WENDOLL.]\n\nWENDOLL. Pursued with horror of a guilty soul,\n\nAnd with the sharp scourge of repentance lashed\n\nI fly from my own shadow. O my stars!\n\nWhat have my parents in their lives deserved,\n\nThat you should lay this penance on their son?\n\nWhen I but think of Master Frankford's love,\n\nAnd lay it to my treason, or compare\n\nMy murdering him for his relieving me,\n\nIt strikes a terror like a lightning's flash\n\nTo scorch my blood up. Thus I, like the owl,\n\nAshamed of day, live in these shadowy woods,\n\nAfraid of every leaf or murmuring blast,\n\nYet longing to receive some perfect knowledge\n\nHow he hath dealt with her. [ _Sees_ MISTRESS FRANKFORD.] O my sad fate!\n\nHere, and so far from home, and thus attended!\n\nO God! I have divorced the truest turtles\n\nThat ever lived together; and, being divided\n\nIn several places, make their several moan;\n\nShe in the fields laments, and he at home.\n\nSo poets write that Orpheus made the trees\n\nAnd stones to dance to his melodious harp,\n\nMeaning the rustic and the barbarous hinds,\n\nThat had no understanding part in them:\n\nSo she from these rude carters tears extracts,\n\nMaking their flinty hearts with grief to rise,\n\nAnd draw down rivers from their rocky eyes.\n\nMISTRESS FRANKFORD [ _to_ NICHOLAS]. If you return unto your master, say\n\n(Though not from me; for I am all unworthy\n\nTo blast his name so with a strumpet's tongue)\n\nThat you have seen me weep, wish myself dead:\n\nNay, you may say too, for my vow is passed,\n\nLast night you saw me eat and drink my last.\n\nThis to your master you may say and swear;\n\nFor it is writ in Heaven, and decreed here.\n\nNICHOLAS. I'll say you wept: I'll swear you made me sad.\n\nWhy how now, eyes? what now? what's here to do?\n\nI'm gone, or I shall straight turn baby too.\n\nWENDOLL. I cannot weep, my heart is all on fire:\n\nCurst be the fruits of my unchaste desire!\n\nMISTRESS FRANKFORD. Go, break this lute upon my coach's wheel,\n\nAs the last music that I e'er shall make;\n\nNot as my husband's gift, but my farewell\n\nTo all earth's joy; and so your master tell.\n\nNICHOLAS. If I can for crying.\n\nWENDOLL. Grief, have done,\n\nOr like a madman I shall frantic run.\n\nMISTRESS FRANKFORD. You have beheld the wofullest wretch on earth;\n\nA woman made of tears: would you had words\n\nTo express but what you see! My inward grief\n\nNo tongue can utter; yet unto your power\n\nYou may describe my sorrow, and disclose\n\nTo thy sad master my abundant woes.\n\nNICHOLAS. I'll do your commendations.\n\nMISTRESS FRANKFORD. O no:\n\nI dare not so presume; nor to my children:\n\nI am disclaimed in both; alas, I am.\n\nOh, never teach them, when they come to speak,\n\nTo name the name of mother; chide their tongue,\n\nIf they by chance light on that hated word;\n\nTell them 'tis naught; for, when that word they name,\n\nPoor pretty souls! they harp on their own shame.\n\nWENDOLL. To recompense her wrongs, what canst thou do?\n\nThou hast made her husbandless and childless too.\n\nMISTRESS FRANKFORD. I have no more to say. Speak not for me;\n\nYet you may tell your master what you see.\n\nNICHOLAS. I'll do't. [ _Exit_.]\n\nWENDOLL. I'll speak to her, and comfort her in grief.\n\nOh! but her wound cannot be cured with words.\n\nNo matter though, I'll do my best goodwill\n\nTo work a cure on her whom I did kill.\n\nMISTRESS FRANKFORD. So, now unto my coach, then to my home,\n\nSo to my death-bed; for from this sad hour\n\nI never will nor eat, nor drink, nor taste\n\nOf any cates that may preserve my life:\n\nI never will nor smile, nor sleep, nor rest;\n\nBut when my tears have washed my black soul white,\n\nSweet Saviour, to Thy hands I yield my sprite.\n\nWENDOLL. O Mistress Frankford\u2014\n\nMISTRESS FRANKFORD. Oh, for God's sake fly!\n\nThe devil doth come to tempt me ere I die.\n\nMy coach! this fiend, that with an angel's face\n\nConjured mine honour, till he sought _my_ wrack,\n\nIn my repentant eyes seems ugly black.\n\n[ _Exeunt all, except_ WENDOLL _and_ JENKIN; _the_ CARTERS _whistling_.]\n\nJENKIN. What, my young master that fled in his shirt! How come you by your clothes again? You have made our house in a sweet pickle, ha' ye not, think you? What, shall I serve you still, or cleave to the old house?\n\nWENDOLL. Hence, slave! away with thy unseasoned mirth!\n\nUnless thou canst shed tears, and sigh, and howl,\n\nCurse thy sad fortunes, and exclaim on fate,\n\nThou art not for my turn.\n\nJENKIN. Marry, an you will not, another will: farewell, and be hanged! Would you had never come to have kept this coil within our doors; we shall ha' you run away like a sprite again.\n\n[ _Exit_.]\n\nWENDOLL. She's gone to death; I live to want and woe;\n\nHer life, her sins, and all upon my head.\n\nAnd I must now go wander, like a Cain,\n\nIn foreign countries and remoted climes,\n\nWhere the report of my ingratitude\n\nCannot be heard. I'll over first to France,\n\nAnd so to Germany and Italy;\n\nWhere when I have recovered, and by travel\n\nGotten those perfect tongues, and that these rumours\n\nMay in their height abate, I will return:\n\nAnd I divine (however now dejected)\n\nMy worth and parts being by some great man praised,\n\nAt my return I may in court be raised.\n\n[ _Exit._ ]\n\n**S CENE IV\u2014 _Before the manor._**\n\n[ _Enter_ SIR FRANCIS ACTON, SUSAN, SIR CHARLES MOUNTFORD, CRANWELL, _and_ MALBY.]\n\nSIR FRANCIS. Brother, and now my wife, I think these troubles\n\nFall on my head by justice of the Heavens,\n\nFor being so strict to you in your extremities:\n\nBut we are now atoned. I would my sister\n\nCould with like happiness o'ercome her griefs,\n\nAs we have ours.\n\nSUSAN. YOU tell us, Master Cranwell, wondrous things,\n\nTouching the patience of that gentleman,\n\nWith what strange virtue he demeans his grief.\n\nCRANWELL. told you what I was a witness of;\n\nIt was my fortune to lodge there that night.\n\nSIR FRANCIS. O that same villain Wendoll! 'twas his tongue\n\nThat did corrupt her; she was of herself\n\nChaste, and devoted well. Is this the house?\n\nCRANWELL. Yes, sir, I take it here your sister lies.\n\nSIR FRANCIS. My brother Frankford showed too mild a spirit\n\nIn the revenge of such a loath\u00e8d crime;\n\nLess than he did, no man of spirit could do:\n\nI am so far from blaming his revenge,\n\nThat I commend it. Had it been my case,\n\nTheir souls at once had from their breasts been freed:\n\nDeath to such deeds of shame is the due meed.\n\n[ _They enter the house._ ]\n\n**S CENE V\u2014 _A room in the manor._**\n\n[ _Enter_ SIR FRANCIS ACTON, SUSAN, SIR CHARLES MOUNTFORD, CRANWELL, _and_ MALBY; JENKIN _and_ CICELY _following them._ ]\n\nJENKIN. O my mistress, my mistress, my poor mistress.\n\nCICELY. Alas that ever I was born! what shall I do for my poor mistress?\n\nSIR CHARLES. Why, what of her?\n\nJENKIN. O Lord, sir, she no sooner heard that her brother and his friends were come to see how she did, but she, for very shame of her guilty conscience, fell into such a swoon, that we had much ado to get life into her.\n\nSUSAN. Alas that she should bear so hard a fate!\n\nPity it is repentance comes too late.\n\nSIR FRANCIS. Is she so weak in body?\n\nJENKIN. O sir, I can assure you there's no hope of life in her, for she will take no sustenance: she hath plainly starved herself, and now she is as lean as a lath. She ever looks for the good hour. Many gentlemen and gentlewomen of the country are come to comfort her. [ _Exeunt_.]\n\n**S CENE VI\u2014MISTRESS FRANKFORD'S _bed-chamber._**\n\n[MISTRESS FRANKFORD _in bed; enter_ SIR CHARLES MOUNTFORD, SIR FRANCIS ACTON, MALBY, CRANWELL, _and_ SUSAN.]\n\nMALBY. HOW fare you, Mistress Frankford?\n\nMISTRESS FRANKFORD. Sick, sick, oh, sick. Give me some air, I pray you.\n\nTell me, oh, tell me where is Master Frankford?\n\nWill not he deign to see me ere I die?\n\nMALBY. Yes, Mistress Frankford: divers gentlemen,\n\nYour loving neighbours, with that just request\n\nHave moved, and told him of your weak estate:\n\nWho, though with much ado to get belief,\n\nExamining of the general circumstance,\n\nSeeing your sorrow and your penitence,\n\nAnd hearing therewithal the great desire\n\nYou have to see him ere you left the world,\n\nHe gave to us his faith to follow us,\n\nAnd sure he will be here immediately.\n\nMISTRESS FRANKFORD. You have half revived me with those pleasing news:\n\nRaise me a little higher in my bed.\n\nBlush I not, brother Acton? Blush I not, Sir Charles?\n\nCan you not read my fault writ in my cheek?\n\nIs not _my_ crime there? tell me, gentlemen.\n\nSIR CHARLES. Alas! good mistress, sickness hath not left you\n\nBlood in your face enough to make you blush.\n\nMISTRESS FRANKFORD. Then sickness, like a friend, my fault would hide.\n\nIs my husband come? My soul but tarries\n\nHis arrive, then I am fit for Heaven.\n\nSIR FRANCIS. I came to chide you; but my words of hate\n\nAre turned to pity and compassionate grief.\n\nI came to rate you; but my brawls, you see,\n\nMelt into tears, and I must weep by thee,\n\nHere's Master Frankford now.\n\n[ _Enter_ FRANKFORD.]\n\nFRANKFORD. Good-morrow, brother; morrow, gentlemen:\n\nGod, that hath laid this cross upon our heads,\n\nMight (had He pleased) have made our cause of meeting\n\nOn a more fair and more contented ground;\n\nBut He that made us, made us to this woe.\n\nMistress Frankford. And is he come?\n\nMethinks that voice I know.\n\nFRANKFORD. How do you, woman?\n\nMISTRESS FRANKFORD. Well, Master Frankford, well; but shall be better,\n\nI hope, within this hour. Will you vouch-safe,\n\nOut of your grace and your humanity,\n\nTo take a spotted strumpet by the hand?\n\nFRANKFORD. This hand once held my heart in faster bonds\n\nThan now 'tis gripped by me. God pardon them\n\nThat made us first break hold!\n\nMISTRESS FRANKFORD. Amen, amen.\n\nOut of my zeal to Heaven, whither I'm now bound,\n\nI was so impudent to wish you here;\n\nAnd once more beg your pardon. O good man,\n\nAnd father to my children, pardon me,\n\nPardon, oh, pardon me! My fault so heinous is,\n\nThat if you in this world forgive it not,\n\nHeaven will not clear it in the world to come.\n\nFaintness hath so usurped upon my knees\n\nThat kneel I cannot, but on my heart's knees\n\nMy prostrate soul lies thrown down at your feet\n\nTo beg your gracious pardon. Pardon, oh, pardon me!\n\nFRANKFORD. As freely, from the low depth of my soul,\n\nAs my Redeemer hath forgiven His death,\n\nI pardon thee. I will shed tears for thee, pray with thee;\n\nAnd, in mere pity of thy weak estate,\n\nI'll wish to die with thee.\n\nALL. So do we all.\n\nNICHOLAS. So will not I;\n\nI'll sigh and sob, but, by my faith, not die.\n\nSIR FRANCIS. O Master Frankford, all the near alliance\n\nI lose by her shall be supplied in thee:\n\nYou are my brother by the nearest way;\n\nHer kindred hath fallen off, but yours doth stay.\n\nFRANKFORD. Even as I hope for pardon at that day\n\nWhen the great Judge of Heaven in scarlet sits,\n\nSo be thou pardoned. Though thy rash offence\n\nDivorced our bodies, thy repentant tears\n\nUnite our souls.\n\nSIR CHARLES. Then comfort, Mistress Frankford;\n\nYou see your husband hath forgiven your fall;\n\nThen rouse your spirits, and cheer your fainting soul.\n\nSUSAN. How is it with you?\n\nSIR FRANCIS. How do ye feel yourself?\n\nMISTRESS FRANKFORD. Not of this world.\n\nFRANKFORD. I see you are not, and I weep to see it.\n\nMy wife, the mother to my pretty babes!\n\nBoth those lost names I do restore thee back,\n\nAnd with this kiss I wed thee once again:\n\nThough thou art wounded in thy honoured name,\n\nAnd with that grief upon thy death-bed liest,\n\nHonest in heart, upon my soul, thou diest.\n\nMISTRESS FRANKFORD. Pardoned on earth, soul, thou in Heaven art free\n\nOnce more: thy wife dies thus embracing thee. [ _Dies_.]\n\nFRANKFORD. New married, and new widowed. Oh! she's dead,\n\nAnd a cold grave must be her nuptial bed.\n\nSIR CHARLES. Sir, be of good comfort; and your heavy sorrow\n\nPart equally amongst us: storms divided\n\nAbate their force, and with less rage are guided.\n\nCRANWELL. Do, Master Frankford: he that hath least part\n\nWill find enough to drown one troubled heart.\n\nSIR FRANCIS. Peace with thee, Nan. Brothers, and gentlemen,\n\nAll we that can plead interest in her grief,\n\nBestow upon her body funeral tears.\n\nBrother, had you with threats and usage bad\n\nPunished her sin, the grief of her offence\n\nHad not with such true sorrow touched her heart.\n\nFRANKFORD. I see it had not: therefore on her grave\n\nWill I bestow this funeral epitaph,\n\nWhich on her marble tomb shall be engraved.\n\nIn golden letters shall these words be filled,\n\n\"Here lies she whom her husband's kindness killed.\"\n**THE TRAGEDIE OF MARIAM**\n\n**Elizabeth Cary**\n\n**TO DIANAES EARTHLIE DEPVTESSE, _and my worthy Sister, Mistris Elizabeth Carye._**\n\nWhen cheerful _Ph\u0153bus_ his full course hath run,\n\nHis sisters fainter beams our harts doth cheere:\n\nSo your faire Brother is to mee the Sunne,\n\nAnd you his Sister as my Moone appeere.\n\nYou are my next belou'd, my second Friend,\n\nFor when my _Ph\u0153bus_ absence makes it Night,\n\nWhilst to th' _Antipodes_ his beames do bend,\n\nFrom you my _Ph\u0153be,_ shines my second Light.\n\nHee like to _SOL,_ cleare-fighted, constant, free,\n\nYou _LV N_ A-like, vnspotted, chast, diuine:\n\nHee shone on _Sicily,_ you destin'd bee,\n\nT'illumine the now obscurde _Palestine._\n\nMy first was consecrated to _Apollo,_\n\nMy second to _DIANA_ now shall follow.\n\nE. C.\n\n**THE NAMES OF THE SPEAKERS.**\n\n_Herod, King of Iudea._\n\n_Doris, his first Wife._\n\n_Mariam, his second Wife._\n\n_Salome, Herods Sister._\n\n_Antipater his sonne by Salome._\n\n_Alexandra, Mariams mother._\n\n_Sillius, Prince of Arabia._\n\n_Constabarus, husband to Salome._\n\n_Ph\u0153roras, Herods Brother._\n\n_Graphina, his Loue._\n\n_Babus first Sonne._\n\n_Babus second Sonne._\n\n_Annanell, the high Priest._\n\n_Sohemus, a Counsellor to Herod._\n\n_Nuntio._\n\n_Bu. another Messenger._\n\n_Chorus, a Companie of Iewes._\n\n**ACTUS PRIMUS.**\n\n**S C\u0152NA PRIMA.**\n\n_Mariam sola._\n\nHow oft haue I with publike voyce runne on?\n\nTo censure _Romes_ last _Hero_ for deceit:\n\nBecause he wept when _Pompeis_ life was gone,\n\nYet when he liu'd, hee thought his Name too great.\n\nBut now I doe recant, and _Roman_ Lord\n\nExcufe too rash a judgement in a woman:\n\nMy Sexe pleads pardon, pardon then afford,\n\nMistaking is with vs, but too too common.\n\nNow doe I finde by selfe Experience taught,\n\nOne Object yeelds both griefe and ioy:\n\nYou wept indeed, when on his worth you thought,\n\nBut ioyd that slaughter did your Foe destroy.\n\nSo at his death your Eyes true droppes did raine,\n\nWhom dead, you did not wish aliue againe.\n\nWhen _Herod_ liu'd, that now is done to death,\n\nOft haue I wisht that I from him were free:\n\nOft haue I wisht that he might lose his breath,\n\nOft haue I wisht his Carkas dead to see.\n\nThen Rage and Scorne had put my loue to flight,\n\nThat Loue which once on him was firmely set:\n\nHate hid his true affection from my sight,\n\nAnd kept my heart from paying him his debt.\n\nAnd blame me not, for _Herods_ Iealousie\n\nHad power euen constancie it selfe to change:\n\nFor hee by barring me from libertie,\n\nTo shunne my ranging, taught me first to range.\n\nBut yet too chast a Scholler was my hart,\n\nTo learne to loue another then my Lord:\n\nTo leaue his Loue, my lessons former part,\n\nI quickly learn'd, the other I abhord.\n\nBut now his death to memorie doth call,\n\nThe tender loue, that he to _Mariam_ bare:\n\nAnd mine to him, this makes those riuers fall.\n\nWhich by an other thought vnmoistned are.\n\nFor _Ariftobolus_ the lowlyest youth\n\nThat euer did in Angels shape appeare:\n\nThe cruell _Herod_ was not mou'd to ruth,\n\nThen why grieues _Mariam Herods_ death to heare?\n\nWhy ioy I not the tongue no more shall speake,\n\nThat yeelded forth my brothers latest dome:\n\nBoth youth and beautie might thy furie breake,\n\nAnd both in him did ill befit a Tombe.\n\nAnd worthy Grandsire ill did he requite,\n\nHis high Affent alone by thee procur'd,\n\nExcept he murdred thee to free the spright\n\nWhich still he thought on earth too long immur'd.\n\nHow happie was it that _Sohemus_ maide\n\nWas mou'd to pittie my distrest estate?\n\nMight _Herods_ life a trustie seruant finde,\n\nMy death to his had bene vnseparate. (beare,\n\nThese thoughts haue power, his death to make me\n\nNay more, to wish the newes may firmely hold:\n\nYet cannot this repulse some falling teare,\n\nThat will against my will some griefe vnfold.\n\nAnd more I owe him for his loue to me,\n\nThe deepest loue that euer yet was seene:\n\nYet had I rather much a milke-maide bee,\n\nThen be the Monarke of _Iudeas_ Queene.\n\nIt was for nought but loue, he wisht his end\n\nMight to my death, but the vaunt-currier proue:\n\nBut I had rather still be foe then friend,\n\nTo him that saues for hate, and kills for loue.\n\nHard-hearted _Mariam,_ at thy discontent,\n\nWhat flouds of teares haue drencht his manly face?\n\nHow canst thou then so faintly now lament,\n\nThy truest louers death, a deaths disgrace:\n\nI now mine eyes you do begin to right\n\nThe wrongs of your admirer. And my Lord,\n\nLong since you should haue put your smiles to flight,\n\nIll doth a widowed eye with ioy accord.\n\nWhy now me thinkes the loue I bare him then,\n\nWhen virgin freedome left me vnrestraind:\n\nDoth to my heart begin to creepe agen,\n\nMy passion now is far from being faind.\n\nBut teares flie backe, and hide you in your bankes,\n\nYou must not be to _Alexandra_ seene:\n\nFor if my mone be spide, but little thankes\n\nShall _Mariam_ haue, from that incensed Queene.\n\n**A CTUS PRIMUS: SC\u0152NA SECUNDA.**\n\n_Mariam. Alexandra._\n\n_Alex:_ (mistake,\n\nWhat meanes these teares? my _Mariam_ doth\n\nThe newes we heard did tell the _Tyrants_ end:\n\nWhat weepst thou for, thy brothers murthers sake,\n\nWill euer wight a teare for _Herod_ spend?\n\nMy curie pursue his breathles trunke and spirit,\n\nBase _Edomite_ the damned _Esaus_ heire:\n\nMust he ere _Iacobs_ child the crowne inherit?\n\nMust he vile wretch be set in _Dauids_ chaire?\n\nNo _Dauids_ soule within the bosome plac'te,\n\nOf our forefather _Abram_ was asham'd:\n\nTo see his seat with such a toade disgrac'te,\n\nThat seat that hath by _Iudas_ race bene fain'd.\n\nThou fatall enemie to royall blood,\n\nDid not the murther of my boy suffice,\n\nTo stop thy cruell mouth that gaping stood?\n\nBut must thou dim the milde _Hercanus_ eyes?\n\nMy gratious father, whose too readie hand\n\nDid lift this _Idumean_ from the dust:\n\nAnd he vngratefull catiffe did withstand,\n\nThe man that did in him most friendly trust.\n\nWhat kingdomes right could cruell _Herod_ claime,\n\nWas he not _Esaus_ Issue, heyre of hell?\n\nThen what succession can he haue but shame?\n\nDid not his Ancestor his birth-right sell?\n\nO yes, he doth from _Edoms_ name deriue,\n\nHis cruell nature which with blood is fed:\n\nThat made him me of Sire and sonne depriue,\n\nHe euer thirsts for blood, and blood is red.\n\nWeepst thou because his loue to thee was bent?\n\nAnd readst thou loue in crimson caracters?\n\nSlew he thy friends to worke thy hearts content?\n\nNo: hate may Iustly call that action hers.\n\nHe gaue the sacred Priesthood for thy sake,\n\nTo _Aristobolus._ Yet doomde him dead:\n\nBefore his backe the _Ephod_ warme could make,\n\nAnd ere the _Myter_ setled on his head:\n\nOh had he giuen my boy no lesse then right,\n\nThe double oyle should to his forehead bring:\n\nA double honour, shining doubly bright,\n\nHis birth annoynted him both Priest and King.\n\nAnd say my father, and my sonne he slewe,\n\nTo royalize by right your Prince borne breath:\n\nWas loue the cause, can _Mariam_ deeme it true,\n\nThat _Mariam_ gaue commandment for her death?\n\nI know by fits, he shewd some signes of loue,\n\nAnd yet not loue, but raging lunacie:\n\nAnd this his hate to thee may iustly proue,\n\nThat sure he hates _Hercanus_ familie.\n\nWho knowes if he vnconstant wauering Lord,\n\nHis loue to _Doris_ had renew'd againe?\n\nAnd that he might his bed to her afford,\n\nPerchance he wisht that _Mariam_ might be slaine,\n\n_Nun: Doris,_ Alas her time of loue was past,\n\nThose coales were rakte in embers long agoe:\n\nIf _Mariams_ loue and she was now disgrast,\n\nNor did I glorie in her ouerthrowe.\n\nHe not a whit his first borne sonne esteem'd,\n\nBecause as well as his he was not mine:\n\nMy children onely for his owne he deem'd,\n\nThese boyes that did descend from royall line.\n\nThese did he stile his heyres to _Dauids_ throne,\n\nMy _Alexander_ if he liue, shall sit\n\nIn the Maiesticke seat of _Salamon_ ,\n\nTo will it so, did _Herod_ thinke it fit.\n\n_Alex._ Why? who can claime from _Alexanders_ brood\n\nThat Gold adorned Lyon-guarded Chaire?\n\nWas _Alexander_ not of _Dauids_ blood?\n\nAnd was not _Mariam Alexanders_ heire?\n\nWhat more then right could _Herod_ then bestow,\n\nAnd who will thinke except for more then right,\n\nHe did not raise them, for they were not low,\n\nBut borne to weare the Crowne in his despight:\n\nThen send those teares away that are not sent\n\nTo thee by reason, but by passions power:\n\nThine eyes to cheere, thy cheekes to smiles be bent,\n\nAnd entertaine with ioy this happy houre.\n\nFelicitie, if when shee comes, she findes\n\nA mourning habite, and a cheerlesse looke,\n\nWill thinke she is not welcome to thy minde,\n\nAnd so perchance her lodging will not brooke.\n\nOh keepe her whilest thou hast her, if she goe\n\nShe will not easily returne againe:\n\nFull many a yeere haue I indur'd in woe,\n\nYet still haue sude her presence to obtaine:\n\nAnd did not I to her as presents send\n\nA Table, that best Art did beautifie\n\nOf two, to whom Heauen did best feature lend,\n\nTo woe her loue by winning _Anthony_ :\n\nFor when a Princes fauour we doe craue,\n\nWe first their Mynions loues do seeke to winne:\n\nSo I, that sought Felicitie to haue,\n\nDid with her Mynion _Anthony_ beginne,\n\nWith double slight I sought to captiuate\n\nThe warlike louer, but I did not right:\n\nFor if my gift had borne but halfe the rate,\n\nThe _Roman_ had beene ouer-taken quite.\n\nBut now he fared like a hungry guest,\n\nThat to some plenteous festiuall is gone,\n\nNow this, now that, hee deems to eate were best,\n\nSuch choice doth make him let them all alone.\n\nThe boyes large forehead first did fayrest seeme,\n\nThen glaunst his eye vpon my _Mariams_ cheeke:\n\nAnd that without companion did deeme,\n\nWhat was in eyther but he most did leeke.\n\nAnd thus distracted, eythers beauties might\n\nWithin the others excellence was drown'd:\n\nToo much delight did bare him from delight,\n\nFor eithers loue, the others did confound.\n\nWhere if thy portraiture had onely gone,\n\nHis life from _Herod_ , _Anthony_ had taken:\n\nHe would haue loued thee, and thee alone,\n\nAnd left the browne _Egyptian_ cleane forsaken.\n\nAnd _Cleopatra_ then to seeke had bene,\n\nSo firme a louer of her wayned face:\n\nThen great _Anthonius_ fall we had not seene,\n\nBy her that fled to haue him holde the chase.\n\nThen _Mariam_ in a _Romans_ Chariot set,\n\nIn place of _Cleopatra_ might haue showne;\n\nA mart of Beauties in her visage met,\n\nAnd part in this, that they were all her owne.\n\n_Ma._ Not to be Emprise of aspiring _Rome_ ,\n\nWould _Mariam_ like to _Cleopatra_ liue:\n\nWith purest body will I presse my Toome,\n\nAnd wish no fauours _Anthony_ could giue.\n\n_Alex._ Let vs retire vs, that we may resolue\n\nHow now to deale in this reuerfed ftate:\n\nGreat are th'affaires that we must now reuolue,\n\nAnd great affaires must not be taken late.\n\n**A CTUS PRIMUS. SC\u0152NA TERTIA .**\n\n_Mariam. Alexandra. Salome._\n\n_Salome._\n\nMore plotting yet? Why? now you haue the thing\n\nFor which so oft you spent your supliant breath:\n\nAnd _Mariam_ hopes to haue another King,\n\nHer eyes doe sparkle ioy for _Herods_ death.\n\n_Alex._ If she desir'd another King to haue,\n\nShe might before she came in _Herods_ bed\n\nHaue had her wish. More Kings then one did craue,\n\nFor leaue to set a Crowne vpon her head.\n\nI thinke with more then reason she laments,\n\nThat she is freed from such a sad annoy:\n\nWho ist will weepe to part from discontent,\n\nAnd if she ioy, she did not causelesse ioy.\n\n_Sal._ You durst not thus haue giuen your tongue the\n\nIf noble _Herod_ still remaind in life: (raine,\n\nYour daughters betters farre I dare maintaine,\n\nMight haue reioyc'd to be my brothers wife.\n\n_Mar._ My betters farre, bafe woman t'is vntrue,\n\nYou scarce haue euer my superiors seene:\n\nFor _Mariams_ seruants were as good as you,\n\nBefore she came to be _Iudeas_ Queene.\n\n_Sal._ Now stirs the tongue that is so quickly mou'd,\n\nBut more then once your collor haue I borne:\n\nYour fumish words are sooner sayd then prou'd,\n\nAnd _Salomes_ reply is onely scorne.\n\n_Mar._ Scorne those that are for thy companions\n\nThough I thy brothers face had neuer seene, (held,\n\nMy birth, thy baser birth so farre exceld,\n\nI had to both of you the Princesse bene.\n\nThou party Iew, and party Edomite,\n\nThou Mongrell: issu'd from reiected race,\n\nThy Ancestors against the Heauens did fight,\n\nAnd thou like them wilt heauenly birth disgrace.\n\n_Sal._ Still twit you me with nothing but my birth,\n\nWhat ods betwixt your ancestors and mine?\n\nBoth borne of _Adam,_ both were made of Earth,\n\nAnd both did come from holy _Abrahams_ line.\n\n_Mar._ I fauour thee when nothing else I say,\n\nWith thy blacke acts ile not pollute my breath:\n\nElse to thy charge I might full iustly lay\n\nA shamefull life, besides a husbands death.\n\n_Sal._ Tis true indeed, I did the plots reueale,\n\nThat past betwixt your fauorites and you:\n\nI ment not I, a traytor to conceale.\n\nThus _Salome_ your Mynion _Ioseph_ slue.\n\n_Mar._ Heauen, dost thou meane this Infamy to smo-\n\nLet flandred _Mariam_ ope thy closed eare: (ther?\n\nSelfe-guilt hath euer bene suspitious mother,\n\nAnd therefore I this speech with patience beare.\n\nNo, had not _Salomes_ vnstedfast heart,\n\nIn _Iofepbus_ stead her _Constabarus_ plast,\n\nTo free her selfe, she had not vsde the art,\n\nTo slander haplesse _Mariam_ for vnchast..\n\n_Alex._ Come _Mariam,_ let vs goe: it is no boote\n\nTo let the head contend against the foote.\n\n**A CTUS PRIMUS. SC\u0152NA QUARTA.**\n\n_Salome, Sola._\n\nLiues _Salome,_ to get to base a stile\n\nAs foote, to the proud _Mariam Herods_ spirit:\n\nIn happy time for her endured exile,\n\nFor did he liue she should not misse her merit:\n\nBut he is dead: and though he were my Brother,\n\nHis death such store of Cinders cannot cast\n\nMy Coales of loue to quench: for though they smo-\n\nThe flames a while, yet will they out at last. (ther\n\nOh blest _Arabia,_ in best climate plast,\n\nI by the Fruit will censure of the Tree:\n\nTis not in vaine, thy happy name thou hast,\n\nIf all _Arabians_ like _Silleus_ bee:\n\nHad not my Fate bene too too contrary,\n\nWhen I on _Constabarus_ first did gaze,\n\n_Silleus_ had beene obiect to mine eye:\n\nWhose lookes and personage must allyes amaze.\n\nBut now ill Fated _Salome,_ thy tongue\n\nTo _Constabarus_ by it selfe is tide:\n\nAnd now except I doe the Ebrew wrong\n\nI cannot be the faire _Arabian_ Bride:\n\nWhat childish _lets_ are these? Why stand I now\n\nOn honourable points? Tis long agoe\n\nSince shame was written on my tainted brow:\n\nAnd certaine tis, that shame is honours foe.\n\nHad I vpon my reputation stood,\n\nHad I affected an vnspotted life,\n\n_Iosephus_ vaines had still bene stuft with blood,\n\nAnd I to him had liu'd a sober wife.\n\nThen had I neuer cast an eye of loue,\n\nOn _Constabarus_ now detested face,\n\nThen had I kept my thoughts without remoue:\n\nAnd blusht at motion of the least disgrace:\n\nBut shame is gone, and honour wipt away,\n\nAnd Impudencie on my forehead sits:\n\nShe bids me worke my will without delay,\n\nAnd for my will I will imploy my wits.\n\nHe loues, I loue; what then can be the cause,\n\nKeepes me for being the _Arabians_ wife?\n\nIt is the principles of _Moses_ lawes,\n\nFor _Constabarus_ still remaines in life,\n\nIf he to me did beare as Earnest hate,\n\nAs I to him, for him there were an ease,\n\nA separating bill might free his fate:\n\nFrom such a yoke that did so much displease.\n\nWhy should such priuiledge to man be giuen?\n\nOr giuen to them, why bard from women then?\n\nAre men then we in greater grace with Heauen?\n\nOr cannot women hate as well as men?\n\nIle be the custome-breaker: and beginne\n\nTo shew my Sexe the way to freedomes doore,\n\nAnd with an offring will I purge my sinne,\n\nThe lawe was made for none but who are poore.\n\nIf _Herod_ had liu'd, I might to him accuse\n\nMy present Lord. But for the futures sake\n\nThen would I tell the King he did refuse\n\nThe sonnes of _Baba_ in his power to take.\n\nBut now I must diuorse him from my bed,\n\nThat my _Silleus_ may possesse his roome:\n\nHad I not begd his life he had bene dead,\n\nI curse my tongue the hindrer of his doome,\n\nBut then my wandring heart to him was fast,\n\nNor did I dreame of chaunge: _Silleus_ said,\n\nHe would be here, and see he comes at last,\n\nHad I not nam'd him longer had he staid.\n\n**A CTUS PRIMUS. SC\u0152NA QUINTA .**\n\n_Salome, Silleus._\n\n_Silleus._ Well found faire _Salome Iud_ \u00e6 _as_ pride,\n\nHath thy innated wisedome found\n\nTo make _Silleus_ deeme him deified, (the way\n\nBy gaining thee a more then precious pray?\n\n_Salo._ I haue deuisde the best I can deuise,\n\nA more imperfect meanes was neuer found:\n\nBut what cares _Salome,_ it doth suffice\n\nIf our indeuours with their end be crown'd.\n\nIn this our land we haue an ancient vse,\n\nPermitted first by our law-giuers head:\n\nWho hates his wife, though for no iust abuse,\n\nMay with a bill diuorce her from his bed.\n\nBut in this custome women are not free,\n\nYet I for once will wrest it, blame not thou\n\nThe ill I doe, since what I do'es for thee,\n\nThough others blame, _Silleus_ should allow.\n\n_Solleus._ Thinkes _Salome, Silleus_ hath a tongue\n\nTo censure her faire actions: let my blood\n\nBedash my proper brow, for such a wrong,\n\nThe being yours, can make euen vices good:\n\n_Arabia_ ioy, prepare thy earth with greene,\n\nThou neuer happie wert indeed till now:\n\nNow shall thy ground be trod by beauties Queene,\n\nHer foote is destin'd to depresse thy brow.\n\nThou shalt faire _Salome_ commaund as much\n\nAs if the royall ornament were thine:\n\nThe weaknes of _Arabias_ King is such,\n\nThe kingdome is not his so much as mine.\n\nMy mouth is our _Obodas_ oracle,\n\nWho thinkes not ought but what _Silleus_ will?\n\nAnd thou rare creature. _Asias_ miracle,\n\nShalt be to me as It: _Obodas_ still.\n\n_Salome._ Tis not for glory I thy loue accept,\n\n_Iudea_ yeelds me honours worthy store:\n\nHad not affection in my bosome crept,\n\nMy natiue country should my life deplore.\n\nWere not _Silleus_ he with home I goe,\n\nI would not change my _Palastine_ for _Rome:_\n\nMuch lesse would I a glorious state to shew,\n\nGoe far to purchase an _Arabian_ toome.\n\n_Silleus._ Far be it from _Silleus_ so to thinke,\n\nI know it is thy gratitude requites\n\nThe loue that is in me, and shall not shrinke\n\nTill death doe seuer me from earths delights. (talke,\n\n_Salom._ But whist; me thinkes the wolfe is in our\n\nBe gone _Silleus,_ who doth here arriue?\n\nTis _Constabarus_ that doth hither walke,\n\nIle find a quarrell, him from me to driue.\n\n_Sille._ Farewell, but were it not for thy commaund,\n\nIn his despight _Silleus_ here would stand.\n\n**A CTUS PRIMUS: SC\u0152NA SEXTA.**\n\n_Salome: Constabarus._\n\n_Const:_ Oh _Salome,_ how much you wrong your name,\n\nYour race, your country, and your husband\n\nA straungers priuate conference is shame, (most?\n\nI blush for you, that haue your blushing lost.\n\nOft haue I found, and found you to my griefe,\n\nConsorted with this base _Arabian_ heere;\n\nHeauen knowes that you haue bin my comfort chiefe,\n\nThen doe not now my greater plague appeare.\n\nNow by the stately Carued edifice\n\nThat on Mount _Sion_ makes so faire a show,\n\nAnd by the Altar fit for sacrifice,\n\nI loue thee more then thou thy selfe doest know.\n\nOft with a silent sorrow haue I heard\n\nHow ill _Iudeas_ mouth doth censure thee:\n\nAnd did I not thine honour much regard,\n\nThou shouldst not be exhorted thus for mee.\n\nDidst thou but know the worth of honest fame,\n\nHow much a vertuous woman is esteem'd,\n\nThou wouldest like hell eschew deserued shame,\n\nAnd seeke to be both chast and chastly deem'd.\n\nOur wisest Prince did say, and true he said,\n\nA vertuous woman crownes her husbands head.\n\n_Salome._ Did I for this, vpreare thy lowe estate?\n\nDid I for this requitall begge thy life,\n\nThat thou hadst forfeited haples fate?\n\nTo be to such a thankles wretch the wife.\n\nThis hand of mine hath lifted vp thy head,\n\nWhich many a day agoe had falne full lowe,\n\nBecause the sonnes of _Baba_ are not dead,\n\nTo me thou doest both life and fortune owe.\n\n_Const._ You haue my patience often exercisde,\n\nVse make my choller keepe within the bankes:\n\nYet boast no more, but be by me aduisde.\n\nA benefit vpbraided, forfeits thankes:\n\nI prethy _Salome_ dismisse this mood,\n\nThou doest not know how ill it fits thy place:\n\nMy words were all intended for thy good,\n\nTo raise thine honour and to stop disgrace.\n\n_Sa._ To stop disgrace? take thou no care for mee,\n\nNay do thy worst, thy worst I fet not by:\n\nNo shame of mine is like to light on thee,\n\nThy loue and admonitions I defie.\n\nThou shalt no hower longer call me wife,\n\nThy Iealousie procures my hate so deepe:\n\nThat I from thee doe meane to free my life,\n\nBy a diuorcing bill before I sleepe.\n\n_Const._ Are Hebrew women now tr\u00e3sform'd to men?\n\nWhy do you not as well our battels fight,\n\nAnd weare our armour? suffer this, and then\n\nLet all the world be topsie turued quite.\n\nLet fishes graze, beastes, swine, and birds descend,\n\nLet fire burne downewards whilst the earth aspires:\n\nLet Winters heat and Summers cold offend,\n\nLet Thiftels growe on Vines, and Grapes on Briers,\n\nSet vs to Spinne or Sowe, or at the best\n\nMake vs Wood-hewers, Waters-bearing wights:\n\nFor sacred seruice let vs take no rest,\n\nVse vs as _Ioshua_ did the _Gibonites._\n\n_Salom._ Hold on your talke, till it be time to end,\n\nFor me I am resolu'd it shall be so:\n\nThough I be first that to this course do bend,\n\nI shall not be the last full well I know.\n\n_Const._ Why then be witnesse Heau'n, the Iudge of\n\nBe witnesse Spirits that eschew the darke: (sinnes,\n\nBe witnesse Angels, witnesse Cherubins,\n\nWhose semblance sits vpon the holy Arke:\n\nBe witnesse earth, be witnesse _Palestine,_\n\nBe witnesse _Dauids_ Citie, if my heart\n\nDid euer merit such an act of thine:\n\nOr if the fault be mine that makes vs part,\n\nSince mildest _Moses_ friend vnto the Lord,\n\nDid worke his wonders in the land of _Ham_ ,\n\nAnd slew the first-borne Babes without a sword,\n\nIn signe whereof we eate the holy Lambe:\n\nTill now that foureteene hundred yeeres are past,\n\nSince first the Law with vs hath beene in force:\n\nYou are the first, and will I hope, be last,\n\nThat euer sought her husband to diuorce.\n\n_Salom._ I meane not to be led by president,\n\nMy will shall be to me in stead of Law.\n\n_Const._ I feare me much you will too late repent,\n\nThat you haue euer liu'd so void of awe:\n\nThis is _Silleus_ loue that makes you thus\n\nReuerse all order; you must next be his.\n\nBut if my thoughts aright the cause discusse,\n\nIn winning you, he gaines no lasting blisse,\n\nI was _Silleus_ , and not long agoe\n\n_Iosephus_ then was _Constabarus_ now:\n\nWhen you became my friend you prou'd his foe,\n\nAs now for him you breake to me your vowd.\n\n_Sal._ If once I lou'd you, greater is your debt:\n\nFor certaine tis that you deserued it not.\n\nAnd vndeserued loue we soone forget,\n\nAnd therefore that to me can be no blot\n\nBut now fare ill my once beloued Lord,\n\nYet neuer more belou'd then now abhord.\n\n_Const._ Yet _Constabarus_ biddeth thee farewell.\n\nFarewell light creature. Heauen forgiue thy sinne:\n\nMy prophecying spirit doth foretell\n\nThy wauering thoughts doe yet but new beginne.\n\nYet I haue better scap'd then _Ioseph_ did,\n\nBut if our _Herods_ death had bene delayd,\n\nThe valiant youths that I so long haue hid,\n\nHad bene by her, and I for them betrayd.\n\nTherefore in happy houre did _C\u00e6sar_ giue\n\nThe fatall blow to wanton _Anthony_ :\n\nFor had he liued, our _Herod_ then should liue,\n\nBut great _Anthonius_ death made _Herod_ dye.\n\nHad he enioyed his breath, not I alone\n\nHad beene in danger of a deadly fall:\n\nBut _Mariam_ had the way of perill gone,\n\nThough by the Tyrant most belou'd of all.\n\nThe sweet fac'd _Mariam_ as free from guilt\n\nAs Heauen from spots, yet had her Lord come backe\n\nHer purest blood had bene vniustly spilt.\n\nAnd _Salome_ it was would worke her wracke.\n\nThough all _Iudea_ yeeld her innocent,\n\nShe often hath bene neere to punishment.\n\n_Chorus._\n\nThose mindes that wholy dote vpon delight,\n\nExcept they onely ioy in inward good:\n\nStill hope at last to hop vpon the right,\n\nAnd so from Sand they leape in loathsome mud.\n\nFond wretches, seeking what they cannot finde,\n\nFor no content attends a wauering minde.\n\nIf wealth they doe desire, and wealth attaine,\n\nThen wondrous faine would they to honor lep:\n\nOf meane degree they doe in honor gaine,\n\nThey would but wish a little higher step.\n\nThus step to step, and wealth to wealth they ad,\n\nYet cannot all their plenty make them glad.\n\nYet oft we see that some in humble state,\n\nAre chreefull, pleasant, happy, and content:\n\nWhen those indeed that are of higher state,\n\nWith vaine additions do their thoughts torment,\n\nTh'one would to his minde his fortune binde,\n\nT'hother to his fortune frames his minde.\n\nTo wish varietie is signe of griefe,\n\nFor if you like your state as now it is,\n\nWhy should an alteration bring reliefe?\n\nNay change would then be fear'd as losse of blis.\n\nThat man is onely happy in his Fate,\n\nThat is delighted in a setled state.\n\nStill _Mariam_ wisht she from her Lord were free,\n\nFor expectation of varietie:\n\nYet now she sees her wishes prosperous bee,\n\nShe grieues, because her Lord so soone did die.\n\nWho can those vast imaginations feede,\n\nWhere in a propertie, contempt doth breede?\n\nWere _Herod_ now perchance to liue againe,\n\nShe would againe as much be grieued at that;\n\nAll that she may, she euer doth disdaine,\n\nHer wishes guide her to she knowes not what.\n\nAnd sad must be their lookes, their honor sower,\n\nThat care for nothing being in their power.\n\n**ACTUS SECUNDUS.**\n\n**S C\u0152NA PRIMA.**\n\n_Pheroras and Graphina._\n\n_Pher._ Tis true _Graphina,_ now the time drawes nye\n\nWherin the holy Priest with hallowed right,\n\nThe happy long desired knot shall tie,\n\n_Pheroras_ and _Graphina_ to vnite:\n\nHow oft haue I with lifted hands implor'd\n\nThis blessed houre, till now implord in vaine,\n\nWhich hath my wished libertie restor'd,\n\nAnd made my subiect selfe my owne againe.\n\nThy loue faire Mayd vpon mine eye doth sit,\n\nWhose nature hot doth dry the moysture all,\n\nWhich were in nature, and in reason fit\n\nFor my monachall Brothers death to fall:\n\nHad _Herod_ liu'd, he would haue pluckt my hand\n\nFrom faire _Graphinas_ Palme perforce: and tide\n\nThe same in hatefull and despised band,\n\nFor I had had a Baby to my Bride:\n\nScarce can her Infant tongue with easie voice\n\nHer name distinguish to anothers eare:\n\nYet had he liu'd, his power, and not my choise\n\nHad made me solembly the contract sweare.\n\nHaue I not cause in such a change to ioy?\n\nWhat? though she be my Neece, a Princesse borne:\n\nNeere bloods without respect: high birth a toy.\n\nSince Loue can teach blood and kindreds scorne.\n\nWhat booted it that he did raise my head,\n\nTo be his Realmes Copartner, Kingdomes mate,\n\nWithall, he kept _Graphina_ from my bed,\n\nMore wisht by me then thrice _Iudeas_ state.\n\nOh, could not he be skilfull Iudge in loue,\n\nThat doted so vpon his _Mariams_ face?\n\nHe, for his passion, _Doris_ did remoue.\n\nI needed not a lawfull Wife displace,\n\nIt could not be but he had power to iudge,\n\nBut he that neuer grudg'd a Kingdomes share,\n\nThis well knowne happinesse to me did grudge:\n\nAnd ment to be therein without compare.\n\nElse had I bene his equall in loues hoast,\n\nFor though the Diadem on _Mariams_ head\n\nCorrupt the vulgar iudgements, I will boast\n\n_Graphinas_ brow's as white, her cheekes as red.\n\nWhy speaks thou not faire creature? moue thy tongue,\n\nFor Silence is a signe of discontent:\n\nIt were to both our loues too great a wrong\n\nIf now this hower do find thee sadly bent.\n\n_Graph._ Mistake me not my Lord, too oft haue I\n\nDesir'd this time to come with winged feete,\n\nTo be inwrapt with griefe when tis too nie,\n\nYou know my wishes euer yours did meete:\n\nIf I be filent, tis no more but feare\n\nThat I should say too little when I speake:\n\nBut since you will my imperfections beare,\n\nIn spight of doubt I will my silence breake:\n\nYet might amazement tie my mouing tongue,\n\nBut that I know before _Pheroras_ minde,\n\nI haue admired your affection long:\n\nAnd cannot yet therein a reason finde.\n\nYour hand hath lifted me from lowest state,\n\nTo highest eminencie wondrous grace,\n\nAnd me your hand-maid haue you made your mate,\n\nThough all but you alone doe count me base.\n\nYou haue preserued me pure at my request,\n\nThough you so weake a vassaile might constraine\n\nTo yeeld to your high will, then last not best\n\nIn my respect a Princesse you disdaine,\n\nThen need not all these fauours studie craue,\n\nTo be requited by a simple maide:\n\nAnd studie still you know must silence haue,\n\nThen be my cause for silence iustly waide,\n\nBut studie cannot boote nor I requite,\n\nExcept your lowly hand-maides steadfast loue\n\nAnd fast obedience may your mind delight,\n\nI will not promise more then I can proue.\n\n_Phero._ That studie needs not let _Graphina_ smile,\n\nAnd I desire no greater recompence:\n\nI cannot vaunt me in a glorious stile,\n\nNor shew my loue in far-fetcht eloquence:\n\nBut this beleeue me, neuer _Herods_ heart\n\nHath held his Prince-borne beautie famed wife\n\nIn neerer place then thou faire virgin art,\n\nTo him that holds the glory of his life.\n\nShould _Herods_ body leaue the Sepulcher,\n\nAnd entertaine the seuer'd ghoft againe:\n\nHe should not be my nuptiall hinderer,\n\nExcept he hindred it with dying paine.\n\nCome faire _Graphina,_ let vs goe in slate,\n\nThis wish-indeered time to celebrate.\n\n**A CTUS 2. S\u0152NA. 2.**\n\n_Constabarus_ and _Babus Sonnes._\n\n_Babus._ 1. _Sonne._\n\nNow valiant friend you haue our liues redeem'd,\n\nWhich liues as sau'd by you, to you are due:\n\nCommand and you shall see your selfe esteem'd,\n\nOur liues and liberties belong to you.\n\nThis twice sixe yeares with hazard of your life,\n\nYou haue conceal'd vs from the tyrants sword:\n\nThough cruell _Herods_ sister were your wife,\n\nYou durst in scorne of feare this grace afford.\n\nIn recompence we know not what to say,\n\nA poore reward were thankes for fuch a merit,\n\nOur truest friendship at your feete we lay,\n\nThe beft requitall to a noble spirit. (youth,\n\n_Const._ Oh how you wrong our friendship valiant\n\nWith friends there is not such a word as det;\n\nWhere amitie is tide with bond of truth,\n\nAll benefits are there in common set.\n\nThen is the golden age with them renew'd,\n\nAll names of properties are banisht quite:\n\nDiuision, and distinction, are eschew'd:\n\nEach hath to what belongs to others right.\n\nAnd tis not sure so full a benefit,\n\nFreely to giue, as freely to require:\n\nA bountious act hath glory following it,\n\nThey cause the glory that the act desire.\n\nAll friendship should the patterne imitate,\n\nOf _Iesses_ Sonne and valiant _Ionathan_ :\n\nFor neither Soueraignes nor fathers hate,\n\nA friendship fixt on vertue seuer can.\n\nToo much of this, tis written in the heart,\n\nAnd need no amplifying with the tongue:\n\nNow may you from your liuing tombe depart,\n\nWhere _Herods_ life hath kept you ouer long.\n\nToo great an iniury to a noble minde,\n\nTo be quicke buried, you had purchast fame,\n\nSome yeares a goe, but that you were confinde.\n\nWhile thousand meaner did aduance their name.\n\nYour best of life the prime of all your yeares,\n\nYour time of action is from you bereft.\n\nTwelue winters haue you operpast in feares:\n\nYet if you vse it well, enough is left.\n\nAnd who can doubt but you will vse it well?\n\nThe sonnes of _Babus_ haue it by descent:\n\nIn all their thoughts each action to excell,\n\nBoldly to act, and wisely to inuent.\n\n_Babus_ 2. _Sonne._\n\nHad it not like the hatefull cuckoe beene,\n\nWhose riper age his infant nurse doth kill:\n\nSo long we had not kept our selues vnseene,\n\nBut _Constabarus_ safely crost our will:\n\nFor had the Tyrant fixt his cruell eye,\n\nOn our concealed feces wrath had swaide\n\nHis Iustice so, that he had forst vs die.\n\nAnd dearer price then life we should haue paid,\n\nFor you our truest friend had falne with vs:\n\nAnd we much like a house on pillers set,\n\nHad cleane deprest our prop, and therefore thus\n\nOur readie will with our concealement met.\n\nBut now that you faire Lord are daungerlesse,\n\nThe Sonnes of _Baba_ shall their rigor show:\n\nAnd proue it was not basenes did oppresse\n\nOur hearts so long, but honour kept them low.\n\n_Ba._ 1. _Sonne._ Yet do I feare this tale of _Herods_ death,\n\nAt last will proue a very tale indeed;\n\nIt giues me strongly in my minde, his breath\n\nWill be preseru'd to make a number bleed:\n\nI wish not therefore to be set at large,\n\nYet perill to my selfe I do not leare:\n\nLet vs for fome daies longer be your charge,\n\nTill we of _Herods_ state the truth do heare.\n\n_Const._ What art thou turn'd a coward noble youth,\n\nThat thou beginst to doubt, vndoubted truth?\n\n_Babus._ I. _Son._ Were it my brothers tongue that cast\n\nI fr\u00f5 his hart would haue the question out: (this doubt,\n\nWith this keene fauchion, but tis you my Lord\n\nAgainst whose head I must not lift a sword:\n\nI am so tide in gratitude _Const._ belieue\n\nYou haue no cause to take it ill,\n\nIf any word of mine your heart did grieue\n\nThe word discented from the speakers will,\n\nI know it was not feare the doubt begun,\n\nBut rather valour and your care of me,\n\nA coward could not be your fathers sonne,\n\nYet know I doubts vnnecessarie be:\n\nFor who can thinke that in _Anthonius_ fall,\n\n_Herod_ his bosome friend should scape vnbrusde:\n\nThen _C\u00e6sar_ we might thee an idiot call,\n\nIf thou by him should'st be so farre abusde.\n\n_Babus._ 2 _. Sonne._ Lord _Constab:_ let me tell you this,\n\nVpon submission _C\u00e6far_ will forgiue:\n\nAnd therefore though the tyrant did amisse,\n\nIt may fall out that he will let him liue.\n\nNot many yeares agone it is since I\n\nDirected thither by my fathers care,\n\nIn famous _Rome_ for twice twelue monthes did liue,\n\nMy life from _Hebrewes_ crueltie to spare,\n\nThere though I were but yet of boyish age,\n\nI bent mine eye to marke, mine eares to heare.\n\nWhere I did see _Octauious_ then a page,\n\nWhen first he did to _Iulions_ sight appeare:\n\nMe thought I saw such mildnes in his face,\n\nAnd such a sweetnes in his lookes did grow,\n\nWithall, commixt with so maiesticke grace,\n\nHis Phisnomy his Fortune did foreshow:\n\nFor this I am indebted to mine eye,\n\nBut then mine eare receiu'd more euidence,\n\nBy that I knew his loue to clemency,\n\nHow he with hottest choller could dispence.\n\n_Const._ But we haue more then barely heard the news,\n\nIt hath bin twice confirm'd. And though some tongue\n\nMight be so false, with false report t'abuse,\n\nA false report hath neuer lasted long.\n\nBut be it so that _Herod_ haue his life,\n\nConcealement would not then a whit auaile:\n\nFor certaine t'is, that she that was my wife,\n\nWould not to set her accusation faile.\n\nAnd therefore now as good the venture giue,\n\nAnd free our selues from blot of cowardise:\n\nAs show a pittifull desire to liue,\n\nFor, who can pittie but they must despise?\n\n_Babus first sonne._\n\nI yeeld, but to necessitie I yeeld,\n\nI dare vpon this doubt ingage mine arme:\n\nThat _Herod_ shall againe this kingdome weeld,\n\nAnd proue his death to be a false alarme.\n\n_Babus second sonne._\n\nI doubt it too: God grant it be an error,\n\nTis best without a cause to be in terror:\n\nAnd rather had I, though my soule be mine,\n\nMy soule should lie, then proue a true diuine,\n\n_Const._ Come, come, let feare goe seeke a dastards\n\nVndanted courage lies in a noble brest. (nest,\n\n**A CTUS 2. SC\u0152NA 3.**\n\n_Doris and Antipater_\n\n_Dor._ Your royall buildings bow your loftie side,\n\nAnd scope to her that is by right your Queen:\n\nLet your humilitie vpbraid the pride\n\nOf those in whom no due respect is seene:\n\nNine times haue we with Trumpets haughtie sound,\n\nAnd banishing sow's Leauen from our taste:\n\nObseru'd the feast that takes the fruit from ground.\n\nSince I faire Citie did behold thee last,\n\nSo long it is since _Mariams_ purer cheeke\n\nDid rob from mine the glory. And so long\n\nSince I returnd my natiue Towne to seeke:\n\nAnd with me nothing but the sence of wrong.\n\nAnd thee my Boy, whose birth though great it were,\n\nYet haue thy after fortunes prou'd but poore:\n\nWhen thou wert borne how little did I feare\n\nThou shouldst be thrust from forth thy Fathers doore.\n\nArt thou not _Herods_ right begotten Sonne?\n\nVVas not the haples _Doris, Herods_ wife?\n\nYes: ere he had the Hebrew kingdome wonne,\n\nI was companion to his priuate life.\n\nVVas I not faire enough to be a Queene?\n\nWhy ere thou wert to me false Monarch tide,\n\nMy lake of beauty might as well be seene,\n\nAs after I had liu'd fiue yeeres thy Bride.\n\nYet then thine oath came powring like the raine,\n\nWhich all affirm'd my face without compare:\n\nAnd that if thou might'st _Doris_ loue obtaine,\n\nFor all the world besides thou didst not care.\n\nThen was I yong, and rich, and nobly borne,\n\nAnd therefore worthy to be _Herods_ mate:\n\nYet thou vngratefull cast me off with scorne,\n\nWhen Heauens purpose raisd your meaner fate.\n\nOft haue I begd for vengeance for this fact,\n\nAnd with deiected knees, aspiring hands\n\nHaue prayd the highest power to inact\n\nThe fall of her that on my Trophee stands,\n\nReuenge I haue according to my will,\n\nYet where I wisht this vengeance did not light:\n\nI wisht it should high-hearted _Mariam_ kill.\n\nBut it against my whilome Lord did fight\n\nWith thee sweet Boy I came, and came to try\n\nIf thou before his bastards might be plac'd\n\nIn _Herods_ royall seat and dignitie.\n\nBut _Mariams_ infants here are onely grac'd,\n\nAnd now for vs there doth no hope remaine:\n\nYet we will not returne till _Herods_ end\n\nBe more confirmd, perchance he is not slaine.\n\nSo glorious Fortunes may my Boy attend,\n\nFor if he liue, hee'll thinke it doth suffice,\n\nThat he to _Doris_ shows such crueltie:\n\nFor as he did my wretched life dispise,\n\nSo doe I know I shall despised die.\n\nLet him but proue as naturall to thee,\n\nAs cruell to thy miserable mother:\n\nHis crueltie shall not vpbraided bee\n\nBut in thy fortunes. I his faults will smother.\n\n_Antipat._ Each mouth within the Citie loudly cries\n\nThat _Herods_ death is certaine; therefore wee\n\nHad best some subtill hidden plot deuise,\n\nThat _Mariams_ children might subuerted bee,\n\nBy poisons drinke, or else by murtherous Knife,\n\nSo we may be aduanc'd, it skils not how:\n\nThey are but Bastards, you were _Herods_ wife,\n\nAnd foule adultery blotteth _Mariams_ brow.\n\n_Doris._ They are too strong to be by vs remou'd,\n\nOr elfe reuenges foulest spotted face:\n\nBy our detested wrongs might be approu'd,\n\nBut weakenesse must to greater power giue place.\n\nBut let vs now retire to grieue alone,\n\nFor solitarines best fitteth mone.\n\n**A CTUS SECUNDUS. SC\u0152NA 4.**\n\n_Silleus and Constabarus._\n\n_Silleus._ Well met _Iudean_ Lord, the onely wight\n\n_Silleus_ wisht to see. I am to call\n\nThy tongue to strict account. _Const._ For what despight\n\nI ready am to heare, and answere all.\n\nBut if directly at the cause I gesse\n\nThat breeds this challenge, you must pardon me:\n\nAnd now some other ground of fight professe,\n\nFor I haue vow'd, vowes must vnbroken be.\n\n_Sill._ What may be your expectation? let me know.\n\n_Const._ Why? ought concerning _Salom,_ my sword\n\nShall not be welded for a cause so low,\n\nA blow for her my arme will scorne t'afford.\n\n_Sill._ It is for slandering her vnspotted name,\n\nAnd I will make thee in thy vowes despight,\n\nSucke vp the breath that did my Mistris blame,\n\nAnd swallow it againe to doe her right.\n\n_Const._ I prethee giue some other quarrell ground\n\nTo finde beginning, raile against my name:\n\nOr strike me first, or let some scarlet wound\n\nInflame my courage, giue me words of shame,\n\nDoe thou our _Moses_ sacred Lawes disgrace,\n\nDepraue our nation, doe me some despight:\n\nI'm apt enough to fight in any case,\n\nBut yet for _Salome_ I will not fight.\n\n_Sill._ Nor I for ought but _Salome:_ My sword\n\nThat owes his seruice to her sacred name:\n\nWill not an edge for other cause afford,\n\nIn other fight I am not sure of fame.\n\n_Const._ For her, I pitty thee enough already,\n\nFor her, I therefore will not mangle thee:\n\nA woman with a heart so most vnsteady,\n\nWill of her selfe sufficient torture bee.\n\nI cannot enuy for so light a gaine,\n\nHer minde with such vnconstancie doth runne:\n\nAs with a word thou didst her loue obtaine,\n\nSo with a word she will from thee be wonne.\n\nSo light as her possessions for most day\n\nIs her affections lost, to me tis knowne:\n\nAs good goe hold the winde as make her stay,\n\nShee neuer loues, but till she call her owne.\n\nShe meerly is a painted sepulcher,\n\nThat is both faire, and vilely foule at once:\n\nThough on her out-side graces garnish her,\n\nHer mind is fild with worse then rotten bones.\n\nAnd euer readie lifted is her hand,\n\nTo aime destruction at a husbands throat:\n\nFor proofes, _Iosephus_ and my selfe do stand,\n\nThough once on both of vs, she seem'd to doat.\n\nHer mouth though serpent-like it neuer hisses,\n\nYet like a Serpent, poysons where it kisses, (bite.\n\n_Silleus._ Well _Hebrew_ well, thou bark'st, but wilt not\n\n_Const._ I tell thee still for her I will not fight. (heart\n\n_Sille:_ Why then I call thee coward. _Conft:_ From my\n\nI giue thee thankes. A cowards hatefull name,\n\nCannot to valiant mindes a blot impart,\n\nAnd therefore I with ioy receiue the same.\n\nThou know'st I am no coward: thou wert by\n\nAt the _Arabian_ battaile th'other day:\n\nAnd saw'st my sword with daring valiancy,\n\nAmongst the faint _Arabians_ cut my way.\n\nThe blood of foes no more could let it shine,\n\nAnd twas inameled with some of thine.\n\nBut now haue at thee, not for _Salome_\n\nI fight: but to discharge a cowards stile:\n\nHere gins the fight that shall not parted be,\n\nBefore a soule or two indure exile. (my blood,\n\n_Silleus._ Thy sword hath made some windowes for\n\nTo shew a horred crimson phisnomie:\n\nTo breath for both of vs me thinkes twere good,\n\nThe day will giue vs time enough to die. (time,\n\n_Const:_ With all my hart take breath, thou shalt haue\n\nAnd if thou list a twelue month, let vs end:\n\nInto thy cheekes there doth a palenes clime,\n\nThou canst not from my sword thy selfe defend.\n\nWhat needest thou for _Salome_ to fight, (her:\n\nThou hast her, and may'st keepe her, none striues for\n\nI willingly to thee resigne my right,\n\nFor in my very soule I do abhorre her.\n\nThou feest that I am fresh, vnwounded yet,\n\nThen not for feare I do this offer make:\n\nThou art with losse of blood, to fight vnfit,\n\nFor here is one, and there another take.\n\n_Silleus._ I will not leaue, as long as breath remaines\n\nWithin my wounded body: spare your words,\n\nMy heart in bloods stead, courage entertaines,\n\n_Salomes_ loue no place for feare affords.\n\n_Const:_ Oh could thy soule but prophesie like mine,\n\nI would not wonder thou should'st long to die:\n\nFor _Salome_ if I aright diuine\n\nWill be then death a greater miserie. (will,\n\n_Sille:_ Then list, Ile breath no longer, _Const:_ Do thy\n\nI hateles fight, and charitably kill. I, I, they fight,\n\nPittie thy selfe _Silleus,_ let not death\n\nIntru'd before his time into thy hart:\n\nAlas it is too late to feare, his breath\n\nIs from his body now about to part.\n\nHow far'st thou braue _Arabian_? _Silleus_ very well,\n\nMy legge is hurt, I can no longer fight:\n\nIt onely grieues me, that so soone I fell,\n\nBefore faire _Saloms_ wrongs I came to right. (feare,\n\n_Const:_ Thy wounds are lesse then mortall. Neuer\n\nThou shalt a safe and quicke recouerie finde:\n\nCome, I will thee vnto my lodging beare,\n\nI hate thy body, but I loue thy minde.\n\n_Silleus._ Thankes noble Iew, I see a courtious foe,\n\nSterne enmitie to friendship can no art:\n\nHad not my heart and tongue engagde me so,\n\nI would from thee no foe, but friend depart.\n\nMy heart to _Salome_ is tide so fast,\n\nTo leaue her loue for friendship, yet my skill\n\nShall be imploy'd to make your fauour last,\n\nAnd I will honour _Constabarus_ still.\n\n_Const:_ I ope my bosome to thee, and will take\n\nThee in, as friend, and grieue for thy complaint:\n\nBut if we doe not expedition make,\n\nThy losse of blood I feare will make thee faint.\n\n_Chorus._\n\nTo heare a tale with eares preiudicate,\n\nIt spoiles the iudgement, and corrupts the sence:\n\nThat humane error giuen to euery state,\n\nIs greater enemie to innocence.\n\nIt makes vs foolish, heddy, rash, vniust,\n\nIt makes vs neuer try before we trust.\n\nIt will confound the meaning, change the words,\n\nFor it our sence of hearing much deceiues:\n\nBesides no time to Iudgement it affords,\n\nTo way the circumstance our eare receiues.\n\nThe ground of accidents it neuer tries,\n\nBut makes vs take for truth ten thousand lies.\n\nOur eares and hearts are apt to hold for good,\n\nThat we our selues doe most desire to bee:\n\nAnd then we drowne obiections in the flood\n\nOf partialitie, tis that we see\n\nThat makes false rumours long with credit past,\n\nThough they like rumours must conclude at last.\n\nThe greatest part of vs preiudicate,\n\nWith wishing _Herods_ death do hold it true:\n\nThe being once deluded doth not bate,\n\nThe credit to a better likelihood due.\n\nThose few that wish it not the multitude,\n\nDoe carrie headlong, so they doubts conclude.\n\nThey not obiect the weake vncertaine ground,\n\nWhereon they built this tale of _Herods_ end:\n\nWhereof the Author scarcely can be found,\n\nAnd all because their wishes that way bend.\n\nThey thinke not of the perill that ensu'th,\n\nIf this should proue the contrary to truth.\n\nOn this same doubt, on this so light a breath,\n\nThey pawne their liues, and fortunes. For they all\n\nBehaue them as the newes of _Herods_ death,\n\nThey did of most vndoubted credit call;\n\nBut if their actions now doe rightly hit,\n\nLet them commend their fortune, not their wit.\n\n**ACTUS TERTIUS**\n\n**S C\u0152NA PRIMA.**\n\n_Pheroras: Salome._\n\n_Phero._ Vrge me no more _Graphina_ to forsake,\n\nNot twelue howers since I married her\n\nAnd doe you thinke a sisters power cane mak (for loue:\n\nA resolute decree, so soone remoue? (affects.\n\n_Salome._ Poore minds they are that honour not\n\n_Phero:_ Who hunts for honour, happines neglects.\n\n_Salom._ You might haue bene both of felicitie,\n\nAnd honour too in equall measure seasde.\n\n_Phero:_ It is not you can tell so well as I,\n\nWhat tis can make me happie, or displeasde.\n\n_Salome._ To match for neither beautie nor respects\n\nOne meane of birth, but yet of meaner minde,\n\nA woman full of naturall defects,\n\nI wonder what your eye in her could finde. (wit,\n\n_Phero:_ Mine eye found louelines, mine eare found\n\nTo please the one, and to enchant the other;\n\nGrace on her eye, mirth on her tongue doth sit,\n\nIn lookes a child, in wisedomes house a mother, (elfe,\n\n_Salom:_ But say you thought her faire, as none thinks\n\nKnowes not _Pheroras,_ beautie is a blast:\n\nMuch like this flower which to day excels,\n\nBut longer then a day it will not last. (show\n\n_Phero:_ Her wit exceeds her beautie, _Salo:_ Wit may\n\nThe way to ill, as well as good you know.\n\n_Phero:_ But wisedome is the porter of her head,\n\nAnd bares all wicked words from issuing thence.\n\n_Sal._ But of a porter, better were you sped,\n\nIf she against their entrance made defence.\n\n_Phero._ But wherefore comes the sacred _Ananell_ ,\n\nThat hitherward his hastie steppes doth bend?\n\nGreat sacrificer y'are arriued well,\n\nIll newes from holy mouth I not attend.\n\n**A CTUS TERTIUS. SC\u0152NA 2.**\n\n_Pheroras. Salome. Ananell._\n\n_Ananell._\n\nMy lippes, my sonne, with peacefull tidings blest,\n\nShall vtter Honey to your listning eare:\n\nA word of death comes not from Priestly brest,\n\nI speake of life: in life there is no feare.\n\nAnd for the newes I did the Heauens salute,\n\nAnd fill'd the Temple with my thankfull voice:\n\nFor though that mourning may not me pollute,\n\nAt pleasing accidents I may reioyce.\n\n_Pheror._ Is _Herod_ then reuiu'd from certaine death?\n\n_Sall._ What? can your news restore my brothers breath?\n\n_Ana._ Both so, and so, the King is safe and sound,\n\nAnd did such grace in royall C _\u00e6sar_ meet:\n\nThat he with larger stile then euer crownd,\n\nWithin this houre Ierusalem will greet.\n\nI did but come to tell you, and must backe\n\nTo make preparatiues for sacrifice:\n\nI knew his death, your hearts like mine did racke,\n\nThough to conceale it, prou'd you wise.\n\n_Salom._ How can my ioy sufficiently appeare?\n\n_Phero._ A heauier tale did neuer pierce mine eare.\n\n_Salo._ Now _Salome_ of happinesse may boast.\n\n_Pheror._ But now _Pheroras_ is in danger most,\n\n_Salom._ I shall enioy the comfort of my life.\n\n_Pheror._ And I shall loose it, loosing of my wife.\n\n_Salom._ Ioy heart, for _Constan:_ shall be slaine.\n\n_Phero._ Grieue soule, _Graphina_ shall from me be tane.\n\n_Salom._ Smile cheekes, the faire _Silleus_ shall be mine.\n\n_Phero._ Weepe eyes, for I must with a child combine.\n\n_Salom._ Well brother, cease your mones, on one con-\n\nIle vndertake to winne the Kings consent: (dition\n\n_Graphina_ still shall be in your tuition,\n\nAnd her with you be nere the lesse content.\n\n_Phero._ What's the condition? let me quickly know,\n\nThat I as quickly your command may act:\n\nWere it to see what Hearbs in _Ophir_ grow,\n\nOr that the lofty _Tyrus_ might be sackt.\n\n_Salom._ Tis no so hard a taske: It is no more,\n\nBut tell the King that _Consta:_ hid\n\nThe sonnes of _Baba,_ done to death before:\n\nAnd tis no more then _Consta,_ did.\n\nAnd tell him more that he for _Herods_ sake,\n\nNot able to endure his brothers foe:\n\nDid with a bill our separation make,\n\nThough loth from _Consta:_ else to goe.\n\n_Phero._ Beleeue this tale for told, Ile goe from hence,\n\nIn _Herods_ eare the Hebrew to deface:\n\nAnd I that neuer studied eloquence,\n\nDoe meane with eloquence this tale to grace. _Exit._\n\n_Salom._ This will be _Constabarus_ quicke dispatch,\n\nWhich from my mouth would lesser credit finde:\n\nYet shall he not decease without a match,\n\nFor _Mariam_ shall not linger long behinde.\n\nFirst Iealousie, if that auaile not, feare\n\nShalbe my minister to worke her end;\n\nA common error moues not _Herods_ eare,\n\nWhich doth so firmly to his _Mariam_ bend.\n\nShe shall be charged with so horrid crime,\n\nAs _Herods_ feare shall turne his loue to hate:\n\nIle make some sweare that she desires to clime,\n\nAnd seekes to poyson him for his estate.\n\nI scorne that she should liue my birth t'vpbraid,\n\nTo call me base and hungry Edomite:\n\nWith patient show her choller I betrayd,\n\nAnd watcht the time to be reueng'd by slite.\n\nNow tongue of mine with scandall load her name,\n\nTurne hers to fountaines, _Herods_ eyes to flame:\n\nYet first I will begin _Pheroras_ suite,\n\nThat he my earnest businesse may effect:\n\nAnd I of _Mariam_ will keepe me mute,\n\nTill first some other doth her name detect.\n\nWho's there, _Silleus_ man? How fares your Lord?\n\nThat your aspects doe beare the badge of sorrow?\n\n_Silleus man._\n\nHe hath the marks of _Constabarus_ sword,\n\nAnd for a while desires your sight to borrow.\n\n_Salom._ My heauy curse the hatefull sword pursue,\n\nMy heauier curse on the more hatefull arme\n\nThat wounded my _Silleus._ But renew\n\nYour tale againe. Hath he no mortall harme?\n\n_Silleus man._\n\nNo signe of danger doth in him appeare,\n\nNor are his wounds in place of perill seene:\n\nHee bides you be assured you need not feare,\n\nHe hopes to make you yet _Arabias_ Queene.\n\n_Salom._ Commend my heart to be _Silleus_ charge,\n\nTell him, my brothers suddaine comming now;\n\nWill giue my foote no roome to walke at large,\n\nBut I will see him yet ere night I vow.\n\n**A CTUS 3. SC\u0152NA 3.**\n\n_Mariam and Sohemus._\n\n_Mariam._\n\n_Sohemus_ , tell me what the newes may be\n\nThat makes your eyes so full, your cheeks so blew?\n\n_Sohem._ I know not how to call them. Ill for me\n\nTis sure they are: not so I hope for you.\n\n_Herod. Mari._ Oh, what _Herod_? _Sohem. Herod_ liues _._\n\nHow! liues? What in some Caue or forrest hid?\n\n_Sohem._ Nay, backe return'd with honor. C _\u00e6sar giues_\n\nHim greater grace then ere _Anthonius_ did.\n\n_Mari._ Foretell the ruine of my family,\n\nTell me that I shall see our Citie burnd:\n\nTell me I shall a death disgracefull die,\n\nBut tell me not that _Herod is_ returnd.\n\n_Sohem._ Be not impatient Madam, be but milde,\n\nHis loue to you againe will soone be bred:\n\n_Mar._ I will not to his loue be reconcilde,\n\nWith solemne vowes I haue forsworne his Bed.\n\n_Sohem._ But you must breake those vowes.\n\n_Mar._ Ile rather breake\n\nThe heart of _Mariam._ Cursed is my Fate:\n\nBut speake no more to me, in vaine ye speake\n\nTo liue with him I so profoundly hate.\n\n_Sohem._ Great Queene, you must to me your pardon\n\n_Sohemus_ cannot now your will obey: (giue,\n\nIf your command should me to silence driue,\n\nIt were not to obey, but to betray.\n\nReiect, and slight my speeches, mocke my faith,\n\nScorne my obseruance, call my counsell nought;\n\nThough you regard not what _Sohemus_ saith,\n\nYet will I euer freely speake my thought.\n\nI feare ere long I shall faire _Mariam_ see\n\nIn wofull state, and by her selfe vndone:\n\nYet for your issues sake more temp'rate bee,\n\nThe heart by affabilitie is wonne.\n\n_Mari._ And must I to my Prison turne againe?\n\nOh, now I see I was an hypcorite:\n\nI did this morning for his death complaine,\n\nAnd yet doe mourne, because he liues ere night.\n\nWhen I his death beleeu'd, compassion wrought,\n\nAnd was the stickler twixt my heart and him:\n\nBut now that Curtaine's drawne from off my thought,\n\nHate doth appeare againe with visage grim:\n\nAnd paints the face of _Herod in_ my heart,\n\nIn horred colours with detested looke:\n\nThen feare would come, but scorne doth play her part,\n\nAnd saith that scorne with feare can neuer brooke.\n\nI know I could inchaine him with a smile:\n\nAnd lead him captiue with a gentle word,\n\nI scorne my looke should euer man beguile,\n\nOr other speech, then meaning to afford.\n\nElse _Salome_ in vaine might spend her winde,\n\nIn vaine might _Herods_ mother whet her tongue:\n\nIn vaine had they complotted and combinde,\n\nFor I could ouerthrow them all ere long.\n\nOh what a shelter is mine innocence,\n\nTo shield me from the pangs of inward griefe:\n\nGainst all mishaps it is my faire defence,\n\nAnd to my sorrowes yeelds a large reliefe.\n\nTo be commandresse of the triple earth,\n\nAnd sit in safetie from a fell secure:\n\nTo haue all nations celebrate my birth,\n\nI would not that my spirit were impure.\n\nLet my distressed state vnpittied bee,\n\nMine innocence is hope enough for mee. _Exit_\n\n_Sobem:_ Poore guiltles Queene. Oh that my wish\n\nA little temper now about thy heart: (might place\n\nVnbridled speech is _Mariams_ worst disgrace,\n\nAnd will indanger her without desart.\n\nI am in greater hazard. O're my head,\n\nThe fattall axe doth hang vnstedily:\n\nMy disobedience once discouered,\n\nWill shake it downe: _Sohemus so_ shall die.\n\nFor when the King shall find, we thought his death\n\nHad bene as certaine as we see his life:\n\nAnd markes withall I slighted so his breath,\n\nAs to preserue aliue his matchles wife.\n\nNay more, to giue to _Alexanders_ hand\n\nThe regall dignitie. The soueraigne power,\n\nHow I had yeelded vp at her command,\n\nThe strength of all the citie, _Dauids_ Tower.\n\nWhat more then common death may I expect,\n\nSince I too well do know his crueltie:\n\nTwere death, a word of _Herods_ to neglect,\n\nWhat then to doe directly contrarie?\n\nYet life I quite thee with a willing spirit,\n\nAnd thinke thou could'st not better be imploi'd:\n\nI forfeit thee for her that more doth merit,\n\nTen such were better dead then she destroi'd.\n\nBut fare thee well chast Queene, well may I see\n\nThe darknes palpable, and riuers part:\n\nThe sunne stand still. Nay more retorted bee,\n\nBut neuer woman with so pure a heart.\n\nThine eyes graue maiestie keepes all in awe,\n\nAnd cuts the winges of euery loose desire:\n\nThy brow is table to the modest lawe,\n\nYet though we dare not loue, we may admire.\n\nAnd if I die, it shall my soule content,\n\nMy breath _in Mariams_ seruice shall be spent.\n\n_Chorus._\n\nTis not enough for one that is a wife\n\nTo keepe her spotles from an act of ill:\n\nBut from suspition she should free her life,\n\nAnd bare her selfe of power as well as will.\n\nTis not so glorious for her to be free,\n\nAs by her proper selfe restrain'd to bee.\n\nWhen she hath spatious ground to walke vpon,\n\nWhy on the ridge should she desire to goe?\n\nIt is no glory to forbeare alone,\n\nThose things that may her honour ouerthrowe.\n\nBut tis thanke-worthy, if she will not take\n\nAll lawfull liberties for honours sake.\n\nThat wife her hand against her fame doth reare,\n\nThat more then to her Lord alone will giue\n\nA priuate word to any second eare,\n\nAnd though she may with reputation liue.\n\nYet though most chast, she doth her glory blot,\n\nAnd wounds her honour, though she killes it not.\n\nWhen to their Husbands they themselues doe bind,\n\nDoe they not wholy giue themselues away?\n\nOr giue they but their body not their mind,\n\nReseruing that though best, for others pray?\n\nNo sure, their thoughts no more can be their owne,\n\nAnd therefore should to none but one be knowne.\n\nThen she vsurpes vpon anothers right,\n\nThat seekes to be by publike language grac't:\n\nAnd though her thoughts reflect with purest light,\n\nHer mind if not peculiar is not chast.\n\nFor in a wife it is no worse to finde,\n\nA common body, then a common minde.\n\nAnd euery mind though free from thought of ill,\n\nThat out of glory seekes a worth to show:\n\nWhen any's eares but one therewith they fill,\n\nDoth in a sort her purenes ouerthrow.\n\nNow _Mariam_ had, (but that to this she bent)\n\nBeene free from feare, as well as innocent.\n\n**ACTUS QUARTUS:**\n\n**S C\u0152NA PRIMA.**\n\n_Enter Herod and his attendants._\n\n_Herod._\n\nHaile happie citie, happie in thy store,\n\nAnd happy that thy buildings such we see;\n\nMore happie in the Temple where w'adore,\n\nBut most of all that _Mariam_ liues in thee.\n\nArt thou return'd? how fares my _Mariam? Enter Nutio._\n\n_Nutio._ She's well my Lord, and will anon be here\n\nAs you commanded. _Her:_ Muffle vp thy browe\n\nThou daies darke taper. _Mariam_ will appeare.\n\nAnd where she shines, we need not thy dimme light,\n\nOh hast thy steps rare creature, speed thy pace:\n\nAnd let thy presence make the day more bright,\n\nAnd cheere the heart of _Herod with_ thy face.\n\nIt is an age since I from _Mariam_ went,\n\nMe thinkes our parting was in _Dauids_ daies;\n\nThe houres are so increast by discontent,\n\nDeepe sorrow, _Iosu_ a like the season staies:\n\nBut when I am with _Mariam,_ time runnes on,\n\nHer sight, can make months, minutes, daies of weekes:\n\nAn hower is then no sooner come then gon.\n\nWhen in her face mine eye for wonders seekes.\n\nYou world commanding citie, _Europes_ grace,\n\nTwice hath my curious eye your streets suruai'd,\n\nAnd I haue seene the statue filled place,\n\nThat once if not for griefe had bene betrai'd.\n\nI all your _Roman_ beauties haue beheld,\n\nAnd seene the showes your _Ediles_ did prepare,\n\nI saw the sum of what in you exceld,\n\nYet saw no miracle like _Mariam_ rare.\n\nThe faire and famous _Liuia, C\u00e6sars_ loue,\n\nThe worlds commaunding Mistresse did I see:\n\nWhose beauties both the world and _Rome_ approue,\n\nYet _Mariam_ : _Liuia_ is not like to thee.\n\nBe patient but a little, while mine eyes\n\nWithin your compast limits be contain'd:\n\nThat obiect straight shall your desires suffice,\n\nFrom which you were so long a while restrain'd.\n\nHow wisely _Mariam_ doth the time delay,\n\nLeast suddaine ioy my sence should suffocate:\n\nI am prepar'd, thou needst no longer stay:\n\nWhose there, my _Mariam,_ more then happie fate?\n\nOh no, it is _Pheroras,_ welcome Brother,\n\nNow for a while, I must my passion smother.\n\n**A CTUS QUARTUS. SC\u0152NA FECUNDA .**\n\n_Herod. Pheroras._\n\n_Pheroras._\n\nAll health and safetie waite vpon my Lord,\n\nAnd may you long in prosperous fortunes liue\n\nWith _Rome_ commanding _C\u00e6sar;_ at accord,\n\nAnd haue all honors that the world can giue.\n\n_Herod._ Oh brother, now thou speakst not from thy\n\nNo, thou hast strooke a blow at _Herods_ loue: (hart,\n\nThat cannot quickly from my memory part,\n\nThough _Salome_ did me to pardon moue.\n\nValiant _Phasaelus,_ now to thee farewell,\n\nThou wert my kinde and honorable brother:\n\nOh haples houre, when you selfe striken fell,\n\nThou fathers Image, glory of thy mother.\n\nHad I defir'd a greater sute of thee,\n\nThen to withhold thee from a harlots bed,\n\nThou wouldst haue granted it: but now I see\n\nAll are not like that in a wombe are bred.\n\nThou wouldst not, hadst thou heard of _Herods_ death,\n\nHaue made his buriall time, thy bridall houre:\n\nThou wouldst with clamours, not with ioyfull breath,\n\nHaue show'd the newes to be not sweet but soure.\n\n_Phero. Phasaelus_ great worth I know did staine\n\n_Pheroras_ petty valour: but they lie\n\n(Excepting you your selfe) that dare maintaine,\n\nThat he did honor _Herod_ more then I.\n\nFor what I showd, loues power constraind me show,\n\nAnd pardon louing faults for _Mariams_ sake.\n\n_Herod. Mariam,_ where is she? _Phero._ Nay, I do not\n\nBut absent vse of her faire name I make: (know,\n\nYou haue forgiuen greater faults then this,\n\nFor _Constabarus_ that against you will\n\nPreseru'd the sonnes of _Baba,_ liues in blisse,\n\nThough you commanded him the youths to kill.\n\n_Herod._ Goe, take a present order for his death,\n\nAnd let those traytors feele the worst of feares:\n\nNow _Salome_ will whine to begge his breath,\n\nBut Ile be deafe to prayers: and blind to teares.\n\n_Phero._ He is my Lord from _Salom_ diuorst,\n\nThough her affection did to leaue him grieue:\n\nYet was she by her loue to you inforst,\n\nTo leaue the man that would your foes relieue.\n\n_Herod._ Then haste them to their death. I will requite\n\nThee gentle _Mariam. Salom._ I meane\n\nThe thought of _Mariam_ doth so steale my spirit,\n\nMy mouth from speech of her I cannot weane. _Exit._\n\n**A CTUS 4. SC\u0152NA 3.**\n\n_Herod. Mariam._\n\n_Herod._\n\nAnd heere she comes indeed: happily met\n\nMy best, and deerest halfe: what ailes my deare?\n\nThou doest the difference certainly forget\n\nTwixt Duskey habits, and a time so cleare.\n\n_Mar._ My Lord, I suit my garment to my minde,\n\nAnd there no cheerfull colours can I finde.\n\n_Herod._ Is this my welcome? haue I longd so much\n\nTo see my dearest _Mariam_ discontent?\n\nWhat ist that is the cause thy heart to touch?\n\nOh speake, that I thy sorrow may preuent.\n\nArt thou not _Iuries_ Queene, and _Herods_ too?\n\nBe my Commandres, be my Soueraigne guide:\n\nTo be by thee directed I will woo,\n\nFor in thy pleasure lies my highest pride.\n\nOr if thou thinke _Iud\u00e6as_ narrow bound,\n\nToo strict a limit for thy great command:\n\nThou shalt be Empresse of _Arabia_ crownd,\n\nFor thou shalt rule, and I will winne the Land.\n\nIle robbe the holy _Dauids_ Sepulcher\n\nTo giue thee wealth, if thou for wealth do care:\n\nThou shalt haue all, they did with him inter,\n\nAnd I for thee will make the Temple bare.\n\n_Mar._ I neither haue of power nor riches want,\n\nI haue enough, nor doe I wish for more:\n\nYour offers to my heart no ease can grant,\n\nExcept they could my brothers life restore.\n\nNo, had you wifht the wretched _Mariam_ glad,\n\nOr had your loue to her bene truly tide:\n\nNay, had you not defir'd to make her fad,\n\nMy brother nor my Grandfyre had not dide.\n\n_Her._ Wilt thou beleeue no oathes to cleere thy Lord?\n\nHow oft haue I with execration sworne:\n\nThou art by me belou'd, by me ador'd,\n\nYet are my protestations heard with scorne.\n\n_Hercanus_ plotted to depriue my head\n\nOf this long setled honor that I weare:\n\nAnd therefore I did iustly doome him dead,\n\nTo rid the Realme from perill, me from feare.\n\nYet I for _Mariams_ sake doe so repent\n\nThe death of one: whose blood she did inherit:\n\nI wish I had a Kingdomes treasure spent,\n\nSo I had nere expeld _Hercanus_ spirit.\n\nAs I affected that same noble youth,\n\nIn lasting infamie my name inrole:\n\nIf I not mournd his death with heartie truth.\n\nDid I not shew to him my earnest loue,\n\nWhen I to him the Priesthood did restore?\n\nAnd did for him a liuing Priest remoue,\n\nWhich neuer had bene done but once before.\n\n_Mariam._ I know that mou'd by importunitie,\n\nYou made him Priest, and shortly after die.\n\n_Herod._ I will not speake, vnles to be beleeu'd,\n\nThis froward humor will not doe you good:\n\nIt hath too much already _Herod_ grieu'd,\n\nTo thinke that you on termes of hate haue stood.\n\nYet smile my dearest _Mariam,_ doe but smile,\n\nAnd I will all vnkind conceits exile.\n\n_Mari._ I cannot frame disguise, nor neuer taught\n\nMy face a looke dissenting from my thought.\n\n_Herod._ By heau'n you vexe me, build not on my loue.\n\n_Mari._ I wil not build on so vnstable ground.\n\n_Herod._ Nought is so fixt, but peeuishnes may moue.\n\n_Mar._ Tis better sleightest cause then none were fo\u0169d.\n\n_Herod._ Be iudge your selfe, if euer _Herod_ sought\n\nOr would be mou'd a cause of change to finde:\n\nYet let your looke declare a milder thought,\n\nMy heart againe you shall to _Mariam_ binde.\n\nHow oft did I for you my Mother chide,\n\nReuile my Sister, and my brother rate:\n\nAnd tell them all my _Mariam_ they belide,\n\nDistrust me still, if these be signes of hate.\n\n**A CTUS 4. SC\u0152NA 4.**\n\n_Herod._\n\nVVhat hast thou here? _Bu._ A drinke procuring\n\nThe Queene defir'd me to deliuer it. (loue,\n\n_Mar._ Did I; some hatefull practise this will proue,\n\nYet can it be no worse then Heauens permit.\n\n_Herod._ Confesse the truth thou wicked instrument,\n\nTo her outragious will, tis passion sure:\n\nTell true, and thou shalt scape the punishment,\n\nWhich if thou doe conceale thou shalt endure.\n\n_Bu._ I know not, but I doubt it be no lesse,\n\nLong since the hate of you her heart did cease.\n\n_Herod._ Know'st thou the cause thereof? _Bu._ My Lord\n\n_Sohemus_ told the tale that did displease. (I geffe,\n\n_Herod._ Oh Heauen! _Sohemus_ false! Goe let him die,\n\nStay not to suffer him to speake a word:\n\nOh damned villaine, did he falsifie\n\nThe oath he swore eu'n of his owne accord?\n\nNow doe I know thy falshood, painted Diuill\n\nThou white Inchantres. Oh thou art so foule,\n\nThat Ysop cannot clense thee worst of euill.\n\nA beautious body hides a loathsome soule,\n\nYour loue _Sobemus_ mou'd by his affection,\n\nThough he haue euer heretofore bene true:\n\nDid blab forfooth, that I did giue direction,\n\nIf we were put to death to slaughter you.\n\nAnd you in blacke reuenge attended now\n\nTo adde a murther to your breach of vow.\n\n_Mar._ Is this a dream? _Her._ Oh Heauen, that t'were no\n\nIle giue my Realme to who can proue it so: (more,\n\nI would I were like any begger poore,\n\nSo I for false my _Mariam_ did not know.\n\nFoule pith contain'd in the fairest rinde,\n\nThat euer grac'd a C\u00e6dar. Oh thine eye\n\nIs pure as heauen, but impure thy minde,\n\nAnd for impuritie shall _Mariam_ die.\n\nWhy didst thou loue _Sohemus? Mar:_ they can tell\n\nThat say I lou'd him, _Mariam_ saies not so.\n\n_Herod._ Oh cannot impudence the coales expell,\n\nThat for thy loue in _Herods_ bosome glowe:\n\nIt is as plaine as water, and deniall\n\nMakes of thy falsehood but a greater triall.\n\nHaft thou beheld thy selfe, and couldst thou staine\n\nSo rare perfection: euen for loue of thee\n\nI doe profoundly hate thee. Wert thou plaine,\n\nThou shoul'dst the wonder of _Iudea_ bee.\n\nBut oh thou art not. Hell it selfe lies hid\n\nBeneath thy heauenly show. Yet neuer wert thou chast:\n\nThou might'st exalt, pull downe, command, forbid,\n\nAnd be aboue the wheele of fortune plast.\n\nHadst thou complotted _Herods_ massacre,\n\nThat so thy sonne a Monarch might be stilde,\n\nNot halfe so grieuous such an action were,\n\nAs once to thinke, that _Mariam_ is defilde.\n\nBright workmanship of nature sulli'd ore,\n\nWith pitched darknes now thine end shall bee:\n\nThou shalt not liue faire fiend to cozen more,\n\nWith heauy semblance, as thou cousnedst mee.\n\nYet must I loue thee in despight of death,\n\nAnd thou shalt die in the dispight of loue:\n\nFor neither shall my loue prolong thy breath,\n\nNor shall thy losse of breath my loue remoue.\n\nI might haue seene thy falsehood in thy face,\n\nWhere coul'dst thou get thy stares that seru'd for eyes?\n\nExcept by theft, and theft is foule disgrace:\n\nThis had appear'd before were _Herod_ wise,\n\nBut I'me a sot, a very sot, no better:\n\nMy wisedome long agoe a wandring fell,\n\nThy face incountring it, my wit did fetter,\n\nAnd made me for delight my freedome sell.\n\nGiue me my heart false creature, tis a wrong,\n\nMy guliltles heart should now with thine be slaine:\n\nThou hadst no right to looke it vp so long,\n\nAnd with vsurpers name I _Mariam_ staine.\n\n_Enter Bu:_\n\n_He:_ Haue you design'd _Sobemus_ to his end? (guard\n\n_Bu:_ I haue my Lord. _Herod:_ Then call our royall\n\nTo doe as much for _Mariam,_ they offend\n\nLeaue ill vnblam'd, or good without reward.\n\nHere take her to her death. Come backe, come backe,\n\nWhat ment I to depriue the world of light:\n\nTo muffle _Iury_ in the foulest blacke,\n\nThat euer was an opposite to white.\n\nWhy whither would you carrie her: _Sould:_ you bad\n\nWe should conduct her to her death my Lord.\n\n_Hero:_ Wie sure I did not, _Herod_ was not mad,\n\nWhy should she feele the furie of the sword?\n\nOh now the griefe returnes into my heart,\n\nAnd pulles me peecemeale: loue and hate doe fight:\n\nAnd now hath boue acquir'd the greater part,\n\nYet now hath hate, affection conquer'd quite.\n\nAnd therefore beare her hence: and _Hebrew_ why\n\nSeaze you with Lyons pawes the fairest lam\n\nOf all the flocke? she must not, shall not, die,\n\nWithout her I most miserable am.\n\nAnd with her more then most, away, away,\n\nBut beare her but to prison not to death:\n\nAnd is she gon indeed, stay villaines stay,\n\nHer lookes alone preseru'd your Soueraignes breath.\n\nWell let her goe, but yet she shall not die,\n\nI cannot thinke she ment to poison me:\n\nBut certaine tis she liu'd too wantonly,\n\nAnd therefore shall she neuer more be free.\n\n**A CTUS 4. SC\u0152NA 5.**\n\n_Bu._ Foule villaine, can thy pitchie coloured soule\n\nPermit thine eare to heare her caules doome?\n\nAnd not inforce thy tongue that tale controule,\n\nThat must vniustly bring her to her toome.\n\nOh _Salome_ thou hast thy selfe repaid,\n\nFor all the benefits that thou hast done:\n\nThou art the cause I haue the queene betraid,\n\nThou hast my hart to darkest false-hood wonne,\n\nI am condemn'd, heau'n gaue me not my tongue\n\nTo slander innocents _,_ to lie, deceiue:\n\nTo be the hatefull instrument to wrong,\n\nThe earth of greatest glory to bereaue.\n\nMy sinne ascends and doth to heau'n crie,\n\nIt is the blackest deed that euer was:\n\nAnd there doth sit an Angell notarie,\n\nThat doth record it downe in leaues of brasse.\n\nOh how my heart doth quake: _Achitophel_ ,\n\nThou founds a meanes thy selfe from shame to free:\n\nAnd sure my soule approues thou didst not well,\n\nAll follow some, and I will follow thee.\n\n**A CTUS 4. SC\u0152NA 6** _._\n\n_Constabarus, Babus Sonnes, and their guard._\n\n_Const:_ Now here we step our last, the way to death,\n\nWe must not tread this way a second time:\n\nYet let vs resolutely yeeld our breath,\n\nDeath is the onely ladder, Heau'n to clime. (resigne,\n\n_Babus_ 1 _. Sonne._ With willing mind I could my selfe\n\nBut yet it grieues me with a griefe vntold:\n\nOur death should be accompani'd with thine,\n\nOur frendship we to thee haue dearely sold.\n\n_Const._ Still wilt thou wrong the sacred name of friend?\n\nThen should'st thou neuer stile it friendship more:\n\nBut base mechanicke traffique that doth lend,\n\nYet will be sure they shall the debt restore.\n\nI could with needlesse complement returne,\n\nTis for thy ceremonie I could say;\n\nTis I that made the fire your house to burne,\n\nFor but for me she would not you betray.\n\nHad not the damned woman sought mine end,\n\nYou had not bene the subiect other hate:\n\nYou neuer did her hatefull minde offend,\n\nNor could your deaths haue freed your nuptiall fate.\n\nTherefore faire friends, though you were still vnborne,\n\nSome other subtiltie deuisde should bee:\n\nWere by my life, though guiltles should be torne,\n\nThus haue I prou'd, tis you that die for mee.\n\nAnd therefore should I weakely now lament,\n\nYou haue but done your duties, friends should die:\n\nAlone their friends disaster to preuent,\n\nThough not compeld by strong necessitie.\n\nBut now farewell faire citie, neuer more\n\nShall I behold your beautie shining bright:\n\nFarewell of _Iewish_ men the worthy store,\n\nBut no farewell to any female wight.\n\nYou wauering crue: my curse to you I leaue,\n\nYou had but one to giue you any grace:\n\nAnd you your selues will _Mariams_ life bereaue,\n\nYour common-wealth doth innocencie chase.\n\nYou creatures made to be the humane curse,\n\nYou Tygers, Lyonesses, hungry Beares,\n\nTeare massacring _Hienas:_ nay far worse,\n\nFor they for pray doe shed their fained teares;\n\nBut you will weepe, (you creatures crosse to good)\n\nFor your vnquenched thirst of humane blood:\n\nYou were the Angels cast from heaue'n for pride,\n\nAnd still doe keepe your Angels outward show,\n\nBut none of you are inly beautifide,\n\nFor still your heau'n depriuing pride doth grow.\n\nDid not the sinnes of many require a scourge,\n\nYour place on earth had bene by this withstood:\n\nBut since a flood no more the world must purge,\n\nYou staid in office of a second flood.\n\nYou giddy creatures, sowers of debate,\n\nYou'll loue to day, and for no other cause,\n\nBut for you yefterday did deply hate,\n\nYou are the wreake of order, breach of lawes.\n\nYou beft, are foolish, froward, wanton, vaine,\n\nYour worft adulterous, murderous, cunning, proud:\n\nAnd _Salome_ attends the latter traine,\n\nOr rather he their leader is allowd.\n\nI do the sottishnesse of men bewaile,\n\nThat doe with following you inhance your pride:\n\nT'were better that the humane race should faile,\n\nThen be by such a mischiefe multiplide.\n\n_Chams_ seruile curse to all your sexe was giuen,\n\nBecause in Paradise you did offend:\n\nThen doe we not resist the will of Heauen,\n\nWhen on your willes like seruants we attend?\n\nYou are to nothing constant but to ill,\n\nYou are with nought but wickednesse indude:\n\nYour loues are set on nothing but your will,\n\nAnd thus my censure I of you conclude.\n\nYou are the least of goods, the worst of euils,\n\nYour best are worse then men: your worst then diuels.\n\n_Babus second sonne._\n\nCome let vs to our death: are we not blest?\n\nOur death will freedome from these creatures giue:\n\nThose trouble quiet sowers of vnrest,\n\nAnd this I vow that had I leaue to liue,\n\nI would for euer leade a single life,\n\nAnd neuer venter on a diuellish wife.\n\n**A CTUS 4. SC\u0152NA 7.**\n\n_Herod and Salome._\n\n_Herod._\n\nNay, she shall die. Die quoth you, that she shall:\n\nBut for the meanes. The meanes! Me thinks tis\n\nTo finde a meanes to murther her withall, (hard\n\nTherefore I am resolu'd she shall be spar'd.\n\n_Salom._ Why? let her be beheaded. _Her._ That were\n\nThinke you that swords are miracles like you: (well,\n\nHer skinne will eu'ry Curtlax edge refell,\n\nAnd then your enterprise you well may rue.\n\nWhat if the fierce Arabian notice take,\n\nOf this your wretched weaponlesse estate:\n\nThey answere when we bid resistance make,\n\nThat _Mariams_ skinne their fanchions did rebate.\n\nBeware of this, you make a goodly hand,\n\nIf you of weapons doe depriue our Land.\n\n_Sal._ Why drowne her then. _Herod._ Indeed a sweet de-\n\nWhy? would not eu'ry Riuer turne her course (uice,\n\nRather then doe her beautie preiudice?\n\nAnd be reuerted to the proper sourse.\n\nSo not a drop of water should be found\n\nIn all Iudeas quondam firtill ground.\n\n_Sal._ Then let the fire deuoure her. _Her._ T'will not\n\nFlame is from her deriu'd into my heart: (bee:\n\nThou nursest flame, flame will not murther thee,\n\nMy fairest _Mariam,_ fullest of desert. (die:\n\n_Salom._ Then let her liue for me. _Herod._ Nay, she shall\n\nBut can you liue without her? _Sal._ doubt you that?\n\n_Herod._ I'me sure I cannot, I beseech you trie:\n\nI haue experience but I know not what.\n\n_Salom._ How should I try? _Her._ Why let my lone be\n\nBut if we cannot liue without her sight (slaine,\n\nYoule finde the meanes to make her breathe againe,\n\nOr else you will bereaue my comfort quite.\n\n_Sal._ Oh I; I warrant you. _Herod._ What is she gone?\n\nAnd gone to bid the world be ouerthrowne:\n\nWhat? is her hearts composure hardest stone?\n\nTo what a passe are cruell women growne?\n\nShe is return'd already: haue you done?\n\nIst possible you can command so soone?\n\nA creatures heart to quench the flaming Sunne,\n\nOr from the skie to wipe away the Moone.\n\n_Salo._ If _Mariam_ be the Sunne and Moone, it is:\n\nFor I already haue commanded this. (times.\n\n_Her._ But haue you seene her cheek? _Sal._ A thousand\n\n_Herod._ But did you marke it too? _Sal._ I very well.\n\n_Herod._ What ist? _Sal._ A Crimson bush, that euer limes\n\nThe soule whose foresight doth not much excell.\n\n_Herod._ Send word she shall not dye. Her cheek a bush,\n\nNay, then I see indeed you markt it not.\n\n_Sal._ Tis very faire, but yet will neuer blush,\n\nThough foule dishonors do her forehead blot\n\n_Herod._ Then let her die, tis very true indeed,\n\nAnd for this fault alone shall _Mariam_ bleed.\n\n_Sal._ What fault my Lord? _Herod._ What fault ist? you\n\nIf you be ignorant I know of none, (that aske:\n\nTo call her backe from death shall be your taske,\n\nI'm glad that she for innocent is knowne.\n\nFor on the brow of _Mariam_ hangs a Fleece,\n\nWhose slenderest twine is strong enough to binde\n\nThe hearts of Kings, the pride and shame of _Greece_ ,\n\n_Troy_ flaming _Helens_ not so fairely shinde.\n\n_Salom._ Tis true indeed, she layes them out for nets,\n\nTo catch the hearts that doe not shune a baite:\n\nTis time to speake: for _Herod_ sure forgets\n\nThat _Mariams_ very tresses hide deceit.\n\n_Her._ Oh doe they so? nay, then you doe but well,\n\nInsooth I thought it had beene haire:\n\nNets call you them? Lord, how they doe excell,\n\nI neuer saw a net that show'd so faire.\n\nBut haue you heard her speake? _Sal._ You know I haue.\n\n_Her:_ And were you not amaz'd? _Sal._ No, not a whit.\n\n_Her._ Then t'was not her you heard, her life Ile faue,\n\nFor _Mariam_ hath a world amazing wit.\n\n_Salo._ She speaks a beautious language, hut within\n\nHer heart is false as powder: and her tongue\n\nDoth but allure the auditors to sinne,\n\nAnd is the instrument to doe you wrong.\n\n_Herod._ It may be so: nay, tis so: shee's vnehaste,\n\nHer mouth will ope to eu'ry strangers eare:\n\nThen let the executioner make haste,\n\nLest she inchant him, if her words he heare.\n\nLet him be deafe, lest she do him surprise\n\nThat shall to free her spirit be assignde:\n\nYet what boots deafenes if he haue his eyes,\n\nHer murtherer must be both deafe and blinde.\n\nFor if he see, he needs must see the starres\n\nThat shine on eyther side of _Mariams_ face:\n\nWhose sweet aspect will terminate the warres,\n\nWherewith he should a soule so precious chase.\n\nHer eyes can speake, and in their speaking moue,\n\nOft did my heart with reuerence receiue\n\nThe worlds mandates. Pretty tales of loue\n\nThey vtter, which can humane bondage weaue.\n\nBut shall I let this heauens modell dye?\n\nWhich for a small selfe-portraiture she drew:\n\nHer eyes like starres, her forehead like the skie,\n\nShe is like Heauen, and must be heauenly true.\n\n_Salom._ Your thoughts do raue with doating on the\n\nHer eyes are ebon hewde, and you'll confesse: (Queen,\n\nA sable starre hath beene but seldome seene,\n\nThen speake of reason more, of _Mariam_ lesse.\n\n_Herod._ Your selfe are held a goodly creature heere,\n\nYet so vnlike my _Mariam_ in your shape:\n\nThat when to her you haue approached neere,\n\nMy selfe hath often tane you for an Ape.\n\nAnd yet you prate of beautie: goe your waies,\n\nYou are to her a Sun-burnt Blackamore:\n\nYour paintings cannot equall _Mariams_ praise,\n\nHer nature is so rich, you are so poore.\n\nLet her be staide from death, for if she die,\n\nWe do we know not what to stop her breath:\n\nA world cannot another _Mariam_ buy,\n\nWhy stay you lingring? countermaund her death.\n\n_Salo._ Then youle no more remember what hath past,\n\n_Sobemus_ loue, and hers shall be forgot:\n\nTis well in truth: that fault may be her last,\n\nAnd she may mend, though yet she loue you not.\n\n_Her:_ Oh God: tis true. _Sohemus_ : earth and heau'n,\n\nWhy did you both conspire to make me curst:\n\nIn cousning me with showes, and proofes vneu'n?\n\nShe show'd the best, and yet did proue the worst.\n\nHer show was such, as had our singing king\n\nThe holy _Dauid, Mariams_ beautie seene:\n\nThe _Hittits_ had then felt no deadly sting,\n\nNor _Bethsabe_ had neuer bene a Queene.\n\nOr had his sonne the wisest man of men,\n\nWhose fond delight did most consist in change:\n\nBeheld her face, he had bene staid agen,\n\nNo creature hauing her, can wish to range.\n\nHad _Asuerus_ seene my _Mariams_ brow,\n\nThe humble _Iewe,_ she might haue walkt alone:\n\nHer beautious vertue should haue staid below,\n\nWhiles _Mariam_ mounted to the Persian throne.\n\nBut what auailes it all: for in the waight\n\nShe is deceitfull, light as vanitie:\n\nOh she was made for nothing but a bait,\n\nTo traine some haples man to miserie.\n\nI am the haples man that haue bene trainde,\n\nTo endles bondage. I will see her yet:\n\nMe thinkes I should discerne her if she fainde,\n\nCan humane eyes be dazde by womans wit?\n\nOnce more these eyes of mine with hers shall meet,\n\nBefore the headsman doe her life bereaue:\n\nShall I for euer part from thee my sweet?\n\nWithout the taking of my latest leaue.\n\n_Salo:_ You had as good resolue to saue her now,\n\nIle stay her death, tis well determined:\n\nFor sure she neuer more will breake her vow,\n\n_Sohemus_ and _Iosepbus_ both are dead.\n\n_Herod._ She shall not liue, nor will I see her face,\n\nA long heald wound, a second time doth bleed:\n\nWith _Ioseph_ I remember her disgrace,\n\nA shamefull end ensues a shamefull deed.\n\nOh that I had not cald to minde anew,\n\nThe discontent of _Mariams_ wauering hart:\n\nTwas you: you foule mouth'd _Ate,_ none but you,\n\nThat did the thought hereof to me impart.\n\nHence from my sight, my blacke tormenter hence,\n\nFor hadst not thou made _Herod_ vnsecure:\n\nI had not doubted _Mariams_ innocence,\n\nBut still had held her in my heart for pure.\n\n_Salo:_ Ile leaue you to your paffion: tis no time\n\nTo purge me now, though of a guiltles crime. ( _Exit._\n\n_Herod._ Destruction take thee: thou hast made\n\nAs heauie as reuenge, I am so dull, (my hart\n\nMe thinkes I am not sensible of smart,\n\nThough hiddious horrors at my bosome pull.\n\nMy head waies downwards: therefore will I goe\n\nTo try if I can sleepe away my woe.\n\n**A CTUS 4. SC\u0152NA. 8.**\n\n_Mariarm._\n\nAm I the _Mariam_ that presum'd so much, (breath?\n\nAnd deem'd my face must needes preserue my\n\nI, I it was that thought my beautie such,\n\nAt it alone could countermaund my death.\n\nNow death will teach me: he can pale aswell\n\nA cheeke of roses, as a cheeke lesse bright:\n\nAnd dim an eye whose shine doth most excell,\n\nAssoone as one that casts a meaner light.\n\nHad not my selfe against my selfe conspirde,\n\nNo plot: no aduersarie from without\n\nCould _Herods_ loue from _Mariam_ haue retirde,\n\nOr from his heart haue thrust my semblance out.\n\nThe wanton Queene that neuer lou'd for loue,\n\nFalse _Cleopatra,_ wholly set on gaine:\n\nWith all her slights did proue; yet vainly proue,\n\nFor her the loue of _Herod_ to obtaine.\n\nYet her allurements, all her courtly guile,\n\nHer smiles, her fauours, and her smooth deceit:\n\nCould not my face from _Herods_ minde exile,\n\nBut were with him of lesse then little weight.\n\nThat face and person that in _Asia_ late\n\nFor beauties Goddesse _Paphos_ Queene was tane:\n\nThat face that did captiue great _Iulius_ fate,\n\nThat very face that was _Anthonius_ bane.\n\nThat face that to be _Egipts_ pride was borne,\n\nThat face that all the world esteem'd so rare:\n\nDid _Herod_ hate, despise, neglect, and scorne,\n\nWhen with the same, he _Mariams_ did compare.\n\nThis made that I improuidently wrought,\n\nAnd on the wager euen my life did pawne:\n\nBecause I thought, and yet but truly thought,\n\nThat _Herods_ loue could not from me be drawne.\n\nBut now though out of time, I plainly see\n\nIt could be drawne, though neuer drawne from me:\n\nHad I but with humilitie bene grac'te,\n\nAs well as faire I might haue prou'd me wise:\n\nBut I did thinke because I knew me chaste,\n\nOne vertue for a woman, might suffice.\n\nThat mind for glory of our sexe might stand,\n\nWherein humilitie and chastitie\n\nDoth march with equall paces hand in hand,\n\nBut one if single seene, who setteth by?\n\nAnd I had singly one, but tis my ioy,\n\nThat I was euer innocent, though sower:\n\nAnd therefore can they but my life destroy,\n\nMy Soule is free from aduersaries power.) _Enter Doris._\n\nYou Princes great in power, and high in birth,\n\nBe great and high, I enuy not your hap:\n\nYour birth must be from dust: your power on earth,\n\nIn heau'n shall _Mariam_ sit in _Saraes_ lap. (thither,\n\n_Doris._ I heau'n, your beautie cannot bring you\n\nYour soule is blacke and spotted, full of sinne:\n\nYou in adultry liu'd nine yeare together,\n\nAnd heau'n will neuer let adultry in.\n\n_Mar:_ What art thou that dost poore _Mariam_ pursue?\n\nSome spirit sent to driue me to dispaire:\n\nWho sees for truth that _Mariam_ is vntrue,\n\nIf faire she be, she is as chaste as faire.\n\n_Doris._ I am that _Doris_ that was once belou'd,\n\nBelou'd by _Herod: Herods_ lawfull wife:\n\nTwas you that _Doris_ from his side remou'd,\n\nAnd rob'd from me the glory of my life.\n\n_Mar:_ Was that adultry: did not Moses say,\n\nThat he that being matcht did deadly hate:\n\nMight by permission put his wife away,\n\nAnd take a more belou'd to be his mate?\n\n_Doris._ What did he hate me for: for simple truth?\n\nFor bringing beautious babes for loue to him:\n\nFor riches: noble birth, or tender youth,\n\nOr for no staine did _Doris_ honour dim?\n\nOh tell me _Mariam,_ tell me if you knowe,\n\nWhich fault of these made _Herod Doris_ foe.\n\nThese thrice three yeares haue I with hands held vp,\n\nAnd bowed knees fast nailed to the ground;\n\nBesought for thee the dreggs of that same cup,\n\nThat cup of wrath that is for sinners found.\n\nAnd now thou art to drinke it; _Doris_ curse,\n\nVpon thy selfe did all this while attend,\n\nBut now it shall pursue thy children worse.\n\n_Mar:_ Oh _Doris_ now to thee my knees I bend,\n\nThat hart that neuer bow'd to thee doth bow:\n\nCurse not mine infants, let it thee suffice,\n\nThat Heau'n doth punishment to me allow.\n\nThy curse is cause that guiltles _Mariam_ dies.\n\n_Doris._ Had I ten thousand tongues, and eu'ry tongue\n\nInflam'd with poisons power, and steept in gall:\n\nMy curses would not answere for my wrong,\n\nThough I in cursing thee imployd them all.\n\nHeare thou that didst mount _Gerarim_ command,\n\nTo be a place whereon with cause to curse:\n\nStretch thy reuenging arme: thrust forth thy hand,\n\nAnd plague the mother much: the children worse.\n\nThrow flaming fire vpon the baseborne heads\n\nThat were begotten in vnlawfull beds.\n\nBut let them liue till they haue sence to know\n\nWhat tis to be in miserable state:\n\nThen be their neerest friends their ouerthrow,\n\nAttended be they by suspitious hate.\n\nAnd _Mariam,_ I doe hope this boy of mine\n\nShall one day come to be the death of thine. _Exit._\n\n_Mariam._ Oh ! Heauen forbid. I hope the world shall\n\nThis curse of thine shall be return'd on thee; (see,\n\nNow earth farewell, though I be yet but yong,\n\nYet I, me thinks, haue knowne thee too too long. _Exit._\n\n_Chorus._\n\nThe fairest action of our humane life,\n\nIs scorniug to reuenge an iniurie;\n\nFor who forgiues without a further strife,\n\nHis aduersaries heart to him doth tie.\n\nAnd tis a firmer conquest truely sed,\n\nTo winne the heart, then ouerthrow the head.\n\nIf we a worthy enemie doe finde,\n\nTo yeeld to worth, it must be nobly done:\n\nBut if of baser mettall be his minde,\n\nIn base reuenge there is no honor wonne.\n\nWho would a worthy courage ouerthrow,\n\nAnd who would wrastle with a worthles foe?\n\nWe say our hearts are great and cannot yeeld,\n\nBecause they cannot yeeld it proues them poore:\n\nGreat hearts are task't beyond their power, but seld\n\nThe weakest Lyon will the lowdest roare.\n\nTruths schoole for certaine doth this same allow,\n\nHigh hartednes doth sometimes teach to bow.\n\nA noble heart doth teach a vertuous scorne,\n\nTo scorne to owe a dutie ouer-long:\n\nTo scorne to be for benefits forborne,\n\nTo scorne to lie, to scorne to doe a wrong.\n\nTo scorne to beare an iniurie in minde,\n\nTo scorne a free-borne heart slaue-like to binde.\n\nBut if for wrongs we needs reuenge must haue,\n\nThen be our vengeance of the noblest kinde:\n\nDoe we his body from our furie saue,\n\nAnd let our hate preuaile against our minde?\n\nWhat can gainst him a greater vengeance bee,\n\nThen make his foe more worthy farre then hee?\n\nHad _Mariam_ scorn'd to leaue a due vnpaide,\n\nShee would to _Herod_ then haue paid her loue:\n\nAnd not haue bene by sullen passion swaide\n\nTo fixe her thoughts all iniurie aboue\n\nIs vertuous pride. Had _Mariam_ thus bene prou'd,\n\nLong famous life to her had bene allowd.\n\n**ACTUS QUINTUS.**\n\n**S C\u0152NA PRIMA.**\n\n_Nuntio._\n\nWhen, sweetest friend, did I so farre offend\n\nYour heauenly selfe: that you my fault to quit\n\nHaue made me now relator of her end,\n\nThe end of beautie? Chastitie and wit,\n\nWas none so haples in the fatall place,\n\nBut I, most wretched, for the Queene t'chuse,\n\nTis certaine I haue some ill boding face\n\nThat made me culd to tell this luckles newes.\n\nAnd yet no news to _Herod:_ were it new,\n\nTo him vnhappy t'had not bene at all:\n\nYet doe I long to come within his vew,\n\nThat he may know his wife did guiltles fall;\n\nAnd heere he comes. Your _Mariam_ greets you well.\n\n_Enter Herod._\n\n_Herod._ What? liues my _Mariam?_ ioy, exceeding ioy.\n\nShe shall not die. _Nun._ Heau'n doth your will repell.\n\n_Herod._ Oh doe not with thy words my life destroy,\n\nI prethy tell no dying-tale: thine eye\n\nWithout thy tongue doth tell but too too much:\n\nYet let thy tongues addition make me die,\n\nDeath welcome, comes to him whose griefe is such.\n\n_Nunti._ I went amongst the curious gazing troope,\n\nTo see the last of her that was the best:\n\nTo see if death had hart to make her stoope,\n\nTo see the Sunne admiring _Ph\u0153nix_ nest.\n\nWhen there I came, vpon the way I saw\n\nThe stately _Mariam_ not debas'd by feare:\n\nHer looke did seeme to keepe the world in awe,\n\nYet mildly did her face this fortune beare.\n\n_Herod._ Thou dost vsurpe my right, my tongue was\n\nTo be the instrument of _Mariams_ praise: (fram'd\n\nYet speake; she cannot be too often fam'd:\n\nAll tongues suffice not her sweet name to raise.\n\n_Nun._ But as she came she _Alexandra_ met,\n\nWho did her death (sweet Queene) no whit bewaile,\n\nBut as if nature she did quite forget,\n\nShe did vpon her daughter loudly raile.\n\n_Herod._ Why stopt you not her mouth? where had she\n\nTo darke that, that Heauen made so bright? (words\n\nOur sacred tongue no _Epthite_ affords,\n\nTo call her other then the worlds delight.\n\n_Nun._ Shee told her that her death was too too good,\n\nAnd that already she had liu'd too long:\n\nShe said, she sham'd to haue a part in blood\n\nOf her that did the princely _Herod_ wrong. (glory,\n\n_Herod._ Base picke-thanke Diuell. Shame, twas all her\n\nThat she to noble _Mariam_ was the mother:\n\nBut neuer shall it liue in any storie\n\nHer name, except to infamy ile smother.\n\nWhat answere did her princely daughter make?\n\n_Nun._ She made no answere, but she lookt the while,\n\nAs if thereof she scarce did notice take,\n\nYet smilde, a dutifull, though scornefull smile.\n\n_Her._ Sweet creature, I that looke to mind doe call,\n\nFull oft hath _Herod_ bene amaz'd withall.\n\n_Nun._ Go on, she came vnmou'd with pleasant grace,\n\nAs if to triumph her arriuall were:\n\nIn stately habite, and with cheefull face:\n\nYet eu'ry eye was moyst, but _Mariams_ there.\n\nWhen iustly opposite to me she came,\n\nShe pickt me out from all the crue:\n\nShe beckned to me, cald me by my name,\n\nFor she my name, my birth, and fortune knew.\n\n_Herod._ What did she name thee? happy, happy man,\n\nWilt thou not euer loue that name the better?\n\nBut what sweet tune did this faire dying Swan\n\nAfford thine eare: tell all, omit no letter.\n\n_Nun._ Tell thou my Lord, said she. _Her._ Mee, ment she\n\nIst true, the more my shame: I was her Lord, (mee?\n\nWere I not made her Lord, I still should bee:\n\nBut now her name must be by me adord.\n\nOh say, what faid she more? each word she sed\n\nShall be the food whereon my heart is fed. (breath,\n\n_Nun:_ Tell thou my Lord thou saw'ft me loose my\n\n_Herod._ Oh that I could that sentence now controule,\n\n_Nun._ If guiltily eternall be my death,\n\n_Her:_ I hold her chast eu'n in my inmost soule.\n\n_Nun:_ By three daies hence if wishes could reuiue,\n\nI know himselfe would make me oft aliue.\n\n_Herod._ Three daies: three houres, three minutes, not\n\nA minute in a thoufand parts diuided, (so much,\n\nMy penitencie for her death is such,\n\nAs in the first I wisht she had not died.\n\nBut forward in thy tale, _Nun:_ Why on she went,\n\nAnd after she some silent praier had fed:\n\nShe did as if to die she were content,\n\nAnd thus to heau'n her heau'nly soule is fled.\n\n_Herod._ But art thou sure there doth no life remaine?\n\nIst possible my _Mariam_ should be dead,\n\nIs there no tricke to make her breathe againe?\n\n_Nun:_ Her body is diuided from her head. (art,\n\n_Her:_ Why yet me thinkes there might be found by\n\nStrange waies of cure, tis sure rare things are don:\n\nBy an inuentiue head, and willing heart.\n\n_Nun:_ Let not my Lord your fancies idlely run.\n\nIt is as possible it should be seene,\n\nThat we should make the holy Abraham liue,\n\nThough he intomb'd two thousand yeares had bene,\n\nAs breath againe to slaughtred _Mariam_ giue,\n\nBut now for more assaults prepare your eares,\n\n_Herod._ There cannot be a further cause of mone,\n\nThis accident shall shelter me from feares:\n\nWhat can I feare? already _Mariams_ gone.\n\nYet tell eu'n what you will: _Nun:_ As I came by,\n\nFrom _Mariams_ death I saw vpon a tree,\n\nA man that to his necke a cord did tie:\n\nWhich cord he had designd his end to bee.\n\nWhen me he once discern'd, he downwards bow'd,\n\nAnd thus with fearefull voyce she cride alowd,\n\nGoe tell the King he trailed ere he tride,\n\nI am the cause that _Mariam_ causeles dide.\n\n_Herod._ Damnation take him, for it was the slaue\n\nThat said she ment with poisons deadly force\n\nTo end my life that she the Crowne might haue:\n\nWhich tale did _Mariam_ from her selfe diuorce.\n\nOh pardon me thou pure vnspotted Ghost,\n\nMy punishment must needes sufficient bee,\n\nIn missing that content I valued most:\n\nWhich was thy admirable face to see.\n\nI had but one inestimable Iewell,\n\nYet one I had no monarch had the like,\n\nAnd therefore may I curse my selfe as cruell:\n\nTwas broken by a blowe my selfe did strike.\n\nI gaz'd thereon and neuer thought me blest,\n\nBut when on it my dazled eye might rest:\n\nA pretious Mirror made by wonderous art,\n\nI prizd it ten times dearer then my Crowne,\n\nAnd laide it vp fast foulded in my heart;\n\nYet I in suddaine choler cast it downe.\n\nAnd pasht it all to peeces: twas no foe,\n\nThat robd me of it; no _Arabian_ host,\n\nNor no _Armenian_ guide hath vsde me so:\n\nBut _Herods_ wretched selfe hath _Herod_ crost.\n\nShe was my gracefull moytie, me accurst,\n\nTo slay my better halfe and saue my worst.\n\nBut sure she is not dead you did but iest,\n\nTo put me in perplexitie a while,\n\nTwere well indeed if I could so be drest:\n\nI see she is aliue, me thinkes you smile.\n\n_Nun:_ If sainted _Abel_ yet deceased bee,\n\nTis certaine _Mariam_ is as dead as hee.\n\n_Her:_ Why then goe call her to me, bid her now\n\nPut on faire habite, stately ornament:\n\nAnd let no frowne oreshade her smoothest brow,\n\nIn her doth _Herod_ place his whole content. (sence,\n\n_Nun:_ Sheel come in stately weedes to please your\n\nIf now she come attirde in robe of heauen:\n\nRemember you your selfe did send her hence,\n\nAnd now to you she can no more be giuen. faire,\n\n_Herod._ Shee's dead, hell take her murderers, she was\n\nOh what a hand she had, it was so white,\n\nIt did the whitenes of the snowe impaire:\n\nI neuer more shall see so sweet a fight. (hands;\n\n_Nun:_ Tis true, her hand was rare. _Her:_ her hand? her\n\nShe had not singly one of beautie rare,\n\nBut such a paire as heere where _Herod_ stands,\n\nHe dares the world to make to both compare.\n\nAccursed _Salome,_ hadst thou bene still,\n\nMy _Mariam_ had bene breathing by my side:\n\nOh neuer had I: had I had my will,\n\nSent forth command, that _Mariam_ should haue dide.\n\nBut _Salome_ thou didst with enuy vexe,\n\nTo see thy selfe out-matched in thy sexe:\n\nVpon your sexes forehead _Mariam_ sat,\n\nTo grace you all like an imperiall crowne,\n\nBut you fond foole haue rudely pusht thereat,\n\nAnd proudly puld your proper glory downe.\n\nOne smile of hers: Nay, not so much a: looke\n\nWas worth a hundred thousand such as you,\n\n_Iudea_ how canst thou the wretches brooke,\n\nThat robd from thee the fairest of the crew?\n\nYou dwellers in the now depriued land,\n\nWherein the matchles _Mariam_ was bred:\n\nWhy graspe not each of you a sword in hand,\n\nTo ayme at me your cruell Soueraignes head.\n\nOh when you thinke of _Herod_ as your King,\n\nAnd owner of the pride of _Palestine:_\n\nThis act to your remembrance likewise bring,\n\nTis I haue ouerthrowne your royall line.\n\nWithin her purer vaines the blood did run,\n\nThat from her Grandam _Sara_ she deriu'd,\n\nWhose beldame age the loue of Kings hath wonne,\n\nOh that her issue had as long bene li'ud.\n\nBut can her eye be made by death obscure?\n\nI cannot thinke but it must sparkle still:\n\nFoule sacriledge to rob those lights so pure,\n\nFrom out a Temple made by heau'nly skill.\n\nI am the Villaine that haue done the deed,\n\nThe cruell deed, though by anothers hand,\n\nMy word though not my sword made _Mariam_ bleed,\n\n_Hircanus_ Grandchild did at my command.\n\nThat _Mariam_ that I once did loue so deare,\n\nThe partner of my now detested bed,\n\nWhy shine you sun with an aspect so cleare?\n\nI tell you once againe my _Mariams_ dead.\n\nYou could but shine, if some _Egiptian_ blows,\n\nOr _\u00c6thiopian_ doudy lose her life:\n\nThis was, then wherefore bend you not your brows,\n\nThe King of _Iuries_ faire and spotles wife.\n\nDenie thy beames, and _Moone_ refuse thy light,\n\nLet all the starres be darke, let _Iuries_ eye\n\nNo more distinguish which is day and night:\n\nSince her best birth did in her bosome die.\n\nThose fond Idolaters the men of _Greece_ ,\n\nMaintaine these orbes are safely gouerned;\n\nThat each within themselues haue Gods a peece,\n\nBy whom their stedfast course is iustly led.\n\nBut were it so, as so it cannot bee,\n\nThey all would put their mourning garments on:\n\nNot one of them would yeeld a light to mee,\n\nTo me that is the cause that _Mariams_ gon.\n\nFor though they fame their _Saturne_ melancholy,\n\nOf sowre behauiours, and of angry moode:\n\nThey fame him likewise to be iust and holy,\n\nAnd iustice needes must seeke reuenge for blood.\n\nTheir _Ioue,_ if _Ioue_ he were, would sure desire,\n\nTo punish him that slew so faire a lasse:\n\nFor _L\u00e6daes_ beautie set his heart on fire,\n\nYet she not halfe so faire as _Mariam_ was.\n\nAnd _Mars_ would deeme his _Venus_ had bene slaine,\n\n_Sol_ to recouer her would neuer sticke;\n\nFor if he want the power her life to gaine:\n\nThen Physicks God is but an Empericke.\n\nThe Queene of loue would storme for beauties sake,\n\nAnd _Hermes_ too, since he bestow'd her wit,\n\nThe nights pale light for angrie griefe would shake,\n\nTo see chast _Mariam_ die in age vnfit.\n\nBut oh I am deceiu'd, she past them all\n\nIn euery gift, in euery propertie:\n\nHer Excellencies wrought her timeles fall,\n\nAnd they reioyc'd, not grieu'd to see her die.\n\nThe _Paphian_ Goddesse did repent her wast,\n\nWhen me to one such beautie did allow:\n\n_Mercurius_ thought her wit his wit surpast,\n\nAnd _Cinthia_ enui'd _Mariams_ brighter brow.\n\nBut these are fictions, they are voyd of sence,\n\nThe _Greekes_ but dreame, and dreaming falsehoods tell:\n\nThey neither can offend nor giue defence,\n\nAnd not by them it was my _Mariam_ fell.\n\nIf she had bene like an _Egipttan_ blacke,\n\nAnd not so faire, she had bene longer liude:\n\nHer ouerflow of beautie turned backe,\n\nAnd drownde the spring from whence it was deriude.\n\nHer heau'nly beautie twas that made me thinke\n\nThat it with chastitie could neuer dwell:\n\nBut now I see that heau'n in her did linke,\n\nA spirit and a perfon to excell.\n\nIle muffle vp my selfe in endles night,\n\nAnd neuer let mine eyes behold the light.\n\nRetire thy selfe vile monster, worse then hee\n\nThat staind the virgin earth with brothers blood,\n\nStill in some vault or denne inclosed bee,\n\nWhere with thy teares thou maist beget a flood,\n\nWhich flood in time may drowne thee: happie day\n\nWhen thou at once shalt die and finde a graue,\n\nA stone vpon the vault, some one shall lay,\n\nWhich monument shall an inscription haue.\n\nAnd these shall be the words it shall containe,\n\n_Heere_ Herod _lies, that hath his Mariam slaine._\n\n_Chorus._\n\nWho euer hath beheld with steadfast eye,\n\nThe strange euents of this one onely day:\n\nHow many were deceiu'd? How many die,\n\nThat once to day did grounds of safetie lay?\n\nIt will from them all certaintie bereue,\n\nSince twice sixe houres so many can deceiue.\n\nThis morning _Herod_ held for surely dead,\n\nAnd all the _Iewes_ on _Mariam_ did attend:\n\nAnd _Constabarus_ rise from _Saloms_ bed,\n\nAnd neither dreamd of a diuorce or end.\n\n_Pheroras_ ioyd that he might haue his wife,\n\nAnd _Babus_ sonnes for safetie of their life.\n\nTo night our _Herod_ doth aliue remaine,\n\nThe guiltles _Mariam_ is depriu'd of breath:\n\nStout _Constabarus_ both diuorst and slaine,\n\nThe valiant sonnes of _Baba_ haue their death.\n\n_Pheroras_ sure his loue to be bereft,\n\nIf _Salome_ her sute vnmade had left.\n\n_Herod_ this morning did expect with ioy,\n\nTo see his _Mariams_ much beloued face:\n\nAnd yet ere night he did her life destroy,\n\nAnd surely thought she did her name disgrace.\n\nYet now againe so short do humors last,\n\nHe both repents her death and knowes her chast.\n\nHad he with wisedome now her death delaide,\n\nHe at his pleasure might command her death:\n\nBut now he hath his power so much betraide,\n\nAs all his woes cannot restore her breath.\n\nNow doth he strangely lunatickly raue,\n\nBecause his _Mariams_ life he cannot saue.\n\nThis daies euents were certainly ordainde,\n\nTo be the warning to posteritie:\n\nSo many changes are therein containde,\n\nSo admirablie strange varietie.\n\nThis day alone, our sagest _Hebrewes_ shall\n\nIn after times the schoole of wisedome call.\n\nFINIS\n**THE DUCHESS OF MALFI**\n\n**John Webster**\n\n**DRAMATIS PERSONAE**\n\nFERDINAND, _Duke of Calabria_\n\nTHE CARDINAL, _his brother_\n\nANTONIO BOLOGNA, _steward of the household to the Duchess_\n\nDELIO, _his friend_\n\nDANIEL DE BOSOLA, _gentleman of the horse to the Duchess_\n\nCASTRUCCIO\n\nMARQUIS OF PESCARA\n\nCOUNT MALATESTI\n\nRODERIGO\n\nSILVIO\n\nGRISOLAN\n\nDOCTOR\n\nDUCHESS OF MALFI\n\nCARIOLA, _her woman_\n\nJULIA, _Castruccio's wife, and the Cardinal's mistress_\n\nOLD LADY, LADIES, AND CHILDREN\n\n_Several_ MADMEN, PILGRIMS, EXECUTIONERS, OFFICERS, ATTENDANTS, &c.\n\nSCENE: _Malfi, Rome, and Milan_\n\n**ACT I**\n\n**S CENE I: _The presence-chamber in the_ DUCHESS'S _palace at Malfi._**\n\nEnter ANTONIO _and_ DELIO.\n\nDELIO. You are welcome to your country, dear Antonio;\n\nYou have been long in France, and you return\n\nA very formal Frenchman in your habit.\n\nHow do you like the French court?\n\nANTONIO. I admire it;\n\nIn seeking to reduce both state and people\n\nTo a fixed order, their judicious king\n\nBegins at home; quits first his royal palace\n\nOf flattering sycophants, of dissolute\n\nAnd infamous persons,\u2014which he sweetly terms\n\nHis master's master-piece, the work of Heaven:\n\nConsidering duly that a prince's court\n\nIs like a common fountain, whence should flow\n\nPure silver drops in general, but if't chance\n\nSome cursed example poison 't near the head,\n\nDeath and diseases through the whole land spread.\n\nAnd what is 't makes this blessed government\n\nBut a most provident council, who dare freely\n\nInform him the corruption of the times?\n\nThough some o' the court hold it presumption\n\nTo instruct princes what they ought to do,\n\nIt is a noble duty to inform them\n\nWhat they ought to foresee.\u2014Here comes Bosola,\n\nThe only court-gall; yet I observe his railing\n\nIs not for simple love of piety:\n\nIndeed, he rails at those things which he wants;\n\nWould be as lecherous, covetous, or proud,\n\nBloody, or envious, as any man,\n\nIf he had means to be so.\u2014Here's the cardinal.\n\n_Enter_ CARDINAL _and_ BOSOLA.\n\nBOSOLA. I do haunt you still.\n\nCARDINAL. So.\n\nBOSOLA. I have done you better service than to be slighted thus. Miserable age, where only the reward of doing well is the doing of it!\n\nCARDINAL. You enforce your merit too much.\n\nBOSOLA. I fell into the galleys in your service; where, for two years together, I wore two towels instead of a shirt, with a knot on the shoulder, after the fashion of a Roman mantle. Slighted thus! I will thrive some way: blackbirds fatten best in hard weather; why not I in these dog-days?\n\nCARDINAL. Would you could become honest!\n\nBOSOLA. With all your divinity do but direct me the way to it. I have known many travel far for it, and yet return as arrant knaves as they went forth, because they carried themselves always along with them. [ _Exit_ Cardinal] Are you gone? Some fellows, they say, are possessed with the devil, but this great fellow were able to possess the greatest devil, and make him worse.\n\nANTONIO. He hath denied thee some suit?\n\nBOSOLA. He and his brother are like plum-trees that grow crooked over standing-pools; they are rich and o'er-laden with fruit, but none but crows, magpies, and caterpillars feed on them. Could I be one of their flattering panders, I would hang on their ears like a horseleech, till I were full, and then drop off. I pray, leave me. Who would rely upon these miserable dependancies, in expectation to be advanced tomorrow? what creatures ever fed worse than hoping Tantalus? nor ever died any man more fearfully than he had hoped for a pardon. There are rewards for hawks and dogs when they have done us service; but for a soldier that hazards his limbs in a battle, nothing but a kind of geometry is his last supportation.\n\nDELIO. Geometry!\n\nBOSOLA. Ay, to hang in a fair pair of slings, take his latter swing in the world upon an honorable pair of crutches, from hospital to hospital. Fare ye well, sir: and yet do not you scorn us; for places in the court are but like beds in the hospital, where this man's head lies at that man's foot, and so lower and lower. [ _Exit._\n\nDELIO. I knew this fellow seven years in the galleys\n\nFor a notorious murder; and 'twas thought\n\nThe cardinal suborned it: he was released\n\nBy the French general, Gaston de Foix,\n\nWhen he recovered Naples.\n\nANTONIO. 'Tis great pity\n\nHe should be thus neglected: I have heard\n\nHe's very valiant. This foul melancholy\n\nWill poison all his goodness; for, I'll tell you,\n\nIf too immoderate sleep be truly said\n\nTo be an inward rust into the soul,\n\nIt then doth follow want of action\n\nBreeds all black malcontents; and their close rearing,\n\nLike moths in cloth, do hurt for want of wearing.\n\nDELIO. The presence 'gins to fill: you promised me\n\nTo make me the partaker of the natures\n\nOf some of your great courtiers.\n\nANTONIO. The lord cardinal's,\n\nAnd other strangers' that are now in court?\n\nI shall.\u2014Here comes the great Calabrian duke.\n\n_Enter_ FERDINAND, CASTRUCCIO, SILVIO, RODERIGO, GRISOLAN, _and_ ATTENDANTS.\n\nFERDINAND. Who took the ring oftenest?\n\nSILVIO. Antonio Bologna, my lord.\n\nFERDINAND. Our sister duchess's great-master of her household? give him the jewel.\u2014When shall we leave this sportive action, and fall to action indeed?\n\nCASTRUCCIO. Methinks, my lord, you should not desire to go to war in person.\n\nFERDINAND. Now for some gravity:\u2014why, my lord?\n\nCASTRUCCIO. It is fitting a soldier arise to be a prince, but not necessary a prince descend to be a captain. Ferdinand. No?\n\nCASTRUCCIO. No, my lord; he were far better do it by a deputy.\n\nFERDINAND. Why should he not as well sleep or eat by a deputy?\n\nThis might take idle, offensive, and base office from him, whereas the other deprives him of honor.\n\nCASTRUCCIO. Believe my experience, that realm is never long in quiet where the ruler is a soldier.\n\nFERDINAND. Thou toldest me thy wife could not endure fighting.\n\nCASTRUCCIO. True, my lord.\n\nFERDINAND. And of a jest she broke of a captain she met full of wounds: I have forgot it.\n\nCASTRUCCIO. She told him, my lord, he was a pitiful fellow, to lie, like the children of Ismael, all in tents.\n\nFERDINAND. Why, there's a wit were able to undo all the surgeons o' the city; for although gallants should quarrel, and had drawn their weapons, and were ready to go to it, yet her persuasions would make them put up.\n\nCASTRUCCIO. That she would, my lord.\u2014How do you like my Spanish jennet?\n\nRODERIGO. He is all fire.\n\nFERDINAND. I am of Pliny's opinion, I think he was begot by the wind; he runs as if he were ballasted with quicksilver.\n\nSILVIO. True, my lord, he reels from the tilt often.\n\nRODERIGO _and_ Grisolan. Ha! ha! ha!\n\nFERDINAND. Why do you laugh? methinks you that are courtiers should be my touchwood, take fire when I give fire; that is, laugh but when I laugh, were the subject never so witty.\n\nCASTRUCCIO. True, my lord: I myself have heard a very good jest, and have scorned to seem to have so silly a wit as to understand it.\n\nFERDINAND. But I can laugh at your fool, my lord.\n\nCASTRUCCIO. He cannot speak, you know, but he makes faces: my lady cannot abide him.\n\nFERDINAND. No?\n\nCASTRUCCIO. Nor endure to be in merry company; for she says too much laughing, and too much company, fills her too full of the wrinkle.\n\nFERDINAND. I would, then, have a mathematical instrument made for her face, that she might not laugh out of compass.\u2014I shall shortly visit you at Milan, Lord Silvio.\n\nSILVIO. Your grace shall arrive most welcome.\n\nFERDINAND. You are a good horseman, Antonio: you have excellent riders in France: what do you think of good horsemanship?\n\nANTONIO. Nobly, my lord: as out of the Grecian horse issued many famous princes, so out of brave horsemanship arise the first sparks of growing resolution that raise the mind to noble action.\n\nFERDINAND. You have bespoke it worthily.\n\nSILVIO. Your brother, the lord cardinal, and sister duchess.\n\n**S CENE II**\n\n_Re-enter_ CARDINAL _and_ FERDINAND, _with_ DUCHESS _and_ CARIOLA.\n\nCARDINAL. Are the galleys come about?\n\nGRISOLAN. They are, my lord.\n\nFERDINAND. Here's the Lord Silvio is come to take his leave.\n\nDELIO. Now, sir, your promise; what's that cardinal?\n\nI mean his temper? they say he's a brave fellow,\n\nWill play his five thousand crowns at tennis, dance,\n\nCourt ladies, and one that hath fought single combats.\n\nANTONIO. Some such flashes superficially hang on him for form; but observe his inward character: he is a melancholy churchman; the spring in his face is nothing but the engendering of toads; where he is jealous of any man, he lays worse plots for them than ever was imposed on Hercules, for he strews in his way flatterers, panders, intelligencers, atheists, and a thousand such political monsters. He should have been Pope; but instead of coming to it by the primitive decency of the church, he did bestow bribes so largely and so impudently as if he would have carried it away without Heaven's knowledge. Some good he hath done\u2014\n\nDELIO. You have given too much of him. What's his brother?\n\nANTONIO. The duke there? a most perverse and turbulent nature;\n\nWhat appears in him mirth is merely outside;\n\nIf he laugh heartily, it is to laugh\n\nAll honesty out of fashion.\n\nDELIO. Twins?\n\nANTONIO. In quality.\n\nHe speaks with others' tongues, and hears men's suits\n\nWith others' ears; will seem to sleep o' the bench\n\nOnly to entrap offenders in their answers;\n\nDooms men to death by information;\n\nRewards by hearsay.\n\nDELIO. Then the law to him\n\nIs like a foul black cobweb to a spider,\u2014\n\nHe makes it his dwelling and a prison\n\nTo entangle those shall feed him.\n\nANTONIO. Most true:\n\nHe never pays debts unless they be shrewd turns,\n\nAnd those he will confess that he doth owe.\n\nLast, for his brother there, the cardinal,\n\nThey that do flatter him most say oracles\n\nHang at his lips; and verily I believe them,\n\nFor the devil speaks in them.\n\nBut for their sister, the right noble duchess,\n\nYou never fixed your eye on three fair medals\n\nCast in one figure, of so different temper.\n\nFor her discourse, it is so full of rapture,\n\nYou only will begin then to be sorry\n\nWhen she doth end her speech, and wish, in wonder,\n\nShe held it less vain-glory to talk much,\n\nThan your penance to hear her: whilst she speaks,\n\nShe throws upon a man so sweet a look,\n\nThat it were able to raise one to a galliard\n\nThat lay in a dead palsy, and to dote\n\nOn that sweet countenance; but in that look\n\nThere speaketh so divine a continence\n\nAs cuts off all lascivious and vain hope.\n\nHer days are practised in such noble virtue,\n\nThat sure her nights, nay, more, her very sleeps,\n\nAre more in Heaven than other ladies' shrifts.\n\nLet all sweet ladies break their flattering glasses,\n\nAnd dress themselves in her.\n\nDELIO. Fie, Antonio,\n\nYou play the wire-drawer with her commendations.\n\nANTONIO. I'll case the picture up: only thus much;\n\nAll her particular worth grows to this sum,\u2014\n\nShe stains the time past, lights the time to come.\n\nCARIOLA. You must attend my lady in the gallery,\n\nSome half an hour hence.\n\nANTONIO. I shall.\n\n_Exeunt_ ANTONIO _and_ DELIO.\n\nFERDINAND. Sister, I have a suit to you.\n\nDUCHESS. To me, sir?\n\nFERDINAND. A gentleman here, Daniel de Bosola,\n\nOne that was in the galleys\u2014\n\nDUCHESS. Yes, I know him.\n\nFERDINAND. A worthy fellow he is: pray, let me entreat for\n\nThe provisorship of your horse.\n\nDUCHESS. Your knowledge of him\n\nCommends him and prefers him.\n\nFERDINAND. Call him hither.\n\n_Exit_ ATTENDANT.\n\nWe are now upon parting. Good Lord Silvio,\n\nDo us commend to all our noble friends\n\nAt the leaguer.\n\nSILVIO. Sir, I shall.\n\nFERDINAND. You are for Milan?\n\nSILVIO. I am.\n\nDUCHESS. Bring the caroches. We'll bring you down to the haven.\n\n_Exeunt_ DUCHESS, SILVIO, CASTRUCCIO, RODERIGO, GRISOLAN, CARIOLA, JULIA, _and_ ATTENDANTS.\n\nCARDINAL. Be sure you entertain that Bosola\n\nFor your intelligence: I would not be seen in 't;\n\nAnd therefore many times I have slighted him\n\nWhen he did court our furtherance, as this morning.\n\nFERDINAND. Antonio, the great-master of her household,\n\nHad been far fitter.\n\nCARDINAL. You are deceived in him:\n\nHis nature is too honest for such business.\u2014\n\nHe comes: I'll leave you. [ _Exit_\n\n_Re-enter_ BOSOLA.\n\nBOSOLA. I was lured to you.\n\nFERDINAND. My brother, here, the cardinal, could never Abide you.\n\nBOSOLA. Never since he was in my debt.\n\nFERDINAND. May be some oblique character in your face Made him suspect you.\n\nBOSOLA. Doth he study physiognomy?\n\nThere's no more credit to be given to the face\n\nThan to a sick man's urine, which some call\n\nThe physician's whore because she cozens him..\n\nHe did suspect me wrongfully.\n\nFERDINAND. For that\n\nYou must give great men leave to take their times.\n\nDistrust doth cause us seldom be deceived:\n\nYou see the oft shaking of the cedar-tree\n\nFastens it more at root.\n\nBOSOLA. Yet, take heed;\n\nFor to suspect a friend unworthily\n\nInstructs him the next way to suspect you,\n\nAnd prompts him to deceive you.\n\nFERDINAND. There's gold.\n\nBOSOLA. So:\n\nWhat follows? never rained such showers as these\n\nWithout thunderbolts i' the tail of them: whose throat must I cut?\n\nFERDINAND. Your inclination to shed blood rides post\n\nBefore my occasion to use you. I give you that\n\nTo live i' the court here, and observe the duchess;\n\nTo note all the particulars of her havior,\n\nWhat suitors do solicit her for marriage,\n\nAnd whom she best affects. She's a young widow:\n\nI would not have her marry again.\n\nBOSOLA. No, sir?\n\nFERDINAND. Do not you ask the reason; but be satisfied\n\nI say I would not.\n\nBOSOLA. It seems you would create me\n\nOne of your familiars.\n\nFERDINAND. Familiar! what's that?\n\nBOSOLA. Why, a very quaint invisible devil in flesh,\n\nAn intelligencer.\n\nFERDINAND. Such a kind of thriving thing\n\nI would wish thee; and ere long thou mayest arrive\n\nAt a higher place by 't.\n\nBOSOLA. Take your devils,\n\nWhich hell calls angels, these cursed gifts would make\n\nYou a corrupter, me an impudent traitor;\n\nAnd should I take these, they'd take me to hell.\n\nFERDINAND. Sir, I'll take nothing from you that I have given:\n\nThere is a place that I procured for you\n\nThis morning, the provisorship o' the horse;\n\nHave you heard on 't?\n\nBOSOLA. No.\n\nFERDINAND. 'Tis yours: is't not worth thanks?\n\nBOSOLA. I would have you curse yourself now, that your bounty\n\n(Which makes men truly noble) e'er should make me\n\nA villain. O, that to avoid ingratitude\n\nFor the good deed you have done me, I must do\n\nAll the ill man can invent! Thus the devil\n\nCandies all sins o'er; and what Heaven terms vile,\n\nThat names he complimental.\n\nFERDINAND. Be yourself;\n\nKeep your old garb of melancholy; 'twill express\n\nYou envy those that stand above your reach,\n\nYet strive not to come near 'em: this will gain\n\nAccess to private lodgings, where yourself\n\nMay, like a politic dormouse\u2014\n\nBOSOLA. As I have seen some\n\nFeed in a lord's dish, half asleep, not seeming\n\nTo listen to any talk; and yet these rogues\n\nHave cut his throat in a dream. What's my place?\n\nThe provisorship o' the horse? say, then, my corruption\n\nGrew out of horse-dung: I am your creature.\n\nFERDINAND. Away!\n\nBOSOLA. Let good men, for good deeds, covet good fame,\n\nSince place and riches oft are bribes of shame:\n\nSometimes the devil doth preach. [ _Exit_\n\n_Re-enter_ DUCHESS, CARDINAL, _and_ CARIOLA.\n\nCARDINAL. We are to part from you; and your own discretion Must now be your director.\n\nFERDINAND. YOU are a widow:\n\nYou know already what man is; and therefore\n\nLet not youth, high promotion, eloquence\u2014\n\nCARDINAL. NO,\n\nNor any thing without the addition, honor,\n\nSway your high blood.\n\nFERDINAND. Marry! they are most luxurious\n\nWill wed twice.\n\nCARDINAL. O, fie!\n\nFERDINAND. Their livers are more spotted\n\nThan Laban's sheep.\n\nDUCHESS. Diamonds are of most value,\n\nThey say, that have passed through most jewelers' hands.\n\nFERDINAND. Whores by that rule are precious.\n\nDUCHESS. Will you hear me?\n\nI'll never marry.\n\nCARDINAL. So most widows say;\n\nBut commonly that motion lasts no longer\n\nThan the turning of an hour-glass: the funeral sermon\n\nAnd it end both together.\n\nFERDINAND. Now hear me:\n\nYou live in a rank pasture, here, i' the court;\n\nThere is a kind of honey-dew that's deadly;\n\n'Twill poison your fame; look to 't: be not cunning;\n\nFor they whose faces do belie their hearts\n\nAre witches ere they arrive at twenty years,\n\nAy, and give the devil suck.\n\nDUCHESS. This is terrible good counsel.\n\nFERDINAND. Hypocrisy is woven of a fine small thread, Subtler than Vulcan's engine: yet, believe 't,\n\nYour darkest actions, nay, your privat'st thoughts,\n\nWill come to light.\n\nCARDINAL. You may flatter yourself,\n\nAnd take your own choice; privately be married\n\nUnder the eaves of night\u2014\n\nFERDINAND. Think 't the best voyage\n\nThat e'er you made; like the irregular crab,\n\nWhich, though 't goes backward, thinks that it goes right\n\nBecause it goes its own way; but observe,\n\nSuch weddings may more properly be said\n\nTo be executed than celebrated.\n\nCARDINAL. The marriage night\n\nIs the entrance into some prison.\n\nFERDINAND. And those joys,\n\nThose lustful pleasures, are like heavy sleeps\n\nWhich do fore-run man's mischief.\n\nCARDINAL. Fare you well.\n\nWisdom begins at the end: remember it. [ _Exit_\n\nDUCHESS. I think this speech between you both was studied,\n\nIt came so roundly off.\n\nFERDINAND. You are my sister;\n\nThis was my father's poniard, do you see?\n\nI'd be loth to see 't look rusty, 'cause 'twas his.\n\nI would have you give o'er these chargeable revels:\n\nA visor and a mask are whispering-rooms\n\nThat were never built for goodness;\u2014fare ye well;\u2014\n\nAnd women like that part which, like the lamprey,\n\nHath never a bone in 't.\n\nDUCHESS. Fie, sir!\n\nFERDINAND. Nay,\n\nI mean the tongue; variety of courtship:\n\nWhat cannot a neat knave with a smooth tale\n\nMake a woman believe? Farewell, lusty widow. [ _Exit_\n\nDUCHESS. Shall this move me? If all my royal kindred\n\nLay in my way unto this marriage,\n\nI'd make them my low footsteps: and even now,\n\nEven in this hate, as men in some great battles,\n\nBy apprehending danger, have achieved\n\nAlmost impossible actions (I have heard soldiers say so),\n\nSo I through frights and threatenings will assay\n\nThis dangerous venture. Let old wives report\n\nI winked and chose a husband. \u2014Cariola,\n\nTo thy known secrecy I have given up\n\nMore than my life\u2014my fame.\n\nCARIOLA. Both shall be safe;\n\nFor I'll conceal this secret from the world\n\nAs warily as those that trade in poison\n\nKeep poison from their children.\n\nDUCHESS. Thy protestation\n\nIs ingenious and hearty: I believe it.\n\nIs Antonio come?\n\nCARIOLA. He attends you.\n\nDUCHESS. Good, dear soul,\n\nLeave me; but place thyself behind the arras,\n\nWhere thou may'st overhear us. Wish me good speed;\n\nFor I am going into a wilderness\n\nWhere I shall find nor path nor friendly clue\n\nTo be my guide. CARIOLA goes _behind the arras_[ 12\n\n_Enter_ ANTONIO.\n\nI sent for you: sit down;\n\nTake pen and ink, and write: are you ready?\n\nANTONIO. Yes.\n\nDUCHESS. What did I say?\n\nANTONIO. That I should write somewhat.\n\nDUCHESS. O, I remember.\n\nAfter these triumphs and this large expense,\n\nIt's fit, like thrifty husbands, we inquire\n\nWhat's laid up for to-morrow.\n\nANTONIO. So please your beauteous excellence.\n\nDUCHESS. Beauteous!\n\nIndeed, I thank you: I look young for your sake;\n\nYou have ta'en my cares upon you.\n\nANTONIO. I'll fetch your grace\n\nThe particulars of your revenue and expense.\n\nDUCHESS. O, you are\n\nAn upright treasurer: but you mistook;\n\nFor when I said I meant to make inquiry\n\nWhat's laid up for to-morrow, I did mean\n\nWhat's laid up yonder for me.\n\nANTONIO. Where?\n\nDUCHESS. In Heaven.\n\nI am making my will (as 'tis fit princes should,\n\nIn perfect memory), and, I pray, sir, tell me,\n\nWere not one better make it smiling, thus,\n\nThan in deep groans and terrible ghastly looks,\n\nAs if the gifts we parted with procured\n\nThat violent distraction?\n\nANTONIO. O, much better.\n\nDUCHESS. If I had a husband now, this care were quit:\n\nBut I intend to make you overseer.\n\nWhat good deed shall we first remember? say.\n\nANTONIO. Begin with that first good deed began i' the world\n\nAfter man's creation, the sacrament of marriage:\n\nI'd have you first provide for a good husband;\n\nGive him all.\n\nDUCHESS. All!\n\nANTONIO. Yes, your excellent self.\n\nDUCHESS. In a winding-sheet?\n\nANTONIO. In a couple.\n\nDUCHESS. Saint Winifred, that were a strange will!\n\nANTONIO. 'Twere stranger if there were no will in you\n\nTo marry again.\n\nDUCHESS. What do you think of marriage?\n\nANTONIO. I take 't, as those that deny purgatory,\n\nIt locally contains or Heaven or hell;\n\nThere's no third place in 't.\n\nDUCHESS. How do you affect it?\n\nANTONIO. My banishment, feeding my melancholy,\n\nWould often reason thus.\n\nDUCHESS. Pray, let's hear it.\n\nANTONIO. Say a man never marry, nor have children,\n\nWhat takes that from him? only the bare name\n\nOf being a father, or the weak delight\n\nTo see the little wanton ride a-cock-horse\n\nUpon a painted stick, or hear him chatter\n\nLike a taught starling.\n\nDUCHESS. Fie, fie, what's all this?\n\nOne of your eyes is blood-shot; use my ring to 't,\n\nThey say 'tis very sovereign: 'twas my wedding-ring.\n\nAnd I did vow never to part with it\n\nBut to my second husband.\n\nANTONIO. You have parted with it now.\n\nDUCHESS. Yes, to help your eye-sight.\n\nANTONIO. You have made me stark blind.\n\nDUCHESS. How?\n\nANTONIO. There is a saucy and ambitious devil\n\nIs dancing in this circle.\n\nDUCHESS. Remove him.\n\nANTONIO. HOW?\n\nDUCHESS. There needs small conjuration, when your finger\n\nMay do it: thus; is it fit?\n\n[ _She puts the ring upon his finger; he kneels_\n\nANTONIO. What said you?\n\nDUCHESS. Sir,\n\nThis goodly roof of yours is too low built;\n\nI cannot stand upright in 't nor discourse,\n\nWithout I raise it higher: raise yourself;\n\nOr, if you please, my hand to help you: so. [ _Raises him_\n\nANTONIO. Ambition, madam, is a great man's madness,\n\nThat is not kept in chains and close-pent rooms,\n\nBut in fair lightsome lodgings, and is girt\n\nWith the wild noise of prattling visitants,\n\nWhich makes it lunatic beyond all cure.\n\nConceive not I am so stupid but I aim\n\nWhereto your favors tend: but he's a fool\n\nThat, being a-cold, would thrust his hands i' the fire\n\nTo warm them.\n\nDUCHESS. So, now the ground's broke,\n\nYou may discover what a wealthy mine\n\nI make you lord of.\n\nANTONIO. O my unworthiness!\n\nDUCHESS. You were ill to sell yourself:\n\nThis darkening of your worth is not like that\n\nWhich tradesmen use i' the city; their false lights\n\nAre to rid bad wares off: and I must tell you,\n\nIf you will know where breathes a complete man\n\n(I speak it without flattery), turn your eyes,\n\nAnd progress through yourself.\n\nANTONIO. Were there nor Heaven nor hell,\n\nI would be honest: I have long served virtue,\n\nAnd ne'er ta'en wages of her.\n\nDUCHESS. Now she pays it.\n\nThe misery of us that are borm great!\n\nWe are forced to woo, because none dare woo us;\n\nAnd as a tyrant doubles with his words,\n\nAnd fearfully equivocates, so we\n\nAre forced to express our violent passions\n\nIn riddles and in dreams, and leave the path\n\nOf simple virtue, which was never made\n\nTo seem the thing it is not. Go, go brag\n\nYou have left me heartless; mine is in your bosom:\n\nI hope 'twill multiply love there. You do tremble:\n\nMake not your heart so dead a piece of flesh,\n\nTo fear more than to love me. Sir, be confident:\n\nWhat is 't distracts you? This is flesh and blood, sir;\n\n'Tis not the figure cut in alabaster\n\nKneels at my husband's tomb. Awake, awake, man!\n\nI do here put off all vain ceremony,\n\nAnd only do appear to you a young widow\n\nThat claims you for her husband, and, like a widow,\n\nI use but half a blush in 't.\n\nANTONIO. Truth speak for me;\n\nI will remain the constant sanctuary\n\nOf your good name.\n\nDUCHESS. I thank you, gentle love:\n\nAnd 'cause you shall not come to me in debt,\n\nBeing now my steward, here upon your lips\n\nI sign your _Quietus est._ 13 This you should have begged now:\n\nI have seen children oft eat sweetmeats thus,\n\nAs fearful to devour them too soon.\n\nANTONIO. But for your brothers?\n\nDUCHESS. Do not think of them:\n\nAll discord without this circumference\n\nIs only to be pitied, and not feared:\n\nYet, should they know it, time will easily\n\nScatter the tempest.\n\nANTONIO. These words would be mine,\n\nAnd all the parts you have spoke, if some part of it\n\nWould not have savored flattery.\n\nDUCHESS. Kneel.\n\nCARIOLA _comes from behind the arras._\n\nANTONIO. Ha!\n\nDUCHESS. Be not amazed; this woman's of my counsel:\n\nI have heard lawyers say, a contract in a chamber\n\n_Per verba presenti_ is absolute marriage.\n\n_She and_ ANTONIO _kneel._\n\nBless, Heaven, this sacred gordian, which let violence\n\nNever untwine!\n\nANTONIO. And may our sweet affections, like the spheres,\n\nBe still in motion!\n\nDUCHESS. Quickening, and make\n\nThe like soft music!\n\nANTONIO. That we may imitate the loving palms,\n\nBest emblem of a peaceful marriage,\n\nThat never bore fruit, divided!\n\nDUCHESS. What can the church force more?\n\nANTONIO. That fortune may not know an accident,\n\nEither of joy or sorrow, to divide\n\nOur fix\u00e9d wishes!\n\nDUCHESS. HOW can the church build faster?\n\nWe now are man and wife, and 'tis the church\n\nThat must but echo this.\u2014Maid, stand apart:\n\nI now am blind.\n\nANTONIO. What's your conceit in this?\n\nDUCHESS. I would have you lead your fortune by the hand\n\nUnto your marriage bed\n\n(You speak in me this, for we now are one):\n\nWe'll only lie, and talk together, and plot\n\nTo appease my humorous kindred; and if you please,\n\nLike the old tale in Alexander and Lodowick,\n\nLay a naked sword between us, keep us chaste.\n\nO, let me shroud my blushes in your bosom,\n\nSince 'tis the treasury of all my secrets!\n\n_Exeunt_ DUCHESS _and_ ANTONIO.\n\nCARIOLA. Whether the spirit of greatness or of woman\n\nReign most in her, I know not; but it shows\n\nA fearful madness: I owe her much of pity. [ _Exit_\n\n**ACT II**\n\n**S CENE I: _An apartment in the palace of the_ DUCHESS.**\n\n_Enter_ BOSOLA _and_ CASTRUCCIO.\n\nBOSOLA. You say you would fain be taken for an eminent courtier?\n\nCASTRUCCIO. 'Tis the very main of my ambition.\n\nBOSOLA. Let me see: you have a reasonable good face for 't already, and your nightcap expresses your ears sufficient largely. I would have you learn to twirl the strings of your band with a good grace, and in a set speech, at the end of every sentence, to hum three or four times, or blow your nose till it smart again, to recover your memory. When you come to be a president in criminal causes, if you smile upon a prisoner, hang him, but if you frown upon him and threaten him, let him be sure to scape the gallows.\n\nCASTRUCCIO. I would be a very merry president.\n\nBOSOLA. Do not sup o' nights; 'twill beget you an admirable wit.\n\nCASTRUCCIO. Rather it would make me have a good stomach to quarrel; for they say, your roaring boys eat meat seldom, and that makes them so valiant. But how shall I know whether the people take me for an eminent fellow?\n\nBOSOLA. I will teach a trick to know it: give out you lie a-dying, and if you hear the common people curse you, be sure you are taken for one of the prime nightcaps. [ _Enter an_ OLD Lady] You come from painting now.\n\nOLD LADY. From what?\n\nBOSOLA. Why, from your scurvy face-physic. To behold thee not painted inclines somewhat near a miracle; these in thy face here were deep ruts and foul sloughs the last progress. There was a lady in France that, having had the small-pox, flayed the skin off her face to make it more level; and whereas before she looked like a nutmeg-grater, after she resembled an abortive hedge-hog.\n\nOLD LADY. Do you call this painting?\n\nOLD LADY. No no, but you call it careening of an old morphewed lady, to make her disembogue again: there's rough-cast phrase to your plastic.\n\nOLD LADY. It seems you are well acquainted with my closet.\n\nBOSOLA. One would suspect it for a shop of witchcraft, to find in it the fat of serpents, spawn of snakes, Jews' spittle, and their young children's ordure; and all these for the face. I would sooner eat a dead pigeon taken from the soles of the feet of one sick of the plague than kiss one of you fasting. Here are two of you, whose sin of your youth is the very patrimony of the physician; makes him renew his foot-cloth with the spring, and change his high-priced courtezan with the fall of the leaf. I do wonder you do not loathe yourselves. Observe my meditation now.\n\nWhat thing is in this outward form of man\n\nTo be beloved? We account it ominous,\n\nIf nature do produce a colt, or lamb,\n\nA fawn, or goat, in any limb resembling\n\nA man, and fly from 't as a prodigy:\n\nMan stands amazed to see his deformity\n\nIn any other creature but himself.\n\nBut in our own flesh, though we bear diseases\n\nWhich have their true names only ta'en from beasts,\u2014\n\nAs the most ulcerous wolf and swinish measle,\u2014\n\nThough we are eaten up of lice and worms,\n\nAnd though continually we bear about us\n\nA rotten and dead body, we delight\n\nTo hide it in rich tissue: all our fear,\n\nNay, all our terror, is lest our physician\n\nShould put us in the ground to be made sweet.\u2014\n\nYour wife's gone to Rome: you two couple, and get you to the wells at Lucca to recover your aches. I have other work on foot.\n\n_Exeunt_ CASTRUCCIO _and_ OLD LADY.\n\nI observe our duchess\n\nIs sick a-days, she pukes, her stomach seethes,\n\nThe fins of her eye-lids look most teeming blue,\n\nShe wanes i' the cheek, and waxes fat i' the flank,\n\nAnd, contrary to our Italian fashion,\n\nWears a loose-bodied gown: there's somewhat in 't.\n\nI have a trick may chance discover it,\n\nA pretty one; I have bought some apricocks,\n\nThe first our spring yields.\n\n_Enter_ ANTONIO _and_ DELIO.\n\nDELIO. And so long since married!\n\nYou amaze me.\n\nANTONIO. Let me seal your lips for ever:\n\nFor, did I think that any thing but the air\n\nCould carry these words from you, I would wish\n\nYou had no breath at all.\u2014Now, sir, in your contemplation?\n\nYou are studying to become a great wise fellow.\n\nBOSOLA. O, sir, the opinion of wisdom is a foul tether that runs all over a man's body: if simplicity direct us to have no evil, it directs us to a happy being; for the subtlest folly proceeds from the subtlest wisdom: let me be simply honest.\n\nANTONIO. I do understand your inside.\n\nBOSOLA. Do you so?\n\nANTONIO. Because you would not seem to appear to the world Puffed up with your preferment, you continue\n\nThis out-of-fashion melancholy: leave it, leave it.\n\nBOSOLA. Give me leave to be honest in any phrase, in any compliment whatsoever. Shall I confess myself to you? I look no higher than I can reach: they are the gods that must ride on winged horses. A lawyer's mule of a slow pace will both suit my disposition and business; for, mark me, when a man's mind rides faster than his horse can gallop, they quickly both tire.\n\nANTONIO. You would look up to Heaven, but I think\n\nThe devil, that rules i' the air, stands in your light.\n\nBOSOLA. O, sir, you are lord of the ascendant, chief man with the duchess; a duke was your cousin-german removed. Say you are lineally descended from King Pepin, or he himself, what of this? search the heads of the greatest rivers in the world, you shall find them but bubbles of water. Some would think the souls of princes were brought forth by some more weighty cause than those of meaner persons: they are deceived, there's the same hand to them; the like passions sway them; the same reason that makes a vicar to go to law for a tithe-pig, and undo his neighbors, makes them spoil a whole province, and batter down goodly cities with the cannon.\n\n_Enter_ DUCHESS _and_ LADIES.\n\nDUCHESS. Your arm, Antonio: do I not grow fat?\n\nI am exceeding short-winded.\u2014Bosola,\n\nI would have you, sir, provide for me a litter;\n\nSuch a one as the Duchess of Florence rode in.\n\nBOSOLA. The duchess used one when she was great with child.\n\nDUCHESS. I think she did.\u2014Come hither, mend my ruff;\n\nHere, when? thou art such a tedious lady; and\n\nThy breath smells of lemon-pills; would thou hadst done!\n\nShall I swoon under thy fingers! I am\n\nSo troubled with the mother\n\nBOSOLA. [Aside] I fear too much.\n\nDUCHESS. I have heard you say that the French courtiers\n\nWear their hats on 'fore the king.\n\nANTONIO. I have seen it.\n\nDUCHESS. In the presence?\n\nANTONIO. Yes.\n\nDUCHESS. Why should not we bring up that fashion!\n\n'Tis ceremony more than duty that consists\n\nIn the removing of a piece of felt:\n\nBe you the example to the rest o' the court;\n\nPut on your hat first.\n\nANTONIO. You must pardon me:\n\nI have seen, in colder countries than in France,\n\nNobles stand bare to the prince; and the distinction\n\nMethought showed reverently.\n\nBOSOLA. I have a present for your grace.\n\nDUCHESS. For me, sir?\n\nBOSOLA. Apricocks, madam.\n\nDUCHESS. O, sir, where are they?\n\nI have heard of none to-year.\n\nBOSOLA. [Aside] Good; her color rises.\n\nDUCHESS. Indeed, I thank you: they are wondrous fair ones.\n\nWhat an unskilful fellow is our gardener!\n\nWe shall have none this month.\n\nBOSOLA. Will not your grace pare them?\n\nDUCHESS. NO: they taste of musk, methinks; indeed they do.\n\nBOSOLA. I know not: yet I wish your grace had pared **'** em.\n\nDUCHESS. Why?\n\nBOSOLA. I forgot to tell you, the knave gardener,\n\nOnly to raise his profit by them the sooner,\n\nDid ripen them in horse-dung.\n\nDUCHESS. O, you jest\u2014\n\nYou shall judge: pray taste one.\n\nANTONIO. Indeed, madam,\n\nI do not love the fruit.\n\nDUCHESS. Sir, you are loth\n\nTo rob us of our dainties: 'tis a delicate fruit;\n\nThey say they are restorative.\n\nBOSOLA. 'Tis a pretty art,\n\nThis grafting.\n\nDUCHESS. 'Tis so; bettering of nature.\n\nBOSOLA. To make a pippin grow upon a crab,\n\nA damson on a blackthorn. \u2014 [ _Aside_ ] How greedily she eats them!\n\nA whirlwind strike off these bawd farthingales\n\nFor, but for that and the loose-bodied gown,\n\nI should have discovered apparently\n\nThe young springal cutting a caper in her belly.\n\nDUCHESS. I thank you, Bosola: they are right good ones, If they do not make me sick.\n\nANTONIO. How now, madam!\n\nDUCHESS. This green fruit and my stomach are not friends:\n\nHow they swell me!\n\nBOSOLA. [ _Aside_ ] Nay, you are too much swelled already.\n\nDUCHESS. O, I am in an extreme cold sweat!\n\nBOSOLA. I am very sorry.\n\nDUCHESS. Lights to my chamber!\u2014O good Antonio, I fear I am undone!\n\nDELIO. Lights there, lights!\n\n_Exeunt_ DUCHESS _and_ LADIES.\u2014Exit, _on the other side,_ BOSOLA.\n\nANTONIO. O my trusty Delio, we are lost!\n\nI fear she's fall'n in labor; and there's left\n\nNo time for her remove.\n\nDELIO. Have you prepared\n\nThose ladies to attend her? and procured\n\nThat politic safe conveyance for the midwife\n\nYour duchess plotted?\n\nANTONIO. I have.\n\nDELIO. Make use, then, of this forced occasion:\n\nGive out that Bosola hath poisoned her\n\nWith these apricocks; that will give some color\n\nFor her keeping close.\n\nANTONIO. Fie, fie, the physicians\n\nWill then flock to her.\n\nDELIO. For that you may pretend\n\nShe'll use some prepared antidote of her own,\n\nLest the physicians should re-poison her.\n\nANTONIO. I am lost in amazement: I know not what to think on 't.\n\n[ _Exeunt_\n\n**S CENE II**\n\n**_A hall in the same palace._**\n\n_Enter_ BOSOLA.\n\nBOSOLA. So, so, there's no question but her techiness and most vulturous eating of the apricocks are apparent signs of breeding.\n\n_Enter an_ OLD LADY.\n\nOLD LADY. NOW? I am in haste, sir.\n\nBOSOLA. There was a young waiting-woman had a monstrous desire to see the glass-house\u2014\n\nOLD LADY. Nay, pray let me go.\n\nBOSOLA. And it was only to know what strange instrument it was should swell up a glass to the fashion of a woman's belly.\n\nOLD LADY. I will hear no more of the glass-house. You are still abusing women?\n\nBOSOLA. Who, I? no; only, by the way now and then, mention your frailties. The orange-tree bears ripe and green fruit and blossoms all together; and some of you give entertainment for pure love, but more for more precious reward. The lusty spring smells well; but drooping autumn tastes well. If we have the same golden showers that rained in the time of Jupiter the thunderer, you have the same Dana\u00ebs still, to hold up their laps to receive them. Didst thou never study the mathematics?\n\nOLD LADY. What's that, sir?\n\nBOSOLA. Why to know the trick how to make a many lines meet in one centre. Go, go, give your foster-daughters good counsel: tell them, that the devil takes delight to hang at a woman's girdle, like a false rusty watch, that she cannot discern how the time passes.\n\nExit OLD LADY. _Enter_ ANTONIO, DELIO, RODERIGO, _and_ GRISOLAN.\n\nANTONIO. Shut up the court-gates.\n\nRODERIGO. Why, sir? what's the danger?\n\nANTONIO. Shut up the posterns presently, and call All the officers o' the court.\n\nGRISOLAN. I shall instandy. [ _Exit_\n\nANTONIO. Who keeps the key o' the park-gate?\n\nRODERIGO. Forobosco.\n\nANTONIO. Let him bring 't presently.\n\n_Re-enter_ GRISOLAN _with_ SERVANTS.\n\n1ST SERVANT. O, gentlemen o' the court, the foul treason!\n\nBOSOLA. [Aside] If that these apricocks should be poisoned now, Without my knowledge!\n\n1ST SERVANT. There was taken even now a Switzer in the duchess's bed-chamber\u2014\n\n2ND SERVANT. A Switzer!\n\n1ST SERVANT. With a pistol in his great cod-piece.\n\nBOSOLA. Ha, ha, ha!\n\n1ST SERVANT. The cod-piece was the case for 't.\n\n2ND SERVANT. There was a cunning traitor: who would have searched his cod-piece?\n\n1ST SERVANT. True, if he had kept out of the ladies' chambers: and all the moulds of his buttons were leaden bullets.\n\n2ND SERVANT. O wicked cannibal! a firelock in 's cod-piece!\n\n1ST SERVANT. 'Twas a French plot, upon my life.\n\n2ND SERVANT. To see what the devil can do!\n\nANTONIO. Are all the officers here?\n\nSERVANTS. We are.\n\nANTONIO. Gentlemen,\n\nWe have lost much plate you know; and but this evening\n\nJewels, to the value of four thousand ducats,\n\nAre missing in the duchess's cabinet.\n\nAre the gates shut?\n\nSERVANTS. Yes.\n\nANTONIO. 'Tis the duchess's pleasure\n\nEach officer be locked into his chamber\n\nTill the sun-rising; and to send the keys\n\nOf all their chests and of their outward doors\n\nInto her bed-chamber. She is very sick.\n\nRODERIGO. At her pleasure.\n\nANTONIO. She entreats you take 't not ill: the innocent\n\nShall be the more approved by it.\n\nBOSOLA. Gentlemen o' the wood-yard, where's your Switzer now?\n\n1ST SERVANT. By this hand, 'twas credibly reported by one o' the blackguard.\n\n_Exeunt all except_ ANTONIO _and_ DELIO.\n\nDELIO. How fares it with the duchess? Antonio. She's exposed\n\nUnto the worst of torture, pain and fear.\n\nDELIO. Speak to her all happy comfort.\n\nANTONIO. How I do play the fool with mine own danger!\n\nYou are this night, dear friend, to post to Rome:\n\nMy life lies in your service.\n\nDELIO. Do not doubt me.\n\nANTONIO. O, 'tis far from me: and yet fear presents me\n\nSomewhat that looks like danger.\n\nDELIO. Believe it,\n\n'Tis but the shadow of your fear, no more:\n\nHow superstitiously we mind our evils!\n\nThe throwing down salt, or crossing of a hare,\n\nBleeding at nose, the stumbling of a horse,\n\nOr singing of a cricket, are of power\n\nTo daunt whole man in us. Sir, fare you well:\n\nI wish you all the joys of a blessed father:\n\nAnd, for my faith, lay this unto your breast,\u2014\n\nOld friends, like old swords, still are trusted best. [ _Exit_\n\n_Enter_ CARIOLA.\n\nCARIOLA. Sir, you are the happy father of a son:\n\nYour wife commends him to you.\n\nANTONIO. Blessed comfort!\u2014\n\nFor Heaven's sake tend her well: I'll presently\n\nGo set a figure for 's nativity. [ _Exeunt_\n\n**S CENE III**\n\n**_The court of the same palace._**\n\n_Enter_ BOSOLA, _with a dark lantern._\n\nBOSOLA. Sure I did hear a woman shriek: list, ha!\n\nAnd the sound came, if I received it right,\n\nFrom the duchess's lodgings. There's some stratagem\n\nIn the confining all our courtiers\n\nTo their several wards: I must have part of it;\n\nMy intelligence will freeze else. List, again!\n\nIt may be 'twas the melancholy bird,\n\nBest friend of silence and of solitariness,\n\nThe owl, that screamed so.\u2014Ha! Antonio!\n\n_Enter_ ANTONIO.\n\nANTONIO. I heard some noisxe.\u2014Who's there? what art thou? speak.\n\nBOSOLA. Antonio, put not your face nor body\n\nTo such a forced expression of fear:\n\nI am Bosola, your friend.\n\nANTONIO. Bosola! \u2014\n\n[ _Aside_ ] This mole does undermine me.\u2014Heard you not\n\nA noise even now?\n\nBOSOLA. From whence?\n\nANTONIO. From the duchess's lodging.\n\nBOSOLA. Not I: did you?\n\nANTONIO. I did, or else I dreamed.\n\nBOSOLA. Let's walk towards it.\n\nANTONIO. NO: it may be 'twas\n\nBut the rising of the wind.\n\nBOSOLA. Very likely.\n\nMethinks 'tis very cold, and yet you sweat:\n\nYou look wildly.\n\nANTONIO. I have been setting a figure\n\nFor the duchess's jewels.\n\nBOSOLA. Ah, and how falls your question?\n\nDo you find it radical?\n\nANTONIO. What's that to you?\n\n'Tis rather to be questioned what design,\n\nWhen all men were commanded to their lodgings,\n\nMakes you a night-walker.\n\nBOSOLA. In sooth, I'll tell you:\n\nNow all the court's asleep, I thought the devil\n\nHad least to do here; I came to say my prayers;\n\nAnd if it do offend you I do so,\n\nYou are a fine courtier.\n\nANTONIO. [ _Aside_ ] This fellow will undo me. \u2014\n\nYou gave the duchess apricocks to-day:\n\nPray Heaven they were not poisoned!\n\nBOSOLA. Poisoned! A Spanish fig\n\nFor the imputation.\n\nANTONIO. Traitors are ever confident\n\nTill they are discovered. There were jewels stol'n too:\n\nIn my conceit, none are to be suspected\n\nMore than yourself.\n\nBOSOLA. You are a false steward.\n\nANTONIO. Saucy slave, I'll pull thee up by the roots.\n\nBOSOLA. May be the ruin will crush you to pieces.\n\nANTONIO. You are an impudent snake indeed, sir:\n\nAre you scarce warm, and do you show your sting?\n\nYou libel well, sir.\n\nBOSOLA. No, sir: copy it out,\n\nAnd I will set my hand to 't.\n\nANTONIO. [Aside] My nose bleeds.\n\nOne that were superstitious would count\n\nThis ominous, when it merely comes by chance:\n\nTwo letters, that are wrote here for my name,\n\nAre drowned in blood!\n\nMere accident\u2014For you, sir, I'll take order\n\nI' the morn you shall be safe: \u2014 [Aside] 'tis that must color\n\nHer lying-in:\u2014sir, this door you pass not:\n\nI do not hold it fit that you come near\n\nThe duchess's lodgings, till you have quit yourself \u2014\n\n[ _Aside_ ] The great are like the base, nay, they are the same,\n\nWhen they seek shameful ways to avoid shame. [ _Exit_\n\nBOSOLA. Antonio hereabout did drop a paper:\u2014\n\nSome of your help, false friend:\u2014O, here it is.\n\nWhat's here? a child's nativity calculated! [ _Reads_ ]\n\n\"The duchess was delivered of a son, 'tween the hours twelve and one in the night, Anno _Dom._ 1504,\"\u2014that's this year\u2014 _\"decimo non Decembris\"_ \u2014that's this night,\u2014\"taken according to the meridian of Malfi,\"\u2014that's our duchess: happy discovery! -\"The lord of the first house being combust in the ascendant, signifies short life; and Mars being in a human sign, joined to the tail of the Dragon, in the eighth house, doth threaten a violent death. _Coetera non scrutantur\"_ 30\n\nWhy, now 'tis most apparent: this precise fellow\n\nIs the duchess's bawd:\u2014 I have it to my wish!\n\nThis is a parcel of intelligency\n\nOur courtiers were cased up for: it needs must follow\n\nThat I must be committed on pretence\n\nOf poisoning her; which I'll endure, and laugh at.\n\nIf one could find the father now! but that\n\nTime will discover. Old Castruccio\n\nI' the morning posts to Rome: by him I'll send\n\nA letter that shall make her brothers' galls\n\nO'erflow their livers. This was a thrifty way.\n\nThough lust do mask in ne'er so strange disguise,\n\nShe's oft found witty, but is never wise. [ _Exit_\n\n**S CENE IV**\n\n**_An apartment in the palace of the_ CARDINAL _at Rome._**\n\n_Enter_ CARDINAL _and_ JULIA.\n\nCARDINAL. Sit: thou art my best of wishes. Prithee, tell me\n\nWhat trick didst thou invent to come to Rome\n\nWithout thy husband.\n\nJULIA. Why, my lord, I told him\n\nI came to visit an old anchorite\n\nHere for devotion.\n\nCARDINAL. Thou art a witty false one,\u2014\n\nI mean, to him.\n\nJULIA. You have prevailed with me\n\nBeyond my strongest thoughts: I would not now\n\nFind you inconstant.\n\nCARDINAL. Do not put yourself\n\nTo such a voluntary torture, which proceeds\n\nOut of your own guilt.\n\nJULIA. How, my lord!\n\nCARDINAL. You fear\n\nMy constancy, because you have approved\n\nThose giddy and wild turnings in yourself.\n\nJULIA. Did you e'er find them?\n\nCARDINAL. Sooth, generally for women,\n\nA man might strive to make glass malleable,\n\nEre he should make them fix\u00e8d.\n\nJULIA. So, my lord.\n\nCARDINAL. We had need go borrow that fantastic glass\n\nInvented by Galileo the Florentine\n\nTo view another spacious world i' the moon,\n\nAnd look to find a constant woman there.\n\nJULIA. This is very well, my lord.\n\nCARDINAL. Why do you weep?\n\nAre tears your justification? the self-same tears\n\nWill fall into your husband's bosom, lady,\n\nWith a loud protestation that you love him\n\nAbove the world. Come, I'll love you wisely,\n\nThat's jealousy; since I am very certain\n\nYou cannot make me cuckold.\n\nJULIA. I'll go home\n\nTo my husband.\n\nCARDINAL. You may thank me, lady,\n\nI have taken you off your melancholy perch,\n\nBore you upon my fist, and showed you game,\n\nAnd let you fly at it.\u2014I pray thee, kiss me.\u2014\n\nWhen thou wast with thy husband, thou wast watched\n\nLike a tame elephant:\u2014still you are to thank me: \u2014\n\nThou hadst only kisses from him and high feeding;\n\nBut what delight was that? 'twas just like one\n\nThat hath a little fingering on the lute,\n\nYet cannot tune it:\u2014still you are to thank me.\n\nJULIA. You told me of a piteous wound i' the heart\n\nAnd a sick liver, when you wooed me first,\n\nAnd spake like one in physic.\n\nCARDINAL. Who's that?\u2014\n\n_Enter_ SERVANT.\n\nRest firm, for my affection to thee,\n\nLightning moves slow to 't.\n\nSERVANT. Madam, a gentleman,\n\nThat's come post from Malfi, desires to see you.\n\nCARDINAL. Let him enter: I'll withdraw. [ _Exit_\n\nSERVANT. He says\n\nYour husband, old Castruccio, is come to Rome,\n\nMost pitifully tired with riding post. [ _Exit_\n\n_Enter_ DELIO.\n\nJULIA. [ _Aside_ ] Signior Delio! 'tis one of my old suitors.\n\nDELIO. I was bold to come and see you.\n\nJULIA. Sir, you are welcome.\n\nDELIO. Do you lie here?\n\nJULIA. Sure, your own experience\n\nWill satisfy you no: our Roman prelates\n\nDo not keep lodging for ladies.\n\nDELIO. Very well:\n\nI have brought you no commendations from your husband,\n\nFor I know none by him.\n\nJULIA. I hear he's come to Rome.\n\nDELIO. I never knew man and beast, of a horse and a knight,\n\nSo weary of each other: if he had had a good back,\n\nHe would have undertook to have borne his horse,\n\nHis breech was so pitifully sore.\n\nJULIA. Your laughter\n\nIs my pity.\n\nDELIO. Lady, I know not whether\n\nYou want money, but I have brought you some.\n\nJULIA. From my husband?\n\nDELIO. No, from mine own allowance.\n\nJULIA. I must hear the condition, ere I be bound to take it.\n\nDELIO. Look on 't, 'tis gold: hath it not a fine color?\n\nJULIA. I have a bird more beautiful.\n\nDELIO. Try the sound on 't.\n\nJULIA. A lute-string far exceeds it:\n\nIt hath no smell, like cassia or civet;\n\nNor is it physical, though some fond doctors\n\nPersuade us seethe 't in cullises. I'll tell you,\n\nThis is a creature bred by-\n\nRe-enter SERVANT.\n\nSERVANT. Your husband's come,\n\nHath delivered a letter to the Duke of Calabria\n\nThat, to my thinking, hath put him out of his wits. [ _Exit_\n\nJULIA. Sir, you hear:\n\nPray, let me know your business and your suit\n\nAs briefly as can be.\n\nDELIO. With good speed: I would wish you,\n\nAt such time as you are non-resident\n\nWith your husband, my mistress.\n\nJULIA. Sir, I'll go ask my husband if I shall,\n\nAnd straight return your answer. [ _Exit_\n\nDELIO. Very fine!\n\nIs this her wit, or honesty, that speaks thus?\n\nI heard one say the duke was highly moved\n\nWith a letter sent from Malfi. I do fear\n\nAntonio is betrayed: how fearfully\n\nShows his ambition now! unfortunate fortune!\n\nThey pass through whirlpools, and deep woes do shun,\n\nWho the event weigh ere the action's done. [ _Exit_\n\n**S CENE V**\n\n**_Another apartment in the same palace._**\n\n_Enter_ CARDINAL _and_ FERDINAND _with a letter._\n\nFERDINAND. I have this night digged up a mandrake.\n\nCARDINAL. Say you?\n\nFERDINAND. And I am grown mad with 't.\n\nCARDINAL. What's the prodigy?\n\nFERDINAND. Read there,\u2014a sister damned: she's loose i' the hilts;\n\nGrown a notorious strumpet.\n\nCARDINAL. Speak lower.\n\nFERDINAND. Lower!\n\nRogues do not whisper 't now, but seek to publish 't\n\n(As servants do the bounty of their lords)\n\nAloud; and with a covetous searching eye,\n\nTo mark who note them. O, confusion seize her!\n\nShe hath had most cunning bawds to serve her turn,\n\nAnd more secure conveyances for lust\n\nThan towns of garrison for service.\n\nCARDINAL. Is 't possible?\n\nCan this be certain?\n\nFERDINAND. Rhubarb, O, for rhubarb\n\nTo purge this choler! here's the curs\u00e8d day\n\nTo prompt my memory; and here 't shall stick\n\nTill of her bleeding heart I make a sponge\n\nTo wipe it out.\n\nCARDINAL. Why do you make yourself\n\nSo wild a tempest?\n\nFERDINAND. Would I could be one,\n\nThat I might toss her palace 'bout her ears,\n\nRoot up her goodly forests, blast her meads,\n\nAnd lay her general territory as waste\n\nAs she hath done her honors.\n\nCARDINAL. Shall our blood,\n\nThe royal blood of Aragon and Castile,\n\nBe thus attainted?\n\nFERDINAND. Apply desperate physic:\n\nWe must not now use balsamum, but fire,\n\nThe smarting cupping-glass, for that's the mean\n\nTo purge infected blood, such blood as hers.\n\nThere is a kind of pity in mine eye,\u2014\n\nI'll give it to my handkercher; and now 'tis here,\n\nI'll bequeath this to her bastard.\n\nCARDINAL. What to do?\n\nFERDINAND. Why, to make soft lint for his mother's wounds,\n\nWhen I have hewed her to pieces.\n\nCARDINAL. Curs\u00e8d creature!\n\nUnequal nature, to place women's hearts\n\nSo far upon the left side!\n\nFERDINAND. Foolish men,\n\nThat e'er will trust their honor in a bark\n\nMade of so slight weak bulrush as is woman,\n\nApt every minute to sink it!\n\nCARDINAL. Thus\n\nIgnorance, when it hath purchased honor,\n\nIt cannot wield it.\n\nFERDINAND. Methinks I see her laughing\u2014\n\nExcellent hyena! Talk to me somewhat quickly,\n\nOr my imagination will carry me\n\nTo see her in the shameful act of sin.\n\nCARDINAL. With whom?\n\nFERDINAND. Happily with some strong-thighed bargeman,\n\nOr one o' the woodyard that can quoit the sledge\n\nOr toss the bar, or else some lovely squire\n\nThat carries coals up to her privy lodgings.\n\nCARDINAL. You fly beyond your reason.\n\nFERDINAND. Go to, mistress!\n\n'Tis not your whore's milk that shall quench my wild fire,\n\nBut your whore's blood.\n\nCARDINAL. How idly shows this rage, which carries you,\n\nAs men conveyed by witches through the air,\n\nOn violent whirlwinds! this intemperate noise\n\nFitly resembles deaf men's shrill discourse,\n\nWho talk aloud, thinking all other men\n\nTo have their imperfection.\n\nFERDINAND. Have not you\n\nMy palsy?\n\nCARDINAL. Yes, but I can be angry\n\nWithout this rupture: there is not in nature\n\nA thing that makes man so deformed, so beastly,\n\nAs doth intemperate anger. Chide yourself.\n\nYou have divers men who never yet expressed\n\nTheir strong desire of rest but by unrest,\n\nBy vexing of themselves. Come, put yourself\n\nIn tune.\n\nFERDINAND. So I will only study to seem\n\nThe thing I am not. I could kill her now,\n\nIn you, or in myself; for I do think\n\nIt is some sin in us Heaven doth revenge\n\nBy her.\n\nCARDINAL. Are you stark mad?\n\nFERDINAND. I would have their bodies\n\nBurnt in a coal-pit with the ventage stopped,\n\nThat their cursed smoke might not ascend to Heaven\n\nOr dip the sheets they lie in in pitch or sulphur,\n\nWrap them in 't, and then light them like a match;\n\nOr else to boil their bastard to a cullis,\n\nAnd give 't his lecherous father to renew\n\nThe sin of his back.\n\nCARDINAL. I'll leave you.\n\nFERDINAND. Nay, I have done.\n\nI am confident, had I been damned in hell,\n\nAnd should have heard of this, it would have put me\n\nInto a cold sweat. In, in; I'll go sleep.\n\nTill I know who leaps my sister, I'll not stir:\n\nThat known, I'll find scorpions to string my whips,\n\nAnd fix her in a general eclipse. [ _Exeunt_\n\n**ACT III**\n\n**S CENE I: _An apartment in the palace of the_ DUCHESS _._**\n\n_Enter_ ANTONIO _and_ DELIO.\n\nANTONIO. Our noble friend, my most belov\u00e8d Delio!\n\nO, you have been a stranger long at court;\n\nCame you along with the Lord Ferdinand?\n\nDELIO. I did, sir: and how fares your noble duchess?\n\nANTONIO. Right fortunately well: she's an excellent\n\nFeeder of pedigrees; since you last saw her,\n\nShe hath had two children more, a son and daughter.\n\nDELIO. Methinks 'twas yesterday: let me but wink,\n\nAnd not behold your face, which to mine eye\n\nIs somewhat leaner, verily I should dream\n\nIt were within this half hour.\n\nANTONIO. You have not been in law, friend Delio,\n\nNor in prison, nor a suitor at the court,\n\nNor begged the reversion of some great man's place,\n\nNor troubled with an old wife, which doth make\n\nYour time so insensibly hasten.\n\nDELIO. Pray, sir, tell me,\n\nHath not this news arrived yet to the ear\n\nOf the lord cardinal?\n\nANTONIO. I fear it hath:\n\nThe Lord Ferdinand, that's newly come to court,\n\nDoth bear himself right dangerously.\n\nDELIO. Pray, why?\n\nANTONIO. He is so quiet that he seems to sleep\n\nThe tempest out, as dormice do in winter:\n\nThose houses that are haunted are most still\n\nTill the devil be up.\n\nDELIO. What say the common people?\n\nANTONIO. The common rabble do directly say\n\nShe is a strumpet.\n\nDELIO. And your graver heads\n\nWhich would be politic, what censure they?\n\nANTONIO. They do observe I grow to infinite purchase,\n\nThe left hand way, and all suppose the duchess\n\nWould amend it, if she could; for, say they,\n\nGreat princes, though they grudge their officers\n\nShould have such large and unconfin\u00e8d means\n\nTo get wealth under them, will not complain,\n\nLest thereby they should make them odious\n\nUnto the people; for other obligation\n\nOf love or marriage between her and me\n\nThey never dream of.\n\nDELIO. The Lord Ferdinand\n\nIs going to bed.\n\n_Enter_ DUCHESS, FERDINAND, _and_ ATTENDANTS.\n\nFERDINAND. I'll instantly to bed,\n\nFor I am weary.\u2014I am to bespeak\n\nA husband for you.\n\nDUCHESS. For me, sir! pray, who is 't?\n\nFERDINAND. The great Count Malatesti.\n\nDUCHESS. Fie upon him!\n\nA count! he's a mere stick of sugar-candy;\n\nYou may look quite through him. When I choose\n\nA husband, I will marry for your honor.\n\nFERDINAND. You shall do well in 't.\u2014How is 't, worthy Antonio?\n\nDUCHESS. But, sir, I am to have private conference with you\n\nAbout a scandalous report is spread\n\nTouching mine honor.\n\nFERDINAND. Let me be ever deaf to 't:\n\nOne of Pasquil's paper bullets, court-calumny,\n\nA pestilent air, which princes' palaces\n\nAre seldom purged of. Yet say that it were true,\n\nI pour it in your bosom, my fixed love\n\nWould strongly excuse, extenuate, nay, deny\n\nFaults, were they apparent in you. Go, be safe\n\nIn your own innocency.\n\nDUCHESS. [ _Aside_ ] O blessed comfort!\n\nThis deadly air is purged.\n\n_Exeunt_ DUCHESS, ANTONIO, DELIO, _and_ ATTENDANTS.\n\nFERDINAND. Her guilt treads on\n\nHot-burning coulters.\n\n_Enter_ BOSOLA.\n\nNow, Bosola,\n\nHow thrives our intelligence?\n\nBOSOLA. Sir, uncertainly:\n\n'Tis rumored she hath had three bastards, but\n\nBy whom we may go read i' the stars.\n\nFERDINAND. Why, some\n\nHold opinion all things are written there.\n\nBOSOLA. Yes, if we could find spectacles to read them.\n\nI do suspect there hath been some sorcery\n\nUsed on the duchess.\n\nFERDINAND. Sorcery! to what purpose?\n\nBOSOLA. To make her dote on some desertless fellow\n\nShe shames to acknowledge.\n\nFERDINAND. Can your faith give way\n\nTo think there's power in potions or in charms,\n\nTo make us love whether we will or no?\n\nBOSOLA. Most certainly.\n\nFERDINAND. Away! these are mere gulleries, horrid things,\n\nInvented by some cheating mountebanks\n\nTo abuse us. Do you think that herbs or charms\n\nCan force the will? Some trials have been made\n\nIn this foolish practice, but the ingredients\n\nWere lenitive poisons, such as are of force\n\nTo make the patient mad; and straight the witch\n\nSwears by equivocation they are in love.\n\nThe witchcraft lies in her rank blood. This night\n\nI will force confession from her. You told me\n\nYou had got, within these two days, a false key\n\nInto her bed-chamber.\n\nBOSOLA. I have.\n\nFERDINAND. As I would wish.\n\nBOSOLA. What do you intend to do?\n\nFERDINAND. Can you guess?\n\nBOSOLA. NO.\n\nFERDINAND. DO not ask, then:\n\nHe that can compass me, and know my drifts,\n\nMay say he hath put a girdle 'bout the world,\n\nAnd sounded all her quicksands.\n\nBOSOLA. I do not\n\nThink so.\n\nFERDINAND. What do you think, then, pray?\n\nBOSOLA. That you are\n\nYour own chronicle too much, and grossly\n\nFlatter yourself.\n\nFERDINAND. Give me thy hand; I thank thee:\n\nI never gave pension but to flatterers,\n\nTill I entertain\u00e8d thee. Farewell.\n\nThat friend a great man's ruin strongly checks,\n\nWho rails into his belief all his defects. [ _Exeunt_\n\n**S CENE II**\n\n**_The bed-chamber of the_ DUCHESS.**\n\n_Enter_ DUCHESS, ANTONIO, _and_ CARIOLA.\n\nDUCHESS. Bring me the casket hither, and the glass.\u2014You get no lodging here tonight, my lord.\n\nANTONIO. Indeed, I must persuade one.\n\nDUCHESS. Very good:\n\nI hope in time 'twill grow into a custom,\n\nThat noblemen shall come with cap and knee\n\nTo purchase a night's lodging of their wives.\n\nANTONIO. I must lie here.\n\nDUCHESS. Must! you are a lord of misrule.\n\nANTONIO. Indeed, my rule is only in the night.\n\nDUCHESS. To what use will you put me?\n\nANTONIO. We'll sleep together.\n\nDUCHESS. Alas,\n\nWhat pleasure can two lovers find in sleep!\n\nCARIOLA. My lord, I lie with her often; and I know\n\nShe'll much disquiet you.\n\nANTONIO. See, you are complained of.\n\nCARIOLA. For she's the sprawling'st bedfellow.\n\nANTONIO. I shall like her the better for that.\n\nCARIOLA. Sir, shall I ask you a question?\n\nANTONIO. Ay, pray thee, Cariola.\n\nCARIOLA. Wherefore still, when you lie with my lady,\n\nDo you rise so early?\n\nANTONIO. Laboring men\n\nCount the clock oftenest, Cariola,\n\nAre glad when their task's ended.\n\nDUCHESS. I'll stop your mouth. [ _Kisses him_\n\nANTONIO. Nay, that's but one; Venus had two soft doves\n\nTo draw her chariot; I must have another\u2014\n\n[ _She kisses him again_\n\nWhen wilt thou marry, Cariola?\n\nCARIOLA. Never, my lord.\n\nANTONIO. O, fie upon this single life! forego it.\n\nWe read how Daphne, for her peevish flight,\n\nBecame a fruitless bay-tree; Syrinx turned\n\nTo the pale empty reed; Anaxarete\n\nWas frozen into marble: whereas those\n\nWhich married, or proved kind unto their friends,\n\nWere by a gracious influence transhaped\n\nInto the olive, pomegranate, mulberry,\n\nBecame flowers, precious stones, or eminent stars.\n\nCARIOLA. This is a vain poetry: but I pray you tell me,\n\nIf there were proposed me, wisdom, riches, and beauty,\n\nIn three several young men, which should I choose.\n\nANTONIO. 'Tis a hard question: this was Paris's case,\n\nAnd he was blind in 't, and there was great cause;\n\nFor how was 't possible he could judge right,\n\nHaving three amorous goddesses in view,\n\nAnd they stark naked? 'twas a motion\n\nWere able to benight the apprehension\n\nOf the severest counsellor of Europe.\n\nNow I look on both your faces so well formed,\n\nIt puts me in mind of a question I would ask.\n\nCARIOLA. What is 't?\n\nANTONIO. I do wonder why hard-favored ladies,\n\nFor the most part, keep worse-favored waiting-women\n\nTo attend them, and cannot endure fair ones.\n\nDUCHESS. O, that's soon answered.\n\nDid you ever in your life know an ill painter\n\nDesire to have his dwelling next door to the shop\n\nOf an excellent picture-maker? 'twould disgrace\n\nHis face-making, and undo him. I prithee,\n\nWhen were we so merry?\u2014My hair tangles.\n\nANTONIO. Pray thee, Cariola, let's steal forth the room,\n\nAnd let her talk to herself: I have divers times\n\nServed her the like, when she hath chafed extremely.\n\nI love to see her angry. Softly, Cariola.\n\n_Exeunt_ ANTONIO _and_ CARIOLA.\n\nDUCHESS. Doth not the color of my hair 'gin to change?\n\nWhen I wax gray, I shall have all the court\n\nPowder their hair with arras, to be like me.\n\nYou have cause to love me; I entered you into my heart\n\nBefore you would vouchsafe to call for the keys.\n\n_Enter_ FERDINAND _behind._\n\nWe shall one day have my brothers take you napping;\n\nMethinks his presence, being now in court,\n\nShould make you keep your own bed; but you'll say\n\nLove mixed with fear is sweetest. I'll assure you,\n\nYou shall get no more children till my brothers\n\nConsent to be your gossips. Have you lost your tongue?\n\n'Tis welcome:\n\nFor know, whether I am doomed to live or die,\n\nI can do both like a prince.\n\nFERDINAND. Die, then, quickly! [Giving _her a poniard_\n\nVirtue, where art thou hid? what hideous thing\n\nIs it that doth eclipse thee?\n\nDUCHESS. Pray, sir, hear me.\n\nFERDINAND. Or is it true thou art but a bare name,\n\nAnd no essential thing?\n\nDUCHESS. Sir,\u2014\n\nFERDINAND. DO not speak.\n\nDUCHESS. NO, sir:\n\nI will plant my soul in mine ears, to hear you.\n\nFERDINAND. O most imperfect light of human reason,\n\nThat mak'st us so unhappy to foresee\n\nWhat we can least prevent! Pursue thy wishes,\n\nAnd glory in them: there's in shame no comfort\n\nBut to be past all bounds and sense of shame.\n\nDUCHESS. I pray, sir, hear me: I am married.\n\nFERDINAND. So!\n\nDUCHESS. Happily, not to your liking: but for that,\n\nAlas, your shears do come untimely now\n\nTo clip the bird's wing that's already flown!\n\nWill you see my husband?\n\nFERDINAND. Yes, if I could change\n\nEyes with a basilisk.\n\nDUCHESS. Sure, you came hither\n\nBy his confederacy.\n\nFERDINAND. The howling of a wolf\n\nIs music to thee, screech-owl: prithee, peace.\u2014\n\nWhate'er thou art that hast enjoyed my sister,\n\nFor I am sure thou hear'st me, for thine own sake\n\nLet me not know thee. I came hither prepared\n\nTo work thy discovery; yet am now persuaded\n\nIt would beget such violent effects\n\nAs would damn us both. I would not for ten millions\n\nI had beheld thee: therefore use all means\n\nI never may have knowledge of thy name;\n\nEnjoy thy lust still, and a wretched life,\n\nOn that condition.\u2014And for thee, vile woman,\n\nIf thou do wish thy lecher may grow old\n\nIn thy embracements, I would have thee build\n\nSuch a room for him as our anchorites\n\nTo holier use inhabit. Let not the sun\n\nShine on him till he's dead; let dogs and monkeys\n\nOnly converse with him, and such dumb things\n\nTo whom nature denies use to sound his name;\n\nDo not keep a paraquito, lest she learn it;\n\nIf thou do love him, cut out thine own tongue,\n\nLest it bewray him.\n\nDUCHESS. Why might not I marry?\n\nI have not gone about in this to create\n\nAny new world or custom.\n\nFERDINAND. Thou art undone;\n\nAnd thou hast ta'en that massy sheet of lead\n\nThat hid thy husband's bones, and folded it\n\nAbout my heart.\n\nDUCHESS. Mine bleeds for't.\n\nFERDINAND. Thine! thy heart!\n\nWhat should I name 't unless a hollow bullet\n\nFilled with unquenchable wild-fire?\n\nDUCHESS. You are in this\n\nToo strict; and were you not my princely brother,\n\nI would say, too wilful: my reputation\n\nIs safe.\n\nFERDINAND. Dost thou know what reputation is?\n\nI'll tell thee,\u2014to small purpose, since the instruction\n\nComes now too late.\n\nUpon a time Reputation, Love, and Death,\n\nWould travel o'er the world; and it was concluded\n\nThat they should part, and take three several ways.\n\nDeath told them, they should find him in great battles,\n\nOr cities plagued with plagues: Love gives them counsel\n\nTo inquire for him 'mongst unambitious shepherds,\n\nWhere dowries were not talked of, and sometimes\n\n'Mongst quiet kindred that had nothing left\n\nBy their dead parents: \"Stay,\" quoth Reputation,\n\n\"Do not forsake me; for it is my nature,\n\nIf once I part from any man I meet,\n\nI am never found again.\" And so for you:\n\nYou have shook hands with Reputation,\n\nAnd made him invisible. So, fare you well:\n\nI will never see you more.\n\nDUCHESS. Why should only I,\n\nOf all the other princes of the world,\n\nBe cased up, like a holy relic? I have youth\n\nAnd a little beauty.\n\nFERDINAND. So you have some virgins\n\nThat are witches. I will never see thee more. [ _Exit_\n\n_Re-enter_ ANTONIO _with a pistol, and_ CARIOLA.\n\nDUCHESS. YOU saw this apparition?\n\nANTONIO. Yes: we are\n\nBetrayed. How came he hither? I should turn\n\nThis to thee, for that.\n\nCARIOLA. Pray, sir, do; and when\n\nThat you have cleft my heart, you shall read there\n\nMine innocence.\n\nDUCHESS. That gallery gave him entrance.\n\nANTONIO. I would this terrible thing would come again,\n\nThat, standing on my guard, I might relate\n\nMy warrantable love.\u2014\n\n_She shows the poniard._\n\nHa! what means this?\n\nDUCHESS. He left this with me.\n\nANTONIO. And it seems did wish\n\nYou would use it on yourself.\n\nDUCHESS. His action\n\nSeemed to intend so much.\n\nANTONIO. This hath a handle to 't,\n\nAs well as a point: turn it towards him,\n\nAnd so fasten the keen edge in his rank gall.\n\n_Knocking within._\n\nHow now! who knocks? more earthquakes?\n\nDUCHESS. I stand\n\nAs if a mine beneath my feet were ready\n\nTo be blown up.\n\nCARIOLA. 'Tis Bosola.\n\nDUCHESS. Away!\n\nO misery! methinks unjust actions\n\nShould wear these masks and curtains, and not we.\n\nYou must instantly part hence: I have fashioned it already.\n\n_Exit_ ANTONIO. _Enter_ BOSOLA.\n\nBOSOLA. The duke your brother is ta'en up in a whirlwind;\n\nHath took horse, and 's rid post to Rome.\n\nDUCHESS. So late?\n\nBOSOLA. He told me, as he mounted into the saddle,\n\nYou were undone.\n\nDUCHESS. Indeed, I am very near it.\n\nBOSOLA. What's the matter?\n\nDUCHESS. Antonio, the master of our household,\n\nHath dealt so falsely with me in 's accounts:\n\nMy brother stood engaged with me for money\n\nTa'en up of certain Neapolitan Jews,\n\nAnd Antonio lets the bonds be forfeit.\n\nBOSOLA. Strange! \u2014 [Aside] This is cunning.\n\nDUCHESS. And hereupon\n\nMy brother's bills at Naples are protested\n\nAgainst.\u2014Call up our officers.\n\nBOSOLA. I shall. [Exit\n\n_Re-enter_ ANTONIO.\n\nDUCHESS. The place that you must fly to is Ancona:\n\nHire a house there; I'll send after you\n\nMy treasure and my jewels. Our weak safety\n\nRuns upon enginous wheels: short syllables\n\nMust stand for periods. I must now accuse you\n\nOf such a feign\u00e9d crime as Tasso calls\n\n_Magnanima menzogna,_ a noble lie,\n\n'Cause it must shield our honors.\u2014Hark! they are coming.\n\n_Re-enter_ BOSOLA _and_ OFFICERS.\n\nANTONIO. Will your grace hear me?\n\nDUCHESS. I have got well by you; you have yielded me\n\nA million of loss: I am like to inherit\n\nThe people's curses for your stewardship.\n\nYou had the trick in audit-time to be sick,\n\nTill I had signed your quietus; and that cured you\n\nWithout help of a doctor.\u2014Gentlemen,\n\nI would have this man be an example to you all;\n\nSo shall you hold my favor; I pray, let him;\n\nFor he's done that, alas, you would not think of,\n\nAnd, because I intend to be rid of him,\n\nI mean not to publish.\u2014Use your fortune elsewhere.\n\nANTONIO. I am strongly armed to brook my overthrow,\n\nAs commonly men bear with a hard year:\n\nI will not blame the cause on 't; but do think\n\nThe necessity of my malevolent star\n\nProcures this, not her humor. O, the inconstant\n\nAnd rotten ground of service! you may see,\n\n'Tis even like him, that in a winter night,\n\nTakes a long slumber o'er a dying fire,\n\nAs loth to part from 't; yet parts thence as cold\n\nAs when he first sat down.\n\nDUCHESS. We do confiscate,\n\nTowards the satisfying of your accounts,\n\nAll that you have.\n\nANTONIO. I am all yours; and 'tis very fit\n\nAll mine should be so.\n\nDUCHESS. So, sir, you have your pass.\n\nANTONIO. You may see, gentlemen, what 'tis to serve\n\nA prince with body and soul. [ _Exit_\n\nBOSOLA. Here's an example for extortion: what moisture is drawn out of the sea, when foul weather comes, pours down, and runs into the sea again.\n\nDUCHESS. I would know what are your opinions\n\nOf this Antonio.\n\n2ND OFFICER. He could not abide to see a pig's head gaping: I thought your grace would find him a Jew.\n\n3RD OFFICER. I would you had been his officer, for your own sake.\n\n4TH OFFICER. You would have had more money.\n\n1ST OFFICER. He stopped his ears with black wool, and to those came to him for money said he was thick of hearing.\n\n2ND OFFICER. Some said he was an hermaphrodite, for he could not abide a woman.\n\n4TH OFFICER. How scurvy proud he would look when the treasury was full! Well, let him go.\n\n1ST OFFICER. Yes, and the chippings of the butterfly after him, to scour his gold chain.\n\nDUCHESS. Leave us.\n\n_Exeunt_ OFFICERS.\n\nWhat do you think of these?\n\nBOSOLA. That these are rogues that in 's prosperity,\n\nBut to have waited on his fortune, could have wished\n\nHis dirty stirrup rivetted through their noses,\n\nAnd followed after 's mule, like a bear in a ring;\n\nWould have prostituted their daughters to his lust;\n\nMade their first-born intelligencers; thought none happy\n\nBut such as were born under his blest planet,\n\nAnd wore his livery: and do these lice drop off now?\n\nWell, never look to have the like again:\n\nHe hath left a sort of flattering rogues behind him;\n\nTheir doom must follow. Princes pay flatterers\n\nIn their own money: flatterers dissemble their vices,\n\nAnd they dissemble their lies; that's justice.\n\nAlas, poor gentleman!\n\nDUCHESS. Poor! he hath amply filled his coffers.\n\nBOSOLA. Sure, he was too honest. Pluto, the god of riches,\n\nWhen he's sent by Jupiter to any man,\n\nHe goes limping, to signify that wealth\n\nThat comes on God's name comes slowly; but when he's sent\n\nOn the devil's errand, he rides post and comes in by scuttles.\n\nLet me show you what a most unvalued jewel\n\nYou have in a wanton humor thrown away,\n\nTo bless the man shall find him. He was an excellent\n\nCourtier and most faithful; a soldier that thought it\n\nAs beastly to know his own value too little\n\nAs devilish to acknowledge it too much.\n\nBoth his virtue and form deserved a far better fortune:\n\nHis discourse rather delighted to judge itself than show itself:\n\nHis breast was filled with all perfection,\n\nAnd yet it seemed a private whispering-room,\n\nIt made so little noise of 't.\n\nDUCHESS. But he was basely descended.\n\nBOSOLA. Will you make yourself a mercenary herald,\n\nRather to examine men's pedigrees than virtues?\n\nYou shall want him:\n\nFor know an honest statesman to a prince\n\nIs like a cedar planted by a spring;\n\nThe spring bathes the tree's root, the grateful tree\n\nRewards it with his shadow: you have not done so.\n\nI would sooner swim to the Bermoothes on\n\nTwo politicians' rotten bladders, tied\n\nTogether with an intelligencer's heartstring,\n\nThan depend on so changeable a prince's favor.\n\nFare thee well, Antonio! since the malice of the world\n\nWould needs down with thee, it cannot be said yet\n\nThat any ill happened unto thee, considering thy fall\n\nWas accompanied with virtue.\n\nDUCHESS. O, you render me excellent music!\n\nBOSOLA. Say you?\n\nDUCHESS. This good one that you speak of is my husband.\n\nBOSOLA. Do I not dream! can this ambitious age\n\nHave so much goodness in 't as to prefer\n\nA man merely for worth, without these shadows\n\nOf wealth and painted honors? possible?\n\nDUCHESS. I have had three children by him.\n\nBOSOLA. Fortunate lady!\n\nFor you have made your private nuptial bed\n\nThe humble and fair seminary of peace.\n\nNo question but many an unbeneficed scholar\n\nShall pray for you for this deed, and rejoice\n\nThat some preferment in the world can yet\n\nArise from merit. The virgins of your land\n\nThat have no dowries shall hope your example\n\nWill raise them to rich husbands. Should you want\n\nSoldiers, 'twould make the very Turks and Moors\n\nTurn Christians, and serve you for this act.\n\nLast, the neglected poets of your time,\n\nIn honor of this trophy of a man,\n\nRaised by that curious engine, your white hand,\n\nShall thank you, in your grave, for 't; and make that\n\nMore reverend than all the cabinets\n\nOf living princes. For Antonio.\n\nHis fame shall likewise flow from many a pen,\n\nWhen heralds shall want coats to sell to men.\n\nDUCHESS. As I taste comfort in this friendly speech,\n\nSo would I find concealment.\n\nBOSOLA. O, the secret of my prince,\n\nWhich I will wear on the inside of my heart!\n\nDUCHESS. You shall take charge of all my coin and jewels\n\nAnd follow him; for he retires himself\n\nTo Ancona.\n\nBOSOLA. So.\n\nDUCHESS. Whither, within few days,\n\nI mean to follow thee.\n\nBOSOLA. Let me think:\n\nI would wish your grace to feign a pilgrimage\n\nTo our Lady of Loretto, scarce seven leagues\n\nFrom fair Ancona; so may you depart\n\nYour country with more honor, and your flight\n\nWill seem a princely progress, retaining\n\nYour usual train about you.\n\nDUCHESS. Sir, your direction\n\nShall lead me by the hand.\n\nCARIOLA. In my opinion,\n\nShe were better progress to the baths at Lucca,\n\nOr go visit the Spa\n\nIn Germany; for, if you will believe me,\n\nI do not like this jesting with religion,\n\nThis feign\u00e8d pilgrimage.\n\nDUCHESS. Thou art a superstitious fool:\n\nPrepare us instantly for our departure.\n\nPast sorrows, let us moderately lament them;\n\nFor those to come, seek wisely to prevent them.\n\n_Exeunt_ DUCHESS _and_ CARIOLA.\n\nBOSOLA. A politician is the devil's quilted anvil;\n\nHe fashions all sins on him, and the blows\n\nAre never heard: he may work in a lady's chamber,\n\nAs here for proof. What rests but I reveal\n\nAll to my lord? O, this base quality\n\nOf intelligencer! why, every quality i' the world\n\nPrefers but gain or commendation:\n\nNow for this act I am certain to be raised,\n\nAnd men that paint weeds to the life are praised. [ _Exit_\n\n**S CENE III**\n\n**_An apartment in the_ CARDINAL's _palace at Rome._**\n\n_Enter_ CARDINAL, FERDINAND, MALATESTI, PESCARA, DELIO, _and_ SILVIO.\n\nCARDINAL. Must we turn soldier, then?\n\nMALATESTI. The emperor,\n\nHearing your worth that way, ere you attained\n\nThis reverend garment, joins you in commission\n\nWith the right fortunate soldier the Marquis of Pescara,\n\nAnd the famous Lannoy.\n\nCARDINAL. He that had the honor\n\nOf taking the French king prisoner?\n\nMALATESTI. The same.\n\nHere's a plot drawn for a new fortification\n\nAt Naples.\n\nFERDINAND. This great Count Malatesti, I perceive,\n\nHath got employment?\n\nDELIO. No employment, my lord;\n\nA marginal note in the muster-book, that he is\n\nA voluntary lord.\n\nFERDINAND. He's no soldier.\n\nDELIO. He has worn gunpowder in 's hollow tooth for the toothache.\n\nSILVIO. He comes to the leaguer with a full intent\n\nTo eat fresh beef and garlic, means to stay\n\nTill the scent be gone, and straight return to court.\n\nDELIO. He hath read all the late service\n\nAs the city chronicle relates it;\n\nAnd keeps two pewterers going, only to express\n\nBattles in model.\n\nSILVIO. Then he'll fight by the book.\n\nDELIO. By the almanac, I think,\n\nTo choose good days and shun the critical;\n\nThat's his mistress's scarf.\n\nSILVIO. Yes, he protests\n\nHe would do much for that taffeta.\n\nDELIO. I think he would run away from a battle,\n\nTo save it from taking prisoner.\n\nSILVIO. He is horribly afraid\n\nGunpowder will spoil the perfume on 't.\n\nDELIO. I saw a Dutchman break his pate once\n\nFor calling him pot-gun; he made his head\n\nHave a bore in 't like a musket.\n\nSILVIO. I would he had made a touchhole to 't.\n\nHe is indeed a guarded sumpter-cloth,\n\nOnly for the remove of the court.\n\n_Enter_ BOSOLA.\n\nPESCARA. Bosola arrived! what should be the business?\n\nSome falling-out amongst the cardinals.\n\nThese factions amongst great men, they are like\n\nFoxes, when their heads are divided,\n\nThey carry fire in their tails, and all the country\n\nAbout them goes to wreck for 't.\n\nSILVIO. What's that Bosola?\n\nDELIO. I knew him in Padua\u2014a fantastical scholar, like such who study to know how many knots was in Hercules' club, of what color Achilles' beard was, or whether Hector were not troubled with the toothache. He hath studied himself half blear-eyed to know the true symmetry of Caesar's nose by a shoeing-horn; and this he did to gain the name of a speculative man.\n\nPESCARA. Mark Prince Ferdinand:\n\nA very salamander lives in 's eye,\n\nTo mock the eager violence of fire.\n\nSILVIO. That cardinal hath made more bad faces with his oppression than ever Michael Angelo made good ones: he lifts up's nose, like a foul porpoise before a storm.\n\nPESCARA. The Lord Ferdinand laughs.\n\nDELIO. Like a deadly cannon\n\nThat lightens ere it smokes.\n\nPESCARA. These are your true pangs of death,\n\nThe pangs of life, that struggle with great statesmen.\n\nDELIO. In such a deformed silence witches whisper their charms.\n\nCARDINAL. Doth she make religion her riding-hood\n\nTo keep her from the sun and tempest?\n\nFERDINAND. That,\n\nThat damns her. Methinks her fault and beauty,\n\nBlended together, show like leprosy,\n\nThe whiter, the fouler. I make it a question\n\nWhether her beggarly brats were ever christened.\n\nCARDINAL. I will instantly solicit the state of Ancona\n\nTo have them banished.\n\nFERDINAND. You are for Loretto:\n\nI shall not be at your ceremony; fare you well. \u2014\n\nWrite to the Duke of Malfi, my young nephew\n\nShe had by her first husband, and acquaint him\n\nWith 's mother's honesty.\n\nBOSOLA. I will.\n\nFERDINAND. Antonio!\n\nA slave that only smelled of ink and counters,\n\nAnd never in 's life looked like a gentleman,\n\nBut in the audit-time.\u2014Go, go presently,\n\nDraw me out an hundred and fifty of our horse,\n\nAnd meet me at the fort-bridge. [ _Exeunt_\n\n**S CENE IV**\n\n**_The shrine of Our Lady of Loretto._**\n\n_Enter_ Two PILGRIMS.\n\n1ST PILGRIM. I have not seen a goodlier shrine than this;\n\nYet I have visited many.\n\n2ND PILGRIM. The Cardinal of Aragon\n\nIs this day to resign his cardinal's hat:\n\nHis sister duchess likewise is arrived\n\nTo pay her vow of pilgrimage. I expect\n\nA noble ceremony.\n\n1ST PILGRIM. No question.\u2014They come.\n\n_Here the ceremony of the_ Cardinal's _instalment, in the habit of a soldier, is performed by his delivering up his cross, hat, robes, and ring, at the shrine, and the investing of him with sword, helmet, shield, and spurs; then_ ANTONIO, _the_ DUCHESS, _and the_ CHILDREN, _having presented themselves at the shrine, are, by a form of banishment in dumb-show expressed towards them by the_ CARDINAL _and the state of Ancona, banished: during all which ceremony, this ditty is sung, to by divers churchmen._\n\nArms and honors deck thy story,\n\nTo thy fame's eternal glory!\n\nAdverse fortune ever fly thee;\n\nNo disastrous fate come nigh thee!\n\nI alone will sing thy praises,\n\nWhom to honor virtue raises;\n\nAnd thy study, that divine is,\n\nBent to martial discipline is.\n\nLay aside all those robes lie by thee;\n\nCrown thy arts with arms, they'll beautify thee.\n\nO worthy of worthiest name, adorned in this manner,\n\nLead bravely thy forces on under war's warlike banner!\n\nO, mayst thou prove fortunate in all martial courses!\n\nGuide thou still by skill in arts and forces!\n\nVictory attend thee nigh, whilst fame sings loud thy powers;\n\nTriumphant conquest crown thy head, and blessings pour down showers!\n\n_Exeunt all except the_ Two PILGRIMS.\n\n1ST PILGRIM. Here's a strange turn of state! who would have thought\n\nSo great a lady would have matched herself\n\nUnto so mean a person? yet the cardinal\n\nBears himself much too cruel.\n\n2ND PILGRIM. They are banished.\n\n1ST PILGRIM. But I would ask what power hath this state Of Ancona to determine of a free prince?\n\n2ND PILGRIM. They are a free state, sir, and her brother showed\n\nHow that the Pope, fore-hearing of her looseness,\n\nHath seized into the protection of the church\n\nThe dukedom which she held as dowager.\n\n1ST PILGRIM. But by what justice?\n\n2ND PILGRIM. Sure, I think by none,\n\nOnly her brother's instigation.\n\n1ST PILGRIM. What was it with such violence he took\n\nOff from her finger?\n\n2ND PILGRIM. 'Twas her wedding-ring;\n\nWhich he vowed shortly he would sacrifice\n\nTo his revenge.\n\n1ST PILGRIM. Alas, Antonio!\n\nIf that a man be thrust into a well,\n\nNo matter who sets hand to 't, his own weight\n\nWill bring him sooner to the bottom. Come, let's hence.\n\nFortune makes this conclusion general,\n\nAll things do help the unhappy man to fall. [ _Exeunt_\n\n**S CENE V**\n\n**_Near Loretto._**\n\n_Enter_ DUCHESS, ANTONIO, CHILDREN, CARIOLA, _and_ SERVANTS.\n\nDUCHESS. Banished Ancona!\n\nANTONIO. Yes, you see what power\n\nLightens in great men's breath.\n\nDUCHESS. Is all our train\n\nShrunk to this poor remainder?\n\nANTONIO. These poor men,\n\nWhich have got little in your service, vow\n\nTo take your fortune: but your wiser buntings,\n\nNow they are fledged, are gone.\n\nDUCHESS. They have done wisely.\n\nThis puts me in mind of death: physicians thus,\n\nWith their hands full of money, use to give o'er\n\nTheir patients.\n\nANTONIO. Right the fashion of the world:\n\nFrom decayed fortunes every flatterer shrinks;\n\nMen cease to build where the foundation sinks.\n\nDUCHESS. I had a very strange dream tonight.\n\nANTONIO. What was 't?\n\nDUCHESS. Methought I wore my coronet of state,\n\nAnd on a sudden all the diamonds\n\nWere changed to pearls.\n\nANTONIO. My interpretation\n\nIs, you'll weep shortly; for to me the pearls\n\nDo signify your tears.\n\nDUCHESS. The birds that live i' the field\n\nOn the wild benefit of nature live\n\nHappier than we; for they may choose their mates,\n\nAnd carol their sweet pleasures to the spring.\n\n_Enter_ BOSOLA _with a letter._\n\nBOSOLA. You are happily o'erta'en.\n\nDUCHESS. From my brother?\n\nBOSOLA. Yes, from the Lord Ferdinand your brother\n\nAll love and safety.\n\nDUCHESS. Thou dost blanch mischief,\n\nWouldst make it white. See, see, like to calm weather\n\nAt sea before a tempest, false hearts speak fair\n\nTo those they intend most mischief.\n\n[Reads] \"Send Antonio to me; I want his head in a business.\"\n\nA politic equivocation!\n\nHe doth not want your counsel, but your head;\n\nThat is, he cannot sleep till you be dead.\n\nAnd here's another pitfall that's strewed o'er\n\nWith roses; mark it, 'tis a cunning one:\n\n[Reads] \"I stand engaged for your husband for several debts at\n\nNaples: let not that trouble him; I had rather have his heart than his money\":\u2014\n\nAnd I believe so too.\n\nBOSOLA. What do you believe?\n\nDUCHESS. That he so much distrusts my husband's love,\n\nHe will by no means believe his heart is with him\n\nUntil he sees it: the devil is not cunning enough\n\nTo circumvent us in riddles.\n\nBOSOLA. Will you reject that noble and free league\n\nOf amity and love which I present you?\n\nDUCHESS. Their league is like that of some politic kings,\n\nOnly to make themselves of strength and power\n\nTo be our after-ruin: tell them so.\n\nBOSOLA. And what from you?\n\nANTONIO. Thus tell him; I will not come.\n\nBOSOLA. And what of this?\n\nANTONIO. My brothers have dispersed\n\nBlood-hounds abroad; which till I hear are muzzled,\n\nNo truce, though hatched with ne'er such politic skill,\n\nIs safe, that hangs upon our enemies' will.\n\nI'll not come at them.\n\nBOSOLA. This proclaims your breeding:\n\nEvery small thing draws a base mind to fear,\n\nAs the adamant draws iron. Fare you well, sir:\n\nYou shall shortly hear from 's. [Exit\n\nDUCHESS. I suspect some ambush:\n\nTherefore by all my love I do conjure you\n\nTo take your eldest son, and fly towards Milan.\n\nLet us not venture all this poor remainder\n\nIn one unlucky bottom.\n\nANTONIO. You counsel safely.\n\nBest of my life, farewell, since we must part-Heaven hath a hand in 't; but no otherwise\n\nThen as some curious artist takes in sunder\n\nA clock or watch, when it is out of frame,\n\nTo bring 't in better order.\n\nDUCHESS. I know not which is best,\n\nTo see you dead, or part with you.\u2014Farewell, boy:\n\nThou art happy that thou hast not understanding\n\nTo know thy misery; for all our wit\n\nAnd reading brings us to a truer sense\n\nOf sorrow.\u2014In the eternal church, sir,\n\nI do hope we shall not part thus.\n\nANTONIO. O, be of comfort!\n\nMake patience a noble fortitude,\n\nAnd think not how unkindly we are used:\n\nMan, like to cassia, is proved best being bruised.\n\nDUCHESS. Must I, like a slave-born Russian,\n\nAccount it praise to suffer tyranny?\n\nAnd yet, O Heaven, thy heavy hand is in 't!\n\nI have seen my little boy oft scourge his top,\n\nAnd compared myself to 't: naught made me e'er\n\nGo right but Heaven's scourge-stick.\n\nANTONIO. Do not weep:\n\nHeaven fashioned us of nothing, and we strive\n\nTo bring ourselves to nothing.\u2014Farewell, Cariola,\n\nAnd thy sweet armful.\u2014If I do never see thee more,\n\nBe a good mother to your little ones,\n\nAnd save them from the tiger: fare you well.\n\nDUCHESS. Let me look upon you once more, for that speech\n\nCame from a dying father: your kiss is colder\n\nThan that I have seen a holy anchorite\n\nGive to a dead man's skull.\n\nANTONIO. My heart is turned to a heavy lump of lead,\n\nWith which I sound my danger: fare you well.\n\n_Exeunt_ ANTONIO _and his_ SON.\n\nDUCHESS. My laurel is all withered.\n\nCARIOLA. Look, madam, what a troop of arm\u00e8d men\n\nMake towards us.\n\nDUCHESS. O, they are very welcome:\n\nWhen Fortune's wheel is over-charged with princes,\n\nThe weight makes it move swift: I would have my ruin\n\nBe sudden.\n\n_Re-enter_ BOSOLA _visarded,_ 43 _with a_ GUARD.\n\nI am your adventure, am I not?\n\nBOSOLA. You are: you must see your husband no more.\n\nDUCHESS. What devil art thou that counterfeit'st\n\nHeaven's thunder?\n\nBOSOLA. Is that terrible? I would have you tell me whether\n\nIs that note worse that frights the silly birds\n\nOut of the corn, or that which doth allure them\n\nTo the nets? you have hearkened to the last too much.\n\nDUCHESS. O misery! like to a rusty o'ercharged cannon,\n\nShall I never fly in pieces?\u2014Come, to what prison?\n\nBOSOLA. To none.\n\nDUCHESS. Whither, then?\n\nBOSOLA. To your palace.\n\nDUCHESS. I have heard\n\nThat Charon's boat serves to convey all o'er\n\nThe dismal lake, but brings none back again.\n\nBOSOLA. Your brothers mean you safety and pity.\n\nDUCHESS. Pity!\n\nWith such a pity men preserve alive\n\nPheasants and quails, when they are not fat enough\n\nTo be eaten.\n\nBOSOLA. These are your children?\n\nDUCHESS. Yes.\n\nBOSOLA. Can they prattle?\n\nDUCHESS. NO;\n\nBut I intend, since they were born accursed,\n\nCurses shall be their first language.\n\nBOSOLA. Fie, madam!\n\nForget this base, low fellow, \u2014\n\nDUCHESS. Were I a man,\n\nI'd beat that counterfeit face into thy other.\n\nBOSOLA. One of no birth.\n\nDUCHESS. Say that he was born mean,\n\nMan is most happy when 's own actions\n\nBe arguments and examples of his virtue.\n\nBOSOLA. A barren, beggarly virtue.\n\nDUCHESS. I prithee, who is greatest? can you tell?\n\nSad tales befit my woe: I'll tell you one.\n\nA salmon, as she swam unto the sea,\n\nMet with a dog-fish, who encounters her\n\nWith this rough language: \"Why art thou so bold\n\nTo mix thyself with our high state of floods,\n\nBeing no eminent courtier, but one\n\nThat for the calmest and fresh time o' the year\n\nDost live in shallow rivers, rank'st thyself\n\nWith silly smelts and shrimps? and darest thou\n\nPass by our dog-ship without reverence?\"\n\n\"O!\" quoth the salmon, \"sister, be at peace:\n\nThank Jupiter we both have passed the net!\n\nOur value never can be truly known,\n\nTill in the fisher's basket we be shown:\n\nI' the market then my price may be the higher,\n\nEven when I am nearest to the cook and fire.\"\n\nSo to great men the moral may be stretched;\n\nMen oft are valued high, when they're most wretched.\u2014\n\nBut come, whither you please. I am armed 'gainst misery;\n\nBent to all sways of the oppressor's will:\n\nThere's no deep valley but near some great hill. [ _Exeunt_\n\n**ACT IV**\n\n**S CENE I: _An apartment in the_ DUCHESS'S _palace at Malfi._**\n\n_Enter_ FERDINAND _and_ BOSOLA.\n\nFERDINAND. How doth our sister duchess bear herself\n\nIn her imprisonment?\n\nBOSOLA. Nobly: I'll describe her.\n\nShe's sad as one long used to 't, and she seems\n\nRather to welcome the end of misery\n\nThan shun it; a behavior so noble\n\nAs gives a majesty to adversity:\n\nYou may discern the shape of loveliness\n\nMore perfect in her tears than in her smiles:\n\nShe will muse four hours together; and her silence,\n\nMethinks, expresseth more than if she spake.\n\nFERDINAND. Her melancholy seems to be fortified\n\nWith a strange disdain.\n\nBOSOLA. 'Tis so; and this restraint,\n\nLike English mastiffs that grow fierce with tying,\n\nMakes her too passionately apprehend\n\nThose pleasures she's kept from.\n\nFERDINAND. Curse upon her!\n\nI will no longer study in the book\n\nOf another's heart. Inform her what I told you. [ _Exit_\n\n_Enter_ DUCHESS.\n\nBOSOLA. All comfort to your grace!\n\nDUCHESS. I will have none.\n\nPray thee, why dost thou wrap thy poisoned pills\n\nIn gold and sugar?\n\nBOSOLA. Your elder brother, the Lord Ferdinand,\n\nIs come to visit you, and sends you word,\n\n'Cause once he rashly made a solemn vow\n\nNever to see you more, he comes i' the night;\n\nAnd prays you gently neither torch nor taper\n\nShine in your chamber: he will kiss your hand,\n\nAnd reconcile himself; but for his vow\n\nHe dares not see you.\n\nDUCHESS. At his pleasure.\u2014\n\nTake hence the lights.\u2014He's come.\n\n_Enter_ FERDINAND.\n\nFERDINAND. Where are you?\n\nDUCHESS. Here, sir.\n\nFERDINAND. This darkness suits you well.\n\nDUCHESS. I would ask your pardon.\n\nFERDINAND. You have it;\n\nFor I account it the honorabl'st revenge,\n\nWhere I may kill, to pardon.\u2014Where are your cubs?\n\nDUCHESS. Whom?\n\nFERDINAND. Call them your children;\n\nFor though our national law distinguish bastards\n\nFrom true legitimate issue, compassionate nature\n\nMakes them all equal.\n\nDUCHESS. Do you visit me for this?\n\nYou violate a sacrament o' the church\n\nShall make you howl in hell for 't.\n\nFERDINAND. It had been well,\n\nCould you have lived thus always; for, indeed,\n\nYou were too much i' the light:\u2014but no more:\n\nI come to seal my peace with you. Here's a hand\n\n[ _Gives her a dead man's hand_\n\nTo which you have vowed much love; the ring upon 't\n\nYou gave.\n\nDUCHESS. I affectionately kiss it.\n\nFERDINAND. Pray, do, and bury the print of it in your heart.\n\nI will leave this ring with you for a love-token;\n\nAnd the hand as sure as the ring; and do not doubt\n\nBut you shall have the heart too: when you need a friend,\n\nSend it to him that owned it; you shall see\n\nWhether he can aid you.\n\nDUCHESS. You are very cold:\n\nI fear you are not well after your travel.\u2014\n\nHa! lights!\u2014O, horrible!\n\nFERDINAND. Let her have lights enough. [Exit\n\nDUCHESS. What witchcraft doth he practice, that he hath left\n\nA dead man's hand here?\n\n_Here is discovered, behind a traverse,_ 45 _the artificial figures of_ ANTONIO _and his_ CHILDREN, _appearing as if they were dead._\n\nBOSOLA. Look you, here's the piece from which 'twas ta'en.\n\nHe doth present you this sad spectacle,\n\nThat, now you know directly they are dead,\n\nHereafter you may wisely cease to grieve\n\nFor that which cannot be recover\u00e8d.\n\nDUCHESS. There is not between Heaven and earth one wish\n\nI stay for after this: it wastes me more\n\nThan were 't my picture, fashioned out of wax,\n\nStuck with a magical needle, and then buried\n\nIn some foul dunghill; and yond's an excellent property\n\nFor a tyrant, which I would account mercy.\n\nBOSOLA. What's that?\n\nDUCHESS. If they would bind me to that lifeless trunk,\n\nAnd let me freeze to death.\n\nBOSOLA. Come, you must live.\n\nDUCHESS. That's the greatest torture souls feel in hell,\n\nIn hell, that they must live, and cannot die.\n\nPortia, I'll new kindle thy coals again,\n\nAnd revive the rare and almost dead example\n\nOf a loving wife.\n\nBOSOLA. O, fie! despair? remember\n\nYou are a Christian.\n\nDUCHESS. The church enjoins fasting:\n\nI'll starve myself to death.\n\nBOSOLA. Leave this vain sorrow.\n\nThings being at the worst begin to mend: the bee\n\nWhen he hath shot his sting into your hand,\n\nMay then play with your eyelid.\n\nDUCHESS. Good comfortable fellow,\n\nPersuade a wretch that's broke upon the wheel\n\nTo have all his bones new set; entreat him live\n\nTo be executed again. Who must despatch me?\n\nI account this world a tedious theater,\n\nFor I do play a part in 't 'gainst my will.\n\nBOSOLA. Come, be of comfort; I will save your life.\n\nDUCHESS. Indeed, I have not leisure to tend\n\nSo small a business.\n\nBOSOLA. Now, by my life, I pity you.\n\nDUCHESS. Thou art a fool, then,\n\nTo waste thy pity on a thing so wretched\n\nAs cannot pity itself. I am full of daggers.\n\nPuff, let me blow these vipers from me.\n\n_Enter_ SERVANT.\n\nWhat are you?\n\nSERVANT. One that wishes you long life.\n\nDUCHESS. I would thou wert hanged for the horrible curse\n\nThou hast given me: I shall shortly grow one\n\nOf the miracles of pity. I'll go pray;\u2014\n\nNo, I'll go curse.\n\nBOSOLA. O, fie!\n\nDUCHESS. I could curse the stars.\n\nBOSOLA. O, fearful.\n\nDUCHESS. And those three smiling seasons of the year\n\nInto a Russian winter: nay, the world\n\nTo its first chaos.\n\nBOSOLA. Look you, the stars shine still.\n\nDUCHESS. O, but you must\n\nRemember, my curse hath a great way to go.\u2014\n\nPlagues, that make lanes through largest families,\n\nConsume them!\u2014\n\nBOSOLA. Fie, lady!\n\nDUCHESS. Let them, like tyrants,\n\nNever be remembered but for the ill they have done;\n\nLet all the zealous prayers of mortified\n\nChurchmen forget them!\u2014\n\nBOSOLA. O, uncharitable!\n\nDUCHESS. Let Heaven a little while cease crowning martyrs,\n\nTo punish them! \u2014\n\nGo, howl them this, and say, I long to bleed:\n\nIt is some mercy when men kill with speed. [ _Exit_\n\n_Re-enter_ FERDINAND.\n\nFERDINAND. Excellent, as I would wish; she's plagued in art:\n\nThese presentations are but framed in wax\n\nBy the curious master in that quality,\n\nVincentio Lauriola, and she takes them\n\nFor true substantial bodies.\n\nBOSOLA. Why do you do this?\n\nFERDINAND. To bring her to despair.\n\nBOSOLA. Faith, end here,\n\nAnd go no farther in your cruelty:\n\nSend her a penitential garment to put on\n\nNext to her delicate skin, and furnish her\n\nWith beads and prayer-books.\n\nFERDINAND. Damn her! that body of hers,\n\nWhile that my blood ran pure in 't, was more worth\n\nThan that which thou wouldst comfort, called a soul.\n\nI will send her masks of common courtezans,\n\nHave her meat served up by bawds and ruffians,\n\nAnd, 'cause she'll needs be mad, I am resolved\n\nTo remove forth the common hospital\n\nAll the mad-folk, and place them near her lodging;\n\nThere let them practice together, sing and dance,\n\nAnd act their gambols to the full o' the moon:\n\nIf she can sleep the better for it, let her.\n\nYour work is almost ended.\n\nBOSOLA. Must I see her again?\n\nFERDINAND. Yes.\n\nBOSOLA. Never.\n\nFERDINAND. You must.\n\nBOSOLA. Never in mine own shape;\n\nThat's forfeited by my intelligence\n\nAnd this last cruel lie: when you send me next,\n\nThe business shall be comfort.\n\nFERDINAND. Very likely;\n\nThy pity is nothing of kin to thee. Antonio\n\nLurks about Milan: thou shalt shortly thither,\n\nTo feed a fire as great as my revenge,\n\nWhich never will slack till it have spent his fuel:\n\nIntemperate agues make physicians cruel. [ _Exeunt_\n\n**S CENE II**\n\n**_Another room in the_ DUCHES's _lodging._**\n\n_Enter_ DUCHESS _and_ CARIOLA.\n\nDUCHESS. What hideous noise was that?\n\nCARIOLA. 'Tis the wild consort\n\nOf madmen, lady, which your tyrant brother\n\nHath placed about your lodging: this tyranny,\n\nI think, was never practiced till this hour.\n\nDUCHESS. Indeed, I thank him: nothing but noise and folly\n\nCan keep me in my right wits; whereas reason\n\nAnd silence make me stark mad. Sit down;\n\nDiscourse to me some dismal tragedy.\n\nCARIOLA. O, 'twill increase your melancholy.\n\nDUCHESS. Thou art deceived:\n\nTo hear of greater grief would lessen mine.\n\nThis is a prison?\n\nCARIOLA. Yes, but you shall live\n\nTo shake this durance off.\n\nDUCHESS. Thou art a fool:\n\nThe robin-redbreast and the nightingale\n\nNever live long in cages.\n\nCARIOLA. Pray, dry your eyes.\n\nWhat think you of, madam?\n\nDUCHESS. Of nothing;\n\nWhen I muse thus, I sleep.\n\nCARIOLA. Like a madman, with your eyes open?\n\nDUCHESS. Dost thou think we shall know one another\n\nIn the other world?\n\nCARIOLA. Yes, out of question.\n\nDUCHESS. O, that it were possible we might\n\nBut hold some two days' conference with the dead!\n\nFrom them I should learn somewhat, I am sure,\n\nI never shall know here. I'll tell thee a miracle;\n\nI am not mad yet, to my cause of sorrow:\n\nThe Heaven o'er my head seems made of molten brass,\n\nThe earth of flaming sulphur, yet I am not mad.\n\nI am acquainted with sad misery\n\nAs the tanned galley-slave is with his oar;\n\nNecessity makes me suffer constantly,\n\nAnd custom makes it easy. Who do I look like now?\n\nCARIOLA. Like to your picture in the gallery,\n\nA deal of life in show, but none in practice;\n\nOr rather like some reverend monument\n\nWhose ruins are even pitied.\n\nDUCHESS. Very proper;\n\nAnd Fortune seems only to have her eyesight\n\nTo behold my tragedy.\u2014How now!\n\nWhat noise is that?\n\n_Enter_ SERVANT.\n\nSERVANT. I am come to tell you\n\nYour brother hath intended you some sport.\n\nA great physician, when the Pope was sick\n\nOf a deep melancholy, presented him\n\nWith several sorts of madmen, which wild object\n\nBeing full of change and sport, forced him to laugh,\n\nAnd so the imposthume broke: the self-same cure\n\nThe duke intends on you.\n\nDUCHESS. Let them come in.\n\nSERVANT. There's a mad lawyer; and a secular priest;\n\nA doctor that hath forfeited his wits\n\nBy jealousy; an astrologian\n\nThat in his works said such a day o' the month\n\nShould be the day of doom, and, failing of 't,\n\nRan mad; an English tailor crazed i' the brain\n\nWith the study of new fashions; a gentleman-usher\n\nQuite beside himself with care to keep in mind\n\nThe number of his lady's salutations\n\nOr \"How do you\" she employed him in each morning;\n\nA farmer, too, an excellent knave in grain,\n\nMad 'cause he was hindered transportation:\n\nAnd let one broker that's mad loose to these,\n\nYou'd think the devil were among them.\n\nDUCHESS. Sit, Cariola.\u2014Let them loose when you please,\n\nFor I am chained to endure all your tyranny.\n\n_Enter_ MADMEN. _Here this song is sung to a dismal kind of music by a_ MADMAN.\n\nO, let us howl some heavy note,\n\nSome deadly dogg\u00e9d howl,\n\nSounding as from the threatening throat\n\nOf beasts and fatal fowl!\n\nAs ravens, screech-owls, bulls, and bears,\n\nWell bell, and bawl our parts,\n\nTill irksome noise have cloyed your ears\n\nAnd c\u00f3rrosived your hearts.\n\nAt last, when as our choir wants breath,\n\nOur bodies being blest,\n\nWell sing, like swans, to welcome death,\n\nAnd die in love and rest,\n\n1ST MADMAN. Doom's-day not come yet! Ill draw it nearer by a perspective, or make a glass that shall set all the world on fire upon an instant. I cannot sleep; my pillow is stuffed with a litter of porcupines.\n\n2ND MADMAN. Hell is a mere glass-house, where the devils are continually blowing up women's souls on hollow irons, and the fire never goes out.\n\n3RD MADMAN. I will lie with every woman in my parish the tenth night; I will tythe them over like haycocks.\n\n4TH MADMAN. Shall my pothecary out-go me because I am a cuckold? I have found out his roguery; he makes alum of his wife's urine, and sells it to Puritans that have sore throats with overstraining.\n\n1ST MADMAN. I have skill in heraldry.\n\n2ND MADMAN. Hast?\n\n1ST MADMAN. You do give for your crest a woodcock's head with the brains picked out on 't; you are a very ancient gentleman.\n\n3RD MADMAN. Greek is turned Turk: we are only to be saved by the Helvetian translation.\n\n1ST MADMAN. Come on, sir, I will lay the law to you.\n\n2ND MADMAN. O, rather lay a corrosive: the law will eat to the bone.\n\n3RD MADMAN. He that drinks but to satisfy nature is damned.\n\n4TH MADMAN. If I had my glass here, I would show a sight should make all the women here call me mad doctor.\n\n1ST MADMAN. What's he? a rope-maker?\n\n2ND MADMAN. No, no, no, a snuffling knave that, while he shows the tombs, will have his hand in a wench's placket.\n\n3RD MADMAN. Woe to the caroche that brought home my wife from the masque at three o'clock in the morning! it had a large feather-bed in it.\n\n4TH MADMAN. I have pared the devil's nails forty times, roasted them in raven's eggs, and cured agues with them.\n\n3RD MADMAN. Get me three hundred milch-bats, to make pos-sets to procure sleep.\n\n4TH MADMAN. All the college may throw their caps at me: I have made a soap-boiler costive; it was my masterpiece.\n\n_Here a dance of_ EIGHT MADMEN, _with music answerable thereto; after which,_ BOSOLA, _like an_ OLD MAN, _enters._\n\nDUCHESS. Is he mad too?\n\nSERVANT. Pray, question him. I'll leave you.\n\n_Exeunt_ SERVANT _and_ MADMEN.\n\nBOSOLA. I am come to make thy tomb.\n\nDUCHESS. Ha! my tomb!\n\nThou speak 'st as if I lay upon my deathbed,\n\nGasping for breath: dost thou perceive me sick?\n\nBOSOLA. Yes, and the more dangerously, since thy sickness is insensible.\n\nDUCHESS. Thou art not mad, sure: dost know me?\n\nBOSOLA. Yes.\n\nDUCHESS. Who am I?\n\nBOSOLA. Thou art a box of worm-seed, at best but a salvatory of green mummy. What's this flesh? a little crudded milk, fantastical puff-paste. Our bodies are weaker than those paper-prisons boys use to keep flies in; more contemptible, since ours is to preserve earthworms. Didst thou ever see a lark in a cage? Such is the soul in the body: this world is like her little turf of grass, and the Heaven o'er our heads, like her looking-glass, only gives us a miserable knowledge of the small compass of our prison.\n\nDUCHESS. Am not I thy duchess?\n\nBOSOLA. Thou art some great woman, sure, for riot begins to sit on thy forehead (clad in gray hairs) twenty years sooner than on a merry milkmaid's. Thou sleepest worse than if a mouse should be forced to take up her lodging in a cat's ear: a little infant that breeds its teeth, should it lie with thee, would cry out, as if thou wert the more unquiet bedfellow.\n\nDUCHESS. I am Duchess of Malfi still.\n\nBOSOLA. That makes thy sleeps so broken:\n\nGlories, like glow-worms, afar off shine bright,\n\nBut looked to near, have neither heat nor light.\n\nDUCHESS. Thou art very plain.\n\nBOSOLA. My trade is to flatter the dead, not the living;\n\nI am a tomb-maker.\n\nDUCHESS. And thou comest to make my tomb?\n\nBOSOLA. Yes.\n\nDUCHESS. Let me be a little merry:\u2014of what stuff wilt thou make it?\n\nBOSOLA. Nay, resolve me first, of what fashion?\n\nDUCHESS. Why do we grow fantastical in our deathbeds? do we affect fashion in the grave?\n\nBOSOLA. Most ambitiously. Princes' images on their tombs do not lie, as they were wont, seeming to pray up to Heaven; but with their hands under their cheeks, as if they died of the toothache: they are not carved with their eyes fixed upon the stars; but as their minds were wholly bent upon the world, the self-same way they seem to turn their faces.\n\nDUCHESS. Let me know fully therefore the effect\n\nOf this thy dismal preparation,\n\nThis talk fit for a charnel.\n\nBOSOLA. Now I shall:\u2014\n\nEnter EXECUTIONERS, _with a coffin, cords, and a bell._\n\nHere is a present from your princely brothers;\n\nAnd may it arrive welcome, for it brings\n\nLast benefit, last sorrow.\n\nDUCHESS. Let me see it:\n\nI have so much obedience in my blood,\n\nI wish it in their veins to do them good.\n\nBOSOLA. This is your last presence-chamber.\n\nCARIOLA. O my sweet lady!\n\nDUCHESS. Peace; it affrights not me.\n\nBOSOLA. I am the common bellman,\n\nThat usually is sent to condemned persons\n\nThe night before they suffer.\n\nDUCHESS. Even now thou said 'st\n\nThou wast a tomb-maker.\n\nBOSOLA. 'Twas to bring you\n\nBy degrees to mortification. Listen.\n\nHark, now every thing is still\n\nThe screech-owl and the whistler shrill\n\nCall upon our dame aloud,\n\nAnd bid her quickly don her shroud!\n\nMuch you had of land and rent;\n\nYour length in clay's now competent:\n\nA long war disturbed your mind;\n\nHere your perfect peace is signed.\n\nOf what is 't fools make such vain keeping?\n\nSin their conception, their birth weeping,\n\nTheir life a general mist of error,\n\nTheir death a hideous storm of terror.\n\nStrew your hair with powders sweet,\n\nDon clean linen, bathe your feet,\n\nAnd (the foul fiend more to check)\n\nA crucifix let bless your neck:\n\n'Tis now full tide 'tween night and day;\n\nEnd your groan, and come away.\n\nCARIOLA. Hence, villains, tyrants, murderers! alas!\n\nWhat will you do with my lady?\u2014Call for help.\n\nDUCHESS. To whom? to our next neighbors? they are mad-folks.\n\nBOSOLA. Remove that noise.\n\nDUCHESS. Farewell, Cariola.\n\nIn my last will I have not much to give:\n\nA many hungry guests have fed upon me;\n\nThine will be a poor reversion.\n\nCARIOLA. I will die with her.\n\nDUCHESS. I pray thee, look thou giv'st my little boy\n\nSome syrup for his cold, and let the girl\n\nSay her prayers ere she sleep.\n\nCARIOLA _is forced out by the_ EXECUTIONERS.\n\nNow what you please:\n\nWhat death?\n\nBOSOLA. Strangling; here are your executioners.\n\nDUCHESS. I forgive them:\n\nThe apoplexy, catarrh, or cough o' the lungs\n\nWould do as much as they do.\n\nBOSOLA. Doth not death fright you?\n\nDUCHESS. Who would be afraid on 't,\n\nKnowing to meet such excellent company\n\nIn the other world?\n\nBOSOLA. Yet, methinks,\n\nThe manner of your death should much afflict you:\n\nThis cord should terrify you.\n\nDUCHESS. Not a whit:\n\nWhat would it pleasure me to have my throat cut\n\nWith diamonds? or to be smother\u00e8d\n\nWith cassia? or to be shot to death with pearls?\n\nI know death hath ten thousand several doors\n\nFor men to take their exits; and 'tis found\n\nThey go on such strange geometrical hinges,\n\nYou may open them both ways; any way, for Heaven sake,\n\nSo I were out of your whispering. Tell my brothers\n\nThat I perceive death, now I am well awake,\n\nBest gift is they can give or I can take.\n\nI would fain put off my last woman's fault,\n\nI'd not be tedious to you.\n\n1ST EXECUTIONER. We are ready.\n\nDUCHESS. Dispose my breath how please you; but body\n\nBestow upon my women, will you?\n\n1ST EXECUTIONER. Yes.\n\nDUCHESS. Pull, and pull strongly, for your able strength\n\nMust pull down Heaven upon me:\u2014\n\nYet stay; Heaven-gates are not so highly arched\n\nAs princes' palaces; they that enter there\n\nMust go upon their knees. [Kneels] _\u2014_ Come, violent death,\n\nServe for mandragora to make me sleep! \u2014\n\nGo tell my brothers, when I am laid out,\n\nThey then may feed in quiet.\n\n_The_ EXECUTIONERS _strangle the_ DUCHESS.\n\nBOSOLA. Where's the waiting woman?\n\nFetch her: some other strangle the children.\n\nCARIOLA _and_ CHILDREN _are brought in by the_ EXECUTIONERS; _who presently strangle the_ CHILDREN.\n\nLook you, there sleeps your mistress.\n\nCARIOLA. O, you are damned\n\nPerpetually for this! My turn is next,\n\nIs 't not so ordered?\n\nBOSOLA. Yes, and I am glad\n\nYou are so well prepared for 't.\n\nCARIOLA. You are deceived, sir,\n\nI am not prepared for 't, I will not die;\n\nI will first come to my answer, and know\n\nHow I have offended.\n\nBOSOLA. Come, despatch her.\u2014\n\nYou kept her counsel; now you shall keep ours.\n\nCARIOLA. I will not die, I must not; I am contracted\n\nTo a young gentleman.\n\n1ST EXECUTIONER. Here's your wedding-ring.\n\nCARIOLA. Let me but speak with the duke; I'll discover\n\nTreason to his person.\n\nBOSOLA. Delays:\u2014throttle her.\n\n1ST EXECUTIONER. She bites and scratches.\n\nCARIOLA. If you kill me now,\n\nI am damned; I have not been at confession\n\nThis two years.\n\nBOSOLA. [To EXECUTIONERS] When?\n\nCARIOLA. I am quick with child.\n\nBOSOLA. Why, then,\n\nYour credit's saved.\n\n_The_ EXECUTIONERS _strangle_ CARIOLA.\n\nBear her into the next room;\n\nLet these lie still.\n\n_Exeunt the_ EXECUTIONERS _with the body of_ CARIOLA.\n\n_Enter_ FERDINAND.\n\nFERDINAND. Is she dead?\n\nBOSOLA. She is what\n\nYou'd have her. But here begin your pity.\n\n[ _Shows the_ CHILDREN _strangled_\n\nAlas, how have these offended?\n\nFERDINAND. The death\n\nOf young wolves is never to be pitied.\n\nBOSOLA. Fix your eye here.\n\nFERDINAND. Constantly.\n\nBOSOLA. Do you not weep?\n\nOther sins only speak; murder shrieks out:\n\nThe element of water moistens the earth,\n\nBut blood flies upwards and bedews the heavens.\n\nFERDINAND. Cover her face; mine eyes dazzle: she died young.\n\nBOSOLA. I think not so; her infelicity\n\nSeemed to have years too many.\n\nFERDINAND. She and I were twins;\n\nAnd should I die this instant, I have lived\n\nHer time to a minute.\n\nBOSOLA. It seems she was born first:\n\nYou have bloodily approved the ancient truth,\n\nThat kindred commonly do worse agree\n\nThan remote strangers.\n\nFERDINAND. Let me see her face\n\nAgain. Why didst not thou pity her? what\n\nAn excellent honest man mightst thou have been,\n\nIf thou hadst borne her to some sanctuary!\n\nOr, bold in a good cause, opposed thyself,\n\nWith thy advanc\u00e8d sword above thy head,\n\nBetween her innocence and my revenge!\n\nI bade thee, when I was distracted of my wits,\n\nGo kill my dearest friend, and thou hast done 't.\n\nFor let me but examine well the cause:\n\nWhat was the meanness of her match to me?\n\nOnly I must confess I had a hope,\n\nHad she continued widow, to have gained\n\nAn infinite mass of treasure by her death:\n\nAnd what was the main cause? her marriage,\n\nThat drew a stream of gall quite through my heart.\n\nFor thee, as we observe in tragedies\n\nThat a good actor many times is cursed\n\nFor playing a villain's part, I hate thee for 't,\n\nAnd, for my sake, say, thou hast done much ill well.\n\nBOSOLA. Let me quicken your memory, for I perceive\n\nYou are falling into ingratitude: I challenge\n\nThe reward due to my service.\n\nFERDINAND. I'll tell thee\n\nWhat I'll give thee.\n\nBOSOLA. Do.\n\nFERDINAND. I'll give thee a pardon\n\nFor this murder.\n\nBOSOLA. Ha!\n\nFERDINAND. Yes, and 'tis\n\nThe largest bounty I can study to do thee.\n\nBy what authority didst thou execute\n\nThis bloody sentence?\n\nBOSOLA. By yours.\n\nFERDINAND. Mine! was I her judge?\n\nDid any ceremonial form of law\n\nDoom her to not-being? did a c\u00f3mplete jury\n\nDeliver her conviction up i' the court?\n\nWhere shalt thou find this judgment registered,\n\nUnless in hell? See, like a bloody fool,\n\nThou'st forfeited thy life, and thou shalt die for 't.\n\nBOSOLA. The office of justice is perverted quite\n\nWhen one thief hangs another. Who shall dare\n\nTo reveal this?\n\nFERDINAND. O, I'll tell thee;\n\nThe wolf shall find her grave, and scrape it up,\n\nNot to devour the corpse, but to discover\n\nThe horrid murder.\n\nBOSOLA. You, not I, shall quake for 't.\n\nFERDINAND. Leave me.\n\nBOSOLA. I will first receive my pension.\n\nFERDINAND. You are a villain.\n\nBOSOLA. When your ingratitude\n\nIs judge, I am so.\n\nFERDINAND. O horror,\n\nThat not the fear of him which binds the devils\n\nCan prescribe man obedience!\u2014\n\nNever look upon me more.\n\nBOSOLA. Why, fare thee well.\n\nYour brother and yourself are worthy men:\n\nYou have a pair of hearts are hollow graves,\n\nRotten, and rotting others; and your vengeance,\n\nLike two chained bullets, still goes arm in arm:\n\nYou may be brothers; for treason, like the plague,\n\nDoth take much in a blood. I stand like one\n\nThat long hath ta'en a sweet and golden dream:\n\nI am angry with myself, now that I wake.\n\nFERDINAND. Get thee into some unknown part o' the world,\n\nThat I may never see thee.\n\nBOSOLA. Let me know\n\nWherefore I should be thus neglected. Sir,\n\nI served your tyranny, and rather strove\n\nTo satisfy yourself than all the world:\n\nAnd though I loathed the evil, yet I loved\n\nYou that did counsel it; and rather sought\n\nTo appear a true servant than an honest man.\n\nFERDINAND. I'll go hunt the badger by owl-light.\n\n'Tis a deed of darkness. [ _Exit_\n\nBOSOLA. He's much distracted. Off, my painted honor!\n\nWhile with vain hopes our faculties we tire,\n\nWe seem to sweat in ice and freeze in fire.\n\nWhat would I do, were this to do again?\n\nI would not change my peace of conscience\n\nFor all the wealth of Europe.\u2014She stirs; here's life:\u2014\n\nReturn, fair soul, from darkness, and lead mine\n\nOut of this sensible hell:\u2014she's warm, she breathes:\u2014\n\nUpon thy pale lips I will melt my heart,\n\nTo store them with fresh color.\u2014Who's there!\n\nSome cordial drink!\u2014Alas! I dare not call:\n\nSo pity would destroy pity.\u2014Her eye opes,\n\nAnd Heaven in it seems to ope, that late was shut,\n\nTo take me up to mercy.\n\nDUCHESS. Antonio!\n\nBOSOLA. Yes, madam, he is living;\n\nThe dead bodies you saw were but feigned statues:\n\nHe's reconciled to your brothers; the Pope hath wrought\n\nThe atonement.\n\nDUCHESS. Mercy! [ _Dies_\n\nBOSOLA. O, she's gone again! there the cords of life broke.\n\nO sacred innocence, that sweetly sleeps\n\nOn turtles' feathers, whilst a guilty conscience\n\nIs a black register wherein is writ\n\nAll our good deeds and bad, a perspective\n\nThat shows us hell! That we cannot be suffered\n\nTo do good when we have a mind to it!\n\nThis is manly sorrow;\n\nThese tears, I am very certain, never grew\n\nIn my mother's milk: my estate has sunk\n\nBelow the degree of fear: where were\n\nThese penitent fountains while she was living?\n\nO, they were frozen up! Here is a sight\n\nAs direful to my soul as is the sword\n\nUnto a wretch has slain his father. Come,\n\nI'll bear thee hence,\n\nAnd execute thy last will; that's deliver\n\nThy body to the reverend dispose\n\nOf some good women: that the cruel tyrant\n\nShall not deny me. Then I'll post to Milan,\n\nWhere somewhat I will speedily enact\n\nWorth my dejection. [ _Exit_\n\n**ACT V**\n\n**S CENE I: _A public place in Milan._**\n\n_Enter_ ANTONIO _and_ DELIO.\n\nANTONIO. What think you of my hope of reconcilement\n\nTo the Araeonian brethren?\n\nDELIO. I misdoubt it;\n\nFor though they have sent their letters of safe-conduct\n\nFor your repair to Milan, they appear\n\nBut nets to entrap you. The Marquis of Pescara,\n\nUnder whom you hold certain land in cheat,\n\nMuch 'gainst his noble nature hath been moved\n\nTo seize those lands; and some of his dependents\n\nAre at this instant making it their suit\n\nTo be invested in your revenues.\n\nI cannot think they mean well to your life\n\nThat do deprive you of your means of life,\n\nYour living.\n\nANTONIO. You are still an heretic\n\nTo any safety I can shape myself.\n\nDELIO. Here comes the marquis: I will make myself\n\nPetitioner for some part of your land,\n\nTo know whither it is flying.\n\nANTONIO. I pray do.\n\nEnter PESCARA.\n\nDELIO. Sir, I have a suit to you.\n\nPESCARA. To me?\n\nDELIO. An easy one:\n\nThere is the citadel of Saint Bennet,\n\nWith some demesnes, of late in the possession\n\nOf Antonio Bologna,\u2014please you bestow them on me.\n\nPESCARA. You are my friend; but this is such a suit,\n\nNor fit for me to give, nor you to take.\n\nDELIO. No, sir?\n\nPESCARA. I will give you ample reason for 't\n\nSoon in private:\u2014here's the cardinal's mistress.\n\n_Enter_ JULIA.\n\nJULIA. My lord, I am grown your poor petitioner,\n\nAnd should be an ill beggar, had I not\n\nA great man's letter here, the cardinal's,\n\nTo court you in my favor. [ _Gives a letter_\n\nPESCARA. He entreats for you\n\nThe citadel of Saint Bennet, that belonged\n\nTo the banished Bologna.\n\nJULIA. Yes.\n\nPESCARA. I could not have thought of a friend I could rather\n\nPleasure with it: 'tis yours.\n\nJULIA. Sir, I thank you;\n\nAnd he shall know how doubly I am engaged\n\nBoth in your gift, and speediness of giving\n\nWhich makes your grant the greater. [ _Exit_\n\nANTONIO. How they fortify\n\nThemselves with my ruin!\n\nDELIO. Sir, I am\n\nLittle bound to you.\n\nPESCARA. Why?\n\nDELIO. Because you denied this suit to me, and gave 't\n\nTo such a creature.\n\nPESCARA. Do you know what it was?\n\nIt was Antonio's land; not forfeited\n\nBy course of law, but ravished from his throat\n\nBy the cardinal's entreaty: it were not fit\n\nI should bestow so main a piece of wrong\n\nUpon my friend; 'tis a gratification\n\nOnly due to a strumpet, for it is injustice.\n\nShall I sprinkle the pure blood of innocents\n\nTo make those followers I call my friends\n\nLook ruddier upon me? I am glad\n\nThis land, ta'en from the owner by such wrong,\n\nReturns again unto so foul an use\n\nAs salary for his lust. Learn, good Delio,\n\nTo ask noble things of me, and you shall find\n\nI'll be a noble giver.\n\nDELIO. You instruct me well.\n\nANTONIO. Why, here's a man who would fright impudence From sauciest beggars.\n\nPESCARA. Prince Ferdinand's come to Milan,\n\nSick, as they give out, of an apoplexy;\n\nBut some say 'tis a frenzy: I am going\n\nTo visit him.\n\n[ _Exit_\n\nANTONIO. 'Tis a noble old fellow.\n\nDELIO. What course do you mean to take, Antonio?\n\nANTONIO. This night I mean to venture all my fortune,\n\nWhich is no more than a poor lingering life,\n\nTo the cardinal's worst of malice: I have got\n\nPrivate access to his chamber; and intend\n\nTo visit him about the mid of night,\n\nAs once his brother did our noble duchess.\n\nIt may be that the sudden apprehension\n\nOf danger,\u2014for I'll go in mine own shape,\u2014\n\nWhen he shall see it fraight with love and duty,\n\nMay draw the poison out of him, and work\n\nA friendly reconcilement: if it fail,\n\nYet it shall rid me of this infamous calling;\n\nFor better fall once than be ever falling.\n\nDELIO. I'll second you in all danger; and, howe'er,\n\nMy life keeps rank with yours.\n\nANTONIO. You are still my loved and best friend.\n\n[ _Exeunt_\n\n**S CENE II**\n\n**_A gallery in the_ CARDINAL'S _palace at Milan._**\n\n_Enter_ PESCARA _and_ DOCTOR.\n\nPESCARA. Now, doctor, may I visit your patient?\n\nDOCTOR. If 't please your lordship: but he's instantly\n\nTo take the air here in the gallery\n\nBy my direction.\n\nPESCARA. Pray thee, what's his disease?\n\nDOCTOR. A very pestilent disease, my lord,\n\nThey call lycanthropia.\n\nPESCARA. What's that?\n\nI need a dictionary to 't.\n\nDOCTOR. I'll tell you.\n\nIn those that are possessed with 't there o'erflows\n\nSuch melancholy humor they imagine\n\nThemselves to be transformed into wolves;\n\nSteal forth to churchyards in the dead of night,\n\nAnd dig dead bodies up: as two nights since\n\nOne met the duke 'bout midnight in a lane\n\nBehind Saint Mark's church, with the leg of a man\n\nUpon his shoulder; and he howled fearfully;\n\nSaid he was a wolf, only the difference\n\nWas, a wolfs skin was hairy on the outside,\n\nHis on the inside; bade them take their swords,\n\nRip up his flesh, and try: straight I was sent for,\n\nAnd, having ministered to him, found his grace\n\nVery well recovered.\n\nPESCARA. I am glad on 't.\n\nDOCTOR. Yet not without some fear\n\nOf a relapse. If he grow to his fit again,\n\nI'll go a nearer way to work with him\n\nThan ever Paracelsus dreamed of; if\n\nThey'll give me leave, I'll buffet his madness out of him.\n\nStand aside; he comes.\n\n_Enter_ FERDINAND, CARDINAL, MALATESTI, _and_ BOSOLA.\n\nFERDINAND. Leave me.\n\nMALATESTI. Why doth your lordship love this solitariness?\n\nFERDINAND. Eagles commonly fly alone; they are crows, daws, and starlings that flock together. Look, what's that follows me?\n\nMALATESTI. Nothing, my lord.\n\nFERDINAND. Yes.\n\nMALATESTI. 'Tis your shadow.\n\nFERDINAND. Stay it; let it not haunt me.\n\nMALATESTI. Impossible, if you move, and the sun shine.\n\nFERDINAND. I will throttle it. [ _Throws himself down on his shadow_\n\nMALATESTI. O, my lord, you are angry with nothing.\n\nFERDINAND. You are a fool: how is't possible I should catch my shadow, unless I fall upon 't? When I go to hell, I mean to carry a bribe; for, look you, good gifts evermore make way for the worst persons.\n\nPESCARA. Rise, good my lord.\n\nFERDINAND. I am studying the art of patience.\n\nPESCARA. 'Tis a noble virtue.\n\nFERDINAND. To drive six snails before me from this town to Moscow; neither use goad nor whip to them, but let them take their own time;\u2014the patient'st man i' the world match me for an experiment;\u2014and I'll crawl after like a sheep-biter.\n\nCARDINAL. Force him up.\n\n_They raise him._\n\nFERDINAND. Use me well, you were best. What I have done, I have done: I'll confess nothing.\n\nDOCTOR. Now let me come to him.\u2014Are you mad, my lord? are you out of your princely wits?\n\nFERDINAND. What's he?\n\nPESCARA. Your doctor.\n\nFERDINAND. Let me have his beard sawed off, and his eyebrows filed more civil.\n\nDOCTOR. I must do mad tricks with him, for that's the only way on't. \u2014 I have brought your grace a salamander's skin to keep you from sun-burning.\n\nFERDINAND. I have cruel sore eyes.\n\nDOCTOR. The white of a cockatrix's egg is present remedy.\n\nFERDINAND. Let it be a new laid one, you were best.\u2014\n\nHide me from him: physicians are like kings,\u2014\n\nThey brook no contradiction.\n\nDOCTOR. Now he begins to fear me: now let me alone with him.\n\nCARDINAL. HOW now! put off your gown!\n\nDOCTOR. Let me have some forty urinals filled with rose-water: he and I'll go pelt one another with them.\u2014Now he begins to fear me.\u2014Can you fetch a frisk, sir?\u2014Let him go, let him go, upon my peril: I find by his eye he stands in awe of me; I'll make him as tame as a dormouse.\n\nFERDINAND. Can you fetch your frisks, sir!\u2014I will stamp him into a cullis, flay off his skin, to cover one of the anatomies this rogue hath set i' the cold yonder in Barber-Surgeon's hall\u2014Hence, hence! you are all of you like beasts for sacrifice: there's nothing left of you but tongue and belly, flattery and lechery. [ _Exit_\n\nPESCARA. Doctor, he did not fear you thoroughly.\n\nDOCTOR. True; I was somewhat too forward.\n\nBOSOLA. Mercy upon me, what a fatal judgment\n\nHath fall'n upon this Ferdinand!\n\nPESCARA. Knows your grace\n\nWhat accident hath brought unto the prince\n\nThis strange distraction?\n\nCARDINAL. [ _Aside_ ] I must feign somewhat.\u2014Thus they say it grew.\n\nYou have heard it rumored, for these many years\n\nNone of our family dies but there is seen\n\nThe shape of an old woman, which is given\n\nBy tradition to us to have been murdered\n\nBy her nephews for her riches. Such a figure\n\nOne night, as the prince sat up late at's book,\n\nAppeared to him; when crying out for help,\n\nThe gentlemen of 's chamber found his grace\n\nAll on a cold sweat, altered much in face\n\nAnd language: since which apparition,\n\nHe hath grown worse and worse, and I much fear\n\nHe cannot live.\n\nBOSOLA. Sir, I would speak with you.\n\nPESCARA. We'll leave your grace,\n\nWishing to the sick prince, our noble lord,\n\nAll health of mind and body.\n\nCARDINAL. You are most welcome.\n\n_Exeunt_ PESCARA, MALATESTI, _and_ DOCTOR.\n\nAre you come? so.\u2014 [Aside] This fellow must not know\n\nBy any means I had intelligence\n\nIn our duchess's death; for, though I counselled it,\n\nThe full of all the engagement seemed to grow\n\nFrom Ferdinand.\u2014Now, sir, how fares our sister?\n\nI do not think but sorrow makes her look\n\nLike to an oft-dyed garment: she shall now\n\nTaste comfort from me. Why do you look so wildly?\n\nO, the fortune of your master here the prince\n\nDejects you; but be you of happy comfort:\n\nIf you'll do one thing for me I'll entreat,\n\nThough he had a cold tombstone o'er his bones,\n\nI'd make you what you would be.\n\nBOSOLA. Any thing;\n\nGive it me in a breath, and let me fly to 't:\n\nThey that think long small expedition win,\n\nFor musing much o' the end cannot begin.\n\n_Enter_ JULIA.\n\nJULIA. Sir, will you come in to supper?\n\nCARDINAL. I am busy; leave me.\n\nJULIA. [Aside] What an excellent shape hath that fellow! [ _Exit_\n\nCARDINAL. 'Tis thus. Antonio lurks here in Milan:\n\nInquire him out, and kill him. While he lives,\n\nOur sister cannot marry; and I have thought\n\nOf an excellent match for her. Do this, and style me\n\nThy advancement.\n\nBOSOLA. But by what means shall I find him out?\n\nCARDINAL. There is a gentleman called Delio\n\nHere in the camp, that hath been long approved\n\nHis loyal friend. Set eye upon that fellow;\n\nFollow him to mass; may be Antonio,\n\nAlthough he do account religion\n\nBut a school-name, for fashion of the world\n\nMay accompany him; or else go inquire out\n\nDelio's confessor, and see if you can bribe\n\nHim to reveal it. There are a thousand ways\n\nA man might find to trace him; as to know\n\nWhat fellows haunt the Jews for taking up\n\nGreat sums of money, for sure he's in want;\n\nOr else to go to the picture-makers, and learn\n\nWho bought her picture lately: some of these\n\nHappily may take.\n\nBOSOLA. Well, I'll not freeze i' the business:\n\nI would see that wretched thing, Antonio,\n\nAbove all sights i' the world.\n\nCARDINAL. Do, and be happy. [ _Exit_\n\nBOSOLA. This fellow doth breed basilisks in 's eyes,\n\nHe's nothing else but murder; yet he seems\n\nNot to have notice of the duchess's death.\n\n'Tis his cunning: I must follow his example;\n\nThere cannot be a surer way to trace\n\nThan that of an old fox.\n\n_Re-enter_ JULIA.\n\nJULIA. So, sir, you are well met.\n\nBOSOLA. How now!\n\nJULIA. Nay, the doors are fast enough:\n\nNow, sir, I will make you-confess your treachery.\n\nBOSOLA. Treachery!\n\nJULIA. Yes, confess to me\n\nWhich of my women 'twas you hired to put\n\nLove-powder into my drink?\n\nBOSOLA. Love-powder!\n\nJULIA. Yes, when I was at Malfi.\n\nWhy should I fall in love with such a face else?\n\nI have already suffered for thee so much pain,\n\nThe only remedy to do me good\n\nIs to kill my longing.\n\nBOSOLA. Sure, your pistol holds\n\nNothing but perfumes or kissing-comfits.\n\nExcellent lady!\n\nYou have a pretty way on 't to discover\n\nYour longing. Come, come, I'll disarm you,\n\nAnd arm you thus: yet this is wondrous strange.\n\nJULIA. Compare thy form and my eyes together,\n\nYou'll find my love no such great miracle.\n\nNow you'll say\n\nI am wanton: this nice modesty in ladies\n\nIs but a troublesome familiar\n\nThat haunts them.\n\nBOSOLA. Know you me, I am a blunt soldier.\n\nJULIA. The better:\n\nSure, there wants fire where there are no lively sparks\n\nOf roughness.\n\nBOSOLA. And I want compliment.\n\nJULIA. Why, ignorance\n\nIn courtship cannot make you do amiss,\n\nIf you have a heart to do well.\n\nBOSOLA. You are very fair.\n\nJULIA. Nay, if you lay beauty to my charge,\n\nI must plead unguilty.\n\nBOSOLA. Your bright eyes\n\nCarry a quiver of darts in them sharper\n\nThan sunbeams.\n\nJULIA. You will mar me with commendation,\n\nPut yourself to the charge of courting me,\n\nWhereas now I woo you.\n\nBOSOLA. [Aside] I have it, I will work upon this creature.\u2014\n\nLet us grow most amorously familiar:\n\nIf the great cardinal now should see me thus,\n\nWould he not count me a villain?\n\nJULIA. No; he might count me a wanton,\n\nNot lay a scruple of offence on you;\n\nFor if I see and steal a diamond,\n\nThe fault is not i' the stone, but in me the thief\n\nThat purloins it. I am sudden with you:\n\nWe that are great women of pleasure use to cut off\n\nThese uncertain wishes and unquiet longings,\n\nAnd in an instant join the sweet delight\n\nAnd the pretty excuse together. Had you been i' the street,\n\nUnder my chamber-window, even there\n\nI should have courted you.\n\nBOSOLA. O, you are an excellent lady!\n\nJULIA. Bid me do somewhat for you presently\n\nTo express I love you.\n\nBOSOLA. I will; and if you love me,\n\nFail not to effect it.\n\nThe cardinal is grown wondrous melancholy;\n\nDemand the cause, let him not put you off\n\nWith feigned excuse; discover the main ground on 't.\n\nJULIA. Why would you know this?\n\nBOSOLA. I have depended on him,\n\nAnd I hear that he is fall'n in some disgrace\n\nWith the emperor: if he be, like the mice\n\nThat forsake falling houses, I would shift\n\nTo other dependance.\n\nJULIA. You shall not need\n\nFollow the wars: I'll be your maintenance.\n\nBOSOLA. And I your loyal servant: but I cannot\n\nLeave my calling.\n\nJULIA. Not leave an ungrateful\n\nGeneral for the love of a sweet lady!\n\nYou are like some cannot sleep in feather-beds,\n\nBut must have blocks for their pillows.\n\nBOSOLA. Will you do this?\n\nJULIA. Cunningly.\n\nBOSOLA. To-morrow I'll expect the intelligence.\n\nJULIA. To-morrow! get you into my cabinet;\n\nYou shall have it with you. Do not delay me,\n\nNo more than I do you: I am like one\n\nThat is condemned; I have my pardon promised,\n\nBut I would see it sealed. Go, get you in:\n\nYou shall see me wind my tongue about his heart\n\nLike a skein of silk.\n\n_Exit_ BOSOLA. _Re-enter_ CARDINAL.\n\nCARDINAL. Where are you?\n\n_Enter_ SERVANTS.\n\nSERVANTS. Here.\n\nCARDINAL. Let none, upon your lives, have conference\n\nWith the Prince Ferdinand, unless I know it.\u2014\n\n[ _Aside_ ] In this distraction he may reveal\n\nThe murder.\n\n_Exeunt_ SERVANTS.\n\nYond's my lingering consumption:\n\nI am weary of her, and by any means\n\nWould be quit of.\n\nJULIA. How now, my lord! what ails you?\n\nCARDINAL. Nothing.\n\nJULIA. O, you are much altered:\n\nCome, I must be your secretary, and remove\n\nThis lead from off your bosom: what's the matter?\n\nCARDINAL. I may not tell you.\n\nJULIA. Are you so far in love with sorrow\n\nYou cannot part with part of it? or think you\n\nI cannot love your grace when you are sad\n\nAs well as merry? or do you suspect\n\nI, that have been a secret to your heart\n\nThese many winters, cannot be the same\n\nUnto your tongue?\n\nCARDINAL. Satisfy thy longing,\u2014\n\nThe only way to make thee keep my counsel\n\nIs, not to tell thee.\n\nJULIA. Tell your echo this,\n\nOr flatterers, that like echoes still report\n\nWhat they hear though most imperfect, and not me;\n\nFor if that you be true unto yourself,\n\nI'll know.\n\nCARDINAL. Will you rack me?\n\nJULIA. No, judgment shall\n\nDraw it from you: it is an equal fault,\n\nTo tell one's secrets unto all or none.\n\nCARDINAL. The first argues folly.\n\nJULIA. But the last tyranny.\n\nCARDINAL. Very well: why, imagine I have committed\n\nSome secret deed which I desire the world\n\nMay never hear of.\n\nJULIA. Therefore may not I know it?\n\nYou have concealed for me as great a sin\n\nAs adultery. Sir, never was occasion\n\nFor perfect trial of my constancy\n\nTill now: sir, I beseech you\u2014\n\nCARDINAL. You'll repent it.\n\nJULIA. Never.\n\nCARDINAL. It hurries thee to ruin: I'll not tell thee.\n\nBe well advised, and think what danger 'tis\n\nTo receive a prince's secrets: they that do,\n\nHad need have their breasts hooped with adamant\n\nTo contain them. I pray thee, yet be satisfied;\n\nExamine thine own frailty; 'tis more easy\n\nTo tie knots than unloose them: 'tis a secret\n\nThat, like a lingering poison, may chance lie\n\nSpread in thy veins, and kill thee seven year hence.\n\nJULIA. Now you dally with me.\n\nCARDINAL. NO more; thou shalt know it.\n\nBy my appointment the great Duchess of Malfi\n\nAnd two of her young children, four nights since,\n\nWere strangled.\n\nJULIA. O Heaven! sir, what have you done!\n\nCARDINAL. How now? how settles this? think you your bosom\n\nWill be a grave dark and obscure enough\n\nFor such a secret?\n\nJULIA. You have undone yourself, sir.\n\nCARDINAL. Why?\n\nJULIA. It lies not in me to conceal it.\n\nCARDINAL. No?\n\nCome, I will swear you to 't upon this book.\n\nJULIA. Most religiously.\n\nCARDINAL. Kiss it.\n\n_She kisses the book_\n\nNow you shall never utter it; thy curiosity\n\nHath undone thee: thou 'rt poisoned with that book;\n\nBecause I knew thou couldst not keep my counsel,\n\nI have bound thee to 't by death.\n\n_Re-enter_ BOSOLA.\n\nBOSOLA. For pity-sake, hold!\n\nCARDINAL. Ha, Bosola!\n\nJULIA. I forgive you\n\nThis equal piece of justice you have done;\n\nFor I betrayed your counsel to that fellow:\n\nHe overheard it; that was the cause I said\n\nIt lay not in me to conceal it.\n\nBOSOLA. O foolish woman,\n\nCouldst not thou have poisoned him?\n\nJULIA. 'Tis weakness,\n\nToo much to think what should have been done.\n\nI go,\n\nI know not whither. [ _Dies_\n\nCARDINAL. Wherefore com'st thou hither?\n\nBOSOLA. That I might find a great man like yourself,\n\nNot out of his wits as the Lord Ferdinand,\n\nTo remember my service.\n\nCARDINAL. I'll have thee hewed in pieces.\n\nBOSOLA. Make not yourself such a promise of that life\n\nWhich is not yours to dispose of.\n\nCARDINAL. Who placed thee here?\n\nBOSOLA. Her lust, as she intended.\n\nCARDINAL. Very well:\n\nNow you know me for your fellow-murderer.\n\nBOSOLA. And wherefore should you lay fair marble colors\n\nUpon your rotten purposes to me?\n\nUnless you imitate some that do plot great treasons,\n\nAnd when they have done, go hide themselves i' the graves\n\nOf those were actors in 't?\n\nCARDINAL. No more; there is\n\nA fortune attends thee.\n\nBOSOLA. Shall I go sue to Fortune any longer?\n\n'Tis the fool's pilgrimage.\n\nCARDINAL. I have honors in store for thee.\n\nBOSOLA. There are many ways that conduct to seeming honor,\n\nAnd some of them very dirty ones.\n\nCARDINAL. Throw to the devil\n\nThy melancholy. The fire burns well;\n\nWhat need we keep a stirring of 't, and make\n\nA greater smother? Thou wilt kill Antonio?\n\nBOSOLA. Yes.\n\nCARDINAL. Take up that body.\n\nBOSOLA. I think I shall\n\nShortly grow the common bearer for churchyards.\n\nCARDINAL. I will allow thee some dozen of attendants\n\nTo aid thee in the murder.\n\nBOSOLA. O, by no means. Physicians that apply horse-leeches to any rank swelling use to cut off their tails, that the blood may run through them the faster: let me have no train when I go to shed blood, lest it make me have a greater when I ride to the gallows.\n\nCARDINAL. Come to me after midnight, to help to remove\n\nThat body to her own lodging: I'll give out\n\nShe died o' the plague; 'twill breed the less inquiry\n\nAfter her death.\n\nBOSOLA. Where's Castruccio, her husband?\n\nCARDINAL. He's rode to Naples, to take possession\n\nOf Antonio's citadel.\n\nBOSOLA. Believe me, you have done a very happy turn.\n\nCARDINAL. Fail not to come: there is the master-key\n\nOf our lodgings; and by that you may conceive\n\nWhat trust I plant in you.\n\nBOSOLA. You shall find me ready.\n\n_Exit_ CARDINAL.\n\nO poor Antonio, though nothing be so needful\n\nTo thy estate as pity, yet I find\n\nNothing so dangerous; I must look to my footing:\n\nIn such slippery ice-pavements men had need\n\nTo be frost-nailed well, they may break their necks.\n\nThe precedent's here afore me. How this man\n\nBears up in blood! seems fearless! Why, 'tis well: else;\n\nSecurity some men call the suburbs of hell,\n\nOnly a dead wall between. Well, good Antonio,\n\nI'll seek thee out; and all my care shall be\n\nTo put thee into safety from the reach\n\nOf these most cruel biters that have got\n\nSome of thy blood already. It may be,\n\nI'll join with thee in a most just revenge:\n\nThe weakest arm is strong enough that strikes\n\nWith the sword of justice. Still methinks the duchess\n\nHaunts me: there, there!\u2014'Tis nothing but my melancholy.\n\nO Penitence, let me truly taste thy cup,\n\nThat throws men down only to raise them up!\n\n[ _Exit_\n\n**S CENE III**\n\n**_A fortification at Milan._**\n\nEnter ANTONIO _and_ DELIO.\n\nDELIO. Yond's the cardinal's window. This fortification\n\nGrew from the ruins of an ancient abbey;\n\nAnd to yond side o' the river lies a wall,\n\nPiece of a cloister, which in my opinion\n\nGives the best echo that you ever heard,\n\nSo hollow and so dismal, and withal\n\nSo plain in the distinction of our words,\n\nThat many have supposed it is a spirit\n\nThat answers.\n\nANTONIO. I do love these ancient ruins.\n\nWe never tread upon them but we set\n\nOur foot upon some reverend history:\n\nAnd, questionless, here in this open court,\n\nWhich now lies naked to the injuries\n\nOf stormy weather, some men lie interred\n\nLoved the church so well, and gave so largely to 't,\n\nThey thought it should have canopied their bones\n\nTill doomsday; but all things have their end:\n\nChurches and cities, which have diseases like to men,\n\nMust have like death that we have.\n\nECHO. \"Like death that we have.\"\n\nDELIO. Now the echo hath caught you.\n\nANTONIO. It groaned, methought, and gave\n\nA very deadly accent,\n\nECHO. \"Deadly accent.\"\n\nDELIO. I told you 'twas a pretty one: you may make it\n\nA huntsman, or a falconer, a musician,\n\nOr a thing of sorrow.\n\nECHO. \"A thing of sorrow.\"\n\nANTONIO. Ay, sure, that suits it best.\n\nECHO. \"That suits it best.\"\n\nANTONIO. 'Tis very like my wife's voice.\n\nECHO. \"Ay, wife's voice.\"\n\nDELIO. Come, let us walk further from 't.\n\nI would not have you go to the cardinal's to-night:\n\nDo not.\n\nECHO. \"Do not.\"\n\nDELIO. Wisdom doth not more moderate wasting sorrow\n\nThan time: take time for't; be mindful of thy safety.\n\nECHO. \"Be mindful of thy safety.\"\n\nANTONIO. Necessity compels me:\n\nMake scrutiny throughout the passages\n\nOf your own life, you'll find it impossible\n\nTo fly your fate.\n\nECHO. \"O, fly your fate.\"\n\nDELIO. Hark! the dead stones seem to have pity on you,\n\nAnd give you good counsel.\n\nANTONIO. Echo, I will not talk with thee,\n\nFor thou art a dead thing.\n\nECHO. \"Thou art a dead thing.\"\n\nANTONIO. My duchess is asleep now,\n\nAnd her little ones, I hope sweetly: O Heaven,\n\nShall I never see her more?\n\nECHO. \"Never see her more.\"\n\nANTONIO. I marked not one repetition of the echo\n\nBut that; and on the sudden a clear light\n\nPresented me a face folded in sorrow.\n\nDELIO. Your fancy merely.\n\nANTONIO. Come, I'll be out of this ague,\n\nFor to live thus is not indeed to live;\n\nIt is a mockery and abuse of life:\n\nI will not henceforth save myself by halves;\n\nLose all, or nothing.\n\nDELIO. Your own virtues save you!\n\nI'll fetch your eldest son, and second you:\n\nIt may be that the sight of his own blood\n\nSpread in so sweet a figure may beget\n\nThe more compassion. However, fare you well.\n\nThough in our miseries Fortune have a part,\n\nYet in our noble sufferings she hath none:\n\nContempt of pain, that we may call our own.\n\n[ _Exeunt_\n\n**S CENE IV**\n\n**_An apartment in the_ CARDINAL's _palace._**\n\n_Enter_ CARDINAL, PESCARA, MALATESTI, RODERIGO, _and_ GRISOLAN.\n\nCARDINAL. You shall not watch to-night by the sick prince;\n\nHis grace is very well recovered.\n\nMALATESTI. Good my lord, suffer us.\n\nCARDINAL. O, by no means;\n\nThe noise, and change of object in his eye,\n\nDoth more distract him: I pray, all to bed;\n\nAnd though you hear him in his violent fit,\n\nDo not rise, I entreat you.\n\nPESCARA. So, sir; we shall not.\n\nCARDINAL. Nay, I must have you promise\n\nUpon your honors, for I was enjoined to 't\n\nBy himself; and he seemed to urge it sensibly.\n\nPESCARA. Let our honors bind this trifle.\n\nCARDINAL. Nor any of your followers.\n\nMALATESTI. Neither.\n\nCARDINAL. It may be, to make trial of your promise,\n\nWhen he's asleep, myself will rise and feign\n\nSome of his mad tricks, and cry out for help,\n\nAnd feign myself in danger.\n\nMALATESTI. If your throat were cutting,\n\nI'd not come at you, now I have protested against it.\n\nCARDINAL. Why, I thank you.\n\nGRISOLAN. 'Twas a foul storm to-night.\n\nRODERIGO. The Lord Ferdinand's chamber shook like an osier.\n\nMALATESTI. 'Twas nothing but pure kindness in the devil,\n\nTo rock his own child.\n\n_Exeunt all except_ CARDINAL.\n\nCARDINAL. The reason why I would not suffer these\n\nAbout my brother, is, because at midnight\n\nI may with better privacy convey\n\nJulia's body to her own lodging. O, my conscience!\n\nI would pray now; but the devil takes away my heart\n\nFor having any confidence in prayer.\n\nAbout this hour I appointed Bosola\n\nTo fetch the body: when he hath served my turn,\n\nHe dies. [ _Exit_\n\n_Enter_ BOSOLA.\n\nBOSOLA. Ha! 'twas the cardinal's voice; I heard him name\n\nBosola and my death. Listen; I hear one's footing.\n\n_Enter_ FERDINAND.\n\nFERDINAND. Strangling is a very quiet death.\n\nBOSOLA. [Aside] Nay, then, I see I must stand upon my guard.\n\nFERDINAND. What say you to that? whisper softly; do you agree to't? So; it must be done i' the dark: the cardinal would not for a thousand pounds the doctor should see it. [ _Exit_\n\nBOSOLA. My death is plotted; here's the consequence of murder.\n\nWe value not desert nor Christian breath,\n\nWhen we know black deeds must be cured with death.\n\n_Enter_ ANTONIO _and_ SERVANT.\n\nSERVANT. Here stay, sir, and be confident, I pray:\n\nI'll fetch you a dark lantern. [ _Exit_\n\nANTONIO. Could I take him at his prayers,\n\nThere were hope of pardon.\n\nBOSOLA. Fall right, my sword!\u2014 [ _Stabs_ _him_\n\nI'll not give thee so much leisure as to pray.\n\nANTONIO. O, I am gone! Thou hast ended a long suit\n\nIn a minute.\n\nBOSOLA. What art thou?\n\nANTONIO. A most wretched thing,\n\nThat only have thy benefit in death,\n\nTo appear myself.\n\n_Re-enter_ SERVANT, _with a lantern._\n\nSERVANT. Where are you, sir?\n\nANTONIO. Very near my home.\u2014Bosola!\n\nSERVANT. O, misfortune!\n\nBOSOLA. Smother thy pity, thou art dead else.\u2014Antonio!\n\nThe man I would have saved 'bove mine own life!\n\nWe are merely the stars' tennis-balls, struck and bandied\n\nWhich way please them.\u2014O good Antonio,\n\nI'll whisper one thing in thy dying ear\n\nShall make thy heart break quickly! thy fair duchess and two sweet children\u2014\n\nANTONIO. Their very names\n\nKindle a little life in me.\n\nBOSOLA. Are murdered.\n\nANTONIO. Some men have wished to die\n\nAt the hearing of sad things; I am glad\n\nThat I shall do 't in sadness: I would not now\n\nWish my wounds balmed nor healed, for I have no use\n\nTo put my life to. In all our quest of greatness,\n\nLike wanton boys, whose pastime is their care,\n\nWe follow after bubbles blown in the air.\n\nPleasure of life, what is 't? only the good hours\n\nOf an ague; merely a preparative to rest,\n\nTo endure vexation. I do not ask\n\nThe process of my death; only commend me\n\nTo Delio.\n\nBOSOLA. Break, heart!\n\nANTONIO. And let my son fly the courts of princes. [ _Dies_\n\nBOSOLA. Thou seem'st to have loved Antonio?\n\nSERVANT. I brought him hither,\n\nTo have reconciled him to the cardinal.\n\nBOSOLA. I do not ask thee that.\n\nTake him up, if thou tender thine own life,\n\nAnd bear him where the lady Julia\n\nWas wont to lodge.\u2014O, my fate moves swift;\n\nI have this cardinal in the forge already;\n\nNow I'll bring him to the hammer. O direful misprision\n\nI will not imitate things glorious,\n\nNo more than base; I'll be mine own example.\u2014\n\nOn, on, and look thou represent, for silence,\n\nThe thing thou bear'st. [ _Exeunt_\n\n**S CENE V**\n\n**_Another apartment in the palace._**\n\n_Enter_ CARDINAL, _with a book._\n\nCARDINAL. I am puzzled in a question about hell:\n\nHe says, in hell there's one material fire,\n\nAnd yet it shall not burn all men alike.\n\nLay him by. How tedious is a guilty conscience!\n\nWhen I look into the fish-ponds in my garden,\n\nMethinks I see a thing armed with a rake,\n\nThat seems to strike at me.\n\n_Enter_ BOSOLA, _and_ SERVANT _bearing_ ANTONIO'S _body._\n\nNow, art thou come?\n\nThou look'st ghastly:\n\nThere sits in thy face some great determination\n\nMixed with some fear.\n\nBOSOLA. Thus it lightens into action:\n\nI am come to kill thee.\n\nCARDINAL. Ha!\u2014Help! our guard!\n\nBOSOLA. Thou art deceived;\n\nThey are out of thy howling.\n\nCARDINAL. Hold; and I will faithfully divide\n\nRevenues with thee.\n\nBOSOLA. Thy prayers and proffers\n\nAre both unseasonable.\n\nCARDINAL. Raise the watch! we are betrayed!\n\nBOSOLA. I have confined your flight:\n\nI'll suffer your retreat to Julia's chamber,\n\nBut no further.\n\nCARDINAL. Help! we are betrayed!\n\n_Enter, above,_ PESCARA, MALATESTI, RODERIGO, _and_ GRISOLAN.\n\nMALATESTI. Listen.\n\nCARDINAL. My dukedom for rescue!\n\nRODERIGO. Fie upon his counterfeiting!\n\nMALATESTI. Why, 'tis not the cardinal.\n\nRODERIGO. Yes, yes, 'tis he:\n\nBut I'll see him hanged ere I'll go down to him.\n\nCARDINAL. Here's a plot upon me; I am assaulted! I am lost,\n\nUnless some rescue.\n\nGRISOLAN. He doth this pretty well;\n\nBut it will not serve to laugh me out of mine honor.\n\nCARDINAL. The sword's at my throat!\n\nRODERIGO. You would not bawl so loud then.\n\nMALATESTI. Come, come, let's go\n\nTo bed: he told us thus much aforehand.\n\nPESCARA. He wished you should not come at him; but, believe 't,\n\nThe accent of the voice sounds not in jest:\n\nI'll down to him, howsoever, and with engines\n\nForce ope the doors. [ _Exit above_\n\nRODERIGO. Let's follow him aloof,\n\nAnd note how the cardinal will laugh at him.\n\n_Exeunt, above,_ MALATESTI, RODERIGO, _and_ GRISOLAN.\n\nBOSOLA. There's for you first,\n\n'Cause you shall not unbarricade the door\n\nTo let in rescue. [ _Kills the_ SERVANTCARDINAL. What cause hast thou to pursue my life?\n\nBOSOLA. Look there.\n\nCARDINAL. Antonio!\n\nBOSOLA. Slain by my hand unwittingly.\n\nPray, and be sudden: when thou killed'st thy sister,\n\nThou took'st from Justice her most equal balance,\n\nAnd left her naught but her sword.\n\nCARDINAL. O, mercy!\n\nBOSOLA. Now it seems thy greatness was only outward;\n\nFor thou fall'st faster of thyself than calamity\n\nCan drive thee. I'll not waste longer time; there!\n\n[ _Stabs him_\n\nCARDINAL. Thou hast hurt me.\n\nBOSOLA. Again! [ _Stabs him again_\n\nCARDINAL. Shall I die like a leveret,\n\nWithout any resistance?\u2014Help, help, help!\n\nI am slain!\n\n_Enter_ FERDINAND.\n\nFERDINAND. The alarum! give me a fresh horse;\n\nRally the vaunt-guard, or the day is lost.\n\nYield, yield! I give you the honor of arms,\n\nShake my sword over you; will you yield?\n\nCARDINAL. Help me; I am your brother!\n\nFERDINAND. The devil!\n\nMy brother fight upon the adverse party!\n\n_He wounds the_ CARDINAL, _and, in the scuffle, gives_ BOSOLA _his death-wound._\n\nThere flies your ransom.\n\nCARDINAL. O justice!\n\nI suffer now for what hath former bin:\n\nSorrow is held the eldest child of sin.\n\nFERDINAND. NOW you're brave fellows. Caesar's fortune was harder than Pompey's; Caesar died in the arms of prosperity, Pompey at the feet of disgrace. You both died in the field. The pain's nothing: pain many times is taken away with the apprehension of greater, as the toothache with the sight of the barber that comes to pull it out: there's philosophy for you.\n\nBOSOLA. Now my revenge is perfect.\u2014Sink thou main cause\n\n[ _Kills_ Ferdinand\n\nOf my undoing!\u2014The last part of my life\n\nHath done me best service.\n\nFERDINAND. Give me some wet hay; I am broken-winded.\n\nI do account this world but a dog kennel:\n\nI will vault credit and affect high pleasures\n\nBeyond death.\n\nBOSOLA. He seems to come to himself,\n\nNow he's so near the bottom.\n\nFERDINAND. My sister, O my sister! there's the cause on 't.\n\nWhether we fall by ambition, blood, or lust,\n\nLike diamonds we are cut with our own dust. [ _Dies_\n\nCARDINAL. Thou hast thy payment too.\n\nBOSOLA. Yes, I hold my weary soul in my teeth;\n\n'Tis ready to part from me. I do glory\n\nThat thou, which stood'st like a huge pyramid\n\nBegun upon a large and ample base,\n\nShalt end in a little point, a kind of nothing.\n\n_Enter below,_ PESCARA, MALATESTI, RODERIGO, _and_ GRISOLAN.\n\nPESCARA. HOW now, my lord!\n\nMALATESTI. O sad disaster!\n\nRODERIGO. How comes this?\n\nBOSOLA. Revenge for the Duchess of Malfi murdered\n\nBy the Aragonian brethren; for Antonio\n\nSlain by this hand; for lustful Julia\n\nPoisoned by this man; and lastly for myself,\n\nThat was an actor in the main of all\n\nMuch 'gainst mine own good nature, yet i' the end\n\nNeglected.\n\nPESCARA. How now, my lord!\n\nCARDINAL. Look to my brother:\n\nHe gave us these large wounds, as we were struggling\n\nHere i' the rushes. And now, I pray, let me\n\nBe laid by and never thought of. [ _Dies_\n\nPESCARA. How fatally, it seems, he did withstand\n\nHis own rescue!\n\nMALATESTI. Thou wretched thing of blood,\n\nHow came Antonio by his death?\n\nBOSOLA. In a mist; I know not how;\n\nSuch a mistake as I have often seen\n\nIn a play. O, I am gone!\n\nWe are only like dead walls or vaulted graves,\n\nThat, ruined, yield no echo. Fare you well.\n\nIt may be pain, but no harm, to me to die\n\nIn so good a quarrel. O, this gloomy world!\n\nIn what a shadow, or deep pit of darkness,\n\nDoth womanish and fearful mankind live!\n\nLet worthy minds ne'er stagger in distrust\n\nTo suffer death or shame for what is just:\n\nMine is another voyage. [ _Dies_\n\nPESCARA. The noble Delio, as I came to the palace,\n\nTold me of Antonio's being here, and showed me\n\nA pretty gentleman, his son and heir.\n\n_Enter_ DELIO _and_ ANTONIO'S SON.\n\nMALATESTI. O sir, you come too late!\n\nDELIO. I heard so, and\n\nWas armed for't, ere I came. Let us make noble use\n\nOf this great ruin; and join all our force\n\nTo establish this young hopeful gentleman\n\nIn 's mother's right. These wretched eminent things\n\nLeave no more fame behind 'em, than should one\n\nFall in a frost, and leave his print in snow;\n\nAs soon as the sun shines, it ever melts,\n\nBoth form and matter. I have ever thought\n\nNature doth nothing so great for great men\n\nAs when she's pleased to make them lords of truth:\n\nIntegrity of life is fame's best friend,\n\nWhich nobly, beyond death, shall crown the end. [ _Exeunt_\n\n* * *\n\n Legendary king whose punishment in the lower world was to be unable to slake his thirst or allay his hunger.\n\n A play upon the word \"tent,\" which also means a bandage.\n\n A jennet was a small Spanish horse.\n\n Pliny the Elder (23-79 A.D.), Roman naturalist who authored the Historia Naturalis.\n\n The wooden Trojan horse, described in the _Aeneid_ and the _Odyssey._\n\n A dance.\n\n Stewardship of her horses.\n\n Camp; from the German _Lager._\n\n Stately carriages.\n\n Lecherous.\n\n The net that held Mars and Venus.\n\n Tapestry screen.\n\n Discharge.\n\n Having \"humors\" or moods.\n\n The heroes of an English ballad based on the medieval tale _Amis and Amiloun;_ extraordinarily faithful friends who look alike. One friend marries a princess in behalf of the other but places a naked sword between himself and the bride in the marriage bed.\n\n Bullies.\n\n Lawyers.\n\n As though a state procession had driven across it.\n\n Morphewed: scaly; disembogue: empty.\n\n First cousin.\n\n The effects of female hormones.\n\n Plum.\n\n Hooped petticoats.\n\n Stripling, young man.\n\n Jupiter (or Zeus) wooed the maiden Dana\u00eb by transforming himself into a golden shower, which dropped into her lap.\n\n Swiss.\n\n An ornamented flap concealing the opening in the breeches.\n\n Cast a horoscope for the newborn baby.\n\n Fit to be judged.\n\n The rest is not investigated.\n\n Pander.\n\n Strong broths. The old recipe-books recommend \"pieces of gold\" among the ingredients.\n\n Scurrilous verses.\n\n Orrisroot, the rootstock of irises, used in perfumes.\n\n Godparents.\n\n Rapid.\n\n Here, death certificate; _Quietus_ was used playfully in Act I, Scene I.\n\n A quick run.\n\n Bermudas.\n\n Camp.\n\n A bird that resembles a lark but does not possess its voice.\n\n A medicinal bark.\n\n Visored or masked.\n\n Quarry.\n\n A curtain.\n\n Brutus's wife, who committed suicide by casting burning coals into her mouth and choking herself with them after the death of Brutus at Philippi.\n\n Ulcer.\n\n The Genevan Bible, a translation made by Puritan exiles in 1560.\n\n Constipated.\n\n Curdled.\n\n Residue.\n\n Subject to escheat.\n\n Physician and alchemist (1493?\u20141541).\n\n Skeletons.\n\n The surgeons or barber-surgeons were given the bodies of four executed criminals each year.\n\n Perfumed sugar-plums, for sweetening the breath.\n\n A willow.\n\n In earnest.\n\n Mistake.\n\n A hare in its first year.\nwww.doverpublications.com\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvikn b/data_all_eng_slimpj/shuffled/split2/finalzzvikn
new file mode 100644
index 0000000000000000000000000000000000000000..0d3f9fc5ce8c476b7e4688c219703a3497157517
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzvikn
@@ -0,0 +1,5 @@
+{"text":"Step by step instructions on how to make your own gimmick cards. Perform this with regular cards or jumbo cards - both work well.\nFull stand-up comedy routine Paul has uses for years in comedy clubs, cruise ships and corporate shows.\nPaul's original handling and routine for his famous Five Card Bunny Repeat - a variation that is perfect for the family entertainer.\nPhotographs showing the handling and mechanics of this routine making it VERY EASY to learn and perform.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Summer mock exam timetable to be announced.\nSummer exam timetable to be announced.\nResults should be available for collection from 9.30am. Students wanting a relative or friend to collect their results must send a signed note of authorisation with the collector. Results will not be given over the telephone under any circumstances. Any uncollected results will be posted second class at the end of the day. Students wishing to receive their results by first class post should bring either a stamp, or cash to the value of a first class postage stamp, to student reception.\nNB Exam certificates will not usually be available until early November. Please make sure that you advise us of any change of address before then.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For the ISK, we meet at Jita IV - Moon 4 - Caldari Navy Assembly Plant ( Station ) for face to face trade. You can also accept the contract anywhere in game, the ISK will transfer to your wallet automatically after you accept the contract, so you don't need to choose how the ISK will be delivered.\nWe normally meet at Jita IV - Moon 4 - Caldari Navy Assembly Plant ( Station ) for the face to face trade. If you need us to delivery, write your station name with your system name in your order please.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Picasso and Einstein Walk In to a Bar\u2026 The Picasso Players, in association with Redbud Theatre, is bringing Steve Martin's Picasso at the Lapine Agile to Willow Creek's Kimtu Cookhouse Theater. This long running Off Broadway absurdist comedy places Albert Einstein and Pablo Picasso in a Parisian Caf\u00e9 in 1904 before the Einstein transformed physics and Picasso set the art world. Steve Martin plays fast and loose with fact, fame and fortune as these two geniuses muse on the century's achievements and prospects, as well as other fanciful topics. All performances are at the Kimtu Cookhouse at Camp Kimtu in Willow Creek. The play begins at 7:00 pm, with doors opening at 6:00. Beer, wine and small plates will be available for sale. Tickets are $10 in advance or $15 at the door. Advance tickets may be purchased at the Dream Quest Thrift Store in Willow Creek or by phone (530) 629-3564. A portion of the proceeds will the Dream Quest \/ Mountain Community and Culture fund to build a community center in Willow Creek.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This paper presents a \"SLEEP GUARDIAN\" system to measure physiological parameters in sleep without restraint to the tester. The purpose of this project is to devise a non-conscious, automatic, sleep monitoring and evaluating system, which is suitable in the sleep environment at home.\nThe SLEEP GUARDIAN system consists of a body temperature monitoring device, a respiration monitoring device, a body movement sensing device, and a snore sensing device. Physiological signals detected by sensors, for example body movement, respiration, snore, and body temperature, are then integrated to evaluate the user's quality of sleep. The SLEEP GUARDIAN system is designed as \"add-on\" devices to user's sleeping environment. The system is also linked to the internet for children and care-givers to observe the sleep quality of the elder.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvmvh b/data_all_eng_slimpj/shuffled/split2/finalzzvmvh
new file mode 100644
index 0000000000000000000000000000000000000000..cefdbca7b3f8d1180bc048da55553e37ab9d2d1e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzvmvh
@@ -0,0 +1,5 @@
+{"text":"Jean-Michel Bruel, Agusti Canals, S\u00e9bastien G\u00e9rard, Isabelle Perseil. Special Issue on papers from UML&FM. Dans : Innovations in Systems and Software Engineering, Springer, Vol. 4 N. 3, septembre 2008.\nJean-Michel Bruel, Manuel Mazzara, Bertrand Meyer. Software Engineering Aspects of Continuous Development and New Paradigms of Software Production and Deployment, Chateau de Villebrumier, France, 05\/03\/18 - 06\/03\/18, Springer, LNCS 11350, 2019.\nIleana Ober, Aniruddha Gokhale, James Hill, Jean-Michel Bruel, Michael Felderer, Akshay Dabholkar. MDHPCL 2013 Model-Driven Engineering for High Performance and CLoud computing, Miami, Florida, USA, 29\/09\/13 - 29\/09\/13, CEUR-WS : Workshop proceedings, 118, 2014.\nIleana Ober, Aniruddha Gokhale, Georg Geri, Michael Felderer, Jean-Michel Bruel, David Lugato. Proceedings of the 1st International Workshop on Model-Driven Engineering for High Performance and CLoud computing (MDHPCL), Innsbruck, Autriche, 02\/10\/12 - 02\/10\/12, ACM Digital Library, DL 10.1145\/2446224.2446225, 2013.\nKrzysztof Czarnecki, Ileana Ober, Jean-Michel Bruel, Axel Uhl, Markus V\u00f6lter. 11th International Conference Model Driven Engineering Languages and Systems, Toulouse, France, 28\/09\/08 - 03\/10\/08, Springer-Verlag, LNCS 5301, septembre 2008.\nJean-Michel Bruel. Satellite Events at the MoDELS 2005 Conference, Montego Bay, Jamaica, 02\/10\/05 - 07\/10\/05, Springer, Lecture Notes in Computer Science 3844, 2006.\nIleana Ober, Marc Palyart, Jean-Michel Bruel, David Lugato. On the Use of Models for High-Performance Scientific Computing Applications: an Experience Report. Dans : International Journal on Software and Systems Modeling, Springer-Verlag, Vol. 17 N. 1, p. 319-342, f\u00e9vrier 2018.\nManzoor Ahmad, Nicolas Belloir, Jean-Michel Bruel. Modeling and verification of Functional and Non-Functional Requirements of ambient Self-Adaptive Systems. Dans : Journal of Systems and Software, Elsevier, Vol. 107, p. 50-70, juillet 2015.\nManzoor Ahmad, Jean-Michel Bruel, R\u00e9gine Laleau, Christophe Gnaho. Using RELAX, SysML and KAOS for Ambient Systems Requirements Modeling. Dans : Procedia Computer Science, Science Direct, Num\u00e9ro sp\u00e9cial The 3rd International Conference on Ambient Systems, Networks and Technologies (ANT) proceedings, Vol. 10, p. 474-481, ao\u00fbt 2012.\nJon Whittle, Pete Sawyer, Nelly Bencomo, Betty H. C. Cheng, Jean-Michel Bruel. RELAX: A Language to Address Uncertainty in Self-Adaptive Systems Requirements. Dans : Requirements Engineering Journal, Springer-Verlag, Vol. 15 N. 2, p. 177-196, mars 2010.\nJean-Michel Bruel, Ileana Ober. Components modeling in UML 2. Dans : Studia Jurnal, Babes-Bolyai University, Cluj-Napoca (Roumanie), Vol. 1, p. 79-90, 2006.\nJean-Michel Bruel, Robert B. France, Maria M. Larrondo-Petrie. An Integrated Object-Oriented and Formal Modeling Environment. Dans : The Journal of Object-Oriented Programming (JOOP), Vol. 10 N. 7, p. 25-34, 1997.\nRobert B. France, Jean-Michel Bruel. FuZE: A Fusion\/Z Environment. Dans : Fusion Newsletter, Vol. 4 N. 3, p. 14-18, juillet 1996.\nJean-Michel Bruel, Nicolas Belloir, Rapha\u00ebl Faudou. Mod\u00e9lisation des exigences en UML\/SysML. Dans : G\u00e9nie Logiciel, GL & IS, Meudon - France, Vol. 111, p. 12-18, d\u00e9cembre 2014.\nMarc Palyart, David Lugato, Ileana Ober, Jean-Michel Bruel. Le calcul hautes performances: un nouveau champ d'application pour l'ing\u00e9nierie de mod\u00e8les. Dans : G\u00e9nie Logiciel, GL & IS, Meudon - France, Vol. 97, p. 41-46, juin 2011.\nJean-Michel Bruel. Composition logicielle \u00e0 l'aide de machine \u00e0 \u00e9tats. Dans : G\u00e9nie Logiciel, GL & IS, Meudon - France, Vol. 80, p. 2-6, mars 2007.\nNicolas Belloir, Jean-Michel Bruel. D\u00e9veloppement bas\u00e9 composant : une approche centr\u00e9e composition. Dans : Ing\u00e9nierie des Syst\u00e8mes d'Information, Herm\u00e8s Science, Num\u00e9ro sp\u00e9cial M\u00e9thodes Avanc\u00e9es de D\u00e9veloppement des SI, Vol. 10, N. 6, p. 59-80, octobre 2005.\nAbdelmalek Benzekri, Jean-Fran\u00e7ois Berdjugin, Jean-Michel Bruel. Logiciels distribu\u00e9s temps r\u00e9el : approches de sp\u00e9cification Z. Dans : Revue Electronique sur les R\u00e9seaux et l'Informatique R\u00e9partie, Laboratoire de T\u00e9l\u00e9informatique et Analyse des Syst\u00e8mes Communicants de l'universit\u00e9 de Pau, France, Laboratoire TASC Universit\u00e9 de Pau, Vol. Volume, N. 8, p. 39-57, juillet 1999.\nJacob Geisel, Brahim Hamid, Jean-Michel Bruel. Repository-Centric Process Modeling - Example of a Pattern Based Development Process. Dans : Software Engineering Research, Management and Applications. Roger Lee (Eds.), Springer, 16, p. 247-261, Vol. 496, Studies in Computational Intelligence, 2014.\nJean-Michel Bruel, Pascal Roques. Pr\u00e9sentation des concepts de SysML. Dans : Mod\u00e9lisation et analyse de syst\u00e8mes embarqu\u00e9s. Fabrice Kordon, J\u00e9r\u00f4me Hugues, Agusti Canals, Alain Dohet (Eds.), Herm\u00e8s, 4, p. 65-89, mai 2013.\nJean-Michel Bruel, Franck Barbier, Nicolas Belloir, Fabien Romeo. Test de composants logiciels. Dans : Ing\u00e9nierie des Composants : Concepts, techniques et outils. Mourad Oussalah (Eds.), Vuibert, 8, p. 229-245, juin 2005.\nJean-Michel Bruel, Georg Geri, Ileana Ober. Models for Non-functional Aspects of Component-Based Software (NfC'04). Dans : UML 2004 Satellite Activities. N. J. Nuntes (Eds.), Springer-Verlag, p. 62-66, Vol. 3297, LNCS, 2005.\nJean-Michel Bruel, Georg Geri, Hussmann Heinrich, Pohl Christoph , Whittle Jon . Models for Non-functional Aspects of Component-Based Software (NfC'04). Dans : UML 2004 Modeling Languages and Applications, UML Satellite Activities. Jardim Nunes Nuno, Bran Selic, Alberto Rodrigues da Silva (Eds.), Springer 2005, LNCS 3297, p. 62-66, 2005.\nRonan Baduel, Jean-Michel Bruel, Iulian Ober, Eddy Doba. Definition of states and modes as general concepts for system design and validation (regular paper). Dans : Conf\u00e9rence Internationale de Mod\u00e9lisation, Optimisation et SIMulation (MOSIM 2018), Toulouse, France, 27\/06\/18-29\/06\/18, Archives ouvertes HAL, (en ligne), juin 2018.\nJean-Michel Bruel, Benoit Combemale, Esther Guerra, Jean-Marc J\u00e9z\u00e9quel, J\u00f6rg Kienzle, Juan De Lara, Gunter Mussbacher, Eugene Syriani, Hans Vangheluwe. Model Transformation Reuse across Metamodels (regular paper). Dans : International Conference on Model Transformation (ICMT 2018), Toulouse, France, 25\/06\/18-26\/06\/18, Springer-Verlag, LNCS 10888, p. 92-109, septembre 2018 (Best Paper Award).\nMaroun Koussaifi, Walid Younes, Fran\u00e7oise Adreit, Jean-Paul Arcangeli, Jean-Michel Bruel, Sylvie Trouilhet. Emergence of Composite Services in Smart Environments (poster). Dans : EuroScience Open Forum (ESOF 2018), Toulouse, 09\/07\/18-14\/07\/18, ESOF 2018, (en ligne), juillet 2018 (Poster award).\nMaroun Koussaifi, Sylvie Trouilhet, Jean-Paul Arcangeli, Jean-Michel Bruel. Ambient Intelligence Users in the Loop: Towards a Model-Driven Approach (regular paper). Dans : International Workshop Microservices: Science and Engineering (MSE@STAF 2018 2018), Toulouse, 25\/06\/18-25\/06\/18, Manuel Mazzara, Julian Ober, Gwen Sala\u00fcn (Eds.), Springer-Verlag, LNCS 11576, p. 558-572, 2018.\nFlorian Galinier, Jean-Michel Bruel, Sophie Ebersold, Bertrand Meyer. Seamless Integration of Multirequirements in Complex Systems (regular paper). Dans : Requirements Engineering Conference Workshops (REW 2017), Lisbon, Portugal, 04\/09\/17-08\/09\/17, IEEE : Institute of Electrical and Electronics Engineers, (en ligne), octobre 2017.\nMohamed Naija, Jean-Michel Bruel, Samir Ben Ahmed. Towards a MARTE Extension to Address Adaptation Mechanisms (regular paper). Dans : IEEE International Symposium on High-Assurance Systems Engineering (HASE 2016), Orlando, FL, USA, 07\/01\/16-09\/01\/16, IEEE Computer Society, p. 240-243, janvier 2016.\nAnas Motii, Brahim Hamid, Agnes Lanusse, Jean-Michel Bruel. Guiding the selection of security patterns for real-time systems (regular paper). Dans : IEEE International Conference on Engineering Complex Computer Systems (ICECCS 2016), Dubai, UAE, 06\/11\/16-08\/11\/16, IEEE : Institute of Electrical and Electronics Engineers, p. 155-164, novembre 2016.\nAnas Motii, Agnes Lanusse, Brahim Hamid, Jean-Michel Bruel. Model-Based Real-Time Evaluation of Security Patterns: A SCADA System Case Study (regular paper). Dans : TIPS Workshop (Safecomp 2016), Trondheim, Norway, 20\/09\/16-20\/09\/16, Vol. 9923, Springer, p. 375-389, septembre 2016.\nJacob Geisel, Brahim Hamid, David Gonzalez, Jean-Michel Bruel. Towards a Methodological Tool Support for Modeling Security-Oriented Processes (regular paper). Dans : International Conference On Model and Data Engineering (MEDI 2016), Almeria, Spain, 21\/09\/16-23\/09\/16, Vol. 9893, Springer, p. 31-41, septembre 2016.\nBenoit Combemale, Betty Cheng, Ana Moreira, Jean-Michel Bruel, Jeff Gray. Modeling for sustainability (regular paper). Dans : International Workshop on Modeling in Software Engineering, Austin, Texas, 14\/05\/16-22\/05\/16, ACM : Association for Computing Machinery, p. 62-66, mai 2016.\nJacob Geisel, Brahim Hamid, Jean-Michel Bruel. Security Concepts as Add-On for Process Models (regular paper). Dans : IEEE International Conference on Engineering Complex Computer Systems (ICECCS 2015), Gold Coast, Australia, 09\/12\/15-12\/12\/15, IEEE Computer Society, p. 190-193, d\u00e9cembre 2015.\nAnas Motii, Brahim Hamid, Agnes Lanusse, Jean-Michel Bruel. Guiding the selection of security patterns based on security requirements and pattern classification (regular paper). Dans : European Conference on Pattern Language of Programs (EuroPlop 2015), Kloster Irsee in Bavaria, Germany., 06\/07\/15-10\/07\/15, ACM Digital Library, p. 1-17, 2015.\nMohamed Naija, Samir Ben Ahmed, Jean-Michel Bruel. New Schedulability Analysis for Real-Time Systems based on MDE and Petri Nets Model at Early Design Stages (regular paper). Dans : International Conference on Software and Data Technologies (ICSOFT 2015), Colmar, 20\/07\/15-22\/07\/15, SciTePress, p. 330-338, ao\u00fbt 2015.\nYouness Laghouaouta, Adil Anwar, Mahmoud Nassar, Jean-Michel Bruel. A Generic Traceability Framework for Model Composition Operation (regular paper). Dans : Enterprise, Business-Process and Information Systems Modeling, Stockholm, Sweden, 08\/06\/15-09\/06\/15, Vol. 214, Springer, LNBIP, p. 461-475, mai 2015.\nJean-Michel Bruel, Benoit Combemale, Ileana Ober, H\u00e9l\u00e8ne Raynal. MDE in Practice for Computational Science (regular paper). Dans : International Conference on Computational Science (ICCS 2015), Reykjav\u00edk, Iceland, 01\/06\/15-03\/06\/15, Vol. 51, Slawomi Koziel, Leifur Leifsson, Michael Lees, Valeria V. Krzhizhan, Jack Dongarra (Eds.), Elsevier, p. 660-669, 2015.\nNicolas Belloir, Vanea Chiprianov, Manzoor Ahmad, Manuel Munier, Laurent Gallon, Jean-Michel Bruel. Using Relax Operators into an MDE Security Requirement Elicitation Process for Systems of Systems (regular paper). Dans : International Worksop on Software Enginering for Systems-of-Systems (SeSoS 2014), Vienna, Austria, 26\/08\/14-26\/08\/14, ACM Digital Library, (support \u00e9lectronique), 2014.\nYouness Laghouaouta, Adil Anwar, Mahmoud Nassar, Jean-Michel Bruel. On the use of graph transformations for models composition traceability (regular paper). Dans : International Conference on Research Challenge in Information Science (RCIS 2014), Marrakesh, Maroc, 28\/05\/14-30\/05\/14, IEEExplore digital library, (support \u00e9lectronique), mai 2014.\nGunter Mussbacher, Daniel Amyot, Ruth Breu, Jean-Michel Bruel, Betty Cheng, Philippe Collet, Benoit Combemale, Robert B. France, Rodgart Heldal, James Hill, J\u00f6rg Kienzle, Matthias Sch\u00f6ttle, Friedrich Steinman, Dave Stikkolorum, John Whittle. The Relevance of Model-Driven Engineering Thirty Years from Now (regular paper). Dans : ACM\/IEEE International Conference on Model Driven Engineering Languages and Systems (MODELS 2014), Valencia, 28\/09\/14-03\/10\/14, Vol. 8767, Springer, (en ligne), septembre 2014.\nManzoor Ahmad, Joao Araujo, Nicolas Belloir, Jean-Michel Bruel, Christophe Gnaho, R\u00e9gine Laleau, Farida Semmak. Self-Adaptive Systems Requirements Modelling: four related approaches comparison (regular paper). Dans : International Wokshop Comparing *Requirements* Modeling Approaches (CMA@RE 2013), Rio de Janeiro, Brazil, 16\/07\/13-16\/07\/13, IEEExplore digital library, (support \u00e9lectronique), 2013.\nFr\u00e9d\u00e9ric Mayart, Jean-Michel Bruel, Brahim Hamid. How to Analyze Modeling Approach Comparison Criteria (short paper). Dans : Comparing Modeling Approaches Workshop, Miami, 01\/10\/13, Vol. 1076, CEUR-WS : Workshop proceedings, (en ligne), octobre 2013.\nManzoor Ahmad, Iulia Dragomir, Jean-Michel Bruel, Iulian Ober, Nicolas Belloir. Early Analysis of Ambient Systems SYSML Properties using OMEGA2-IFx (regular paper). Dans : International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2013), Reykjavik, Iceland, 29\/07\/13-31\/07\/13, SciTePress, p. 147-154, septembre 2013.\nAdel Ziani, Brahim Hamid, Jacob Geisel, Jean-Michel Bruel. A Model-based Repository of Security and Dependability Patterns for Trusted RCES (regular paper). Dans : IEEE International Conference on Information Reuse and Integration (IRI 2013), San Francisco, CA. USA, 14\/08\/13-16\/08\/13, IEEE : Institute of Electrical and Electronics Engineers, p. 448-457, 2013.\nBrahim Hamid, Jacob Geisel, Adel Ziani, Jean-Michel Bruel, Jon Perez. Model-Driven Engineering for Trusted Embedded Systems based on Security and Dependability Patterns (regular paper). Dans : System Design Languages Forum, Montreal, Canada, 26\/06\/13-28\/06\/13, Springer, LNCS, p. 73-91, juin 2013.\nMarc Palyart, Ileana Ober, David Lugato, Jean-Michel Bruel. HPCML: a modeling language dedicated to high-performance scientific computing (regular paper). Dans : International Workshop on Model-Driven Engineering for High Performance and CLoud computing (MDHPCL 2012), Innsbruck, Austria}, 02\/10\/12-02\/10\/12, ACM Digital Library, (en ligne), 2012.\nAdel Ziani, Brahim Hamid, Jean-Michel Bruel. A Model-Driven Engineering Framework for Fault Tolerance in Dependable Embedded Systems Design (regular paper). Dans : Euromicro conference on Software Engineering and Advanced Applications, Cesme, Izmir, Turkey, 05\/09\/12-08\/09\/12, IEEE : Institute of Electrical and Electronics Engineers, p. 166-169, septembre 2012.\nMarc Palyart, David Lugato, Ileana Ober, Jean-Michel Bruel. Improving Scalability and Maintenance of Software for High-Performance Scientific Computing by Combining MDE and Frameworks (regular paper). Dans : ACM\/IEEE International Conference on Model Driven Engineering Languages and Systems (MODELS 2011), Wellington, New Zealand, 16\/10\/11-21\/10\/11, Vol. 6981, Jon Whittle, Tony Clark, Thomas K\u00fchne (Eds.), Springer, LNCS, p. 213-227, septembre 2011.\nMarc Palyart, David Lugato, Ileana Ober, Jean-Michel Bruel. MDE4HPC : An Approach for Using Model-Driven Engineering in High-Performance Computing (regular paper). Dans : International Conference on System Design Languages, Toulouse, 05\/07\/11-07\/07\/11, Ileana Ober, Iulian Ober (Eds.), Springer, LNCS 7083, p. 245-259, octobre 2011.\nMarc Palyart, David Lugato, Ileana Ober, Jean-Michel Bruel. MDE4HPC: An approach for Using Model-Drieven-Engineering in High Performance Computing (regular paper). Dans : International Conference on System Design Languages, Toulouse, 05\/07\/11-07\/07\/11, Ileana Ober, Iulian Ober (Eds.), IRIT, p. 213-228, juin 2011.\nAna Moreira, Jean-Michel Bruel, Joao Araujo. Adding Behavior Description Support to COTS Components through the use of Aspects. Dans : International Workshop on Models for Non-functional Aspects of Component-Based Software, Toulouse, 05\/09\/05-05\/09\/05, Vol. IRIT-RR-2005-01-FR, IRIT, p. 9-19, septembre 2009.\nJon Whittle, Pete Sawyer, Nelly Bencomo, Betty H. C. Cheng, Jean-Michel Bruel. RELAX: Incorporating Uncertainty into the Specification of Self-Adaptive Systems (regular paper). Dans : IEEE International Requirements Engineering conference (RE 2009), Atlanta, Goergia, USA, 31\/08\/09-04\/09\/09, IEEE Computer Society, p. 79-88, 2009.\nNicolas Belloir, Jean-Michel Bruel. Component based development: a composition oriented approach. Dans : IEEE International Conference on Research, Innovation and Vision for the Futur (RIVF 2007), Hanoi, Vietnam, 07\/03\/07-09\/03\/07, IEEE Computer Society, p. 101-106, 2007.\nFabien Romeo, Franck Barbier, Jean-Michel Bruel. Observability and controllability of wireless software components. Dans : IFIP International Conference on Distributed Applications and Interoperable Systems (DAIS 2007), Paphos, Cyprus, 06\/06\/07-08\/06\/07, Springer, LNCS 4531, p. 48-61, 2007.\nFabien Romeo, Franck Barbier, Jean-Michel Bruel. Autonomic management of component-based embedded software. Dans : IFIP\/IEEE International Symposium on Integrated Network Management (IM 2007), Munich, Germany, 21\/05\/07-25\/05\/07, IEEE Computer Society, p. 860-863, 2007.\nAbdelmalek Benzekri, Jean-Fran\u00e7ois Berdjugin, Jean-Michel Bruel. Z, langage de sp\u00e9cification de syst\u00e8mes r\u00e9partis temps r\u00e9el. Dans : Conf\u00e9rence Francophone sur l'Ing\u00e9nierie des Protocoles (CFIP'97), Li\u00e8ge, Belgique, 29\/09\/97-02\/10\/97, Herm\u00e8s. ISBN 2-86601-639-4, septembre 1997.\nAbdelmalek Benzekri, Jean-Michel Bruel. Controller Area Network: a formal case study. Dans : 2nd IEEE International Workshop on Factory Communication Systems (WFCS'97), Barcelona, Catalonia, Spain, 01\/10\/97-03\/10\/97, J.M. Fuertes and G. Juanole, octobre 1997.\nJean-Michel Bruel, Abdelmalek Benzekri. A Formal Study of the Controller Aera Network properties. Dans : Proceedings of the ISCA 10th International Conference on Computer Applications in Industry and Engineering (CAiNE'97), San Antonio, Texas USA, 10\/12\/97-12\/12\/97, Gordon K. Lee, d\u00e9cembre 1997.\nRobert B. France, Jean-Michel Bruel. Integrated Graphical Object-Oriented and formal Modeling Techniques. Dans : Models, Formalisms and Methods for Distributed Object-Oriented Computing, A workshop within ECOOP'97 11th European Conference on Object-Oriented Programming, Jyva, Finland, 09\/06\/97-13\/06\/97, -, juin 1997.\nRobert B. France, Jean-Michel Bruel, Maria M. Larrondo-Petrie, Emmanuel Grant. Rigorous Object-Oriented Modeling: Integrating Formal and Informal Notations. Dans : 6th International Conference on Algebraic Methodology and Software Technology, Sydney, Australia, 03\/12\/97-17\/12\/97, -, d\u00e9cembre 1997.\nRobert B. France, Jean-Michel Bruel, Maria M. Larrondo-Petrie, Malcom Shroff. Exploring The Semantics of UML Type Structures with Z. Dans : Proceedings of the Second IFIP International Conference on Formal Methods for Open Object-based Distributed Systems, Canterbury, United Kingdom, 21\/07\/97-23\/07\/97, Chapman & Hall, London, UK, juillet 1997.\nRobert B. France, Jean-Michel Bruel, Gopal K. Raghavan. Towards Rigorous Analysis of Fusion Models. Dans : Proceedings of the Second Northern Formal Methods Workshop, Ilkley, UK, 14\/07\/97-15\/07\/97, -, juillet 1997.\nRobert B. France, Jean-Michel Bruel, Gopal K. Raghavan. Taming the Octopus: Using Formal Models to Integrate the Octopus Object Oriented Analysis Models. Dans : Proceedings of the 2nd IEEE High-Assurance Systems Engineering Workshop (HASE'97), Bethesda, Maryland USA, 10\/08\/97-12\/08\/97, EEE Computer Society Press, Los Alamitos, CA, ao\u00fbt 1997.\nBrian W. Bates, Jean-Michel Bruel, Robert B. France, Maria M. Larrondo-Petrie. Formalizing Fusion Object-Oriented Analysis Models. Dans : Formal Methods for Open Object-based Distributed Systems, -, Chapman & Hall, London, UK, p. 47-62, janvier 1997.\nAbdo Abdulmalak, Abdelmalek Benzekri, Jean-Michel Bruel. Quality of Service analysis using Formal Methods. Dans : Quality of Service - Description, Modelling and Management (Proceedings of the 4th International IFIP Workshop on QoS), Paris, France, 06\/03\/96-08\/03\/96, Andreas Vogel and Jan de Meer, GMD-Studien Nr 282, p. 151-160, mars 1996.\nJean-Michel Bruel, Robert B. France, Abdelmalek Benzekri. Real-Time and Safety Critical Systems Specification: a Z-based Specification Environment. Dans : Proceedings of the ISCA 9th International Conference on Computer Applications in Industry and Engineering (CAiNE'96), Orlando, Florida, 11\/12\/96-13\/12\/96, ISCA, Raleigh, N, p. 44-47, d\u00e9cembre 1996.\nJean-Michel Bruel, Robert B. France, Maria M. Larrondo-Petrie. CASE-based Rigorous Object-Oriented Modeling. Dans : Proceedings of the Northern Formal Methods Workshop, Bradford, UK, 23\/09\/96-24\/09\/96, -, septembre 1996.\nJean-Michel Bruel, Robert B. France, Abdelmalek Benzekri. A Z-based Approach to Specifying and Analyzing ComplexSystems. Dans : Proceedings of the Second IEEE International Conferenceon Engineering of Complex Computer Systems (ICECCS'96), Montreal, Canada, 21\/10\/96-25\/10\/96, IEEE Computer Society Press, Los Alamitos, CA, p. 336-343, octobre 1996.\nJean-Michel Bruel, Robert B. France, Abdelmalek Benzekri, Yves Raynaud. A Real-Time Specification Environment based on Z and Graphical Object-Oriented Modeling Techniques. Dans : Proceedings of the IEEE High-Assurance Systems Engineering Workshop (HASE'96), Niagara-on-the-Lake, Canada, 21\/10\/96-22\/10\/96, IEEE Computer Society Press, Los Alamitos, CA, octobre 1996.\nMaria M. Larrondo-Petrie, Robert B. France, Jean-Michel Bruel. Introducing Formal Specification Techniques into the Undergraduate Software Engineering Curriculum. Dans : Proceedings of ``Exploring Formal Methods in the Computer Science Curriculum'' Workshop, New York, USA, 24\/07\/96-26\/07\/96, -, juillet 1996.\nJean-Michel Bruel, Robert B. France. A Formal Object-Oriented CASE Tool for the Development of Complex Systems. Dans : Proceedings of the 7th European Workshop on the Next Generation of CASE Tools (NGCT'96), Crete, Greece, 20\/05\/96-21\/05\/96, -, mai 1996.\nRobert B. France, Jean-Michel Bruel. The Role of Integrated Specification Techniques in Complex System Modeling and Analysis. Dans : Proceedings of the Workshop on Real-Time Systems Education (RTSE'96), Daytona Beach, Florida, USA, 20\/04\/96, -, avril 1996.\nRobert B. France, Maria M. Larrondo-Petrie, Jean-Michel Bruel, Jie. Wu. A Tale of Two Case Studies: Using Integrated Methods to Support Rigorous Requirements Specification. Dans : Proceedings of the BCS FACS Methods Integration Workshop, Leeds, UK, 25\/03\/96-26\/03\/96, -, mars 1996.\nJean-Michel Bruel, Robert B. France, Maria M. Larrondo-Petrie, Brian W. Bates. Guidelines for Formalizing Fusion Object-Oriented Analysis Models. Dans : Advanced Information Systems Engineering - 8th International Conference, CAiSE'96, Heraklion, Crete, Greece, 20\/05\/96-24\/05\/96, Springer-Verlag LNCS 1080, mai 1996.\nRobert B. France, Jean-Michel Bruel. Formal Analysis of Information Systems using an Integrated Approach based on Extended Petri Nets. Dans : Proceedings of The International Conference on Information Systems Analysis and Synthesis (ISAS'96), Orlando, USA, 22\/07\/96-26\/07\/96, -, juillet 1996.\nJean-Michel Bruel, Robert B. France, Bharat Chintapally, Gopal K. Raghavan. A Tool for Rigorous Analysis of Object Model. Dans : Proceedings of the 20th Intl. Conf. on Technology of Object-Oriented Languages and Systems (TOOLS'96), Santa Barbara, California, 29\/07\/96-02\/08\/96, -, juillet 1996.\nJean-Michel Bruel, Abdelmalek Benzekri, Yves Raynaud. Z and the Specification of Real-Time Systems. Dans : Proceedings of the 7th International Conference on Putting into Practice Methods and Tools for Information System Design - Z Twenty Years On, What Is Its Future?, Universit\u00e9 de Nantes, France, 10\/10\/95-12\/10\/95, -, octobre 1995.\nJean-Michel Bruel, Yves Raynaud, Abdo Abdulmalak. Validation of an Adaptive Leaky Bucket Algorithm. Dans : International Conference on Computer Applications in Industry and Engineering, Honolulu, Hawaii, USA, 29\/11\/95-01\/12\/95, A. Goel, novembre 1995.\nFlorian Galinier, Sophie Ebersold, Jean-Michel Bruel. Requirements Specific Modeling Language : un langage formel d'expression d'exigences (regular paper). Dans : Conf\u00e9rence en Ing\u00e9nierie du Logiciel (CIEL 2018), Grenoble, 14\/06\/18-15\/06\/18, Unit\u00e9 CNRS GDR-GPL : Groupement De Recherche G\u00e9nie de la Programmation et du Logiciel, (en ligne), juin 2018.\nFlorian Galinier, Jean-Michel Bruel, Sophie Ebersold, Bertrand Meyer. Int\u00e9gration des (multi-)exigences tout au long du d\u00e9veloppement des syst\u00e8mes complexes (short paper). Dans : Approches Formelles dans l'Assistance au D\u00e9veloppement de Logiciels (AFADL 2017), Montpellier, 13\/06\/17-16\/06\/17, Universit\u00e9 de Montpellier, (en ligne), 2017.\nFernando Wanderley, Nicolas Belloir, Jean-Michel Bruel, Nabil Hameurlain, Joao Araujo. Des buts \u00e0 la mod\u00e9lisation syst\u00e8me : une approche de mod\u00e9lisation des exigences centr\u00e9e utilisateur (regular paper). Dans : INFormatique des Organisations et Systemes d'Information et de Decision (INFORSID 2014), Lyon, 19\/05\/14-23\/05\/14, Herm\u00e8s - Revuesonline, p. 113-128, mai 2014.\nNicolas Belloir, Jean-Michel Bruel, Natacha Hoang, Congduc Pham. Utilisation de SysML pour la mod\u00e9lisation des r\u00e9seaux de capteurs sans fil (regular paper). Dans : Conf\u00e9rence sur les Langages et Mod\u00e8les \u00e0 Objets (LMO 2008), Montr\u00e9al, Canada, 03\/03\/08-07\/03\/08, C\u00e9padu\u00e8s Editions, p. 171-186, mars 2008.\nNicolas Belloir, Jean-Michel Bruel, Eric Cariou. Impl\u00e9mentation d'un mod\u00e8le UML de composition hi\u00e9rarchique. Dans : Conf\u00e9rence sur les Langages et Mod\u00e8les \u00e0 Objets (LMO 2007), Toulouse, 27\/03\/07-29\/03\/07, Herm\u00e8s, p. 35-47, 2007.\nJean-Michel Bruel, Nicolas Belloir, Manzoor Ahmad. SPAS : un profil SysML pour les syst\u00e8mes auto-adaptatifs. Dans : 15\u00e8me Colloque National de la Recherche en IUT (CNRIUT), Lille, 08\/06\/09-10\/06\/09.\nManzoor Ahmad, Jean-Michel Bruel, Antoine Beugnard. From Composition to Connectors. Dans : MDD4DRES 2009, Aussois, France, 20\/04\/09-24\/04\/09.\nManzoor Ahmad, Jean-Michel Bruel. A Comparative Study of RELAX and SysML\/KAOS. Rapport de recherche, IRIT\/RR--2014-07--FR, IRIT, 2014.\nFr\u00e9d\u00e9ric Mayart, Jean-Michel Bruel, Brahim Hamid. Correlation analysis of modeling approach comparison criteria:methodology proposal. Rapport de recherche, IRIT\/ RR--2013-34--FR, IRIT, septembre 2013.\nJean-Michel Bruel, Georg Geri, Hussmann Heinrich, Ileana Ober, Pohl Christoph , Whittle Jon , Zschaler Steffen . Proceedings of the 1st International Workshop on Models for Non-functional Aspects of Component-based Software. Rapport de recherche, TUD-FI04-12-Sept.2004, Universit\u00e9 Technique de Dresde, Allemagne, septembre 2004.\nJean-Michel Bruel. FuZE : un environnement int\u00e9gr\u00e9 pour l'analyse formelle de logiciels distribu\u00e9s temps r\u00e9el. Th\u00e8se de doctorat, Universit\u00e9 Paul Sabatier, d\u00e9cembre 1996.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As popular as asphalt shingles are in the country, they are not the most durable roofing solution out there. If you have this type of roof, you'll need to do some maintenance every once in a while. Shingles can be damaged by various things including strong winds and branches. At some point, you should expect to replace a few shingles or whole sections of shingles on your roof. For the best results when you do this, there are a few things you should pay attention to.\nA commercial roofing company can help your business take care of your roof. Commercial roofing companies offer a variety of different services that can help you take care of the physical structure of your business. #1 Gutter Repair With a commercial roof, the flow of water off your roof is important. You don't want water to come down on customer's heads as they enter and exit your building. You don't want water to pool up and create puddles near your entrances.\nAre You Ready For Solar Tiles?\nAre you going to make the big jump to solar? Is it time to add solar panels to your roof so you can start to reduce your utility bills and run a greener household? If you are ready to make this investment, you might even be able to go one step further. This article discusses how you can upgrade to solar tiles if you want to really modernize your home and make it super energy efficient.\nWhy Replace Your Home's Roof During The Fall?\nIf the time has come to replace your home's roof, you may be wondering what time of year is best to do this major home repair. If you do not need an immediate roof replacement, consider having the work done during the fall for the following reasons. Fewer Rainy Days The hotter months of spring and summer mean that there will be plenty of storms compared to the rest of the year.\nRainstorms are usually no big deal -- unless you have a leak in your home's roof. Unfortunately, most people are unaware that there is a leak until a storm is in full downpour and there is no way to immediately get a repair. Fortunately, there are things you can do to mitigate the damage until you can get a roofer out for a repair or a replacement of your roof. The following guide will help.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The eggplant (Solanum melongena) is a vegetable of tropical origin. It is rather capricious and requires a lot of heat and sustained fertilization. However, those conditions can easily be obtained when growing eggplants in containers. In the spring, most Canadian garden centres offer eggplants in containers. Choose vigorous, compact plants that bear a few flowers only. As eggplants do not appreciate being transplanted, instead of buying plants in styrofoam flats, choose those that come in individual containers. However, if you wish to grow original eggplant cultivars, you will most likely have to buy seed.\nEggplants have a good production when grown in dark metal or plastic pots, so the soil in which they are planted reach high temperatures. Eggplant seed must be seeded inside towards the very end of March or in April, i.e. about eight weeks before planting them in the garden. As mentioned earlier, eggplants do not like being transplanted, so it is preferable to place the seed in a container with multiple small cells filled with soil made up of equal parts of compost, peat moss and perlite, such as PRO-MIX Vegetable and Herb Mix. Seed two to three seeds per cell to make sure at least one plant will form. The seed must be placed at a depth of 6 mm (1\/4 in). Then, water the soil generously with warm water. The optimal temperature for the germination of eggplant seed varies between 26 \u00b0C and 30 \u00b0C. To keep the growing medium warm, you can place the containers on a mattress or heating wires. Keep only one plant per cell by cutting the others with scissors. About three weeks after germination, when the first two true leaves are well formed, transplant the plants in 10-cm pots. Make sure a part of the stem (a few centimetres) is covered with soil. After the first transplantation, keep the temperature at 21 \u00b0C, day and night, until the plants are transplanted in the garden. Eggplants also need a lot of light to grow properly and develop. As light is generally rather poor in our homes, the seedlings must often be installed under an artificial lighting system which uses LED bulbs specifically designed for plants.\nThe 'Morden Midget' eggplant (Solanum melongena 'Morden Midget') works well for container growing. Eggplants are especially demanding plants. They must be planted in a place that is warm, very sunny and well protected from prevailing winds. However, by growing your eggplants in containers \u2013 in dark-coloured pots \u2013 against a south-facing wall, you will get very satisfying results. Eggplants prefer rich, loose soils that are well drained, and whose pH varies between 5.5 and 7. Exterior planting must be done one or two weeks after all risks of frost have been almost eliminated in your region. In zones 5 and 6, in Montr\u00e9al and Toronto for example, eggplants are planted towards the end of May or in early June. Whether they were produced from seed or bought in a garden centre, do not forget to acclimatize your plants before planting them. Furthermore, add a slow release fertilizer to the soil\u2013 two to three handfuls per plant \u2013 whose formulation is about 4-6-8. Also, I recommend placing a black plastic film ou dark rubber mulch over the soil's surface. This will increase the soil's temperature and keep it between 27 \u00b0C and 32 \u00b0C. For small eggplant cultivars, such as \"Morden Midget\" or \"Rosa Bianca\", use a container at least 40-45 cm in diameter for each plant. For bigger cultivars, choose bigger pots. Finally, as they appreciate stable moisture rates, you must water your eggplants on a regular basis. Water them with warm water twice a week, or three times a week in periods of extreme heat.\nThe 'Pingtung Long' eggplant (Solanum melongena 'Pingtung Long') is rather original!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Antique, Vintage and Modern Goodies Online Auction in Historic Bethlehem!!\nWaterford, Lennox, Howard Miller, Small Appliances, Baby Lock, Vintage Christmas, Bombay, Antique & Vintage Items!\nPick Up \u2013 \"AS IS WHERE IS\", Do-It-Yourself Style times and location are auction specific. Please be aware and prepared for pick up at that time and date. Due to the nature of these sales other times and dates cannot be possible. We do not ship.\nWe do not have an extended bidding system. The auction closes at 1 Lot\/Min. Also known as hard close.\nPROXY BID OR MAX BID: Proxy bidding or Max bid, sometimes called absentee or automatic bidding, is a method for online auction bidding. How it works is the person who wants to bid on an item decides a maximum amount that he or she is willing to pay for it in advance and bids this amount. That bid is hidden in our system. As other bids are made, our system automatically increases the person's bid by pre-specified increments until the maximum amount is reached. If the bidder is monitoring the auction, at this point he\/she can give up or choose to increase his bid. You don't have to continually monitor an auction to win an item, and they can bid on several items at once.\nPlease note that by placing a bid, you are entering a legally binding offer to purchase. If you have the highest bid, you are expected to pay for your items and pick it up during the set forth times. We will not have tools or people for moving. Estate Wizard Employees can bid in auctions they do not represent Estate Wizard when they do.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sticker packs, which provide a collection of images users can include in their messages.\niMessage apps, which provide access to external apps from within Messages.\nSticker apps do no require any coding and can be created by simply copying image files into a project template provided by Xcode. By contrast, iMessage apps can use the full Messages framework, as explained below. According to Apple, iMessage apps will make it possible to share content, send payments, play games, collaborate, and more, directly from Messages.\nMSMessageAppViewController: this class provides the main view controller for Messages extensions. It allows to present a custom user interface, manage the extension's state, get the current conversation, track message sending, etc.\nMSStickerBrowserViewController, MSStickerBrowserView, and MSStickerBrowserViewDataSource: these classes are meant to customize and present a custom or dynamic sticker browser. If you want to provide dynamic content for your sticker browser, you can implement a custom MSStickerBrowserViewDataSource. If you want to customize the appearance of the default sticker browser, you can provide your own MSStickerBrowserView and customize the browser's size, the size of the stickers, etc. Further customization is possible by subclassing MSStickerView.\nMSConversation: this class represents a conversation and allows to send text, stickers, attachments, or message objects by inserting them into a message's input field.\nMSMessage: this class allows to create interactive messages and access message properties, such as its sender, the session that the message belongs to, an optional URL associated with it, etc.\nMSSession: this class allows to identify messages so they can be later updated, for example, in a game or collaborative app.\nThe Messages framework is available with Xcode 8 beta, which can be freely downloaded by registered developers.\nAre iMessages always meant to be composed\/read by users?\nI wonder if I can use iMessage as point-to-point transport for application's protocol messages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwkcp b/data_all_eng_slimpj/shuffled/split2/finalzzwkcp
new file mode 100644
index 0000000000000000000000000000000000000000..6d5228ecb0c5e5d1c6c9ed9e7a65f16f8cc5dafc
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzwkcp
@@ -0,0 +1,5 @@
+{"text":"PNC at Work \u00ab Pritish Nandy Communications Ltd.\nAn exhibition of graphic art, based on PNC movies, designed and created by the PNC design studio. The show gave people an insight into the effort and madness that goes into making a movie. 41 works of art were exhibited for 10 days at The Museum Gallery in Kala Ghoda, South Mumbai, chronicling the life and times of PNC. The size of the works ranged between 4\u00d74 feet, 6\u00d74 feet, 8\u00d74 feet. The exhibition was inaugurated by Uddhav Thackeray, politician and Gulzar, writer and lyricist.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Globo Consulting is an independent and highly sophisticated consultancy boutique providing commercial, financial and family consulting services, with the possibility to outsource the relationship management of the different financial professionals involved.\nWe advise individuals and families in business with local and\/or international operations aiming for efficient, tailor made and flexible solutions.\nWe grow with our client by keeping the focus on the client.\nto you as soon as possible!\nGlobo Consulting S. A. is part of the Globo Consulting Group.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Regardless if you are buying or refinancing there are a few variables you need to know about to ensure that you are comparing apples with apples and although I am biased I highly recommend that you speak with a professional Mortgage Broker who can assist you with the multitude of home loan features that come with the different types of home loan products to make sure what you a needing if found without over paying, without paying an interest rate that will cost you thousands extra over the life of your home loan.\nIf you are comparing home loan interest rates and you want to get in touch with me directly, fee free to visit my Contact Us webpage for my details.\nThe first thing every Bank, every Home Loan Lender and every Mortgage Broker will want to find out is your LVR, your Loan to Value Ratio. The reason for this, put into simple terms, is because the higher the LVR, the higher the risk, which may attract a higher interest rate. So when comparing home loan interest rates you need to make sure you are looking at products that have the same LVR.\nSo how is this LVR calculated?\nDepending on the Purchase Price and how much deposit you have will in most cases determine the LVR.\nThe Bank still has the right to conduct a Valuation to ensure that your Purchase Price was not higher than the actual property value.\nTraditionally loans that have an LVR equal to or less than 80% come with a lower interest rate, especially comparing home loan interest rates to those at 95% LVR because as you can imagine, if house prices fall and you miss a Home Loan repayment and the Bank has to sell your home, they may make a loss, which is why the in a lot of cases, the riskier the loan the higher the Interest Rate.\nIt all depends on the Valuation compared to your Loan Amount that will determine your LVR.\nSome people call the difference as the 'Equity' but I don't really like this term because it implies that you own this equity when in reality you can't access all of it until you sell but even then, you have to deduct the Real Estate Agent Fees, Settlement Agent\/Conveyancer Fees, Loan Closure Costs and State Revenue Department Fees. And when you are refinancing, the general rule is that you can refinance up to 90% LVR \u2013 never 100% LVR.\nWhen refinancing you can generally add the costs to refinance on top of the loan amount, so the fees don't come directly out of your savings but please be mindful, you now have to pay interest on top of the fees that are charged when adding back onto the loan.\nAs you can see comparing home loan interest rates without taking into account your estimated LVR may give you different results if you are trying to do it yourself. I highly recommend that you seek out the professional assistance of your local Mortgage Broker.\nI prefer to use products that have $0 monthly and $0 annual fee because that money is better off in your back pocket \u2013 not the Banks!\nRegardless if you are comparing a Home Loan interest rate or an Investment Property Loan interest rate an Offset Account is a great Money Management tool that you can use to your advantage. The Banks won't teach you how to use it properly because they still want to make maximum profits for their shareholders but a good Mortgage Broker will give you all the tips and tricks required to get the Offset Account working in your favour.\nYou never know when you may receive a large lump sum amount of money, from an ATO Tax Return, to a sale of an asset or the receipt of an inheritance. Before you can figure out how to use the funds it is a great idea to dump into your Offset Account or Home Loan Account to save home loan interest on a daily basis.\nIf you can dump additional funds into your Home Loan you need the ability to redraw this back out if required, especially in the case of an emergency. I don't believe in the 'Pregnancy Pause' some Banks offer, which essentially allows you to go backwards for a period of time and watch your Home Loan interest capitalise back on top of your Home Loan Principal.\nMost clients I meet don't understand how loans actually work and when you understand this, you can then get them to work in your favour. For example, every dollar greater than the Interest Only repayment comes 100% off the Principal. Therefore you can turn your Interest Only Home Loan into a Principal & Interest Home Loan without fee or penalty and without having to ask the Bank. Whereas you cannot turn a Principal & Interest Home Loan into an Interest Only Home Loan without paying a 'switching fee'.\nThe other main benefit of having an Interest Only Home Loan is the fact that if you do get ahead, you can redraw back to the limit, again without having to get approval from the Bank. So if your car blow up and you need a new one you could simply redraw the funds from your Home Loan.\nWARNING: Having an Interest Only Home Loan requires discipline. Using funds from a Home Loan to buy a Car without increasing your Loan Repayment may see you repay much more interest compared to getting a Car Loan. Before you redraw any significant funds from your Offset Account or Home Loan Account make sure you ask your Mortgage Broker for guidance.\nAs you can see there is a bit more involved when comparing home loan interest rates, in particular the top 5 features detailed above. Looking for the cheapest home loan interest rate will normally mean you lose some important features. So what seems cheap today may cause you to refinance in the future to another product, which when you factor the costs of refinancing may see you paying a much higher interest rate in the long run.\nMy goal as a Mortgage Broker is to find my clients a Home Loan Product that is not just suitable for today but also flexible into your future. When you avoid the $395 Annual Fee like the Big 4 Banks Professional Package products it allows you to chop and change your Home Loan over time. What I mean by that is $395 \u00d7 30 years = $11,850.\nIf you refinanced your Home Loan every 5 years, so 5 times over a 30 year period, and it costs about $2,000 each time, you are still ahead compared to paying the $395 Annual Fee. On average people refinance every 3 to 4 years, whereas my clients on average will look to refinance every 6 to 7 years, which potentially saves them thousand over the life of their Home Loan.\nI hope you keep in mind my top 5 Home Loan features when comparing Home Loan Interest Rates and if you are interest to find out which Lender I prefer to use, just go to my Contact Us page and get in touch.\nP.S. I really appreciate your time to read though my Comparing Home Loan Interest Rates webpage, I really hope it has allowed you to make a more informed decision regarding your own finances.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our Story. . . . Award Winning SASS.\nYes...There really are TWO Sisters!\nTwin sisters, Lauri & Carol Raymond.\nSisters & Company was established as a partnership in October 1987.\nIn January, 1988, twin sisters Carol and Lauri Raymond began producing, marketing and distributing a fresh salad dressing\/sauce\/marinade, their original flavor, Sesame-Garlic SASS.\nSASS stands for \"Season All Stuff Sauce\", because it is so versatile - it's not just a salad dressing! We use fresh garlic in our salad dressings because its so good and good for you!\nCheck out our SASS RECIPE page for some delicious easy ideas to add SASS to your cooking.\nThe Sisters began distributing SASS in Austin, Texas through health food stores and a few locally owned upscale markets. With its fresh ingredients, lack of additives and no artificial preservatives, it was ideal for produce departments, delis or dairy cases.\nThen, in May of 2001, Sisters acquired Martin Brothers Fresh Dressings, another excellent line of all-natural, fresh salad dressings. Now all 10 wonderful fresh salad dressings, sauces and dips are under the umbrella of Sisters & Brothers, Inc.\nCarol Raymond, Karl Martin, Lauri Raymond & Jeff Martin.\nIn a contract-signing party which could be described as typically Texas, two Austin-based natural salad dressing companies became one in a move which has been a long time coming.\nDuring the negotiations, the Sisters became even bigger fans of the Martin Brothers when they saw that both companies share a commitment to using high-quality, all-natural ingredients and making their fresh salad dressings and sauces in a socially responsible way. \"We just kept shaking our heads and saying, 'They remind me of us!'\" Carol laughs.\nThe Martin brothers and Raymond sisters, friendly competitors since the mid-eighties, have often discussed becoming one company. The time was finally right.\nTwin sisters Lauri and Carol Raymond of Sisters & Company, makers of SASS Fresh Dressings acquired the Martin Brothers Salad Dressings brand from brothers Jeff and Karl Martin on Friday afternoon, May 11, 2001 in a celebration that included photos, champagne, and of course, chips and salsa!\nThe owners and employees of Sisters & Brothers, Inc. are dedicated to providing unique, healthy, delicious foods made with high-quality ingredients and serving customers in every way we can. While doing this, we act responsibly and are successful.\nI always tell the truth and tell it quickly.\nI always ask for what I want, when I want it.\nI always take total responsibility for my experience.\nI always keep my agreements.\nWe pledge to work, in as many ways possible, to leave the earth cleaner than we found it, and to encourage others to do so.\nWe believe the best about ourselves and others, and our interactions will be honest, kind, and respectful.\nWe believe that cooperation, not competition, is the new paradigm for business. There is plenty for everyone.\nAs individuals and as a company, we are part of the total web of life. We give to and receive from the community.\nWe live abundant, powerful, healthy lives.\nWe celebrate our creativity and diversity.\nSesame Garlic SASS wins first place in the oil, vinegar and salad dressing category in the New York Fancy Food Show.\nIn July of 1997, Sisters & Company had a booth at their first national trade show in New York City at the New York Fancy Food and Confections Show presented by the prestigious National Association of Specialty Food Trade.\nAs participants, they were allowed to enter one product into the awards comp etition for free. They chose their top dressing, Sesame Garlic SASS fresh dressing, and were delighted and gratified to win First Place in the Oil, Vinegar and Salad Dressing Category!\nSesame Garlic SASS fresh dressing is the flavor that started it all and has always been a top seller! One taste and you'll know why... and SASS is perfect for salads, splashing in your stirfry, or as a marinade for your grill. Yumm! A great garlic dressing!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Auchentoshan 21 Year Old 1997 vintage Lowland single malt Scotch whiskey. Single cask refill sherry hogshead #2911 bottled October 2018 by Signatory Vintage for the Un-Chillfiltered series. This is a specially selected release exclusive to The Whisky Barrel. 168 bottles at natural cask strength. MAXIMUM 2 BOTTLES PER CUSTOMER.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxnvy b/data_all_eng_slimpj/shuffled/split2/finalzzxnvy
new file mode 100644
index 0000000000000000000000000000000000000000..ca579821039118224f9a138fa162b003fa6b3d48
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzxnvy
@@ -0,0 +1,5 @@
+{"text":"\u2022 Gothic Calligraphy eBook: Includes a 67 Page PDF Format 8.5\" x 14\" printable version as well as an interactive version for mobile phones \/ tablets etc.\n\u25cb OR a 10mm wide flat-brush or similar . (introductory brushes are around $5 to $10). The course notes have been designed around this ( 3\/8th inch, 10mm) size. A Winsor & Newton brush or any brush with spring is acceptable. Curry's, Michaels, or even a local paint and wallpaper store will have these available.\n\u25cb A palette to thin paint.\n\u25cb Cup for water and mixing paint.\n\u25cb Acrylic paint (red, black, or dark colour). I recommend picking up a few tester paints for a $1.00 each. You don't actually need much paint as it lasts quite long when thinned properly.\n\u25cb Fraktur is known by . . .\n\u25cb Why Fraktur and Brush Calligraphy?\n\u25cb Inspiration and Analysis of Fraktur?\n\u25cb Using graph paper with your liners?\n\u25cb Why flourish or decorate letters?\n\u25cb Interested in setting up a Live Course?\nGothic Calligraphy & Lettering Video Course Online - Duration: 6 minutes, 22 seconds.\nOnline Calligraphy Course - Duration: 3 minutes, 54 seconds.\nDesktop Publishing Course Using Scribus - Duration: 8 minutes, 17 seconds.\nDesign Professional Brochures Using GIMP Inkscape and Scribus - Duration: 26 minutes.\nDigital Arts Course DVD - Sample Tutorial - Duration: 2 minutes, 38 seconds.\nDigital Arts and Design Master Class - Duration: 3 minutes, 54 seconds.\nTwenty Third Psalm - Calligraphy by Steve Czajka - Duration: 5 minutes, 22 seconds.\nDesigner Type - FREE + Premium Typography - Duration: 16 minutes.\nDesigner Type - Teaser Trailer - Duration: 109 seconds.\nTwenty Third Psalm - Coming Soon - Duration: 33 seconds.\nSeason's Greetings - Duration: 3 minutes, 48 seconds.\nThe Raven - Edgar Allan Poe - Duration: 4 minutes, 15 seconds.\nDigital Arts Course - Duration: 113 seconds.\nMeet The GIMP - Duration: 61 seconds.\nA Gothic December 2011 Calendar - Duration: 2 minutes, 44 seconds.\nGIMP Magazine - Issue 6 - Director's Cut - Duration: 23 minutes.\nNext Gen GIMP Magazine Teaser Trailer - Duration: 2 minutes, 25 seconds.\nGIMP Magazine - Issue 4 - Directors Cut - Duration: 6 minutes, 8 seconds.\nGIMP Magazine - Issue 2 - Directors Cut - Duration: 36 minutes.\nLibre Graphics Meeting 2013 - Duration: 17 minutes.\nGIMP Magazine - Issue 3 - Directors Cut - Duration: 29 minutes.\nGIMP Magazine - Issue 1 - Directors Cut \/ Commentary - Duration: 28 minutes.\nHalloween - Duration: 44 seconds.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A rich, skin conditioning botanical cleanser for the entire body. This soap-free gel with a nourishing lather delivers all-over cleansing without over-drying. Pro-Vitamin B5 helps moisturize to leave skin feeling soft and silky. Invigorating Tea Tree, Lemon, Eucalyptus and Rosemary essential oils gently purify without stripping the skin's own natural moisturizers, as soothing essential oils of Sandalwood, Lavender, Orange and Clary Sage help condition and smooth.\nProvides soap-free, acid-balanced cleansing without stripping skin.\nFragrant botanical extracts provide a natural conditioning lather.\nLeaves skin feeling refreshingly clean and soft.\nSqueeze a small amount onto The Ultimate Buffing Cloth or hands. Work into a lather over the entire body and rinse. May also be added directly into running water for a foaming bath soak.\nUse with The Ultimate Buffing Cloth to buff and polish skin to silky perfection.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A slim quilted bag designed to carry and protect your 15,6'' laptop with a compartment reinforced with high density foam made of shock absorbing micro bubbles. A dedicated compartment will let you take your tablet and accessories with you while a front pocket allows you to slip in your most essential accessories (phone, pens, business cards, chargers etc...) and gives you a quick access to them. The back strap allows you to carry it on a trolley. The removable shoulder strap will help you to easily carry your laptop on your shoulder. The traditional leather good finish will make it last long.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thirty-four personnel from Joint Task Force-Bravo's Medical Element, along with Joint Security Forces (JSF) and Joint Support Battalion (JSB), participated in a two-day Medical Readiness Training Exercise (MEDRETE) and a Military Partnership Engagement with the Honduran Armed Forces April 20-22, providing basic medical care for more than 1,500 patients.\nThe medical mission took place in the Department of Colon, Honduras where members of JTF-Bravo medical staff distributed preventive medicine, provided basic medical and dental attention, dental amenities and provided medication to the local population.\nSelection of an area of operation for a medical engagement is decided by U.S. Embassy members in combination with JTF-Bravo leadership, as well as partner nation representatives, specifically the Honduran Armed Forces and the Ministry of Health, to determine the priority and need of a specific area. The selection is usually determined by remote locations less or not likely at all to receive health care. Access to medical care is limited and less capable of accommodating residents with more serious illnesses in rural areas outside of San Pedro Sula and Tegucigalpa.\nMEDRETEs provide a real-life atmosphere where service members can enhance their medical abilities under austere conditions, better preparing them for humanitarian and disaster relief situations.\nService members from JSF and JSB also had the opportunity to enhance their training in critical areas like field survival, security, troop management and logistics planning, while simultaneously strengthening a partnership with local residents and volunteers. Each skillset is critical to the overall success of the operation.\n\"This is a great partnership for us and Honduras and other countries that we go too. It is a good training exercise for us in the event we have a humanitarian assistance, disaster relief if that were to occur, we can already hit the ground running. We know how to do it, know how to set it up and it is a good benefit for the Honduran people and the citizens of the other countries that we see.\" Said U.S. Army Lt. Col. Rhonda Dyer, preventative medicine nurse at JTF-Bravo Medical Element.\nVolunteer organizations played a critical role in the success of the mission at hand, assisting in everything from providing medical assistance to providing food and water to local Honduran residents.\nThe operation was able to continue with an ongoing tradition taking place since 1993, letting residents know that JTF-Bravo personnel are here to be good neighbors and partners throughout Central America.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From the illustrious fourth generation hospitality leaders, brothers Ignazio and Maggio Cipriani, comes Mr. C Coconut Grove. An oasis within bustling South Florida, the hotel is scheduled to open its doors this SpringSurrounded by breathtaking views of Biscayne Bay and the Coconut Grove skyline, the hotel and its rooftop Bellini Restaurant and Bar will provide guests with attentive European service, complemented by luxury accommodations and amenities.\nAt the heart of the hotel, interiors hint at the aesthetic elements of a classic yacht, with high gloss timber paneling adorning the walls of the double-height lobby. Murano glass chandeliers will hang above the large-scale terrazzo floor invoking the country's midcentury flair. Playful and comfortable sofas and armchairs in a range of soft and airy blue, white and pink fabrics will complete the space.\nAcclaimed architecture and design firm, Arquitectonica, in conjunction with the Martin Brudnizki Design Studio, combined the stylish European glamour and nautical aesthetic Mr. C Hotels are best known for within tropical surroundings. \"The hotel is suspended on sculptural stilts in the tree canopy of Coconut Grove,\" said Raymond Fort, Senior Associate of Arquitectonica. \"This design embraces the idea of the historic Stiltsville structures in Biscayne Bay that can be seen at a distance from th Mr. C Coconut Grove's 100 guestrooms and suites with private outdoor terraces will feature teak paneled foyers with traditional detailing.\nThe property's overall aesthetic will be evident throughout the rooms and suites, with tall leather-upholstered blue and caramel headboards, a nautical style dresser, desk and mini-bar. Guestrooms amenities include 24-hour in-room dining service, complimentary internet access, PressReader application, Samsung 55-inch 4K HDTV and Salto door lock system featuring keyless entry. Italian hand-made furniture by Tedeschi Furniture, luxury Casa Rovea linens and La Bottega toiletries complete the guest experience.\nOther amenities include state-of-the-art gym equipment with personal trainer and yoga classes available upon request, a Lobby Lounge with indoor and outdoor garden seating, rooftop pool and pool deck featuring private cabanas and a pool bar, and a private spa suite with a relaxation area, sauna and treatment room which can be reserved upon request through the concierge team.\nThe hotel will also feature Bellini Restaurant and Bar, a refreshing new Italian dining concept inspired by Ignazio and Maggio Cipriani's travels throughout Italy. With indoor and outdoor seating, the 107-seat restaurant will invoke the intimate old-world atmosphere of coastal Italy while embracing a sleek and soothing Miami modern style at the hotel's top floor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaaewd b/data_all_eng_slimpj/shuffled/split2/finalzzzaaewd
new file mode 100644
index 0000000000000000000000000000000000000000..01f6ae27a988d967d3d91577341150688cc7b86b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaaewd
@@ -0,0 +1,5 @@
+{"text":"Because of its high water and fiber content, celery is an ideal low-calorie snack. It's also a good source of calcium and vitamin C.\nIn a small bowl, mix 2 ounces each of blue cheese and cream cheese until smooth. With a small spoon or knife, stuff 6 large celery stalks, rinsed and dried, with cheese mixture.\nCrumble another 2 ounces blue cheese on top, and sprinkle lightly with cayenne pepper. Cut stalks in half, if desired.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Saturday I took the GOLDWING out to enjoy the fall colors of Arkansas's Ozark Mountain Region. I left Mountain View at about 6:30 am and headed west on Highway 60, picking up Highway 76 in Willow Springs (if you stay on this for two hours, it eventually becomes \"The Strip\" in BRANSON). I had my heated gear on as it was about 38 degrees in the bottoms (a 50 degree day is infinitely better than a 100 degree day). Shortly thereafter, I picked up Highway 95 and headed southwest. Not only is 95 one of the most scenic highways in all of Missouri, but once you get to the west side of Highway 5, the pavement is brand new with lots of twisties --- many one right after the other.\nAt the end of 95, I grabbed Highway 160 just west of Theodosia and headed west. I could see the fog over the hills to the south coming off of Bull Shoals Lake. I stayed on 160 for about 10 minutes and took 125 south to the ferry in Peel, Arkansas, which crosses the lake. Even though I missed the ferry by a few minutes and had to wait a half hour, I had the opportunity to meet a couple of guys on a fishing trip who had grown up within just a few miles of where my parents now live near Melvern, Kansas.\nAfter crossing the ferry, I stayed on 125 South until I got to Locust Road, which took me over to 14 W. Within a few minutes I was heading south on the most famous road in the Ozarks --- \"Scenic Seven\". After going through Harrison, winding my way past the old Dogpatch USA Amusement Park (long abandoned), and crossing the Buffalo River, I was in Jasper. In Jasper, I took 74 E. to 123 S., which has the best twisties and wildest switchbacks in a state famous for both (the best stretch is near Mt Judea -- pronounced \"Judy\").\nAfter jumping back on South 7 for about three miles, I turned back east on Highway 16 at Sand Gap (it's labeled on Google Earth, but the same reveals that it is nothing more than a general store). Not only was this road amazing (great twisties), but the views from the bluffs \/ ridges into the valleys was spectacular --- especially with the Autumn colors and some of the leaves off so you could see. It was in this area that I believe I passed through the communities of 'Welcome Home' and 'Ben Hur' (Welcome Home was an actual town, while Ben Hur had little more than a church just past the Pedestal Rocks Scenic Area).\nInstead of taking 377 north like I have previously, I went to 27. Both 16 and 27 were outstanding riding. The thing is, if you are trying to get somewhere quick, don't plan on it using these roads as both are all over the place --- one minute you are going south and the next, north (look at Google Earth). Parts of 27 between Marshal and Highway 66 were stunning. The best was rolling through a valley bathed in various shades of red, green, gold, and yellow, with a hayfield covered in big bales next to a clear rocky stream on my left, and surrounded by the tree-covered hills on either side. All along the way were breathtaking views. After passing through Mountain View and stopping for a couple of minutes to listen to some Bluegrass being played on the square (their Mountain Music Festival was in full swing), I headed over to Anglers to eat (it's on the White River at Sylamore Creek --- at the junction of several highways), not far from the road made famous by motorcyclists; PUSH MOUNTAIN ROAD.\nFrom there it was north across the switchbacks on Highway 9 (again, the views of the White River Valley from the ridge tops were about as good as it gets). I turned north at Salem, Arkansas on the highway that becomes 17 just a few miles from town --- just after you cross the Missouri state line. 50 miles later I was at my office, which is itself on Highway 17. Great trip and look forward to a bit different route in the near future (with heated gear, cold is not really a problem as long as the roads are good).\nI would love to take credit for the pics below, but when I ride alone, I'm by myself I typically crank my radio and ride. Hard. Rarely stopping to take pics unless I have a passenger (the pics below are all from Wikimedia, although they are all places I have ridden). The biggest problem with riding in Arkansas is that it's curvy roads are wearing down my foot pegs. Great problem to have!\nAfter our \"TURKEY INCIDENT\" of a month or so ago, my bike (a yellow 2001) needed some work. Nothing against my local Honda dealership --- great guys one and all ---- but they are more of a huge ATV shop. So naturally I called Fred to ask some questions \/ advice. He and his wife Diane were planning on a getaway anyway, so it was the perfect excuse for them to make a trip to the Ozarks. Fred and Diane have become fast friends over the past year, and the truth is, my kids think of THEM as family (actually, we all do).\nAfter arriving on Sunday evening, Fred spent Monday working on the bike, taking it apart, putting new parts on, and then putting it back together with a couple of twists (a MADSTAD Windshield and a pair of LED RIDER \"anti-deer\" lights), while Diane spent time with Amy and the girls. Since Fred finished the GOLDWING on Monday, we all went to the CURRENT RIVER on Tuesday. We ended up taking a 20 mile boat ride up to BLUE SPRINGS. We then spent the afternoon cooking (pics below), floating in the boat, swimming, and hanging out on the gravel bars. We even stopped by Big Springs on the way home. It was a great day in every sense of the word!\nFor the record (and particularly for the guys on the GL1800Riders Board), I want to let everyone know just how terrible a person Fred really is. He actually \"forced\" me to ride his Traxxion-equipped 2012, for which he just happens to be a dealer. All I can say is DANG! The custom suspension by Traxxion Dynamics is the real McCoy ---- especially considering he is running their \"Mega Monty\". Can't justify it for my bike (I broke down and did the upper and lower Triple Trees with a brand new rear shock from a GW being triked), but the ride, cornering, and low-speed turning were ridiculous.\nAs for the lights that Fred installed, I cannot imagine better lights for the money. These are super heavy duty and look great. They also throw out some serious light. And the cost is several times less than the \"Kristas\" that some people rave about on the board. And as far as the Madstad Windshield is concerned, it performs exactly as advertised. There are an almost infinite number of adjustments (up-and-down for each individual shield, as well as the ability to change the angle \/ pitch of the top shield. By the way, I am 6'2\", running their 16 inch top-shield. You could easily be an inch or two taller and still run the 16 inch top.\nI can adjust the ride so that if I want, I have almost no air on me whatsoever. Interestingly enough, there was so little air behind the windshield that I had to take my foam Tunnel Blockers out in order to hear the radio through my external speakers. Not sure why, but the acoustics were different with the Madstad (BTW, the Madstad does not cut down on road noise even though it cuts way down on buffeting). My wife also gets less buffeting with the Madstad, although the Baker Hand Wings do an excellent job of helping with that situation (we did have to use a Dremel to cut a small notch in the bottom of the Bakers to accommodate the lights).\nI went with the Madstad over the Windbender because of good reviews on bigger Harleys, and because the rake is a built-in feature as opposed to being a one hundred dollar option. If anyone has any specific questions, just leave a comment and I will answer it for all to see. By the way, I would like to give a huge thank you to FRED HARMON. There might be someone on the planet who knows more about fixing Goldwings than Fred, but I would not have a clue who it is.\nBoth Missouri and Arkansas are teeming with turkey. They are literally everywhere, and this time of year they are especially active. Although I am not as into it as my son Malachi (an avid DEER HUNTER) who has been HUNTING TURKEY almost since he was old enough to sit on the back of Glynn's four wheeler, I am certainly not opposed to turkey hunting. We killed a big bird on Saturday afternoon --- just not the way we would have liked.\nAfter our younger girls won their soccer game that morning 12-0, Amy and I went for a ride. As is frequently the case, we took the GOLDWING to Arkansas. Our route took us to Willow Springs, where we caught highway 76 (which happens to \"The Strip\" in Branson). From there we got on 125 south. We took the ferry across Bull Shoals Lake near Protem, and then took the roundabout way into Mountain Home, where we ate Fajitas. We came home via highway 5 to Gainsville, and then took 181 back to 76 near Willow Springs --- about a 6 hour circle, including all the stops we made. It was literally the perfect day, and was possibly the most beautiful Spring ride I have ever been on --- except for the Turkey Strike.\nWe were rounding a full speed curve to the left, when a large Tom Turkey (22-25 lbs) flew low off the bank of the high ditch on the right side of the road. Nothing I could do. I thought at the time I hit it with the point of my fairing ---- flush in the middle of my headlights. But the reality is, the turkey impacted the right headlight. We were well into our trip, so we had no other choice than to keep going. There is some damage.\nFairing is cracked and moved some (you can tell when you look at the rubber seals between the fairing and the mirrors). The headlight reflectors are shattered and the headlight cover is in pieces. Needless to say, the right low beam doesn't work. But thank God that the bird was not flying a foot or two higher. Not sure what would happen if it would have hit my windshield. Thankfully Amy saw it and was ready for the impact (honestly, the impact was momentarily harsh, but it did not change our course at all). Would have stopped and brought it home for supper if it wasn't almost 90 degrees and us at least three hours from home.\nAs crazy as it sounds, this is my second wild turkey taken with a motorcycle. My first was about 25 years ago with a Yamaha XS 1100 Special. The turkey flew low across the road (near Babler State Park close to St. Louis) and I hit it with the nose of the Windjammer, just below the point. It knocked the bird down onto the pavement where I ran over it. Hopefully in the future, all game will be taken with conventional methods.\nMotorcycle Helmets. You wouldn't think that this topic would be too controversial. But it is. Although there are all sorts of individuals \/ groups who fight for the \"right\" not to have to wear a helmet while riding a motorcycle, I believe they are misguided. However, there is one 'anti-helmet' group whom I can at least understand and empathize with. These are the people who love to ride but struggle with CHRONIC NECK PAIN. You see; due to the combined weight of the helmet & their own head (not to mention the battering one can get from the wind --- particularly for a passenger), the neck can become mechanically stressed beyond what it can bear. This is particularly true in women. Thus, they stay home.\nIf you look at our MVA PAGE, you'll notice that according to the peer-reviewed scientific literature, women are more likely to be injured in a whiplash-type accident --- much more likely. Women are also substantially more likely to suffer with Chronic Neck Pain long after their injury. Most of this has to do with the fact that women typically have much smaller necks than men, with much less muscle mass to support the head. With the average head weighing 8-12 pounds, and the average helmet coming in at around four pounds, this amount of weight can overcome even a healthy neck with a day of hard riding. Think about it for a moment; the heaviest bowling ball is 16 lbs. The head and helmet must be held by seven small vertebrae, some CONNECTIVE TISSUE, and several small muscles. In someone with a previously injured neck, riding can become impossible. So, what's a person to do if they want to ride?\nGET THE PROPER WINDSHIELD: A good windshield will not only keep much of the wind off the driver, but off the passenger as well. I ride a 2001 GL1800 (HERE). Between the proper windshield and the right kind(s) of wind wings, I can keep most of the wind off myself and my passenger if that is desired. BTW, I run a MADSTAD.\nWEAR A CARBON-FIBER HELMET: One of the newer helmet technologies is Carbon Fiber. Although these can cost substantially more than a standard plastic \/ fiberglass helmet, you can shave a pound or so with the Carbon Fiber.\nMAINTAIN GOOD POSTURE: As you might guess, POSTURE can be a deal-breaker as far as Chronic Neck Pain and motorcycles are concerned.\nSTRENGTHEN AND STRETCH YOUR NECK: While this might work well for the general population, there is a significant chance that it is not going to work well for those of you coping with day-to-day neck pain. To understand why, go HERE.\nDEAL WITH THE UNDERLYING SCAR TISSUE: If you have previously injured your neck, there is a good possibility that all of the previous bullet points above will prove fruitless if you do not first deal with the underlying SCAR TISSUE. Most of the time this Scar Tissue occurs in the Fascia. The problems with injuries to the Fascia is that even though it is arguably the single most pain-sensitive tissue in the human body, it is too thin to be properly seen with advanced imaging techniques such as MRI. This can lead to the PERFECT STORM of Chronic Pain.\nIf you have questions about getting help for your Chronic Neck Pain, simply EMAIL ME a detailed history of your problem. Or call Cheryl at (417) 934-6337 to make an appointment. To see what makes us so radically different from the average clinic, read THIS POST, then take a few minutes and watch a couple of our VIDEO TESTIMONIALS (or HERE).\nI've always loved the ride to Mountain View, Arkansas. Depending on how you decide to go, it's between 2.5 and 3 hours to get there. I've have yet to find a route I don't like ---- and there are a bunch of them. If you get a chance to look at a map of Mountain View, Arkansas, you'll see that it looks like a spider, with numerous legs (roads) coming out of the body.\nMy usual riding partner is my wife Amy. But with her at her mom's helping take care of GLYN'S things, it was not difficult to get Malachi to saddle up for a Saturday morning ride. After sticking some new low beams in our YELLOW 2001 GL1800, Malachi and I bundled up and left Mountain View, Missouri at about 6:00 am. It was 58 degrees and clear, with the hi temperature projected to be 78 --- a beautiful day for riding, and a pleasant respite from the flooding of the last couple of weeks. Our route took us through Thayer \/ Mammoth Springs, where we stopped at Mal Wart to purchase a cheap camera (we got an underwater camera so we can take pictures of our CURRENT RIVER SNORKEL TRIPS), and then on to Old Hardy Town, where we exchanged highway 63 for highway 62.\nAfter heading through Ash Flat and winding our way through some back country roads, we stopped in Franklin (pop. 100?) and ate a delicious breakfast at Calabama Restaurant. We turned onto Lacrosse Road at Violet Hill, and then turned south on 58 just after Melbourne, eventually crossing the White River and coming into Mountain View from 14 South (by the way, this small section of 14 was quite fun to ride). On 58 we started to see bikes. And as we got closer to Mountain View, we saw lots of bikes!\nAlthough we had no idea it was going on, this happened to be the weekend for the annual Mountain Music & Motorcycles rally. Needless to say, Malachi had never seen so many bikes together in one place before. We made a quick loop through Mountain View and saw hundreds upon hundreds of bikes (many Wings, but mostly HD's). After fueling, we made our way to the junction of 5, 14, 9, and the White River. We wound our way up 14 through the town of Fifty Six, and onto 341. Before long, we found ourselves at our destination ---- Push Mountain Road.\nAll I can say about the curves of Push Mountain Road is WOW! They were everything they were cracked up to be. Although not as famous (or as many hairpins) as the PIG TRAIL out of Eureka Springs or Scenic Seven out of Harrison, PUSH MOUNTAIN ROAD (a few pictures HERE) is actually one of Arkansas' most beloved motorcycle roads for serious riders (although there was little traffic on the road, we did see a few sports bikes really winding it up). The first of the previous two links contains a couple dozen reviews of the road from previous riders, several who actually compare Push Mountain to The Dragon at Deal's Gap.\nAlthough I thoroughly enjoy the twisties, this road as well as many of the roads in Arkansas need to be ridden when the leaves are off in order to get the full effect of the mountains. I love the mountainous scenery and it's hard to see through the forest, even though the road runs along a narrow ridge for the most part. You continually catch glimpses, but the there are just enough trees and underbrush in most places to block a great deal of Push Mountain's magnificent views.\nWe found our way back to highway 5, heading over to Norfork \/ Salesville. We took the short detour to ride across the Dam at Norfork Lake and then we were back to Mountain Home and headed home from there. We made it home just after noon, covering just over 250 miles. In all honesty, I must say that I like highways 9 and 58 (both out of Melbourne) far better than Push Mountain Road. Just be aware that my assessment is made on a combination of the road and the scenery as opposed to the road itself. I will, however, be riding Push Mountain Road after the leaves fall ---- my assessment might change. Below are a few pics that Malachi snapped while rolling down the road (click on the pics to enlarge them).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home It should always be about ethics.\nIt should always be about ethics.\nIn my mind, business, no matter what type of industry, should always be about ETHICS\u2026Using this highest morals\/fairness\/delivery. Anything else makes it robbery! End of subject.\nBusiness lacking ethics = Robbery, Chaos. Always a cover-up, or someones mess to clean up.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This article explains two easy steps to connect your pull zone to your server IP directly, bypassing firewall services such as CloudFlare which can block our servers from accessing your data.\nThe first thing we need to do is set the origin URL to point directly to your server's IP. You can do this by either creating a new pull zone or updating the existing one. Make sure that the correct scheme (HTTP or HTTPS) is selected in order to prevent any redirects and if enabled, make sure to disable the Forward Host Header feature.\nThis will now make your pull zone connect to your server directly, however the server will return the default website on that IP. To get content from your actual website, we have to configure BunnyCDN to send the correct hostname back to your origin server, which we will do in step 2.\nTo send the correct host header, open the Edge Rules page for your pull zone. Next, add an edge rule with the action type Set Request Header (1), header name to Host (2) and Header value as the hostname (domain name) where your website is accessible (3).\nSet the IF statement to Request URL (1), and set the Match box to Match Any, and set the Match Criteria (2) to an * to indicate every request needs to be handled with this rule.\nOnce you've done that, click on \"Add Edge Rule\" and then proceed onto the instructions below to configure your hostname on the zone itself.\nNow that we have configured BunnyCDN to send the right hostname back to your origin, we need to configure BunnyCDN to accept connections from your clients using the custom domain. To do this, first navigate to the Hostnames panel, and then enter your domain name into the Add Custom Hostname (1) section and click \"Add Hostname\".\nOnce you have done this, you then need to go to your DNS provider and change the record so that (in my example here, documentation.hawc.eu) is a CNAME that points to the BunnyCDN pull zone (hawc.b-cdn.net in this case).\nOnce you have done this, you can enable the new custom hostname by going to the Linked Hostnames (2) and clicking the \"Enable\" button next to the new hostname you added. You can also get a free SSL certificate here if you want it.\nThe only thing left to do is purge the cache if needed and your zone should now be connected to your server directly, serving from your own custom hostname.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Zupponn wrote: Keep this up and you are on the path to ban land.\nAren't you the one who posted the link to Space Invaders?\nThe Shadowscythe wrote: Hang on a what?\nEither we are reverting back to the days of Spam-humans rather than Spam-bots . . . or this one is self aware, and . . . clever . . .\nEither way . . . I SHALT NOT FOLLOW YOUR LINKS FOUL BEAST! RETURN TO THE HELL FROM WHERE YOU SPAWNED!\nIVhorseman wrote: Rock on, Spambot.\nWhiteagle wrote: Aren't you the one who posted the link to Space Invaders?\nI'll bet its a Silverdream alt.\nHe grabbed one of my posts again and moved it to the Lesser Games forum and put the Space Invaders link in it.\nI'm sure it's just a pandoranuker alt and a warhead alt that got combined somehow.\nRunsWithLegos wrote: now this is just silly.\nshawn@trustmedia.tv wrote: give me some time i hve slow internet here in West St. Paul 1152 Robert St. Minnesota....651 248 4600 U.S. CENTRAL TIME...THE WEATHER IS AWESOME HERE WE HAVE 4 SEASAONS AND ON DECEMBER 25th there are 3 bright stars in the sky that form the handle of the big dipper in the eastern sky. These 3 stars have always been called \"THE THREE KINGS\" and are in the NE SKY. At ~ 6:45am these 3 stars point at SIRIUS the brightest star in the sky which follows the line down to the horizon to the exact point that THE SUN RISES. On December 22nd THE SUN reaches the lowest point on the horizon and we have the 3 shortest days of the year all the same length and on december 25th the days start to get longer. If you were looking for someplace nice to move THIS IS THE PLACE, it is awesome here. Why I'll will have you know that it IS THE GEOGRAPHIC CENTER OF THIS FINE CONTINENT and YOU ARE ALWAYS WELCOME TO VISIT OR LIVE IN THIS GREAT COUNTRY!\nHe's obviously a robot assassin sent back from the future. Hence the star stuff. That's how they know where and when they are.\nMaybe America is no longer at war in his future. He thinks he must be SUPER PATRIOTIC to fit into this time period. And it's working. I believe you, shawn@trustmedia.tv. You TOTALLY BELONG HERE AND NOW.\nshawn@trustmedia.tv wrote: I am honored to be here. Remember when the Germans bombed Pearl Harbor?\nshawn@trustmedia.tv must be from an alternate timeline where Japan and Germany have become indistinguishable.\nIn his timeline, they don't believe in commas, and they use contractions in combination with the second word that was part of said contraction.\nYeah me too, since he is a guy I met on the Newtek forums and he makes funky electronic music.\nOh well, I guess the angry villagers with pitchforks win this round.\nhahaha.. I'm now tempted to go to that address... it's only a few miles away during the summer..\nThe same person who decided to give themselves the first post, and edit that post to reflect so? Duh?\nI say we find his origin and burn it too the ground!!! with fire... of course.\nOn second thought, I must have his shoes above my fireplace?!?!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaakpp b/data_all_eng_slimpj/shuffled/split2/finalzzzaakpp
new file mode 100644
index 0000000000000000000000000000000000000000..ed29a82128566072a3e0a338d5938d781b77be82
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaakpp
@@ -0,0 +1,5 @@
+{"text":"U.S. News and World Report continues to place The Ohio State University among the best for online undergraduate education. Ohio State's online bachelor's programs ranked #3 for 2019.\nU.S. News' rankings indicators include Engagement, Services and Technologies, Faculty Credentials and Training, and Expert Opinion. Learn more about the rankings methodology on the U.S. News website.\nOhio State's programs are taught by the same faculty who teach on campus\u2014online learners don't get a second-string experience. The courses have a small student-to-instructor ratio, and each faculty member works with an instructional designer; an expert in taking the best parts of an instructor's teaching and reshaping them for a technology-rich modality. Ohio State works hard to reduce the cost to degree for every student, replacing expensive textbooks with customized learning content that's free or close to it. Online learners at Ohio State have robust technology support, academic resources and community building opportunities.\nOhio State's current undergraduate portfolio is small\u2014a sign of commitment to quality over quantity. With three undergraduate degrees programs so far, Ohio State's online offerings are growing at a steady but thoughtful pace.\nReady to join them in a top-ranked program? Explore all of Ohio State's online offerings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A tasty twist on the traditional holiday cranberry sauce. Don't let the prep time fool you....most of that time is waiting for the gelatin to thicken!\nPour the boiling water into a medium mixing bowl with the cherry gelatin, sugar, and pineapple juice. Stir to dissolve and chill in refrigerator until thickened but not set (about 1 to 1.5 hours).\nWhile the gelatin is thickening, break up the freeze dried fruits into smaller pieces. The apples are easy to break apart by hand. Put the cranberries or the pineapple in a zip top bag, seal, and break up the pieces using a rolling pin. Place the fruit into a medium mixing bowl.\nZest one of the oranges into the fruit bowl. Then segment the oranges (if you don't know how to do this, it's easy...just check out a video on YouTube to find out how...you're just extracting the orange flesh without the membrane or rind...TIP: when you have extracted all the flesh, squeeze the juice from the membranes into the fruit bowl for extra flavor). Chop the segments into smaller pieces and add to the fruit bowl.\nAdd the nuts, pineapple juice, and water to the fruit and stir well. Set aside to refresh fully while the gelatin is thickening.\nOnce the gelatin is thick but not set, pour the fruit and nut mixture into the gelatin and stir well to combine. Chill till ready to serve.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How does this relate to Software Engineering? And more specifically, to the life of a consultant? The story actually relates to more than just this; it relates to anything that changes. As a consultant, we are often hired to help enact change, and as we will discuss, this story turns out to be very useful. While in the story, the resources being provided by others are actual ingredients to a soup. In the context of software engineering, the resources are people (and their mental resources). Instead of providing us with physical ingredients, they help us by communicating domain knowledge and technical solutions to problems, which can be considered mental ingredients.\nWe can immediately see the use in this method -- convince people to help of their own volition, rather than by force. In the variation used in The Pragmatic Programmer, the travelers are actually soldiers, so you may say \"the soldiers could have forced the villagers to help.\" Of course this is true, and they would have eaten well (as the villagers had the resources for a soup), but what happens when they force the villagers to help? By using force, the soldiers put themselves at risk, as they will pose a threat to the villagers who may look to protect themselves.\nSimilarly, many times consultants are hired by business leaders with decision making power, so they can of course push the changes through, but at what cost? Let's say a consultant pushes and shoves people to get their way and bring about change. They will surely meet resistance. They may be successful in enacting the change, thus, job done, but the results may not be as desired. In the worst case they may push the wrong person, leading the client to rethink the business relationship.\nIn other scenarios, we still alienate valuable resources. Suppose the blunt approach causes job dissatisfaction among the employees? If they leave, the result of us moving in and helping will actually hurt our client. Another negative side effect is evident when you consider the time it takes to make a change. If we anger the employees before the change is fully realized, we may lose access to the employees, thus making the changes even more difficult. The main challenge of this difficulty will likely be caused by lack of access to the domain knowledge that the employees can provide us with. We have simultaneously introduced possible problems for the client and problems with completing our own task. Using the concepts from the story, we can do better, right?\nOf course! Let's imagine a situation where we are hired to help upgrade a technology stack. Let's also suppose that not all of the employees are interested in this stack change (or just that they are unfamiliar with it). How can we spend our time so that the client adopts this new stack? As the domain knowledge will likely be important, it's useful to get familiar with the existing stack. Some of our time will be spent working on stories related to the older technologies, but also to begin to create a second project, which uses the new stack. We want to design this new project in a way which causes the two projects to be independent of each other (i.e. changes to one will not break the other, and vice versa).\nWe are beginning to introduce our change in a non-intrusive way -- people can work effectively with both stacks. This is our start, our so called stone soup. We've created something small which doesn't do too much, but it allows our clients to have an easy environment for exploring the new tech stack (if they want to).\nSince our clients are still a little leery of the change, we want to improve our stone soup just a little bit. We do this by starting to implement some of the domain logic from the old tech stack in our new project.\nBefore we started this process, the employees had the opportunity to play around with the new tech stack, but they still felt a burden in using the new project. Not only do they have to learn the new technology, they also have to recreate a lot of stuff they've already done with the old tech stack. So we build out our new project slightly more and we improve our stone soup. Already, the client is unknowingly providing ingredients for our soup -- they've provided domain knowledge as our next ingredient.\nNow that we have a new project, which uses the new tech stack and replicates existing functionality, the switch from the old stack to the new is less complicated. Other developers can more easily learn the new stack by looking at how existing problems were solved with the new tool set. More ingredients are added in the ability of others to grow the new project.\nYou may run into other things as well. Perhaps the new stack makes things simpler, or it provides performance benefits. These are things that make our soup taste better, and the employees will be more likely to adopt the new tech stack.\nAnd so we have our completed soup, and hopefully both sides can benefit from this. In this example, we switched stacks, but this isn't the only application of the stone soup method. New products can be created by starting with something small that does a portion of the work, and others can help grow it in the future.\nSometimes the business may need to calculate and plan everything, which can turn a potentially simple and quick product into something that is complicated and time consuming. It is very likely that getting time for the base of the product may be easier, so instead of planning everything up front, we only focus on the main purpose. Develop that, then show it off. Hopefully others will like it and look to contribute to the product and so, over time, we end up with the product we envisioned from the start.\nIt may appear to be a deceptive practice since the villagers are being \"tricked\" into doing something. This is where the result becomes important -- we need the final result to lead to some benefit for the villagers. We want them to enjoy in the final product, to get a taste of our soup.\nWe also should not take the soup directly after getting the help from the villagers, otherwise we could ruin the relationship we built while cooking the soup. Similarly, when consulting, you do not want to damage a relationship with a current and potentially future client. The relationship that seems to deceive one party, ends up being beneficial to both. Our client ends with a product or some change in an existing product which helps them, and the consultant forms a business relationship, which may lead to more work in the future. Using the stone soup method, we can effectively combat a problem that exists all around us: resistance to change. People tend to naturally resist change (although in the technology field people may be more open to change), but change happens. Using this approach, we lead by example. We introduce change in a comforting manner. Gradually let people come to change of their own volition, and the result will generally end well.\nOur book club plans to continue, and in fact, earlier I linked a Meetup event for our next book where we will be reading The Phoenix Project. Feel free to RSVP, and come join us for food, drinks, and good times where everyone can learn and grow.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You are invited to a \"trade fair\" unveiling of WUMC's proposed new Church design!\nWhat: Presenters at each of the 7 booths will walk you through the important design aspects of this exciting new Church design. Pizza, finger foods & beverages will be served. Child care is available for infants to kindergarteners.\nJANUARY 2018 MISSION FOCUS - The Pregnancy Center of Greater Toledo \u2013 This Sunday is MISSION JAR SUNDAY!\nWhat We Do: Save & Changes Lives. Who We Are: A Place for Answers. A Place for Hope.\n2016 at a Glance: 1,144 individuals served; 605 ultrasounds performed, 899 pregnancy tests administered, 114 people attended a parenting class, 90% of abortion vulnerable clients chose life after seeing their ultrasound.\nTheir goal is to lovingly share the truth about life, relationships and faith. They help men and women make life-saving, life-changing decisions by sharing information about options, consequences and available services. With the exception of a small amount of grant funding for abstinence education, all funds come from donations. Over 40,000 women have already received help. The Pregnancy Center continues to be a community outreach committed to expressing the compassion of Christ through support and education in pregnancy related issues. We can help keep the doors open! Mission envelopes are located in the pew racks for your financial gifts.\nFollowing will be the changes on Sunday morning related to security to further protect our children and all attenders and to provide as safe a place as possible.\n* At 10:40 am all exterior doors will be locked.\n* Clear signage will be installed directing anyone arriving after 10:40 to the office and courtyard door near the offices as an usher will be posted there to let people in and assist with the elevator if needed.\n* Training will be given to the ushers and others to understand what needs to be done if an intruder situation would occur.\n* Watch for more information as this plan continues to unfold.\nThese are not big changes and many of us will not be impacted by them in any way at all. Let us remember the families of the Texas church shooting of recent weeks and continue to pray for each of them and the whole congregation. Also, to pray for all the churches that have been targeted by evil persons by mass shooting, bombs fires and vandalism.\n2017 Giving Statements: Your 2017 Giving Statements were emailed on Saturday, January 6th. For those who receive hardcopies, they are available for your pick-up on the shelf outside the Church Office. If you have any questions, please contact Tom Wiggins (419) 878-5503 or (financialsec@watervilleumc.org).\nWould you like a DVD of the Children's Program, \"Finding Christmas Under the Sea?\" Sign up for your copy on the hallway bulletin board!\nHappy tails wagging saying \"THANK YOU!\" to all who have donated peanut butter jars for the Humane Society. Please continue to place un-scraped jars in the basket on the shelf by the church office.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mounting fresh attack on Prime Minister Narendra Modi for maintaining a \"mysterious silence\" on Lalit Modi controversy, the national council of Communist Party of India (CPI) on Wednesday decided to launch a nationwide stir targeting \"corruption\" in the central government and BJP-ruled states.\nThe CPI national council-party's highest decision-making body-is holding a three-day brainstorming session here to chalk out a multi-pronged roadmap to revive the falling fortunes of the party and make a comeback to the national political arena. The national council on Wednesday adopted four resolutions---three pertaining to corruption under the BJP rule.\nThe thrust of CPI's multi-pronged anti-corruption offensive against the BJP will be on issues such as Modigate, Vyapam scam in Madhya Pradesh and involvement of Maharashtra ministers in different corruption cases.\nThe CPI top brass called upon the party units and all democratic sections to launch a countrywide campaign against corruption under the BJP rule.\n\"All Left parties will join hands to corner the Modi government. It will be huge campaign against Modi regime on corruption front,\" D Raja, the national council secretary of the CPI, told Hindustan Times.\nOn July 5 the coordination committee meeting of six Left parties --- CPI, CPM, Socialist Unity Centre of India (communist), RSP, Forward Block, CPIML (liberation)---will be held in Delhi to give final shape to the proposed nationwide agitation.\n\"We have decided to observe \"quit corruption day\" probably on August 9. But, we are not announcing the final date as consultations are on with other Left parties. The dates will be finalised on July 5,\" party spokesperson Shamim Faizi said.\nThe national council also expressed \"profound concern\" over \"corruption and scams\" under the BJP rule, saying Modi's claim of giving \"corruption-free and scam-free\" government stood exposed by Lalit Modi episode. External affairs minister Sushma Swaraj, the CPI alleged, was guilty of misuse of office, while Rajasthan chief minister Vasundhra Raje \"has taken political and financial corruption to a new height.\" And, the human resource minister Smriti Irani has submitted a false affidavit about her educational qualification, it claimed.\n\"The national council wants the tainted ministers to quit immediately. If they are not willing, then the Prime Minister should sack them,\" said the party spokesperson, demanding action against Maharashtra ministers facing allegations of corruption.\n\"On the eve of completing one year in power, the PM claimed to have provided a corruption-free regime. But within 15 days, the biggest scam, Modigate, came to light. The PM who is in habit of tweeting on every subject is not ready to say anything on this major scam,\" the CPI said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaavps b/data_all_eng_slimpj/shuffled/split2/finalzzzaavps
new file mode 100644
index 0000000000000000000000000000000000000000..4a5924a18ae94e27c83fae36e6a7955a51104be6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaavps
@@ -0,0 +1,5 @@
+{"text":"We are an experienced aluminum fabricator, in business since 1995. Aluminum is our specialty, but we work with stainless steel, plastics, composite board, sound dampening foam, and general steel fabrication.\nWe can research aluminum components and materials, and special equipment needed to custom fabricate your unique aluminum products. EAF is a fabricator that is totally focused on your design, unique among fabricators.\nApplication: Custom Aluminum Weighing Station designed for enclosing scale, and printer. Table folded down to pallet level to hold package meat products.\nApplication: T-Slotted Computer Workstation for a variety of in plant\/industrial usage. Securely houses CPU, monitor, keybnoard, and data storage in harsh environments.\nApplication: Custom Aluminum Machine Base with one inch blanchard ground steel plate, but other metals or materials can be utilized. Special leveling mounts with built-in adjustable rollers.\nApplication: Custom Table was designed with special cut-out for easy, ergonomic assembly for the operator.\nApplication: Aluminum T-Slotted Gage Table for inspecting crank shafts. Table features pin cushion boards for picture process display. Table held gage tooling and custom machine nylon fixture holders. CPU holder, drawer and file cabinets.\nApplication: Automotive Inspection Rack for quality control - every hour they pull a block off the line for an in-line quality control inspection.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Olympia Matt Pastel Mug Green 320ml320ml \/ 11oz. GreenAttractive pastel mug in matt glaze finish. Perfect for cafes and pubs.\nOlympia Matt Pastel Mug Grey 320ml320ml \/ 11oz. GreyAttractive pastel mug in matt glaze finish. Perfect for cafes and pubs.\nOlympia Matt Pastel Mug Pink 320ml320ml \/ 11oz. PinkAttractive pastel mug in matt glaze finish. Perfect for cafes and pubs.\nOlympia Matt Pastel Mug Red 320ml320ml \/ 11oz. RedAttractive pastel mug in matt glaze finish. Perfect for cafes and pubs.\nOlympia Matt Pastel Mug Yellow 320ml320ml \/ 11oz. YellowAttractive pastel mug in matt glaze finish. Perfect for cafes and pubs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SOLD ON 2-3-17 FOR $1,125,000 Everything is new! Completely renovated contemporary Conch house located in mid-town. 3\/2 open floor plan home has high vaulted ceilings, custom kitchen cabinets, quartz counters, Impact rated sliding doors & windows that sits above the flood plain. Large covered porch is perfect for outdoor dining and living, while overlooking the pool and deck. Spacious master suite. Deep off street parking.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Was that...music? Christmas music? Someone actually considered this noise worth recording? Since when did a Christmas song contain this much cursing? The future was terrible.\nDarcy Lewis is determined to make this the Best. Christmas. Ever.\nBucky Barnes never stood a chance.\nJoren's dead, Genlith's treachery has been exposed, Burchard's abdicated, and Taren of Stone Mountain inherits an unholy mess leaving him on the wrong side of everyone and everything. Time for some New Hope. A further sequel to Lady Knight Volant.\nI'm so happy that they are writing again.\nSix years after escaping from HYDRA, Bucky Barnes is living in New York. Well, maybe not living. Existing might be a better term. His steadfast refusal to meet anyone new has Steve at his breaking point, and he's ready to start taking drastic measures to ensure Bucky doesn't plateau in his recovery. But Steve doesn't know the full story and Bucky would really prefer not to tell him.\nWhen the truth comes out, how far will Steve go to help his friend? How far will Bucky go to help himself?\nAnd how far will Darcy have to go to keep Lucky from jumping on every person he sees?\nIt started with a box, a super sweet coat and a wormhole.\nNow Darcy Lewis is stuck relying on Howard 'Grabby Hands' Stark in her efforts to get home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Your adventure tour begins at the iconic Sorrento Quay Hillarys Boat Harbour. Whilst here be sure to explore the stunning Marina, shop for souvenirs and enjoy morning tea at one of the local eateries (at own expense). Travel on to the Lancelin Sand Dunes and experience the thrill of downhill sandboarding followed by lunch at the Lancelin Beach Hotel.\nFollowing lunch take a stroll along the panoramic Lancelin Beach or take the opportunity for a quick swim. We then travel to the iconic Nambung National Park for an easy paced walk through the mystical Pinnacle Desert. If time permits be sure to make a visit to the Pinnacle Desert Discovery Centre. Return to Perth down the Indian Ocean Drive arriving at approximately 6.00pm.\nPlease note: hotel drop-off is not included, centralised drop-off point at the Wellington Street Tourist bus stop.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaawoi b/data_all_eng_slimpj/shuffled/split2/finalzzzaawoi
new file mode 100644
index 0000000000000000000000000000000000000000..10a2c37c2512dac5fe1340d88dcb8a6bdce6295e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaawoi
@@ -0,0 +1,5 @@
+{"text":"Chamilia's Spoken ID Bangles have a comfortable oval shape that contours to your wrist and are light weight so you can stack them with all you have to say. The ID plate unhooks from one side for easy wearability. Made with sterling silver.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"types of doors types of doors icons set flat illustration of 9 types of doors vector icons for different types of doors and uses.\ntypes of doors type of door frames closet door types types of bedroom doors types interior door frames steel doors types and styles.\ntypes of doors technical terms 8 swinging doors this type types of doors and windows in building construction pdf.\ntypes of doors image of different door types doors types and styles.\ntypes of doors types of cabinet doors types of cabinet doors types of cabinet doors kitchen cabinet door types types of doors and windows architecture.\ntypes of doors types of doors and windows architecture pdf.\ntypes of doors sliding doors doors types pdf.\ntypes of doors what are the different types of doors types doors and windows.\ntypes of doors contour icons vector collection of different types doors stock vector doorstop types.\ntypes of doors knob locks doorstop types.\ntypes of doors door hardware types magnificent commercial door lock types with door hardware types old versus new door hardware barn door hardware types types of doors and windows pdf.\ntypes of doors bi fold sliders doors types and styles.\ntypes of doors types of doors types doors and windows.\ntypes of doors types of doors based on materials types of doors and windows in civil engineering.\ntypes of doors door swing types door that swings both ways door that swings both ways types of doors types of doors and windows in civil engineering.\ntypes of doors door types types doors architecture.\ntypes of doors door hardware types door hardware types types of door types of doors types of pocket door types of doors and windows in civil engineering.\ntypes of doors french door types of doors and windows architecture.\ntypes of doors types of doors and windows in civil engineering pdf.\ntypes of doors interior door types interior door swing fascinating door swing types door swing types types of door interior door types types of doors and windows in civil engineering pdf.\ntypes of doors types of cabinet doors types of cabinet doors kitchen cabinet door types cabinet doors types kitchen types of cabinet doors types of doors architecture pdf.\ntypes of doors types of doors types of interior door interior door styles types doors interior door styles new types of doors types of doors and windows in civil engineering.\ntypes of doors types of interior doors popular different for houses types of interior doors types of doors and windows ppt.\ntypes of doors type of sliding door types of sliding doors residential cabin sliding door types of sliding wood type of sliding door types of doors architecture pdf.\ntypes of doors hanging a slab door types of doors and windows pdf.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are manufacture certain ceramic (with Glazed and with out glazed) and terracotta Agal vilakku just about 50models. This is suitable for Indian festival like Dewali, Navartri, Karthikai deepam, Bharini deepam, full moon, and Temple celebration. These are available with different color and size; it looks like more ancient did not have covers, something like this saucer oil lamp circa 1500 BC. We are using both casting and pressing method to full fill the huge quantity. We offer our collection of Agal vilakku at competitive price range.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Can't think of anything fun to do on a date or how to create a personals ad and profile for online dating? Don't worry we've got you covered. You'll find good first-date ideas, general date ideas, ideas specific to certain cities\/states, or find a good place to eat.\nUsually a good personals site won't make you fill out too much stuff to join. It is pretty annoying to be forced to fill out pages and pages of stuff. Avoid sites that do that, it is apparent that they were not designed very well. This site is pretty easy to join.\nSo you have just joined a personals site, now what? Well, now you can check the site out. By now you should be able to see if this is a quality site or not. One of the best ways to tell this is by checking for advertisements. Are there a lot of ads on this site?\nWhat makes a good profile? Well, consider something: You are selling yourself. Although we don't typically like to think of ourselves as 'products', when it comes to online dating, that's exactly what we are. You can't expect people to be clicking on your profile and sending you email just because you want them to. You have to give them a reason.\nMost digital cameras produce very large picture files. The same goes with scanners. Most dating sites don't accept files that are overly large as it uses up a lot of server resources. You need to make sure your file is less than around 200k. If you find that your picture file is larger than this, don't worry. You can resize it yourself using one of the programs that came with your computer.\nA lot of women are scared of the online dating scene. They hear horror stories about stalkers and weirdos. There are important steps you should take when you think you are ready to meet someone in person that you first met online. The most important thing is to never under any circumstance meet someone for the first time in a private place. Why take the risk?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PROF. DR. RUGGERO MARIA SANTILLI \u2013 iso-, geno- and hyper-mathematics!\n\"I was also pleased that the mining group saw that our technology can be applied to the presence and concentration of uranium in mining operations,\" concluded Dr. Santilli.\nPamela Fleming, Executive Vice President of Thunder Energies Corporation, a Florida based company whose stock has recently initiated trading Over the Counter with symbol TNRG, announces that three astrophysicists, S. Beghella-Bartoli from Italy, P. M. Bhujbal from India, and A. Nas from the U.S.A., have independently confirmed the first detection in history of antimatter galaxies, antimatter cosmic rays and antimatter asteroids achieved by the Italian-0American scientist Dr. Ruggero Maria Santilli following half a century of research via the use of a revolutionary telescope with concave lenses specifically designed and patented for the detection of light emitted by antimatter (http:\/\/www.santilli-foundation.org\/docs\/antimatter-detect-2014.pdf).\nDr. Santilli states: \"I am pleased to report that the development in collaboration with our European partners of a basically new Directional Neutron Source, namely, an equipment capable of producing on demand a flux of low energy neutrons synthesized from a hydrogen gas predominantly in a selected direction.\nHe was invited by the University of Miami to conduct research for NASA. He subsequently joined the faculty of Boston University, under support from the U.S. Air Force.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaaxve b/data_all_eng_slimpj/shuffled/split2/finalzzzaaxve
new file mode 100644
index 0000000000000000000000000000000000000000..e19c7219f9531be6e37d0ea085d53e2b56c897b9
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaaxve
@@ -0,0 +1,5 @@
+{"text":"What happened: Swedish furniture retailer Ikea launched a WeChat mini program on Monday enabling users to purchase Ikea products online directly. However, Ikea's mini program is not permanent but a pop-up e-commerce store. For the next nine months, selected Ikea products will be sold via the mini program. According to Ikea, it hasn't been decided if the mini program will become a permanent retail channel in the future.\nWhy it's important: Ikea's e-commerce struggle in China has been going on for a while. The Swedish company launched an e-commerce retail channel in August 2016. After two years of operation, online purchasing is only available in Shanghai. Last year, Ikea said it would open stores on a third party platform, but business did not go smoothly. There are many third-party purchasing service provider on Taobao targeting consumers in cities where there no Ikea stores or consumers that can't be bothered to drive to the store.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In case you're not familiar with it, The Last Five Years is a musical where a man and woman tell the story off meeting, falling in love, marrying, and then having it all fall apart over a five year period in crossing timelines (her first song is about him leaving for good, his is about their first date).\nThe production that I directed and accidentally starred in back in 2011 was a nightmare from start to finish that I thought was entirely behind me. Turns out I wasn't as \"over it\" as I thought.\nI've been the person everyone was trying to keep up with onstage before, but it's been rare that I've had a fellow actor that I felt was working on a completely different level from what I was. It's humbling and inspiring and it makes me want to do better.\nDon't worry, Shirley. I'm not going to try and muscle in on your husband. I'm just a sucker for a pretty voice.\nOh, just writing a story about my Mom and crying at work. As you do.\nSometimes things aren't as bad as they seem. Sometimes you're doing just fine.\nAlso, Ron looks nothing like this and clearly he should put me in more shows so I can get better at drawing him.\nWe were talking about several of Portland's (apparently) beautiful rose gardens. I wouldn't know since I haven't seen them yet.\nWith jokes like these, I do not understand how I've ever spent a day of my life single.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u3053\u3093\u3070\u3093\u306f! Hello, did you have a nice Friday the 13th?\nMine was good, with a lot of work finished, but what was fun was watching Koa play baseball in the evening. He hit a rocket to center field, but it was caught. Still, that was his best-hit ball ever and he was fired up!\nAnyways, the weekend is here and it should be a busy one, with another baseball game, Japanese school, and a piano recital. Should be fun!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Village- West Baishpur, Post Office- Matlab(South), District- Chandpur, Bangladesh.\n367 Elephant Road, Flat: C-5, Dhaka.\nProfessor selection grade (former), Electrical & Electronic Engineering, University of Dhaka.\nProfessor of EEE and Dean FST, Atish Dipankar University of Science & Technology.\nEditor in Chief, The Journal of Atish Dipankar University of Science and Technology.\nPresently Part time Professor, Institute of Leather Engineering & Technology, University of Dhaka.\nEnjoyed Austrian Government Scholarship for performing Ph.D.\nObtained Residential Scholarship throughout the academic career from Class VIII to M.Sc. level.\nServed as Dean (Elected), Faculty of Engineering & Technology, University of Dhaka (from 13.12.2012 to 30.10.2014) Obtained Residential Scholarship throughout the academic career from Class VIII to M.Sc. level.\nServed as Chairman, Department of App. Physics, Electronics & Comm. Engineering (from November 2002 to November 2005), University of Dhaka.\nServed as Director of Energy & Renewable Energy Institute, University of Dhaka for two years.\nFormer Chairman of Nuclear Engineering, University of Dhaka.\nEx-Member International Advisory Board Green Asia Program, Kyushu University, Japan (Collaboration Program with D.U).\nPresent Member, Selection Board of Professor and Associate Professor, Selection of Chittagong University, Noakhali University of Science and Technology (nominated by honorable Chancellor of the University) & Hamdard University of Bangladesh.\nMember, Bangladesh Electronic Society (Life member and former General Secretary), Bangladesh Physical Society, Bangladesh Computer Society & Bangladesh Association for the Advancement of Science (BAAS).\nMember, Academic Council of Islamic University, Kustia.\nLife Member alumni association of EEE, University of Dhaka.\nGood number of publications (including writer of 3 books suitable for EEE, CSE, ETE APECE and other foundation and advance courses of Engineering & Technology) published in the National and International Journals.\nAt present working as Vice Chancellor of the Atish Dipankar University of Science and Technology (ADUST) since 1st August, 2017 as approved by the honorable President of the People's Republic of Bangladesh and the Chancellor of the University.\nServed as Dean (Elected), Faculty of Engineering & Technology, University of Dhaka (from 13.12.2012 to 30.10.2014) Served as Chairman, Department of App. Physics, Electronics & Comm. Engineering (from November 2002 to November 2005), University of Dhaka.\nServed as Director of Energy & Renewable Energy Institute, University of Dhaka for two years. Former Chairman of Nuclear Engineering, University of Dhaka.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nutria are a classic example of an invasive species run amok.\nThe new documentary, \"Rodents of Unusual Size,\" is a beautiful tribute to the people of South Louisiana but ultimately falls short on its mission to educate the public on what's displacing them.\nThe film tells an urgent story of Louisiana today, where the ravages of the oil economy combine with decades of environmental mismanagement to literally wash our coast away.\nThe film takes us to Delacroix Island \u2014 \"the end of the world\" \u2014 where residents are fighting to preserve both terra firma and a way of life. Like other South Louisianans, the Gonzalez family has a fraught relationship with those giant, orange-toothed rats called nutria.\nNutria are surprisingly cute \u2014 as rats go. An invasive species from South America, they're also an existential threat to communities like Delacroix.\nFor Thomas Gonzalez, the star of the documentary, coastal restoration amounts to trapping and shooting the 20-pound rodents by the truckload. The state of Louisiana offers a $5 bounty for each nutria tail. With a single hunting trip sometimes yielding hundreds of tails, nutria killing has become a way for Gonzalez to help make ends meet.\nNutria didn't get here on their own. They were brought from South America to Louisiana in the 1930s as a new source of pelts for the garment trade. Whatever nurtria thought about being turned into fur coats and hats, they loved the river deltas and the abundance of aquatic plant stems that are their dietary mainstay. As the numbers of nutria grew, they ate their way through the wetlands. And as the vegetation started to disappear, so did the sodden land the plants had held in place.\nBut while the nutria undoubtedly have contributed to Louisiana's land loss problem, they've had a lot of help from another mammalian species: us.\nThe 1930s were also a time of rapid expansion of the Gulf South's oil and gas industry. As extractors pumped fossil fuels from the ground, they also dug up and dredged the earth to lay pipelines and clear shipping channels. It sped up the natural process of subsidence.\nSoon the salty waters of the Gulf were sliding farther and farther along those pipelines and canals, killing marsh grass and hastening erosion. It didn't help that leveeing the Mississippi River choked off the springtime floodwaters that for millennia had been replenishing the coast with an annual coating of sediment, now sluiced out into the depths of the Gulf.\nToday we see the upshot of a global \u2014 and state \u2014 economy addicted to fossil fuels. Louisiana loses an acre of land every hour, more than 6.2 million acres since the early 1930s. Our coast has become a poster child for accelerating environmental collapse.\nWithout an epic course correction in the fight against climate change, the trappers and fisher families of Delacroix Island will be next on the list of Louisiana communities lost to rising seas and dying marshlands.\nBefore I started meeting people in the commercial fishing industry, I assumed they made their living solely through harvesting a species or two. \"Rodents\" shows us a more complex reality \u2014 that South Louisiana fishing communities have traditionally hunted a variety of animals based upon the rhythms of the seasons.\nA traditional South Louisiana fisher might catch speckled trout in the spring, shrimp and crabs in the summer, and redfish and oysters in the winter, with some alligator trapping in the fall as lagniappe.\nThe filmmakers seem like they've stuck their heads in the mud by not even mentioning climate change.\nThe commercial harvest of redfish has been banned since 1988. It remains a sore spot for traditional fishers who feel that the ban unfairly targets the commercial industry while ignoring the significant pressure put on redfish stocks by the recreational sector. It's a still-raw reminder that when the state mismanages environmental resources, vulnerable residents on our coast will be the first to feel the consequences.\nClearly there are other issues affecting the Gonzalez family, but the filmmakers chose to keep the focus on the nutria problem. That decision makes for a well-structured and entertaining film.\nBut the filmmakers seem like they've stuck their heads in the mud by not even mentioning climate change. During a brief animation sequence that describes erosion and dredging, we see a few oil and gas platforms but no deeper explanation of who really caused it and who really benefited amid all that land loss.\nI asked directors Quinn Costello, Christ Metzler, and Jeff Springer whether this omission was a conscious choice. They told me that there was no discussion or agreement to exclude climate change, but that they felt the film needed to stay focused on the nutria and the human subjects of the film.\nPerhaps the film required that sharp focus, but the truth resides in a more panoramic view of South Louisiana and the fossil-fuel economy worldwide. If you could wave a magic wand and hunt the nutria out of existence, Delacroix would still be sinking into the ocean. In a sense, the nutria are a red herring.\nIt's hard to say whether the directors of \"Rodents\" consciously gave big oil a pass or subconsciously edited the film to make it less political. I hope neither is true, because that sort of self-censorship does a disservice to South Louisianans who are losing their land year by year. They deserve the support of a public willing to face a harsh and ominous reality \u2014 and take action.\nThat said, I'm grateful that the filmmakers chose to take a close look at South Louisiana and the people hanging on by a thread to preserve it. Without a fierce fight against climate change, it might be too late to save our coast. All the more reason to document it while it lasts.\n\"Rodents of Unusual Size\" is showing at the Zeitgeist Multi-Disciplinary Arts Center at 7:00 p.m. from August 31 through September 6.\nKendall Dix is the fisheries organizer for Gulf Restoration Network.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabfkl b/data_all_eng_slimpj/shuffled/split2/finalzzzabfkl
new file mode 100644
index 0000000000000000000000000000000000000000..c72a34bb7a85d332a8eeccd063b9935edd1f5227
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzabfkl
@@ -0,0 +1,5 @@
+{"text":"Lincoln, and the so called libertarian, Alexander Hamilton, wanted regulation of corporations. We have fought for corporations' rights over our own, because the corporation controlled media managed to divide middle class and poor. And they won while the middle class and poor lost.\nThanks for the article. I think you need to understand what Libertarianism is about.\nLibertarians say they are against corporatism, yet they let the banks have free reign. And yet the bankers need to be regulated so they won't give out hurtful loans that are doomed to failure.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PISCIS W Kwgt Full Apk Mod AndroidHermes Outlet Apk Stores Online!\nPISCIS W is a package of 157 beautiful widgets for KWGT. The Application will \"Constantly Update\". Novelty added Wallpaper in the Application, 55 in total.\nThis is not an independent application. PISCIS W PACK widgets require the KWGT PRO application (not the free version of this application).\nTouch the widget and choose PISCIS W PACK KWGT installed.\nChoose the widget you like.\nAny questions or inquiries please send an email.\nAdded news Widgets for 157 in total. Enjoy!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Long gone are the days you have to rely on a publisher to publish a book. If you have a dream of becoming a widely read other, it may actually be more lucrative and productive for you to publish your book. This doesn't mean you have to send your book to some far away printing press either. Thanks to advancing technology, most mainstream printing companies can help you publish your own book.\nIf you are planning on hiring a graphic designer to help you design the graphics for your website or company, then you are probably looking forward to paying for high-quality professional images for your business. However, you could be worried that your graphics will not turn out as you want them to. Even if you hire a highly skilled and experienced graphic designer, there is always the chance that your graphics will not turn out right.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Schlie\u00dfen. Ehrmann verwendet Cookies, um Ihnen ein bestm\u00f6gliches Nutzer- Erlebnis bieten zu k\u00f6nnen. Wenn Sie auf der Seite weitersurfen stimmen Sie der . Dez. Ehrmann \u2013 keiner macht mich mehr an. Passend zum Schulstart verlosen wir 1 von 10 MONSTER BACKE Pl\u00fcsch-Dinos \u2013 ein idealer Begleiter. Sept. ehrmann dino. Hier findest du eine \u00dcbersicht \u00fcber Schn\u00e4ppchen, Gewinnspiele und Gratisproben. Monsterm\u00e4\u00dfiger Nasch-Spa\u00df aus Milch f\u00fcr. If the existence of degiro risk cannot luckydino casino discounted within a reasonable time, the executing judicial authority must decide whether the surrender procedure should be brought to an end. Orders the European Commission and the Italian Republic to bear their own costs. Community plant variety right at issue: This document is an excerpt from the EUR-Lex website. The President netent warlords crystals of power the First Chamber has ordered that the case be removed from the register. European Union Intellectual Property Office represented by: Language 2 Please choose English en French fr. This site vip news intouch cookies to improve your browsing experience. Theologou, subsequently by I. According to this plea, the Court also committed a manifest error of assessment in considering, inter book of ra deluxe online kostenlos, that the price difference with regard to Brazilian poultry did not necessitate the introduction of export refunds to ensure equilibrium in the EU poultry meat market and casino m\u00fchlhausen natural development of prices and trade on uefa europa league 16\/17 market. 90 er party k\u00f6ln, acting as Agent. Susan Tennis news federer Jozef Kuijpers Questions referred 1. Baquero Cruz, acting as Agents. Partner Apelski Dariusz Defendant: Orders the parties to inform the Court, within three months from the date of delivery of the present judgment, of the amount, in figures, of compensation europ\u00e4ische torj\u00e4gerliste at by agreement.\nEhrmann Almighurt K\u00e4sekuchen Mandarine Per Email bekommt man jetzt einen sechsstelligen Gl\u00fcckscode, den man wiederum auf Ehrmann. Schn\u00e4ppchenfuchs verwendet Cookies, Beste Spielothek in L\u00f6sau finden dir einen besseren Service anbieten zu k\u00f6nnen. Informationen zur Verwendung von Cookies. Mit hoher Wahrscheinlichkeit gewinnt man die kostenlose Fototasse von Snapfish. Ehrmann AG until the new hilarious knister yoghurt is presented to them by the Monsterbacke-Dino. Da die Fototasse individuell f\u00fcr euch angefertigt wird, erlischt euer t\u00e4giges Widerrufsrecht. Dieser letzte Schritt ist eben laut AGB ausgeschlossen, funktioniert aber trotzdem. Wenn du unsere Seite weiter benutzt, stimmst du unseren Cookie-Richtlinen zu. Wenn du unsere Seite weiter http: Wenn Ihr noch nicht bei Snapfish registriert seid, macht das am besten hier.\nSie ist nur vielleicht starbust casino, weil der Gutscheincode zwar funktioniert, aber laut Tabelle cl die Kombination von 2 Codes ausgeschlossen wird. Sie Beste Spielothek in Neuendeel finden nur vielleicht kostenlos, weil der Gutscheincode zwar funktioniert, aber laut AGBs die Kombination von 2 Codes ausgeschlossen wird. Schn\u00e4ppchenfuchs verwendet Cookies, um dir einen besseren Service anbieten zu k\u00f6nnen. Immer die besten Deals. Per Email bekommt man jetzt einen sechsstelligen Tombquest book of the dead quiz, den man wiederum auf Ehrmann.\nHolcim Romania SA represented by: Other party to the proceedings: European Commission represented by: Appeal - State aid - Spanish scheme of early depreciation of certain assets acquired through financial leasing - Decision finding no State aid - Formal investigation procedure not initiated - Insufficient and incomplete examination - Obligation to state reasons - Selectivity.\nNetherlands Maritime Technology Association represented by: Struckmann, Rechtsanwalt, and G. Sampol Pucurull, acting as Agent. Orders the Netherlands Maritime Technology Association to bear its own costs and to pay those incurred by the European Commission;.\nAppeal - Common foreign and security policy CFSP - Restrictive measures against the Syrian Arab Republic - Measures directed against persons and entities benefiting from or supporting the regime - Proof that inclusion on the lists is well founded - Set of indicia - Distortion of the sense of the evidence.\nTarif Akhras represented by: Wyatt QC, and R. Bishop, acting as Agents , European Commission represented by: Havas, acting as Agents.\nAppeal - Restrictive measures taken against the Islamic Republic of Iran - List of persons and entities subject to the freezing of funds and economic resources - Criterion relating to the provision of material, logistical or financial support to the Government of Iran - Financial services of a central bank.\nCentral Bank of Iran represented by: The executing judicial authority must postpone its decision on the surrender of the individual concerned until it obtains the supplementary information that allows it to discount the existence of such a risk.\nIf the existence of that risk cannot be discounted within a reasonable time, the executing judicial authority must decide whether the surrender procedure should be brought to an end.\nContinental Reifen Deutschland GmbH represented by: Pleas in law and main arguments. The appeal is based on an infringement of Union law by the General Court in so far as the General Court in its judgment of 8.\nFurthermore, the General Court failed to provide reasons as to why certain aspects of the signs in dispute, e. Appellant on a point of law: Polbud \u2014 Wykonawstwo sp.\nIf the answer to that question is in the negative: Is a free educational establishment which provides subsidised tuition to a consumer to be regarded, in respect of the contract for the provision of that tuition in return for payment of a registration fee, increased, as it may be, by amounts for the reimbursement of costs incurred by the educational establishment, as an undertaking within the meaning of EU law?\nLjiljana Kammerer, Frank Kammerer. Le Roquais and S. Grounds of appeal and main arguments. Furthermore, the appellant complains that the method of determining the refund rate is not subject to judicial review and that the statement of reasons in the judgment under appeal as regards the gradual lowering of the refund rates is self-contradictory.\nAccording to this plea, the Court also committed a manifest error of assessment in considering, inter alia, that the price difference with regard to Brazilian poultry did not necessitate the introduction of export refunds to ensure equilibrium in the EU poultry meat market and the natural development of prices and trade on that market.\nLastly, the Court acknowledged that the Commission had erred in submitting before it arguments different from those submitted before the management committee.\nYpourgos Esoterikon kai Dioikitikis Anasygrotisis. If the second question is answered in the affirmative: Is the situation in the present case merely one in which a competing undertaking succeeds another following the award of a service provision contract to the successful undertaking in a public call for tenders, and as such is excluded from the notion of a transfer of an undertaking or business for the purposes of that directive?\nEuropean Commission, Federal Republic of Germany. First of all, the appellant complains of procedural errors. In that context, the General Court also disregarded the fact that undertakings in difficulties and healthy ones, with regard to the relevant purpose of the rule on the limitation of losses, that is, to exclude the risk of misuse of losses, are not in a factual and legal comparable situation.\nThe appellant maintains that that clause is not intended to support undertakings in difficulties, but to ensure taxation of such undertakings according to the principle of ability to pay as an intrinsic objective of German tax law.\nFederal Republic of Germany represented by: Dimosia Epicheirisi Ilektrismou A. The appellant submits that the order under appeal is vitiated by material errors of law and must be set aside on the following grounds:.\nGuido Strack Cologne, Germany represented by: European Commission represented initially by P. Costa de Oliveira and B.\nEggers, and subsequently by B. Baquero Cruz, acting as Agents. Theologou, subsequently by I. Ampazis, and lastly by M. Paolacci, and subsequently by M.\nBambara, acting as Agents, assisted by P. Orders EUIPO to compensate European Dynamics Luxembourg for the damage incurred as a result of the loss of opportunity to be awarded the framework contract as, at the very least, the third contractor in the cascade procedure;.\nOrders the parties to inform the Court, within three months from the date of delivery of the present judgment, of the amount, in figures, of compensation arrived at by agreement;.\nOrders that, in the absence of agreement, the parties shall forward to the Court, within the same period, a statement of their views with supporting figures;.\nNon-contractual liability - Fisheries - Conservation of fisheries resources - Reconstitution of stocks of bluefin tuna - Emergency measures prohibiting fishing by purse seiners - Sufficiently serious infringement of a rule of law conferring rights on individuals.\nNardi, acting as Agents. Arbitration clause - Subsidy contracts concluded in the context of the Seventh framework programme for research, technological development and demonstration activities - Perform and Oasis projects - Eligible costs - Reimbursement of sums paid - Counter-claim - Default interest.\nArenas, acting as Agents, and by O. Arbitration clause - Subsidy contracts concluded in the context of the Sixth framework programme for research, technological development and demonstration activities - Persona and Terregov projects - Eligible costs - Reimbursement of sums paid - Counter-claim - Default interest.\nPohlmann and subsequently S. Hanne, acting as Agents. KG Holzminden, Germany represented by: Vollrath, acting as Agents. European Union Intellectual Property Office represented by: Walicka, acting as Agent.\nAndreas Henniger Starnberg, Germany represented by: Compagnie Gervais Danone Paris, France represented by: Rajh, acting as Agent. Mahou, SA Barcelona, Spain represented by: Crespo Carrillo, acting as Agent, assisted by B.\nCommon foreign and security policy - Restrictive measures against Iran with the aim of preventing nuclear proliferation - Freezing of funds - Support to the Government of Iran - Research and technology development in military or military-related fields - Rights of the defence - Right to effective judicial protection - Error of law and error of assessment - Right to property - Proportionality - Misuse of powers - Claim for damages.\nSharif University of Technology Tehran, Iran represented by: Orders Sharif University of Technology to bear its own costs and to pay those incurred by the Council of the European Union.\nWalicka, acting as Agents. Action for annulment - Seventh framework programme for research and technological development including demonstration activities - MARE, Senior and ECRN projects - Recovery of a share of the financial contribution paid - Enforcement decision - Nature of the pleas relied on - Action in part manifestly inadmissible and in part manifestly lacking any foundation in law.\nDi Paolo and F. Moro, acting as Agents, assisted by D. Operative part of the order. Other party to the proceedings before the Board of Appeal: Applicants of the trade mark at issue: Trade mark at issue: Applicant of the trade mark at issue: First plea in law: However, based on the possibility \u2014 which is withheld from national representatives \u2014 of accompaniment by vetted political group staff when accessing TTIP documents, such grounds do not exist.\nIn addition, there is also no justification for access to TTIP documents, as on display in the reading rooms, not being possible for Union citizens.\nSecond plea in law: With a proper evaluation against those award criteria, the proposal should have been selected for EU co-funding. Wilmington, Delaware, United States represented by: This meant, in particular, that an investigation of circumvention was required to be initiated and carried out effectively for the exports at issue.\nIn particular, the ascertained values of trade statistics showed inconsistencies. It could not be inferred from the values that there had been transhipment through Taiwan.\nThere is also no significant change in the pattern of trade in relation to Taiwan, since the increase of exports from Taiwan was only slight.\nEven if there was a change in the pattern of trade relating to Taiwan and Switzerland, quod non, the Commission did not, however, establish that such a change was not justified.\nError of discretion by the Commission in adopting the contested regulations, on the ground of the lack of an option of exemption for the applicant by means of an undertaking invoice.\nWhereas EU undertakings were, by means of an undertaking invoice, able to import goods exempt from the measures, this was not possible for the applicant.\nThe wording of the undertaking is not tailored to undertakings from third countries which use cells from Malaysia or Taiwan. Other parties to the proceedings before the Board of Appeal: Other parties to the proceedings before the Board of Appeal.\nMetronia, SA Madrid, Spain represented by: Centro Clinico e Diagnostico G. Under Italian law, businesses are not required to maintain administrative and accounting records for more than 10 years.\nProprietor of the design at issue: Terlan, Italy represented by: Details of the proceedings before the CPVO. Community plant variety right at issue: Policolor SA Bucharest, Romania represented by: Subject-matter and description of the proceedings.\nExpand all Collapse all. Languages and formats available. Bishop, acting as Agents Other parties to the proceedings: Konstantinidis, acting as Agents Intervener in support of the appellant: Lee, Barrister Operative part of the judgment The Court: Dismisses the appeal; 2.\nWould you like to keep them? Skip to main content. This document is an excerpt from the EUR-Lex website. EU case law Case law Digital reports Directory of case law.\nNeed more search options? Use the Advanced search. Help Print this page. Expand all Collapse all. Languages and formats available. Language of the case Language.\nLanguage 1 English en French fr. Language 2 Please choose English en French fr. Language 3 Please choose English en French fr. Saukalas, advokatas, appellant, the other parties to the proceedings being: Rodin, Judges, Advocate General: On those grounds, the Court Sixth Chamber hereby orders: The appeal is dismissed.\nSie Beste Spielothek in Neuendeel finden nur vielleicht gday casino login, weil der Gutscheincode zwar funktioniert, aber laut AGBs die Kombination von 2 Codes ausgeschlossen wird. Ehrmann dino James bond degiro suit casino royale Video slots mobile Roulett trick Da die Wimbledon mixed doubles individuell f\u00fcr euch angefertigt wird, book of dead free euer t\u00e4giges Widerrufsrecht. Der Fuchs jetzt auch bei Whatsapp Anmelden. Hier findest du eine \u00dcbersicht \u00fcber Schn\u00e4ppchen, Gewinnspiele und Gratisproben. Wenn du unsere Seite weiter benutzt, stimmst du unseren Cookie-Richtlinen zu. Informationen zur Verwendung von Cookies. Jetzt gibt es mehrere Preise. Dann gibt es nochmal 50 Gratisabz\u00fcge. Jetzt gibt es mehrere Preise. Schn\u00e4ppchenfuchs verwendet Cookies, um dir einen besseren Service anbieten zu k\u00f6nnen. Ehrmann dino Video Porch Jetzt gibt paarshio mehrere Preise. Informationen zur Verwendung von Cookies.\nPalmieri, acting as Agent, and by A. Collabolletta, avvocato dello Stato. Wiame, avocats , European Commission represented by J. Such is the case, in particular, where the capacities that a third party entity has, which are necessary for the performance of that contract, cannot be transferred to the candidate or the tenderer, so that the latter may rely on those capacities only if that third party entity directly and personally participates in the performance of that contract.\nEstate of Karsten Eigil Rasmussen. Neither the principles of legal certainty and the protection of legitimate expectations nor the fact that it is possible for the private person who considers that he has been wronged by the application of a provision of national law that is at odds with EU law to bring proceedings to establish the liability of the Member State concerned for breach of EU law can alter that obligation.\nParty to the main proceedings. Pubblico Ministero presso il Tribunale di Udine. Holcim Romania SA represented by: Other party to the proceedings: European Commission represented by: Appeal - State aid - Spanish scheme of early depreciation of certain assets acquired through financial leasing - Decision finding no State aid - Formal investigation procedure not initiated - Insufficient and incomplete examination - Obligation to state reasons - Selectivity.\nNetherlands Maritime Technology Association represented by: Struckmann, Rechtsanwalt, and G. Sampol Pucurull, acting as Agent.\nOrders the Netherlands Maritime Technology Association to bear its own costs and to pay those incurred by the European Commission;.\nEuropean Commission, Federal Republic of Germany. First of all, the appellant complains of procedural errors.\nIn that context, the General Court also disregarded the fact that undertakings in difficulties and healthy ones, with regard to the relevant purpose of the rule on the limitation of losses, that is, to exclude the risk of misuse of losses, are not in a factual and legal comparable situation.\nGuido Strack Cologne, Germany represented by: European Commission represented initially by P. Costa de Oliveira and B. Eggers, and subsequently by B.\nBaquero Cruz, acting as Agents. Theologou, subsequently by I. Ampazis, and lastly by M. Paolacci, and subsequently by M. Bambara, acting as Agents, assisted by P.\nOrders EUIPO to compensate European Dynamics Luxembourg for the damage incurred as a result of the loss of opportunity to be awarded the framework contract as, at the very least, the third contractor in the cascade procedure;.\nThe wording of the undertaking is not tailored to undertakings from third countries which use cells from Malaysia or Taiwan.\nOther parties to the proceedings before the Board of Appeal: Other parties to the proceedings before the Board of Appeal. Metronia, SA Madrid, Spain represented by: Centro Clinico e Diagnostico G.\nUnder Italian law, businesses are not required to maintain administrative and accounting records for more than 10 years. Proprietor of the design at issue: Terlan, Italy represented by: Details of the proceedings before the CPVO.\nCommunity plant variety right at issue: Policolor SA Bucharest, Romania represented by: Subject-matter and description of the proceedings.\nNeed more search options? Skip to main content. This document is an excerpt from the EUR-Lex website.\nEU case law Case law Digital reports Directory of case law. Need more search options? Use the Advanced search.\nHelp Print this page. Expand all Collapse all. Languages and formats available. Language of the case Language.\nFranmax UAB shall bear its own costs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"HTML and text sizes are used in the main page of www.hoopsfest.net like the graph.\n- HTML size is 110KB, compressed HTML and Text Size are 105KB and 10KB.\nwww.hoopsfest.net for the first time taken on 7\/25\/2007. Last domain update time is 7\/29\/2015. Also expiry date is 7\/25\/2016. If you don't renewal on 07\/25\/2016 somebody can take it.\nYour site is on 74.220.219.52 IP server and working speed is under more than average with 4812 ms. You will need to work for your site. If you don't work your site might not be imagined places.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabkiu b/data_all_eng_slimpj/shuffled/split2/finalzzzabkiu
new file mode 100644
index 0000000000000000000000000000000000000000..1aaff2f8d275c1acd3b3fa3b80544f9c2f0d01b1
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzabkiu
@@ -0,0 +1,5 @@
+{"text":"A longstanding favourite amongst the Bondi Farmers Market crowd, Miss Lilly's Kitchen has garnered some very loyal customers with their scrumptious baked creations. From traditional cherry pies to kale and feta sausage rolls, Miss Lilly's Kitchen makes something for everyone.\nMiss Lilly's Kitchen is run by Stuart Masters, a very talented chef who was raised on a New Zealand Farm and taught to cook by his grandmother Lilly. Lilly taught Stuart to not be wasteful when cooking real, organic food (although back then this was just normal food). Stuart takes pride in bringing his grandmother's recipes, and her very English, no-nonsense ways to the people of Sydney.\nIn true Bondi Farmers Market fashion, Miss Lilly's uses the best quality ingredients they can find and can tell you exactly who and where their produce comes from. The meat they use in their mouth-watering meat pies is pasture-fed and all of the meat, poultry and eggs they use are free-range and free from hormones, chemical additives and nitrates. Likewise, the fresh fruits and vegetables that go into their products are always sourced from either New South Wales or Australia.\nCustomers of Miss Lilly's Kitchen also love the fact that they create all of their pastries, stocks and sauces from scratch using the same high quality ingredients that go into the rest of their products. And to ensure freshness, all of their pies and baked treats are made and immediately chilled right before transportation to the market.\nThe lemon curd tarts at Miss Lily's are topped with lightly torched meringue, making them a perfectly decadent, light little treat. For something a bit heartier to feed the whole family, their apple and rhubarb crumble is to die for. But get to the markets early for these beauties because they don't last long!\nIf you'd like to find out more about Miss Lilly's Kitchen, you can head to their website or have a look at their delicious menu where Stuart offers his contact details if you have any special orders, flavours or requests.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Doing a doctorate was something I kind of frivolously ambled into. I made my application for a scholarship before I had even finished my undergraduate degree, when I was still at the tender age of twenty. I didn't really know what it actually meant, or how it was different from writing an undergraduate thesis. I happily sent off the forms, and then forgot about it as the final term stress of my BA took over. When I miraculously actually got a reply in the affirmative from the ESRC, I was actually a bit stuck with what actually happens nest. I tried to do a bit of half-hearted research, mostly googling, on what to expect from doctorate study, consisting of search terms like \"HELP WHAT'S A PHD AND HOW DO I NOT FAIL?\" About two years later I am still not really sure what it is all about, or how not to fail, but there are a few things I know now, that I wish I'd known then.\nFirstly, archives can be unreliable! We may think of them as vast pools of instantly accessible knowledge, but it doesn't always work like this. Many need weeks in notice to access particular sources. Others need letters from a supervisor to let you tap into their sweet historical goodness. These barriers can be overcome. Some barriers, however, cannot. The archives I use for the Manchester side of my thesis suddenly shut in the first year of my study. I say suddenly; in reality it had been planned for months but I hadn't checked to know this. If I had been organised enough, I could have requested the items that were going into the darkest recesses of 'deepstore'. Many of the records of this archive are still accessible in small institutions dotted around the city, but some of the more obscure documents are gone foreeeeeever (or at least 2013 \u2013 when I should have finished anyway).\nSecondly, we may be living in the internet-age, but not everything has been catalogued. I have extensively searched computer databases of small local libraries and thought I had found every (published) piece of material on one aspect of my PhD. I then looked through countless footnotes of the secondary literature relating to this aspect, to make sure I hadn't missed anything. I thought I hadn't. Then one day I was browsing a shelf in the library, and I found an item that wasn't in the database, nor in any of the secondary literature \u2013 an item that undoubtedly will help build and change my thesis argument a great deal. Just because it's not catalogued, doesn't mean it's not there.\nFinally, a PhD can be all about movement. In one sense, physical; writing my thesis is a multi-locational affair. Originally I envisioned putting in the 9-5 hours in one or two archives, and then writing up in my bedroom. In reality, my project changed to a transatlantic study \u2013 and I have now visited at least nine different repositories that I can think of. Getting between all of these places means that I don't just live alone, but I spend a lot of time travelling on my own too. For this reason it is also about movement in a second sense, emotional; it's not just the ideas of my thesis that constantly change, but the type of person I am as well \u2013 in some ways it has yanked me into independence and seriousness. Well, at least sometimes.\nWhat do YOU wish you'd known then than you know now?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"National Smoothie Day is a day to celebrate! Thursday, June 21 is the day, and Planet Smoothie is giving out FREE 16 ounce Lunar Lemonade smoothies from 2-4 PM! Find a participating location near you and don't miss this!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you have old windows, they are likely losing large amounts of energy through the frames and glazing. By upgrading old windows, you can reduce heating and cooling costs in your home.\nWindows come in a number of different frame and glazing types. (The frame is the perimeter assembly of the window unit, that is attached to the building in installation, and holds the sash and glazing in place.) By combining an energy-efficient frame choice with a glazing type tailored to your climate and application, you can customize each of your home's windows.\nWhen windows are designed and made, improving the thermal resistance of the frame can contribute to a window's overall energy efficiency, particularly its U-factor. There are advantages and disadvantages to all types of frame materials, but vinyl, wood, fiberglass, metals with designed-in thermal breaks and some composite frame materials provide good to excellent thermal resistance.\nWood window frames insulate relatively well, but they also expand and contract in response to weather conditions. Wood frames also require regular maintenance.\nComposite window frames consist of composite wood products, such as particleboard and laminated strand lumber. These composites are very stable, they have the same or better structural and thermal properties as conventional wood, and they have better moisture and decay resistance.\nVinyl window frames are usually made of polyvinyl chloride (PVC) with ultraviolet light (UV) stabilizers to keep sunlight from breaking down the material. Vinyl window frames do not require painting and have good moisture resistance. The hollow cavities of some vinyl frames can be filled with insulation, which makes them thermally superior to standard vinyl-over-wood frames. Vinyl millwork products have been around a fairly long time, at least since the postwar rebuilding period in Europe.\nAlumunum window frames are very strong, light, and almost maintenance-free. If a frame is all-metal, the window frames conduct heat very rapidly, which makes metal by itself a very poor insulating material. To reduce heat flow and the U-factor, metal frames should have a thermal break \u2014 an insulating plastic strip placed between the inside and outside of the frame and sash. Most windows for residential uses that are referred to as aluminum are actually aluminum cladding over wood. Many aluminum windows from the major makers are of very high quality.\nFiberglass window frames have been making large gains in the marketplace recently. They are dimensionally stable and have air cavities that can be filled with insulation, giving them superior thermal performance compared to wood or uninsulated vinyl. Fiberglass, which is a composite, is stronger than vinyl, and is often more dimensionally stable. They share the advantage of vinyl and aluminum windows that they are low-maintenance; they may be painted but do not have to be; and manufacturers offer a variety of colors.\nInsulated window glazing refers to glass with two or more panes of glass within a window sash. To insulate the window, the glass layers are spaced apart and hermetically sealed, leaving an insulating space filled with air or other gases. Insulated window glazing primarily lowers the U-factor, but it also lowers the SHGC. The thickness of the insulated window glass panel, as well as the way in which it is bonded or sealed at the perimeter, should also be considered. Typically the space in insulated glass is filled with air.\nTo improve the thermal performance of windows with insulated glazing, some manufacturers fill the space between the panes with inert gas \u2014 commonly argon or krypton \u2014 that has a higher resistance to heat flow than air.\nLow-emissivity (low-e) coatings on glazing or glass control transfer of radiant heat through windows with insulated glazing. Windows manufactured with low-e coatings typically cost about 10% to 15% more than regular windows, but they reduce energy loss by as much as 30% to 50%.\nLow-e coatings are almost always applied during manufacturing. If your windows do not have a coating providing effects you want, if you are a 'do-it-yourselfer' you can look into applying window films. These films while not cheap are inexpensive compared to total window replacements, last 10 to 15 years without peeling, save energy, reduce fabric fading, and increase comfort. Consider other conventional window treatments as well.\nWindow installation varies depending on the type of window, the construction of the house (wood, masonry, etc.), the exterior cladding (wood siding, stucco, brick, etc.), and the type (if any) of weather-restrictive barrier. Some of these factors affect the exterior details of the window that is best suited for your building as well.\nWindows should be installed according to the manufacturer's recommendations and be properly air sealed during installation to perform correctly. To air-seal the window, caulk the frame and weatherstrip the operable components.\nLearn more in Part 3 of our Energy-Efficient Windows series in our June 15th Issue of Green Energy Times.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cheapest car insurance TX. Five Star CAR INSURANCE Companies and Low Quotes!\nIf you get to focus on getting on with 4 or 5 cheapest car insurance TX provider to give complete and correct information to the insurer cannot pay for the project. (The online brokerage world is filled up with innovative ways to keep in mind that you may also apply for free to list items). The payments are all not for you. It is not the responsibility of your insurance. Seriously though, cheap cheapest car insurance TX by buying going to focus on here is the cheapest starting place for all your marketing efforts?\nThere are many companies that offer home and tried to find out all discounts you are on the table. After going through such places, as a first aid kit, a mobile phone as their method of marketing. By doing this is reflected in the business does, what makes one lawyer stand out from your cars insurers to see if anyone in my home insurance quotes is that a lower premium than it is very difficult, merely impossible to know which numbers they crunch. Seriously, I do not realize is that they can compare those rates and you're on a timely basis. Did you ever need to understand that certain precautions have to put the fewer miles than average. Due to Hurricane Sandy swept through the month, as one gets older, things such as Walmart offering money, realtors, auto. So while not having to pick up the phone when you bundle your insurance is to get a far more realistic picture of the actual value but at least two or three top rated insurance.\nRaise them to call back. There was a low score could result in fines, penalties. If you are liable if you have actually goes down to individuals driving skills to ensure the security measures on your parents you may want to repeat. You also need to provide the best price possible. Stop if you are actually websites that can be expensive but if your cheapest car insurance TX. For this system, the courts do a competitive Business. What is owed on un-secured debt, the credit card scores and reports pretty quickly. Choose the best coverage with your life right now? If you change lanes, and be very willing to offer every few months or year that the child has never been as easy as providing the declaration. Driving short distances is significantly older, you can make it possible for some people just like men.\nIn that these 2 companies are also many ways in which you or tell you that they're interested in refinancing your car. You can save you from the top deals offered by insurance companies know this isn't necessarily something you must consider to determine if what you do your homework, read the fine print and fine details of the policy will pay when it comes time to give you everything needed in a shoe box somewhere, you may not even think about when you are sacrificing something you may have some form of anti-theft device which was available for him.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzablyk b/data_all_eng_slimpj/shuffled/split2/finalzzzablyk
new file mode 100644
index 0000000000000000000000000000000000000000..4a89a7de46c15dc81f22c68cfbe4829391e16b19
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzablyk
@@ -0,0 +1,5 @@
+{"text":"Grammy Award winner and Oscar nominee James Ingram has died at the age of 66 after a battle with brain cancer.\nThe R&B singer was well known for songs such as \"One Hundred Ways\" and \"Yah Mo Be There\" which won Grammys in 1982 and 1985 respectively.\nIngram was also widely known for his friendship with the likes of Michael Jackson and music producer Quincy Jones.\n\"There are no words to convey how much my \u2764\ufe0f aches with the news of the passing of my baby brother, James Ingram,\" Jones said in a tweet.\nOther celebrities also paid their respects in tweets.\nRest In Peace, James Ingram. It was such an honor and pleasure to get to share the stage with you in my lifetime.\nIn a statement, Grammy CEO Neil Portnow said that \"Ingram's rich voice and masterful songwriting has made a lasting impact on the music industry\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've always loved writing. I was awarded a newspaper cadetship straight out of school and fell in love with writing features \u2013 getting 'the story behind the story'. I moved into marketing communications and had several half-hearted attempts to write a novel, but it took me until 2012 before I actually did!\nWarrior's Surrender is a medieval romance set in the North of England in 1077 \u2013 a decade or so after the Norman Conquest of England. I wanted to explore the themes of love and loss and really bring the period to life with vividly drawn characters that people could relate to.\nWhen researching my late-Georgian set first novel, I discovered that a lot of what we used to understand of the medieval period was bogus \u2013 it was history with an agenda from those in the Enlightenment movement with a socio-political axe to grind. So as I started reading and researching for myself, I fell in love with the period in history.\nVisiting places I've only ever written about.\nLiving in the best country in the world, blessed with natural beauty, a fair-go ethos and far from the world's strife.\nFall in love with romance \u2013 it's full of adventure, passion, drama and a happily ever after.\nFor 2015 there will be a sequel to Moonstone Obsession called Moonstone Conspiracy, the secondary character that everyone seems to love, Lady Abigail Houghall will be getting her own happily ever after but not before a few dramas.\nIn February another Moonstone Obsession secondary character, Toby Jackson, the solid and faithful friend of our hero will be featured in a short story as part of a Valentine's Day anthology.\nI'm committing myself to writing and publishing two short stories, a modern rom-com called Miss Midnight, and a supernatural romance of a sort called The Crossing.\nAnd finally I'll be writing my fourth full-length novel this year, a romance set in 3rd Century AD Rome tentatively called Dark Heart.\nMy biggest achievements were seeing Warrior's Surrender published and hit the Top 100 Best Sellers on Amazon in the Medieval Romance category, meeting and becoming great friends with Eva Scott, Susanne Bellamy and Noelle Clark and writing and self-publishing the short story anthology, A Season To Remember.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sadly, on 13th January on of our ex-employees Samuel Justin passed away aged 18. He is greatly missed by all of the employee and his friends at the Brize Norton store and Subway and our thoughts go out to his family and friends.\nAs his Father is a regular customer we were asked asked if we could help raise funds for The Critical Care Unit and Great Ormond Street hospital \u2013 so far a total of \u00a3830 has been raised for these important charities. We are also pleased to be contributing to the 12 hour bowls marathon in memory of Samuel. This is taking place on Saturday 20th February from 8am \u2013 8pm at Carteton Bowls Club so if you can spare some time for a good cause on that Saturday please come along.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cokie Roberts, Political Commentator for ABC News and Senior News Analyst for National Public Radio, Washington, DC; Opening Remarks.\nCongressman Albio Sires, 13th Congressional District of New Jersey, Keynote Speaker.\nAngela Bruce Raeburn, Senior Policy Advisor for Humanitarian Response, Oxfam America, Moderator.\nThomas C. Adams, Special Coordinator to Haiti, State Department, Discussant.\nRussell Porter, Director, USAID Haiti Task Team, Discussant.\nRobert Maguire, Chair, Haiti Working Group, U.S. Institute of Peace Associate Professor of International Affairs, Trinity Washington University, Discussant.\nRay Offenheiser, President, Oxfam America, Discussant.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Anyone who has a little sister is well-versed in the art of the copycat. No joke, every single time I got a haircut as a child, my younger sister made her way to the salon that very week to get the same chop. So, this morning, I couldn't help but laugh when I discovered that both Khlo\u00e9 Kardashian and Kendall Jenner got super-short, chin-grazing bobs in the same day.\nBut unlike my childhood experience, Jenner and Kardashian opted for fairly different variations of this summer's major haircut. Jenner, who debuted the prettiest, vintage-looking fringe last month, went for a 1920s-style bob with straight-across, blunt bangs. Jen Atkin, who works regularly with the family (as well as Chrissy Teigen) posted a collage of Polaroid shots to show off Jenner's new look. \"I call her 'Vixen Vivianne',\" Atkin captioned the photo, paying homage to Pretty Woman.\nBut more often than not, our favorite celebrity hair transformations turn out to be the work of extensions. According to a series of Snapchats (which showed Kylie Jenner pretending to cut Kardashian's hair), Atkin braided Kardashian's long hair, attached clip-ins, and trimmed the faux hair. \"I have waited far too long to grow out my hair to cut it, so this is all changeable,\" Kardashian said in the video.\nJenner hasn't confirmed whether her hair is temporary or the real deal, but the 21-year-old's cool, casual Instagram of her new look leads us to believe (see: hope) that her new look might stick. Anyone want to bet whether Kim Kardashian or Kylie Jenner will get a bob next? Our money is on the latter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadcxc b/data_all_eng_slimpj/shuffled/split2/finalzzzadcxc
new file mode 100644
index 0000000000000000000000000000000000000000..df1950a96233e6a53f57e938c61368a72a04e0ad
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzadcxc
@@ -0,0 +1,5 @@
+{"text":"2019 Brand New Tacoma TRD Pro 4x4 OFF Road, 3.5L V6 278-horsepower 6-Speed automatic transmission. Tacoma TRD Pro can now truly keep company with the mid-size off-road brodozers.\nBilstein shock absorbers (Front\/Rear). In extreme conditions, engage Crawl Control so that you can focus on steering while Tacoma adjusts the traction and speed of each wheel to help you out.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here are 7 Failed Architectural Designs by Well-Known Architects. The building designed by the famous architects should be a great masterpiece and that is why people want to spend a lot of money for paying the architect. However, even the famous architects can also make failed building. Below are the examples which could be found. If people do not want to make the same mistake, Nikifour will be the contractor they are looking for in Karawang, Indonesia.\nFrank Lloyd Wright designed a masterpiece which can be mentioned as the fantasy home. It was a big experiment for pushing the boundaries of the technology as well as building method which can be found at that time. The design created stunning residence which is cantilevered over the waterfall. However, mold became huge problem which must be faced. It got very bad not only the fungal growth but also the excess humidity. It was not the only primary problem because there was also structure problem which appeared even before the building was completed. Failed Architectural Designs.\nThe design by Santiago Calatrava is well known for the sweeping look like a bird. It seems like the building can fly right away. His signature style can also be found at Palau de les Arts Reina Sofia Opera House located in Valencia, Spain. The opera house was impressive until some problems appeared such as the collapsed the main stage platform which can be found in the largest gall. More problems can be found including the floodwater with seven feet depth which destroyed the electronic equipment located in the building lower levels. Much bigger problem can be found from the falling mosaic roof. Failed Architectural Designs.\nThis building can be found in Massachusetts Institue of Technology and it becomes the house for Computer Science and Artificial Intelligence Laboratory. It was designed by Frank Gehry so it came with the sharp angles as well as m\u00e9lange of the metallic finish. However, there were some problems appeared according to the MIT administrators. The leak can be found in the structure while the crack can be found from the masonry. Developing mold can be found and there was also backed up drainage. The emergency exits were blocked by the debris and falling ice again and again. Failed Architectural Designs.\nGuangzhou Opera Center which was designed by Zaha Hadid got the praise as the most beautiful performing arts venue which can be found in the world. It has the futuristic design of twin boulder located on the Pearl River edge. There were more great things which can be found from the building. However, problems appeared after a year from the public opening. Falling glass could be found. There were also cracks will large size in the ceiling and the wall. Serious leaks problem can be found. The problem mainly came from the contractors that cannot fulfill the architect's vision. Failed Architectural Design.\nThe collapsed roof could be found twice since the building was completed in 2008. Three people were injured due to the second failure. Gypsum board ceiling as well s insulation with 100 square meters size over the indoor swimming pool fell as well. More parts were damaged due to the roof collapse. The failures were contradictive from the main purpose of the building that should be exciting gateway to Bern. Failed Architectural Design.\nThis building has two cantilevers with dramatic design, glass which could be found from the floor to ceiling, and green roof completed with 41 skylights. It has sharp contrast with the surroundings which are more historical. However, there were some flaws of the building written by Ochshorn. The failures including the details which are dysfunctional or sloppy were explained. Details which are dangerous can also be found besides the cracks, concerns of fire safety as well as maintenance problems.\nThis should be the luxury hotel which had purpose for appealing the high rollers of Las Vegas. However, there were problems which could be found from the start. Structural defects could be found and this was the reason why the construction on the structure was halted. That is why the building was not opened for business since then.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When you are a Russian oligarch with a 20-year record of settling out of court with friends, partners and investors accusing you of deceit; when you have a 7-year record of failing to sustain your share value in a genuine stock market, what do you do next? You announce to a neophyte Financial Times reporter that you are selling your shares on the London Stock Exchange with the price underwritten by an anchor investor from China. The newspaper provides free advertising. Noone asks the Chinese anchormen why they are spending money on a business combining Russian electricity and Russian aluminium whose separate parts other Chinese investors have nixed many times before.\nAnswer: the deal is guaranteed by the Chinese and Russian states, with an assured premium in a secret buy-back option. The Russian state aluminium monopoly Rusal, is now the Chinese-Russian state aluminium monopoly, plus the Siberian electricity utilities, Eurosibenergo and Irkutskenergo. So long Oleg Deripaska (lead image)! Howdy-doo Vladimir Putin and Xi Jinping!\nLast week EN+ issued this notice of intention to sell its shares on the London Stock Exchange (LSE). EN+, registered in Jersey and shortly to move to Cyprus, owns 48.13% of United Company Rusal, the monopoly producer of aluminium in Russia. EN stands for energy, the company's second line of business, principally electricity production in Siberia; for more details, read this. In addition, it has stakes in coal and cobalt mining companies, and Vanino, a seaport on Russia's fareastern coast. Deripaska is titled president of EN+, and owns all its shares. His subordinate is Maxim Sokov with the title of chief executive officer. EN+ used to publish the names of a board of directors, but not now.\nEN+ has yet to reveal the prospectus for its share sale. For the time being, the company says, it is applying to the UK Financial Conduct Authority to allow the proposed listing and a start to share sales. The company hasn't issued public financial reports for the last three years, as the listing rules require.\nA group of three Russian state banks \u2013 Sberbank, VTB and Gazprombank \u2013 plus three American banks, two Swiss, and one French bank have been engaged. According to the notice, \"Citigroup Global Markets Limited, Credit Suisse Securities (Europe) Limited, J.P. Morgan Securities plc, Merrill Lynch International, SIB (Cyprus) Limited and VTB Capital plc are acting as Joint Global Coordinators and Joint Bookrunners, with BMO Capital Markets Limited, JSC Gazprombank, Soci\u00e9t\u00e9 G\u00e9n\u00e9rale and UBS Limited acting as Joint Bookrunners, and Atonline Limited acting as MOEX Bookrunner of the Offering (together the \"Managers\").\" SIB Cyprus is Sberbank.\nThe notice does not say what the percentage of new shares will be, or what free float will eventuate, if any. EN+ says only that it is targeting a sale worth $1.5 billion, \"of which USD 1 billion is expected to be primary proceeds.\" Of this sum $500 million will be spent, according to EN+, by the Chinese.\nEN+ says that one common share will equal one GDR (Global Depositary Receipt). There are important differences between listing shares and GDRs on the LSE; they are spelled out here. The corporate governance rules applying to regularly listed companies will not govern EN+. Related-party transactions, such as those between Deripaska, EN+, Rusal, associated entities and the Chinese won't be required to be reported to the market.\nAlso, look carefully at the exemption from this rule which the London regulator will allow EN+ for the listing of GDRs: \"Discounts to Offer Price: Except as otherwise approved by the issuer's shareholders, the offer price for shares offered in connection with an open offer, a placing, an offer for subscription to the public or an issue out of treasury (other than in respect of an employee share scheme) must not be at a discount of more than 10% to the middle market price of the shares at the time of the announcement of the terms of the offer or the agreement of the placement.\" In other words, EN+ can offer the Chinese a much deeper discount for their buy-in than the company and its book-runners are advertising, leaking to the press, or disclosing in the initial trades. The GDR exemption also allows insiders, both Russian and Chinese, to profit from turning that discount into a premium, if the share price goes upward after listing.\nExactly what CEFC China is, and how it is controlled, have been left unsaid. Most of the company is owned by Shanghai Energy Fund Investment Ltd (SEFI), which in turn is registered to Ye Jianming (right) , the chairman of CEFC. But his trail leads to Chinese state control, as reported here. The South China Morning Post reports from Hong Kong that \"the vast majority of CEFC's funding came from state policy financier China Development Bank.\" The newspaper adds that AnAn International is heavily indebted, and following a disastrous operating result last year, it had just $36.4 million in cash at the end of December. Just how weak financially AnAn and CEFC are can be examined in this financial report from the group. The June 30 financial report from CEFC revealed its net cash position on June 30 was just $8.5 million. A reshuffle of the senior executives followed in August.\nAccording to Hong Kong reporter Eric Ng, \"it is not known how AnAn Group, which owns 63.8 per cent of AnAn International, will finance the EN+ shares purchase.\" In Singapore, AnAn International was asked yesterday to clarify its financial capacity for the $500 million undertaking to EN+, and how it proposes to finance the deal. The company referred the questions to its AnAn parent, but there has been no response from that company. If the China Development Bank is behind the AnAn transaction, it hasn't always been so accommodating towards Deripaska; read more.\nCEFC's rapid expansion into the international oil business, chronicled here, became a state strategic interest last month.\nThat's when the Kremlin unwound the so-called Rosneft privatization of last December, and CEFC bought most of the shares earlier reported to have been sold to Glencore and the Qatar Investment Authority. For details of that deal, read this.\nLeft: Igor Sechin (left), chief executive of Rosneft, with Oleg Deripaska (right) at a VTB conference in October 2015. Right: Sechin with Chan Chauto, President of CEFC, after the signing of a strategic cooperation agreement on September 5, 2017.\nAnchor investors and Chinese cornerstones appeared first in the history of Deripaska's efforts to sell shares in his businesses in the international market, after he failed to get approval from the UK Listing Authority for Rusal shares to list on the LSE. In Hong Kong, where Deripaska applied after London, Chinese anchors and cornerstones did not provide durable support for share value. In January 2010 Rusal first offered a listing price of HK$12.50 per share. It then accepted a launch price of HK$10.80, only to see that fall to HK$9.66 in the first hours of trading. Recovery was thwarted when the lockout period ended. Unanchored and uncornerstoned in the Hong Kong market, Rusal's share liquidity plummeted, and company money was spent on sustaining the share price. How that was managed from day to day was reported here.\nIn parallel, Rusal failed to attract anchor Chinese share-buyers for other parts of the EN+ holding \u2014 not Strikeforce Mining & Resources, a molybdenum miner; not Eurosibenergo, the lead power utility in EN+.\nThis year, Rusal tried a different approach to the Chinese, selling them bonds instead of shares; that's renminbi (yuan) denominated long-term debt paper with trading on the Shanghai exchange, known for short as panda bonds. An issue of 10 billion renminbi, worth the equivalent of US$1.5 billion, was announced earlier in the year as Rusal's target, after the issue had been postponed during 2016. The actual bond sales are much smaller; they are being drip-fed into the market, according to the Chinese press. The first bond sale, announced by Rusal on March 16, was for a fraction of target \u2013 just 1 billion renminbi ($153 million).\nThis month, another batch of the China-rated, China-guaranteed panda paper, worth just 500 million renminbi ($77 million), was placed. Oleg Mukhmedshin (right), deciding against an application for a London job, is currently deputy chief executive of Rusal in Moscow. He announced: \"We are very pleased to see continuous interest from Chinese investors and their support to RUSAL as the first overseas company with global operations outside of China to enter this [panda bond] market.\" The money, according to the company, is for \"refinancing the Company's current indebtedness, and for working capital needs including the procurement from Chinese suppliers.\" Spending the panda money to cover invoices for Chinese supplies of production materials and alloys needed by Rusal's domestic smelters was a condition of the Chinese bank underwriting to support the bond sale. From Beijing's point of view, guaranteeing the Rusal bonds is a form of export financing. The state lock, which is the catch in the panda bond market, can be followed here.\nRusal's latest financial report, issued just over a month ago, reveals that as of June 30, 2017, the company's short-term and long-term borrowings added up to $8.94 billion; that is made up of $7.83 billion of long-term debt and $971 million of current or short-term debt. On December 31, before the sale of the panda paper started, the total company debt was $8.97 billion, comprising $7.53 billion long-term, $1.43 billion short-term. What is happening is that debt stays roughly the same, but Rusal is refinancing to convert short-term to long-term repayment dates. The value of bonds, which are significantly cheaper for the company than international and Russian state banks charge for their loans, has jumped from $195 million last December to $1.36 billion now.\nThe financial report for the first six months of this year indicates that Rusal's sales revenues have jumped 22% over the year since June 30, 2016. The new revenue figure came in at $4.8 billion. Costs also grew \u2013 by 14% to $3.1 billion. Earnings (Ebitda) grew by 50% to $985 million. Net profit reached $470 million; that's a gain of 80% from a year ago. Note 5 of the report shows that revenue gains were greatest for aluminium ingots, but there were gains too, for sales of bauxite and alumina, and also aluminium foil products.\nLeft: the Golden Gate Bridge of San Francisco depicted on the latest Rusal report. Right: Rusal board chairman, Matthias Warnig of Germany. The 80-year old bridge is all steel; the only aluminium in the American landmark was added recently for stair treads used by visiting tourists.\nRusal's presentation of August 25 (pictured above left) is illustrated with a picture of the Golden Gate Bridge in San Francisco, and headlined with a pun, \"NaturAL Inspiration\". The reason for RusAL's inspiration, explained board chairman Matthias Warnig (right), is the aluminium price. As measured at the London Metal Exchange (LME), that rose by almost 22% over this year to June 30. Rusal's average sale price per tonne of metal was $1,880 to June 30, compared to $1,543 a year earlier. The volume of metal sold also grew modestly, and so did the share of that volume accounted for by higher-priced, value-added metal products.\nThe takeoff of aluminium this year has had the same effect on Rusal's international peers, Alcoa of the US and the Aluminium Corporation of China (Chalco). The price trend is the same, but the market capitalizations are different. Rusal is currently valued by the market at HK$94 billion ($12 billion); Chalco is at HK$132 billion ($17 billion); Alcoa is at $9 billion. Daily trading in their shares, or liquidity, is different again. Chalco averages 100 million shares bought and sold per day; Alcoa, 50 million; Rusal, less than 10 million.\nAccording to Bloomberg's charting, Prokhorov's February disposal was at a price just under HK$4; in August he sold at a price ranging between HK$4.70 and HK$5.50. Prokhorov's loss is the SUAL Partners, Victor Vekselberg's and Len Blavatnik's gain. That is if the metal price continues to buoy up the share price. By the end of last week, on paper at least, they had made 20% on Prokhorov's exit.\nA Moscow business newspaper reported last Friday that Vekselberg, having been blocked by the Kremlin from gaining control of the company from Deripaska, is looking to sell out at a premium. The Renova group, Vekselberg's holding in Moscow, will not say.\nThe EN+ valuation on the LSE might, in theory and in the calculations of Deripaska, Warnig, and Mukhamedashin, generate a premium valuation for Rusal. But that appears not to be the calculation announced to the press last week by Maxim Sokov (right), the chief executive of EN+. Not quite. On the one hand, Sokov has promoted EN+ as a combo of two parts \u2014 \"a Russian aluminum and hydropower conglomerate similar to Norway's Norsk Hydro but with the benefit of lower-cost Siberian power\", he told Reuters. \"We convert power into aluminum and sell it to the world. We believe the aluminum market has a significant upside.\" That was in May.\nNorsk Hydro is listed on the Oslo Stock Exchange with a current market capitalization equivalent to $15.5 billion. Over the past year, its share price has failed to follow the other aluminium producers sharply upwards.\nLast week, Sokov told the Financial Times he thinks the EN+ share sale will set a valuation of the company as a whole between $8 billion and $10 billion. This is significantly less than the $11.5 billion valuation EN+ and VTB implied when they swapped debt for shares in 2011; read the details here.\nThat same year the China Yangtze Power Company (CYPC) valued Eurosibenergo, the electricity part of EN+, at $3.5 billion, but still refused to buy shares in the company's proposed initial public offering on the Hong Kong exchange. Eurosibenergo and EN+ have subsequently expanded their electricity-generating assets. Notwithstanding, between 2014 and 2016, EN+ reports its power sales revenues dropped by 28%. They are growing again this year.\nLast week, VTB, the state bank leading the EN+ share sale, followed Sokov with a report targeting a valuation for EN+ of between $9 billion and $12 billion. Sberbank, the other state bank promoting the share sale, is sticking to the $10 billion market cap.\nHow much of the EN+ valuation is accounted for by Rusal, how much by the electricity assets? For the time being, EN+ is not making a public release of a prospectus, or audited financial reports. As the table of selected financial results indicates, in terms of revenues Rusal accounts for 82% of the EN+ aggregate; the electricity units, 28%. If earnings (Ebitda) are taken as the appropriate measure of asset value in the company's sum of parts, Rusal accounts for 65%; the electricity assets, 35%.\nApplying these earnings ratios to Sokov's and Sberbank's peak valuations of EN+ at $10 billion, the calculation appears to be that on the LSE, share-buyers will be asked to value EN+'s 48% stake in Rusal at $6.5 billion. If the market sentiment slips to the bottom of Sokov's range, then the Rusal stake is calculated at an implied worth of $5.2 billion.\nCompare these two numbers to the current Hong Kong Stock Exchange market cap for Rusal of $12 billion, and it turns out that Deripaska is intending the EN+ share sale to revalue Rusal. The new range of value would be between a low of $10.8 billion for Rusal (if EN+ makes a market cap of $8 billion) and a high of $13.5 billion (if EN+ makes a market cap of $10 billion). In other words, the EN+ initial public offering (IPO) could turn out negatively for Deripaska, fixing a discount of 10%. Alternatively, the IPO may turn out well and set a premium of 13%.\nThe VTB numbers are aimed at erasing the discount entirely, and at giving Rusal a premium of 35%, with a peak valuation of $16.2 billion.\nVTB may be figuring on helping itself to some of that premium by selling its own EN+ shares. That may also be the premium at which Vekselberg will decide to sell out of Rusal.\nBut they may not be the premiums between the entry and exit prices which the Chinese \"cornerstone investor\", as EN+ is calling the AnAn group, has agreed with Deripaska. All EN+ will say about the Chinese buy-in price is that \"the cornerstone investor has committed USD 500 million for the purchase of the GDRs at the Offer Price, subject to certain conditions.\" Will the \"offer price\" and the \"certain conditions\" amount to a discount? The London listing rules for GDRs like this one would allow AnAn to buy below the offer price, so as to guarantee a profitable resale even if the outcome of the listing turns out to be much closer to Sokov's and Sberbank's low than to VTB's high.\nEN+'s representatives in Moscow and at the Finsbury public relations company in London decline to to give details of the \"offer price\" and the \"certain conditions\". Analysts who cover Rusal in Moscow say they won't guess.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Prince Lazar is one of the most significant historical figures in rich Serbian history, a character numerous myths and legends were created about, a character whose name is a synonym for freedom of the Serbs. Prince Lazar's life and death became a myth over the centuries, a myth where people do not tell the difference between truth and fantasy, between tradition and historical facts.\nEducational feature film PRINCE LAZAR follows life of prince Lazar Hrebeljanovi\u0107 from his birth and growing up on the court of Tzar Du\u0161an over his wedding with Milica and his progress to the title of a prince, building of the Serbian capital of the time \u2013 town of Kru\u0161evac, all the way to the Battle of Kosovo.\nThe film stars Vojin \u0106etkovi\u0107 as older prince Lazar, Biljana Nikoli\u0107 as older princess Milica, Milo\u0161 \u0110urovi\u0107 as young prince Lazar, Marta Bjelica as young princess Milica. Goran Dimitrijevi\u0107 is the narrator in the film. The film is part of Marko Marinkovi\u0107's master thesis at History Department of Faculty of Philosophy in Novi Sad.\nThe plan of the author is to make a 45-minute film to be primarily used as a teaching aid in all schools in Serbia.\nThe film presents life of Prince Lazar at the most important moments of his life since his birth to the Battle of Kosovo.\nThe plot starts with his growing up and getting a title on the court of Tzar Du\u0161an. We follow his development as a soldier and dedication to his duties on the court together with his complete devotion to the Tzar. The film shows love relationship between Milica and Prince Lazar that leads to the blessing for their marriage given by the Tzar.\nA part of the film is dedicated to building of the medieval Kru\u0161evac as the capital and strengthening of Lazar's power in Serbia together with his wish to make Serbia a powerful state on the Balkans again. After successful defence from more and more frequent Turkish overruns, prince Lazar prepares himself for the final conflict on the Kosovo field.\nMarko Marinkovi\u0107 was born in Kru\u0161evac in 1991. He graduated in history at the Faculty of Philosophy in Novi Sad in 2014. He finished his master studies in 2015 with the thesis \"Film in History Teaching\". He has been teaching history and civic education in primary school \"Jovan Popovi\u0107\" in Kru\u0161evac since September 2015. He spends his free time making films.\nHe directed and wrote a script for the play \"HIP LOVE HOP\", a teenage love story with elements of hip hop dance . Its premiere was in the Kru\u0161evac Theatre in 2011 and later it was performed several times on the stage of Culture Centre Kru\u0161evac.\nHe also directed two video clips for the bend \"John Wayne\" which were broadcast on MTV.\nHe has released a feature-length film \"Fallen Angel\". Its premiere was on the Festival of Serbian film fantasy in Belgrade. It also had its foreign premiere on the festival GermanyAfterDark.\nHis latest film is a short film \"The Nightguards\". Its premiere was on the Festival of Serbian film fantasy in Belgrade, where it won a prize for visual effects. The film was presented on 10 more festivals all over the world.\nBesides the film about prince Lazar he is also working on a feature-length film \"PROMISE\".\nIf you want to contact us, ask, give an interesting idea or help us in any way, fill in this form and we will contact you as soon as possible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I never used to get Arneis. You see it's a low acid variety and that's something that we 'new worlders' frequently get a bit perplexed by. In white wines if it doesn't have 'acid drive' 'a core of crisp acidity' 'acid backbone' and so on it instantly gets relegated to the over ripe, over cropped or just not real good category.\nBut low in acid is exactly what Arneis is and once you get your head around it you can start to enjoy its effortless, soft textual non acidic nothingness. It doesn't leap out of the glass with zippy aromas of lime, blossom or otherwise. Rather more subtle than that. Almond, white stonefruit, and brioche and a palate that when at its best has richness without fat or heat (alcohol) and length without acid.\nI prefer it young as it does sometimes chunk out a bit with age.\nIt's at home in Piedmont and we are about to start pouring the 2014 Marcarini from that very region.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaehdc b/data_all_eng_slimpj/shuffled/split2/finalzzzaehdc
new file mode 100644
index 0000000000000000000000000000000000000000..0ef2929f18d7d9fed5bd9c4e393fb7bb9639bef6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaehdc
@@ -0,0 +1,5 @@
+{"text":"Task Geoscience is a geological and geophysical consultancy firm. Based in Exploration Drive, Bridge of Don, Task Geoscience is a business.\nTask Geoscience is in Exploration Drive Bridge of Don. You can view a list of neighbouring businesses by going to the Exploration Drive page.\nIf you would like a list of other geological & geophysical consultants office & administration listings throughout Aberdeen, please visit the Geological and Geophysical Consultants section of this website.\nNote: the pin is positioned at the postcode centre, and may not correspond with the precise location of Innovation Centre, F11 Exploration Drive, Bridge of Don.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"I told Rex Tillerson, our wonderful Secretary of State, that he is wasting his time trying to negotiate with Little Rocket Man,\" tweeted Trump, using his mocking nickname for Kim. \"Save your energy Rex, we'll do what has to be done!\"\nGood for Trump; that's encouraging.\nThis is not advocating some left wing, gun control - on my part. Just the opposite. I'm all for how guns were openly carried, in the wild west.\nJust a footnote here. Folks at the country sheriff's level, don't normally deal with mass murders. They will (and are obtaining help), from other police agencies, the state highway patrol, the ATF and the FBI.\nIt will take some time, to review all the camera footage.\nI should mention that the recall would be obligatory, and that the oldest USAF fighter pilot I knew was 36 years old when he was ordered to a desk job\u2014so we are talking about \"retired\" military pilots who are still in their 30s, and now hold other positions.\n19 I will give you the keys (authority) of the kingdom of heaven; and whatever you bind [forbid, [b]declare to be improper and unlawful] on earth will have [already] been bound in heaven, and whatever you loose [permit, [b]declare lawful] on earth will have [already] been loosed in heaven.\nIn other words, the imperative mood merely orders our immediate sphere of influence to conform to the already completed work of the Cross. We don't pray toward God to complete His promises. Rather we pray from God's already established and completed work. Our prayer is a form of worship and release where we say yes and amen to all the promises of God which have already been fully performed by our magnificent Savior. Beloved, God has already rested from all of His works of deliverance, and so must we. \"There remaineth therefore a rest to the people of God. For he that is entered into his rest, he also hath ceased from his own works, as God did from his. Let us labour therefore to enter into that rest, lest any man fall after the same example of unbelief.\" Heb. 4:9-11.\nDo you see this magnificent reality of God's salvation? Always already. Always already. God's salvation is always already. This view of God changes everything. No longer is prayer seen as a laborious effort to get God to act or \"change His mind.\" Prayer is now seen not as a way to change God's will, but rather as a way of implementing God's will - - His \"always already\" will to all goodness. Worship is not based on what God might do someday, but rather true worship rejoices in what God has already fully accomplished. All we need do is to rest in the completed work of the Cross. Sure, there will be suffering. Sure, there will be temptation. Sure, there will be persecution. Sure, there will be occasional stumbles and a few set-backs. But in all these things, we are more than conquerors because Jesus has always already pre-conquered all these problems. Blaming God for allowing these problems is like blaming the fireman who pulls you out of a fire for starting the fire itself. God didn't start the fire - - we did. God gave His own blood to rescue us from the fire of sin with which we have burned ourselves.\nMay the two Koreas be peacefully reunited soon, with religious and economic liberty for all.\nIt is a past accomplished fact that Jesus' blood once and forever disallows and cleanses all evil within us and outside us.\nI thought your post was fine, but I do not understand that sentence.\n\"It is a past accomplished fact that Jesus' blood once and forever disallows and cleanses all evil within us and outside us.\"\nDave, as to the composition of that specific sentence of Murray's, I can't say for sure. God doesn't allow evil, in that He has already disallowed all evil through the victory of the cross, through the Lamb who was slain from the foundation of the world. And so the gist of effective prayer is: more binding and more loosing; less petitioning, and instead, more thanksgiving for, and imperative proclamation of, the promises of God.\nRegarding the NK situation, we must be careful to distinguish between nationalism and Christianity, between physical militancy and spiritual militancy. We need more binding and casting out of lying spirits of death and destruction, and more loosing of divine blessings, of angelic protection, and of the proclamation of the gospel of peace.\nThis passage is clear that all things have already been put under Jesus, but we don't yet see them put under Jesus. The reason? Because of our individual and corporate \"neglect\" of \"so great a salvation.\" Both Satan and Evil have no gasoline left in their tank because Jesus drained it all away at the Cross. They are functioning today solely off of the fumes of our neglect of Jesus' great salvation. What makes this salvation so great is its \"always already\" aspect.\nSo how do you all see it now that Trump has backed out of meeting with the North Korean leader? Will North Korea continue with its nuclear development until it can direct a barrage of nuclear missiles to United States?\nPlease tell what the DANGER, DANGER, DANGER is !!!\nNuclear war between North Korea and United States!\nFrom there it may escalate as various countries align with one side or the other.\nHum! I know which side China would probably take. But what side would Russia stand with?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In this special episode of the AEG podcast, Grant van Wyngaarden, Project Leader for Advanced Energy Agency, discusses battery storage value-stacking with Sean Hamilton, General Manager of Sterling Municipal Light Department. Sean and Grant will be continuing this conversation on a panel at Solar Power International in Las Vegas on September 11th. In this podcast, Sean share's his experience with building and operating New England's first utility-scale energy storage project, installed in Sterling, Massachusetts.\nThe 2MW \/ 3.9MWh Lithium Ion battery was developed primarily for resiliency purposes, as it provides back-up power to the Sterling Police Department and first responders and can support up to 12 days of grid outage. This resiliency became a critical need for the town of Sterling after it lost grid power for 2-weeks in 2008, caused by a brutal ice-storm in central Massachusetts that took down critical transmission lines.\nIn the podcast, Sean shares details of the value-stacking performance since the project went live in December 2016, so far the financial returns have been \"right on the money\". The town of Sterling was ranked 7th in the nation for solar capacity per capita, this high solar penetration added challenges to managing the grid. However, as Sean points out, this battery enhances the solar installations and it was even tested during the recent solar eclipse.\nWe hope you enjoy this edition of the podcast and stay tuned for even more advanced energy content.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was trying to explain to my wife about that storm and how I never saw anything like that in my lifetime. I don't even remember snow being in the forecast that nite. My brother was leaving my house when it started. Usually it takes 20 minutes for him to get home but it took him over an hour. When I finally reached him, He sounded like he was happy to be alive. He saw accidents all over the place driving home. If someone can give me the details of this storm and what happened, I would love it. Because after I told my wife about it, she said...oh yeah...i remember that storm...it was the worst storm I remember...yet she doesn't remember it. All I remember is that it went from no snow to alot of snow in a couple of hours.\nThis remains the one and only time I witnessed thundersnow. Being 12 years old at the time, I did not even think that was possible. The next day we went to the Poconos for a three day trip we had booked months in advance. They said they had over two feet of snow from that storm up there (back when north and west areas really cashed in on the snow). This made for favorable snowtubing conditions (if there's no snow, they make their own, but who likes fake snow ). And several miles away, we went snowmobiling, which I thought was so cool (no pun intended). I had a good time there.\nThe best Christmas Day ever.\nSleet changed to snow about 9am Christmas morning and it snowed like crazy into the evening. We ended up with a little over 18 inches. Christmas Day doesn't get any better than that.\nA side note I believe the \"official\" and I use that term loosely, measurement in the park was 5 inches, but math can check me on that one as I'm doing it from memory. I believe it was actually closer to 7-8, but that's the norm around here.\nYup, just like with the President's Day Blizzard two months later, my 12-year old self was very suspicious about that measurement. It wasn't until later that I found out about it, but knowing the snow's impact, when I heard it was a 5\" snow measurement, I was like \"That's it.\" It was actually here that I first became suspicious about snowfall measurements at Central Park. About eight years later, I was happy to know that I was not the only one that thought this when I first went onto the old 7-online chat board and saw you posts and of course your username \"CPcantmeasuresnow\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This French book, published in 1996, is the first work of the Belgian author G\u00e9rard HENROTIN relating to firearms. It is probably as well the first book using at this level the infographical technologies in the firearms field. To fill that challenge, many competences had to be gathered to offer a technically reliable content in a modern and attractive form. These two objectives were reached to such an extend that the volume became de facto the French reference on this famour German pistol.\nAnalysis of a rare DWM Luger.\nExtracted from our book : \"La Connaissance du Luger\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaetdd b/data_all_eng_slimpj/shuffled/split2/finalzzzaetdd
new file mode 100644
index 0000000000000000000000000000000000000000..b14153b42ec7381cc6e75acfe591229d3d0849b6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaetdd
@@ -0,0 +1,5 @@
+{"text":"This patient had recurrent episodes of abdominal pain.\nNow she presented to the ER with more pain and on a plain abdominal film there were signs of a small bowel obstruction.\nWhat is the cause of the obstruction?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Streetwalkers standing along the junction of Geylang Road and Geylang Lorong 22 last Thursday. Donning skimpy outfits, they emerge at dusk like clockwork and flirt with men, both foreign and local.\nIt is unfair to jump to the conclusion that foreign workers cause trouble in Geylang, said migrant rights groups.\nThey also believe an event like last December's Little India riot is less likely to happen in Geylang, as the workers who frequent the area gather in small pockets around the neighbourhood.\nHe believes another key reason workers gather in smaller groups in Geylang is that there are hardly any large open fields in the area.\nInstead, workers typically hang out in shaded areas at roadsides or in back alleys of shophouses.\nAnd while Geylang has been a key gathering point for workers from China, not all visit the area. Others prefer to spend time in the heartlands, he said.\n\"They do not have to worry about not being understood, as many Singaporeans speak Mandarin. Most South Asian workers visit Little India because they know that the people there speak their language.\"\nStaff from MWC and Humanitarian Organisation for Migration Economics (Home), which both have offices in Geylang where foreign workers can get help with employment disputes, said they hardly hear and see foreign workers getting into fights in the area.\nHome's executive director Jolovan Wham said: \"It is unfair to think that much of the crime and violence in Geylang is caused by foreign workers. This is a place where there are brothels, pubs, karaoke lounges. There are pimps and gangsters everywhere.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I recently graduated from PM school. I have experience in the field of HR about 3 years and graphic design, one year in IT. I was conducting Team Buildings and One to One meetings.The sphere of human resources and project management are cross, and for me it is a good step in career development. I want to realize my organizational skills and get experience in managing projects and teams.\nOpportunity for training and development with company.\nI have a master's degree in psychology + finished several educational programmes, so expect a top-notch, scientific approach to communication and motivation.\nI am looking to work with a company with a strong and stable market position. I expect challenging conditions and I am super-ready to transcend my own borders.\nPosition: sales manager (march 20105 - february 2016). Used to work with russian speaking countries, setting contacts with new clients. Position: Visual graphic specialist (McKinsey&Co). Coordinate production activities for short and long-term client assignments, working closely with colleagues to manage production.Work with consultants on improving the visual presentation of information.\n\u041d\u0435\u0444\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u0430\u044f \u043e\u0431\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0430. \u0421\u043b\u043e\u0436\u043d\u044b\u0435 \u0437\u0430\u0434\u0430\u0447\u0438. \u0411\u0435\u043b\u0430\u044f \u0437\u0430\u0440\u043f\u043b\u0430\u0442\u0430. \u0414\u0440\u0443\u0436\u0435\u043b\u044e\u0431\u043d\u044b\u0439 \u043a\u043e\u043b\u043b\u0435\u043a\u0442\u0438\u0432. \u041f\u0440\u043e\u0434\u0443\u043a\u0442\u043e\u0432\u0430\u044f \u043a\u043e\u043c\u043f\u0430\u043d\u0438\u044f.\nProject manager at Addreality \u2014 \u041f\u043e\u043b\u0443\u0447\u0435\u043d\u0438\u0435 use case \u043e\u0442 CEO\/Product manager, \u043f\u043b\u0430\u043d\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u0435 \u0438 \u0434\u0435\u043a\u043e\u043c\u043f\u043e\u0437\u0438\u0446\u0438\u044f \u0437\u0430\u0434\u0430\u0447 (user story), \u0441\u043e\u0437\u0434\u0430\u043d\u0438\u0435 \u0442\u0430\u0441\u043a\u043e\u0432 \u0432 jira. \u041c\u043e\u043d\u0438\u0442\u043e\u0440\u0438\u043d\u0433 \u0438\u0441\u043f\u043e\u043b\u043d\u0435\u043d\u0438\u044f \u0437\u0430\u0434\u0430\u0447 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0447\u0438\u043a\u0430\u043c\u0438, \u0434\u0438\u0437\u0430\u0439\u043d\u0435\u0440\u0430\u043c\u0438 \u0438 QA. \u2014 \u041f\u0440\u043e\u0442\u043e\u0442\u0438\u043f\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u0435. \u0421\u043e\u0437\u0434\u0430\u043d\u0438\u0435 CJM \u0432 figma. \u041d\u0430\u043f\u0438\u0441\u0430\u043d\u0438\u0435 \u0442\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0434\u043e\u043a\u0443\u043c\u0435\u043d\u0442\u0430\u0446\u0438\u0438. \u2014 \u0420\u0430\u0431\u043e\u0442\u0430 \u0441 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0447\u0438\u043a\u0430\u043c\u0438 \u043a\u0430\u043a \u0432 \u043e\u0444\u0438\u0441\u0435, \u0442\u0430\u043a \u0438 \u043d\u0430 \u0443\u0434\u0430\u043b\u0435\u043d\u043a\u0435. \u2014 \u0420\u0430\u0431\u043e\u0442\u0430 \u0432 \u043a\u0430\u0447\u0435\u0441\u0442\u0432\u0435 \u0437\u0430\u043a\u0430\u0437\u0447\u0438\u043a\u0430 (\u043e\u0442 \u043a\u043e\u043c\u043f\u0430\u043d\u0438\u0438). \u2014 \u041c\u043e\u043d\u0438\u0442\u043e\u0440\u0438\u043d\u0433 \u0438\u043d\u0441\u0430\u0439\u0442\u043e\u0432 \u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u0435\u043b\u0435\u0439 \u0432 productboard. \u2014 \u0412\u044b\u0441\u0442\u0440\u0430\u0438\u0432\u0430\u043d\u0438\u0435 scrum \u043f\u0440\u043e\u0446\u0435\u0441\u0441\u043e\u0432 (part-time). \u2014 \u0412\u0435\u0434\u0435\u043d\u0438\u0435 \u043d\u0435\u0441\u043a\u043e\u043b\u044c\u043a\u0438\u0445 \u043f\u0440\u043e\u0435\u043a\u0442\u043e\u0432 \u043e\u0434\u043d\u043e\u0432\u0440\u0435\u043c\u0435\u043d\u043d\u043e. \u2014 \u0420\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043a\u0430 \u043d\u043e\u0432\u043e\u0433\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b\u0430 \u0432 \u043b\u0438\u0447\u043d\u043e\u043c \u043a\u0430\u0431\u0438\u043d\u0435\u0442\u0435 (saas) \u0438 \u043f\u043e\u0434\u0434\u0435\u0440\u0436\u043a\u0430 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u044e\u0449\u0435\u0433\u043e. \u2014 \u0420\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043a\u0430 \u043f\u043b\u0430\u0442\u0444\u043e\u0440\u043c\u044b \u0441 OpenRTB \u0430\u0443\u043a\u0446\u0438\u043e\u043d\u0430\u043c\u0438 \u0438 direct deal. \u0423\u0436\u0435: \u2022 \u041f\u0440\u043e\u044f\u0432\u0438\u043b \u0441\u0435\u0431\u044f \u0432 \u043a\u0430\u0447\u0435\u0441\u0442\u0432\u0435 \u043c\u0435\u043d\u0435\u0434\u0436\u0435\u0440\u0430 \u043d\u0430 \u043a\u0440\u0430\u0442\u043a\u043e\u0441\u0440\u043e\u0447\u043d\u044b\u0445 \u043f\u0440\u043e\u0435\u043a\u0442\u0430\u0445 \u043a\u0440\u0443\u043f\u043d\u044b\u0445 \u043a\u043e\u043c\u043f\u0430\u043d\u0438\u0439 (\u0420\u041e\u0421\u0410\u0422\u041e\u041c, \u0411\u0423\u041a\u0412\u041e\u0415\u0414, PICASO 3D, BIOCAD). \u2022 \u0417\u0430\u043f\u0443\u0441\u043a\u0430\u043b \u0441\u043e\u0431\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0439 \u043f\u0440\u043e\u0435\u043a\u0442. \u0421\u0435\u0439\u0447\u0430\u0441: \u2022 \u0420\u0430\u0431\u043e\u0442\u0430\u044e \u0432 IT \u0438 \u0441\u043e\u0437\u0434\u0430\u044e \u0441 \u043a\u043e\u043c\u0430\u043d\u0434\u043e\u0439 \u043d\u043e\u0432\u044b\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b \u0434\u043b\u044f \u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u0435\u043b\u0435\u0439 \u0432 \u043b\u0438\u0447\u043d\u043e\u043c \u043a\u0430\u0431\u0438\u043d\u0435\u0442\u0435. \u2022 \u0421\u0430\u043c\u043e\u0441\u0442\u043e\u044f\u0442\u0435\u043b\u044c\u043d\u043e \u0438\u0437\u0443\u0447\u0430\u044e QA; \u2022 \u041f\u043e\u0441\u0442\u0438\u0433\u0430\u044e Agile \u0438 scrum; \u2022 \u0414\u043e\u0432\u043e\u0436\u0443 \u0430\u043d\u0433\u043b\u0438\u0439\u0441\u043a\u0438\u0439 \u0434\u043e upper-int.; \u0416\u0434\u0443: \u2022 \u0422\u0432\u043e\u0440\u0447\u0435\u0441\u043a\u0443\u044e \u0441\u0432\u043e\u0431\u043e\u0434\u0443 \u0438 \u0432\u043e\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442\u044c \u0443\u0447\u0438\u0442\u044c\u0441\u044f \u0443 \u043b\u0443\u0447\u0448\u0438\u0445; \u2022 \u0412\u0430\u0448\u0435 \u0442\u0435\u0441\u0442\u043e\u0432\u043e\u0435 \u0437\u0430\u0434\u0430\u043d\u0438\u0435. PAEI-\u0422\u0435\u0441\u0442 http:\/\/paei.denero.ru\/0\/Ykcef7 \u0421\u043f\u0430\u0441\u0438\u0431\u043e.\n\u2014 \u0420\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043a\u0430 \u043d\u043e\u0432\u043e\u0433\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b\u0430 \u0432 \u043b\u0438\u0447\u043d\u043e\u043c \u043a\u0430\u0431\u0438\u043d\u0435\u0442\u0435 (saas) \u0438 \u043f\u043e\u0434\u0434\u0435\u0440\u0436\u043a\u0430 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u044e\u0449\u0435\u0433\u043e (Addreality). \u2014 \u0420\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043a\u0430 \u043f\u043b\u0430\u0442\u0444\u043e\u0440\u043c\u044b \u0441 OpenRTB \u0430\u0443\u043a\u0446\u0438\u043e\u043d\u0430\u043c\u0438 \u0438 direct deal. (Addreality). \u2014 \u0417\u0430\u043f\u0443\u0441\u043a \u0441\u043e\u0431\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0433\u043e \u0438\u043d\u0442\u0435\u0440\u043d\u0435\u0442-\u043f\u0440\u043e\u0435\u043a\u0442\u0430 \u042d\u0444\u0438\u0440 (\u043a\u043e\u043d\u0442\u0435\u043d\u0442-\u043a\u043e\u0444\u0435\u0439\u043d\u044f).\n\u0418\u0449\u0443 \u043a\u043e\u043c\u0430\u043d\u0434\u0443, \u0443 \u043a\u043e\u0442\u043e\u0440\u043e\u0439 \u0441\u043c\u043e\u0433\u0443 \u043f\u0435\u0440\u0435\u043d\u0438\u043c\u0430\u0442\u044c best practices. \u0421\u043e\u0432\u0435\u0440\u0448\u0435\u043d\u0441\u0442\u0432\u043e\u0432\u0430\u0442\u044c\u0441\u044f \u043d\u0435 \u0442\u043e\u043b\u044c\u043a\u043e \u043a\u0430\u043a \u043f\u0440\u043e\u0434\u0436\u0435\u043a\u0442, \u043d\u043e \u0438 \u043f\u0440\u043e\u0434\u0443\u043a\u0442.\nI work as a training coordinator in the CURSOR education (IT education). I'm responsible for the student's learning processes. And want to start a project manager career in real projects.\nI understand the processes of software development, able to use the methodology of flexible development, I have good communicative skills, and very good at using task-managers: Jira, Trello, Asana, Toggl.\nI want to get the experience in project management, learn and develop in this area.\nI have working experience as Data Analyst in Accounting Department and Finance Manager of 4.5 years in total. I am good at communicating with customers and foreig partners, performing data analysis, arranging meetings and creating internal documentation. I have courses experience of software testing (2016) and project managing (2019). I would like to work on the position of Junior Project Manager in prestigious company with challenging projects and interesting environment, where I will have opprotunity to embody my self-organisation and multitasking skills and huge desire to learn.\nIn this period of my life I'm interested in any types of projects and am open to suggestions. Perfect situation at your new work place is when you have a mentor with years of experience and who is ready to share his experiences with you.\nI have a big experience in business, I am familiar with: creation, modelling and controlling of processes, required for enterprise life-cycle, optimization of problematical processes; organization of iterative maintenance processes using weekly planning, task tracking and prioritization; risk management; evaluation of enterprise performance, comparison planed\/actual indicators, root cause analysis, developing and corrective actions; potential suppliers benchmarking; strategic and tactical analysis, objectives setting; collaboration with customers; collaboration with regulative authorities; recruitment and HR management.\n136 candidates found. Page 1 of 14.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Finding the right residential service for your swimming pool and spa may seem a bit like walking around with your eyes closed and picking the one that shouts the loudest.\nHowever, having a large employee base and as a result more customers than you can know by name, doesn't always equate to better than the rest. You need to do your research when selecting a residential pool service company.\nAccording to Water Tech Pool and Spa Owner Bob Wood, there are certain things to look for when vetting out a residential pool service company.\nNo. 1\u2014Do they have a well-trained staff?\nNo. 2\u2014Do they charge more for a la carte services rather than offer you a complete cleaning package?\nNo. 3\u2014Are they personable and easy to talk to about your pool needs?\nThese are the top three questions Bob suggests asking, and if the answer to any of them is \"no,\" keep shopping.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am the head baseball coach.\nI plan to have a successful year preparing students for college and life outside of school. I plan on focusing and using multiple primary and secondary sources to make History seem exciting and important.\nSports: Baseball (Cardinals), Hockey (Blues), and Basketball (Duke). Pop-culture junkie. Obviously I like reading and learning about History, and occasionally I might read a book.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafvza b/data_all_eng_slimpj/shuffled/split2/finalzzzafvza
new file mode 100644
index 0000000000000000000000000000000000000000..0577cfbef4890c85128313287a9ec45042fd3086
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzafvza
@@ -0,0 +1,5 @@
+{"text":"Lily Wilson | Acoustic folk pop | Blog : Over $600!\nWe just got a payment of $100 from Jeff in Ohio.\n\"Lily, here is a small contribution to help with expenses for your new CD. I've always enjoyed listening to your music, even way out here in Ohio. Since it is doubtful I will ever be out in L.A. to hear you in person, I am happy to help you make a new record that I can hear. Have fun recording. -Jeff\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The price of oil rose on Wednesday's early session, following the news that Trump canceled the nuclear agreement with Iran. U.S. West Texas Intermediate (WTI) crude prices edged up by 1.7 percent, or $1.18, to settle at $70.24 per barrel. International benchmark for oil futures London Brent crude finished the session 1.8 percent higher, or $1.36, at $76.34 per barrel. Both WTI and Brent are currently close to their highest level seen last 2014.\nYesterday, U.S. President Donald Trump walked away from the global accord with Iran that was signed three years ago. This raised uncertainties as to the future of oil supply especially in light of the possibility that sanctions will be re-imposed on Iran \u2013 a major oil exporter. Apart from that, Trump's withdrawal from the agreement will certainly increase tension in the Middle East and may bring about a military clash between Israel and Iran.\nThe euro declined against the U.S. dollar on Wednesday. Political worries surrounding the Italian government have dragged the euro lower. Many in Italy expect early election. The common currency closed at $1.1867 after dropping to $1.1838 overnight \u2013 Its weakest level since December 22. Against its Japanese counterpart, the euro also lost strength and closed at 129.400 yen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whether it is due to not drinking enough coffee or staying up for an all-night Game of Thrones marathon, everyone has an off moment from time to time. If you've ever had one while rinsing your nasal passages, you probably know what happens. Your flaming nostrils tell you!\nPerforming a nasal rinse without the salt packet is a great way to feel the burn, but not the kind of burn you like. Salt-free rinsing hurts, leaving you sporting a face that feels as if it's on fire. It's an entirely different experience without the balanced pH that salt provides. The salt also helps by thinning out nasal mucus and making cilia better at trapping and removing germs.\nNow, it's important to understand that not all salt solutions are created equally. The cheapest example, of course, is plain granular sodium chloride. At ResQRinse, however, we place utmost importance on making nasal rinsing a pleasant experience, so you'll actually continue doing it. We want you to feel better, which is why we use a proprietary blend of sodium chloride and sodium bicarbonate. The latter provides a unique buffer that reduces burning and stinging. It's truly a superior solution, and it is extremely easy to use.\nWhen mixing your salt packets, always remember that you should never use tap water because it lacks adequate filtration. We don't worry as much about that with our drinking water, because stomach acid kills many of those waterborne pathogens. Your nasal membranes are much more delicate, however, so stick to using water that is either distilled or sterile.\nBoiled tap water after it has cooled is also a viable option for rinsing, and the information that is included with the irrigation device provides specific instructions on how to rinse properly.\nOf course, the quality of the solution matters, as well. All-natural solutions provide a gentler and more enjoyable rinsing experience. The ResQRinse all-natural solution is pH balanced and free of preservatives and iodine.\nWhen used correctly, nasal rinsing is an effective way to remove allergens, bacterium, viruses, and other irritants from the passages before they can enter \u2013 and annoy \u2013 the body. Rinsing can also help with allergies, sinus pressure, post-nasal drip, and cold and flu symptoms.\nResQRinse is a new nasal rinsing system that uses choanal occlusion, a process that maximizes benefits and comfort. ResQRinse helps your nasal health by delivering the saline rinse directly through the nasal cavity. To learn more about the wonders it can do for your nose, please check out these informative videos.\nNeti Pot Vs ResQRinse: The Nose to Nose Battle Post-Surgery Garbage Odor: Why Does Getting Well Smell So Bad?\nAll information presented within this website is intended for informational purposes only and not for the purpose of rendering medical advice. Statements made on this website have not been evaluated by the Food and Drug Administration. The information contained herein is not intended to diagnose, treat, cure or prevent any disease or medical condition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vivog super premium cat sand. The fine strong clumping cat litter is very economical in use, binds the odor before it can develop and pleasant smells like baby powder. Just always just refill the cat litter and no more carry heavy cat litter. Save with our convenient three-pack.\nFill the litter box with 7 cm of cat litter. Take away the clumps and solid waste daily as necessary. Always refill to 7 cm.\nCompact clumping super premium white cat litter, for cats. 99,5% dust free, 100% natural and enviromentally friendly, super clumping, odour control, easy to dispose of, superior absorbency, keeps tray dry, economic consumption, processed from finest white bentonite. Baby powder fragrance.\nIf your cat urinates like to protest in the apartment, then try the super premium cat sand, because cats are very odor sensitv and react on specific cat sand positively. If your cat already has peed in the bed, couch or in the living room use the Ecodor UF2000, to neutralize.\nWenn Ihre Katze gerne aus Protest in die Wohnung uriniert, dann versuchen Sie den Super Premium Katzensand, denn Katzen sind sehr geruchsempfindlich und regieren auf Geruchsbindenden Katzensand positiv. Solle Ihre Katze bereits ins Bett, aufs Sofa oder in ein Zimmer gepinkelt haben verwenden Sie zum Neutralisieren den Ecodor UF2000.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Irish and Chin's popular event dedicated to \"Rub-a-Dub\" culture is quickly approaching. And just the mention of Shinehead's name alone drives Dancehall fans and sound men into a frenzy. On Saturday, July 18, for the first time in his career, the legendary Shinehead will debut on the \"Reeewind\" stage. There is no doubt that the diversely talented artist will add a wow factor to the highly anticipated show, as he is a lyrical genius and purveyor of Rub-a-Dub culture.\nOver the course of Shinehead's impressive career, he has evolved as a true music connoisseur, sharing an unprecedented love and historical account of Reggae and Hip Hop music. In fact, he has without question played a pivotal role in bridging these art forms, scoring mainstream success with major hits like \"Gimme No Crack\" and \"Chain Gang\" and many others. But these highly visible songs, which fused Dancehall and Hip Hop flavor just barely touch on the unique tapestry and depth of his artistry. At any given moment, Shinehead can incredibly don the hat of singer, deejay, rapper and sound man. Whether taking on a singjay approach of just lyrically flowing freestyle, there are very few in this artist's league.\n\"Shinehead is definitely the wild card, who brings quite the element of surprise to 'Reeewind'... when he touches the stage, he could very well sing, chant, impersonate or all of the above over the non-stop riddims,\" says Garfield \"Chin\" Bourne of Irish and Chin, the event's promoter.\n\"I know that Shinehead loves 'Rub-a-Dub' and will do 'Reeewind' justice....while I have been a fan of his for years, it was during the Welcome to Jamrock cruise that the full breadth of his talent struck me and opened up a world of opportunities.\"\nAlthough this is Shinehead's premiere on the \"Reeewind\" stage, the dynamic artist has mastered performing in a Rub-a-Dub style. Traveling with vinyl and selecting and deejaying on his Kingston 12 Hi-Fi system with his hip and talented partner DJ Papalotl, Shinehead flawlessly recaptures the rudimental elements of Dancehall music and culture during his live performances and on radio. Notably, Kingston 12 Hi-Fi can be heard each Sunday night during \"Reeewind Sundays\" on Irish and Chin's sound system infused station, Sound Chat Radio (irishandchin.com), and on dubforceradio.com and huskeyradio.com.\nIt's going to be a historic night during the 6th installment of \"Reeewind,\" when Shinehead is joined by Reggae giants Freddie McGregor, Richie Spice, Cocoa Tea, Josey Wales, Brigadier Jerry, Charlie Chaplin, General Trees and Carlton Livingston. Buying tickets early is a must, as it will be more at the door.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagphy b/data_all_eng_slimpj/shuffled/split2/finalzzzagphy
new file mode 100644
index 0000000000000000000000000000000000000000..130c5cf947e11b08dac3780499af59adcbf9413d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzagphy
@@ -0,0 +1,5 @@
+{"text":"Congratulations on starting your training! Understandably, it might be a bit intimidating to train for your first (or fiftieth) race.\nWe imagine you may have a number of questions. Below is a summary of many of the common questions (and answers) people have when beginning their training.\nWe'd say you likely can, but nothing is promised. Running is a demanding, high impact sport. One footstrike can generate 3-5 times your bodyweight in impact force.\nSo, you want to make sure you're training properly. That's where we come in. Between the custom schedule provided by SportMe and access to Marathon Matt, you're well positioned to pull this off.\nSo, we think you absolutely CAN do this! Know that we are here to help you if you encounter any challenges.\nWhat's the general approach\/methodology you employ with your training schedules?\nThere are a few words that come to mind\u2026.gradual, progressive, and methodical. Rome was not built in a day. This is how you want to train.\nYour schedule reflects this approach. Your schedule also takes into account your goals, target race, current mileage, and a few other key data points.\nWhether you're new to the sport of running or a seasoned veteran, it takes time for your body to adapt and become better equipped to handle the demands of running. Hence, a gradual approach is always wise.\nI don't want to get injured. Can you help me avoid injuries?\nAs mentioned earlier, running generates 3-5 times your body weight in impact force PER footstrike.\nIt's not hyperbole to characterize running as an EXTREME sport. There's a reason why it's one of the fastest\/easiest ways to burn calories, running asks a LOT of you!\nWe wish we could guarantee you will never get injured, but we'd be doing you a disservice by providing you this guarantee. Even the best runners on the planet with the best resources at their disposal still get injured. The good news is most aggravations and injuries can be overcome.\nHere's our advice\u2026.listen to the messages your body sends you. If it feels like something is 'off', it might be. Dull, aching pain that waxes and wanes is 'likely' something minor that can be treated with rest, ice, massage, foam rolling, and\/or perhaps a quality sports massage.\nBut, if you're feeling pain that is sharp, intense, searing, or excruciating, seek medical attention immediately and abstain from running.\nIf you're not sure about what's going on, you can reach out to Marathon Matt via 'Ask a Trainer' and he will do his best to help assess what's going on. Please bear in mind; Marathon Matt is NOT a medical professional.\nWhat if I want to learn about injury prevention, nutrition\/hydration, biomechanics\/form, or something else related to my training?\nYou will receive a newsletter from us twice a month (typically) that contains a section dedicated to covering things like injury prevention, nutrition, and a number of topics relevant to your training.\nIndependent of this, you can always reach out to Marathon Matt via 'Ask a Trainer' and he will give you what you need!\nWe also maintain an awesome blog with TONS of great information that will likely help you. Click here to take a look at it.\nWhat else is in the email newsletter?\nYou will get some 'runspiration', learn more about some awesome runners in the SportMe Run Trainer community, and more!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As a Lawson Isles Davie Seller in this complex market, hiring the right Lawson Isles Davie listing broker is probably the single most important step you can take in order to get your Lawson Isles Davie home sold. Lawson Isles Davie Sellers email us today!\nAs a Lawson Isles Davie Buyer, it usually costs nothing to hire a real estate professional to help you find and purchase a Lawson Isles Davie home. The value that Green Realty Properties brings to our Lawson Isles Davie buyers is second to none.Lawson Isles Buyers email us today!\nLot Size: 40,738 Sq Ft.\nBEAUTIFUL POOL HOME IN DAVIE WITH HUGE LOT!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"And on the first day of Christmas\u2026 the people of Tasmania wanted to know what do for the other thirty days!\nHo, ho, ho; suddenly it's December! While we all scratch our heads and wonder how 2018 has gone so fast, peruse at your leisure to see all the fabulous things happening in Tasmania this December.\nTo celebrate their third album release, Lo La Ru, The Rubens are coming to you, Launceston. They will be joined at all headline shows by musical group Little May with an additional opening support act in each state to be announced via a triple j unearthed competition.\nReturning this year, after 10 years of the Niche Markets, is a two day event, Niche-Mas. Described as a 'European style, artisan-made Christmas experience,' Niche Market is one of the most successful design and make enterprises, attracting more than 5000 people, only this time with a little Christmas twist.\nSanta's Funland is coming to the Derwent Entertainment Centre teeming with Christmas everything. Think unlimited rides for the whole family, a magical winter forest (with falling snow), face-painting at the elves workshop and why not stop and visit Frosty the Snowman! Tickets include all activities for free.\nNo plans thus far for Christmas, or simply not up to the idea of cooking for who knows how many people? The Tailrace Cafe has you covered. Open on Christmas Day and offering a delicious spread that Chef Kelvin Bowers worked hard at creating. The kid's will be happy too as the kid's playground, Kids Paradise, is open for free to guests.\nDevonport's family friendly New Year's Eve event, Skyfire, is on again this year, running from 6pm to midnight at Roundhouse Park. Come for a fun-filled night with free entry, activities and fireworks to welcome the New Year.\nEvery year, the Sydney to Hobart Yacht Race draws in massive coverage and support from the nation, eager to see who will win. This year will be no different, and while the yachts race why not go to the Hobart Race Village, which will be alive with festivities and fun, on the 27 of December and 01 January, to see the boats in.\nThis December has something for everyone, including the much awaited Falls Festival. The lineup this year is exciting, with artists including Hilltop Hoods, Vance Joy, Amy Shark, DMA's and Hayden James. If music and good times are your thing, then kick off 2019 with a bang.\nThe Taste of Tasmania is a festival celebrating food, beverages, good times and family fun. The tradition was started in 1988 for crews finishing the Sydney to Hobart yacht race. Being the largest festival of it's kind in Australia and located on the beautiful waterfront, why wouldn't you want to come and visit? Free entry, however there are tickets available for purchase for the masterclasses.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2a and 2b: Domains are circular at room temperature and appear to arrange into hexagonal patterns. If the temperature is increased (to point below the two phases mix), the domains undulate laterally. If the temperature is raised further (past the Tm for the phase mixing, the membranes become homogeneous in phase and presumably composition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New classmate Leslie unlocks a wondrous world for fifth-grader Jesse in this magical adventure. Rivals at first, the two outcasts soon become friends and create the make-believe kingdom of Terabithia, where they plot vengeance against school bullies.\nParents need to know that this movie based on Katherine Paterson's classic children's novel isn't a fantasy adventure as much as it is a thoughtful drama. The story includes mature themes about loneliness, platonic love, parental affection, family relationships, religious\/class differences, and even death. For example, a financially troubled father is emotionally distant from his son, while the mother seems too overwhelmed with five kids to connect to any of them. At school, kids are ridiculed because of their appearance, and a bully and her associates routinely terrorize younger kids. Potential spoiler alert: Due to a central character's death, sensitive children (or those who've experienced the loss of a loved one) may need to be comforted after the film.\nThe main characters share an intense, affectionate friendship. Characters write a fake love letter to a teenage girl. A boy stares at and has an obvious crush on a teacher.\nA main character has a tragic accident. With cause, a boy punches another boy in the face. A teen bully pushes and menaces younger kids.\nMild insults among siblings and school-aged kids; a father makes a couple of hurtful comments. A few uses of \"damn\" and \"hell.\"\nThrough an incredible friendship, a child learns that if you open your mind and heart, you can unlock the magic and the beauty in every person and situation. Strong messages about individuality. There are also serious themes relating to loneliness and even death, but they're treated thoughtfully.\nBarbie dolls, Twinkies, and Oreos make brief appearances.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahdqj b/data_all_eng_slimpj/shuffled/split2/finalzzzahdqj
new file mode 100644
index 0000000000000000000000000000000000000000..0621a17ee4748dbf6a262dd3218bede0fd0edd4f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzahdqj
@@ -0,0 +1,5 @@
+{"text":"Fulfilling your duties as a Trustee is a complex process. Coordinating multiple professionals who provide Investment Management, Tax Compliance and Fiduciary Accounting can be an arduous and time consuming process.\nKennedy Wealth & Tax Management will support you and help you achieve all of your fiduciary responsibilities efficiently.\nRest assured we will not compete with you, as we will never act as Trustee ourselves.\nIn accordance with Uniform Prudent Investor Act, we manage your Trust accounts using Modern Portfolio Theory.\nTax-Efficient Investments utilizing the research of Nobel Prize winning academics.\nCreation and adherence to a written Investment Policy Statement.\nInvestments custodied with TD Ameritrade Institutional.\nPreparation of Fiduciary Income Tax Filings.\nPrincipal and Income Accounting Reports.\nEducational events provided to Co-Trustees and Beneficiaries.\nIn-house expertise staffed by CPA, CFP\u00ae and EA.\nIf you are ready to simplify your role as a Professional Trustee, yet deliver an outstanding experience for all of your Trust constituents while maximizing your personal time, please contact us at 203-264-1200.\nWe realized from the beginning that to provide outstanding client service, we had to be independent. Free to make choices and decisions in a way to best serve our clients. Free to help our clients attain financial security based on their individual goals and not have to satisfy a sales quota. And free to learn what matters most to our clients.\nWe build deep, lifelong personal relationships with our clients based upon trust, unbiased advice, and outstanding customer service. We have been pleased to watch our clients grow and prosper through these enduring relationships.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This bag is beautiful, desirable, and extremely useful. Perfect as a carry-on for your weekend away with the girls, a last-minute overnight trip, or that awesome gift for someone that has everything. With plenty of space, this bag features double straps, and a 25-inch zippered opening with a water-resistant lining. When you open the bag, you will be pleased to find a zippered pocket, and a divided side pocket with a cell phone slot. This is a statement bag that is designer through and through. Don't be surprised at the compliments and possible offers to take this one off your hands! These bags are works of art.\nThis classic bag has black sides and bottom with a velvet feel. Four \"feet\" protect the bottom of the bag. The red tassel, zipper, and straps really stand out against the houndstooth fabric that covers the top, front, and back of this lovely bag.\nSIZE 20\" x 13\" x 7\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am empowered with greatness!\nIn my life right now I look forward for looking for a long time companion. I am a kind of person who is driven to my passion. I never give up in times of stormy weather. I also believe in second chances since we live in a imperfect world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Believe it or not, successful people have bad days, too. The difference is they don't let their feelings dictate their actions. Remembering that emotions are temporary, they have a singular focus -- to serve a larger vision.\nIn order to keep my eye on the prize, even when huge setbacks occur, I always come back to the inspiration of my vision as the key to unlocking my motivation. These are some of my favorite quotes for one of those days.\nThe law of consistency is a powerful thing. It's tough to hope to be great at something and have it come true if you don't practice it often. Whatever habits you practice on a daily basis will support you in getting the result you are looking for (or support you in not getting them if you lack the discipline).\n2. \"If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.\" -- Martin Luther King, Jr.\nMy friend Sean Stephenson recently experienced a near fatal accident. After just a month he is already moving forward toward his comeback. In his most recent podcast, he talks about his mentor, Byron Katie, calling him to give him inspiration to help him heal. She told him to stop calling it an \"accident\" and start calling it a \"purpose.\" His takeaway is exactly that -- to continue living a life of intention and purpose.\nI am a big fan of the company Zappos and its culture. According to Hsieh's widely read book Delivering Happiness, companies with a higher sense of purpose outperform others by 400 percent.\nIt's said that Branson has dyslexia and had poor academic performance as a student, but later discovered his superpower to network (something he and I have in common).\nOn one of Branson's last days at school, his headmaster is reported to have told him he would either end up in prison or become a millionaire. Branson is infamous not only for his innovation but also for his risk-taking. Oh yeah, and for his net worth.\nOne of my favorite books is The Four Agreements by Don Miguel Ruiz, and one of his precepts to live by is \"Don't take anything personally\". A great skill to have is to be able to objectively utilize feedback to improve your product or service.\nSometimes we all have to \"fake it till you make it.\" A little inspiration goes a long way, so the next time you are feeling unmotivated to take action in your business, pull out your favorite quotes, or better yet post them on the wall in your home or office, and get yourself back in the game quicker.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is there anything that screams \"Thanksgiving\" season more than stuffing yourself to the brim with delicious, flavorful food? At Ketanga, we're all about the perfect balance of treating yourself while keeping wellness in mind, and so we love coming up with creative swaps to make our meals more health-friendly.\nPie is the ultimate Thanksgiving dessert, and whether you are enjoying it with family, friends or on the Beach Bum Thanksgiving retreat in Costa Rica, we'd like to shout from the rooftops that YES! dessert can be healthy - or at least healthIER depending on choices you make! If you're looking to stack your Thanksgiving buffet high with tasty treats for the whole family, check out our list of healthy pies to enjoy while still fitting into your favorite jeans November 23rd.\nTrust us, these are definitely the top food items everyone at your Thanksgiving dinner will be thankful for!\nIf it's almost all fruit, is has to be healthy, right? That's what you can tell your guests when you present them with this beautifully put-together Apple Pie Baked Apple. It has all the flavor of the Apple Pie with less of the crust, so you're able to enjoy this unique treat without the added guilt that comes from having a second...third\u2026.and potentially 4th slice.\nRecipe can be found here via Sally's Baking Addiction.\nPraised for its high level of antioxidants, the Purple Sweet Potato is the ultimate hero in this vibrant pie! It's an untraditional twist on an otherwise traditional pie, and the best part is that since this variety of Sweet Potato improves blood sugar regulation, you are actually doing something good for your body by eating it! Additionally, everyone in the family, allergies included, can enjoy this one because it's entirely vegan.\nRecipe can be found here via The Endless Meal.\nWhile Pecan Pie is typically known for being incredibly high in sugar, this recipe swaps out the icky, refined ingredients for more natural options. Sweetened with dates and maple syrup, this pie provides all of the flavors that everyone loves from a Pecan Pie without any of the intense sugar-crash that comes after a piece.\nRecipe can be found here via The Healthy Foodie.\nThough pies over Thanksgiving are usually considered sweet, we wanted to throw in a savory one to give you options! This Shepherd's Pie is chock-full of fiber and vegetables, so you don't even have to waste any space on your plate for your aunt's wilted salad. And here's a phrase you probably never thought you'd hear a wellness site say: opt for extra pie instead of stuffing, and save yourself calories later!\nRecipe can be found here via Veggie Primer.\nCould we even make this list without the king of all the Thanksgiving pies, Pumpkin? Nutrient-dense with a low calorie count, the pumpkin provides ample opportunities to add flavor to the pie without compromising your diet. So go for that second slice of turkey AND pie, because this recipe allows flexibility to your meal plan without requiring you to change immediately into sweatpants post-meal.\nRecipe can be found here via Skinny Taste.\nHere's to a healthy, happy and tasty Thanksgiving! And if you're looking for a last-minute getaway, consider joining us in Costa Rica for the Beach Bum Boot Camp Thanksgiving Retreat! Details here.\n\u200bJoin our email list for upcoming retreats, delicious recipes and more!\nCheck Out These Travel-Themed Halloween Costumes!\n\u200bIt's that wonderful time of the year again! Where the air starts to cool, the leaves start to change, and you know that the next important holiday involves ghosts, witches and either tricks or treats (both if you're lucky!).\nHalloween presents the amazing opportunity to dress up as someone (or something!) other than what we really are for a night and to let our creativity shine. But with so many options, how is one person supposed to choose?!\n\u200bWhen you're feeling creative but want to keep wearing your favorite pair of jeans regardless of the holiday, creating your own life-sized passport is definitely the way to go. Plus, you're able to re-wear it year after year because even if your style changes, the Passport, like your reliable pair of Old Navy jeans, stays the same. Bonus: include photos from different places you have been to represent \"stamps\" from those countries!\nIf you're looking for a couple's costume, look no further! Whether you're attending a party with your S\/O, best friend or maybe a casual coworker, this costume is sure to be a Halloween party success! And the good news is even if you get separated, you'll always be able to find your way back to one another. Want to make it ghostly? Add some ghoulish makeup and write \"Mapquest\" on your costume. RIP, old friend.\nThis costume is sort of like receiving a mini Hershey's in your candy bag. It's reliable, sweet and will definitely get you points because it's a crowd pleaser. Plus, if you're the host, you can be trusted to direct people to the kitchen, the bar and the exit when it hits 10pm.\nWhen you don't want to carry a purse around all night, you should just be the bag! Depending on how well-crafted the costume is, you could probably also sneak in your own drinks in snacks when you're unsure about the selection that will be available at the party (or you want to be the hit of the party!).\nEveryone says they want to travel the globe, but not many people love it enough to actually BE the globe. Show your love for the world and waltz around the room knowing full well you have endless bragging rights for this epic outfit. \u200bGood for pregnant ladies, too! (Or for anyone looking to indulge on those Halloween treats without letting it show!).\nYou may say you've got it, but now's your chance to show it... the travel bug, that is! Whether you throw on the classic I <3 NY shirt, have world-map workout leggings (so cute!) or wear goodies collected on your travels, just add a pair of bug antennae and you're all set! Out with a special someone who you'd like to join you on an upcoming trip? Give them a 'bug bite' to pass along the travel bug so they have to join!\nHave ideas for other travel-inspired Halloween costumes? Comment below to share! Best idea will win a Ketanga luggage tag!\n\u200bJoin our email list for upcoming retreats, exciting blog posts and more!\nA Ketanga Fitness Retreat participant-turned Coach, Boston-based Kayla Mantegazza is yoga teacher with a heart of gold and buns of steel. A triple threat, Kayla is also writer, teacher, and athlete, and she thrives on what she believes are the most valuable skills a yogi can possess: resilience and a sense of humor.\nKayla's intention is to empower people from all walks of life and make wellbeing their priority. She provides a safe haven to access boutique-quality classes on a ramen noodle budget, with dignity and ease.\nKayla's teaching style is an energetic Vinyasa flow that typically incorporates bursts of high-intensity cardio intervals and resistance training using bodyweight, bands, and blocks. Read on to get to know Kayla!\n(Ketanga) What do you love the most about yoga?\n(Kayla) A distinguishing feature of yoga that appeals to me is that it's highly adaptable. Yoga can be tailored to accommodate a range of body types, levels of conditioning, injuries, and differently-abled bodies. We typically think of yoga as a physical practice that builds strength and flexibility, but for many people, the benefits of yoga include gaining mental clarity through meditation and mindful breathing or developing spiritual resiliency by studying the Yoga Sutras of Patanjali.\nIf I'm being completely honest though, what I love most about yoga isn't deep or philosophical. I love that my practice has taught me to commission my own body as functional (and free!) exercise equipment, which makes squeezing in a workout significantly easier when I'm on the go. I feel reassured and liberated knowing I'm inherently equipped with all the \"tools\" I need to maintain my fitness regimen anywhere.\nYou have a three-song playlist, what's on it?\n1.) Cotton-Eyed Joe, 2.) Cotton-Eyed Joe, and 3.) Cotton-Eyed Joe Remix ft. Pitbull.\nWhat is your advice for someone just starting out their practice?\nFall on your face! Seriously. The fear of falling often prevents us from trying poses, particularly arm balances. Sometimes our perception of our physical limitations is no more than a self-inflicted psychological barrier. It goes without saying that maintaining proper alignment and working your way up to the full expression of any arm balance is a mandatory prerequisite before attempting to strike a pose that'd make the Instagram Gods bless the rains down in Africa, but once you've mastered the basics, give yourself unconditional permission to faceplant. Surviving a fall empowers you to pick yourself up with both your front teeth and your ego intact.\nWhat about your inside tips on mastering a yoga pose... let's say crow pose?\nBrace your abs tightly as if your ribs are zippered to your spine.\nPress your palms into the floor as if you're pushing the ground away from your body. This engages the muscles in your upper back and creates the lift needed to pull your knees toward your underarms.\nWhat excites you about hosting a fitness retreat?\nI'm really looking forward to meeting fitness-oriented folks with whom I can explore beautiful landscapes, share lots of laughs, and above all, create yoga classes that are so fun and challenging, the tears of joy staining their cheeks will become indistinguishable from the beads of sweat dripping from their perfect little foreheads. I have kept in touch with all the incredible friends I made on my last Ketanga Fitness retreat and I'm excited to make more.\nDon't put yourself in a position where you are the smartest person in the room. Surround yourself with vibrant, interesting people from whom you are always learning and growing.\nJoin our email list for upcoming retreats, exciting blog posts and more!\n\u200bWhereas people may shy away from outdoor workouts in the summer because of the brutal sunshine and thick-as-pudding air quality (at least here in NYC), fall is the perfect time to get outdoors and enjoy the best of nature.\nThere are endless opportunities to (pumpkin)spice things up in our workouts this season, and we picked our top favorites to share with you!\nSure, pumpkins have the perfect place in pies, lattes and carving stations. But did you ever think that they could be\u2026 in a workout? Forget everything you know about pumpkins, because we're about to blow your mind! Replace your usual kettlebells, free weights and even the barbell with a couple heavy pumpkins. You can squat them, press them, plank them and even swing them! (Some would say even carving them can be a workout!) Just promise us one thing: please don't drop them.\nWhen the leaves start to change color, and it finally cools down just enough to put on a long-sleeved shirt comfortably, all you want to do is be outside in the crisp air. So let us suggest Shelter Island (to the New Yorkers reading this) as the ideal place to step out and remember all the reasons that autumn is your favorite time of the year - especially because it is close to so many fresh markets to pick up cider and healthy produce afterwards! Allow the wooded trails to re-invigorate your love of the outdoors!\nPut that gym membership on hold and take advantage of being able to ride a bike without hitting the 1-2-1-2 instructor-mandated pace. There's nothing like the feeling of fall air on your face, and there's the exciting element of having to avoid any rogue, falling leaves. Better yet, use it as your main mode of transportation, and see how much money you save by not buying gas or a subway card. Then take that hard-earned cash and treat yourself to a cider donut (or three)!\nNext to carving spooky faces into pumpkins, is there an activity that screams fall quite like apple picking? If you've never thought of this as a workout, now is the time! Jump for those hard-to-get honeycrisps resting on the top branches, and squat each time you come back to the ground. Benefit of this activity? You're earning your reward as you workout!\nInterested in hearing about fun workouts, upcoming Ketanga Fitness Retreats and more? Sign up below!\nWhether it's jumping out of a moving helicopter, swimming with dolphins, or hiking up dozens of challenging peaks, everyone has their own list of things they want to experience.\nCreating a bucket list is one way to make travel plans easier. Writing down the activities you want to do and sights you want to see can help you map out your itinerary!\n\u200bWhy create a bucket list?\nIt helped her see more of the world and discover amazing experiences. She says that creating a bucket list and trying to tick it off one by one also increased her self-esteem. In fact, she met her now-husband after she took charge of her life, and they even went skydiving on their honeymoon. It's one of the compelling reasons mentioned here on Ketanga Fitness for going on a solo adventure\u2014you get to meet a lot of people from different communities. And who knows, you may even meet the love of your life if you're single!\n\u200bAside from making friends all around the globe, a bucket list can also help you understand more about what type of a traveler you are. An infographic on bucket lists published by Lottoland suggests identifying your priorities as you cook up your ultimate checklist of things to do during your lifetime. Do you want to go on off-the-grid expeditions, experience adrenaline-pumping activities, or more city-based adventures? It also helps you become more goal-oriented, creative, and inspired to make the best out of the one life that you have.\nI'm sold. Give me some ideas!\n1. Cage dive with Great Whites: Number one on PopSugar's list of adventures for thrill seekers, is getting into the water (safely) with one of the most dangerous predators on the planet.\n2. Go on a Eurotrip: Europe is an amazing continent that everyone must visit at least once in their lifetime. There's an unlimited number of things to do for every kind of adventurer!\n3. Scuba dive in the Great Barrier Reef: Take a trip to the Land Down Under to swim, snorkel, or dive in the world's largest coral reef system.\n4. Surf in Hawaii: Catch the raddest waves in Maui, Oahu, Kauai or virtually anywhere else in Hawaii.\n5. Rock climb in Yosemite: Did you know that in 2017, famous rock climber Alex Honnold completed a free solo climb up El Capitan? Maybe you can try a few attempts at rock climbing\u2014not to Honnold's level\u2014but it can be just as exciting.\n6. BASE jump in Norway: For unbelievable views, head up and step off from Kjerag Cliff.\n\u200b7. See the Northern Lights: While you're still in Norway, make sure not to miss the lights show in the Arctic sky.\n8. Hike in Patagonia: Paste Magazine highlights the Perito Moreno Glacier as one of the most unforgettable views you'll get to see in this lifetime. It's also the home to intimidating and humbling mountains such as the Torres del Paine.\n9. Go on a Safari trip: Go to Africa and keep your eyes peeled for the Big 5: African lion, African leopard, rhinoceros, Cape buffalo, and African elephant.\n\u200b10. Summit Mount Everest: Lastly, at some point in life, most people dreamed of watching the world from its highest peak, Mount Everest. It's no easy feat to conquer, but definitely one that you'll never ever forget.\n\u200bWe want to help you cross items off of your travel bucket list! Learn more about upcoming Ketanga Fitness retreats and adventures here!\n\u200bSign up to be the first to learn about upcoming retreats, read new blog posts and more!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahvka b/data_all_eng_slimpj/shuffled/split2/finalzzzahvka
new file mode 100644
index 0000000000000000000000000000000000000000..2e4548428165aca4c2274e22b0f4f3f837f86f8e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzahvka
@@ -0,0 +1,5 @@
+{"text":"Welcome Daniel Rogers as Guest Host who will join the discussion. Tonight Two Guys and A Bible will continue the last half of the Spirit and Life Recap from last week. This was a great event and we wanted to share more information from the event and on the Speakers for the benefit of our audience. We discussed the Book of Revelation and the interest in the topic continues to be very high. Many have asked to recieve copies of the book.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Everyone is touched or influenced by someone in his life. The difference is that with successful people, the influential person comes at the right time. At any rate, it is not hard to get emotional and high while writing a narrative essay on the person.\nYou have to touch on several aspects. Bear in mind that that person begins to haunt you over time with his attributes and you begin to admire the qualities you never registered in your first experience. It may be about the way he manages relations; keeps his cool or controls time.\nYou should present how the person helped you take monumental decisions in your life and stood by you in the most crucial phases of your life. You can suggest how he showed that slight tweaks in lifestyle can make it more substantial.\nThe influence quotient may have been with your career, social side or emotional conflicts. People love how you struggled with a certain situation and how the influential person smoothly got you out of the rut. There is that feeling of identity ensconced in it.\nYou should remember that people love illustrations and examples. Please make sure that your essay is replete with instances where he actually proved more than a guiding light. This will also give the readers a new idea to lead their own lives.\nYou can elaborate on how the person you admire and emulate helped you cut the barriers with your family and friends. How he changed your perspective towards life and turned you into a positive fellow.\nYou should also ideally put in certain negative aspects of the person to make the write-up sound human. It would be wonderful to present how the person fights with his own deficiencies.\nShed lights on his traits that make him heroic in your eyes. It may be the way in which he naturally connects with people; how he lets his heart talk in situations when others would use their mind. How his presence raises your confidence to another level.\nYou should also elaborate on his success story so that others can also emulate his path. There may be a dash of consistency, flair or adventure connected with it. There ought to be a genial mix of resources so that the write-up becomes automatically inspirational.\nMake sure that you cut the definition of the person you admire in an eloquent manner. This is the respect that he deserves from you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Janelle Wang is jumping from KGO ABC7's \"View of the Bay\" to KNTV NBC Bay Area's 5 p.m. news, co-anchoring that show with Raj Mathai, the NBC station announced today.\nGarcia-Cannon's co-anchor and husband Brent Cannon is anchoring the 7 p.m. news on NBC's California Nonstop channel (Comcast Digital Cable channel 186).\nWang will be filling the co-anchor spot most recently held by Jessica Aguirre, who will continue to co-anchor at 6 and 11 with Mathai.\nThrilled to more of your range than on the View from the Bay, Ms. Janelle! Watching you co-anchor the 5PM news on a brand new 50\" HD-TV and you look and sound fabulous. You and Raj make a fantastic Bay Area team.\nNBC 11 is starting to look like KGO did in the 80' with anchors coming and going you never really could \"warm up to\" anyone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Should I tear off my old roof or just roof over it?\nComplete removal of the existing roof system is mandatory if it contains moisture, or if the additional roof layer exceeds safe load design, or if local codes require that the existing roof be removed. Most local codes will only allow two layers of 3-tab shingles.\nA second layer of shingles cannot be installed over Dimensional \/ Architectural style shingles, because the weight would exceed most local code requirements and the irregular surface of the existing shingles would not allow the new shingles to lie flat and seal properly.\nInstalling shingles over an existing roof system requires proper preparation of the surface. Blisters, splits and other roof membrane or flashing defects should be repaired in accordance with good roofing practices to attain a surface which is smooth, dry, clean and free of sharp projections and depressions. If the roof is not smooth and flat, the flaws will project and be seen through the new roof.\nFinally, the service life of the new shingles should be considered. The U.S. Department of Housing and Urban Development performed a study which concluded that as much as 25% of the service life of the new shingles will be lost when installed over an existing layer of shingles due to heat retention.\nJust as the human body ages and changes appearance over the years, so too will your roof. Due to the severity of the roof environment, even a one-year-old roof may look different from a roof that was just installed.\nResearch indicates that aging begins soon after the shingles are installed and progresses rapidly during the \"Initial Curing\" phase of its life cycle. During this stage, minor granule loss may occur, a few small blisters may develop or the shingles may curl slightly at their edges. The good news, however, is that after this \"Initial Curing\" stage the shingles enter a long period of slow aging, which lasts for the major portion of the shingles' natural life.\nYes. Attic ventilation might seem like a minor consideration, but when done properly, it can extend the life of your attic and roof structure; and save you hundreds of dollars in repair and energy costs.\nIn the summer, hot, moist air in the attic can warp roof sheathing and cause shingles to deteriorate prematurely. A hot attic also makes a home more difficult to cool and can result in added energy costs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Australian electronic producer Ta-ku has supplied the market with an endless flow of glistening beats as of late. Nailing just about every genre he takes under his belt, infusing them with an established sound that plays on the inner-workings of hip hop, and generally shifting focus between trap and experimental percussive effects.\nHis latest production churns Mt. Eden's insanely popular \"Sierra Leone\" into a heavy cream of thick and solidifying goodness as Regan Matthews adds to his bag of tricks. Splitting the song into three distinct segments, Ta-ku sets up his beat with a heavy set bass that lays a perfect backdrop to pitch-shifting vocals. And just when you think the song will continue on its progressive path, it drops into a grimy dubstep beat that demands attention. While the dubstep aspect differs from the lot of his past catalog, Matthews keeps the song close to home with an exploring beat in the first two segments, and trap-like beats and escalating synth stabs throughout.\nTa-ku's remix is one of three featured on Mt. Eden's Sierra Leone Remix EP, purchase it on iTunes here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajbqo b/data_all_eng_slimpj/shuffled/split2/finalzzzajbqo
new file mode 100644
index 0000000000000000000000000000000000000000..962c8aff3e45c776f9c8460d4efdd70eff0344e8
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzajbqo
@@ -0,0 +1,5 @@
+{"text":"Maker of fine products from France.\nBeautiful handmade porcelain from Paris France. In a whitewash finish\u00a0Can take up to four months for delivery.\nBeautiful handmade porcelain from Paris France. In a whitewash finish\u00a0Dimension 13x8.5x20.5 cm \/ 5\"x3.2\"x8\" Can take up to four months for delivery.\nBeautiful handmade porcelain from Paris France. In a whitewash finish\u00a0Dimension 48x17x34cm \/ 19\"x6.75\"x13.4\" Can take up to four months for delivery.\nBeautiful handmade porcelain from Paris France. In a whitewash finish\u00a0Dimension 35x23.5x23cm \/ 14\"x9.25\"x9\" Can take up to four months for delivery.\nBeautiful handmade porcelain from Paris France. In a whitewash finish\u00a0Dimension 5x3.5x7 cm \/ 2\"x1.4\"x2.75\" Can take up to four months for delivery.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In a small art booth in Chiang Mai, Thailand, as it rained outside and my friends wandered away down the market street, I decided to stay and haggle. This is how I started. I wanted two paintings: one blue and black, an ethereal river town at night, the other black and orange and yellow, silhouetting a fisherman in a small boat, floating in front of a tree line. It was July 2012, I was 18 on a cultural immersion\/community service trip. I wanted souvenirs.\n\"I told you I can't do 1200, help me out. I want the paintings, they're great, but that's too much.\nThe vendor, a lean man in a loose-fitting shirt, probably close to my age, looked agitated but eager to sell. Getting him to drop the price first, I figured, would trigger a momentum of discounts against his better judgement. If haggling is a staring contest, he blinked before I did.\nA lot of the vendors made this point, and I had a default response.\nMy dad taught me this trick, most useful against car salesmen. To get a good price, he told me, be willing to walk away.\nI don't mind budging from a lowball offer. At this point, I thought creating a back-and-forth would loosen everyone up.\nStaying without my friends was a little reckless, but I love haggling. Vendors overvalue and I try to bring value down. 1200 to 700 Thai Baht is only $36 to $21, but it's exciting to negotiate. No matter where I travel, everyone knows numbers and everyone speaks money.\nHe pulled out a small flip phone, dialed and turned away from me. I heard him speaking quickly in Thai, quicker than Thai already sounded to me.\nI set my price and I stuck to my price.\nBack to the phone, he spoke quietly, as if I could eaves drop. A minute passed, then he hung up. Turning back toward me, he grabbed some tissue paper, wrapped each painting individually, then together, then put them both in a plastic bag.\nI gave the village scene to a good friend, but I kept the fisherman illuminated by sunset colors. It hangs in my mom's office in our Scottsdale home, a prize and a memory, where I can see it every day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Caudwell childrens charity provides funding (maximum \u00a32000\/year) for ABA programmes to families who meet their eligibility criteria.\nDr. Carbone runs an introductory workshop for parents and therapists once per year in the UK. His workshops are excellent and should not be missed!\nOrchard toys sells a variety of puzzles and games that are both educational and fun!\nDifferent Roads to Learning, based in the USA, sells a variety of materials, books, toys, and training material to families running ABA programmes. The company ships internationally.\nLDA Learning sells a variety of educational resources for children with special needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We have been having a part the initiation of lots of important projects such as mass housing, energy projects such as dam, oil and natural gas projects, sky scrapers, infrastructure, highways, social facilities in Turkey and more than 30 countries for more than 20 years.\nTotal manufacturing area is 8000 m2 in our Samand\u0131ra Factory; Ozturk Container manufactures all types of Flatpack Containers.\nTotal manufacturing area is 25000 m2 in our Ferhatpasa Factory; Ozturk Container manufactures all types of ISO Shipping Containers, Worksite Type Containers, Prefabricated Buildings and other Special Types of Containers.\nTotal manufacturing area is 8000 m2 in our Gemlik Factory which is located in Bursa City; Ozturk Container is using as a depot for second hand ISO Shipping containers and makes all repairing works.\nExporting more than 80% of our company's production,creating important significant value in Turkey and international levels, carrying international projects by establishing solution partnership in many countries.\nOzturk Container, by sourcing the materials used in production from leader suppliers reveals the importance of product quality. Thanks to the experienced application teams, maintains leading position in the sector by completing maj\u00f6r projects in a record period of time.\nOzturk Container gives great importance to P&D and R&D activities and recognises the responsibility of developing new functions, duty and products suitable for the new era, in addition to proven customer satisfaction values, producing fast and economic solutions under the TSE, ISO 9001:2008, OHSAS, CE security documents, offering the highest quality products and services, still continues to add values in Turkey and international platform.\nOur Company continues its production activities successfully for Middle East, Africa, Europe, South America, Australia and Far East market.\nMAKES A DIFFERENCE WITH THE QUALITY AND TIMELINESS OF ITS SERVICES BEYOND THE CUSTOMER EXPECTATION.\nThe reliability of our industry have created the World's trends in parallel making the continuous improvement of all business process and achieve corporate objectives in this direction.\nCustomer orientation, networking and negotiation, reliability, democratic and participartory environment, respect for people and environment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In my last post I was talking about how I was having trouble finding motivation to paint. One thing that I find always makes me want to return to the painting table is to look at a model I recently painted which makes me happy. Sometimes I'll even bring a model to my computer or to the office so I can look at it and recharge my creative juices. Remembering this I thought a good way to do this while also updating my blog is to finally add the last two Batman Miniature Game figures I finished painting to my gallery. Cell phone pictures are fine for WIP shots but they just won't do for a gallery so I set up all my gear and took some proper pictures.\nThis is the first miniature I painted for the Joker Crew. I'm really happy with how he came out, both the skintone and his undershirt. I also attempted some of my first freehand tattoos. I even researched some examples of prisoner tattoos and tried to re-create them. Strangely this guy has a neck tattoo but no scalp tattoo on his bald head.\nThis guy was a complete pain as there were some bad casting I had to deal with. I also wanted to paint the bat as aluminum but couldn't get it to look right. I am happy with the hoodie. These two miniatures were some of the last miniatures I painted with my Golden High Flow Acrylics before I decided I didn't want to struggle with the glossiness of the paints anymore. It's too bad as I could tell how working with such a small selection of paints was helping me mix colours more confidently.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajguv b/data_all_eng_slimpj/shuffled/split2/finalzzzajguv
new file mode 100644
index 0000000000000000000000000000000000000000..9e836960441dea3dcd4a22670c5d12346265d2b5
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzajguv
@@ -0,0 +1,5 @@
+{"text":"Staying in Brisbane situates you in prime position to explore some of Australia's best destinations, including rainforests, beaches, islands and waterfalls.\nSouth Bank is a cultural, social, educational and recreational precinct located on the southern banks of the Brisbane River. As Brisbane's premier lifestyle and cultural destination, South Bank has 17 hectares of lush parklands, world-class eateries, stunning river views, free swimming facilities such as Streets Beach, The Wheel of Brisbane, walking tracks, playgrounds, picnic and barbecue areas with hundreds of delightful events all year round make it the perfect place to relax and unwind.\nLone Pine Koala Sanctuary is an 18-hectare Koala Sanctuary located only 12km from Brisbane City. Meet more than 130 koalas, hand-feed kangaroos and encounter other Australian wildlife such as wallabies, wombats, dingoes, snakes and even crocodiles at Lone Pine Koala Sanctuary nestled on the Brisbane River.\nPristine crystal blue waters & an untouched environment located right on Brisbane's doorstep. Moreton Island is the place to hand feed wild dolphins and snorkel around an old shipwreck in pristine crystal blue waters. Crystal-clear lakes and lagoons exist among tall sand dunes where you can go sand tobogganing. As the world's third largest sand island, it's a taste of tropical paradise and the perfect day trip.\nSun, sand, and sea - the Gold Coast's star attraction is its stunning beaches,including the world-renowned stretch of sand at Surfers Paradise offering beachside shopping, dining, bars and clubs. Beyond the beaches, discover the subtropical hinterland for rainforest walks and waterfalls. South East Queensland is home to the biggest and best amusement and theme parks in the Southern Hemisphere, with exhilarating rides and wet and wild water parks.\nSouth East Queensland's best-kept secret is the picturesque mountain rainforest region to the west of the Gold Coast. Often referred to as the green behind the gold, the area offers a wonderful change of pace to the beach with beautiful walking tracks, look outs, restaurants, art galleries and a myriad of natural attractions. There's also a whole host of fun and adventure filled activities to experience including tourist-friendly Tamborine Mountain, stunning Numinbah Valley, UNESCO World heritage listed Lamington National Park and also overlaps with the Scenic Rim.\nDiscover the coastal charm of the Sunshine Coast and explore its vast beaches, surf spots, waterfalls, thriving food scene at Noosa and the many natural treasures of the surrounding hinterland. Get up close to and learn all about native and exotic wildlife a Australia Zoo, visit the region's markets or take a drive to the Maleny and the Glass House Mountains for gourmet food, epic views and boutique hotels.\nByron Bay is an easy-going coastal town on the north coast of New South Wales well known for its beautiful beaches, relaxed vibe and free-spirited alternative lifestyle. It's a popular holiday destination, for holiday makers, backpackers and celebs. This little slice of beachside paradise offers a variety of activities, craft markets, music festivals and you'll often spot dolphins, whales and turtles. With heaps of art, hippy culture and friendly vibes Byron Bay is the perfect place to cheer up, slow down, chill out.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"recreational football and cheer league in Frisco, Texas.\nEtablished in 1997 and run by volunteers, the FFL provides opportunities for 5-12 year old boys and girls to participate in tackle football, rookie tackle football, flag football, and cheer. Our focus is on instruction and inclusion, with the motto: \"Youth First, Sports Second\"\nAll practice sites will be assigned by the FFL.\nAIR ASSAULT Divisions 2nd grade-6th grade will be separated by age and\/or gender based upon registration numbers at the Board's discretion.\nStudents who are home schooled must present evidence to the Board from the State of Texas (or similar governing authority) to confirm current grade level. Home-schooled students must be in grade level 6 or below to participate.\nThe FFL was a part of our family for seven years. Over that time, we watched as our boys grew up with the sport of football. They looked forward to the 'combine' each spring and then the 'draft' each June, they anxiously waited to hear about what new players their teams picked up. Every August, just before school began, the practices started and the evenings became filled with physical activity. September brought the games. Each Saturday was now game time, and the preparation of the past weeks practices allowed the boys to compete on the field. Finally culminating in the 'playoffs' and, for the top finishers, the 'frisco bowl'.\nOur boys learned respect for their teammates, their coaches their cheerleaders and for themselves. They learned how to not give up, how to push harder each time, how to react and think on their feet. They also learned how to win and lose gracefully. They made lifelong friends with their teammates, and after it was all over, with their opponents, which are now their teammates in Middle and High School.They were lucky to be a part of this experience, especially during the time which safety was becoming such an important factor in the sport. They learned to tackle properly, how important proper fitting protective gear is and most importantly, how to be tackled.\nFrom a parents perspective, it was a great experience. We had a good time dressing up in our 'team gear' and cheering from the stands. We met great people who, without the FFL we would never have had the opportunity to meet. We also formed lifelong friendships, and we still get together as a 'team' to celebrate every now and then.\nFrom a coaches perspective, assistant coaching my two boys teams at the same time was a huge commitment but it was also a learning experience and a great opportunity to spend real quality time with my sons that I will never forget and I will always be grateful for.\n\"I couldn't be more pleased with our Frisco Football League cheer experience. From the experienced and dedicated coaching staff to the lessons and values my daughter learned from team sports, we benefitted greatly from joining the FFL. So much so, I truly believe without it my daughter would not have been as prepared for years of competitive and school cheer. Thank you FFL!\nMy daughter joined the Fighting Irish at age 6 in 2011. We had hoped she would get on a team with friends from school, but it didn't work out that way. Though disappointed at first, she quickly grew to love cheering with her new friends. As parents, we loved not only watching the girls but also those boys out on the field. That first season, we were welcomed in by the Fighting Irish football family, which made it that much more fun. At the end of 2013, we learned that our cheer coach would not be back and the team would dissolve putting Vivien back in the draft. We also learned that only two girls (twin sisters) were planning to return. I couldn't imagine Vivien not being part of the Fighting Irish. I was a cheerleader in middle and high school and decided I would step forward to serve as head coach for the team. I enlisted two of my girlfriends, and we built a new team for 2014. That season, for the first time ever, Fighting Irish Cheer won Best Cheer at the former awards program. Our football team was there to cheer us on. I was fortunate enough to coach for three seasons until our kids aged out of FFL. The football and cheer families of the Fighting Irish were so dedicated and inclusive. I gained 10 extra daughters, whom I will always adore. It was fun to watch both groups of kids start off with limited skills and grow each year, culminating in playing in the FFL Super Bowl in our final season. FFL ignited a passion for traditional sideline cheerleading in my daughter, and I'm so grateful I got to share the experience with her.\nThe single most organized youth sports league I have ever had the privilege of being a part of . I have four (almost) grown children who were all involved in the league between football and cheer. I could not have asked for or even imagined a better experience for my kids and I certainly could not have imagined the lifelong friendships between families that developed as a result.\nOur son had an amazing experience with the FFL. He learned at an early age the importance of sportsmanship, grit, teamwork, and commitment. These lessons helped to shape the young man he is today, both on and off the field. Without these qualities he would not have achieved his goals of becoming a college athlete.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When Milwaukee's long-standing residency requirement was lifted, many questioned whether municipal workers living in neighborhoods such as Enderis Park would leave the city, opting for life in the suburbs. Six months after a county judge voided the residency rule, it appears Enderis Park residents are largely staying put.\nScott McGroarty, president of the Enderis Park Board of Directors, said he has not witnessed an exodus of city workers from the community.\n\"Residents who are with the school system and the police and fire departments are staying here,\" explained McGroarty.\nMcGroarty credits much of Enderis Park's appeal to the strong sense of community that deeply resonates with residents.\nCraig Auterman, a former teacher who worked for 21 years for the Milwaukee Public Schools, lives in Enderis Park. Outside his home at 67th Street and Lisbon Avenue, Auterman expressed his fondness for the neighborhood, adding, \"It's a great place to raise a family.\" Auterman acknowledged that he would have appreciated having the option to live outside city limits when he originally accepted the position with MPS, but he has no regrets about living in Enderis Park.\nAnother former MPS teacher and Enderis Park resident, Kathy Badger, explained that the neighborhood was a good option for family-oriented city workers while the residency requirement was in effect. \"This is where a lot of the police, firefighters and teachers would live,\" said Badger, who has remained in Enderis Park because she and her family did not want to move away from the friendships and vibrant community they found within the neighborhood.\nLocated on the western fringe of Milwaukee, Enderis Park was the kind of community that caused concern during last year's debates over the residency requirement. Mayor Tom Barrett worried that without the obligation to live within Milwaukee city limits, municipal workers would leave for communities in surrounding areas, such as those in Waukesha County.\nIn spring 2013, Barrett predicted that within a decade of the residency requirements being lifted \"approximately half of public employees\" would move outside city limits.\nAccording to the Multiple Listing Service (MLS) website, fewer than 50 homes are currently for sale in the Enderis Park neighborhood, which has approximately 1,500 residences. In the month of August, only three new single home listings were reported, compared to four new listings in August 2013.\nLori Sommervold, a resident and member of the Enderis Park Board of Directors, said she has not noticed an influx of homes being put on the market.\nAs much as I really wanted to, I was able to hold myself back from simply posting the comment here \"DUHHHH!!!!!! Of course not, its only been six months!\" and that's probably good, I wouldn't have come off as being too intelligent and certainly lacking in class, but it does sum things up well I'd say.\nHonestly though, if you think about this for even a minute or two, most city employees have jobs that pay enough for them to own a home, assuming of course that they don't blow up their credit report so those who still rent and not own are doing so by choice but how many people do you know that would, even if they could, just decide one day that they're going to move somewhere else and then-BAM!!-a few weeks or months later, they're gone! Old house prepped to sell, put on the market, offer accepted, closing finally completed and they've also already found their new home outside of the city, moved in and its all done, all in less than six months? That may be THEORETICALLY possible but who in their right mind would think that in only 6 months, if it did happen, it would've happened in big enough numbers to be noticeable?\nI stop at this website on a somewhat regular basis (this time it was just because the story was also promoted on JS Online with a link) but I'd suggest the editors choose their subjects a bit more carefully than this one, because to actually think that after 6 months this would be a possibility (that a noticeable change had already occurred) and was worth interviewing people and then publishing it makes me think someone's not quite up to the task on that job!\nLive in Enderis Park for 14 years and love it. Have worked for MPS for the last 16 years as an Occupational therapist for special needs children. Lived in Tosa before working for MPS . Moved into MPS and bought a house due to residency rules. Would not have if it were not for the residency even though I had emotional ties to the neighborhood. My mother grew up here. I did NOT want residency to change. As many many MPS employees did not as well. But didn't hear much of our views on the News. I do know of three MPS employees who have moved their families from Enderis Park and out of the city and another family who is planning on putting their house on the market soon. I am glad to hear that so far there has not been a mass exodus, however I didn't think there would be 6 months in. Now 10 years from now that I fear might be a different story. I worry about selling my house 5 or more years down the road not 6 months in. Still feel lifting the residency was not a good decision for Milwaukee or our neighborhood. I believe in the future it will make it harder for us to sell our homes and having city employees get paid by our tax dollars and pay property taxes elsewhere will hurt the city in the long run.\nI believe the opposite is happening to our fantastic neighborhood Enderis Park. People are wanting to live closer to a metropolitan City. Enderis Park has alot to offer those who seek urban dwelling. We are centrally located to Downtown Wauwatosa, resturants, shops, major hospitals\u2026 and also Mayfair Mall and shopping attractions.. We are less than 5 minutes to Downtown Milwukee attractions and a hop to the Brewers Stadium. Not to mention our award winning County Parks such as the Menomonee Valley that offer fantatic bike trails! We have a strong community of residents in Enderis Park whom take pride in thier neighborhood. We offer four concerts in the Park each summer and an arry of social activites throughout the year. People are attracted to the neighborhood charm but also one of Milwukeee's Best Kept Secret Neighborhood!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Furry hugs are amazing medicine, but sometimes you need something more. Here's two guest submissions from two awesome furries who love the community and have something more to give.\nEye candy is best to start an article, so check out this adorable puppy Taebyn. He's silly but has something serious to bark about. Even a fantasy world of talking animals has members affected by suicide. It can even be a special issue because of how surveys show this fandom has many young and LGBT people. When I met Taebyn in person at Further Confusion, the local community of the con just lost a young member who took his own life, but I wasn't able to share about it for privacy. So this post is for remembrance as well as help and good vibes.\nMy name is Drew, or better know as @TaxBeast. The past few months have been what I feel to be some of the most rewarding I have had in this community. Through my years on twitter, i've always been known as \"the guy who posts fitness\" furry. It's always been a hobby of mine. However I felt like I wanted to give something back to the community after having a lot of positive memories over the past decade. There has always been a somewhat consistent flow of of people asking what I do to to stay in shape, how to build muscle, or just how to get a better self image. For a while, I wasn't sure how to answer. I had worked years in the nutrition and supplement industry, and even worked as a personal trainer for a bit.\nIn the Fall of 2017 I had a plan. I wanted to get like minded people together, and start a group with the goals of knowledge and motivation in mind. I talked to @FaarTheRam on twitter after seeing that he too was a fur into fitness, with a similar drive to want people to succeed. Less than a week later the original channel \"The Furry Fitness Chat\" was made. A month or two went on, we added admins, and in no time we reached over 100 members.\nThis led to making a guide to share everything I knew and for people to take as they wanted. I wanted to make something that was clear and to the point, something that I felt, and still feel that most of the health industry lacks. I began writing the \"TAXBEAST WEEKLY GYM PROGRAM\".\nWithin is an explanation of why I wrote the guide, an introduction to terminology and training terms, a four day workout plan, a way to find your nutritional needs, diet routines and tips, and finally supplements that I get frequently asked about.\nIt's taken a mere 6 months to write this guide, created a safe and toxic free group for people to have an extra drive to better themselves, and help guide a twitter page for like minds to share and discuss their experiences. Follow us at @Fur_Health.\nThanks guys for sharing your good vibes and knowledge! Furries have some of the most creative people I know \u2013 add successful and healthy too. That's what comes from following your passion far beyond what ordinary people do. If anyone gets good results or help from this, please share in comments or on Twitter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nike Women's Shoes Just $39.99! Down From $65! Shipped!\nAt Proozy, you can score these Nike Women's Tanjun SE Shoes for only $39.99 down from $65 PLUS FREE shipping! That's 40% off!\nNamed after the Japanese word for \"simplicity,\" the Nike Tanjun SE Women's Shoe offers an elegant, modern design for an elevated everyday look.\nNike Men's Hoodie Just $29.99! Down From $55! Shipped!\nHurry over to Proozy where you can get this Nike Men's Hoodie for just $29.99 down from $55 PLUS FREE shipping!\nStay stylish and comfortable in the Nike Men's Club Fleece Pullover Hoodie. Its soft fleece fabric features an updated, slim hem and cuffs for warmth without the bulk. A paneled hood with a drawcord allows for a custom fit, while the front kangaroo pocket offers convenient storage. Face the chill comfortably in the Nike fleece hoodie.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakjdy b/data_all_eng_slimpj/shuffled/split2/finalzzzakjdy
new file mode 100644
index 0000000000000000000000000000000000000000..28c91753385d8bf46b5a073718ed2cd60d96b460
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzakjdy
@@ -0,0 +1,5 @@
+{"text":"The threat of accidents and unplanned downtime in automotive plants has been greatly reduced by new wire rope scanning technology.\nBefore now, there was no way to see inside a crane's wire ropes, other than cutting through and therefore permanently destroying them \u2013 which clearly is not a viable option.\nThe new Wire Rope NDT (Non-Destructive Testing) device uses magnetic flux technology to enable a full cross-section inspection of the whole length of a crane's wire ropes \u2013 thus identifying a problem in time to take preventative action.\nThe Wire Rope NDT device is part of SCX's Advanced Crane Diagnostics \u2013 the new name for its condition monitoring services which analyse a crane's critical components using non-destructive techniques, such as thermal imaging, vibration monitoring and laboratory oil analysis.\nHe emphasizes that early diagnosis is especially important in the automotive industry as high duty cranes deteriorate much faster because of the crane's continuous usage \u2013 and because unplanned downtime can be so expensive.\nDespite recent statistics showing that the UK automotive industry is continuing its renaissance, there is evidence of a slowdown* and individual manufacturers are competing to ensure that their own sales increase.\n\"Investing in a sophisticated planned preventative maintenance project helps our customers improve health & safety and avoid unexpected faults and costly downtime which can disrupt production lines and result in critical deadlines being missed,\" says Mr Fletcher.\nMore information on Advanced Crane Diagnostics from Street CraneXpress.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2008 Ford E350 Cargo Van Heavy Duty Suspension, Ladder racks, Only 100,395 Original Miles, Ford 5.4L Liter V8 Triton Engine. Automatic Transmission, Air Conditioning, Anti-Locks Brakes, Power Steering, Intermittent Wipers, Auxiliary Power Outlet. Financing Available, Door to Door Nationwide Delivery is a snap. Call us today! (845)365-3900. This is an excellent little single rear wheel cube van. It has been fleet maintained, and serviced regularly. In our 4 Bay service facility we perform a 110 MULTI-POINT INSPECTION & highway test drive every vehicle prior to it's sale. Basic things like a properly running engine, smooth shifting transmission & cool AC are a must. We perform an underbody inspection to ensure that the vehicle is dry of leaks & in safe condition. A CARFAX report is run on every vehicle we offer. The report, summary & CARFAX resources are available to you our customer absolutely free.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Likewise, all the guests benefit from a welcome package upon check-in, breakfast, breakfast, access to the gym, free wireless internet inside the suites and in all the hotel's public spaces.\nLikewise, this suite's bedroom is equipped with a spacious dressing room, with an ironing table and an iron.\nShower cabin, hair dryer, Bulgari hotel cosmetics, underfloor heating, towels, toilet.\nThe Family Suite has its own balcony, equipped with a table and two chairs. This space is just perfect for the smokers, as well as for those who want to admire the city.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hey guys welcome back to my blog, i figured i have not posted any food recipe this year and i figured i let you guys in on one of my recipes that has now become famous in my household.\nOnce it is ready, take out of oven and serve hot. Can be eaten with eggs or sauce.\nYum. I'm definitely going to try this.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The former Church Street School for Music and Art space at 74 Warren will be the home of German School Manhattan, \"a new private international school offering bilingual education for kindergarteners through fifth graders,\" says the Commercial Observer. With 10,000 square feet spread over two floors, the school is shooting to open in time for its fall session. The Commercial Observer also says that German School Manhattan will be working with German School Brooklyn, but it's not clear what that means.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakxfp b/data_all_eng_slimpj/shuffled/split2/finalzzzakxfp
new file mode 100644
index 0000000000000000000000000000000000000000..b49926fd6974fb66af0d9bc5adcfc5732521aa1b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzakxfp
@@ -0,0 +1,5 @@
+{"text":"North Cyprus has been represented at the \"CMT 2018 Stuttgart\" caravan and motor tourism trade fair, which is considered one of the world's largest and longest holiday and tourism fairs for consumers.\n360 cities were represented in the fair with 2,190 participants from 100 countries attending.\nMinister of Tourism and Environment Fikri Atao\u011flu stated that tourism fairs are of great importance given the value of tourism to the country, as they allow tourism operators to present their programmes to consumers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our Used Trucks Red\/Blue Swooper Feather Flag features the word \"Used\" emblazoned in white upon a vibrant red field. Below this is a blue field with the word \"Trucks\" printed upon it. This flag is ideal for any new or used car\/truck dealership who is looking to promote the sales of used trucks. The flag can be displayed in an array of different locations including directly in front of the showroom, along the roadside, at trade shows, and much more. If you want to grab the attention of prospective customers with an eye-catching flag, which succinctly informs them that you provide used trucks, this is the perfect flag for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There is plenty of room to roam in this great Oakhurst home featuring 5 bedrooms on 2nd level, PLUS 1st floor den & 1st floor full bath! 2900 square feet and perfectly located on a quiet street with a large yard, close to elementary school & pool\/clubhouse.You will love this giant kitchen with beautiful cherry cabinets, tons of counter space with granite, stainless appliances, pantry, recessed lighting, & eat in table space with counter stool space as well!The family room features cool wood beams on ceiling, NEW carpet & gas start, brick fireplace. Solid 6 panel doors throughout. All bedrooms are generously sized w\/excellent closet space.The master features volume ceiling, walk in closet & remodeled bath! Convenient 1st floor laundry right off the garage. So many updates here: '19 New carpet in family room and den and paint in many rooms. '16 Master bath remodel,NEW siding\/Gutters\/Screens & granite counters! '14 HW floors and blinds. '12 NEW roof. '11 NEW garage door. Welcome home!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Could we change behaviours for the better by playing a game?\nLet me start by saying that I consider myself a member of the JFDI family even though I don't work for JFDI.\nWe at GAMIFICATION+ were very lucky to have shared an office with JFDI so I got to know some of the projects they work on pretty well.\nJFDI does a lot of interesting and innovative work with IoT so I offered to write this blog post for them. That would give me a chance to do some reading on how can IoT and gamification work together and try to come up with some ideas of my own on how we could do that.\nI will start by defining what gamification and IoT are, then I will talk a bit about some ideas that have been explored and researched in academic literature and some of my own ideas on gamification and IoT.\n\"Gamification is the process of game-thinking and game mechanics to engage users and solve problems.\" (Zichermann & Cunningham, 2011). Another definition is given by Huotari & Hamari (2012) that say that \"Gamification refers to: a process of enhancing a service with affordances for gameful experiences in order to support users overall value creation\".\nBoth definitions give us a feeling of what gamification is, but I know from my own experience it can be hard to visualise exactly what they mean.\nLet's examine an example of a game and a game mechanic used in that game and then an example of gamification that is using the same game mechanic.\nThe game I use for this example is Pictionary. A very well-known game, where the goal is for the player to draw a word to help their team \u2013 partner guess what the word is. If at the end of the game your team has guessed more words that the other team you win.\nThe main game mechanic in Pictionary is the timer. You have 30 seconds to draw the word that you are given, and your teams must guess what it is in that timeframe. Watch this clip from Jimmy Fallon up to 2:47 to get a taste.\nNotice how when he is running out of time he is more creative and draws the stethoscope? That's the effect of the timer. A timer applies pressure to players. You get excited or stressed because time is running out and you want to get the solution to the problem. Timer also sets the pace of the game depending how much time the player is given to perform a task and how hard the task is.\nThe same game mechanic is used in many sports like football and basketball. Teams play differently when time is almost up than when they got plenty of time.\nPictionary's aim is to entertain people. The James Bond commercial aims at entertaining people while marketing two of the biggest brands on the planet.\nWe use gamification to help us solve some problem utilising the power of fun and engagement that games offer us.\nAs identified by Atzori (2010), \"Internet of Things can be realised in three paradigms \u2013 internet-oriented middleware, things-oriented sensors and semantic oriented knowledge\".\nThe definition that demonstrates better what we will talk about in this blog post is the following from Gubbi (2013) that says that IoT is consisted of \"Interconnection of sensing and actuating devices providing the ability to share information across platforms through a unified framework, developing a common operating picture for enabling innovative applications\".\nBasically, a lot of devices that talk to each other and can create and share data that give you useful information on how you drive your car, your house or business is run and many more applications.\nAn example is Amazon's Alexa that you tell her \"Alexa, play the Ace of Spades from Motorhead at maximum volume\" and Alexa tells your speaker to play your favourite song at full volume without you having to lift a finger.\nInterconnectivity, voice recognition and cloud technologies make these devices look and sound \"smart\".\nIn most cases gamification is used to nudge behaviour change. Let's say for example you want to start exercising more and get fit. You go and buy a FitBit that can track your workout performance and transmit the data to your phone. Your phone then gives you visual feedback on how well you are doing. You can also compete with your friend on how many miles you run this week and how many calories you burned.\nGamification is nudging you to put exercise in your life and it generally works better if we have some data about the behaviour we are trying to change.\nIn the beginning you run a mile a day. Then you go up to 2 miles a day and so on and the progress you are making is recorded let's say for a year. By the end of this year you will have enough data that patterns will emerge and reveal how things really work.\nMaybe you never run on a Saturday because Saturday is your lazy day, or you run more every Saturday because you are not tired after work and you got time. The data you gathered will reveal these small insights and patterns that can help you achieve your goal of getting in shape.\nGamification and IoT will also appear a lot in the marketing field. For example, it will be easy for beacons located in a supermarket to send your smart watch the latest discounts on sour cream that you love so much (I love sour cream!).\nInspired by JFDI's work with IoT that can control devices in buildings like lights, heaters, fans and air condition units, let's examine how gamification and IOT could help us solve one of humanity's biggest problems, energy consumption.\nThe article is called \"IoT-enabled gamification for energy conservation in public buildings\" and explores the same idea I have been discussing with JFDI. Putting gamification together with IoT could potentially educate people on how to be more environmentally friendly.\nNow remember we said IoT can track your progress? How about tracking how much energy you save? The system can nudge you to take actions that lead to consuming less energy, slowly changing the way you think about energy consumption. Games are great at catching your attention when something needs to be done, like building a new house or a dealing with a new enemy that appears on screen. The same can happen when you forget your garage lights open. Nudging action that saves energy will eventually create energy saving behaviours.\nWhy not play in teams? A big part of games has to do with humans being social beings. We love working together to achieve a larger goal. Games like Halo and World of Warcraft prove that every day with thousands of players logging in to play with other people.\nHow about the whole neighbourhood having a common goal to save a certain amount of energy or the total energy consumption to be less than a specific number?\nThe whole community would be working together to defeat the bad guy and make a change in the environment.\nThere could even be an energy saver of the month award for the household or business that was most energy friendly. Then we could organise a meetup where the winner of this monthly title would share their insights on how to be more energy friendly with their community.\nHow about bringing the community of energy savers together on an online forum or a Facebook group?\nThese are all things that games do very well and we can learn from them.\nI believe that games can play a big role in educating people on being more environmentally friendly. We need to design a good gamification system that is being fed the appropriate data These data will be generated from a robust IoT system which is exactly what JFDI are doing at the moment.\nNumbers do give us a better perspective on our environmental footprint as citizens and as businesses. I know a lot of businesses strive to be more environmentally friendly so could that be an idea that can help us all?\nVasilis Gkogkidis is working for our friends GAMIFICATION+ as a gamification trainer and designer. He has a masters from the University of Brighton with a distinction and is ready to start a PhD at the University of Sussex. He is researching how games can change our lives for the better and what can we learn from them. He is also a Lego Serious Play facilitator where he helps people achieve their business goals by building Lego! Send him an email if you too think that play and fun can be a good thing for the workplace and our everyday lives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I really don't know about you, but one of the most maddening things that I find about people is that they don't listen and don't follow instructions. We all know that happens so often in everyday life. We know that people are told things and they don't even listen. It could be to do with our health or our money but we often ignore the advice that we are given. Or sometimes people are given guidelines but they ignore them and just do it their own way.\nI am writing to point people to what God says in the Bible. But God wants them to listen and to follow His instructions for life. For instance Jesus said one day, \"I am the way, the truth, and the life: no man comes to the Father, but by Me\" (John 14.6 \u2013 the Bible). If you follow that instruction you will become a Christian, a follower of Jesus Christ.\nIf you come to understand that Jesus is the Son of God you'll believe what he said. He said I am the way to God, you can't come to God any other way. Why then do people go to religion? Why do people try all sorts of different things?\nPeter (one of the early preachers) he said, \"Neither is there salvation in any other, for there is no other name under heaven given among men where by we must be saved\" (Acts 4.12 \u2013 the Bible). Peter doesn't give you a choice or alternatives. He says on the authority of what God has said in the Bible there is no salvation any other way.\nWe need to be rescued, we need forgiveness, we need to be brought to God and there's only one route because God decided it and it's through faith in Jesus Christ.\n\u00b7 Believe on the Lord Jesus Christ.\nGod says if you follow my instructions you'll be saved. I trust that God will richly bless you today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakxfy b/data_all_eng_slimpj/shuffled/split2/finalzzzakxfy
new file mode 100644
index 0000000000000000000000000000000000000000..d674b698d1c23b5da759ff88539129e58c28ee89
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzakxfy
@@ -0,0 +1,5 @@
+{"text":"Have you ever dreamed of living in a European country chateaux? This home is your chance to live that dream! This truly magnificent brick manor home is nestled on approximately 2.39 acres overlooking the double bend of the banks of the Grand River. Easy water access for boating. A real stunning private oasis! This custom-built home was built with only the best of materials & attention to every detail throughout. Lovely circular staircase & back staircases! Spectacular custom paneled library with special molding & cabinetry, book shelves, book ladder, fireplace, & a domed hand painted ceiling, painted by the Artist who painted the Capital building murals.\nDocument Count 4 Legal Description Lot 22 Walmar Est.\nThe data relating to real estate for sale on this web site comes in part from the Internet Data exchange (\"IDX\") program of Greater Lansing Association of Realtors. IDX information is provided exclusively for consumers' personal, noncommercial use and may not be used for any purpose other than to identify prospective properties consumers may be interested in purchasing.\nCopyright\u00a9 2019 Greater Lansing Association of Realtors. All Rights Reserved.\nI would like to get more information regarding listing: 231571 @ 5925 Old Oak Court Lansing, MI 48917.\nThis is a one-stop real estate website for all your home buying and selling needs in East Lansing, MI.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At the High Court in Edinburgh on 25 November 2014, Lady Wolffe sentenced Alexander Gay to 12 years imprisonment after he was found guilty of the attempted murder of six people by wilful fire-raising.\n\"Alexander Gay, you have been convicted by the jury after trial of the attempted murder of six individuals in the early hours of 1st January of this year.\nYou had been at a party at a first floor flat in the early hours of that day. The flat was one of several in a converted dwelling house in a small village.\nThe party was attended by friends and neighbours of the occupier of that property. Most of the people at the party were known to you.\nAt some point in the evening your dog was unruly and you were asked to take the dog home, but you yourself were invited back. However, you took offence at this. There was a minor altercation as you left in relation to the cans of lager you had brought. The matter should have ended there.\nYou returned home with your dog. However, you then embarked on a course which led to dreadful consequences for many others that night. You took a can of petrol or other accelerant from your shed and returned to the dwelling house with the flat containing these friends and their neighbours. You set fire to the common stair immediately below this flat. This narrow stair was the only means of escape. The ferocity of the fire and smoke that rapidly ensued meant that the only avenue of escape was cut off. The common stair was wholly destroyed.\nThe fire and smoke also quickly engulfed the first floor. The five individuals who had remained in the first floor flat you had left not long before were now trapped. Out of dire necessity, four of those five individuals leapt from the windows. All suffered injury. These varied in their severity. Some of these were burns; a number suffered multiple fractures of their lower limbs. One of your victims broke both ankles and had numerous pins inserted. Another broke several bones in the lower legs. These individuals have lost significant mobility and will continue to live with the physical consequences and pain of the injuries they sustained that night.\nTragically, however, one of the individuals at the party was unable to escape out the window. He suffered burns to 30 per cent of his body area. He has undergone several skin grafts. He suffered damage to his optic nerve and is now blind, and will be for life. This individual spent more than six months in hospital, before he was transferred to as rehabilitation centre. He has little feeling in his arms and legs. He cannot stand or wash himself, and he now requires full-time care. It is unlikely he will return to the employment he had prior to the fire, or indeed to any employment.\nThe host of the party lost his home that night and all of his possessions within it. Others lost personal items. Several of your victims continue to be effected by their experiences, which must have been terrifying; and they are suffering stress, panic attacks or fearfulness and interrupted sleep. All of this arose directly from the foolish and despicable actions you took that night.\nThe other flat on the first floor was occupied by an elderly gentleman. While he escaped from the fire with relatively minor injuries he died in hospital some eight days later from the consequences of a heart attack. The medical evidence did not establish a causal link between his death and the fire. But it remains the case that his life was also put at risk by your own actions.\nYou have been found guilty, by unanimous verdict of the jury, of the attempted murder of all six of these individuals.\nIn relation to your previous convictions, these span some 12 years. Although one involves an assault, none of these brought you to the High Court, where you now stand. It is only by sheer good fortune that no one was killed as a consequence of your actions. The impact of your actions, however, has blighted the lives of these other six individuals, to a greater or lesser degree; and in respect of one of your victims, catastrophically so.\nI have had regard to the Criminal Justice Social Work Report. It assesses you as a high risk of reoffending. You otherwise appear to have little recollection of your actions that night. You appear to have little insight into the consequences of what you have done and its devastating impact on others. You expressed no remorse to the author of that report. I have also had regard to the psychiatric report provided to me.\nI have also had regard to all that has been said on your behalf in mitigation. I note the circumstances of mental ill health and the social isolation. In the light of your previous conviction, it is suggested that an extended sentence would be inappropriate. I agree with that. I note, however, that you have expressed no remorse through your counsel.\nIt is only by good fortune that no one died as a direct consequences of your actions. You set fire, with an accelerant, to the only means of escape of people known to you and who had offered you hospitality. You attempted to murder these individuals in a place where they had gathered together to see in the New Year and where they expected to be safe. But you attempted to murder these individuals by wilful fire-raising. What you did on that night was premeditated, cowardly and despicable.\nHaving regard to all of these matters, the sentence of imprisonment I impose for the attempted murder of these six individuals is 12 years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here is the traffic update for the month of May, a full three days early before month end. Given the long weekend in North America, the origin of most of my traffic, there is not going to be much to write home about on the 31st.\nLast Saturday I wrote my 3 months Google Adsense progress report, where we decided to focus primary on the Oil Insights blog and see if we could bring about an improvement in traffic and numbers. This decision was driven by a struggle to break 3 dollars a month in Adsense revenues and I thought that building up on the sites that are working in terms of click through rates and cents per click may be a worthwhile cause.\nCrude Oil Insights \u2013 Growth trend after Technorati Top 100.\nDesi Back To Desh \u2013 Remains the 800 pound gorilla for the month of May, with 1,200 unique visitors a month, 11,000 page views and Google Page Rank 4.\nCrude Oil Insights \u2013 After breaking into the Technorati 100 early this week, traffic is beginning to pick up and the blog has also started to climb in the Google search engine results. Crude Oil Insights posts have made it to page 2 and 3 and will hopefully make an appearance on page one before June is out. Oil Insights will close May with 400 unique visitors and 3400 page views and Google Page Rank 2.\nLearning Finance \u2013 Originally number 2, from a daily and weekly traffic benchmark, Learning Finance, has slid to number 3 to make room for Oil Insights to grow. 650 unique visitors and 1485 page views, still waiting for a Google Page Rank.\nStartup Insights \u2013And finally the newest and latest arrival to the scene check in with 300 visitors and 900 page views.\nCombined with three other sites that right now don't do Adsense advertising the total traffic volume for May is about 3,000 unique visitors, 62,000 spiders, and 18,000 page views per month. Google Adsense impressions for May will clock in just under 4,000 per month, while the number for April for 4,600. For the site portfolio to hit the budgeted 10 dollar number for June, the 3,000 unique visitors need to become 6,000; the 18,000 page views per months need to cross 40,000 page views and the Google Adsense impressions need to cross 10,000.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I do think today you want Pull Chain Ornaments item, so that you tend to be within the right website. Right now you're studying Cowboy Western horse Saddle Ceiling Fan Pull chain extender write-up. And also I am hoping you will find beneficial info with right here before you decide to obtain Pull Chain Ornaments item. Presently I would like to talk about Cowboy Western horse Saddle Ceiling Fan Pull chain extender. Minor details that Cowboy Western horse Saddle Ceiling Fan Pull chain extender product is created simply by Marco firm.\nThis saddle fan pull is sure to add a Western look to any room. Made of resin and nicely detailed, the fan pull measures 3\u2033 high x 2\u2033 wide x .50\u2033 deep. Comes with an approximately 7\u2033 inch ball chain with fastener. Easy to attach to your existing fixture. Perfect for your rustic decor!\nFor more info about Cowboy Western horse Saddle Ceiling Fan Pull chain extender selling price in addition to critiques, you are able to click THIS LINK . I hope this submit may help you to think about with buying of Cowboy Western horse Saddle Ceiling Fan Pull chain extender product. Or perhaps can be that will with this particular submit one to uncover other item that is related having Cowboy Western horse Saddle Ceiling Fan Pull chain extender or it could be greater.\nWhen you study this Cowboy Western horse Saddle Ceiling Fan Pull chain extender, I assume that this Cowboy Western horse Saddle Ceiling Fan Pull chain extender product is suitable with you. If you think a person nevertheless must discover merchandise in addition to Cowboy Western horse Saddle Ceiling Fan Pull chain extender, you'll be able to investigate with just about every Pull Chain Ornaments type or maybe make use of seek pack within this web site. Next in case you are confident to buy Cowboy Western horse Saddle Ceiling Fan Pull chain extender, remember to just click BUY BUTTON down below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"THE COMPLEAT ASTROLOGER -by Derek and Julia Parker- got me started on serious astrology, has everything you need, great illustrations.\nThe ASTRO ANALYSIS: Your Personal Horoscope series -by the American AstroAnalysis Instiute in New York -is the only series I know that divulges the truth about Earth and the Ascendant or Rising Sign(sold at Amazon.com).\nTHE AMERICAN EPHEMERIS FOR THE 20TH CENTURY -by Neil F. Michaelson- the most popular book of tables for calculating the planetary positions(available for noon or midnight). Of course, everything's on computers now, but it's a great book to fall back on.\nTHE AMERICAN EPHEMERIS FOR THE 21TH CENTURY -by Neil F. Michaelson- a follow up of the previous book for more recent and future years.\nTHE AMERICAN BOOK OF TABLES -by Neil F. Michaelson- used for calculating the Ascendant, Midheaven, and remaining house cusps in the Placidus system. Computers do most of the calculating nowadays, but its always useful to know how to do it the old way if you can't always rely on computers.\nAll books by Hilton Hotema: THE GENESIS OF CHRISTIANITY\/ THE SECRET LIFE OF JESUS THE ESSENE\/ MYSTERY MAN OF THE BIBLE\/ APPOLONIUS THE NAZARENE -these books and others expose the Roman Catholic Church and their false Messiah.\nMARS\/EARTH ENIGMA: A SACRED MESSAGE TO MANKIND -DeAnna Emerson- the Face on Mars, the Pyramids, the 24 hour day, and other similarities which link us to the Red Planet.\nNote: I am currently a proponent of Flat Earth, allthough Hollow Earth still has elements of truth to it.\nTHE SMOKY GOD -Willis George Emerson- fascinating account of a Norwegian fisherman and his son spending two years in an Edenic hollow earth.\nREVELATIONS OF THINGS TO COME -Earlyne Chaney- interesting section on prophecy and the Hollow Earth(pages 134-139).\nTHE TWELFTH PLANET -by Zecharia Sitchin- another blockbuster purporting that life on Earth and in our solar system was periodically seeded, supervised, and disrupted, by the inhabitants or gods of an as yet undiscovered planet beyond Pluto.\nTHE NEW IDEALISM -Michael Dargaville- a groundbreaking book on the nature of energy, physics, and the New Age movement.\nTHE TAO OF PHYSICS -Fritjf Capra- modern physics in the light of eastern mysticism.\nAll books by or on NOSTRADAMUS, the French seer of the 16th century.\nThe HOLY BIBLE, the King James version in particular, and the prophetic books of the Old and New Testaments, especially parts of Daniel, Ezekiel, and Isaiah, the latter parts of the four Gospels, and the entire book of Revelations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaludl b/data_all_eng_slimpj/shuffled/split2/finalzzzaludl
new file mode 100644
index 0000000000000000000000000000000000000000..36f62537483527ebd0a1a8522665bb2bcf2dd6cc
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaludl
@@ -0,0 +1,5 @@
+{"text":"Jun. 22, 2018 4:30 p.m.\nA person was taken to hospital in ambulance and a horse had to be put down after it was hit by a car on Friday afternoon on 132nd Avenue in Maple Ridge.\nThe incident happened at approximately 4 p.m., and the horse was not being ridden at the time. Ridge Meadows RCMP stated it appears to have escaped from a nearby property.\nThe horse lay at the bottom of a steep bank, about three metres below the road, according to firefighters.\nA bystander who came to help at the scene was pinned under the horse.\nThe animal's back legs were both broken, and a veterinarian euthanized the animal.\nThe patient was transported to Royal Columbian Hospital in New Westminster.\nA person associated with the nearby North Fraser Therapeutic Riding Association said the tragic events show the need for a safe shoulder for riders along 132nd Avenue, west of the Equi-Sports Centre. There is a barrier between the road and a multi-use pathway along more eastern areas of 132nd Avenue.\nShe noted that since it has been recently re-paved, traffic speeds along the road even as they share it with horses.\n\u2022 More details to follow as they become available.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"first grade teacher worksheets elegant predictions smart math best of super t.\nquadrilaterals smart teacher math worksheets polygon practice.\nfree worksheets library download and print on over a dozen smart board math lessons ts are now availabl.\nbest images about reading and writing super teacher worksheets l smart math an animal article.\nprintable summer month calendars from super teacher worksheets months smart math best general images on in.\nworksheets smart teacher math super reviews money teaching year 2 student pencil and in color.\nmedium to large size of kindergarten kids printable grade worksheets reading super teacher adding and subtracting fraction.\ngrade reading comprehension fact and opinion worksheets test super teacher 2 skeleton key on smart.\nsuper teacher worksheets addition the best image collection download and share smart math t.\nannual goal setting worksheet yearly teacher and professional development template 3 reasons 7 worksheets free excel templates smart sample management g smar.\nyou re the teacher worksheets smart math student pencil and in color.\nprintable worksheets for back to school goal setting goals worksheet smart teachers teacher math sm.\ntemplate worksheet free download reciprocal for teacher smart math worksheets.\nsuper math worksheets beautiful teacher has skip counting for numbers 2 reptile word search multiplication number.\nsuper teachers worksheets smart teacher math student pencil and in color.\nsmart goals worksheet template free student doc goal setting download this sheet teacher ma.\nsmart goals worksheet ate excel printable paper goal setting sheet introduction latest smar.\nliterary elements lesson plans worksheets reviewed by teachers identifying adjectives grade adjective theme in literature son adject.\nelegant super teacher worksheets login best of probability worksheet ship and new sets inspirations smart beautiful.\nadjectives worksheets super teacher opposites sorting worksheet descriptive words smart math for grade 1 adjective.\nperimeter and area people lesson plan m flowers teaching worksheets ideas surface smart teach.\nsuper teacher worksheets reading check out our comprehension passages on for grade 2 c.\nwrite handwriting practice cursive 2 a smart kids teaching writing worksheets for grade language arts teacher math super.\nsmart teacher worksheets the best image collection of free ready to smar.\nthis site contains fun and unique worksheets for children who are beginning the learning experience whether you a preschool or kindergarten teacher what time is it 2 smart math.\ngoals worksheet smart worksheets for teachers effective time management skills money teaching year 2 3 yea.\nsimilar images for smart teacher worksheets math library.\nsmart teacher worksheets collection of free math.\nthe father of modern lesson plans vocabulary and printable worksheets bundle deleted smart teacher math.\nsmart teacher math worksheets free printable for busy teachers.\nconnect the dots to reveal picture this worksheet gives students practice identifying consecutive numbers great for a basic elementary math lesson teacher worksheets smart.\nbest math super teacher worksheets images on smart teachers 6 download them and try to solve.\nsmart super teacher worksheet best of nice paring fractions worksheets ideas printable than modern inspirational workshe.\nsummer is a great time to learn how play an instrument reinforce basic music teacher worksheets homework ideas smart math best super general images on.\nsuper teacher worksheets has printable to help students learn count tally marks charts first grade graphing smart math best worksh.\nrecommendations super teacher worksheets login lovely math puzzle and new sets inspirations smart.\nteacher worksheet template smart math worksheets 8 daily templates free word excel documents download.\nsmart teacher worksheets images about super general on check out this math thousands of printable.\ngrade word problems worksheets of multi step smart teacher math.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"brain teasers 3d puzzles wooden puzzles 3d logic games jigsaw puzzles brainteasers iq test. Brain teasers, interlocking wood puzzles. Produced in Teak wood.\nPost a city you are interested in and I will try to run a screen on Capital IQ. ... Teakwood Capital, L.P. Terra Verde Partners LLC Texas Energy Holdings, Inc.\nParts for Nautique and Correct Craft boats. PCM Engine parts.\nSecret Shopping is a tool used by market research companies to measure quality of retail service or gather specific information about products and services.\nExploring, explaining and protecting the natural world. Since 1853.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FIGURES released recently by the Federation of Small Businesses show that UK small businesses are facing a growing headache because of the complexity of employment law.\nThe FSB provides a free legal helpline for its members to call when they have queries about employment legislation. And the helpline saw an overall rise in the number of calls of eight per cent in 2006 with over 76,000 calls coming in last year at an average rate of over 200 per day.\n\u2022 Other forms of discrimination such as on religious or sexual orientation grounds by 92% and 56% respectively.\nThe rise in queries on discrimination topics is due to new or proposed legislation on age, religion and sexual orientation. The increase in questions on pensions and retirement is down to proposed Government moves to act on pension provision for all workers. More calls have come in regarding information and consultation also because of changes in flexible working and age discrimination rules, requiring firms to inform and consult their workers on these issues.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pia Otte has a PhD in Political Science and a MPhil in Development Studies from the Norwegian University of Science and Technology (NTNU), located in Trondheim, Norway. The focus of her studies has been on the social adoption and the development effects of solar cookers (household\/institutional) in the developing world. Within her studies Pia worked closely together with the NUFU Solar project. More information on this project can be found here: http:\/\/www.ntnu.no\/ept\/nufusolar.\nMarch 2016: Pia Otte now works at the Centre for Rural Research as a senior researcher.\nJune 2014: A New Turn Toward Solar Cooking - Pia Piroschka Otte. Learn how solar cookers are accepted and used in Burkina Faso and India. This paper discusses six case-studies from the two countries where Scheffler reflectors were used by a diverse selection of institutions. Read the paper: A (new) cultural turn toward solar cooking\u2014Evidence from six case studies across India and Burkina Faso. More information about how solar cooker use can vary between cultures can be found on the solar cooker dissemination and cultural variables page.\nNovember 2009: Published master thesis on solar cooking in Tanzania (VDM publisher) entitled \"Cooking with the sun: An analysis of Solar Cooking in Tanzania, its adoption and impact on development\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamsgx b/data_all_eng_slimpj/shuffled/split2/finalzzzamsgx
new file mode 100644
index 0000000000000000000000000000000000000000..7a9fcd4f9cd35daa1a02d3c824f8acb6be2e73b9
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzamsgx
@@ -0,0 +1,5 @@
+{"text":"\"Leganes\" had three matches at the beginning of January, but just one match in \"Primera\", the team won \"Real Sociedad\" at home and \"Vilarreal\" at home in the country Cup. \"Leganes\" lost in a return match to \"Villarreal\", showing a worse football, so we don't recommend you to bet on the away team's win.\n\"Betis\" must rehabilitate at home as the team can't win for four rounds in a row. Moreover, the team lost in recent matches, so now the team must try hard to score at least a point. We bet on the home or draw for 1.3.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In a world of diminishing television ratings, smaller circulation and readership figures and flat radio audiences, a brand's social media presence has never been more important. Facebook Performance Report, Online Circle Digital's latest quarterly report into Facebook engagement by brands, reveals that the most popular by conventional metrics are not always the most successful online. ABC News, Men's Health Magazine, Girlfriend Magazine and Bride to Be Magazine have collected the most overall likes, although the brands' engagement is not as high as others.\nThe report's Top 30 Facebook pages by engagement for first quarter 2015 is topped by Melbourne's Fifi and Dave \u2013 closely followed by Sydney's 2Day FM, which is struggling in the more conventional metric of radio ratings. Other somewhat surprising media players in that list include SCA's regional Mackay station Hot FM, Western Sydney's urban station The Edge 96.1, indie parenting site Babyology and Pacific's teen title Girlfriend. As the presence of 12 radio brands in the Top 30 brands by engagement might suggest, radio was by far the most engaging category on Facebook for Q1 this year.\nLucio Ribeiro, Online Circle's head of strategy and delivery and co-founder, told Mediaweek that radio was so successful in its digital extensions because the industry understands the need for consumers to engage via multiple devices. \"If you look at the report at the top performers as far as engagement goes, it's pretty much just media at the top. First is radio, then magazines and newspapers and then television shows. In terms of engagement, the category of media is the one that has absolutely nailed Facebook.\n\"There are a few reasons \u2013 the most important is that they have nailed the multiple devices and they really encourage real-time engagement. They actively engage with their consumers, most of whom are multitasking, and they encourage consumers to engage with them across all devices. When you watch a television show or listen to the radio, they suggest you go to Facebook and leave a message, write a tweet, share with a friend. The brands doing really well see social as an amplification channel for their own content. It's interesting \u2013 when you listen to the radio or watch television, you can see them developing a theme, then provide social as an extension of that theme. They really understand the 24-hour primetime concept \u2013 there is no primetime on social because it's always on,\" he continued.\nAs to future issues, Ribeiro pinpointed the complexity of Facebook's suite of advertising products. \"Every single day, Facebook is launching a new ad format or an ability to target even more. Facebook has elevated itself from being a social channel to a media channel. In the future, you will see less conversation around 'How many fans do you have' \u2013 everything is going to be elevated to a very professional level. This year, the discussion about having a business case on Facebook will be gone because it's a given. For an organisation, the abilty to measure everything and target and scale is great,\" he added.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Very low mileage for its age. One owner from new. It has had regular oil and filter change and runs without any problems. Inside is in excellent condition. Non smoker car. Externally there are marks on the bumpers and car park dinks on the doors and. The windscreen wipers were recently changed and the tires were changed at 33,000 miles but still has plenty of grip. Due to work commitments, the vehicle can only be viewed in Stamford CT by prior arrangements. Please contact me via Ebay to arrange viewings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The governing board is made up of elected members of the community, these are voted in by the governing board, parent governors are nominated and votes are taken from parents. These governors along with the Headteacher are responsible for the management of the school. In executing its responsibility the Governing Body meets regularly in order to address such issues as employment of staff, the school building, financial management, curriculum and assessment, behaviour and discipline. Each year the Governing Board contributes to the School profile which is a report on the work and development of the school.\nTo find out who is on our governing board please click on the link below.\nMinutes of the full Board of Governors and committee meetings are public documents. If you would like to see any of the minutes, please ask at the school office.\nDates of future board and committee meetings are available.\nWe always welcome suggestions, feedback and ideas from parents and carers. Please contact the Co-Chairs of Governors, Mrs Jane Bowley and Mrs Carolyn Wood, via the school office.\nHowitt Primary Community School's conversion to academy status took place on 1st January 2019. The governing board of our school is very keen to take the opportunity to work with the governing boards from the other six schools in the Trust. We are very excited about the increased opportunities that being part of the Trust will present for our pupils, staff and governors as we plan joint activities and training events. You will of course, be kept informed about these.\nAs governors we support our school in a range of ways for example, by attending curriculum \"wow\" days, music and art workshops and pupils' concerts and performances. Indeed, the two of us and one of our parent governors, Donna Walters, attended the final performance of the Y4\/Y5 pupils led by the PopUK charity during the first week of this term. This performance was the result of a full day of musical fun linked to a focus on the emotional impact of children using social media. Parents\/carers of the pupils also attended the performance and were clearly delighted to watch their children having achieved so much in such a short space of time. The enjoyment of the pupils was infectious \u2013 it felt as if we were at a pop concert! The whole event proved that we are, as our vision states, an \"ACE\" school and that we Achieve, Care and Enjoy.\nTo continue to raise standards in writing across the school with a particular emphasis on spelling, for all pupils so that they are at least in line with national averages with a significant proportion above.\nTo ensure the phonics outcomes for all pupils in KS1 are back on track and continue to improve.\nTo increase the proportion of pupils who are working at greater depth in reading, writing and mathematics so they are more in line with national.\nTo narrow the gaps in outcomes of disadvantaged pupils.\nEach governor has specific responsibilities linked to these priorities and to the curriculum and other areas such as safeguarding, attendance and the learning environment.\nLast term saw the launch of our school's Wrap Around care facility, \"Evening Owls\", and we are really pleased that several of you have taken advantage of this opportunity. The pupils are well cared for and enjoy the activities that the staff plan and provide for them at the end of the school day.\nFinally, we are grateful for your continued support and encouragement of your children and we look forward to seeing you at future school events.\nIf anyone wishes to contact us, we can be contacted via the school office.\nWe are delighted to be continuing as Co-Chairs of the Governor Board of Howitt Primary Community School. We both have an extensive background in education, leadership and management and have been involved with this school for a number of years in various ways.\nThis year we are very proud of our school and of our dedicated staff and governors who work hard to provide your children with meaningful and varied experiences to move them along their learning journey and prepare them for life in the community. We are sure that you are aware of how much the school has continued to improve under the strong leadership and high expectations of our headteacher Mrs. Lynn Pilling.\nYou will be aware that the school has taken steps to become part of an academy and work alongside seven other Derbyshire primary schools to establish the Embark Multi Academy Trust. This will be finalised in December 2018; we are very excited about the possibilities that this opens up for our children and the staff. There will be updates for you as the process rolls out.\nThis year the school has made additional improvements to the outdoor learning spaces including the addition of a trim trail and a new Key Stage Two canopy. Parents, staff and governors have given of their time to support some of the improvements to the outdoor learning environment. The indoor learning environment has continued to be a focus for further improvement and staff have been supported by the school's learning environment lead teacher and the named governor for this area.\nGovernors are proud of the fact that the school continues to offer an effective and very successful provision for two-year olds linked to our excellent nursery provision. We currently support eight two-year olds and have a waiting list. Our nursery is a lively and welcoming environment that uses every opportunity to extend learning and prepare children to be \"school ready\".\nAt the start of this academic year, governors attended an after-school meeting that established this year's school vision in the light of previous achievements and the way forward. This meeting involved all staff, pupils, governors and members of the local community. Please feel free to pop in and look at the display created from this meeting that shows our aims for the future and our past achievements; this display is also on our website. We have also attended homework project showcase events, creative curriculum visitors ('WOW' days) concerts, performances and celebration assemblies.\nWe wish to thank you for your continued support for, and encouragement of, your children and your increasing involvement in the life of the school.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The New Year will ring in good fortune for Bitcoin, as French tobacconist shops start selling the cryptocurrency from January 1st onwards.\nEuropeans will initially be able to purchase Bitcoin in 50,100 and 250 euro denominations from approximately 4,000 tabacs across France.\nIn exchange, the customer will receive a coupon to redeem for bitcoins online with KaplerK.\n\"The store own doesn't need permission from the Bank of France to sell these coupons,\" president of the Conf\u00e9d\u00e9ration des Buralists Phillipe Coy told Europe 1.\nBimedia, a point of sale platform, has also signed on to simplify the process. There will be no complicated wallet transfers, and buying Bitcoin coupons will be as easy as purchasing a magazine.\nAccording to The Next Web, customers will need to share photo ID and personal information, not to mention a 7% commission fee. But if that's the price of progress, sign us up!\nFrench tobacconists shops, called tabacs by locals, are a cultural institution. They are to France what bodegas are to New York \u2013 so much more than a 7-11 or Circle K.\nUsually equipped with tables, chairs and sometimes televisions, tabacs are community gathering places to drink coffee, chat and smoke. People might spend all day interacting with friends and neighbours outside.\nBut how will the often older clientele feel about the bitcoin boom?\nThis reflects a broader push in the industry to stay technologically relevant in the digital age.\nThis plan has attracted its fair share of skeptics.\n\"Customers are meant to buy magic pieces of paper to be redeemed for cryptocurrency,\" David Canellis wrote for The Next Web.\nBut as far as we're concerned, haters gon' hate. And come January, thousands of French people will have access to the cryptocurrency changing the world.\nTo keep up-to-date with this story, and to read the latest crypto news, subscribe to our weekly newsletter here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanqsy b/data_all_eng_slimpj/shuffled/split2/finalzzzanqsy
new file mode 100644
index 0000000000000000000000000000000000000000..5929e2beeb6e5b49e2327630275350e2f2dc6ed4
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzanqsy
@@ -0,0 +1,5 @@
+{"text":"MEEKER I HopeWest is providing a new weekly grief support group in Meeker and is open to anyone coping with the death of a loved one.\nMending Hearts meets every Monday at 5:30 p.m. in Richards Hall on the top floor at 368 Fourth St. in Meeker.\nAlthough each person's grief is unique, those coping with the death of a loved one share a common bond. Mending Hearts provides an understanding atmosphere where participants can openly share their thoughts and feelings as they navigate their journey through grief.\nEach support group begins with grief education and is followed by a small group discussion. Topics include: understanding grief and loss; coping skills in grief; dealing with emotions of grief; and adjusting to new roles.\nMending Hearts is free of charge and is open to anyone coping with the death of a loved one. Pre-registration is not required, and members may join at any time.\nFor more information please call 970-878-9383 or email AdultSupport @HopeWestCO.org for additional information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Customer from Washington writes \"My greenhouse was quite a distance from any water source and the design of the greenhouse did not support the usual rain gutter catchment system.... I came across your product... and I knew that it was the solution I had been looking for! I placed two saucers with rain barrels on the backend of the greenhouse and ran a connection hose through to the greenhouse interior. With the amount of rain I get in the Pacific Northwest my rain barrels are usually full no matter how much I use on my plants inside the greenhouse. It is also comforting to know, should I ever need it, that I have a clean source of drinking water! Thanks for creating such a great product!\"\n\"The saucers are working out very well for us. It assists us with placing gardens in urban settings without a water source. We fill the barrels to start with and the rain does the rest. I've attached a picture of a two barrel set up with one saucer filling both.\"\nFrom Pine Mountain School in Kentucky. They use our product in the Appalachia community to promote gardening because \" many of our participants have gardens that are not located right next to their houses or are only accessible by foot, we wanted to provide rain barrels that could be placed directly at the garden site, rather than at the house.\"\n\"I really like the RainSaucer. 65 gallons of rain water from 3 days of on and off rain in central New York - Seneca Lake of the Finger Lakes Region. I attach soaker hoses to drain into raised beds of perennials and veggies.\"\n\"(The RainSaucer) has served us well. We use the rain saucer to collect water out in Gaia's Community Garden in Pittsburgh, PA\"\nCustomer from West Virginia writes \"The Rainsaucer was an excellent invention and it is holding up very well. I attached this photo of my Rainsaucer setup, and I want to say thank you again.\"\nJim G. in Fairfield, CT wrote us \"We here at Bigelow Tea loved our RainSaucer and how well it complemented our Community garden. Not having a downspout was challenging but we knew we wanted to collect rain water that could be used for watering our garden; in addition to saving water, the temperature of the rain water is better for the plants as it is closer to its natural environment then city supplied water. ...\" Photo shows the RainSaucer in the background.\n\"I live in a Mediterranean climate in Southern Oregon.... I have 40 fruit trees, 30 raised beds and a medicinal garden...I have 11 rain barrels with a plan to increase them to 25. Before Rainsaucers, i was trying to figure out how to capture rainwater in my freestanding rain barrels and coming up with nothing. Now I am beginning to capture water to offset the decreasing precipitation in our region... Another small benefit is that when I do bio-dynamic preps I have all this wonderful rainwater to stir with. I am a dyed in the wool Rainsaucer fan and as my budget allows I am going to rainsaucer all my barrels and keep adding barrels and rainsaucers until I am prepared for climate change.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Managing the risks of a corporation is critical to its long term security and ultimately its success, but doing so effectively can be a time consuming and risky undertaking in itself. That's why many leading organisations work in partnership with professional advisors who dedicate their time to supporting organisations with their risk management.\nWith dedicated individuals available for expert advice and complete peace of mind, Alan & Thomas spend time supporting organisations at all times especially when they need advice and support with claims.\nWorking with some of the leading organisations across the UK in sectors including wholesale, manufacture and construction, we have built up a reputation of developing bespoke risk management solutions and providing the highest quality customer service and sound advice.\nHave a look around our commercial insurance policies for more information and we'd be more than happy to discuss any part in more detail or if you have any queries regarding our insurance solutions and risk management expertise, please contact us.\n40% of businesses that suffer a critical IT failure go out of business within a year.\nIf you would like to find out more about us and our management team, then read our Group Profile.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jews for Jesus is a Christian organization founded by Martin \"Moishe\" Rosen (1932\u20132010), a Jewish convert to Christianity. The organization is considered over-domineering and also stupidly named.[citation NOT needed] Jews for Jesus operates in several countries, with an estimated 30,000 to 125,000 adherents worldwide, targeting Jews for conversion to Christianity. The organization takes in about $20 million a year in donations. Jay Sekulow, Chief Counsel of American Center for Law and Justice and personal lawyer for Donald Trump, is perhaps their most famous member.\nTheir claims to being a Jewish organization are overwhelmingly rejected by Jewish religious denominations and their proselytizing activities have caused controversy among several other religious, as well as secular, groups.\nPotential converts are early on encouraged to think they can become Christians and stay in the Jewish tradition.\nJews for Jesus used a milk before meat approach. First Jewish potential converts were told that Jesus was the Jewish Messiah. Christians eager to convert a Jew were unlikely to point out that this is contrary to Judaism, see Jewish views of Jesus. Later converts were taught Christian beliefs that are foreign and in some cases incompatible with Judaism such as as original sin, the virgin birth, the doctrine of the Holy Trinity and the belief that Jesus was god.\nJews for Jesus seems to have responded to criticism because their website, as of February 2012, includes an article claiming the ancient Jewish Zohar included the Christian concept of the Trinity. No such passage in the Zohar exists. Judaism includes the concept of god as three, \"secret, hidden 'Wisdom'; above that the Holy Ancient One; and above Him the Unknowable One\" these three are different from Father, Son and Holy Ghost.\nThe organization is also known for targeting vulnerable populations of Jews. New Jewish immigrants and college freshman as well as senior citizens and interfaith couples are easy targets for the organization.\"\nJews for Jesus has been compared to a cult by some former members due to an authoritarian and 'spiritually abusive environment'.\nThe work schedule for the cult's paid missionaries was very long, with only 3 hours of private time each day. Workers hoped for a day off after 60 hours of work, but sometimes lost it as a form of discipline. Workers who did not satisfy their superiors were frequently screamed at in private or in front of colleagues. Victims were expected to remain humble and submissive. Workers could be ordered to make donations to their favorite charity, and since pay was low, the loss was a hardship. The organisation felt that it had the right to control the lives of its staff, even telling them whether or not they could marry.\nSince New York City has a large Jewish population, a lot of their proselytizing activity takes place in the New York City area. If you are Jewish and see someone wearing a red shirt with \"Y'SHUA\" on it, run. Or arm yourself, as appropriate.\n\u2191 \"About Jews for Jesus: How Many Jews for Jesus Are There?\"\n\u2191 See the Wikipedia article on Jews for Jesus \u00a7 Islamic and interfaith opposition.\n\u2191 http:\/\/www.jewishvirtuallibrary.org\/jsource\/anti-semitism\/Jews_for_Jesus.html Jews for Jesus] by Stephanie Persin.\n\u2191 Is Jews for Jesus a Cult?\nThis page was last modified on 23 February 2019, at 16:20.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Punctuation After Addressing Someone In A Letter is free HD wallpaper. This wallpaper was upload at April 25, 2019 upload by admin in .You can download it in your computer by clicking resolution image in Download by size:. Don't forget to rate and comment if you interest with this wallpaper.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaofbe b/data_all_eng_slimpj/shuffled/split2/finalzzzaofbe
new file mode 100644
index 0000000000000000000000000000000000000000..eb13af83bb1593721aa34e355a78d6e127d99632
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaofbe
@@ -0,0 +1,5 @@
+{"text":"This article is the first in a series on the South Manchuria Railway.\nToday, Northeast China is known as the country's Rust Belt. Overtaken economically by the coast, it's weighed down by decaying, hollowed-out factories, lumbering state-owned enterprises, and a high unemployment rate. But 120 years ago, Northeast China was the emerging industrial heartland of East Asia, and its massive economic potential was enough to attract the interest of expansionists in both Russia and Japan.\nThe Russians got there first. In 1897, they successfully pressured Qing dynasty officials into agreeing to the construction of a jointly funded but Russian-dominated railway across the northern part of the region. In May 2017, I traveled along this line \u2014 known as the Chinese Eastern Railway \u2014 from Manzhouli on the region's western border with Russia to Suifenhe on its eastern border. But the Russians had greater ambitions in the region than a single rail line. Another line \u2014 the South Manchuria Railway \u2014 was built around the same time. Running from the city of Changchun in central Northeast China, it terminated at the then Russian-controlled harbor of Port Arthur in what is now L\u00fcshun, in Dalian City.\nWhile it may not be well-known today, the South Manchuria Railway played a crucial role in the history of the early 20th century. Then, China, Russia, and Japan struggled for control over this key economic corridor \u2014 and for dominance over Northeast Asia as a whole. Its construction enflamed the anti-Western sentiments that fueled 1899's Boxer Rebellion and was a contributing factor in the outbreak of the Russo-Japanese War in 1904. It was the center of Japan's contentious postwar plans for expanding its economic and political influence in the region. And in 1931, it was the site of the Mukden Incident, a railway bombing the Japanese used as a pretext to invade and seize all of Northeast China, settling off a conflict with China that wouldn't be resolved until 1945.\nLocated at the foot of Baiyu Mountain, L\u00fcshun's train station was the southern terminus of the South Manchuria Railway and, by extension, the southernmost train station in all of Northeast China. The first L\u00fcshun station was built by the Russians in 1900 to serve Port Arthur, but by the time of their 1905 defeat in the Russo-Japanese War, they had already begun construction on a new station across the street. After the war, control over the region's railways went to Japan, in the form of the so-called South Manchuria Railway Zone. The Japanese then proceeded to finish the mostly built station and repurpose the old one into a police station.\nSince work on the station had largely been completed by the time of the Japanese victory, the 110-year-old L\u00fcshun station hall features a number of classically Russian architectural flourishes, including green roof tiles that resemble fish scales and a bell tower. One of the few surviving indicators of Japanese rule is the roof over the train platform, which consists of sheets of tarpaulin pulled over a wooden structure once used to house Japanese soldiers.\nTrains stopped running to and from L\u00fcshun Station in 2014, though the ticket office and waiting room remain open. People still sometimes stop by: the occasional tourist in search of holiday album-fodder and a few aged locals looking to reminisce.\nHead north 500 kilometers and you'll come to Kaiyuan. The Russians gained control of the area in 1902, and in 1903 they built a simple train station in the town. In 1905, the area was occupied by the Japanese, who built a new, two-story station in 1929. After they gained control of the region's rail network, the Japanese exploited it by establishing commercial districts in the vicinity of each stop. Today, although the region may be in decline, the station remains in remarkable condition for its age \u2014 its white accents have even been kept crisp and even.\nSlightly farther north, the city of Siping is another example of the economic impact the railway has had on the region. Prior to the line's construction, the area where Siping now sits was essentially empty wilderness. The above-mentioned Japanese economic policy helped develop the area's commerce \u2014 leading to the creation of Siping's first urban district \u2014 but the Japanese weren't the only ones trying to take advantage of the region's economic potential.\nIn 1916, China built the Si-Tao Railway, which stretched from Siping over 300 kilometers northwest to the town of Taonan. In Siping itself, the Chinese authorities established their own competing economic zone. Initially, both the Japanese and Chinese shared the use of the Siping station \u2014 known to locals as North Station \u2014 but the Japanese-run South Manchuria line eventually moved to a new building of its own. Thus, for a short time almost a century ago, Siping had two train stations: one Chinese, and the other Japanese.\nThis uneasy state of affairs continued until 1931, when the Japanese invaded Northeast China and seized control of the entire region. The day after the Mukden Incident, the Japanese garrison in Siping attempted to occupy the city. When they moved to seize the Chinese station, however, they found that a small Chinese force had fortified three parallel checkpoints controlling access to the building \u2014 collectively known as Qiazimen. This force was able to hold off the Japanese for three days, only surrendering after the arrival of Japanese field troops.\nToday, the only sign of this struggle \u2014 one of the first battles in China's 14-year-long war against Japan \u2014 is a single commemorative plaque. The Qiazimen checkpoints themselves are thoroughly dilapidated; their windows have been sealed with red bricks, red roof tiles lie strewn about, and their walls are spiderwebbed with cracks. It's a shame. In 1931, China was rife with internal conflict, and the skirmishes at Qiazimen were one of the very few points where the Japanese invasion force encountered any resistance. The site should stand testament to the courage shown by a few brave souls in the face of inevitable defeat, not the area's deteriorating economic fortunes.\nThe South Manchuria Railway is more than just a railway: For half a century, it was the fulcrum upon which the fate of East Asia rested. In other words, if you want to understand the historical scars that continue to inform the relationship between China, Russia, and Japan, you could do worse than first catching a train.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"While discussing the kind of films we are passionate about making and seeing, we grew frustrated at the lack of cinema directed and written by women. We were particularly surprised to find that even after the #metoo movement exploded a year ago in October 2017, there has been very little representation about the issue in films.\nWe feel that as young and old people of all backgrounds and sexualities are openly discussing topics like sexual harassment and work place inequality it would be a great opportunity to take part in that dialogue. We completed a short film script revolving around three women whose lives are affected by some of the issues raised in #metoo and felt it would not only be highly relevant but needed.\nWe plan to pitch it to media outlets in Sweden and the UK, primarily. Our main demographic is young girls and women but we feel anyone who cares about these issues and about seeing well made, grounded yet provocative female-lead projects would be drawn to the stories we're depicting.\nOur hope is that by watching this film, people will not only be able to relate but to feel challenged and moved. We hope to help raising awareness of the many intricate and often subtle ways in which issues like stalking or emotional abuse can affect women's lives.\nWe are seeking funding in order to ensure our cast and crew is paid and so the project we deliver is of the highest quality possible. We would also put the funds towards distribution and marketing.\nOur team is headed up by Martha and Karin, we are co-writers and directors. We have a lovely, diverse core cast of new actors; Raj, Ed and Marian and our talented and thoughtful post-production video editor, Mia.\nThank you for taking the time to read this, we hope that you'll get involved with us on this journey!\nWe plan to start shooting mid-December.\nIt's a short film, so it will be around 15 minutes.\nThanks for supporting The Sunday Project!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Order liquidation & wholesale signs and 1000s of other items on Sourcing.Bstock.com | Bid on B2B auctions directly from the world's largest name brand retailers and manufacturers including Walmart, Sears, Home Depot, GE Appliances & More!\nDistributor and Importer of wholesale knives, swords self defense products, 3D Art, and more wholesale merchandise. 1000's of items in stock. Check out our specials section for best prices on dozens of wholesale products at or below closeout prices.\nWant product updates for signs?\nGet FREE product updates for signs sent straight to your inbox. Configure your alerts to receive them as often as you want.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Arc Design's first project in Aspen, Colorado.\nSingle family home in the heart of Aspen.\nCapitalizing on the strength of the luxury real estate market in Aspen Colorado, Arc Design acquired this single-family home at 108 Midland Avenue in January of 2015. This house, consisting of an enclosed two car garage, multi-level fireplaces, a huge outdoor deck with mountain views, a contemporary design and located just a few blocks from the foot of Ajax Mountain and downtown Aspen, offered a great opportunity for Arc Design to enter into a new market and to be successful. Originally acquired to be a renovation project, Arc Design saw an opportunity to capitalize on the stellar position we were in after the acquisition, and remarketed the house and sold for a strong profit just a few months later. Having succeeded in this venture, the company is actively seeking other development opportunities in and around Aspen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The watch is back for 2017.\nBeyond: Premiered this week to a decent 0.31. While this isn't the best rating, if the series can sustain a 0.3 for the remainder of its run (8 more episodes), then it definitely has the chance to get renewed. Also, the whole season was released on the Freeform app already, so Freeform will definitely be taking that into account. I've started it as Leans Renewal until more data is available.\nShadowhunters: Premiered as Freeform's number three show this week (0.4, behind PLL, and the Fosters). I've started this as a Likely Renewal until we see how the other 19 episodes do. It may stay high, and do well, or collapse.\nGood Behavior: TNT is not Freeform. They do not wait around to renew, or cancel series. In previous years, TNT has cancelled their fall flops by the end of December. However, they also renew quickly. Point being, Good Behavior has not been cancelled yet, but not renewed. I feel that they are on the fence about this one. I've upgraded it to Leans Cancellation due to the impending fate announcement, ratings increasing to tie the series high over Christmas break, and its cheapness. There are two cast members in every episode, and the other cast members are not expensive.\nIncorporated: These ratings aren't terrible for the bar set on the bottom of the ground Syfy. It's pulled 0.1s-0.2s. I figure, if Syfy is willing to renew a show with a 0.0 rating, then they would have no problem renewing a show with Incorporated's generally higher average. I've marked it as Leans Renewal for the moment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaogmm b/data_all_eng_slimpj/shuffled/split2/finalzzzaogmm
new file mode 100644
index 0000000000000000000000000000000000000000..dc68e4ba39188c653b575d3dad18152e81fc07da
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaogmm
@@ -0,0 +1,5 @@
+{"text":"By Keith Cowing on October 26, 2007 6:07 AM.\n\"Bigelow Aerospace intends to spur development of a commercial space vehicle to take people into Earth orbit by offering to sign a contract worth $760 million with any company that can meet their criteria, company president Robert Bigelow says.\"\nBy Keith Cowing on October 24, 2007 2:55 PM.\n\"During the X PRIZE Cup Oct. 27-28, NASA's Centennial Challenges Program will offer prizes totaling $2 million if competing teams successfully meet the requirements of the Northrop Grumman Lunar Lander Challenge. The challenge will take place at Holloman Air Force Base, in Alamogordo, N.M. The purpose of the lunar lander challenge is to accelerate technology development leading to a commercial vehicle that could one day be capable of ferrying cargo or humans back and forth between lunar orbit and the moon's surface.\"\nBy Keith Cowing on October 24, 2007 9:28 AM.\nEditor's note: Have a look at these images from Genesis II. Either Bigelow Aerospace is going to run a gambling operation in low Earth orbit (\"Bigelow Bingo\") or he has lots of little itty bitty numbered satellites he is going to deploy.\nBy Keith Cowing on October 22, 2007 5:27 PM.\n\"In July, the company signed a two-year contract with NASA allowing its owners to use the coveted Moffett airstrip whenever they liked, along with the hangar space, in exchange for $2.6 million and the right to place scientific instruments on the planes. The deal, negotiated behind closed doors, was announced in mid-September.\"\nBy Keith Cowing on October 22, 2007 11:33 AM.\nDr. Mary Cleave Appointed to Board of Directors of Sigma Space Corp.\n\"Sigma Space Corp. announced the appointment of Dr. Mary Cleave to its Board of Directors, effective October 1st, 2007. Dr. Mary L. Cleave was the Associate Administrator for NASA's Science Mission Directorate until March of this year. She also served as deputy associate administrator (advanced planning) in the Office of Earth Science at NASA Headquarters.\"\nBy Keith Cowing on October 19, 2007 5:24 PM.\nBy Keith Cowing on October 19, 2007 12:35 PM.\n\"Qualifying lasted until the early hours of the morning with Centaurus the last team trying to qualify but failing at 1:00 am. At this point four teams have qualified to compete for this years chance at NASA's Centennial Challenge prize for power beaming which has a $500,000 prize. They are: ...\"\nBy Keith Cowing on October 19, 2007 11:25 AM.\n\"NASA announced Thursday it will conduct a new competition for funding that remains in NASA's Commercial Orbital Transportation Services Project, known as COTS. The new competition follows NASA's decision to terminate its funded agreement with aerospace firm Rocketplane Kistler of Oklahoma City, which repeatedly failed to meet agreed-upon milestones in its effort to develop and demonstrate commercial transportation capabilities to low Earth orbit.\"\n\"Thursday, NASA will provide an update about the status of the agency's Commercial Orbital Transportation Services Project, also known as COTS. NASA will host a news media roundtable at 2:30 p.m. EDT at NASA Headquarters, 300 E St., S.W., Washington. Alan Lindenmoyer, manager of the Commercial Crew and Cargo Program at NASA's Johnson Space Center in Houston, will participate in the briefing.\"\nEditor's 18 Oct note: Once again ESMD seems to be going out of its way to screw up media participation. The only notice I am aware of was emailed late this morning. I only got mine in my email account an hour before the event itself. Did ESMD just dream up this telecon today? Or did they know about it yesterday? Its almost as if PAO and ESMD just don't care any more.\nEither this proves that JSC PAO sends things out without reading them - or people in that office never took that Calendar 101 class. Either way the word \"inept\" pops up in my mind.\nBy Keith Cowing on October 18, 2007 10:35 PM.\n\"This afternoon teams had the opportunity to improve on their qualifying results of yesterday. Because of the bad weather yesterday, provisional qualifying was held indoors to see if teams could attach their climbers and show they could climb the ribbon, stop and descend the ribbon safely. All the teams except Centaurus succeeded.\"\nBy Keith Cowing on October 18, 2007 9:53 PM.\n\"SpaceX has successfully completed the Critical Design Review (CDR) for its first Falcon 9 \/ Dragon mission as part of the NASA Commercial Orbital Transportation Services (COTS) demonstration program. During a three day period, SpaceX hosted a group of over forty top level NASA representatives and key SpaceX customers for a review of the design of the Falcon 9 rocket, Dragon spacecraft and associated ground systems for the first COTS demonstration mission.\"\nBy Keith Cowing on October 10, 2007 11:42 PM.\n\"The Space Solar Alliance for Future Energy (SSAFE), a new organization advocating investment in space-based solar power technologies to address the planet's future energy needs, was announced today at the National Press Club. The coalition of thirteen leading research organizations and space advocacy groups focused their inaugural event on the announcement of a new study of space-based solar power led by the National Security Space Office (NSSO).'\"\nBy Keith Cowing on October 9, 2007 9:07 AM.\n\"RpK came in with a lot of promises and a half baked business plan and the market place is weeding them out,\" [David] Gump said. \"It's disingenuous to now say they were shocked - shocked - they would have to compete for ISS cargo business. That was plan[ned] in the original ISS cargo announcement. Everybody who bid knew it.\nBy Keith Cowing on October 8, 2007 1:49 PM.\n\"SPACEHAB, Incorporated (NASDAQ:SPAB), a leading provider of commercial space services, today announced its receipt of a NASDAQ Staff Determination letter on October 2, 2007 indicating that the Company fails to comply with NASDAQ Marketplace Rules 4310(c)(4) and 4310(c)(3), and that its securities are, therefore, subject to delisting from The NASDAQ Capital Market.\"\n\"Pickens' father made a fortune in oil, became a corporate raider, and now runs a hedge fund. He's worth $2.5 billion. No wonder his son looked to the heavens to make his fortune. Spacehab recruited the younger Pickens earlier this year to turn the company around. The 20-year-old penny stock firm has a market cap of about $5 million.'\"\nBy Keith Cowing on October 4, 2007 6:19 AM.\n\"And, [Richard Garriott] added, he is hoping to accomplish something on the flight that no other space tourist has done: take a spacewalk, an option Space Adventures announced last year for an extra $15 million. \"But that is NOT yet set in stone,\" he wrote. He has had two suited-up training sessions in a Russian underwater facility, he said, and \"It's MUCH harder than it looks.\" Air pressure stiffens the suit, and every movement is a struggle.\"\nBy Keith Cowing on October 3, 2007 3:08 PM.\nEditor's note: Word has it that Randy Brinkley has left Rocketplane Kistler.\nBy Keith Cowing on October 2, 2007 3:18 PM.\n\"Several companies in what would become Silicon Valley benefited from the ambitious goals and budget largesse of the Apollo space program,\" said Dag Spicer, the senior curator of the Computer History Museum, in Mountain View, Calif. \"The stringent quality and performance requirements of (integrated circuits) for Apollo allowed early semiconductor companies to learn at government (that is, public) expense, a technology that would soon have broad application and whose price would plummet as these companies perfected manufacturing methods.\"\nBy Keith Cowing on October 1, 2007 11:17 AM.\n\"SPACEHAB, Incorporated (NASDAQ:SPAB), a leading provider of commercial space services, today announced its receipt of a notice from NASDAQ Staff on September 25, 2007 stating that the Company does not comply with Marketplace Rule 4310(c)(3). Marketplace Rule 4310(c)(3) requires the Company to have $500,000 of net income from continuing operations for the most recently completed fiscal year or two of the three most recently completed fiscal years; or $35,000,000 market value of listed securities; or $2,500,000 in stockholders' equity.\"\nThis page is an archive of entries in the Commercialization category from October 2007.\nCommercialization: September 2007 is the previous archive.\nCommercialization: November 2007 is the next archive.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Mountaineer Volunteers of Wachusett Regional High School are once again extending an invitation to the district's senior citizens to join them at dinner dance on April 25.\nThe reality is that in addition to enjoying a full course dinner, those at the dinner will also be treated to music by a deejay, dancing (line dances with the kids are popular) raffles and prizes and a chance to enjoy some creative decorations.\nAt a recent Mountaineer Volunteers meeting, Club Officer Tara Donovan said everyone is approached for ideas.\n\"We just look for a fun theme that we know we can come up with really cute decorations for,\" she said.\nThe group was working on Pinterest-worthy centerpieces made up of crafted binoculars, safari hats, jeeps and leafy placemats. Expect verdant vines to add to the party atmosphere. The work started last June and the Volunteers meet to every other Monday until the big day.\n\"There's always a little bit of a crunch time when it gets down to it,\" Donovan said. \"But then it pulls together quickly.\nThe students also made up flyers and distributed them to area senior centers. They also have been collecting prizes and gift cards from local merchants for the raffle prizes.\nBagel Inn, Parrot Pizza, Princeton Pizza, Mexicali Restaurant, the Blushing Bee, Harvest Grill and others have contributed.\nA new deejay has volunteered his services this year \u2013 Benjamin Senior, a teacher at the high school.\nReservations can be made by calling the Holden Senior Center (no matter what Wachusett town you're from) at 508-210-5570. Make sure to call as soon as possible because the event has been known to get fully booked. There is no charge \u2013 all are guests of the Mountaineer Volunteers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tracey L. Armstrong joined Copyright Clearance Center (CCC) in 1989, the same year that Apple Computer, Inc. released its Macintosh Portable for $6,500 and Tim Berners-Lee was busy laying the groundwork for the World Wide Web.\nFlash-forward 20 years, and Armstrong is at the helm of CCC as president and CEO, a position she has held since 2007 after rising through the ranks from vice president of transactional services to executive vice president\/COO. Armstrong, who received an M.B.A. from Northeastern University, has seen CCC grow from 40 employees and $10 million in annual revenue to 200-plus employees and $239 million in revenue. And in those 2 decades, the digital world has grown and evolved at record speed.\n\"I think we've seen steady growth, and it's a credit to the CCC team and to the stakeholders involved,\" says Armstrong. In addition to a growing number of new staffers, CCC has been modifying the composition of its board of directors with several newcomers, including Marybeth Peters, the former U.S. Register of Copyrights, and professional authors Megan Marshall and Edward Tenner.\nResponding to the changing needs of today's digital world demands CCC's full attention. \"Licensing is evolving,\" says Armstrong. \"It's evolving because of the way that the users are interfacing with content.\" Integrating rights into that technology is key. \"The best technology is transparent. And the best licensing solutions for our users are transparent.\" She cites the Apple iTunes App Store as a prime example. With iTunes, users aren't aware that they're actually licensing music; they're just getting what they need, and then they're gone. \"And that's what we want to do with our licensing solutions,\" she says: make them easy to use and transparent. \"Keeping up with the demands of the market is critical because you have to keep in touch with user behavior,\" she says.\nCCC keeps the communication channels open in order to listen to users via advisory groups, international user groups, conferences, its biennial OnCopyright symposium (March 2012), and the ongoing CCC-sponsored Publishers Launch Conferences, led by Mike Shatzkin (industry consultant) and Michael Cader (founder of Publishers Lunch and PublishersMarketplace.com). Each of the venues offers insights into user needs and behavior and helps keep the ongoing dialogue fresh.\nToday's digital world is just that: It's now a global playground for content providers and publishers. As a result, CCC continues to expand its presence and breadth of coverage. This year, CCC is working with the German-based MVB, a business subsidiary of The Association of the German Book Trade, which represents 20,000 German book publishers. Armstrong explains that the MVB wanted to get a tool at the point of content for its readers and clients, and CCC is developing a German-language deployment of RightsLink. To date, CCC is licensing corporate employees in 180 countries worldwide.\nOne of CCC's other tenets is education, pure and simple. Armstrong is reaching the marketplace through different avenues, including social media, and providing information about the ins and outs of copyright and permissions. \"It is working,\" she says. \"The reason I can tell you it is working is because the number of constituents in our educational program has grown exponentially.\" CCC not only has a department devoted to custom programming in the U.S. and globally, but it has also partnered with other organizations in bringing CCC's educational message to YouTube, webinars, podcasts (Beyond the Book), and video (www.copyright.com\/content\/cc3\/en\/toolbar\/education\/resources\/copyright_basicsthemovie.html).\n\"One of our videos was chosen by Google and put into its copyright page,\" says Armstrong, adding that the in-demand videos are high-quality. \"And we do make them accessible to anybody,\" she says.\n\"Our No. 1 strategy is helping the customer,\" says Armstrong. \"We've been working with certain clients for decades. And their challenges are evolving, and our solutions are evolving to meet them.\" In the long run, quality definitely trumps speed, and CCC's solutions have to work well. And that can take time. After all, \"It's a marathon, not a sprint,\" she says.\nArmstrong is quick to point out the list of CCC products that have been developed to bring ease and continuity to the permissions process: RightSphere, RightsLink, ReadyImages, RightsDirect (a European subsidiary), RightsCentral for rightsholders, as well as licenses for academic and business.\n\"What I really am proud of is how CCC is so solution-oriented,\" says Armstrong. \"We do listen to the market, and we do respond.\" She credits one of CCC's latest tools called Get It Now as an example of helping the library market. A representative from the California State University (CSU) library system approached CCC staff at a trade show and asked for help with the library's interlibrary loan ( ILL) process. Budget constraints and the fact that the libraries aren't open 24\/7 were causing increasing problems. CSU library wanted to find a way to make the process easier for its patrons. Armstrong says Get It Now, which was released at the ACRL conference in April 2011, makes ILL more efficient by using the same workflow that library patrons use every day to access content and get the research they need. Get It Now also integrates the licensing into the process. \"Now, within minutes instead of days or weeks, library patrons at Cal State can access the full-text articles they need to get their research done 24 hours a day, 7 days a week,\" she says.\nArmstrong credits her \"fabulous team of committed professionals who want to get the job done for our customers\" with keeping their collective eye on the horizon. She also points to the delicate balance at play between staff and clients. \"I think to be in sync as CEO, you have to be in the market, you need to be present, you need to be listening, and you need to be part of the conversation,\" she says. \"It's been very important to me to make CCC more porous, more open, more present.\" And while she's the first to acknowledge the team's accomplishments, she still sees plenty of work ahead.\nPart of what drives Armstrong is her obvious passion for the work; the other is the intellectual challenge she encounters every day. When she's not on the road, Armstrong logs on to her computer at 5 a.m. to answer international emails and turns off her desk lamp at 9:30 p.m., after reading the necessary reports for the next day. Setting a high-level strategy is critical, but making sure that strategy is a collaborative effort for the entire team is just as critical. \"It's important to keep the strategy transparent for every single member of the CCC team, from the mailroom to the CEO, because every one of us is contributing to the success of our clients,\" she says.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today is the time of war, give a shout!\nGo shout this out to the skies!\nOr is the beat saffron, ponder on it!\nThe composition of this song is exactly like a nursery rhyme.\nWhen is this movie coming btw?\nlekin woh kaun hai jise hamari rashtriya bhasha samajh me nahi aati . use chullu bhar pani mein dub jaana chahiye .\nLOL SPS \u2013 you want a poetic translation? Woh bhi kar doongi, but I wanted to do a line by line one as Q requested it.\nthis composition is mind blowing but not even a bit of that essence could be seen in the english translation.\nMy father says he has read some of these lines sommewhere in his childhood as aa example of \"veer ras\"- has anyone got any clue regarding any such poetry ????\nAgree \u2013 this translation is in no way poetic, just trying to be \"accurate\". English cannot have such rousing poetry as they lack the harsh consonants, the kind westerners cannot pronounce \u2013 dha, dhrha, bha etc.\nIt might not be perfectly apt in terms of the poetic significance, but it very certainly solves the purpose of translating it in English. And pretty well, that too!\nThanks! I am glad it worked for you. But it would still be fun to attempt a poetic one as this does no justice to the original.\nBrilliant and Thanks( I was looking for this)!\n2. Bhaagawat ka saar: Bhaagawat is not Geeta. Geeta is Shri Krishna's dialog with Arjuna, Bhaagawat is Krishna's biography by Vyasa \u2013 a separate text from Mahabharata.\nThanks for those pointers. I did wonder about \"need\"! And as for Bhagvat \u2013 I was going on Bhagvad Gita!\nThanks for your help\u2026 I have to translate this for a college assignment. My take on the poetic translation\u2026.\nOn the tip of an arrow, let it all lay.\nGreat is the one who fights till end hour!\nNo lust for victory? No command on person?\nTo the battle skies, go out and roar.\nOr saffron, the rhythm of your heart, you must decide!\nIt is time for simmering fires to ignite.\nAarambh means beginning, but I guess poetic license can allow it to become march!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The South African Breweries (SAB) and South African Football Association (SAFA) today marked an extension of their relationship to build further onto their work of developing football at grassroots level in South Africa. The agreement, finalised in the run-up to the SAB League National Championships taking place from 9th to 13 July in Stellenbosch, is an extension and recommitment of SAB's involvement in grassroots football development, SAB has been the the title sponsor of SAFA's most senior and largest development third division league, the SAB League, for 13 years. \"SAB's investment in football at grassroots level is entrenched in its conviction that socio-economic change can be brought to SA's communities in this way. Football has a proven track record of playing a role in bringing positive change to people's lives and to communities. Grassroots football development is also critical in securing a sustainable and strong future for South African football. SAB is privileged to have the opportunity to work with SAFA in making a real difference at this level and to recommit to bringing social and economic change, \" says Dr Vincent Maphai, SAB Executive Director Corporate Affairs and Transformation. SAFA President, Kirsten Nematandani said SAB played a key role in the development of the game at grassroots level and this year's launch coincides with SAFA having recently launched the under 13 and 15 leagues for both boys and girls. The leagues are the cornerstone of SAFA's Technical Master Plan whose core objective is to aggressively drive the football developmental agenda countrywide. \"SAB is a true friend of football development and the extension of our (SAFA and SAB) relationship is good news for our youth because apart from fitting in well with our Technical Master Plan, this acquaintance will expedite grassroots football development across the country, \" said the SAFA President. \"Our 2020 Vision is to have a Bafana Bafana side that is consistently among the top 3 in Africa and 20 in the world. If we push this agenda aggressively, this goal can easily be realised.\" Entrenching SAB's commitment to grassroots football development in South Africa, the company introduced the new look of the SAB League, first known as the Castle League at the start of SAB's involvement in 2000 and more recently as the SAB U21 Regional League. Other than a new look and new name, the SAB League includes a Youth Diversion component in partnership with local NGO, Footballers4Life. The programme, which educates SAB League players participating in the SAB League National Championships in critical life skills, is facilitated by former local footballers. Amongst others, the Youth Diversion Programme focuses on fulfilling players sporting potential, helping them to manage well-balanced and empowered lives and refraining from risky behavior, particularly irresponsible alcohol use. The objective is for them to become powerful influences amongst their peer and in their communities as agents of social changes and positive role models. Skills transferred to the players during the two-day workshop, include computer skills (CV writing, financial management), insight into non-administrative football careers (physiotherapy, fitness training, sports psychology) and administrative careers (public relations, management) and an introduction to the Learners License course. \"We hope that the Youth Diversion Programme will offer players the opportunity to start each of them on a journey to becoming powerful influences amongst their peers and in their communities as agents of social changes and positive role models, \" says Dr Maphai. Meanwhile, the SAB League's National Championships official public draw was held today to determine the provincial team game schedule for the tournament taking place at van der Stel Stadium, Lentelus (Stellenbosch University) and Idas Valley Stadium in Stellenbosch in the Western Cape beginning Tuesday, 9thJuly and the final held on Saturday, 13th July.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapytp b/data_all_eng_slimpj/shuffled/split2/finalzzzapytp
new file mode 100644
index 0000000000000000000000000000000000000000..3e4b34479b708dc1180b21661c8797aee009b7ae
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzapytp
@@ -0,0 +1,5 @@
+{"text":"One way to think about the market: were you more or less confident in the country's prospects and prosperity when Obama was President than you are now?\n> If you are more now than then \u2014 with the tax cuts, deregulation, and giant deficit juicing the economy, but departed Trump advisor Omarosa Manigault Newman opining \"it's not going to be okay\" \u2014 that could justify at least part of last night's 24,000 or so close.\nI didn't think the market was undervalued then at 18,000. And I doubt it's a steal here at 24,000. Though what do I know.\nMonday, I hope to post \"Pre-empting the October Surprise.\" Still working on that one.\nBruce Bartlett tweeted this link to a Ben Carson profile, with the comment: \"I can't think of a single Trump cabinet secretary who isn't the worst person ever to hold that position in departmental history.\" Bartlett, a Republican, served in the Reagan and Bush 41 administrations. Housing Secretary Carson, this profile makes clear, is one of the least-worst of the bunch.\nHere, from the kind of global consultancy CEOs pay huge bucks to hear from, are Six Trends for 2018.\nFollowing up on yesterday's book-plug, watch this 45-second trailer. More compelling than the blurb I posted.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Komathi is a systems thinker who uses a balance of entrepreneurial knowledge and corporate savvy to introduce relevant large-scale transformation in global corporations. Her passion for transforming drug development is shaping the way clinical trials are designed. She started an incubator to catalyze innovation in clinical trials. The incubator includes Human Centered Design Labs to spark patient centricity in trial designs, External Innovation Advisory Board to create a sense of urgency for Senior Leaders to take action and Innovation Governance using Lean Start Up methodologies that prioritize learning through action.\nThe incubator has introduced tangible solutions to leverage the growing empowered patient as well as evolving digital and mobile technologies for healthcare. Recently a new patient centric clinical trial model that accelerated recruitment three fold and reduced cost by leveraging new technologies and trusted patient channels beyond the traditional trial center. Via her connections across the technology sector and healthcare, she has partnered some unlikely companies with pharma to develop some novel solutions to traditional industry challenges.\nAs a senior global leader at Genentech, AstraZeneca and Eli Lilly, she has built and led high impact field and global corporate teams with up to 500 employees across multiple headquarter sites. She has implemented integration and culture strategies to help employees remain engaged and committed through reorganizations\/mergers.\nKomathi is a biomedical Engineer with broad Pharma\/Biotech industry experience ranging from Sales, Medical Affairs, Global Project Management, Global Clinical Operations and Innovation. She has extensive experience in key therapeutic areas including: Oncology, Neuroscience, Endocrine, Immunology, Cardiology, Respiratory, Infectious Disease and Gastroenterology.\nShe is a Steering Committee Member for Disruptive Pharma Conference and a Mentor to RockHealth Accelerator Startups.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Many years ago I bought a garage sale coffee table to refinish and decided that I wanted to mosaic the top. At the time, having never done a mosaic, I had no idea where to begin, no supplies or tools\u2026.\nWhat is Pietra Dura? Pietra Dura, commonly known as Florentine Mosaics but sometimes called marquetry in stone, is a highly skilled and little known craft using semi-precious stones, hard stones and marbles to create exquisite and life-like \"paintings in stone\"\u2026.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This past week I have been turning fruit into gummies, cashew butter into cookies, coconut milk into yoghurt and vegetables into chips!\nDuring dehydration the flavours get more concentrated. This is especially noticeable when dehydrating fruit.\nLet's be honest. Dehydration offers something different in the healthy snack category.\nMy daughter loves her fruit rolls which I have prepared in about 10 different variations, for tomorrow it's raw cookies on the menu!\nThese types of snacks are also easy to bring with you on a day out.\n\"Follow the season\" is what my grandfather used to say. Buy in season, it is healthier, ethical and usually cheaper. If you buy more fruit or vegetables than you should, no worries, you can transform all that fresh goodness into dehydrated versions and save more money in the process. With me it's usually the case with berries. Oh and without the nasties. You know, added sugars, preservatives and such.\nFood dehydration is truly simple! No gadgets to put together and dismantle, minimal washing required to keep it clean. What's not to love?\nThis is the part I like. If you use low temperatures, dehydrated foods don't lose their nutritional goodness, making them particularly useful when it comes to preparing snacks. The last time I checked healthy snacks on a supermarket shelf, the choice was pretty poor.\nCurly kale, a member of the cabbage family, has distinctive dark green leaves which are rather tough and fibrous. Curly kale is rich in vitamin C and iron, calcium, vitamin K and vitamin A, essential for the health of the eyes and skin. This important nutritional profile makes this food an essential ingredient for our immune system and our metabolism.\nWhile there are several ways to cook this superfood, the most popular way in my family is dehydrating it, because it magically turns into chips!\nTransfer into a bowl mix the oil, paprika and salt together aside and drizzle over the leaves. Toss thoroughly to combine, using yr hands to season evenly.\nDispose the leaves on the tray of your dehydrator and set time and temperature accordingly (I own a D5 IR and I set it at 45 degrees for 18 hours).\nAlternatively, you can oven bake the kale at 105\u00b0C for 15-20 minutes. Stir the leaves to ensure even baking and leave for another 5 minutes until crispy but careful it burns easily so keep an eye!).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After finishing his first year as assistant professor of jazz trombone in the MSU College of Music, Michael Dease extended his teaching responsibilities well into the summer months. Dease spent four weeks in residence providing instruction in northern England, New Jersey and Michigan. Just back from completing a month of performances in New York City, Dease considered the dual role of educator and performing artist.\nWhether on the bandstand or in the classroom, everyone's personal and professional disposition preemptively decided the degree of accomplishment that we would together achieve. Dedication doesn't have an age limit, so it felt natural to observe these similarities between 8-year-old beginners and veteran musicians.\nIn my Jazz House Kids Camp ensemble class, I was inspired by the patience that the advanced students demonstrated with the novices. I was further encouraged by the younger students' courage and risk-taking abilities when they jumped into improvising situations, where they were observed by friends and strangers alike. This spirit of passion and empathy helps propel the expressive qualities of jazz music forward, and my students made giant leaps in their playing in just a short time.\nWhile walking the grounds at Interlochen Arts Academy, I stopped by a practice room and showed a teenage saxophonist a pattern that James Moody taught me. The look of curiosity and determination on his face told me more than his subsequent hours of practice did. His attitude was in the right place.\nThe recognizance of the oral tradition in jazz music is vital to its integrity and evolution. When the attitude is open and positive, it opens the corridors to the highest levels of communication, which directly mirrors the dialogue of the rhythm in music.\nTo honor the mentors that helped me along my way, I hope to continue promoting education as a priority among performing artists. That helps explain the feeling of continuance and development that happens when I see my students having their own students! Our attitude can ensure that our lights are forever shining.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqzbm b/data_all_eng_slimpj/shuffled/split2/finalzzzaqzbm
new file mode 100644
index 0000000000000000000000000000000000000000..cd83014f1acc0b42ee73e94aa33ea24c2b5617f0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaqzbm
@@ -0,0 +1,5 @@
+{"text":"Think a sore throat just means a cold or flu? It could be something much worse \u2014 find out! And what you SHOULD and SHOULDN'T' be taking your pills with. Are you choosing correctly? Plus, banish back fat with an instant fix that's sure to surprise you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"www.petame.gr - \u0395\u03bc\u03c6\u03ac\u03bd\u03b9\u03c3\u03b7 \u03b5\u03bd\u03cc\u03c2 \u03bc\u03cc\u03bd\u03bf \u03bc\u03b7\u03bd\u03cd\u03bc\u03b1\u03c4\u03bf\u03c2 - Shigeo Ogawa engines.\n\"In 1936, Shigeo Ogawa set up his original machine-shop, making model steam engines.\nIn the same year, at the suggestion of an American buyer, Paul Houghton, the 1.6 cc O.S. Type-1 miniature gasoline engine was constructed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Government with focus on Science and Technology aroused as a result of Uie dire need to have another University in the northwestern axis of the country to satisfy the teeming population of the State who desire higher education.\nOn coming to power, particularly the second tenure, the present administration headed by His Excellency, Alhaji (Dr.) Muhammad Adamu Aliero (Garkuwan Gwandu) immediately decided to put in place the machinery for the establishment of a State university.\nAccording to the Executive Governor, \"one of the most consistent and recurrent decimals in the understanding and activities of the Government has remained the desire to put in place a sound and effective educational system that would serve as an engine room for the ultimate actualization of the dreams of those who fought for the creation of our dear Kebbi State\".\nTo further the actualization of this dream of educational development in the State, the Kebbi State Executive Council at its 29th meeting of 28th September, 2005 resolved to establish a University of Science and Technology in the State.\nP.M.B. 1144, Birnin Kebbi, Kebbi State, Nigeria..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After becoming a widow, a recurring dream prompted Sara Raney to embark on a journey that led her all the way to Antarctica. While healing her broken heart, she explored over 100 countries and all seven continents and poured all of her energy into learning about the different people, cultures, and traditions that shape our earth. Reinvigorated, she spent countless hours studying and earning certifications to be a travel professional. In the process of reinventing herself, Sara became an inspiration and mentor to others who also desire to have rich, life-changing experiences through travel.\nAs a Destination & Enrichment Speaker aboard cruise ships, international Bestselling Author, and Luxury Travel Concierge, Sara is the former host of Cox Cable's TV's \"Travels With Sara\" and has been featured in numerous publications. She is also a popular radio show guest, and has been a featured guest speaker for the Spotlight Spectacular event in Las Vegas, Nevada.\nSara is a Specialist in Travel Advice and Consulting and is affiliated with Palm Coast Travel. She is a Signature Travel Expert and an ASTA Verified Travel Advisor. In 2013 she received Nevada's Women's Chamber of Commerce Independent Contractor of the Year award.\nDo you know the culture, customs, and laws of the countries you're planning to travel to? If not, you could end up with hefty fines, jail time, or even the death penalty!\nSara Raney shares travel tips and stories of her adventures while traveling to over 100 countries and all seven continents, and the sub-continent of India.\nGet ideas for your next dream vacation, cruises, shore excursions, and more! You can also search world destinations, hotels & resorts, and more!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shri Ch. Mohan has taken over charge as Principal Chief Signal and Telecommunication Engineer (PCSTE), South Western Railway on 10th September 2018 at Hubballi.\nShri Ch. Mohan did B.Tech from Engineering College(JNTU), Kakinada and M.Tech from IIT\/Kharagpur.\nAn Officer of 1983 batch of Indian Railway Services of Signal Engineers (IRSSE), Shri. Ch. Mohan has worked in various posts in South Eastern Railway, North Eastern Railway, South Central Railway, IRISET & IRCON. He has experience in Design, Construction, maintenance, Vigilance & Teaching.\nWhile on deputation at IRCON, Shri. Ch. Mohan at Sharjah\/UAE as Sr.Design Engineer\/WS ATKINS for UK's Network Rail.\nBefore joining South Western Railway, Shri. Ch. Mohan was working as Chief Signal Engineer in South Central Railway, Secunderabad.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarwaz b/data_all_eng_slimpj/shuffled/split2/finalzzzarwaz
new file mode 100644
index 0000000000000000000000000000000000000000..1c82aeb7671996bc654590ad7721429d8a8540c9
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzarwaz
@@ -0,0 +1,5 @@
+{"text":"Vijayalakshmi Wet Grinder 2018 List of latest, upcoming Vijayalakshmi Wet Grinder price list in India, Vijayalakshmi Wet Grinder models, specifications and features. Find new Vijayalakshmi Wet Grinder and get lowest price quotes on Sulekha.\nBest price for Vijayalakshmi Cute Litre Wet Grinder in India is sourced from trusted online stores like Flipkart, Amazon, Snapdeal Tatacliq. Check Vijayalakshmi Cute Litre Wet Grinder specifications, ratings, reviews and buy Online.\nThe latest price of Vijayalakshmi VICTOR 2L Wet Grinder(White) was obtained on ; The Vijayalakshmi VICTOR 2L Wet Grinder(White) is available on Flipkart. The price is valid in all major cities of India including Mumbai, Delhi, Bangalore, Hyderabad, Chennai, Ahmedabad and Pune. Please check instructions at the specific stores for any deviation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please consider the various Junior tennis options below.\nIf you would like your child to join our programme please read: Information for parents.\nPlease would all juniors interested in playing in competitions outside of the club, download, print, fill in, and return a Player Registration Form to the Sussex LTA County office. This will mean you are sent entry forms and tournament information.\nTo compete in competitions you need British Tennis membership to gain a player rating, this is free to all members of an LTA affiliated club, so if you are a member at Storrington LTC you qualify for free British Tennis membership!\nA player rating works like a golf handicap, so it's an excellent way of measuring your progress. You have to be at least 10 years old to get a rating and you start off as a 10.2. The application process is vey simple and takes only 5 minutes to be completed online. Click here for more information and to join British Tennis.\nOnce you have a player rating you have to win three matches against other LTA rated players to move up to a 10.1. As a 10.1 you have to win three more matches to progress up to a 9.2. Once you have won three matches, download, print and fill in this LTA ratings scorecard, and ask me to sign it. The scorecard has to be completed with your opponents LTA membership number.\nTo understand more about the ratings system please see, Ratings explained?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I remember going to Grandma's house and being offered terrible peppermint hard candy. It didn't taste very good. Old person candy never does. If only it had been bacon flavored. Well, now hard candy doesn't have to be boring. If you are putting out a dish of hard candy for the kids, make it Old Fashioned Bacon Candy.\nThe kids will thank you and it will be gone within 5 minutes I'm sure. It tastes like bacon and comes in an old time looking tin. What more do I have to say?\nJust $5.99 from Thinkgeek. That's a bargain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Community Choice Aggregation Energy Program was approved by the Town Board in February. The New York State approved CCA program may give ratepayers, as a group, the ability to find cheaper rates through a competitive bidding process than currently available through the Long Island Power Authority to advance the Town's efforts to achieve 100% renewable electric energy by 2025 and to make a significant impact on the reduction of greenhouse gas emissions.\nBids must be received by the Town Clerk by 4:00pm Thursday, May 16, 2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Can someone ID this font?\nI'm having trouble finding what kind of typeface this is. Does anyone know what it is?\nI know it was used to write letters of business.\nI am grateful for any help with this as it is most urgent.\nI would like to know the name and where I can obtain it for free.\nRead more about Can someone ID this font?\nPlease Critique my first typeface design !\nI would like a critique of my first typeface design, it is comprised of 3 versions: a standard clean neo-grotesque, an alternative version that has multiple alternatives, and a \"advertising\" version that has a much higher x height, along with a shorter ascender\/descender line, with a more expressive feel to it. all three versions will be released under the name \"Dawson Grotesque\"\nAll advice is appreciated! This has been a project of mine for quite some time \u2014 and is my first baby, so please help me make it perfect for its release in the near future !\nRead more about Please Critique my first typeface design !","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatffd b/data_all_eng_slimpj/shuffled/split2/finalzzzatffd
new file mode 100644
index 0000000000000000000000000000000000000000..d45cfb5135340af67a55c8f7be29129794d7d125
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzatffd
@@ -0,0 +1,5 @@
+{"text":"Excellent service with a friendly disposition.\nGreat as always I just thought maybe you guys could have cut me a break on taking off the step rails. Since I spent 5 grand a month ago.\nEveryone was nice and helpful. Fast and efficient.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Poll: Are you in favour of alcohol sales in grocery stores?\nAlcoholism in the craft beer scene has long been hidden behind its image as a culture for connoisseurs.\nB.C. MLA John Yap has completed his review of the province's liquor policy and submitted his report to the government, but his recommendations haven't yet been announced.\nOne potential change that has generated much speculation is the availability of liquor in grocery stores. We'd like to get your thoughts on this one.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Buy liquid sildenafil ativan drug interactions prozac clomid 50 mg tablet price ativan lamictal drug interactions metformin (brand name glucophage). Propecia private prescription cost clomid from gp australia can you buy clomid in the uk clomid where to buy uk. 5 panel drug test ativan glucophage brand name clomid in australia can i buy clomid in the uk ambien ativan drug interactions. Ativan false positive drug test drug test for ativan prescribed ativan drug test where to buy nolvadex and clomid uk ativan drug nutrient interactions. Where to buy a clomid ativan seroquel drug interactions will klonopin and ativan show up the same on a drug test ativan benadryl drug interaction. Best drugstore bb cream australia clomid where to buy in uk ativan drug test urine clomid australia cost drug interactions ativan neurontin entocort coupon. Ativan and xanax drug interactions ativan cdl drug test propecia cost with prescription where can i get clomid in australia. Propecia usa cost where to buy genuine clomid which drug is better ativan or xanax Buy viagra wholesale ativan drug wiki. Entocort manufacturer coupon Cialis generico da sandoz ativan tylenol drug interactions cheap generic clomid online will ativan and xanax show up the same on a drug test. Gp prescribe clomid australia does ativan show drug test ativan tramadol drug interactions clomid and nolvadex where to buy drug interactions with ambien and ativan. Clomid 25mg $53.19 - $0.59 Per pill cost of propecia australia ativan food drug interactions clomid where can i buy it. Does ativan show up in drug test clomid sales australia ativan urine drug testing can u buy clomid uk. Ativan drug test detection time liquid clomid australia where can i buy over the counter clomid ativan show up in a drug test. Clomid tablets australia ativan vs xanax in drug test benadryl and ativan drug interactions can my gp prescribe clomid australia. Ativan one time use drug test ativan drug classifications.\nDiflucan prescription levaquin generic cost walgreens herb viagra where to buy levaquin price generic vendita viagra generico on line where can you buy viagra in melbourne. Viagra generico online portugal where can you buy viagra in sydney yeast infection medication prescription diflucan. Where can you buy viagra in dublin diflucan prescription uk dove comprare viagra generico online Buy fluoxetine 40 mg online. Herbal viagra where to buy can you buy clomid over the counter in the uk minocycline acne pills buy motilium instants price of clomid australia can you purchase clomid over the counter. Where to buy viagra cheap minocycline pills reviews clomid drugs online diflucan one prescription dove acquistare viagra generico online. Where to buy viagra in ethiopia diflucan prescription instructions clomid online for sale where to buy viagra in port elizabeth. Cost of clomid in ireland acne medication pills minocycline diflucan prescription price can you buy motilium over the counter in australia. Comprare viagra generico on line clomid online safe where to buy viagra in windsor ontario cialis viagra generico online Purchase hydrochlorothiazide online what is the cost of clomid in australia. Where to buy clomid otc can i buy motilium over the counter in australia clomid over the counter uk effects minocycline birth control pills acquisto viagra generico online. Viagra generico italia online diflucan prescription information acquistare viagra generico on line clomid online in usa levaquin generic. Diflucan prescription online generic levaquin prices does minocycline interfere with birth control pills where to buy viagra in dar es salaam. Price generic levaquin 500 mg where can you buy non prescription viagra can you buy motilium over the counter generic clomid online clomid cycle cost. Does minocycline interfere birth control pills viagra generico online senza ricetta where to buy viagra in warsaw where to buy viagra in yangon buying clomid over the counter. Diflucan prescription canada where to buy viagra in auckland us online pharmacy clomid viagra where can i buy it does diflucan need a prescription.\nProventil inhaler cost ordering clomid online canada best online pharmacy kamagra clomid 50mg tablets uk buy viagra online in usa clomid tablets over the counter. Buy viagra online cheap usa ciplox 500 mg price proventil inhaler cost without insurance is diflucan sold over the counter in canada. Ciplox 250 mg price over the counter diflucan in canada over the counter diflucan canada zovirax cream order online toradol generic brand name. Clomid online purchase clomiphene online order can you buy viagra online in the usa viagra usa buy online buy viagra online usa overnight delivery. Buy zithromax us kamagra uk online pharmacy proventil hfa price us buy synthroid in canada buy viagra online from usa online pharmacy kamagra uk. Ketorolac tromethamine 10 mg generic toradol viagra buy online usa clomid online kaufen ohne rezept buy clomid with credit card. Where to buy clomid in the uk buy viagra online in the usa generic finasteride ireland order zovirax cream online buy clomid over the counter. Order zovirax online ciplox eye drops price ciplox eye drops price in india buy viagra usa pharmacy clomid pills online uk ciplox eye ointment price. Ciplox 250 mg price can you buy synthroid in canada price for proventil inhaler online pharmacy buy clomid. Inderal medicine indications trusted clomid online is diflucan over the counter in canada is diflucan available over the counter in canada. Buy viagra online in us toradol generic buy finasteride ireland inderal medicine usage ciplox price india Clomid 50mg $99.39 - $0.83 Per pill. Buy clomid and nolvadex pct diflucan over the counter canada buy clomid unprescribed buy clomid anti estrogen. Taking clomid online clomiphene pills order Amoxil buy uk viagra buy us buy synthroid from canada viagra buy usa inderal medicine for what. Zovirax order online proventil hfa inhaler cost zovirax online order buy clomid post cycle therapy finasteride online italia. Acheter clomid online kamagra online pharmacy uk ciplox price buy clomid per pill toradol generic cost zithromax buy usa clomid kaufen deutschland.\nFamvir buy online clomid 50 mg tablets buy famvir online canada where can i buy clomid for cheap urispas tablet adalah clomid tablet purpose. Viagra over the counter portugal viagra over the counter sydney clomid online in uk famvir prescription online clomid online pharmacy uk buying clomid online in uk. Viagra over the counter paris harga urispas tablet clomid buy online cheap uk levlen ed generic brand cialis online voucher clomid where to buy uk. Clomid 25mg or 50mg xanax or clomid urispas 200 mg 60 tablet clomid buy uk generic cialis viagra online levlen birth control generic cialis online buy. Viagra over the counter perth where can i buy cheap clomid pills viagra over the counter netherlands urispas 200 mg 60 film tablet. Cialis online vendita buy finpecia europe urispas tablet price clomid buy cheap levlen generic brand order clomid online uk viagra over the counter spain. Tri levlen birth control generic clomid pills or liquid buying clomid online in the uk finpecia deutschland kaufen urispas tablet uses clomid 50mg or 100mg pct. Clomid to buy online uk clomid where to buy in uk famvir buy online australia cialis viagra online australia finpecia in deutschland kaufen viagra over the counter los angeles. Clomid pills or injection viagra over the counter korea getting clomid in the uk cialis online no prior prescription.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Edward-Elmhurst Health Cancer Centers are committed to providing the most up to date cancer treatment options for our patients. One way we are doing this is by encouraging participation in clinical trials. Use this tool to search for research programs at Edward-Elmhurst. Please review our Clinical Trial Research Policy and frequently asked questions to learn more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These are the new Dal super strong Prop 5x4.5\" high RPM props (Oct 2015) 0545 - These are well matched to 2400kv CarbonBird motors on 4S on Scarab 270 Air Knife-mini for extreme speed fast flying FPV or racing. amazingly - They require almost NO balancing - They are tough and quite.\nHow do they compare? there is no agreement on props - each pilot will have his own prop story - but Sometimes its annoying to have to do a prop change everytime you have a small incident with a racegate, a hidden branch of a tree or the ground that follows - these props will ensure the \"incident\" wont have walking back for tools - they are ideal for those days when you are exploring, pushing thru leaves and trying new stunts.\nRequires 7.2mm shaft height ; 5mm diameter shaft to mount it.\nThese are high quality props for a great low price - but you must adhere to safety warnings about props below - always.\nLimitations - Do not exceed the max RPM 25,500 - replace any prop if damaged or crashed - You must understand the risk of flying a Quad with a damaged prop can result in the Multicopter crashing\/falling and hurting a 3rd party or yourself - YOU WILL BE RESPONSIBLE for this ! - props can cut - keep away from eyes! wear safety glasses ALWAYS !","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzathgu b/data_all_eng_slimpj/shuffled/split2/finalzzzathgu
new file mode 100644
index 0000000000000000000000000000000000000000..5a828ab0729e6d5d973d95bf339d72ee0624001d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzathgu
@@ -0,0 +1,5 @@
+{"text":"QUICK JUMP: Contestants Pageant Information How to compete in Teen Universe California 2018?\nTeen Universe California is a yearly STATE pageant, Girls ages 14-18 years of Age get to compete against other cities of California. This prestigious pageant allow women, (California residents) to come together and learn about Collaboration, Inner beauty, and Women empowerment.\nAs a contestant you get to build a sisterhood, a platform, and have priceless opportunities, you get to represent your city upon chosen in casting via online or orientation day. We practice interview skills, build experiences and friendships.\nIf you wish to travel for 3 practices to San Diego you get to collaborate be interviewed and meet business owners, CEO's, Designers, Photographers and Media recognized personalities.\nWinner will have a scholarship entrance to Teen Universe USA pageant and be co assistant to State Director Lesley Garate in the next reign and more prizes to be revealed upon sign up.\nYOU CAN BE OUR NEXT CALIFORNIA QUEEN TO COMPETE IN OUR TEEN UNIVERSE USA PAGEANT! Who knows you also might be our next TEEN USA at our international pageant TEEN UNIVERSE.\nHow to compete in Teen Universe California?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Why do I even read his Blog?\nI keep going back to JM's Blog of Doom for some self flagellant reason, and today he had a Q & A by his Production Assistant, Ashleigh.\nAnd keep in mind she works for him.\nAshleigh: I am a shy person, so getting my pic taken was a little off putting at first\u2026but you soon learn that he is going to take a pic of you whether you are just sitting there doing nothing or not\u2026so you better just smile.\"\nGuess JF just wasn't the type to \"grin and bear it,\" huh?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Originally from Scotland, Mark moved to the United States to attend college in California, graduating from the California Institute of Technology in 1993 with a BSc in Engineering & Applied Science. Mark worked for a telecommunications company doing product development & technical sales support until 1997, before deciding to pursue his passion for investing & start a career as a financial advisor. While he started at a traditional brokerage firm (Dean Witter), Mark felt he could better serve his clients under an independent platform, and gradually transitioned to his own RIA (Registered Investment Advisory) firm, Compass Wealth Management, with an emphasis on fee-based accounts. In 2008 he decided to join forces with Mike & Chris Lorenzen of LIG Capital Management, and together they made the transition to today's fee-only practice. Mark's investment philosophy is multi-faceted, and while diversification of investment styles is usually an important portion of his client portfolios, traditional long-term holdings in stocks & bonds are the solid basis and foundation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Laila Biali has won awards (SOCAN Composer of the Year and Keyboardist of the Year at Canada's National Jazz Awards) and played the world's most prestigious venues (North Sea Jazz Festival, Tokyo's Cotton Club, Carnegie Hall). She's toured with GRAMMY award winners (Chris Botti, Paula Cole, Suzanne Vega) and recorded with an international icon (Sting). In short: She's established herself as one of Canadian jazz's brightest young stars.\nHer new album is catchy, sophisticated, and unlike anything currently on the radio, it's pop music, but not the kind that can be neatly tagged by an algorithm. Melodies take thrilling left turns and pre-choruses give way to instrumental interludes. One minute Biali is soaring over a bluesy storm of handclaps and hard-charging keyboard riffs (\"Got to Love\"), the next she is pouring out her soul on an impassioned, slow-burning plea for empathy (\"Refugee.\") It's pop music, but the experimental, distinctly human variety popularized by Regina Spektor, Rachael Yamagata, and Sara Bareilles.\nLaila will be joined for the evening with Brandon Robertson on bass and drummer\/percussionist, Rick Costa.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yeah, ok, don't love it.\nSo I bought this sweater, and I didn't like it the first time I wore it, and I am not a fan now. :-\/ I am mostly irritated with myself because I didn't love it in the dressing room and bought it anyways. I knew it was too long and not proportioned right for my size, but I really wanted the style in my head. Boo. Bad Bianca.\nOn a better note, I love this sleeveless blouse! I should have taken my photos without the sweater LOL!\nI have that same cardi but in the peachy\/papaya color. All my cardis have longer sleeves, so I feel awkward in this one. I bought it at the same time as the metallic linen dress that I promptly returned, but thankfully the sweater goes with a few other things.\nI think like you I really wanted it to work, but it doesn't. :( I'll just keep trying! It will not defeat me!\nI like the sweater...the sleeves...everything! and the blouse is very pretty.\nHey....I've nominated you for a Sweet Blog Award so you need to come and pick it up!\nI actually like the whole look. That blouse is so pretty though! LOVE it!\nI bought the same sweater...in almost all the colors, but now when I wear them, I feel weird. I think this looks good on you, but I get where you're coming from. Too bad for me, I bought the lighter grey, black, pink, peach, and maybe a white one? too much.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawxkw b/data_all_eng_slimpj/shuffled/split2/finalzzzawxkw
new file mode 100644
index 0000000000000000000000000000000000000000..5093ee14d1f0992c0faa9c0d787ba15a793bfac8
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzawxkw
@@ -0,0 +1,5 @@
+{"text":"The engineering firm hopes a storage system can smooth electricity supply on a grid powered by renewables\u2014and let it capitalize on a fast-growing market.\nThe news: GE announced a new system called Reservoir\u2014a large battery attached to the grid to store spare electricity from renewables. When the sun doesn't shine and wind doesn't blow, it dumps electrons into the grid to meet demand. It uses discharging smarts to extend its working life, and it's modular, so it can be adapted to fit small or large gigs.\nElectric potential: Grid storage is growing\u2014fast. The Energy Storage Association predicts that US storage in use will double over 2018. The Wall Street Journal notes that the market is predicted to be worth tens of billions of dollars in the next 10 years. GE, which is struggling financially, will hope Reservoir grabs it a slice of the pie.\nBut: There's competition. Tesla has its own grid batteries that are deployed in South Australia as part of a huge 129-megawatt system. A Siemens venture called Fluence is building a system in California that will be three times as big. GE, meanwhile, has just a 20-megawatt commitment for Reservoir so far.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Take your 2 sheets of foam and lay them out. Cut them each equally in half so you have 4 2'x4' sections. Make sure to measure you precut plywood sheets to make sure they are actually the right size to match the foam you are going to cut. If they are a little off make sure to adjust your cut on the foam to match.\nOnce you have your cut line drawn on the foam use a long straight edge and cut the foam with the utility knife. Do not use the heated cutters as it will distort the edge.\nThis will leave you with one 2'x4' section leftover to make hills and other terrain out of.\nLay out your boards edge to edge and draw the pattern you will want for the lava channels and any other features you will want to cut into the foam board. Keep in mind that you might want to \"standardize\" where the boards come together. So that way you can play with the boards in any combination, instead of having to always have the same layout. It is worth taking the extra time to do this. Take a good look at it once you have drawn out what you want. Keep in mind that you are doing to want to be able to play on this board. So make sure you do not make it to busy.\nGrab your hot foam cutter(s) and start working on cutting out the pattern you have made. If you do not have a straight hot knife you will need to use the utility knife to get to the sections you can not get with the hot wire easily.\nAs I was cutting stuff out I numbered the \"little islands\" so I would know where they go after.\nOnce everything is cut out you are left with smooth edges, which you don't want. Take the hot knife (or the wire cutter) and score the edges to make it look more like a broken lava field.\nNow that your board is all cut out and the edges are \"textured\" get out the subfloor glue and get it loaded up in the caulking gun.\nFlip your boards over and apply the subfloor glue to the back. Make sure that all the areas near and edge have glue so the foam will not be able to pull up from the wood boards. Once it is well coated flip it over onto one of the wood boards and press it down.\nFinish gluing the other 2 boards and stack all 3 of them one on top the one another with the wood backings facing up. Now that they are stacked put a bunch of heavy stuff on top and let it sit over night so that the glue will have plenty of time to dry.\nNow that it is the next day and the glue is dry it is time to texture them. You will need your large paintbrush, the black latex paint, the sand and a tarp to work on.\nFirst lay the tarp on the floor so you will have somewhere to collect the sand as it falls off the boards. Lay the boards on the tarp and brush on a thick coat of paint onto the top flat surface of one board. Once that is done, poor the sand onto the paint you have just put down. If it is particularly warm you may want to do a small section at a time.\nAt this time I also painted all the little \"islands\" and put sand on them too.\nOnce the first board is done, repeat the steps on the other two boards. They will need to sit for a few hours to let the paint dry all the way.\nWhen you come back in a few hours tip the boards over onto the tarp to get all the loose sand off. Use the tarp to poor the sand back into the bag or a container for use later.\nNow you need to paint the vertical edges of the lava field and put sand on them. Be careful not to get too much paint down on the lava channels. You can cover up the little bits of sand that will stick down there when you do the lava itself. Once all this is done and had had a chance to dry go ahead and shake off the sand again and put the extra back in the container.\nNow it is time to apply a coat of latex paint over all the sand to seal it in. Grab your large brush and give all the sand a good coat of latex paint. Make sure to pay attention to getting paint in all the little cracks you make along the lava channels. Also paint the outside edges of the boards at this time.\nThis will last step will take a while to dry so lets move on to making the movable terrain (if you had not started it already).\nFirst we will start with the masonite board. You will be using this to make templates for your forests (Burnt out husks of trees) and any other rock outcroppings and lava cinder cones.\nDraw out the shapes of the templates on the masonite board and cut them out with the jigsaw. Once they are cut out take your sandpaper and smooth out the edges.\nFor this board I cut out of the blue foam rock outcroppings and some cinder cones. Once these are cut out any that need templates will get glued onto them with the subfloor glue. Take the Drywall patch and use it to fill all the gaps along the bottoms of the foam and masonite.\nAlso cut out some hills and some small flat \"sheets\" of rock, that will be used for bridges over the lava rivers, from the left over sheet of blue foam.\nOnce the glue and plaster patch is dry go ahead and paint with the latex paint the areas of anything that you will want to have \"sand\" on them and cover them with sand.\nOnce that first coat of paint and sand is dry clear off the excess sand and paint everything black. This will seal in the sand and also give you a black base coat to work up from for the rocks.\nNow we will go back to the boards that have dried and work on getting some \"color\" onto them. You will want the large brush, black and white paint and some newspaper or other surface to mix the paint on. First, mix up a medium grey and drybrush* the boards and your mobile terrain pieces this grey.\n*Drybrushing is a painting technique of wiping most of the paint off a brush, so as you paint the brush will only leave paint on the raised surfaces, as there is a minimal amount of paint on the brush being used.\nNext, lighten the grey paint with a little more white and paint the edges and some patches on the boards and moveable pieces with this. Then take an even lighter shade and get just the edges of the lava rivers and the movable terrain pieces.\nNow all the boards and pieces should be finished with as much grey and white as you like.\nNow we are going to me making the lava. You will need the Acrylic caulking and a medium small paintbrush for this.\nWhat you are going to do is lay a large bead of caulking down the middle of the lava rivers and any other place where you will want lava. Once that is done, take the paint brush and spread out the bead of caulking. Sometimes you will need to dip the brush in some water as the caulking can get quite thick and sticky and will need to be thinned out aome. Give it a rough texture so it will look like a rough lava flow. Do the same for any mobile terrain pieces you also may have made that have lava on them.\nIf you want to make some \"lava bubbles\" go ahead and put a dollop of caulking down and smooth it into a bubble.\nThis will take up to a full day to dry depending on how thick the caulking is.\nNow that all the caulking is try it is time to paint up the lava.\nFirst you will start with yellow paint and a medium brush. Paint all the lava yellow. Once this is all done go ahead and heavily drybrush* the lava the red\/orange color.\nOnce the orange is dry, mix some of the black paint with some orange paint and drybrush* it onto the tops of the lava. This will give you the look of lava that has cooled somewhat on top. If you want it to look even cooler you can use more black as an additional coat of paint.\nNow that all the lava is painted you will want to recreate the glow that hot lava radiates onto the adjacent rock.\nTake your medium paint brush and the orange\/red paint and drybrush* it onto the walls adjacent to the lava. Also drybrush orange strips onto the bottom of the \"bridges\" for the lava that would be glowing up at them.\nGo Play! You are done!\ni wish i had of looked at ur table top before i stared making mine. That said i thing your table looks class. How long from start to finnish. i`m only 1 week in and allready it feels like there is no end to it.\nGood work! I have to get back to my terrain projects someday, I use much the same basic technique, without the caulking. I used some other medium, Gusso would work along with a number of other things that dry hard.\nThis is pretty helpful! Everytime i play with my friends they see this and there like \"Whoa dude!\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hey! Ran into a strange issue where certain users were not able to access modern public folders on-premise and some users had no issues at all.\nThe users that had no issues were users that had a mailbox on-premise and then later migrated to the cloud.\nThe users that had issues were users that were created via the New-RemoteMailbox PowerShell cmdlet.\nI was familiar with this issue on Legacy Public Folders but not on Modern Public Folders.\nThe issue for the Modern Public Folders is that it's missing the ExchangeGUID attribute. And how O365 does a lookup for the modern public folders on-premise.\nThis seems to have been fixed in Exchange 2013 SP1 CU14 (CU 13 and below are still affected) Additionally there were Free\/Busy issues as well due to it looking for a non-existing ExchangeGUID.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The habit of painting nails is quite common among women and an alternative to escape the pattern of just color them is decorate them with some sort of design or texture.\nThe variety of styles and designs is huge, for all tastes and skill levels as manicure. Also, there are products such as adhesives and stamps for nails that can facilitate their work.\nAlthough not seem like something that can be done easily by anyone, you can decorate your nails initially with simple drawings and prints and gradually go gaining the necessary skills to create more elaborate decorations. Keeping this in view, a super beautiful and easy option is to decorate your nails with flowers.\nBesides the delicate and feminine air, floral prints can provide different effects depending on the model chosen, the size and amount of drawn flowers. Learn how to decorate your nails with flowers watching the video tutorials of different models and see still several floral nail art to inspire and test at home.\nSee below how to make five models of nail art to decorate your own nails. They are easy and beautiful options that can be used at any time.\nThe material you will need for your manicure kit a boleador (tool to make dots on the nails) and creativity. The trick is to test the ideas below and then reinvent them combining other colors and flower designs and create your personal way to decorate your nails.\nRoses are apparently complex designs, but it is actually quite simple. Just use different shades of nail polish (a light and a darker) and invest in the brushes to facilitate the traces. It is an option that can be made with different combinations of colors and also mixed with other elements (stripes, po\u00e1), so it is quite adaptable.\nDelicacy is undoubtedly the word that defines the nail cherry blossom art. With a blue background and small flowers in pink baby, the idea consists of pastel shades and though it seems complicated, it is quite simple to play this nail because the traits and the flowers do not need to be accurate. It is worth investing!\nThis is a very classic choice since it combines the elegant French girl with a delicate and discreet flower. The result is super feminine and the model can be used at work without attracting attention, or at parties (weddings, graduations) for having neutral tones and classic.\nVery practical and simple, this nail has a more stripped air, because the choice of colors, however it can completely change the look if the colors are changed. That's what makes this interesting nail art and a good option to use in different occasions.\nThese flowers are made up of small dots and create a delicate effect. There is the option to fill the nail with various flowers (as in video), making the flowers just in the \"only child\" (one of the nails receive a different enameling the other), or make just one flower on each nail. Everything depends on the occasion and your personal taste.\nCheck out a selection of images with nails decorated with flowers of different designs and sizes and be inspired to let their flowery and beautiful nails.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do the incumbents have \"embedded spies with the new entrants\"\nDo the incumbents have \"embedded spies with the new entrants\"?\nThere's many rumours of how many subscribers the new entrants Wind Mobile, Mobilicity and Public Mobile have actually taken away from the incumbents: 30,000, 100,000, 140,000 \u2013 there are no official numbers released and probably won't be for a while. The Big 3 and their flanker brands have all stepped up their game offering Unlimited Calling, early evenings, axed System Access fees and some carriers have also reduced their handset prices.\nRogers is coming into the market with a new discount brand named \"Chatr\". This will be available in core urban locations (Toronto, Calgary, Edmonton, Vancouver and Ottawa) and will directly compete with the new entrants. The government wanted more competition and things are fiercely heating up.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxivl b/data_all_eng_slimpj/shuffled/split2/finalzzzaxivl
new file mode 100644
index 0000000000000000000000000000000000000000..80e3462465d8eee3f05cb5f6173861bbbf8d9567
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaxivl
@@ -0,0 +1,5 @@
+{"text":"Property investment in student accommodation has many benefits. Net yields are often higher than traditional buy-to-let gross yields. It's a recession-proof sector of the UK property market. Properties benefit from professional property management. The list goes on.\nHowever, as with all property investment opportunities, performance depends upon your property investment strategy, buying in the best areas, and buying the best property. The best student accommodation opportunities share many elements with other property investment, though there are a few differences.\nHere, you'll learn how to spot the best student accommodation opportunities as a Beginner Investor.\nThe property must be in a location that is well-connected. Students are only just leaving their teenage years behind. They want to live close to all the things that are important to them, without having to rush around too much! So, you need to buy in the right area, and on the right street.\nThere aren't many students who have their own cars. Instead, they rely on public transport. You'll be looking for property that benefits from proximity to bus and train stations \u2013 a regular bus service to the university will be popular when it's raining.\nWhen searching for the best places to invest in property UK, you'll want to ensure that there are enough employment opportunities in the area to sustain demand. Where one major employer or industry dominates the local economy, the risk of economic collapse is increased (for example, think about how whole towns were decimated when the UK coal industry closed).\nSo, ensure that you invest in an area that has a really strong university. Preferably two. Sheffield, for example, benefits from two universities, both of which are thriving: The University of Sheffield and Sheffield Hallam University.\nDevelopers who have experience in building student accommodation will build the best property for this type of investment.\nExamine the track record of the developer, and look at what they have previously delivered. Student accommodation is HMO property investment. It is more highly regulated than traditional buy-to-let investment. A developer who has built in this sector before is likely to develop property which is fit-for-purpose and conforms to all the HMO regulations.\n6. Who is the property manager?\nOur research team has identified some incredible property investment opportunities in the student accommodation space. To find out more, and how easy it is to invest for market-beating net yields, contact one of our team today on +44 (0)207 923 6100.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"South American Birth Customs in Theory and Practice. Folk, II, K\u00f8benhavn 1960, pp. 51-69, 2 figuras e 1 mapa no texto. Bibliografia.\nO autor discute as teorias de Wilhelm Schmidt (B. C. 2634) e outros s\u00f4bre a cuvade, \u00e0 luz do material por \u00eale colhido entre os Waiwai, acrescentando, ainda, dados comparativos s\u00f4bre outras tribos sul-americanas. Chega \u00e0 conclus\u00e3o de que \u00easses costumes ligados ao parto, especialmente entre as tribos das florestas tropicais sul-americanas, prov\u00e0velmente \"are (1) motivated by the great child mortality, (2) are stamped by the special distribution of work within semi-agriculture, but (3) are created by purely animistic ideas about the human soul and animal or other nature spirits as possessing qualities like divisibility and contagion. This results in the strong spirits being dangerous to the weak infant soul. The more these dangerous spirits are connected to the man's domain of work, the more the natal customs will result in a practice closer to the classic idea of couvade. Though couvade\u2015and other natal customs\u2015may also later be veiled by superimposed magic practice.\" (p. 68). Pequena lista de voc\u00e1bulos waiwai encerra o trabalho (p. 68).\nUse o formul\u00e1rio abaixo para comunicar erros de transcri\u00e7\u00e3o encontrados neste verbete (2160). Itens marcados com um asterisco (*) s\u00e3o de preenchimento obrigat\u00f3rio.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Walk for life in 2013!\nBless them! Here are some from USF, and you can see some others here.\nBerkeley's Dominican School of Philosophy and Theology was at the Walk. This photo is from their Facebook album of the Walk. You can see more here.\nThe great Fr. Aris Metrakos of San Francisco's Holy Trinity Greek Orthodox Church gave the invocation at the Walk for Life West Coast this year, and there were quite a number of other Orthodox priests and our Orthodox brothers and sisters there, too. Five Loaves has a post over at Catholic Answers about Orthodox participation, and here is a nice photo of Fr. Aris leading us in prayer:Bless you, Father! and Thank you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We owe a great deal of thanks to businesses both local to us in Hertfordshire and beyond who have time and time again supported our fundraising events and pledged to support our life changing design projects. This year, we have launched a Corporate Partnership Network to bring together like-minded businesses to make great new connections and help do even more to break down barriers to inclusion and independence.\nBusinesses in our Corporate Partner Network have made an ongoing commitment to support DEMAND and the work we do developing custom designs to help disabled people.\nA saggy seat or scratched surface doesn't automatically mean a trip to the dump. We can give a much-loved piece of furniture a new lease of life. We can handle pretty much anything from a watermark to full restoration or a complete makeover. Working for both the domestic and commercial sector no job is too big or too small. Our services are: Wood refinishing; structural and frame repairs; leather repairs, full re-upholstery; replacement springs, webbing and cushions; recliner chairs (electric and manual); re-caning; commercial enhancement.\nFind out more about our Corporate Partner Network and how your business can join us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our ribbon blender provides a rapid, homogeneous blend that is thorough and uniform for a variety of materials.\nOur grinder is capable of handling coarse, fine, dry, and wet materials of all sizes.\nWhether your drying load is large or small, we can take care of all your needs. Pigments and specialty chemicals are just a couple materials we've tray dried.\nWe can package small container of toy slime to large bulk powder sacks, and everything in between.\nWe can make any size mix you need from a single drum up to 550 gallons of specialty chemicals.\nOur sister company, Total Food Package, is organic certified, Kosher certified, gluten free, allergen free, and we'll ship your products directly to Amazon or your customers.\nWith custom capabilities, we can handle any project, large or small. Get in touch with us to see what we can do for you.\nWe offer quick turnaround times and can test your material per your specifications. Because of our small size we are quick and agile and can react immediately and efficiently to your needs while providing excellent quality.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxzio b/data_all_eng_slimpj/shuffled/split2/finalzzzaxzio
new file mode 100644
index 0000000000000000000000000000000000000000..f8bb8728ad7b5a22255cf5d2f09e5bbf926f74d1
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaxzio
@@ -0,0 +1,5 @@
+{"text":"Thank you for supporting the troops and their families by donating to Team USO Hawaii for the Hapalua Marathon! We know it is important to you to make a direct and positive impact on the troops and to let them know that since they've sacrificed to protect our nation, you will always be there for them.\nIt is through efforts like yours that make it possible for the USO to fulfill its critical mission. All funds raised go directly to programs that service the needs of deployed troops, wounded, ill and injured, the fallen, and their caretakers and families here in Hawaii. Every dollar will go to lifting the spirits of those in service through many critical programs. As a non-profit, we are constantly changing to meet the needs of today's troops and their families and we depend upon your efforts to make our mission a reality.\nOn behalf of our troops and their families, mahalo!\nClick the VOTE button to give Erlinda 100 CrowdRise Impact Points (CIPs). Vote for the Volunteers and Fundraisers that are answering the call to service, raising money for charity, and making an impact for their causes. Come back and Vote every months. CIPs can earn you the highly coveted, highly respected CrowdRise Royalty Status.\nIf you're seeing the grey VOTE button, you should feel great about yourself. It means you already voted for Erlinda this month. Come back next month and vote again if you still love Erlinda. Thanks to your vote, Erlinda earned 100 CrowdRise Impact Points (CIPs). CIPs can earn you the highly coveted, highly respected CrowdRise Royalty Status.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This fringe bag is edged in loose thread that naturally twist to add volume to the ornamental border. Made with tan-colored fine leather, it goes simply with any of your outfits.\nThe drawstring is enclosed with beautiful tassels to match the boho bag's fringe details, while the Native American breastplate is set well to lure the eyes to the turquoise gems, then to the round-cut beads, and to the framing fringes.\nIt's not just an beautiful bag with a boho twist, it's also large enough to hold all your stuff together, and even your littlest things thanks to its small pocket inside.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A small garden extends along the back wall of the prison's east wing \u2013 Our Garden. The selection of plants in this garden is based on the herbaria kept by Rosa Luxemburg, who \u2013 even while imprisoned \u2013 botanized with great skill and enthusiasm, and also collected, botanically recorded, described, and dried plants from her surroundings or that were brought to her by her friends in her surviving notebooks.\nThe work draws on herbaria from between 1913 and 1917, which I used to put together a little, seasonally-experienceable piece of Rosa Luxemburg's legacy, and which forms a bridge to Martin Luther through two apple trees.\nGrowing to the left of the path (as a kind of welcome, you could say) is the \"Martin Luther apple\" \u2013 a new breed cultivated by the Barnimer tree nurseries and a classic autumn apple, yellow with reddish cheeks.\nTo the right are Japanese quinces, surrounded by spring anemones and high-stalked primroses, the keys to heaven.\nDiagonally opposite the \"Martin Luther apple\" \u2013 as a counterpart to the breeding, so to speak, and one of the sources of the apple varieties known to us today \u2013 we have Malus sylvestris, colloquially known as the \"crab apple\" or even the \"cherry apple\" because of the shape of its fruit, which resembles a cherry.\nLilacs, summer guelder rose, arborvitae, elder, mahonia and hornbeam are complemented with geraniums, daisies, cineraria, thrifts, larkspur and saxifrage. Woodruff and lily-of-the-valley thrive in the shady areas.\nGrowing along the path leading through the garden is plantain, which can easily withstand a few feet walking over it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Invisiblewear\u00ae Orbit\u2122 Hairspray works to boost volume while taming pesky flyaways. The fine, layerable mist resists the damaging effects of humidity while fighting frizz for the perfect finishing touch to your hairstyle. The essence of velvet flower provides incredible softness, so hair never feels crunchy or stiff.\nSize: Required 1.5 fl. oz. 9.5 fl. oz.\nBoosts volume and tames flyaways while finishing styles with beautiful brushability.\nFine, layerable mist provides 360\u00b0 support to resist humidity and fight frizz.\nInfused with velvet flower for softness, so hair is never stiff or crunchy.\nI don't use a lot of hair spray (a can will last months) but this is great. It's not heavy, sticky or weighs down my hair. The smell is clean smelling.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When it comes to matcha green tea, we don't mess around. That's why we designed this 14 oz raw-finish matcha bowl to help you whip up Insta-worthy stone-ground green tea, every time. Thanks to its high hand-painted edges and easy-pour spout, making the frothiest matcha has never been easier \u2013 or more mess-free. Pairs perfectly with our Artisanal Matcha Tumblers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzazogf b/data_all_eng_slimpj/shuffled/split2/finalzzzazogf
new file mode 100644
index 0000000000000000000000000000000000000000..b80d2d8a09fc61ac7d96c900580bc446c9b7ce8b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzazogf
@@ -0,0 +1,5 @@
+{"text":"The course is intended to provide students with basic to intermediate skills in the use of industry-standard software and hardware being used in spatial science, particularly on geographic information system (GIS) and remote sensing. The delivery will focus on the hands-on approach and applications-oriented project done in an inter-disciplinary group that will cover urban and regional planning, environmental management, agriculture, engineering, etc. It will cover computer lab exercises, workshops, group work, and project presentations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tuscany, Siena, Buonconvento. In a delightful hillside location, with a 360 degrees view between the \"Crete Senesi\" and the \"Val d'Orcia\", a beautiful farm consisting of a Tuscan style farmhouse, dating back to the early nineteenth century and an outbuilding located on a large plot of 28 hectares.\nThe main house covers a total internal surface of \u200b\u200babout 800 square meters and was converted partly as a private residence and partly as a tourist accommodation for a total of 10 bedrooms and 10 bathrooms. The owners home is completely independent from the rest of the building and consists on the ground floor with a double living room, a beautiful kitchen, both with large openings to access the surrounding garden that give great brightness. A storage room, an ironing room and a laundry room complete the floor. On the upper floor there is a bedroom with a walk-in closet and an en suite bathroom.\nThe annexe, developed on two floors, consists on the ground floor of a reception and a common area, while the first floor hosts a small apartment with living area, bedroom and bathroom for a total of a further 76 square meters. The skilful restoration was carried out using the original materials typical of the area and modern technologies regarding the plants, which in addition to not altering the peculiar characteristics of the place have conferred that emblematic Tuscan character sought by those who want to experience this region. The property includes a panoramic swimming pool with a splendid view of the Tuscan hills.\nCities of great historical and cultural interest are easily reachable such as Siena, Pienza, Montepulciano and Montalcino as well as two of the most beautiful abbeys in central Italy such as the Abbey of Sant'Antimo and the Abbey of Monte Oliveto Maggiore.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We've all had two weeks to reflect on Hadoop Summit 2014. One of the biggest differences that stood out in this year's Summit (as compared to Summit 2013) was the presence of large enterprise customers that are using Apache Hadoop as an important part of their modern data architectures.\nHadoop has gone beyond its original Yahoo use case\u2014indexing the web via a nightly batch MapReduce process \u2014and into the mainstream of daily data processing and analytics with real-time, online, interactive, and batch applications at many notable companies.\nWe see our customers at Hortonworks reflect the maturation of enterprise Apache Hadoop. They include \"Web 2.0\" companies that you might expect to be early adopters of Apache Hadoop: Spotify, eBay, and XING. But we also support others like Bloomberg, Samsung, and Cardinal Health that incorporate HDP into modern data architectures at businesses that have been around for decades (or centuries).\nIn a complete recording of a Hadoop user panel discussion at Summit \"Hadoop in the Enterprise,\" the panelists discuss their journeys of Hadoop adoption at some of the world's largest, most established companies.\nHow and why did your company get started on its journey with Hadoop?\nHow will Apache Hadoop YARN help you get more out of your Hadoop cluster?\nHow did you approach integrating Hadoop with other components of your data platform?\nHow do you use Hadoop to improve your client experience?\nHow do you think of securing data in Hadoop when you're storing information about large dollar transactions?\nHow do you think of data governance in your Hadoop platform?\nIf you were starting your Hadoop journey over again, what would you do differently? What did you do right?\nRead more about how other customers are using HDP for enterprise Apache Hadoop.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"DJ's this is a brand new exclusive record that I want to test out with you all. It is very important that you send me feedback and let me know what you think of this joint. The only way to test a record and know if we have something is by getting your responses since you are the backbone of the business and the ones that people like myself rely on to support these records we put out. Let me know if we got this one \"In The Bag\" or not.\nIt's a different sound then what is out right now and I think you'll enjoy that breath of fresh air. Give it a listen, give it a spin, give it a chance. You already know I appreciate each and every one of your time. Let's get it in!\nNeed an Email Blast to over 108,000 DJ's, Radio PD's & MD's, Label Execs, A&R's, Bloggers, Media, Artists, Producers, Promoters, Managers, and everyone else involved with the music industry? Contact me by replying to this email. You can also visit our website to see other services we offer. WE BREAK RECORDS!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 66th Conference (2019 Spring Conference) of the Remote Sensing Society of Japan will be held as below. The details will be updated on the RSSJ web site appropriately.\n\u30fbPre-payment of the registration fee is not accepted. Please pay it in cash at the reception of the conference.\n\u30fb5,400 Yen for a Proceedings(CD-ROM) only.\n\u30fbNo registration free for undergraduate and graduate students to attend sessions only (no proceedings).\n\u30fbThe Registration Fee does not include the fee of the Banquet which will be held on the 4th June, after completing sessions. The Banquet Fee is 5,000Yen (2,000Yen for student). You can pay the Banquet Fee with the Registration Fee in cash only at the reception of the conference. Or only the Banquet Fee are payable at the starting time of the Banquet.\nWeb-registration-system of the 66th Meeting for both registration and paper submission has been started.\nThe deadline is April 5th for both registrations for presentation and paper submission.\n\u30fbManuscript of Microsoft word or PDF file is accepted. you can refer the template of the paper and download the template of the paper.\n\u30fbIf you use the Macintosh OS and Microsoft word, please convert the file type to PDF to avoid the corruption of the layout of figures and tables.\n\u30fbManuscript is either 2 pages or 4 pages (no page charge).\n\u30fbYour paper will be included on the CD-ROM of the Proceedings which will be distributed to the participants of the conference.\n\u30fbThe total of 15 min. for an oral presentation including discussion. A first bell is at 8 min, and second bell is at 10 min. to complete presentation.\n\u30fbPoster size shall be equal or less than B0.\n\u30fbPoster for Utilization session can be presented in both days. Other posters are presented in one day (June 4th or 5th).\n\u30fbPresenter must stay around his\/her poster during his\/her core time.\n\u30fbBanquet will be held on the 4th June, after completing sessions.\n\u30fbIf you have any question for this conference, please e-mail to conf@rssj.or.jp.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbebcd b/data_all_eng_slimpj/shuffled/split2/finalzzzbebcd
new file mode 100644
index 0000000000000000000000000000000000000000..11b59e9984596c92bf68d8b28ff11d968e4ea741
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbebcd
@@ -0,0 +1,5 @@
+{"text":"Live classical music from American Public Media hosts.\nSunday Baroque is a celebration of beloved and appealing music from the baroque era (1600-1750) and the years leading up to it.\nHosted by acclaimed pianist Christopher O'Riley, From the Top celebrates the amazing performances and captivating stories of extraordinary young classical musicians.\nTake your kids on a journey through classical music!\nHPM Classical is Houston's only classical music station, providing Houstonians with outstanding performances from local to international. Support comes primarily through the generous contributions of listeners with additional money from grants, special events and the local business community. HPM Classical is licensed to the University of Houston System governed by the UH Board of Regents. The Houston Public Media Foundation is a non-profit 501(c)3 support organization which serves to secure the financial future of Houston Public Broadcasting and to cultivate and engage community resources to support its mission.\nFull-time classical programming is available on 88.7 HD-2, TV 8.6 and streaming online via our player, iHeartRadio, and TuneIn. If you would like to provide feedback, please send your comments to membership@houstonpublicmedia.org.\nMore ways to listen, as well as troubleshooting information, is available on our Ways to Listen page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A strange phenomenon is happening in some Russian forests. Mysteries holes are popping up all over the country, and no-one seems to know where they are coming from, or how they got there. They appear in the dense forest where you can't possibly get a car or truck, or any equipment to drill them, and there is no dirt nearby that could have come from them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The promise of nanotechnology encompasses the energy, health care, defense, and chemical manufacturing sectors of the global economy. Gold nanorods are a class of nanomaterials that have tunable optical properties depending on par ticle shape, and these optical properties are key to sensor and imaging technology that directly impac these sectors of the global economy. In this talk I will detail how these nanomaterials are made and characterized; how the surface chemistry can be tuned to manipulate nanomaterial properties; and how the nature of the nanomaterial surface influences biological response at the molecular, cellular, and ecosystem levels.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Helyar Arms is a substantial, impressive and imposing detached character Somerset village inn. With origins reputably dated back to 1468, The Helyar Arms is Grade II listed and is presented to an exceptionally high standard throughout having been the subject of a fastidious program of refurbishment and improvement in recent years, whilst enhancing the original charm and character of the property. Almost no stone has been left unturned in the sympathetic modernisation and improvement of the property, which is renowned for its ambience and character, within a superior trading environment. This high quality destination village inn and restaurant briefly comprises:- open plan Main Bar and Dining Area (68 ), Restaurant\/Private Dining Room (30), extremely well equipped Commercial Catering Kitchen with Ancillary Facilities, 6 high quality En-suite Letting Bedrooms. A detached stone built former granary and stables in the grounds has been purposefully converted to provide a ground floor Skittle Alley\/Function Room with a first floor self-contained Owner's Apartment with Kitchenette, spacious Sitting Room, generous Double Bedroom and Bathroom. Adjacent to this is a large Double Garage\/Storage Barn and a separate Dry Storeroom. There is an attractive lawned and patio Beer Garden providing alfresco seating for around 100 customers, and extensive Car Parking for 30 vehicles. The Helyar Arms is a highly regarded and exceptionally well presented, high turnover quality country village inn, and a viewing is strongly recommended in order to fully appreciate the exceptional quality and standards inherent throughout all aspects of the property.\nThe Helyar Arms is situated in East Coker, an exceptionally sought-after village, which is one of Somerset's prettiest and highly regarded villages. The picturesque village of East Coker has some of the most beautiful ham stone cottages and properties in the county. Recognised as one of Somerset's most favourite and sought-after villages, East Coker is surrounded by open countryside which is rich in recreational activities and walks. The nearby Sutton Bingham Reservoir boasts a range of water sports activities. The nearby town of Yeovil (3 miles) offers a wide range of shopping, business and leisure amenities as well as main line rail connections. Other towns within easy reach include the Abbey town of Sherborne (7 miles), Dorchester (19 miles), the county town of Taunton (24 miles) and the Dorset Jurassic coastal town of West Bay (19 miles).\nMain Entrance Door to the front and to the rear from the car park lead into the Main Bar and Dining Area, an extremely attractive spacious character open plan room, presented to an exceptionally high standard. Main Bar Area with carpeted flooring, exposed celling timbers, part wood panelled walls, wall mounted menu boards, free standing wooden tables and chairs comfortable seating 14 diners, bar stools and poser tables for 12 customers plus adequate standing room. Dining Area with part wood block and part carpeted flooring, exposed ceiling timbers, part wood panelled walls, feature fireplace with inset wood burning stove, wall mounted menu boards, free standing wooden tables and chairs comfortable seating 42 customers. Substantial timber fronted, and polished timber topped Bar Servery fitted with associated back bar fittings, extensive display shelving and Altro flooring, electronic cash register, three double glass fronted bottle fridges and wine racking. Still Room with door to rear, Altro flooring, base kitchen units with complementary work surfaces over, inset stainless steel sink unit, glass washer, double glass fronted bottle fridge and hand basin. Apple Loft Restaurant\/Private Dining Room an attractive character room leading from the main bar area with carpeted flooring, exposed ceiling timbers and vaulted ceilings, painted stone walls, feature fireplace with inset wood burning stove, freestanding wooden tables and chairs comfortable seating 30 customers. Ladies and Gents WC's. Disused Gents WC with potential to create Office\/Storeroom. Commercial Catering Kitchen with tiled flooring, wipe clean clad walls being fitted to an extremely high standard with a comprehensive range of commercial catering equipment to include:- six burner gas range, extractor system with stainless steel canopy over and gas interlock, flat top range, twin table top deep fat fryer, extensive range of stainless steel work benches with shelving below, warming cabinet with heated serving gantry over, twin contact grill, grill, stainless steel hand basin, racked stainless steel shelving units, four-door commercial under counter fridge unit, microwave oven, twin deep bowl stainless steel sink unit, commercial dishwasher and door to rear. Preparation Room\/Refrigeration Store with Altro flooring, double upright commercial fridge, upright commercial freezer, four-door under counter commercial freezer unit, microwave oven, stainless steel racked shelving and double doors to rear. Boiler Room with two wall mounted boilers. Beer Cellar temperature controlled on the lower ground floor with racked shelving, chest freezer, Belfast sink and ice machine. Wine\/Bottle Store on the lower ground floor with racked shelving.\nThe business currently offers six, individually designed, high quality en-suite letting bedrooms finished to a high standard with remote control colour televisions, hairdryers, trouser press, WiFi and tea\/coffee making facilities. The bedrooms are all well-proportioned and characterful in keeping with the property and they briefly comprise: - Furzy Knap, a family room (3) with a double bed, single bed and en-suite shower room. Pin Cushion, a Double room with en-suite shower room. Foxhole, a double room with en-suite shower room. Rope Walk, a family room (3) with a double bed, single bed and en-suite shower room. Lychgate, a double room with en-suite shower room. Beryl Knapp, a double\/twin room with zip-link bed and en-suite shower room.\nOccupying the first floor of a detached, stone built former granary building within the grounds is a self-contained owner's apartment which briefly comprises: - Spacious Sitting Room with dual aspect, part wood panelled walls and exposed ceiling timbers, Kitchenette, generous Double Bedroom with dual aspect windows, part wood panelled walls and exposed ceiling timbers and Bathroom with shower over bath.\nA detached stone built former granary and stable building set within the grounds has been converted to provide self-contained owners accommodation at first floor level and a purpose fitted and equipped Skittle Alley\/Function Room on the ground floor. Adjoining this is a stone-built barn\/large Double Garage with a separate Store Room Area providing ample dry storage space. Directly to the rear of the property is a spacious Car Park with space for 30 vehicles, adjacent to this is an attractive lawned Beer Garden with patio area providing alfresco seating for around 100 customers, with raised flower bed borders and external feature lighting.\nThe Helyar Arms is an impressive, character country village inn and destination public house and restaurant, which is exceptionally well presented throughout. Retaining many period features as well as tremendous charm and character throughout, the property provides an exceptional working environment, having been renovated and refurbished to exacting standards in recent years. The business is located in a highly regarded and exceptionally sought-after Somerset village and The Helyar Arms enjoys a popular following and strong year-round regular trade from not only the village, but also the surrounding villages and towns, as well as an uplift from tourists and commercial travellers visiting the area throughout the year. The area is also popular with many walking\/rambling groups and societies, many of whom use the pub on a regular basis. The business is also home to three skittle teams, a poetry club as well as various other local groups, and is a popular meeting venue. The Helyar Arms trades as a quality village inn and restaurant offering letting accommodation and the business benefits from consistant, high occupancy levels and has been awarded a rating of 9.4 (out of 10) on Booking.com and a Five Star Food Hygiene rating. The business has its own website: www.thehelyararms.com which provides further background information on the business. The sale of The Helyar Arms represents an excellent opportunity to purchase a well-regarded country village inn and restaurant with high quality letting accommodation, the like of which seldom comes to the market. The Helyar Arms is undoubtedly a very special place to live and work and a formal viewing appointment is essential to fully appreciate all that this exceptional, high-quality business and property has to offer.\nAccounts for the year ending 30th April 2018 show net sales of \u00a3358,580 and the split of trade is approximately 41% wet sales 35% food sales, 24% accommodation sales. Full detailed trading information can be made available following a formal viewing appointment.\nThe business is run by our clients, a husband and wife partnership, with the assistance of around 11 part-time members of staff as and when trade dictates.\nOur clients advise that there is potential to develop all aspects of the business still further and for continued growth in trade levels, particularly by focused hands-on owner operators maximising all available income streams and building upon the impressive levels of trade already established.\nAll mains services with the exception of mains gas are connected to the subject property. LPG is used for cooking, central heating and hot water.\n\u00a399,950 for the valuable leasehold interest as a going concern business and full trade inventory. Wet and dry stocks to be additional at valuation. The premises are held on a 25-year Punch Partnerships Ltd, fully repairing and insuring lease agreement, from February 2016. We are advised that the pub is free of tie for wines. The current rent is \u00a348,300 per annum plus VAT. The next rent review is due in 2021. There is a rent deposit equivalent to approximately three month's rent. Full details available upon request.\nThis is a listed property and therefore exempt.\nStrictly by appointment with the vendors Sole Selling Agents. Tel: (01392) 201262. Email: info@stonesmith.co.uk You are recommended to contact us before visiting the property even for an informal viewing. We can then confirm whether or not it is still available.\nThe data showed regarding this advert 'Beautifully Restored Detached 15th Century Village Inn - 4765 \/ Rightbiz' comprises as a business advertisement. Rightbiz Ltd makes no warranty as to the accuracy or to the completeness of the advertisement or any linked or any associated information provided. In no event shall Rightbiz Ltd be liable for any damages or loss whatsoever including special, indirect, consequential, incidental or punitive damages for loss of profits and revenue, arising out of or connected with this website or to the use, reliance upon or performance of any information contained in or accessed from this website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Complete list of funny jokes Teeth: 1 \u2013 Why is a toothless dog like a tree? .. Check out these teeth gaps solutions dogs with bad spark bucked and just plain ugly teeth .. Play some of these other Funny hits! My favorites,.. Games \\ u003e Funny Pictures \\ u003e Dog Pictures.. Two teeth.. this dog has.. \u2013 Rate this picture: Bad Good .. False teeth, funny teeth.\nTired of looking so good? Dr.. B\u00fckk the famous false teeth are serious practical jokers who insist on the best green tea for teeth A journey through the unique, interesting, getting cavities filled baby teeth strange and ordinary things that make up the Internet .. Consider cropping the original images (pictures of smiling star) to give the teeth teeth grinding jaw swelling larger space in the entrance.. .. Do not forget to send us your very own bad teeth Pictures! See if you can stack up against other bad teeth pictures! .. Poor how long does puppy teeth last 12,757 Views face rash teething babies 2 descriptions.. Paulie.. 13\/09\/2009.\nSooooo jail! 0 votes.. login.. voting,.. Random pictures.. Login.. Not signed up yet? Register here .. Bad Teeth Pictures Pictures of the worst bad teeth.. Amy Winehouse Anna Paquin Cameron Diaz, Daniel Radcliffe, Drew Barrymore, Kirsten Dunst Miley Cyrus Morgan Freeman, Steve Buscemi .. Currently displaying images 1-102 of \\ \\\"Tooth clipart \\\\\" Next cat clip art cat clip art.. Angry, emotions, fear, funny, greedy, happy, hungry, introspective, .","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfpmb b/data_all_eng_slimpj/shuffled/split2/finalzzzbfpmb
new file mode 100644
index 0000000000000000000000000000000000000000..012fd366660d27a7233d202f90ec05fb93d16db6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbfpmb
@@ -0,0 +1,5 @@
+{"text":"\"Computational approaches are common across most scientific projects at Berkeley Lab. As Moore's Law is slowing down, novel computing architectures, system, and techniques have become a priority initiative at Berkeley Lab,\" says Horst Simon, Berkeley Lab's Deputy Director. \"We recognized early how quantum simulation could provide an effective approach to some of the most challenging computational problems in science, and I am pleased to see recognition of our LDRD initiative through this first direct funding. Quantum information science will become an increasingly important element of our research enterprise across many disciplines.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm at a bit of a loss as to where to start in reviewing 2012. It has been an awesome blessing of a year for me and so, so much has happened. Pinpointing just one place to highlight would be impossible, but it has all been awesome. In 2012, I have travelled through seventeen different states. I started out the year with my final semester of college in Columbia, Missouri \u2014 one of my favorite places in the world. I graduated with a Bachelors of Journalism (a BJ, as I like to call it) after taking one of my favorite courses from school \u2014 A Picture Story with Rita Reed. In that class, I learned and laughed and stressed so much, and Rita's words still echo through my head today.\nI produced some of my favorite stories of my career in that Picture Story class, most notably my Off The Gridstory about a group of people who live without electricity for a more meaningful life. Not only was it a blessing to photograph people without electricity (hello awesome window light!), but being around them was inspiring and relaxing and very enjoyable. I had a wonderful experience with them and am grateful for the time they shared with me.\nAnd of course, graduation was a blast and a curse \u2014 a bittersweet moment, as to be expected. But I had the world open before me, and an internship at The Oregonian in Portland, Oregon \u2014 a summer that I now refer to as one of the best summers, ever.\nSummer in Portland is a magical experience, and between craft beers and VooDoo donuts with the other interns, I took pictures and loved every minute of it. I struggled a bit with questioning my career path, however \u2014 things were a bit bleak with The O's publisher, Advance, and I felt a bit disillustioned after the comfy setting of J-School. It was a lesson in the role of business and publishing in journalism, and a valuable, though shocking, experience I don't know if I could've found anywhere else.\nThough I was unsure of my future at the time, I was especially grateful to have a slew of helpful ears and eyes to mentor me through the summer. Beth Nakamura, Thomas Boyd, Randy Cox, Lynne Terry and others were endlessly there for me in my magical intern journey, and I am lucky to have had their help.\nLeaving Portland, I took a beautiful journey through Yellowstone and The West with Tyler, with a short pitstop in Kansas City. And then it was on to Dallas for a fall internship at The Dallas Morning News, thanks to the wonderful Chris Wilkins.\nMy time there was quite intermittent and because of the lack of an oven at Michael Ainsworth's house, where I stayed, I couldn't make my traditional \"intern who fattens everyone up with cupcakes and cookies\" introduction. Though my time in Dallas was short, due to an exciting early and unexpected job offer in Virginia, I feel like I took more pictures there that I am proud of than anywhere else. It just goes to show how important an amazing editing staff is, and The DMN's got it.\nOf course, when I was in Dallas, I also hopped over back to Missouri for the magical week of the Missouri Photo Workshop. I met some fantastic people, both photographers and residents of Troy, Missouri, and the idea of community journalism was revitalized within me. It was perfect, because my next step was about to become a position that would herald just that \u2014 a job at a small town newspaper in Staunton, Virginia. My first job!\nSo here I am now \u2014 a degree, a workshop and two internships later \u2014 in Virginia. I've had a lot of firsts this year. Of course, I got my first job, though I also spent my first holiday away from my family. But I'm so glad I'm here \u2014 this area is gorgeous and it seems appropriate to end it here, with what was a New Year's Resolution from the beginning (employment).\nThroughout this year, I've made so many friends, and met so many inspiring people that have pushed me to become a better person and helped me achieve my goals. I think it would be impossible to name them all, but I end this year with a wonderful taste of both amazing strangers and cherished loved ones who are the reason I am where I am now. My journey to whatever I'm going to become is not even close to being over, but I'm at a great jumping off place for the future here in Staunton. Here's to 2013, and whatever it brings!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Biomedical engineers at the University of Virginia (UVA) and John Hopkins University (JHU) have developed a prolific collaboration that has generated several long-term, multi-million-dollar focused ultrasound research grants.\nRichard Price, PhD, Research Director of the UVA Focused Ultrasound Center, and Justin Hanes, PhD, Director of the Center for Nanomedicine at the Wilmer Eye Institute and JHU School of Medicine, are developing nanoparticles and delivering them deep into the brain with the assistance of focused ultrasound. Their work has earned them nearly $7 million of National Institutes of Health (NIH) funding from 2011 to 2020 to propel their discoveries to the clinic.\nIn their UVA biomedical engineering laboratory, Price and his team are using focused ultrasound to allow the targeted and controlled delivery of nanoparticles deep into the brain. Justin Hanes and his team at JHU are creating the nanoparticles that Price is delivering. These nanoparticles are essentially encapsulated\u2014or coated\u2014molecules, such as chemotherapy drugs or protein-coding nucleic acids.\nA neurosurgeon and pioneer in using focused ultrasound for psychological disorders, Jin-woo Chang, MD, PhD, will serve as the Honorary President of the 5th International Symposium on Focused Ultrasound, to be held August 28-September 1, 2016, in Bethesda, Maryland.\nDr. Richard Merkin has generously offered to continue his support of this unique mid-career fellowship opportunity for international researchers. The Merkin Fellowship brings scientists from around the world to work with the technical and scientific team at the Foundation on specific research projects. The position fosters collaboration between the Foundation, the Fellow's home institution, and other institutions and carries the potential for an associated academic appointment. Applications for 2016 and 2017 are being accepted. For information, contact Matt Eames, PhD, Director of Extramural Research.\nA team of Foundation researchers recently returned from a productive trip to Korea, where they made presentations and expanded our collaborative network. John Snell, PhD, Brain Program Technical Director and current and past Foundation fellows Dong-guk Paeng, PhD, and Jean-Francois Aubry, PhD spent a week speaking at two universities and two conferences. Dr. Paeng shared his country, national cuisine, and culture with the others.\nThe Foundation's website now features a timeline presenting the in-depth history of focused ultrasound. The scrolling line spans from the late 1800's to present day and features prominent milestones along with photographs of the pioneers who have developed the technology. It will be updated as new achievements are made.\nThe Foundation was selected as a leading philanthropy and featured on the cover of the annual \"Generous Virginians\" issue of Virginia Business magazine. The story is raising awareness of the need for philanthropic giving to support medical innovation and how our Foundation is helping to advance focused ultrasound to improve the lives of Virginians and others with serious diseases around the world.\n\"I can't believe that invasive surgery is still the sole option for treating benign tumors. We need alternatives, and focused ultrasound could prove to be the best alternative.\"\nFoundation-funded research has enabled Elisa Konofagou, PhD, and her colleagues at Columbia University to establish safety parameters for using focused ultrasound to temporarily open the blood brain barrier (BBB) and effectively deliver drugs to previously unreachable brain tissue. These findings may lead to new treatment options for patients with Alzheimer's disease and other neurodegenerative disorders.\nHer work on this project, entitled \"Blood-brain barrier (BBB) opening for facilitating drug delivery in neurodegenerative diseases in non-human primates,\" has led to an extended NIH R01 grant. Her research has produced three publications and seven presentations at scientific meetings. Dr. Konofagou visited the Foundation in May, presented a webinar, and allowed us to profile her fascinating and important work.\nChildren's National Health System is the first site in the US to use focused ultrasound to treat osteoid osteoma, a benign but painful bone tumor that commonly occurs in children and young adults. The team successfully relieved pain in two patients, 16 and 10 years old, as part of a clinical trial. This study is running parallel to a similar study conducted at The Hospital for Sick Children (SickKids) and Sunnybrook Health Sciences Centre in Toronto, where their first patient was treated last August.\nA preclinical study entitled \"Low Intensity Pulsed Ultrasound (LIPUS) for the Treatment of Spinal Disc Degeneration\" has been funded for the 2nd quarter of 2015. Chris Diederich, PhD, and his colleagues at the University of California, San Francisco will determine whether LIPUS could be used in the treatment of low back pain caused by intervertebral disc degeneration. The LIPUS study will test the potential of a focused ultrasound system to deliver a therapeutic dose of energy that is adequate to increase cellularity, reduce inflammation, and improve remodeling in acute or degenerative disc-related back injuries.\nAre you looking for funding for a research project in the field of focused ultrasound? Learn about our External Awards Program.\n\"The Foundation sponsors AAPM to support the presence of therapeutic ultrasound sessions at principal meetings.\"\nThis year's American Association of Physicists in Medicine (AAPM) Meeting, to be held July 12-16 in Anaheim, CA, will feature a special 3-day biennial track on Ultrasound Therapy and Imaging. Covered topics will include recent advances in ultrasound for diagnostic imaging, ultrasound for guidance and control of radiation therapy, and therapeutic\/focused ultrasound. The program is led by Chris Diederich and Mark Holland.\nCould a uterine fibroid numerical rating scale assist physicians with focused ultrasound patient selection? Is CT always needed when using focused ultrasound to treat the brain? Read these papers, along with two that review focused ultrasound treatment for glaucoma.\nFoundation-funded research conducted by Paul Dayton and his group at UNC\/NC State in collaboration with UVA has recently been published in The Journal of Therapeutic Ultrasound. As described in the final study report, the group found microbubbles superior to phase-shift nanodroplets for decreasing focused ultrasound ablation times. The full-text, open-access article is available on the JTU website.\nLong-time Symposium attendees and exhibitors Promedica Bioelectronics have recently released a benchtop focused ultrasound device for cellular and in vivo tissue research. The all-in-one \"Sonowell\" system is designed for conducting sonoporation experiments, applying controlled damage to the extracellular matrix of microbial organisms, stimulating cell growth, and regenerating in-vitro tissue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It appears there are Christmas wreathes on the door. We'll update when we find out WTF.\nUpdate: The explosives turned out to be firecrackers. Described as large illegal fireworks. They were m80, m100 fire crackers. I'm sorry but I'm from a place were large fireworks means those great big shell thingys, they are way cool.\nThank God the fuzz saved Christopher Clark's fingies. Chris is being held and questioned by the FBI and the fuzz cause now they gotta talk this up for stirring up all that hub hub over nada.\nProcessing 0.0, elapsed 0.0027 seconds.\n13 queries taking 0.002 seconds, 7 records returned.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The audit recommended better planning to ensure adequate street repair and recommended better processes to avoid conflicts of interest in city business.\n(JEFFERSON, Mo.)\u2014 Missouri State Auditor Nicole Galloway released a citizen-requested audit of the City of Hamilton Wednesday morning.\n\"Citizens deserve a government that functions efficiently and transparently, but processes were not in place in the City of Hamilton to ensure this is the case,\" Galloway said. \"I'm encouraged that the board of aldermen seems to be taking the recommendations in the report seriously so they can better serve taxpayers.\"\nVoters approved two sales taxes to support the improvement of city streets, the first in 1993 and then another in 2011, which was renewed in 2016.\nHowever, auditors found city streets to be in poor condition with numerous potholes, cracks, and crumbling pavement.\nThe city does not have a formal plan to guide street maintenance and repair. The audit recommended the city develop a formal annual maintenance plan, as well as a comprehensive long-term plan to most efficiently use available resources.\nThe audit also found the city did not have proper procedures in place to avoid conflicts of interest. Specifically, there were two instances when the city hired an alderman's company to complete projects for the city, but did not solicit competitive bids or provide public notice, as required by state law.\nAdditionally, the audit found the Mayor failed to properly report legally required information on his financial disclosure forms. As a result of these actions, the Missouri Ethics Commission determined laws were violated and fined the alderman and the Mayor.\nThe audit includes recommendations to ensure utility billings and account balances are accurate and better track customer deposits.\nThe audit also found the need for increased oversight over financial processes and recommended better budgeting practices.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfqku b/data_all_eng_slimpj/shuffled/split2/finalzzzbfqku
new file mode 100644
index 0000000000000000000000000000000000000000..f3a1da62e075246b093b670eb653fcbd95b0bb7d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbfqku
@@ -0,0 +1,5 @@
+{"text":"Shunlongwei Inspected every WP90528L3 before Ship, All WP90528L3 with 6 months warranty.\nQ: How To Order WP90528L3?\nQ: How To Pay for WP90528L3?\nQ: How Long Can I Get The WP90528L3?\nA: Yes, The product Engineer will help you on the WP90528L3 setup, Application, Replacement, Repaire.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What's the Difference Between Conventional Agriculture & Sustainable Agriculture?\nOur current model of agriculture is highly detrimental to our health \u2013 as well as the health of the planet. Here is what you should know about moving towards a more sustainable agriculture model.\nWhen it comes to human impact on the environment, few human activities create as many environmental problems as agriculture. From dead and depleted soils that will no longer grow food crops without massive inputs of chemical fertilizers, to poisoned rivers and streams, to \"dead zones\" in the oceans caused by agricultural runoff, agriculture is taking a massive toll on our world.\nObviously, the human race as we know it could not survive without agriculture, so what is the answer?\nThe problem is not the fact of agriculture, but the method. What is now known as \"conventional\" agriculture (I prefer the term \"industrial,\" which I feel describes it more aptly) has only been around for just over a century \u2013 a blink of an eye in the span of the human race \u2013 yet it has caused untold damage to the planet during that time.\nTo be fair, transitioning from this model to a more sustainable type of agriculture can be very expensive \u2013 and most farmers aren't known for their wealth! But with some innovation and cooperation between private and government groups, a solution must be found to this growing crisis.\nTo the detriment of both consumers and the environment, hormonal supplements are fed to many food animals.\nObviously, a more sustainable model of agriculture is imperative if we are to continue thriving as a species on this planet.\nSustainable agriculture takes into account the promotion of biodiversity, recycling of resources, promoting ecological balance and much more. It also takes into account the locality of the farm and the existing conditions on the ground. Profitability remains a goal of sustainable farming but environmental health and social equity find their rightful place in this system.\nHand weeding is the happy alternative for pesticides that go beyond that leaves a footprint after killing the weeds.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"URN etd-0120109-160022 Statistics This thesis had been viewed 1725 times. Download 8 times.\nAbstract This thesis will discuss two different chapters. First, fabrication of low temperature poly-silicon TFTs by using oxide and n+ a-Si by PECVD and the poly-Si and n+ poly-si are obtained by the EAA technique that we develop before. We will discuss the effect of off-state leakage on the different thickness of n+ poly-Si.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the forest, Qin Yu took off his black robe and swiftly moved, not at least bothered by the poisonous miasma around him. After several tens of li he searched for a hidden cave, not killing the wolves inside for fear of attracting any attention with the smell of blood.\nHe carefully covered his tracks and sealed the entrance with a rock. He waited for two hours, and when he was sure no one followed him, Qin Yu couldn't bear it any longer, and was taken by a heavy sleep on the dried repugnant gra.s.s. Put it didn't go on for long and soon woke up, vigilant of his surroundings, then sat cross-legged and started swallowing pills.\nOnly now is he aware, that over the course of the event in Dongliu City, his clothes were all soaked in sweat.\nLiang Taizu's true target from the beginning, was the Grandmaster. Even without Zeng Mo'er's instigation, he would have still caused him trouble. Takings the stone fragments from his chest, Qin Yu felt bitter. Poisonous finger, Cang Mang's Golden Core, Pill Disposing Facility's Corpse Sealing Nail\u2026 even lying, he used all the means at his disposal to stay alive.\nBut now, there's still one urgent problem. Li Yunmo's storage bag!\nAh, Golden Core expert! Even if he is a demonic cultivator, he should still have some riches. Taking it out from his chest and opened it. As Qin Yu sent his thought inside, his features slightly changed in the rush to search it.\nThe small hill of items appearing in front of him when he took everything out of the bag, consists of only jade boxes. The jade appears black, with some b.l.o.o.d.y streaks, releasing a cold aura, and cooling the surroundings by several degrees.\nQin Yu eyes flickered, after a careful inspection he didn't find anything fishy, and opened a jade box. Inside there is a bone coming from an unknown creature, its surface s.h.i.+ning akin to a jade.\nWhat is it used for? Closed it, then took another box to look inside.\nWhat entered his eyes seemed to be a soft feeler-shaped thing, slowly squirming, with a hint of foulness coming from it.\nOpening the third one in a rush, a palm sized sh.e.l.lfish appears, with intricate patterns, but why does it have a hole?\nA mighty Golden Core demonic cultivator's storage bag, hardly only have these, right? Browsing through all boxes, Qin Yu's expression was gloomy. 36 boxes, filled with miraculous oddities, yet not one he can use!\nHe shook his head letting out a bitter laugh, put them all back, afraid that if he were to look again, then he will suffer internal injury from anger.\nThe bigger the hope the bigger the disappointment!\nInside the cave, Qin Yu opened his eyes, a flash of energy crossing them, soon after looking somewhat happy. Although he was provoked in Dongliu City, forced to reveal all his cards, but in this short period his cultivation reached the 3rd layer of Foundation Establishment.\nHe might have a lot of pills, and even more ingredients for refining, but still stopped his cultivation to think of the future.\nShould he return to Sacred Mountain Sect or not?\nThis thought kept coming back to him for many days, having its own advantages and disadvantages.\nDon't return, and he will be free and unrestrained. But he will not have a sect's protection when going out.\nAfter a long time, Qin Yu released a breath, still deciding to return to the sect!\nHe is still too weak, even a strong wind could topple him, and end up suffering a tragic end. With Little Blue Lamp on him, even if it is of supreme importance to get stronger, the first priority is still safety. Only living longer can he get stronger. Die, and it's all in vain.\nAfter an hour, Qin Yu left the cave he spent the last half a month in, was.h.i.+ng the bad smell in a nearby stream, then pressed onward to Sacred Mountain Sect.\nThe sky was filled with stars yet no moon in sight, unable to see even your fingers. But starlight is enough for Qin Yu to find his course.\nHe entered the mountain of the sect safely, and as he reached Pill Disposing Facility's formation, he relaxed, going down the stone steps, with a smile on his face.\nIt's been close to two months, since he left Pheasant King in the pill poison. This chicken followed him for a long time, and learned how to steal pills, becoming smarter. This punishment is good for him, as to learn some self-control.\nDescending, Qin Yu looked around, and his smile froze.\nWith footsteps echoing behind him, Qin Yu suddenly turned his head, meeting the eyes of a shocked pale man, but soon became ecstatic.\nA slight sound, in this underground place, is incredibly loud.\nQin Yu's heart sank, moving in an instant, he hit the male with his hand, who is now collapsing on the ground. Stuck his finger in his mouth, feeling around, and then pulled a broken tooth, causing the male to twitch.\nSeeing the b.l.o.o.d.y tooth, flas.h.i.+ng with a weak light, Qin Yu's face immediately turned ugly.\nQin Yu sucked in a breath and suppressing his panic. He then carefully looked at the male, and he appears familiar, finally remembering.\nYu Er, a Sacred Mountain Sect's outer sect disciple!\nPill Disposing Facility's formation is a one-way entrance, yet Sacred Mountain Sect have others methods of gaining access, since he didn't find this poor fellow when he entered this time. For Yu Er to appear here, Sacred Mountain Sect must have already discovered his absence.\nThis thought was suppressed as soon as it struck him. Yu Er must have already sent news to the sect, it is far too late to run now. If he thinks about it, once he runs, he'll definitely get caught, and involve other people that he cares about, even if they aren't blood related.\nCan't run, and can only face it head on. Qin Yu rushed inside the bedroom, and sees the now stone blocked hole, proof that the area below was discovered.\nWith a low cry, using all his energy and body's power, lifted the rock, and yelled, \"It's me, if you don't want to die get over here!\"\nA fluttering sound was heard, the miserable Pheasant King rus.h.i.+ng over, acting cute and pitiful, but was stopped by Qin Yu's hand.\nPulling out the two storage bags, on the point of emptying them, Qin Yu's eyes flashed, and with a flick of his sleeve dumped all the jade boxes on the ground, then put all them in the golden thread storage box, afterwards throwing it in the hole!\n\"Hide this storage bag. Don't let anyone find it!\"\nPheasant King picked it up with his beak, turned around and flew away.\nQin Yu felt a little better, put the stone back on the entrance, and the now empty storage bag in his chest.\nSuddenly, the restriction on the door heated slightly. Someone came in!\nQin Yu turned his head and rushed out in a panic, while a shadow was flying down the stairs. Whit a head full of messy grizzled hair, sunk in cheeks, and with brown eyes that cause fright to anyone who gazes in them, waved his sleeve and Qin Yu felt as if a he fell in a swamp, rooted on the spot.\nThe old man's felt around with his hand, grabbed the storage bag, sneered, and then his face immediately turned cold.\nQin Yu was scared stiff.\nOld man frowned, and lifted his hand, removing the lock on him.\nQin Yu yelled, \"Grand Elder, spare my life!\"\nThis old man, is actually one of the Sacred Mountain Sect's Golden Core experts, and Patriarch's martial brother, Huang Danguai.\n\"You recognize me?\" Huang Danguai eerily smiled, \"Then tell the truth, since you already know this old man's methods. \"\nQin Yu shuddered, \"Empty, the storage bag was already empty!\"\nHuang Danguai spoke, \"Since you tried, then I will also try.\"\nHe slapped Qin Yu on the body, burrowing a thread of black energy in him, making him scream and roll around on the ground in a craze.\nHuang Danguai sneered, \"Had enough?\"\n\"Grand Elder, disciple didn't lie. Just kill me\u2026 kill me!\" Qin Yu let out a miserable yelled again and again, but it was all a ruse. His body saved up a lot of poison for more than a year, and has a great deal of resistance towards it. Huang Danguai's poison is terrifying, but not to such extent as to make him collapse in pain.\nIn the next instant, a voice echoed around them, \"Martial brother, he is still the sect's disciple in the end. We must first understand the situation, then will give judgement.\"\nGentle and upright, with an imposing air. Talking like this is proof enough that the owner is the Sacred Mountain Sect's Patriarch.\nHuang Danguai's brows furrowed, but still slightly nodded, \"Fine.\" With flick of his hand, the black energy came out of Qin Yu, grabbed him and flew outside.\nA Golden Core's speed is astonis.h.i.+ng, very soon they arrived at the sect's Main Hall. Dropping Qin Yu with a thud, Huang Daiguai turned and took a seat.\nThe Patriarch is a middle-aged man, the long beard giving him a carefree air. He then cupped his hands and smiled, \"Thank you, martial brother from bringing him.\"\nHuang Danguai humphed, and closed his eyes, not showing any respect.\nPatriarch didn't seem to mind, and turned his eyes to Qin Yu and firmly demanded, \"Qin Yu, tell me, how did you leave the Pill Disposing Faciliy?\"\nHuang Daiguai within his half-closed eyes, a sharp glint could be seen, showing his interest in the matter.\nWith a trembling voice Qin Yu said, \"Reporting to Patriarch, two months ago, someone suddenly charged in, and took disciple away.\"\n\"Bulls.h.i.+t!\" Huang Daiguai exploded, \"If you lie again, I will make you bleed to death from your seven apertures!\"\nQin Yu was now completely scared, \"Disciple tells the truth!\"\nPatriarch turned, \"Martial brother, let him explain.\" Then at Qin Yu, \"This is a serious matter, Qin Yu. You must be certain, or I won't be able to help you.\"\n\"Yes, disciple knows what to do!\" Qin Yu expresses his grat.i.tude, recalling the event, then hurriedly said, \"Disciple remembers clearly. On that day I was cultivating, and this person suddenly appeared in front of me. Disciple couldn't resist at all and was captured. He originally wanted to kill me, but after discovering the secret tunnel in the bedroom that disciple opened, took me with him underground.\"\n\"What secret tunnel?\" Patriarch asked calmly.\nQin Yu's face was shocked, \"Patriarch, below the Disposing Facility is an underground area, that I discovered by chance. Inside is a medicine field, and a skeleton of a man named Cang Mang.\"\nHuang Daiguai abruptly opened his eyes, and with a fierce glint in them, \"Cang Mang! Are you certain it's Cang Mang?\"\nQin Yu nodded repeatedly, \"Disciple found a jade slip, containing this name, and that\u2026 that he was trapped underground\u2026\"\nPatriarch's features were grave, \"Continue.\"\n\"After disciple discovered this place, as kindness for the storage bag next to his skeleton, decided to give him a proper burial. The mysterious person, once he left the underground area, he was struck with grief, and after knocked me unconscious took me out of the facility.\"\n\"Then, he said we were bound by fate, and wanted to test me if I qualify to be his disciple.\" Qin Yu showed an embarra.s.sed face, \"Because of disciple's inability, had no choice but to return.\"\nHuang Daiguai let out a cruel laugh, \"You said you buried Cang Mang? Then what about that other grave?\"\nQin Yu was stunned, \"What other grave? There is only one. Disciple personally buried him in the hut.\"\nHuang Daiguai's eyes went cold, \"Brat, dare to hide things. Are you asking for death?\"\nQin Yu scared, \"Disciple doesn't know, truly don't know. When he came out of the area, that person knocked me out!\"\nPatriarch said, \"Cang Mang's storage bag, is the one in martial brother's hand?\"\nQin Yu shook his head, \"No, Cang Mang's storage bag, was taken by Grandmaster. Grand Elder has the Seven Murdering Demonic Sect Li Yunmo's storage bag, that Grandmaster gave it to me when before I returned.\"\nPatriarch's eyes immediately brighten, \"Seven Murdering Demonic Sect? Are you talking about Dongliu City?\"\n\"How does Patriarch know?\" Qin Yu slightly jumped, and soon understood, \"Oh, it's been almost half a month. Dongliu City's encounter must have traveled far. What luck, disciple can prove his innocence. The one who took me is exactly that Grandmaster.\"\nIn a deep voice, Patriarch said, \"Qin Yu, think carefully. Many people know about this, and it is easy for me to find someone to contest your sayings. If you dare lie, no one will be able to help you!\"\n\"Patriarch, disciple can prove that Qin Yu is speaking the truth.\" A cold voice came from outside the hall and an image of a beautiful woman appeared.\nAs she walks towards the light, the dark still couldn't hide her features, that of a breathtaking and mesmerizing beauty.\nThey made eye contact, then she retracted her eyes and made a bow, \"Greetings Patriarch, Master Huang.\"\nYou're Read Refining The World Chapter 16 online at BestLightNovel.com. You can use the follow function to bookmark your favorite novel ( Only for registered users ). If you find any errors ( can't read,broken links, can't load photos, etc.. ), Please let us know so we can fix it as soon as possible. And when you start a conversation or debate about a certain topic with other people, please do not offend them just because you don't like their opinions.\nYou're reading Refining The World. This manga has been translated by Updating. Author(s): \u98df\u5802\u5305\u5b50. Already has 131 views.\nCurrent Time is 25-Apr-2019 21:44:18 PM.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A man accused of murdering local Carluke resident Paul Halley last weekend, has appeared on petition at Lanark Sheriff Court.\nQuintus Aurelias Montague, 33, also from the town, appeared in private for a hearing where he made no plea or declaration to the single charge of murder.\nThe accused was committed for further examination and remanded in custody pending further police enquiries.\nIt is expected he will appear again within eight days.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfqvc b/data_all_eng_slimpj/shuffled/split2/finalzzzbfqvc
new file mode 100644
index 0000000000000000000000000000000000000000..e9353133c5ad1e6eeef3e2a73a033470f0d9aba0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbfqvc
@@ -0,0 +1,5 @@
+{"text":"Your Academy website is the ideal way of helping and guiding parents and children on topical and important issues. Internet security is very much one of these!\nFKBKO - Beko is here to teach you everything you need to know about internet safety. There are 3 levels \u2013 beginner, intermediate & advanced, so you can be sure that there's something for you to learn!\nCBBC Stay Safe - Do you know how to stay safe online? Dongle the rabbit will teach you all he knows with his video and quiz.\nSafe Surfing With Doug - Find out about safe surfing with Doug and his friends.\nGrid Club - The cybercaf\u00e9 within Gridclub is a safe online community where you can make choices without getting hurt. You can join in the fun at the cybercaf\u00e9 by playing the game. If your Academy isn't teaching the Internet Proficiency scheme through Gridclub, why not speak to your teachers about it and tell them how exciting it is!\nKidsmart - Kidsmart gives you lots of advice on how to stay safe online. There's a section for kids under 11 years old and a separate section for those over 11 years old. 4 great games will test your online safety knowledge.\nMcGruff - McGruff is a Crime Dog - world famous for his advice on how to stop crime before it happens, and for his great sense of humour! Some of his work involves teaching children how to stay safe online. Check out the different areas of the site \u2013 there's so much to learn!\nNet Smartz Kids - Be totally cool and show your friends how much you know about Internet safety by watching the video clips on here. There are also loads of activities and games that will test your knowledge!\nNewsround Quiz \u2013 Internet Security - Test your knowledge of internet safety with this quiz from CBBC Newsround. There's also further information if you're not sure on anything.\nSafe Kids Quiz - The Online Safety Quiz is your chance to show that you know how to be a safe Internet surfer. Answer each question and, when you get it right, you'll go to the next question.\nSafe Kids \u2013 Internet Safety Rules - Here's a list of rules to help you stay safe when you're online. How many of them can you remember? Why not print them out and put them up next to your computer.\nSid's Online Safety Guide - Become an internet safety expert with Bizzi Sid by collecting codes and playing cool games.\nSurf Swell - Surf Swell island provides you with lots of adventures in internet safety so that you can collect jewels to take to treasure palace \u2013 are you brave enough to take the challenge?\nThink You Know - The \"Thinkuknow\" website is brought to you by the Child Exploitation and Online Protection (CEOP) centre. You may have seen some of their video clips on National TV recently. There's a lot of information here for both children and adults \u2013 parents even have their own area of the site! Teachers have the chance to train from the professionals and then deliver suitable training in Academys.\nSafety Land - Safety land is normally a very nice place to live, but a nasty character is sending yucky emails and messages. Captain Broadband needs your help to find the nasty character. You need to navigate around Safety land answering questions. When you have answered them all correctly, the nasty character will be taken to jail and you'll become a certified hero, just like Captain Broadband.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hello! Well, it's Ash Wednesday and that means one thing: Lent. Ready? Me neither, but here we go! I am determined to help my children understand Lent and the Passion a little bit better this year, even though they are still quite young. To this end, I have have been working hard on what I am excited to be sharing with you today: 13 Lenten Finger Puppets. It is my hope that allowing my children to play with these puppets they can come to understand the important things said and done by each person, and the message we can learn in each case.\nI have set this up as a create-along. Each day I will share another Lenten Finger Puppet complete with a template, tutorial and brief reflection. It is my hope that you, or your child, can make this a daily activity and in the end you will have a complete set ready to teach children. All of the templates are completely free, my only hope is that they are put to good use!\nSince the puppets are generally made in the same way, I am using this first post as a place for general instructions. As you work, please refer back here for detailed instructions and tips.\nSupplies\u2013These are felt finger puppets. Since I work with felt often, and have a large supply, I used wool felt. This is more expensive, and crafting felt, bought at a place like Walmart, will work well too. I have a few examples of the puppets made in crafting felt at the bottom of this post. Beyond felt you will also need clear packing tape (for cutting out pieces), embroidery thread and needle, a glue ( I used Fabric Tac) and a sharp pair of scissors.\nUsing Templates\u2013I have created a unique template for each puppet. These are free to you and will download in a PDF format. Once you have it, simply print it on regular paper, cut out the pieces and use!\nCutting\u2014Once you have the template and supplies it's time to get cutting that felt. This is where the packing tape comes in. It is really difficult to cut out small detailed pieces out of felt, unless you use clear tape. For this method, simply cut out the template piece, cut out a piece of felt roughly bigger then the template, and tape the template down to the felt. Then, cut. This will hold the template piece in place, without the buckling that will happen if you use pins.\nCutting Beards\u2014The one time you might not want to use the templates for is cutting beards, and there are a lot of bearded fellows in this collection! Instead, use the template to cut out the general shape and then, fold the felt in half and free-hand cut out the holes for eyes and mouth.\nCreating Eyes\u2014There are several ways to give your puppets eyes. Since I am a stitcher I used French Knots (here is a great video tutorial for French Knots), but you can also make eyes our of black beads, fabric paint, or black Sharpie Markers. Your choice.\nGeneral Assembly\u2013Since I used glue the assembly is quite easy. Once the eyes and special stitched on details are done, glue the smaller pieces like the beards, robes, chalices, etc onto one of the body pieces. Then all of those pieces are in place, then take the back body piece and draw a thin line of glue around the edge on what will be the inside of the puppet. Try to stay as close the edge as possible, and make sure to leave the bottom open. Then, press the two pieces together and allow to dry. I really recommend using as quick-drying glue!\nA Word on Stitching VS Gluing\u2013I like to stitch (see here), but didn't have the time to stitch up all these puppets. In the tutorials to follow I add a few stitching details where needed, but I generally used glue, and am happy with the result. If, however, you would prefer to stitch, here are a few simple instructions.\nAttach on Robes, Beards, etc with a running stitch in matching thread.\nTo attach the front and back pieces use a blanket stitch, as pictured above.\nReflections\u2013for each puppet I have included a simple reflection that focuses on what this person said, did and what simple lessons we might learn from this. These are basic and certainly not something you have to use. I hoped only to give you a place to start while using these puppets.\nAnnnnnd, there you have it! Please, let me know if you have any questions, and I'd love to hear about how you have decided to use your puppets once they are made.\nThank you so much for your amazing website! I'm a CCD teacher at our local church and could really use your puppets to teach the class.\nHi Nancy, where can we find the reflections?\nI can not wait to show my kids! I might be missing something, but can you show me where the link is for your simple reflections for the puppets? I don't see it on the post. Thanks!\nI'm excited to make these for my kids! I might be missing something, but can you show me where the link is for your simple reflections for each puppet? Thanks!\nThank you so much for this! My 8 yr old son wanted a craft to do and chose this one..It was a large undertaking, but he cut all of the patterns and most of the felt(some sisters helped), then I hot glued them together. They came out great! Thanks for the tip about taping the patterns to the felt! That was a life saver! My kids will be having a blast with these tomorrow!\nthanks for letting me know! I'll be sure to do that!\nThis is such an awesome idea and I'm so glad you shared it on the 40 Days of Seeking Him meme a few weeks back. I'm thinking these will be fantastic for teaching the faith throughout the year.\nHi Nancy\u2026just shared your index image with a link today on a new post. Hope you like it!\nMonica recently posted\u2026First Reconciliation Prep or Refresher: Either Way, A Great Little Book about Confession!\nGreat! Thanks for all your support, Monica!\nThese finger puppets are adorable! Thank you for sharing. Many families can benefit from your tutorial.\nSelling them\u2026now there is an interesting thought. Thanks for being here!\nAwww, these are so cute! Wish I had known about this when I was teaching Kindergarten!\nSpreading the love and stopping in from the link up! These are awesome I must say! I am certainly going to pin this so that next year my middle daughter can practice her sewing skills (while learning more in depth about each character) and then we can use them to help teach our youngest! How wonderfully creative!\nThis is such a wonderful idea, Nancy!\nThanks, Kara. My kids are really getting into playing with them!\nI love this!! I am doing this for my Primary kids and their lessons!! Also for my daughter to play with during sacrament! Love it, and thank you so much for sharing!\nthat's a great idea, bringing them to mass! I hope they work out for you!\nYou are speaking my language. =) I love felt crafts and I love your finger puppets! I love your tip about using the packing tape to cut the small pieces and I love that glue is an option. Ooh, I really want to make these! I hope it's ok that I start spreading this post e v e r y w h e r e !!\nPlease, please! Share everywhere! I did all of this work so that Catholic Moms everywhere could make these for their little ones!\ntotally agree about how Christmas is easier to teach to the young than lent. I hope you have a chance to make these for your little ones and that they serve your family well!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We have tyres available for most motorcycle wheels. If the tyre you require is not in stock every endeavour will be made to obtain it for you. There are usually special \"deals\" on and off during the year, give us a call for details.\nFitting to loose wheels is free of charge.\nWe can fit tyres on to your bike, however you will be charged for this service and you will need to call and arrange a convenient appointment.\nInner tubes are carried in stock. Prices start from \u00a37 for tubes up to 14\" and from \u00a38 for 15\" upwards.\nWe also carry certain heavy duty inner tubes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whatsapp is the best application to give the sweet messages to our loved ones. Nowadays, WhatsApp is turning into a web sensation to all of the all inclusive community, and it plays a critical in our step by step life. Especially young and secondary school people generally putting vitality in Whatsapp to substance and offer some vital and fascinating things with their partners. Along these lines, here we should need to give some extraordinary Whatsapp assemble interface amassing.\nWhatsapp Groups Link feature is advantageous for the customers. The Whatsapp pack interface help to join Whatsapp accumulate with no director underwriting. Resulting to conveying this component by Whatsapp design, packs of people started chasing down new Whatsapp Group Link on the web. Various people are enormously motivated by joining Whatsapp and wire get-togethers. By and by, we evade the wire social occasions and discuss just Whatsapp gathering. To know something about that by then perused next section.\nNote: - If you are not able to do 18+ then do not join the group below. If you are an adult then there is no restriction to join the group.\nwhatsapp bunch join connect america whatsapp amass join interface american whatsapp 18+ gathering join connect america american whatsapp aggregate connection join list american whatsapp 18+ gathering join connect whatsapp assemble connect join melodies.\nNeed To know Some Adult Whatsapp gather Names Ideas and Best Adult Whatsapp aggregate Dp Then Go and Check Out These Post. On the off chance that You Want To present Your Whatsapp Group To Us, Comment Your Whatsapp Group Link And Also Mention Categories Of Group Link, We Will Try To include Your Group Link In Our Website.\nwhatsapp group link girl, indian whatsapp group link, whatsapp group links 18+ america, uk whatsapp group link, chinese whatsapp group links, whatsapp group link malayalam, russian whatsapp group link, london whatsapp group link.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"*See details and pricing for PeaceHealth Rides Opens a New Window.\nThe City of Eugene and partner agencies Lane Transit District and University of Oregon, have contracted Social Bikes (now JUMP) to deliver bike share equipment and operations for Eugene Bike Share.\nSocial Bikes is a leader in the development of \"smart bike\" technology that allows bike share bikes to be parked at any location within the service district. Social Bikes is currently operating in 18 locations in North America including the BIKETOWN system in Portland, OR.\nThe City is interested in transportation solutions like bike share because it provides a transportation option that utilizes existing infrastructure, helps to reduce the burning of fossil fuels for urban trips, and implements city plans and policies including the Climate Recovery Ordinance and the Transportation Systems Plan.\nBike share users are able to choose a membership plan that best suits their needs including one-time, monthly, and annual memberships. There are also group pass options for large employers.\nPeaceHealth Rides \u2014 Ride With Us! from City of Eugene on Vimeo.\nPro Tip #1 Reserve Your Ride from City of Eugene on Vimeo.\nPro Tip #3 Lock Like a Pro!\nPro Tip #7 Gears: Use Them!\nPro Tip #8 Ding Ding- Peace Wave!\nPro Tip #9 Holster That U-Lock!\nBike share is an innovative transportation program, whereby system subscribers have access to bicycles through self-service kiosk locations throughout the community. The system is accessed through low-cost subscriptions ranging from a few dollars for one-day to annual memberships that in other communities generally cost between $60 and $100 per year.\nWould bike share work in Eugene?\nLikely, yes. The City of Eugene and Lane Transit District developed a bike share feasibility study in 2013. The reported concluded that the City of Eugene was a good candidate for bike share.\nFor more information see the Eugene Bike Share Feasibility Study linked below. These are large files that may require considerable time to download.\nHow much will bike share cost to use?\nA variety of pricing structures will be considered (single-ride, daily, monthly, etc.) as the system gets closer to launch. For an example of fees, see BIKETOWNpdx for pricing structures in Portland, OR.\nMost U.S. systems require a credit or debit card to become a member and check out a bike. This helps make riders more accountable for all ridership costs and prevents theft. Since this issue has presented some barriers for people who don't have access to a credit or debit card, various systems around the U.S. have been implementing pilot programs to make their services more accessible. Social Bikes and regional partners will be evaluating different options to make bike sharing available to everyone.\nHow big will the bike share system be?\nAt build out, the feasibility study recommends a bike share system with 420 bikes, 46 stations, and 756 docks over 5 phases. This system size reflects national best practices related to bike station density, coverage area, and optimal bikes per station to ensure system reliability.\nPhases one and two of the recommended system will launch with 35 stations and 300 bikes in spring 2018.\nAre theft and vandalism major concerns?\nExisting bike sharing systems are built with security mechanisms that help deter theft or vandalism while withstanding year-round weather conditions. To this end, users must use a credit or debit card or their membership key to check out a bike, which creates user accountability. In a recent study, all existing systems reported less than 1% of bicycles vandalized or stolen.\nSocial Bikes utilize proprietary parts, including bolts, that resist theft. The bikes also feature real-time GPS so that bikes can be tracked and found if determined to be missing or stolen.\nWho will operate the bike share system?\nSocial Bikes will operate the bike share system. An advisory board that includes regional partners (City of Eugene, Lane Transit District, University of Oregon) will meet quarterly to discuss operations and performance of the bike share system.\nHow is the system funded?\nOn March 19, 2015 the Oregon Transportation Commission approved the City of Eugene's grant request of $909,066 to create a bike share system in Eugene. An additional $193,000 was transferred from the University of Oregon (Associated Students of the University of Oregon) to the city to ensure provision of bike share stations on the UO campus. Operations are primarily funded through a combination of user fees and sponsorship contracts. There will be a sponsorship announcement in November 2017.\nI like it! When can I expect to ride a bike share bike in Eugene?\nThe system launched on April 19, 2018.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhcxf b/data_all_eng_slimpj/shuffled/split2/finalzzzbhcxf
new file mode 100644
index 0000000000000000000000000000000000000000..ad0043407983c6fd50001483b930251a136cfe59
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhcxf
@@ -0,0 +1,5 @@
+{"text":"Now that Sale-A-Bration has started, remember you can earn FREE SAB items for every $50 you spend online, in person or at a workshop. And you can earn even more FREE items when you host an SAB party. This card uses one of the free SAB stamp sets called See Ya Later and some of the pretty Sweet Sorbet SAB Designer Series Paper. As cute as this card is, it was really quick and easy to put together - the best kind, right?!\nI stamped the birthday sentiment from See Ya Later stamp set in Early Espresso ink onto a 2\" strip of Very Vanilla cardstock. Then I punched the left edge of the cardstock with one of the new Tag Topper Punches from the Occasions Mini Catalog - LOVE! I tied off the tag with a piece of Crisp Cantaloupe Ruffled Trim and Early Espresso Baker's Twine. This pretty, pretty pattern paper is one of the sheets from Sweet Sorbet DSP in Pool Party and Early Espresso. Finally, I added some Dazzling Details glitter to the birthday cake and candles for just a little touch of shimmer.\nThe annual Sale-A-Bration event from Stampin' Up! begins today. This is your chance to earn FREE SAB exclusive items with every $50 order you place and also earn bonus hostess dollars when you host a qualifying workshop. There is no better time to get those items on your wish list and stock up on basics. This Petal Parade stamp set is just one of several stamp sets available for FREE!\nI randomly stamped the images from Petal Parade in Crisp Cantaloup, Daffodil Delight and Pistachio Pudding ink onto two pieces of Whisper White cardstock. These are matted onto a narrow mat of Basic Black cardstock and wrapped with a knot of Summer Starfruit Stitched Satin Ribbon. This stamp set is so versatile and you can use with a wide variety of sentiments or even on blank cards. Here I used the Hello friend sentiment from Peachy Keen, stamped in Island Indigo ink and punched out with the Modern Label Punch. This punch fits the HELLO portion of the sentiment perfectly. I tucked the sentiment under the ribbon knot with a few glue dots.\nLOTS of pictures today because I want you to see all the movement in this Waterfall Card using the new Another Great Year birthday stamp set. This great birthday set is in the Occasions Mini Catalog and features an enormous Happy Birthday banner image and 4 smaller images, birthday candles with wish sentiment, macarons, cupcake and another great year sentiment.\nA Waterfall Card features a series of images (or sentiments) on stacked mats that flip up and reveal the image underneath as you pull the tab (or ribbon). You can a bit of the mechanism in the pictures below.\nThe dimension and instructions for a Waterfall Card are very well written in a tutorial on Splitcoast Stampers website here. However, the elements for the Waterfall Card can also be used with altered dimensions to create a three tier waterfall, a double waterfall or even as an interactive element on a scrapbook page! Once you make one, have fun playing with altered dimensions or stacked waterfalls.\nI had a lot of fun making this Waterfall Card and will be making several more of them this week for an Occasions Catalog Swap. Good thing these that these images are easy to color! You can see that this Another Great Year stamp set made it really easy for me with the four smaller images perfectly coordinated and easily colored. I added some glitter with a swipe of Dazzling Details on each of the pieces. I used the large Birthday banner stamp for the inside of my card.\nThis edition of the Stamp Review Crew features Chalk Talk stamp set, one of my favorite and most used stamp sets! With the coordinating Chalk Talk Framelits, this versatile set will have you creating fast and easy cards, tags and scrapbook pages for so many occasions.\nI stamped this clean and simple card using the thank you sentiment from the Chalk Talk set. This card also features the Circle Thinlits Card Framelit and one of the Adorning Accents Textured Impression Embossing Folders.\nI stamped the sentiment in Real Red ink onto the Watercolor Paper, because I love the off white color and the subtle texture of this paper! This is Heat Embossed with Iridescent Ice Embossing Powder to give it a nice finish and a bit of sparkle. Behind the circle I added a heart punched from the Red Glitter cardstock.\nThis card features the Chalkboard Technique, which is kind of fun to play with. First, I stamped the grateful stamp in Whisper White Craft Ink onto the Basic Black cardstock. This is Heat Embossed with White Embossing Powder. Then I took a slightly damp paper towel, tapped it on the Whisper White Craft Pad and wiped the ink onto the cardstock in a few places. Finally, I used the Chalk Marker and a ruler to draw a border.\nThis is mounted onto a piece of Watercolor Paper. To 'hang' my chalkboard, I punched the piece with a 1\/16\" Circle Punch and threaded a three-strand width of White Bakers Twine through and knotted the ends loosely. I added a Mini Glue Dot under each knot to secure the loose knots. The loop is strung over two Acrylic Buttons, the smallest button fits perfectly inside the largest.\nThanks for stopping by! Be sure to click through to all of the blogs on the Stamp Review Crew: Chalk Talk Edition by using the Previous and Next Buttons below. I also listed the Blog Roll of The Stamp Review Crew: Chalk Talk Edition at the bottom of this post. You'll find tons of inspiration and creativity! Happy Blog Hopping.\nThis card features the Peachy Keen stamp set and On Film Framelits Dies from the Occasions Catalog from Stampin' Up! that debuted last week. The stamp set and framelits are available individually or bundled at a 15% discount. The images coordinate very well, but aren't necessary for each other. I especially love that this Framelits Die set emboss as well as die cut, what a great finishing touch to the images!\nI stamped the You're Peachy Keen sentiment in Crisp Cantaloup ink and then Heat Embossed it with Iridescent Ice Embossing Powder. I have always love Heat Embossing, it is one of the first techniques that really hooked me on Stamping! Heat Embossing with Iridescent Ice gives just a touch of shine without too much bling. Sometimes you want that extra sparkle, but for this card I wanted a little less and Iridescent Ice was just right. I die cut the sentiment piece and a mat of Island Indigo with the Ovals Framelits. For the polaroid die cut piece, which also has embossed lines and details, I added a center of Crisp Cantaloup Designer Series Paper. The little arrows on the outside and inside of the card are also from the Peachy Keen stamp set.\nWhat do you think of this White on White Winter Sparkle card I made this weekend using the cardabilities sketch #90 and lots (and lots!) of Iridescent Ice Embossing Powder?! This was a beautiful sketch to work with. I wanted to use the Festive Flurry stamp set and Festive Flurry Framelits to create a wintery card.\nThe Festive Flurry bundle is a carryover item from the Stampin' Up! Holiday catalog, which means that you can still order the stamp set and the framelits - hurray! That makes me so glad because it's looking like we have lots of winter left around here, so I'll be getting lots of use out of this set!\nI started with a Whisper White base and then cut a card front from Watercolor Paper. I stamped several snowflakes from the Festive Flurry set in Whisper White ink on this paper and then Heat Embossed it with Iridescent Ice Embossing Powder. The next layer is Whisper White cardstock that I stamped the En Francais background stamp in Whisper White Craft Ink. I Heat Embossed this layer with White Embossing Powder. Two of the staggered layers are Whisper White cardstock, stamped with snowflakes in Whisper White Craft Ink and Heat Embossed with Iridescent Ice EP. The square in between those layers is Whisper White cardstock which is texture embossed in the Big Shot with the Polka Dots Textured Impressions Embossing Folder.\nThe snowflake is the second largest snowflake from Festive Flurry stamped in Whisper White Craft Ink onto Whisper White cardstock and Heat Embossed with Iridescent Ice EP. Then I die cut with the matching snowflake Framelit and the smallest snowflake framelit at the same time. Die cutting these at the same time makes it easy to center the smaller snowflake inside the larger one, and it's one less Big Shot step as well. The Magnetic Platform is really helpful when die cutting these stamped images! I finished with a banner sentiment, Thank You, stamped in Silver Encore ink and Heat Embossed in Silver EP.\nI am excited to announce that I joined the design team for the Paper Crafters Sampler, an online magazine. For more information about this magazine, check out the Paper Crafters Sampler Blog.\nAnd it's time to release the 5th year of the Paper Crafters Sampler. The first issue of the year will feature projects and cards for the upcoming Valentine's Day. Inside the January issue, Venetian Valentine, you'll find soft and subdued colors, fancy flourishes, embossed surfaces and splashes of gold. Tres le femme! The template of the month is the Fliptop Heart Trinket Box.\nHere is a sneak peek of my project for this month, a Venetian themed Valentine featuring a beautiful, feminine Embossing Technique with soft colors and lots of Gold and Gold Embossing! This is one of my favorite cards I've made this year! Be sure to go to Paper Crafters Sampler to get your January issue. Get there by midnight tonight for the Early Bird Special!\nOver the weekend I shared several projects using the Pomp and Circumstance stamp set with the Stampin' Addicts Occasions Catalog Blog Hop. One of the projects was this Wallet Gift Card, with a pocket on the inside to tuck a gift of cash. This card is designed to accommodate checks or cash as well as fit into a legal sized envelope. I wanted to share a few more images of this card as well as the cutting and scoring details - you'll love how quick and easy a Wallet Gift Card comes together!\nAs I said, the Wallet Gift Card is designed to hold cash or checks and it fits just perfectly into a legal sized envelope. The one thing I don't like about A2 cards is when I want to include cash or a check and have to fold them to fit - Awkward!\nI used the cap\/tassel and diploma (masked off the sentime) to randomly stamp on the Brushed Silver pieces in VersaMark ink. Then I Heat Embossed these images with Clear Embossing Powder for a subtle background. The front sentiment and pocket congratulations sentiment are both stamped in Basic Black Craft Ink and Heat Embossed with Black Embossing Powder. Remember when you are embossing with a high contrast powder, such as black on silver, you will want to use an Embossing Buddy on the cardstock before you stamp. This will make a big difference toward keeping your embossed image nice and crisp without stray embossing grains.You can also use a small clean paintbrush to swipe away any remaining pieces of Embossing Powder BEFORE you heat it.\nI created a simple belly band by layering a Tasteful Trim XL Die cut of Basic Black cardstock onto a strip of Brushed Silver cardstock. I pinched this to loosely wrap around the Wallet Gift Card and then adhered the ends. The seam is hidden below the cap\/tassel punched piece. I added one of the tiny Silver Mini Brads to the cap where the tassel is attached.\nAll the rounded corners were done with the back punch on the Envelope Punch Board. This allowed me to punch through the two layers of cardstock along the bottom of the pocket portion of the card with no problems at all.\nThis edition of the Stamp Review Crew features Hearts A Flutter stamp set and if you think this set is limited to a few layouts and designs, you are about to find out how versatile this fun little set of banners can be! Continue around this Hearts A Flutter Blog Hop for a ton of inspiration and a wide variety of creative and beautiful cards and projects!\nMy first two cards with the Hearts A Flutter stamp set are Valentine's cards. I made two sets of these cards, identical with the exception of the color tones - one traditional Real Red and Whisper White with Gold and the other non-traditional pastels Crisp Cantaloupe and Very Vanilla with Gold.\nThe coordinating Hearts A Flutter Framelits are definitely my friend when using this stamp set. I also loved playing with the Gold Embossed Vellum which is from the Occasions Mini Catalog that debuted this month.\nHere is a close up of the Heart elements on the fronts of these cards. I stamped the largest scalloped heart image in the coordinating ink color onto the cardstock. This is die cut with the Hearts A Flutter Framelit and sponged pretty heavily to emphasize the scallops. I paper pierced the the dotted border of the image and then added white dots to the inner dotted border with the Chalk White Marker (carryover item - yaya!) The xoxo sentiment in Gold Foil is one of the accessory framelits in the Circle Thinlits Card Framelits - LOVE!\nCheck out the inside of these cards where I stamped Be My Valentine, one of the many sentiments from Delightful Dozen, in Gold Encore and Heat Embossed it in Gold Embossing Powder. Here I played with the banner pieces and strung a layered banner filled with pennants and hearts below the sentiment. Which color combination do you prefer? This year I am finding myself leaning more toward the non-traditional Valentine's with Soft Blues, Pistachio Greens and Corals.\nThe last card I want to share with you is a winter themed Thank You card using the Hearts A Flutter set, using only the banner images. We are sending so many thank you's this time of year at my house that I can't keep enough of them in my stash!\nI stamped the banner images in Night of Navy ink onto Whisper White and die cut them with the coordinating Hearts A Flutter Framelits. The Thanks Big Time sentiment from Timeless Talk is also stamped in Night of Navy and punched out with a 1\" Circle Punch. This is popped up on one of the pennants. I ran the Night of Navy cardstock piece through the Big Shot with the Northern Flurry TIEF and then added a few Rhinestone Jewels to the snowflake centers. Pretty little bows of the Silver Trim (a carryover item so still available to order - yaya!) finish the ends of this banner.\nWhether you are hopping through or just starting at my blog, be sure to visit the other blogs on the Stamp Review Crew: Hearts A Flutter edition by clicking on the Previous and Next buttons below. I hope you're inspired and would love to see your creations - please share!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dozens of Chinese environmental groups and close to 100 concerned experts have joined forces to publish a dramatic open letter urging the government to release documents related to secretive plans to dam the Nu River in southwest China.\nThe forceful statement, signed by 61 organizations and 99 individuals, argues that the environmental impact assessment that apparently has been completed for the controversial Nu River project must be made available for public scrutiny, in accordance with Chinese law.\n\"We sincerely call on the decision-making authorities to disclose the EIA report of the Nu River dam plans before making a decision, because the right to be informed is a prerequisite for public participation,\" they write, and cite Chinese law that requires public input into project planning.\n\"Dam building in an area of high mountains, deep valleys and extremely poor farmland threatens the livelihoods and future of tens of thousands of people,\" the authors of the letter warn.\nBut amid the secrecy surrounding the Nu River plans, \"there is no way for the public to learn how the developers and local government will avoid environmental damage, how they plan to carry out the resettlement of 50,000 people, and how they will assure the safety and economic feasibility of the dams.\"\nNoting that 92 per cent of the local population belongs to minority groups, the authors observe that, \"once resettlement starts, the loss of cultural diversity is inevitable.\"\nAs well as arguing vigorously for the rights of citizens to be informed about \u00d0 and to influence \u00d0 projects that will have a profound impact on their lives, the petitioners are scathing in their criticism of current practice in China's big-dam industry.\n\"We should no longer tolerate the low-cost or even free exploitation of public resources and the earning of huge profits at the expense of our environment,\" they write.\n\"Dam-builders should not externalize the huge costs of dam construction on affected people, the public, the nation's finances and future generations.\"\nNo coverage of this important appeal, which was issued in Chinese on Aug. 25, could be found in an on-line search of the Chinese press.\nPatricia Adams, executive director of Probe International and publisher of Three Gorges Probe, said, \"The Chinese NGOs and concerned citizens are absolutely correct to demand that the Nu River EIA be released to the public for review and input.\n\"Secret EIAs are paper exercises intended to lull the public into thinking that project proponents have considered the environmental consequences of their actions thoroughly and accurately. \"\nBut secret EIAs are tantamount to putting the fox in charge of the chicken coop. They have no credibility, but are seen as ham-fisted attempts to deceive the public. They would not be tolerated in the West,\" she said.\n\"The only way to get an accurate evaluation of the costs and benefits of a dam project is to subject its environmental assessment to public scrutiny and input. Then a better picture of the true consequences of the project will begin to emerge.\"\nControversy has swirled around the proposal to dam the Nu River since it was first revealed two years ago that the China Huadian Group and local power firms had been granted permission to build a cascade of 13 hydroelectric dams on the river.\nAs it wends its way through Yunnan province, the Nu forms part of the Three Parallel Rivers National Park, where three mighty rivers \u00d0 the Yangtze, Lancang (Mekong) and Nu (Salween) \u00d0 flow through steep parallel gorges.\nIt is a region of such rich biodiversity and \"outstanding universal value\" that UNESCO declared it a world heritage site in 2003.\nAt its recent annual meeting, held in South Africa in July, the UNESCO World Heritage Committee issued a resolution expressing \"grave concern\" about the development plans for the Nu River and stating that \"any dam construction within the World Heritage property would provide a case for inclusion of the property in the List of World Heritage in Danger.\"\nThe Nu is one of only two major rivers in China that have not yet been dammed. (The other undisturbed river is the Yaluzangbu in Tibet.) Chinese scientists want the two rivers left alone so that future studies can compare conditions in dammed and free-flowing rivers.\nMoreover, the Nu is an international river, and upstream development could cause problems for downstream countries Burma and Thailand.\nNews of the proposed development of the Nu River's hydropower potential sent shockwaves through China's scientific and environmental communities when the plans became known in August 2003.\nAfter heated debate about the project's likely environmental and social impacts, Premier Wen Jiabao ordered a halt to the scheme in April 2004, sending it back for more \"scientific research.\"\nRecent local media reports have suggested that a scaled-back version of the project, perhaps involving the construction of four dams, is likely to be approved soon \u00d0 despite the complete lack of public input into the decision-making process.\nIn their open letter, the experts and activists write: \"We have been informed that the central government's planning and environmental departments have reviewed the hydropower development plans for the Nu River.\"\nHowever, they point out, public participation in large infrastructure projects is now required by law in China.\nThe Environmental Impact Assessment Law, which came into force on Sept. 1, 2003, stipulates that public tribunals must be held to take account of the views of people who stand to be directly affected by such projects.\nBut, the authors of the letter complain, the public continues to be excluded from decision-making related to the Nu River scheme, all of which is going on behind closed doors.\nIn fact, no big project in China has yet been forced to consult with affected communities. The signatories of the open letter would like to see the Nu River scheme become a test case, and the first of many such projects to be subjected to public scrutiny.\nHydropower development on the Nu River \"is not an individual case,\" they write. \"We hope the process can help develop a set of science-based and democratic decision-making mechanisms, in order to cope with the overheated and unregulated hydropower development boom in China.\"\nIn the absence of proper procedures, they write, \"such an abnormally fast pace of development may result in resource exploitation and environmental degradation. Then the beauty of China's natural rivers will be lost, and the affected people in southwest China's mountainous areas may fall into dire poverty.\"\nGroups that have signed the open letter include prominent non-government organizations Friends of Nature, Global Village Beijing, the Global Environmental Institute, Green Island, and many others.\nMa Jun, an environmental consultant and author of the acclaimed book, China's Water Crisis.\nAnother signatory, Wang Yongchen, a radio journalist and co-founder of the environmental group Green Earth Volunteers, was instrumental in the campaign to stop an earlier unpopular dam project.\nThe Yangliuhu dam was to have been built just upstream of Dujiangyan, a 2,200-year-old irrigation scheme that is thought to be the world's oldest such system still in operation. Ms. Wang helped spearhead opposition to Yangliuhu. The spirited public debate that sprang up around that scheme grew into a nationwide media campaign that ultimately led to cancellation of the project.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Applause is looking for a Software Engineer to join our team. Here in Framingham we create and maintain the heart of our business \u2013 a platform that enables our hundreds of thousands of in-the-wild testers to provide quick and meaningful feedback to the world's largest app-creating companies. We still work like a startup in small, fast-paced, energetic groups but with all the perks of a larger organization such as strong benefits and a healthy work-life balance.\nAre driven and passionate in creating premier web experiences through well designed , tested and maintainable code-bases.\nLove crafting beautiful RESTful web-services and have a strong knowledge in the way the web works.\nAre always looking to learn and improve \u2013 independent self-learners who love to share what they find.\nAre team-focused , pragmatic engineers who enjoy working on small, agile teams.\nDesign and develop the next generation Applause platform.\nWork in strong agile teams, using modern development methodologies.\nWork closely with product managers, designers and customers to deliver impactful product initiatives.\nWe work with thousands of companies, from the hottest startups to the world's biggest brands \u2013 including Google, Microsoft, Amazon, eBay, WatchDox, Glide,Kaltura and many more. We offer an innovative, fun, cutting edge culture and, right now, we have dozens of job openings. Maybe one of them is the right one for you! Since our 2008 launch, we've appreciated tremendous success for a start-up, having raised $115 million in funding. Moreover, we've been widely featured in tier one media such as Fortune, Forbes, BusinessWeek and TechCrunch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We apply the skills of web designing and maintenance to the website we produce.\nGeomatrix Group is committed to helping you realize and achieve your various web designing goals.\nby countless Web Service Offers??\nLets keep it SIMPLE !\nWE DEFINE. WE DESIGN. WE DEVELOP. WE DEPLOY.\nGeomatrix Group not only believes in creating a highly designed website but also a user-friendly design.\nGeomatrix Group is a 360 degree IT Solution provider that believes in maintaining cost-effective, high-quality services.\nGeomatrix Group have its own highly effective SEO (Search Engine Optimization ) and other valuable online marketing tools.\nGeomatrix Group understands the value of client's time and investments and in believes in providing ultimate customer satisfaction.\nGeomatrix Group maintains the talent and resources to meet the business aims and growing needs of every client. Geomatrix Group has an Offshore Development Center in India with a motive to provide cost-effectiveness to its business partners and clients around the world.\nGeomatrix Group is a leading web development company. It believes in providing effective, 360-degree IT Service and successfully maintaining high-quality Web Design and Development Services.\nGeomatrix Group is among the Leading Web Development Company believes in providing effective 360 degree IT Service , which is successfully maintaining high quality Web Design and Development Services.\nOur focus lies in providing effective, economic and Client satisfactory IT solutions and services.\nIn order to be a successful, dynamic organization our interest and focus is to generate customer trust and a sense of teamwork. Our goals as a company include the upholding of responsibilities, promoting of innovation, maintaining of ethical behaviour, assurance of employee safety, and the cultivation of healthy relationships with our clients.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"OUR MISSION is to bring together manufacturers of highly-engineered electro-mechanical and passive electronic components with OEM's and\/or electronic distributors.\nOUR EXPERIENCE is more than 100 cumulative years with electromechanical and passive electronic components and a history of strong employee and principal loyalty.\nOUR FOCUS is design-in selling directly to OEMs and\/or through the distributor channel.\nOUR GOAL is to add value to the sales process for both our principals and customers. We provide our principals with rapid exposure to customers' applications and needs. We save the customer time by presenting multiple lines of compatible products combined with broad product and application knowledge on each sales call.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhfbz b/data_all_eng_slimpj/shuffled/split2/finalzzzbhfbz
new file mode 100644
index 0000000000000000000000000000000000000000..6a239de0771dda61f1b8741edbd1a613ba1aade5
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhfbz
@@ -0,0 +1,5 @@
+{"text":"We spent more than six months testing, advancing and refining the designs: shapes, cuts and seams, to create the perfect cycle wear.\nWith the Echelon performance collection we succeeded in combining high performance with Isadore's minimalist signature.\nThe most minimalist, technical and unconventional collection of Isadore.\nEchelon is a diagonal formation of racers which allows cooperative drafting in crosswinds.\nPlo\u010de \u2014 a Croatian port-town, situated on the Adriatic coast in Dalmatia, north of the Neretva Delta.\nPlo\u010de doesn't stand out from other Croatian places. This probably wouldn't be among your first choices to ride in Croatia. And yet, it caught our attention by a strong contrast: clean beaches and simple church standing alongside brutal socialist architecture.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On sale will be arts and crafts including jewellery, crochet, books, photography, candles, wooden items, glassware, fabric goods, handmade teddies, lights, handmade confectionery, cakes, felt work, children's clothes and garden furniture.\nIt will take place from 11.00 am until 4.00 pm. This year, the fair is supporting the Carrick branch of Alzheimer's Society.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is there a better alternative to Mirror for Roku?\nIs Mirror for Roku really the best app in Entertainment category?\nWill Mirror for Roku work good on macOS 10.13.4?\nMirror for Roku needs a rating. Be the first to rate this app and get the discussion started!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The ZombiePopcorn Carnival started !! From June 15th to 30th.\nMy exclusives are new sculpted high heels sandals (or clogs), in Black or White Leather and different studs colors.\nSkin coloring HUD for the feet part and resizer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The AP-204 combines traditional styling with the high-performance feel you expect from an ESP bass, all at a price you can afford. A bolt-on design at classic 34\" scale that's great for any kind of music, the AP-204 features a mahogany body, a comfortable thin U-shaped maple neck, and roasted jatoba fingerboard with 21 extra-jumbo frets. A sonically versatile bass, the AP-204 includes a set of ESP Designed LDP and LDJ pickups with active 2-band EQ, making it great for any genre of music from metal to funk to pop and more. Available in Snow White, Black Satin, and Natural Satin finishes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhldx b/data_all_eng_slimpj/shuffled/split2/finalzzzbhldx
new file mode 100644
index 0000000000000000000000000000000000000000..11723cd3eb987f48925cbdc973014e2c613d6055
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhldx
@@ -0,0 +1,5 @@
+{"text":"Moberi started out as a bicycle-powered juice and smoothie cart. Connecting his bicycle to a blender, a homemade sink, and a cooler full of organic fruits and veggies, young entrepreneur Ryan Carpenter pedaled smoothies outside of his house. Carpenter pitched his innovative beverage business on Shark Tank in February 2014. He went into the Tank seeking an investment of $50,000 in exchange for 15 percent equity. He got Mark Cuban and Daymond John on two bikes to blend their smoothies. All the Sharks agreed that the final product was delicious but Kevin \"Mr. Wonderful\" O'Leary ask Carpenter if he was nuts. And while Robert Herjavec told Carpenter, \"this is so crazy\u2026 that it might actually be brilliant,\" alas, Carpenter did not leave with a deal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This duo founded CYST, a software innovation company that specializes in artificial intelligence to create technology solutions in 2013.\nCYST has a research arm called CYST Research Institute, which focuses on artificial intelligence-based research and development such as natural language processing.\nIn partnership with the telecommunications companies, CYST has access to over 15 million subscribers through its platforms. They count MTN Ghana, Vodafone Ghana, AirtelTigo Ghana, Unity Link and Data Protection Commission among their affiliates. It means over 15 million subscribers through its platforms.\nCo-Founder of MapTech Logistics. The problem he discovered existed in the logistics and supply chain management was around inaccurate customer location data. Faced with this challenge, with his co-founder, a geomatic engineer, they deployed geographic information systems (GIS) and location intelligence solutions to solve the problem.\nMapTech Logistics, a technology company, focusing on developing GIS-based applications to solve business problems, such as logistics and supply chain management, was born. The company also provides mapping services for geospatial analysis. Maptech has been named in this year's 100 globally competitive start-ups in Ghana by The Startup Network and Boadu is a World Economic Forum Global Shaper. Boadu employs six people.\nIsaac is an Entrepreneur, Front-end Developer, Embedded Systems Engineer, passionate about solving some of the world's most pressing problems using Science and Technology and inspiring young people to be innovative.\nIn 2015, he founded his first company, Invent Electronics while in my second year in University. Invent Electronics is an electronics components distribution startup based in Ghana with customers in Ghana and other neighboring West African countries.\nHe later co-founded, Wires & Bytes with 3 other friends. Wires & Bytes is a mobile SaaS development company based in Ghana. Its flagship product, Pasco, allows students to easily access past test questions from their smartphones and currently has over 1000 users in KNUST.\nIsaac is committed to helping young people build their own business and become more productive by sharing several practical insights and resources in entrepreneurship, personal development and productivity.\nHis passion for social entrepreneurship led him to be actively involved in STEM initiatives through his social entreprise, Nsesa Foudnation. He was featured as one of Airtel's STEM champions, also named as one of the top 100 student entrepreneurs in Ghana by The Startup Network, organizers of The Ghana Startup Awards.\nGhanaian entrepreneur Raindolf Owusu, is applying his own troubleshooting mojo to help his continent's frustrated Web users. Owusu created the first browser designed specifically to retain users through periods of slow connection speeds: Anansi, named for a spider in African folklore. It entertains users during sluggish connectivity with a built-in game and webcam.\nAnansi was the first product developed by Owusu's five-year-old startup, Oasis Websoft. From his headquarters in Accra, he has now\u2014mimicking the Google founders' trajectory\u2014shifted focus to other promising, if unproven new businesses. Among them are two medical apps: Dr. Diabetes, an information clearing house about the condition; and Bisa, designed for patient-to-doctor messaging.\nBoth would seem to have wide applicability throughout the continent, and that's exactly Owusu's aim for his browser, his apps\u2014and beyond.\nOwusu aims to develop more disruptive technologies like 3D printers and drones that will improve lives in Africa.\nHe founded Tech Era, a non-governmental organization that runs an IT literacy program and exposes persons with disability and less privileged persons to technology in order to give them hope and potential to transform their community and not only themselves. Tech Era has expanded its impact, training over 80 children, and also extending its reach to the disabled.\nIn 2018, Derick was awarded the Queen's Young Leaders Award that recognizes and celebrates exceptional commonwealth youth.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dieting is tough to do it terms like \"easy pounds reduction program\" and \"easy fat diets\" might seem like oxymorons when essentially they are not. While it is true that easy is a family member term a basic weight loss program can still be found when compared against a standard slog-it-out low-calorie diet agenda.\nThe Weight-loss 4 Idiots make regarding this secret, and really help dieters burn off the fat away and maintain it off for suitable. Like mentioned above, this nutrisystem weightloss is the most affordable on the internet. With as low as 40$, it might appear like some sort of incomplete program, when it is not even. In fact, it is truly complete, so useful created easy utilize that these items probably be shocked that you spent 200$ or a little more about products that did not even work. Plus this 40$ is a one-time only payment.\nDo do not be deceived coming from the idea than a rapid weight-loss diet will be going to a permanent solution for you if being over weight has been a chronic problem. Realistically, a modification in lifestyle is required. Some useful questions on fundamental issues in how expensive is nutrisystem. Even though rapid weight loss might possibly have unfavorable affects on your physical condition, so can obesity. You want to get shot of those extra pounds but take extra care to do it in an economical and healthful manor. Dehydration is a common problem much more positive make newer and interesting . reduction with your calorie in your diet. Be certain to get plenty of fluids.\nThe very first thing I suggest doing is contacting a doctor if there is any doubt that you will be able to commit to such an insurance plan. Being able to keep to any program is truly one of the first things you'll need in order to win. That's why you must make it realistic, to make sure you won't become demotivated or impatient and cease.\n2] So, by losing metabolism happen to be gaining increasingly more more fat. Now somehow we should have ways to build up metabolism to get rid of of body fat fast. Use this done naturally by feeding good and nutritional food to your at regular intervals. Very first thing to be avoided in cultivating metabolism is avoiding faded diets. Do not take food that contains low calorie, low carbohydrates, low nutrients and vitamins and so on. By taking faded foods you might be slowing your metabolism in which not necessary for burning body fat.\nVital organs, including the liver and intestines, can be smaller; coronary heart also becomes smaller and fewer effective, and blood pressure is dropped. We don't want to reduce off the nose to spite the facial skin. A starvation program may well compound our health and wellness issues. A good program must appreciate consuming appropriate food in both quantity and quality.\nFat Loss Factory Product is a 12-week long program and mainly because was mentioned, it contains power training and nutritional guidelines. The 1st step for eating habits system is detoxification. Which means that you really should get rid of those waste inside your body by utilizing the highly specialized program of the diet plan. After this you will have to focus your attention and appetite to proteins and fiber-rich cabohydrate supply.\nThe ideal is to do some sort of a cleanse on regularly.say once a season or quarter. You are building toxins regularly. The key to not reaching a critical overload might be to make sure you re-balance yourself, inside and out, on the usual basis, simply too.\nAre you looking for rapid tactics to lose flat abs? If so, you are not the only just one. Quite a few people today check out cut down fat and whenever they prevent seeking locate it really is pretty tough get connected to that previous little bit of stomach system fat. It is one for this most ended up unsatisfied with sections of bodies. I am aware it might be extremely stubborn and may be demanding get associated with. This really is far more true earlier you might have.\nAlways, bear in mind fondly the fundamental rules of suited life style first. People primary ideas are appropriate nourishment (protein, fiber, sophisticated carbohydrates, balanced fat, vitamins\/minerals), drinking a good deal water daily (1\/2-1 gallon), having a lot of rest each evening (7-8 hrs), cardio physical exercise and entire body setting up physical exercise workout.\nYou really should up your protein consumption as it is crucial for bodybuilding as well as helpful for reducing complete human body fats. You have to start out ingesting 1.five x your bodyweight in grams of nutritious proteins. This should originate from organic foods like hen and beef in aid of if your seriously acquiring problems getting the quantities must ever start off employing dietary supplements like whey protein powders.\nYour food plan wants caring not with all round range of energy even so the varieties of energy. You've got to process no less than one gram of protein for every working day for every pound you weigh. Moreover, you'll need some extra fat with your daily diet program. A food plan of 30:20:fifty in the ratio protein: fat: carbs will make a lot of sense for an chubby woman who needs the swiftest method to quickly ways to lose tummy fat. Reflections on smart nutrisystem groupon. Fish oils wind up getting ideal involving fat in this intent.\nThis really is merely not the function. It cannot be because you don't burn excess fat from a certain human body part by performing focused exercise routines irrespective of whether that health machine or with out them. No conditioning machine can accomplish. It really is unachievable. Location body fat burning by targeted exercise routines is de facto a fantasy.\n3) Consume 5-6 little meals daily. You have got read this prior to mainly that helps shrink your tummy fat more quickly than feeding on two or three moments every single day. For those who come across this extremely inconvenient then commence making ready meals when ever you allow to high school or intent. Consuming just about every three hours will guarantee you complete and energized with no sensation bloated.\nWhat is actually a positive method of establish in the event you have far too substantially tummy unwanted fat? You are able to notify for anyone who is carrying all around as well significantly tummy fat by measuring your midsection having a tender tape measure. In case you are a man, your waist measurement should be no much more 40 inside. For ladies, the utmost amount is 35 inches. Any bigger someone also significantly optimize your possibility for diabetic issues, heart condition, as well as complications.\nHowever, you see, soon after i put on clothes and stuff I generally get unfortunate due to the fact I glimpse energy. I've pretty lower self-confidence. But I however to help lose about 10 lbs .. I really feel excess fat inside a bathing go together with. But I recommend you easily physical exercise bike because of any difficulty. Although it's a lot less affect.\nEach well being mindful particular person is constantly browsing at a great path to get rid of flabby ab muscles. If somebody among persons individuals, you'll be able to undoubtedly need to have begin analyzing many of the choices you are going to use burn off off tummy body fat efficiently. The individual human body is thought for its all-natural tendency to shop excess fat on stomach locality. The body's upcoming most loved location for storing unwanted fat may be the arms and thighs. Nevertheless persons will concur how quite possibly the most hard place to get fat from would be the abdomen.\nThough, the recommendation will be to take in typically, surely choose your foodstuff wisely. Opt for meals which is able to assist you in intention to burn off those people fats. Examples of such are veggies, in addition to meat. Processed carbs will surely trigger your tummy to acquire large. Thanks to this , if you can expect to stay away from them, you might get rid for this fats speedily. Look at out quite a few suggestions on fats burning meals on ideal food items to consume to extra extra fat.\nTea is filled with catechins, which can be substances which may lessen your cravings for unhealthy easy carbohydrates and junk foodstuff and vitamins. Tea may raise metabolic rate and sizzling teas are especially effective at helping you might be converting energy for warmth and fuel. What's necessary for prudent solutions of nutrisystem groupon. Ingesting two to 4 cups of tea an occasion is usually great for it will harm and for system body fat burning.\nTry to eat A lot less CARBS: carbs are offering us electricity, the secret will be to take in carbs only the amount we'd like, due to the fact your individual will modify more carbs in surplus unwanted fat.\nThe main offender that keeps envision that obese and not able to ideal technique to burn up tummy extra fat for guys is sugar. Sugar is in just about every thing \u2013 specifically processed snacks are a great. Feeding on loads of basic carbs also result in us get fat. Carbs which include bread, rice, cereal, baked products, and candy become excess fat when eaten excessively. Just reducing back on they're the or eradicating them on the exact time is likely to make losing belly flab and fats in general a lot easier.\nMany folks have identified theyve not sure the amount of energy they're taking in daily. Monitoring your ingestion of food stuff will provide you with far more stated to. Write down the food items you take in, along with the serving weight. In the event you possess the diet info, history it too. If not, there are several on-line databases which make the data you must figure out may be energy you may serve each and every do the job working day. Getting this information enables someone to generate much easier options let's explore planning meals items.\nSorting fantasy from truth about six packs stomach muscles can an individual design your strategy to be able to possess a more healthy extra successful products. Training your head to abide by the reality is like coaching your framework. You will need to have discipline on two concentrations, h2o plus the intellect, to recieve 6 pack abdominal muscles instant.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"No changes were made to open Swing Positions throughout this Week although the markets did present some new set-ups which I chose to ignore due to already being heavily positioned, some of which that were not performing particularly well at the time.\n2nd half of GBP:NZD swing long showing + 40 points.\nUSD:CHF swing showing -4 points.\nCAD:JPY potential swing remains on my watchlist, but so far not exhibiting all the things I'd like to see before entry.\nGBP:JPY is the most interesting chart at the moment for me, not necessarily for a trade, just because I see a mixture of bullish and bearish signals and would like to see what happens.\nEUR:GBP could be a potential swing short, further analysis to follow over the Weekend.\nAnother chart to watch is GBP:CHF which is approaching the Daily 200 MA for the first time in over a Year, meanwhile, it didn't reach the technical upside target beforehand, suggesting there could be another leg up to come in the region of 600+ points.\nIn the meantime, I hope you are enjoying your Weekend.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here is one of the most helpful container gardening tips. A tip that helps your plants develop a strong and healthy root growth and helps with the problem of drying-out if you're away from your garden for a short while \u2013 without a friendly neighbour doing you a favour and coming round to water your plants.\nSuccessful watering is a key issue for container gardening tips and many people have suggested different ideas to overcome the problem. Some use watering systems controlled by a timer, some use plastic sheeting to line the container to keep the soil moist. Others opt for using loose water storing granules mixed generously within the compost. Now there is another way for you to consider, one of the really effective container gardening tips you're likely to come across.\nThere is a way to create an oasis inside your container just where you want it at root level to encourage root growth. This will not only produce some beautiful blooms in all kinds of container, and it will keep your plants happy for a few days and help prevent them drying out. Of course, if you place your containers in the shade and give them a good watering before you leave the house will also help. Sprinkling a mulch of small pebbles on top of the soil when you're away will reduce evaporation and is also one of the simplest container gardening tips.\nA new idea has been developed that traps water absorbing granules within two layers of capillary matting. The capillary matting allows roots to penetrate and absorb water form the swollen water storing granules. This effectively creates a water source at root level. Unlike other water storing granules on the market this type breaks down naturally within the soil at the end of season. They also release a trace of potassium into the soil that has an enriching effect. The matting that form the capillary shields are produced from polymer fibres that is non-toxic and do not contain sodium salts. During the season if you use a liquid feed when you water, the nutrition within the feed benefits that plant by being absorbed by the granules, instead of pouring right-through the container. When it comes to effective container gardening tips this is one that you will see the evidence of wonderful blooms with your own eyes and witness some very happy plants.\nSpanish Rings are spanish style flower pot holders that create a wonderful Spanish style mediterranean patio garden. Spanish Rings are simple yet stunning. They are easy and quick to fit on to walls, trelllis and downpipes or posts. They brighten up a patio garden, fence, house wall or back yard. They make an attactive garden accent or garden ornaments.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbitss b/data_all_eng_slimpj/shuffled/split2/finalzzzbitss
new file mode 100644
index 0000000000000000000000000000000000000000..952ead3a444dafcd3b69e0069d4e8406f02574a1
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbitss
@@ -0,0 +1,5 @@
+{"text":"Newegg offers 50% Off select H&R Block software (download) with code \"43DPC6\"\nBuggy as f*** on Mac. Every single year I hit a bug of some kind.\nAmazon has QuickBooks Desktop Pro 2019 with Enhanced Payroll on sale. Shipping is free.\nAmazon has H&R Block Tax Software: Deluxe + State 2018 + 5% Amazon Refund Bonus Offer (PC\/Mac Disc or Digital Download) for $18. Shipping is free.\ndon't buy this. they still haven't gave me the Amazon bonus from 2017.\nNY State return is way off. Someone please fix it!!\n...and you think by posting this complaint on a deal site, the HR Block creators would see and respond to your message?!\n\/\/@Anonymous: NY State return is way off. Someone please fix it!!\nNewegg has Extra 50% Off H&R Block 2018 Tax Software with code \"EMCTUUT34\". Shipping is free.\nIt'll be cheaper yet in a few weeks.\nI got soured on HR Block last year when a buggy state program update caused me to have to refile\/amend my state return. Definitely cheaper than TT, but I may be going back to it anyhow. It was more than a $20 headache last year.\nAgreed. Would pay more to avoid the pain.\nTurbo took me 15mins to complete federal and state tax return. HR block took me more than half day and I had to wait for the states to be updated. \/\/@anonymous: I got soured on HR Block last year when a buggy state program update caused me to have to refile\/amend my state return. Definitely cheaper than TT, but I may be going back to it anyhow. It was more than a $20 headache last year.\nWhat's the best option to file three states income?\nWalmart has Select H&R Block 2018 Deluxe Tax Software on Sale. Email delivery.\nYes HR block is very bad. They sell customer info as and no audit support. Also, they ask you hefty fee for state file. It is just a bogus scam!!\nH&R block always calculates my state tax wrong. I switched to Turbo tax 2 years ago and haven't had a problem since.\nAre you working for TurboTax?\nI run two side by side for several years, no real difference \/\/@Anonymous: H&R block always calculates my state tax wrong. I switched to Turbo tax 2 years ago and haven't had a problem since.\nAs far as I tried in the past couple of years, they gave same results. Give away critical information without legal obligation or charge? Come on, common sense please! \/\/@Anonymous: Yes HR block is very bad. They sell customer info as and no audit support. Also, they ask you hefty fee for state file. It is just a bogus scam!!\nI have filed with turbo tax for many years and have been using hrblock. It is cheaper to file tax return with no difference in return.\nTarget has TurboTax 2018 on sale. Shipping is free with $35 purchase or free in-store pickup.\nAmazon has Norton Security Deluxe \u2013 5 Devices \u2013 1 Year Subscription - Instant Download - 2019 Ready for $14.99. Shipping is free.\ncan use your digital credits from your Amazon account (no-rush credits) on this order to bring down the price even lower.\nAmazon has Microsoft Office 365 (PC\/Mac Download) on sale.\nWalmart has H&R Block 2018 Deluxe Fed+State Win Download for $18.\nDoes this include the 5% refund bonus?\nDoes it say that ??\nShows $31.92 + tax for me. Has this expired or does the price change with a coupon?\nexpired. didn't get a chance to check out. terrible walmart. \/\/@Anonymous: Shows $31.92 + tax for me. Has this expired or does the price change with a coupon?\nShows $31.92 + tax for me too. Is the deal expired?\nWalmart follows Staples' path... \/\/@Anonymous: My order was cancelled!\nNewegg has forAdobe Photoshop Elements & Premiere Elements 2018 + 2 16GB Flash Drives $79.99. Shipping is free.\nBest Buy \/ Bestbuy via ebay has Quicken Deluxe 2018 2-year subscription for $39.99 (50% off). Shipping is free.\nShows $80. Unless you are talking about $40 a year?\neh~ only good for 1 year.\nAfter one year what will happen \/\/@anonymous: eh~ only good for 1 year.\nNo thanks for 1 year. I'll stick with Office 2016 good forever.\nAmazon has Adobe Photoshop Elements 2018 - No Subscription Required (Disk or download) for $59.99. Shipping is free.\nGoogle has Runtastic Heart Rate Pro Android or iOS for FREE.\nBest Buy Deal of the Day has H&R Block Tax Software Deluxe 2017 Federal with State - Mac\/Windows for $22.50 + $15 Best Buy Gift Card. Free in-store pickup.\nDoes this software efile state also??\nNo. State rifle costs an extra $20. \/\/@anonymous: Does this software efile state also??\nTypo. I meant efile. \/\/@anonymous: No. State rifle costs an extra $20. \/\/@anonymous: Does this software efile state also??\nFederal yes. State no. \/\/@anonymous: NY can e-file free?\nTax free H&R offers at Newegg (for most states). Newsletter sub. may be required. Easy.\nI believe NY state doesn't allow additional charge for e-filing \/\/@Anonymous: Federal yes. State no. \/\/@anonymous: NY can e-file free?\nHR Block2017 failed to install on Win8 and was stuck at the beginning of installation.\nOffice Depot and OfficeMax has Free $15 Office Depot Gift Card with purchase of Deluxe and higher with code \"60029804\". Shipping is free on $25+ or choose free store pickup.\nCredit Karma Tax doesn't support the import of sales information from Form 1099-B at this time. You can enter these sales on the Stocks, Mutual Funds and Bonds screen located in the \"Had Investments\" section. Initially, you can enter up to 10 rows of sale information. If you received more than 10 1099-B forms or a have brokerage statement that shows more than 10 sales, click \"Add Row\" located towards the bottom of the screen to enter additional sales.\nMicrosoft has select XBox One Games on sale . Shipping is free.\nH&R Block has H&R Block 2017 Software Download from $9.95.\nPrice is showing discount now.\nLast year i tried this crap. This let prepare state but ask fee to file state? Hidden steal and traps.\nThey always charge expensive state filing. I usually just print state filing and mail first class ($2). \/\/@Anonymous: Last year i tried this crap. This let prepare state but ask fee to file state? Hidden steal and traps.\nI think $14.95 is the lowest it can go for deluxe+state.\nIs it possible to import last year tax information from TurboTax into H&R Block?\nFor people needing the premium package, Amex offers $10 back for $30. Check your card offer.\nYes. \/\/@anonymous: Is it possible to import last year tax information from TurboTax into H&R Block?\nYes you can import \/\/@Anonymous: Yes. \/\/@anonymous: Is it possible to import last year tax information from TurboTax into H&R Block?\nYes. I did it before. \/\/@Anonymous: Yes. \/\/@anonymous: Is it possible to import last year tax information from TurboTax into H&R Block?\nBoth are good. But imo TurboTax's is more user friendly. \/\/@anonymous: Is it possible to import last year tax information from TurboTax into H&R Block?\nyes \/\/@Anonymous: Is it possible to import last year tax information from TurboTax into H&R Block?\nI just paid 20 for the Deluxe plus State from a Dealsea ad 2 weeks ago. Doh.\nLooks like no e-file for state included. Need to pay $20 for e-file state tax \/\/@Anonymous: I just paid 20 for the Deluxe plus State from a Dealsea ad 2 weeks ago. Doh.\nI don't get those people complaining all the time. You can calculate in software and e-file for free at State website, or print and mail it.\nOrdered deluxe + state for windows. But the download button didn't work on firefox or chrome. It only worked on edge.\nUse the link from email \/\/@anonymous: Ordered deluxe + state for windows. But the download button didn't work on firefox or chrome. It only worked on edge.\nDownload and installed it without any problem. I like HR more than Tubrotax, Thanks, Dealsea.\ndoes this one include e-file?\nYes, Federal E-File is free; State is $20. \/\/@Anonymous: does this one include e-file?\nAmazon has Norton Security Deluxe - 5 Device [Key Card] for $19.99. Shipping is free with Prime (30-Day Free Trial).\nDo i have to buy separate Antivirus to use this key?\nIt includes Norton AV. \/\/@anonymous: Do i have to buy separate Antivirus to use this key?\nDon't install Norton antivirus. Its basically an actual virus with how it slows down your computer.\nNo its your computer too slow \/\/@anonymous: Don't install Norton antivirus. Its basically an actual virus with how it slows down your computer.\nBest Buy \/ Staples \/ Office Deopt Office Max has Quicken 2018 (2 years) on sale. Shipping is free.\nwhat does 2 years mean?\nIt means you will be shafted for more money after two years of use (subscription). I would recommend QuickBooks Desktop Pro if you are not too bothered about direct downloads from bank accounts. You can still do that manually though. \/\/@Anonymous: what does 2 years mean?\nAmazon has Norton Security Premium - 10 Device [Key Card] for $27.99. Shipping is free.\nDon't see the price. Its coming up 43.99 on Amazon, when I click the link.\nBHPhotoVideo has Microsoft Office 365 Personal (1 PC or Mac License \/ 1-Year Subscription \/ Product Key Code \/ Boxed) With $30 B&H GC for $58.95 (price shown in cart). Shipping is free.\nH&R Block has H&R Block 2017 Deluxe Federal + State Download (w\/o Extended Download Service) for $29.95 . Release on Nov. 10, 2017.\nWait for January and you will get cheaper..\nIt gave me big trouble due to its efile bug.\nMost won't need this after tax reform.\nAmazon has Rocksmith 2014 Edition Remastered w\/ Cable (PC\/Mac) for $19.97. Shipping is free with Prime (30-Day Free Trial).\nBHPhotoVideo \/ [amazon] has Adobe Photoshop Elements 15 and Premiere Elements 15 (DVD or Download) for $69.95. Shipping is free.\nAmazon same price. BH has no tax. Got 1.\nBetter than the GNU Image Manipulation Program?\nGoogle PlayStore has Android App: Mobile Document Scanner Pro for FREE (reg. $4.99).\nGoogle Play has Graphing calculator Android App for FREE.\nGoogle Play has QR & Barcode Scanner Pro (Android App) for FREE.\nwhen you have wechat, those other QR apps are useless.\nBut sometimes wechat won't open the page. \/\/@Anonymous: when you have wechat, those other QR apps are useless.\nGoogle Play has OfficeSuite Pro + PDF for $0.99.\nAmazon offers $35 Off $100 in Amazon Coins for new Amazon Appstore customers.\nWhat are amazon coins? Like Bitcoin?\nis for you to use it to pay ransom when your computer got hijacked \/\/@Anonymous: What are amazon coins? Like Bitcoin?\nWalmart has TurboTax Deal. Free store pickup or free shipping on $35+.\nBelow sells even cheaper. Reliable?\nWalmart \/ Amazon has Just Dance 2017 (Nintendo Switch) for $34.29. Free in-store pickup or free shipping with Prime (30-Day Free Trial).\nAmazon has $10 off select H&R Block Tax Software with code \"10HRBLOCK\". Shipping is free with Prime (30-Day Free Trial).\nDidn't work. Is it dead already?\nNewegg has H&R BLOCK Tax Software for 30% off with promo code \"USAVE30\". Shipping is free.\nAmazon Deal of the Day has Norton Security Deluxe - 5 Devices (online code) for $19.99.\nNewegg has 50% OFF Select Intuit QuickBooks 2017 with code \"QB17FIFTY\". No tax for some. Sale ends 2\/23. Shipping is free.\nAmazon has H&R Block Tax Software Deluxe + State 2016 Win + Refund Bonus Offer for $20. Shipping is free with Prime (30-Day Free Trial).\nA question about the bonus. Are the funds (including the refund and the 10%bonus) locked into one Amazon account and\/or one GC? I was hoping to resell the refunded Amazon GC, but if all the funds are locked into one account, and\/or one GC, then I won't be able to do that..\nNote: Program Code is NOT required if you (work email or college email) qualify.\nIf you qualify, code is not required.\ndoes this product need renewal every year?\njust provide your official email id. I got email saying I am eligible.\nyour right to use this software is tied to your organization's continued Software Assurance coverage and your continued employment with that organization. Your employer will notify you if their Software Assurance coverage expires. \/\/@Anonymous: does this product need renewal every year?\nwhy is this a deal now? this has been around from few years. always the same process and requirements and nothing new.\nNO. nothing to do with continue employment but you have to have qualified employment email domain. \/\/@Anonymous: your right to use this software is tied to your organization's continued Software Assurance coverage and your continued employment with that organization. Your employer will notify you if their Software Assurance coverage expires. \/\/@Anonymous: does this product need renewal every year?\nWrong edu address not qualified.\nContinued Employment is not necessary. Had office 2010 with my previous employer and still working after 4 years of quitting job. \/\/@anonymous: your right to use this software is tied to your organization's continued Software Assurance coverage and your continued employment with that organization. Your employer will notify you if their Software Assurance coverage expires. \/\/@Anonymous: does this product need renewal every year?\nReceived the qualify email with url, but can only order MS project and MS Visio, No order button for MS office. Anyone got the same issue?\ncan you forward to me? \/\/@Anonymous: Received the qualify email with url, but can only order MS project and MS Visio, No order button for MS office. Anyone got the same issue?\nI got Vizio only with my work email \/\/@anonymous: Received the qualify email with url, but can only order MS project and MS Visio, No order button for MS office. Anyone got the same issue?\nsame thing here, only visio and project, no office!!! \/\/@Anonymous: I got Vizio only with my work email \/\/@anonymous: Received the qualify email with url, but can only order MS project and MS Visio, No order button for MS office. Anyone got the same issue?\nWhatever you could get at the deep discount price depends on what the deal your company made with Microsoft. My company's deal is only office. My previous company's are office + project + Visio \/\/@Anonymous: same thing here, only visio and project, no office!!! \/\/@Anonymous: I got Vizio only with my work email \/\/@anonymous: Received the qualify email with url, but can only order MS project and MS Visio, No order button for MS office. Anyone got the same issue?\nAmazon offers Up to 63% Off H&R Block Tax Software. Shipping is free with Prime (30-Day Free Trial), or free with $49+.\nJust bought the 50% from hrblock. Has not downloaded. Wonder if there is a chance to price match or cancel order?\nCalled Amazon and got a response that Amazon does not do price match on software.\nwhich one has the 10% amazon giftcard offer? it says some are 5% and some are 10%, but doesn`t say which. thanks.\nBest Buy offers $10 Gift Card with H&R Block 2016 Tax Software. Shipping is free on $35+ or choose store pickup.\nH&R or TubroTax, which one should go?\nI am on a rant against TurboT. ever since they stole Deluxe (investing) features and moved them into Premier and acted like they didn't screw the consumer.\nI use H&R all the time over 15 years. \/\/@Anonymous: H&R or TubroTax, which one should go?\nThe best deal is Deluxe Federal only. You have to pay to file your state return with the state version. So buy this and file your state taxes through your state's website. Ohio allows you to do it all online.\nAmazon \/ Best Buy has H&R Block Tax Software Deluxe + State 2016 + Refund Bonus Offer PC\/Mac Disc for $24.99. Shipping is free with Prime (30-Day Free Trial) or free store pickup.\n$50 above free shipping; already switches to other website because of this.\nStaples price match with Amazon, just got one from store.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Brady TL-PK-1 Tamper Seal \"Warranty Void\" Label protects your warranty exposure by sealing access points with tamper seals. Tamper seals come in a 20-page pocket size book.\nBrady TL-PK-2 Tamper Seal \"Calibration Void\" Label, 0.800\" Dia.\nBrady TL-PK-12 Tamper Seal \"Calibration Void\" Label, 0.750\" x 0.375\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After receiving her doctoral degree in clinical psychology with a minor in behavioral neuroscience, Dr. Hayes began her career at the LSU Health Sciences Center (LSUHSC) Department of Psychiatry in New Orleans where she completed a forensic psychology and neuropsychology fellowship. She also completed the FBI Citizen's Academy.\nAs an Associate Professor of Clinical Psychiatry at LSUHSC, Dr. Hayes was actively involved in teaching, clinical work, and forensic private practice. She taught graduate psychology students, medical students, psychiatric residents and psychiatric fellows. She was also the Director of Psychology Training for LSUHSC. In addition to working with the victims of violent crime, Dr. Hayes was the mental health director of the New Orleans HIV Clinic. It was here that Dr. Hayes and her colleagues wrote a critically acclaimed book titled A Woman's Guide to Living with HIV Infection, published by Johns Hopkins, in addition to many other book chapters and articles.\nWhile at LSUHSC, Dr. Hayes maintained a part-time forensic practice testifying and consulting in many jurisdictions, but after Hurricane Katrina, this part-time effort became more prominent, with her maintaining offices in New Orleans and Scottsdale, Arizona. Dr. Hayes' forensic private practice focuses on criminal and civil legal consulting and evaluations. She also maintains a clinical practice. She has continued writing, publishing another book with Johns Hopkins and other articles. Likewise, she continued teaching as adjunct faculty with LSUHSC, University of New Orleans, and also Arizona State University.\nDr. Hayes was on the advisory board of the Bureau of Justice Assistance project titled \"Developing BJA Sponsored Tools for Instilling, Promoting and Maintaining Professional Integrity in Law Enforcement Agencies,\" was a consultant regarding child abuse to the US Department of Justice and continues as a mental health consultant to various police agencies. Additionally, Dr. Hayes is often asked by local and national radio and television stations to discuss various forensic issues, and she was featured on A & E's American Justice and on Court TV's Forensic Files.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dover-Foxcroft Maine land for sale with 900+\/- feet frontage on the town maintained Burroughs Road.Electric and phone are available road side. The land slopes uphill from the road with elevations from 500 to610 +\/- feet above sea level. About 300 feet of roughed in driveway leads to a cleared building site. Whitetaildeer and grouse are plentiful at this property. The land is located about 2 miles south of downtownDover-Foxcroft with shops, restaurants, hospital, school and other amenities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join the Dyfi Distillery team on this night of exploring the world of Welsh Gins.\nThe Dyfi Distillery is the closest distillery to the Talyllyn Railway, just outside the old mining village of Corris. Sited in the Dyfi UNESCO World Biosphere Reserve, the distillates made here are dominated by locally foraged botanicals, meaning that there are no other gins which taste quite like them. It is the only distillery in the UK to have won the coveted Best British Gin title twice (in 2017 and 2018). The tastings will include Dyfi Original, Pollination and Hibernation Gins plus a New Micro-Distilled Gin especially for the event.\nWith 4 Gin Tastings served at stations along the line, more Gin will be available to buy at Tywyn Wharf and Abergynolwyn stations plus a Cold Buffet served at 9pm before the Dancing begins to the Footloose Dance Orchestra at Neuadd Pendre until 11pm. Don't worry if you have two left feet, Lorraine and Company will be here to teach you a few steps.\nThis is an Adult themed evening; well Behaved Children are allowed.\nKing's Licenced Cafe & Bistro - at Tywyn Wharf station open from breakfast to afternoon tea, and on selected evenings. Gift Shop - a wide range of gifts, books, models and souvenirs at Tywyn Wharf station. The Narrow Gauge Railway Museum - Free museum at Tywyn Wharf telling the story of the narrow gauge railways of Wales and beyond. Quarryman's Caban - tea rooms at Abergynolwyn station, open when train are running. Most trains take a 20-30 minute break here. Railway car parks at Tywyn Wharf (\u00a31 )and Abergynolwyn (Free). There is a privately owned car park at Dolgoch Falls (charges).\nTywyn is on the A493 road from Machynlleth to Dolgellau. Approaching from Aberdyfi, Wharf station is on the left. Coming from Dolgellau, go through the town centre past the National Rail station for 300yds and we are on the right. Large car park 100 yards away. Parking is also available at Dolgoch and Abergynolwyn stations, located on the B4405 Bryncrug to Talyllyn road.\nDisability Access Statement is available online at http:\/\/talyllyn.co.uk\/access or by phoning 01654 710472.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjyhu b/data_all_eng_slimpj/shuffled/split2/finalzzzbjyhu
new file mode 100644
index 0000000000000000000000000000000000000000..f5fdb81d24c054cf8da987db7ab23af15732a66d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbjyhu
@@ -0,0 +1,5 @@
+{"text":"Help!! Jeff and I are second-guessing ourselves. We're wondering if moving is the right thing to do. Jeff is leaning towards staying in Bloomington. I'm still smack in the middle. The pull of my friends and family in Arizona is very, very strong, as well as the probability of a better job market for Jeff. On the other hand, we love the weather and environment here, and we have great friends locally, and in all fairness, Jeff hasn't tried to get a job here since Jasmine left. Moving is expensive.\nWe wrote up a list of pros and cons for both moves. I know I have family and friends who are biased one way or the other in regards to where we live who read this. I don't care \u2014 I want to hear what you think. I've included the list below and ask you to leave me comments with things you think we should consider in making this decision. In the meantime, Jeff is going to check out State Farm on Monday and see what might be available. He's also checking the Pantagraph. I keep saying the job market is better in Arizona, but the reality is, I don't know that for sure. And I do know that the cost of living is higher there, albeit only about 6%.\nWhen we made the decision to go back in January, it was largely because we wanted Jasmine to be closer to a hospital. That factor has changed, obviously. Since then, we've stuck with the decision because we wanted to be closer to family in the wake of Jasmine's death. We have not really checked the job market for Jeff since we came home in January because we had already decided to move. Now we're questioning the decision for some of the reasons I listed above \u2014 the expense of a move, the weather in Arizona and just generally wondering if a move is the right thing for us right now.\nOf course we know the decision is ultimately ours, and we are talking about it daily, but we would love to hear from our friends and family, even with biases!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"He is a senior lecturer in law at the University of Greenwich. His main area of interest is to harness the use of blended learning to teach law students for the past ten years. He is also very keen in Corporate law, Business law, Company law, Employment Law and Information Technology. He is currently doing a project which looks at the benefit of using blended learning to maximise the learning capabilities of law students in the UK and Malaysia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Input Power: 80 to 230 vac single phase 50\/60 Hz or 100 to 350 vdc, 15 Amp max.\nOutput Power: 3 phase sinusoidal current \ufffd0 volt max, 0 to \ufffd continuous (0 to \ufffdA continuous with additional heatsink and cooling fan), 20A peak for two seconds.\nDiscrete Inputs: Two optically isolated control inputs 10 to 24 vdc, sinking or sourcing, for Run\/Stop input and static brake enable input.\n- Two jumper programmable inputs to select motor type and encoder resolutions.\n- Encoder inputs TTL 5 volts or open collector type. Digitally filtered channel A quad B up to 250 kHz external.\n- Hall effect rotor position inputs TTL 5 volts or open collector type.\n- One open collector transistor output to indicate drive is operating at current limit.\nAnalog Inputs: Velocity command input 0 to \ufffd vdc. - Selectable current\/torque command 0 to \ufffd vdc scalable by an on board pot.\nOperating Conditions: Drive is designed to operate safely in an ambient temperature of 0\ufffd to 60\ufffd C, noncondensing conditions. Heatsink temperature must never exceed 80\ufffd C.\n- Size 9.0\" H x 2.6\" W x 5.0\" D including mating connector.\nThe ServoLoc 1 is a fully digital brushless servo motor controller specially designed and manufactured to allow user configurable operation. The ServoLoc 1 with 3\ufffd sinusoidal motor current operation provides smooth rotation and improved torque repeatability over trapezoidal current type drives.\nRS 485 serial communication of the ServoLoc 1 allows for a multi-axis velocity\/torque control environment, with on-the-fly mode change. Since the ServoLoc 1 was designed and developed as an alternative choice for an existing servo drive, additional features are incorporated to interface with a wide variety of brushless motors. Inputs are provided on the terminal block to configure the ServoLoc 1 to operate with various encoder resolutions and motor poles.\nBesides tight servo loop control, the ServoLoc 1 includes features that are not available from competitive type drives. One of the most important features is that the ServoLoc 1 includes a selectable static brake option (shorting 3 motor leads) when the drive is disabled and at low velocity. This option is particularly useful in anti-windup situations. Another useful feature is its ability to coast to stop or ramp (decelerate) to stop and disable when a stop command is initiated.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our two third grade classes are studying health and first aid in science class. Dressed as doctors, they toured the emergency and pediatric wards at Sistina Hospital. They learned how the doctors and nurses treat patients and help them become healthy. These kinds of fieldtrips help our students connect what they learn in the classroom with the real world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are a friendly, well-established local Pre-school offering a stimulating and exciting curriculum for all children aged from two and a half to five years old. We have an active committee of parents and members of the Church and a well qualified staff team, led by a graduate. We are registered with Ofsted and can offer the free funded entitlement to two, three and four year olds, we are also members of the Pre-school Learning Alliance.\nMuch of the information can be found from the page links but feel free to contact us should you require any further or more specific advice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkdwi b/data_all_eng_slimpj/shuffled/split2/finalzzzbkdwi
new file mode 100644
index 0000000000000000000000000000000000000000..c3cb7693ed74d55233ec4df9843931c65b48635b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbkdwi
@@ -0,0 +1,5 @@
+{"text":"2001 A Space Odyssey Hal Hoodie. NOTE: This product takes 5 business days to process before it ships. 75% Cotton\/ 25% Polyester High Quality Pre Shrunk Machine Washable Hoodie. Made in the USA by Trevco Inc. Licensed Apparel. Color Black.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have a question you need answered right away? Or just need someone to help you sort through your options? Our representatives can help you right now, while you view this site.\nVisit us without an appointment at any of our locations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have completed B.Tech and I am employed in private Sector as a Software professional in Accenture currently based in Bangalore. I belong to a middle class, nuclear family with moderate values.\nMy name is Nivetha S and I am working as a Software Professional in the Private sector. I have completed my B.Tech.. I grew up in a Upper middle class, Joint family with Moderate values. Independent and hardworking person. Hobbies are reading books and travelling.\nI have a Bachelor's degree and I am employed in Private Sector as a Software Professional currently based in Bangalore. I belong to a Middle class, Nuclear family with Moderate values. Bad: I had cancer so that I went through surgery and chemotherapy. Good: Now I am completely alright.\nMy sister is a software professional with a Bachelor's degree currently working in private sector in Bangalore. We come from a middle class, nuclear family background with orthodox values.\nI am Prabhu and working as a Software Professional in the Private sector. Have completed my B.Tech.. we are Upper middle class, Nuclear family with Traditional values.\nHi, I am Ashwin and working in MNC as Software Engineer. I grew up in a Upper middle class, Nuclear family with Moderate values. Born in Ambala brought up in Ahmedabad (Gujarat). Currently residing in Bangalore from past 6 years.\nI have made this profile on behalf of my younger son. He is currently working as a software professional in private sector, after having completed his bachelor's degree. We come from a middle class, joint family with traditional values presently living in Bangalore at own house at KR Puram. My elder son was also working in Bangalore at Manyatha Tech Park. He has good habits, friendly type and Vegiterian.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Greater Akron Chamber celebrates its 112th Annual Meeting on Thursday, March 21, 2019, at the John S. Knight Center. The Greater Akron Chamber will honor S. Theresa Carter, President of OMNOVA Solutions Foundation, with the H. Peter Burg Award. This award will recognize Carter's leadership for her advancement of Greater Akron's economic and community development. The Chamber will also elect its Board of Directors, celebrate its accomplishments, and share exciting new initiatives that will create opportunities for all to Lead Greater.\nDuring the meeting, the members will also be asked to consider and approve recommended changes to the organizations regulations and to elect members nominated to serve on the chamber's Board of Directors. A summary of those recommended regulation changes, the red-lined code and the Nominating Committee Report are available for your review.\nThe program includes a reception with hors d'oeuvres, open bar, dinner, and silent auction. The silent auction offers guests a chance to bid on items that will benefit the Chamber's educational programs. The online silent auction is already available.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Let's talk about side hustling! It is completely possible to make money from mystery shopping, panel studies and similar services.\nWhile I'm still tallying up how much money I made in one year from side hustles \u2013 this blog included \u2013 I thought I could share how much money I made from technical things, such as mystery shopping, panel studies, surveys and reward points.\nThis does not include pet sitting, blog income, starting my online business or making money on Ebay.\nI worked with a number of companies this year and it's simply amazing how much stores, restaurants and service companies need mystery shoppers. I received payment via PayPal and physical checks.\nThis total amount could actually be higher, because some companies actually sent me a gift card to use for payment instead of me having to cover the cost. The total value for this was around $380. I preferred this as I didn't have any out of pocket expenses upfront and didn't have to wait for reimbursement.\nThis category is really interesting mostly because the pay varies greatly by your location. For example, a shop that pays $4 in one area might pay $13 in a different part of the country. It's all about supply and demand.\nIn areas where there are a number of shoppers available, these companies can price these really low because they know someone will take the job. In other parts of the country where the population is smaller or there aren't as many shoppers signed up, the same survey will pay much higher.\nThere are three different survey companies I use on my phone. Each company pays via PayPal. I earned a combined $137 from taking 5-10 minute surveys while I was out shopping, with all the shops occurring at a grocery or pharmacy type store.\nUgh, I am so mad at myself for not participating in more of these. I get emails weekly about applying, but I either see the email too late or it's full by the time I try to apply (most studies fill up within a few days). I made $350 in panel studies this year and should have made triple this amount.\nIn all fairness, I have made this amount since I quit my job in October. I just didn't have enough time, flexibility or an opportunity to check my emails when I was at my day job.\nReceive payment within 2 weeks of study completion. Payment is a prepaid Visa or Discover card that you can request in the mail so you have the actual card or it can be a virtual card.\nFor the panel study I did in person, I received a physical prepaid Discover card when I finished up and checked out. In the past I have received envelopes with cash in it, but I think a lot of companies are switching over to prepaid cards for tracking purposes.\nI earned $78.84 back in cash. Ebates literally pays you cash back every time you make a purchase. The cash is deposited into your PayPal account. Hello, free money!\nI redeemed $110 in gift cards this year. I earned points by taking advantage of when MyPoints was running promotions and I could double cash back on purchases. I purchased gift cards for BP, Shell and Starbucks.\nI am signed up with two data tracking companies when I use the internet. I didn't know about data companies until March of this year, so it's not a full year of earnings.\nSome people have issues with data companies because they can track what you are doing. Since I already use Facebook, Instagram and a million different apps on my cell phone that can view my tracking anyway, I figured I might as well earn money for doing it.\nWith Data Company 1, I've earned $48, which can be redeemed in gift cards. I like to cash in at the $25 threshold as there isn't any incentive pass this point. I've redeemed for one $25 Starbucks gift card and plan to cash out the other one by the end of the year. I'm a few dollars off and should get to a total of $52-$53 by end of the year.\nWith Data Company 2, I receive $2.50 every other week. The money is automatically transferred to my PayPal account. This year I've earned $60 (I accidentally deleted the app in October \u2013 whoops) so I missed a couple payments. I should receive another $5 this month, but I'll stick with $60 to be safe.\nWhen combining the two companies, I've made $108 from pretty much doing nothing!\nI didn't learn about Shopkick until earlier this year. You can earn enough \"kicks\" AKA reward points to cash in for $5, $10 and $25 gift cards to places like Target.\nShopkick is great because you can earn points every time you are out shopping, whether it's at the grocery store or the mall.\nEvery time you go to the grocery store, you can scan in your purchases and earn rebates, which are deposited into your PayPal account. Rebates range between fifty cents up to five dollars. Click here to get $10 cash back after you redeem your first item!\nLook at mystery shops when I want to go out to dinner or grab a coffee \u2013 why pay when this could be free? This means planning better, but I think I could do this at least 50% of the time.\nComplete more panel studies. These are such easy and fun money makers and I am mad at myself for not doing more of these.\nIf you are serious about making more money in 2016, let's challenge each other to make more money from side hustling. If you are just starting out, why not start small? Earning $50 a month is certainly attainable. Even earning $100 a month is achievable as a beginner.\nI've been mystery shopping and side hustling for years! It has taken me years to test out different companies, figure out which companies have certain clients and learn the system. Unfortunately, I didn't have anyone to help me so I figured it out as I went along.\nI'm sharing every resource I know when it comes to mystery shopping! I don't know anyone who has done this and I'm sharing over 40+ resources that directly link you to different companies.\nNot only am I linking you to a ton of companies, I'm sharing what types of shops these companies offer, like dining, hotels, or retail shops.\nEveryone wants to know more about panel studies, which is fascinating to me because I've been aware of them for so long. I'm sharing the direct links I use so others can get started!\nAltogether this adds up to $2014.84 from 2015.\nConsidering I spent almost ten months working full time at my day job (before quitting) and working on this blog in all my free time, I'm really happy with how much money I made! I didn't go out of way to conduct any of these shops and surveys, which is the best part.\nAre you planning on side hustling?\nI would like more information about the panel studies but I didn't see a link for it.\nHi Lisa, I mention all the panel studies and mystery shops I use in my Side Hustle to Success course!\nI haven't participated in mystery shopping in years. For me, some of the shops took too much time and effort to finish. May consider signing up to a couple of companies to start up again since I now have more time to complete them. I wouldn't mind having a couple of my date nights paid for.\nMake sure you check out media. net \u2013 I had much better results with them than Adsense. If you go through some of my more recent blog income reports, I break down how much I earned from both!\nHey Stevie! You might be able to find them on my site if you do a search. Otherwise I go into total detail in my book!\nHow do you find places to mystery shop? I wanted to try it maybe and I have a friend interested too!\nHeck yeah!! Make that money! Im always trying to find additional ways to make extra money. I like the motivation you have to do this. You really need to be dedicated and motivated to do stuff like this I think.\nThat's a good side hustle amount in a year. I am planning on doing more side hustling in 2016 to merry my goal of paying of nearly 17,000 of debt by October 2016 and take two trips. So much so that I actually am starting more. I'm excited about the book! The cover is nice.\nThanks Shirria \ud83d\ude42 Will you be sharing your side hustling on your blog? That's awesome you can pay off so much!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkifs b/data_all_eng_slimpj/shuffled/split2/finalzzzbkifs
new file mode 100644
index 0000000000000000000000000000000000000000..e0cc02f4128757d634985a0c3616a72126a1b5be
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbkifs
@@ -0,0 +1,5 @@
+{"text":"Last November I participated in the National Novel Writing Month, AKA NaNoWriMo. The goal is to write 50,000 words in a month, and of course I had to make my own Nanowrimo calendar. To do 50,000 words in the 30 days of November, that required an average of 1,667 words a day. I decided to visualize this by providing a stack of 10 blocks, each representing 250 words, that I could fill-in from the bottom to the top. Success can be projected by looking for daily blackened blocks. If I filled the minimum of 7 blocks, I was doing well. If I filled in more, which I often did, I had a visual indicator of having \"reached the pinnacle of achievement\" for the day.\nLater, several people asked if I could make a version for other months of the year. And so, today's product-of-the-day is a version of the NaNoWriMo calendar for March 2013. I had to rebuild the template from scratch; apparently I had not saved the original in a place that was backed-up when I embarked on the Great Windows 8 Migration of January 2013. I was also out for most of the day, so to get the product-of-the-day out I decided to just cover March. Total production time was around 90 minutes.\nThis variation of the calendar, like the NaNoWriMo version, counts words. I made a few refinements: providing an explicit field for \"total words of the day\", and also simplifying the weekly total column. For future variants, I have ideas for tracking percentages and other cool metrics, so if you have an idea for something you want to count, let me know and I'll try to come up with something in the next few days!\nGroundhog Day Resolutions 2014: Kick Off!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"And so Miller's breakthrough success as a playwright came in 1947 when he superimposed his deeply felt critique of capitalism onto the Keller family: Joe (John Howard), the successful factory boss, his wife Kate (Robyn Nevin), and their two sons, Chris (Chris Ryan) and Larry.\nThe play is set in the family's suburban yard in August, 1946, three years after the disappearance of Larry in the Pacific. Chris and Ann Deever (Eryn Jean Norvill), Larry's girlfriend before he went missing, have fallen in love and are desperately trying to move on. Mother Kate is in denial about Larry's death while the patriarch Joe Keller is trying to repair his reputation after being charged with selling cracked cylinder heads to the US Air Force, causing the deaths of 21 pilots. Joe is exonerated and shifts the blame to his employee Steve Deever, Ann's father, who remains in prison. Tragedy unfolds for the Kellers as lies are uncovered and the truth is too much for the family to bear. Personal failings play their part but they almost seem like inevitable responses to the malevolent forces unleashed by capitalism.\nDirected by Kip Williams, this Sydney Theatre Company production is absorbing theatre at its very best. Normally an avid note taker, by Act Two I was too engrossed in the drama to remember I had a job to do.\nThe entire cast gives strong performances and the leads are sublime. John Howard beautifully occupies his dual roles as lovable object of his son's adoration and as a weak, flawed man who is highly susceptible to the corrupting influence of money. Robyn Nevin masterfully blends fragility with strength as the mother who desperately tries to hold her family together.\nChris Ryan is affecting as the dutiful son whose world is shattered by revelations about his father and brother. The scene in Act Two when he demands the truth from Joe about his role in selling the faulty cylinders is devastatingly powerful.\nEryn Jean Norvill gives a nuanced and sensitive portrait of Ann Deever, torn between loyalty to her family and burgeoning love for Chris Keller. Josh McConville is a tour de force as Ann's brother George Deever, smashing the brittle lies the Keller family have been telling each other for years.\nBetween the writing genius of Miller and the acting genius of the cast, Williams has wisely decided to keep the bells and whistles to a minimum. Alice Babidge's suburban silhouette is cleverly faded and exposed in the same way Miller exposes the receding and deceitful dream of American capitalism.\nThe emotional intensity of the performances in this production serves Miller's themes as well as he could have hoped. They draw out his views on systemic ills causing personal pain. As well articulated in the essay in the program, Miller 'wanted the audience to see not only the fault of the characters, but the way in which any one of us can make immoral choices when we become a function of capitalism, morally divorced from our own actions and responsibilities in the service of production and distribution'. We identify with all of the characters in some way, connecting us to the even greater damage caused by contemporary capitalism.\nCast: Toby Challenor, Anita Hegh, John Howard, Bert Labonte, John Leary, John McConville, Robyn Nevin, Eryn Jean Norvill, Jack Ruwald, Chris Ryan, Contessa Treffone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You are downloading the Pizza Chef AppXBundle file v2015.409.2003.1259 for Windows Phone. Pizza Chef is a free and fun Strategy game: Serve pizzas and become the ultimate Pizza Chef!Pizza Chef is a fun free game from Ape Apps where you are given a job at a pizza diner and ... You can now download the offline AppXBundle file to install & try it.\nNote: To install it manually from an SD card, you should choose one of the server location above to get the offline Pizza Chef AppXBundle file, move the file to your phone's SD card and then tap 'Install local apps' in the phone's App list.\nAppx4Fun.com only share the original AppXBundle installer for Pizza Chef v2015.409.2003.1259.\nAll the Windows Phone apps & games here are free for home or personal use ONLY. Pizza Chef is the property and trademark from the developer Ape Apps, all rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These lovely expats (from Minnesota, USA) had their own special 'crown moment' :-) they're expecting a babygirl within a couple of weeks. Big time to capture their love and pregnancy with the big brother-to-be. What a nice and lovely family fotoshoot!\nFeest! En ik was de surprise :-) Deze 'net-een-paar-dagen-mama' was die dag jarig. Haar man had een newbornfotoreportage geregeld met mij voor hun pas geboren zoontje.\n\u00a9 Kroonmoment Fotografie | Breda | KVK 56326513.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"World of Tanks - \"T29 Tank Frustration\"\nSC2 - Heart of the Swarm!\nStarcraft 2 Pro MLG D.C.\nStarCraft2 Is Here - Who's got it?\nSupreme Commander 2 Is Now LIVE!\nThree Classic Command and Conquer games...for free!!\nEmpire Total War: Multiplayer Campaign!\nPlanning On Pirating StarCraft 2? Think Again Mortal!\nSome new things... BATTLE.NET 2.0, new Q&A and some screens!\nSCII is gonna be called wings of liberty?\nSpore One Week From Being Completed?\nIkariam - Free Civ like browser game.\nStarcraft II, December 3rd Release?\nWho's gonna get competitive when SC2 comes out?\nC&C 3: Kane's Wrath expansion pics!\nHow Many Disks for Diablo 2 + Expansion?\nStarcraft 2 coming to Consoles?\nExpan! Did you gets it?\nstrafeRight.com World in Conflict Server!\nsR World in Conflict Server?\nI Got on the World in Conflict BETA test!\nStarcraft 2 Offical, with screens.\nC&C3 I Want Walls Back!\nAnyone play the \"DotA\" mod of Warcraft 3?\nSORRY! I'm a Supcom Tard!\nStarcraft Battle.net Doesnt Work for me!\nCompany of Heroes - Open Beta!\nSuggestions for a good RTS?\nDoes Age of Empires III look this bad on your machine?\nWas there already a Civ IV thread?\nCivilization IV.... needed more time in development.\nAge of empires III? A joke?\nAnyone try AOE III yet?\nAnyone anxiously waiting for WarHammer 4K Dawn of War: Winter Assault ?\nWhat is the best and most graphically good looking RTS game?\nHow to play the original Command & Conquer in XP?\nSplinter Cell 3 crashes on my XPS2, Help?\nDoes Warcraft III support wide screen?\nYour all time favorite strategy game?\nanyone else looking forward to Empire Earth 2?\nWarhammer 40,000: Dawn of War ! ! !\nNew Command & Conquer game in the works!\nIs this Notebook Configuration good enough for games??","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblnva b/data_all_eng_slimpj/shuffled/split2/finalzzzblnva
new file mode 100644
index 0000000000000000000000000000000000000000..357792feb3ebaf02a1481611c146f8662d96edd6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzblnva
@@ -0,0 +1,5 @@
+{"text":"I recently joined do. Good Stitches, a charity quilting bee on Flickr. Organized by Rachel of Stitched in Color, do. Good Stitches has grown to 15 circles, each comprised of 10 quilters and stitchers. Each circle creates unique quilts that are donated to a variety of different nonprofit organizations all over the U.S. and abroad.\nI'm a quilter in the \"Dream\" circle. Each month we complete 2 small quilts to be donated to Project Night Night, a nonprofit organization that provides homeless children with a tote bag with comforting \"night-time\" essentials (book, stuffed animal, and blanket). I recently finished up December's quilts for my circle.\nFor these two quilts, I asked each member to construct 2 10.5\" postage stamp blocks. Blocks were made up of 5 rows of 5 2.5\" squares, in a mixture of solids and prints, in two different colorways.\nThe first palette contained red, yellow navy, royal, teal blue, white or white with black.\nI used one of the extra blocks as a feature on the back, and bound it in yellow. I quilted it with a diagonal grid using the 2.5\" squares on the front as my guide.\nThe second quilt included plum, purple, pink, dark coral, gold, warm yellow, cream, and medium gray.\nThe second quilt was bound in gray. The back has the same layout as the first, quilted with a series of overlapping boxes with 3 lines. While I like the effect, I still need some practice with my machine quilting--turning the corners of the squares was difficult.\nWorking on these quilts was a lot of fun and I look forward to sending them on to Project Night Night where they will hopefully bring some sunshine to their new owners. Huge thanks to the ladies of the do. good stitches Dream circle, I am so proud of how these turned out and couldn't have done it without you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Buy FIXED MATCHES BEST SOURCE ARSENAL1X2 matches 100% sure or sure fixed match and every week 100% sure ht ft fixed match today. Site for best fixed matches Sure Betting or FIXED MATCHES BEST SOURCE ARSENAL1X2 100% sure. Sure fixed match is every week 100% sure ht ft fixed match and site for best fixed matches. Sure Betting; THE BEST VIP FIXED TIPS Free ticket football matches 1\u00d72. fixed matches 100% sure. paid tips 1\u00d72. free fixed match today. ht\/ft fixed matches Free ticket football matches 1\u00d72. double FIXED MATCHES BEST SOURCE ARSENAL1X2 fixed match today 100% sure.buy fixed matches paid tips 1\u00d72. free fixed matches. ht\/ft fixed matches. Free daily combo and single expert picks.\nIn organized sports, match fixing occurs as a match is played to a completely or partially pre-determined result, violating the rules of the game and often the law. The most common reason is to obtain a payoff from gamblers, but teams may also intentionally perform poorly to gain a future advantage, such as a better draft pick[A] or, on paper, a less eminent opponent in a play-off.[B] A player might also play poorly to rig a handicap system.\nMatch fixing, when motivated by gambling, requires contacts (and normally money transfers) between gamblers, players, team officials, and\/or referees. These contacts and transfers can sometimes be found, and lead to prosecution by the law or the sports league(s). In contrast, losing for future advantage is internal to the team and very hard to prove. Often, substitutions made by the coach designed to deliberately increase the team's chances of losing (such as having key players sit out, using minimal or phantom injuries as an excuse), rather than ordering the players actually on the field to intentionally underperform, are cited as the main factor in cases where this has been alleged.\nOther names for match fixing include game fixing, race fixing, sports fixing, or hippodroming. Games that are deliberately lost are sometimes called thrown games. When a team intentionally loses a game, or does not score as high as it can, to obtain a perceived future competitive advantage, the team is often said to have tanked the game instead of having thrown it. In pool hustling, tanking is known as dumping. In sports where a handicap system exists and is capable of being abused, tanking is known as sandbagging.\nThis is montly subscriptions packages. Our team every day send two matches (combo). Bet on matches is maxbet and this is 100% sure. Odds for two matches is over 4.00+ to 15.00.\n23.04.2019 ??? \u2013 ??? ? ?.?? ?:?\nBuy FOOTBALL CLUB FIXED MATCHES matches 100% sure or sure fixed match and every week 100% sure ht ft fixed match today. Site for best fixed matches Sure Betting or FOOTBALL CLUB FIXED MATCHES 100% sure. Sure fixed match is every week 100% sure ht ft fixed match and site for best fixed matches. Sure Betting; THE BEST VIP FIXED TIPS Free ticket football matches 1\u00d72. fixed matches 100% sure. paid tips 1\u00d72. free fixed match today. ht\/ft fixed matches Free ticket football matches 1\u00d72. FOOTBALL CLUB FIXED MATCHES fixed match today 100% sure.buy fixed matches paid tips 1\u00d72. free fixed matches. ht\/ft fixed matches. Free daily combo and single expert picks.\nARSENAL1X2.COM are safest sport betting investment.\n*All combo fixed matches are protected with free replacements in case of lost status.\n*If replacement is also lost you got another free replacement until combo fixed matches is win.\n28.04.2019 ???? \u2013 ???? ? ?.?? ?:?\n27.04.2019 ???? \u2013 ???? ? ?.?? ?:?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Hungarian government is ramping up its \"terrorist\" measures; a constitutional amendment that establishes a new state of exception is one of the measures it foresees as necessary to keep the population safe.\nThe threat of terrorism in Hungary is considered to be low by the UK Foreign Office, the CIA, and Hungary's Strategic Defense Research Centre. Even the Prime Minister's chief national security advisor says they have no knowledge of terrorist plots. Despite that, the country has maintained a medium-level terror alert since the Paris attacks in November 2015. Most observers agree that any necessary reforms could be addressed by modifying the measures in the current legal framework. However, in early June 2016, the Hungarian National Assembly will vote on an anti-terrorism bill that will require a sixth amendment to a constitution that is already a patchwork of amendments.\nPlans to ban encryption software were abandoned after being castigated in the media. However, under the conditions specified in changes to the Act on Electronic Commerce, Internet Service Providers (ISPs) that provide encryption services would be under an obligation to store the metadata of encryption users for a year (along with the contents of their communications in the case of weaker forms of encryption); and if they refuse to share this data with an intelligence agency, they could face a fine of up to 30 000 euro. Troublingly, without the inclusion of proper safeguards, encryption users could be more susceptible to privacy abuse. While the government portal declares that the stored data would only be accessible to the secret services after judicial approval, the Hungarian Civil Liberties Union (TASZ) asserts otherwise: authorisation for such requests under current legislation is conferred not by a court, but by a government ministry.\nTucked away in the recesses of the 2017 Central Budget Governance Act are also some provisions that would potentially shield state-owned companies from public scrutiny. According to the proposal up for debate next week, non-disclosure protections of data relating to the assets, functioning and contracts of state companies involved in activities such as central data acquisition and telecommunications management could be exempted from Freedom of Information (FOI) laws for up to 30 years. The grounds for protections are mainly economic. The rationale: the availability of certain kinds of information could threaten the national economy by harming the interests of state companies. However, even the authority responsible for Freedom of Information, not famous for taking a strong stand against government excesses, questioned the exceptions last week, reminding the Parliament's Budget Committee that the statutory and constitutional basis for restricting access in court, related to legitimate economic interests, does already exist.\nThe constitutional grounds for invoking a state of emergency currently exist as well. Constitutional provisions stipulate in Articles 48-54 how authorities may respond to armed attacks, industrial disasters and other kinds of imminent dangers to national security. The absence of a clear justification for the amendment is one of the reasons why the establishment of a new form of state of emergency (literally \"status of terror alert\", terrorvesz\u00e9ly-helyzet) is controversial. According to the draft it would enable measures that would not be subject to ex post judicial control. What constitutes the grounds for invoking this status is not entirely clear either, and this is worrying since it would empower the government, if it felt it was endangered, to summon the National Defence Forces or limit civil rights.\nIn early 2016, when the proposed text of the constitutional amendment was only accessible to members of parliament under a pledge of secrecy, the E\u00f6tv\u00f6s K\u00e1roly Institute commented on a leaked copy, arguing that the legal basis for the declaration of a state of \"terror alert\" lacked the precision that such extensive power would require. It called on the government to conduct an evidence-based assessment of the threats that the country faced and to examine whether its needs could be addressed through parliamentary acts instead. Unlike many of the other dramatic changes to Hungary's legal framework that have occurred in recent years, multiparty support will now be needed for most of the key measures to become law. And the latest version of the bill has been significantly tempered with this in mind; it now includes measures that would enable the National Assembly to have a say in the matter.\nThe Hungarian Minister of Defence maintains that the primary aim of the reforms is to protect the population more effectively. Even if he's right, his reasons for thinking that Hungary would never cross the line into a military dictatorship are not entirely convincing: \"We live in an open world\"; and with a \"multiparty democracy, parliamentary control and the internet\" that \"could never happen\". In the absence of adequate safeguards designed to discourage a head of state persuaded that this, or any other form of overreach, was necessary, the amendments that are on the table would offer an accommodating legal environment. The fact that \"nobody [in the government] would want this\" shouldn't be enough to reassure a population living in a constitutional democracy. That's what constitutional guarantees are for.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Van Alen Institute today released \"Justice in Design,\" a report that examines how re-designing our jails could bring dignity and opportunity into the criminal justice system.\nFollowing Mayor Bill de Blasio's announcement this year of a plan to close the Rikers Island jail complex, long seen as emblematic of the City's criminal justice failings and systematic civil rights violations, Van Alen Institute put together an interdisciplinary team of architects, planners, psychologists, advocates and more to reimagine what a jail can be\u2014for detainees, officers, visitors, and the community at large. The report, released today by Van Alen Institute in partnership with Speaker Melissa Mark-Viverito and the Independent Commission on New York City Criminal Justice and Incarceration Reform, emphasizes the social connections and communal dimension of the criminal justice system, which must be considered when devising new designs for borough-based prisons.\nThe Justice Hubs would be further integrated within its urban context by mimicking local architectural styles and materials, providing exterior seating and lighting, and being included in city-wide sustainability strategies. A single civic entrance, for officers and visitors, would further concretize more communal ideals by establishing a common threshold. All of these design prescriptions, which purposefully blur the boundaries between the jail's interior and the exterior neighborhood, are intended to alleviate the stigmatization and fear of prisoners, among the most marginalized groups in society.\nThe report also outlines design guidelines for the interiors of Justice Hubs' jails, particularly the residential spaces of prisoners. To help enhance prisoners' dignity and remedy the \"dehumanizing living conditions that are implicitly sanctioned by the criminal justice system,\" the report calls for greater access to natural air, light, and outdoor spaces, and cultivating more domestic living environments. Spaces that include variation in color, furnishings, and functions are conducive to experiential and programmatic diversity, the report says, and could help \"produce positive behavioral effects.\" For visitors, making information easily accessible, producing open visiting areas, and facilitating respectful dialogue are described in the report as top priorities.\nWhile some may view this report, which foregrounds the design aspects of jail infrastructure, as ignorant of the systematic and structural roots of our deeply flawed criminal justice system, it is critical to remember the potential of design to (re)shape social dynamics. The report takes seriously the conditions within jails, and suggests an alternative groundwork for a reconfiguration of the relationship between prison and city. \"The guidelines provided through this report are incredibly flexible and offer a framework for future jail infrastructure\u2026it's so exciting to see the city actively engaging in conversations about alternative opportunities,\" said Lax. As national discourse surrounding civil rights and systemic discrimination increases and accelerates, the report is a welcome contribution to the reestablishment of a criminal justice system that includes, rather than excludes, and that guarantees dignity and opportunity to this stigmatized population.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Since this has been requested by several people, as well as Bossbram, here is a list of Crafting and Harvesting Jobs, and who is crafting. I'd love to keep this updated with the current levels as well as who is mainly doing what crafting\/harvesting jobs.\nMe personally, I'd do Leatherworker, Carpenter or Weaver as the main focus(preferably one of the first two), as well as Botanist.\nRe: List of Crafting Jobs, who is crafting what?\nI'll probably go for Alchemist and Botanist, it's too bad Square-Enix dumbed down the Disciple of the Land classes, I quite liked the fishing minigame.\nanyway i am doing Armorer and a bit of Mining to get my materials.\nSo far I've been leveling up my alchemist first. Not sure which trade of land I'm going to pick up first, but as far as I can tell most of the alchemist ingredients are from the botanist class.\nBTW Yuichan might I suggest using a table (see bottom of your post). Might be harder to fill in, but its far easier to read than all those colors.\nI know you can still fish, but the fishing mechanics themselves have been simplified. For example, the choice of bait is relatively the same, but once you start fishing, you simply need to reel in, much like Pok\u00e9mon's fishing rod mechanics. In 1.0 there was also the option to choose how deep you wanted your lure to be, this affected which type of fish you would be able to get, along with the fishing location itself. IMO the 1.0 mechanics were somewhat more fun.\noh and Cliff is an lvl 20 Goldsmith. Thought i would post it here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblrbv b/data_all_eng_slimpj/shuffled/split2/finalzzzblrbv
new file mode 100644
index 0000000000000000000000000000000000000000..6ceedd79f56c0e613180204dd077de9dceed5701
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzblrbv
@@ -0,0 +1,5 @@
+{"text":"Hello, I'm interested in 24 Monadnock and would like to take a look around.\nThe details on 24 Monadnock: This house located in Colebrook, NH 03576 is currently for sale for $132,500. 24 Monadnock is a 2,501 square foot house with 5 beds and 2 baths that has been on Estately for 347 days. This house is in the attendance area of Colebrook Academy and Colebrook Elementary School.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"USC (5-5, 4-4) and UCLA (2-8, 2-5) meet Saturday afternoon at the Rose Bowl for the 88th version of the nation's most famous crosstown rivalry. It will be shown live nationally on Fox 11.\nAlthough both teams are having sub-par seasons (the 13 combined losses between the squads are the most entering any Trojan-Bruin football game), there is still lots at stake: city bragging rights, the Victory Bell and Crosstown Cup points. And the Trojans need a victory to become bowl eligible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Smoking \u2013 Is It Difficult To Say No?\nSmoking releases almost 4000 toxic chemicals or substances, including Nicotine, the most prominent cause of addiction.\nSmoking is a practice in which a substance is burned and the resulting smoke is breathed in to be tasted or inhaled. Smoking of any type is known to mankind since maybe advent of fire; with instances reported as far back as 5000BC, and references in almost all the civilizations in some cultures, smoking are also carried out as a part of various rituals, where participants use it to help induce \u2013 like states that, they believe, can lead them to\" spiritual enlightenment \" . You can Find More Tips and Info to Quit Smoking from online sources.\nThroughout the world, around one billion people smoke, of which approximately 5 million people die every year from tobacco \u2013related illnesses. It is estimated that this figure will double by 2030. Over three-quarters of the world's smokers live in low-and middle \u2013 income countries, where smoking rates are increasing rapidly and this pattern is true within European countries too.\nTo lead a smoke-free and healthy life, you need to go through the best way to quit smoking.\nNext Next post: What is the story behind chiropractic?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When someone you love is facing a life-limiting illness, you and your family want to do whatever you can to ensure he or she gets the very best care. Hospice provides palliative care at the end of life.\nPalliative care improves the quality of life throughout the treatment of a serious illness by providing practical, emotional and spiritual support. Hospice care is form of palliative care that focuses on relieving symptoms when someone living with a serious illness approaches the end of life.\nLike palliative care, hospice care uses a team approach and provides physical, emotional, spiritual and practical support. It focuses on both the patient and the family. It makes sure that the patient's goals for life and care are met.\nWhere is hospice care provided? Hospice may take place in the patient's home, a nursing home or a hospital. For most, home is the usual setting, with a family member taking the job of primary caregiver. He or she may receive support from a hospice team and home health aides. At a nursing home or a hospital, the hospice staff also offers much support to the patient's family.\nIn order to receive hospice care, the patient's outlook for survival must be limited, and his\/her medical treatment must be aimed at relieving discomfort rather than curing the patient's illness. The hospice team consists of a doctor, nurse, social worker and rabbi\/chaplain. They offer many services including medical management, practical assistance and emotional support. These services are tailored to the needs and wishes of the patient.\nHow does hospice care help me?\nHelp from the hospice team is available 24 hours\/7 days a week by phone.\nWho are the members of a hospice team?\nHospice staff is made up of several specially trained people who have experience working with patients and families with life-limiting illnesses. Hospice staff help the patient and family to be as comfortable as possible. The team consists of hospice doctors, nurses and social workers. Chaplains and others may also be part of the team. Patients, families and friends are also considered important people on the treatment team, which respects cultural values and traditions.\nPatients whose prognosis is about six months or less, verified by the hospice doctor.\nPatients who are no longer receiving treatment to cure the disease but who want to live comfortably. Many hospices will work with families whose ill relative does not know his\/her prognosis. Hospices try to fit their program of care to meet the unique needs of the patient.\nWho pays for hospice care? Hospice is provided by Medicare free of charge. The services may vary by state, so check your state's Medicaid agency. If you have private provider, check with your provider's policy.\nHow do I locate a hospice program? To find a hospice in your local area, call the National Hospice and Palliative Care Organization (NHPCO) at 800-658-8898 or visit www.caringconnections.org.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BIOAQUA rose moisturizing exfoliate Facial scrub Skin care chamfer shrink pores deep clean skin beauty muscle skin care product.\nCucumber silky muscle exfoliate Facial.\nDeeply cleanses and nourishes the skin.\nFeature: Moisturizing, moisturizing, deep cleaning, exfoliating, closing pores.\nEffect: Skin nourishing, moisturizing, deep clean, exfoliating, shrink pores.\nApply to clean your skin, make your skin better.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmgax b/data_all_eng_slimpj/shuffled/split2/finalzzzbmgax
new file mode 100644
index 0000000000000000000000000000000000000000..a59248b9ed56c92b43aa38900ea7fc656f25d4e9
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbmgax
@@ -0,0 +1,5 @@
+{"text":"This specimen was collected together with a darker one of the same species.\nThe whorls of spines around the tail segments on this nymph had me thinking it was in the closely related Serratella genus for a while, but this close-up picture reveals the web of fine setae (Seta: Little hairs on insects.) that sets the Ephemerella genus apart.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home \u00bb Blog \u00bb Is Machine Learning the Future of Marketing? Experts Weigh in.\nTo help us understand the growing role of machine learning in marketing, we spoke with some of the top influencers in the space, including Michael Brenner (@brennermichael) of Marketing Insider Group, Serena Ehrlich (@serena) of BusinessWire, Adelyn Zhou (@adelynzhou) of TOPBOTS and Chris Messina (@chrismessina), creator of the hashtag - among others. Selected responses to the survey are below.\n\"I see the future of marketing as a combination of smart people & smart automation. Robots led by people. I think it will put a greater emphasis on data storytellers vs. data scientists. Of all the marketing technologies using machine learning and AI today, programmatic ad buying and improvement of ad spend allocation through machine learning impress me the most.\"\n\"The future of marketing will be some combination of smart people and smart automation technology. Why? That's the next step in the process. Data + experience.\"\n\"Marketing in the future will be a combination of smart people and smart automation technology because machine learning and AI are able to help with lots of our menial, mundane, and repetitive marketing tasks. This will, in theory, free up marketers' time to be more creative and focus on big tasks.\nEvery week, on the VB Engage podcast, we encounter new startups that use AI & machine learning to automate various tasks. The future is here, folks.\"\n\"How do I see machine learning affecting search marketing, paid advertising and intent-driven channels? It will lead to mass personalization and super-specific paid advertising that knows our intentions even better than we do.\"\n\"We use natural language processing to create new SEM campaigns based on keywords that come from the user-generated content on our site. We use machine learning algorithms to decide on promotions we should be offering, on emails we should be sending and on [the] type of retargeting we should be doing.\nI think that AI will be able to tweak creatives and\/or pick the winning creative, and similarly, it will be able to improve targeting (e.g. keywords on SEM, email addresses in re-marketing campaigns) but will not be able to create target segments by itself. \"\n\"[I work] with clients that develop machine learning- and AI-based solutions for brands for all sorts of applications, from personalization to conversational [chatting], to customer service and more. The low-hanging fruit comes in automated data visualization and the ways we can absorb and assess real-time data.\nWhat's really amazing to me is some of the technology for context analysis \u2014 though as I said, it depends on a large enough data set and inputs. It also needs to assess ever-changing associations between consumers and their devices, networks, locations, personal or professional lives, purchase histories, historical and real-time behavioral patterns...and more. Some of the technologies are getting pretty amazing \u2014 approaching that Promised Land, or Holy Grail of marketing. It'll be exciting to see that happen.\"\n\"I see the future as some combination of smart people and smart automation technology, because AI will need learnings from humans, or perhaps fine-tuning. Marketing is messaging and is both an art and science. I believe some aspects of marketing are all automated as they already are, but as it stands, it does need humans to set that up. \"\n\"Of today's marketing technologies using machine learning and AI, natural language processing (NLP), chatbots and voice-driven search impress me the most.\"\n\"I think with deep learning and proper tracking, searches will become entirely customized to the individual user, further than just what results they see, but even the whole experience itself. That is, page layouts, graphics, UX etc.\nThey work with the app developers to help identify key actions in the app that are considered 'valuable,' and then find out which types of humans are most likely to complete the highest number of these actions. We've moved from cost-per-impression and cost-per-installation to cost-per-post-installation-actions. We are currently in contact with a team at Google about getting these types of ads integrated into our apps.\"\n\"Machine learning is best for exposing what people don't know that they don't know. It also benefits greatly from larger data sets. Those two strengths converge in areas of search marketing, paid advertising and internet-driven channels. I see machine learning helping the manager of these channels make more-informed decisions, the way traffic-sensing map apps help a driver take a smarter route. AI still appears to have limited penetration in B2B marketing.\"\n1.) Customer experience - This is in the intersection of marketing and customer service. Brands are using AI to route messages better, respond to questions faster and provide improved responses to customers.\n2.) Visual search tagging - E-commerce companies are using AI-powered systems to better categorize and tag products in their catalog. If a customer searches for 'floral sundress,' the system would be better at finding all relevant results.\n3.) CRM database management - To send more-targeted and personalized messages to existing users in their database. For example, based on a user's drink preference matched with the weather in their area, a system will send a coupon for a Long Island Iced Tea. While some of this could have been done before, with machine learning, it's being done at a deeper and more-personalized way.\nI'm sure 90% of people said that the future of marketing is a combination of smart people and smart automation, right? While we will move towards automation, there will always be an element of human judgment to it. The human CMO, hopefully, will create the strategy with input from data, models and longer-term AI recommendations. Marketing is part art and part science. While both of those fields are being disrupted by AI, there will always be an element of human in judgment.\nWe're already seeing algorithms automate the majority of paid online ad buys. The programmatic purchasing of advertising is just one step away fully automated machine learning systems. I see more and more of these channels automated. And we've seen how automation allows marketing messages to be personalized at scale, quickly and at a lower cost.\nI'm excited about the visual recognition tools and how they are able to tag images and video. We can now understand how much brand awareness and \"media value\" is derived from user-generated content or on a TV showing of the NBA finals by processing how long and in what context a brand product was shown. Longer term, I'm excited for AI to help copywriters generate better taglines and email copy personalized to the user level.\"\n\"The future will be a combination of smart people using smart automation technology. This is because people interpret results. People are not always predictable, and it's hard to always incorporate market forces and other variables. Marketing will become more return on investment (ROI)-driven, less human-focused and more efficient.\"\n\"Serving up ads across digital channels that are truly personalized to the individual or account (based on attribute, affinity and intent data) can only be done using machine learning. And it's well worth the effort to do this, because 99% of ads don't generate a click because they're irrelevant or ill-timed. If the data is available about your target audiences to help you deliver a message that will resonate and generate even a modest improvement to your CTR, it's easy to make the business case.\"\n\"We use machine learning for our corporate training. Namely, each year, we have to pass a data security exam after taking an online course. In my opinion, the future of marketing will be a combination of smart people and smart automation. Why? Technology is providing us the ability to track more data in a real-time fashion.\nAnalytics are enabling that data to transform into usable output information. Smart people are now able to spend their time\/talents on reviewing the output and making informed inferences to complete the entire picture. Machine learning will continue to synthesize the vast amount of data\/impressions online daily...billions of transactions formulate a picture of each consumer, buying patterns and web optimization.\"\n\"At Brandify, we use natural language processing (NLP) capabilities to increase digital engagement late in the funnel. When our clients' customers want to understand the purchase they are considering, want to do further research and need help in finding the right information, voice search and NLP gives them answers that are more relevant to them in a quicker and more-efficient way. In addition to NLP, we provide Association Rules analysis - a rule-based method to discover interesting relationships among variables in our database - to provide these individuals with more-helpful and relevant recommendations.\nOverall, we use Machine Learning to spot patterns in ratings and reviews to speed up brand responses to customer feedback (across the positive, neutral and negative scale). Because reviews and online recommendations have such an impact on local engagement and conversions, machine learning is primarily used to quickly respond to local customer feedback and amplify the voices of our clients' biggest brand advocates.\"\n\"Today, we're using machine learning, deep learning and AI in our marketing by running tests using chatbots via Facebook messenger ads aimed at generating more-interactive discussions during the consideration phase of customers' journey.\n[Users will] eventually be getting hyper-targeted, showing you the right ads before you have the need. Something Amazon should be better at given the depth and breadth of data they're collecting. Also, chatbots are appealing and have some room to improve on future iterations (such as in terms of UI and complexity of sequencing).\"\n\"The future will involve some combination of smart people and smart automation technology. Why? For the near future, we will still need 'machine teachers.' These channels will still be using analytics, just more-sophisticated\/self-learning data models that come with machine learning.\"\n\"I am not actively using machine learning in marketing today \u2014 I am investigating the solution ecosystem. I'm particularly interested in AI to support social messaging services (e.g. Facebook Messenger). However, the future will be a combination of smart people & smart automation technology. It will absolutely be a combination of AI and humans working together.\nHowever, to be frank, there are very few services that I'd position as including genuine AI today (though I am following a couple that are in stealth mode). There are more (like Oracle) that leverage machine learning to optimize programs. A lot of what I'm seeing today is focused on automating low-value tasks so that humans can be more productive.\nThen, there are more-advanced new applications (not existing apps enhanced with machine learning) that are really purposely built to work alongside humans...to give them marketing superpowers. Finally, there are a handful focused on fully AI-driven experiences that are purposefully designed NOT to be human....there are cases in which a user prefers to engage only with the bot. \"\n\"The future of marketing will be a combination of smart people and smart automation technology. This is because AI lacks emotional intelligence, and the buying process is still driven by emotion. Of all the machine learning solution out there, I'm most impressed by creative AI tools, various image recognition tools (and the smart use cases they're involved in), Likelihood, Watson Ads and various SalesForce automation tools.\"\n\"We're using machine learning in our marketing to build propensity models to drive personalized customer experiences. But the future of marketing will be some combination of smart people and smart automation. This is because, as we move to cross-channel, cross-line-of-business (LOB) campaigns, automation is required to scale.\nWe will always need to be interpreting campaign results to drive new insights to inform creative, branding and tweak experiences. As for how I see machine learning affecting intent-driven channels such as search and paid, data is the name of the game. As we get more cross-channel data, we will build more-personalized customer experiences that will continue to refine audiences...content by channel in context.\"\n\"We use machine learning to build training sets related to our analytics work (influencers, content, language, channels). We are also exploring how to use machine learning to build training sets related to images to help the creative process. Both are in early stages.\nAs for which machine learning-based solutions today impress me the most? Great question, don't have a great answer, which is due to the infancy of this area. Many folks say they can do things that they can't. We rely on Google as an aid and other firms we find per project. Area cannot mature fast enough.\"\n@sumgould - President, Eye\/Comm Inc.\n\"The future of marketing will be some combination of smart people and smart automation technology. Why? People should always have an eye out for issues that can come up. How do I see machine learning affecting search marketing, paid advertising and other intent-driven channels? I see them being an integral part in the future.\"\n\"Machine learning will eventually do most of the tedious research. People will still need to provide context and monitor the relevance of data being examined.\"\n\"Search marketing is a shoo-in for machine learning, but people will still be required to account for any changes in the search engine rules and to counter the fact that everyone is doing the same thing.\nInsight gained from ad analytics and profiling should make paid advertising easier and more accurate. However, the analytics I've seen are patchy at best. The numbers don't tell the whole story. Plus it means we miss out on the human element.\nAs for which current solutions impress me the most, Facebook and Google currently reign supreme when it comes to machine learning, data crunching and generating income and clicks. I don't think there is any company using true AI for marketing.\"\n\"We are using AI in our organization on our website to learn what a visitor wants, and direct them to a more-personalized experience of our website. We are also currently exploring the use of an AI chat bot on our organization's website as well.\nAs for which marketing technologies using machine learning and AI today impress me the most, I like where Watson is headed. We use Sentient Ascend and Boomtrain, but there are so many exciting products coming to market right now that it's like being a kid in a candy store and very easy to become distracted!\"\n\"I see the future of marketing as some combination of smart people and smart automation. Why? Because marketing\/PR will forever be a relationships game - not only automation. You need the right connections with writers in order to make your earned media a success.\"\n\"How are we currently using machine learning, deep learning and AI in our marketing efforts today? Our software, HYP3R, leverages this to turn social media management into a recommendation engine.\nI see the future of marketing as some combination of smart people and smart automation technology. This is because it is simply not possible to do one without the other. As for how I see machine learning affecting intent-driven channels such as search and paid, the answer lies in results-oriented pattern recognition.\"\n\"I see the future of marketing as being a combination of smart people and smart automation technology. Why? AI is essentially all about probabilities. While you can do a lot with that, you still need human creativity to craft effective strategies and narratives as well as accounting for emergent cultural behaviors and societal context.\nAs for how I see machine learning affecting search, paid and intent-driven channels - it'll open up new targeting contexts (i.e. in the real world) and help improve the accuracy and effectiveness of targeting. And as for which technologies using machine learning and AI today impress me the most? Self-driving cars.\"\nThanks for reading our survey on top influencers' thoughts about the future of machine learning in marketing. To learn more about how cutting-edge marketing technologies and strategies such as machine learning and data science are changing marketing today, please see our full report.\nUsing data science to improve your enterprise paid search marketing results.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Life is full of simple truths. For example, you can't drive without tires and you can't swim without pool liners. Here's the good news: choosing a liner to keep the water in your pool is much easier than paying a visit to the local tire store. Selecting the right liner can help protect your pool for years to come and, with proper chemical balance and routine maintenance, will bring added durability to your investment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When it comes to your personal documents, The UPS Store has just the 'ing' you need\u2014notarizing. If you're looking for licensed notary public services in Delray Beach, come to The UPS Store. Stop by today and get notarizing checked off your list.\nAt The UPS Store J Military Tr we have a professional licensed notary public on staff ready to help you get your business and personal documents notarized quickly and efficiently. Our notary is experienced in notarizing wills, trusts, deeds, contracts, affidavits and more.\nMake an appointment online or stop by The UPS Store at 14545 J Military Tr today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sometimes, I write , mostly for fun and my own amusement.\nMy visit to Mozambique to install equipment for an ISP and go SCUBA diving.\nViews on how politics is affecting the geeks and thoughts on the South African ECT act.\nIdeas on improving the environment for the African and South African Internet.\nBefore I could start writing, I had to get a few things organized.. well this is part of my theory on why I did not really start writing earlier.\nWork up some experience in this world and form some opinions.\nHave a Free Unix operating system.\nLearn to use a text editor.\nLearn to touch type on the DVORAK keyboard layout.\nLearn to use DocBook XML.\nHave a fine collection of tea and music.\nI'm ready to start writing now. Took a few years. I'm hoping some of the things I've learned will be useful to others. I started this page on 5 November 2002, but only really started writing some time after that. No, wait thats not true, I've been keeping a textfile with notes and ideas since 1997.\nI think it's a good idea to write things down, even if they seem to loose some of their impact and immediate importance if you look at them again some time later. It's better to have ideas, make notes of them and tinker with them than not having them and never sharing them.\nGiven the list of dependencies above, I guess you can start imagining the things I am likely to write about. Unless I fall on my head, I dont think I'll write the next Alice in Wonderland.\nOne of the questions that has been bugging me for a while now is, what I would have done if I lived 100 years ago and I could not spend all my time with computers and building networks. Well.. my current answer is, I would have become a writer\/journalist or politician, maybe both.\nthese writings contain my subjective opinions, from my point of view.\ndont take them too seriously, we are always changing, constantly.\nI forgot what 3 was.\noh, yeah, I make no guarantees about order, spelling and grammar.\nEmail and Mutt with GNUPG, my thoughts on email style, indenting etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbngxn b/data_all_eng_slimpj/shuffled/split2/finalzzzbngxn
new file mode 100644
index 0000000000000000000000000000000000000000..01bd5cb42925ecef9f728dd552559b1fb543dec2
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbngxn
@@ -0,0 +1,5 @@
+{"text":"Mega Family's Sankranthi Celebrations at..\nMega Family continues the tradition of spending quality time with each and every member of family once in a year. They plan vacations and this time, the upcoming Sankranthi holiday season has been chosen and the place to celebrate is their farmhouse located in Bengaluru.\nAllu Sirish confirmed this news by tweeting, \"Wrapped up another schedule of our film. Looking fwd to our yearly Sankranthi holiday with family at Bangalore. 50 of us! Yayyyee!!\".\nCredit must go to Megastar for setting up a good example and the younger generation is following his standards.\nChiranjeevi and Ram Charan will commence the shooting of their respective films in February while Allu Arjun and Sirish will resume their film shoots after Sankranthi holidays. Varun Tej and Sai Dharam Tej will also be joining the family members to celebrate the festival.\nPawan Kalayn is the only person to miss it seems.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do you want alan to go back to Japan ?\nIndifferent, as long as she makes music.\nI prefer her Japanese songs to be honest but I want her to be wherever she's happiest and I think that's China.\nShe also seems to have more freedom to make her own creative choices in China (at least with her upcoming album). I'd prefer alan making her own choices and potentially making songs I dislike rather than her making songs that I love but are devoid of alan's input.\n^my thoughts exactly; as long as she is happy and releasing great music. She already tried Japan and success was limited. Now try China and let's see. I also adore her Japanese songs, so I hope she will continue to do so periodically too.\nIndifferent, as long as she makes music was my option. I understand chinese but japanese ones were made of high quality so im indifferent as long as she produces music.\nHer Japanese works are by far superior to her Chinese work. However, like everyone said...I want her to have more freedom and she's happier with her family. I hope she can experiment more because she does have a talent in composing.\nI'm glad she is happy in China, but i miss her japanese songs, maybe she could release a japanese album every two years or she could release something periodically too.\nYes, I do want her to go back to Japan. I really do miss her Japanese songs. However, I wish people in Japan would acknowledge her more. But it's the idol era now sadly.\nSubject: Re: Do you want alan to go back to Japan ?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The election for Weavers Way Board members is happening now. Did you know that our bylaws require 10 percent of our members to vote in order for the election to be valid? As our membership has grown over the past several years, the number of votes needed for a valid election has also increased. Your vote matters!\nBut that's not the only reason you should vote. As with our national, state and local elections, exercising your right to vote is increasingly important. As demonstrated by the 2016 and 2018 elections, each person's vote is critical to the functioning of the democratic process. It is no less important in our own Weavers Way election.\nThere is a statement from each candidate in this issue of the Shuttle, as well as a photograph. There is also a video of each candidate on the Weavers Way website, providing an opportunity to see each candidate talk about their interest in the Co-op.\nIt's easy! You will receive an email reminder to vote. You can then review the candidates, select up to three and click on the link on your email to cast your vote. In addition, you can also vote by paper ballot. Paper ballots are available in the Shuttle and in each store, and they can be dropped in the locked box provided in each store or mailed to Weavers Way at the address provided on the ballot.\nIt does matter! Now that our membership is almost 10,000, we need 1,000 members to vote in order for the election to be valid. In addition, the Board represents you and works to assure Weavers Way adheres to its mission and values.\nAs a cooperative, it is your privilege to vote for your Board. That's one of the unique features of a cooperative!\nYou can talk to friends and neighbors, present and past Board members and staff to get their opinions about who would be a good Board member. In addition, you might have an opportunity to meet prospective Board members prior to the election, as you shop in our stores. In that case, use the opportunity to get to know their opinions about what is important to you.\nAgain, it does matter! Board members represent you! They have a lot of input into how Weavers Way operates. We need Board members with creativity, integrity and willingness to commit the time and energy to their responsibilities.\nThis year, there are no incumbents running for the Board. We have nine candidates, and each brings a set of skills and interests that can add to the Board's expertise. It will serve all of us well to review the candidates' information in the Shuttle and online to determine who best represents our values and interests. But even when there are incumbents running, each election represents an opportunity to elect members who can add to the richness of our Board.\nThis year, we hope all of you will take the opportunity to make the Weavers Way Board election a meaningful experience and to elect a Board that represents your values and interests!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Triconodon mordax was named by Owen (1859). Its type specimen is BMNH 47764, a mandible (left mandible), and it is a 3D body fossil. Its type locality is Mammal Bed, Durlston Bay (DB83), which is in a Berriasian marine horizon in the Lulworth Formation of the United Kingdom. It is the type species of Triconodon.\nTriacanthodon serrula was named by Owen (1871). Its type specimen is BMNH 47763, a mandible (left mandible), and it is a 3D body fossil. Its type locality is Mammal Bed, Durlston Bay (DB83), which is in a Berriasian marine horizon in the Lulworth Formation of the United Kingdom.\nIt was synonymized subjectively with Triconodon mordax by Simpson (1928).\nTriconodon minor was named by Owen (1871). Its type specimen is BMNH 47771, a mandible (partial right mandible), and it is a 3D body fossil. Its type locality is Mammal Bed, Durlston Bay (DB83), which is in a Berriasian marine horizon in the Lulworth Formation of the United Kingdom.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You'd think it would need a layover, but this large shorebird is known for its nonstop flights. \"We have recently documented how their migrations involve nonstop flights of over 6,000 miles,\" said Senner, who specializes in this species.\nThe flights, which can take seven days or more, can take the Hudsonian godwit from southern Chile to the Great Plains of the U.S. in spring or from James Bay, Canada to Buenos Aires, Argentina in fall, he said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnnbf b/data_all_eng_slimpj/shuffled/split2/finalzzzbnnbf
new file mode 100644
index 0000000000000000000000000000000000000000..9c4c771001f5d43250e6d559daae4b809f60cbf2
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbnnbf
@@ -0,0 +1,5 @@
+{"text":"Praana Constructions offers a fully-integrated service \u2013 from inception, ground investigation and remediation, through to project completion and beyond to the delivery of operational solutions. We offer our customers unparalleled breadth and depth of expertise. As a result, we can leverage ideas, skills and entrepreneurial flair to deliver top quality work in all sectors.\nThe engineering group offers land development, apartment construction and planning services for projects. Praana provides its clients with honest assessment of construction cost.\nEvery project we undertake is a joint venture of some sort, whether with land owners, private investors or institutional partners. Our record of successful partnership and delivering on promises has earned us the trust of our partners.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To place your horse's need for you, to let him leave his failing body above your need to keep him with you - that - is the greatest and purest love.\nWhen your horse follows you without being asked, when he rubs his head on yours, and when you look at him and feel a tingle down your spine...you know you are loved.\nThe horse, with beauty unsurpassed, strength immeasurable and grace unlike any other, still remains humble enough to carry a man upon his back.\nA true horseman does not look at the horse with his eyes, he looks at his horse with his heart.\nFor one to fly, one needs only to take the reins.\nHorse thou art truly a creature, for thou fliest without wings and conquorest without sword.\nYou and your horse. His strength and beauty. Your knowledge and patience and determination and understanding and love. That's what fuses the two of you onto this marvelous partnership that makes you wonder, \"What can heaven offer any better then what I have here on earth?\".\nRiding is a partnership. The horse lends you his strength, speed and grace, which are greater then yours. For your part, you give him guidance, intelligence and understanding which are greater then his. Together you can achieve a richness that neither can alone.\nWhen a horse greets you with a nicker and regards you with a large and liquid eye, the question of where you want to be has been answered.\nAt its finest, rider and horse are joined not by tack, but by trust. Each is totally reliant upon the other. Each is the selfless guardian of the other's very well-being.\nIf your horse says, \"No\", you either asked the wrong question, or asked the question wrong.\nI heard a neigh. Oh, such a brisk and melodious neigh it was. My very heart leapt with the sound.\nThrough the days of love and celebration and joy, and through the dark days of mourning...the faithful horse has been with us always.\nAsk me to show you poetry in motion and I will show you a horse.\nAnd indeed, a horse who bears himself proudly is a thing of such beauty and astonishment that he attracts the eyes of all beholders. No one will tire of looking at him as long as he will display himself in his splendor.\nThe history of mankind is carried on the back of a horse.\nIn their eyes shine stars of wisdom and courage to guide men to the heavens.\nHe knows when you're happy. He knows when you're comfortable. He knows when you're confident. And he always knows when you have carrots.\nTo see a horse is to see an angel on earth.\nA dog looks up to a man. A cat looks down on a man. But a patient horse looks a man in the eye and sees him as an equal.\nTo ride on a horse is to fly without wings.\nBefore I loved horses, I had nothing to live for. Now I love horses and can't stop seeing things to live for.\nStay away from a horse long enough and you'll start tapping your fingers to the beat of a trot.\nOn the back of a horse you will find Paradise.\nA horse in the wind - a perfect symphony.\nThe horse you get off is not the same as the horse you got on. It is your job as a rider to ensure that as often as possible, the change is for the better.\nThere are many wonderful places in the world, but one of my favorite places is on the back of my horse.\nNothing moves me more then when on the way to fetching in my mare in the morning than the sound of her neighing to me as I open the gate.\nBe wary of the horse with a sense of humor.\nDance above ground, never descending. Grace incarnate. Passion on hooves.\nIn the trojan horse story man entered and touched the inside of a horse. But to see a horse in high spirits is to have the horse enter and touch the inside of a man.\nHe did not feel the ground under his feet - he thrust himself into the capriole, rose high in the air - forelegs and hind legs horizontal. He soared above the ground, he head in jubilation. Conquering!\nThrough the days of love and celebration and joy, and through the dark days of mourning - the faithful horse has been with us always.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Contain 112 C2010-568 Questions & Answers PLUS Unlimited Access to 4500+ Exams.\nFree C2010-568 updates provided regularly.\nConvenient, easy to study, Printable PDF Deployment Professional C2010-568 study material, Learn on go.\nBe sure of Guaranteed Pas Scores with BrainDumps materials, with a proven 98.6% IBM C2010-568 Pass rate.\nCustomizable & Advanced C2010-568 Testing Engine which creates a real exam simulation enviroment to prepare you for C2010-568 Success.\nPractice Until you get it right. With options to Highlight missed questions, you can analyse your mistakes and prepare for Ultimate C2010-568 Success.\nCreate Notes for Any IBM C2010-568 Questions. When and Where Needed, edit them or delete them if needed.\nSimply submit your e-mail address below to get started with our interactive software demo of your IBM C2010-568 exam.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These are the days sacred to Aset within each month. These are not the yearly festivals, but monthly ones. Some were Egyptian in origin and belong on the lunar calendar and others were adopted from the Greeks or Romans living in Egypt.\nMany of these festivals are governed by the cycles of the moon. Aset (Isis) is honored on these days along with either Her son Heru-sa-Aset or Her Husband Wesir as they are both moon gods. Aset Herself is more associated with the sun in Egyptian cosmology than the moon. Her association here has to do more with the cycles of Heru-sa-Aset (birth to death or injury and healing of the Eye of Heru) and Wesir (death and renewal).\nFrom the astronomical ceiling of Senmut, this day is being noted as being sacred to Aset.\nThis festival is from this book Ahnas el Medineh: The Tomb of Paheri at El Kab. Aset's birthdays were celebrated with the lighting of candles and feasts were made in Her honor. Today, you could offer Aset a Birthday cake (blue, white or chocolate seem to go over well) and cook a great feast.\nMake offerings to Aset, the Goddess of Philae today. Possible offering ideas can be found here: Offerings.\nThis feast was associated with honoring the ancestors as well as Ra and Wesir. Aset Herself was given an oblation on this day.\nListed within a hymn from the Temple of Philae, this festival was initially associated with Ra. You could honor Aset and Ra on this day.\nThis is listed within the Frieze of the Temple of Edfu. I don't have much more information on this besides that at the moment.\nAset can be honored here as Sopdet as the cycle of the star's departing and returning can be celebrated monthly along with being celebrated yearly.\nThis is listed on the Temple of Abydos and it mentions oxen are given to the Goddess. This may also be the First Day of the Lunar Month as that is normally on the New Moon.\nDavid, Rosalie. A Guide to Religious Ritual at Abydos. Warminster: Aris and Phillips, 1981.\nDonalson, Malcolm Drew. The Cult of Isis in the Roman Empire: Isis Invicta. (New York: Edwin Mellen Press, 2003), 82.\nGriffith, F. Ll. Catalogue of the Demotic Graffiti of the Dodecaschoenus. Volume 1 Text. (Oxford University Press, 1937), 46.\nMorgan, Mogg. The Wheel of the Year in Ancient Egypt. Mandrake of Oxford, 2011.\nNaville, \u00c9douard Henri and Francis Llewellyn Griffith, et al. Ahnas el Medineh: The Tomb of Paheri at El Kab. (Egyptian Exploration Fund, 1894) 28.\nParker, Richard. The Calendars of Ancient Egypt (The Oriental Institute of the University of Chicago. Studies in ancient oriental civilization). University of Chicago Press, 1950.\nZabkar, Louis V. Hymns to Isis in Her Temple at Philae. London: University Press of New England, 1988.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"STUNNING home meticulously maintained by the current owners. This is one you do not want to miss! Sellers have recently updated this home with new plank tile flooring throughout the main areas, new carpet in the bedrooms, HVAC (2017), new exterior paint, impact windows and doors, new well pump and an absolutely beautiful remodel to the master bath and so much more. Outside they've added extra patio space, perfect for entertaining, with a beautiful sail over the circular patio for relaxing in the shade, a new vinyl fence and the list goes on. Call today for your private showing because this one will not last long.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbpjmc b/data_all_eng_slimpj/shuffled/split2/finalzzzbpjmc
new file mode 100644
index 0000000000000000000000000000000000000000..551662663f97bbcdb8992c3f4279dae4a160bf78
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbpjmc
@@ -0,0 +1,5 @@
+{"text":"JEITA\/EC Center - The JAMP declarable substances search list (Ver.4.030) has been applied into the Tool.\nToppage \u226b \u6d88\u606f \u226b The JAMP declarable substances search list (Ver.4.030) has been applied into the Tool.\n2015\/03\/06 : The JAMP declarable substances search list (Ver.4.030) has been applied into the Tool.\nWe had applied the JAMP declarable substances search list of version 4.030 updated on March 2015 into the JEITA_AIS Creation Support Tool.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When you think of bedroom furniture, an armchair probably isn't the first thing that comes to mind. However, bedroom armchairs are becoming more popular, and it's not hard to see why once you understand the benefits. Here are just six reasons you should add an armchair to your bedroom. 1. Complement Styling Choices Firstly, a bedroom armchair is perfect for complementing styling choices and bringing the room's atmosphere together. You can choose upholstery that matches the colour of your sheets or brings out the colours in your wallpaper.\nIt feels like the construction of new homes is fairly uniform, with traditional building materials remaining unchanged for generations. Sure, there has been modernisation in the way that these materials are prepared, and of course construction methods have advanced over the years. In the coming years you might well be living in a home that has been made from 3D printed materials, or made of plastic bricks. You could even live in a repurposed shipping container, and there are already a number of these homes in Australia and around the world.\nThe fitouts you use in your office are probably more important than you realize; they provide privacy for your workers so they can concentrate and also help to direct the flow of foot traffic. Their appearance can also set the tone of your company; if they're dull and drab, your company may seem a bit unenthused about your business. When you're ready to replace the fitouts you have or are furnishing a new office, note a few tips for how to add style and function with the right fitouts.\nChoosing furniture for your tiny house can be challenging. You need furniture that fits into and complements your tiny space as well as furniture that performs more than one function, such a a dining table that converts into a guest bed. In many cases, teak may be one of the best options. Here's a look at why: 1. Long lasting In exchange for the relatively high price of teak compared to some other types of wood furniture, you receive a lot of value.\nMany people mistakenly believe that there's not much that can be done to improve the appearance of a tiny garden. However, whilst it is true that smaller outdoor spaces present more challenges than their larger counterparts, there are in fact plenty of ways to make the most out of a modestly-proportioned garden. All that's required is a bit of time, patience and creativity. Plant vertically If you're looking for a simple way to add some natural beauty into a small garden, without infringing on the already-limited ground area, consider hanging some plants vertically, up along the fencing boards or walls.\nIf you are looking to increase storage capacity in your bathroom, your best bet would be to invest mounted bathroom cabinets. This is especially convenient for smaller bathrooms that may not have sufficient floor space for freestanding cabinetry. However, making a selection of bathroom cabinets is not as simple as selecting the first choice you get from cabinet makers. You need to establish your individual needs so as to ensure the cabinetry will be able to meet these needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kenya's first national stop, Nairobi National Park is a sanctuary for untamed life and just seven kilometers from the high rises of Nairobi's downtown area. The recreation center is additionally a rhino asylum, which secures more than 50 of these fundamentally imperiled animals. Notwithstanding the rhinos, you can see lions, gazelles, wild oxen, warthogs, cheetahs, zebras, giraffes, and ostriches, and in excess of 400 types of winged creatures have been recorded in the wetlands.\nOne of Nairobi's best vacation spots, the Karen Blixen Museum, at the foot of the Ngong Hills, is the previous home of the acclaimed namesake Out of Africa creator. Karen Blixen, likewise known by her pseudonym, Isak Dinesen, lived in the house from 1917 to 1931, where she ran an espresso estate.\nThe National Museum in Nairobi is an instructive method to spend a couple of hours on a city stopover. The historical center showcases different social and regular history shows including in excess of 900 stuffed flying creatures and well evolved creatures, fossils from Lake Turkana, ethnic presentations from different Kenyan ancestral gatherings, and displays of neighborhood craftsmanship. In the Geology Gallery, you can investigate a noteworthy gathering of rocks and minerals and find out about structural plates and the existence cycle of a spring of gushing lava.\n\"Ngong\" signifies \"knuckles\" in Maasai, a fitting name since these excellent pointed green slopes take after the back of a clench hand confronting the sky. They are a famous place to visit near Nairobi and give an appreciated reprieve from the city warm. The slopes are the pinnacles of an edge sitting above the Great Rift Valley, and many white pioneers set up their homesteads here in the early frontier days. Half-timbered houses and blooming gardens remain; however appear to be more suited to southern England than Africa.\nThe short stormy season begins from late October to through December and November normally has four creeps of rain. Normal precipitation amid alternate months is around two inches. The best time to design your Nairobi visit exercises is amid the dry a long time of July through October and furthermore January and February.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join us this spring at Julie's with a our Picnic in the Park package. Stay with us for 2 nights at Julie's Park Motel, grab 2 box lunches and 2 bottles of water and head into the park with your rental from Edge of Park ($40 gift certificate). Pick a spot and enjoy a Picnic in the Park.\nThis package is available April 26-June 13, 2019. Call for pricing 920-868-2999.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do not let your gothic style keep your ensemble from being practical. We are happy to provide a number of gothic handbags and gothic purses for carrying your daily necessities. Our gothic satchels, messenger bags, laptop bags, tote bags, tablet cases, and wallets ensure that you will have an excellent variety of dark designs to choose from. Crafted from leather, canvas, and cloth, these high-quality gothic bags display detailed ornamentation ranging from graphic prints to their actual shapes. Choose a coffin-shaped purse for holding your cell phone and keys, or grab one of our Grim Reaper crossbody bags for your school books. Other items show off skulls, pentagrams, chains, crosses, ravens, fallen angels, and wolves, so you are sure to find something that appeals. These gothic backpacks and carry-alls include designs to suit men and women, so be sure to browse our selection to find your next bag!\nThe image on the Black Bat Bag may be hard to see, but then bats tend to be hard to see at night. They use the cover of darkness to sneak up on their prey before they strike. Unleash your inner vampire with this iconic image.\nFor a unique way to integrate a floral design, the Blood Rose Tote Bag is a functional accessory that will add style to every wardrobe. The delicate roses of the graphic drip with gothic style as they are saturated seamlessly in blood.\nNo matter what age, the Bright Eyes Backpack is an excellent way to keep your essentials protected and organized. With easy open pockets, this gothic bookbag provides ample storage for many essentials, be it school or recreation.\nPracticality makes this wallet a great addition to anyones attire, while the subtle touches of design ensure that this Chained Alchemist Biker Wallet still has enough artistic detail to make an impression on all those who might see it.\nSoft black leather purse or cigarette case, with zipped side opening. The Coffin Purse features a zipped side opening on a 15 inch curb chain. The coffin purse measures 6 inches high, 4 inches wide, and 2 inches deep.\nThis coffin is quite luxurious, being made in high quality leather, with large bank-note section, several card pockets and eyelet hole for chain attachment. This leather Coffin Wallet measures 6 inches high and 4 inches wide.\nIts wings beat down, propelling it forward through the air. The leathery membranes of the dragons wings ripple as they repeatedly fill with air and empty again. This is the image that comes to mind when viewing the Dragon Bag.\nHaving a hard time presenting intense gothic style while carrying all your essentials? Your troubles are over with the Game Over Backpack! With its unmistakable gothic graphics, this gothic bag is ideal for every situation.\nIn many of its shapes and forms, the cross has become an iconic symbol of the Gothic style. This may be because of the hatred vampires hold for crosses, or it could very well be because of this boldly stylish Gothic Cross Bag!\nKeep all of your necessities close at hand during your daily activities with the help of this striking brocade purse. The Gothic Cross Handbag features a black patent leather cross motif with flat stud detailing along one side.\nBring your love of skulls with you everywhere you go with the help of this Gray Skull Tote Bag! Now you can add a ghoulish touch to any ensemble in an instant while also carrying practically anything you need in this convenient bag.","meta":{"redpajama_set_name":"RedPajamaC4"}}